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\begin{document}
\title{\LARGE \bf
Differentially private Nash equilibrium seeking for networked aggregative games}
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
This paper considers the privacy-preserving Nash equilibrium seeking strategy design for a class of networked aggregative games, in which the players' objective functions are considered to be sensitive information to be protected. In particular, we consider that the networked game is free of central node and the aggregate information is not directly available to the players. As there is no central authority to provide the aggregate information required by each player to update their actions, a dynamic average consensus protocol is employed to estimate it. To protect the players' privacy, we perturb the transmitted information among the players by independent random noises drawn from Laplace distributions. By synthesizing the perturbed average consensus protocol with a gradient algorithm, distributed privacy-preserving Nash equilibrium seeking strategies are established for the aggregative games under both fixed and time-varying communication topologies. With explicit quantifications of the mean square errors, the convergence results of the proposed methods are presented. Moreover, it is analytically proven that the proposed algorithm is ${\Greekmath 010F}$-differentially private, where ${\Greekmath 010F}$ depends on the stepsize of the gradient algorithm and the scaling parameter of the random variables. The presented results indicate that there is a tradeoff between the convergence accuracy and the privacy level. Lastly, a numerical example is provided for the verification of the proposed methods.
\end{abstract}
\begin{keywords}
Nash equilibrium seeking; privacy protection; differential privacy; random variable.
\end{keywords}
\section{INTRODUCTION}
Privacy has become a critical concern for many practical systems that involve sensitive data transmission and collection. Wireless sensor networks \cite{LiAHN}, smart grids \cite{LiangTSG13}, social networks \cite{Wu10}, just to name a few, are typical examples that are in urgent need of privacy protection techniques. Inspired by the fact that privacy preservation is pivotal in information-sensitive systems, privacy protection methods have gained increasing attention in recent years. For instance, in the field of database and data mining, cryptographic secure multi-party computation methods, random perturbation techniques and $l$-diversity, $k$-anonymity based algorithms were adopted for information sharing systems, data collection systems and data publishing systems, respectively \cite{LiAHN}. Homomorphic encryption was adapted for privacy protection in smart grids in \cite{LiangTSG13}. Randomization, $k$-anonymity and generalization-based approaches can be utilized for privacy protection in social networks \cite{Wu10}. Motivated by the importance of privacy protection, this paper aims to achieve the distributed Nash equilibrium seeking for networked aggregative games with privacy guarantees.
In many practical situations, the utilities associated with the interacting decision-makers rely both on the decision-maker's own action and an aggregate of all the decision-makers' actions. For example, in the energy consumption model described in \cite{YEcyber17}, the utilities of the electricity users are determined by the user's own energy consumption as well as the total energy consumption in the electricity market. Cournot price/quantity competitions among multiple oligopolistic firms fall into similar scenario \cite{Agiza03}. In factory production, the utility of each manufactory relies on the averaged output of all the engaged manufactories and the manufactory's own production \cite{WangIET}. In addition, public good provision models and many other examples are also of aggregative nature, i.e., the interacting participants affect each other through a specific aggregate of their actions rather than in an arbitrary fashion \cite{Cornes}. Aggregative games serve as powerful game theoretic models to accommodate these competitive circumstances with aggregative interactions among multiple decision-makers. Motivated by the wide applications of aggregative games in distributed systems, Nash equilibrium seeking for aggregative games on communication graphs is attracting increasing attention in recent years \cite{KoshalOR16}-\cite{PariseCDC15}.
A discrete-time method was proposed in \cite{KoshalOR16} for networked aggregative games and a continuous-time counterpart was provided in \cite{YEcyber17} considering their applications for demand response in smart grids. A gossip-based algorithm was proposed to achieve distributed Nash equilibrium seeking in an asynchronous fashion in \cite{Salehisadaghiani}. Coupled constraints were further addressed in \cite{LiangAT17}. The authors in \cite{PariseCDC15} considered quadratic quasi-aggregative games, in which the players' objective functions depend on the players' actions and an aggregate of its neighbors' actions. From the perspective of privacy issues, the works in \cite{Cummings}\cite{ZhouJSA} shed some light on the privacy protection for aggregative games. However, in the mechanisms of the game, there is a mediator/weak mediator, that can receive information from the players and give suggested actions for the players, to induce the players' behaviours. Hence, the methods in \cite{Cummings}\cite{ZhouJSA} are not distributed. To achieve distributed Nash equilibrium seeking for networked games, the players usually need to broadcast their local information to their neighbors via local communication networks. The information dissemination among the players may raise privacy concern for the players. \RIfM@\expandafter\text@\else\expandafter\mbox\fibf{Nevertheless, to the best of the authors' knowledge, privacy issues have rarely been explored for distributed Nash equilibrium seeking schemes though it is a problem of significant interest.} Motivated by the above observation, this paper considers privacy-preserving distributed Nash equilibrium seeking for networked aggregative games by utilizing the notion of differential privacy \cite{Dwork}.
Differential privacy has been widely adopted to describe the privacy level in many situations that include sensitive information. For example, the differential privacy of the agents engaged in distributed optimization problems was established in \cite{Huang15}-\cite{DingCDC18}. In \cite{Huang15}, random noises were utilized for the protection of the agents' objective functions. Based on the notion of differential privacy, the tradeoff between the convergence accuracy and the privacy level was analyzed. In \cite{HanTAC17}, the constraints of the optimization problem were considered to be the private information to be protected. Based on the stochastic gradient method, a private optimization algorithm was developed by introducing additive noises. Instead of probing the transmitted messages, the authors in \cite{Nozari} proposed a functional perturbation algorithm which ensures that the inaccuracy of the optimization algorithm is only resulted from the introduced noises. In \cite{DingCDC18}, both the agents' states and moving directions were masked by random noises to achieve the privacy protection of the agents' local objective functions. Differentially private average consensus protocols have also been widely studied (see e.g., \cite{MoTAC17}-\cite{Manitara13}, to mention just a few). For instance, the agents' initial states were considered to be private information to be protected in the average consensus problems in \cite{MoTAC17}, where the authors developed an asymptotically convergent algorithm to achieve privacy protection. Both a non-exact convergent algorithm and an almost surely convergent algorithm were established to achieve differentially private average consensus in \cite{NozariAT}. Moreover, differential privacy has been extensively investigated in the area of database and data mining. Interested readers are referred to \cite{Dwork} for a survey on differential privacy.
To shed some light on privacy-protected Nash equilibrium seeking for aggregative games under distributed communication networks, this paper proposes a distributed algorithm by perturbing the transmitted information among the players using random variables. Moreover, motivated by the observation that differential privacy is robust against the auxiliary information exposed to the adversary, we adopt it to describe the players' privacy level in this paper. In brief, the main contributions of the paper are summarized as follows.
\begin{enumerate}
\item This paper considers privacy-preserving distributed Nash equilibrium seeking for networked aggregative games. To achieve the goal, we employ a dynamic average consensus protocol for the distributed estimation of the players' aggregate actions. To protect the players' privacy, the information transmitted among the players in the consensus part is perturbed by independent random noises drawn from Laplace distributions. By utilizing the perturbed consensus protocol and a gradient method with a decaying stepsize, distributed privacy-preserving Nash equilibrium strategies are established for games under fixed and time-varying communication graphs, respectively.
\item The convergence result and the privacy level of the proposed methods are analytically investigated. In particular, the mean square error bound and privacy parameter are explicitly quantified. The presented results illustrate that the privacy level depends on the selection of the initial stepsize, the decaying rate of the stepsize as well as the scaling parameter of the random variables. Moreover, it is shown that there is a tradeoff between the convergence accuracy and the privacy level.
\end{enumerate}
The rest of the paper is organized as follows. Section \ref{NP} provides some preliminaries and formulates the considered problem. The main results are given in Section \ref{main}, where the distributed algorithm is presented with its convergence accuracy and privacy level successively investigated. Moreover, the results under fixed communication topologies are extended to time-varying communication topologies in Section \ref{time}. In Section \ref{num_ex}, a numerical example is provided to verify the effectiveness of the proposed method. Lastly, conclusions are drawn in Section \ref{conc}.
\emph{Notations:} In this paper, we use $\mathbb{R}$ and $\mathbb{R}_{+}$ to denote the set of real numbers and positive real numbers, respectively. Moreover, $\mathbb{Z}^+$ denotes the set of non-negative integers. Let $v$ be a vector or matrix, then $||v||$ denotes the $\ell_2$-norm of $v$. Moreover, $||v||_{\infty}$ denotes the $\ell_{\infty}$-norm of $v$. We say that a random variable $w\sim Lap(b)$, where $b\in \mathbb{R}_{+}$, if its probability density function is
\begin{equation}\nonumber
\mathcal{L}(w,b)=\frac{1}{2b}e^{-\frac{|w|}{b}}.
\end{equation}
In addition, $\mathbb{E}(\Phi)$ is the expectation of $\Phi.$ The partial derivative of $f_i(\mathbf{x})$ with respect to $x_i$ is denoted as ${\Greekmath 0272}_i f_i(\mathbf{x}),$ where $\mathbf{x}=[x_1,x_2,\cdots,x_N]^T.$ Moreover, $\mathbf{\mathbf{1}}_N$ denotes an $N$-dimensional column vector whose elements are all $1$ and the transpose of $\Psi$, where $\Psi$ is either a matrix or a vector, is denoted as $\Psi^T.$ The notation $[g_i]_{vec}$ for $i\in\{1,2,\cdots,N\}$ denotes a column vector whose $i$th element is $g_i.$ For a sequence $g(k)$, where $k\in\{1,2,3,\cdots\}$ and $g(k)$ is either a vector or a scalar, $g(\infty)=\lim_{k\rightarrow \infty}g(k)$ given that $\lim_{k\rightarrow \infty}g(k)$ exists. The maximum and minimum eigenvalues of a symmetric matrix $P$ are denoted as ${\Greekmath 0115}_{max}(P)$ and ${\Greekmath 0115}_{min}(P),$ respectively. For a matrix $Q$, $[Q]_{ij}$ denotes the entry on the $i$th row and $j$th column of $Q$. In addition, for $l_i\in\mathbb{R},i\in\{1,2,\cdots,N\},$ the maximum and minimum values of $l_i$ are denoted as $\max_{i\in\{1,2,\cdots,N\}}\{l_i\}$ and $\min_{\{1,2,\cdots,N\}}\{l_i\},$ respectively.
\section{Preliminaries and Problem Formulation}\label{NP}
\subsection{Preliminaries}
In the following, we provide some preliminaries on game theory and differential privacy.
\subsubsection{Game Theory}
The following game related definitions are adopted from \cite{YEcyber17}.
\begin{definition}
\emph{(A Normal Form Game)} A game in a normal form is defined as a triple $\Gamma=\{\mathcal{V},X,f\},$ where $\mathcal{V}=\{1,2,\cdots,N\}$ is the set of players, $X=X_1\times X_2\cdots \times X_N, X_i\subseteq \mathbb{R}$ is the action set of player $i$ and $f=(f_1,f_2,\cdots,f_N),$ where $f_i$ is the cost function of player $i$.
\end{definition}
\begin{definition}
\emph{(Nash Equilibrium)} Nash equilibrium is an action profile on which no player can reduce its cost by unilaterally changing its own action, i.e., an action profile $\mathbf{x}^*=(x_i^*,\mathbf{x}_{-i}^*)\in X$ is a Nash equilibrium if for $i\in\mathcal{V},$
\begin{equation}
f_i(x_i^*,\mathbf{x}_{-i}^*)\leq f_i(x_i,\mathbf{x}_{-i}^*),
\end{equation}
for $x_i\in X_i,$ where $\mathbf{x}_{-i}=[x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_N]^T.$
\end{definition}
\begin{definition}
(\emph{Aggregative Game}) A game $\Gamma$ is aggregative if there exists an aggregative function $l(\mathbf{x}):X\in \mathbb{R}$, which is continuous, additive and separable, such that there are functions $\tilde{f}_i(x_i,l(\mathbf{x})),i\in\mathcal{V}$ that satisfy
\begin{equation}
\tilde{f}_i(x_i,l(\mathbf{x}))=f_i(x_i,\mathbf{x}_{-i}),\forall \mathbf{x}\in X.
\end{equation}
\end{definition}
Without loss of generality, we consider that $l(\mathbf{x})=\frac{1}{N}\sum_{i=1}^N x_i$ in this paper.
\subsubsection{Differential privacy}
Differential privacy serves as a mathematical quantification on the level of the engaged individuals' privacy guarantee in a statistical database. It provides a rigorous and formal mathematical formulation on the privacy of the sensitive data. We refer readers to \cite{Dwork} for more detailed elaborations on differential privacy and give the subsequent definitions for clarity of presentation.
\begin{definition}
\emph{(Adjacency)} Two function sets $F^{(1)}=\{f_i^{(1)}\}_{i=1}^N$, $F^{(2)}=\{f_i^{(2)}\}_{i=1}^N$ are said to be adjacent if there exists some $i_0\in\mathcal{V}$ such that $f_i^{(1)}=f_i^{(2)},\forall i\neq i_0,$
and $f_{i_0}^{(1)}\neq f_{i_0}^{(2)}$ \cite{Huang15}\cite{DingCDC18}.
\end{definition}
\begin{definition}
\emph{(${\Greekmath 010F}$-Differential Privacy)} Given a positive constant ${\Greekmath 010F},$ adjacent function sets $F^{(1)}$, $F^{(2)}$ and any observation $\mathcal{O},$ the algorithm is ${\Greekmath 010F}$-differentially private if
\begin{equation}
\mathbb{P}\{F^{(1)}|\mathcal{O}\}\leq e^{{\Greekmath 010F}} \mathbb{P}\{F^{(2)}|\mathcal{O}\},
\end{equation}
where $\mathbb{P}\{F^{(j)}|\mathcal{O}\}$ for $j\in\{1,2\}$ is the conditional probability representing the probability of inferring $F^{(j)}$ from the observation $\mathcal{O}$ \cite{Huang15}\cite{DingCDC18}.
\end{definition}
\begin{Remark}
The above defined ${\Greekmath 010F}$-differential privacy illustrates that from the sequences of observations, the adversary could not distinguish between the two function sets with a high probability. Hence, it is challenging for the adversary to identify the players' sensitive information, which further indicates that the players are protected from information leakage. Note that a smaller ${\Greekmath 010F}$ indicates a higher level privacy.
\end{Remark}
\subsection{Problem formulation}
\begin{Problem}
Consider an aggregative game in which player $i$ intends to
\begin{equation}\label{eqform}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{min}_{x_i}\ \ f_i(x_i,\mathbf{x}_{-i})
\end{equation}
where
\begin{equation}
f_i(x_i,\mathbf{x}_{-i})=\tilde{f}_i(x_i,\bar{x}),
\end{equation}
and $\bar{x}=\frac{1}{N}\sum_{j=1}^N x_j$
for $i\in\mathcal{V}.$ Suppose that there is no central authority to broadcast $\bar{x}$ and the players can communicate with each other via a communication graph $\mathcal{G},$ the objective of this paper is to design a Nash equilibrium seeking strategy for the aggregative game such that
\begin{enumerate}
\item Given any positive constant ${\Greekmath 010F}$, the strategy can be ${\Greekmath 010F}$-differentially private by tuning the control parameters.
\item The players' actions can be driven to a neighborhood of the Nash equilibrium point in the mean square sense, $\lim_{k\rightarrow\infty}\mathbb{E}(||\mathbf{x}(k)-\mathbf{x}^*||^2)\leq D$, where $D$ is a positive constant that is as small as possible;
\end{enumerate}
\end{Problem}
\begin{Remark}
Note that the considered aggregative games are practically inspired by the fact that in many decision-making processes (e.g., energy consumption control, Cournot quantity competitions, public good provision, factory production, to mention just a few), the decision-makers affect the others via their averaged/aggregate behaviors. Moreover, different from the previous works on distributed Nash equilibrium seeking for aggregative games \cite{KoshalOR16}-\cite{PariseCDC15}, \RIfM@\expandafter\text@\else\expandafter\mbox\fibf{the objective of this paper includes privacy protection for the players' cost functions $f_i(x_i,\mathbf{x}_{-i}),i\in\mathcal{V}$.}
\end{Remark}
For notational convenience, let $\mathbf{x}=[x_1,x_2,\cdots,x_N]^T$ and with a slight abuse of notation, $f_i(x_i,\mathbf{x}_{-i})$ and ${\Greekmath 0272}_i f_i(x_i,\mathbf{x}_{-i})$ might be written as $f_i(\mathbf{x})$ and ${\Greekmath 0272}_i f_i(\mathbf{x})$, respectively in the rest of the paper.
The following assumptions (see e.g., \cite{YETAC17}) will be utilized to establish the main results of the paper.
\begin{Assumption}\label{a2}
The players' objective functions are twice continuously differentiable functions and ${\Greekmath 0272}_i f_i(\mathbf{x})$ is globally Lipschitz for $i\in\mathcal{V}$, i.e., there exists a positive constant $l_i$ such that
\begin{equation}
||{\Greekmath 0272}_i f_i(\mathbf{x})-{\Greekmath 0272}_i f_i(\mathbf{y})||\leq l_i||\mathbf{x}-\mathbf{y}||,
\end{equation}
for $\mathbf{x},\mathbf{y}\in \mathbb{R}^N,i\in\mathcal{V}.$
\end{Assumption}
\begin{Assumption}\label{a3}
There exists a positive constant $m$ such that for $\mathbf{x},\mathbf{y}\in \mathbb{R}^N,$
\begin{equation}
(\mathbf{x}-\mathbf{y})^T (g(\mathbf{x})-g(\mathbf{y}))\geq m||\mathbf{x}-\mathbf{y}||^2,
\end{equation}
where $g(\mathbf{x})=\left[{\Greekmath 0272}_1 f_1(\mathbf{x}),{\Greekmath 0272}_2 f_2(\mathbf{x}),\cdots,{\Greekmath 0272}_N f_N(\mathbf{x})\right]^T.$
\end{Assumption}
\begin{Remark}
The strong monotonicity condition in Assumption \ref{a3} characterizes a global Nash equilibrium, i.e., the Nash equilibrium is unique under Assumption \ref{a3}. In addition, by Assumption \ref{a3},
\begin{equation}
g(\mathbf{x})=\mathbf{0}_N,
\end{equation}
if and only if $\mathbf{x}=\mathbf{x}^*.$
Note that it is a widely adopted assumption in the existing literature and we refer interested readers to \cite{Facchinei03} for more insights on the assumption.
\end{Remark}
\begin{Assumption}\label{ass}
${\Greekmath 0272}_i f_i(\mathbf{x})$ for $i\in\mathcal{V}$ are uniformly bounded, i.e., there exists a positive constant $C$ such that for $\mathbf{x}\in \mathbb{R}^N,$
$|{\Greekmath 0272}_i f_i(\mathbf{x})|\leq C,$ $\forall i\in\mathcal{V}.$
\end{Assumption}
\section{Privacy-preserving Nash equilibrium seeking under fixed communication graphs}\label{main}
In this section, a privacy-preserving distributed Nash equilibrium seeking strategy will be proposed for games under a fixed undirected communication graph $\mathcal{G}$, which is defined as $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}.$ Moreover, $\mathcal{V}=\{1,2,\cdots,N\}$ denotes the set of vertices and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges. In this way, player $j$ can communicate with player $i$ if and only if $(j,i)\in \mathcal{E}.$ Associate with $\mathcal{G}$ a weight matrix $\mathcal{A}=[a_{ij}]$ whose element on the $i$th row and $j$th column is $a_{ij}$. In this section, we suppose that $\mathcal{G}$ and $\mathcal{A}$ satisfy the following assumption:
\begin{Assumption}\label{comm}
The communication graph $\mathcal{G}$ is undirected and connected. Moreover, $\mathcal{A}$ satisfies
\begin{enumerate}
\item $a_{ii}>0,$ $a_{ij}>0$ if $(j,i)\in \mathcal{E}$ and $a_{ij}=0$ if $(j,i)\notin \mathcal{E}.$
\item $\mathcal{A}$ is doubly stochastic, i.e., $\mathcal{A}\mathbf{1}_N=\mathbf{1}_N$ and $\mathbf{1}_N^T\mathcal{A}=\mathbf{1}_N^T$.
\end{enumerate}
\end{Assumption}
Let $\{{\Greekmath 0115}_i\}_{i=1}^N$ be the eigenvalues of $\mathcal{A}$ and suppose that ${\Greekmath 0115}_1\geq {\Greekmath 0115}_2\geq \cdots, \geq {\Greekmath 0115}_N.$ Then, by Assumption \ref{comm}, ${\Greekmath 0115}_1=1,$ ${\Greekmath 0115}_2<1$, ${\Greekmath 0115}_N>-1$.
Moreover, the following lemma holds.
\begin{Lemma}\label{lemam1}
\cite{Xiao}\cite{Johansson} Suppose that Assumption \ref{comm} is satisfied. Then,
\begin{equation}
\left|\left|\left(\mathcal{A}^k-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right)\right|\right|=\left|\left|\left(\mathcal{A}-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right)^k\right|\right|
\leq {\Greekmath 010D}^k,
\end{equation}
where ${\Greekmath 010D}$ is a positive constant such that
\begin{equation}
{\Greekmath 011A}\left(\mathcal{A}-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right)\leq {\Greekmath 010D} <1,
\end{equation}
and ${\Greekmath 011A}\left(\mathcal{A}-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right)$ is the spectral radius of $\mathcal{A}-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}.$
\end{Lemma}
\subsection{Method development}
As there is no central authority to broadcast $\bar{x}$ to the players, we design a strategy by utilizing a synthesis of consensus protocols and optimization algorithms as in \cite{KoshalOR16}-\cite{YEcyber17}. Moreover, to protect the players' privacy, we utilize independent random variables to perturb the transmitted information among the players in the average consensus protocol. More specifically, each player $i,i\in\mathcal{V}$ can update its action according to
\begin{equation}\label{al1}
x_i(k+1)=x_i(k)-{\Greekmath 010B}_kg_i(x_i(k),y_i(k)),
\end{equation}
where $k\in \mathbb{Z}^+$, ${\Greekmath 010B}_k=cq^k$, $c\in \mathbb{R}_+,$ $q\in(0,1)$ and $g_i(x_i,y_i)=\left(\frac{\partial \tilde{f}_i(x_i,\bar{x})}{\partial x_i}+\frac{\partial \tilde{f}_i(x_i,\bar{x})}{\partial \bar{x}}\frac{\partial \bar{x}}{\partial x_i}\right)\left.\right|_{\bar{x}=y_i}$. Moreover, $y_i(k)$ is an intermediate variable updated according to
\begin{equation}\label{al2}
y_i(k+1)=\sum_{j=1}^{N}a_{ij}p_j(k)+x_i(k+1)-x_i(k),
\end{equation}
in which $y_i(0)=x_i(0)$, $p_i(k)=y_i(k)+w_i(k)$, $w_i(k)\sim Lap({\Greekmath 0112}_{k})$ for $i\in\mathcal{V}$ are independent random variables, ${\Greekmath 0112}_{k}=d \bar{q}^k$, $d\in \mathbb{R}_+$ and $\bar{q}\in (q,1)$.
The steps of the designed algorithm are described as follows.\\
\ \\
\begin{tabular}{l}
\toprule
Privacy-preserving distributed Nash seeking:\\
\midrule
\RIfM@\expandafter\text@\else\expandafter\mbox\fibf{Initialization:} Choose $x_i(0)\in \mathbb{R}$ and $y_i(0)= x_i(0)$.\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fibf{Iterations:}\\
1. Define $p_i(k)=y_i(k)+w_i(k)$ \\
2. Update $x_i(k)$ according to\\
$\ \ \ \ x_i(k+1)=x_i(k)-{\Greekmath 010B}_k g_i(x_i(k),y_i(k))$\\
3. Update $y_i(k)$ according to\\
$\ \ \ \ y_i(k+1)=\sum_{j=1}^{N}a_{ij}p_j(k)+x_i(k+1)-x_i(k)$\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fibf{end}\\
\bottomrule
\end{tabular}
\\
Let $\mathbf{x}(k)=[x_1(k),x_2(k),\cdots,x_N(k)]^T,\mathbf{y}(k)=[y_1(k),y_2(k),\cdots,y_N(k)]^T,$ $\mathbf{w}(k)=[w_1(k),w_2(k),\cdots,w_N(k)]^T$ and $\mathbf{p}(k)=[p_1(k),p_2(k),\cdots,p_N(k)]^T.$
Then, the concatenated vector form of \eqref{al1}-\eqref{al2} is
\begin{equation}\label{cotn}
\begin{aligned}
\mathbf{x}(k+1)&=\mathbf{x}(k)-{\Greekmath 010B}_k [g_i(x_i(k),y_i(k))]_{vec}\\
\mathbf{y}(k+1)&=\mathcal{A}\mathbf{p}(k)+\mathbf{x}(k+1)-\mathbf{x}(k).
\end{aligned}
\end{equation}
\begin{Remark}
In this paper, we suppose that there is no central authority to broadcast $\bar{x}(k)$ to the players. Hence, player $i,i\in\mathcal{V}$ would generate a local variable $y_i(k)$ to estimate $\bar{x}(k)$. Moreover, the update of $y_i(k)$ in \eqref{al2} is motivated by the dynamic average consensus protocol in \cite{YEcyber17}\cite{ZhuAT10}. However, in the dynamic average consensus protocol of \cite{YEcyber17}, the players communicate with their neighbors on their estimates of $\bar{x}(k),$ which may raise privacy concern. Hence, in \eqref{al2}, we perturb their transmitted information by utilizing independent random variables. Moreover, different from the continuous-time scenario proposed in our previous work in \cite{YEcyber17}, we adopt a decaying stepsize in the presented algorithm to release the side-effect of the random noises on the convergence properties to some extent.
\end{Remark}
\begin{Remark}
In \cite{HeACC17}, the authors provided necessary and sufficient conditions for general noise adding mechanisms to achieve differential privacy. It was shown that the probability density function of the added noises should have zero measure for the set of zero-points and the relative probability density in the considered adjacent sets should be upper bounded by a positive constant (see the conditions in Theorem 3.1 of \cite{HeACC17} for accurate mathematical descriptions on the conditions). In addition, Laplace distribution was shown to satisfy the given conditions. Hence, we follow the existing works to adopt noises drawn from Laplace distributions with a decaying parameter to mask the transmitted information.
\end{Remark}
\subsection{Analysis on the disagreement of estimates}
In this section, we provide a bound for the expectation of the absolute difference between $y_i(k)$ and the actual aggregate action $\bar{x}(k).$
The following lemma is given to support the quantification of the estimation error.
\begin{Lemma}\label{lemm}
Suppose that Assumption \ref{comm} is satisfied. Then, for each nonnegative integer $k,$
\begin{equation}
\mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k)-\mathbf{1}_N^T\mathbf{x}(k)|)\leq \frac{Nd(1-\bar{q}^{k})}{1-\bar{q}}.
\end{equation}
\end{Lemma}
\begin{proof}
By \eqref{cotn}, we see that
\begin{equation}\label{ex}
\mathbf{1}_N^T\mathbf{y}(k+1)=\mathbf{1}_N^T (\mathbf{y}(k)+\mathbf{w}(k))+\mathbf{1}_N^T(\mathbf{x}(k+1)-\mathbf{x}(k)).
\end{equation}
Hence,
\begin{equation}
\begin{aligned}
&\mathbf{1}_N^T\mathbf{y}(k+1)-\mathbf{1}_N^T\mathbf{x}(k+1)\\
=&\mathbf{1}_N^T (\mathbf{y}(k)-\mathbf{x}(k))+\mathbf{1}_N^T\mathbf{w}(k),
\end{aligned}
\end{equation}
and
\begin{equation}\label{expp}
\begin{aligned}
&|\mathbf{1}_N^T\mathbf{y}(k+1)-\mathbf{1}_N^T\mathbf{x}(k+1)|\\
\leq &|\mathbf{1}_N^T (\mathbf{y}(k)-\mathbf{x}(k))|+|\mathbf{1}_N^T\mathbf{w}(k)|.
\end{aligned}
\end{equation}
Taking expectations on both sides of \eqref{expp} gives
\begin{equation}
\begin{aligned}
&\mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k+1)-\mathbf{1}_N^T\mathbf{x}(k+1)|)\\
\leq & \mathbb{E}(|\mathbf{1}_N^T (\mathbf{y}(k)-\mathbf{x}(k))|)+\mathbb{E}(|\mathbf{1}_N^T\mathbf{w}(k)|)\\
\leq &\mathbb{E}(|\mathbf{1}_N^T (\mathbf{y}(0)-\mathbf{x}(0))|) +\sum_{j=0}^k \mathbb{E}(|\mathbf{1}_N^T\mathbf{w}(j)|).
\end{aligned}
\end{equation}
Noticing that $\mathbf{y}(0)=\mathbf{x}(0),$
\begin{equation}
\begin{aligned}
&\mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k+1)-\mathbf{1}_N^T\mathbf{x}(k+1)|)\\
\leq &\sum_{j=0}^k \mathbb{E}(|\mathbf{1}_N^T\mathbf{w}(j)|)= \frac{Nd (1-\bar{q}^{k+1})}{1-\bar{q}},
\end{aligned}
\end{equation}
where the last inequality is derived by utilizing $\mathbb{E}(|w_i(j)|)=d\bar{q}^j$ for $i\in\mathcal{V}.$
\end{proof}
\begin{Remark}
Actually, by mathematical induction, it can be obtained that
\begin{equation}
\mathbb{E}(\mathbf{1}_N^T\mathbf{y}(k)-\mathbf{1}_N^T\mathbf{x}(k))=0,
\end{equation}
for all nonnegative integer $k$. However, due to the effect of the added noises, we can only conclude that $\mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k)-\mathbf{1}_N^T\mathbf{x}(k)|)$ is bounded by $\frac{Nd(1-\bar{q}^{k})}{1-\bar{q}}$ as indicated in Lemma \ref{lemm}. The bias of $\mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k)-\mathbf{1}_N^T\mathbf{x}(k)|)$ would result in a convergence error as indicated in the upcoming theorems.
\end{Remark}
Based on Lemma \ref{lemm}, the following theorem which establishes the estimation error bound can be obtained.
\begin{Theorem}\label{th3}
Suppose that Assumptions \ref{ass}-\ref{comm} are satisfied. Then, for each positive integer $k,$
\begin{equation}
\begin{aligned}
&\mathbb{E}(|y_i(k)-\bar{x}(k)|)\\
\leq &\frac{2\sqrt{N}(N-1)C_1}{N}{\Greekmath 010D}^{k}+\frac{2(N-1)\sqrt{N} d{\Greekmath 010D}}{({\Greekmath 010D}-\bar{q})N}({\Greekmath 010D}^{k}-\bar{q}^{k})\\
&+\frac{2 (N-1)\sqrt{N} C c ({\Greekmath 010D}^{k}-q^{k})}{({\Greekmath 010D}-q)N}+\frac{d (1-\bar{q}^{k})}{1-\bar{q}},
\end{aligned}
\end{equation}
where $i\in\mathcal{V},$ and $C_1=\max_{n\in\mathcal{V}}|y_n(0)|,$ $n\in\mathcal{V}.$
\end{Theorem}
\begin{proof}
For each positive integer $k$,
\begin{equation}
\begin{aligned}
&\mathbb{E}(|y_i(k)- \bar{x}(k)|)\\
= &\mathbb{E}(|y_i(k)-\frac{\mathbf{1}_N^T\mathbf{y}(k)}{N}+\frac{\mathbf{1}_N^T\mathbf{y}(k)}{N}-\frac{\mathbf{1}_N^T \mathbf{x}(k)}{N}|)\\
\leq & \frac{1}{N}\sum_{j=1,j\neq i}^N\mathbb{E}(|y_i(k)-y_j(k)|)\\
&+\frac{1}{N} \mathbb{E}(|\mathbf{1}_N^T\mathbf{y}(k)-\mathbf{1}_N^T\mathbf{x}(k)|).
\end{aligned}
\end{equation}
By \eqref{cotn}, we get that
\begin{equation}\label{ex22}
\begin{aligned}
&\mathbf{y}(k+1)=\mathcal{A}(\mathbf{y}(k)+\mathbf{w}(k))+\mathbf{x}(k+1)-\mathbf{x}(k)\\
=&\mathcal{A}^2 \mathbf{y}(k-1)+\mathcal{A}^2\mathbf{w}(k-1)+\mathcal{A} \mathbf{w}(k)\\
&+\mathcal{A}(\mathbf{x}(k)-\mathbf{x}(k-1))+\mathbf{x}(k+1)-\mathbf{x}(k).
\end{aligned}
\end{equation}
By repeating the above process, it can be obtained that
\begin{equation}\label{ex2}
\begin{aligned}
\mathbf{y}(k+1)=&\mathcal{A}^{k+1}\mathbf{y}(0)+\sum_{j=0}^k \mathcal{A}^{k+1-j}\mathbf{w}(j)\\
&+\sum_{j=1}^{k+1}\mathcal{A}^{k+1-j}(\mathbf{x}(j)-\mathbf{x}(j-1)).
\end{aligned}
\end{equation}
Hence, by \eqref{ex2},
\begin{equation}
\begin{aligned}
&|y_i(k+1)-y_j(k+1)|\\
\leq &\sum_{n=1}^N|[\mathcal{A}^{k+1}]_{in}-[\mathcal{A}^{k+1}]_{jn}||y_n(0)|\\
&+\sum_{n=1}^N\sum_{l=0}^k|[\mathcal{A}^{k+1-l}]_{in}-[\mathcal{A}^{k+1-l}]_{jn} | |w_n(l)|\\
&+\sum_{n=1}^N\sum_{l=1}^{k+1}|[\mathcal{A}^{k+1-l}]_{in}-[\mathcal{A}^{k+1-l}]_{jn}||x_n(l)-x_n(l-1)|.
\end{aligned}
\end{equation}
Note that
\begin{equation}
\begin{aligned}
&\sum_{n=1}^N\left|[\mathcal{A}^k]_{in}-\frac{1}{N}\right|\\
=&\sum_{n=1}^N\left|\left[\mathcal{A}^k-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right]_{in}\right|\leq \left|\left|\mathcal{A}^k-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right|\right|_{\infty}\\
\leq& \sqrt{N}\left|\left|\mathcal{A}^k-\frac{\mathbf{1}_N\mathbf{1}_N^T}{N}\right|\right|
\leq \sqrt{N}{\Greekmath 010D}^k,
\end{aligned}
\end{equation}
where the last inequality is obtained by Lemma \ref{lemam1}.
Hence,
\begin{equation}\label{exptt}
\begin{aligned}
&|y_i(k+1)-y_j(k+1)|\\
\leq & 2\sqrt{N}C_1 {\Greekmath 010D}^{k+1}+2\sqrt{N}C \sum_{l=1}^{k+1}{\Greekmath 010D}^{k+1-l}{\Greekmath 010B}_{l-1}\\
&+\sum_{n=1}^N\sum_{l=0}^k|[\mathcal{A}^{k+1-l}]_{in}-[\mathcal{A}^{k+1-l}]_{jn} | |w_n(l)|,
\end{aligned}
\end{equation}
where $C_1=\max_{n\in\mathcal{V}}|y_n(0)|$ as $|y_n(0)|$ for $n\in\mathcal{V}$ are bounded.
Taking expectations on both sides of \eqref{exptt} gives
\begin{equation}\label{exptt1}
\begin{aligned}
&\mathbb{E}(|y_i(k+1)-y_j(k+1)|)\\
\leq & 2\sqrt{N}C_1 {\Greekmath 010D}^{k+1}+2\sqrt{N}C \sum_{l=1}^{k+1}{\Greekmath 010D}^{k+1-l}{\Greekmath 010B}_{l-1}\\
&+\sum_{n=1}^N\sum_{l=0}^k|[\mathcal{A}^{k+1-l}]_{in}-[\mathcal{A}^{k+1-l}]_{jn} | \mathbb{E}(|w_n(l)|).
\end{aligned}
\end{equation}
Recalling that $\mathbb{E}(|w_n(l)|)=d\bar{q}^l,$ for $n\in\mathcal{V},$
\begin{equation}
\begin{aligned}
&\sum_{n=1}^N\sum_{l=0}^k|[\mathcal{A}^{k+1-l}]_{in}-[\mathcal{A}^{k+1-l}]_{jn} | \mathbb{E}(|w_n(l)|)\\
=&\frac{2\sqrt{N} d{\Greekmath 010D}}{{\Greekmath 010D}-\bar{q}}({\Greekmath 010D}^{k+1}-\bar{q}^{k+1}).
\end{aligned}
\end{equation}
Similarly,
\begin{equation}
2\sqrt{N} C \sum_{l=1}^{k+1}{\Greekmath 010D}^{k+1-l}{\Greekmath 010B}_{l-1}
=\frac{2 \sqrt{N} C c ({\Greekmath 010D}^{k+1}-q^{k+1})}{{\Greekmath 010D}-q}.
\end{equation}
Hence,
\begin{equation}
\begin{aligned}
&\mathbb{E}(|y_i(k)-\bar{x}(k)|)\\
\leq &\frac{2\sqrt{N}(N-1)C_1}{N}{\Greekmath 010D}^{k}+\frac{2(N-1)\sqrt{N} d{\Greekmath 010D}}{({\Greekmath 010D}-\bar{q})N}({\Greekmath 010D}^{k}-\bar{q}^{k})\\
&+\frac{2 (N-1)\sqrt{N} C c ({\Greekmath 010D}^{k}-q^{k})}{({\Greekmath 010D}-q)N}+\frac{d (1-\bar{q}^{k})}{1-\bar{q}},
\end{aligned}
\end{equation}
by further utilizing the results in Lemma \ref{lemm}.
\end{proof}
\subsection{Convergence analysis}
In this section, we establish the convergence results for the proposed method.
\begin{Theorem}\label{th1}
Suppose that Assumptions \ref{a2}-\ref{comm} are satisfied. Then,
\begin{equation}\label{hou}
\begin{aligned}
&\lim_{k\rightarrow \infty} \mathbb{E}(|| \mathbf{x}(k)-\mathbf{x}^*||^2)\\
\leq & C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1+\Phi_2,
\end{aligned}
\end{equation}
where
\begin{equation}\label{phi1}
\begin{aligned}
\Phi_1=&4\max_{i\in\mathcal{V}}\{l_i\}(N-1)\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\times\\
&\left(\frac{C_1c}{1-q{\Greekmath 010D}}+\frac{Cc^2q}{(1-q{\Greekmath 010D})(1-q^2)}\right),
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\Phi_2=&2\max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right)dcq \times\\
&\left(\frac{2(N-1){\Greekmath 010D}}{(1-q{\Greekmath 010D})(1-\bar{q}q)}+\frac{\sqrt{N}}{(1-q)(1-\bar{q}q)}\right),
\end{aligned}
\end{equation}
where $C_2=||\mathbf{x}(0)-\mathbf{x}^*||.$
\end{Theorem}
\begin{proof}
By \eqref{cotn}, we can get that
\begin{equation}
\mathbf{x}(k+1)-\mathbf{x}^*=\mathbf{x}(k)-\mathbf{x}^*-{\Greekmath 010B}_k [g_i(x_i(k),y_i(k))]_{vec}.
\end{equation}
Hence,
\begin{equation}
\begin{aligned}
&||\mathbf{x}(k+1)-\mathbf{x}^*||^2\\
=&||\mathbf{x}(k)-\mathbf{x}^*-{\Greekmath 010B}_k [g_i(x_i(k),y_i(k))]_{vec}||^2\\
=&||\mathbf{x}(k)-\mathbf{x}^*||^2-2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T [g_i(x_i(k),y_i(k))]_{vec}\\
&+ {\Greekmath 010B}_k^2 ||[g_i(x_i(k),y_i(k))]_{vec}||^2.
\end{aligned}
\end{equation}
Noticing that $g_i(x_i(k),y_i(k))$ is uniformly bounded by $C$, we can get that $||[g_i(x_i(k),y_i(k))]_{vec}||^2\leq NC^2.$ Hence
\begin{equation}
{\Greekmath 010B}_k^2 ||[g_i(x_i(k),y_i(k))]_{vec}||^2\leq {\Greekmath 010B}_k^2NC^2.
\end{equation}
Moreover,
\begin{equation}
\begin{aligned}
&-2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T [g_i(x_i(k),y_i(k))]_{vec}\\
=& -2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T g(\mathbf{x}(k))\\
&+2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T (g(\mathbf{x}(k))-[g_i(x_i(k),y_i(k))]_{vec})\\
\leq & -2{\Greekmath 010B}_km ||\mathbf{x}(k)-\mathbf{x}^*||^2\\
&+2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T (g(\mathbf{x}(k))-[g_i(x_i(k),y_i(k))]_{vec}).
\end{aligned}
\end{equation}
Noticing that $g_i(\mathbf{x}(k))$ is globally Lipschitz with constant $l_i,$ we get that
\begin{equation}
\begin{aligned}
&||g(\mathbf{x}(k))-[g_i(x_i(k),y_i(k))]_{vec}||\\
\leq& \max_{i\in\mathcal{V}}\{l_i\}||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||.
\end{aligned}
\end{equation}
Therefore,
\begin{equation}
\begin{aligned}
&-2{\Greekmath 010B}_k(\mathbf{x}(k)-\mathbf{x}^*)^T [g_i(x_i(k),y_i(k))]_{vec}\\
\leq & -2{\Greekmath 010B}_km ||\mathbf{x}(k)-\mathbf{x}^*||^2\\
&+2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}||\mathbf{x}(k)-\mathbf{x}^*||||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||,
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&||\mathbf{x}(k+1)-\mathbf{x}^*||^2\\
\leq &||\mathbf{x}(k)-\mathbf{x}^*||^2-2{\Greekmath 010B}_km ||\mathbf{x}(k)-\mathbf{x}^*||^2+{\Greekmath 010B}_k^2NC^2\\
&+2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}||\mathbf{x}(k)-\mathbf{x}^*||||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||.
\end{aligned}
\end{equation}
Moreover, as $\mathbf{x}(k+1)=\mathbf{x}(k)-{\Greekmath 010B}_k[g_i(x_i(k),y_i(k))]_{vec}$,
\begin{equation}
\begin{aligned}
||\mathbf{x}(k+1)-\mathbf{x}^*||\leq &||\mathbf{x}(k)-\mathbf{x}^*|| +{\Greekmath 010B}_k \sqrt{N} C\\
\leq & ||\mathbf{x}(0)-\mathbf{x}^*||+\frac{cC\sqrt{N}}{1-q}.
\end{aligned}
\end{equation}
Let $C_2=||\mathbf{x}(0)-\mathbf{x}^*||$, we have
\begin{equation}\label{expe}
\begin{aligned}
&||\mathbf{x}(k+1)-\mathbf{x}^*||^2\\
\leq &||\mathbf{x}(k)-\mathbf{x}^*||^2-2{\Greekmath 010B}_km ||\mathbf{x}(k)-\mathbf{x}^*||^2+{\Greekmath 010B}_k^2NC^2\\
&+2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||.
\end{aligned}
\end{equation}
Taking expectations on both sides of \eqref{expe} gives
\begin{equation}\label{exp}
\begin{aligned}
&\mathbb{E}(||\mathbf{x}(k+1)-\mathbf{x}^*||^2)\\
\leq & (1-2{\Greekmath 010B}_km)\mathbb{E}(||\mathbf{x}(k)-\mathbf{x}^*||^2)+{\Greekmath 010B}_k^2NC^2\\
&+2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\times\\
&\mathbb{E}(||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||).
\end{aligned}
\end{equation}
By Theorem \ref{th3},
\begin{equation}
\begin{aligned}
&2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\mathbb{E}(||\mathbf{y}(k)-\mathbf{1}_N\bar{x}(k)||)\\
\leq &2{\Greekmath 010B}_k\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)(2(N-1){\Greekmath 010D}^{k}C_1\\
&+\frac{2(N-1) d{\Greekmath 010D}}{{\Greekmath 010D}-\bar{q}}({\Greekmath 010D}^{k}-\bar{q}^{k})+\frac{2 (N-1) C c ({\Greekmath 010D}^{k}-q^{k})}{{\Greekmath 010D}-q}\\
&+\frac{\sqrt{N}d (1-\bar{q}^{k})}{1-\bar{q}}).
\end{aligned}
\end{equation}
Therefore,
\begin{equation}\label{expp}
\begin{aligned}
&\mathbb{E}(||\mathbf{x}(k+1)-\mathbf{x}^*||^2)\\
\leq & \prod_{j=0}^k(1-2{\Greekmath 010B}_jm)\mathbb{E}(||\mathbf{x}(0)-\mathbf{x}^*||^2)+\sum_{j=0}^k{\Greekmath 010B}_j^2NC^2\\
&+\sum_{j=0}^k 2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\left(2(N-1){\Greekmath 010D}^{j}C_1\right.\\
&+\frac{2(N-1) d{\Greekmath 010D}}{{\Greekmath 010D}-\bar{q}}({\Greekmath 010D}^{j}-\bar{q}^{j})+\frac{2 (N-1) C c ({\Greekmath 010D}^{j}-q^{j})}{{\Greekmath 010D}-q}\\
&\left.+\frac{\sqrt{N}d (1-\bar{q}^{j})}{1-\bar{q}}\right).
\end{aligned}
\end{equation}
Let $k\rightarrow \infty$, then
\begin{equation}\label{expp1}
\begin{aligned}
&\lim_{k\rightarrow \infty}\mathbb{E}(||\mathbf{x}(k)-\mathbf{x}^*||^2)\\
\leq & \prod_{j=0}^{\infty}(1-2{\Greekmath 010B}_jm)\mathbb{E}(||\mathbf{x}(0)-\mathbf{x}^*||^2)+\sum_{j=0}^{\infty}{\Greekmath 010B}_j^2NC^2\\
&+\sum_{j=0}^{\infty} 2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\left(2(N-1){\Greekmath 010D}^{j}C_1\right.\\
&+\frac{2(N-1) d{\Greekmath 010D}}{{\Greekmath 010D}-\bar{q}}({\Greekmath 010D}^{j}-\bar{q}^{j})+\frac{2 (N-1) C c ({\Greekmath 010D}^{j}-q^{j})}{{\Greekmath 010D}-q}\\
&\left.+\frac{\sqrt{N}d (1-\bar{q}^{j})}{1-\bar{q}}\right),
\end{aligned}
\end{equation}
in which
\begin{equation}
\begin{aligned}
&\sum_{j=0}^{\infty}2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\left(2(N-1){\Greekmath 010D}^{j}C_1\right)\\
= & 4(N-1) C_1 \max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right) \frac{c}{1-q {\Greekmath 010D}},
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\sum_{j=0}^{\infty} 2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}(C_2+\frac{cC\sqrt{N}}{1-q})\frac{2(N-1) d{\Greekmath 010D}}{{\Greekmath 010D}-\bar{q}}({\Greekmath 010D}^{j}-\bar{q}^{j})\\
= & 4(N-1)\max_{i\in\mathcal{V}}\{l_i\} d{\Greekmath 010D} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\frac{cq}{(1-q{\Greekmath 010D})(1-\bar{q}q)}.
\end{aligned}
\end{equation}
In addition,
\begin{equation}
\begin{aligned}
&\sum_{j=0}^{\infty}2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\frac{2 (N-1) C c ({\Greekmath 010D}^{j}-q^{j})}{{\Greekmath 010D}-q}\\
&=4\max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\left(\frac{(N-1)Cc^2q}{(1-q{\Greekmath 010D})(1-q^2)}\right),
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\sum_{j=0}^\infty 2{\Greekmath 010B}_j\max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\frac{\sqrt{N}d (1-\bar{q}^{j})}{1-\bar{q}}\\
=& 2 \max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right) \frac{\sqrt{N}dcq}{(1-q)(1-q\bar{q})}.
\end{aligned}
\end{equation}
Rearranging these terms gives
\begin{equation}
\begin{aligned}
&\lim_{k\rightarrow \infty} \mathbb{E}(|| \mathbf{x}(k)-\mathbf{x}^*||^2)\\
\leq & \prod_{j=0}^{\infty} (1-{\Greekmath 010B}_jm) \mathbb{E}(||\mathbf{x}(0)-\mathbf{x}^*||^2)\\
&+\frac{c^2NC^2}{1-q^2}+\Phi_1+\Phi_2\\
\leq & C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1+\Phi_2,
\end{aligned}
\end{equation}
in which the last inequality is derived by utilizing $1-{\Greekmath 010B}_jm\leq e^{-{\Greekmath 010B}_jm}$ and $\mathbb{E}(||\mathbf{x}(0)-\mathbf{x}^*||^2)\leq C_2^2.$ Moreover,
\begin{equation}
\begin{aligned}
\Phi_1=&4\max_{i\in\mathcal{V}}\{l_i\}(N-1)\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\times\\
&\left(\frac{C_1c}{1-q{\Greekmath 010D}}+\frac{Cc^2q}{(1-q{\Greekmath 010D})(1-q^2)}\right),
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\Phi_2=&2\max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right)dcq \times\\
&\left(\frac{2(N-1){\Greekmath 010D}}{(1-q{\Greekmath 010D})(1-\bar{q}q)}+\frac{\sqrt{N}}{(1-q)(1-\bar{q}q)}\right).
\end{aligned}
\end{equation}
\end{proof}
\begin{Remark}
From the above analysis, it is clear that $\lim_{k\rightarrow \infty} \mathbb{E}(|| \mathbf{x}(k)-\mathbf{x}^*||^2)$ is bounded by $C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1+\Phi_2,$ where $C_2^2e^{-\frac{mc}{1-q}}$ is resulted from the initial error. Moreover, $\frac{c^2NC^2}{1-q^2}$ and $\Phi_1$ depend on stepsize selection. In addition, $\Phi_2$ is resulted from the added noises $w_i(k),i\in\mathcal{V}$.
\end{Remark}
\subsection{Differential privacy}
In this section, we show that the proposed method is ${\Greekmath 010F}$-differentially private. To analyze the privacy level for the proposed Nash equilibrium seeking strategy, we consider two adjacent function sets $F^{(1)}=\{f_i^{(1)}\}_{i=1}^N$, $F^{(2)}=\{f_i^{(2)}\}_{i=1}^N$ where $f_i^{(1)}=f_i^{(2)},\forall i\neq i_0,$ and $f_i^{(1)}\neq f_i^{(2)}$ for some $i_0\in\mathcal{V}.$
Note that if ${\Greekmath 0272}_i f_i^{(1)}={\Greekmath 0272}_i f_i^{(2)}$ for $i=i_0,$ the two sequences generated by the proposed method under $F^{(1)}$ and $F^{(2)}$ would be the same by enforcing the added noises to be the same, indicating that the proposed method is of complete privacy. Hence, in the rest, we only consider the case in which ${\Greekmath 0272}_i f_i^{(1)}\neq {\Greekmath 0272}_i f_i^{(2)}$ for $i=i_0.$ In addition, the worst case scenario is considered, i.e., the adversary knows $\mathcal{A},\mathbf{x}(0),\{f_i\}_{i\neq i_0},{\Greekmath 010B}_k$ and the distributions of the random variables. Moreover, the observations are $\mathcal{O}=\{\mathcal{O}_k\}_{k=0}^{\infty},$ where $\mathcal{O}_k=\{\mathbf{p}(k)\}.$
Then, the following theorem can be derived.
\begin{Theorem}\label{the2}
Suppose that Assumption \ref{ass} is satisfied. Then, the proposed method in \eqref{al1}-\eqref{al2} is ${\Greekmath 010F}$-differentially private, where
\begin{equation}\label{priv}
{\Greekmath 010F}=\frac{2 cC\bar{q}}{d(\bar{q}-q)}.
\end{equation}
\end{Theorem}
\begin{proof}
For any fixed initial state $\mathbf{x}(0)$, we see from \eqref{al1}-\eqref{al2} that
\begin{equation}
x_i(1)=x_i(0)-{\Greekmath 010B}_0g_i(x_i(0),y_i(0)),
\end{equation}
and hence $x_i(1),$ and $x_i(1)-x_i(0)$ are fixed. Moreover,
\begin{equation}
y_i(1)=\sum_{j=1}^{N}a_{ij}p_j(0)+x_i(1)-x_i(0),
\end{equation}
and hence if $p_j(0)$ for $j\in\mathcal{V}$ are fixed, we get that $w_l(0)$ and $y_i(1)$ are fixed. Repeating the above analysis, we get that for any set of objective functions, if we fix the observation sequence, then, there exists a unique sequence $\mathbf{x}(k),\mathbf{y}(k),\mathbf{w}(k)$ for $k\in\{0,1,2,\cdots\}$ that can generate the sequence of observation. Hence, under the set of objective functions $F^{(l)},l\in\{1,2\}$, there exists a bijective mapping from the noise sequence to the set of observations. For notational convenience, denote the mapping as $\Omega^{(l)}(\mathbf{w}),l\in\{1,2\},$ respectively.
Moreover, for presentation clarity, we denote the sequence generated by the proposed method under $F^{(1)}$ as
\begin{equation}\label{a5l2}
\begin{aligned}
x_i^{(1)}(k+1)=&x_i^{(1)}(k)-{\Greekmath 010B}_kg_i^{(1)}(x_i^{(1)}(k),y_i^{(1)}(k)),\\
y_i^{(1)}(k+1)=&\sum_{j=1}^{N}a_{ij}p_j^{(1)}(k)+x_i^{(1)}(k+1)-x_i^{(1)}(k).
\end{aligned}
\end{equation}
Correspondingly, the sequence generated by the proposed method under $F^{(2)}$ is given by
\begin{equation}\label{a5l3}
\begin{aligned}
x_i^{(2)}(k+1)&=x_i^{(2)}(k)-{\Greekmath 010B}_kg_i^{(2)}(x_i^{(2)}(k),y_i^{(2)}(k)),\\
y_i^{(2)}(k+1)&=\sum_{j=1}^{N}a_{ij}p_j^{(2)}(k)+x_i^{(2)}(k+1)-x_i^{(2)}(k).
\end{aligned}
\end{equation}
As $x_i^{(1)}(0)=x_i^{(2)}(0),y_i^{(1)}(0)=y_i^{(2)}(0),$ it can be easily obtained that for $i\neq i_0,$
\begin{equation}
x_i^{(1)}(k)=x_i^{(2)}(k),y_i^{(1)}(k)=y_i^{(k)}(k),
\end{equation}
for all nonnegative integer $k,$ given that the two function sets generate the same observations.
Therefore, to ensure that $y_i^{(1)}(k)+w_i^{(1)}(k)=y_i^{(2)}(k)+w_i^{(2)}(k),$ we only need to enforce that,
\begin{equation}\label{dp1}
w_i^{(1)}(k)=w_i^{(2)}(k),
\end{equation}
for $i\neq i_0.$
Moreover, for $i=i_0$, we enforce that
\begin{equation}\label{dp2}
\Delta w_i(k)=-\Delta y_i(k),
\end{equation}
in which $\Delta w_i(k)=w_i^{(1)}(k)-w_i^{(2)}(k),\Delta y_i(k)=y_i^{(1)}(k)-y_i^{(2)}(k)$ and
\begin{equation}\label{dp3}
\begin{aligned}
\Delta y_i(k+1)&=\Delta x_i(k+1)-\Delta x_i(k)\\
&=- {\Greekmath 010B}_k \Delta g_i(k),
\end{aligned}
\end{equation}
where $\Delta g_i(k)=g_i^{(1)}(x^{(1)}_i(k),y_i^{(1)}(k))-g_i^{(2)}(x_i^{(2)}(k),y_i^{(2)}(k)),
$
and $\Delta x_i(k)=x_i^{(1)}(k)-x_i^{(2)}(k).$
Let $\mathbf{w}^{(l)}(k)=[w_1^{(l)}(k),w_2^{(l)}(k),\cdots,w_N^{(l)}(k)]^T$ and $\mathbf{w}^{(l)}=\{\mathbf{w}^{(l)}(k)\}_{k=0}^{\infty}$ for $l\in\{1,2\}$. Moreover, let $\mathcal{O}^{(l)}$ be the sequence of observations under the function set $F^{(l)}.$ According to \eqref{dp1}-\eqref{dp3}, we know that for each $\mathbf{w}^{(1)}$, there exists a unique $\mathbf{w}^{(2)}$ such that $\mathcal{O}^{(1)}=\mathcal{O}^{(2)}.$ Let $\mathcal{B}(\cdot)$ be a mapping such that $\mathbf{w}^{(2)}=\mathcal{B}(\mathbf{w}^{(1)})$ if and only if $\mathcal{O}^{(1)}=\mathcal{O}^{(2)}.$ Then, it is clear that $\mathcal{B}(\cdot)$ is bijective.
Let $\Gamma^{(l)}=\{\mathbf{w}^{(l)}|\Omega^{(l)}(\mathbf{w})\in\mathcal{O}\},$ then, following the analysis in \cite{DingCDC18}, it can be obtained that
\begin{equation}
\frac{\mathbb{P}\{F^{(1)}|\mathcal{O}\}}{\mathbb{P}\{F^{(2)}|\mathcal{O}\}}=
\frac{\int_{\Gamma^{(1)}}\prod_{i=1}^N \prod_{k=0}^{\infty} \mathcal{L}(w_i^{(1)}(k),{\Greekmath 0112}_k)d\mathbf{w}^{(1)} }{\int_{\Gamma^{(2)}}\prod_{i=1}^N \prod_{k=0}^{\infty} \mathcal{L}(\mathcal{B}(w_i^{(1)}(k)),{\Greekmath 0112}_k)d\mathbf{w}^{(1)}},
\end{equation}
in which
\begin{equation}
\begin{aligned}
&\frac{\prod_{i=1}^N \prod_{k=0}^{\infty} \mathcal{L}(w_i^{(1)}(k),{\Greekmath 0112}_k)}{\prod_{i=1}^N \prod_{k=0}^{\infty} \mathcal{L}(\mathcal{B}(w_i^{(1)}(k)),{\Greekmath 0112}_k)}\\
\leq &\prod_{i=1}^N \prod_{k=0}^{\infty} e^{\frac{|p_i^{(1)}(k)-y_i^{(1)}(k)-(p_i^{(2)}(k)-y_i^{(2)}(k))|}{{\Greekmath 0112}_k}}\\
=&\prod_{k=0}^{\infty}e^{\frac{|\Delta w_{i_0}(k)|}{{\Greekmath 0112}(k)}}\leq e^{\sum_{k=0}^{\infty}\frac{|\Delta y_{i_0}(k)|}{{\Greekmath 0112}(k)}}.
\end{aligned}
\end{equation}
By the boundedness of the gradient value, we get that
$|\Delta y_{i_0}(k)|\leq 2 C {\Greekmath 010B}_k.$
Therefore,
\begin{equation}
e^{\sum_{k=0}^{\infty}\frac{|\Delta y_{i_0}(k)|}{d\bar{q}^k}}\leq e^ {\sum_{k=0}^{\infty}\frac{{2 C {\Greekmath 010B}_k}}{d\bar{q}^k}}.
\end{equation}
By further noticing that $\bar{q}\in(q,1),$ we get that
\begin{equation}
e^{\sum_{k=0}^{\infty}\frac{|\Delta y_{i_0}(k)|}{d\bar{q}^k}}\leq e^{\frac{2Cc\bar{q}}{d(\bar{q}-q)}}.
\end{equation}
Note that $i_0\in \mathcal{V}$ stands for any player engaged in the game, we can conclude that the privacy level of the whole system is ${\Greekmath 010F}=\frac{2Cc\bar{q}}{d(\bar{q}-q)},$ i.e., the proposed method is ${\Greekmath 010F}$-differentially private.
\end{proof}
\begin{Remark}
In many existing works on distributed optimization and Nash equilibrium seeking, the decaying stepsize is required to satisfy
\begin{equation}
\sum_{k=0}^{\infty} {\Greekmath 010B}_k=\infty, \sum_{k=0}^{\infty} {\Greekmath 010B}_k^2<\infty,
\end{equation}
to establish the convergence results (see, e.g., \cite{KoshalOR16}\cite{Ram}). However, in this paper, we adopt a decaying stepsize that satisfies
\begin{equation}
\sum_{k=0}^{\infty} {\Greekmath 010B}_k<\infty.
\end{equation}
Note that this is required to establish the ${\Greekmath 010F}$-differential privacy of the proposed method, i.e., differential privacy is not achievable with not summable stepsize in the proposed method. Therefore, the convergence accuracy is sacrificed to some extent for differential privacy. Note that this is in accordance with the ``impossibility result for $0$-LAS message perturbing algorithms" in \cite{Nozari} and some other existing results on differential privacy (see, e.g., \cite{Huang15}). The tradeoffs between the convergence property and differential privacy will be discussed in more details later.
\end{Remark}
\begin{Remark}
In reality, communication channels are often subject to various kinds of noises during information dissemination (see e.g., \cite{LiTAC2010}-\cite{LongIJRNC} and the references therein). Therefore, if we can do some experiments to investigate the characteristics and properties of communication noises in real communication channels, it would be an interesting open question to study whether it is possible to employ the noises in the communication channels to achieve differentially private Nash equilibrium seeking or not.
\end{Remark}
\subsection{Tradeoffs between the accuracy and privacy level}
From \eqref{hou}, it can be seen that
\begin{equation}
\lim_{k\rightarrow \infty} \mathbb{E}(|| \mathbf{x}(k)-\mathbf{x}^*||^2)\leq D,
\end{equation}
where $D=C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1+\Phi_2$ and $\Phi_2$ is resulted from the added noises.
By Theorem \ref{the2}, ${\Greekmath 010F}=\frac{2 cC\bar{q}}{d(\bar{q}-q)}$. Therefore,
\begin{equation}
d=\frac{2 cC\bar{q}}{{\Greekmath 010F}(\bar{q}-q)}.
\end{equation}
To see the tradeoff between the convergence accuracy and the privacy level, the accuracy bound $D$ can be restated in terms of ${\Greekmath 010F}$ as:
\begin{equation}\label{hou}
\begin{aligned}
D= & C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1+\tilde{\Phi}_2,
\end{aligned}
\end{equation}
where $\Phi_1$ is defined in \eqref{phi1} and,
\begin{equation}
\begin{aligned}
\tilde{\Phi}_2=&2\max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right) \frac{2C c^2q\bar{q}}{{\Greekmath 010F}(\bar{q}-q)}\times\\
&\left(\frac{2(N-1){\Greekmath 010D}}{(1-q{\Greekmath 010D})(1-\bar{q}q)}+\frac{\sqrt{N}}{(1-q)(1-\bar{q}q)}\right).
\end{aligned}
\end{equation}
From \eqref{hou}, it can be seen that with $q,\bar{q},c$ being fixed, $D=O(\frac{1}{{\Greekmath 010F}}),$ indicating that the accuracy of the proposed method becomes arbitrarily bad when the method is of complete privacy. Hence, there is a tradeoff between the convergence accuracy and the privacy level. However, the presented result preserves the following properties:
\begin{enumerate}
\item For any given ${\Greekmath 010F}>0,$ $c,d,\bar{q},q$ can be tuned such that the proposed method is ${\Greekmath 010F}$-differentially private;
\item If privacy is not concerned, then, for any given positive constant $D$, $c,\bar{q},q$ can be tuned such that $\lim_{k\rightarrow\infty} \mathbb{E}(||\mathbf{x}-\mathbf{x}^*||^2)\leq D.$
\end{enumerate}
The first property can be easily obtained by \eqref{priv}. The second property is derived following the subsequent observations.
\begin{itemize}
\item For any bounded initial condition, $C_2^2e^{-\frac{mc}{1-q}}$ can be tuned to be arbitrarily small by adjusting $c$ and $q$ such that $\frac{c}{1-q}$ is sufficiently large;
\item Denote $\frac{c}{1-q}={\Greekmath 0110}$. Then, for fixed and sufficiently large ${\Greekmath 0110}$,
\begin{equation}
\frac{c}{1-q{\Greekmath 010D}}=\frac{{\Greekmath 0110} (1-q)}{1-q{\Greekmath 010D}},
\end{equation}
and
\begin{equation}
\frac{c^2}{1-q^2}=\frac{{\Greekmath 0110}^2(1-q)}{1+q}.
\end{equation}
Noticing that the partial derivatives of $\frac{{\Greekmath 0110} (1-q)}{1-q{\Greekmath 010D}}$ and$\frac{{\Greekmath 0110}^2(1-q)}{1+q}$ with respect to $q$ are negative for $q\in(0,1)$ and $\lim_{q\rightarrow 1}\frac{{\Greekmath 0110} (1-q)}{1-q{\Greekmath 010D}}=0$, $\lim_{q\rightarrow 1}\frac{{\Greekmath 0110}^2(1-q)}{1+q}=0,$ it can be concluded that $c$ and $q$ can be adjusted such that $\frac{c}{1-q{\Greekmath 010D}}$ and $\frac{c^2}{1-q^2}$ are sufficiently small with fixed ${\Greekmath 0110}$. Therefore, by such a tuning rule, $C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\Phi_1$ can be adjusted to be arbitrarily small.
\item $\tilde{\Phi}_2$ depends on the level of privacy ${\Greekmath 010F}$ and if privacy is not of concern, ${\Greekmath 010F}$ can be chosen to be sufficiently large such that $\tilde{\Phi}_2$ is sufficiently small.
\end{itemize}
From \eqref{hou}, we see that the accuracy of the proposed method depends on $c, {\Greekmath 010F}, \bar{q}$ and $q$. Hence, we write $D$ as $D(c,q,\bar{q},{\Greekmath 010F}).$ With fixed requirement of privacy level (i.e., ${\Greekmath 010F}$), one can tune $c,q,\bar{q}$ by solving
\begin{equation}\label{mini}
\begin{aligned}
&\min_{c,q,\bar{q}} \ \ \ \ D(c,q,\bar{q}).\\
&\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s.t.} \ \ \ \ c>0,q\in(0,1),\bar{q}\in(q,1).
\end{aligned}
\end{equation}
Moreover, noticing that for any fixed $c$ and $q$,
\begin{equation}
\frac{\partial D}{\partial \bar{q}}<0,
\end{equation}
indicating that $D$ is monotonically decreasing with any fixed $c,q$. Hence, one can choose $\bar{q}$ to be sufficiently close to $1$ as $\bar{q}\in(q,1)$ and solve
\begin{equation}\label{mini1}
\begin{aligned}
&\min_{c,q} \ \ \ \ D(c,q)\\
&\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s.t.} \ \ \ \ c>0,q\in(0,\bar{q}),
\end{aligned}
\end{equation}
to get the optimal $c,q.$
\section{Extensions to time-varying communication topologies}\label{time}
In the previous section, we suppose that the communication topology among the players is fixed, undirected and connected for presentation clarity. Actually, the proposed method can be slightly adapted to accommodate time-varying communication graphs that satisfy the following conditions:
\begin{Assumption}\label{as5}
There exists a positive integer $z$ such that $(\mathcal{V},\bigcup_{l=1}^z \mathcal{E}_{l+k})$ is connected for all nonnegative integer $k$, in which $\mathcal{G}_l=(\mathcal{V},\mathcal{E}_l)$ is the undirected communication graph at time $l$ and $\mathcal{E}_l$ is the corresponding edge set at time $l$.
\end{Assumption}
Note that in Section \ref{main}, the communication graph is supposed to be connected at each time instant $k=\{0,1,2,\cdots\}.$ However, in this section, the communication condition is generalized to be time-varying and it is only required that there exists a positive integer $z$ such that the joint graph $\bigcup_{l=1}^z \mathcal{G}_{l+k}$ is connected for nonnegative integer $k$.
\begin{Assumption}\label{as6}
Let $\mathcal{A}(l)$ be the weight matrix associated with $\mathcal{G}_l$ that satisfies:
\begin{enumerate}
\item There exits a positive constant ${\Greekmath 010E}$ such that $a_{ii}(l)>{\Greekmath 010E}$, $a_{ij}(l)>{\Greekmath 010E}$ if $(j,i)\in\mathcal{E}_l$ and $a_{ij}(l)=0,$ if $(j,i)\notin\mathcal{E}_l$;
\item $\mathcal{A}(l)$ is doubly stochastic, i.e., $\mathbf{1}_N^T\mathcal{A}(l)=\mathbf{1}_N^T,$ $\mathcal{A}(l)\mathbf{1}_N=\mathbf{1}_N.$
\end{enumerate}
\end{Assumption}
Let $\Psi(k,s)=\mathcal{A}(k)\mathcal{A}(k-1)\cdots \mathcal{A}(s)$ where $k\geq s\geq 0.$ Then, under Assumptions \ref{as5}-\ref{as6}, the following lemma can be obtained.
\begin{Lemma}
\cite{NEDIC08} Let Assumptions \ref{as5}-\ref{as6} be satisfied. Then,
\begin{enumerate}
\item $\lim_{k\rightarrow \infty} \Psi(k,s)=\frac{\mathbf{1}_N\mathbf{1}_N^T}{N},$ $\forall s\geq 0;$
\item For $k\geq s \geq 0,$ $ |[\Psi(k,s)]_{ij}-\frac{1}{N}|\leq {\Greekmath 0112} {\Greekmath 010C}^{k+1-s},$
where ${\Greekmath 0112}=(1-\frac{{\Greekmath 010E}}{4N^2})^{-2},$ ${\Greekmath 010C}=(1-\frac{{\Greekmath 010E}}{4N^2})^{\frac{1}{z}}.$
\end{enumerate}
\end{Lemma}
With time-varying communication topologies, the Nash equilibrium seeking strategy should be adapted as
\begin{equation}\label{cotn1}
\begin{aligned}
\mathbf{x}(k+1)&=\mathbf{x}(k)-{\Greekmath 010B}_k [g_i(x_i(k),y_i(k))]_{vec}\\
\mathbf{y}(k+1)&=\mathcal{A}(k)\mathbf{p}(k)+\mathbf{x}(k+1)-\mathbf{x}(k).
\end{aligned}
\end{equation}
Note that the variations of the communication graph will only affect the convergence results but not the differential privacy of the proposed method. Hence, in the following, only the convergence results are presented.
First, following the proof of Theorem \ref{th3}, the subsequent result can be obtained.
\begin{Theorem}\label{col1}
Suppose that Assumptions \ref{as5}-\ref{as6} are satisfied. Then, for each positive integer $k,$
\begin{equation}
\begin{aligned}
&\mathbb{E}(|y_i(k)-\bar{x}(k)|)\\
\leq &2(N-1){\Greekmath 0112} {\Greekmath 010C}^{k}C_1+\frac{2(N-1){\Greekmath 0112} d{\Greekmath 010C}}{{\Greekmath 010C}-\bar{q}}({\Greekmath 010C}^{k}-\bar{q}^{k})\\
&+\frac{2{\Greekmath 0112} (N-1) C c ({\Greekmath 010C}^{k}-q^{k})}{{\Greekmath 010C}-q}+\frac{d (1-\bar{q}^{k})}{1-\bar{q}}.
\end{aligned}
\end{equation}
for $i\in\mathcal{V}.$
\end{Theorem}
\begin{Remark}
From Theorem \ref{col1}, it can be seen that compared with the results under fixed communication topologies, the time-varying communication topology would affect the estimation speed.
\end{Remark}
Moreover, following Theorem \ref{th1}, the subsequent convergence result can be obtained.
\begin{Theorem}\label{col2}
Suppose that Assumptions 1-3 and \ref{as5}-\ref{as6} are satisfied. Then,
\begin{equation}
\begin{aligned}
&\lim_{k\infty} \mathbb{E}(|| \mathbf{x}(k)-\mathbf{x}^*||^2)\\
\leq & C_2^2e^{-\frac{mc}{1-q}}+\frac{c^2NC^2}{1-q^2}+\bar{\Phi}_1+\bar{\Phi}_2,
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
\bar{\Phi}_1=&4(N-1)\sqrt{N}{\Greekmath 0112} \max_{i\in\mathcal{V}}\{l_i\}\left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\\
&\left(\frac{C_1c}{1-q{\Greekmath 010C}}+\frac{Cc^2q}{(1-{\Greekmath 010C} q)(1-q^2)}\right),
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\bar{\Phi}_2=&2\max_{i\in\mathcal{V}}\{l_i\} \left(C_2+\frac{cC\sqrt{N}}{1-q}\right)\sqrt{N}dcq\times\\ &\left(\frac{2(N-1){\Greekmath 0112}{\Greekmath 010C}}{(1-q{\Greekmath 010C})(1-\bar{q}q)}+\frac{1}{(1-q)(1-\bar{q}q)}\right).
\end{aligned}
\end{equation}
\end{Theorem}
\section{A Numerical example}\label{num_ex}
In this section, we consider the $5$-player energy consumption game studied in \cite{YEcyber17}. In the energy consumption game, player $i$'s objective function is given by
\begin{equation}
f_i(\mathbf{x})=(x_i-\hat{x}_i)^2+(0.04\sum_{i=1}^5x_i+5)x_i,
\end{equation}
in which $\hat{x}_1=50,\hat{x}_2=55,\hat{x}_3=60,\hat{x}_4=65,\hat{x}_5=70.$
As demonstrated in \cite{YEcyber17}, the game has a unique pure-strategy Nash equilibrium at
$\mathbf{x}^*=(41.5,46.4,51.3,56.2,61.1).$ In the following, fixed communication topologies and time-varying communication topologies will be considered successively.
\subsection{Nash equilibrium seeking under fixed communication topologies}\label{fix}
In the simulation, the players are supposed to communicate via a cycle depicted in Fig. \ref{comm1}.
\begin{figure}
\caption{The communication graph for the players.}
\label{comm1}
\end{figure}
Correspondingly, the weight matrix is given as
\begin{equation}\nonumber
\mathcal{A}=\left[
\begin{array}{ccccc}
0.5 & 0.2 & 0 & 0 & 0.3 \\
0.2 & 0.5 & 0.3 & 0 & 0 \\
0 & 0.3 & 0.5 & 0.2 & 0 \\
0 & 0 & 0.2 & 0.5 & 0.3 \\
0.3 & 0 & 0 & 0.3 & 0.4 \\
\end{array}
\right],
\end{equation}
and $c=1,q=0.9,d=1,\bar{q}=0.99.$ Moreover, the proposed method is run for $2000$ times for the observation of the simulation results. Fig. \ref{simula1} shows the expectations of the players' squared Nash equilibrium seeking errors, i.e., $\mathbb{E}((x_i(k)-x_i^*)^2)$ for $i\in\{1,2,\cdots,5\}$ from which we see that the proposed method drives $\mathbb{E}((x_i(k)-x_i^*)^2)$ to a small neighborhood of zero.
Fig. \ref{simulas2} (1) plots $\mathbb{E}((y_i(k)-\bar{x}^*)^2)$ for $i\in\{1,2,\cdots,5\}$ generated by the proposed method and Fig. \ref{simulas2} (2) shows $\bar{y}_i(k)-\bar{x}^*$ for $i\in\{1,2,\cdots,5\}$, where $\bar{y}_i(k)$ denotes the averaged value of $y_i(k)$ for the $2000$ running times and $\bar{x}^*=\frac{1}{5}\sum_{i=1}^5x_i^*.$
\begin{figure}
\caption{The expectations of the players' squared Nash equilibrium seeking errors, i.e., $\mathbb{E}
\label{simula1}
\end{figure}
\begin{figure}
\caption{(1): The expectations of the players' squared estimation errors on $\bar{x}
\label{simulas2}
\end{figure}
To show the tradeoff between the convergence accuracy and $d$, we vary $d$ from $0$ to $3$ and observe the simulation results. Fig. \ref{simulas3} plots $\mathbb{E}(||\mathbf{x}(\infty)-\mathbf{x}^*||^2).$ Roughly speaking, the figure shows that as $d$ increases, the upper bound of $\mathbb{E}(||\mathbf{x}(\infty)-\mathbf{x}^*||^2)$ increases, which illustrates the tradeoff between the privacy level and the convergence accuracy.
\begin{figure}
\caption{The plot of $\mathbb{E}
\label{simulas3}
\end{figure}
To observe the data distributions generated by the proposed method, we fix $d=1,\bar{q}=0.99$ and run the proposed method for $20000$ times. Fig. \ref{simulas4} plots the data distributions of $x_i(\infty)$ and its corresponding fitted density functions for $i\in\{1,2,\cdots,5\}$. Fig. \ref{simulas5} shows the data distributions of $y_i(\infty)$ and the plots of the corresponding fitted density functions for players $1$-$5,$ respectively. From Figs. \ref{simulas4}-\ref{simulas5}, we see that the players' actions converge to a small neighborhood of the Nash equilibrium with a high probability under the given parameters.
\begin{figure}
\caption{The plots of data distributions of $x_i(\infty)$ and the corresponding fitted density functions for players $1$-$5$, generated by the proposed method in \eqref{al1}
\label{simulas4}
\end{figure}
\begin{figure}
\caption{The plots of data distributions of $y_i(\infty)$ and the corresponding fitted density functions for players $1$-$5$, generated by the proposed method in \eqref{al1}
\label{simulas5}
\end{figure}
\subsection{Nash equilibrium seeking under time-varying communication topologies}
In the simulation, we suppose that the communication topology among the players switches between the two graphs depicted in Fig. \ref{comm2}. Correspondingly, we define
\begin{equation}\nonumber
\mathcal{A}(k)=\left[
\begin{array}{ccccc}
0.3 & 0.3 & 0 & 0 & 0.4 \\
0.3 & 0.5 & 0.2 & 0 & 0 \\
0 & 0.2 & 0.5 & 0 & 0.3 \\
0 & 0 & 0 & 1 & 0 \\
0.4 & 0 & 0.3 & 0 & 0.3 \\
\end{array}
\right],
\end{equation}
for $k\in\{0,2,4,6,\cdots\}$. In addition,
\begin{equation}\nonumber
\mathcal{A}(k)=\left[
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0.7 & 0.3 \\
0 & 0 & 0 & 0.3 & 0.7 \\
\end{array}
\right],
\end{equation}
for $k\in\{1,3,5,7\cdots\}$. For comparison convenience, the parameters are chosen as those in Section \ref{fix}. Moreover, the proposed method is run for $2000$ times for the observation of the simulation results. Fig. \ref{simula1_time_varying} plots $\mathbb{E}((x_i(k)-x_i^*)^2),i\in\{1,2,\cdots,5\},$ and the two sub-figures in Fig. \ref{simulas2_time_varying} depict $\mathbb{E}((y_i(k)-\bar{x}^*)^2)$ and $\bar{y}_i(k)-\bar{x}^*$ for $i\in\{1,2,\cdots,5\},$ respectively. From Figs. \ref{simula1_time_varying}-\ref{simulas2_time_varying}, it can be seen that driven by the proposed method, $\mathbb{E}((x_i(k)-x_i^*)^2),$ $\mathbb{E}((y_i(k)-\bar{x}^*)^2)$ and $\bar{y}_i(k)-\bar{x}^*$ for $i\in\{1,2,\cdots,5\}$ would converge to a small neighborhood of zero. In addition, comparing Fig. \ref{simula1_time_varying}. (2) with Fig. \ref{simulas2}. (2), it is clear that the players can achieve the aggregate estimation at a faster speed under the fixed communication topology given in Fig. \ref{comm1}.
\begin{figure}
\caption{(a) is the communication graph for the players at $k=\{0,2,4,6,\cdots\}
\label{comm2}
\end{figure}
\begin{figure}
\caption{The expectations of the players' squared Nash equilibrium seeking errors, i.e., $\mathbb{E}
\label{simula1_time_varying}
\end{figure}
\begin{figure}
\caption{(1): The expectations of the players' squared estimation errors on $\bar{x}
\label{simulas2_time_varying}
\end{figure}
To show the tradeoff between the privacy level and the convergence accuracy under the time-varying communication topologies, we vary $d$ from $0$ to $3$. Correspondingly, the plot of $\mathbb{E}(||\mathbf{x}(\infty)-\mathbf{x}^*||^2)$ with different values of $d$ is given in Fig. \ref{simulas3_timevying}, from which we see that $\mathbb{E}(||\mathbf{x}(\infty)-\mathbf{x}^*||^2)$ increases with $d$.
\begin{figure}
\caption{The plot of $\mathbb{E}
\label{simulas3_timevying}
\end{figure}
\begin{figure}
\caption{The plots of data distributions of $x_i(\infty)$ and the corresponding fitted density functions for players $1$-$5,$ generated by the proposed method in \eqref{cotn1}
\label{simulas4_time_varying}
\end{figure}
\begin{figure}
\caption{The plots of data distributions of $y_i(\infty)$ and the corresponding fitted density functions for players $1$-$5,$ generated by the proposed method in \eqref{cotn1}
\label{simulas5_time_varying}
\end{figure}
Likewise, to observe the data distributions generated by the proposed method, we fix $d=1,\bar{q}=0.99$ and run the proposed method for $20000$ times. The data distributions of $x_i(\infty)$ and $y_i(\infty)$ for $i\in\{1,2,\cdots,5\}$, together with their corresponding fitted density functions are plotted in Figs. \ref{simulas4_time_varying}-\ref{simulas5_time_varying}. Figs. \ref{simulas4_time_varying}-\ref{simulas5_time_varying} show that the players' actions converge to a small neighborhood of the Nash equilibrium with a high probability under the given parameters.
\section{Conclusions}\label{conc}
This paper considers privacy-preservation in the distributed Nash equilibrium seeking problem for networked aggregative games. To estimate the averaged value of the players' actions, a dynamic average consensus protocol is employed in which the transmitted information is masked by independent random noises drawn from Laplace distributions. The random noises are included for the protection of the players' objective functions. Moreover, with the estimated information, the gradient descent method with a decaying stepsize is implemented to optimize the players' objective functions. The convergence property as well as the privacy level of the proposed method are analytically investigated. It is shown that there is a tradeoff between the convergence accuracy and the privacy level. Fixed communication topologies and time-varying communication topologies are addressed successively in the paper. Privacy-preserving Nash equilibrium seeking for more general games (see e.g., the games considered in \cite{YETAC17}-\cite{YETAC19}) will be included in our future works.
\end{document} |
\begin{document}
\title[Numerical study of the zeta derivative at zeros]{Numerical study of the derivative of the Riemann zeta function at zeros}
\author[G.A. Hiary]{Ghaith A. Hiary}
\thanks{Preparation of this material was partially supported by the National Science Foundation
under agreements No. DMS-0757627 (FRG grant) and DMS-0635607. Computations were carried out
at the Minnesota Supercomputing Institute.}
\address{School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW.}
\email{[email protected]}
\author[A.M. Odlyzko]{Andrew M. Odlyzko}
\thanks{}
\address{School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, 55455.}
\curraddr{}
\email{[email protected]}
\subjclass[2000]{Primary, Secondary}
\keywords{Riemann zeta function, derivative at zeros}
\maketitle
\begin{center}
{\em Dedicated to Professor Akio Fujii on his retirement.}
\end{center}
\begin{abstract}
The derivative of the Riemann zeta function
was computed numerically on several large sets of zeros at large heights.
Comparisons to known and conjectured
asymptotics are presented.
\end{abstract}
\section{Introduction}
Throughout this paper, we assume the truth of the Riemann Hypothesis (RH),
and we let $\gamma_n >0$ denote the ordinate of the $n$-th non-trivial zero of $\zeta(s)$.
Hejhal~\cite{He} assumed the RH and a weak consequence of Montgomery's~\cite{Mo}
pair-correlation conjecture,
namely that for some $\tau>0$, there is a constant $B$ such that
\begin{equation}
\limsup_{N\to \infty}\frac{1}{N}\left|\{n \,:\, N\le n\le 2N\,,\, (\gamma_{n+1}-\gamma_n)\log\gamma_n < c\}\right|\le B c^{\tau}\,,
\end{equation}
\noindent
holds for all $c\in (0,1)$. Under these assumptions, he
proved the following central limit theorem: for $\alpha <\beta$,
\begin{equation} \label{eq:derclt1}
\lim_{N\to \infty} \frac{1}{N}\,\left|\left\{ n\,:\, N\le n\le 2N\,,\, \frac{\log \left|\displaystyle \frac{2\pi \zeta'(1/2+i\gamma_n)}{\log (\gamma_n/2\pi)}\right|}{\sqrt{\frac{1}{2} \log \log N}}\in(\alpha,\beta)\right\}\right| =
\frac{1}{\sqrt{2\pi}} \int_{\alpha}^{\beta} e^{-x^2/2}\,dx\,,
\end{equation}
\noindent
So under these assumptions, $\log|\zeta'(1/2+i\gamma_n)|$,
suitably normalized, converges in distribution over fixed ranges to a standard normal variable.
To obtain more precise information about the tails of the distribution,
we consider the moments
\begin{equation} \label{eq:dermo1}
J_{\lambda}(T) := \frac{1}{N(T)} \sum_{0<\gamma_n\le T} |\zeta'(1/2+i\gamma_n)|^{2\lambda}\,,
\end{equation}
\noindent
where $N(T):=\sum_{0<\gamma_n\le T} 1= \frac{T}{2\pi}\log \frac{T}{2\pi e}+ O(\log T)$
is the zero counting function.
Notice that $J_{\lambda}(T)$ is defined for all $\lambda$ provided the zeros of $\zeta(s)$
are simple, as is widely believed.
Gonek~\cite{Go1}~\cite{Go2} carried out an extensive study of $J_{\lambda}(T)$.
He proved, under the assumption of the RH, that $J_1(T)\sim \frac{1}{12} (\log T)^3$ as
$T\to\infty$. It was suggested by Gonek~\cite{Go2}, and independently by Hejhal~\cite{He},
that $J_{\lambda}(T)$ is on the order of $(\log T)^{\lambda(\lambda+2)}$.
Ng~\cite{Ng} proved, under the RH, that $J_2(T)$ is order of $(\log T)^8$,
which is in agreement with that suggestion.
Hughes, Keating, and O'Connell~\cite{HKO}, applied the random matrix
philosophy (e.g. see~\cite{KS}), which
predicts that certain behaviors of $L$-functions are mimicked statistically by characteristic
polynomials of large matrices from the classical compact groups. This led them to
predict that for Re($\lambda$) $>-3/2$,
\begin{equation}\label{eq:hkc}
J_{\lambda}(T)\sim a(k)\frac{G^2(\lambda+2)}{G(2\lambda+3)}
\left(\log \frac{T}{2\pi}\right)^{\lambda(\lambda+2)} ~~~~~~{\hbox{as}} ~~~~~~~~~~T \to \infty,
\end{equation}
\noindent
where $G(z)$ is the Barnes G-function, and $a(k)$ is an ``arithmetic factor.''
The conjecture (\ref{eq:hkc}) is consistent with previous theorems and conjectures.
Recently, Conrey and Snaith~\cite{CS}, assuming the ratios conjecture,
gave lower order terms in asymptotic expansions for $J_1(T)$ and $J_2(T)$.
They conjectured the existence of certain polynomials $P_{\lambda}(x)$,
for $2\lambda=2$ and $2\lambda=4$, such that
\begin{equation}
\sum_{0<\gamma_n< T}|\zeta'(1/2+i\gamma_n)|^{2\lambda}\sim\int_0^T P_{\lambda}\left(\log \frac{t}{2\pi}\right)\,dt\,,
\end{equation}
\noindent
The conjecture for the case $2\lambda=2$ was subsequently proved by Milinovich~\cite{Mi}, assuming the RH.
It is expected that such polynomials exist for other integer values of $\lambda>0$ as well.
The purpose of this article is to study numerically various statistics
of the derivative of the zeta function at its zeros. In particular, we consider
the distribution of $\log|\zeta'(1/2+i\gamma_n)|$,
moments of $|\zeta'(1/2+i\gamma_n)|$,
and correlations among moments.
The goal is to obtain more detailed information about
the derivative at zeros, and to enable comparison with various
conjectured and known asymptotics. Our computations rely on large sets of zeros at large
heights that are described in detail in \cite{HO}.
We find that the empirical distribution of $\log|\zeta'(1/2+i\gamma_n)|$,
normalized to have mean zero and standard deviation one, agrees
generally well with the limiting normal distribution proved by Hejhal, as shown in Figure~\ref{distder23}.
But the empirical mean and standard deviation pre-normalization
are noticeably different from predicted ones.
Also, as shown in Figure~\ref{tailder23}, the frequency
of very small normalized values of $\log|\zeta'(1/2+i\gamma_n)|$ is higher than predicted by
a standard normal distribution, while the frequency of very large normalized values is lower than predicted.
Since these differences appear to
decrease steadily with height, however, they are probably not significant.
To examine the tails of the distribution of $\log |\zeta'(1/2+i\gamma_n)|$,
we present data for the moments of $|\zeta'(1/2+i\gamma_n)|$ over short ranges:
\begin{equation}
J_{\lambda}(T,H):= \frac{1}{N(T+H)-N(T)} \sum_{T\le \gamma_n\le T+H} |\zeta'(1/2+i\gamma_n)|^{2\lambda} .
\end{equation}
\noindent
For large $\lambda$, the empirical values of $J_{\lambda}(T,H)$
deviate substantially from the values suggested by the
leading term prediction (\ref{eq:hkc}).
This is not surprising. Because
for $\lambda$ large relative to $T$, the contribution of lower order
terms is likely to dominate, and so the leading term asymptotic on its own
may not suffice.
Furthermore, the said deviations decrease steadily with height
and they occur in a generally uniform way for roughly $2\lambda \le 6$,
so they are consistent with the effect of ``lower order terms'' still
being felt even at such relatively large heights.
In the specific cases of the second and fourth moments of $|\zeta'(1/2+i\gamma_n)|$,
the conjectures of Conrey and Snaith~\cite{CS} supply lower order terms,
and the agreement with the data is much better, as shown in Table~\ref{csmomentder23}.
\footnote{It might be worth mentioning that
we attempted to calculate the coefficients of lower order terms
in the \cite{CS} conjectures by calculating $J_{\lambda}(T)$
for sufficiently many values of $T$, then solving the resulting system of equations. However,
this did not yield good approximations of the coefficients (even for small $\lambda$),
which is not surprising, since the scale is logarithmic and the Conrey and Snaith expansion
is only asymptotic.}
As $\lambda$ increases, the observed variability in the moments of $|\zeta'(1/2+i\gamma_n)|$
is more extreme, but it is still
significantly less than we previously encountered in the moments of $|\zeta(1/2+it)|$ (see \cite{HO}).
To illustrate, our computations of the twelfth moment of $|\zeta'(1/2+i\gamma_n)|$ over
15 separate sets of $\approx 10^9$ zeros each (near the $10^{23}$-rd zero)
show that the ratio of highest to lowest moment among the 15 twelfth moments thus obtained was 2.36.
In contrast, that ratio
for the twelfth moment of $|\zeta(1/2+it)|$
was 16.34, which is significantly larger
(see \cite{HO}).
In general, the variability in statistical data for $|\zeta'(1/2+i\gamma_n)|$
is considerably less than the variability in statistical data for $|\zeta(1/2+it)|$.
It is not immediately clear why this should be so, considering, for instance, that
the central limit theorem for $\log|\zeta'(1/2+i\gamma_n)|$ is only conditional, while that for $\log|\zeta(1/2+it)|$ is not,
and both theorems scale by the same asymptotic variance.
In the case of negative moments, our data is in agreement with Gonek's conjecture (\cite{Go1})
$J_{-1}(T) \sim \frac{6}{\pi^2}(\log T/(2\pi))^{-1}$ as $T\to\infty$.
But starting at $2\lambda=-3$,
and as $\lambda$ decreases, the empirical behavior of
negative moments becomes rapidly more erratic.
For example, using the same 15 zero sets near the $10^{23}$-rd zero mentioned previously,
the ratio of highest to lowest negative moment among them
gets very large as $\lambda$ decreases; we obtain: 1.03, 8.45, 178.49, and 17240.99, for $2\lambda= -2, -3,-4,$
and $-6$, respectively (this can be deduced easily from Table~\ref{Nmomentder23}).
Notice that the point $2\lambda=-3$ is special because it is where the leading term prediction
(\ref{eq:hkc}) first breaks down
due to a pole of order 1 in the ratio of Barnes G-functions.
Extreme values of negative
moments are caused by very few zeros. When $2\lambda=-3$, for instance,
the largest observed moment
among our 15 sets is $0.178047$.
About 87\% of this value is contributed by 4 zeros where
$|\zeta'(1/2+i\gamma_n)|$ is small and equal to
0.002439, 0.002453, 0.004388, and 0.004365.\footnote{We checked such small values of $|\zeta'(1/2+i\gamma)|$
by computing them in two ways, using the Odlyzko-Sch\"onhage algorithm,
and using the straightforward Riemann-Siegel formula;
the results from the two methods agreed to within $\pm 10^{-6}$}
Such small values of $|\zeta'(1/2+i\gamma)|$
typically occur at pairs of consecutive zeros that are close to each other.
For example,
the values 0.002439 and 0.002453 occur
at the following two consecutive zero ordinates:
\begin{equation}
\begin{split}
& 1.30664344087942265202071895041619 \times 10^{22}\,,\\
& 1.30664344087942265202071898265199 \times 10^{22}\,.
\end{split}
\end{equation}
\noindent
The above pair of zeros is separated by 0.00032, which is about 1/400 times
the average spacing of zeros at that height (which is $\approx 0.128$).
To investigate possible correlations among values of $|\zeta'(1/2+i\gamma_n)|^{2\lambda}$,
we studied numerically the (shifted moment) function:
\begin{equation}
S_{\lambda}(T,H,m):= \sum_{T\le \gamma_n\le T+H} |\zeta'(1/2+i\gamma_n)\zeta'(1/2+i\gamma_{n+m})|^{2\lambda}\,.
\end{equation}
\noindent
We plotted $S_{\lambda}(T,H,m)$, for several choices of $\lambda$, $T$, and $H$,
and as $m$ varies. The resulting plots indicate there are
long-range correlations among the values of the derivative at zeros.
Unexpectedly, the tail of $S_2(T,H,m)$ (Figure~\ref{corrder23}; right plot) strongly resembles
the tail for the shifted fourth moment of $|\zeta(1/2+it)|$ (Figure~4 in \cite{HO}).
To better understand these correlations, we considered the ``spectrum'' of $\log|\zeta'(1/2+i\gamma_n)|$;
see (\ref{eq:specf}) for a definition.
A plot of the spectrum reveals sharp spikes, shown in Figure~\ref{fftder23}.
These spikes can be explained heuristically by applying techniques already used
by Fujii \cite{Fu,Fu2} and Gonek \cite{Go1} to estimate sums involving $\zeta'(1/2+i\gamma_n)$.
\section{Numerical results}
Conjecture (\ref{eq:derclt1}) suggests the mean and standard deviation
of $\log |\zeta'(1/2+i\gamma)|$ for
zeros from near $T=1.3066434\times 10^{22}$ (i.e. near the $10^{23}$-rd zero)
are about 2.0 and 1.4, respectively. This is far from the empirical mean and standard deviations
listed in Table~\ref{sum0}, which are 3.4907 and 1.0977.
\footnote{The mean and standard deviations listed in Table~\ref{sum0} change very little across different
zero sets near the same height. For example, using a different set of $10^8$ zeros near
the $10^{23}$-rd zero, the empirical mean is 3.4907 and the empirical
standard deviation is 1.0978, which are very close the numbers listed in Table~\ref{sum0}.
We note that the empirical mean and standard deviation
are closer to the values suggested by the central limit theorem for characteristic polynomials of
unitary matrices (see \cite{HKO}), which are 3.47 and 1.12.}
Since these quantities grow very slowly (like $\log \log T$),
these differences are probably not significant.
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Summary statistics for $\log|\zeta'(1/2+i\gamma_n)|$ using sets of $10^7$ zeros from different heights
The column "Zero" lists the zero number near which the set is located.
SD stands for standard deviation. \label{sum0}}
\begin{tabular}{|c|c|c|c|c|}
\hline
Zero &Min & Max & Mean & SD \\
\hline
$10^{16}$ & -3.7371 & 7.3920 & 3.1211 & 1.0135 \\
$10^{20}$ & -3.2181 & 8.0085 & 3.3458 & 1.0653 \\
$10^{23}$ & -2.9602 & 8.2836 & 3.4907 & 1.0977 \\
\hline
\end{tabular}
\end{table}
We normalize the sequence $\{\log |\zeta'(1/2+i\gamma_n)|\,:\, N \le n \le N + 10^7\}$, where $N\approx 10^{23}$,
to have mean zero and variance one.
The distribution of the normalized sequence is illustrated in Figure~\ref{distder23}, which contains
two plots, one of the empirical density function, and another of the difference between the empirical
density and the predicted (standard Gaussian) density $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$.
The fit in the first plot is visibly good, but there is a slight shift to the right about the center.
This shift is made more visible in
the second plot, which shows that the empirical density is generally larger than expected for $x>0$,
and is smaller than expected for $x<0$.
\begin{figure}
\caption{\footnotesize Empirical density of $\log |\zeta'(1/2+i\gamma_n)|$, after being normalized to have mean
0 and standard deviation 1, using $10^7$ values of $\log |\zeta'(1/2+i\gamma_n)|$ from near
the $10^{23}
\label{distder23}
\end{figure}
Near the tails, however, the situation is reversed. Figure~\ref{tailder23}
shows there is a deficiency in the occurrence of very large values of $|\zeta'(1/2+i\gamma_n)|$, and an abundance in the occurrence of very small values.
For instance, conjecture (\ref{eq:derclt1}) suggests that about
0.1462\% of the values of $|\zeta'(1/2+i\gamma_n)|$ near the $10^{23}$-rd zero
should satisfy $|\zeta'(1/2+i\gamma_n)| > 860$,
which is noticeably larger than the observed 0.1056\%.
The conjecture also suggests about 0.0736\% of the values should satisfy $|\zeta'(1/2+i\gamma_n)|<1$, which
is smaller than the observed 0.1051\%.
We remark the behavior near the tails becomes more consistent with
expectation as height increase . For example, only 0.0025\% of the time
do we have $\log|\zeta'(1/2+i\gamma_n)|>3.2$ near the $10^{16}$-th zero,
which is far from the expected 0.068\%, but the percentage increases to 0.040\% near the $10^{23}$-rd zero.
\begin{figure}
\caption{\footnotesize Distribution at the tails using $1.5\times 10^{10}
\label{tailder23}
\end{figure}
For another measure of the quality of the fit to the standard Gaussian in Figure~\ref{distder23},
we compare moments of both distributions. Table~\ref{gaussder23} shows the first few moments (the even moments in particular)
agree reasonably well. Notice the odd moments tend to be negative, which is likely due to the aforementioned bias in the
frequency of very small and very large values.
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Moments of $\log|\zeta'(1/2+i\gamma_n)|$, after being normalized to have mean zero and variance one,
calculated using $10^7$ zeros from near the $10^{23}$-rd zero. The third column is the moment of a standard Gaussian.}\label{gaussder23}
\begin{tabular}{c|cc}
Moment & Derivatives & Gaussian \\
\hline
3rd& -0.02728 & 0\\
4th& 3.01364 & 3\\
5th& -0.49120& 0\\
6th& 15.3053& 15\\
7th& -7.43073& 0 \\
8th& 112.013& 105 \\
9th& -118.588& 0\\
10th & 1116.64 & 945
\end{tabular}
\end{table}
To better understand the tails of the distribution of $\log |\zeta'(1/2+i\gamma_n)|$,
we consider the moments $J_\lambda(T)$ defined in (\ref{eq:dermo1}).
Since we are interested in the asymptotic behavior of $J_{\lambda}(T)$,
we compare against the leading term prediction (\ref{eq:hkc}).
We calculated ratios of the form
\begin{equation}\label{eq:appder1}
\frac{\frac{1}{|B|}\sum_{\gamma\in B} |\zeta'(1/2+i\gamma)|^{2\lambda}}{a(\lambda) \frac{G^2(\lambda+2)}{G(2\lambda+3)} \left(\log \frac{T}{2\pi}\right)^{\lambda(\lambda+2)}}\,,
\end{equation}
\noindent
where $B$ is a block of consecutive zeros, $|B|$ denotes the number of zeros in $B$,
and $T$ is the height where block $B$ lies.
If $T$ is large enough, one expects the value of (\ref{eq:appder1})
to approach 1 as the block size $|B|$ increases.
Table~\ref{momentder23}, which uses blocks of size $|B|\approx 10^9$ (except for the first set, which uses the first $10^8$ zeta zeros),
shows that the empirical moments are significantly larger than the corresponding predictions, even for low moments.
For example, the empirical second moments ($2\lambda = 2$) near the $10^{23}$-rd zero
are generally off from expectation by about 9.6\%.
Nevertheless, the ratios (\ref{eq:appder1}) appear to decrease towards the expected 1 as the height increases,
and there is relatively little variation in the moment data for sets from near the same height when $2\lambda\le 6$.
Both of these observations are consistent with the ``lower order terms'' still contributing significantly.
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Ratio (\ref{eq:appder1}) calculated with $|B|\approx 10^9$,
except for the first set, which uses the first $10^8$ zeros. The column ``Zero'' lists
the approximate zero number near which block $B$ is located.} \label{momentder23}
\begin{tabular}{|l||l|l|l|l|l|l|}
\hline
Zero & $2\lambda=2$ & $2\lambda=4$ & $2\lambda=6$ & $2\lambda=8$ & $2\lambda=10$ & $2\lambda=12$ \\
\hline
$10^8$ & 1.1247 & 3.1579 & 91.856 & 78341 & $4.1016\times 10^9$ & $2.3478\times 10^{16}$ \\
$10^{16}$ & 1.1424 & 2.2087 & 17.686 & 1266.9 & $1.5057\times 10^6$ & $4.9628\times 10^{10}$\\
$10^{20}$ & 1.1123 & 1.9102 & 10.943 & 422.72 & $1.9904\times 10^5$ & $1.8362\times 10^9$ \\
$10^{23}$ & 1.0964 & 1.7645 & 8.4406 & 233.63 & $6.4583\times 10^4$ & $2.7127\times 10^8$ \\
- & 1.0964 & 1.7603 & 8.1602 & 199.18 & $4.1647\times 10^4$ & $1.1369\times 10^8$ \\
- & 1.0964 & 1.7598 & 8.1879 & 202.40 & $4.3355\times 10^4$ & $1.2325\times 10^8$ \\
- & 1.0964 & 1.7629 & 8.3221 & 217.58 & $5.2539\times 10^4$ & $1.7809\times 10^8$ \\
- & 1.0964 & 1.7630 & 8.3861 & 228.51 & $6.2549\times 10^4$ & $2.6614\times 10^8$ \\
- & 1.0964 & 1.7600 & 8.2022 & 206.36 & $4.6423\times 10^4$ & $1.4200\times 10^8$ \\
- & 1.0965 & 1.7642 & 8.3321 & 218.38 & $5.3663\times 10^4$ & $1.8923\times 10^8$ \\
- & 1.0965 & 1.7612 & 8.1862 & 201.43 & $4.3256\times 10^4$ & $1.2547\times 10^8$\\
- & 1.0963 & 1.7590 & 8.2176 & 209.97 & $4.8853\times 10^4$ & $1.5596\times 10^8$ \\
- & 1.0964 & 1.7654 & 8.3856 & 217.09 & $4.8781\times 10^4$ & $1.4148\times 10^8$ \\
- & 1.0963 & 1.7616 & 8.3009 & 218.92 & $5.4691\times 10^4$ & $1.9491\times 10^8$ \\
- & 1.0964 & 1.7585 & 8.1576 & 204.55 & $4.6872\times 10^4$ & $1.5134\times 10^8$ \\
- & 1.0965 & 1.7615 & 8.2380 & 209.26 & $4.7946\times 10^4$ & $1.5078\times 10^8$ \\
- & 1.0963 & 1.7586 & 8.1764 & 203.00 & $4.4241\times 10^4$ & $1.2904\times 10^8$ \\
- & 1.0964 & 1.7603 & 8.2037 & 208.39 & $4.9019\times 10^4$ & $1.6822\times 10^8$ \\
\hline
\end{tabular}
\end{table}
The full moment prediction of \cite{CS}, which takes lower order terms into account,
might lead one to expect that for $2\lambda=2$, $2\lambda=4$,
as $T\to \infty$, and for blocks $B$ not too small compared to $T$,
\begin{equation}\label{eq:dercs}
\sum_{\gamma\in B} |\zeta'(1/2+i\gamma)|^{2\lambda}\sim\int_{B} P_{\lambda}(\log (t/2\pi))\,dt\,,
\end{equation}
\noindent
where $P_{\lambda}(x)$ is as given in \cite{CS},
and $\int_{B}$ is short for integrating over the interval spanned by the block $B$.
To test this, we calculated ratios of the form
\begin{equation}\label{eq:appder2}
\frac{\sum_{\gamma\in B} |\zeta'(1/2+i\gamma)|^{2\lambda}}{\int_{B} P_{\lambda}(\log (t/2\pi))\,dt}\,.
\end{equation}
\noindent
As the block size increases, we expect (\ref{eq:appder2}) to be significantly closer to 1 than (\ref{eq:appder1}) since
it relies on a more accurate prediction.
This is indeed what Table~\ref{csmomentder23} illustrates, where we see the fit to moment data is much better
than we found in Table~\ref{momentder23}.
\footnote{Notice if $T$ is large compared to the length of the interval spanned by block $B$,
the denominator in ratio (\ref{eq:appder2}) is largely a function of $T$ multiplied
by the length of the interval spanned by $B$.} (We point out that in the case $2\lambda=4$
only the first three terms in the full moment conjecture were used, because these were the only terms
provided explicitly in \cite{CS}. It is likely the fit to the data
will be even better if the missing terms are included.)
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Ratio (\ref{eq:appder2}) calculated with $|B|\approx 10^9$,
except for the first set, which uses the first $10^8$ zeros. The column ``Zero'' lists
the approximate zero number near which block $B$ is located.}\label{csmomentder23}
\begin{tabular}{|l||l|l|}
\hline
Zero & $2\lambda=2$ & $2\lambda=4$ \\
\hline
$10^8$ & 1.0000 & 1.0924 \\
$10^{16}$ & 1.0000 & 1.0144 \\
$10^{20}$ & 1.0000 & 1.0087 \\
$10^{23}$ & 1.0000 & 1.0074 \\
~~~~`` & 1.0000 & 1.0050 \\
~~~~`` & 0.9999 & 1.0047 \\
~~~~`` & 1.0000 & 1.0064 \\
~~~~`` & 0.9999 & 1.0065 \\
~~~~`` & 0.9999 & 1.0048 \\
~~~~`` & 1.0000 & 1.0072 \\
~~~~`` & 1.0000 & 1.0055 \\
~~~~`` & 0.9998 & 1.0042 \\
~~~~`` & 1.0000 & 1.0079 \\
~~~~`` & 0.9999 & 1.0057 \\
~~~~`` & 0.9999 & 1.0039 \\
~~~~`` & 1.0000 & 1.0057 \\
~~~~`` & 0.9999 & 1.0040 \\
~~~~`` & 1.0000 & 1.0049 \\
\hline
\end{tabular}
\end{table}
We remark that the five largest values of $|\zeta'(1/2+i\gamma_n)|$ in our data set
are $\approx$ 7057, 6907, 6658, 6636, and 6399.
The cumulative contribution of these large values to the $2\lambda$-th moment, as a percentage of the overall $2\lambda$-th moment, is listed
in Table~\ref{accumder23} for several $\lambda$.
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Cumulative contribution percentage of the 5 largest values of $|\zeta'(1/2+i\gamma_n)|$ to
the empirical $2\lambda$-th moment
for $1.5\times 10^{10}$ zeros near the $10^{23}$-rd zero.}\label{accumder23}
\begin{tabular}{|l|l|l|}
\hline
$2\lambda=8$ & $2\lambda=10$ & $2\lambda=12$ \\
\hline
0.50 & 1.84 & 4.51 \\
0.92 & 3.32 & 7.99 \\
1.24 & 4.35 & 10.2 \\
1.54 & 5.35 & 12.3 \\
1.77 & 6.04 & 13.7 \\
\hline
\end{tabular}
\end{table}
In the case of negative moments, the conjecture $J_{-1}(T) \sim \frac{6}{\pi^2}(\log T/(2\pi))^{-1}$ as $T\to\infty$, due to Gonek~\cite{Go2},
suggests the negative second moment
should be $\approx 0.01808$ near zero number $10^{16}$,
$\approx 0.01436$ near zero number $10^{20}$, and $\approx 0.01238$ near zero number $10^{23}$.
These predictions are in good agreement with the values listed in Table~\ref{Nmomentder23}.
For $2\lambda \le -3$, the behavior
is much less predictable because, empirically, their sizes are determined by
a few zeros where $|\zeta'(1/2+i\gamma_n)|$ is small.
In fact, the particularly large fluctuations in the size of the negative
sixth moment ($2\lambda = -6$), near the $10^{23}$-rd zero in Table~\ref{Nmomentder23}, are essentially due to 8 zeros (out of $1.5\times 10^{10}$)
where $|\zeta'(1/2+i\gamma_n)|$ is equal to 0.002439, 0.002453, 0.002719, 0.002737, 0.003094, 0.003108, 0.004365, and 0.004388.
\begin{table}[ht]
\footnotesize
\caption{\footnotesize Ratio (\ref{eq:appder2}) calculated with $|B|\approx 10^9$,
except for the first set, which uses the first $10^8$ zeros. The column ``Zero'' lists
the approximate zero number near which block $B$ is located.} \label{Nmomentder23}
\begin{tabular}{|l||l|l|l|l|}
\hline
Zero & $2\lambda=-2$ & $2\lambda=-3$ & $2\lambda=-4$ & $2\lambda=-6$\\
\hline
$10^8$ & 0.041129 & 0.059025 & 1.04212 & 2935.6 \\
$10^{16}$ & 0.018057 & 0.030660 & 0.55588 & 1488.1 \\
$10^{20}$ & 0.014341 & 0.028403 & 0.73586 & 2873.2 \\
$10^{23}$ & 0.012347 & 0.022040 & 0.41441 & 1106.5 \\
~~~~`` & 0.012365 & 0.022605 & 0.43869 & 1314.6 \\
~~~~`` & 0.012462 & 0.037677 & 2.76255 & 63336 \\
~~~~`` & 0.012321 & 0.021618 & 0.42275 & 1431.0 \\
~~~~`` & 0.012776 & 0.178047 & 59.6610 & 9288238 \\
~~~~`` & 0.012326 & 0.021062 & 0.33853 & 665.29 \\
~~~~`` & 0.012515 & 0.052929 & 7.46570 & 412318 \\
~~~~`` & 0.012334 & 0.022429 & 0.56305 & 4157.4 \\
~~~~`` & 0.012376 & 0.025800 & 0.81652 & 5414.6 \\
~~~~`` & 0.012541 & 0.089163 & 21.5695 & 2174342 \\
~~~~`` & 0.012411 & 0.039415 & 4.32860 & 185114 \\
~~~~`` & 0.012329 & 0.022729 & 0.55154 & 2723.6 \\
~~~~`` & 0.012386 & 0.027487 & 1.08706 & 11563 \\
~~~~`` & 0.012605 & 0.117993 & 35.4067 & 4686740 \\
~~~~`` & 0.012334 & 0.021217 & 0.33424 & 538.73 \\
\hline
\end{tabular}
\end{table}
Starting with the investigations of \cite{Od2}, several long-range correlations
have been found experimentally in zeta function statistics. Such correlations are
not present in random matrices, but do appear in some dynamical systems that
for certain ranges are modeled by random matrices. So far all the zeta function
correlations of this nature have been explained (at least numerically and heuristically)
by relating them to known properties of the zeta function, such as explicit
formulas that relate primes to zeros. A natural question is whether such
correlations arise among values of $\zeta'(1/2+i\gamma_n)$.
In order to detect correlations among values of $|\zeta'(1/2+i\gamma_n)|$,
consider
\begin{equation}
S_{2}(T,H,m):= \sum_{T\le \gamma_n\le T+H} |\zeta'(1/2+i\gamma_n)\zeta'(1/2+i\gamma_{n+m})|^4\,.
\end{equation}
\noindent
We computed this shifted moment function for various choices of $m$, $T$, and $H$.
(We also considered similar sums with exponents other than 4, but for simplicity
do not discuss them here.)
Figure~\ref{corrder23} presents some of our results
near the $10^{16}$-th and $10^{23}$-rd zeros, and with $H$ spanning about $10^7$ zeros
in both cases.
The figure shows that correlations do exist and persist over long ranges.
Also, the shape of $S_2(T,H,m)$ near the $10^{16}$-th zero is similar to that near the $10^{23}$-rd zero,
except the former has higher peaks, and covers the range $3\le m\le 222$, as opposed to $3\le m\le 325$,
which suggests oscillations scale as $1/\log (T/2\pi)$.
We remark the plot of $S_2(T,H,m)$ in Figure~\ref{corrder23} (right plot)
is similar to a plot in \cite{HO} of the shifted fourth moment of the zeta function on the
critical line:
\begin{equation}\label{eq:malpha}
M(T, H; \alpha) := \int_T^{T+H} |\zeta(1/2+it)|^2\,|\zeta(1/2+it +i\alpha)|^2\,dt\,,
\end{equation}
\noindent
which we reproduce here in Figure~\ref{smg2} for the convenience of the reader.
\begin{figure}
\caption{\footnotesize Plots of $S_2(T,H,m)/S_2(T,H,0)$ using $10^7$ zeros near the $10^{16}
\label{corrder23}
\end{figure}
\begin{figure}
\caption{\footnotesize Plot of $M(T,H,\alpha)/M(T,H,0)$, with $H\approx 6.5\times 10^5$, near the $10^{23}
\label{smg2}
\end{figure}
To explain observed correlations, we numerically calculated the function:
\begin{equation}\label{eq:specf}
f(T,H,x)=\left|\sum_{T\le \gamma_n\le T+H} \zeta'(1/2+i\gamma_n) e^{2\pi i nx}\right|\,,
\end{equation}
\noindent
which is related to long-range periodicities in $\zeta'(1/2+i\gamma_n)$.
Assuming the RH, Fujii~\cite{Fu} supplied the
following asymptotic formula in the case $x=0$:
\begin{equation}\label{eq:fuj}
\sum_{0<\gamma_n\le T} \zeta'(1/2+i\gamma_n)= \frac{T}{4\pi} \log^2\frac{T}{2\pi}+(c_0-1)\frac{T}{2\pi}\log\frac{T}{2\pi} -(c_1+c_0)\frac{T}{2\pi}+O\left(T^{1/2}\log^{7/2} T\right)\,,
\end{equation}
\noindent
where $c_0=0.5772\ldots$ (the Euler constant) and $c_1=-0.0728\ldots$.
Empirical values of $f(T,H,0)$ agree well with formula (\ref{eq:fuj}).
For example, with $H$ spanning $10^6$ zeros, we obtain $f(T,H,0)= 21766088 - 14579 i$
near the $10^{20}$-zero, and we obtain $f(T,H,0)= 25137126+ 61663 i$ near the $10^{23}$-rd zero.
But as $x$ increases, $f(T,H,x)$ experiences sharp spikes for certain $x$, as shown in Figure~\ref{fftder23},
which depicts the segment $0\le x\le 0.05$ (in the remaining portion $0.05< x<1$, the spikes get progressively denser).
The sharp spikes in Figure~\ref{fftder23} show the
existence of long-range periodicities among values of $\zeta'(1/2+i\gamma_n)$.
These spikes, as well as the correlations described above,
are not unexpected. They can be demonstrated to follow from the
properties of the zeta function, by estimating proper contour
integrals. Such methods were used for continuous averages by Ingham \cite{Ingh}
and even others before him, and for discrete averages over zeros by
Gonek~\cite{Go1} and Fujii~\cite{Fu,Fu2}. The main step involves integration
of ${\zeta'(s)}^2/{\zeta (s)}$, and estimates of such integrals.
Applying such methods to ${\zeta '(s)}^2 e^{x s \log\frac{T}{2\pi}} /{\zeta (s)}$
suggests that the function
\begin{displaymath}
\tilde{f}(T,H,x)=\left|\sum_{T\le \gamma_n \le T+H} \zeta'(1/2+ i \gamma_n) e^{2\pi i \tilde{\gamma}_n x}\right|\,,
\qquad \tilde{\gamma}_n := \frac{\gamma_n}{2\pi}\,\log\frac{T}{2\pi}\,,
\end{displaymath}
\noindent
experiences large spikes at approximately $x=\log(k) /\log(T/(2\pi))$.
For by a heuristic argument
involving the (very) regular spacing of zeros
one expects that $\tilde{\gamma}_n$ in the definition of $\tilde{f}(T,H,x)$ can be replaced by $n$ without too much error
(see \cite{Od2} for a similar argument in the context of long-range correlations in zero spacings).
Therefore, $f(T,H,x)$ should behave similarly to $\tilde{f}(T,H,x)$.\footnote{Indeed,
the plots in Figure~\ref{fftder23} are almost unchanged if instead of plotting $f(T,H,x)$ we plot
$\tilde{f}(T,H,x)$.}
In particular, we expect the $k$-th spike in Figure~\ref{fftder23} to occur
at approximately $\log(k) /\log(T/(2\pi))$,
and that agrees well with the evidence of the graphs.
\begin{figure}
\caption{\footnotesize Plots of $f(T,H,x)$, defined in (\ref{eq:specf}
\label{fftder23}
\end{figure}
\section{Numerical methods}
As usual, define the rotated zeta function on the critical line by
\begin{equation}
Z(t)= e^{i\theta}\zeta(1/2+it)\,,\qquad e^{i\theta(t)}= \left(\frac{\Gamma(1/4+it/2)}{\Gamma(1/4-it/2)}\right)^{1/2} \pi^{-it/2}\,.
\end{equation}
\noindent
The rotation factor $e^{i\theta(t)}$ is chosen so that $Z(t)$ is real.
In our numerical experiments, $t < 1.31 \times 10^{22}$.
Since $|Z'(\gamma_n)|=|\zeta'(1/2+i\gamma_n)|$,
it suffices to compute $Z'(\gamma_n)$.
To do so, we used
the numerical differentiation formula (Taylor expansion)
\begin{equation}\label{eq:numdiff}
Z'(t) = \frac{Z(t + h) - Z(t - h)}{2 h} + R(t,h)\,,
\end{equation}
\noindent
where the remainder term in (\ref{eq:numdiff}) satisfies
\begin{equation}
|R(t,h)| \le \max_{t - h \le t_1 \le t + h} \frac{|Z'''( t_1 )|}{6} h^2\,,
\end{equation}
\noindent
We chose $h = 10^{-5}$, and approximated the derivative by
\begin{equation}\label{eq:numdiff1}
Z'(t) \approx \frac{Z(t + h) - Z(t - h)}{2 h}\,.
\end{equation}
To evaluate $Z(t)$ at individual
points, we used a version of the Odlyzko-Sch\"onhage algorithm~\cite{OS}
implemented by the second author~\cite{Od1}.
If the point-wise evaluations
of $Z(t+h)$ and $Z(t-h)$
via this implementation
are accurate to within
$\pm \epsilon$ each, then the approximation (\ref{eq:numdiff1})
is accurate to within $\pm (10^5\epsilon + |R(t,h)|)$.
Numerical tests suggested $\epsilon$
is normally distributed with mean zero and standard deviation $10^{-9}$.
Therefore, $\epsilon$ is typically around $10^{-9}$.
Also, varying the choice of $h$ in (\ref{eq:numdiff1}) suggested
the approximation is accurate to about 4
decimal digits with $h=10^{-5}$ and $t\approx 10^{22}$.
In principle, our computations of $\zeta'(1/2+i\gamma_n)$
can be made completely rigorous
by carrying them out in sufficient precision.
If one plans on calculating $\zeta'(1/2+i\gamma_n)$ with very high
precision, however, it will likely be better to first derive a
Riemann-Siegel type formula for $Z'(t)$ itself, with explicit estimates for
the remainder. Such a formula will be useful on its own as it can be
be used to check other conjectures about $\zeta'(1/2+it)$.
\section{Conclusions}
Numerical data from high zeros of the zeta function generally
agrees well with the asymptotic results that have been proved,
as well as with several conjectures. There are some systematic
differences between observed and expected distributions, but
the discrepancies decline with growing heights.
The results of this paper provide additional evidence for the
speed of convergence of the zeta function to its asymptotic
limits. They also demonstrate the importance of outliers,
and thus the need to collect extensive data in order
to obtain valid statistical results. The long-range correlations
that have been found among values of the derivative of the zeta
function at zeros can be explained by known analytic techniques.
\end{document} |
\begin{document}
\markboth{Sl. Shtrakov and I. Damyanov}
{On the complexity of finite valued functions}
\title{On the complexity of finite valued functions}
\author{Slavcho Shtrakov}
\address{Department of Computer Science,\\ South-West University, Blagoevgrad, Bulgaria\\
}
\author{Ivo Damyanov}
\address{Department of Computer Science,\\ South-West University, Blagoevgrad, Bulgaria\\
}
\begin{abstract}
The essential variables in a finite function $f$ are defined as variables which occur in $f$ and weigh with the values of that function.
The number of essential variables is an important measure of complexity for discrete functions.
When replacing some variables in a function with constants the resulting functions are called subfunctions, and when replacing all essential variables in a function with constants we obtain an implementation of this function.
Such an implementation corresponds with a path in an ordered decision diagram (ODD) of the function which connects the root with a leaf of the diagram. The sets of essential variables in subfunctions of $f$ are called separable in $f$.
In this paper we study several properties of separable sets of variables in functions which directly impact on the number of implementations and subfunctions in these functions.
We define equivalence relations which classify the functions of $k$-valued logic into classes with same number of implementations, subfunctions or separable sets. These relations induce three transformation groups which are compared with the lattice of all subgroups of restricted affine group (RAG). This allows us to solve several important computational and combinatorial problems.
\end{abstract}
\keywords{Ordered decision diagram; implementation; subfunction; separable set.}
\maketitle
\section{Introduction}\langleabel{sec1}
Understanding the complexity of $k$-valued functions is still one of the fundamental tasks
in the theory of computation. At present, besides classical methods like substitution
or degree arguments a bunch of combinatorial and algebraic techniques have been
introduced to tackle this extremely difficult problem. There has been significant progress analysing the power of
randomness and quantum bits or multiparty communication protocols that help to
capture the complexity of switching functions. For tight estimations concerning the
basic, most simple model switching circuits there still seems a long way to go (see \cite{comp_sem}).
In Section \rangleef{sec2} we introduce the basic notation and give definitions of separable sets, subfunctions, etc.
The properties of distributive sets of variables with their s-systems are also discussed in Section \rangleef{sec2}. In Section \rangleef{sec3} we study the ordered decomposition trees (ODTs), ordered decision diagrams (ODDs), and implementations of discrete functions.
We also discuss several problems with the complexity of representations of functions with respect to their ODDs, subfunctions and separable sets. In Section \rangleef{sec5} we classify discrete functions by transformation groups and equivalence relations concerning the number of implementations, subfunctions and separable sets in functions. In Section \rangleef{sec6} we use these results to classify all boolean (switching) functions with "small" number of its essential variables. Here we calculate the number of equivalence classes and cardinalities of equivalence classes of boolean functions depending on 3, 4 and 5 variables.
\section{Separable and Distributive Sets of Variables}\langleabel{sec2}
We start this section with basic definitions and notation.
A discrete function is defined as a mapping: $f:A\to B$ where the domain $A={\times}_{i=1}^n A_i$ and range $B$ are non-empty finite or countable sets.
To derive the means and methods to represent, and calculate with finite valued functions, some algebraic structure is imposed on the domain $A$ and the range $B$. Both $A$ and $B$ throughout the present paper will be finite ring of integers.
Let $X=\{x_1,x_2,\langledots \}$ be a countable set of variables and $X_n=\{x_1,x_2,\langledots,x_n\}$ be a finite subset of $X$. Let $k$, $k\geq 2$ be a natural number
and let us denote by $Z_k=\{0,1,\langledots,k-1\}$ the set (ring) of
remainders modulo $k$. The set $Z_k$ will identify the ring of residue classes $mod\ k$, i.e. $Z_k=Z/_{kZ}$, where $Z$ is the ring of all integers. An {\it $n$-ary $k$-valued function
(operation) on $Z_k$ } is a mapping $f: Z_k^n\to Z_k$ for some natural
number $n$, called {\it the arity} of $f$. $P_k^n$ denotes the set of all $n$-ary $k$-valued functions on $Z_k$. It is well known fact that there are $k^{k^n}$ functions in $P_k^n$. The set of all $k$-valued
functions
$P_k=\bigcup_{n=1}^\infty P_k^n$
is called {\it the algebra of $k$-valued logic}.
All results obtained in the present paper can be easily extended to arbitrary algebra of finite operations.
For a given variable $x$ and $\alpha\in Z_k$, $x^\alpha$ is defined as follows:
$$
x^\alpha=\langleeft\{\begin{array}{ccc}
1 \ &\ if \ &\ x=\alpha \\
0 & if & x\neq\alpha.
\end{array}
\rangleight.
$$
We use {\it sums of
products (SP)} to represent the functions from $P_k^n$. This is the most natural representation and it is based on so called operation tables of the functions.
Thus each function $f\in P_k^n$ can be uniquely represented of SP-form as
follows
\[ f=a_0.x_1^{0}\langledots x_n^{0}\overlineplus\langledots\overlineplus
a_{m}.x_1^{\alpha_1}\langledots
x_n^{\alpha_n}\overlineplus\langledots\overlineplus a_{k^n-1}.x_1^{k-1}\langledots
x_n^{k-1}\]
with ${\mathbf{\alpha}}={({\alpha_1,\langledots,\alpha_n})}\in Z_k^n$, where $m=\sum_{i=0}^n\alpha_ik^i\langleeq k^n-1$. $"\overlineplus"$ and
$"."$ denote the operations addition (sum) and multiplication (product) modulo $k$ in
the ring $Z_k$. Then $(a_0,\langledots,a_{k^n-1})$ is the vector of output values of $f$ in its table representation.
Let $f\in P_k^n$ and $var(f)=\{x_1,\langledots,x_n\}$ be the set of all variables, which occur in $f$.
We say that the $i$-th variable $x_i\in var(f)$ is {\it essential} in $f$, or $f$ {\it
essentially} {\it depends} on $x_i$, if there exist values
$a_1,\langledots,a_n,b\in Z_k$, such that
\[
f(a_1,\langledots,a_{i-1},a_{i},a_{i+1},\langledots,a_n)\neq f(a_1,\langledots,a_{i-1},b,a_{i+1},\langledots,a_n).
\]
The set of all essential variables in the function $f$ is denoted by
$Ess(f)$ and the number of essential variables in $f$ is denoted by
$ess(f)=|Ess(f)|$.
The variables from $var(f)$ which
are not essential in
$f$ are called {\it inessential} or {\it fictive}.
The set of all output values of a function $f\in P_k^n$ is called {\it the range} of $f$, which is denoted as follows:
\[range(f)=\{c\in Z_k\ |\ \exists (a_1,\langledots,a_n)\in Z_k^n, \quad such\ that\quad f(a_1,\langledots,a_n)=c\}.\]
\begin{definition}\langleabel{d1.2}
Let $x_i$ be an essential variable in $f$ and $c\in Z_k$ be a constant from $Z_k$. The
function $g=f(x_i=c)$ obtained from $f\in P_{k}^{n}$ by replacing the variable $x_i$ with $c$ is
called a {\it simple subfunction of $f$}.
When $g$ is a simple subfunction of $f$ we shall write $g\prec f$. The transitive closure of $\prec$ is denoted by $\preceq$. $Sub(f)=\{g\ | \ g\preceq f\}$ is the set of all subfunctions of $f$ and $sub(f)=|Sub(f)|$.
\end{definition}
For each $m=0,1,\langledots, n$ we denote by $sub_m(f)$ the number of subfunctions in $f$ with $m$ essential variables, i.e. $sub_m(f)=|\{g\in Sub(f)\ |\ ess(g)=m\}|$.
Let $g\preceq f$, $\mathbf{c}=(c_1,\langledots,c_m)\in Z_k^m$ and $M=\{x_1,\langledots,x_m\}\subset X$ with \[g\prec g_1\prec\langledots\prec g_m=f,\quad
g=g_1(x_1=c_1)\quad and \quad g_i=g_{i+1}(x_{i+1}=c_{i+1})\]
for $i=1,\langledots,m-1$. Then we shall write $g=f(x_1=c_1,\langledots,x_m=c_m)$ or equivalently, $g\preceq_M^{\mathbf{c}} f$ and we shall say that the vector $\mathbf{c}$ {\it determines} the subfunction $g$ in $f$.
\begin{definition}\langleabel{d1.3}
Let $M$ be a non-empty set of essential variables in the function $f$.
Then $M$ is called {\it a separable set} in $f$ if there exists a subfunction $g$, $g\preceq f$ such that $M=Ess(g)$.
$Sep(f)$ denotes the set of the all separable sets in $f$ and $sep(f)=|Sep(f)|$.
\end{definition}
The sets of essential variables in $f$ which are not separable are called {\em inseparable} or {\em non-separable}.
For each $m= 1,\langledots, n$ we denote by $sep_m(f)$ the number of separable sets in $f$ which consist of $m$ essential variables, i.e. $sep_m(f)=|\{M\in Sep(f)\ |\ |M|=m\}|$.
The numbers $sub(f)$ and $sep(f)$ characterize the computational complexity of the function $f$ when calculating its values. Our next goal is to classify the functions from $P_k^n$ under these complexities.
The initial and more fundamental results concerning essential variables and separable sets were obtained in the work of Y. Breitbart \cite{bre}, K. Chimev \cite{ch51}, O. Lupanov \cite{lup}, A. Salomaa \cite{sal}, and others.
~
\begin{remark}\langleabel{r1.1} Note that if $g\preceq f$ and $x_i\notin Ess(f)$ then $x_i\notin Ess(g)$ and also if $M\notin Sep(f)$ then $M\notin Sep(g)$.
\end{remark}
\begin{definition}\langleabel{d2.1} Let $M$ and $J$ be two non-empty sets of essential variables in the function $f$ such that $M\cap J=\emptyset.$
The set $J$ is called {\it a
distributive set of $M$ in $f$,
} if for every $|J|$-tuple of constants $\mathbf{c}$ from $Z_k$ it holds $M\not\subseteq Ess(g)$, where $g\preceq_J^{\mathbf{c}} f$
and $J$ is minimal with respect to the order $\subseteq$.
$Dis(M,f)$ denotes the set of the all distributive sets of $M$ in $f.$
\end{definition}
It is clear that if $M\notin Sep(f)$ then $Dis(M,f)\neq\emptyset$. So, the distributive sets ``dominate'' on the inseparable sets of variables in a function. We are interested in the relationship between the structure of the distributive sets of variables and complexity of functions concerning $sep(f)$ and $sub(f)$, respectively, which is illustrated by the following example.
\begin{example}\langleabel{ex2.1}
Let $k=2$, $f=x_1x_2\overlineplus x_1x_3$ and $g=x_1x_2\overlineplus x_1^0x_3.$
It is easy to verify that the all three pairs of variables $\{x_1,x_2\}$, $\{x_1,x_3\}$ and $\{x_2,x_3\}$ are separable in $f$, but $\{x_2,x_3\}$ is inseparable in $g$. {Figure} \rangleef{f1} presents graphically, separable pairs in $f$ and $g$.
The set $\{x_1\}$ is distributive of $\{x_2,x_3\}$ in $g$.
\end{example}
\begin{figure}
\caption{Separable pairs.}
\end{figure}
\begin{definition}\langleabel{d2.2}
Let $\mathcal F=\{P_1,\langledots,P_m\}$ be a family of non-empty sets. A set $\beta=\{x_1,\langledots,x_p\}$ is called
{\it an $s$-system} of $\mathcal F$, if for all
$P_i\in \mathcal F$, $1\langleeq i\langleeq m$ there exists $x_j\in\beta$ such that
$x_j\in P_i$ and for all $x_s\in\beta$ there exists $P_l\in
\mathcal F$ such that $\{x_s\}=P_l\cap\beta.$
$Sys(\mathcal F)$ denotes the set of the all $s$-systems of the family $\mathcal F$.
\end{definition}
Applying the results concerning $s$-systems of distributive sets is one of the basic tools for studying inseparable pairs and inseparable sets in functions. These results are deeply discussed in \cite{ch51,s27,s23}.
\begin{theorem}\langleabel{t2.1}\ Let $M\subseteq Ess(f)$ be a non-empty inseparable set of essential variables in $f\in P_k^n$ and $\beta\in Sys(Dis(M,f))$. Then the following statements hold:
\begin{enumerate}
\item[(i)] $M\cup \beta \in Sep(f)$;
\item[(ii)] $(\ \forall \alpha,\ \alpha\subseteq\beta,\ \alpha\neq\beta)\quad M\cup \alpha\notin Sep(f)$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(i)$\
First, note that $M\notin Sep(f)$ implies $|M|\geq 2$. Without loss of generality assume that
$\beta=\{x_1,\langledots,x_m\}\in Sys(Dis(M,f))$ and $M=\{x_{m+1},\langledots,x_{p}\}$
with $1\langleeq m<p\langleeq n$.
Let us denote by $L$ the following set of variables
$L=Ess(f)\setminus(M\cup\beta )=\{x_{p+1},\langledots,x_{n}\}.$
Since $\beta\in Sys(Dis(M,f))$ it follows that for each $Q\subseteq L$ we have $Q\notin Dis(M,f)$. Hence there is a vector $\mathbf{c}=(c_{p+1},\langledots,c_n)\in Z_k^{n-p}$ such that $M\subseteq Ess(g)$ where $g\preceq_L^{\mathbf{c}} f$.
Next, we shall prove that
$\beta\subset Ess(g).$
For suppose this were not the case and without loss of generality, assume
$x_1\notin Ess(g)$, i.e. $g=g(x_1=d_1)$ for each
$d_1\in Z_k.$ Let $J\in Dis(M,f)$ be a distributive set of $M$ such that
$J\cap\beta=\{x_1\}.$
The existence of the set $J$ follows because $\beta$ is an $s$-system of $Dis(M,f)$ (see Definition \rangleef{d2.2}).
Since $J\cap M=\emptyset$ and
$x_1\notin
Ess(g)$ it follows that $J\cap Ess(g)=\emptyset.$
Now $M\subset Ess(g)$ implies that
$J\notin Dis(M,f)$, which is a contradiction. Thus we have $x_1\in Ess(g)$ and $\beta\subset Ess(g)$.
Then
$Ess(f)\setminus L=M\cup\beta$
shows that $M\cup\beta=Ess(g)$ and hence $M\cup\beta\in
Sep(f).$
$(ii)$\ Let $\alpha$, $\alpha\subseteq\beta,\ \alpha\neq\beta$ be a proper subset of $\beta$.
Let $x_i\in \beta\setminus \alpha$. Then $\beta\in Sys(Dis(M,f))$ implies that there
is a distributive set $P\in Dis(M,f)$ of $M$ such that $P\cap\beta=\{x_i\}$. Hence
$P\cap\alpha=\emptyset$
which shows that there is an non-empty distributive set $P_1$ for $M\cup\{\alpha\}$ with $P_1\subseteq P$. Hence
$M\cup\alpha\notin Sep(f)$.
\end{proof}
\begin{corollary}\langleabel{c2.1}
Let $\emptyset\neq M\subset Ess(f)$ and $M\notin Sep(f)$. If $\beta\in Sys(Dis(M,f))$ and $x_i\in\beta$ then
$M\setminus Ess(f(x_i=c))\neq\emptyset$ for all $c\in Z_k$.
\end{corollary}
\begin{theorem}\langleabel{t2.2}\cite{s23}
For each finite family $\mathcal F$ of non-empty sets there exists at least one
$s-$system of $\mathcal F$.
\end{theorem}
\begin{theorem}\langleabel{t2.3}
Let $M$ be an inseparable set in $f$. A set $\beta\subset Ess(f)$ is an $s$-system of $Dis(M,f)$ if and only if $\beta\cap J\neq\emptyset$ for all $J\in Dis(M,f)$ and $\alpha\subseteq\beta,\ \alpha\neq\beta$ implies $\alpha\cap P=\emptyset$ for some $P\in Dis(M,f)$.
\end{theorem}
\begin{proof}
"$\Leftarrow$" Let $\beta\cap J\neq \emptyset$ for all $J\in Dis(M,f)$ and $\alpha\varsubsetneqq \beta$ implies $\alpha\cap P=\emptyset$ for some $P\in Dis(M,f)$. Since $\beta\cap J\neq \emptyset$ it follows that there is a set $\beta'$, $\beta'\subset \beta\subset Ess(f)$ and $\beta'\in Sys(Dis(M,f))$. If we suppose that $\beta'\neq\beta$ then there is $P\in Dis(M,f)$ with $\beta'\cap P=\emptyset$. Hence $M\cup\beta'\notin Sep(f)$ because of $P\in Dis(M\cup\beta',f)$ which contradicts Theorem \rangleef{t2.1}.
"$\Rightarrow$" Let $\beta$ be an $s$-system of $Dis(M,f)$ and $\alpha\varsubsetneqq \beta$. Let $x\in \beta\setminus\alpha$ and $P\in Dis(M,f)$ be a distributive set of $M$ for which $\beta\cap P=\{x\}$. Hence $\alpha\cap P=\emptyset$ and we have $P\in Dis(M\cup \alpha,f)$ and $M\cup\alpha\notin Sep(f)$ which shows that $\alpha\notin Sys(Dis(M,f))$.
~~~~~
\end{proof}
\section{Ordered Decision Diagrams and Complexity of Functions}\langleabel{sec3}
The distributive sets are also important when constructing efficient procedures for simplifying in analysis and synthesis of functional schemas.
In this section we discuss {\it ordered decision diagrams} (ODDs) for the functions obtained by restrictions on their {\it ordered decomposition trees} (ODTs).
{Figure} \rangleef{f2} shows an ordered decomposition tree for the function $g=x_1x_2\overlineplus x_1^0x_3\in P_2^3$ from Example \rangleef{ex2.1}, which essentially depends on all its three variables $x_1, x_2$ and $x_3$.
The node at the top, labelled $g$ - is the {\it function} node.
The nodes drawn as filled circles labelled with variable names are the {\it internal (non-terminal)} nodes, and the rectangular nodes (leaves of the tree) are the {\it terminal} nodes. The terminal nodes are labelled by the numbers from $Z_k$. Implementation of $g$ for a given values of $x_1, x_2$ and $x_3$ consists of selecting a path from the function node to a terminal node. The label of the terminal node is the sought value. At each non-terminal node the path follows the solid edge if the variable labelling the node evaluates to $1$, and the dashed edge if the variable evaluates to $0$. In the case of $k>2$ we can use colored edges with $k$ distinct colors.
The ordering in which the variables appear is the same along all paths of an ODT. {Figure} \rangleef{f2} shows the ODT for the function $g$ from Example \rangleef{ex2.1}, corresponding to the variable ordering $x_1,x_2,x_3$ (denoted briefly as $\langle 1; 2; 3\rangle$). It is known that for a given function $g$ and a given ordering of its essential variables there is a unique ODT.
We extend our study to ordered decision diagrams for the functions from $P_k^n$ which were studied by D. Miller and R. Drechsler \cite{mil1,mil2}.
\begin{figure}
\caption{Decomposition tree for $g=x_1x_2\overlineplus x_1^0x_3.$}
\end{figure}
An {\it ordered decision diagram} of a function $f$ is obtained from the corresponding ODT by {\it reduction} of its nodes applying of the following two rules starting from the ODT and continuing until neither rule can be applied:
{\bf Reduction rules}
\begin{enumerate}
\item[$\bullet$] If two nodes are terminal and have the same label, or are non-terminal and have the same children, they are merged.
\item[$\bullet$] If an non-terminal node has identical children it is removed from the graph and its incoming edges are redirected to the child.
\end{enumerate}
When $k=2$ ODD is called {\it a binary decision diagram} (BDD). BDDs are extensively used in the theory of {\it switching circuits} to represent and manipulate Boolean functions and to measure the complexity of binary terms.
The size of the ODD is determined both by the function being represented and the chosen ordering of the variables.
It is of crucial importance to care about variable ordering when applying ODDs in practice. The problem of finding the best variable ordering is NP-complete (see \cite{bra}).
{Figure} \rangleef{f3} shows the BDDs for the functions from Example \rangleef{ex2.1} obtained from their decomposition trees under the natural ordering of their variables - $\langle 1; 2; 3\rangle$. The construction of the ODT for $f$ under the natural ordering of the variables is left to the reader.
\begin{figure}
\caption{BDD for $f$ and $g$ under the natural ordering of variables.}
\end{figure}
The BDD of $f$ is more complex than the BDD of $g$. This reflects the fact that
$f$ has more separable pairs.
Thus we have $M=\{x_2,x_3\}\notin Sep(g)$, $\{x_1\}\in Dis(M,g)$ and $\{x_1\}\in Sys(Dis(M,g))$. Additionally, the diagram of $g$ starts with $x_1$ - a variable which belongs to an $s$-system of $Dis(M,g)$. In this simple case we have $Sys(Dis(M,g))= Dis(M,g)=\{x_1\}$.
Figure \rangleef{f3} shows that when constructing the ODD of a function, it is better to start with the variables from an $s$-system of the family of distributive sets of an inseparable set $M$ in this function. In \cite{ivo2} it is shown that the BDDs of functions have to be most simple when starting with variables from $Sys(Dis(M,f))$.
Consequently, the inseparable sets with their distributive sets are important in theoretical and applied computer science concerning the computational complexity of the functions.
Next, we define and explore complexity measures of the functions in $P_k^n$ which are directly connected with the computational complexity of functions.
We might think that the complexity of a function $f$ depends on the complexity of its ODDs.
Let ${f\in P_k^n}$ and let $DD(f)$ be the set of the all ODDs for $f$ constructed under different variable orderings in $f$.
\begin{definition}\langleabel{d3.2}
Each path starting from the function node and finishing into a terminal node is called an {\it implementation} of the function $f$ under the given variable ordering.
The set of the all implementations of $D_f$ we denote by $Imp(D_f)$ and
\[Imp(f)=\bigcup_{D_f\in DD(f)}Imp(D_f).\]
\end{definition}
Each implementation of the function ${f\in P_k^n}$, obtained from the diagram $D_f$ of $f$ by the non-terminal nodes
$x_{i_1},\langledots,x_{i_r}$ and corresponding constants $c_{1},\langledots,c_{r},c\in Z_k$ with
$f(x_{i_1}=c_{1},\langledots,x_{i_r}=c_{r})=c,\quad r\langleeq ess(f)$,
can be represented as a pair $\mathbf{(i,c)}$ of two words (strings) over $\mathbf{n}=\{1,\langledots,n\}$ and $Z_k$ where $\mathbf{i}=i_1i_2\langledots i_r\in \mathbf{n}^*$ and $\mathbf{c}=c_1c_2\langledots c_rc\in Z_k^*$.
There is an obvious way to define a measure of complexity of a given ordered decision diagram $D_f$, namely as the number ${imp}(D_f)$ of all paths in $D_f$ which starts from the function node and finish in a terminal node of the diagram.
The {\it implementation complexity} of a function ${f\in P_k^n}$ is defined as the number of all implementations of $f$, i.e.
$imp(f)=|Imp(f)|.$
We shall study also two other measures of computational complexity of functions as $sub(f)$ and $sep(f)$.
\begin{example}\langleabel{ex3.1} Let us consider again the functions $f$ and $g$ from Example \rangleef{ex2.1}, namely
$f=x_1x_2\overlineplus x_1x_3\quad and\quad g=x_1x_2\overlineplus x_1^0x_3.$
Then $(123, 1011)$ is an implementation of $f$ obtained by the diagram $D_f$ presented in
{Figure} \rangleef{f3}, following the path $\pi= (f; x_1: 1; x_2: 0; x_3: 1;terminal\ node: 1)$.
It is easy to see that there are six distinct BDDs for $f$ and five distinct BDDs for $g$. We shall calculate the implementations of $f$ and $g$ for the variable orderings $\langle 1; 2; 3\rangle$ (see Figure \rangleef{f3}) and $\langle 2; 1; 3\rangle$, only. Thus for $f$ we have:
\noindent
\begin{tabular}{|l|c|}
\hline
ordering & implementations\\ \hline
$\langle 1; 2; 3\rangle$ &
$(1,00);\ (123,1000);\ (123,1011);\ (123,1101);\ (123, 1110)$ \\ \hline
$\langle 2; 1; 3\rangle$& $(21,000);\ (213,0100);\ (213,0111);\ (21,100); (213,1101);\ (213,1110)$ \\ \hline
\end{tabular}
For the function $g$ we obtain:
\noindent
\begin{tabular}{|l|c|}
\hline
ordering &implementations\\ \hline
$\langle 1; 2; 3\rangle$ &
$(13,000);\ (13,011);\ (12,100);\ (12,111)$\\ \hline
$\langle 2; 1; 3\rangle$& $(21,010);\ (213,0000);\ (213,0011);\ (213,1000); (213,1011);\ (21,111)$\\ \hline
\end{tabular}
For the diagrams in
{Figure} \rangleef{f3} we have
${imp}(D_f)=5$ and ${imp}(D_g)=4$.
Since $f$ is a symmetric function with respect to $x_2$ and $x_3$ one can count that $imp(f)=33$.
Note that the implementation $(1,00)$ occurs in two distinct diagrams of $f$, namely under the orderings $\langle 1; 2; 3\rangle$ and $\langle 1; 3; 2\rangle$. Hence, it has to be counted one time and we obtain that $imp(f)$ is equal to $33$ instead of $34$.
For the function $g$, the diagrams under the orderings $\langle 1; 2; 3\rangle$ and $\langle 1; 3; 2\rangle$ have the same implementations, i.e. the diagrams are identical (isomorphic). This fact is a consequence of inseparability of the set $\{x_2,x_3\}$. Hence $g$ has five (instead of six for $f$) distinct ordered decision diagrams. Then, one might calculate that $imp(g)=28$.
For the other two measures of complexity we obtain:
$sub(f)=13$ because of
$Sub(f)=\{0,1,x_1,x_2,x_3,x_2^0,x_3^0,x_2\overlineplus x_3,x_1x_2,$ $ x_1x_2^0,x_1x_3,x_1x_3^0, f\}$
and $sub(g)=11$ because of
$Sub(g)=\{0,1,x_1,x_2,x_3,x_1^0,x_1 x_2,x_1^0x_3,x_1\overlineplus x_1^0x_3,x_1x_2\overlineplus x_1^0,g\}$.
Furthermore, $sep(f)=7$ because of
$M\in Sep(f)$ for all $M$, $\emptyset\neq M\subseteq \{x_1,x_2,x_3\}$ and $sep(g)=6$ because of
$Sep(g)=\{\{x_1\},\{x_2\},\{x_3\},\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_2,x_3\}\}$.
\end{example}
\begin{lemma}\langleabel{l17}
A variable $x_i$ is essential in $f\in P_k^n$ if and only if $x_i$ occurs as a label of an non-terminal node in any ODD of $f$.
\end{lemma}
\begin{proof}
$"\Rightarrow"$
Let us assume that $x_i$ does not occur as a label of any non-terminal node in an ordered decision diagram $D_f$ of $f$. Since all values of the function $f$ can be obtained by traversal walk-trough all paths in $D_f$ from function node to leaf nodes this will mean that $x_i$ will not affect the function value and hence $x_i$ is an inessential variable in $f$.
$"\Leftarrow"$ Let $x_i\notin Ess(f)$ be an inessential variable in $f$. It is obvious that for each subfunction $g$ of $f$ we have $x_i\notin Ess(g)$. Then we have $f(x_i=c)=f(x_i=d)$ for all $c,d\in Z_k$. Consequently, if there is a non-terminal node labelled by $x_i$ in an ODT of $f$ then its children have to be identical, which shows that this node has to be removed from the ODT, according to the reduction rules, given above.
\end{proof}
An essential variable $x_i$ in a function $f$ is called {\it a strongly essential variable } in $f$ if there is a constant $c\in Z_k$ such that $Ess(f(x_i=c))=Ess(f)\setminus\{x_i\}$.
\begin{fact}\langleabel{fc1}
If $ess(f)\geq 1$ then there is at least one strongly essential variable in $f$.
\end{fact}
This fact was proven by O. Lupanov \cite{lup} in case of Boolean functions and by A. Salomaa \cite{sal} for arbitrary functions.
Later, Y. Breitbart \cite{bre} and K. Chimev \cite{ch51} proved that if $ess(f)\geq 2$ then there exist at least two strongly essential variables in $f$.
We need Fact \rangleef{fc1} to prove the next important theorem.
\begin{theorem}\langleabel{t3.2} A non-empty set $M$ of essential variables is separable in $f$ if and only
if there exists an implementation $\mathbf{(j,c)}$ of the form \[\mathbf{(j,c)}=(j_1j_2\langledots j_{r-m}j_{r-m+1}\langledots j_r, c_1c_2\langledots c_{r-m}c_{r-m+1}\langledots c_r c)\in Imp(f)\] where $M=\{x_{j_{r-m+1}},\langledots, x_{j_r}\}$ and $1\langleeq m\langleeq r\langleeq ess(f)$.
\end{theorem}
\begin{proof}
"$\Leftarrow$" Let \[\mathbf{(j,c)}=(j_1\langledots j_{r-m}j_{r-m+1}\langledots j_r, c_1\langledots c_{r-m}c_{r-m+1}\langledots c_rc)\in Imp(f)\] be an implementation of $f$ and let $M=\{x_{j_{r-m+1}},\langledots, x_{j_r}\}$. Hence the all variables from $\{x_{j_{r-m+1}},\langledots,x_{j_r}\}$ are essential in the following subfunction of $f$
\[g=f(x_{j_1}=c_{1},\langledots,x_{j_{r-m}}=c_{{r-m}})\] which shows that $M\in Sep(f)$.
"$\Rightarrow$" Without loss of generality let us assume that $M=\{x_1,\langledots,x_m\}$ is a non-empty separable set in $f$ and $n=ess(f)$. Then there are constants $d_{m+1},\langledots,d_n\in Z_k$ such that $M=Ess(h)$ where $h=f(x_{m+1}=d_{m+1},\langledots,x_{n}=d_{n})$. From Fact \rangleef{fc1} it follows that there is a variable $x_{i_1}\in M$ and a constant $d_{1}\in Z_k$ such that $Ess(h_1)=M\setminus \{x_{i_1}\}$ where $h_1=h(x_{i_1}=d_{1})$. Consequently, we might inductively obtain that there are variables $x_{i_r}\in M$ and constants $d_{r}\in Z_k$ for $r=2,\langledots,m$, such that $Ess(h_r)=M\setminus \{x_{i_1},\langledots,x_{i_r}\}$ where $h_r=h_{r-1}(x_{i_r}=d_{r})$. Hence, the string $m+1m+2\langledots n$ has a substring $j_1\langledots j_s$ such that $(j_1\langledots j_si_1\langledots i_m, d_{j_1}\langledots d_{j_s}d_{1}\langledots d_{m}d)$ is an implementation of $f$ with $M=\{x_{i_1},\langledots,x_{i_m}\}$ and $d=h_m$.
\end{proof}
\begin{corollary}\langleabel{c3.2} For each variable $x_i\in Ess(f)$ there is an implementation $\mathbf{(j,c)}$ of $f$ whose last letter of $\mathbf{j}$ is $i$, i.e. $\mathbf{(j,c)}=(j_1\langledots j_{m-1}i, c_{j_1}\langledots c_{j_{m-1}}c_ic)\in Imp(f)$, $m\langleeq ess(f)$.
\end{corollary}
Note that there exists an ODD of a function whose non-terminal nodes are labelled by the variables from a given set, but this set might not be separable. For instance, the implementation $(231,0101)\in Imp(g)$ of the function $g$ from Example \rangleef{ex3.1} shows that the variables from the set $M=\{x_2,x_3\}$ occur as labels of the starting two non-terminal nodes in the BDD of $g$ under the ordering $\langle 2; 3; 1 \rangle$, but $M\notin Sep(g)$.
\begin{lemma}\langleabel{l2.1}
If $ess(f)=n$, $g\preceq f$ with
$ess(g)= m<n$ then there exists a variable $x_t\in Ess(f)\setminus Ess(g)$ such that $Ess(g)\cup\{x_t\}\in Sep(f)$.
\end{lemma}
\begin{proof}
Let $M=Ess(g)$. Then $M\in Sep(f)$ and from Theorem \rangleef{t3.2} it follows that there is an implementation $\mathbf{(j,c)}$ of the form $\mathbf{(j,c)}=(j_1j_2\langledots j_{r-m}j_{r-m+1}\langledots j_r,$ $c_1c_2\langledots c_{r-m}c_{r-m+1}\langledots c_r c)\in Imp(f)$ where $M=\{x_{j_{r-m+1}},\langledots, x_{j_r}\}$ and $1\langleeq m\langleeq r\langleeq ess(f)$. Since $m<n$ it follows that $r-m>0$ and Lemma \rangleef{l17} shows that there is $x_{j_i}\in Ess(h)$ where \[h=f(x_{j_1}=c_1,\langledots,x_{j_{i-1}}=c_{i-1},x_{j_{i+1}}=c_{i+1},\langledots x_{j_{r-m}}=c_{r-m}).\] It is easy to see that $Ess(h)=M\cup\{x_{j_i}\}$.
\end{proof}
Now, as an immediate consequence of the above lemma we obtain Theorem \rangleef{t2.4} which
was inductively proven by K. Chimev.
\begin{theorem}\langleabel{t2.4} \cite{ch51}
If $ess(f)=n$, $g\preceq f$ with
$ess(g)= m\langleeq
n$ then there exist $n-m$ subfunctions $g_1,\langledots,g_{n-m}$ such that
\[g \prec g_1 \prec g_2\prec \langledots\prec g_{n-m}= f\]
and $ess(g_i)=m+i$ for $i=1,\langledots,n-m$.
\end{theorem}
The {\it depth}, (denoted by $Depth(D_f)$) of an ordered decision diagram $D_f$ for a function $f$ is defined as the number of the edges in a longest path from the function node in $D_f$ to a leaf of $D_f$.
Thus for the diagrams in Figure \rangleef{f3} we have $Depth(D_f)=4$ and $Depth(D_g)=3$.
Clearly, if $ess(f)=n$ then $Depth(D_f)\langleeq n+1$ for all ODDs of $f$.
\begin{theorem}\langleabel{l3.2}
If $ess(f)=n\geq 1$ then there is an ordered decision diagram $D_f$ of $f$ with $Depth(D_f)=n+1$.
\end{theorem}
\begin{proof}
Let $Ess(f)=\{x_1,\langledots,x_n\}$, $n\geq 1$. Since $x_1$ is an essential variable it follows that $\{x_1\}\in Sep(f)$. Theorem \rangleef{t2.4} implies that there is an ordering $\langle i_1; i_2; \langledots; i_{n-1}\rangle$ of the rest variables $x_{2}, \langledots, x_{{n}}$ such that for each $j$, $1\langleeq j\langleeq n-1$ we have $g_j\prec_J^{\mathbf{c}} f$ where $J=\{x_{i_1},\langledots,x_{i_j}\}$, $\mathbf{c}\in Z_k^{J}$ and $Ess(g_j)=\{x_1,x_{i_{j+1}},\langledots,x_{i_{n-1}}\}$. This shows that the all variables from $J$ have to be labels of non-terminal nodes in a path $\pi$ of the ordered decision diagram $D_f$ of $f$ under the variable ordering $\langle i_1; i_2; \langledots; i_{n-1}; 1\rangle$. Hence $\pi$ has to contain all essential variables in $f$ as labels at the non-terminal nodes of $\pi$. Hence $Depth(D_f)=n+1$.
\end{proof}
\begin{theorem}\langleabel{t3.3}
Let $f\in P_k^n$ and $Ess(f)=\{x_1,\langledots,x_n\}$, $n\geq 1$. If $M\neq \emptyset$, $M\subset Ess(f)$ and $M\notin Sep(f)$ then there is a decision diagram $D_f$ of $f$ with $Depth(D_f)<n+1$.
\end{theorem}
\begin{proof}
Without loss of generality, let us assume that $M=\{x_1,\langledots,x_m\}$, $m<n$. Since $M$ is inseparable in $f$, the family $Dis(M,f)$ of the all distributive sets of $M$ is non-empty. According to Theorem \rangleef{t2.2} there is a non-empty $s$-system $\beta=\{x_{i_1},\langledots,x_{i_t}\}$ of $Dis(M,f)$. Since $f(x_{i_1}=c_1)\neq f(x_{i_1}=c_2)$ for some $c_1,c_2\in Z_k$ it follows that there exists an ODD $D_f$ for $f$ under a variable ordering with $x_{i_1}$ as the label of the first non-terminal node of $D_f$. According to Corollary \rangleef{c2.1} for all $c\in Z_k$ there is a variable $x_j\in M$ which is inessential in $f(x_{i_1}=c)$. Hence, each path of $D_f$ does not contain at least one variable from $M$ among its labels of non-terminal nodes. Hence $Depth(D_f)<n+1$.
\end{proof}
\section{Equivalence Relations and Transformation Groups in $P_k^n$}\langleabel{sec5}
Many of the problems in applications of the $k$-valued functions are compounded because of the large number of the functions, namely $k^{k^n}$. Techniques which involve enumeration of functions can only be used if $k$ and $n$ are trivially small. A common way for extending the scope of such enumerative methods is to classify the functions into equivalence classes under some natural equivalence relation.
In this section we define equivalence relations in $P_k^n$ which classify functions with respect to number of their implementations, subfunctions and separable sets. We are intended to determine several numerical invariants of the transformation groups generated by these relations. The second goal is to compare these groups with so called classical subgroups of the Restricted Affine Group(RAG) \cite{lech} which have a variety of applications such as coding theory, switching theory, multiple output combinational logic, sequential machines and other areas of theoretical and applied computer sciences.
Let us denote by $S_A$ the symmetric group of all permutations of a given no-empty set $A$. $S_m$ denotes the symmetric group $S_{\{1,\langledots,m\}}$ for a natural number $m$, $m\geq 1$.
Let us define the following three equivalence relations: $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$.
\begin{definition}\langleabel{d5.1} Let $f,g\in P_k^n$ be two functions.
\begin{enumerate}
\item[(i)] If $ess(f)=ess(g)\langleeq 1$ then $f\simeq_{imp} g$;
\item[(ii)] Let $ess(f)=n>1$. We say that $f$ is $imp$-equivalent to $g$ (written $f\simeq_{imp} g$) if there are $\pi\in S_n$ and $\sigma_i\in S_{Z_k}$ such that $f(x_i=j)\simeq_{imp} g(x_{\pi(i)}=\sigma_i(j))\quad\mathbfox{for all}\quad i=1,\langledots,n\quad\mathbfox{and}\quad j\in Z_k.$
\end{enumerate}
\end{definition}
Hence two functions are ${imp}$-equivalent if they produce same number of implementations, i.e.
$imp(f)=imp(g)$ and there are $\pi\in S_n$, and $\sigma$, $\sigma_i\in S_{Z_k}$ such that
$(i_1\langledots i_m,c_{1},\langledots,c_{m}c)\in Imp(f)\iff $ $ (\pi(i_1)\langledots \pi(i_m), \sigma_1(c_{1})\langledots\sigma_m(c_{m})\sigma(c))\in Imp(g).$
Table \rangleef{tb1} shows the classification of Boolean functions of two variables into four classes, called {\it imp-classes} under the equivalence relation $\simeq_{imp}$.
The second column shows the number of implementations of the functions from the $imp$-classes given at the first column. The third column presents the number of functions per each $imp$-class.
\begin{table}[h]
\caption{$Imp$-classes in $P_2^2$.} \langleabel{tb1}
\centering
\begin{tabular}{l|l|l}
\hline\hline
$[\ 0,\ 1\ ]$ & \ 1& 2\\ \hline
$[\ x_1,\ x_2,\ x_1^0,\ x_2^0\ ]$&\ 2& 4\\ \hline
$[\ x_1x_2,\ x_1x_2^0,\ x_1^0x_2,\ x_1^0x_2^0,\ x_1\overlineplus x_1x_2,$ &&\\
$ x_2^0\overlineplus x_1x_2,\ x_1^0\overlineplus x_1x_2,\ x_1^0\overlineplus x_1x_2^0\ ]$& \ 6 & 8\\ \hline
$[\ x_1\overlineplus x_2,\ x_1\overlineplus x_2^0\ ]$&\ 8& 2\\ \hline\hline
\end{tabular}
\end{table}
\begin{definition}\langleabel{d5.2} Let $f,g\in P_k^n$ be two functions.
\begin{enumerate}
\item[(i)] If $ess(f)=ess(g)=0$ then $f\simeq_{sub} g$;
\item[(ii)] If $ess(f)=ess(g)=1$ then $f\simeq_{sub} g\iff range(f)=range(g)$;
\item[(iii)] Let $ess(f)=n>1$. We say that $f$ is $sub$-equivalent to $g$ (written $f\simeq_{sub} g$) if $sub_m(f)=sub_m(g)$
for all $m=0,1,\langledots, n$.
\end{enumerate}
\end{definition}
It is easy to see that the equivalence relation $\simeq_{sub}$ partitions the algebra of Boolean functions of two variables in the same equivalence classes (called {\it the sub-classes}) as the relation $\simeq_{imp}$ (see Table \rangleef{tb1}).
\begin{definition}\langleabel{d5.3} Let $f,g\in P_k^n$ be two functions.
\begin{enumerate}
\item[(i)] If $ess(f)=ess(g)\langleeq 1$ then $f\simeq_{sep} g$;
\item[(ii)]Let $ess(f)=n>1$. We say that $f$ is $sep$-equivalent to $g$ (written $f\simeq_{sep} g$) if $sep_m(f)=sep_m(g)$
for all $m=1,\langledots, n$.
\end{enumerate}
\end{definition}
The equivalence classes under $\simeq_{sep}$ are called {\it sep-classes}.
Since $P_k^n$ is a finite algebra of $k$-valued functions each equivalence relation $\simeq$ on $P_k^n$ makes a partition of the algebra in the set of disjoint equivalence classes $Cl(\simeq)=\{P_1^\simeq,\langledots,P_r^\simeq\}$. Then, in the set of all equivalence relations a partial order is defined as follows: $\simeq_1\ \langleeq\ \simeq_2$ if for each $P\in Cl(\simeq_1)$ there is a $Q\in Cl(\simeq_2)$ such that $P\subseteq Q$. Thus $\simeq_1\ \langleeq\ \simeq_2$ if and only if $f\simeq_1 g\ \Rightarrow f\simeq_2 g$, for $f,g\in P_k^n$.
\begin{theorem}\langleabel{t5.1}
~~~
\begin{enumerate}
\item[(i)] $ \simeq_{imp}\ \langleeq\ \simeq_{sep}$; \quad (iii) $\simeq_{imp}\ \not\langleeq\ \simeq_{sub}$;
\item[(ii)] $ \simeq_{sub}\ \langleeq\ \simeq_{sep}$; \quad (iv) $ \simeq_{sub}\ \not\langleeq\ \simeq_{imp}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i)\
Let $f,g\in P_k^n$ be two ${imp}$-equivalent functions, i.e. $f\simeq_{imp} g$.
We shall proceed by induction on the number $n=ess(f)$ of essential variables in $f$ and $g$.
Clearly, if $n\langleeq 1$ then $f\simeq_{sub} g$, which is our inductive basis.
Let us assume that $f\simeq_{imp} g$ implies $f\simeq_{sep} g$ if $n< r$ for some natural number $r$, $r\geq 2$.
Let $f$ and $g$ be two functions with $f\simeq_{imp}g$ and $ess(f)=ess(g)=r$. Then there are $\pi\in S_r$ and $\sigma_i\in S_{Z_k}$ for $i=1,\langledots,r$ such that $f(x_i=j)\simeq_{imp} g(x_{\pi(i)}=\sigma_i(j))$. Let $M$, $\emptyset\neq M\in Sep(f)$ be a separable set of essential variables in $f$ with $|M|=m$, $1\langleeq m\langleeq r$. Theorem \rangleef{t3.2} implies that there is an implementation
\[\mathbf{(j,c)}=(j_1\langledots j_{r-m} i_1\langledots i_m, c_{j_1}\langledots c_{j_{r-m}}c_{i_1}\langledots c_{i_m}c)\]
of $f$ obtained under an ODD whose variable ordering finishes with the variables from $M$, i.e. $M=\{x_{i_1}\langledots,x_{i_m}\}$. Then $f(x_{j_1}=c_{j_1})\simeq_{imp} g(x_{\pi(j_1)}=\sigma_{j_1}(c_{j_1}))$ implies that
\[({\pi({j_1})}\langledots {\pi({j_{r-m}})}{\pi({i_1})}\langledots {\pi({i_m})},\sigma_{j_1}(c_{j_1})\langledots \sigma_{j_{r-m}}(c_{j_{r-m}})\sigma_{i_1}(c_{i_1})\langledots \sigma_{i_m}(c_{i_m})\sigma(c))\]
is an implementation of $g$, for some $\sigma\in S_{Z_k}$. Again, from Theorem \rangleef{t3.2} it follows that
$\pi(M)=\{x_{\pi(i_1)},\langledots,x_{\pi(i_m)}\}\in Sep(g).$ Since $\pi$ is a permutation of $S_r$ it follows that $sep_m(f)=sep_m(g)$ for $m=1,\langledots,r$ and hence $\simeq_{imp}\ \langleeq\ \simeq_{sep}.$
(ii) \
Definition \rangleef{d1.3} shows that $M\in Sep(f)$ if and only if there is a subfunction $g\in Sub(f)$ with $g\prec_Q^{\mathbf{c}}f$ where $Q=Ess(f)\setminus M$ and $\mathbf{c}\in Z_k^{n-|M|}$. Hence
\[\forall f,g\in P_k^n,\quad Sub(f)=Sub(g)\implies Sep(f)=Sep(g),\]
which implies that
$sub_m(f)=sub_m(g)\implies sep_m(f)=sep_m(g)$ and
$\simeq_{sub}\langleeq \simeq_{sep}$.
(iii)\
Let us consider the functions
\[f=x_1^0x_2x_3\overlineplus x_1x_2^0x_3^0\ (mod\ 2)\quad\mathbfox{and}\quad g=x_2x_3\overlineplus x_1x_2^0x_3\overlineplus x_1x_2x_3^0\ (mod\ 2).\]
The set of the all simple subfunctions in $f$ is: $\{x_1x_2^0,
x_1^0x_2, x_1x_3^0, x_1^0x_3, x_2x_3, x_2^0x_3^0\}$ and in $g$ is: $\{x_1x_2,
x_1x_3, x_2x_3, x_2\overlineplus x_1x_2^0, x_3\overlineplus x_1x_3^0, x_2^0x_3^0\overlineplus 1\}$.
Hence $f$ and $g$ have six simple subfunctions, which depends essentially on two variables. Table \rangleef{tb1} shows that all these subfunctions belong to same $imp$-class and the number of their implementations is $6$. Thus we might calculate that $imp(f)=imp(g)=36$ and $f\simeq_{imp}g$.
The set of the all subfunctions with one essential variable in the function $f$ is: $\{ x_1, x_2, x_3, x_1^0, x_2^0, x_3^0\}$ and in $g$ is: $\{x_1, x_2, x_3\}$.
Then we have $sub_0(f)=sub_0(g)=2$, $sub_1(f)=6$, $sub_1(g)=3$ and $sub_2(f)=sub_2(g)=6$ and hence $f\not\simeq_{sub}g$.
It is clear that $sub(f)=15$, $sub(g)=12$ and $\simeq_{imp}\ \not\langleeq\ \simeq_{sub}$.
(iv)\
Let us consider the functions
\[f=x_1x_2^0x_3^0\overlineplus x_1\ (mod\ 2)\quad\mathbfox{and}\quad g= x_1x_2x_3\ (mod\ 2).\]
The simple subfunctions in $f$ and $g$ are:\\
\begin{tabular}{lll}
$f(x_1=0)=0,$&$f(x_3=0)=x_1x_2^0\overlineplus x_1,$& $g(x_2=0)=0,$ \\
$f(x_1=1)=x_2^0x_3^0\overlineplus 1,$& $f(x_3=1)=x_1,$ & $g(x_2=1)=x_1x_3,$\\
$f(x_2=0)=x_1x_3^0\overlineplus x_1,$ & $g(x_1=0)=0,$&$g(x_3=0)=0,$\\
$f(x_2=1)=x_1,$ & $g(x_1=1)=x_2x_3,$&$g(x_3=1)=x_1x_2$.\\
\end{tabular}
Now, using Table \rangleef{tb1}, one can easily calculate that $imp(f)=23$ and $imp(g)=21$, and hence $ f \not\simeq_{imp} g$.
On the other side we have
$Sub(f)=\{0, 1, x_1, x_2, x_3, x_2^0x_3^0\overlineplus 1, x_1x_3^0\overlineplus x_1, x_1x_2^0\overlineplus x_1, f\}$
and
$Sub(g)=\{0, 1, x_1, x_2, x_3, x_2x_3, x_1x_3, x_1x_2, g\}$
which show that
$sub_m(f)=sub_m(g)\quad\mathbfox{for}\quad m=0,1,2,3$ and $f\simeq_{sub} g$.
Hence $ \simeq_{sub}\ \not\langleeq\ \simeq_{imp}$.
\end{proof}
A {\it transformation} $\psi:P_k^n\langleongrightarrow P_k^n$ can be viewed as an $n$-tuple of functions
\[\psi=(g_1,\langledots,g_n),\quad g_i\in P_k^n,\quad i=1,\langledots,n\]
acting on any function $f=f(x_1,\langledots,x_n)\in P_k^n$ as follows
$\psi(f)=f(g_1,\langledots,g_n)$.
Then the composition of two transformations $\psi$ and $\phi=(h_1,\langledots,h_n)$ is defined as follows
\[\psi\phi=(h_1(g_1,\langledots,g_n),\langledots,h_n(g_1,\langledots,g_n)).\]
Thus the set of all transformations of $P_k^n$ is the {\it universal monoid $\Omega_k^n$} with unity - the identical transformation. When taking only invertible transformations we obtain the {\it universal group} $C_k^n$ isomorphic to the symmetric group $S_{Z_k^n}$.
Throughout this paper we shall consider invertible transformation, only. The groups consisting of invertible transformations of $P_k^n$ are called {\it transformation groups}.
Let $\simeq$ be an equivalence relation in $P_k^n$.
A mapping $\varphi:P_k^n\langleongrightarrow P_k^n$ is called {\it a transformation, preserving $\simeq$} if $f\simeq \varphi(f)$ for all $f\in P_k^n$.
Taking only invertible transformations which preserve $\simeq$, we get the group $G$ of all transformations preserving $\simeq$, whose {\it orbits} (also called {\it $G$-types}) are the equivalence classes $P_1,\langledots,P_r$ under $\simeq$.
The number of orbits of a group $G$ of transformations in finite algebras of functions is denoted by $t(G)$.
Next, we relate groups to combinatorial problems trough the following obvious, but important definition:
\begin{definition}\langleabel{d5.4}~Let $G$ be a transformation group acting on the algebra of functions $P_k^n$and suppose that $f,g\in P_k^n$. We say that $f$ is $G$-equivalent to $g$ (written $f\simeq_G g$) if there exists $\psi\in G$ so that $g=\psi(f)$.
\end{definition}
Clearly, the relation $\simeq_G $ is an equivalence relation. We summarize and extend the results for the "classical" transformation groups, following \cite{har2,lech,str3}, where these notions are used to study classification and enumeration in the algebra of boolean functions. Such groups are induced under the following notions of equivalence: complementation and/or permutation of the variables; any linear or affine function of the variables.
Since we want to classify functions from $P_k^n$ into equivalence classes, three natural problems occur.
\begin{enumerate}
\item[$\bullet$] We ask for the number $t(G)$ of such equivalence classes. This problem will be partially discussed for the family of ``natural'' equivalence relations in the algebra of boolean functions.
\item[$\bullet$] We ask for the cardinalities of the equivalence classes. This problem is important in applications as functioning the switching gates, circuits etc. For boolean functions of 3 and 4 variables we shall solve these two problems, also concerning $imp$-, $sub$- and $sep$-classes.
\item[$\bullet$] We want to give a method which will decide the class to which an arbitrary function belongs. In some particular cases this problem will be discussed below. We also develop a class of algorithms for counting the complexities $imp$, $sub$ and $sep$ for each boolean function which allow us to classify the algebras $P_2^n$ for $n=2,3,4$ with respect to these complexities as group invariants.
\end{enumerate}
These problems are very hard and for $n\geq 5$ they are practically unsolvable.
We use the denotation $\langleeq$ also, for order relation ``subgroup''. More precisely, $H\langleeq G$ if there is a subgroup $G'$ of $G$ which is isomorphic to $H$.
Let us denote by $IM_k^n$, $SB_k^n$ and $SP_k^n$ the transformation groups induced by the equivalence relations $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$, respectively.
Now, as a direct consequence of Theorem \rangleef{t5.1} we obtain the following proposition.
\begin{proposition}\langleabel{c5.1}~
\begin{enumerate}\item[(i)] $IM_k^n\langleeq SP_k^n$; \quad (iii) $IM_k^n\not\langleeq SB_k^n$;
\item[(ii)] $SB_k^n\langleeq SP_k^n$; \quad (iv) $SB_k^n\not\langleeq IM_k^n$.
\end{enumerate}
\end{proposition}
We deal with "natural" equivalence relations which involve variables in some functions. Such relations induce permutations on the domain $Z_k^n$ of the functions. These mappings form a transformation group whose number of equivalence classes can be determined.
The restricted affine group (RAG) is defined as a subgroup of the symmetric group on the direct sum of the vector space $Z_k^n$ of arguments of functions and the vector space $Z_k$ of their outputs. The group RAG permutes the direct sum $Z_k^n+Z_k$ under restrictions which preserve single-valuedness of all functions from $P_k^n$. The equivalence relation induced by RAG is called {\it prototype equivalence relation}.
In the model of RAG an affine transformation operates on the domain or space of inputs $\mathbf{x}=(x_1,\langledots,x_n)$ to produce the output $\mathbf{y}=\mathbf{xA}\overlineplus \mathbf{c}$, which might be used as an input in a function $g$. Its output $g(\mathbf{y})$ together with the function variables $x_1,\langledots,x_n$ are linearly combined by a range transformation which defines the image $f(\mathbf{x})$ as follows:
\begin{equation}\langleabel{eq2}
f(\mathbf{x})=g(\mathbf{y})\overlineplus a_1x_1\overlineplus\langledots\overlineplus a_nx_n\overlineplus d=g(\mathbf{xA}\overlineplus \mathbf{c})\overlineplus \mathbf{a^tx}\overlineplus d
\end{equation}
where $d$ and $a_i$ for $i=1,\langledots,n$ are constants from $Z_k$.
Such a transformation belongs to RAG if $\mathbf{A}$ is a non-singular matrix. The name RAG was given to this group by R. Lechner in 1963 (see \cite{lech1}) and it was studied by Ninomiya (see \cite{nin}) who gave the name "prototype equivalence" to the relation it induces on the function space $P_k^n$.
We want to extract basic facts about some of the subgroups of RAG which are "neighbourhoods" or "relatives" of our transformation groups - $IM_k^n$, $SB_k^n$ and $SP_k^n$.
First, we consider a group which is called $CA_k^n$ (complement arguments) and each transformation $\mathbf{j}\in CA_k^n$ is determined by an $n$-tuple from $Z_k^n$, i.e.
$CA_k^n=\{(j_1,\langledots,j_n)\in Z_k^n\}.$
Intuitively, $CA_k^n$ will complement some of the variables of a function. If $\mathbf{j}=(j_1,\langledots,j_n)$ is in $CA_k^n$, define
$\mathbf{j}(x_1,\langledots,x_n)=(x_1\overlineplus j_1,\langledots,x_n\overlineplus j_n).$
The group operation is sum mod $k$ and written $\overlineplus$. For example if $n=k=3$ and $\mathbf{j}=(2,1,0)$ then $\mathbf{j}(x_1,x_2,x_3)=(x_1\overlineplus 2,x_2\overlineplus 1,x_3)$ and $\mathbf{j}$ induces a permutation on $Z_3^3=\{0,1,2\}^3$. Then the following sequence of images: $\mathbf{j}: 000\rangleightarrow 210\rangleightarrow 120\rangleightarrow 000$
determines the cycle $(0,21,15)$ and if we agree to regard each triple from $Z_3^3$ as a ternary number, then the permutation induced by $\mathbf{j}$ can be written in cyclic notation as $(0,21,15)(1,22,16)(2,23,17)(3,24,9)(4,25,10)(5,26,11)$ $(6,18,12)(7,19,13)(8,20,14).$
In \cite{har2} M. Harrison showed that the boolean functions of two variables are grouped into seven classes under the group $CA_2^2$.
Another classification occurs when permuting arguments. If $\pi\in S_n$ then $\pi$ acts on variables by:
$\pi(x_1,\langledots,x_n)=(x_{\pi(1)},\langledots,x_{\pi(n)}).$
Each permutation induces a map on the domain $Z_k^n$. For instance the permutation $\pi=(1,2)$ induces a permutation on $\{0,1,2\}^3$ when considering the algebra $P_3^3$. Then we have $\pi: 010\rangleightarrow 100 \rangleightarrow 010$ and in cyclic notation it can be written as
\[(3,9)(4,10)(5,11)(6,18)(7,19)(8,20)(15,21)(16,22)(17,23).\]
$S_k^n$ denotes the transformation group induced by permuting of variables. It is clear that $S_k^n$ is isomorphic to $S_n$.
If we allow both complementations and permutations of the variables, then a transformation group, called $G_k^n$, is induced. The group action on variables is represented by
$((j_1,\langledots,j_n),\pi)(x_1,\langledots,x_n)=(x_{\pi(1)}\overlineplus j_1,\langledots, x_{\pi(n)}\overlineplus j_n)$
where $j_m\in Z_k$ for $1\langleeq m\langleeq n$ and $\pi\in S_n$. The group $G_2^n$ is especially important in switching theory and other areas of discrete mathematics, since it is the symmetry group of the $n$-cube. The classification of the boolean functions under $G_2^2$ into six classes is shown in \cite{har2}.
Let us allow a function to be equivalent to its complements as well as using equivalence under $G_k^n$. Then the transformation group which is induced by this equivalence relation is called the {\it genera} of $G_k^n$ and it is denoted by $GE_k^n$. Thus the equivalence relation $\simeq_{gen}$ which induces genera of $G_k^n$ is defined as follows $f\simeq_{gen}g\iff f\simeq_{G_k^n}g$ or $f=g\overlineplus j$ for some $j\in Z_k$. Then there exist only four equivalence classes in $P_2^2$, induced by $GE_2^2$.
These classes are the same as the classes induced by the group $IM_2^2$ in the algebra $P_2^2$ (see \cite{har2} and Table \rangleef{tb1}, given above).
Next important classification is generated by equivalence relations which allow adding linear or affine functions of variables. In order to preserve the group property we shall consider invertible linear transformations and assume that $k$ is a prime number such that $LG_k^n$ the general linear group on an $n$-dimensional vector space is over the field $Z_k$. The transformation groups $LG_2^n$ and $A_2^n$ of linear and affine transformations in the algebra of boolean functions are included in the lattice of the subgroups of RAG. We extend this view to the functions from $P_k^n$. The algebra of boolean functions in the simplest case of two variables is classified in eight classes under $LG_2^2$ and in five classes under $A_2^2$. Table \rangleef{tb1_1} presents both equivalence classes of boolean functions from $P_2^2$ under the transformation group $RAG$.
\begin{table}
\caption{Classes in $P_2^2$ under $RAG$.} \langleabel{tb1_1}
\centering
\begin{tabular}{l}
\hline\hline
$[\ 0,\
1,\
x_1,\ x_2,\ x_1^0,\ x_2^0,\ x_1\overlineplus x_2,\ x_1\overlineplus x_2^0\ ]$\\ \hline
$[\ x_1x_2,\ x_1x_2^0,\ x_1^0x_2,\ x_1^0x_2^0,\
x_1\overlineplus x_1x_2,\ x_2^0\overlineplus x_1x_2,\
x_1^0\overlineplus x_1x_2,\ x_1^0\overlineplus x_1x_2^0\ ]$\\ \hline\hline
\end{tabular}
\end{table}
The subgroups of RAG defined above are
determined by equivalence relations as it is shown in Table \rangleef{tb2}, where $\mathbf{P}$ denotes a permutation matrix, $\mathbf{I}$ is the identity matrix, $\mathbf{b\mathbfox{ and }c}$ are vectors from $Z_k^n$ and $d\in Z_k$.
\begin{table}
\caption{Subgroups of RAG}\langleabel{tb2}
\begin{tabular}{||l|l|l||}\hline\hline
Subgroup& Equivalence relations& Determination\\ \hline
RAG & Prototype equivalence& $\mathbf{A}$-non-singular\\
$GE_k^n$ & genus & $\mathbf{A}=\mathbf{P}$, $\mathbf{a}=\mathbf{0}$\\
$CF_k^n$ & complement function & $\mathbf{A}=\mathbf{I}$, $\mathbf{a}=\mathbf{0}$, $\mathbf{c}=\mathbf{0}$\\
$A_k^n$ &affine transformation & $\mathbf{a}=\mathbf{0}$, $d=0$\\
$G_k^n$ & permute \& complement & \\
& variables (symmetry types) & $\mathbf{A}=\mathbf{P}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$LF_k^n$ & add linear function & $\mathbf{A=I}$, $\mathbf{c=0}$, $d=0$\\
$CA_k^n$ & complement arguments & $\mathbf{A}=\mathbf{I}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$LG_k^n$ & linear transformation & $\mathbf{c}=\mathbf{0}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\
$S_k^n$ & permute variables & $\mathbf{A}=\mathbf{P}$, $\mathbf{c}=\mathbf{0}$, $\mathbf{a}=\mathbf{0}$, $d=0$\\ \hline\hline
\end{tabular}
\end{table}
It is naturally to ask which subgroups of RAG are subgroups of the groups $IM_k^n$ or $SB_k^n$. The answer of this question is our next goal.
\begin{example}\langleabel{ex5.1}
Let
$f=x_1x_2^0x_3\overlineplus x_1^0\quad\mathbfox{and}\quad g=x_1x_2^0x_3\overlineplus x_1x_2$ be two boolean functions.
Then
\[sub_1(f)=sub_1(g)=3,\ sub_2(f)=sub(g)=3\quad\mathbfox{and}\quad sub_3(f)=sub_3(g)=1.\]
Hence $f\simeq_{sub} g$.
In a similar way, it can be shown that $f\simeq_{imp} g$.
The details are left to the reader.
On the other side, one can prove that there is no transformation $\varphi\in RAG$ such that $\varphi(x_1^0)=x_1x_2$ (see Table \rangleef{tb1_1}) and hence there is no affine transformation $\varphi\in RAG$ for which $g=\varphi(f)$.
Consequently, each group among $IM_k^n$, $SB_k^n$ and $SP_k^n$ can not be a subgroup of $RAG$.
\end{example}
Table \rangleef{tb2} allows us to establish the following fact.
\begin{fact}\langleabel{fc2}
If $f$ and $g$ satisfy (\rangleef{eq2}) with $\mathbf{A\notin \{0,P,I\}}$ or $\mathbf{a\neq 0}$ then $f\not\simeq_{imp} g$, $f\not\simeq_{sub} g$ and $f\not\simeq_{sep} g$.
\end{fact}
\begin{proposition}\langleabel{p5.1}~~
\begin{enumerate}
\item[(i)] $LG_k^n\not\langleeq SP_k^n$;\quad (ii) $LF_k^n\not\langleeq SP_k^n$;
\item[(iii)] $IM_k^n\not\langleeq RAG$;\quad (iv) $SB_k^n\not\langleeq RAG$.
\end{enumerate}
\end{proposition}
\begin{proof}
Immediate from Fact \rangleef{fc2} and Example \rangleef{ex5.1}.
\end{proof}
Let $\sigma:Z_k\langleongrightarrow Z_k$ be a mapping and $\psi_\sigma:P_k^n\langleongrightarrow P_k^n$ be a transformation of $P_k^n$ determined by $\sigma$ as follows
$\psi_\sigma(f)(\mathbf{a})=\sigma(f(\mathbf{a}))$
for all $\mathbf{a}=(a_1,\langledots,a_n)\in Z_k^n$.
\begin{theorem}\langleabel{t5.2} $\psi_\sigma\in IM_k^n$ and $\psi_\sigma\in SB_k^n$ if and only if $\sigma$ is a permutation of $Z_k$, $k>2$.
\end{theorem}
\begin{proof} "$\Leftarrow$" Let $\sigma\in S_{Z_k}$ be a permutation of $Z_k$ and let $f$ be an arbitrary function with $ess(f)=n\geq 0$. We shall proceed by induction on $n$, the number of essential variables in $f$.
If $n=0$ then clearly $\psi_\sigma(f)$ is a constant and hence $f\simeq_{imp} \psi_\sigma(f)$ and $f\simeq_{sub} \psi_\sigma(f)$.
Assume that if $n<p$ then $f\simeq_{imp} \psi_\sigma(f)$ and $f\simeq_{sub} \psi_\sigma(f)$ for some natural number $p, p>0$. Hence $f(x_i=j)\simeq_{imp}\psi_\sigma(f(x_i=j))$ and $sub_m(f(x_i=j))=sub_m(\psi_\sigma(f(x_i=j)))$ for all $i\in\{1,\langledots,n\}$, $m\in\{1,\langledots,n-1\}$ and $j\in Z_k$.
Let $n=p$. Let $x_i\in\{x_1,\langledots,x_n\}=Ess(f)$ and $j\in Z_k$, and let us set $g=f(x_i=j)$. Then $\psi_\sigma(g)=\psi_\sigma(f(x_i=j)$ and $ess(g)=n-1<p$. Hence our inductive assumption implies
$g\simeq_{imp}\psi_\sigma(g)$ and $g\simeq_{sub}\psi_\sigma(g)$. Consequently, we have
\[f(x_i=j)\simeq_{imp}\psi_\sigma(f(x_i=j))
\quad\mathbfox{and}\quad sub_m(f(x_i=j))=sub_m(\psi_\sigma(f(x_i=j)))\]
for all $x_i\in\{x_1,\langledots,x_n\}$ and $j\in Z_k$, which shows that $f\simeq_{imp}\psi_\sigma(f)$ and $f\simeq_{sub}\psi_\sigma(f)$.
"$\Rightarrow$" Let us assume that $\sigma$ is not a permutation of $Z_k$. Hence there exist two constants $a_1$ and $a_2$ from $Z_k$ such that $a_1\neq a_2$ and $\sigma(a_1)=\sigma(a_2)$. Let us fix the vector $\mathbf{b}=(b_1,\langledots,b_n)\in Z_k^n$. Then we define the following function from $P_k^n$:
\[
f(x_1,\langledots,x_n)=\langleeft\{\begin{array}{ccc}
a_1 \ &\ if \ &\ x_i=b_i\ for\ i=1,\dots,n \\
a_2 & & otherwise.
\end{array}
\rangleight.
\]
Clearly, $Ess(f)=X_n$. On the other hand the range of $f$ is $range(f)=\{a_1,a_2\}$ and $\sigma(range(f))=\{\sigma(a_1)\}$, which implies that
$\psi_\sigma(f)(c_1,\langledots,c_n)=\sigma(a_1)$
for all $(c_1,\langledots,c_n)\in Z_k^n$. Hence $\psi_\sigma(f)$ is the constant $\sigma(a_1)\in Z_k$ and $Ess(\psi_\sigma(f))=\emptyset$.
Thus we have $f\not\simeq_{imp} \psi_\sigma(f)$ and $f\not\simeq_{sub} \psi_\sigma(f)$.
\end{proof}
\begin{theorem}\langleabel{t5.3}
Let $\pi\in S_n$ and $\sigma_i\in S_{Z_k}$ for $i=1,\langledots,n$. Then
$f(x_1,\langledots,x_n)\simeq_{imp} f(\sigma_1(x_{\pi(1)}),\langledots,\sigma_n(x_{\pi(n)}))$
and
$f(x_1,\langledots,x_n)\simeq_{sub} f(\sigma_1(x_{\pi(1)}),\langledots,\sigma_n(x_{\pi(n)}))$.
\end{theorem}
\begin{proof}
Let $f\in P_k^n$ be an arbitrary function and assume $Ess(f)=X_n$.
First, we shall prove that
\[f(x_1,\langledots,x_n)\simeq_{imp} f(x_{\pi(1)},\langledots,x_{\pi(n)})\] {and} \[f(x_1,\langledots,x_n)\simeq_{sub} f(x_{\pi(1)},\langledots,x_{\pi(n)}).\]
Let $g=f(x_{\pi(1)},\langledots,x_{\pi(n)})$.
Clearly, if $n\langleeq 1$ then $f\simeq_{imp} g$ and $f\simeq_{sub} g$.
Assume that if $n<p$ then $f\simeq_{imp} g$ and $f\simeq_{sub} g$ for some natural number $p$, $p\geq 1$.
Let us suppose $n=p$. Let $x_i\in Ess(f)$ be an arbitrary essential variable in $f$ and let $c\in Z_k$ be an arbitrary constant from $Z_k$. Then we have
\[f(x_i=c)(x_1,\langledots,x_{i-1},x_{i+1},\langledots,x_p)=\]
\[=g(x_{\pi^{-1}(i)}=c)(x_{\pi^{-1}(1)},\langledots,x_{\pi^{-1}({i-1})},x_{\pi^{-1}({i+1})},\langledots,x_{\pi^{-1}(p)}).\]
Our inductive assumption implies
$f(x_i=c)\simeq_{imp}g(x_{\pi(i)}=c)$
{and} $sub_m(f(x_i=c))=sub_m(g(x_{\pi(i)}=c))$ for all $x_i\in X_n$, $m\in\{1,\langledots,p-1\}$ and $c\in Z_k$. Hence
$f\simeq_{imp}g
\ \mathbfox{and}\ f\simeq_{sub}g$.
Second, let us prove that
\[f(x_1,\langledots,x_n)\simeq_{imp} f(\sigma_1(x_{1}),\langledots,\sigma_n(x_{n}))\]
{and}
\[f(x_1,\langledots,x_n)\simeq_{sub} f(\sigma_1(x_{1}),\langledots,\sigma_n(x_{n})).\]
Let $h=f(\sigma_1(x_{1}),\langledots,\sigma_n(x_{n}))$. Then we have
\[f(a_1,\langledots,a_n)=h(\sigma_1^{-1}(a_1),\langledots,\sigma_n^{-1}(a_n)).\]
Hence, if $(i_1\langledots i_r, a_{i_1}\langledots a_{i_r}c)\in Imp(f)$ then $(i_1\langledots i_r, \sigma_{i_1}^{-1}(a_{i_1})\langledots \sigma_{i_r}^{-1}(a_{i_r})c)\in Imp(h)$ for some $r$, $1\langleeq r\langleeq n$.
Since $\sigma_i$ is a permutation of $Z_k$ for $i=1,\langledots,n$ it follows that
$f\simeq_{imp}h$. By similar arguments it follows that
$f\simeq_{sub}h$.
\end{proof}
\begin{corollary}\langleabel{c5.4} (i) $GE_k^n\langleeq IM_k^n$; \quad (ii) $GE_k^n\langleeq SB_k^n$; \quad
(iii) $GE_k^n\langleeq SP_k^n$.
\end{corollary}
\section{Classification of Boolean Functions}\langleabel{sec6}
In this section we compare a collection of subgroups of RAG with the groups of transformations preserving the relations $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$ and to obtain estimations for the number of equivalence classes, and for the cardinalities of these classes in the algebra of Boolean functions.
Our results are based on Proposition \rangleef{p5.1}, Theorem \rangleef{t5.2} and Theorem \rangleef{t5.3}. Thus we have
\begin{equation}\langleabel{eq3}
GE_2^n\langleeq IM_2^n,\quad GE_2^n\langleeq SB_2^n, \quad LG_2^n\not\langleeq SP_2^n \quad\mathbfox{and}\quad LF_2^n\not\langleeq SP_2^n.
\end{equation}
These relationships determine the places of the groups $IM_2^n$, $SB_2^n$ and $SP_2^n$ with respect to the subgroups of RAG. Figure \rangleef{f4} shows the location of these groups together with the subgroups of RAG.
M. Harrison \cite{har2} and R. Lechner \cite{lech} counted the number of equivalence classes and the cardinalities of the classes under some transformation subgroups of RAG for Boolean functions of 3 and 4 variables.
The relations (\rangleef{eq3}) show that if we have the values of $t(GE_2^n)$ then we can count the numbers $t(IM_2^n)$, $t(SB_2^n)$ and $t(SP_2^n)$ because the equivalence classes under these transformation groups are union of equivalence classes under $GE_2^n$ and hence we have $t(IM_2^n)\langleeq t(GE_2^n)$ and $t(SB_2^n)\langleeq t(GE_2^n)$. Moreover, if we know the factor-set $P_2^n/_{\simeq_{gen}}$ of representative functions under $\simeq_{gen}$ then we can effectively calculate the sets $P_2^n/_{\simeq_{imp}}$, $P_2^n/_{\simeq_{sub}}$ and $P_2^n/_{\simeq_{sep}}$ because of $P_2^n/_{\simeq_{imp}}\subseteq P_2^n/_{\simeq_{gen}}$ and $P_2^n/_{\simeq_{sub}}\subseteq P_2^n/_{\simeq_{gen}}$.
The next theorem allows us to count the number $imp(f)$ of the implementations of any function $f$ by a recursive procedure. Such a procedure is realized and its execution is used when calculating the number of the implementations and classifying the functions under the equivalence $\simeq_{imp}$.
\begin{theorem} \langleabel{t35} Let $f\in P_2^n$ be a boolean function. The number of all implementations in $f$ is determined as follows:
\[imp(f) = \langleeft\{\begin{array}{ccc}
1 \ &\ if \ &\ ess(f)=0 \\
&&\\
2 & if\ & ess(f)=1\\
&&\\
\sum_{x\in Ess(f)}[imp(f(x=0)) + imp(f(x=1))] & if\ & ess(f)\geq 2.
\end{array}
\rangleight.
\]
\end{theorem}
\begin{proof}
We shall proceed by induction on $n=ess(f)$ - the number of essential variables in $f$.
The lemma is clear if $ess(f)=0$. If $f$ depends essentially on one variable $x_1$, then there is a unique BDD of $f$ with one non-terminal node which has two outcoming edges. These edges together with the labels of the corresponding terminal nodes form the set $Imp(f)$ of all implementations of $f$, i.e. $imp(f)=2$.
Let us assume that
\[imp(f)= \sum_{i=1}^n[imp(f(x_i=0)) + imp(f(x_i=1))]\]
if $n< s$ for some natural number $s$, $1\langleeq s$.
Next, let us consider a function $f$ with $ess(f)=s$. Without loss of generality, assume that $Ess(f)=\{x_1,\langledots,x_n\}$ with $n=s$.
Since $x_i\in Ess(f)$ for $i=1,\langledots,n$ it follows that $f(x_i=0)\neq f(x_i=1)$ and there exist BDDs of $f$ whose label of the first non-terminal node is $x_i$. Let $D_f$ be a such BDD of $f$ and let $(ij_2\langledots j_m,c_1c_2\langledots c_mc)\in Imp(f)$ with $m\langleeq n$. Hence
\[(j_2\langledots j_m,c_2\langledots c_mc)\in Imp(g)\] where $g=f(x_i=c_1)$. On the other side it is clear that if
$(j_2\langledots j_m,d_2\langledots d_md)\in Imp(g)$ then $(ij_2\langledots j_m,c_1d_2\langledots d_md)\in Imp(f)$. Consequently, there is an one-to-one mapping between the set of implementations of $f$ with first variable $x_i$ and first edge labelled by $c_1$, and $Imp(g)$, which completes the proof.
\end{proof}
We also develop recursive algorithms to count $sub_m(f)$ and $sep_m(f)$ for $f\in P_2^n$, presented below.
Table \rangleef{tb4} shows the number of equivalence classes under the equivalence relations induced by the transformation groups $G_2^n$, $IM_2^n$, $SB_2^n$ and $SP_2^n$ for $n=1,2,3,4$. M. Harrison found from applying Polya's counting theorem (see \cite{har2}) the numbers $t(G_2^5)$ and $t(G_2^6)$, which are upper bounds of $t(IM_2^n)$, $t(SB_2^n)$ and $t(SP_2^n)$ for $n=5,6$.
Figure \rangleef{f4} and Table \rangleef{tb3} show that for the algebra $P_2^3$ there are only 14 different generic equivalent classes, 13 imp-classes, 11 sub-classes and 5 sep-classes. Hence three mappings that converts each generic class into an imp-class, into a sub-class and into a sep-class are required. Each generic class is a different row of Table \rangleef{tb3}. For example, the generic class \textnumero 12 (as it is numbered in Table VIII, \cite{lech}) is presented by 10-th row of Table \rangleef{tb3}. It consists of 8 functions obtained by complementing function $f$ and/or permuting and/or complementing input variables in all possible ways, where $f=x_1x_2^0x_3\overlineplus x_1x_2x_3^0\overlineplus x_2x_3$. This generic class \textnumero 12 is included in imp-class \textnumero 9, sub-class \textnumero 8 and sep-class \textnumero 5 which shows that $imp(f)=36$, $sub(f)=12$ and $sep(f)=7$. The average cardinalities of equivalence classes and complexities of functions are also shown in the last row of Table \rangleef{tb3}.
Table \rangleef{tb5} shows the $sep$-classes of boolean functions depending on at most five variables. Note that there are $2^{32}=4294967296$ functions in $P_2^5$. All calculations were performed on a computer with two Intel Xeon E5/2.3 GHz CPUs. The execution with total exhaustion took 244 hours.
\begin{figure}
\caption{Transformation groups in $P_2^n$ ($n=3/n=4$)}
\end{figure}
\begin{table}
\centering
\caption{Number of classes under $symmetry$ $type$, $\simeq_{imp}$, $\simeq_{sub}$ and $\simeq_{sep}$}\langleabel{tb4}
\begin{tabular}{rrrrr}
$n$ & $t(G_2^n)$ & $t(IM_2^n)$ &$t(SB_2^n)$ &$t(SP_2^n)$ \\ \hline\hline
1&3&2&2&2\\
2&6&4&4&3 \\
3&22&13&11&5 \\
4&402&104&74&11 \\
5&1\ 228\ 158&1606&{$<$ 1228158}& 38 \\
6&400\ 507\ 806\ 843\ 728&\multicolumn{3}{c}{$<$ 400\ 507\ 806\ 843\ 728} \\
\hline\hline
\end{tabular}
\end{table}
\begin{sidewaystable}[p]\centering
\caption{Classification of $P_2^3$ under $\simeq_{sep}$, $\simeq_{sub}$, $\simeq_{imp}$ and genus.} \langleabel{tb3}
\begin{tabular}{||c|c|c||c|c|c||c|c|c||c|c||c||} \hline
\hline
sep- & $sep(f)$ & func. & sub- & $sub(f)$ & func. & imp- & $imp(f)$ & func.& Generic & func. & representative \\
class & & per & class & & per & class & & per &class \cite{lech}& per & function $f$ \\
\textnumero &&class&\textnumero &&class&\textnumero &&class&\textnumero &class&\\
\hline\hline
1 & 0 & 2 & 1 & 1 & 2 & 1 & 1 & 2&1&2 & $0$ \\ \hline
2 & 1 & 6 & 2 & 3 & 6 & 2 & 2 & 6&9&6& $x_1$ \\ \hline
\multirow{2}{*}{3} & \multirow{2}{*}{3} & \multirow{2}{*}{30} & 3 & 5 & 24 & 3 & 6 & 24&3&24& $x_1x_2$ \\ \cline{4-12}
& & & 4 & 7 & 6 & 4 & 8 & 6&10&6& $x_1\overlineplus x_2$ \\ \hline
4 & 6 & 24 & 5 & 11 & 24 & 5 & 28 & 24&13&24& $x_1\overlineplus x_1x_3\overlineplus$ \\
& & & & & & & & &&& $ x_2x_3$ \\
\hline
\multirow{9}{*}{5} & \multirow{9}{*}{7} & \multirow{9}{*}{194} & \multirow{2}{*}{6} & \multirow{2}{*}{9} & \multirow{2}{*}{64} & 6 & 21 & 16&2&16 & $x_1x_2x_3$ \\ \cline{7-12}
& & & & & & 7 & 23 & 48&6&48& $x_1x_2^0x_3^0\overlineplus x_1$ \\ \cline{4-12}
& & & 7 & 12 & 48 & 8 & 30 & 48&7&48 & $x_1x_2^0x_3^0\overlineplus $ \\ & & & & & & & & & & & $x_2x_3$ \\ \cline{4-12}
& & & 8 & 12 & 8 & \multirow{2}{*}{} & \multirow{2}{*}{} & \multirow{2}{*}{}&12&8& $x_1x_2^0x_3\overlineplus$ \\ & & & & & & \multirow{2}{*}{9} & \multirow{2}{*}{36} & \multirow{2}{*}{16}&& & $ x_1x_2x_3^0 \overlineplus x_2x_3$ \\ \cline{4-6} \cline{10-12}
& & & \multirow{3}{*}{9} & \multirow{3}{*}{15} & \multirow{3}{*}{26} & & & &5&8& $x_1^0x_2x_3\overlineplus x_1x_2^0x_3^0$ \\ \cline{7-12}
& & & & & & 10 & 42 & 16&8&16& $x_1x_2^0x_3\overlineplus $ \\
& & & & & & & & & && $ x_1x_2x_3^0\overlineplus x_1^0x_2x_3$ \\
\cline{7-12}
& & & & & & 11 & 48 & 2& 11&2& $x_1\overlineplus x_2\overlineplus x_3$ \\ \cline{4-12}
& & & 10 & 13 & 24 & 12 & 32 & 24&14&24 & $x_1\overlineplus x_2x_3$ \\ \cline{4-12}
& & & 11 & 13 & 24 & 13 & 33 & 24&4&24 & $x_1x_2^0x_3\overlineplus x_1x_2x_3^0$ \\ \hline
aver.&6.2&51.2&&10.6&23.3& &26.0&19.7&&18.3&\\ \hline\hline
\end{tabular}
\end{sidewaystable}
\begin {table}
\centering
\caption{Classes in $P_2^5$ under $\simeq_{sep}$ }\langleabel{tb5}
\noindent
\begin{tabular}{||c|c|c|c|c|c|c|c||} \hline\hline
sep- & \multirow{3}{*}{\rangleotatebox{00}{$sep_5(f)$}} & \multirow{3}{*}{\rangleotatebox{0}{$sep_4(f)$}} & \multirow{3}{*}{\rangleotatebox{0}{$sep_3(f)$}} & \multirow{3}{*}{\rangleotatebox{0}{$sep_2(f)$}} & \multirow{3}{*}{\rangleotatebox{0}{$sep_1(f)$}} & \multirow{3}{*}{\rangleotatebox{90}{$sep(f)$}} & functions \\
class & & & & & & & per \\
\textnumero & & & & & & & class \\ \hline
1 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \hline
2 & 0 & 0 & 0 & 0 & 1 & 1 & 10\\ \hline
3 & 0 & 0 & 0 & 1 & 2 & 3 & 100\\ \hline
4 & 0 & 0 & 1 & 2 & 3 & 6 & 240\\ \hline
5 & 0 & 0 & 1 & 3 & 3 & 7 & 1940\\ \hline
6 & 0 & 1 & 2 & 5 & 4 & 12 & 1920\\ \hline
7 & 0 & 1 & 3 & 4 & 4 & 12 & 2400\\ \hline
8 & 0 & 1 & 3 & 5 & 4 & 13 & 8160\\ \hline
9 & 0 & 1 & 4 & 4 & 4 & 13 & 120\\ \hline
10 & 0 & 1 & 4 & 5 & 4 & 14 & 8400\\ \hline
11 & 0 & 1 & 4 & 6 & 4 & 15 & 301970\\ \hline
12 & 1 & 2 & 7 & 9 & 5 & 24 & 20480\\ \hline
13 & 1 & 3 & 5 & 7 & 5 & 21 & 3840\\ \hline
14 & 1 & 3 & 5 & 8 & 5 & 22 & 9600\\ \hline
15 & 1 & 3 & 6 & 6 & 5 & 21 & 1920\\ \hline
16 & 1 & 3 & 6 & 7 & 5 & 22 & 1920\\ \hline
17 & 1 & 3 & 6 & 8 & 5 & 23 & 38400\\ \hline
18 & 1 & 3 & 7 & 7 & 5 & 23 & 1920\\ \hline
19 & 1 & 3 & 7 & 8 & 5 & 24 & 38400\\ \hline
20 & 1 & 3 & 7 & 9 & 5 & 25 & 130560\\ \hline
21 & 1 & 4 & 6 & 6 & 5 & 22 & 3000\\ \hline
22 & 1 & 4 & 7 & 7 & 5 & 24 & 34720\\ \hline
23 & 1 & 4 & 7 & 8 & 5 & 25 & 177120\\ \hline
24 & 1 & 4 & 7 & 9 & 5 & 26 & 274560\\ \hline
25 & 1 & 4 & 8 & 7 & 5 & 25 & 7680\\ \hline
26 & 1 & 4 & 8 & 8 & 5 & 26 & 274560\\ \hline
27 & 1 & 4 & 8 & 9 & 5 & 27 & 1847280\\ \hline
28 & 1 & 5 & 7 & 9 & 5 & 27 & 81920\\ \hline
29 & 1 & 5 & 8 & 8 & 5 & 27 & 600\\ \hline
30 & 1 & 5 & 8 & 9 & 5 & 28 & 1013760\\ \hline
31 & 1 & 5 & 8 & 10 & 5 & 29 & 38400\\ \hline
32 & 1 & 5 & 9 & 7 & 5 & 27 & 1200\\ \hline
33 & 1 & 5 & 9 & 8 & 5 & 28 & 449040\\ \hline
34 & 1 & 5 & 9 & 9 & 5 & 29 & 4093200\\ \hline
35 & 1 & 5 & 9 & 10 & 5 & 30 & 5443200\\ \hline
36 & 1 & 5 & 10 & 8 & 5 & 29 & 13680\\ \hline
37 & 1 & 5 & 10 & 9 & 5 & 30 & 5826160\\ \hline
38 & 1 & 5 & 10 & 10 & 5 & 31 & 4274814914\\ \hline\hline
\end{tabular}
\end{table}
\end{document} |
\begin{document}
\maketitle
\begin{abstract}
We study a polymer model on hierarchical lattices very close to the one introduced and studied in
\cite{DGr, CD}. For this model, we prove the existence of free energy and derive the necessary and sufficient condition for which very strong disorder holds for all $\beta$, and give some accurate results on the behavior of the free energy at high-temperature.
We obtain these results by using a combination of fractional moment method and change of measure over the environment to obtain an upper bound, and second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of $\log Z_n$, and study the infinite polymer measure in the weak disorder phase.
\\
\\
2000 \textit{Mathematics Subject Classification: 60K35,}
\\
\\
\textit{Keywords: Hierarchical Models, Free Energy, Dynamical System, Directed Polymers}
\end{abstract}
\tableofcontents
\section{Introduction and presentation of the model}
The model of directed polymers in random environment appeared first in the physics literature as
an attempt to modelize roughening in domain wall in the $2D$-Ising model due to impurities \cite{HH}.
It then reached the mathematical community in \cite{IS}, and in \cite{B}, where the author applied
martingale techniques that have became the major technical tools in the study of this model since then.
A lot of progress has been made recently in the mathematical understanding of directed polymer model (see for example \cite{J,CY, CSY, CY_cmp, CH_al, CH_ptrf, CV} and \cite{CSY_rev} for a recent review).
It is known that there is a phase transition from a delocalized phase at high temperature, where the behavior
of the polymer is diffusive, to a localized phase, where it is expected that the influence of the media is relevant in order to produce nontrivial phenomenons, such as super-diffusivity. These two different situations are
usually referred to as weak and strong disorder, respectively. A simple characterization of this dichotomy is
given in terms of the limit of a certain positive martingale related to the partition function
of this model.
It is known that in low dimensions ($d=1$ or $2$), the polymer is essentially
in the strong disorder phase (see \cite{Lac}, for more precise results), but for $d\;\geqqslant\; 3$, there
is a nontrivial region of temperatures where weak disorder holds. A weak form of invariance principle is
proved in \cite{CY}.
However, the exact value of the critical temperature which separates the two regions (when it is finite) remains an open question.
It is known exactly in the case of directed polymers on the tree, where a complete analysis is available
(see \cite{Buf, Fr, KP}). In the case of $\mathbb{Z}^d$, for $d\geq 3$, an $L^2$ computation yields an upper bound on the critical temperature, which is however known not to coincide with this bound (see \cite{BS, Birk} and \cite{CC}).
We choose to study the same model of directed polymers on diamond hierarchical lattices. These lattices present
a very simple structure allowing to perform a lot of computations together with a richer geometry
than the tree (see Remark \ref{convexity-tree} for more details). They have been introduced in physics in order to perform exact renormalization group computations for spin systems (\cite{Migd, Kad}). A detailed treatment of more general hierarchical lattices can be found
in \cite{KG1} and \cite{KG2}. For an overview of the extensive literature on Ising and Potts models on hierarchical lattices, we refer the reader to \cite{BZ, DF} and references therein. Whereas statistical mechanics model on trees have to be considered as mean-field versions of the original models, the hierarchical lattice models are in many sense very close to the models on $\mathbb{Z}^d$; they are a very powerful tool to get an intuition for results and proofs on the more complex $\mathbb{Z}^d$ models (for instance, the work on hierarchical pinning model in \cite{GLT} lead to a solution of the original model in \cite{DGLT}. In the same manner, the present work has been a great source of inspiration for \cite{Lac}).
Directed polymers on hierarchical lattices (with bond disorder) appeared in \cite{CD, DG,DGr, DHV}
(see also \cite{R_al} for directed first-passage percolation).
More recently, these lattice models raised the interest of mathematicians in the study of random resistor networks (\cite{W}), pinning/wetting transitions (\cite{GLT,L}) and diffusion on a percolation cluster (\cite{HK}).
We can also mention \cite{Ham} where the authors consider a random analogue of the hierarchical lattice, where
at each step, each bond transforms either into a series of two bonds or into two bonds in parallel, with probability
$p$ and $p-1$ respectively.
Our aim in this paper is to describe the properties of the quenched free energy of directed polymers on
hierarchical lattices with site disorder at high temperature:
\begin{itemize}
\item First, to be able to decide, in all cases, if the quenched and annealed free energy differ at low temperature.
\item If they do, we want to be able to describe the phase transition and to compute the critical exponent.
\end{itemize}
We choose to focus on the model with site disorder, whereas \cite{GM_h, CD} focus on the model with \textsl{bond disorder} where computations are simpler.
We do so because we believe that this model is closer to the model of directed polymer in $\mathbb{Z}^{d}$ (in particular, because of the inhomogeneity of the Green Function), and because there exists a nice recursive construction of the partition functions in our case, that leads to a martingale property. Apart from that, both models are very similar, and we will shortly talk about the bound disorder model in section \ref{bddis}.
The diamond hierarchical lattice $D_n$ can be constructed recursively:
\begin{itemize}
\item $D_0$ is one single edge linking two vertices $A$ and $B$.
\item $D_{n+1}$ is obtained from $D_n$ by replacing each edges by $b$ branches of $s-1$ edges.
\end{itemize}
\begin{figure}
\caption{\label{fig:Dn}
\label{fig:Dn}
\end{figure}
\noindent We can, improperly, consider $D_n$ as a set of vertices, and, with the above construction, we have $D_n\subset D_{n+1}$. We set $D=\bigcup_{n\geq 0} D_n$.
The vertices introduced at the $n$-th iteration are said to belong to the $n$-th generation $V_n=D_n\setminus D_{n-1}$.
We easily see that $|V_n|= (bs)^{n-1} b (s-1)$.
\noindent We restrict to $b\;\geqqslant\; 2$ and $s\;\geqqslant\; 2$. The case $b=1$ (resp. $s=1$) is not interesting as it just corresponds to a familly of edges
in serie (resp. in parallel)
We introduce disorder in the system as a set of real numbers associated to vertices $\omega=(\omega_z)_{z\in D\setminus{\{A,B\}}}$.
Consider $\Gamma_n$ the space of directed paths in $D_n$ linking $A$ to $B$. For each
$g \in \Gamma_n$ (to be understood as a sequence of connected vertices in $D_n$, $(g_0=A, g_1, \dots, g_{s^n}=B)$),
we define the Hamiltonian
\begin{eqnarray}
H^{\omega}_n(g) := \sum^{s^{n}-1}_{t=1} \omega(g_t).
\label{eq: Hn}
\end{eqnarray}
\noindent For $\beta > 0$, $n\;\geqqslant\; 1$, we define the (quenched) polymer measure on $\Gamma_n$ which chooses a path $\gamma$ at random with law
\begin{eqnarray}
\mu^{\omega}_{\beta, n}(\alphamma=g) := \frac{1}{Z_{n}(\beta)} \exp( \beta H^{\omega}_n(g)),
\label{eq:polymer}
\end{eqnarray}
\noindent where
\begin{eqnarray}
Z_{n}(\beta)= Z_{n}(\beta, \omega):=\sum_{g \in \Gamma_n} \exp (\beta H^{\omega}_n(g)),
\label{eq:Zn}
\end{eqnarray}
\noindent is the partition function, and $\beta$ is the inverse temperature parameter.
In the sequel, we will focus on the case where $\omega=( \omega_z,\, z\in D\setminus\{A,B\} )$ is a collection of i.i.d.\ random variables and denote the product measure by $Q$. Let $\omega_0$ denote a one dimensional marginal of $Q$, we assume that $\omega_0$ has expectation zero, unit variance, and that
\begin{eqnarray}
\lambda(\beta):= \log Q e^{\beta \omega_0}<\infty \quad \forall \beta>0.
\label{eq:lambda}
\end{eqnarray}
As usual, we define the quenched free energy (see Theorem \ref{thermo}) by
\begin{eqnarray}
p(\beta) := \lim_{n\to +\infty} \frac{1}{s^{n}} Q \log Z_n(\beta),
\label{eq:quenched}
\end{eqnarray}
\noindent and its annealed counterpart by
\begin{eqnarray}
f(\beta) := \lim_{n\to +\infty} \frac{1}{s^{n}} \log Q Z_n(\beta).
\label{eq:annealed}
\end{eqnarray}
\noindent This annealed free energy can be exactly computed. We will prove
\begin{eqnarray}
f(\beta):= \lambda(\beta)+ \frac{\log b}{s-1}.
\label{eq:annealedexact}
\end{eqnarray}
This model can also be stated as a random dynamical system: given two integer parameters $b$ and $s$ larger than $2$, $\beta>0$, consider the following recursion:
\begin{align}
W_0&\stackrel{\mathcal L}{=}1\notag\\
W_{n+1}&\stackrel{\mathcal L}{=}\frac{1}{b}\sum_{i=1}^b \prod_{j=1}^{s}W_n^{(i,j)}\prod_{i=1}^{s-1}A^{(i,j)}_n\label{eq:rec},
\end{align}
\noindent where equalities hold in distribution, $W_n^{(i,j)}$ are independent copies of $W_n$, and $A^{(i,j)}_n$ are i.i.d.\ random variables, independent of the $W_n^{(i,j)}$ with law
\begin{align*}
A\stackrel{\mathcal L}{=}\exp(\beta\omega-\lambda(\beta)).
\end{align*}
\noindent In the directed polymer setting, $W_n$ can be interpretative as the normalized partition function
\begin{eqnarray}
W_n(\beta)=W_n(\beta,\omega) = \frac{Z_n(\beta,\omega)}{Q Z_n(\beta, \omega)}.
\label{eq:W}
\end{eqnarray}
\noindent Then, (\ref{eq:rec}) turns out to be an almost sure equality if we interpret $W_n^{(i,j)}$ as the partition function of the
$j$-th edge of the $i$-th branch of $D_1$.
\noindent The sequence $(W_n)_{n\geq 0}$ is a martingale with respect to $\mathcal{F}_n = \sigma(\omega_z:\, z\in \cup^n_{i=1}V_i)$ and
as $W_n>0$ for all $n$, we can define the almost sure limit
$W_{\infty}= \lim_{n\to +\infty} W_n$. Taking limits in both sides of (\ref{eq:rec}), we obtain a functional equation for $W_{\infty}$.
\section{Results}
Our first result is about the existence of the free energy.
\begin{theorem}\label{thermo}
For all $\beta$, the limit
\begin{eqnarray}
\lim_{n\to +\infty} \frac{1}{s^n} \log Z_n(\beta),
\label{eq:free}
\end{eqnarray}
\noindent exists a.s. and is a.s. equal to the quenched free
energy $p(\beta)$. In fact for any $\varepsilon>0$, one can find $n_0(\varepsilon,\beta)$ such that
\begin{eqnarray}\label{eq:concentrate}
Q\leqft(\leqft|Z_n-Q\log Z_n\right|> s^n \varepsilon\right)\leq \exp\leqft(-\frac{\varepsilon^{2/3}s^{n/3}}{4}\right), \quad \text{for all } n\geq n_0
\end{eqnarray}
Moreover, $p(\cdot)$ is a strictly convex function of $\beta$.
\end{theorem}
\begin{rem}\rm
The inequality (\ref{eq:concentrate}) is the exact equivalent of \cite[Proposition 2.5]{CSY},
and the proof given there can easily be adapted to our case. It applies concentration results for
martingales from \cite{LV}.
It can be improved in order to obtain the same bound as for Gaussian
environments stated in \cite{CH_ptrf} (see \cite{Cnotes} for details). However, it is believed that it is
not of the optimal order, similar to the case of directed polymers on $\mathbb{Z}^d$.
\end{rem}
\begin{rem}\rm \label{convexity-tree}
The strict convexity of the free energy is an interesting property. It is known that it holds also for the directed polymer on $\mathbb{Z}^d$ but not on the tree. In the
later case, the free energy is strictly convex only for values of $\beta$ smaller than the critical value $\beta_c$
(to be defined latter) and it is linear on $[\beta_c,+\infty)$. This fact is related to the particular structure of
the tree that leads to major simplifications in the 'correlation' structure of the model (see \cite{Buf}). The strict convexity, in
our setting, arises essentially from the property that two path on the hierarchical lattice can re-interesect after being separated at some step. This underlines once more, that $\mathbb{Z}^d$ and the hierarchical lattice have a lot of features in common, which they do not share with the tree.
\end{rem}
We next establish the martingale property for $W_n$ and the zero-one law for its limit.
\begin{lemma}\label{martingale}
$(W_n)_n$ is a positive $\mathcal{F}_n$-martingale. It converges
$Q$-almost surely to a non-negative limit $W_{\infty}$ that satisfies
the following zero-one law:
\begin{eqnarray}
Q \leqft( W_{\infty} > 0 \right)\,\in\, \{0,1\}.
\label{zeroone}
\end{eqnarray}
\end{lemma}
\noindent Recall that martingales appear when the disorder is displayed on sites, in contrast with disorder on bonds
as in \cite{CD,DG}.
Observe that
\begin{eqnarray*}
p(\beta)-f(\beta)=\lim_{n\to +\infty} \frac{1}{s^n} \log W_n(\beta),
\end{eqnarray*}
\noindent so, if we are in the situation $Q(W_{\infty}>0)=1$, we have
that $p(\beta)=f(\beta)$. This motivates the following definition:
\begin{definition}\label{disorder}
If $Q(W_{\infty}>0)=1$, we say that weak disorder holds. In the opposite situation,
we say that strong disorder holds.
\end{definition}
\begin{rem}\rm
Later, we will give a statement(Proposition \ref{th:fracmom}) that guarantees that strong disorder is equivalent to $p(\beta)\neq f(\beta)$, a situation that is sometimes called very strong disorder. This is believed to be true for polymer models on $\mathbb{Z}^d$ or $\mathbb{R}^d$ but it remains an unproved and challenging conjecture in dimension $d\geq 3$ (see \cite{CH_al}).
\end{rem}
The next proposition lists a series of partial results that in some sense clarify
the phase diagram of our model.
\begin{proposition}\label{misc}
$(i)$ There exists $\beta_0\in[0, +\infty]$ such that strong disorder holds
for $\beta > \beta_0$ and weak disorder holds for $\beta\leq \beta_0$.
$(ii)$ If $b>s$, $\beta_0>0$. Indeed, there exists $\beta_2\in(0,\infty]$ such that for all $\beta<\beta_2$,
$\sup_n Q(W^2_{n}(\beta))<+\infty$, and therefore weak disorder holds.
$(iii)$ If $\beta \lambda'(\beta)-\lambda(\beta)> \frac{2\log b}{s-1}$, then strong
disorder holds.
$(iv)$
In the case where $\omega_z$ are gaussian random variables, $(iii)$ can be improved for $b>s$: strong disorder holds as soon as $\beta>\sqrt{\frac{2(b-s)\log b}{(b-1)(s-1)}}$.
$(v)$ If $b\;\leqqslant\; s$, then strong disorder holds for all $\beta$.
\end{proposition}
\begin{rem}\rm
On can check that the formula in $(iii)$ ensures that $\beta_0<\infty$ whenever the distribution of $\omega_z$ is unbounded.
\end{rem}
\begin{rem}\rm
An implicit formula is given for $\beta_2$ in the proof and this gives a lower bound for $\beta_0$. However, when $\beta_2<\infty$, it never coincides with the upper bound given by $(iii)$ and $(iv)$, and therefore knowing the exact value of the critical temperature when $b>s$ remains an open problem.
\end{rem}
We now provide more quantitative information for the regime considered in $(v)$:
\begin{theorem} \label{th:bs}
When $s>b$, there exists a constant $c_{s,b}=c$ such that for any $\beta\leq 1$ we have
\begin{align*}
\frac{1}{c}\beta^{\frac{2}{\alpha}}\leq \lambda(\beta)-p(\beta) \leq c\beta^{\frac{2}{\alpha}}
\end{align*}
where $\alpha=\frac{\log s-\log b}{\log s}$.
\end{theorem}
\begin{theorem} \label{th:ss}
When $s=b$, there exists a constant $c_s=c$ such that for any $\beta\leq 1$ we have
\begin{align*}
\exp\leqft(-\frac{c}{\beta^2}\right)\leq \lambda(\beta) - p(\beta) \leq c\exp\leqft(-\frac{1}{c\beta}\right)
\end{align*}
\end{theorem}
In the theory of directed polymer in random environment, it is believed that, in low dimension, the quantity $\log Z_n$ undergoes large fluctuations around its average (as opposed to what happens in the weak disorder regime where the fluctuation are of order $1$). More precisely: it is believed that there exists exponents $\xi>0$ and $\chi\geq 0$ such that
\begin{equation}
\log Z_n-Q\log Z_n \asymp N^{\xi} \text{ and } \var_Q \log Z_n\asymp N^{2\chi},
\end{equation}
where $N$ is the length of the system ($=n$ on $\mathbb{Z}^d$ and $s^n$ one our hierarchical lattice).
In the non-hierarchical model this exponent is of major importance as it is closely related to the {\sl volume exponent} $\xi$ that gives
the spatial fluctuation of the polymer chain (see e.g.\ \cite{J} for a discussion on fluctuation exponents). Indeed it is conjectured for the $ \mathbb{Z}^d$ models that
\begin{equation}
\chi=2\xi-1.
\end{equation}
This implies that the polymer trajectories are superdiffusive as soon as $\chi>0$.
In our hierarchical setup, there is no such geometric interpretation but having a lower bound on the fluctuation allows to get a significant localization result.
\begin{proposition}\label{fluctuations}
When $b<s$, there exists a constant $c$ such that for all $n\geq 0$ we have
\begin{equation}
\var_Q\leqft(\log Z_n\right)\geq \frac{c (s/b)^{n}}{\beta^2}.
\end{equation}
Moreover, for any $\varepsilon>0$, $n\geq 0$, and $a\in {\ensuremath{\mathbb R}} $,
\begin{equation}
Q\leqft\{\log Z_n \in [a, a+\varepsilon(s/b)^{n/2}]\right\}\leq \frac{8\varepsilon}{\beta}.
\end{equation}
\end{proposition}
\noindent This implies that if the fluctuation exponent $\chi$ exists, $\chi\geq \frac{\log s -\log b}{2\log s}$.
We also have the corresponding result for the case $b=s$
\begin{proposition}\label{fluc2}
When $b=s$, there exists a constant $c$ such that for all $n\geq 0$ we have
\begin{equation}
\var_Q \leqft(\log Z_n\right)\geq \frac{c n}{\beta^2}.
\end{equation}
Moreover for any $\varepsilon>0$, $n\geq 0$, and $a\in {\ensuremath{\mathbb R}} $,
\begin{equation}
Q\leqft\{\log Z_n \in [a, a+\varepsilon\sqrt{n}]\right\}\leq \frac{8\varepsilon}{\beta}.
\end{equation}
\end{proposition}
From the fluctuations of the free energy we can prove the following:
For $g \in \Gamma_n$ and $m<n$, we define $g|_m$ to be the restriction of $g$ to $D_m$.
\begin{cor}\label{locloc}
If $b\leq s$, and $n$ is fixed we have
\begin{equation}
\lim_{n\to\infty} \sup_{g\in \Gamma_m} \mu_n(\gamma|_m=g)=1,
\end{equation}
\noindent where the convergence holds in probability.
\end{cor}
Intuitively this result means that if one look on a large scale, the law of $\mu_n$ is concentrated in the neighborhood of a single path. Equipping $\Gamma_n$ with a natural metric (two path $g$ and $g'$ in $\Gamma_n$ are at distance $2^{-m}$ if and only if $g|_m\ne g'|_m$ and $g|_{m-1}=g|_{m-1}$) makes this statement rigorous.
\begin{rem}\rm
Proposition \ref{misc}$(v)$ brings the idea that $b\leq s$ for this hierarchical model is equivalent to the $d\leq 2$ case for the model in $\mathbb{Z}^{d}$ (and that $b>s$ is equivalent to $d>2$). Let us push further the analogy: let $\gamma^{(1)}$ , $\gamma^{(2)}$ be two paths chosen uniformly at random in $\Gamma_n$ (denote the uniform-product law by $P^{\otimes 2}$), their expected site overlap is of order $(s/b)^n$ if $b<s$, of order $n$ if $b=s$, and of order $1$ if $b>s$.
If one denotes by $N= s^n$ the length of the system, one has
\begin{equation}
P^{\otimes 2}\leqft[\sum_{t=0}^{N} \mathbf{1}_{\{\alphamma^{(1)}_t= \alphamma^{(2)}_t\}}\right]\, \asymp\, \begin{cases}N^{\alpha} \quad \text{ if } b < s,\\
\log N \quad \text{ if } b=s,\\
1 \text{ if } b> s,
\end{cases}
\end{equation}
(where $\alpha=(\log s-\log b)/\log s$).
Comparing this to the case of random walk on $\mathbb{Z}^d$, we can infer that the case $b=s$ is just like $d=2$ and that the case $d=1$ is similar to $b=\sqrt{s}$ ($\alpha=1/2$). One can check in comparing \cite[Theorem 1.4, 1.5, 1.6]{Lac} with Theorem \ref{th:bs} and \ref{th:ss}, that this analogy is relevant.
\end{rem}
The paper is organised as follow
\begin{itemize}
\item In section \ref{mtricks} we prove
some basic statements about the free energy, Lemma \ref{martingale} and
the first part of Proposition \ref{misc}.
\item Item $(ii)$ from Proposition \ref{misc} is proved in Section $5.1$.
Item $(v)$ is a consequence of Theorems \ref{th:bs} and \ref{th:ss}.
\item Items $(iii)$ and $(iv)$ are proved in Section $6.3$.
Theorems \ref{th:bs} and \ref{th:ss} are proved in Section $6.1$ and $6.3$
respectively.
\item In section \ref{flucloc} we prove Propositions \ref{fluctuations} and \ref{fluc2} and Corrolary \ref{locloc}.
\item In section \ref{weakpol} we define and investigate the properties of the infinite volume polymer measure in the weak disorder phase.
\item In section \ref{bddis} we shortly discuss about the bond disorder model.
\end{itemize}
\section{Martingale tricks and free energy}\label{mtricks}
We first look at to the existence of the quenched free energy
\begin{eqnarray*}
p(\beta)= \lim_{n\to +\infty} \frac{1}{n} Q \leqft( \log Z_n(\beta) \right),
\end{eqnarray*}
\noindent and its relation with the annealed free energy. The case $\beta=0$ is
somehow instructive. It gives the number of paths in $\Gamma_n$ and is
handled by the simple recursion:
\begin{eqnarray*}
Z_n(0)=b \leqft( Z_{n-1}(0) \right).
\end{eqnarray*}
\noindent This easily yields
\begin{eqnarray}
\label{paths}
|\Gamma_n| = Z_n(\beta=0)= b^{\frac{s^{n}-1}{s-1}}.
\end{eqnarray}
Much in the same spirit than (\ref{eq:rec}), we can find a recursion for
$Z_n$:
\begin{eqnarray}
Z_{n+1}= \sum^b_{i=1} Z^{(i,1)}_n \cdots Z^{(i,s)}_n \times e^{\beta \omega_{i,1}}\cdots e^{\beta \omega_{i,s-1}}.
\label{eq:recz}
\end{eqnarray}
The existence of the quenched free energy follows by monotonicity: we have
$$Z_{n+1} \;\geqqslant\; Z^{(1,1)}_n Z^{(1,2)}_n \cdots Z^{(1,s)}_n \times e^{\beta \omega_{1,1}}\cdots e^{\beta \omega_{1,s-1}},$$
\noindent so that (recall the $\omega$'s are centered random variables)
$$\frac{1}{s^{n+1}} Q \log Z_{n+1} \;\geqqslant\; \frac{1}{s^n} Q \log Z_n.$$
\noindent The annealed free energy provides an upper bound:
\begin{eqnarray*}
\frac{1}{s^n} Q \log Z_n &\;\leqqslant\;& \frac{1}{s^n} \log Q Z_n\\
&=& \frac{1}{s^n} \log e^{\lambda(\beta)(s^n-1)}Z_n(\beta=0)\\
&=& \leqft( 1 - \frac{1}{s^n} \right) \leqft( \lambda(\beta) + \frac{\log b}{s-1} \right)\\
&=& \leqft( 1 - \frac{1}{s^n} \right) f(\beta).
\end{eqnarray*}
We now prove the strict convexity of the free energy. The proof
is essentially borrowed from \cite{CPV}, but it is remarkably simpler
in our case.
\begin{proof}[Proof of the strict convexity of the free energy]
We will consider a Bernoulli environment ($\omega_z=\pm 1$ with probability $p$, $1-p$; note that our assumptions on the variance and expectation for $\omega$ are violated but centering and rescaling $\omega$ does not change the argument).
We refer to \cite{CPV} for generalization to more general environment.
\noindent An easy computation yields
\begin{eqnarray*}
\frac{d^2}{d\beta^2} Q \log Z_n = Q {\rm Var}_{\mu_n} H_n (\alphamma).
\end{eqnarray*}
\noindent We will prove that for each $K>0$, there exists a constant
$C$ such that, for all $\beta \in[0,K]$ and $n\;\geqqslant\; 1$,
\begin{eqnarray}\label{lowerboundenergy}
{\rm Var}_{\mu_n} H_n (\alphamma) \;\geqqslant\; C s^n
\end{eqnarray}
For $g \in \Gamma_n$ and $m<n$, we define $g|_m$ to be the restriction of $g$ to $D_m$.
By the conditional variance formula,
\begin{eqnarray}
\nonumber
{\rm Var}_{\mu_n} H_n &=& \mu_n \leqft( {\rm Var}_{\mu_n} (H_n(\alphamma)\, |\, \alphamma_{|_{n-1}}) \right) +
{\rm Var}_{\mu_n} \leqft( \mu_n(H_n(\alphamma)\, |\, \alphamma_{|_{n-1}})\right)\\
\label{condvar}
&\;\geqqslant\;& \mu_n \leqft( {\rm Var}_{\mu_n} (H_n(\alphamma) \,|\, \alphamma_{|_{n-1}}) \right)
\end{eqnarray}
\noindent Now, for $l=0,...,s^{n-1}-1$, $g \in \Gamma_n$, define
\begin{eqnarray*}
H^{(l)}_n (g) = \sum^{(l+1)s-1}_{t=ls+1} \omega(g_t),
\end{eqnarray*}
\noindent so (\ref{condvar}) is equal to
\begin{eqnarray*}
\mu_n {\rm Var}_{\mu_n} \leqft( \sum^{s^{n-1}-1}_{l=0} H^{(l)}_{n}(\alphamma) | \alphamma_{|_{n-1}} \right)
= \sum^{s^{n-1}-1}_{l=0} \mu_n {\rm Var}_{\mu_n} \leqft( H^{(l)}_{n}(\alphamma) | \alphamma_{|_{n-1}} \right),
\end{eqnarray*}
\noindent by independence. Summarizing,
\begin{equation}\label{varcond}
{\rm Var}_{\mu_n} H_n\;\geqqslant\; \sum^{s^{n-1}}_{l=1} \mu_n {\rm Var}_{\mu_n} \leqft( H^{(l)}_{n}(\alphamma) | \alphamma_{|_{n-1}} \right).
\end{equation}
\noindent The rest of the proof consists in showing that each term of the sum is bounded from below by a positive constant,
uniformly in $l$ and $n$.
For any $x\in D_{n-1}$ such that the graph distance between $x$ and $A$ is $ls$ in $D_n$ (i.e.\ $x\in D_{n-1}$), we define the set of environment
\begin{eqnarray*}
M(n,l,x)= \leqft\lbrace \omega: \leqft|\{H^{(l)}_n(g,\omega)\,:\, g\in \Gamma_n, g_{ls}=x\} \right| \;\geqqslant\; 2 \right\rbrace.
\end{eqnarray*}
\noindent These environments provide the fluctuations in the energy needed for the
uniform lower bound we are searching for.
One second suffices to convince oneself
that $Q(M(n,l,x)>0$, and does not depend on the parameters $n,\, l$ or $x$. Let $Q(M)$ denote improperly the common value of $Q( M(n,l,x))$.
Now, it is easy to see (from \eqref{varcond}) that there exists a constant $C$ such that for all $\beta<K$,
\begin{eqnarray*}
Q \leqft[{\rm Var}_{\mu_n} H_n\right] &\geq& C Q \leqft[\sum^{s^{n-1}-1}_{l=1} \sum_{x\in D_{n-1}} {\bf 1}_{M(n, l, x)}\mu_n ({\alphamma_{ls} = x})\right].
\end{eqnarray*}
\noindent Define now $\mu^{(l)}_n$ as the polymer measure in the environment obtained from
$\omega$ by setting $\omega(y)=0$ for all sites $y$ which distance to $0$ is between $ls$ and $(l+1)s$.
One can check that for all $n$, and all path $g$,
\begin{eqnarray*}
\exp(-2\beta(s-1)) \mu^{(l)}_n(\alphamma=g) \;\leqqslant\; \mu_n(\alphamma=g) \;\leqqslant\; \exp(2\beta (s-1)) \mu^{(l)}_n(\alphamma).
\end{eqnarray*}
\noindent We note that under $Q$, $\mu_n^{(l)}(\alphamma_{ls} = x)$ and $\mathbf{1}_{M(n, l, x)}$ are random variables, so that
\begin{eqnarray*}
Q\leqft[ {\rm Var}_{\mu_n} H_n \right]&\;\geqqslant\;& C\exp(-2\beta(s-1)) Q \leqft[\sum^{s^{n-1}-1}_{l=0} \sum_x {\bf 1}_{M(n, l, x)}\mu^{(l)}_n ({\alphamma_{ls} = x})\right]\\
&=& C\exp(-2\beta(s-1))\sum^{s^{n-1}}_{l=1} \sum_{x\in D_{n-1}} Q(M(n,l,x)) Q\leqft[ \mu^{(l)}_n ({\alphamma_l = x})\right]\\
&=& C\exp(-2\beta(s-1)) Q(M) s^{n-1}.
\end{eqnarray*}
\end{proof}
\noindent We now establish the martingale property for the normalized free energy.
\begin{proof}[Proof of Lemma \ref{martingale}]
Set $z_n=Z_n(\beta=0)$. We have already remarked that this is just the number of (directed) paths in $D_n$,
and its value is given by (\ref{paths}).
Observe that
$g\in \Gamma_n$ visits $s^n(s-1)$ sites of $n+1$-th generation. The restriction of paths in $D_{n+1}$
to $D_n$ is obviously not one-to-one as for each path $g'\in \Gamma_n$, there are $b^{s^n}$ paths
in $\Gamma_{n+1}$ such that $g|_n=g'$. Now,
\begin{eqnarray*}
Q\leqft(Z_{n+1}(\beta)| \mathcal{F}_n\right)
&=& \sum_{g \in D_{n+1}} Q\leqft( e^{\beta H_{n+1}(g)}|\mathcal{F}_n\right)\\
&=&\sum_{g' \in D_n} \sum_{g \in D_{n+1}}Q\leqft( e^{\beta H_{n+1}(g)}|\mathcal{F}_n\right){\bf 1}_{g|_n=g'}\\
&=&\sum_{g' \in D_n}\sum_{g \in D_{n+1}} e^{\beta H_{n}(g')}e^{s^n(s-1)\lambda(\beta)}{\bf 1}_{g|_n=g'}\\
&=&\sum_{g' \in D_n} e^{\beta H_{n}(g')}e^{s^n(s-1)\lambda(\beta)}\sum_{g \in D_{n+1}}{\bf 1}_{g|_n=g'}\\
&=&e^{s^n(s-1)\lambda(\beta)}b^{s^n}\sum_{g' \in D_n} e^{\beta H_{n}(g')}\\
&=&Z_n(\beta) \frac{z_{n+1}e^{s^{n+1}\lambda(\beta)}}{z_n e^{s^n \lambda(\beta)}}.
\end{eqnarray*}
\noindent This proves the martingale property. For (\ref{zeroone}), let's generalize a little
the preceding restriction procedure. As before, for a path $g \in D_{n+k}$, denote by
$g |_n$ its restriction to $D_n$. Denote by $I_{n,n+k}$ the set of time indexes that have been
removed in order to perform this restriction and by $N_{n,n+k}$ its cardinality. Then
\begin{eqnarray*}
Z_{n+k}= \sum_{g \in D_n} e^{\beta H_n(g)} \sum_{g' \in D_{n+k}, g'|_n=g}
\exp \leqft\lbrace \beta \sum_{t\in I_{n,n+k}} \omega(g'_t) \right\rbrace.
\end{eqnarray*}
\noindent Consider the following notation, for $g\in \Gamma_n$,
\begin{eqnarray*}
\widetilde{W}_{n,n+k}(g)= c^{-1}_{n,n+k}\sum_{g' \in D_{n+k}, g'|_n=g}\exp \leqft\lbrace \beta \sum_{t\in I_{n,n+k}} \omega(g'_t) - N_{n,n+k} \lambda(\beta)\right\rbrace,
\end{eqnarray*}
\noindent where $c_{n,n+k}$ stands for the number paths in the sum. With this notations, we have,
\begin{eqnarray}
W_{n+k} = \frac{1}{z_n} \sum_{g \in D_n} e^{\beta H_n(g)-(s^{n}-1)\lambda(\beta)}\widetilde{W}_{n,n+k}(g),
\end{eqnarray}
\noindent and, for all $n$,
\begin{eqnarray}
\leqft\lbrace W_{\infty}=0 \right\rbrace =
\leqft\lbrace \widetilde{W}_{n,n+k}(g) \to 0,\, {\rm as}\, k\to +\infty,\, \forall \, g \in D_n \right\rbrace.
\label{eq:WW}
\end{eqnarray}
\noindent The event in the right hand side is measurable with respect to the disorder of generation
not earlier than $n$. As $n$ is arbitrary, the right hand side of (\ref{eq:WW}) is in the tail $\sigma$-algebra
and its probability is either $0$ or $1$.
\end{proof}
This, combined with FKG-type arguments (see \cite[Theorem 3.2]{CY} for details), proves part $(i)$ of
Proposition \ref{misc}. Roughly speaking, the FKG inequality is used to insure that there is no reentrance
phase.
\section{Second moment method and lower bounds}
This section contains all the proofs concerning coincidence of annealed and quenched free--energy for $s>b$ and lower bounds on the free--energy for $b\leq s$ (i.e. half of the results from Proposition \ref{misc} to Theorem \ref{th:ss}.)
First, we discuss briefly the condition on $\beta$ that one has to fulfill to to have $W_{n}$ bounded in ${\ensuremath{\mathbb L}} _2(Q)$. Then for the cases when strong disorder holds at all temperature ($b\leq s$), we present a method that combines control of the second moment up to some scale $n$ and a percolation argument to get a lower bound on the free energy.
\\
First we investigate how to get the variance of $W_n$ (under $Q$).
From \eqref{eq:rec} we get the induction for the variance $v_n=Q\leqft[(W_n-1)^2\right]$:
\begin{eqnarray}
v_{n+1}&=&\frac{1}{b}\leqft(e^{(s-1)\gamma(\beta)}(v_n+1)^s-1\right), \label{eq:var}\\
v_0&=&0.\label{eq:v00}
\end{eqnarray}
where $\gamma(\beta):=\lambda(2\beta)-2\lambda(\beta)$.
\subsection{The $L^2$ domain: $s<b$}
If $b>s$, and $\gamma(\beta)$ is small, the map
\begin{align*}
g:\ x\mapsto\frac{1}{b}\leqft(e^{(s-1)\gamma(\beta)}(x+1)^s-1\right)
\end{align*}
possesses a fixed point. In this case, \eqref{eq:var} guaranties that $v_n$ converges to some finite limit. Therefore, in this case, $W_n$ is a positive martingale bounded in ${\ensuremath{\mathbb L}} ^2$, and therefore converges almost surely to $W_\infty \in {\ensuremath{\mathbb L}} ^2(Q)$ with $Q W_{\infty}=1$, so that
\begin{align*}
p(\beta)-\lambda(\beta)=\lim_{n\rightarrow\infty}\frac{1}{s^n}\log W_n=0,
\end{align*}
\noindent and weak disorder holds.
One can check that $g$ has a fixed point if and only if
\begin{align*}
\gamma(\beta)\leq \frac{s}{s-1}\log \frac{s}{b}-\log \frac{b-1}{s-1}
\end{align*}
\subsection{Control of the variance: $s>b$}
For $\epsilon>0$, let $n_0$ be the smallest integer such that $v_n\geq \varepsilon$.
\begin{lemma}\label{th:bss}
For any $\varepsilon>0$, there exists a constant $c_{\varepsilon}$ such that for any $\beta\leq 1$
\begin{align*}
n_0\geq \frac{2 |\log \beta|}{\log s - \log b}-c_{\varepsilon}.
\end{align*}
\end{lemma}
\begin{proof}
Expanding \eqref{eq:var} around $\beta=0$, $v_n=0$, we find a constant $c_1$ such that, whenever $v_n\leq 1$ and $\beta\leq 1$,
\begin{align}
v_{n+1}\leq \frac{s}{b}(v_n+c_1\beta^2)(1+c_1v_n) \label{eq:varmod}.
\end{align}
Using \eqref{eq:varmod}, we obtain by induction
\begin{align*}
v_{n_0}\leq \prod_{i=0}^{n_0-1}(1+c_1v_i)\leqft[c_1\beta^2\leqft(\sum_{i=0}^{n_0-1}(s/b)^{i}\right)\right].
\end{align*}
From \eqref{eq:var}, we see that $v_{i+1}\geq (s/b)v_i$. By definition of $n_0$, $v_{n_0-1}<\epsilon,$ so that $v_{i} <\varepsilon (s/b)^{i-n_0+1}$. Then
\begin{align*}
\prod_{i=0}^{n_0-1}(1+c_1v_i)\leq \prod_{i=0}^{n_0-1}(1+c_1\varepsilon(s/b)^{i-n_0+1})\leq \prod_{k=0}^{\infty}(1+c_1\varepsilon(s/b)^{-k})\leq 2,
\end{align*}
where the last inequality holds for $\varepsilon$ small enough.
In that case we have
\begin{align*}
\varepsilon\leq v_{n_0}\leq 2 c_1 \beta^2 (s/b)^{n_0},
\end{align*}
so that \begin{equation*}
n_0\geq \frac{\log(\varepsilon/2c_1\beta^2)}{\log(s/b)}.
\end{equation*}
\end{proof}
\subsection{Control of the variance: $s=b$}
\begin{lemma}\label{th:sss}
There exists a constant $c_2$ such that, for every $\beta\leq 1$,
\begin{align*}
v_n\leq \beta, \quad \forall \, n\ \leq \ \frac{c_2}{\beta}.
\end{align*}
\end{lemma}
\begin{proof}
By (\ref{eq:varmod}) and induction we have, for any $n$ such that $v_{n-1}\leq 1$ and $\beta\;\leqqslant\; 1$,
\begin{align*}
v_{n}\leq n\beta^2 \prod_{i=0}^{n-1} (1+c_1v_i).
\end{align*}
Let $n_0$ be the smallest integer such that $v_{n_0}>\beta$. By the above formula, we have
\begin{align*}
v_{n_0}\leq n_0\beta^2(1+c_1\beta)^{n_0}
\end{align*}
Suppose that $n_0\leq (c_2/\beta)$, then
\begin{align*}
\beta\leq v_{n_0}\leq c_2c_1\beta(1+c_1\beta)^{c_2/\beta}.
\end{align*}
If $c_4$ is chosen small enough, this is impossible.
\end{proof}
\subsection{Directed percolation on $D_n$} For technical reasons, we need to get some understanding on directed independent bond percolation on $D_n$.
Let $p$ be the probability that an edge is open (more detailed considerations about edge disorder
are given in the last section).
The probability of having an open path from $A$ to $B$ in $D_n$ follows the recursion
\begin{align*}
p_0&=p,\\
p_n&=1-(1-p_{n-1}^s)^b.
\end{align*}
On can check that the map $x\mapsto 1-(1-x^s)^b$ has a unique unstable fixed point on $(0,1)$; we call it $p_c$.
Therefore if $p>p_c$, with a probability tending to $1$, there will be an open path linking $A$ and $B$ in $D_n$. If $p<p_c$, $A$ and $B$ will be disconnected in $D_n$ with probability tending to $1$. If $p=p_c$, the probability that $A$ and $B$ are linked in $D_n$ by an open path is stationary. See \cite{HK} for a deep investigation of percolation on hierarchical lattices.
\subsection{From control of the variance to lower bounds on the free energy}
Given $b$ and $s$, let $p_c=p_c(b,s)$ be the critical parameter for directed bond percolation.
\begin{proposition}\label{th:perco}
Let $n$ be an integer such that $v_n=Q (W_n-1)^2 <\frac{1-p_c}{4}$ and $\beta$ such that $p(\beta)\leq (1-\log2)$.
Then
\begin{align*}
\lambda(\beta)-p(\beta)\;\geqqslant\; s^{-n}
\end{align*}
\end{proposition}
\begin{proof}
If $n$ is such that $Q\leqft[ (W_n-1)^2\right] <\frac{1-p_c}{4}$, we apply Chebycheff inequality to see that
\begin{align*}
Q(W_n<1/2)\leq 4 v_n< 1-p_c.
\end{align*}
Now let be $m\geq n$. $D_m$ can be seen as the graph $D_{m-n}$ where the edges have been replaced by i.i.d.\ copies of $D_n$ with its environment (see fig. \ref{perco}).
To each copy of $D_n$ we associate its renormalized partition function; therefore, to each edge $e$ of $D_{m-n}$ corresponds an independent copy of $W_n$, $W_n^{(e)}$. By percolation (see fig. \ref{perco2}), we will have, with a positive probability not depending on $n$, a path in $D_{m-n}$ linking $A$ to $B$, going only through edges which associated $W_n^{(e)}$ is larger than $1/2$.
\begin{figure}
\caption{\label{perco}
\label{perco}
\end{figure}
\begin{figure}
\caption{\label{perco2}
\label{perco2}
\end{figure}
When such paths exist, let $\alphamma_0$ be one of them (chosen in a deterministic manner, e.g.\ the lowest such path for some geometric representation of $D_n$).
We look at the contribution of these family of paths in $D_m$ to the partition function.
We have
\begin{align*}
W_m\geq (1/2)^{s^{m-n}}\exp\leqft(\sum_{z\in \alphamma_0} \beta\omega_z-\lambda(\beta)\right)
\end{align*}
Again, with positive probability (say larger than $1/3$), we have $\sum_{z\in \alphamma_0} \omega_z\geq 0$ (this can be achieved the the central limit theorem). Therefore with positive probability we have
\begin{align*}
\frac{1}{s^m}\log W_m\geq -\frac{1}{s^{n}}(\log 2+\lambda(\beta)).
\end{align*}
As $1/s^m \log W_m$ converges in probability to the free energy this proves the result.
\end{proof}
\begin{proof}[Proof of the right-inequality in Theorems \ref{th:bs} and \ref{th:ss}].
The results now follow by combining Lemma \ref{th:bss} or \ref{th:sss} for $\beta$ small enough, with Proposition
\ref{th:perco}.
\end{proof}
\section{Fractional moment method, upper bounds and strong disorder}\label{fm}
In this section we develop a way to find an upper bound for $\lambda(\beta)-p(\beta)$, or just to find out if strong disorder hold. The main tool we use are fractional moment estimates and measure changes.
\subsection{Fractional moment estimate}
In the sequel we will use the following notation. Given a fixed parameter $\theta\in(0,1)$, define
\begin{eqnarray}
u_n&:=&Q W_n^{\theta},\\ \label{un}
a_\theta&:=&Q A^{\theta}=\exp(\lambda(\theta\beta)-\theta\lambda(\beta)) \label{aomega}.
\end{eqnarray}
\begin{proposition}\label{th:fracmom}
The sequence $(f_n)_n$ defined by
$$f_n:=\theta^{-1}s^{-n}\log \leqft(a_\theta b^{\frac{1-\theta}{s-1}}u_n\right)$$
\noindent is decreasing and we have
\begin{align*}
\lim_{n\rightarrow\infty} f_n\geq p(\beta)-\lambda(\beta).
\end{align*}
\noindent $(i)$ In particular, if for some $n\in\mathbb{N}$, $u_n< a_{\theta}^{-1}b^{\frac{\theta-1}{s-1}}$, strong disorder holds.
\noindent $(ii)$ Strong disorder holds in particular if $a_\theta< b^{\frac{\theta-1}{s-1}}$.
\end{proposition}
\begin{proof}
The inequality $\leqft(\sum a_i\right)^{\theta}\leq \sum a_i^{\theta}$ (which holds for any $\theta\in (0,1)$ and any collection of positive numbers $a_i$) applied to
\eqref{eq:rec} and averaging with respect to $Q$ gives
\begin{align*}
u_{n+1}\leq b^{1-\theta}u_n^{s}a_{\theta}^{s-1}
\end{align*}
\noindent From this we deduce that the sequence
\begin{align*}
s^{-n}\log\leqft( a_{\theta} b^{\frac{1-\theta}{s-1}}u_n\right)
\end{align*}
\noindent is decreasing. Moreover we have
\begin{align*}
p(\beta)-\lambda(\beta)=\lim_{n\rightarrow\infty}\frac{1}{s^n}Q \log W_n\leq \lim_{n\rightarrow \infty} \frac{1}{\theta s^n}\log Q W_n^{\theta}=\lim_{n\to\infty}f_n.
\end{align*}
As a consequence very strong disorder holds if $f_n<0$ for any $f_n$. As a consequence, strong disorder and very strong disorder are equivalent.
\end{proof}
\subsection{Change of measure and environment tilting}
The result of the previous section assures that we can estimate the free energy if we can bound accurately some non integer moment of $W_n$.
Now we present a method to estimate non-integer moment via measure change, it has been introduced to show disorder relevance in the case of wetting on non hierarchical lattice \cite{GLT} and used since in several different contexts since, in particular for directed polymer models on $\mathbb{Z}^d$, \cite{Lac}.
Yet, for the directed polymer on hierarchical lattice, the method is remarkably simple to apply, and it seems to be the ideal context to present it.
\\
Let $\widetilde Q$ be any probability measure such that $Q$ and $\widetilde Q$ are mutually absolutely continuous.
Using H\"older inequality we observe that
\begin{equation}
Q W_n^{\theta}
= \widetilde Q \frac{\,\text{\rm d} Q}{\,\text{\rm d} \widetilde Q} W_n^{\theta}
\leq \leqft[\widetilde Q\leqft(\frac{\,\text{\rm d} Q}
{\,\text{\rm d} \widetilde Q}\right)^{\frac{1}{1-\theta}}\right]^{(1-\theta)}\leqft(\widetilde Q W_n\right)^{\theta}.
\label{holder}
\end{equation}
Our aim is to find a measure $\widetilde Q$ such that the term $\leqft[\widetilde Q\leqft(\frac{\,\text{\rm d} Q}
{\,\text{\rm d} \widetilde Q}\right)^{\frac{1}{1-\theta}}\right]^{(1-\theta)}$ is not very large (i.e.\ of order $1$), and which significantly lowers the expected value of $W_n$. To do so we look for $\widetilde Q$ which lowers the value of the environment on each site, by exponential tilting. For $b<s$ it sufficient to lower the value for the environment uniformly of every site of $D_n\setminus \{A,B\}$ to get a satisfactory result, whereas for the $b=s$ case, on has to do an inhomogeneous change of measure. We present the change of measure in a united framework before going to the details with two separate cases.
Recall that $V_i$ denotes the sites of $D_i\setminus D_{i+1}$, and that the number of sites in $D_n$ is
\begin{align}\label{Dnn}
|D_n\setminus \{A,B\}|=\sum_{i=1}^n |V_i|=\sum_{i=1}^n (s-1)b^is^{i-1}=\frac{(s-1)b((sb)^n-1)}{sb-1}
\end{align}
We define $\widetilde Q= \widetilde Q_{n,s,b}$ to be the measure under which the environment on the site of the $i$-th generation for $i\in\{1,\dots,n\}$ are standard gaussians with mean $-\delta_i=\delta_{i,n}$, where $\delta_{i,n}$ is to be defined.
The density of $\widetilde Q$ with respect to $Q$ is given by
\begin{align*}
\frac{\,\text{\rm d} \widetilde Q}{\,\text{\rm d} Q}(\omega)=\exp\leqft(-\sum^n_{i=1}\sum_{z\in V_i} (\delta_{i,n}\omega_z+\frac{\delta^2_{i,n}}{2})\right).
\end{align*}
\noindent As each path in $D_n$ intersects $V_i$ on $s^{i-1}(s-1)$ sites, this change of measure lowers the value of the Hamiltonian \eqref{eq: Hn} by $\sum_{i=1}^n s^{i-1}(s-1)\delta_{i,n}$ on any path. Therefore,
both terms can be easily computed,
\begin{eqnarray}
\widetilde Q\leqft(\frac{\,\text{\rm d} Q}
{\,\text{\rm d} \widetilde Q}\right)^{\frac{1}{1-\theta}} =
\exp \leqft\lbrace \frac{\theta}{2(1-\theta)} \sum^n_{i=1}|V_i| \delta^2_{i,n} \right\rbrace.
\label{cost}
\end{eqnarray}
\begin{eqnarray}
\leqft(\widetilde Q W_n\right)^{\theta} = \exp \leqft\lbrace -\beta \theta \sum^n_{i=1} s^{i-1}(s-1) \delta_{i,n}\right\rbrace.
\label{gain}
\end{eqnarray}
\noindent Replacing \eqref{gain} and \eqref{cost} back into \eqref{holder} gives
\begin{eqnarray}
u_n \;\leqqslant\; \exp \leqft\lbrace \theta \sum^n_{i=1} \leqft( \frac{|V_i|\delta^2_{i,n}}{2(1-\theta)}- \beta s^{i-1}(s-1) \delta_{i,n} \right) \right\rbrace.
\label{eq:ineq0}
\end{eqnarray}
\noindent When $\delta_{i,n}= \delta_n$ (i.e. when the change of measure is homogeneous on every site) the last expression becomes simply
\begin{eqnarray}
u_n \;\leqqslant\; \exp \leqft\lbrace \theta \leqft( \frac{|D_n\setminus\{A,B\}|\delta^2_{n}}{2(1-\theta)}- (s^n-1)\beta \delta_{n} \right) \right\rbrace.
\label{ineq1}
\end{eqnarray}
\noindent In either case, the rest of the proof then consists in finding convenient values for $\delta_{i,n}$ and
$n$ large enough to insure that $(i)$ from Proposition \ref{th:fracmom} holds.
\subsection{Homogeneous shift method: $s>b$}
\begin{proof}[Proof of the left inequality in Theorem \ref{th:bs}, in the gaussian case]
Let $0<\theta<1$ be fixed (say $\theta=1/2$) and $\delta_{i,n}= \delta_n :=(sb)^{-n/2}$.
\noindent Observe from \eqref{Dnn} that $|D_n\setminus\{A,B\}| \delta^2_n \;\leqqslant\; 1$, so that \eqref{ineq1} implies
\begin{align*}
u_n\leq \exp\leqft(\frac{\theta}{2(1-\theta)}-\theta \beta (s/b)^{n/2}\frac{s-1}{s}\right).
\end{align*}
\noindent Taking $n=\frac{2(|\log \beta|+\log c_3)}{\log s-\log b}$, we get
\begin{align*}
u_n\leq \exp\leqft(\frac{\theta}{2(1-\theta)}-\frac{\theta c_5s}{s-1}\right).
\end{align*}
\noindent Choosing $\theta=1/2$ and $c_3$ sufficiently large, we have
\begin{align}
f_n=s^{-n}\log a_{\theta}b^{\frac{1-\theta}{s-1}}u_n\leq -s^{-n},
\end{align}
so that Proposition \ref{th:fracmom} gives us the conclusion
\begin{align*}
p(\beta)-\lambda(\beta) \leq -s^{-n}=-(\beta/c_3)^{\frac{2\log s}{\log s -\log b}}.
\end{align*}
\end{proof}
\subsection{Inhomogeneous shift method: $s=b$} \label{inshm}
One can check that the previous method does not give good enough results for the marginal case $b=s$.
One has to do a change of measure which is a bit more refined and for which the intensity of the tilt in proportional to the Green Function on each site. This idea was used first for the marginal case in pinning model on hierarchical lattice (see \cite{L}).
\begin{proof}[Proof of the left inequality in Theorem \ref{th:ss}, the gaussian case]
This time, we set $\delta_{i,n}:=n^{-1/2}s^{-i}$.
Then (recall \eqref{Dnn}), \eqref{eq:ineq0} becomes
\begin{align*}
u_n\leq \exp\leqft(\frac{\theta}{2(1-\theta)}\frac{s-1}{s}-\theta\beta n^{-1/2}\frac{s-1}{s}\right).
\end{align*}
\noindent Taking $\theta=1/2$ and $n=(c_4/\beta)^2$ for a large enough constant $c_4$, we get that $f_n\leq -s^n$ and applying Proposition \ref{th:fracmom}, we obtain
\begin{align*}
p(\beta) - \lambda(\beta) \leq -s^{-n}=-s^{-(c_4/\beta)^2}=\exp\leqft(-\frac{c_4^2\log s}{\beta^2}\right).
\end{align*}
\end{proof}
\subsection{Bounds for the critical temperature}
From Proposition \ref{th:fracmom}, we have that strong disorder holds if
$a_{\theta}< b^{(1-\theta)/(s-1)}$. Taking logarithms, this condition
reads
\begin{eqnarray*}
\lambda(\theta \beta)-\theta \lambda(\beta)< (1-\theta)\frac{ \log b}{s-1}.
\end{eqnarray*}
\noindent We now divide both sides by $1-\theta$ and let $\theta \to 1$. This
proves part $(iii)$ of Proposition \ref{misc}.
\noindent For the case $b>s$, this condition can be improved by the inhomogeneous shifting method; here, we perform it just in the gaussian case. Recall that
\begin{eqnarray}
u_n \;\leqqslant\; \exp \leqft\lbrace \theta \sum^n_{i=1} \leqft( \frac{|V_i|\delta^2_{i,n}}{2(1-\theta)}- \beta s^{i-1}(s-1)\delta_{i,n} \right) \right\rbrace.
\end{eqnarray}
\noindent We optimize each summand in this expression taking $\delta_{i,n}=\delta_i = (1-\theta) \beta / b^i$. Recalling
that $|V_i|= (bs)^{i-1}b(s-1)$, this yields
\begin{eqnarray}
\nonumber
u_n &\;\leqqslant\;& \exp \leqft\lbrace -\theta (1-\theta) \frac{\beta^2}{2} \frac{s-1}{s} \sum^n_{i=1} \leqft( \frac{s}{b}\right)^i \right\rbrace\\
\nonumber
&\;\leqqslant\;& \exp \leqft\lbrace -\theta (1-\theta) \frac{\beta^2}{2} \frac{s-1}{s} \frac{s/b-(s/b)^{n+1}}{1-s/b} \right\rbrace.
\end{eqnarray}
\noindent Because $n$ is arbitrary, in order to guaranty strong disorder it is enough to have ( cf. first condition in Proposition \ref{th:fracmom}) for some $\theta\in(0,1)$
\begin{eqnarray}\nonumber
\theta (1-\theta) \frac{\beta^2}{2} \frac{s-1}{s} \frac{s/b}{1-s/b}
> (1-\theta)\frac{\log b}{s-1}+\log a_\theta.
\end{eqnarray}
\noindent In the case of gaussian variables $\log a_\theta=\theta(\theta-1)\beta^2/2$. This is equivalent to
\begin{eqnarray*}
\frac{\beta^2}{2} > \frac{(b-s)\log b}{(b-1)(s-1)}.
\end{eqnarray*}
\noindent This last condition is an improvement of the bound in part $(iii)$ of
Proposition \ref{misc}.
\subsection{Adaptation of the proofs for non-gaussian variables}
\begin{proof}[Proof of the left inequality in Theorem \ref{th:bs} and \ref{th:ss}, the general case]
To adapt the preceding proofs to non-gaussian variables, we have to investigate the consequence of exponential tilting on non-gaussian variables. We sketch the proof in the inhomogeneous case $b=s$, we keep $\delta_{i,n}:=s^{-i}n^{-1/2}$.
Consider $\widetilde Q$ with density
\begin{align*}
\frac{\,\text{\rm d} \widetilde Q}{\,\text{\rm d} Q}(\omega):=\exp\leqft(-\sum_{i=1}^n\sum_{z\in V_i}\leqft(\delta_{i,n}\omega_z+\lambda(-\delta_{i,n})\right)\right),
\end{align*}
(recall that $\lambda(x):=\log Q \exp(x\omega)$).
The term giving cost of the change of measure is, in this case,
\begin{eqnarray*}
\leqft[\widetilde Q\leqft(\frac{\,\text{\rm d} Q}{\,\text{\rm d} \widetilde Q}\right)^{\frac{1}{1-\theta}}\right]^{(1-\theta)}&=&\exp\leqft((1-\theta)\sum_{i=1}^{n}|V_i|\leqft[\lambda\leqft(\frac{\theta\delta_{i,n}}{1-\theta}\right)+\frac{\theta}{1-\theta}\lambda(-\delta_{i,n})\right]\right)\\
&\;\leqqslant\;& \exp\leqft(\frac{\theta}{(1-\theta)}\sum_{i=1}^n|V_i|\delta_{i,n}^2\right)\, \leq \, \exp\leqft(\frac{\theta}{(1-\theta)}\right)
\end{eqnarray*}
Where the inequality is obtained by using the fact the $\lambda(x)\sim_0 x^2/2$ (this is a consequence of the fact that $\omega$ has unit variance) so that if $\beta$ is small enough, one can bound every $\lambda(x)$ in the formula by $x^2$.
We must be careful when we estimate $\widetilde Q W_n$. We have
\begin{align*}
\widetilde Q W_n=\exp \leqft(\sum_{i=1}^n (s-1)s^{i-1}\lambda (\beta-\delta_{i,n})-\lambda(\beta)-\lambda(-\delta_{i,n})\right) Q W_n
\end{align*}
By the mean value theorem
\begin{align*}
\lambda (\beta-\delta_{i,n})-\lambda(\beta)-\lambda(-\delta_{i,n})+\lambda(0)=-\delta_{i,n}\leqft(\lambda'(\beta-t_0)-\lambda'(-t_0)\right)=-\delta_{i,n}\beta\lambda''(t_1),
\end{align*}
for some $t_0\in(0,\delta_{i,n})$ and some $t_1\in (\beta,-\delta_{i,n})$. As we know that $\lim_{\beta\to 0}\lambda''(\beta)=1$,
when $\delta_i$ and $\beta$ are small enough, the right-hand side is less than $-\beta\delta_{i,n}/2$.
Hence,
\begin{align*}
\widetilde Q W_n\leq \exp \leqft(\sum_{i=1}^n(s-1)s^{i-1}\frac{\beta\delta_{i,n}}{2}\right).
\end{align*}
We get the same inequalities that in the case of gaussian environment, with different constants, which do not affect the proof. The case $b<s$ is similar.
\end{proof}
\section{Fluctuation and localisation results}\label{flucloc}
In this section we use the shift method we have developed earlier to prove fluctuation results
\subsection{Proof of Proposition \ref{fluctuations}}
The statement on the variance is only a consequence of the second statement. Recall that the random variable $\omega_z$ here are i.i.d. centered standard gaussians, and that the product law is denoted by $Q$.
We have to prove
\begin{equation}
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon(s/b)^{n/2}]\right\}\leq 4\varepsilon \quad \forall \varepsilon>0, n\geq 0, a\in {\ensuremath{\mathbb R}} \label{fluc}
\end{equation}
Assume there exist real numbers $a$ and $\varepsilon$, and an integer $n$ such that \eqref{fluc} does not hold, i.e.
\begin{equation}
Q\leqft\{ \log \bar Z_n \in [a, a+\beta\varepsilon(s/b)^{n/2})\right\}> 4\varepsilon.
\end{equation}
Then one of the following holds
\begin{equation}\begin{split}\label{abcd}
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon(s/b)^{n/2})\right\}\cap\leqft\{\sum_{z\in D_n} \omega_z\geq 0\right\}&> 2\varepsilon,\\
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon(s/b)^{n/2})\right\}\cap\leqft\{\sum_{z\in D_n} \omega_z\leq 0\right\}&> 2\varepsilon.
\end{split}\end{equation}
We assume that the first line is true. We consider the events related to $Q$ as sets of environments $(\omega_z)_{z\in D_n\setminus \{A,B\}}$. We define
\begin{equation}
A_\varepsilon=\leqft\{ \log Z_n \in [a, a+\beta\varepsilon b^{-n/2})\right\}\cap\leqft\{\sum_{z\in D_n} \omega_z\geq 0\right\},
\end{equation}
and
\begin{equation}
A^{(i)}_\varepsilon=Q\leqft\{ \log Z_n \in [a-i\beta\varepsilon(s/b)^{n/2}, a-(i-1)\beta\varepsilon(s/b)^{n/2})\right\}.
\end{equation}
Define $\delta=\frac{s^{n/2}}{(s^n-1)b^{n/2}}$. We define the measure $\widetilde Q_{i,\varepsilon}$ with its density:
\begin{equation}
\frac{\,\text{\rm d} \widetilde Q_{i,\varepsilon}}{\,\text{\rm d} Q}(\omega):=\exp\leqft(\leqft[i\varepsilon\delta^2 \sum_{z\in D_n} \omega_z\right]-\frac{i^2\varepsilon^2\delta^2|D_n\setminus\{A,B\}|}{2}\right).
\end{equation}
If the environment $(\omega_z)_{z\in D_n}$ has law $Q$ then $(\widehat \omega_z^{(i)})_{z\in D_n}$ defined by
\begin{equation}
\widehat \omega_z^{(i)}:= \omega_z+\varepsilon i \delta,
\end{equation}
has law $ \widetilde Q_{i,\varepsilon}$. Going from $\omega$ to $\widehat \omega^{(i)}$, one increases the value of the Hamiltonian by $\varepsilon i(s/b)^{n/2}$ (each path cross $s^n-1$ sites).
Therefore if $(\widehat\omega^{(i)}_z)_{z\in D_n}\in A_{\varepsilon}$, then $(\omega_z)_{z\in D_n}\in A^{(i)}_{\varepsilon}$.
From this we have $\widetilde Q_{i,\varepsilon} A_\varepsilon \leq Q A^{(i)}_{\varepsilon}$, and therefore
\begin{equation}
Q A^{(i)}_{\varepsilon}\, \geq \int_{A_\varepsilon} \frac{\,\text{\rm d} \widetilde Q_{i,\varepsilon}}{\,\text{\rm d} Q}Q(\,\text{\rm d} \omega)\geq \exp(-(\varepsilon i)^2/2)Q(A_\varepsilon).
\end{equation}
The last inequality is due to the fact that the density is always larger than $\exp(-(\varepsilon i)^2/2)$ on the set $A_\varepsilon$ (recall its definition and the fact that $|D_n\setminus\{A,B\}|\delta^2\leq 1$).
Therefore, in our setup, we have
\begin{equation}
Q A^{(i)}_{\varepsilon}> \varepsilon, \quad \forall i\in[0,\varepsilon^{-1}].
\end{equation}
As the $A^{(i)}_{\varepsilon}$ are disjoints, this is impossible.
If we are in the second case of \eqref{abcd}, we get the same result by shifting the variables in the other direction. \qed
\subsection{Proof of Proposition \ref{fluc2}}
Let us suppose that there exist $n$, $\varepsilon$ and $a$ such that
\begin{equation}
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon\sqrt{n})\right\}> 8\varepsilon.
\end{equation}
We define $\delta_{i,n}=\delta_i:=\varepsilon s^{1-i}(s-1)^{-1}n^{-1/2}$.
Then one of the following inequality holds (recall the definition of $V_i$)
\begin{equation}\begin{split} \label{fghi}
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon\sqrt{n})\right\}\cap\leqft\{\sum_{i=1}^{n}\delta_i\sum_{z\in V_i}\omega_z\geq 0\right\}&> 4\varepsilon,\\
Q\leqft\{ \log Z_n \in [a, a+\beta\varepsilon\sqrt{n})\right\}\cap\leqft\{\sum_{i=1}^{n}\delta_i\sum_{z\in V_i}\omega_z\leq 0\right\}&> 4\varepsilon.
\end{split}\end{equation}
We assume that the first line holds and define
\begin{equation}
A_{\varepsilon}=\leqft\{ \log Z_n \in [a, a+\beta\varepsilon\sqrt{n})\right\}\cap\leqft\{\sum_{i=1}^{n}\delta_i\sum_{z\in V_i}\omega_z\geq 0\right\}
\end{equation}
And
\begin{equation}
A_{\varepsilon}^{(j)}=\leqft\{ \log Z_n \in [a-j\beta\varepsilon\sqrt{n}, a-(j-1)\beta\varepsilon\sqrt{n})\right\}
\end{equation}
\begin{equation}
j\beta\sum_{i=1}^n\delta_i (s-1)s^{i-1}=j\beta\varepsilon \sqrt{n}.
\end{equation}
Therefore, an environment $\omega\in A_{\varepsilon}$ will be transformed in an environment in $A^{(j)}_{\varepsilon}$.
We define $\widetilde Q_{j,\varepsilon}$ the measure whose Radon-Nicodyn derivative with respect to $Q$ is
\begin{equation}
\frac{\,\text{\rm d} \widetilde Q_{i,\varepsilon}}{\,\text{\rm d} Q}(\omega):=\exp\leqft(\leqft[j\sum_{i=1}^n\delta_i \sum_{z\in V_i} \omega_z\right]-\sum_{i=1}^{n}\frac{j^2\delta_i^2|V_i|}{2}\right).
\end{equation}
We can bound the deterministic term.
\begin{equation}
\sum_{i=1}^{n}\frac{j^2\delta_i^2|V_i|}{2}=j^2\varepsilon^2 \sum_{i=1}^n \frac{s}{2(s-1)}\leq j^2\varepsilon^2.
\end{equation}
For an environment $(\omega_z)_{z\in D_n\setminus\{A,B\}}$, define $(\widehat \omega_z^{(j)})_{z\in D_n\setminus\{A,B\}}$ by
\begin{equation}
\widehat \omega_z^{(j)}:= \omega_z+j\varepsilon\delta_i, \quad \forall z \in V_i.
\end{equation}
If $(\omega_z)_{z\in D_n\setminus\{A,B\}}$ has $Q$, then $(\widehat \omega_z^{(j)})_{z\in D_n\setminus\{A,B\}}$ has law $\widetilde Q_{j,\varepsilon}$.
When one goes from $\omega$ to $\widehat\omega^{(j)}$, the value of the Hamiltonian is increased by
\begin{equation*}
\sum_{i=1}^n j\delta_i s^{i-1}(s-1)=\varepsilon \sqrt{n}.
\end{equation*}
Therefore, if $\widehat \omega^{(j)}\in A_{\varepsilon}$, then $\omega\in A^{(j)}_{\varepsilon}$, so that
\begin{equation*}
Q A^{(j)}_{\varepsilon}\geq \widetilde Q_{j,\varepsilon} A_{\varepsilon}.
\end{equation*}
Because of the preceding remarks
\begin{equation}
Q A^{(j)}_{\varepsilon}\geq \widetilde Q_{j,\varepsilon} A_{\varepsilon}=\int_{A_{\varepsilon}}\frac{\,\text{\rm d} \widetilde Q_{i,\varepsilon}}{\,\text{\rm d} Q}Q(\,\text{\rm d}\omega) \geq \exp\leqft(-j^2\varepsilon^2\right)Q A_{\varepsilon}.
\end{equation}
The last inequality comes from the definition of $A_{\varepsilon}$ which gives an easy lower bound on the Radon-Nicodyn derivative.
For $j\in [0,(\varepsilon/2)^{-1}]$, this implies that $Q A^{(j)}_{\varepsilon}> 2\varepsilon$. As they are disjoint events this is impossible. The second case of \eqref{fghi} can be dealt analogously. \qed
\subsection{Proof of Corollary \ref{locloc}}
Let $g\in \Gamma_n$ be a fixed path. For $m\geq n$, define
\begin{equation}
Z_m^{(g)}:=\sum_{\{ g'\in \Gamma_m: g|_n=g\}}\exp\leqft(\beta H_m(g')\right).
\end{equation}
With this definition we have
\begin{equation}
\mu_m(\alphamma|_n=g)=\frac{Z_m^{(g)}}{Z_m}.
\end{equation}
To show our result, it is sufficient to show that for any constant $K$ and any distinct $g,g'\in \Gamma_n$
\begin{equation}
\lim_{m\to\infty}Q\leqft( \frac{\mu_m(\alphamma|_n=g)}{\mu_m(\alphamma|_n=g')}\in[K^{-1},K]\right)=0.
\end{equation}
For $g$ and $g'$ distinct, it is not hard to see that
\begin{equation}
\log\leqft( \frac{\mu_m(\alphamma|_n=g)}{\mu_m(\alphamma|_n=g')}\right)=\log Z_m^{(g)}-\log Z_m^{(g')}=: \log Z^{(0)}_{m-n}+X,
\end{equation}
where $Z^{(0)}_{m-n}$ is a random variable whose distribution is the same as the one of $Z_{m-n}$, and $X$ is independent of $Z^{(0)}_{m-n}$.
We have
\begin{multline}
Q\leqft(\log \leqft(\frac{\mu_m(\alphamma|_n=g)}{\mu_m(\alphamma|_n=g')}\right)\in [-\log K, \log K]\right)\\
=Q \leqft[ Q\leqft(\log Z^{(0)}_{m-n}\in [-\log K -X, \log K -X] \, \big| \, X\right)\right]
\\
\leq \max_{a\in \mathbb{R}} Q \leqft(\log Z_{m-n}\in [a,a+2\log K]\right).
\end{multline}
Proposition \ref{fluctuations} and \ref{fluc2} show that the right--hand side tends to zero. \qed
\section{The weak disorder polymer measure}\label{weakpol}
Comets and Yoshida introduced in \cite{CY} an infinite
volume Markov chain at weak disorder that corresponds in some sense to the limit of the polymers
measures $\mu_n$ when $n$ goes to infinity. We perform the same construction here.
The notation is more cumbersome in our setting.
Recall that $\Gamma_n$ is the space of directed paths from $A$ to $B$ in $D_n$. Denote by $P_n$ the uniform law on $\Gamma_n$. For $g\in \Gamma_n$,
$0\;\leqqslant\; t \;\leqqslant\; s^n-1$, define $W_{\infty}(g_t,g_{t+1})$ by performing the same construction that leads to $W_{\infty}$, but taking $g_t$ and $g_{t+1}$ instead of $A$ and $B$ respectively. On the classical directed polymers on $\mathbb{Z}^d$, this would be equivalent to take the $(t,g_t)$ as the initial point of the polymer.
We can now define the weak disorder polymer measure for $\beta < \beta_0$. We define $\Gamma$ as the projective limit of $\Gamma_n$ (with its natural topology), the set of path on $D:=\bigcup_{n\geq 1} D_n$. As for finite path, we can define, for $\bar g\in \Gamma$, its projection onto $\Gamma_n$, $\bar g|_n$.
We define
\begin{eqnarray}\label{wdpol}
\mu_{\infty}(\bar \alphamma|_n =g):= \frac{1}{W_{\infty}} \exp \{ \beta H_n(g) - (s^n-1) \lambda(\beta) \} \prod^{s^n-1}_{i=0} W_{\infty}({g}_i,{g}_{i+1}) \, P_n({\bar \alphamma|_n = g}).
\end{eqnarray}
Let us stress the following:
\begin{itemize}
\item Note that the projection on the different $\Gamma_n$ are consistent (so that our definition makes sense)
\begin{eqnarray*}
\mu_{\infty}(\bar \alphamma|_n = g)= \mu_{\infty}\leqft((\bar \alphamma|_{n+1})|_n = g\right).
\end{eqnarray*}
\item Thanks to the martingale convergence for both the numerator and the
denominator, for any ${\bf s} \in \Gamma_n$,
\begin{eqnarray*}
\lim_{k\to +\infty} \mu_{k+n}(\alphamma|_n = g) = \mu_{\infty}(\bar \alphamma|_n = g).
\end{eqnarray*}
Therefore, $\mu_{\infty}$ is the only reasonable definition for the limit of $\mu_n$.
\end{itemize}
It is an easy task to prove the law of large numbers for the time-averaged quenches mean of the energy.
This follows as a simple consequence of the convexity of $p(\beta)$.
\begin{proposition}
At each point where $p$ admits a derivative,
\begin{eqnarray*}
\lim_{n\to +\infty} \frac{1}{s^n} \mu_{n}(H_n(\alphamma)) \to p'(\beta), \quad Q-a.s..
\end{eqnarray*}
\end{proposition}
\begin{proof}
It is enough to observe that
\begin{eqnarray*}
\frac{d}{d \beta} \log Z_n = \mu_n(H_n(\alphamma)),
\end{eqnarray*}
\noindent then use the convexity to pass to the limit.
\end{proof}
We can also prove a quenched law of large numbers under our infinite volume
measure $\mu_{\infty}$, for almost every environment. The proof is very easy,
as it involves just a second moment computation.
\begin{proposition}
At weak disorder,
\begin{eqnarray*}
\lim_{n\to +\infty}\frac{1}{s^n}H_n(\bar \alphamma|_n)=\lambda'(\beta),\quad \mu_{\infty}-a.s., \, Q-a.s..
\end{eqnarray*}
\end{proposition}
\begin{proof}
We will consider the following auxiliary measure (size biased measure) on the environment
\begin{eqnarray*}
\overline{Q}(f(\omega)) = Q(f(\omega) W_{+\infty}).
\end{eqnarray*}
\noindent So, $Q$-a.s. convergence will follow from $\overline{Q}$-a.s. convergence.
This will be done by a direct computation of second moments. Let us write
$\Delta = Q(\omega^2 e^{\beta \omega - \lambda(\beta)})$.
\begin{eqnarray*}
&\ &\overline{Q} \leqft( \mu_{\infty}(|H_n(\bar \alphamma|_n)|^2) \right)\\
&=& Q \leqft[ P_n\leqft( |H_n(\alphamma)|^2 \exp \{ \beta H_n(\alphamma)- (s^n-1) \lambda(\beta)\} \prod^{s^n-1}_{i=0} W_{\infty}(\alphamma_i,\alphamma_{i+1})\right)\right]\\
&=& Q \leqft[ P_n( |H_n(\alphamma)|^2 \exp \{ \beta H_n(\alphamma)- (s^n-1) \lambda(\beta)\})\right]\\
&=& Q \leqft[ P_n( |\sum^{s^n}_{t=1}\omega(\alphamma_t)|^2 \exp \{ \beta H_n(\alphamma)- (s^n-1) \lambda(\beta)\})\right]\\
&=& Q\leqft[ \sum^{s^n-1}_{t=1}P_n\leqft(|\omega(\alphamma_t)|^2\exp \{ \beta H_n(\alphamma)- (s^n-1) \lambda(\beta)\}\right)\right]\\
&\ &+ Q \leqft[\sum_{1\leq t_1\neq t_2\leq s^{n}-1}P_n\leqft(\omega(\alphamma_{t_1})\omega(\alphamma_{t_2})\exp \{ \beta H_n(\alphamma)- (s^n-1) \lambda(\beta)\}\right)\right]\\
&=& (s^n-1) \Delta+(s^n-1)(s^n-2)(\lambda'(\beta))^2,
\end{eqnarray*}
\noindent where we used independence to pass from line two to line three. So, recalling that $\overline{Q} ( \mu_{\infty}(H_n(\bar \alphamma_n))= (s^n-1) \lambda'(\beta)$, we
have
\begin{eqnarray*}
&\ &\overline{Q} \leqft( \mu_{\infty}(|H_n(\alphamma^{(n)}) - (s^n-1) \lambda'(\beta)|^2) \right)\\
&=& (s^n-1) \Delta+(s^n-1)(s^n-2)(\lambda'(\beta))^2
- 2 (s^n-1) \lambda'(\beta)\overline{Q} \leqft( \mu_{\infty}(H_n(\bar \alphamma_n)) \right)\\
&\ & \quad + \, (s^{n}-1)^2(\lambda'(\beta))^2\\
&=& (s^n-1) \leqft( \Delta - (\lambda'(\beta)^2)\right).
\end{eqnarray*}
\noindent Then
\begin{eqnarray*}
\overline{Q} \mu_{\infty}\leqft( \leqft|\frac{H_n(\bar\alphamma_n) - (s^n-1) \lambda'(\beta)}{s^n} \right|^2\right)
\;\leqqslant\; \frac{1}{s^n}\leqft( \Delta - (\lambda'(\beta)^2)\right),
\end{eqnarray*}
\noindent so the result follows by Borel-Cantelli.
\end{proof}
\section{Some remarks on the bond--disorder model}\label{bddis}
In this section, we shortly discuss, without going through the details, how the methods we used in this paper could be used (or could not be used) for the model of directed polymer on the same lattice with disorder located on the bonds.
\\
In this model to each bond $e$ of $D_n$ we associate i.i.d.\ random variables $\omega_e$.
We consider each set $g\in \Gamma_n$ as a set of bonds and define the Hamiltonian as
\begin{equation}
H^{\omega}_n(g) = \sum_{e\in g} \omega_e,
\end{equation}
\noindent The partition function $Z_n$ is defined as
\begin{equation}
Z_n:= \sum_{g\in \Gamma_n} \exp(\beta H_n(g)).
\end{equation}
One can check that is satisfies the following recursion
\begin{equation}\begin{split}
Z_0& \, \stackrel{ \mathcal L}{=}\, \exp(\beta\omega)\\
Z_{n+1}&\, \stackrel{\mathcal L)}{=}\, \sum_{i=1}^{b}Z_n^{(i,1)}Z_n^{(i,2)}\dots Z_n^{(i,s)}.
\end{split}
\end{equation}
\noindent where equalities hold in distribution and and $Z_n^{i,j}$ are i.i.d.\ distributed copies of $Z_n$.
Because of the loss of the martingale structure and the homogeneity of the Green function in this model
(which is equal to $b^{-n}$ on each edge), Lemma \ref{martingale} does not hold, and we cannot prove part $(iv)$ in Proposition \ref{misc}, Theorem \ref{th:ss} and Proposition \ref{fluc2} for this model. Moreover we have to change $b\leq s$ by $b<s$ in $(v)$ of Proposition \ref{misc}.
Moreover, the method of the control of the variance would give us a result similar to \ref{th:ss} in this case
\begin{proposition}\label{ssbd}
When $b$ is equal to $s$, on can find constants $c$ and $\beta_0$ such that for all $\beta\leq \beta_0$
\begin{equation}
0\leq \lambda(\beta)-p(\beta) \leq \exp\leqft(-\frac{c}{\beta^2}\right).
\end{equation}
\end{proposition}
However, we would not be able to prove that annealed and free energy differs at high temperature for $s=b$ using our method. The techniques used in \cite{GLT_marg} or \cite{Lac} for dimension $2$ should be able to tackle this problem, and show marginal disorder relevance in this case as well.
\noindent {\bf Acknowledgements:} The authors are very grateful to Francis Comets and Giambattista Giacomin for their suggestion to work on this subject and many enlightning discussions. This work was partially supported by CNRS, UMR $7599$ ``Probabilit\'es et
Mod\`eles Al\'eatoires". H.L.\ acknowledges the support of ANR grant POLINTBIO. G.M.\ acknowledges the support of Beca Conicyt-Ambassade de France.
\end{document} |
\begin{document}
\title{On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping}
\titlerunning{Perturbations of inertial systems with Hessian driven damping}
\author{Hedy Attouch \and Jalal Fadili \and Vyacheslav Kungurtsev}
\authorrunning{H. Attouch, J. Fadili, V. Kungurtsev}
\institute{
H. Attouch \at IMAG, Univ. Montpellier, CNRS, Montpellier, France\\
\email{[email protected]}
\and J. Fadili
\at{Normandie Univ, ENSICAEN, CNRS, GREYC, Caen, France}\\
\email{[email protected]}
\and V. Kungurtsev
\at Department of Computer Science and Engineering, Czech Technical University, Prague.\\\email{[email protected]}
}
\date{}
\maketitle
\begin{abstract}
Second-order continuous-time dissipative dynamical systems with viscous and Hessian driven damping have inspired effective first-order algorithms for solving convex optimization problems. While preserving the fast convergence properties of the Nesterov-type acceleration, the Hessian driven damping makes it possible to significantly attenuate the oscillations. To study the stability of these algorithms with respect to perturbations, we analyze the behaviour of the corresponding continuous systems when the gradient computation is subject to exogenous additive errors. We provide a quantitative analysis of the asymptotic behaviour of two types of systems, those with implicit and explicit Hessian driven damping. We consider convex, strongly convex, and non-smooth objective functions defined on a real Hilbert space and show that, depending on the formulation, different integrability conditions on the perturbations are sufficient to maintain the convergence rates of the systems. We highlight the differences between the implicit and explicit Hessian damping, and in particular point out that the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case.
\end{abstract}
\keywords{Hessian driven damping; damped inertial dynamics; accelerated convex optimization; convergence rates; Lyapunov analysis; perturbation; errors.}
\noindent \textbf{AMS subject classification} 37N40, 46N10, 49M30, 65B99, 65K05, 65K10, 90B50, 90C25
\section{Introduction}
The continuous-time dynamic perspective of optimization algorithms, which can be viewed as temporal discretization schemes thereof, offers an insightful and powerful framework for the study of the behaviour of these algorithms. In this paper, we study inertial systems involving both viscous and Hessian-driven damping, where the first-order gradient information is only accessible up to some {\emph{exogenous additive error}}.
\subsection{Problem statement}
Throughout the paper, we make the following standing assumptions:
\[
\boxed{
\text{
$f$ is a convex function on a real Hilbert space ${\mathcal H}$, and $S := \argmin_{\mathcal H} f\neq \emptyset$.\hspace{1cm}}
}
\]
\noindent We will study perturbed versions of two second-order ordinary differential equations (ODE). {They differ from each other in that the Hessian driven damping appears explicitly in one and implicitly in the other}.
\subsubsection{Explicit Hessian}
The first system we look at, which was proposed in \cite{attouch2019first} (see also \cite{APR1}), takes the form
\begin{equation}\tag{ISEHD}\label{eq:origode}
\ddot{x}(t)+\gamma(t)\dot{x}(t)+\beta(t)\frac{d}{dt}(\nabla f(x(t)))+b(t)\nabla f(x(t)) = 0,
\end{equation}
where $f \in {\mathcal C}^1({\mathcal H})$, $\gamma,\beta,b:[t_0,+\infty[ \to {\mathbb R}_+$ are continuous functions, and $t_0>0$ is the initial time. The coefficients $(\gamma,\beta,b)$ have a physical interpretation corresponding to natural phenomena:
\begin{enumerate}[label=$\bullet$]
\item $\gamma(t)$ is the viscous damping coefficient,
\item $\beta(t)$ is the Hessian-driven damping coefficient (which will be made clear),
\item $b(t)$ is the time scaling coefficient (see \cite{ACR-SIOPT}).
\end{enumerate}
\noindent We term the above ODE an Inertial System with {\emph{Explicit}} Hessian Damping (ISEHD for short), since
\[
\frac{d}{dt}(\nabla f(x(t))) = \nabla^2 f(x(t)) \dot x(t) ,
\]
when $f$ is of class ${\mathcal C}^2({\mathcal H})$. Throughout the paper, we consider \eqref{eq:origode} with the particular choice of parameters
\begin{center}
$\gamma(t) = \displaystyle{\frac{\alpha}{t}}$, \; $\alpha \geq 0$,\; $\beta(t) \equiv \beta > 0$ and $b(t) \equiv 1$.
\end{center}
This choice of the viscous damping parameter $\gamma(t) = \frac{\alpha}{t}$ is justified by its direct link with the accelerated gradient method of Nesterov \cite{Nest1,Nest2}, as shown in \cite{AC10}, \cite{attouch2018fast}, \cite{ADR}, \cite{CD}, \cite{SBC}.
Related systems have been considered in \cite{LJ} from the closed loop control perspective and in \cite{SDJS} by means of
high-resolution of differential equations.
\subsubsection{Implicit Hessian}
The second system we consider, inspired by~\cite{alecsa2019extension} (see also \cite{MJ} for a related autonomous system in the case of a strongly convex function $f$), is
\begin{equation}\tag{ISIHD}\label{eq:odetwo}
\ddot{x}(t)+\frac{\alpha}{t} \dot{x}(t)+\nabla f\Big(x(t)+\beta(t)\dot{x}(t)\Big)=0 ,
\end{equation}
where $\alpha \ge 3$ and $\beta(t)=\gamma+\frac{\beta}{t}$, $\gamma, \; \beta \geq 0$. We coin this ODE an Inertial System with {\emph{Implicit}} Hessian Damping (ISIHD for short). The rationale justifying our use of the term ``implicit" comes from the observation that by a Taylor expansion (as $t \to +\infty$ we have $\dot{x}(t) \to 0$ which justifies using Taylor expansion), one has
\[
\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}\approx \nabla f (x(t)) + \beta(t)\nabla^2 f(x(t))\dot{x}(t) ,
\]
hence making the Hessian damping appear indirectly in (\ref{eq:odetwo}). This ODE was found to have a smoothing effect on the energy error and oscillations.
\subsubsection{Exogenous additive error}
We are interested in the situation where $\nabla f(x(t))$ is always evaluated with an exogenous additive error $e(t)$. With the choice of parameters made above, the perturbed dynamics of~\eqref{eq:origode} and \eqref{eq:odetwo} are written
\begin{equation}
\boxed{
\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\beta\frac{d}{dt}\Big(\nabla f(x(t))+e(t)\Big)+ \nabla f(x(t))+e(t)=0 ,\tag{ISEHD-\textsc{Pert}} \hspace{1cm}\label{eq:origode_a}
}
\end{equation}
\begin{equation}
\boxed{
\ddot{x}(t)+\frac{\alpha}{t} \dot{x}(t)+\nabla f\Big(x(t)+\beta(t)\dot{x}(t)\Big)+e(t)=0. \hspace{2.9cm}\tag{{ISIHD-\textsc{Pert}}} \label{eq:odetwo_a}
}
\end{equation}
For system \eqref{eq:origode_a}, the overall perturbation error affecting the system is
$\beta\dot{e}(t) + e(t).$
Because the Hessian appears explicitly, both the error on the gradient and its derivative appear. It can then be anticipated that assumptions regarding both $e(t)$ and $\dot{e}(t)$, in particular their integrability, will be instrumental in deriving any convergence guarantees. On the other hand, in the system \eqref{eq:odetwo_a} with implicit Hessian damping, the error perturbation $e(t)$ appears without its time derivative. Naturally, we will see in this case that convergence results will be derived without any assumptions on the time derivative of the error. While this may be seen as an advantage at first glance, this comes at a price. Indeed, as we will also see, to maintain fast convergence guarantees, the integrability requirements on the error $e(t)$ will be more stringent for \eqref{eq:odetwo_a} than for \eqref{eq:origode_a}, {\it i.e.}, higher-order moments of $e(t)$ will be required to be finite. We anticipate that when it comes to discrete algorithms, the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case. We plan to study these questions in a future work. Note that similar questions arise when the perturbation is attached to a Tikhonov regularization term with an asymptotically vanishing coefficient \cite{BCL}.
One of our motivations for the above additive perturbation model originates from optimization, where the gradient may be accessed only inaccurately, either because of physical or computational reasons. The prototype example we think of is
\[
f(x) = {\mathbb E}_{\xi}[F(x,\xi] ,
\]
where ${\mathbb E}_{\xi}[\cdot]$ is expectation with respect to the random variable $\xi$, and $F(\cdot,\xi) \in {\mathcal C}^1({\mathcal H})$ for any $\xi$. This is a popular setting in numerous applications (imaging, statistical learning, etc.), where computing $\nabla f(x)$ (or even $f(x)$) is either impossible or computationally very expensive. Rather, one draws $m$ independent samples of $\xi$, say $(\xi_i)_{1 \leq i \leq m}$, and compute the average estimate
\[
\widehat{\nabla f}(x) = \frac{1}{m} \sum_{i=1}^m \nabla F(x,\xi_i) .
\]
In our notation, the error is then $e(t) = \nabla f(x(t)) - \widehat{\nabla f}(x(t))$. Under independence and mild assumptions, one has by the law of iterated logarithm that $\|e(t)\|={\mathcal O}\pa{\sqrt{\frac{\log(\log(m))}{m}}}$ almost surely. Thus, to make this error vanish or even integrable, one has to take $m(.)$ an increasing function of $t$ at least at the rate ${\mathcal O}(t^{2(1+\delta)})$ for $\delta > 0$. At this stage, some readers may have expected a smallness condition on the perturbation rather than integrability conditions, i.e., of $\|e(t)\|=\mathcal{O}(r(t))$ for some $r(t)$ such that $\lim\limits_{t\to\infty} r(t)=0$. Of course, integrability is stronger as it obviously implies that the error function, whenever it converges, will vanish asymptotically. However, one has to keep in mind that our goal is not only to establish convergence (and rate of convergence) of the objective values to a "perturbation dominated region" around the optimal value, but to show additional convergence guarantees, in particular the important matter of weak convergence of the trajectory. Deriving convergence of trajectories is typically much more challenging than the objective and cannot be proved under a mere smallness condition on the error.
\subsection{Contributions}
In \cite{attouch2019first} (resp. \cite{alecsa2019extension}), which studied the unperturbed system \eqref{eq:origode} (resp. \eqref{eq:odetwo}), fast convergence rates were obtained for the objective, velocities and gradients.
Our main contribution in this paper is to analyze the robustness and stability of these systems, by quantifying their convergence properties in the presence of errors. We do this both in the general convex case and in the strongly convex case. We also study the case where the function $f$ is non-smooth convex by proposing a first order formulation in time and space, with existence results and Lyapunov analysis. The main motivation for our work is to pave the way for the design and study of provably accelerated optimization algorithms that appropriately discretize the above dynamics while handling inexact evaluations of the gradient with deterministic and/or stochastic errors. The extension to the discrete setting of the results here will be the focus of a forthcoming paper.
\subsection{Related Works}
Due to the importance of the subject in optimization and control, several articles have been devoted to the study of perturbations in dissipative inertial systems and in the corresponding
accelerated first order algorithms. The subject was first considered in the case of a fixed viscous damping, \cite{acz,HJ1}. Then it was studied within the framework of the accelerated gradient method of Nesterov, and of the corresponding inertial dynamics with vanishing viscous damping, see \cite{SDJS,AC2R-JOTA,attouch2018fast,AD15,SLB,VSBV}. In the presence of the additional Hessian driven damping, first results have been obtained in \cite{APR1,attouch2019first,attouchiterates2021} in the case of a smooth function. To the best of our knowledge, our work is the first to consider these questions in full generality and in presence of perturbations.
\subsection{Contents}
In Section~\ref{s:wellposed}, we prove that the two systems are well-posed both in the smooth and non-smooth cases. In Section~\ref{s:convex}, we study the convex case, and establish convergence rates for both systems under appropriate integrability assumptions on the error. In Section~\ref{s:sconvex}, we consider the strongly convex case. Section~\ref{s:nonsmooth}\; is devoted to studying non-smooth $f$. In Section~\ref{s:num}, we present some numerical illustrations of the results. In Section~\ref{s:conc}, we draw key conclusions and present some perspectives.
\subsection{Main notations}
${\mathcal H}$ is a real Hilbert space, $\dotp{\cdot}{\cdot}$ is the scalar product on ${\mathcal H}$ and $\norm{\cdot}$ is the corresponding norm. $\Gamma_0({\mathcal H})$ is the class of proper, lower semicontinuous (lsc) and convex functions from ${\mathcal H}$ to ${\mathbb R}b$. A function $g: {\mathcal H} \to {\mathbb R}b$ is $\mu$-strongly convex ($\mu >0$) if $g - \frac{\mu}{2}\| \cdot\|^2$ is convex. For $g \in \Gamma_0({\mathcal H})$, its domain is $\dom(g) := \setcond{x\in{\mathcal H}}{g(x) < +\infty}$. $\partial g$ denotes the (convex) subdifferential operator of $g$. When $g$ is differentiable at $x\in{\mathcal H}$, then $\partial g(x)=\bra{\nabla g(x)}$. We also denote $\dom(\partial g) := \setcond{x \in {\mathcal H}}{\partial g(x) \neq \emptyset}$.
${\mathcal C}^s({\mathcal D})$ is the class of $s$-continuously differentiable functions on ${\mathcal D}$, ${\mathcal D}$ will be specified in the context. For $T > t_0 $ and $p \geq 1$, $L^p(t_0,T; {\mathcal H})$ is the Lebesgue space of measurable functions $x: t \in [t_0,T] \mapsto x(t) \in {\mathcal H}$ such that $\int_{t_0}^T \norm{x(t)}^p dt < +\infty$. ${\mathcal W}^{1,1} (t_0,T; {\mathcal H})$ is the Sobolev space of functions $x(.) \in L^1(t_0,T; {\mathcal H})$ with distributional derivative $\dot{x}(\cdot)\in L^{1} (t_0,T;{\mathcal H})$. We will also invoke the notion of a strong solution to a differential inclusion, see \cite[Definition 3.1]{Bre1}, that will be given precisely in Section \ref{section-gen-existence}. The reader interested mostly in quantitative convergence estimates can skip the corresponding section.
We take $t_0 >0$ as the origin of time. This is justified by the singularity of the viscous damping coefficient $\frac{\alpha}{t}$ at the origin. This is not restrictive since we are interested in asymptotic analysis. We denote $S$ to be the set of minimizers of $f$, {\it i.e.}, $S := \argmin\limits_{x\in{\mathcal H}} f(x)$ which is assumed nonempty, and $x^\star$ to be an arbitrary element of $S$ (unique in the case of strongly convex $f$). We denote $\bar{f} := \inf\limits_{x\in{\mathcal H}} f(x)$.
\section{Well-posedness}\label{s:wellposed}
When $\beta > 0$, the presence of Hessian driven damping in the inertial dynamics makes it possible to reformulate the equations as first-order systems both in time and in space, without explicit evaluation of the Hessian. This will allow us to extend the existence of trajectories and the convergence results to the case $f \in \Gamma_0({\mathcal H})$, by simply replacing the gradient of $f$ with the subdifferential $\partial f$. This approach was initiated in \cite{aabr} and used in \cite{APR1} for the unperturbed case.
\subsection{Explicit Hessian Damping}
\subsubsection{Formulation as a first-order system}
Let us start by establishing this equivalence in the case of a smooth function $f$.
\begin{theorem}\label{Thm-first-order-system}
Let $f:{\mathcal H}\to{\mathbb R}$ be a ${\mathcal C}^2({\mathcal H})$ function and $e: [t_0,+\infty[ \to {\mathcal H}$ be ${\mathcal C}^1({\mathcal H})$. Suppose that $\alpha\geq0$, $\beta>0$. Let $(x_0,\dot x_0)\in {\mathcal H} \times {\mathcal H}$. The
following statements are equivalent:
\begin{enumerate}[label=\arabic*.]
\item \label{item:Thm-first-order-system_1}
$x:[t_0,+\infty [ \to {\mathcal H}$ is a solution trajectory of \eqref{eq:origode_a}
with the initial conditions $x(t_0)=x_0$, $\dot x(t_0)=\dot x_0$.
\item \label{item:Thm-first-order-system_2}
$(x,y):[t_0,+\infty [ \to {\mathcal H} \times {\mathcal H}$ is a solution trajectory of the first-order system
\begin{equation}\label{eq:ISEHDequiv1stode}
\begin{cases}
\dot x(t) + \beta (\nabla f (x(t))+ e(t)) - \pa{\frac{1}{\beta} - \frac{\alpha}{t}} x(t) + \frac{1}{\beta}y(t) &= 0 \\
\dot{y}(t)-\pa{\frac{1}{\beta} - \frac{\alpha}{t} + \frac{\alpha \beta}{t^2}} x(t) + \frac{1}{\beta} y(t) &= 0,
\end{cases}
\end{equation}
\end{enumerate}
with initial conditions $x(t_0)=x_0$,
$y(t_0)=-\beta(\dot x_0+\beta\nabla f(x_0))+(1-\beta\alpha/t_0)x_0 -\beta^2 e(t_0)$.
\end{theorem}
\begin{proof}
\textit{{\ref{item:Thm-first-order-system_2} }}${\mathbb R}ightarrow$ \textit{{\ref{item:Thm-first-order-system_1} }}
Differentiating the first equation of \eqref{eq:ISEHDequiv1stode} gives
\begin{equation}\label{eq:fos01}
\ddot x(t) + \beta \pa{\nabla^2 f (x(t)) \dot{x}(t) + \dot{e}(t)} -
\frac{\alpha}{t^2}x(t)- \pa{\frac{1}{\beta} - \frac{\alpha}{t}} \dot x(t)+\frac{1}{\beta}\dot y(t) = 0 .
\end{equation}
Replacing $\dot y(t)$ by its expression as given by the second equation of \eqref{eq:ISEHDequiv1stode} gives
\begin{equation}
\ddot x(t) + \beta \pa{\nabla^2 f (x(t)) \dot{x}(t) + \dot{e}(t)} -
\frac{\alpha}{t^2}x(t)- \pa{ \frac{1}{\beta} - \frac{\alpha}{t} } \dot x(t)
+\frac{1}{\beta}\pa{\pa{ \frac{1}{\beta} - \frac{\alpha}{t} + \frac{\alpha \beta}{t^2}} x(t)
-\frac{1}{\beta} y(t)} = 0 .\label{eq:fos02}
\end{equation}
Then replace $y(t)$ by its expression as given by the first equation of \eqref{eq:ISEHDequiv1stode}
\begin{multline*}
\ddot x(t) + \beta \pa{\nabla^2 f (x(t)) \dot{x}(t) + \dot{e}(t)} -
\frac{\alpha}{t^2}x(t)- \pa{ \frac{1}{\beta} - \frac{\alpha}{t} } \dot x(t) \\
+\frac{1}{\beta}\pa{\pa{ \frac{1}{\beta} - \frac{\alpha}{t} + \frac{\alpha \beta}{t^2} } x(t)
+ \dot x(t) + \beta (\nabla f (x(t))+ e(t)) - \pa{ \frac{1}{\beta} - \frac{\alpha}{t}} x(t)} = 0 .\label{eq:fos03}
\end{multline*}
After simplification of the above expression, we obtain \eqref{eq:origode_a}.
\textit{{\ref{item:Thm-first-order-system_1}}} ${\mathbb R}ightarrow$ \textit{{\ref{item:Thm-first-order-system_2}}} Define $y(t)$ by the first equation of \eqref{eq:ISEHDequiv1stode}. Differentiating $y(t)$ and using equation \eqref{eq:origode_a} allows one to eliminate $\ddot{x}(t)$, which finally gives the second equation of \eqref{eq:ISEHDequiv1stode}. \qed
\end{proof}
\subsubsection{Existence and uniqueness of a solution}\label{section-gen-existence}
Capitalizing on the result of Theorem~\ref{Thm-first-order-system}, the following first order formulation assists in providing a meaning to our system when $f \in \Gamma_0({\mathcal H})$.
It is obtained by substituting the subdifferential $\partial f$ for the gradient $\nabla f$ in the first-order formulation \eqref{eq:ISEHDequiv1stode}.
\begin{definition}
Let $\alpha\geq0$, $\beta>0$ and $f \in \Gamma_0({\mathcal H})$. {Given
$(x_0, y_0) \in \dom(f) \times {\mathcal H} $, the Cauchy problem for the perturbed inertial system with explicit generalized Hessian driven damping is defined by}
\begin{equation}\label{eq:fos2}
\begin{cases}
\dot x(t)+\beta(\partial f(x(t)) + e(t)) -\pa{\frac{1}{\beta}-\frac{\alpha}{t}}x(t)+\frac{1}{\beta}y(t) & \ni 0 \\
\dot{y}(t) -\pa{\frac{1}{\beta}-\frac{\alpha}{t}+\frac{\alpha\beta}{t^2}}x(t) +\frac{1}{\beta} y(t) & = 0 \\
x(t_0)=x_0, y(t_0)=y_0 .
\end{cases}
\end{equation}
\end{definition}
Let us formulate \eqref{eq:fos2} in a condensed form as an evolution equation in the product space ${\mathcal H} \times {\mathcal H}$.
Setting $Z(t) = (x(t), y(t)) \in {\mathcal H} \times {\mathcal H}$, \eqref{eq:fos2}
can be equivalently written
\begin{equation}\label{eq:fos3}
\dot{Z}(t) + \partial {\mathcal G}(Z(t)) + {\mathcal D}(t, Z(t)) \ni 0, \quad {Z(t_0)=(x_0,y_0)},
\end{equation}
where ${\mathcal G} \in \Gamma_0({\mathcal H} \times {\mathcal H})$ is the function defined by ${\mathcal G} (Z) = \beta f (x)$, and the time-dependent operator ${\mathcal D}:\ [t_0,+\infty[ \times {\mathcal H} \times {\mathcal H} \to {\mathcal H} \times {\mathcal H}$ is given by
\begin{equation}\label{eq:fos5}
{\mathcal D}(t,Z)=
\pa{\beta e(t)-\pa{\frac{1}{\beta}-\frac{\alpha}{t}}x+\frac{1}{\beta}y,
-\pa{\frac{1}{\beta}-\frac{\alpha}{t}+\frac{\alpha\beta}{t^2}}x
+\frac{1}{\beta} y } .
\end{equation}
The differential inclusion \eqref{eq:fos3} is governed by the sum of the maximal monotone operator $\partial {\mathcal G}$ (a convex subdifferential) and the time-dependent affine continuous operator ${\mathcal D} (t,\cdot)$. The existence and uniqueness of a global solution for the corresponding Cauchy problem is a consequence of the general theory of evolution equations governed by maximally monotone operators. In this setting, we need to invoke the notion of strong solution that we make precise now.
\begin{definition}\label{defsolforte}
Given $g \in \Gamma_0({\mathcal H})$, and an operator $D:[t_0,+\infty[\times {\mathcal H}\to{\mathcal H}$, we say that $z:[t_0,T]\to {\mathcal H}$
is a strong solution trajectory on $[t_0,T]$ of the differential inclusion
\begin{equation}\label{eq:fosdef}
\dot z(t)+\partial g(z(t))+D(t,z(t)) \ni 0,
\end{equation}
if the following properties are satisfied:
\begin{enumerate}[label=\arabic*.]
\item \label{defsolforte-1}
$z$ is continuous on $[t_0,T]$ and absolutely continuous on any compact subset of $]t_0,T]$;
\item \label{defsolforte-3}
$z(t)\in\dom(\partial g)$ for almost every $t\in]t_0,T]$, and
\eqref{eq:fosdef} is verified for almost every $t\in]t_0,T]$.
\end{enumerate}
\noindent $z:[t_0,+\infty[\to {\mathcal H}$ is a global strong solution of \eqref{eq:fosdef}, if it is a strong solution on $[t_0,T]$ for all $T>t_0$.
\end{definition}
The existence and uniqueness of a global strong solution of the Cauchy problem \eqref{eq:fos2} is established in the following theorem.
\begin{theorem} \label{Thm-existence}
Let $f \in \Gamma_0({\mathcal H})$, $\alpha \geq 0$ and $\beta >0$.
Suppose that $e \in L^2 (t_0, T; {\mathcal H})$ for every $T > t_0$. Then, for any Cauchy data $(x_0, y_0) \in \dom(f) \times {\mathcal H} $, there exists a unique global strong solution $(x,y):[t_0, +\infty[ \to {\mathcal H} \times {\mathcal H} $ of \eqref{eq:fos2} satisfying the initial condition $x(t_0)=x_0$, $y(t_0)= y_0$. Moreover, this solution exhibits the following properties:
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{existence-1}
$y \in {\mathcal C}^1([t_0,+\infty[)$, and
$
\dot{y}(t)
-\pa{\frac{1}{\beta}-\frac{\alpha}{t}+\frac{\alpha\beta}{t^2}}x(t)
+\frac{1}{\beta} y(t) =0,$
for $t\geq t_0$;
\item \label{existence-2}
$x$ is absolutely continuous on $[t_0,T]$ and $\dot x\in L^2(t_0,T; {\mathcal H})$ for
all $T>t_0$;
\item \label{existence-3}
$x(t)\in\dom(\partial f)$ for all $t>t_0$;
\item \label{existence-4}
$x$ is Lipschitz continuous on any compact subinterval of $]t_0,+\infty[$;
\item
the function $ t\mapsto f(x(t))$ is absolutely continuous on
$[t_0,T]$ for all $T>t_0$;
\item \label{existence-6}
there exists a function $\xi:[t_0,+\infty[\to {\mathcal H}$ such that
\begin{enumerate}[label={\rm (\alph*)}]
\item \label{existence-6:a}
$\xi(t)\in\partial f(x(t))$ for all $t>t_0$;
\item \label{existence-6:b}
$\dot x(t)+\beta\xi(t) + \beta e(t)
-\pa{\frac{1}{\beta}-\frac{\alpha}{t}}x(t)+\frac{1}{\beta}y(t)=0$
for almost every $t>t_0$;
\item \label{existence-6:c}
$\xi\in L^2(t_0,T; {\mathcal H})$ for all $T>t_0$;
\item \label{existence-6:d}
$\displaystyle{\frac{d}{dt}f(x(t))}=\langle\xi(t),\dot x(t)\rangle$ for almost every
$t>t_0$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
It is sufficient to prove that $(x,y)$ is a strong solution of \eqref{eq:fos2} on $[t_0,T]$ and that the properties hold on $[t_0,T]$ for all $T>t_0$. So let us fix $T>t_0$. As we have already noticed, \eqref{eq:fos2} can be written in the form \eqref{eq:fos3} which is a Lipschitz perturbation of the differential inclusion governed by the subdifferential of a proper lsc convex function. A direct application of \cite[Proposition~3.12]{Bre1} gives the existence and uniqueness of a strong global solution $Z=(x,y):[t_0,T]\to{\mathcal H} \times {\mathcal H}$ to \eqref{eq:fos3}, or equivalently to \eqref{eq:fos2}, with initial condition $Z(t_0)=(x(t_0),y(t_0))=(x_0,y_0)$. Verification of items \ref{existence-3} to \ref{existence-6} follows the same lines as the proof of \cite[Theorem~4.4]{APR1}. Of particular importance is the generalized derivation chain rule given in \ref{existence-6}\ref{existence-6:d}, which follows from \cite[Lemma~3.3]{Bre1} after checking that the corresponding assumptions are met thanks to \ref{existence-2}, \ref{existence-6}\ref{existence-6:a} and \ref{existence-6}\ref{existence-6:c}.\qed
\end{proof}
{
Under sufficient differentiability properties of the data,
we recover a classical solution, {\it i.e.} \; $x(\cdot)$ is a ${\mathcal C}^2([t_0, +\infty[)$ function, all the derivatives involved in the equation \eqref{eq:origode_a} are taken in the sense of classical differential calculus, and the equation \eqref{eq:origode_a} is satisfied for all $t \geq t_0$. This is made precise in the following statement.
\begin{corollary}\label{corr-existence-origode_a}
Assume that $f$ is a convex ${\mathcal C}^2({\mathcal H})$ function and $e$ belongs to ${\mathcal C}^1([t_0, +\infty[)$. Then, for any $t_0 > 0$, and any Cauchy data $(x_0, \dot{x}_0)$, the system \eqref{eq:origode_a} with $\alpha, \beta \geq 0$ admits a unique classical global solution $x: [t_0,+\infty[ \to {\mathcal H}$ satisfying $(x(t_0),\dot{x}(t_0)) = (x_0, \dot{x}_0)$.
\end{corollary}
\begin{proof}
Under the above regularity assumptions, the first equation of the first order system \eqref{eq:fos2}
$$
\dot x(t)+\beta(\nabla f(x(t)) + e(t)) -\pa{\frac{1}{\beta}-\frac{\alpha}{t}}x(t)+\frac{1}{\beta}y(t) = 0
$$
implies that $\dot x$ is a ${\mathcal C}^1([t_0, +\infty[)$ function, and hence $x\in{\mathcal C}^2([t_0, +\infty[)$.
Then, combining Theorem~\ref{Thm-first-order-system} with Theorem~\ref{Thm-existence} with $y(t_0) = -\beta(\dot x_0+\beta\nabla f(x_0))+(1-\beta\alpha/t_0)x_0 -\beta^2 e(t_0)$, we obtain the existence and uniqueness of a classical solution to the Cauchy problem associated with \eqref{eq:origode_a}.
\end{proof}
}
\subsection{Implicit Hessian Damping}
\subsubsection{Formulation as a first-order system}
Let us now turn to \eqref{eq:odetwo_a}. We use the shorthand notation $\alpha(t)=\alpha/t$. Here and in the rest of the paper, we assume that $\beta(\cdot)$ is ${\mathcal C}^1([t_0, +\infty[, {\mathbb R}^+)$ and
$\inf\limits_{t\in[t_0,+\infty[} \beta(t) > 0$.\\
Let us introduce the new function
\begin{equation}\label{eq:odetwo-2}
y(t) := x(t)+\beta(t)\dot{x}(t),
\end{equation}
whose time derivation gives
\begin{equation}\label{eq:odetwo-3}
\dot{y}(t) = \dot{x}(t)+\beta(t)\ddot{x}(t)+ \dot{\beta}(t) \dot{x}(t).
\end{equation}
From \eqref{eq:odetwo_a} we know that
\begin{equation}\label{eq:odetwo-4}
\ddot{x}(t)= -\alpha (t) \dot{x}(t)-\nabla f (y(t))-e(t).
\end{equation}
By combining \eqref{eq:odetwo-3} and \eqref{eq:odetwo-4} we obtain
\begin{align}
\dot{y}(t) &= \dot{x}(t)+\beta(t)\pa{-\alpha(t)\dot{x}(t)-\nabla f(y(t))-e(t)} + \dot{\beta}(t) \dot{x}(t) \nonumber \\
&= \pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}\dot{x}(t)-\beta(t)\pa{\nabla f(y(t) + e(t)}.\label{eq:odetwo-5}
\end{align}
From \eqref{eq:odetwo-2} and the fact that $\inf\limits_{t\in[t_0,+\infty[} \beta(t) > 0$ we get
$
\dot{x}(t) = \frac{1}{\beta (t)}(y(t) -x(t)).
$
Replacing $\dot{x}(t)$ in \eqref{eq:odetwo-5} with this expression gives
\begin{eqnarray*}
\dot{y}(t)
&=& \pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}\frac{1}{\beta (t)}(y(t) -x(t))-\beta(t)\pa{\nabla f (y(t)) +e(t)} \\
&=& -\frac{1}{\beta (t)}\pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}x(t) + \frac{1}{\beta (t)}\pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}y(t) - \beta(t)\pa{\nabla f (y(t)) + e(t)} .
\end{eqnarray*}
The reverse implication is obtained in a similar way. Let us summarize the results.
\begin{theorem}\label{Thm-first-order-system_implicit}
Let $f \in {\mathcal C}^1({\mathcal H})$. Suppose that $\alpha \geq 0$ and $\inf\limits_{t\in[t_0,+\infty[} \beta(t) > 0$. The following statements are equivalent:
\begin{enumerate}[label=\arabic*.]
\item $x:[t_0,+\infty [ \to {\mathcal H}$ is a solution trajectory of \eqref{eq:odetwo_a} with initial conditions $x(t_0)=x_0$, $\dot x(t_0)=\dot x_0$.
\item $(x,y):[t_0,+\infty [ \to {\mathcal H} \times {\mathcal H}$ is a solution trajectory of the first-order system
\begin{align}\label{eq:ISIHDequiv1stode}
\begin{cases}
\dot{x}(t) + \frac{1}{\beta(t)}x(t) - \frac{1}{\beta(t)}y(t) = 0.
\\
\dot{y}(t) + \beta(t)\pa{\nabla f (y(t)) + e(t)} + \frac{1}{\beta (t)}\pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}(x(t) - y(t)) = 0
\end{cases}
\end{align}
with initial conditions $x(t_0)=x_0$,
$y(t_0)=x_0 + \beta(t_0)\dot{x}_0$.
\end{enumerate}
\end{theorem}
\subsubsection{Existence and uniqueness of a solution}
Existence and uniqueness of a global strong solution for the {Cauchy problem associated with the} unperturbed problem \eqref{eq:odetwo} was shown in \cite{alecsa2019extension} when $\nabla f$ is Lipschitz continuous using the Cauchy-Lipschitz theorem. This result can be easily extended to \eqref{eq:odetwo_a}. Rather, we take a different path here and proceed as in Section~\ref{section-gen-existence}, so that we can extend the above formulation to the case where $f \in \Gamma_0({\mathcal H})$, by replacing the gradient $\nabla f$ with the subdifferential $\partial f$.
\begin{definition}
Let $\alpha(t) \geq 0$, $\beta (t) > 0$, $f \in \Gamma_0({\mathcal H})$. {Given
$(x_0, y_0) \in {\mathcal H} \times \dom(f) $, the Cauchy problem associated with the} perturbed inertial system with implicit generalized Hessian driven damping is defined by
\begin{align}
\begin{cases}
\dot{x}(t) + \frac{1}{\beta(t)}x(t) - \frac{1}{\beta(t)}y(t) = 0 \\
\dot{y}(t) + \beta(t)\pa{\partial f (y(t)) + e(t)} + \frac{1}{\beta (t)}\pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}(x(t) - y(t)) \ni 0 \label{eq:fos2_implicit}\\
x(t_0) = x_0, y(t_0)=y_0 .
\end{cases}
\end{align}
\end{definition}
We reformulate \eqref{eq:fos2_implicit} in the product space ${\mathcal H} \times {\mathcal H}$ by setting $Z(t) = (x(t), y(t)) \in {\mathcal H} \times {\mathcal H}$, and thus \eqref{eq:fos2_implicit} can be equivalently written as
\begin{equation}\label{eq:fos3_implicit}
\dot{Z}(t) + \beta(t) \partial {\mathcal G} (Z(t)) + {\mathcal D}(t, Z(t)) \ni 0,
\end{equation}
where ${\mathcal G} \in \Gamma_0({\mathcal H} \times {\mathcal H})$ is the function defined as
$ {\mathcal G} (Z) = f (y)$, and the time dependent operator ${\mathcal D}:\ [t_0,+\infty[ \times {\mathcal H} \times {\mathcal H} \to {\mathcal H} \times {\mathcal H}$ is given by
\begin{equation}\label{eq:fos5_implicit}
{\mathcal D}(t,Z)=
\pa{\frac{1}{\beta (t)}(x - y) ,
\beta(t)e(t) + \frac{1}{\beta (t)}\pa{1 - \alpha(t)\beta(t) + \dot{\beta}(t)}(x - y)} .
\end{equation}
\myparagraph{Constant $\beta$} When $\beta$ is independent of $t$, the differential inclusion \eqref{eq:fos3_implicit} is governed by the sum of the convex subdifferential operator $\beta \partial {\mathcal G}$ and the time-dependent affine continuous operator ${\mathcal D} (t,\cdot)$. The existence and uniqueness of a global strong solution for the Cauchy problem associated to \eqref{eq:fos2_implicit} follows exactly from the same arguments as those for Theorem~\ref{Thm-existence}. {In turn, if $f\in {\mathcal C}^1({\mathcal H})$, $e\in {\mathcal C}([t_0, +\infty[)$, and $\beta \in {\mathcal C}^1([t_0, +\infty[)$, then \eqref{eq:ISIHDequiv1stode} admits a unique ${\mathcal C}^1([t_0, +\infty[)$ global solution $(\dot{x},\dot{y})$. It then follows from the first equation in \eqref{eq:ISIHDequiv1stode} that $\dot x$ is a ${\mathcal C}^1([t_0, +\infty[)$ function, and hence $x\in{\mathcal C}^2([t_0, +\infty[)$. Existence and uniqueness of a classical global solution to the Cauchy problem associated to \eqref{eq:odetwo_a} is then obtained thanks to the equivalence in Theorem~\ref{Thm-first-order-system_implicit}.}
\myparagraph{Time-dependent $\beta$} When $\beta$ depends on time, one cannot invoke directly the results of \cite{Bre1}. Instead, one can appeal to the theory of evolution equations governed by general time-dependent subdifferentials as proposed in \cite{AD} for example. {In fact, for a system in the simpler form \eqref{eq:fos3_implicit}, one can argue more easily, by making the change of time variable $t = \tau(s)$ with $\beta(\tau(s))\dot{\tau}(s)=1$. Lemma~\ref{lem:timerescale} then shows that \eqref{eq:fos3_implicit} is equivalent to
\begin{equation}\label{eq:fos3_implicit_rescale}
\dot{W}(s) + \partial {\mathcal G} (W(s)) + {\mathcal F}(s, W(s)) \ni 0,
\end{equation}
where $W(s)= Z(\tau(s))$, and ${\mathcal F}(s,W(s))= \frac{1}{\beta(\tau(s))}{\mathcal D}(\tau(s),W(s))$ is affine continuous in its second argument. Provided that $\beta \not\in L^1(t_0,+\infty;{\mathbb R})$, this defines a proper change of variable in time}. With the formulation \eqref{eq:fos3_implicit_rescale}, we are brought back to the appropriate form to argue as before and invoke the results of \cite{Bre1}. We leave the details to the reader for the sake of brevity.
\section{Smooth Convex Case}\label{s:convex}
\subsection{Explicit Hessian Damping}\label{s:secondorderconv}
Consider first the explicit Hessian system~\eqref{eq:origode_a}, where we assume that $f \in {\mathcal C}^2({\mathcal H})$, and recall the specific choices of $\gamma(t)=\frac{\alpha}{t}$, $\alpha > 0$, $\beta(t)\equiv\beta$ and $b(t)\equiv 1$.
We will develop a Lyapunov analysis to study the dynamics of~\eqref{eq:origode_a}. Some of our arguments are inspired by the works of~\cite{attouch2018fast} and \cite{attouch2019first}. Throughout this section we use the shorthand notation
\begin{equation}\label{def_g}
g(t):= e(t) + \beta \dot{e}(t)
\end{equation}
{for the overall contribution of the errors terms. We will first establish the minimization property which is valid by simply assuming the integrability of the error term and its derivative. Then, by reinforcing these hypotheses, we will obtain rapid convergence results, and the convergence of trajectories.}
\subsubsection{Minimizing properties}
{Define $u: [t_0,+\infty[ \to {\mathcal H}$ by
\[
u(t) := x(t) + \beta \int_{t_0}^t \nabla f(x(s)) ds,
\]
which will be instrumental in the proof of the following theorem. Note that, in the following statement, it is simply assumed that $f$ is bounded from below, the set $S := \argmin_{{\mathcal H}} f$ may be empty.
\begin{theorem}\label{lem:smoothexplicitestimates}
Let $f: {\mathcal H} \to {\mathbb R}$ be a ${\mathcal C}^2({\mathcal H})$ function which is bounded from below. Assume that $e: [t_0,+\infty[ \to {\mathcal H}$ is a ${\mathcal C}^1({\mathcal H})$ function which satisfies the integrability properties $\DS{\int_{t_0}^{+\infty} \norm{e(t)}dt<+\infty}$ and $\DS{\int_{t_0}^{+\infty} \norm{\dot{e}(t)}dt<+\infty}$. Suppose that $\alpha,\beta>0$. Then, for any solution trajectory $x: [t_0,+\infty[ \to {\mathcal H}$ of \eqref{eq:origode_a}, we have
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{lem:smoothexplicitestimates1}
$\sup\limits_{t \geq t_0} \norm{\dot{u}(t)}<+\infty$;
\item \label{lem:smoothexplicitestimates2}
$\DS{\int_{t_0}^{+\infty} \frac{1}{t} \|\dot{x}(t)\|^2 dt < +\infty}$, $\DS{\int_{t_0}^{+\infty} \norm{\nabla f(x(t))}^2 dt < +\infty}$, $\DS{\int_{t_0}^{+\infty} \frac{1}{t} \|\dot{u}(t)\|^2 dt < +\infty}$;
\item \label{lem:smoothexplicitestimates3}
$\lim\limits_{t\to+ \infty} \norm{\dot{u}(t)}=0$; \, $\lim\limits_{t\to+ \infty} \norm{\dot{x}(t)}=0$; $\lim\limits_{t\to+ \infty} \norm{\nabla f(x(t))}=0$;
\item \label{lem:smoothexplicitestimates4}
$\lim\limits_{t\to +\infty} f(x(t))=\inf_{{\mathcal H}} f$.
\end{enumerate}
\end{theorem}
}
\begin{proof}
Recall $\bar{f} := \inf_{{\mathcal H}} f$. Since our analysis is asymptotic, {there is no restriction in assuming} that $t \geq t_1 := \max(t_0,2\alpha\beta)$. We will then prove the statements in terms of $t_1$ and passing to $t_0$ is immediate thanks to the properties of the solution $x(t)$ in Theorem~\ref{Thm-existence}.
\paragraph{Claim~\ref{lem:smoothexplicitestimates1}}
For $T \geq t \geq t_1$, define the function
\[
W_T(t) := \frac{1}{2}\norm{\dot{u}(t)}^2 + \pa{f(x(t))-\bar{f}} - \int_t^T \dotp{\dot{u}(\tau)}{g(\tau)} d\tau .
\]
Observe that $W_T$ is well-defined under our assumptions. Thus, taking the derivative in time and using \eqref{eq:origode_a}, we get,
\begin{eqnarray}
\dot{W}_T(t)
&=& \dotp{\dot{u}(t)}{\ddot{u}(t)+g(t)} + \dotp{\dot{x}(t)}{\nabla f(x(t))} \nonumber\\
&=& \dotp{\dot{x}(t)+\beta \nabla f(x(t))}{\ddot{x}(t)+\beta\nabla^2 f(x(t))\dot{x}(t)+g(t)} + \dotp{\dot{x}(t)}{\nabla f(x(t))} \nonumber\\
&=& \dotp{\dot{x}(t)+\beta \nabla f(x(t))}{-\frac{\alpha}{t}\dot{x}(t) -\nabla f(x(t))} + \dotp{\dot{x}(t)}{\nabla f(x(t))} \nonumber\\
&=&-\frac{\alpha}{t} \norm{\dot{x}(t)}^2 - \beta \norm{\nabla f(x(t))}^2 - \frac{\alpha\beta}{t}\dotp{\dot{x}(t)}{\nabla f(x(t))} \nonumber\\
&\leq& -\frac{\alpha}{2t} \norm{\dot{x}(t)}^2 - \beta\pa{1-\frac{\alpha\beta}{2t}} \norm{\nabla f(x(t))}^2 \nonumber\\
&\leq& -\frac{\alpha}{2t} \norm{\dot{x}(t)}^2 - \frac{\beta}{2} \norm{\nabla f(x(t))}^2 , \label{eq:WTbound}
\end{eqnarray}
where we used Young inequality and the fact that $t \geq t_1 > \alpha\beta$. This implies that $W_T$ is non-increasing and in turn that $W_T(t)\le W_T(t_1)$ for $t \in [t_1,T]$, {\it i.e.}
\begin{equation*}
\frac{1}{2}\norm{\dot{u}(t)}^2+\pa{f(x(t))-\bar{f}}-\int_t^T \dotp{\dot{u}(\tau)}{g(\tau)} d\tau
\le \frac{1}{2} \|\dot{u}(t_1)\|^2+\pa{f(x(t_1))-\bar{f}}-\int_{t_1}^T \dotp{\dot{u}(\tau)}{g(\tau)} d\tau .
\end{equation*}
Therefore,
\[
\frac{1}{2}\norm{\dot{u}(t)}^2 \le \frac{1}{2}\|\dot{u}(t_1)\|^2+\pa{ f(x(t_1))-\bar{f}}+\int_{t_1}^t \|\dot{u}(\tau)\|\|g(\tau)\|d\tau .
\]
{Applying the Gronwall} Lemma~\ref{lem:BrezisA5}, we get
\[
\sup\limits_{t \geq t_1} \norm{\dot{u}(t)}\le \pa{\|\dot{u}(t_1)\|^2+2(f(x(t_1))-\bar{f})}^{1/2}+\int_{t_1}^{+\infty} \|e(\tau)\| d\tau + \beta\int_{t_1}^{+\infty} \|\dot{e}(\tau)\| d\tau < +\infty ,
\]
hence proving the first claim.
\paragraph{Claim~\ref{lem:smoothexplicitestimates2}}
Now define {for all $t\geq t_1$ }
\begin{align*}
W(t) := \frac{1}{2} \norm{\dot{u}(t)}^2+(f(x(t))-\bar{f}) - \int_t^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)} d\tau .
\end{align*}
This is again a well-defined function thanks to the first claim,
{and the integrability of $g$}. Moreover, $W(\cdot )$ is bounded from below,
\begin{equation}\label{W_minorized}
\inf_{t \geq t_1} W(t) \ge -\brac{\sup\limits_{t \geq t_1} \norm{\dot{u}(t)}} \brac{\int_{t_1}^{+\infty} \|e(\tau)\|d\tau+\beta\int_{t_1}^{+\infty} \|\dot{e}(\tau)\|d\tau} > -\infty .
\end{equation}
Observe that $\dot{W}(t)=\dot{W}_T(t)$. This together with \eqref{eq:WTbound} yields
\begin{equation}\label{W_decrease}
\dot{W}(t) + \frac{\alpha}{2t} \norm{\dot{x}(t)}^2 + \frac{\beta}{2} \norm{\nabla f(x(t))}^2 \leq 0 .
\end{equation}
{Integrating and using that $W$ is bounded from below, we obtain the first two claims. From $\dot{u}(t)= \dot{x}(t) + \beta \nabla f(x(t))$, we deduce that $\|\dot{u}(t)\|^2 \leq 2( \|\dot{x}(t)\|^2 + \beta^2 \| \nabla f(x(t))\|^2 )$. After integration,
we get the last claim
\begin{align*}
\int_{t_1}^{+\infty} \frac{1}{t} \|\dot{u}(t)\|^2 dt
&\leq 2 \pa{\int_{t_1}^{+\infty} \frac{1}{t} \|\dot{x}(t)\|^2 dt + \int_{t_1}^{+\infty} \frac{\beta^2}{t} \|\nabla f(x(t))\|^2 dt} \\
&\leq 2\pa{\int_{t_1}^{+\infty} \frac{1}{t} \|\dot{x}(t)\|^2 dt + \frac{\beta^2}{t_1} \int_{t_1}^{+\infty} \|\nabla f(x(t))\|^2 dt} < +\infty .
\end{align*}}
\paragraph{Claim~{\ref{lem:smoothexplicitestimates3}} and~\ref{lem:smoothexplicitestimates4}}
Define $h: t \in [t_1,+\infty[ \mapsto \frac{1}{2}\|u(t)-z\|^2$ for arbitrary $z\in{\mathcal H}$. We then have
\begin{align*}
\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t)
&= \norm{\dot{u}(t)}^2 + \dotp{u(t)-z}{\ddot{u}(t)+\frac{\alpha}{t}\dot{u}(t)} \\
&= \norm{\dot{u}(t)}^2 + \dotp{u(t)-z}{\ddot{x}(t)+\beta\nabla^2 f(x(t))\dot{x}(t) + \frac{\alpha}{t}\dot{x}(t)+\frac{\alpha\beta}{t}\nabla f(x(t))} \\
&= \norm{\dot{u}(t)}^2 - \dotp{u(t)-z}{g(t)+\pa{1-\frac{\alpha\beta}{t}}\nabla f(x(t))} \\
&= \norm{\dot{u}(t)}^2 - \pa{1-\frac{\alpha\beta}{t}}\dotp{x(t)-z}{\nabla f(x(t))} - \dotp{u(t)-z}{g(t)} \\
&- \beta\pa{1-\frac{\alpha\beta}{t}}\dotp{\int_{t_1}^t \nabla f(x(s)) ds}{\nabla f(x(t))} \\
&= \norm{\dot{u}(t)}^2 - \pa{1-\frac{\alpha\beta}{t}}\dotp{x(t)-z}{\nabla f(x(t))} - \dotp{u(t)-z}{g(t)} - \beta\pa{1-\frac{\alpha\beta}{t}} \dot{I}(t) ,
\end{align*}
where $\DS{I(t) := \frac{1}{2}\norm{\int_{t_1}^t \nabla f(x(s)) ds}^2}$. {From the convexity of $f$ and Cauchy-Schwarz inequality we get}
\begin{equation*}
\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t) + \pa{1-\frac{\alpha\beta}{t}}\pa{f(x(t))-f(z)} + \beta\pa{1-\frac{\alpha\beta}{t}}\dot{I}(t)
\leq \norm{\dot{u}(t)}^2 + \norm{u(t)-z}\norm{g(t)} .
\end{equation*}
Inserting $W(t)$ into this expression, we get,
\begin{eqnarray}
&&\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t) + \pa{1-\frac{\alpha\beta}{t}}\pa{W(t)+ \bar{f} -f(z)} + \beta\pa{1-\frac{\alpha\beta}{t}}\dot{I}(t) \nonumber \\
&& \leq \pa{\frac{3}{2}-\frac{\alpha\beta}{2t}}\norm{\dot{u}(t)}^2
+ \norm{u(t)-z}\norm{g(t)} - \pa{1-\frac{\alpha\beta}{t}} \int_{t}^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)} d\tau .\label{basic-est-1}
\end{eqnarray}
According to \eqref{W_decrease} and \eqref{W_minorized} $W(\cdot)$ is nonincreasing and bounded from below. Therefore, it converges to some $W_\infty \in {\mathbb R}$ as $t \to +\infty$.
{Since $\dot{u}$ is bounded, and $g$ is integrable, we have that $\tau \to \dotp{\dot{u}(\tau)}{g(\tau)} $ is integrable on $[t_0, +\infty[$. Therefore
$$
\lim_{t\to +\infty} \int_t^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)} d\tau =0.
$$
By definition of $W(t)$, this implies that, as $t \to +\infty$
$$
\frac{1}{2} \norm{\dot{u}(t)}^2+(f(x(t))-\bar{f}) \to W_\infty.
$$
If $W_\infty =0$, since the
two terms that enter the above expression (potential energy and kinetic energy) are nonnegative, we obtain that each of them tends to zero as $t \to +\infty$.
This gives the claims ~{\ref{lem:smoothexplicitestimates3}} and~\ref{lem:smoothexplicitestimates4}.
To prove that $W_\infty =0$, we argue by contradiction, and show that assuming $W_\infty >0$ leads to a contradiction. Since $W(\cdot)$ is nonincreasing, we then have
$W(t) \geq W_\infty >0$. Take $z\in {\mathcal H}$ such that $f(z) < \bar{f} + \demi W_\infty$. Then
$$
W(t)+ \bar{f} -f(z) > W_\infty - \demi W_\infty = \demi W_\infty.
$$
Returning to (\ref{basic-est-1}) we deduce that, for $t\geq t_1$
\begin{eqnarray}
&&\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t) + \demi\pa{1-\frac{\alpha\beta}{t}} W_\infty + \beta\pa{1-\frac{\alpha\beta}{t}}\dot{I}(t) \nonumber \\
&& \leq \pa{\frac{3}{2}-\frac{\alpha\beta}{2t}}\norm{\dot{u}(t)}^2
+ \norm{u(t)-z}\norm{g(t)} - \pa{1-\frac{\alpha\beta}{t}} \int_{t}^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)} d\tau .\label{basic-est-2}
\end{eqnarray}
Since $t>2\alpha\beta$, we have $1-\frac{\alpha\beta}{t} >\demi$. Therefore, after rearranging the terms in \eqref{basic-est-2}, we obtain
$$
\frac{1}{4} W_{\infty}
\le \frac{3}{2}\norm{\dot{u}(t)}^2+\norm{u(t)-z}\norm{g(t)}
+\pa{\sup\limits_{t\ge t_1} \norm{\dot{u}(t)}}\int_t^{+\infty} \|g(s)\|ds -\frac{1}{t^\alpha}\frac{d}{dt}(t^\alpha \dot{h}(t))
-\beta\pa{1-\frac{\alpha\beta}{t}}\dot{I}(t).
$$
}
Multiplying both sides by $\frac{1}{t}$, and integrating between $t_1$ and $\tau > t_1$,
\begin{eqnarray}
\frac{1}{4} W_{\infty} \log\pa{\frac{\tau}{t_1}}
&\le& \frac 32\int_{t_1}^\tau \frac{1}{t}\norm{\dot{u}(t)}^2 dt +\int_{t_1}^\tau\frac{\|g(t)\|\|u(t)-z\|}{t}dt
+ \pa{\sup\limits_{t\ge t_1} \norm{\dot{u}(t)}}\int_{t_1}^\tau \pa{\frac1t\int_t^{+\infty}\|g(s)\|ds} dt \nonumber \\
&&- \int_{t_1}^\tau \frac{1}{t^{\alpha+1}}\frac{d}{dt}(t^\alpha \dot{h}(t)) dt
- \beta\int_{t_1}^\tau \pa{\frac{1}{t}-\frac{\alpha\beta}{t^2}}\dot{I}(t) dt . \label{eq:binf}
\end{eqnarray}
Throughout the rest of the proof, we will use the inequality
\[
\norm{u(t)-z} \leq \norm{u(t_1)-z} + \int_{t_1}^t \norm{\dot{u}(s)} ds \leq \norm{u(t_1)-z} + t \sup_{s\ge t_1} \norm{\dot{u}(s)} .
\]
{Let us examine successively the different terms which enter the second member of \eqref{eq:binf}. The first term is bounded according to claim~{\ref{lem:smoothexplicitestimates2}}}. The second term is also bounded since
\[
\int_{t_1}^\tau\frac{\|g(t)\|\|u(t)-z\|}{t}dt \le \pa{\frac{\|u(t_1)-z\|}{t_1}+\sup\limits_{t \ge t_1} \norm{\dot{u}(t)}}\int_{t_1}^{+\infty}
\pa{\|e(t)\|+\beta\|\dot{e}(t)\|}dt < +\infty .
\]
The third term can be handled by integration by parts,
\begin{equation*}
\int_{t_1}^\tau \pa{\frac 1t\int_t^{+\infty}\|g(s)\|ds} dt
=\log\tau\int_\tau^\infty \|g(s)\| ds\
-\log t_1 \int_{t_1}^{+\infty} \|g(s)\| ds
+\int_{t_1}^\tau \|g(t)\|\log t ~ dt .
\end{equation*}
For the fourth term, set $\DS{K(\tau)=-\int_{t_1}^\tau \frac{1}{t^{\alpha+1}}\frac{d}{dt}(t^\alpha \dot{h}(t))dt}$ and integrate by parts twice to get,
\begin{align*}
K(\tau)&=-\brac{\frac{1}{t}\dot{h}(t)}_{t_1}^\tau-(\alpha+1)\int_{t_1}^\tau \frac{1}{t^2}\dot{h}(t)dt
= -\brac{\frac{1}{t}\dot{h}(t)}_{t_1}^\tau-\frac{(1+\alpha)}{\tau^2} h(\tau) + \frac{(1+\alpha)}{t_1^2} h(t_1)
-2(1+\alpha)\int_{t_1}^\tau \frac{h(t)}{t^3} dt \\
&\leq -\brac{\frac{1}{t}\dot{h}(t)}_{t_1}^\tau + \frac{(1+\alpha)}{t_1^2} h(t_1)
\leq \frac{1}{t_1}\abs{\dotp{\dot{u}(t_1)}{u(t_1)-z}}+\frac{1}{\tau}\abs{\dotp{\dot{u}(\tau)}{u(\tau)-z}} + \frac{(1+\alpha)}{t_1^2} h(t_1) \\
&\leq C+ \sup\limits_{t\ge t_1}\norm{\dot{u}(t)}\pa{\frac{\|u(t_1)-z\|}{t_1}+\sup\limits_{t \ge t_1} \norm{\dot{u}(t)}} < +\infty.
\end{align*}
For the last term, we infer from Lemma~\ref{L:int_bounded} that
\[
\sup_{\tau > t_1} -\beta\int_{t_1}^\tau \pa{\frac{1}{t}-\frac{\alpha\beta}{t^2}}\dot{I}(t) dt < +\infty .
\]
Overall, we have shown that there exists a constant $C > 0$ such that \eqref{eq:binf} reads
\[
\frac{1}{4} W_{\infty} \log\pa{\frac{\tau}{t_1}} \le C+\pa{\sup\limits_{t\ge t_1}\norm{\dot{u}(t)}}\pa{\log \tau\int_\tau^\infty\|g(s)\| ds+\int_{t_1}^\tau \|g(t)\| \log t dt} .
\]
Observe that $\lim_{\tau \to +\infty} \int_\tau^\infty\|g(s)\| ds = 0$ since $g$ is integrable. Then, divide the last inequality by $\log\pa{\frac{\tau}{t_1}}$ and let $\tau\to\infty$. According to Lemma~\ref{basic-int}, we get that $W_\infty \le 0$. Thus $W_\infty=0$, hence the contradiction.
{The last statement, $\lim\limits_{t\to+ \infty} \norm{\dot{x}(t)}=0$, is obtained by following an argument similar to the one above, which now uses the perturbed version of the classical energy function, namely
\[
W_0 (t) := \frac{1}{2} \norm{\dot{x}(t)}^2+(f(x(t))-\bar{f}) - \int_t^{+\infty} \dotp{\dot{x}(\tau)}{g(\tau)} d\tau .
\]
We do not detail this proof for the sake of brevity.
Then, according to $\nabla f (x(t)) = \frac{1}{\beta}\left(\dot{u}(t)-\dot{x}(t)\right)$, we obtain the convergence of $\nabla f (x(t))$ to zero.
}
\qed
\end{proof}
\begin{remark}
The result in \cite[Lemma~4.1]{attouch2018fast} is a particular case of Theorem~\ref{lem:smoothexplicitestimates} when $\beta=0$. Theorem~\ref{lem:smoothexplicitestimates} is also a generalization of \cite[Theorem~1.3 and Proposition~1.5]{APR1} to the perturbed case. Performing this generalization necessitates a different Lyapunov function together with particular novel estimates in the course of derivation.
\end{remark}
\subsubsection{Fast convergence rates}
We now move on to showing fast convergence of the objective. For this, we will need to strengthen the integrability assumption on the errors. We will denote in this section the two functions
\begin{equation}\label{def:w_delta}
w(t) := 1-\frac{\beta}{t} \enskip \text{and} \enskip \delta(t) := t^2 w(t) .
\end{equation}
{
\begin{theorem}\label{fast_conv_smooth}
Let $f \in {\mathcal C}^2({\mathcal H})$, $e\in{\mathcal C}^1([t_0, +\infty[, {\mathcal H})$. Suppose that the damping parameters satisfy $\alpha > 3$, $\beta > 0$. Suppose that $\DS{\int_{t_0}^{+\infty} t \|e(t)\| dt<+\infty}$ and $\DS{\int_{t_0}^{+\infty} t \|\dot e(t)\| dt<+\infty}$. Then, for any solution trajectory $x$ of~\eqref{eq:origode_a} the following holds:
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{item:fast_conv_smooth1}
$
\DS{f(x(t))-\inf_{{\mathcal H}} f = o\pa{\frac{1}{t^2}}} \mbox{ as } t \to +\infty.
$
\item \label{item:fast_conv_smooth2}
$
\DS{\int_{t_0}^{+\infty} t^2 \|\nabla f(x(t))\|^2 dt < +\infty} .
$
\item \label{item:fast_conv_smooth3}
$
\DS{\int_{t_0}^{+\infty} t\pa{f(x(t))-\min_{{\mathcal H}} f} dt < +\infty} .
$
\item \label{item:fast_conv_smooth4a}
for any $x^\star \in S$,
$
\DS{\int_{t_0}^{+\infty} t\dotp{\nabla f(x(t))}{x(t)-x^\star} dt < +\infty} .
$
\item \label{item:fast_conv_smooth4b}
$
\DS{\int_{t_0}^{+\infty} t\|\dot{x}(t)\|^2 dt < +\infty}.
$
\item \label{item:fast_conv_smooth4c}
$
\DS{\norm{\dot{x}(t)} = o\pa{t^{-1}}} \mbox{ as } t \to +\infty.
$
\end{enumerate}
\end{theorem}
}
\begin{proof}
{Following an argument similar to that of Theorem \ref{lem:smoothexplicitestimates}, since our analysis is asymptotic, there is no restriction in assuming that $t_0 > \frac{\beta(\alpha-2)}{\alpha-3}$.
This gives $w(t) \geq \frac{1}{\alpha-2} > 0$ for all $t \geq t_0$.} Define,
\[
v(t):= (\alpha-1)(x(t)-x^\star)+t(\dot x(t)+\beta\nabla f(x(t))).
\]
Take $T >t_0$, and define for all $t_0 \leq t \leq T$
\[
{\mathcal E}(t) := \delta(t)(f(x(t))-f(x^\star))+\frac{1}{2}\|v(t)\|^2-\int_t^T \tau\dotp{v(\tau)}{g(\tau)} d\tau.
\]
This is a well-defined differentiable function. Taking its derivative in time yields
\[
\dot{{\mathcal E}}(t)=\dot{\delta}(t)(f(x(t))-f(x^\star))+\delta(t)\dotp{\nabla f(x(t))}{\dot{x}(t)}+\dotp{v(t)}{\dot{v}(t) + t g(t)} .
\]
From \eqref{eq:origode_a}, we have,
\begin{align*}
\dot{v}(t)
&=\alpha \dot{x}(t)+\beta \nabla f(x(t))+t\pa{\ddot{x}(t)+\beta \nabla^2 f(x(t))\dot{x}(t)} \\
&= \alpha \dot{x}(t)+\beta \nabla f(x(t))+t\pa{-\frac{\alpha}{t}\dot{x}(t)-\nabla f(x(t))-g(t)} \\
&= -t\pa{1-\frac{\beta}{t}}\nabla f(x(t))-t g(t) .
\end{align*}
{Let us inject this expression into the {scalar} product $\dotp{v(t)}{\dot{v}(t) + t g(t)}$. After developing and rearranging, and taking into account the definition of $w$ and $\delta$ (see \eqref{def:w_delta}), we obtain}
\begin{equation}\label{eq:dotExp}
\dot{{\mathcal E}}(t) = \dot{\delta}(t) (f(x(t))-f(x^\star)) - (\alpha-1)t w(t) \dotp{\nabla f(x(t))}{x(t)-x^\star} - \beta\delta(t)\|\nabla f(x(t))\|^2 .
\end{equation}
Convexity of $f$ then yields
\begin{equation}\label{eq:dotEbnd}
\dot{{\mathcal E}}(t)+\beta\delta(t)\|\nabla f(x(t))\|^2+\pa{(\alpha-1)t w(t)-\dot{\delta}(t)} (f(x(t))-f(x^\star))\le 0 .
\end{equation}
{By assumption on the parameters, we have for any $t \geq t_0 > \frac{\beta(\alpha-2)}{\alpha-3}$,
\begin{equation}\label{eq:dotEbnd_bis}
(\alpha-1)t w(t)-\dot{\delta}(t)=t\left ((\alpha-3) -\frac{\beta}{t}(\alpha-2) \right)
\geq ct
\end{equation}
with $c= (\alpha-3) -\frac{\beta}{t_0}(\alpha-2) > 0$. Thefore,} \eqref{eq:dotEbnd} implies that ${\mathcal E}$ is non-increasing. In turn, {by definition of ${\mathcal E}$}
\begin{equation}\label{eq:Ebnd}
\delta(t)\left(f(x(t))-f(x^\star)\right)+\frac{1}{2} \|v(t)\|^2 \le C - \int_{t_0}^t \tau \dotp{v(\tau)}{g(\tau)}d\tau
\end{equation}
with $C=\delta(t_0)(f(x_0)-f(x^\star))+\frac{1}{2}\|v(t_0)\|^2$.
{ The above argument is valid for all $t_0 \leq t \leq T$ and arbitrary $T$, therefore for all $ t \geq t_0$.
Neglecting the nonnegative term $\delta(t)\left(f(x(t))-f(x^\star)\right)$, we infer from \eqref{eq:Ebnd} }
\[
\frac{1}{2}\|v(t)\|^2\le C+\int_{t_0}^t \|v(\tau)\| \pa{\tau\|g(\tau)\|} d\tau .
\]
Lemma~\ref{lem:BrezisA5} {(Gronwall lemma)} then gives
\begin{equation}\label{eq:boundvexplicit}
\|v(t)\| \le \pa{2C}^{1/2}+\int_{t_0}^t \tau \pa{\|e(\tau)\|+\beta \|\dot{e}(\tau)\|} d\tau ,
\end{equation}
and thus
\begin{equation}\label{eq:supboundv}
\sup\limits_{t \ge t_0} \|v(t)\|<+\infty.
\end{equation}
{By reinjecting this inequality into \eqref{eq:Ebnd} we obtain}
\begin{equation}\label{eq:boundobjexplicit}
\frac{t^2}{\alpha-2}\pa{f(x(t))-f(x^\star)} \leq \delta(t)\pa{f(x(t))-f(x^\star)}
\leq C + \sup\limits_{t\ge t_0} \|v(t)\| \int_{t_0}^t \pa{\tau\|e(\tau)\|+\beta \tau\|\dot{e}(\tau)\|}d\tau < +\infty
\end{equation}
hence proving
$$
\DS{f(x(t))-\inf_{{\mathcal H}} f = {\mathcal O} \pa{\frac{1}{t^2}}} \mbox{ as } t \to +\infty.
$$
{We will see a little later how to refine this estimate and go from a capital ${\mathcal O}$ to a small $o$ to prove the statement {\ref{item:fast_conv_smooth1}}.
Then}, integrating~\eqref{eq:dotEbnd}, and using the fact that ${\mathcal E}(t)$ is bounded from below by \eqref{eq:supboundv} and the assumptions on the errors, we get
\[
\beta \int_{t_0}^{+\infty} t^2 w(t)\|\nabla f(x(t))\|^2 dt \leq C ,
\]
and
\[
c \int_{t_0}^{+\infty} t (f(x(t))-f(x^\star)) dt \leq \int_{t_0}^{+\infty} \pa{(\alpha-1)tw(t)+\dot{\delta}(t)} (f(x(t))-f(x^\star))dt \leq C ,
\]
for some constant $C > 0$. This shows the integral estimates {\ref{item:fast_conv_smooth2}} and {\ref{item:fast_conv_smooth3}}.
\noindent Let us turn to statement
~\ref{item:fast_conv_smooth4a}. We embark from \eqref{eq:dotExp} to write, for some $\rho \in ]0,1[$ to be chosen shortly,
\begin{align*}
&\dot{{\mathcal E}}(t)
= \dot{\delta}(t) (f(x(t))-f(x^\star)) - (1-\rho)(\alpha-1)t w(t) \dotp{\nabla f(x(t))}{x(t)-x^\star} \\
&\qquad\qquad - \rho(\alpha-1)t w(t) \dotp{\nabla f(x(t))}{x(t)-x^\star} - \beta\delta(t)\|\nabla f(x(t))\|^2 \\
&\leq -\pa{(1-\rho)(\alpha-1)t w(t)-\dot{\delta}(t)} (f(x(t))-f(x^\star)) - \rho(\alpha-1)t w(t) \dotp{\nabla f(x(t))}{x(t)-x^\star} - \beta\delta(t)\|\nabla f(x(t))\|^2.
\end{align*}
To conclude, it remains to check that $\pa{(1-\rho)(\alpha-1)t w(t)-\dot{\delta}(t)}$ is non-negative.
{Since $t_0 > \frac{\beta(\alpha-2)}{\alpha-3}$, we deduce by a continuity argument the existence of some $\varepsilon >0$ such that
$t_0 > \frac{\beta(\alpha-2-\varepsilon)}{\alpha-3-\varepsilon}$.}
Then, take $\rho=\varepsilon/(\alpha-1) \in ]0,1[$. In view of the assumption on the parameters, we have
\begin{multline*}
(1-\rho)(\alpha-1)t w(t)-\dot{\delta}(t)
= (\alpha-1-\varepsilon)t w(t) - \dot{\delta}(t) = t((\alpha-3-\varepsilon) w(t) - t\dot{w}(t)) \\
= t\pa{(\alpha-3-\varepsilon) - \frac{(\alpha-2-\varepsilon)\beta}{t}} \geq t_0\pa{(\alpha-3-\varepsilon) - \frac{(\alpha-2-\varepsilon)\beta}{t_0}} \geq 0.
\end{multline*}
For claim~\ref{item:fast_conv_smooth4b}, we multiply~\eqref{eq:origode_a} by $t^2\dot{x}(t)$ to get
\begin{equation*}
t^2 \dotp{\ddot{x}(t)}{\dot{x}(t)}+\alpha t\norm{\dot{x}(t)}^2 + t^2 \beta \dotp{\dot{x}(t)}{\nabla^2 f(x(t))\dot{x}(t)} + t^2 \dotp{\nabla f(x(t))}{\dot{x}(t)} + t^2\dotp{g(t)}{\dot{x}(t)} = 0 .
\end{equation*}
With the chain rule, Cauchy-Schwarz inequality and convexity of $f$, we obtain
\begin{equation}\label{eq:derivodea}
\frac{1}{2}t^2 \frac{d}{dt} \norm{\dot{x}(t)}^2+\alpha t \norm{\dot{x}(t)}^2
+ t^2 \frac{d}{dt} (f(x(t))-\bar{f}) \le \|tg(t)\|\|t\dot{x}(t)\| .
\end{equation}
Integrating by parts on $[t_0,t]$ we get,
\begin{equation}\label{eq:mainbound}
\frac{t^2}{2}\norm{\dot{x}(t)}^2+(\alpha-1)\int_{t_0}^t s\|\dot{x}(s)\|^2 ds
\le C_0+2\int_{t_0}^t s(f(x(s))-\bar{f}) ds+\int_{t_0}^t \|s g(s)\|\|s \dot{x}(s)\|ds
\end{equation}
for some non-negative constant $C_0$, where we have used claim~{\ref{item:fast_conv_smooth1}} of Theorem~\ref{fast_conv_smooth}. Now by claim~{\ref{item:fast_conv_smooth3}} of Theorem~\ref{fast_conv_smooth} and ignoring the non-negative terms since $\alpha > 1$, we obtain
\begin{equation*}
\frac{1}{2}\norm{t\dot{x}(t)}^2 \le C_1 + \int_{t_0}^t \|s g(s)\|\|s \dot{x}(s)\|ds ,
\end{equation*}
for another non-negative constant $C_1$. {Applying Lemma~\ref{lem:BrezisA5} again then gives}
\begin{equation}\label{eq:boundtdotx}
\sup_{t \geq t_0} t\|\dot{x}(t)\| < +\infty .
\end{equation}
Using this in \eqref{eq:mainbound}, {we also get that}
\begin{equation}\label{eq:intbounds}
\begin{aligned}
\int_{t_0}^{+\infty} t\|\dot{x}(t)\|^2 dt < +\infty .
\end{aligned}
\end{equation}
We finally turn to statement~\ref{item:fast_conv_smooth4c}. We embark from \eqref{eq:derivodea}, use \eqref{eq:boundtdotx}, and integrate on $[s,t]$ to see that
\begin{multline*}
t^2\pa{\frac{1}{2}\norm{\dot{x}(t)}^2+(f(x(t))-\bar{f})} - s^2\pa{\frac{1}{2}\norm{\dot{x}(s)}^2+(f(x(s))-\bar{f})} \\
+ (\alpha-1)\int_{s}^t \tau\|\dot{x}(\tau)\|^2 d\tau - 2\int_{s}^t \tau(f(x(\tau))-\bar{f}) d\tau - C\int_{s}^t \|\tau g(\tau)\|d\tau \leq 0,
\end{multline*}
where $C = \sup_{t \geq t_0} t\|\dot{x}(t)\|$. This means that the function
\begin{equation*}
{\mathcal G}(t) = t^2\pa{\frac{1}{2}\norm{\dot{x}(t)}^2+(f(x(t))-\bar{f})}
+ (\alpha-1)\int_{t_0}^t \tau\|\dot{x}(\tau)\|^2 d\tau
- 2\int_{t_0}^t \tau(f(x(\tau))-\bar{f}) d\tau - C\int_{t_0}^t \|\tau g(\tau)\|d\tau
\end{equation*}
is non-increasing on $[t_0,+\infty[$. Since it is bounded from below by assumption on the errors and claim~{\ref{item:fast_conv_smooth3}}, $\lim_{t \to +\infty} {\mathcal G}(t)$ exists. This together with assertions~{\ref{item:fast_conv_smooth3} }and \ref{item:fast_conv_smooth4b} shows that the limit
\[
0 \leq L := \lim_{t \to +\infty} t^2\pa{\frac{1}{2}\norm{\dot{x}(t)}^2+(f(x(t))-\bar{f})}
\]
exists. Suppose that $L > 0$. Then, there exists $s \geq t_0$ such that
\begin{equation*}
\int_{t_0}^{+\infty}\pa{\frac{t}{2}\norm{\dot{x}(t)}^2+t(f(x(t))-\bar{f})} dt \geq \int_{s}^{+\infty}t^2\pa{\frac{1}{2}\norm{\dot{x}(t)}^2+(f(x(t))-\bar{f})} t^{-1} dt
\geq \int_{s}^{+\infty}\frac{L}{2t} dt = +\infty ,
\end{equation*}
leading to a contradiction with claims {\ref{item:fast_conv_smooth3}} and \ref{item:fast_conv_smooth4b}. {This proves ~\ref{item:fast_conv_smooth4c} and completes the proof of \ref{item:fast_conv_smooth1} with small $o$ instead of capital $\mathcal O$}.\qed
\end{proof}
\begin{remark}
The first three claims (resp. fourth claim) of Theorem~\ref{fast_conv_smooth} are a non-trivial generalization of \cite[Theorem~3]{attouch2019first} (resp. \cite[Theorem~2.1]{attouchiterates2021}) to the perturbed case. The presence of perturbations necessitates a careful analysis of several bounds and new estimates to handle the presence of errors and eventually preserve the convergence rates.
\end{remark}
\begin{remark}
The choice of the viscous damping parameter $\alpha$ is important for optimality of the convergence rates obtained. For the subcritical case $\alpha \leq 3$, $\beta=0$ and $e \equiv 0$, it has been shown by \cite{AAD} and \cite{ACR-subcrit} that the convergence rate of the objective values is $\displaystyle{{\mathcal O}\pa{t^{-\frac{2\alpha}{3}}}}$, and these rates are optimal, that is, they can be attained, or approached arbitrarily closely. For $\alpha \geq 3$, the optimal rate $\displaystyle{{\mathcal O}\pa{t^{-2}}}$ is achieved for the function $f(x) = \|x\|^{r}$ with $r \to +\infty$ \cite{ACPR}, and for $\alpha < 3 $, the optimal rate $\displaystyle{{\mathcal O}\pa{t^{-\frac{2\alpha}{3}}}}$ is achieved by taking $f(x) = \|x\|$ \cite{AAD}. Theorem~\ref{fast_conv_smooth} is consistent with these optimality results. The condition $\alpha > 3$ is important to get the asymptotic rate $o(1/t^2)$.
\end{remark}
\subsubsection{Convergence of the trajectories}
We complete our analysis by showing weak convergence of the trajectories.
\begin{theorem}\label{T:weak_convergence-smooth}
{Assume that $e\in {\mathcal C}^1([t_0, +\infty[; {\mathcal H})$ with $\DS{\int_{t_0}^{+\infty} t \|e(t)\| dt<+\infty}$ and $\DS{\int_{t_0}^{+\infty} t \|\dot e(t)\| dt<+\infty}$. Let $x(t)$ be a solution trajectory to~\eqref{eq:origode_a} for $\alpha > 3$ and $\beta > 0$. Then $x(t)$ converges weakly to a minimizer of $f$.}
\end{theorem}
\begin{proof}
Keeping in mind that the goal is to apply Opial's Lemma (see Lemma~\ref{Opial}), we will now show that $\lim_{t \to +\infty}\|x(t)-x^\star\|$ exists.
{Following an argument similar to that of Theorem \ref{lem:smoothexplicitestimates} and \ref{fast_conv_smooth}, since our analysis is asymptotic, there is no restriction in assuming that $t_0 > \frac{\beta(\alpha-2)}{\alpha-3}$. Hence the existence of $\varepsilon >0$ such that
$t_0 \geq \frac{\beta(\alpha-2-\varepsilon)}{\alpha-3-\varepsilon}$ for some $\varepsilon \in ]0,\alpha-3[$.}
Recall the Lyapunov function ${\mathcal E}$ from the proof of Theorem~\ref{fast_conv_smooth}, and define its generalized version
\begin{equation}\label{eq:Eeps}
{\mathcal E}_\varepsilon(t) :=
\pa{\delta(t)+\varepsilon\beta t}(f(x(t))-f(x^\star))+\frac{1}{2}\|v_\varepsilon(t)\|^2\\
+\frac{\varepsilon(\alpha-1-\varepsilon)}{2}\norm{x(t)-x^\star}^2-\int_t^T \tau\dotp{v_\varepsilon(\tau)}{g(\tau)} d\tau ,
\end{equation}
where
\[
v_\varepsilon(t) = (\alpha-1-\varepsilon)(x(t)-x^\star)+t(\dot x(t)+\beta\nabla f(x(t))) .
\]
One can check, arguing as for ${\mathcal E}$, that for $t_0 \leq t\leq T$
\begin{equation*}
\dot{{\mathcal E}}_\varepsilon(t) = \pa{\dot{\delta}(t)+\varepsilon\beta}(f(x(t))-f(x^\star)) - (\alpha-1-\varepsilon)t w(t) \dotp{\nabla f(x(t))}{x(t)-x^\star}
- \beta\delta(t)\|\nabla f(x(t))\|^2
- \varepsilon t \norm{\dot{x}(t)}^2.
\end{equation*}
Convexity of $f$ then entails
\begin{equation*}
\dot{{\mathcal E}}_\varepsilon(t) + \pa{(\alpha-1-\varepsilon)t w(t) - \dot{\delta}(t)}(f(x(t))-f(x^\star)) +\beta\delta(t)\|\nabla f(x(t))\|^2
+ \varepsilon t \norm{\dot{x}(t)}^2 \leq 0 .
\end{equation*}
The assumption on the parameters gives
\begin{equation*}
(\alpha-1-\varepsilon)t w(t) - \dot{\delta}(t) = t((\alpha-3-\varepsilon) w(t) - t\dot{w}(t)) \geq t_0\pa{(\alpha-3-\varepsilon) - \frac{(\alpha-2-\varepsilon)\beta}{t_0}} \geq 0.
\end{equation*}
Thus, ignoring the non-negative terms in this inequality entails that ${\mathcal E}_\varepsilon(\cdot)$ is a decreasing function on $[t_0,T[$.
{According to the boundedness of ${\mathcal E}_\varepsilon(\cdot)$, an argument similar to that developed in Theorem 5 gives that
\begin{equation}\label{eq:supboundv_epsilon}
\sup\limits_{t \ge t_0} \|v_\varepsilon(t)\|<+\infty,
\end{equation}
with a bound which is independent of $\varepsilon$ and $T$.
Comparing with \eqref{eq:supboundv} gives
\begin{equation}\label{eq:supbound_x}
\sup\limits_{t \ge t_0} \|x(t)\|<+\infty,
\end{equation}
where we use that the previous argument is valid for arbitrary $t\leq T$, hence for all $t\geq t_0$.}
As a consequence, the energy functions ${\mathcal E}(\cdot)$ and ${\mathcal E}_{\varepsilon}(\cdot)$ with $T=+\infty$ are well-defined on $[t_0,+\infty[$, and are then Lyapunov functions for the dynamical system \eqref{eq:origode_a}. Both ${\mathcal E}(t)$ and ${\mathcal E}_{\varepsilon}(t)$ thus have limits as $t \to +\infty$, and so does their difference
\begin{multline*}
{\mathcal E}_{\varepsilon}(t) - {\mathcal E}(t) = \varepsilon\beta t(f(x(t))-f(x^\star))
-\frac{\varepsilon(\alpha-1)}{2}\norm{x(t)-x^\star}^2 - \varepsilon t\dotp{\dot{x}(t)}{x(t)-x^\star} \\
- \varepsilon t\dotp{\nabla f(x(t))}{x(t)-x^\star} + \varepsilon \int_t^{+\infty} \tau\dotp{x(\tau)-x^\star}{g(\tau)} d\tau .
\end{multline*}
{By Theorem~\ref{fast_conv_smooth}\ref{item:fast_conv_smooth1}, the first term converges to $0$ as $t \to +\infty$. By the integrability assumptions on the errors and boundedness of $x(t)$ (see ~\eqref{eq:supbound_x}), the last term also converges to $0$ as $t \to +\infty$. We have then shown that the limit as $t$ goes to infinity of
\[
p(t) := \frac{\alpha - 1}{2}\norm{x(t) -x^\star}^{2} + t \dotp{\dot{x}(t)}{x(t) -x^\star} + t \dotp{\nabla f(x(t))}{x(t) -x^\star}
\]
exists. Set
\[
q(t):= \frac{\alpha - 1}{2}\norm{x(t) -x^\star}^{2} + (\alpha - 1)\int_{t_0}^t \dotp{\nabla f(x(s))}{x(s) - x^\star} ds.
\]
We obviously have
\[
p(t)= q(t) + \frac{t}{\alpha - 1}\dot{q}(t) - (\alpha - 1)\int_{t_0}^t\dotp{\nabla f(x(s))}{x(s) -x^\star} ds.
\]
By Theorem~\ref{fast_conv_smooth}\ref{item:fast_conv_smooth4a}, and since $\dotp{\nabla f(x(s))}{x(s) -x^\star}$ is non-negative, we have that
\begin{equation}\label{eq:limiprodgradf}
\lim_{t\to +\infty} \int_{t_0}^t \dotp{\nabla f(x(s))}{x(s) -x^\star} ds
\end{equation}
exists. Overall, we have shown that
\[
\lim_{t \to +\infty}\pa{q(t) + \frac{t}{\alpha-1}\dot{q}(t)}
\]
exists. Since $\alpha > 1$, it follows from \cite[Lemma~7.2]{APR1} that $\lim_{t \to +\infty} q(t)$ exists. Using again \eqref{eq:limiprodgradf}, we deduce that $\lim_{t \to +\infty}\norm{x(t)-x^\star}$ exists for any $x^\star \in S$.} From claim {\ref{item:fast_conv_smooth1}} of Theorem~\ref{fast_conv_smooth} (see also Lemma~\ref{lem:smoothexplicitestimates}{\ref{lem:smoothexplicitestimates4}}), it follows that
for any sequence $\pa{x(t_n)}_{n \in {\mathbb N}}$ which converges weakly to, say, $\bar{x}$, we have
\[
f(\bar{x}) \leq \liminf_{n \to +\infty} f(x(t_n)) = \lim_{t \to +\infty} f(x(t)) = \bar{f} ,
\]
{\it i.e.}, $\bar{x} \in S$. Consequently, all the conditions of Lemma~\ref{Opial} are satisfied, hence the weak convergence of the trajectories. \qed
\end{proof}
\begin{remark}
In \cite[Theorem~2.2]{attouchiterates2021}, the authors proved weak convergence of the trajectory for the perturbation-free system \eqref{eq:origode}. Theorem~\ref{T:weak_convergence-smooth} shows that weak convergence is preserved under perturbations provided that they verify reasonable integrability results. Again, the proof necessitates new estimates and bounds to cope with the presence of errors.
\end{remark}
\begin{remark}
The condition $\alpha > 3$ is known to play an important role to show that each trajectory converges weakly to a minimizer. The case $\alpha = 3$, which corresponds to Nesterov's historical algorithm when $\beta=0$ and $e \equiv 0$, is critical. In fact, even for those inertial systems with $\alpha = 3$, convergence of the trajectories remains an open problem (except in one dimension where it holds as shown in \cite{ACR-subcrit}).
\end{remark}
\subsection{Implicit Hessian Damping}\label{s:approxconv}
{We now turn to the second-order ODE \eqref{eq:odetwo_a} where $f \in {\mathcal C}^1({\mathcal H})$, {$e \in {\mathcal C}([t_0,+\infty[)$} and $\beta(t)=\gamma+\frac{\beta}{t}$, $\gamma,\beta \geq 0$. Let us denote for brevity $\bar{f} := \inf_{{\mathcal H}} f$. Given $x^\star \in S$, we consider the function}
\begin{multline}\label{eq:lyap}
{\mathcal E}(t) = a(t)\pa{f\pa{x(t)+\beta(t)\dot{x}(t)}-\bar{f}}+\frac{1}{2}\|b(t)(x(t)-x^\star)+c(t)\dot{x}(t)\|^2
+\frac{d(t)}{2}\|x(t)-x^\star\|^2 \\ -\int_t^{+\infty} c(\tau)\dotp{b(\tau)(x(\tau)-x^\star)+c(\tau) \dot{x}(\tau)}{e(\tau)} d\tau
-\int_t^{+\infty} a(\tau)\beta(\tau) \dotp{\nabla f\pa{x(\tau)+\beta(\tau)\dot{x}(\tau)}}{e(\tau)} d\tau
\end{multline}
parametrized by some functions $a(t)$, $b(t)$, $c(t)$ and $d(t)$ to be specified later.
\subsubsection{Lyapunov function}
We first show that for proper choices of $(a(t),b(t),c(t),d(t))$ as a function of the problem parameters $(\alpha,\gamma,\beta)$, ${\mathcal E}$ can serve as a Lyapunov function for \eqref{eq:odetwo_a}. We will denote for short $\alpha(t)=\frac{\alpha}{t}$.
\begin{lemma}\label{lem:decreaselyap}
Assume that {$f \in {\mathcal C}^1({\mathcal H})$, $e \in {\mathcal C}([t_0,+\infty[)$, and}
\begin{equation}\label{eq:conditionsscalarfunc}
\begin{cases}
\dot{a}(t)-b(t)c(t) &\le 0,\\
-a(t)\beta(t) &\leq 0,\\
-a(t)\alpha(t)\beta(t)+a(t)\dot{\beta}(t)+a(t)-c(t)^2+b(t)c(t)\beta(t) &= 0,\\
\dot{b}(t)b(t)+\frac{\dot{d}(t)}{2} &\le 0,\\
\dot{b}(t)c(t)+b(t)(b(t)+\dot{c}(t)-c(t)\alpha(t))+d(t) &=0,\\
c(t)(b(t)+\dot{c}(t)-c(t)\alpha(t)) &\le 0 .
\end{cases}
\end{equation}
Then
\begin{eqnarray}\label{eq:edotfour}
\dot{{\mathcal E}}(t) &\le& (\dot{a}(t)-b(t)c(t))(f(x(t)+\beta(t)\dot{x}(t))-\bar{f})-a(t)\beta(t)\|\nabla f(x(t)+\beta(t)\dot{x}(t))\|^2 \nonumber \\
&&+\pa{\dot{b}(t)b(t)+\frac{\dot{d}(t)}{2}}\|x(t)-x^\star\|^2+c(t)(b(t)+\dot{c}(t)-c(t)\alpha(t))\norm{\dot{x}(t)}^2 \le 0 .
\end{eqnarray}
\end{lemma}
\begin{proof}
{Recall that since $f \in {\mathcal C}^1({\mathcal H})$ and $e \in {\mathcal C}([t_0,+\infty[)$, \eqref{eq:odetwo_a} has a unique classical global solution $x$; see paragraph after \eqref{eq:fos5_implicit}}. {We now proceed as in the proof of Theorems~\ref{lem:smoothexplicitestimates} and \ref{fast_conv_smooth}, and first consider the function ${\mathcal E}_T$ where the integrals involving the error terms are calculated on $[t,T]$, $T<+\infty$. This shows that
${\mathcal E}$ is well-posed under our assumptions. We can then compute the time derivative of ${\mathcal E}$ and use the chain rule to get}
\begin{eqnarray}
\dot{{\mathcal E}}(t) &=& \dot{a}(t)\pa{f\pa{x(t)+\beta(t)\dot{x}(t)}-\bar{f}}+
a(t)\dotp{\nabla f \pa{x(t)+\beta(t)\dot{x}(t)}}{\dot{x}(t)+\dot{\beta}(t)\dot{x}(t)+\beta(t)\ddot{x}(t)} \nonumber \\
&&+ \dotp{(b(t)+\dot{c}(t))\dot{x}(t)+c(t)\ddot{x}(t)+\dot{b}(t)(x(t)-x^\star)}{b(t) (x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber \\
&&+ \frac{\dot{d}(t)}{2}\|x(t)-x^\star\|^2+d(t)\dotp{\dot{x}(t)}{x(t)-x^\star} \nonumber \\
&&+ c(t)\dotp{b(t)(x(t)-x^\star)+c(t) \dot{x}(t)}{e(t)}+a(t)\beta(t)\dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{e(t)}\label{eq:edot}
\end{eqnarray}
Using \eqref{eq:odetwo_a} in the second term of \eqref{eq:edot}, we get
\begin{multline}\label{eq:term1}
a(t)\dotp{\nabla f \pa{x(t)+\beta(t)\dot{x}(t)}}{\dot{x}(t)+\dot{\beta}(t)\dot{x}(t)+\beta(t)\ddot{x}(t)} \\
= a(t)\dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{\pa{1+\dot{\beta}(t)-\alpha(t) \beta(t)} \dot{x}(t)-\beta(t)\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}-\beta(t)e(t)} \\
= -a(t) \beta(t) \norm{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}^2+\pa{1+\dot{\beta}(t)-\alpha(t) \beta(t)} a(t) \dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{\dot{x}(t)}
\\-\beta(t)a(t)\dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{e(t)} .
\end{multline}
We expand the third term in \eqref{eq:edot} as
\begin{eqnarray}
&&\dotp{(b(t)+\dot{c}(t))\dot{x}(t)+c(t)\ddot{x}(t)+\dot{b}(t)(x(t)-x^\star)}{b(t) (x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber \\
&& =
\dotp{(b(t)+\dot{c}(t)-c(t)\alpha(t))\dot{x}(t)-c(t)\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{b(t) (x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber \\
&&\quad +\dotp{-c(t)e(t)+\dot{b}(t)(x(t)-x^\star)}{b(t) (x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber \\
&&= -c(t)\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{b(t)(x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber \\
&&\quad +\dotp{\dot{b}(t)(x(t)-x^\star)}{b(t)(x(t)-x^\star)+c(t)\dot{x}(t)} \nonumber\\
&&\quad +c(t)(b(t)+\dot{c}(t)-c(t)\alpha(t))\norm{\dot{x}(t)}^2-c(t)\dotp{e(t)}{b(t)(x(t)-x^\star)+c(t)\dot{x}(t)} . \label{eq:term2}
\end{eqnarray}
Plugging~\eqref{eq:term1} and~\eqref{eq:term2} into~\eqref{eq:edot}, we get,
\begin{eqnarray}
\dot{{\mathcal E}}(t) &=& \dot{a}(t)\pa{ f\pa{x(t)+\beta(t)\dot{x}(t)}-\bar{f}}-a(t) \beta(t) \norm{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}^2 \nonumber \\
&+&c(t)(b(t)+\dot{c}(t)-c(t)\alpha(t))\norm{\dot{x}(t)}^2 + \pa{\dot{b}(t)b(t)+\frac{\dot{d}(t)}{2}}\|x(t)-x^\star\|^2 \nonumber\\
&+&\pa{b(t)^2+b(t)\dot{c}(t)+\dot{b}(t)c(t)-b(t)c(t)\alpha(t)+d(t)}\dotp{\dot{x}(t)}{x(t)-x^\star} \nonumber \\
&+& \pa{-a(t)\alpha(t)\beta(t)+a(t)\dot{\beta}(t)+a(t)-c(t)^2} \dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{\dot{x}(t)}\nonumber
\\
&-& b(t)c(t)\dotp{\nabla f\pa{x(t)+\beta(t)\dot{x}(t)}}{x(t)-x^\star} .\label{eq:edottwo}
\end{eqnarray}
Since,
\begin{equation*}
\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{x(t)-x^\star} = \dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{x(t)+\beta(t)\dot{x}(t)-x^\star} \\
-\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{\beta(t)\dot{x}(t)}
\end{equation*}
and using the convex (sub)differential inequality on $f$,
we can write
\begin{multline*}
-b(t)c(t)\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t)}{x(t)-x^\star} \\
\le -b(t)c(t)( f(x(t)+\beta(t)\dot{x}(t))-\bar{f})+b(t)c(t)\beta(t)\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{\dot{x}(t)}
\end{multline*}
and we arrive at
\begin{eqnarray}
&&\dot{{\mathcal E}}(t) \le (\dot{a}(t)-b(t)c(t))(f(x(t)+\beta(t)\dot{x}(t))-\bar{f})-a(t)\beta(t)\|\nabla f(x(t)+\beta(t)\dot{x}(t))\|^2 \nonumber \\
&& \; +(-a(t)\alpha(t)\beta(t)+a(t)\dot{\beta}(t)+a(t)-c(t)^2+b(t)c(t)\beta(t))\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{\dot{x}(t)} \nonumber\\
&&\; +\pa{\dot{b}(t)b(t)+\frac{\dot{d}(t)}{2}}\|x(t)-x^\star\|^2 \nonumber\\
&& \;+(b(t)^2+b(t)\dot{c}(t)+\dot{b}(t)c(t)-b(t)c(t)\alpha(t)+d(t))\dotp{\dot{x}(t)}{x(t)-x^\star}
+c(t)(b(t)+\dot{c}(t)-c(t)\alpha(t))\norm{\dot{x}(t)}^2 .\nonumber
\end{eqnarray}
Thus, conditions~\eqref{eq:conditionsscalarfunc} guarantee that $\dot{{\mathcal E}}(t)\le 0$, in particular they imply \eqref{eq:edotfour}. \qed
\end{proof}
Following the discussion of \cite[Remark~11]{alecsa2019extension}, in the rest of the section, we take
\begin{equation}\label{eq:odetwo_aparamchoice}
\begin{gathered}
\beta(t)=\gamma+\frac{\beta}{t}, \quad \gamma,\beta \geq 0, \\
b(t) \equiv b \in ]0,\alpha-1], \alpha > 1, \qquad c(t)=t \enskip \text{and} \enskip d(t) \equiv b(\alpha-1-b).
\end{gathered}
\end{equation}
Such a choice is reminescent of that in \eqref{eq:Eeps}. The choices of $d(t)$ and $b(t)$ comply with the fourth, fifth and sixth conditions of \eqref{eq:conditionsscalarfunc}. To satisfy the third condition, one has to take
\begin{equation}\label{eq:odetwo_choice_a}
a(t) = t^2\pa{1+\frac{(\alpha-b)\gamma t - \beta(\alpha+1-b)}{t^2-\alpha\gamma t - \beta(\alpha+1)}} .
\end{equation}
Clearly, for $t$ large enough, one has $a(t) \geq t^2$ and $\beta(t) \geq \gamma/2$. Thus, the second condition is in force. One can also verify that the first inequality is satisfied for $t$ large enough provided that $b > 2$ (and thus $\alpha > 3$) when $\gamma > 0$, and $b=2$ (with $\alpha=3$) when $\gamma=0$.
\subsubsection{Fast convergence rates}
We start with the following boundedness properties.
\begin{lemma}\label{lem:boundconv}
Let
\[
E(t) = a(t)\pa{f\pa{x(t)+\beta(t)\dot{x}(t)}-\bar{f}}+\frac{1}{2}\|b(x(t)-x^\star)+t\dot{x}(t)\|^2
+\frac{b(\alpha-1-b)}{2}\|x(t)-x^\star\|^2 .
\]
Choose the parameters according to \eqref{eq:odetwo_aparamchoice}-\eqref{eq:odetwo_choice_a} with $\alpha > 3$, $\gamma > 0$. Define, for $t \geq t_0$,
\begin{equation}\label{eq:asm}
m(t) :=
\max\pa{t,L|a(t)\beta(t)|,L|a(t)|\beta(t)^2}.
\end{equation}
Assume that $\nabla f$ is $L$-Lipschitz continuous, {$e \in {\mathcal C}([t_0,+\infty[)$ and} $m(\cdot)e(\cdot) \in L^1(t_0,+\infty;{\mathcal H})$. Then, we have
\begin{center}
$\sup_{t \geq t_0} E(t) < +\infty$, $\sup_{t \geq t_0} t\norm{\dot{x}(t)} < +\infty$ and $\sup_{t \geq t_0} \|x(t)-x^\star\| < +\infty$.
\end{center}
\end{lemma}
\begin{proof}
Consider the function ${\mathcal E}(t)$ in \eqref{eq:lyap} with the choices \eqref{eq:odetwo_aparamchoice}-\eqref{eq:odetwo_choice_a} for $c(t)$, $d(t)$, $b(t)$ and $a(t)$, with $b \in ]2,\alpha-1[$. For such a choice, there exists $t_1 \geq t_0$ such that for all $t\ge t_1$, $a(t) > 0$, $\beta(t) > 0$ (and in turn, $m(t) > 0$), and all conditions of \eqref{eq:conditionsscalarfunc} are satisfied. Thus, ${\mathcal E}(t)$ is monotonically decreasing on $[t_1,+\infty[$ according to Lemma~\ref{lem:decreaselyap}. Since the solution $x(t)$ is continuous, it is bounded on $[t_0,t_1]$ and so without loss of generality we can assume that $t_1=t_0$ and proceed to show,
\begin{equation}\label{eq:bounde}
\begin{aligned}
&E(t)
\leq E(t_0)+\int_{t_0}^t \dotp{\tau\pa{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+a(\tau) \beta(\tau)\nabla f\pa{x(\tau)+\beta(\tau)\dot{x}(\tau)}}{e(\tau)} d\tau \\
&= E(t_0)+\int_{t_0}^t \dotp{\tau\pa{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+a(\tau) \beta(\tau)\pa{\nabla f\pa{x(\tau)+\beta(\tau)\dot{x}(\tau)}-\nabla f(x^\star)}}{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{\tau\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+a(\tau) \beta(\tau)\norm{\nabla f\pa{x(\tau)+\beta(\tau)\dot{x}(\tau)}-\nabla f(x^\star)}}\norm{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{\tau\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+a(\tau) \beta(\tau)L\norm{x(\tau)-x^\star+\beta(\tau)\dot{x}(\tau)}}\norm{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{\tau\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+La(\tau) \beta(\tau)\norm{x(\tau)-x^\star}+L a(\tau)\beta(\tau)^2\norm{\dot{x}(\tau)}}\norm{e(\tau)} d\tau .
\end{aligned}
\end{equation}
Denote $d = b(\alpha-1-b)$. We have $d > 0$. Moreover, $a(t) > 0$ for $t \geq t_0$. One can then drop the first term in $E(t)$, and \eqref{eq:bounde} becomes, for any $t \geq t_0$,
\begin{align*}
&\frac{1}{2}\|b(x(t)-x^\star)+t\dot{x}(t)\|^2+\frac{d}{2}\|x(t)-x^\star\|^2\\
&\leq E(t_0)+\int_{t_0}^t \bpa{\tau\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+\sqrt{d}\frac{La(\tau) \beta(\tau)}{\sqrt{d}}\norm{x(\tau)-x^\star}+\tau\frac{L a(\tau)\beta(\tau)^2}{t_0}\norm{\dot{x}(\tau)}}\norm{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+\sqrt{d}\norm{x(\tau)-x^\star}+\tau\norm{\dot{x}(\tau)}}\max\pa{1,d^{-1/2},t_0^{-1}}m(\tau)\norm{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{2\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+(b+\sqrt{d})\norm{x(\tau)-x^\star}}\max\pa{1,d^{-1/2},t_0^{-1}}m(\tau)\norm{e(\tau)} d\tau \\
&\leq E(t_0)+\int_{t_0}^t \bpa{\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+\sqrt{d}\norm{x(\tau)-x^\star}} C m(\tau)\norm{e(\tau)} d\tau ,
\end{align*}
for some constant $C \geq \max\pa{1,d^{-1/2},t_0^{-1}}\max\pa{2,1+\sqrt{\frac{b}{\alpha-1-b}}}$. Now, Jensen's inequality yields
\begin{align*}
&\frac{1}{4}\pa{\|b(x(t)-x^\star)+t\dot{x}(t)\| + \sqrt{d}\|x(t)-x^\star\|}^2 \leq \frac{1}{2}\|b(x(t)-x^\star)+t\dot{x}(t)\|^2+\frac{d}{2}\|x(t)-x^\star\|^2\\
&\leq E(t_0)+\int_{t_0}^t \bpa{\norm{b(x(\tau)-x^\star)+\tau \dot{x}(\tau)}+\sqrt{d}\norm{x(\tau)-x^\star}} C |m(\tau)|\norm{e(\tau)} d\tau .
\end{align*}
Using {the Gronwall} Lemma~\ref{lem:BrezisA5}, we conclude that, for all $t \geq t_0$
\begin{equation}\label{eq:integrability}
\|b(x(t)-x^\star)+t\dot{x}(t)\| + \sqrt{d}\|x(t)-x^\star\| \leq 2\sqrt{|E(t_0)|} + 2 C \int_{t_0}^{+\infty} |m(\tau)| \|e(\tau)\|d\tau < +\infty ,
\end{equation}
whence we get boundedness of $\|x(t)-x^\star\|$ and $\|b(x(t)-x^\star)+t\dot{x}(t)\|$. The triangle inequality then shows that $t\norm{\dot{x}(t)}$ is also bounded. Using this into \eqref{eq:bounde} together with Cauchy-Schwarz inequality and our integrability assumption, we deduce boundedness of $E(t)$.\qed
\end{proof}
\begin{remark}\label{rem:sumbound}
Recall our discussion on the parameters in \eqref{eq:odetwo_aparamchoice}-\eqref{eq:odetwo_choice_a}. Notice that we have $t^2 \leq a(t) \leq t^2 + \kappa_1$ and $\gamma/2 \leq \beta(t) \leq \beta_0$ for $t$ large enough, where $\beta_0 = \gamma + |\beta|/t_0$ and $\kappa$ is a non-negative constant. In turn, for $t$ large enough, we have
\[
\max\pa{1,\gamma/2}L\gamma/2t^2 \leq m(t) \leq \max\pa{1,\beta_0}L\beta_0(t+\kappa_1)^2 .
\]
Clearly the condition $m(\cdot)e(\cdot) \in L^1(t_0,+\infty;{\mathcal H})$ is equivalent to $t^2e(t) \in L^1(t_0,+\infty;{\mathcal H})$.
\end{remark}
\noindent From Lemma~\ref{lem:boundconv}, we obtain the following convergence rates and integral estimates.
\begin{theorem}\label{th:int}
Under the assumptions of Lemma~\ref{lem:boundconv}, the following holds:
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{th:intclaim1}
{$\DS{f\left(x(t)+\beta(t)\dot{x}(t)\right) - \min_{{\mathcal H}} f = {\mathcal O}\pa{\frac{1}{t^2}}}$ as $t \to +\infty$};
\item \label{th:intclaim2}
$\DS{\norm{\dot{x}(t)} = {\mathcal O}\pa{\frac{1}{t}}}$ as $t \to +\infty$;
\item \label{th:intclaim3}
$\DS{
\int_{t_0}^{+\infty} t \pa{f(x(t)) - \min_{{\mathcal H}} f }dt < +\infty
}
$;
\item \label{th:intclaim4}
$\DS{
\int_{t_0}^{+\infty} t^2 \norm{\nabla f\left( x(t)+\beta(t)\dot{x}(t)\right)}^2dt < +\infty
}
$;
\item \label{th:intclaim5}
$
\DS{\int_{t_0}^{+\infty} t\norm{\dot{x}(t)}^2 dt < +\infty
}
$.
\end{enumerate}
\end{theorem}
\begin{proof} {
Claim~{\ref{th:intclaim2}} follows from Lemma~\ref{lem:boundconv}. Discarding the non-negative terms in $E(t)$, Lemma~\ref{lem:boundconv} together with the fact that $a(t) \geq t^2$ for $t$ large enough, also gives
\[
f\left(x(t)+\beta(t)\dot{x}(t)\right) - \min_{{\mathcal H}} f = {\mathcal O}\pa{\frac{1}{t^2}}.
\]
To show the remaining integral estimates, consider the function ${\mathcal E}(\cdot)$ in \eqref{eq:lyap} with the choices \eqref{eq:odetwo_aparamchoice}-\eqref{eq:odetwo_choice_a} of $c(t)$, $d(t)$, $b(t)$ and $a(t)$, where $b \in ]2,\alpha-1[$. We first argue similarly to \cite{alecsa2019extension} to show that for $t$ large enough, we have $\dot{a}(t)-bt \leq -\frac{(\alpha-3)t}{2}$, since $\alpha > 3$ and $b > 2$. In addition, for (a possibly different) $t$ large enough, it is straightforward to see that $a(t)\pa{\gamma+\beta/t} \geq t^2\gamma/2$. With these bounds, \eqref{eq:edotfour} reads, for $t$ large enough,
\begin{equation}\label{eq:edotfourspecial}
\dot{{\mathcal E}}(t) \leq -\frac{(\alpha-3)t}{2}(f(x(t)+\beta(t)\dot{x}(t))-\bar{f})
-t^2\gamma/2\norm{\nabla f(x(t)+\beta(t)\dot{x}(t))}^2 \\ -t(\alpha-1-b)\norm{\dot{x}(t)}^2 .
\end{equation}
Integrating \eqref{eq:edotfourspecial}, and using that ${\mathcal E}$ is bounded thanks to Lemma~\ref{lem:boundconv}, we get statements {\ref{th:intclaim4}-\ref{th:intclaim5}} and
\begin{equation}\label{eq:claim3}
\int_{t_0}^{+\infty} t \pa{f(x(t)+\beta(t)\dot{x}(t)) - \min_{{\mathcal H}} f} dt < +\infty .
\end{equation}
Let $\beta_0 = \gamma+\abs{\beta}/t_0$. By the gradient descent lemma,
\begin{align}
f(x(t)) - f(x(t)+\beta(t)\dot{x}(t))
&\leq -\beta(t)\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{\dot{x}(t)} + \frac{L}{2} \beta(t)^2\norm{\dot{x}(t)}^2 \label{lem:descent}\\
&\leq \beta_0 \norm{\nabla f(x(t)+\beta(t)\dot{x}(t))}\norm{\dot{x}(t)} + \frac{L}{2} \beta_0^2\norm{\dot{x}(t)}^2.\nonumber
\end{align}
By Cauchy-Schwarz inequality, we have
\begin{eqnarray*}
&&\int_{t_0}^{+\infty} t \pa{f(x(t)) - \min_{{\mathcal H}} f}dt
\leq \int_{t_0}^{+\infty} t \pa{f(x(t)+\beta(t)\dot{x}(t)) - \min_{{\mathcal H}} f}dt \\
&&+\beta_0 \pa{\int_{t_0}^{+\infty} t \norm{\nabla f(x(t)+\beta(t)\dot{x}(t))}^2 dt}^{1/2} \pa{\int_{t_0}^{+\infty} t\norm{\dot{x}(t)}^2 dt}^{1/2}
+ \frac{L\beta_0^2}{2}\int_{t_0}^{+\infty} t\norm{\dot{x}(t)}^2 dt.
\end{eqnarray*}
In view of \eqref{eq:claim3} and claims {\ref{th:intclaim4}-\ref{th:intclaim5}}, statement {\ref{th:intclaim3}} follows. }\qed
\end{proof}
\subsubsection{Convergence of the trajectories}
We now turn to showing weak convergence of the trajectories to a minimizer.
\begin{theorem}\label{T:weak_convergence-smooth-implicit}
Suppose that the assumptions of Lemma~\ref{lem:boundconv} hold. Then $x(t)$ converges weakly to a minimizer of $f$.
\end{theorem}
\begin{proof}
As in the explicit case, we invoke {Opial}'s Lemma~\ref{Opial}. Recall that the trajectory $x(\cdot)$ is bounded by Lemma~\ref{lem:boundconv}. Therefore, for any sequence $\pa{x(t_n)}_{n \in {\mathbb N}}$ which converges weakly to, say, $\bar{x}$, as $t_n \to +\infty$, {Theorem~\ref{th:int}\ref{th:intclaim1}-\ref{th:intclaim2} entails that
\[
f(\bar{x}) \leq \liminf_{n \to +\infty} f(x(t_n)+ \beta(t_n)\dot{x}(t_n)) = \lim_{t \to +\infty} f\left(x(t)+\beta(t)\dot{x}(t)\right) = \bar{f} ,
\]
}
{\it i.e.}, each weak cluster point of $x(t_n)$ belongs to $S$. To get weak convergence of the trajectory, it remains to show that $\lim_{t \to +\infty}\|x(t)-x^\star\|$ exists.
Let $h: t \in [t_0,+\infty[ ~ \mapsto \frac{1}{2}\|x(t)-x^\star\|^2$. {Under the assumptions on $f$ and $e$, $x$ is the unique classical global solution to \eqref{eq:odetwo_a}, {\it i.e.}, $x \in {\mathcal C}^2([t_0,+\infty[)$. Thus so is $h$ and}
\[
\dot{h}(t)=\dotp{\dot{x}(t)}{x(t)-x^\star} \enskip \text{and} \enskip \ddot{h}(t)=\dotp{\ddot{x}(t)}{x(t)-x^\star}+\norm{\dot{x}(t)}^2 .
\]
From \eqref{eq:odetwo_a}, we obtain
\begin{align*}
&\ddot{h}(t)+\frac{\alpha}{t} \dot{h}(t)
=\dotp{\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)}{x(t)-x^\star}+\norm{\dot{x}(t)}^2 \\
&= -\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))+e(t)}{x(t)-x^\star} + \norm{\dot{x}(t)}^2 \\
&= -\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{x(t)+\beta(t)\dot{x}(t)-x^\star} -\dotp{e(t)}{x(t)-x^\star} +\norm{\dot{x}(t)}^2 \\
&\quad+\beta(t)\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{\dot{x}(t)} .
\end{align*}
Convexity of $f$ implies,
\[
-\dotp{\nabla f(x(t)+\beta(t)\dot{x}(t))}{x(t)+\beta(t)\dot{x}(t)-x^\star} \leq \bar{f}-f(x(t)+\beta(t)\dot{x}(t)) \leq 0 ,
\]
and thus,
\begin{align*}
\ddot{h}(t)+\frac{\alpha}{t} \dot{h}(t)
&\leq \norm{x(t)-x^\star}\norm{e(t)} + \norm{\dot{x}(t)}^2 + \beta_0\norm{\nabla f(x(t)+\beta(t)\dot{x}(t))}\norm{\dot{x}(t)} \\
&\leq C\norm{e(t)} + \norm{\dot{x}(t)}^2 + \beta_0\norm{\nabla f(x(t)+\beta(t)\dot{x}(t))}\norm{\dot{x}(t)}
\end{align*}
where $C = \sup_{t \geq t_0}\norm{x(t)-x^\star} < +\infty$ thanks to Lemma~\ref{lem:boundconv}, and we denoted $\beta_0 = 1+|\beta|/t_0$. Multiplying both sides by $t$, we arrive at
\begin{align*}
t\ddot{h}(t)+\alpha \dot{h}(t)
&\leq C t\norm{e(t)} + t\norm{\dot{x}(t)}^2 + \frac{\beta_0}{\sqrt{t_0}} (t\norm{\nabla f(x(t)+\beta(t)\dot{x}(t))})(\sqrt{t}\norm{\dot{x}(t)}) .
\end{align*}
The right-hand side of this inequality belongs to $L^1(t_0,+\infty;{\mathbb R})$ by assumption on the error, and using the Cauchy-Schwarz inequality and Theorem~\ref{th:int}\ref{th:intclaim4}-\ref{th:intclaim5} for the last two terms. {Since $h \in {\mathcal C}^2([t_0,+\infty[$),} it then follows from Lemma~\ref{lem:convw} that $\lim_{t \to +\infty}\|x(t)-x^\star\|$ exists. We have now shown that all conditions of Lemma~\ref{Opial} are satisfied, {hence the weak convergence of the trajectories}. \qed
\end{proof}
\begin{remark}
For the unperturbed case, similar rates to ours in Theorem~\ref{th:int} and weak convergence of the trajectory were proved in \cite{alecsa2019extension}. Again, handling errors necessitates new estimates and bounds, for instance those established in Lemma~\ref{lem:boundconv}.
\end{remark}
\subsection{Discussion}\label{s:discussion}
We now discuss the main differences between the two systems in terms of their stability to perturbations and the corresponding assumptions. Recall from Remark~\ref{rem:sumbound}, the integrability assumption $m(\cdot)e(\cdot) \in L^1(t_0,+\infty;{\mathcal H})$ required to ensure stability for system \eqref{eq:odetwo_a} is equivalent to ensuring that the second-order moment of the error $e(\cdot)$ is finite. One may wonder whether this is more stringent than the integrability assumptions for the explicit Hessian system \eqref{eq:origode_a} involving the control of the first-order moments of the error and its derivative (see Section~\ref{s:secondorderconv}). The answer is clearly affirmative in the scalar case with a simple integration by parts argument. {Indeed, supposing without loss of generality that $e(\cdot)$ is a non-increasing and non-negative function, one has
\[
\int_{t_0}^{+\infty} t |\dot{e}(t)| dt = -\int_{t_0}^{+\infty} t \dot{e}(t) dt \leq t_0e(t_0) + \int_{t_0}^{+\infty}e(t) dt \leq t_0 e(t_0) + t_0^{-2}\int_{t_0}^{+\infty}t^2|e(t)| dt .
\]
}
Another intuitive way to understand this is to look at what happens if the system is discretized with finite differences. In this case, the integrability assumptions on the errors for system \eqref{eq:origode_a} boil down to controlling only the first-order moment of the (discretized) error.
{
Indeed, temporal discretization with fixed step size of $t\|\dot{x}(t)\|$ gives $k \|x_{k+1} -x_k\|$ whose summability is clearly implied by the summability of $k \|x_{k} \|$.
}
We conclude this discussion by noting that Lipschitz continuity of the gradient is not needed for the estimates and convergence analysis of \eqref{eq:origode_a} while it is used extensively to analyze \eqref{eq:odetwo_a}. This is a distinctive avantage of \eqref{eq:origode_a} compared to \eqref{eq:odetwo_a}. This will be even more notable when extending to the non-smooth case; see Section~\ref{s:nonsmooth}.
\section{Smooth Strongly Convex Case}\label{s:sconvex}
We will successively examine the Explicit Hessian Damping, then the Implicit Hessian Damping.
\subsection{Explicit Hessian Damping}\label{s:secondordersconv}
In this section we consider the explicit Hessian system under the assumption of strong convexity of $f$. Following Polyak's heavy ball system \cite{BP}, consider the second-order perturbed system
\begin{equation}\label{dyn-sc}
\ddot{x}(t) + 2\sqrt{\mu} \dot{x}(t) + \beta \nabla^2 f (x(t))\dot{x}(t) + \beta \dot{e}(t)+ \nabla f (x(t)) + e(t)=0 ,
\end{equation}
which has a fixed positive damping coefficient that is adjusted to the modulus $\mu$ of strong convexity of $f$. To study
\eqref{dyn-sc}, we define the function ${\mathcal E} : [t_0, +\infty[ \to {\mathbb R}_{+}$
\begin{align}
t \mapsto {\mathcal E} (t) := f(x(t))- \min_{{\mathcal H}} f + \frac{1}{2} \| v(t) \|^2,
\label{dyn-sc-c}
\end{align}
where
\begin{equation}\label{dyn-sc-d}
v(t)= \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)+ \beta \nabla f (x(t)) .
\end{equation}
\begin{theorem}\label{strong-conv-thm}
Suppose that $f: {\mathcal H} \to {\mathbb R}$ is $\mu$-strongly convex for some $\mu >0$, let $x^\star$ be the unique minimizer of $f$. Let $x(\cdot): [t_0, + \infty[ \to {\mathcal H}$ be a solution trajectory of \eqref{dyn-sc}. Suppose that
\begin{enumerate}[label=\alph*)]
\item $\DS{ 0 \leq \beta \leq \frac{1}{2\sqrt{\mu}}}$.
\item $\DS{\int_{t_0}^{+\infty} \|e(t)\| dt< +\infty}$ and $\DS{\int_{t_0}^{+\infty} \|\dot e(t)\| dt< +\infty}$.
\end{enumerate}
{Then the following properties are satisfied:}
\begin{enumerate}[label={\rm (\roman*)}]
\item Minimizing properties: for all $t\geq t_0$
\[
{\mathcal E} (t) \leq {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\int_{t_0}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| e(\tau)+\beta \dot{e}(\tau) \| d\tau,
\]
where $M := \sqrt{2 {\mathcal E}(t_0)} + \DS{\int_{t_0}^{+\infty} \| e(\tau)+\beta \dot{e}(\tau) \| d \tau}$.
As a consequence,
\begin{eqnarray*}
&& \lim_{t\to +\infty} {\mathcal E} (t) =0; \; \lim_{t\to +\infty} f(x(t))= \min_{{\mathcal H}} f \\
&& \lim_{t\to +\infty}\| x(t) -x^\star\|=
\lim_{t\to +\infty}\| \nabla f (x(t))\|=
\lim_{t\to +\infty}\| \dot{x}(t)\|=0.
\end{eqnarray*}
\item Convergence rates: suppose moreover that for some $p>0$,
$
\DS{\|e(t)+\beta \dot{e}(t)\| = {\mathcal O}\pa{\frac{1}{t^p}}},
$ as $t \to +\infty$.
Then
$
{\mathcal E} (t)= {\mathcal O} \pa{ \frac{1}{t^p} },
$
{\it i.e.} \; ${\mathcal E} (t)$ inherits the decay rate of the error terms. As a consequence, as $t \to +\infty$
\begin{eqnarray*}
&& f(x(t))- \min_{{\mathcal H}} f = {\mathcal O} \pa{ \frac{1}{t^p} };\\
&& \| x(t) -x^\star\|^2= {\mathcal O} \pa{ \frac{1}{t^p} }; \;
\| \dot{x}(t)\|^2= {\mathcal O} \pa{ \frac{1}{t^p} }.
\end{eqnarray*}
In addition, when $\beta >0$
\[
e^{- \sqrt{\mu}t} \int_{t_0}^t e^{ \sqrt{\mu}s}\| \nabla f (x(s))\|^2 ds = {\mathcal O} \pa{ \frac{1}{t^p} }.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
Recall $\bar{f} := \min_{{\mathcal H}} f = f(x^\star)$. Define $g(t) := e(t)+\beta \dot{e}(t)$, so that the constitutive equation is written in the compact form
\begin{equation}\label{dyn-sc-b}
\ddot{x}(t) + 2\sqrt{\mu} \dot{x}(t) + \beta \nabla^2 f (x(t))\dot{x}(t) + \nabla f (x(t)) + g(t)=0.
\end{equation}
Derivation of ${\mathcal E} (\cdot) $ gives
\begin{eqnarray*}
\dot{{\mathcal E}}(t)
&=& \dotp{\nabla f (x(t))}{\dot{x}(t)} + \dotp{v(t)}{\dot{v}(t)}\\
&=& \dotp{\nabla f (x(t))}{\dot{x}(t)} + \dotp{v(t)}{\sqrt{\mu}\dot{x}(t) + \ddot{x}(t) + \beta \nabla^2 f (x(t))\dot{x}(t)}.
\end{eqnarray*}
Using the definition of $v(t)$ and \eqref{dyn-sc-b}, we get
\begin{multline*}
\dot{{\mathcal E}} (t)=\dotp{\nabla f (x(t))}{\dot{x}(t)} + \dotp{\sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)+ \beta \nabla f (x(t))}{-\sqrt{\mu}\dot{x}(t)- \nabla f (x(t))} - \dotp{v(t)}{g(t)} .
\end{multline*}
After developing and simplifying, we obtain
\begin{eqnarray*}
&\dot{{\mathcal E}}(t) + \sqrt{\mu}\dotp{\nabla f (x(t))}{x(t)-x^\star} + \mu\dotp{x(t)-x^\star}{\dot{x}(t)} + \sqrt{\mu} \| \dot{x}(t) \|^2 \\
&+ \beta\sqrt{\mu}\dotp{\nabla f (x(t))}{\dot{x}(t)} + \beta \|\nabla f (x(t))\|^2 = -\dotp{ v(t)}{g(t)}.
\end{eqnarray*}
According to strong convexity of $f$, we have
\[
\dotp{\nabla f(x(t))}{x(t) - x^\star} \geq f(x(t))- \bar{f} + \frac{\mu}{2} \|x(t) -x^\star\|^2 .
\]
Thus, by combining the last two relations, and by the Cauchy-Schwarz inequality, we obtain
\[
\dot{{\mathcal E}} (t) + \sqrt{\mu}A(t) \leq \| v(t)\| \| g(t) \| ,
\]
where
\begin{equation*}
A(t) := f(x(t))- \bar{f} + \frac{\mu}{2} \|x(t) - x^\star\|^2 + \sqrt{\mu} \dotp{ x(t) -x^\star }{\dot{x}(t)} + \| \dot{x}(t) \|^2 \\
+ \beta \dotp{ \nabla f (x(t))}{ \dot{x}(t) } + \frac{\beta}{\sqrt{\mu}} \| \nabla f (x(t)) \|^2 .
\end{equation*}
Let us make appear ${\mathcal E}(t)$ in $A(t)$,
\begin{multline*}
A(t)={\mathcal E} (t) - \frac{1}{2} \| \dot{x}(t)+ \beta \nabla f (x(t)) \|^2 - \sqrt{\mu} \dotp{ x(t) -x^\star }{\dot{x}(t) + \beta \nabla f (x(t))} + \sqrt{\mu} \dotp{ x(t) -x^\star }{\dot{x}(t)} \\
+ \| \dot{x}(t) \|^2 + \beta \dotp{\nabla f (x(t))}{\dot{x}(t)} + \frac{\beta}{\sqrt{\mu}} \| \nabla f (x(t)) \|^2 .
\end{multline*}
After developing and simplifying, we obtain
\begin{equation*}
\dot{{\mathcal E}} (t) + \sqrt{\mu}\pa{{\mathcal E} (t)
+ \frac{1}{2} \| \dot{x}(t) \|^2 + \pa{\frac{\beta}{\sqrt{\mu}}-\frac{\beta^2}{2}} \| \nabla f (x(t)) \|^2 - \beta\sqrt{\mu} \dotp{ x(t) -x^\star }{ \nabla f (x(t))}}
\leq \| v(t)\| \| g(t) \| .
\end{equation*}
Since $ 0 \leq \beta \leq \frac{1}{\sqrt{\mu}}$, it holds that
$\frac{\beta}{\sqrt{\mu}}-\frac{\beta^2}{2} \geq \frac{\beta}{2\sqrt{\mu}}$. Hence
\begin{equation*}
\dot{{\mathcal E}} (t) + \sqrt{\mu}\pa{{\mathcal E} (t)
+ \frac{1}{2} \| \dot{x}(t) \|^2 + \frac{\beta}{2\sqrt{\mu}}\| \nabla f (x(t)) \|^2 - \beta\sqrt{\mu} \dotp{ x(t) -x^\star }{ \nabla f (x(t))}}
\leq \| v(t)\| \| g(t) \| .
\end{equation*}
Let us use again the strong convexity of $f$ to write
\[
{\mathcal E} (t) = \frac{1}{2}{\mathcal E} (t) + \frac{1}{2}{\mathcal E} (t) \geq \frac{1}{2}{\mathcal E} (t)+ \frac{1}{2} \pa{f(x(t))- \bar{f}} \geq
\frac{1}{2}{\mathcal E} (t) + \frac{\mu}{4} \| x(t) -x^\star \|^2 .
\]
By combining the two inequalities above, we obtain
\[
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) + \frac{\sqrt{\mu}}{2}\| \dot{x} (t)\|^2 + \sqrt{\mu} B(t) \leq \| v(t)\| \| g(t) \|,
\]
where $B(t) = \frac{\mu}{4} \| x(t) -x^\star \|^2 + \frac{\beta}{2\sqrt{\mu}} \|\nabla f(x(t))\|^2 - \beta\sqrt{\mu} \|x(t) -x^\star\| \|\nabla f(x(t)) \| $.
Set $X=\| x -x^\star\|$, $Y= \| \nabla f (x) \|$. Elementary algebraic computation gives that, under the condition $ 0 \leq \beta \leq \frac{1}{2\sqrt{\mu}}$
\[
\frac{\mu}{4} X^2 + \frac{\beta}{2\sqrt{\mu}} Y^2 - \beta\sqrt{\mu}XY \geq 0.
\]
Hence for $ 0 \leq \beta \leq \frac{1}{2\sqrt{\mu}}$
\begin{equation}\label{basic_diff_ineq_1}
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) + \frac{\sqrt{\mu}}{2}\| \dot{x} (t)\|^2 \leq \| v(t)\| \| g(t) \|.
\end{equation}
\begin{enumerate}[label={\rm (\roman*)},leftmargin=3ex]
\item From \eqref{basic_diff_ineq_1}, we first deduce that
\[
\dot{{\mathcal E}} (t) \leq \| v(t)\| \| g(t) \|,
\]
which by integration gives
\[
{\mathcal E} (t) \leq {\mathcal E}(t_0) + \int_{t_0}^t \| v(\tau)\| \| g(\tau) \|d\tau.
\]
By definition of ${\mathcal E}(t)$, we have ${\mathcal E}(t) \geq \demi \|v(t)\|^2 $, which gives
\[
\demi \|v(t)\|^2 \leq {\mathcal E}(t_0) + \int_{t_0}^t \| v(\tau)\| \| g(\tau) \|d\tau.
\]
According to Lemma~\ref{lem:BrezisA5}, we obtain
\[
\|v(t)\| \leq \sqrt{2 {\mathcal E}(t_0)} + \int_{t_0}^t \| g(\tau) \|d\tau.
\]
Set $M := \sqrt{2 {\mathcal E}(t_0)} + \int_{t_0}^{+\infty} \| g(\tau) \|d\tau$. By assumption, $\DS{\int_{t_0}^{+\infty} \| g(\tau) \|d\tau <+\infty}$, and thus $\sup_{t \geq t_0} \|v(t)\| \leq M < +\infty$.
Returning to \eqref{basic_diff_ineq_1} we deduce that
\begin{equation}\label{basic_diff_ineq_2}
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) + \frac{\sqrt{\mu}}{2}\| \dot{x} (t)\|^2\leq M \| g(t) \|.
\end{equation}
Therefore
\begin{equation}\label{basic_diff_ineq_3}
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) \leq M \| g(t) \|.
\end{equation}
By integrating the differential inequality above, we obtain
\begin{equation}\label{basic_diff_ineq_4}
{\mathcal E} (t) \leq {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\int_{t_0}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| g(\tau) \| d\tau.
\end{equation}
We now use Lemma~\ref{basic-int}, which is the continuous version of Kronecker's Theorem for series, with $f(t)= \|g(t)\| $ and $\varphi(t)= e^{\frac{\sqrt{\mu}}{2}t}$. By assumption we have $ \int_{t_0}^{+\infty} \| g(\tau) \|d\tau <+\infty$.
We deduce that
\[
\lim_{t\to +\infty} \frac{1}{e^{\frac{\sqrt{\mu}}{2}t}}\int_{t_0}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| g(\tau) \| d\tau =0.
\]
Therefore, from \eqref{basic_diff_ineq_4} we obtain
\[
\lim_{t\to +\infty} {\mathcal E} (t) =0.
\]
By definition of ${\mathcal E} (t)$ this implies
\begin{eqnarray}
&&\lim_{t\to +\infty} f(x(t))- \min_{{\mathcal H}} f =0 \label{conv1} \\
&&\lim_{t\to +\infty} \| \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)+ \beta \nabla f (x(t)) \|=0. \label{conv2}
\end{eqnarray}
Acoording to \eqref{conv1} and the strong convexity of $f$ we deduce that
\[
\lim_{t\to +\infty}\|x(t) -x^\star\|=0
\]
By continuity of $\nabla f $, and since $\nabla f (x^\star)=0$, we deduce that
\[
\lim_{t\to +\infty}\| \nabla f (x(t))\|=0.
\]
Combining the above results with \eqref{conv2}, we deduce that
\[
\lim_{t\to +\infty}\| \dot{x}(t)\|=0.
\]
\item Let us make precise the argument developed above, and assume that, as $t \to +\infty$
\[
\| g(t) \| = {\mathcal O} \pa{ \frac{1}{t^p} },
\]
where $p>0$. Then, from \eqref{basic_diff_ineq_4} we get
\begin{eqnarray*}
{\mathcal E} (t) &\leq& {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\Big( \int_{t_0}^{\frac{t}{2}} e^{\frac{\sqrt{\mu}}{2}\tau} \| g(\tau) \| d\tau
+ \int_{\frac{t}{2}}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| g(\tau) \| d\tau \Big)\\
&\leq& {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\Big( C_1 e^{\frac{\sqrt{\mu}t}{4}}
+ \int_{\frac{t}{2}}^t e^{\frac{\sqrt{\mu}}{2}\tau} \frac{C_2}{\tau^p} d\tau \Big)\\
&\leq& {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\Big( C_1 e^{\frac{\sqrt{\mu}t}{4}}
+ \frac{C_2}{t^p} e^{\frac{\sqrt{\mu}}{2}t} \Big)\\
&\leq & {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M\Big( C_1 e^{-\frac{\sqrt{\mu}t}{4}}
+ \frac{C_2}{t^p} \Big) \\
&=& {\mathcal O}\pa{\frac{1}{t^p} }.
\end{eqnarray*}
By definition of ${\mathcal E} (t)$ and strong convexity of $f$, we infer
\[
\frac{\mu}{2}\norm{x(t)-x^\star}^2 \leq f(x(t)) - \min f({\mathcal H}) = {\mathcal O}\pa{\frac{1}{t^p} } \enskip \text{and} \enskip
\|\sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)+ \beta \nabla f (x(t)) \|^2 = {\mathcal O}\pa{\frac{1}{t^p} } .
\]
Developing the left-hand side of the last expression, we obtain
\begin{multline*}
\mu \| x(t) -x^\star\|^2 + \| \dot{x}(t)\|^2 + \beta^2 \|\nabla f (x(t))\|^2 + 2\beta \sqrt{\mu}\dotp{ x(t) -x^\star}{\nabla f (x(t))} \\
+ \dotp{\dot{x}(t)}{ 2 \beta\nabla f (x(t)) + 2\sqrt{\mu} ( x(t) -x^\star)} \leq \frac{C}{t^p}.
\end{multline*}
By convexity of $f$, we have $\dotp{x(t) -x^\star}{\nabla f (x(t))} \geq f(x(t)) - \bar{f}$. Moreover,
\begin{equation*}
\dotp{\dot{x}(t)}{ 2 \beta\nabla f (x(t)) + 2\sqrt{\mu} ( x(t) -x^\star)}
= \frac{d}{dt} \pa{2 \beta (f(x(t)) - \bar{f} )+ \sqrt{\mu} \| x(t) -x^\star\|^2 }.
\end{equation*}
Combining the above results, we obtain
\begin{equation*}
\sqrt{\mu} \pa{2 \beta (f(x(t)) - \bar{f}) +\sqrt{\mu} \| x(t) -x^\star\|^2} + \beta^2 \| \nabla f (x(t))\|^2
+ \frac{d}{dt} \pa{2 \beta (f(x(t)) - \bar{f})+ \sqrt{\mu} \| x(t) -x^\star\|^2 } \leq \frac{C}{t^p}.
\end{equation*}
Set $Z(t):= 2 \beta (f(x(t)) - \bar{f}) +\sqrt{\mu} \| x(t) -x^\star\|^2$. We have
\[
\frac{d}{dt} Z(t) + \sqrt{\mu} Z(t) + \beta^2 \| \nabla f (x(t))\|^2
\leq \frac{C}{t^p}.
\]
By integrating this differential inequality, elementary computation gives
\[
e^{- \sqrt{\mu}t} \int_{t_0}^t e^{ \sqrt{\mu}s}\| \nabla f (x(s))\|^2 ds \leq \frac{C}{t^p}.
\]
This completes the proof. \qed
\end{enumerate}
\end{proof}
\subsection{Implicit Hessian Damping}\label{s:approxsconv}
We now turn to the implicit Hessian system, and take in the Polyak heavy ball system a fixed positive damping coefficient which is adjusted to the modulus of strong convexity of $f$. This gives the system
\begin{equation}\label{dyn-sc-implicit}
\ddot{x}(t) + 2\sqrt{\mu} \dot{x}(t) + \nabla f \pa{x(t)+ \beta\dot{x}(t)} + e(t)=0.
\end{equation}
To analyze \eqref{dyn-sc-implicit}, we define the function ${\mathcal E} : [t_0, +\infty[ \to {\mathbb R}_+ $
\begin{align}
t \mapsto {\mathcal E} (t) := f\pa{x(t)+ \beta\dot{x}(t))} - \min_{{\mathcal H}} f + \frac{1}{2} \| \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)\|^2.
\end{align}
\begin{theorem}\label{strong-conv-thm-implicit}
Suppose that $f: {\mathcal H} \to {\mathbb R}$ is $\mu$-strongly convex for some $\mu >0$, and let $x^\star$ be the unique minimizer of $f$. Let $x(\cdot): [t_0, + \infty[ \to {\mathcal H}$ be a solution trajectory of \eqref{dyn-sc-implicit}.
Suppose that
\begin{enumerate}[label=\alph*)]
\item $\DS{ 0 \leq \beta \leq \frac{1}{2\sqrt{\mu}}}$.
\item$\DS{\int_{t_0}^{+\infty} \|e(t)\| dt< +\infty}$.
\end{enumerate}
{Then the following properties are satisfied:}
\begin{enumerate}[label={\rm (\roman*)}]
\item Minimizing properties: there exists a positive constant $M$ such that for all $t\geq t_0$
\[
{\mathcal E} (t) \leq {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\int_{t_0}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| e(\tau) \| d\tau.
\]
More precisely,
\[
M := \sqrt{\frac{{\mathcal E}(t_0)}{c}} + \frac{1}{2c}\int_{t_0}^{+\infty} \| e(\tau) \|d\tau \; \mbox{ with } \; c = \frac{\min\{\mu,1\}}{4 \max\{\beta^2L^2,1\}}
\]
and $L$ is the Lipschitz constant of $\nabla f$. Consequently,
\begin{eqnarray*}
&& \lim_{t\to +\infty} {\mathcal E} (t) =0; \; \lim_{t\to +\infty} f(x(t)) = \min_{{\mathcal H}} f \\
&& \lim_{t\to +\infty}\| x(t) - x^\star\|=
\lim_{t\to +\infty}\| \nabla f (x(t))\|=
\lim_{t\to +\infty}\| \dot{x}(t)\|=0.
\end{eqnarray*}
\item Convergence rates: suppose moreover that for some $p>0$,
$
\DS{\|e(t) \| = {\mathcal O}\pa{ \frac{1}{t^p}}},
$ as $t \to +\infty$.
Then
$
{\mathcal E} (t)= {\mathcal O} \pa{ \frac{1}{t^p} },
$
{\it i.e.} ${\mathcal E} (t)$ inherits the decay rate of the error terms. In turn, as $t \to +\infty$
\begin{eqnarray*}
&& f\pa{x(t)} - \min_{{\mathcal H}} f = {\mathcal O}\pa{\frac{1}{t^p} };\\
&& \|x(t) -x^\star\|^2= {\mathcal O}\pa{\frac{1}{t^p} }; \;
\| \dot{x}(t)\|^2= {\mathcal O}\pa{\frac{1}{t^p} }; \; \|\nabla f(x(t))\|^2 = {\mathcal O} \pa{\frac{1}{t^p}}.
\end{eqnarray*}
\end{enumerate}
\end{theorem}
\begin{proof}
Let us define
\begin{equation}\label{dyn-sc-d-implicit}
v(t)= \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t).
\end{equation}
and thus, ${\mathcal E}$ equivalently reads
\begin{equation}\label{dyn-sc-c-implicit}
{\mathcal E} (t) = f\pa{x(t)+ \beta\dot{x}(t)} - \min_{{\mathcal H}} f + \frac{1}{2} \| v(t) \|^2 .
\end{equation}
Taking the derivative in time of ${\mathcal E} (\cdot) $ gives
\begin{align*}
\dot{{\mathcal E}} (t)&=\dotp{\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{\dot{x}(t) + \beta \ddot{x}(t)} + \dotp{ v(t)}{\dot{v}(t)}\\
&= \dotp{\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{ \dot{x}(t) + \beta \ddot{x}(t)} + \dotp{ \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)}{\sqrt{\mu} \dot{x}(t) + \ddot{x}(t)}.
\end{align*}
Using the constitutive equation \eqref{dyn-sc-b}, we get
\begin{multline*}
\dot{{\mathcal E}} (t) = \dotp{ \nabla f\pa{x(t)+ \beta\dot{x}(t)}}{ (1-2\beta \sqrt{\mu})\dot{x}(t) - \beta \nabla f\pa{x(t)+ \beta\dot{x}(t)}-\beta e(t) } \\
+ \dotp{ \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)}{-\sqrt{\mu} \dot{x}(t) - \nabla f\pa{x(t)+ \beta\dot{x}(t)} -e(t) }.
\end{multline*}
After developing and simplifying, we obtain
\begin{multline*}
\dot{{\mathcal E}} (t) +2\beta \sqrt{\mu}\dotp{ \nabla f\pa{x(t)+ \beta\dot{x}(t)}}{ \dot{x}(t)} + \sqrt{\mu}\dotp{ \nabla f\pa{x(t)+ \beta\dot{x}(t)}}{x(t) -x^\star } + \beta \| \nabla f\pa{x(t)+ \beta\dot{x}(t)}\|^2 \\
+ \sqrt{\mu} \| \dot{x}(t) \|^2 + \mu \dotp{ x(t) -x^\star }{ \dot{x}(t)} = -\dotp{ \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)
+\beta\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{e(t)}.
\end{multline*}
In view of strong convexity of $f$, we have
\begin{multline*}
\dotp{\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{ x(t) -x^\star} = \dotp{ \nabla f\pa{x(t)+ \beta\dot{x}(t)}}{x(t)+\beta\dot{x}(t) - x^\star} - \dotp{\nabla f\pa{x(t) + \beta\dot{x}(t)}}{ \beta\dot{x}(t)} \\
\geq f\pa{x(t)+ \beta\dot{x}(t)}- \bar{f} + \frac{\mu}{2} \| x(t) -x^\star + \beta\dot{x}(t) \|^2 - \dotp{\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{\beta\dot{x}(t)} .
\end{multline*}
Thus, by combining the last two relations, we obtain
\begin{multline}
\dot{{\mathcal E}} (t) +\beta \sqrt{\mu}\dotp{ \nabla f\pa{x(t)+ \beta\dot{x}(t)}}{\dot{x}(t)} + \sqrt{\mu}\pa{f\pa{x(t)+ \beta\dot{x}(t)}- \bar{f} + \frac{\mu}{2} \| x(t) -x^\star +\beta\dot{x}(t) \|^2} \\
+ \beta \|\nabla f\pa{x(t)+ \beta\dot{x}(t)}\|^2 + \sqrt{\mu} \| \dot{x}(t) \|^2 + \mu \dotp{ x(t) -x^\star }{\dot{x}(t)} \leq \| w(t) \| \|e(t)\|, \label{Lyap_22}
\end{multline}
where we have used Cauchy-Schwarz inequality, and we set
\[
w(t) := \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)
+\beta\nabla f\pa{x(t)+ \beta\dot{x}(t)}.
\]
Let us make ${\mathcal E} (t)$ appear on the left-hand side of \eqref{Lyap_22}. We get
\[
\dot{{\mathcal E}} (t) + \sqrt{\mu}{\mathcal E} (t) + B(t) \leq \| w(t)\| \| e(t) \|
\]
where
\begin{equation*}
B(t) := \beta \| \nabla f\pa{x(t)+ \beta\dot{x}(t)}\|^2
+ \frac{\sqrt{\mu}}{2} (\beta^2 \mu +1)\| \dot{x}(t) \|^2+ \beta \sqrt{\mu}\dotp{\nabla f\pa{x(t)+ \beta\dot{x}(t)}}{\dot{x}(t)} \\
+ \beta \mu \sqrt{\mu} \dotp{ x(t) -x^\star }{ \dot{x}(t)} .
\end{equation*}
Let us use again the strong convexity of $f$ to write
\[
{\mathcal E} (t) = \frac{1}{2}{\mathcal E} (t) + \frac{1}{2}{\mathcal E} (t) \geq \frac{1}{2}{\mathcal E} (t)+ \frac{1}{2} \pa{f(x(t)+ \beta\dot{x}(t))- \bar{f}} \geq
\frac{1}{2}{\mathcal E} (t) + \frac{\mu}{4} \| x(t) -x^\star + \beta\dot{x}(t)\|^2 .
\]
By combining the inequalities above, we obtain
\[
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t)+ C(t) \leq \| w(t)\| \| e(t) \|,
\]
where
\begin{multline*}
C(t) :=
\beta \| \nabla f (y(t) \|^2
+ \beta \sqrt{\mu}\dotp{\nabla f(y(t))}{\dot{x}(t)} + \frac{\sqrt{\mu}}{2} (\beta^2 \mu +1)\| \dot{x}(t) \|^2 + \beta \mu \sqrt{\mu} \dotp{x(t) -x^\star }{\dot{x}(t)} \\
+\frac{\mu\sqrt{\mu}}{4} \| x(t) -x^\star + \beta\dot{x}(t)\|^2 ,
\end{multline*}
and we set $y(t) := x(t)+ \beta\dot{x}(t) $.
Let us show that, for an adequate choice of the parameters, $C(t)$ is non-negative. Let us reformulate $C(t)$ as follows:
Young's inequality gives the following minorization for the two first terms of $C(t)$
\[
\beta \| \nabla f (y(t)) \|^2
+ \beta \sqrt{\mu}\langle \nabla f (y(t)), \dot{x}(t) \rangle \geq -\frac{1}{4}\beta \mu \| \dot{x}(t) \|^2 .
\]
By using this inequality in $C(t)$, and after simplification, we arrive at
\begin{align*}
&C(t)\geq
\pa{\frac{\sqrt{\mu}}{2} (\beta^2 \mu +1)-\frac{1}{4}\beta \mu}\| \dot{x}(t) \|^2 + \beta \mu \sqrt{\mu} \dotp{ x(t) -x^\star }{\dot{x}(t)}
+\frac{\mu\sqrt{\mu}}{4} \| x(t) -x^\star + \beta\dot{x}(t)\|^2 \\
&= \frac{\mu\sqrt{\mu}}{4} \| x(t) -x^\star + \beta\dot{x}(t)\|^2 + \pa{\frac{\sqrt{\mu}}{2} (\beta^2 \mu +1)-\frac{1}{4}\beta \mu
- \beta^2 \mu \sqrt{\mu}}\| \dot{x}(t) \|^2 + \beta \mu \sqrt{\mu} \dotp{ x(t) -x^\star + \beta\dot{x}(t)}{ \dot{x}(t) } \\
&= \frac{\mu\sqrt{\mu}}{4} \| x(t) -x^\star + \beta\dot{x}(t)\|^2 + \sqrt{\mu}\pa{-\frac{\beta^2 \mu }{2} -\frac{1}{4}\beta \sqrt{\mu}
+ \frac{1}{2}}\| \dot{x}(t) \|^2 + \beta \mu \sqrt{\mu} \dotp{ x(t) -x^\star + \beta\dot{x}(t)}{ \dot{x}(t)}.
\end{align*}
Elementary algebra gives that
$
-\frac{\beta^2 \mu }{2} -\frac{1}{4}\beta \sqrt{\mu} + \frac{1}{2} \geq 0
$
if and only if
$
\beta \sqrt{\mu} \leq \frac{\sqrt{17}-1}{4}.
$
According to the classical rule for the sign of a quadratic function of a real variable, we get that $C(t) \geq 0$ under the condition
\[
(\beta \mu \sqrt{\mu})^2 \leq \mu^2\pa{-\frac{\beta^2 \mu }{2} -\frac{1}{4}\beta \sqrt{\mu}
+ \frac{1}{2}}.
\]
Setting $Z= \beta \sqrt{\mu}$, the latter inequality is equivalent to ensuring
$$
\frac{3}{2} Z^2 + \frac{1}{4}Z - \demi \leq 0.
$$
which is satisfied for $0 \leq Z \leq \demi$, implying
$
\beta \leq \frac{1}{2 \sqrt{\mu}}.
$
Since $\demi < \frac{\sqrt{17}-1}{4}$, we get as a final condition
\[
\beta \leq \frac{1}{2 \sqrt{\mu}}.
\]
Thus under this condition we get
\begin{equation}\label{basic-equ-implit-1}
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) \leq \| w(t)\| \| e(t) \|.
\end{equation}
From \eqref{basic-equ-implit-1}, we first deduce that
\[
\dot{{\mathcal E}} (t) \leq \| w(t)\| \| e(t) \|,
\]
which, after integration, gives
\[
{\mathcal E} (t) \leq {\mathcal E}(t_0) + \int_{t_0}^t \| w(\tau)\| \| e(\tau) \| d\tau.
\]
By definition of $w$ we have
\begin{align*}
\| w(t)\|
&\leq \| v(t)\| + \beta \| \nabla f\pa{x(t)+ \beta\dot{x}(t)}- \nabla f (x^\star) \| \\
&\leq \| v(t)\| + \beta L \| x(t)- x^\star + \beta\dot{x}(t) \| ,
\end{align*}
where $L$ is the Lipschitz constant of $\nabla f$. On the other hand, strong convexity of $f$ entails
\[
{\mathcal E} (t) \geq \frac{\mu}{2} \| x(t) -x^\star + \beta\dot{x}(t) \|^2 + \demi \| v(t)\|^2.
\]
Hence, there exists a positive constant $c$ such that\footnote{One can take $c = \frac{\min\{\mu,1\}}{4 \max\{\beta^2L^2,1\}}$.}
\[
{\mathcal E} (t) \geq c \|w(t)\|^2.
\]
This in turn gives
\[
c \|w(t)\|^2 \leq {\mathcal E}(t_0) + \int_{t_0}^t \| w(\tau)\| \| e(\tau) \|d\tau.
\]
According to Lemma~\ref{lem:BrezisA5}, and $ \int_{t_0}^{+\infty} \| e(\tau) \|d\tau <+\infty$, we deduce that
\[
\sup_{t \geq t_0} \|w(t)\| \leq M := \sqrt{\frac{{\mathcal E}(t_0)}{c}} + \frac{1}{2c}\int_{t_0}^{+\infty} \| e(\tau) \|d\tau < +\infty.
\]
Returning to \eqref{basic-equ-implit-1} we deduce that
\begin{equation}\label{basic_diff_ineq_3_implicit}
\dot{{\mathcal E}} (t) + \frac{\sqrt{\mu}}{2}{\mathcal E} (t) \leq M \| e(t) \|.
\end{equation}
By integrating the differential inequality above, we obtain
\begin{equation}\label{basic_diff_ineq_4_implicit}
{\mathcal E} (t) \leq {\mathcal E} (t_0) e^{-\frac{\sqrt{\mu}}{2}(t-t_0)} + M e^{-\frac{\sqrt{\mu}}{2}t}\int_{t_0}^t e^{\frac{\sqrt{\mu}}{2}\tau} \| e(\tau) \| d\tau .
\end{equation}
\begin{enumerate}[label={\rm (\roman*)},leftmargin=3ex]
\item We first deduce from \eqref{basic_diff_ineq_4_implicit} that ${\mathcal E} (t) $ tends to zero as $t\to +\infty$. This implies that
\begin{eqnarray}
&&\lim_{t\to +\infty} f( x(t)+ \beta\dot{x}(t))) = \min_{{\mathcal H}} f , \label{conv10}\\
&& \lim_{t\to +\infty} \| \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t)\| =0 . \label{conv11}
\end{eqnarray}
From \eqref{conv10} and strong convexity of $f$ we deduce that
\begin{equation}
\lim_{t\to +\infty} \| (x(t)-x^\star)+ \beta\dot{x}(t)\| =0 \label{conv12} .
\end{equation}
From \eqref{conv11} and \eqref{conv12}, and $\beta \neq \frac{1}{\sqrt{\mu}}$ (a consequence of the assumption $\beta \leq \frac{1}{2\sqrt{\mu} }$), elementary algebra gives
\[
\lim_{t\to +\infty} \| x(t)-x^\star\| = \lim_{t\to +\infty} \| \dot{x}(t)\| =0.
\]
In turn, continuity of $f$ and $\nabla f$ imply
\[
\lim_{t\to +\infty} \|\nabla f(x(t))\| = 0 \enskip \text{and} \enskip \lim_{t\to +\infty} f(x(t)) = \min_{{\mathcal H}} f .
\]
\item Let us now assume that, as $t \to +\infty$, we have
$
\| e(t) \| = {\mathcal O} \pa{ \frac{1}{t^p} },
$
where $p>0$. Based on \eqref{basic_diff_ineq_4_implicit}, {a similar argument as in the explicit case} (see the proof of Theorem~\ref{strong-conv-thm}) gives
$
{\mathcal E} (t) = {\mathcal O}\pa{\frac{1}{t^p}}.
$
By definition of ${\mathcal E} (t)$, we infer that
\begin{equation}\label{eq:energy_6_02}
f(x(t)+ \beta\dot{x}(t)) - \min_{{\mathcal H}} f = {\mathcal O}\pa{\frac{1}{t^p}}
\end{equation}
and
\begin{equation}\label{eq:energy_6_02_b}
\| \sqrt{\mu} (x(t) -x^\star) + \dot{x}(t) \|^2 = {\mathcal O}\pa{\frac{1}{t^p}}.
\end{equation}
From \eqref{eq:energy_6_02} and strong convexity of $f$ we deduce that
\begin{eqnarray}\label{conv12_b}
\| (x(t)-x^\star)+ \beta\dot{x}(t)\|^2 = {\mathcal O}\pa{\frac{1}{t^p}} .
\end{eqnarray}
Combining \eqref{eq:energy_6_02_b} and \eqref{conv12_b}, and recalling that $\beta \sqrt{\mu} \neq 1$ we immediately obtain
\begin{equation}\label{eq:convratevel}
\|x(t)-x^\star\|^2 \leq \frac{C}{t^p} \enskip \text{and} \enskip \| \dot{x}(t)\|^2 = {\mathcal O}\pa{\frac{1}{t^p}}.
\end{equation}
According to the Lipschitz continuity of $\nabla f$, and
$\nabla f (x^\star) =0 $ we deduce that
\[
\|\nabla f(x(t))\|^2 \leq L^2 \| x(t)-x^\star\|^2 = {\mathcal O}\pa{\frac{1}{t^p}} .
\]
Now, combining the descent lemma with \eqref{eq:energy_6_02}, \eqref{conv12_b} and \eqref{eq:convratevel} shows that
\begin{align*}
f(x(t))- \min_{{\mathcal H}} f
&\leq f(x(t)+\beta\dot{x}(t)) - \min_{{\mathcal H}} f - \beta\dotp{\nabla f(x(t)+\beta\dot{x}(t))}{\dot{x}(t)} + \frac{L\beta^2}{2}\norm{\dot{x}(t)}^2 \\
&\leq f(x(t)+\beta\dot{x}(t))-\inf_{{\mathcal H}} f + L\beta\norm{x(t)-x^\star+\beta\dot{x}(t)}\norm{\dot{x}(t)} + \frac{L\beta^2}{2}\norm{\dot{x}(t)}^2 \\
&= {\mathcal O}\pa{\frac{1}{t^p}} ,
\end{align*}
which completes the proof. \qed
\end{enumerate}
\end{proof}
\begin{remark}
The results of Theorem~\ref{strong-conv-thm-implicit} appear new. Even for the unperturbed case of system \eqref{dyn-sc-implicit}, where $e \equiv 0$, we are not aware of any guarantees for these dynamics in the literature.
\end{remark}
\section{The Non-smooth Case}\label{s:nonsmooth}
\subsection{Explicit Hessian Damping}
In the sequel, we will show that most properties obtained in the smooth case still hold for the global strong solution of \eqref{eq:fos2} (and in particular, all properties that do not require $x(t)$ to be twice differentiable).
\subsubsection{Minimizing properties}
{From now on, we assume that, for all $T >t_0$}, \; $e(\cdot)\in {\mathcal W}^{1,1} (t_0,T; {\mathcal H})$. Let $(x,y):[t_0,+\infty[\to{\mathcal H}\times{\mathcal H}$ be the global strong solution to \eqref{eq:fos2} with Cauchy data $(x(t_0),y(t_0))=(x_0,y_0)\in\dom(f)\times{\mathcal H}$.
For $t\geq t_0$ define
\begin{equation} \label{u0}
u(t)=\int_{t_0}^t \pa{-\beta e(s)+\pa{\frac{1}{\beta}-\frac{\alpha}{s}}x(s)-\frac{1}{\beta}y(s)}ds.
\end{equation}
Thus $u$ is continuously differentiable, with derivative satisfying
\begin{align}
\dot u(t)
& = -\beta e(t) +\pa{\frac{1}{\beta}-\frac{\alpha}{t}}x(t)-\frac{1}{\beta}y(t), \quad \forall t\geq t_0, \label{guniter} \\
& = \dot x(t)+\beta \xi(t),\mbox{\hspace{6em}for almost all }t>t_0, \label{gubis}
\end{align}
where $\xi(t) \in \partial f(x(t))$, and the last equality follows from Theorem~\ref{Thm-existence}\ref{existence-6}-\ref{existence-6:b}. Therefore, $u$ can be also written equivalently as
\[
u(t)=x(t)-x_0+ \beta \DS{ \int_{t_0}^t \xi(s) ds}.
\]
With parts {\ref{existence-1} }and {\ref{existence-2}} of Theorem~\ref{Thm-existence}, equality \eqref{guniter} shows that $\dot u$ is absolutely continuous on
any compact subinterval of $[t_0,+\infty[$, hence differentiable almost everywhere on $[t_0,+\infty[$. Therefore,
\[
\ddot u(t)= -\beta \dot{e}(t)+\frac{\alpha}{t^2}x(t)+\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot x(t) -\frac{1}{\beta}\dot y(t).
\]
The equality above, combined with
$\dot y(t)=\frac{\alpha\beta}{t^2} x(t) +\dot x(t)+\beta(\xi(t) +e(t))$ (which is obtained by taking the difference of the two equations in \eqref{eq:fos2}), yields
\begin{equation} \label{dguniter0}
\ddot u(t)=-\frac{\alpha}{t}\dot x(t)-\xi(t) -(e(t)+ \beta \dot{e}(t)),
\end{equation}
for almost all $t>t_0$. Using \eqref{gubis}, we obtain
\begin{align}
\ddot u(t)
&=\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot x(t)-\frac{1}{\beta}\dot u(t) -(e(t)+ \beta \dot{e}(t))
\label{dguniter}
\end{align}
for almost all $t>t_0$. We will need the following energy function of the system, defined for all
$T\geq t\geq t_0$ (recall \eqref{guniter} for the definition of $\dot u(t)$):
\begin{equation}\label{gE:W}
W_T(t) := \frac{1}{2}\|\dot u(t)\|^2 + f (x(t)) - \int_t^T \langle \dot{u}(\tau),e(\tau) +\beta \dot{e}(\tau)\rangle d\tau,
\end{equation}
and when the following expression is well-defined (we will prove it later)
\begin{equation}\label{gE:W_b}
W(t) := \frac{1}{2}\|\dot u(t)\|^2 + f (x(t)) - \int_t^{+\infty} \langle \dot{u}(\tau),e(\tau) +\beta \dot{e}(\tau)\rangle d\tau.
\end{equation}
\begin{theorem} \label{Thm-g-weak-conv2}
{Let $\alpha>0$. Suppose that $\inf_{{\mathcal H}} f > -\infty$. Suppose that $e(\cdot)\in {\mathcal W}^{1,1} (t_0,T; {\mathcal H})$ for all $T > t_0$, with
$\DS{\int_{t_0}^{+\infty} \| e(t)\| <+\infty}$ and $\DS{\int_{t_0}^{+\infty} \| \dot{e}(t)\| <+\infty}$. Then
for any global strong solution of \eqref{eq:fos2}, $(x,y):[t_0,+\infty[\to{\mathcal H}\times{\mathcal H}$
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{gweak-conv2-1}
$W$ is well-defined and non-increasing on $[t_1,+\infty[$ for some $t_1 \geq t_0$.
\item \label{gweak-conv2-5}
$\DS{\int_{t_0}^{+\infty}\frac{1}{t}\|\dot x(t)\|^2dt<+\infty}$,
$\DS{\int_{t_0}^{+\infty}\frac{1}{t}\|\xi(t)\|^2dt<+\infty}$.
\item \label{gweak-conv2-2}
$\lim_{t\to+\infty}W(t)=\lim_{t\to+\infty}f(x(t))=\inf_{{\mathcal H}} f \in {\mathbb R}\cup\{-\infty\}$, \ $\lim_{t\to+\infty}\|\dot x(t) +\beta \xi(t)\|=0$.
\item \label{gweak-conv2-3}
As $t\to+\infty$, every sequential weak cluster point of $x(t)$ belongs to $S$.
\item\label{gweak-conv2-6}
If, moreover, the solution set $S\neq\emptyset$ and $\DS{\int_{t_0}^{+\infty} \log t ~ \| e(t)\| <+\infty}$ and $\DS{\int_{t_0}^{+\infty} \log t ~ \| \dot{e}(t)\| < +\infty}$, then
\begin{enumerate}[label={\rm (\alph*)}]
\item \label{gweak-conv2-6a}
$f(x(t))-\inf_{{\mathcal H}} f=\DS{{\mathcal O}\pa{\frac{1}{\log t}}}$ and
$\|\dot u(t)\|=\DS{{\mathcal O}\pa{\frac{1}{\sqrt{\log t}}}}$ as $t \to +\infty$.
\item \label{gweak-conv2-6b}
$\DS{\int_{t_0}^{+\infty}\frac{1}{t}(f(x(t))-\inf_{{\mathcal H}} f)dt < +\infty}$.
\end{enumerate}
\end{enumerate}
}
\end{theorem}
\begin{proof}
Since we are interested in asymptotic analysis, we can assume $t \geq t_1=\max\pa{t_0,2\alpha\beta}$.
\paragraph{Claim~\ref{gweak-conv2-1}}
According to Theorem \ref{Thm-existence}, $W_T$ is absolutely continuous. Taking the derivative and using the chain rule we get
\[
\dot W_T(t)=\langle\dot u(t),\ddot u(t)\rangle+\langle\xi(t),\dot x(t)\rangle +\langle \dot{u}(t), e(t)+ \beta \dot{e}(t)\rangle,
\]
for almost every $T> t>t_0$. Now use \eqref{gubis} and \eqref{dguniter} to obtain
\begin{align*}
\dot W_T(t)& =
\dotp{\dot u(t)}{\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot x(t)
-\frac{1}{\beta}\dot u(t) -(e(t)+ \beta \dot{e}(t))} +\dotp{\xi(t)}{\dot x(t)} +\dotp{ \dot{u}(t)}{e(t)+ \beta \dot{e}(t)} \nonumber \\
& =
\dotp{\dot u(t)}{\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot x(t)
-\frac{1}{\beta}\dot u(t)} +\dotp{\xi(t)}{\dot x(t)} \nonumber \\
& =
-\frac{1}{\beta}\|\dot u(t)\|^2
+ \dotp{\dot x(t)}{\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot u(t) + \xi(t) } \nonumber \\
& =
-\frac{1}{\beta}\|\dot u(t)\|^2
+ \dotp{\dot x(t)}{\pa{\frac{1}{\beta}-\frac{\alpha}{t}}\dot u(t) + \frac{1}{\beta}(\dot u(t)-\dot x(t))} \nonumber \\
& =
-\frac{1}{\beta}\|\dot u(t)\|^2
+ \dotp{\dot x(t)}{\pa{\frac{2}{\beta}-\frac{\alpha}{t}}\dot u(t) - \frac{1}{\beta}\dot x(t)} \nonumber \\
& =
-\frac{1}{\beta}\|\dot u(t)\|^2 -\frac{1}{\beta}\|\dot x(t)\|^2
+ \pa{\frac{2}{\beta}-\frac{\alpha}{t}} \dotp{\dot x(t)}{\dot u(t)} \nonumber \\
& \leq
-\frac{\alpha}{2t}\|\dot x(t)\|^2-\frac{\alpha}{2t}\|\dot u(t)\|^2,
\end{align*}
for almost every
$t \geq t_1$. So $W_T$ is non-increasing on $[t_1,+\infty[$, because it is absolutely continuous and its derivative is non-positive therein.
Therefore $W_T (t) \leq W_T (t_1)$ for all $t\in [t_1, T]$.
Equivalently
\[
\frac{1}{2}\|\dot u(t)\|^2 + f (x(t)) - \int_t^T \langle \dot{u}(\tau),e(\tau) +\beta \dot{e}(\tau)\rangle d\tau
\leq \frac{1}{2}\|\dot u(t_1)\|^2 + f (x(t_1)) - \int_{t_1}^T \langle \dot{u}(\tau),e(\tau) +\beta \dot{e}(\tau)\rangle d\tau.
\]
After simplification, and setting $C= \frac{1}{2}\|\dot u(t_1)\|^2 + f (x(t_1))- \inf f({\mathcal H})$, we obtain
\[
\frac{1}{2}\|\dot u(t)\|^2
\leq C - \int_{t_1}^t \langle \dot{u}(\tau),e(\tau) + \beta \dot{e}(\tau)\rangle d\tau.
\]
By Cauchy-Schwarz inequality we get
\[
\frac{1}{2}\|\dot u(t)\|^2
\leq C + \int_{t_0}^t \|\dot{u}(\tau)\| \|e(\tau) +\beta \dot{e}(\tau)\| d\tau.
\]
According to {Gronwall's} Lemma~\ref{lem:BrezisA5}
\begin{equation}\label{gE:u_b}
\|\dot u(t)\|
\leq \sqrt{2C} + \int_{t_0}^t \|e(\tau) +\beta \dot{e}(\tau)\| d\tau \leq M := \sqrt{2C} + \int_{t_0}^{+\infty} \|e(\tau) +\beta \dot{e}(\tau)\| d\tau.
\end{equation}
So, $\|\dot u(t)\|$ is bounded on $[t_0, +\infty[$, which allows us to define
\begin{equation}\label{gE:W_bb}
W(t)=\frac{1}{2}\|\dot u(t)\|^2 + f (x(t)) - \int_t^{+\infty} \dotp{\dot{u}(\tau)}{e(\tau) +\beta \dot{e}(\tau)} d\tau.
\end{equation}
Noticing that $W$ and $W_T$ have the same derivative we conclude that
\begin{equation}\label{gE:W_bbb}
\dot{W}(t) +\frac{\alpha}{2t}\|\dot x(t)\|^2 +\frac{\alpha}{2t}\|\dot u(t)\|^2 \leq 0,
\end{equation}
and thus $W$ is non-increasing on $[t_1,+\infty[$.
\paragraph{Claim~\ref{gweak-conv2-5}}
Integrating \eqref{gE:W_bbb}, and using that $f$, and hence $W$, is bounded from below, we obtain,
\begin{equation}\label{gE:u_bb}
\int_{t_0}^{+\infty}\frac{1}{t}\|\dot x(t)\|^2dt<+\infty, \enskip \text{and} \enskip
\int_{t_0}^{+\infty}\frac{1}{t}\|\dot u(t)\|^2dt<+\infty .
\end{equation}
Using Jensen's inequality, we get the integrability claim on $\xi(t)$.
\paragraph{Claim~\ref{gweak-conv2-2}}
Given $z\in{\mathcal H}$, let us define $h:[t_0,+\infty [ \to{\mathbb R}_+$ by
$
h(t)=\frac{1}{2}\|u(t)-z\|^2.
$
The function $h$ is continuously differentiable with
\[
\dot h(t) = \dotp{ u(t) - z }{\dot{u}(t) },
\]
and $\dot h$ is absolutely continuous on compact subintervals of $[t_0,+\infty[$ (since $\dot u$ is) and satisfies
\[
\ddot h(t) = \dotp{ u(t) - z }{\ddot{u}(t) } + \| \dot{u}(t) \|^2
\]
for almost every $t>t_0$. Using \eqref{gubis} and \eqref{dguniter0}
we get
\[
\ddot u(t) + \frac{\alpha}{t} \dot u(t) = -\pa{1-\frac{\alpha\beta}{t}} \xi(t) -(e(t) +\beta \dot{e}(t)) .
\]
Therefore, for almost every $t>t_0$
\begin{eqnarray*}
&&\ddot h(t) + \frac{\alpha}{t} \dot h(t)
=
\|\dot u(t)\|^2
- \dotp{ u(t)-z}{\pa{1-\frac{\alpha\beta}{t}} \xi(t)} -\dotp{ u(t)-z}{e(t) +\beta \dot{e}(t)}
\\
&& =
\|\dot u(t)\|^2
- \pa{1-\frac{\alpha\beta}{t}} \dotp{x(t)-z -x_0 + \beta \int_{t_0}^t\xi(s)ds}{\xi(t)} -\dotp{ u(t)-z}{e(t) +\beta \dot{e}(t)}
\\
&& \leq
\|\dot u(t)\|^2
-\pa{1-\frac{\alpha\beta}{t}} \dotp{x(t)-z}{\xi(t)}
-\pa{1-\frac{\alpha\beta}{t}}
\dotp{-x_0+ \beta \int_{t_0}^t\xi(s)ds}{\xi(t)}
+ \| e(t) +\beta \dot{e}(t) \| \| u(t)-z \| .
\end{eqnarray*}
To interpret $ \left\langle-x_0+ \beta \int_{t_0}^t\xi(s)ds\,,\,\xi(t)\right\rangle$ as a temporal derivative, let us introduce
\begin{center}
$
I(t)=\frac{1}{2\beta}\left\|-x_0+ \beta\int_{t_0}^t\xi(s)ds\right\|^2.
$
\end{center}
Then $I(\cdot)$ is locally absolutely continuous and
$
\dot I(t)=\dotp{-x_0+\DS{\int_{t_0}^t\xi(s)ds}}{\xi(t)}
$
almost everywhere, because $\xi\in L^2(t_0,T;{\mathcal H})\subseteq L^1(t_0,T;{\mathcal H})$ for all $T >t_0$; see part {\ref{existence-6}-\ref{existence-6:c}} of Theorem \ref{Thm-existence}. So,
\[
\ddot h(t) + \frac{\alpha}{t} \dot h(t)\leq
\|\dot u(t)\|^2
-\pa{1-\frac{\alpha\beta}{t}}\dotp{x(t)-z}{\xi(t)}
-\pa{1-\frac{\alpha\beta}{t}}\dot I(t) +\| e(t) +\beta \dot{e}(t) \| \| u(t)-z \|,
\]
for almost every $t>t_0$. On the other hand, by convexity of $f$ and $\xi(t) \in \partial f(x(t))$
\[
\dotp{x(t)-z}{\xi(t)} \geq f(x(t) - f(z).
\]
Therefore
\[
\ddot h(t) + \frac{\alpha}{t} \dot h(t) + \pa{1-\frac{\alpha\beta}{t}}(f(x(t)) - f(z)) +\pa{1-\frac{\alpha\beta}{t}}\dot I(t)\leq \|\dot u(t)\|^2 + \| e(t) +\beta \dot{e}(t) \| \| u(t)-z \|.
\]
Using the definition \eqref{gE:W_bb} of $W$, we get
\begin{multline*}
\ddot h(t) + \frac{\alpha}{t} \dot h(t) +\pa{1-\frac{\alpha\beta}{t}}(W(t) - f(z)) +\pa{1-\frac{\alpha\beta}{t}}\dot I(t) \leq \pa{\frac{3}{2}-\frac{\alpha\beta}{2t}}\|\dot u(t)\|^2 \\
+ \| e(t) +\beta \dot{e}(t) \| \| u(t)-z \| -\pa{1-\frac{\alpha\beta}{t}}\int_t^{+\infty} \dotp{ \dot{u}(\tau)}{e(\tau) +\beta \dot{e}(\tau)} d\tau .
\end{multline*}
According to \eqref{gE:W_bbb}, we have $\|\dot u(t)\|^2 \leq -\frac{2t}{\alpha} \dot{W}(t)$. Therefore,
\begin{multline*}
\ddot h(t) + \frac{\alpha}{t} \dot h(t) +\pa{1-\frac{\alpha\beta}{t}}(W(t) - f(z)) +\pa{1-\frac{\alpha\beta}{t}}\dot I(t) \leq -\pa{\frac{3t}{\alpha}-\beta } \dot W(t) \\
+ \| e(t) +\beta \dot{e}(t) \| \| u(t)-z \| -\pa{1-\frac{\alpha\beta}{t}}\int_t^{+\infty} \dotp{ \dot{u}(\tau)}{e(\tau) +\beta \dot{e}(\tau)} d\tau .
\end{multline*}
Dividing by $t$ and rearranging the terms, we have with $g(t) := e(t) +\beta \dot{e}(t) $
\begin{multline*}
\frac{1}{t}\ddot h(t)+\pa{\frac{1}{t}-\frac{\alpha\beta}{t^2}}\pa{W(t)-f(z)}\leq -\pa{\frac{3}{\alpha}-\frac{\beta}{t}}\dot W(t)-\left[\frac{\alpha}{t^2}\dot h(t)+ \pa{\frac{1}{t}-\frac{\alpha\beta}{t^2}} \dot I(t)\right]\\
+\frac{1}{t} \| g(t) \| \| u(t)-z \| -\pa{\frac{1}{t}-\frac{\alpha\beta}{t^2}}\int_t^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)} d\tau.
\end{multline*}
After integration, and using Lemma \ref{L:int_bounded}, we get
\begin{equation} \label{E:h_dot_pert}
\frac{1}{t}\dot h(t)+\int_{t_1}^t\pa{\frac{1}{s}-\frac{\alpha\beta}{s^2}}\big(W(s)-f(z)\big)\,ds \le -\int_{t_1}^t\pa{\frac{3}{\alpha}-\frac{\beta}{s}}\dot W(s)\,ds+C + K_1(t) + K_2 (t),
\end{equation}
where
\[
K_1(t)= \int_{t_1}^t \frac{1}{s} \| g(s) \| \| u(s)-z \| ds\ \enskip \text{and} \enskip K_2 (t)= \int_{t_1}^t \pa{\frac{1}{s}-\frac{\alpha\theta}{s^2}} \int_s^{\infty} \| \dot u (\tau)\| \| g(\tau) \| d\tau ds.
\]
Let us majorize $K_1 (t)$ and $K_2 (t)$. The relation
\[
\norm{u(s)-z} \leq \norm{u(t_1)-z} + \int_{t_1}^{s}\norm{\dot{u}(\tau)} d\tau,
\]
and $\dot{u}(\cdot)$ bounded (see \eqref{gE:u_b}) give
\[
K_1(t)\leq \int_{t_{1}}^{t}\frac{1}{s} \| g(s)\| \|u (s) -z \| ds \leq \pa{ \frac{\Vert u(t_1)-z\Vert}{t_{1}}+
\sup_{t\ge t_1}\Vert \dot{u}(\tau)\Vert} \int_{t_{1}}^{+\infty}\Vert g(s)\Vert ds\leq C < +\infty.
\]
For $K_2 (t)$, we use again $\dot{u}(\cdot)$ bounded (see \eqref{gE:u_b}) and integration by parts to obtain
$$
K_2 (t)\leq C \int_{t_{1}}^{t} \pa{ \frac{1}{s}\int_{s}^{\infty}\norm{g(\tau)} d\tau} ds \leq C\pa{ \log t \int_{t}^{\infty}\norm{g(\tau)} d\tau + \int_{t_{1}}^{t} \norm{g(\tau)}\log \tau \ d\tau+1 } .
$$
Let us examine the integral terms that enter \eqref{E:h_dot_pert}.
Since $W(\cdot)$ is non-increasing
\begin{eqnarray} \label{E:int_W_pert}
\int_{t_1}^t\pa{\frac{1}{s}-\frac{\alpha\beta}{s^2}}\big(W(s)-f(z)\big)\,ds
& \ge & \pa{W(t)-f(z)}\int_{t_1}^t\pa{\frac{1}{s}-\frac{\alpha\beta}{s^2}}\,ds \nonumber \\
& = & \pa{W(t)-f(z)}\pa{\log t - \log t_1+\frac{\alpha\beta}{t}-\frac{\alpha\beta}{t_1}}.
\end{eqnarray}
In turn, integration by parts gives
\begin{eqnarray} \label{E:int_dot_W_pert}
&&-\int_{t_1}^t\pa{\frac{3}{\alpha}-\frac{\beta}{s}}\dot W(s)\,ds
= \pa{\frac{3}{\alpha}-\frac{\beta}{t_1}}\big(W(t_1)-f(z)\big)
-\pa{\frac{3}{\alpha}-\frac{\beta}{t}}\pa{W(t)-f(z)} + \beta\int_{t_1}^t\frac{W(s)-f(z)}{s^2}\,ds \nonumber \\
&& \hspace{2.5cm}\le \pa{\frac{3}{\alpha}-\frac{\beta}{t_1}}\big(W(t_1)-f(z)\big)
-\pa{\frac{3}{\alpha}-\frac{\beta}{t}}\pa{W(t)-f(z)}
+\beta\big(W(t_1)-f(z)\big)\pa{\frac{1}{t_1}-\frac{1}{t}},\nonumber\\
&& \hspace{2.5cm} \leq \frac{3}{\alpha}\big|W(t_1)-f(z)\big|-\pa{\frac{3}{\alpha}-\frac{\beta}{t}}\pa{W(t)-f(z)}
\end{eqnarray}
since $t\mapsto W(t)-f(z)$ is non-increasing and $t\ge t_1\ge\alpha\beta$.
Combining \eqref{E:h_dot_pert} with \eqref{E:int_W_pert} and \eqref{E:int_dot_W_pert}, we obtain
\[
\frac{1}{t}\dot h(t)
+\pa{W(t)-f(z)}\pa{\log t+D+\frac{E}{t}}
\leq C \pa{ \log t \int_{t}^{\infty}\norm{g(\tau)} d\tau + \int_{t_{1}}^{t} \norm{g(\tau)}\log \tau \ d\tau+1 }
\]
for appropriate constants $C,D,E\in{\mathbb R}$.
Now, take $t_2\ge t_1$ such that $\log s+D+\frac{E}{s}\ge 0$ for all $s\ge t_2$. Integrate from $t_2$ to $t$ and use again that $W$ is non-increasing to obtain
\begin{multline*}
\frac{h(t)}{t}-\frac{h(t_2)}{t_2}+\int_{t_2}^t\frac{h(s)}{s^2}ds +\pa{W(t)-f(z)}
\int_{t_2}^t\pa{\log s+D+\frac{E}{s}}ds\\
\leq C' \int_{t_2}^t \pa{ \log s \int_{s}^{\infty}\norm{g(\tau)} d\tau + \int_{t_{1}}^{s} \norm{g(\tau)}\log \tau \ d\tau + 1 }ds.
\end{multline*}
Since $h$ is non-negative, this implies
\begin{multline}\label{eq:bndWnonsmooth}
\pa{W(t)-f(z)} \pa{t\log t+(D-1)t+E\log t+ F} \\
\leq C'\pa{t + t \log t \int_{t}^{\infty}\norm{g(\tau)} d\tau + \int_{t_{2}}^{t} \norm{ g(\tau)} \tau \log \tau \ d\tau + t \int_{t_{1}}^{t}\norm{g(\tau)} \log \tau d\tau } + G,
\end{multline}
for some appropriate constants $D,E,F,G\in{\mathbb R}$. Divide by $t \log t$, let $t\to+\infty$, and use Lemma \ref{basic-int}, to obtain $\lim_{t\to+\infty}W (t) \leq f(z)$.
The integrability of $g$ and $\dot{u}(\cdot)$ bounded (see \eqref{gE:u_b}) yield $\DS{\lim_{t\to+\infty} \int_t^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)}d\tau =0}$. As a consequence,
\[
\lim_{t\to+\infty}\pa{ f(x(t))+\frac{1}{2}\|\dot x(t)+\beta \xi(t)\|^2 } \leq f(z)
\]
for each $z \in {\mathcal H}$. Thus
\[
\inf_{{\mathcal H}} f \leq \liminf_{t \to +\infty} f(x(t)) \leq \limsup_{t \to +\infty} f(x(t)) \leq \lim_{t \to +\infty} \pa{f(x(t))+\frac{1}{2}\|\dot x(t)+\beta \xi(t)\|^2} \leq \inf_{{\mathcal H}} f,
\]
whence we get $\lim_{t\to+\infty}f (x(t))= \inf_{{\mathcal H}} f $, and thus $\lim_{t\to+\infty} \|\dot x(t)+\beta \xi(t)\| =0$.
\paragraph{Claim~\ref{gweak-conv2-3}} This follows from claim~{\ref{gweak-conv2-2}} and lower semicontinuity of $f$.
\paragraph{Claim~\ref{gweak-conv2-6}-\ref{gweak-conv2-6a}} Let $x^\star \in S$. We start from \eqref{eq:bndWnonsmooth} with $z=x^\star$ and divide by $t$. To conclude, we note that
\begin{gather*}
\log t \int_{t}^{\infty}\norm{g(\tau)} d\tau \leq \int_{t}^{\infty}\log \tau \norm{g(\tau)} d\tau < +\infty, \\
\int_{t_{2}}^{t} \norm{ g(\tau)} \frac{\tau}{t} \log \tau \ d\tau \leq \int_{t_{2}}^{t} \norm{ g(\tau)} \log \tau \ d\tau < +\infty \enskip \text{and} \enskip \\
\log t \int_t^{+\infty} \dotp{\dot{u}(\tau)}{g(\tau)}d\tau \leq C \int_t^{+\infty} \log \tau\norm{g(\tau)}d\tau,
\end{gather*}
where $C = \sup_{t \geq t_0} \norm{\dot{u}(t)} < +\infty$ (see \eqref{gE:u_b}).
\paragraph{Claim~\ref{gweak-conv2-6}-\ref{gweak-conv2-6b}} Putting together \eqref{E:h_dot_pert} and \eqref{E:int_dot_W_pert} with $z=x^\star \in S$, and using non-negativity of $h$, we infer that for some positive constant $C$
\[
\pa{1-\frac{\alpha\beta}{t_1}}\int_{t_1}^t\frac{1}{s}\pa{W(s)-f(z)} ds \leq C + K_2 (t).
\]
Arguing similarly as for proving part {\ref{gweak-conv2-6}-\ref{gweak-conv2-6a}}, we can show that $K_2(\cdot)$ is bounded. Thus
\[
\int_{t_1}^t\frac{1}{s}\pa{f(x(s))-f(z) + \norm{\dot{u}(s)}^2} ds \leq \int_{t_1}^t\frac{1}{s}\pa{W(s)-f(z)} ds + \frac{1}{t_
1}\int_{t_1}^t\norm{\dot{u}(s)}\norm{g(s)} ds < +\infty,
\]
which completes the proof. \qed
\end{proof}
\subsubsection{Fast convergence rates}
When $\alpha \geq 3$, under a reinforced integrability assumption on the perturbation term, we will show fast convergence results. The following theorem is the non-smooth counterpart of Theorem \ref{fast_conv_smooth}.
\begin{theorem}\label{fastconv-thm}
Suppose that $\alpha \geq 3$.
Let $f \in \Gamma_0({\mathcal H})$ such that $S \neq \emptyset$. Suppose that $e(\cdot) \in {\mathcal W}^{1,1} (t_0,T; {\mathcal H})$ for all $T >t_0$, with $\DS{\int_{t_0}^{+\infty} t \| e(t) + \beta \dot{e}(t)\| dt<+\infty}$. Then,
for any global strong solution $(x,y)$ of \eqref{eq:fos2}
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{gfast-conv2-1} $f(x(t))- \min_{{\mathcal H}} f = {\mathcal O}\pa{ t^{-2}}.$
\item\label{gfast-conv2-2}
$\DS{\int_{t_0}^{+\infty} t(f(x(t))-\min_{{\mathcal H}} f)dt<+\infty}$,
$\DS{\int_{t_0}^{+\infty} t^2\|\xi(t)\|^2dt<+\infty}$,
$\DS{\int_{t_0}^{+\infty} t\|\dot{x}(t)\|^2dt<+\infty}$.
\item\label{gfast-conv2-3}
$\|\dot x(t)+\beta\xi(t)\|={\mathcal O}(t^{-1})$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $(x,y):[t_0,+\infty[\to{\mathcal H}\times{\mathcal H}$ be a global strong solution of \eqref{eq:fos2}. Take $\alpha\geq3$ and $x^\star \in S$. Recall $\bar{f} := \min_{{\mathcal H}} f$ and $g(t)= e(t)+\beta \dot{e}(t)$. Our analysis relies on the non-smooth version of the Lyapunov function in \eqref{eq:Eeps}, which is defined for $\lambda\in[2,\alpha-1]$, as ${\mathcal E}_{\lambda,T}:[t_0,T] \to {\mathbb R}$ by
\begin{equation}
{\mathcal E}_{\lambda,T}(t)=
t(t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})+
\frac{1}{2}\|v_\lambda(t)\|^2
+
\lambda(\alpha-\lambda-1)\frac{1}{2}\|x(t)-x^\star\|^2
- \int_t^{T} \tau \dotp{v_\lambda(\tau)}{g(\tau)} d\tau, \label{grfast2}
\end{equation}
where $v_\lambda(t) := \lambda(x(\tau)-x^\star)+\tau\dot u(t)$, and $u$ is defined on $[t_0,+\infty[$ by \eqref{u0} and $\dot u$ is given by \eqref{guniter}.\\
${\mathcal E}_{\lambda,T}(\cdot)$ is the sum of four terms,
each of which is absolutely continuous on $[t_0,T]$ for all $T>t_0$. Hence ${\mathcal E}_{\lambda,T}$ is differentiable almost everywhere. We first differentiate each term of ${\mathcal E}_{\lambda,T}$:
\begin{equation*}
\frac{d}{dt}\brac{t(t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})}
= (2t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})
+ t(t-\beta(\lambda+2-\alpha))\dotp{\xi(t)}{\dot x(t)}.
\end{equation*}
Using \eqref{dguniter0}, we have
\begin{align*}
&\frac{d}{dt}\frac{1}{2}\|v_\lambda(t)\|^2
= \dotp{\lambda(x(t)-x^\star)+t\dot u(t)}{
\lambda\dot x(t)+\dot u(t)+t\ddot u(t)} \\
&=
\dotp{\lambda(x(t)-x^\star)+t\dot u(t)}{
(\lambda+1-\alpha)\dot x(t)-(t-\beta)\xi(t)-t g(t)} \\
&=
\lambda(\lambda+1-\alpha)\dotp{ x(t)-x^\star}{\dot x(t)}
-t(\alpha-\lambda-1)\|\dot x(t)\|^2
-\beta t(t-\beta)\|\xi(t)\|^2 \\
&-\lambda(t-\beta)\dotp{ x(t)-x^\star}{\xi(t)}
-t(t-\beta(\lambda+2-\alpha))\dotp{\xi(t)}{\dot x(t)}
-t \dotp{v_\lambda(t)}{g(t)}, \\
&\frac{d}{dt}\lambda(\alpha-\lambda-1)\frac{1}{2}\|x(t)-x^\star\|^2
=\lambda(\alpha-\lambda-1)\dotp{x(t)-x^\star}{\dot x(t)}, \enskip \text{and} \enskip \\
&\frac{d}{dt}\pa{ - \int_t^{T} \tau \dotp{v_\lambda(\tau)}{g(\tau)} d\tau}
= t \dotp{v_\lambda(t)}{g(t)} .
\end{align*}
By collecting these results, the perturbation terms cancel each other out. We get
\begin{eqnarray}
\frac{d}{dt}{\mathcal E}_{\lambda,T}(t)
&=& (2t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})
-\lambda(t-\beta)\dotp{x(t)-x^\star}{\xi(t)} \nonumber \\
&& -t(\alpha-\lambda-1)\|\dot x(t)\|^2
-\beta t(t-\beta)\|\xi(t)\|^2, \label{gdrfast2}
\end{eqnarray}
for almost all $t>t_0$. Since $\xi(t)\in\partial f(x(t))$ for all $t>t_0$, we have
$
\dotp{\xi(t)}{x(t)-x^\star}\geq f(x(t)-f(x^\star),
$
and we deduce from \eqref{gdrfast2}, that
\begin{equation} \label{grfastoche}
\frac{d}{dt}{\mathcal E}_{\lambda,T}(t)\leq
-((\lambda-2)t-\beta(\alpha-2))(f(x(t))-\bar{f})
-t(\alpha-\lambda-1)\|\dot x(t)\|^2-\beta t(t-\beta)\|\xi(t)\|^2,
\end{equation}
for almost all $t\geq t_1=\max\pa{t_0,\beta}$. It follows that ${\mathcal E}_{\lambda,T}$ is non-increasing on $[t_1,T]$. In particular, $ {\mathcal E}_{\lambda,T}(t) \leq {\mathcal E}_{\lambda,T}(t_1)$ for $t_1 \leq t \leq T$. This gives the existence of a constant $C$
such that
\begin{align}\label{Gronw2}
\frac{1}{2}\|v_\lambda(t)\|^2 \leq C + \int_{t_0}^t \|v_\lambda(t)\| \| \tau g(\tau) \| d\tau .
\end{align}
Applying Lemma~\ref{lem:BrezisA5} to \eqref{Gronw2}, and using the integrability of $t\mapsto tg(t)$, it follows that
\begin{equation} \label{energy-033}
\sup_{t \geq t_0} \|v_\lambda(t)\| \leq \sqrt{2C} + \int_{t_0}^\infty \| \tau g(\tau) \| d\tau< +\infty.
\end{equation}
As a consequence, we can define the energy function
\begin{equation*}
{\mathcal E}_{\lambda} (t) := \ t(t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})+
\frac{1}{2}\|v_\lambda(t)\|^2+
\lambda(\alpha-\lambda-1)\frac{1}{2}\|x(t)-x^\star\|^2 \\
- \int _t^{\infty} \tau \dotp{v_\lambda(t)}{g(\tau)} d\tau,
\end{equation*}
which has the same derivative as ${\mathcal E}_{\lambda,T}$. Hence ${\mathcal E}_{\lambda} (t) \leq {\mathcal E}_{\lambda} (t_0)$. Combined with \eqref{energy-033}, this gives
\[
t(t-\beta(\lambda+2-\alpha))(f(x(t))-\bar{f})
\leq C + \sup_{t \geq t_0} \|v_\lambda(t) \| \int _{t_0}^{\infty} \|\tau g(\tau) \| d\tau < + \infty ,
\]
whence statement {\ref{gfast-conv2-1}}. Claim~{\ref{gfast-conv2-3}} is obtained by letting $\lambda=0$ in \eqref{energy-033}. Integration of \eqref{grfastoche} gives the integral estimates of {\ref{gfast-conv2-2}}, which completes the proof. \qed
\end{proof}
\subsubsection{Convergence of the trajectories and faster asymptotic rates}
Similar argument as in the smooth case (see Theorem~\ref{T:weak_convergence-smooth}), but now using the Lyapunov function \eqref{grfast2}, gives weak convergence of the trajectories of \eqref{eq:fos2}. Moreover, in the same vein as Theorem~\ref{fast_conv_smooth}, $o(\cdot)$ rates can also be obtained. We leave the details to the readers for the sake of brevity.
\begin{theorem}\label{T:weak_convergence-nonsmooth}
Let $\alpha>3$. Let $f \in \Gamma_0({\mathcal H})$ and assume that $S =\argmin f \neq \emptyset$. Suppose that $e(\cdot)\in {\mathcal W}^{1,1} (t_0,T; {\mathcal H})$ for all $T >t_0$, with
$\DS{\int_{t_0}^{+\infty} t \| e(t) + \beta \dot{e}(t)\| dt<+\infty}$. Then,
for any global strong solution $(x,y)$ of \eqref{eq:fos2}
\begin{enumerate}[label={\rm (\roman*)}]
\item $x(t)$ converges weakly, as $t\to+\infty$ to a point in $S$;
\item $f(x(t))- \min_{{\mathcal H}} f = o\pa{t^{-2}}$
and $\|\dot x(t)+\beta\xi(t)\| = o\pa{t^{-1}}$ as $t\to+\infty$.
\end{enumerate}
\end{theorem}
\begin{remark}
In the special case where $e \equiv 0$ in \eqref{eq:fos2}, {\it i.e.}, unperturbed case, Theorem~\ref{strong-conv-thm-implicit}, Theorem~\ref{fastconv-thm} and Theorem~\ref{T:weak_convergence-nonsmooth} recover the result of \cite[Section~4]{APR1}. In a nutshell, our result demonstrates that the properties of the unperturbed system are preserved under reasonable integrability conditions on the errors.
\end{remark}
\subsection{Implicit Hessian Damping}\label{sec:nonsmooth-implicit}
As we have already discussed in the smooth case (see Section~\ref{s:discussion}), the analysis of the convergence properties of the system with implicit Hessian driven damping heavily relies on Lipschitz continuity of the gradient. As shown above, such a property was not needed to analyze system \eqref{eq:fos2}. Therefore, the study of the convergence properties for the non-smooth system \eqref{eq:fos2_implicit} (even without perturbations) is an open challenging topic.
\section{Numerical Experiments}\label{s:num}
To support our theoretical claims, we consider numerical examples in ${\mathcal H}={\mathbb R}^2$ with two real-valued functions:
\begin{enumerate}[label=$\bullet$]
\item The first one is given by $f(x_1,x_2)) = (x_1-1)^4+(x_2-5)^2$. This function is obviously convex (but not strongly so) and smooth, and has a unique minimizer at $(1,5)$. For this function, we consider the continuous time dynamical system \eqref{eq:origode_a} with parameters $(\alpha,\beta)=(3.1,1)$, and \eqref{eq:odetwo_a} with parameters $(\alpha,\gamma,\beta)=(3.1,1,1)$.
\item The second example we consider is with the convex non-smooth function $f(x_1,x_2)=(x_1-1)^4+(x_2-5)^2+0.1 (|x_1|+|x_2|)$. For this function, we use the continuous time non-smooth system \eqref{eq:fos2} with parameters $(\alpha,\beta)=(3.1,1)$. Although we have no theoretical guarantee for system \eqref{eq:fos2_implicit}, we do report the corresponding numerical results with parameters $(\alpha,\gamma,\beta)=(3.1,1,1)$.
\end{enumerate}
\begin{figure}
\caption{Example on a smooth function: Evolution of the objective error and distance to the minimizer as a function of $t$ for different error decay exponents.}
\label{fig:smoothhessianpert}
\end{figure}
For both examples, we take as an exogenous perturbation
$$
e(t) = \displaystyle{\frac{\cos(2\pi t)}{t^{\delta}}} \mbox{ with } \delta \in \{0.1, 1.1, 3.1\}.
$$
All systems are solved numerically with a Runge-Kutta adaptive method in MATLAB on the time interval $[1,50]$ with initial data $(x_0,\dot{x}_0)= (-10,20,5,-5)$. The results are displayed in Figure~\ref{fig:smoothhessianpert} and Figure~\ref{fig:nonsmoothhessianpert}.
Let us first comment on the results for the smooth function. For $\delta=3.1$, all required moment assumptions on the errors are fulfilled (for the explicit Hessian, the term $\dot{e}$ is dominated by $e$ and can then be discarded). Hence the fast rates predicted by Theorem~\ref{fast_conv_smooth}\ref{item:fast_conv_smooth1} and Theorem~\ref{th:int}\ref{th:intclaim1} as well as convergence of the trajectories (see Theorem~\ref{T:weak_convergence-smooth} and Theorem~\ref{T:weak_convergence-smooth-implicit}) hold true. For the value $\delta=0.1$, since the error is not even integrable, neither the convergence of the objective value nor that of the trajectories is ensured, with large oscillations appearing. The implicit Hessian damping seems also less stable as anticipated from our discussion in Section~\ref{s:discussion}. For $\delta=1.1$, though there is no convergence guarantee for the trajectory, the objective value for \eqref{eq:origode_a} decreases but at a rate which is dominated by the error decrease. This can be explained in light of the proof of Theorem~\ref{fast_conv_smooth}\ref{item:fast_conv_smooth1}, where a close inspection of \eqref{eq:boundvexplicit} and \eqref{eq:boundobjexplicit} shows that the bound on the objective error decomposes as
\begin{center}
$
f(x(t))-\bar{f} \leq {\mathcal O}\pa{\frac{1}{t^2}} + \frac{C\pa{\DS{\int_{t_0}^t\tau \norm{e(\tau)}} d\tau}^2}{t^2} .
$
\end{center}
For $\delta \in ]1,2]$, the second term indeed dominates the first one and decreases at the slower rate $t^{-2(\delta-1)}$. This confirms the known rule that there is a trade-off between fast convergence of the methods and their robustness to perturbations.
Similar observations remain true for the non-smooth function with system \eqref{eq:fos2} where we now invoke Theorem~\ref{fastconv-thm} and Theorem~\ref{T:weak_convergence-nonsmooth}. As for system \eqref{eq:fos2_implicit}, it seems that it has a behaviour similar to what we observed in the smooth case for system \eqref{eq:odetwo_a}. As we argued in Section~\ref{sec:nonsmooth-implicit}, supplementing the numerical observations for system \eqref{eq:fos2_implicit} with theoretical guarantees is an open problem that we leave to a future work.
\begin{figure}
\caption{Example on a non-smooth function: Evolution of the objective error and distance to the minimizer as a function of $t$ for different error decay exponents.}
\label{fig:nonsmoothhessianpert}
\end{figure}
\section{Conclusion and Perspectives}\label{s:conc}
The introduction of the correction term attached to the damping driven by the Hessian in first-order accelerated optimization algorithms makes it possible to considerably dampen the oscillations in the trajectory.
The study of the robustness of these algorithms with respect to error perturbations is crucial for their further development in a stochastic framework.
Our systematic study of these questions for the dynamics underlying these algorithms is a fundamental first step in this direction. We paid particular attention to the explicit and the implicit forms of the Hessian driven damping, showing several advantages of the explicit form.
Our study concerns the dynamics with damping driven by the Hessian within the framework of the Nesterov acceleration gradient method. It shows that the convergence of the values still holds when the error terms satisfy an appropriate integrability condition, and fast convergence is satisfied when the (first or second-order) moment of the error is finite.
Indeed, as a general rule, there is a balance between the rate of convergence of the methods and their robustness with respect to error disturbances.
An interesting technique studied in \cite{AA1,AA2} is the introduction of a dry friction term. This makes it possible to have errors which do not necessarily go to zero, they must not exceed a certain threshold, but on the other hand we only obtain an approximate solution.
Finding the right balance between the convergence rate and robustness is an important issue that should be the subject of further study.
Another important aspect of our study is the fact that several results are valid in the case of a non-smooth function. This opens the door to the study of similar topics with respect to structured composite optimization problems involving a non-smooth term.
These are some of the many facets of these flexible dynamics and algorithms which, in the unperturbed case, have been applied in various fields including PDE's and mechanical shocks \cite{AMR}, deep learning \cite{CBFP}, non-convex optimization \cite{ABC}, monotone inclusions \cite{AL1,AL2} to mention a few important applications.
\appendix
\section{Auxiliary results}
Let us first recall the continuous form of the Opial's Lemma \cite{Op}, a key ingredient to establish convergence of the trajectories.
\begin{lemma}\label{Opial} Let $S$ be a nonempty subset of ${\mathcal H}$ and let {$x:[t_0,+\infty[\to {\mathcal H}$}. Assume that
\begin{enumerate}[label={\rm (\roman*)}]
\item for every $z\in S$, $\lim_{t\to\infty}\|x(t)-z\|$ exists;
\item every weak sequential cluster point of $x(t)$, as $t\to\infty$, belongs to $S$.
\end{enumerate}
Then $x(t)$ converges weakly as $t\to\infty$ to a point in $S$.
\end{lemma}
\begin{lemma}[{\cite[Lemma~7.3]{APR1}}]
\label{L:int_bounded}
Let $\tau,p>0$ and let $\psi:]\tau,+\infty[\to{\mathbb R}$ be ${\mathcal C}^2(]\tau,+\infty[)$ and bounded from below. Then,
\[
\inf_{t>\tau}\int_{\tau}^{t}\frac{\dot\psi(s)}{s^p}\,ds>-\infty\quad\hbox{and}\quad \inf_{t>\tau}\int_{\tau}^{t}\frac{\ddot\psi(s)}{s^p}\,ds-\frac{\dot\psi(t)}{t^p}>-\infty.
\]
\end{lemma}
\begin{lemma}[{\cite[Lemma~A.5]{Bre1}}]\label{lem:BrezisA5}
Let $m : [t_0; T] \to [0,+\infty[$ be integrable. Suppose $w : [t_0,T] \to {\mathbb R}$ is continuous and
\[
\frac{1}{2} w(t)^2 \leq \frac{1}{2} c^2 + \int_{t_0}^t m(s)w(s) ds,
\]
for some $c\geq 0$ and for all $t \in [t_0,T]$. Then
\[
|w(t)| \leq c + \int_{t_0}^t m(s) ds, \qquad t \in [t_0,T] .
\]
\end{lemma}
\begin{lemma}\label{lem:timerescale}
Let $\beta$ be a positive function on $[t_0,+\infty[$ such that $\beta \not\in L^1(t_0,+\infty;{\mathbb R}_+)$. Then, the differential inclusion
\begin{equation}
\dot{z}(t) + \beta(t) \partial \Phi(z(t)) + F(t,z(t)) \ni 0 ,
\end{equation}
is equivalent to
\begin{equation}
\dot{w}(s) + \partial \Phi (w(s)) + G(s, w(s)) \ni 0,
\end{equation}
with
\[
G(s, w(s))= \frac{1}{\beta(\tau(s))}F(\tau(s),w(s)), \quad t = \tau(s), \enskip \text{and} \enskip \beta(\tau(s))\dot{\tau}(s)=1 .
\]
\end{lemma}
\begin{proof}
Make the change of time variable
$
t = \tau(s) \enskip \text{and} \enskip z(t)= z \circ \tau(s) = w(s) .
$
We then have
\begin{equation*}
\frac{1}{\beta(\tau(s))\dot{\tau}(s)}\dot{w}(s) + \partial \Phi (w(s)) + \frac{1}{\beta(\tau(s))}F(\tau(s),w(s)) \ni 0 ,
\end{equation*}
Choose $\tau(\cdot)$ such that
\begin{equation}\label{basic1}
\beta(\tau(s))\dot{\tau}(s)=1 .
\end{equation}
Introduce a primitive of $\beta$,
$
p(t) = \int_{t_0}^t \beta(r)dr
$
Therefore, \eqref{basic1} can be equivalently written
\begin{center}
$
\frac{d}{ds} p(\tau (s))=1 \iff p(\tau (s)) = \int_{t_0}^{\tau(s)} \beta(r)dr = s + C,
$
\end{center}
for any constant $C$. Thus, $\tau$ defines a change of variable if and only if
$
\int_{t_0}^{+\infty} \beta(r)dr = +\infty ,
$
hence our assumption on $\beta$. \qed
\end{proof}
\begin{lemma}\label{basic-int}
Take $t_0 >0$, and let $f \in L^1 (t_0 , +\infty;{\mathbb R})$ be continuous. Consider a nondecreasing function $\varphi:[t_0,+\infty[\to{\mathbb R}_+$ such that $\lim\limits_{t\to+\infty}\varphi(t)=+\infty$. Then,
$
\lim_{t \to + \infty} \frac{1}{\varphi(t)} \int_{t_0}^t \varphi(s)f(s)ds =0.
$
\end{lemma}
\begin{proof}
Given $\varepsilonilon >0$, fix $t_\varepsilonilon$ so that $\int_{t_\varepsilonilon}^{\infty} |f(s)| ds \leq \varepsilonilon$.
Then, for $t \geq t_\varepsilonilon$, split the integral $\DS{\int_{t_0}^t \varphi(s)f(s) ds}$ into two parts to obtain
\begin{eqnarray*}
\abs{\frac{1}{\varphi(t)} \int_{t_0}^t \varphi(s)f(s)ds} =
\abs{\frac{1}{\varphi(t)}\int_{t_0}^{t_\varepsilonilon} \varphi(s) f(s) ds
+ \frac{1}{\varphi(t)}\int_{t_\varepsilonilon}^t \varphi(s) f(s) ds}
\leq \frac{1}{\varphi(t)}\int_{t_0}^{t_\varepsilonilon} \varphi(s)|f(s)| ds
+ \int_{t_\varepsilonilon}^t |f(s)| ds.
\end{eqnarray*}
Let $t\to+\infty$ to deduce that
$
0\leq\limsup_{t\to+\infty}\abs{\frac{1}{\varphi(t)}\int_{t_0}^t \varphi(s)f(s)ds} \le \varepsilonilon.
$
Since this is true for any $\varepsilonilon>0$, the result follows.\qed
\end{proof}
\begin{lemma}[{\cite[Lemma~5.9]{attouch2018fast}}]\label{lem:convw}
Let $t_0 > 0$, and let $w : [t_0,+\infty[ \to {\mathbb R}$ be a twice differentiable \footnote{In \cite[Lemma~5.9]{attouch2018fast}, twice differentiability was not stated, but is actually needed for the statement to make sense.} function which is bounded from below. Assume that
\[
t\ddot{w}(t) + \alpha \dot{w}(t) \leq g(t),
\]
for some $\alpha > 1$, almost every $t > t_0$, and some non-negative function $g \in L^1(t_0,+\infty;{\mathbb R})$. Then, the positive part $[\dot w]_+$ of $\dot w$ belongs to $L^1(t_0,+\infty;{\mathbb R})$ and $\lim_{t \to +\infty} w(t)$ exists.
\end{lemma}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{A totally Eulerian Finite Volume solver for multi-material
fluid flows : Enhanced Natural Interface Positioning (ENIP)}
\author[Raph]{Raphaël Loubère\corref{cor1}}
\ead{[email protected]}
\address[Raph]{CNRS et Universit\'e de Toulouse
IMT (Institut de Math\'ematiques de Toulouse),
31062 Toulouse, France}
\author[Pilou]{Jean-Philippe Braeunig}
\ead{[email protected]}
\address[Pilou]{INRIA Nancy Grand-Est, Equipe CALVI, 615 rue du Jardin Botanique 54600 Villers-l\`es-Nancy, France\\
CEA DIF Bruy\`eres-le Ch\^atel, 91297 Arpajon, France}
\author[JMG]{Jean-Michel Ghidaglia}
\ead{[email protected]}
\address[JMG]{CMLA, CNRS et ENS de Cachan
61 Av. du Pr{\'e}sident Wilson
Cachan Cedex 94235, France}
\cortext[cor1]{Corresponding author}
\begin{abstract}
This work concerns the simulation of compressible multi-material fluid flows and follows the method FVCF-NIP described in the former paper \cite{Braeunig09}. This Cell-centered Finite Volume method is totally Eulerian since the mesh is not moving and a sharp interface, separating two materials, evolves through the grid. A sliding boundary condition is enforced at the interface and mass, momentum and total energy are conserved. Although this former method performs well on 1D test cases, the interface reconstruction suffers of poor accuracy in conserving shapes for instance in linear advection. This situation leads to spurious instabilities of the interface. The method Enhanced-NIP presented in the present paper cures an inconsistency in the former NIP method that improves strikingly the results. It takes advantage of a more consistent description of the interface in the numerical scheme. Results for linear advection and compressible Euler equations for inviscid fluids are presented to assess the benefits of this new method.
\end{abstract}
\begin{keyword}
Multi-material fluid flow {\cal S}ep Finite Volume {\cal S}ep Natural Interface Positioning
\MSC[2010] 65M08 {\cal S}ep 76M12 {\cal S}ep 76N99
\end{keyword}
\end{frontmatter}
\tableofcontents
{\cal S}ection{Introduction} \label{sec:introduction}
The two-material compressible hydrodynamics equations (Euler equations) are considered
in this work.
The flow regime is such that molecular viscosity within materials is
neglected: materials are supposed immiscible and separated by sharp
interfaces, with perfect sliding between materials.
Each material is characterized by its own equation of state (EOS).
The formalism of finite volume methods is close to the mechanical
viewpoint, and generic for different types of physical models.
Thus, it might be easier to add such models; surface tension
or turbulent diffusion for instance.
The discretization order is limited, but this method is accurate
to simulate hydrodynamic shock waves, because of the consistency between numerical
treatment and mechanics.\\
The extension of Eulerian schemes to multi-material fluid flows can be
obtained by various techniques.
One is to introduce the cell mass fraction $c_\alpha$ of material
$\alpha$ and let it evolve according to material velocity.
The cell is called pure if a material $\alpha$ satisfies
$c_\alpha=1$ and is called mixed if $c_\alpha \in ]0,1[$.
Pure cells filled by material $\alpha$ are calculated in the same
manner as for the single material method.
Mixed cell evolution is computed using a mixing equation
of state that takes into account material mass fractions, see {\it e.g.} \cite{int_diff1}.
One drawback of this approach is the numerical diffusion
of the interface that obviates sharp interface capturing. It turns out that for some applications, this drawback is not acceptable since the diffusion of one material into another one will correspond to a different physics. For example the two material could react when a molecular mixture is formed. Hence such a diffusion should occur only for physical reasons and not for numerical ones.
In the case of sharp interface capturing methods, the interface is approximated
in a mixed cell by a segment by most authors.
However more complex curves than straight line or more complex theory (see \cite{MOF} for instance)
might be used.
A famous method using sharp interface reconstruction is the
Lagrange+Remap Finite Volume scheme, initiated in \cite{NohWood} and
further improved in \cite{DLY1}. It belongs to the family of so called
Volume of Fluid (VOF) methods.
The first step of this method is a Lagrangian scheme, resulting in a mesh displacement with
material velocity.
The second step is a multi-material remapping of Lagrangian mesh onto the
original Eulerian mesh, by exchanging volume fluxes between cells
related to the Lagrangian motion of cell edges.
The new interface position in mixed cells is determined using
the partial volumes of the materials and the interface normal vector.
The later is
calculated using volume fractions from neighboring cells.
Thus the ratio of each material in volume fluxes is
deduced from the multi-material remapping.
Some methods with the same kind of operator splitting are used
for incompressible multi-material fluid flows as in
\cite{Zaleski}.
These methods provide sharp interface between materials and discontinuous
quantities in mixed cells, allowing large deformations and
transient flows.
In this context, the drawback of these Lagrange+Remap methods is the limited accuracy of
the underlying single phase scheme due to diffusion
induced by the remapping step.
Moreover, more complex physics at material interfaces such as
sliding effects, is not possible.
The FVCF scheme (Finite Volume with Characteristic Flux) has been introduced in \cite{GKL1} for simulating single phase compressible flows or multi-phase models without sharp interface capturing.
The method described in \cite{Braeunig09}, so called NIP method (Natural Interface Positioning),
is an add-on to the FVCF method in order to deal with multi-material fluid flows with sharp interface capturing.
It is a cell centered totally Eulerian scheme, in which
material interfaces are represented by a discontinuous piecewise
linear curve. A treatment for interface evolution is proposed on Cartesian
structured meshes which is locally conservative in mass, momentum and total energy and
allow the materials to slide on each others.
Discrete conservation laws are written on
partial volumes as well as on pure cells, considering the interface in the cell as a
moving boundary without any diffusion between materials.
A specific data structure called
{\em condensate} is introduced in order to write a finite volume scheme even when the
considered volume is made of moving boundaries, i.e. interfaces. This treatment includes
an explicit computation of pressure and velocity at interfaces.
In \cite{Braeunig09} are shown $2D$
results illustrating the capability of the method to deal with perfect sliding, high pressure
ratios and high density ratios.
This former method however produces non satisfactory results in the context of advection of geometrical shapes
especially when dealing with low Mach numbers.
It is however a classical misbehavior of most of advection and reconstruction methods which have a tendency
to destroy the shape of advected objects due to numerical approximations.
However, this former method gives very poor results when advecting geometrical shapes especially when dealing with low Mach number flows.
In this work we propose a new method called ENIP (Enhanced NIP) that is an improvement of the NIP method by a more accurate treatment of condensates. On a very simple
example: the advection of a square, an inconsistency in the NIP interface reconstruction method will be exhibited. We will then introduce ENIP that cures this situation.
Numerical examples are presented in the last Section to assess the validity and efficiency
of this new approach.
{\cal S}ection{FVCF-ENIP: Finite Volume Characteristics Flux with
Enhanced Natural Interface Positioning technique}
\label{sec:VFFC-NIP}
{\cal S}ubsection{Governing equations}
The model addressed in this work is the compressible Euler equations in space dimension $d$ that can be
written in a conservative form as follows:
\begin{eqnarray}
\label{euler_equation_rho}
\partialt (\rho) + \hbox{div\,} (\rho u) &=& 0, \\
\label{euler_equation_rhou}
\partialt (\rho u)
+ \hbox{div\,} (\rho u \otimes u + p I) &=& 0, \\
\label{euler_equation_rhoE}
\partialt (\rho E)
+ \hbox{div\,} ((\rho E+p)u ) &=& 0,
\end{eqnarray}
where $\rho$ denotes the density, $u \in \mathbb{R}^d$ the velocity field,
$p$ the pressure, $E=e+| u |^2/2$
the specific total energy and $e$ the specific internal energy.
An equation of state of the form $EOS(\rho,e,p)=0$ or $p=p(\rho, e)$
is provided in order to close the system.\\
Let us consider a generic
conservative form with ${{\bm V}}=(\rho,\rho u,\rho E)^t$ the unknown
vector of conservative variables and flux ${\bm F}$ is a matrix
valued function defined as:
\begin{equation}
\begin{array}{llcl}
{\bm F} : & \mathbb{R}^{d+2} & \longrightarrow & \mathbb{R}^{d+2} \times \mathbb{R}^{d}\\
& {\bm V} & \longmapsto & {\bm F}({\bm V}),
\end{array}
\end{equation}
for all direction $n \in \mathbb{R}^{d}$, ${\bm F}({\bm V}) \cdot n$ is given
in terms of ${\bm V}$ by:
\begin{equation}
{\bm F}({\bm V}) \cdot n= \left( \rho \left(u \cdot n\right),
\rho u \left(u \cdot n\right) + p n,
\left(\rho E +p\right) \left(u \cdot n\right) \right).
\end{equation}
The compressible Euler equations (\ref{euler_equation_rho}-\ref{euler_equation_rhoE}) can then
be rewritten as:
\begin{equation}
\partial_t {\bm V} + \hbox{div\,} {\bm F}({\bm V}) = 0.
\end{equation}
{\cal S}ubsection{FVCF: Single material scheme}
FVCF method uses a directional splitting on Cartesian
structured meshes. The method is thus detailed for only one generic direction denoted by $x$.
In $d$
dimensions of space, the algorithm described for direction $x$ has
to be replicated $d$ times, one for each direction. However, this directional splitting does
not modify at all the underlying single material scheme FVCF for pure cells.
In \textit{2D}:
\begin{itemize}
\item[-] variables at $t^{n,x}$ are
calculated from those at $t^n$ by the $x$ direction step,
\item[-] variables at $t^{n+1}$ are calculated
from those at $t^{n,x}$ by the $y$ direction step.
\end{itemize}
\begin{eqnarray}
\label{2step1}
{\bm V}ol_i \frac{{\bm V}_i^{n,x }-{\bm V}_i^{n }}{\Delta t} + A_x \left(\bm{\phi}^n_\ell+\bm{\phi}^n_r \right) &=& 0,\\
\label{2step2}
{\bm V}ol_i \frac{{\bm V}_i^{n+1}-{\bm V}_i^{n,x}}{\Delta t} + A_y \left(\bm{\phi}^n_d +\bm{\phi}^n_u \right) &=& 0,
\end{eqnarray}
where the cell volume is ${\bm V}ol_i$, the cell face area are $A_x$ and $A_y$ respectively
normal to $x$ and $y$ directions,
up, down, right and left direction fluxes $\bm{\phi}^n_u$, $\bm{\phi}^n_d$, $\bm{\phi}^n_r$, $\bm{\phi}^n_\ell$
calculated with respect of the outgoing normal direction $n_d$ of cell face
$\Gamma_d$ in direction $d$ using variables at time $t^n$, i.e.
\begin{eqnarray}
\bm{\phi}^n_d=\frac{1}{A_d} \int_{\Gamma_d} {\bm F}({\bm V}^n) \cdot n_d dS.
\end{eqnarray}
This flux is further approximated using the finite volume scheme FVCF described in \cite{GKL1}.
{\cal S}ubsection{FVCF-NIP: Multi-material scheme}
One considers multi-material flows. The subcell model addressed here for the multi-material representation is
a cell $C$ of volume ${\bm V}ol_C$ containing $n_m$ different materials, each of them filling
a partial volume ${\bm V}ol^k_{C}$ such that
\begin{eqnarray}
\displaystyle {\cal S}um^{n_m}_{k=1} {\bm V}ol^k_{C}={\bm V}ol_{C}.
\end{eqnarray}
Cell $C$ is referred to as pure if $n_m=1$,
and as mixed if $n_m>1$. The interfaces in mixed cells are approximated by segments separating materials into two partial volumes which are pure on both sides of the interface. \\
A partial volume cell-centered variable vector
${\bm V}_{k}=(\rho_k,\rho_k u_k,\rho_k E_k)^t$ and an equation of state
$EOS_{k} (\rho_k,e_k,p_k)=0$ are also associated with each
material labeled by $k \leq n_m$ in the mixed cell.\\
FVCF-NIP method uses a directional
splitting scheme for the interface evolution without loosing the accuracy of the
Eulerian scheme in the bulk of materials.
Consequently this scheme is restricted to structured Cartesian
mesh.
The multi-material extension proposed in \cite{Braeunig09} considers the finite
volume scheme (\ref{2step1}-\ref{2step2}) on each partial volume in a mixed
cell. The obtained scheme is conservative by construction and is constrained
with the same \textit{CFL} condition as the single material
scheme\footnote{Without such a special treatment the time step would be constrained
by the smallest partial volume, which is arbitrarily small.}.
NIP method consists in removing cell edges when this cell contains
an interface.
Therefore each partial volume is merged with the
neighbor pure cells filled with the same material, see
Figure \ref{fig_evol_cond}.
\begin{figure}
\caption{ \label{fig_evol_cond}
\label{fig_evol_cond}
\end{figure}
Variables in these enlarged partial
volumes are obtained by writing the conservation laws on the merged volumes
\begin{eqnarray}
\overline{{\bm V}ol_1} &=& {\bm V}ol_1+{\bm V}ol_{pure~1}, \\
\overline{{\bm V}ol_2} &=& {\bm V}ol_2+{\bm V}ol_{pure~2},
\end{eqnarray}
then on the conserved variables
\begin{eqnarray}
\overline{{\bm V}}_1 &=& \frac{{\bm V}ol_1 ~{\bm V}_1 + {\bm V}ol_{pure~1}~ {\bm V}_{pure~1}}{\overline{{\bm V}ol_1}} \\
\overline{{\bm V}}_2 &=& \frac{{\bm V}ol_2 ~{\bm V}_2 + {\bm V}ol_{pure~2}~ {\bm V}_{pure~2}}{\overline{{\bm V}ol_2}}.
\end{eqnarray}
This set of cells is associated with its left and
right single material fluxes $\bm{\phi}_\ell$ and $\bm{\phi}_r$.
Internal cell edges are forgotten, considering only
enlarged volumes $\overline{{\bm V}ol_1}$ and $\overline{{\bm V}ol_2}$ and
averaged variables $\overline{{\bm V}}_1$ and $\overline{{\bm V}}_2$,
separated by an interface; this system is called a condensate.
Actually, this numerical strategy consists in condense
neighboring mixed cells in one direction of the Cartesian mesh, in which
interfaces are considered as mono dimensional objects, namely they are
considered vertical during $x$ direction step and horizontal during $y$ direction step.
A condensate then contains layers of successive different
materials that are separated by straight interfaces. The thickness of
these layers is calculated through volume conservation. The ordering of
layers is given by the \textit{2D} description from the
previous time step. It is determined thanks to the volume fractions of
neighboring cells.
The layer evolution is calculated in a Lagrangian fashion
which implies that layers can be as thin as partial volumes are small.
Once quantities and interface positions inside the
condensate are known at time $t^{n+1}$, they are remapped back
onto the original Eulerian mesh. Finally a 2D normal in each mixed cell
is computed as described in \cite{DLY1}: the method is based on an approximation
of the gradient of the volume fraction function
in mixed cells.
It provides the normal to materials interface in each cell that is further used to locate materials
within mixed cells.
The numerical scheme used in a condensate
is presented in great details in \cite{Braeunig09}
and we omit this description in this work and rather focus on the interface reconstruction method.
As shown in \cite{Braeunig09} this numerical method has several attractive properties as
conservation and perfect sliding of materials as instance.
Moreover $\Delta t$ is not restricted by small partial volume thanks to a tight control of density
and pressure \cite{Braeunig10}.
The numerical experiments carried out in \cite{GKL1,these_jpb,Braeunig09,benchmark}
have confirmed the efficiency of such a method for compressible multi-material computation.
Although very promising, the method suffers from the way interfaces are dealt with.
In order to illustrate the interface reconstruction method
NIP let us consider a square like interface cutting the Eulerian cells,
as in Figure \ref{fig:nip}-(A). These interfaces are indeed defined by their normals within each cell.
\begin{figure}
\caption{ \label{fig:nip}
\label{fig:nip}
\end{figure}
NIP method consists of the following steps assuming the condensate is in the $x$ direction:
\begin{itemize}
\item {\em Representation} Figure \ref{fig:nip}-(B).
The representation step can be seen as the way of determining on which side (left or right)
of the mixed cell the material is to be put. This is done by comparing the direction of the
interface normal at time $t^n$ with the vertical direction.
\item {\em Condensate construction} Figure~\ref{fig:nip}-(C).
The construction of the condensate consists in discarding any cell edges in the mixed cells considered.
Then the partial volumes of the same contiguous materials are glued together into so called condensate layers.
As instance cell $2$ and $3$ dark materials are merged into one stand-alone layer with associated volume averaged values.
\item {\em Condensate evolution} Figure~\ref{fig:nip}-(D).
The condensate layers evolution is computed from $t^n$ to $t^{n+1}$
thanks to the numerical scheme developed in \cite{Braeunig09}. In short, each vertical interface
is assigned a velocity and, consequently, a new position of each layer within the condensate
is determined in a Lagrangian way. Any conserved variable is computed accordingly.
\item {\em Reconstruction} Figure~\ref{fig:nip}-(E).
This phase consists in ``guessing'' the shape of each layer in the condensate
before remapping.
The reconstruction phase was not originally considered as a true phase of the algorithm
as the author used the same shapes as the ones produced in phase {\em Condensate construction}, i.e. only vertical interfaces.
\item {\em Projection} Figure~\ref{fig:nip}-(F).
The projection step consists in remapping the shapes obtained from the reconstruction phase onto
the Eulerian grid. This step produces updated partial volumes in mixed cells. Volume
fractions are deduced.
\end{itemize}
When all mixed cells in the domain are treated for direction $x$, the interface normals are computed using the updated volume fractions. This concludes the system evolution in direction $x$, as we are back to a similar situation as the one described in Figure~\ref{fig:nip}-(A).\\
In the case where the normal is almost vertical, positioning the material
on either side of the cell might be, at least inaccurate, or, worse, incorrect.
Furthermore the reconstruction phase is here clearly
inconsistent: the interfaces are initially horizontal in cell 3 and 4 (Figure~\ref{fig:nip}-(A)), while in the Reconstruction Figure~\ref{fig:nip}-(E) and in the Projection Figure~\ref{fig:nip}-(F) phase interfaces are set vertical for any initial geometry. This situation of a horizontal interface is the worst case, but it illustrates the lack of geometrical consistency of NIP. This inaccurate reconstruction
step leads to a lack of accuracy of the volume fractions obtained after the remapping step.
Ultimately, it impacts the whole numerical method in any advection process. \\
As an illustration let us consider the diagonal advection of a square back and forth as shown in
Figure~\ref{fig:advection_square}. We omit the exhaustive description of this test as it will be done
in the numerical Section of this paper. On the right panel it is obvious that the shape of the square is not
well approximated. More important the horizontal and vertical edges of the square
do not remain so. This behaviour is less pronounced if one refines the mesh but still remains.
\begin{figure}
\caption{ \label{fig:advection_square}
\label{fig:advection_square}
\end{figure}
Our goal is to improve the reconstruction step so that the new method, denoted ENIP standing for Enhanced Natural Interface Positioning, cure this geometrical inconsistency during the advection phase of the algorithm.
{\cal S}ubsection{FVCF-ENIP}
The main idea of the new interface reconstruction method ENIP emanates from the following remarks:
\begin{enumerate}
\item At time $t^n$ any interface normal in mixed cell $i$ denoted $\vec{n}_i$ is known. It is
used to locate the partial volumes within cell $i$ when the condensate is constructed (phase
(B) and (C) of Figure~\ref{fig:nip}). However $\vec{n}_i$ is never taken into account in the reconstruction
and projection phases (E) and (F) from the same figure.
\item Any layer of the condensate evolves as a Lagrangian object in the original method. Consequently the cell faces could evolve in an almost Lagrangian manner within this condensate. This makes possible to conserve the initial geometry of partial volumes during this Lagrangian motion.
\end{enumerate}
Therefore ENIP modifies several steps of NIP as depicted in Figure~\ref{fig:nip-2}.
Once a patch of neighbor mixed cells in $x$ direction\footnote{The $y$ direction is treated likewise.} are agglomerated,
The same five steps as for NIP method are performed.
The first two steps are kept unmodified. The last three are modified as described
in the following.
\begin{figure}
\caption{ \label{fig:nip-2}
\label{fig:nip-2}
\end{figure}
{\cal S}ubsubsection{Lagrangian {\em Condensate evolution} step}
\paragraph{Cell interface Lagrangian velocity}
After the condensate at $t^n$ is constructed,
each layer labeled $c$ is located thanks to the
left and right interface position respectively called $x_c^-, x_c^+$. The numerical scheme
provides the layer evolution, and as a by-product, the velocity of these interface positions,
$u_c^-,u_c^+$ are given by
\begin{eqnarray}
x_c^{-,n+1} = x_c^- + \Delta t \ u_c^- ,\ \ \ \
x_c^{+,n+1} = x_c^+ + \Delta t \ u_c^+ .
\end{eqnarray}
We make the following fundamental linear displacement assumption:
\textit{The velocity linearly varies within any layer}, see Figure~\ref{fig:vitesse}
for a sketch.
\begin{figure}
\caption{ \label{fig:vitesse}
\label{fig:vitesse}
\end{figure}
This assumption implies that any point $x_i \in \left[x_c^-;x_c^+\right]$
characterized by its 1D barycentric coordinates
\begin{eqnarray}
\lambda_i^- = \frac {x_c^+-x_i}{x_c^+-x_c^-}, \ \ \ \ \ \lambda_i^+ = \frac {x_i-x_c^-}{x_c^+-x_c^-},
\end{eqnarray}
moves to location
\begin{eqnarray}\label{vit_noeud}
x_i^{n+1} = \lambda_i^- x_c^{-,n+1} + \lambda_i^+ x_c^{+,n+1}
= x_i + \Delta t \left( \lambda_i^- u_c^{-} + \lambda_i^+ u_c^{+} \right) .
\end{eqnarray}
Then the point velocity is naturally set to
$u_i= \lambda_i^- u_c^{-} + \lambda_i^+ u_c^{+}$.
Using this previous formula one can associate a ``Lagrangian''
velocity to any cell interface.
As instance in Figure~\ref{fig:nip-2}-(C) cell interface located at $x_i^n$ moves
to the position $x_i^{n+1}=x_i^n + \Delta t \ u_i$ with
$u_i$ being the linear combination between $u_c^{-}$ and $u_c^{+}$ {\it via} the
barycentric coordinates of point $x_i$ in $[x_c^-;x_c^+]$.
With the same formula one gets $x_{i+1}^{n+1} = x_{i+1}^n + \Delta \ u_{c+1}^-$
in the next layer as $x_{i+1}^n \equiv x_{c+1}^-$.
\paragraph{Compression/expansion rates}
The global rate of compression/expansion in layer $c$ during $\Delta t$ is given by
\begin{eqnarray}
\delta \textrm{Vol}_c = \frac{x_c^{+,n+1}-x_c^{-,n+1}}{x_c^{+}-x_c^{-}}
= 1 + \Delta t \frac{u_c^{+}-u_c^{-}}{x_c^{+}-x_c^{-}}.
\end{eqnarray}
The linearity assumption provides a simple way to determine
the rates of compression/expansion at left/right of a point
$x_i\in \left[x_c^-;x_c^+\right]$
\begin{eqnarray}
\delta \textrm{Vol}_c^- = \frac{x_i^{n+1}-x_c^{-,n+1}}{x_c^{+}-x_c^{-}},\ \ \
\delta \textrm{Vol}_c^+ = \frac{x_c^{+,n+1}-x_i^{n+1}}{x_c^{+}-x_c^{-}},
\end{eqnarray}
that fulfil $\delta \textrm{Vol}_c^- + \delta \textrm{Vol}_c^+ = \delta \textrm{Vol}_c$.
Moreover the substitution of $x_i^{n+1}$ in the previous equations
yields
\begin{eqnarray}
\delta \textrm{Vol}_c^- = \frac{x_i-x_c^-}{x_c^+ -x_c-} +
\Delta t \frac{u_i-u_c^{-}}{x_c^{+}-x_c^{-}}
= \lambda_i^++ \Delta t \frac{u_i-u_c^{-}}{x_c^{+}-x_c^{-}},
\end{eqnarray}
where $u_i-u_c^- = (\lambda_i^- u_c^{-} + \lambda_i^+ u_c^{+}) - u_c^-
= \lambda^+_i (u_c^+-u_c^{-})$, therefore the compression rates simply
writes
\begin{eqnarray}
\delta \textrm{Vol}_c^- &=& \lambda_i^+ \left( 1 + \Delta t \frac{u_c^+-u_c^{-}}{x_c^{+}-x_c^{-}} \right)
= \lambda_i^+ \delta \textrm{Vol}_c ,\\
\delta \textrm{Vol}_c^+ &=& \lambda_i^- \left( 1 + \Delta t \frac{u_c^+-u_c^{-}}{x_c^{+}-x_c^{-}} \right)
=\lambda_i^- \delta \textrm{Vol}_c .
\end{eqnarray}
Each $\delta \textrm{Vol}_c^+$ or $\delta \textrm{Vol}_c^-$ is associated to a unique Eulerian cell; as instance
in Figure~\ref{fig:nip-2}, $\delta \textrm{Vol}_c^-$ is associated to cell $2$, $\delta \textrm{Vol}_c^+$ to
cell $3$, $\delta \textrm{Vol}_{c+1}^+$ to cell $4$ and so on.
Therefore $\delta \textrm{Vol}_c^\pm$ provides {\it de facto}
the compression/expansion of the partial volume originating from its associated Eulerian
cell motion. Furthermore, as any Eulerian mixed cell $i$ possesses a unique normal denoted
$\vec{n}_i$, this last is associated to the corresponding partial volume $\delta \textrm{Vol}_c^\pm$;
this normal is consequently labeled $\vec{n}_c^\pm$.
These rates are then used to reconstruct the material topology into the
Lagrangian cell.
{\cal S}ubsubsection{{\em Reconstruction} step}
The Lagrangian cell $i+1/2$ at $t^{n+1}$ the interfaces of which moved as
\begin{eqnarray}
x_i^{n+1} = x_i + \Delta t \ u_i , \ \ \ \ x_{i+1}^{n+1} = x_{i+1} + \Delta t \ u_{i+1},
\end{eqnarray}
changed its volume as
\begin{eqnarray}
\delta \textrm{Vol}_{i+1/2} = \frac{V_{i+1/2}^{n+1}}{V_{i+1/2}} = \frac{x_{i+1}^{n+1} - x_i^{n+1} }{x_{i+1} - x_i}
= 1 + \Delta t \frac{ u_{i+1} - u_i }{x_{i+1} - x_i}.
\end{eqnarray}
The velocity $u_i$ depends on $u_{c-1}^-, u_{c-1}^+$ and $u_{i+1}$ depends on
$u_{c}^-, u_{c}^+$.
Moreover $u_{c-1}^+ \equiv u_c^-$ by definition. \\
The second fundamental assumption states that the interface normals $\vec{n}_c^\pm$
do not change their direction during their Lagrangian evolution.
The goal is to locate the partial volume into the Lagrangian cell at $t^{n+1}$ and construct the linear interface, knowing its normal $\vec{n}_c^\pm$. Necessarily this partial volume is either in contact with
cell interface $x_i$ (superscript $+$) or $x_{i+1}$ (superscript $-$).
Its volume at $t^{n+1}$ is given by
\begin{eqnarray}
V_c^{\pm,n+1} = V_c^\pm \ \delta \textrm{Vol}_c^\mp
= V_c^\pm + \Delta t \ \lambda_i^\mp (u_c^+ - u_c^-).
\end{eqnarray}
If $V_c^{\pm,n+1} \leq V_{i+1/2}^{n+1}$ then there exists a unique line oriented
by the normal $\vec{n}_c^\pm$ and separating the cell volume into
two sub-volumes $V_c^{\pm,n+1}$ and $(V_{i+1/2}^{n+1}-V_c^{\pm,n+1})$ respectively by the
PLIC (``\textit{Piecewise Linear Interface Construction}'' \cite{DLY1}) method.
As the displacement velocity $u(x)$ is supposed to be piecewise
linear (by the first assumption see Figure~\ref{fig:vitesse}), then, if $x_i < x_c^- < x_{i+1}$ one deduces
$x_i^{n+1} < x_c^{-,n+1} < x_{i+1}^{n+1}$. Therefore the sub-volume at $t^{n+1}$
is strictly included into the Lagrangian cell volume $V_{i+1/2}^{n+1}$.
This phase is depicted in Figure~\ref{fig:nip-2}-(E)
{\cal S}ubsubsection{{\em Projection} step}
The projection step performs the exact intersection between the Lagrangian condensate
obtained after the reconstruction step in Figure~\ref{fig:nip-2}-(E) and the
Eulerian mesh (bold line squares in Figure~\ref{fig:nip-2}-(A)).
This step is depicted in Figure~\ref{fig:nip-2}-(F). The exact intersection consists
in projecting each partial volume that is accurately located into the condensate, onto some Eulerian fixed cell(s). As instance in
Figure~\ref{fig:nip-2}-(F) the first partial volume is projected onto Eulerian
cells $2$ (green cell) and $3$ (red cell). Contrarily the last partial volume is
totally projected into Eulerian cell $5$ (brown cell).
This projection provides the quantity of material per Eulerian cell, or, equivalently
its volume fraction.
Once volume fractions in the mixed cells are updated though the evolution of
condensates, 2D normals are computed using the same technique as in original
NIP method.
{\cal S}ection{Numerical results} \label{sec:numerics}
In this Section we present a set of test cases to assess the
efficiency of the approach described in the previous Sections.
First, one validates the technique on pure advection test cases
that often present excessive smearing of interfaces due to the
numerical inaccuracy embedded into the scheme.
A square shaped object is advected with constant velocity
in a diagonal direction in a first test, then into a rotating flow.
Finally an hydrodynamics test case is presented.
{\cal S}ubsection{Advection context} \label{ssec:advection}
An initial square $[0.1;0.1]\times [0.2;0.2]$
is located into the domain $\Omega= [0:0.4]\times [0;0.6]$.
The density into the square is set
to $\rho_0(x)=1$ whereas it is set to $\rho_0(x)=0$ outside.
In the pure advection context this square shape
should be perfectly conserved through the equation
\begin{eqnarray}
\partialt \rho + u\partialx \rho + v \partialx \rho = 0,
\end{eqnarray}
where $(u,v)$ is a constant velocity field. The exact solution
at any point $x$ and any time $t$ is
$\rho^{ex}(x,y,t) = \rho_0(x - u \ t, y- v\ t)$. If the numerical
method provides an approximated solution called
$\rho_i^n$ in cell $i$ at time $t^n$ then the error
in $L_\alpha$ norm is evaluated by ($\alpha=1,2$)
\begin{eqnarray}
\varepsilon_\alpha = \frac{{\cal S}um_i | \rho^n_i - \rho^{ex}(x_i,t^n) |^\alpha}{{\cal S}um_i | \rho^{ex}(x_i,t^n) |^\alpha}.
\end{eqnarray}
The first test consists in advecting the square with the constant velocity field $u=1$, $v=3$ up to the time $t=0.1$ then reversing the
advection field by setting $u=-1$, $v=-3$ up to final time $t=0.2$ so that the final configuration perfectly fits the initial one.
Any method (NIP and ENIP included) introduces some error that we intend to measure with this test.
In Figure~\ref{fig:adv_square} are shown the exact solution (top-left) and the results obtained with a $60\times 60$ mesh for NIP (top-right) and ENIP (bottom-right).
ENIP is visibly able to preserve the shape of the square whereas NIP is not. A mesh refinement of NIP computation ($120\times 120$ mesh for the bottom-left panel) does not improve the situation.
\begin{figure}
\caption{ \label{fig:adv_square}
\label{fig:adv_square}
\end{figure}
In table~\ref{tab:adv} we gather the errors for the $L_1$, $L_2$ norms for successively refined meshes for the
NIP and the proposed ENIP method on this advection problem. Systematically ENIP over-tops NIP.
\begin{table}
\begin{tabular}{|c||cc|cc|}
{\cal H}line
$\Delta x=\Delta y$ & \textbf{$L_1$ NIP} & \textbf{$L_1$ ENIP} & \textbf{$L_2$ NIP} & \textbf{$L_2$ ENIP} \\
{\cal H}line
{\cal H}line
0.02 & 3.652 & 0.196 & 2.575 & 0.079 \\
0.0133 & 0.389 & 0.165 & 0.318 & 0.081 \\
0.01 & 0.339 & 0.111 & 0.284 & 0.053 \\
0.005 &0.221 & 0.042 & 0.195 & 0.017 \\
0.0033 &0.155 & 0.025 & 0.138 & 0.010 \\
{\cal H}line
\end{tabular}
\caption{ \label{tab:adv} Error in $L_1, L_2$ norms for the advection problem --- NIP versus ENIP methods.}
\end{table}
In Figure~\ref{fig:convergence} we display the log-log scale results for the error in $L_2$ norm for both methods showing
the improvement gained by ENIP; indeed the slope which represents a measure of the numerical order of convergence is improved by a factor $2.5$ ($0.6$ for NIP and $1.5$ for ENIP).
\begin{figure}
\caption{ \label{fig:convergence}
\label{fig:convergence}
\end{figure}
The next test consists in the rigid rotation of a square $[0.06;0.46]\times [0.3;0.7]$ (density $1$) into the unit square domain, see
Figure~\ref{fig:rotation} top-left panel.
A $100\times 100$ uniform mesh is considered and the rotation is given by the velocity field
\begin{eqnarray} \nonumber
u = -100(y-0.5), & & v=100(x-0.5).
\end{eqnarray}
In Figure~\ref{fig:rotation} we display the density after $5/8$ of the full rotation, after one and three rotations.
The square shape is almost preserved. Contrarily the classical NIP method would totally lose the shape after one rotation.
\begin{figure}
\caption{ \label{fig:rotation}
\label{fig:rotation}
\end{figure}
{\cal S}ubsection{Hydrodynamics context} \label{ssec:physics}
We run an idealized 2D test case that corresponds to the free drop of a liquid rectangle within a 2D rectangular tank filled with gas \cite{benchmark}.
This context is inspired by the problem of sloshing that may appear in the tanks of Liquid-Natural-Gas (LNG) carriers.
The study focuses on the ability for the numerical simulations to take properly into account the physics that is of major importance during the liquid impact such as the escape of the gas underneath and its compression.
As a strong sliding process occurs between the compressed gas and the falling liquid. The ability of the method to properly deal with sliding conditions at the interface has a major effect on the final numerical compression and shape of the trapped air. This has ultimately a strong influence on the impact pressure. \\
The test case consists in a domain $\Omega = [0.0;0.0]\times[10m;15m]$ filled with air. The liquid is initially at rest in the rectangle $[0;2]\times[5;10]$ and is falling under the gravity that is pointing downward with magnitude $g=9.81m.s^{-2}$.
A free fall of the liquid into vacuum would impact at $t_{\textrm{impact}}=0.64s$ however due to the presence of the gas this theoretical value is not correct for our simulation however some critical phenomena still occur in the vicinity of this time. As instance around $t_{\textrm{impact}}$ a pocket of gas is trapped under the falling liquid and this strongly impacts the numerical impact pressure by decelerating and damping the free fall of the liquid. Therefore a good interface reconstruction method should qualitatively improve the numerical results.
One considers a mesh made of $100\times 150$ uniform cells on the domain.
One shows the results for NIP and ENIP at time $t=0.6s$ Figure~\ref{fig:liquid}-\textbf{(a)-(b)} and $t=0.64s$ in Figure~\ref{fig:liquid}-\textbf{(c)-(d)}.
The classical NIP method was already able to deal with such sliding effects.
However the interface reconstruction method employed is not accurate and stable enough to be free of oscillation that one suspects to be only a numerical artifacts (see panels \textbf{(a-c)}).
Contrarily the new reconstruction method ENIP on this very same test case is able to produce a smooth interface that permits to obtain a more realistic simulation. Indeed this simulation prominently displays the fact that the ``bubbling'' effects of NIP is of pure numerical origin and that ENIP cures this drawback.
\begin{figure}
\caption{ \label{fig:liquid}
\label{fig:liquid}
\end{figure}
{\cal S}ection{Conclusion and perspectives} \label{sec:conclusion}
This paper deals with the improvement of the so-called NIP (Natural Interface Positioning) method.
The NIP method described in \cite{Braeunig09} is an add-on to the FVCF method in order to treat multi-material fluid flows
uses the concept of condensate. A condensate is the association of contiguous mixed cells in either $x$ or $y$ direction. They are further treated as an entity to make possible the treatment of each mixed cell taken individually. NIP is the method based on the following steps:
{\em Representation}, {\em Condensate construction}, {\em Condensate evolution}, {\em Reconstruction}, and {\em Projection}.
The present paper points the weakness of the NIP method in pure advection context and, consequently, in a full multi-material hydrodynamics one. An enhanced NIP method is proposed (ENIP). It modifies several of the previous listed steps. More precisely the condensate is assumed to evolved in an almost-Lagrangian fashion. The reconstruction step assumes that the condensate keeps the same form modulo some expansion/compression that the numerical scheme already provides. So the displacement of the condensate is performed either with the true computed velocity or with an interpolation of it. \textit{In fine} the condensate preserves its topology contrarily to the original NIP method for which the condensate has no recollection of its shape from the beginning of the time step. \\
The capability of the full numerical method is now dramatically improved as seen on advection test cases (advection and rigid rotation of a square). Moreover we ran ENIP on a difficult mutli-material hydrodynamics tests simulating the free drop of a liquid rectangle within a 2D rectangular tank filled with gas
in the context of sloshing that may appear in the tanks of Liquid-Natural-Gas carrier (see \cite{benchmark}). The accuracy, stability and robustness of the ENIP method is clearly seen especially at the time some air is trapped under the water.
In the near future we plan to investigate the evolution of this method to the case of mixed cells with more than two materials. In this case the only difficulty lays in the positioning of the different materials in the cell, but their evolution within the condensate follows exactly the same algorithm ENIP with no modification of the numerical scheme. We also plan to investigate the evolution of the method in 3D.
\end{document} |
\begin{document}
\title{An efficient descent method for locally Lipschitz multiobjective optimization problems}
\author[1]{Bennet Gebken}
\author[1]{Sebastian Peitz}
\affil[1]{\normalsize Department of Mathematics, Paderborn University, Germany}
\maketitle
\begin{abstract}
In this article, we present an efficient descent method for locally Lipschitz continuous multiobjective optimization problems (MOPs). The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth MOPs with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method to the multiobjective proximal bundle method by M{\"a}kel{\"a}. The results indicate that our method is competitive while being easier to implement. While the number of objective function evaluations is larger, the overall number of subgradient evaluations is lower. Finally, we show that our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth MOPs.
\end{abstract}
\section{Introduction}
In many scenarios in real life, the problem of optimizing multiple objectives at the same time arises. In engineering for example, one often wants to steer a physical system as close as possible to a desired state while minimizing the required energy cost at the same time. These problems are called \emph{multiobjective optimization problems} (MOPs) and generally do not possess a single optimal solution. Instead, the solution is the set of all optimal compromises, the so-called \emph{Pareto set} containing all \emph{Pareto optimal} points. Due to this, the numerical computation of solutions to MOPs is more challenging than to single-objective problems. On top of that, there are numerous applications where the objectives are nonsmooth, for example contact problems in mechanics, which adds to the difficulty. In this article, we will address both difficulties combined by considering nonsmooth MOPs.
When addressing the above-mentioned difficulties, i.e., multiple objectives and nonsmoothness, separately, there exists a large number solution methods. For smooth MOPs, the most popular methods include evolutionary \cite{D2001, DPAM2002} and scalarization methods \cite{M1998}. Additionally, some methods from single-objective optimization have been generalized, like gradient descent methods \cite{FS2000,SSW2002,GPD2017} and Newton's method \cite{FGS2008}. For the nonsmooth single-objective case, commonly used methods include subgradient methods \cite{S1985}, bundle methods \cite{K1990} and gradient sampling methods \cite{BLO2005}. More recently, in \cite{MY2012}, a generalization of the steepest descent method to the nonsmooth case was proposed, which is based on an efficient approximation of the subdifferential of the objective function. For nonsmooth multiobjective optimization, the literature is a lot more scarce. Since classical scalarization approaches do not require the existence of gradients, they can still be used. In \cite{AGG2015}, a generalization of the steepest descent method was proposed for the case when the full subdifferentials of the objectives are known, which is rarely the case in practice. In \cite{B2013,NSFL2013}, the subgradient method was generalized to the multiobjective case, but both articles report that their method is not suitable for real life application due to inefficiency. In \cite{MKW2014} (see also \cite{K1985b,M2003}), a multiobjective version of the proximal bundle method was proposed, which currently appears to be the most efficient solver.
In this article, we develop a new descent method for locally Lipschitz continuous MOPs by combining the descent direction from \cite{AGG2015} with the approximation of the subdifferentials from \cite{MY2012}. In \cite{AGG2015} it was shown that the element with the smallest norm in the negative convex hull of the subdifferentials of the objective functions is a common descent direction for all objectives. In \cite{MY2012}, the subdifferential of the objective function was approximated by starting with a single subgradient and then systematically computing new subgradients until the element with the smallest norm in the convex hull of all subgradients is a direction of (sufficient) descent. Combining both approaches yields a descent direction for locally Lipschitz MOPs and together with an Armijo step length, we obtain a descent method. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare the performance of our method to the multiobjective proximal bundle method from \cite{MKW2014}. The results indicate that our method is inferior in terms of function evaluations, but superior in terms of subgradient evaluations.
The structure of this article is as follows: we start with a short introduction to nonsmooth and multiobjective optimization in Section \ref{sec:introduction}. In Section \ref{sec:descent_method}, we derive our descent method by replacing the Clarke subdifferential for the computation of the descent direction by the Goldstein $\varepsilon$-subdifferential and then showing how the latter can be efficiently approximated. In Section \ref{sec:numerical_examples}, we apply our descent method to numerical examples. We first visualize and discuss the typical behavior of our method before comparing it to the multiobjective proximal bundle method from \cite{MKW2014} using a set of test problems. Afterwards, we show how our method can be combined with a subdivision algorithm to approximate entire Pareto sets. Finally, in Section \ref{sec:conclusion}, we draw a conclusion and discuss possible future work.
\section{Nonsmooth multiobjective optimization} \label{sec:introduction}
We consider the nonsmooth multiobjective optimization problem
\begin{align} \tag{MOP} \label{eq:MOP}
\min_{x \in \mathbb{R}^n} f(x) = \min_{x \in \mathbb{R}^n}
\begin{pmatrix}
f_1(x) \\
\vdots \\
f_k(x)
\end{pmatrix},
\end{align}
where $f : \mathbb{R}^n \rightarrow \mathbb{R}^k$ is the \emph{objective vector} with components $f_i : \mathbb{R}^n \rightarrow \mathbb{R}$, $i \in \{1,...,k\}$, called \emph{objective functions}. We assume the objective functions to be \emph{locally Lipschitz continuous}, i.e., for each $i \in \{1,...,k\}$ and $x \in \mathbb{R}^n$, there is some $L_i > 0$ and $\varepsilon > 0$ with
\begin{align*}
|f_i(y)- f_i(z)| \leq L_i \| y - z \| \quad \forall y, z \in \{ y \in \mathbb{R}^n : \| x - y \| < \varepsilon \},
\end{align*}
where $\| \cdot \|$ denotes the Euclidean norm in $\mathbb{R}^n$. Since \eqref{eq:MOP} is an optimization problem with a vector-valued objective function, the classical concept of optimality from the scalar case can not directly be conveyed. Instead, we are looking for the \emph{Pareto set}, which is defined in the following way:
\begin{definition}
A point $x \in \mathbb{R}^n$ is called \emph{Pareto optimal}, if there is no $y \in \mathbb{R}^n$ such that
\begin{align*}
f_i(y) &\leq f_i(x) \quad \forall i \in \{1,...,k\}, \\
f_j(y) &< f_j(x) \quad \text{for some } j \in \{1,...,k\}.
\end{align*}
The set of all Pareto optimal points is the \emph{Pareto set}.
\end{definition}
In practice, to check if a given point is Pareto optimal, we need optimality conditions. In the smooth case, there are the well-known KKT conditions (cf.~\cite{M1998}), which are based on the gradients of the objective functions. In case the objective functions are merely locally Lipschitz, the KKT conditions can be generalized using the concept of \emph{subdifferentials}. In the following, we will recall the required definitions and results from nonsmooth analysis. For a more detailed introduction, we refer to \cite{C1983}.
\begin{definition}
Let $\Omega_i \subseteq \mathbb{R}^n$ be the set of points where $f_i$ is not differentiable. Then
\begin{align*}
\partial f_i(x) = \conv ( \{ \xi \in \mathbb{R}^n : &\exists (x_j)_j \in \mathbb{R}^n \setminus \Omega_i \text{ with } x_j \rightarrow x \text{ and } \\
&\nabla f_i(x_j) \rightarrow \xi \text{ for } j \rightarrow \infty \} )
\end{align*}
is the \emph{(Clarke) subdifferential of} $f_i$ \emph{in} $x$. $\xi \in \partial f_i(x)$ is a \emph{subgradient}.
\end{definition}
It is easy to see that if $f_i$ is continuously differentiable, then the Clarke subdifferential is the set containing only the gradient of $f_i$. We will later use the following technical result on some properties of the Clarke subdifferential (cf.~\cite{C1983}, Prop.~2.1.2).
\begin{lemma} \label{lem:subdiff}
$\partial f_i(x)$ is nonempty, convex and compact.
\end{lemma}
Using the subdifferential, we can state a necessary optimality condition for locally Lipschitz MOPs (cf.~\cite{MEK2014}, Thm.~12).
\begin{theorem} \label{thm:KKT}
Let $x \in \mathbb{R}^n$ be Pareto optimal. Then
\begin{align} \label{eq:KKT}
0 \in \conv \left( \bigcup_{i=1}^k \partial f_i(x) \right).
\end{align}
\end{theorem}
In the smooth case, \eqref{eq:KKT} reduces to the classical multiobjective KKT conditions. Note that in contrast to the smooth case, the optimality condition \eqref{eq:KKT} is numerically challenging to work with, as subdifferentials are difficult to compute. Thus, in numerical methods, \eqref{eq:KKT} is only used implicitly.
The method we are presenting in this paper is a \emph{descent method}, which means that, starting from a point $x_1 \in \mathbb{R}^n$, we want to generate a sequence $(x_j)_j \in \mathbb{R}^n$ in which each point is an improvement over the previous point. This is done by computing directions $v_j \in \mathbb{R}^n$ and step lengths $t_j \in \mathbb{R}^{>0}$ such that $x_{j+1} = x_j + t_j v_j$ and
\begin{align*}
f_i(x_{j+1}) < f_i(x_j) \quad \forall j \in \mathbb{N}, \ i \in \{1,...,k\}.
\end{align*}
For the computation of $v_j$, we recall the following basic result from convex analysis that forms the theoretical foundation for descent methods in the presence of multiple (sub)gradients. Let $\| . \|$ be the Euclidean norm in $\mathbb{R}^n$.
\begin{theorem} \label{thm:steepest_descent_direction}
Let $W \subseteq \mathbb{R}^n$ be convex and compact and
\begin{align} \label{eq:min_norm_problem}
\bar{v} := \argmin_{\xi \in -W} \| \xi \|^2.
\end{align}
Then either $\bar{v} \neq 0$ and
\begin{align} \label{eq:steepest_descent_ineq}
\langle \bar{v}, \xi \rangle \leq - \| \bar{v} \|^2 < 0 \quad \forall \xi \in W,
\end{align}
or $\bar{v} = 0$ and there is no $v \in \mathbb{R}^n$ with $\langle v, \xi \rangle < 0$ for all $\xi \in W$.
\end{theorem}
\begin{proof}
Since $\bar{v}$ is the projection of the origin onto the closed and convex set $-W$, we have
\begin{align*}
& 0 \leq \langle -\xi - \bar{v}, \bar{v} - 0 \rangle = -\langle \bar{v}, \xi \rangle - \| \bar{v} \|^2 \\
\Leftrightarrow \quad & \langle \bar{v}, \xi \rangle \leq - \| \bar{v} \|^2
\end{align*}
for all $\xi \in W$ (cf.~\cite{CG1959}, Lem.). In particular, if $\bar{v} \neq 0$ then $\langle \bar{v}, \xi \rangle \leq - \| \bar{v} \|^2 < 0$.
Conversely, $\bar{v} = 0$ implies $0 \in W$, so in this case there can not be any $v \in \mathbb{R}^n$ with $\langle v, \xi \rangle < 0$ for all $\xi \in W$.
\end{proof}
Roughly speaking, Theorem \ref{thm:steepest_descent_direction} states that the element of minimal norm in the convex and compact set $-W$ is directionally opposed to all elements of $W$. To be more precise, $\bar{v}$ is contained in the intersection of all half-spaces induced by elements of $-W$. In the context of optimization, this result has several applications:
\begin{itemize}
\item[(i)] In the smooth, single-objective case, $W = \{ \nabla f(x) \}$ trivially yields the classical steepest descent method.
\item[(ii)] In the smooth, multiobjective case, $W = \conv(\{ \nabla f_1(x), ..., \nabla f_k(x) \})$ yields the descent direction from \cite{FS2000} (after dualization) and \cite{SSW2002}.
\item[(iii)] In the nonsmooth, single-objective case, $W = \partial f(x)$ yields the descent direction from \cite{C1983}, Prop.~6.2.4.
\item[(iv)] In the nonsmooth, multiobjective case, $W = \conv \left( \bigcup_{i = 1}^k \partial f_i(x) \right)$ yields the descent direction from \cite{AGG2015}.
\end{itemize}
In (i) and (ii), the solution of problem \eqref{eq:min_norm_problem} is straightforward, since $W$ is a convex polytope with the gradients as vertices. In (iii), the solution of \eqref{eq:min_norm_problem} is non-trivial due to the difficulty of computing the subdifferential. In subgradient methods \cite{S1985}, the solution is approximated by using a single subgradient instead of the entire subdifferential. As a result, it can not be guaranteed that the solution is a descent direction and in particular, \eqref{eq:min_norm_problem} can not be used as a stopping criterion. In gradient sampling methods \cite{BLO2005}, the subdifferential is approximated by the convex hull of gradients of the objective function in randomly sampled points around the current point. Due to the randomness, it can not be guaranteed that the resulting direction yields sufficient descent. Additionally, a check for differentiability of the objective is required, which can pose a problem \cite{HSS2016}. In (iv), the solution of \eqref{eq:min_norm_problem} gets even more complicated due to the presence of multiple subdifferentials. So far, the only methods that deal with \eqref{eq:min_norm_problem} in this case are multiobjective versions of the subgradient method \cite{B2013,NSFL2013}, which were reported unsuitable for real life applications.
In the following section, we will describe a new way to compute descent directions for nonsmooth MOPs by systematically computing an approximation of $\conv \left( \cup_{i = 1}^k \partial f_i(x) \right)$ that is sufficient to obtain a "good enough" descent direction from \eqref{eq:min_norm_problem}.
\section{Descent method for nonsmooth MOPs} \label{sec:descent_method}
In this section, we will present a method to compute descent directions of nonsmooth MOPs that generalizes the method from \cite{MY2012} to the multiobjective case. As described in the previous section, when computing descent directions via Theorem \ref{thm:steepest_descent_direction}, one has the problem of having to compute subdifferentials. Since these are difficult to come by in practice, we will instead replace $W$ in Theorem \ref{thm:steepest_descent_direction} by an approximation of $\conv \left( \cup_{i = 1}^k \partial f_i(x) \right)$ such that the resulting direction is guaranteed to have sufficient descent. To this end, we will first replace the Clarke subdifferential by the so-called $\varepsilon$-\emph{subdifferential}, and then take a finite approximation of the latter.
\subsection{The epsilon-subdifferential}
By definition, $\partial f_i(x)$ is the convex hull of the limits of the gradient of $f_i$ in all sequences near $x$ that converge to $x$. Thus, if we evaluate $\nabla f_i$ in a number of points close to $x$ (where it is defined) and take the convex hull, we expect the resulting set to be an approximation of $\partial f_i(x)$. To formalize this, we introduce the following definition \cite{G1977,K2010}.
\begin{definition}
Let $\varepsilon \geq 0$, $x \in \mathbb{R}^n$ and $B_\varepsilon(x) := \{ y \in \mathbb{R}^n : \| x - y \| \leq \varepsilon\}$. Then
\begin{align*}
\partial_\varepsilon f_i(x) := \conv \left( \bigcup_{y \in B_\varepsilon(x)} \partial f_i(y) \right)
\end{align*}
is the \emph{(Goldstein)} $\varepsilon$-\emph{subdifferential of} $f_i$ \emph{in} $x$. $\xi \in \partial_\varepsilon f_i(x)$ is an $\varepsilon$-\emph{subgradient}.
\end{definition}
Note that $\partial_0 f_i(x) = \partial f_i(x)$ and $\partial f_i(x) \subseteq \partial_\varepsilon f_i(x)$. For $\varepsilon \geq 0$ we define for the multiobjective setting
\begin{align*}
F_\varepsilon(x) := \conv \left( \bigcup_{i=1}^k \partial_\varepsilon f_i(x) \right).
\end{align*}
To be able to choose $W = F_\varepsilon(x)$ in Theorem \ref{thm:steepest_descent_direction}, we first need to establish some properties of $F_\varepsilon(x)$.
\begin{lemma}
$\partial_\varepsilon f_i(x)$ is nonempty, convex and compact. In particular, the same holds for $F_\varepsilon(x)$.
\end{lemma}
\begin{proof}
For $\partial_\varepsilon f_i(x)$, this was shown in \cite{G1977}, Prop.~2.3. For $F_\varepsilon(x)$, it then follows directly from the definition.
\end{proof}
We immediately get the following corollary from Theorems \ref{thm:KKT} and \ref{thm:steepest_descent_direction}.
\begin{corollary}
Let $\varepsilon \geq 0$.
\begin{itemize}
\item[a)] If $x$ is Pareto optimal, then
\begin{align} \label{eq:eps_critical}
0 \in F_\varepsilon(x).
\end{align}
\item[b)] Let $x \in \mathbb{R}^n$ and
\begin{align} \label{eq:eps_min_norm_problem}
\bar{v} := \argmin_{\xi \in -F_\varepsilon(x)} \| \xi \|^2.
\end{align}
Then either $\bar{v} \neq 0$ and
\begin{align} \label{eq:eps_descent_ineq}
\langle \bar{v}, \xi \rangle \leq - \| \bar{v} \|^2 < 0 \quad \forall \xi \in F_\varepsilon(x),
\end{align}
or $\bar{v} = 0$ and there is no $v \in \mathbb{R}^n$ with $\langle v, \xi \rangle < 0$ for all $\xi \in F_\varepsilon(x)$.
\end{itemize}
\end{corollary}
The previous corollary states that when working with the $\varepsilon$-subdifferential instead of the Clarke subdifferential, we still have a necessary optimality condition and a way to compute descent directions, although the optimality conditions are weaker and the descent direction has a less strong descent. This is illustrated in the following example.
\begin{example} \label{exam:epssubdiff}
Consider the locally Lipschitz function
\begin{align*}
f : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \quad x \mapsto
\begin{pmatrix}
(x_1 - 1)^2 + (x_2 - 1)^2 \\
x_1^2 + | x_2 |
\end{pmatrix}.
\end{align*}
The set of nondifferentiable points of $f$ is $\mathbb{R} \times \{ 0 \}$. For $\varepsilon > 0$ and $x \in \mathbb{R}^2$ we have
\begin{align*}
\nabla f_1(x) =
\begin{pmatrix}
2(x_1 - 1) \\
2(x_2 - 1)
\end{pmatrix}
\quad \text{and} \quad
\partial_\varepsilon f_1(x) = 2 B_\varepsilon(x) -
\begin{pmatrix}
2 \\
2
\end{pmatrix}.
\end{align*}
For $x \in \mathbb{R} \times \{ 0 \}$ we have
\begin{align*}
\partial f_2(x) = \{ 2 x_1 \} \times [-1,1] \quad \text{and} \quad \partial_\varepsilon f_2(x) = \{ 2 x_1 + [-2\varepsilon,2\varepsilon] \} \times [-1,1].
\end{align*}
Figure \ref{fig:example_epssubdiff_descent} shows the Clarke subdifferential (a), the $\varepsilon$-subdifferential (b) for $\varepsilon = 0.2$ and the corresponding sets $F_\varepsilon(x)$ for $x = (1.5, 0)^\top$.
\begin{figure}
\caption{Clarke subdifferentials (a), $\varepsilon$-subdifferentials (b) for $\varepsilon = 0.2$ and the corresponding sets $F_\varepsilon(x)$ for $x = (1.5, 0)^\top$ in Example \ref{exam:epssubdiff}
\label{fig:example_epssubdiff_descent}
\end{figure}
Additionally, the corresponding solutions of \eqref{eq:eps_min_norm_problem} are shown. In this case, the predicted descent $-\| \bar{v} \|^2$ (cf.~\eqref{eq:steepest_descent_ineq}) is approximately $-3.7692$ in (a) and $-2.4433$ in (b).
Figure \ref{fig:example_epssubdiff_critical} shows the same scenario for $x = (0.5,0)^\top$.
\begin{figure}
\caption{Clarke subdifferentials (a), $\varepsilon$-subdifferentials (b) for $\varepsilon = 0.2$ and the corresponding sets $F_\varepsilon(x)$ for $x = (0.5, 0)^\top$ in Example \ref{exam:epssubdiff}
\label{fig:example_epssubdiff_critical}
\end{figure}
Here, the Clarke subdifferential still yields a descent, while $\bar{v} = 0$ for the $\varepsilon$-subdifferential. In other words, $x$ satisfies the necessary optimality condition \eqref{eq:eps_critical} but not \eqref{eq:KKT}.
\end{example}
The following lemma shows that for the direction from \eqref{eq:eps_min_norm_problem}, there is a lower bound for a step size up to which we have guaranteed descent in each objective function $f_i$.
\begin{lemma} \label{lem:step_size_bound}
Let $\varepsilon \geq 0$ and $\bar{v}$ be the solution of \eqref{eq:eps_min_norm_problem}. Then
\begin{align*}
f_i(x + t \bar{v}) \leq f_i(x) - t \| \bar{v} \|^2 \quad \forall t \leq \frac{\varepsilon}{\| \bar{v} \|}.
\end{align*}
\end{lemma}
\begin{proof}
Let $t \leq \frac{\varepsilon}{\| \bar{v} \|}$. Since $f_i$ is locally Lipschitz continuous on $\mathbb{R}^n$, it is in particular Lipschitz continuous on an open set containing $x + [0,t] \bar{v}$. By applying the mean value theorem (cf.~\cite{C1983}, Thm.~2.3.7), we obtain
\begin{align*}
f_i(x + t \bar{v}) - f_i(x) \in \langle \partial f_i(x + r \bar{v}), t \bar{v} \rangle
\end{align*}
for some $r \in (0,t)$. Since $\| x - (x + r \bar{v}) \| = r \| \bar{v} \| < \varepsilon$ we have $\partial f_i(x + r \bar{v}) \subseteq \partial_\varepsilon f_i(x)$. This means that there is some $\xi \in \partial_\varepsilon f_i(x)$ such that
\begin{align*}
f_i(x + t \bar{v}) - f_i(x) = t \langle \xi, \bar{v} \rangle.
\end{align*}
Combined with \eqref{eq:eps_descent_ineq} we obtain
\begin{align*}
& f_i(x + t \bar{v}) - f_i(x) \leq - t \| \bar{v} \|^2 \\
\Leftrightarrow \quad & f_i(x + t \bar{v}) \leq f_i(x) - t \| \bar{v} \|^2.
\end{align*}
\end{proof}
In the following, we will describe how we can obtain a good approximation of \eqref{eq:eps_min_norm_problem} without requiring full knowledge of the $\varepsilon$-subdifferentials.
\subsection{Efficient computation of descent directions}
In this part, we will describe how the solution of \eqref{eq:eps_min_norm_problem} can be approximated when only a single subgradient can be computed at every $x \in \mathbb{R}^n$. Similar to the gradient sampling approach, the idea behind our method is to replace $F_\varepsilon(x)$ in \eqref{eq:eps_min_norm_problem} by the convex hull of a finite number of $\varepsilon$-subgradients $\xi_1,...,\xi_m \in F_\varepsilon(x)$, $m \in \mathbb{N}$. Since it is impossible to know a priori how many and which $\varepsilon$-subgradients are required to obtain a good descent direction, we solve \eqref{eq:eps_min_norm_problem} multiple times in an iterative approach to enrich our approximation until a satisfying direction has been found. To this end, we have to specify how to enrich our current approximation $\conv(\{ \xi_1, ..., \xi_m \})$ and how to characterize an acceptable descent direction.
Let $W = \{\xi_1, ..., \xi_m\} \subseteq F_\varepsilon(x)$ and
\begin{align} \label{eq:approx_desc_dir}
\tilde{v} := \argmin_{v \in -\conv(W)} \| v \|^2.
\end{align}
Let $c \in (0,1)$. Motivated by Lemma \ref{lem:step_size_bound}, we regard $\tilde{v}$ as an \emph{acceptable} descent direction, if
\begin{align} \label{eq:accept_direction}
f_i \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) \leq f_i(x) - c \varepsilon \| \tilde{v} \| \quad \forall i \in \{1,...,k\}.
\end{align}
If the set $I \subseteq \{1,...,k\}$ for which \eqref{eq:accept_direction} is violated is non-empty then we have to find a new $\varepsilon$-subgradient $\xi' \in F_\varepsilon(x)$ such that $W \cup \{ \xi' \}$ yields a better descent direction. Intuitively, \eqref{eq:accept_direction} being violated means that the local behavior of $f_i$, $i \in I$, in $x$ in the direction $\tilde{v}$ is not sufficiently captured in $W$. Thus, for each $i \in I$, we expect that there exists some $t' \in (0,\frac{\varepsilon}{\| \tilde{v} \|}]$ such that $\xi' \in \partial f_i(x + t' \tilde{v})$ improves the approximation of $F_\varepsilon(x)$. This is proven in the following lemma.
\begin{lemma} \label{lem:infeasible_direction}
Let $c \in (0,1)$, $W = \{ \xi_1, ..., \xi_m \} \subseteq F_\varepsilon(x)$ and $\tilde{v}$ be the solution of \eqref{eq:approx_desc_dir}. If
\begin{align*}
f_i \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) > f_i(x) - c \varepsilon \| \tilde{v} \|,
\end{align*}
then there is some $t' \in (0,\frac{\varepsilon}{\| \tilde{v} \|}]$ and $\xi' \in \partial f_i(x + t' \tilde{v})$ such that
\begin{align} \label{eq:new_subgrad_condition}
\langle \tilde{v}, \xi' \rangle > - c \| \tilde{v} \|^2.
\end{align}
In particular, $\xi' \in F_\varepsilon(x) \setminus \conv(W)$.
\end{lemma}
\begin{proof}
Assume that for all $t' \in (0,\frac{\varepsilon}{\| \tilde{v} \|}]$ and $\xi' \in \partial f_i(x + t' \tilde{v})$ we have
\begin{align} \label{eq:c_descent_contr}
\langle \tilde{v}, \xi' \rangle \leq - c \| \tilde{v} \|^2.
\end{align}
By applying the mean value theorem as in Lemma \ref{lem:step_size_bound}, we obtain
\begin{align*}
f_i \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) - f_i(x) \in \langle \partial f_i(x + r \tilde{v}), \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \rangle
\end{align*}
for some $r \in (0,\frac{\varepsilon}{\| \tilde{v} \|})$. This means that there is some $\xi' \in \partial f_i(x + r \tilde{v})$ such that
\begin{align*}
f_i \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) - f_i(x) = \langle \xi', \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \rangle = \frac{\varepsilon}{\| \tilde{v} \|} \langle \xi', \tilde{v} \rangle.
\end{align*}
By \eqref{eq:c_descent_contr} it follows that
\begin{align*}
& f_i \left(x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) - f_i(x) \leq -c \varepsilon \| \tilde{v} \| \\
\Leftrightarrow \quad & f_i \left(x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) \leq f_i(x) - c \varepsilon \| \tilde{v} \|,
\end{align*}
which is a contradiction. In particular, \eqref{eq:steepest_descent_ineq} yields $\xi' \in F_\varepsilon(x) \setminus \conv(W)$.
\end{proof}
The following example visualizes the previous lemma.
\begin{example} \label{exam:bad_descent}
Consider $f$ as in Example \ref{exam:epssubdiff}, $\varepsilon = 0.2$ and $x = (0.75, 0)^\top$. The dashed lines in Figure \ref{fig:example_epssubdiff_false_descent} show the $\varepsilon$-subdifferentials, $F_\varepsilon(x)$ and the resulting descent direction (cf.~Figure \ref{fig:example_epssubdiff_descent} and \ref{fig:example_epssubdiff_critical}).
\begin{figure}
\caption{Approximations of $F_\varepsilon(x)$ for $\varepsilon = 0.2$ and $x = (0.75, 0)^\top$ in Example \ref{exam:bad_descent}
\label{fig:example_epssubdiff_false_descent}
\end{figure}
Let $y = (0.94,-0.02)^\top$. Then $\| x - y \| \approx 0.191 \leq \varepsilon$, so $y \in B_\varepsilon(x)$ and
\begin{align*}
\partial_\varepsilon f_1(x) &\supseteq \partial f_1(y) = \left\{
\begin{pmatrix}
-0.12 \\
-2.04
\end{pmatrix}
\right\}
=: \{ \xi_1 \},\\
\partial_\varepsilon f_2(x) &\supseteq \partial f_2(y) = \left\{
\begin{pmatrix}
1.88 \\
-1
\end{pmatrix}
\right\}
=: \{ \xi_2 \}.
\end{align*}
Let $W := \{ \xi_1, \xi_2 \}$ and $\conv(W)$ be the approximation of $F_\varepsilon(x)$, shown as the solid line in Figure \ref{fig:example_epssubdiff_false_descent}(a). Let $\tilde{v}$ be the solution of \eqref{eq:approx_desc_dir} for this $W$ and choose $c = 0.25$. Checking \eqref{eq:accept_direction}, we have
\begin{align*}
f_2 \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) &\approx 0.6101 > 0.4748 \approx f_2(x) - c \varepsilon \| \tilde{v} \|.
\end{align*}
By Lemma \ref{lem:infeasible_direction}, this means that there is some $t' \in (0, \frac{\varepsilon}{\| \tilde{v} \|}]$ and $\xi' \in \partial f_2(x + t' \tilde{v})$ such that
\begin{align*}
\langle \tilde{v}, \xi' \rangle > -c \| \tilde{v} \|^2.
\end{align*}
In this case, we can take for example $t' = \frac{1}{2} \frac{\varepsilon}{\| \tilde{v} \|}$, resulting in
\begin{align*}
\partial f_2(x + t'v) \approx \left\{
\begin{pmatrix}
1.4077 \\
1
\end{pmatrix}
\right\} =: \{ \xi' \}, \\
\langle \tilde{v}, \xi' \rangle \approx 0.4172 > -0.7696 \approx -c \| \tilde{v} \|^2.
\end{align*}
Figure \ref{fig:example_epssubdiff_false_descent}(b) shows the improved approximation $W \cup \{ \xi' \}$ and the resulting descent direction $\tilde{v}$. By checking \eqref{eq:accept_direction} for this new descent direction, we see that $\tilde{v}$ is acceptable. (Note that in general, a single improvement step like this will not be sufficient to reach an acceptable direction.)
\end{example}
Note that Lemma \ref{lem:infeasible_direction} only shows the existence of $t'$ and $\xi'$ without stating a way how to actually compute them. To this end, let $i$ be the index of an objective function for which \eqref{eq:accept_direction} is not satisfied, define
\begin{align} \label{eq:def_h}
h_i : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto f_i(x + t \tilde{v}) - f_i(x) + c t \| \tilde{v} \|^2
\end{align}
(cf.~\cite{MY2012}) and consider Algorithm \ref{algo:new_subgradient}. If $f_i$ is continuously differentiable around $x$, then \eqref{eq:new_subgrad_condition} is equivalent to $h_i'(t') > 0$, i.e., $h_i$ being monotonically increasing around $t'$. Thus, the idea of Algorithm \ref{algo:new_subgradient} is to find some $t$ such that $h_i$ is monotonically increasing around $t$, while checking if \eqref{eq:new_subgrad_condition} is satisfied for a subgradient $\xi \in f_i(x + t \tilde{v})$.
\begin{algorithm}
\caption{Compute new subgradient}
\label{algo:new_subgradient}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE Current point $x \in \mathbb{R}^n$, direction $\tilde{v}$, tolerance $\varepsilon$, Armijo parameter $c \in (0,1)$.
\STATE Set $a = 0$, $b = \frac{\varepsilon}{\| \tilde{v} \|}$ and $t = \frac{a+b}{2}$.
\STATE Compute $\xi' \in \partial f_i(x + t \tilde{v})$.
\STATE If $\langle \tilde{v}, \xi' \rangle > - c \| \tilde{v} \|^2$ then stop.
\STATE If $h_i(b) > h_i(t)$ then set $a = t$. Otherwise set $b = t$.
\STATE Set $t = \frac{a+b}{2}$ and go to step 2.
\end{algorithmic}
\end{algorithm}
Although in the general setting, we can not guarantee that Algorithm \ref{algo:new_subgradient} yields a subgradient satisfying \eqref{eq:new_subgrad_condition}, we can at least show that after finitely many iterations, a factor $t$ is found such that $\partial f_i(x + t \tilde{v})$ contains a subgradient that satisfies \eqref{eq:new_subgrad_condition}.
\begin{lemma} \label{lem:conv_new_subgradient}
Let $(t_k)_k$ be the sequence generated in Algorithm \ref{algo:new_subgradient}. If $(t_k)_k$ is finite, then some $\xi'$ was found such that \eqref{eq:new_subgrad_condition} is satisfied. If $(t_k)_k$ is infinite, then it converges to some $\bar{t} \in [0,\frac{\varepsilon}{\| \tilde{v} \|}]$ such that there is some $\xi' \in \partial f_i(x + \bar{t} \tilde{v})$ which satisfies \eqref{eq:new_subgrad_condition}. Additionally, there is some $N \in \mathbb{N}$ such that for all $k > N$ there is some $\xi' \in \partial f_i(x + t_k \tilde{v})$ satisfying \eqref{eq:new_subgrad_condition}.
\end{lemma}
\begin{proof}
Let $(t_k)_k$ be finite with last element $\bar{t} \in (0,\frac{\varepsilon}{\| \tilde{v} \|})$. Then Algorithm \ref{algo:new_subgradient} must have stopped in step 3, i.e., some $\xi' \in \partial f_i(x + \bar{t} \tilde{v})$ satisfying \eqref{eq:new_subgrad_condition} was found. \\
Now let $(t_k)_k$ be infinite. By construction, $(t_k)_k$ is a Cauchy sequence in the compact set $[0,\frac{\varepsilon}{\| \tilde{v} \|}]$, so it has to converge to some $\bar{t} \in [0,\frac{\varepsilon}{\| \tilde{v} \|}]$. Additionally, since \eqref{eq:accept_direction} is violated for the index $i$ by assumption, we have
\begin{align*}
h_i(0) = 0 \quad \text{and} \quad h_i \left( \frac{\varepsilon}{\| \tilde{v} \|} \right) > 0.
\end{align*}
Let $(a_k)_k$ and $(b_k)_k$ be the sequences corresponding to $a$ and $b$ in Algorithm \ref{algo:new_subgradient} (at the start of each iteration). Then $h_i(a_k) < h_i(b_k)$ for all $k \in \mathbb{N}$. Thus, by the mean value theorem, there has to be some $r_k \in (a_k,b_k)$ such that
\begin{align*}
0 < h_i(b_k) - h_i(a_k) \in \langle \partial h_i(r_k), b_k - a_k \rangle = \partial h_i(r_k) (b_k - a_k).
\end{align*}
In particular, $\lim_{k \rightarrow \infty} r_k = \bar{t}$ and since $a_k < b_k$, $\partial h_i(r_k) \cap \mathbb{R}^{> 0} \neq \emptyset$ for all $k \in \mathbb{N}$. By upper semicontinuity of $\partial h$ there must be some $\theta \in \partial h_i(\bar{t})$ with $\theta > 0$. By the chain rule, we have
\begin{align} \label{eq:h_chain_rule}
0 < \theta \in \partial h_i(\bar{t}) \subseteq \langle \tilde{v}, \partial f_i(x + \bar{t} \tilde{v}) \rangle + c \| \tilde{v} \|^2.
\end{align}
Thus, there must be some $\xi' \in \partial f_i(x + \bar{t} \tilde{v})$ with
\begin{align*}
& 0 < \langle \tilde{v}, \xi' \rangle + c \| \tilde{v} \|^2 \\
\Leftrightarrow \quad & \langle \tilde{v}, \xi' \rangle > - c \| \tilde{v} \|^2.
\end{align*}
By upper semicontinuity of $\partial h$ it also follows that there is some $N \in \mathbb{N}$ such that $\partial h_i(t_k) \cap \mathbb{R}^{> 0} \neq \emptyset$ for all $k > N$. Applying the same argument as above completes the proof.
\end{proof}
In the following remark, we will briefly discuss the implication of Lemma \ref{lem:conv_new_subgradient} for practical use of Algorithm \ref{algo:new_subgradient}.
\begin{remark}
Let $N \in \mathbb{N}$ be as in Lemma \ref{lem:conv_new_subgradient}.
\begin{itemize}
\item[a)] Note that if $k > N$ and $h$ is differentiable in $t_k$, then we have
\begin{align*}
0 < \nabla h_i(t_k) = \langle \tilde{v}, \nabla f_i(x + t_k \tilde{v}) \rangle + c \| \tilde{v} \|^2,
\end{align*}
i.e., the stopping criterion in step 3 is satisfied. Thus, if Algorithm \ref{algo:new_subgradient} generates an infinite sequence, $h$ must be nonsmooth in $t_k$ for all $k > N$. In particular, $f_i$ must be nonsmooth in $x + t_k \tilde{v}$ for all $k > N$.
\item[b)] If $f$ is regular (cf.~\cite{C1983}, Def.~2.3.4), then equality holds when applying the chain rule to $h$ (cf.~\cite{C1983}, Thm. 2.3.10), i.e.,
\begin{align*}
\partial h_i(t_k) = \langle \tilde{v}, \partial f_i(x + t_k \tilde{v}) \rangle + c \| \tilde{v} \|^2.
\end{align*}
Thus, if Algorithm \ref{algo:new_subgradient} generates an infinite sequence, then for all $k > N$ there must be both positive and negative elements in $\partial h_i(t_k)$. By convexity of $\partial h_i(t_k)$, this implies that $0 \in \partial h_i(t_k)$ for all $k > N$, i.e., $h$ must have infinitely many (nonsmooth) stationary points in $[0,\frac{\varepsilon}{\| \tilde{v} \|}]$.
\end{itemize}
\end{remark}
Motivated by the previous remark, we will from now on assume that Algorithm \ref{algo:new_subgradient} stops after finitely many iterations and thus yields a new subgradient satisfying \eqref{eq:new_subgrad_condition}. We can use this method of finding new subgradients to construct an algorithm that computes descent directions of nonsmooth MOPs, namely Algorithm \ref{algo:descent_direction}.
\begin{algorithm}
\caption{Compute descent direction}
\label{algo:descent_direction}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE Current point $x \in \mathbb{R}^n$, tolerances $\varepsilon, \delta > 0$, Armijo parameter $c \in (0,1)$.
\STATE Compute $\xi^i_1 \in \partial_\varepsilon f_i(x)$ for all $i \in \{1,...,k\}$. Set $W_1 = \{ \xi^1_1, ..., \xi^k_1 \}$ and $l = 1$.
\STATE Compute $v_l = \argmin_{v \in -\conv(W_l)} \| v \|^2$.
\STATE If $\| v_l \| \leq \delta$ then stop.
\STATE Find all objective functions for which there is insufficient descent:
\begin{align*}
I_l = \left\{ j \in \{1,...,k\} : f_j \left( x + \frac{\varepsilon}{\| v_l \|} v_l \right) > f_j(x) - c \varepsilon \| v_l \| \right\}.
\end{align*}
If $I_l = \emptyset$ then stop.
\STATE For each $j \in I_l$, compute $t \in (0,\frac{\varepsilon}{\| v_l \|}]$ and $\xi^j_l \in \partial f_j(x + t v_l)$ such that
\begin{align*}
\langle v_l, \xi^j_l \rangle > -c \| v_l \|^2
\end{align*}
via Algorithm \ref{algo:new_subgradient}.
\STATE Set $W_{l+1} = W_l \cup \{ \xi^j_l : j \in I_l \}$, $l = l+1$ and go to step 2.
\end{algorithmic}
\end{algorithm}
The following theorem shows that Algorithm \ref{algo:descent_direction} stops after a finite number of iterations and produces an acceptable descent direction (cf.~\eqref{eq:accept_direction}).
\begin{theorem} \label{thm:descent_dir_conv}
Algorithm \ref{algo:descent_direction} terminates. In particular, if $\tilde{v}$ is the last element of $(v_l)_l$, then either $\| \tilde{v} \| \leq \delta$ or $\tilde{v}$ is an acceptable descent direction, i.e.,
\begin{align*}
f_i \left( x + \frac{\varepsilon}{\| \tilde{v} \|} \tilde{v} \right) \leq f_i(x) - c \varepsilon \| \tilde{v} \| \quad \forall i \in \{1,...,k\}.
\end{align*}
\end{theorem}
\begin{proof}
Assume that Algorithm \ref{algo:descent_direction} does not terminate, i.e., $(v_l)_{l \in \mathbb{N}}$ is an infinite sequence. Let $l > 1$ and $j \in I_{l-1}$. Since $\xi^j_{l-1} \in W_l$ and $-v_{l-1} \in W_{l-1} \subseteq W_l$ we have
\begin{align} \label{eq:proof_algo_est}
\| v_l \|^2 &\leq \| -v_{l-1} + s (\xi^j_{l-1} + v_{l-1}) \|^2 \notag \\
&= \| v_{l-1} \|^2 - 2 s \langle v_{l-1}, \xi^j_{l-1} + v_{l-1} \rangle + s^2 \| \xi^j_{l-1} + v_{l-1} \|^2 \notag \\
&= \| v_{l-1} \|^2 - 2 s \langle v_{l-1}, \xi^j_{l-1} \rangle - 2 s \| v_{l-1} \|^2 + s^2 \| \xi^j_{l-1} + v_{l-1} \|^2
\end{align}
for all $s \in [0,1]$. Since $j \in I_{l-1}$ we must have
\begin{align} \label{eq:est_1}
\langle v_{l-1}, \xi^j_{l-1} \rangle > -c \| v_{l-1} \|^2
\end{align}
by step 5. Let $L$ be a common Lipschitz constant of all $f_i$, $i \in \{1,...,k\}$, on the closed $\varepsilon$-ball $B_\varepsilon(x)$ around $x$. Then by \cite{C1983}, Prop.~2.1.2, and the definition of the $\varepsilon$-subdifferential, we must have $\| \xi \| \leq L$ for all $\xi \in F_\varepsilon(x)$. So in particular,
\begin{align} \label{eq:est_2}
\| \xi^j_{l-1} + v_{l-1} \| \leq 2 L.
\end{align}
Combining \eqref{eq:proof_algo_est} with \eqref{eq:est_1} and \eqref{eq:est_2} yields
\begin{align*}
\| v_l \|^2 &< \| v_{l-1} \|^2 - 2 s c \| v_{l-1} \|^2 - 2 s \| v_{l-1} \|^2 + 4 s^2 L^2 \\
&= \| v_{l-1} \|^2 - 2 s (c+1) \| v_{l-1} \|^2 + 4 s^2 L^2.
\end{align*}
Let $s := \frac{c+1}{4 L^2} \| v_{l-1} \|^2$. Since $c + 1 \in (1,2)$ and $\| v_{l-1} \| \leq L$ we have $s \in (0,1)$. We obtain
\begin{align*}
\| v_l \|^2 &< \| v_{l-1} \|^2 - 2 \frac{(c+1)^2}{4 L^2} \| v_{l-1} \|^4 + \frac{(c+1)^2}{4 L^2} \| v_{l-1} \|^4 \\
&= \left( 1 - \frac{(c+1)^2}{4 L^2} \| v_{l-1} \|^2 \right) \| v_{l-1} \|^2.
\end{align*}
Since Algorithm \ref{algo:descent_direction} did not terminate, it holds $\| v_{l-1} \| > \delta$. It follows that
\begin{align*}
\| v_l \|^2 < \left( 1 - \left( \frac{c+1}{2 L} \delta \right)^2 \right) \| v_{l-1} \|^2.
\end{align*}
Let $r = 1 - \left( \frac{c+1}{2 L} \delta \right)^2$. Note that we have $\delta < \| v_l \| \leq L$ for all $l \in \mathbb{N}$, so $r \in (0,1)$. Additionally, $r$ does not depend on $l$, so we have
\begin{align*}
\| v_l \|^2 < r \| v_{l-1} \|^2 < r^2 \| v_{l-1} \|^2 < ... < r^{l-1} \| v_1 \|^2 \leq r^{l-1} L^2.
\end{align*}
In particular, there is some $l$ such that $\| v_l \| \leq \delta$, which is a contradiction.
\end{proof}
\begin{remark}
The proof of Theorem \ref{thm:descent_dir_conv} shows that for convergence of Algorithm \ref{algo:descent_direction}, it would be sufficient to consider only a single $j \in I_j$ in step 5. Similarly, for the initial approximation $W_1$, a single element from $\partial_\varepsilon f_i(x)$ for any $i \in \{1,...,k\}$ would be enough. A modification of either step can potentially reduce the number of executions of step 5 (i.e., Algorithm \ref{algo:new_subgradient}) in Algorithm \ref{algo:descent_direction} in case the $\varepsilon$-subdifferentials of multiple objective functions are similar. Nonetheless, we will restrain the discussion in this article to Algorithm \ref{algo:descent_direction} as it is, since both modifications also introduce a bias towards certain objective functions, which we want to avoid.
\end{remark}
To highlight the strengths of Algorithm \ref{algo:descent_direction}, we will consider an example where standard gradient sampling approaches can fail to obtain a useful descent direction.
\begin{example} \label{exam:compl_subdiff}
For $a, b \in \mathbb{R} \setminus \{ 0 \}$ consider the locally Lipschitz function
\begin{align*}
f : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \quad x \mapsto
\begin{pmatrix}
(x_1 - 1)^2 + (x_2 - 1)^2 \\
| x_2 - a |x_1|| + b x_2
\end{pmatrix}.
\end{align*}
The set of nondifferentiable points is
\begin{align*}
\Omega_f = (\{ 0 \} \times \mathbb{R}) \cup \{ (\lambda, a | \lambda| )^\top : \lambda \in \mathbb{R} \},
\end{align*}
separating $\mathbb{R}^2$ into four smooth areas (cf.~Figure \ref{fig:example_compl_subdiff}(a)). For large $a > 0$, the two areas above the graph of $\lambda \mapsto a | \lambda |$ become small, making it difficult to compute the subdifferential close to $(0,0)^\top$.
Let $a = 10$, $b = 0.5$, $\varepsilon = 10^{-3}$ and $x = (10^{-4}, 10^{-4})^\top$. In this case, $(0,0)^\top$ is the minimal point of $f_2$ and
\begin{align*}
\partial_\varepsilon f_2(x) &= \conv\left\{
\begin{pmatrix}
-a \\
b - 1
\end{pmatrix},
\begin{pmatrix}
a \\
b + 1
\end{pmatrix},
\begin{pmatrix}
a \\
b - 1
\end{pmatrix},
\begin{pmatrix}
-a \\
b + 1
\end{pmatrix}
\right\} \\
&= \conv\left\{
\begin{pmatrix}
-10 \\
-0.5
\end{pmatrix},
\begin{pmatrix}
10 \\
1.5
\end{pmatrix},
\begin{pmatrix}
10 \\
-0.5
\end{pmatrix},
\begin{pmatrix}
-10 \\
1.5
\end{pmatrix}
\right\}.
\end{align*}
In particular, $0 \in \partial_\varepsilon f_2(x)$, so the descent direction with the exact $\varepsilon$-subdifferentials from \eqref{eq:eps_min_norm_problem} is zero. When applying Algorithm \ref{algo:descent_direction} in $x$, after two iterations we obtain
\begin{align*}
\tilde{v} = v_2 \approx (0.118, 1.185) \cdot 10^{-9},
\end{align*}
i.e., $\| \tilde{v} \| \approx 1.191 \cdot 10^{-11}$. Thus, $x$ is correctly identified as (potentially) Pareto optimal. The final approximation $W_2$ of $F_\varepsilon(x)$ is
\begin{align*}
W_2 = \left\{ \xi^1_1, \xi^2_1, \xi^2_2 \right\} = \left\{
\begin{pmatrix}
10 \\
-0.5
\end{pmatrix},
\begin{pmatrix}
-1.9998 \\
-1.9998
\end{pmatrix},
\begin{pmatrix}
-10 \\
1.5
\end{pmatrix}
\right\}.
\end{align*}
The first two elements of $W_2$ are the gradients of $f_1$ and $f_2$ in $x$ from the first iteration of Algorithm \ref{algo:descent_direction}, and the last element is the gradient of $f_2$ in $x' = x + tv_1 = (0.038, 0.596)^\top \cdot 10^{-3} \in B_\varepsilon(x)$ from the second iteration. The result is visualized in Figure \ref{fig:example_compl_subdiff}.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:example_compl_subdiff}
\end{figure}
\end{example}
Building on Algorithm \ref{algo:descent_direction}, it is now straightforward to construct the descent method for locally Lipschitz continuous MOPs given in Algorithm \ref{algo:nonsmooth_descent_method}. In step 4, the classical Armijo backtracking line search was used (cf.~\cite{FS2000}) for the sake of simplicity. Note that it is well defined due to step 4 in Algorithm \ref{algo:descent_direction}.
\begin{algorithm}
\caption{Nonsmooth descent method}
\label{algo:nonsmooth_descent_method}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE Initial point $x_1 \in \mathbb{R}^n$, tolerances $\varepsilon, \delta > 0$, Armijo parameters $c \in (0,1), t_0 > 0$.
\STATE Set $j = 1$.
\STATE Compute a descent direction $v_j$ via Algorithm \ref{algo:descent_direction}.
\STATE If $\| v_j \| \leq \delta$ then stop.
\STATE Compute
\begin{align*}
\bar{s} = \inf(\{ s \in \mathbb{N} \cup \{ 0 \} : f_i(x_j + 2^{-s} t_0 v_j) \leq f_i(x_j) - 2^{-s} t_0 c \| v_j \|^2 \ \forall i \in \{1,...,k\} \})
\end{align*}
and set $\bar{t} = \max( \{ 2^{-\bar{s}} t_0, \frac{\varepsilon}{\| v_j \|} \} )$.
\STATE Set $x_{j+1} = x_j + \bar{t} v_j$, $j = j+1$ and go to step 2.
\end{algorithmic}
\end{algorithm}
Since we introduced the two tolerances $\varepsilon$ (for the $\varepsilon$-subdifferential) and $\delta$ (as a threshold for when we consider $\varepsilon$-subgradients to be zero), we can not expect that Algorithm \ref{algo:nonsmooth_descent_method} computes points which satisfy the optimality condition \eqref{eq:KKT}. This is why we introduce the following definition, similar to the definition of $\varepsilon$-stationarity from \cite{BLO2005}.
\begin{definition}
Let $x \in \mathbb{R}^n$, $\varepsilon > 0$ and $\delta > 0$. Then $x$ is called $(\varepsilon, \delta)$\emph{-critical}, if
\begin{align*}
\min_{v \in -F_\varepsilon(\bar{x})} \| v \| \leq \delta.
\end{align*}
\end{definition}
It is easy to see that $(\varepsilon, \delta)$-criticality is a necessary optimality condition for Pareto optimality, but a weaker one than \eqref{eq:KKT}. The following theorem shows that convergence in the sense of $(\varepsilon, \delta)$-criticality is what we can expect from our descent method.
\begin{theorem}
Let $(x_j)_j$ be the sequence generated by Algorithm \ref{algo:nonsmooth_descent_method}. Then either $(f_i(x_j))_j$ is unbounded below for each $i \in \{1,...,k\}$, or $(x_j)_j$ is finite with the last element being $(\varepsilon,\delta)$-critical.
\end{theorem}
\begin{proof}
Assume that $(x_j)_j$ is infinite. Then $\| v_j \| > \delta$ for all $j \in \mathbb{N}$. By step 4 and Lemma \ref{lem:step_size_bound} we have
\begin{align*}
f_i(x_j + \bar{t} v_j) -f_i(x_j) \leq - \bar{t} \| v_j \|^2 \leq - \varepsilon \| v_j \| < -\varepsilon \delta < 0
\end{align*}
for all $i \in \{1,...,k\}$. This implies that $(f_i(x_j))_j$ is unbounded below for each $i \in \{1,...,k\}$. \\
Now assume that $(x_j)_j$ is finite, with $\bar{x}$ and $\bar{v}$ being the last elements of $(x_j)_j$ and $(v_j)_j$, respectively. Since the algorithm stopped, we must have $\| \bar{v} \| \leq \delta$. From the application of Algorithm \ref{algo:descent_direction} in step 2, we know that there must be some $\overline{W} \subseteq F_\varepsilon(\bar{x})$ such that $\bar{v} = \argmin_{v \in -\overline{W}} \| v \|^2$. This implies
\begin{align*}
\min_{v \in -F_\varepsilon(\bar{x})} \| v \| \leq \min_{v \in -\conv(\overline{W})} \| v \| = \| \bar{v} \| \leq \delta.
\end{align*}
\end{proof}
\section{Numerical examples} \label{sec:numerical_examples}
In this section we will apply our nonsmooth descent method (Algorithm \ref{algo:nonsmooth_descent_method}) to several examples. We will begin by discussing its typical behavior before comparing its performance to the \emph{multiobjective proximal bundle method} \cite{MKW2014}. Finally, we will combine our method with the \emph{subdivision algorithm} \cite{DSH2005} in order to approximate the entire Pareto set of nonsmooth MOPs.
\subsection{Typical behavior}
In smooth areas, the behavior of Algorithm \ref{algo:nonsmooth_descent_method} is almost identical to the behavior of the multiobjective steepest descent method \cite{FS2000}. The only difference stems from the fact that, unlike the Clarke subdifferential, the $\varepsilon$-subdifferential does not reduce to the gradient when $f$ is continuously differentiable. As a result, Algorithm \ref{algo:nonsmooth_descent_method} may behave differently in points $x \in \mathbb{R}^n$ where
\begin{align*}
\max \{ \| \nabla f_i(x) - \nabla f_i(y) \| : y \in B_\varepsilon(x), \ i \in \{1,...,k\} \}
\end{align*}
is large. (If $f$ is twice differentiable, this can obviously be characterized in terms of second order derivatives.) Nevertheless, if $\varepsilon$ is chosen small enough, this difference can be neglected. Thus, in the following, we will focus on the behavior with respect to the nonsmoothness of $f$.
To show the typical behavior of Algorithm \ref{algo:nonsmooth_descent_method}, we consider the objective function
\begin{align} \label{eq:MOP_14_15}
f : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \quad x \mapsto
\begin{pmatrix}
\max \{ x_1^2 + (x_2 - 1)^2 + x_2 - 1, -x_1^2 - (x_2 - 1)^2 + x_2 + 1 \} \\
-x_1 + 2 (x_1^2 + x_2^2 - 1) + 1.75 | x_1^2 + x_2^2 - 1 |
\end{pmatrix}
\end{align}
from \cite{MKW2014} (combining \emph{Crescent} from \cite{K1985a} and \emph{Mifflin 2} from \cite{MN1992}). The set of nondifferentiable points is $\Omega_f = S^1 \cup (S^1 + (0,1)^\top)$. We consider the starting points
\begin{align*}
x^1 = (0,-0.3)^\top, \quad x^2 = (0.6,1.0)^\top, \quad x^3 = (-1,-0.2)^\top,
\end{align*}
the tolerances $\varepsilon = 10^{-3}$, $\delta = 10^{-3}$ and the Armijo parameters $c = 0.25$, $t_0 = 1$. The results are shown in Figure \ref{fig:example_typical_behavior}.
\begin{figure}
\caption{Result of Algorithm \ref{algo:nonsmooth_descent_method}
\label{fig:example_typical_behavior}
\end{figure}
We will briefly go over the result for each starting point:
\begin{itemize}
\item For $x^1$, the sequence moves through the smooth area like the steepest descent method until a point is found with a distance less or equal $\varepsilon$ to the set of nondifferentiable points $\Omega_f$. In that point, more than one $\varepsilon$-subgradient is required to obtain a sufficient approximation of the $\varepsilon$-subdifferentials. Since this part of $\Omega_f$ is Pareto optimal, no acceptable descent direction (cf.~\eqref{eq:accept_direction}) is found and the algorithm stops (in a $(\varepsilon,\delta)$-critical point).
\item For $x^2$, the sequence starts zig-zagging around the non-optimal part of $\Omega_f$, since the points are too far away from $\Omega_f$ for the algorithm to notice the nondifferentiability. When a point is found with distance less or equal $\varepsilon$ to $\Omega_f$, a better descent direction is found, breaking the zig-zagging motion.
\item For $x^3$, the sequence has a similar zig-zagging motion to the previous case. The difference is that this time, the sequence moves along $\Omega_f$ until a Pareto optimal point in $\Omega_f$ is found.
\end{itemize}
As described above, the zig-zagging behavior when starting in $x^2$ is caused by the fact that $\varepsilon$ was too small for the method to notice the nondifferentiability. To circumvent problems like this and quickly move through problematic areas, it is possible to apply Algorithm \ref{algo:nonsmooth_descent_method} consecutively with decreasing values of $\varepsilon$. The result is Algorithm \ref{algo:eps_decr_descent_method}. (A similar idea was implemented in \cite{MY2012}.)
\begin{algorithm}
\caption{$\varepsilon$-decreasing nonsmooth descent method}
\label{algo:eps_decr_descent_method}
\begin{algorithmic}[1]
\mathbb{R}EQUIRE Initial point $x_1 \in \mathbb{R}^n$, tolerances $\delta, \varepsilon_1, ..., \varepsilon_K > 0$ , Armijo parameters $c \in (0,1), t_0 > 0$.
\STATE Set $y_1 = x_1$.
\FOR{$i = 1, ..., K$}
\STATE Apply Algorithm \ref{algo:nonsmooth_descent_method} with initial point $y_i$ and tolerance $\varepsilon = \varepsilon_i$. Let $y_{i+1}$ be the final element in the generated sequence.
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Comparison to the multiobjective proximal bundle method}
We will now compare Algorithms \ref{algo:nonsmooth_descent_method} and \ref{algo:eps_decr_descent_method} to the \emph{multiobjective proximal bundle method} (MPB) by M\"akel\"a, Karmitsa and Wilppu from \cite{MKW2014} (see also \cite{M2003}). As test problems, we consider the 18 MOPs in Table \ref{table:test_problems}, which are created on the basis of the scalar problems from \cite{MKW2014}. Problems 1 to 15 are convex (and were also considered in \cite{MKM2018}) and problems 16 to 18 are nonconvex. Due to their simplicity, we are able to differentiate all test problems by hand to obtain explicit formulas for the subgradients. For each test problem, we choose 100 starting points on a 10 $\times$ 10 grid in the corresponding area given in Table \ref{table:test_problems}.
\begin{table}
\centering
\caption{Test problems (using objectives from \cite{MKW2014})}
\label{table:test_problems}
\begin{small}
\begin{tabular}{| c | l | c || c | l | c |}
\hline
Nr. & $f_i$ & Area & Nr. & $f_i$ & Area \\
\hline
1. & CB3, DEM & $[-3,3]^2$ & 10. & QL, LQ & $[-3,3]^2$ \\
2. & CB3, QL & $[-3,3]^2$ & 11. & QL, Mifflin 1 & $[-3,3]^2$ \\
3. & CB3, LQ & $[0.5,1.5]^2$ & 12. & QL, Wolfe & $[-3,3]^2$ \\
4. & CB3, Mifflin 1 & $[-3,3]^2$ & 13. & LQ, Mifflin 1 & $[0.5,1.5] \times [-0.5,1]$ \\
5. & CB3, Wolfe & $[-3,3]^2$ & 14. & LQ, Wolfe & $[-3,3]^2$ \\
6. & DEM, QL & $[-3,3]^2$ & 15. & Mifflin 1, Wolfe & $[-3,3]^2$ \\ \cline{4-6}
7. & DEM, LQ & $[-3,3]^2$ & 16. & Crescent, Mifflin 2 & $[-0.5,1.5]^2$ \\
8. & DEM, Mifflin 1 & $[-3,3]^2$ & 17. & Mifflin 2, WF & $[-3,3]^2$ \\
9. & DEM, Wolfe & $[-3,3]^2$ & 18. & Mifflin 2, SPIRAL & $[-3,3]^2$ \\
\hline
\end{tabular}
\end{small}
\end{table}
For the MPB, we use the Fortran implementation from \cite{M2003} with the default parameters. For Algorithm \ref{algo:nonsmooth_descent_method}, we use $\varepsilon = 10^{-3}$, $\delta = 10^{-3}$, $c = 0.25$ and $t_0 = \max\{ \| v_j \|^{-1}, 1 \}$ (i.e., the initial step size $t_0$ is chosen depending on the norm of the descent direction $v_j$ in the current iteration). For Algorithm \ref{algo:eps_decr_descent_method}, we additionally use $\varepsilon_1 = 10^{-1}$, $\varepsilon_2 = 10^{-2}$, $\varepsilon_3 = 10^{-3}$. By this choice of parameters, all three methods produce results of similar approximation quality.
To compare the performance of the three methods, we count the number of evaluations of objectives $f_i$, their subgradients $\xi \in \partial f_i$ and the number of iterations (i.e., descent steps) needed. (This means that one call of $f$ will account for $k$ evaluations of objectives.) Since the MPB always evaluates all objectives and subgradients in a point, the value for the objectives and the subgradients are the same here. The results are shown in Table \ref{table:performance} and are discussed in the following.
\begin{table}
\centering
\caption{Performance of MPB, Algorithm \ref{algo:nonsmooth_descent_method} and Algorithm \ref{algo:eps_decr_descent_method} for the test problems in Table \ref{table:test_problems} for $100$ starting points}
\label{table:performance}
\begin{small}
\begin{tabular}{|c !{\vrule width 2pt} c | c | c !{\vrule width 2pt} c | c | c !{\vrule width 2pt} c | c | c |}
\hline
& \multicolumn{3}{c !{\vrule width 2pt}}{\#$f_i$} & \multicolumn{3}{c !{\vrule width 2pt}}{\#$\partial f_i$} & \multicolumn{3}{c |}{\# Iter} \\
\hline
Nr. & MPB & Alg. \ref{algo:nonsmooth_descent_method} & Alg. \ref{algo:eps_decr_descent_method} & MPB & Alg. \ref{algo:nonsmooth_descent_method} & Alg. \ref{algo:eps_decr_descent_method} & MPB & Alg. \ref{algo:nonsmooth_descent_method} & Alg. \ref{algo:eps_decr_descent_method} \\
\hline
1. & \textbf{1780} & 6924 & 7801 & 1780 & \textbf{1102} & 1751 & 761 & \textbf{492} & 695 \\
2. & \textbf{2522} & 14688 & 12263 & 2522 & \textbf{1906} & 2351 & 1151 & \textbf{842} & 914 \\
3. & \textbf{880} & 5625 & 6447 & \textbf{880} & 921 & 1534 & \textbf{340} & 448 & 662 \\
4. & \textbf{4416} & 103826 & 17664 & 4416 & 11774 & \textbf{3415} & 1832 & 4644 & \textbf{1242} \\
5. & \textbf{2956} & 30457 & 16877 & \textbf{2956} & 3479 & 3037 & 1377 & 1616 & \textbf{1161} \\
6. & \textbf{1640} & 8357 & 8684 & 1640 & \textbf{1209} & 1802 & 706 & \textbf{552} & 736 \\
7. & \textbf{1702} & 8736 & 8483 & 1702 & \textbf{1307} & 1832 & 723 & \textbf{595} & 739 \\
8. & \textbf{4226} & 8283 & 8620 & 4226 & \textbf{1318} & 1914 & 1204 & \textbf{582} & 759 \\
9. & \textbf{1828} & 8201 & 8794 & 1828 & \textbf{1194} & 1805 & 793 & \textbf{536} & 732 \\
10. & \textbf{1782} & 6799 & 7201 & 1782 & \textbf{1101} & 1722 & 684 & \textbf{543} & 733 \\
11. & \textbf{4426} & 52096 & 17594 & 4426 & 6311 & \textbf{3189} & 1964 & 2442 & \textbf{1206} \\
12. & \textbf{2482} & 15146 & 12446 & 2482 & \textbf{1992} & 2401 & 1140 & \textbf{967} & 1010 \\
13. & \textbf{2662} & 36570 & 9513 & 2662 & 4958 & \textbf{2247} & 1221 & 1692 & \textbf{787} \\
14. & \textbf{4264} & 95303 & 12227 & 4264 & 9524 & \textbf{2571} & 1774 & 4379 & \textbf{921} \\
15. & \textbf{3594} & 85936 & 15669 & 3594 & 9329 & \textbf{3124} & 1444 & 3963 & \textbf{1125} \\
\hline
16. & \textbf{2206} & 20372 & 11094 & \textbf{2206} & 2596 & 2400 & \textbf{884} & 1194 & 947 \\
17. & \textbf{2388} & 7920 & 5852 & 2388 & \textbf{1272} & 1556 & 868 & \textbf{626} & 706 \\
18. & \textbf{11430} & 166707 & 31528 & 11430 & 16676 & \textbf{6902} & 2789 & 8291 & \textbf{2412} \\
\hline
Avg. & \textbf{3176.9} & 37885.9 & 12153.2 & 3176.9 & 4331.6 & \textbf{2530.7} & 1203.1 & 1911.3 & \textbf{971.5} \\
& 100\% & 1192.5\% & 382.5\% & 100\% & 136.3\% & 79.7\% & 100\% & 158.9\% & 80.8\% \\
\hline
\end{tabular}
\end{small}
\end{table}
\begin{itemize}
\item \textbf{Function evaluations:} When considering the number of function evaluations, it is clear that the MPB requires far less evaluations than both of our algorithms. In our methods, these evaluations are used to check if a descent direction is acceptable (cf.~\eqref{eq:accept_direction}) and for the computation of the Armijo step length. One reason for the larger total amount is the fact that unlike the MPB, our methods are autonomous in the sense that they do not reuse information from previous iterations, so some information is potentially gathered multiple times. Additionally, the step length we use is fairly simple, so it might be possible to lower the number of evaluations by using a more sophisticated step length. When comparing our methods to each other, we see that Algorithm \ref{algo:eps_decr_descent_method} is a lot more efficient than Algorithm \ref{algo:nonsmooth_descent_method} when the number of evaluations is high and is slightly less efficient when the number of evaluations is low. The reason for this is that for simple problems (i.e., where the number of evaluations is low), some of the iterations of Algorithm \ref{algo:eps_decr_descent_method} will be redundant, because the $(\varepsilon_{i-1},\delta)$-critical point of the previous iteration is already $(\varepsilon_i,\delta)$-critical.
\item \textbf{Subgradient evaluations:} For the subgradient evaluations, we see that MPB is slightly superior to our methods on problems 3, 5 and 16, but inferior on the rest. Regarding the comparison of Algorithms \ref{algo:nonsmooth_descent_method} and \ref{algo:eps_decr_descent_method}, we observe the same pattern as for the function evaluations: Algorithm \ref{algo:nonsmooth_descent_method} is superior for simple and Algorithm \ref{algo:eps_decr_descent_method} for complex problems.
\item \textbf{Iterations:} For the number of iterations, besides problem 5, we see the exact same pattern as for the number of subgradient evaluations. Note that the MPB can perform \emph{null steps}, which are iterations where only the bundle is enriched, while the current point in the descent sequence stays the same.
\end{itemize}
For our set of test problems, this leads us to the overall conclusion that in terms of function evaluations, the MPB seems to be superior to our methods, while in terms of subgradient evaluations, our methods seem to be (almost always) more efficient. Furthermore, we remark that the implementation of the MPB is somewhat challenging, whereas our method can be implemented relatively quickly.
\subsection{Combination with the subdivision algorithm}
Note that so far, we have a method where we can put in some initial point from $\mathbb{R}^n$ and obtain a single $(\varepsilon,\delta)$-critical point (close to an actual Pareto optimal point) as a result. But ultimately, we are not interested in one, but all Pareto optimal points. The intuitive and straightforward approach to extend our method would be to just take a large set of well-spread initial points and apply our method to each of them. The problem with this is that we have no guarantee that this results in a good approximation of the Pareto set. To solve this issue, we combine our method with the \emph{subdivision algorithm} which was developed for smooth problems in \cite{DSH2005}. Since we only have to do minor adjustments for the nonsmooth case, we will only sketch the method here and refer to \cite{DSH2005} for the details.
The idea is to interpret the nonsmooth descent method as a discrete dynamical system
\begin{align} \label{eq:dyn_system}
x_{j+1} = g(x_j), \quad j = 0, 1, 2, ..., \quad x_0 \in \mathbb{R}^n,
\end{align}
where $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is the map that applies one iteration of Algorithm \ref{algo:nonsmooth_descent_method} to a point in $\mathbb{R}^n$. (For the sake of brevity, we have omitted the rest of the input of the algorithm here.) Since no information is carried over between iterations of the algorithm, the trajectory (i.e., the sequence) generated by the system \eqref{eq:dyn_system} is the same as the one generated by Algorithm \ref{algo:nonsmooth_descent_method}. In particular, this means that the Pareto set of the MOP is contained in the set of fixed points of the system \eqref{eq:dyn_system}. Thus, the subdivision algorithm (which was originally designed to compute attractors of dynamical systems) can be used to compute (a superset of) the Pareto set.
The subdivision algorithm starts with a large hypercube (or \emph{box}) in $\mathbb{R}^n$ that contains the Pareto set and mainly consists of two steps:
\begin{enumerate}
\item \textbf{Subdivision:} Divide each box in the current set of boxes into smaller boxes.
\item \textbf{Selection:} Compute the image of the union of the current set of boxes under $g$ and remove all boxes that have an empty intersection with this image. Go to step 1.
\end{enumerate}
In practice, we realize step 1 by evenly dividing each box into $2^n$ smaller boxes and step 2 by using the image of a set of sample points. The algorithm is visualized in Figure~\ref{fig:GAIO}.
\begin{figure}
\caption{Subdivision algorithm. \textbf{(a)}
\label{fig:GAIO}
\end{figure}
Unfortunately, the convergence results of the subdivison algorithm only apply if $g$ is a diffeomorphism. If the objective function $f$ is smooth, then the descent direction is at least continuous (cf.~\cite{FS2000}) and the resulting dynamical system $g$, while not being a diffeomorphism, still behaves well enough for the subdivision algorithm to work. If $f$ is nonsmooth, then our descent direction is inherently discontinuous close to the nonsmooth points. Thus, the subdivision algorithm applied to \eqref{eq:dyn_system} will (usually) fail to work. In practice, we were able to solve this issue by applying multiple iterations of Algorithm \ref{algo:nonsmooth_descent_method} in $g$ at once instead of just one. Roughly speaking, this smoothes $g$ by pushing the influence of the discontinuity further away from the Pareto set and was sufficient for convergence (in our tests).
Figures \ref{fig:subdiv_9_10} to \ref{fig:subdiv_15_16} show the result of the subdivision algorithm for some of the problems from Table \ref{table:test_problems}. For each problem, we used $15$ iterations of Algorithm \ref{algo:nonsmooth_descent_method} in $g$, $[-3.1,3]^2$ as the starting box and applied $9$ iterations of the subdivision algorithm. For the approximation of the Pareto front (i.e., the image of the Pareto set), we evaluated $f$ in all points of the image of $g$ of the last selection step in the subdivision algorithm. In all of these examples, the algorithm produced a tight approximation of the Pareto set.
\begin{figure}
\caption{\textbf{(a)}
\label{fig:subdiv_9_10}
\end{figure}
\begin{figure}
\caption{\textbf{(a)}
\label{fig:subdiv_10_13}
\end{figure}
\begin{figure}
\caption{\textbf{(a)}
\label{fig:subdiv_15_16}
\end{figure}
\section{Conclusion and outlook} \label{sec:conclusion}
In this article, we have developed a new descent method for locally Lipschitz continuous multiobjective optimization problems, which is based on the efficient approximation of the Clarke subdifferentials of the objective functions from \cite{MY2012}. In \cite{AGG2015}, it was shown that the element with the smallest norm in the negative convex hull of the union of the subdifferentials is a descent direction for all objectives at the same time. In practice, the entire subdifferentials which are required to compute this direction are rarely known and only single subgradients can be computed. To solve this issue, we presented a method to obtain an approximation of the subdifferentials which is sufficient to obtain a descent direction. The idea is to start with a rough approximation of the subdifferentials by only a few subgradients and then systematically enrich the approximation with new subgradients until a direction of sufficient descent is found. By combining the descent direction with an Armijo step length, we obtained a descent method for nonsmooth MOPs and showed convergence to points which satisfy a necessary condition for Pareto optimality. We then compared the performance to the multiobjective proximal bundle method from \cite{MKW2014}. For the 18 test problems we considered, the MPB was superior in terms of objective function evaluations, but our method required less subgradient evaluations and iterations. Finally, we showed that our descent method can be combined with the subdivision algorithm from \cite{DSH2005} to compute approximations of entire Pareto sets.
For future work, we believe that it is straightforward to extend our method to constrained MOPs by adding constraints to the problem \eqref{eq:approx_desc_dir} that ensure that the descent direction maintains the feasibility of the descent sequence (similar to \cite{GPD2017} for smooth problems). Additionally, in \cite{CQ2013}, the classical gradient sampling method for scalar nonsmooth optimization was generalized by allowing variable norms in the direction finding problem, increasing its efficiency. We expect that a similar generalization can be performed for problem \eqref{eq:approx_desc_dir}, which potentially yields a similar increase in efficiency. Additional potential for increased performance lies in more advanced step length schemes as well as descent directions with memory (for instance, conjugate-gradient-like). Furthermore, it might be possible to extend our method to infinite-dimensional nonsmooth MOPs \cite{M2008,CM2020}. Finally, in the context of nonsmooth many-objective optimization, we believe that considering subsets of objectives is a very promising and efficient approach (cf.~\cite{GPD2019} for smooth problems). However, theoretical advances are required for locally Lipschitz continuous problems.
\end{document} |
\begin{document}
\markboth{R. Rajkumar and P. Devi}{Intersection graph of subgroups}
\title{\bf Toroidality and projective-planarity of intersection graphs of subgroups of finite groups}
\author{R. Rajkumar\footnote{e-mail: {\tt [email protected]}},\ \ \
P. Devi\footnote{e-mail: {\tt [email protected]}}\\
{\footnotesize Department of Mathematics, The Gandhigram Rural Institute -- Deemed University,}\\ \footnotesize{Gandhigram -- 624 302, Tamil Nadu, India.}\\[3mm]
}
\date{}
\maketitle
\begin{abstract}
Let $G$ be a group. \textit{The intersection graph of subgroups of $G$}, denoted by $\mathscr{I}(G)$, is a graph with all the
proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if
the corresponding subgroups having a non-trivial intersection in $G$.
In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar.
In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also we classify the finite groups whose intersection graph of subgroups does not contain one of $K_5$, $K_4$, $C_5$, $C_4$, $P_4$, $P_3$, $P_2$, $K_{1,3}$, $K_{2,3}$ or $K_{1,4}$ as a subgraph.
We estimate the girth of the intersection graph of subgroups of finite groups. Moreover, we characterize some finite groups by using their intersection graphs. Finally, we obtain the clique cover number of the intersection graph of subgroups of groups and show that intersection graph of subgroups of groups are weakly $\alpha$-perfect.
\paragraph{Keywords:}Intersection graph, finite groups, genus, toroidal graph, nonorientable genus, projective-planar graph.
\paragraph{2010 Mathematics Subject Classification:} 05C25, 05C10, 05E15, 20E99.
\end{abstract}
\section{Introduction} \label{sec:int}
Let $\mathcal{F}=\{S_i~|~i\in I\}$ be an arbitrary family of sets. The intersection graph of $\mathcal{F}$ is a graph having the elements of $\mathcal{F}$ an
its vertices and two vertices $S_i$ and $S_j$ are adjacent if and only if $i\neq j$ and $S_i\cap S_j\neq \{\emptyset\}$. For the properties of these graphs and some
special class of intersection graphs, we refer the reader to \cite{mckee}. In the past fifty years,
it has been an interesting topic for mathematicians, when the members of $\mathcal{F}$ have some specific algebraic
structures. Especially, they investigate on the interplay between the algebraic properties of algebraic structures, and the graph theoretic properties of their
intersection graphs. In this direction, in 1964 Bosak~\cite{bosak} initiated the study of the intersection graphs of semigroups. Later,
Cs$\acute{a}$k$\acute{a}$ny and Poll$\acute{a}$k ~\cite{csak} defined the intersection graph of subgroups of a finite group.
Let $G$ be a group. \textit{The intersection graph of subgroups of $G$}, denoted by $\mathscr{I}(G)$, is a graph with all the
proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if
the corresponding subgroups have a non-trivial intersection in $G$.
In~\cite{zelinka} Zelinka
made some investigations on the intersection graphs of subgroups of finite abelian groups. Motivated by these, many authors have defined, and studied the intersecting graphs on several algebraic structures, viz., rings, vector spaces, modules, and contributed interesting results. See, for instance \cite{ akbari_2, akbari_1, akbari, Chakara, laison, Shen, yaraneri} and the references therein.
Embeddability of graphs, associated with algebraic structures, on topological surfaces is considered in several recent papers \cite{hung, zoran, zoran1, raj2015}. Planarity of intersection graphs of subgroups finite groups were studied by Sel\c{c}¸uk Kayacan \emph{et al.} in \cite{sel}, and by H. Ahmedi \emph{et al.} in \cite{hadi}. Planarity of intersection graphs of ideals of rings, and submodules of modules were studied in \cite{jafari1, yaraneri}.
A natural question arise in this direction is the following: Which groups have their intersection graph of subgroups is of genus one, that is toroidal, or of nonorientable genus one, that is projective-planar ?. In this paper, we answer this question in the case of finite groups by classifying the finite groups whose intersection graph of subgroups is toroidal or projective-planar (see Theorem~\ref{intersecting graph t12} in Section~\ref{sec: 5} below). As a consequence of this research, we also classify finite groups whose intersection graph of subgroups is in some class of graphs (see Theorem ~\ref{intersecting graph t233} and Corollary~\ref{intersecting graph c1} in Section~\ref{sec: 5} below) and characterize some finite groups by using their intersection graphs (see Corollary~\ref{intersecting graph c2} in Section~\ref{sec: 5} below), which are some of the main applications of these results for the group theory. Also we estimate the grith of the intersection graph of finite groups. Finally, we obtain the clique covering number of the intersection graph of subgroups of groups and show that intersection graph of subgroups of groups are weakly $\alpha$-perfect.
\section{Preliminaries and notations}\label{sec:pre}
In this section, we first recall some notation, and results in graph theory, which are used later in the subsequent sections. We use standard basic graph theory terminology and notation (e.g., see \cite{arthur}).
Let $G$ be a simple graph with a vertex set $V$ and an edge set $E$.
$G$ is said to be \textit{bipartite} if $V$ can be partitioned into two subsets $V_1$ and $V_2$ such that every edge of $G$ joins a vertex of $V_1$ to a vertex of $V_2$. Then $(V_1, V_2)$ is called a \textit{bipartition} of $G$.
Moreover, if every vertex of $V_1$ is adjacent to every vertex of $V_2$, then $G$ is called \textit{complete bipartite} and is denoted by $K_{m,n}$, where $|V_1|=m$, $|V_2|=n$. In particular, $K_{1,n}$ is a \emph{star graph}. $G$ is said to be \emph{complete} if each pair of distinct vertices in $G$ are adjacent. The complete graph on $n$ vertices is denoted by $K_n$. $G$ is said to be \textit{totally disconnected} if its edge set is empty. A \textit{path} connecting two vertices $u$ and $v$ in $G$ is a finite sequence $(u=) v_0, v_1, \ldots, v_n (=v)$ of distinct vertices (except,
possibly, $u$ and $v$) such that $u_i$ is adjacent to $u_{i+1}$ for all $i=0, 1, \ldots , n-1$.
A path is a \emph{cycle} if $u=v$. The length of a path or a cycle is the number of edges in it. A path or a cycle of length $n$ is denoted by $P_n$ or $C_n$, respectively. A graph is \emph{unicyclic} if it has exactly one cycle, and is \emph{acyclic} if it has no cycles. The \emph{girth} of a graph $G$, denoted by $girth(G)$, is the length of the shortest cycle in $G$, if it exist; otherwise $girth(G) = \infty$.
We define a graph $G$ to be $X$-\textit{free} if $G$ has no subgraph isomorphic to a given graph $X$. An \emph{independent set} of a graph $G$ is a subset of the vertices of $G$ such that no two vertices in the subset are adjacent. The \emph{independence
number} of $G$, denoted by $\alpha(G)$, is the cardinality of a maximum independent set of $G$. A \emph{clique} of a graph $G$ is a complete subgraph of $G$. The \emph{clique cover number} of $G$, denoted by $\theta(G)$, is the minimum number of cliques in $G$ which cover all the vertices of $G$ (not necessarily all the edges of $G$). $G$ said to be \emph{weakly $\alpha$-perfect} if $\alpha(G)=\theta(G)$. For two graphs $G$ and $H$, $G \cup H$ denotes disjoint union of $G$ and $H$, $G+H$ denotes a graph with the vertex set consist of vertices of $G$ and $H$ and edge set having all the lines
joining vertices of $G$ to vertices of $H$. $\overline{G}$ denotes the complement of a graph $G$, and, for an integer $n\ge 1$, $nG$ denotes the graph having $q$ disjoint copies of $G$.
Let $\mathbb{S}_n$ denote the surface obtained from the sphere by attaching $n$ handles. A graph is said to be \textit{embeddable} on a topological surface if it can be drawn on the surface such that no two edges cross.
The (orientable) \textit{genus} of a graph $G$, denoted by $\gamma(G)$, is the smallest non-negative integer $n$ such that $G$ can be embedded on $\mathbb{S}_n$.
$G$ is \textit{planar} if $\gamma(G)=0$ and \textit{toroidal} if $\gamma(G)=1$.
A crosscap is a circle (on the surface) such that all its pairs of opposite points are identified, and
the interior of this circuit is removed. Let $\mathbb{N}_k$ denote the sphere with $k$ added crosscaps.
For non-orientable topological surfaces (e.g., the projective plane, Klein bottle, etc.),
the \textit{nonorientable genus} of $G$ is the smallest integer $k$ such that $G$ can be embedded on $\mathbb{N}_k$, and it is denoted by $\overline{\gamma}(G)$.
$\mathbb{N}_1$ is the projective plane. Respectively, a graph $G$ is \textit{projective-planar} if $\overline{\gamma}(G) =1$.
A \textit{topological obstruction} for a surface is a graph $G$ of minimum vertex degree at least three such that $G$ does not embed on the surface,
but $G-e$ is embeddable on the surface for every edge $e$ of $G$. A \emph{minor-order obstruction} $G$ is a topological obstruction with the additional
property that, for each edge $e$ of $G$, $G$ with the edge $e$ contracted embeds on the surface.
The following results are used in the subsequent sections.
\begin{thm}\label{genus 110}(\cite[Theorems 6.37, 6.38 and 11.19, 11.23]{arthur})
\
\begin{enumerate}[{\normalfont (1)}]
\item $\gamma(K_n)=\displaystyle\left\lceil \frac{(n-3)(n-4)}{12}\right\rceil$, $n\geq 3$;
$\gamma(K_{m,n})=\displaystyle\left\lceil \frac{(m-2)(n-2)}{4}\right\rceil$, $m$, $n\geq 2$.
\item $\overline{\gamma}(K_n)=\left\{
\begin{array}{ll}
\displaystyle\left\lceil \frac{(n-3)(n-4)}{6} \right\rceil, & \mbox{~~if~ } n\geq 3, n\neq 7; \\
3, & \mbox{~~if~ } n=7;
\end{array}
\right.$
$\overline{\gamma}(K_{m,n})=\displaystyle\left\lceil \frac{(m-2)(n-2)}{2}\right\rceil$, $m$, $n\geq 2$.
\end{enumerate}
\end{thm}
By Theorem~\ref{genus 110}, one can see that $\gamma(K_n)>1$ for $n\geq 8$, $\overline{\gamma}(K_n)>1$ for $n\geq 7$, $\gamma(K_{m,n})>1$ if either $m\geq4$, $n\geq 5$ or $m\geq 3$, $n\geq 7$, and $\overline{\gamma}(K_{m,n})>1$ if either $m\geq3$, $n\geq5$ or $m=n=4$. Neufeld and Myrvold \cite{pra} have shown the following.
\begin{thm}(\cite[Figure 4, p. 578]{pra})\label{genus 501}
There are exactly three eight-vertex obstructions $\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3$ for the torus, each of them being topological and minor-order (see Figure~\ref{fig:f2}).
\end{thm}
\begin{figure}
\caption{The eight vertex obstructions for the torus.}
\label{fig:f2}
\end{figure}
Note that the graph $\mathcal {A}_1$ in Figure~\ref{fig:f2} has $K_{3,5}$ as a subgraph, so it follows that if a graph $G$ has $\mathcal {A}_1$ as a subgraph, then $\gamma(G)>1$ and $\overline{\gamma}(G)>1$. In this paper, we repeatedly use this fact.
Gagarin \emph{et al.}~\cite{and} have found all the toroidal obstructions for the graphs containing no subdivisions of $K_{3,3}$ as a subgraph. These graphs coincide with the graphs containing no $K_{3,3}$-minors and are called \emph{with no $K_{3,3}$'s}.
\begin{thm}(\cite[Theorem~3, p. 3628]{and}) \label{genus 500}
There are exactly four minor-order obstructions with no $K_{3,3}$'s for the torus, precisely,
$\mathcal{B}_1,\mathcal{B}_2,\mathcal{B}_3,\mathcal{B}_4$ shown in Figure~\ref{fig:int f3}
as a minor.
\end{thm}
\begin{figure}
\caption{The minor-order obstructions with no $K_{3,3}
\label{fig:int f3}
\end{figure}
Notice that all the obstructions in Theorems~\ref{genus 501} and \ref{genus 500} are obstructions for toroidal graphs in general, which are very numerous (e.g., see \cite{and}).
In \cite{Chakara}, Chakarabarthy \emph{et al.} proved the following results on the intersection graph of ideals of the ring $\mathbb Z_n$.
\begin{thm}(\cite{Chakara})\label{l1}
Let $p$, $q$, $r$ be distinct primes. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(\mathbb Z_n)$ is planar if and only if $n$ is one of $p^\alpha (\alpha=2,3,4,5)$, $p^\alpha q (\alpha=1, 2)$, or $pqr$.
\item $\mathscr{I}(\mathbb Z_n)$ is $K_5$-free if and only if it is planar.
\item $\mathscr{I}(\mathbb Z_n)$ is bipartite if and only if $n$ is either $p^3$ or $pq$.
\item $\mathscr{I}(\mathbb Z_n)$ is $C_3$-free if and only if $n$ is either $p^\alpha$ $(\alpha=2,3)$ or $pq$.
\end{enumerate}
\end{thm}
In \cite{sel} Sel\c{c}¸uk Kayacan \emph{et al.} have classified all the finite groups whose intersection graphs of subgroups are planar.
\begin{thm}(\cite{sel})\label{1000}
Let $G$ be a finite group and $p$, $q$, $r$ be distinct primes. Then $\mathscr{I}(G)$ is planar if and only if $G$ is one of the following groups:
\begin{enumerate}[\normalfont (1)]
\item $\mathbb Z_{p^\alpha} (\alpha=2,3,4,5)$, $\mathbb Z_{p^\alpha q} (\alpha=1, 2)$, $\mathbb Z_{pqr}$, $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_4 \times \mathbb Z_2$, $\mathbb Z_6\times \mathbb Z_2$;
\item $Q_8$, $M_8$, $\mathbb Z_q \rtimes \mathbb Z_p$, $\mathbb Z_q \rtimes_2 \mathbb Z_{p^2}$, $A_4$;
\item $\mathcal{G}_1 \cong \langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, where $q~|~(p+1)$;
\item $\mathcal{G}_2 \cong\langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=b^{-1}, cbc^{-1}=a^1b^l\rangle$,
where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, where $q^2~|~(p+1)$;
\item $\mathcal{G}_3 \cong\langle a,b,c~|~a^p=b^q=c^r=1, b^{-1}ab=a^\mu, c^{-1}ac=a^v, bc=cb\rangle$, $q$, $r$ are divisors of $(p-1)$, $v$, $\mu\neq 1$.
\end{enumerate}
\end{thm}
The intersection graph of subgroups of the groups listed in Theorem~\ref{1000}, are given below for use in the subsequent sections.
\begin{equation}\label{intersection graph e1}
\mathscr{I}(\mathbb Z_{p^\alpha})\cong K_{\alpha-1}, \alpha=2, 3, 4, 5.
\end{equation}
\begin{equation}\label{intersection graph eqn2}
\mathscr{I}(\mathbb Z_{pq})\cong \overline{K}_2.
\end{equation}
\begin{equation}\label{intersection graph e2}
\mathscr{I}(\mathbb Z_{p^2q})\cong K_1+(K_2\cup K_1).
\end{equation}
\begin{equation}\label{e4}
\mathscr{I}(\mathbb Z_p\times \mathbb Z_p) \cong \overline{K}_{p+1}.
\end{equation}
\begin{equation}\label{5}
\mathscr{I}(\mathbb Z_4\times \mathbb Z_2) \cong K_1 +(K_4 \cup \overline{K}_2).
\end{equation}
\begin{equation}\label{23}
\mathscr{I}(Q_8)\cong K_4.
\end{equation}
\begin{equation}\label{e8}
\mathscr{I}(\mathbb Z_q \rtimes \mathbb Z_p) \cong \overline{K}_{q+1}
\end{equation}
\begin{equation}\label{e9}
\mathscr{I}(\mathbb Z_q \rtimes_2 \mathbb Z_{p^2}) \cong K_1+ (K_1 \cup q K_2),
\end{equation}
\begin{equation}\label{e11}
\Gamma(\mathcal{G}_1)\cong K_{1,p+1}\cup \overline{K}_{p^2}.
\end{equation}
\begin{equation}\label{e10}
\mathscr{I}(A_4) \cong K_{1,3} \cup \overline{K}_4.
\end{equation}
\begin{equation}\label{33}
\mathscr{I}(\mathcal{G}_2)\cong K_1+(K_{1,p+1}\cup p^2K_2).
\end{equation}
\begin{figure}
\caption{(a) $\mathscr{I}
\label{int f1}
\end{figure}
We summarize some of the results of group theory which we use in forthcoming sections.
\begin{thm}\label{501}
\begin{enumerate}[\normalfont (i)]
\item (\cite[Theorem IV, p.129]{burn}) If $G$ is a $p$-group of order $p^n$, then the number of subgroups of order $p^s$, $1\leq s\leq n$ is congruent to 1 (mod p).
\item (\cite[Proposition 1.3]{scott}) If $G$ is a $p$-group of order $p^n$ and it has a unique subgroup of order $p^s$, $1<s\leq n$, then $G$ is cyclic or $s=1$ and $p=2$, $G\cong Q_{2^\alpha}$.
\item (\cite[Corollary IV, p. 53]{burn}) If $G$ has a normal subgroup $H$ of order $mn$, where $m$ and $n$ are relatively prime and $N$ is a normal subgroup of $H$ of order $n$, then $N$ is also normal in $G$.
\end{enumerate}
\end{thm}
\section{Finite abelian groups}\label{sec:3}
In this section, first we classify the finite abelian groups whose intersection graphs of subgroups are toroidal or projective-planar, and next we classify the finite non-cyclic abelian groups whose intersection graphs of subgroups are one of $K_5$ free, $C_3$-free, acyclic or bipartite.
\begin{pro}\label{intersecting graph t100}
Let $G$ be a finite cyclic group and $p$, $q$ be distinct primes. Then
\begin{enumerate}
\item $\mathscr{I}(G)$ is toroidal if and only if
$G$ is one of $\mathbb Z_{p^\alpha}(\alpha=6, 7, 8)$, $\mathbb Z_{p^\alpha q}(\alpha=3, 4)$ or $\mathbb Z_{p^2q^2}$;
\item $\mathscr{I}(G)$ is projective-planar if and only if $G$ is one of $\mathbb Z_{p^\alpha}$$(\alpha=6, 7)$ or $\mathbb Z_{p^3q}$.
\end{enumerate}
\end{pro}
\begin{proof}
Let $|G|=p_1^{\alpha_ 1}p_2^{\alpha_ 2}\dots p_k^{\alpha_ k}$, where $p_i$'s are distinct primes, and $\alpha_i \geq 1$ are integers.
Note that to prove this result, it is enough to investigate the finite cyclic groups other than those listed in Theorem~\ref{l1}(1). We have to deal with the following cases.
\noindent \textbf{Case 1:} If $k=1$, then the number of proper subgroups of $G$ is $\alpha_1-1$ and so
\begin{equation}\label{200}
\mathscr{I}(G)\cong K_{\alpha_1-1}.
\end{equation}
\noindent It follows that
$\gamma(\mathscr{I}(G))=1$ if and only if $\alpha_1=6$, 7, 8. and $\overline{\gamma}(\mathscr{I}(G))=1$ if and only if $\alpha_1=6$, 7.
\noindent \textbf{Case 2:} If $k=2$ and $\alpha_2= 1$, then we consider the following subcases:
\noindent \textbf{Sub case 2a:} $\alpha_1=3$. Let $H_i$, $i=1$, $\ldots$, 6 be the subgroups of $G$ of order $p_1$, $p_1^2$, $p_1^3$, $p_2$, $p_1p_2$,
$p_1^2p_2$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_3$, $H_5$, $H_6$; $H_4$ is a subgroup of $H_5$, $H_6$; $H_4$ has trivial intersection with $H_i$,
$i=1$, 2, 3. Therefore,
\begin{equation}\label{201}
\mathscr{I}(G)\cong K_2+(K_3\cup K_1),
\end{equation} which is a subgraph of $K_6$ and it contains $K_5$. So $\gamma(\mathscr{I}(G))=1$ and $\overline{\gamma}(\mathscr{I}(G))=1$.
\noindent\textbf{Sub case 2b:} $\alpha_1=4$. Let $H_i$, $i=1$, $\ldots$, 8 be the subgroups of $G$ of order
$p_1$, $p_1^2$, $p_1^3$, $p_1^4$, $p_2$, $p_1p_2$,
$p_1^2p_2$, $p_1^3p_2$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_3$, $H_4$, $H_6$, $H_7$, $H_8$; $H_5$ is a subgroup of $H_i$, $i=6$, 7, 8;
$H_5$ has trivial intersection with $H_i$, $i=1$, 2, 3, 4.
It follows that
\begin{equation}\label{2000}
\mathscr{I}(G)\cong K_3+(K_4\cup K_1).
\end{equation}
Also $\gamma(\mathscr{I}(G))=1$ and the corresponding toroidal embedding is shown in Figure~\ref{fig:int f4}. Since
$\mathscr{I}(G)$ contains $K_7$ as a subgraph, it follows that $\overline{\gamma}(\mathscr{I}(G))>1$.
\begin{figure}
\caption{An embedding of $\mathscr{I}
\label{fig:int f4}
\end{figure}
\noindent\textbf{Sub case 2c:} $\alpha_1\geq5$. Let $H_i$, $i=1$, $\ldots$, 8 be the subgroups of $G$ of order
$p_1$, $p_1^2$, $p_1^3$, $p_1^4$, $p_1^\alpha$, $p_1p_2$,
$p_1^2p_2$, $p_1^3p_2$ respectively. Since $H_1$ is a subgroup of $H_i$, $i=2$, $\ldots$, 8, it follows that
they form $K_8$ as a subgraph of $\mathscr{I}(G)$ and so $\gamma(\mathscr{I}(G))>1$,
$\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 3:} $k=2$ and $\alpha _2\geq 2$. Now we have to deal with the following sub cases:
\noindent\textbf{Sub case 3a:} $\alpha_1=\alpha_2=2$. Let $H_i$, $i=1$, $\ldots$, 7 be the subgroups of $G$ of order
$p_1$, $p_1^2$, $p_2$, $p_2^2$, $p_1p_2$, $p_1^2p_2$,
$p_1p_2^2$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_5$, $H_6$, $H_7$; $H_3$ is a subgroup of $H_4$, $H_5$, $H_6$, $H_7$; $H_1$, $H_2$ have trivial
intersection with $H_3$, $H_4$. It follows that
\begin{equation}\label{202}
\mathscr{I}(G)\cong K_3+2K_2.
\end{equation}
Moreover $\gamma(\mathscr{I}(G))=1$, since $\mathscr{I}(G)$ is a subgraph of $K_7$ and it contains $K_5$.
Also $\overline{\gamma}(\mathscr{I}(G))>1$, since $\mathscr{I}(G)$ has a subgraph isomorphic to the graph shown in Figure~\ref{fig:int f5}, which is one of the obstruction for projective-plane (e.g., see Theorem 0.1 and graph
$B_1$ of case (3.6) on p. 340 in \cite{glov}).
\begin{figure}
\caption{An obstruction for the projective-plane.}
\label{fig:int f5}
\end{figure}
\noindent\textbf{Sub case 3b:} Either $\alpha>2$ or $\beta>2$. Let
$H_i$, $i=1$, $\ldots$, 8 be subgroups of $G$ of order
$p_1$, $p_1^2$, $p_1^3$, $p_2$, $p_2^2$, $p_1p_2$,
$p_1^2p_2$, $p_1^3p_2$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_3$, $H_6$, $H_7$, $H_8$; $H_4$ is a subgroup of $H_5$, $H_6$, $H_7$, $H_8$.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
Also $\mathscr{I}(G)$ contains $K_{3,5}$ as a subgraph with bipartition $X:=\{H_3$, $H_6$ $H_7\}$ and $Y:=\{H_1$, $H_2$, $H_4$, $H_5$, $H_8\}$ and so
$\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 4:} If $k=3$, then we need to consider only for $\alpha_1\geq2$, $\alpha_2$, $\alpha_3\geq1$. Let $H_i$, $i=1$, $\ldots$, 8
be subgroups of $G$ of order
$p_1$, $p_1^2$, $p_1p_2p_3$, $p_1p_2$, $p_1p_3$,
$p_2p_3$, $p_1^2p_2$, $p_1^2p_3$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_3$, $H_4$, $H_5$, $H_7$, $H_8$; $H_i$, $i=3$, $\ldots$, 8
intersect non-trivially with each other. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$
and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 5:} If $k\geq 4$, then let $H_i$, $i=1$, $\ldots$, 8 be subgroups of $G$ of orders
$p_1$, $p_1p_2$, $p_1p_2p_3$, $p_1p_2p_4$, $p_1p_3p_4$,
$p_2p_3p_4$, $p_1p_3$, $p_1p_4$ respectively. Here $H_1$ is a subgroup of $H_2$, $H_3$, $H_4$, $H_6$, $H_7$, $H_8$; $H_5$, $H_3$, $H_4$, $H_6$, $H_7$, $H_8$
intersect non-trivially with each other. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$
Combining all the cases together the proof follows.
\end{proof}
\begin{pro}\label{intersecting graph t1}
Let $G$ be a finite non-cyclic abelian group and $p$, $q$ be distinct primes. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G$ is one of $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_4 \times \mathbb Z_2$, or $\mathbb Z_6\times \mathbb Z_2$;
\item The following are equivalent:
\begin{enumerate}
\item $G\cong\mathbb Z_p\times \mathbb Z_p$;
\item $\mathscr{I}(G)$ is $C_3$-free;
\item $\mathscr{I}(G)$ is acyclic;
\item $\mathscr{I}(G)$ is bipartite.
\end{enumerate}
\item $\mathscr{I}(G)$ is toroidal if and only if $G$ is one of $\mathbb Z_{p^2}\times \mathbb Z_p$$(p=3,5)$ or $\mathbb Z_{3q}\times \mathbb Z_3$;
\item $\mathscr{I}(G)$ is projective-planar if and only if $G$ is either $\mathbb Z_9\times \mathbb Z_3$ or $\mathbb Z_{3q}\times \mathbb Z_3$.
\end{enumerate}
\end{pro}
\begin{proof}
We split the proof in to several cases:
\noindent\textbf{Case 1}: $G \cong \mathbb Z_p \times \mathbb Z_p$. Then by Theorem~\ref{1000}, $\mathscr{I}(G)$ is planar and by \eqref{e4}, it is acyclic.
\noindent\textbf{Case 2:} $G\cong{\mathbb Z_{p^2}} \times{\mathbb Z_p}$. Here $\langle(1,0)\rangle$, $\langle(1,1)\rangle$, $\ldots$, $\langle(1,p-1)\rangle$,
$\langle(p,0), (0,1)\rangle$, $\langle(p,0)\rangle$, $\langle(p,1)\rangle$, $\ldots$, $\langle(p,p-1)\rangle$, $\langle(0,1)\rangle$ are the only
proper subgroups of $G$. Note that $\langle(p,0)\rangle$ is a subgroup of $\langle(1,0)\rangle$, $\langle(1,1)\rangle$, $\ldots$, $\langle(1,p-1)\rangle$,
$\langle(p,0), (0,1)\rangle$; $\langle(p,0)\rangle$, $\langle(p,1)\rangle$, $\ldots$, $\langle(p,p-1)\rangle$, $\langle(0,1)\rangle$ are
subgroups of $\langle(p,0), (0,1)\rangle$; no two remaining subgroups intersect non-trivially. It follows that
\begin{equation}\label{e5}
\mathscr{I}(G) \cong K_1 +(K_{p+1} \cup \overline{K}_p).
\end{equation}
So $\mathscr{I}(G)$ contains $C_3$ as a subgraph. Note that $\mathscr{I}(G)$ is a graph obtained by attaching $p$ pendent edges to any one of the vertices of $K_{p+2}$. So $\gamma(\mathscr{I}(G))=1$ if and only if $p=3,5$; $\overline{\gamma}(\mathscr{I}(G))=1$ if and only if $p=3$; $\mathscr{I}(G)$ is $K_5$-free if and only if $p=2$.
\noindent\textbf{Case 3:} $G\cong \mathbb Z_{pq}\times \mathbb Z_p$. If $p=2$, then
by Theorem~\ref{1000}, $\mathscr{I}(G)$ is planar and by Figure~3(b), it contains $C_3$.
If $p=3$, then $H:=\mathbb Z_p\times \mathbb Z_p$ is a subgroup of $G$. $H$ has four proper subgroups, let them be $H_1$, $H_2$, $H_3$, $H_4$. Now $H$ and its proper subgroups, $H_5:=\mathbb Z_q\times \{e\}$, $H_iH_5$, $i=1,2,3,4$ are the only proper subgroups of $G$. Here $H_5$ is a subgroup of $H_iH_5$,
$i=1,2,3,4$; $H_iH_5$, $i=1,2,3,4$ has non-trivial intersection with $H$; $H_i=1,2,3,4$ are subgroups of $H$; no two remaining subgroups intersect nontrivially.
A toroidal embedding of $\mathscr{I}(G)$ is shown in Figure~\ref{fig:int f6},
and an embedding of $\mathscr{I}(G)$ in the projective-plane is shown in Figure~\ref{fig:int f7}. Note that $\mathscr{I}(G)$ contains $K_5$.
\begin{figure}
\caption{An embedding of $\mathscr{I}
\label{fig:int f6}
\end{figure}
\begin{figure}
\caption{An embedding of $\mathscr{I}
\label{fig:int f7}
\end{figure}
If $p=5$, then $H:=\mathbb Z_p\times \mathbb Z_p$ is a subgroup of $G$. $H$ has six proper subgroups, let them be $H_i$, $i=1$, $\ldots$, $6$. Now $H$ and its proper subgroups, $H_7:=\mathbb Z_q\times \{e\}$, $B_i=H_iH_7$, $i=1$, $\ldots$, 6 are the only proper subgroups of $G$. Here $H_7$ is a subgroup of $B_i$,
$i=1$, $\ldots$, 6; $A_i$, $i=1$, $\ldots$, 6 and $H$ intersect non-trivially; $H_i$, $i=1$, $\ldots$, 6 are subgroups of $H$;
no two remaining subgroups have non-trivial intersection.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
If $p\geq 7$, then $H:=\mathbb Z_p\times \mathbb Z_p$, $H_1:=\langle (1,0)\rangle$, $H_2:=\langle (1,1)\rangle$, $H_3:=\langle (2,1)\rangle$,
$H_4:=\langle (3,1)\rangle$, $H_5:=\langle (4,1)\rangle$, $H_6:=\langle (5,1)\rangle$, $H_7:=\langle (0,1)\rangle$, $H_8:=\mathbb Z_q\times \{e\}$, $H_iH_8$,
$i=1$, $\ldots$, 7 are proper subgroups of $G$. Here $H_8$ is a subgroup of $H_iH_8$,
$i=1$, $\ldots$, 7, so they form $K_8$ as a subgraph of $\mathscr{I}(G)$ and hence $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 4:} $G\cong \mathbb Z_{p^2q}\times \mathbb Z_p$. Here $H:=\mathbb Z_{p^2}\times \mathbb Z_p$ is a subgroup of $G$.
As in Case 2, $H$ has at least three proper subgroups of order $p^2$, say $H_i$, $i=1$, 2, 3 and has at least three subgroups of order $p$, say $H_i$, $i=4$, 5, 6; $H_j$, $j=4,5,6$ are subgroups of $H_i$ for some $i\in\{1$, 2, $3\}$, let it be $H_3$; also for some $j\in\{4$, 5, $6\}$, $H_j$ is a
subgroup of $H_i$, for every $i=1$, 2, 3, let it be $H_4$. Let $H_7$ be a subgroup of $G$ of order $q$. Here
$H_iH_7$, $H_jH_7$, $i=1$, 2, 3 and $j=4$, 5, 6 are subgroups of $G$ and they have $H_7$ as their intersection, so $H_7$, $H_iK$, $H_jK$, $i=1$, 2,
$j=4$, 5, 6 form $K_6$ as a subgroup of $\mathscr{I}(G)$; $H$ and $H_3H_7$ intersect non-trivially and they also intersect with $H_jH_7$, $j=4$, 5, 6
non-trivially. It follows
that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case 5:} $G \cong \mathbb Z_{p^2} \times \mathbb Z_{p^2}=\langle a,b~|~a^{p^2}=b^{p^2}=1, ab=ba\rangle$.
Then $H_1:=\langle a\rangle$, $H_2:=\langle ab^p\rangle$, $H_3:=\langle a^p, b^p\rangle$, $H_4:=\langle a,b^p\rangle$, $H_5:=\langle a^p, a^pb\rangle$,
$H_6:=\langle a^p\rangle$, $H_7:=\langle b\rangle$, $H_8:=\langle b^p\rangle$ are subgroups of $G$. Also $H_6$ is a subgroup of $H_i$, $i=1$, $\ldots$, 5;
$H_8$ is a subgroup of $H_3$, $H_4$, $H_5$, $H_7$. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case 6:} $G \cong \mathbb Z_{p^k} \times \mathbb Z_{p^l}$, $k$, $l$ $\geq 3$. Then $\mathbb Z_{p^2} \times \mathbb Z_{p^2}$
is a proper subgroup of $G$ and so by Case 4, we have $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 7:} $G\cong \mathbb Z_p\times \mathbb Z_{pqr}$. Then $H:=\mathbb Z_p\times \mathbb Z_p$ is a subgroup of $G$ and
as in Case 1,
$H$ has at least three proper subgroups of order $p$, say $H_i$, $i=1,2,3$. Let $H_4$, $H_5$ be subgroups of $G$ of orders $q$, $r$ respectively.
Here $H_iH_j$, $i=1$, 2, 3, $j=4$, 5 are also proper subgroups of $G$. So $H_iH_4$, $i=1$, 2, 3, $HH_4$, $H_4$ forms
$K_5$ as a subgraph of $\mathscr{I}(G)$; $H_iH_5$, $i=1$, 2, 3, $HH_5$, $H_5$ also forms $K_5$ as a subgraph of $\mathscr{I}(G)$. Thus $\mathscr{I}(G)$ has a subgraph $\mathcal B_1$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$. $\mathcal B_1$ is also a topological obstruction for projective-plane embedding (e.g., see Theorem. 0.1 and the graph $A_5$ of case (3.15) on p. 343 in \cite{glov}), so $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case 8:} $G \cong\mathbb Z_p\times\mathbb Z_p\times\mathbb Z_p$. Here $\langle(1,0,0)$, $(0,1,0)\rangle$, $\langle(1,0,0)$, $(0,0,1)\rangle$,
$\langle(1,0,0)$, $(1,1,0)\rangle$, $\langle(1,0,0)$, $(0,1,1)\rangle$ are proper subgroups of $G$, which have $\langle(1,0,0)\rangle$ as their intersection.
It follows that these five subgroups form $K_5$ as a subgraph of $\mathscr{I}(G)$.
Also $\langle(1,1,1)$, $(0,1,0)\rangle$, $\langle(0,1,0)$, $(0,0,1)\rangle$,
$\langle(0,1,0)$, $(1,1,0)\rangle$, $\langle(0,1,0)$, $(0,1,1)\rangle$ are proper subgroups of $G$, which have $\langle(0,1,0)\rangle$ as their intersection
and so they also form $K_5$ as a subgraph of $\mathscr{I}(G)$. Thus $\mathscr{I}(G)$ has a subgraph $\mathcal B_1$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case 9:} $G \cong {\mathbb Z}_{p_1^{\alpha_1}} \times {\mathbb Z}_{p_2^{\alpha_2}} \times \ldots \times {\mathbb Z_{p_k^{\alpha_k}}}$, where
$p_i$'s are primes with at least two $p_i$'s are equal and $\alpha_i\geq1$. If $k\geq 2$, then $G$ has one of the following groups as its subgroup: $\mathbb Z_{p_i^2}\times \mathbb Z_{p_i^2}$,
$\mathbb Z_{p_i^2p_j}\times \mathbb Z_{p_i}$, $\mathbb Z_{p_i}\times \mathbb Z_{p_i}\times \mathbb Z_{p_i}$, for some $i$, $j$.
So by Cases 3, 4, 7, the intersection graph of subgroups of these subgroups are non-toroidal, non-projective planar and contains $K_5$. So it follows that $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph.
The result follows by combining all the above cases.
\end{proof}
\section{Finite non-abelian groups}\label{sec:4}
In this section, we classify the finite non-abelian groups whose intersection graphs of subgroups are one of toroidal, projective-planar, $K_5$ free, $C_3$-free, acyclic or bipartite. We first investigate the non-abelian solvable groups and then we deal with the nonsolvable groups.
For any integer $n\geq 3$, the dihedral group of order $2n$ is given by $D_{2n}=\langle a, b~|~a^n=b^2=1, ab=ba^{-1}\rangle$;
For any integer $n \geq 2$, the generalized quaternion group of order $2^n$ is given by
$Q_{2^n} = \big < a, b ~|~a^{2^{n-1}} = b^4 = 1, a^{2^{n-2}} = b^2 = 1,
bab^{-1} = a^{-1}\big >$; For any $\alpha\geq 3$ and $p$ is a prime, the modular group of order $p^\alpha$ is given by
$M_{p^\alpha}=\langle a,b~|~a^{p^{\alpha-1}}=b^p=1, bab^{-1}=a^{p^{\alpha-2}+1}\rangle$; $S_n$ and $A_n$ are symmetric and alternating groups of degree $n$
acting on $\{1,2, \ldots, n\}$ respectively.
We denote the order of an element $a \in \mathbb Z_n$ by $\text{ord}_n(a)$. The number of Sylow $p$-subgroups of a group $G$
is denoted by $n_p(G)$.
\subsection{Finite non-abelian solvable groups}\label{sec:4a}
\begin{pro}\label{intersecting graph t2}
Let $G$ be a non-abelian group of order $p^{\alpha}$, where $p$ is a prime and $\alpha \geq 3$. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G$ is either $Q_8$ or $M_8$;
\item $\mathscr{I}(G)$ contains $C_3$;
\item $\mathscr{I}(G)$ is toroidal if and only if $G$ is one of $M_{p^3}(p=3$, $5)$ or $M_{16}$.
\item $\mathscr{I}(G)$ is projective-planar if and only if $G\cong M_{27}$.
\end{enumerate}
\end{pro}
\begin{proof} We prove the result in the following cases:
\noindent\textbf{Case 1:} $\alpha =3$. If $p=2$, then the only non-abelian groups of order 8 are $Q_8$ and $M_8$. By Theorem~\ref{1000}, the intersection graphs of subgroups of these two groups are planar and by \eqref{23} and Figure~\ref{int f1}(c), they contains $C_3$.
If $p \neq 2$, then up to isomorphism the only non-abelian groups of order $p^3$ are $M_{p^3}$ and $(\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p$.
\begin{enumerate}[{\normalfont (i)}]
\item If $G \cong M_{p^3}$, then the subgroup lattices of $M_{p^3}$ and $\mathbb Z_{p^2}\times \mathbb Z_p$ are isomorphic, so their intersection graphs of subgroups are also isomorphic.
By Case 2 in the proof of Proposition~\ref{intersecting graph t1}, we have
\begin{equation}\label{e100}
\mathscr{I}(G)\cong K_1+(K_{p+1}\cup \overline{K}_p).
\end{equation}
Also, $\gamma(\mathscr{I}(G))=1$ if and only if $p=3,5$; $\overline{\gamma}(\mathscr{I}(G))=1$ if only if $p=3$; $\mathscr{I}(G)$ contains $K_5$.
\item If $G \cong (\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p$, then consider
its proper subgroups $H_1:=\langle a, b \rangle$, $H_2:=\langle a,c \rangle$, $H_3:=\langle ab, c \rangle$, $H_4:=\langle ab^2, c\rangle$,
$H_5:=\langle b, ac \rangle$, $H_6:=\langle b, a^2c\rangle$, $H_7:=\langle b, c\rangle$, $H_8:=\langle b\rangle$, $H_9:=\langle c\rangle$.
Here $H_9$ is a subgroup of $H_2$, $H_3$, $H_4$, $H_7$, so these five subgroups forms $K_5$ as a subgraph of $\mathscr{I}(G)$; $H_8$ is a subgroup of
$H_1$, $H_5$, $H_6$, so these four subgroups forms $K_4$ as a proper subgraph of $\mathscr{I}(G)$;
$H_1$, $H_4$ intersects non-trivially; $H_8$ is a subgroup of $H_7$, $H_5$, $H_6$. It follows that
$\mathscr{I}(G)$ has a subgraph $\mathcal B_3$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
Also $\mathscr{I}(G)$ has a
subgraph as shown in Figure~\ref{fig:int f8}, which is an obstruction for projective-plane (e.g., see Theorem 0.1 and graph
$C_7$ of case (3.20) on p. 344 in \cite{glov}). Therefore, $\overline{\gamma}(\mathscr{I}(G))>1$.
\begin{figure}
\caption{An obstruction for the projective-plane.}
\label{fig:int f8}
\end{figure}
\end{enumerate}
\noindent\textbf{Case 2:} $\alpha=4$. According to Burnside~\cite{burn}, up to isomorphism, there are fifteen groups of order $p^4$, $p>2$ and there are nine groups
of order $2^4$.
In following we investigate the genus of the intersection graphs of subgroups of each of them:
\noindent \textbf{Sub case 2a:} $p>2$.
\begin{enumerate}[{\normalfont (i)}]
\item $G\cong M_{p^4}$. Here $H_1:=\langle a\rangle$, $H_2:=\langle ab\rangle$, $H_3:=\langle ab^2\rangle$, $H_4:=\langle a^p\rangle$,
$H_5:=\langle a^pb\rangle$, $H_6:=\langle a^pb^2\rangle$, $H_7:=\langle a^p, b\rangle$, $H_8:=\langle a^{p^2}\rangle$ are proper subgroups of $G$. Also
$H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7. It follows that they form $K_8$ as a
subgraph of $\mathscr{I}(G)$, so $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^p=1, cb=a^pbc, ab=ba, ac=ca\rangle$. Here $H_1:=\langle a,b\rangle$,
$H_2:=\langle a,c\rangle$, $H_3:=\langle a\rangle$, $H_4:=\langle a^p, c\rangle$, $H_5:=\langle a^p,b\rangle$,
$H_6:=\langle ab\rangle$, $H_7:=\langle ac\rangle$, $H_8:=\langle a^p\rangle$ are proper subgroups of $G$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7.
It follows that they form $K_8$ as a subgraph of $\mathscr{I}(G)$, so $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b~|~a^{p^2}=b^{p^2}=1, bab^{-1}=a^{1+p}\rangle$. Here $H_1:=\langle a\rangle$, $H_2:=\langle a^p\rangle$, $H_3:=\langle a^p,b\rangle$,
$H_4:=\langle a^p, b^p\rangle$, $H_5:=\langle a, b^p\rangle$, $H_6:=\langle b\rangle$, $H_7:=\langle b^p\rangle$, $H_8:=\langle ab, b^p\rangle$,
$H_9:=\langle a^2b, b^p\rangle$ are proper subgroups of $G$. Here $H_i$, $i=1$, $\ldots$, 5 intersect with each other non-trivially
and so they form $K_5$ as a subgraph of
$\mathscr{I}(G)$; $H_7$ is a subgroup of $H_4$, $H_6$, $H_8$, $H_9$; $H_6$, $H_8$, $H_9$ intersect with $H_5$ non-trivially.
It follows that
$\mathscr{I}(G)$ has a subgraph $\mathcal B_3$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^p=1, ca=a^{1+p}c, ab=ba, cb=bc\rangle$. We can take the proper subgroups of $G$ as in (ii), except by taking
$H_7:=\langle a, bc\rangle$ instead of $H_7=\langle ac\rangle$. Then
$\mathscr{I}(G)$ has $K_8$ as a subgraph and so $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^p=1, ca=abc, ab=ba, cb=bc\rangle$. We can use a similar argument as in (iv) to show
$\mathscr{I}(G)$ has $K_8$ as a subgraph and so $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^p=1, ba=a^{1+p}b, ca=abc, cb=bc\rangle$. Here $H_1:=\langle a,b\rangle$, $H_2:=\langle a,c\rangle$,
$H_3:=\langle b,c\rangle$,
$H_4:=\langle a^p, c\rangle$, $H_5:=\langle a^p,b\rangle$, $H_6:=\langle a^p\rangle$, $H_7:=\langle a\rangle$, $H_8:=\langle a^p, bc\rangle$,
$H_9:=\langle a^p, b^2c\rangle$ are proper subgroups of $G$. Also $H_i$, $i=1$, $\ldots$, 5 intersect with each other non-trivially
and so they form $K_5$ as a subgraph of
$\mathscr{I}(G)$; $H_6$ is a subgroup of $H_7$, $H_8$, $H_9$; $H_6$ is a subgroup of $H_1$; $H_7$, $H_8$, $H_9$ intersect with $H_4$ non-trivially.
It follows that
$\mathscr{I}(G)$ has a subgraph $\mathcal B_3$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
\item If $p=3$, then $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^{p^2}=1, c^p=a^p, ab=ba^{1+p}, ac=cab^{-1}, cb=bc\rangle$. Here
$H_1:=\langle a,b\rangle$, $H_2:=\langle a,c\rangle$, $H_3:=\langle b,c\rangle$,
$H_4:=\langle a^p, b\rangle$, $H_5:=\langle c\rangle$, $H_6:=\langle a\rangle$, $H_7:=\langle c^p\rangle$, $H_8:=\langle bc\rangle$,
$H_9:=\langle b^2c\rangle$ are subgroups of $G$. Also $H_i$, $i=1$, $\ldots$, 5 intersect with each other non-trivially and so they form $K_5$ as a subgraph of
$\mathscr{I}(G)$; $H_7$ is a subgroup of $H_6$, $H_8$, $H_9$; $H_7$ is a subgroup of $H_4$; $H_6$, $H_8$, $H_9$ intersect with $H_5$ non-trivially.
It follows that
$\mathscr{I}(G)$ has a subgraph $\mathcal B_3$ as shown in Figure~\ref{fig:int f3},
so $\gamma(\mathscr{I}(G))>1$.
If $p>3$, then $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^{p^2}=1, ba=a^{1+p}b, ca=a^{1+p}bc, cb=a^pbc\rangle$.
Here $\langle a,b\rangle\cong M_{p^3}$ is a subgroup of $G$,
so by \eqref{e100}, $M_{p^3}$ together with its proper subgroups forms $K_8$ as a subgraph of $\mathscr{I}(G)$. Hence $\gamma(\mathscr{I}(G))>1$.
\item If $p=3$, then $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^{p^2}=1, c^p=a^{-p}, ab=ba^{1+p}, ac=cab^{-1}, cb=bc\rangle$.
We can use a similar argument as in (vii), to show $\gamma(\mathscr{I}(G))>1$.
If $p>3$, then $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^{p^2}=1, ba=a^{1+p}b, ca=a^{1+dp}bc, cb=a^{dp}bc, d\ncong 0,1 (\mbox {mod} p)\rangle$.
Here $\langle a,b\rangle\cong M_{p^3}$ is a subgroup of $G$,
so by \eqref{e100}, $M_{p^3}$ together with its proper subgroups forms $K_8$ as a subgraph of $\mathscr{I}(G)$. Thus $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b,c,d~|~a^p=b^p=c^p=d^p=1, dc=acd, bd=db, ad=da, bc=cb, ac=ca\rangle$. Here $\langle a,b,c\rangle\cong
\mathbb Z_p\times \mathbb Z_p\times \mathbb Z_p$ is a subgroup of $G$ and so by Case 7 in the proof of Proposition~\ref{intersecting graph t1},
it follows that $\gamma(\mathscr{I}(G))>1$.
\item If $p=3$, then $G\cong \langle a,b,c~|~a^{p^2}=b^p=c^p=1, ab=ba, ac=cab, cb=ca^{-p}b\rangle$.
Here $H_1:=\langle a,b\rangle$, $H_2:=\langle a,c\rangle$, $H_3:=\langle b,c\rangle$,
$H_4:=\langle a^p, b\rangle$, $H_5:=\langle a^p, c\rangle$, $H_6:=\langle a\rangle$, $H_7:=\langle a^p\rangle$, $H_8:=\langle ab\rangle$,
$H_9:=\langle a^2b\rangle$ are proper subgroups of $G$. Also $H_i$, $i=1$, $\ldots$, 5 intersect with each other non-trivially
and so they form $K_5$ as a subgraph of
$\mathscr{I}(G)$; $H_7$ is a subgroup of $H_4$, $H_6$, $H_8$, $H_9$; $H_6$, $H_8$, $H_9$ intersect with $H_5$ non-trivially.
It follows that
$\mathscr{I}(G)$ has a subgraph $\mathcal B_3$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
If $p>3$, then $G\cong \langle a,b,c,d~|~a^p=b^p=c^p=d^p=1, dc=acd, bd=db, ad=da, bc=cb, ac=ca\rangle$. Here $\langle a,b,c\rangle\cong
\mathbb Z_p\times \mathbb Z_p\times \mathbb Z_p$ is a subgroup of $G$ and so by Case 7 in the proof of Proposition~\ref{intersecting graph t1},
it follows that $\gamma(\mathscr{I}(G))>1$.
\end{enumerate}
\noindent \textbf{Sub case 2b:} $p=2$.
\begin{enumerate}[{\normalfont (i)}]
\item $G\cong M_{2^4}$. Here $H_1:=\langle a\rangle$, $H_2:=\langle ab\rangle$, $H_3:=\langle a^2,b\rangle$,
$H_4:=\langle a^2\rangle$, $H_5:=\langle a^2b, c\rangle$, $H_6:=\langle a^4,b\rangle$, $H_7:=\langle a^4\rangle$, $H_8:=\langle a^4b\rangle$,
$H_9:=\langle b\rangle$ are the only subgroups of $M_{2^4}$. Further $H_7$ is a subgroup of $H_i$, $i=1$, $\ldots$, 6;
$H_8$, $H_9$ are proper subgroups of $H_6$;
no two remaining subgroups intersect non-trivially. It follows that
\begin{equation}\label{e101}
\mathscr{I}(G)\cong K_1+(K_6\cup \overline{K}_2).
\end{equation}
Note that $\mathscr{I}(G)$ is a graph obtained by attaching 2 pendent edges to any one of the vertices of $K_7$, so $\gamma(\mathscr{I}(G))=1$.
\item $G\cong \langle a,b,c~|~a^4=b^4=c^2=1, bab^{-1}=a^{-1}, a^2=b^2, bc=cb, ac=ca\rangle$. Here
$H_1:=\langle a, c\rangle$, $H_2:=\langle a\rangle$, $H_3:=\langle ac\rangle$,
$H_4:=\langle a^2\rangle$, $H_5:=\langle a^2, c\rangle$, $H_6:=\langle bc\rangle$, $H_7:=\langle b\rangle$, $H_8:=\langle b, c\rangle$
are proper subgroups of
$G$ and these subgroups intersect with each other non-trivially, so they form $K_8$ as a subgraph of $\mathscr{I}(G)$. It follows that $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b~|~a^8=b^2=1, bab^{-1}=a^{-1}\rangle$. Here
$H_1:=\langle a\rangle$, $H_2:=\langle a^2\rangle$, $H_3:=\langle a^4\rangle$,
$H_4:=\langle a^2, b\rangle$, $H_5:=\langle a^4, b\rangle$, $H_6:=\langle a^4, ab\rangle$, $H_7:=\langle a^4, a^2b\rangle$, $H_8:=\langle a^2, ab\rangle$
are proper subgroups of
$G$ and these eight subgroups intersect with each other non-trivially, so they form $K_8$ as a subgraph of $\mathscr{I}(G)$. It follows that $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b~|~a^8=b^2=1, b^{-1}ab=a^3\rangle$. Here
$H_1:=\langle a\rangle$, $H_2:=\langle a^2\rangle$, $H_3:=\langle a^4\rangle$,
$H_4:=\langle a^2, b\rangle$, $H_5:=\langle a^4, b\rangle$, $H_6:=\langle a^4, ab\rangle$, $H_7:=\langle a^4, a^2b\rangle$, $H_8:=\langle a^4, a^3b\rangle$
are proper subgroups of
$G$ and these eight subgroups intersect with each other non-trivially. Hence they form $K_8$ as a subgraph of $\mathscr{I}(G)$.
It follows that $\gamma(\mathscr{I}(G))>1$.
\item $G\cong \langle a,b~|~a^8=b^2=1, b^{-1}ab=a^{-1}, b^2=a^4\rangle$. Here
$H_1:=\langle a\rangle$, $H_2:=\langle a^2\rangle$, $H_3:=\langle a^4\rangle$,
$H_4:=\langle b\rangle$, $H_5:=\langle a^2, b\rangle$, $H_6:=\langle ab\rangle$, $H_7:=\langle a^2b\rangle$, $H_8:=\langle a^3b\rangle$
are proper subgroups of
$G$ and these eight subgroups intersect with each other non-trivially, so they form $K_8$ as a subgraph of $\mathscr{I}(G)$. It follows that $\gamma(\mathscr{I}(G))>1$.
\end{enumerate}
The remaining groups in this subcase are identical with the groups described in (ii), (iii), (iv), (v) of Subcase 2a.
\noindent\textbf{Case 3:} $\alpha=5$. By Theorem~\ref{501} (i),
$G$ has at least three subgroups of order $p^4$, say $H_i$, $i=1, 2, 3$ and at least three subgroups of order $p^3$, say
$H_i$, $i=4, 5, 6$. Here for each $i=1$, 2, $3$ and $j=4$, 5, $6$, $H_i$ and $H_j$ have a non-trivial intersection. For otherwise, $|H_iH_j|=p^k$, $k=7$ or
8, which is not possible. Let $H_7$ be a common subgroup of order $p^2$ for both $H_4$ and $H_5$. Let $H_8$ be a subgroup of $H_7$ of order $p$.
Here $H_1$, $H_4$, $H_5$, $H_7$, $H_8$
intersects with each other non-trivially, so they form $K_5$ as a subgraph of $\mathscr{I}(G)$.
Now $H_2$ and $H_3$ have a common subgroup of order
$p^3$, say $H_9$; let $H_{10}$ be a subgroup of $H_9$ of order $p^2$; let $H_{11}$ be a subgroup of $H_{10}$ of order $p$.
Then $H_2$, $H_3$, $H_9$, $H_{10}$, $H_{11}$ intersect with each other non-trivially.
Therefore, $\mathscr{I}(G)$ contains a subgraph $\mathcal B_1$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
\noindent\textbf{Case 4:} $\alpha\geq6$. By Theorem~\ref{501} (i), (ii),
$G$ has at least two subgroups of order $p^{\alpha-1}$, say $H_1$, $H_2$. Let $H_i$, $i=3$, 4, 5, 6 be subgroups of $H_1$ of order $p^{\alpha-2}$,
$p^3$, $p^2$, $p$ respectively. Also let $H_i$, $i=7$, 8, 9, 10 be subgroups of $H_2$ of orders $p^{\alpha-2}$, $p^3$, $p^2$, $p$ respectively.
It follows that $H_1$, $H_3$, $H_4$, $H_5$, $H_6$ forms $K_5$ as a subgraph of $\mathscr{I}(G)$ and $H_2$, $H_7$, $H_8$, $H_9$, $H_{10}$ forms another copy of $K_5$ as a subgraph of $\mathscr{I}(G)$.
Thus $\mathscr{I}(G)$ has a subgraph $\mathcal B_1$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$.
Now we investigate the projective-plane embedding of the intersection graphs of subgroups of $G$, when $\alpha\geq 4$. We need to consider the following cases:
\noindent\textbf{Case a:} $p>2$. Then by Theorem~\ref{501} (i),
$G$ has at least four subgroups of order $p^3$, let them be $H_i$, $i=1, 2, 3, 4$; and has at least
four subgroups of order $p^2$, say $B_i$, $i=1, 2, 3, 4$. Here $H_i\cap B_j\neq \emptyset$ for all $i,j=1, 2, 3, 4$ with $i \neq j$ and so they form $K_{4,4}$ as a subgraph of $\mathscr{I}(G)$ with
bipartition $X:=\{H_1$, $H_2$, $H_3$, $H_4\}$ and $Y:=\{B_1$, $B_2$, $B_3$, $B_4\}$. It follows that
$\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case b:} $p=2$. If $G$ has an unique subgroup of order $2$, then by Theorem~\ref{501} (i), (ii), $G\cong Q_{2^\alpha}$. Then $G$ has at least seven proper subgroups and they have a unique subgroup of order 2 in common. It follows that
$\mathscr{I}(G)$ has $K_7$ as a subgraph, so $\overline{\gamma}(\mathscr{I}(G))>1$. If $G\neq Q_{2^\alpha}$, then it has atleast three subgroups of order $p^{\alpha-1}$,
let them be $H_1$, $H_2$, $H_3$;
$G$ has atleast three subgroups, say $H_4$, $H_5$, $H_6$ of order $p^{\alpha-2}$; atleast two subgroups of order $p^{\alpha-3}$, say $H_7$, $H_8$. Here
$H_i\cap H_j\neq \emptyset$, for every $i=1$, 2, 3 and $j=4$, 5, 6, $i\neq j$; $H_7$, $H_8$ are subgroups of $H_i$, $i=1$, 2, 3. It follows that
$K_{3,5}$ is a subgraph of $\mathscr{I}(G)$ with bipartition $X:=\{H_1$, $H_2$, $H_3\}$ and $Y:=\{H_4$, $H_5$, $H_6$, $H_7$, $H_8\}$, so
$\overline{\gamma}(\mathscr{I}(G))>1$.
The proof follows by putting together all the cases.
\end{proof}
If $G$ is the non-abelian group of order $pq$, where $p< q$ and $p$, $q$ are two distinct primes,
then by Theorem~\ref{1000}, $\gamma(\mathscr{I}(G))$ is planar and by \eqref{e8} it is acyclic.
Next we investigate the groups of order greater than $pq$.
Consider the semi-direct product $\mathbb Z_q \rtimes_{t} \mathbb Z_{p^{\alpha}} = \langle a,b | a^q= b^{p^{\alpha}}= 1, bab^{-1}= a^i,
{ord_{q}}(i)= p^t \rangle$, where $p$, $q$ are distinct primes with $p^t~|~(q-1)$, $t \geq 0$.
Then every semi-direct product $Z_q \rtimes Z_{p^{\alpha}}$ is one of these types \cite[Lemma 2.12]{boh-reid}. So here after, when $t = 1$ we will suppress the subscript.
\begin{pro}\label{intersecting graph t4}
Let $G$ be a non-abelian group of order $p^2q$, where $p$ and $q$ are distinct primes. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G$ is either $\mathbb Z_q \rtimes_{2} \mathbb Z_{p^2}$, $\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$ or $A_4$.
\item The following are equivalent:
\begin{enumerate}[\normalfont (a)]
\item $G\cong \langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^{-1}b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$ or $A_4$;
\item $\mathscr{I}(G)$ is $C_3$-free;
\item $\mathscr{I}(G)$ is acyclic;
\item $\mathscr{I}(G)$ is bipartite.
\end{enumerate}
\item $\mathscr{I}(G)$ is toroidal if and only if $G$ is one of $\mathbb Z_3\rtimes \mathbb Z_4$, $\mathbb Z_5\rtimes \mathbb Z_4$ or
$\mathbb Z_9\rtimes \mathbb Z_2$, $\mathbb Z_{25}\rtimes \mathbb Z_2$.
\item $\mathscr{I}(G)$ is projective-planar if and only if $G$ is either $\mathbb Z_3\rtimes \mathbb Z_4$ or $\mathbb Z_9\rtimes \mathbb Z_2$.
\end{enumerate}
\end{pro}
\begin{proof}
To prove the result we use the classification of groups of order $p^2q$ given in \cite[p.~76-80]{burn}. We have the following cases to consider.
\noindent\textbf{Case 1:} $p<q$:
\noindent\textbf{Case 1a:} $p \nmid (q-1)$. By Sylow's Theorem, there is no non-abelian group in this case.
\noindent\textbf{Case 1b:} $p \mid (q-1)$ but $p^2 \nmid (q-1)$. In this case, there are two non-abelian groups.
The first group is $G_1:= \mathbb Z_q \rtimes \mathbb Z_{p^2} = \langle a, b | a^q= b^{p^2}=1, bab^{-1}=a^i, ord_q(i)=p \rangle$.
We have $\langle a\rangle$, $\langle a^ib\rangle$, $i=1,2,\ldots,q$, $\langle b^p\rangle$ and
$\langle ab^p\rangle$ are the only proper subgroups of $G_1$. Here $\langle b^p\rangle$ is a subgroup of the remaining proper subgroups, except
$\langle a\rangle$. Also $\langle a\rangle$ is a subgroup of $\langle ab^p\rangle$. It follows that
\begin{equation}\label{e1}
\mathscr{I}(G_1)\cong K_1+(K_1\cup K_{q+1}).
\end{equation}
Note that $q=5$ is not possible here, since $p=2$ is such that $p \mid (q-1)$ but $p^2 \mid (q-1)$.
Note that $\mathscr{I}(G)$ is a graph obtained by attaching one pendent edge to any one of the vertices of $K_{q+2}$. So $\gamma(\mathscr{I}(G))=1$ if and only if $q=3$; $\overline{\gamma}(\mathscr{I}(G))=1$ if and only if $q=3$; $\mathscr{I}(G)$ contains $K_5$.
The second group is $G_2:= \langle a, b, c| a^q= b^p= c^p=1, bab^{-1}=a^i, ac=ca, bc=cb, ord_q(i)=p \rangle$. Here $H_1:=\langle a,b\rangle$,
$H_2:=\langle a,c\rangle$, $H_3:=\langle b,c\rangle$, $H_4:=\langle ab,c\rangle$, $H_5:=\langle a^2b,c\rangle$, $H_6:=\langle a^3b,c\rangle$,
$H_7:=\langle c\rangle$, $H_8:=\langle a^4b,c\rangle$ are proper subgroups of $G$. Also $H_i$, $i=3,4$,$\ldots , 8$ intersect with each other non-trivially
and so they form $K_6$
as a subgraph of $\mathscr{I}(G_2)$; $H_1$ and $H_2$ intersect non-trivially; also they intersect non-trivially with $H_3$, $H_4$, $H_5$.
It follows that $\mathscr{I}(G_2)$ has a
subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G_2))>1$ and $\overline{\gamma}(\mathscr{I}(G_2))>1$.
\noindent\textbf{Case 1c:} $p^2 | (q-1)$. In this case, we have both groups $G_1$ and $G_2$ from Case 1b together with the group $G_3:= \mathbb Z_q \rtimes_2 \mathbb Z_{p^2} = \langle a, b | a^q= b^{p^2}=1, bab^{-1}=a^i, ord_q(i)=p^2 \rangle$. Note that here $q=5$, $p=2$ is possible for the group $G_1$. By~\eqref{e1}, $\gamma(\mathscr{I}(G_1))=1$, $\overline{\gamma}(\mathscr{I}(G_1))>1$.
$\mathscr{I}(G_2)$ is already discussed in Case 1b. By Theorem~\ref{1000}, $\mathscr{I}(G_3)$ is planar and by \eqref{e9}, it contains $C_3$.
\noindent\textbf{Case 2:} $p > q$
\noindent\textbf{Case 2a:} $q \nmid (p^2 -1)$. Then there is no non-abelian subgroups.
\noindent\textbf{Case 2b:} $q | (p-1)$. In this case, we have two groups.
The first one is $G_4:=\langle a,b | a^{p^2}= b^q=1, bab^{-1}, ord_{p^2}(i)=q \rangle$. By Sylow's Theorem, $G_4$ has a unique subgroup say $N$,
of order $p^2$ and has a unique subgroup $N'$ of order $p$; $G_4$ has $p^2$ Sylow $q$-subgroups of order $q$, say
$H_i$, $i=1,2\ldots,p^2$; it has $p$ subgroups of order $pq$, say $A_i=\langle a^p,a^ib\rangle,i=0,1,\ldots p-1$. These are the
only proper subgroups of $G_4$. Here $N'$ is a subgroup
of $N$ and $A_i$, $i=0,1,\ldots,p-1$; $H_i,H_{i+p},\ldots,H_{i+p(p-1)}$ are subgroups of $A_i$ for each $i=0,1,\ldots,p-1$. It follows that
$\mathscr{I}(G_4)$ is the graph obtained by attaching a single pendant edge to any $p$ vertices of $K_{p+2}$.
Thus, $\gamma(\mathscr{I}(G_4))=1$ if and only if $p=3$, $5$; $\overline{\gamma}(\mathscr{I}(G_4))=1$ if and only if $p=3$;$\mathscr{I}(G_4)$ contains $K_5$.
Next we have the family of groups $\langle a, b, c | a^p=b^p=c^q=1, cac^{-1}=a^i, cbc^{-1}=b^{i^t}, ab=ba, ord_p(i)=q \rangle$. There are $(q+3)/2$
isomorphism types in this family (one for $t=0$ and one for each pair $\{ x, x^{-1} \}$ in ${F_p}^{\times}$. We will refer to all of these groups as $G_{5(t)}$
of order $p^2q$. Here $H_1:=\langle a,b\rangle$,
$H_2:=\langle a,c\rangle$, $H_3:=\langle a,bc\rangle$, $H_4:=\langle a,b^2c\rangle$, $H_5:=\langle ab, c\rangle$,
$H_6:=\langle b,c\rangle$, $H_7:=\langle a^2b, c\rangle$, $H_8:=\langle c\rangle$ are proper subgroups of $G$. Further $H_i$, $i=1$, $\ldots$, 6 intersect non-trivially with each other and so they form $K_6$
as a subgraph of $\mathscr{I}(G_{5(t)})$; $H_7$, $H_8$ intersect non-trivially and they intersect with $H_2$, $H_5$, $H_6$.
It follows that $\mathscr{I}(G_{5(t)})$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2},
so $\gamma(\mathscr{I}(G_{5(t)}))>1$, $\overline{\gamma}(\mathscr{I}((G_{5(t)}))$ and $\mathscr{I}(G_{5(t)})$ contains $K_5$ as a subgraph.
\noindent\textbf{Case 2c:} $q| (p+1)$. In this case, we have only one group of order $p^2q$, given by $G_6:= (\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_q
= \langle a, b,c| a^p=b^p=c^q=1, ab=ba, cac^{-1}=a^ib^j, cbc^{-1}=a^kb^l\rangle$, where $\bigl(\begin{smallmatrix}
i & j\\ k & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$.
\begin{itemize}
\item [(i) ] If $G_6$ has a subgroup of order $pq$, then $H_1:=\langle a,b\rangle$,
$H_2:=\langle a,c\rangle$, $H_3:=\langle a,bc\rangle$, $H_4:=\langle a,b^2c\rangle$, $H_5:=\langle ab, c\rangle$,
$H_6:=\langle b,c\rangle$, $H_7:=\langle a^2b, c\rangle$, $H_8:=\langle c\rangle$ are proper subgroups of $G$. Also $H_i$, $i=1$, $\ldots$, 6 intersect non-trivially with each other and so they form $K_6$
as a subgraph of $\mathscr{I}(G)$; $H_7$, $H_8$ intersect non-trivially and they intersect with $H_2$, $H_5$, $H_6$. Thus
$\mathscr{I}(G_6)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G_6))>1$ and
$\overline{\gamma}(\mathscr{I}(G))>1$.
\item [(ii) ]If $G_6$ has no subgroup of order $pq$, then $G_6:=\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^{1}b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$. By Theorem~\ref{1000}, $\mathscr{I}(G_6)$ is planar and by \eqref{e11}, it is acyclic.
\end{itemize}
Note that if $(p, q)= (2, 3)$, the Cases 1 and 2 are not mutually exclusive. Up to isomorphism, there are three non-abelian groups of order 12:
$\mathbb Z_3 \rtimes \mathbb Z_4$, $D_{12}$ and $A_4$. Here the intersection graph of subgroups of $\mathbb Z_3 \rtimes \mathbb Z_4$ (the group $G_1$)
and $D_{12}$ (the group $G_2$) contains $K_5$. But by Theorem~\ref{1000}, $\mathscr{I}(A_4)$ is planar and by \eqref{e10}, it is acyclic.
Combining all the cases together, the proof follows.
\end{proof}
\begin{pro}\label{intersecting graph t5}
If $G$ is a non-abelian group of order $p^ \alpha q$, where $p$, $q$ are distinct primes and $\alpha \geq 3$, then $\gamma(\mathscr{I}(G))>1$,
$\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Let $P$ denote a Sylow p-subgroup of $G$. We shall prove this result by induction on $\alpha$. First we prove this result when
$\alpha =3$. If $p > q$, then $n_p(G)=1$, by Sylow's theorem and our group $G \cong P \rtimes \mathbb Z_q$. Suppose $\gamma(\mathscr{I}(P))>1$,
$\overline{\gamma}(\mathscr{I}(P))>1$ and
$\mathscr{I}(P)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(P)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free or bipartite.
By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1}, \ref{intersecting graph t2} and Theorem~\ref{1000},
$P$ is isomorphic to one of $\mathbb Z_{p^3}$, $\mathbb Z_{p^2}\times \mathbb Z_p$ ($p=2$, $3$, $5$) or $M_{p^3}$ ($p=3$, $5$).
\begin{itemize}
\item If $P \cong \mathbb Z_{p^3}$, then $G \cong \mathbb Z_{p^3} \rtimes \mathbb Z_q=\langle a,b~|~a^{p^3}=b^q=1, bab^{-1}, ord_{p^3}(i)=q\rangle$.
Here $H_1:=\langle a \rangle$, $H_2:=\langle a^p, b \rangle$, $H_3:=\langle a^{p^2}, b\rangle$, $H_4:=\langle a^p \rangle$,
$H_5:=\langle a^p, ab\rangle$, $H_6:=\langle a^{p^2}, ab\rangle$, $H_7:=\langle a^p, a^2b\rangle$, $H_8:=\langle a^{p^2} \rangle$ are proper subgroups of $G$.
Further $H_8$ is a subgroup of $H_i$, for every $i=1$, $\ldots$, 7.
It follows that they form $K_8$ as a subgraph of $\mathscr{I}(G)$, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P\cong \mathbb Z_{p^2}\times \mathbb Z_p$, $p=2$, $3$, $4$, then $G\cong (\mathbb Z_{p^2}\times \mathbb Z_p)\rtimes \mathbb Z_q$. Here $H_1:=\langle a,c\rangle$, $H_2:=\langle ab,c\rangle$, $H_3:=\langle a^p,b,c\rangle$,
$H_4:=\langle a^p,c\rangle$, $H_5:=\langle a\rangle$, $H_6:=\langle ab\rangle$, $H_7:=\langle a^p,b\rangle$, $H_8:=\langle a^p\rangle$ are subgroups
of $G$, where $\langle a, b\rangle=P$ and $\langle c\rangle=\mathbb Z_q$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7. Thus $\mathscr{I}(G)$ has $K_8$
as a subgraph, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P\cong M_{p^3}$, $p=3$, $5$,
then $G\cong M_{p^3}\rtimes \mathbb Z_q$. Here $H_1:=\langle a,c\rangle$, $H_2:=\langle ab,c\rangle$, $H_3:=\langle a^p,b,c\rangle$,
$H_4:=\langle a^2,c\rangle$, $H_5:=\langle a\rangle$, $H_6:=\langle ab\rangle$, $H_7:=\langle a^p,b\rangle$, $H_8:=\langle a^p\rangle$ are subgroups
of $G$, where $\langle a, b\rangle=P$ and $\langle c\rangle=\mathbb Z_q$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7. Therefore, $\mathscr{I}(G)$ has $K_8$
as a subgraph and so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\end{itemize}
Now, let us consider the case $p<q$ and $(p,q)\neq (2,3)$. $n_q(G)=p$ is not possible. If $n_q= p^2$, then $q|(p+1)(p-1)$, which implies that $q|(p+1)$ or
$q|(p-1)$. But this only leaves $p^3q-p^3(q-1)=p^3$ elements and our Sylow p-subgroup must be normal, a case we already considered. Therefore,
the only remaining possibility is that $G \cong \mathbb Z_q \rtimes P$. Suppose $\gamma(\mathscr{I}(P))>1$,
$\overline{\gamma}(\mathscr{I}(P))>1$ and
$\mathscr{I}(P)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(P)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite.
By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1}, \ref{intersecting graph t2} and Theorem~\ref{1000},
$P$ is isomorphic to one of $\mathbb Z_{p^3}$, $\mathbb Z_{p^2}\times \mathbb Z_p$ ($p=2$, $3$, $5$), $Q_8$, $M_8$ or $M_{p^3}$.
\begin{itemize}
\item If $P\cong \mathbb Z_{p^3}$, then $G \cong \mathbb Z_q \rtimes \mathbb Z_{p^3}=\langle a,b~|~a^q=b^3=1, bab^{-1}=a^i, ord_q(i)=p\rangle$.
Here $H_1:=\langle ab^p\rangle$, $H_2:=\langle b\rangle$, $H_3:=\langle ab\rangle$,
$H_4:=\langle a^2b\rangle$, $H_5:=\langle a^3b\rangle$, $H_6:=\langle ab^{p^2}\rangle$, $H_7:=\langle b^p\rangle$,
$H_8:=\langle b^{p^2}\rangle$ are proper subgroups
of $G$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, $7$. Thus $\mathscr{I}(G)$ has $K_8$
as a subgraph and so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P\cong \mathbb Z_{p^2}\times \mathbb Z_p$ or $M_{p^3}$, then $G\cong \mathbb Z_q\rtimes (\mathbb Z_{p^2}\times \mathbb Z_p)$ or $\mathbb Z_q\rtimes M_{p^3}$.
Here $H_1:=\langle a,c\rangle$, $H_2:=\langle ab,c\rangle$, $H_3:=\langle a^p,b,c\rangle$,
$H_4:=\langle a^p,c\rangle$, $H_5:=\langle a\rangle$, $H_6:=\langle ab\rangle$, $H_7:=\langle a^p,b\rangle$, $H_8:=\langle a^p\rangle$ are subgroups
of $G$, where $\langle a, b\rangle=P$ and $\langle c\rangle=\mathbb Z_q$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7. Thus $\mathscr{I}(G)$ has $K_8$
as a subgraph, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P \cong Q_8$, then $G\cong \mathbb Z_q\rtimes Q_8$. Here $H_1:=\langle a,c\rangle$, $H_2:=\langle b,c\rangle$, $H_3:=\langle ab,c\rangle$,
$H_4:=\langle a^2,c\rangle$, $H_5:=\langle a\rangle$, $H_6:=\langle b\rangle$, $H_7:=\langle ab\rangle$, $H_8:=\langle a^2\rangle$ are proper subgroups
of $G$, where $\langle a, b\rangle=P$ and $\langle c\rangle=\mathbb Z_q$. Also $H_8$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7. Thus these eight subgroups
forms $K_8$
as a subgraph of $\mathscr{I}(G)$ and this implies that $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P \cong M_8$. Here $\mathbb Z_q\rtimes M_8$,
$H_1:=\langle a,c\rangle$, $H_2:=\langle a^2,ab,c\rangle$, $H_3:=\langle a^2,b,c\rangle$,
$H_4:=\langle a^2,c\rangle$, $H_5:=\langle a\rangle$, $H_6:=\langle a^2,b\rangle$, $H_7:=\langle a^2,ab\rangle$, $H_8:=\langle a^2\rangle$ are proper subgroups
of $G$, where $\langle a, b\rangle=P$ and $\langle c\rangle=\mathbb Z_q$.
Also $H_8$ is a subgroup of $H_i$, $i=1$ to 7. Thus these eight subgroups form $K_8$
as a subgraph of $\mathscr{I}(G)$ and so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$.
\end{itemize}
If $(p,q)= (2, 3)$,
then $G \cong S_4$. In this case, $G$ has at least two copies of $D_8$. So by Figure~\ref{int f1}(c), $D_8$ together with its proper subgroups form $K_5$ as a subgraph of $\mathscr {I}(G)$. Then $\mathscr{I}(G)$ has a subgraph $\mathcal B_1$ as shown in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
Thus from the above arguments the result is true when $\alpha =3$.
Assume that the result is true for all non-abelian group of order $p^mq$, with $m< \alpha$. We prove the result when $\alpha >3$. If $n_p(G)=1$,
then $G\cong P \rtimes \mathbb Z_q$. Suppose $\gamma(\mathscr{I}(P))>1$,
$\overline{\gamma}(\mathscr{I}(P))>1$ and
$\mathscr{I}(P)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(P)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite.
By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1},
\ref{intersecting graph t2} and Theorem~\ref{1000}, $P \cong
\mathbb Z_{p^ \alpha}$ or $M_{2^4}$.
\begin{itemize}
\item If $P\cong \mathbb Z_{p^\alpha}$, then $G$ has a subgroup
$\langle a^p, b \rangle \cong \mathbb Z_{p^3} \rtimes \mathbb Z_q$. So by induction hypothesis
$\gamma(\mathscr{I}( \langle a^p, b \rangle))>1$, $\overline{\gamma}(\mathscr{I}( \langle a^p, b \rangle))>1$, and so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item If $P\cong M_{2^4}$, then by \eqref{e101},
$P$ together with its proper subgroups form $K_8$ as a subgraph of $\mathscr{I}(G)$ and so $\overline{\gamma}(\mathscr{I}(G))>1$ and $\gamma(\mathscr{I}(G))>1$.
\end{itemize}
Let $n_p(G) \neq 1$. Since $G$ is solvable, $G$ has a normal subgroup
$N$ of order $p^{\alpha -1}q$.
Suppose $\gamma(\mathscr{I}(N))>1$,
$\overline{\gamma}(\mathscr{I}(N))>1$ and
$\mathscr{I}(N)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(N)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite.
By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1},
\ref{intersecting graph t4} and Theorem~\ref{1000},
$N\cong \mathbb Z_{p^3q}$.
Suppose $G$ has two elements, say $a$, $b$ of orders $p$, $q$ respectively with $b\in N$, $a\notin N$, then
$H:=\langle a,b\rangle$ is a proper subgroup of $G$. Consider $N$ together with its proper subgroups
$H_1$, $H_2$, $H_3$, $H_4$, $H_5$, $H_6$, $H_7$, of order $p$, $p^2$, $p^3$, $pq$, $p^2q$, $p^3q$, $q$ respectively.
Here $H_i\cap H_j\neq \{e\}$, for every $i$, $j=1$ to $6$; $H_7$ is a subgroup of $H_i$, $i=4$, 5, 6,
$H$; $H\cap H_i=H_7$, $i=4$, 5, 6. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2},
so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
Suppose every subgroup of $G$ of order $p$ is contained in $N$, then $G$ has at least three Sylow $p$-subgroups, let them be $P_1$, $P_2$, $P_3$. Consider $N$ together with its proper subgroups
$H_1$, $H_2$, $H_3$, $H_4$, $H_5$, $H_6$ of order $p$, $p^2$, $p^3$, $pq$, $p^2q$, $p^3q$, respectively. Here $H_1$ is a subgroup of $H_i$, $i=2$, $\ldots$, 6, $N$, $P_1$, so they form $K_8$ as a subgraph of $\mathscr{I}(G)$. Therefore, $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
Thus the result is true when $\alpha>3$, it follows that the result is true for all $\alpha\geq 3$.
\end{proof}
\begin{pro}\label{intersecting graph t6}
If $G$ is a non-abelian group of order $p^2q^2$, where $p$, $q$ are distinct primes, then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $C_3$. Also $\mathscr{I}(G)$ is $K_5$-free if and only if $G \cong \langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=ab^{-1}, cbc^{-1}=ab^l\rangle$,
where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2~|~(p+1)$.
\end{pro}
\begin{proof}
We use the classification of groups of order $p^2q^2$ given in \cite{lin}.
Let $P$ and $Q$ denote a Sylow $p$, $q$-subgroups of $G$ respectively,
with out loss of generality, we assume that $p > q$. By Sylow's theorem, $n_p(G)=1, q, q^2$. But $n_p(G)= q$ is not possible,
since $p> q$. If $n_p(G)= q^2$, then $p~|~(q+1)(q-1)$, this implies that $p~|~(q+1)$, which is true only
when $(p, q)= (3, 2)$.
\noindent \textbf{Case 1:} $(p, q) \neq (3, 2)$. Then $G \cong P \rtimes Q$.
\noindent\textbf{Subcase 1a:} If $G \cong \mathbb Z_{p^2} \rtimes \mathbb Z_{q^2}= \langle a, b| a^{p^2}=b^{q^2}=1, bab^{-1}=a^i, i^{q^2} \equiv 1 (\mbox{mod}~ p^2) \rangle$, then we have
$H:=\langle a^p, b\rangle\cong \mathbb Z_p\rtimes \mathbb Z_{q^2}$, so by \eqref{e1}, $H$ together with its proper subgroups forms $K_8$ as a subgraph of
$\mathscr{I}(G)$. Hence $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Subcase 1b:} If $G \cong \mathbb Z_{p^2} \rtimes (\mathbb Z_q \times \mathbb Z_q)$, then $H_1:=\langle a \rangle=\mathbb Z_{p^2}$, $H_2:=\langle a^p \rangle$,
$H_3:=\langle a, c \rangle$, $H_4:=\langle a^p, b \rangle$, $H_5:=\langle a, b \rangle$, $H_6:=\langle a^p, c\rangle$,
$H_7:=\langle b,c\rangle=\mathbb Z_q\times \mathbb Z_q$,
$H_8:=\langle a^p,b,c\rangle$ are proper subgroups of $G$, where $\langle a\rangle=\mathbb Z_{p^2}$ and $\langle b,c\rangle=\mathbb Z_q\times \mathbb Z_q$.
Here $H_i$, $i=3$, $\ldots$, $8$ intersect with each other nontrivially;
$H_2$ is a subgroup of $H_1$, $H_3$, $H_4$;
$H_1\cap H_4=H_2$; $H_1$ is a subgroup of $H_3$.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\Gamma_(G))>1$.
\noindent \textbf{Subcase 1c:} If $G \cong (\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_{q^2}:=\langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=a^ib^j, cbc^{-1}=a^kb^l\rangle$,
where $\bigl(\begin{smallmatrix}
i & j\\ k & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2~|~(p+1)$.
Then we have two possibilities:
\begin{itemize}
\item Suppose $G$ has a subgroups of order $pq^2$ and $pq$, then $H_1:=\langle c \rangle=\mathbb Z_{q^2}$, $H_2:=\langle c^p \rangle$,
$H_3:=\langle a, c \rangle$, $H_4:=\langle b, c^p \rangle$, $H_5:=\langle b, c\rangle$, $H_6:=\langle a,c^p\rangle$,
$H_7:=\langle a,b\rangle=\mathbb Z_p\times \mathbb Z_p$,
$H_8:=\langle a,b,c^p\rangle$ are proper subgroups of $G$. Here $H_i$, $i=3$, $\ldots$, $8$ intersect with each other nontrivially;
$H_2$ is a subgroup of $H_1$, $H_3$, $H_4$;
$H_1$ is a subgroup of $H_3$; $H_1\cap H_4=H_2$.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$.
\item Suppose $G$ has a no subgroups of order $pq^2$ and $pq$, then by Theorem~\ref{1000}, $\mathscr{I}(G)$ is planar and by \eqref{33}, $\mathscr{I}(G)$ contains $C_3$.
\end{itemize}
\noindent \textbf{Subcase 1d:} If $G \cong (\mathbb Z_p \times \mathbb Z_p) \rtimes (\mathbb Z_q \times \mathbb Z_q)$, then $H_1:=\langle a,b \rangle=\mathbb Z_p\times \mathbb Z_p$,
$H_2:=\langle a,c\rangle$,
$H_3:=\langle a,d\rangle$, $H_4:=\langle b,c \rangle$, $H_5:=\langle a,b,c \rangle$, $H_6:=\langle a,b,d\rangle$, $H_7:=\langle a,c,d\rangle$,
$H_8:=\langle b,c,d\rangle$ are proper subgroups of $G$, where $\langle a,b\rangle:=\mathbb Z_p\times \mathbb Z_p$ and
$\langle c,d\rangle=\mathbb Z_q\times \mathbb Z_q$.
Here $H_i$, $i=1$, $\ldots$, $8$ intersect with each other nontrivially.
It follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph and hence $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent \textbf{Case 2:} $(p, q)= (3, 2)$. Up to isomorphism, there are nine groups of order 36. In the following we consider each of these groups.
\begin{enumerate}[{\normalfont (i)}]
\item $G\cong D_{18}$. Here $H_1:=\langle a\rangle$, $H_2:=\langle a^2\rangle$,
$H_3:=\langle a^6\rangle$, $H_4:=\langle a^6,b\rangle$, $H_5:=\langle a^6, ba\rangle$, $H_6:=\langle a^6, ba^2\rangle$, $H_7:=\langle a^6,ba^3\rangle$,
$H_8:=\langle a^6, ba^4\rangle$ are proper subgroups of $G$. Also $H_3$ is a subgroup of $H_i$, $i=1$, $\ldots$, $8$.
It follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong S_3\times S_3$. Here $H_1:=S_3\times \{e\}$, $H_2:=S_3\times \langle (123)\rangle$,
$H_3:=S_3\times \langle (12)\rangle$, $H_4:=S_3\times \langle (13)\rangle$, $H_5:=S_3\times \langle (23)\rangle$,
$H_6:=\langle (13)\rangle\times S_3$, $H_7:=\langle (123)\rangle\times S_3$,
$H_8:=\langle (12)\rangle\times S_3$ are proper subgroups of $G$.
Also $H_i$, $i=1$, $\ldots$, $8$ intersect with each other nontrivially, it follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph and so
$\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong \mathbb Z_3\times A_4$. Here $H_1:=\mathbb Z_3\times \{e\}$, $H_2:=\mathbb Z_3\times \langle (123)\rangle$,
$H_3:=\mathbb Z_3\times \langle (124)\rangle$, $H_4:=\mathbb Z_3\times \langle (134)\rangle$, $H_5:=\mathbb Z_3\times \langle (234)\rangle$,
$H_6:=\mathbb Z_3\times \langle (12)(34)\rangle$, $H_7:=\mathbb Z_3\times \langle (13)(24)\rangle$,
$H_8:=\mathbb Z_3\times \langle (14)(23)\rangle$ are proper subgroups of $G$.
Also $H_1$ is a subgroup of $H_i$, $i=2$, $\ldots$, $8$, it follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph and so $\gamma(\mathscr{I}(G))>1$
and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong \mathbb Z_6\times S_3$. Here $H_1:=\mathbb Z_6\times \langle (123)\rangle$, $H_2:=\mathbb Z_6\times \langle (12)\rangle$,
$H_3:=\mathbb Z_6\times \langle (13)\rangle$, $H_4:=\mathbb Z_6\times \langle (23)\rangle$, $H_5:=\mathbb Z_3\times \langle (123)\rangle$,
$H_6:=\mathbb Z_3\times \langle (12)\rangle$, $H_7:=\mathbb Z_3\times \langle (13)\rangle$,
$H_8:=\mathbb Z_3\times \langle (23)\rangle$ are proper subgroups of $G$.
Also $H_i$, $i=1$, $\ldots$, $8$ intersect with each other nontrivially, it follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph and hence
$\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong \mathbb Z_9\rtimes\mathbb Z_4=\langle a,b~|~a^9=b^4=1, bab^{-1}=a^i, i^4\equiv 1(\mbox{mod}~ 9)\rangle$. Here $H_1:=\langle a\rangle$,
$H_2:=\langle a, b^2\rangle$,
$H_3:=\langle a^3\rangle$, $H_4:=\langle a^3,b\rangle$, $H_5:=\langle a^3, b^2\rangle$, $H_6:=\langle a^3, ab^2\rangle$, $H_7:=\langle a^3, a^2b^2\rangle$,
$H_8:=\langle b\rangle$ are proper subgroups of $G$. Also $H_3$ is a subgroup of $H_i$, $i=1$, $\ldots$, 7; $H_8$ intersect with $H_2$, $H_4$, $H_5$ non-trivially. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong \mathbb Z_3\times(\mathbb Z_3\rtimes \mathbb Z_4)=\langle a,b,c~|~a^3=b^3=c^4=1, ab=ba, ac=ca, cbc^{-1}=b^i,
\mbox{ord}_2(i)=3\rangle$. Here $H:=\mathbb Z_3\rtimes \mathbb Z_4$
is a subgroup of $G$, and as in Case 1b, in the proof of Theorem~\ref{intersecting graph t4}, $H$ has a unique subgroup of order $2$, let it be $H_1$;
$H_i$, $i=2$, 3, 4 be subgroups of $H$ of order $4$; $H_5$ be a subgroup of $H$ of order $6$; it follows that $H$ together with its proper subgroups forms $K_6$
as a subgraph of $\mathscr{I}(H)$.
Moreover $H_6:=\mathbb Z_3\times H_1$ and
$H_7:=\mathbb Z_3\times H_2$ are subgroups of $G$. Thus $H_1$ is a subgroup of $H$, $H_i$, $i=1$, 2, 3, 4, 6, 7. It follows that $G$
contains $K_8$ as a subgraph, and so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong (\mathbb Z_3\times \mathbb Z_3)\rtimes \mathbb Z_4:=\langle a,b,c~|~a^3=b^3=c^4=1, ab=ba, cac^{-1}=a^ib^j, cbc^{-1}=a^kb^l\rangle$,
where $\bigl(\begin{smallmatrix}
i & j\\ k & l
\end{smallmatrix} \bigr)$ has order $4$ in $GL_2(3)$. We already discussed the intersection graph of subgroups of this group in Subcase 1c.
\item $G\cong \mathbb Z_2\times (\mathbb Z_3\times \mathbb Z_3)\rtimes \mathbb Z_2$. Here $H_1:=\langle a,b, c\rangle$,
$H_2:=\langle a, b,d\rangle$,
$H_3:=\langle a,c,d\rangle$, $H_4:=\langle b,c,d\rangle$, $H_5:=\langle a, b\rangle$, $H_6:=\langle a, c\rangle$, $H_7:=\langle a,d\rangle$,
$H_8:=\langle b,c\rangle$ are proper subgroups of $G$. Also these subgroups intersect with each other non-trivially.
It follows that $\mathscr{I}(G)$ contains $K_8$ as a subgraph, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item $G\cong (\mathbb Z_2\times \mathbb Z_2)\rtimes \mathbb Z_9$. Here $H_1:=\langle a,b \rangle$, $H_2:=\langle a,c\rangle$,
$H_3:=\langle a, c^3 \rangle$, $H_4:=\langle b,c\rangle$, $H_5:=\langle b,c^3\rangle$, $H_6:=\langle a,b,c^3\rangle$, $H_7:=\langle c\rangle$,
$H_8:=\langle c^3\rangle$ are proper subgroups of $G$. Also $H_i$, $i=1$, $\ldots$, $6$ intersect with each other nontrivially;
$H_7$ is a subgroup of $H_8$, $H_3$, $H_4$;
$H_1$ intersect with $H_3$, $H_4$ non-trivially.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2},
so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\end{enumerate}
The proof follows by combining all the above cases.
\end{proof}
\begin{pro}\label{intersecting graph t7}
If $G$ is a non-abelian group of order $p^{\alpha}q^{\beta}$, where $p$, $q$ are distinct primes and $\alpha$, $\beta \geq 2$,
then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
We prove the result by induction on
$\alpha + \beta$. If $\alpha+\beta=5$, then $|G|=p^3q^2$. Since $G$ is solvable,
it has a normal subgroup $N$ of prime index.
\noindent\textbf{Case 1:} If $[G:N]=p$, then $|N|=p^2q^2$. Suppose $\gamma(\mathscr{I}(N))>1$,
$\overline{\gamma}(\mathscr{I}(N))>1$ and
$\mathscr{I}(N)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(N)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite.
By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1}, \ref{intersecting graph t6} and Theorem~\ref{1000}, $N\cong \mathbb Z_{p^2q^2}$ or $\langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=b^{-1}, cbc^{-1}=a^1b^l\rangle$,
where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2~|~(p+1)$.
If $N\cong \mathbb Z_{p^2q^2}$, then $N$ together with its proper subgroups forms a subgraph in $\mathscr{I}(G)$, which is isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2} and so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$.
If $N$ is isomorphic to the second group, then $N$ has unique subgroups of order $p^2$ and $p^2q$, let them be $H_1$, $H_2$; also there are $p+1$ subgroups of order $p$, let them be $B_i$, $i=1$, 2, $\ldots$, $p+1$. Here $p>q$. Suppose $p\geq 5$, then $\mathscr{I}(G)$ contains $K_{4,5}$ as a subgraph with bipartition $X:=\{P$, $H_1$, $H_2$, $N\}$ and $Y:=\{B_1$, $B_2$, $B_3$, $B_4$, $B_5\}$, where $P$ is a Sylow $p$-subgroup of $G$ containing $H_1$ and $H_1\cong \mathbb Z_p\times \mathbb Z_p$. Here $B_1$ is a subgroup of $N$, $H_1$, $H_2$, $P$ and so they form $K_5$ as a subgraph of $\mathscr{I}(G)$. If $p=3$, then $P\cong \mathbb Z_{p^2}\times \mathbb Z_p$, $M_{p^3}$ or $(\mathbb Z_p\times \mathbb Z_p)\rtimes \mathbb Z_p$. If $P\cong \mathbb Z_{p^2}\times \mathbb Z_p$ or $M_{p^3}$, then by \eqref{e5}, \eqref{e100}, $H_1$, $H_2$ together with the proper subgroups of $H_1$ forms a subgraph of $\gamma(\mathscr{I}(G))$, which is isomorphic to $\mathcal A_1$ shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$. If $P\cong (\mathbb Z_p\times \mathbb Z_p)\rtimes \mathbb Z_p$, then by Proposition~\ref{intersecting graph t2}, $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case 2:} If $[G:N]=q$, then $|N|=p^3q$. Suppose $\gamma(\mathscr{I}(N))>1$,
$\overline{\gamma}(\mathscr{I}(N))>1$ and
$\mathscr{I}(N)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(P)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1},
\ref{intersecting graph t5} and Theorem~\ref{1000}, $N\cong \mathbb Z_{p^3q}$. Let $H_i$, $i=1$, 2, 3, 4, 5, 6 be subgroups of $N$ of order $p$, $p^2$, $p^3$, $pq$, $p^2q$, $q$ respectively. Let $H$ be a subgroup of $G$ of order $q^2$ such that $H$ contains $H_6$. By \eqref{201}, $N$ together with its proper subgroups and $H$ forms a subgraph in $\mathscr{I}(G)$, which is isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
Now we assume that the
result is true for all non-abelian groups of order $p^mq^n$, where $m+n < \alpha + \beta$ ($m+n\geq 5$, $m,n\geq 2$). We prove the result when $\alpha + \beta >5$.
Since $G$ is solvable, $G$ has a subgroup $H$ of prime index, with out loss of generality, say $q$, and so $|H|=p^{\alpha}q^{\beta -1}$. If $H$ is
cyclic, then $H$ together with its proper subgroups forms $K_6$ as a subgraph of $\mathscr{I}(G)$. Now let $K$ be the subgroup of $H$ of order $q^{\beta-1}$
and let $Q$ be a $q$-Sylow subgroup of $G$ containing $K$. Also let $H_i$, $i=1$, 2, 3 be subgroups of $H$ of order $pq$, $p^2q$, $p^3q$ respectively.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
If $H$ is non-cyclic
abelian, then by Proposition~\ref{intersecting graph t1}, $\gamma(\mathscr{I}(H))>1$, $\overline{\gamma}(\mathscr{I}(H))>1$
and $\mathscr{I}(H)$ contains $K_5$ as a subgraph, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$
and $\mathscr{I}(G)$ contains $K_5$.
If $H$ is non-abelian, then we have the following cases to consider:
\noindent\textbf{Case a:} If $\beta = 2$, then $\alpha > 3$. So by Proposition~\ref{intersecting graph t2}, $\gamma(\mathscr{I}(H))>1$, $\overline{\gamma}(\mathscr{I}(H))>1$
and $\mathscr{I}(H)$ contains $K_5$ as a subgraph.
\noindent\textbf{Case b:} If $\beta > 2$, then by induction hypothesis, $\gamma(\mathscr{I}(H))>1$, $\overline{\gamma}(\mathscr{I}(H))>1$
and $\mathscr{I}(H)$ contains $K_5$ as a subgraph.
\noindent\textbf{Case c:} If $\alpha = 2$, then $\beta > 2$. By Case b, $\gamma(\mathscr{I}(H))>1$, $\overline{\gamma}(\mathscr{I}(H))>1$
and $\mathscr{I}(H)$ contains $K_5$ as a subgraph.
\noindent\textbf{Case d:} If $\alpha >2$, then by induction hypothesis, $\gamma(\mathscr{I}(H))>1$, $\overline{\gamma}(\mathscr{I}(H))>1$
and $\mathscr{I}(H)$ contains $K_5$ as a subgraph.
It follows that $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph.
\end{proof}
\begin{pro}\label{intersecting graph t8}
Let $G$ be a non-abelian solvable group of order $pqr$, where $p$, $q$, $r$ are distinct primes, $p>q>r$. Then
\begin{enumerate}[\normalfont (1)]
\item $\mathscr{I}(G)$ contains $C_3$;
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G \cong \langle a,b,c~|~a^p=b^q=c^r=1, b^{-1}ab=a^\mu, c^{-1}ac=a^v, bc=cb\rangle$, where $r$, $q$ are divisor of $(p-1)$ and $v$, $\mu\neq 1$;
\item $\mathscr{I}(G)$ is non-toroidal and non-projective-planar;
\end{enumerate}
\end{pro}
\begin{proof}
\noindent By~\cite[p. 215]{cole},
up to isomorphism there are four groups of order $pqr$. In the following we deal with each of these groups.
\noindent\textbf{Case a:} If $r\nmid (p-1)$ and $q\nmid (p-1)$, then $G\cong \mathbb Z_{pqr}$, which is not possible.
\noindent\textbf{Case b:} If $r |(p-1)$, $r \nmid (q-1)$ and $q\nmid (p-1)$, then $G\cong\langle a,b,c~|~a^p=b^q=c^r=1, ab=ba, ac=ca, c^{-1}bc=b^v\rangle$, $v\neq 1$.
Here $H_1:=\langle a\rangle$, $H_2:=\langle b\rangle$, $H_3:=\langle a,b\rangle$, $H_4:=\langle b,c\rangle$,
$I_1:=\langle c\rangle$, $I_2:=\langle bc\rangle$, $I_3:=\langle b^2c\rangle$,$\ldots$, $I_q:=\langle b^{q-1}c\rangle$, $B_1:=\langle a,c\rangle$,
$B_2:=\langle a,bc\rangle$, $\ldots$, $B_q:=\langle a,b^{q-1}c\rangle$ are the only subgroups of $G$.
If $q=3$, then $r=2$, and so $r$ divides $(q-1)$, which is not possible.
If $q\geq 5$, then $B_i$, $i=1 \ldots , 5$, $H_1$ intersect with each other non-trivially; $H_2$ is a subgroup of $H_3$, $H_4$;
$H_3\cap B_i=H_1$. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so
$\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case c:} If $r |(p-1)$, $r \nmid (q-1)$ and $q\nmid (p-1)$, then the group is different from the group given in Case b only in the exchange the roles of $a$, $b$, so $G\cong\langle a,b,c~|~a^p=b^q=c^r=1, ab=ba, bc=cb, c^{-1}ac=a^\mu\rangle$, $\mu\neq 1$.
Here $H_1:=\langle a\rangle$, $H_2:=\langle b\rangle$, $H_3:=\langle a,b\rangle$, $H_4:=\langle a,c\rangle$,
$I_1:=\langle c\rangle$, $I_2:=\langle ac\rangle$, $I_3:=\langle a^2c\rangle$,$\ldots$, $I_q:=\langle a^{q-1}c\rangle$, $B_1:=\langle a,c\rangle$,
$B_2:=\langle b,ac\rangle$, $\ldots$, $B_q:=\langle b,a^{q-1}c\rangle$ are the only subgroups of $G$.
Here also $q=3$ not possible.
If $q\geq 5$, then $B_i$, $i=1$ to 5, $H_1$ intersect with each other non-trivially; $H_2$ is a subgroup of $H_3$, $H_4$;
$H_3\cap B_i=H_1$. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case d:} If $r|(p-1)$, $r | (q-1)$ and $q\nmid (p-1)$, then
$G\cong \langle a,b,c~|~a^p=b^q=c^r=1, ab=ba, c^{-1}ac=a^\mu, c^{-1}bc=b^v\rangle$, $v$, $\mu\neq 1$.
Here $H_1:=\langle a\rangle$, $H_2:=\langle a,b\rangle$,
$H_3:=\langle a,c\rangle$, $H_4:=\langle a,bc\rangle$, $H_5:=\langle a, b^2c\rangle$, $H_6:=\langle b,c\rangle$, $H_7:=\langle b,ac\rangle$,
$H_8:=\langle b,a^2c\rangle$, $H_9:=\langle b,a^4c\rangle$, $H_{10}:=\langle b,a^{p-1}c\rangle$ are proper subgroups of $G$.
Also $H_1$ is a subgroup of $H_i$, $i=2$, 3, 4, 5
and so they form $K_5$ as a subgraph of $\mathscr{I}(G)$. $\langle b\rangle$ is a subgroup of $H_i$, $i=6$, 7, 8, 9, 10, so they form $K_5$ as a subgraph
of $\mathscr{I}(G)$. It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal B_1$ as in Figure~\ref{fig:int f3}, so $\gamma(\mathscr{I}(G))>1$ and
$\overline{\gamma}(\mathscr{I}(G))>1$.
\noindent\textbf{Case e:} If $q|(p-1)$, then we have the group is of essentially the same form as Case 1b and Case 1c. The elements c, a, b
here playing the same role as a, b, c in Case 1b. The toroidality and the projective-planarity of the intersection graphs of subgroups of this group are as described in Case 1b and Case 1c.
\noindent\textbf{Case f:} If $r |(p-1)$ and $q|(p-1)$, then
$G\cong \langle a,b,c~|~a^p=b^q=c^r=1, b^{-1}ab=a^\mu, c^{-1}ac=a^v, bc=cb\rangle$, $v$, $\mu\neq 1$. By Theorem~\ref{1000}, $\mathscr{I}(G)$ is planar and by Figure~\ref{int f1}(d), it contains $C_3$.
Combining together all the cases, the proof follows.
\end{proof}
\begin{pro}\label{intersecting graph t10}
Let $G$ be a non-abelian solvable group of order $p^2qr$, where $p$, $q$, $r$ are distinct primes. Then
\begin{enumerate}[\normalfont (1)]
\item $\mathscr{I}(G)$ contains $C_3$;
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G \cong \langle a, b, c~|~a^p=b^p=c^{qr}=1, ab=ba, cac^{-1}=b, cbc^{-1}=ab^l\rangle$, where $l$ is any integer with $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $qr$ in $GL_2(p)$, $qr|(p+1)$.
\item $\mathscr{I}(G)$ is toroidal if and only if $G \cong \langle a, b, c~|~a^5=b^5=c^6=1, ab=ba, cac^{-1}=b, cbc^{-1}=ab^l\rangle$, where $l$ is any integer with $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $6$ in $GL_2(5)$.
\item $\mathscr{I}(G)$ is non projective-planar.
\end{enumerate}
\end{pro}
\begin{proof}
\noindent Since $G$ is solvable and it has a Sylow basis $\{P,Q,R\}$, where $P$, $Q$, $R$ denotes the Sylow p,q,r-subgroups of $G$ respectively. Then $PQ$, $PR$, $QR$ are subgroups of $G$ of order $p^2q$, $p^2r$, $qr$ respectively.
\noindent \textbf{Case 1:} A subgroup of $G$ of index $r$ is normal in $G$:
With out loss of generality, we may assume that $PQ$ is normal in $G$.
\noindent \textbf{Subcase 1a:} Suppose $P$ is normal in $PQ$, then by Theorem~\ref{501}, $P$ is normal in $G$ . Then two possibilities arise:
\begin{enumerate}[\normalfont (i)]
\item $P\cong \mathbb Z_{p^2}$: Let $H_1$ be the subgroup of $P$ of order $p$. Then $H_1$ is also normal in $G$. Here $H_1R$, $H_1Q$, $H_1QR$ are subgroups of $G$. Also $H_1$ is a subgroup of $H_1Q$, $H_1R$, $P$, $PQ$, $PR$; $QR$ intersect with $PQ$, $PR$, $H_1Q$, $H_1QR$ non-trivially. It follows that $\mathscr{I}(G)$ has a subgraph isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\Gamma(G))>1$.
\item $P\cong \mathbb Z_p\times \mathbb Z_p$: Then $P$ has atleast three subgroups of order $p$, let them be $H_i$, $i= 1, 2, 3$. Suppose $\gamma(\mathscr{I}(PQ))>1$,
$\overline{\gamma}(\mathscr{I}(PQ))>1$ and
$\mathscr{I}(PQ)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(PQ)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t1} and \ref{intersecting graph t4}, the only possibilities for $PQ$ such that $\mathscr{I}(PQ)$ satisfying the above properties are $\mathbb Z_{pq}\times \mathbb Z_p$ or $\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^{-1}b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q~|~(p+1)$. By a similar argument, we can show that the only possibilities for $PR$ are $\mathbb Z_{pr}\times \mathbb Z_p$ or $\langle a, b, c~|~ a^p=b^p=c^r=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^{-1}b^{k} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & k
\end{smallmatrix} \bigr)$ has order $r$ in $GL_2(p)$, $q~|~(p+1)$.
\begin{itemize}
\item Suppose either $PQ$ or $PR$ is isomorphic to the respective first group, with out loss of generality, let us take $PQ$. Then $PQ$ has at least three subgroups $H_iQ$, $i=1$, 2, 3 of order $pq$. Here $PQ$, $PR$, $QR$, $H_1Q$, $H_2Q$, $H_3Q$ intersect with each other non-trivially; $H_1$ is a subgroup of $PQ$, $PR$, $H_1Q$, $P$. It follows that these eight subgroups forms a subgraph of $\mathscr{I}(G)$, which is isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}. Therefore, $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\item Now assume that $PQ$ and $PR$ are isomorphic to the respective second group.
\hspace{.7cm} Suppose that $PQ$ is the only subgroup of order $p^2q$, but the subgroup of order $p^2r$ is not unique. If $n_r(G)=1$, then $R$ is normal in $G$. Then $H_1Q$, $H_1R$, $H_1QR$ are subgroups of $G$ of order $pq$, $pr$, $pqr$ respectively, where $H_1$ is subgroup of $P$ of order $p$. Here $PQ$, $PR$, $P$, $H_1Q$, $H_1R$, $H_1QR$ intersect non-trivially; $Q$ is a subgroup of $PQ$, $H_1Q$, $H_1QR$, $QR$. It follows that $\mathscr{I}(G)$ has a subgraph isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}. If $n_r(G)\neq 1$, then by Sylow's theorem $n_r(G)= p$, $q$, $pq$, $p^2$ or $p^2q$. Here $n_r(G)=p$, $q$, $pq$ are not possible, since $G$ has at least $2p^2$ subgroups of order $r$. Assume that $n_r(G)=p^2q$. Then for $q\geq 5$, $G$ has at least four subgroups of order $p^2r$, let them be $PR_1$, $PR_2$, $PR_3$, $PR_4$. It follows that $\mathscr{I}(G)$ contains $K_{5,4}$ as a subgraph with bipartition $X:=\{PQ$, $PR_1$, $PR_2$, $PR_3$, $PR_4\}$ and $Y:=\{P$, $H_1$, $H_2$, $H_3\}$, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$, where $H_i$, $i=2$, 3 are subgroups of $P$ of order $p$. Also $PQ$, $P$, $H_1$, $PR_1$, $PR_2$ intersect non-trivially and so they form $K_5$ as a subgraph of $\mathscr{I}(G)$. If $p>2$, $q=3$, then $G$ has at least three subgroups of order $p^2r$ and so $\mathscr{I}(G)$ contains $K_{5,4}$ as a subgraph with bipartition $X:=\{PQ$, $PR_1$, $PR_2$, $PR_3$, $P\}$ and $Y:=\{H_1$, $H_2$, $H_3$, $H_4\}$, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$, where $H_4$ is a subgroup of $P$ of order $p$. Also $PQ$, $P$, $H_1$, $PR_1$, $PR_2$ intersect non-trivially and so they form $K_5$ as a subgraph of $\mathscr{I}(G)$. If $p=2$, $q=3$, then $n_r(G)=12$, which implies $r=11$. Thus we have $n_p(G)=1$, $n_q(G)=4$, $n_r(G)=12$, and this implies that $G$ has subgroup $QR$ of order $33$. Suppose $QR$ is cyclic, then $n_q(G)=n_r(G)$, which is not possible. Suppose $QR$ is non-cyclic, then $G$ has at least 11 subgroups of order $3$, which is also not possible. If $q=2$, $p\geq 5$, then $\mathscr{I}(G)$ contains $K_{5,4}$ as a subgraph with bipartition $X:=\{PQ$, $PR_1$, $PR_2$, $P\}$ and $Y:=\{H_1$, $H_2$, $H_3$, $H_4$, $H_5\}$, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$, where $H_i$, $i=2$, 3 of $P$ of order $p$. Also $PQ$, $P$, $H_1$, $PR_1$, $PR_2$ intersect non-trivially and so they form $K_5$ as a subgraph of $\mathscr{I}(G)$. If $q=2$, $p=3$, then $n_r(G)=18$, which implies $r=17$. Here $n_r(G)=18$, $n_q(G)=9$ and by a similar argument as above, we can show that such a group does not exist. If $n_r(G)=p^2$, then $G$ has a unique subgroup of order $p^2r$, which is a contradiction to our assumption.
\hspace{.7cm} Suppose the subgroup of order $p^2q$ and $p^2r$ in $G$ are not unique, then we can use the previous argument to show $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\hspace{.7cm} Suppose the subgroup of order $p^2q$ and $p^2r$ in $G$ are unique, then the subgroup structure of $G$ is as follows: $G$ has unique subgroup of order $p$; $G$ has $p+1$ subgroups of order $p$, let them be $H_i$, $i=1$, 2, $\ldots$, $p+1$.
Now $PQ$ has $p^2$ subgroups of order $q$, say $Q_i$, $i=1$, $\ldots$, $p^2$ and $PR$ has $p^2$ subgroups of order $r$, say $R_i$, $i=1$, $\ldots$, $p^2$. Now we show that the subgroup of order $qr$ in $G$ is not unique. If it is unique, then this subgroup is isomorphic to one of $\mathbb Z_{qr}$, $\mathbb Z_q\rtimes \mathbb Z_r$ or $\mathbb Z_r\rtimes \mathbb Z_q$. But in either case, $G$ has a unique subgroup of order either $q$ or $r$, which is not possible.
Note that the normalizer in $G$ of $Q$, $N_G(Q)$ is a subgroup of $G$ of order $qr$. Similarly consider $R$, then $N_G(R)$
is a subgroup of G of order $qr$. If $N_G(Q)$ is non-abelian, then necessarily $r< q$. Likewise if $N_G(R)$ is non-abelian, then necessarily $q< r$. As these inequalities cannot both be true, it follows that $G$ has an abelian (and hence cyclic) subgroup of order $qr$, and we may choose $Q$ and $R$ so that $N_G(Q)=N_G(R)= Q\times R$ is this cyclic subgroup. It follows that $G$ has $p^2$ subgroups of order $qr$, let them be $L_i$, $i=1, \ldots , p^2$. This completes the subgroup structure of $G$.
By the presentation of the subgroup of order $p^2 q$, an element $c$ of order $q$ acts on the $p$-Sylow subgroup $P$
via a matrix of determinant 1 which is not $- 1$. Hence $q$ is not 2. Likewise $r$ is not 2. So, $Q \times R$ is uniquely determined up to conjugacy in $GL_2(p)$. Hence, $G$ is uniquely determined up to isomorphism.
\hspace{.7cm} A presentation of this group $G$ is $\langle a, b, c~|~a^p=b^p=c^{qr}=1, ab=ba, cac^{-1}=b, cbc^{-1}=ab^l\rangle$, where $l$ is an integer with $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $qr$ in $GL_2(p)$, $qr~|~(p+1)$.
The structure of $\mathscr{I}(G)$ is shown in Figure~\ref{fig:int f111}. Note that $\mathscr{I}(G)$ is $K_5$-free. If $p\geq 7$, then $\mathscr{I}(G)$ contains $K_{3,7}$ as a subgraph with bipartition $X:=\{PQ, PR, P\}$ and $Y:=\{H_1, H_2, H_3, H_4, H_5, H_6, H_7\}$, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$. If $p = 5$, then $\mathscr{I}(G)$ is toroidal and the corresponding toroidal embedding is shown in Figure~\ref{fig:int f11}.
Moreover, $\mathscr{I}(G)$ contains $K_{3,5}$ as a subgraph with bipartition $X:=\{PQ, PR, P\}$ and $Y:=\{H_1, H_2, H_3, H_4, H_5\}$, so $\overline{\gamma}(\mathscr{I}(G))>1$.
Since $qr | (p+1)$, so $p\leq 3$ is not possible.
\begin{figure}
\caption{The structure of $\mathscr{I}
\label{fig:int f111}
\end{figure}
\begin{figure}
\caption{An embedding of $\mathscr{I}
\label{fig:int f11}
\end{figure}
\end{itemize}
\end{enumerate}
\noindent \textbf{Subcase 1b:} Suppose $P$ is non-normal in $PQ$, then $Q$ is normal in $PQ$. So, by Theorem~\ref{501}(iii), $Q$ is normal in $G$. Then $H_1Q$, $H_1R$, $H_1QR$ are subgroups of $G$ of order $pq$, $pr$, $pqr$ respectively, where $H_1$ is subgroup of $P$ of order $p$. Here $PQ$, $PR$, $P$, $H_1Q$, $H_1R$, $H_1QR$ intersect non-trivially; $Q$ is a subgroup of $PQ$, $H_1Q$, $H_1QR$, $QR$. It follows that $\mathscr{I}(G)$ has a subgraph isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}.
\noindent \textbf{Case 2:} A subgroup of $G$ of index $q$ is normal in $G$: With out loss of generality, we may assume that, $PR$ is normal in $G$. Here we can use a similar argument as in Subcase 1a, by taking $R$ instead of $Q$ and end up with the same group which we have obtained in Subcase 1a, whose intersection graph is toroidal, non-projective-planar and $K_5$-free.
\noindent\textbf{Case 3:} A subgroup of $G$ of index $p$ is normal in $G$: With out loss of generality, we may assume that, $H_1QR$ is normal in $G$, where $H_1$ is a subgroup of $P$ of order $p$. Then by Theorem~\ref{501}(iii), $H_1$ is normal in $G$. So, $H_1Q$, $H_1R$, $H_1QR$ are subgroups of $G$. Here $PQ$, $PR$, $P$, $H_1Q$, $H_1R$, $H_1QR$ intersect non-trivially; $Q$ is a subgroup of $PQ$, $H_1Q$, $H_1QR$, $QR$. It follows that $\mathscr{I}(G)$ has a subgraph isomorphic to $\mathcal A_1$ as shown in Figure~\ref{fig:f2}.
Putting together all the cases, the result follows.
\end{proof}
\begin{pro}\label{intersection graph t11}
Let $G$ be a non-abelian solvable group of order $p^3qr$, where $p$, $q$, $r$ are distinct primes. Then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Since $G$ is solvable, so it has a Sylow basis $ \{P , Q, R \}$, where $p, Q, R$ are Sylow $p,q,r$-subgroup of $G$ respectively. Then $PQ$, $PR$ are subgroups of $G$. Suppose $\gamma(\mathscr{I}(PQ))>1$,
$\overline{\gamma}(\mathscr{I}(PQ))>1$ and
$\mathscr{I}(PQ)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(PQ)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000},
Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1} and \ref{intersecting graph t5}, $PQ\cong \mathbb Z_{p^3q}$. By a similar argument, we can show that the only possibility of $PR$ is $\mathbb Z_{p^2r}$.
Let $H_1$, $H_2$, $H_3$, $H_4$, $H_5$ be subgroups of $PQ$ of orders $p$, $p^2$, $p^3$, $pq$, $p^2q$ respectively; $H$ be a subgroup of $PR$ of order $p^2r$. Here $H_1$ is a
subgroup of $H$, $PQ$, $PR$, $H_i$, $i=2$ to 5. It follows that, these five subgroups intersect with each other non-trivially and so $\mathscr{I}(G)$ contains
$K_8$ as a subgraph. Hence, $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
\end {proof}
\begin{pro}\label{intersection graph t12}
Let $G$ be a non-abelian solvable group of order $p^2q^2r$, where $p$, $q$, $r$ are distinct primes. Then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Since $G$ is a solvable group, it has subgroups of order $p^2r$ and $q^2r$. Now we split the proof in to the following cases:\\
\noindent \textbf{Case 1:} A subgroup of $G$ of order either $p^2r$ or $q^2r$ is unique: Then $G$ has a subgroup of order either $p^2qr$ or $pq^2r$, let it be $H$, and without loss of generality, we assume that $|H|=p^2qr$. Suppose $\gamma(\mathscr{I}(H))>1$,
$\overline{\gamma}(\mathscr{I}(H))>1$ and
$\mathscr{I}(H)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(H)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1} and \ref{intersecting graph t10}, $H\cong \langle a, b, c~|~ a^p=b^p=c^{qr}=1, ab=ba, cac^{-1}=b, cbc^{-1}=ab^l\rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $qr$ in $GL_2(p)$, $qr|(p+1)$. So by Figure~\ref{fig:int f111}, $H$ together with its proper subgroups forms $K_{3,7}$ as a subgraph of $\mathscr{I}(G)$ and it contains $K_5$ as a subgraph.
\noindent \textbf{Case 2:} Subgroups of $G$ of order $p^2r$, $q^2r$ are not unique:\\
Since $G$ is a solvable group, so it has a subgroup, say $N$ of prime index. Now we need to consider the following subcases:
\noindent \textbf{Subcase 2a:} Let $[G:N]=p$ or $q$, without loss of generality, we assume that $[G:N]=q$, and so $|N|=p^2qr$. Then by the argument mentioned in Case 1, $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph.
\noindent \textbf{Subcase 2b:} Let $[G:N]=r$. Then $|N|=p^2q^2$. Suppose $\gamma(\mathscr{I}(N))>1$,
$\overline{\gamma}(\mathscr{I}(N))>1$ and
$\mathscr{I}(N)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(N)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1} and \ref{intersecting graph t6}, $N\cong \mathbb Z_{p^2q^2}$ or $\langle a, b, c~|~ a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=b, cbc^{-1}=ab^l\rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2|(p+1)$. If $N\cong \mathbb Z_{p^2q^2}$, then $N$ together with its proper subgroups forms a subgraph isomorphic to $\mathcal A_1$ shown in Figure~\ref{fig:f2}. Therefore, $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
If $N$ isomorphic to the second group, then it has a subgroup of order $p^2q$, let it be $I$.
Suppose $G$ has unique subgroup of order $p^2r$, then $G$ has a subgroup of order $p^2qr$, let it be $H$. Suppose $\gamma(\mathscr{I}(H))>1$,
$\overline{\gamma}(\mathscr{I}(H))>1$ and
$\mathscr{I}(H)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(H)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1} and \ref{intersecting graph t10}, $H\cong \langle a, b, c~|~a^p=b^p=c^{qr}=1, ab=ba, cac^{-1}=b^{-1}, cbc^{-1}=ab^l\rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $qr$ in $GL_2(p)$, $qr|(p+1)$. By Figure~\ref{fig:int f111}, $H$ together with its subgroups forms $K_{3,7}$ as a subgraph of $\mathscr{I}(G)$ and it contains $K_5$. Suppose the subgroup of order $p^2r$ is not unique, then $G$ has at least two subgroups of order $p^2r$, let them be $PR_1$, $PR_2$. Let $L_1$ be a subgroup of $Q$ of order $q$. Here $H_1$ is a subgroup of $P$, $PR_1$, $PR_2$, $N$; $L_1$ is a subgroup of $Q$, $N$, $I$, $QR$. It follows that $\mathscr{I}(G)$ has a subgraph isomorphic to $\mathcal B_2$ as shown in Figure~\ref{fig:f2}. Therefore, $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph.
Proof follows by combining all the cases together.
\end{proof}
\begin{pro}\label{intersection graph t13}
Let $G$ be a non-abelian solvable group of order $p^\alpha q^\beta r^\delta$, where $p$, $q$, $r$ are distinct primes and $\alpha+\beta+\delta=6$. Then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Here $|G|=p^4qr$, $p^3q^2r$ or $p^2q^2r^2$. First let us assume that $|G|$ is either $p^4qr$ or $p^3q^2r$. Since $G$ is solvable, it has a subgroup of order either $p^4q$ or $p^3q^2$ respectively, let it be $H$. Suppose $\gamma(\mathscr{I}(H))>1$,
$\overline{\gamma}(\mathscr{I}(H))>1$ and
$\mathscr{I}(H)$ contains $K_5$,
then the same holds for $\mathscr{I}(G)$ also. So it is enough to consider the cases when $\mathscr{I}(H)$ is one of planar, toroidal, projective-planar,
$K_5$-free, $C_3$-free and bipartite. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1}, \ref{intersecting graph t5} and \ref{intersecting graph t7}, we have $H\cong \mathbb Z_{p^4q}$. Here $H$ together with its proper subgroups forms $K_8$, so $\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
If $|G|=p^2q^2r^2$, then $G$ has a normal subgroup $N$ with prime index, without loss of generality, say $r$. Since $|N|=p^2q^2r$, and so by Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100}, \ref{intersecting graph t1} and \ref{intersection graph t12}, $\gamma(\mathscr{I}(N))>1$, $\overline{\gamma}(\mathscr{I}(N))>1$ and $\mathscr{I}(N)$ contains $K_5$. Therefore, the same holds for $\mathscr{I}(G)$ also.
Hence $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph. This completes the proof.
\end{proof}
\begin{pro}\label{intersection graph t14}
Let $G$ be a non-abelian solvable group of order $p^\alpha q^\beta r^\delta$, where $p$, $q$, $r$ are distinct primes and $\alpha+\beta+\delta\geq 7$. Then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Without loss of generality, we assume that $\alpha\geq \beta\geq \delta$. Since $G$ is solvable, $G$ has a Sylow basis
containing $P$, $Q$, $R$, where $P$, $Q$, $R$ are Sylow $p,q,r$-subgroups of $G$ respectively and so $PQ$ is a subgroup of $G$. By Theorem~\ref{1000}, Propositions~\ref{intersecting graph t100},~\ref{intersecting graph t1},~\ref{intersecting graph t5} and \ref{intersecting graph t7},
$\gamma(\mathscr{I}(PQ))>1$, $\overline{\gamma}(\mathscr{I}(PQ))>1$, $\mathscr{I}(PQ)$ contains $K_5$, and so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$ as a subgraph.
\end{proof}
\begin{pro}\label{intersecting graph t9}
If $G$ is a solvable group, whose order has more than three distinct prime factors, then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof}
Since $G$ is solvable, it has a Sylow basis containing $P$, $Q$, $R$, $S$, where $P$, $Q$, $R$, $S$ are Sylow $p$, $q$, $r$, $s$-subgroups of $G$ respectively. Then $P$, $PQ$,
$PR$, $PS$, $PQR$, $PQS$, $PRS$, $QRS$ are subgroups of $G$. Here $P$ is a subgroup of $PQ$,
$PR$, $PS$, $PQR$, $PQS$, $PRS$; $QRS$, $PR$, $PS$, $PQR$, $PQS$, $PRS$, $QRS$ intersect with each other nontrivially.
It follows that $\mathscr{I}(G)$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{proof}
\subsection{Finite non-solvable groups}\label{sec:4b}
If $G$ is a group and $N$ is a normal subgroups of $G$, then $\mathscr{I}(G/N)$ is isomorphic (as a graph) to a subgraph of $\mathscr{I}(G)$.
It is well known that any non-solvable group has a simple group as a sub-quotient and every simple group has a minimal simple group as a sub-quotient.
Therefore, if we can show that the minimal simple groups have non-toroidal (non-projective-planar) intersection graphs,
then the intersection graph of a non-solvable group is non-toroidal (resp., non-projective-planar).
Recall that $SL_m(n)$ is the group of $m \times m$ matrices having determinant 1, whose entries are lie in a field with $n$ elements and that
$L_m(n)=SL_m(n)/ H$, where $H=\{kI| k^m=1 \}$. For any prime $q > 3$, the Suzuki group is denoted by $Sz(2^q)$
\begin{lemma}\label{intersecting graph l1}
If $n>2$, then $\gamma(\mathscr{I}(D_{4n}))>1$, $\overline{\gamma}(\mathscr{I}(D_{4n}))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{lemma}
\begin{proof}
Here $H_1:=\langle a \rangle$, $H_2:=\langle a^n, b\rangle$, $H_3:=\langle a^n, ba\rangle$, $H_4:=\langle a^n, ba^2\rangle$, $H_5:=\langle a^n\rangle$,
$H_6:=\langle a^{2n},b\rangle$, $H_7:=\langle a^{2n},ba\rangle$, $H_8:=\langle a^{2n},ba^2\rangle$, $H_9:=\langle a^{2n},ba^3\rangle$,
$H_{10}:=\langle a^{2n}\rangle$ are proper subgroups of $D_{4n}$. Also $H_5$ is a subgroup of $H_i$, $i=1$, 2, 3, 4; $H_{10}$ is a subgroup of $H_i$,
$i=6$, 7, 8, 9. It follows that $\mathscr{I}(D_{4n})$ has a subgraph $\mathcal A_1$ as shown in Figure~\ref{fig:f2}, so $\gamma(\mathscr{I}(D_{4n}))>1$,
$\overline{\gamma}(\mathscr{I}(D_{4n}))>1$.
\end{proof}
\begin{pro}\label{intersecting graph t11}
If $G$ is a finite non-solvable group, then $\gamma(\mathscr{I}(G))>1$, $\overline{\gamma}(\mathscr{I}(G))>1$ and $\mathscr{I}(G)$ contains $K_5$.
\end{pro}
\begin{proof} The classification of minimal simple groups is given in \cite[Corollary 1]{thomp}.
As mentioned above, to prove the result, it is enough to show that these minimal simple groups are non-toroidal, not projective-planar and contains $K_5$.
Now we investigate each of these groups. Her we will denote the image of a matrix $A$ in $L_m(n)$ by $\overline{A}$.
\noindent\textbf{Case 1:} $G \cong L_2(q^p)$, where $q=2,3$ and $p$ is any prime.
If $p=2$, then the only non-solvable group is $ L_2(4)$. Also $ L_2(4) \cong A_5$ (see \cite{atlas}). $A_5$ has five copies of $A_4$ and any two $A_4$ in $A_5$ have
non-trivial intersection, for otherwise $|A_4A_4|=144$, which is not possible. Also $H_1:=\langle (12, 34)\rangle$, $H_2:=\langle (12,34), (13, 24)\rangle$,
$H_3:=\langle (12,34), (12354)\rangle$, $H_4:=\langle (12,34), (12453)\rangle$, $H_5:=\langle (12,34),(345)\rangle$ are proper subgroups of $A_5$. Here
$H_1$ is a subgroup of $H_i$, for every $i=1$, 2, 3, 4. It follows that $\mathscr{I}(A_5)$ has a subgraph $\mathcal B_1$ as in Figure~\ref{fig:int f3}, so
$\gamma(\mathscr{I}(G))>1$ and $\overline{\gamma}(\mathscr{I}(G))>1$.
If $p> 2$, then $ L_2(q^p)$ contains a subgroup isomorphic to $(\mathbb Z_q)^p$,
namely the subgroup of matrices of the form $\overline{
\bigl( \begin{smallmatrix}
1 & a\\
0 & 1
\end{smallmatrix}\bigr)
}$ with $a \in \mathbb F_{q^p}$.
By Proposition~\ref{intersecting graph t1}, $\gamma(\mathscr{I}((\mathbb Z_q)^p ))>1$, and $\overline{\gamma}(\mathscr{I}((\mathbb Z_q)^p ))>1$, $p >2$
and $\mathscr{I}((\mathbb Z_q)^p )$ contains $K_5$.
\noindent\textbf{Case 2:} $G \cong L_3(3)$. In $SL_3(3)$ the only matrix in the subgroup $H$ is the identity matrix, so $L_3(3) \cong SL_3(3)$.
Let us consider the subgroup consisting of matrices of the form
$\left(\begin{smallmatrix}
1 & a & b\\
0 & 1 & c\\
0 & 0 & 1
\end{smallmatrix}\right)$
with $a, b, c \in \mathbb F_3$. This subgroup is isomorphic to the group
$(\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p$ with $p=3$.
By Proposition~\ref{intersecting graph t2}, $\gamma(\mathscr{I}((\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p))>1$,
$\overline{\gamma}(\mathscr{I}((\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p))>1$ and
$\mathscr{I}((\mathbb Z_p \times \mathbb Z_p) \rtimes \mathbb Z_p)$ contains $K_5$.
\noindent\textbf{Case 3:} $G \cong L_2(p)$, where $p$ is any prime exceeding $3$ such that $p^2 +1 \equiv 0~(\text{mod}~ 5)$. We have to consider two subcases:
\noindent\textbf{Subcase 3a:} $p \equiv 1 ~(\text{mod}~ 4)$. It is shown in \cite[p.~222]{boh-reid} that $L_2(p)$
has a subgroup isomorphic to $D_{p-1}$. So by Lemma~\ref{intersecting graph l1}, $\gamma(\mathscr{I}(D_{p-1}))>1$,
$\overline{\gamma}(\mathscr{I}(D_{p-1}))>1$ and
$\mathscr{I}(D_{p-1})$ contains $K_5$.
If $p=5$, then $L_2(5) \cong A_5 \cong L_2(4)$. By Case 1,
$\gamma(\mathscr{I}(A_5))>1$, $\overline{\gamma}(\mathscr{I}(A_5))>1$ and $\mathscr{I}(A_5)$ contains $K_5$.
\noindent\textbf{Subcase 3b:} $p \equiv 3 ~(\text{mod}~ 4)$. $L_2(p)$
has a subgroup isomorphic to $D_{p+1}$\cite[p.~222]{boh-reid}. By Lemma~\ref{intersecting graph l1}, $\gamma(\mathscr{I}(D_{p+1}))>1$,
$\overline{\gamma}(\mathscr{I}(D_{p+1}))>1$ and
$\mathscr{I}(D_{p+1})$ contains $K_5$.
If $p=7$, then $S_4$ is a maximal subgroup of $L_2(7)$\cite{atlas}. Also by Theorem~\ref{intersecting graph t5}, $\gamma(\mathscr{I}(S_4))>1$,
$\overline{\gamma}(\mathscr{I}(S_4))>1$ and
$\mathscr{I}(S_4)$ contains $K_5$.
\noindent\textbf{Case 4:} $G \cong Sz(2^q)$, where $q$ is any odd prime. Then $ Sz(2^q)$ has a subgroup isomorphic to
$(\mathbb Z_2)^q, q \geq 3$ \cite[p.~466]{goren}. But by Proposition~\ref{intersecting graph t1}, $\gamma(\mathscr{I}((\mathbb Z_2)^q))>1$,
$\overline{\gamma}(\mathscr{I}((\mathbb Z_2)^q))>1$, $q \geq 3$ and
$\mathscr{I}((\mathbb Z_2)^q)$ contains $K_5$.
Combining all these cases together, the proof follows.
\end{proof}
\section{Main results}\label{sec: 5}
By combining all the results obtained in Sections~\ref{sec:3} and \ref{sec:4} above, we have the following general main results,
which classifies the finite groups whose intersection graphs of subgroups are toroidal or projective-planar, and classifies the finite non-cyclic groups whose intersection graphs of subgroups are one of $K_5$ free, $C_3$-free, acyclic or bipartite.
\begin{thm}\label{intersecting graph t12}
Let $G$ be a finite group and $p$, $q$, $r$ be distinct primes. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(G)$ is toroidal if and only if $G$ is isomorphic to one of the following:
\begin{enumerate}[\normalfont (a)]
\item $\mathbb Z_{p^\alpha} (\alpha=6,7,8)$, $\mathbb Z_{p^\alpha q}(\alpha=3,4)$, $\mathbb Z_{p^2q^2}$, $\mathbb Z_{p^2qr}$, $\mathbb Z_9\times \mathbb Z_3$, $\mathbb Z_{25}\times \mathbb Z_5$, $\mathbb Z_{3q}\times \mathbb Z_3$, $M_{p^3} (p= 3, 5)$, $M_{16}$, $\mathbb Z_3\rtimes \mathbb Z_4$, $\mathbb Z_5\rtimes \mathbb Z_4$, $\mathbb Z_9\rtimes \mathbb Z_2$, $\mathbb Z_{25}\rtimes \mathbb Z_2$;
\item $\langle a, b, c~|~ a^5=b^5=c^6=1, ab=ba, cac^{-1}=b,
cbc^{-1}= ab^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $6$ in $GL_2(5)$.
\end{enumerate}
\item $\mathscr{I}(G)$ is projective-planar if and only if $G$ is isomorphic to one of
$\mathbb Z_{p^\alpha}$$(\alpha=6, 7)$, $\mathbb Z_{p^3q}$, $\mathbb Z_9\times \mathbb Z_3$, $\mathbb Z_{3q}\times \mathbb Z_3$, $M_{27}$, $\mathbb Z_3\rtimes \mathbb Z_4$, $\mathbb Z_9\rtimes \mathbb Z_2$.
\end{enumerate}
\end{thm}
\begin{thm}\label{intersecting graph t233}
Let $G$ be a finite non-cyclic group and $p$, $q$, $r$ be distinct primes. Then
\begin{enumerate}[{\normalfont (1)}]
\item $\mathscr{I}(G)$ is $K_5$-free if and only if $G$ is isomorphic to one of the following groups:
\begin{enumerate}[{\normalfont(a)}]
\item $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_4 \times \mathbb Z_2$, $\mathbb Z_6\times \mathbb Z_2$, $Q_8$, $M_8$, $\mathbb Z_q \rtimes \mathbb Z_p$, $\mathbb Z_q \rtimes_2 \mathbb Z_{p^2}$, $A_4$;
\item $\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
\item $\langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=b^{-1}, cbc^{-1}=a^1b^l\rangle$, where
$\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2|(p+1)$;
\item $\langle a,b,c~|~a^p=b^q=c^r=1, b^{-1}ab=a^\mu, c^{-1}ac=a^v, bc=cb\rangle$, where $r$, $q$ are divisor of $p$ and $v$, $\mu\neq 1$;
\item $\langle a, b, c~|~ a^p=b^p=c^{qr}=1, ab=ba, cac^{-1}=b,
cbc^{-1}= ab^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $qr$ in $GL_2(p)$, $qr|(p+1)$.
\end{enumerate}
\item The following are equivalent:
\begin{enumerate}[{\normalfont(a)}]
\item $G$ is isomorphic to one of the following:
$\mathbb Z_{p^\alpha}(\alpha=2,3)$, $\mathbb Z_{pq}$, $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_q \rtimes \mathbb Z_p$, $A_4$ or $\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
\item $\mathscr{I}(G)$ is $C_3$-free;
\item $\mathscr{I}(G)$ is acyclic;
\item $\mathscr{I}(G)$ is bipartite.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{cor}\label{intersecting graph c1}
Let $G$ be a finite group and $p,q,r$ are distinct primes. Then
\begin{enumerate}[{\normalfont(1)}]
\item $\mathscr{I}(G)$ is unicyclic if and only if $G$ is either $\mathbb Z_{p^4}$ or $\mathbb Z_{p^2q}$;
\item $\mathscr{I}(G)\cong C_n$ if and only if $n=3$ and $G \cong \mathbb Z_{p^4}$;
\item $\mathscr{I}(G)\cong P_n$ if and only if $n=1$ and $G \cong \mathbb Z_{p^3}$;
\item $\mathscr{I}(G)$ is $C_5$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha} (\alpha=2,3,4,5)$, $\mathbb Z_{p^\alpha q} (\alpha=1,2)$,
$\mathbb Z_p\times \mathbb Z_p$, $\mathbb Z_4\times \mathbb Z_2$, $Q_8$, $\mathbb Z_q\rtimes \mathbb Z_p$, $\mathbb Z_q\rtimes_2 \mathbb Z_{p^2}$, $A_4$,
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q~|~(p+1)$ or $\langle a,b,c~|~a^p=b^p=c^{q^2}=1, ab=ba, cac^{-1}=b^{-1}, cbc^{-1}=a^1b^l\rangle$,
where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q^2$ in $GL_2(p)$, $q^2|(p+1)$;
\item $\mathscr{I}(G)$ is $C_4$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3,4)$, $\mathbb Z_{p^\alpha q}(\alpha= 1,2)$,
$\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_q \rtimes \mathbb Z_p$, $\mathbb Z_q \rtimes_2 \mathbb Z_{p^2}$, $A_4$ or
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
\item $\mathscr{I}(G)$ is $P_4$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3,4,5)$, $\mathbb Z_{p^\alpha q}(\alpha= 1,2)$,
$Q_8$, $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_q \rtimes \mathbb Z_p$ or
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$ or $A_4$;
\item $\mathscr{I}(G)$ is $P_3$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3,4)$, $\mathbb Z_{pq}$,
$\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_q \rtimes \mathbb Z_p$, $A_4$ or
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
\item $\mathscr{I}(G)$ is $P_2$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3)$, $\mathbb Z_{pq}$,
$\mathbb Z_p \times \mathbb Z_p$ or $\mathbb Z_q \rtimes \mathbb Z_p$;
\item $\mathscr{I}(G)$ is totally disconnected if and only if $G$ is isomorphic to one of $\mathbb Z_{p^2}$, $\mathbb Z_{pq}$, $\mathbb Z_p\times \mathbb Z_p$ or $\mathbb Z_q\rtimes \mathbb Z_p$;
\item $\mathscr{I}(G)$ is $K_{2,3}$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3,4,5)$, $\mathbb Z_{p^\alpha q}(\alpha=1,2)$,
$\mathbb Z_{pqr}$,
$\mathbb Z_p\times \mathbb Z_p$, $\mathbb Z_4\times \mathbb Z_2$, $Q_8$, $\mathbb Z_q\rtimes Z_p$, $\mathbb Z_q\rtimes_2 \mathbb Z_{p^2}$, $A_4$ or
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
\item $\mathscr{I}(G)$ is $K_4$-free if and only if $G$ is one of
$\mathbb Z_{p^\alpha} (\alpha=2,3,4)$, $\mathbb Z_{p^\alpha q} (\alpha=1,2)$, $\mathbb Z_{pqr}$,
$\mathbb Z_p\times \mathbb Z_p$, $\mathbb Z_q\rtimes_2 \mathbb Z_p$, $\mathbb Z_q\rtimes \mathbb Z_{p^2}$, $A_4$ or
$\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$ ;
\item $\mathscr{I}(G)$ is $K_{1,4}$-free if and only if $G$ is one of $\mathbb Z_{p^\alpha} (\alpha=2,3,4,5)$, $\mathbb Z_{p^\alpha q} (\alpha=1,2)$,
$\mathbb Z_p\times \mathbb Z_p$, $Q_8$, $\mathbb Z_q\rtimes \mathbb Z_p$ or $A_4$;
\item $\mathscr{I}(G)$ is claw-free if and only if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3,4)$, $\mathbb Z_{pq}$,
$\mathbb Z_p \times \mathbb Z_p$ or $\mathbb Z_q \rtimes \mathbb Z_p$;
\item The following are equivalent:
\begin{enumerate}[{\normalfont(a)}]
\item $G\cong \mathbb Z_{p^\alpha}(\alpha=2,3)$;
\item $\mathscr{I}(G)$ is a tree;
\item $\mathscr{I}(G)$ is a star graph;
\item $\mathscr{I}(G)$ is complete bipartite.
\end{enumerate}
\item $girth(\mathscr{I}(G))$ is $\infty$, if $G$ is one of $\mathbb Z_{p^\alpha}(\alpha=2,3)$, $\mathbb Z_{pq}$, $\mathbb Z_p \times \mathbb Z_p$, $\mathbb Z_q \rtimes \mathbb Z_p$, $A_4$ or $\langle a, b, c~|~ a^p=b^p=c^q=1, ab=ba, cac^{-1}=b,
cbc^{-1}= a^1b^{l} \rangle$, where $\bigl(\begin{smallmatrix}
0 & -1\\ 1 & l
\end{smallmatrix} \bigr)$ has order $q$ in $GL_2(p)$, $q|(p+1)$;
otherwise $girth(\mathscr{I}(G))=3$.
\end{enumerate}
\end{cor}
\begin{proof}
Note that if the intersection graph of subgroups of a group has $K_5$ as a subgraph, then it is none of the following graphs: unicyclic, cycle, path, claw-free, $C_5$-free, $C_4$-free, $P_4$-free, $P_3$-free, $P_2$-free, totally disconnected, $K_{2,3}$-free, $K_4$-free, $K_{1,4}$-free.
So to classify the finite groups whose intersection graph of subgroups is one of
outerplanar, unicyclic, claw-free, path, cycle, $C_4$-free, $C_5$-free, totally disconnected, $P_2$-free, $P_3$-free, $P_4$-free, $K_{1,4}$-free,
$K_{2,3}$-free,
it is enough to consider the finite groups whose intersection graph of subgroups are $K_5$-free. Thus, we need to investigate these properties only for groups
given in Theorems~\ref{l1}(2) and \ref{intersecting graph t233}(1).
\noindent By Theorem~\ref{l1}(2) and using \eqref{intersection graph e1}, \eqref{intersection graph e2}, the only groups $G$ such that $\mathscr{I}(G)$ is unicyclic are $\mathbb Z_{p^4}$ and $\mathbb Z_{p^2q}$. By Theorem~\ref{intersecting graph t233}(1) and using \eqref{e4}-\eqref{33}, Figure~\ref{int f1}(b),~\ref{int f1}(c),~\ref{int f1}(d), Figure~\ref{fig:int f11}, there is no group $G$ such that $\mathscr{I}(G)$ is unicyclic. Thus, the proof of $(1)$ follows.
\noindent Proof of parts $(2)$ to $(14)$ of this Corollary are similar to the proof of part $(1)$.
If $\mathscr{I}(G)$ contains $C_3$, then obviously $girth(G)=3$. Now assume that $\mathscr{I}(G)$ is $C_3$-free. If $G$ is cyclic, then by Theorem~\ref{l1}(4), $\mathscr{I}(G)$ is acyclic, since $\mathscr{I}(Z_{p^\alpha})\cong K_{p^{\alpha -1}}(\alpha = 2,3)$ and $\mathscr{I}(Z_{pq}) \cong \overline{K}_2$. If $G$ is non-cyclic, then by Theorem~\ref{intersecting graph t233}(2), $\mathscr{I}(G)$ is also acyclic. So in both these cases $girth(G)=\infty$. This completes the proof.
\end{proof}
\begin{rem}\normalfont
\begin{enumerate}[\normalfont (i)]
\item In Theorem~\ref{intersecting graph t12}, we show that all the projective-planar intersection graphs of subgroups of groups are toroidal, which is not the case for arbitrary graphs (e.g., see pp.\,367-368 and Figure~13.33 in \cite{KK2005}).
\item In \cite{sel}, Selc¸uk Kayacan \emph{et al.} showed that the intersection graph of subgroups of groups other than those listed in Theorem~\ref{1000}, the group $G_2$ given in Case 1b in the proof of Proposition~\ref{intersecting graph t4}, and the group of order $p^2qr$ given in part (2) of Proposition~\ref{intersecting graph t8} contains $K_5$ as a subgraph. Also they showed that the intersection graph of subgroups of these two groups contains $K_{3,3}$ as a subgraph. In this paper, we proved by using another method that the intersection graph of subgroups of the first group contains $K_5$ as a subgraph, but not the second group. Thus it follows that the class of all groups having $K_5$-free intersection graph of subgroups properly contains the class of all groups having planar intersection graph of subgroups.
\item In \cite{akbari_2} Akbari \emph{et al.} classified all groups whose intersection graphs of subgroups are one of $C_3$-free, tree, totally disconnected. Also they obtained the girth of the intersection graph of groups. In this paper, we proved these results for finite groups in a different method.
\end{enumerate}
\end{rem}
In the next result we characterize some finite groups by using their intersection graph of subgroups.
\begin{cor}\label{intersecting graph c2}
Let $G$ be a group and $p, q, r$ are distinct primes. Then
\begin{enumerate}[\normalfont (1)]
\item $\mathscr{I}(G)\cong \mathscr{I}(M_8)$ if and only if $G\cong M_8$.
\item $\mathscr{I}(G)\cong \mathscr{I}(\mathbb Z_q\rtimes_2 \mathbb Z_{p^2})$ if and only if $G\cong \mathbb Z_q\rtimes_2 \mathbb Z_{p^2}$;
\item $\mathscr{I}(G)\cong \mathscr{I}(\mathcal{G}_1)$ if and only if $G\cong \mathcal{G}_1$, where $\mathcal {G}_1$ is the group described in Theorem~\ref{1000}.
\item $\mathscr{I}(G)\cong \mathscr{I}(A_4)$ if and only if $G\cong A_4$;
\item $\mathscr{I}(G)\cong \mathscr{I}(\mathcal{G}_3)$ if and only if $G\cong \mathcal{G}_2$, where $\mathcal {G}_2$ is the group described in Theorem~\ref{1000};
\item $\mathscr{I}(G)\cong \mathscr{I}(\mathcal{G}_2)$ if and only if $G\cong \mathcal{G}_3$, where $\mathcal {G}_3$ is the group described in Theorem~\ref{1000};
\item $\mathscr{I}(G)\cong \mathscr{I}(G_1)$ if and only if $G\cong G_1$, where $G_1$ is the group of order $p^2qr$ described in Theorem~\ref{intersecting graph t233}(1).
\end{enumerate}
\end{cor}
\begin{proof} (1)-(7): By Theorem~\ref{intersecting graph t233}(1), we can see that the groups mentioned in this Corollary are having $K_5$-free intersection graph of subgroups. Also by
\eqref{e4}-\eqref{33}, Figure~\ref{int f1}(b)-(d), Figure~\ref{fig:int f11}, the intersection graphs of subgroups of these groups are unique. Since an infinite group has infinite number of subgroups, so its intersection graph of subgroups can not be isomorphic to any of the intersection graphs of subgroups of these groups.
If the intersection graph of a given finite group is isomorphic to any of these intersection graphs of subgroups of these groups, then by the uniqueness of these graphs, the given group must be isomorphic to that corresponding group. Proof of the result follows from this fact.
\end{proof}
\begin{thm}\label{intersection graph 300}
Let $G$ be a group. Then $\theta(\mathscr{I}(G))=m$, where $m$ is the number of prime order subgroups of groups of $G$.
\end{thm}
\begin{proof} Let $G$ be finite and $|G|=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$, where $p_i$'s are distinct primes, and $\alpha_i\geq 1$.
For each $i=1$, 2, $\ldots$, $k$, let $t_i$ be the number of subgroups of order $p_i$ and let $H(j,p_i)$, $j=1$, 2, $\ldots$, $t_i$ be a
subgroup of $G$
of order $p_i$. For each $j=1$, 2, $\ldots$, $t_i$, let $\mathcal S(j,p_i)$ be the set of all subgroups of $G$ having $H(j,p_i)$ in common.
Clearly $\mathcal S(j, p_i)$ forms a clique in $\mathscr{I}(G)$ and $\mathcal S:=\{\mathcal S(j,p_i)~|~i=1$, 2, $\ldots$, $k$ and $j=1$, 2, $\ldots$, $t_i\}$
forms a clique cover of $\mathscr{I}(G)$ with $|\mathcal S|=t_1+t_2+\ldots+t_k$. Therefore, $\Theta(\mathscr{I}(G))\leq |\mathcal S|$. Let $\mathcal{T}$ be a clique cover of $\mathscr{I}(G)$ such that $\Theta(\mathscr{I}(G))=|\mathcal{T}|$. If $|\mathcal{T}|<|\mathcal S|$, then by pigeonhole principle, $\mathcal{T}$ has a clique which contains atleast two subgroups, say $H(j,p_i)$, $H(l,p_r)$, for some $i\neq r$,
$j\in \{1$, 2, $\ldots$, $t_i\}$, $l\in \{1$, 2, $\ldots$, $t_r\}$, which is not possible, since $H(j,p_i)$ and $H(l,p_r)$ are not adjacent in $\mathscr{I}(G)$. So $|\mathcal{T}|=|\mathcal S|=$ the number of prime order subgroups of $G$. A similar argument also works when $G$ is infinite.
\end{proof}
Zelinka shown in \cite{zelinka} that for any group $G$, $\alpha(\mathscr(G))=m$, where $m$ is the number of prime order proper subgroups of $G$. This fact together with Theorem~\ref{intersection graph 300} gives the following result.
\begin{cor}\label{124}
If $G$ is a group, then $\mathscr{I}(G)$ is weakly $\alpha$-perfect.
\end{cor}
\end{document} |
\begin{document}
\title{On the local fluctuations of last-passage percolation models}
\alphauthor{Eric Cator and Leandro P. R. Pimentel}
\date{\today}
\maketitle
\begin{abstract}
Using the fact that the Airy process describes the limiting fluctuations of the Hammersley last-passage percolation model, we prove that it behaves locally like a Brownian motion. Our method is quite straightforward, and it is based on a certain monotonicity and good control over the equilibrium measures of the Hammersley model (local comparison).
\end{abstract}
\section{Introduction and results}
In recent years there has been a lot of research on the Airy process and related processes such as the Airy sheet \cite{CQ}. Most papers use analytic methods and exact formulas given by Fredholm determinants to prove properties of these processes, but some papers use the fact that these processes are limiting processes of last-passage percolation models or random polymer models, and they use properties of these well studied models to prove the corresponding property of the limiting process. A nice example of these different approaches can be found in two recent papers, one by H\"agg \cite{H} and one by Corwin and Hammond \cite{CH}. H\"agg proves in his paper that the Airy process behaves locally like a Brownian motion, at least in terms of convergence of finite dimensional distributions. He uses the Fredholm determinant description of the Airy process to obtain his result. Corwin and Hammond on the other hand, use the fact that the Airy line process, of which the top line corresponds to the Airy process, can be seen as a limit of Brownian bridges, conditioned to be non-intersecting. They show that a particular resampling procedure, which they call the Brownian Gibbs property, holds for the system of Brownian bridges and also in the limit for the Airy line process. As a consequence, it follows that the local Brownian behavior of the Airy process holds in a stronger functional limit sense. Our paper will prove the same theorem, also using the fact that the Airy process is a limiting process, but in a much more direct way: we will consider the Hammersley last-passage percolation model, and show that we can control local fluctuations of this model by precisely chosen equilibrium versions of this model, which are simply Poisson processes. Then we show that in the limit this control suffices to prove the local Brownian motion behavior of the Airy process, as well as tightness of the models approaching the Airy process. We also extend the control of local fluctuations of the Hammersley process to scales smaller than the typical cube-root scale.
Our method is quite straightforward, yet rather powerful, mainly because we have a certain monotonicity and good control over the equilibrium measures. In fact, we think that we can extend our result to the more illustrious Airy sheet, the two dimensional version of the Airy process. We address the reader to \cite{CQ}, for a description of this process in terms of the renormalization fixed point of the KPZ universality class. However, here we run in to much more technical problems, and this will still require a lot more work, beyond the scope of this paper.
We will continue the introduction by developing notation, introducing all relevant processes and stating the three main theorems. In Section 2 we introduce the local comparison technique and in each of the following three sections one theorem is proved.
\subsection{The Hammersley Last Passage Percolation model}
The Hammersley last-passage percolation model \cite{AD} is constructed from a two-dimensional homogeneous Poisson point process of intensity $1$. Denote $[x]_t:=(x,t)\in{\mathbb R}^2$ and call a sequence $[x_1]_{t_1},[x_2]_{t_2},\dots,[x_k]_{t_k}$ of planar points increasing if $x_j<x_{j+1}$ and $t_j<t_{j+1}$ for all $j=1,\dots,k-1$. The last-passage time $L([x]_s,[y]_t)$ between $[x]_s<[y]_t$ is the maximal number of Poisson points among all increasing sequences of Poisson points lying in the rectangle $(x, y ]\times(s, t]$. Denote $L[x]_t:=L([0]_0,[x]_t)$ and define ${\mathcal A}_n$ by
$$u\in{\mathbb R}\,\mapsto\,{\mathcal A}_n(u):=\frac{L[n+2un^{2/3}]_n-(2n+2un^{2/3})+u^2n^{1/3}}{n^{1/3}}\,.$$
Pr\"ahofer and Spohn \cite{PS} proved that
\begin{equation}{\lambda}bel{eq:Airy}
\lim_{n\to\infty}{\mathcal A}_n(\cdot)\stackrel{dist.}{=}{\mathcal A}(\cdot)\,,
\end{equation}
in the sense of finite dimensional distributions, where ${\mathcal A}\equiv({\mathcal A}(u))_{u\in{\mathbb R}}$ is the so-called Airy process. This process is a one-dimensional stationary process with continuous paths and finite dimensional distributions given by a Fredholm determinant \cite{J}:
$${\mathbb P}\left({\mathcal A}(u_1)\leq \xi_1,\dots,{\mathcal A}(u_m)\leq \xi_m\right):=\det\left(I-f^{1/2}Af^{1/2}\right)_{L^2\left(\{u_1,\dots,u_m\}\times{\mathbb R}\right)}\,.$$
The function $A$ denotes the extended Airy kernel, which is defined as
$$A_{s,t}(x,y):=\left\{\begin{array}{ll}\int_0^\infty e^{-z(s-t)}{\rm Ai}(x+z){\rm Ai}(y+z)dz\,,& \mbox{ if } s\geq t\,,\\&\\
-\int_{-\infty}^0 e^{-z(t-s)}{\rm Ai}(x+z){\rm Ai}(y+z)dz\,,& \mbox{ if } s< t\,,\end{array}\right.$$
where ${\rm Ai}$ is the Airy function, and for $\xi_1,\dots,\xi_m\in{\mathbb R}$ and $u_1<\dots<u_m$ in ${\mathbb R}$,
\begin{eqnarray*}
f\,:\,\{u_1,\dots,u_m\}\times{\mathbb R}&\to&{\mathbb R}\\
(u_i,x)&\mapsto&1_{(\xi_i,\infty)}(x)\,.
\end{eqnarray*}
The main contribution of this paper is the development of a local comparison technique to study the local fluctuations of last-passage times and its scaling limit. The ideas parallel the work of Cator and Groeneboom \cite{CG1,CG2}, where they studied local convergence to equilibrium and the cube-root asymptotic behavior of $L$. This technique consists of bounding from below and from above the local differences of $L$ by the local differences of the equilibrium regime (Lemma \ref{lem:LocalComparison}), with suitable parameters that will allow us to handle the local fluctuations in the right scale (Lemma \ref{lem:ExitControl}). For the Hammersley model the equilibrium regime is given by a Poisson process. We have strong indications that the technique can be applied to a broad class of models, as soon as one has Gaussian fluctuations for the equilibrium regime. Although this is a very natural assumption, one can only check that for a few models. As a first application, we will prove tightness of ${\mathcal A}_n$.
\begin{thm}{\lambda}bel{thm:Tight}
The collection $\{{\mathcal A}_n\}$ is tight in the space of cadlag functions on $[a,b]$. Furthermore, any weak limit of ${\mathcal A}_n$ lives on the space of continuous functions.
\end{thm}
The local comparison technique can be used to study local fluctuations of last-passage times for lengths of size $n^{\gamma}$, with $\gamma\in(0,2/3)$ (so smaller than the typical scale $n^{2/3}$). Let ${\mathcal B}\equiv({\mathcal B}(u))_{u\geq 0}$ denote the standard two-sided Brownian motion process.
\begin{thm}{\lambda}bel{thm:LocalFluct}
Fix $\gamma\in(0,2/3)$ and $s>0$ and define $\Delta_n$ by
$$u\in{\mathbb R}\,\mapsto\, \Delta_n(u):=\frac{L[sn+un^{\gamma}]_{n}-L[sn]_n-\mu un^{\gamma}}{\sigma n^{\gamma/2}}\,,$$
where $\mu:=s^{-1/2}$ and $\sigma:=s^{-1/4}$. Then
$$\lim_{n\to\infty}\Delta_n(\cdot)\stackrel{dist.}{=}{\mathcal B}(\cdot)\,,$$
in the sense of weak convergence of probability measures in the space of cadlag functions.
\end{thm}
As we mentioned in the previous section, the Airy process locally behaves like Brownian motion \cite{CH,H}. By applying the local comparison technique again, we will present an alternative proof of the functional limit theorem for this local behavior.
\begin{thm}{\lambda}bel{thm:LocalAiry}
Define ${\mathcal A}^{\varepsilon }ilon$ by
$$u\in{\mathbb R}\,\mapsto\,{\mathcal A}^{{\varepsilon }ilon}(u):={\varepsilon }ilon^{-1/2}\left({\mathcal A}({\varepsilon }ilon u)-{\mathcal A}(0)\right)\,.$$
Then
$$\lim_{{\varepsilon }ilon\to 0}{\mathcal A}^{{\varepsilon }ilon}(\cdot)\stackrel{dist.}{=}\sqrt{2}{\mathcal B}(\cdot)\,,$$
in the sense of weak convergence of probability measures in the space of continuous functions.
\end{thm}
\subsection{The lattice model with exponential weights}
Consider a collection $\{\omega_{[x]_t}\,:\,[x]_t\in{\mathbf Z}^2\}$ of i.i.d. non negative random variables with an exponential distribution of parameter one. Let ${\mathbb P}i([x]_t,[y]_u)$ denote the collection of all lattice paths $\varpi=([z]_{v_j})_{j=1,\dots,k}$ such that:
\begin{itemize}
\item $[z]_{v_1}\in\{[x]_t+[1]_0,[x]_t+[0]_1\}$ and $[z]_{v_k}=[y]_u$;
\item $[z]_{v_{j+1}}-[z]_{v_j}\in\{[1]_0,[0]_1\}$ for $j=0,1\dots,,k-1$.
\end{itemize}
The (lattice) last-passage percolation time between $[x]_t <[y]_u$ is defined by
$$L^l([x]_t,[y]_u):=\max_{\varpi\in{\mathbb P}i([x]_t,[y]_u)}\big\{\sum_{[z]_v\in\varpi}\omega_{[z]_v}\big\}\,.$$
Denote $L^l[x]_t:=L^l([0]_0,[x]_t)$ and define ${\mathcal A}^l_n$ by
$$u\in{\mathbb R}\,\mapsto\,{\mathcal A}^l_n(u):=\frac{L^l[n+2^{5/3}un^{2/3}]_n-(4n+2^{8/3}un^{2/3})+2^{4/3}u^2n^{1/3}}{2^{4/3}n^{1/3}}\,.$$
Corwin, Ferrari and P\'ech\'e \cite{CFP} proved that
\begin{equation}{\lambda}bel{eq:Airy}
\lim_{n\to\infty}{\mathcal A}^l_n(\cdot)\stackrel{dist.}{=}{\mathcal A}(\cdot)\,,
\end{equation}
in the sense of finite dimensional distributions. The local comparison method can be used in this context as well. (The lattice version of Lemma \ref{lem:LocalComparison} is straightforward. For exponential weights, the analog to Lemma \ref{lem:ExitControl} was proved in \cite{BCS}.)
\begin{thm}{\lambda}bel{thm:LTight}
The collection $\{{\mathcal A}^l_n\}$ is tight in the space of cadlag functions on $[a,b]$. Furthermore, any weak limit of ${\mathcal A}_n$ lives on the space of continuous functions.
\end{thm}
\begin{thm}{\lambda}bel{thm:LLocalFluct}
Fix $\gamma\in(0,2/3)$ and $s>0$ and define $\Delta_n$ by
$$u\in{\mathbb R}\,\mapsto\, \Delta^l_n(u):=\frac{L^l[sn+un^{\gamma}]_{n}-L^l[sn]_n-\mu un^{\gamma}}{\sigma n^{\gamma/2}}\,,$$
where $\mu=\sigma:=s^{-1/2}(1+s^{1/2})$. Then
$$\lim_{n\to\infty}\Delta^l_n(\cdot)\stackrel{dist.}{=}{\mathcal B}(\cdot)\,,$$
in the sense of weak convergence of probability measures in the space of cadlag functions.
\end{thm}
To avoid repetitions, we will not present a proof of the lattice results. We hope that the reader can convince his (or her) self that the method that we will describe in detail for the Hammersley last-passage percolation model can be easily adapted to the lattice models with exponentials weights.
\begin{rem}
We also expect that the local comparison method can be used in the log-gamma polymer model, introduced by Sepp\"al\"ainen \cite{T}. The polymer versions of Lemma \ref{lem:LocalComparison} and Lemma \ref{lem:ExitControl} were proved in \cite{T}.
\end{rem}
\section{Local comparison and exit points}
The Hammersley last-passage percolation model has a representation as an interacting particle system, called the Hammersley process \cite{AD,CG1}. We will use notations used in \cite{CP}. In the positive time half plane we have the same planar Poisson point process as before. On the $x$-axis we take a Poisson process of intensity ${\lambda}mbda>0$. The two Poisson process are assumed to be independent of each other. For $x\in {\mathbb R}$ and $t>0$ we define
$$L_{{\lambda}}[x]_t\equiv L_{\nu_{\lambda}mbda}[x]_t:=\sup_{z\in (-\infty,x]} \left\{ \nu_{\lambda}(z) + L([z]_0,[x]_t)\right\}\,,$$
where, for $z\leq x$,
$$\nu_{\lambda}(z)=\left\{\begin{array}{ll}\mbox{ the number of Poisson points in }(0,z]\times\{0\}& \mbox{ for } z> 0\\
\mbox{ minus the number of Poisson points in }(z,0]\times \{0\} & \mbox{ for } z\leq0\,.\end{array}\right.$$
The process $(M^t_{\lambda}mbda)_{t\geq 0}$, given by
$$M^t_{\nu_{\lambda}}(x,y]\equiv M^t_{\lambda}(x,y]:=L_{\lambda}[y]_t-L_{\lambda}[x]_t\,\,\,\mbox{ for }x<y\,,$$
is a Markov process on the space of locally finite counting measures on ${\mathbb R}$. The Poisson process is the invariant measure of this particle system in the sense that
\begin{equation}{\lambda}bel{eq:equilibrium}
M^t_{\lambda}\stackrel{dist.}{=}\mbox{ Poisson process of intensity ${\lambda}mbda$ for all $t\geq 0$}\,.
\end{equation}
Notice that the last-passage time $L=L_{\nu_0}$ can be recovered in the positive quadrant by choosing a measure $\nu_0$ on the axis that has no points to the right of $0$, and an infinite amount of points in every interval $(-{\varepsilon },0)$, $\forall {\varepsilon }>0$ (this could be called a ``wall'' of points). Thus
$$L_{\lambda}\equiv L_{\nu_{\lambda}mbda}({\cal P})\,\mbox{ and }\, L\equiv L_{\nu_0}({\cal P})\,$$
are coupled by the same two-dimensional Poisson point process $\cal P$, which corresponds to the basic coupling between $M^t_{\nu_{\lambda}mbda}$ and $M^t_{\nu_0}$.
Define the exit points
$$Z_{\lambda}[x]_t :=\sup\left\{z\in(-\infty,x]\,:\,L_{\lambda}[x]_t=\nu_{\lambda}mbda(z)+L([z]_0,[x]_t)\right\}\,,$$
and
$$Z'_{\lambda}[x]_t :=\inf\left\{z\in(-\infty,x]\,:\,L_{\lambda}[x]_t=\nu_{\lambda}mbda(z)+L([z]_0,[x]_t)\right\}\,.$$
By using translation invariance and invariance under the map $(x,t)\mapsto({\lambda}mbda x,t/{\lambda}mbda)$, we have that
\begin{equation}{\lambda}bel{eq:sym}
Z_{\lambda}mbda[x+h]_t\stackrel{dist.}{=}Z_{\lambda}mbda[x]_t+h\,\,\,\mbox{ and }\,\,\, Z_{\lambda}[x]_t\stackrel{dist.}{=}{\lambda}mbda Z_1[{\lambda}mbda x]_{t/{\lambda}mbda}.
\end{equation}
We need to use one more symmetry. In \cite{CG1}, the Hammersley process was set up as a process in the first quadrant with sources on the $x$-axis and sinks on the $t$-axis. In our notation this means that the process $t\mapsto L_{\lambda}mbda[0]_t$ is a Poisson process with intensity $1/{\lambda}mbda$ which is independent of the Poisson process $\nu_{\lambda}mbda$ restricted to the positive $x$-axis and independent of the Poisson process in the positive quadrant. We can now use reflection in the diagonal to see that the following equality holds:
\begin{equation}{\lambda}bel{eq:diagsym}
{\mathbb P}\left(Z'_{\lambda}mbda[x]_t < 0\right) = {\mathbb P}\left(Z_{1/{\lambda}mbda}[t]_x > 0\right).
\end{equation}
We use that $Z'_{\lambda}mbda[x]_t<0$ is equivalent to the fact that the maximizing path to the left-most exit point crosses the positive $t$-axis, and not the positive $x$-axis.
The local comparison technique consists of bounding from above and from below the local differences of $L$ by the local differences of $L_{\lambda}mbda$. These bounds depend on the position of the exit points. It is precisely summarized by the following lemma.
\begin{lem}{\lambda}bel{lem:LocalComparison}
Let $0\leq x\leq y$ and $t\geq 0$. If $Z'_{\lambda}[x]_t\geq0$ then
$$L[y]_t-L[x]_t\leq L_{\lambda}[y]_t-L_{\lambda}[x]_t\,,$$
and if $Z_{\lambda}[y]_t\leq 0$ then
$$L[y]_t-L[x]_t\geq L_{\lambda}[y]_t-L_{\lambda}[x]_t\,.$$
\end{lem}
\noindent{\bf Proof\,\,} When we consider a path $\varpi$ from $[x]_s$ to $[y]_t$ consisting of increasing points, we will view $\varpi$ as the lowest increasing continuous path connecting all the points, starting at $[x]_s$ and ending at $[y]_t$. In this way we can talk about crossings with other paths or with lines. The geodesic between $[x]_s$ and $[y]_t$ is given by the lowest path (in the sense we just described) that attains the maximum in the definition of $L([x]_s,[y]_t)$. We will denote this geodesic by $\varpi([x]_s,[y]_t)$. Notice that
$$L([x]_s,[y]_t)=L([x]_s,[z]_r)+L([z]_r,[y]_t)\,,$$
for any $[z]_r\in\varpi([x]_s,[y]_t)$.
Assume that $Z'_{\lambda}[x]_t\geq 0$ and let ${\mathbf c}$ be a crossing between the two geodesics $\varpi([0]_0,[y]_t)$ and $\varpi([z']_0,[x]_t)$, where $z':=Z'_{\lambda}[x]_t$. Such a crossing always exists because $x\leq y$ and $z'=Z'_{\lambda}[x]_t\geq 0$. We remark that, by superaddivity,
$$L_{\lambda}[y]_t \geq \nu_{\lambda}(z') + L([z']_0,[y]_t) \geq \nu_{\lambda}(z') + L([z']_0,{\mathbf c}) + L({\mathbf c},[y]_t)\,.$$
We use this, and that (since ${\mathbf c}\in\varpi([z']_0,[x]_t)$)
$$ \nu_{\lambda}(z') + L([z']_0,{\mathbf c})-L_{\lambda} [x]_t= -L({\mathbf c},[x]_t)\,,$$
in the following inequality:
\begin{eqnarray*}
L_{\lambda}[y]_t - L_{\lambda}[x]_t & \geq & \nu_{\lambda}\big(z'\big)+L([z']_0,{\mathbf c}) + L({\mathbf c},[y]_t) - L_{\lambda}[x]_t\\
& = & L({\mathbf c},[y]_t) - L({\mathbf c},[x]_t)\,.
\end{eqnarray*}
By superaddivity,
$$ - L({\mathbf c}\,,\,[x]_t)\geq L([0]_0,{\mathbf c})-L[x]_t\,,$$
and hence (since ${\mathbf c}\in\varpi([0]_0,[y]_t)$)
\begin{eqnarray*}
L_{\lambda}[y]_t - L_{\lambda}[x]_t & \geq & L({\mathbf c},[y]_t) - L({\mathbf c},[x]_t)\\
& \geq & L({\mathbf c},[y]_t) + L([0]_0,{\mathbf c})-L([0]_0,[x]_t)\\
& = & L[y]_t-L[x]_t\,.
\end{eqnarray*}
The proof of the second inequality is very similar. Indeed, denote $z:=Z_{\lambda}[y]_t$ and let ${\mathbf c}$ be a crossing between $\varpi([0]_0,[x]_t)$ and $\varpi([z]_0,[y]_t)$. By superaddivity,
$$L_{\lambda}[x]_t \geq \nu_{\lambda}(z) + L([z]_0,[x]_t) \geq \nu_{\lambda}(z) + L([z]_0,{\mathbf c}) + L({\mathbf c},[x]_t)\,.$$
Since ${\mathbf c}\in\varpi([z]_0,[y]_t)$ we have that
$$L_{\lambda}[y]_t-\nu_{\lambda}(z) - L([z]_0,{\mathbf c})=L({\mathbf c},[y]_t)\,,$$
which implies that
\begin{eqnarray*}
L_{\lambda}[y]_t - L_{\lambda}[x]_t & \leq &L_{\lambda}[y]_t- \nu_{\lambda}(z)-L([z]_0,{\mathbf c}) - L({\mathbf c},[x]_t)\\
& = & L({\mathbf c},[y]_t) - L({\mathbf c},[x]_t)\\
& \leq & L[y]_t-L([0]_0,{\mathbf c})- L({\mathbf c},[x]_t)\\
& = & L[y]_t-L[x]_t\,,
\end{eqnarray*}
where we have used that ${\mathbf c}\in\varpi([0]_0,[x]_t)$ in the last step.
$\Box$\\
\begin{rem} In fact the first statement of the lemma is also true when $Z_{\lambda}mbda[x]_t\geq 0$ and the second statement is true when $Z'_{\lambda}mbda[y]_t\leq 0$, both without any change to the given proof. This is a stronger statement, since $Z'_{\lambda}mbda[x]_t\leq Z_{\lambda}mbda[x]_t$, but we will only need the lemma as it is formulated.
\end{rem}
In order to apply Lemma \ref{lem:LocalComparison} and extract good bounds for the local differences one needs to control the position of exit points. This is given by the next lemma.
\begin{lem}{\lambda}bel{lem:ExitControl}
There exist constant $C>0$ such that,
$${\mathbb P}\left(Z_1[n]_n > r n^{2/3}\right)\leq \frac{C}{r^3}\,,$$
for all $r\geq 1$ and all $n\geq 1$.
\end{lem}
\noindent{\bf Proof\,\,} See Corollary 4.4 in \cite{CG2}.
$\Box$\\
\section{Proof of Theorem \ref{thm:Tight}}
For simple notation, and without loss of generality, we will restrict our proof to $[a,b]=[0,1]$.
\begin{lem}{\lambda}bel{lem:tight}
Fix $\beta\in(1/3,1)$ and for each $\delta\in(0,1)$ and $n\geq 1$ set
$${\lambda}mbda_\pm={\lambda}mbda_\pm(n,\delta):=1\pm\frac{\delta^{-\beta}}{n^{1/3}}\,.$$
Define the event
$$E_n(\delta):=\left\{Z'_{{\lambda}mbda_{+}}[n]_n\geq 0\,\,\mbox{ and }\,\,Z_{{\lambda}mbda_{-}}[n+2n^{2/3}]_n\leq 0\right\}\,.$$
Then
there exists a constant $C>0$ such that, for sufficiently small $\delta>0$,
$$\limsup_{n\to\infty}{\mathbb P}\left(E_n(\delta)^c\right)\leq C\delta^{3\beta}\,.$$
\end{lem}
\noindent{\bf Proof\,\,} Denote $r:=\delta^{-\beta}$ and let
$$n_+:={\lambda}mbda_+ n< 2n\,\,\mbox{ and }\,\,h_{+}:=\left({\lambda}mbda_+-\frac{1}{{\lambda}mbda_+}\right) n> rn^{2/3}> r n_+^{2/3}/2 \,$$
(for all sufficiently large $n$). By \eqref{eq:sym} and \eqref{eq:diagsym},
\begin{eqnarray*}
{\mathbb P}\left(Z'_{{\lambda}mbda_+}[n]_n<0\right)&=&{\mathbb P}\left(Z_{1/{\lambda}mbda_+}[n]_n>0\right)\\
&=&{\mathbb P}\left(Z_1[n/{\lambda}mbda_+]_{{\lambda}mbda_+ n}>0\right)\\
&=&{\mathbb P}\left(Z_1[{\lambda}mbda_+n-h_+]_{{\lambda}mbda_+ n}>0\right)\\
&=&{\mathbb P}\left(Z_1[{\lambda}mbda_+n]_{{\lambda}mbda_+ n}>h_+\right)\\
&\leq &{\mathbb P}\left(Z_1[n_+]_{n_+}>r n_+^{2/3}/2\right)\,.
\end{eqnarray*}
Analogously, for
$$n_-:= \frac{n}{{\lambda}mbda_-}<2n\,\,\mbox{ and }\,\,h_{-}:=\left(\frac{1}{{\lambda}mbda_-}-{\lambda}mbda_-\right) n> rn^{2/3}>r n_-^{2/3}/2 \,,$$
we have that
\begin{eqnarray*}
{\mathbb P}\left(Z_{{\lambda}mbda_-}[n+2 n^{2/3}]_n>0\right)&=&{\mathbb P}\left(Z_{{\lambda}mbda_-}[n]_n>-2 n^{2/3}\right)\\
&=&{\mathbb P}\left({\lambda}mbda_-Z_{1}[{\lambda}mbda_-n]_{n/{\lambda}mbda_-}>-2 n^{2/3}\right)\\
&\leq&{\mathbb P}\left(Z_{1}[n_- - h_-]_{n/{\lambda}mbda_-}>-2 n^{2/3}\right)\\
&=&{\mathbb P}\left(Z_{1}[n_-]_{n_-}>h_- -2 n^{2/3}\right)\\
&\leq& {\mathbb P}\left(Z_1[n_-]_{n_-}>(r-4) n_-^{2/3}/2\right)\,.
\end{eqnarray*}
Now one can use Lemma \ref{lem:ExitControl} to finish the proof.
$\Box$\\
\begin{lem}{\lambda}bel{lem:compa}
Let $\delta\in(0,1)$ and $u\in[0,1-\delta)$. Then, on the event $E_n(\delta)$, for all $v\in[u,u+\delta]$ we have that
$$ {\mathcal B}_{n,-}(v)-{\mathcal B}_{n,-}(u)-2\delta^{1-\beta}\leq{\mathcal A}_n(v)-{\mathcal A}_n(u)\,\leq\, {\mathcal B}_{n,+}(v)-{\mathcal B}_{n,+}(u)+4\delta^{1-\beta}\,,$$
where
$${\mathcal B}_{n,\pm}(u):\frac{L_{{\lambda}mbda\pm}[n+2un^{2/3}]_n-L_{{\lambda}mbda\pm}[n]_n-{\lambda}mbda_\pm 2un^{2/3}}{n^{1/3}}\,.$$
\end{lem}
\noindent{\bf Proof\,\,} For fixed $t$, $Z_{\lambda}mbda'[x]_t$ and $Z_{\lambda}mbda[x]_t$ are non-decreasing functions of $x$. Thus, on the event $E_n(\delta)$,
$$Z'_{{\lambda}mbda_+}[n+2un^{2/3}]_n\geq 0\,\mbox{ and }\,Z_{{\lambda}mbda_-}[n+2(u+\delta)n^{2/3}]_n\leq 0\,.$$
By Lemma \ref{lem:LocalComparison}, this implies that, for all $v\in[u,u+\delta]$,
$$L[n+vn^{2/3}]_{n}-L[n+un^{2/3}]_n\leq L_{{\lambda}mbda_+}[n+vn^{2/3}]_{n}-L_{{\lambda}mbda_+}[n+un^{2/3}]_n\,,$$
and
$$L[n+vn^{2/3}]_{n}-L[n+un^{2/3}]_n\geq L_{{\lambda}mbda_-}[n+vn^{2/3}]_{n}-L_{{\lambda}mbda_-}[n+un^{2/3}]_n\,.$$
Since
$$({\lambda}mbda_+-1)(2v-2u)n^{1/3}+v^2-u^2\leq 2\delta^{1-\beta}+2\delta\leq 4\delta^{1-\beta}\,,$$
and
$$({\lambda}mbda_- -1)(2v-2u)n^{1/3}+v^2-u^2\geq -2\delta^{1-\beta}\,,$$
we have that, on the event $E_n(\delta)$,
$${\mathcal A}_n(v)-{\mathcal A}_n(u)\,\leq\, {\mathcal B}_{n,+}(v)-{\mathcal B}_{n,+}(u)+4\delta^{1-\beta}\,.$$
and
$${\mathcal A}_n(v)-{\mathcal A}_n(u)\,\geq\, {\mathcal B}_{n,-}(v)-{\mathcal B}_{n,-}(u)-2\delta^{1-\beta}\,,$$
for all $v\in[u,u+\delta]$.
$\Box$\\
\noindent{\bf Proof of Theorem \ref{thm:Tight}\,\,}
For fixed $u\in[0,1)$ take $\delta>0$ such that $u+\delta\leq1$. By Lemma \ref{lem:compa},
$$\sup_{v\in[u,u+\delta]}|{\mathcal A}_n(v)-{\mathcal A}_n(u)|\leq \max\left\{\sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,\pm}(v)-{\mathcal B}_{n,\pm}(u)| \right\}+4\delta^{1-\beta}\,,$$
on the event $E_n(\delta)$. Hence, for any $\eta>0$,
\begin{eqnarray*}
{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal A}_n(v)-{\mathcal A}_n(u)|>\eta\right)&\leq&{\mathbb P}\left(E_n(\delta)^c\right)\\
&+&{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,+}(v)-{\mathcal B}_{n,+}(u)|>\eta-4\delta^{1-\beta}\right)\\
&+&{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,-}(v)-{\mathcal B}_{n,-}(u)|>\eta-4\delta^{1-\beta}\right)\,.
\end{eqnarray*}
By \eqref{eq:equilibrium},
$$P_n(x):=L_{\lambda}mbda\big([n+x]_n\big)-L_{\lambda}mbda\big([n]_n\big)\,,\,\mbox{ for }\,\,\,x\geq 0\,,$$
is a Poisson process of intensity ${\lambda}mbda$. Since ${\lambda}mbda^{\pm}\to1$ as $n\to\infty$, ${\mathcal B}_{n,-}(u/2)$ and ${\mathcal B}_{n,+}(u/2)$ converge in distribution to a standard Brownian motion ${\mathcal B}$. Thus, by Lemma \ref{lem:tight}, for $\delta<(\eta/8)^{1/(1-\beta)}$,
\begin{eqnarray*}
\limsup_{n\to\infty}{\mathbb P}\left(\sup_{v\in[u,u+\delta]}|{\mathcal A}_n(v)-{\mathcal A}_n(u)|>\eta\right)&\leq& C\delta^{3\beta}+2{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}(2v)-{\mathcal B}(2u)|>\eta-4\delta^{1-\beta}\right)\\
&\leq&C\delta^{3\beta}+2{\mathbb P}\left( \sup_{v\in[0,1]}|{\mathcal B}(v)|>\frac{\eta}{2\sqrt{2\delta}}\right)\,,
\end{eqnarray*}
which implies that (recall that $\beta\in(1/3,1)$)
\begin{equation}{\lambda}bel{tight}
\limsup_{\delta\to 0^+}\frac{1}{\delta}\left(\limsup_{n\to\infty}{\mathbb P}\left(\sup_{v\in[u,u+\delta]}|{\mathcal A}_n(v)-{\mathcal A}_n(u)|>\eta\right)\right)=0\,.
\end{equation}
Since \cite{CG2}
$$\limsup_{n\to\infty}\frac{{\mathbb E} |L[n]_n-2n|}{n^{1/3}}<\infty\,,$$
we have that $\{{\mathcal A}_n(0)\,,\,n\geq 1\}$ is tight. Together with \eqref{tight}, this shows tightness of the collection $\{{\mathcal A}_n\,,\,n\geq 1\}$ in the space of cadlag functions on $[0,1]$, and also that every weak limit lives in the space of continuous functions \cite{Bi}.
$\Box$\\
\section{Proof of Theorem \ref{thm:LocalFluct}}
For simple notation, we will prove the statement for $s=1$ and restrict our selves to $[0,1]$. The reader can then check that rescaling gives the result for general $s>0$, since
\[ L[sn]_n \stackrel{dist.}{=} L[s^{1/2}n]_{s^{1/2}n}.\]
\begin{lem}{\lambda}bel{lem:PoissonTight}
Fix $\gamma'\in(\gamma,2/3)$ and let
$${\lambda}_{\pm}={\lambda}_{\pm}(n):=1\pm\frac{1}{n^{\gamma'/2}}\,.$$
Define the event
$$E_n:=\left\{Z'_{{\lambda}_+}[n]_n\geq0\,\,\mbox{ and }\,\,Z_{{\lambda}_-}[n+n^{\gamma}]_n\leq0\right\}\,.$$
There exists a constant $C>0$ such that
$${\mathbb P}\left(E_n^c\right)\leq \frac{C}{n^{1-3\gamma'/2}}\,$$
for all sufficiently large $n$.
\end{lem}
\noindent{\bf Proof\,\,} Denote $r:=n^{1/3-\gamma'/2}$ and let
$$n_+:={\lambda}mbda_+ n< 2n\,\,\mbox{ and }\,\,h_{+}:=\left({\lambda}mbda_+-\frac{1}{{\lambda}mbda_+}\right) n> rn^{2/3}> r n_+^{2/3}/2 \,,$$
(for all sufficiently large $n$). By \eqref{eq:sym} and \eqref{eq:diagsym},
\begin{eqnarray*}
{\mathbb P}\left(Z'_{{\lambda}mbda_+}[n]_n<0\right)&=&{\mathbb P}\left(Z_{1/{\lambda}mbda_+}[n]_n>0\right)\\
&=&{\mathbb P}\left(Z_1[n/{\lambda}mbda_+]_{{\lambda}mbda_+ n}>0\right)\\
&=&{\mathbb P}\left(Z_1[{\lambda}mbda_+n-h_n]_{{\lambda}mbda_+ n}>0\right)\\
&=&{\mathbb P}\left(Z_1[{\lambda}mbda_+n]_{{\lambda}mbda_+ n}>h_+\right)\\
&\leq&{\mathbb P}\left(Z_1[n_+]_{n_+}>r n_+^{2/3}/2\right)\,.
\end{eqnarray*}
Analogously, for
$$n_-:= \frac{n}{{\lambda}mbda_-}<2n\,\,\mbox{ and }\,\,h_{-}:=\left(\frac{1}{{\lambda}mbda_-}-{\lambda}mbda_-\right) n> rn^{2/3}> r n_-^{2/3}/2 \,,$$
we have that
\begin{eqnarray*}
{\mathbb P}\left(Z_{{\lambda}mbda_-}[n+n^{\gamma}]_n>0\right)&=&{\mathbb P}\left(Z_{{\lambda}mbda_-}[n]_n>-n^{\gamma}\right)\\
&=&{\mathbb P}\left({\lambda}mbda_-Z_{1}[{\lambda}mbda_-n]_{n/{\lambda}mbda_-}>-n^{\gamma}\right)\\
&\leq&{\mathbb P}\left(Z_{1}[n_-]_{n_-}>h_- - n^{\gamma}\right)\\
&\leq& {\mathbb P}\left(Z_1[n_-]_{n_-}>(r-n^{\gamma-2/3}) n_-^{2/3}/2\right)\,,
\end{eqnarray*}
Now one can use Lemma \ref{lem:ExitControl} to finish the proof.
$\Box$\\
\begin{lem}{\lambda}bel{lem:PoissonComparison}
On the event $E_n$, for all $u<v$ in $[0,1]$,
$$\Gamma_n^{-}(v)-\Gamma_n^{-}(u)-\frac{1}{n^{(\gamma'-\gamma)/2}}\leq \Delta_n(v)-\Delta_n(u)\leq \Gamma_n^+(v)-\Gamma_n^+(u)+\frac{1}{n^{(\gamma'-\gamma)/2}}\,,$$
where
$$\Gamma_n^{\pm}(u):=\frac{L_{{\lambda}_\pm}[n+un^{\gamma}]_{n}-L_{{\lambda}_\pm}[n]_n-{\lambda}mbda_\pm un^{\gamma}}{n^{\gamma/2}}\,.$$
\end{lem}
\noindent{\bf Proof\,\,} By Lemma \ref{lem:LocalComparison}, if $Z'_{{\lambda}mbda_+}[n]_n\geq 0$ then
$$L[n+vn^{\gamma}]_{n}-L[n+un^{\gamma}]_n\leq L_{{\lambda}mbda_+}[n+vn^{\gamma}]_{n}-L_{{\lambda}mbda_+}[n+un^{\gamma}]_n\,,$$
and if $Z_{{\lambda}mbda_-}[n+n^{\gamma}]_n\leq 0$ then
$$L[n+vn^{\gamma}]_{n}-L[n+un^{\gamma}]_n\geq L_{{\lambda}mbda_-}[n+vn^{\gamma}]_{n}-L_{{\lambda}mbda_-}[n+un^{\gamma}]_n\,.$$
Using that ${\lambda}mbda_{\pm}:=1\pm n^{-\gamma'/2}$, one can finish the proof of the lemma.
$\Box$\\
\noindent{\bf Proof of Theorem \ref{thm:LocalFluct}\,\,} By Lemma \ref{lem:PoissonComparison}, on the event $E_n^c$,
$$| \Delta_n(v)-\Delta_n(u)|\leq\max \left\{|\Gamma_n^{\pm}(v)-\Gamma_n^{\pm}(u)|\right\}+\frac{1}{n^{(\gamma'-\gamma)/2}}\,.$$
Thus, by Lemma \ref{lem:PoissonTight},
\begin{eqnarray*}
{\mathbb P}\left(\sup_{v\in[u,u+\delta]} | \Delta_n(v)-\Delta_n(u)|>\eta\right)&\leq&{\mathbb P}\left(\sup_{v\in[u,u+\delta]} |\Gamma^+_n(v)-\Gamma^+_n(u)|+\frac{1}{n^{(\gamma'-\gamma)/2}}>\eta\right)\\
&+&{\mathbb P}\left(\sup_{v\in[u,u+\delta]} |\Gamma^-_n(v)-\Gamma^-_n(u)|+\frac{1}{n^{(\gamma'-\gamma)/2}}>\eta\right)\\
&+&{\mathbb P}\left(E_n^c\right)\,.
\end{eqnarray*}
As before, ${\lambda}mbda^{\pm}\to1$ as $n\to\infty$, which implies that
$$\limsup_{n\to\infty}{\mathbb P}\left(\sup_{v\in[u,u+\delta]} | \Delta_n(v)-\Delta_n(u)|>\eta\right)\leq 2{\mathbb P}\left(\sup_{v\in[0,\delta]} |B(v)|>\eta\right)=2{\mathbb P}\left(\sup_{v\in[0,1]} |B(v)|>\frac{\eta}{\sqrt{\delta}}\right)\,,$$
and hence
\begin{equation}{\lambda}bel{eq:tight}
\limsup_{\delta\to0^+}\frac{1}{\delta}\left(\limsup_{n\to\infty}{\mathbb P}\left(\sup_{v\in[u,u+\delta]} | \Delta_n(v)-\Delta_n(u)|>\eta\right)\right)=0\,.
\end{equation}
Since $\Delta_n(0)=0$, \eqref{eq:tight} implies tightness of the collection $\{\Delta_n\,,\,n\geq 1\}$ in the space of cadlag functions on $[0,1]$, and also that every weak limit lives in the space of continuous functions \cite{Bi}.
The finite dimensional distributions of the limiting process can be obtained in the same way. Indeed, by Lemma \ref{lem:PoissonTight} and Lemma \ref{lem:PoissonComparison}, for $u_1,\dots,u_k\in[0,1]$ and $a_1,\dots,a_k,\in{\mathbb R}$,
$${\mathbb P}\left(\cap_{i=1}^k\left\{\Delta_n(u_i)\leq a_i\right\}\right)\geq {\mathbb P}\left(\cap_{i=1}^k\left\{\Gamma^+_n(u_i)\leq a_i -\frac{1}{n^{(\gamma'-\gamma)/2}}\right\}\right)-{\mathbb P}\left(E_n^c\right)\,,$$
and
$${\mathbb P}\left(\cap_{i=1}^k\left\{\Delta_n(u_i)\leq a_i\right\}\right)\leq{\mathbb P}\left(\cap_{i=1}^k\left\{\Gamma^-_n(u_i)\leq a_i+\frac{1}{n^{(\gamma'-\gamma)/2}}\right\}\right)+{\mathbb P}\left(E_n^c\right)\,,$$
which shows that the finite dimensional distributions of $\Delta_n$ converge to the finite dimensional distributions of the standard Brownian motion process.
$\Box$\\
\section{Proof of Theorem \ref{thm:LocalAiry}}
\begin{lem}{\lambda}bel{lem:local}
Fix $\beta\in(0,1/2)$ and for ${\varepsilon }ilon\in(0,1)$ let
$${\lambda}mbda_\pm={\lambda}mbda_\pm(n,{\varepsilon }ilon):=1\pm\frac{{\varepsilon }ilon^{-\beta}}{n^{1/3}}\,.$$
Define the event
$$E_n({\varepsilon }ilon):=\left\{Z'_{{\lambda}mbda_{+}}([n]_n)\geq 0\,\,\mbox{ and }\,\,Z_{{\lambda}mbda_{-}}([n+n^{2/3}]_n)\leq 0\right\}\,.$$
There exists a constant $C>0$ such that, for all sufficiently small ${\varepsilon }ilon>0$,
$$\limsup_{n\to\infty}{\mathbb P}\left(E_n({\varepsilon }ilon)^c\right)\leq C{\varepsilon }ilon^{3\beta}\,.$$
\end{lem}
\noindent{\bf Proof\,\,} The same proof as in Lemma \ref{lem:tight} applies.
$\Box$\\
\begin{lem}{\lambda}bel{lem:localcompa}
On the event $E_n({\varepsilon }ilon)$, for all $u\in[0,1-\delta)$ and $v\in[u,u+\delta]$, we have that
$$ {\mathcal B}_{n,-}({\varepsilon }ilon v)-{\mathcal B}_{n,-}({\varepsilon }ilon u)-2\delta{\varepsilon }ilon^{1-\beta}\leq{\mathcal A}_n({\varepsilon }ilon v)-{\mathcal A}_n({\varepsilon }ilon u)\,\leq\, {\mathcal B}_{n,+}({\varepsilon }ilon v)-{\mathcal B}_{n,+}({\varepsilon }ilon u)+4\delta{\varepsilon }ilon^{1-\beta}\,.$$
\end{lem}
\noindent{\bf Proof\,\,} The same proof as in Lemma \ref{lem:compa} applies. Note that in this case we have
\[ ({\lambda}mbda_+ - 1)(2{\varepsilon } v- 2{\varepsilon } u) \leq 2{\varepsilon }^{1-\beta}\delta.\]
$\Box$\\
\noindent{\bf Proof of Theorem \ref{thm:LocalAiry}\,\,}
For $u\in[0,1]$, let
$${\mathcal A}_n^{\varepsilon }ilon(u):={\varepsilon }ilon^{-1/2}\left({\mathcal A}_n({\varepsilon }ilon u)-{\mathcal A}_n(0)\right)\,\mbox{ and }\,{\mathcal B}^{\varepsilon }ilon_{n,\pm}(u):={\varepsilon }ilon^{-1/2}{\mathcal B}_{n,\pm}({\varepsilon }ilon u)\,.$$
By Lemma \ref{lem:localcompa}, on the event $E_n({\varepsilon }ilon)$, for all $v\in[u,u+\delta]$,
$$ {\mathcal B}^{\varepsilon }ilon_{n,-}(v)-{\mathcal B}^{\varepsilon }ilon_{n,-}(u)-2\delta{\varepsilon }ilon^{1/2-\beta}\leq{\mathcal A}^{\varepsilon }ilon_n(v)-{\mathcal A}^{\varepsilon }ilon_n(u)\,\leq\, {\mathcal B}^{\varepsilon }ilon_{n,+}(v)-{\mathcal B}^{\varepsilon }ilon_{n,+}(u)+4\delta{\varepsilon }ilon^{1/2-\beta}\,,$$
which shows that
$$\sup_{v\in[u,u+\delta]}|{\mathcal A}^{\varepsilon }ilon_n(v)-{\mathcal A}^{\varepsilon }ilon_n(u)|\leq \max\left\{\sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,\pm}^{\varepsilon }ilon(v)-{\mathcal B}_{n,\pm}^{\varepsilon }ilon(u)| \right\}+4\delta{\varepsilon }ilon^{1/2-\beta}\,.$$
Therefore,
\begin{eqnarray*}
{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal A}^{\varepsilon }ilon_n(v)-{\mathcal A}^{\varepsilon }ilon_n(u)|>\eta\right)&\leq&{\mathbb P}\left(E_n({\varepsilon }ilon)^c\right)\\
&+&{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,+}^{\varepsilon }ilon(v)-{\mathcal B}_{n,+}^{\varepsilon }ilon(u)|>\eta-4\delta{\varepsilon }ilon^{1/2-\beta}\right)\\
&+&{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}_{n,-}^{\varepsilon }ilon(v)-{\mathcal B}_{n,-}^{\varepsilon }ilon(u)|>\eta-4\delta{\varepsilon }ilon^{1/2-\beta}\right)\,.
\end{eqnarray*}
Since ${\mathcal A}_n^{\varepsilon }ilon$ is converging to $\cal A^{\varepsilon }ilon$, and ${\mathcal B}_{n,\pm}^{\varepsilon }ilon$ is converging to a Brownian motion ${\mathcal B}$, the preceding inequality implies that
$${\mathbb P}\left(\sup_{v\in[u,u+\delta]}|{\mathcal A}^{\varepsilon }ilon(v)-{\mathcal A}^{\varepsilon }ilon(u)|>\eta\right)\leq C{\varepsilon }ilon^{3\beta}+2{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal B}(2v)-{\mathcal B}(2u)|>\eta-4\delta{\varepsilon }ilon^{1/2-\beta}\right)\,.$$
Hence
$$\limsup_{{\varepsilon }ilon\to 0^+}{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal A}^{\varepsilon }ilon(v)-{\mathcal A}^{\varepsilon }ilon(u)|>\eta\right)\leq 2{\mathbb P}\left( \sup_{v\in[0,1]}|{\mathcal B}(v)|>\frac{\eta}{\sqrt{2\delta}}\right)\,,$$
which shows that
\begin{equation}{\lambda}bel{eq:BlowTight}
\limsup_{\delta\to 0^+}\frac{1}{\delta}\left(\limsup_{{\varepsilon }ilon\to 0^+}{\mathbb P}\left( \sup_{v\in[u,u+\delta]}|{\mathcal A}^{\varepsilon }ilon(v)-{\mathcal A}^{\varepsilon }ilon(u)|>\eta\right)\right)=0\,.
\end{equation}
Since ${\mathcal A}^{{\varepsilon }ilon}(0)=0$, by \eqref{eq:BlowTight} we have that $\left\{{\mathcal A}^{\varepsilon }ilon\,,\,{\varepsilon }ilon\in(0,1]\right\}$ is tight \cite{Bi}.
The finite dimensional distributions of the limiting process can be obtained in the same way. Indeed, by Lemma \ref{lem:localcompa}, for $u_1,\dots,u_k\in[0,1]$ and $a_1,\dots,a_k,\in{\mathbb R}$,
$${\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal A}^{\varepsilon }ilon_n(u_i)\leq a_i\right\}\right)\leq{\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal B}^{\varepsilon }ilon_{-,n}(u_i)\leq a_i+4{\varepsilon }ilon^{1/2-\beta}\right\}\right)+{\mathbb P}\left(E_n({\varepsilon }ilon)^c\right)\,,$$
and
$${\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal A}^{\varepsilon }ilon_n(u_i)\leq a_i\right\}\right)\geq {\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal B}^{\varepsilon }ilon_{+,n}(u_i)\leq a_i-4{\varepsilon }ilon^{1/2-\beta}\right\}\right)-{\mathbb P}\left(E_n({\varepsilon }ilon)^c\right)\,.$$
Thus, by Lemma \ref{lem:local},
$$ {\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal A}^{\varepsilon }ilon(u_i)\leq a_i\right\}\right)\leq {\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal B}(2u_i)\leq a_i+4{\varepsilon }ilon^{1/2-\beta}\right\}\right)+C{\varepsilon }ilon^{3\beta}\,,$$
and
$$ {\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal A}^{\varepsilon }ilon(u_i)\leq a_i\right\}\right)\geq {\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal B}(2u_i)\leq a_i-4{\varepsilon }ilon^{1/2-\beta}\right\}\right)-C{\varepsilon }ilon^{3\beta}\,,$$
which proves that,
\begin{equation}{\lambda}bel{eq:BlowDist}
\lim_{{\varepsilon }ilon\to 0^+}{\mathbb P}\left(\cap_{i=1}^k\left\{{\mathcal A}^{\varepsilon }ilon(u_i)\leq a_i\right\}\right)={\mathbb P}\left(\cap_{i=1}^k\left\{\sqrt{2}{\mathcal B}(u_i)\leq a_i\right\}\right)\,.
\end{equation}
$\Box$\\
\end{document} |
\begin{document}
\title{Generalized Euler classes, differential forms and commutative DGAs}
\begin{abstract}
In the context of commutative differential graded algebras over $\mathbb Q$, we show that an iteration of ``odd spherical fibration" creates a ``total space" commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its ``fibrations" with ``fiber" of finite cohomological dimension, the induced map on cohomology is injective.
\end{abstract}
\section{Introduction}
In geometry, one would like to know which rational cohomology classes in a base space can be annihilated by pulling up to a fibration over the base with finite dimensional fiber. One knows that if $[x]$ is a $2n$-dimensional rational cohomology class on a finite dimensional CW complex $X$, there is a $(2n-1)$-sphere fibration over $X$ so that $[x]$ pulls up to zero in the cohomology groups of the total space. In fact there is a complex vector bundle $V$ over $X$ of rank $n$ whose Euler class is a multiple of $[x]$. Thus this multiple is the obstruction to a nonzero section of $V$, and vanishes when pulled up to the part of $V$ away from the zero section, which deformation retracts to the unit sphere bundle.
Rational homotopy theory provides a natural framework to study this type of questions, where topological spaces are replaced by commutative differential graded algebras (commutative DGAs) and topological fibrations replaced by algebraic fibrations. This will be the context in which we work throughout the paper. The reader can read more in \cite{MR1802847, MR2355774, MR0646078} about the topological meaning of the results of this paper from the perspective of rational homotopy theory of manifolds and general spaces.
The first theorem (Theorem $\ref{thm:itoddsphere}$) of the paper states that the above construction, when iterated, creates a ``total space" commutative DGA with only odd degree cohomology.
\begin{thma}
For each commutative DGA $(A, d)$, there exists an iterated odd algebraic spherical fibration $(TA, d) $ over $(A, d)$ so that all even cohomology [except dimension zero] vanishes.
\end{thma}
Our next theorem (Theorem $\ref{thm:oddsphere}$) then limits the odd degree classes that can be annihilated by fibrations whose fiber has finite cohomological dimension.
\begin{thmb}
Let $(B, d)$ be a connected commutative DGA such that $H^{2k}(B)=0$ for all $0 < 2k \leq 2N$. If $\iota \colon (B, d) \to (B\otimes \Lambda V, d) $ is an algebraic fibration whose algebraic fiber has finite cohomological dimension, then the induced map
\[ \iota_\ast \colon \bigoplus_{i\leq 2N} H^i(B)\to \bigoplus_{i\leq 2N} H^i(B\otimes \Lambda V)\]
is injective.
\end{thmb}
It follows from the two theorems above that the iterated odd spherical fibration construction is universal for cohomology classes that pull back to zero by any fibrations whose fiber has finite cohomological dimension.
The paper is organized as follows. In Section $\ref{sec:pre}$, we recall some definitions from rational homotopy theory. In Section $\ref{sec:gysin}$, we use iterated algebraic spherical fibrations to prove Theorem A. In Section $\ref{sec:algsphere}$, we define bouquets of algebraic spheres and analyze their minimal models. In Section $\ref{sec:main}$, we prove Theorem B.
The work in this paper started during a visit of the authors at IHES. The authors would like to thank IHES for its hospitality and for providing a great environment.
\section{Preliminaries} \label{sec:pre}
We recall some definitions related to commutative differential graded algebras. For more details, see \cite{MR1802847, MR2355774, MR0646078}.
\begin{definition}
A commutative differential graded algebra (commutative DGA) is a graded algebra $B = \oplus_{i\geq 0} B^{i}$ over $\mathbb Q$ together with a differential $d\colon B^i \to B^{i+1}$ such that $d^2 = 0$, $xy = (-1)^{ij} yx,$ and $d(xy) = (d x) y + (-1)^{i} x (d y)$, for all $x\in B^{i}$ and $y\in B^{j}$.
\end{definition}
\begin{definition}
\begin{enumerate}[(1)]
\item
A commutative DGA $(B, d)$ is called connected if $B^0 = \mathbb Q$.
\item A commutative DGA $(B, d) $ is called simply connected if $(B, d)$ is connected and $H^1(B) = 0$.
\item
A commutative DGA $(B, d)$ is of finite type if $H^k(B)$ is finite dimensional for all $k\geq 0$.
\item A commutative DGA $(B,d )$ has finite cohomological dimension $d$, if $d$ is the smallest integer such that $H^k(B) = 0 $ for all $k> d$.
\end{enumerate}
\end{definition}
\begin{definition}
A connected commutative DGA $(B, d)$ is called a model algebra if as a commutative graded algebra it is free on a set of generators $\{x_\alpha\}_{\alpha\in \Lambda}$ in positive degrees, and these generators can be partially ordered so that $d x_\alpha$ is an element in the algebra generated by $x_\beta$ with $\beta < \alpha$.
\end{definition}
\begin{definition}
A model algebra $(B, d)$ is called minimal if for each generator $x_\alpha$, $d x_\alpha$ has no linear term, that is,
\[ d(B) \subset B^+ \cdot B^+, \textup{ where } B^+ = \oplus_{k>0} B^k.\]
\end{definition}
\begin{remark}
For every connected commutative DGA $(A, d_A)$, there exists a minimal model algebra $(\mathcal M(A), d)$ and a morphism $\varphi\colon (\mathcal M(A), d)\to (A, d_A)$ such that $\varphi$ induces an isomorphism on cohomology. $(\mathcal M(A), d)$ is called a minimal model of $(A, d)$, and is unique up to isomorphism. See page $288$ of \cite{MR0646078} for more details, cf. \cite{MR1802847, MR2355774}.
\end{remark}
\begin{definition}\label{def:algfib}
\begin{enumerate}[(i)]
\item An algebraic fibration (also called \textit{relative model algebra}) is an inclusion of commutative DGAs $(B, d) \hookrightarrow (B\otimes \Lambda V, d)$ with $V = \oplus_{k\geq 1} V^k$ a graded vector space; moreover, $V = \bigcup_{n=0}V(n)$, where $V(0) \subseteq V(1) \subseteq V(2) \subseteq \cdots $ is an increasing sequence of graded subspaces of $V$ such that
\[ d : V(0) \to B \quad \textup{ and } \quad d: V(n) \to B\otimes \Lambda V(n-1), \quad n\geq 1,\]
where $\Lambda V$ is the free commutative DGA generated by $V$.
\item An algebraic fibration is called minimal if
\[ \textup{Im}(d) \subset B^+\otimes \Lambda V + B\otimes \Lambda^{\geq 2} V.\]
\end{enumerate}
\end{definition}
Let $\iota\colon (B, d) \hookrightarrow (B\otimes \Lambda V, d)$ be an algebraic fibration. Suppose $B$ is connected. Consider the canonical augmentation morphism $\varepsilon: (B, d) \to (\mathbb Q, 0)$ defined by $\varepsilon (B^+) = 0$. It naturally induces a commutative DGA:
\[ (\Lambda V, \bar d) := \mathbb Q\otimes_B (B\otimes \Lambda V, d).\]
We call $(\Lambda V, \bar d) $ the algebraic fiber of the given algebraic fibration.
\section{Iterated odd spherical algebraic fibrations}\label{sec:gysin}
In this section, we show that for each commutative DGA, there exists an iterated odd algebraic spherical fibration over it such that the total commutative DGA has only odd degree cohomology.
Let $(B, d)$ be a connected commutative DGA. An \emph{odd algebraic spherical fibration} over $(B, d)$ is an inclusion of commutative DGAs of the form
\[ \varphi: (B, d) \to (B\otimes \Lambda (x), d), \]
such that $d x \in B$, where $x$ has degree $2k-1$ and $\Lambda(x)$ is the free commutative graded algebra generated by $x$. The element $e = d x \in B$ is called the Euler class of this algebraic spherical fibration.
\begin{proposition}\label{prop:gysin}
Let $(B, d)$ be a commutative DGA. For every even dimensional class $\beta\in H^{2k}(B)$ with $k>0$, there exists an odd algebraic spherical fibration $\varphi\colon (B, d) \to (B\otimes \Lambda(x), d)$ such that its Euler class is equal to $\beta$ and the kernel of the map $\varphi_\ast\colon H^{i+2k}(B) \to H^{i+2k}(B\otimes \Lambda (x) )$ is $H^i(B)\cdot\beta = \{a\cdot \beta \mid a\in H^i(B)\}$.
\end{proposition}
\begin{proof}
Let $(B\otimes \Lambda (x), d)$ be the commutative DGA obtained from $(B, d)$ by adding a generator $x$ of degree $2k-1$ and defining its differential to be $d x = \beta$. We have the following short exact sequence
\[ 0 \to (B, d) \to (B\otimes \Lambda (x), d) \to ( B \otimes (\mathbb Q\cdot x), d\otimes \textup{Id}) \to 0,\]
which induces a long exact sequence \[ \cdots \to H^{i-1}(B\otimes (\mathbb Q \cdot x)) \to H^i(B) \to H^i(B\otimes \Lambda (x)) \to H^i(B\otimes (\mathbb Q \cdot x)) \to \cdots.
\]
Applying the identification $H^{i + (2k-1)}(B\otimes (\mathbb Q \cdot x)) \cong H^i(B)$, we obtain the following Gysin sequence
\[ \cdots \to H^{i}(B) \xrightarrow{\cup e} H^{i+2k}(B) \xrightarrow{\varphi_\ast} H^{i+2k}(B\otimes \Lambda (x)) \xrightarrow{\partial_{i+1}} H^{i+1}(B) \to \cdots. \]
This finishes the proof.
\end{proof}
\begin{definition}
An \emph{iterated odd algebraic spherical fibration} over $(B, d)$ is algebraic fibration $(B, d) \hookrightarrow (B\otimes \Lambda V, d)$
such that $V^k=0$ for $k$ even. This fibration is called \emph{finitely iterated odd algebraic spherical fibration} if $\dim V < \infty$.
\end{definition}
Now let us prove the main result of this section.
\begin{theorem}\label{thm:itoddsphere}
For each commutative DGA $(A, d)$, there exists an iterated odd algebraic spherical fibration $(TA, d) $ over $(A, d)$ such that all even cohomology [except dimension zero] vanishes.
\end{theorem}
\begin{proof}
We will construct $TA$ by induction. In the following, for notational simplicity, we shall omit the differential $d$ from our notation.
Let $\mathcal A_0 = A$. Suppose we have defined the iterated odd algebraic spherical fibration $\mathcal A_{m-1}$ over $A$. Fix a basis of $H^{2k}(\mathcal A_{m-1})$ for each $k>0$. Denote the union of all these bases by $\{a_i\}_{i\in I}$. Define $W_{m-1}$ to be a $\mathbb Q$ vector space with basis $\{x_i\}_{i\in I}$, where $|x_i|= |a_i| -1$.
We define $\mathcal A_{m}$ to be the iterated odd algebraic spherical fibration $ \mathcal A_{m-1} \otimes \Lambda (W_{m-1})$ over $\mathcal A_{m-1}$ with $d x_i = a_i$ for all $i\in I$. The inclusion map $\iota\colon \mathcal A_{m-1} \hookrightarrow \mathcal A_{m}$ induces the zero map $\iota_\ast = 0 \colon H^{2k}(\mathcal A_{m-1}) \to H^{2k}(\mathcal A_{m})$ for all $k>0$.
By construction, $\mathcal A_m$ is also an iterated odd algebraic spherical fibration.
Finally, we define $TA$ to be the direct limit of $\mathcal A_m$ under the inclusions $\mathcal A_m \hookrightarrow \mathcal A_{m+1}$. Clearly, $TA$ is an iterated odd algebraic spherical fibration over $A$. More precisely, let $V = \bigcup_{i=0}^\infty W_{i}$. We have $TA = A\otimes \Lambda V$ with the filtration of $V$ given by $V(n) = \bigcup_{i=0}^n W_{i}$. Moreover, we have $H^{2k}(TA) = 0$ for all $2k>0$. This completes the proof.
\end{proof}
\begin{remark}\label{rm:finsphere}
If an element $\alpha \in H^\bullet(A) $ maps to zero in $H^{\bullet}(TA)$, then there exists a subalgebra $S_\alpha$ of $TA$ such that $S_{\alpha}$ is a \emph{finitely} iterated odd algebraic spherical fibration over $A$ and $\alpha$ maps to zero in $H^\bullet(S_{\alpha})$.
\end{remark}
\section{Bouquets of algebraic spheres}
\label{sec:algsphere}
In this section, we introduce a notion of bouquets of algebraic spheres. It is an algebraic analogue of usual bouquets of spheres in topology.
\begin{definition}\label{def:algsphere}
For a given set of generators $X= \{x_i\}$ with $x_i$ having odd degree $|x_i|$, we define the bouquet of odd algebraic spheres labeled by $X$ to be the following commutative DGA
\[\mathcal S(X) = \left( \bigwedge_{x_i\in X} \mathbb Q[x_i]\right)\Big/\langle x_ix_j = 0 \mid \textup{all } i, j \rangle \]
with the differential $d = 0$.
\end{definition}
\begin{proposition}\label{prop:algsphere}
Let $\mathcal S(X)$ be a bouquet of odd algebraic spheres, and $\mathcal M(X) = (\Lambda V, d)$ be its minimal model. Then $\mathcal M(X)$ satisfies the following properties:
\begin{enumerate}[(i)]
\item $\mathcal M$ has no even degree generators, that is, $V$ does not contain even degree elements;
\item each element in $H^{\geq 1}(\mathcal M(X))$ is represented by a generator, that is, an element in $V$.
\end{enumerate}
\end{proposition}
\begin{proof}
This is a special case of Koszul duality theory, cf. \cite[Chapter 3, 7 \& 13]{MR2954392}. Since $\mathcal S = \mathcal S(X)$ has zero differential, we may forget its differential and view it as a graded commutative algebra. An explicit construction of a minimal model of $\mathcal S$ is given as follows: first take the Koszul dual coalgebra $\mathcal S^{\text{\textexclamdown}}$ of $\mathcal S$; then apply the cobar construction to $\mathcal S^{\text{\textexclamdown}}$, and denote the resulting commutative DGA by $\Omega \mathcal S^{\text{\textexclamdown}}$. By Koszul duality, $\mathcal M(X) \coloneqq \Omega \mathcal S^{\text{\textexclamdown}}$ is a minimal model of $\mathcal S$.
More precisely, set $W = \bigoplus_{i\geq 0} W_{i}$ to be the graded vector space spanned by $X$. Let $sW$ (resp. $s^{-1}W$) be the suspension (resp. desuspension) of $W$, that is, $(sW)_{i-1} = W_i$ (resp. $(s^{-1}W)_{i} = W_{i-1}$). Let $\mathcal L^c = \mathcal L^c(sW)$ be the cofree Lie coalgebra generated by $sW$. More explicitly, let $T^c(sW) = \bigoplus_{n\geq 0}(sW)^{\otimes n}$ be the tensor coalgebra, and $T^c(sW)^+ = \bigoplus_{n\geq 1} (sW)^{\otimes n}$. The coproduct on $T^c(sW)$ naturally induces a Lie cobraket on $T^c(sW)$. Then we have $\mathcal L^c(sW) = T^c(sW)^+/T^c(sW)^+\ast T^c(sW)^+$, where $\ast$ denotes the shuffle multiplication. With the above notation, we have $\mathcal S^{\text{\textexclamdown}} \cong \mathcal L^c$. The cobar construction of $\mathcal L^c$ is given explicitly by
\[ \mathbb Q\to s^{-1}\mathcal L^c \xrightarrow{\ d \ } \Lambda^2 (s^{-1}\mathcal L^c) \to \cdots \to \Lambda^n(s^{-1}\mathcal L^c) \to \cdots \]
with the differential $d$ determined by the Lie cobraket of $\mathcal L^c$. Now the desired properties of $\mathcal M(X)$ follow from this explicit construction.
\end{proof}
\begin{remark}
In the special case of a bouquet of odd algebraic spheres where the cohomology of a commutative DGA model is that of a circle or the first Betti number is zero, this was discussed by Baues \cite[Corollary 1.2]{MR0442922} and by Halperin and Stasheff \cite[Theorem 1.5]{MR539532}.
\end{remark}
\section{Main theorem} \label{sec:main}
In this section, we show that if a commutative DGA has cohomology, up to a certain degree, isomorphic to that of a bouquet of odd algebraic spheres, then its minimal model is isomorphic to that of the bouquet of odd algebraic spheres, up to that given degree. Then we apply it to prove that if a commutative DGA has only odd degree cohomology up to a certain degree, then all nonzero cohomology classes
up to that degree will never pull back to zero by any algebraic fibration whose fiber has finite cohomological dimension.
Suppose $B$ is a connected commutative DGA of finite type such that $H^{2k}(B)=0$ for all $ 0 < 2k \le 2N$. Let $X_i$ be a basis of $H^{i}(B)$ and $X = \bigcup_{i=1}^{2N+1}X_i$. Let $M = \mathcal M(X)$ be the bouquet of odd algebraic spheres labeled by $X$ from Definition $\ref{def:algsphere}$. Then we have $H^{i}(M) \cong H^i(B)$ for all $0\leq i \leq 2N$. Let $M_k \subset M$ be the subalgebra generated
by the generators of degree $\leq k$.
\begin{lemma}Let $k$ be an odd integer. Then $H^{k+2}(M_k)=H^{k+1}(M_k) =0$.
\end{lemma}
\begin{proof}
$H^{k+1}(M_k)=0$ as $H^{k+1}(M_k) \to H^{k+1}(M)=0$ is injective.
By Proposition $\ref{prop:algsphere}$ above, $M$ has no even-degree generators. In particular, we have $M_k=M_{k+1}$. Moreover,
$H^{\geq 1}(M)$ is spanned by odd-degree generators. From the first observation it
follows that the map $H^{k+2}(M_k) \to H^{k+2}(M)$ is injective, and from the second that its range is $0$.
\end{proof}
It follows that for an odd $k$, we have $M_{k+2} =M_k \otimes \Lambda (V[k+2])$ as an algebra, where the vector space $V = V_1 \oplus V_2 $ is placed at degree $(k+2)$, with $V_1 \cong H^{k+2}(M)$ and $V_2 = H^{k+3}(M_{k})$.
The differential can be described as follows. It suffices to define $d \colon V \to M_{k}$. We define $d=0$ on $V_1$.
To define $d$ on $V_2$, let us choose a basis $\{a_i\}_{i\in I}$ of $H^{k+3}(M_k)$. Let $\{\tilde a_i\}_{i\in I}$ be the corresponding basis of $V_2$. Then we define $d \tilde a_i =a_i$.
\begin{proposition} \label{prop:constr}
For each odd integer $k\le 2N$, there exists a morphism $\varphi_k \colon M_k \to B$ such that
the induced map on cohomology $H^i(M_k)\cong H^i(M) \to H^i(B)$ is an isomorphism for $i \le k$.
\end{proposition}
\begin{proof}We construct the maps $\varphi_k$ by induction. By the previous lemma and the fact that $M$ has no even degree generators, it suffices to define $\varphi_k$ for odd integers $k$ . The case where $k=1$ is clear.
Now assume that we have constructed $\varphi_n$, with $n$ an odd integer $\leq 2N-3$. We shall extend $\varphi_n$ to a morphism $\varphi_{n+2}$ on $M_{n+2} = M_n\otimes \Lambda (V[n+2])$, where the vector space $V = V_1 \oplus V_2 $ is placed at degree $(n+2)$, with $V_1 \cong H^{n+2}(M)$ and $V_2 = H^{n+3}(M_{n})$. It suffices to define $\varphi_{n+2}$ on $V$. Let $\{b_j\}_{j\in J}$ be a basis of $H^{n+2}(B)$. Since $H^{n+2}(M)\cong H^{n+2}(B)$, let $\{\tilde b_j\}_{j\in J}$ be the corresponding basis of $V_1$. We define $\varphi_{n+2}$ on $V_1$ by setting $\varphi_{n+2}(\tilde b_j) = b_j$. Similarly, choose a basis $\{c_\lambda\}_{\lambda \in K}$ of $H^{n+3}(M_n)$, and let $\{\tilde c_\lambda\}_{\lambda \in K}$ be the corresponding
basis of $V_2$. Since $H^{n+3}(B) = 0$, for each $c_\lambda \in M_n$, there exists $\theta_\lambda\in B$ such that $\varphi_n(c_\lambda) = d \theta_\lambda$. We define $\varphi_{n+2}$ on $V_2$ by setting $\varphi_{n+2}(\tilde c_\lambda) =\theta_\lambda$.
By construction, the induced map $(\varphi_{n+2})_\ast$ on $H^i$ agrees with $(\varphi_n)_\ast$ for $i\leq n+1$ and $(\varphi_{n+2})_\ast$ is an isomorphism on $H^{2n+2}$. This finishes the proof.
\end{proof}
Now let $\mathcal{M}_B$ be a minimal model of $B$ and $(\mathcal{M}_B)_k$ be the subalgebra generated by the generators of degree $\le k$. Combining the above results, we have proved the following proposition.
\begin{proposition} \label{prop:sw} The commutative DGAs $(\mathcal{M}_B)_{2N-1}$ and $M_{2N-1}$ are isomorphic.
\end{proposition}
Moreover, we have the following result, which is an immediate consequence of the construction in Proposition $\ref{prop:constr}$.
\begin{corollary}\label{cor:tosphere}
Let $B$ be a connected commutative DGA such that $H^{2i}(B)=0$ for all $ 0 < 2i \le 2N$. Let $\alpha$ be a nonzero class in $H^{2k+1}(\mathcal{M}_B)$ with $2k+1<2N$.
Then there exists a morphism $\psi \colon \mathcal{M}_B \to (\Lambda(\eta), 0)$ such that $\psi_*(\alpha)=[\eta]$, where $\eta$ has degree $2k+1$ and $\Lambda(\eta)$ is the free commutative graded algebra generated by $\eta$.
\end{corollary}
\begin{proof} From the description of the minimal model $\mathcal M_B$ of $B$, it follows that $\mathcal M_B$ has a set of generators such that
all the cohomology groups up to degree $(2N-1)$ is generated by the cohomology classes of these generators; moreover we can choose these generators
so that the given class $\alpha$ is represented by a generator, say, $a$. Then we define $\psi$ by mapping $a$ to $\eta$ and the other generators to $0$.
\end{proof}
An inductive application of the same argument above proves the following.
\begin{proposition}
Suppose $(C, d)$ is a connected commutative DGA with $H^{2k}(C) = 0 $ for all $2k>0$. Let $X_i$ be a basis of $H^i(C)$ and $X_C = \bigcup_{i=1}^\infty X_i$. Then the bouquet of odd algebraic spheres $\mathcal M(X_C)$ is a minimal model of $(C, d)$.
\end{proposition}
Applying the above proposition to the commutative DGA $(TA, d)$ from Theorem $\ref{thm:itoddsphere}$ immediately gives us the following corollary.
\begin{corollary}\label{cor:ta}
With the same notation as above, the minimal model of $(TA, d)$ is isomorphic to a bouquet of odd algebraic spheres.
\end{corollary}
Before proving the main theorem of this section, we shall prove the following special case first.
\begin{theorem}\label{thm:oddsphere}
Let $(\Lambda (x), d)$ be the commutative DGA generated by $x$ of degree $2k+1\geq 1$ such that $d x=0$. For any algebraic fibration $\varphi: (\Lambda(x), d) \to (\Lambda(x)\otimes \Lambda V, d)$ whose algebraic fiber $(\Lambda V, \bar d )$ has finite cohomological dimension, the map $\varphi_\ast: H^j(\Lambda (x)) \to H^j(\Lambda(x)\otimes \Lambda V)$ is injective for all $j$.
\end{theorem}
\begin{proof}
The case where $2k+1 = 1$ is trivial. Let us assume $2k+1>1$ in the rest of the proof.
Let $\varphi\colon (\Lambda V, d) \hookrightarrow (\Lambda(x)\otimes\Lambda V, d)$ be any algebraic fibration whose algebraic fiber has finite cohomological dimension. It suffices to show that $\varphi_\ast\colon H^{2k+1}(\Lambda(x)) \to H^{2k+1}(\Lambda(x)\otimes \Lambda V)$ is injective, since the induced map $\varphi_\ast$ on $H^i$ is automatically injective for $i\neq 2k+1$.
Now suppose to the contrary that
\[\varphi_\ast(x) = 0 \textup{ in } H^{2k+1}(\Lambda(x)\otimes \Lambda V). \]
Then we have $ x = d (w\cdot x + v)$
for some $w, v\in \Lambda V$. By inspecting the degrees of the two sides, one sees that $w = 0$. Therefore, we have $x = d v$ for some $v\in \Lambda V$. It follows that $\bar d v = 0$.
Now let $n\in \mathbb N$ be the smallest integer such that $[v^n] = 0$ in $H^{\bullet}(\Lambda V, \bard)$. Such an integer exists since $(\Lambda V, \bard)$ has finite cohomological dimension. Then there exists $u\in \Lambda V$ such that $v^n = \bar d u$. Equivalently, we have
\[ v^n = u_0\cdot x + d u,\]
for some $u_0\in \Lambda V$. It follows that
\[ 0 = d^2 u = d( v^n - u_0\cdot x) = nv^{n-1}\cdot x - (du_0)\cdot x. \]
Therefore, $v^{n-1} = \frac{1}{n} d u_0$, which implies that $[v^{n-1}] = 0$ in $H^{\bullet}(\Lambda V, \bard)$. We arrive at a contradiction. This completes the proof.
\end{proof}
Now let us prove the main result of this section.
\begin{theorem} \label{thm:inj} Let $(B, d)$ be a connected commutative DGA such that $H^{2k}(B)=0$ for all $0 < 2k \leq 2N$. If $\iota \colon (B, d) \to (B\otimes \Lambda V, d) $ is an algebraic fibration whose algebraic fiber has finite cohomological dimension, then the induced map
\[ \iota_\ast \colon \bigoplus_{i< 2N} H^i(B)\to \bigoplus_{i< 2N} H^i(B\otimes \Lambda V)\]
is injective.
\end{theorem}
\begin{proof}
Let $f\colon (\mathcal M_B, d)\to (B, d)$ be a minimal model algebra of $B$.
\begin{claim*}
For any algebraic fibration $\iota \colon (B, d) \to (B\otimes \Lambda V, d) $, there exist an algebraic fibration $\varphi\colon (\mathcal M_B, d) \to (\mathcal M_B\otimes \Lambda V, d)$ and a quasi-isomorphism $g\colon (\mathcal M_B\otimes \Lambda V, d) \to (B\otimes \Lambda V, d)$ such that the following diagram commutes:
\[ \xymatrix{ \mathcal M_B \ar@{^{(}->}[d]_{\varphi} \ar[r]^f & B \ar@{^{(}->}[d]^\iota \\
\mathcal M_B\otimes \Lambda V \ar[r]^-{g} & B\otimes \Lambda V.} \]
\end{claim*}
We construct $\varphi$ and $g$ inductively. Consider the filtration $V = \cup_{n=0}^\infty V(k)$ from Definition $\ref{def:algfib}$. Choose a basis $\{x_i\}_{i\in I_0}$ of $V(0)$. Let $x = x_i$ be a basis element. If $d x = a \in B$, then $d a = d^2 x = 0$. It follows that there exists $\tilde a \in \mathcal M_B$ such that $f(\tilde a) = a + d c$ for some $c\in B$. We define an algebraic fibration $\varphi_0\colon (\mathcal M_B, d) \hookrightarrow (\mathcal M_B\otimes \Lambda(x), d)$ by setting $d x = \tilde a$. Moreover, we extend $f\colon (\mathcal M_B, d) \to (B, d)$ to a morphism (of commutative DGAs) $g_0\colon (\mathcal M_B\otimes \Lambda (x), d) \to (B\otimes \Lambda (x), d)$ by setting $g(x) = x + c$. By the Gysin sequence from Section $\ref{prop:gysin}$, we see that $g_0$ is a quasi-isomorphism. Now apply the same construction to all basis elements $\{x_i\}_{i\in I_0}$. We still denote the resulting morphisms by $\varphi_0 \colon (\mathcal M_B, d) \to (\mathcal M_B\otimes \Lambda (V(0)), d)$ and $g_0\colon (\mathcal M_B\otimes \Lambda (V(0)), d) \to (B\otimes \Lambda (V(0)), d)$.
Now suppose we have constructed an algebraic fibration
\[ \varphi_{k} \colon (\mathcal M_B\otimes \Lambda (V(k-1)), d) \to (\mathcal M_B\otimes \Lambda (V(k)), d)\] and a quasi-isomorphism $g_k\colon (\mathcal M_B\otimes \Lambda (V(k)), d) \to (B\otimes \Lambda (V(k)), d)$ such that
the following diagram commutes:
\[ \xymatrixcolsep{4pc}\xymatrix{ \mathcal M_B\otimes \Lambda (V(k-1)) \ar@{^{(}->}[d]_{\varphi_k} \ar[r]^{g_{k-1}} & B\otimes \Lambda(V(k-1)) \ar@{^{(}->}[d]^\iota \\
\mathcal M_B\otimes \Lambda (V(k)) \ar[r]^-{g_k} & B\otimes \Lambda (V(k)).} \]
Let $\{y_i\}_{i\in I_{k+1}}$ be a basis of $V(k+1)$ that extends the basis $\{x_i\}_{i\in I_{k}}$ of $V(k)\subseteq V(k+1)$. Apply the same construction above to elements in $\{y_i\}_{i\in I_{k+1}} \backslash \{x_i\}_{i\in I_{k}}$, but with $B\otimes \Lambda (V(k))$ in place of $B$, and $\mathcal M_B\otimes \Lambda (V(k))$
in place of $\mathcal M_B$.
We define $(\mathcal M_B\otimes \Lambda V, d)$ to be the direct limit of $(\mathcal M_B\otimes \Lambda (V(k)), d)$ with respect to the morphisms $\varphi_{k} \colon (\mathcal M_B\otimes \Lambda (V(k-1)), d)$. We define $\varphi$ to be the natural inclusion morphism $ (\mathcal M_B, d) \hookrightarrow (\mathcal M_B\otimes \Lambda V, d)$. The morphisms $g_k$ together also induce a quasi-isomorphism $g\colon (\mathcal M_B\otimes \Lambda V, d) \to (B\otimes \Lambda V, d)$, which makes the diagram in the claim commutative. This finishes the proof of the claim.
Now assume to the contrary that there exists $0\neq \alpha\in H^{2k+1}(B)$ with $2k+1 < 2N$ such that $\iota_\ast(\alpha) = 0$. Let $\tilde \alpha \in H^{2k+1}(\mathcal M_B)$ be the class such that $f_\ast(\tilde \alpha) = \alpha$. In particular, we have $\varphi_{\ast}(\tilde \alpha) = 0$. By Corollary $\ref{cor:tosphere}$, there exists a morphism $\psi\colon (\mathcal M_B, d) \to (\Lambda (\eta), 0)$ such that $\psi_{\ast}(\tilde \alpha) = \eta$. Now let \[ \tau\colon (\Lambda(\eta), 0) \to (\Lambda(\eta) \otimes \Lambda V, d) = (\Lambda(\eta) \otimes_{\mathcal M_B} (\mathcal M_B\otimes \Lambda V), d)\] be the push-forward algebraic fibration of $\varphi\colon (\mathcal M_B, d) \to (\mathcal M_B\otimes \Lambda V, d)$. It follows that
\[ \tau_\ast(\eta) = \tau_\ast \psi_{\ast}(\tilde \alpha) = (\psi\otimes 1)_\ast\varphi_{\ast}(\tilde \alpha) = 0 \]
which contradicts Theorem $\ref{thm:oddsphere}$. This completes the proof.
\end{proof}
\end{document} |
\begin{document}
\title{Cauchy-Davenport Theorem for linear maps: Simplification and Extension}
\author{
John Kim\thanks{Department of Mathematics, Rutgers University. Research supported in part by NSF Grant Number DGE-1433187. {\tt [email protected]}.} \and
Aditya Potukuchi\thanks{Department of Computer Science, Rutgers University. {\tt [email protected]}.}}
\maketitle
\begin{abstract}
We give a new proof of the Cauchy-Davenport Theorem for linear maps given by Herdade et al., (2015) in~\cite{HKK}. This theorem gives a lower bound on the size of the image of a linear map on a grid. Our proof is purely combinatorial and offers a partial insight into the range of parameters not handled in~\cite{HKK}.
\end{abstract}
\section{Introduction}
Let $\mathbb{F}_p$ be the field containing $p$ elements, where $p$ is a prime, and let $A,B \subseteq \mathbb{F}_p$. The Cauchy-Davenport Theorem gives a lower bound on the size of the sumset $A + B \defeq \{a + b \;\ifnum\currentgrouptype=16 \middle\fi|\; a \in A,b \in B\}$ (for more on sumsets, see, for example,~\cite{TV}). The size of the sumset can be thought of as the size of the image of the linear map $(x,y) \rightarrow x + y$, where $x \in A$, and $y \in B$. Thus the theorem can be restated as follows:
\begin{theorem}[Cauchy-Davenport Theorem]
Let $p$ be a prime, and $L:\mathbb{F}_p \times \mathbb{F}_p \rightarrow \mathbb{F}_p$ be a linear map that takes $(a,b)$ to $a+b$. For $A,B \subseteq \mathbb{F}_p$, Let $L(A,B)$ be the image of $L$ on $A \times B$. Then,
$$
|L(A,B)| \geq \min(|A|+|B|-1,p)
$$
\end{theorem}
In~\cite{HKK}, this notion was extended to study the sizes of images of general linear maps on product sets. A lower bound was proved using the polynomial method (via a nonstandard application of the Combinatorial Nullstellensatz~\cite{Alon}). In this paper, we give a simpler, and combinatorial proof of the same using just the Cauchy-Davenport Theorem. \\
Notation: For a linear map $L : \mathbb{F}_p^{n} \rightarrow \mathbb{F}_p^m$, and for $S_1, S_2, \ldots S_n \subseteq \mathbb{F}_p$, we use $L(S_1, S_2 \ldots S_n)$ to denote the image of $L$ on $S_1 \times S_2 \times \cdots S_n$. The \emph{support} of a vector is the set of nonzero entries in the vector. A \emph{min-support vector} in a set $V$ of vectors is a nonzero vector of minimum support size in $V$.
\begin{theorem}[Main Theorem]
\label{the:main}
Let $p$ be a prime, and $L:\mathbb{F}_p^{m+1} \rightarrow \mathbb{F}_p^m$ be a linear map of rank $m$. Let $A_1, A_2, \ldots A_{m+1} \subseteq \mathbb{F}_p$ with $|A_i| = k_i$. Further, suppose that $\min_i(k_i) + \max_i(k_i) < p$. Let $S$ be the support of $\ker(L)$, and $S' = [n] \setminus S$. Then
$$
|L(A_1,A_2, \ldots, A_n)| \geq \left( \prod_{j \in S'}k_j \right) \cdot \left( \prod_{i\in S}k_i - \prod_{i \in S}(k_i - 1) \right)
$$
\end{theorem}
As noted in~\cite{HKK}, this bound is tight for every $m$ and $p$. We restrict our theorem to study only maps from $\mathbb{F}_p^{m+1}$ to $\mathbb{F}_p^m$ of rank $m$ for two reasons mainly:(1) It is simpler to state, and contains the tight case and (2) We are unable prove any better bounds if the rank is not $m$. It is not clear to us what the correct bound for the general case is. \\
We also show the following result for the size of the image for certain full rank linear maps from $\mathbb{F}_p^n \rightarrow \mathbb{F}_p^{n-1}$ when the size of the sets it is evaluated on are all large enough.
\begin{theorem}
\label{cover}
Let $L:\mathbb{F}_p^n \rightarrow \mathbb{F}_p^{n-1}$ be a linear map given by $L(x_1,\ldots x_n) = (x_1 + x_n, x_2 + x_n \ldots x_{n-1} + x_n)$. Let $S_1, \ldots S_n \subseteq \mathbb{F}_p$ with $|S_i| = k$ for $i \in [n]$ such that $k > \frac{(n-1)p}{n}$, then $|L(S_1, \ldots S_n)| = p^{n-1}$ (i.e., $L(S_1, \ldots S_n) = \mathbb{F}_p^{n-1}$).
\end{theorem}
The theorems do not, however, give tight bounds for all set sizes, for example if $\min_i |A_i| > p/2$. It would be interesting to obtain a tight bound even for the simple linear map $(x,y,z) \rightarrow (x+z, y+z)$ on the product set $A_1 \times A_2 \times A_3 \subseteq \mathbb{F}_p^3$ which holds for all sizes of the $A_i$'s.
\section{The Theorem}
\subsection{The Main Lemma}
The idea is that since the size of the image is invariant under row operations of $L$, we perform row operations to isolate a `hard' part, which gives the main part of the required lower bound \\
Our proof proceeds by induction on the dimension of the linear map. The base case is given by the Cauchy Davenport Theorem.
\begin{lemma}
\label{lem:main}
Let $L:\mathbb{F}_p^n \rightarrow \mathbb{F}_p^{n-1}$ be a linear map such that $L(x_1,\ldots, x_n) = (x_1 + x_n, x_2 + x_n, \ldots, x_{n-1} + x_n)$. Let $S_1, \ldots S_n \subseteq \mathbb{F}_p$ with $|S_i| = s_i$ such that $\min_i(s_i) + \max_i(s_i) \leq p+1$. Then $|L(S_1, \ldots S_n)| \geq \prod_{i=1}^n s_i-\prod_{i=1}^n (s_i-1)$
\end{lemma}
\begin{proof}
We use the shorthand notation $|L| \defeq |L(S_1,S_2 \ldots S_n)|$. W.L.O.G, let $S_1$ be such that $|S_1| = \min_{i \in [n-1]}(|S_i|)$. \\
A preliminary observation is that $|S_1| + |S_n| \leq p+1$, and therefore, by the Cauchy-Davenport Theorem,
\begin{equation} \label{CD}
|S_1 + S_n| \geq s_1 + s_n - 1
\end{equation}
The proof proceeds by induction on $n$. If $n = 2$, the result $|L| \geq s_1 \cdot s_2 - (s_1-1) \cdot (s_2-1) = s_1 + s_2 - 1$ is given by the Cauchy-Davenport Theorem. \\
For every $a \in \mathbb{F}_p$, we have $T_a \defeq \{x_n \in S_n \;\ifnum\currentgrouptype=16 \middle\fi|\; \exists x_1 \in S_1, x_1 + x_n = a\}$, and $t_a \defeq |T_a|$. We now look at the restricted linear map $L|_{x_1 + x_n = a}$. In this case, the induction is on sets $S_2, \ldots S_{n-1} \times T_a$. This is equivalent to restricting $S_n$ to the set $T_a$, and dropping $S_1$, since for every $x_n \in S_n$, there is a unique $x_1 \in S_1$ such that $x_1 + x_n = a$. \\
We first observe that the conditions are satisfied, i.e., $\min_i(|S_i|) + \max_i(|S_i|) \leq p + 1$, since $t_a \leq \min(|S_1|,|S_n|)$. Also the resulting linear map is of the same form, i.e., $L|_{x_1 + x_n = a}(x_2,\ldots x_n) = (x_2 + x_n \ldots x_{n-1} + x_n)$. (In reality, $L|_{x_1 + x_n = a}$ is a map from $\mathbb{F}_p^n$ to $\mathbb{F}_p^{n-1}$, given by $L|_{x_1 + x_n = a}(x_1,x_2,\ldots x_n) = (a, x_2 + x_n \ldots x_{n-1} + x_n)$ but we drop the first coordinate because it is fixed, i.e., $a$)\\
By induction hypothesis, the number of points in the image of $L_{x_1 + x_n = a}$ is at least:
$$
\left( \prod_{i=2}^{n-1}s_i \right) t_a - \left( \prod_{i=2}^{n-1}(s_i - 1) \right)(t_a - 1)
$$
Summing over all $a \in \mathbb{F}_p$, we get a bound on the number of points in the image:
\begin{eqnarray*}
|L| &\geq& \sum_{a \in \mathbb{F}_p, t_a \neq 0} \left( \left( \prod_{i=2}^{n-1}s_i \right) t_a - \left( \prod_{i=2}^{n-1}(s_i - 1) \right)(t_a - 1) \right) \\
&=& \left( \prod_{i=2}^{n-1}s_i \right)\sum_{a \in \mathbb{F}_p}t_a - \left( \prod_{i=2}^{n-1}(s_i - 1) \right)\sum_{a\in \mathbb{F}_p, t_a \neq 0}(t_a - 1) \\
&\geq& \prod_{i=1}^{n}s_i - \prod_{i=1}^n(s_i - 1)
\end{eqnarray*}
The last inequality comes from observing that $\sum_{a \in \mathbb{F}_p}t_a = s_1s_n$, and an upper bound on $\sum_{a \in \mathbb{F}_p, t_a \neq 0}(t_a - 1)$, by using~\ref{CD}. We have $\sum_{a \in \mathbb{F}_p, t_a \neq 0}(t_a - 1) = \sum_{a \in \mathbb{F}}t_a - \sum_{a \in \mathbb{F}_p}\mathbbm{1}_{t_a \neq 0} = \sum_{a \in \mathbb{F}}t_a + |S_1 + S_n| \leq s_1s_n - (s_1 + s_n - 1)$.
\end{proof}
\subsection{Arriving at the Main Theorem}
The first step in arriving at the main theorem is exactly as in~\cite{HKK}. For completeness, we describe it here. The idea is to transform a general linear map into a specific form, without reducing the size of the image (in fact, here it remains the same). This step is very intuitive, but describing it requires some setup. \\
Let $L:\mathbb{F}_p^{m+1} \rightarrow \mathbb{F}_p^m$ be an $\mathbb{F}_p$-linear map of rank $m$. Let $v$ be a non-zero min-support vector of $\ker(L)$. So, we have $Lv = 0$. The main observation is that under row operations, two quantities remain unchanged: the size of the image of $L$, and the size of the support of the min-support vector in the kernel. \\
Let $r_1, \ldots r_m$ be the rows, and $c_1, c_2, \ldots c_{m+1}$ be the columns of associated to $L$ with respect to the standard basis. We show that one can perform elementary row operations, and some column operations on $L$ while preserving the size of the image.
\begin{lemma}
\label{lem:linop}
The size of the image of $L$ does not change under
\begin{enumerate}
\item Elementary row operations.
\item Scaling any column $c_i$ by some $d \in \mathbb{F}_p \setminus \{0\}$ and scaling every element of $A_i$ by $d$.
\item Swapping any two columns $c_i$ and $c_j$, and swapping sets $A_i$ and $A_j$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove this by considering each given operation separately.
\begin{enumerate}
\item Suppose $L'$ was obtained from $L$ by elementary row operations. There is an invertible linear map $M$ such that $M \cdot L = L'$. This gives the bijection from every vector $v$ in the image of $L$, to the vector $M \cdot v$ in the image of $L'$.
\item Suppose $L'$ was obtained from $L$ by scaling column $c_i$ by $d \in \mathbb{F}_p \setminus \{0\}$, and scaling the set $A_i$ by $d^{-1}$. We map every vector $(u_1, \ldots, u_m) \in$ $L(A_1,\ldots ,A_i, \ldots, A_{m+1})$, to the vector $(u_1, \ldots, u_m) \in L'(A_1,\ldots , d^{-1} \cdot A_i, \ldots, A_{m+1})$. Here $d^{-1} \cdot A_i \defeq \{d^{-1}a_i \;\ifnum\currentgrouptype=16 \middle\fi|\; a_i \in A_i\}$ This map is invertible.
\item Suppose $L'$ was obtained from $L$ by switching columns $c_i$ and $c_j$, and swapping the sets $A_i$ and $A_j$. We map every vector $(u_1,\ldots, u_m) \in L(A_1,\ldots ,A_i, \ldots, A_j, \ldots, A_{m+1})$ to the identical vector $(u_1, \ldots, u_m) \in$
$ L'(A_1,\ldots ,A_j, \ldots, A_i, \ldots, A_{m+1})$. This map is invertible.
\end{enumerate}
For every given operation, we have a bijection between the images of $L$ before and after the operation.
\end{proof}
\begin{observation}
After the operations stated in Lemma~\ref{lem:linop}, the size of the support of the min-support vector in $\ker(L)$ does not change.
\end{observation}
To see this, we first observe that the kernel has rank $1$, and is orthogonal to the row span of $L$. Therefore, all nonzero vectors in $\ker(L)$ have the same support. Since, row operations do not change the row span of $L$, the resulting kernel spans the same subspace of $\mathbb{F}^{m+1}$, and therefore, the size of the support of the vectors in $\ker(L)$ does not change. \\
Next, we do the following operations, each of which preserves the size of the image.
\begin{enumerate}
\item Perform row operations so that the last $m$ columns form an identity matrix.
\item Scale the rows so that the first column of every row is $1$.
\item Scale the last $m$ columns so that every nonzero entry in $L$ is $1$.
\end{enumerate}
After we perform these operations, we have a linear map where the first column consists of $1$'s and $0$'s and the remaining $m$ columns form an identity matrix. Let the $S'$ be the set of indices of rows containing $1$'s in the first column. Consider the vector $v = -e_1 + \sum_{i \in S'}e_{i+1}$. This vector has support $|S'| + 1$, and lies in the kernel of $L$. Therefore, $|S| = |S'| + 1$.
\begin{proof}[Proof of Theorem~\ref{the:main}]
Apply the transformation from Lemma~\ref{lem:linop} to $L$ to reduce it to the simple form. Let $S'$ be the set of rows where the first column is nonzero. Consider the restriction of $L$ on the the coordinates given by $S$. By Lemma~\ref{lem:main}, the size of this image is at least $\left( \prod_{i\in S}k_i - \prod_{i \in S}(k_i - 1) \right)$.
The linear map restricted to the coordinates $[m] \setminus S$ is nothing but the identity map, so the size of the image is $\prod_{i \not \in S} |A_i|$, and is independent of the linear map restricted to $S$. Putting them together, we have the desired result.
\end{proof}
\section{The case when $2k > p+1$}
The proof of Lemma~\ref{lem:main} breaks down when $s_1+s_n > p + 1$ and, unfortunately, we do not know how to fix this issue. Consider, for example, the simplest nontrivial case where $m = 2$, i.e., $L(x,y,z) = (x+z, y+z)$, and we are interested in the size of the image of $L$ on $X \times Y \times Z$, further suppose, for simplicity, that $|X|=|Y|=|Z| = k$. If $k < \frac{p + 1}{2}$, then the above bound holds, and is tight. If $k>\frac{2p}{3}$, then $L$ covers $\mathbb{F}_p^2$, i.e., $|L(A,B,C)| = p^2$. This makes the case in between the interesting one. We conjecture that the correct lower bound is the size of the image of $L$ when $X = Y = Z = \{1,2,\ldots k\}$. Towards this, we are able to prove a partial result (Lemma~\ref{2d}) using the above method.
We will need the following Lemma:
\begin{lemma}
\label{boundtx}
Let $X,Y\subseteq \mathbb{F}_p$ and $t_a = |\{(x,y)\in X\times Y: x+y = a\}|$. Then for every $a \in \mathbb{F}_p$:
$$|X|+|Y|-p\leq t_a \leq \min(|X|,|Y|).$$
\end{lemma}
\begin{proof}
The bounds follow from the fact that $t_a$ can be written as the size of the
intersection of two sets of sizes $|X|$ and $|Y|$:
$$t_a = |X \cap (a-Y)|.$$
\end{proof}
Now we state the partial result:
\begin{theorem}
\label{2d}
Let $L:\mathbb{F}_p^3\rightarrow \mathbb{F}_p^2$ be the linear map defined by $L(x,y,z) = (x+z,y+z)$.
Let $X,Y,Z\subset F_p$ be sets of size $k$, where $k \geq \frac{p+1}{2}$.
Then we have the following lower bound:
$$|L(X,Y,Z)| \geq \min(p^2 + 3k^2 - (2p+1)k,p^2).$$
\end{theorem}
\begin{proof}
Let $T_a \defeq \{z \in Z \;\ifnum\currentgrouptype=16 \middle\fi|\; \exists x \in X, x+z = a\}$, with $t_a\defeq |T_a|$.
Looking at this restriction, $L|_{x+z = a}$, by Cauchy-Davenport Theorem, there are at least $\min(t_a + k-1,p)$ points of $L(X,Y,Z)$ on $L_{x+z=a}(Y,T_a)$. By summing over all $a\in \mathbb{F}_p$,
we get a lower bound on the size of $L(X,Y,Z)$:
\begin{eqnarray*}
|L(X,Y,Z)| &\geq& \displaystyle\sum_{a\in\mathbb{F}_p}{\min(t_a + k-1,p)} \\
&=& \displaystyle\sum_{a\in\mathbb{F}_p}{\min(t_a,p-k+1)} + p(k-1) \\
&=& \displaystyle\sum_{a:t_a \leq p-k+1}{t_a} + \displaystyle\sum_{a:t_a > p-k+1}{(p-k+1)} + p(k-1).
\end{eqnarray*}
We now want to remove the dependence of the lower bound on the $t_a$ by considering
the worst case scenario, where the $t_a$ take values that minimize the lower bound.
First, we observe $\sum_{a\in\mathbb{F}_p}{t_a} = k^2$, a fixed quantity. So to minimize the above lower bound for $|L(X,Y,Z)|$, we need $t_a$ to be maximal for as many $a\in\mathbb{F}_p$ as possible.
By Lemma~\ref{boundtx}, we know that $2k-p\leq t_a \leq k$. We set $t_a = k$ for as many
$a\in\mathbb{F}_p$ as possible, and the remainder of the $t_a = 2k-p$. This gives:
\begin{eqnarray*}
|L(X,Y,Z)| &\geq& \displaystyle\sum_{a:t_a \leq p-k+1}{t_a} + \displaystyle\sum_{a:t_a > p-k+1}{(p-k+1)} + p(k-1) \\
&\geq& k(2k-p) + (p-k)(p-k+1) + p(k-1) \\
& = & 3k^2 + p^2 - (2p-1)k.
\end{eqnarray*}
\end{proof}
As a corollary, we get, independent of theorem~\ref{cover}, the following corrolary:
\begin{corollary}
If the linear map $L$, and the sets $A$, $B$, $C$ were as above, with $|A|=|B|=|C|=k$, and $k > \frac{2p}{3}$, then $L(A,B,C) = p^2$.
\end{corollary}
We would like to point out that at the two extremes, i.e., when $k = \frac{p+1}{2}$, and when $k =\lceil\frac{2p}{3} \rceil$, the above bound matches the `correct' lower bound.
\subsection{Proof of Theorem~\ref{cover}}
We prove theorem~\ref{cover} via a slightly stronger claim
\begin{claim}
\label{lem:cover}
Let $L:\mathbb{F}_p^n \rightarrow \mathbb{F}_p^{n-1}$ be a linear map given by $L(x_1,\ldots x_n) = (x_1 + x_n, x_2 + x_n \ldots x_{n-1} + x_n)$. Let $S_1, \ldots S_n \subseteq \mathbb{F}_p$ with $|S_i| = k$ for $i \in [n-1]$, and $|S_n| = k'$. Further, suppose that $(n-1)k + k' \geq (n-1)p + 1$, then $|L(S_1, \ldots S_n)| = p^{n-1}$.
\end{claim}
\begin{proof}
We prove this by induction on $n$, analogous to Lemma~\ref{lem:main}. The case where $n = 2$ is, again, given by the Cauchy-Davenport Theorem. \\
For $a \in \mathbb{F}_p$, $T_a \defeq \{x_n \in S_n \;\ifnum\currentgrouptype=16 \middle\fi|\; \exists x_1 \in S_1, x_1 + x_n = a\}$ with $t_a \defeq |T_a|$. Looking at this restriction of $L$ (i.e., $x_1 + x_n = a$), we have a linear map, $L_{x_1 + x_n = a}$ on the sets $S_2 \times S_3 \times \cdots T_a$, given by the $L_{x_1 + x_n = a}(x_2 ,\ldots x_n) = (x_2 + x_n, \ldots x_{n-1} + x_n)$. (similar to Lemma~\ref{lem:main}, we drop the first coordinate). \\
Here, $|S_i| = k$ for $i = 2,\ldots n-1$, and $|T_a| \geq k+k'-p$, by Lemma~\ref{boundtx}. Further, the required condition holds, i.e.,:
$$(n-2)k + t_a \geq (n-2)k + k + k' - p = (n-1)k + k' - p \geq (n-2)p+1.$$
Therefore, by induction hypothesis $|L|_{x_1 + x_n = a}(S_2, \ldots S_{n-1}, T_a)| = p^{n-2}$. Since this holds for every $a \in \mathbb{F}_p$, we have $|L(S_1, \ldots S_n)| = p^{n-1}$.
\end{proof}
In particular, Lemma~\ref{lem:cover} tells that for the linear map $L$ given by $L(x_1,\ldots x_n) = (x_1 + x_n, x_2 + x_n, \ldots, x_{n-1} + x_n)$ on $S_1 \times S_2 \times \cdots S_n$, if $|S_i| \geq \frac{(n-1)p}{n}$, then $L(S_1, \ldots, S_n) = \mathbb{F}_p^{n-1}$.
\end{document} |
\begin{document}
\title{Rubidium resonant squeezed light from a diode-pumped optical-parametric
oscillator}
\author{A. Predojevi\'{c}}
\affiliation{ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park,
08860 Castelldefels (Barcelona), Spain}
\author{Z. Zhai}
\affiliation{ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park,
08860 Castelldefels (Barcelona), Spain}
\author{J. M. Caballero}
\affiliation{ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park,
08860 Castelldefels (Barcelona), Spain}
\author{M. W. Mitchell}
\affiliation{ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park,
08860 Castelldefels (Barcelona), Spain}
\date{27 July 2008}
\begin{abstract}
We demonstrate a diode-laser-pumped system for generation of quadrature
squeezing and polarization squeezing. Due to their excess phase noise,
diode lasers are challenging to use in phase-sensitive quantum optics
experiments such as quadrature squeezing. The system we present overcomes
the phase noise of the diode laser through a combination of active
stabilization and appropriate delays in the local oscillator beam.
The generated light is resonant to the rubidium D1 transition at 795$\,$nm
and thus can be readily used for quantum memory experiments.
\end{abstract}
\maketitle
Interaction of quantum states of light is of interest both for quantum
communications, for improved sensitivity in measurements limited by
quantum noise, and for understanding light-matter interactions at
the most fundamental level. Our interest is in quadrature squeezing
and polarization squeezing, which are phase-dependent quantum features.
A proven technique for generation of squeezing is phase-sensitive
amplification in a subthreshold optical parametric oscillator (OPO)
\citep{Polzik1992}. For strong interaction with atoms the squeezed
light needs to be atom-resonant, which limits the choice of lasers,
nonlinear crystals, and detectors. Several experiments have demonstrated
squeezing at the rubidium resonance \citep{Akamatsu2004,Appel2008,Hetet2007,Tanimura:06}.
These experiments use distinct methods: squeezing in a waveguide \citep{Akamatsu2004}
and downconversion in an OPO \citep{Tanimura:06,Hetet2007,Appel2008}.
In the latter case, the squeezing at the rubidium D1 line using the
non-linear interaction in a subthreshold OPO was achieved by using
a Ti:Sapphire laser and periodically poled potassium titanium oxide
phosphate (PPKTP) as nonlinear medium \citep{Tanimura:06,Hetet2007,Appel2008}.
The noise suppression of this method was shown to be more than -5dB
\citep{Hetet2007}. To our knowledge the only experiment that generated
squeezing in an OPO pumped by a diode laser system worked at 1080$\,$nm
\citep{Zhang:06}, far from any useful atomic resonance and produced
relative intensity squeezing, a phase-independent property.
Compared to other laser systems, diode lasers are easy to operate,
compact, and inexpensive. It has long been suspected that the excess
phase noise of the diode laser, which results in a relatively large
linewidth, would be an obstacle for production of phase-sensitive
quantum states such as quadrature squeezing. The spectral distribution
of diode laser phase noise over different frequencies was investigated
in \citep{Lax1967}. There is was shown that the main contribution
in the noise comes from the low frequency part of the spectrum, as
expected for a process of phase diffusion. This suggests that the
laser output can be treated as quasi-stationary, with the laser frequency
drifting slowly (on the time-scale of propagation and cavity relaxation)
within the laser linewidth.
Here we show that cavity stabilization of the diode laser frequency,
in combination with appropriate delays for the local oscillator beam,
allows squeezing to be observed with diode laser based systems. The
laser, doubling system, and stabilization use standard techniques
and could be applied to a variety of other wavelengths. The use of
appropriate delays makes the system robust against frequency fluctuations,
and could be incorporated into non-diode-laser pumped systems as well.
We present our experimental system, a PPKTP-based subthreshold OPO
pumped by a frequency-doubled diode laser, and expected and observed
squeezing performance. We then consider the effect of the frequency
fluctuations on the observable squeezing in the regime of the quasi-stationary
fluctuations, an analysis which indicates that the system can be made
immune to random frequency drifts for appropriate local oscillator
delay. Finally, we present measurements of squeezing versus delay
in agreement with the theory.
\section{Experimental setup}
\begin{figure}
\caption{Experimental apparatus. Light from the diode laser is amplified in
the tapered amplifier and fed into the doubling cavity. The blue output
light is mode-matched into fiber F2 and fed into the OPO cavity. Both
doubling and OPO cavity are only resonant to the red light. The length
of the local oscillator beam path can be changed by fiber F1. The
modes of squeezed vacuum and local oscillator are then overlapped
on a beamsplitter (PBS1) where power balancing is performed by a waveplate
(WP) and a beamsplitter (PBS2). Light is collected onto diodes D1
and D2 of the balanced detector. The obtained electrical signal is
recorded using a spectrum analyzer (SA).}
\label{fig:experiment}
\end{figure}
The schematic of the experiment is shown in the Fig.\ref{fig:experiment}.
Our laser system (Toptica TA-SHG) consists of a grating stabilized
795$\,$nm diode laser which is amplified by an optical tapered amplifier
and injected into a frequency doubler with lithium triborate crystal
as nonlinear medium. A 20$\,$MHz modulation is applied to the laser
current, resulting in frequency modulation sidebands of 5\%. The reflection
from the cavity is demodulated to provide an error signal (the Pound-Drever-Hall
technique (PDH)). The laser and cavity are locked in frequency by
a proportional-integral-derivative (PID) circuit acting on the cavity
piezo and a fast proportional component acting on the current of the
diode. At the same time, the absolute laser frequency is stabilized
by frequency-modulation spectroscopy of a saturated-absorption signal,
fed back by digital PID to the piezo-electric transducer of the laser
grating. For the experiments described here, the laser was locked
to the $F=2\rightarrow F'=1$ transition of $^{\text{87}}$Rb. Residual
fluctuations of the FM spectroscopy signal indicate that the fast
cavity lock reduces the linewidth to 400$\,$kHz full width half maximum
(FWHM).
The generated 397$\,$nm light is passed through a single-mode fiber
for spatial filtering and pumps the sub-threshold degenerate optical
parametric oscillator. The nonlinear material used in the OPO is a
10$\,$mm long PPKTP crystal, temperature tuned for the maximum second-harmonic
generation efficiency. The OPO cavity is a 64$\,$cm long bow-tie
configuration resonator which consists of two spherical mirrors (R=10$\,$cm)
and two flat mirrors. The distance between the spherical mirrors is
11.6$\,$cm yielding to the beam waist in the crystal of 42$\,$\textgreek{m}m.
The output coupling mirror of the OPO has a transmission of 7.8\%,
and the measured intracavity losses are 0.55\%. The measured cavity
linewidth is $\delta\nu$=8$\,$MHz (FWHM) and the output coupling
efficiency $\eta=0.93$. The free spectral range of the OPO cavity
is 504$\,$MHz. The cavity is locked using the Pound-Drever-Hall technique
performed on the transmission signal of a counter-propagating beam
fed into cavity through the high reflecting flat mirror. The error
signal is digitized and fed into a PID circuit programmed within a
National Instruments field-programmable gate array (FPGA) board type
NI 7833R. It controls the OPO cavity length by moving the position
of one cavity mirror with a piezo-electric transducer.
A local oscillator beam is derived from the diode laser by passing
through single-mode fibers whose combined lengths can be chosen to
give a desired group delay. The vertically-polarized OPO output is
overlapped with 400$\,$\textmu{}W of this horizontally-polarized
beam on polarizing beamsplitter (PBS1). Optimized overlap results
in a measured homodyne efficiency of $\eta_{hom}=0.98$. Local oscillator
and squeezed vacuum beams are mixed and balanced in power on a second
polarizing beamsplitter (PBS2) and detected with a ThorLabs (PDB150)
switchable-gain balanced detector. The quantum efficiency of this
detector at detection wavelength of 795$\,$nm is 88\% by manufacturer
specifications. Losses are mainly caused by the reflection of the
surface of the protective window and diode surface. We use two spherical
mirrors (R=10$\,$mm), to retro-reflect the reflected light onto the
detector improving the quantum efficiency by 7\%, i.e., to 95\%. Quarter-wave
plates are used to prevent the returning light from reaching the OPO
cavity. For the local oscillator power of 400$\,$\textmu{}W electronic
noise of the detector is 14$\,$dB below the standard quantum limit.
Electronic noise was subtracted from all the traces.
\begin{figure}
\caption{Squeezed vacuum generation: (a) squeezing trace when scanning the
phase of the local oscillator (b) shot noise level. Electronic noise
is subtracted. Spectrum analyzer at zero span, resolution bandwidth
= 30kHz, video bandwidth = 30Hz}
\label{fig:sq}
\end{figure}
As described in the theory section, when fluctuations in frequency
are included, the degree of squeezing is expected to depend on the
relative delay through two paths: From laser to PBS1 through the local
oscillator fiber, and from laser to PBS1 through amplifier, doubler,
pump fiber and OPO. Insensitivity to these fluctuations is expected
to occur at a {}``white light'' condition of equal delays. Initial
measurements were taken with the local oscillator fiber chosen to
achieve this condition, as described in detail in section \ref{sec:delay-considerations}.
Noise measurements were performed at fixed frequency, zero-span of
the spectrum analyzer. The degree of squeezing we observe matches
the above mentioned gain and loss parameters for the demodulation
frequencies 3MHz and higher. The highest level of squeezing of -2.5$\,$dB
we observe at the demodulation frequency of 2$\,$MHz shown in Fig.
\ref{fig:sq}.
\section{relative lock quality}
We note that the achieved linewidth for the stabilized diode laser
(400$\,$kHz) is an order of magnitude below the linewidths of the
doubling cavity (14$\,$MHz) and OPO cavity (8$\,$MHz). This justifies
treating the frequency fluctuations of the laser as quasi-stationary
when determining the effect on squeezing. Another treatment of phase
noise has been discussed \citep{Gea-Banacloche1990}, but is far more
involved and does not consider group delay effects. At the same time,
while fast feedback to the laser current allows a high-bandwidth lock
of the laser and doubling cavity, there is no corresponding fast control
for locking of the OPO cavity to the laser frequency. Also, the PDH
scheme, which achieves a very good signal by injecting through the
cavity output coupler, cannot be used in many squeezing experiments
because it would contaminate the squeezed light. For these reasons,
the active stabilization of the OPO cavity may be an important factor
in the performance of squeezing experiments.
\section{Theory}
\begin{figure}
\caption{Squeezing vs.delay for three different detection frequencies 1$\,$MHz,
2$\,$MHz, and 3$\,$MHz depicted black, red, and blue respectively;
a) models the laser spectrum as Gaussian of linewidth 700$\,$kHz
full width half maximum (FWHM), b) models the laser spectrum as Lorentzian
of linewidth 300$\,$kHz (FWHM).}
\label{fig:gauss}
\end{figure}
The theoretical description adapts the treatment of Collet and Gardiner
\citep{PhysRevA.30.1386} to model the nonlinear interaction inside
the OPO cavity. Here we assume that the frequency drift of the diode
laser is slow on the time scale of the decay of light inside the OPO
cavity. This quasi monochromatic treatment describes a single mode
laser drifting slowly within a finite linewidth. In such a system
frequency fluctuations lead to a fluctuating phase shift between the
squeezed mode and the local oscillator mode. Our calculation modifies
\citep{PhysRevA.30.1386} by including a relative detuning $\Delta\omega$
between pump laser and OPO cavity caused by the random frequency drifts.
As in \citep{PhysRevA.30.1386}, we start from the quantum Langevin
equation of the OPO cavity\begin{equation}
\dot{a}=-\frac{i}{\hbar}[a,H_{sys}]-\left(k_{1}+k_{2}\right)a+\sqrt{2k_{1}}a_{v1}+\sqrt{2k_{2}}a_{v2}\end{equation}
where $a$ and $a^{\dagger}$ denote annihilation and creation operators
of the cavity mode with frequency $\omega_{0}$, $k_{1}$ and $k_{2}$
denote the loss rates due to output coupler and intracavity losses,
and $a_{v1}$ and $a_{v2}$ denote the annihilation operators of the
(vacuum) field entering the cavity due to output coupler and intracavity
losses. The Hamiltonian operator of the system is\begin{equation}
H_{sys}=\hbar\omega_{0}a^{\dagger}a+\frac{i\hbar}{2}(\epsilon e^{-i\omega_{P}t}(a^{\dagger})^{2}-\epsilon^{*}e^{i\omega_{P}t}a^{2})\end{equation}
where the first term describes the energy of photons inside the cavity
while the second term models the non-linear interaction induced by
the pump field with frequency $\omega_{P}$. The phase of the non-linear
coupling constant $\epsilon=\left|\epsilon\right|e^{i\phi}$ is determined
by the phase of the pump field, $\phi$. Furthermore, we assume that
the squeezed mode is detuned from the cavity resonance by $2\Delta\omega\equiv\omega_{P}-2\omega_{0}$.
By performing the equivalent calculation as in \citep{PhysRevA.30.1386}
we finally reach the Bogoliubov transformation from input to output
fields
\begin{eqnarray}
\tilde{a}_{out}\left(\omega+\Delta\omega\right) & = & \left[A_{1}\tilde{a}_{v1}\left(\omega+\Delta\omega\right)+A_{2}\tilde{a}_{v2}\left(\omega+\Delta\omega\right)\right.\nonumber \\
& & +\left.C_{1}\tilde{a}_{v1}^{\dagger}\left(-\omega+\Delta\omega\right)\right.\nonumber \\
& & +\left.C_{2}\tilde{a}_{v2}^{\dagger}\left(-\omega+\Delta\omega\right)\right]B^{-1}\label{eq:outputfield}\end{eqnarray}
where \begin{eqnarray}
A_{1} & = & \eta^{2}-\left(1-\eta-i\Omega\right)^{2}+\Delta\Omega\left(2\eta i-\Delta\Omega\right)+\left|\alpha\right|^{2}\\
A_{2} & = & 2\sqrt{\eta\left(1-\eta\right)}\left[i\left(-\Omega+\Delta\Omega\right)+1\right]\\
C_{1} & = & 2\eta\alpha\\
C_{2} & = & 2\alpha\sqrt{\eta\left(1-\eta\right)}\\
B & = & \left(1-i\Omega\right)^{2}+\Delta\Omega^{2}-\left|\alpha\right|^{2}\end{eqnarray}
We have introduced $\tilde{a}$ as the operators in rotating frame,
and scaled all frequencies and rates to the cavity linewidth, i.e.
demodulation (detection) frequency $\Omega=\frac{\omega}{k_{1}+k_{2}}$,
detuning $\Delta\Omega=\frac{\Delta\omega}{k_{1}+k_{2}}$, cavity
escape efficiency $\eta=\frac{k_{1}}{k_{1}+k_{2}}$, $1-\eta=\frac{k_{2}}{k_{1}+k_{2}}$,
and pump amplitude $\alpha=\frac{\varepsilon}{k_{1}+k_{2}}$. The
squeezing spectrum $S(\Omega)$ can be deduced from equation (\ref{eq:outputfield})
using $q_{\theta}=\frac{1}{\sqrt{2}}(\tilde{a}_{out}e^{-i\theta}+\tilde{a}_{out}^{\dagger}e^{i\theta})$
\begin{eqnarray}
S(\Omega) & = & 1+2\eta_{det}\left\langle :q_{\theta},q_{\theta}:\right\rangle \nonumber \\
& = & 1+\frac{8\eta_{det}\eta\left|\alpha\right|^{2}}{\left|B\right|^{2}}\left[1+\frac{B}{2|\alpha|}\cos\left(\Delta\phi+2\Delta\omega\tau_{D}\right)\right.\nonumber \\
& & +\left.\frac{\Delta\Omega}{|\alpha|}\sin\left(\Delta\phi+2\Delta\omega\tau_{D}\right)\right]\end{eqnarray}
where $\theta=\theta_{0}+\Delta\omega\tau_{D}$ denotes the phase
of the local oscillator, with $\theta_{0}$ being the phase of the
local oscillator in the white light configuration and $2\Delta\omega\tau_{D}$
being the phase shift for detuned local oscillator when the light
is delayed for $\tau_{D}$ from the white light configuration. Furthermore,
$\Delta\phi=2\theta_{0}-\phi$ denotes the relative phase between
the phase of the local oscillator in the white light configuration
and of the pump laser of the OPO, $\alpha=|\alpha|e^{i\phi}$. Best
squeezing is obtained for the phase that gives
\begin{equation}
\tan(\Delta\phi+2\Delta\omega\tau_{D})=\frac{2\Delta\Omega}{1-\Delta\Omega^{2}+\Omega^{2}+|\alpha|^{2}}\end{equation}
which due to the cavity dispersion depends on the detuning of the
pump laser $\Delta\Omega$. The right side of the equation represents
the delay in the OPO cavity. In first order of the detuning the squeezing
phase is
\begin{equation}
\Delta\phi+2\Delta\omega\tau_{D}=\pi+\frac{2}{1+\Omega^{2}+|\alpha|^{2}}\Delta\Omega\end{equation}
This dispersion can be compensated by delaying the local oscillator
before the homodyne detection. A delay line of length $l$ and group
index $n_{g}$ will introduce the phase shift
\begin{equation}
2\Delta\omega\tau_{D}=2(k_{1}+k_{2})\frac{ln_{g}}{c}\Delta\Omega\end{equation}
Thus for delay length
\begin{equation}
l=\frac{c}{n_{g}(k_{1}+k_{2})(1+\Omega^{2}+|\alpha|^{2})}\label{eq:delaylength}\end{equation}
the homodyne detection will be performed, to first order, at the correct
squeezing phase $\Delta\phi=\pi$ even for detuned pump. The dispersion
in the OPO cavity and therefore also the compensation length depends
on the detection frequency $\Omega$. For higher detection frequency
a shorter compensation delay is necessary. Assuming a slowly drifting
laser with power spectral density of $\rho(\Delta\Omega)d\Delta\Omega$
the obtained squeezing can be modeled by averaging the homodyne power
spectrum $S(\Omega)$ for phase $\Delta\phi=\pi$ over $\Delta\Omega$.
The averaged squeezing spectrum
\begin{equation}
\bar{S}(\Omega)=\int_{-\infty}^{+\infty}S(\Omega,\Delta\Omega)\rho(\Delta\Omega)\, d\Delta\Omega\end{equation}
is plotted in Fig.\ref{fig:gauss} for a Gaussian and a Lorentzian
linewidth $\rho(\Delta\Omega)$. We note that physically $\Delta\Omega$
is the mismatch between half the pump frequency and the OPO cavity
frequency scaled to the cavity linewidth, and thus both laser frequency
fluctuations and OPO cavity fluctuations will contribute to $\rho(\Delta\Omega)$.
The shift of optimum squeezing to positive delay is due to the existence
of the delay introduced by OPO.
\section{delay considerations\label{sec:delay-considerations}}
We note that in Eqs. (\ref{eq:outputfield}) and (\ref{eq:delaylength}),
$\tau_{D}$ is the group delay between the local oscillator and the
pump light at the cavity. As both local oscillator and pump are ultimately
derived from the same laser, we can identify $\tau_{D}=0$ as a {}``white-light''
condition in a Mach-Zehnder-topology interferometer. The light in
the squeezing path passes the tapered amplifier, doubling cavity,
mode matching fiber, lengths of free-space propagation, and the OPO
cavity. The light in the local oscillator path passes lengths of free-space
propagation and a mode-cleaning fiber (which we use to introduce the
desired delays).
In presenting the results, the {}``zero'' of $\tau_{D}$ is taken
to be when the total delay in the local oscillator path, as calculated
from measurements of fiber and free-space lengths, is equal to the
combined delays in the amplifier, doubling cavity, fiber and free
space. We do not include the OPO cavity delay because this depends
on $\Omega$, as presented in the theory and shown in fig. \ref{fig:gauss}.
The delay introduced by the doubling cavity is the cavity group delay
at line center \begin{equation}
\tau=\frac{1}{\pi\cdot\delta\nu}\end{equation}
The measured doubling cavity linewidth is $\delta\nu=14$$\,$MHz.
To delay the local oscillator we have used fibers with group index
$n_{g}=1.5$.
We note that, as the the laser and doubling cavity are mutually locked,
it is not obvious how the doubling cavity delay should be included.
While the light is obviously propagating from laser through doubling
cavity, a frequency fluctuation in the doubling cavity will, via the
current feedback, affect the laser frequency. We choose to include
the cavity delay, in the squeezing path because it gives best agreement
with the data presented below.
\section{measured squeezing vs. delay}
We have performed a series of measurement where a controllable delay
was introduced in the path of the local oscillator with intention
to: (i) measure level of squeezing in white light configuration (ii)
see the effect of the change of delay on the level of squeezing. The
results are presented in the Fig. \ref{fig:delay}.
We performed the measurements of the quadrature variance for every
four meters added in the local oscillator path starting from the proximity
of the balanced delay configuration. Final fiber length was 60$\,$m
longer than the balanced configuration. Due to the limited pump power
and large fiber losses for the blue light measurements at negative
delay were not feasible. We measured squeezing vs. delay for three
different demodulation frequencies 1$\,$MHz, 2$\,$MHz and 3$\,$MHz
(Fig. \ref{fig:delay}).
\begin{figure}
\caption{Squeezing dependent on the path mismatch measured for three
different detection frequencies: a) at 1$\,$MHz demodulation frequency,
b) at 2$\,$MHz demodulation frequency c) at 3$\,$MHz demodulation
frequency. The points show the experimental data, the solid lines
the predicted level of squeezing for the parameters measured in the
experiments using a Gaussian profile of 700$\,$kHz linewidth (FWHM),
the dashed lines show the theoretical level of squeezing for the same
parameters as the solid line with additional technical noise independent
of the relative delay. The error bars represent standard deviation
over series of identical measurements.}
\label{fig:delay}
\end{figure}
The experimental results show minima at positive delay as predicted
by theory. Eqn. \ref{eq:delaylength} predicts $l=\{7.3,6.0,4.7\}$$\,$m
shift for demodulation frequencies $\{1,2,3\}$$\,$MHz, respectively.
Here we assume that the doubling cavity delay is equal to the delay
which the cavity introduces at the resonance. Naturally this delay
does not depend on the demodulation frequency.
The theoretical curves in Fig. \ref{fig:delay} are obtained using
all experimental parameters as stated above, but varying the width
of $\rho(\Delta\Omega)$ as the only free parameter. Of two different
profiles treated in the theory the comparison with the Gaussian reflects
the shape of the experimental curve more closely than the Lorentzian
profile. We see good agreement, especially at 3$\,$MHz demodulation
frequency, for a Gaussian spectrum of 700$\,$kHz (FWHM). Using the
in-loop signal from the laser lock to a saturated-absorption reference,
we find a 400$\,$kHz laser linewidth. A similar measurement of the
distribution of $\omega_{0}-\omega_{laser}$ can be made using the
OPO cavity locking signal. Under the conditions of the squeezing measurements,
however, the locking signal was too weak to extract a meaningful signal,
largely because we cannot inject through the output mirror as in the
PDH technique. We can place a lower limit of 300$\,$kHz on the width
of $\rho(\Delta\Omega)$ based on PDH locking of the same cavity,
and the 700$\,$kHz estimate for the width of $\rho(\Delta\Omega)$
appears reasonable.
On the other hand, the level of squeezing we observe in the 1$\,$MHz
and 2$\,$MHz measurements is smaller than predicted by theory. This
might be caused by the light back reflected form the n-faces of the
nonlinear crystal contaminating the squeezed light. If we assume that
this noise is independent of the relative delay, it can be modeled
by a constant offset to our theoretical squeezing curves. With an
offset of (+0.07,+0.03) relative to the standard quantum limit for
(1, 2) MHz, respectively, the theory for a Gaussian laser spectrum
of 700$\,$kHz fits well in shape and amplitude to our measured data
as shown in fig. \ref{fig:delay}. By solving the problem of noise
which causes the decrease of squeezing in the 1$\,$MHz and 2$\,$MHz
measurements one could in agreement with the theory detect more than
5$\,$dB of noise reduction at the OPO output.
\section{conclusion}
We have demonstrated quadrature and polarization squeezing using a
sub-threshold OPO and a frequency-doubled diode laser for a pump.
We have investigated and optimized the squeezing properties by using
a delayed local oscillator. We adapted the theoretical description
of Collet and Gardiner \citep{PhysRevA.30.1386} under the assumption
of slow frequency fluctuations. The theoretical description can be
used to model random frequency fluctuations of the laser but also
the problem of optimization of the OPO cavity stabilization. This
approach showed that the OPO cavity exhibits dispersive behavior which
causes a delay of the squeezed light. Optimum squeezing is observed
if the squeezed light is in white light configuration with respect
to the local oscillator. Experimental results confirmed that the vacuum
mode of the OPO is also taking part in the delay line. This investigation
shows that, by taking into account the balancing and the delay lines,
diode laser sources can be used for producing quadrature and polarization
squeezing in an OPO. Since diode lasers are much cheaper and simpler
to operate our work brings portable inexpensive squeezing devices
for application in e.g.$\,$ precisions measurements into reach.
We gratefully acknowledge inspiring discussions and motivating support
by Eugene S. Polzik , Professor (NBI Copenhagen).
This investigation was supported by the Departament d'Universitats,
Recerca i Societat de la Informació of the Generalitat de Catalunya,
the European Social Fund and the Ministerio de Educación y Ciencia
under the FLUCMEM project (Ref. FIS2005-03394), the Consolider-Ingenio
2010 Project {}``QOIT'' and by Marie Curie RTN {}``EMALI.''
\end{document} |
\begin{document}
\title{Permutation-based uncertainty quantification about a mixing distribution}
\begin{abstract}
Nonparametric estimation of a mixing distribution based on data coming from a mixture model is a challenging problem. Beyond estimation, there is interest in uncertainty quantification, e.g., confidence intervals for features of the mixing distribution. This paper focuses on estimation via the predictive recursion algorithm, and here we take advantage of this estimator's seemingly undesirable dependence on the data ordering to obtain a permutation-based approximation of the sampling distribution which can be used to quantify uncertainty. Theoretical and numerical results confirm that the proposed method leads to valid confidence intervals, at least approximately.
\emph{Keywords and phrases:} Confidence interval; density estimation; mixture model; nonparametric; predictive recursion.
\end{abstract}
\section{Introduction}
\label{S:intro}
At a high-level, statistical analysis aims to separate signal from noise, and one of the more challenging problems is deconvolution or, more generally, estimation of a mixing distribution based on samples from the mixture. Suppose that data $Y^n = (Y_1,\ldots,Y_n)$ are independent and identically distributed from a density $f$ with respect to Lebesgue measure on $\mathbb{Y}$, which we model as a mixture
\begin{equation}
\label{eq:mixture}
f(y) = \int_\mathcal{X}X k(y \mid x) \, p(x) \, \mu(dx), \quad y \in \mathbb{Y},
\end{equation}
where $k$ is a known kernel, i.e., $y \mapsto k(y \mid x)$ is a density on $\mathbb{Y}$ for each $x \in \mathcal{X}X$, and $p$ is a unknown mixing density with respect to a known $\sigma$-finite measure $\mu$ on $\mathcal{X}X$. Alternatively, one can view this model hierarchically by assuming that $X_1,\ldots,X_n$ are independent and identically distributed from $p$, and $Y_i$, given $X_i$, are independently distributed from $k(y \mid X_i)$, $i=1,\ldots,n$. So if we think of $X_1,\ldots,X_n$ as ``signals'' with distribution $p$, and $Y_i$ the version corrupted by noise, then our goal---inference about $p$ based on data from model \eqref{eq:mixture}---can be viewed as separation of signal from noise.
Often, $p$ is assumed to be discrete with finitely many points in $\mathcal{X}X$. For these so-called finite mixture models, standard modes of inference can be applied. For example, when the number of support points of $p$ is known, there is a likelihood function with relatively simple form that can be optimized, often using the EM algorithm \citep{dempetal, teeletal}, to produce the corresponding maximum likelihood estimator, to which the classical asymptotic distribution theory applies \citep[e.g.,][]{rednerwalker}; for a comprehensive treatment, see \citet{mcpeel}. Similarly, with this same likelihood and a corresponding prior distribution for $p$, one can apply an EM-like data-augmentation strategy \citep[e.g.,][]{vm} to carry out a Bayesian analysis. In the more realistic scenario where the number of components in the finite mixture model is unknown, those methods described above can be modified by introducing a penalty term or a prior distribution on the number of mixture components, as in \citet{leroux} and \citet{richardgreen}.
For the case considered here, where $p$ is a smooth mixing density, a number of methods for nonparametric estimation have appeared in the literature. When $k(y \mid x) = k(y - x)$ is a location-shift kernel, so that the mixture density is just a convolution, estimation of $p$ is referred to as deconvolution, a case that has been studied in \citet{fan}, \citet{sc}, and \citet{zhang}. For general mixtures, there are a variety of different approaches. Maximizing the likelihood will almost surely produce a discrete estimate of the mixing distribution \citep{lindsay}, which is not a satisfactory estimate of a smooth mixing density. Various approaches aim to smooth the discrete nonparametric maximum likelihood estimator, either directly \citep[e.g.,][]{eggermont} or by introducing a smoothness penalty \citep[e.g.,][]{liuetall}. \citet{chaeetall} investigate an iterative algorithm that generates a sequence of smooth mixing density estimates that converge to the nonparametric maximum likelihood estimator. Another interesting and related method, which is the focus of the present paper, is that based on the {\em predictive recursion} algorithm first described in \citet{nqz}, \citet{newtonzhang}, and \citet{newton02}, with extensions and theoretical properties developed in \citet{ghoshtokdar}, \citet{martinghosh}, \citet{tmg}, and \citet{mt-rate, mt-prml, mt-test}; for a recent review of these developments, see \citet{pr.jkg.review}.
Beyond estimation, a goal is to quantify uncertainty about the mixing density $p$ and, for this, the literature is scarce. The work that has been done is as listed below. \citet{bd} discuss asymptotic and bootstrap confidence bands for deconvolution problems, building on ideas first presented in \citet{br}. \citet{ls} discuss empirical Bayes estimation of an empirical mixing distribution with emphasis on construction of interval estimates, using the nonparametric maximum likelihood estimator. \citet{fortpetrone} have developed asymptotically approximate credible intervals for the cumulative distribution function based on a quasi-Bayesian interpretation of the predictive recursion algorithm.
A seemingly undesirable feature of the predictive recursion estimator is that it depends on the order in which the data is processed. In particular, this means that the estimator is not a function of the sufficient statistic---which, in this setting, is the empirical distribution---and, hence, the estimator cannot be Bayesian. In previous literature on predictive recursion, the focus has been on reducing its dependence on the order. For example, \citet{newton02} suggested elimination of the order-dependence by averaging the estimators over a number of randomly chosen permutations; see \citet{tmg} for details. The idea in the present paper is to leverage predictive recursion's order-dependence for the purpose of uncertainty quantification. Specifically, we propose to generate multiple copies of the predictive recursion estimator by permuting the data sequence, and then use this permutation-based distribution as an approximation of the estimator's sampling distribution. After a review of the predictive recursion estimator and its properties in Section~\ref{S:pr}, we show in Section~\ref{S:perm} that, for a given feature of the mixing distribution, the estimator's sampling distribution variance can be approximately unbiasedly estimated by the corresponding permutation-based distribution variance. Being able to accurately estimate the spread of the relevant sampling distribution immediately suggests using the same permutation distribution quantiles as an approximate confidence interval for that mixing distribution feature. Numerical results presented in Section~\ref{S:examples} reveal that this permutation-based approach gives approximately valid confidence intervals in finite samples across a range of mixture models and for various features of the mixing distribution, including the density function at a point.
\section{Predictive recursion}
\label{S:pr}
Recall that we observe independent data $Y^n = (Y_1,\ldots,Y_n)$ from the mixture in \eqref{eq:mixture}, and the goal is to estimate the mixing density $p$. The following predictive recursion algorithm returns a computationally efficient nonparametric estimator of $p$.
\begin{pralg}
Start with an initial estimate $p_0$ of the mixing density and a sequence of weights $\{w_i : i \geq 1\} \subset (0,1)$. Using the observations $Y_1,\ldots,Y_n$ from the mixture model, in that order, compute
\begin{equation}
\label{eq:pr}
p_i(x) = (1-w_i) \, p_{i-1}(x) + w_i \, \frac{k(Y_i \mid x)p_{i-1}(x)}{f_{i-1}(Y_i)}, \quad i=1,\ldots,n,
\end{equation}
where $f_{i-1}(y) = \int k(y \mid x)p_{i-1}(x) \, \mu(dx)$ is the mixture corresponding to $p_{i-1}$. Finally, return $p_n$ and $f_n=f_{p_n}$ as the estimates of $p$ and $f$, respectively.
\end{pralg}
An interesting observation is that, if $p_0$ is a smooth density with respect to the measure $\mu$, then the output, $p_n$, of the predictive recursion algorithm will also be a smooth density. Compare this to the nonparametric MLE which is almost surely discrete, regardless of the smoothness of $p$ in \eqref{eq:mixture}. Therefore, there is no need for post hoc smoothing of the predictive recursion estimator. And the ability to specify the dominating measure in the predictive recursion algorithm proved to be a useful property in the multiple testing application considered in \citet{mt-test}.
Aside from estimating the mixing density itself, one can readily estimate various features of the mixing distribution. That is, if $\psi$ is a suitable function, then $\int \psi \, p \, d\mu$ can be estimated by $\int \psi \, p_n \, d\mu$. For example, we can estimate the mixing distribution function at a point $x_0$ by taking $\psi$ to be the indicator function corresponding to $(-\infty,x_0]$.
Asymptotic convergence properties of the predictive recursion estimator were investigated in \citet{tmg} and \citet{mt-rate}. To summarize, under suitable tail conditions on the kernel, if $p$ is identifiable from the mixture model \eqref{eq:mixture} and if the weights $(w_i)$ in the predictive recursion algorithm satisfy
\[ \sum_{i=1}^\infty w_i = \infty \quad \text{and} \quad \sum_{i=1}^\infty w_i^2 < \infty, \]
then $p_n$ converges to $p$ almost surely in the weak topology, that is, if $\psi: \mathcal{X}X \to \mathbb{R}$ is a bounded and continuous function, then $\int \psi \, p_n \,d\mu \to \int \psi \, p \, d\mu$ almost surely.
\section{Leveraging order-dependence}
\label{S:perm}
It is clear from \eqref{eq:pr} that $p_n$ depends on the order in which the data are processed. However, since the data are assumed to be independent and identically distributed, the ordering should be irrelevant. To alleviate predictive recursion's seemingly undesirable order-dependence, \citet{newton02} and others have suggested to average the final estimate, $p_n$, over a number of randomly chosen permutations of the data sequence. \citet{tmg} describe this as a sort of Rao--Blackwellization, replacing $p_n$ by a Monte Carlo approximation of its conditional expectation given the order statistics. Here, instead of trying to remove predictive recursion's order-dependence, we propose to leverage it for the purpose of uncertainty quantification.
Let $\mathcal{S}_n$ denote the permutation group on integers $\{1,2,\ldots,n\}$, i.e., the set of all bijections from $\{1,2,\ldots,n\}$ to itself. If $p_n$ is the predictive recursion estimator based on data $(Y_1,\ldots,Y_n)$, in its given order, write $p_n^s$ for the corresponding estimator based on the data $Y_{s(1)},\ldots,Y_{s(n)}$, permuted according to $s \in \mathcal{S}_n$. If $\psi$ is a suitable function, write $\Psi_n = \int \psi \, p_n \, d\mu$ and $\Psi_n^s = \int \psi \, p_n^s \,d\mu$ as the estimators of $\Psi = \int \psi \, p \, d\mu$ based on the original and permuted data, respectively.
Our proposal is to approximate the sampling distribution of $\Psi_n$, as a function of $Y^n$ sampled from the mixture in \eqref{eq:mixture}, by the distribution of $\Psi_n^S$, for fixed $Y^n$, as a function of $S \sim {\sf Unif}(\mathcal{S}_n)$. The justification for our claim that this provides an accurate approximation, at least asymptotically, is the following simple identity,
\begin{equation}
\label{eq:identity}
\mathsf{V}_{Y^n}( \Psi_n ) = \mathsf{V}_{Y^n,S}( \Psi_n^S ),
\end{equation}
where $\mathsf{V}_{Y^n}$ is the variance with respect to the distribution of $Y^n$ from the mixture model \eqref{eq:mixture}, and $\mathsf{V}_{Y^n,S}$ is the variance with respect to the joint distribution of $Y^n$ and $S \sim {\sf Unif}(\mathcal{S}_n)$, assumed to be independent. The identity \eqref{eq:identity} holds because, when data $Y^n$ are independent and identically distributed, the extra layer of permutations changes nothing. In other words, if we imagine enumerating all possible realizations of $Y^n$ to evaluate the left-hand side of \eqref{eq:identity}, then we would get no new realizations if we also enumerated permutations for evaluating the right-hand side.
The practical value of the identity \eqref{eq:identity} is that the right-hand side suggests a familiar total-variance decomposition:
\begin{equation}
\label{eq:decomp}
\mathsf{V}_{Y^n, S}(\Psi_n^S) = \mathsf{E}_{Y^n}\{ \mathsf{V}_{Y^n,S}( \Psi_n^S \mid Y^n)\} + \mathsf{V}_{Y^n}\{ \mathsf{E}_{Y^n, S}( \Psi_n^S \mid Y^n) \}.
\end{equation}
By the asymptotic consistency property of predictive recursion, discussed in Section~\ref{S:pr}, if $\Psi$ is bounded and continuous, then $\Psi_n$ and, hence, $\mathsf{E}_{Y^n,S}(\Psi_n^S \mid Y^n)$ are consistent estimates of $\Psi$, so its variance, the second term on the right-hand side of \eqref{eq:decomp}, should be near 0 when $n$ is large. Therefore,
\begin{equation}
\label{eq:unbiased}
\mathsf{V}_{Y^n, S}(\Psi_n^S) \approx \mathsf{E}_{Y^n}\{ \mathsf{V}_{Y^n,S}( \Psi_n^S \mid Y^n)\},
\end{equation}
in other words, $\mathsf{V}_{Y^n,S}( \Psi_n^S \mid Y^n)$ is an approximately unbiased estimator of $\mathsf{V}_{Y^n}(\Psi_n)$. The key point, of course, is that the left-hand side of \eqref{eq:identity}, namely, $\mathsf{V}_{Y^n}(\Psi_n)$, the variance of the sampling distribution of $\Psi_n$, is relevant for uncertainty quantification, but it is not readily available. However, the quantity on the right-hand side of \eqref{eq:identity}, namely, $\mathsf{V}_{Y^n,S}(\Psi_n^S \mid Y^n)$, can be readily computed by repeatedly sampling $S \sim {\sf Unif}(\mathcal{S}_n)$, permuting the data according to $S$, and re-evaluating the predictive recursion estimator. This provides us with a relatively simple and fast approach to construct a data-dependent distribution for $p$ or $\Psi$ from which valid uncertainty quantification can be achieved. And beyond variance estimation, to get an approximate $100(1-\alpha)$\% confidence interval for $\Psi$, we can easily extract the $\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$ quantiles of $\Psi_n^S$ based on repeated sampling of $S \sim {\sf Unif}(\mathcal{S}_n)$ with data $Y^n$ fixed.
\section{Numerical results}
\label{S:examples}
Here we carry out an empirical investigation into the performance of our proposed permutation-based approach to uncertainty quantification. We begin with pointwise evaluation of the mixing distribution function. The specific scenario we consider here is one where the true mixing density in \eqref{eq:mixture} is a gamma with shape 2 and rate 1, and the kernel $k(y \mid x)$ is also a gamma with shape $20x$ and rate 20; this is Example~3-3 below. We generate samples of size $n=500$ from this mixture model and consider estimating the mixing distribution function, in particular, at the fixed points $x \in \{2, 5, 8\}$. For predictive recursion, we take initial guess $p_0 = {\sf Unif}(0,10)$ and weight sequence $w_i = (i+1)^{-0.67}$ suggested by \citet{mt-rate}.
Before presenting the results, we have to address a relevant and practically important question, namely, how many permutations? In our examples, we are using 200 randomly generated permutations on which to evaluate the predictive recursion estimator. Of course, it does not hurt to do more than 200, and this is still computationally feasible thanks to the algorithm's efficiency; however, at least in the examples we tried, there were no substantial differences in the results based on more than 200 random permutations.
A plot of distribution function estimates for this gamma mixture example, based on 200 random permutations, is shown in Figure~\ref{fig:cdf}. Note that the span of these estimates over permutations hugs the true distribution relatively closely across the entire range of $x$. The vertical bars at $x \in \{2, 5, 8\}$ correspond to the central 95\% interval of the sampling distribution of the predictive recursion estimator, based on repeated sampling from the gamma mixture. The goal of uncertainty quantification is to match this interval as closely as possible, so it is notable that, as predicted by the arguments leading up to \eqref{eq:unbiased}, our permutation-based intervals are comparable to this ``gold-standard'' across various $x$. Moreover, the permutation distribution takes only about 10 seconds in R running on an ordinary laptop computer. \citet{fortpetrone} present asymptotically approximate credible intervals for the same cumulative distribution function based on the predictive recursion algorithm. But their analysis is based on a dependent and non-stationary model for $Y^n$---one that makes the predictive recursion estimator ``quasi-Bayes''---and, since their model and perspective on uncertainty quantification is different from ours, a direct comparison is not appropriate.
\begin{figure}
\caption{Plot of the predictive recursion estimates of the distribution function for each of 200 random permutations (gray), along with the true distribution function (black); dashed line corresponds to the predictive recursion estimate averaged over permutations. Vertical bars correspond to the central 95\% interval of the sampling distribution of the predictive recursion estimator.}
\label{fig:cdf}
\end{figure}
Next we consider pointwise estimation of the mixing density. This is not strictly covered by the theoretical arguments discussed above because the functional $p \mapsto p(x)$ for a fixed $x$ cannot be expressed as $\int \psi \, p \, d\mu$ for a bounded and continuous $\psi$. However, intuition and prior experience suggests that the predictive recursion density estimate ought to satisfy a pointwise consistency property, i.e., $p_n(x) \to p(x)$ for fixed $x$; see, also, Section~\ref{S:discuss}. Therefore, we can apply the same total-variance decomposition as in \eqref{eq:decomp} and reason that
\[ \mathsf{V}_{Y^n}\{p_n(x)\} \approx \mathsf{E}_{Y^n}\bigl[ \mathsf{V}_{Y^n,S}\{ p_n^S(x) \mid Y^n\} \bigr] \]
and, moreover, that suitable quantiles from the permutation distribution can be used in the obvious way to construct approximate confidence intervals for $p(x)$.
Like in \citet{chaeetall}, we consider nine examples corresponding to different combinations of the following three kernels and mixing densities.
\begin{description}
\item[\sc Kernel 1.] $k(y \mid x) = {\sf N}(y \mid x, 0.5)$;
\item[\sc Kernel 2.] $k(y \mid x) = \frac{1}{0.3}{\sf t}(\frac{y-x}{0.3} \mid \text{df}=5)$;
\item[\sc Kernel 3.] $k(y \mid x) = {\sf Gamma}(x \mid \text{shape}=20x,\, \text{rate}=20)$;
\item[\sc Mixing Density 1.] $p(x) = \frac{1}{10}{\sf Beta}(\frac{x}{10} \mid 5, 5)$;
\item[\sc Mixing Density 2.] $p(x) = \frac{3}{4}{\sf N}(x \mid 3,0.8^{2}) + \frac{1}{4}{\sf N}(x \mid 7,0.8^{2})$;
\item[\sc Mixing Density 3.] $p(x) = {\sf Gamma}(x \mid \text{shape}=2, \, \text{rate}=1)$.
\end{description}
In what follows, Example~$a$-$b$ will refer to the case with kernel $a$ and mixing density $b$, where $a=1,2,3$ and $b=1,2,3$.
As a first visualization, we simulate $n=500$ observations from each of the above mixture models. For each data set, we run the predictive recursion algorithm for 200 randomly sampled permutations. Each panel in Figure~\ref{fig:density} shows these 200 density estimates, $p_n^S$, the true density, $p$. As we expect, the cluster of permutation-based densities hugs the true mixing density rather closely throughout the range, with more variability in regions where the true density has more curvature. Also displayed in these panels is a central 95\% interval from the sampling distribution of $p_n(x)$, at $x \in \{2,5,8\}$, based on 500 samples of size $n=500$ from the mixture model. As suggested by \eqref{eq:unbiased}, the spread of the permutation distribution matches that of the sampling distribution relatively accurately at all three $x$ values and across all 9 of the examples.
\begin{figure}
\caption{Plots of the predictive recursion mixing density estimates (gray) based on 200 random permutations of the data sequence, with the true mixing density (black) overlaid. Dashed line corresponds to the predictive recursion estimate averaged over permutations. Vertical lines correspond to the central 95\% interval from the sampling distribution of $p_n(x)$, for $x \in \{2,5,8\}
\label{fig:density}
\end{figure}
To assess our claim that the permutation-based uncertainty quantification is approximately valid, we repeat the above experiment 500 times, extract the nominal 95\% confidence interval for $p(x)$ based on the permutation distribution and check its coverage probability. Table~\ref{tab:coverage} shows the estimated coverage probabilities for the all 9 examples, at each of the three $x$ values, and for two different sample sizes. Note, first, that the coverage probability increases from $n=500$ to $n=1000$. Major departures from the targeted 95\% level are at $x$ values around which $p$ has considerable curvature; in regions where $p$ is smoother, the coverage probability estimate tends to be closer to the 95\% level. The overall message is that the permutation-based distribution gives approximately valid uncertainty quantification across a variety of mixing densities and kernels.
\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc}
\hline
& \multicolumn{3}{c}{$n=500$} & \multicolumn{3}{c}{$n=1000$}\\
\cline{2-7}
Example & $x=2$ & $x=5$ & $x=8$ & $x=2$ & $x=5$ & $x=8$\\
\hline
1-1 & 0.914 & 1.000 & 0.904 & 0.964 & 1.000 & 0.976\\
2-1 & 0.882 & 0.990 & 0.882 & 0.956 & 1.000 & 0.952\\
3-1 & 0.880 & 1.000 & 0.888 & 0.950 & 1.000 & 0.962\\
1-2 & 0.994 & 0.476 & 0.948 & 1.000 & 0.488 & 0.982\\
2-2 & 0.986 & 0.914 & 0.938 & 0.998 & 0.968 & 0.970\\
3-2 & 0.972 & 0.910 & 0.918 & 0.998 & 0.966 & 0.972\\
1-3 & 0.998 & 0.930 & 0.550 & 1.000 & 0.982 & 0.710\\
2-3 & 0.994 & 0.862 & 0.378 & 1.000 & 0.938 & 0.540\\
3-3 & 0.996 & 0.906 & 0.554 & 1.000 & 0.984 & 0.644\\
\hline
\end{tabular}
\end{center}
\caption{Estimated coverage probabilities for the mixing density $p(x)$ in the nine examples across different sample sizes and $x$ values.}
\label{tab:coverage}
\end{table}
\section{Conclusion}
\label{S:discuss}
This paper describes a simple permutation-based approach to uncertainty quantification about a mixing distribution by leveraging the built-in dependence of the predictive recursion estimator on the data ordering. The development and numerical results presented here suggest that the uncertainty quantification achieved by this approach, e.g., a confidence interval for the mixing distribution or density function, are valid in the sense that the frequentist coverage probability is approximately equal to the interval's nominal level.
We will end with two concluding remarks. First, as noted in Section~\ref{S:examples}, we are currently lacking results on the pointwise consistency of the predictive recursion estimator of the mixing density estimate. At present, we have only results on almost sure convergence of mixing measure estimator to the truth in the weak topology. One idea is to prove the pointwise convergence directly using the structure of the predictive recursion estimator. Another idea is to check the available sufficient conditions, namely, equicontinuity \citep[e.g.,][]{boos}, to convert weak convergence of measures into uniform convergence of densities. Unfortunately, we were unable to push through either of these approaches, but this does not shake our confidence in the convergence conjecture.
Second, the idea of leveraging data ordering dependence employed herein is not specific to the predictive recursion estimator. That is, when data are independent and identically distributed or, more generally, exchangeable, and the density estimator in consideration depends on the data ordering, then the total-variance argument in \eqref{eq:unbiased} could be applied and approximately valid uncertainty quantification could be achieved. A natural question is: are there any density estimators that depend on the data ordering? Interestingly, while off-the-shelf estimators tend to be permutation invariant, one can consider Ces\'aro averages of these order-independent estimators, which retain the estimator's good asymptotic properties while simultaneously creating order-dependence that can be leveraged using the techniques presented here for valid uncertainty quantification.
\section*{Acknowledgments}
This work is partially supported by the U.S.~National Science Foundation, under grants DMS--1737929 and DMS--1811802.
\ifthenelse{1=0}{
}
{
}
\end{document} |
\begin{document}
\title{Mean value formulas for classical solutions to uniformly parabolic equations in divergence form}
\author{{\sc{Emanuele Malagoli}
\thanks{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} degli Studi di Modena e Reggio Emilia, via Campi 213/b, 41125 Modena (Italy). E-mail: [email protected]}, \
\sc{Diego Pallara}
\thanks{Dipartimento di Matematica e Fisica ``Ennio De Giorgi'', Universit\`{a} del Salento and INFN, Sezione di Lecce, Ex Collegio Fiorini - Via per Arnesano - Lecce (Italy). E-mail: [email protected]} \
\sc{Sergio Polidoro}
\thanks{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41125 Modena (Italy). E-mail: [email protected]}
}}
\date{ }
\maketitle
\begin{abstract}
We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form.
We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality.
We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with {\em almost $C^1$-boundary}, the other stated for sets with finite perimeter.
\end{abstract}
\setcounter{equation}{0}
\section{Introduction}\label{secIntro}
Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. We consider classical solutions $u$ to the equation $\L u = f$ in $\Omega$, where $\L$ is a parabolic operator in divergence form defined for $z=(x,t) \in \mathbb{R}^{N+1}$ as follows
\begin{equation} \label{e-L}
\L u (z) := \sum_{i,j=1}^{N}\tfrac{\partial}{\partial x_i} \left( a_{ij} (z) \tfrac{\partial u}{\partial x_j}(z) \right) +
\sum_{i=1}^{N} b_{i} (z) \tfrac{\partial u}{\partial x_i}(z) + c(z) u(z) - \, \tfrac{\partial u}{\partial t}(z).
\end{equation}
In the following we use the notation
$A(z) := \left( a_{ij}(z) \right)_{i,j=1,\dots, N}, b(z) := \left( b_{1} (z), \dots, b_{N} (z) \right)$ and we write $\L u $ in the short form
\begin{equation} \label{e-LL}
\L u (z) := \div \left( A (z) \nabla_x u(z) \right) + \langle b(z), \nabla_x u(z)\rangle + c(z) u(z) -
\, \tfrac{\partial u}{\partial t}(z).
\end{equation}
Here $\div, \nabla_x$ and $\langle \, \cdot \, , \, \cdot \, \rangle$ denote the divergence, the gradient and the inner product in $\mathbb{R}^N$, respectively. We assume that the matrix $A(z)$ is symmetric and that the coefficients of the operator $\L$ are H\"older continuous functions with respect to the parabolic distance. This means that there exist two constants $M>0$ and $\alpha \in ]0,1]$, such that
\begin{equation} \label{e-hc}
|c (x,t) - c (y,s)| \le M \left(|x-y|^\alpha + |t-s|^{\alpha/2}\right),
\end{equation}
for every $(x,t), (y,s) \in \mathbb{R}^{N+1}$. We require that the above condition is satisfied not only by $c$,
but also by $a_{ij}, \frac{\partial a_{ij}}{\partial x_i}, b_{i}, \frac{\partial b_{i}}{\partial x_i}$, for $i, j = 1, \dots, N$,
with the same constants $M$ and $\alpha$. We finally assume that the coefficients of $\L$ are bounded and
that $\L$ is uniformly parabolic, \emph{i.e.}, there exist two constants $\lambda, \Lambda$, with $0 < \lambda < \Lambda$, such that
\begin{equation} \label{e-up}
\lambda |\xi|^2 \le \langle A(z) \xi, \xi \rangle \le \Lambda |\xi|^2,
\quad \left|\tfrac{\partial a_{ij}}{\partial x_i}\right| \le \Lambda,
\quad |b_i(z)| \le \Lambda, \quad |c(z)| \le \Lambda,
\end{equation}
for every $\xi \in \mathbb{R}^N$, for every $z \in \mathbb{R}^{N+1}$, and for $i,j=1, \dots, N$. Under the above assumptions, the classical parametrix method provides us with the existence of a fundamental solution $\Gamma$. In Section \ref{secFundSol} we shall quote from the monograph of Friedman \cite{Friedman} the results we need for our purposes.
The main achievements of this note are some mean value formulas for the solutions to $\L u = f$ that are written in terms of the level and super-level sets of the fundamental solution $\Gamma$. We extend previous results of Fabes and Garofalo \cite{FabesGarofalo} and Garofalo and Lanconelli \cite{GarofaloLanconelli-1989} in that we weaken the regularity requirement on the coefficients of $\L$ that in \cite{FabesGarofalo, GarofaloLanconelli-1989} are assumed to be $C^\infty$ smooth. As applications of the mean value formulas we give an elementary proof of the parabolic strong maximum principle.
We note that the conditions on the functions $\frac{\partial a_{ij}}{\partial x_i}$'s are needed in order to deal with classical solutions to the adjoint equation $\L^* v = 0$, as the mean value formulas rely on the divergence theorem applied to the function $(\x,\tau) \mapsto \Gamma(x,t,\xi,\tau)$.
We introduce some notation in order to state our main results. For every $z_0=(x_0, t_0) \in \mathbb{R}^{N+1}$ and
for every $r>0$, we set
\begin{equation} \label{e-Psi}
\begin{split}
\partialsi_r(z_0) & := \left\{ z \in \mathbb{R}^{N+1} \mid \Gamma(z_0; z ) = \tfrac{1}{r^N} \right\},
\\
\Omega_r(z_0) & := \left\{ z \in \mathbb{R}^{N+1} \mid \Gamma(z_0; z) > \tfrac{1}{r^N} \right\}.
\end{split}
\end{equation}
\begin{center}
\begin{tikzpicture}
\clip (-.51,7.21) rectangle (6.31,1.99);
\partialath[draw,thick] (-.5,7.2) rectangle (6.3,2);
\begin{axis}[axis x line=middle, axis y line=middle,
xtick=\empty,ytick=\empty,
ymin=-1.2, ymax=1.2, xmin=-.2,xmax=1.7, samples=121, rotate= -90]
\addplot [ddddblue,line width=.7pt,domain=.001:.1111] {sqrt(- 3 * x * ln(9*x)};
\addplot [ddddblue,line width=.7pt,domain=.001:.1111] {- sqrt(- 3 * x * ln(9*x))};
\addplot [dddblue,line width=.7pt,domain=.001:.25] {sqrt(- 2 * x * ln(4*x)};
\addplot [dddblue,line width=.7pt,domain=.001:.25] {- sqrt(- 2 * x * ln(4*x))};
\addplot [ddblue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))};
\addplot [ddblue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))};
\addplot [black,line width=1pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {\quad $z_0$};
\addplot [black,line width=1pt, domain=-.01:.01] {-sqrt(.0001 - x * x)}
node[below]{\qquad \qquad \qquad \qquad \quad \quad \quad \quad {\color{ddddblue} $\partialsi_r(z_0)$}};
\end{axis}
\draw [<-,line width=.4pt] (2.8475,7) -- (2.8475,2);
\end{tikzpicture}
{{\sc Fig.1}} - $\partialsi_r(z_0)$ for three different values of $r$.
\end{center}
Similarly to the elliptic case, we call $\partialsi_r(z_0)$ and $\Omega_r(z_0)$ respectively the \emph{parabolic sphere} and the \emph{parabolic ball} with radius $r$ and ``center'' at $(x_0,t_0)$. Note that, unlike the elliptic setting, $z_0$ belongs to the topological boundary of $\Omega_r(z_0)$. Because of the properties of the fundamental solution of uniformly parabolic operators, the parabolic balls $\Omega_r(z_0)$ are bounded sets and shrink to the center $z_0$ as $r \to 0$. We finally introduce the following kernels
\begin{equation} \label{e-kernels}
\begin{split}
K (z_0; z) & :=
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|},
\\
M(z_0; z) & :=
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0; z) \rangle }
{\Gamma(z_0; z)^2}.
\end{split}
\end{equation}
Here $\nabla_x \Gamma(z_0; z)$ and $|\nabla_{(x,t)}\Gamma(z_0; z)|$ denote the gradient with respect to the space variable $x$ and the norm of the gradient with respect to the variables $(x,t)$ of $\Gamma$, respectively. Moreover, we agree to set $K (z_0; z) = 0$ whenever $\nabla_{(x,t)}\Gamma(z_0;z)=0$. In the following, $\H^{N}$ denotes the $N$-dimensional Hausdorff measure. The first achievements of this note are the following mean value formulas.
\begin{theorem} \label{th-1}
Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, $f\in C(\Omega)$ and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Then, for every $z_0 \in \Omega$ and for almost every $r>0$ such that $\overline{\Omega_r(z_0)} \subset \Omega$ we have
\begin{align*}
u(z_0) = \int_{\partialsi_r(z_0)} K (z_0; z) u(z) \, d \H^{N} (z)
+ & \int_{\Omega_r(z_0)} f (z) \left( \tfrac{1}{r^N} - \Gamma(z_0; z) \right)\ dz \\
+ & \frac{1}{r^N} \int_{\Omega_r(z_0)} \left( \div \, b(z) - c(z) \right) u(z) \ dz,
\\
u(z_0) = \frac{1}{r^N} \int_{\Omega_r(z_0)} \!\!\!\!\! M (z_0; z) u(z) \, dz
+ \frac{N}{r^N} & \int_0^{r} \left(\r^{N-1}
\int_{\Omega_\r (z_0)} f (z) \left( \tfrac{1}{\r^N} - \Gamma(z_0;z) \right) dz \right) d \r \\
+ & \frac{N}{r^N} \int_0^{r} \left(\frac{1}{\r}
\int_{\Omega_\r (z_0)} \left( \div \, b(z) - c(z) \right) u(z) \, dz \right) d \r.
\end{align*}
The second statement holds for \emph{every} $r>0$ such that ${\Omega_r(z_0)} \subset \Omega$.
\end{theorem}
Note that $\tfrac{1}{r^N} - \Gamma(z_0; z) < 0$ in the set $\Omega_r(z_0)$, because of its very definiton \eqref{e-Psi}. This fact, together with the non-negativity of the kernels \eqref{e-kernels} will be used in the sequel to obtain the strong maximum principle from Theorem \ref{th-1}.
We next put Theorem \ref{th-1} in its context. It restores the mean value formulas first proved by Pini in \cite{Pini1951} for the heat equation $\partial_t u = \partial_x^2 u$, then by Watson in \cite{Watson1973} for the heat equation in several space variables. We also recall the mean value formulas first proved by Fabes and Garofalo in \cite{FabesGarofalo} for the equation $\L u = 0$, then extended by Garofalo and Lanconelli \cite{GarofaloLanconelli-1989} to the equation $\L u = f$, where the operator $\L$ has the form \eqref{e-L} and its coefficients are assumed to be $C^\infty$ smooth. This extra regularity assumption on the coefficients of $\L$ is due to the fact that the mean value formula relies on the divergence theorem applied to the parabolic ball $\Omega_r(z_0)$. Since the explicit epression of the fundamental solution $\Gamma$ is not available when the coefficients of $\L$ are variable, the authors of \cite{FabesGarofalo} and \cite{GarofaloLanconelli-1989} rely on the Sard theorem (see \cite{Sard}) which guarantees that $\partialsi_r(z_0)$ is a manifold for \emph{almost every} positive $r$, provided that the fundamental solution $\Gamma$ is $N+1$ times differentiable. The smoothness of the coefficients of the operator $\L$ is used in \cite{FabesGarofalo} and \cite{GarofaloLanconelli-1989} in order to have the needed regularity on $\Gamma$.
The main goal of this note is the restoration of natural regularity hypotheses for the existence of classical solutions to $\L u =f$. These assumptions can be further weakened, since the existence of a fundamental solution has been proved for operators with Dini continuous coefficients. We prefer to keep our treatment in the usual setting of H\"older continuous functions for the sake of simplicity. The unnecessary regularity conditions on the coefficients of $\L$ can be removed in two ways. Following an approach close to the classical one, it is possible to rely on a result due to Dubovicki\u{\i} \cite{Dubovickii} (see also Bojarski, Ha{j\l}asz, and Strzelecki \cite{BojHajStrz}) which allows to reduce the regularity requirement on $\Gamma$ in order to apply a generalized divergence theorem for {\em sets with almost $C^1$ boundary}. This is presented in Section \ref{SectionDivergence} and applied in Section \ref{sectionProof}. The other approach relies on geometric measure theory and is presented in the last section: we show how the proof of Theorem \ref{th-1} can be modified relying on the generalized divergence theorem proved by De Giorgi \cite{DeGiorgi2,DeGiorgi3}
in the framework of finite perimeter sets. As said before, this deep theory is not necessary in the present context, but it is more flexible and its generalization to Carnot groups (where the analogue of Dubovicki\u{\i}'s Theorem is not available) will allow us to extend the results of the present paper to {\em degenerate} parabolic operators. We have presented the application to uniformly parabolic operators to pave the way to this generalization, which will be the subject of a forthcoming paper.
The mean value formulas stated in Theorem \ref{th-1} provide us with a simple proof of the strong maximum (minimum) principle for the operator $\L$ when $c = 0$. Note that, in this case, the constant function $u(x,t) = 1$ is a solution to $\L u = 0$, so that the mean value formula gives $\frac{1}{r^N}\int_{\Omega_r(z_0)}M(z_0;z)dz=1$. In order to state this result we first introduce the notion of \emph{attainable set}. We say that a curve $ \g: [0,T] \rightarrow \mathbb{R}^{N+1}$ is \emph{$\L$-admissible} if it is absolutely continuous and
\begin{equation*}
\dot{\g}(s) = \left( \dot {x}_1(s), \dots, \dot {x}_N(s), -1 \right)
\end{equation*}
for almost every $s \in [0,T]$, with $\dot{x}_1, \dots, \dot{x}_N \in L^{2}([0,T])$.
\begin{definition} \label{def-prop-set}
Let $\O$ be any open subset of $\mathbb{R}^{N+1}$, and let $z_0 \in \O$. The \emph{attainable set} is
\begin{equation*}
\AS ( \O ) =
\begin{Bmatrix}
z \in \O \mid \hspace{1mm} \text{\rm there exists an} \ \L - \text{\rm admissible curve}
\ \g : [0,T] \rightarrow \O \hspace{1mm} \\
\text{\rm such that} \ \g(0) = z_0 \hspace{1mm} {\rm and}
\hspace{1mm} \g(T) = z
\end{Bmatrix}.
\end{equation*}
Whenever there is no ambiguity on the choice of the set $\O$ we denote $\AS = \AS ( \O )$.
\end{definition}
\begin{proposition} \label{prop-smp}
Let $\O$ be any open subset of $\mathbb{R}^{N+1}$, and suppose that $c = 0$. Let $z_0=(x_0,t_0) \in \O$ and let $u$
be a classical solution to
$\L u = f$. If $u (z_0) = \max_\Omega u$ and $f \ge 0$ in $\Omega$, then
\begin{equation*}
u(z) = u(z_0) \quad \text{and} \quad f(z) = 0 \qquad \text{for every} \ z \in \overline {\AS ( \O )}.
\end{equation*}
The analogous result holds true if $u (z_0) = \min_\Omega u$ and $f \le 0$ in $\Omega$.
\end{proposition}
If we remove the assumption $c =0$ we obtain the following weaker result.
\begin{proposition} \label{prop-smp1}
Let $\O$ be any open subset of $\mathbb{R}^{N+1}$. Let $u \le 0$ ($u \ge 0$, respectively) be a classical solution to
$\L u = f$ with $f \ge 0$ ($f \le 0$, respectively) in $\Omega$. If $u (z_0) = 0$ for some $z_0 \in \Omega$, then
\begin{equation*}
u(z) = 0 \quad \text{and} \quad f(z) = 0 \qquad \text{for every} \ z \in \overline {\AS ( \O )}.
\end{equation*}
\end{proposition}
In the remaning part of this intoduction we focus on some modified mean value formulas useful in the proof of parabolic Harnack inequality. As already noticed, the main difficulty one encounters in the proof of the Harnack inequality is due to the unboundedness of the kernels introduced in \eqref{e-kernels}. In order to overcome this issue, we can rely on the idea introduced by Kupcov in \cite{Kupcov4}, and developed by Garofalo and Lanconelli in \cite{GarofaloLanconelli-1989} in the case of parabolic operators with smooth coefficients. This method provides us with some bounded kernels and gives us a useful tool for a direct proof of the Harnack inequality. We outline here the procedure. Let $m$ be a positive integer, and let $u$ be a solution to $\L u = f$ in $\mathbb{R}^{N+1}$. We set
\begin{equation*}
\widetilde u(x,y,t) := u(x,t), \qquad \widetilde f(x,y,t) := f(x,t), \qquad (x,y,t) \in \mathbb{R}^{N}\times \mathbb{R}^{m} \times \mathbb{R},
\end{equation*}
and we note that
\begin{equation*}
\widetilde \L \ \widetilde u(x,y,t) = \widetilde f(x,y,t) \qquad \widetilde \L
= \L + \sum_{j=1}^{m}\tfrac{\partialartial^2}{\partialartial y^2_j} = \L + \Delta_y.
\end{equation*}
Moreover, if $\Gamma$ and $K_m$ denote fundamental solutions of $\L$ and of the heat equation in $\mathbb{R}^m$, respectively, then the function
\begin{equation*}
\widetilde \Gamma(\xi, \eta, \tau ;x,y,t) = \Gamma (\xi, \tau ;x,t) K_m (\eta,\tau ;y,t)
\end{equation*}
is a fundamental solution of $\widetilde \L$. Then, integrating with respect to $y$ in the mean value formulas of Theorem \ref{th-1}, applied to $\widetilde u$ and to the operator $\widetilde \L$, gives new kernels, that are bounded whenever $m>2$. We intoduce further notations.
\begin{equation} \label{e-Omegam}
\begin{split}
\Omega^{(m)}_r(z_0) := & \left\{ z \in \mathbb{R}^{N+1} \mid (4 \partiali (t_0-t))^{-m/2}\Gamma(z_0; z) >
\tfrac{1}{r^{N+m}} \right\}, \\
N_r(z_0;z) := & 2 \sqrt{t_0-t}\sqrt{\log\left(\tfrac{r^{N+m}}{ (4 \partiali (t_0-t))^{m/2}} \Gamma(z_0;z) \right)},
\\
M_r^{(m)} (z_0;z):= & \omega_m N_r^m(z_0;z) \left( M(z_0;z) + \frac{m}{m+2} \cdot \frac {N_r^2(z_0;z)}{4(t_0-t)^2} \right),\\
W_r^{(m)} (z_0;z):= & \frac{\omega_m}{r^{N+m}} N_r^m(z_0;z) - \frac{m}{2} \cdot \frac{\omega_m}{(4 \partiali)^{m/2}} \,
\Gamma(z_0,z) \cdot \widetilde \gamma \left(\frac{m}{2}; \frac {N_r^2(z_0;z)}{4(t_0-t)} \right),
\end{split}
\end{equation}
where $M(z_0;z)$ is the kernel introduced in \eqref{e-kernels}, $\omega_m$ denotes the volume of the $m$-dimensional unit ball
and $\widetilde \gamma$ is the lower incomplete gamma function
\begin{equation*}
\widetilde \gamma(s;w) := \int_0^w \tau^{s-1} e^{- \tau} d \tau.
\end{equation*}
Note that the function $N(z_0,z)$ is well defined for every $z \in \Omega^{(m)}_r(z_0)$, as the argument of
the logarithm is positive, and that we did not point out the dependence of $N_r$ on the space dimension
$m$ to avoid a possible confusion with its powers appearing in the definitions of $M_r^{(m)}$ and
$W_r^{(m)}$.
\begin{proposition} \label{prop-2}
Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Then, for every $z_0 \in \Omega$ and for every $r>0$ such that ${\Omega^{(m)}_r(z_0)} \subset \Omega$ we have
\begin{equation*}
\begin{split}
u(z_0) = & \frac{1}{r^{N+m}} \int_{\Omega^{(m)}_r(z_0)} \! \! \! \! \! \! M_r^{(m)} (z_0; z) u(z) \, dz \,
\\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left( \r^{N+m-1} \int_{\Omega_\r^{(m)} (z_0)}
\! \! \! \! W_\r^{(m)} (z_0;z) f (z) \, dz \right) d \r \\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left( \frac{\omega_m}{\r} \int_{\Omega_\r^{(m)} (z_0)} \! \! \! \! N_\r^m(z_0;z)
\left( \div \, b(z) - c(z) \right) u(z) \, dz \right) d \r.
\end{split}
\end{equation*}
\end{proposition}
We conclude this introduction with two statements of the parabolic Harnack inequality. The first one is given in terms of the parabolic ball $\Omega_{r}^{(m)}(z_0)$, the second one is the usual invariant parabolic Harnack inequality. We emphasize that our proof is elementary, as it is based on the mean value formula, however some accurate estimates of the fundamental solution are needed in order to control the Harnack constant and the size of the cylinders appearing in its statement.
For every $z_0 = (x_0,t_0) \in \mathbb{R}^{N+1}, r >0$, and $m \in \mathbb{N}$ we set
\begin{equation} \label{e-Kr}
K^{(m)}_{r}(z_0) := \overline{\Omega_{r}^{(m)}(z_0)} \cap \Big\{ t \le t_0 - \frac{1}{4 \partiali \lambda^{N/(N+m)}} \,r^2 \Big\}.
\end{equation}
We note that, as a consequence of Lemma \ref{lem-localestimate*} below, for every sufficiently small $r$ the compact set $K^{(m)}_{r}(z_0)$ is non empty.
\begin{proposition} \label{prop-Harnack}
For every $m \in \mathbb{N}$ with $m > 2$, there exist two positive constants $r_0$ and $C_K$, only depending on $\L$ and $m$, such that the following inequality holds. Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le r_0$ and $\Omega_{5r}^{(m)}(z_0) \subset \Omega$ we have that
\begin{equation} \label{e-H0}
\sup_{K^{(m)}_{r}(z_0)} u \le C_K u(z_0)
\end{equation}
for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$.
\end{proposition}
We introduce some further notation in order to state an \emph{invariant} Harnack inequality. For every $z_0 = (x_0,t_0) \in \mathbb{R}^{N+1}$ and for every $r>0$ we set
\begin{equation} \label{e-Q}
\Q_{r}(z_0) := B_r(x_0) \times ]t_0- r^2,t_0[,
\end{equation}
where $B_r(x_0)$ denotes the Euclidean ball with center at $x_0$ and radius $r$. Moreover, for $0 < \iota < \kappa < \mu < 1$ and $0 < \vartheta < 1$ we set
\begin{equation} \label{e-QPM}
\Q^-_{r}(z_0) := B_{\vartheta r}(x_0) \times ]t_0 - \kappa r^2,t_0 - \mu r^2[, \qquad
\Q^+_{r}(z_0) := B_{\vartheta r}(x_0) \times ]t_0 - \iota r^2,t_0[.
\end{equation}
We have
\begin{theorem} \label{th-Harnack-inv}
Choose positive constants $R_0$ and $\iota, \kappa, \mu, \vartheta$ as above and let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$. Then there exists a positive constant $C_H$, only depending on $\L$, on $R_0$ and on the constant that define the cylinders $\Q, \Q^+, \Q^-$, such that the following inequality holds. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le R_0$ and ${\Q_{r}(z_0)} \subset \Omega$ we have that
\begin{equation} \label{e-H1}
\sup_{\Q^-_{r}(z_0)} u \le C_H \inf_{\Q^+_{r}(z_0)} u
\end{equation}
for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$.
\end{theorem}
\begin{center}
\begin{tikzpicture}
\partialath[draw,thick] (-1,.7) rectangle (9,6.8);
\filldraw [fill=blue!20!white, draw=blue, line width=.6pt] (1.5,6) rectangle node {{\color{blue} $Q^+_r(x_0,t_0)$}} node [below=.7cm,right=1.95cm] {{\color{blue} $t_0 - \iota r^2$}} (5.5,4.5);
\filldraw [fill=red!20!white, draw=red, line width=.6pt] (1.5,2) rectangle node {{\color{red} $Q^-_r(x_0,t_0)$}} node [above=.7cm,right=1.95cm] {{\color{red} $t_0 - \kappa r^2$}} node [below=.7cm,right=1.95cm] {{\color{red} $t_0 - \mu r^2$}} (5.5,3.5);
\draw [line width=.6pt] (0,6) rectangle node [above=2cm,right=3.6cm] {$Q_r(x_0,t_0)$} node [below=2.2cm,right=3.6cm] {$t_0-r^2$} (7,1);
\draw [line width=.6pt] (3.5,6) circle (.6pt) node[above] {$(x_0,t_0)$} node[above= 7pt, right=1.2cm] {{\color{blue}$\vartheta r$}} node[above=7pt, right=2.8cm] {$r$};
\end{tikzpicture}
{\sc \qquad Fig.2} - The set $Q_r(x_0,t_0)$.
\end{center}
We conclude this introduction with some comments about our main results. Mean value formulas don't require the uniqueness of the fundamental solution $\Gamma$. In Section \ref{secFundSol} we recall the main results we need on the existence of a fundamental solution together with some known facts about its uniqueness. We also recall in Proposition \ref{prop-localestimate} an asymptotic bound of $\Gamma$ which allows us to use a direct procedure in a part of the proof of the mean value formulas stated in Theorem \ref{th-1}. We point out that recent progresses on mean value formulas and their applications can be found e.g. in \cite{CMPCS}. Moreover, an alternative and more general approach has been introduced by Cupini and Lanconelli in \cite{CupiniLanconelli2021}, where a wide family of differential operators with smooth coefficients is considered. We continue the outline of this article. Section \ref{SectionDivergence} contains the statement of a generalized divergence theorem for {sets with almost $C^1$ boundary} that is used in Section \ref{sectionProof} for the proof of the mean values formulas. Section \ref{sectionHarnack} is devoted to the proof of the Harnack inequality. In the last section we present an alternative approach for the mean value formula that relies on geometric measure theory.
We finally remark that our method also applies to uniformly elliptic equations. Moreover, mean value formulas and Harnack inequality are fundamental tools in the development of the Potential Theory for the operator $\L$.
\setcounter{equation}{0}
\section{Fundamental solution}\label{secFundSol}
In this Section we recall some notations and some known results on the classical theory of uniformly parabolic equations that will be used in the sequel. Points of $\mathbb{R}^{N+1}$ are denoted by $z=(x,t), \z = (\x,\t)$ and $\Omega$ denotes an open subset of $\mathbb{R}^{N+1}$.
Let $u$ be a real valued function defined on $\Omega$. We say that $u$ belongs to $C^{2,1}(\O)$ if
$u, \frac{\partial u}{\partial x_j}, \frac{\partial^2 u}{\partial x_i\partial x_j}$ for $i,j= 1, \dots, N$ and $\frac{\partial u}{\partial t}$
are continuous functions, it belongs to $C^{2+ \alpha,1 + \alpha/2}(\O)$ if $u$ and all the derivatives
of $u$ listed above belong to the space $C^{\alpha}(\O)$ of the H\"older continuous functions defined by
\eqref{e-hc}. A function $u$ belongs to $C^\alpha_\loc(\Omega)$ ($C^{2+ \alpha,1 + \alpha/2}_\loc(\O)$,
respectively) if it belongs to $C^\alpha(K)$ (resp. $C^{2+ \alpha,1 + \alpha/2}(K)$) for every compact set
$K \subset \O$. Let $f$ be a continuous function defined on $\O$. We say that $u \in C^{2,1}(\O)$ is
a classical solution to $\L u = f$ in $\Omega$ if the equation \eqref{e-L} is satisfied at every point
$z \in \O$.
According to Friedman \cite{Friedman}, we say that a fundamental solution $\G$ for the operator $\L$ is a function $\G = \G(z; \z)$ defined for every $(z; \z) \in \mathbb{R}^{N+1} \times \mathbb{R}^{N+1}$ with $t> \tau$, which satisfies the following contitions:
\begin{enumerate}
\item For every $\z = (\xi, \t) \in \mathbb{R}^{N+1}$ the function $\G( \, \cdot \, ; \z)$ belongs to
$C^{2,1}(\mathbb{R}^{N} \times ]\t, + \infty[)$ and is a classical solution to
$\L \, \G(\cdot \,; \z) = 0$ in $\mathbb{R}^{N} \times ]\t, + \infty[$;
\item for every $\partialhi \in C_c(\mathbb{R}^N)$ the function
$$
u(z)=\int_{\mathbb{R}^{N}}\Gamma(z;\xi, \t)\partialhi(\x)d \x,
$$
is a classical solution to the Cauchy problem
\begin{equation*}
\left\{
\begin{array}{ll}
\L u = 0, & \hbox{$z \in \mathbb{R}^{N} \times ]\t, + \infty[$} \\
u( \cdot,\t) = \varphi & \hbox{in} \ \mathbb{R}^N.
\end{array}
\right.
\end{equation*}
\end{enumerate}
Note that $u$ is defined for $t>\t$, then the above identity is understood as follows: for every $\x \in \mathbb{R}^N$ we have $\lim_{(x,t) \to (\x,\t)}u(x,t) = \varphi (\x)$. We also point out that the two above conditions do not guarantee the uniqueness of the fundamental solution. However, as we shall see in the following, estimates \eqref{upper-bound} and \eqref{e-deriv-bds} hold for the fundamental solution $\Gamma$ built by the parametrix method and the fundamental solution verifying such estimates is unique. Indeed, it follows from the proof of Theorem 15 in Ch.1 of \cite{Friedman} that there is only one fundamental solution under the further assumptions that $\Gamma(x,t;\x,\t) \to 0$ as $|x| \to + \infty$ and $|\partialartial_{x_j}\Gamma(x,t;\x,\t)| \to 0$ as $|x| \to + \infty$, for $j=1, \dots,N$, uniformly with respect to $t$ varying in bounded intervals of the form $]\tau, \tau + T]$.
We outline here the parametrix method for the construction of a fundamental solution $\G$ of $\L$.
We first note that, if the matrix $A$ in the operator $\L$ is constant, then the fundamental solution of
$\L$ is explicitly known
\begin{equation} \label{eq-fundsol-A}
\G_A(z;\z) = \frac{1}{\sqrt{(4 \partiali (t-\t))^{N}\det A}}
\exp \left( - \frac{\langle A^{-1}(x-\x),x-\x \rangle}{4(t-\t)} \right),
\end{equation}
and moreover the \emph{reproduction property} holds:
\begin{equation} \label{eq-rep-prop-A}
\G_A(z;\z) = \int_{\mathbb{R}^N} \G_A(x,t;y,s) \G_A(y,s;\x,\t) d y,
\end{equation}
for every $z=(x,t), \z=(\x,\t) \in \mathbb{R}^{N+1}$ and $s \in \mathbb{R}$ with $\t < s < t$.
A direct computation shows that, for every $T>0$, and $\Lambda^+ > \Lambda$ as in \eqref{e-up},
there exists a positive constant $C^+= C^+(\lambda, \Lambda, \Lambda^+,T)$ such that
\begin{equation} \label{eq-fundsol-bd-2}
\begin{split}
\left| \frac{\partialartial \G_A}{\partialartial {x_j}}(z;\z) \right| \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z),
\qquad
\left| \frac{\partialartial^2 \G_A}{\partialartial{x_i x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z)
\end{split}
\end{equation}
for any $i,j= 1, \dots, N$ and for every $z, \z \in \mathbb{R}^{N+1}$ such that $ 0 < t - \t \le T$.
Here the function
\begin{equation} \label{eq-fundsol+}
\G^+(z;\z) = \frac{1}{(\Lambda^+4\partiali (t-\t))^{N/2}} \exp \left( - \frac{|x- \x|^2}{4 \Lambda^+(t-\t)} \right),
\end{equation}
is the fundamental solution of $\Lambda^+ \Delta - \frac{\partialartial}{\partialartial t}$.
The \emph{parametrix} $Z$ for $\L$ is defined as
\begin{equation} \label{eq-fundsol-Z}
Z(z;\z) := \G_{A(\z)}(z;\z) =
\frac{1}{\sqrt{(4 \partiali (t-\t))^{N}\det A(\z)}} \exp \left( -
\frac{\langle A(\z)^{-1}(x-\x),x-\x \rangle}{4(t-\t)} \right).
\end{equation}
More specifically, for every fixed $(\x,\tau) \in \mathbb{R}^{N+1}, Z( \, \cdot \, ; \x,\t)$ is the fundamental solution of the operator $\L_{\z}$ obtained by \emph{freezing} the coefficients $a_{ij}$'s of the operator $\L$ at the point $\z$:
\begin{equation} \label{e-L0}
\L_{\z} := \div \left( A (\z) \nabla_x \right) - \, \tfrac{\partial }{\partial t}.
\end{equation}
Note that
\begin{equation} \label{eq-LZ}
\L Z(z;\z) := \div \left[\left( A(z) - A(\z) \right) \nabla_x Z(z;\z) \right],
\end{equation}
which vanishes as $z \to \z$, by the continuity of the matrix $A$. The fundamental solution $\G$ for $\L$ is obtained from $Z$ by an iterative procedure. We define the sequence of functions
$(\L Z)_1 (z;\z) := \L Z(z;\z)$,
\begin{equation} \label{eq-LkZ}
(\L Z)_{k+1}(z;\z) := \int_\t^t \bigg( \int_{\mathbb{R}^N}(\L Z)_{k} (x,t;y,s) \L Z(y,s;\x,\t) d y \bigg) d s,
\qquad k \in \mathbb{N}.
\end{equation}
Note that estimates \eqref{eq-fundsol-bd-2} also apply to $Z$ then, by using the H\"older continuity of the coefficients of $\L$, we obtain
\begin{equation*}
\left|\L Z(z;\z) \right| \le \frac{\widetilde C}{(t-\t)^{1 - \alpha/2}} \G^+(z;\z),
\end{equation*}
for a positive constant $\widetilde C$ depending on $\lambda, \Lambda, \Lambda^+, T$ and on the constant $M$ in \eqref{e-hc}.
This inequality and the reproduction property \eqref{eq-rep-prop-A} applied to $\Gamma^+$ imply that, for every $k \ge 2$, the integral that defines $(\L Z)_{k}$ converges and
\begin{equation*}
\left| (\L Z)_{k}(z;\z) \right| \le \frac{ (\Gamma_E(\alpha/2) \widetilde C)^k }{\Gamma_E(\alpha k /2)(t-\t)^{1 - k\alpha/2}}
\G^+(z;\z), \qquad k \in \mathbb{N},
\end{equation*}
were $\Gamma_E$ denotes the Euler's Gamma function. Theorem 8 in \cite[Chapter 1]{Friedman} states that, under the assumption that the coefficients $a_{ij}, \frac{\partial a_{ij}}{\partial x_i}, b_{i}, \frac{\partial b_{i}}{\partial x_i}$, for $i, j = 1, \dots, N$ and $c$ belong to the space $C^\alpha(\mathbb{R}^N \times ]T_0, T_1[)$ with $T_0 < T_1$ and satisfy \eqref{e-up}, the series
\begin{equation} \label{eq-Gamma}
\G (z;\z) := Z(z;\z) + \sum_{k=1}^{\infty} \int_\t^t \bigg( \int_{\mathbb{R}^N}Z(x,t;y,s) (\L Z)_k (y,s;\x,\t) d y \bigg) d s
\end{equation}
converges in $\mathbb{R}^N \times ]T_0, T_1[$ and it turns out that its sum $\G$ is a fundamental solution for $\L$. We next list some properties of the function $\G$ defined in \eqref{eq-Gamma}. We mainly refer to Chapter I in the monograph \cite{Friedman} by Friedman.
\begin{enumerate}
\item Theorem 8 in \cite{Friedman}: for every $\z \in \mathbb{R}^{N+1}$ the function $\G(\cdot \, ; \z)$ belongs to $C^{2,1}(\mathbb{R}^N \times ]\t, + \infty[)$ and it is a classical solution to
$\L \, \G = 0$ in $\mathbb{R}^N \times ]\t, + \infty[$.
\item Theorem 9 in \cite{Friedman}: for every bounded functions $\partialhi \in C(\mathbb{R}^N)$ and
$f \in C^\alpha(\mathbb{R}^N \times ]\t, T_1[)$, with $T_0 < \t < T_1$, the function
$$
u(z)=\int_{\mathbb{R}^{N}}\Gamma(z;\z)\partialhi(\x)d \x - \int_\t^t \bigg( \int_{\mathbb{R}^{N}}\Gamma(x,t;\x, s) f(\x, s)d \x \bigg) d s
$$
is a classical solution to the Cauchy problem
\begin{equation} \label{cauchyproblem}
\left\{
\begin{array}{ll}
\L u = f, & \hbox{$z \in \mathbb{R}^{N} \times ]\t, + \infty[$} \\
u( \cdot,\t) = \varphi & \hbox{in} \ \mathbb{R}^N.
\end{array}
\right.
\end{equation}
\item Theorem 15 in \cite{Friedman}: The function $\G^*(z;\z) := \G(\z;z)$ is the fundamental solution of the transposed operator $\L^*$ acting on a suitably smooth function $v$ as follows
\begin{equation} \label{e-L*}
\L^* v (z) := \div \left( A (z) \nabla_x v(z) \right) - \langle b(z), \nabla_x v(z)\rangle +
(c(z) - \div \, b(z)) v(z) + \, \tfrac{\partial u}{\partial t}(z).
\end{equation}
\item Inequalities (6.10) and (6.11) in \cite{Friedman}: for every positive $T$ and $\Lambda^+ > \Lambda$ there exists a positive constant $C^{+}$ such that
\begin{equation} \label{upper-bound}
\G(z; \z) \le C^{+} \, \G^{+} (z; \z),
\end{equation}
for every $z = (x,t), \z = (\x, \t) \in \mathbb{R}^{N+1}$ with $0 < t- \t < T$.
Moreover, the following bounds for the derivatives hold
\begin{equation} \label{e-deriv-bds}
\begin{split}
\left| \frac{\partialartial \G}{\partialartial {x_j}}(z;\z) \right| & \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z), \quad
\left| \frac{\partialartial^2 \G}{\partialartial{x_i x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z),
\\
\left| \frac{\partialartial \G}{\partialartial {\x_j}}(z;\z) \right| & \le \frac{C^+}{\sqrt{t-\t}} \G^+(z;\z), \quad
\left| \frac{\partialartial^2 \G}{\partialartial{\x_i \x_j}}(z;\z) \right| \le \frac{C^+}{t-\t} \G^+(z;\z),
\end{split}
\end{equation}
for any $i,j=1, \dots, N$ and for every $z, \z \in \mathbb{R}^{N+1}$ with $0 < t- \t < T$.
\end{enumerate}
We recall that the monograph \cite{Friedman} also contains an existence and uniqueness result for the Cauchy problem under the assumptions that the functions $\varphi$ and $f$ in the Cauchy problem \eqref{cauchyproblem} do satisfy the following growth condition:
\begin{equation*}
| \varphi(x)| + |f(z)| \le C_0 \exp \left( h |x|^2 \right) \qquad \text{for every} \ x\in \mathbb{R}^N \ \text{and} \ t \in ]\tau, T_1],
\end{equation*}
for some positive constants $C_0$ and $h$. The reproduction property \eqref{eq-rep-prop-A} for $\G$ holds as a direct consequence of the uniqueness of the solution to the Cauchy problem. We also have
\begin{equation*}
e^{-\Lambda(t-\t)} \le \int_{\mathbb{R}^{N}} \G(x,t;\x, \t) \; d \x \le e^{\Lambda(t-\t)}
\end{equation*}
for every $(x, t), (\x, \t) \in \mathbb{R}^{N+1}$ with $\t < t$, where $\Lambda$ is the constant introduced in \eqref{e-up}.
We conclude this section by quoting a statement on the asymptotic behavior of fundamental solutions, which in the stochastic theory is referred to as \emph{large deviation principle}. In our setting it is useful in the description of the \emph{parabolic ball} $\Omega_r(z_0)$ introduced in \eqref{e-Psi}. The first large deviation theorem is due to Varhadhan \cite{Varadhan1967behavior, Varadhan1967diffusion}, who considers parabolic operators $\L$ whose coefficients only depend on $x$ and are H\"older continuous. It states that
\begin{equation}
4 (t-\tau) \log (\Gamma(x,t;\xi,\tau)) \longrightarrow - d^2(x,\xi) \quad \text{as} \ t \to \tau,
\end{equation}
uniformly with respect to $x,\xi$ varying on compact sets. Here $d(x,\xi)$ denotes the Riemannian distance (induced by the matrix $A$) of $x$ and $\xi$. Several extensions of the large deviation principle are available in literature, under different assumption on the regularity of the coefficients of $\L$. Azencott considers in \cite{Azencott84} operators with smooth coefficients and proves more accurate estimates for the asymptotic behavior of $\log\big(\Gamma(x,t;\x,\t)\big)$. Garofalo and Lanconelli prove an analogous result by using purely PDEs methods in \cite{GarofaloLanconelli-1989}. We recall here a version of this result which is suitable for our purposes.
\begin{proposition} \label{prop-localestimate}
{\sc [Theorem 1.2 in \cite{Polidoro2}]} \ For every $\eta \in ]0,1[$ there exists $C_\eta>0$ such that
\begin{equation} \label{eq:approximation}
(1 - \eta) Z(z;\z) \le \G(z;\z) \le (1 + \eta) Z(z;\z)
\end{equation}
for every $z, \z \in \mathbb{R}^{N+1}$ such that $Z(z;\z) > C_\eta $.
\end{proposition}
We finally prove a simple consequence of Proposition \ref{prop-localestimate} that will be used in the following. We introduce some further notation in order to give its statement. We first note that the function $\Gamma^*$ can be built by using the parametrix method, starting from the expression of the parametrix relevant to $\L^*$, that is
\begin{equation} \label{eq-fundsol-Z*}
Z^*(z;\z) := \G^*_{A(\z)}(z;\z) = \frac{1}{\sqrt{(4 \partiali (\t-t))^{N}\det A(\z)}} \exp \left( -
\frac{\langle A(\z)^{-1}(x-\x),x-\x \rangle}{4(\t-t)} \right).
\end{equation}
We set
\begin{equation} \label{eq-Omega_r*}
\Omega_r^*(z_0) := \bigg\{ z \in \mathbb{R}^{N+1} \mid Z^*(z;z_0) \ge \frac{2}{r^N} \bigg\},
\end{equation}
and we point out that its explicit expression is:
\begin{equation} \label{eq-Omega_r*-exp}
\begin{split}
\Omega_r^*(z_0) = & \Big\{ (x,t) \in \mathbb{R}^{N+1} \mid \langle A^{-1}(z_0)(x-x_0), x-x_0 \rangle \le \\
& \qquad - 4 (t_0 - t) \left( \log \big( \tfrac{2}{r^N} \big) + \tfrac{1}{2} \log (\text{det} A(z_0) )+ \tfrac{N}{2} \log (4 \partiali (t_0 - t)) \right) \Big\}.
\end{split}
\end{equation}
We have
\begin{lemma} \label{lem-localestimate*}
There exists a positive constant $r^*$, only depending on the operator $\L$, such that
\begin{equation*}
\Omega_r^*(z_0) \subset \Omega_r(z_0) \subset \Omega_{3r}^*(z_0)
\end{equation*}
for every $z_0 \in \mathbb{R}^{N+1}$ and $r \in ]0,r^*]$.
\end{lemma}
\begin{proof}
As said before, the function $\Gamma^*$ can be built by using the parametrix $Z^*$ defined in \eqref{eq-fundsol-Z*}. In particular, Proposition \ref{prop-localestimate} applies to $\Gamma^*$. Then, if we apply the estimate \eqref{eq:approximation} with $\eta = \frac12$ and we use \eqref{e-L*}, we find that there exists $C^*>0$ such that
\begin{equation*}
\frac12 Z^*(\z;z_0) \le \G(z_0;\z) \le \frac32 Z^*(\z;z_0)
\end{equation*}
for every $z_0, \z \in \mathbb{R}^{N+1}$ such that $Z^*(\z;z_0) > C^*$. The claim then follows from \eqref{e-Psi} and \eqref{eq-Omega_r*} by choosing $r^*:=\left(\frac{2}{C^*}\right)^{1/N}$.
\end{proof}
We conclude this section with a further result useful in the proof of the Harnack inequality.
\begin{lemma} \label{lem-localestimategradient} {\sc [Proposition 5.3 in \cite{Polidoro2}]} \
Let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*}. There exists a positive constants $C$, only depending on the operator $\L$, such that
\begin{equation*}
\left| \partialartial_{x_j} \Gamma(z_0, z) \right| \le C \left( \frac{|x_0-x|}{t_0-t} + 1 \right) \Gamma(z_0, z),
\qquad j=1, \dots, N,
\end{equation*}
for every $z_0 \in \mathbb{R}^{N+1}$ and $z \in \Omega_r(z_0)$ with $r \in ]0,r^*]$.
\end{lemma}
\setcounter{equation}{0}
\section{A generalized divergence theorem}\label{SectionDivergence}
Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $\Phi \in C^1 \partialr{\Omega;\mathbb{R}^{n}}$. The classical divergence formula reads
\begin{equation} \label{eq-div}
\int_{ E } \mathrm{div}\,\Phi\ dz =-\int_{ \partial E} \scp{\nu,\Phi}\ d \H^{n-1},
\end{equation}
where $E$ is a bounded set such that $\overline E \subset \Omega$ and its boundary is $C^1$.
We are interested in the situation in which $E$ is the \emph{super-level} set of a real valued
function $F \in C^1\partialr{\O}$, that is $E = \left\{F>y\right\}$ for some $y \in \mathbb{R}$. At every point
$z\in \partialartial E$ such that $\nabla F(z)\neq 0$ the \emph{inner} unit normal vector $\nu = \nu(z)$
appearing in \eqref{eq-div} is defined as $\nu(z) = \frac{1}{|\nabla F(z)|}\nabla F(z)$ and $\partialartial E$
is a $C^1$ manifold in a neighborhood of $z$. But, if we denote
$$
\mathrm{Crit} \partialr{F}:=\left\{z \in \mathbb{R}^n : \nabla F =0\right\},
$$
the set of \emph{critical points} and $F \partialr{\mathrm{Crit}\partialr{F}}$ the set of \emph{critical values} of
$F$, under our hypotheses we cannot apply the classical Sard theorem to state that
``for almost every $y \in \mathbb{R}$ the level set $\{F=y\}$ is globally a $C^1$ manifold''. Indeed, Whitney proves in \cite{whitney1935} that there exist functions $F \in C^1\partialr{\O}$ having the property that $\{F=y\} \cap \mathrm{Crit} \partialr{F}$ is not empty for every $y$.
Therefore, the purpose of this section is to discuss a version of \eqref{eq-div}
when the boundary of $E$ is $C^1$ up to a closed set of null Hausdorff measure and to see
how it can be applied in our framework. We first introduce the class of sets with the relevant
regularity and state the corresponding divergence formula.
We draw this definition and the following theorem from \cite[Section 9.3]{Maggi}.
\begin{definition}\label{def-almost-C1}
An open set $E\subset \mathbb{R}^n$ has {\em almost $C^1$-boundary} if there is a closed set $M_0\subset \partialartial E$
with $\H^{n-1}(M_0)=0$ such that, for every $z_0\in M = \partialartial E \setminus M_0$ there exist $s > 0$ and
$F \in C^1 (B(z_0, s))$ with the property that
\begin{align*}
B(z_0,s) \cap E & = \{z\in B(z_0,s):\ F(z) > 0\} ,
\\
B(z_0,s) \cap \partialartial E & = \{z\in B(z_0,s):\ F(z) = 0\}
\end{align*}
and $\nabla F(z)\neq 0$ for every $z\in B(z_0,s)$.
We call $M$ the {\em regular part} of $\partialartial E$ (note that $M$ is a $C^1$-hypersurface).
The inner unit normal to $E$ is the continuous vector field
$\nu\in C^0 (M;{\mathbb S}^{n-1})$ given by
\[
\nu(z) = \frac{\nabla F(z)}{|\nabla F(z)|}, \quad
z \in B(z_0,s) \cap M .
\]
\end{definition}
Let us state the divergence theorem for sets with almost $C^1$-boundary.
\begin{theorem}\label{gen-div-thm}
If $E\subset\mathbb{R}^n$ is an open set with almost $C^1$-boundary and $M$ is the regular part of its boundary,
then for every $\Phi\in C^1_c(\mathbb{R}^n;\mathbb{R}^n)$ the following equality holds
\begin{equation}\label{eq-div-almost}
\int_{ E } \mathrm{div}\,\Phi\ dz =-\int_{M} \scp{\nu,\Phi}\ d \H^{n-1} .
\end{equation}
\end{theorem}
If $F \in C^1\partialr{\O}$ and $E = \left\{F>y\right\}$ for some $y \in \mathbb{R}$, we can apply
Theorem \ref{gen-div-thm} thanks to the following result due to
A.~Ya.~Dubovicki\v{\i} \cite{Dubovickii}, that generalizes Sard's theorem.
\begin{theorem}[\sc Dubovicki\v{\i}] \label{th-Dubo}
Assume that $\mathbb{N}^n$ and $\M^m$ are two smooth Riemannian manifolds of dimension $n$ and $m$, respectively.
Let $F:\mathbb{N}^n \rightarrow \M^m$ be a function of class $C^k$. Set $s=n-m-k+1$, then for
$\H^m-$a.e.~$y \in \M^m$
\begin{equation}\label{e-Du}
\H^s \partialr{\left\{F=y \right\} \cap \mathrm{Crit} \partialr{F}}=0.
\end{equation}
\end{theorem}
Notice that if $m=k=1$ and $\M^m=\mathbb{R}$, then $s=n-1$ and for $\H^1-$a.e.~$y \in \mathbb{R}$ the critical part
of $\left\{F=y \right\}$ is an $\H^{n-1}$ null set, while its regular part is an $\partialr{n-1}-$manifold of
class $C^1$. In other words, $\{F=y\}$ is a set with almost $C^1$-boundary and we cannot apply
the classical divergence theorem \eqref{eq-div}, but rather Theorem \ref{gen-div-thm}.
Summarizing, we have the following result, that immediately follows from the above discussion.
\begin{proposition} \label{prop-1}
Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $F \in C^1 \partialr{{\Omega};\mathbb{R}}$.
Then, for $\H^1$-almost every $y \in \mathbb{R}$, we have:
$$
\int_{\left\{F>y\right\}} \mathrm{div}\,\Phi\ dz =
-\int_{\left\{F=y\right\}\setminus \mathrm{Crit} \partialr{F}} \scp{\nu,\Phi}\ d \H^{n-1},
\quad \forall \: \Phi \in C_c^1 \partialr{\Omega;\mathbb{R}^{n}},
$$
were $\nu=\tfrac{\nabla F}{\abs{\nabla F}}$.
\end{proposition}
\begin{proof} By Dubovicki\v{\i} Theorem \ref{th-Dubo} for $\H^1-$almost every $y \in \mathbb{R}$
the set $\{F>y\}$ has almost $C^1$-boundary, hence Theorem \ref{eq-div-almost} applies.
Moreover, as $F$ is continuous, for any such $y$ we have $\partialartial\{F>y\}\subset\{F=y\}$,
$\H^{n-1}(\{F=y\}\setminus\partialartial\{F>y\})=0$ and the regular part of $\partialartial\{F>y\}$ is
$\{F=y\}\setminus\{\nabla F=0\}$ and has full $\H^{n-1}$ measure.
\end{proof}
In order to prove Theorem \ref{th-1} we apply Proposition \ref{prop-1} to the super-level set
$\Omega_r(z_0)$ of the fundamental solution $\Gamma(z_0,\cdot)$ of $\L$. Then, as explained
in the Introduction, we have to cut at a time less than $t_0$ to avoid the singularity of the kernels
at $z_0$. Therefore, we specialize Proposition \ref{prop-1} as follows.
\begin{proposition}\label{prop-1bis}
Let $G \in C^1 \partialr{\mathbb{R}^{N+1} \setminus \left\{ \partialr{x_0,t_0} \right\};\mathbb{R}}$. Then for $\H^1-$almost
every $w, \varepsilon \in \mathbb{R}$
$$
\int_{\left\{ G > w \right\} \cap \left\{ t<t_0-\varepsilon \right\}}
\!\!\!\!\!\!\mathrm{div}\Phi\, dz
=-\int_{(\{ G = w \} \setminus \mathrm{Crit}\partialr{G}) \cap \{ t<t_0-\varepsilon \}}
\!\!\!\!\!\!\scp{\nu,\Phi}d \H^{N}
+\int_{\left\{ G > w \right\} \cap \left\{ t=t_0-\varepsilon \right\}}
\!\!\!\!\!\!\scp{e,\Phi}d\H^{N},
$$
for every $\Phi \in C^1_c \partialr{\Omega;\mathbb{R}^{N+1}}$, where $\nu=\tfrac{\nabla G }{\abs{\nabla G}}$ and
$e=\partialr{0,\ldots,0,1}$.
\end{proposition}
\begin{proof}
Notice that for $\H^1-$a.e. $w\in\mathbb{R}$ the level set $\{G>w\}$ has almost-$C^1$ boundary and fix
such a value.
Let $S$ be the $\H^N$-negligible singular set of $\partialartial\{G>w\}$: by Fubini theorem, for
$\H^1-$a.e.
$\varepsilon>0$ the set $S\cap\{t=t_0-\varepsilon\}$ is in turn $\H^{N-1}-$negligible,
and out of this set the unit normal is given $\H^N-$a.e. by $\nu$ in
$\{ G = w \} \setminus \mathrm{Crit}\partialr{G} \cap \{ t<t_0-\varepsilon \}$ and by $e$ in
$\{ G > w \} \cap \{ t=t_0-\varepsilon \}$. Therefore, Proposition \ref{prop-1} applies
with $n=N+1$, $\Omega=\mathbb{R}^{N+1}\setminus \left\{ \partialr{x_0,t_0} \right\}$,
\[
F(x,t)= (G(x,t)- w)\wedge (t-t_0+\varepsilon) ,
\]
$y=0$ and the set
\begin{equation*}
\Sigma=(\partialartial\{ G > w \} \cap \{ t<t_0-\varepsilon \} \cap \mathrm{Crit}\partialr{G} \bigr)
\cup \bigl(\{ G = w \} \cap \{ t=t_0-\varepsilon \}\bigr)
\end{equation*}
is $\H^N$-negligible.
\end{proof}
The last result we need to prove Theorem \ref{th-1} is the coarea formula for Lipschitz functions.
We refer to \cite[3.2.12]{federer1969geometric} or \cite{AmbrosioFuscoPallara}, Theorem 2.93 and
formula (2.74) for the proof.
\begin{theorem}[\sc Coarea formula for Lipschitz functions] \label{th-co}
Let $G:\mathbb{R}^n \rightarrow \mathbb{R}$ be a Lipschitz function, and let $g$ be a non-negative measurable function. Then
\begin{equation} \label{e-co}
\int_{\mathbb{R}^n} g\partialr{z}\abs{\nabla G \partialr{z}}dz=
\int_{\mathbb{R}}\partialr{\int_{\left\{ G=y \right\}}g\partialr{z}d\H^{n-1} \partialr{z}}dy.
\end{equation}
\end{theorem}
\setcounter{equation}{0}
\section{Proof of the mean value formulas and maximum principle}\label{sectionProof}
In this Section we give the proof of the mean value formulas and of the strong maximun principle.
\begin{proof} {\sc of Theorem \ref{th-1}.} Let $\Omega$ be an open subset of $\mathbb{R}^{N+1}$, and let $u$ be a classical solution to $\L u = f$ in $\Omega$. Let $z_0=(x_0,t_0) \in \Omega$ and let $r_0>0$ be such that $\overline{\Omega_{r_0}(z_0)} \subset \Omega$. We prove our claim by applying Proposition \ref{prop-1bis} with $G(z) = \Gamma(z_0; z)$ and $w = \frac{1}{r^N}$, where $r \in ]0,r_0]$ is such that the statement of Proposition \ref{prop-1bis} holds true with $w = \frac{1}{r^N}$, and $\varepsilon := \varepsilon_k$ for some monotone sequence $\big(\varepsilon_k\big)_{k \in \mathbb{N}}$ such that $\varepsilon_k \to 0$ as $k \to + \infty$ (see Figure 3).
\begin{center}
\begin{tikzpicture}
\clip (-.5,7.5) rectangle (6.7,2);
\shadedraw [top color=blue!10] (-2,6) rectangle (7,1);
\begin{axis}[axis y line=middle, axis x line=middle,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90]
\addplot [black,line width=.7pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {\hskip12mm $(x_0,t_0)$};
\addplot [black,line width=.7pt, domain=-.01:.01] {-sqrt(.0001 - x * x)};
\addplot [blue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))};
\addplot [blue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))}
node[below] { \hskip20mm $\Omega_r(x_0,t_0)$};
\end{axis}
\draw [<-,line width=.4pt] (2.8475,7) -- (2.8475,2);
\draw [red, line width=.6pt,] (-1,6) -- (6.7,6) node[below] { \hskip-18mm $t = t_0 - \varepsilon_k$};
\partialath[draw,thick] (-.49,7.49) rectangle (6.69,2.01);
\end{tikzpicture}
{\sc Fig.3} - The set $\Omega_r(x_0,t_0) \cap \big\{ t < t_0 - \varepsilon_k \big\}$.
\end{center}
For this choice of $r$, we set $v(z) := \Gamma(z_0;z) - \frac{1}{r^N}$, and we note that
\begin{equation} \label{eq-div-L*}
\begin{split}
u(z) \L^* v(z) - v(z) \L u(z) =
& \div_x \big( u(z) A(z) \nabla_x v(z) - v(z) A(z) \nabla_x u(z) \big) - \\
& \div_x \big( u(z) v(z) b(z) \big) + \partialartial_t (u(z)v(z))
\end{split}
\end{equation}
for every $z \in \Omega \backslash \big\{z_0 \big\}.$ We then recall that $\L^* v = \frac{1}{r^N}
\left( \div \, b - c \right)$ and $\L u = f$ in $\Omega \backslash \big\{z_0 \big\}$.
Then \eqref{eq-div-L*} can be written as follows
\begin{equation*}
\frac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z) =
\div \, \Phi (z), \qquad \Phi(z) := \big( u A \nabla_x v - v A \nabla_x u - uv b, uv \big)(z).
\end{equation*}
We then apply Proposition \ref{prop-1bis} to the set
$\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}$ and we find
\begin{equation} \label{eq-div-k}
\begin{split}
\int_{\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz
= & \\
- \int_{\partialsi_r(z_0)\setminus \mathrm{Crit}\partialr{\Gamma} \cap \left\{ t<t_0-\varepsilon_k \right\}}
\!\!\scp{\nu,\Phi} & d \H^{N} + \int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}}
\!\!\scp{e,\Phi}d\H^{N},
\end{split}
\end{equation}
where $\nu(z)=\tfrac{\nabla_{(x,t)} \Gamma(z_0,z) }{\abs{\nabla_{(x,t)} \Gamma(z_0,z)}}$ and
$e=\partialr{0,\ldots,0,1}$. We next let $k \to + \infty$ in the above identity. As $f$ is continuous on
$\overline{\Omega_r(z_0)}$ and $v \in L^1({\Omega_r(z_0)})$, we find
\begin{equation} \label{eq-div-1}
\begin{split}
\lim_{k \to + \infty} \int_{\Omega_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}}
&\left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz = \\
& \int_{\Omega_r(z_0)} \left( \tfrac{1}{r^N} \left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz.
\end{split}
\end{equation}
We next consider the last integral in the right hand side of \eqref{eq-div-k}. We have
$\scp{e,\Phi} (z) = u(z) v(z)$, then
\begin{equation} \label{eq-psir}
\int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}} \!\!\scp{e,\Phi}d\H^{N} =
\int_{\mathcal{I}_r^k (z_0)} \!\! \!\!
u(x,t_0- \varepsilon_k) \left( \Gamma(x_0, t_0; x,t_0 - \varepsilon_k) - \frac{1}{r^N} \right) d x,
\end{equation}
where we have denoted
\begin{equation*}
\mathcal{I}_r^k (z_0) := \left\{ x \in \mathbb{R}^N \mid (x,t_0-\varepsilon_k) \in \overline{\Omega_r (z_0)} \right\}.
\end{equation*}
We next prove that the right hand side of \eqref{eq-psir} tends to $u(z_0)$ as $k \to + \infty$. Since
$\Gamma$ is the fundamental solution to $\L$ we have
\begin{equation*}
\lim_{k \to + \infty} \int_{\mathbb{R}^N} \!\! \!\! \Gamma(x_0, t_0; x,t_0 - \varepsilon_k)
u(x,t_0- \varepsilon_k) d x = u(x_0,t_0),
\end{equation*}
then, being $u$ continuous on $\overline{\Omega_r (z_0)}$, we only need to show that
\begin{equation} \label{eq-claim-2}
\lim_{k \to + \infty} \H^{N} \left( \mathcal{I}_r^k (z_0) \right) = 0, \qquad
\lim_{k \to + \infty} \int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} \!\! \!\!
\Gamma(x_0, t_0; x,t_0 - \varepsilon_k) d x = 0.
\end{equation}
With this aim, we note that the upper bound \eqref{upper-bound} and \eqref{eq-fundsol+} imply
\begin{equation*}
\mathcal{I}_r^k (z_0) \subset \left\{ x \in \mathbb{R}^N \mid | x - x_0|^2 \le 4 \Lambda^+ \varepsilon_k
\left( \log \left(C^+ r^N \right) - \tfrac{N}{2} \log (4 \partiali \Lambda^+ \varepsilon_k) \right) \right\}.
\end{equation*}
The first assertion of \eqref{eq-claim-2} is then a plain consequence of the above inclusion. In order to prove the second statement in \eqref{eq-claim-2}, we rely on Lemma \ref{lem-localestimate*}. We let $r_0 := \min (r, r^*)$, so that
\begin{equation*}
\Omega_{r_0}^*(z_0) \subset \Omega_{r_0}(z_0) \subset \Omega_r(z_0),
\end{equation*}
thus
\begin{equation*}
\begin{split}
\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0) \subset & \left\{ x \in \mathbb{R}^N \mid Z^*(x,t_0- \varepsilon_k;x_0,t_0) \le \tfrac{2}{r_0^N} \right\} \\
= & \Big\{ x \in \mathbb{R}^N \mid \langle A(z_0) (x - x_0), x-x_0 \rangle \\
& \quad \ge - 4 \varepsilon_k \left( \log \left( \tfrac{2}{r_0^N} \right) + \tfrac{1}{2} \log (\det A(z_0) ) + \tfrac{N}{2} \log (4 \partiali \varepsilon_k) \right) \Big\}.
\end{split}
\end{equation*}
By using again \eqref{upper-bound}, the above inclusion, and the change of variable $x = x_0 + 2 \sqrt{\Lambda^+ \varepsilon_k} \, \xi$, we find
\begin{equation*}
\begin{split}
\int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} & \!\! \!\! \Gamma(x_0, t_0; x,t_0 - \varepsilon_k) d x \le C^+ \int_{\mathbb{R}^N \backslash \mathcal{I}_r^k (z_0)} \!\! \!\! \Gamma^+ (x_0, t_0; x,t_0 - \varepsilon_k) d x \\
& \le \frac{C^+}{\partiali^{N/2}} \int_{ \left\{ \langle A(z_0) \xi, \xi \rangle \ge - \tfrac{1}{\Lambda^+} \left( \log \left( \tfrac{2}{r_0^N} \right) + \tfrac{1}{2} \log (\det A(z_0) ) + \tfrac{N}{2} \log (4 \partiali \varepsilon_k) \right) \right\} }
\!\! \exp\left( - |\xi|^2 \right) d \xi.
\end{split}
\end{equation*}
The second assertion of \eqref{eq-claim-2} then follows. Thus, we have shown that
\begin{equation} \label{eq-div-2}
\lim_{k \to + \infty} \int_{\Omega_r(z_0) \cap \left\{ t=t_0-\varepsilon_k \right\}} \scp{e,\Phi} d\H^{N} = u(z_0).
\end{equation}
We are left with the first integral in the right hand side of \eqref{eq-div-k}. We preliminarily note that its limit, as $k \to + \infty$, does exist. Moreover, for every $z \in \partialsi_r(z_0)$ we have $v(z) = 0$, then $\Phi(z) = \big( u (z) A(z) \nabla_x v(z), 0 \big)$, so that
\begin{equation*}
\int_{\partialsi_r(z_0) \setminus \mathrm{Crit}\partialr{\Gamma} \cap \left\{ t<t_0-\varepsilon_k \right\}}
\!\!\scp{\nu,\Phi}d \H^{N} = \int_{\partialsi_r(z_0) \setminus \mathrm{Crit}\partialr{\Gamma} \cap \left\{ t<t_0-z\varepsilon_k \right\}} \!\! u(x,t) K (z_0;z) d \H^{N},
\end{equation*}
where
\begin{equation*}
K (z_0; z) = \frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0; z)|}
\end{equation*}
is the kernel defined in \eqref{e-kernels}. Note that $K$ is non-negative and, if we consider the function
$u = 1$ and we let $k \to + \infty$, we find
\begin{equation*}
\lim_{k \to + \infty} \int_{\partialsi_r(z_0) \setminus \mathrm{Crit}\partialr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}}
\!\! K (z_0; z) d \H^{N} = \int_{\partialsi_r(z_0)\setminus \mathrm{Crit}\partialr{\Gamma}} \!\!\!
K (z_0;z) d \H^{N} < + \infty.
\end{equation*}
Thus, if $u$ is a classical solution to $\L u = 0$, we obtain
\begin{equation} \label{eq-div-3i}
\lim_{k \to + \infty} \int_{\partialsi_r(z_0) \setminus \mathrm{Crit}\partialr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}}
\!\!\scp{\nu,\Phi}d \H^{N} = \int_{\partialsi_r(z_0)\setminus \mathrm{Crit}\partialr{\Gamma}} \!\!\!
K (z_0;z) u(z) d \H^{N}.
\end{equation}
We recall that Dubovicki\v{\i}'s theorem implies that
$\H^{N} \left(\partialsi_r(z_0) \cap \mathrm{Crit}\partialr{\Gamma}\right) = 0$ for $\H^{1}$ almost every
$r$, so that we can equivalently write
\begin{equation} \label{eq-div-3}
\lim_{k \to + \infty} \int_{\partialsi_r(z_0) \setminus \mathrm{Crit}\partialr{\Gamma} \cap
\left\{ t<t_0-\varepsilon_k \right\}}
\!\!\scp{\nu,\Phi}d \H^{N} = \int_{\partialsi_r(z_0)} \!\!\! K (z_0;z) u(z) d \H^{N}.
\end{equation}
The proof of the first assertion of Theorem \ref{th-1} then follows by using \eqref{eq-div-1}, \eqref{eq-div-2} and \eqref{eq-div-3} in \eqref{eq-div-k}.
The proof of the second assertion of Theorem \ref{th-1} is a direct consequence of the first one and of the
coarea formula stated in Theorem \ref{th-co}. Indeed, fix a positive $r$ as above, multiply by $\frac{N}{r^N}$
and integrate over $]0,r[$. We find
\begin{equation} \label{e-meanvalue-step1}
\begin{split}
\frac{N}{r^N} \int_0^r \varrho^{N-1} u(z_0) d \varrho = &
\frac{N}{r^N} \int_0^r \varrho^{N-1} \bigg(\int_{\partialsi_\varrho(z_0)}
K (z_0;z) u(z) \, d \H^{N} (x,t) \bigg) d \varrho\,
\\
& + \frac{N}{r^N} \int_0^r \varrho^{N-1} \bigg( \int_{\Omega_\r(z_0)} f (z)
\left( \tfrac{1}{r^N} - \Gamma(z_0;z) \right) dz \bigg) d \varrho
\\
& + \frac{N}{r^N} \int_0^r \frac{1}{\varrho} \bigg(
\int_{\Omega_\r (z_0)} \left( \div \, b(z) - c(z) \right) u(z) \, dz \bigg) d \varrho.
\end{split}
\end{equation}
The left hand side of the above equality equals $u(z_0)$, while the last two terms agree with the last two terms appearing in the statement of Theorem \ref{th-1}. In order to conclude the proof we only need to show that
\begin{equation} \label{e-meanvalue-step2}
\begin{split}
\int_0^r \varrho^{N-1} \bigg(\int_{\left\{\Gamma(z_0;z) =
\tfrac{1}{\varrho^N}\right\} } & K (z_0;z) u(z) \, d \H^{N} (z) \bigg) d \varrho
\\
& = \frac{1}{N} \int_0^{r} \r^{N-1}\int_{\Omega_\r (z_0)} M (z_0;z) u(z) dz.
\end{split}
\end{equation}
With this aim, we substitute $y = \frac{1}{\varrho^N}$ in the left hand side of \eqref{e-meanvalue-step2} and we recall the definition of the kernel $K$. We find
\begin{equation} \label{e-meanvalue-step3}
\begin{split}
\int_0^r \varrho^{N-1} & \bigg( \int_{\left\{\Gamma(z_0;z) = \tfrac{1}{\varrho^N}\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|} u(z) \, d \H^{N} (z) \bigg) d \varrho \\
& = \frac{1}{N}
\int_{\frac{1}{r^N}}^{+ \infty} \frac{1}{y^{2}} \bigg(\int_{\left\{\Gamma(z_0;z) = y\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0; z), \nabla_x \Gamma(z_0;z) \rangle }
{|\nabla_{(x,t)}\Gamma(z_0;z)|}u(z) \, d \H^{N} (z) \bigg) d y \\
& =\frac{1}{N} \int_{\frac{1}{r^N}}^{+ \infty} \bigg(\int_{\left\{\Gamma(z_0;z) = y\right\} }
\frac{\langle A(z) \nabla_x \Gamma(z_0;z), \nabla_x \Gamma(z_0;z) \rangle }
{\Gamma^2(z_0;z) {|\nabla_{(x,t)}\Gamma(z_0;z)|}} u(z) \, d \H^{N} (z) \bigg) d y.
\end{split}
\end{equation}
We conclude the proof of \eqref{e-meanvalue-step2} by applying the coarea formula stated in Theorem
\ref{th-co}.
\end{proof}
\begin{proof} {\sc of Proposition \ref{prop-smp}.} We prove our claim under the additional assumption $\div \, b \ge 0$. At the end of the proof we show that this assumption is not restrictive.
We first note that, as a direct consequence of our assumption $c=0$, we have that $\L \, 1 = 0$, then Theorem \ref{th-1} yields
\begin{equation*}
\frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) \, dz +
\frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \div \, b(z) \, dz \Big) d s = 1
\end{equation*}
for every $z_1 \in \Omega$ and $\r >0$ such that $\overline{\Omega_\varrho(z_1)} \subset \Omega$.
We claim that, if $u(z_1) = \max_\Omega u$, then
\begin{equation} \label{eq-claim-smp}
u(z) = u(z_1) \qquad \text{for every} \quad z \in \overline{\Omega_\varrho(z_1)}.
\end{equation}
By using again Theorem \ref{th-1} and the above identity we obtain
\begin{align*}
0 = & \frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) \big((u(z)- u(z_1)\big) \, dz
\\
& + \frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \div \, b(z)
\big((u(z)- u(z_1)\big) \, dz \Big) d s
\\
& + \frac{N}{\r^N} \int_0^{\r} \left(s^{N-1}
\int_{\Omega_s (z_1)} f (z) \left( \tfrac{1}{s^N} - \Gamma(z_1;z) \right) dz \right) d s \le 0,
\end{align*}
since $f \ge 0$, $\div \, b \ge 0$ and $u(z) \le u(z_1)$, being $u(z_1) = \max_{\Omega} u$. We have also used the fact that $M(z_1;z) \ge 0$ and $\Gamma(z_1;z) \ge \tfrac{1}{s^N}$ for every $z \in \Omega_s(z_1)$. Hence, $M (z_1; z) \big((u(z)- u(z_1)\big)=0$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$. As already noticed, Dubovicki\v{\i}'s theorem implies that $\H^{N} \left( \partialsi_s (z_1) \cap \mathrm{Crit}\partialr{\Gamma} \right) = 0$, for almost every $s \in ]0,\r]$, then $M (z_1; z) \ne 0$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$. As a consequence $u(z)= u(z_1)$ for $\H^{N+1}$ almost every $z \in \Omega_\r(z_1)$, and the claim \eqref{eq-claim-smp} follows from the continuity of $u$.
We are in position to conclude the proof of Proposition \ref{prop-smp}. Let $z$ be a point of $\AS ( \O )$, and let $\gamma: [0,T] \to \O$ be an $\L$--admissible path such that $\g(0)= z_0$ and $\g(T) = z$. We will prove that $u(\gamma(t)) = u(z_0)$ for every $t \in [0,T]$. Let
\begin{equation*}
I := \big\{ t \in [0,T] \mid u(\gamma(s)) = u(z_0) \ \text{for every} \ s \in[0,t] \big\}, \qquad \overline t := \sup I.
\end{equation*}
Clearly, $I \ne \emptyset$ as $0 \in I$. Moreover $I$ is closed, because of the continuity of $u$ and $\gamma$, then $\overline t \in I$. We now prove by contradiction that $\overline t = T$.
Suppose that $\overline t < T$. Let $z_1 := \gamma(\overline t)$ and note that $z_1 \in \Omega$, $u(z_1) = \max_\Omega u$. We aim to show that there exist positive constants $r_1$ and $s_1$ such that $\overline{\Omega_{r_1}(z_1)} \subset \O$ and
\begin{equation} \label{eq-claim2-smp}
\gamma(\overline t + s) \in \Omega_{r_1}(z_1)\quad \text{for every} \quad s \in [0, s_1[.
\end{equation}
As a consequence of \eqref{eq-claim-smp} we obtain $u(\gamma(\overline t + s)) = u(z_1) = u(z_0)$ for every $s \in [0, s_1[$, and this contradicts the assumption $\overline t < T$.
The proof of \eqref{eq-claim2-smp} is a consequence of Lemma \ref{lem-localestimate*}. It is not restrictive to assume that $r_1 \le r^*$, then it is sufficient to show that there exists a positive $s_1$ such that
\begin{equation} \label{eq-claim4-smp}
\gamma(\overline t + s) \in \Omega^*_{r_1}(z_1) \quad \text{for every} \quad s \in [0, s_1[.
\end{equation}
Recall the definition of $\gamma(\overline t + s) = (x(\overline t + s), t(\overline t + s))$. We have $\gamma(\overline t) = z_1 = (x_1, t_1), t(\overline t + s) = t_1-s$ and, for every positive $s$
\begin{equation*}
\begin{split}
|x(s + \overline t) - x_1 | & = \left| \int_0^s \dot x(\overline t + \sigma) d \sigma \right| \le \int_0^s \left| \dot x(\overline t + \sigma) \right| d \sigma \\
& \le \left( \int_0^s \left| \dot x(\overline t + \sigma) \right|^2 d\sigma \right)^{1/2} s^{1/2} \le \|\dot x \|_{L^2([0,T])} \sqrt{s},
\end{split}
\end{equation*}
then
\begin{equation*}
\langle A^{-1}(z_1 )(x(\overline t + s)-x_1), x(\overline t + s)-x_1 \rangle \le
s \cdot \| A^{-1}(z_1)\| \cdot \|\dot x \|_{L^2([0,T])}^2.
\end{equation*}
By using the above inequality in \eqref{eq-Omega_r*-exp} we see that there exists a positive constant $s_1$ such that \eqref{eq-claim4-smp} holds. This proves \eqref{eq-claim2-smp}, and then $u(z) = u(z_0)$ for every $z \in \AS ( \O )$. By the continuity of $u$ we conclude that $u(z) = u(z_0)$ for every $z \in \overline{\AS ( \O )}$. Eventually, since $u$ is constant in $\overline{\AS ( \O )}$ and $c= 0$, we conclude that $\L u = 0$.
We finally prove that the additional assumption $\div \, b \ge 0$ is not restrictive. Let $k$ be any given constant such that $k>\Lambda$, where $\Lambda$ is the quantity appearing in \eqref{e-up}, recall that $z_0 = (x_0,t_0)$ and define the function
\begin{equation}
v(y,t):= u\left( e^{-k(t-t_0)}y, t\right), \qquad (y,t) \in \widehat \Omega,
\end{equation}
where $(y,t) \in \widehat \Omega$ if, and only if $\left( e^{-k(t-t_0)}y, t\right) \in \Omega$. Then $v$ is a solution to
\begin{equation*}
\widehat \L v (y,t) :=
\div \left( \widehat A (y,t) \nabla_y v(y,t) \right) + \langle \widehat b(y,t) + k y, \nabla_y v(y,t)\rangle -
\, \tfrac{\partial v}{\partial t}(y,t) = f\left( e^{-k(t-t_0)}y, t\right),
\end{equation*}
where $\widehat A (y,t) = \left( \widehat a_{ij} (y,t) \right)_{i,j=1, \dots, N}$, $\widehat b(y,t) = \left( \widehat b_1(y,t), \dots, \widehat b_N(y,t) \right)$, are defined as $\widehat a_{ij} (y,t) = e^{-2k(t-t_0)} a_{ij} \left( e^{-k(t-t_0)}y, t\right), \widehat b_{j} (y,t) = e^{-k(t-t_0)} b_{j} \left( e^{-k(t-t_0)}y, t\right)$, for $i,j=1, \dots, N$. Note that from the assumption \eqref{e-up} it follows that $|\div \, b| \le N \Lambda$, then
\begin{equation*}
\div \left( \widehat b(y,t) + k y \right) \ge N \left(k - \Lambda e^{-2k(t-t_0)} \right).
\end{equation*}
In particular, there exists a positive $\delta$, depending on $k$ and $\Lambda$, such that the right hand side of above expression is non-negative as $t \ge t_0 - \delta$. Then, if we set $\widehat t_0 := t_0- \delta$, we have
\begin{equation*}
\div \left( \widehat b(y,t) + k y \right) \ge 0 \qquad \text{for every} \quad (y,t) \in \widehat \Omega
\cap \big\{t \ge \widehat t_0 \big\}.
\end{equation*}
We also note that $\widehat \L$ satisfies the same assumptions as $\L$, with a possibily different constant $\widehat \Lambda$, in every set of the form $\widehat \Omega \cap \big( \mathbb{R}^N \times I \big)$, where $I$ is any bounded open interval of $\mathbb{R}$. We then apply the above argument to prove that, if $v$ reaches its maximum at some point $(y_0,t_0) \in \widehat \Omega$, then it is constant in its propagation set in $\widehat \Omega \cap \big\{t \ge \widehat t_0 \big\}$. Note that $u$ reaches its maximum at some point $(x,t)$ if, and only if, $v$ reaches its maximum at $\left( e^{k(t-t_0)}x, t\right)$. Moreover, $(x(s), t-s)$ is an admissible curve for $\L$ if, and only if, $(e^{-k(t-s-t_0)}x(s), t-s)$ is an admissible curve for $\widehat \L$. We conclude that
\begin{equation*}
u(z) = u(z_0) \qquad \text{for every} \ z \in \overline {\AS ( \O )} \cap \big\{t \ge \widehat t_0 \big\}.
\end{equation*}
We then repeat the above argument. Assume that $u$ reaches its maximum at some point $\left( \widehat x_0, \widehat t_0\right)$, we define a new function $\widehat v(y,t):= u\left( e^{-k(t-\hat t_0)}y, t\right)$ and we find a new constant $\widehat t_1 := t_0 - 2 \delta$ such that
\begin{equation*}
u(z) = u(z_0) \qquad \text{for every} \ z \in \overline {\AS ( \O )} \cap \big\{t \ge \widehat t_1 \big\}.
\end{equation*}
As we can use the same constant $\delta$ at every iteration, we conclude that the above identity holds for every $z \in \overline {\AS ( \O )}$.
\end{proof}
\begin{proof} {\sc of Proposition \ref{prop-smp1}.} Let $k$ be a constant such that $\div \, b - c -k \ge 0$ and note that the function $v(x,t) := e^{k t}u(x,t)$ is a non negative solution to the equation
\begin{equation*}
\L v(x,t) + k v(x,t) = e^{k t}f(x,t), \qquad (x,t) \in \Omega.
\end{equation*}
Then Theorem \ref{th-1} yields
\begin{align*}
0 = & \frac{1}{\varrho^N} \int_{\Omega_\varrho(z_1)} M (z_1; z) v(z) \, dz
\\
& + \frac{N}{\r^N} \int_0^{\r} \Big(\frac{1}{s} \int_{\Omega_s (z_1)} \big(\div \, b(z) - c(z) - k \big)
v(z) \, dz \Big) d s
\\
& + \frac{N}{\r^N} \int_0^{\r} \left(s^{N-1}
\int_{\Omega_s (z_1)} e^{k t} f (z) \left( \tfrac{1}{s^N} - \Gamma(z_1;z) \right) dz \right) d s \le 0,
\end{align*}
for every $z_1 \in \Omega$ and $\r >0$ such that $\overline{\Omega_\varrho(z_1)} \subset \Omega$. Here we have sued the facts that
$f \ge 0$, and $\div \, b - c -k \ge 0$. By following the same argument used in the proof of Proposition \ref{prop-smp} we find that $v \ge 0$ in $\overline {\AS ( \O )}$. This concludes the proof of Proposition \ref{prop-smp1}.
\end{proof}
\begin{proof} {\sc of Proposition \ref{prop-2}.}
Let $m$ be a positive integer, and let $u$ be a solution to $\L u = f$ in $\Omega \subset \mathbb{R}^{N+1}$. As said in the Introduction, we set
\begin{equation*}
\widetilde u(x,y,t) := u(x,t), \qquad \widetilde f(x,y,t) := f(x,t),
\end{equation*}
for every $(x,y,t) \in \mathbb{R}^{N}\times \mathbb{R}^{m} \times \mathbb{R}$ such that $(x,t) \in \Omega$, and we note that
\begin{equation*}
\widetilde \L \ \widetilde u(x,y,t) = \widetilde f(x,y,t)
\qquad \widetilde \L := \L + \sum_{j=1}^{m}\tfrac{\partialartial^2}{\partialartial y_j^2}.
\end{equation*}
Moreover, the function
\begin{equation} \label{eq-tildegamma}
\widetilde \Gamma(x_0, y_0, t_0 ;x,y,t) := \Gamma (x_0, t_0 ;x,t) \cdot
\frac{1}{(4 \partiali (t_0-t))^{m/2}}\exp \left( \frac{-|y_0-y|^2}{4(t_0-t)} \right)
\end{equation}
is a fundamental solution of $\widetilde \L$. We then use $\widetilde \Gamma$ to represent the solution $u$ in accordance with Theorem \ref{th-1} as follows
\begin{equation*}
\begin{split}
u(z_0) = & \, \widetilde u(x_0,y_0,t_0) =
\frac{1}{r^{N+m}} \int_{\widetilde\Omega_r(x_0,y_0,t_0)} \! \! \! \!
\widetilde M(x_0,y_0,t_0; x,y,t) u(x,t) \, dx \, dy\, dt \\
& +\frac{N+m}{r^{N+m}} \int_0^{r} \left(\r^{N+m-1}
\int_{\widetilde\Omega_\r(x_0,y_0,t_0)} \! \! \! \!
f (x,t) \left( \tfrac{1}{\r^{N+m}} - \widetilde \Gamma(x_0,y_0,t_0;x,y,t) \right) \, dx \, dy\, dt \right) d \r \\
& + \frac{N+m}{r^{N+m}} \int_0^{r} \left(\frac{1}{\r}
\int_{\widetilde\Omega_\r(x_0,y_0,t_0)} \! \! \! \!
\left( \div \, b(x,t) - c(x,t) \right) u(x,t) \, dx \, dy\, dt \right) d \r.
\end{split}
\end{equation*}
where $\widetilde\Omega_r(x_0,y_0,t_0)$ is the parabolic ball relevant to $\widetilde \Gamma$ and
\begin{equation*}
\widetilde M(x_0,y_0,t_0; x,y,t) = M(x_0,t_0;x,t) + \frac{|y_0-y|^2}{4(t_0-t)^2}.
\end{equation*}
The proof is accomplished by integrating the above identity with respect to the variable $y$.
\end{proof}
\setcounter{equation}{0}
\section{Proof of the Harnack inequalities}\label{sectionHarnack}
In this Section we use the mean value formula stated in Proposition \ref{prop-2} to give a simple proof of the parabolic Harnack inequality.
\begin{proof} {\sc of Proposition \ref{prop-Harnack}.} We first prove our claim under the additional assumption that $\div \, b - c = 0$. This assumption simplifies the proof as in this case we only need to use the first integral in the representation formula given in Proposition \ref{prop-2}. It will be removed at the end of the proof.
Let $m \in \mathbb{N}$ with $m > 2$, let $\Omega$ be an open subset of $\mathbb{R}^{N+1}, z_0 \in \Omega$ and $r>0$ such that $\Omega_{4r}^{(m)}(z_0) \subset \Omega$. We claim that there exist four positive constants $r_0, \vartheta, M^+, m^-$ such that the following assertions hold for every $r \in ]0,r_0]$.
\begin{description}
\item[{\it i)}] $K^{(m)}_{r}(z_0) \ne \emptyset$;
\item[{\it ii)}] $M_{\vartheta r}^{(m)} (z; \zeta) \le M^+$ for every $\zeta \in \Omega_{\vartheta r}^{(m)}(z)$;
\item[{\it iii)}] $\Omega_{\vartheta r}^{(m)}(z) \subset \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}\big\}$ for every $z \in K^{(m)}_{r}(z_0)$;
\item[{\it iv)}] $M_{5 r}^{(m)} (z_0; \zeta) \ge m^-$ for every $\zeta = (\xi, \tau) \in \Omega_{4 r}^{(m)}(z_0)$ such that $\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}$. \end{description}
By using Proposition \ref{prop-2} and the above claim it follows that, for every $z \in K^{(m)}_{r}(z_0)$, it holds
\begin{equation} \label{e-core-h}
\begin{split}
u(z) = & \frac{1}{(\vartheta r)^{N+m}} \int_{\Omega^{(m)}_{\vartheta r}(z)}
\! \! \! \! \! \! M_{\vartheta r}^{(m)} (z; \zeta) u(\zeta) \, d\zeta \\
& (\text{by {\it ii}}) \le \frac{M^+}{(\vartheta r)^{N+m}}
\int_{\Omega^{(m)}_{\vartheta r}(z)} \!\!\! u(\zeta) \, d\zeta \\
& (\text{by {\it iii}}) \le \frac{M^+}{(\vartheta r)^{N+m}}
\int_{\Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}\big\}}
\!\!\! u(\zeta) \, d\zeta \\
& (\text{by {\it iv}}) \le \frac{M^+}{m^- (\vartheta r)^{N+m}}
\int_{\Omega_{5r}^{(m)}(z_0)} \! \! \! \! \! \! M_{5 r}^{(m)} (z_0; \zeta)u(\zeta) \, d\zeta =
\frac{5^{N+m}M^+}{\vartheta^{N+m} m^-} u(z_0).
\end{split}
\end{equation}
This proves Proposition \ref{prop-Harnack} with $C_K := \frac{5^{N+m}M^+}{ \vartheta^{N+m}m^-}$.
We are left with the proof of our claims. We mainly rely on Lemma \ref{lem-localestimate*}, applied to the function $\widetilde \Gamma$ introduced in \eqref{eq-tildegamma}. In the sequel we let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*} and relative to $\widetilde \Gamma$, and in accordance with \eqref{eq-Omega_r*},
\begin{equation} \label{eq-Omega_rm*}
\Omega_r^{(m)*}(z_0) := \bigg\{ z \in \mathbb{R}^{N+1} \mid (4 \partiali (t_0-t))^{-m/2}Z^*(z;z_0) \ge \frac{2}{r^{N+m}} \bigg\}.
\end{equation}
Moreover, we choose $r_0 := r^*/2$.
\begin{center}
\begin{tikzpicture}
\clip (-.52,7.02) rectangle (6.52,1.78);
\partialath[draw,thick] (-.5,7) rectangle (6.5,1.8);
\begin{axis}[axis y line=none, axis x line=none,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90]
\addplot [black,line width=1pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {$z_0$};
\addplot [black,line width=1pt, domain=-.01:.01] {-sqrt(.0001 - x * x)};
\addplot [lblue,line width=.7pt,domain=.001:.051] {sqrt(- 3 * x * ln(9*x)};
\addplot [lblue,line width=.7pt,domain=.001:.051] {- sqrt(- 3 * x * ln(9*x))};
\addplot [blue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))};
\addplot [blue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))} node[below=-10pt] { \hskip35mm $\Omega^{(m)}_{4r}(z_0)$};
\addplot [ddddblue,line width=.7pt,domain=.051:.1111] {sqrt(- 3 * x * ln(9*x)};
\addplot [ddddblue,line width=.7pt,domain=.051:.1111] {- sqrt(- 3 * x * ln(9*x))};
\addplot [red,line width=.7pt,domain=.0001:.0625,below=15pt,right=9pt] {sqrt(- 4 * x * ln(16*x)};
\addplot [red,line width=.7pt,domain=.0001:.0625,below=15pt,right=9pt] {- sqrt(- 4 * x * ln(16*x))};
\addplot [black,line width=1pt, domain=-.01:.01,below=15pt,right=9pt] {sqrt(.0001 - x * x)} node[below=-3pt] {$z$};
\addplot [black,line width=1pt, domain=-.01:.01,below=15pt,right=9pt] {-sqrt(.0001 - x * x)} node[below=15pt,right=10pt] {\hskip-20mm \color{red} $\Omega^{(m)}_{\vartheta r}(z)$};
\end{axis}
\draw [ddddblue,line width=.7pt] (1.92,6) -- node [below=5pt] { \hskip25mm $K^{(m)}_r(z_0)$} (3.75,6);
\end{tikzpicture}
{\sc \qquad Fig.4} - The inclusion \emph{(iii)}.
\end{center}
\noindent {\it Proof of i)} \ By the definition \eqref{e-Kr} of $K^{(m)}_{r}(z_0)$ we only need to show that there exists at least a point $(x,t) \in \overline{\Omega^{(m)}_{r}(z_0)}$ with $t \le t_0 - \frac{1}{4 \partiali \lambda^{N/(N+m)}} \,r^2$. From Lemma \ref{lem-localestimate*} it follows that $\Omega_r^{(m)*}(z_0) \subset \Omega^{(m)}_r(z_0)$, then we only need to show that that the point $\big(x_0, t_0 - \frac{1}{4 \partiali \lambda^{N/(N+m)}} \,r^2\big)$ belongs to $\overline{\Omega^{(m)}_{r}(z_0)}$. In view of \eqref{eq-fundsol-Z*}, this is equivalent to $\det A(z_0)\ge \lambda^N/4$, which directly follows from the parabolicity assumption \eqref{e-up}.
\noindent {\it Proof of ii)} \ We first note that \eqref{eq-fundsol+} and the defintion of $N_{\vartheta r}$ directly give
\begin{equation} \label{eq-boundN}
N_{\vartheta r} (z;\z) \le 2 \sqrt{t-\tau}\sqrt{\log\left(
\tfrac{C^+ (\vartheta r)^{N+m}}{ (\Lambda^+)^{N/2}(4 \partiali (t-\tau))^{(N+m)/2}} \right)} = 2 \sqrt{t-\tau}
\sqrt{C_1 + \tfrac{N+m}{2}\log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right)},
\end{equation}
where $C_1$ is a positive constant that only depends on $\L$. Moreover, Lemma \ref{lem-localestimategradient} implies that there exists a positive constant $M_0$, only depending on the operator $\L$, such that
\begin{equation*}
M(z,\z) \le M_0 \left( \frac{|x-\xi|^2}{(t-\tau)^2} + 1 \right), \qquad \text{for every} \
\zeta \in \Omega_{\vartheta r}^{(m)}(z).
\end{equation*}
Moreover, Lemma \ref{lem-localestimate*} implies that there exists another positive constant $M_1$ such that
\begin{equation} \label{eq-boundM}
M(z,\z) \le M_1 \left( \frac{1}{t-\tau} \log\left( \frac{\vartheta^2 \, r^2}{t-\tau} \right) + 1 \right),
\qquad \text{for every} \
\zeta \in \Omega_{\vartheta r}^{(m)}(z).
\end{equation}
We point out that the constants $C_1$ and $M_1$ depend neither on the choice of $\vartheta \in ]0,1[$, that will be specified in the following proof of the point {\it iii)}, nor on the choice of $r \in ]0,r_0[$. By using \eqref{eq-boundN} and \eqref{eq-boundM} we conclude that there exists a positive constant $M_2$, that only depends on $\L$ and on $m$, such that
\begin{equation*}
M_{\vartheta r}^{(m)} (z; \zeta) \le M_2 (t-\tau)^{m/2}
\left( 1 + \left| \log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right) \right| \right)^{m/2}
\left( 1 + \tfrac{1}{t-\tau} \left| \log\left( \tfrac{\vartheta^2 \, r^2 }{t-\tau} \right) \right| \right)
\end{equation*}
for every $\zeta \in \Omega_{\vartheta r}^{(m)}(z)$. The right hand side of the above inequality is bounded whenever $m>2$, uniformly with respect to $r \in ]0,r_0[$ and $\vartheta \in ]0,1[$. This concludes the proof of {\it ii)}.
\noindent {\it Proof of iii)} \ We prove the existence of a constant $\vartheta \in ]0,1[$ as claimed by using a compactness argument and the parabolic scaling. We first observe
that Lemma \ref{lem-localestimate*} implies that
\begin{equation*}
K_r^{(m)}(z_0) \subset \overline{\Omega^{(m)*}_{3r}(z_0)} \cap
\Big\{ t \le t_0 - \tfrac{1}{4 \partiali \lambda^{N/(N+m)}} \,r^2 \Big\},
\end{equation*}
which is a compact subset of $\Omega^{(m)*}_{4r}(z_0)$. We now show that there exists $\vartheta \in ]0,1[$ such that
\begin{equation} \label{eq-claim*}
\Omega_{3 \vartheta r}^{(m)*}(z) \subset \Omega_{4r}^{(m)*}(z_0)
\cap \Big\{\tau \le t_0 - \tfrac{r^2}{4 \partiali \lambda^{N/(N+m)}}\Big\}
\end{equation}
for every $z \in \overline{\Omega^{(m)*}_{3r}(z_0)} \cap \Big\{ t \le t_0 - \frac{1}{4 \partiali \lambda^{N/(N+m)}} \,r^2 \Big\}$. Our claim {\it iii)} will follow from \eqref{eq-claim*} and from Lemma \ref{lem-localestimate*}.
We next prove \eqref{eq-claim*} by using the parabolic scaling. We note that
\begin{equation} \label{eq-parabolicscaling}
(x_0 + r \xi,t_0 + r^2 \tau) \in {\Omega^{(m)*}_{k r}(z_0)} \quad \Longleftrightarrow \quad
(x_0+\xi,t_0 + \tau) \in {\Omega^{(m)*}_{k}(z_0)},
\end{equation}
for every positive $k$. We will need to use $k = 3, 4$ and $\vartheta$.
We next show that \eqref{eq-claim*} holds for $r=1$. The result for every $r \in ]0,r_0[$ will follow from \eqref{eq-parabolicscaling}. Let $\delta(z_0)$ be the distance of the compact set $\overline{\Omega_{3}^{(m)*}(z_0)} \cap \big\{t \le t_0 - \frac{1}{4 \partiali \lambda^{N/(N+m)}}\big\}$ from the boundary of $\Omega_{4}^{(m)*}(z_0)$. We have that $\delta$ is a strictly positive function which depends continuously on $z_0$ through the coefficients of the matrix $A(z_0)$. Moreover, the condition \eqref{e-up} is satisfied, then there exists a positive constant $\delta_0$, only depending on $\lambda, \Lambda$ and $N$, such that
\begin{equation*}
\delta(z_0) \ge \delta_0 \quad \text{for every} \ z_0 \in \Omega.
\end{equation*}
On the other hand, the diameter of the set $\Omega^{(m)*}_{1}(z)$ is bounded by a constant that doesn't depend on $z$. Then, by \eqref{eq-parabolicscaling}, it is possible to find $\vartheta \in ]0,1[$ such that the diameter of $\Omega^{(m)*}_{\vartheta}(z)$ is not greater than $\delta_0$. This concludes the proof of \eqref{eq-claim*} in the case $r=1$. As said above, the case $r \in ]0,r_0[$ does follow from \eqref{eq-parabolicscaling}. This concludes the proof of {\it iii)}.
\noindent {\it Proof of iv)} \ From \eqref{e-Omegam} it directly follows that
\begin{equation*}
M_{5r}^{(m)} (z_0;\z) \ge \frac{m \, \omega_m }{m+2} \cdot \frac {N_{5r}^{m+2}(z_0;\z)}{4(t_0-\t)^2} \ge
\frac{m \, \omega_m}{m+2} (2(t_0-\t))^{m-2} \left( (N+m) \log(5/4)\right)^{(m+2)/2}.
\end{equation*}
The last claim then follows by choosing
\begin{equation*}
m^-:= \lambda^{- N(m-2)/(N+m)}\frac{m \, \omega_m}{m+2} \frac{r^{2m-4}}{(2 \partiali)^{m-2} } \left( (N+m) \log(5/4)\right)^{(m+2)/2}.
\end{equation*}
We eventually remove the assumption $\div \, b - c = 0$. We mainly rely on the steps in the display \eqref{e-core-h} and we point out the needed changes for this more difficult situation. With this aim, we recall that $| \div \, b(z) - c(z)| \le k := (N+1) \Lambda $ because of \eqref{e-up}. We next introduce two auxiliary functions
\begin{equation*}
\widehat u(x,t) := e^{k(t-t_0)} u(x,t), \qquad \widetilde u(x,t) := e^{-k(t-t_0)} u(x,t).
\end{equation*}
Note that, as $\L u = 0$, we have that
\begin{equation*}
\widehat \L \, \widehat u : = \L \widehat u + k \widehat u = 0, \qquad
\widetilde \L \, \widetilde u : = \L \widetilde u - k \widetilde u = 0.
\end{equation*}
Note that $\widehat \L, \widetilde \L$ satisfy the condition \eqref{e-up} with $\Lambda$ replaced by $k$. In particular, the statemets \emph{i)}-\emph{iv)} hold for $\L, \widehat \L$ and $\widetilde \L$ with the same constants $r_0, \vartheta, M^+, m^-$.
We denote by $\widehat M^{(m)}, \widetilde M^{(m)}$ the kernels relative to $\widehat \L, \widetilde \L$, respectively, and $\widehat \Omega^{(m)}, \widetilde \Omega^{(m)}$ the superlevel sets we use in the representation formulas appearing in \eqref{e-core-h}. As we did before, we let $r^*$ be the constant appearing in Lemma \ref{lem-localestimate*} and we choose $r_0 := r^*/2$. As a direct consequence of the definiton of $\widehat u$ and $\widetilde u$, there exist two positive constants $\widehat c$ and $\widetilde c$ such that
\begin{equation} \label{e-bounds-u}
\widehat c u(z) \le \widehat u(z) \le u(z), \qquad u(z) \le \widetilde u(z) \le \widetilde c u(z),
\end{equation}
for every $z \in \Omega_{5r}^{(m)}(z_0)$.
We are now in position to conclude the of proof Proposition \ref{prop-Harnack}. Let's consider the first two lines of \eqref{e-core-h}. Since $\widehat u$ is a solution to $\widehat \L \, \widehat u = 0$, for every $z \in K^{(m)}_{r}(z_0)$, it holds
\begin{equation*}
\widehat u(z) \le \frac{1}{(\vartheta r)^{N+m}} \int_{\widehat \Omega^{(m)}_{\vartheta r}(z)}
\! \! \! \! \! \! \widehat M_{\vartheta r}^{(m)} (z; \zeta) \widehat u(\zeta) \, d\zeta \le
\frac{M^+}{(\vartheta r)^{N+m}}
\int_{\widehat \Omega^{(m)}_{\vartheta r}(z)} \!\!\! \widehat u(\zeta) \, d\zeta.
\end{equation*}
The first inequality follows from the fact that $\div \, b(\zeta) - c(\zeta) - k \le 0$ for every $\zeta$. From \eqref{e-bounds-u} it then follows that
\begin{equation*}
u(z) \le \frac{M^+}{\widehat c \, (\vartheta r)^{N+m}}
\int_{\widehat \Omega^{(m)}_{\vartheta r}(z)} \!\!\! u(\zeta) \, d\zeta.
\end{equation*}
Continuing along the next lines of \eqref{e-core-h}, we note that {\it iii)} also holds in this form: $\widehat \Omega_{\vartheta r}^{(m)}(z) \subset \widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}\big\}$ for every $z \in K^{(m)}_{r}(z_0)$, so that
\begin{equation*}
u(z) \le \frac{M^+}{\widehat c \, (\vartheta r)^{N+m}}
\int_{\widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}\big\}}
\!\! u(\zeta) \, d\zeta.
\end{equation*}
On the other hand, using the fact that $\widetilde \L \, \widetilde u = 0$, and $\div \, b(\zeta) - c(\zeta) + k \ge 0$, we find
\begin{equation*}
\frac{m^-}{(5 r)^{N+m}} \int_{\widetilde \Omega_{4r}^{(m)}(z_0) \cap \big\{\tau \le t_0 - \frac{r^2}{4 \partiali \lambda^{N/(N+m)}}\big\}}
\!\!\! \widetilde u(\zeta) \, d\zeta \le
\frac{1}{(5 r)^{N+m}} \int_{\widetilde \Omega_{5r}^{(m)}(z_0)} \! \! \! \! \! \! \widetilde M_{5 r}^{(m)} (z_0; \zeta) \widetilde u(\zeta) \, d\zeta \le
\widetilde u(z_0).
\end{equation*}
Thus, recalling that $u \le \widetilde u$ and $u (z_0) = \widetilde u(z_0)$, we conclude that
\begin{equation*}
u(z) \le
\frac{5^{N+m} M^+}{\widehat c \, \vartheta^{N+m} m^-} u(z_0),
\end{equation*}
for every $z \in K^{(m)}_{r}(z_0)$. This concludes the proof of Proposition \ref{prop-Harnack}.
\end{proof}
As a simple consequence of Proposition \ref{prop-Harnack} we obtain the following result.
\begin{corollary} \label{cor-Harnack-inv}
There exist four positive constants $r_1, \kappa_1, \vartheta_1$ and $C_D$, with $\kappa_1, \vartheta_1 < 1$, such that the following inequality holds. For every $z_0 \in \Omega$ and for every positive $r$ such that $r \le r_1$ and ${\Q_{r}(z_0)} \subset \Omega$ we have that
\begin{equation} \label{e-HD}
\sup_{D_{r}(z_0)} u \le C_D u(z_0)
\end{equation}
for every $u \ge 0$ solution to $\L u = 0$ in $\Omega$. Here
\begin{equation*}
D_{r}(z_0) := B_{\vartheta_1 r}(x_0) \times \{t_0 - \kappa_1 r^2\}.
\end{equation*}
\end{corollary}
\begin{center}
\begin{tikzpicture}
\clip(-.52,6.82) rectangle (7.02,1.48);
\partialath[draw,thick] (-.5,6.8) rectangle (7,1.5);
\begin{axis}[axis y line=none, axis x line=none,
xtick=\empty,ytick=\empty,
ymin=-1.1, ymax=1.1,
xmin=-.2,xmax=1.8, samples=101, rotate= -90]
\addplot [black,line width=.7pt, domain=-.01:.01] {sqrt(.0001 - x * x)} node[above] {$z_0$};
\addplot [black,line width=.7pt, domain=-.01:.01] {-sqrt(.0001 - x * x)};
\addplot [ddblue,line width=.7pt, domain=.001:1] {sqrt(- 2 * x * ln(x))};
\addplot [ddblue,line width=.7pt,domain=.001:1] {- sqrt(- 2 * x * ln(x))} node[below] { \hskip30mm {\color{dddblue} $\Omega_r(z_0)$}};
\addplot [bblue,line width=.7pt,domain=.001:.1111] {sqrt(- 3 * x * ln(9*x)};
\addplot [bblue,line width=.7pt,domain=.001:.1111] {- sqrt(- 3 * x * ln(9*x))};
\end{axis}
\draw [line width=.6pt] (0,6.165) rectangle node [above=1.5cm,right=2.8cm] {$Q_r(z_0)$} (5.65,2);
\draw [red,line width=1.2pt] (2.1,5.9) -- node [below=5pt] { \hskip20mm $D_r(z_0)$} (3.6,5.9);
\end{tikzpicture}
{\sc \qquad Fig.5} - The set $D_r(z_0)$.
\end{center}
The above assertion follows from the fact that there exists a positive constant $\delta_1$ such that $\Omega_{r}^{(m)}(z_0) \subset \Q_{\delta_1 r}(z_0)$ for every $r \in 0], r_0[$ and that $D_{r}(z_0) \subset K^{(m)}_{r}(z_0)$, for some positive $\kappa_1, \vartheta_1$. Note that Corollary \ref{cor-Harnack-inv} and Theorem \ref{th-Harnack-inv} differ in that, unlike the cylinders $\Q^+_{r}(z_0)$ and $\Q^-_{r}(z_0)$, the set $D_{r}(z_0)$ is not arbitrary. We next prove Theorem \ref{th-Harnack-inv} by using iteratively the Harnack inequality proved in Corollary \ref{cor-Harnack-inv}.
\begin{proof} {\sc of Theorem \ref{th-Harnack-inv}.} As a first step we note that, up to the change of variable $v(x,t) := u(x_0 + r t, t_0 +r^2 t)$, it is not restrictive to assume that $z_0 = 0$ and $r = 1$. Indeed, the function $v$ is a solution to an equation $\widehat \L v = 0$, where the coefficients of the operator $\widehat \L$ are $\widehat a_{ij}(x,t) = a_{ij}(x_0 + r t, t_0 +r^2 t)$ satisfy all the assumptions made for $\L$, with the constants $M$ and $\Lambda$ appearing in \eqref{e-hc} and \eqref{e-up} replaced by $r^\alpha M$ and $r^\alpha \Lambda$, respectively, and the same constant $\lambda$ in \eqref{e-up}. Then, as $r \in ]0, R_0]$, the H\"older constant in \eqref{e-hc} of $\widehat \L$ is $R_0^{\, \alpha} M$ and the parabolicity constants in \eqref{e-up} are $\lambda$ and $R_0^{\, \alpha} \Lambda$, for every $r \in ]0, R_0]$. In the following we then assume that $z_0 = 0$ and $r=1$. Moreover, $r_1$ denotes the constant appearing in Corollary \ref{cor-Harnack-inv} and relative to $\widehat \L$, which depends on the constants $M, \lambda, \Lambda$ and $R_0$. We then choose four positive constants $\iota, \kappa, \mu, \vartheta$ with $0 < \iota < \kappa < \mu < 1$ and $0 < \vartheta < 1$ and we consider the cylinders $\Q^+ := \Q_1^+(0)$ and $\Q^- := \Q_1^-(0)$ as defined in \eqref{e-QPM}. We let
\begin{equation*}
r_0 := \min \big\{r_1, 1 - \vartheta, \sqrt{1 - \mu} \big\}
\end{equation*}
and we note that $\Q_r(z) \subset \Q_1(0)$ whenever $z \in B(0,\vartheta) \times ]- \mu, 0[$ and $0 < r < r_0$.
We next choose any $z^- = (x^-,t^-) \in \Q^-, z^+ = (x^+,t^+) \in \Q^+$ and we rely on Corollary \ref{cor-Harnack-inv} to construct a \emph{Harnack chain}, that is a finite sequence $w_0, w_1, \dots, w_m$ in $\Q_1(0)$ such that
\begin{equation} \label{eq-HC}
w_0 = z^+, \qquad w_k = z^-, \qquad u(w_j) \le C_D u(w_{j-1}), \quad j=1, \dots, m.
\end{equation}
\begin{center}
\begin{tikzpicture}
\partialath[draw,thick] (-1,.7) rectangle (8.7,6.5);
\filldraw [fill=black!4!white, line width=.6pt] (1.5,6) rectangle node[right=2cm] {$Q^+_r(z_0)$} (5.5,4.5);
\filldraw [fill=black!4!white, line width=.6pt] (1.5,2) rectangle node[right=2cm] {$Q^-_r(z_0)$} (5.5,3.5);
\draw [line width=.6pt] (0,6) rectangle node[right=3.5cm] {$Q_r(z_0)$}(7,1);
\draw [line width=.6pt] (3.5,6) circle (1pt) node[above] {$z_0$};
\foreach \x in {0,1,...,5}
\draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,bblue,line width=.6pt] (.4,3.6) rectangle (3.2,4.6);
\foreach \x in {0,1,...,5}
\draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,red,line width=1.2pt] (1.4,4.4) -- (2.2,4.4);
\foreach \x in {0,1,...,5}
\draw [xshift=\x*.4 cm,yshift=-\x*.2 cm,line width=1pt] (2.2,4.4) circle (1pt);
\draw [line width=1pt] (1.8,4.6) circle (1pt) node[above] {$z^+$};
\draw [line width=1pt] (4.2,3.4) circle (1pt) node[above=2pt] {$\, z^-$};
\end{tikzpicture}
{\sc \qquad Fig.6} - A Harnack chain.
\end{center}
We build a Harnack chain as follows. For a positive integer $m$ that will be fixed in the sequel, we choose a positive $r$ and the vector $y \in \mathbb{R}^N$ satisfying
\begin{equation} \label{eq-y}
m \kappa_1 r^2 = t^+-t^-, \qquad m r y = x^+ - x^-.
\end{equation}
Let $\kappa_1,\ \vartheta_1$ be the constants in Corollary \ref{cor-Harnack-inv}. We define
\begin{equation} \label{eq-wj}
w_j := (x^+ + j r y, t^+ - j \kappa_1 r^{2}), \qquad j=0,1, \dots, m.
\end{equation}
Clearly, if $r \le r_0$, then $\Q_r(w_j) \subset \Q_1(0)$ for $j=0, 1, \dots, m$. If moreover $|y| \le \vartheta_1$, then $w_{j} \in D_{r}(w_j-1)$ for $j=1, \dots, m$. This proves that \eqref{eq-HC} holds, and we conclude that
\begin{equation} \label{eq-H+-}
u(z^-) \le C_D^{\, m} u(z^+).
\end{equation}
We next choose $m$ in order to have both condtions $r \le r_0$ and $|y| \le \vartheta_1$ satisfied.
The choice of $m$ is different in the case $|x^+ - x^-|$ is \emph{small} or \emph{large} with respect to $t^+-t^-$. If
\begin{equation} \label{eq-case1}
\frac{|x^+-x^-|}{t^+-t^-} \le \frac{\vartheta_1}{\kappa_1 r_0},
\end{equation}
we let $m$ be the positive integer satisfying
\begin{equation} \label{eq-mm}
(m-1) \kappa_1 r_0^{\, 2} < t^+ - t^- \le m \kappa_1 r_0^{\, 2},
\end{equation}
and, in accordance with \eqref{eq-y}, we choose $r$ as the unique positive number satisfying $m \kappa_1 r^2 = t^+-t^-$. From \eqref{eq-mm} it directly follows $r \le r_0$, while from \eqref{eq-mm} and \eqref{eq-case1} we obtain $|y| \le \vartheta_1$.
Suppose now that
\begin{equation} \label{eq-case2}
\frac{|x^+-x^-|}{t^+-t^-} > \frac{\vartheta_1}{\kappa_1 r_0}.
\end{equation}
In view of \eqref{eq-y}, in this case we choose $m$ as the integer satisfying
\begin{equation} \label{eq-m}
m - 1 < \frac{\kappa_1 |x^+-x^-|^2}{\vartheta_1^{\, 2} (t^+-t^-)} \le m,
\end{equation}
and we let $y$ be the vector parallel to $x^- - x^+$ and such that
\begin{equation*}
m |y| = \frac{\kappa_1 |x^+-x^-|^2}{{\vartheta_1 (t^+-t^-)}}.
\end{equation*}
Clearly, $|y| \le \vartheta_1$, and \eqref{eq-case2} implies $r \le r_0$.
We next find a bound for the integer $m$, which is uniform with respect to $z^- \in \Q^-$ and $z^+ \in \Q^+$, and we rely on \eqref{eq-H+-} to conclude the proof. In the first case \eqref{eq-case1} we obtain from \eqref{eq-mm} that $m \le \frac{t^+-t^-}{\kappa_1 r_0^{\, 2}}$. In the second case \eqref{eq-case2} we rely on \eqref{eq-m} and we note that $t^+-t^- \ge \kappa- \iota$, by our choiche of $\Q^-$ and $\Q^+$. Then in this case we have $m < \frac{ 4\kappa_1}{\vartheta_1^{\, 2} (\kappa - \iota)}$
Summarizing, we have proved that the inequality \eqref{e-H1} holds with
\begin{equation*}
C_H := \exp \left( \max \big\{ \tfrac{1}{\kappa_1 r_0^{\, 2}},
\tfrac{4\kappa_1}{\vartheta_1^{\, 2} (\kappa - \iota)} \big\} \log C_D \right).
\end{equation*}
\end{proof}
\setcounter{equation}{0}
\section{An approach relying on sets of finite perimeter}\label{SectionBV}
In this section we present another approach to the generalized divergence theorem, relying on
De Giorgi's theory of perimeters, see \cite{DeGiorgi2,DeGiorgi3} or
\cite{AmbrosioFuscoPallara,Maggi},
and we show how this leads to a slightly different proof of Theorem \ref{th-1}.
This approach requires more prerequisites than that used in Section \ref{SectionDivergence}, but,
as explained in the Introduction, is more flexible and avoids the Dubovicki\v{\i} theorem.
In this section, if $\mu$ is a Borel measure and $E$ is a Borel set, we use the notation
$\mu\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E(B)=\mu(E\cap B)$.
As before, $C^1_c \partialr{\Omega}$ denotes the set of $C^1$ functions compactly supported in the open set $\Omega \subset \mathbb{R}^n$.
\begin{definition}[\sc $BV$ Functions]
Let $u \in L^1 \partialr{\Omega}$; we say that $u$ is a function of bounded variation in $\Omega$ if its
distributional derivative $Du=\partialr{D_1 u,\ldots,D_n u}$ is an $\mathbb{R}^n$-valued Radon measure in $\Omega$,
{\em i.e.}, if
\[
\int_{\Omega}u \frac{\partialartial \varphi}{\partialartial z_i}\ dz=-\int_{\Omega}\varphi \, dD_i u,
\quad \forall \: \varphi \in C_c^1 \partialr{\Omega}, \; i=1,\ldots,n
\]
or, in vectorial form,
\begin{equation} \label{e-bv-div}
\int_{\Omega}u\, \mathrm{div}\,\Phi\ dz=-\sum_{i=1}^n \int_{\Omega}\Phi_i\ d D_i u
= -\int_{\Omega}\langle\Phi, Du\rangle ,
\quad \forall \: \Phi \in C_c^1 \partialr{\Omega;\mathbb{R}^n}.
\end{equation}
The vector space of all functions of bounded variation in $\Omega$ is denoted by $BV \partialr{\Omega}$.
The variation $V \partialr{u,\Omega}$ of $u$ in $\Omega$ is defined by:
$$
V \partialr{u,\Omega}:=\sup \left\{ \int_{\Omega}u\, \mathrm{div}\,\Phi\ dz:
\Phi \in C^1_c \partialr{\Omega;\mathbb{R}^{n}},\, \norm{\Phi}_{\infty} \leq 1 \right\}.
$$
\end{definition}
We recall that ${V \partialr{u,\Omega}=\abs{Du}\partialr{\Omega}}<\infty$ for any $u \in BV \partialr{\Omega}$, where
$\abs{Du}$ denotes the total variation of the measure $Du$. We also recall that if
$u \in C^1 \partialr{\Omega}$ then
$$
V \partialr{u,\Omega}=\int_{\Omega} \abs{\nabla u} dz.
$$
When the function $u$ is the characteristic functions $\chi_E$ of some measurable set, its variation is said \emph{ perimeter of $E$}.
\begin{definition}[\sc Sets of finite perimeter]
Let $E$ be an $\mathcal{L}^n-$measurable subset of $\mathbb{R}^n$. For any open set $\Omega \subset \mathbb{R}^n$ the perimeter of $E$ in $\Omega$ is denoted by $P \partialr{E,\Omega}$ and it is the variation of $\rchi_{E}$ in
$\Omega$, {\em i.e.},
\begin{equation*}
P \partialr{E,\Omega}:=\sup \left\{\int_{E}\mathrm{div}\, \Phi\ dz:\Phi \in C^1_c \partialr{\Omega;\mathbb{R}^n},\, \norm{\Phi}_{\infty}\leq 1 \right\}.
\end{equation*}
We say that $E$ is a set of finite perimeter in $\Omega$ if $P \partialr{E,\Omega}<\infty$.
\end{definition}
Obviously, several properties of the perimeter of $E$ can be stated in terms of the variation of $\chi_E$. In particular, if $\mathcal{L}^n(E \cap \Omega)$ is finite, then $\rchi_{E} \in L^1 \partialr{\Omega}$ and $E$ has finite perimeter in $\Omega$ if and only if $\rchi_{E}\in BV \partialr{\Omega}$ and $P \partialr{E,\Omega} = \abs{D{\rchi_{E}}}\partialr{\Omega}$. Both the notations $|D\chi_E|(B)$ and $P(E,B)$, $B$ Borel, are used
to denote the total variation measure of $\chi_E$ on a Borel set $B$ and we say that $E$ is a set of locally
finite perimeter in $\Omega$ if $P(E,K)<\infty$ for every compact set $K \subset \Omega$. Finally, formula \eqref{e-bv-div} looks like a divergence theorem:
\begin{equation} \label{e-per-div}
\int_{E}\, \mathrm{div}\, \Phi\ dz=- \int_{\Omega}\langle\Phi,D\chi_E\rangle,
\quad \forall \: \Phi \in C_c^1 \partialr{\Omega;\mathbb{R}^n},
\end{equation}
but it becomes more readable if some precise information is given on the set where the measure $D\chi_E$
is concentrated. Therefore, we introduce the notions of {\em reduced boundary} and of {\em density} and
recall the structure theorem for sets with finite perimeter due to E. De Giorgi, see \cite{DeGiorgi3}
and \cite[Theorem 3.59]{AmbrosioFuscoPallara}, and the characterization due to H. Federer.
\begin{definition}[\sc Reduced boundary]
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $E$ be a set of locally finite perimeter in $\Omega$.
We say that $z \in \Omega$ belongs to the reduced boundary ${\mathcal F}E$ of $E$ if $|D\chi_E|(B_\r(z))>0$
for every $\r>0$ and the limit
$$
\nu_E\partialr{z}:=\lim_{\r \to 0^+}\frac{D{\rchi_{E}}\partialr{B_{\r}\partialr{z}}}{\abs{D{\rchi_{E}}}\partialr{B_{\r}\partialr{z}}}
$$
exists in $\mathbb{R}^n$ and satisfies $\abs{\nu_E \partialr{z}}=1$. The function
$\nu_E:{\mathcal F}E \rightarrow {\mathbb S}^{n-1}$
is Borel continuous and it is called the {\em generalized (or measure-theoretic) inner normal} to $E$.
\end{definition}
Notice that the reduced boundary is a subset of the topological boundary. The Besicovitch differentiation
theorem, see e.g. \cite[Theorem 2.22]{AmbrosioFuscoPallara}, yields $D\rchi_{E} = \nu_E\abs{D\rchi_{E}}$,
and $\abs{D\rchi_{E}}(\Omega\setminus{\mathcal F}E)=0$, hence \eqref{e-per-div} becomes
\begin{equation} \label{e-3}
\int_{E}\mathrm{div}\, \Phi\ dz=-\int_{{\mathcal F}E}\scp{\nu_E,\Phi}\ d \abs{D\rchi_{E}},
\quad \forall \: \Phi \in C^1_c \partialr{\Omega;\mathbb{R}^N}.
\end{equation}
\begin{comment}
The following celebrated \emph{De Giorgi’s structure theorem} gives a further improvement of the above identity. Let us recall that a set $S\subset\mathbb{R}^n$ is {\em countably $(n-1)-$rectifiable} if there
exist countably many Lipschitz functions $\partialsi_j:\mathbb{R}^{n-1}\to\mathbb{R}^n$ such that
\[
{\cal H}^{n-1}\left(S\setminus\bigcup_{j=0}^\infty \partialsi_j(\mathbb{R}^{n-1})\right)=0
\]
and ${\cal H}^{n-1}(S)<\infty$.
\begin{theorem}[\sc De Giorgi]
Let $E$ be an $\mathcal{L}^n-$measurable subset of $\mathbb{R}^n$. Then ${\mathcal F}E$ is countably
$\partialr{n-1}-$rectifiable and $\abs{D{\rchi_{E}}}=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \mathcal{F}E$.
In addition, for any $z_0 \in {\mathcal F}E$ the following statements hold:
\begin{description}
\item[$(a)$] the sets $\frac{1}{\r}\partialr{E-z_0}$ locally converge in measure in $\mathbb{R}^n$ as $\r \to 0^+$
to the half-space $H$ orthogonal to $\nu_E \partialr{z_0}$ and containing $\nu_E \partialr{z_0}$;
\item[$(b)$] $\mathrm{Tan}^{n-1} \partialr{\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} {\mathcal F} E,z_0}=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \nu_E^\partialerp \partialr{z_0}$, {\em i.e.},
\[
\lim_{\varrho\to 0+}\frac{1}{\varrho^{n-1}}
\int_{{\mathcal F}E}\partialhi\Bigl(\frac{z-z_0}{\varrho}\Bigr) d\H^{n-1}(z)
=\int_{\nu_E^\partialerp(z_0)} \Phi(z)\ d\H^{n-1}(z).
\]
In particular,
\begin{equation*}
\lim_{\r \to 0^+}\frac{\H^{n-1}\partialr{{\mathcal F}E \cap B_{\r}\partialr{z_0}}}{\omega_{n-1}\r^{n-1}}=1,
\end{equation*}
where $\omega_{n-1}$ is the Lebesgue measure of the unit ball in $\mathbb{R}^{n-1}$.
\end{description}
\end{theorem}
Here $\mathrm{Tan}^{n-1}\partialr{\mu,z}$ denotes the \emph{approximate tangent space} to a Radon measure $\mu$ at point $z$ while $\mu \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E$ denotes the \emph{restriction} of a measure $\mu$ to $E$ (see Definitions 1.65 and 2.79 in \cite{AmbrosioFuscoPallara}).
\end{comment}
The relation between the topological boundary and the reduced boundary can be further analyzed by
introducing the notion of {\em density} of a set at a given point.
\begin{definition}[\sc Points of density $\alpha$]
For every $\alpha \in \left[0,1\right]$ and every $\mathcal{L}^n-$measurable set $E \subset \mathbb{R}^n$ we denote by $E^{\partialr{\alpha}}$ the set
$$
E^{\partialr{\alpha}}=\left\{ z \in \mathbb{R}^n: \lim_{\r \to 0^+}
\frac{\mathcal{L}^n(E \cap B_\r \partialr{z})}{\mathcal{L}^n(B_\r \partialr{z})}=\alpha\right\}.
$$
\end{definition}
Thus $E^{\partialr{\alpha}}$, which turns out to be a Borel set, is the set of all points where $E$ has density
$\alpha$. The sets $E^{\partialr{0}}$ and $E^{\partialr{1}}$ are called \emph{the measure-theoretic exterior} and \emph{interior} of $E$ and, in general, strictly contain the topological exterior and interior of the set $E$, respectively.
We recall the well known \emph{Lebesgue's density theorem}, that asserts that for every
$\mathcal{L}^n-$measurable set $E \subset \mathbb{R}^n$
$$
\mathcal{L}^n(E \triangle E^{\partialr{1}})=0, \quad \mathcal{L}^n(\partialr{\mathbb{R}^n \setminus E} \triangle E^{\partialr{0}})=0,
$$
{\em i.e.}, the density of $E$ is $0$ or $1$ at $\mathcal{L}^n-$almost every point in $\mathbb{R}^n$.
This notion allows to introduce the
{\em essential} or {\em measure-theoretic} boundary of $E$ as
$\partialartial^*E=\mathbb{R}^n\setminus (E^{\partialr{0}}\cup E^{\partialr{1}})$, which is contained in the topological boundary
and contains the reduced boundary. Finally, the De Giorgi structure theorem says that
$|D\rchi_E|=\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}{\mathcal F}E$ and a deep result due to Federer
(see \cite[4.5.6]{federer1969geometric} or
\cite[Theorem 3.61]{AmbrosioFuscoPallara}) states that if $E$ has finite perimeter in $\mathbb{R}^n$ then
\[
\mathcal{F}E\subset E^{\partialr{1/2}}\subset \partialartial^*E \quad \text{and}\quad
\H^{n-1}(\mathbb{R}^n\setminus (E^{\partialr{0}}\cup\mathcal{F}E\cup E^{\partialr{1}}))=0
\]
hence, in particular, $\nu_E$ is defined $\H^{n-1}-$a.e. in $\partialartial^*E$. Notice also (see
\cite[Theorem 3.62]{AmbrosioFuscoPallara}) that if $\H^{n-1}(\partialartial E)<\infty$ then $E$ has finite
perimeter. The results of De Giorgi and Federer imply that if $E$ is a set of finite perimeter in
$\Omega$ then $D \rchi_E=\nu_E \H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \mathcal{F}E$ and the divergence
theorem \eqref{e-3} can be rewritten in the form:
\begin{equation}\label{e-rem-1}
\int_E \mathrm{div}\, \Phi\ dz=
-\int_{{\mathcal F}E}\scp{\nu_E,\Phi}\ d \H^{n-1}=
-\int_{\partialartial^*E}\scp{\nu_E,\Phi}\ d \H^{n-1}, \quad \forall \: \Phi \in C^1_c \partialr{\Omega;\mathbb{R}^n},
\end{equation}
much closer to the classical formula \eqref{eq-div}. Indeed, the only difference is that the inner normal and the boundary are understood in a measure-theoretic sense and not in the topological one; in particular, for a generic set of finite perimeter, ${\mathcal F}E$ needs not to be closed and $\nu_E$ needs not to be continuous.
Moreover, $\nu_E$ is defined $\H^{n-1}-$a.e. in $\partialartial^*E$.
Let us see now how we can rephrase the results of Section \ref{SectionDivergence} in terms of
perimeters and how we can modify the proof of Theorem \ref{th-1}.
We first recall the Fleming--Rischel formula
(see \cite{fleming} or \cite[Theorem 3.40]{AmbrosioFuscoPallara}), {\em i.e.}, the coarea
formula for $BV$ functions.
\begin{theorem}[\sc Coarea formula in $BV$] \label{th-cobv}
For any open set $\Omega \subset \mathbb{R}^n$ and $G \in L^1_{\mathrm{loc}}\partialr{\Omega}$ one has
$$
V \partialr{G,\Omega}=\int_{\mathbb{R}}P\partialr{\{z \in \Omega:G \partialr{z}>y\},\Omega}\ dy.
$$
In particular, if $G \in BV \partialr{\Omega}$ the set $\left\{G >y \right\}$ has finite perimeter in $\Omega$ for
$\H^1-$a.e.~$y \in \mathbb{R}$ and
$$
\abs{DG}\partialr{B}=\int_{\mathbb{R}}\abs{D{\rchi_{\left\{G>y\right\}}}}\partialr{B}\ dy, \quad
DG\partialr{B}=\int_{\mathbb{R}}D{\rchi_{\left\{G>y\right\}}}\partialr{B}\ dy, \quad
\forall \: B \in \mathcal{B}\partialr{\Omega}.
$$
\end{theorem}
Now we are ready to state the analogue of Proposition \ref{prop-1} and to prove Theorem \ref{th-1}
again.
\begin{proposition} \label{prop-1BV}
Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $F \in BV \partialr{{\Omega};\mathbb{R}}\cap C \partialr{{\Omega};\mathbb{R}}$.
Then, for $\H^1-$almost every $y \in \mathbb{R}$, we have:
\begin{equation}\label{cobv}
\int_{\left\{F>y\right\}} \mathrm{div}\, \Phi\ dz =
-\int_{\partialartial^*\{F>y\}} \scp{\nu,\Phi}\ d \H^{n-1},
\quad \forall \: \Phi \in C_c^1 \partialr{\Omega;\mathbb{R}^{n}},
\end{equation}
were $\nu$ is the generalized inner normal to $\{F>y\}$.
\end{proposition}
\begin{proof} By Theorem \ref{th-cobv},
the set $\left\{F>y\right\}$ has finite perimeter in $\Omega$ for $\H^1-$a.e. $y \in \mathbb{R}$,
hence we may apply \eqref{e-rem-1} with $E=\{F>y\}$ and conclude.
\end{proof}
As in Section \ref{SectionDivergence}, we have to cut the integration
domain: therefore, we study the intersection between the super-level set of a
generic function $G \in BV\partialr{\Omega;\mathbb{R}}\cap C\partialr{\Omega;\mathbb{R}}$ and a half-space
$H_t=\left\{ x \in \mathbb{R}^n: \scp{x,e}<t \right\}$,
for some $e \in {\mathbb S}^{n-1}$, $t \in \mathbb{R}$. First, we present a general formula that characterizes the
intersection of two sets of finite perimeter for which we refer to Maggi's book, see \cite[Theorem 16.3]
{Maggi}.
\begin{theorem}[\sc Intersection of sets of finite perimeter] \label{th-int}
If $A$ and $B$ are sets of locally finite perimeter in $\Omega$, and we let
$$
\left\{ \nu_A = \nu_B \right \} =
\left\{ x \in {\mathcal F}A \cap {\mathcal F}B: \nu_A \partialr{x}=\nu_B \partialr{x} \right\},
$$
then $A \cap B$ is a set of locally finite perimeter in $\Omega$, with
\begin{equation} \label{e-9}
D \rchi_{A \cap B}=D \rchi_A \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} B^{\partialr{1}}+D \rchi_B \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} A^{\partialr{1}}
+ \nu_A \H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \left\{ \nu_A = \nu_B \right\}.
\end{equation}
\end{theorem}
In the case in which $B$ is a half-space, formula \eqref{e-9} can be greatly simplified;
indeed we can prove the following corollary.
\begin{corollary}[\sc Intersections with a half-space]\label{cor-1}
Let $E$ be a set of locally finite perimeter in $\Omega$ and let
$H_t=\left\{ z \in \mathbb{R}^n : \scp{z,e} < t\right\}$ for some $e \in {\mathbb S}^{n-1}$, $t \in \mathbb{R}$. Then, for every
$t \in \mathbb{R}$, $E \cap H_t$ is a set of locally finite perimeter in $\Omega$ and moreover,
for $\H^{1}-$almost every $t \in \mathbb{R}$,
$$
D \rchi_{E \cap H_t}=D \rchi_E \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} H_t-e\H^{n-1}\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} \partialr{E \cap \left\{ \scp{x,e}=t\right\}}.
$$
\end{corollary}
\begin{proof}
The half-space $H_t$ is clearly a set of locally finite perimeter in $\Omega$ for every $t\in\mathbb{R}$,
and for every $t \in \mathbb{R}$ we have, $H_t^{\partialr{1}}=H_t$,
$\mathcal{F}H_t=\partialartial H_t =\left\{ \scp{x,e}=t \right\}$ and $\nu_{H_t} \equiv -e$.
Then, applying Theorem \ref{th-int} we see that $E \cap H_t$ is a set of locally finite perimeter
in $\Omega$ for every $t \in \mathbb{R}$ and \eqref{e-9} reads
\[
D \rchi_{E \cap H_t}=D \rchi_E \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} H_t - e \H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E^{\partialr{1}}\cup
\left\{ \nu_E = \nu_{H_t} \right\}) .
\]
Since by Fubini theorem
\[
0=\mathcal{L}^n (E\triangle E^{(1)}) =
\int_{\mathbb{R}} \H^{n-1}\partialr{(E\triangle E^{(1)}) \cap \left\{ \scp{x,e}=t \right\}}\ dt ,
\]
for $\H^1-$a.e. $t \in \mathbb{R}$ we have
\[
\H^{n-1}\partialr{E \triangle E^{\partialr{1}} \cap \left\{ \scp{x,e}=t \right\}}=0.
\]
Therefore,
\[
D \rchi_{H_t} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E = D \rchi_{H_t} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} E^{\partialr{1}} = -e\H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E\cap\{\scp{x,e}=t\})
= -e\H^{n-1} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} (E\cap\{\nu_E=\nu_{H_t}\})
\]
for $\H^1-$a.e. $t \in \mathbb{R}$ and the thesis follows.
\end{proof}
The following corollary allows us to perform (with some modifications) the last part of the
proof of our main result.
\begin{corollary}\label{prop-2-BV}
Let $\Omega=\mathbb{R}^{N+1} \setminus \left\{ \partialr{z_0} \right\}$, $G \in BV\partialr{\Omega;\mathbb{R}}
\cap C \partialr{\Omega;\mathbb{R}}$,
$H_t=\left\{ z \in \mathbb{R}^{N+1} : \scp{z,e} < t\right\}$ for some $e \in {\mathbb S}^N$, $t \in \mathbb{R}$. Then,
for $\H^{1}-$almost every $w \in \mathbb{R}$ and for every $t < t_0$ the set
$E \cap H_t$ has locally finite perimeter in $\Omega$ and
$$
\int_{\left\{ G > w \right\} \cap H_t}\mathrm{div}\, \Phi\ dz=
-\int_{\partialartial^*\{ G > w \} \cap H_t}\scp{\nu,\Phi}\ d \H^N
+\int_{\left\{ G >w \right\} \cap \left\{ \scp{x,e}=t \right\}} \scp{e,\Phi}\ d\H^N,
$$
for every $\Phi \in C^1_c \partialr{\Omega;\mathbb{R}^n}$, where $\nu$ is the generalized inner normal to
$\partialartial^*(\{G>w\})$.
In particular, if $e=\partialr{0,\ldots,0,1}$, for every $\varepsilon > 0$
$$
\int_{\left\{ G > w \right\} \cap \left\{ t<t_0-\varepsilon \right\}}
\!\!\!\mathrm{div}\, \Phi\ dz
=-\int_{\partialartial^*\{ G>w \} \cap \left\{ t<t_0-\varepsilon \right\}}
\!\!\scp{\nu,\Phi}d \H^{N}
+\int_{\left\{ G>w \right\} \cap \left\{ t=t_0-\varepsilon \right\}}
\!\!\scp{e,\Phi}d\H^{N}.
$$
\end{corollary}
Notice that the difference between Proposition \ref{prop-1bis} and
Corollary \ref{prop-2-BV} is that in the former we can exclude the set of critical points of $G$ from
the surface integral, thanks to Dubovicki\v{\i} theorem, and we know that $\nu$ is given by the
normalized gradient of $G$ {\em everywhere} in the integration set, whereas in the latter we don't
need to know any estimate on the size of ${\rm Crit}(G)$ and $\nu$ is defined $\H^N-${\em a.e.} on
the integration set (still coinciding with the normalized gradient of $G$ out of ${\rm Crit}(G)$,
of course). First, notice that we apply Corollary \ref{prop-2-BV} to $G(z)=\Gamma(z_0;z)$, which is
$C^1(\Omega)$, hence Lipschitz on bounded sets. As a consequence, $\partialartial\{G>w\}\subseteq\{G=w\}$
and comparing the coarea formulas \eqref{e-co} and \eqref{cobv}, we deduce that
$\H^N(\{G=w\}\setminus\partialartial^*\{G>w\})=0$ for $\H^{1}$-a.e. $w$.
Let us see how this entails modifications of the
proof of Theorem \ref{th-1}: the proof goes in the same vein until \eqref{eq-div-3i},
\eqref{eq-div-3}, which in the present context are replaced by
\begin{align*}
\lim_{k \to +\infty} \int_{\partialsi_r(z_0) \cap \left\{ t<t_0-\varepsilon_k \right\}}
\scp{\nu,\Phi}d \H^{N} & = \int_{\partialsi_r(z_0)} K (z_0;z) u(z) d \H^{N}
\\
& = \int_{\partialsi_r(z_0)\setminus \mathrm{Crit}\partialr{\Gamma}} K (z_0;z) u(z) d \H^{N}
\end{align*}
where the first equality follows from Corollary \ref{prop-2-BV}, as explained, and the last equality
follows from the fact that the kernel $K$ vanishes in ${\rm Crit}(\Gamma)$.
The rest of the proof needs no modifications.
\def$'${$'$} \def$'${$'$} \def$'${$'$}
\def\lfhook#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{'}\hidewidth\crcr\unhbox0}}} \def$'${$'$}
\def$'${$'$}
\end{document} |
\begin{document}
\title{Analytic Continuation of Holomorphic Correspondences and
Equivalence of Domains in ${\mathbb C}^n$.}
\author{Rasul Shafikov \\{\small \tt [email protected]}}
\date{\today}
\maketitle
\begin{abstract}
The following result is proved: Let $D$ and $D'$ be bounded domains in
$\mathbb C^n$, $\partial D$ is smooth, real-analytic, simply
connected, and $\partial D'$ is connected, smooth, real-algebraic. Then there
exists a proper holomorphic correspondence $f:D\to D'$ if and only if
$\partial D$ and $\partial D'$ a locally CR-equivalent. This leads
to a characterization of the equivalence relationship between bounded domains in
$\mathbb C^n$ modulo proper holomorphic correspondences in terms of
CR-equivalence of their boundaries.
\end{abstract}
\section{Definitions and Main Results.}
Following Stein \cite{st1} we say that a holomorphic correspondence
between two domains
$D$ and $D'$ in ${\mathbb C}^n$ is a complex-analytic set $A \subset
D\times D'$ which satisfies: (i) $A$ is of pure complex dimension $n$,
and (ii) the natural projection $\pi:A\to D$ is proper. A
correspondence $A$ is called proper, if in addition the natural projection
$\pi': A\to D'$ is proper. We may also think of $A$ as the graph of
a multiple valued mapping defined by $F:=\pi'\circ\pi^{-1}$.
Holomorphic correspondences were studied for instance, in \cite{bb},
\cite{bb2}, \cite{be2}, \cite{bsu}, \cite{dp1}, \cite{dfy}, \cite{p3},
and \cite{v}.
In this paper we address the following question: Given two domains $D$
and $D'$, when does there exist a proper holomorphic correspondence
$F:D\to D'$?
Note (see \cite{st2}) that the existence of a correspondence $F$
defines an equivalence relation $D \sim D'$. This equivalence relation
is a natural generalization of biholomorphic equivalence of domains in
${\mathbb C}^n$.
To illustrate the concept of equivalence modulo holomorphic
correspondences, consider domains of the form
$\Omega_{p,q}= \{|z_1|^{2p}+|z_2|^{2q}<1 \}, \ \ p,q\in{\mathbb Z}^+$.
Then $f(z)=({z_1}^{p/s},{z_2}^{q/t})$ is a proper holomorphic
correspondence between $\Omega_{p,q}$ and $\Omega_{s,t}$, while a
proper holomorphic map from $\Omega_{p,q}$ to $\Omega_{s,t}$ exists
only if $s|p$ and $t|q$, or $s|q$ and $t|p$. For details see \cite{bd}
or \cite{l}.
The main result of this paper is the following theorem.
\begin{theorem}\label{t1}
Let $D$ and $D'$ be bounded domains in ${\mathbb C}^n$, $n>1$. Let
$\partial D$ be smooth, real-analytic, connected and simply connected,
and let $\partial D'$ be connected, smooth, real-algebraic. Then there
exists a proper holomorphic correspondence $F:D\to D'$ if and only if
there exist points $p\in\partial D$, $p'\in\partial D'$, neighborhoods
$U\ni p$ and $U'\ni p'$, and a biholomorphic map $f:U\to U'$, such
that $f(U\cap\partial D)= U'\cap\partial D'$.
\end{theorem}
In other words, we show that a germ of a biholomorphic mapping between
the boundaries extends analytically to a holomorphic correspondence of
the domains. By the example above, the extension will not be in
general single-valued. Note that we do not require pseudoconvexity of
either $D$ or $D'$. Also $\partial D'$ is not assumed to be simply
connected. If both $D$ and $D'$ have real-algebraic boundary,
i.e. each is globally defined by an irreducible real polynomial, then
we can drop the requirement of simple connectivity of $\partial D$.
\begin{theorem}\label{t2}
Let $D$ and $D'$ be bounded domains in ${\mathbb C}^n$, $n>1$, with
connected smooth real-algebraic boundaries. Then there exists a proper
holomorphic correspondence between $D$ and $D'$
if and only if there exist points $p\in\partial D$,
$p'\in\partial D'$, neighborhoods $U\ni p$ and $U'\ni p'$, and a
biholomorphic map $f:U\to U'$, such that $f(U\cap\partial D)=
U'\cap\partial D'$.
\end{theorem}
The proof of Theorem \ref{t2} is simpler than that of Theorem \ref{t1}
due to the fact that when both domains are algebraic, Webster's
theorem \cite{w} can be applied. We give a separate complete proof of
Theorem \ref{t2} in Section 3 to emphasize the ideas of the proof of
Theorem \ref{t1} and the difficulties that arise in the case when one
of the domains is not algebraic.
Local CR-equivalence of the boundaries, which is used in the above
theorems to characterize correspondence equivalence, is a well-studied
subject. Chern and Moser \cite{cm} gave the solution to local
equivalence problem for real-analytic hypersurfaces with
non-degenerate Levi form both in terms of normalization of Taylor
series of the defining functions and in terms of intrinsic
differential-geometric invariants of the hypersurfaces. See also
\cite{se}, \cite{ca} and \cite{t}. Note that for a bounded domain in
${\mathbb C}^n$ with smooth real-analytic boundary, the set of points
where the Levi form is degenerate is a closed nowhere dense set. Thus
we may reformulate Theorem \ref{t1} in the following way.
\begin{theorem}
Let $D$ and $D'$ be as in Theorem \ref{t1}. Then $D$ and $D'$ are
correspondence equivalent if and only if there are points
$p\in\partial D$ and $p'\in\partial D'$ such that the Levi form of the
defining functions are non-degenerate at $p$ and $p'$, and the
corresponding Chern-Moser normal forms are equivalent.
\end{theorem}
Theorem \ref{t1} generalizes the result in \cite{p1}, which
states that a bounded, strictly pseudoconvex domain $D\subset{\mathbb
C}^n$ with connected, simply-connected, real-analytic boundary is
biholomorphically equivalent to ${\mathbb B}^n={\mathbb B}(0,1)$, the
unit ball centered at the origin, if and only if there
exists a local biholomorphism of the boundaries. It was later
shown in \cite{cj} (see also \cite{ber}) that simple connectivity of
$\partial D$ can be replaced by simple connectivity of the domain $D$
and connectedness of $\partial D$. In \cite{p2} Pinchuk also established
sufficient and necessary conditions of equivalence for
bounded, strictly pseudoconvex domains with
real-analytic boundaries. Two such domains $D$ and $D'$ are
biholomorphically equivalent if and only if there exist points
$p\in\partial D$ and $p'\in\partial D'$ such that $\partial D$ and
$\partial D'$ are locally equivalent at $p$ and $p'$.
If in Theorem \ref{t1} $\partial D$ is not assumed to be simply
connected, then the result is no longer true. Indeed, in a famous
example Burns and Shnider \cite{bs} constructed a domain $\O$ with the
boundary given by
\begin{equation}\label{bad}
\partial \O=\{z\in{\mathbb C}^2 : \sin(\ln|z_2|) + |z_1|^2=0, \
e^{-\pi}\le|z_1|\le 1\},
\end{equation}
which is real-analytic and strictly pseudoconvex but not simply
connected. There exists a mapping $f:{\mathbb B}^2\to \O$ such
that $f$ does not extend even continuously to a point on $\partial
{\mathbb B}^2$. The inverse to $f$ gives a local biholomorphic map
from $\partial \O$ to $\partial {\mathbb B}^2$, but nevertheless
$f^{-1}$ does not extend to a global proper holomorphic correspondence
between $\O$ and ${\mathbb B}^2$. Furthermore, suppose that there exists a
proper correspondence $g:\O\to {\mathbb B}^2$. Since ${\mathbb
B}^2$ is simply connected, $g^{-1}$ is a proper holomorphic mapping,
which extends holomorphically to a neighborhood of $\overline{ {\mathbb
B}^2}$. Let $p\in \partial \O$, $q^1\in f^{-1}(p)$, and $q^2\in
g(p)$. By the result in \cite{bb} $g$ splits at every point in
${\overline \O}$.
Therefore, with a suitable choice of the branch of $g$ near $p$,
$\phi:=g\circ f$ defines a local biholomorphic mapping from a
neighborhood $U_1\ni q^1$ to some neighborhood $U_2\ni q^2$. Moreover,
$\phi(U_1\cap \partial {\mathbb B}^2)\subset (U_2\cap \partial
{\mathbb B}^2)$. By the theorem of Poincar\'e-Alexander
(see e.g. \cite{a}), $\phi$ extends to
a global automorphism of ${\mathbb B}^2$. Thus after a biholomorphic
change of coordinates, by the uniqueness theorem $f$ and $g^{-1}$ must
agree in ${\mathbb B}^2$. But $f:{\mathbb B}^2\to \Omega$ is not a
proper map. This contradiction shows that there are no
proper holomorphic correspondences between $\O$ and ${\mathbb
B}^2$. Thus the condition of simple connectivity of $\partial D$ in
Theorem \ref{t1} cannot be in general weakened.
One direction in the proof of Theorems \ref{t1} and \ref{t2} is
essentially contained in the work of Berteloot and Sukhov
\cite{bsu}. The proof in the other direction is based on
the idea of extending the mapping $f$ along $\partial D$ as a
holomorphic correspondence.
It is not known whether Theorem \ref{t1} holds if $\partial D'$ is
real-analytic. The main difficulty is to prove local analytic
continuation of holomorphic mappings along real-analytic hypersurfaces
in the case when hypersurfaces are not strictly pseudoconvex. In
particular, Lemma \ref{l:alongSV} cannot be directly established for
real-analytic hypersurfaces.
In Section 2 we present background material on Segre varieties and
holomorphic correspondences. In Section 3 we prove a technical
result, important for the proof of the main theorems. Section 4
contains the proof of Theorem~\ref{t2}. In Section 5 we prove local
extendability of holomorphic correspondences along hypersurfaces. Theorem \ref{t1}
is proved in Section 6.
\section{Background Material.}
Let $\G$ be an arbitrary smooth real-analytic hypersurface with a defining
function $\rho(z,\overline z)$ and let $z^0\in\G$. In a suitable
neighborhood $U\ni z^0$ to every point $w\in U$ we can associate its
so-called Segre variety defined as
\begin{equation}
Q_w=\left\{ z\in U: \rho(z,\overline w)=0 \right\},
\end{equation}
where $\rho(z,\overline w)$ is the complexification of the defining
function of $\G$. After a local biholomorphic change of coordinates
near $z^0$, we can find neighborhoods $U_1 \Subset U_2$ of $z^0$, where
\begin{equation}\label{e:1.00}
U_2={'U_2}\times {{''U_{2}}}\subset {\mathbb C}^{n-1}_{{\ 'z}}\times
{\mathbb C}_{z_n},
\end{equation}
such that for any $w\in U_1$, $Q_w$ is a closed smooth complex-analytic
hypersurface in $U_2$. Here $z=({'z},z_n)$. Furthermore, a Segre variety
can be written as a graph of a holomorphic function,
\begin{equation}\label{e:implicit}
Q_w=\left\{({'z},z_n)\in {'U_2}\times {''U_{2}} : \ z_n=h({'z},
\overline w)\right\},
\end{equation}
where $h(\cdot,\overline w)$ is holomorphic in ${'U_{2}}$. Following
\cite{dp1} we call $U_1$ and $U_2$ a \textsl{standard} pair of
neighborhoods of $z^0$. A detailed discussion of elementary properties
of Segre varieties can be found in \cite{dw}, \cite{df2},
\cite{dp1} or \cite{ber1}.
The map $\lambda:z\to Q_z$ is called the Segre map. We define
$I_w=\l^{-1}\circ \l(w)$. This is equivalent to
\begin{equation}
I_w=\left\{z\in U_1: Q_z=Q_w\right\}.
\end{equation}
If $\G$ is of finite type in the sense of D'Angelo, or more generally,
if $\G$ is essentially finite, then there exists a neighborhood $U$ of
$\G$ such that for any $w\in U$ the set $I_w$ is finite. Due to the
result in \cite{df1}, this is the case for compact smooth real-analytic
hypersurfaces, in particular for the boundaries of $D$ and $D'$ in
Theorem \ref{t1}. The last observation is crucial for the construction
of proper holomorphic correspondences used throughout this paper.
We remark that our choice of a standard pair of neighborhoods of
any point $z\in\G$ is always such that for any $w\in U_2$, the set $I_w$ is
finite.
For the proof of Theorem \ref{t1} we will need the following lemma.
\begin{lemma}\label{l-finite}
Let $\G$ be a compact, smooth, real-algebraic hypersurface. Then there
exist a neighborhood $U$ of $\G$ and an
integer $m\ge 1$ such that for almost any $z\in U$, $\#I_z = m$.
\end{lemma}
\noindent{\it Proof.} Let $P(z,\overline z)$ be the defining
polynomial of $\G$ of degree $d$ in $z$.
The complexified defining function can be written in the form
\begin{equation}
P(z,\overline w) = \sum_{|K|=0}^d{a_K(\overline w) z^K},\ \ K=(k_1,\dots, k_n).
\end{equation}
We may consider the projectivized version of the polynomial
$P:\mathbb P^n \times\mathbb C^n \to \mathbb C$:
\begin{equation}\label{proj-poly}
\tilde P(\tilde\zeta,\overline w)=\zeta_0^d
\sum_{|K|=0}^d{a_K(\overline w) \left(\frac{\zeta}{\zeta_0}\right)^K}=
\sum_{|K|\le d} a_K(\overline w)\tilde \zeta^K,
\end{equation}
where $z_j=\frac{\zeta_j}{\zeta_0}$, $\zeta=(\zeta_1,\dots,\zeta_n)$,
and $\tilde\zeta = (\zeta_0,\zeta)$. Let $\tilde Q_w = \{\tilde\zeta\in
\mathbb P^n : \tilde P(\tilde\zeta,\overline w)=0\}$. Then $Q_w=\tilde
Q_w\cap \{\zeta_0=1\}$.
Define the map $\hat \l$ between the set of points in $\mathbb C^n$
and the space of complex hypersurfaces in $\mathbb P^n$, by letting
$\hat \l (w) = \{\tilde\zeta\in\mathbb P^n : \tilde P(\tilde\zeta,
\overline w)=0\}$. Hypersurfaces in $\mathbb P^n$ can be parametrized
by points in $\mathbb P^N$, where $N$ is some large integer, thus
$\hat \l: \mathbb C^n\to \mathbb P^{N}$. Note that each component
of $\hat\l$ is defined by a (antiholomorphic) polynomial.
It follows that $\hat\l^{-1}\circ\hat\l (w)=I_w$ for every
$w\in\mathbb C^n$, for which $Q_w$ is defined. Indeed, suppose $\xi\in
I_w$. Then
$\{P(z,\overline\xi)=0\}=\{P(z,\overline w)=0\}$, and therefore,
\begin{equation}\label{p=p}
\{\tilde P(\tilde\zeta,\overline\xi)=0\}=
\{\tilde P(\tilde\zeta,\overline w)=0\}.
\end{equation}
Thus $\hat\l(\xi)=\hat\l(w)$. The converse clearly also holds.
From the reality condition on $P(z,\overline z)$ (see \cite{dw}),
\begin{equation}
I_w\subset \G, {\rm\ \ for\ } w\in\G.
\end{equation}
Since $\G$ is compact, $I_w=\hat\l^{-1}\circ\hat\l (w)$ is finite. Let
$Y=\hat\l(\mathbb C^n)$. Then $\dim Y = n$, and $\hat\l$
is a dominating regular map. It follows (see e.g. \cite{m}), that there
exists an algebraic variety $Z\subset Y$ such that for any $q\in Y\setminus Z$,
\begin{equation}\label{deg}
\#\hat\lambda^{-1} (q)=\deg(\hat\lambda)=m,
\end{equation}
where $m$ is some positive integer. From (\ref{deg}) the assertion
follows. $\square$
The following statement describes the invariance property of Segre
varieties under holomorphic mappings. It is analogous to the classical
Schwarz reflection principle in dimension one.
Suppose that $\G$ and $\G'$ are real-analytic hypersurfaces in
${\mathbb C}^n$, $(U_1, U_2)$ and $(U'_1, U'_2)$ are standard pairs
of neighborhoods for $z_0\in\G$ and $z'_0\in\G'$ respectively. Let $f:
U_2 \to U'_2$ be a holomorphic map, $f(U_1)\subset U'_1$ and $f(\G\cap
U_2)\subset(\G'\cap U'_2)$. Then
\begin{equation}\label{e-invar}
f(Q_w\cap U_2)\subset Q'_{f(w)}\cap U'_2,\ \ {\rm for\ all\ } w\in U_1.
\end{equation}
Moreover, a similar invariance property also holds for a proper
holomorphic correspondence $f:U_2 \to U'_2$, $f(\G\cap
U_2)\subset(\G'\cap U'_2)$. In this case
(\ref{e-invar}) indicates that any branch of $f$ maps any point from
$Q_w$ to $Q_{w'}$ for any $w'\in f(w)$. For details, see \cite{dp1} or
\cite{v}.
Let $f:D\to D'$ be a holomorphic correspondence. We say
that $f$ is {\it irreducible}, if the corresponding analytic set
$A\subset D\times D'$ is irreducible. The condition that $f$ is proper
is equivalent to the condition that
\begin{equation}
\sup \{ {\rm dist}(f(z),\partial D') \}\to 0,\ \ {\rm as}\ \
{\rm dist}(z,\partial D)\to 0.
\end{equation}
Recall that if $A\subset D\times D'$ is a proper holomorphic
correspondence, then $\pi: A\to D$ and $\pi':A\to D'$ are proper. There
exists a complex subvariety $S\subset D$ and a number $m$ such that
\begin{equation}
f:=\pi'\circ \pi^{-1}=(f^1(z),\dots,f^m(z)), \ \ z\in D,
\end{equation}
where $f^j$ are distinct holomorphic functions in a neighborhood of
$z\in D\setminus S$. The set $S$ is called the {\it branch locus} of
$f$. We say that the correspondence $f$ {\it splits} at
$z\in {\overline D}$ if there is an open subset $U\ni z$ and
holomorphic maps $f^j:D\cap U\to D'$, $i=1,2,\dots,m$ that
represent $f$.
Given a proper holomorphic correspondence $A$, one can find the system
of canonical defining functions
\begin{equation}\label{e-canon}
\Phi_I(z,z') = \sum_{|J|\le m}\phi_{IJ}(z){z'}^J,\ \ |I|=m, \ \
(z,z')\in{\mathbb C}^n\times{\mathbb C}^n,
\end{equation}
where $\phi_{IJ}(z)$ are holomorphic on $D$, and $A$ is precisely
the set of common zeros of the functions $\Phi_I(z,z')$. For details
see e.g. \cite{c}.
We define analytic continuation of analytic sets as follows. Let $A$
be a locally complex analytic set in ${\mathbb C}^n$ of pure dimension
$p$. We say that $A$ {\it extends analytically} to an open set
$U\subset{\mathbb C}^n$ if there
exists a (closed) complex-analytic set $A^*$ in $U$ such that (i)
$\dim A^*=p$, (ii) $A\cap U\subset A^*$ and (iii) every irreducible
component of $A^*$ has a nonempty intersection with $A$ of dimension
$p$. Note that if conditions (i) and (ii) are satisfied, then the last
condition can always be met by removing certain components of $A^*$.
It follows from the uniqueness theorem that such analytic continuation
of $A$ is uniquely defined. From this we define analytic continuation
of holomorphic correspondences:
\begin{definition}
Let $U$ and $ U'$ be open sets in ${\mathbb C}^n$. Let $f:U\to
U'$ be a holomorphic correspondence, and let $A\subset U\times
U'$ be its graph. We say that $f$ extends as a holomorphic
correspondence to an open set $V$, $U\cap V\not=\varnothing$,
if there exists an open set
$V'\subset {\mathbb C}^n$ such that $A$ extends analytically to a
set $A^*\subset V\times V'$ and $\pi:A^*\to V$ is proper.
\end{definition}
Note that we can always choose $V'={\mathbb C}^n$ in the definition
above. In general, correspondence $g=\pi'\circ\pi^{-1}:V\to V'$,
where $\pi':A^*\to V'$ is the natural projection, may have more
branches in $U\cap V$ than $f$. The following lemma gives a simple criterion
for the extension to have the same number of branches.
\begin{lemma}\label{l-intersection}
Let $A^*\subset V\times{\mathbb C}^n$ be a holomorphic correspondence
which is an analytic extension of the correspondence $A\subset U\times
{\mathbb C}^n$. Suppose that for any $z\in (V\cap U)$,
\begin{equation}\label{pre}
\#\{\pi^{-1}(z)\}=\#\{\pi^{*-1}(z)\},
\end{equation}
where $\pi:A\to U$ and $\pi^*:A^*\to V$. Then $A\cup A^*$ is a
holomorphic correspondence in $(U\cup V)\times {\mathbb C}^n$.
\end{lemma}
\noindent{\it Proof.} We only need to check that $A\cup A^*$ is closed
in $(U\cup V)\times {\mathbb C}^n$. If not, then there exists a
sequence $\{q^j\}\subset A^*$ such that $q^j\to q^0$ as $j\to\infty$,
$q^0\in U\times {\mathbb C}^n$, and $q^0\not\in A$. Then $q^j\not\in A$
for $j$ sufficiently large. Since by the definition of analytic
continuation of correspondences
$A\cap (U\cap V)\times {\mathbb C}^n\subset A^*$, we have
$$
\#\{\pi^{-1}(\pi^*(q^j))\} < \#\{\pi^{*-1}(\pi^*(q^j))\}.
$$
But this contradicts (\ref{pre}). $\square$
\section{Extension along Segre Varieties.}
Before we prove the main results of the paper we need to establish a
technical result of local extension of holomorphic
correspondence along Segre varieties. This will be used on several
occasions in the proof of the main theorems.
\begin{lemma}\label{l:alongSV}
Let $\G\subset{\mathbb C}^n$ be a smooth, real-analytic, essentially
finite hypersurface, and let $\G'\subset{\mathbb C}^n$ be a smooth,
real-algebraic, essentially finite hypersurface. Let $0\in\G$, and let
$U_1$, $U_2$ be a sufficiently small standard pair of neighborhoods of
the origin. Let
$f:U\to{\mathbb C}^n$ be a germ of a holomorphic correspondence such
that $f(U\cap\G)\subset\G'$, where $U$ is some neighborhood of the
origin. Then there exists a neighborhood $V$ of $Q_0\cap U_1$ and an
analytic set $\L\subset V$, $\dim_{\mathbb C} \L\le n-1$, such that $f$
extends to $V\setminus \L$ as a holomorphic correspondence.
\end{lemma}
\noindent{\it Proof.} In the case when $\G'$ is strictly pseudoconvex
and $f$ is a germ of a biholomorphic mapping, the result was
established in \cite{s1}. Here we prove the lemma in a more general
situation.
Lemma \ref{l:alongSV} only makes sense if $U\subset U_1$. We shrink
$U$ and choose $V$ in such a way that for any $w\in V$, the set
$Q_w\cap U$ is non-empty and connected. Note that if $w\in Q_0$, then
$0\in Q_w$ and $Q_w\cap U\not=\varnothing$. Let $S\subset U$ be the branch
locus of $f$, and let
\begin{equation}
\S=\{z\in V: (Q_z\cap U)\subset S\}
\end{equation}
Since $\dim_{\mathbb C} S=n-1$ and $\G$ is essentially finite, $\S$
is a finite set. Define
\begin{equation}\label{e:A1}
A=\left\{ (w,w')\in (V\setminus\S)\times {\mathbb C}^n :
f\left(Q_w\cap U\right)\subset Q'_{w'} \right\}
\end{equation}
We establish the following facts about the set $A$:
\begin{list}{}{}
\item[(i)] $A$ is not empty
\item[(ii)] $A$ is locally complex analytic
\item[(iii)] $A$ is closed
\item[(iv)] $\S\times{\mathbb C}^n$ is a removable singularity for $A$.
\end{list}
(i) $A\not=\varnothing$ since by the invariance property of Segre
varieties, $A$ contains the graph of $f$.
(ii) Let $(w,w')\in A$. Consider
an open simply connected set $\O\in(U\setminus S)$ such that
$Q_w\cap\Omega\not=\varnothing$. Then the branches of $f$ are
correctly defined in $\O$. Since $Q_w\cap U$ is connected, the inclusion
$f(Q_w\cap U)\subset Q'_{w'}$ is equivalent to
\begin{equation}\label{e:omega1}
f^j(Q_w\cap\Omega)\subset Q'_{w'}, \ j=1,\dots,m,
\end{equation}
where $f^j$ denote the branches of $f$ in $\Omega$. Note that such
neighborhood $\Omega$ exists for any $w\in V\setminus\S$.
Inclusion (\ref{e:omega1}) can be written as a system of holomorphic equations as
follows. Let $\rho'(z,\overline z)$ be the defining function of $\G'$. Then
\begin{equation}
\rho'(f^j(z), \overline{w'})=0, \ {\rm\ for\ any}\ z\in (Q_w\cap
\Omega), \ \ j=1,2,\dots,m.
\end{equation}
We can choose $\Omega$ in the form
\begin{equation}
\O={'\O}\times{\O_n}\subset{\mathbb C}^{n-1}_{'z}\times{\mathbb C}_{z_n}
\end{equation}
Combining this with (\ref{e:implicit}) we obtain
\begin{equation}\label{e-system1}
\rho'(f^j({'z},h('z,\overline w)), \overline{w'})=0
\end{equation}
for any $'z\in {'\Omega}$. Then (\ref{e-system1}) is an infinite system of holomorphic
equations in $(w,w')$ thus defining $A$ as a locally complex analytic
variety in $(V\setminus\S)\times{\mathbb C}^n$.
(iii) Let us show now that $A$ is a closed set. Suppose that $(w^j, {w'}^j)\to(w^0,
{w'}^0)$, as $j\to\infty$, where $(w^j, {w'}^j)\in A$ and $(w^0,{w'}^0)\in
(V\setminus\S)\times{\mathbb C}^n$. Then by the definition of $A$,
$f(Q_{w^j}\cap U)\subset Q'_{{w'}^j}$. Since $Q_{w^j}\to
Q_{w^0}$, and $Q'_{{w'}^j}\to Q'_{w^0}$ as $j\to\infty$, by analyticity
$f(Q_{w^0}\cap U)\subset Q'_{{w'}^0}$, which implies that
$(w^0,{w'}^0)\in A$ and thus $A$ is a closed set. Since $A$ is locally
complex-analytic and closed, it is a complex variety in
$(V\setminus\S)\times{\mathbb C}^n$. We now may restrict considerations
only to the irreducible component of $A$ which coincides with the
graph of $f$ at the origin. Then $\dim A = n$.
(iv) Let us show now that $\overline A$ is a complex variety in $V\times{\mathbb
C}^n$. Let $q\in \S$, then
\begin{equation}
\overline {A}\cap(\{q\}\times{\mathbb C}^n) \subset \{q\}\times
\{z': f(Q_q)\subset Q'_{z'}\}.
\end{equation}
Notice that if $w'\in f(Q_q)\subset Q'_{z'}$, then $z'\in
Q'_{w'}$. Hence the set $\{z': f(Q_q)\subset Q'_{z'}\}$ has dimension
at most $2n-2$, and $\overline{A}\cap(\S\times{\mathbb C}^n)$ has
Hausdorff $2n$-measure zero. It follows from Bishop's theorem on
removable singularities of analytic sets (see e.g. \cite{c}) that
$\overline{A}$ is an analytic set in $V\times {\mathbb C}^n$.
Thus from (i) - (iv) we conclude that (\ref{e:A1}) defines a
complex-analytic set in $V\times {\mathbb C}^n$ which we denote again
by $A$. Also we observe that since $\G'$ is algebraic, the system of
holomorphic equations in (\ref{e-system1}) is algebraic in $w'$ and
thus we can define the closure of $A$ in $V\times {\mathbb P}^n$. Let
$\pi:A\to V$ and $\pi': A\to {\mathbb P}^n$ be the natural
projections. Since ${\mathbb P}^n$ is compact, $\pi^{-1}(K)$ is
compact for any compact set $K\subset V$, and thus $\pi$ is
proper.
This, in particular, implies that $\pi(A)=V$. We let
$\L_1=\pi({\pi'}^{-1}(H_0))$, where $H_0\subset{\mathbb P}^n$
is the hypersurface at infinity. It is easy to see that $\L_1$ is a
complex analytic set of dimension at most $n-1$. We also consider
the set
$\L_2:=\pi\{ (w,w')\in A : \dim_{\mathbb C} \pi^{-1}(w) \ge 1\}$.
It was shown in \cite{s1} Prop.~3.3, that $\L_2$ is a complex-analytic set of
dimension at most $n-2$. Let $\L=\L_1\cup\L_2$. Then $\pi'\circ
\pi^{-1}|_{V\setminus\L}$ is the desired extension of $f$ as a holomorphic
correspondence. $\square$
\section{Proof of Theorem \ref{t2}.}
For completeness let us repeat the argument of \cite{bsu} to prove the
``only if'' direction of Theorems \ref{t1} and \ref{t2}. Suppose that
$f:D\to D'$ is a proper holomorphic correspondence. Let us show
that $\partial D$ and $\partial D'$ are locally CR-equivalent.
If $D$ is not pseudoconvex, then for $p\in\widehat D$, there exists
a neighborhood $U\ni p$ such that all the functions in the
representation (\ref{e-canon}) of $f$ extend holomorphically to
$U$. Here $\widehat D$ refers to the envelope of holomorphy of $D$.
Moreover, we can replace $p$ by a nearby point $q\in
U\cap\partial D$ so that $f$ splits at $q$ and at least one of the
holomorphic mappings of the splitting is biholomorphic at $q$.
If $D$ is pseudoconvex, then $D'$ is also pseudoconvex. By \cite{bsu}
$f$ extends continuously to $\partial D$ and we can choose $p\in
\partial D$ such
that $f$ splits in some neighborhood $U\ni p$ to holomorphic mappings
$f^{j}: D\cap U\to D'$, $j=1,\dots,m$. Since $f^{-1}:D'\to D$
also extends continuously to $\partial D'$, the set $\{f^{-1}\circ
f(p)\}$ is finite. Therefore, by \cite{be3}, $f^j$ extend smoothly to
$\partial D\cap U$. It follows that $f^j$ extend holomorphically to a
neighborhood of $p$ by \cite{be} and \cite{df2}. Finally, choose $q\in
U\cap\partial D$ such that $f^j$ is biholomorphic at $q$ for some $j$.
To prove Theorem \ref{t2} in the other direction, consider a
neighborhood $U$ of $p\in\partial D$ and a biholomorphic map
$f:U\to{\mathbb C}^n$ such that $f(U\cap\partial D)\subset \partial
D'$. Let us show that $f$ extends to a proper holomorphic
correspondence $F:D\to D'$.
Let $\G=\partial D$ and $\G'=\partial D'$.
Since the set of Levi non-degenerate points is dense in $\G$, by
Webster's theorem \cite{w}, $f$ extends to an algebraic mapping,
i.e. the graph of $f$ is contained in an algebraic variety
$X\subset{\mathbb C}^n \times {\mathbb C}^n$ of dimension $n$. Without
loss of generality assume that $X$ is irreducible, as otherwise
consider only the irreducible component of $X$ containing $\G_f$, the
graph of the mapping $f$.
Let $E=\{ z\in \mathbb C^n : \dim \pi^{-1}(z)>0\}$, where $\pi:X\to
\mathbb C^n$ is the coordinate projection to the first component. Then
$E$ is an algebraic variety in $\mathbb C^n$. Let $f:{\mathbb C}^n
\setminus E \to {\mathbb C}^n$
now denote the multiple valued map corresponding to $X$. Let $S\subset{\mathbb
C}^n\setminus E$ be the branch locus of $f$, in other words, for any
$z\in S$ the coordinate projection onto the first component is not
locally biholomorphic near $\pi^{-1}(z)$. To prove Theorem \ref{t2} it is
enough to show that $E\cap \G=\varnothing$.
\begin{lemma}\label{l2.1}
Let $p\in\G$. If $Q_p\not\subset E$, then $p\not\in E$.
\end{lemma}
\noindent{\it Proof.} Suppose, on the contrary, that $p\in E$.
Since $Q_p\not\subset E$, there exist a point $b\in Q_p$ and
a small neighborhood $U_b$ of $b$ such that $U_b\cap E = \varnothing$.
Choose neighborhoods $U_b$ and $U_p$ such that for any $z\in U_p$, the
set $Q_z\cap U_b$ is non-empty and connected. Let
\begin{equation}
\S=\{z\subset U_p: Q_z\cap U_b\subset S\}.
\end{equation}
Similar to (\ref{e:A1}), consider the set
\begin{equation}
A=\left\{ (w,w')\in (U_p\setminus\S)\times {\mathbb C}^n :
f\left(Q_w\cap U_b\right)\subset Q'_{w'} \right\}.
\end{equation}
Then $A\not=\varnothing$. Indeed, since $\dim_{\mathbb C} E\le n-1$,
there exists a sequence of points
$\{p^j\}\subset (U_p\cap\G)\setminus (E\cup\S)$
such that $p^j\to p$ as $j\to\infty$. By the invariance property
of holomorphic correspondences, for every $p^j$ there exists a
neighborhood $U_j\ni p^j$ such that $f(Q_{p^j}\cap U_j)\subset
Q'_{{p'}^j}$, where ${p'}^j\in f(p^j)$. But this implies that
$f(Q_{p^j}\cap U_b)\subset Q'_{{p'}^j}$, and therefore
$(p^j,{p'}^j)\in A$. Moreover, it follows that
\begin{equation}\label{e:A=X}
A|_{U_{j}\times{\mathbb C}^n}=X|_{U_{j}\times{\mathbb C}^n}, \ \
j=1,2,\dots,m.
\end{equation}
Similar to the proof of Lemma \ref{l:alongSV}, one can show that $A$ is
a complex analytic variety in $(U_p\setminus\S)\times{\mathbb C}^n$,
and that $\S\times{\mathbb C}^n$ is a removable singularity for
$A\subset U_p\times{\mathbb C}^n$. Denote the closure of $A$
in $U_p\times{\mathbb C}^n$ again by $A$.
Without loss of generality we assume that $A$ is irreducible,
therefore in view of (\ref{e:A=X}) we conclude that
$A|_{U_{p}\times{\mathbb C}^n}=X|_{U_{p}\times{\mathbb C}^n}$.
Let $\hat f$ be a multiple valued mapping corresponding to $A$. Then
by analyticity, there exists $p'\in\G'\cap \hat f(p)$. Moreover, by
construction, $\hat f(p) = I'_{p'}$. By \cite{dw},
\begin{equation}\label{e-inG}
I'_{z'}\subset\G', \ \ {\rm for\ any\ } z'\in\G'.
\end{equation}
Now choose $U_p$ so small
that $\overline A \cap (U_p \times \partial U') =\varnothing$, where
$U'$ is a neighborhood of $\G'$.
This is always possible, since otherwise there exists a
sequence of points $\{(z^j, {z'}^j), j=1,2,\dots\}\subset A$, such that $z^j\to p$ and
${z'}^j\to {z'}^0 \in\partial U'$ as $j\to\infty$. Then $(p,{z'}^0)\in A$ and
${z'}^0\not\in \G'$. But this contradicts (\ref{e-inG}).
This shows that $\hat f:U_p\to U'$ is a holomorphic correspondence
extending $f$, which contradicts the assumption $p\in E$. $\square$
\begin{lemma}\label{l2.2}
Let $p\in\G$. Then there exists a change of variables, which is
biholomorphic near $\overline {D'}$, such that in the new coordinate system
$Q_p\not\subset E$.
\end{lemma}
{\it Proof.} Suppose that $Q_p\subset E$. Then we find a point
$a\in(\G\setminus E)$ such that $Q_a\cap Q_p\not=\varnothing$. The
existence of such $a$ follows, for example, from \cite{s1}
Prop~4.1. (Note, that $\dim E\cap \G \le 2n-3$). By Lemma
\ref{l:alongSV} the germ of the correspondence $f$
defined at $a$, extends holomorphically to a neighborhood $V$ of
$Q_a$. Let $\L_1$ and $\L_2$ be as in Lemma~\ref{l:alongSV}.
Since $\dim \L_2<n-1$, we may assume that
$(Q_p\cap V)\not\subset \L_2$. If $(Q_p\cap V)\subset\L_1$, we can
perform a linear-fractional transformation such that $H_0$ is mapped
onto another complex hyperplane $H\subset{\mathbb P}^n$ and such that
$H\cap\G'=\varnothing$. Note that after such transformation $\G'$
remains compact in ${\mathbb C}^n$. Then we may also assume that
$(Q_p\cap V)\not\subset \L_1$. Thus holomorphic extension along $Q_a$
defines $f$ on a non-empty set in $Q_p$. $\square$
Theorem \ref{t2} now follows. Indeed, from Lemmas \ref{l2.1} and
\ref{l2.2} we conclude that $E\cap\G=\varnothing$. Since $D$ is
bounded, $D\cap E=\varnothing$, and $X\cap(D\times D')$ defines a
proper holomorphic correspondence from $D$ to $D'$.
\section{Local Extension.}
To prove Theorem \ref{t1} we first establish local extension of
holomorphic correspondences.
\begin{definition}
Let $\G$ and $\G'$ be smooth, real-analytic hypersurfaces in ${\mathbb
C}^n$. Let $f: U\to \mathbb C^n$ be a holomorphic correspondence such that
$f(U\cap\G)\subset \G'$. Then $f$ is called {\it complete} if for any
$z\in U\cap\G$, $f(Q_z\cap U)\subset Q'_{z'}$ and $f(z)=I_{z'}$.
\end{definition}
By the invariance property of Segre varieties, $f(_{z}Q_z)\subset
Q_{f(z)}$, where $_{z}Q_z$ is the germ of $Q_z$ at $z$, for any $z\in
U\cap \G$. The condition $f(Q_z\cap U)\subset Q'_{z'}$ in the
definition is somewhat stronger: it states that every connected
component of $Q_z\cap U$ is mapped by $f$ into the same Segre
variety. Note that in general Segre varieties are defined only
locally, while the set $U$ can be relatively large. In this case the
inclusion $f(Q_z\cap U)\subset Q'_{z'}$ should be considered only in a
suitable standard pair of neighborhoods of $z$.
The condition $f(z)=I_{z'}$ in the definition above indicates that $f$
has the maximal possible number of branches. It is convenient to
establish analytic continuation of complete correspondences, as such
continuation does not introduce additional branches.
\begin{lemma}\label{l3.2}\label{l:sublocal}
Let $f:U\to {\mathbb C}^n$ be a complete holomorphic correspondence,
$f(\G\cap U)\subset \G'$, where $\G=\partial D$ and $\G'=\partial D'$,
$D$ and $D'$ are as in Theorem \ref{t1}. Suppose $p\in \partial U\cap\G$ is
such that $Q_p\cap U\ne\varnothing$. Then there exists a neighborhood
$U_p$ of $p$ such that $f$ extends to a holomorphic correspondence
$\hat f: U_p\to {\mathbb C}^n$.
\end{lemma}
\noindent{\it Proof.} The proof of this lemma repeats that of Lemma \ref{l2.1}.
Let $b\in Q_p\cap U$. Consider a small neighborhood $U_b$ of $b$, $U_b\subset U$,
and a neighborhood $U_p$ of $p$ such that for any $z\in U_b$, the set $Q_z\cap
U_p$ is non-empty and connected. As before, let $S\subset U$ be the branch
locus of $f$, and $\S=\{z\subset U_p: Q_z\cap U_b\subset S\}$.
Define
\begin{equation}\label{e:A}
A=\left\{ (w,w')\in (U_p\setminus\S)\times {\mathbb C}^n :
f\left(Q_w\cap U_b\right)\subset Q'_{w'} \right\}.
\end{equation}
Observe that since $f$ is complete, for any $w\in U \cap\G$, the inclusion
$f(Q_w\cap U_b)\subset Q'_{w'}$ implies that $f(Q_w\cap U)\subset
Q'_{w'}$. In particular, this holds for any $w$ arbitrary close to $p$.
Therefore $A$ is
well-defined if the neighborhood $U_p$ is chosen sufficiently small.
Analogously to Lemma \ref{l2.1}, one can show that $A$ is a non-empty
closed complex analytic set in $(U_p\setminus\S)\times{\mathbb C}^n$.
Similar argument also shows that
$\S\times{\mathbb C}^n$ is a removable singularity for $A$, and thus
$\overline A$ defines a closed-complex analytic set in
$U_p\times{\mathbb C}^n$.
Let us show now that $A$ defines a holomorphic correspondence
$\hat f:U_p\to U'$, where $U'$ is a suitable neighborhood of
$\G'$.
Consider the closure of $A$ in $U_p\times {\mathbb P}^n$. Recall, that
since ${\mathbb P}^n$ is compact, the projection $\pi:\overline A \to
U_p$ is proper. In particular, $\pi (\overline A)= U_p$.
Let $U'$ be a neighborhood of $\G'$ as in Lemma
\ref{l-finite}. To simplify the notation, denote the restriction of
$\overline A$ to $U_p \times U'$ again by $A$. Let $\pi: A\to U_p$ and
$\pi': A\to U'$ be the natural projections, and let $\hat f=\pi'\circ
\pi$.
Let $Z={\pi'}^{-1}(\G')$. Then since $f(\G\cap U)\subset \G'$,
$\pi^{-1}(\G\cap U\cap U_p)\subset Z$. Therefore there exists at least
one irreducible component of $\pi^{-1}(\G\cap U_p)$ which is contained in
$Z$. Thus for any $z\in\G\cap U_p$, there exists $z'\in\G'$ such that
$z'\in \hat f(z)$. By construction, if $z\in U_p$ and $z'\in \hat
f(z)$, then $\hat f(z) = I'_{z'}$. In view of (\ref{e-inG}) we
conclude that $\hat f(\G\cap U_p)\subset \G'$. Now the same argument
as in Lemma \ref{l2.1} shows that $U_p$ can be chosen so small that
$\hat f$ is a holomorphic correspondence. $\square$
\begin{theorem}[Local extension] \label{t:local}
Let $D$ and $D'$ be as in Theorem \ref{t1}, $\G=\partial D$ and
$\G'=\partial D'$. Let
$f:U\to {\mathbb C}^n$ be a complete holomorphic correspondence,
such that $f(\G\cap U)\subset \G'$, where $\G\cap U$ is connected, and
$\G\cap \partial U$ is a smooth submanifold. Let $p\in\partial U\cap \G$. Then
there exists a neighborhood $U_p$ of the point $p$ such that $f$
extends to a holomorphic correspondence $\hat f:U_p\to {\mathbb
C}^n$. Moreover, $\hat f|_{U\cap U_p} = f|_{U\cap U_p}$, and the
resulting correspondence $F:U\cup U_p \to {\mathbb C}^n$ is
complete.
\end{theorem}
\noindent{\it Proof.}
We call a point $p\in\partial U\cap \G$ {\it regular}, if $\partial
U\cap \G$ is a generic submanifold of $\G$, i.e.
$T^c_p(\partial U\cap\G)=n-2$. We prove the theorem in three
steps. First we prove the result under the assumption $Q_p\cap
U\not=\varnothing$, then for regular points in $\partial U\cap\G$, and
finally for arbitrary $p\in \partial U\cap\G$.
{\it Step 1.} Suppose that $Q_p\cap U\not=\varnothing$.
Then by Lemma \ref{l:sublocal}
$f$ extends as a holomorphic correspondence $\hat f$ to some
neighborhood $U_p$ of $p$. It follows from the construction that for any
$z\in U\cap U_p$ the number of preimages of $f(z)$ and $\hat f(z)$ is
the same. Thus by Lemma~\ref{l-intersection}, $f$ and $\hat f$ define a
holomorphic correspondence in $U\cup U_p$. Denote this correspondence
by $F$.
We now show that $F$ is also complete in $U\cup U_p$.
Since $f$ is complete, for any $z\in U\cap U_p\cap\G$, arbitrarily close to $p$,
$f(Q_z\cap U)\subset Q'_{f(z)}$. Thus if $U_p$ is chosen sufficiently
small, then for any $z\in U_p\cap\G$,
\begin{equation}
F(Q_z\cap(U\cup U_p))\subset Q'_{F(z)}.
\end{equation}
Suppose now that there exists some
point $z$ in $(U\cup U_p)\cap\G$ such that not all components of
$Q_z\cap (U\cup U_p)$ are mapped by $F$ into the same Segre
variety. From the argument above, $z\notin U_p$. Since $U\cap\G$ is
connected, there exists a simple smooth curve $\gamma\subset \G\cap U$
connecting $z$ and $p$. By Lemma \ref{l:alongSV} for every point
$\zeta\in\g$, the germ of a correspondence $F$ at $\zeta$ extends as a
holomorphic correspondence along the Segre variety $Q_\zeta$.
Moreover, for $\zeta\in\g$ which are close to $p$, the extension of $F$
along $Q_\zeta$ coincides with the correspondence $f$ in $U$
(even if $Q_\zeta\cap U$ is disconnected). Since
$\cup_{\zeta\in\g}Q_\zeta$ is connected, this property holds for all
$\zeta\in\g$. The extension of $F$ along $Q_\zeta$ clearly maps $Q_\zeta$
into $Q'_{F(\zeta)}$, and therefore $Q_\zeta\cap U$ is mapped by $f$ into the same
Segre variety. But this contradicts the assumption that the components
of $Q_z\cap U$ are mapped into different Segre varieties. This shows
that $F$ is also a complete correspondence.
{\it Step 2.}
Suppose now that $Q_p\cap U=\varnothing$, but $p$ is a regular point.
Then by \cite{s1} Prop. 4.1, there exists a point $a\in U$ such that
$Q_a\cap Q_p\not=\varnothing$. We now apply Lemma \ref{l:alongSV} to
extend the germ of the correspondence at $a$ along $Q_a$. We note that
such extension along $Q_a$ may not in general define a complete
correspondence, since apriori $Q_a\cap \G$ may be disconnected from $U\cap\G$.
Let $\L$ be as in Lemma \ref{l:alongSV}. Then after performing, if
necessary, a linear-fractional transformation in the target space, we
can find a point $b\in Q_p\cap Q_a$, such that $b\notin \L$. Let $U_b$
be a small neighborhood of $b$ such that $U_b\cap\L=\varnothing$ and
$f$ extends to $U_b$ as a holomorphic correspondence $f_b$. Then for any
$z\in U\cap\G$ such that $Q_z\cap U_b\not=\varnothing$, the sets
$f(Q_z\cap U)$ and $f_b(Q_z\cap U_b)$ are contained in the same Segre
variety. Indeed, if not, then we can connect $a$ and $z$ by a smooth
path $\g\subset\G\cap U$ and apply the argument that we used to prove
completeness of $F$ in Step 1.
Now the same proof as in Step 1 shows that $f$ extends as a
holomorphic correspondence to some neighborhood of $p$, and that the
resulting extension is also complete.
{\it Step 3.} Suppose now that $p\in \partial U\cap \G$ is not a
regular point. Let
\begin{equation}
M=\left\{z\in\partial U\cap\G: T_z(\partial U\cap\G)=T^c_z(\G)\right\}.
\end{equation}
It is easy to see
that $M$ is a locally real-analytic subset of $\G$. Moreover, since
$\G$ is essentially finite, $\dim M < 2n-2$. Choose the coordinate
system such that $p=0$ and the defining function of $\G$ is given
in the so-called normal form (see \cite{cm}):
\begin{equation}
\rho(z,\overline z)= 2x_n+\sum_{|k|,|l|\ge
1}\rho_{k,l}(y_n)('z)^k(\overline{'z})^l,
\end{equation}
where $'z=(z_1,\dots,z_{n-1})$. Since the extendability of $f$
through regular points is already established, after possibly an
additional change of variables,
we may assume that $f$ extends as a holomorphic
correspondence to the points $\{z\in\partial U\cap\G: x_1>0\}$. Let
$L_c$ denote the family of real hyperplanes in the form $\{z\in
{\mathbb C}^n : x_1=c\}$. Then there exists $\epsilon>0$ such for any
$c\in[-\epsilon,\epsilon]$,
\begin{equation}\label{good-c}
T^c_z(\G) \not= T_z(L_c\cap\G), {\rm \ \ for\ any\ } z\in
L_c\cap\G\cap {\mathbb B}(0,\epsilon).
\end{equation}
Let $\Omega_{c,\delta}$ be the intersection of $\G$, the
$\delta$-neighborhood of $x_1$-axis and the set bounded by $L_c$ and
$L_{c+\delta}$, that is
\begin{equation}
\Omega_{c,\delta}=\left\{
z\in\G\cap {\mathbb B}(0,\epsilon) :
c<x_1<c+\delta, \ y_1^2+\sum_{j=2}^n|z_j|^2<\delta
\right\}.
\end{equation}
Then there exist $\delta>0$ and $c>0$, such that $f$ extends as a
holomorphic correspondence to a neighborhood of the set
$\Omega_{c,\delta}$. Since $L_c\cap\G$ consists only of regular
points, from Steps 1 and 2 we conclude that $f$ extends to a
neighborhood of any point in $L_c\cap\G$ that belongs to the boundary
of $\Omega_{c,\delta}$. Let $c_0$ be the smallest number such that $f$
extends past $L_c\cap\G$. Then from (\ref{good-c}) and previous steps,
$c_0<0$, and therefore, $f$ extends to a neighborhood of the
origin. $\square$
\section{Proof of Theorem \ref{t1}.}
The proof of the local equivalence of boundaries $\partial D$ and
$\partial D'$ is equivalent to that of Theorem \ref{t2}.
To prove the theorem in the other direction let us first show that a germ
of a biholomorphic map $f:U\to {\mathbb C}^n$, $f(U\cap\G)\subset\G'$
can be replaced by a complete correspondence. Without loss of
generality we assume that $0\in U\cap \G$. We choose a neighborhood
$U_0$ of the origin and shrink $U$ in such a way, that $Q_w\cap U$ is
non-empty and connected for any $w\in U_0$. Define
\begin{equation}\label{comp}
A=\left\{
(w,w')\in U_0\times{\mathbb C}^n: f(Q_w\cap U)\subset Q'_{w'}
\right\}.
\end{equation}
Then (\ref{comp}) defines a holomorphic correspondence, which in particular
contains the germ of the graph of $f$ at the origin. Let $\hat f$ be
the multiple valued mapping corresponding to $A$. Then by construction and
from (\ref{e-inG}), $w'\in\hat f(w)$ implies $\hat f(w)=I'_{w'}$. Thus
$\hat f$ is a complete correspondence.
If $\partial U_0$ is smooth, then By Theorem \ref{t:local} we can
locally extend $\hat f$ along $\G$ past the boundary of
$U_0\cap\G$ to a larger open set $\Omega$. However, local extension in general does
not imply that $A$ is a closed set in $\Omega\times{\mathbb C}^n$.
Indeed, there may exist a point $p\in \G\cap\partial \Omega$ such
that for any sufficiently small neighborhood
$V$ of $p$, $\Omega\cap V\cap\G$ consists of two connected components, say
$\G_1$ and $\G_2$. Then local extension from $\G_1$ to a neighborhood
of $p$ may not coincide with the correspondence $\hat f$ defined in
$\G_2$. Therefore, local extension past the boundary of $\Omega\cap\G$
does not lead to a correspondence defined globally in a neighborhood of
$\overline\Omega\times \mathbb C^n$.
Note that this cannot happen if $\Omega$ is sufficiently small.
We now show that $\hat f$ extends analytically along any path on $\G$.
\begin{lemma}\label{along-paths}
Let $\g:[0,1]\to\G$ be a simple curve without self-intersections, and
$\g(0)=0$. Then there exist a neighborhood $V$ of $\g$ and a
holomorphic correspondence $F:V\to\mathbb C^n$ which extends $\hat f$.
\end{lemma}
\noindent{\it Proof.} Suppose that $\hat f$ does not extend along
$\g$. Then let $\zeta\in\gamma$ be the first point to which
$\hat f$ does not extend. Let $\epsilon_0>0$ be so small that $\mathbb
B(\zeta,\epsilon)\cap\G$ is connected and simply connected for any
$\epsilon \le \epsilon_0$. Choose a point $z\in B(\zeta,\epsilon_0/2)\cap
\g$ to which $\hat f$ extends. Let $\delta$ be the largest positive
number such that $\hat f$ extends holomorphically to $\mathbb B(z,\delta)\cap\G$.
By Theorem \ref{t:local} $\hat f$ extends to a
neighborhood of every point in $\partial\mathbb
B(z,\delta)\cap\G$. Moreover, if $\mathbb B(z,\delta)\subset \mathbb
B(\zeta,\epsilon_0)$, then the extension of $\hat f$ is a closed
complex analytic set. Thus $\delta>\epsilon/2$. This shows that $\hat
f$ also extends to $\zeta$, and therefore extends along $\g$. $\square$
Note that analytic continuation of $\hat f$ along $\g$ in Lemma
\ref{along-paths} always yields a complete correspondence.
The Monodromy theorem cannot be directly applied for multiple valued
mappings, and we need to show that analytic continuation is
independent of the choice of a curve connecting two points on $\G$.
\begin{lemma}
Suppose that $\g\subset\G$ is a Jordan curve $\g(0)=\g(1)=0$.
Let $F$ be the holomorphic correspondence defined near the origin and
obtained by analytic continuation of $\hat f$ along $\g$. Then $F=\hat f$ in some
neighborhood of the origin.
\end{lemma}
\noindent{\it Proof.}
Since $\G$ is simply connected and compact, there exists $\epsilon_0 >0$
such that for any $z\in\G$, $\mathbb B(z,\epsilon)\cap\G$ is connected
and simply connected for any $\epsilon\le\epsilon_0$.
Let $\phi$ be the homotopy map, that is
$\phi(t,\tau):I\times I\to\G$, $\phi(t,0)=\g(t)$, $\phi(t,1)\equiv 0$,
$I=[0,1]$. Let
$\{(t_j,\tau_k)\in I\times I$, $j,k=0,1,2,\dots,m\}$ be the set of
points satisfying:
\begin{enumerate}
\item[(i)] $t_0=\tau_0=0$, $t_m=\tau_m=1$,
\item[(ii)]
$\{\phi(t,\tau_k): t_j\le t \le t_{j+1}\} \subset
\mathbb B(\phi(t_{j},\tau_{k}),\epsilon_0/2)$, \\
$\{\phi(t_{j},\tau): \tau_k \le \tau \le \tau_{k+1}\} \subset
\mathbb B(\phi(t_{j},\tau_{k}),\epsilon_0/2)$,
for any $j,k<m$.
\end{enumerate}
Suppose that $f$ is a complete holomorphic correspondence
defined in a ball $B$ of small radius centered at
$\phi(tj,\tau_k)\in\G$. By Theorem \ref{t:local}, $f$ extends
holomorphically past every boundary point of $\partial B\cap\G$. Since
$B(\phi(t_j,\tau_k),\epsilon_0)$ is connected and simply connected, $f$
extends at least to a ball of radius $\epsilon_0/2$. Consider the closed path
$\g_{j,k}=\{\phi(t,\tau_k): t_j\le t \le t_{j+1}\}\cup \{\phi(t_{j+1},\tau):
\tau_k\le \tau \le \tau_{k+1}\}\cup
\{\phi(t,\tau_{k+1}): t_j\le t \le t_{j+1}\}\cup \{\phi(t_{j},\tau):
\tau_k\le \tau\le \tau_{k+1}\},$
where the second and fourth pieces are traversed in the opposite
direction. Then $\g_{j,k}$ is entirely contained in $\mathbb
(B(\phi(tj,\tau_k),\epsilon_0/2)$. Therefore, analytic continuation of $f$
along $\g_{j,k}$ defines the same correspondence at $\phi(t_j,\tau_k)$.
Analytic continuation of $\hat f$ along $\g$ can be reduced to
continuation along paths $\g_{j,k}$. Since continuation
along each path $\g_{j,k}$ does not introduce new branches of $\hat
f$, $F=\hat f$. $\square$
Thus simple connectivity of $\G$ implies that the process of local
extension of $\hat f$ leads to a global extension of $\hat f$ to some
neighborhood of $\G$. Since $\hat f(\G)\subset \G'$, there exist
neighborhoods $U$ of $\G$ and $U'$ of $\G'$ such that $\hat f: U\to U'$ is a proper
holomorphic correspondence. Let $A$ be the analytic set corresponding to
$\hat f$. By (\ref{e-canon}) there exist functions
$\phi_{IJ}$ holomorphic in $U$ such that $A$ is determined from the
system $\sum_{|J|\le m}\phi_{IJ}(z){z'}^J=0$. By Hartog's theorem all
$\phi_{IJ}$ extend holomorphically to a neighborhood of $\overline D$,
(recall that $\G=\partial D$). This defines a proper holomorphic
correspondence $f:D\to D'$. $\square$
\begin{small}
\end{small}
\end{document} |
\begin{document}
\newtheorem{de}{Definition}
\newtheorem{ex}[de]{\emph{Example}}
\newtheorem{thm}[de]{Theorem}
\newtheorem{lemma}[de]{Lemma}
\newtheorem{cor}[de]{Corollary}
\newtheorem{con}[de]{Conjecture}
\newtheorem{prop}[de]{Proposition}
\title{An upper bound of the numbers of minimally intersecting filling coherent pairs}
\author{Hong Chang}
\address{
\noindent
Hong Chang, [email protected], Department of Mathematics, University at Buffalo--SUNY}
\keywords{curve graph, origami, coherent pair, origami pair of curves.}
\begin{comment}
It's known that a pair of coherent minimal filling pairs $\alpha$ and $\beta$ will form an $1-1-$origami. With $\alpha$, $\beta$ and another curve $\gamma$ in standard position, $\gamma$ will avoid triple points or singularities (i.e. vertices of the squares). So there are following case:
A real crossing create intersections with $\beta$ while a fake crossing does not. Note that a non-coherent fake crossing is actually nugatory. Last semester I actually proved if $\alpha$ and $\gamma$ also coherent and minimal filling, $\beta$ has to intersect with $\gamma$ and thus have at least one real crossing.
We can confine a real crossing will not pass two or more sections of curve $\beta$, that is actually to say for all segments $y\in\gamma\backslash\alpha$ we have $i(y,\beta)<2$. And notice this condition is very similar to ``initial efficient'' . It's actually not since $\alpha$ and $\beta$ does intersect.
Question: if $\alpha$ and $\gamma$ are also coherent and minimal filling, can we make it ``initial efficient''?
If $\alpha$ and $\gamma$ are also coherent and minimal filling, then the $i(\alpha,\gamma)$ is the same as the number of the squares. And the complement of $\alpha$ and $\gamma$ could have only one component.
See the picture for an example of $i(\alpha,\gamma)$ for genus 5 surface. According to my program, it's very interesting that 1 intersection is impossible, while $2-9$ are all possible (similar happens for genus 7).
\begin{prop}
$i(\alpha,\gamma)>1$
\end{prop}
In fact, in my proof of $i(\alpha,\gamma)>0$, either a square upside or downside would cause a contradict. If there are only one crossing then there are still problem for either side.
\end{comment}
\begin{abstract}
Let $S_g$ denoting the genus $g$ closed orientable surface. An {\em origami} (or flat structure) on $S_g$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one.
Coherent filling pairs of simple closed curves, $(\alpha,\beta)$ in $S_g$ are pairs for which their minimal intersection is equal to their algebraic intersection. And, a minimally intersecting filling of $(\alpha,\beta)$ in $S_g$ is a pair whose intersection number is the minimal among all filling pairs of $S_g$.
A coherent pair of curves is naturally associated with an origami on $S_g$, and a minimally intersecting filling coherent pair of curves has the smallest number of squares in all origamis on $S_g$.
Our main result introduces an algorithm to count the numbers of minimal filling pairs on $S_g$, and establish a new upper bound of this count using M\'enage Problem by \'Edouard Lucas in \cite{lucas}.
\end{abstract}
\maketitle
\section{Introduction}
Let $S_g$ denote the closed orientable surface of genus $g$. A finite collection of curves $\Gamma = {\gamma_1, ...\gamma_n}$ in
pairwise minimal position on $S_g$ is said to fill if the complement $S_g\backslash\gamma$ is a disjoint union of topological
disks. When each $\gamma_i$ is simple---no self intersections---and when n = 2, we call $\Gamma$ a filling pair.
Let $\alpha$ and $\beta$ be a filling pair on $S_g$. The pair, $(\alpha, \beta)$, is a \emph{minimally intersecting} filling pair if the intersecting number $i(\alpha,\beta)$ is minimal among all filling pairs on $S_g$.
Aougab and Huang showed in \cite{AMN} that for all $g >2$ there exists filling pairs of curves whose intersection achieves the $2g-1$ minima, and when $g=2$, the minima is 4. In fact, the minima can be obtained such that the absolute value of the algebraic intersection of $\alpha$ and $\beta$ is equal to the geometric intersection---$\alpha$ and $\beta$ \emph{intersect coherently}. So, \emph{minimally intersecting coherent filling pairs} is a non-empty category for all $S_{g\ge 2}$.
Let $u$ and $v$ be the isotopy classes of $\alpha$ and $\beta$, and let $\alpha$ and $\beta$ be the minimal intersection number over all isotopic
representatives. We will abuse notation by having $i(u,v)=i(\alpha,\beta)$. For convenience, we will also say $(u,v)$ is a \emph{filling pair} when $(\alpha, \beta)$ is a filling pair. In this paper, when counting the number of filling pairs, we see $(u,v)$ as an ordered pair of classes. Necessarily, curves and classes have no orientation.
The \emph{mapping} class group of S, denoted $Mod(S)$, is the group of isotopy classes of orientation preserving self-homeomorphisms of S. The \emph{extended mapping class group} of S, denoted $Mod^{\pm}(S)$, is the group of isotopy classes of self-homeomorphisms of S, including the orientation-reversing ones. We say $\alpha$, $\beta$ and $\alpha'$, $\beta'$ are two filling pairs in the same $Mod(S)$ (or $Mod^{\pm}(S)$) orbit if there exist $g\in Mod(S)$ (or $Mod^{\pm}(S)$) such that $g(\alpha)=\alpha'$ and $g(\beta)=\beta'$.
Before we go to the bounds, we give a definition for asymptotic notation ``$\sim''$:
\begin{de}
If $f_n$ and $g_n$ are sequences of real valued function and $g_i\not=0$ for all $i$. We call $f_n\sim\ g_n$ if $\underset{n\rightarrow\infty}{\lim}f_n/g_n=1$.
\end{de}
Use $\mathcal{MI}$ for the set of all $Mod(S)$ orbit of minimally intersecting pairs. Then the size of the set is $$f(g)\leq |\mathcal{MI}| \leq 2^{2g-2}(4g-5)(2g-3)!$$Where $f(g)\sim 3^{g/2}/g^2$. This result is provided in Aoubag--Huang \cite{AH}.
Use $\mathcal{CMI}$ for $Mod^{\pm}(S)$ orbit of coherently minimally intersecting pairs. Then the size of the set is $$p(g)\leq |\mathcal{CMI}| \leq h(g)$$Where $p(g)=\begin{cases} (g-2)! & \text{ if } g>2 \text{ odd} \\(g-5)(g-3)! & \text{ if } g>2 \text{ even} \end{cases}$ and $h(g)\sim\frac{(g-1)(2g-2)!}{e^2}$. The left side is provided in \cite{AMN} and the right side will be proved in this paper.
\begin{thm} \label{main}
Let $\mathcal{CMI}$ be $Mod^{\pm}(S)$ orbit of coherently minimally intersecting pairs on $S_g$, then $|\mathcal{CMI}| \leq h(g)$ where $$h(g)\sim\frac{(g-1)(2g-2)!}{e^2}$$.
\end{thm}
\subsection{Origamis}
An \emph{origami}, or a square-tiled surface for a closed surface $S_{g \geq 2}$ is obtained from a finite number of Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. A \emph{$[1,1]$-origami} will be an origami that has exactly one horizontal annulus and one vertical annulus. In particular, we have the following well known result associating a $[1,1]$-origami to an origami pair of curves.
\begin{thm}[Theorem 1.1 of \cite{CJM}]
A coherent filling pair of curves (origami pair of curves) naturally corresponds to an origami on $S_g$.
\end{thm}
As mentioned in \cite{AMN}, an abelian differential on a surface of genus $g > 1$ must have $2g - 2$ zeros, counted with multiplicity. In the moduli space of abelian differentials we denote the stratum $H(m_1, ..., m_n)$ of the moduli space consists of those abelian differentials which have $n$ zeros of degrees $m_1+...+m_n=2g-2$.
Eskin-Okounkov \cite{Esk} and Zorich \cite{Zor} are using square-tiled surfaces (origamis) to calculate the volume of a stratum of abelian differentials. And for this reason, counting the number of origamis has connections to Teichm\"uller dynamics and to the study of flat surfaces. In this fashion of origamis, Theorem \ref{main} can be also expressed as:
\emph{In the minimal stratum $H(2g-2)$, there exist at most $h(g)$ $[1,1]$-origamis.}
\subsection{Construction of surface from polygons}
Since $\alpha$ and $\beta$ is minimally filling, when $g\ge 3$, $S_g\backslash\alpha\cup\beta$ will consist of only one component, which is a $4(2g-1)$-gon with 2 copies of $\alpha$ and 2 copies of $\beta$ as the sides.
\begin{figure}
\caption{Coherent intersecting sides of $4(2g-1)$-gons, where $g$ is odd(left) and even(right). The red is $\alpha$ while green is $\beta$.}
\label{gon}
\end{figure}
Please notice the edges for the $4(2g-1)$-gon are one-to-one correspond with subarcs of $\alpha$ and $\beta$, thus the horizontal and vertical lines of the origami.
\subsection{M\'enage Problem}
M\'enage Problem is asked by \`Edouard Lucas in \cite{lucas} as ``the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner'', or by Peter Tait in \cite{tait} as ``numbers of arrangements such that there are of n letters, when A cannot be in the first or second place, B not in the second or third'' (such arrangements can be regarded as permutations called \emph{m\'enage permutation}). The first one will be equivalent to the second one if we already seat, say wives, on the table, which there are $2n!$ ways to do that.
Gilbert, E. N. modified Tait's problem further in \cite{gen}. Let $P$ be a m\'enage permutation and $C$ a cyclic permutation $i\rightarrow i+a$ (mod $n$), and let $P'=C^{-1}PC$, then $P'$ is another m\'enage permutation and we call $P\sim P'$. In fact, $P'$ is just a relabel of $P$ where we replace $i$ with $i+a$. In the modified problem, we counted the numbers of equivalence classes from the construction above.
For additional reading on the progression of the number of m\'enage permutations, we refer the reader to \cite{oeis}.
In \S \ref{section: filling pairs and origami}, we show how m\'enage problem, especially the modified one, is associated with the problem constructing surfaces by identifying the edges of the polygon.
\subsection{Outline}
In \S \ref{section: filling pairs and origami}, we discuss some fundamental properties for a minimally intersecting filling coherent pair. In section 3, we give an explicit algorithm to construct such pairs on $S_g$. In \S \ref{section: filling pairs and origami}, we give an upper bound for the number using the construction from \S \ref{section: example}. In \S \ref{section: searching}, we give an alternative algorithm to search for such pairs with a program. And finally in section \S \ref{section: genus 2}, we discuss the case with $g=2$, where we will have a minimal of 4 intersections.
\section{Minimal filling pairs and origami}
\label{section: filling pairs and origami}
Suppose further $\alpha$ and $\beta$ intersect coherently, then the sides will also be coherent, and it's a $1-1$ origami with $2g-1$ squares.
\begin{prop} \label{oppo}
There exists a side of $\alpha$ (or $\beta$) such that it's not identified with the opposite.
\end{prop}
Now the opposite sides are in the different direction, so if every side of $\alpha$ is identified with the opposite, we have two pairs of $\alpha$: $a_1,a_1'$ and $a_2, a_2'$ adjacent to each other in the origami (note it's not necessary in the polygon), sharing the same section of $\beta$, with $b, b'$ as the image of that arc in the sides of the polygon. Then $b$ and $b'$ is in the right (or left) of $a_1$ and $a_2$. Let $a_3$ be
the other side of $b$ and $a_4$ be the other side of $b'$. Then since $b$ and $b'$ is actually the same in the origami, $a_3$ and $a_4$ has to be adjacent to each other. (See figure above) Since every side of $\alpha$ is identified with the opposite, we are actually playing the marking game on a circle with $n$ points on it: if the number of two vertices is adjacent, then the ``outer'' and ``inner'' vertices should also be adjacent. So it should look like an ``onion'' :$\cdots,3,1,2,4, \cdots$, but it's a circle with odd vertices, so it's impossible.
\begin{figure}
\caption{One example of possible orientation, the left is part of the origami while the right is part of the polygon. Please note there are other possible orientations, but $a_3$ should always be adjacent to $a_4$ in the origami.}
\label{orientation}
\end{figure}
\begin{figure}
\caption{The game of putting numbers on vertices: if the number of two vertices (red)is adjacent, then the ``outer'' and ``inner'' vertices (blue and green) should also be adjacent. See Fig. \ref{orientation}
\label{outer and inner}
\end{figure}
\begin{de}
Let $a_i$ (i=1,2,...,2(2g-1)) to be the edges in clockwise direction from the curve $\alpha$ of the $4(2g-1)$ gon mentioned above. We define the distance between two edges, $a_i$ and $a_j$, to be$$
d(a_i,a_j)=
\begin{cases}
|i-j|, |i-j|\le 2g-1\\
2(2g-1)-|i-j|, |i-j|>2g-1
\end{cases}$$If $a_i$ and $a_j$ ($i\not=j$) is identified in the curve $\alpha$, we define the distance of the identified segment as $d(a_i,a_j)$.
\end{de}
\begin{lemma} \label{odd}
For the distance of two identified edges we have :\\
(1) The distance is always odd.\\
(2) The distance cannot be 1.
\end{lemma}
Since $\alpha$ and $\beta$ are intersecting coherently, in the polygon if the distance between two edges is even, they are in the same direction and vice versa. If the direction is same, the ``next'' edge following also have to identified with each other. With a simple induction, the distance of every pair of identified edges will be the same and even, which will be impossible with the fact that $2g-1$ is odd. If the distance between identified $a$ and $a'$ of $\alpha$ is 1, then there will be an arc $b$ of $\beta$ lies between them, which is also impossible.
\begin{figure}
\caption{Left: if two edges with same direction are identified, the next one will also be identified; right: if the distance is 1, there will be one arc intersecting two identified arcs in the same direction.}
\label{identified edges}
\end{figure}
The algorithm to transfer a polygon into an origami is simple, we are just following the way of connection for each intersection. However, things will be harder if we are transforming an origami into a polygon.
\section{An example}
\label{section: example}
We are now showing how to find minimal filling pairs for $S_g$, we take $g=3$ as an example:
\begin{prop} \label{g3}
There's a unique pair of filling curves in $S_3$ that is coherent with minimal intersecting number.
\end{prop}
\subsection{Step 1: Find possible ways to identify edges from curve $\alpha$ on the 20-gon.}
If we check-board the edges of $\alpha$ in the $4(2g-1)$-gon (there are $2(2g-1)$ of them), according to Lemma \ref{odd}, two identified edges have to be one white and one black, and their distance cannot be $1$. And if we rotate the table and relabel them, we are actually getting the same way to identify the edges since the edges are not ordered. So, this is actually the modified m\'enage problem discussed in introduction, and as a result, when $n=5$, there are $5$ equivalent classes. However, the case that all labels are opposite is impossible due to Proposition \ref{oppo}, so there are $4$ cases in total.
\begin{figure}
\caption{An example of rotating and relabelling, red are ladies that are fixed and green are gentlemen. Arcs belong to $\beta$ are not shown.}
\label{rotation}
\end{figure}
We notice we are actually finding ``permutations'' of the m\'enage problem, which is introduced in \cite{gen}. According to the Table.1 of that paper, and notice Proposition \ref{oppo}, the case that all couples are seated opposite is impossible. So we have:
\begin{prop}
The only possible ways to identify edges of $\alpha$ will be one of the four following cases up to symmetry.
\end{prop}
\begin{figure}
\caption{Four possible cases. Arcs belong to $\beta$ are not shown.}
\label{4 possible cases}
\end{figure}
\subsection{Step 2: Find possible labels of edges in $\beta$ for edges in $\alpha$}
We give an orientation of $\beta$ and label the arcs in $\beta$ with the orientation. Suppose the curves are orientated as the left of the picture, and two $a$s are identified as an arc of $\alpha$ like the middle, then the right shows it will behave like that in the origami and thus the label of $b'$ has to be 1 bigger than $b$.
\begin{comment}
\begin{figure}\label{}
\end{figure}
\end{comment}
We take the loop (1 2 3 4 5 1 4 5 2 3) as an example (the reader can check this is a just a relabel of the second case in \ref{4 possible cases}), first we put the labels of $\alpha$ on the polygon, see Figure \ref{loop: step 1}
\begin{figure}
\caption{Putting the labels of $\alpha$ on the polygon.}
\label{loop: step 1}
\end{figure}
According to the discussion above, we notice that the label of $b'$ have to be the next of $b$. Without loss of generality, we let the label of $b$ to be 1, so we continue this process until we label five edges according to one direction of $\alpha$:
\begin{figure}
\caption{Labelling five edges according to one direction of $\alpha$.}
\label{loop: step 2}
\end{figure}
We continue on labelling the other sides, notice $b'$ have to be the next of $b$, however since the other direction is already labelled, we label $b$ as $x$, so $b'$ will be $x+1$.
\begin{figure}
\caption{Labelling the other sides.}
\label{loop: step 3}
\end{figure}
Similarly, we finished labelling all the edges in $\beta$.
\subsection{Step 3: Identify edges in $\beta$.}
We want to find what $x$ is in the last step, where $x$ can be an integer from 1 to 5. Notice that in the lower half of the origami and the upper half origami will be like the following:
\begin{figure}
\caption{Identifying edges in $\beta$.}
\label{identified edges in example}
\end{figure}
We say $p$ is \emph{the permutation of the $1-1$ origami} if the top edge of $i$-th square is identified with the bottom edge $p(i)$-th square.
So if $x=1$, it is not an origami since edge 1 in $\alpha$ is followed by itself, same happen with $x=2$ where edge 2 is followed by itself, and when $x=5$, edge 3 is followed by itself. When $x=3$, it's an origami with permutation (1 2 5 3 4), and when $x=4$, it's an origami with permutation (1 5 2 4 3). For (1 2 5 3 4), the different between two adjacent entries are $1,3,3,1,2$ while for (1 5 2 4 3) they are $-1,-3,-3,-1,-2$ so they are just the mirror reflection of each other, so there's one (up to relabelling) possible labelling of the 20-gon with the loop (1 2 3 4 5 1 4 5 2 3). Similarly we can prove there's no possible labelling with the other three loops and this finish the proof of the proposition.
We can use similar algorithm on higher genus surfaces, however the way to identify $\beta$ may not be unique. For example, in $S_5$, if the top of $\beta$ is $1,2,9,3,4,6,7,5,8$ and the bottom is $x,x+1,x+9,x+3,x+4,x+6,x+7,x+5,x+8$, then when $x=8$ and when $x=1$, they will form two origamis with permutation (1 2 4 5 8 6 3 7 9) and (1 3 9 7 6 8 5 4 2). Taking the different between two adjacent entries we get $1,2,1,3,7,1,2,3,7$ and $-7,-3,-2,-1,-7,-3,-1,-2,-1$. They are neither the same nor mirror reflection.
\section{An estimate on upper bound for the \\ number of ordered filling pairs}
\label{section: estimate}
Notice in Step 1, according to \cite{oeis}, the number of m\'enage permutations $h(g)\sim\frac{(2g-2)!}{e^2}$.
In Step 2, for fixed $\alpha$, the $\beta$ in the two half squares are uniquely determined.
In Step 3, when all edges in $\beta$ are labelled, we may have up to $2g-2$ ways to identify them and they are actually symmetric (notice 1 cannot be identified with 1).
So according to the steps, notice we counted twice for mirror reflections, there will be at most $(2g-2)h(g)/2$ possibilities and this finish the proof of Theorem \ref{main}.
\section{Searching for possible origamis}
\label{section: searching}
In the discussion above, given an origami with $2g-1$ squares, we have an algorithm to find the arcs of $\alpha$ on the corresponding $[4(2g-1)]$-gon. Suppose the origami is already oriented, we start from an arc of $\alpha$ in the origami and locate it on the polygon. According to the orientation of the polygon and the origami, we can find the location on the polygon for the adjacent arc of $\alpha$. Then we repeat this process until all the polygon is filled. Notice the orientation of the polygon is coherent, so if the last step is finding the right neighbour of an arc on the upper half of the origami, the next will be finding the left neighbour of the new arc on the lower half, and vice versa.
Take the origami below as example, we start from arc 1 in $\alpha$ and locate it in one of the arcs in polygon as shown in the picture. We notice the two blue areas in the origami and the polygon have the same direction, so we move one grid right in the lower half of the origami and find the next arc is 4. Again, we find that the direction for two light blue area in the origami and the polygon are also the same, so we move one grid left in the upper half of the origami and the result is 5.
\begin{figure}
\caption{Example of an origami and its corresponding polygon.}
\label{search}
\end{figure}
\begin{prop}
An origami determines a minimal filling pair if and only if they can fill the arcs of $\alpha$ in the $4(2g-1)-gon$ with the algorithm above and the distance of two identified edges are always odd.
\end{prop}
With the proposition above, we have an algorithm to check whether an origami determines a minimal filling pair. And if we check all the possible $1-1$ origamis with $2g-1$ squares, we will be able to find all ordered minimal filling pairs. Notice if we change the start of $\beta$, we will have $2g-1$ different origamis but they all determine the same origami; and if we change the orientation of $\beta$, we will find 2 origamis. So each ordered minimal filling pair will be correspond to $2(2g-1)$ origamis. With the discussion above I made a searching program \url{https://github.com/expectedid/countorigami}. Please notice Figure 3 in \cite{AMN} counted the number of $SL(2, \mathbb{Z})$ orbits which include shearing and rotating but my program is counting the number of all possible origamis up to mirror symmetry and relabelling (no shearing), so the result may be different and mine will be larger for the same genus.
The results with smaller genera are:
\begin{center}
\begin{tabular}{cccc}
\hline
\makecell[c]{Genus of\\the surface}&\makecell[c]{Number of squares\\in origami}&\makecell[c]{Number of\\coherent minimal\\filling pairs}&$[\frac{(g-1)(2g-2)!}{e^2}]$\\
\hline
3&5&1&6\\
4&7&8&292\\
5&9&436&21826\\
6&11&23904&2455523\\
7&13&2448720&388954903\\
\hline
\end{tabular}
\end{center}
\section{Case with genus 2}
\label{section: genus 2}
When $g=2$, \cite{AH} suggests that for the number of squares, the bound $2g-1=3$ cannot be obtained and instead the minimal number of squares for the filling pairs is 4 with 2 boundary components in $S_2-(\alpha\cup\beta)$. There are, $(4-1)!=6$, $1-1$ origamis with four squares and we check the Euler characteristic for all of them. As a result, there are 4 out of 6 whose Euler characteristic is $-2$, which means the surface is $S_2$. However, up to symmetry, they are actually the same pair of curves on the surface. So, we can improve the proposition into following:
\begin{thm}
There's a unique pair of minimal coherent filling curves in $S_2$ or $S_3$ that is coherent with minimal intersecting number.
\end{thm}
We have discussed the genus $3$ case in \ref{g3}. Figure below is one of the genus $2$ origami and all genus $2$ origami is in the same $Mod^{\pm}(S_2)$ orbit.
\begin{figure}
\caption{Example of an genus 2 origami, the violet is one disk in $S_2-(\alpha\cup\beta)$ while the blue one is the other.}
\end{figure} \label{g2}
\end{document} |
\begin{document}
\author[a]{Galit Ashkenazi-Golan}
\author[b]{J\'{a}nos Flesch}
\author[c]{Arkadi Predtetchinski}
\author[d]{Eilon Solan}
\affil[a]{London School of Economics and Political Science,
Houghton Street
London
WC2A 2AE, UK}
\affil[b]{Department of Quantitative Economics,
Maastricht University, P.O.Box 616, 6200 MD, The Netherlands}
\affil[c]{Department of Economics, Maastricht University, P.O.Box 616, 6200 MD,
The Netherlands}
\affil[d]{School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel, 6997800}
\title{Regularity of the minmax value and equilibria in multiplayer Blackwell games\thanks{Ashkenazi-Golan acknowledges the support of the Israel Science Foundation, grants \#217/17 and \#722/18, and the NSFC-ISF Grant \#2510/17. Solan acknowledges the support of the Israel Science Foundation, grant \#217/17. This work has been partly supported by COST Action CA16228 European Network for Game Theory.}}
\maketitle
\begin{abstract}
\noindent
A real-valued function $\varphi$ that is defined over all Borel sets of a topological space is \emph{regular} if for every Borel set $W$, $\varphi(W)$ is the supremum of $\varphi(C)$, over all closed sets $C$ that are contained in $W$, and the infimum of $\varphi(O)$, over all open sets $O$ that contain $W$.
We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular.
We then use the regularity of the minmax value to establish the existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. One is the class of $n$-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
For the latter class, we obtain a characterization of the set of equilibrium payoffs.
\end{abstract}
\noindent\textbf{Keywords:} Blackwell games, determinacy, value, equilibrium, regularity.
\noindent\textbf{AMS classification code:} Primary: \textsc{91A44} (Games involving topology, set theory, or logic). Secondary: \textsc{91A20} (Multistage and repeated games).
\section{Introduction}
Blackwell games (Blackwell \cite{Blackwell69}) are dynamic multiplayer simultaneous-move games where the action sets of the players may be history dependent, and the payoff function is an arbitrary Borel-measurable function of the play. When the payoff function of a player is given by the characteristic function of a given set $W$, we say that $W$ is the \emph{winning set} of the player. These games subsume several familiar classes of dynamic games: repeated games with the discounted payoff or the limiting average payoff (e.g., Sorin \cite{Sorin92}, Mailath and Samuelson \cite{{Mailath06}}), games with perfect information (e.g., Gale and Stewart \cite{Gale53}), and graph games arising in the computer science applications (e.g., Apt and Gr\"{a}del \cite{AptGradel12}, Bruy\`{e}re \cite{Bruyere17, Bruyere21}, Chatterjee and Henzinger \cite{Chatterjee12}).
While two-player zero-sum Blackwell games and Blackwell games with perfect information are quite well understood (see, e.g., Martin \cite{Martin75, Martin98}, Mertens \cite{Mertens86}, Kuipers, Flesch, Schoenmakers, and Vrieze \cite{Kuipers21}), general multiplayer nonzero-sum Blackwell games have so far received relatively little attention.
The goal of this paper is to introduce a new technique to the study of multiplayer Blackwell games: regularity of the minmax value, along with a number of related approximation results. In a nutshell, the technique amounts to the approximation of the minmax value of a winning Borel set using a closed subset. This approach allows us to establish existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games.
\noindent\textsc{Regularity and approximation results:} A real-valued function $\varphi$ that is defined over all Borel sets of a certain space is \emph{inner regular} if for every Borel set $W$, $\varphi(W)$ is the supremum of $\varphi(C)$, over all closed sets $C$ that are contained in $W$. The function $\varphi$ is \emph{outer regular} if for every Borel set $W$ it is the infimum of $\varphi(O)$, over all open sets $O$ that contain $W$.
The function $\varphi$ is \emph{regular} if it is both inner regular and outer regular.
Borel probability measures on metric spaces are one example of a regular function (see, e.g., Kechris \cite[Theorems 17.10 and 17.11]{Kechris95}).
When restricted to two-player zero-sum Blackwell games with finite action sets and Borel-measurable winning set for Player~1, the value function is known (Martin \cite{Martin98}) to be regular. This result was extended to two-player zero-sum stochastic games by Maitra, Purves, and Sudderth \cite{Maitra92}.
We show that in multiplayer Blackwell games with countable action sets and Borel winning sets, the minmax value of all players is regular. We thus extend the regularity result of Martin \cite{Martin98} in terms of both the number of actions (countable versus finite) and the number of players (finite versus two).
A related approximation result concerns the case when a player's objective is represented by a bounded Borel-measurable payoff function. Denote by $v_i(f)$ player~$i$'s minmax value when her payoff function is $f$. We show that $v_i(f)$ is the supremum of $v_i(g)$ over all bounded limsup functions $g \leq f$, and the infimum of $v_i(g)$ over all bounded limsup function $g \geq f$. A \emph{limsup function} is a function that can be written as the limit superior of a sequence of rewards assigned to the nodes of the game tree.
This too, is an extension of results by Maitra, Purves, and Sudderth \cite{Maitra92} and Martin \cite{Martin98} for two-player games to multiplayer games. If, moreover, the player's minmax value is the same in every subgame, one obtains an approximation from below by an upper semi-continuous function, and an approximation from above by a lower semi-continuous function.
\noindent\textsc{Existence of $\varepsilon$-equilibria:} The main contribution of the paper is the application of the regularity of the minmax value to the problem of existence of an $\varepsilon$-equilibrium in multiplayer Blackwell games. We establish the existence in two distinct classes of Blackwell games.
One is the class of $n$-player Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. The latter assumption means that every player's minmax value is the same in each subgame. Under these assumptions, for each $\varepsilon > 0$, there is an $\varepsilon$-equilibrium with a pure path of play.
A prominent sufficient condition for the minmax value to be history-independent the is that payoff be tail-measurable. Roughly speaking, tail-measurability amounts to the requirement that the payoff is unaffected by a change of the action profile in any finite number of stages. We thus obtain the existence of $\varepsilon$-equilibria in Blackwell games with history-independent finite action spaces and bounded, upper semi-analytic, and tail-measurable payoff functions.
The second class of games for which we derive an existence result is $n$-player Blackwell games where each player has a finite action space at each history, her objective is represented by an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. Under these conditions we show that there exists a play that belongs to each player's winning set; any such play induces a $0$-equilibrium. At the heart of the proof is an approximation of each player's minmax value by the minmax value of a closed subset of the player's winning set.
The key idea of the proof of the first result is to consider an auxiliary Blackwell game with winning sets, where the winning set of player $i$ is the set of player $i$'s $\varepsilon$-individually rational plays: the plays that yield player $i$ a payoff no smaller then her minmax value minus $\varepsilon$. We show that, in the thus-defined auxiliary Blackwell game, each player's minmax value equals 1, and apply the second\textbf{} result.
The question whether $\varepsilon$-equilibria exist in multiplayer Blackwell games is a largely uncharted territory. An important benchmark is the result of Mertens and Neyman (see Mertens \cite{Mertens86}): all games of perfect information with bounded Borel-measurable payoff functions admit an $\varepsilon$-equilibrium for every $\varepsilon > 0$. Zero-sum Blackwell games (where at least one of the two players has a finite set of actions) are known to be determined since the seminal work of Martin \cite{Martin98}. Shmaya \cite{Shmaya11} extends the latter result by showing the determinacy of zero-sum games with eventual perfect monitoring, and Arieli and Levy \cite{Arieli15} extend Shmaya's result to stochastic signals.
Only some special classes of multiplayer dynamic games have been shown to have an $\varepsilon$-equilibrium. These include stochastic games with discounted payoffs (see, e.g., the survey by Ja\'{s}kiewicz and Nowak \cite{Nowak16}), two-player stochastic games with the limiting average payoff (Vieille \cite{Vieille00I,Vieille00II}), and graph games with classical computer science objectives (e.g., Secchi and Sudderth \cite{Sechi}, Chatterjee \cite{Chatterjee04,Chatterjee05}, Bruy\`{e}re \cite{Bruyere21}, Ummels, Markey, Brenguier, and Bouyer \cite{Ummels15}).
A companion paper (\cite{AFPS}) establishes the existence of $\varepsilon$-equilibria in Blackwell games with countably many players, finite action sets, and bounded, Borel-measurable, and tail-measurable payoff functions.
The present paper departs from \cite{AFPS} in two dimensions. Firstly, it invokes a new proof technique, the regularity of the minmax value. Secondly, it makes different assumptions on the primitives. The second of our two existence results (Theorem \ref{theorem:sumofprob}) has, in fact, no analogue in \cite{AFPS}. The first (Theorem \ref{theorem:minmax_indt}) applies to a larger class of payoff functions than does the main result in \cite{AFPS}: it only requires players' minmax values to be history-independent. While tail-measurability of the payoff functions is a sufficient condition for history-independence of the minmax values, it is by no means a necessary condition. Furthermore, Borel-measurability imposed in \cite{AFPS} is relaxed here to upper semi-analyticity. On the other hand, \cite{AFPS} has a countable rather than a finite set of players, something that the methods developed here do not allow for.
\noindent\textsc{Characterisation of equilibrium payoffs:} An equilibrium payoff is an accumulation point of the expected payoff vectors of $\varepsilon$-equilibria, as $\varepsilon$ tends to $0$. We establish a characterisation of equilibrium payoffs in games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
In repeated games with patient players the folk theorem asserts that under proper conditions, the set of limiting average equilibrium payoffs (or the limit set of equilibrium payoffs, as the discount factor goes to 0 or the horizon increases to infinity) is the set of all vectors that are individually rational and lie in the convex hull of the range of the stage payoff function (see, e.g., Aumann and Shapley \cite{Aumann94}, Sorin \cite{Sorin92}, or Mailath and Samuelson \cite{{Mailath06}}). Our result identifies the set of equilibrium payoffs of a Blackwell game as the set of all vectors that lie in the convex hull of the set of feasible and individually rational payoffs. The intuition for this discrepancy is that in standard repeated games, a low payoff in one stage can be compensated by a high payoff in another stage, therefore payoff vectors that are convex combinations of the stage payoff function can be equilibrium payoffs as long as this convex combination of payoffs is individually rational. In particular, these combinations can place some positive weight on payoff vectors that are not individually rational. In Blackwell games, however, the payoff is obtained only at the end of the game, hence only plays that generate individually rational payoffs can be taken into account when constructing equilibria.
Our characterization of the set of equilibrium payoffs is related to the rich literature on the folk theorem, and the study of the minmax value is instrumental to this characterizaion (see, e.g., the folk theorems in Fudenberg and Maskin \cite{Fudenberg86}, Mailath and Samuelson \cite{Mailath06}, or H\"{o}rner, Sugaya, Takahashi, and Vieille \cite{Horner11}). The minmax value of a player would often be used in the proofs of equilibrium existence to construct suitable punishments for a deviation from the supposed equilibrium play (as is done, for instance, in Aumann and Shapley \cite{Aumann94}, Rubinstein \cite{Rubinstein94}, Fudenberg and Maskin \cite{Fudenberg86}, and Solan \cite{Solan01}).
The paper is structured as follows. Section \ref{secn.games} describes the class of Blackwell games. Section \ref{secn.approx} is devoted to the regularity of the minmax value and related approximation theorems. Section \ref{secn.appl} applies these tools to the problem of existence of equilibrium. Section~\ref{secn.folk} is devoted to the characterisation of equilibrium payoffs. Section \ref{secn.tail} discusses the implications of the results for games with tail-measurable payoffs. Section \ref{secn.disc} contains a discussion, concluding remarks, and open questions.
\section{Blackwell games}\label{secn.games}
\noindent\textbf{Blackwell games:} An $n$-\textit{player Blackwell game} is a tuple $\Gamma = (I, A, H, (f_{i})_{i \in I})$. The elements of $\Gamma$ are as follows.
The set of players is $I$, a finite set of cardinality $n$. For a player $i \in I$ we write $-i$ to denote the set of $i$'s opponents, $I \setminus \{i\}$.
The set $A$ is a countable set and $H \subseteq \cup_{t \in \mathbb{N}}A^{t}$ is the game tree (throughout the paper $\mathbb{N} = \{0,1,\ldots\}$). Elements of $H$ are called histories. The set $H$ is assumed to have the following properties: (a) $H$ contains the empty sequence, denoted $\oslash$; (b) a prefix of an element of $H$ is an element of $H$; that is, if for some $h \in \cup_{t \in \mathbb{N}}A^{t}$ and $a \in A$ the sequence $(h,a)$ is an element of $H$, so is $h$; (c) for each $h \in H$ there is an element $a \in A$ such that $(h, a) \in H$; we define $A(h) := \{a \in A: (h,a) \in H\}$; and (d) for each $h \in H$ and each $i \in I$ there exists a set $A_{i}(h)$ such that $A(h) = \prod_{i \in I}A_{i}(h)$. The set $A_{i}(h)$ is called player $i$'s set of actions at history $h$, and $A(h)$ the set of action profiles at $h$.
Conditions (a), (b), and (c) above say that $H$ is a pruned tree on $A$.
Condition (c) implies that the game has infinite horizon.
Let $H_{t} := H \cap A^{t}$ denote the set of histories in stage $t$.
An infinite sequence $(a_0,a_1,\ldots) \in A^{\mathbb{N}}$ such that $(a_0,\ldots,a_t) \in H$ for each $t \in \mathbb{N}$ will be called a \emph{play}. The set of plays is denoted by $[H]$.
This is the set of infinite branches of $H$. For $h \in H$ let $O(h)$ denote the set of all plays of $\Gamma$ having $h$ as a prefix. We endow $[H]$ with the topology generated by the basis consisting of the sets $\{O(h):h \in H\}$. The space $[H]$ is Polish. For $t \in \mathbb{N}$ let $\mathcal{F}_{t}$ be the sigma-algebra on $[H]$ generated by the sets $\{O(h):h \in H_{t}\}$. The Borel sigma-algebra of $[H]$ is denoted by $\mathscr{B}$. It is the minimal sigma-algebra containing the topology. A subset $S$ of $[H]$ is \emph{analytic} if it is the image of a continuous function from the Baire space $\mathbb{N}^\mathbb{N}$ to $[H]$. Each Borel set is analytic.
Each analytic set is universally measurable. Recall that a set $S \subseteq [H]$ is said to be universally measurable if (Kechris \cite[Section 17.A]{Kechris95}), for every Borel probability measure $\mathbb{P}$ on $[H]$, there exist Borel sets $B,Z \in \mathscr{B}$ such that $S \bigtriangleup B \subseteq Z$ and $\mathbb{P}(Z) = 0$; here $S \bigtriangleup B = (S \setminus B) \cup (B \setminus S)$ is the symmetric difference of the sets $S$ and $B$.
The last element of the game is a vector $(f_i)_{i \in I}$, where $f_i : [H] \to \mathbb{R}$ is player $i$'s \emph{payoff function}. The most general class of payoff functions we allow for are bounded upper semi-analytic functions. A function $f_i : [H] \to \mathbb{R}$ is said to be \emph{upper semi-analytic} if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is analytic. In particular, the indicator function $1_{S}$ of a subset $S \subseteq [H]$ of plays is upper semi-analytic if and only if $S$ is an analytic set. Each Borel-measurable function in upper semi-analytic. Note that a bounded upper semi-analytic function is universally measurable, i.e., for each open set $U \subseteq \mathbb{R}$, the set $f^{-1}(U) \subseteq [H]$ is universally measurable (see, e.g., Chapter 7 in Bertsekas and Shreve \cite{Bertsekas96}).
The play of the game starts at the empty history $h_0 = \oslash$. Suppose that by a certain stage $t \in \mathbb{N}$ a history $h_t \in H_t$ has been reached. Then in stage $t$, the players simultaneously choose their respective actions; thus player $i \in I$ chooses an action $a_{i,t} \in A_{i}(h_{t})$. This results in the stage $t$ action profile $a_{t} = (a_{i,t})_{i \in I} \in A(h_{t})$. Once chosen, the actions are revealed to all players, and the history $h_{t+1} = (h_{t},a_{t})$ is reached. The result of the infinite sequence of choices is the play $p = (a_0,a_1,\ldots)$, an element of $[H]$. Each player $i \in I$ receives the corresponding payoff $f_i(p)$.
Given a Blackwell game $\Gamma$ and a history $h \in H$, the \textit{subgame} of $\Gamma$ starting at $h$ is the Blackwell game $\Gamma_h = (I, A, H_h, (f_{i,h})_{i \in I})$. The set $H_h$ of histories of $\Gamma_h$ consists of finite sequences $g \in \bigcup_{t \in \mathbb{N}} A^{t}$ such that $hg \in H$, where $hg$ is the concatenation of $h$ and $g$. The payoff function $f_{i,h} : [H_h] \to \mathbb{R}$ is the composition $f_i \circ s_h$, with $s_h : [H_h] \to [H]$ given by $p \mapsto hp$, where $hp$ is the concatenation of $h$ and $p$. Note that $\Gamma_\oslash$ is just the game $\Gamma$ itself.
The Blackwell game $\Gamma$ is said to have \textit{history-independent action sets} if $A_{i}(h) = A_{i}(\oslash)$ for each history $h \in H$ and each player $i \in I$; the common action set is simply denoted by $A_i$. If $\Gamma$ has history-independent action sets, then the set of its histories is $H = \cup_{t \in \mathbb{N}}A^{t}$, and the set of plays in $\Gamma$ is $[H] = A^{\mathbb{N}}$. A Blackwell game with history-independent action sets can be described as a tuple $(I,(A_{i},f_{i})_{i \in I})$.
\noindent\textbf{Strategies and expected payoffs:} A strategy for player $i\in I$ is a function $\sigma_i$ assigning to each history $h \in H$ a probability distribution $\sigma_{i}(h)$ on the set $A_{i}(h)$. The set of player $i$'s strategies is denoted by $\Sigma_i$. We also let $\Sigma_{-i} := \prod_{j \in -i} \Sigma_{j}$ and $\Sigma := \prod_{i \in I} \Sigma_{i}$. Each strategy profile $\sigma=(\sigma_i)_{i\in I}$ induces a unique probability measure on the Borel sets of $[H]$, denoted $\mathbb{P}_{\sigma}$. The corresponding expectation operator is denoted $\mathbb{E}_{\sigma}$. In particular, $\mathbb{E}_{\sigma}[f_{i}]$ denotes an expected payoff to player $i$ in the Blackwell game under the strategy profile $\sigma$. It is well defined under the maintained assumptions, namely boundedness and upper semi-analyticity of $f_{i}$.
Take a history $h \in H_t$ in stage $t$. A strategy profile $\sigma \in \Sigma$ in $\Gamma$ induces the strategy profile $\sigma_h$ in $\Gamma_h$ defined as $\sigma_h(g)=\sigma(hg)$ for each history $g \in H_h$. Let us define $\mathbb{E}_{\sigma}(f_i \mid h)$ as the expected payoff to player $i$ in the Blackwell game $\Gamma_h$ under the strategy profile $\sigma_h$: that is,
$\mathbb{E}_{\sigma}(f_i \mid h) := \mathbb{E}_{\sigma_h}(f_{i,h})$. Note that $\mathbb{E}_{\sigma}(f_i \mid h)$, when viewed as an $\mathcal{F}_{t}$-measurable function on $[H]$, is a conditional expectation of $f_i$ with respect to the measure $\mathbb{P}_{\sigma}$ and the sigma-algebra $\mathcal{F}_{t}$; whence our choice of notation.
\noindent\textbf{Minmax value:} Consider a Blackwell game $\Gamma$, and suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic. Player $i$'s \textit{minmax value} is defined as
\[v_i(f_i) := \inf_{\sigma_{-i} \in \Sigma_{-i}}\sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_i).\]
Whenever $f_i = 1_{W_i}$ is an indicator of an analytic set $W_i \subseteq [H]$ we write $v_i(W_i)$ for $v_i(1_{W_i})$.
Player $i$'s minmax value is said to be \textit{history-independent} if her minmax value in the subgame $\Gamma_h$ equals that in the game $\Gamma$, for each history $h \in H$.
\section{Regularity and approximation theorems}\label{secn.approx}
In this section we state the regularity property of the minmax: the minmax value of a Borel winning set can be approximated from below by the minmax value of closed subset and from above by the minmax value of an open superset. We also describe two related approximation results: the minmax value of a bounded Borel-measurable payoff function can be approximated from below and from above by limsup functions. If, in addition, the minmax values are history-independent, then one can choose the approximation from below to be upper semicontinuous, and the approximation from above to be lower semicontinuous.
The proofs of all results are detailed in the appendix.
\begin{theorem} {\rm (Regularity of the minmax value)}\label{thrm:reg}
Consider a Blackwell game. Suppose that player $i$'s objective is given by a winning set $W_i \subseteq [H]$. Suppose that $W_i$ is Borel. Then
\begin{align*}
v_i(W_i) &= \sup\{v_i(C):C\subseteq W_i, C \text{ is closed}\}=\inf\{v_i(O):O\supseteq W_i, O \text{ is open}\}.
\end{align*}
\end{theorem}
One implication of Theorem~\ref{thrm:reg} concerns the complexity of strategies of player~$i$ that ensures that her probability of winning is close to her minmax value.
Suppose, for example, that $v_i(W_i) = \frac{1}{2}$.
Then for every strategy profile $\sigma_{-i}$ of the opponents of player~$i$ and every $\varepsilon > 0$, she has a response $\sigma_i$ such that $\mathbb{P}_{\sigma_{-i},\sigma_i}(W_i) \geq \frac{1}{2}-\varepsilon$. The strategy profile $\sigma_{-i}$ and the winning set $W_i$ may be complex, and accordingly the good response $\sigma_i$ may be complex as well.
However, take now a closed subset $C \subseteq W_i$ such that $v_i(C) > v_i(W_i) - \varepsilon = \frac{1}{2}-\varepsilon$. The complement of $C$, denoted $C^c$, is open, hence it is the union of basic open sets; that is, it can be presented as a union $C^c = \bigcup_{h \in H'} O(h)$,
for some subset $H' \subseteq H$ of histories. A strategy $\sigma'_i$ that satisfies
$\mathbb{P}_{\sigma_{-i},\sigma_{i}'}(C_i) \geq \frac{1}{2}-\varepsilon$ must aim at avoiding $C^c$, that is, at avoiding histories in $H'$. In that sense, $\sigma'$ may have a simple structure.
\begin{exl}\label{exl.io}\rm
Here we consider a Blackwell game where the same stage game is being played at every stage. The stage game specifies a stage winning set for each player. A player's objective in the Blackwell game is to win the stage game infinitely often.
Thus let $\Gamma = (I,(A_{i},1_{W_{i}})_{i \in I})$ be a Blackwell game with history-independent countable action sets, where player $i$'s winning set is
\[W_i = \{(a_0,a_1,\ldots) \in A^\mathbb{N}:a_t \in U_i\text{ for infinitely many }t \in \mathbb{N}\};\]
here $U_{i}$, called player $i$'s \textit{stage winning set}, is a given subset of $\prod_{i \in I}A_{i}$. If $a_t \in U_i$, we say that player $i$ \textit{wins stage $t$}. Thus, player~$i$'s objective is to win infinitely many stages of the Blackwell game. The set $W_{i}$ is a $G_\delta$-set, i.e., an intersection of countably many open subsets of $A^\mathbb{N}$.
Fix a player $i \in I$.
Let
\begin{equation}\label{eqn.stageminmax}
d_{i} := \inf_{x_{-i} \in X_{-i}} \sup_{x_i \in X_i} \mathbb{P}_{x_{-i},x_i}(U_i)
\end{equation}
be player~$i$'s minmax value in the stage game. As follows from the arguments below, $v_{i}(W_{i})$ is either $0$ or $1$, and it is $1$ exactly when $d_{i} > 0$. In either case, there are intuitive approximations of player $i$'s wining sets by a closed set from below and an open set from above.
First assume that $d_{i} > 0$. Take an $\varepsilon > 0$. Let us imagine that player $i$'s objective is not merely to win infinitely many stages in the course of the Blackwell game, but to make sure that she wins at least once in every block of stages $t_{n},\ldots,t_{n+1}-1$, where the sequence of stages $t_0 < t_1 < \cdots$ is chosen to satisfy
\[(1 - \tfrac{1}{2}d_{i})^{t_{n+1} - t_{n}} < 2^{-n-1}\cdot\varepsilon\]
for each $n \in \mathbb{N}$. This, more demanding condition, defines an approximating set. Formally, define
\[C_{i} := \bigcap_{n \in \mathbb{N}}\bigcup_{t_{n} \leq k < t_{n+1}} \{(a_0,a_1,\ldots) \in A^{\mathbb{N}}: a_k \in U_i\}.\]
As the intersection of closed sets, $C_{i}$ is a closed subset of $W_{i}$. Moreover, $1 - \varepsilon \leq v_{i}(C_{i})$. To see this, fix any strategy $\sigma_{-i}$ for $i$'s opponents. At any history $h$, player $i$ has a mixed action $\sigma_{i}(h)$ that, when played against $\sigma_{-i}(h)$, guarantees a win at history $h$ with probability of at least $\tfrac{1}{2}d_{i}$. Thus, under the measure $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$ the probability for player $i$ not to win at least once in a block of stages $t_{n},\ldots,t_{n+1}-1$ is at most $2^{-n-1}\cdot\varepsilon$, for any history of play up to stage $t_{n}$. And hence the probability that there is a block within which player $i$ does not win once is at most $\varepsilon$.
Suppose that $d_{i} = 0$. Let us imagine that player $i$'s objective is merely to win the stage game at least once. This modest objective defines the approximating set:
\[O_{i} = \bigcup_{t \in \mathbb{N}}\{(a_{0},a_{1},\ldots)\in A^\mathbb{N}: a_{t} \in U_i\}.\]
As the union of open sets,
$O_{i}$ is an open set containing $W_{i}$. Moreover, $v_{i}(O_{i}) \leq \varepsilon$. To see this, let $\sigma_{-i}$ be the strategy for $i$'s opponents such that, at any stage $t \in \mathbb{N}$ and any history $h \in A^{t}$ of stage $t$, the probability that the action profile $a_{t}$ is an element of $U_{i}$ is not greater than $2^{-t-1}\cdot\varepsilon$ regardless of the action of player $i$. Then, for any player $i$'s strategy $\sigma_{i}$, the probability that $i$ wins at least once is not greater than $\varepsilon$. $\Box$
\end{exl}
We turn to two related approximation results for Blackwell games with Borel payoff functions. A function $f : [H] \to \mathbb{R}$ is said to be a \textit{limsup function} if there exists a function $u : H \to \mathbb{R}$ such that for each play $(a_0,a_1,\ldots) \in [H]$,
\[f(a_0,a_1,\ldots) = \limsup_{t \to \infty} u(a_0,\dots,a_t).\]
The function $f : [H] \to \mathbb{R}$ is a \textit{liminf function} if $-f$ is a limsup function.
Limsup and liminf payoff functions are ubiquitous in the literature on infinite dynamic games. At least since the work of Gillette \cite{Gillette57}, the so-called limiting average payoff (that is, the limit superior or the limit inferior of the average of the stage payoffs) is a standard specification of the payoffs in a stochastic game (see for example Mertens and Neyman \cite{Mertens81}, or Levy and Solan \cite{Levy20}). Stochastic games with limsup payoff functions have been studied in Maitra and Sudderth \cite{Maitra93}.
Limsup functions have relatively ``low" set-theoretic complexity. Various characterizations of the limsup functions can be found in Hausdorff \cite{Hausdorff05}. In particular, $f$ is a limsup function if and only if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is a $G_{\delta}$-set.
We now state a result on the approximation of the minmax value for Blackwell games where a player's objective is represented by a bounded Borel-measurable payoff function.
\begin{theorem}\label{thrm:regfunc}
Consider a Blackwell game. Suppose that player $i$'s payoff function $f_i : [H] \to \mathbb{R}$ is bounded and Borel-measurable. Then:
\[\begin{aligned}
v_i(f_i) &= \sup\{v_i(g): g\text{ is a bounded limsup function and }g \leq f_i\}\\
&= \inf\{v_i(g): g \text{ is a bounded limsup function and }f_i \leq g\}.
\end{aligned}\]
\end{theorem}
Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc} have been proven by Martin \cite{Martin98}
for the case $n=2$, see \cite[Theorem 5, and Remark (b)]{Martin98} and
\cite[Remark (c)]{Martin98}. They have been extended to two-player stochastic games by
Maitra and Sudderth \cite{Maitra98}. Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc}
extend the known results in two respects. First, they allow for more than two players,
and second, they allow for countably many actions.
The proof of Theorem \ref{thrm:reg} combines and fine-tunes the arguments in Martin \cite{Martin98} and Maitra and Sudderth \cite{Maitra98}. The key element of the proof is a zero-sum perfect information game,
denoted $G_i(f_i,c)$, where the aim of Player I is to ``prove" that the minmax value of $f_i$ is at least $c$. Roughly speaking, the game proceeds as follows.
Player~I commences the game by proposing a fictitious continuation payoff, which one could think of as a payoff player $i$ hopes to attain, contingent on each possible stage $0$ action profile. The number $c$ serves as the initial threshold: player $i$'s minmax value of the proposed continuation payoffs is required to be at least $c$. Player II then chooses a stage $0$ action profile, and the corresponding continuation payoff serves as the new threshold.
Player I then proposes a fictitious continuation payoff contingent on each possible stage $1$ action profile,
and Player II chooses the stage $1$ action profile, etc.
Player I wins if the sequence of continuation payoffs is ``justified" by the actual payoff on a play produced by Player II. Ultimately the proof rests on the determinacy of the game $G_i(f_i,c)$, which follows by Martin \cite{Martin75}.
The perfect information game $G_i(f_i,c)$ is a version of the games used in Martin \cite{Martin98}. The main difference is in the use of player $i$'s minmax value that constrains Player I's choice of fictitious continuation payoffs. The details of our proof are slightly closer to those in Maitra and Sudderth \cite{Maitra98}. Like them we invoke martingale convergence and the Fatou lemma.
Finally, we state an approximation result for a Blackwell game with history-independent minmax values. Recall that a function $g : A^\mathbb{N} \to \mathbb{R}$ is \emph{upper semicontinuous} if, for each $r \in \mathbb{R}$, the set $\{p \in [H]: r \leq f(p)\}$ is a closed set, and $g$ is \emph{lower semicontinuous} if $-g$ is upper semicontinuous. When $g = 1_B$ for some $B \subseteq A^\mathbb{N}$, $g$ is upper semicontinuous (resp.~lower semicontinuous) if and only if $B$ is closed (resp.~open).
\begin{theorem}\label{thrm:tailapprox}
Consider a Blackwell game. Suppose that player $i$'s payoff function $f_i$ is bounded and Borel-measurable, and player $i$'s minmax values are history-independent. Then
\[\begin{aligned}
v_i(f_i)
&= \sup\{v_i(g): \text{g is a bounded upper semicontinuous function and }g \leq f_i\}\\
&= \inf\{v_i(g): \text{g is a bounded lower semicontinuous function and }f_i \leq g\}.
\end{aligned}\]
\end{theorem}
As the proof reveals, both the upper semicontinuous and the lower semicontinuous functions can be chosen to be two-valued. Recall that (Hausdorff \cite{Hausdorff05}) an upper semicontinuous function and a lower semicontinuous function are both a limsup and a liminf function. Consequently, in comparison to Theorem \ref{thrm:regfunc}, an additional assumption of history-independence of the minmax values in Theorem \ref{thrm:tailapprox} leads to a stronger approximation result. The latter condition cannot be dropped; see Section~\ref{secn.disc} for an example of a game with a limsup payoff function such that the minmax value cannot be approximated from below by an upper semicontinuous function.
\section{Existence of equilibria}\label{secn.appl}
In this section, we employ the results of the previous section to establish existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. Theorem~\ref{theorem:sumofprob} concerns $n$-player Blackwell games where each player has a finite action space at each history, her objective is represented by an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. Theorem~\ref{theorem:minmax_indt} concerns for Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values.
Consider a Blackwell game $\Gamma$ and let $\varepsilon \geq 0$. A strategy profile $\sigma \in \Sigma$ is \emph{an $\varepsilon$-equilibrium} of $\Gamma$ if for each player $i \in I$ and each strategy $\eta_i \in \Sigma_i$ of player $i$,
\[\mathbb{E}_{\sigma_{-i},\eta_{i}}(f_i) \leq \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_i) + \varepsilon.\]
We state our first existence result.
\begin{theorem}\label{theorem:sumofprob}
Consider an $n$-player Blackwell game $\Gamma = (I, A, H, (1_{W_{i}})_{i \in I})$. Suppose that for each player $i \in I$ player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, and that her winning set $W_i$ is analytic. If $v_1(W_1)+ \cdots +v_n(W_n) > n-1$, then the set $W_1 \cap \cdots \cap W_n$ is not empty. Consequently, $\Gamma$ has a 0-equilibrium.
\end{theorem}
Note that any play $p \in W_1 \cap \cdots \cap W_n$ is in fact a 0-equilibrium, or more precisely, any strategy profile that requires all the players to follow $p$ is a 0-equilibrium, because it yields all players the maximal payoff 1.
The key step of the proof is the approximation of the minmax value of a player using a closed subset of her winning set.
To prove Theorem~\ref{theorem:sumofprob} we need the following technical observation.
\begin{lemma}\label{lemma:intersect}
Let $(X,\mathscr{B},P)$ be a probability space, and let $Q_1,\ldots,Q_n\in \mathscr{B}$ be $n$ events. Then
\[P(Q_1\cap\cdots\cap Q_n)\geq P(Q_1)+\cdots+P(Q_n)-n+1.\]
\end{lemma}
\begin{proof}
For $n=1$ the statement is obvious, and for $n=2$ we have
\begin{equation}\label{indineq}
P(Q_1\cap Q_2)=P(Q_1)+P(Q_2)-P(Q_1\cup Q_2)\geq P(Q_1)+P(Q_2)-1.
\end{equation}
Assume that the statement holds for some $n-1$. Then for $n$ we have
\begin{align*}
P(Q_1\cap\cdots\cap Q_n)\,&=\,P((Q_1\cap\cdots\cap Q_{n-1})\cap Q_n)\\
&\geq\, P(Q_1\cap\cdots\cap Q_{n-1}) + P(Q_n)-1\\
&\geq\, \big(P(Q_1)+\cdots+P(Q_{n-1})-n+2\big)+ P(Q_n)-1\\
&=\,P(Q_1)+\cdots+P(Q_n)-n+1,
\end{align*}
where the first inequality follows from Eq.~\eqref{indineq} and the second by the induction hypothesis.
\end{proof}
\noindent\textbf{Proof of Theorem \ref{theorem:sumofprob}:} We first establish the theorem in the special case of Borel winning sets, and then generalize it to analytic winning sets.
\noindent\textsc{Part I:} Suppose that for each $i \in I$ the set $W_{i} \subseteq [H]$ is Borel.
By Theorem \ref{thrm:reg} there are closed sets $C_1\subseteq W_1,\ldots,C_n\subseteq W_n$ such that $v_1(C_1)+\cdots+v_n(C_n) > n-1$. We show that the intersection $C_1 \cap\cdots\cap C_n$ is not empty.
Given $m \in \mathbb{N}$ consider the $n$-player Blackwell game $\Gamma^{m} = (I, A, H, (1_{C_{i}^{m}})_{i \in I})$, where player $i$'s winning set is defined by
\[C_i^{m} := \bigcup\{O(h): h \in H_{m}\text{ such that }O(h) \cap C_{i} \neq \oslash\}.\]
The game $\Gamma^{m}$ essentially ends after $m$ stages: by stage $m$ each player $i$ knows whether the play is an element of her winning set $C_{i}^{m}$ or not. In $\Gamma^{m}$, player $i$ wins if after $m$ stages there is a continuation play that leads to $C_i$.
Note that this continuation play might be different for different players.
The set $C_i^{m}$ is a clopen set. For each $m \in \mathbb{N}$ and $i \in I$ we have the inclusion $C_i^{m} \supseteq C_i^{m+1}$ (winning in $\Gamma^{m+1}$ is more difficult than winning in $\Gamma^{m}$). Moreover, $\bigcap_{m \in \mathbb{N}} C_i^{m} = C_i$. Indeed, the inclusion $C_i^{m} \supseteq C_i$ is evident from the definition. Conversely, take an element $q$ of the set $[H] \setminus C_i$. Since $[H] \setminus C_i$ is an open set, there exists a history $h \in H$ such that $q \in O(h)$ and $O(h) \subseteq [H] \setminus C_{i}$. But then $q \in [H] \setminus C_i^{m}$, where $m$ is the length of the history $h$.
Define $C^m := C_1^m \cap\cdots\cap C_n^m$. Thus $\{C^{m}\}_{m \in \mathbb{N}}$ is a nested sequence of closed sets converging to $C_1 \cap\cdots\cap C_n$. Note that, since by the assumption of the theorem $H$ is a finitely branching tree, the space $[H]$ is compact. Thus $C^{m}$ is a compact set. Consequently, to prove that $C_{1} \cap \cdots \cap C_{n}$ is not empty, we only need to argue that $C^{m}$ is not empty for each $m \in \mathbb{N}$.
The game $\Gamma^m$ being finite, it has a $0$-equilibrium (Nash \cite{Nash50}), say $\sigma^m$. By the definition of $0$-equilibrium,
the equilibrium payoff is not less than the minmax value:
\[\mathbb{P}_{\sigma_{-i}^m,\sigma_{i}^{m}} (C_i^m) \,=\, \sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{P}_{\sigma_{-i}^m,\sigma_{i}} (C_i^m)\, \geq\, \inf_{\sigma_{-i} \in \Sigma_{-i}} \sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{P}_{\sigma_{-i},\sigma_{i}} (C_i^m) = v_i(C_i^m).\]
Moreover, since $C_i^{m} \supseteq C_{i}$, it holds that $v_i(C_i^m) \geq v_i(C_i)$. We conclude that
\[\mathbb{P}_{\sigma^m}(C_1^m)+ \cdots +\mathbb{P}_{\sigma^m}(C_n^m) > n-1.\]
Finally, we apply Lemma \ref{lemma:intersect} to conclude that $\mathbb{P}_{\sigma^m}(C^m) > 0$, hence $C^{m}$ is not empty.
\noindent\textsc{Part II:} Now let $\Gamma$ be any game as in the statement of the theorem. Suppose by way of contradiction that $W_1 \cap \cdots \cap W_n$ is empty. By Novikov's separation theorem (Kechris \cite[Theorem 28.5]{Kechris95}) there exist Borel sets $B_1, \ldots, B_n$ such that $W_{i} \subseteq B_{i}$ for each $i \in I$ and $B_1 \cap \cdots \cap B_n = \oslash$. But since $v_{i}(W_{i})\leq v_{i}(B_{i})$ for each $i \in I$, the game $\Gamma = (I, A, H, (1_{B_{i}})_{i \in I})$ satisfies the assumptions of the theorem, and Part I of the proof yields a contradiction. $\Box$
We state our second and main existence result.
\begin{theorem}\label{theorem:minmax_indt}
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Then for every $\varepsilon>0$ the game admits an $\varepsilon$-equilibrium.
\end{theorem}
The key idea behind the proof is to consider an auxiliary Blackwell game with winning sets, the winning set of a player consisting of that player's $\varepsilon$-individually rational plays. We show that in the thus-defined auxiliary Blackwell game each player's minmax value equals 1, and apply Theorem \ref{theorem:sumofprob}.
Given $\varepsilon > 0$ we define the set of \textit{player $i$'s $\varepsilon$-individually rational plays}:
\[Q_{i,\varepsilon}(f_i) := \{p\in [H] : f_i(p) \geq v_i(f_i) - \varepsilon\}.\]
Also define the set
\[U_{i,\varepsilon}(f_i) := \{p\in [H] : f_i(p) \geq v_i(f_i) + \varepsilon\}.\]
Note that under the assumptions of Theorem \ref{theorem:minmax_indt} both sets are analytic.
\begin{proposition}\label{prop:v(Q)=1}
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$ and a player $i \in I$. Suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic, and that her minmax values are history-independent. Let $\varepsilon > 0$. Then
\begin{enumerate}
\item $v_i(Q_{i, \varepsilon}(f_i)) = 1$. In fact, for each strategy profile $\sigma_{-i} \in \Sigma_{-i}$ of players $-i$ there is a strategy $\sigma_i \in \Sigma_{i}$ for player $i$ such that $\mathbb{P}_{\sigma_{-i},\sigma_{i}}(Q_{i, \varepsilon}(f_i)) = 1$.
\item $v_i(U_{i, \varepsilon}(f_i)) = 0$. In fact, there exists a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ of players $-i$ such that for each strategy $\sigma_i \in \Sigma_{i}$ for player $i$ it holds that $\mathbb{P}_{\sigma_{-i},\sigma_{i}}(U_{i, \varepsilon}(f_i)) = 0$.
\end{enumerate}
\begin{proof}
\noindent\textsc{Claim 1:} It suffices to prove the second statement. Take a strategy profile $\sigma_{-i}$ of players $-i$. It is known that player $i$ has a strategy $\sigma_i$ that is an $\varepsilon/2$-best response to $\sigma_{-i}$ in each subgame (see, for example, Mashiah-Yaakovi \cite[Proposition 11]{Ayala15}, or Flesch, Herings, Maes, and Predtetchinski \cite[Theorem 5.7]{JJJJ}), and therefore
\[\mathbb{E}_{\sigma_{-i},\sigma_i}(f_i \mid h) \geq
v_i(f_{i,h}) -\varepsilon/2\,=\,v_i(f_i)-\varepsilon/2,\]
for each history $h\in H$. Since the payoff function $f_i$ is bounded, it follows that there is $d>0$ such that
\[\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \mid h) \geq d\]
for each $h \in H$. Indeed, it is easy to verify that one can choose
\[d\,=\,\frac{\varepsilon}{2(\sup_{p \in [H]}f_i(p)-v_i(f_i)+\varepsilon)}.\]
Since $Q_{i,\varepsilon}(f_i)$ is an analytic set, there is a Borel set $B$ such that $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \bigtriangleup B) = 0$, where $\bigtriangleup$ stands for the symmetric difference of two sets. It follows that $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i) \bigtriangleup B \mid h) = 0$, and consequently $\mathbb{P}_{\sigma_{-i},\sigma_i}(B \mid h) \geq d$ for each history $h \in H$ that is reached under $\mathbb{P}_{\sigma_{-i},\sigma_i}$ with positive probability. L\'{e}vy's zero-one law implies that $\mathbb{P}_{\sigma_{-i},\sigma_i}(B) = 1$, and hence $\mathbb{P}_{\sigma_{-i},\sigma_i}(Q_{i, \varepsilon}(f_i)) = 1$.
\noindent\textsc{Claim 2:} By an argument similar to that
in Mashiah-Yaakovi \cite[Proposition 11]{Ayala15} or Flesch, Herings, Maes, and Predtetchinski \cite[Theorem 5.7]{JJJJ}, one shows that there is a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ such that
\[\mathbb{E}_{\sigma_{-i},\sigma_i}(f_{i} \mid h) \leq v_{i}(f_{i,h}) + \varepsilon/2 = v_{i}(f_{i}) + \varepsilon/2,\]
for each history $h \in H$ and each strategy $\sigma_{i} \in \Sigma_{i}$. Fix any $\sigma_{i} \in \Sigma_{i}$. The rest of the proof of the claim is similar to that of Claim 1.
\end{proof}
\end{proposition}
\noindent\textbf{The proof of Theorem \ref{theorem:minmax_indt}:} Fix an $\varepsilon > 0$. By Proposition \ref{prop:v(Q)=1}, $v_{i}(Q_{i,\varepsilon}(f_{i})) = 1$.
Let $\Gamma^\varepsilon = (I, A, H, (1_{Q_{i, \varepsilon}(f_i)})_{i \in I})$ be an auxiliary Blackwell game where player $i$'s winning set is $Q_{i, \varepsilon}(f_i)$, the set of player $i$'s $\varepsilon$-individually rational plays in $\Gamma$. Each player's minmax value in the game $\Gamma^\varepsilon$ equals $1$. Therefore, the auxiliary game $\Gamma^\varepsilon$ satisfies the hypothesis of Theorem \ref{theorem:sumofprob}. We conclude that the intersection $\bigcap_{i \in I}Q_{i, \varepsilon}(f_i)$ is not empty, and hence there is a play $p^* \in [H]$ such that $f_i(p^*) \geq v_i(f_i) - \varepsilon$, for every $i \in I$.
The following strategy profile is a $2\varepsilon$-equilibrium of $\Gamma$ (see also Aumann and Shapley \cite{Aumann94}):
\begin{itemize}
\item The players follow the play $p^*$, until the first stage in which one of the players deviates from this play. Denote by $i$ the minimal index of a player who deviates from $p^*$ at that stage.
\item From the next stage and on, the players in $-i$ switch to a strategy profile that reduces player~$i$'s payoff to $v_i(f_i) + \varepsilon$. A strategy profile with this property does exist by the assumption of history-independence of the minmax values.
\end{itemize}
This completes the proof of the theorem. $\Box$
We illustrate the construction of the $\varepsilon$-equilibrium with the following example.
\begin{exl}\label{exl.eq}\rm
We consider a 2-player Blackwell game with history-independent action sets where the same stage game is being played at each stage, and a player's objective is to maximize the long-term frequency of the stages she wins. Specifically, $\Gamma = (\{1,2\}, A_{1}, A_{2}, f_{1}, f_{2})$, where $A_1$ and $A_2$ are finite, and
\[f_{i}(a_{0},a_{1},\ldots) = \limsup_{t \to \infty}\tfrac{1}{t} \cdot \#\{k < t : a_{k} \in U_{i}\},\]
for each $(a_{0},a_{1},\ldots) \in A^{\mathbb{N}}$. Here $U_i$ is player $i$'s stage winning set. We assume that $U_1$ and $U_2$ are disjoint, and let $d_i$ denote player $i$'s minmax value in the stage game.
Note that $f_i$ is a tail function (see Section \ref{secn.tail}), and it is a limsup function (in the sense of the definition in Section \ref{secn.approx}). We have $d_{i} = v_{i}(f_{i})$, i.e., player~$i$'s minmax value in the stage game is also player~$i$'s minmax value in the Blackwell game.
Take any Nash equilibrium $x \in \prod_{i \in I}\Delta(A_{i})$ of the stage game. Playing $x$ at each stage is certainly a $0$-equilibrium of the Blackwell game $\Gamma$, but typically it is \textbf{not} of the type that appears in the proof of Theorem \ref{theorem:minmax_indt}. An important feature of the $\varepsilon$-equilibrium constructed in the proof is that the equilibrium play is pure; only off the equilibrium path might a player be requested to play a mixed action. In this particular example we can even choose the equilibrium play to be periodic. This can be done as follows.
First note that $d_{1} + d_{2} \leq 1$. This follows since $\mathbb{P}_{x}(U_{1}) + \mathbb{P}_{x}(U_{2}) \leq 1$ (because $U_1$ and $U_2$ are disjoint by supposition) and since $d_i \leq \mathbb{P}_{x}(U_{i})$ for $i=1,2$ (because the Nash equilibrium payoff is at least the minmax value). Let $\varepsilon > 0$. Choose natural numbers $m$, $m_{1}$, and $m_{2}$ such that $d_{i} - \varepsilon \leq \tfrac{m_{i}}{m} \leq d_{i}$, for $i=1,2$. Note that $m_{1} + m_{2} \leq m$. Pick a point $a_{1} \in U_{1}$ and a point $a_{2} \in U_{2}$, and let $p^*$ be the periodic play with period $m_1+m_2$
obtained by repeating $a_{1}$ for the first $m_{1}$ stages, and repeating $a_{2}$ for the next $m_{2}$ stages.
We have
\[d_{i} - \varepsilon \leq \tfrac{m_{i}}{m} \leq \tfrac{m_{i}}{m_{1} + m_{2}} = f_{i}(p^*),\]
for $i \in \{1,2\}$. One can support $p^*$ as an $\varepsilon$-equilibrium play by a threat of punishment: in case of a deviation by player $1$, player $2$ will switch to playing the minmax action profile from the stage game for the rest of the game, thus reducing $i$'s payoff to $d_{i}$. A symmetric punishment is imposed on player 2 in case of a deviation.
Under the periodic play $p^*$, the sum of the players' payoffs is $1$. There are alternative plays where the payoff to \textit{both} players is 1, which can support 0-equilibria.
For example, consider the non-periodic play $p$ that is played in blocks of increasing size: for each $k \in \mathbb{N}$, the length of block $k$ is $2^{2^k}$. In even (resp.~odd) blocks the players play the action profile $a_1$ (resp.~$a_2$). The reader can verify that since the ratio between the length of block $k$ and the total length of the first $k$ blocks goes to $\infty$, the payoff to both players at $p$ is 1.
\end{exl}
\section{Regularity and the folk theorem}\label{secn.folk}
A payoff vector $w\in \mathbb{R}^{|I|}$, assigning a payoff to each player, is called \emph{an equilibrium payoff} of the Blackwell game $\Gamma$ if for every $\varepsilon>0$ there exists an $\varepsilon$-equilibrium $\sigma^{\varepsilon}$ of $\Gamma$ such that $\|w-\mathbb{E}_{\sigma^{\varepsilon}}(f)\|_{\infty}\leq \varepsilon$. In other words, an equilibrium payoff is an accumulation point of $\varepsilon$-equilibrium payoff vectors as $\varepsilon$ goes to $0$. We let $\mathcal{E}$ denote the set of equilibrium payoffs. Our goal here is to provide a description of $\mathcal{E}$.
In repeated games with stage payoffs,
where the total payoff is some average (discounted average with low discounting, liminf of average, limsup of average, etc.) of the stage game payoffs,
the folk theorem states that the set of equilibrium payoffs coincide
with the set of all individually rational vectors
that are in the convex hull of the feasible payoff vectors,
see, e.g., Aumann and Shapley \cite{Aumann94}, Sorin \cite{Sorin92}, and Mailath and Samuelson \cite{{Mailath06}}. As we will see, when the payoff functions are general, the set of equilibrium payoffs is the convex hull of the set of feasible payoff vectors that are individually rational. The reason for the difference is that in repeated games with stage payoffs, getting a low payoff in one stage can be compensated by getting a high payoff in the following stage; when the payoff is obtained only at the end of the game, there is no opportunity to compensate low payoffs.
Define
\begin{align*}
Q^\varepsilon(f)&:=\bigcap_{i\in I} Q_{i, \varepsilon}(f_i),\\
W^\varepsilon(f)&:=\{f(p):p\in Q^\varepsilon(f)\}.
\end{align*}
The set $Q^\varepsilon(f)$ is the set of $\varepsilon$-individually rational plays, and $W^\varepsilon(f)$ is the set of feasible and $\varepsilon$-individually rational payoffs vectors.
Whenever convenient, we write simply $Q^\varepsilon$ and $W^\varepsilon$. For every set $X$ in a Euclidean space we denote its closure by $\textnormal{cl}(X)$ and its convex hull by $\textnormal{conv}(X)$.
\begin{theorem}\label{thrm:folk}
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Then
\[\mathcal{E}=\bigcap_{\varepsilon>0}\textnormal{conv}(\textnormal{cl}(W^\varepsilon(f))).\]
\end{theorem}
To prove Theorem~\ref{thrm:folk} we need the following result, which states that every $\varepsilon$-equilibrium assigns high probability to plays in $Q^{\varepsilon^{1/3}}(f)$.
\begin{lemma}\label{lemma:largeprob}
Consider an $n$-player Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$. Suppose that for each player $i \in I$, player $i$'s action set $A_i(h)$ at each history $h \in H$ is finite, her payoff function $f_i$ is bounded and upper semi-analytic, and her minmax value is history-independent. Let $\varepsilon > 0$ be sufficiently small, and let $\sigma^\varepsilon$ be an $\varepsilon$-equilibrium. Then
\[\mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q^{\varepsilon^{1/3}}(f)) < n\varepsilon^{1/3}.\]
\end{lemma}
\begin{proof}
Set $\eta := \varepsilon^{1/3}$. It suffices to show that for every $i \in I$,
\begin{equation}\label{equ:231}
\mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q_{i, \eta} (f_i)) < \eta.
\end{equation}
Fix a player $i \in I$ and suppose to the contrary that Eq.~\eqref{equ:231} does not hold.
We derive a contradiction by showing that player~$i$ has a deviation from $\sigma^{\varepsilon}$ that yields her a gain higher than $\varepsilon$.
For $t \in \mathbb{N}$, denote by $X_t := \mathbb{P}_{\sigma^{\varepsilon}} (Q_{i,\eta}(f_i) |\mathcal{F}_t)$ the conditional probability of the event $Q_{i,\eta}(f_i)$ under the strategy profile $\sigma^\varepsilon$ given the sigma-algebra $\mathcal{F}_t$. By Doob's martingale convergence theorem, $(X_{t})_{t \in \mathbb{N}}$ converges to the indicator function of the event $Q_{i,\eta}(f_i)$, almost surely under $\mathbb{P}_{\sigma^{\varepsilon}}$. Since by supposition $\mathbb{P}_{\sigma^{\varepsilon}}(A^{\mathbb{N}} \setminus Q_{i, \eta} (f_i)) > \eta$, we know that $\mathbb{P}_{\sigma^\varepsilon}(X_{t} \to 0) > \eta$.
Let $K$ be a bound on the game's payoffs, and let $\rho := \varepsilon^2/K$. Let us call a history $h \in H_{t}$ a \textit{deviation history} if under $h$, stage $t$ is the first one such that $X_t < \rho$. On the event $\{X_{t} \to 0\}$, a deviation history arises at some point during play. Consequently, under $\mathbb{P}_{\sigma^{\varepsilon}}$, a deviation history arises with probability of at least $\eta$.
Consider the following strategy $\sigma_{i}'$ of player $i$: play according to $\sigma_i^\varepsilon$ until a deviation history, say $h$, occurs (and forever if a deviation history never occurs). At $h$, switch to playing a strategy which guarantees player $i$ a payoff of at least $v_i(f_i) - \varepsilon$ against $\sigma_{-i}^{\varepsilon}$ in $\Gamma_h$. Such a strategy exists by our supposition of history-independence of the minmax values. To conclude the argument, we compute the gain from the deviation to $\sigma_{i}'$.
For every deviation history $h \in H_{t}$,
\begin{eqnarray}\label{equ:232}
\mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}'}(f_{i}\mid h) &\geq& v_{i}(f_i) - \varepsilon,\\
\mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}^{\varepsilon}}(f_{i}\mid h) &\leq& \rho K + (1- \rho)(v_i(f_i) - \eta).
\label{equ:233}
\end{eqnarray}
Eq.~\eqref{equ:232} holds by the choice of $\sigma_{i}'$. To derive Eq.~\eqref{equ:233}, suppose that, following the history $h$, player $i$ conforms to $\sigma_{i}^{\varepsilon}$. Then, conditional on $h$, with probability at most $\rho$ the play belongs to $Q_{i,\eta}(f_i)$, and player $i$'s payoff is at most $K$, and with probability at least $1-\rho$ the play does not belong to $Q_{i,\eta}(f_i)$, and player~$i$'s payoff is at most $v_i(f_i)-\eta$.
We can now compute the gain from the deviation to $\sigma_{i}'$: If a deviation history never arises, $\sigma_{i}'$ recommends the same actions as $\sigma_{i}^{\varepsilon}$, and therefore the gain is 0.
A deviation history occurs with a probability of at least $\eta$, and thus
\begin{align*}
\mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}'}(f_{i}) - \mathbb{E}_{\sigma_{-i}^{\varepsilon},\sigma_{i}^{\varepsilon}}(f_{i}) &\geq
\eta \bigl(v_i(f_i)-\varepsilon - \rho K - (1 - \rho)(v_i(f_i)-\eta)\bigr)\\
&= \eta(-\varepsilon - \varepsilon^2 + \rho v_i(f_i) + \eta - \rho \eta)\\
&= \varepsilon^{\frac{2}{3}}(1 - \varepsilon^{\frac{2}{3}} - \varepsilon^{\frac{5}{3}} + \tfrac{v_i(f_i)}{K} \varepsilon^{\frac{5}{3}} - \tfrac{1}{K} \varepsilon^{2}),
\end{align*}
which behaves like $\varepsilon^{\frac{2}{3}}$ when $\varepsilon$ is small, and therefore exceeds $\varepsilon$. \end{proof}
\noindent\textbf{Proof of Theorem~\ref{thrm:folk}}: Let $|I| = n$. Let $w \in \mathbb{R}^{n}$ be an equilibrium payoff. Assume by contradiction that there is an $\alpha > 0$ such that $w \not\in \textnormal{conv}(\textnormal{cl}(W^{\alpha}))$. For a vector $z \in \mathbb{R}^{n}$ write ${\rm dist}(z)$ to denote the distance from $z$ to the set $\textnormal{conv}(\textnormal{cl}(W^{\alpha}))$ under the $\|\cdot\|_{\infty}$ metric on $\mathbb{R}^{n}$.
By assumption, $\delta := \tfrac{1}{4}{\rm dist}(w) > 0$. Denote $\varepsilon: = \min(\delta, \alpha^3, (\frac{\delta}{Kn})^3) > 0$, where $K$ is a bound on the game payoff.
From $w$ being an equilibrium payoff, there exits an $\varepsilon$-equilibrium, say $\sigma^\varepsilon$, such that $\|w-\mathbb{E}_{\sigma^\varepsilon}(f)\|_{\infty} \leq \varepsilon \leq \delta$. We have the following chain of inequalities:
\[{\rm dist}(\mathbb{E}_{\sigma^\varepsilon}(f)) \leq \mathbb{E}_{\sigma^\varepsilon}({\rm dist}(f)) \leq 2K \cdot \mathbb{P}_{\sigma^\varepsilon}(A^{\mathbb{N}} \setminus Q^{\alpha}(f)) \leq 2K \cdot n \cdot \varepsilon^{\frac{1}{3}} \leq 2\delta,\]
where the first inequality follows from the fact that ${\rm dist}:\mathbb{R}^{n} \to \mathbb{R}$ is a convex function, the second from the fact that $f(p) \in W^{\alpha}$ whenever $p \in Q^{\alpha}(f)$, the third follows since $Q^{\varepsilon^{1/3}}(f) \subseteq Q^{\alpha}(f)$ and
by Lemma~\ref{lemma:largeprob}, and the last holds by the choice of $\varepsilon$.
But then
\[{\rm dist}(w) \leq \|w-\mathbb{E}_{\sigma^\varepsilon}(f)\|_{\infty} + {\rm dist}(\mathbb{E}_{\sigma^\varepsilon}(f)) \leq 3\delta,\]
contradicting the choice of $\delta$.
We turn to prove the other direction. Let $w\in \bigcap_{\varepsilon>0}\textnormal{conv}(\textnormal{cl}(W^\varepsilon))$. We need to show that $w$ is an equilibrium payoff. Fix an $\varepsilon>0$.
Carath\'eodory's Theorem (Carath\'eodory, \cite{Caratheodory07}) implies that $\textnormal{cl} (\textnormal{conv}(W^{\varepsilon})) = \textnormal{conv} (\textnormal{cl}(W^{\varepsilon}))$, hence $w$ is an element of $\textnormal{cl} (\textnormal{conv}(W^{\varepsilon}))$, and thus we can choose a vector $w_{\varepsilon}\in \textnormal{conv}(W^{\varepsilon})$ such that $\|w-w_{\varepsilon}\|_\infty \leq \varepsilon$. We argue that $w_{\varepsilon}$ is a vector of expected payoffs in some $3\varepsilon$-equilibrium.
The payoff $w_{\varepsilon}$ can be presented as a convex combination of $n + 1$ vector payoffs, say \linebreak $f(p^1), \ldots, f(p^{n+1})$, with each $p^{k}$ an element of $Q^{\varepsilon}(f)$. Using jointly controlled lotteries as done, e.g., in Forges \cite{Forges}, Lehrer \cite{Lehrer1996}, or Lehrer and Sorin \cite{LehrerSorin1997}, the players can generate the required randomization over the plays $p^1, \dots, p^{n+1}$ during the first stages of the game. Once a specific play $p^k$ has been chosen, the construction of the $3\varepsilon$-equilibrium is standard: the players play $p^k$, and if player $i$ deviates, her opponents revert to playing a strategy profile that gives player $i$ at most $v_i(f_i)+\varepsilon$. Such a strategy exists by the assumption of history-independence of the minmax values. $\Box$
\begin{exl}\label{exl.folk}\rm
Consider the 2-player Blackwell game $\Gamma = (\{1,2\}, A_{1}, A_{2}, f_{1}, f_{2})$, where the action sets are $A_1=\left\{{\rm T},{\rm M},{\rm B}\right\}$ and $A_2=\left\{{\rm L},{\rm C},{\rm R}\right\}$, and for a play $p = (a_0,a_1,\ldots)$ the payoffs are
\[(f_1(p),f_2(p)) = \begin{cases}
(1,1)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm T},{\rm L})\text{ or } a_k = ({\rm M},{\rm C})\} >\frac{1}{2},\\
(4,-1)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm B},{\rm L})\} = 1,\\
(-1,4)&\text{if }\displaystyle\liminf_{n\to \infty} \tfrac{1}{t} \cdot \#\{k < t: a_k = ({\rm T},{\rm R})\} = 1,\\
(0,0)&\text{otherwise}.
\end{cases}\]
Thus the payoff is $(1,1)$ if the (liminf) frequency of the stages where either (T,L) or (M,C) is played is larger than $\tfrac{1}{2}$. It is $(4,-1)$ if (B,L) is played with frequency of 1, and $(-1,4)$ if (T,R) is played with the frequency of 1. All other cases result in a payoff of $(0,0)$.
Observe that when player 1 plays B repeatedly, the maximal payoff that player 2 can achieve is 0, and this is player 2's minmax value. Similarly, player 1's minmax value is 0. For each $\varepsilon \in (0,1)$, the set $W^{\varepsilon}(f)$ consists of the two points $(0,0)$ and $(1,1)$. By Theorem~\ref{thrm:folk}, the set of equilibrium payoffs $\mathcal{E}$ is the line segment connecting $(0,0)$ and $(1,1)$, see Figure \ref{figure}.
Naturally, all equilibrium payoffs $w$ are (a) convex combinations of the feasible payoffs vectors $(0,0)$, $(1,1)$, $(4,-1)$, and $(-1,4)$, and (b) individually rational, i.e., they satisfy $w_{1} \geq 0$ and $w_{2} \geq 0$. The set of all payoff vectors satisfying (a) and (b) is represented in Figure~\ref{figure} by the shaded triangle. The point we wish to make here is that the properties (a) and (b) are not sufficient for a payoff vector to be an equilibrium payoff.
Take for concreteness the point $(3,0)$. This payoff vector is in the convex hull of the feasible payoff vectors and is individually rational. Yet, for $\varepsilon < \tfrac{2}{3}$, there is no $\varepsilon$-equilibrium with the payoff (close to) the vector $(3,0)$. We give a heuristic argument.
Suppose to the contrary that $\sigma$ is such an $\varepsilon$-equilibrium. The strategy profile $\sigma$ necessarily assigns a probability of at least $\tfrac{2}{3}$ to the set of plays that yield the payoff vector $(4,-1)$. But this implies that Player 2 has a deviation that would improve her payoff over the candidate $\varepsilon$-equilibrium by at least $\tfrac{2}{3}$. Player 2 needs to deviate to playing R forever (for example), at any history of the game where her conditional expected payoff under $\sigma$ is close enough to $-1$. Since playing R would yield at least $0$, by such a deviation, she would improve her conditional expected payoff by at least $1$. Levy's zero-one law guarantees that the histories where player 2 is called to deviate in this way arise with a probability close to $\tfrac{2}{3}$, so that the expected gain from the deviation is also close to $\tfrac{2}{3}$.
\begin{figure}
\caption{The set of equilibrium payoffs (the segment connecting $(0,0)$ and $(1,1)$) vs. the set of convex combinations of feasible payoffs that are individually rational (the dark triangle).}
\label{figure}
\end{figure}
\end{exl}
The above discussion of Example~\ref{exl.folk} leads to a slightly more general conclusion: if the set of feasible payoffs is finite, then the set of equilibrium payoffs is the convex-hull of the feasible payoffs that are individually rational (equal or larger than the minmax).
For each player the minmax value is within the finite set of feasible payoffs, and placing any probability on a payoff that is not individually rational enables profitable deviations.
\section{Blackwell games with tail-measurable payoffs}\label{secn.tail}
An important class of games with history-independent minmax values are those where the payoff functions are tail-measurable.
In this section we concentrate on games with tail-measurable payoffs.
Consider a Blackwell game with history-independent action sets, $\Gamma = (I,(A_{i},f_{i})_{i \in I})$. A set $Q \subseteq A^{\mathbb{N}}$ is said to be a \textit{tail set} if whenever a play $p = (a_{0},a_{1},\ldots)$ is an element of $Q$ and $q = (b_{0},b_{1},\ldots)$ is such that $a_t = b_t$ for all $t \in \mathbb{N}$ sufficiently large, then $q$ is also an element of $Q$. Let $\mathscr{T}$ denote the sigma-algebra of the tail subsets of $A^{\mathbb{N}}$. We note that the tail sigma-algebra $\mathscr{T}$ and the Borel sigma-algebra $\mathscr{B}$ are not nested. For constructions of tail sets that are not Borel, see Rosenthal \cite{Rosenthal75} and Blackwell and Diaconis \cite{Diaconis96}.
Examples of tail sets are: (1) the winning sets of Example \ref{exl.io}, (2) the set of plays in which a certain action profile $a\in A$ is played with limsup-frequency at most $\tfrac{1}{2}$, and (3) the set of plays in which a certain action profile $a^*\in A$ is played at most finitely many times at even stages (with no restriction at odd stages).
An important class of tail sets are the shift invariant sets. A set $Q \subseteq A^{\mathbb{N}}$ is a \textit{shift invariant set} if for each play $p = (a_0,a_1,\ldots)$, $p \in Q$ if and only if $(a_1,a_2,\ldots) \in Q$. Equivalently, shift invariant sets are the sets that are invariant under the backward shift operator on $A^{\mathbb{N}}$. Shift invariant sets are tail sets. The converse is not true: while the sets in examples (1) and (2) above are shift invariant, that of example (3) is not.
A function $f:A^{\mathbb{N}} \to \mathbb{R}$ is called tail-measurable if, for each
$r\in\mathbb{R}$, the set $\{p\in A^{\mathbb{N}} : r \leq f(p)\}$ is an element of $\mathscr{T}$. Intuitively, a payoff function is tail measurable if an action taken in any particular stage of the game has no impact on the payoff. The payoff function in Example \ref{exl.eq} is tail-measurable.
\begin{remark}\rm
The assumption that the set of actions of each player is history-independent is required so that the tail-measurability of the payoff functions has a bite.
If the sets of actions were history-dependent,
then by having a different set of actions at each history,
any function could be turned into tail-measurable.
\end{remark}
We now state one key implication of tail-measurability, namely the history-independence of minmax values.
\begin{proposition}\label{prop:minmaxtail}
Let $\Gamma = (I,(A_{i},f_{i})_{i \in I})$ be a Blackwell game with history-independent action sets, and let $i \in I$ be a player. If player $i$'s payoff function is bounded, upper semi-analytic, and tail-measurable, then her minmax value is history-independent.
\end{proposition}
\begin{proof}
It suffices to show that $v_{i}(f_{i,a}) = v_{i}(f_{i})$ for each $a \in A$, where, with a slight abuse of notation, we write $a$ for a history in stage $1$. Since $f_i$ is tail-measurable, all the functions $f_{i,a}$ for $a \in A$ are identical to each other. Hence, fixing any particular action profile $\bar{a} \in A$, letting $X_i := \Delta(A_i)$ and $X_{-i} := \prod_{j \in -i}X_{j}$, we have
\begin{align*}
v_{i}(f_{i}) &= \inf_{x_{-i} \in X_{-i}}\sup_{x_{i} \in X_{i}} \sum_{a \in A} \prod_{j \in I}x_{j}(a_{j}) \cdot \Big(\inf_{\sigma_{-i} \in \Sigma_{-i}}\sup_{\sigma_{i} \in \Sigma_{i}} \mathbb{E}_{\sigma_{-i},\sigma_{i}}(f_{i,a})\Big)\\
&= \inf_{x_{-i} \in X_{-i}}\sup_{x_{i} \in X_{i}} \sum_{a \in A} \prod_{j \in I}x_{j}(a_{j}) \cdot v_{i}(f_{i,\bar{a}}) = v_{i}(f_{i,\bar{a}}).
\end{align*}
\end{proof}
If the payoff functions of all the players in a game $\Gamma$ are tail-measurable, then, for each fixed stage $t \in \mathbb{N}$, all the subgames of $\Gamma$ starting at stage $t$ are identical. On the other hand, the subgames starting, say, at stage $1$, are not identical to the game itself (see example (3) of a tail-measurable payoff function above). Nonetheless, as Proposition \ref{prop:minmaxtail} implies, the players' minmax values \textit{are} the same in every subgame.
The condition of history-independence of the minmax values is more inclusive than that of tail-measurability of the payoffs; the examples that follow illustrate the point.
\begin{exl}\rm
Consider a one-player Blackwell game where the player's payoff function is $1_{S}$, the indicator of a set $S \subseteq [H]$.
If $S$ is dense in $[H]$, then the minmax value of the player is $1$ in each subgame. A dense set may or may not be a tail set.
\end{exl}
\begin{exl}\rm
We consider a Blackwell game similar to that of Example \ref{exl.io}, but where the stage game may depend on the history, as long as each player's stage minmax value is the same.
Specifically, let $\Gamma = (I, A, H, (1_{W_i})_{i \in I})$. Suppose that at each history $h \in H$, each player $i \in I$ has a stage winning set $U_{i}(h) \subseteq A(h)$, and her winning set in the Blackwell game $\Gamma$ is
\[W_i = \{(a_0,a_1,\ldots) \in [H]:a_t \in U_i(a_0,\ldots,a_{t-1})\text{ for infinitely many }t \in \mathbb{N}\}.\]
Assume that the stage minmax value of player $i$ is the same at each history: there is a number $d_{i}$ such that
\[d_{i} = \inf_{x_{-i} \in \Delta(A_{-i}(h))} \sup_{x_i \in \Delta(A_i(h))} \mathbb{P}_{x_{-i},x_i}(U_i(h))\]
for every $h \in H$. Then player $i$'s minmax value in each subgame of $\Gamma$ is $0$ if $d_{i} = 0$, and is $1$ if $d_{i} > 0$. Thus player $i$'s minmax value is history-independent.
Note that the game $\Gamma$ need not have history-independent action sets. Even when the action sets \textit{are} history-independent, the winning sets need not necessarily be tail-measurable.
To illustrate the last claim, suppose that there are two players playing matching pennies at each stage. At stage 0, player 1 wants to match the choice of player 2 (and player 2 wants to mismatch the choice of player~1). Subsequently the roles of the two players swap as follows: the player to win stage $t$ wants to match her opponent's action at stage $t+1$, while the loser at stage $t$ wants to mismatch the action of her opponent at stage $t+1$. Formally, we let $\Gamma = (\{1,2\}, A_1, A_2, 1_{W_{1}}, 1_{W_{2}})$ be the 2-player Blackwell game with history-independent action sets, where $A_1 = A_2 = \{{\rm H},{\rm T}\}$, the winning sets $W_1$ and $W_2$ are as above, and the stage winning sets are defined recursively as follows:
\[U_1(\oslash) = \{({\rm H},{\rm H}), ({\rm T},{\rm T})\}\quad\text{and}\quad
U_2(\oslash) = \{({\rm H},{\rm T}), ({\rm T},{\rm H})\},\]
and
\[U_1(h,a) = \begin{cases}
U_1(\oslash) &\text{if }a \in U_1(h),\\
U_2(\oslash) &\text{if }a \in U_2(h),
\end{cases}\quad\text{and}\quad
U_2(h,a) = \begin{cases}
U_2(\oslash) &\text{if }a \in U_1(h),\\
U_1(\oslash) &\text{if }a \in U_2(h),
\end{cases}\]
for each $h \in H$ and $a \in A$. The sets $W_1$ and $W_2$ are not tail: out of the two plays
\begin{align*}
&(({\rm H},{\rm H}), ({\rm H},{\rm H}),({\rm H},{\rm H}),\ldots) \hbox{ and}\\
&(({\rm H},{\rm T}), ({\rm H},{\rm H}), ({\rm H},{\rm H}),\ldots),
\end{align*}
the first is an element of $W_1 \setminus W_2$, while the second is an element of $W_2 \setminus W_1$.
\end{exl}
\begin{exl}\rm
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$, where player $i$'s objective is (as in Example \ref{exl.eq}) to maximize the long-term frequency of the stages she wins:
\[f_{i}(a_{0},a_{1},\ldots) = \limsup_{t \to \infty}\tfrac{1}{t}\cdot\#\{k < t : a_{k} \in U_{i}(a_0,\ldots,a_{k-1})\}.\]
As in the previous example, $U_{i}(h) \subseteq A(h)$ is player $i$'s stage winning set at history $h \in H$. Assume, as above, that player $i$'s minmax value in each stage game is $d_i$. Then also her minmax value in each subgame of $\Gamma$ is $d_{i}$.
\end{exl}
\begin{exl}\rm
Start with a Blackwell game with tail-measurable payoff functions. Suppose that the minmax values of all the players in the game are $0$. Take any history $h$, and redefine the payoff functions so that any play having $h$ as a prefix has a payoff of $0$. In the resulting game, the minmax value of each player in each subgame remains $0$, but the payoff functions are no longer tail-measurable (unless the original payoff functions are constant). A similar modification can be performed with any subset of histories, not just one.
\end{exl}
From the results above we now deduce a number of implications for Blackwell games with tail-measurable payoffs.
\begin{corollary}\label{cor.0-1law}
Consider a Blackwell game $\Gamma = (I, (A_{i}, 1_{W_{i}})_{i \in I})$ with history-independent action sets. If player $i$'s winning set $W_i$ is an analytic tail set, then $v_i(W_i)$ is either $0$ or $1$.
\end{corollary}
\begin{proof}
Suppose that $v_i(W_i) > 0$. Let $\varepsilon := v_i(W_i)/2$. In view of Proposition \ref{prop:minmaxtail}, player $i$'s minmax value in $\Gamma$ is history-independent. Applying Proposition \ref{prop:v(Q)=1}, we conclude that $v_i(Q_{i, \varepsilon}(1_{W_{i}})) \linebreak = 1$. But $Q_{i, \varepsilon}(1_{W_{i}}) = W_{i}$ by the choice of $\varepsilon$.
\end{proof}
The following conclusion follows directly from Proposition \ref{prop:minmaxtail} and Theorem \ref{theorem:minmax_indt}.
\begin{corollary}\label{cor:eq}
Suppose that the game $\Gamma = (I, (A_{i}, 1_{W_{i}})_{i \in I})$ has history-independent action sets. Suppose, furthermore, that for each player $i \in I$, player $i$'s action set $A_i$ is finite and her payoff function $f_i$ is bounded, upper semi-analytic, and tail-measurable. Then for every $\varepsilon>0$ the game admits an $\varepsilon$-equilibrium.
\end{corollary}
\section{Concluding remarks}\label{secn.disc}
\noindent\textbf{Approximations by compact sets.} Any Borel probability measure on
$[H]$ (recall that $[H]$ is Polish under the maintained assumptions), is not merely regular, but is tight: the probability of a Borel set $B \subseteq [H]$ can be approximated from below by the probability of a compact subset $K \subseteq B$ (Kechris \cite[Theorem 17.11]{Kechris95}). The minmax value is not tight in this sense. To see this, consider any 2-player Blackwell game where player 1's winning set $W_{1}$ is the entire set of plays $[H]$, so that $v_1(W_1) = 1$,
and where $A_{2}(\o)$, player 2's action set at the beginning of the game, is $\mathbb{N}$.
We argue that $v_1(K) = 0$ for every compact set $K \subseteq W_1$. Indeed,
the projection of a compact set $K \subseteq W_1$ on $A_{2}(\o)$ is a compact, and hence a finite set.
Therefore, player 2 can guarantee that the realized play is outside $K$ by choosing a sufficiently large action at stage $0$. Thus $v_1(K) = 0$, as claimed.
\noindent\textbf{Approximations by semicontinuous functions.} The conclusion of Theorem \ref{thrm:tailapprox} would no longer be true without the assumption of history-indepenence of the minmax values. Here we give an example of a game with a limsup payoff function where the minmax value cannot be approximated from below by an upper semicontinuous function.
Consider a zero-sum game $\Gamma$ where $A_{1} = A_{2} = \{0,1\}$, and player 1's payoff function is
\[f(a_0,a_1,\ldots) = \begin{cases}
\displaystyle\limsup_{t \to \infty}\tfrac{1}{t}\#\{k < t: a_{2,k} = 0\},&\text{if }\tau = \infty,\\
2, &\text{if } \tau < \infty\text{ and }a_{2,\tau} = 1,\\
0, &\text{if } \tau < \infty\text{ and }a_{2,\tau} = 0,
\end{cases}\]
where $\tau = \tau(a_0,a_1,\ldots) \in \mathbb{N} \cup \{\infty\}$ is the first stage where player 1 chooses action $1$. The game was analyzed in Sorin \cite{Sorin86}, who showed that $v_1(f) = 2/3$.
Let $g \leq f$ be a bounded upper semicontinuous function. We argue that $v_1(g) \leq1/2$.
For $t \in \mathbb{N}$, let $S_t$ denote the set of plays $p$ such that $t \leq \tau(p)$. Note that $S_{t}$ is closed.
We argue that
\[\inf_{t \in \mathbb{N}} \sup\{g(p):p \in S_{t}\} \leq 1.\]
Suppose this is not the case. Take an $\varepsilon > 0$ such that $1 + \varepsilon < \sup\{g(p):p \in S_{t}\}$ for each $t \in \mathbb{N}$. Let $U_{0} := \{1 + \varepsilon \leq g\}$, and for each $t \geq 1$ let $U_{t} := U_{0} \cap S_{t}$.
The set $U_{t}$ is not empty for each $t \in \mathbb{N}$. Moreover, it is a closed, and hence a compact subset of $A^{\mathbb{N}}$. Thus $U_{0} \supseteq U_{1} \supseteq \cdots$ is a nested sequence of non-empty compact sets. Therefore, there is a play $p \in \bigcap_{t \in \mathbb{N}} U_{t}$. It holds that $\tau(p) = \infty$, and consequently $f(p) \leq 1 < g(p)$, a contradiction.
Take an $\varepsilon > 0$. Find a $t \in \mathbb{N}$ such that $\sup\{g(p):p \in S_{t}\} \leq 1 + \varepsilon$. Suppose that player 2 plays $0$ for the first $t$ stages, and thereafter plays $0$ with probability $1/2$ at each stage. This guarantees that the payoff under the function $g$ is at most $(1 + \varepsilon)/2$.
\noindent\textbf{On the assumption of finiteness of the action sets.} The hypothesis of Theorem \ref{theorem:sumofprob} requires that the action sets at each history be finite, and its conclusion is not true without this assumption. Indeed, consider the 2-player Blackwell game $\Gamma = (\{1,2\}, A_{1}, A_{2}, W_{1}, W_{2})$ with history-independent action sets $A_1 = A_2 = \mathbb{N}$. Player 1's winning set $W_1$ consists of all plays $(a_{1,t}, a_{2,t})_{t\in\mathbb{N}}$ such that $a_{1,t} > a_{2,t}$ holds for all sufficiently large $t \in \mathbb{N}$, and player 2's winning set $W_2$ consists of all plays $(a_{1,t},a_{2,t})_{t\in\mathbb{N}}$ such that $a_{1,t} < a_{2,t}$ holds for all sufficiently large $t \in \mathbb{N}$. Then $W_1$ and $W_2$ are Borel-measurable and tail-measurable, and $v_1(W_1) = v_2(W_2)=1$, but $W_1\cap W_2=\emptyset$. Hence, the game has no $\varepsilon$-equilibrium for any $\varepsilon < 1/2$. Indeed, an $\varepsilon$-equilibrium $\sigma$ would need to satisfy $\mathbb{P}_{\sigma}(W_i) \geq v_i(W_i) - \varepsilon > 1/2$ for both $i \in \{1,2\}$.
As discussed above, the assumption that the sets of actions are history-dependent is intertwined with the assumption that the payoffs are tail-measurable.
\noindent\textbf{Continuity of the minmax.} Unlike Borel probability measures, the minmax value is in general not continuous in the following sense: there is an increasing sequence of Borel sets $C_0 \subseteq C_1\subseteq \ldots$ such that $\lim_{n\to \infty} v_i(C_n) < v_i(\bigcup_{n \in \mathbb{N}} C_n)$. In fact, one can construct an example of this kind where $C_{n}$ is both a $G_{\delta}$ and an $F_{\sigma}$ set, as follows. Consider a 2-player Blackwell game with history-independent action sets where $A_1$ is a singleton (player 1 is a dummy) while $A_2$ contains at least two distinct elements. Let $\{p_{0},p_{1},\ldots\}$ be a converging (with respect to any compatible metric on $A^{\mathbb{N}}$) sequence of plays, no two members of which are the same. Let $C_{n} := A^{\mathbb{N}}\setminus\{p_{n},p_{n+1},\ldots\}$. Then $v_1(C_n) = 0$ for each $n \in \mathbb{N}$ while $v_1(\bigcup_{n \in \mathbb{N}}C_n) = v_1(A^{\mathbb{N}}) = 1$.
\noindent\textbf{Maxmin value.} Consider a Blackwell game $\Gamma$, and suppose that player $i$'s payoff function $f_i$ is bounded and upper semi-analytic. Player $i$'s \textit{maxmin value} is defined as
\[z_i(f_i) = \sup_{\sigma_i \in \Sigma_{i}}\inf_{\sigma_{-i} \in \Sigma_{-i}}\mathbb{E}_{\sigma_{-i},\sigma_i}(f_i).\]
The minmax value is not smaller than the maxmin value: $z_i(f_i) \leq v_i(f_i)$. If $I = \{1,2\}$, player 1's payoff function $f_1$ is bounded and Borel-measurable, and for every $h \in H$ either the set $A_1(h)$ of player $1$'s actions or the set $A_2(h)$ of player 2's actions at $h$ is finite, then in fact $z_1(f_1) = v_1(f_1)$, as follows from the determinacy of zero-sum Blackwell games (Martin \cite{Martin98}). Strict inequality might arise for at least two reasons.
The first is the failure of determinacy. The results of Section \ref{secn.approx} are established under the assumption that the action sets be countable, an assumption that is insufficient to guarantee determinacy of a two-player zero-sum Blackwell game even if player 1's winning set is clopen. Wald's game provides an illustration. Suppose that each of the two players chooses a natural number; player 1 wins provided that his choice is at least as large as player 2's. Formally, consider a Blackwell game with $I = \{1,2\}$, where the action sets at $\o$ are $A_1(\o) = A_2(\o) = \mathbb{N}$, and player 1's winning set $W_1$ consists of plays such that player 1's stage $0$ action is at least as large as player 2's stage 0 action: $a_{1,0} \geq a_{2,0}$. Then player 1's minmax value is $v_1(W_1) = 1$ while her maxmin value is $z_1(W_1) = 0$.
The second possibility for a maxmin and the minmax values to be different arises in games with three or more players. The reason is that the definitions of both the maxmin and the minmax values impose that the opponents of player $i$ choose their actions independently after each history. The point is illustrated by Maschler, Solan, and Zamir \cite[Example 5.41]{Maschler13}, which can be seen as a 3-player Blackwell game with binary action sets, where the player's payoff function only depends on the stage 0 action profile.
Analogues of Theorems ~\ref{thrm:reg}, \ref{thrm:regfunc}, and \ref{thrm:tailapprox} could be established for the maxmin values using the same approach.
\noindent\textbf{Open problems.} Existence of an $\varepsilon$-equilibrium in dynamic games with general \linebreak (Borel-measurable) payoffs has been, and still is, one of the Holy Grails of game theory. A more modest approach, also pursued in this paper, is to establish existence in some special classes of games. Blackwell games, as they are defined here, do not include moves of nature. An interesting avenue for a follow up research is to extend the methods developed in this paper to the context of stochastic games with general Borel-measurable payoff functions.
Theorems \ref{thrm:reg} and \ref{thrm:regfunc} provide two distinct approximation results, and neither seems to be a consequence of the other. This raises the question of whether there is a natural single generalization that would encompass both these results as two special cases.
\section{Appendix: The proof of Theorems~\ref{thrm:reg}, \ref{thrm:regfunc}, and \ref{thrm:tailapprox}}\label{subsecn.proof}
The proofs of Theorems~\ref{thrm:reg} and~\ref{thrm:regfunc} are adaptations of the corresponding arguments in Maitra and Sudderth \cite{Maitra98} and in Martin \cite{Martin98} and are provided here for completeness. Theorem \ref{thrm:tailapprox} follows easily from Theorem~\ref{thrm:reg} and Proposition \ref{prop:v(Q)=1}.
Consider a Blackwell game $\Gamma = (I, A, H, (f_{i})_{i \in I})$, fix a player $i \in I$, and suppose that player $i$'s payoff function $f_{i}$ is bounded and Borel-measurable. Also assume w.l.o.g. that $0 \leq f_{i} \leq 1$. When we will consider Theorem~\ref{thrm:reg} we will substitute $f_i = 1_{W_i}$.
Given $h \in H$, let $R(h)$ denote the set of one-shot payoff functions $r: A(h) \to [0,1]$. Let $X_i(h) := \Delta(A_i(h))$ denote player $i$'s set of mixed actions at history $h$, and let $X_{-i}(h) := \prod_{j \in -i}X_{j}(h)$. For $x \in \prod_{i \in I}X_{i}(h)$ we write $r(x)$ to denote $\mathbb{E}_{x}(r)$, the expectation of $r$ with respect to $x$. Player $i$'s minmax value of the function $r \in R(h)$ is
\[d_i(r) := \inf_{x_{-i} \in X_{-i}(h)}\sup_{x_i \in X_i(h)} r(x_{-i},x_i).\]
We next introduce the main tool of the proof, an auxiliary two-player game of perfect information denoted by $G_i(f_i,c)$. This is a variation of the games $G_v$ and $G'_v$ in Martin \cite[pp. 1575]{Martin98}.
Given $c \in (0,1]$ and a Borel measurable function $f_i \colon [H] \to [0,1]$, define the game $G_i(f_i,c)$ as follows:
\begin{itemize}
\item Let $h_0 := \oslash$. Player~I chooses a one-shot payoff function $r_0:A(h_0)\to[0,1]$ such that $d_i(r_0)\geq c$.
\item Player~II chooses an action profile $a_0 \in A(h_0)$ such that $r_0(a_0) > 0$.
\item Let $h_1 := (a_0)$. Player~I chooses a one-shot payoff function $r_1:A(h_1)\to[0,1]$ such that $d_i(r_1)\geq r_0(a_0)$.
\item Player~II chooses an action profile $a_1 \in A(h_1)$ such that $r_1(a_1)>0$.
\item Let $h_2 := (a_0,a_1)$. Player~I chooses a one-shot payoff function $r_2:A(h_2)\to[0,1]$ such that $d_i(r_2)\geq r_1(a_1)$. And so on.
\end{itemize}
This results in a run\footnote{To distinguish histories and plays of $\Gamma$ from those of $G_i(f_i,c)$, we refer to the latter as \textit{positions} and \textit{runs}. To distinguish the players of $\Gamma$ from those of $G_i(f_i,c)$, we refer to the latter as Player I and Player II, using the initial capital letters.} $(r_0,a_0,r_1,a_1,\ldots)$. Player I wins the run if
\[\limsup_{t \to \infty}r_{t}(a_{t}) \leq f_{i}(a_0,a_1,\ldots) \quad\text{and}\quad 0 < f_{i}(a_0,a_1,\ldots).\]
Let $T$ be the set of all legal positions in the game $G_i(f_i,c)$. This is a tree on the set $R \cup A$ where $R := \cup_{h \in H}R(h)$. Sequences of even (odd) length in $T$ are Player I's (Player II's) positions. The tree $T$ is pruned: an active player has a legal move at each legal position of the game. Indeed, consider Player I's legal position in the game $G_i(f_i,c)$ and let $h_t$ denote, as above, the sequence of action profiles produced, to date, by Player II. Then the function $r_t$ which is identically equal to $1$ on the set $A(h_t)$ is a legal move for Player I. Consider now Player II's legal position in $G_i(f_i,c)$, let $h_{t}$ denote the sequence of action profiles produced to date by Player II, and let $r_{t}$ be Player I's latest move. Then $d_i(r_t) > 0$. Therefore, there exists an action profile $a_{t} \in A(h_{t})$ such that $r_{t}(a_{t}) > 0$, and thus $a_{t}$ is Player II's legal move at the given position.
The set $[T]$ is the set of all runs of the game $G_i(f_i,c)$, a subset of $(R \cup A)^{\mathbb{N}}$.
A run is \emph{consistent} with a pure strategy $\sigma_{\rm I}$ of Player~I if it is generated by the pair $(\sigma_{\rm I},\sigma_{\rm I}I)$, for some pure strategy $\sigma_{\rm I}I$ of Player~II.
Runs that are consistent with pure strategies of Player~II are defined analogously.
Player I's pure strategy $\sigma_{{\rm I}}$ in $G_i(f_i,c)$ is said to be \emph{winning} if Player I wins all runs of the game that are consistent with $\sigma_{{\rm I}}$.
\begin{proposition}\label{prop:PlayerI}
Let $c \in (0,1]$ and let $f_i : [H] \to [0,1]$ be a Borel-measurable function. If Player {\rm I}~ has a winning strategy in the game $G_i(f_i,c)$, then there exists a closed set $C \subseteq [H]$ and a limsup function $g : [H] \to [0,1]$ such that $g \leq f_i$, $\{g > 0\} \subseteq C \subseteq \{f_i > 0\}$, and $c \leq v_{i}(g)$. In particular, $c \leq v_i(C)$; and if $f_i = 1_{W_i}$, then $C \subseteq W_i$.
\end{proposition}
\begin{proof}
Fix Player I's winning strategy $\sigma_{{\rm I}}$ in $G_i(f_i,c)$.
\noindent\textsc{Step 1:} Defining $C \subseteq [H]$ and $g : [H] \to [0,1]$.
Let $T_{\rm I} \subseteq T$ denote the set of positions in the game $G_i(f_i,c)$ of even length (i.e., Player I's positions) that are consistent with $\sigma_{\rm I}$, i.e., those positions that can be reached under a strategy profile $(\sigma_{{\rm I}},\sigma_{{\rm I}I})$ for some pure strategy of $\sigma_{{\rm I}I}$ of Player II. Let $\pi_{\rm I} : T_{\rm I} \to H$ be the projection that maps a position of length $2t$ in $G_i(f_i,c)$ to a history of length $t$ in $\Gamma$: Formally, $\pi_{\rm I}(\oslash) := \oslash$, $\pi_{\rm I}(r_0,a_0) := (a_0)$, etc. Let $H_{\rm I} \subseteq H$ be the image of $T_{\rm I}$ under $\pi_{\rm I}$. Since in the tree $T_{\rm I}$ Player I's moves are uniquely determined by $\sigma_{\rm I}$, the map $\pi_{\rm I}$ is in fact a bijection between $T_{\rm I}$ and $H_{\rm I}$. We write $\phi : H_{\rm I} \to T_{\rm I}$ for the inverse of $\pi_{\rm I}$. The map $\phi$ induces a continuous bijection $[H_{\rm I}] \to [T_{\rm I}]$, which we also denote by $\phi$.
We say that positions in $H_{\rm I}$ are \emph{$\sigma_{\rm I}$-acceptable}, and define $C$ to be the set $[H_{\rm I}]$.
For each $t \in \mathbb{N}$, define the function $\rho_t : H_{t} \to \mathbb{R}$ as follows: $\rho_0(\oslash) := c$. Let $t \in \mathbb{N}$ and consider a history $h_{t} \in H_{t}$. If $h_{t}$ is not $\sigma_{\rm I}$-acceptable, we define $\rho_{t+1}(h_{t},a_{t}) := 0$ for each $a_{t} \in A(h_{t})$. Suppose that $h_{t}$ is $\sigma_{\rm I}$-acceptable, and let $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. For each $a_{t} \in A(h_{t})$ define $\rho_{t+1}(h_{t},a_{t}) := r_{t}(a_{t})$. Note that if $h_{t}$ is $\sigma_{\rm I}$-acceptable while $(h_{t},a_{t})$ is not, we have $\rho_{t+1}(h_{t},a_{t}) = r_{t}(a_{t}) = 0$.
Also define $g : [H] \to [0,1]$ by letting
\[g(a_{0},a_{1},\ldots) := \limsup_{t \to \infty}\rho_t(a_{0},\dots,a_{t-1}).\]
\noindent\textsc{Step 2:} Verifying that $g \leq f_i$ and $\{g > 0\} \subseteq C \subseteq \{f_i > 0\}$.
Since $\sigma_{\rm I}$ is Player I's winning strategy in $G_i(f_i,c)$, all runs in $[T_{\rm I}]$ are won by Player I, and hence $[H_{\rm I}] \subseteq \{f_i > 0\}$. For a play $p = (a_{0},a_{1},\ldots)$ in $[H_{\rm I}]$, if $\phi(p) = (r_{0},a_{0},r_{1},a_{1},\ldots)$, then $g(p)$ equals $\limsup_{t \to \infty}r_{t}(a_{t})$. Since the run $\phi(p)$ is won by Player I, we conclude that $g(p) \leq f_i(p)$. Thus $g \leq f_i$ on $[H]$.
\noindent\textsc{Step 3:} Verifying that $c \leq v_{i}(g)$. Since $g \leq 1_{C}$ it will then follow that $c \leq v_i(C)$.
Fix a strategy profile $\sigma_{-i} \in \Sigma_{-i}$ for the players in $-i$ in the game $\Gamma$. Take any $\varepsilonsilon > 0$. We define a strategy $\sigma_i$ for player $i$ in the game $\Gamma$ with the property that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \geq c - 2\varepsilonsilon$.
\noindent\textsc{Step 3.1:} Defining player $i$'s strategy $\sigma_{i}$.
Let $r_0 := \sigma_{\rm I}(\oslash)$, Player I's first move in $G_i(f_i,c)$ according to her strategy $\sigma_{\rm I}$. Define $\sigma_{i}(\oslash)$ to be a mixed action on $A_i(\oslash)$ such that
\[r_0(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) \geq c - \varepsilonsilon.\]
Let $t \geq 1$ and consider a history $h_{t} = (a_{0},\ldots,a_{t-1}) \in H_{t}$ of $\Gamma$. If $h_{t}$ is not $\sigma_{\rm I}$-acceptable, then $\sigma_{i}(h_{t})$ is arbitrary. If $h_{t}$ is $\sigma_{\rm I}$-acceptable, let $\phi(h_{t}) := (r_{0}, a_{0}, \ldots, r_{t-1}, a_{t-1})$ and $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. Define $\sigma_{i}(h_{t})$ to be a mixed action on $A_i(h_{t})$ such that
\[r_t(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) \geq r_{t-1}(a_{t-1})-\varepsilonsilon \cdot 2^{-t}.\]
\noindent\textsc{Step 3.2:} Verifying that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \geq c - 2\varepsilonsilon$.
For each $t \in \mathbb{N}$ let us define $\rho_t^{\varepsilonsilon} := \rho_t - \varepsilonsilon \cdot 2^{-t+1}$. One can think of the functions $\rho_0^{\varepsilonsilon}, \rho_1^{\varepsilonsilon}, \dots$ as a stochastic process on $[H]$ that is measurable with respect to the filtration $\{\mathcal{F}_{t}\}_{t \in \mathbb{N}}$. We now argue that this process is a submartingale with respect to the measure $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$.
Letting $r_0 := \sigma_{\rm I}(\oslash)$ we have
\[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{1}^{\varepsilonsilon}) = \mathbb{E}_{\sigma_{-i},\sigma_{i}}(r_{0}(a_{0})) - \varepsilonsilon=
r_{0}(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) - \varepsilonsilon \geq c-2\varepsilonsilon = \rho_{0}^{\varepsilonsilon}(\oslash).\]
Consider a $\sigma_{\rm I}$-acceptable history $h_{t} = (a_{0},\dots,a_{t-1}) \in H_t$ of length $t \geq 1$. Let $(r_{0},a_{0},\ldots,r_{t-1},a_{t-1}) := \phi(h_{t})$ and $r_{t} := \sigma_{\rm I}(\phi(h_{t}))$. We have
\begin{align*}
\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{t+1}^{\varepsilonsilon} | h_{t}) &= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(r_{t}(a_{t}) | h_{t}) - \varepsilonsilon \cdot 2^{-t}\\ &= r_{t}(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) - \varepsilonsilon \cdot 2^{-t}\\ &\geq r_{t-1}(a_{t-1}) - \varepsilonsilon \cdot 2^{-t} - \varepsilonsilon \cdot 2^{-t}\\ &= \rho_t^{\varepsilonsilon}(h_{t}).
\end{align*}
On the other hand, if $h_{t}$ is not $\sigma_{\rm I}$-acceptable, then
\[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_{t+1}^{\varepsilonsilon} | h_{t}) = -\varepsilonsilon \cdot 2^{-t} > -\varepsilonsilon \cdot 2^{-t+1} = \rho_t^{\varepsilonsilon}(h_{t}).\]
This establishes the submartingale property for $\rho_0^{\varepsilonsilon}, \rho_1^{\varepsilonsilon}, \dots$.
The submartingale property implies that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_t^{\varepsilonsilon}) \geq \rho_0^{\varepsilonsilon}(\oslash) = c - 2\varepsilonsilon$ for each $t \in \mathbb{N}$. Using Fatou lemma we thus obtain
\[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) =\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\limsup_{t \to \infty}\rho_t) \geq \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\limsup_{t \to \infty}\rho_t^{\varepsilonsilon}) \geq \limsup_{t \to \infty}\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\rho_t^{\varepsilonsilon}) \geq c - 2\varepsilonsilon,\]
as desired.
\end{proof}
\begin{proposition}\label{prop:PlayerII}
Let $c \in (0,1]$ and let $f_i : [H] \to [0,1]$ be a Borel-measurable function. If Player {\rm I}I~has a winning strategy in the game $G_i(f_i,c)$, then for every $\varepsilonsilon > 0$ there exists an open set $O \subseteq [H]$ and a limsup function $g : [H] \to [0,1]$ such that $f_i \leq g$, $\{f_i = 1\} \subseteq O \subseteq \{g = 1\}$, and $v_{i}(g) \leq c + \varepsilonsilon$. In particular, $v_i(O) \leq c + \varepsilonsilon$; and if $f_i = 1_{W_i}$, then $W_i \subseteq O$.
\end{proposition}
\begin{proof}
Fix Player II's winning strategy $\sigma_{\rm I}I$ in $G_i(f_i,c)$.
\noindent\textsc{Step 1:} Defining $O \subseteq [H]$ and $g : [H] \to [0,1]$.
We recursively define (a) the notion of a \textit{$\sigma_{\rm I}I$-acceptable} history in the game $\Gamma$, (b) for each $\sigma_{\rm I}I$-acceptable history $h$ in $\Gamma$, Player~I's position $\psi(h)$ in the game $G_i(f_i,c)$, and (c) for each $\sigma_{\rm I}I$-acceptable history $h$ of $\Gamma$, a function $u_{h} : A(h) \to [0,1]$.
The empty history $\oslash$ of $\Gamma$ is $\sigma_{\rm I}I$-acceptable. We define $\psi(\oslash) := \oslash$, the empty history in $G_i(f_i,c)$. Let $t \in \mathbb{N}$ and consider a history $h_{t} \in H_{t}$ of the game $\Gamma$. If $h_{t}$ is not $\sigma_{\rm I}I$-acceptable, so is the history $(h_{t},a_{t})$ for each $a_{t} \in A(h_{t})$. Suppose that $h_{t}$ is $\sigma_{\rm I}I$-acceptable and that Player I's position $\psi(h_{t})$ in $G_i(f_i,c)$ has been defined. Take $a_{t} \in A(h_{t})$. Let $R^{*}(h_{t},a_{t})$ denote the set of Player~I's legal moves at position $\psi(h_{t})$ to which $\sigma_{\rm I}I$ responds with $a_{t}$:
\[R^{*}(h_{t},a_{t}) := \{r_{t} \in R(h_t): (\psi(h_{t}),r_{t}) \in T\text{ and }\sigma_{\rm I}I(\psi(h_{t}),r_{t}) = a_{t}\}.\]
The history $(h_{t},a_{t})$ is defined to be $\sigma_{\rm I}I$\emph{-acceptable} if $R(h_{t},a_{t})$ is not empty. In this case we define
\[u_{h_{t}}(a_{t}) := \inf\{r_{t}(a_{t}): r_{t} \in R^{*}(h_{t},a_{t})\}.\]
Choose $r_{t} \in R^{*}(h_{t},a_{t})$ with the property that
\begin{equation}\label{eqn:proprt}
u_{h_{t}}(a_{t}) \leq r_{t}(a_{t}) \leq u_{h_{t}}(a_{t}) + \varepsilonsilon \cdot 3^{-t-2},
\end{equation}
and define $\psi(h_{t},a_{t}) := (\psi(h_{t}),r_{t},a_{t})$.
Finally, extend the definition of $u_{h}$ to all histories $h$ of $\Gamma$ by setting $u_{h}(a) := 1$ whenever $(h,a)$ is not $\sigma_{\rm I}I$-acceptable.
Let $H_{{\rm I}I}$ be the set of $\sigma_{\rm I}I$-acceptable histories of $\Gamma$. We define the set $O$ to be the complement of $[H_{\rm I}I]$, that is $O := [H] \setminus [H_{\rm I}I]$. Since $[H_{\rm I}I]$ is a closed subset of $[H]$ (e.g. Kechris \cite[Proposition 2.4]{Kechris95}), $O$ is an open subset of $[H]$. Let $T_{{\rm I}I} \subseteq T$ be the image of $H_{\rm I}I$ under $\psi$. The function $\psi_{{\rm I}I} : H_{{\rm I}I} \to T_{{\rm I}I}$ induces a continuous function $\psi_{{\rm I}I} : [H_{{\rm I}I}] \to [T_{{\rm I}I}]$. Note that all runs in $[T_{{\rm I}I}]$ are consistent with Player II's winning strategy $\sigma_{\rm I}I$.
For $t \in \mathbb{N}$ define a function $\upsilon_{t} : H_{t} \to \mathbb{R}$ by letting $\upsilon_0(\oslash) := c$; and for each $t \in \mathbb{N}$ and each history $(h_{t},a_{t}) \in H_{t+1}$, by letting $\upsilon_{t+1}(h_{t},a_{t}) := u_{h_{t}}(a_{t})$. Note that, for $t \in \mathbb{N}$ and $h_{t} \in H_{t}$, we have $\upsilon_{t}(h_{t}) = 1$ whenever $h_{t}$ is not $\sigma_{\rm I}I$-acceptable.
Also define $g : [H] \to [0,1]$ by letting
\[g(a_0,a_1,\ldots) := \limsup_{t \to \infty}\upsilon_{t}(a_0,\ldots,a_{t-1}).\]
\noindent\textsc{Step 2:} Verifying that $f_i \leq g \leq 1$ and $\{f_i = 1\} \subseteq O \subseteq \{g = 1\}$.
The function $g$ is equal to $1$ on the set $O$; thus $O \subseteq \{g = 1\}$. Consider a play $p = (a_0,a_1,\ldots) \in [H_{\rm I}I]$, and let $\psi(p) := (r_{0}, a_{0}, r_{1}, a_{1}, \ldots)$. It follows by \eqref{eqn:proprt} that
\[g(p) := \limsup_{t \to \infty}r_{t}(a_{t}).\]
Since the run $\psi(p)$ is won by Player II, it must hold that either $f_i(p) < g(p)$ or $0 = f_i(p)$; in either case $f_i(p) < 1$ and $f_i(p) \leq g(p)$. We conclude that $[H_{{\rm I}I}] \subseteq \{f_i < 1\}$, or equivalently that $\{f_i = 1\} \subseteq O$, and that $f_i \leq g$ on $[H]$.
\noindent\textsc{Step 3:} Verifying that $v_i(g) \leq c + \varepsilonsilon$. Since $1_{O} \leq g$, it then follows that $v_i(O) \leq c + \varepsilonsilon$.
\noindent\textsc{Step 3.1:} Defining a strategy profile for player~$i$'s opponents.
First we argue that
\begin{equation}\label{eqn argue}
d_i(u_{\oslash}) \leq c.
\end{equation}
Suppose to the contrary that $c \leq d_i(u_{\oslash}) - \lambda$ for some $\lambda > 0$. Define $r_0 \in R(\oslash)$ by letting $r_0(a) := \max\{u_{\oslash}(a) - \lambda,0\}$.
Since $u_{\oslash} - \lambda \leq r_0$, it holds that $c \leq d_{i}(u_{\oslash}) - \lambda \leq d_{i}(r_0)$. Consequently, $r_0$ is a legal move of Player~I in the game $G_i(f_i,c)$ at position $\oslash$. Denote $a_0 := \sigma_{\rm I}I(r_{0})$. As $a_0$ is Player~II's legal move in $G_i(f_i,c)$ at position $(r_0)$, it must be the case that $r_0(a_0) > 0$, and hence $r_0(a_0) = u_{\oslash}(a_{0}) - \lambda$. On the other hand, $r_{0} \in R^{*}(\oslash,a_{0})$, so the definition of $u_{\oslash}$ implies that $u_{\oslash}(a_{0}) \leq r_{0}(a_{0})$, a contradiction.
Take $t \geq 1$, let $h_{t} := (h_{t-1},a_{t-1}) \in H_{t}$ be a $\sigma_{\rm I}I$-acceptable history, and let $r_{t-1}$ be such that $\psi(h_{t}) = (\psi(h_{t-1}),r_{t-1},a_{t-1})$. Then
\begin{equation}\label{eqn:argue1}
d_i(u_{h_{t}}) \leq r_{t-1}(a_{t-1}).
\end{equation}
Indeed, suppose to the contrary that $r_{t-1}(a_{t-1}) \leq d_i(u_{h_{t}}) - \lambda$ for some $\lambda > 0$. Define $r_{t} \in R(h_{t})$ by letting $r_{t}(a) := \max\{u_{h_{t}}(a) - \lambda,0\}$. Since $u_{h_{t}} - \lambda \leq r_{t}$, it holds that $r_{t-1}(a_{t-1}) \leq d_{i}(u_{h_{t}}) - \lambda \leq d_{i}(r_{t})$. Consequently, $r_{t}$ is a legal move of Player~I at position $\psi(h_{t})$. Let $a_{t} := \sigma_{\rm I}I(\psi(h_{t}), r_{t})$. As $a_{t}$ is Player~II's legal move at position $(\psi(h_{t}), r_{t})$, it must be the case that $r_{t}(a_{t}) > 0$, and hence $r_{t}(a_{t}) = u_{h_{t}}(a_{t}) - \lambda$. On the other hand, $r_{t} \in R^{*}(h_{t}, a_{t})$, so the definition of $u_{h_{t}}$ implies that $u_{h_{t}}(a_{t}) \leq r_{t}(a_{t})$, a contradiction.
We now define a strategy profile $\sigma_{-i}$ of $i$'s opponents in $\Gamma$ as follows: For a history $h_{t} \in H_{t}$ of $\Gamma$ let $\sigma_{-i}(h_{t}) \in X_{-i}(h)$ be such that
\begin{equation}\label{eqn str opponent}
u_{h_{t}}(\sigma_{-i}(h_{t}),x_{i}) \leq d_{i}(u_{h_{t}}) + \varepsilonsilon \cdot 3^{-t-1}\text{ for each }x_{i} \in \Delta(A_i(h_{t})).
\end{equation}
\noindent\textsc{Step 3.2:} Verifying that $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) \leq c + \varepsilonsilon$ for each strategy $\sigma_{i} \in \Sigma_{i}$ of player $i$ in $\Gamma$.
Fix a strategy $\sigma_{i} \in \Sigma_{i}$. For $t \in \mathbb{N}$ define a function $\upsilon_{t}^\varepsilonsilon := \upsilon_{t} + \varepsilonsilon \cdot 3^{-t}$. The sequence $\upsilon_0^\varepsilonsilon, \upsilon_1^\varepsilonsilon, \dots$ could be thought of as a process on $[H]$, measurable with respect to the filtration $\{\mathcal{F}_{t}\}_{t \in \mathbb{N}}$. We next show that the process is a supermartingale w.r.t $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$.
By Eqs. \eqref{eqn str opponent} and \eqref{eqn argue},
\begin{align*}
\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_1^\varepsilonsilon)
&= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(u_{\oslash}(a_0)) + \varepsilonsilon\cdot 3^{-1}\\
&= u_{\oslash}(\sigma_{-i}(\oslash),\sigma_{i}(\oslash)) + \varepsilonsilon\cdot 3^{-1}\\
&\leq d_{i}(u_{\oslash}) + \varepsilonsilon\cdot 2 \cdot 3^{-1}\\
&\leq c + \varepsilonsilon = \upsilon_{0}^\varepsilonsilon(\oslash).
\end{align*}
Take $t \geq 1$, let $h_{t} = (h_{t-1},a_{t-1}) \in H_{t}$ be a $\sigma_{\rm I}I$-acceptable history, and let $r_{t-1}$ be such that $\psi(h_{t}) = (\psi(h_{t-1}),r_{t-1},a_{t-1})$. We have by Eqs. \eqref{eqn str opponent}, \eqref{eqn:argue1}, and \eqref{eqn:proprt}:
\begin{align*}
\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_{t+1}^\varepsilonsilon \mid h_{t})
&= \mathbb{E}_{\sigma_{-i},\sigma_{i}}(u_{h_{t}}(a_t) \mid h_{t}) + \varepsilonsilon\cdot3^{-t-1}\\
&= u_{h_{t}}(\sigma_{-i}(h_{t}),\sigma_{i}(h_{t})) + \varepsilonsilon\cdot3^{-t-1}\\
&\leq d_{i}(u_{h_{t}}) + \varepsilonsilon \cdot 2 \cdot 3^{-t-1}\\
&\leq r_{t-1}(a_{t-1}) + \varepsilonsilon \cdot 2 \cdot 3^{-t-1}\\
&\leq u_{h_{t-1}}(a_{t-1}) + \varepsilonsilon \cdot 3 \cdot 3^{-t-1}\\
&= \upsilon_{t}^\varepsilonsilon(h_{t-1},a_{t-1}) = \upsilon_{t}^\varepsilonsilon(h_{t}).
\end{align*}
If, on the other hand, the history $h_{t}$ is not $\sigma_{\rm I}I$-acceptable, then
\[\mathbb{E}_{\sigma_{-i},\sigma_{i}}(\upsilon_{t+1}^\varepsilonsilon \mid h_{t}) = 1+\varepsilonsilon\cdot 3^{-t-1} \leq 1+\varepsilonsilon\cdot 3^{-t} = \upsilon_{t}^\varepsilonsilon(h_{t}).\]
Since the process $\upsilon_0^\varepsilonsilon,\upsilon_1^\varepsilonsilon,\dots$ is bounded below (by 0), by the Martingale Convergence Theorem, it converges pointwise $\mathbb{P}_{\sigma_{-i},\sigma_{i}}$-almost surely; whenever the process converges, its limit is $g$. Hence $\mathbb{E}_{\sigma_{-i},\sigma_{i}}(g) = \mathbb{E}_{\sigma_{-i},\sigma_{i}}(\lim_{t \to \infty}\upsilon_t^\varepsilonsilon) \leq \upsilon_0^\varepsilonsilon(\oslash) = c + \varepsilonsilon$, as desired.
\end{proof}
We now invoke the result of Martin \cite{Martin75} on Borel determinacy of perfect information games. To do so, we endow $[T]$ with its relative topology as a subspace of the product space $(R \cup A)^{\mathbb{N}}$, where $R \cup A$ is given its discrete topology. One can then check that Player I's winning set in $G_i(f_i,c)$ is a Borel subset of $[T]$. It follows that for each $c \in (0,1]$ the game $G_i(f_i,c)$ is determined: either Player~I has a winning strategy in the game or Player~II does. We arrive at the following conclusion.
\begin{proposition}\label{prop:det}
If $v_i(f_i) < c$, then Player II has a winning strategy in $G_i(f_i,c)$. If $c< v_i(f_i)$, then Player I has a winning strategy in $G_i(f_i,c)$.
\end{proposition}
Theorems \ref{thrm:reg} and \ref{thrm:regfunc} follow from Propositions \ref{prop:PlayerI}, \ref{prop:PlayerII}, and \ref{prop:det}.
\noindent\textbf{Proof of Theorem \ref{thrm:tailapprox}:} Take an $\varepsilon > 0$. Without loss of generality, suppose that $f_i$ takes values in $[0,1]$.
By Proposition \ref{prop:v(Q)=1} we know that $v_{i}(Q_{i,\varepsilon}(f_i)) = 1$. To obtain an approximation from below, use Theorem \ref{thrm:reg} to choose a closed set $C \subseteq Q_{i, \varepsilon}(f_i)$ such that $1 - \varepsilon \leq v_{i}(C)$, and define the function $g := (v_i(f_i) - \varepsilon) \cdot 1_{C}$. Then $g \leq f_i$ and $v_i(f_i) - 2\varepsilon \leq (v_i(f_i) - \varepsilon)\cdot(1-\varepsilon) \leq v_{i}(g)$. Since $C$ is closed, $g$ is upper semicontinuous.
By Proposition \ref{prop:v(Q)=1} we know that $v_{i}(U_{i,\varepsilon}(f_i)) = 0$. To obtain an approximation from above, use Theorem \ref{thrm:reg} to choose an open set $O \supseteq U_i^{\varepsilon}(f_i)$ such that $v_{i}(O) \leq \varepsilon$, and define the function $g := v_i(f_i) + \varepsilon + (1 - v_i(f_i) - \varepsilon) \cdot 1_{O}$. Then $f_i \leq g \leq 1$ and $v_{i}(g) \leq v_i(f_i) + 2\varepsilon$. Since $O$ is open, $g$ is lower semicontinuous. $\Box$
\end{document} |
\begin{enumerate}gin{document}
\title{Legendrian $\theta-$graphs}
\author{Danielle O'Donnol$^ \dagger$}
\address{Department of Mathematics, Imperial College London, London SW7 2AZ, UK}
\email{[email protected]}
\author{Elena Pavelescu}
\address{Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA}
\email{[email protected]}
\sigmaubjclass[2010]{Primary 57M25, 57M50; Secondary 05C10}
\thanks{$\dagger$ supported in part by an AMS-Simons Travel Grant}
\date{\today}
\keywords{Legendrian graph, Thurston-Bennequin number, rotation number, $\theta-$graph}
\begin{enumerate}gin{abstract}
In this article we give necessary and sufficient conditions for two triples of integers to be realized as the \textcolor{blue}n and the rotation number of a Legendrian $\theta-$graph with all cycles unknotted.
We show that these invariants are not enough to determine the Legendrian class of a topologically planar $\theta-$graph.
We define the transverse push-off of a Legendrian graph and we determine its self linking number for Legendrian $\theta-$graphs.
In the case of topologically planar $\theta-$graphs, we prove that the topological type of the transverse push-off is that of a pretzel link.
\end{abstract}
\maketitle
\sigmaection{Introduction}\label{intro}
In this paper, we continue the systematic study of Legendrian graphs in $(\mathbb{R}^3, \xi_{std})$ initiated in \cite{ODPa} .
Legendrian graphs have appeared naturally in several important contexts in the study of contact manifolds.
They are used in Giroux's proof of existence of open book decompositions compatible with a given contact structure \cite{G}.
Legendrian graphs also appeared in Eliashberg and Fraser's proof of the Legendrian simplicity of the unknot \cite{EF}.
In this article we focus on Legendrian $\theta-$graphs.
We predominantly work with topologically planar embeddings and embeddings where all the cycles are unknots.
In the first part we investigate questions about realizability of the classical invariants and whether the Legendrain type can be determined by these invariants.
In the second part we introduce the transverse push-off a Legendrian graph and investigate its properties in the case of $\theta-$graphs.
In \cite{ODPa}, the authors extended the classical invariants Thurston-Bennequin number, $tb$, and rotation number, $rot$, from Legendrian knots to Legendrian graphs.
Here we prove that all possible pairs of $(tb, rot)$ for a $\theta-$graph with unknotted cycles are realized.
It is easily shown that all pairs of integers $(tb, rot)$ of different parities and such that $tb+|rot|\le -1$ can be realized as the Thurston-Bennequin number and the rotation number of a Legendrian unknot.
We call a pair of integers \textit{acceptable} if they satisfy the two restrictions above.
For $\theta-$graphs, we show the following:
\begin{enumerate}gin{theorem}
Any two triples of integers $(tb_1, tb_2, tb_3)$ and $(rot_1, rot_2, rot_3)$ for which $(tb_i, rot_i)$ are acceptable and $R=rot_1- rot_2 +rot_3 \in\{0, -1\}$ can be realized as the Thurston-Bennequin number and the rotation number of a Legendrian $\theta-$graph with all cycles unknotted.
\end{theorem}
It is known that certain Legendrian knots and links are determined by the invariants $tb$ and $rot$: the unknot \cite{EF}, torus knots and the figure eight knot \cite{EH}, and links consisting of an unknot and a cable of that unknot \cite{DG}.
To ask the same question in the context of Legendrian graphs, we restrict to topologically planar Legendrian $\theta-$graphs.
A \textit{topologically planar graph} is one which is ambient isotopic to a planar embedding.
The answer is no, the \textcolor{blue}n and the rotation number do not determine the Legendrian type of a topologically planar $\theta-$graph.
The pair of graphs in Figure~\ref{fig-not-determined} provides a counterexample.
The second part of this article is concerned with Legendrian ribbons of Legendrian $\theta-$graphs and their boundary.
Roughly, a ribbon of a Legendrian graph $g$ is a compact oriented surface $R_g$ containing $g$ in its interior, such that the contact structure is tangent to $R_g$ along $g$, transverse to $R_g\sigmamallsetminus g$, and $\partial R_g$ is a transverse knot or link.
We define the \textit{transverse push-off of} $g$ to be the boundary of $R_g $.
This introduces two new invariants of Legendrian graphs, the transverse push-off and its self linking number.
In the case of a Legendrian knot, this definition gives a two component link consisting of both the positive and the negative transverse push-offs.
However, with graphs the transverse push-off can have various numbers of components, depending on connectivity and Legendrian type.
We show the push-off of a Legendrian $\theta-$graph is either a transverse knot $K$ with $sl=1$ or a three component transverse link whose three components are the positive transverse push-offs of the three Legendrian cycles given the correct orientation.
For topologically planar graphs, the topological type of $\partial R_g$ is determined solely by the Thurston-Bennequin number of $g$, as per the following:
\begin{enumerate}gin{theorem} Let $G$ represent a topologically planar Legendrian $\theta-$graph with $tb = (tb_1, tb_2, tb_3)$.
Then the boundary of its attached ribbon is an $(a_1, a_2, a_3)-$pretzel, where
$a_1= tb_1+tb_2-tb_3$,
$a_2= tb_1+tb_3-tb_2$,
$a_3= tb_2+tb_3-tb_1$.
\label{thm-pretzel}
\end{theorem}
\noindent This elegant relation is specific to $\theta-$graphs and does not generalize to $n\theta-$graphs for $n>3$.
We give examples to sustain this claim in the last part of the article.
This phenomenon is due to the relationship between flat vertex graphs and pliable vertex graph in the special case of all vertices of degree at most three.
\sigmaubsection*{Acknowledgements}
The authors would like to thank Tim Cochran and John Etnyre for their continued support, and Chris Wendel and Patrick Massot for helpful conversations.
\sigmaection{Background}
We give a short overview of contact structure, Legendrian and transverse knots and their invariants. We recall how the invariants of Legendrian knots can be extended to Legendrian graphs.
Let $M$ be an oriented 3-manifold and $\xi$ a 2-plane field on $M$.
If $\xi =\ker \alpha$ for some $1-$form $\alpha$ on $M$ satisfying $\alpha\wedge d\alpha > 0,$ then $\xi$ is a \textit{contact structure} on $M$.
On $\mathbb{R}^3$, the $1-$form $\alpha = \, dz - y\, dx$ defines a contact structure called the standard contact structure, $\xi_{std}$.
Throughout this article we work in $(\mathbb{R}^3, \xi_{std})$.
A knot $K \sigmaubset (M, \xi)$ is called \textit{Legendrian} if for all $p\in K$ and $\xi_p$ the contact plane at $p$, $T_pK\sigmaubset \xi_p$.
A spatial graph $G$ is called \textit{Legendrian} if all its edges are Legendrian curves that are non-tangent to each other at the vertices.
If all edges around a vertex are oriented outward, then no two tangent vectors at the vertex coincide in the contact plane.
However, two tangent vectors may have the same direction but different orientations resulting in a smooth arc through the vertex.
It is a result of this structure that the order of the edges around a vertex in a contact plane is not changed up to cyclic permutation under Legendrian isotopy.
We study Legendrian knots and graphs via their front projection, the projection on the $xz-$plane.
Two generic front projections of a Legendrian graph are related by Reidemeister moves I, II and III, together with three moves given by the mutual position of vertices and edges \cite{BI}. See Figure~\ref{moves}.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(400, 188)
\put(0,0){\includegraphics[width=5.5in]{fig-moves}}
\put(84, 153){I}
\put(82,99){II}
\put(80,33){III}
\put(256, 150){IV}
\put(328, 151){IV}
\put(295,93){V}
\put(291,30){VI}
\end{picture}
\caption{\sigmamall Legendrian isotopy moves for graphs: Reidemeister moves I, II and III, a vertex passing through a cusp (IV), an edge passing under or over a vertex (V), an edge adjacent to a vertex rotates to the other side of the vertex (VI). Reflections of these moves that are Legendrian front projections are also allowed.}\label{moves}
\end{center}
\end{figure}
Apart from the topological knot class, there are two classical invariants of Legendrian knots, the Thurston-Bennequin number, $tb$, and the rotation number, $rot$.
The Thurston-Bennequin number is independent of the orientation on $K$ and measures the twisting of the contact framing on $K$ with respect to the Seifert framing.
To compute the Thurston-Bennequin number of a Legendrian knot $K$, consider a non-zero vector field $v$ transverse to $\xi$, take $K'$ the push-off of $K$ in the direction of $v$, and define $tb(K):= lk(K,K').$
For a Legendrian knot $K$, $tb(K)$ can be computed in terms of the writhe and the number of cusps in its front projection $\tilde{K}$ as $$tb(K) = w(\tilde{K})-\frac{1}{2}\textrm{cusps}(\tilde{K}).$$
To define the rotation number, $rot(K)$, assume $K$ is oriented and $K=\partial \Sigma$, where $\Sigma\sigmaubset \mathbb{R}^3$ is an embedded oriented surface.
The contact planes when restricted to $\Sigma$ form a trivial $2-$dimensional bundle and the trivialization of $\xi | _{\Sigma}$ induces a trivialization on $\xi | _L = L\times \mathbb{R}^2$.
Let $v$ be a non-zero vector field tangent to $K$ pointing in the direction of the orientation on $K$.
The winding number of $v$ about the origin with respect to this trivialization is the rotation number of $K$, denoted $rot(K)$ .
Taking the positively oriented trivialization $\{d_1=\frac{\partial}{\partial y}, d_2=-y\, \frac{\partial }{\partial z}-\frac{\partial}{\partial x} \}$ for $\xi_{std}$, one can check that for $\tilde{K}$ the front projection for $K$,
$$ rot(K) = \frac{1}{2}(\downarrow\textrm{cusps}(\tilde{K})-\uparrow\textrm{cusps}(\tilde{K})).$$
Given a Legendrian knot $K$, Legendrian knots in the same topological class as $K$ can be obtained by stabilizations.
A \textit{stabilization} means replacing a strand of $K$ in the front projection of $K$ by one of the zig-zags in Figure~\ref{stabilizations}.
The stabilization is said to be positive if down cusps are introduced and negative if up cusps are introduced.
The Legendrian isotopy type of $K$ changes through stabilization and so do the Thurston-Bennequin number and rotation number : $tb(S_{\pm}(K)) = tb(K) -1$ and $rot(S_{\pm}(K)) = rot(K) \pm 1$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(300, 119)
\put(0,0){\includegraphics[width=4.3in]{fig-stabilizations}}
\put(53,69){$K$}
\put(135,85){\sigmamall $S_+(K)$}
\put(135,28){\sigmamall $S_-(K)$}
\end{picture}
\caption{Positive and negative stabilizations in the front
projection.}\label{stabilizations}
\end{center}
\end{figure}
Both the Thurston-Bennequin number and the rotation number can be extended to piece-wise smooth Legendrian knots and to Legendrian graphs \cite{ODPa}.
For a Legendrian graph $G$, fix an order on the cycles of $G$ and define $tb(G)$ as the ordered list of the Thurston-Bennequin numbers of the cycles of $G$.
Once we fix an order on the cycles of $G$ with orientation, we define $rot(G)$ to be the ordered list of the rotation numbers of the cycles of $G$.
If $G$ has no cycles, define both $tb(G)$ and $rot(G)$ to be the empty list.
An oriented knot $t \sigmaubset (\mathbb{R}^3, \xi_{std})$ is called \textit{transverse} if for all $p\in t$ and $\xi_p$ the contact plane at $p$, $T_pt$ is positively transverse to $\xi_p$.
If $t$ is transverse, we let $\Sigma$ be an oriented surface with $t=\partial \Sigma$.
As above, $\xi|_\Sigma$ is trivial, so there is a non-zero vector field $v$ over $\Sigma$ in $\xi$.
If $t'$ is obtained by pushing $t$ slightly in the direction of $v$, then the \textit{self linking number} of $t$ is $sl(t)= lk(t, t')$.
It is easily seen that if $\tilde{t}$ is the front projection of $t$, then $sl(t)=writhe(\tilde{t})$.
For an embedded surface $\Sigma \sigmaubset (\mathbb{R}^3, \xi_{std})$, the intersection $l_x=T_x\Sigma \cap \xi_x$ is a line for most $x\in \Sigma$, except where the contact plane and the plane tangent to $\Sigma$ coincide.
We denote by $\l:=\cup l_x \sigmaubset T\Sigma$ this singular line field, where the union includes lines of intersection only.
Then, there is a singular foliation $\mathcal{F}$, called \textit{the characteristic foliation on $\Sigma$}, whose leaves are tangent to $l$.
\sigmaection{realization theorem}
In this section we find which triples of integers can be realized as $tb$ and $rot$ of Legendrian $\theta-$graphs with all cycles unknotted.
Both the structure of the $\theta-$graph and the required unknotted cycles impose restrictions on these integers.
We also investigate whether $tb$ and $rot$ uniquely determine the Legendrian type.
The following lemma identifies restrictions on the invariants of Legendrian unknots.
\begin{enumerate}gin{lemma} All pairs of integers $(tb, rot)$ of different parities and such that $$tb+|rot|\le -1$$ can be realized as the Thurston-Bennequin number and the rotation number of a Legendrian unknot.
\end{lemma}
\begin{enumerate}gin{proof}
We know from \cite{El} that for a Legendrian unknot $K$ in $(\mathbb{R}^3, \xi_{std})$, $tb(K)+|rot(K)|\le -1$.
Eliashberg and Fraser \cite{EF} showed that a Legendrian unknot $K$ is Legendrian isotopic to a unique unknot in standard form.
The standard forms are shown in Figure~\ref{fig-standard_unknotEF}.
The front projection in Figure~\ref{fig-standard_unknotEF}(a) represents two distinct Legendrian classes, depending on the chosen orientation.
For the front projection shown in Figure~\ref{fig-standard_unknotEF}(b) both orientations give the same Legendrian class.
The number of cusps and the number of crossings of the unknot in standard form are uniquely determined by $tb(K)$ and $rot(K)$ as follows:
\begin{enumerate}
\item If $rot(K)\ne 0$ (Figure~\ref{fig-standard_unknotEF}(a)), then
$$ tb(K) = -(2t+1+s) $$
$$ rot(K) = \left\{
\begin{enumerate}gin{array}{rl}
s, & \mbox{ if the leftmost cusp is a down cusp} \\
-s, & \mbox{ if the leftmost cusp is an up cusp.} \end{array}
\right. $$
\item If $rot(K) =0$ (Figure~\ref{fig-standard_unknotEF}(b)), then
$$ tb(K) = -(2t+1). $$
\end{enumerate}
Notice that in both cases the $tb$ and $rot$ have different parities.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(330, 70)
\put(0,0){\includegraphics{fig-standard_unknotEF}}
\put(55,-6){(a)}
\put(249, -6){(b)}
\put(35,65){\sigmamall $2t+1$}
\put(145, 31){\sigmamall $s$}
\put(250,63){\sigmamall $2t$}
\end{picture}
\caption{\sigmamall Legendrian unknot in standard form: (a) $rot (K) > 0$ [reverse orientation gives $rot(K)< 0$], (b) $rot(K) =0$.}
\label{fig-standard_unknotEF}
\end{center}
\end{figure}
For a pair $(tb, rot)$, the integers $s$ and $t$ are determined as follows:
\begin{enumerate}gin{itemize}
\item If $rot>0$, the pair $(tb, rot)$ is realized via the Legendrian unknot with front projection as in Figure~\ref{fig-standard_unknotEF}(a), for $(t,s)=(-\frac{tb+rot+1}{2}, rot)$.
\item If $rot<0$, the pair $(tb, rot)$ is realized via the Legendrian unknot with front projection as in Figure~\ref{fig-standard_unknotEF}(a), for $(t,s)=(-\frac{tb-rot+1}{2}, -rot)$.
\item If $rot=0$, the pair $(tb, rot)$ is realized via the Legendrian unknot with front projection as in Figure~\ref{fig-standard_unknotEF}(b), for $t=-\frac{tb+1}{2}$.
\end{itemize}
\end{proof}
\noindent We have described the pairs $(tb, rot)$ that can occur for the unknot.
Towards the proof of Theorem \ref{theorem-realize}, we show in the next lemma that Legendrian $\theta-$graphs can be standardized near their two vertices.
\begin{enumerate}gin{lemma}\label{NearVer}
Any Legendrian $\theta-$graph $G$, can be Legendrian isotoped to a graph $\tilde{G}$ whose front projection looks as in Figure ~\ref{fig-two-vertices} in the neighborhood of its two vertices.
\end{lemma}
\begin{enumerate}gin{proof}
Label the vertices of $G$ by $a$ and $b$.
In the front projection of $G$, use the Reidemeister VI move if necessary, to move the three strands on the right of vertex $a$ while near $a$ and on the left of vertex $b$ while near $b$.
Then, small enough neighborhoods of the two vertices look as in Figure ~\ref{fig-two-vertices}.
\end{proof}
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(200, 70)
\put(0,0){\includegraphics{fig-two-vertices}}
\put(15, 33){\sigmamall $a$}
\put(173, 32){\sigmamall $b$}
\end{picture}
\caption{Legendrian $\theta-$graph near its two vertices.}
\label{fig-two-vertices}
\end{center}
\end{figure}
For the remainder of this section, we assume that near its two vertices, $a$ and $b$, the front projection of the graph looks as in Figure~\ref{fig-two-vertices}.
We fix notation: $e_1$ is the top strand at $a$ in the front projection, $e_2$ is the middle strand at $a$, $e_3$ is the lower strand at $a$,
$\mathcal{C}_1$ is the oriented cycle exiting vertex $a$ along $e_1$ and entering vertex $a$ along $e_2$,
$\mathcal{C}_2$ is the oriented cycle exiting vertex $a$ along $e_1$ and entering vertex $a$ along $e_3$,
$\mathcal{C}_3$ is the oriented cycle exiting vertex $a$ along $e_2$ and entering vertex $a$ along $e_3$.
We note that there is no consistent way of orienting the three edges which gives three oriented cycles.
It should also be noted that the above notation is a labelling given after the graph is embedded.
If a labelled graph is embedded relabelling of the graph and reorienting of the cycles may be necessary in order to have the following lemma apply.
In the next lemma we show what additional restrictions occur as a result of the structure of the $\theta-$graph.
\begin{enumerate}gin{lemma}
Let $rot_1, rot_2$ and $rot_3$ be integers representing rotation numbers for cycles $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$, in the above notation.
Then $rot_1- rot_2 +rot_3 \in \{0, -1\}$.
\end{lemma}
\begin{enumerate}gin{proof}
For $i=1, 2, 3$, let $k_i$ ($k_i'$) represent the number of positive (negative) stabilizations along the edge $e_i$ when oriented from vertex $a$ to vertex $b$.
Let $s_i:=k_i-k_i'$, for $i=1, 2, 3$.
Then,
$$ rot_1 = \left\{
\begin{enumerate}gin{array}{rl}
s_1-s_2, & \mbox{ if $\mathcal{C}_1$ has a down cusp at $b$} \\
s_1-s_2-1, & \mbox{ if $\mathcal{C}_1$ has an up cusp at $b$} \end{array}
\right. $$
$$ rot_2 = \left\{
\begin{enumerate}gin{array}{rl}
s_1-s_3, & \mbox{ if $\mathcal{C}_2$ has a down cusp at $b$} \\
s_1-s_3-1, & \mbox{ if $\mathcal{C}_2$ has an up cusp at $b$} \end{array}
\right. $$
$$ rot_3 = \left\{
\begin{enumerate}gin{array}{rl}
s_2-s_3, & \mbox{ if $\mathcal{C}_3$ has a down cusp at $b$} \\
s_2-s_3-1, & \mbox{ if $\mathcal{C}_3$ has an up cusp at $b$} \end{array}
\right. $$
This gives eight different possible combinations and the possible values of $R=rot_1- rot_2 +rot_3$ are given in Table 1.
\begin{enumerate}gin{table}[htdp]
\begin{enumerate}gin{center}
\begin{enumerate}gin{tabular}{|c|c|c|c|r|}
\hline
\multirow{2}{*}{Case} & \multicolumn{3}{|c|}{Cusp at $b$} & \multirow{2}{*}{$R$}\\ \cline{2-4}
& $\mathcal{C}_1$ & $\mathcal{C}_2$ & $\mathcal{C}_3$ & \\
\hline
1 & $\downarrow$ & $\downarrow$ & $\downarrow$ & 0 \\
\hline
2 & $\downarrow$ & $\downarrow$ & $\uparrow$ & $-1$ \\
\hline
3 & $\downarrow$ & $\uparrow$ & $\downarrow$ & 1 \\
\hline
4 & $\downarrow$ & $\uparrow$ & $\uparrow$ & 0 \\
\hline
5 & $\uparrow$ & $\downarrow$ & $\downarrow$ & $-1$ \\
\hline
6 & $\uparrow$ & $\downarrow$ & $\uparrow$ & $-2$ \\
\hline
7 & $\uparrow$ & $\uparrow$ & $\downarrow$ & 0 \\
\hline
8 & $\uparrow$ & $\uparrow$ & $\uparrow$ & $-1$ \\
\hline
\end{tabular}
\end{center}
\label{table-sums}
\caption{Possible values for $R=rot_1-rot_2+rot_3.$}
\end{table}
Case 3 cannot occur. If both $\mathcal{C}_1$ and $\mathcal{C}_3$ have a down cusp at $b$, edge $e_3$ sits below edge $e_1$ at $b$, hence $\mathcal{C}_2$ has a down cusp at $b$.
Also, Case 6 cannot occur. If both $\mathcal{C}_1$ and $\mathcal{C}_3$ have an up cusp at $b$, edge $e_3$ sits above edge $e_1$ at $b$, hence $\mathcal{C}_2$ has an up cusp at $b$.
Among the six remaining cases, three of them give $rot_1- rot_2 +rot_3 =-1$ (when there is an odd number of up cusps at $b$ between $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$).
The other three cases give $rot_1- rot_2 +rot_3 = 0$ (when there is an even number of up cusps at $b$ between $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$).
\end{proof}
\begin{enumerate}gin{theorem}
Any two triples of integers $(tb_1, tb_2, tb_3)$ and $(rot_1, rot_2, rot_3)$ for which $tb_i+|rot_i|\le -1$, $tb_i$ and $rot_i$ are of different parities for $i=1,2,3$ and $R=rot_1- rot_2 +rot_3 \in\{0, -1\}$ can be realized as the Thurston-Bennequin number and the rotation number of a Legendrian $\theta-$graph with all cycles unknotted.
\label{theorem-realize}
\end{theorem}
\begin{enumerate}gin{proof} Let $tb=(tb_1, tb_2, tb_3)$ and $rot=(rot_1, rot_2, rot_3)$ be triples of integers as in the hypothesis.
We give front projections of Legendrian $\theta-$graphs realizing these triples.
Let $r_i:= |rot_i|$, for $i=1, 2, 3$.
We differentiate our examples according to the values of $rot_1$, $rot_2$ and $rot_3$ and the relationship between $r_1, r_2$ and $r_3$.
\begin{enumerate}gin{table}[htdp]
\begin{enumerate}gin{center}
\begin{enumerate}gin{tabular}{|c|c|c|c|c|c|}
\hline
Case & $rot_1$ & $rot_2$ & $rot_3$ & $R=0$ & $R=-1$\\ \hline
(i) & + & + & + & $r_1-r_2+r_3=0$ & $r_1+r_3+1=r_2$\\
\hline
(ii) & + & + & $-$& $r_1-r_2-r_3=0$ & $r_1+1=r_2+r_3$\\
\hline
(iii) & + & $-$ & + & $r_1+r_2+r_3=0$ & $r_1+r_2 +r_3 +1=0$\\
\hline
(iv) & + & $-$ & $-$& $r_1+r_2-r_3=0$ & $r_1+r_2+1=r_3$\\
\hline
(v) & $-$ & + & + & $-r_1-r_2+r_3=0$ & $r_1+r_2=r_3+1$\\
\hline
(vi) & $-$ & + & $-$ & $-r_1-r_2-r_3=0$ & $r_1+r_2+r_3=1$\\
\hline
(vii) & $-$ & $-$ & + & $-r_1+r_2+r_3=0$ & $r_1=r_2+r_3+1$\\
\hline
(viii) & $-$ & $-$ & $-$ & $-r_1+r_2-r_3=0$ & $r_1+r_3=r_2+1$\\
\hline
\end{tabular}
\end{center}
\label{sums1}
\caption{ $+$ stands for $rot_i\ge 0$ and $-$ stands for $rot_i<0$}
\end{table}
When $R=0$, for each case (i)--(viii), there is a choice of indices $i,j,k$ with $\{i,j,k\}=\{1,2,3\}$
such that $r_i \ge r_j+r_k$ (in fact, $r_i = r_j+r_k$).
When $R=-1$, for each case (i), (iv), (vi) and (vii) there is a choice of indices $i,j,k$ with $\{i,j,k\}=\{1,2,3\}$ such that $r_i \ge r_j+r_k$;
case (iii) is not realized; and for each case (ii), (v) and (viii), there is a choice of indices $i,j,k$ with $\{i,j,k\}=\{1,2,3\}$ such that $r_i + 1=r_j+r_k$.
Thus any realizable $(rot_1, rot_2, rot_3)$ falls under at least one of the following six conditions: (1) $r_1 \ge r_2+r_3$, (2) $r_2 \ge r_1+r_3$, (3) $r_3 \ge r_1+r_2$, (4) $r_1 + 1=r_2+r_3$, (5) $r_2 + 1=r_1+r_3$ and (6) $r_3 + 1=r_1+r_2$.
We describe ways of realizing the invariants for these six cases.
The cycles $\mathcal{C}_1, \mathcal{C}_2$ and $\mathcal{C}_3$ are as described earlier.
The choice of orientations for the three cycles implies that $e_1$ is oriented from $a$ to $b$ in both $\mathcal{C}_1$ and $\mathcal{C}_2$, while $e_3$ is oriented from $b$ to $a$ in both $\mathcal{C}_2$ and $\mathcal{C}_3$.
A box along a single strand designates number of stabilizations along the strand.
We take
\begin{enumerate}gin{itemize}
\item $r_i$ positive stabilizations if $rot_i\ge 0$
\item $r_i$ negative stabilizations if $rot_i< 0$,
\end{itemize}
when edges $e_1$, $e_2$ and $e_3$ are oriented as in cycle $\mathcal{C}_i$.
A box along a pair of strands designates number of crossings between the two strands.
All the crossings are as those in Figure~\ref{fig-standard_unknotEF}.\\
\noindent {\bf Case 1.}$(\mathbf{ r_1\ge r_2+r_3})$ Figure~\ref{fig-r1} represents the front projection of a Legendrian $\theta-$graph with the prescribed $tb$ and $rot$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(380, 40)
\put(0,0){\includegraphics{r1}}
\put(57,37){\sigmamall $r_2$}
\put(57, 22){\sigmamall $r_3$}
\put(37,43){\sigmamall $e_1$}
\put(37,28){\sigmamall $e_2$}
\put(37,7){\sigmamall $e_3$}
\put(217,42){\sigmamall $e_1$}
\put(307,5){\sigmamall $e_3$}
\put(103,6){\sigmamall $-tb_2-r_2-1$}
\put(195,16){\sigmamall $-tb_3-r_3-1$}
\put(266,28){\sigmamall $-tb_1-r_2-r_3-1$}
\end{picture}
\caption{\sigmamall
Case 1: $ r_1\ge r_2+r_3$.
}\label{fig-r1}
\end{center}
\end{figure}
\noindent Since $tb_i+|rot_i|\le -1$, the integers $-tb_2-r_2-1$ and $-tb_3-r_3-1$ are non-negative.
Since $ r_1\ge r_2+r_3$, then $-tb_1-r_2-r_3-1\ge -tb_1-r_1-1\ge 0$.
So all of the indicated number of half twists are non-negative integers as needed.
The number $-tb_1-r_2-r_3-1$ changes parity, according to whether $rot_1- rot_2 +rot_3$ equals $-1$ or $0$.
We check that the Thurston-Bennequin number and the rotation number for this embedding have the correct values.
For a cycle $\mathcal{C}$ we use $$ tb(\mathcal{C})= w(\mathcal{C})-\frac{1}{2}\textrm{cusps}(\mathcal{C})$$
$$ rot(\mathcal{C}) = \frac{1}{2}(\downarrow\textrm{cusps}(\mathcal{C})-\uparrow\textrm{cusps}(\mathcal{C})),$$
where $w=$ writhe, $\textrm{cusps}=$ total number of cusps, $\downarrow\textrm{cusps}=$ number of down cusps, $\uparrow\textrm{cusps}=$ number of up cusps.
\begin{enumerate}gin{itemize}
\item $ tb(\mathcal{C}_1)= w(\mathcal{C}_1)-\frac{1}{2}\textrm{cusps}(\mathcal{C}_1)= (tb_1+r_2+r_3+3)-(r_2+r_3+3)=tb_1$
\item $ tb(\mathcal{C}_2)= w(\mathcal{C}_2)-\frac{1}{2}\textrm{cusps}(\mathcal{C}_2) = (tb_2+r_2+3)-(r_2+3)=tb_2$
\item $ tb(\mathcal{C}_3)= w(\mathcal{C}_3)-\frac{1}{2}\textrm{cusps}(\mathcal{C}_3)= (tb_3+r_3+1)-(r_3+1)=tb_3$
\end{itemize}
\noindent If $rot_1- rot_2 +rot_3=0$, then $-tb_1-r_2-r_3-1$ has the same parity as $-tb_1-r_1-1$.
They are both even, since $tb_1$ and $rot_1$ have different parities.
This implies that at vertex $b$ the upper strand is $e_1$ and the middle strand is $e_2$.
\begin{enumerate}gin{itemize}
\item $ rot(\mathcal{C}_1)=\frac{1}{2}(\downarrow\textrm{cusps}(\mathcal{C}_1)-\uparrow\textrm{cusps}(\mathcal{C}_1))$\\
$=\frac{1}{2}(2\cdot {\rm sgn}(rot_2)\cdot r_2 +3- 2\cdot {\rm sgn}(rot_3)\cdot r_3 -3) = rot_2-rot_3=rot_1$
\item $ rot(\mathcal{C}_2) = \frac{1}{2}(\downarrow\textrm{cusps}(\mathcal{C}_2)-\uparrow\textrm{cusps}(\mathcal{C}_2))= \frac{1}{2}(2\cdot {\rm sgn}(rot_2)\cdot r_2 + 3 - 3)=rot_2 $
\item $rot(\mathcal{C}_3) = \frac{1}{2}(\downarrow\textrm{cusps}(\mathcal{C}_3)-\uparrow\textrm{cusps}(\mathcal{C}_3)) = \frac{1}{2}(2\cdot {\rm sgn}(rot_3)\cdot r_3 +1 - 1) =rot_3 $\\
\end{itemize}
\noindent If $rot_1- rot_2 +rot_3=-1$, then$-tb_1-r_2-r_3-1$ has different parity than $-tb_1-r_1-1$.
Since $tb_1$ and $rot_1$ have different parities, $-tb_1-r_1-1$ is even and $-tb_1-r_2-r_3-1$ is odd.
This implies that at vertex $b$ the upper strand is $e_2$ and the middle strand is $e_1$.
Computations for $ rot(\mathcal{C}_2) $ and $ rot(\mathcal{C}_3) $ are the same as above.
\begin{enumerate}gin{itemize}
\item $ rot(\mathcal{C}_1) =\frac{1}{2}(\downarrow\textrm{cusps}(\mathcal{C}_1)-\uparrow\textrm{cusps}(\mathcal{C}_1))$\\
$= \frac{1}{2}( 2\cdot {\rm sgn}(rot_2)\cdot r_2 +2- 2\cdot {\rm sgn}(rot_3)\cdot r_3 -4) = rot_2-rot_3 -1 =rot_1 $\\
\end{itemize}
In the remaining cases, a similar check may be done to verify that they have the correct $tb$ and $rot$.
\noindent {\bf Case 2.}$\mathbf{(r_2\ge r_1+r_3)}$ Figure~\ref{fig-r2} represents the front projection of a Legendrian $\theta-$graph with the prescribed $tb$ and $rot$.
Since $r_2\ge r_1+r_3$, then $-tb_2-r_1-r_3-1\ge -tb_2-r_2-1\ge 0$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(410, 60)
\put(0,0){\includegraphics{r2}}
\put(51,48){\sigmamall $r_1$}
\put(52, 18){\sigmamall $r_3$}
\put(96,38){\sigmamall $-tb_1-r_1-1$}
\put(188,24){\sigmamall $-tb_3-r_3-1$}
\put(276,12){\sigmamall $-tb_2-r_1-r_3-1$}
\end{picture}
\caption{\sigmamall
Case 2: $r_2\ge r_1+r_3.$
}
\label{fig-r2}
\end{center}
\end{figure}
\noindent {\bf Case 3.}$\mathbf{(r_3\ge r_1+r_2)}$ Figure~\ref{fig-r3} represents the front projection of a Legendrian $\theta-$graph with the prescribed $tb$ and $rot$.
As $r_3\ge r_1+r_2$, then $-tb_3-r_1-r_2-1\ge -tb_3-r_3-1\ge 0$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(410, 55)
\put(0,0){\includegraphics{r3}}
\put(50,30){\sigmamall $r_1$}
\put(50, 10){\sigmamall $r_3$}
\put(94,35){\sigmamall $-tb_1-r_1-1$}
\put(205,6){\sigmamall $-tb_2-r_2-1$}
\put(295,24){\sigmamall $-tb_3-r_1-r_2-1$}
\end{picture}
\caption{\sigmamall
Case 3: $r_3\ge r_1+r_2.$
}
\label{fig-r3}
\end{center}
\end{figure}
\noindent {\bf Case 4.}$\mathbf{(r_1+1=r_2+r_3)}$ In this case the graph in Figure~\ref{fig-B} realizes $(tb, rot)$.
Since $r_2+r_3=r_1+1$, we have $-tb_1-r_2-r_3=-tb_1-r_1-1\ge 0$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(430, 55)
\put(0,0){\includegraphics{rB}}
\put(64,30){\sigmamall $r_2$}
\put(55, 13){\sigmamall $r_3-1$}
\put(113,20){\sigmamall $-tb_1-r_2-r_3$}
\put(215,7){\sigmamall $-tb_3-r_3$}
\put(302,20){\sigmamall $-tb_2-r_2-1$}
\end{picture}
\caption{\sigmamall Case 4: $r_1+1=r_2+r_3.$
}
\label{fig-B}
\end{center}
\end{figure}
\noindent {\bf Case 5.}$\mathbf{(r_2+1=r_1+r_3)}$ For this case the graph in Figure~\ref{fig-A} realizes $(tb, rot)$. Given $r_1+r_3=r_2+1$, we have that $-tb_2-r_1-r_3+1=-tb_2-r_2> 0$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(430, 55)
\put(0,0){\includegraphics{rA}}
\put(44,34){\sigmamall $r_1-1$}
\put(44, 6){\sigmamall$r_3-1$}
\put(108,26){\sigmamall$-tb_1-r_1$}
\put(193,11){\sigmamall $-tb_2-r_1-r_3+1$}
\put(323,28){\sigmamall $-tb_3-r_3$}
\end{picture}
\caption{\sigmamall Case 5: $r_2+1=r_1+r_3.$
}
\label{fig-A}
\end{center}
\end{figure}
\noindent {\bf Case 6.}$\mathbf{(r_3+1=r_1+r_2)}$ In this case the graph in Figure~\ref{fig-C} realizes $(tb, rot).$
Since $r_1+r_2=r_3+1$, we have $-tb_3-r_1-r_2=-tb_3-r_3-1\ge 0$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(4300, 55)
\put(0,0){\includegraphics{rC}}
\put(41,23){\sigmamall $r_1-1$}
\put(50, 5){\sigmamall $r_2$}
\put(100,13){\sigmamall $-tb_3-r_2-r_1$}
\put(205,29){\sigmamall $-tb_1-r_1$}
\put(275,15){\sigmamall $-tb_2-r_2-1$}
\end{picture}
\caption{\sigmamall Case 6: $r_3+1=r_1+r_2.$
}
\label{fig-C}
\end{center}
\end{figure}
This completes the proof.
\end{proof}
\sigmaubsection{Topologically planar $\theta-$graphs are not Legendrian simple}We ask whether the invariants $tb$ and $rot$ determine the Legendrian type of a planar $\theta-$graph.
If we do not require that the cyclic order of the edges around the vertex $a$ (or $b$) is the same in both embeddings, the answer is negative. The following is a counterexample in this case.
\begin{enumerate}gin{example}
The two graphs in Figure~\ref{fig-not-determined} have the same invariants but they are not Legendrian isotopic.
Let $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$ be the three cycles of $G$ determined by the pairs of edges \{$e_1$, $e_2$\}, \{$e_1$, $e_3$\} and \{$e_2$, $e_3$\}, respectively.
Let $\mathcal{C}'_1$, $\mathcal{C}'_2$ and $\mathcal{C}'_3$ be the three cycles of $G'$ determined by \{$f_2$, $f_1$\}, \{$f_2$, $f_3$\} and \{$f_1$, $f_3$\}, respectively.
The cycles have $tb(\mathcal{C}_1)=tb(\mathcal{C}'_1)=-1$, $tb(\mathcal{C}_2)=tb(\mathcal{C}'_2)=-5 $, $tb(\mathcal{C}_3)=tb(\mathcal{C}'_3)=-3$ and $rot(\mathcal{C}_i)=rot(\mathcal{C}'_i)=0$ for $i=1,2,3$.
Assume the two graphs are Legendrian isotopic.
Since the cycles with same invariants should correspond to each other via the Legedrian isotopy (which we denote by $\iota$), the edges correspond as $e_1\leftrightarrow \iota(e_1)=f_2$, $e_2\leftrightarrow \iota(e_2)=f_1$ and $e_3\leftrightarrow \iota(e_3)=f_3$.
But at both vertices of $G$ the (counterclockwise) order of edges in the contact plane is $e_1-e_2-e_3$ and at both vertices of $G'$ the (counterclockwise) order of edges in the contact plane is $\iota(e_1)-\iota(e_3)-\iota(e_2)$.
This leads to a contradiction, since a Legendrian isotopy preserves the cyclic order of edges at each vertex.
\label{example-not-determined}
\end{example}
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(390, 55)
\put(0,0){\includegraphics{fig-not-determined}}
\put(80,-5){$G$}
\put(300, -5){$G'$}
\put(19,54){\tiny $e_1$}
\put(27, 43){\tiny$e_2$}
\put(20,27){\tiny$e_3$}
\put(230,55){\tiny $f_1$}
\put(233,43){\tiny $f_2$}
\put(225,24){\tiny $f_3$}
\end{picture}
\caption{\sigmamall Non-Legendrian isotopic graphs with the same invariants.
}
\label{fig-not-determined}
\end{center}
\end{figure}
\begin{enumerate}gin{corollary}
The invariants $tb$ and $rot$ are not enough to distinguish the Legendrian class of an $n\theta-$graph for $n\ge 3$.
\end{corollary}
\begin{enumerate}gin{proof}
For $n\ge 4$, a pair of graphs with the same invariants but of different Legendrian type can be otained from $(G, G')$ in Example \ref{example-not-determined} by adding $n-3$ unknotted edges at the top of the three existing ones.
\end{proof}
\sigmaection{Legendrian Ribbons and transverse push-offs}
\label{sec-ribbon}
In this section we work with Legendrian ribbons of $\theta-$graphs.
We examine the relationship between the Legendrian graph and the boundary of its ribbon, the transverse push-off.
The transverse push-off is another invariant of Legendrian graphs.
We explore whether it contains more information than the classical invariants rotation number and Thurston-Bennequin number.
We determine the number of components and the self linking number for the push-off of a Legendrian $\theta-$graph.
In the special case of topologically planar graphs, we prove that the topological type of the transverse push-off of a $\theta-$graph is that of a pretzel-type curve whose coefficients are determined by the Thurston-Bennequin invariant of the graph.\\
Let $g$ be a Legendrian graph. A \textit{ribbon for $g$} is a compact oriented surface $R_g$ such that:
\begin{enumerate}
\item $g$ in contained in the interior of $R_g$;
\item there exists a choice of orientations for $R_g$ and for $\xi$ such that $\xi$ has no negative tangency with $R_g$;
\item there exists a vector field $X$ on $R_g$ tangent to the characteristic foliation whose time flow $\phi_t$ satisfies
$\cap_{t\geq 0}\,\phi_t (R_g)=g$; and
\item the boundary of $R_g$ is transverse to the contact structure.\\
\end{enumerate}
The following is a construction which takes a graph in the front projection and produces its ribbon viewed in the front projection.
Portions of this construction were previously examined by Avdek in \cite{A} (algorithm 2, steps 4--6).
Starting with a front projection of the graph, we construct a ribbon surface containing the graph as described in Figure~\ref{fig-ribbon}.
\begin{enumerate}
\item[(a)] to a cusp free portion of an edge we attach a band with a single negative half twist,
\item[(b)] to each left and right cusp along a strand we attach disks containing a positive half twist,
\item[(c,d)] to each vertex we attach twisted disks as in Figure~\ref{fig-ribbon}(c,d),
\item[(e)] crossings in the diagram of the graph are preserved.
\end{enumerate}
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(400, 110)
\put(0,0){\includegraphics{fig-ribbon}}
\end{picture}
\caption{\sigmamall Attaching a ribbon surface to a Legendrian graph. The two sides of the surface are marked by different colors.}
\label{fig-ribbon}
\end{center}
\end{figure}
Legendrian ribbons were first introduced by Giroux \cite{G} to have a well-defined way to contract a contact handlebody onto the Legendrian graph at the core of the handlebody.
We are interested in some particular features of Legendrian ribbons.
The boundary of a Legendrian ribbon is an oriented transverse link with the orientation inherited from the ribbon surface.
The ribbon associated with a given Legendrian graph is unique up to isotopy and therefore gives a natural way to associate a transverse link to the graph.
\begin{enumerate}gin{definition} The \textit{transverse push-off} of a Legendrian graph is the boundary of its ribbon.
\end{definition}
In the case of Legedrian knots the above definition gives a two component link of both the positive and negative transverse push-offs.
However, with graphs the transverse push-off can have various numbers of components, depending on connectivity and Legendrian type.
The transverse push-off is a new invariant of Legendrian graphs.
\sigmaubsection{Self-linking of transverse push-offs.}
Here we determine possible self-linking numbers and the number of components of the transverse push-off of a Legendrian $\theta-$graph.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(460, 120)
\put(0,0){\includegraphics{fig-transverse-pushoff}}
\end{picture}
\caption{Transverse push-off of a Legendrian $\theta-$graph with (b) one component or (c) three components.}
\label{fig-transverse-pushoff}
\end{center}
\end{figure}
\begin{enumerate}gin{theorem}\label{trans-push-off-sl}
The transverse push-off of a Legendrian $\theta-$graph is either
a transverse knot $K$ with $sl=1$ or a three component transverse link whose three components are the transverse push-offs of the three Legendrian cycles given the correct orientation.
\end{theorem}
\begin{enumerate}gin{proof}
Before working with the transverse push-off of a Legendrian $\theta-$graph, we will move the graph into a form that will simplify our argument.
Given an arbitrary Legendrian $\theta-$graph, by Lemma \ref{NearVer}, it can be isotoped to an embedding where near the vertices it has a projection like that shown in Figure \ref{fig-two-vertices}.
Label the arcs at the left vertex I, II, III from top to bottom.
Then move the edges around the right vertex (using a combination of RVI and RIV) so that arc I is also in the top position.
There are two possibilities for the order at the right vertex.
The first case, where the arcs are I, II, III from top to bottom at the right vertex, shown in Figure \ref{fig-transverse-pushoff}(b), we will call \textit{parallel vertices}.
The second case, where the arcs are I, III, II from top to bottom at the right vertex, shown in Figure \ref{fig-transverse-pushoff}(c), we will call \textit{antiparallel vertices}.
Now we will focus on the number of components of the transverse push-off.
For simplicity of book keeping we will place the negative half twists that occur on each cusp free portion of an edge to the left on that portion of the edge.
For the projections shown in Figure \ref{fig-transverse-pushoff}(b,c) the portion of the graph not pictured could have any number of crossings and cusps.
Along each edge, the top (resp. bottom) position of the strands is preserved through cusps and crossings.
See Figure \ref{fig-transverse-pushoff}(a).
So we see that the arc of the transverse push-off which lies above (resp. below) the Legendrian arc in the projection on one side of the diagram still lies above (resp. below) on the other side.
Thus the number of components in the transverse push-off can be determined by a careful tracing of the diagrams in Figure \ref{fig-transverse-pushoff}(b,c).
Therefore graphs with parallel vertices have a transverse push-off with one component, and graphs with antiparallel vertices have a transverse push-off with three components.
If the boundary of the Legendrian ribbon is a knot $T$, then $sl(T)$ equals the signed count of crossings in a front diagram for $T$.
Crossings in the diagram of the graph and cusps along the three edges do not contribute to this count.
A cusp contributes a canceling pair of positive and negative crossings.
A crossing contributes two negative and two positive crossings. See Figure~\ref{fig-transverse-pushoff}(a).
Apart from these, there is one positive crossing along each edge and one negative crossing for every disk at each vertex, giving $sl(T)=1$. See Figure \ref{fig-transverse-pushoff}(b).
If the boundary has three components $T_1$, $T_2$ and $T_3$, then they have the same self linking as the transverse push-offs of the cycles of the Legendrian graph with the correct orientation.
Let $\bar{\mathcal{C}_i}$ be the cycle $\mathcal{C}_i$ with the opposite orientation.
Then $T_1$, $T_2$ and $T_3$, are the positive transverse push-offs of $\bar{\mathcal{C}_1}$, $\mathcal{C}_2$ and $\bar{\mathcal{C}_3}$, respectively.
\end{proof}
\sigmaubsection{Topologically Planar Legendrian $\theta-$graphs}
To be able to better understand the topological type of a Legendrian ribbon and the transverse
push-off (its boundary) we will model the ribbon with a flat vertex graph.
A \textit{flat vertex graph} (or \textit{rigid vertex graph}) is an embedded graph where the vertices are rigid disks with the edges being flexible tubes or strings between the vertices.
This is in contrast with pliable vertex graphs (or just spatial graphs) where the edges have freedom of motion at the vertices.
Both flat vertex and pliable vertex graphs are studied up to ambient isotopy and have sets of five Reidemeister moves.
For both of them the first three Reidemeister moves are the same as those for knots and links and Reidemeister move IV consists of moving an edge over or under a vertex.
See Figure \ref{fig-ReidemeisterIVandV}.
For flat vertex graphs, Reidemeister move V is the move where the flat vertex is flipped over.
For pliable vertex graphs, Reidemeister move V is the move where two of the edges are moved near the vertex in such a way that their order around the vertex is changed in the projection.
For a \textit{Legendrian ribbon, the associated flat vertex graph} is given by the following construction: a vertex is placed on each twisted disk -- where the original vertices were, and an edge replaces each band in the ribbon.
The information that is lost with this model is the amount of twisting that occurs on each edge.
The flat vertex graph model is particularly useful when working with the $\theta-$graph because it is a trivalent graph.
We see with the following Lemma, the relationship between trivalent flat vertex and trivalent pliable vertex graphs.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(220, 145)
\put(0,0){\includegraphics{fig-RedemeisterIVandV}}
\end{picture}
\caption{\sigmamall Reidemeister moves IV and V for pliable and flat vertex graphs.}
\label{fig-ReidemeisterIVandV}
\end{center}
\end{figure}
\begin{enumerate}gin{lemma}
For graphs with all vertices of degree 3 or less, the set of equivalent diagrams is the same for both pliable and flat vertex spatial graphs.
\label{same-moves}
\end{lemma}
\begin{enumerate}gin{proof} We follow notation in \cite[pages 699, 704]{K}. The lemma can be reformulated to say, given the diagrams of two ambient isotopic pliable vertex graphs with maximal degree 3, these are also ambient isotopic as flat vertex graphs, and vice versa. The Reidemeister moves for pliable vertex graphs and flat vertex graphs differ only in Reidemeister move V. See Figure~\ref{fig-ReidemeisterIVandV}. For pliable vertex graphs, Reidemeister move V is the move where two of the edges are moved near the vertex in such a way that this changes their order around the vertex in the projection. For flat vertex graphs, Reidemeister move V is the move where the flat vertex is flipped over. For vertices of valence at most 3, these two moves give the same diagrammatic results. Thus the same sequence of Reidemeister moves can be used in the special case of graphs with maximal degree 3.
\end{proof}
Here we set up the notation that will be used in the following theorem.
For a Legendrian $\theta-$graph $G$, we consider a front projection in which the neighborhoods of the two vertices are as those in Figure~\ref{fig-two-vertices} and we denote its three cycles by
$\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$, following the notation of Section 2.
Let $\textrm{cr}[e_i,e_j]$ be the signed intersection count of edges $e_i$ and $e_j$ in the cycle $\mathcal{C}_1$, $\mathcal{C}_2$ or $\mathcal{C}_3$ which they determine.
Let $\textrm{cr}[e_i]$ be the signed self-intersection count of $e_i$.
Let $tb_1$, $tb_2$ and $tb_3$ be the Thurston-Bennequin numbers of $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$.
\begin{enumerate}gin{theorem} Let $G$ represent a topologically planar Legendrian $\theta-$graph with $tb = (tb_1, tb_2, tb_3)$.
Then the boundary of its attached ribbon is an $(a_1, a_2, a_3)-$pretzel, where
$a_1= tb_1+tb_2-tb_3$,
$a_2= tb_1+tb_3-tb_2$ and
$a_3= tb_2+tb_3-tb_1$.
\label{pretzel}
\end{theorem}
\begin{enumerate}gin{proof}
The proof will be done in two parts.
First, the transverse push-off will be shown to be a pretzel knot or link.
Second, it will be shown to be of the particular type of pretzel, an $(a_1, a_2, a_3)-$pretzel knot or link.
We first look at the ribbon as a topological object.
If the ribbon can be moved through ambient isotopy to a projection where the three bands do not cross over each other and come together along a flat disk, then the boundary of the ribbon would be a pretzel link with crossings only occurring as twists on each band.
If we model the ribbon with a flat vertex graph this simplifies our question to whether the resulting flat vertex graph can be moved so that it is embedded in the plane.
The resulting graph is topologically planar because it is coming from a topologically planar Legendrian graph.
Thus by Lemma \ref{same-moves}, it can be moved to a planar embedding.
In order to show the pretzel knot (or link) is an $(a_1, a_2, a_3)-$pretzel, we will look at what happens to the ribbon as the associated flat vertex graph is moved to a planar embedding.
We will work with the Legendrian $\theta-$graph in the form shown in Figure~\ref{fig-two-vertices} near its vertices.
We need to count the number of twists in the bands of the Legendrian ribbon once it has been moved to the embedding where the associated flat vertex graph is planar.
We will prove $a_1= tb_1+tb_2-tb_3$ by writing each of these numbers in terms of the number of cusps and the number of singed crossings between the edges of the Legendrian graph.
The proofs for $a_2$ and $a_3$ are similar.
We will use the following observations to be able to write $a_1$, the number of half twists in the band associated with edge $e_1,$ in terms of the number of cusps, $\textrm{cr}[e_i] $ and $\textrm{cr}[e_i,e_j].$
\begin{enumerate}
\item Based on the construction of the ribbon surface, $c$ cusps on one of the edges contribute with $c+1$ negative half twists to the corresponding band.
\item We look at each of the Reidemeister moves for flat vertex graphs and see how they change the number of twists on the associated band of the ribbon surface.
\begin{enumerate}
\item A positive (negative) Reidemeister I move adds a full positive (negative) twist to the band. See Figure~\ref{fig-RedemeisterI-V}(a,b).
\item Reidemeister moves II, III and IV do not change the number of twists in any of the bands.
\item A Reidemeister V move adds a half twist on each of the three bands. See Figure~\ref{fig-RedemeisterI-V}(c,d).
The sign of the half twists depends on the crossing, and which bands are crossed.
If the bands have a positive (resp. negative) crossing, then they each have the addition of a positive (resp. negative) half twist, and the other band has the addition of a negative (resp. positive) half twist.
\end{enumerate}
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(500, 180)
\put(0,0){\includegraphics[width=6in]{fig-RedemeisterI-V}}
\end{picture}
\caption{\sigmamall (a) a positive Reidemeister I move adds a full positive twist to the band, (b) a negative Reidemeister I move adds a full negative twist to the band,
(c,d) a Reidemeister V move adds a half twist on each of the three bands.
}
\label{fig-RedemeisterI-V}
\end{center}
\end{figure}
\end{enumerate}
Since we proved earlier that the flat vertex graph can be moved to a planar embedding, we know that all of the crossings between edges will be eventually removed through Reidemeister moves.
Thus this gives:
\[ a_1 = -[\textrm{cusps on }e_1]-1+2\, \textrm{cr}[e_1] +\textrm{cr}[e_1,e_2]+ \textrm{cr}[e_1,e_3] -\textrm{cr}[e_2,e_3]\]
This count is easily seen to be invariant under moves RII and RIII, since these do not change the signed crossing of the diagram.
We show it is invariant under move RIV at the end of the proof.
Next, we describe $tb_1+tb_2-tb_3$ in terms of the number of cusps and the crossings between the edges.
Recall, for a cycle $\mathcal{C}$ we use $$ tb(\mathcal{C})= w(\mathcal{C})-\frac{1}{2}\textrm{cusps}(\mathcal{C}).$$
Thus,
\begin{enumerate}gin{eqnarray*}
tb_1+tb_2-tb_3&=&w(\mathcal{C}_1)-\frac{1}{2}\textrm{cusps}(\mathcal{C}_1)+w(\mathcal{C}_2)-\frac{1}{2}\textrm{cusps}(\mathcal{C}_2)-w(\mathcal{C}_3)+\frac{1}{2}\textrm{cusps}(\mathcal{C}_3)\\
&=& \textrm{cr}[e_1, e_2] + \textrm{cr}[e_1] + \textrm{cr}[e_2] -\frac{1}{2}\big([\textrm{cusps on }e_1]+[\textrm{cusps on }e_2]+2\big)\\
& &+\textrm{cr}[e_1, e_3] + \textrm{cr}[e_1] + \textrm{cr}[e_3] -\frac{1}{2}\big([\textrm{cusps on }e_1]+[\textrm{cusps on }e_3]+2\big)\\
& &- \big( \textrm{cr}[e_2, e_3] + \textrm{cr}[e_2] + \textrm{cr}[e_3]\big) +\frac{1}{2}\big([\textrm{cusps on }e_2]+[\textrm{cusps on }e_3]+2\big)\\
&=& -[\textrm{cusps on }e_1]-1 + 2\, \textrm{cr}[e_1] +\textrm{cr}[e_1,e_2]+ \textrm{cr}[e_1,e_3] -\textrm{cr}[e_2,e_3].
\end{eqnarray*}
Thus, $a_1= tb_1+tb_2-tb_3$.
\textit{\textbf{Claim}: The sum $2\, \textrm{cr}[e_1] +\textrm{cr}[e_1,e_2]+ \textrm{cr}[e_1,e_3] -\textrm{cr}[e_2,e_3]$ is unchanged under Reidemeister move IV. }
\textit{Proof of claim. }
Let $b_1=2\, \textrm{cr}[e_1] +\textrm{cr}[e_1,e_2]+ \textrm{cr}[e_1,e_3] -\textrm{cr}[e_2,e_3]$.
Let $d$ represent the strand that is moved past the vertex.
We distinguish two cases, $(a)$ and $(b)$, according to the number of crossings on each side of the vertex. See Figure~\ref{fig-RademeisterIV}.
We check that the contributions to $b_1$ of the crossing before the move (left) is the same as the contribution to $b_1$ of the crossings after the move (right).
The strand $d$ can be part of $e_1$, $e_2$ or $e_3$.
For both cases (a) and (b), the equality is shown step by step for $d=e_1$ and $d=e_3$. In a similar way $b_1$ is unchanged if $d=e_2$.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(340, 50)
\put(0,0){\includegraphics{fig-RedemeisterIV}}
\put(-9,40){$e_1$}
\put(-7,15){$e_2$}
\put(20,45){$d$}
\put(40,23){$e_3$}
\put(100,43){$e_1$}
\put(100,12){$e_2$}
\put(145,45){$d$}
\put(145,23){$e_3$}
\put(198,41){$e_1$}
\put(195,30){$e_2$}
\put(198,15){$e_3$}
\put(292,41){$e_1$}
\put(290,28){$e_2$}
\put(292,15){$e_3$}
\put(225,45){$d$}
\put(345,45){$d$}
\end{picture}
\caption{\sigmamall Reidemeister IV moves change crossings between different pairs of edges. }
\label{fig-RademeisterIV}
\end{center}
\end{figure}
\noindent Case (a-1) If $d$ is part of $e_1$,
then $b_{1,\textrm{left}}= 2\textrm{cr}[e_1]+\textrm{cr}[e_1,e_2]=
\textrm{cr}[e_1]$, since the two crossings have opposite sign when seen in the cycle determined by $e_1$ and $e_2$;
and $b_{1,\textrm{right}}= \textrm{cr}[e_1,e_3]= \textrm{cr}[e_1].$\\
Case (a-2) If $d$ is part of $e_3$, then
$b_{1,\textrm{left}}= \textrm{cr}[e_1,e_3]-\textrm{cr}[e_2,e_3]=\textrm{cr}[e_2,e_3]-\textrm{cr}[e_3]=0$ and $b_{1,\textrm{right}}= 0$.\\
\noindent Case (b-1) If $d$ is part of $e_1$,
then $b_{1,\textrm{left}}= 2\textrm{cr}[e_1]+\textrm{cr}[e_1,e_2]+\textrm{cr}[e_1,e_3] = 0$ and $b_{1,\textrm{right}}= 0$. \\
\noindent Case (b-2) If $d$ is part of $e_3$,
then $b_{1,\textrm{left}}=\textrm{cr}[e_1,e_3]-\textrm{cr}[e_2,e_3]=0$, since both these crossings have sign opposite to $\textrm{cr}[e_3]$; and $b_{1,\textrm{right}}= 0$. \\
This complete the proof of the claim and the theorem.
\end{proof}
The combination of Theorem \ref{trans-push-off-sl} and Theorem \ref{pretzel} gives a complete picture of the possible transverse push-offs of topologically planar Legendrian $\theta-$graphs.
In this case, the transverse push-off is completely described by the $tb$ of the graph.
So while this does not add to our ability to distinguish topologically planar Legendrian $\theta-$graphs, it does add to our understanding of the interaction between a Legendrian graph and its transverse push-off.
It is worth noting that Theorem \ref{pretzel} also implies that the transverse push-off will either have one or three components.
The possible transverse push-offs of a topologically planar Legendrian $\theta$-graph are more restricted than it may first appear.
Not all pretzel links will occur in this way.
In Theorem \ref{pretzel}, we found the pretzel coefficients as linear combinations with coefficients +1 or $-1$ of the $tb$'s.
We note that the three pretzel coefficients have the same parity, restricting the number of components the transverse push-off can have.
If exactly one of or all three of $tb_1$, $tb_2$ and $tb_3$ are odd, then all pretzel coefficients are odd and the pretzel curve is a knot.
If none or exactly two of $tb_1$, $tb_2$ and $tb_3$ are odd, then all pretzel coefficients are even and the pretzel curve is a three component link.
The pairwise linking between its components is equal to the number of full twists between the corresponding pair of strands in the pretzel presentation, i.e. $a_1/2$, $a_2/2$ and $a_3/2$.
\sigmaubsection{The transverse push-off of $n\theta-$graphs.}
We give examples showing the boundary of the Legendrian ribbon associated to an $n\theta-$graph, $n>3$, is not necessarily a pretzel-type link.
Independent of $n$, each component of an $n-$pretzel type link is linked with at most two other components.
The tranverse push-offs of the graphs in Figure~\ref{fig-ntheta} have at least one component linking more than two other components of the link.
The characterization as a pretzel curve of the topological type of the push-off is therefore exclusive to the case $n=3$, that of $\theta-$graphs.
\begin{enumerate}gin{figure}[htpb!]
\begin{enumerate}gin{center}
\begin{enumerate}gin{picture}(370, 230)
\put(0,0){\includegraphics{fig-ntheta}}
\put(267,190){\textcolor{red}{$L_k$}}
\put(267,110){\textcolor{red}{$L_k$}}
\end{picture}
\caption{\sigmamall The $n\theta-$graphs in (a1), (b1) and (c1) have transverse push-offs (a2), (b2) and (c2) which do not have the topological type of a pretzel-type curve.}
\label{fig-ntheta}
\end{center}
\end{figure}
For $n=2k, k\ge 2$, let $L_{2k}$ be the Legendrian $2k\theta-$graph whose front projection is the one in Figure~\ref{fig-ntheta}(a1).
Then the transverse push-off has the topological type of the link $L\cup L_k$ in Figure~\ref{fig-ntheta}(a2).
If $k$ is odd, $L$ has one component and it links all $k\ge 3$ components of $L_k$.
If $k$ is even, then $L$ has two components where each of the two components links all $k\ge 2$ components of $L_k$ and the other component of $L$.
For $n=2k+1, k\ge 3$, let $L_{2k+1}$ be the Legendrian $(2k+1)\theta-$graph whose front projection is the one in Figure~\ref{fig-ntheta}(b1).
Then the transverse push-off has the topological type of the link $L\cup L_k$ in Figure~\ref{fig-ntheta}(b2).
If $k$ is even, then $L$ has one component and it links all $k\ge 3$ components of $L_k$.
If $k$ is odd, then $L$ has two components where each of the two components links all $k\ge 3$ components of $L_k$ and the other component of $L$.
For $n=5$, the link in Figure~\ref{fig-ntheta}(b2) is a pretzel link and we give a different example in this case, the one in Figure~\ref{fig-ntheta}(c1,c2).
The highlighted component of the transverse push-off links three other components.
\begin{enumerate}gin{thebibliography}{99}
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\bibitem{BI} Baader, Sebastian and Ishikawa, Masaharu. \textit{Legendrian graphs and quasipositive diagrams.} Ann. Fac. Sci. Toulouse Math. {\bf 18} (2009), 285--305
\bibitem{DG} Ding, Fan and Geiges, Hansj$\ddot{\rm o}$rg. \textit{ Legendrian knots and links classified by classical invariants.} Commun. Contemp. Math. {\bf 9} (2007), No. 2, 135--162
\bibitem{EF} Eliashberg, Yakov and Fraser, Maia. \textit{Topologically trivial legendrian knots.} J. Symplectic Geom. {\bf 7} (2009), No. 2, 77--127
\bibitem{El} Eliashberg, Yakov. \textit{Contact $3$-manifolds twenty years since J. Martinet's work.} Ann. Inst. Fourier (Grenoble) {\bf 42} (1992), No. 1-2, 165--192
\bibitem{EH} Etnyre, John and Honda, Ko. \textit{Knots and contact geometry I torus knots and the figure eight knot.} J. Sympl. Geom. {\bf 1} (2001), No. 1, 63--120.
\bibitem{G} Giroux, Emmanuel. \textit{Contact geometry: from dimension three to higher dimensions.} Proceedings of the International Congress of Mathematicians, Vol. II, Higher Ed. Press, Beijing (2002), 405--414
\bibitem{K} Kauffman, Louis. \textit{Invariants of graphs in three-space.} Trans. Amer. Math. Soc. {\bf 311} (1989), No. 2, 697--710
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\end{thebibliography}
\end{document} |
\begin{document}
\begin{abstract}
A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains.
For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\alpha=(\alpha_1,\cdots,\alpha_m)$, with $0\le \alpha_i \le n$ for each $i=1,\cdots,m$. Define order $<$ as follow, $\forall \alpha,\beta \in N(m,n)$, $\beta < \alpha$ if and only if $\beta_i \le \alpha_i(i=1,\cdots,m)$ and $\sum\limits_{i=1}^{m}\beta_i <\sum\limits_{i=1}^{m}\alpha_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.
\end{abstract}
<^saketitle
<^sathfrak{n}oindent
\begin{small}
\emph{Mathematic subject classification}: Primary 05A19; Secondary 05E99.
\end{small}
<^sathfrak{n}oindent
\begin{small}
\emph{Keywords}: composition; chain; symmetric chain decomposition.
\end{small}
\section{Introduction}
For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\alpha=(\alpha_1,\cdots,\alpha_m)$, with $0\le \alpha_i \le n$ for each $i=1,\cdots,m$. Define order $<$ as follow, $\forall \alpha,\beta \in N(m,n)$, $\beta < \alpha$ if and only if $\beta_i \le \alpha_i(i=1,\cdots,m)$ and $\sum\limits_{i=1}^{m}\beta_i <\sum\limits_{i=1}^{m}\alpha_i$.
For $\alpha \in N(m,n)$, define: $rank(\alpha)=\sum\limits_{i=1}^{m}\alpha_i$.
Let $(P,<)$ be a graded poset of rank $n$. We say that these elements $x_1, x_2, \cdots, x_k$ of $P$ has a symmetric chain if
(1) $x_{i+1}$ cover $x_i$, $i<k-1$;
(2) $rank(x_{i+1}) + rank(x_i)= rank(P)$, where $rank(P)$ is the maximum rank.
A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains.
Many beautiful results have been derived by symmetric chain since de Bruijn introduced it in 1949, see \cite{DeBruijn}. The elegant proof of Lubell appears in \cite{Lubell}. Symmetric chain decompositions requires each chain is rank symmetric. Its existence for $L(m,n)$ was not confirmed for $m\ge 5$.
Symmetric chain decompositions was only constructed for $m=3$ \cite{B.Lindstrom,Wen} and for $m=4$ \cite{Nathan,Wen}. A kind of chain decomposition of $L(m,n)$ for $m=3,4$ is also given in \cite{GuoceXin}. For more related symmetry chain decomposition, see \cite{Dhand-necklacePosets,Hersh-Boolean,Jordan-necklacePoset,Muhle-noncrossingPartition}.
In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method, as the following Theorem <^sathbb{S'}ot{r}ef{theo2}.
\begin{theo}\label{theo2}
For any positive integers $m,n$, the poset $(N(m,n),<)$ has a symmetric chain decomposition into $\{C_\alpha: \alpha\in S_{m,n}\}$.
That is, i) $N(m,n)=\cup_{\alpha\in S_{m,n}} C_{\alpha}$; ii) the above is a disjoint union; iii) $\operatorname{rank}(S(C_\alpha))+\operatorname{rank}(E(C_\alpha))=mn$.
(Note: $C_\alpha$, $S(C_\alpha)$ and $E(C_\alpha)$ as defined in <^sathbb{S'}ot{r}ef{Def:Calpha}).
\end{theo}
This paper is organized as follows:
In Section <^sathbb{S'}ot{r}ef{sec:Nmn}, firstly, we give the construction of $C_\alpha$ as Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}.
The feasibility of our construction is discussed in the second subsection.
Finally, we prove that these $C_\alpha$ do give a symmetric chain decomposition of $N(m,n)$. This result is given by Theorem <^sathbb{S'}ot{r}ef{theo2}.
In Section <^sathbb{S'}ot{r}ef{sec:involution}, we give an beautiful property of $C_{\alpha}$: an induced involution as Theorem <^sathbb{S'}ot{r}ef{theo:involution}.
\section{The symmetric chain decomposition of the $N(m,n)$\label{sec:Nmn}}
In this section, we will give a symmetric chain decomposition of $N(m,n)$ into $C_\alpha$.
This is divided into 3 subsections. Firstly, we give the construction of $C_\alpha$.
The feasibility of our construction is discussed in the second subsection.
Finally, we prove that these $C_\alpha$ do give a symmetric chain decomposition of $N(m,n)$.
\subsection{The construction of the symmetric chains $C_{\alpha}$}\label{Def:Calpha}
The symmetric chains $C_\alpha$ are indexed by $S_{m,n}$, which is defined by
\begin{align*}
S_{m,n}=&\{\alpha\in N(m,n): \alpha=(\alpha_1,\cdots,\alpha_{m-1},0), \alpha<^sodels k, 0\leq k \leq \lfloor\frac{mn}{2}<^sathbb{S'}ot{r}floor, \\
&\qquad \sum\limits_{i=t}^{m-1}\alpha_i \le \sum\limits_{i=t+1}^{m}(n-\alpha_i),1\le t \le m-1\}.
\end{align*}
For each ordered partition $\alpha\in S_{m,n}$, we construct a symmetric chain tableau
$C_\alpha$ as follows.
\begin{algor}
\label{alg-CTabCalpah}<^sathfrak{n}ormalfont
<^sathfrak{n}oindent
\textbf{Input:} An ordered partition $\alpha \in S_{m,n}$.
<^sathfrak{n}oindent
\textbf{Output:} The chain tableau $C_\alpha$.
\begin{enumerate}
\item[\textbf{Step 1.}] Draw an empty tableau $T$ of shape $m\times n$.
\item[\textbf{Step 2.}] For each $i$, color the first $\alpha_i$ cells in the $i$-th row of $T$ by green. These cells are called \emph{fixed cells} for $\alpha$.
\item[\textbf{Step 3.}] For $i$ from $1$ to $m-1$, we successively color some cells by gray as follows and call them \emph{forbidden cells} for $\alpha$.
\begin{enumerate}
\item For $i=1$, read the unfixed cells starting at row $i+1=2$, from right to left, and top to bottom. Color the first $\alpha_1$ cells by gray.
These are the \emph{forbidden cells} for $\alpha_1$.
\item Suppose the forbidden cells for $\alpha_1,<^sathbb{S'}ots, \alpha_{i-1}$ have been colored by gray
and we are going to construct the forbidden cells for $\alpha_i$. Starting at row $i+1$, we read the unfixed and unforbidden cells from right to left and from top to bottom. Color the first $\alpha_i$ cells by gray. These are the \emph{forbidden cells} for $\alpha_i$.
\item Repeat the above step until the forbidden cells for $\alpha_{m-1}$ are colored gray.
\end{enumerate}
\item[\textbf{Step 4.}] Call the unfixed and unforbidden cells fillable cells. Successively fill $1,2,3,<^sathbb{S'}ots$ in the fillable cells from left to right and from bottom to top.
The resulting tableau is the chain tableau $C_\alpha$ of $\alpha$.
\end{enumerate}
\end{algor}
The number of \emph{forbidden cells} in the $i$-th row of $C_\alpha$ is denoted $\alpha^E_i$ for each $i=1,\cdots,m$. Clearly $\alpha^E_1=0$. Let $\alpha^E=(\alpha^E_1,\alpha^E_2,\cdots,\alpha^E_m)$. It is convenient to denote by $S(C_\alpha)=\alpha$ the starting point of $C_{\alpha}$ and by $E(C_\alpha)=
(n-\alpha^E_1,n-\alpha^E_2,\cdots,n-\alpha^E_m)$ the end point of $C_{\alpha}$. The set $S_{m,n}$ is called the \emph{starting set} of our chain decomposition of $N(m,n)$.
For the sake of clarity, we give two examples to illustrate the Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}.
\begin{exam}\label{exa1}
The construction of $C_\alpha$ for $\alpha=(2,0,5,0)\in S_{4,6}$ is given in Figure <^sathbb{S'}ot{r}ef{fig:exa1}. The tableau $C_\alpha$ has natural correspondence with a situated chain, still denoted $C_{\alpha}:(2,0,5,0)<(2,0,5,1)<(2,0,6,1)<(2,1,6,1)<(2,2,6,1)<(2,3,6,1)<(2,4,6,1)<(3,4,6,1)<(4,4,6,1)<(5,4,6,1)<(6,4,6,1)$.
The chain starts at $S(C_\alpha)=(2,0,5,0)$ and ends at $E(C_\alpha)=(6,4,6,1).$ We get $\alpha^E=(0,2,0,5)$.
\begin{figure}
\caption{The construction of $C_{(2,0,5,0)}
\label{fig:exa1}
\end{figure}
\end{exam}
\begin{exam}
The construction of $C_\alpha$ for $\alpha=(1,3,2,0)\in S_{4,4}$ is given in Figure <^sathbb{S'}ot{r}ef{fig:exa2}. The tableau $C_\alpha$ has natural correspondence with a situated chain, still denoted $C_{\alpha}:(1,3,2,0)<(1,3,2,1)<(2,3,2,1)<(3,3,2,1)<(4,3,2,1)$.
The chain starts at $S(C_\alpha)=(1,3,2,0)$ and ends at $E(C_\alpha)=(4,3,2,1).$ We get $\alpha^E=(0,1,2,3)$.
\begin{figure}
\caption{The construction of $C_{(1,3,2,0)}
\label{fig:exa2}
\end{figure}
\end{exam}
\subsection{The feasibility of Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}}
To show the feasibility of Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}, it suffices to show the third step is feasible, since the other steps are obviously feasible.
This is achieved by the following Lemma <^sathbb{S'}ot{r}ef{lem:splitting}. We need the following definition first.
\begin{defn}
Let $\alpha\in S_{m,n}$. We recursively define the splitting rows for $\alpha$ as follows.
Row 1 is splitting.
If row $p$ is splitting, then row $q+1$ is splitting, where $q$ is the smallest positive integer with $p\le q\le m-1$ such that $\sum\limits_{i=p}^{q}\alpha_i\le \sum\limits_{i=p+1}^{q+1}(n-\alpha_i)$, which condition is satisfied for $q=m-1$ for any $p$ by
definition of $\alpha$.
\end{defn}
The following lemma justifies the name of splitting row.
\begin{lem}\label{lem:splitting}
Suppose $\alpha \in S_{m,n}$ has splitting row indices $1=q_{1}<q_{2}<\cdots<q_{s}=m$.
According Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}, if row $i$ is not splitting, then the forbidden cells for $\alpha_{i-1}$ has at least one cell in row $i+1$; otherwise
if row $i$ is splitting, then the forbidden cells for $\alpha_{i-1}$ has no cell in row $i+1$.
\end{lem}
\begin{proof}
We prove by induction on $k$ the following claim: the lemma holds for $i=1,2,<^sathbb{S'}ots, q_k$.
The base case $k=1$ is simple: since there is no forbidden cells for $\alpha_{0}$ and row $1$ is always splitting.
Assume the claim holds true for $k$, and we want to show that it holds for $k+1$.
Now row $i=q_k$ is splitting. By induction hypothesise, there is no forbidden cells for $\alpha_{q_k-1}$ in row $q_k+1$.
The forbidden cells for $\alpha_i$ starts at the rightmost cell in row $i+1$.
Then row $i+1$ is splitting if and only if $n-\alpha_{i+1}\ge \alpha_i$. This shows the claim holds when $q_{k+1}=i+1$.
If row $i+1$ is not splitting, then $n-\alpha_{i+1}> \alpha_i$, so that the forbidden cells for $\alpha_i$ occupies all
the rightmost $n-\alpha_{i+1}$ cells and at least one cell in row $i+2$.
This gives $\alpha_{i+1}^E=n-\alpha_{i+1}$.
By similar reasons, the forbidden cells for $\alpha_j$ has at least one
cell in row $j+2$ for $q_k \le q_{k+1}-2$, and we have
$\alpha^E_j=n-\alpha_j$ for $j=q_{k}+1,q_{k}+2,\cdots,q_{k+1}-1$.
Next consider the forbidden cells for $\alpha_i$ with $i=q_{k+1}-1$. They starts in row $i+2$ and end in the same row,
because of the inequality $\alpha_{q_k} + \alpha_{q_k+1} +\cdots + \alpha_{q_{k+1}-1} \le (n-\alpha_{{q_k}+1}) + (n-\alpha_{{q_k}+2}) + \cdots + (n-\alpha_{q_{k+1}})$
by the definition of splitting row $q_{k+1}$. The claim then holds for $k+1$.
\end{proof}
\begin{cor}\label{cor:aEnd}
Suppose $\alpha \in S_{m,n}$ has splitting row indices $1=q_{1}<q_{2}<\cdots<q_{s}=m$.
Then the following conditions hold for all $1\le k<s$. \[
\begin{cases}
\alpha_{q_k} > n-\alpha_{{q_k}+1} ,\\
\alpha_{q_k} + \alpha_{q_k+1} > (n-\alpha_{{q_k}+1}) + (n-\alpha_{{q_k}+2}) ,\\
\qquad \qquad \vdots\\
\alpha_{q_k} + \alpha_{q_k+1} +\cdots + \alpha_{q_{k+1}-2} > (n-\alpha_{{q_k}+1}) + (n-\alpha_{{q_k}+2}) + \cdots + (n-\alpha_{q_{k+1}-1}) ,\\
\alpha_{q_k} + \alpha_{q_k+1} +\cdots + \alpha_{q_{k+1}-1} \le (n-\alpha_{{q_k}+1}) + (n-\alpha_{{q_k}+2}) + \cdots + (n-\alpha_{q_{k+1}}) .\\
\end{cases}\]
Furthermore, we have $\alpha^E_j=n-\alpha_j$ for each $j=q_{k}+1,q_{k}+2,\cdots,q_{k+1}-1$ and $\alpha^E_{q_{{k+1}}}=\sum\limits_{j=q_{k}}^{q_{k+1}-1}\alpha_j-\sum\limits_{j=q_{k}+1}^{q_{{k+1}}-1}\alpha^E_j$.
\end{cor}
Corollary <^sathbb{S'}ot{r}ef{cor:aEnd} provides a better way to compute $\alpha^E$. See Figure <^sathbb{S'}ot{r}ef{fig:alphaE} for an example.
\begin{figure}
\caption{Computation of $\alpha^E$ by Corollary <^sathbb{S'}
\label{fig:alphaE}
\end{figure}
\subsection{The symmetric chain decomposition of $N(m,n)$}
Given $\alpha\in S_{m,n}$. Elements $c=(c_1,c_2,\cdots,c_m)\in C_{\alpha}$ in $C_\alpha$ can be described by the vector
$a=(a_1,a_2,\cdots,a_m)=c-\alpha$, called a \emph{fillable vector}. We will use the following equivalent definition, which follows easily from Algorithm
<^sathbb{S'}ot{r}ef{alg-CTabCalpah}: Let $d^{\alpha}_i=(n-\alpha_i)-\alpha^E_i$ be the number of cells in row $i$ that are neither fixed nor forbidden.
A vector $a_{\alpha}=(a_1,a_2,\cdots,a_m)$ is a {fillable vector} with respect to $\alpha$ if and only if there exists an integer $0 \le i_0 \le m$ such that $a_i=0$ for each $i=1,\cdots, i_0-1$, $0<a_{i_0}< d^{\alpha}_{i_0}$ and $a_j=d^{\alpha}_j$ for each $j=i_0+1,\cdots,m$. The $i_0=0$ case corresponds to $a_i=d^{\alpha}_i$ for $i=1,\cdots,m$.
According to the Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah}, we have the following properties:
\begin{prop}\label{prop1}
Assume $c=(c_1,c_2,\cdots,c_m)\in C_{\alpha}$ for some $\alpha=(\alpha_1,\cdots,\alpha_{m-1},0)\in S_{m,n}$. Then
there exists a unique fillable vector $a=(a_1,a_2,\cdots,a_m)$ such that $c=\alpha+a$. Moreover,
denote by $P=\{1\le i\le m: a_i>0\} \cup \{m\}$. Then
i) If $i\in P$ then row $i$ is splitting, and $c_t+\alpha^E_t=n$ for $t\ge i+1$;
ii) Suppose $j\in P$ and $a_t=0$ for all $i+1\le t \le j-1$. Then we have
\begin{align}
\sum\limits_ {t=i}^{j-1}\alpha_t=\sum\limits_{t=i+1}^{j}\alpha^E_t=\sum\limits_{t=i+1}^{j}(n-c_t), \qquad \text{ if } a_i>0;\\
\sum\limits_{t=i}^{j-1}\alpha_t \le \sum\limits_{t=i+1}^{j}\alpha^E_t=\sum\limits_{t=i+1}^{j}(n-c_t), \qquad \text{ if } a_i=0.
\end{align}
\end{prop}
Next, we show that the $\biguplus_{\alpha\in S_{m,n}}C_{\alpha}$ by the above Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah} is a symmetric chain decomposition of $N(m,n)$ as described by the Theorem <^sathbb{S'}ot{r}ef{theo2}. Now we give the proof of the Theorem <^sathbb{S'}ot{r}ef{theo2} as follows.
\begin{proof}
Part i) and ii) follows from Lemmas <^sathbb{S'}ot{r}ef{lem:part1} and <^sathbb{S'}ot{r}ef{lem:part2} below. Part iii) is obvious.
\end{proof}
\begin{lem}\label{lem:part2}
Suppose $\alpha$ and $\beta$ are distinct element of $S_{m,n}$. Then $C_\alpha \cap C_\beta=\emptyset$.
\end{lem}
\begin{proof}
Write $\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_{m-1},0)$ and $\beta=(\beta_1,\beta_2,\cdots,\beta_{m-1},0)$.
Since $\alpha<^sathfrak{n}eq \beta$, there exists a positive integer $1 \le i_0 \le m-1$ such that $\alpha_{i_0}<^sathfrak{n}eq\beta_{i_0}$ and $\alpha_{i}=\beta_{i}$ for each
$i$ with $i_0<i\le m-1$. Without loss of generality, we may assume $\alpha_{i_0}<\beta_{i_0}$.
Assume to the contrary that $c\in C_{\alpha}\cap C_{\beta}$. Then there exists fillable vectors $a_{\alpha}=(a_1,\cdots,a_m)$ and $b_{\beta}=(b_1,\cdots,b_m)$ such that
$c=\alpha+a_{\alpha}=\beta+b_{\beta}$.
Now $a_{i_0}+\alpha_{i_0}=b_{i_0}+\beta_{i_0}$ implies that $a_{i_0}=b_{i_0}+\beta_{i_0}-\alpha_{i_0}>0$. It follows that
\[
\begin{cases}
\text{$\textcircled{1}$\ $\alpha_{i}=\beta_{i}$ and $a_i=b_i$, where $i_0\le i\le m$.}\\
\text{$\textcircled{2}\ X:=\alpha_{i_0}+\alpha_{i_0+1}+\cdots+\alpha_{m-1}$}\\
\text{$=(n-\alpha_{i_0+1}-a_{i_0})+\cdots+(n-\alpha_{m-1}-a_{m-1})+(n-a_m)$, where $a_{i_0}>0$.}\\
\text{$\textcircled{3}\ \beta_{i_0}+\beta_{i_0+1}+\cdots+\beta_{m-1}$}\\
\text{$\le(n-\beta_{i_0+1}-b_{i_0})+\cdots+(n-\beta_{m-1}-b_{m-1})+(n-b_m)$.}
\end{cases}\]
Thus we get
$X=\alpha_{i_0}+\alpha_{i_0+1}+\cdots+\alpha_{m-1}<\beta_{i_0}+\beta_{i_0+1}+\cdots+\beta_{m-1}\le X$,
where $X=(n-\alpha_{i_0+1}-a_{i_0})+\cdots+(n-\alpha_{m-1}-a_{m-1})+(n-a_m)=(n-\beta_{i_0+1}-b_{i_0})+\cdots+(n-\beta_{m-1}-b_{m-1})+(n-b_m)$.
Therefore $X<X$, which is a contradiction. This proves the lemma.
\end{proof}
\begin{lem}\label{lem:part1}
For any $c\in N(m,n)$, there is an $\alpha\in S_{m,n}$ such that $c\in C_\alpha$.
\end{lem}
\begin{proof}
For given $c=(c_1,c_2,\cdots,c_m)$, we assume the existence of an undetermined
$\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_{m-1},0)\in S_{m,n}$ and an undetermined fillable vector $a=(a_1,a_2,\cdots,a_m)$ such that $c=\alpha+a$.
We recursively compute $P$, $\alpha$, $\alpha^E$, and $a$ backwardly as follows.
\begin{enumerate}
\item Set $P=\{m\}$, $a_m=c_m$, $\alpha_m=0$, and $t=m$. Treat $a_0>0$.
\item Iteratively using the following Claim to find a $r$ such that
$a_{r+1}=a_{r+2}=\cdots = a_{t-1}=0$ and $a_r>0$ for some $r\le t-1$. Add the element $r$ into $P$.
\item If $r>0$ then $a_{r+1}=a_{r+2}=\cdots = a_{t-1}=0$ which implies $\alpha_i=c_i$ for $r+1\le i \le t-1$,
and set $\alpha_r= \sum\limits_{i=r+1}^{t}(n-c_i)-\sum\limits_{i=r+1}^{t-1}c_i$.
Set $t=r$ and go to step 2.
\item If $r=0$ then $a_{t-1}=a_{t-2}=\cdots = a_{1}=0$, and we have $\alpha_i=c_i$ for $1\le i \le t-1$.
\end{enumerate}
The lemma clearly follows if we prove the following claim.
\textbf{Claims:}
Suppose $t\in P$, $r<t$ and $a_{r+1}=\cdots=a_{t-1}=0$. Then
i) if $\sum\limits_{i=r}^{t-1}c_i\le \sum\limits_{i=r+1}^{t}(n-c_i)$ then $a_r=0$;
ii) if $\sum\limits_{i=r}^{t-1}c_i> \sum\limits_{i=r+1}^{t}(n-c_i)$ then $a_r>0$. Consequently
$\alpha_i=c_i$ for each $i=r+1,\cdots,t-1$ . Hence $\sum\limits_{i=r}^{t-1}\alpha_i = \sum\limits_{i=r+1}^{t}(n-\alpha_i)$.
For part i), assume to the contrary that $a_r>0$ holds for possible $\alpha$ and $a$.
By Property <^sathbb{S'}ot{r}ef{prop1}(ii)(1), we get $\alpha_r+\cdots+\alpha_{t-1} = \alpha^E_{r+1}+\cdots+\alpha^E_t$.
Since $a_{r+1}=\cdots=a_{t-1}=0$, we have $\alpha_i=c_i$, and $\alpha^E_i=n-c_i$ for each $i=r+1,\cdots,t-1$. Note that $\alpha^E_t=n-c_t$.
It follows that
$$\sum\limits_{i=r}^{t-1}c_i\le (n-c_{r+1})+\cdots+(n-c_t) =\alpha_r+c_{r+1}+\cdots+c_{t-1}<c_r+\cdots+c_{t-1}. $$
This is a contradiction. Therefore $a_r=0$.
For part ii), we have $\alpha_i=c_i$ and $\alpha^E_i=n-c_i$ for each $i=r+1,<^sathbb{S'}ots,t-1$.
Assume to the contrary that $a_r=0$ holds for possible $\alpha$ and $a$.
Then $\alpha_r=c_r$ and $\alpha^E_r=n-c_r$.
By Property <^sathbb{S'}ot{r}ef{prop1}(ii)(2), we get $\alpha_r+\cdots+\alpha_{t-1} \le \alpha^E_{r+1}+\cdots+\alpha^E_t$,
which is rewritten as
$$c_r+\cdots+c_{t-1} \le \alpha^E_{r+1}+\cdots+\alpha^E_t\le (n-c_{r+1})+\cdots+(n-c_t).$$
This contradicts the hypothesis. Therefore $a_r>0$. Then by Corollary <^sathbb{S'}ot{r}ef{cor:aEnd}, we should have
$\sum\limits_{i=r}^{t-1}\alpha_i = \sum\limits_{i=r+1}^{t}(n-\alpha_i)$. Solving for $\alpha_r$ gives
$\alpha_r=\sum\limits_{i=r+1}^{t}(n-c_i)-\sum\limits_{i=r+1}^{t-1}c_i$. Then $a_r=c_r-\alpha_r$. Note that $\alpha^E_r$
cannot be determined yet.
\end{proof}
\begin{exam}
For $c=(5,2,3,6,4,1,5,3)\in N(8,7)$, we can find the chain $C_{\alpha}$ with $c\in C_{\alpha}$, where $\alpha=(5,2,1,6,4,1,4,0)$, as follows in Figure <^sathbb{S'}ot{r}ef{fig:cToalpha}.
\begin{figure}
\caption{Computing $\alpha$ from $c$.\label{fig:cToalpha}
\label{fig:cToalpha}
\end{figure}
\end{exam}
\begin{exam}
We give the symmetric chain decomposition of $N(3,2)$ by Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah} in Figure <^sathbb{S'}ot{r}ef{fig:N32} as follows.
\begin{figure}
\caption{The symmetric chain decomposition of $N(3,2)$.\label{fig:N32}
\label{fig:N32}
\end{figure}
\end{exam}
\section{An induced involution\label{sec:involution}}
There is a natural involution $*$ on $N(m,n)$ defined by $c^*=(n-c_m,n-c_{m-1},<^sathbb{S'}ots, n-c_1)$.
It induces a map $\psi(S(C_\alpha))= (E(C_{\alpha}))^*$, or equivalently
$$ \psi(\alpha)= \alpha^E_{rev}:=(\alpha^E_m,\alpha^E_{m-1},\cdots,\alpha^E_2,0).$$
\begin{theo}\label{theo:involution}
The map $\psi$ is an involution on $S_{m,n}$. More precisely,
$C_{\psi(\alpha)}$ is obtained by rotating the tableau $C_\alpha$ 180 degree.
\end{theo}
\begin{proof}
Suppose Algorithm <^sathbb{S'}ot{r}ef{alg-CTabCalpah} produces the tableau $C_{\alpha}$. Let $1=q_{1}<q_{2}<\cdots<q_{s}=m$ be all the indices of splitting rows of $C_{\alpha}$.
Define $\bar{q_k}:=m-q_{m-k+1}+1$, $k=1,2,\cdots,s$.
For any $k\in \{1,2,\cdots,s-1\}$, by Corollary <^sathbb{S'}ot{r}ef{cor:aEnd}, we obtain $\alpha^E_j=n-\alpha_j$ for each $j=q_{k}+1,q_{k}+2,\cdots,q_{k+1}-1$ and $\alpha^E_{q_{{k+1}}}=\sum\limits_{j=q_{k}}^{q_{k+1}-1}\alpha_j-\sum\limits_{j=q_{k}+1}^{q_{{k+1}}-1}\alpha^E_j$.
The reason that $\alpha^E_{rev} \in S_{m,n}$ is as follows:
The condition of $\alpha \in S_{m,n}$ is equivalent to
\begin{align*}
&\alpha_{q_k}+\alpha_{q_k}^E\le n, \quad 1\le k \le s;\\
&\begin{cases}\label{formula3}\tag{3}
\alpha_{q_k} > \alpha^E_{{q_k}+1},\\
\alpha_{q_k} + \alpha_{q_k+1} > \alpha^E_{{q_k}+1} + \alpha^E_{{q_k}+2},\\
\qquad \qquad \vdots\\
\alpha_{q_k} + \alpha_{q_k+1} +\cdots + \alpha_{q_{k+1}-2} > \alpha^E_{{q_k}+1} + \alpha^E_{{q_k}+2} + \cdots + \alpha^E_{q_{k+1}-1},\\
\alpha_{q_k} + \alpha_{q_k+1} +\cdots + \alpha_{q_{k+1}-1} = \alpha^E_{{q_k}+1} + \alpha^E_{{q_k}+2} + \cdots + \alpha^E_{q_{k+1}}.\\
\end{cases} 1\le k \le s-1;
\end{align*}
The above formula (<^sathbb{S'}ot{r}ef{formula3}) holds if and only if the formula
\begin{align*}
&\alpha_{q_k}^E+\alpha_{q_k}\le n, \quad 1\le k \le s;\\
&\begin{cases}\label{formula4}\tag{4}
\alpha^E_{q_{k+1}} > \alpha_{q_{k+1}-1},\\
\alpha^E_{q_{k+1}} + \alpha^E_{q_{k+1}-1} > \alpha_{q_{k+1}-1} + \alpha_{q_{k+1}-2},\\
\qquad \qquad \vdots\\
\alpha^E_{q_{k+1}} + \alpha^E_{q_{k+1}-1} + \cdots + \alpha^E_{{q_k}} > \alpha_{q_{k+1}-1} + \alpha_{q_{k+1}-2} + \cdots + \alpha_{q_k-1},\\
\alpha^E_{q_{k+1}} + \alpha^E_{q_{k+1}-1} + \cdots + \alpha^E_{{q_k}+1} = \alpha_{q_{k+1}-1} + \alpha_{q_{k+1}-2} + \cdots + \alpha_{q_k}.\\
\end{cases} 1\le k \le s-1;
\end{align*}
holds. The above formula (<^sathbb{S'}ot{r}ef{formula4}) holds if and only if the formula
\begin{align*}
&(\alpha_{rev})_{\bar{q_k}}+(\alpha_{rev})_{\bar{q_k}}^E\le n, \quad 1\le k \le s;\\
&\begin{cases}
(\alpha_{rev})_{\bar{q_k}} >(\alpha_{rev})^E_{{\bar{q_k}}+1},\\
(\alpha_{rev})_{\bar{q_k}} + (\alpha_{rev})_{\bar{q_k}+1} > (\alpha_{rev})^E_{\bar{{q_k}}+1} + (\alpha_{rev})^E_{\bar{{q_k}}+2},\\
\qquad \qquad \vdots\\
(\alpha_{rev})_{\bar{q_k}} + (\alpha_{rev})_{\bar{q_k}+1} +\cdots + (\alpha_{rev})_{\bar{q_{k+1}}-2} > (\alpha_{rev})^E_{{\bar{q_k}}+1} + (\alpha_{rev})^E_{{\bar{q_k}}+2} + \cdots + (\alpha_{rev})^E_{\bar{q_{k+1}}-1},\\
(\alpha_{rev})_{\bar{q_k}} + (\alpha_{rev})_{\bar{q_k}+1} +\cdots + (\alpha_{rev})_{\bar{q_{k+1}}-1} = (\alpha_{rev})^E_{\bar{{q_k}}+1} + (\alpha_{rev})^E_{{\bar{q_k}}+2} + \cdots + (\alpha_{rev})^E_{\bar{q_{k+1}}},\\
\text{where $1\le k \le s-1$.}
\end{cases}
\end{align*}
holds. This is equivalent to $\psi(\alpha)=\alpha^E_{rev} \in S_{m,n}$.
\end{proof}
\begin{exam}
We give an example for Theorem <^sathbb{S'}ot{r}ef{theo:involution}.
\begin{figure}
\caption{Obtaining the $\alpha^E_{rev}
\label{exa:alphaEndSpoint}
\end{figure}
\begin{figure}
\caption{$C_{\alpha}
\label{exa:alphaInvolution}
\end{figure}
\end{exam}
\textbf{Acknowledgements:} The author would like to thank Professor Guoce Xin for helpful discussions.
\end{document} |
\begin{document}
\title{Information complexity of the AND function in the two-party, and multiparty settings}
\author{
Yuval Filmus\inst{1}
\and Hamed Hatami\thanks{Supported by an NSERC grant.}\inst{2}
\and Yaqiao Li\inst{3}
\and Suzin You\inst{4}
}
\institute{
Technion --- Israel Institute of Technology, \email{[email protected]}
\and
McGill University, \email{[email protected]}
\and
McGill University, \email{[email protected]}
\and
McGill University, \email{[email protected]}
}
\maketitle
\begin{abstract}
In a recent breakthrough paper [M. Braverman, A. Garg, D. Pankratov, and O. Weinstein, From information to exact communication, STOC'13 Proceedings of the 2013 ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 151--160.] Braverman {\it et al.}\ developed a local characterization for the zero-error information complexity in the two party model, and used it to compute the exact internal and external information complexity of the $2$-bit AND function.
In this article, we extend their result to the multiparty number-in-hand model by proving that the generalization of their protocol has optimal internal and external information cost for certain distributions. Our proof has new components, and in particular it fixes some minor gaps in the proof of Braverman {\it et al}.
\end{abstract}
\section{Introduction}
Although communication complexity has since its birth been witnessing steady and rapid progress, it was not until recently that a focus on an information-theoretic approach resulted in new and deeper understanding of some of the classical problems in the area. This gave birth to a new area of complexity theory called \emph{information complexity}. Recall that communication complexity is concerned with minimizing the amount of communication required for players who wish to evaluate a function that depends on their private inputs. Information complexity, on the other hand, is concerned with the amount of information that the communicated bits reveal about the inputs of the players to each other, or to an external observer.
One of the important achievements of information complexity is the recent result of~\cite{MR3210776} that determines the exact asymptotics of the randomized communication complexity of one of the oldest and most studied problems in communication complexity, set disjointness:
\begin{equation}
\label{eq:DISJ}
\lim_{\varepsilonilon \to 0} \lim_{n\to\infty} \frac{R_\varepsilonilon(\mathrm{DISJ}_n)}{n} \approx 0.4827.
\end{equation}
Here $R_\varepsilonilon(\cdot)$ denotes the randomized communication complexity with an error of at most $\varepsilonilon$ on every input, and $\mathrm{DISJ}_n$ denotes the set disjointness problem. In this problem, Alice and Bob each receive a subset of $\{1,\ldots,n\}$, and their goal is to determine whether their sets are disjoint or not. Prior to the discovery of these information-theoretic techniques, proving the lower bound $R_\varepsilonilon(\mathrm{DISJ}_n) = \Omega(n)$ had already been a challenging problem, and even Razborov's~\cite{MR1192778} short proof of that fact is intricate and sophisticated.
Note that the set disjointness function is nothing but an $\OR$ of $\AND$ functions. More precisely, for $i=1,\ldots,n$, if $x_i$ is the Boolean variable which represents whether $i$ belongs to Alice's set or not, and $y_i$ is the corresponding variable for Bob, then $\bigvee_{i=1}^n (x_i \wedge y_i)$ is true if and only if Alice's input intersects Bob's input. Braverman {\it et al.}~\cite{MR3210776} exploited this fact to prove \eqref{eq:DISJ}. Roughly speaking, they first determined the exact information cost of the $2$-bit AND function for any underlying distribution $\mu$ on the set of inputs $\{0,1\} \times \{0,1\}$, and then used the fact that amortized communication equals information cost~\cite{MR3265014} to relate this to the communication complexity of the set disjointness problem. The constant $0.4827$ in (\ref{eq:DISJ}) is indeed the maximum of the information complexity of the $2$-bit AND function over all
measures $\mu$ that assign a zero mass to $(1,1) \in \{0,1\} \times \{0,1\}$. That is
$$
\max_{\mu:\mu(11)=0} \IC_\mu(\AND) \approx 0.4827,
$$
where $\IC_\mu(\AND)$ denotes the information cost of the $2$-bit $\AND$ function with respect to the distribution $\mu$ with no error (See Definition~\ref{def:infocost} below). These results show the importance of knowing the exact information complexity of simple functions such as the $\AND$ function.
Although obtaining the asymptotics of $R_\varepsilonilon(\mathrm{DISJ}_n)$ from the information complexity of the $\AND$ function is not straightforward and a formal proof requires overcoming some technical difficulties, the bulk of~\cite{MR3210776} is dedicated to computing the exact information complexity of the $2$-bit AND function. This rather simple-looking problem had been studied previously by Ma and Ishwar~\cite{MaIshwar2011,MaIshwar2013}, and some of the key ideas of~\cite{MR3210776} originate from their work. In~\cite{MR3210776} Braverman {\it et al} introduced a protocol to solve the $\AND$ function, and proved that it has optimal internal and external information cost. Interestingly this protocol is not a conventional communication protocol as it has access to a continuous clock, and the players are allowed to ``buzz'' at randomly chosen times. However, one can approximate it by conventional communication protocols through dividing the time into
finitely many discrete units. Indeed, it is known~\cite{MR3210776} that no protocol with a bounded number of rounds can have optimal information cost for the $\AND$ function, and hence the infinite number of rounds, implicit in the continuous clock, is essential. We shall refer to this protocol as the \emph{buzzers} protocol.
\subsection{Our contributions}
{\bf Fixing the argument of \cite{MR3210776}:}
In order to show that the \emph{buzzers} protocol has optimal information cost, inspired by the work of Ma and Ishwar~\cite{{MaIshwar2011,MaIshwar2013}}, Braverman {\it et al} came up with a local concavity condition, and showed that if a protocol satisfies this condition, then it has optimal information cost. This condition, roughly speaking, says that it suffices to verify that one does not gain any advantage over the conjectured optimal protocol if one of the players starts by sending a bit $B$. In the original paper~\cite{MR3210776}, it is claimed that it suffices to verify this condition only for signals $B$ that reveal arbitrarily small information about the inputs. As we shall see, however, this is not true, and one can easily construct counter-examples to this statement.
In Theorem~\ref{thm:LocalCharNoErrorWeak} we prove a variant of the local concavity condition that allows one to consider only signals $B$ with small information leakage, and then apply it to fix the argument in \cite{MR3210776}. We have been informed through private communication that Braverman {\it et al} have also independently fixed the argument in \cite{MR3210776}.
{\bf \noindent Extension of \cite{MR3210776} to the multi-party setting:} We then apply Theorem~\ref{thm:LocalCharNoErrorWeak} to extend the result of~\cite{MR3210776} to the multiparty number-in-hand model by defining a generalization of the \emph{buzzers} protocol, and then prove in Theorem~\ref{thm:mainMultiparty} that it has optimal internal and external information cost when the underlying distribution satisfies the following assumption:
\begin{assumption}
\label{assumption}
The support of $\mu$ is a subset of $\{\vec{0},\vec{1},\e_1,\ldots,\e_k\}$, where $\e_i$ is the usual $i^\text{th}$ basis vector $(0,\dots,0,1,0,\dots,0)$.
\end{assumption}
Note that in the two-party setting, every distribution satisfies this assumption and thus our results are complete generalizations of the results of~\cite{MR3210776} in the two party setting. The distribution in Assumption~\ref{assumption} arise naturally in the study of the set disjointness problem, and as a result they have been considered previously in~\cite{specialdistribution}.
This extension is not straightforward since in \cite{MR3210776}, a large part of the calculations for verifying the local concavity conditions are carried out by the software Mathematica. However, in the number-in-hand model, having an arbitrary number of players, one cannot simply rely on a computer program for those calculations. Instead, first we had to analyze and understand what happens at different stages of the protocol, and once we reduced the problem to sufficiently simple equations (with a constant number of variables), then we used a computer program to verify them. We believe our proof provides some new insights even for the two-party setting.
\section{Preliminaries}
\subsection{Notation}
We typically denote the random variables by capital letters (e.g $A,B,C,X,Y,\Pi$). For the sake of brevity, we shall write $A_1\ldots A_n$ to denote the random variable $(A_1,\ldots,A_n)$ and \emph{not} the product of the $A_i$'s. We use $[n]$ to denote the set $\{1,\ldots,n\}$, and $\supp(\mu)$ to denote the support of a measure $\mu$. We denote the statistical distance (a.k.a. total variation distance) of two measures $\mu$ and $\nu$ on the sample space $\Omega$ by $|\mu-\nu| := \frac{1}{2}\sum_{a \in \Omega} |\mu(a)-\nu(a)|$.
For every $\varepsilonilon \in [0,1]$, $\entropy(\varepsilonilon) = -\varepsilonilon \log\varepsilonilon - (1-\varepsilonilon) \log(1-\varepsilonilon)$ denotes the binary entropy, where here and throughout the paper $\log(\cdot)$ is in base $2$, and $0 \log 0 = 0$.
Recall $\Dvg(\mu || \nu)$ means the divergence (a.k.a. relative entropy, or Kullback Leibler distance) between two distributions $\mu$ and $\nu$. Let $X$ and $Y$ be two random variables, the standard notation $\MI(X,Y)$ means the mutual information between $X$ and $Y$, sometimes we use the notation $\Dvg(X||Y)$ to denote the divergence between the distributions of $X$ and $Y$. For definitions and basic facts regarding divergence and mutual information, see~\cite{cover2012elements}.
\subsection{Communication complexity}
The notion of two-party communication complexity was introduced by Yao~\cite{Yao:1979} in 1979. In this model there are two players (with unlimited computational power), often called Alice and Bob, who wish to collaboratively compute a given function $f\colon \mathcal X \times \mathcal Y \to \mathcal Z$. Alice receives an input $x \in \mathcal X$ and Bob receives $y \in \mathcal Y$. Neither of them knows the other player's input, and they wish to communicate in accordance with an agreed-upon protocol $\pi$ to compute $f(x,y)$. The protocol $\pi$ specifies as a function of (only) the transmitted bits whether the communication is over, and if not, who sends the next bit. Furthermore $\pi$ specifies what the next bit must be as a function of the transmitted bits, and the input of the player who sends the bit. The \emph{cost} of the protocol is the total number of bits transmitted on the worst case input. The \emph{transcript} $\Pi$ of a protocol $\pi$ is the list of all the transmitted bits during the execution of the protocol.
In the randomized communication model, the players might have access to a shared random string (\emph{public randomness}), and their own private random strings (\emph{private randomness}). These random strings are independent, but they can have any desired distributions individually. In the randomized model the transcript also includes the public random string in addition to the transmitted bits. Similar to the case of deterministic protocols, the \emph{cost} is the total number of bits transmitted on the worst case input and random strings. The \emph{average cost} of the protocol is the expected number of bits transmitted on the worst case input.
For a function $f\colon \mathcal X \times \mathcal Y \to \mathcal Z$ and a parameter $\varepsilonilon>0$, we denote by $R_\varepsilonilon(f)$ the cost of the best randomized protocol for computing $f$ with probability of error at most $\varepsilonilon$ on \emph{every} input.
\subsection{Information complexity} \label{sec:IC-definition}
The setting is the same as in communication complexity, where Alice and Bob (having infinite computational power) wish to mutually compute a function $f\colon \mathcal X \times \mathcal Y \to \mathcal Z$. To be able to measure information, we also need to assume that there is a prior distribution $\mu$ on $\mathcal X \times \mathcal Y$.
For the purpose of communication complexity, once we allow public randomness, it makes no difference whether we permit the players to have private random strings or not. This is because the private random strings can be simulated by parts of the public random string. On the other hand, for information complexity, it is crucial to permit private randomness, and once we allow private randomness, public randomness becomes inessential. Indeed, one of the players can use her private randomness to generate the public random string, and then transmit it to the other player. Although this might have very large communication cost, it has no information cost, as it does not reveal any information about the players' input.
Probably the most natural way to define the information cost of a protocol is to consider the amount of information that is revealed about the inputs $X$ and $Y$ to an external observer who sees the transmitted bits and the public randomness. This is known as the \emph{external information cost} and is formally defined as the mutual information between $XY$ and the transcript of the protocol (recall that the transcript $\Pi_{XY}$ contains the public random string $R$). While this notion is interesting and useful, it turns out there is another way of defining information cost that has some very useful properties. This is called the \emph{internal information cost} or just the \emph{information cost} for short, and is equal to the amount of information that Alice and Bob learn about each other's input from the communication. Note that Bob knows $Y$, the public randomness $R$ and his own private randomness $R_B$, and thus what he learns about $X$ \emph{from the communication} can be measured by $I(X;\Pi|YRR_B)$. Similarly, what Alice learns about $Y$ from the communication can be measured by $I(Y;\Pi|XRR_A)$ where $R_A$ is Alice's private random string. It is not difficult to see~\cite{MR2743255} that conditioning on the public and private randomness does not affect these quantities. In other words $I(X;\Pi|YRR_B)=I(X;\Pi|Y)$ and $I(Y;\Pi|XRR_A)= I(Y;\Pi|X)$. We summarize these in the following definition.
\begin{definition}
\label{def:infocost}
The \emph{external information cost} and the \emph{internal information cost} of a protocol $\pi$ with respect to a distribution $\mu$ on inputs from $\mathcal X \times \mathcal Y$ are defined as
$$\IC_\mu^\ext(\pi) = I(\Pi; XY),$$
and
$$\IC_\mu(\pi) = I(\Pi; X|Y)+I(\Pi; Y|X),$$
respectively, where $\Pi=\Pi_{XY}$ is the transcript of the protocol when it is executed on the inputs $XY$.
\end{definition}
We will be interested in certain \emph{communication tasks}. Let $[f,\varepsilon]$ denote the task of computing the value of $f(x,y)$ correctly with probability at least $1-\varepsilon$ for \emph{every} $(x,y)$. Thus a protocol $\pi$ performs this task if
\begin{equation}
\ProbOp[\pi(x,y) \neq f(x,y)] \leqslant \varepsilonilon, \quad \forall\ (x,y) \in \mathcal X \times \mathcal Y.
\end{equation}
Given another distribution $\nu$ on $\mathcal X \times \mathcal Y$, let $[f, \nu, \varepsilonilon]$ denote the task of computing the value of $f(x,y)$ correctly with probability at least $1-\varepsilon$ if the input $(x,y)$ is sampled from the distribution $\nu$. A protocol $\pi$ performs this task if
\begin{equation}
\ProbOp_{(x,y) \sim \nu}[\pi(x,y) \neq f(x,y)] \leqslant \varepsilonilon.
\end{equation}
Note that a protocol $\pi$ performs $[f, 0]$ if it computes $f$ correctly on \emph{every} input while performing $[f, \nu,0]$ means computing $f$ correctly on the inputs that belong to the support of $\nu$.
The \emph{information complexity} of a communication task $T$ with respect to a measure $\mu$ is defined as
\begin{equation} \IC_\mu(T) = \inf_{\pi :\ \pi \text{\ performs\ } T} \IC_\mu(\pi). \end{equation}
It is essential here that we use infimum rather than minimum as there are tasks for which there is no protocol that achieves $\IC_\mu(T)$ while there is a sequence of protocols whose information cost converges to $\IC_\mu(T)$. The \emph{external information complexity} of a communication task $T$ is defined similarly. We will abbreviate $\IC_\mu(f,\varepsilon)=\IC_\mu([f,\varepsilon])$, $\IC_\mu(f,\nu,\varepsilon)=\IC_\mu([f,\nu,\varepsilon])$, etc.
It is important to note that when $\mu$ does not have full support, $\IC_\mu(f,\mu,0)$ can be strictly smaller than $\IC_\mu(f,0)$. We sometimes also abbreviate $\IC_\mu(f) = \IC_\mu(f,0)$.
\begin{remark}[A warning regarding our notation]
In the literature of information complexity it is common to use ``$\IC_\mu(f,\varepsilonilon)$'' to denote the distributional error case, i.e. what we denote by $\IC_\mu(f,\mu,\varepsilonilon)$. Unfortunately this has become the source of some confusions in the past, as sometimes ``$\IC_\mu(f,\varepsilonilon)$'' is used to denote both of the distributional error $[f,\mu,\varepsilonilon]$ and the point-wise error $[f,\varepsilonilon]$. To avoid ambiguity we distinguish the two cases by using the different notations $\IC_\mu(f,\mu,\varepsilonilon)$ and $\IC_\mu(f,\varepsilonilon)$.
\end{remark}
\subsection{The continuity of information complexity} \label{sec:IC-continuity}
The information complexities $\IC_\mu(f,\varepsilonilon)$ and $\IC_\mu(f,\nu,\varepsilonilon)$ are both continuous with respect to $\varepsilonilon$. The following simple lemma from~\cite{braverman2015interactive} proves the continuity for $\varepsilonilon \in (0,1]$. The continuity at $0$ is more complicated and is proven in~\cite{MR3210776}.
\begin{lemma}\cite{braverman2015interactive}
For every $f\colon\mathcal{X} \times \mathcal{Y} \to \mathcal{Z}$, $\varepsilonilon_2 >\varepsilonilon_1>0$ and measures $\mu,\nu$ on $\mathcal{X} \times \mathcal{Y}$, we have
\begin{equation}
\label{eq:continuityEps}
\IC_\mu(f,\nu,\varepsilonilon_1)- \IC_\mu(f,\nu,\varepsilonilon_2) \leqslant (1-\varepsilonilon_1/\varepsilonilon_2) \log |\mathcal{X} \times \mathcal{Y}|,
\end{equation}
\end{lemma}
\begin{proof}
Let $f\colon\mathcal{X} \times \mathcal{Y} \to \mathcal{Z}$, and consider a protocol $\pi$ with information cost $I$, and error $\varepsilonilon_2>0$. Set $\delta= 1-\varepsilonilon_1/\varepsilonilon_2$, and let $\tau$ be the following protocol
\begin{itemize}
\item With probability $1-\delta$ run $\pi$.
\item With probability $\delta$ Alice and Bob exchange their inputs and compute $f(x,y)$.
\end{itemize}
The theorem follows as the new protocol has error at most $(1-\delta)\varepsilonilon_2=\varepsilonilon_1$, and information cost at most $I + \delta \log |\mathcal{X} \times \mathcal{Y}|$.
\end{proof}
\begin{remark}
The same proof implies
$\IC_\mu(f,\varepsilonilon_1)- \IC_\mu(f,\varepsilonilon_2) \leqslant (1-\varepsilonilon_1/\varepsilonilon_2) \log |\mathcal{X} \times \mathcal{Y}|$.
\end{remark}
Note that $\IC_\mu(f,\mu,0)$ is not always continuous with respect to $\mu$. For example, let
\begin{equation} \label{eq:measure-delta}
\mu_\varepsilonilon =
\begin{pmatrix}
\frac{1-\varepsilonilon}{3} & \frac{1-\varepsilonilon}{3} \\
\frac{1-\varepsilonilon}{3} & \varepsilonilon
\end{pmatrix}, \qquad
\mu = \lim_{\varepsilonilon \to 0} \mu_\varepsilonilon = \begin{pmatrix}
\frac{1}{3} & \frac{1}{3} \\
\frac{1}{3} & 0
\end{pmatrix}.
\end{equation}
Now for the $2$-bit $\AND$ function, we have $\IC_{\mu}(\AND,\mu,0)=0$, while $\IC_{\mu_\varepsilonilon}(\AND,\mu_\varepsilonilon,0)=\IC_{\mu_\varepsilonilon}(\AND)$ as $\mu_\varepsilonilon$ has full support. Thus
$$\lim_{\varepsilonilon \to 0} \IC_{\mu_\varepsilonilon}(\AND, \mu_\varepsilonilon,0) =\lim_{\varepsilonilon \to 0} \IC_{\mu_\varepsilonilon}(\AND) = \IC_{\mu}(\AND),$$
which is known to be bounded away from $0$. The same example also shows that $\IC_\mu(f, \mu, \varepsilonilon)$ is not always continuous with respect to $\varepsilonilon$ at $\varepsilonilon = 0$ if $\mu$ depends on $\varepsilonilon$. In fact, $\IC_{\mu_\varepsilonilon}(\AND, \mu_\varepsilonilon, \varepsilonilon) = 0$ while $\IC_{\mu_\varepsilonilon}(\AND, \mu_\varepsilonilon, 0) = \IC_{\mu_\varepsilonilon}(\AND)$ for all $\varepsilonilon > 0$, hence when $\varepsilonilon > 0$ is sufficiently small we find $\IC_{\mu_\varepsilonilon}(\AND, \mu_\varepsilonilon, 0)$ is close to $\IC_\mu(\AND)$ which is bounded away from $0$.
However it turns out that $\IC_\mu(f,\nu,\varepsilonilon)$ is continuous with respect to $\mu$ for all $\varepsilonilon \geqslant 0$ when $\nu$ is fixed. This follows from the fact, established in~\cite[Lemma 4.4]{SelfRed}, that for every protocol $\pi$ and every two measures $\mu_1$ and $\mu_2$ with $|\mu_1-\mu_2| \leqslant \delta$ (the distribution metric is statistical distance), we have
\begin{equation}
\label{eq:continMu}
|\IC_{\mu_1}(\pi)-\IC_{\mu_2}(\pi)| \leqslant 2 \log(|\mathcal X \times \mathcal Y|) \delta + 2 H(2\delta).
\end{equation}
Consequently
$$|\IC_{\mu_1}(f,\nu,\varepsilonilon)-\IC_{\mu_2}(f,\nu,\varepsilonilon)| \leqslant 2 \log(|\mathcal X \times \mathcal Y|) \delta + 2 H(2\delta),$$
as $\IC_{\mu_1}(f,\nu,\varepsilonilon) = \inf_{\pi} \IC_{\mu_1}(\pi)$ and $\IC_{\mu_2}(f,\nu,\varepsilonilon)=\inf_\pi \IC_{\mu_2}(\pi)$ where both infimums are over all protocols $\pi$ that computes $[f, \nu, \varepsilonilon]$. In particular, by taking $\varepsilonilon = 0$ and $\nu$ to be a measure with full support, we get the following theorem.
\begin{theorem}[{\cite[Lemma 4.4]{SelfRed}}] \label{thm:uniform-continuity-without-error}
$\IC_\mu(f)$ is uniformly continuous with respect to $\mu$.
\end{theorem}
Finally, note that if $|\nu_1-\nu_2| \leqslant \delta \leqslant \varepsilonilon$, then
\begin{equation}
\label{eq:continNu}
\IC_{\mu}(f,\nu_1,\varepsilonilon+\delta) \leqslant \IC_{\mu}(f,\nu_2,\varepsilonilon) \leqslant \IC_{\mu}(f,\nu_1,\varepsilonilon-\delta).
\end{equation}
This proves the continuity with respect to $\nu$ when $\varepsilonilon>0$. The following theorem summarizes the continuity of $\IC_{\mu}(f,\nu,\varepsilonilon)$ with respect to its parameters.
\begin{theorem}[Uniform continuity with error] \label{thm:uniform-continuity-with-error}
Consider $\delta>0$. For each $f\colon\mathcal{X} \times \mathcal{Y} \to \mathcal{Z}$, real numbers $\varepsilonilon_2 \geqslant \varepsilonilon_1 \geqslant \delta$, and measures $\mu_1,\mu_2,\nu_1,\nu_2$ on $\mathcal{X} \times \mathcal{Y}$ with $|\mu_1-\mu_2| \leqslant \delta$ and $|\nu_1-\nu_2| \leqslant \delta$, we have
$$|\IC_{\mu_1}(f,\nu_1,\varepsilonilon_1)- \IC_{\mu_2}(f,\nu_2,\varepsilonilon_2)| \leqslant \leqslantft(1-\frac{\varepsilonilon_1}{\varepsilonilon_2}+\frac{4\delta}{\varepsilonilon_1}\right) \log |\mathcal{X} \times \mathcal{Y}|+2 H(2\delta).$$
\end{theorem}
\begin{proof}
By (\ref{eq:continuityEps}), we have
$$|\IC_{\mu_2}(f,\nu_2,\varepsilonilon_2)- \IC_{\mu_2}(f,\nu_2,\varepsilonilon_1)| \leqslant (1-\frac{\varepsilonilon_1}{\varepsilonilon_2}) \log(|\mathcal X \times \mathcal Y|).$$
By (\ref{eq:continNu}) and (\ref{eq:continuityEps}), we have
$$|\IC_{\mu_2}(f,\nu_2,\varepsilonilon_1)- \IC_{\mu_2}(f,\nu_1,\varepsilonilon_1)| \leqslant(1-\frac{\varepsilonilon_1-\delta}{\varepsilonilon_1+\delta}) \log(|\mathcal X \times \mathcal Y|) \leqslant \frac{2\delta}{\varepsilonilon_1} \log(|\mathcal X \times \mathcal Y|) .$$
By (\ref{eq:continMu}), we have
$$|\IC_{\mu_2}(f,\nu_1,\varepsilonilon_1)- \IC_{\mu_1}(f,\nu_1,\varepsilonilon_1)| \leqslant 2 \log(|\mathcal X \times \mathcal Y|) \delta + 2 H(2\delta).$$
These three inequalities imply the theorem.
\end{proof}
\subsection{The multiparty number-in-hand model} \label{sec:multipartyModel}
The number-in-hand model is the most straightforward generalization of Yao's two-party model to the settings where more than two players are present. In this model there are $k$ players who wish to collaboratively compute a function $f\colon\mathcal X_1 \times \ldots \times \mathcal X_k \to \mathcal Z$. The communication is in the shared blackboard model, which means that all the communicated bits are visible to all the players. Let $\mu$ be a probability distribution on $\mathcal X_1 \times \ldots \times \mathcal X_k$, and let $X=(X_1,\ldots, X_k)$ be sampled from $\mathcal X_1 \times \ldots \times \mathcal X_k$ according to $\mu$. Definition~\ref{def:infocost} generalizes in a straightforward manner to
$$\IC_\mu^\ext(\pi) = I(\Pi; X),$$
and
$$\IC_\mu(\pi) = \sum_{i=1}^k I(\Pi; X_{-i}|X_i),$$
where $X_{-i}:=(X_1,\ldots ,X_{i-1},X_{i+1},\ldots, X_k)$. Note also that $I(\Pi; X|X_i) = I(\Pi;X_{-i}|X_i)$, and thus we have
$$\IC_\mu(\pi) = \sum_{i=1}^k I(\Pi; X|X_i).$$
The notations $\IC_\mu(f)$, $\IC_\mu(f,\varepsilonilon)$, and $\IC_\mu(f,\nu,\varepsilonilon)$, and the continuity results in Section~\ref{sec:IC-continuity} also generalize in a straightforward manner to this setting.
\section{The local characterization of the optimal information cost\label{sec:LocalConcav}}
We start by some definitions. Let $B$ be a random bit sent by one of the players, and let $\mu_0 = \mu|_{B=0}$ and $\mu_1 = \mu|_{B=1}$, or in other words
$$\mu_b(xy):= \ProbOp[XY=xy|B=b],$$
for $b=0,1$. Denote $\ProbOp[\cdot |xy] := \ProbOp[\cdot | XY=xy]$.
\begin{definition}
Let $\mu$ be a distribution and $B$ be a signal sent by one of the players.
\begin{itemize}
\item $B$ is called \emph{unbiased} with respect to $\mu$ if $\ProbOp[B=0]=\ProbOp[B=1]=\frac{1}{2}$.
\item $B$ is called \emph{non-crossing} if $\mu(xy) < \mu(x'y')$ implies that $\mu_b(xy) \leqslant \mu_b(x'y')$ for $b=0,1$.
\item $B$ is called \emph{$\varepsilonilon$-weak} if $|\ProbOp[B=0|xy]- \ProbOp[B=1|xy] | \leqslant \varepsilonilon$ for every input $xy$.
\end{itemize}
A protocol is said to be in \emph{normal form} with respect to $\mu$ if all its signals are unbiased and non-crossing with respect to $\mu$.
\end{definition}
Let $\Delta(\mathcal X \times \mathcal Y)$ denote the set of distributions on $\mathcal X \times \mathcal Y$. A measure $\mu \in \Delta(\mathcal X \times \mathcal Y)$ is said to be \emph{internal-trivial} (\emph{resp. external-trivial}) for $f$ if $\IC_\mu(f)=0$ (resp. $\IC_\mu^\ext(f)=0$). These measures are characterized in \cite{DFHL}.
\subsection{The Local Characterization} \label{sec:local-char}
Suppose that after a random bit $B$ is sent, if $B=0$, the players continue by running a protocol that is (almost) optimal for $\mu_0$, and if $B=1$, they run a protocol that is (almost) optimal for $\mu_1$. Note that the amount of information that $B$ reveals about the inputs to an external observer is $\MI(B;XY)$. This shows
\begin{equation}
\label{eq:localExternalUpper}
\IC_\mu^\ext(f) \leqslant \MI(B;XY) + \Ex_B [\IC^\ext_{\mu_B}(f)],
\end{equation}
and similarly
\begin{equation}
\label{eq:localInternalUpper}
\IC_\mu(f) \leqslant \MI(B;X|Y)+\MI(B;Y|X) +\Ex_B [\IC_{\mu_B}(f)].
\end{equation}
In~\cite{MR3210776} it is shown that these inequalities essentially characterize $\IC_\mu^\ext(f,\mu,0)$ and $\IC_\mu(f,\mu,0)$. Denote $\MI^\ext_B:=\MI(B;XY)$, and $\MI_B:=\MI(B;X|Y)+\MI(B;Y|X)$.
\begin{theorem}[\cite{MR3210776}]
\label{thm:LocalCharDistError}
Suppose that $C\colon\Delta(\mathcal X \times \mathcal Y) \to [0,\log(| \mathcal X \times \mathcal Y|)]$ satisfies
\begin{enumerate}[(i)]
\item $C(\mu)=0$ for every measure $\mu$ such that $\IC_\mu(f,\mu,0) = 0$, and
\item for every signal $B$ that can be sent by one of the players
$$
\label{eq:localConc}
C(\mu) \leqslant \MI_B +\Ex_B [ C(\mu_B)].
$$
\end{enumerate}
Then $C(\mu) \leqslant \IC_\mu(f,\mu,0)$. Similarly if $\IC_\mu(f,\mu,0)$ is replaced by $\IC^\ext_\mu(f,\mu,0)$, and $\MI_B$ is replaced by $\MI^\ext_B$, then $C(\mu) \leqslant \IC_\mu^\ext(f,\mu,0)$. Furthermore, in both of the external and the internal cases, it suffices to verify (ii) only for non-crossing unbiased signals $B$.
\end{theorem}
In light of Theorem~\ref{thm:LocalCharDistError}, in order to determine the values of $\IC_\mu(f,\mu,0)$, one has to first prove an upper bound by constructing a protocol (or a sequence of protocols) for every measure $\mu$. Then it suffices to verify that the bound satisfies the conditions of Theorem~\ref{thm:LocalCharDistError}.
In~\cite{MR3210776}, $\IC_\mu(\AND_2)$ is determined using Theorem~\ref{thm:LocalCharDistError}. However, the proof presented in~\cite{MR3210776} contains some gaps.
One error is the claim that it suffices to verify (\ref{eq:localConc}) for sufficiently weak signals. While it is not difficult to see that indeed it suffices to verify (\ref{eq:localConc}) for signals $B$ which are $\varepsilonilon$-weak for an absolute constant $\varepsilonilon>0$, in ~\cite{MR3210776} the condition (ii) is only verified for $\varepsilonilon$ that is smaller than a function of $\mu$. This is not sufficient, and one can easily construct a counter-example by allowing the signal $B$ to become increasingly weaker as $\mu$ moves closer to the boundary (by boundary we mean the set of measures $\mu$ that satisfy Theorem~\ref{thm:LocalCharDistError}~(i)). Indeed, for example, set $C(\mu)=K$ for a very large constant $K$ if $\mu$ does not satisfy Theorem~\ref{thm:LocalCharDistError}~(i), and otherwise set $C(\mu)=0$. Obviously (\ref{eq:localConc}) holds if $\mu$ is on the boundary. On the other hand, if $\mu$ is not on the boundary, then by taking $B$ to be sufficiently weak as a function of $\mu$, we can guarantee that $\mu_0$ and $\mu_1$ are not on the boundary either, and thus (\ref{eq:localConc}) holds in this case as well. However, taking $K$ to be sufficiently large violates the desired conclusion that $C(\mu) \leqslant \IC_\mu(f,\mu,0)$.
To fix these errors, we start by observing that a straightforward adaptation of the proof of Theorem~\ref{thm:LocalCharDistError} yields an identical characterization of $\IC_\mu(f)$, however, with a different boundary condition.
\begin{theorem}
\label{thm:LocalCharNoError}
Suppose that $C\colon\Delta(\mathcal X \times \mathcal Y) \to [0,\log(| \mathcal X \times \mathcal Y|)]$ satisfies
\begin{enumerate}[(i)]
\item $C(\mu)=0$ for every measure $\mu$ such that $\IC_\mu(f) = 0$, and
\item for every signal $B$ that can be sent by one of the players
$$C(\mu) \leqslant I_B +\Ex_B [ C(\mu_B)]. $$
\end{enumerate}
Then $C(\mu) \leqslant \IC_\mu(f)$. Similarly, if we replace $\IC_\mu(f)$ by $\IC^\ext_\mu(f)$, and $I_B$ by $I^\ext_B$, then $C(\mu) \leqslant \IC_\mu^\ext(f)$.
Furthermore, in both cases, it suffices to verify (ii) only for non-crossing unbiased signals $B$.
\end{theorem}
\begin{proof}
We only prove the internal case as the external case is similar. Let $\pi$ be a $c$-bit protocol in normal form and $\Pi$ be its transcript. For every possible transcript $t$, let $\mu_t:=\mu|_{\Pi=t}$. Condition (ii) says that for every $1$-bit protocol $\pi$,
\begin{equation}
\label{eq:Lemma5.6II}
C(\mu) \leqslant \IC_\mu(\pi) + \Ex_{t \sim \Pi}[C(\mu_t)].
\end{equation}
By a simple induction (see \cite[Lemma 5.6]{MR3210776}) this implies that (\ref{eq:Lemma5.6II}) holds for every $c$-bit protocol $\pi$ in a normal form where $c<\infty$.
Now consider an arbitrary protocol $\tau$ that computes $[f, 0]$. Lemma~\ref{lem:signal-simulation} below shows that one can simulate $\tau$ with a protocol $\pi$ that is in normal form and terminates with probability $1$. Note that $\pi$ also computes $[f, 0]$ and by Proposition~\ref{prop:diff-protocol-same-cost} we have $\IC_\mu(\tau)=\IC_\mu(\pi)$.
Consider a large integer $c$, and let $\pi_c$ be the protocol that is obtained by truncating $\pi$ after $c$ bits of communication, clearly $\IC_\mu(\pi_c) \leqslant \IC_\mu(\pi)$ as $\pi_c$ is a truncation of $\pi$. Let $G_c$ denote the set of leaves of $\pi_c$ in which the protocol is forced to terminate, and had we run $\pi$ instead, the communication would have continued. Let $\Pi_c$ denote the transcript of $\pi_c$. For any given $\delta>0$, one can guarantee that for every $xy$,
$$\ProbOp[\Pi_c(xy) \in G_c]<\delta$$
by choosing $c$ to be sufficiently large. As $\pi$ computes $[f, 0]$, for every leaf $t$ in $\pi_c$ such that $t \not\in G_c$, $\mu_t$ is an internal-trivial distribution, hence Condition (i) is satisfied on $\mu_t$ implying $C(\mu_t)=0$. Therefore (\ref{eq:Lemma5.6II}) shows
$$
C(\mu) \leqslant \IC_\mu(\pi_c) + \delta \log(|\mathcal X \times \mathcal Y|) \leqslant \IC_\mu(\pi) + \delta \log(|\mathcal X \times \mathcal Y|) = \IC_\mu(\tau) + \delta \log(|\mathcal X \times \mathcal Y|).
$$
Letting $\delta \to 0$ one obtains the desired bound.
\end{proof}
We use the uniform continuity of $\IC_\mu(f)$ with respect to $\mu$ to prove that it suffices to verify Theorem~\ref{thm:LocalCharNoError}~(ii) for signals $B$ that are weaker than quantities that can depend on $\mu$. This as we shall see suffices to fix the proof of~\cite{MR3210776}.
\begin{theorem}[Main Theorem 1]
\label{thm:LocalCharNoErrorWeak}
Let $w\colon(0,1] \to (0,1]$ be a non-decreasing function, $\Omega \subseteq \Delta(\mathcal X \times \mathcal Y)$ be a subset of measures containing the internal trivial distributions for function $f$. Let $\delta(\mu)$ denote the distance of $\mu$ from $\Omega$. Suppose that $C\colon\Delta(\mathcal X \times \mathcal Y) \to [0,\log(| \mathcal X \times \mathcal Y|)]$ satisfies
\begin{enumerate}[(i)]
\item $C(\mu)$ is uniformly continuous with respect to $\mu$;
\item $C(\mu)=\IC_\mu(f)$ if $\delta(\mu)=0$, and
\item for every non-crossing unbiased $w(\delta(\mu))$-weak signal $B$ that can be sent by one of the players,
\begin{equation}
\label{eq:localConcIII}
C(\mu) \leqslant I_B +\Ex_B [ C(\mu_B)].
\end{equation}
\end{enumerate}
Then $C(\mu) \leqslant \IC_\mu(f)$. Similarly, if we replace $\Omega$ by a subset containing the external trivial distributions for function $f$, in Condition (ii) replace $\IC_\mu(f)$ by $\IC^\ext_\mu(f)$, and in Condition (iii) replace $I_B$ by $I^\ext_B$, then $C(\mu) \leqslant \IC_\mu^\ext(f)$.
\end{theorem}
The proof of Theorem~\ref{thm:LocalCharNoErrorWeak} is presented in Section~\ref{sec:fixed-proof}.
\subsection{The local characterization in a different form} \label{sec:local-char-different}
Information cost measures the amount of information that is revealed by communicated bits. The local concavity conditions in Section~\ref{sec:local-char} become more natural if they are represented in terms of the amount of information that is \emph{not} revealed. Define the \emph{concealed information} and \emph{external concealed information} of a protocol $\pi$ with respect to $\mu$, respectively, as
$$\mathbb{C}I_\mu(\pi) =H(X|\Pi Y)+H(Y|\Pi X) = H(X|Y)+H(Y|X)-\IC_\mu(\pi),$$
and
$$\mathbb{C}I^\ext_\mu(\pi) = H(XY|\Pi) = H(XY)-\IC^\ext_\mu(\pi),$$
where $\Pi$ is the transcript of $\pi$.
\begin{remark}
In the setting of multi-party number-in-hand model, we have
$$\mathbb{C}I_\mu(\pi) = \sum_{i=1}^k H(X|\Pi X_i) =\leqslantft(\sum_{i=1}^k H(X|X_i)\right) -\IC_\mu(\pi),$$
and
$$\mathbb{C}I^\ext_\mu(\pi) = H(X|\Pi) = H(X)-\IC^\ext_\mu(\pi),$$
where $X=(X_1,\ldots,X_k)$ is the random vector of all inputs.
\end{remark}
By using concealed information rather than information cost, the local characterization turns into a condition about the local concavity of the function.
\begin{lemma}
Inequalities \eqref{eq:localExternalUpper} and \eqref{eq:localInternalUpper} are respectively equivalent to
$$ \mathbb{C}I_\mu^\ext(f) \geqslant \Ex_B [ \mathbb{C}I^\ext_{\mu_B}(f)], \qquad \mbox{and} \qquad \mathbb{C}I_\mu(f) \geqslant \Ex_B[ \mathbb{C}I_{\mu_B}(f)].$$
\end{lemma}
\begin{proof} Substituting $I(B;XY)=H(XY)-H(XY|B)$ in $\IC_\mu^\ext(f) \leqslant I(B;XY) + \Ex_B [\IC^\ext_{\mu_B}(f)]$ leads to
$$\mathbb{C}I_\mu^\ext(f) \geqslant H(XY|B) - \Ex_B [H(XY|B=b) - \mathbb{C}I^\ext_{\mu_B}(f)]$$
which simplifies to the desired $\mathbb{C}I_\mu^\ext(f) \geqslant \Ex_B [\mathbb{C}I^\ext_{\mu_B}(f)]$.
Similarly substituting $I(B;X|Y)+I(B;Y|X)=H(X|Y)- H(X|YB) + H(Y|X) -H(Y|XB)$ in $\IC_\mu(f) \leqslant I(B;X|Y)+I(B;Y|X) +\Ex_B [\IC_{\mu_B}(f)]$ leads to
$$\mathbb{C}I_\mu(f) \geqslant H(X|YB)+ H(Y|XB) - \Ex_B[H(X|YB=b)+H(Y|XB=b)-\IC_{\mu_B}(f)]$$
which simplifies to $\mathbb{C}I_\mu(f) \geqslant \Ex_B[ \mathbb{C}I_{\mu_B}(f)]$.
\end{proof}
\section{Communication protocols as random walks on $\Delta(\mathcal X \times \mathcal Y)$} \label{sec:randomwalk}
Consider a protocol $\pi$ and a prior distribution $\mu$ on the set of inputs $\mathcal X \times \mathcal Y$. Suppose that in the first round Alice sends a random signal $B$ to Bob. We can interpret this as a random update of the prior distribution $\mu$ to a new distribution $\mu_0 = \mu|_{B=0}$ or $\mu_1 = \mu|_{B=1}$ depending on the value of $B$. It is not difficult to see that $\mu_b(x,y) =p_b(x) \mu(x,y)$ for $b=0,1$, where $p_b(x)=\frac{\ProbOp[B=b|x]}{\ProbOp[B=b]}$. In other words, $\mu_b$ is obtained by multiplying the rows of $\mu$ by non-negative numbers. Similarly if Bob is sending a message, then $\mu_b$ is obtained by multiplying the columns of $\mu$ by the numbers $p_b(y)=\frac{\ProbOp[B=b|y]}{\ProbOp[B=b]}$. That is $\mu_b(x,y) = \mu(x,y)p_b(y)$. Therefore, we can think of a protocol as a random walk on $\Delta(\mathcal X \times \mathcal Y)$ that starts at $\mu$, and every time that a player sends a message, it moves to a new distribution. Note further that this random walk is without drift as $\mu=\Ex_B [ \mu_B]$.
Let $\Pi$ denote the transcript of the protocol. When the protocol terminates, the random walk stops at $\mu_\Pi := \mu|_\Pi$. Since $\Pi$ itself is a random variable, $\mu_\Pi$ is a random variable that takes values in $\Delta(\mathcal X \times \mathcal Y)$. Interestingly, both the internal and external information costs of the protocol depend only on the distribution of $\mu_\Pi$ (this is a distribution on the set $\Delta(\mathcal X \times \mathcal Y)$, which itself is a set of distributions). To see this, note $\MI(X;\Pi|Y) = \Ex_{\pi \sim \Pi, y \sim Y}\Dvg(X|_{\Pi=\pi, Y=y} \| X|_{Y=y})$ and $\MI(XY;\Pi) = \Ex_{\pi \sim \Pi} \Dvg(XY|_{\Pi=\pi} \| XY)$, and thus both of these quantities are determined by $\mu$ and $\mu_\Pi$. This immediately leads to the following observation:
\begin{proposition} \cite{MarkComputable} \label{prop:diff-protocol-same-cost}
\label{prop:randomWalk}
Let $\pi$ and $\tau$ be two communication protocols with the same input set $\mathcal X \times \mathcal Y$ endowed with a probability measure $\mu$. Let $\Pi$ and $T$ denote the transcripts of $\pi$ and $\tau$, respectively. If $\mu_\Pi$ has the same distribution as $\mu_T$, then $\IC_\mu(\pi)=\IC_\mu(\tau)$ and $\IC^\ext_\mu(\pi)=\IC^\ext_\mu(\tau)$.
\end{proposition}
Proposition~\ref{prop:randomWalk} shows that in the context of information complexity, it does not matter how different the steps of two protocol are, and as long as they both yield the same distribution on $\Delta(\mathcal X \times \mathcal Y)$, they have the same internal and external information cost. Consequently, one can directly work with this random walk (or the resulting distribution on $\Delta(\mathcal X \times \mathcal Y)$) instead of working with the actual protocols. Indeed, let $\mathcal{C}^T_\mu(\Delta(\mathcal X \times \mathcal Y))$ denote the set of all probability distributions on $\Delta(\mathcal X \times \mathcal Y)$ that can be obtained, starting from the distribution $\mu$, through communication protocols that perform a given communication task $T$. The information cost of performing the task $T$ is the infimum of the information costs of the distributions in $\mathcal{C}^T_\mu(\Delta(\mathcal X \times \mathcal Y))$. Although, as mentioned earlier, this infimum is not always attained, if one takes the closure of $\mathcal{C}^T_\mu(\Delta(\mathcal X \times \mathcal Y))$ (under weak convergence) then one can replace the infimum with
minimum. For the $2$-bit $\AND$ function, the buzzers protocol of \cite{MR3210776} yields the distribution in the closure of $\mathcal{C}^T_\mu(\Delta(\mathcal X \times \mathcal Y))$ that achieves the minimum information cost. The buzzers protocol is not a communication protocol, but one can consider it as the limit of a sequence of communication protocols. We believe that the following is an important open problem.
\begin{problem}
Define a paradigm such that for every communication task $T$ and every measure $\mu$ on an input set $\mathcal X \times \mathcal Y$, the set of distributions on $\Delta(\mathcal X \times \mathcal Y)$ resulting from the protocols performing the task $T$ in this paradigm is exactly equal to the closure of $\mathcal{C}^T_\mu(\Delta(\mathcal X \times \mathcal Y))$.
\end{problem}
Partial progress towards resolving this problem has been made in ~\cite{DaganFilmus}, see also~\cite{DFHL}.
\subsection{A signal simulation lemma} \label{sec:signal-simulation-lemma}
Here we prove a simulation lemma that will be useful in the proof of the local characterization theorems. We start by restating a splitting lemma from \cite{MR3210776}. We use the notation $[\mu_0,\mu_1]$ for the set of all convex combinations $\alpha \mu_0 + (1-\alpha) \mu_1$, where $\alpha \in [0,1]$.
\begin{lemma}[Splitting Lemma,~\cite{MR3210776}]
\label{lem:Splitting}
Consider $\mu \in \Delta(\mathcal X \times \mathcal Y)$ and a signal $B$ sent by one of the players, and let $\mu_b=\mu|_{B=b}$ for $b=0,1$. Consider $\rho_0,\rho_1 \in [\mu_0,\mu_1]$ and $\rho \in (\rho_0,\rho_1)$. There exists a signal $B'$ that the same player can send starting at $\rho$ such that $\rho_b=\rho|_{B'=b}$ for $b=0,1$.
\end{lemma}
Lemma \ref{lem:Splitting} is proved in~\cite[Lemma 7.11]{MR3210776}, there is a minor error in the original statement as it is claimed that the lemma holds for $\rho \in [\rho_0,\rho_1]$ where the interval is closed.
We are now ready to prove the signal simulation lemma, which says every signal can be perfectly simulated by a non-crossing unbiased $\varepsilonilon$-weak signal sequence. This lemma generalizes \cite[Lemma 5.2]{MR3210776}.
\begin{lemma}[Signal Simulation] \label{lem:signal-simulation}
Let $\varepsilonilon>0$, and consider $\mu \in \Delta(\mathcal X \times \mathcal Y)$ and a signal $B$ sent by one of the players. There exists a sequence of non-crossing unbiased $\varepsilonilon$-weak random signals $\mathcal{B}=(B_1 B_2 \ldots)$ that with probability $1$ terminates, and furthermore $\mu|_{\mathcal{B}}$ has the same distribution as $\mu|_B$.
\end{lemma}
\begin{proof}
Let $\mu_0=\mu|_{B=0}$ and $\mu_1=\mu|_{B=1}$. The following protocol explains how the sequence $(B_1 B_2 \ldots)$ is constructed from the signal $B$.
\begin{framed}
\begin{itemize}
\item Set $\mu_c = \mu$ and $i=1$;
\item Repeat until $\mu_c=\mu_0$ or $\mu_c=\mu_1$;
\item \qquad If $\mu_c \in [\mu_0,\mu]$, then
\item \qquad \qquad Set $\lambda$ to be the largest value in $[0,1]$ satisfying
\item \qquad \qquad \qquad $\lambda \max_{xy} \frac{|\mu_c(x,y)-\mu_0(x,y)|}{\mu_c(x,y)} \leqslant \varepsilonilon$, and
\item \qquad \qquad \qquad $\lambda |\mu_0(x,y)-\mu_0(x',y')-\mu_c(x,y)+\mu_c(x',y')| \leqslant \mu_c(x',y') - \mu_c(x,y)$ if $\mu_c(x,y)<\mu_c(x',y')$.
\item \qquad \qquad Send a signal $B_i$ that splits $\mu_c$ to $(1-\lambda) \mu_c+\lambda \mu_0$ and $(1+\lambda) \mu_c -\lambda \mu_0$;
\item \qquad If $\mu_c \in (\mu,\mu_1]$, then
\item \qquad \qquad Set $\lambda$ to be the largest value in $[0,1]$ satisfying
\item \qquad \qquad \qquad $\lambda \max_{xy} \frac{|\mu_c(x,y)-\mu_1(x,y)|}{\mu_c(x,y)} \leqslant \varepsilonilon$, and
\item \qquad \qquad \qquad $\lambda |\mu_1(x,y)-\mu_1(x',y')-\mu_c(x,y)+\mu_c(x',y')| \leqslant \mu_c(x',y') - \mu_c(x,y)$ if $\mu_c(x,y)<\mu_c(x',y')$.
\item \qquad \qquad Send a signal $B_i$ that splits $\mu_c$ to $(1-\lambda) \mu_c+\lambda \mu_1$ and $(1+\lambda) \mu_c -\lambda \mu_1$;
\item \qquad Update $\mu_c$ to the current distribution;
\item \qquad Increase $i$;
\end{itemize}
\end{framed}
Note that every signal $B_i$ sent in the above protocol is $\varepsilonilon$-weak and non-crossing. Indeed, if $B_i$ splits $\mu_c$ into $(1-\lambda) \mu_c+\lambda \mu_0$ and $(1+\lambda) \mu_c -\lambda \mu_0$, then
\begin{align*}
|\ProbOp[B_i=0 | xy]-\ProbOp[B_i=1 | xy]|
&= \leqslantft|\frac{\mu_c(xy | B_i=0)}{2\mu_c(xy)} - \frac{\mu_c(xy | B_i=1)}{2\mu_c(xy)} \right| \\
&= \lambda \frac{|\mu_c(xy)-\mu_0(xy)|}{\mu_c(xy)},
\end{align*}
and the choice of $\lambda$ guarantees that this is bounded by $\varepsilonilon$. The same calculation shows the $\varepsilonilon$-weakness for $\mu_c \in [\mu,\mu_1]$. It can also be easily verified that the signal is non-crossing.
To see that this sequence terminates with probability $1$, define
\begin{align*}
\Omega
&= \{ \nu \in [\mu_0,\mu_1] : \exists\ (x,y),(x',y') \ \text{s.t.} \ \nu(x,y)=\nu(x',y'), \phantom{\}} \\
&\phantom{\quad \{} \text{while}\ \mu_0(x,y) \neq \mu_0(x',y') \ \text{or}\ \mu_1(x,y) \neq \mu_1(x',y') \},
\end{align*}
and notice that $\Omega$ is a finite set. Consider $\mu_c \in [\mu_0,\mu]$. If the value of $\lambda$ is set by the first condition, then there is a uniform lower-bound for $\lambda$:
$$\lambda \geqslant \lambda_0:= \varepsilonilon /\max_{xy} \frac{|\mu(x,y)-\mu_0(x,y)|}{\mu(x,y)} =\varepsilonilon / \max_{xy} \frac{|\mu(x,y)-\mu_1(x,y)|}{\mu(x,y)}>0.$$
Moreover if $\lambda$ is set by the other condition, then it means $\mu_c(x,y)<\mu_c(x',y')$, and at least one of $\mu_c|_{B_i=0}$ or $\mu_c|_{B_i=1}$ belongs to $\Omega$. Hence starting at any point $\mu_c$, the random walk terminates with probability at least $2^{-\mathrm{lc}eil 1/\lambda_0 \rceil + |\Omega|}$ after $\mathrm{lc}eil 1/\lambda_0 \rceil+ |\Omega|$ steps. It follows that with probability $1$, the random walk terminates.
\end{proof}
\subsection{Proofs of Theorem~\ref{thm:LocalCharNoErrorWeak}} \label{sec:fixed-proof}
We present the proof for the internal case only as the external case is similar.
\begin{lemma} \label{lem:induction-lemma}
Let $w, \delta(\mu)$ and $C$ be as in Theorem \ref{thm:LocalCharNoErrorWeak}, and suppose $C$ satisfies Conditions (i), (ii) and (iii). Let $\tau$ be a protocol that terminates with probability $1$, and further assume $\tau$ is in normal form and every signal sent in $\tau$ is $\varepsilonilon$-weak. Given a probability distribution $\mu \in \Delta(\mathcal X \times \mathcal Y)$, for every node $u$ in the protocol tree of $\tau$, let $\mu_u$ be the probability distribution conditioned on the event that the protocol reaches $u$. If $\mu$ satisfies $w(\delta(\mu_u)) \geqslant \varepsilonilon$ for every internal node $u$, then
$$C(\mu) \leqslant \IC_\mu(\tau) + \Ex_{\ell} [C(\mu_\ell)],$$
where the expected value is over all leaves $\ell$ of $\tau$ chosen according to the distribution (on the leaves) when the inputs are sampled according to $\mu$.
\end{lemma}
\begin{proof}
For every internal node $u$, the assumption in the statement of the lemma implies that the signal sent from $u$ is $w(\delta(\mu_u))$-weak. Hence Condition (iii) shows that the claim is true if $\tau$ is a $1$-bit protocol, and thus by a simple induction (See \cite[Lemma 5.6]{MR3210776}) it is true if $\tau$ is a $c$-bit protocol for any $c < \infty$.
Now assume $\tau$ has infinite depth. Consider a large integer $c$, obtain $\tau_c$ by truncating $\tau$ after $c$ bits of communication, trivially $\IC_\mu(\tau_c) \leqslant \IC_\mu(\tau)$. Let $G_c$ denote the set of the leaves of $\tau_c$ in which the protocol is forced to terminate. Let $\mathcal L_c$ be the set of leaves in $\tau$ with depth at most $c$. Clearly, the set of leaves in $\tau_c$ is exactly $G_c \cup \mathcal L_c$. As $\tau_c$ has a bounded depth, we have
$$
C(\mu) \leqslant \IC_\mu(\tau_c) + \Ex_{\ell \in \mathcal L_c \cup G_c} [C(\mu_\ell)] \leqslant \IC_\mu(\tau) + \Ex_{\ell \in \mathcal L_c \cup G_c} [C(\mu_\ell)].
$$
Let $\Pi_c$ denote the transcript of $\tau_c$. As $\tau$ terminates with probability $1$, given any $\alpha > 0$, one can guarantee $\ProbOp[\Pi_c(xy) \in G_c]<\alpha$ for every $xy$ by choosing $c$ to be sufficiently large. Hence
$$
C(\mu) \leqslant \IC_\mu(\tau) + \Ex_{\ell \in \mathcal L_c} [C(\mu_\ell)] + \alpha \log(|\mathcal X \times \mathcal Y|).
$$
Taking the limit $\alpha \to 0$ shows $C(\mu) \leqslant \IC_\mu(\tau) + \Ex_{\ell \in \mathcal L} [C(\mu_\ell)]$ where $\mathcal L$ is the set of all leaves of $\tau$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:LocalCharNoErrorWeak}]
Firstly by (ii), $\delta(\mu) = 0$ implies $C(\mu) = \IC_\mu(f) \leqslant \IC_\mu(f)$. Hence assume $\delta(\mu) > 0$. Consider an arbitrary signal $B$ sent by Alice. As we discussed before, one can interpret $B$ as a one step random walk that starts at $\mu$ and jumps either to $\mu_0$ or to $\mu_1$ with corresponding probabilities $\ProbOp[B=0 | X=x]$ and $\ProbOp[B=1 | X=x]$. The idea behind the proof is to use Lemma \ref{lem:signal-simulation} to simulate this random jump with a random walk that has smaller steps so that we can apply the concavity assumption of the theorem to those steps.
Let $\pi$ be a protocol that computes $[f, 0]$. For $0< \eta < \delta(\mu)$, applying Lemma~\ref{lem:signal-simulation} one gets a new protocol $\tilde{\pi}$ by replacing every signal sent in $\pi$ with a random walk consisting of $w(\eta)$-weak non-crossing unbiased signals. Note $\tilde{\pi}$ terminates with probability $1$. Moreover, since $\tilde{\pi}$ is a perfect simulation of $\pi$, by Proposition~\ref{prop:diff-protocol-same-cost} we have $\IC_\mu(\pi)=\IC_\mu(\tilde{\pi})$.
For every node $v$ in the protocol tree of $\tilde{\pi}$, let $\mu_v$ be the measure $\mu$ conditioned on the event that the protocol reaches the node $v$. Obtain $\tau$ from $\tilde{\pi}$ by terminating at every node $v$ that satisfies $\delta(\mu_v) \leqslant \eta$. Note that by the construction, Condition (iii) is satisfied on every internal node $v$ of $\tau$, as every such node satisfies $\eta < \delta(\mu_v)$, thus $w(\eta) \leqslant w(\delta(\mu_v))$ implying the signal sent on node $v$ is $w(\delta(\mu_v))$-weak. Hence by Lemma~\ref{lem:induction-lemma},
$$C(\mu) \leqslant \IC_\mu(\tau) + \Ex_{\ell} [C(\mu_\ell)],$$
where the expected value is over all leaves of $\tau$. For every $\mu_\ell$, let $\mu'_\ell \in \Omega$ be a distribution such that $\delta(\mu_\ell) = |\mu_\ell - \mu'_\ell|$. By Conditions (i) and (ii), and the uniform continuity of $\IC_\mu(f)$, we have that for every $\varepsilonilon > 0$ there exists $\eta > 0$, such that for all $\mu_\ell$, as long as $\delta(\mu_\ell) = |\mu_\ell - \mu'_\ell| \leqslant \eta$, then
\begin{equation} \nonumber
C(\mu_\ell) \leqslant C(\mu'_\ell) + \varepsilonilon = \IC_{\mu'_\ell}(f) + \varepsilonilon \leqslant \IC_{\mu_\ell}(f) + \varepsilonilon + \varepsilonilon = \IC_{\mu_\ell}(f) + 2\varepsilonilon.
\end{equation}
As a result,
\begin{equation} \nonumber
C(\mu) \leqslant \IC_\mu(\tau) + \Ex_{\ell} [\IC_{\mu_\ell}(f) + 2\varepsilonilon] = \IC_\mu(\tau) + \Ex_{\ell} [\IC_{\mu_\ell}(f)] + 2\varepsilonilon.
\end{equation}
Since $\mu_\ell$ is generated by truncating $\tilde{\pi}$, we have
\begin{equation} \nonumber
\IC_\mu(\tau) + \Ex_{\ell} [\IC_{\mu_\ell}(f)] \leqslant \IC_\mu(\tilde{\pi}) = \IC_\mu(\pi).
\end{equation}
Therefore $C(\mu) \leqslant \IC_\mu(\pi) + 2\varepsilonilon$. As this holds for arbitrary $\varepsilonilon$, we must have $C(\mu) \leqslant \IC_\mu(\pi)$.
\end{proof}
\section{The multiparty $\AND$ function in the number-in-hand model}
In~\cite{MR3210776} it is shown that in the two-player setting, a certain (unconventional) protocol that we refer to as the buzzers protocol, has optimal information and external information cost for the $2$-bit $\AND$ function. In this section we show that the buzzers protocol can be generalized to an optimal protocol for the multi-party $\AND$ function in the number-in-hand model (assuming Assumption~\ref{assumption}).
For the sake of brevity, we denote $\mu_x:=\mu(\{x\})$ for every $x \in \{0,1\}^k$. Furthermore we assume that $\mu_{\e_1} \leqslant \ldots \leqslant \mu_{\e_k}$. The protocol is given by having buzzers with waiting times which have independent exponential distributions, and start at times $t_1,\ldots,t_k$ for players $1,\ldots,k$, respectively. Although the protocol $\pi_\mu^\wedge$ described in Figure~\ref{fig:prot} is not a conventional communication protocol, it can be easily approximated by discretization and truncation of time.
\begin{figure}
\caption{The protocol $\pi_\mu^\wedge$ for solving the $\AND$ function on a distribution $\mu$.\label{fig:prot}
\label{fig:prot}
\end{figure}
Recall that the exponential distribution is memoryless, and intuitively can be generated in the following manner: Consider a buzzer starting at time $t=0$. At every infinitesimal time interval of length $dt$, independently of the past, it buzzes with probability $dt$, and then stops. The waiting time for a buzz to happen has exponential distribution.
Thus we can describe $\pi_\mu^\wedge$ as in the following: For every $i \in [k]$, if $x_i=0$, the $i$-th player activates a buzzer at time $t_i$. When the first buzz happens the protocol terminates, and the players decide that $\bigwedge_{i=1}^k x_i = 0$. If the time reaches $\infty$ without anyone buzzing, they decide $\bigwedge_{i=1}^k x_i = 1$. In Theorem~\ref{thm:mainMultiparty} we show that for the measures $\mu$ that satisfy Assumption~\ref{assumption}, the protocol $\pi^\wedge_\mu$ has optimal external and internal information cost.
\begin{theorem}[Main Theorem 2]
\label{thm:mainMultiparty}
For every $\mu$ satisfying Assumption~\ref{assumption}, the protocol $\pi^\wedge_\mu$ has the smallest external and internal information cost.
\end{theorem}
In order to prove Theorem~\ref{thm:mainMultiparty}, we need to verify that the concavity conditions of Theorem~\ref{thm:LocalCharNoErrorWeak} are satisfied. Consider the measure $\mu$ that is uniformly distributed over $\e_1,\ldots,\e_k$. That is $\mu_{\vec{0}} = 0$ and $\mu_{\e_1} = \ldots = \mu_{\e_k} = 1/k$. Note that when the protocol $\pi^\wedge$ is executed on this measure, all the players become active at time $0$, and as time proceeds, they do not obtain any new information about the inputs of the other players until one of the players buzzes. Then at that point the input is revealed to all the players. Due to this discrete nature of the corresponding random-walk on $\Delta(\mathcal X \times \mathcal Y)$, we need to analyze this particular measure separately, and afterwards when verifying the concavity conditions, we can let $\Omega$ in the statement of Theorem~\ref{thm:LocalCharNoErrorWeak} include this measure. Claim~\ref{claim:protocol-is-opt-when-mu-i-are-all-equal} below verifies Theorem~\ref{thm:mainMultiparty} for this particular measure.
\begin{myclaim} \label{claim:protocol-is-opt-when-mu-i-are-all-equal}
Let $\mu$ be the measure that $\mu_{\vec{0}} = 0$ and $\mu_{e_1} = \ldots = \mu_{e_k} = 1/k$.
The internal and external information cost of the protocol $\pi^\wedge$ is optimal with respect to $\mu$.
\end{myclaim}
\begin{proof}
First we present the proof for the external information complexity. Let $\pi$ be any protocol that solves the multi-party AND function correctly, and let $t$ be a possible transcript of the protocol. First note that it is not possible to have $\ProbOp[\Pi_{e_m} = t] > 0$ for all $1 \leqslant m \leqslant k$. Indeed by rectangle property this would imply $\ProbOp[\Pi_{\vec{1}} = t] > 0$, and since the correct output for $\vec{1}$ is different from that of $e_1,\ldots,e_k$, we would get a contradiction with the assumption that $\pi$ solves AND correctly on all inputs. Hence to every transcript $t$, we can assign a $j(t) \in \{1,\ldots,m\}$ with $\ProbOp[\Pi_{e_j}=t]=0$. Now for a random $X \sim \mu$, denote $J=j(\Pi_X)$, and notice that conditioned on $J=j$, $X$ is supported on the set $\{e_1,\ldots,e_k\} \setminus \{e_j\}$ of size $k-1$, and thus $\entropy(X|_J) \leqslant \log (k-1)$. Consequently, we have
$$\IC_{\mu}^\ext(\pi)=\MI(X;\Pi_X) = \MI(X;\Pi_X J) \geqslant \MI(X; J) = \entropy(X)-\entropy(X|J) \geqslant \log k - \log (k-1).$$
On the other hand, consider our protocol $\pi_\mu^\wedge$. Note that under $\mu$, all players are activated at the same time, and consequently by symmetry, for every termination time $\tau$ and player $j \in \{1,\ldots,k\}$, the random variable $X|_{\Pi_X=(\tau,j)}$ is uniformly distributed on $\{e_1,\ldots,e_k\} \setminus \{e_j\}$. Hence $\entropy(X| \Pi_X)= \log(k-1)$. We conclude that
$$\IC_{\mu}^\ext(\pi^\wedge)=\MI(X;\Pi_X) = \entropy(X) - \entropy(X|\Pi_X) = \log k - \log (k-1).$$
Next we turn to the internal case. Again, let $\pi$ be any protocol that solves the multi-party AND correctly, and let $J$ be defined as above. First note that for $i \in [k]$, $X|_{X_i=1}$ is supported on the single point $\{e_i\}$ and $X|_{X_i=0}$ is uniformly distributed on $\{e_1,\ldots,e_k\} \setminus \{e_i\}$. Hence
$$\entropy(X|X_i)= \frac{1}{k} \entropy(X|X_i=1)+ \frac{k-1}{k} \entropy(X|X_i=0)=\frac{k-1}{k} \log(k-1).$$
Moreover for $i,j \in [k]$, $X|_{J=j,X_i=0}$ is supported on $\{e_1,\ldots,e_k\} \setminus \{e_i,e_j\}$. Hence using $\ProbOp[J=i]=\ProbOp[J=i, X_i=0]$, we have
\begin{align*}
\entropy(X|JX_i) &= \sum_{j=1}^k \ProbOp[J=j, X_i=0] \entropy(X|_{J=j,X_i=0}) \\
&\leqslant \sum_{j=1}^k \ProbOp[J=j, X_i=0] \log |\{e_1,\ldots,e_k\} \setminus \{e_i,e_j\}| \\
&= \frac{k-1}{k} \log(k-2) + \ProbOp[J=i] (\log(k-1)-\log(k-2)).
\end{align*}
Summing over $i$, we obtain
$$ \sum_{i=1}^k \entropy(X|JX_i) = (k-2) \log(k-2) + \log(k-1), $$
and thus
\begin{align*}
\IC_{\mu}(\pi)
&=\sum_{i=1}^k \MI(X;\Pi_X|X_i) = \sum_{i=1}^k \MI(X;\Pi_X J|X_i)\\
&\geqslant \sum_{i=1}^k \MI(X; J|X_i) = \sum_{i=1}^k \entropy(X|X_i)-\entropy(X|JX_i) \\
&\geqslant (k-1) \log(k-1) - ((k-2) \log(k-2)+\log(k-1)) \\
&= (k-2)(\log(k-1)-\log(k-2)).
\end{align*}
On the other hand, for the protocol $\pi_\mu^\wedge$, by symmetry, for every termination time $\tau$ and player $j \in \{1,\ldots,k\}$, the random variable $X|_{\Pi_X=(\tau,j), X_i=0}$ is uniformly distributed on $\{e_1,\ldots,e_k\} \setminus \{e_j,e_i\}$. Hence
$$ \entropy(X| \Pi_X X_i)=\frac{1}{k} \log(k-1) + \frac{k-2}{k} \log(k-2).$$
We conclude that
\begin{align*}
\IC_{\mu}(\pi^\wedge)
&= \sum_{i=1}^k \MI(X;\Pi_X|X_i) = (k-1)\log(k-1) - \sum_{i=1}^k \entropy(X|\Pi_X X_i)\\
&= (k-2)(\log(k-1)-\log(k-2)). \qedhere
\end{align*}
\end{proof}
\begin{myclaim}
\label{claim:No1}
It suffices to verify Theorem~\ref{thm:mainMultiparty} for measures $\mu$ with $\mu(\vec{1})=0$.
\end{myclaim}
\begin{proof}
Let $\mu$ be a measure satisfying Assumption~\ref{assumption}, and let $\pi$ be a protocol that solves the multiparty AND function correctly on all the inputs. Let $\Pi$ denote the transcript of this protocol, and let $B=1_{[X = \vec{1}]}$. Since $\pi$ solves the AND function correctly, $\Pi$ determines the value of $B$. We have
\begin{align*}
\IC_\mu^\ext(\pi)&=\MI(X;\Pi_X) = \MI(X B_X;\Pi_X) = \MI(B_X;\Pi_X)+ \MI(X;\Pi_X|B_X) \\
&=0+ \ProbOp[X=\vec{1}]\MI(X;\Pi_X|X=\vec{1})+ \ProbOp[X \neq \vec{1}]\MI(X;\Pi_X|X \neq \vec{1}) \\
&=\ProbOp[X \neq \vec{1}]\MI(X;\Pi_X|X \neq \vec{1})=(1-\mu_{\vec{1}}) \IC_{\mu'}^\ext(\pi),
\end{align*}
where $\mu'$ is the measure $\mu$ conditioned on the event that the input is not equal to $\vec{1}$. Similarly
$$\IC_\mu(\pi)=(1-\mu_{\vec{1}})\IC_{\mu'}(\pi).$$
Finally, to conclude the claim, note that $\pi_\mu^\wedge$ and $\pi_{\mu'}^\wedge$ are identical as $\mu_{\e_i}/\mu_{\e_1}=\mu'_{\e_i}/\mu'_{\e_1}$ for all $i=1,\ldots,k$.
\end{proof}
\section{Proof of Theorem~\ref{thm:mainMultiparty}}
In this section we prove Theorem~\ref{thm:mainMultiparty} by verifying the concavity conditions of Theorem~\ref{thm:LocalCharNoErrorWeak}. Let $\mu$ be a measure satisfying Assumption~\ref{assumption}, and $X=(X_1,\dots, X_k)$ denote the random $k$-bit input. Let $\Pi$ be the random variable corresponding to the transcript of the protocol $\pi^\wedge_{\mu}$. Let $\Pi_x=\Pi|_{X=x}$.
To verify the concavity condition, we consider a signal $B$ with parameter $\varepsilonilon$ sent by the player $s$. That is
$$\ProbOp[B=0 | X_s=0]=\frac{1+\varepsilonilon \ProbOp[X_s=1]}{2},$$
and
$$\qquad \ProbOp[B=1 | X_s=1]=\frac{1+\varepsilonilon \ProbOp[X_s=0]}{2}.$$
Note that $\ProbOp[B=0]=\ProbOp[B=1]=\frac{1}{2}$, i.e., the signal $B$ is unbiased. Let $\mu^0$ and $\mu^1$ respectively denote the distributions of $X^0 := X|_{B=0}$ and $X^1 := X|_{B=1}$. We have $\mu= \frac{\mu^0 + \mu^1}{2}$. Let $\Pi^0$ and $\Pi^1$ denote the random variables corresponding to the transcripts of $\pi^\wedge_{\mu^0}$ and $\pi^\wedge_{\mu^1}$, respectively.
Note that the transcript of $\pi_\mu^\wedge$ contains the termination time $t$, and if $t<\infty$, also the name of the player who first buzzed. We denote by $\pi_\infty$ the transcript corresponding to termination time $t=\infty$, and by $\pi_t^m$ the termination time $t<\infty$ with the $m$-th player buzzing.
For $t \in [0,\infty)$, let $\Phi_x(t)$ denote the total amount of active time spent by all players before time $t$ if the input is $x$. For $t_r \leqslant t < t_{r+1}$, we have
$$\Phi_x(t) =\sum_{i: x_i=0} \max(t-t_i,0) = \sum_{i \in [1,r], x_i=0} t-t_i.$$
The probability density function $f_x$ of $\Pi_x$ is given by
$$f_{x}(\pi_t^m) = \leqslantft\{
\begin{array}{lcl}
0 &\qquad& \mbox{$t_m>t$ or $x_m=1$} \\
e^{-\Phi_x(t)} & & \mbox{otherwise}
\end{array}
\right.,
$$
and $\ProbOp(\Pi_{\vec{1}}=\pi_\infty)=1$. The distribution of the transcript $\Pi$ is then
$$f(\pi_t^m) =\sum_x \mu_x f_{x}(\pi_t^m).$$
Define $f^0$, $f^0_x$ and $f^1$, $f^1_x$ analogously for $\pi^{\wedge}_{\mu^0}$ and $\pi^{\wedge}_{\mu^1}$, respectively.
\subsection{Probability distributions $\mu^0$ and $\mu^1$} \label{sec:distribution-mu0-mu1}
Denote $\beta_s :=\ProbOp[X_s=1]$, and $\zeta_s :=\ProbOp[X_s=0]$. For $B=0$, we have,
\begin{align*}
\mu^{0}_x =& \leqslantft\{
\begin{array}{lcl}
(1+\varepsilonilon \ProbOp[x_s=1]) \mu_x = (1+\varepsilonilon \beta_s) \mu_x &\qquad& x_s=0 \\
(1-\varepsilonilon \ProbOp[x_s=0]) \mu_x = (1-\varepsilonilon \zeta_s) \mu_x & & x_s=1
\end{array}
\right.
\end{align*}
Consequently, the new starting times are $t^0_i=t_i$ for $i \neq s$, and $t^0_s= t_s - \gamma_0$ where
\begin{equation} \label{eqn:gamma-0}
\gamma_0=\ln\leqslantft(\frac{1+\varepsilonilon \beta_s}{1-\varepsilonilon \zeta_s}\right).
\end{equation}
Hence
$$\mu^{0}_x f^0_x(\pi_t^m) =
\leqslantft\{
\begin{array}{lcl}
\mu^{0}_x f_x(\pi_t^m) &\qquad & t<t_s-\gamma_0 \\
(1+\varepsilonilon \beta_s)\mu_x f_x(\pi_t^m) e^{-(t-t_s+\gamma_0)} & & t \in [t_s-\gamma_0,t_s), x_s=0, m \neq s \\
(1-\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m) & & t \in [t_s-\gamma_0,t_s), x_s=1, m \neq s \\
\mu^{0}_x f^0_x(\pi_t^s) & & t \in [t_s-\gamma_0,t_s), m=s \\
(1-\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m) & & t\geqslant t_s \\
\end{array}
\right.
$$
On the other hand, for $B=1$, we have,
\begin{align*}
\mu^{1}_x =& \leqslantft\{
\begin{array}{lcl}
(1-\varepsilonilon \beta_s) \mu_x &\qquad& x_s=0 \\
(1+\varepsilonilon \zeta_s) \mu_x & & x_s=1
\end{array}
\right.
\end{align*}
Consequently the new starting times are $t^1_i=t_i$ for $i \neq s$, and $t^1_s= t_s + \gamma_1$ where
\begin{equation} \label{eqn:gamma-1}
\gamma_1=\ln\leqslantft(\frac{1+\varepsilonilon \zeta_s}{1-\varepsilonilon \beta_s}\right).
\end{equation}
Hence when $m \neq s$,
$$\mu^{1}_x f^1_x(\pi_t^m) =
\leqslantft\{
\begin{array}{lcl}
\mu^{1}_x f_x(\pi_t^m) &\qquad & t\leqslant t_s \\
(1- \varepsilonilon \beta_s)\mu_x f_x(\pi_t^m) e^{t-t_s} & & t \in [t_s,t_s+\gamma_1), x_s=0 \\
(1+\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m) & & t \in [t_s,t_s+\gamma_1), x_s=1 \\
(1+\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m) & & t\geqslant t_s+\gamma_1 \\
\end{array}
\right.
$$
For $m = s$,
$$\mu^{1}_x f^1_x(\pi_t^s) =
\leqslantft\{
\begin{array}{lcl}
(1+\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^s) &\qquad& t > t_s + \gamma_1 \text{\ and\ } x_s=0 \\
0 & & \text{otherwise}.
\end{array}
\right.
$$
\subsection{Setting up and first reductions} \label{sec:Some-reductions}
We set $\Omega$ to be the set of all external (resp. internal) trivial measures together with the measure in Claim~\ref{claim:protocol-is-opt-when-mu-i-are-all-equal}, and in the external case we set $w(x) = ck^{-20} x^4$ for some fixed constant $c > 0$ (one may need to pick a different $w(x)$ for internal case). Using the memoryless property of exponential distribution, we can shift the activation time of all the players by $-\ln(\mu_{\e_s}/\mu_{\e_1})$, and assume that $t_1=-\ln(\mu_{\e_s}/\mu_{\e_1}), \ldots, t_s=0, \ldots, t_k=\ln(\mu_{\e_k}/\mu_{\e_s})$.
Let $\phi(x) := x \ln (x)$. Using the notion of concealed information from Section~\ref{sec:local-char-different}, the concavity conditions of Theorem~\ref{thm:LocalCharNoErrorWeak} reduce to verifying
\begin{align}
\int_{-\infty}^\infty &\sum_m \leqslantft( \phi(f(\pi^m_t)) - \frac{\phi(f^0(\pi^m_t))+\phi(f^1(\pi^m_t))}{2} \right) \nonumber \\
&-\sum_m \sum_x \leqslantft( \phi(\mu_x f_x(\pi^m_t))-\frac{\phi(\mu^0_x f^0_x(\pi^m_t))+\phi(\mu^1_x f^1_x(\pi^m_t))}{2} \right) dt \geqslant 0, \label{eq:ExternalGoal}
\end{align}
for the external case, and
\begin{align}
\sum_{j=1}^k \int_{-\infty}^\infty &\sum_m \sum_{b=0}^1 \leqslantft( \phi(f_{x_j=b}(\pi^m_t)) - \frac{\phi(f_{x_j=b}^0(\pi^m_t))+\phi(f_{x_j=b}^1(\pi^m_t))}{2} \right) \nonumber \\
& -\sum_m \sum_x \leqslantft( \phi(\mu_x f_x(\pi^m_t))-\frac{\phi(\mu^0_x f^0_x(\pi^m_t))+\phi(\mu^1_x f^1_x(\pi^m_t))}{2} \right) dt \geqslant 0, \label{eq:InternalGoal}
\end{align}
for the internal case, where
$$f_{x_j=b}(\pi^m_t):= \sum_{X: X_j=b} \mu_X f_X(\pi^m_t),$$
and
$$\qquad f^0_{x_j=b}(\pi^m_t):= \sum_{X: X_j=b} \mu^0_X f^0_X(\pi^m_t), \qquad f^1_{x_j=b}(\pi^m_t):= \sum_{X: X_j=b} \mu^1_X f^1_X(\pi^m_t).$$
Denote the function inside the integral of (\ref{eq:InternalGoal}) by $\concav_\mu(t,j)$, and the function inside the integral of (\ref{eq:ExternalGoal}) by $\concav_\mu^\ext(t)$. Note further that by Claim~\ref{claim:No1} we can assume that $\mu_{\vec{1}}=0$. Hence our goal reduces to show the following:
\begin{statement}[First reduction]
\label{stat:1st}
To prove Theorem~\ref{thm:mainMultiparty} it suffices to assume $\mu$ satisfies $\mu(\vec{1})=0$, and verify
$$\int_{-\infty}^\infty \concav_\mu^\ext(t) dt \geqslant 0 \qquad \mbox{and} \qquad \sum_{j=1}^k \int_{-\infty}^\infty \concav_\mu(t,j) dt \geqslant 0.$$
\end{statement}
Recall we assumed $t_s=0$ by shifting the time. The next two claims show that one only needs to focus on the interval $[-\gamma_0,\gamma_1]$.
\begin{myclaim} \label{claim:less-than-ts-1-contributes-nonnegative}
We have $ \int_{-\infty}^{- \gamma_0} \concav_\mu^\ext(t) dt \geqslant 0$ and $\sum_{j=1}^k \int_{-\infty}^{- \gamma_0} \concav_\mu(t,j)dt \geqslant 0$.
\end{myclaim}
\begin{proof}
Observe that $\Pi, \Pi^0$ and $\Pi^1$ are identical up to time $- \gamma_0$. Let $\Pi_P$ denote a similar protocol, with the only difference that in $\Pi_P$ at time $t=- \gamma_0$ all the players reveal their inputs. Then,
$$\int_{-\infty}^{- \gamma_0} \concav_\mu^\ext(t) dt
= \entropy(X|\Pi_P)-\entropy(X|\Pi_P, B) \geqslant 0,$$
and
\begin{equation} \nonumber
\sum_{j=1}^k \int_{-\infty}^{- \gamma_0} \concav_\mu(t,j)dt
= \sum_{j=1}^k (\entropy(X|X_j, \Pi_P)-\entropy(X|X_j, \Pi_P, B)) \geqslant 0. \tag*{\qedhere}
\end{equation}
\end{proof}
\begin{myclaim} \label{claim:greater-than-gamma1-contributes-0}
We have $ \int_{ \gamma_1}^\infty \concav_\mu^\ext(t) dt = 0$ and $\sum_{j=1}^k \int_{ \gamma_1}^\infty \concav_\mu(t,j)dt = 0$.
\end{myclaim}
\begin{proof}
We use the formula in \eqref{eq:ExternalGoal} by integrating in the corresponding range $[\gamma_1, \infty)$. As $t \geqslant \gamma_1$, plug in $\mu_x^0 f_x^0(\pi_t^m) = (1-\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m)$ and $\mu_x^1 f_x^1(\pi_t^m) = (1+\varepsilonilon \zeta_s) \mu_x f_x(\pi_t^m)$, a simple calculation shows $ \int_{\gamma_1}^\infty \concav_\mu^\ext(t) dt = 0$. Similarly one can calculate the internal case.
\end{proof}
\begin{statement}[Second reduction]
\label{stat:2nd}
To prove Theorem~\ref{thm:mainMultiparty} it suffices to assume $\mu$ satisfies $\mu(\vec{1})=0$, and verify
$$
\int_{-\gamma_0}^{\gamma_1} \concav_\mu^\ext(t) dt \geqslant 0
\qquad
\mbox{and}
\qquad
\sum_{j=1}^k \int_{-\gamma_0}^{\gamma_1} \concav_\mu(t,j) dt \geqslant 0.$$
\end{statement}
\begin{remark}
The computation result of the two-party $\AND$ done in \cite[Section 7.7]{MR3210776} shows the concavity term (the one that we want to verify its non-negativity) can be of order $\varepsilonilon^2$. One will see in Section \ref{sec:multiparty-code} that our computation gives only an order $\varepsilonilon^3$. This is because we choose to focus our computation, as allowed by Statement \ref{stat:2nd}, on the interval $[-\gamma_0, \gamma_1]$ only. Claim \ref{claim:epsilon-2-contribution} below shows an order $\varepsilonilon^2$ term can appear if the whole range is considered.
\end{remark}
\begin{myclaim} \label{claim:epsilon-2-contribution}
Suppose $s \geqslant 2$ and $L = |t_{s-1}| > 0$. If $\gamma_0 \leqslant L/2$, then
\begin{equation} \label{eqn:epsilon-2-bound-Ext}
\int_{t_{s-1}}^{-\gamma_0} \concav_\mu^\ext(t) dt
\geqslant
\frac{(1 - e^{-(s-1)L/2}) \mu_{\vec{0}} \mu_{e_s} }{2(s-1)} \varepsilonilon^2 \geqslant 0,
\end{equation}
and
\begin{equation} \label{eqn:epsilon-2-bound-Int}
\sum_{j=1}^k \int_{t_{s-1}}^{-\gamma_0} \concav_\mu(t,j) dt
\geqslant
\frac{(k-1)(1 - e^{-(s-1)L/2}) \mu_{\vec{0}} \mu_{e_s} }{2(s-1)} \varepsilonilon^2 \geqslant 0.
\end{equation}
\end{myclaim}
\begin{proof}
Consider external case first. Let $\mu'$ be defined as
$$
\mu'_x =
\begin{cases}
\beta \mu_x, &\quad x_s = 0, \\
-\zeta \mu_x, &\quad x_s = 1.
\end{cases}
$$
Then $\mu^0= \mu + \varepsilonilon \mu'$ and $\mu^1= \mu - \varepsilonilon \mu'$, hence $f^0(\pi^m_t)= f(\pi^m_t)+ \varepsilonilon \sum \mu'_x f_x(\pi^m_t)$ and $f^1(\pi^m_t)= f(\pi^m_t)- \varepsilonilon \sum \mu'_x f_x(\pi^m_t)$.
Note that $f(\pi^m_t) = 0$ (in our case this happens when $m \geqslant s$) implies $\concav_\mu^\ext(t) = 0$. On the other hand, when $f(\pi^m_t) \neq 0$ (i.e., $1 \leqslant m \leqslant s-1$), using Taylor expansion at the point $f(\pi^m_t)$ for functions $\phi(f^0(\pi^m_t))$ and $\phi(f^1(\pi^m_t))$, and expansion at $\mu_x f_x(\pi^m_t)$ for functions $\phi(\mu^0_x f^0_x(\pi^m_t))$ and $\phi(\mu^1_x f^1_x(\pi^m_t))$, we obtain (note here we won't have $\varepsilonilon^3$)
\begin{align} \label{eqn:epsilon-square-term-Ext}
\begin{split}
\concav_\mu^\ext(t)
&\geqslant
\sum_{m = 1}^{s-1} \leqslantft(\sum_x \frac{(\mu'_x f_x(\pi^m_t))^2}{2\mu_x f_x(\pi^m_t)} - \frac{(\sum \mu'_x f_x(\pi^m_t))^2}{2f(\pi^m_t)} \right)\varepsilonilon^2 \\
&=
\frac{1}{2} \sum_{m = 1}^{s-1} \leqslantft(\mu_{e_s} f_{e_s}(\pi^m_t) \leqslantft(1 - \frac{\mu_{e_s} f_{e_s}(\pi^m_t)}{f(\pi^m_t)} \right) \right) \varepsilonilon^2 \\
&\geqslant \frac{1}{2} \sum_{m = 1}^{s-1} \leqslantft(\mu_{e_s} f_{e_s}(\pi^m_t) \frac{\mu_{\vec{0}} f_{\vec{0}}(\pi^m_t)}{f(\pi^m_t)} \right) \varepsilonilon^2 \geqslant 0.
\end{split}
\end{align}
By Statement \ref{stat:3rd} one can assume $\mu_{e_s} = \cdots = \mu_{e_k}$ and $\mu_{e_1} = \cdots = \mu_{e_{s-1}} = \mu_{e_s} e^{-L}$, thus $\mu_{\vec{0}} = 1 - (k-s+1) \mu_{e_s} - (s-1) \mu_{e_s} e^{-L}$. Then $t_{s-1} = 0$ and $t_s = L$. As $\gamma_0 \leqslant L/2$ implies $t_s - \gamma_0 = L - \gamma_0 \geqslant L/2$, and \eqref{eqn:epsilon-square-term-Ext} says the integrand is non-negative, hence a lower bound is given by the integration of \eqref{eqn:epsilon-square-term-Ext} in the range $[0, L/2]$. For $t \in [0, L/2]$ and $1 \leqslant m \leqslant s-1$, we have $f_{\vec{0}} (\pi_t^m) = f_{e_s} (\pi_t^m) = \ldots =f_{e_k} (\pi_t^m) = e^{-(s-1)t}$, and $f_{e_1} (\pi_t^m) = \ldots =f_{e_{s-1}} (\pi_t^m) = e^{-(s-2)t}$, thus $f(\pi_t^m) = (1 - (s-1) \mu_{e_s} e^{-L} + (s-2) \mu_{e_s} \e^{-L} e^t) e^{-(s-1)t} \leqslant (s-1)e^{-(s-1)t}$. Hence,
\begin{equation} \nonumber
\eqref{eqn:epsilon-square-term-Ext}
\geqslant
\frac{s-1}{2} \mu_{e_s} f_{e_s} \frac{\mu_{\vec{0}} f_{\vec{0}} (\pi_t^m)}{(s-1)e^{-(s-1)t}} \varepsilonilon^2
= \frac{\mu_{\vec{0}} \mu_{e_s} }{2} e^{-(s-1)t} \varepsilonilon^2.
\end{equation}
Integrating in the range $[0, L/2]$ with respect to $t$ gives the desired bound \eqref{eqn:epsilon-2-bound-Ext}.
Similarly, in the internal case one has
\begin{align*}
\sum_{j=1}^k \concav_\mu(t,j)
&\geqslant
\frac{1}{2} \sum_{m = 1}^{s-1} \sum_{j=1,j\neq s}^k \leqslantft(\mu_{e_s} f_{e_s}(\pi^m_t) \frac{\mu_{\vec{0}} f_{\vec{0}}(\pi^m_t)}{f(\pi^m_t) - \mu_{e_j} f_{e_j}(\pi^m_t)} \right) \varepsilonilon^2 \\
&\geqslant
\frac{k-1}{2} \sum_{m = 1}^{s-1} \leqslantft(\mu_{e_s} f_{e_s}(\pi^m_t) \frac{\mu_{\vec{0}} f_{\vec{0}}(\pi^m_t)}{f(\pi^m_t)} \right) \varepsilonilon^2.
\end{align*}
Hence we get the bound \eqref{eqn:epsilon-2-bound-Int} after integration.
\end{proof}
\subsection{Futher reductions} \label{sec:Further-reductions}
In this section we obtain a futher reduction of Statement~\ref{stat:2nd} that will have a constant number of variables and so one can finally verify it using Wolfram Mathematica:
\begin{statement}[Third reduction]
\label{stat:3rd}
To prove Theorem~\ref{thm:mainMultiparty} it suffices to assume $\mu$ satisfies
\begin{equation} \label{eqn:measure-mu}
\mu_{e_1}=\dots=\mu_{e_{s-1}}=\beta, \quad
\mu_{e_{s}}= \dots =\mu_{e_k}= e^{\gamma_0} \beta, \quad
\mu_{\vec{0}} = 1-(s-1) \beta - (k-s+1) e^{\gamma_0} \beta,
\end{equation}
where $0 < \beta < 1$, and verify
$$\int_{-\gamma_0}^{\gamma_1} \concav_\mu^\ext(t) dt \geqslant 0 \qquad \mbox{and} \qquad \sum_{j=1}^k \int_{-\gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt \geqslant 0.$$
\end{statement}
Statement~\ref{stat:3rd} follows from Claim \ref{claim:mu-k-Equals-mu-s} showing that it suffices to consider measures $\mu$ such that $\mu_{e_j} = \mu_{e_s}$ for all $j \geqslant s$, together with the observation that conditioned on the buzz time $t \in [-\gamma_0, \gamma_1]$, we have $\mu_{e_1}|_{t \geqslant t_s - \gamma_0} = \cdots = \mu_{e_{s-1}}|_{t \geqslant t_s - \gamma_0}$.
\begin{myclaim} \label{claim:SameAverage}
For every $z$,
\begin{align*}
&\ProbOp\leqslantft[X=z \wedge t(\Pi_z) \in [-\gamma_0,\gamma_1] \right] \\
&=\frac{\ProbOp\leqslantft[X^0=z \wedge t(\Pi^0_z) \in [-\gamma_0,\gamma_1] \right]+\ProbOp\leqslantft[X^1=z \wedge t(\Pi^1_z) \in [-\gamma_0,\gamma_1] \right]}{2}.
\end{align*}
\end{myclaim}
\begin{proof}
We need to show
$$\mu_z \sum_m \int_{-\gamma_0}^{\gamma_1} f_z(\pi_t^m) dt= \frac{1}{2} \leqslantft(\mu^0_z \sum_m \int_{-\gamma_0}^{\gamma_1} f_z^0(\pi_t^m) dt+ \mu_z^1 \sum_m \int_{-\gamma_0}^{\gamma_1} f_z^1(\pi_t^m) dt\right).$$
Recall that $\Phi_z(t)$ denotes the total amount of active time spent by all players before time $t$. The probability that $\Pi_z$ finishes in the interval $[-\gamma_0,\gamma_1]$ is equal to $$e^{-\Phi_z(-\gamma_0)} - e^{-\Phi_z(\gamma_1)}.$$
Denoting by $\Phi^0_z(t)$ and $\Phi^1_z(t)$ the total active time for the protocols $\pi^\wedge_{\mu_0}$ and $\pi^\wedge_{\mu_1}$ on the input $z$, the claim is equivalent to
$$\mu_z \cdot (e^{-\Phi_z(-\gamma_0)} - e^{-\Phi_z(\gamma_1)}) = \frac{\mu^0_z \cdot (e^{-\Phi^0_z(-\gamma_0)} - e^{-\Phi^0_z(\gamma_1)})}{2} + \frac{\mu^1_z \cdot (e^{-\Phi^1_z(-\gamma_0)} - e^{-\Phi^1_Z(\gamma_1)})}{2}.$$
Since $\mu_z = \frac{\mu^0_z+\mu^1_z}{2}$ and $\Phi_z(-\gamma_0)=\Phi_z^0(-\gamma_0)=\Phi_z^1(-\gamma_0)$, the equality reduces to
$$\mu_z e^{-\Phi_z(\gamma_1)} = \frac{\mu^0_z e^{-\Phi^0_z(\gamma_1)}+\mu^1_z e^{-\Phi^1_z(\gamma_1)}}{2}.$$
When $z_s=1$, $\Phi_z=\Phi_z^0=\Phi_z^1$, and thus $\mu_z = \frac{\mu^0_z+\mu^1_z}{2}$ verifies the equality. In the case of $z_s=0$, we have that $\Phi^0_z(\gamma_1)=\Phi_z(\gamma_1)+\gamma_0$, and $\Phi^1_z(\gamma_1)=\Phi_z(\gamma_1)-\gamma_1$. Substituting $\gamma_0 = \ln\leqslantft(\frac{1+\varepsilonilon \beta_s}{1-\varepsilonilon \zeta_s}\right)$, $\gamma_1 = \ln\leqslantft(\frac{1+\varepsilonilon \zeta_s}{1-\varepsilonilon \beta_s}\right)$, $\mu^0_z=(1+\varepsilonilon \beta_s) \mu_z$ and $\mu^1_z=(1-\varepsilonilon \beta_s) \mu_z$ verifies the equality.
\end{proof}
\begin{myclaim} \label{claim:mu-k-Equals-mu-s}
Suppose $\mu$ satisfies $\mu_{e_s} = \ldots = \mu_{e_{s+a}} < \mu_{e_{s+a+1}}$ for some $a \geqslant 0$ with $s+a+1 \leqslant k$. Assume $\varepsilonilon$ is sufficiently small so that $\gamma_1 \leqslant t_{s+a+1}$. Let $\mu'$ be a measure for the $(s+a)$-player AND function, defined as: $\mu'_{\vec{0}} = \mu_0 + \sum_{j > s+a} \mu_{e_j}$, and $\mu'_{e_j} = \mu_{e_j}$ for $1 \leqslant j \leqslant s+a$. Then the following hold.
\begin{enumerate}[(1).]
\item $\int_{- \gamma_0}^{\gamma_1} \concav_\mu^\ext(t) dt = \int_{- \gamma_0}^{\gamma_1} \concav_{\mu'}^\ext(t) dt$;
\item If $\sum_{j=1}^{s+a} \int_{- \gamma_0}^{\gamma_1} \concav_{\mu'}(t,j) dt \geqslant 0$, then $\sum_{j=1}^{k} \int_{- \gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt \geqslant 0$.
\end{enumerate}
\end{myclaim}
This claim shows that to verify the concavity conditions \eqref{eq:ExternalGoal} and \eqref{eq:InternalGoal} it suffices to consider only those measures satisfying $\mu_{e_j} = \mu_{e_s}$ for all $j > s$.
\begin{proof}
We confine ourselves in the interval $t \in [- \gamma_0, \gamma_1]$ throughout the proof. Let $f, f'$ and $\Pi, \Pi'$ denote the pdf and protocols for $\pi_\mu^\wedge$ and $\pi_{\mu'}^\wedge$, respectively.
\begin{enumerate}[(1).]
\item Obviously we have
\begin{equation} \label{eqn:temp1}
f'_{\vec{0}}(\pi_t^m) = f_{\vec{0}}(\pi_t^m), \quad\text{and\ }\quad f'_{e_j}(\pi_t^m) = f_{e_j}(\pi_t^m), \quad 1 \leqslant j \leqslant s+a.
\end{equation}
For the protocol $\pi_\mu^\wedge$, observe we have
\begin{equation} \label{eqn:temp2}
f_{\vec{0}}(\pi_t^m) = f_{e_j}(\pi_t^m), \quad j > s+a,
\end{equation}
for all $m = 1, \ldots, k$. Hence \eqref{eqn:temp1} and \eqref{eqn:temp2} imply that $f(\pi_t^m) = f'(\pi_t^m)$. Clearly similar results hold for $\Pi^0, \Pi^1$ and $\Pi'^0, \Pi'^1$ . This imply that the first integral in \eqref{eq:ExternalGoal} does not change from $\mu$ to $\mu'$.
It remains to show that the second integral in \eqref{eq:ExternalGoal} does not change either. Expand this integral gives,
\begin{equation} \label{eqn:temp3}
\begin{aligned}
&\int_{-\gamma_0}^{\gamma_1} \sum_X \sum_m \leqslantft( f_X(\pi_t^m) \mu_X \log(\mu_X) -\frac{\mu^0_X f^0_X(\pi_t^m) \log(\mu_X)+\mu^1_X f^1_X(\pi_t^m) \log(\mu_X)}{2} \right) dt \\
+&\int_{-\gamma_0}^{\gamma_1} \sum_X \sum_m \Bigg(\mu_X \phi(f_X(\pi_t^m)) - \phantom{\Bigg)} \\
&\phantom{\Bigg(} \frac{\mu^0_X \leqslantft(\phi(f^0_X(\pi_t^m))+ f^0_X(\pi_t^m) \log(1+\varepsilonilon \beta)\right) + \mu^1_X \leqslantft(\phi(f^1_X(\pi_t^m))+ f^1_X(\pi_t^m) \log(1-\varepsilonilon \beta)\right)}{2} \Bigg) dt.
\end{aligned}
\end{equation}
By Claim~\ref{claim:SameAverage} the first integral in \eqref{eqn:temp3} is $0$. Hence it only remains to show the second integral in \eqref{eqn:temp3} does not change. But this is again a direct consequence of \eqref{eqn:temp1} and \eqref{eqn:temp2} with corresponding facts for $\Pi_X^0$ and $\Pi_X^1$.
\item By definition of the measures $\mu, \mu'$, one has
\begin{equation} \nonumber
\sum_{j=1}^{k}\int_{- \gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt
-
\sum_{j=1}^{s+a} \int_{- \gamma_0}^{\gamma_1} \concav_{\mu'}(t,j) dt
=
\sum_{j=s+a+1}^{k} \int_{- \gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt.
\end{equation}
Hence it suffices to show $\sum_{j=s+a+1}^{k} \int_{- \gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt \geqslant 0$.
Let $\mu_{X_j = b}$ denote the distribution $\mu$ of $X$ conditioned on $X_j = b$, one can check that
\begin{equation} \label{eqn:Connection-Ext-Int}
\int_{-\gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt =
\Ex_{b} \int_{-\gamma_0}^{\gamma_1} \concav^\ext_{\mu_{X_j = b}} (t) dt.
\end{equation}
In section \ref{sec:ExtComputation} we show the external concavity condition indeed holds, hence \eqref{eqn:Connection-Ext-Int} implies $\int_{- \gamma_0}^{\gamma_1} \concav_{\mu}(t,j) dt \geqslant 0$, as desired. \qedhere
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:mainMultiparty}]
We use $\wedge$ to denote the multiparty $\AND$ function.
Consider the external case first. Recall we set $\Omega$ to be the set of all external trivial measures together with the measure in Claim~\ref{claim:protocol-is-opt-when-mu-i-are-all-equal}, hence Condition (i) and (ii) are satisfied. Picking $w(x) = ck^{-20} x^4$ for some fixed constant $c > 0$, we verify Condition (iii) in Section \ref{sec:ExtComputation}. Hence $\IC^\ext_\mu(\pi^\wedge) \leqslant \IC^\ext_\mu(\wedge)$, as $\IC^\ext_\mu(\pi^\wedge)$ is also an upper bound, hence we proved $\IC^\ext_\mu(\pi^\wedge) = \IC^\ext_\mu(\wedge)$.
Similarly for the internal case the concavity Condition (iii) is verified in Section \ref{sec:IntComputation}.
\end{proof}
\subsection{Information cost of multiparty AND function} \label{sec:multiparty-code}
To simplify the notation, since every function has the argument $\pi_t^m$, we sometimes omit it from the writing while knowing it was there, such as we write $f$ to mean $f(\pi_t^m)$. We will use $\mu_0, \mu_j, f_0, f_j$ instead of $\mu_{\vec{0}}, \mu_{\e_{j}}, f_{\vec{0}}, f_{e_j}$ when there is no ambiguity, similar notations are used for measures $\mu^0, \mu^1$ and functions $f^0, f^1$.
\subsubsection{Taylor expansions} \label{sec:multi-party-Taylor}
Recall $\beta_s = \mu_s$ and $\zeta_s = 1 - \beta_s$. By the claims in Section \ref{sec:Further-reductions}, we can assume that
\begin{equation} \label{eqn:measure-mu}
\mu_1=\dots=\mu_{s-1}=\beta, \quad
\mu_s=\dots =\mu_k= e^{\gamma_0} \beta, \quad
\mu_0 = 1-(s-1) \beta - (k-s+1) e^{\gamma_0} \beta.
\end{equation}
Observe that $0 < \beta < 1/k$ (the measure when $\beta=0$ is both external and internal trivial). Furthermore, viewing $\gamma_0$ and $\gamma_1$ as functions of $\varepsilonilon$, plugging $\beta_s = e^{\gamma_0} \beta$ and $\zeta_s = 1 - \beta_s = 1 - e^{\gamma_0}\beta$ into \eqref{eqn:gamma-0} and \eqref{eqn:gamma-1}, by implicit differentiation, we have
\begin{equation} \label{eqn:derivatives-of-gamma}
\begin{cases}
\gamma_0(0) = 0, \gamma_0'(0) = 1, \gamma_0''(0) = 1 - 2\beta, \gamma_0'''(0) = 2 - 10 \beta + 8 \beta^2; \\
\gamma_1(0) = 0, \gamma_1'(0) = 1, \gamma_1''(0) = 2\beta - 1, \gamma_1'''(0) = 2 + 6 \beta^2.
\end{cases}
\end{equation}
These derivatives are used in the Mathematica computation. To simplify the notation we let $\zeta = \zeta_s = 1 - e^{\gamma_0}\beta$.
Assuming $\varepsilonilon<1/2$, then $\gamma_0+\gamma_1 \leqslant 2 \ln (1+2\varepsilonilon) \leqslant 4 \varepsilonilon$. Also note $|e^{-x}-1| \leqslant x$ for $x \geqslant 0$, which together with the fact that $\Phi_x(t) \leqslant k(\gamma_0+\gamma_1) \leqslant 4 k \varepsilonilon$ implies the following:
\begin{itemize}
\item $\mu_0 f_0(\pi^m_t)$ is either $0$ or close to $1-k\beta$ with distance bounded by $4k\varepsilonilon$;
\item for $j\neq 0$, we have $\mu_j f_j(\pi^m_t)$ is either $0$ or close to $\beta$ with distance bounded by $4(k+1)\varepsilonilon$;
\item $f(\pi^m_t)$ is either $0$ or close to $1-\beta$ with distance bounded by $16k^2\varepsilonilon$;
\item for $j\neq 0$ and $j \neq m$, we have $f(\pi_t^m) - \mu_j f_j(\pi_t^m)$ is either $0$ or close to $1-2\beta$ with distance bounded by $16k^2\varepsilonilon$.
\end{itemize}
Note that $f(\pi_t^m) = 0$ implies $\mu_x f_x(\pi_t^m) = 0$ for every $x$, hence $\phi(\cdot) = 0$ for all functions under consideration. On the other hand when $f(\pi_t^m) \neq 0$, then all $\mu_x f_x(\pi_t^m)$ are nonzero except when $x_m = 1$. In this case these functions $\phi(\cdot)$ have the following Taylor expansions at corresponding points as follows,
\begin{align*}
&\phi(\mu_0 f_0(\pi^m_t))
= -\frac{1-k\beta}{2} + (\ln(1-k\beta)) \mu_0 f_0(\pi^m_t) + \frac{(\mu_0 f_0(\pi^m_t))^2}{2(1-k\beta)} + O(\varepsilonilon^3), \\
&\phi(\mu_j f_j(\pi^m_t))
= -\frac{\beta}{2} + (\ln\beta) \mu_j f_j(\pi^m_t) + \frac{(\mu_j f_j(\pi^m_t))^2}{2\beta} + O(\varepsilonilon^3), \ j \neq 0, j\neq m \\
&\phi(f(\pi^m_t))
= -\frac{1-\beta}{2} + (\ln(1-\beta)) f(\pi_t^m) + \frac{f(\pi_t^m)^2}{2(1-\beta)} + O(\varepsilonilon^3), \\
&\phi(f(\pi_t^m) - \mu_j f_j(\pi_t^m))
= -\frac{1-2\beta}{2} + (\ln(1-2\beta)) f(\pi_t^m) + \frac{f(\pi_t^m)^2}{2(1-2\beta)} - \\
&\phantom{=\ } (\ln(1-2\beta)) \mu_j f_j(\pi_t^m) + \frac{(\mu_j f_j(\pi_t^m))^2}{2(1-2\beta)} - \frac{\mu_j f_j(\pi_t^m) f(\pi_t^m)}{1-2\beta} + O(\varepsilonilon^3), \ j\neq 0, j\neq m.
\end{align*}
Recall Taylor's theorem with the remainder in Lagrange form says that the error term $O(\varepsilonilon^3)$ in the expansion of $\phi(\mu_0 f_0(\pi_t^m))$ equals $\frac{|\phi^{(3)}(\xi)|}{6} |\mu_0 f_0(\pi_t^m) - (1-k\beta)|^3$ for some $\xi$ between $\mu_0 f_0(\pi_t^m)$ and $1-k\beta$. Since $|\mu_0 f_0(\pi_t^m) - (1-k\beta)| \leqslant 4k\varepsilonilon$, we have $|\xi - (1-k\beta)| \leqslant 4k\varepsilonilon$, hence $\xi \geqslant (1-k\beta) - 4k\varepsilonilon > 0$ if $\varepsilonilon < \frac{1-k\beta}{4k}$. Furthermore, we have $0 < \frac{1}{(1-k\beta) - 4k\varepsilonilon} \leqslant \frac{2}{1-k\beta}$ as long as $\varepsilonilon \leqslant \frac{1-k\beta}{8k}$. Therefore,
\begin{align*} \nonumber
\frac{|\phi^{(3)}(\xi)|}{6} |\mu_0 f_0(\pi_t^m) - (1-k\beta)|^3
&= \frac{1}{6\xi^2} |\mu_0 f_0(\pi_t^m) - (1-k\beta)|^3 \\
&\leqslant \frac{1}{6(1-k\beta - 4k\varepsilonilon)^2} (4k\varepsilonilon)^3 \\
&\leqslant \frac{4}{6(1-k \beta)^2} (4k)^3 \varepsilonilon^3 \leqslant \frac{k^{11}}{6(1-k \beta)^2} \varepsilonilon^3,
\end{align*}
when $0 < \varepsilonilon \leqslant \frac{1-k\beta}{k^7} < \frac{1-k\beta}{8k}$. Denote the constant in this upper bound by $R_1$.
Similarly, let $R_2, R_3$ and $R_4$ denote the constants that we can get as upper bounds of the absolute values of error terms in the expansions of $\phi(\mu_j f_j(\pi^m_t)), \phi(f(\pi^m_t))$ and $\phi(f(\pi_t^m) - \mu_j f_j(\pi_t^m))$, respectively. We have
\begin{equation} \label{eqn:Remainders-bound}
\begin{cases}
R_1
\leqslant \frac{k^{11}}{6(1-k \beta)^2}, &\text{when\ } 0 < \varepsilonilon \leqslant \frac{1-k\beta}{k^7}; \\
R_2
\leqslant \frac{k^{14}}{6 \beta^2} , &\text{when\ } 0 < \varepsilonilon \leqslant \frac{\beta}{k^7}; \\
R_3
\leqslant \frac{k^{20}}{6 (1-\beta)^2}, &\text{when\ } 0 < \varepsilonilon \leqslant \frac{1-\beta}{k^7}; \\
R_4
\leqslant \frac{k^{20}}{6 (1-2\beta)^2}, &\text{when\ } 0 < \varepsilonilon \leqslant \frac{1-2\beta}{k^7}.
\end{cases}
\end{equation}
Observe that $\mu_x^0 f_x^0$ and $\mu_x^1 f_x^1$ are both close to $\mu_x f_x$ with distance bounded by $3\varepsilonilon$, hence the corresponding functions $\phi(\mu_x^0 f_x^0)$ and $\phi(\mu_x^1 f_x^1)$ have the same expansions as above, the same holds for functions $\phi(f^0), \phi(f^1)$ and $\phi(f^0 - \mu^0_x f^0_x), \phi(f^1 - \mu^1_x f^1_x)$.
We continue to use the Taylor expansions to expand the concavity conditions \eqref{eq:ExternalGoal} and \eqref{eq:InternalGoal}.
\begin{itemize}
\item Taylor expansion of external concavity condition \eqref{eq:ExternalGoal}.
When $f(\pi_t^m) \neq 0$, we have the following expansion,
\begin{align} \label{eqn:extExpand}
\begin{split}
&\phi(f(\pi_t^m)) - \sum_x \phi(\mu_x f_x(\pi_t^m)) \\
&= \phi(f(\pi_t^m)) - \phi(\mu_0 f_0(\pi_t^m)) - \sum_{j=1, j\neq m}^k \phi(\mu_j f_j(\pi_t^m)) \\
&= (\ln(1-\beta)) f + \frac{1}{2(1-\beta)} f^2 - (\ln(1-k\beta)) \mu_0 f_0 - \frac{1}{2(1-k\beta)} (\mu_0 f_0)^2 \\
&\phantom{=\ } - \ln\beta \sum_{j=1, j\neq m}^k \mu_j f_j - \frac{1}{2\beta} \sum_{j=1, j\neq m}^k (\mu_j f_j)^2 + O(\varepsilonilon^3),
\end{split}
\end{align}
where constant in $O(\varepsilonilon^3)$ can be bounded by $R_1 + (k-1)R_2 + R_3$ according to \eqref{eqn:Remainders-bound}. Using Claim \ref{claim:SameAverage}, we see that the first, third and fifth terms in \eqref{eqn:extExpand} become $0$ in \eqref{eq:ExternalGoal}. Let
\begin{equation} \nonumber
F^{\ext}_m (t) = \frac{1}{2(1-\beta)} f^2 - \frac{1}{2(1-k\beta)} (\mu_0 f_0)^2 - \frac{1}{2\beta} \sum_{j=1, j\neq m}^k (\mu_j f_j)^2.
\end{equation}
Define $F_m^{\ext,0}(t)$ and $F_m^{\ext,1}(t)$ with $f$ replaced by $f^0$ and $f^1$, respectively, etc. Observe that $F^{\ext}_m (t) = 0$ when $f(\pi_t^m) = 0$, hence in general $F^{\ext}_m (t)$ is a correct representation of $\phi(f(\pi_t^m)) - \sum_x \phi(\mu_x f_x(\pi_t^m))$. Therefore, what we want to verify in \eqref{eq:ExternalGoal} becomes
\begin{equation} \label{eqn:ExtGoalCompute}
\int_{-\gamma_0}^{\gamma_1} \sum_{m=1}^k \leqslantft( F^{\ext}_m (t) - \frac{F_m^{\ext,0} (t) + F_m^{\ext,1} (t)}{2} \right) dt + O(\varepsilonilon^4).
\end{equation}
As $\gamma_0 + \gamma_1 \leqslant 4\varepsilonilon$, by \eqref{eqn:Remainders-bound}, the constant in $O(\varepsilonilon^4)$ in \eqref{eqn:ExtGoalCompute} can be bounded by
\begin{equation} \nonumber
8k(R_1 + (k-1)R_2 + R_3) \leqslant 4 k^{21} \leqslantft( \frac{1}{(1-k\beta)^2} + \frac{1}{\beta^2} \right),
\end{equation}
when $\varepsilonilon \leqslant \frac{1}{k^7} \min\{\beta, 1-k\beta\}$.
\item Taylor expansion of internal concavity condition \eqref{eq:InternalGoal}.
A direct calculation gives,
\begin{align} \label{eqn:Int-temp1}
\begin{split}
&\phantom{=} \sum_{j=1}^k \sum_{b=0,1} \phi(f_{x_j=b}(\pi_t^m)) - k \sum_x \phi(\mu_x f_x(\pi_t^m)) \\
&= \phi(f(\pi_t^m)) - k \phi(\mu_0 f_0(\pi_t^m)) \\
&+ \sum_{j=1, j \neq m}^k \Big( \phi(f(\pi_t^m) - \mu_j f_j(\pi_t^m)) - (k-1) \phi(\mu_j f_j(\pi_t^m)) \Big).
\end{split}
\end{align}
As did for the external case, when $f(\pi_t^m) \neq 0$ the above formula expands as follows,
\begin{align} \label{eqn:intExpand}
\begin{split}
\eqref{eqn:Int-temp1} &= \leqslantft(\ln(1-\beta) + (k-2) \ln(1-2\beta) - (k-1) \ln\beta \right) f \\
&\phantom{=\ } + \leqslantft( \frac{1}{2(1-\beta)} + \frac{k-3}{2(1-2\beta)} \right) f^2 \\
&\phantom{=\ } + \leqslantft( \ln(1-2\beta) + (k-1)\ln\beta - k \ln(1-k\beta) \right) \mu_0 f_0 \\
&\phantom{=\ } - \frac{k}{2(1-k\beta)} (\mu_0 f_0)^2 + \leqslantft( \frac{1}{2(1-2\beta)} - \frac{k-1}{2\beta} \right) \sum_{j=1, j\neq m}^k (\mu_j f_j)^2\\
&\phantom{=\ }
+ \frac{1}{1-2\beta} \mu_0 f_0 f + O(\varepsilonilon^3).
\end{split}
\end{align}
Claim \ref{claim:SameAverage} implies the first and third terms in \eqref{eqn:intExpand} become $0$ in \eqref{eq:InternalGoal}. Let
\begin{align*}
F_m (t)
&= \leqslantft( \frac{1}{2(1-\beta)} + \frac{k-3}{2(1-2\beta)} \right) f^2 - \frac{k}{2(1-k\beta)} (\mu_0 f_0)^2 \\
&\phantom{=\ } + \leqslantft( \frac{1}{2(1-2\beta)} - \frac{k-1}{2\beta} \right) \sum_{j=1, j\neq m}^k (\mu_j f_j)^2
+ \frac{1}{1-2\beta} \mu_0 f_0 f.
\end{align*}
Define $F_m^0(t)$ and $F_m^1(t)$ similarly. Then $F_m (t)$ is a correct representation for \eqref{eqn:Int-temp1}. Therefore what we want to verify in \eqref{eq:InternalGoal} becomes
\begin{equation} \label{eqn:IntGoalCompute}
\int_{-\gamma_0}^{\gamma_1} \sum_{m=1}^k \leqslantft( F_m (t) - \frac{F_m^0 (t) + F_m^1 (t)}{2} \right) dt + O(\varepsilonilon^4).
\end{equation}
\end{itemize}
\subsubsection{Density functions in explicit form} \label{sec:multi-party-density-functions}
We continue to calculate functions explicitly that will be used for computing.
\begin{itemize}
\item In the protocol $\pi_\mu^\wedge$.
Consider the interval $t \in [-\gamma_0, 0)$. Let $A = (s-1)(t+\gamma_0) = (s-1)t + (s-1)\gamma_0$, The total active time $\Phi_0(t) = \Phi_j(t) = A$ for $s \leqslant j \leqslant k$, and $\Phi_j(t) = A-(t+\gamma_0)$ for $1 \leqslant j \leqslant s-1$. Hence for $1 \leqslant m \leqslant s-1$, we have,
\begin{equation} \nonumber
\mu_j f_j(\pi_t^m) =
\begin{cases}
\mu_0 e^{-A}, &\quad j=0,\\
\mu_j e^{t+\gamma_0} e^{-A} = e^{\gamma_0} \beta e^t e^{-A}, &\quad 1 \leqslant j \leqslant s-1 \text{\ and \ } j \neq m, \\
\mu_j e^{-A} = e^{\gamma_0} \beta e^{-A}, &\quad s \leqslant j \leqslant k, \\
0, &\quad j=m.
\end{cases}
\end{equation}
For $m \geqslant s$, we have $\mu_x f_x (\pi_t^m) = 0$ for all $x$. Therefore when $t \in [-\gamma_0, 0)$,
\begin{equation} \nonumber
f(\pi_t^m) =
\begin{cases}
0, &\quad m \geqslant s, \\
(1 - (s-1)\beta + (s-2)e^{\gamma_0} \beta e^t) e^{-A}, &\quad 1 \leqslant m \leqslant s-1.
\end{cases}
\end{equation}
Similarly for the interval $t \in [0, \gamma_1)$, let $B= (s-1)(t+\gamma_0) + (k-s+1)t = kt + (s-1)\gamma_0$, the total active time is $\Phi_0(t) = B$, $\Phi_j(t) = B-(t+\gamma_0)$ for $1 \leqslant j \leqslant s-1$, and $\Phi_j(t) = B-t$ for $s \leqslant j \leqslant k$. Hence for all $1 \leqslant m \leqslant k$ we have,
\begin{equation} \nonumber
\mu_j f_j(\pi_t^m) =
\begin{cases}
\mu_0 e^{-B}, &\quad j=0, \\
\mu_j e^{t+\gamma_0} e^{-B} = e^{\gamma_0} \beta e^t e^{-B}, &\quad 1 \leqslant j \leqslant s-1 \text{\ and\ } j \neq m, \\
\mu_j e^t e^{-B} = e^{\gamma_0} \beta e^t e^{-B}, &\quad s \leqslant j \leqslant k \text{\ and\ } j \neq m, \\
0, &\quad j=m.
\end{cases}
\end{equation}
Therefore when $t \in [0, \gamma_1)$,
\begin{equation} \nonumber
f(\pi_t^m) = (1-(s-1)\beta - (k-s+1) e^{\gamma_0} \beta + (k-1) e^{\gamma_0} \beta e^t) e^{-B}.
\end{equation}
\item In the protocol $\pi^{\wedge}_{\mu^0}$.
Using results from Section \ref{sec:distribution-mu0-mu1}, we have,
\begin{equation} \nonumber
\mu_x^0 f_x^0(\pi_t^m) =
\begin{cases}
(1-\varepsilonilon \zeta)e^{-t} \mu_x f_x(\pi_t^m), &\quad t \in [-\gamma_0, 0), x_s = 0, m\neq s, \\
(1-\varepsilonilon \zeta) \mu_x f_x(\pi_t^m), &\quad t \in [-\gamma_0, 0), x_s = 1, m\neq s, \\
(1-\varepsilonilon \zeta) \mu_x f_x(\pi_t^m), &\quad t \in [0, \gamma_1).
\end{cases}
\end{equation}
For the special case $m=s$ and $t \in [-\gamma_0, 0)$, we have,
\begin{equation} \nonumber
\mu_j^0 f_j^0(\pi_t^s) =
\begin{cases}
(1-\varepsilonilon \zeta) \mu_0 e^{-t} e^{-A}, &\quad t \in [-\gamma_0, 0), j=0, \\
(1-\varepsilonilon \zeta) e^{\gamma_0} \beta e^{-A}, &\quad t \in [-\gamma_0, 0), 1 \leqslant j \leqslant s-1, \\
0, &\quad t \in [-\gamma_0, 0), j=s\\
(1-\varepsilonilon \zeta) e^{\gamma_0} \beta e^{-t} e^{-A}, &\quad t \in [-\gamma_0, 0), s+1 \leqslant j \leqslant k.
\end{cases}
\end{equation}
Therefore when $t \in [-\gamma_0, 0)$,
\begin{equation} \nonumber
f^0(\pi_t^m) =
\begin{cases}
(1-\varepsilonilon \zeta) ((1 - e^{\gamma_0} \beta - (s-1) \beta)e^{-t} + (s-1) e^{\gamma_0} \beta) e^{-A}, &1 \leqslant m \leqslant s, \\
0, &s+1 \leqslant m \leqslant k.
\end{cases}
\end{equation}
When $t \in [0, \gamma_1)$, it is simply,
\begin{equation} \nonumber
f^0(\pi_t^m) = (1-\varepsilonilon \zeta) f(\pi_t^m).
\end{equation}
\item In the protocol $\pi^{\wedge}_{\mu^1}$.
Using results from Section \ref{sec:distribution-mu0-mu1}, when $m=s$, then $\mu_x^1 f_x^1(\pi_t^m) = 0$ for all $x$ for $t \in [-\gamma_0, \gamma_1]$. Therefore $f^1(\pi_t^s) = 0$ for all $t \in [-\gamma_0, \gamma_1]$.
When $m \neq s$, we have,
\begin{equation} \nonumber
\mu_x^1 f_x^1(\pi_t^m) =
\begin{cases}
\mu^1_x f_x(\pi_t^m), &\quad t \in [-\gamma_0, 0), \\
(1-\varepsilonilon e^{\gamma_0} \beta) e^t \mu_x f_x(\pi_t^m), &\quad t \in [0, \gamma_1), x_s = 0, \\
(1 + \varepsilonilon \zeta) \mu_x f_x(\pi_t^m), &\quad t \in [0, \gamma_1), x_s = 1.
\end{cases}
\end{equation}
Hence when $t \in [-\gamma_0, 0)$, we have $f^1(\pi_t^m) = (1-\varepsilonilon e^{\gamma_0} \beta) f(\pi_t^m) + \varepsilonilon \mu_s f_s(\pi_t^m)$,
and when $t \in [0, \gamma_1)$ we have $f^1(\pi_t^m) = (1-\varepsilonilon e^{\gamma_0} \beta) e^t f(\pi_t^m) + (1+\varepsilonilon\zeta -(1-\varepsilonilon e^{\gamma_0} \beta)e^t) \mu_s f_s(\pi_t^m)$. Plug in $f$ we get, when $t \in [-\gamma_0, 0)$,
\begin{equation} \nonumber
f^1(\pi_t^m) =
\begin{cases}
0, &m \geqslant s, \\
(1 + e^{\gamma_0} \beta(1-\varepsilonilon e^{\gamma_0} \beta) ((s-2)e^t - (s-1)e^{-\gamma_0})) e^{-A}, &1 \leqslant m \leqslant s-1.
\end{cases}
\end{equation}
When $t \in [0, \gamma_1)$,
\begin{equation} \nonumber
f^1(\pi_t^m) =
\begin{cases}
0, &m = s, \\
(1 + e^{\gamma_0} \beta(1-\varepsilonilon e^{\gamma_0} \beta)((k-2)e^t - (s-1)e^{-\gamma_0} - k + s)) e^t e^{-B}, &m \neq s.
\end{cases}
\end{equation}
\end{itemize}
\subsubsection{External information cost} \label{sec:ExtComputation}
Using Wolfram Mathematica with results from Section \ref{sec:multi-party-Taylor} and \ref{sec:multi-party-density-functions}, we obtain
\begin{equation} \label{eqn:Ext-numeric}
\eqref{eqn:ExtGoalCompute} = \frac{(k+5s-6)(1-2\beta)\beta}{12(1-\beta)\ln 2} \varepsilonilon^3 + O(\varepsilonilon^4).
\end{equation}
Therefore, using the bound of the error term given in Section \ref{sec:multi-party-Taylor}, one finds $\eqref{eqn:Ext-numeric} > 0$ as long as
\begin{equation} \nonumber
\varepsilonilon < \min\leqslantft\{\frac{(k+5s-6)(1-2\beta)\beta}{12(1-\beta)\ln 2} {\bigg/} 4 k^{21} \leqslantft( \frac{1}{(1-k\beta)^2} + \frac{1}{\beta^2} \right), \frac{1}{k^7} \min\{\beta, 1-k\beta\} \right\}.
\end{equation}
Note that $\frac{2}{1/x + 1/y} = \frac{2xy}{x+y} \geqslant \min\{x, y\}$ for all $x, y > 0$. Simplifying the above formula, one obtains $\eqref{eqn:Ext-numeric} > 0$ as long as
\begin{equation} \nonumber
\varepsilonilon < c k^{-20} \min\{\beta, 1-k\beta\}^3,
\end{equation}
for some constant $c > 0$. So we have verified the concavity condition \eqref{eq:ExternalGoal} is satisfied for all $\varepsilonilon$-weak signals such that $\varepsilonilon$ is no greater than $c k^{-20} \min\{\beta, 1-k\beta\}^3$.
Let $\mu^E$ denote the distribution in Claim~\ref{claim:protocol-is-opt-when-mu-i-are-all-equal}, we have $|\mu - \mu^E| \leqslant 1-k\beta$. Let $\mu'$ be defined as $\mu'_{s-1} = 0$, $\mu'_{s} = e^{\gamma_0} \beta + \beta$, and $\mu'_j = \mu_j$ for all other $j$, then $|\mu - \mu'| = \beta$.
Observe that $\mu'$ is external trivial, hence $\mu^E, \mu' \in \Omega$ (the $\Omega$ we chose at the beginning of Section \ref{sec:Some-reductions}). Therefore we have $\delta(\mu) \leqslant \min\{\beta, 1-k\beta\}$. Thus as we choose $w(x) = ck^{-20} x^4$, the concavity condition \eqref{eq:ExternalGoal} is satisfied for all $w(\delta(\mu))$-weak signals because
\begin{equation} \nonumber
w(\delta(\mu)) \leqslant ck^{-20} \min\{\beta, 1-k\beta\}^4 < c k^{-20} \min\{\beta, 1-k\beta\}^3.
\end{equation}
By Theorem~\ref{thm:LocalCharNoErrorWeak}, we have proved the protocol $\pi^\wedge$ in Figure \ref{fig:prot} is optimal for external information cost.
\subsubsection{Internal information cost} \label{sec:IntComputation}
Similarly, using Wolfram Mathematica, we obtain
\begin{equation} \label{eqn:Int-numeric}
\eqref{eqn:IntGoalCompute} =
\begin{cases}
\frac{(k+5s-6)(1-2\beta)\beta}{12(1-\beta)\ln 2} \varepsilonilon^3 + O(\varepsilonilon^4), &\quad k=2, \\
\frac{(k+5s-6)((3k-2)\beta^2 - 4(k-1)\beta + k-1)\beta}{12(1-\beta)(1-2\beta)\ln 2} \varepsilonilon^3 + O(\varepsilonilon^4), &\quad k \geqslant 3.
\end{cases}
\end{equation}
As did in Section \ref{sec:ExtComputation}, one can show \eqref{eqn:Int-numeric} is positive when $\varepsilonilon$ is sufficiently small. And furthermore one can pick an appropriate function $w$ to verify that the concavity condition \eqref{eq:InternalGoal} is satisfied for all $w(\delta(\mu))$-weak signals. Hence by Theorem~\ref{thm:LocalCharNoErrorWeak}, our protocol is optimal for internal information cost.
\end{document} |
\begin{document}
\title[Local Finiteness of the Twisted Bruhat Orders]{Local Finiteness of the Twisted Bruhat Orders\\ on Affine Weyl Groups}
\author{Weijia Wang}
\address{School of Mathematics (Zhuhai)
\\ Sun Yat-sen University \\
Zhuhai, Guangdong, 519082 \\ China}
\email{[email protected]}
\date{\today}
\begin{abstract}
In this paper we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer in J. Algebra, 163, 861--879 (1994). We also show that for a non-finite and non-cofinite biclosed set $B$ in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed $B$-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.
\end{abstract}
\maketitle
\section{Introduction}
Twisted strong and weak Bruhat orders on a Coxeter group were introduced by Dyer and the author respectively.
These generalizations of the ordinary strong and weak Bruhat orders have found connections with problems related to reflection orders,
representation theory and combinatorics of infinite reduced words.
In \cite{DyerTwistedBruhat} it is shown that Kazhdan-Lusztig type polynomials can be defined for certain intervals of the twisted strong
Bruhat orders and these polynomials are used to formulate a conjecture regarding the representation of Kac-Moody Lie algebras.
In \cite{Gobet}, twisted strong Bruhat order is studied for the twisted filtration of the Soergel bimodules.
In \cite{chenyu}, twisted strong Bruhat order is related to the poset of $B\times B$-orbits on the wonderful compactification of algebraic groups.
In \cite{orderpaper}, the semilattice property of twisted weak Bruhat order is shown to characterize the biclosed sets arising from the infinite reduced words in affine cases.
In this paper, we prove several properties regarding the structure of twisted Bruhat orders which are not previously known. First we study the finite intervals of twisted strong Bruhat orders. Finite intervals are of particular importance as the Kazhdan-Lusztig type polynomials in \cite{DyerTwistedBruhat} can only be defined for finite intervals with other favorable properties. It is not previously known whether for any infinite biclosed set $B$, the $B$-twisted strong Bruhat order on an affine Weyl group is locally finite, i.e. any interval of this poset is finite. In \cite{quotient}, Dyer gives a partial answer to the question, providing a technical condition guaranteeing the local finiteness. In section \ref{localfinite} we solve the problem in the positive, showing that such a poset is always locally finite. Indeed we prove a stronger fact for affine Weyl groups: the set $\{y|y\leq_B x, l_B(x)-l_B(y)=n\}$ is finite for any $x$ and biclosed set $B$. The proof exploits the explicit description of biclosed sets in the affine root system first given in \cite{DyerReflOrder}.
In Section \ref{hyperinterval}, we consider the question whether for some biclosed set $B$, the strong $B$-twisted Bruhat order on a nonaffine Coxeter group is not locally finite. We propose a procedure to obtain an infinite interval for certain $B.$
In Section \ref{fixlength}, we show that while a twisted strong Bruhat order on an affine Weyl group is locally finite, the set of elements with a fixed twisted length is always infinite provided the twisting biclosed set is neither finite nor cofinite and the rank of the affine Weyl group is greater than 2. This result implies that the twisted strong and weak Bruhat order has an infinite antichain in those cases. The proof makes use of the properties of twisted weak Bruhat order and an explicit description of the biclosed sets in the affine root system.
In Section \ref{examplesec}, we present the structure of a twisted strong Bruhat order on the affine Weyl group $\widetilde{W}$ of type $\widetilde{A}_2$. Such a description of the structure of a twisted Bruhat order (which is not isomorphic to ordinary Bruhat order or its opposite) was only known for $\widetilde{A}_1$ previously.
In Section \ref{omsec}, we study the tope poset of the infinite oriented matroid from an affine root system. We first describe all hemispaces (topes) of such an oriented matroid and then show that twisted weak Bruhat orders show up in the tope poset. These results allow us to prove that certain finite intervals in the tope poset are lattices.
\section{Preliminaries}
Let $B$ be a set. Denote by $|B|$ the cardinality of $B$. We denote the disjoint union of sets by $\uplus$.
We refer the reader to \cite{bjornerbrenti} and \cite{Hum} for the basic notions of Coxeter groups and their root systems. We call a Coxeter system without braid relations a universal Coxeter system.
\subsection{Biclosed Sets of Coxeter Groups}
Let $(W,S)$ be a Coxeter system. Denote by $T$ the set of reflections. Given a root $\alpha$, denote by $s_{\alpha}$ the corresponding reflection. Let $t\in T$ be a reflection. Denote by $\alpha_t$ the corresponding positive root.
For $w\in W$, the inversion set of $w^{-1}$ is defined to be $\{\alpha\in \Phi^+|w^{-1}(\alpha)\in \Phi^-\}$ and is denoted by $N(w)$.
For a general Coxeter system $(W,S)$, $\Phi,\Phi^+,\Phi^-$ denote the set of roots, positive roots and negative roots respectively. We denote by $l(w)$ the (usual) length of $w\in W.$
A set $\Gamma\subset\Phi$ is 2 clousre closed if for any $\alpha,\beta\in \Gamma$ and $k_1\alpha+k_2\beta\in\Phi, k_1,k_2\in \mathbb{R}_{\geq 0}$ one has that $k_1\alpha+k_2\beta\in\Gamma$. A set $B\subset \Gamma$ such that both $B$ and $\Gamma\backslash B$ are closed is called a biclosed set in $\Gamma$. Finite biclosed sets in $\Phi^+$ are precisely inversion sets $N(x)$ for some $x\in W$ (\cite{DyerWeakOrder} Lemma 4.1(d)). If $s_1s_2\cdots s_k$ is a reduced expression of $x$, then $N(x)=\{\alpha_{s_1},s_1(\alpha_{s_2}),\cdots,s_1s_2\cdots,s_{k-1}(\alpha_{s_k})\}$.
Denote by $\widetilde{N}(x)=\{t\in T|\alpha_t\in N(x)\}$.
We say a positive root $\beta$ dominates another positive root $\alpha$ if $\beta\in N(w)$ for some $w\in W$ implies $\alpha\in N(w).$
Suppose that $W$ is infinite. An infinite sequence $s_1s_2s_3\cdots, s_i\in S$ is called an infinite reduced word of $W$ provided that $s_1s_2\cdots s_j$ is reduced for any $j\geq 1$. For an infinite reduced word $x=s_1s_2\cdots$, define the inversion set $N(x)=\cup_{i=1}^{\infty}N(s_1s_2\cdots s_i)$. Two infinite reduced words $x,y$ are considered equal if $N(x)=N(y)$. The inversion set of an infinite reduced word is biclosed in $\Phi^+$. A (finite) prefix of an infinite reduced word $w$ is an element $u$ in the Coxeter group $W$ such that $N(u)\subset N(w).$ The set of infinite reduced words of $(W,S)$ is denoted by $W_l$. For an inversion set $N(w)$ we denote by $N(w)'$ the complement of $N(w)$ in $\Phi^+$. An element $w\in W$ is said to be straight if $l(w^n)=|nl(w)|.$ It is shown that in \cite{speyer} that if $W$ is infinite any Coxeter element is straight. Choose a reduced expression $\underline{w}$ of a straight element $w$, then $\underline{w}\underline{w}\underline{w}\cdots$ well defines an infinite reduced word indepenent of the choice of $\underline{w}$ and we denote it by $w^{\infty}$.
There exists a $W$-action on the set of all biclosed sets in $\Phi^+$ given by $w\cdot B:=(N(w)\backslash w(-B))\cup (w(B)\backslash
(-N(w)))$. In particular $u\cdot N(v)=N(uv)$ for $u\in W, v\in W\cup W_l.$
We shall call a set $B$ in $\Phi^+$ cofinite if $\Phi^+\backslash B$ is finite.
\subsection{Twisted Bruhat Order and Twisted Weak Bruhat Order}
Let $B$ be a biclosed set in $\Phi^+$. The left (resp. right) ($B$-)twisted length of an element of $w\in W$ is defined as $l(w)-2|N(w^{-1})\cap B|$ (resp. $l(w)-2|N(w)\cap B|$) and is denoted by $l_B(w)$ (resp. $l'_B(w)$). Clearly $l_B(w)=l_B'(w^{-1})$ and $l_{\emptyset}(w)=l'_{\emptyset}(w)=l(w)$.
Fix a biclosed set $B$ in $\Phi^+$, the left (resp. right) ($B$-)twisted strong Bruhat order $\leq_B$ on $W$ is defined as follow: $x\leq_B y$ if and only if $y=t_kt_{k-1}\cdots t_1x$ with $l_B(t_i\cdots t_1x)=l_B(t_{i-1}\cdots t_1x)+1, t_i\in T, 1\leq i\leq k$ (resp. $l_B'(xt_1\cdots t_{i})=l_B'(xt_1\cdots t_{i-1})+1, t_i\in T, 1\leq i\leq k$).
One easily sees that the left twisted strong Bruhat order is isomorphic to the right twisted strong Bruhat order under the map $w\mapsto w^{-1}$. The left (resp. right) $\emptyset-$twisted strong Bruhat order is the ordinary Bruhat order. For $x\in W$ it is known that the left (resp. right) $N(x)-$twisted strong Bruhat order is isomorphic to the ordinary Bruhat order. Following the convention in the literature, we consider the left twisted strong Bruhat orders instead of the right ones and often refer to left twisted strong Bruhat order as twisted Bruhat orders. Left $B-$twisted strong Bruhat order can also be characterized equivalently as the unique partial order on $W$ with the following property:
for $t\in T, w\in W$, if $\alpha_t\in w\cdot B$, $tw\leq_B w$ and if $\alpha_t\not\in w\cdot B$ then $tw\geq_B w.$
A closed interval $[x,y]$ in a twisted (strong) Bruhat order is said to be spherical if any length 2 subinterval of it contains 4 elements. It is known that for a closed spherical interval $[x,y]$, the order complex of the corresponding open interval $(x,y)$ is a sphere.
The left (resp. right) ($B$-)twisted weak Bruhat order $\leq_B'$ on $W$ is defined as follow: $x\leq_B' y$ if and only if $N(x^{-1})\backslash N(y^{-1})\subset B$ and $N(y^{-1})\backslash N(x^{-1})\subset \Phi^+\backslash B$ (resp. $N(x)\backslash N(y)\subset B$ and $N(y)\backslash N(x)\subset \Phi^+\backslash B$). The left $B$-twisted weak Bruhat order is isomorphic to the right $B$-twisted weak Bruhat order under the map $w\mapsto w^{-1}$. For $x\in W$ it is known that the left (resp. right) $N(x)-$twisted weak Bruhat order is isomorphic to the ordinary left (resp. right) weak order.
If $x\leq_B' y$ (under the left (resp. right) $B$-twisted weak Bruhat order), one has that $x\leq_B y$ (under the left (resp. right) $B$-twisted strong Bruhat order).
The left (resp. right) $\emptyset-$twisted weak Bruhat order is the ordinary left (resp. right) weak Bruhat order. Following the convention in the literature, we consider the right twisted weak Bruhat orders instead of the left ones and often refer to right twisted weak Bruhat orders as twisted weak orders.
The (left) twisted (strong) Bruhat order (resp. (right) twisted weak (Bruhat) order) is graded by the left twisted length function (resp. right twisted length function). Suppose that $u\leq_B' v$ (under the right $B$-twisted weak order). It is shown in \cite{orderpaper} that one has a chain: $$u=u_0<_B'u_1<_B'u_2<_B'\cdots <_B' u_t=v$$
such that $u_{i}s=u_{i+1}$ for some $s\in S$ and $l_B'(u_i)+1=l_B'(u_{i+1})$. We will refer to this property as the chain property of the twisted weak order and write $u_i\vartriangleleft u_{i+1}$.
Let $B$ be an inversion set of an element in $W$. Then $(W,\leq_B')$ is isomorphic to $(W,\leq_{\emptyset}')$. It is known that $(W,\leq_{\emptyset}')$ is a complete meet semilattice (Chapter 3 of \cite{bjornerbrenti}). Let $B$ be the inversion set of an infinite reduced word.
It is shown in \cite{orderpaper} that $(W,\leq_B')$ is a non-complete meet semilattice
and for an affine Weyl group $W$, $(W,\leq_B')$ is a non-complete meet semilattice only if $B$ is the inversion set of an infinite reduced word.
\subsection{Construction of Affine Weyl Groups and Their Root Systems}
Let $W$ be an irreducible Weyl group with the crystallographic root system $\Phi$ contained in the Euclidean space $V$. For these notions, see Chapter 1 of \cite{Hum}. The root system of an (irreducible) affine Weyl group $\widetilde{W}$ (corresponding to $W$) can be constructed as the ``loop extension" of $\Phi$.
We describe such a construction. Let $\Phi^+$ be the chosen standard positive system of $\Phi$ and let $\Delta$ be the simple system of $\Phi^+.$
Define a $\mathbb{R}-$vector space $V'=V\oplus\mathbb{R}\delta$ where $\delta$ is an indeterminate. Extend
the inner product on $V$ to $V'$ by requiring $(\delta,v)=0$ for any $v\in V'$.
If
$\alpha\in \Phi^+$, define
$\{\alpha\}^{\wedge}=\{\alpha+n\delta|n\in \mathbb{Z}_{\geq 0}\}\subset
V'$. If $\alpha\in \Phi^-$, define
$\{\alpha\}^{\wedge}=\{\alpha+(n+1)\delta|n\in \mathbb{Z}_{\geq
0}\}\subset V'$.
For a set $\Lambda\subset \Phi$, define
$\Lambda^{\wedge}=\bigcup_{\alpha\in\Lambda}\{\alpha\}^{\wedge}\subset
V'$. The set of roots of the affine Weyl group $\widetilde{W}$, denoted by $\widetilde{\Phi}$, is $\Phi^{\wedge}\uplus-\Phi^{\wedge}$.
The set of positive roots and the set of negative roots are $\Phi^{\wedge}$ and $-\Phi^{\wedge}$ respectively. The set of simple roots is $\{\alpha|\alpha\in \Delta\}\cup\{\delta-\rho\}$ where $\rho$ is the highest root in $\Phi^+$. Let $\alpha\in \widetilde{\Phi}$ be a root. The reflection in $\alpha$, denoted by $s_{\alpha}$, is an $\mathbb{R}-$linear map $V'\rightarrow V'$ defined by
$v\mapsto v-2\frac{(v,\alpha)}{(\alpha,\alpha)}\alpha.$
The (irreducible) affine Weyl group $\widetilde{W}$ is generated by
$s_{\alpha},\alpha\in \widetilde{\Phi}$. The simple reflections are the reflections in the simple roots of $\Phi^{\wedge}$. For $v\in V$, define the $\mathbb{R}-$linear map $t_v$ which acts on $V'$ by $t_v(u)=u+(u,v)\delta.$ For $\alpha\in \Phi,$ define the coroot $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$. Note that $s_{\alpha+n\delta}=s_{\alpha}t_{-n\alpha^{\vee}}$. Let $T$ be the free Abelian group generated by $\{t_{\gamma^{\vee}}|\gamma\in \Delta\}$. Then it is known that $\widetilde{W}=W\ltimes T.$ Let $\pi$ be the canonical projection from $\widetilde{W}$ to $W$.
If $\alpha\in \Phi^+$, let $(\alpha)_0=\alpha\in \Phi^{\wedge}$. If $\alpha\in \Phi^-$, let $(\alpha)_0=\alpha+\delta\in \Phi^{\wedge}$. We also introduce the following compact notation: $\alpha_a^b:=\{\alpha+k\delta|a\leq k\leq b\}$ for $a,b\in \mathbb{Z}\cup \{\pm\infty\}.$ We also denote $\cup_{i=1}^t(\alpha_i)_{a_i}^{b_i}$ by $(\alpha_1)_{a_1}^{b_1}(\alpha_2)_{a_2}^{b_2}\cdots(\alpha_t)_{a_t}^{b_t}$.
\subsection{Biclosed Sets of an Affine Weyl Group}\label{biclosedaffine}
Let $\Phi$ be a finite irreducible crystallographic root system with $\Phi^+$ the standard positive system and $\Delta$ the simple system. Suppose that $\Delta'\subset \Phi$. Denote by $\Phi_{\Delta'}$ the root subsystem generated by $\Delta'$. It is shown in \cite{biclosedphi} that the biclosed sets in $\Phi$ are those $(\Psi^+\backslash \Phi_{\Delta_1})\cup \Phi_{\Delta_2}$ where $\Psi^+$ is a positive system of $\Phi$ and $\Delta_1,\Delta_2$ are two orthogonal subsets (i.e. $(\alpha,\beta)=0$ for any $\alpha\in \Delta_1,\beta\in \Delta_2$) of the simple system of $\Psi^+.$ For simplicity we denote the set $(\Psi^+\backslash \Phi_{\Delta_1})\cup \Phi_{\Delta_2}$ by $P(\Psi^+,\Delta_1,\Delta_2)$.
The biclosed sets in $\Phi^{\wedge}$ is determined in \cite{DyerReflOrder}.
Let $W'$ be the reflection subgroup of $\widetilde{W}$ generated by $(\Delta_1\cup\Delta_2)^{\wedge}$. Then any biclosed set in $\widetilde{\Phi}^+(=\Phi^{\wedge})$ is of the form $w\cdot P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$ for some $\Psi^+,\Delta_1,\Delta_2$ and $w\in \widetilde{W}.$
In particular a biclosed set $w\cdot P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$ where $w\in W'$ differs from $P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$ by finite many roots.
Suppose that $B$ is a biclosed set in $\widetilde{\Phi}^+=\Phi^{\wedge}$.
Let $I_B=\{\alpha\in \Phi|\,\{\alpha\}^{\wedge}\cap B$ is infinite$\}$ and
$A_B=\{\alpha\in \Phi|\,|\{\alpha\}^{\wedge}\cap B\neq \emptyset\}$. It is known that $I_B$ is biclosed in $\Phi$. Those biclosed sets $B$ such that $I_B=P(\Psi^+,\Delta_1,\Delta_2)$ are precisely the biclosed sets of the form $w\cdot P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$ where $w\in W'$. It is shown in \cite{wang} that if $B$ is an inversion set of an element of the affine Weyl group $\widetilde{W}$ or an infinite reduced word in $\widetilde{W}_l$, then $A_B\cap -A_B=\emptyset.$
The following theorem is proved in \cite{wang}
\begin{theorem}\label{infiniteword}
For an affine Weyl group, the biclosed sets which are inversion sets of infinite reduced words are precisely those of the form $w\cdot P(\Psi^+,\Delta_1,\emptyset)^{\wedge}, w\in \widetilde{W},\Delta_1\subsetneq \Delta$.
\end{theorem}
\section{Twisted Bruhat Orders on an Affine Weyl Group Are Locally Finite}\label{localfinite}
Throughout this paper unless otherwise specified, $W$ is a finite irreducible Weyl group and $\widetilde{W}$ is the corresponding irreducible affine Weyl group. Denote by $\Phi$ the crystallographic root system of $W$ and denote by $\Phi^+$ a chosen standard positive system. Let $\alpha\in \Phi$. A finite $\delta$-chain through $\alpha$ is a set $\{\alpha+d\delta|p\leq d\leq q\}$ where $p\leq q$ are two integers.
\begin{lemma}\label{dominance}
(1) Suppose that $\alpha\in \Phi$ and $k\in \mathbb{Z}_{\geq 0}$. Then in $\widetilde{\Phi}^+$, $\alpha_0+k\delta$ dominates the roots $\alpha_0,\alpha_0+\delta,\cdots,\alpha_0+(k-1)\delta.$
(2) Let $w\in \widetilde{W}$. Then $w$ carries a finite $\delta$-chain through $\alpha\in \Phi$ to another finite $\delta$-chain through some $\beta\in \Phi.$ In particular $\beta=\pi(w)(\alpha)$
\end{lemma}
\begin{proof}
(1) is well known. See for example Lemma 4.1 in \cite{wang}. (2) follows from the formula
\begin{equation}\label{action}
s_{\alpha+p\delta}(\beta+q\delta)=s_{\alpha}(\beta)+(q-(\beta,\alpha^{\vee})p)\delta
\end{equation}
and the fact $\pi(s_{\alpha+p\delta})=\pi(s_{\alpha}t_{-p\alpha^{\vee}})=s_{\alpha}$.
\end{proof}
\begin{lemma}\label{structure}
(1) $N(s_{\alpha+n\delta})$ has the following structure:
(i) Suppose that $\beta\in \Phi^+, s_{\alpha}(\beta)\in \Phi^+$. Then for $0\leq
k<\frac{2(\alpha,\beta)}{(\alpha,\alpha)}n$, $\beta+k\delta\in
N(s_{\alpha+n\delta}).$
(ii) Suppose that $\beta\in \Phi^+, s_{\alpha}(\beta)\in \Phi^-$. Then for
$0\leq k\leq\frac{2(\alpha,\beta)}{(\alpha,\alpha)}n$,
$\beta+k\delta\in N(s_{\alpha+n\delta}).$
(iii) Suppose that $\beta\in \Phi^-, s_{\alpha}(\beta)\in \Phi^+$. Then for
$0<k<\frac{2(\alpha,\beta)}{(\alpha,\alpha)}n$, $\beta+k\delta\in
N(s_{\alpha+n\delta}).$
(iv) Suppose that $\beta\in \Phi^-, s_{\alpha}(\beta)\in \Phi^-$. Then for
$0<k\leq\frac{2(\alpha,\beta)}{(\alpha,\alpha)}n$, $\beta+k\delta\in
N(s_{\alpha+n\delta}).$
(2) If there exists a $\delta$-chain through $\beta$ beginning from $\beta_0$ in $N(s_{\alpha+n\delta})$ then there also exists a $\delta$-chain through $-s_{\alpha}(\beta)$ beginning from $(-s_{\alpha}(\beta))_0$ in $N(s_{\alpha+n\delta})$. In addition, these two chains are of equal length.
\end{lemma}
\begin{proof}
(1) follows from Equation \eqref{action}. To see (2), one notes that $(-s_{\alpha}(\beta),\alpha)=(s_{\alpha}(\beta),-\alpha)=(s_{\alpha}(\beta),s_{\alpha}(\alpha))=(\beta,\alpha)$ and then uses (1).
\end{proof}
\begin{lemma}
When $n$ is sufficiently large,
the inversion set $N(ws_{\alpha+n\delta})$ consists of some finite $\delta$-chains (through roots in $\Phi$) whose lengths are independent of $n$ and the following $\delta$-chains:
$$(\pi(w)(\alpha))_0, (\pi(w)(\alpha))_0+\delta,\cdots, (\pi(w)(\alpha))_0+k_0\delta,$$
$$(\beta_1)_0, (\beta_1)_0+\delta,\cdots, (\beta_1)_0+k_1\delta,$$
$$(\beta_2)_0, (\beta_2)_0+\delta,\cdots, (\beta_2)_0+k_2\delta,$$
$$\cdots$$
$$(\beta_t)_0, (\beta_t)_0+\delta,\cdots, (\beta_t)_0+k_t\delta$$
where $\beta_i\in \Phi, 1\leq i\leq t$ and one can pair each $\beta_i$ with some $\beta_j, j\neq i$ such that $p\beta_i+p\beta_j=\pi(w)(\alpha), p>0$. Furthermore when $n$ is changed to $n+1$, the increment of the length of the $\delta$-chain through $\beta_i$ and that of the $\delta$-chain through $\beta_j$ coincides.
\end{lemma}
\begin{proof}
By Lemma 4.1(e) of \cite{DyerWeakOrder} $$N(ws_{\alpha+n\delta})=(N(w)\backslash (-wN(s_{\alpha+n\delta})))\uplus (wN(s_{\alpha+n\delta})\backslash (-N(w))).$$
By Lemma \ref{dominance} (1) $N(w)$ consists some finite $\delta$-chains through certain roots in $\Phi$. Therefore we conclude that $(N(w)\backslash (-wN(s_{\alpha+n\delta})))$ consists of some finite $\delta$-chains (through roots in $\Phi$) whose lengths are independent of $n$ (when $n$ is sufficiently large).
By Lemma \ref{structure} (1) and Lemma \ref{dominance} (2), one sees that $wN(s_{\alpha+n\delta})\cap \widetilde{\Phi}^+=wN(s_{\alpha+n\delta})\backslash (-N(w))$ consists of $\delta$-chains through certain roots in $\Phi$ and that when $n$ grows the upper bound of the coefficient of $\delta$ in each of these chains grows accordingly.
Again Lemma \ref{dominance} (1) forces that any $\delta$-chain through a root $\beta$ must start from $\beta_0$ in $N(ws_{\alpha+n\delta})$.
Therefore we have shown that $N(ws_{\alpha+n\delta})$ consists of roots in the form as listed in the lemma.
We still need to show the existence of the pairing described in the Lemma.
Now suppose $\beta_i=\pi(w)(\beta)$. Then the root $\beta_j=\pi(w)(-s_{\alpha}(\beta))$ satisfies the condition in this lemma thanks to Lemma \ref{structure} (2). Note that the positivity of $p$ follows from the fact $(\beta,\alpha)>0$ (note that $(\beta,\alpha)>0$ because otherwise the $\delta$-chain through $\beta$ will not appear in the inversion set of $s_{\alpha+n\delta}$ by Lemma \ref{structure} (1)).
\end{proof}
\begin{lemma}\label{limit}
$$\lim_{n\rightarrow \infty} |l_B(s_{\alpha+n\delta}w)|=+\infty $$
for any $B$ biclosed in $\widetilde{\Phi}^+$, $w\in \widetilde{W}, \alpha\in \Phi.$
\end{lemma}
\begin{proof}
We examine the set $N(ws_{\alpha+n\delta})$ for sufficiently large $n$.
By Lemma \ref{structure} besides some finite $\delta$-chain which will stay stationary when $n$ grows, this set consists some finite $\delta$-chain through roots in $\Phi$ which will grow when $n$ grows, i.e.
$$(\pi(w)(\alpha))_0, (\pi(w)(\alpha))_0+\delta,\cdots, (\pi(w)(\alpha))_0+k_0\delta,$$
$$(\beta_1)_0, (\beta_1)_0+\delta,\cdots, (\beta_1)_0+k_1\delta,$$
$$(\beta_2)_0, (\beta_2)_0+\delta,\cdots, (\beta_2)_0+k_2\delta,$$
$$\cdots$$
$$(\beta_t)_0, (\beta_t)_0+\delta,\cdots, (\beta_t)_0+k_t\delta$$
Also one can pair those $\beta_i$'s, for each $\beta_i$ one can find $\beta_j, j\neq i$ such that $p\beta_i+p\beta_j=\pi(w)(\alpha), p>0$.
Furthermore when $n$ is changed to $n+1$ the increment of the length of the $\delta$-chain through $\beta_i$ and that of the $\delta$-chain through $\beta_j$ coincides.
Now we consider the set $I_B=\{\alpha\in \Phi|\{\alpha\}^{\wedge}\cap B$ is infinite$\}$. Then $I_B$ is a biclosed set in $\Phi.$
Also by Lemma 5.10 in \cite{DyerReflOrder} for any $\alpha\in I_B$, there exists $N_{\alpha}\in \mathbb{Z}_{>0}$ such that $\alpha+n\delta\in B$ for all $n>N_{\alpha}$.
Suppose that $\pi(w)(\alpha)\in I_B$. Then for a pair of $\beta_i,\beta_j$ either one of them is in $I_B$ and one of them is in $\Phi\backslash I_B$ or both of them are in $I_B$ since otherwise $p\beta_i+p\beta_j=\pi(w)(\alpha)$ has to be in $\Phi\backslash I_B$ by coclosedness of $I_B.$
So when $n$ is changed to $n+1$ more added roots in the inversion set $N(ws_{\alpha+n\delta})$ are in $B$ than in $\widehat{\Phi}\backslash B$, making the twisted length decreasing.
Similarly if $\pi(w)(\alpha)\not\in I_B$ one can deduce that when $n$ is changed to $n+1$ more added roots in the inversion set $N(ws_{\alpha+n\delta})$ are in $\widehat{\Phi}\backslash B$ than in $B$, making the length increasing. Therefore we see that the twisted length tends to infinity as $n\rightarrow \infty.$
\end{proof}
\begin{proposition}\label{finiteunder}
For any $x\in \widetilde{W}$, $n\in \mathbb{N}$ and a biclosed set $B$ in $\widetilde{\Phi}^+$, the set
$$\{y\in \widetilde{W}|y\leq_B x, l_B(x)-l_B(y)=n\}$$
is finite.
\end{proposition}
\begin{proof}
We prove this by induction. Let $n=1$ and $\alpha\in \Phi$. The set $\{k|l_B(s_{\alpha+k\delta}x)=l_B(x)-1\}$ is finite by Lemma \ref{limit}. Note that $\Phi$ is finite and therefore the proposition holds when $n=1.$ Now let $n=k$. Then
$$\{y\in \widetilde{W}|y\leq_B x, l_B(x)-l_B(y)=k\}=$$$$\bigcup_{\alpha,k:l_B(s_{\alpha+k\delta}x)=l_B(x)-1}\{y\in \widetilde{W}|y\leq_B s_{\alpha+k\delta}x, l_B(s_{\alpha+k\delta}x)-l_B(y)=k-1\}$$
since twisted Bruhat order is graded by twisted length function.
Now the proposition follows from induction.
\end{proof}
\begin{theorem}
For any biclosed set $B$ in $\widetilde{\Phi}^+$ and any $x,y\in \widetilde{W}$ such that $x<_B y$, the interval $[x,y]$ in the poset $(W,\leq_B)$ is finite.
\end{theorem}
\begin{proof}
If $l_B(y)-l_B(x)=1$ then the assertion is trivial (as $[x,y]=\{x,y\}$). Now suppose that the assertion holds for all pairs of $x,y$ with the properties $x<_B y$ and $l_B(y)-l_B(x)\leq k-1$. Take $x,y\in \widetilde{W}, x<_B y$ and $l_B(y)-l_B(x)=k$.
By Proposition \ref{finiteunder}, the set $\{z|x<_B z<_B y, l_B(y)-l_B(z)=1\}$ is finite. For any $z$ such that $x<_B z<_B y, l_B(y)-l_B(z)=1$ the interval $[x,z]$ is finite by induction assumption. Then $[x,y]$ is also finite since the twisted Bruhat order is graded by the left twisted length function.
\end{proof}
\section{Construction of infinite interval in a non-affine Coxeter system}\label{hyperinterval}
It is natural to ask that given a non-affine infinite Coxeter group $W$ whether infinite intervals always show up in $(W,\leq_B)$ for some biclosed set $B\subset \Phi^+.$ In \cite{DyerTwistedBruhat} subsection 1.10, it is shown that this is true for a universal Coxeter group of rank $>2$ as follow.
Let $(W,S)$ be a universal Coxeter system of rank $3$ with $S=\{s_1,s_2,s_3\}$. Let $B=N((s_1s_2s_3)^{\infty})$. Then the closed interval $[1,s_1s_2s_3]$ is infinite in $(W,\leq_B)$.
By \cite{UniversalRefl}, every irreducible infinite non-affine Coxeter group $W$ has a reflection subgroup $W'$ which is universal of rank $3$. Therefore by above for certain biclosed set $B\subset \Phi_{W'}^+$ (the set of positive roots of $W'$) the left $B-$twisted strong Bruhat order on such a reflection subgroup (denoted $\leq_{W',B}$) has an infinite interval (with bottom element $e$) which we denote by $[e,w]$.
Suppose that there exists a biclosed set $A\subset \Phi^+$ such that $A\cap \Phi_{W'}^+=B$. Then by Proposition 1.4 in \cite{DyerTwistedBruhat} (or see subsection 1.12 of \cite{quotient}) we have a poset injection $([e,w],\leq_{W',B})\rightarrow ([e,w],\leq_A)$.
Therefore the closed interval $[e,w]$ is also infinite in $(W,\leq_A)$.
Now we illustrate this process through a concrete example.
Let $(W,S)$ be of $(2,3,\infty)$ type, i.e. $W=\{s_1, s_2, s_3|s_1^2=s_2^2=s_3^2=(s_1s_2)^3=(s_1s_3)^2=e\}$ and $S=\{s_1, s_2, s_3\}.$ Denote $\alpha_{s_i}$ by $\alpha_i$. For a reflection subgroup $W'$ of $W$, denote $T\cap W'$ by $T_{W'}$. The root system of $W'$ is $\Phi_{W'}=\{\alpha\in \Phi|s_{\alpha}\in W'\}.$
Consider the reflection subgroup $W'$ generated by $S':=\{s_1, s_2s_3s_2, s_3s_2s_3s_2s_3\}$. Write $r_1=s_1, r_2=s_2s_3s_2, r_3=s_3s_2s_3s_2s_3.$ Note that $(\alpha_{r_i},\alpha_{r_j})=-1$ for $i,j\in \{1,2,3\},i\neq j.$
Also we note that $$\widetilde{N}(s_1)\backslash \{s_1\}=\emptyset,$$ $$\widetilde{N}(s_2s_3s_2)\backslash \{s_2s_3s_2\}=\{s_2,s_2s_3s_2s_3s_2\},$$ $$\widetilde{N}(s_3s_2s_3s_2s_3)\backslash\{s_3s_2s_3s_2s_3\}=\{s_3,s_3s_2s_3, s_3s_2s_3s_2s_3s_2s_3, s_3s_2s_3s_2s_3s_2s_3s_2s_3\}.$$
One easily checks that $r_i$ and $r_j, i\neq j, i,j\in \{1,2,3\}$ generate an infinite dihedral subgroup with $r_i$ and $r_j$ being the canonical simple generators.
Then apply Proposition 3.5 of \cite{DyerReflSubgrp} we see that $\widetilde{N}(s_1)\cap T_{W'}=\{s_1\}, \widetilde{N}(s_2s_3s_2)\cap T_{W'}=\{s_2s_3s_2\},$ $\widetilde{N}(s_3s_2s_3s_2s_3)\cap T_{W'}=\{s_3s_2s_3s_2s_3\}.$
Therefore $(W',S')$ is indeed a rank 3 universal Coxeter system.
Since the only braid moves that can be applied to $(s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3))^k$ are short braid moves between $s_1$ and $s_3$, the element $s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3)$ is straight. One can also check that $\widetilde{N}(s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3))\cap T_{W'}=\{r_1, r_1r_2r_1, r_1r_2r_3r_2r_1\}$. Consequently $\widetilde{N}((s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3))^{\infty})\cap T_{W'}=\newline$
$\{r_1,r_1r_2r_1,r_1r_2r_3r_2r_1, r_1r_2r_3r_1r_3r_2r_1,r_1r_2r_3r_1r_2r_1r_3r_2r_1\cdots\}$. Based on the above discussion, one concludes that the interval $[1, s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3)]$ in $(W,\leq_A)$ is infinite where $A=N((s_1(s_2s_3s_2)(s_3s_2s_3s_2s_3))^{\infty})$.
\section{Set of elements of a given twisted length}\label{fixlength}
In this section we consider the set $\{w\in \widetilde{W}|l_B'(w)=n\}$. In contrast to the situation for the ordinary Bruhat order, we will show that such a set is infinite in most cases. Since $l_B(w^{-1})=l_B'(w)$, our result also shows that the set $\{w\in \widetilde{W}|\,l_B(w)=n\}$ is infinite.
It is convenient to investigate this question in the context of (right) $B$-twisted weak Bruhat order $<_B'$. In the following two lemmas $(W,S)$ is a general finite rank infinite Coxeter system and $\Phi,\Phi^+$ are its root system and its set of positive roots.
\begin{lemma}\label{nomaxminone}
Let $w\in W_l$ (the set of infinite reduced words), then $(W,\leq_{N(w)}')$ (resp. $(W,\leq_{\Phi^+\backslash N(w)}')$) has no maximal element or minimal element.
\end{lemma}
\begin{proof}
Suppose to the contrary that $u$ is a maximal element of $(W,\leq_{N(w)}')$ (i.e. for any element $p\in W$ either $u$ and $p$ are not comparable or $u\geq'_{N(w)} p$). Let $S=\{s_1,s_2,\cdots,s_n\}$ be the set of simple reflections. Then $us_i\lhd u$ for all $s_i\in S$. Let $U_i=N(us_i)\backslash N(u)$. These sets are contained in $N(w)$ by definition. Therefore there exists a (finite) prefix $v$ of $w$ such that $U_i\subset N(v)$ for all $i$. Also note that $N(u)\backslash N(us_i)\subset \Phi^+\backslash N(w)\subset \Phi^+\backslash N(v)$. Hence under the twisted weak order $(W,\leq_{N(v)}')$, one has $us_i\lhd u$ for all $i$. Therefore $u$ is a maximal element in $(W,\leq_{N(v)}')$ since if there exists some $u'\geq u$ then by the chain property of the twisted weak order there must be some $s_i$ such that $us_i\vartriangleright u.$ But $\leq_{N(v)}'$ is isomorphic to the standard right weak order which has no maximal element when $W$ is infinite. A contradiction. Therefore we have shown that $(W,\leq_{N(w)}')$ has no maximal elements.
Now we claim that it cannot happen that there exists $u\in W$ such that $u\leq_{N(w)}' x, \forall x\in W$. Since $N(u)$ is finite, there exists $\alpha\in \Phi^+$ such that $\alpha\in N(w)\backslash N(u)$. Then $N(s_{\alpha})\backslash N(u)\not\subset (\Phi^+\backslash N(w))$. So $u\nleq_{N(w)}' s_{\alpha}$ and we establish the claim. Now for any $u\in W$ we take $x\in W$ such that $u\not\leq_{N(w)}' x$, and then consider the meet of $u$ and $x$ (it exists since $(W,\leq_{N(w)}')$ is a meet semilattice). Such a meet is clearly not equal to $u$ as $u\not\leq_{N(w)}' x$. Therefore $(W,\leq_{N(w)}')$ has no minimal element.
The dual assertion about $(W,\leq_{\Phi^+\backslash N(w)}')$ can be proved in the same fashion and thus is omitted.
\end{proof}
\begin{lemma}\label{dotactioniso}
Let $B$ be a biclosed set in $\Phi^+$ and $w\in W$. Then as posets $(W,\leq_B')\simeq (W,\leq_{w\cdot B}')$ under the map $u\mapsto wu.$
\end{lemma}
\begin{proof}
Suppose that $u\leq_B'v$. We show that $N(wu)\backslash N(wv)$ is contained in $w\cdot B=(N(w)\backslash -w(B))\uplus (w(B)\backslash -N(w))$.
Note that $N(wu)\backslash N(wv)=((N(w)\backslash -wN(u))\backslash (N(w)\backslash -wN(v)))\uplus ((wN(u)\backslash -N(w)\backslash (w(N(v))\backslash -N(w)))).$
Take $\alpha\in (N(w)\backslash -wN(u))\backslash (N(w)\backslash -wN(v))$. This implies that $\alpha\in N(w)$ and $-w^{-1}(\alpha)\in N(v)\backslash N(u)$. Since $N(v)\subset N(u)\subset \Phi^+\backslash B$, $\alpha\not\in -w(B).$
So $\alpha\in N(w)\backslash -w(B).$
Take $\alpha\in (wN(u)\backslash -N(w)\backslash (w(N(v))\backslash -N(w)))$. This implies that $\alpha\not\in -N(w)$ and $\alpha\in w(N(u)\backslash N(v))$. Therefore $\alpha\in w(B)\backslash -N(w).$
Therefore we see that $N(wu)\backslash N(wv)\subset w\cdot B.$
Next we show that $N(wv)\backslash N(wu)\subset \Phi^+\backslash w\cdot B$. Since $v\leq_{\Phi^+\backslash B}'u$, the above argument shows that
$N(wv)\backslash N(wu)\subset w\cdot (\Phi^+\backslash B)$. It follows from the formula $w\cdot B:=(N(w)\backslash w(-B))\cup (w(B)\backslash
(-N(w)))$ that $w\cdot (\Phi^+\backslash B)=\Phi^+\backslash w\cdot B.$
Therefore we conclude that $wu\leq_{w\cdot B}' wv$.
The above arguments also prove that the inverse map $u\mapsto w^{-1}u$ preserves the order since $w^{-1}\cdot(w\cdot B)=B$ and hence the lemma is proved.
\end{proof}
Now let $\Phi,\Psi^+$ and $\Delta$ be a finite crystallographic root system, a positive system of it and the corresponding simple system respectively. Suppose that $\Delta_1\subset \Delta$. Let $\Phi_{\Delta_1}=\mathbb{R}\Delta_1\cap \Phi$ be the root subsystem generated by $\Delta_1$. Then $\mathbb{R}_{\geq 0}\Delta_1\cap \Phi=\Phi_{\Delta_1}^+$ is a positive system of $\Phi_{\Delta_1}$.
Let $W$ be the finite Weyl group with the root system $\Phi$ and $\widetilde{W}$ be the corresponding affine Weyl group. A subset $\Gamma$ of $\Phi$ is said to be $\mathbb{Z}$-closed if for $\alpha,\beta\in \Gamma$ such that $\alpha+\beta\in \Phi$ one has $\alpha+\beta\in \Gamma.$
\begin{lemma}\label{assemble}
Let $\Gamma^+$ be another positive system of $\Phi_{\Delta_1}$. Then $(\Psi^+\backslash \Phi_{\Delta_1}^+)\cup \Gamma^+$ is a positive system of $\Phi$. If $\Delta_2\subset \Delta$ such that $(\Delta_1,\Delta_2)=0$, $\Delta_2$ is also a subset of the simple system of $(\Psi^+\backslash \Phi_{\Delta_1}^+)\cup \Gamma^+$.
\end{lemma}
\begin{proof}
Find the element $w$ in the parabolic subgroup generated by $s_{\alpha},\alpha\in \Delta_1$ such that $w\Phi_{\Delta_1}^+=\Gamma^+$. Act $w$ on $\Psi^+$ and we get $(\Psi^+\backslash \Phi_{\Delta_1}^+)\cup \Gamma^+$. Note that we have $w(\Delta_2)=\Delta_2.$
\end{proof}
Let $B$ be a biclosed set in $\widetilde{\Phi}^+$.
\begin{lemma}\label{nomaxmintwo}
If $B$ and $\widetilde{\Phi}^+\backslash B$ are both infinite, then $(\widetilde{W},\leq_B')$ has no maximal element or minimal element.
\end{lemma}
\begin{proof}
By the characterization of the biclosed sets in $\widetilde{\Phi}$, $I_B=P(\Psi^+,\Delta_1,\Delta_2)$ for some positive system $\Psi^+$ and some orthogonal subsets $\Delta_1, \Delta_2$ of its simple system.
By Lemma \ref{nomaxminone} and Theorem \ref{infiniteword}, it suffices to treat the case where $I_B=P(\Psi^+,\Delta_1,\Delta_2)$ where $\Delta_1$ and $\Delta_2$ are both nonempty. Further it suffices to check the case where $B=P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$ as all $(W,\leq_B')$ with $I_B=P(\Psi^+,\Delta_1,\Delta_2)$ are isomorphic by Lemma \ref{dotactioniso}. We suppose to the contrary that there exists $u\in W$ such that $us_i\rhd u$ for all $s_i$ simple reflection of $\widetilde{W}$ (so $u$ is minimal by the chain property of the twisted weak order).
We write $A$ for $A_{N(u)}$. It is $\mathbb{Z}-$closed in $\Phi$ by Lemma 3.13 in \cite{wang}. We also have $A\cap -A=\emptyset$ by subsection \ref{biclosedaffine}. Then we deduce that $A':=\Phi_{\Delta_2}\cap A$ is $\mathbb{Z}$-closed in $\Phi_{\Delta_2}$
and that $A'\cap -A'=\emptyset$. Then by \cite{Bourbaki} Chapter IV, $\S 1$, Proposition 22, $A'$ is contained in a positive system $\Gamma^+$ in $\Phi_{\Delta_2}.$ Let $\mathbb{R}_{\geq 0}\Delta_2\cap \Phi=\Phi_{\Delta_2}^+.$
Then $A'$ is contained in $(\Psi^+\backslash \Phi_{\Delta_2}^+\cup \Gamma^+)\backslash \Phi_{\Delta_1}$ where $\Psi^+\backslash \Phi_{\Delta_2}^+\cup \Gamma^+$ is a positive system in $\Phi$ by Lemma \ref{assemble}. We denote this positive system by $\Xi^+$. Also by Lemma \ref{assemble} $\Delta_1$ is a subset of $\Xi^+$'s simple system.
Since $A_{N(u)\backslash N(us_i)}\cap \Phi_{\Delta_2}\subset A_{N(u)}\cap \Phi_{\Delta_2}(=A')$, so
\begin{equation}\label{partone}
A_{N(u)\backslash N(us_i)}\cap \Phi_{\Delta_2}\subset \Xi^+\backslash \Phi_{\Delta_1}.
\end{equation}
Because $N(u)\backslash N(us_i)\subset B$, $A_{N(u)\backslash N(us_i)}\subset P(\Psi^+,\Delta_1,\Delta_2)=(\Psi^+\backslash \Phi_{\Delta_1})\cup \Phi_{\Delta_2}$. Hence
\begin{equation}\label{parttwo}
A_{N(u)\backslash N(us_i)}\backslash \Phi_{\Delta_2}\subset (\Psi^+\backslash \Phi_{\Delta_1})\backslash \Phi_{\Delta_2}\subset (\Psi^+\backslash \Phi_{\Delta_2}^+\cup \Gamma^+)\backslash \Phi_{\Delta_1}=\Xi^+\backslash \Phi_{\Delta_1}.
\end{equation}
Hence by equation \ref{partone} and \ref{parttwo} we conclude that $A_{N(u)\backslash N(us_i)}\subset \Xi^+\backslash \Phi_{\Delta_1}$. Consequently $N(u)\backslash N(us_i)\subset (\Xi^+\backslash \Phi_{\Delta_1})^{\wedge}$ for all $i$. The right hand side is the inversion set of an infinite reduced word $w$ by Theorem \ref{infiniteword}.
On the other hand, one has the inclusion $N(us_i)\backslash N(u)\subset \widetilde{\Phi}^+\backslash P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}\subset \widetilde{\Phi}^+\backslash (\Xi^+\backslash \Phi_{\Delta_1})^{\wedge}$. Thus we conclude $u$ is minimal under $(W,\leq_{N(w)}')$. This contradicts Lemma \ref{nomaxminone}.
Similarly one can also prove that the poset has no maximal element.
\end{proof}
\begin{theorem}\label{setisinfinite}
Suppose that $\widetilde{W}$ is irreducible and of rank $\geq 3.$ Let $k\in \mathbb{Z}$ and let $B$ be an infinite biclosed set in $\widetilde{\Phi}^+$ such that $\widetilde{\Phi}^+\backslash B$ is also infinite. The set $\{w\in \widetilde{W}|\,l_B'(w)=k\}$ is infinite.
\end{theorem}
\begin{proof}
Suppose that to the contrary there exists $k\in \mathbb{Z}$, the set $\{w\in \widetilde{W}|\,l_B'(w)=k\}$ is finite. Suppose that this set is $\{w_1,w_2,\cdots,w_t\}$. We show that under the right $B$-twisted weak order $\leq_B'$ any element is either greater than one of $w_i$ or less than one of $w_i$ (depending whether its twisted length is $\geq$ or $\leq k$).
Let $u$ be an element in $W$ and assume without loss of generality that $l_B'(u)>k.$ By Lemma \ref{nomaxmintwo} $u$ is not minimal. So one can find $s\in S$ such that $us<_B'u$ and $l_B'(us)=l_B'(u)-1$. Proceed this way one sees that $u$ must be greater than some element with twisted length $k$.
Now by Lemma \ref{dotactioniso} we only need to consider the case where $B=P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}$. If $\Delta_1=\Delta$, $B$ is empty. Let $\alpha\in \Delta\backslash \Delta_1$ and $\rho$ be the highest root in $\Psi^+$. Since $\cup_{i}N(w_i)$ is finite, there must exist some positive roots $\alpha+p\delta\not\in \cup_{i}N(w_i)$ and $-\rho+q
\delta\not\in \cup_{i}N(w_i)$. We claim that there exists some $v$ such that $N(v)$ contains $-\rho+q\delta, \alpha+p\delta.$
Proof of the claim. Since the rank of $\Phi$ is at least two, $\alpha\neq \rho$. So $-\rho,\alpha\in \Omega^+:=\Psi^-\backslash \{-\alpha\}\cup \{\alpha\}$, which is another positive system. So $-\rho+q\delta, \alpha+p\delta$ are contained in $(\Omega^+)^{\wedge}$, which is the inversion set of an infinite reduced word. Consequently they are contained in the inversion set of a finite prefix $v$ of this infinite reduced word.
Note that $\Delta_2\neq \Delta$ since that will cause $B=\widetilde{\Phi}^+=\Phi^{\wedge}$ and then $B$ is cofinite. Therefore $-\rho+q\delta\not\in B$.
However $\alpha+p\delta\in B$.
Therefore $v$ is not comparable to any of $w_i$ under the right $B$-twisted weak order. Contradiction.
\end{proof}
\begin{corollary}
Suppose that $\widetilde{W}$ is irreducible and of rank $\geq 3.$ Let $B$ be an infinite biclosed set in $\widetilde{\Phi}^+$ such that $\widetilde{\Phi}^+\backslash B$ is also infinite. Then the $B$-twisted weak Bruhat order $(\widetilde{W},\leq_B')$ (resp. the $B$-twisted strong Bruhat order $(\widetilde{W},\leq_B)$) has an infinite antichain.
\end{corollary}
\begin{proof}
Note that if $u\lneq'_B v$ (resp. $u\lneq_B v$) then $l_B'(v)>l_B'(u)$ (resp. $l_B(v)>l_B(u)$). Therefore the set $\{w\in \widetilde{W}|\,l_B'(w)=k\}$ (resp. $\{w\in \widetilde{W}|\,l_B(w)=k\}$) forms an infinite antichain by Theorem \ref{setisinfinite}.
\end{proof}
We remark that in \cite{antichain} it is proved that if $B$ is finite or cofinite, the $B$-twisted weak order (and consequently the $B$-twisted strong Bruhat order) on an affine Weyl group does not have an infinite antichain.
\section{An example: alcove order of $\widetilde{A}_2$}\label{examplesec}
In this section we provide a clear picture of a specific twisted (strong) Bruhat order.
Let $W$ and $\Phi$ be of type $A_2$. The two simple roots are denoted by $\alpha, \beta$.
We consider the $B$-twisted (strong) Bruhat order $(\widetilde{W}, \leq_B)$ where $B=\{\alpha,\alpha+\beta,\beta\}^{\wedge}$. Such an order is also called alcove order in \cite{quotient}. We describe all of its spherical intervals.
In the following lemma we characterize all covering relations in $(\widetilde{W}, \leq_B)$.
\begin{lemma}\label{sixtype}
For $(\widetilde{W},\leq_B)$ one has the following results:
(1) Suppose that $w\in T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)-2k-1, l_B(s_{\beta+k\delta}w)=l_B(w)-2k-1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)-4k-3.$
(2) Suppose that $w\in s_{\alpha}s_{\beta}T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)+4k+1, l_B(s_{\beta+k\delta}w)=l_B(w)-2k-1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)+2k+1.$
(3) Suppose that $w\in s_{\beta}s_{\alpha}T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)-2k-1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)+2k+1, l_B(s_{\beta+k\delta}w)=l_B(w)+4k+1.$
(4) Suppose that $w\in s_{\beta}s_{\alpha}s_{\beta}T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)+2k+1, l_B(s_{\beta+k\delta}w)=l_B(w)+2k+1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)+4k+3.$
(5) Suppose that $w\in s_{\beta}T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)-4k-1, l_B(s_{\beta+k\delta}w)=l_B(w)+2k+1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)-2k-1.$
(6) Suppose that $w\in s_{\alpha}T$.
Then $l_B(s_{\alpha+k\delta}w)=l_B(w)+2k+1, l_B(s_{\alpha+\beta+k\delta}w)=l_B(w)-2k-1, l_B(s_{\beta+k\delta}w)=l_B(w)-4k-1.$
\end{lemma}
\begin{proof}
It follows from the fact $\widetilde{W}=W\ltimes T$ that every element is in one of the six subsets and these six subsets are pairwise disjoint.
One can verify by induction easily that
$$N(s_{\alpha+k\delta})=\alpha_0^{2k}(-\beta)_1^k(\alpha+\beta)_0^{k-1}, k\geq 0,$$
$$N(s_{\alpha+k\delta})=(-\alpha)_1^{-2k-1}(\beta)_0^{-k-1}(-\alpha-\beta)_1^{-k}, k<0,$$
$$N(s_{\alpha+\beta+k\delta})=(\alpha+\beta)_0^{2k}\alpha_0^k\beta_0^k, k\geq 0,$$
$$N(s_{\alpha+\beta+k\delta})=(-\alpha-\beta)_1^{-2k-1}(-\alpha)_1^{-k-1}(-\beta)_1^{-k-1}, k<0,$$
$$N(s_{\beta+k\delta})=\beta_0^{2k}(-\alpha)_1^k(\alpha+\beta)_0^{k-1}, k\geq 0,$$
$$N(s_{\beta+k\delta})=(-\beta)_1^{-2k-1}(\alpha)_0^{-k-1}(-\alpha-\beta)_1^{-k}, k<0.$$
Now by using the identity $N(wu)=(N(w)\backslash w(-N(u)))\cup (wN(u)\backslash
(-N(w)))$, we compute
$$N(t_{k_1\alpha+k_2\beta})$$
$$=(\alpha_0^{k_1-\frac{k_2}{2}-1}(-\alpha)_1^{-k_1+\frac{k_2}{2}}\beta_0^{-\frac{k_1}{2}+k_2-1}(-\beta)_1^{\frac{k_1}{2}-k_2}
(\alpha+\beta)_0^{\frac{k_1}{2}+\frac{k_2}{2}-1}(-\alpha-\beta)_1^{-\frac{k_1}{2}-\frac{k_2}{2}}).$$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha+k\delta})$$
$$=\alpha_{0}^{k_1-\frac{k_2}{2}+2k}(-\alpha)_1^{-2k-1-k_1+\frac{k_2}{2}}\beta_0^{-k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{k-1+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{-k-\frac{k_1+k_2}{2}}.$$
$l_B(s_{\alpha+k\delta}t_{k_1\alpha+k_2\beta})=-2k-1-k_1+\frac{k_2}{2}+k+\frac{k_1}{2}-k_2-k-\frac{k_1+k_2}{2}=-2k-k_1-k_2-1.$
$l_B(t_{k_1\alpha+k_2\beta})=-k_1+\frac{k_2}{2}+\frac{k_1}{2}-k_2-\frac{k_1}{2}-\frac{k_2}{2}=-k_1-k_2$.
Hence $l_B(t_{k_1\alpha+k_2\beta})-2k-1=l_B(s_{\alpha+k\delta}t_{k_1\alpha+k_2\beta})$.
The map $\tau: s_{\alpha}\mapsto s_{\beta}, s_{\beta}\mapsto s_{\alpha}, s_{\delta-\alpha-\beta}\mapsto s_{\delta-\alpha-\beta}$ induces an automorphism of $\widetilde{W}$.
From this and the above calculation, one sees that
$l_B(t_{k_1\alpha+k_2\beta})-2k-1=l_B(t_{k_1\alpha+k_2\beta}s_{\beta+k\delta})$.
We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha+\beta+k\delta})$$
$$=\alpha_0^{k+k_1-\frac{k_2}{2}}(-\alpha)_1^{-k-1-k_1+\frac{k_2}{2}}\beta_0^{k-\frac{k_1}{2}+k_2}(-\beta)_1^{-k-1+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{2k+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{-2k-1-\frac{k_1+k_2}{2}}.$$
$l_B(s_{\alpha+\beta+k\delta}t_{k_1\alpha+k_2\beta})=-k-1-k_1+\frac{k_2}{2}-k-1+\frac{k_1}{2}-k_2-2k-1-\frac{k_1+k_2}{2}=-4k-k_1-k_2-3.$
Hence $l_B(s_{\alpha+\beta+k\delta}t_{k_1\alpha+k_2\beta})=l_B(t_{k_1\alpha+k_2\beta})-4k-3.$ So we have proved (1).
We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha})$$
$$=(\alpha_0^{k_1-\frac{k_2}{2}}(-\alpha)_1^{-k_1+\frac{k_2}{2}-1}\beta_0^{-\frac{k_1}{2}+k_2-1}(-\beta)_1^{\frac{k_1}{2}-k_2}
(\alpha+\beta)_0^{\frac{k_1}{2}+\frac{k_2}{2}-1}(-\alpha-\beta)_1^{-\frac{k_1}{2}-\frac{k_2}{2}}).$$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\alpha+k\delta})$$
$$=\alpha_0^{-2k-1+k_1-\frac{k_2}{2}}(-\alpha)_1^{2k-k_1+\frac{k_2}{2}}\beta_0^{k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{-k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{-k-1+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{k-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+k\delta}s_{\alpha}t_{k_1\alpha+k_2\beta})=2k-k_1+\frac{k_2}{2}-k+\frac{k_1}{2}-k_2+k-\frac{k_1+k_2}{2}=2k-k_1-k_2$.
$l_B(s_{\alpha}t_{k_1\alpha+k_2\beta})=-k_1+\frac{k_2}{2}-1+\frac{k_1}{2}-k_2-\frac{k_1}{2}-\frac{k_2}{2}=-k_1-k_2-1$.
Therefore
$l_B(s_{\alpha+k\delta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\alpha}t_{k_1\alpha+k_2\beta})+2k+1.$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta+k\delta})$$
$$=\alpha_0^{k+k_1-\frac{k_2}{2}}(-\alpha)_1^{-k-1-k_1+\frac{k_2}{2}}(\beta)_0^{k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{-k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{2k+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{-2k-1-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\beta+k\delta}s_{\alpha}t_{k_1\alpha+k_2\beta})=-k-1-k_1+\frac{k_2}{2}-k+\frac{k_1}{2}-k_2-2k-1-\frac{k_1+k_2}{2}=-4k-2-k_1-k_2=l_B(s_{\alpha}t_{k_1\alpha+k_2\beta})-4k-1.$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\alpha+\beta+k\delta})$$
$$=\alpha_0^{-k-1+k_1-\frac{k_2}{2}}(-\alpha)_1^{k-k_1+\frac{k_2}{2}}\beta_0^{2k-\frac{k_1}{2}+k_2}(-\beta)_1^{-2k-1+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{k+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{-k-1-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+\beta+k\delta}s_{\alpha}t_{k_1\alpha+k_2\beta})=k-k_1+\frac{k_2}{2}-2k-1+\frac{k_1}{2}-k_2-k-1-\frac{k_1+k_2}{2}=-2k-k_1-k_2-2=l_B(s_{\alpha}t_{k_1\alpha+k_2\beta})-2k-1.$
So we have proved (6).
Using the map $\tau$ and the calculation for the case (6), one sees (5).
Now we prove (3). We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta})$$
$$=(\alpha_0^{k_1-\frac{k_2}{2}}(-\alpha)_1^{-k_1+\frac{k_2}{2}-1}\beta_0^{-\frac{k_1}{2}+k_2-1}(-\beta)_1^{\frac{k_1}{2}-k_2}
(\alpha+\beta)_0^{\frac{k_1}{2}+\frac{k_2}{2}}(-\alpha-\beta)_1^{-\frac{k_1}{2}-\frac{k_2}{2}-1}).$$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\alpha+k\delta})$$
$$=\alpha_0^{-k+k_1-\frac{k_2}{2}}(-\alpha)_1^{k-1-k_1+\frac{k_2}{2}}\beta_{0}^{2k-\frac{k_1}{2}+k_2}(-\beta)_1^{-2k-1+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{k+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{-k-1-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=k-1-k_1+\frac{k_2}{2}-2k-1+\frac{k_1}{2}-k_2-k-1-\frac{k_1+k_2}{2}=-2k-k_1-k_2-3.$
$l_B(s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=-k_1+\frac{k_2}{2}-1+\frac{k_1}{2}-k_2-\frac{k_1}{2}-\frac{k_2}{2}-1=-k_1-k_2-2$. Therefore
$l_B(s_{\alpha+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})-2k-1.$
We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\beta+k\delta})$$
$$=\alpha_0^{-k+k_1-\frac{k_2}{2}}(-\alpha)_1^{k-1-k_1+\frac{k_2}{2}}\beta_0^{-k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{-2k-1+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{2k-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\beta+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=k-1-k_1+\frac{k_2}{2}+k+\frac{k_1}{2}-k_2+2k-\frac{k_1+k_2}{2}=4k-k_1-k_2-1.$
Therefore $l_B(s_{\beta+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})+4k+1$.
We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\alpha+\beta+k\delta})$$
$$=\alpha_0^{-2k-1+k_1-\frac{k_2}{2}}(-\alpha)_1^{2k-k_1+\frac{k_2}{2}}\beta_0^{k-\frac{k_1}{2}+k_2}(-\beta)_1^{-k-1+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{-k-1+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{k-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+\beta+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=2k-k_1+\frac{k_2}{2}-k-1+\frac{k_1}{2}-k_2+k-\frac{k_1+k_2}{2}=2k-k_1-k_2-1.$
Therefore $l_B(s_{\alpha+\beta+k\delta}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})+2k+1$.
Using the map $\tau$ and the calculation for the $s_{\beta}s_{\alpha}T$ case, one sees (2).
Now we prove (4). We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\alpha})$$
$$=(\alpha_0^{k_1-\frac{k_2}{2}}(-\alpha)_1^{-k_1+\frac{k_2}{2}-1}\beta_0^{-\frac{k_1}{2}+k_2}(-\beta)_1^{\frac{k_1}{2}-k_2-1}
(\alpha+\beta)_0^{\frac{k_1}{2}+\frac{k_2}{2}}(-\alpha-\beta)_1^{-\frac{k_1}{2}-\frac{k_2}{2}-1}).$$
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\alpha}s_{\alpha+k\delta})$$
$$=\alpha_0^{k+k_1-\frac{k_2}{2}}(-\alpha)_1^{-k-1-k_1+\frac{k_2}{2}}\beta_0^{-2k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{2k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{-k+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{k-1-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+k\delta}s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=-k-1-k_1+\frac{k_2}{2}+2k+\frac{k_1}{2}-k_2+k-1-\frac{k_1+k_2}{2}=2k-2-k_1-k_2.$
$l_B(s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=-k_1+\frac{k_2}{2}-1+\frac{k_1}{2}-k_2-1-\frac{k_1}{2}-\frac{k_2}{2}-1=-k_1-k_2-3.$
Therefore $l_B(s_{\alpha+k\delta}s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})+2k+1.$
Using the map $\tau$ and the above calculation, one sees that $l_B(s_{\beta+k\delta}s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})+2k+1.$
We compute
$$N(t_{k_1\alpha+k_2\beta}s_{\alpha}s_{\beta}s_{\alpha}s_{\alpha+\beta+k\delta})$$
$$=\alpha_0^{-k-1+k_1-\frac{k_2}{2}}(-\alpha)_1^{k-k_1+\frac{k_2}{2}}\beta_0^{-k-1-\frac{k_1}{2}+k_2}(-\beta)_1^{k+\frac{k_1}{2}-k_2}$$
$$(\alpha+\beta)_0^{-2k-1+\frac{k_1+k_2}{2}}(-\alpha-\beta)_1^{2k-\frac{k_1+k_2}{2}}.$$
So $l_B(s_{\alpha+\beta+k\delta}s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=k-k_1+\frac{k_2}{2}+k+\frac{k_1}{2}-k_2+2k-\frac{k_1+k_2}{2}=4k-k_1-k_2.$
Therefore $l_B(s_{\alpha+\beta+k\delta}s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})=l_B(s_{\alpha}s_{\beta}s_{\alpha}t_{k_1\alpha+k_2\beta})+4k+3.$
\end{proof}
We draw below part of the Hasse diagram of the poset $(\widetilde{W},\leq_B)$. The numeric sequence $a_1a_2\cdots a_t, a_i\in \{1,2,3\}$ stands for the reduced expression $s_{a_1}s_{a_2}\cdots s_{a_t}$ where $s_1:=s_{\alpha}, s_2:=s_{\beta}$ and $s_3:=s_{\delta-\alpha-\beta}.$ The black edge corresponds to a covering relation in the corresponding $B$-twisted left weak Bruhat order. The blue edge corresponds to the additional covering relation in $B$-twisted strong Bruhat order.
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We collect in the following proposition some properties and the symmetry of this poset.
\begin{proposition}\label{posetprop}
(1) The sphericity of the intervals is characterized as follow.
(a) The length 2 intervals of $(\widetilde{W},\leq_B)$ are either of the form $[x,s_1s_2x]$ where $s_1\neq s_2, s_1,s_2\in \{s_{\alpha},s_{\beta},s_{\delta-\alpha-\beta}\}$ or of the form $[x,s_{\delta-\gamma}s_{\gamma}x]$ where $\gamma\in \{\alpha,\beta,\alpha+\beta, \delta-\alpha, \delta-\beta, \delta-\alpha-\beta\}$. The former is spherical and the latter is non-spherical.
(b) The length 3 intervals of $(\widetilde{W},\leq_B)$ are all non-spherical except those of the following types:
$[x,s_{\alpha}s_{\beta}s_{\alpha}x]$ where $x\in s_{\alpha}s_{\beta}s_{\alpha}T$; $[x, s_{\delta-\alpha-\beta}s_{\beta}s_{\delta-\alpha-\beta}x]$ where $x\in s_{\beta}T$; $[x, s_{\alpha}s_{\delta-\alpha-\beta}s_{\alpha}x]$ where $x\in s_{\alpha}T$.
(c) All intervals of $(\widetilde{W},\leq_B)$ of length greater than 3 are non-spherical.
(2) Let $x=(s_{\beta}s_{\delta-\alpha-\beta}s_{\alpha})^{\infty}$ and $y=(s_{\alpha}s_{\delta-\alpha-\beta}s_{\beta})^{\infty}$ be two infinite reduced words. For $i\geq 0$, denote $x(i)$ and $x(-i)$ the unique length $i$ prefixes of $x$ and $y$ respectively.
Let $U$ be the (infinite dihedral) reflection subgroup of $\widetilde{W}$ generated by $v:=s_{\delta-\alpha-\beta}$ and $u:=s_{\alpha}s_{\beta}s_{\alpha}$. For $x\in \widetilde{W}$, $y$ is the unique minimal element of the left coset $xU$ if and only if $y=w(i)^{-1}$ for some $i\in \mathbb{Z}$. Consequently every element $w\in \widetilde{W}$ can be written in the form $w(i)^{-1}u$ for some $u\in U$ and $i\in \mathbb{Z}$.
Furthermore for every $w\in \widetilde{W}$, $l_B(w)$ is determined as follow:
(a) $l_B(w(i)^{-1}u(vu)^k)=-4k-3$ if $i$ is even and $l_B(w(i)^{-1}u(vu)^k)=-4k-2$ if $i$ is odd.
(b) $l_B(w(i)^{-1}(vu)^k)=-4k$ if $i$ is even and $l_B(w(i)^{-1}(vu)^k)=-4k-1$ if $i$ is odd.
(c) $l_B(w(i)^{-1}v(uv)^k)=4k+1$ if $i$ is even and $l_B(w(i)^{-1}v(uv)^k)=4k+2$ if $i$ is odd.
(d) $l_B(w(i)^{-1}(uv)^k)=4k$ if $i$ is even and $l_B(w(i)^{-1}(uv)^k)=4k-1$ if $i$ is odd.
(3) Suppose that $x,y\in \widetilde{W}$ such that $l_B(x)$ and $l_B(y)$ are of the same parity. Then there exists a poset automorphism $\gamma: (\widetilde{W},\leq_B)\rightarrow (\widetilde{W},\leq_B)$ such that $\gamma(x)=y.$
\end{proposition}
\begin{proof}
(1) This can be verified routinely by using the covering relations in Lemma \ref{sixtype}.
(2) By \cite{DyerReflSubgrp} $y$ is an element that satisfies $N(y^{-1})\cap\Phi_{U}^+=\emptyset.$ Hence $N(y^{-1})\subset \{\alpha,-\beta\}^{\wedge}\uplus \{-\alpha,\beta\}^{\wedge}$. Note that $\{\alpha,-\beta\}^{\wedge}=(s_{\beta}s_{\delta-\alpha-\beta}s_{\alpha})^{\infty}$ and $ \{-\alpha,\beta\}^{\wedge}=(s_{\alpha}s_{\delta-\alpha-\beta}s_{\beta})^{\infty}$.
Assume that $N(y^{-1})\not\subset N(s_{\beta}s_{\delta-\alpha-\beta}s_{\alpha})^{\infty})$ and $N(y^{-1})\not\subset N((s_{\alpha}s_{\delta-\alpha-\beta}s_{\beta})^{\infty})$. Without loss of generality assume that $\alpha\in N(y^{-1})$.
Then $\{-\alpha\}^{\wedge}\cap N(y^{-1})=\emptyset$. This forces some $\beta+k\delta\in N(y^{-1})$. Then $\alpha+\beta+k\delta\in N(y^{-1})$, contradicting $N(y^{-1})\cap\Phi_{U}^+=\emptyset.$ Hence either $N(y^{-1})\subset N(s_{\beta}s_{\delta-\alpha-\beta}s_{\alpha})^{\infty})$ or $N(y^{-1})\subset N((s_{\alpha}s_{\delta-\alpha-\beta}s_{\beta})^{\infty})$.
Hence given $w\in \widetilde{W}$, then $w(i)^{-1}=wu$ for some $i\in \mathbb{Z}$ and $u\in U.$
Now we check the results on twisted length. We prove (a) and (b)-(d) can be verified similarly. One computes that
$N((uv)^kuw(i))=\alpha_0^{k-\lceil\frac{i}{2}\rceil}\beta_0^{k+\lfloor\frac{i}{2}\rfloor}(\alpha+\beta)_0^{2k}(-\alpha)_1^{\lceil\frac{i}{2}\rceil-k-1}(-\beta)_1^{\lceil-\frac{i}{2}\rceil-k-1}.$
There are six possiblities depending on $k$ and $i$.
(i) $N((uv)^kuw(i))=\alpha_0^{k-\lceil\frac{i}{2}\rceil}\beta_0^{k+\lfloor\frac{i}{2}\rfloor}(\alpha+\beta)_0^{2k}.$ Then $l_B((uv)^kuw(i))=-4k-3+\lceil\frac{i}{2}\rceil-\lfloor\frac{i}{2}\rfloor$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(ii) $N((uv)^kuw(i))=\alpha_0^{k-\lceil\frac{i}{2}\rceil}(\alpha+\beta)_0^{2k}(-\beta)_1^{\lceil-\frac{i}{2}\rceil-k-1}.$ Then $l_B((uv)^kuw(i))=-4k-3+\lceil\frac{i}{2}\rceil+\lceil-\frac{i}{2}\rceil$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(iii) $N((uv)^kuw(i))=\beta_0^{k+\lfloor\frac{i}{2}\rfloor}(\alpha+\beta)_0^{2k}(-\alpha)_1^{\lceil\frac{i}{2}\rceil-k-1}.$ Then $l_B((uv)^kuw(i))=-4k-3+\lceil\frac{i}{2}\rceil-\lfloor\frac{i}{2}\rfloor$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(iv) $N((uv)^kuw(i))=(\alpha+\beta)_0^{2k}(-\alpha)_1^{\lceil\frac{i}{2}\rceil-k-1}(-\beta)_1^{\lceil-\frac{i}{2}\rceil-k-1}.$
Then $l_B((uv)^kuw(i))=-4k-3+\lceil\frac{i}{2}\rceil+\lceil-\frac{i}{2}\rceil$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(v) If $k-\lceil\frac{i}{2}\rceil=-1$, $N((uv)^kuw(i))=\beta_0^{k+\lfloor\frac{i}{2}\rfloor}(\alpha+\beta)_0^{2k}.$ Then $l_B((uv)^kuw(i))=-3k-2-\lfloor\frac{i}{2}\rfloor=-4k-3+\lceil\frac{i}{2}\rceil-\lfloor\frac{i}{2}\rfloor$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(vi) If $k+\lfloor\frac{i}{2}\rfloor=-1$, $N((uv)^kuw(i))=\alpha_0^{k-\lceil\frac{i}{2}\rceil}(\alpha+\beta)_0^{2k}.$ Then $l_B((uv)^kuw(i))=-3k-2+\lceil\frac{i}{2}\rceil=-4k-3+\lceil\frac{i}{2}\rceil-\lfloor\frac{i}{2}\rfloor$. If $i$ is even the twisted length is $-4k-3$. If $i$ is odd, the twisted length is $-4k-2.$
(3) Let $\sigma$ be the group automorphism of $\widetilde{W}$ induced by the map $s_{\delta-\alpha-\beta}\mapsto s_{\alpha}, s_{\beta}\mapsto s_{\delta-\alpha-\beta}, s_{\alpha}\mapsto s_{\beta}$.
Note that $\sigma(t_{\alpha^{\vee}})=\sigma(s_{\alpha}s_{\beta}s_{\delta-\alpha-\beta}s_{\beta})=s_{\beta}s_{\delta-\alpha-\beta}s_{\alpha}s_{\delta-\alpha-\beta}=t_{\beta^{\vee}}$ and $\sigma(t_{\beta^{\vee}})=\sigma(s_{\beta}s_{\alpha}s_{\delta-\alpha-\beta}s_{\alpha})=s_{\delta-\alpha-\beta}s_{\beta}s_{\alpha}s_{\beta}=t_{-(\alpha+\beta)^{\vee}}$. So we conclude $\sigma(T)=T$.
We show that there exists a poset automorphism $\eta$ of $(\widetilde{W},\leq_B)$ given by $w\mapsto \sigma(w)s_{\alpha}s_{\beta}$ and a poset automorphism $\eta'$ of $(\widetilde{W},\leq_B)$ given by $w\mapsto \sigma^{-1}(w)s_{\beta}s_{\alpha}$.
Using the fact that $T$ is a normal subgroup one obtains that
$$\eta(w)\in s_{\alpha}s_{\beta}T,$$
$$\eta(s_{\alpha}s_{\beta}w)\in s_{\beta}s_{\alpha}T,$$
$$\eta(s_{\beta}s_{\alpha}w)\in T,$$
$$\eta(s_{\beta}s_{\alpha}s_{\beta}w)\in s_{\beta}T,$$
$$\eta(s_{\beta}T)\in s_{\alpha}T,$$
$$\eta(s_{\alpha}T)\in s_{\beta}s_{\alpha}s_{\beta}T.$$
By above and the covering relations given in Lemma \ref{sixtype} one can easily checks that $\eta$ preserves covering relations. Similarly one verifies that $\eta'$ also preserves covering relations. Let $\rho=\eta(\eta')^{-1}$ be the poset automorphism of $(\widetilde{W},\leq_B)$.
For $k\in \mathbb{Z},$ $\rho^k(w)=\sigma^{2k}(w)(x(2k))^{-1}$.
For any $z\in U$ (as in (2)) we have
$$\rho^2(w(i)^{-1}z)=\sigma^{-1}(w(i)^{-1}z)x(2)^{-1}=w(i+2)^{-1}(z).$$
By (2) if $l_B(x)=l_B(y)$, $x=w(j)^{-1}z$ and $y=w(k)^{-1}z$ where $j$ and $k$ are of the same parity and $z\in U$. Then for some $t$, $\rho^t(x)=y$.
Now assume $l_B(x)\neq l_B(y)$ but they are of the same parity. Let $v,u$ be as in (2).
We compute
$$\eta(w(i)^{-1}u(vu)^k)=w(i+1)^{-1}(vu)^{k+1},$$
$$\eta(w(i)^{-1}(vu)^k)=w(i+1)^{-1}u(vu)^k,$$
$$\eta(w(i)^{-1}v(uv)^k)=w(i+1)^{-1}(uv)^k,$$
$$\eta(w(i)^{-1}(uv)^k)=w(i+1)^{-1}v(uv)^{k-1}.$$
Hence for any $z\in \widetilde{W}.$ $l_B(\eta(z))=l_B(z)-2.$
Therefore for some $t,$ $l_B(x)=l_B(\eta^t(y))$.
\end{proof}
\begin{remark}
One of the classical theme in the study of Coxeter group (or more generally combinatorics) is the enumeration problem.
Given the local finiteness of the twisted Bruhat order on affine Weyl groups, one can
define the Poincar\'e series associated to the twisted Bruhat order $\leq_B$ and an element $w$ to be $F_{w,B}(t)=\sum_{u\leq_B w}t^{l(w)-l(u)}$. This generalizes the usual Poincar\'e polynomial associated to the ordinary Bruhat order.
For type $\widetilde{A}_2$ and the above biclosed set $B$, we have the following result.
If $l(w)$ is even, $F_{w,B}(t)=\frac{t^3+2t^2+2t+1}{t^4-2t^2+1}$. If $l(w)$ is odd, $F_{w,B}(t)=\frac{2t^2+3t+1}{t^4-2t^2+1}$.
To see this, one can apply the Principle of Inculsion-Exclusion and Proposition \ref{posetprop} (3) to obtain
$$F_1(t)=1+2tF_2(t)-2t^2F_1(t)+t^3F_2(t)$$
$$F_2(t)=1+3tF_1(t)-2t^2F_2(t)$$
where $F_1(t)=F_{w,B}(t)$ for $w$ even and $F_2(t)=F_{w,B}(t)$ for $w$ odd.
Solve for $F_1(t)$ and $F_2(t)$ one obtains the closed formlae.
\end{remark}
\section{Tope Poset of the affine root system and a Bj\"{o}rner-Edelman-Ziegler type theorem}\label{omsec}
The root system of a Coxeter group can be naturally regarded as an oriented matroid. The affine root systems provide interesting and accessible examples of infinite oriented matroids in the sense of \cite{largeconvex}.
In this section we will use twisted weak order as a tool to investigate the tope poset of the oriented matroid arising from an affine root system.
An (possibly infinite) oriented matroid is a triple $(E,*,cx)$ where $E$ is a set with an involution map $*: E\rightarrow E$ (i.e. $x^{**}=x, x\neq x^*$) and $cx$ a closure operator on $E$ such that
(i) if $x\in cx(X)$ there exists a finite set $Y\subseteq X$ such that $x\in cx(Y)$,
(ii) $cx(X)^*=cx(X^*)$,
(iii) if $x\in cx(X\cup \{x^*\})$ then $x\in cx(X)$,
(iv) if $x\in cx(X\cup \{y^*\})$ and $x\not\in cx(X)$ then $y\in cx(X\backslash\{y\}\cup\{x^*\})$.
Given a root system $\Phi$, $(\Phi,-,\text{cone}_{\Phi})$ is an oriented matroid where $\text{cone}_{\Phi}=\text{cone}(\Gamma)\cap \Phi$ and $\cone(\Gamma)=\{\sum_{i\in I}k_iv_i|v_i\in \Gamma\cup\{0\}, k_i\in \mathbb{R}_{\geq 0}, |I|<\infty\}$.
A $cx$-closed set in $E$ is said to be convex.
A convex set $H\subset E$ is called a hemispace or a tope if $H\cap H^*=\emptyset, H\cup H^*=E$.
Suppose that $\Phi$ is the root system of a finite Coxeter system. Then every positive system is a hemispace of the oriented matroid $(\Phi,-,\text{cone}_{\Phi})$ and vice versa. We denote the set of hemispaces by $\mathcal{H}(E).$
For a finite oriented matroid $(E,*,cx)$
and a hemispace $H$, there exists a unique minimal subset of $H$ whose $cx-$closure is $H$. We denote such a set by $\mathrm{ex}(H)$ and call it the set of extreme elements of $H.$
A hemispace is said to be simplicial if $|\mathrm{ex}(H)|$ equals the rank of the underlying (unoriented) matroid of $(E,*,cx)$. A finite oriented matroid is said to be simplicial if all of its hemispaces are simplicial.
Denote by $\Delta$ the set symmetric difference. Fix a hemispace $H$ of $(E,*,cx)$ and one can define a poset structure on the set of hemispaces by $F\leq G\Longleftrightarrow F\Delta H \subset G\Delta H, F,G\in \mathcal{H}(E).$ We call such a poset the tope poset based at $H.$ In a tope poset $P$ of an infinite oriented matroid, we say two hemisapces $H_1, H_2$ are in the same block of $P$ if their symmetric difference is finite.
In \cite{hyperplane}, Bj\"{o}rner, Edelman and Ziegler proved the following
\begin{theorem}
For a finite simplicial oriented matroid, the tope poset for any choice of base hemispace is a lattice.
\end{theorem}
It is well known that for the root system of a finite Coxeter system, the corresponding oriented matroid is simplicial. In fact all tope posets are isomorphic to the ordinary right weak order of the Coxeter group (which is a lattice). In this section we prove the following analogous theorem for irreducible affine root systems.
\begin{theorem}\label{intervallattice}
Let $\Phi$ be a finite irreducible crystallographic root system and let $H$ be any hemispace of the oriented matroid $(\widetilde{\Phi},-,cx)$ where $cx:=\mathrm{cone}_{\widetilde{\Phi}}$. Let $H_1, H_2$ be two hemispaces in a same block and $H_1\leq H_2$ in the tope poset based at $H$. Then $[H_1, H_2]$ is a lattice.
\end{theorem}
We will first classify all hemispaces of $(\widetilde{\Phi},-,cx)$.
Let $\Gamma$ be a subset of a real vector space $V$. A subset $A$ of $\Gamma$ is called biconvex in $\Gamma$ if $A$ and $\Gamma\backslash A$ are both convex in the oriented matroid $(\Gamma,-,\mathrm{cone}_{\Gamma})$.
For the next lemma, let $\Phi$ be a finite irreducible crystallographic root system and let $\Psi^+$ be a positive system of $\Phi$.
\begin{lemma}\label{lem:coneintersectzero}
Let $P(\Psi^+,\Delta_1,\Delta_2)$ be a 2 closure biclosed set in $\Phi$. Then $\mathbb{R}\Delta_1\cap \mathrm{cone}(P(\Psi^+,\Delta_1,\Delta_2))=0$ and the 2 closure biclosed set $P(\Psi^+,\Delta_1,\Delta_2)$ is biconvex in $\Phi$.
\end{lemma}
\begin{proof}
Take a linear function $f:\mathbb{R}\Phi\rightarrow \mathbb{R}$ such that $f$ is positive on $\Delta\backslash (\Delta_1\cup\Delta_2)$ and zero on $\Delta_1\cup \Delta_2.$ Then $f$ is non-negative on $P(\Psi^+,\Delta_1,\Delta_2)$.
If $\gamma_1,\cdots,\gamma_k\in P(\Psi^+,\Delta_1,\Delta_2), c_1,\cdots,c_k\in \mathbb{R}_{>0}$ and $\gamma=\sum_{i=1}^kc_i\gamma_i\in \mathbb{R}\Delta_1,$ then apply $f$ on both sides we see all $\gamma_i$ are in $\mathbb{R}(\Delta_1\cup \Delta_2)$ as $f(\gamma)=0.$ Then they are in either $\mathbb{R}\Delta_1$ or $\mathbb{R}\Delta_2$ as $\Delta_1\perp\Delta_2.$ But since they are in $P(\Psi^+,\Delta_1,\Delta_2)$, they have to be in $\mathbb{R}\Delta_2.$ Then this forces $\gamma=0.$
To prove the second assertion, again let $\gamma_1,\cdots,\gamma_k\in P(\Psi^+,\Delta_1,\Delta_2), c_1,\cdots,c_k\in \mathbb{R}_{>0}$ and $\gamma=\sum_{i=1}^kc_i\gamma_i$. Note that $\Phi\backslash P(\Psi^+,\Delta_1,\Delta_2)=\mathbb{R}\Delta_1\uplus -P(\Psi^+,\Delta_1\cup\Delta_2,\emptyset).$ Suppose $\gamma\in -P(\Psi^+,\Delta_1\cup\Delta_2,\emptyset)$, apply $f$ on $\gamma$ one gets a negative number, which is a contradiction. Therefore combining this with the first assertion, one sees $P(\Psi^+,\Delta_1,\Delta_2)$ is convex.
Note $\Phi\backslash P(\Psi^+,\Delta_1,\Delta_2)=P(\Psi^-,-\Delta_1,-\Delta_2)$ and therefore is convex as well. Hence we see that $P(\Psi^+,\Delta_1,\Delta_2)$ is biconvex in $\Phi$.
\end{proof}
Similarly one can define the notion of a 2 closure hemispace in a root system $\Phi$. A 2 closure hemispace in $\Phi$ is a 2 closure closed set $H$ such that $H\uplus -H=\Phi$. A hemispace of the oriented matroid $(\Phi,-,cx)$ is necessarily a 2 closure hemispace of $\Phi$ but not vice versa.
There exists a bijection between the set of 2 closure biclosed sets in $\Phi^+$ and the set of 2 closure hemispaces under the map $B\mapsto B\uplus -(\Phi^+\backslash B)$ for $B$ a 2 closure biclosed set in $\Phi^+.$ Since the 2 closure biclosed sets in $\widetilde{\Phi}^+$ are classified, the corresponding 2 closure hemispaces are therefore known via the bijection. We will next examine which of these 2 closure hemispaces are indeed hemispaces of the oriented matroid $(\widetilde{\Phi},-,cx)$.
\begin{lemma}\label{lem:finitehemispace}
For any Coxeter system $(W,S)$ and $w\in W$,
(1) the 2-closure hemispace $N(w)\uplus -(\Phi^+\backslash N(w))$ is convex,
(2) the 2-closure hemispace $N(w)'\uplus -(\Phi^+\backslash N(w)')$ is convex.
\end{lemma}
\begin{proof}
(1) follows from the fact that $N(w)\uplus -(\Phi^+\backslash N(w))=-w\Phi^+$ and that $\Phi^+$ is convex.
(2) follows from the fact that $N(w)'\uplus -(\Phi^+\backslash N(w)')=w\Phi^+$ and that $\Phi^+$ is convex.
\end{proof}
Now let $P(\Psi^+,\Delta_1,\Delta_2)$ be a 2 closure biclosed set (also biconvex by Lemma \ref{lem:coneintersectzero}) in $\Phi$. Any 2 closure biclosed set in $(\Phi)^{\wedge}$ is of the form $(P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}\backslash B_2)\cup B_1$
where $B_1$ (resp. $B_2$) be a finite 2 closure biclosed set in $(\Phi_1)^{\wedge}$ (resp. $(\Phi_2)^{\wedge}$) and $\Phi_i=\mathbb{R}\Delta_i\cap \Phi$.
Then we can associate to it a 2 closure hemispace $H:=B\uplus -((\Phi)^{\wedge}\backslash B)$. We investigate when this 2 closure hemispace is a cone closure (i.e. oriented matroidal) hemispace.
\begin{proposition}\label{hemispacecoincide}
(1) $H$ is convex if $\Delta_2$ is empty or $\Delta_1$ is empty.
(2) If both $\Delta_1$ and $\Delta_2$ are non-empty, $H$ is not convex.
\end{proposition}
\begin{proof}
(1) Assume $\Delta_2=\emptyset.$
Suppose $\{\pm\alpha_1,\pm\alpha_2,\cdots,\pm\alpha_m\}=\mathbb{R}\Delta_1\cap \Phi=\Phi_1.$
Let $\{\gamma_1,\gamma_2,\cdots,\gamma_p\}=\Psi^+\backslash \Phi_1$.
By the construction $H$ contains $(\gamma_1)_{-\infty}^{\infty},(\gamma_2)_{-\infty}^{\infty},\cdots,(\gamma_p)_{-\infty}^{\infty}$.
Suppose $\alpha_{i_1}+k_1\delta, \alpha_{i_2}+k_2\delta, \cdots, \alpha_{i_t}+k_t\delta, \gamma_{j_1}+l_1\delta, \gamma_{j_2}+l_2\delta,\cdots, \gamma_{j_s}+l_s\delta\in H$ where
$\alpha_{i_p}\in \Phi_1, 1\leq p\leq t.$ Take $a_1, a_2, \cdots, a_t, b_1, b_2, \cdots, b_s\in \mathbb{R}_{>0}$. Assume that
\begin{equation}\label{imp}
\sum_{u=1}^t a_u(\alpha_{i_u}+k_u\delta)+\sum_{v=1}^s b_v(\gamma_{j_v}+l_v\delta)=\alpha'+q\delta
\end{equation}
where $\alpha'\in \Phi_1$ and $\alpha'+q\delta\not\in H.$ Then one has
$$\alpha'-\sum_{u=1}^t a_u\alpha_{i_u}=\sum_{v=1}^s b_v\gamma_{j_v}.$$
By Lemma \ref{lem:coneintersectzero} both sides have to be $0$. This forces all $b_v=0$ (as all $\gamma_i$ are in a positive system). So we have
$$\sum_{u=1}^t a_u(\alpha_{i_u}+k_u\delta)=\alpha'+q\delta.$$
On the other hand, note that $H\cap \widetilde{\Phi_1}$ is a hemispace in $(\widetilde{\Phi_1},-,\mathrm{cone}_{\widetilde{\Phi_1}})$. Indeed it is $B_1\cup -((\Phi_1)^{\wedge}\backslash B_1)$.
Since $B_1$ is finite and biclosed in $(\Phi_1)^{\wedge}$, it is of the form $\Phi_w$ where $w$ is an element of the reflection subgroup generated by the reflections corresponding to the roots in $(\Phi_1)^{\wedge}$. So Lemma \ref{lem:finitehemispace} (1) ensures the above equation \ref{imp} is not possible.
Now assume that
\begin{equation}\label{imp2}
\sum_{u=1}^t a_u(\alpha_{i_u}+k_u\delta)+\sum_{v=1}^s b_v(\gamma_{j_v}+l_v\delta)=-\gamma_w+q\delta.
\end{equation}
Then one has
$$-\sum_{u=1}^t a_u\alpha_{i_u}=\sum_{v=1}^s b_v\gamma_{j_v}+\gamma_w.$$
Again by Lemma \ref{lem:coneintersectzero} both sides have to be $0$. But all $\gamma_v$ are in a positive system of $\Phi.$ So the equation \ref{imp2} is not possible.
Hence $H$ is indeed convex.
Now assume $\Delta_1=\emptyset$. Then same type of reasoning as above (using Lemma \ref{lem:finitehemispace} (2)) shows $H$ is convex.
(2) Now we assume that both $\Delta_1$ and $\Delta_2$ are non-empty.
Again suppose that $\{\pm\alpha_1,\pm\alpha_2,\cdots,\pm\alpha_m\}=\mathbb{R}\Delta_1\cap \Phi=\Phi_1$ and assume that $\alpha_1,\alpha_2,\cdots, \alpha_m\in \Phi^+.$
Suppose that $\{\pm\beta_1, \pm\beta_2, \cdots, \pm\beta_l\}=\mathbb{R}\Delta_2\cap \Phi=\Phi_2$ and assume that $\beta_1, \beta_2, \cdots, \beta_l\in \Phi^+.$
By Lemma 4.1 of \cite{wang}, $(\{\beta_i+t\delta|t\in \mathbb{Z}\}\cup \{-\beta_i+t\delta|t\in \mathbb{Z}\})\cap (P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}\backslash B_2)\cup B_1$
has four possibilities:
(i) $(\beta_i)_p^{\infty}\cup (-\beta_i)_1^{\infty}, p\geq 0,$
(ii) $(\beta_i)_p^{\infty}\cup (-\beta_i)_0^{\infty}, p\geq 1,$
(iii) $(-\beta_i)_p^{\infty}\cup (\beta_i)_1^{\infty}, p\geq 0,$
(iv) $(-\beta_i)_p^{\infty}\cup (\beta_i)_0^{\infty}, p\geq 1.$
Correspondingly $(\{\beta_i+t\delta|t\in \mathbb{Z}\}\cup \{-\beta_i+t\delta|t\in \mathbb{Z}\})\cap H$ can be one of the following four possibilities:
(i) $(\beta_i)_p^{\infty}\cup (-\beta_i)_{1-p}^{\infty}, p\geq 0,$
(ii) $(\beta_i)_p^{\infty}\cup (-\beta_i)_{1-p}^{\infty}, p\geq 1,$
(iii) $(-\beta_i)_p^{\infty}\cup (\beta_i)_{1-p}^{\infty}, p\geq 0,$
(iv) $(-\beta_i)_p^{\infty}\cup (\beta_i)_{1-p}^{\infty}, p\geq 1.$
We note in each of these four cases for sufficiently large $s,t\in \mathbb{Z}_{>0}, \beta_i+s\delta$ and $-\beta_i+t\delta\in H$. It follows that $\delta\in \cone(H)$.
Similarly by Lemma 4.1 of \cite{wang}, $(\{\alpha_i+t\delta|t\in \mathbb{Z}\}\cup \{-\alpha_i+t\delta|t\in \mathbb{Z}\})\cap\newline (P(\Psi^+,\Delta_1,\Delta_2)^{\wedge}\backslash B_2)\cup B_1$
has four possibilities:
(i) $(\alpha_i)_0^p, p\geq 0$,
(ii) $(-\alpha_i)_0^p, p\geq 0,$
(iii) $(\alpha_i)_1^p, p\geq 1,$
(iv) $(-\alpha_i)_1^p, p\geq 1.$
Correspondingly $(\{\alpha_i+t\delta|t\in \mathbb{Z}\}\cup \{-\alpha_i+t\delta|t\in \mathbb{Z}\})\cap H$ can be one of the following four possibilities:
(i) $(\alpha_i)_{-\infty}^p\cup (-\alpha_i)_{-\infty}^{-p-1}, p\geq 0,$
(ii) $(-\alpha_i)_{-\infty}^p\cup (\alpha_i)_{-\infty}^{-p-1}, p\geq 0,$
(iii) $(\alpha_i)_{-\infty}^p\cup (-\alpha_i)_{-\infty}^{-p-1}, p\geq 1,$
(iv) $(-\alpha_i)_{-\infty}^p\cup (\alpha_i)_{-\infty}^{-p-1}, p\geq 1.$
We note in each of these four cases it follows that there exist $s,t\in \mathbb{Z}_{<0}, \beta_i+s\delta$ and $-\beta_i+t\delta\in H$. Hence $-\delta\in \cone(H)$.
Thus we see that $(\{\beta_i+t\delta|t\in \mathbb{Z}\}\cup\{-\beta_i+t\delta|t\in \mathbb{Z}\})$ (resp. $(\{\alpha_i+t\delta|t\in \mathbb{Z}\}\cup\{-\alpha_i+t\delta|t\in \mathbb{Z}\})$) is completely contained in $\cone(H).$ Therefore $H$ is not convex.
\end{proof}
Now by Theorem \ref{infiniteword} we immediately obtain
\begin{corollary}\label{twotypes}
For an irreducible affine Weyl group $\widetilde{W}$, any hemispace of the oriented matroid $(\widetilde{\Phi},-,\mathrm{cone}_{\widetilde{\Phi}})$ is of the form $N(w)\uplus -(\Phi^{\wedge}\backslash N(w))$ or $-N(w)\uplus (\Phi^{\wedge}\backslash N(w))$ where $w\in \widetilde{W}$ or is an infinite reduced word.
\end{corollary}
\begin{remark}
Let $(W,S)$ be a rank 3 universal Coxeter system. The three simple roots are denoted by $\alpha_1,\alpha_2,\alpha_3.$ Then $B:=\{\alpha\in \Phi^+|\alpha=k_1\alpha_1+k_2\alpha_2\}$ is a biclosed set in $\Phi^+$. One easily sees that $H=B\uplus -(\Phi^+\backslash B)=\{\alpha\in \Phi^+|\alpha=k_1\alpha_1+k_2\alpha_2\}\uplus \{\alpha\in \Phi^-|\alpha=k_1\alpha_1+k_2\alpha_2+k_3\alpha_3, k_3<0\}$ is a matroidal hemispace of the oriented matroid $(\Phi,-,cone_{\Phi})$. However in this case neither $B$ nor $\Phi^+\backslash B$ is an inversion set.
\end{remark}
\begin{lemma}\label{torder}
Let $B=P(\Psi^+,\Delta_1,\emptyset)^{\wedge}$ be a biclosed set in $(\Phi)^{\wedge}$.
Let $H=B\uplus -((\Phi)^{\wedge}\backslash B)$ be a hemispace of the oriented matroid $(\widetilde{\Phi},-,cx)$ and $G$ is an arbitrary hemispace of $(\widetilde{\Phi},-,cx)$.
Then the block containing $H$ of the tope poset based at $G$ is isomorphic to the twisted weak order $(W',G\cap \widetilde{\Phi_1})$ where $W'$ is the reflection subgroup generated by $\widetilde{\Phi_1}$ and $\widetilde{\Phi_1}=\pm(\mathbb{R}\Delta_1)^{\wedge}.$
\end{lemma}
\begin{proof}
First note that since $G$ is a hemispace in the oriented matroid, it is biclosed in $\widetilde{\Phi}$. Therefore $G\cap \widetilde{\Phi_1}$ is indeed a biclosed set in $\widetilde{\Phi_1}.$
Let $H_1=B_1\uplus -(\Phi^{\wedge}\backslash B_1)$ be another hemispace. If $H$ and $H_1$ are in the same block, then clearly the symmetric difference of $B$ and $B_1$ is finite.
By Proposition 5.11 and Corollary 5.12 of \cite{DyerReflOrder} (See also Theorem 1.3 of \cite{wang}), $B_1=w\cdot B$ for some unique $w\in W'$. Then $H_1=w\cdot B\uplus -(\Phi^{\wedge}\backslash w\cdot B)=wH$ by \cite{BPK} 3.2(e).
Now define the map from the block containing $H$ of the tope poset based at $G$ to $(W',G\cap \widetilde{\Phi_1})$ by $wH\mapsto w$.
We must show that such a map and its inverse preserve the orders.
Take $w_1H, w_2H$ such that $w_1H<w_2H$ in the block containing $H$ of the tope poset based at $G$.
Note that $w_iH=w_i\cdot P(\Psi^+,\Delta_1,\emptyset)^{\wedge} \uplus -((\Phi)^{\wedge}\backslash w_i\cdot P(\Psi^+,\Delta_1,\emptyset)^{\wedge}), i=1,2.$ by \cite{BPK} 3.2(e).
Further one has that $w_i\cdot P(\Psi^+,\Delta_1,\emptyset)^{\wedge}=P(\Psi^+,\Delta_1,\emptyset)^{\wedge}\uplus (N(w_i)\cap \widetilde{\Phi_1})$ by Proposition 5.11 of \cite{DyerReflOrder}.
Hence $w_iH=H\uplus (N(w_i)\cap \widetilde{\Phi_1})\backslash -((N(w_i)\cap \widetilde{\Phi_1}))$. Then $w_1H\Delta G\subset w_2H\Delta G$ if and if the following two containments hold:
$$(N(w_1)\cap \widetilde{\Phi_1})\backslash G\subset (N(w_2)\cap \widetilde{\Phi_1})\backslash G,$$
$$(N(w_2)\cap \widetilde{\Phi_1})\cap G\subset (N(w_1)\cap \widetilde{\Phi_1})\cap G.$$
One sees that these two containments are equivalent to the conditions $(N(w_1)\cap \widetilde{\Phi_1})\backslash (N(w_2)\cap \widetilde{\Phi_1})\subset G$ and $(N(w_2)\cap \widetilde{\Phi_1})\backslash (N(w_1)\cap \widetilde{\Phi_1})\subset -G$, i.e. $w_1\leq_{G\cap \widetilde{\Phi_1}} w_2$.
\end{proof}
\begin{corollary}\label{block}
Let $P$ be a block of a tope poset of $(\widetilde{\Phi},-,cx)$. Then $P$ is isomorphic to the twisted weak order on some Coxeter group $W'$.
\end{corollary}
\begin{proof}
By Corollary \ref{twotypes}, there are two possible types of the base tope (hemisapce).
Case I: The base hemispace is of the form $N(w)\uplus -(\widehat{\Phi}\backslash N(w))$ where $w\in \widetilde{W}$ or $w\in \widetilde{W}_l$. Then by Theorem \ref{infiniteword}, $N(w)=u\cdot P(\Psi^+,\Delta_1,\emptyset)^{\wedge}$ where $u\in \widetilde{W}$. (Note if $w\in \widetilde{W}$, then $\Delta_1=\Delta$ and $P(\Psi^+,\Delta_1,\emptyset)^{\wedge}=\emptyset.$) Suppose that $u=e$, the assertion follows from Lemma \ref{torder}.
Now assume that $u\neq e$. Note that the map $H\mapsto uH$ is a bijection from the set of hemispaces to itself. One easily sees that the tope poset based at $H$ is isomorphic to the tope poset based at $wH$ by noting $G_1\Delta H\subset G_2\Delta H\Leftrightarrow uG_1\Delta uH\subset uG_2\Delta uH$. Therefore the assertion also holds in this situation.
Case II. The base hemispace is of the form $-N(w)\uplus (\widehat{\Phi}\backslash N(w))$ where $w\in \widetilde{W}$ or $w\in \widetilde{W}_l$. One easily sees that such a tope poset is the opposite poset of the tope poset based at $N(w)\uplus -(\widehat{\Phi}\backslash N(w))$. Since the opposite poset of the twisted weak order $(W,\leq_B')$ is isomorphic to $(W,\leq_{\Phi^+\backslash B'})$, the assertion also holds in this case.
\end{proof}
\noindent
\emph{Proof of Theorem \ref{intervallattice}}. By Corollary \ref{block}, $[H_1, H_2]$ is isomorphic to a closed interval of the twisted weak order of some Coxeter group $W'$. By Corollary 4.2 of \cite{orderpaper}, it is also isomorphic to a closed interval of the ordinary weak order of $W'$. Since the ordinary weak order $(W',\leq_{\emptyset}')$ is a complete meet semilattice, such an interval is a lattice.
\begin{remark}
Note that the example in Section 4 of \cite{wang} shows that in general the tope poset of the oriented matroid coming from an affine root system is not a lattice.
\end{remark}
To end this section, we sketch (part of) the Hasse diagram of a specific tope poset of $\widetilde{\Phi}$ of type $\widetilde{A}_2$ with the base hemispace $\widetilde{\Phi}^+$. This particular tope poset is isomorphic to the poset of 2 closure biclosed sets in $\widetilde{\Phi}^+$ (but in general there is no such isomorphism by our results above) and thus is a lattice by \cite{wang}. The two simple roots in $\Phi^+$ are denoted by $\alpha,\beta.$
\begin{tikzpicture}[scale=.15]
\tiny
\node (a) at (0,0) {$H_1$};
\node (b) at (-12,4) {$H_2$};
\node (c) at (0,4) {$H_3$};
\node (d) at (16,4) {$H_4$};
\node (e) at (-20,8) {$H_5$};
\node (f) at (-10,8) {$H_6$};
\node (g) at (-2,8) {$H_7$};
\node (h) at (3,8) {$H_8$};
\node (i) at (10,8) {$H_9$};
\node (j) at (20,8) {$H_{10}$};
\node (k) at (-20,12) {$H_{11}$};
\node (l) at (-14,12) {$H_{12}$};
\node (m) at (-8,12) {$H_{13}$};
\node (n) at (-2,12) {$H_{14}$};
\node (o) at (4,12) {$H_{15}$};
\node (p) at (10,12) {$H_{16}$};
\node (q) at (16,12) {$H_{17}$};
\node (r) at (22,12) {$H_{18}$};
\node (s) at (28,12) {$H_{19}$};
\node (t) at (0,16) {$\cdots\cdots\cdots$};
\node (u) at (-17,20) {$T_1$};
\node (v) at (-6,20) {$T_2$};
\node (w) at (4,20) {$T_3$};
\node (x) at (12,20) {$T_4$};
\node (y) at (22,20) {$T_5$};
\node (z) at (31,20) {$T_6$};
\node (aa) at (-20,24) {$T_{11}$};
\node (ab) at (-14,24) {$T_{12}$};
\node (ac) at (-20,28) {$T_{13}$};
\node (ad) at (-14,28) {$T_{14}$};
\node (ae) at (-10,24) {$T_{21}$};
\node (af) at (-3,24) {$T_{22}$};
\node (ag) at (-10,28) {$T_{23}$};
\node (ah) at (-3,28) {$T_{24}$};
\node (ai) at (1,24) {$T_{31}$};
\node (aj) at (6,24) {$T_{32}$};
\node (ak) at (1,28) {$T_{33}$};
\node (al) at (6,28) {$T_{34}$};
\node (am) at (10,24) {$T_{41}$};
\node (an) at (15,24) {$T_{42}$};
\node (ao) at (10,28) {$T_{43}$};
\node (ap) at (15,28) {$T_{44}$};
\node (aq) at (19,24) {$T_{51}$};
\node (ar) at (24,24) {$T_{52}$};
\node (as) at (19,28) {$T_{53}$};
\node (at) at (24,28) {$T_{54}$};
\node (au) at (28,24) {$T_{61}$};
\node (av) at (33,24) {$T_{62}$};
\node (aw) at (28,28) {$T_{63}$};
\node (ax) at (33,28) {$T_{64}$};
\node (ay) at (0,32) {$\cdots\cdots\cdots$};
\node (az) at (-17,36) {$U_1$};
\node (ba) at (-6,36) {$U_2$};
\node (bb) at (4,36) {$U_3$};
\node (bc) at (12,36) {$U_4$};
\node (bd) at (20,36) {$U_5$};
\node (be) at (27,36) {$U_6$};
\node (bf) at (0,40) {$\cdots\cdots\cdots$};
\node (bg) at (-20,48) {$-T_{11}$};
\node (bh) at (-14,48) {$-T_{12}$};
\node (bi) at (-20,44) {$-T_{13}$};
\node (bj) at (-14,44) {$-T_{14}$};
\node (bk) at (-10,48) {$-T_{21}$};
\node (bl) at (-3,48) {$-T_{22}$};
\node (bm) at (-10,44) {$-T_{23}$};
\node (bn) at (-3,44) {$-T_{24}$};
\node (bo) at (1,48) {$-T_{31}$};
\node (bp) at (6,48) {$-T_{32}$};
\node (bq) at (1,44) {$-T_{33}$};
\node (br) at (6,44) {$-T_{34}$};
\node (bs) at (10,48) {$-T_{41}$};
\node (bt) at (15,48) {$-T_{42}$};
\node (bu) at (10,44) {$-T_{43}$};
\node (bv) at (15,44) {$-T_{44}$};
\node (bw) at (19,48) {$-T_{51}$};
\node (bx) at (24,48) {$-T_{52}$};
\node (by) at (19,44) {$-T_{53}$};
\node (bz) at (24,44) {$-T_{54}$};
\node (ca) at (28,48) {$-T_{61}$};
\node (cb) at (33,48) {$-T_{62}$};
\node (cc) at (28,44) {$-T_{63}$};
\node (cd) at (33,44) {$-T_{64}$};
\node (uu) at (-17,52) {$-T_1$};
\node (vv) at (-6,52) {$-T_2$};
\node (ww) at (4,52) {$-T_3$};
\node (xx) at (12,52) {$-T_4$};
\node (yy) at (22,52) {$-T_5$};
\node (zz) at (31,52) {$-T_6$};
\node (dots) at (0,56) {$\cdots\cdots\cdots$};
\node (aaa) at (0,72) {$-H_1$};
\node (bbb) at (-12,68) {$-H_2$};
\node (ccc) at (0,68) {$-H_3$};
\node (ddd) at (16,68) {$-H_4$};
\node (eee) at (-20,64) {$-H_5$};
\node (fff) at (-10,64) {$-H_6$};
\node (ggg) at (-2,64) {$-H_7$};
\node (hhh) at (3,64) {$-H_8$};
\node (iii) at (10,64) {$-H_9$};
\node (jjj) at (20,64) {$-H_{10}$};
\node (kkk) at (-20,60) {$-H_{11}$};
\node (lll) at (-14,60) {$-H_{12}$};
\node (mmm) at (-8,60) {$-H_{13}$};
\node (nnn) at (-2,60) {$-H_{14}$};
\node (ooo) at (4,60) {$-H_{15}$};
\node (ppp) at (10,60) {$-H_{16}$};
\node (qqq) at (16,60) {$-H_{17}$};
\node (rrr) at (22,60) {$-H_{18}$};
\node (sss) at (28,60) {$-H_{19}$};
\draw (ccc) -- (aaa);
\draw (bbb) -- (aaa);
\draw (ddd) -- (aaa);
\draw (eee)--(bbb);
\draw (fff)--(ccc);
\draw (ggg)--(bbb);
\draw (hhh)--(ddd);
\draw (iii)--(ccc);
\draw (jjj)--(ddd);
\draw (kkk)--(eee);
\draw (kkk)--(fff);
\draw (lll)--(eee);
\draw (mmm)--(fff);
\draw (nnn)--(ggg);
\draw (ooo)--(ggg);
\draw (ooo)--(hhh);
\draw (ppp)--(hhh);
\draw (qqq)--(iii);
\draw (rrr)--(iii);
\draw (rrr)--(jjj);
\draw (sss)--(jjj);
\draw (ca)--(zz);
\draw (cb)--(zz);
\draw (bw)--(yy);
\draw (bx)--(yy);
\draw (bs)--(xx);
\draw (bt)--(xx);
\draw (bo)--(ww);
\draw (bp)--(ww);
\draw (bk)--(vv);
\draw (bl)--(vv);
\draw (bg)--(uu);
\draw (bh)--(uu);
\draw (by)--(bw);
\draw (bz)--(bx);
\draw (cc)--(ca);
\draw (cd)--(cb);
\draw (bu)--(bs);
\draw (bv)--(bt);
\draw (bq)--(bo);
\draw (br)--(bp);
\draw (bm)--(bk);
\draw (bn)--(bl);
\draw (bi)--(bg);
\draw (bj)--(bh);
\draw (c) -- (a);
\draw (b) -- (a);
\draw (d) -- (a);
\draw (e)--(b);
\draw (f)--(c);
\draw (g)--(b);
\draw (h)--(d);
\draw (i)--(c);
\draw (j)--(d);
\draw (k)--(e);
\draw (k)--(f);
\draw (l)--(e);
\draw (m)--(f);
\draw (n)--(g);
\draw (o)--(g);
\draw (o)--(h);
\draw (p)--(h);
\draw (q)--(i);
\draw (r)--(i);
\draw (r)--(j);
\draw (s)--(j);
\draw (aa)--(u);
\draw (ab)--(u);
\draw (ac)--(aa);
\draw (ad)--(ab);
\draw (ae)--(v);
\draw (af)--(v);
\draw (ag)--(ae);
\draw (ah)--(af);
\draw (ai)--(w);
\draw (aj)--(w);
\draw (ak)--(ai);
\draw (al)--(aj);
\draw (am)--(x);
\draw (an)--(x);
\draw (ao)--(am);
\draw (ap)--(an);
\draw (aq)--(y);
\draw (ar)--(y);
\draw (as)--(aq);
\draw (at)--(ar);
\draw (au)--(z);
\draw (av)--(z);
\draw (aw)--(au);
\draw (ax)--(av);
\end{tikzpicture}
\tiny
$H_1=-\widehat{\Phi};$
$H_2=\{\alpha\}\uplus -(\widehat{\Phi}\backslash \{\alpha\})$;
$H_3=\{\beta\}\uplus -(\widehat{\Phi}\backslash \{\beta\});$
$H_4=\{\delta-\alpha-\beta\}\uplus -(\widehat{\Phi}\backslash \{\delta-\alpha-\beta\})$;
$H_5=\{\alpha, \alpha+\beta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\alpha+\beta\})$;
$H_6=\{\beta, \alpha+\beta\}\uplus -(\widehat{\Phi}\backslash \{\beta,\alpha+\beta\});$
$H_7=\{\alpha, \delta-\beta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\delta-\beta\})$;
$H_8=\{\delta-\alpha-\beta,\delta-\beta\}\uplus -(\widehat{\Phi}\backslash \{\delta-\alpha-\beta,\delta-\beta\})$;
$H_9=\{\beta, \delta-\alpha\}\uplus -(\widehat{\Phi}\backslash \{\beta,\delta-\alpha\})$;
$H_{10}=\{\delta-\alpha-\beta,\delta-\alpha\}\uplus -(\widehat{\Phi}\backslash \{\delta-\alpha-\beta,\delta-\alpha\})$;
$H_{11}=\{\alpha, \alpha+\beta, \beta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\alpha+\beta, \beta\})$;
$H_{12}=\{\alpha, \alpha+\beta, \alpha+\delta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\alpha+\beta, \alpha+\delta\})$;
$H_{13}=\{\beta, \alpha+\beta, \beta+\delta\}\uplus -(\widehat{\Phi}\backslash \{\beta,\alpha+\beta, \beta+\delta\})$;
$H_{14}=\{\alpha, \delta-\beta, \alpha+\delta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\delta-\beta, \alpha+\delta\})$;
$H_{15}=\{\alpha, \delta-\beta, \delta-\alpha-\delta\}\uplus -(\widehat{\Phi}\backslash \{\alpha,\delta-\beta,\delta-\alpha-\delta\})$;
$H_{16}=\{2\delta-\alpha-\delta, \delta-\beta, \delta-\alpha-\delta\}\uplus -(\widehat{\Phi}\backslash \{2\delta-\alpha-\delta,\delta-\beta,\delta-\alpha-\delta\})$;
$H_{17}=\{\beta,\delta-\alpha, \beta+\delta\}\uplus -(\widehat{\Phi}\backslash \{\beta,\delta-\alpha, \beta+\delta\})$;
$H_{18}=\{\beta,\delta-\alpha, \delta-\beta-\delta\}\uplus -(\widehat{\Phi}\backslash \{\beta,\delta-\alpha, \delta-\beta-\delta\})$;
$H_{19}=\{2\delta-\beta-\delta,\delta-\alpha, \delta-\beta-\delta\}\uplus -(\widehat{\Phi}\backslash \{2\delta-\beta-\delta,\delta-\alpha, \delta-\beta-\delta\})$;
$T_1=\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{\alpha+\beta}))$;
$T_2=\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{\alpha+\beta}))$;
$T_3=\widehat{\beta}\uplus\widehat{-\alpha}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{-\alpha}))$;
$T_4=\widehat{-\beta-\alpha}\uplus\widehat{-\alpha}\uplus -(\widehat{\Phi}\backslash (\widehat{-\beta-\alpha}\uplus\widehat{-\alpha}))$;
$T_5=\widehat{-\beta-\alpha}\uplus\widehat{-\beta}\uplus -(\widehat{\Phi}\backslash (\widehat{-\beta-\alpha}\uplus\widehat{-\beta}))$;
$T_6=\widehat{\alpha}\uplus\widehat{-\beta}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{-\beta}))$;
$T_{11}=\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\beta\}))$;
$T_{12}=\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\beta\}))$
$T_{13}=\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\beta, \beta+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\beta, \beta+\delta\}))$;
$T_{14}=\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\beta, 2\delta-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\beta, 2\delta-\beta\}))$;
$T_{21}=\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\alpha\}))$;
$T_{22}=\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\alpha\}))$;
$T_{23}=\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\alpha, \alpha+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\alpha, \alpha+\delta\}))$;
$T_{24}=\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\alpha, 2\delta-\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{\alpha+\beta}\uplus \{\delta-\alpha, 2\delta-\alpha\}))$;
$T_{31}=\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\alpha+\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\alpha+\beta\}))$;
$T_{32}=\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\alpha-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\alpha-\beta\}))$;
$T_{33}=\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\alpha+\beta, \alpha+\beta+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\alpha+\beta, \alpha+\beta+\delta\}))$;
$T_{34}=\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\alpha-\beta, 2\delta-\alpha-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\alpha-\beta, 2\delta-\alpha-\beta\}))$;
$T_{41}=\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\beta\}))$;
$T_{42}=\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\beta\}))$;
$T_{43}=\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\beta, \beta+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\beta, \beta+\delta\}))$;
$T_{44}=\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\beta, 2\delta-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\alpha}\uplus \{\delta-\beta, 2\delta-\beta\}))$;
$T_{51}=\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\alpha\}))$;
$T_{52}=\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\delta-\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{\-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\delta-\alpha\}))$;
$T_{53}=\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\alpha, \alpha+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\alpha, \alpha+\delta\}))$;
$T_{54}=\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\delta-\alpha, 2\delta-\alpha\}\uplus -(\widehat{\Phi}\backslash (\widehat{-\alpha-\beta}\uplus\widehat{-\beta}\uplus \{\delta-\alpha, 2\delta-\alpha\}))$;
$T_{61}=\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\alpha+\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\alpha+\beta\}))$;
$T_{62}=\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\delta-\alpha-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\delta-\alpha-\beta\}))$;
$T_{63}=\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\alpha+\beta, \alpha+\beta+\delta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\alpha+\beta, \alpha+\beta+\delta\}))$;
$T_{64}=\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\delta-\alpha-\beta, 2\delta-\alpha-\beta\}\uplus -(\widehat{\Phi}\backslash (\widehat{\alpha}\uplus\widehat{-\beta}\uplus \{\delta-\alpha-\beta, 2\delta-\alpha-\beta\}))$;
$U_1=\widehat{\{\alpha,\beta,\alpha+\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{\alpha,\beta,\alpha+\beta\}})$;
$U_2=\widehat{\{-\alpha,\beta,\alpha+\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{-\alpha,\beta,\alpha+\beta\}})$;
$U_3=\widehat{\{-\alpha,\beta,-\alpha-\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{-\alpha,\beta,-\alpha-\beta\}})$;
$U_4=\widehat{\{-\alpha,-\beta,-\alpha-\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{-\alpha,-\beta,-\alpha-\beta\}})$;
$U_5=\widehat{\{\alpha,-\beta,-\alpha-\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{\alpha,-\beta,-\alpha-\beta\}})$;
$U_6=\widehat{\{\alpha,-\beta,\alpha+\beta\}}\uplus -(\widehat{\Phi}\backslash \widehat{\{\alpha,-\beta,\alpha+\beta\}})$;
\normalsize
\end{document} |
\begin{document}
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\def{\cal L}{{\cal L}}
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\title[Pascal Triangles of Configurations]{
Hyperplanes in Configurations, decompositions, and Pascal Triangle of Configurations}
\author{Krzysztof Pra{\.z}mowski}
\maketitle
\begin{abstract}
An elegant procedure which characterizes a decomposition of some class of binomial configurations
into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}.
In essence, this construction was already presented in \cite{perspect}. We show that
such a procedure is a result of fixing in configurations in some class $\mathcal K$ suitable hyperplanes
which both: are in this class, and deleting such a hyperplane results in a configuration
in this class. By a way of example we show two more (added to that of \cite{gevay})
natural classes of such configurations, discuss some other, and propose some open questions
that seem also natural in this context.
\par\noindent
Mathematics Subject Classification: 05B30, 51E30 (51E20)
\par\noindent
Keywords: Pascal Triangle (of binomials), binomial, configuration, hyperplane, combinatorial
Grassmannian, combinatorial Veronesian, Pascal Triangle of Configurations
\end{abstract}
\section*{Introduction}\label{sec:intro}
On one hand, ``Pascal Triangle'' is a term which is known to all mathematicians: it characterizes
an arrangement of binomial coefficients in a form of a `pyramid' such that each item is the sum
of items placed immediately above it. In another view: the sum of each neighbour items in a row equals to
the item which is their common neighbour (in the row below).
Clearly, binomial coefficients are simply values of a two-argument function $b(n,k)$ defined on nonnegative integers
($n=0,1,\ldots$, $k=0,\ldots,n$)
and nothing `magic' is in the pyramid defined above. It is a visual presentation of recursive
equation which these coefficients satisfy. Clearly, the sequences of boundary values $b(n,0)$ and $b(n,n)$
uniquely determine then the function $b$. Nevertheless the recurrence in question is extremely simple...
Quite recently, Gabor G{\'e}vay in \cite{gevay} noted that there is family of point-line configurations
which can be arranged in such a pyramid, with a suitably defined ``sum'' of the configurations in question.
Or: each (nontrivial, non-boundary) configuration in this family can be decomposed into two other members
of this family. In essence, this decomposition (even in a more general form) was presented also earlier
in \cite[Representation 2.12]{perspect};
the class in question consists of configurations which generalize Desargues configuration considered as schemes of
mutual perspectives between several simplexes. On other hand, such systems of (geometrical) perspectives can be found
even in the classical book of Veblen and Young \cite{class:proj} (G{\'e}vay quotes also explicitly Danzer and Cayley)
and its combinatorial schemes are special instances
of so called binomial graphs, investigated in the context of association schemes (cf. e.g. \cite{klin}),
and associated incidence structures. Combinatorial schemes characterizing these configurations can be found
already in \cite{levi} and \cite{coxet}.
So:
\begin{quotation}
the subject was known, but its regular nature was not known -- was not stated explicitly until \cite{gevay}.
\end{quotation}
But then it appeared that the ``sum'' of two configurations is not a well defined operation that depends
solely on the summands, and the associated decomposition is, in fact, associated with a choice of a hyperplane
in the decomposed configuration. After that become clear (we present these observation in
Section \ref{sec:binconfy}, Theorem \ref{thm:decompo0} and equation \eqref{eq:decomp0}) there appeared
that there are other natural known classes of configurations that can be arranged into respective
triangles. These are, in particular, so called combinatorial Veronesians (defined originally in \cite{combver},
without any connections with studying hyperplanes in configurations). In Section \ref{sec:exm}
we discuss some of the classes which appear within this theory.
\section{Notations, standard constructions}\label{sec:nota}
\subsection{Elementary combinatorics}\label{ssec:intro:combin}
There are well known formulas concerning binomial coefficients, frequently
referred to as ``Pascal Triangle of Binomials". To be more precise,
these formulas correspond
to the arrangement of the binomial coefficients in a pyramid with consecutive rows:
\begin{ctext}
$\Big( \left(\binom{n}{k}\colon k=0,\ldots,n \right)\colon n = 0,1,2,\ldots \Big)$.
\end{ctext}
Then the corresponding recursive formula is the following
\begin{eqnarray}\label{pyramid:0}
\binom{n}{k} & = & \binom{n-1}{k-1} + \binom{n-1}{k};
\end{eqnarray}
equation \eqref{pyramid:0} yields immediately next two:
\begin{eqnarray}\label{pyramid:1}
\textstyle{\binom{n}{k} - \binom{n-1}{k}} & = & \textstyle{\binom{n-1}{k-1}}, \text{ and}
\\ \label{pyramid:2}
\textstyle{\binom{n}{k} - \binom{n-1}{k-1}} & = & \textstyle{\binom{n-1}{k}}.
\end{eqnarray}
For purposes of our next investigations it will be more convenient to arrange
binomial coefficients into a (infinite) matrix:
\begin{ctext}
$\big[ \ginomx(m,k)\colon m,k = 0,1,\ldots \big]$,
\end{ctext}
where
\begin{equation}
\ginomx(m,k) = \ginom(m,k);
\end{equation}
clearly, $\ginomx(m,k) = \ginomx(k,m)$; the fundamental recursive formula for
the binomial coefficients takes the form
\begin{equation}\label{rec:ginom}
\ginomx(m,k) = \ginomx(m,k-1) + \ginomx(m-1,k).
\end{equation}
\subsection{Rudiments of geometry of configurations}\label{ssec:intro:config}
We say that a structure
${\goth K} = \struct{U,{\cal L},\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}}$ with $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}\; \subset U\times{\cal L}$
is a {\em $\konftyp(\nu,\rho,\beta,\kappa)$-configuration}
if $\goth K$ is a partial linear space
(i.e. $a,b \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A,B$ yields $a =b$ or $A = B$)
such that $|U| = \nu$, $|{\cal L}| = \beta$, exactly $\rho$ elements of ${\cal L}$
are in the relation $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}$ with $a\in U$, for each $a\in U$, and exactly
$\kappa$ elements of $U$ are in the relation $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}$ with $A \in {\cal L}$, for each
$A\in{\cal L}$.
Let $\goth K$ be a configuration as above, then the following equation (a specialized form
of the so called fundament equation of partial linear spaces) holds
\begin{equation}\label{equ:pls}
\nu \cdot \rho = \beta \cdot \kappa.
\end{equation}
The elements of $U$ are called {\em points} of $\goth K$, the elements of ${\cal L}$
are called {\em lines} of $\goth K$, and the relation $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}$ is {\em the incidence}.
The numbers $\rho$ and $\kappa$ are referred to as {\em point rank} and
{\em line size/rank} resp.
It is a folklore, that every configuration as above with $\kappa\geq 2$ is isomorphic
to a configuration, whose lines are sets of points, and the incidence is
the standard membership relation $\in$.
If this will not cause a confusion (as it may happen in particular examples)
we shall frequently assume that
the incidence of $\goth K$ is the membership relation.
A subset $\cal H$ of the set of points of $\goth K$ is called {\em a hyperplane} of $\goth K$
when
\begin{itemize}\def--{--}\itemsep-2pt
\item $\cal H$ is {\em a subspace} of $\goth K$, i.e. if
the conditions $a,b \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A\in{\cal L}$ and $a,b\in{\cal H}$, $a\neq b$
yield $x \in {\cal H}$ for every $x$ such that $x \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$,
\item[\strut] and
\item
each line of $\goth K$ crosses $\cal H$, i.e. for each $A\in{\cal L}$ there is
$x\in {\cal H}$ such that $x\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$.
\end{itemize}
Let $\cal H$ be a hyperplane of $\goth K$.
Then, for each line $A$ of $\goth K$ either there is a unique $x\in{\cal H}$ with
$x \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$ (we write $x = A^\infty$ in that case) or every point incident with $A$
belongs to $\cal H$: the set of such lines will be denoted by ${\cal L}[{\cal H}]$.
Clearly,
\begin{ctext}
${\goth K}\restriction{\cal H} := \struct{{\cal H},{\cal L}[{\cal H}],
\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} \cap \big({\cal H}\times{\cal L}[{\cal H}]\big) }$
\end{ctext}
is a partial linear space; quite frequently in the sequel we shall make no distinction between
$\cal H$ and ${\goth K}\restriction{\cal H}$.
Clearly, the set $U$ of all the points of $\goth K$ is a hyperplane of $\goth K$.
In what follows we shall assume that a hyperplane means a {\em proper} (i.e. ${\cal H}\neq U$)
subspace that satisfies suitable conditions.
Given a hyperplane $\cal H$ of $\goth H$ we define {\em the reduct}
\begin{ctext}
${\goth K} \setminus {\cal H} :=
\struct{ U\setminus {\cal H},\, {\cal L}\setminus {\cal L}[{\cal H}],\,
\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} \cap \big((U\setminus {\cal H})\times({\cal L}\setminus {\cal L}[{\cal H}])\big)}$;
\end{ctext}
if $\kappa \geq 3$ then ${\goth K}\setminus {\cal H}$ is a partial linear
space with all the lines of size (rank) $\kappa -1$.
Let us write, for symmetry, ${\goth K}_1 = {\goth K}\setminus {\cal H}$
and ${\goth K}_2 = {\goth K}\restriction{\cal H}$.
Recall, that we have a function $\infty$ from the lines
of ${\goth K}_1$ into the points of ${\goth K}_2$.
Let us try to ``reverse'' this decomposition:
\begin{constr}\label{def:zlepka}
Let ${\goth K}_i = \struct{U_i,{\cal L}_i,\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}_i}$ be a partial linear space for $i=1,2$.
Assume that $U_1 \cap U_2 = \emptyset = {\cal L}_1 \cap {\cal L}_2$.
Let $\infty\colon{\cal L}_1\longrightarrow U_2$ be a map such that the following
holds
\begin{ctext}
if $U_1\ni x \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A,B\in{\cal L}_1$ and $A^\infty = B^\infty$ then $A = B$.
\end{ctext}
We define
\begin{description}\itemsep-2pt
\item[$U:$] $= U_1 \cup U_2$,
\item[${\cal L}:$] $= {\cal L}_1\cup{\cal L}_2$,
\item[$\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}:$] $= \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}_1 \cup \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}_2 \cup \{(x,A) \colon U_2\ni x=A^\infty, A\in{\cal L}_1\}$.
\end{description}
Finally, we set
\begin{equation}\label{def:zlepka0}
{\goth K}_1 \rtimes_\infty {\goth K}_2 := \struct{U,{\cal L},\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}}.
\end{equation}
It is evident that
{\em ${\goth K}_1 \rtimes_\infty {\goth K}_2$ is a partial linear space}.
\end{constr}
\begin{prop}
Let ${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$ with ${\goth K}_i$
as in \ref{def:zlepka}.
Then $U_2$ is a hyperplane in $\goth K$ and ${\cal L}_2 = {\cal L}[U_2]$.
\end{prop}
\begin{proof}
It suffices to state directly that if $A\in{\cal L}$ then either
$A\in{\cal L}_2$ and then $x\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$ gives $x\in U_2$, or $A\in{\cal L}_1$ and then $x\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$
yields $x\in U_1$ or $U_2 \ni x = \infty(A)$.
\end{proof}
The construction of the type \ref{def:zlepka} is quite frequent in geometry.
One particular case let us mention below:
\begin{note}\normalfont
Let ${\goth K}_1 = \struct{U_1,{\cal L}_1,\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}_1,\parallel_1}$ be a partial linear
space with parallelism of lines; we write $[A]_{\parallel_1}$ for the equivalence
class of $A\in{\cal L}_1$ w.r.t. the relation $\parallel_1$
(i.e. simply for the direction of $A$).
Suppose that there is a formula $\Phi$ in the
language of ${\goth K}_1$ such that the relation
\begin{ctext}
$\{ ([A_1]_{\parallel_1},[A_2]_{\parallel_1},[A_3]_{\parallel_1})\colon
(A_1,A_2,A_3)\in{\cal L}_1^3,\ \Phi(A_1,A_2,A_3) \}$
\end{ctext}
is a ternary equivalence relation on the set $\big( {\cal L}_1\diagup\parallel_1 \big)^3$
(cf. \cite{ternequiv});
let ${\cal L}_2$ be the set of its equivalence classes, and
${\goth K}_2 = \struct{{{\cal L}_1\diagup\parallel_1},{\cal L}_2,\in}$.
With $A^\infty = [A]_{\parallel_1}$ for $A\in{\cal L}_1$ we obtain the structure
${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$ which is called, in that context, the
{\em closure of an affine structure} ${\goth K}_1$.
\par
In particular cases of this construction, practically, the structures
${\goth K}_1$ and $\goth K$ are given, and
{\em we search for an appropriate formula $\Phi$}
(see \cite{afclos}: affine completion, \cite{polarclos}, \cite{segreclos}).
{}
{\small$\bigcirc$}
\end{note}
Other examples of this construction will appear in the next Section.
\subsection{Dualization}\label{ssec:duale}
Let ${\goth K} = \struct{U,{\cal L},\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}}$ be an incidence structure;
we call the structure
\begin{ctext}
$\dual{{\goth K}} = \struct{{\cal L},U,\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1}}$
\end{ctext}
the dual of $\goth K$. It is evident that $\dual{{\goth K}}$ is a partial linear space
whenever $\goth K$ is so. In particular
\begin{ctext}
if $\goth K$ is a $\konftyp(\nu,\rho,\beta,\kappa)$-configuration then
$\dual{\goth K}$ is a $\konftyp(\beta,\kappa,\nu,\rho)$-configuration.
\end{ctext}
\begin{prop}\label{prop:dual-hypy}
Let $\cal H$ be a hyperplane of a partial linear space ${\goth K} = \struct{U,{\cal L}}$
such that the induced correspondence $\infty$ is bijective.
Then ${\cal L}\setminus{\cal L}[{\cal H}]$ is a hyperplane of $\dual{\goth K}$.
\end{prop}
\begin{proof}
Let $L_1,L_2 \in {\cal L}\setminus{\cal L}[{\cal H}]$. Assume that
$L_1,L_2\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1} x\in U$ and ${\cal L}\ni L \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1} x$.
Suppose that $L\notin{\cal L}\setminus{\cal L}[{\cal H}]$, then $L\in{\cal L}[{\cal H}]$
and, consequently, $x\in{\cal H}$. This gives $x = \infty(L_1) = \infty(L_2)$;
we have $L_1 = L_2$ then. This proves that ${\cal L}[{\cal H}]$
is a subspace of $\dual{\goth K}$.
\par
Let $L$ be an arbitrary line of $\dual{\goth K}$, then $L \in U$. If $L\notin{\cal H}$
then each line of $\goth K$ (each point of $\dual{\goth K}$) that passes through $L$
is in ${\cal L}\setminus{\cal L}[{\cal H}]$. If $L\in{\cal H}$ then
$\infty^{-1}(L) \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1} L$. This suffices for the proof.
\end{proof}
Standard examples show that the condition {\em $\infty$ is bijective}
assumed in \ref{prop:dual-hypy} cannot be removed. Indeed, the plane in a
projective 3-space $\goth P$
is a hyperplane, but the family of lines of the resulting affine 3-space is not
even a subspace of $\dual{\goth P}$.
However, \ref{prop:dual-hypy} appears useful when we deal with (binomial) configurations.
Proposition \ref{prop:dual-hypy} can be easily (re)formulated in a more `constructive'
fashion:
\begin{cor}\label{cor:dual-hipy}
Let ${\goth K}_i$ be configurations as in \ref{def:zlepka} with a suitable map
$\infty$ defined. Assume that $\infty$ is a bijection and
${\goth K} = {\goth K}_1\rtimes_\infty {\goth K}_2$.
Then
\begin{equation}\label{wzor:dual-hipy}
\dual{\goth K}\quad = \quad \dual{{\goth K}_2} \rtimes_{\infty^{-1}} \dual{{\goth K}_1}
\end{equation}
\end{cor}
\section{Binomial configurations}\label{sec:binconfy}
\subsection{Generalities}
The main subject of this section consists in investigations
on the family of {\em binomial configurations}
i.e. of configurations of the type
$\ginconf(k,m)$ for some positive integers $k,m$.
It is easily seen that each parameters of this form satisfy \eqref{equ:pls}.
Let us write
\begin{ctext}
$\ginkonfx(k,m)$ for the class of all $\ginconf(k,m)$-configurations.
\end{ctext}
\begin{thm}\label{thm:decompo0}
Let ${\goth K}\in\ginkonfx(k,m)$ and let $\cal H$ be a hyperplane of $\goth K$.
Assume that
\begin{sentences}\itemsep-2pt
\item\label{war1}
$\cal H$ is a configuration (in this case this means simply that
${\goth K}\restriction{\cal H}$ has constant point rank),
and
\item\label{war2}
${\goth K}\setminus{\cal H}$ is a binomial configuration.
\setcounter{zdanie}{\value{sentence}}
\end{sentences}
Then
\begin{sentences}\itemsep-2pt\setcounter{sentence}{\value{zdanie}}
\item\label{war3}
$\cal H$ is a binomial configuration, more precisely:
${\goth K}_2 = {\goth K}\restriction{\cal H}\in\ginkonfx(k-1,m)$;
\item\label{war4}
${\goth K}_1 = {\goth K}\setminus{\cal H}\in\ginkonfx(k,m-1)$;
\item\label{war5}
there is a 1-1 correspondence
$\infty\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$
such that
${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$.
\setcounter{zdanie}{\value{sentence}}
\end{sentences}
\end{thm}
\begin{proof}
Recall that, right from the definition, the points of $\goth K$ have rank $k$,
and the lines of $\goth K$ have size $m$.
Set $n=m+k-1$.
Then,
from the definition we get immediately that
the points of ${\goth K}_1$ are all of the same rank $k$ and
the lines are all of the size $m-1$,
so, in accordance with \eqref{war2},
${\goth K}_1$ is a $\ginconf(k,m-1)$-configuration, which justifies \eqref{war4}.
The number of points in ${\goth K}_1$ is $\binom{n-1}{k}$ and
the number of points of $\goth K$ is $\binom{n}{k}$;
from the Pascalian equations the number of points
of ${\goth K}_2$ is $\binom{n}{k} - \binom{n-1}{k} = \binom{n-1}{k-1}$.
Similarly we compute the number of lines of ${\goth K}_2$:
it equals to $\binom{n}{m} - \binom{n-1}{m-1} = \binom{n-1}{m}$.
The size of the lines in $\cal H$ is $m$;
from assumption \eqref{war1} and \eqref{equ:pls}
applied to ${\goth K}_2$ we get that the point rank in ${\goth K}_2$ equals to $k-1$.
So, ${\goth K}_2$ is a $\ginconf(k-1,m)$-configuration.
This justifies \eqref{war3}.
Finally, since each point in ${\goth K}_2$ has its rank on one less than in $\goth K$
we get that through each one of these points there passes exactly one line of ${\goth K}_1$,
so $\infty$ is a bijection, as required in \eqref{war5}.
\end{proof}
Informally speaking, \ref{thm:decompo0} gives a decomposition
\begin{equation}\label{eq:decomp0}
\ginconfx(k,m) = \ginconfx(k,m-1) \rtimes_\infty \ginconfx(k-1,m),
\end{equation}
which resembles reverent Pascalian equation \eqref{rec:ginom}.
But note, that the ``operation'' $\rtimes_\infty$ is not commutative, and
it depends essentially on the parameter $\infty$.
\begin{rema}\normalfont
{\it Not every hyperplane of a binomial configuration is a (binomial) configuration.}
Indeed, it suffices to have a look on hyperplanes in binomial partial Steiner
triple systems, either in a more general approach of \cite{hypinbin:psts} or
in a more particular case of \cite{hypingendes} and note that in the Desargues
configuration
a line accomplished with a point not joinable with any point on this line
is a hyperplane, it contains three points of rank $3$ and one point of rank $0$
so, it is not a configuration.
{}
{\small$\bigcirc$}
\end{rema}
\begin{rema}\normalfont
Let us consider the smallest sensible and possible case:
$\ginconfx(2,3) \rtimes \ginconfx(3,2) = \ginconfx(3,3)$.
If ${\goth K}\in\ginconfx(3,3)$ then $\goth K$ is a
$\ginconf(3,3)= \konftyp(10,3,10,3)$-configuration: one of
ten possible.
If ${\goth K}_1$ is a $\ginconf(3,2)=\konftyp(4,3,6,2)$-configuration then
it is the complete graph $K_4$.
If ${\goth K}_2$ is a $\ginconf(2,3)=\konftyp(6,2,4,3)$-configuration then
it is simply the Pasch-Veblen configuration $\goth V$.
It was shown in \cite{klik:VC} that there are exactly six maps $\infty$ which yield
pair wise non isomorphic configurations $K_4 \rtimes_\infty {\goth V}$.
So, {\em there are binomial configurations ${\goth K}_1,{\goth K}_2$
and bijections
$\infty',\infty''\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$
such that
${\goth K}_1\rtimes_{\infty'}{\goth K}_2 \not\cong {\goth K}_1\rtimes_{\infty''}{\goth K}_2$.}
Consequently, the symbol $\rtimes$ {\em is not a well defined operation,
without the argument $\infty$ defined explicitly}.
\end{rema}
\begin{rema}\normalfont
Let ${\goth K}_1,{\goth K}_2$ be binomial configurations, let a map
$\infty\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$
be a bijection.
\par
From assumption, ${\goth K}_i\in \ginconfx(k_i,m_i)$ for some integers $k_i,m_i$, $i=1,2$.
Moreover, the two numbers: of lines of ${\goth K}_1$ and of points of ${\goth K}_2$
coincide. This means than
$\binom{k_1+m_1-1}{m_1} = \binom{k_2+m_2-1}{k_2}$.
Then $k_1+m_1-1 = k_2+m_2-1$ and one of the following holds:
\begin{sentences}
\item
either $m_1 = m_2-1$ -- in this case $k_2 = k_1-1$ and
{\it ${\goth K} = {\goth K}_1 \rtimes_\infty{\goth K}_2$ is a binomial configuration},
\item
or $m_1 = k_2$ and then $k_1 = m_2$. Consider e.g. the case $k_1=m_1=k_2=m_2=3$, then
${\goth K}_i$ are $\konftyp(10,3,10,3)$-configurations. But then
${\goth K} = {\goth K}_1 \rtimes_\infty{\goth K}_2$ has $20$ points and $20$
lines. Ten lines have size $3$, and ten have size $4$. So, in this case
{\it $\goth K$ is not even a configuration.}
\end{sentences}
This shows that a `sum' of two binomial configurations, even determined by constructing `improper
points', may be not a binomial configuration.{}
{\small$\bigcirc$}
\end{rema}
In the next Section we present two remarkable families of binomial configurations
which yield families indexed by positive integers and which yield ``a Pascal Triangle".
\section{Examples}\label{sec:exm}
\subsection{Example: the family of combinatorial Grassmannians}\label{exm:grasy}
For an integer $k$ and a set $X$ we write $\sub_k(X)$ for the family of $k$-subsets
of $X$.
Nowadays the notation $\binom{X}{k}$ instead of $\sub_k(X)$ becomes widely used.
We prefer, however, not to mix integers and sets.
Let ${\goth K}\in\ginconfx(k,m)$; then the points of $\goth K$ can be identified with
the $k$-subsets of a fixed $n$-element set $X$, where $n = m+k-1$.
Let us identify the lines of $\goth K$ with the elements of $\sub_m(X)$ and
define
\begin{equation}\label{def:inc:gras0}
a \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A :\iff a \in \sub_k(X) \land A \in \sub_m(X) \land |a \cap A|=1.
\end{equation}
Suppose that $a\neq b$ and $a,b \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} A$ with $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}$ defined by \eqref{def:inc:gras0}.
Then $a\cap b = X\setminus A$ and therefore $A$ is uniquely determined by its
two points $a$ and $b$. So, the structure
\begin{ctext}
${\goth G}(k,m) := \struct{\sub_k(X),\sub_m(X),\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}}$
\end{ctext}
is a partial linear space. It is not too hard to verify that it is
a configuration with the lines of size $m$ and the points of rank $k$,
so ${\goth G}(k,m)\in\ginconfx(k,m)$.
In practice, the above presentation is not so easy to handle with and not too intuitive.
\begin{sentences}
\item
There is a one-to-one correspondence between the elements of $\sub_m(X)$ and
the elements of $\sub_{k-1}(X)$: indeed, $n = m + (k-1)$ so, the boolean complementation
$\varkappa$ is a bijection in question. Then we see that the pair of maps
$(\mathrm{id},\varkappa)$ maps ${\goth G}(k,m)$ onto the structure
$\struct{\sub_k(X),\sub_{k-1}(X),\supset}$, which coincides with the $DCD(n,k)$
introduced in \cite{gevay}.
\item
Analogously, there is a one-to-one correspondence between the elements of
$\sub_{m-1}(X)$ and the elements of $\sub_k(X)$; set $k_0 = m-1$, then
$(\varkappa,\mathrm{id})$ maps ${\goth G}(m,k)$ onto the structure
$\struct{\sub_{k_0}(X),\sub_{k_0+1}(X),\subset}$, which coincides with the
{\em combinatorial Grassmannian} $\GrasSpace(X,k_0)$ defined in \cite{perspect}.
\end{sentences}
Let us concentrate upon the presentation given in \cite{perspect}, let us
drop out the superfluous index $0$ and let
${\goth K} = \GrasSpace(X,k)$, $|X| = n$; remember that
$\GrasSpace(X,k) \in \ginconfx(n-k,k+1)$.
We write $\GrasSpace(n,k)$ for the type of $\GrasSpace(X,k)$ where $|X| = n$.
Let us fix an element $i\in X$, then $\sub_k(X)$ is the disjoint union
$\sub_k(X) = {\cal X}_1 \cup {\cal X}_2$, where
${\cal X}_1 = \{ a\in \sub_k(X)\colon i\in a \}$ and
${\cal X}_2 = \{ a\in\sub_k(x)\colon i \notin a\} = \sub_k(X\setminus\{i\})$.
The following is easily seen:
\begin{sentences}
\item
${\cal X}_2$ is a hyperplane of $\goth K$,
${\goth K}_2 := {\goth K}\restriction{{\cal X}_2} = \GrasSpace(X\setminus\{i\},k)$
\item
${\goth K}_1 = {\goth K}\setminus{\cal X}_2$, with the point-set ${\cal X}_1$,
is isomorphic under the map
${\cal X}_1\ni a \longmapsto a\setminus\{ i \}\in\sub_{k-1}(X\setminus\{ i \})$
to the structure $\GrasSpace(X\setminus\{i\},k-1)$.
\item\label{jawne1}
Let $A$ be a line of ${\goth K}_1$, so $A\in\sub_{k+1}(X)$ where $i\in A$.
Then $A \setminus \{ i \}\in \sub_k(A)\cap {\cal X}_2$, so $A^\infty = A\setminus\{i\}$.
\end{sentences}
In view of the above and \ref{thm:decompo0} we get that
\begin{prop}
If $i \in X$ is arbitrary then
\begin{equation}
\GrasSpace(X,k) \cong \GrasSpace(X\setminus\{i\},k-1)\rtimes_\infty\GrasSpace(X\setminus\{i\},k)
\end{equation}
with $\infty$ defined by \eqref{jawne1} above.
\end{prop}
In numerical symbols we can write:
$$\GrasSpace(n,k) = \GrasSpace(n-1,k-1)\rtimes_\infty\GrasSpace(n-1,k).$$
This decomposition was studied in many details in \cite{gevay}, it was also
noticed in \cite[Representation 2.12]{perspect}.
While expressed in terms of ${\goth G}(k,m)$ it assumes the form
$${\goth G}(k_0,m_0) = {\goth G}(k_0,m_0-1)\rtimes_\infty{\goth G}(k_0-1,m_0),$$
where $k_0 = n-k$, $m_0 = k+1$.
\subsection{Example: the family of combinatorial Veronesians}\label{exm:very}
Let $X$ be an $m$-element set; we write $\mbox{\large$\goth y$}_k(X)$ for the $k$-element
multisets with the elements in $X$. In naive words, a multiset is a `set'
whose elements belong to $X$, and each one of them can occur several times.
Formally, it is a function $f$ defined on $X$ with values in the set of natural numbers
(with zero); this function `counts' how many times given item from $X$ occurs in $f$.
It is a convenient way to symbolize such a function $f$ in the form
$f = \prod_{x\in X} x^{f(x)}$ (with the natural relations like $x^ix^j = x^{i+j}$,
$x^i y^j = y^j x^i$, $x^0 = 1$, $1 x = x$, etc...).
Then the cardinality of $f$ is $|f| = \sum_{x\in X}f(x)$.
We write ${\mathrm{supp}}(f) = \{ x\in X\colon f(x) > 0 \}$;
clearly, $|f| = \sum_{x\in{\mathrm{supp}}(f)} f(x)$
Let us write $\bigcup_{i=0}^{i=k-1} \mbox{\large$\goth y$}_i(X) =: \mbox{\large$\goth y$}_{<k}(X)$.
On the set $\mbox{\large$\goth y$}_k(X) \times \mbox{\large$\goth y$}_{<k}(X)$ we define the incidence relation $\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}$
by the formula:
\begin{equation}\label{def:inc:ver0}
e \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} f :\iff f = e\, x^{k-|e|} \text{ for some } x\in X.
\end{equation}
The structure
\begin{ctext}
$\VerSpace(m,k) = \struct{\mbox{\large$\goth y$}_k(X),\mbox{\large$\goth y$}_{<k}(X),\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}}$
\end{ctext}
is called a {\em combinatorial Veronesian}; the class of combinatorial Veronesians
was introduced in \cite{combver}.
It was proved that $\VerSpace(m,k)$ is a partial linear space with the points of rank
$k$ and the lines of size $m$; the formulas counting the cardinality of $\mbox{\large$\goth y$}_k(X)$ and
of $\mbox{\large$\goth y$}_{<k}(X)$ are known in the elementary combinatorics; summing up we get that
$\VerSpace(m,k)\in\ginconfx(k,m)$.
Let us fix $a\in X$ and define
${\cal X}_2 = \{f\in\mbox{\large$\goth y$}_k(X)\colon a\in{\mathrm{supp}}(f)\}$
and
${\cal X}_1 = \{f\in\mbox{\large$\goth y$}_k(X)\colon a\notin{\mathrm{supp}}(f)\}$;
then $\mbox{\large$\goth y$}_k(X)$ is the disjoint union ${\cal X}_1 \cup {\cal X}_2$.
\begin{sentences}
\item
It is seen that the map $\mbox{\large$\goth y$}_{k-1}(X)\ni f\longmapsto f\,a^1\in {\cal X}_2$
is a bijection. Suppose that $f' a^1, f'' a^1 \mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} e$ where $e\in\mbox{\large$\goth y$}_{<k}(X)$.
Then $a\in{\mathrm{supp}}(e)$ and $f',f''\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} \frac{e}{a}\in\mbox{\large$\goth y$}_{<k-1}(X)$.
Finally, $a\in{\mathrm{supp}}(f)$ for every $f$ with $f\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} e$, which yields that
${\cal X}_2$ is a subspace of $\VerSpace(X,k)$;
as we noted, it is isomorphic to $\VerSpace(X,k-1)$.
\item
Let $e\in\mbox{\large$\goth y$}_{<k}(X)$ be a line of $\VerSpace(m,k)$. If $a \in{\mathrm{supp}}(e)$ then
$f\in{\cal X}_2$ for every $f$ with $f\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} e$.
If $a\notin{\mathrm{supp}}(e)$ then $e^\infty = e\,a^{k-|e|}$ is the unique element incident with
$e$ which belongs to ${\cal X}_2$.
\item\label{jawne2}
Evidently, the points in ${\cal X}_1$ can be considered as the points
of $\VerSpace(X\setminus\{a\},k)$.
Let $e\in\mbox{\large$\goth y$}_{<k}(X\setminus\{ a \})$ be a line of $\VerSpace(X\setminus\{a\},k)$;
then $e^\infty = e\,a^{k-|e|}\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut} e$ is well defined.
\item
In particular, the above yields that ${\cal X}_2$ is a hyperplane of $\VerSpace(m,k)$.
\end{sentences}
Summing up, we obtain
\begin{prop}
Let $a\in X$ be arbitrary.
\begin{equation}
\VerSpace(X,k) = \VerSpace(X\setminus\{ a\},k) \rtimes_\infty \VerSpace(X,k-1),
\end{equation}
where $\infty$ is defined by \eqref{jawne2} above.
\end{prop}
In (numerical) symbols we can express this fact by
$$
\VerSpace(m,k) = \VerSpace(m-1,k)\rtimes_\infty \VerSpace(m,k-1).
$$
As a consequence of \cite[Cor. 4.8, Thm. 4.5]{combver}, $\VerSpace(m,k)$
is a combinatorial Grassmannian only for $k=2$ or $m=2$ so,
Grassmannians and Veronesians are essentially distinct families.
\subsection{Example: the family of dual combinatorial Veronesians}\label{exm:duvery}
In Subsections \ref{exm:grasy} and \ref{exm:very},
we have found decompositions of the scheme
$\ginconfx(k,m) = \ginconfx(k,m-1)\rtimes\ginconfx(k-1,m)$. Clearly,
$\ginconfx(m,k)$ are dual to $\ginconfx(k,m)$;
therefore, in view of \ref{cor:dual-hipy} one can expect that each of these
decompositions determines a decomposition of the scheme
\begin{ctext}
$\ginconfx(m,k) =\dual{\ginconfx(k,m)} =
\dual{\ginconfx(k-1,m)} \rtimes \dual{\ginconfx(k,m-1)} =
\ginconfx(m,k-1)\rtimes \ginconfx(m-1,k)$
\end{ctext}
In case of combinatorial Grassmannians the dualization procedure does not yield any
new family of configurations:
\begin{fact}
Let $n = |X|$ for a set $X$. Then
$\dual{\GrasSpace(X,k)} \cong \GrasSpace(X,n-k)$.
\end{fact}
However, the dual Veronesians yield another, third family: if $\dual{\VerSpace(m,k)}$
is (isomorphic to) a combinatorial Grassmannian then either $k=2$ or $m=2$;
if it is isomorphic to a combinatorial Veronesian then $k=2$, or $m=2$, or $k=3=m$.
Even $\dual{\VerSpace(k,k)} \cong \VerSpace(k,k)$ is not valid for $k > 3$
(see \cite[Thm.'s 4.14, 4.15]{combver})!
Let us adopt notation of Subsection \ref{exm:very} and let
${\goth K} = \struct{U,{\cal L}} = \VerSpace(X,k)$; let us remind that
${\cal X}_2 = \{ f\in\mbox{\large$\goth y$}_k(X)\colon a \in {\mathrm{supp}}(f) \}$ is a hyperplane of $\goth K$
and then
${\cal L}[{{\cal X}_2}] = \{ e\in \mbox{\large$\goth y$}_{<k}(X)\colon a \in{\mathrm{supp}}(e) \} =:{\cal L}_2$.
Consequently, ${\cal L}_1 := {\cal L}\setminus{\cal L}_2 = \mbox{\large$\goth y$}_{<k}(X\setminus\{ a\})$ is
a hyperplane of $\dual{\goth K}$; set
${\cal X}_1 := U\setminus{\cal X}_2 = \mbox{\large$\goth y$}_k(X\setminus\{ a \})$.
Consider a line $f\in\mbox{\large$\goth y$}_{k}(X)$ of $\struct{{\cal L}_2,{\cal X}_2}$;
then $a \in {\mathrm{supp}}(f)$: let $dg(a,f)$ be the greatest integer $s$ such that $f=a^s g$
for a multiset $g$.
We associate with such an $f$ the point $f^\infty = \frac{f}{a^{dg(a,f)}}\in{\cal L}_1$,
it is seen that we obtain
\begin{prop}
$\dual{\VerSpace(m,k)} =
\struct{{\cal L}_2,{\cal X}_2,\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1}} \rtimes_{\infty}\struct{{\cal L}_1,{\cal X}_1,\mathrel{\strut\rule{3pt}{0pt}\rule{1pt}{9pt}\rule{3pt}{0pt}\strut}^{-1}}
\cong \dual{\VerSpace(m,k-1)} \rtimes_{\infty} \dual{\VerSpace(m-1,k)}$.
\end{prop}
With the symbols $\VerSpacex(m,k) = \dual{\VerSpace(m,k)}\in\ginconfx(m,k)$
we arrive to
$$
\VerSpacex(m,k) = \VerSpacex(m,k-1) \rtimes_\infty \VerSpacex(m-1,k)
$$
Consequently, following \ref{cor:dual-hipy} we can explicitly characterize the Pascal
Triangle of Configurations consisting of dual combinatorial Veronesians.
\section{Comments and problems}
We have shown three families $\mathscr K$ of configurations
${\goth K}(m,k)\colon m,k=1, 2\ldots$ such that the formula
${\goth K}(m,k) = {\goth K}(m,k-1)\rtimes_{\infty_{m,k}}{\goth K}(m-1,k)$
is valid for all $m,k$ and suitable maps $\infty_{m,k}$.
One can expect that there are more such families: the point is to find a
suitable family
\begin{ctext}
$\big[\infty_{m,k}\colon\sub_{m-1}(m+k-2)\longrightarrow\sub_{k-1}(m+k-2)\colon
m,k=1,2,\ldots \big]$
\end{ctext}
It is seen how huge variety of binomial partial triple systems can be obtained
via `completing' complete graphs (see \cite{hypinbin:psts}):
one can expect that our procedure produces
much more required configurations (cf. Problem \ref{prob:rozmnoz}).
However, one essential question appears: which of them can be realized
in a Desarguesian projective space: we call them {\em projective} then.
It is known that all the combinatorial Grassmannians
are projective. It is also known that (practically all) combinatorial Veronesians are not
projective (only $\VerSpace(3,3)$ and $\VerSpace(2,k)$, $\VerSpace(m,2)$ are realizable).
Similarly, dual of combinatorial Veronesians are also not projective (besides
the exceptions indicated before), \cite[Thm.'s 6.9, 6.10]{combver}.
The statement like
{\em if ${\goth K}_1$ and ${\goth K}_2$ are realizable then
${\goth K}_1\rtimes{\goth K}_2$, if it is a (binomial) configuration then is realizable as well}
is false, in general. It suffices to present $\VerSpace(4,3)$ as the ``sum''
of projectively realizable structures $\VerSpace(3,3)$ and $\VerSpace(4,2)$.
So, a natural question arises
\begin{prob}\label{prob:rozmnoz}
Assume that ${\goth K}_1$ and ${\goth K}_2$ are projective (binomial) configurations
which satisfy corresponding `recursive equation'
\begin{equation}
{\goth K}_1\in\ginconfx(k,m-1) \text{ and } {\goth K}_2\in\ginconfx(k-1,m)
\text{ for some } k,m\geq 2.
\end{equation}
Then there is a bijection
$\infty\colon\text{ lines of }{\goth K}_1\longrightarrow \text{ points of }{\goth K}_2$
so as
${\goth K}_1 \rtimes_\infty{\goth K}_2 \in\ginconfx(k,m)$.
This observation enables us to construct `Pascal Triangle of Configurations' from, practically,
arbitrary boundary sequences of configurations, considering arbitrary $\infty$'s.
\par
For which maps $\infty$ (is there
necessarily at least one) the structure ${\goth K}_1\rtimes_\infty{\goth K}_2$
is projective?{}
{\small$\bigcirc$}
\end{prob}
Note that ``boundary'' sequences $\ginconfx(2,k)$ and $\ginconfx(k,2)$ are known:
$\ginconfx(2,k) = \{ \dual{K_{k+1}} \}$ and
$\ginconfx(k,2) = \{ K_{k+1} \}$,
and these two sequences consist of projective configurations.
So, considering configurations decomposed with the following schemes
\begin{ctext}
$\ginconfx(3,k) = \ginconfx(3,k-1)\rtimes \ginconfx(2,k) =
\ginconfx(3,k-1)\rtimes \dual{K_{k+1}}$,
$\ginconfx(k,3) = \ginconfx(k,2)\rtimes \ginconfx(k-1,3) =
K_{k+1} \rtimes \ginconfx(k-1,3)$.
\end{ctext}
the real problem lies in the classification/choice of bijections $\infty$!
In particular, there are known binomial partial Steiner triple systems not in the
families $\VerSpace(?,?)$ nor among $\GrasSpace(?,?)$, and nor among $\dual{\VerSpace(?,?)}$
which are projective,
for example, so called quasi-Grassmannians of \cite{skewgras}.
Each such structure ${\goth R}_n$ has parameters as the corresponding $\GrasSpace(n,2)$.
So, there arises a very particular, but intriguing
\begin{prob}
Is there a map $\infty$ such that the structure
${\goth R}_{n-1}\rtimes_\infty\GrasSpace(n-1,3)$ (which has the parameters of
$\GrasSpace(n,3)$) is realizable in a Desarguesian projective space.{}
{\small$\bigcirc$}
\end{prob}
\section*{Addendum}
The paper is a result of discussions during Combinatorics 2018 in Arco.
\begin{small}
\noindent
Authors' address:
\\
Krzysztof Pra{\.z}mowski,
\\
Institute of Mathematics, University of Bia{\l}ystok
\\
K. Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok, Poland
\\
e-mail: \[email protected]+,
\end{small}
\end{document} |
\begin{document}
\title{Description and complexity of Non-markovian open quantum dynamics}
\author{Rahul Trivedi$^{1, 2}$}
\email{[email protected], [email protected]}
\address{Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str.~1, 85748 Garching, Germany,\\
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 Munich, Germany,\\
Department of Electrical and Computer Engineering, University of Washington, Seattle, WA - 98195, USA.}
\date{\today}
\begin{abstract}
Understanding and simulating non-Markovian quantum dynamics remains an important challenge in open quantum system theory. A key advance in this endeavour would be to develop a unified mathematical description of non-Markovian dynamics, and classify its complexity in the many-body setting. In this paper, we identify a general class of non-Markovian memory kernels, described by complex-valued radon measures, and define their dynamics through a regularization procedure constructing the corresponding system-environment unitary groups. Building on this definition, we then consider $k-$local many-body non-Markovian systems with physically motivated assumptions on the total variation and smoothness of the memory kernels. We establish that their dynamics can be efficiently approximated on quantum computers, thus providing a rigorous verification of the Extended Church-Turing thesis for this general class of non-Markovian open quantum systems.
\end{abstract}
\maketitle
\section{Introduction}
Quantum systems invariably interact with their environment, and any model describing their behaviour should model this interaction. Often such systems are approximated as Markovian, wherein the environment does not retain any memory of the system. Markovian open quantum systems have been extensively studied in quantum information theory and quantum optics --- a mathematically rigorous description of a unitary group over the system-environment Hilbert space which generates Markovian open quantum dynamics is provided in the theory of quantum stochastic calculus \cite{hudson1984quantum, kr1992introduction} as the solution of a quantum stochastic differential equation. When evolved with this unitary group, the system's reduced state satisfies the Lindbladian master equation \cite{breuer2002theory}. Furthermore, Markovian open quantum systems have also been considered in the many-body setting, whereing the system has an exponentially large Hilbert space --- several quantum algorithms have been developed to simulate their dynamics on quantum computers \cite{zanardi2016dissipative,chenu2017quantum,cleve2016efficient, kliesch2011dissipative}.
However, a number of quantum systems arising in solid-state physics \cite{finsterholzl2020nonequilibrium, chin2011generalized, groeblacher2015observation, de2008matter}, quantum optics \cite{calajo2019exciting, andersson2019non, gonzalez2019engineering, aref2016quantum, leonforte2021vacancy} as well as quantum biology and chemistry \cite{ishizaki2005quantum, chin2012coherence, ivanov2015extension, caycedo2021exact} are not Markovian and the environment's memory needs to be explicitly taken into account. Furthermore, these systems are also many-body, and simulating their dynamics requires the development of quantum algorithms that can account for the environment's memory. This opens up the question of formulating a mathematically rigorous description of non-Markovian dynamics, as well as studying their computational complexity.
In this paper, we provide rigorous answers to both of these questions. We first consider the problem of formulating a well-defined non-Markovian model --- while it is generically expected that non-Markovian open quantum systems satisfy a generalized Nakajima-Zwazig \cite{xu2018convergence, ivanov2015extension} or time-convolutionless master equation \cite{smirne2010nakajima, pereverzev2006time,kidon2015exact}, it is usually hard to obtain such a master equation explicitly except only for weakly interacting environments \cite{vacchini2010exact, schroder2007reduced, timm2011time, mukamel1978statistical}. Given that generalizing the Markovian master equation to the non-Markovian setting is hard, a natural question is if we can formulate a well-defined system-environment unitary group that is general enough to capture all physically relevant systems. This question is non-trivial to rigorously answer for most physical models, since the environment is often described by a quantum field theory as opposed to a finite-dimensional system.
This question has been addressed in several specific settings in open quantum systems literature --- using standard theory of second quantization, unitary groups have been constructed for non-Markovian normalizable spin boson models \cite{leggett1987dynamics} as well as for weakly interacting non-normalizable spin boson models \cite{lonigro2021generalized}. Furthermore, for time-delayed feedback systems, non-Markovian dynamics can be rigorously defined by expanding the system Hilbert space with time \cite{grimsmo2015time, whalen2017open}. In this paper, we identify a general class of non-perturbative non-Markovian models with memory kernels described by tempered radon measures, which have been studied in classical probability and distribution theory as generalization of the delta function \cite{grigis1994microlocal}. We rigorously define a unitary group, which in general generates non-Markovian system dynamics, associated with a tempered radon measure. Our generalization ties together disparate models used for open-system dynamics within the same mathematical framework --- as special cases, we recover Markovian dynamics, quantum systems with time-delay and feedback \cite{calajo2019exciting, grimsmo2015time, pichler2016photonic} and spin-boson models described by spectral density functions with vanishing high-frequency response \cite{leggett1987dynamics}. A major difficulty in constructing this unitary group is that the Schroedinger's equation for the system-environment state cannot be naively used since it is not guaranteed to have a solution. Our key technical contribution is to construct this unitary group via a regularization procedure --- we use standard mollifiers to regularize the radon measure to obtain a non-Markovian model where the Schroedinger's equation has a guaranteed solution, show that the limit of this solution as the regularization is removed exists and hence defines the dynamics associated with the radon measure.
We next consider the simulability of non-Markovian many-body dynamics, thus defined, on quantum computers. While the quantum simulability of Markovian many-body open quantum systems \cite{zanardi2016dissipative,chenu2017quantum,cleve2016efficient, kliesch2011dissipative}, and many-body closed quantum systems\cite{lloyd1996universal, berry2014exponential, low2019hamiltonian, berry2015hamiltonian} have been extensively studied, non-Markovian open quantum systems have remain relatively unexplored. \edit{Physical intuition suggests non-Markovian dynamics should be efficiently simulable on quantum computer since a non-Markovian open quantum system should be approximable by a Markovian dilation i.e.~a larger Markovian open quantum system which includes some environment modes. However, to the best of our knowledge, the current bounds on the approximation error incurred by a Markovian dilation grow exponentially with evolution time in the worst case \cite{trivedi2021convergence, mascherpa2017open}, and are known to grow at-most polynomially with time only for non-Markovian models with sufficiently rapidly decaying spectral density functions \cite{woods2015simulating, woods2016dynamical}. This implies that the number of environment modes needed to be retained in the Markovian dilation would need to be increased exponentially with time to control the approximation error. Consequently, any quantum algorithm based on such Markovian dilations would cease to be efficient after evolution time that scales polynomially with system size, thus leaving open the question of quantum simulability of non-Markovian dynamics. }
Here, we establish that the general class of non-Markovian many-body models with memory kernels described by tempered radon measures can be efficiently simulated on a quantum computer under physically motivated assumptions on the growth and smoothness of the memory kernel. The quantum algorithm for simulating non-Markovian dynamics relies on a Markovian dilation of non-Markovian dynamics using a Lanczos iteration, also referred to the star-to-chain transformation \cite{chin2010exact, woods2014mappings}, which has been previously analyzed for Pauli-Fierz Hamiltonians \cite{woods2015simulating, woods2016dynamical, gualdi2013renormalization}, as well as for general models with distributional memory kernels but under the assumption of a finite particle emission rate into the environment \cite{trivedi2021convergence}. The key technical contribution in our work that allows us to prove that the quantum algorithm is efficient (i.e. with run-time polynomial in system size \emph{and} the evolution time) is to establish error bounds between the non-Markovian dynamics and its Markovian dilation which grow polynomially with system size and evolution time, as opposed to the previously known bounds which in the worst case grow exponentially with evolution time \cite{trivedi2021convergence, mascherpa2017open}.
\section{Overview of results}
\subsection{Informal Summary}
\edit{\emph{Setup}: We consider non-Markovian open quantum system models where a quantum system, described by a finite-dimensional Hilbert space $\mathcal{H}_S$, interacts with $M$ bosonic baths. Each bath is individually a symmetric Fock spaces over $L^2(\mathbb{R})$ i.e.~the Hilbert space of the $\alpha^\text{th}$ bath can be described within second quantization with an annihilation operator $a_{\alpha, \omega}$ for $\omega \in \mathbb{R}$ with the canonical commutation relations (CCR)
\[
[a_\omega, a_\nu] = 0, [a_\omega, a_\nu^\dagger] = \delta(\omega - \nu).
\]
Physically, it is convenient to interpret each bath as a continuum of Harmonic oscillators with $a_{\alpha, \omega}$ as the annihilation operator corresponding to the mode of the $\alpha^\text{th}$ bath at frequency $\omega$. We will consider a system-environment Hamiltonian that can be formally written as
\begin{subequations}\label{eq:hamiltonian_heuristic}
\begin{align}
H = H_S(t) + \sum_{\alpha = 1}^M H_{\alpha, E} + \sum_{\alpha = 1}^M \bigg(L_\alpha^\dagger A_\alpha+ \text{h.c.} \bigg),
\end{align}
where
$H_S(t)$ is the Hamiltonian describing the system dynamics in the absence of its interaction with the environment and $H_{\alpha, E}$ is the Hamiltonian corresponding to the environment's bath and is given by
\begin{align}
H_{\alpha, E} = \int_{-\infty}^\infty \omega a_{\alpha, \omega}^\dagger a_{\alpha, \omega} d\omega.
\end{align}
\end{subequations}
The interaction between the $\alpha^\text{th}$ bath and the system is specified by a system operator $L_\alpha$ (which we will refer to as `jump operators' to be consistent with the terminology for Markovian systems) and $A_\alpha$ is an operator that acts on the $\alpha^\text{th}$ bath can be formally expressed as
\begin{align}\label{eq:mode_annihilation_operator}
A_\alpha = \int_{-\infty}^\infty \hat{v}_\alpha(\omega) a_{\alpha, \omega} \frac{d\omega}{\sqrt{2\pi}},
\end{align}
where $\hat{v}_\alpha$, referred to as a `coupling function' throughout this paper, describes the frequency-dependence of the interaction between the system and the $\alpha^\text{th}$ bath. It will often be convenient to analyze this Hamiltonian in the interaction picture with respect to the bath, in which case the Hamiltonian is given by
\[
H(t) = H_S(t) + \sum_{\alpha = 1}^M \bigg(L_\alpha^\dagger A_\alpha(t) + \text{h.c.}\bigg),
\]
where
\begin{align*}
A_\alpha(t) &= e^{iH_{\alpha, E} t} A_\alpha e^{-i H_{\alpha, E} t}=\int_{-\infty}^\infty \hat{v}_{\alpha}(\omega) a_{\alpha, \omega}e^{-i\omega t} \frac{d\omega}{\sqrt{2\pi}}.
\end{align*}
\begin{figure}
\caption{Schematic depiction of a quantum system interacting with an environment of $M$ bosonic bath. The system dynamics is specified by, a possibly time-dependent, Hamiltonian $H_S(t)$. The interaction of the system with the individual bosonic baths is described by a jump operator $L_\alpha$ and a coupling function $\hat{v}
\end{figure}
\emph{Well definition of non-Markovian dynamics}: While one can formally write down a Hamiltonian for the system-environment interaction (Eq.~\ref{eq:hamiltonian_heuristic}), it is a time-dependent unbounded operator over an infinite-dimensional Hilbert space, and thus it is not immediate that a propagator corresponding to this Hamiltonian can be defined using the Schroedinger's equation (since its solution might not exist). Furthermore, the environment's Hilbert space is a Fock space over a continuum of modes and, in general, the operator $A_\alpha$ (Eq.~\ref{eq:mode_annihilation_operator}) does not correspond to the annihilation operator of a normalizable mode in the environment's Hilbert space, e.g.,~when the coupling function $\hat{v}_\alpha$ is the Fourier transform of a distribution as opposed to a square-integrable function. This is not just a mathematical consideration, several models of practical interest (such as that of Markovian quantum optical systems or quantum optical systems with time-delay and feedback) are modelled by distributional coupling functions.
To make further progress on this problem, we first identify a class of coupling functions $\hat{v}_\alpha$ for which we can rigorously define a dynamical unitary group. It is useful to consider a very simple setting --- the single particle spontaneous emission problem into a single bath, i.e.~$\mathcal{H}_S \cong \mathbb{C}^2 \cong \text{span}\{ \ket{0}, \ket{1}\} $, $L = \ket{0}\bra{1}$. For an initial system-environment state $\ket{\psi(0)} = \ket{1}\otimes \ket{\textnormal{vac}}$. The amplitude of the system being in $\ket{1}$, $\varepsilon(t) = \bra{1} \otimes \bra{\text{vac}} \psi(t)\rangle$, is governed by
\begin{align}\label{eq:decay_equation}
\frac{d}{dt}\varepsilon(t) = -\int_0^t \mu(s) \varepsilon(t - s) ds
\end{align}
where $\mu$ is the non-Markovian memory kernel for the bath given by
\[
\mu(t) = [A(t), A^\dagger(0)] = \int_{-\infty}^\infty \abs{\hat{v}(\omega)}^2 e^{-i\omega t} \frac{d\omega}{2\pi} .
\]
As we previously alluded to, $\mu$ can be a distribution. For Eq.~\ref{eq:decay_equation} to have a meaningful solution, we expect that the right hand side should be finite, else the derivative of $\varepsilon$ will be ill-defined. This expectation suggests that the memory kernel $\mu$ introduced above should be bounded i.e.~its application on a function $f$, that isnt large, should not be large. We can formalize this by demanding that for any continuous and compactly supported function $f$ with support $\Omega \subseteq \mathbb{R}$,
\begin{align}\label{eq:radon_measure_bound}
\abs{\int_{-\infty}^\infty \mu(t) f(t) dt}\leq c_\Omega \norm{f}_{L^\infty},
\end{align}
where $\norm{f}_{L^\infty} = \sup_{x\in \mathbb{R}} \abs{f(x)}$ and $c_\Omega$ is a constant that only depends on the support $\Omega$. The class of memory kernels that satisfy such a constraint have been well studied in analysis, and are called radon measures \cite{grigis1994microlocal}. Furthermore, in most physical problems of interest, the coupling function $\hat{v}$ and all of its derivatives grow at-most polynomially with $\omega$ --- since $\mu$ is a Fourier transform of $\abs{\hat{v}}^2$ for such a $\hat{v}$, it can additionally be assumed to be a tempered distribution. We will refer to such kernels as tempered radon measures throughout this paper.
Several kernels corresponding to models of common physical systems are tempered radon measures. For instance, Markovian open quantum systems are described by the delta function kernel, $\mu(t) = \gamma\delta(t)$ --- it can be immediately seen that this kernel is bounded in the sense of Eq.~\ref{eq:radon_measure_bound} with the constant $c_\Omega = \gamma$ for all $\Omega$. Non-Markovian open quantum systems with time-delays \cite{calajo2019exciting, grimsmo2015time, pichler2016photonic} and feedback are often modelled with kernels that are expressible as a sum of delta function i.e.~$\mu(t) = \sum_{j = 1}^P a_j \delta(t- \tau_j)$, and hence Eq.~\ref{eq:radon_measure_bound} is again satisfied with $c_\Omega = \sum_{j = 1}^P \abs{a_j}$. Finally, a very large class of physical models are described by memory kernels corresponding to square-integrable coupling functions ($\hat{v} \in L^2(\mathbb{R})$) --- examples of such models include lossy cavity QED systems in quantum optics \cite{tamascelli2018nonperturbative, pleasance2020generalized, mazzola2009pseudomodes, dalton2001theory, garraway2006theory, dalton2012quasimodes}, non-Markovian spin-boson models \cite{leggett1987dynamics} and non-Markovian models with ohmic baths \cite{shi2016bound}. If $\hat{v}$ is square integrable, then it follows that the corresponding kernel $\mu(t)$ is a bounded continuous function of $t$ i.e.~for all $t$,
\[
\abs{\mu(t)} \leq {\int_{-\infty}^\infty \abs{\hat{v}(\omega)}^2 \frac{d\omega}{2\pi}} = \frac{\norm{\hat{v}}_{L^2}^2}{2\pi}.
\]
Consequently, it follows that Eq.~\ref{eq:radon_measure_bound} is again satisfied with
\[
c_\Omega = \text{diam}(\Omega) \sup_{t \in \mathbb{R}}\abs{\mu(t)} \leq \text{diam}(\Omega)\frac{\norm{\hat{v}}_{L^2}^2}{2\pi},
\]
where $\text{diam}(\Omega) = \sup_{x, y \in \Omega} \abs{x - y}$ is the diameter of $\Omega \in \mathbb{R}$. These examples suggest that the class of tempered radon measure is general enough to encompass models for most physically relevant open-quantum systems.
Having identified a class of physically relevant coupling functions or memory kernels, we now turn to the question of the existence of the solution to the Schroedinger equation for the Hamiltonian in Eq.~\ref{eq:hamiltonian_heuristic} i.e.~we are interested in defining a unitary group $U(t, s)$ that satisfies
\[
i \frac{d}{dt} U(t, s) = H(t) U(t, s) \ \text{with} \ U(s, s) = \text{id}.
\]
If the coupling functions $\hat{v}_\alpha$ are square integrable, then the well-definition of the solution to the Schroedinger's equation follows from the standard theory of time-dependent Hamiltonians over infinite-dimensional Hilbert spaces \cite{kato1953integration, kato1956linear}. Physically, this is an easy case since the annihilation operators $\hat{A}_\alpha$ appearing in the Hamiltonian in Eq.~\ref{eq:hamiltonian_heuristic} correspond to a well defined (normalizable) mode in the bath's Hilbert space. In order to treat the more difficult cases where the coupling functions are not square integrable, but correspond to memory kernels that are tempered distributions, we proceed via a regularization procedure. We first regularize the coupling function $\hat{v}_\alpha$ to a square integrable coupling function --- this accomplished by the transformation
\begin{align}\label{eq:reg_informal}
\hat{v}_\alpha(\omega) \to \hat{\rho}(\omega \varepsilon) \hat{v}_\alpha(\omega),
\end{align}
where $\varepsilon$ is the regularization parameter and $\hat{\rho}$ is the fourier transform of a smooth function $\rho$ that approximates the delta function. Such functions are also called mollifiers --- common examples include a gaussian or a standard mollifier \cite{grigis1994microlocal}. Physically, this regularization damps the high-frequency components of $\hat{v}_\alpha(\omega)$ --- it suppresses amplitudes of coupling functions at frequencies larger than $\sim 1 / \varepsilon$. The existence of the solution to the Schroedinger equation for the regularized coupling function is guaranteed since the coupling function is square integrable --- we then study the limit of this solution as the regularization is removed (i.e.~as $\varepsilon \to 0$). Our main result is to show that if the memory kernels corresponding to the coupling functions are tempered radon measures then this limit exists, and hence defines the solution of the Schroedinger equation.
\begin{theorem}[Informal]
\label{theorem:non_mkv_exis}
A unitary group corresponding to the Hamiltonian in Eq.~\ref{eq:hamiltonian_heuristic} can be defined for non-Markovian model s for which the coupling functions $\hat{v}_\alpha$ correspond to memory kernels that are tempered radon measures.
\end{theorem}
\noindent We provide a more technically precise statement of this theorem in the next section. \\
\emph{Simulability of non-Markovian dynamics on quantum computers}: Now that we have a well-defined model for described non-Markovian open quantum systems, we can turn to another fundamentally important question --- is this model, in the many-body setting, simulable on a quantum computer? The quantum extended church turing thesis suggests that for physically reasonable models, this should indeed be the case else we would have a system that is potentially more powerful than a quantum computers. Apart from being a fundamental question, it could also be practically interesting to develop quantum algorithms for simulating many-body non-Markovian open quantum systems on quantum computers, since they are a model of a variety of interesting physical systems in solid-state physics \cite{finsterholzl2020nonequilibrium, chin2011generalized, groeblacher2015observation, de2008matter}, quantum optics \cite{calajo2019exciting, andersson2019non, gonzalez2019engineering, aref2016quantum, leonforte2021vacancy} and quantum biology and chemistry \cite{ishizaki2005quantum, chin2012coherence, ivanov2015extension, caycedo2021exact} .
Our next result, under some further assumptions on the growth and smoothness of the memory kernels of the non-Markovian system, provides a quantum algorithm for simulating many-body quantum dynamics whose run-time scales polynomially with the system-size i.e.~it is provably efficient. Our key contribution that enables this result is to show that the regularization of the non-Markovian environment, even when the memory kernel is a tempered distribution, results in an error that increases only polynomially (as opposed to exponentially) with system size and evolution time. While we do not focus on finding an optimal quantum algorithm for non-Markovian dynamics, the basic steps and analysis procedure that we develop in this paper could lay the foundations of future works improving the algorithm's run-time.
The general strategy behind the quantum algorithm is schematically depicted in Fig.~\ref{fig:reg_method}(a). First the bath Hilbert space, which is a continuum of modes, is discretized into a finite set of bosonic modes. Second, the infinite-dimensional Hilbert space of the resulting bosonic modes are truncated to obtain a finite-dimensional many-body problem. Then, any standard quantum algorithm for many-body Hamiltonian simulation \cite{lloyd1996universal, berry2014exponential, low2019hamiltonian, berry2015hamiltonian} to simulate the non-Markovian dynamics. To rigorously analyze the run-time of the quantum algorithm, errors in both the approximation steps need to be analyzed --- the discretization step is the more challenging one to analyze and we focus most of this paper on that step. The truncation of the Hilbert space is more straightforward and follows from a simple bound on the environment's particle number moments.
\begin{figure}
\caption{(a) Schematic depiction of the steps involved in approximating a non-Markovian environment. First, the environment is discretized, which is done in two steps --- regularization with introduction of frequency cutoff followed an application of the star-to-chain transformation. Then, the Hilbert space of the environment, which is still infinite dimensional, is truncated to finite dimensions. (b) Transformation of a system-environment coupling function before applying the star-to-chain transformation --- the coupling function is first regularized to dampen the high frequency contribution and then a frequency cut-off is introduced.}
\label{fig:reg_method}
\end{figure}
Consider the problem of discretizing the continuum of modes --- this problem has been previously studied for square-integrable coupling functions that additionally have a hard high frequency cutoff (i.e.~$\hat{v}(\omega) = 0$ for $\abs{\omega} \geq \omega_c$ for some cut-off frequency $\omega_c$). For this case, a discrete set of modes equivalent to the continuum can be computed using a Lanczos iteration, also known as the star-to-chain transformation --- this transformation, which is described in detail in Ref.~\cite{woods2016dynamical}, approximates the bath by a semi-infinite nearest-neighbour 1D chain of bosonic modes where the first mode couples to the local system through the operator $L$. If all the infinite number of modes are retained, then this mapping is exact --- for practical simulations, only a finite number of modes is retained. An application of the Lieb-Robinson bound for 1D bosonic lattices \cite{woods2015simulating} can then be used to show that with $N_b$ modes, local system dynamics at time $t$ can be approximated within a $\text{poly}(M, g, t, N_b^{-1}, \omega_c)$ total variation error. Here $M$ is the number of baths in the environment (Eq.~\ref{eq:hamiltonian_heuristic}), and $g$ is such that $\norm{H_S(t)}, \norm{L_\alpha} \leq g$.
The more general class of tempered radon measures contains coupling functions that do not have a frequency cutoff. Furthermore, several coupling functions [e.g. memory kernels that are expressible as a sum of delta functions ($\mu(t) = \sum_{j = 1}^P a_j \delta(t - \tau_j)$)], have appreciable high frequency amplitudes. A frequency cutoff must be introduced to apply the star-to-chain transformation to these cases --- this again is done in two steps depicted schematically in Fig.~\ref{fig:reg_method}(b). First, we use the regularization defined in Eq.~\ref{eq:reg_informal} --- this leads to a coupling function that rapidly goes to 0 at frequency scales $\sim 1 / \varepsilon$, where $\varepsilon$ is the regularization parameter. We rigorously show that the error incurred in this regularization scales as $\text{poly}(M, g, t, \varepsilon)$ --- a notable contribution in our work is to prove an error bound that grows as $\text{poly}(t)$ for a very large class of memory kernels (tempered radon measures), as opposed to exponentially with $t$ as had been shown previously \cite{trivedi2021convergence, mascherpa2017open}. We numerically verify this prediction for a single-particle problem in Fig.~\ref{fig:benchmark_fig}. on a 1D XY spin chain with periodic boundary conditions. Here, the system has $n$ spins, and the system Hamiltonian is
\[
H_S = \sum_{i = 1}^n J \big(\sigma_i^\dagger \sigma_{i + 1} + \sigma_{i + 1}^\dagger \sigma_i \big),
\]
where $\sigma_{n + 1} \cong \sigma_1$. Furthermore, the spins at sites $k = 1, \sqrt{n} + 1, 2\sqrt{n} + 1 \dots$ couple to a bath with jump operator $\sqrt{\gamma}\sigma_k$ and coupling function $v_k(\omega) = 1 + e^{i\varphi}e^{i\omega t_d}$ [which corresponds to $\mu(t) = 2\delta (t) + e^{i\varphi} \delta(t + t_d) + e^{-i\varphi} \delta(t - t_d)$]. We consider exciting the first spin, and compare the error, in the reduced density matrix of the spins, for the true and regularized models after $t = n / \gamma$. Fig.~\ref{fig:reg_method}(b) shows the dynamics of the true and regularized models and we see that the regularized model reproduces the true dynamics as $\varepsilon \to 0$ --- in Fig.~\ref{fig:reg_method}(c), we study the scalings of the regularization error with both $\varepsilon$ and system size $n$ --- as theoretically predicted, this error scales polynomially with $\varepsilon$ and $n$.
Having regularized the coupling function, the high frequency tails of such a decaying coupling function can then be ignored to introduce a hard frequency cutoff, enabling an application of the star-to-chain transformation. The analysis of the frequency truncation of the regularized model used a bound on the moments of high-frequency particle number in the environment, and can be shown to scale as $\text{poly}(t, \varepsilon^{-1}, \omega_c^{-1})$. Combining this with the estimate of the regularization step and the truncated star-to-chain transformation, we obtain that the total error in discretizing the continuum of modes in the bath to a set of $N_b$ discrete modes is given by
\[
\text{poly}\bigg(\omega_c, \frac{1}{N_b}\bigg) + \text{poly}\bigg(\frac{1}{\varepsilon}, \frac{1}{\omega_c}\bigg) + \text{poly}\big(\varepsilon\big),
\]
where we have hidden the polynomial dependence on $M, g$ and $t$ for brevity. From this expression, we note that a choice of $\varepsilon^{-1}, \omega_c$ and $N_b$ as $\text{poly}(M, g, t, 1/\epsilon)$ ensures that this error is $O(\epsilon)$. This result provides us with a bound on the number of discrete modes of the bath needed to accurately simulate the non-Markovian model, and this bound scales at-most polynomially with $t$ and with the parameters $M, g$.
\begin{figure}
\caption{Numerical validation study of the regularization step for a single-particle problem. (a) We consider a system which is a 1D XY model with periodic boundary conditions , $n$ spins, hopping strength $J$, and every $\sqrt{n}
\label{fig:benchmark_fig}
\end{figure}
This result has an immediate consequence on the complexity classification of many-body non-Markovian quantum dynamics. For a more precise statement of this result, we specifically analyze a physically motivated many-body setting --- the problem of $k-$local non-Markovian many-body dynamics. We consider a system of $n$ spins with the local-system Hamiltonian $H_S(t)$ which is $k-$local i.e. $H_S(t) = \sum_{i} H_i(t)$, where $H_i(t)$ act on at-most $k-$spins and has a normalization $\norm{H_i(t)} \leq O(1)$. Furthermore, we assume that the jump operators individually also act on at-most $k$ spins, and that the memory kernel corresponding to each jump operator is a tempered radon measure (with some additional reasonable conditions on its growth and smoothness, that were also required earlier). This problem can be considered to be a generalization of the $k-$local Hamiltonian dynamics \cite{lloyd1996universal, berry2014exponential, low2019hamiltonian, berry2015hamiltonian} and $k-$local Markovian dynamics \cite{zanardi2016dissipative,chenu2017quantum,cleve2016efficient, kliesch2011dissipative}, simulation problems. Applying the discretization procedure described above, followed by a truncation of the Hilbert space of the resulting bosonic modes, we show that
\begin{theorem}[Informal]
\label{theorem:bqp_non_mkv} The $k-$local non-Markovian many body dynamics problem can be efficiently simulated on a quantum computer.
\end{theorem}
We note that that establishing error bounds on the discretization procedure that grow as $\text{poly}(t)$, as opposed to the previously proven bounds which were exponential in $t$, is central to establishing this result. This is so because, if the exponential in $t$ growth of errors was indeed tight, then it would indicate that using this discretization procedure would not yield an efficient algorithm for simulating the non-Markovian many-body system after $t = \text{poly}(n)$. To the best of our knowledge, our work is the first that establishes error bounds on this discretization that grows as $\text{poly}(t)$ for memory kernels that are tempered radon measures, and thus enables a proof of the theorem above.
}
\subsection{Formal Statements and proof ideas}\label{subsec:proof_ideas}
\emph{Well definition of non-Markovian dynamics}: As discussed in the previous section, a {radon measure} $\mu$ is a map from the space of continuous and compactly supported functions ($\textnormal{C}_c^0(\mathbb{R})$) to complex numbers which is bounded in the sense that $\forall f\in \textnormal{C}_c^0(\mathbb{R})$ with support $\text{supp}(f)\subseteq \Omega \subseteq \mathbb{R}$
\[
\abs{\langle \mu, f\rangle} \leq \textnormal{TV}_{\Omega}(\mu) \sup_{x \in \Omega}\abs{f(x)},
\]
with $\textnormal{TV}_\Omega(\mu)$ is defined to be the total variation of $\mu$ within the compact set $\Omega$. Furthermore, we will call a radon measures that has a fourier transform which is of at-most polynomial growth in frequency a tempered radon measure. Since most physical systems have a spectral density function that do not grow very rapidly with frequency, and since the fourier transform of $\mu_\alpha$ describes the environment's spectral density function, it is reasonable to assume it to be a tempered radon measure.
A non-Markovian model can thus be specified by the system Hamiltonian, jump operators and the coupling functions in between system and the baths. These coupling functions are provided as a tempered radon transform, which specifies the magnitude of the coupling function, and the phase of the coupling function.
\begin{definition}[Non-Markovian model]\label{def:model}
A non-Markovian open system model for a quantum system with finite-dimensional Hilbert space $\mathcal{H}_S$ is specified by
\begin{enumerate}
\item[(a)] A time-dependent system Hamiltonian $H_S(t) $ which is Hermitian, norm continuous and differentiable in $t$,
\item[(b)] A set of coupling functions $\{(\mu_\alpha, \varphi_\alpha) \}_{\alpha \in \{1, 2 \dots M\}}$, where $\mu_\alpha$ are tempered radon measures and $\varphi_\alpha : \mathbb{R}\to \mathbb{C}$ specify the phase of the coupling functions,
\item[(c)] A set of bounded jump operators \\ $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$.
\end{enumerate}
\end{definition}
We now turn to the question of defining the quantum dynamics corresponding to a non-Markovian model. In general, this cannot be done simply through the Schroedinger's equation, since for general radon measure kernel it isnt clear if a meaningful solution with the Hamiltonian in Eq.~\ref{eq:hamiltonian_heuristic} exists. We approach the problem of defining the associated quantum dynamics through a regularization procedure. An elementary but important observation that enables this regularization is that if the coupling functions are square integrable ($v_\alpha \in L^2(\mathbb{R})$), then the solution to Schroedinger's equation with Hamiltonian in Eq.~\ref{eq:hamiltonian_heuristic} can be shown to exist using standard tools from the theory of non-autonomous differential equations on Banach spaces \cite{kato1953integration, kato1956linear}. Now, a coupling function $(\mu, \varphi)$ can be approximated by a square integrable function by using a mollifier\footnote{A mollifier $\rho$ is a smooth compact function ($\rho \in \textnormal{C}_c^\infty(\mathbb{R})$) which is positive and with support $\textnormal{supp}(\rho) \subseteq [-1, 1]$ with
\[
\int_{[-1, 1]} \rho(x) dx = 1.
\]
Unless otherwise mentioned, we will assume $\rho$ to be a symmetric (even) function.
}(smoothing function) $\rho$.
\begin{definition}[Regularization]\label{def:regularization}
For $\varepsilon > 0$ and given a symmetric mollifier $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$, an $\varepsilon, \rho-$regularization of a distributional coupling function $(\mu, \varphi)$ is a square integrable function $v_\varepsilon \in L^2(\mathbb{R})$ whose fourier transform\footnote{We will assume the following convention for the fourier transform $\hat{v}$ of $v \in L^2(\mathbb{R})$: \[
\hat{v}(t) = \int_{\mathbb{R}} {v(\omega)}e^{-i\omega t} \frac{d\omega}{\sqrt{2\pi}}.
\]} is given by $\hat{v}_\varepsilon \in L^2(\mathbb{R})$ is given by
\[
\hat{v}_\varepsilon(\omega) = \sqrt{ \hat{\mu}(\omega)} \hat{\rho}({\omega}{\varepsilon}) e^{i\varphi(\omega)} \ \forall \omega \in \mathbb{R},
\]
where $\hat{\rho}$ is the fourier transform of $\rho$ and $\hat{\mu}$ is the fourier transform of $\mu$ \footnote{For a tempered distribution $\mu \in \mathcal{S}'(\mathbb{R})$, the fourier transform $\hat{\mu}$ (if it exists as a function from $\mathbb{R}$ to $\mathbb{C}$) is defined by demanding that
\[
\langle \mu, f\rangle = \int_{\mathbb{R}} \hat{\mu}(\omega) \hat{f}(\omega) \frac{d\omega}{\sqrt{2\pi}}
\]
for all smooth compact functions $f \in \textnormal{C}_c^\infty(\mathbb{R})$.
}.
\end{definition}
We note that since $\rho$ is smooth and compact, its fourier transform $\hat{\rho}$ falls off faster than any polynomial in $\omega$ as $\omega \to \infty$ and thus $v_\varepsilon$ is indeed a square integrable function. Equivalently, this regularization step can be considered as approximating the kernel $\mu$ with another radon measure $\mu_\varepsilon$ whose action on a continuous compact function $f$ is given by
\[
\langle \mu_\varepsilon, f\rangle = \langle \mu, f\star \rho_\varepsilon \star \rho_\varepsilon \rangle,
\]
where $\rho_\varepsilon(x) = \varepsilon^{-1}\rho(\varepsilon^{-1}x)$ and $\star$ denotes a convolution operation. It can be seen that as $\varepsilon \to 0$, $f \star \rho_\varepsilon \star \rho_\varepsilon $ becomes an increasingly better approximation to $f$, and thus $\mu_\varepsilon$ becomes an increasingly better approximation of $\mu$.
Since the regularized coupling functions are square integrable, their associated dynamics can be computed by solving the Schroedinger's equation. We can now study the limit of this dynamics on removing the regularization ($\varepsilon \to 0$) --- our first result shows that this limit exists, and is independent of the choice of the mollifier. The proof of this result, provided in section \ref{sec:well_def}, relies on an upper bound bound on the rate of change of two-point correlation functions of system observables that is uniform in the regularization parameter $\varepsilon$.\\
\begin{reptheorem} {theorem:non_mkv_exis}[Non-markovian dynamics]
Given a non-Markovian open system model (definition \ref{def:model}) with $U_{\varepsilon, \rho}(t, s)$ for $t, s \in \mathbb{R}$ being the propagator corresponding to an $\varepsilon, \rho-$regularization of its coupling functions, $\lim_{\varepsilon \to 0} U_{\varepsilon, \rho}(t, s)$ exists weakly\footnote{A single parameter family of operators $\{O_x:\mathcal{D} \to \mathcal{H}\}_{x\in [0, \infty)}$ is said to converge weakly (or converge in the weak topology) to $O:\mathcal{D} \to \mathcal{H}$ as $x\to 0$ if $\forall \ket{\psi} \in \mathcal{D}$, $\lim_{x\to 0} O_x\ket{\psi} = O\ket{\psi}$. } as an isometry from a dense subspace $\mathcal{D}$ of the system-environment Hilbert space and is independent of the choice of the mollifier $\rho$.
\end{reptheorem}
\emph{Simulability of non-Markovian dynamics on quantum computers}: Our next result considers the simulability of non-Markovian dynamics of many-body systems on quantum computers. To make further progress, we need two additional assumptions --- one on the radon measures describing the coupling functions between the many-body system and the bath, and the second on the initial state in the environment.
\begin{assumption}
\label{assump:radon_measure}
The radon measure $\mu$ corresponding to the coupling function should satisfy:
\begin{enumerate}
\item[(a)] For any compact interval $[a, b] \subseteq \mathbb{R}$, $\textnormal{TV}_{[a, b]}(\mu) \leq \textnormal{poly}(\abs{a}, \abs{b})$ and
\item[(b)] Given a compact interval $[a, b] \subseteq \mathbb{R}$, $\exists\Delta^0_{\mu; [a, b]}(\varepsilon)$, $\Delta^1_{\mu; [a, b]}(\varepsilon) = \textnormal{poly}(a, b, \varepsilon)$ which are locally integrable with respect to $a, b$, tend to $0$ as $\varepsilon \to 0$ and for any differentiable function $f \in \textnormal{C}^1(\mathbb{R})$
\begin{align*}
&\abs{\langle \mu, f_{[a, b]} \star \rho_\varepsilon \rangle - \lim_{\varepsilon \to 0} \langle \mu, f_{[a, b]} \star \rho_\varepsilon \rangle } \leq \nonumber \\
&\ \Delta_{\mu; [a, b]}^0(\varepsilon) \sup_{t \in [a, b]}\abs{f(t)} + \Delta_{\mu; [a, b]}^1(\varepsilon)\sup_{t \in [a, b]}\abs{f'(t)}
\end{align*}
where $f_{[a, b]}(t) = f(t)$ if $t \in [a, b]$ and $0$ otherwise.
\end{enumerate}
\end{assumption}
Assumption \ref{assump:radon_measure}(a), which constrains the growth of the total variation of the memory kernel, can be physically interpreted as limiting the amount of ``memory" that the non-Markovian system can accumulate. Assumption \ref{assump:radon_measure}(b) is a constraint on the smoothness of the memory kernel (i.e.~the error incurred on smoothing the memory kernel with a mollifier), and it limits how rapidly the memory kernel diverges from its smooth approximations. Both of these assumptions are satisfied coupling functions encountered in most experimentally relevant physical systems.
\begin{assumption}
\label{assump:initial_state}
The initial environment state $\ket{\phi_1}\otimes \ket{\phi_2} \dots \ket{\phi_M}$ where for $\alpha \in \{1,2 \dots M\}$, $\ket{\phi_\alpha} \in \textnormal{Fock}[L^2(\mathbb{R})]$ and for its $n-$particle wavefunctions $\phi_{\alpha, n} \in L^2(\mathbb{R}^n)$, and any $j, k \geq 0$, $\exists \mathcal{N}_{j, k} > 0$ such that
\[
\sum_{n = 0}^\infty n^j \int_{\mathbb{R}^n} (1 + \omega_1^2)^k \abs{\phi_{\alpha, n}(\omega)}^2 d\omega < \mathcal{N}_{j, k}.
\]
\end{assumption}
Assumption \ref{assump:initial_state} demands that both high particle number or high frequency amplitude of the initial wavefunction vanishes superpolynomially. This assumption is reminiscent of assumption on particle number and energies of initial states made in studying the simulatability of quantum field theories \cite{jordan2012quantum, jordan2018bqp}. A number of commonly used initial environment states (such as thermal states) in physically relevant open systems have exponentially vanishing high energy amplitudes, and satisfy this assumption.
The computational problem of simulating $k-$local many-body non-Markovian dynamics can now be stated as
\begin{problem}[$k-$local non-Markovian dynamics]
\label{prob:k_local_non_mkv}
Consider a system of $n$ qudits $(\mathcal{H}_S = \big(\mathbb{C}^d\big)^{\otimes n})$ interacting with $M = \textnormal{poly}(n)$ baths with
\begin{enumerate}
\item[(a)] System Hamiltonian $H_S(t)$ is $k-$local i.e. $H_S(t) = \sum_{i = 1}^{N} H_i(t)$, where $N = \textnormal{poly}(n)$, and $H_i(t)$ is an operator acting on atmost $k$ qudits and $\norm{H_i(t)} \leq 1$.
\item[(b)] Jump operators $\{L_\alpha \}_{\alpha \in \{1, 2 \dots M\}}$ such that $L_\alpha$ acts on at-most $k$ qudits and $\norm{L_\alpha} \leq 1$.
\item[(c)] Coupling functions $\{(\mu_\alpha, \varphi_\alpha) \}_{\alpha \in \{1, 2 \dots M\}}$ such that $\mu_\alpha$ satisfies assumption \ref{assump:radon_measure}.
\item[(d)] An initial state $\ket{\Psi} = \ket{0}^{\otimes n} \otimes \ket{\Phi}$, where $\ket{\Phi} = \ket{\phi_1} \otimes \ket{\phi_2}\otimes \dots \ket{\phi_M}$ is an initial state which satisfies assumption \ref{assump:initial_state}. Furthermore, the initial state is computable in the sense that for $v_1, v_2 \dots v_m \in L^2(\mathbb{R})$ and $P \in\mathbb{Z}_{>0}$, all the amplitudes
\begin{align*}
\bra{\textnormal{vac}}\prod_{i = 1}^m \bigg(\int_{\mathbb{R}}v_i(\omega)a_\omega d\omega\bigg)^{n_i} \ket{\phi_\alpha}
\end{align*}
with $n_1 + n_2 \dots n_m \leq P$ can be computed in $\textnormal{poly}(m, P)$ time on a classical or quantum computer.
.
\end{enumerate}
Denoting by $\rho_S(t)$ the reduced state of the system at time $t$ for this non-Markovian model, then for $t = \textnormal{poly}(n)$, prepare a quantum state $\hat{\rho}$ such that $\norm{\hat{\rho} - \rho_S(t)}_\textnormal{tr} \leq 1 / \textnormal{poly}(n)$.
\end{problem}
The key ingredient to analyzing this problem is a Markovian dilation of the non-Markovian model, which identifies a finite number of modes in the environment's Hilbert space and then approximates the non-Markovian model by a Hamiltonian simulation of the system only interacting with these modes. We make this precise in the definition below.
\begin{definition}[Chain Dilation] Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, coupling functions $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$ and jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$. A chain dilation, with $N_m$ modes and bandwidth $B$, of this model is described by the following Hamiltonian over the system-environment Hilbert space $(\mathcal{H}_S \otimes \textnormal{Fock}[L^2(\mathbb{R})]):$
\begin{align*}
&H(t) = \nonumber\\
& H_S(t) + \sum_{\alpha = 1}^M\bigg(g_\alpha a_{\alpha, 1} L_\alpha^\dagger + \sum_{i = 1}^{N_m - 1} t_{\alpha, i} a_{\alpha, i}^\dagger a_{\alpha, i + 1} + \textnormal{h.c.}\bigg),
\end{align*}
where for $\alpha \in \{1, 2 \dots M\}, i \in \{1, 2 \dots N_m\}$,
\begin{enumerate}
\item[(a)] The operator $a_{\alpha, i} = \int_{\mathbb{R}} \varphi_{\alpha, i}(\omega) a_{\alpha, \omega} d\omega$ is the annihilation operator corresponding to the $i^\textnormal{th}$ mode of the $\alpha^\textnormal{th}$ bath described by the orthonormal mode functions $\varphi_{\alpha, i} \in \textnormal{L}^2(\mathbb{R})$ $($i.e. $\langle \varphi_{\alpha, i}, \varphi_{\alpha, i'}\rangle = \delta_{i, i'})$.
\item[(b)] The coupling constants $g_\alpha, t_{\alpha, i}$ are upper bounded by the bandwidth $B$ i.e.~$\abs{g_\alpha}, \abs{t_{\alpha, i}} \leq B$.
\end{enumerate}
\end{definition}
Our next lemma, which is used in the analysis of problem \ref{prob:k_local_non_mkv}, uses the well-known star-to-chain transformation \cite{chin2010exact, woods2014mappings} to systematically construct a Markovian dilation to the non-Markovian system. We analyze the error between the dynamics of the non-Markovian system and its Markovian dilation and estimate the number of modes and bandwidth of a Markovian dilation needed to approximate the non-Markovian model.
\begin{proposition}[Chain dilation]
\label{theorem:final_dilation}
Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$ and coupling functions $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$ where $\mu_\alpha$ satisfy assumption \ref{assump:radon_measure} and $\hat{\mu}_\alpha(\omega) < O(\omega^{2k}$) for some $k > 0$. For $\ket{\Psi_0} := \ket{\sigma} \otimes \ket{\Phi_0} \in \mathcal{H}_S\otimes \textnormal{Fock}[L^2(\mathbb{R})]^{\otimes M}$, where $\ket{\sigma} \in \mathcal{H}_S$ and $\ket{\Phi_0}$ is an initial environment state that satisfies assumption \ref{assump:initial_state}, then $\exists$ a Markovian dilation of the non-Markovian model with
\begin{align*}
&N_m, B \leq O\bigg(\textnormal{poly}\bigg(\frac{1}{\epsilon}, t, M, \sup_{\alpha}\norm{L_\alpha},\nonumber \\
&\qquad \qquad \qquad \qquad \sup_{\alpha, s \in [0,t]}\norm{[H_S(s), L_\alpha]},\nonumber\\
&\qquad \qquad \qquad \qquad \mathcal{N}_{1, k + 1}, \mathcal{N}_{1, k + 2}, \mathcal{N}_{1, 0}\bigg)\bigg)
\end{align*}
whose system-environment state at time $t$ is within $\epsilon$ norm distance of the exact state.
\end{proposition}
Our analysis of the Markovian dilation, detailed in the supplement, is performed in three steps. First, we analyze the error incurred in regularizing the non-Markovian model with a mollifier $\rho$, then we introduce a sharp frequency cutoff on the resulting regularized square-integrable functions. A key technical contribution in our analysis of the regularization and frequency cutoff is to prove bounds on error that grow only polynomially with time, which improves previous bounds that, in the worst case, grow exponentially with time \cite{trivedi2021convergence, mascherpa2017open}. After introduction of this cut-off, we perform a star-to-chain transformation --- the analysis of this step closely follows previous works that have studied the convergence of the star-to-chain transformation for the spin-boson models with a hard frequency cutoff \cite{woods2015simulating, woods2016dynamical, gualdi2013renormalization}.
Using this lemma, we can map problem \ref{prob:k_local_non_mkv} into a Hamiltonian simulation problem with a finite number of modes. This problem is still infinite-dimensional --- however, we can easily show that the moments of the particle number operator for the environment can grow at-most polynomially with the problem size $n$. Therefore, we can truncate the Hilbert space of this model and obtain a finite-dimensional Hilbert space --- an application of the sparse Hamiltonian lemma \cite{aharonov2003adiabatic} then yields the the second main result of our paper. A detailed proof is provided in section \ref{sec:complexity}.
\begin{reptheorem}{theorem:bqp_non_mkv}[$k-$local Non-Markovian dynamics $\in$ BQP]
Problem \ref{prob:k_local_non_mkv} can be solved in $\textnormal{poly}(n)$ time on a quantum computer.
\end{reptheorem}
\section{Notation and preliminaries}\label{sec:notation_prelim}
This section describes the notation used throughout this paper. For the convenience of the reader, and for the sake of completeness, we also collect some basic definitions and facts from the theory of function spaces and analysis that we use in this paper --- the interested reader can refer to Refs.~\cite{reed1972methods, reed1975ii} for more detailed discussion. The reader can choose to skip this section, and refer back to it as and when it is referenced in the following sections.\\
\noindent \emph{General}: For $x := (x_1, x_2 \dots x_n) \in \mathbb{R}^n, y:= (y_1, y_2 \dots y_m) \in \mathbb{R}^m$ we will denote by $(x, y) \in \mathbb{R}^{n + m}$ defined by $(x, y) = (x_1, x_2 \dots x_n, y_1, y_2 \dots y_m)$. For an ordered subset $B$ of $\{1, 2 \dots n\}$ and $x \in \mathbb{R}^n$, $Bx = (x_{B(1)}, x_{B(2)} \dots )$. We will denote by $\alpha^n$ the $n-$element constant vector $(\alpha, \alpha \dots \alpha)$, and by $\alpha^\infty$ the constant sequence $(\alpha, \alpha, \alpha \dots)$. \\
\noindent\emph{Function spaces and analysis}: Throughout this paper, all integrals over $\mathbb{R}^n$ will be Lesbesgue integrals with respect to the Lesbesgue measure over $\mathbb{R}^n$. Two measurable functions $f, g:\mathbb{R}^n \to \mathbb{C}$ are equal almost everywhere, denoted by $f =_{a.e.} g$, if the set $\{x | f(x) \neq g(x)\}$ is a zero measure set. For a measurable function $f:\mathbb{R}^n \to \mathbb{R}$ and a measurable set $\Omega \subseteq \mathbb{R}^n$, $\text{ess sup}_{x \in \Omega} f(x) = c$ if the set $\{x \in \Omega | f(x) > c\}$ is a zero measure set. We will use the standard $L^p$ function spaces throughout this paper --- for $p \geq 1$, we will denote by
\[
L^p(\mathbb{R}^n) =\bigg\{ f:\mathbb{R} \to \mathbb{C} \bigg | \norm{f}_{L^p} < \infty\bigg\} \bigg/ =_{a.e.}
\]
{where }
\[
\norm{f}_{L^p} := \bigg(\int_{\mathbb{R}^n}|f(x)|^p dx\bigg)^{1/p}.
\]
and
\[
L^\infty(\mathbb{R}^n) = \bigg\{ f:\mathbb{R} \to \mathbb{C} \bigg | \norm{f}_{L^\infty} < \infty\bigg\}\bigg/ =_{a.e.},
\]
\text{where }
\[
\norm{f}_{L^\infty} = \textnormal{ess\ sup}_{x \in \mathbb{R}^n} |f(x)|.
\]
Recall that for $g:\mathbb{R}^n \to \mathbb{R}$, $\textnormal{ess\ sup}_{x\in \mathbb{R}^n} g(x)$ is the smallest $M \in \mathbb{R}$ such that the set $\{x \in \mathbb{R}^n | g(x) \geq M\}$ has measure 0.
A map $f:\mathbb{R} \to \mathbb{C}$ is said to be a compactly supported function with support $\textnormal{supp}(f) \subseteq \mathbb{R}$ if $\overline{\textnormal{supp}(f)}$ is compact, and the set $\{ x \in \mathbb{R}\setminus \textnormal{supp}(f) | f(x) \neq 0\}$ is a zero measure set. For $k \in \mathbb{Z}_{\geq 0}$, we will denote by $\textnormal{C}^k(\mathbb{R})$ the set of $k-$differentiable functions (with $k = 0$ being continuous functions, and $k = \infty$ being smooth) from $\mathbb{R}$ to $\mathbb{C}$, and by $\textnormal{C}^k_c(\mathbb{R})$ the set of such functions with compact support.
A function $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$ is said to be a mollifier if $\rho(x)\geq 0 \ \forall x \in \mathbb{R}$ and $\int_{\mathbb{R}} \rho(x) dx = 1$ --- unless otherwise mentioned, we will assume that $\textnormal{supp}(\rho) = [-1, 1]$. The mollifier is symmetric if $\rho(x) = \rho(-x) \ \forall x \in \mathbb{R}$. Given a mollifier $\rho$ and $\varepsilon > 0$, we will denote by $\rho_\varepsilon \in \textnormal{C}_c^\infty(\mathbb{R})$ the map
\[
\rho_\varepsilon(x) = \frac{1}{\varepsilon} \rho\bigg(\frac{x}{\varepsilon}\bigg) \ \forall \ x \in \mathbb{R}.
\]
Note that $\rho_\varepsilon$ is also a mollifier with $\textnormal{supp}(\rho_\varepsilon) \subseteq [-\varepsilon, \varepsilon]$. Given a subset $\Omega \subseteq \mathbb{R}$, its indicator function $\mathcal{I}_\Omega : \mathbb{R} \to \mathbb{C}$ is defined by
\[
\mathcal{I}_\Omega(x) = \begin{cases}
1 & \text{ if } x \in \Omega, \\
0 & \text{ otherwise}.
\end{cases}
\]
A linear map $\mu: \textnormal{C}^0_c(\mathbb{R}) \to \mathbb{C}$ is a radon measure if $\forall \Omega \subset \mathbb{R}$ which are compact, $\exists u_\Omega > 0$ such that
\[
\langle \mu, f\rangle \leq u_\Omega \sup_{x \in \mathbb{R}} |f(x)| \ \forall f \in \textnormal{C}^0_c(\mathbb{R}) \textnormal{ with }\textnormal{supp}(f) \subseteq \Omega.
\]
The smallest such $u_\Omega$ is defined to be the total variation of $\mu$ in $\Omega$ and will be denoted by $\textnormal{TV}_\Omega(\mu)$. The set of all radon measures will be denoted by $\mathcal{M}(\mathbb{R})$. By the Lesbesgue decomposition theorem \cite{ambrosio2000functions}, any $\mu \in \mathcal{M}(\mathbb{R})$ can be uniquely expressed as
\[
\mu = \mu_c + \mu_d,
\]
where $\mu_c \in \mathcal{M}(\mathbb{R})$ is called the continuous part of $\mu$ and $\mu_a$ is called its atomic part. The continuous part can be characterized by a function $\phi_c \in \textnormal{C}^0(\mathbb{R})$ where $\forall f \in \textnormal{C}^1_c(\mathbb{R})$,
\[
\langle \mu_c , f\rangle = -\int_\mathbb{R} f'(x) \phi_c(x) dx.
\]
The function $\phi_c$ is also often denoted by $\phi_c(x) = \mu_c((-\infty, x])$ to be consistent with the `cumulative function' of $\mu_c$ in the measure-theoretic definition of the radon measure. $\mu_c$ can further be decomposed into an absolutely continuous part, which can be described by a density function, and a Cantor part --- we will not require this decomposition in this paper. The atomic part, $\mu_a$, which can be expressed as
\[
\mu_a \cong \sum_{i \in I} a_i \delta(x - x_i),
\]
for some finite or countably infinite sequence $\{a_i \in \mathbb{C}\}_{i\in I}$ and $\{x_i \in \mathbb{R}\}_{i \in I}$ such that for any compact $\Omega \subseteq \mathbb{R}$,
\[
\sum_{i \in I | x_i \in \Omega} \abs{a_i} < \infty.
\]
For any compact $\Omega \subseteq \mathbb{R}$, it can be shown that
\[
\textnormal{TV}_\Omega(\mu) = \textnormal{TV}_\Omega(\mu_c) + \textnormal{TV}_\Omega(\mu_a)
\]
\text{where }
\begin{align*}
&\textnormal{TV}_\Omega(\mu_c) = \sup_{\substack{f\in \textnormal{C}_c^1(\mathbb{R}) \norm{f}_{L^\infty} = 1}} \abs{\int_\mathbb{R} \phi_c(x) f'(x) dx}\text{ and} \\ &\textnormal{TV}_\Omega(\mu_d) = \sum_{i \in I | x_i \in \Omega} |a_i| .
\end{align*}
We will use standard notation for Schwartz space, $\mathcal{S}(\mathbb{R})$ and tempered distributions by $\mathcal{S}'(\mathbb{R})$. Note that every radon measure is a distribution (i.e.~a continuous map from compact smooth function to complex numbers) --- a radon measure $\mu \in \mathcal{M}(\mathbb{R})$ which is additionally a tempered distribution will be called a tempered radon measure. From the Schwartz representation theorem \cite{adams2003sobolev}, it follows that any tempered distribution can be expressed as
\[
\langle \mu , f\rangle = \sum_{\alpha = 0}^m \int_{\mathbb{R}} u_{\alpha}(\omega) \frac{\partial^\alpha}{\partial \omega^\alpha} \hat{f}(\omega) \frac{d\omega}{\sqrt{2\pi}} \ \forall \ f \in \mathcal{S}(\mathbb{R}),
\]
where $\hat{f}$ is the fourier transform of $f$, and $u_{0}, u_1 \dots u_m$ are continuous functions of at-most polynomial growth. Of particular interest will be distributions which contain only the term corresponding to $\alpha = 0$ i.e.
\[
\langle \mu , f\rangle = \int_{\mathbb{R}} \hat{\mu}(\omega) \hat{f}(\omega) \frac{d\omega}{\sqrt{2\pi}} \ \forall \ f\in \mathcal{S}(\mathbb{R}),
\]
where $\hat{\mu}$ is a continuous function of at-most polynomial growth. Such a distribution will be called a tempered distribution with a fourier transform being a function of at-most polynomial growth, and $\hat{\mu}$ will be referred to as the fourier transform of $\mu$.
Given a Banach space $X$ and an operator $O: X \to X$, the operator norm will be denoted by $\norm{O} = \sup_{x \in X} \norm{Ox} / \norm{x}$. The space of bounded linear operators on a Banach space $X$ will be denoted by $\mathfrak{L}(X)$ i.e. $\mathfrak{L}(X) = \{O: X\to X | \norm{O} < \infty\}$. A map $F: \mathbb{R} \to \mathfrak{L}(X)$ is norm continuous at $t$ if
\[
\lim_{s\to t} \norm{F(t) - F(s)}=0
\]
\text{ and strongly continuous at} $t$ if
\[
\forall x \in X, \lim_{s \to t}\norm{F(t) x - F(s)x} = 0.
\]
Similarly, it is \text{norm differentiable at }$t$ if
\begin{align*}
\exists F'(t): \lim_{s \to t} \norm{F'(t) - \frac{F(t) - F(s)}{t - s}} = 0,
\end{align*}
and strongly differentiable at $t$ if
\begin{align*}
\exists F'(t): \forall x \in X \lim_{s\to t} \norm{F'(t) x - \frac{F(t) x - F(s) x}{t - s}} = 0.
\end{align*}
Note that if $X$ is finite dimensional, then the notion of norm continuity/differentiability and strong continuity/differentiability are equivalent.
A sequence $\{\mu_n : X \to \mathbb{C}\}_{n \in \mathbb{N}}$ weakly converges to $\mu^* : X \to \mathbb{C}$, denoted by $\mu^* = \textnormal{wlim}_{n \to \infty} \mu_n$, if $\forall x \in X, \mu^* x = \lim_{n \to \infty} \mu_n x$.
Given a Hilbert space $\mathcal{H}$, a densely defined operator $O:\textnormal{dom}[O] \to \mathcal{H}$ is said to be closed if $\forall \psi \in \textnormal{dom}[O]$, such that $\forall$ sequences $\{\psi_n \in \textnormal{dom}[O]\}_{n\in \mathbb{N}}$ which converge to $0$ such that the sequence $\{O\psi_n \in \mathcal{H}\}_{n \in \mathbb{N}}$ also converges, $\lim_{n \to \infty} O\psi_n = 0$. A densely defined operator $O$ is said to be closable if it has a closed extension, called the closure of the operator and denoted by $\overline{O}$. We will use the following property of the domain of the closure, $\textnormal{dom}[\overline{O}]$: $\psi \in \textnormal{dom}[\overline{O}]$ if and only if $\exists$ a sequence $\{\psi_n \in \textnormal{dom}[O]\}_{n \in \mathbb{N}}$ such that $\lim_{n \to \infty} \psi_n = \psi$, and the sequence $\{O\psi_n \in \mathcal{H}\}_{n \in \mathbb{N}}$ also converges. Furthermore if $O$ is closable, the limit of the sequence $\{O\psi_n \in \mathcal{H}\}_{n \in \mathbb{N}}$ is independent of the sequence $\{\psi_n \in \mathcal{H}\}_{n \in \mathbb{N}}$, and is equal to $\overline{O}\psi$. The adjoint of a densely defined operator $O: \textnormal{dom}[O] \to \mathcal{H}$ is an operator $O^\dagger : \textnormal{dom}[O^\dagger] \to \mathcal{H}$ where $\textnormal{dom}[O^\dagger] = \{\psi \in \mathcal{H} | \langle \psi, O\cdot\rangle : \textnormal{dom}[O] \to \mathbb{C} \text{ is bounded}\}$ and by the Riesz' representation theorem, $\forall \psi \in \mathcal{H}, O^\dagger \psi$ is identified as the unique vector which satisfies $\langle O^\dagger \psi, \phi\rangle = \langle \psi, O\phi\rangle \ \forall \phi \in \textnormal{dom}[O].$ An operator is self adjoint if $\textnormal{dom}[O] = \textnormal{dom}[O^\dagger]$. A closable operator is essentially self adjoint if it has a self adjoint extension, which then coincides with its closure.
\noindent\emph{Fock Spaces}: For a separable Hilbert space $\mathcal{H}$, and $n\in \mathbb{Z}_{\geq 1}$, we will denote by $\textnormal{Sym}_n(\mathcal{H}) \subseteq \mathcal{H}^{\otimes n}$ the set of symmetric (permutationally invariant) states in $\mathcal{H}^{\otimes n}$. We will denote by $\textnormal{Fock}[\mathcal{H}] := \mathbb{C}\oplus \bigoplus_{n \in \mathbb{Z}_{\geq 1}}\textnormal{Sym}_n(\mathcal{H}) $ the symmetric (bosonic) Fock space generated by $\mathcal{H}$. We will denote by $\Pi_{n}:\textnormal{Fock}[\mathcal{H}] \to \textnormal{Fock}[\mathcal{H}]$ the projector onto $\textnormal{Sym}_{n}(\mathcal{H})$ (i.e.~the $n$ particle sector), by $\Pi_{\leq n} := \sum_{i = 0}^{n}\Pi_i$ and by $\Pi_{>n} = \textnormal{id} - \Pi_{\leq n}$.
We will denote by $\textnormal{F}_\infty[\mathcal{H}] \subseteq \textnormal{Fock}[\mathcal{H}]$ the space of all states with a finite number of particles i.e.~
\[
\textnormal{F}_\infty[\mathcal{H}] = \big\{\ket{\Psi} \in \textnormal{Fock}[\mathcal{H}] \big| \exists N_0 \in \mathbb{N}: \Pi_n\ket{\Psi} = 0 \ \forall n > N_0 \big\}.
\]
The space of states with finite $k^\text{th}$ particle number moment will be denoted by $\textnormal{F}_k(\mathcal{H}) \subseteq \textnormal{Fock}[\mathcal{H}]$ i.e.~for $k \in \mathbb{Z}_{\geq 1}$
\[
\textnormal{F}_k[\mathcal{H}] = \bigg\{\ket{\Psi} \in \textnormal{Fock}[\mathcal{H}] \bigg| \sum_{n = 0}^\infty n^k \bra{\Psi}\Pi_n \ket{\Psi} < \infty \bigg\},
\]
and by $\textnormal{F}_\mathcal{S}[\mathcal{H}]$ we will denote the space of states where all the particle number moments are finite i.e.
\begin{align*}
&\textnormal{F}_{\mathcal{S}}[\mathcal{H}] := \bigcap_{k = 1}^\infty \textnormal{F}_k[\mathcal{H}] \nonumber\\
&\qquad= \bigg\{\ket{\Psi} \in \textnormal{Fock}[\mathcal{H}] \bigg| \sum_{n = 0}^\infty n^k \bra{\Psi}\Pi_n \ket{\Psi} < \infty \ \forall k \in \mathbb{Z}_{\geq 1}\bigg\}.
\end{align*}
We remark that $\textnormal{F}_\infty[\mathcal{H}]$, $\textnormal{F}_\mathcal{S}[\mathcal{H}]$ and $\textnormal{F}_k[\mathcal{H}]$ (for any $k \in \mathbb{Z}_{\geq 1}$) are dense in $\textnormal{Fock}[\mathcal{H}]$. Note also the inclusions $\textnormal{F}_{\infty}[\mathcal{H}] \subseteq \textnormal{F}_\mathcal{S}[\mathcal{H}] \subseteq \textnormal{F}_k[\mathcal{H}]$. For $\ket{\Psi} \in \textnormal{F}_k[\mathcal{H}]$, we will denote by $\mu^{(k)}_{\ket{\Psi}}$ the $k^\textnormal{th}$ moment of the photon number operator i.e.
\[
\mu^{(k)}_{\ket{\Psi}} = \sum_{n = 0}^\infty n^k \bra{\Psi} \Pi_n \ket{\Psi}.
\]
For any $v \in \mathcal{H}$, we will denote by $a^{-}_{v}$ and $a^{+}_v$ the corresponding annhilation and creation operator. These operators can be explicitly defined over the domain $\textnormal{F}_\infty[\mathcal{H}]$ --- $a^{-}_v:\textnormal{F}_\infty[\mathcal{H}] \to \textnormal{Fock}[\mathcal{H}]$ is an operator defined by
\begin{align*}
&a_v^{-}(\alpha, 0^\infty) = 0 \ \forall \ \alpha \in \mathbb{C},
\end{align*}
and for all $\ u \in \mathcal{H}, n \in \mathbb{Z}_{\geq 1}$,
\begin{align*}
&a_v^{-}(0^n, u^{\otimes n}, 0^\infty) = \langle v, u \rangle(0^{n - 1}, \sqrt{n} u^{\otimes {n - 1}}, 0^\infty) .
\end{align*}
Since for every $n \in \mathbb{Z}_{\geq 1}$, the set $\textnormal{span}\{u^{\otimes n} | u \in \mathcal{H} \}$ is dense in $\textnormal{Sym}_n(\mathcal{H})$, and when domain-restricted to $\textnormal{span}\{u^{\otimes n} | u \in \mathcal{H} \}$, $a_v^{-}$ as defined above is a bounded operator, it can be uniquely extended to $\textnormal{Sym}_n(\mathcal{H})$ as a consequence of the bounded linear transformation theorem, and then extended to $\textnormal{F}_\infty[\mathcal{H}]$ by linearity. Similarly, $a_v^+: \textnormal{F}_\infty[\mathcal{H}] \to \textnormal{Fock}[\mathcal{H}]$ is defined via
\begin{align*}
&a_v^+(\alpha, 0^\infty ) = (0, \alpha v, 0^\infty) \ \forall \ \alpha \in \mathbb{C},
\end{align*}
and for all $u \in \mathcal{H}, n \in \mathbb{Z}_{\geq 1}$,
\begin{align*}
&a_v^+(0^n, u^{\otimes n}, 0^\infty) =\nonumber\\
&\qquad \bigg(0^{n + 1}, \frac{1}{\sqrt{n + 1}} \sum_{i = 0}^{n} u^{\otimes i}\otimes v \otimes u^{\otimes (n - i)}, 0^\infty\bigg).
\end{align*}
As with $a_v^{-}$, this definition of $a_v^{+}$ can be extended uniquely to $\textnormal{F}_0$.
In this paper, we will encounter finite tensor products of Fock spaces. Given a Hilbert space $\mathcal{H}$, $\textnormal{Fock}[\mathcal{H}]^{\otimes M} \simeq \textnormal{Fock}\big[\mathcal{H}^{\oplus M}\big]$, where the tensor product is taken as a tensor product over Hilbert spaces. We will use the notation $\textnormal{F}_\infty^M[\mathcal{H}] = \textnormal{F}_\infty[\mathcal{H}^{\oplus M}] $ and $\textnormal{F}_k^M[\mathcal{H}] = \textnormal{F}_k[\mathcal{H}^{\oplus M}]$ for $k \in \mathbb{Z}_{\geq 1}$. For $\alpha \in \{1, 2 \dots M\}$ and $v\in \mathcal{H}$, we define $a_{\alpha, v}^{-}: \textnormal{F}^M_\infty[\mathcal{H}] \to \textnormal{Fock}[\mathcal{H}]^{\otimes M}$ via
\begin{align*}
&a_{\alpha, v}^{-}(\alpha, 0^\infty) = 0 \ \forall \ \alpha \in \mathbb{C},
\end{align*}
and for all $u \in \mathcal{H}^{\oplus M}, n \in \mathbb{Z}_{\geq 1}$,
\begin{align*}
&a_{\alpha, v}^{-}(0^n, u^{\otimes n}, 0^\infty) = (0^{n - 1}, \sqrt{n}\langle v_\alpha, u \rangle u^{\otimes (n - 1)}, 0^\infty),
\end{align*}
where $v_\alpha = 0^{\oplus (\alpha - 1)} \oplus v \oplus 0^{\oplus(M - \alpha)}$. Similarly, we define $a_{\alpha, v}^+:F_\infty^M[\mathcal{H}] \to \textnormal{Fock}[\mathcal{H}]^{\otimes M}$ via
\begin{align*}
&a_{\alpha, v}^+(c, 0^\infty ) = (0, c v_\alpha, 0^\infty) \ \forall \ c \in \mathbb{C},
\end{align*}
and for all $u \in \mathcal{H}^{\oplus M}, n \in \mathbb{Z}_{\geq 1}$,
\begin{align*}
&a_{\alpha, v}^+(0^n, u^{\otimes n}, 0^\infty) =\\
&\qquad \bigg(0^{n + 1}, \frac{1}{\sqrt{n + 1}} \sum_{i = 0}^{n} u^{\otimes i}\otimes v_\alpha \otimes u^{\otimes (n - i)}, 0^\infty\bigg).
\end{align*}
We will denote by $\Pi_n : \textnormal{Fock}[\mathcal{H}]^{\otimes M} \to \textnormal{Fock}[\mathcal{H}]^{\otimes M}$ the projector onto $\textnormal{Sym}_n(\mathcal{H}^{\oplus M})$, by $\Pi_{\leq n} := \sum_{i = 0}^n \Pi_i$ and by $\Pi_{> n} := \textnormal{id} - \Pi_{\leq n}$.
\section{Well-definition of non-Markovian models}
\label{sec:well_def}
\subsection{Square-integrable coupling functions}
We first state some simple results about the non-Markovian model for the case where the coupling functions $v_\alpha$ are square integrable. As described in the previous section, we will assume the system to be finite-dimensional with Hilbert space $\mathcal{H}_S$, and the environment described by $M$ fock spaces $\text{Fock}[L^2(\mathbb{R})]^{\otimes M}$. The system-environment Hamiltonian is assumed to be of the form:
\begin{align}\label{eq:sq_int_hamil}
H = H_S(t) + \sum_{\alpha = 1}^M H_{\alpha, E} + \sum_{\alpha = 1}^M \big(a_{\alpha, v_\alpha} L_\alpha^\dagger +a_{\alpha, v_\alpha}^\dagger L_\alpha \big).
\end{align}
where $H_{\alpha, E}$ is the Hamiltonian describing the dynamics of the $\alpha^\text{th}$ bath and $a_{\alpha, f}$ for $f \in L^2(\mathbb{R})$ is the annihilation operator corresponding to $f$ (see section \ref{sec:notation_prelim}). Assuming the environment to be non-interacting and particle number conserving, we can specify $H_{\alpha, E}$ by a strongly-continuous single-parameter unitary group $\uptau_{\alpha, t} : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ such that $\forall f \in L^2(\mathbb{R})$,
\[
e^{iH_{\alpha, E}t } a_{\alpha, f} e^{-iH_{\alpha, E}t } = a_{\alpha, \uptau_{\alpha, t} f}.
\]
The unitary group $\uptau_{\alpha, t}$ can also be physically interpreted as specifying the dynamics within the single-particle sector of the non-interacting bath. In this paper, we will mostly consider $\uptau_{\alpha, t}$ to be the time-translation unitary group [i.e. $(\uptau_{\alpha, t} f)(\omega) = f(\omega)e^{-i\omega t}$ for all $f \in L^2(\mathbb{R})$)], which is equivalent to $H_{\alpha, E}$ as given by Eq.~\ref{eq:hamiltonian_heuristic}. However, the results stated below for square-integrable $v_\alpha$ hold for more general unitary groups.
The basic data needed to specify a non-Markovian model with square-integrable coupling functions is provided in the definition below.
\begin{definition}
\label{def:non_markovian_sq_int}
A non-Markovian open system model for a quantum system with Hilbert space $\mathcal{H}_S$ with square integrable system-environment coupling functions is specified by
\begin{enumerate}
\item[(a)] A time-dependent system Hamiltonian $H_S(t) \in \mathfrak{L}(\mathcal{H}_S)$ which is Hermitian, norm continuous and differentiable in $t$.
\item[(b)] $M$ square integrable functions $\{v_\alpha \in L^2(\mathbb{R}) \}_{\alpha \in \{1,2 \dots M\}}$,
\item[(c)] $M$ bounded operators on the system Hilbert space $\{L_\alpha \in \mathfrak{L}(\mathcal{H}_S)\}_{\alpha \in \{1, 2 \dots M\}}$.
\item[(d)] $M$ strongly continuous single-parameter unitary groups on $L^2(\mathbb{R})$, $\{\uptau_{\alpha, t} : L^2(\mathbb{R}) \to L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$.
\end{enumerate}
\end{definition}
\noindent It is convenient to consider the Hamiltonian in Eq.~\ref{eq:sq_int_hamil} in the interaction picture with respect to the enivornment Hamiltonian, as provided in the definition below.
\begin{definition}
\label{def:hamil}
For a non-Markovian model with square integrable function as specified by definition \ref{def:non_markovian_sq_int} and for $t \in \mathbb{R}$, $H(t): \mathcal{H}_S\otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \to \mathcal{H}$ is defined via
\[
H(t) = H_S(t) + \sum_{\alpha = 1}^M \big(L^\dagger_\alpha a_{\alpha, \uptau_{\alpha, t} v_\alpha}^- + L_\alpha a^{+}_{\alpha, \uptau_{\alpha, t} v_\alpha}\big).
\]
\end{definition}
\noindent We note that we have been careful in specifying the domain of $H(t)$, since it is an unbounded operator and cannot be defined over the entire system-environment Hilbert space. Following the standard approach to treat quadratic field theories \cite{reed1972methods}, we restrict it to only states which have a finite-number of particles in the environment (i.e.~$\text{F}^M_\infty[L^2(\mathbb{R})]$). Then, using standard results from the theory of unbounded operators, we can ``close" the operator $H(t)$ and extend this domain. While precisely charaterizing the domain of the closure of $H(t)$ is in general a difficult problem, we only need the characterization of its domain provided in the following lemma (proved in appendix \ref{app:sq_int}).
\begin{lemma}\label{lemma:domain}
For all $t \in \mathbb{R}$,
\begin{enumerate}
\item[(a)] $H(t): \mathcal{H}_S\otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \to \mathcal{H}$ is essentially self adjoint.
\item[(b)] $H(t)$ is closable, and if $\overline{H(t)}:\textnormal{dom}[\overline{H(t)}] \to \mathcal{H}$ is its closure then $\mathcal{H}_S\otimes \textnormal{F}_1^M[L^2(\mathbb{R})] \subseteq \textnormal{dom}[\overline{H(t)}]$ and $\forall \ket{\Psi} \in \textnormal{dom}[\overline{H(t)}]$,
$\overline{H(t)} \ket{\Psi} = \sum_{n = 0}^\infty H(t) \big(\Pi_n \ket{\Psi}\big).$
\end{enumerate}
\end{lemma}
\noindent For square integrable coupling function, we can then show the well definition of non-Markovian dynamics, as given in the following proposition (proved in appendix \ref{app:sq_int}).
\noindent\begin{proposition}\label{thm:schr_sq_int}
Given a non-Markovian model with square integrable coupling functions and for $t, s \in \mathbb{R}$, there exists a unique isometry $U(t, s) : \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \to \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \subseteq \mathcal{H}$ which is strongly continuous and differentiable in both $t, s$ and satisfies
\begin{align}\label{eq:prop_schr_eq}
\frac{d}{dt} U(t, s) = -i \overline{H(t)} U(t, s),\ \frac{d}{ds} U(t, s) = i U(t, s) \overline{H(s)},
\end{align}
with $U(s, s) = \textnormal{id} \ \forall s \in \mathbb{R}$.
\end{proposition}
\noindent In the following lemmas, we provide some useful properties of $U(t, s)$ corresponding to non-Markovian models with square integrable functions which will be useful for the analysis of distributional coupling functions. The next lemma can be considered to be an input-output equation for the bath modes \cite{gardiner1985input}.
\begin{lemma}\label{lemma:inp_out_eq}
Given $u \in L^2(\mathbb{R})$ and a non-Markovian model with square integrable coupling functions, $\forall \alpha \in \{1, 2 \dots M\}, s, t \in [0, \infty)$,
\begin{align*}
[a^{-}_{\alpha, u}, U(t, s)] = -i \int_s^t \langle u, \uptau_{\alpha, \tau} v_\alpha\rangle U(t, \tau) L_\alpha U(\tau, s)d\tau
\end{align*}
over the domain $\mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})]$, where $U(t, s)$ is the propagator corresponding to the non-Markovian model as defined in lemma \ref{thm:schr_sq_int}.
\end{lemma}
\noindent \emph{Proof}: Throughout this proof, all operators are considered to be over the domain $\mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})]$ --- in particular, we extend $a_{\alpha, u}^\pm$ from the domain $\mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})]$ to $\mathcal{H}_S \otimes \textnormal{F}_{\infty}^M[L^2(\mathbb{R})]$ via
\[
a_{\alpha, u}^\pm \ket{\Psi} = \sum_{n = 0}^\infty a_{\alpha, u}^\pm \Pi_n \ket{\Psi} \ \forall \ket{\Psi} \in \mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})].
\]
Note that since $U(t, s)$ is strongly differentiable with respect to $t$, and it maps $\mathcal{H}_S \otimes \textnormal{F}_{\infty}^M[L^2(\mathbb{R})]$ to $\mathcal{H}_S \otimes \textnormal{F}_{\infty}^M[L^2(\mathbb{R})]$, it follows that the operator $a_{\alpha, u}^{-}(t, s) = U(s, t) a_{\alpha, u}^{-} U(t, s)$ is strongly differentiable in both $t$ and $s$. Differentiating it with respect to $t$, and using the characterization of $\overline{H(t)}$ when acting on $\mathcal{H}_S \otimes \textnormal{F}_{\infty}^M[L^2(\mathbb{R})]$ as provided in lemma \ref{lemma:domain}, we obtain
\[
\frac{d}{dt}a_{\alpha, u}^{-}(t, s) = -i \langle u, \uptau_{\alpha, t} v_\alpha\rangle U(s, t) L_{\alpha} U(t, s).
\]
Noting that since $U(t, s)$ is strongly continuous in both of its arguments and since $L_\alpha$ is a bounded operator, the right hand side in the above equation is strongly continuous in $t$ and thus the equation can be integrated to obtain
\begin{align*}
&a_{\alpha, u}^{-} U(t, s) \noindent\\
&\qquad= U(t, s) a_{\alpha, u}^{-} - i\int_s^t \langle u, \uptau_{\alpha, \tau} v_\alpha\rangle U(t, \tau) L_\alpha U(\tau, s) d\tau,
\end{align*}
which proves the lemma.
\(\square\)
\noindent We now define an object that we will use repeatedly in the following sections --- the system Green's function. For an open quantum system, it is traditionally defined as the expectation value of a time-ordered product of system operators \cite{xu2015input, trivedi2018few}. We generalize its definition slightly to allow for the propagators appearing in its definition to be from different non-Markovian models. The motivation behind this generalization will be clear from the following lemma \ref{lemma:pert_theory_bound} --- it arises from the fact that while deriving error bounds in between dynamics of two non-Markovian models (e.g. a non-Markovian model and its regularization), we often need to evolve an operator as per the propagator of one model followed by evolving it backwards in time with the propagator of another model.
\begin{definition}[System Green's functions]\label{def:gfunc}
Consider $k + 1$ non-Markovian models specified by coupling functions $v_i = \{v_{i, \alpha} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ for $i \in \{1, 2 \dots k+ 1\}$, but with the same time-dependent system Hamiltonian $H_S(t)$ and jump operators $\{L_\alpha \in \mathfrak{L}(\mathcal{H}_S)\}_{\alpha \in \{1, 2 \dots M\}}$. For $O_1, O_2 \dots O_k \in \mathfrak{L}(\mathcal{H}_S)$, $\ket{\Psi_1}, \ket{\Psi_2} \in \mathcal{H}$ and $t_1, t_2 \dots t_k \in [0, \infty)$, then the Green's function $G^{v_1, v_2 \dots v_{k + 1}}_{O_1, O_2 \dots O_k}(t_1, t_2 \dots t_k)$ is defined by
\begin{align*}
&G^{v_1, v_2 \dots v_{k + 1}}_{O_1, O_2 \dots O_k; \ket{\Psi_1}, \ket{\Psi_2}}(t_1, t_2 \dots t_k) =\nonumber\\
&\qquad \bra{\Psi_1} U_{v_{k + 1}}(0, t_{k + 1}) \prod_{j = k}^1 O_j U_{v_j}(t_j, t_{j - 1}) \ket{\Psi_2},
\end{align*}
where $t_0 = 0$ and $U_{v_i}(\cdot, \cdot)$ is the propagator, as defined in lemma \ref{thm:schr_sq_int}, for the $i^\textnormal{th}$ model for $i \in \{1, 2 \dots k + 1\}$.
\end{definition}
\begin{lemma}
\label{lemma:pert_theory_bound}
Consider two non-Markovian models with coupling functions $v = \{v_\alpha \in L^2(\mathbb{R}) \}_{\alpha \in \{1, 2\dots M\}}$ and $u = \{u_\alpha \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ respectively but with the same system Hamiltonian $H_S(t)$, jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$ and single-particle environment dynamics described by the time-translation unitary group. Let $\ket{\Psi_v(t)} = U_v(t, 0)\ket{\Psi_0}, \ket{\Psi_u(t)} = U_u(t, 0) \ket{\Psi_0}$, where $U_v(t, s), U_u(t, s)$ are the propagators corresponding to the two non-Markovian models, and $\ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})]$ , then
\begin{align*}
&\norm{\ket{\Psi_u(t)} - \ket{\Psi_v(t)}}^2 \leq \nonumber\\
&\qquad\sum_{\alpha = 1}^M \bigg(\int_0^t \mathcal{D}^{u, v}_\alpha(\tau)d\tau + \int_0^t \mathcal{E}^{u, v}_\alpha(\tau) d\tau\bigg),
\end{align*}
where for $\tau \in [0, t]$ and $\alpha \in \{1, 2 \dots M\}$,
\begin{align*}
&\mathcal{D}^{u, v}_\alpha(\tau) = 4 \norm{L}_\alpha \norm{a^-_{\alpha, \uptau_\tau(u_\alpha - v_\alpha)} \ket{\Psi_0}}, \text{ and }\\
&\mathcal{E}^{u, v}_\alpha(\tau) = \nonumber\\
&\qquad 2 \bigg| \int_0^\tau \langle \uptau_\tau (u_\alpha - v_\alpha), \uptau_s v_\alpha\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v, v, u}(s, \tau)ds\bigg| +\nonumber \\
&\qquad 2\bigg | \int_0^\tau \langle\uptau_s u_\alpha, \uptau_\tau(u_\alpha - v_\alpha)\rangle G^{u, u, v}_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(\tau, s) ds \bigg|,
\end{align*}
where the Green's functions $G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v, v, u}(s, \tau)$ and $G^{u, u, v}_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(\tau, s)$ are defined in definition \ref{def:gfunc}.
\end{lemma}
\noindent\emph{Proof}: Note that
\[
\norm{\ket{\Psi_u(t)} - \ket{\Psi_v(t)}}^2 = 2\bigg(1 - \ \text{Re}[\bra{\Psi_u(t)}\Psi_v(t)\rangle\bigg).
\]
Consider now the inner product $\bra{\Psi_u(t)}\Psi_v(t)\rangle$ --- differentiating this with respect to $t$, we obtain that
\begin{align*}
&\frac{d}{dt} \bra{\Psi_u(t)}\Psi_v(t)\rangle = i\sum_{\alpha = 1}^M \bigg(\bra{\Psi_u(t)} L_\alpha^\dagger a^-_{\alpha, \uptau_t(u_\alpha - v_\alpha)}\ket{\Psi_v(t)} + \nonumber\\
&\qquad \qquad \bra{\Psi_u(t)} L_\alpha a^+_{\alpha, \uptau_t(u_\alpha - v_\alpha)}\ket{\Psi_v(t)}\bigg).
\end{align*}
We note that from lemma \ref{lemma:inp_out_eq}, it follows that $\forall \alpha $,
\begin{align*}
&\bra{\Psi_u(t)} L_\alpha^\dagger a^-_{\alpha, \uptau_t(u_\alpha - v_\alpha)}\ket{\Psi_v(t)} \nonumber\\
& =\bra{\Psi_0} U_u(0, t) L_\alpha^\dagger a^-_{\alpha, \uptau_t (u_\alpha - v_\alpha)}U_v(t, 0)\ket{\Psi_0}, \nonumber\\
& =G^{v, u}_{L_\alpha^\dagger; a^-_{\alpha, \uptau_t (u_\alpha - v_\alpha)}\ket{\Psi_0}, \ket{\Psi_0}} -\nonumber\\
&\qquad \quad i \int_0^t \langle \tau_t (u_\alpha - v_\alpha), \tau_s v_\alpha\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v, v, u}(s, t)ds.
\end{align*}
Similarly,
\begin{align*}
&\bra{\Psi_u(t)} L_\alpha a^+_{\alpha, \uptau_t(u_\alpha - v_\alpha)}\ket{\Psi_v(t)} \nonumber \\
&= \bra{\Psi_0} U_u(0, t)L_\alpha a^+_{\alpha, \uptau_t(u_\alpha - v_\alpha)} U_v(t, 0)\ket{\Psi_0} ,\nonumber\\
&=G^{v, u}_{L_\alpha; \ket{\Psi_0}, a_{\alpha, \uptau_t(u_\alpha - v_\alpha)}^{-}\ket{\Psi_0}} + \nonumber\\
&\qquad i\int_0^t \langle\tau_s u_\alpha, \tau_t(u_\alpha - v_\alpha)\rangle G^{u, u, v}_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(t, s) ds.
\end{align*}
Here, we have used the notation for Green's functions defined in definition \ref{def:gfunc}. Furthermore, we can note that $\forall \alpha \in \{1,2 \dots M\}$,
\begin{align*}
&\bigg| G^{v, u}_{L_\alpha^\dagger; a^-_{\alpha, \uptau_t (u_\alpha - v_\alpha)}\ket{\Psi_0}, \ket{\Psi_0}}\bigg |, \bigg |G^{v, u}_{L_\alpha; \ket{\Psi_0}, a_{\alpha, \uptau_t(u_\alpha - v_\alpha)}^{-}\ket{\Psi_0}} \bigg |\nonumber\\
&\qquad\leq \norm{L_\alpha} \norm{a^-_{\alpha, \uptau_t (u_\alpha - v_\alpha)}\ket{\Psi_0}},
\end{align*}
and thus we obtain
\begin{align}\label{eq:bound_olap_deriv}
&\bigg|\frac{d}{dt} \bra{\Psi_u(t)}\Psi_v(t)\rangle \bigg| \leq 2\sum_{\alpha = 1}^M \norm{L_\alpha} \norm{a^-_{\alpha, \uptau_t (u_\alpha - v_\alpha)}\ket{\Psi_0}} +\nonumber\\
&\quad\sum_{\alpha = 1}^M \bigg( \bigg| \int_0^t \langle \tau_t (u_\alpha - v_\alpha), \tau_s v_\alpha\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v, v, u}(s, t)ds\bigg | \nonumber\\
&\quad + \bigg | \int_0^t \langle\tau_s v_\alpha, \tau_t(u_\alpha - v_\alpha)\rangle G^{u, u, v}_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(t, s) ds \bigg |\bigg).
\end{align}
Finally, note that $\norm{\ket{\Psi_u(t)} - \ket{\Psi_v(t)}}^2 = 2(1 - \bra{\Psi_u(t)} \Psi_v(t)\rangle)$ and therefore,
\begin{align}\label{eq:bound_norm_olap}
& \norm{\ket{\Psi_u(t)} - \ket{\Psi_v(t)}}^2 \leq 2 \int_0^t \bigg| \frac{d}{d\tau} \bra{\Psi_u(\tau)}\Psi_v(\tau)\rangle\bigg| d\tau.
\end{align}
Combining the estimates in Eq.~\ref{eq:bound_olap_deriv} and \ref{eq:bound_norm_olap}, we obtain the lemma statement.
\(\square\).
\noindent In preparation for the next subsection, we now provide a lemma that follows from a straightforward application of lemma \ref{lemma:inp_out_eq} and characterizes the rate of change of this Green's function with respect to its time arguments.
\begin{lemma}
\label{lemma:gfunc_deriv}
Consider two non-Markovian models described by coupling functions $v = \{v_{ \alpha} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}, u = \{u_{\alpha} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$, but with the same system Hamiltonian $H_S(t)$, jump operators $\{L_\alpha \in \mathfrak{L}(\mathcal{H}_S)\}_{\alpha \in \{1, 2 \dots M\}}$ and environment single-particle unitary group $\{\uptau_{\alpha, t}: L^2(\mathbb{R}) \to L^2(\mathbb{R})\}$. For $A, B \in \mathfrak{L}(\mathcal{H}_S)$, $\ket{\Psi}, \ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_{\mathcal{S}}^M[L^2(\mathbb{R})]$ and $t, s \in [0, \infty)$,
\begin{widetext}
\begin{align*}
&\frac{d}{ds} G^{v, v, u}_{A, B; \ket{\Psi}, \ket{\Phi}}(s, t) = iG_{[H_S(s), A], B; \ket{\Psi}, \ket{\Phi}}^{v, v, u}(s, t) + i\sum_{\alpha = 1}^M \bigg(G^{v, v, u}_{[L_\alpha^\dagger, A], B; \ket{\Psi}, a^-_{\alpha, \tau_{s}v_{\alpha}} \ket{\Phi}}(s, t) +z G^{v, v, u}_{[L_\alpha, A], B; a^-_{\alpha, \tau_{s}v_{\alpha}}\ket{\Psi}, \ket{\Phi}}(s, t) \bigg) - \nonumber\\
&\quad \quad i\sum_{\alpha = 1}^M \bigg(\int_0^{s} \langle \uptau_{s}v_{\alpha}, \uptau_\tau v_{\alpha}\rangle G^{v, v, v, u}_{L_\alpha, [L_\alpha^\dagger, A], B; \ket{\Psi}, \ket{\Phi}}(s, \tau, t) d\tau + \int_0^{s} \langle \uptau_\tau u_{\alpha}, \uptau_{s}v_{\alpha}\rangle G^{v, v, u, u}_{[L_\alpha, A], B, L_\alpha^\dagger; \ket{\Psi}, \ket{\Phi}}(s, t, \tau) d\tau -\nonumber\\
&\qquad \quad \qquad \int_{s}^{t}
\langle \uptau_\tau u_{\alpha}, \uptau_{s}v_{\alpha} \rangle G^{v, v, u, u}_{[L_\alpha, A], L_\alpha^\dagger, B; \ket{\Psi}, \ket{\Phi}}(s, \tau, t) d\tau\bigg)
\end{align*}
\end{widetext}
\end{lemma}
\noindent\emph{Proof}: Differentiating $G^{v, v, u}_{A, B; \ket{\Psi}, \ket{\Phi}}(s, t)$ with respect to $s$ and using lemma \ref{lemma:domain}(b), we obtain that
\begin{align*}
&\frac{d}{ds} G^{v, v, u}_{A, B; \ket{\Psi}, \ket{\Phi}}(s, t) = \nonumber\\
&\quad i\bra{\Psi} U_u(0, t) B U_v(t, s) [H_S(s), A] U_v(s, 0) \ket{\Phi} + \nonumber\\
&\quad i\sum_{\alpha = 1}^M \bigg( \bra{\Psi} U_u(0, t) B U_v(t, s) a^+_{\alpha,\uptau_{s} v_{\alpha}}[L_\alpha, A] U_v(s, 0)\ket{\Phi} +\nonumber\\
&\quad \bra{\Psi} U_u(0, t) B U_v(t, s) [L_\alpha^\dagger, A]a^-_{\alpha,\uptau_{s} v_{\alpha}} U_v(s, 0)\ket{\Phi} \bigg).
\end{align*}
We note that
\begin{align*}
&\bra{\Psi} U_u(0, t) B U_v(t, s) [H_S(s), A] U_v(s, 0) \ket{\Phi} = \\
&\qquad \qquad \qquad \qquad \qquad \quad G_{[H_S(s), A], B; \ket{\Psi}, \ket{\Phi}}^{v, v, u}(s, t).
\end{align*}
Using lemma \ref{lemma:inp_out_eq}, we obtain that
\begin{align*}
&\bra{\Psi} U_u(0, t) B U_v(t, s) [L_\alpha^\dagger, A]a^-_{\alpha,\uptau_{s} v_{\alpha}} U_v(s, 0)\ket{\Phi} =\\
&\qquad \quad G^{v, v, u}_{[L_\alpha^\dagger, A], B; \ket{\Psi}, a^-_{\alpha, \tau_{s}v_{\alpha}} \ket{\Phi}}(s, t) -\nonumber \\
&\qquad \quad i\int_0^{s} \langle \uptau_{s}v_{\alpha}, \uptau_\tau v_{\alpha}\rangle G^{v, v, v, u}_{L_\alpha, [L_\alpha^\dagger, A], B; \ket{\Psi}, \ket{\Phi}}(s, \tau, t) d\tau,
\end{align*}
and
\begin{align*}
&\bra{\Psi} U_u(0, t) B U_v(t, s) a^+_{\alpha,\uptau_{s} v_{\alpha}}[L_\alpha, A] U_v(s, 0)\ket{\Phi} =\\
&\qquad \quad G^{v, v, u}_{[L_\alpha, A], B; a^-_{\alpha, \uptau_{s }v_{\alpha}} \ket{\Psi}, \ket{\Phi}}(s, t) + \nonumber\\
& \qquad \quad i\int_{s}^{t}
\langle \uptau_\tau u_{\alpha}, \uptau_{s}v_{\alpha} \rangle G^{v, v, u, u}_{[L_\alpha, A], L_\alpha^\dagger, B; \ket{\Psi}, \ket{\Phi}}(s, \tau, t) d\tau - \nonumber\\
&\qquad \quad i\int_0^{s} \langle \uptau_\tau u_{\alpha}, \uptau_{s}v_{\alpha}\rangle G^{v, v, u, u}_{[L_\alpha, A], B, L_\alpha^\dagger; \ket{\Psi}, \ket{\Phi}}(s, t, \tau) d\tau,
\end{align*}
which completes the proof of the lemma.
\(\square\)
\subsection{Extension to general radon measures}
\label{sec:ext_gen_rad}
n this section, we analyze the regularization procedure used to define the dynamics for non-Markovian models that have coupling functions that are not necessarily square integrable. We first begin by analyzing the regularization of a single radon measure and establishing some of its general properties. While it is possible that some of the lemmas relating radon measures that we prove below can be found in analysis literature, we could not find a proof of the precise statements that we needed and so we include our own proof of these statements. After that, we analyze the full unitary group corresponding to the regularized radon measure and study its limiting behaviour as the regularization is removed.
\subsubsection{Some generalities about radon measures}
Recall that a radon measure $\mu$ is a map from the space of continuous compact functions to complex numbers i.e.~$\mu: \textnormal{C}_c^0(\mathbb{R}) \to \mathbb{C}$. However, we need to extend its domain to apply it to certain discontinuous functions --- for e.g.~for the single-particle case described previously (Eq.~\ref{eq:decay_equation}), at time $t$, the radon measure is applied to the excited state amplitude restricted to the interval $[0, t]$ which is discontinuous as a function over $\mathbb{R}$. To extend $\mu$ to a space of discontinuous functions we use a mollifier to first smoothen the discontinuous function to a continuous function, and then apply $\mu$.
The space of discontinuous functions that we will focus on is one which is formed by windowing a differentiable (and thus continuous) function within a given interval. More specifically, we will consider the space $\textnormal{PWC}^1(\mathbb{R})$ is space of all functions which are expressible as $g \cdot \mathcal{I}_{[a, b]}$, where $\mathcal{I}_{[a,b]}(x) = 1$ if $x \in [a, b]$ and $0$ otherwise is the indicator function for the interval $[a, b]$, for some $g \in \textnormal{C}^1(\mathbb{R})$ and $[a, b] \subseteq \mathbb{R}$. The functions belonging to this space thus have discontinuities at two points --- the end points of the windowing intervals. Given a symmetric mollifier $\rho$ and $\varepsilon >0$, we can then define a map $\mu_\varepsilon: \text{PWC}^1(\mathbb{R}) \to \mathbb{C}$ via
\[
\langle \mu_\varepsilon, f\rangle = \langle \mu, \rho_\varepsilon \star f\rangle \ \forall \ f\in \textnormal{PWC}^1(\mathbb{R}),
\]
i.e.~given a discontinuous function from $\text{PWC}^1(\mathbb{R})$, we first smooth the discontinuities by convolving it with a mollifier $\rho_\varepsilon$ and then apply $\mu$ to the resulting continuous function. We can now take the limit of $\varepsilon \to 0$ and attempt to define $\mu^*:\text{PWC}^1(\mathbb{R}) \to \mathbb{C}$ via
\begin{align}\label{eq:mu_star}
\langle \mu^*, f\rangle = \lim_{\varepsilon \to 0} \langle \mu_\varepsilon, f\rangle \forall \ f\in \textnormal{PWC}^1(\mathbb{R}).
\end{align}
In the following lemmas, we establish that $\mu^*$ is well defined (i.e.~the limit defining $\mu^*$ exists), is independent of the precise choice of the mollifier and that when restricted to the function space $ \textnormal{C}_c^1(\mathbb{R})$ (i.e.~the space of continuously differentiable compactly supported functions), its action coincides with that of the radon measure $\mu$. We also derive certain properties of the map $\mu^*$ which will be useful in the following subsection. We first present a technical lemma, whose proof is in appendix \ref{app:distributional_mem_ker}.
\begin{lemma}\label{lemma:main_error_bound_mu}
Consider $\mu \in \mathcal{M}(\mathbb{R})$ with the Lesbesgue decomposition $\mu = \mu_c + \mu_d$ with $\phi_c \in \textnormal{C}^0(\mathbb{R})$ given by $\phi_c(x) = \mu_c((-\infty, x]) \ \forall \ x \in \mathbb{R}$, and $\mu_d \cong \sum_{i \in I} \alpha_i \delta(x - y_i)$ for some $\{\alpha_i \in \mathbb{C}\}_{i \in I}, \{y_i \in \mathbb{R}\}_{i \in I}$ and finite and countably infinite index set $I$. Given a compact interval $[a, b] \subseteq \mathbb{R}$ and $f \in \textnormal{C}^1(\mathbb{R})$, define $\langle\mu_{[a, b]}^*, f\rangle $ by
\begin{align*}
&\langle \mu^*_{[a, b]}, f\rangle =\langle \mu_{c, [a, b]}^*, f\rangle + \langle \mu_{d, [a, b]}^*, f\rangle \ \text{where} \\
&\langle\mu_{c, [a, b]}^*, f\rangle =\nonumber\\
&\qquad f(b) \phi_c(b) - f(a) \phi_c(a) - \int_a^b \phi_c(x) f'(x) dx \ \text{ and}\\
&\langle\mu_{d, [a, b]}^*, f\rangle = \frac{1}{2} \sum_{i \in I | y_i \in \{a, b\}} \alpha_i f(y_i) + \sum_{i \in I | y_i \in (a, b)} \alpha_i f(y_i).
\end{align*}
Then, for every compact intervals $[a, b] \subseteq \mathbb{R}$, $\exists \Delta^0_{\mu; [a, b]}(\varepsilon), \Delta^1_{\mu; [a,b]}(\varepsilon) > 0 $ where $ \Delta^0_{\mu; [a, b]}(\varepsilon), \Delta^1_{\mu; [a,b]}(\varepsilon) \to 0$ as $\varepsilon \to 0$ such that $\forall \varepsilon \in (0, (b- a) / 2)$ and for any even (symmetric about $0$) positive function $\alpha \in \textnormal{C}_c^\infty(\mathbb{R})$ with $\textnormal{supp}(\alpha) \subseteq [-\varepsilon,\varepsilon]$ and $\int_{[-\varepsilon, \varepsilon]}\alpha(x) dx=1$,
\begin{align*}
&\abs{ \langle \mu^*_{ [a, b]}, f\rangle - \langle \mu, \alpha \star (f\cdot \mathcal{I}_{[a, b]}) \rangle} \leq \nonumber\\
&\qquad \Delta^0_{\mu; [a, b]}(\varepsilon) \sup_{x \in [a, b]} \abs{f(x)} + \Delta^1_{\mu; [a, b]}(\varepsilon)\sup_{x \in [a, b]} \abs{f'(x)}.
\end{align*}
The functions $\Delta^0_{\mu; [a, b]}, \Delta^1_{\mu; [a, b]}$ will be called the error functions corresponding to $\mu$.
\end{lemma}
This lemma characterizes the regularization of a radon measure $\mu$. Furthermore, it shows that the rate of convergence of the error between the radon measure and its regularization result as the regularization is removed is dependent only on the maximum value of the function within its support, and the maximum value of its derivative. This lemma now allows us to prove the existence of the limit defining $\mu^*$ (Eq.~\ref{eq:mu_star}).
\begin{lemma}[Existence of $\mu^*$]
\label{lemma:well_def_mu_star}
Given a $\mu \in \mathcal{M}(\mathbb{R})$, consider the map $\mu_\varepsilon : \textnormal{PWC}^1(\mathbb{R}) \to \mathbb{C}$ given by $\langle \mu_\varepsilon, \cdot\rangle = \langle \mu , \rho_\varepsilon \star (\cdot)\rangle$ with $\rho$ being a symmetric mollifier, then
\begin{enumerate}
\item[(a)] $\forall f\in \textnormal{PWC}^1(\mathbb{R})$, $\langle \mu^*, f\rangle := {\lim_{\varepsilon \to 0}} \langle\mu_\varepsilon, f\rangle$ exists and is independent of the choice of the mollifier,
\item[(b)] $\forall f \in \textnormal{C}_c^1(\mathbb{R})$, $\langle \mu^*, f \rangle = \langle \mu, f \rangle$.
\end{enumerate}
\end{lemma}
\noindent\emph{Proof}: Suppose that $f \in \textnormal{PWC}^1(\mathbb{R})$ has the representation $f = g\cdot \mathcal{I}_{[a, b]}$ for some $g \in \textnormal{C}^1(\mathbb{R})$ and compact $[a, b] \subseteq \mathbb{R}$. Part \emph{(a)} of the lemma follows directly from lemma \ref{lemma:main_error_bound_mu} from which it follows that $\lim_{\varepsilon \to 0} \langle \mu, \rho_\varepsilon \star f\rangle = \langle \mu^*_{[a, b]}, g\rangle$, which can be identified as $\langle \mu^*, f\rangle$. Furthermore, we note that by construction, $\mu^*$ is independent of the choice of the mollifier.
For part \emph{(b)}, we note that for $f = g\cdot \textnormal{I}_{[a, b]} \in \textnormal{C}^1_c(\mathbb{R}) \subseteq \textnormal{PWC}^1(\mathbb{R})$, $g(a) = g(b) = 0$, and therefore from the definition of $\langle \mu^*_{[a, b]}, g\rangle$ in lemma \ref{lemma:main_error_bound_mu}
\begin{align*}
&\langle \mu^*, f\rangle = \langle \mu^*_{[a, b]}, g\rangle \nonumber\\
&\qquad =-\int_a^b \phi_c(x) g'(x) dx + \sum_{i \in I}\alpha_i g(y_i) \nonumber\\
&\qquad= -\int_a^b \phi_c(x) f'(x) dx + \sum_{i \in I}\alpha_i f(y_i) = \langle \mu, f\rangle,
\end{align*}
which proves the lemma statement.
\(\square\)
We next provide some examples of distributional coupling functions that appear in a number of physical problems. All of these coupling functions are radon measures, and we also provide their extensions $\mu^*$ (Eq.~\ref{eq:mu_star}) to the space of discontinuous functions as well as the error functions defined in lemma \ref{lemma:main_error_bound_mu}.
\begin{example}[Square integrable coupling functions] Although we analyzed square-integrable coupling functions separately in the previous section, they can also be represented as, and thus are a special case of, distributional coupling functions. In particular, for $v \in L^2(\mathbb{R}) $, with Fourier transform $\hat{v} \in L^2(\mathbb{R}) $, note that
\[
\kappa(t) = \int_{-\infty}^\infty |\hat{v}(\omega)|^2 e^{-i\omega t} d\omega,
\]
is a continuous function with $\norm{\kappa}_{L^\infty} \leq \norm{v}_{L^2}^2$, and thus can be described by the radon measure $\mu$ defined by
\begin{align}\label{eq:eq_mu_st}
\langle \mu, f\rangle = \int_{-\infty}^\infty f(t) \kappa(t) dt \ \forall \ f \in \textnormal{C}^0_c(\mathbb{R})
\end{align}
is a Radon measure. Furthermore, its extension to $\textnormal{PWC}^1(\mathbb{R})$, $\mu^*$ (Eq.~\ref{eq:mu_star}), is given by
\[
\langle\mu^*, f\rangle= \int_{-\infty}^\infty f(t) \kappa(t) dt \ \forall \ f \in \textnormal{PWC}^1(\mathbb{R}),
\]
Consider now $f = g\cdot \mathcal{I}_{[a, b]} \in \textnormal{PWC}^1(\mathbb{R})$ for some compact interval $[a, b] \subseteq \mathbb{R}$ and $g \in \textnormal{C}^1(\mathbb{R})$. Since $\kappa \in \textnormal{C}^1(\mathbb{R})$, $\exists \delta_{\kappa, [a, b]}(\varepsilon) > 0,$ where $\delta_{\kappa; [a, b]}(\varepsilon)\to 0$ as $\varepsilon \to 0$, such that $\forall x, x' \in [(3a - b) / 2, (3b - a)/2]$, with $\abs{x - x'} < \varepsilon$, $\abs{f(x) - f(x')} \leq \delta_{\kappa; [a,b]}(\varepsilon)$. Consequently,
\[
\abs{\langle \mu^*, f \rangle - \langle \mu, f\star \rho\rangle} \leq \delta_{\kappa; [a,b]}(\varepsilon)\sup_{x \in [a, b]}\abs{f(x)},
\]
and thus the error functions (lemma \ref{lemma:main_error_bound_mu}) for $\mu$ defined in Eq.~\ref{eq:eq_mu_st} are $\Delta_{\mu; [a, b]}^0 = \delta_{\kappa; [a, b]}, \Delta_{\mu; [a, b]}^1= 0$. As a specific example, we consider coupling functions which in frequency-domain are expressible as sum of lorentzians i.e.~
\begin{align*}
&\abs{\hat{v}(\omega)}^2 = \sum_{i = 1}^M \frac{\alpha_i}{(\omega - \omega_i)^2 + \gamma_i^2} \ \forall \omega\in \mathbb{R}, \text{ or } \noindent\\
&\kappa(t) = \sum_{j = 1}^M \frac{\alpha_j}{2\gamma_j} e^{-\gamma_j \abs{t}} e^{-i\omega_j t}
\end{align*}
for some $\{\alpha_i \in \mathbb{R}_{> 0}\}_{i \in \{1, 2 \dots M\}}, \{\omega_i \in \mathbb{R}\}_{i \in \{1, 2 \dots M\}}$ and $\{\gamma_i \in \mathbb{R}_{> 0}\}_{i\in \{1, 2 \dots M\}}$. Such coupling functions arise commonly in modelling resonant light-matter interactions in quantum optics \cite{}. For this model, since $\kappa$ is differentiable almost everywhere and $\norm{\kappa'}_{L^\infty} \leq \sum_{j = 1}^M \alpha_j \sqrt{\gamma_j^2 + \omega_j^2} / 2\gamma_j$ and hence $\delta_{\kappa; [a, b]}(\varepsilon) = \varepsilon\sum_{j = 1}^M \alpha_j \sqrt{\gamma_j^2 + \omega_j^2} / 2\gamma_j$.
\end{example}
\begin{example}[Delta trains] This class of coupling functions arise frequently in models studying quantum systems with time-delay and feedback. Consider the coupling function specified by radon measure $\mu$
\[
\mu \cong \sum_{i =1}^M \alpha_i \delta(x - x_i)
\]
for some $\{x_i \in \mathbb{R}\}_{i \in \{1, 2 \dots M\}}, \{\alpha_i \in \mathbb{C}\}_{i \in \{1, 2 \dots M\}}$ with $x_1 < x_2 \dots <x_M$. Furthermore, its extension to $\textnormal{PWC}^1(\mathbb{R})$, $\mu^*$ (Eq.~\ref{eq:mu_star}), is given by
\[
\langle \mu^*, f\rangle = \sum_{i = 1}^M \frac{\alpha_i}{2}\bigg(\lim_{x \to x_i^+} f(x) + \lim_{x\to x_i^-} f(x)\bigg).
\]
The error functions (lemma \ref{lemma:main_error_bound_mu}) for $\mu$ can be chosen to be
\begin{align*}
&\Delta^0_{\mu, [a, b]}(\varepsilon) = \sum_{\substack{i \in \{1, 2 \dots M\} | \\ y_i \in [a - \varepsilon, a)\cup (b, b+ \varepsilon]}} \abs{\alpha_i} + \sum_{\substack{i \in \{1, 2 \dots M\} | \\ y_i \in (a , a + \varepsilon]\cup (b - \varepsilon, b]}} 2 \abs{\alpha_i}, \nonumber\\
&\Delta^1_{\mu, [a, b]}(\varepsilon) = \varepsilon\bigg( \sum_{\substack{i \in \{1, 2 \dots M\} | \\ y_i \in (a + \varepsilon, b - \varepsilon)}} \abs{\alpha_i}+ \frac{1}{2} \sum_{\substack{i \in \{1, 2 \dots M\} | \\ y_i \in \{a, b\}} }\abs{\alpha_i} \bigg).
\end{align*}
We refer the reader to the proof of lemma \ref{lemma:main_error_bound_mu} (appendix \ref{app:distributional_mem_ker}) for a derivation of these error functions in a more general setting of a delta train with a countably finite number of delta functions.
\end{example}
\begin{example} [Complex gaussian] Consider the coupling function specified by the radon measure $\mu \in \mathcal{M}(\mathbb{R})$
\[
\langle \mu, f\rangle = \sum_{j =1}^M c_j \int_{-\infty}^\infty e^{ik_j x^2} f(x) dx \ \forall \ f \in \textnormal{C}_c^0(\mathbb{R}),
\]
where $k_j \in \mathbb{R}, c_j \in \mathbb{C}$ for $j \in \{1, 2 \dots M\}$. Such a coupling function arises frequently in the study of quantum optical systems where the bath is a wire or channel with group velocity dispersion. Furthermore, its extension to $\textnormal{PWC}^1(\mathbb{R})$, $\mu^*$ defined in Eq.~\ref{eq:mu_star}, is given by
\[
\langle \mu^*, f\rangle = \sum_{j =1}^M c_j \int_{-\infty}^\infty e^{ik_j x^2} f(x) dx \ \forall \ f \in \textnormal{C}_c^0(\mathbb{R}),
\]
independent of the mollifier $\rho$. The error functions (lemma \ref{lemma:main_error_bound_mu}) for $\mu$ can be chosen to be
\begin{align*}
&\Delta^0_{\mu, [a, b]}(\varepsilon) =\varepsilon \sum_{j = 1}^M \abs{c_j k_j} \max\bigg(\abs{\frac{3b - a}{2}}, \abs{\frac{3a - b}{2}}\bigg) \ \text{ and } \nonumber\\
&\Delta^1_{\mu, [a, b]}(\varepsilon) = 0.
\end{align*}
\end{example}
\subsubsection{Analyzing the quantum dynamics}
We now turn our attention to analyzing non-Markovian quantum dynamics --- the basic data needed to specify a non-Markovian model is repeated in the definition below.
\begin{repdefinition}{def:model} [Non-Markovian model] A non-Markovian open system model for a quantum system with Hilbert space $\mathcal{H}_S$ is specified by
\begin{enumerate}
\item[(a)] A time-dependent system Hamiltonian $H_S(t) \in \mathfrak{L}(\mathcal{H}_S)$ which is Hermitian, norm continuous and differentiable in $t$,
\item[(b)] A set of distributional coupling functions $\{(\mu_i, \varphi_i) \}_{i \in \{1, 2 \dots M\}}$, where each coupling function is specified by a tempered radon measure (for its magnitude) and a phase.
\item[(c)] A set of bounded jump operators $\{L_i \in \mathfrak{L}(\mathcal{H}_S)\}_{i \in \{1, 2 \dots M\}}$.
\end{enumerate}
\end{repdefinition}
As we mentioned previously, the regularization of the non-Markovian model amounts to approximating the coupling functions, specified by tempered radon measures, by square integrable functions. We repeat this definition below.
\begin{repdefinition}{def:regularization}[Regularization]
For $\varepsilon > 0$ and given a symmetric mollifier $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$, an $\varepsilon, \rho-$regularization of a distributional coupling function $(\mu, \varphi)$ is a square integrable function $v_\varepsilon \in L^2(\mathbb{R})$ whose fourier transform $\hat{v}_\varepsilon \in L^2(\mathbb{R})$ is given by
\[
\hat{v}_\varepsilon(\omega) = \sqrt{\hat{\mu}(\omega)} \hat{\rho}({\omega}{\varepsilon}) e^{i\varphi(\omega)} \ \forall \omega \in \mathbb{R}.
\]
\end{repdefinition}
It is easily seen that $\hat{v} \in L^2(\mathbb{R})$, since $\hat{\mu}(\omega)$ has atmost polynomial growth in $\omega$ due to $\mu$ being a tempered distribution, and $\hat{\rho} \in \mathcal{S}(\mathbb{R})$ decays faster than any polynomial since it is the fourier transform of a smooth compact function. For square integrable coupling functions, proposition \ref{thm:schr_sq_int} guarantees the existence of the solution to the Schroedinger's equation --- we can then study whether the solution to the Schroedinger's equation converges as $\varepsilon \to 0$ and define the limit as the dynamics associated with the non-Markovian model.
In order to proceed further with this convergence study, we need to impose some suitable restrictions on the initial environment state since in general it can have amplitudes at arbitrarily high energies and would thus incur a large error upon regularization. We restrict ourselves to a dense subspace of the environment Hilbert space in which every state has particles only in modes whose high frequency amplitudes decrease faster than any polynomial in $\omega$. This is formalized by constructing this dense subspace using modal functions from the Schwartz spaces, $\mathcal{S}(\mathbb{R})$, which are functions that decay superpolynomially at large values of its argument --- the subspace is provided in the definition below.
\begin{definition}
For $M \in \mathbb{Z}_{\geq 1}$, define $\textnormal{F}^M_{\infty, \mathcal{S}} \subset \textnormal{F}^M_{\infty}[L^2(\mathbb{R})]$ as the set of vectors $\ket{\Phi}$ such that
\[
\forall n \in \mathbb{Z}_{\geq 1}: \Pi_n \ket{\Phi} \in \textnormal{span}\bigg(\bigg\{u^{\otimes n} \bigg| u = \bigoplus_{\alpha = 1}^M u_\alpha, u_\alpha \in \mathcal{S}(\mathbb{R}) \bigg\} \bigg).
\]
\end{definition}
The following lemma establishes error bounds on the action of the annihilation operator corresponding to a regularized coupling function, when applied on the states from $\textnormal{F}^M_{\infty, \mathcal{S}}$. These simply follow from the rapid decay properties of the mode functions corresponding to these states --- we provide a proof of this lemma in appendix \ref{app:lemma_inc_state_trunc}.
\begin{lemma}
\label{lemma:inc_state_trunc}
Let $(\mu,\varphi)$ be a distributional coupling function. Given two mollifiers $\rho, \sigma \in \textnormal{C}_c^\infty(\mathbb{R})$ and $\varepsilon, \delta > 0$, let $v_\varepsilon$ and $v_\delta \in L^2(\mathbb{R})$ be the $\varepsilon, \rho-$ and $\delta, \sigma-$regularization of $(\mu, \varphi)$ respectively. Let $\mathcal{H}_S$ be Hilbert space, then,
\begin{enumerate}
\item[(a)]$\forall \ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_{\infty, \mathcal{S}}^M$, $\exists c_{\mu, \ket{\Phi}} > 0$, $\forall \tau \geq 0$, $\forall \alpha \in \{1, 2 \dots M\}, \varepsilon > 0$ such that $\norm{a_{\alpha, \uptau_\tau v_\varepsilon}^- \ket{\Phi}} \leq c_{\mu, \ket{\Phi}}$.
\item[(b)] $\forall \ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_{\infty, \mathcal{S}}^M$, $\exists c_{\mu,\rho, \ket{\Phi}}, d_{\mu, \sigma, \ket{\Phi}} > 0$, $\forall \tau \geq 0, \forall \alpha \in \{1, 2 \dots M\}$, $\varepsilon, \delta > 0$ such that $\norm{a_{\alpha, \uptau_\tau(v_\varepsilon - v_\delta)}^- \ket{\Phi}} \leq c_{\mu,\rho, \ket{\Phi}}\varepsilon + c_{\mu, \sigma, \ket{\Phi}} \delta$.
\end{enumerate}
\end{lemma}
Our next lemma, which builds on lemma \ref{lemma:pert_theory_bound}, is key to showing convergence of the non-Markovian dynamics on removing regularization.
\begin{lemma}
\label{lemma:gfunc_error_bound}
Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, $M$ jump operators $\{L_\alpha \}_{\alpha \in \{1, 2 \dots M\}}$ and coupling functions, $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$. Given two mollifiers $\rho, \sigma \in \textnormal{C}_c^\infty(\mathbb{R})$ and $\varepsilon, \delta \in (0, 1/2)$, consider two non-Markovian models with the same system Hamiltonian and jump operators, but with square integrable coupling functions given by $v_\varepsilon = \{v_{\alpha, \varepsilon} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ and $v_\delta = \{v_{\alpha, \delta} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$, where $v_{\alpha, \varepsilon}$ and $v_{\alpha, \delta}$ are $\varepsilon, \rho-$ and $\delta, \sigma-$regularizations of $(\mu_\alpha, \varphi)$ respectively. Given an initial state $\ket{\Psi_0} \in \mathcal{H}_S\otimes \textnormal{F}_{\infty, \mathcal{S}}^M[L^2(\mathbb{R})]$, then the errors $\mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(t)$ and $\mathcal{D}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(t)$ defined in lemma \ref{lemma:pert_theory_bound} satisfy the estimates
\begin{enumerate}
\item[(a)]For all $\alpha \in \{1, 2 \dots M\}$ and $t > 0$,
\begin{align*}
\mathcal{E}_{\alpha, \ket{\Psi_0}}^{v_\delta, v_\varepsilon}(t) \leq 4\norm{L_\alpha}^2\textnormal{TV}_{[-1, t + 1]}(\mu_\alpha).
\end{align*}
\item[(b)]For all $\alpha \in \{1, 2 \dots M\}$ and $t > 0$,
\begin{widetext}
\begin{align*}
&\mathcal{E}_{\alpha, \ket{\Psi_0}}^{v_\delta, v_\varepsilon}(t) \leq 2\bigg(2\Delta^1_{\mu_\alpha; [0, t]}(\varepsilon + \delta) + \Delta^1_{\mu_\alpha; [0, t]}(2\varepsilon) + \Delta^1_{\mu_\alpha; [0, t]}(2\delta)\bigg)\times \nonumber\\
&\qquad \bigg( \norm{L_\alpha}\sup_{s\in[0, t]}\norm{[H_S(s), L_\alpha]} + 4\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_\alpha'} c_{\mu_{\alpha'}, \ket{\Psi_0}} + 6\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_{\alpha'}}^2 \textnormal{TV}_{[-1, t + 1]}(\mu_{\alpha'}).\bigg) + \nonumber\\
&\qquad 2\bigg(2\Delta^0_{\mu_\alpha; [0, t]}(\varepsilon + \delta) + \Delta^0_{\mu_\alpha; [0, t]}(2\varepsilon) + \Delta^0_{\mu_\alpha; [0, t]}(2\delta)\bigg)\norm{L_\alpha}^2,
\end{align*}
\end{widetext}
where $\Delta^0_{\mu_\alpha; [-t, 0]}, \Delta^1_{\mu_\alpha; [-t, 0]}$ are the error functions corresponding to $\mu_\alpha$ $($defined in lemma \ref{lemma:main_error_bound_mu}$)$ and $c_{\mu_\alpha, \ket{\Psi_0}}$ is the constant introduced in lemma \ref{lemma:inc_state_trunc}(a).
\item[(c)] For all $\alpha \in \{1, 2 \dots M\}$ and $t > 0$,
\begin{align*}
\mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\delta, v_\varepsilon}(t) \leq 4\norm{L_\alpha} \big(c_{\mu_\alpha, \rho, \ket{\Psi_0}}\varepsilon + c_{\mu_\alpha, \sigma, \ket{\Psi_0}}\delta\big),
\end{align*}
where $c_{\mu_\alpha, \rho, \ket{\Psi_0}}, c_{\mu_\alpha, \sigma, \ket{\Psi_0}}$ are constants introduced in lemma \ref{lemma:inc_state_trunc}(b).
\end{enumerate}
\end{lemma}
\noindent\emph{Proof}: For this proof, it is convenient to note that for $\alpha \in \{1, 2 \dots M\}$ and any $s, \in [0, t]$,
\[
\langle \uptau_{\tau} v_{\alpha, \delta}, \uptau_{s} v_{\alpha, \varepsilon}\rangle =2\pi \int_{\mathbb{R}}\hat{\mu}_{\alpha}(\omega) \hat{\rho}(\varepsilon \omega)\hat{\sigma}^*(\delta \omega) e^{-i\omega(s - t)} d\omega.
\]
Since if $\varepsilon, \delta \in (0, 1/2)$, $\text{supp}(\rho_\varepsilon \star \rho_\varepsilon), \text{supp}(\sigma_\delta \star \sigma_\delta), \text{supp}(\sigma_\delta \star \rho_\varepsilon) \subseteq [-1, 1]$. Consequently, $\forall f \in \textnormal{C}^0(\mathbb{R})$ and $t_1, t_2 \in (0, t]$,
\begin{subequations}\label{eq:tv_norm_estimate}
\begin{align}
&\abs{\int_{t_1}^{t_2} \langle \uptau_t v_{\alpha, \delta}, \uptau_s v_{\alpha, \varepsilon} \rangle f(s) ds} \nonumber\\
&\qquad= \abs{\langle \mu_\alpha, \sigma_\delta \star \rho_\varepsilon \star \uptau_t\big(f \cdot \mathcal{I}_{(t_1, t_2]}\big)\rangle}\nonumber\\
& \qquad \leq \textnormal{TV}_{[-1, t + 1]}(\mu_\alpha) \sup_{s \in [0, t]} \abs{f(s)}.
\end{align}
Similarly,
\begin{align}
&\abs{\int_{t_1}^{t_2} \langle \uptau_t v_{\alpha, \varepsilon},\uptau_s v_{\alpha, \delta} \rangle f(s) ds}, \abs{\int_{t_1}^{t_2} \langle \uptau_t v_{\alpha, \varepsilon}, \uptau_s v_{\alpha, \varepsilon} \rangle f(s) ds},\nonumber\\
& \abs{\int_{t_1}^{t_2} \langle \uptau_t v_{\alpha, \delta}, \uptau_s v_{\alpha, \delta} \rangle f(s) ds} \leq \textnormal{TV}_{[-1, t + 1]}(\mu_\alpha) \sup_{s \in [0, t]} \abs{f(s)}.
\end{align}
\end{subequations}
\noindent\emph{(a)} We note that
\begin{align*}
&\abs{\int_0^t \langle \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon}), \uptau_s v_{\alpha, \varepsilon}\rangle G^{v_\varepsilon, v_\varepsilon, v_\delta}_{L_\alpha, L_\alpha^\dagger}(s, t) ds} \nonumber\\
&\qquad \leq \abs{\int_0^t \langle \uptau_t v_{\alpha, \delta}, \uptau_s v_{\alpha, \varepsilon}\rangle G^{v_\varepsilon, v_\varepsilon, v_\delta}(s, \tau) ds} + \nonumber\\
&\qquad \quad \abs{\int_0^t \langle \uptau_t v_{\alpha, \delta}, \uptau_s v_{\alpha, \varepsilon}\rangle G^{v_\varepsilon, v_\varepsilon, v_\delta}(s, \tau) ds}.
\end{align*}
Since $\forall s \in [0, t] : \abs{G^{v_\varepsilon, v_\varepsilon, v_\delta}_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(s, \tau)}, \leq \norm{L_\alpha}^2$, and using Eq.~\ref{eq:tv_norm_estimate}, we obtain that
\begin{align*}
&\abs{\int_0^t \langle \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon}), \uptau_s v_{\alpha, \varepsilon}\rangle G^{v_\varepsilon, v_\varepsilon, v_\delta}_{L_\alpha, L_\alpha^\dagger}(s, t) ds} \leq \nonumber\\
&\qquad \qquad \qquad \qquad \qquad \qquad 2 \norm{L_\alpha}^2\textnormal{TV}_{[-1, t + 1]}(\mu_\alpha).
\end{align*}
Similarly, we can obtain that
\begin{align*}
&\abs{\int_0^t \langle \uptau_s v_{\alpha, \delta}, \uptau_t(v_{\alpha, \delta} - v_{\alpha, \varepsilon}) \rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\delta, v_\delta}} \leq \nonumber\\
&\qquad \qquad \qquad \qquad \qquad \qquad 2 \norm{L_\alpha}^2\textnormal{TV}_{[-1, t + 1]}(\mu_\alpha).
\end{align*}
Combining these two estimates, we obtain the part \emph{(a)} of the lemma statement.
\noindent\emph{(b)} We begin by noting that
\begin{align*}
&\langle \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon}), \uptau_s v_{\alpha, \varepsilon}\rangle = \nonumber\\
&\qquad 2\pi\int_{-\infty}^\infty \hat{\mu}_\alpha(\omega) \hat{\rho}(\varepsilon \omega) \big(\hat{\rho}^*(\varepsilon \omega) - \hat{\sigma}^*(\delta \omega) \big) e^{-i\omega(s - t)}d\omega,
\end{align*}
and therefore,
\begin{align*}
&\int_0^t \langle \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon}), \uptau_s v_{\alpha, \varepsilon }\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\varepsilon, u_\delta}(s, t) ds =\nonumber\\
&\qquad\big\langle\mu_\alpha, \rho_\varepsilon \star \rho_\varepsilon \star g_{\alpha, t}^{\varepsilon, \delta} - \rho_\varepsilon \star \sigma_\delta \star g^{\alpha, t}_{\varepsilon, \delta} \big\rangle,
\end{align*}
where
\[
g^{\varepsilon, \delta}_{\alpha, t} = \uptau_t \bigg(G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\varepsilon, u_\delta}(\cdot, t)\cdot \mathcal{I}_{[0, t]}\bigg).
\]
We note that $g^{\varepsilon, \delta}_{\alpha, t}$ is a continuous and differentiable function when restricted to $[-t, 0]$ and hence $ \in \textnormal{PWC}^1(\mathbb{R})$. Consequently,
\begin{align}\label{eq:error_g}
&\abs{\big\langle\mu_\alpha, \rho_\varepsilon \star \rho_\varepsilon \star g_{\alpha, t}^{\varepsilon, \delta} - \rho_\varepsilon \star \sigma_\delta \star g^{\alpha, t}_{\varepsilon, \delta} \big\rangle}\nonumber\\
& \leq \abs{\big\langle\mu_\alpha^*, g_{\alpha, t}^{\varepsilon, \delta} \big\rangle - \big\langle \mu_\alpha, \rho_\varepsilon \star \rho_\varepsilon \star g_{\alpha, t}^{\varepsilon, \delta}\big\rangle} +\nonumber\\
&\qquad \qquad \abs{\big\langle \mu_\alpha^*, g_{\alpha, t}^{\varepsilon, \delta} \big\rangle-\big\langle\mu_\alpha, \rho_\varepsilon \star \sigma_\delta \star g^{\alpha, t}_{\varepsilon, \delta} \big\rangle} \nonumber \\
& \leq \bigg(\Delta_{\mu_\alpha; [-t, 0]}^0(2\varepsilon) + \Delta_{\mu_\alpha; [-t, 0]}^0(\delta + \varepsilon)\bigg) \sup_{s \in [0, t]} \abs{g_{\alpha, t}^{\varepsilon, \delta}(s)} + \nonumber\\
&\qquad \bigg(\Delta_{\mu_\alpha; [-t, 0]}^1(2\varepsilon) + \Delta_{\mu_\alpha; [-t, 0]}^1(\delta + \varepsilon)\bigg) \sup_{s \in [0, t]} \abs{\partial_s g_{\alpha, t}^{\varepsilon, \delta}(s)}
\end{align}
Furthermore, note that
\begin{align}\label{eq:sup_norm_g}
\sup_{s \in [-t, 0]} \abs{g_{\alpha, t}^{\varepsilon, \delta}(s)} \leq \norm{L_\alpha}^2,
\end{align}
We next provide a bound on the derivative ($\sup_{s \in [-t, 0]} |\partial_s g^{\varepsilon, \delta}_{\alpha, t}(s)|$) which is uniform in $\varepsilon, \delta$. An application of lemma \ref{lemma:gfunc_deriv}, yields
\begin{align*}
&\abs{\partial_s g^{\varepsilon, \delta}_{\alpha, t}(s)}\leq \nonumber\\
&\abs{G^{v_\varepsilon, v_\varepsilon, v_\delta}_{[H_S(s + t), L_\alpha], L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(s+t, t)} + \nonumber\\
& \sum_{\alpha' = 1}^M \bigg(\abs{G^{v_\varepsilon, v_\varepsilon, v_\delta}_{[L_{\alpha'}^\dagger, L_\alpha], L_\alpha^\dagger; \ket{\Psi_0}, a^{-}_{\alpha', \uptau_{(s + t)}v_{\alpha}}\ket{\Psi_0}}(s + t, t)} + \nonumber\\
&\qquad \quad \abs{G^{v_\varepsilon, v_\delta, v_\delta}_{[L_{\alpha'}, L_\alpha], L_\alpha^\dagger; \ket{\Psi_0}, a^{-}_{\alpha', \uptau_{(s + t)}v_{\alpha}}\ket{\Psi_0}}(s + t, t)}\bigg) + \nonumber\\
&\sum_{\alpha' = 1}^M \bigg(\bigg|\int_0^{s + \tau}\langle \uptau_{s + \tau} v_{\alpha', \varepsilon}, \uptau_{s'} v_{\alpha', \varepsilon}\rangle \times \nonumber\\
& \qquad \qquad \qquad G^{v_\varepsilon, v_\varepsilon, v_\varepsilon, v_\delta}_{L_{\alpha'}, [L_{\alpha'}^\dagger, L_\alpha], L_{\alpha}^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}(s', s + \tau, \tau)ds'\bigg| + \nonumber\\
&\qquad \quad \bigg|\int_0^\tau \langle \uptau_{s'} v_{\alpha', \delta}, \uptau_{s + \tau} v_{\alpha', \varepsilon}\rangle \times \nonumber\\
&\qquad \qquad \qquad G_{[L_{\alpha'}, L_\alpha], L_{\alpha}^\dagger, L_{\alpha'}^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\varepsilon, v_\delta, v_\delta}(s + \tau, \tau, s')ds' \bigg|+\nonumber \\
&\qquad \quad \bigg|\int_{s + \tau}^\tau \langle \uptau_{s'} v_{\alpha', \delta}, \uptau_{s + \tau} v_{\alpha', \varepsilon}\rangle \times \nonumber\\
&\qquad \qquad \qquad G_{[L_{\alpha'}, L_\alpha], L_{\alpha'}^\dagger, L_{\alpha}^\dagger;\ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\varepsilon, v_\delta, v_\delta}(s + \tau, s', \tau)ds'\bigg|\bigg).
\end{align*}
Using Eq.~\ref{eq:tv_norm_estimate}, we thus obtain that
\begin{align*}
&\sup_{s\in(-t, 0)} \abs{\partial_s g_{\alpha, t}^{\varepsilon, \delta}(s)} \leq \norm{L_\alpha} \sup_{s \in [0, t]}\norm{[H_S(s), L_\alpha]} + \nonumber\\
&\qquad 4\norm{L_\alpha}^2 \sum_{\alpha' = 1} ^M \norm{L_{\alpha'}}\sup_{s \in [0, t]} \norm{a^-_{\alpha', \uptau_s v_{\alpha'}} \ket{\Psi_0}} + \nonumber\\
&\qquad 6\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_\alpha'}^2 \textnormal{TV}_{[-1, t + 1]}(\mu_\alpha)
\end{align*}
From lemma \ref{lemma:inc_state_trunc}(a), it follows that $\forall \alpha' \in \{1, 2 \dots M\}, s \in [0, t]$, $\norm{a^-_{\alpha', \uptau_s v_{\alpha'}}\ket{\Psi_0}}\leq c_{\mu_{\alpha'}, \ket{\Psi_0}}$ and therefore, we obtain that
\begin{align}\label{eq:sup_norm_g_prime}
&\sup_{s\in(-t, 0)} \abs{\partial_s g_{\alpha, t}^{\varepsilon, \delta}(s) } \leq \sup_{s\in[0, \tau]} \norm{[H_S(s), L_\alpha]} \norm{L_\alpha} + \nonumber\\
&\qquad 4\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_\alpha'} c_{\mu_{\alpha'}, \ket{\Psi_0}} +\nonumber\\
&\qquad 6\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_{\alpha'}}^2 \textnormal{TV}_{[-1, t + 1]}(\mu_{\alpha'}).
\end{align}
From Eqs.~\ref{eq:error_g}, \ref{eq:sup_norm_g} and \ref{eq:sup_norm_g_prime}, we obtain that
\begin{align*}
&\abs{\int_0^t \langle \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon}), \uptau_s v_{\alpha, \varepsilon }\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\varepsilon, v_\varepsilon, u_\delta}(s, t) ds } \leq \nonumber\\
&\qquad \bigg(\Delta_{\mu_\alpha; [-t, 0]}^0(2\varepsilon) + \Delta_{\mu_\alpha; [-t, 0]}^0(\delta + \varepsilon)\bigg) \norm{L_\alpha}^2 +\nonumber\\
&\qquad \bigg(\Delta_{\mu_\alpha; [-t, 0]}^1(2\varepsilon) + \Delta_{\mu_\alpha; [-t, 0]}^1(\delta + \varepsilon)\bigg)\times \\
&\qquad \bigg(\sup_{s\in[0, \tau]} \norm{[H_S(s), L_\alpha]} \norm{L_\alpha} + \nonumber\\
&\qquad \qquad 4\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_{\alpha'}} c_{\mu_{\alpha'}, \ket{\Psi_0}} + \nonumber\\
&\qquad \qquad 6\norm{L_\alpha}^2 \sum_{\alpha' = 1}^M \norm{L_{\alpha'}}^2 \textnormal{TV}_{[-1, t + 1]}(\mu_{\alpha'})\bigg).
\end{align*}
A similar bound can be obtained for the term $\int_0^t \langle \uptau_s v_{\alpha, \delta }, \uptau_t (v_{\alpha, \delta} - v_{\alpha, \varepsilon})\rangle G_{L_\alpha, L_\alpha^\dagger; \ket{\Psi_0}, \ket{\Psi_0}}^{v_\delta, v_\delta, v_\varepsilon}(t, s) ds $ in the expression for $\mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(t)$ provided in lemma \ref{lemma:pert_theory_bound}. Combining these estimates, we obtain part (d) of the lemma statement.
\noindent (c) This follows from a direct application of lemma \ref{lemma:inc_state_trunc}.
\begin{reptheorem}{theorem:non_mkv_exis}[Formal, Non-markovian dynamics] Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, $M$ jump operators $\{L_\alpha \}_{\alpha \in \{1, 2 \dots M\}}$ and distributional coupling functions, $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$. Construct a square-integral non-Markovian model with the same system Hamiltonian and jump operators, but with coupling functions $v_\varepsilon := \{v_{\alpha, \varepsilon} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$, where for $\alpha \in \{1, 2 \dots M\}$, $v_{\alpha, \varepsilon}$ is an $\varepsilon, \rho-$regularization of $(\mu_\alpha, \varphi_\alpha)$ for a symmetric mollifier $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$, $\varepsilon > 0$ and let $U_{v_\varepsilon}(\cdot, \cdot)$ be its propagator. Then, for $t > 0$, $U(t): \textnormal{F}^M_{\infty, \mathcal{S}}\otimes \mathcal{H}_S \to \mathcal{H}$ defined via $U(t)\ket{\Psi_0} = \lim_{\varepsilon \to 0} U_{v_\varepsilon}(t, 0) \ket{\Psi_0}$ exists, is an isometry and is independent of the choice of mollifier $\rho$.
\end{reptheorem}
\noindent\emph{Proof}: For simplicity, we will assume that $\varepsilon, \delta \in (0, 1)$. Consider two symmetric mollifiers $\rho, \sigma \in \textnormal{C}_c^\infty(\mathbb{R})$ --- let $v_\varepsilon:= \{v_{\alpha, \varepsilon} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ and $v_\delta:= \{v_{\alpha, \delta} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ be the $\varepsilon, \rho-$ and $\delta, \sigma-$regularizations of the distributional coupling functions. For $\ket{\Psi_0} \in \textnormal{F}^M_{\infty, \mathcal{S}}\otimes \mathcal{H}_S $, let $\ket{\Psi_{v_\varepsilon}(t)} = U_{v_\varepsilon}(t, 0) \ket{\Psi_0}$ and $\ket{\Psi_{v_\delta}(t)} = U_{v_\delta}(t, 0) \ket{\Psi_0}$, where $U_{v_\varepsilon}(\cdot, \cdot), U_{v_\delta}(\cdot, \cdot)$ are the propagators corresponding to the two models. We note that from lemma \ref{lemma:gfunc_error_bound}(c) that $\forall \alpha \in \{1, 2 \dots M\}$ and $\tau \in [0, t]$
\begin{align*}
&\lim_{\varepsilon, \delta \to 0} \mathcal{D}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) = 0 \ \text{ and } \\
&\quad \mathcal{D}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) \leq 4\norm{L_\alpha} \big(c_{\mu_\alpha, \rho, \ket{\Psi_0}} + c_{\mu_\alpha, \sigma, \ket{\Psi_0}}\big).
\end{align*}
From the dominated convergence theorem, we then obtain that
\[
\lim_{\varepsilon, \delta \to 0} \int_{0}^t \mathcal{D}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) d\tau = \int_0^t \lim_{\varepsilon, \delta \to 0} \mathcal{D}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) d\tau = 0.
\]
Similarly, we note from lemma \ref{lemma:gfunc_error_bound}(b) that $\forall \alpha \in \{1, 2 \dots M\}$ and $\tau \in [0, t]$
\[
\lim_{\varepsilon, \delta \to 0} \mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) = 0.
\]
From lemma \ref{lemma:gfunc_error_bound}(a), we obtain that $\forall \alpha \in \{1, 2 \dots M\}$ and $\tau \in [0, t]$
\begin{align}
&\mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) \leq 4\norm{L_\alpha}^2 \textnormal{TV}_{[-1, \tau + 1]}(\mu_\alpha)\nonumber\\
&\qquad \qquad \leq 4\norm{L_\alpha}^2 \textnormal{TV}_{[-1, t + 1]}(\mu_\alpha).
\end{align}
Hence, again by dominated convergence theorem, we obtain that
\[
\lim_{\varepsilon, \delta \to 0} \int_0^t \mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) d\tau = \int_0^t \lim_{\varepsilon, \delta \to 0} \mathcal{E}^{v_\delta, v_\varepsilon}_{\alpha, \ket{\Psi_0}}(\tau) d\tau = 0.
\]
We thus obtain from lemma \ref{lemma:pert_theory_bound} that
\begin{align}\label{eq:error_between_two_diff_seq}
\lim_{\varepsilon, \delta\to 0} \norm{\ket{\Psi_{v_\varepsilon}(t)} - \ket{\Psi_{v_\delta}(t)}} = 0,
\end{align}
for all symmetric mollifiers $\rho, \sigma$. From this condition, using $\rho = \sigma$, we obtain that $\lim_{\varepsilon \to 0} \ket{\Psi_{v_\varepsilon}(t)}$ exists. Furthermore, $\norm{\lim_{\varepsilon \to 0} \ket{\Psi_{v_\varepsilon}(t)}} = \lim_{\varepsilon \to 0}\norm{ \ket{\Psi_{v_\varepsilon}(t)}} = \norm{\ket{\Psi_0}}$, and hence the operator mapping $\ket{\Psi_0}$ to $\lim_{\varepsilon \to 0} \ket{\Psi_{v_\varepsilon}(t)}$ is an isometry. Furthermore, since the limit exists, Eq.~\ref{eq:error_between_two_diff_seq} additionally implies that the limit is independent of the choice of the mollifier.
\(\square\)
\section{Complexity of non-Markovian dynamics}
\label{sec:complexity}
\subsection{Certifiable Markovian dilations}
\label{subsec:dilation}
In this section, we develop a certifiable dilation of the non-Markovian problem i.e.~we provide a systematic method to approximate the continuum of modes in the environment by a finite number of modes. More specifically, we will construct a `chain dilation' of the non-Markovian model as defined in section \ref{subsec:proof_ideas} and repeated below.
\begin{definition}[Chain dilation] Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, coupling functions $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$ and jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$. A chain dilation, with $N_m$ modes and bandwidth $B$, of this model is described by the following Hamiltonian over the system-environment Hilbert space $(\mathcal{H}_S \otimes \textnormal{Fock}[L^2(\mathbb{R})]):$
\begin{align*}
&H(t) = \nonumber\\
&\qquad H_S(t) + \sum_{\alpha = 1}^M\bigg(g_\alpha a_{\alpha, 1} L_\alpha^\dagger + \sum_{i = 1}^{N_m - 1} t_{\alpha, i} a_{\alpha, i}^\dagger a_{\alpha, i + 1} + \text{h.c.}\bigg),
\end{align*}
where for $\alpha \in \{1, 2 \dots M\}, i \in \{1, 2 \dots N_m\}$,
\begin{enumerate}
\item[(a)] The operator $a_{\alpha, i} = \int_{\mathbb{R}} \varphi_{\alpha, i}(\omega) a_{\alpha, \omega} d\omega$ is the annihilation operator corresponding to the $i^\textnormal{th}$ mode of the $\alpha^\textnormal{th}$ bath described by the orthonormal mode functions $\varphi_{\alpha, i} \in \textnormal{L}^2(\mathbb{R})$ $($i.e. $\langle \varphi_{\alpha, i}, \varphi_{\alpha, i'}\rangle = \delta_{i, i'})$.
\item[(b)] The coupling constants $g_\alpha, t_{\alpha, i}$ are upper bounded by the bandwidth $B$ i.e.~$\abs{g_\alpha}, \abs{t_{\alpha, i}} \leq B$.
\end{enumerate}
\end{definition}
Thus, in a chain dilation of the non-Markovian model, each bath is approximated by a 1D chain of bosonic modes with nearest neighbour interactions, with the first mode coupling to the system. In this subsection, we will provide both the construction of such a dilation, as well as bounds on the error incurred in this dilation in terms of the number of modes in the environment. We remark that the error bounds that we provide grow only polynomially with the time for which the non-Markovian system is evolved as opposed to previously obtained bounds which grow exponentially with time {\cite{trivedi2021convergence, mascherpa2017open}. } Apart from being of independent interest to simulating non-Markovian dynamics, the result of this section is central to the proof of non-Markovian many-body dynamics being simulable on a quantum computer that is provided in the following subsection.
One of the key ingredients in our proof is the previously studied star-to-chain mapping of non-Markovian models --- this can be used to systematically construct a chain dilation when the coupling functions are square integrable and have finite-frequency support (i.e. $v_\alpha(\omega) = 0$ for $\omega$ outside $[-\omega_c, \omega_c]$). The lemma below provides a bound on the error incurred by the star-to-chain transformation --- while this result is known in the literature \cite{woods2015simulating, trivedi2021convergence}, we include a proof of this in appendix \ref{app:chain_appx}. Our proof closely follows Ref.~\cite{trivedi2021convergence}.
\begin{lemma}[Star-to-chain transformation]
\label{lemma:star_to_chain}
Consider a non-Markovian model with system Hamiltonian $H_S(t)$, square-integrable coupling functions $\{v_\alpha \in \text{L}^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ and jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$. Furthermore, assume that $\exists\ \omega_c > 0$ such that for $\abs{\omega} \geq \omega_c, v_\alpha(\omega) = 0$. Then there exists a chain dilation of this non-Markovian model with $N_m$ modes and bandwidth $\leq \omega_c$ such that
\begin{align*}
&\norm{\ket{\Psi(t)} - \ket{\hat{\Psi}(t)}}^2 \leq \nonumber\\
&\qquad 4\ell t \bigg(1 + \ell^2 t^2 + \mu^{(1)}_{\ket{\Psi_0}}\bigg)^{1/2} N_m \bigg(\frac{2e\omega_c t}{N_m}\bigg)^{N_m/2},
\end{align*}
where $\ket{\Psi(t)}$ and $\ket{\hat{\Psi}(t)}$ are the system-environment states obtained from the model and its chain dilation, $\ket{\Psi(0)} = \ket{\hat{\Psi(0)}} = \ket{\Psi_0}$, $\ell = \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \norm{L_\alpha}$ and $\mu^{(1)}_{\ket{\Psi_0}}$ is the initial expectation value of the particle number operator of the environment.
\end{lemma}
\noindent From this lemma, it follows that a choice of $N_m = \textnormal{poly}(\ell, t, \omega_c, 1/\epsilon)$ ensures that the approximation error incurred using the star-to-chain transformation can be made smaller than $\epsilon$.
To use this lemma for more general non-Markovian models, we need to approximate them with coupling functions that have finite frequency support. As we noted previously, several coupling functions of interest are tempered distributions and do not necessarily fall off to zero at high frequencies --- this complicates the introduction of the frequency cutoff. We perform this in two steps --- first we regularize the coupling function, which damps that the high frequency components, and then we introduce a frequency cutoff. The analysis of the regularization error heavily uses lemma \ref{lemma:gfunc_error_bound} --- however, to predict the scalings of the regularization error with problem parameters, we need two additional assumptions. The first assumption is on the radon measures describing the memory kernels, and restricts their growth and smoothness and the second assumption ensures that the initial environment state has particle number moments that decay sufficiently rapidly with frequency. We state them mathematically below.
\begin{repassumption}{assump:radon_measure}
The radon measure $\mu$ corresponding to the coupling function should satisfy:
\begin{enumerate}
\item[(a)] For any interval $[a, b] \subseteq \mathbb{R}$, $\textnormal{TV}_{[a, b]}(\mu) \leq \textnormal{poly}(\abs{a}, \abs{b})$ and
\item[(b)] The error functions corresponding to $\mu$, $\Delta^0_{\mu; [a, b]}(\varepsilon)$ and $\Delta^1_{\mu; [a, b]}(\varepsilon)$ as specified in lemma \ref{lemma:main_error_bound_mu} are individually locally integrable with respect to $a, b$ and grow at most polynomially with $\abs{a}, \abs{b}$, and fall off polynomially with $\varepsilon$ i.e.~
\[
\Delta^0_{\mu; [a, b]}(\varepsilon), \Delta^1_{\mu; [a, b]}(\varepsilon) \leq \textnormal{poly}(|a|, |b|, \varepsilon).
\]
\end{enumerate}
\end{repassumption}
\begin{repassumption}{assump:initial_state}
The initial environment state $\ket{\phi_1}\otimes \ket{\phi_2} \dots \ket{\phi_M}$ where for $\alpha \in \{1,2 \dots M\}$, $\ket{\phi_\alpha} \in \textnormal{Fock}[L^2(\mathbb{R})]$ and for its $n-$particle wavefunctions $\phi_{\alpha, n} \in L^2(\mathbb{R}^n)$, and any $j, k \geq 0$, $\exists \mathcal{N}_{j, k} > 0$ such that
\[
\sum_{n = 0}^\infty n^j \int_{\mathbb{R}^n} (1 + \omega_1^2)^k \abs{\phi_{\alpha, n}(\omega)}^2 d\omega < \mathcal{N}_{j, k}.
\]
\end{repassumption}
We next provide a lemma that, subject to these assumptions, bounds the regularization error.
\begin{lemma}[Regularization error]\label{lemma:reg_error}
Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$ and coupling functions $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1,2 \dots M\}}$. Furthermore, assume that $\mu_\alpha$ satisfy assumption \ref{assump:radon_measure}, and its fourier transform satisfies $\hat{\mu}_\alpha(\omega) \leq O(\omega^{2k})$ for some $k > 0$ and the initial environment state $\ket{\Psi_0}$ satisfies assumption \ref{assump:initial_state}. Then, if $\ket{\Psi(t)}$ is the system-environment state at time $t$, and $\ket{\Psi_\varepsilon(t)}$ is the system environment state of the regularized non-Markovian model, then
\begin{align*}
&\norm{\ket{\Psi(t)} - \ket{\Psi_\varepsilon(t)}} \leq \nonumber\\
&\qquad O\bigg(\varepsilon^q \textnormal{poly}\bigg(t, M, \norm{L_\alpha}, \sup_{\alpha, s\in[0, t]} \norm{[H_S(s), L_\alpha]}, \nonumber\\
&\qquad \qquad \qquad \ \ \mathcal{N}_{1, k + 1}, \mathcal{N}_{1, k + 2}\bigg)\bigg),
\end{align*}
for some $q > 0$.
\end{lemma}
\noindent\emph{Proof}: Suppose $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$ is a symmetric mollifier, and $\hat{\rho}$ be its fourier transform. Consider a regularized non-Markovian model which has the same system Hamiltonian and jump operator but square integrable coupling functions $v_\varepsilon := \{v_{\alpha, \varepsilon} \in L^2(\mathbb{R})\}_{\alpha \in \{1, 2\dots M\}}$, where $v_{\alpha, \varepsilon}$ is the $\varepsilon, \rho-$regularization of $(\mu_\alpha, \varphi_\alpha)$ (definition \ref{def:regularization}). Denote the system-environment state at time $t$ corresponding to the actual model by $\ket{\Psi(t)}$, and the regularized model by $\ket{\Psi_\varepsilon(t)}$. We note from theorem \ref{thm:} that $\ket{\Psi(t)} = \lim_{\varepsilon'\to 0}\ket{\Psi_\varepsilon(t)}$. Thus, we obtain from lemma \ref{lemma:pert_theory_bound} that
\begin{align*}
&\norm{\ket{\Psi(t)} - \ket{\Psi_\varepsilon(t)}}^2 \leq \nonumber\\
&\qquad \lim_{\varepsilon' \to 0}\sum_{\alpha = 1}^M \bigg(\int_0^t \mathcal{E}^{v_\varepsilon, v_{\varepsilon'}}_{\alpha, \ket{\Psi_0}}(\tau) d\tau + \int_0^t \mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) d\tau\bigg).
\end{align*}
Let us first bound $\lim_{\varepsilon'\to 0}\int_0^t \mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau)d\tau$ --- note from lemma~\ref{lemma:pert_theory_bound} that $\mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) = 4\norm{L_\alpha} \Vert{a_{\alpha, \uptau_\tau(v_{\alpha, \varepsilon} - v_{\alpha, \varepsilon'})}^-\ket{\Psi_0}}\Vert$, where $\uptau_\tau : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is the time-translation unitary group. Now, from the Cauchy-Schwarz inequality it follows that
\begin{align}\label{eq:lemma_reg_init_state_error_1}
&\norm{a_{\alpha, \uptau_\tau(v_{\alpha, \varepsilon} - v_{\alpha, \varepsilon'})}^-\ket{\Psi_0}}^2 =\sum_{n = 0}^\infty n\times \nonumber\\
&\quad \int_{\mathbb{R}^{n - 1}}\abs{\int_{\mathbb{R}} \big(\hat{v}^*_{\alpha, \varepsilon'}(\omega) - \hat{v}_{\alpha, \varepsilon}(\omega)\big) e^{i\omega_1 \tau} \phi_{\alpha, n}((\omega_1, \omega)) d\omega_1}^2d\omega, \nonumber\\
&\quad \leq \mathcal{N}_{1, k + 2} \int_{\mathbb{R}} \frac{\abs{\hat{v}_{\alpha, \varepsilon}(\omega) - \hat{v}_{\alpha, \varepsilon'}(\omega)}^2}{(1 + \omega^2)^{k + 2}} d\omega.
\end{align}
Furthermore, we note from Taylor's theorem that
\begin{align}\label{eq:lemma_reg_init_state_error_2}
&\abs{\hat{v}_\varepsilon(\omega) - \hat{v}_{\varepsilon'}(\omega)}^2, \nonumber\\
&\qquad =\hat{\mu}(\omega) \abs{\hat{\rho}(\omega \varepsilon) - \hat{\rho}(\omega \varepsilon')}^2, \nonumber \\
&\qquad \leq \hat{\mu}(\omega) \omega^2 (\varepsilon - \varepsilon')^2 \sup_{\nu \in \mathbb{R}} \abs{\partial_\nu \hat{\rho}(\nu)}^2, \nonumber \\
&\qquad \leq \hat{\mu}(\omega) \omega^2 (\varepsilon - \varepsilon')^2,
\end{align}
where we have used that since $\partial_\nu \hat{\rho}(\nu)$ is the fourier transform of $x \rho(x)$,
\[
\sup_{\nu \in \mathbb{R}} \abs{\partial_\nu \hat{\rho}(\nu)} \leq \int_{[-1, 1]} \abs{x} \abs{\rho(x)} dx \leq 1.
\]
It now follows from Eqs.~\ref{eq:lemma_reg_init_state_error_1} and \ref{eq:lemma_reg_init_state_error_2}
\begin{align*}
&\lim_{\varepsilon' \to 0} \int_0^t \mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) d\tau \leq \nonumber\\
&\qquad 4\norm{L_\alpha} t \sqrt{\mathcal{N}_{1, k + 2}} \bigg(\int_{\mathbb{R}} \frac{\omega^2 \hat{\mu}(\omega)}{(1 + \omega^2)^{k + 2}} d\omega\bigg)^{1/2} \abs{\varepsilon },
\end{align*}
or equivalently,
\begin{align}\label{eq:lemma_reg_d_bound_final}
&\lim_{\varepsilon' \to 0} \sum_{\alpha = 1}^M \int_0^t \mathcal{D}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) d\tau \leq \nonumber\\
&\qquad \qquad O\bigg(\varepsilon\ \text{poly}\bigg(t, M, \sup_{\alpha}\norm{L_\alpha}, \mathcal{N}_{1, k + 2}\bigg)\bigg).
\end{align}
Next, we consider $\lim_{\varepsilon'\to 0} \int_0^t \mathcal{E}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) d\tau$ --- for this, we use lemma \ref{lemma:gfunc_error_bound}(b). To use this lemma, we need to provide a bound on $c_{\mu_\alpha, \ket{\Psi_0}}$ which appears in the lemma statement and is defined in lemma \ref{lemma:inc_state_trunc}. Note from its definition that $c_{\mu_\alpha, \ket{\Psi_0}}$ bounds $\norm{a_{\alpha, \uptau_\tau v_\varepsilon}^- \ket{\Psi_0}}$ and
\begin{align*}
&\norm{a^-_{\alpha, \uptau_\tau v_{\alpha, \varepsilon}} \ket{\Psi_0}}^2 \leq \sum_{n = 0}^\infty n \times \nonumber\\
&\qquad \int_{\mathbb{R}^{n - 1}} \abs{\int_{\mathbb{R}} \hat{v}^*_{\varepsilon}(\omega) e^{i\omega \tau} \phi_{\alpha, n}((\omega_1, \omega)) d\omega_1}^2 d\omega, \nonumber\\
&\leq \mathcal{N}_{1, k + 1} \int_{\mathbb{R}}\frac{\abs{\hat{v}_{\alpha, \varepsilon}(\omega)}^2}{(1 + \omega^2)^{k + 1}} d\omega.
\end{align*}
Also, since $\sup_{\nu \in \mathbb{R}} \abs{\hat{\rho}(\nu)} \leq \int_{[-1, 1]} \abs{\rho(x)}dx = 1$, $\abs{\hat{v}_{\alpha, \varepsilon}(\omega)}^2 \leq \hat{\mu}(\omega)$, we obtain that
\[
\norm{a^-_{\alpha, \uptau_\tau v_{\alpha, \varepsilon}} \ket{\Psi_0}} \leq \sqrt{\mathcal{N}_{1, k + 1}} \bigg(\int_{\mathbb{R}} \frac{\hat{\mu}(\omega)}{(1 + \omega^2)^{k + 1}}d\omega\bigg)^{1/2}.
\]
Since assumption \ref{assump:radon_measure} holds, $\Delta^0_{\mu_\alpha, [0, t]}(\varepsilon), \Delta^1_{\mu_\alpha, [0, t]}(\varepsilon) \leq O(\varepsilon^p \text{poly}(t))$ for some $p > 0$. It then follows from lemma \ref{lemma:gfunc_error_bound}(b) that
\begin{align}\label{eq:lemma_reg_e_bound_final}
&\lim_{\varepsilon' \to 0} \sum_{\alpha = 1}^M \int_0^t \mathcal{E}_{\alpha, \ket{\Psi_0}}^{v_\varepsilon, v_{\varepsilon'}}(\tau) d\tau\leq \nonumber \\
& O\bigg(\varepsilon^p \text{poly}\bigg(t, M, \text{sup}_\alpha \norm{L_\alpha}, \sup_{\alpha, s \in [0, \tau]} \norm{[H_S(\tau), L_\alpha]}, \mathcal{N}_{1, k + 1}\bigg)\bigg)
\end{align}
Combining Eqs.~\ref{eq:lemma_reg_d_bound_final} and \ref{eq:lemma_reg_e_bound_final}, we obtain the lemma statement. $
\square$
Next, we consider frequency truncation of the coupling functions --- after regularization, the coupling functions decay at high frequencies, and consequently a frequency cut-off can be introduced. To make this precise, we note that for a coupling function specified by $(\mu, \varphi)$ then the regularized coupling function $\hat{v}_\varepsilon$ falls off at least as $\omega^{-1}$ --- this is so because the fourier transform of a mollifier $\hat{\rho}$, $\hat{\rho}(\omega)$ falls off faster than any polynomial in $\omega$. Since $\hat{\mu}(\omega)$ grows atmost polynomially with $\omega$, the $\omega^{-1}$ decay of the regularized coupling function follows. The next lemma considers non-Markovian models with coupling functions fall off as $\omega^{-1}$, and bounds the error incurred on introducing a frequency cut-off.
\begin{lemma}\label{lemma:frequency_truncation_schwartz} Consider a non-Markovian model described by coupling functions
\[
\bigg\{v_\alpha \in L^2(\mathbb{R}) \bigg| \norm{(\cdot) {\hat{v}_\alpha(\cdot)}}_{L^\infty} = \sup_{\omega \in \mathbb{R}} \abs{\omega \hat{v}_{\alpha}(\omega)} < \infty\bigg\}_{\alpha \in \{1, 2 \dots M\}},
\]
jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$ and system Hamiltonian $H_S(t)$. For $\omega_c > 0$, consider a non-Markovian model described by coupling functions $v_{\omega_c} = \{v_{\omega_c} \in L^2(\mathbb{R}) | \hat{v}_{\alpha, \omega_c} := \hat{v}_\alpha \mathcal{I}_{[-\omega_c, \omega_c]}\}_{\alpha \in \mathbb{Z}_{\geq 0}}$ but with the same jump operators and system Hamiltonian. Let $\ket{\Psi(t)}$ and $\ket{\Psi_{\omega_c}(t)}$ be the state at time $t$ for both of these models starting with initial state $\ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_\infty^M(L^2(\mathbb{R})) $, then ,
\begin{align*}
&\norm{\ket{\Psi(t)} - \ket{\Psi_{\omega_c}(t)}}^2 \nonumber \\
&\leq \frac{2}{\omega_c^{1/2}}\sum_{\alpha = 1}^M { \norm{L_\alpha} \norm{(\cdot) \hat{v}_\alpha(\cdot)}_{L^\infty}}\bigg(\norm{L_\alpha}\norm{v_\alpha}_{L^2} t^2 + 2\mu_{\ket{\Psi_0}}^{(1)} t \bigg).
\end{align*}
\end{lemma}
\noindent\emph{Proof}:It is useful to note that since $\abs{ \hat{v}_\alpha(\omega)} \leq \norm{(\cdot)\hat{v}_\alpha(\cdot)}_{L^\infty} / \omega$, we obtain that
\begin{align}\label{eq:norm_cutoff}
\norm{v_\alpha - v_{\alpha, \omega_c}}_{L^2}^2 = \int_{|\omega| \geq \omega_c} |\hat{v}_{\alpha}(\omega)|^2 d\omega \leq \frac{\norm{(\cdot)\hat{v}_\alpha(\cdot)}_{L^\infty}^2}{\omega_c}.
\end{align}
The proof of this lemma follows from lemma \ref{lemma:pert_theory_bound}. Consider the two error terms defined in lemma \ref{lemma:pert_theory_bound}. First, note that
\begin{align*}
&\mathcal{D}_\alpha^{v, v_{\omega_c}}(\tau) = 4\norm{L_\alpha} \norm{a_{\alpha, \uptau_\tau(v_\alpha - v_{\alpha, \omega_c})}\ket{\Psi_0}}, \nonumber\\
&\qquad \leq 4 \mu^{(1)}_{\ket{\Psi_0}}\norm{L_\alpha} \norm{v_\alpha - v_{\alpha, \omega_c}}_{L^2}, \nonumber\\
&\qquad \leq {4\mu_{\ket{\Psi_0}}^{(1)}}\frac{\norm{L_\alpha}\norm{(\cdot)\hat{v}_\alpha(\cdot)}_{L^\infty}}{\sqrt{\omega_c}}.
\end{align*}
Furthermore, since $v_\alpha, v_{\alpha, \omega_c} \in L^2(\mathbb{R})$, we obtain that $\forall \tau, s \in [0, t]$
\begin{align*}
&\abs{\langle \uptau_\tau(v_\alpha - v_{\alpha, \omega_c}), \uptau_s v_\alpha\rangle}\nonumber\\
&\qquad = \abs{\int_{\mathbb{R}} (\hat{v}_\alpha^*(\omega) - \hat{v}_{\alpha, \omega_c}^*(\omega))\hat{v}_\alpha(\omega)e^{-i\omega (s - \tau)} d\omega}\nonumber\\
&\qquad \leq \norm{v_\alpha}_{L^2} \norm{v_\alpha - v_{\alpha, \omega_c}}_{L^2} \nonumber\\
&\qquad \leq \frac{\norm{v_\alpha}_{L_2} \norm{(\cdot) \hat{v}_\alpha(\cdot)}_{L^\infty}}{ \omega_c^{1/2}},
\end{align*}
and
\begin{align*}
&\abs{\langle \uptau_s v_{\alpha, \omega_c}, \uptau_\tau(v_\alpha - v_{\alpha, \omega_c})\rangle}\nonumber\\
&\qquad= \abs{\int_{\mathbb{R}} (\hat{v}_\alpha(\omega) - \hat{v}_{\alpha, \omega_c}(\omega))\hat{v}^*_{\alpha, \omega_c}(\omega)e^{i\omega (s - \tau)} d\omega} \nonumber\\
&\qquad \leq \norm{v_{\alpha, \omega_c}}_{L^2} \norm{v_\alpha - v_{\alpha, \omega_c}}_{L^2} \nonumber\\
&\qquad \leq \frac{\norm{v_\alpha}_{L_2} \norm{(\cdot) \hat{v}_\alpha(\cdot)}_{L^\infty}}{\sqrt{\omega_c}}.
\end{align*}
Therefore, we obtain that $\forall \tau \in [0, t]$,
\[
\mathcal{E}_\alpha^{v, v_{\omega_c}}(\tau) \leq 4 t \frac{\norm{L_\alpha}^2\norm{v_\alpha}_{L_2} \norm{(\cdot) \hat{v}_\alpha(\cdot)}_{L^\infty}}{\omega_c^{1/2}}
\]
Substituting these estimates into lemma \ref{lemma:pert_theory_bound}, we then obtain the lemma statement.
\(\square\)
Finally, we turn to the proof of the full proposition. The proof of this proposition simply puts together the error bounds provided in lemmas \ref{lemma:star_to_chain}, \ref{lemma:reg_error} and \ref{lemma:frequency_truncation_schwartz}.
\begin{repproposition}{theorem:final_dilation}
Consider a non-Markovian model specified by a system Hamiltonian $H_S(t)$, jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$ and coupling functions $\{(\mu_\alpha, \varphi_\alpha)\}_{\alpha \in \{1, 2 \dots M\}}$ where $\mu_\alpha$ satisfy assumption \ref{assump:radon_measure} with $\hat{\mu}_\alpha(\omega) < O(\omega^{2k}$) for some $k > 0$. For $\ket{\Psi_0} := \ket{\sigma} \otimes \ket{\Phi_0} \in \mathcal{H}_S\otimes \textnormal{Fock}[L^2(\mathbb{R})]^{\otimes M}$, where $\ket{\sigma} \in \mathcal{H}_S$ and $\ket{\Phi_0}$ is an initial environment state that satisfies assumption \ref{assump:initial_state}, then $\exists$ a Markovian dilation of the non-Markovian model with
\begin{align*}
&N_m, B \leq O\bigg(\textnormal{poly}\bigg(\frac{1}{\epsilon}, t, M, \sup_{\alpha}\norm{L_\alpha},\nonumber \\
&\qquad \qquad \qquad \qquad \sup_{\alpha, s \in [0,t]}\norm{[H_S(s), L_\alpha]},\nonumber\\
&\qquad \qquad \qquad \qquad \mathcal{N}_{1, k + 1}, \mathcal{N}_{1, k + 2}, \mathcal{N}_{1, 0} \bigg)\bigg)
\end{align*}
whose system-environment state at time $t$ is within $\epsilon$ norm distance of the exact state.
\end{repproposition}
\noindent\emph{Proof}: Throughout this proof, for notational convenience, we will hide several polynomial factors with respect to $t, M, \sup_\alpha \norm{L_\alpha}, \sup_{\alpha, s\in[0, t]} \norm{[H_S(s), L_\alpha]}, \mathcal{N}_{1, 0}, \mathcal{N}_{1, k},$ $\mathcal{N}_{1, k + 1}$ in the big-$O$ notation i.e.~we will use $\tilde{O}$ to denote
\begin{align*}
&\tilde{O}(f) = O\bigg(f \ \textnormal{poly}\bigg(t, M, \sup_{\alpha}\norm{L_\alpha},\nonumber \\
&\qquad \qquad \qquad \qquad \sup_{\alpha, s \in [0,t]}\norm{[H_S(s), L_\alpha]},\nonumber\\
&\qquad \qquad \qquad \qquad \mathcal{N}_{1, k + 1}, \mathcal{N}_{1, k + 2},\mathcal{N}_{1, 0}\bigg)\bigg)
\end{align*}
It would also be convenient to note in the following proof that $\mu^{(1)}_{\ket{\Psi_0}} \leq \mathcal{N}_{1, 0}$.
Consider first the regularization step --- we will denote the mollifier used $\rho$, regularization parameter used $\varepsilon$ and the regularized coupling functions $v_\varepsilon = \{v_{\alpha, \varepsilon} \in L^2(\mathbb{R}) \}_{\alpha \in \{1, 2 \dots M\}}$ here, from lemma \ref{lemma:reg_error}, it follows that a choice of the regularization parameter $\varepsilon = \tilde{O}(\epsilon)$ ensures that the regularization error is $O(\epsilon)$. Next, we consider the error incurred on introducing the frequency-cutoff (lemma \ref{lemma:frequency_truncation_schwartz}) in the regularized model. We note that since $\hat{\rho}(\omega)$, the fourier transform of the mollifier $\rho$, falls of with $\omega$ faster than any polynomial in $\omega$ and since $\hat{\mu}_\alpha(\omega) \leq O(\omega^{2k})$, then $\norm{(\cdot) \hat{v}_\alpha(\cdot)}_{L^\infty} \leq \sup_{\omega \in \mathbb{R}} \abs{\omega} \abs{\hat{\rho}(\omega \varepsilon)} \sqrt{\hat{\mu}_\alpha(\omega)} \leq O(\varepsilon^{-(k + 1)})$. Furthermore, $\norm{v_{\alpha, \varepsilon}}_{L^2} \leq O(\varepsilon^{-(k + 1)})$ Now, from lemma \ref{lemma:frequency_truncation_schwartz}, it then follows that for a frequency cutoff $\omega_c$, the cutoff error scales as $\tilde{O}(\varepsilon^{-(k + 1)} / \omega_c^{1/4})$. Then, choosing $\omega_c = \tilde{O}(\text{poly}(\epsilon^{-1}, \varepsilon^{-1})) = \tilde{O}(\text{poly}(\varepsilon^{-1}))$ ensures that the frequency cut-off error is $O(\epsilon)$. Finally, from lemma \ref{lemma:star_to_chain}, it follows that $N_m = \tilde{O}(\text{poly}(\epsilon^{-1}, \omega_c)) = \tilde{O}(\epsilon)$ modes are needed in a chain dilation of the resulting model to reduce the error of the chain-dilation step to $O(\epsilon)$. This completes the proof.
\(\square\)
\subsection{$k-$local Non-Markovian open system dynamics is in BQP}
\label{subsec:alg}
\noindent We next consider the $k-$local Non-Markovian open system problem.
\begin{repproblem}{prob:k_local_non_mkv}[$k-$local non-Markovian dynamics] Consider a system of $n$ qudits $(\mathcal{H}_S = \big(\mathbb{C}^d\big)^{\otimes n})$ interacting with $M = \textnormal{poly}(n)$ baths with
\begin{enumerate}
\item[(a)] System Hamiltonian $H_S(t)$ is $k-$local i.e. $H_S(t) = \sum_{i = 1}^{N} H_i(t)$, where $N = \textnormal{poly}(n)$, and for $i \in \{1, 2 \dots N\}$, $H_i(t)$ is an operator acting on atmost $k$ qudits and $\norm{H_i(t)} \leq 1$.
\item[(b)] Jump operators $\{L_\alpha \}_{\alpha \in \{1, 2 \dots M\}}$ such that for $\alpha \in \{1, 2\dots M\}$, $L_\alpha$ acts on at-most $k$ qudits and $\norm{L_\alpha} \leq 1$.
\item[(c)] Coupling functions $\{(\mu_\alpha, \varphi_\alpha) \}_{\alpha \in \{1, 2 \dots M\}}$ such that for $\alpha \in \{1, 2 \dots M\}$, $\mu_\alpha$ satisfies the polynomial growth conditions (assumption \ref{assump:radon_measure}).
\item[(d)] Initial state $\ket{\Psi} = \ket{0}^{\otimes n} \otimes \ket{\Phi}$, where $\ket{\Phi}$ satisfies assumption \ref{assump:initial_state}. Furthermore, the initial state is computable in the sense that for $v_1, v_2 \dots v_m \in L^2(\mathbb{R})$ and $P \in\mathbb{Z}_{>0}$, all the amplitudes
\begin{align*}
\bra{\textnormal{vac}}\prod_{i = 1}^m \bigg(\int_{\mathbb{R}}v_i(\omega)a_\omega d\omega\bigg)^{n_i} \ket{\phi_\alpha}
\end{align*}
with $n_1 + n_2 \dots n_m \leq P$ can be computed in $\textnormal{poly}(m, P)$ time on a classical or quantum computer.
\end{enumerate}
Denoting by $\rho_S(t)$ the reduced state of the system at time $t$ for this non-Markovian model, then for $\varepsilon > 0$ and $t = \textnormal{poly}(n)$, prepare a quantum state $\hat{\rho}$ such that $\norm{\hat{\rho} - \rho_S(t)}_\textnormal{tr} \leq \varepsilon$.
\end{repproblem}
To prove that this problem can be efficiently solved on a quantum computer, we proceed in three steps. First, we compute a Markovian dilation of the non-Markovian system with $N_m$ modes --- from theorem \ref{theorem:final_dilation} it follows that $N_m = \text{poly}(n, \varepsilon^{-1})$ modes are needed to ensure that the error between the dynamics of the non-Markovian and its Markovian dilation is $< \varepsilon$. Next, we simulate the Markovian dilation on a quantum computer --- we thus consider a different problem, defined below.
\begin{problem}[$k-$local non-Markovian chain model]
\label{prob:chain_model}
Consider a system of $n$ qudits $(\mathcal{H}_S = \big(\mathbb{C}^d\big)^{\otimes n})$ interacting with $M = \textnormal{poly}(n)$ baths with
\begin{enumerate}
\item[(a)] System Hamiltonian $H_S(t)$ is $k-$local i.e. $H_S(t) = \sum_{i = 1}^{N} H_i(t)$, where $N = \textnormal{poly}(n)$, and for $i \in \{1, 2 \dots N\}$, $H_i(t)$ is an operator acting on atmost $k$ qudits and $\norm{H_i(t)} \leq 1$.
\item[(b)] Jump operators $\{L_\alpha \}_{\alpha \in \{1, 2 \dots M\}}$ such that for $\alpha \in \{1, 2\dots M\}$, $L_\alpha$ acts on at-most $k$ qudits and $\norm{L_\alpha} \leq 1$.
\item[(c)] Square-integrable coupling functions $\{v_\alpha \in L^2(\mathbb{R}) | \norm{v_\alpha}_{L^2} = \textnormal{poly}(n), \ \textnormal{supp}(\hat{v}_\alpha) \subseteq [-\omega_c, \omega_c]\}_{\alpha \in \{1, 2 \dots M\}}$ where $\omega_c = \textnormal{poly}(n)$.
\item[(d)] Single-particle environment dynamics specified by $\{\upnu_{\alpha, t} : L^2(\mathbb{R}) \to L^2(\mathbb{R}) \}_{\alpha \in \{1, 2 \dots M\}}$ where $\upnu_{\alpha, t}$ is the chain unitary group with $N_m = \textnormal{poly}(n)$ modes generated by $v_\alpha$.
\item[(e)] Initial state $\ket{\Psi} = \ket{0}^{\otimes n} \otimes \ket{\Phi}$, where $\ket{\Phi} = \ket{\phi_1} \otimes \ket{\phi_2}\otimes \dots \ket{\phi_M}$ satisfies assumption \ref{assump:initial_state}. Furthermore, the initial state is computable in the sense that for $v_1, v_2 \dots v_m \in L^2(\mathbb{R})$ and $P \in\mathbb{Z}_{>0}$, all the amplitudes
\begin{align*}
\bra{\textnormal{vac}}\prod_{i = 1}^m \bigg(\int_{\mathbb{R}}v_i(\omega)a_\omega d\omega\bigg)^{n_i} \ket{\phi_\alpha}
\end{align*}
with $n_1 + n_2 \dots n_m \leq P$ can be computed in $\textnormal{poly}(m, P)$ time on a classical or quantum computer.
\end{enumerate}
Denoting by $\rho_S(t)$ the reduced state of the system at time $t$ for this non-Markovian model, then $t = \textnormal{poly}(n)$, prepare a quantum state $\hat{\rho}$ such that $\norm{\hat{\rho} - \rho_S(t)}_\textnormal{tr} \leq 1 /\textnormal{poly}(n)$.
\end{problem}
Since the Markovian dilation is still an infinite dimensional system (with an effective Hilbert space $\mathcal{H}_S\otimes \textnormal{Fock}[\mathbb{C}^{N_m}]^{\otimes M}$), this requires a truncation of Hilbert space to a finite-dimensional one, followed by simulating the finite-dimensional quantum dynamics. To show that the approximating finite-dimensional quantum system can be efficiently simulated, we use the Hamiltonian simulatability lemma from Ref.~\cite{aharonov2003adiabatic}, which we restate below
\begin{lemma}[Hamiltonian simulatability, Ref.~\cite{aharonov2003adiabatic}]
\label{lemma:hamil_simul}
Given a Hamiltonian $H(t)$ over a system of $n$ qudits such that for every $t \geq 0$
\begin{enumerate}
\item[(a)] $\forall$ computational basis element $\ket{a}$, the the set of computational basis elements $\ket{b}$ such that $\bra{a} H(t) \ket{b} \neq 0$ together with the elements $\bra{a} H(t)\ket{b}$ can be computed in $\textnormal{poly}(n)$ time, and
\item[(b)] $\int_0^t \norm{H(t')}dt' = \textnormal{poly}(n)$,
\end{enumerate}
then $\exists$ a quantum circuit over $n$ qudits of depth $\textnormal{poly}(n)$ which implements a unitary $\hat{U}$ such that $\norm{\hat{U} - U(t, 0)} \leq 1 / \textnormal{poly}(n)$, where $U(\cdot, \cdot)$ is the propagator corresponding to $H(t)$.
\end{lemma}
The next lemma, which shows the quantum simulability of problem \ref{prob:chain_model}, follows from the application of lemma~\ref{lemma:hamil_simul} together with a bound on the error incurred in truncating the infinite-dimensional Hilbert space of the environment. A detailed proof of this is in appendix \ref{app:hspace_trunc}
\begin{lemma} \label{lemma:chain_model_bqp}Problem \ref{prob:chain_model} can be solved on a quantum computer in run time $\textnormal{poly}(n)$.
\end{lemma}
\begin{reptheorem}{theorem:bqp_non_mkv}[$k-$local Non-Markovian dynamics $\in$ BQP]
Problem \ref{prob:k_local_non_mkv} can be solved in $\textnormal{poly}(n)$ time on a quantum computer.
\end{reptheorem}
\noindent\emph{Proof}: An application of lemma \ref{theorem:final_dilation} approximates problem \ref{prob:k_local_non_mkv} to an instance of problem \ref{prob:chain_model}, and then the theorem statement follows from lemma \ref{lemma:chain_model_bqp}.
\(\square\)
\twocolumngrid
\section{Conclusion}
In conclusion, our work identifies the class of tempered radon measures as memory kernels for which a unitary group generating non-Markovian system dynamics can be constructed. We therefore generalize the unitary group for Markovian dynamics (i.e.~with a delta function memory kernel) described in the theory of quantum stochastic calculus. We then consider the $k-$local many-body non-Markovian dynamics, and show that it can be efficiently simulated on quantum computers, thus establishing that this generalization is consistent with the Extended Church-Turing thesis.
Our work points to several important open questions about non-Markovian dynamics. The first is to understand if the growth conditions on the radon measure describing the memory kernel (assumption \ref{assump:radon_measure}) are necessary --- while there are radon measures that violate these conditions, it is possible that these growth conditions hold for any \emph{tempered} radon measure. Alternatively, violating these growth condition could lead to unphysical situations, such as ``infinitely long" memory times in the non-Markovian system. Formalizing these ideas would allow us to further sharpen the mathematical definition of a physically reasonable non-Markovian model.
Second, can the unitary group describing non-Markovian dynamics constructed in this paper be further characterized? An important characterization would be to understand if this unitary group is strongly continuous (i.e.~is it associated with a self-adjoint Hamiltonian)? Similar questions have been previously answered for the unitary group for Markovian dynamics provided by a quantum stochastic differential equation \cite{chebotarev1997quantum, gregoratti2000hamiltonian, gregoratti2001hamiltonian}.
Finally, it would also be of interest to develop quantum algorithms for non-Markovian dynamics with better dependence on the problem size as well as the incurred approximation error by exploiting further structure in the non-Markovian model (e.g.~spatial locality, or availability of the Hamiltonian/jump operators as linear combination of unitaries) and using similar techniques that have been used in Hamiltonian or Lindbladian simulation problems \cite{berry2014exponential, berry2015hamiltonian, cleve2016efficient, haah2021quantum}.
\onecolumngrid
\appendix
\section{Proof of proposition \ref{thm:schr_sq_int}}
\label{app:sq_int}
In this section, we sketch a detailed proof of proposition 1, which deals with the well definition of non-Markovian dynamics with square integrable coupling functions. For convenience throughout this section, we will define $\ell$ to be the constant
\begin{align}
\ell = \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \norm{L_\alpha}.
\end{align}
\begin{replemma}{lemma:domain}
For all $t \in \mathbb{R}$,
\begin{enumerate}
\item[(a)] $H(t): \mathcal{H}_S\otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \to \mathcal{H}$ is essentially self adjoint.
\item[(b)] $H(t)$ is closable, and if $\overline{H(t)}:\textnormal{dom}[\overline{H(t)}] \to \mathcal{H}$ is its closure then $\mathcal{H}_S\otimes \textnormal{F}_1^M[L^2(\mathbb{R})] \subseteq \textnormal{dom}[\overline{H(t)}]$ and $\forall \ket{\Psi} \in \textnormal{dom}[\overline{H(t)}]$,
$\overline{H(t)} \ket{\Psi} = \sum_{n = 0}^\infty H(t) \big(\Pi_n \ket{\Psi}\big).$
\end{enumerate}
\end{replemma}
\noindent\emph{Proof}:\\
\noindent\emph{(a)} is shown using Nelson's analytic vector theorem (Theorem X.39 of Ref.~\cite{reed1975ii}), and showing that all the vectors in $\mathcal{H}_S\otimes \textnormal{F}_0[L^2(\mathbb{R})]^{\otimes M}$ are analytic vectors of $H_S(t)$. We note that it follows easily from the definition of $H(t)$ that for every $n \in \mathbb{Z}_{\geq 0}$, $H(t)(\textnormal{id}\otimes\Pi_{\leq n})$ is a bounded operator and
\[
\norm{H(t) \big(\textnormal{id}\otimes \Pi_{\leq n}\big)} \leq \norm{H_S(t)} + 2 \ell \sqrt{n + 1} .
\]
Recall that given an operator $O:\textnormal{dom}[O] \to \mathcal{H}$, a vector $x$ is an analytic vector of $O$ if $\sum_{n = 0}^\infty t^n\norm{O^n x} / n! < \infty \ \forall \ t\in \mathbb{R}$. Let $\ket{\Psi} \in \mathcal{H}_S\otimes \textnormal{F}_0[L^2(\mathbb{R})]^{\otimes M}$ and let $N_0 \in \mathbb{Z}_{\geq 1}$ be the number of particles in the environment (i.e.~$\Pi_{>N_0}\ket{\Psi} = 0$). It then immediately follows that for any $k \in \mathbb{Z}_{\geq 0}$, $H^k(t) \ket{\Psi_0}$ has at most $N_0 + k$ particles, and thus
\[
\norm{H^k(t) \ket{\Psi_0}} \leq \big(\norm{H_S(t)} + 2\sqrt{N_0 + k + 1} \ell\big)^k \norm{\ket{\Psi_0}} \leq 2^k \big(\norm{H_S(t)}^k + (N_0 + k + 1)^{k / 2}\ell^k\big) \norm{\ket{\Psi_0}} ,
\]
and thus for $t \geq 0$
\begin{align*}
&\sum_{k = 0}^\infty \frac{t^k}{k!} \norm{H_S^k(t)\ket{\Psi_0}} \leq e^{2t \norm{H_S(t)}} \norm{\ket{\Psi_0}} + \sum_{k = 0}^\infty \frac{(2\ell t)^k}{k!} (N_0 + k + 1)^{k/2} \norm{\ket{\Psi_0}},\nonumber\\
&\qquad\leq e^{2t \norm{H_S(t)}} \norm{\ket{\Psi_0}} + \norm{\ket{\Psi_0}} + \sum_{k = 1}^\infty \frac{(2e\ell t)^k}{k^{k/2}} \bigg(1 + \frac{N_0 + k}{k}\bigg)^{k/2} \norm{\ket{\Psi_0}}.
\end{align*}
Wherein we have used the Stirling's approximation in the last estimate. The summation can now be seen to converge for any $t$ and hence $\ket{\Psi_0}$ is an analytic vector of $H(t)$.
\\
\noindent\emph{(b)} We first consider a sequence $\{ \ket{\Psi_i} \in \mathcal{H}_S \otimes \textnormal{F}_\infty^M(L^2(\mathbb{R}))\}_{i \in \mathbb{N}}$ such that $\lim_{i \to \infty} \ket{\Psi_i} = 0$ and the sequence $\{H(t) \ket{\Psi_i} \}_{i \in \mathbb{N}}$ converges. We first show that under these conditions, $\lim_{i \to \infty} H(t) \ket{\Psi_i} = 0$ --- to see this, assume the contrary i.e.~$\lim_{i \to \infty} H(t) \ket{\Psi_i} = \ket{\Phi} \neq 0$. Since $\ket{\Phi} \neq 0$, $\exists N > 0,\big(\textnormal{id}\otimes\Pi_{\leq N}\big)\ket{\Phi} \neq 0$. Note that
\[
\norm{\big(\textnormal{id}\otimes \Pi_{\leq N} \big)H(t)} \leq \norm{H_S(t)} + 2\ell \sqrt{N + 1},
\]
and therefore $\Pi_{\leq N} \ket{\Phi} = \lim_{i \to \infty} \Pi_{\leq N} H(t)\ket{\Psi_i} = \Pi_{\leq N} H(t) \lim_{i \to \infty} \ket{\Psi_i} = 0$, where we have used that $\Pi_{\leq N}, \Pi_{\leq N} H(t)$ are bounded operators to swap the order of limits. Thus, we contradict our original assumption of $\ket{\Phi}\neq 0$ and hence $\ket{\Phi} = 0$. This shows that the operator $H(t)$ is closable.
Next, we consider $\ket{\Psi} \in \mathcal{H}_S\otimes \textnormal{F}_1^M[L^2(\mathbb{R})]$ --- we consider the sequence $\{\ket{\Psi_n} := \big( \textnormal{id}\otimes \Pi_{\leq n}\big) \ket{\Psi}\}_{n \in \mathbb{N}}$ which converges to $\ket{\Psi}$. Furthermore, we note that the sequence $\{H(t) \ket{\Psi_n}\}_{n \in \mathbb{N}}$ also converges, and converges to $\sum_{m = 0}^\infty H(t) \Pi_m \ket{\Psi}$ since
\[
\norm{H(t) \ket{\Psi_n} - \sum_{m = 0}^\infty H(t) \Pi_m \ket{\Psi}} \leq \norm{H_S(t)} \bigg({\sum_{m = n + 1}^\infty} \norm{\Pi_m \ket{\Psi}}^2\bigg)^{1/2} + 2\ell \bigg(\sum_{m = n + 1}^\infty (m + 1) \norm{\Pi_m \ket{\Psi}}^2 \bigg)^{1/2},
\]
and since $\norm{\ket{\Psi}} < \infty$ and $\mu^{(1)}_{\ket{\Psi}} < \infty$, $\sum_{m = n}^\infty \norm{\Pi_m \ket{\Psi}}^2, \sum_{m = n}^\infty m\norm{\Pi_m \ket{\Psi}}^2 \to 0$ as $n \to \infty$. Consequently, we obtain that $\ket{\Psi} \in \textnormal{dom}[\overline{H(t)}]$ and $\overline{H(t)} \ket{\Psi} = \sum_{n = 0}^\infty H(t) \Pi_{n} \ket{\Psi}$.
\(\square\) \\
\noindent We are now poised to first investigate the existence of solution of the Schr\"odinger's equation for the time-dependent Hamiltonian defined in Definition \ref{def:hamil}. We restrict ourselves to initial states with only a finite number of particles in the environment i.e.~$\ket{\Psi(0)} \in \mathcal{H}_S\otimes \text{F}_\infty^M[L^2(\mathbb{R})]$, and use the density of $\ket{\Psi(0)} \in \mathcal{H}_S\otimes \text{F}_\infty^M[L^2(\mathbb{R})]$ to extend it to the entire system-environment Hilbert space. While $H(t)$, for every $t$, admits a self adjoint extension, since the equation under consideration is non-autonomous, this by itself does not imply that the solution of this equation exists. Instead, we analyze this equation by first truncating the number of particles in the environment, and analyzing the convergence of the obtained solution with the truncation.
\begin{definition}\label{def:hamil_p_particle}
For $p \in \mathbb{Z}_{\geq 0}$ and $t \in \mathbb{R}$, define $H^p(t):\mathcal{H} \to \mathcal{H}$ via $H^p(t) = \Pi_{\leq p} H(t) \Pi_{\leq p}$.
\end{definition}
Several properties of $H^p(t)$ follows trivially from its definition.
\begin{lemma} $H^p(t)$ has the following properties
\begin{enumerate}
\item[(a)] $H^p(t)$ is a bounded operator for all $t \in \mathbb{R}$.
\item[(b)] $H^p(t)$ is norm continuous with respect to $t$.
\end{enumerate}
\end{lemma}
\noindent\emph{Proof}: \\
\noindent (a) This follows straightforwardly by noting that $\forall t \in \mathbb{R}, \alpha \in \{1, 2 \dots M\}$, $\norm{a_{\alpha, \tau_t v_\alpha}^-\Pi_{\leq p}} \leq \sqrt{p}\norm{v_\alpha}_{L^2}$ and $\norm{a_{\alpha, \tau_t v_\alpha}^+ \Pi_{\leq p}} \leq \sqrt{p + 1}\norm{v_\alpha}_{L^2}$.\\
\noindent(b) For any $\delta > 0$, note that
\[
\norm{H^p(t + \delta) - H^p(t)} \leq \norm{H_S(t + \delta) - H_S(t)} + \sum_{\alpha = 1}^M \sqrt{p}\norm{L_\alpha} \norm{(\uptau_{t + \delta} - \uptau_{t})v_\alpha}_{L^2}.
\]
Since $H_S(t)$ is norm continuous, $\norm{H_S(t+ \delta) - H_S(t)} \to 0$ as $\delta \to 0$, and since $\uptau_t$ is strongly continuous in $t$, $\norm{(\uptau_{t + \delta} - \uptau_t)v_\alpha}_{L^2} \to 0$ as $\delta \to 0$, thus showing from the above estimate that $H^p(t)$ is norm continuous.
\(\square\)
\begin{lemma}\label{lemma:particle_num_bound}
For any $p \in \mathbb{N}$ and $\tau, s \in \mathbb{R}$, there exists a unitary operator $U^p(\tau, s):\mathcal{H}\to \mathcal{H}$ which is norm continuous and differentiable with respect to both $s$ and $\tau$ and which satisfies
\[
i \frac{d}{d\tau}U^p(\tau, s) = H^p(\tau) U^p(\tau, s) \text{ with } U^p(s, s) = \textnormal{id}.
\]
Furthermore, let $\ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}[L^2(\mathbb{R})]$, and for $t > 0$, consider $M^{(k)}_{\ket{\Phi}}(t) $ defined by
\[
M^{(0)}_{\ket{\Phi}}(t) = \norm{\ket{\Phi}}^2, M^{(k)}_{\ket{\Phi}}(t) = 2 \mu^{(k)}_{\ket{\Phi}} + 2^{2k - 3} \ell^2 t^2 \bigg(\norm{\ket{\Phi}}^2 + M^{(k - 1)}_{\ket{\Phi}}(t)\bigg)^{2} \ \text{for } k \geq 1,
\]
then $\forall \tau, s \in [0, t]$, $\mu^{(k)}_{U^p(\tau, s)\ket{\Phi}} \leq M^{(k)}(t) \ \forall p \in \mathbb{Z}_{\geq 0}$.
\end{lemma}
\noindent\emph{Proof}: Since $H^p(\tau)$ is both norm continuous in $\tau$ and bounded, the existence, norm continuity and differentiability of $U^p(\tau, s)$ follows follows directly from Dyson expansion (see theorem X.69 of Ref.~\cite{reed1975ii}). For part (b), we use the Sch\"odinger equation. Note that
\[
U^p(\tau, s)\ket{\Phi} = \Pi_{> p} \ket{\Phi} + U^p(\tau, s)\Pi_{\leq p} \ket{\Phi},
\]
and furthermore, $\mu^{(k)}_{\Pi_{\leq p}\ket{\Phi}} \leq \mu^{(k)}_{\ket{\Phi}} \ \forall k \in \mathbb{Z}_{\geq 1}$. For convenience of notation, we set $\ket{\Psi^p(\tau, s)} = U^p(\tau, s) \Pi_{\leq p}\ket{\Phi}$. From the Schroedinger's equation, it follows that
\begin{align*}
\frac{d}{d\tau}\mu^{(k)}_{\ket{\Psi^p(\tau, s)}} = \sum_{\alpha = 1}^M \sum_{n = 0}^{p - 1} \big((n + 1)^k - n^k\big) \textnormal{Im}\bra{\Psi^p(\tau, s)} \Pi_{n + 1}a^+_{\alpha, \uptau_\tau v_\alpha}L_\alpha \Pi_{n}\ket{\Psi^p(\tau, s)}.
\end{align*}
and therefore
\begin{align*}
&\bigg|\frac{d}{d\tau}\mu^{(k)}_{\ket{\Psi^p(\tau, s)}} \bigg|\leq 2 \ell \sum_{n=0}^{p-1}\sqrt{n+1}\big((n + 1)^l - n^l\big)\norm{\Pi_{n + 1}\ket{\Psi^p(\tau, s)}}\norm{\Pi_n\ket{\Psi^p(\tau, s)}},
\end{align*}
Since $(n + 1)^l - n^l = (n + 1)^{l - 1} + n (n + 1)^{l - 2} + n^2 (n + 1)^{l - 3} \dots n^{l - 1}$ for $l \in \mathbb{Z}_{\geq 1}$, we obtain that
\begin{align*}
\bigg|\frac{d}{d\tau}\mu^{(k)}_{\ket{\Psi^p(\tau, s)}} \bigg|& \leq 2 \ell \sum_{n=0}^{p-1}\sum_{s = 1}^k (n + 1)^{k-s + 1/2}n^{s-1} \norm{\Pi_{n + 1}\ket{\Psi^p(\tau, s)}}\norm{\Pi_n\ket{\Psi^p(\tau, s)}}
\end{align*}
An application of the Cauchy-Schwarz inequality yields that $\forall s \in \{1, 2 \dots k\}$
\begin{align*}
&\sum_{n = 0}^{p - 1}(n + 1)^{k - s + 1/2}n^{s - 1}\norm{\Pi_{n + 1}\ket{\Psi^p(\tau, s)}}\norm{\Pi_n\ket{\Psi^p(\tau, s)}} \nonumber\\
&\qquad \qquad \leq \bigg(\sum_{n = 0}^{p- 1}(n + 1)^k \norm{\Pi_{n + 1}\ket{\Psi^p(\tau, s)}}^2 \bigg)^{1/2} \bigg(\sum_{n = 0}^{p - 1}n^{2s-2}(n + 1)^{k + 1 - 2s}\norm{\Pi_n \ket{\psi^p(\tau, s)}}^2 \bigg)^{1/2} \nonumber \\
&\qquad\qquad\leq \big(\mu^{(k)}_{\ket{\Psi^p(t, s)}}\big)^{1/2} \bigg(\sum_{n = 0}^{p - 1} (n + 1)^{k - 1}\norm{\Pi_n \ket{\Psi^p(\tau, s)}}^2\bigg)^{1/2}
\end{align*}
Noting that
\[
\sum_{n = 0}^{p - 1}(n + 1)^{k - 1}\norm{\Pi_n \ket{\Psi^p(\tau, s)}}^2 \leq 2^{k - 2} \sum_{n = 0}^{p - 1} (n^{k - 1} + 1) \norm{\Pi_n \ket{\Psi^p(\tau, s)}}^2 = 2^{k - 2} \bigg( \mu^{(0)}_{\ket{\Psi^p(\tau, s)}} + \mu^{(k - 1)}_{\ket{\Psi^p(\tau, s)}}\bigg).
\]
Noting that $\mu^{(0)}_{\ket{\Psi^p(\tau, s)}} = \mu^{(0)}_{\Pi_{\leq p}\ket{\Phi}} \leq \mu^{(0)}_{\ket{\Phi}}$, we thus obtain
\[
\abs{\frac{d}{dt}\mu^{(k)}_{\ket{\Psi^p(\tau, s)}}} \leq 2^{k - 1}\ell \big(\mu^{(k)}_{\ket{\Psi^p(\tau, s)}}\big)^{1/2} \bigg( \mu^{(0)}_{\ket{\Phi}} + \mu^{(k - 1)}_{\ket{\Psi^p(\tau, s)}}\bigg),
\]
Integrating which we obtain that for $\tau \geq s$
\[
\big(\mu^{(k)}_{\ket{\Psi^p(\tau, s)}}\big)^{1/2} - \big(\mu^{(k)}_{\Pi_{\leq p}\ket{\Phi}}\big)^{1/2} \leq 2^{k -2} \ell\bigg( (\tau - s)\mu^{(0)}_{\ket{\Phi}} + \int_s^{\tau} \mu^{(k - 1)}_{\ket{\Psi^p(\tau', s)}} d\tau'\bigg) \leq 2^{k -2} \ell\bigg( t\mu^{(0)}_{\ket{\Phi}} + \int_s^{\tau} \mu^{(k - 1)}_{\ket{\Psi^p(\tau', s)}} d\tau'\bigg),
\]
and for $\tau < s$
\[
\big(\mu^{(k)}_{\ket{\Psi^p(\tau, s)}}\big)^{1/2} - \big(\mu^{(k)}_{\Pi_{\leq p}\ket{\Phi}}\big)^{1/2} \leq 2^{k -2} \ell\bigg( (s - \tau)\mu^{(0)}_{\ket{\Phi}} + \int_\tau^{s} \mu^{(k - 1)}_{\ket{\Psi^p(\tau', s)}} d\tau'\bigg) \leq 2^{k -2} \ell\bigg( t\mu^{(0)}_{\ket{\Phi}} + \int_\tau^{s} \mu^{(k - 1)}_{\ket{\Psi^p(\tau', s)}} d\tau'\bigg),
\]
Since $\forall k \in \mathbb{Z}_{\geq 0}$, $\mu^{(k)}_{U^p(\tau, s)\ket{\Phi}} = \mu^{(k)}_{\Pi_{> p}\ket{\Phi}} + \mu^{(k)}_{\ket{\Psi^p(\tau, s)}} $These equations can be recursively solved to obtain the functions $M_{\ket{\Phi_0}}^{(k)}(t)$. Note that $\mu^{(0)}_{\ket{\Psi^p(\tau, s)}} = \mu^{(0)}_{\Pi_{\leq p}\ket{\Phi}} \leq \bra{\Phi}\Phi\rangle$, and hence $M^{(0)}_{\ket{\Phi}}(t)$ can be set to $\norm{\ket{\Phi}}^2$. Assuming that $\mu^{(k - 1)}_{U^p(\tau, s)\ket{\Phi}} \leq M^{(k - 1)}_{\ket{\Phi}}(t) \ \forall \tau, s \in [0, t]$, we then obtain that
\[
\mu^{(k)}_{\ket{\Psi^p(\tau, s)}} \leq \bigg(\big(\mu^{(k)}_{\Pi_{\leq p}\ket{\Phi}}\big)^{1/2} + 2^{k-2}\ell t\bigg(\bra{\Phi}\Phi\rangle + M^{(k - 1)}_{\ket{\Phi}}(t)\bigg)\bigg)^2 \leq 2\mu^{(k)}_{\Pi_{\leq p}\ket{\Phi}} + 2^{2k - 3}\ell^2 t^2 \bigg(\bra{\Phi}\Phi\rangle + M^{(k - 1)}_{\ket{\Phi}}(t)\bigg)^2,
\]
and since $\mu^{(k)}_{U^p(\tau, s)\ket{\Phi}} \leq \mu^{(k)}_{\ket{\Phi}} + \mu^{(k)}_{\ket{\Psi^p(\tau, s)}}$, we can choose $M^{(k)}_{\ket{\Phi}} = 2\mu^{(k)}_{\ket{\Phi}} + 2^{2k-3}\ell^2 t^2\big(\bra{\Phi}\Phi\rangle + M^{(k - 1)}_{\ket{\Phi}}(t)\big)^2$, which proves the lemma statement.
\(\square\)
It is important to note that the bounds on $\mu_l^p(t)$ are uniform in $p$ --- we will exploit this in the following proofs to show the existence and differentiability of $\ket{\Psi^p(t)}$.
\begin{lemma}
\label{lemma:useful}
Let $\ket{\Psi_0} \in \mathcal{H}_S\otimes \textnormal{F}_\mathcal{S}^\infty[L^2(\mathbb{R})]$, then
\begin{enumerate}
\item[(a)] $\forall t, s \geq 0$, $\lim_{p\to\infty}U^p(t, s) \ket{\Psi_0}$ exists and $\in \mathcal{H}_{S}\otimes \textnormal{F}_\mathcal{S}^\infty[L^2(\mathbb{R})]$.
\item[(b)] $\forall t, s \geq 0$, $\lim_{p\to\infty} H^p(t)U^p(t, s) \ket{\Psi_0} = \overline{H(t)}\lim_{p \to \infty}U^p(t, s)\ket{\Psi_0}$ and $\overline{H(t)} \lim_{p \to \infty} U^p(t, s) \ket{\Psi_0}$ is strongly continuous in $t$.
\item[(c)] $\forall t, s \geq 0$, $\lim_{p \to \infty} U^p(t, s) H^p(s) \ket{\Psi_0} = \lim_{p \to \infty} U^p(t, s) \overline{H(s)} \ket{\Psi_0}$ and $ \lim_{p \to \infty} U^p(t, s) \overline{H(s)} \ket{\Psi_0}$ is strongly continuous in $s$.
\item[(d)] $\exists g, h \in \textnormal{C}^0(\mathbb{R})$ such that $\norm{H^p(t) U^p(t, s)\ket{\Psi_0}} \leq g(t)$ and $\norm{H^p(t) \ket{\Psi_0}} \leq h(t) \ \forall \ t\geq 0$ and $p \in \mathbb{Z}_{\geq 0}$.
\end{enumerate}
\end{lemma}
\noindent\emph{Proof}:\\
\noindent\emph{(a)} To prove the existence of limit, we appeal to the completeness of $\mathcal{H}$ and show that the sequence $\{U^p(t, s) \ket{\Psi_0}\}_{p \in \mathbb{Z}_{\geq 0}}$ is Cauchy. Consider $p, q \in \mathbb{N}$ with $p > q$ --- we note that
\[
\norm{U^p(t, s) \ket{\Psi_0} - U^q(t, s) \ket{\Psi_0}} \leq \norm{U^p(t, s) \Pi_{\leq q} \ket{\Psi_0} - U^q(t, s) \Pi_{\leq q} \ket{\Psi_0}} + 2 \norm{\Pi_{> q} \ket{\Psi_0}}.
\]
Furthermore, since both $U^p(t, s)$ and $U^q(t, s)$ are norm (and thus strongly) differentiable with respect to $t$ and $s$, we obtain that
\begin{align}\label{eq:pert_theory_bd}
&\norm{U^p(t, s) \Pi_{\leq q} \ket{\Psi_0} - U^q(t, s) \Pi_{\leq q} \ket{\Psi_0}} = \norm{\int_s^t \frac{d}{d\tau}\big(U^p(s, \tau) U^q(\tau, s)\big) \Pi_{\leq q} \ket{\Psi_0} d\tau} \nonumber\\
&\qquad \qquad \leq \int_s^t \norm{\big(H^p(\tau) - H^q(\tau)\big) U^q(\tau, s) \Pi_{\leq q} \ket{\Psi_0}}d\tau .
\end{align}
Furthermore, since $p > q$, we obtain that
\begin{align}\label{eq:pert_theory_state}
\ket{\Phi_{p, q}(\tau, s)} := \big(H^p(\tau) - H^q(\tau)\big)U^q(\tau, s)\Pi_{\leq q}\ket{\Psi_0} = \big(H^p(s) - H^q(s)\big)\Pi_{q}U^q(\tau, s)\Pi_{\leq q}\ket{\Psi_0},
\end{align}
and thus
\begin{align}\label{eq:norm_estimate}
\norm{\ket{\Phi_{p, q}(s)}} \leq \ell \sqrt{q + 1} \norm{\Pi_{q} U^q(\tau, s)\Pi_{\leq q}\ket{\Psi_0}}.
\end{align}
Using the bound from lemma \ref{lemma:particle_num_bound}, we obtain that
$\norm{\Pi_q U^q(\tau, s) \Pi_{\leq q}\ket{\Psi_0}} \leq \sqrt{M_{\Pi_{\leq q}\ket{\Psi(0)}}^{(2)}(\max(s, t))} / q \leq \sqrt{M_{\ket{\Psi_0}}^{(2)}(\max(s, t))} / q$ and thus
\[
\int_{s}^t \norm{\ket{\Phi_{p, q}(\tau, s)}} d\tau \leq \frac{\ell \abs{t - s} \sqrt{q + 1}}{q} \sqrt{M^{(2)}_{\Pi_{\leq q}\ket{\Psi_0}}(\max(s, t))} \leq \frac{\ell \abs{t - s} \sqrt{q + 1}}{q} \sqrt{M^{(2)}_{\ket{\Psi_0}}(\max(s, t))},
\]
and hence
\[
\norm{U^p(t, s) \ket{\Psi_0} - U^q(t, s) \ket{\Psi_0}} \leq \frac{\ell \abs{t - s} \sqrt{q + 1}}{q} \sqrt{M^{(2)}_{\ket{\Psi_0}}(\max(s, t))} + 2\norm{\Pi_{\leq q}\ket{\Psi_0}}.
\]
Thus, $\norm{U^p(t, s) \ket{\Psi_0} - U^q(t, s) \ket{\Psi_0}} \to 0$ as $p, q \to \infty$, thus implying that the sequence $\{U^p(t, s)\ket{\Psi_0}\}_{p \in \mathbb{N}}$ is Cauchy, and hence converges. Furthermore, from lemma \ref{lemma:particle_num_bound}, the moments $\mu^{(k)}_{U^p(t, s) \ket{\Psi_0}}$ are bounded uniformly in $p$ for all $k \in \mathbb{Z}_{\geq 0}$, and hence from the dominated convergence theorem it follows that all the particle number moments of $\lim_{p \to \infty}U^p(t, s) \ket{\Psi_0}$ are also bounded --- this shows that $\lim_{p \to \infty} U^p(t, s) \ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_{\mathcal{S}}^M[L^2(\mathbb{R})]$. \\ \ \\
\noindent\emph{(b)} For $\ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})]$, since $\norm{\lim_{p\to \infty} U^p(t, s) \Pi_{> p} \ket{\Psi_0}} = \lim_{p \to \infty} \norm{\Pi_{> p}\ket{\Psi_0}} = 0$, we obtain that
\[
\overline{H(t)} \lim_{p\to \infty} U^p(t, s) \ket{\Psi_0} = \overline{H(t)} \lim_{p \to \infty} U^p(t, s) \Pi_{\leq p} \ket{\Psi_0}.
\]
We already established in part (a) that the sequence $\{U^p(t, s) \ket{\Psi_0}\}_{p \in \mathbb{Z}_{\geq 0}}$, and hence the sequence $\{U^p(t, s) \Pi_{\leq p}\ket{\Psi_0}\}_{p \in \mathbb{Z}_{\geq 0}}$, converges. We now show that the sequence $\{H(t) U^p(t, s) \Pi_{\leq p}\ket{\Psi_0} \}_{p \in \mathbb{Z}_{\geq 0}}$ also converges. To see this, we note that for $p, q \in \mathbb{Z}_{\geq 0}$ with $q \leq p$,
\begin{align}\label{eq:hamiltonian_diff_lemma_useful}
\norm{H(t) U^p(t, s) \Pi_{\leq p} \ket{\Psi_0} - H(t) U^q(t, s) \Pi_{\leq q} \ket{\Psi_0} } \leq \int_s^t \norm{H(t) U^p(t, \tau)\ket{\Phi_{p, q}(\tau, s)} } d\tau+ \norm{H(t)U^p(t, s)\ket{\Gamma_{p, q}(t, s)}},
\end{align}
where $\ket{\Phi_{p, q}(\tau, s)}$ is defined in Eq.~\ref{eq:pert_theory_state} and $\ket{\Gamma_{p, q}(t, s)} = \big(\Pi_{\leq p} - \Pi_{\leq q}\big) \ket{\Psi_0}$. Now, from the definition of $H(t)$, it follows that
\[
\norm{H(t) U^p(t, s) \ket{\Gamma_{p, q}(t, s)}} \leq \norm{H_S(t)} \norm{ \ket{\Gamma_{p, q}(t, s)}} + 2 \ell \bigg( \norm{\ket{\Gamma_{p, q}(t, s)}}^2 + \mu^{(1)}_{U^p(t, s) \ket{\Gamma_{p, q}(t, s)}}\bigg)^{1/2}.
\]
Furthermore, using lemma \ref{lemma:particle_num_bound}, we obtain that
\[
\mu^{(1)}_{U^p(t, s) \ket{\Gamma_{p, q}(t, s)}} \leq 2\bigg(\mu^{(1)}_{\Gamma_{p, q}(t, s)} + \ell^2 \textnormal{max}^2(s, t) \norm{\ket{\Gamma_{p, q}(t, s)}}^4\bigg).
\]
Since $\ket{\Psi_0} \in \mathcal{H}_S\otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})]$, we obtain that
\begin{align*}
&\norm{\ket{\Gamma_{p, q}(t, s)}} \leq \norm{\Pi_{> q} \ket{\Psi_0}} + \norm{\Pi_{> p}\ket{\Psi_0}} \to 0 \textnormal{ as } p, q \to \infty \ \text{and}, \\
&\mu^{(1)}_{\ket{\Gamma_{p, q}(t, s)}} \leq \mu^{(1)}_{\ket{\Pi_{>q} \ket{\Psi_0}}} + \mu^{(1)}_{\ket{\Pi_{>p} \ket{\Psi_0}}} \to 0 \textnormal{ as } p, q \to \infty,
\end{align*}
and therefore it follows from the previous estimates that $\norm{H(t) U^p(t, s) \ket{\Gamma_{p, q}(t, s)}} \to 0$ as $p, q \to \infty$. Consider now the second term in Eq.~\ref{eq:hamiltonian_diff_lemma_useful} --- since $U^p(t, \tau) \ket{\Phi_{p, q}(\tau, s)} \in \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})]$, we obtain that
\[
\int_s^t \norm{H(\tau)U^p(t, \tau) \ket{\Phi_{p, q}(\tau, s)}}d\tau \leq \int_s^t \norm{H_S(\tau)} \norm{\ket{\Phi_{p, q}(\tau, s)}} d\tau + 2\ell \int_s^t \bigg(\norm{\ket{\Phi_{p, q}(\tau, s)}}^2 + \mu_{U^p(t, \tau)\ket{\Phi_{p, q}(\tau, s)}}^{(1)}\bigg)^{1/2} d\tau.
\]
Using Eq.~\ref{eq:norm_estimate} and the bound from lemma \ref{lemma:particle_num_bound}, we obtain that
\[
\int_s^t \norm{H_S(\tau)} \norm{\ket{\Phi_{p, q}(\tau, s)}} d\tau \leq \frac{\ell \sqrt{q + 1}}{q} \sqrt{M^{(2)}_{\ket{\Psi_0}}(t_\textnormal{max})} \int_s^t \norm{H_S(\tau)}d\tau.
\]
It follows from lemma \ref{lemma:particle_num_bound} that for $\tau \in [\textnormal{min}(t, s), \textnormal{max}(t, s)]$,
\[
{\mu^{(1)}_{U^p(t, \tau)\ket{\Phi_{p, q}(\tau, s)}}} \leq 2 \bigg(\mu^{(1)}_{\ket{\Phi_{p, q}(\tau, s)}} + \ell^2 \textnormal{max}^2(s, t)\norm{\ket{\Phi_{p, q}(\tau, s)}}^4\bigg),
\]
and using Eq.~\ref{eq:norm_estimate} it follows that for $\tau \in [\min(t, s), \max(t, s)]$,
\[
\mu^{(1)}_{\ket{\Phi_{p, q}(\tau, s)}} \leq \frac{(q + 1)^2}{q^3} \ell^2 M^{(3)}_{\Pi_{\leq q}\ket{\Psi_0}}(\max(s, t)) \leq \frac{(q + 1)^2}{q^3} \ell^2 M^{(3)}_{\ket{\Psi_0}}(\max(s, t)).
\]
From these estimates, it thus follows that
\[
\int_s^t \norm{H(\tau) U^p(t, \tau) \ket{\Phi_{p, q}(\tau, s)}} d\tau \to 0 \textnormal{ as } p, q \to \infty.
\]
Therefore, from Eq.~\ref{eq:hamiltonian_diff_lemma_useful}, it follows that the sequence $\{H(t)U^p(t, s) \Pi_{\leq p}\ket{\Psi_0}\}_{p \in \mathbb{N}}$ converges --- since $H(t)$ is a closable operator, it then follows that
\[
\lim_{p \to \infty} H(t) U^p(t, s) \Pi_{\leq p}\ket{\Psi_0} = \overline{H(t)} \lim_{p \to \infty} U^p(t, s) \Pi_{\leq p}\ket{\Psi_0} = \overline{H(t)} \lim_{p \to \infty} U^p(t, s) \ket{\Psi_0}.
\]
Finally, we show that $\lim_{p\to \infty} H^p(t) U^p(t, s) \ket{\Psi_0} = \lim_{p\to \infty} H(t) U^p(t, s) \Pi_{\leq p} \ket{\Psi_0}$. We begin by noting that
\[
H^p(t) U^p(t, s) \ket{\Psi_0} = H^p(t) U^p(t, s) \Pi_{\leq p} \ket{\Psi_0} + \Pi_{> p} \ket{\Psi_0} \implies \lim_{p \to \infty} H^p(t) U^p(t, s) \ket{\Psi_0} = \lim_{p \to \infty} H^p(t) U^p(t, s) \Pi_{\leq p} \ket{\Psi_0} .
\]
Furthermore,
\begin{align*}
&\norm{(H(t) - H^p(t)) U^p(t, s) \Pi_{\leq p} \ket{\Psi_0}} ^2 \leq \nonumber \\
&\qquad \qquad (p +1)\ell^2 \norm{\Pi_p U^p(t, s) \ket{\Psi_0}}^2 \leq \frac{p + 1}{p^2}\ell^2 M^{(2)}_{\Pi_{\leq p}\ket{\Psi_0}}(\max(t, s)) \leq \frac{p + 1}{p^2}\ell^2 M^{(2)}_{\ket{\Psi_0}}(\max(t, s)).
\end{align*}
and thus $\lim_{p \to \infty} H^p(t) U^p(t, s) \Pi_{\leq p}\ket{\Psi_0} = \lim_{p \to \infty} H(t) U^p(t, s) \Pi_{\leq p}\ket{\Psi_0}$. Hence, we obtain that $\overline{H(t)} \lim_{p \to \infty} U^p(t, s) \ket{\Psi_0} = \lim_{p \to \infty} H^p(t) U^p(t, s) \ket{\Psi_0}$.
Now, we investigate the continuity of $\overline{H(t)} \lim_{p\to \infty} U^p(t, s) \ket{\Psi_0} = \lim_{p \to \infty} H^p(t) U^p(t, s) \Pi_{\leq p}\ket{\Psi_0}$ with respect to $t$. For $p \in \mathbb{Z}_{\geq 0}$ and $\delta > 0$, define $\Delta^p(\delta)$ va
\[
\Delta^p(\delta) = \norm{\bigg(H^p(t + \delta) U^p(t + \delta, s) - H^p(t) U^p(t, s) \bigg)\Pi_{\leq p}\ket{\Psi_0}}.
\]
We need to show that $\lim_{\delta \to 0} \lim_{p\to \infty}\Delta^p(\delta) = 0$. To see this, we note that
\[
\Delta^p(\delta) \leq \norm{\bigg(H^p(t + \delta) - H^p(t)\bigg) U^p(t, s)\Pi_{\leq p} \ket{\Psi_0}} + \norm{H^p(t + \delta)\bigg(U^p(t + \delta, s) - U^p(t, s)\bigg) \Pi_{\leq p} \ket{\Psi_0}}.
\]
Now
\begin{align*}
&\norm{\bigg(H^p(t + \delta) - H^p(t)\bigg) U^p(t, s) \ket{\Psi_0}} \nonumber \\
&\qquad \qquad \leq \norm{H_S(t + \delta) - H_S(t)} \norm{\ket{\Psi_0}} +\bigg(\norm{\ket{\Psi_0}}^2 + \mu^{(1)}_{U^p(t, s) \Pi_{\leq p}\ket{\Psi_0}}\bigg)^{1/2} \sum_{\alpha = 1}^M \norm{L_\alpha} \norm{\bigg(\uptau_{\alpha, t + \delta} - \uptau_{\alpha, t}\bigg) v_\alpha}_{L^2}, \nonumber \\
&\qquad \qquad \leq \norm{H_S(t + \delta) - H_S(t)} \norm{\ket{\Psi_0}} +\bigg(\norm{\ket{\Psi_0}}^2 + M^{(1)}_{ \ket{\Psi_0}}\bigg)^{1/2} \sum_{\alpha = 1}^M \norm{L_\alpha} \norm{\bigg(\uptau_{\alpha, t + \delta} - \uptau_{\alpha, t}\bigg) v_\alpha}_{L^2},
\end{align*}
and consequently by the strong continuity of $\uptau_{\alpha, t}$,
\[
\lim_{\delta \to 0} \lim_{p \to \infty}\norm{\bigg(H^p(t + \delta) - H^p(t)\bigg) U^p(t, s) \ket{\Psi_0}} = 0.
\]
Furthermore,
\[
\norm{H^p(t + \delta)\bigg(U^p(t + \delta, s) - U^p(t, s)\bigg) \Pi_{\leq p} \ket{\Psi_0}} \leq \int_{t}^{t+ \delta} \norm{H^p(\tau + \delta) H^p(\tau) U^p(\tau, s) \Pi_{\leq p}\ket{\Psi_0}}d\tau.
\]
It follows from lemma \ref{lemma:particle_num_bound} that $\norm{H^p(\tau + \delta) H^p(\tau) U^p(\tau, s) \ket{\Psi_0}}$ is bounded above by a constant independent of $p$ and continuous in $\tau$. Thus, we obtain that
\[
\lim_{\delta \to 0 }\lim_{p \to \infty} \norm{H^p(t + \delta)\bigg(U^p(t + \delta, s) - U^p(t, s)\bigg) \Pi_{\leq p} \ket{\Psi_0}} = 0.
\]
Thus, we obtain that $\lim_{\delta \to 0} \lim_{p \to \infty} \Delta^p(\delta) = 0$.
\\ \ \\
\emph{(c)} The proof of this part closely follows that of part (b), with only minor modifications which we outline here. We can show that $\lim_{p\to \infty} \norm{H^p(s) \ket{\Psi_0} - \overline{H(s)}\ket{\Psi_0}} = 0$, which would imply that $\lim_{p\to \infty} U^p(t, s) H^p(s) \ket{\Psi_0} = \lim_{p\to \infty} U^p(t, s) \overline{H(s)}\ket{\Psi_0}$, in two steps --- first, we establish that $\lim_{p \to \infty} H^p(s) \ket{\Psi_0} = \lim_{p \to \infty} H^p(s)\Pi_{\leq p} \ket{\Psi_0}$ using the fact that all the particle-number moments of $\ket{\Psi_0}$ are finite. Then, we can show that $\lim_{p\to \infty} H^p(s) \Pi_{\leq p} \ket{\Psi_0} = \lim_{p \to \infty} H(s) \Pi_{\leq p} \ket{\Psi_0}$ by analyzing the norm $\norm{(H(s) - H^p(s)) \Pi_{\leq p}\ket{\Psi_0}}$. By showing that the sequence $\{H(s) \Pi_{\leq p} \ket{\Psi_0} \}_{p \in \mathbb{N}}$ converges, and using the closability of $H(t)$, we then obtain that $\lim_{p \to \infty} H(s) \Pi_{\leq p} \ket{\Psi_0} = \overline{H(s)} \lim_{p \to \infty} \Pi_{\leq p} \ket{\Psi_0}$. Finally, using lemma \ref{lemma:domain}b, you obtain that $\overline{H(s)}\lim_{p \to \infty} \Pi_{\leq p}\ket{\Psi_0} = \overline{H(s)} \ket{\Psi_0}$. To prove that $\lim_{p\to \infty}U^p(t, s) \overline{H(s)} \ket{\Psi_0} = \lim_{p\to \infty} U^p(t, s) H^p(s) \ket{\Psi_0}$ is strongly continuous, we can again analyze $\Delta^p(\delta)$, where
\begin{align*}
\Delta^p(\delta) &= \norm{\bigg(U^p(t, s+\delta) H^p(s+ \delta) - U^p(t, s) H^p(s)\bigg) \ket{\Psi_0}}, \nonumber\\
&\leq \norm{\bigg(U^p(t, s + \delta) - U^p(t, s)\bigg) H^p(s + \delta) \ket{\Psi_0}} + \norm{U^p(t, s)\bigg(H^p(t, s + \delta) -H^p(s)\bigg) \ket{\Psi_0}}.
\end{align*}
Using lemma \ref{lemma:particle_num_bound}, we can show that $\lim_{p\to \infty} \lim_{\delta \to 0}\Delta^p(\delta) = 0$.
\\ \ \\
\emph{(d)} This follows straightforwardly from lemma \ref{lemma:particle_num_bound}, and noting that
\[
\norm{H^p(t) U^p(t, s) \ket{\Psi_0)}} \leq \norm{H_S(t)} \norm{\ket{\Psi_0}} + 2\ell \bigg(\norm{\ket{\Psi_0}}^2 + M^{(1)}_{\ket{\Psi_0}}(\max(t, s)) \bigg)^{1/2}.
\]
which yields the upper bound $g(t)$. Similarly,
\[
\norm{H^p(t) \ket{\Psi_0}} \leq \norm{H_S(t)} \norm{\ket{\Psi_0}} + 2\ell \bigg(\norm{\ket{\Psi_0}}^2 +\mu^{(1)}_{\ket{\Psi_0}}\bigg)^{1/2},
\]
which yields the upper bound $h(t)$.
\(\square\)
\begin{repproposition}{thm:schr_sq_int}
Given a non-Markovian model with square integrable coupling functions and for $t, s \in \mathbb{R}$, there exists a unique isometry $U(t, s) : \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \to \mathcal{H}_S \otimes \textnormal{F}_\infty^M[L^2(\mathbb{R})] \subseteq \mathcal{H}$ which is strongly continuous and differentiable in both $t, s$ and satisfies
\begin{align}\label{eq:prop_schr_eq}
\frac{d}{dt} U(t, s) = -i \overline{H(t)} U(t, s),\ \frac{d}{ds} U(t, s) = i U(t, s) \overline{H(s)},
\end{align}
with $U(s, s) = \textnormal{id} \ \forall s \in \mathbb{R}$.
\end{repproposition}
\noindent\emph{Proof}: We first construct the unitary group $U(t, s)$ --- given a state $\ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})]$, we let
\[
U(t, s) \ket{\Psi_0} = \lim_{p \to \infty} U^p(t, s) \ket{\Psi_0}.
\]
It follows from lemma \ref{lemma:useful} that $U(t, s)$ is well defined, that $U(t, s) \ket{\Psi_0} \in \mathcal{H}_S \otimes \textnormal{F}_\mathcal{S}^M[L^2(\mathbb{R})]$ and that $U(t, s)$ is an isometry. Now, we note that since $U^p(t, s)$ is the propagator corresponding to $H^p(t)$,
\[
U(t, s) \ket{\Psi_0} = \lim_{p\to \infty} U^p(t, s) \ket{\Psi_0} = \ket{\Psi_0}- i \lim_{p\to \infty} \int_s^t H^p(\tau) U^p(\tau, s) \ket{\Psi_0}d\tau.
\]
From lemma \ref{lemma:useful}(d), it follows that $\norm{H^p(\tau) U^p(\tau, s) \ket{\Psi_0}}$ is bounded above by a continuous (and thus integrable) function of $\tau$, and hence from the dominated convergence theorem, we obtain that
\[
U(t, s)\ket{\Psi_0} = \ket{\Psi_0} - i \int_s^t \lim_{p \to \infty} H^p(\tau) U^p(\tau, s)\ket{\Psi_0}d\tau.
\]
Finally, using lemma \ref{lemma:useful}(b), we obtain that $\lim_{p\to \infty} H^p(\tau) U^p(\tau, s) \ket{\Psi_0} = \overline{H(\tau)}U(\tau, s) \ket{\Psi_0}$, and since $ \overline{H(\tau)}U(\tau, s) \ket{\Psi_0}$ is strongly continuous in $\tau$, $U(t, s) \ket{\Psi_0}$ is strongly differentiable in $t$. Thus,
\[
U(t, s) \ket{\Psi_0} = \ket{\Psi_0} - i \int_s^t \overline{H(\tau)} U(\tau, s)\ket{\Psi_0}d\tau \implies \frac{d}{dt}U(t, s) \ket{\Psi_0} = -i \overline{H(t)} U(t, s)\ket{\Psi_0}.
\]
which shows that $d U(t, s)/dt = -i\overline{H(t)} U(t, s)$ (where derivatives are understood as strong derivatives) with $U(s, s) = \textnormal{id}$. A similar argument can be made using lemmas \ref{lemma:useful}(c) and \ref{lemma:useful}(d) to show that $d U(t,s) \ket{\Psi_0} / ds = iU(t, s) \overline{H(s)}$.
Since $\forall t \in \mathbb{R}$, $H(t)$ is essentially self adjoint, $\overline{H(t)}$ is self adjoint --- from this, it immediately follows that if the solution to Eq.~\ref{eq:prop_schr_eq} exists, the it must be unique. To see this, we simply note that the self-adjointness of $\overline{H(t)}$ implies that $\norm{U(t, s)\ket{\Psi_0}} = \norm{U(s, s)\ket{\Psi_0}}$, and hence $\ket{\Psi_0} = 0 \implies \ket{\Psi_0} = 0 \ \forall t \geq 0$. Now if there were two distinct solutions $U_1(t, s), U_2(t, s)$ to Eq.~\ref{eq:prop_schr_eq}, then $\big(U_1(t, s) - U_2(t, s)\big)\ket{\Psi_0}$ would be a non-zero vector, which leads to a contradiction since by essential self adjointness of $H(t)$, $\norm{\big(U_1(t, s) - U_2(t, s)\big)\ket{\Psi_0}} = 0$.
\(\square\)
\section{Proof of lemma \ref{lemma:main_error_bound_mu}}\label{app:distributional_mem_ker}
\begin{replemma}{lemma:main_error_bound_mu}
Consider $\mu \in \mathcal{M}(\mathbb{R})$ with the Lesbesgue decomposition $\mu = \mu_c + \mu_d$ with $\phi_c \in \textnormal{C}^0(\mathbb{R})$ given by $\phi_c(x) = \mu_c((-\infty, x]) \ \forall \ x \in \mathbb{R}$, and $\mu_d \cong \sum_{i \in I} \alpha_i \delta(x - y_i)$ for some $\{\alpha_i \in \mathbb{C}\}_{i \in I}, \{y_i \in \mathbb{R}\}_{i \in I}$ and finite and countably infinite index set $I$. Given a compact interval $[a, b] \subseteq \mathbb{R}$ and $f \in \textnormal{C}^1(\mathbb{R})$, define $\langle\mu_{[a, b]}^*, f\rangle $ by
\begin{align*}
&\langle \mu^*_{[a, b]}, f\rangle =\langle \mu_{c, [a, b]}^*, f\rangle + \langle \mu_{d, [a, b]}^*, f\rangle \ \text{where} \\
&\langle\mu_{c, [a, b]}^*, f\rangle =\nonumber\\
&\qquad f(b) \phi_c(b) - f(a) \phi_c(a) - \int_a^b \phi_c(x) f'(x) dx \ \text{ and}\\
&\langle\mu_{d, [a, b]}^*, f\rangle = \frac{1}{2} \sum_{i \in I | y_i \in \{a, b\}} \alpha_i f(y_i) + \sum_{i \in I | y_i \in (a, b)} \alpha_i f(y_i).
\end{align*}
Then, for every compact intervals $[a, b] \subseteq \mathbb{R}$, $\exists \Delta^0_{\mu; [a, b]}(\varepsilon), \Delta^1_{\mu; [a,b]}(\varepsilon) > 0 $ where $ \Delta^0_{\mu; [a, b]}(\varepsilon), \Delta^1_{\mu; [a,b]}(\varepsilon) \to 0$ as $\varepsilon \to 0$ such that $\forall \varepsilon \in (0, (b- a) / 2)$ and for any even (symmetric about $0$) positive function $\alpha \in \textnormal{C}_c^\infty(\mathbb{R})$ with $\textnormal{supp}(\alpha) \subseteq [-\varepsilon,\varepsilon]$ and $\int_{[-\varepsilon, \varepsilon]}\alpha(x) dx=1$,
\begin{align*}
&\abs{ \langle \mu^*_{ [a, b]}, f\rangle - \langle \mu, \alpha \star (f\cdot \mathcal{I}_{[a, b]}) \rangle} \leq \nonumber\\
&\qquad \Delta^0_{\mu; [a, b]}(\varepsilon) \sup_{x \in [a, b]} \abs{f(x)} + \Delta^1_{\mu; [a, b]}(\varepsilon)\sup_{x \in [a, b]} \abs{f'(x)}.
\end{align*}
The functions $\Delta^0_{\mu; [a, b]}, \Delta^1_{\mu; [a, b]}$ will be called the error functions corresponding to $\mu$.
\end{replemma}
\noindent\emph{Proof}: Let $\rho \in \textnormal{C}_c^\infty(\mathbb{R})$ be a symmetric and positive-valued function with $\textnormal{supp}(\rho) \subseteq [-\varepsilon, \varepsilon]$ for some $\varepsilon > 0$. Let $f \in \textnormal{C}^1(\mathbb{R})$. For $[a, b] \subseteq \mathbb{R}$, define ${f}^\rho_{[a, b]} = \big(f \cdot \mathcal{I}_{[a, b]}\big)\star \rho$.
\\ \emph{Analysis of the continuous part.} Now, we consider $\langle \mu_c, f^\rho_{[a, b]}\rangle$ --- since $f^\rho_{[a, b]} \in \textnormal{C}^1_c(\mathbb{R})$, we note that
\[
\langle \mu_c, f^\rho_{[a, b]}\rangle = - \int_{\mathbb{R}} \phi_c(x) f^{\rho'}_{[a, b]}(x) dx.
\]
We note that $\forall x \in \mathbb{R}$,
\[
f^{\rho'}_{[a, b]}(x) = \int_a^b \rho'(x - y) f(y) dy = f(a) \rho(x - a) - f(b) \rho(x - b) + \int_a^b f'(y)\rho(x - y) dy.
\]
Therefore,
\[
\langle \mu_c, f^\rho_{[a, b]}\rangle = \int_{\mathbb{R}} f(b)\phi_c(y)\rho(y - b) dy - \int_{\mathbb{R}} f(a)\phi_c(y) \rho(y - a) dy - \int_{\substack{y \in \mathbb{R} \\ x \in [a, b]}} \phi_c(y) f'(x) \rho(y - x) dx dy.
\]
Since $\varepsilon < (b - a) / 2 \implies \textnormal{supp}(\rho) \subseteq [-(b - a)/2, (b - a)/2]$, this can be rewritten with integrals being only over compact intervals,
\[
\langle \mu_c, f^\rho_{[a, b]}\rangle = \int_{\frac{3a - b}{2}}^{\frac{3b - a}{2}} f(b)\phi_c(y)\rho(y - b) dy - \int_{\frac{3a - b}{2}}^{\frac{3b - a}{2}} f(a)\phi_c(y) \rho(y - a) dy - \int_{y = \frac{3a - b}{2}}^{\frac{3b - a}{2}} \int_{x = a}^b \phi_c(y) f'(x) \rho(y - x) dx dy.
\]
Now, since $\phi_c$ is continuous, it is uniformly continuous over the compact interval $[(3a - b) / 2, (3b - a)/2]$. Thus, $\exists \delta_{\mu_c; a, b}(\varepsilon)$ where $\delta_{\mu_c; [a, b]}(\varepsilon) \to 0$ as $\varepsilon\to 0$ such that $\forall y, y' \in [(3a - b) / 2, (3b - a)/2]$ with $\abs{y - y'}< \varepsilon$, $\abs{\phi_c(y) - \phi_c(y')} < \delta_{\mu_c; [a, b]}(\varepsilon)$. Using this, we obtain that $\forall x \in [a, b]$
\[
\abs{\phi_c(x) - \int_{y \in[(3a - b) / 2, (3b - a)/2]} \phi_c(y) \rho(x - y) dy} ={\int_{y \in [x-\varepsilon, x + \varepsilon]} \abs{\big(\phi_c(x) - \phi_c(y)\big)}\rho(x - y) dy} \leq \delta_{\mu_c; [a, b]}.
\]
It then follows that
\begin{align*}
&\abs{\langle \mu_c^*, f\rangle - \langle \mu_c, f^\rho_{[a, b]}\rangle} \leq \sum_{x \in \{a, b\}} \abs{f(x)}\abs{\phi_c(x) - \int_{[(3a - b) / 2, (3b - a)/2]} \phi_c(y)\rho(y - x)dy} + \nonumber\\
&\qquad \qquad \qquad \int_{x\in[a, b]}\abs{\phi_c(y) - \int_{y \in [(3a - b) / 2, (3b - a)/2]}\phi_c(y)\rho(x - y) dy} \abs{f'(x)} dx,
\end{align*}
and consequently,
\begin{subequations}\label{eq:mu_c_error_estimate}
\begin{align}
\abs{\langle \mu_c^*, f\rangle - \langle \mu_c, f^\rho_{[a, b]}\rangle} \leq \delta_{\mu_c; [a, b]}(\varepsilon)\bigg(2\sup_{x \in [a, b]}|f(x)| + \int_a^b \abs{f'(x)}dx\bigg) \leq \Delta^0_{\mu_c; [a, b]}(\varepsilon)\sup_{x \in [a, b]}|f(x)| + \Delta^1_{\mu_c; [a, b]}\sup_{x \in [a, b]} \abs{f'(x)},
\end{align}
where
\begin{align}
\Delta^0_{\mu_c; [a, b]}(\varepsilon) = 2\delta_{\mu_c; [a, b]}(\varepsilon) \text{ and }\Delta^1_{\mu_c; [a, b]}(\varepsilon) = (b - a)\delta_{\mu_c; [a,b]}(\varepsilon),
\end{align}
\end{subequations}
both of which $\to 0$ as $\varepsilon \to 0$.
\\
\emph{Analysis of the atomic part.} We now consider $\langle \mu_d, f^\rho_{[a, b]}\rangle$ --- since $f^\rho_{[a, b]} \in \textnormal{C}_c^0(\mathbb{R})$, we obtain that
\[
\langle \mu_d, f^\rho_{[a, b]}\rangle = \sum_{i \in I} \alpha_i f^\rho_{[a, b]}(y_i).
\]
Therefore,
\[
\abs{\langle \mu_d^*, f\rangle - \langle \mu_d, f^\rho_{[a, b]}\rangle } \leq \sum_{i \in I | y_i \notin [a, b]} \abs{\alpha_i} \abs{f^\rho_{[a, b]}(y_i)} + \sum_{i \in I | y_i \in(a, b)} \abs{\alpha_i} \abs{f(y_i) - f^\rho_{[a, b]}(y_i)} + \sum_{i \in I | y_i \in \{a, b\}} \abs{\alpha_i} \abs{\frac{f(y_i)}{2} - f^\rho_{[a, b]}(y_i)}.
\]
Since $\text{supp}(f^\rho_{[a, b]}) \subseteq [a - \varepsilon, a + \varepsilon]$, and $\norm{f^\rho_{[a, b]}}_{L^\infty} \leq \sup_{x \in [a, b]}|f(x)|$, we obtain that
\[
\sum_{i \in I | y_i \notin [a, b]} \abs{\alpha_i} \abs{f^\rho_{[a, b]}(y_i)} \leq \bigg(\sup_{x\in [a, b]} |f(x)|\bigg) \bigg(\sum_{i \in I | y_i \in [a - \varepsilon, a)} |\alpha_i| + \sum_{i \in I | y_i \in (b , b + \varepsilon]} \abs{\alpha_i} \bigg).
\]
Furthermore, we note that for $x \in [a + \varepsilon, b - \varepsilon]$, we obtain that
\[
\abs{f(x) - f^\rho_{[a, b]}(x)} \leq {\int_{[-\varepsilon, \varepsilon]} \abs{f(x) - f(x - y)}\rho( y) dy} \leq \sup_{y \in [a, b]}\abs{f'(y)} \int_{[-\varepsilon, \varepsilon]}|y|\rho(y) dy \leq \varepsilon \sup_{y \in [a, b]}\abs{f'(y)},
\]
and thus we obtain that
\begin{align*}
&\sum_{i \in I | y_i \in (a, b)}\abs{\alpha_i} \abs{f(y_i) - f_{[a, b]}^\rho(y_i)} \leq 2\sup_{y \in [a, b]} \abs{f(y)} \sum_{\substack{i \in I | y_i \in (a, a + \varepsilon] \\ \text{or } y_i \in (b - \varepsilon, b] }} \abs{\alpha_i} + \varepsilon \sup_{y \in [a, b]} \abs{f'(y)} \sum_{i \in I | y_i \in (a + \varepsilon, b - \varepsilon)} \abs{\alpha_i}.
\end{align*}
Similarly, we can note that
\begin{align*}
&\abs{\frac{1}{2}f(a) - f^\rho_{[a, b]}(a)} \leq \int_{[0, \varepsilon]} \abs{f(a) - f(a+ y)} \rho(y) dy \leq \sup_{y \in [a, b]}\abs{f'(y)} \int_{[0, \varepsilon]} y\rho(y) \leq \frac{\varepsilon}{2} \sup_{y \in [a, b]}\abs{f'(y)} \ \text{and}, \\
&\abs{\frac{1}{2}f(b) - f^\rho_{[a, b]}(b)} \leq \int_{[0, \varepsilon]} \abs{f(b) - f(b - y)} \rho(y) dy \leq \sup_{y \in [a, b]}\abs{f'(y)} \int_{[0, \varepsilon]} y\rho(y) \leq \frac{\varepsilon}{2} \sup_{y \in [a, b]}\abs{f'(y)},
\end{align*}
and thus we obtain that
\begin{align*}
\sum_{i \in I | y_i \in \{a, b\}} \abs{\alpha_i} \abs{\frac{f(y_i)}{2} - f^\rho_{[a, b]}(y_i)} \leq \frac{\varepsilon}{2} \sup_{y \in [a, b]}\abs{f'(y)} \sum_{i \in I | y_i \in \{a, b\}} \abs{\alpha_i}.
\end{align*}
Therefore, we obtain that
\begin{subequations}\label{eq:mu_d_error_estimate}
\begin{align}
\abs{\langle \mu_d^*, f\rangle - \langle \mu_d, f^\rho_{[a, b]}\rangle } \leq \Delta_{\mu_d; [a,b]}^0(\varepsilon) \sup_{y \in [a, b]} \abs{f(y)} + \Delta_{\mu_d; [a,b]}^1(\varepsilon) \sup_{y \in [a, b]} \abs{f'(y)},
\end{align}
where
\begin{align}
\Delta_{\mu_d; [a,b]}^0(\varepsilon) = \sum_{\substack{i \in I | y_i \in [a-\varepsilon, a) \text{ or} \\ y_i \in (b, b+\varepsilon]}} \abs{\alpha_i} + 2\sum_{\substack{i \in I | y_i \in (a, a + \varepsilon] \\ \text{or } y_i \in (b - \varepsilon, b] }} \abs{\alpha_i} \text{ and } \Delta^1_{\mu_d; [a, b]}(\varepsilon) = \varepsilon\bigg(\sum_{i \in I | y_i \in (a+ \varepsilon, b - \varepsilon)}\abs{\alpha_i} + \frac{1}{2}\sum_{i \in I | y_i \in \{a, b\}} \abs{\alpha_i}\bigg).
\end{align}
\end{subequations}
We can note that, by construction, $\Delta^0_{\mu_d; [a, b]}(\varepsilon), \Delta^1_{\mu_d; [a, b]}(\varepsilon) \to 0$ as $\varepsilon \to 0$. Using Eqs.~\ref{eq:mu_c_error_estimate} and \ref{eq:mu_d_error_estimate}, we obtain the lemma statement.
\(\square\)
\section{Proof of lemma \ref{lemma:inc_state_trunc}} \label{app:lemma_inc_state_trunc}
\begin{replemma}{lemma:inc_state_trunc}
Let $(\mu,\varphi)$ be a distributional coupling function. Given two mollifiers $\rho, \sigma \in \textnormal{C}_c^\infty(\mathbb{R})$ and $\varepsilon, \delta > 0$, let $v_\varepsilon$ and $v_\delta \in L^2(\mathbb{R})$ be the $\varepsilon, \rho-$ and $\delta, \sigma-$regularization of $(\mu, \varphi)$ respectively. Let $\mathcal{H}_S$ be Hilbert space, then,
\begin{enumerate}
\item[(a)]$\forall \ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_{\infty, \mathcal{S}}^M$, $\exists c_{\mu, \ket{\Phi}} > 0$, $\forall \tau \geq 0$, $\forall \alpha \in \{1, 2 \dots M\}, \varepsilon > 0$ such that $\norm{a_{\alpha, \uptau_\tau v_\varepsilon}^- \ket{\Phi}} \leq c_{\mu, \ket{\Phi}}$.
\item[(b)] $\forall \ket{\Phi} \in \mathcal{H}_S \otimes \textnormal{F}_{\infty, \mathcal{S}}^M$, $\exists c_{\mu,\rho, \ket{\Phi}}, d_{\mu, \sigma, \ket{\Phi}} > 0$, $\forall \tau \geq 0, \forall \alpha \in \{1, 2 \dots M\}$, $\varepsilon, \delta > 0$ such that $\norm{a_{\alpha, \uptau_\tau(v_\varepsilon - v_\delta)}^- \ket{\Phi}} \leq c_{\mu,\rho, \ket{\Phi}}\varepsilon + c_{\mu, \sigma, \ket{\Phi}} \delta$.
\end{enumerate}
\end{replemma}
\noindent\emph{Proof}: Any state $\ket{\Psi} \in \mathcal{H}_S \otimes \textnormal{F}_{\infty, \mathcal{S}}^M$ can be expressed as
\[
\ket{\Psi} = \sum_{j = 1}^N \ket{\sigma_j} \otimes \ket{u_j}^{\otimes n_j},
\]
for some $N \in \mathbb{Z}_{\geq 1}$, and
\[
\{\ket{\sigma_j} \in \mathcal{H}_S\}_{j \in \{1, 2 \dots N\}}, \bigg\{\ket{u_j} = \bigoplus_{\alpha = 1}^M \ket{u_{\alpha, j}}, u_{\alpha, j} \in \mathcal{S}(\mathbb{R}) \ \forall \ \alpha \in \{1, 2 \dots M\}\bigg \}_{j \in \{1, 2 \dots N\}}\ \text{and } \{n_j \in \mathbb{Z}_{\geq 0}\}_{j \in\{1,2 \dots N\}}.
\]
\noindent\emph{(a)} We obtain that
\begin{align*}
&\norm{a^{-}_{\alpha, \uptau_\tau v_\varepsilon} \ket{\Phi}} \leq\sum_{j = 1}^N \sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \bigg| \int_{\mathbb{R}} \sqrt{\hat{\mu}(\omega)} \hat{\rho}(\omega \varepsilon) u_{\alpha, j}(\omega) d\omega\bigg|, \nonumber\\
&\qquad\leq \sum_{j = 1}^N \sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \int_{\mathbb{R}} \bigg|\sqrt{\hat{\mu}(\omega)} \hat{\rho}(\omega \varepsilon) u_{\alpha, j}(\omega)\bigg| d\omega.
\end{align*}
Note that by assumption, $|\sqrt{\hat{\mu}(\omega)}|$ is a continuous function of at-most polynomial growth. Since $\forall j \in \{1, 2\dots N\}, \alpha \in \{1, 2 \dots M\}$, $u_{\alpha, j} \in \mathcal{S}(\mathbb{R})$,
\[
\norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega) ( 1 + \omega^2)}_{L^\infty} < \infty.
\]
Furthermore, note that since $\rho$ is a mollifier,
\[
\norm{\hat{\rho}}_{L^\infty} \leq \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} |\rho(s)| ds = \frac{1}{\sqrt{2\pi}}.
\]
Therefore,
\begin{align*}
&\norm{a^{-}_{\alpha, \uptau_\tau v_\varepsilon} \ket{\Phi}} \leq \sum_{j = 1}^N \sqrt{\frac{n_j}{2\pi} } \norm{\sigma_j}\norm{u_j}^{n_j - 1}\norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega) ( 1 + \omega^2)}_{L^\infty} \int_{\mathbb{R}} \frac{d\omega}{1 + \omega^2}, \nonumber\\
&\qquad \leq \sup_{\alpha \in \{1, 2 \dots M\}} \bigg(\sum_{j =1}^N \sqrt{\frac{\pi n_j}{2}}\norm{\sigma_j}\norm{u_j}^{n_j - 1}\norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega) ( 1 + \omega^2)}_{L^\infty} \bigg),
\end{align*}
where the bound can be identified as the constant $c_{\mu, \ket{\Psi}}$ in the lemma statement. \\
\noindent\emph{(b)} We obtain that
\begin{align*}
&\norm{a^{-}_{\alpha, \uptau_\tau(v_\varepsilon - v_{\delta})} \ket{\Phi}} \leq\sum_{j = 1}^N \sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \bigg| \int_{\mathbb{R}} u_{\alpha, j}(\omega) \sqrt{\hat{\mu}(\omega)} \big({\hat{\rho}(\omega\varepsilon) }- {\hat{\sigma}(\omega \delta)}\big)d\omega \bigg| \nonumber\\
&\qquad\leq \sum_{j = 1}^N \sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \int_{\mathbb{R}} \bigg|\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega)\big(\hat{\rho}(\omega \varepsilon) - \hat{\sigma}(\omega \delta)\big) \bigg| d\omega.
\end{align*}
Again, since by assumption $|\sqrt{\hat{\mu}(\omega)}|$ is a function of at-most polynomial growth, and $\forall \alpha \in \{1, 2 \dots M\}, j \in \{1, 2 \dots N\}$, $u_{\alpha, j} \in \mathcal{S}(\mathbb{R})$ and therefore,
\[
\norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega)(1 + \omega^2)^2}_{L^\infty} < \infty.
\]
Furthermore, since $\rho, \sigma \in \textnormal{C}_c^\infty(\mathbb{R}) \subseteq \mathcal{S}(\mathbb{R})$, $\hat{\rho}, \hat{\sigma} \in \mathcal{S}(\mathbb{R})$. In particular, $\norm{\hat{\rho}'}_{L^\infty} , \norm{\hat{\sigma}'}_{L^\infty} < \infty$. Furthermore,
\[
\hat{\rho}(0) - \hat{\sigma}(0) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \rho(s)ds - \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\sigma(s) ds = 0,
\]
since the mollifiers are, by definition, normalized to have unit area. Thus, using the Taylor's remainder theorem, we obtain that
\[
\big|\hat{\rho}(\omega \varepsilon) - \hat{\sigma}(\omega \delta)\big | \leq \norm{\hat{\rho}'}_{L^\infty}\big|\omega \big| \varepsilon + \norm{\hat{\sigma}'}_{L^\infty}\big|\omega \big| \delta \ \forall \ \omega \in \mathbb{R}.
\]
We thus obtain that
\[
\norm{a^-_{\alpha, \uptau_\tau(v_\varepsilon - v_\delta)} \ket{\Phi}} \leq \max_{\alpha\in\{1, 2 \dots M\}}\sum_{j = 1}^N \sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega)(1 + \omega^2)^2}_{L^\infty} \int_{\mathbb{R}}\frac{\norm{\hat{\rho}'}_{L^\infty}\big|\omega \big| \varepsilon + \norm{\hat{\sigma}'}_{L^\infty}\big|\omega \big| \delta}{(1 + \omega^2)^2} d\omega.
\]
Noting that $\int_{\mathbb{R}} |\omega| / (1 + \omega^2)^2 d\omega <\infty$, we obtain the lemma statement with the constants
\[
c_{\mu, \rho, \ket{\Phi}} = \max_{\alpha \in \{1, 2\dots M\}} \sum_{j = 1}^N\sqrt{n_j} \norm{\sigma_j}\norm{u_j}^{n_j - 1} \norm{\sqrt{\hat{\mu}(\omega)} u_{\alpha, j}(\omega)(1 + \omega^2)^2}_{L^\infty} \norm{\hat{\rho}'}_{L^\infty} \int_{\mathbb{R}}\frac{|\omega|}{(1 + \omega^2)^2}d\omega. \ \ \ \ \ \ \ \ \ \ \square
\]
\section{Chain approximation}\label{app:chain_appx}
We first analyze the unitary group describing the single-particle dynamics of an environment that has been approximated by a 1D chain of a discrete set of modes generated from the star-to-chain transformation. This unitary group should approximate the time-translation unitary group --- following Ref.~\cite{trivedi2021convergence}, we provide a bound on the distance between two unitary groups as a function of the number of modes in the chain.
\begin{definition}[Chain unitary group on $L^2(\mathbb{R})$]\label{def:chain_unitary}
Given $v \in L^2(\mathbb{R})$ with $\textnormal{supp}(\hat{v}) \in [-\omega_c, \omega_c]$ for some $\omega_c > 0$, a chain unitary group with $N_m$ modes generated by $v$ is the strongly continuous single parameter unitary group $\upnu_t : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ defined by
\[
\upnu_t f = \sum_{\beta = 1}^{N_m} c_\beta(t) \varphi_\beta + \bigg(f - \sum_{\beta = 1}^N \langle \varphi_\beta, f\rangle \varphi_\beta \bigg),
\]
where
\begin{enumerate}
\item[(a)] $\{\varphi_\beta \in L^2(\mathbb{R}) \}_{\beta \in \{1, 2 \dots N_m\}}$, called the mode functions, are a set of orthonormal functions (i.e.~$\langle \varphi_\alpha, \varphi_\beta \rangle = \delta_{\alpha, \beta}$ that are given by
\[
\hat{\varphi}_\alpha(\omega) = \frac{p_\alpha(\omega) \hat{v}(\omega)}{\big[\int_{-\omega_c}^{\omega_c} p_\alpha^2(\omega) |\hat{v}(\omega)|^2 d\omega \big]^{1/2}} \ \forall \ \alpha \in \{1, 2 \dots N_m\},
\]
where $p_\alpha$ is a degree $\alpha - 1$ polynomial generated by the following recursion starting from $p_1(\omega) = 1, B_1 = 0$,
\begin{align}\label{eq:lanczos_recurrence}
a_{\alpha} = \frac{\int_{-\omega_c}^{\omega_c} \omega p^2_\alpha(\omega) |\hat{v}(\omega)|^2d\omega}{\int_{-\omega_c}^{\omega_c}p_\alpha^2(\omega)|\hat{v}(\omega)|^2 d\omega}, p_{\alpha + 1}(\omega) = (\omega - A_\alpha) p_\alpha(\omega) - B_{\alpha - 1} p_{\alpha - 1}(\omega), B_\alpha = \frac{\int_{-\omega_c}^{\omega_c}p^2_{\alpha - 1}(\omega) |\hat{v}(\omega)|^2 d\omega}{\int_{-\omega_c}^{\omega_c}p^2_\alpha(\omega)|\hat{v}(\omega)|^2 d\omega}.
\end{align}
\item[(b)] The coefficients $\{c_\beta(t) \in \mathbb{C}\}_{\beta \in \{1, 2 \dots M\}}$ are given by the dynamical law: $c_\beta(0) = \langle \varphi_\beta, f\rangle$ for $\beta \in \{1, 2 \dots M\}$, together with
\begin{align}
i\frac{d}{dt} \begin{bmatrix}
c_1(t) \\
c_2(t) \\
c_3(t) \\
\vdots \\
c_M(t)
\end{bmatrix} =
\begin{bmatrix}
\omega_1 & t_1 & 0 & 0 &\dots & 0 \\
t_1 & \omega_2 & t_2 & 0 &\dots & 0 \\
0 & t_2 & \omega_3 & t_3 & \dots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \dots & \omega_M
\end{bmatrix}
\begin{bmatrix}
c_1(t) \\
c_2(t) \\
c_3(t) \\
\vdots \\
c_M(t)
\end{bmatrix}
\end{align}
where $\omega_\alpha = A_\alpha \ \forall \alpha \in \{1, 2 \dots M\}$ and $t_\alpha = \sqrt{B_{\alpha + 1}} \ \forall \ \alpha \in \{1, 2 \dots N_m\}$,
\end{enumerate}
\end{definition}
For completeness, we provide a simple and well-known upper bound on the coefficients $\{\omega_\alpha\}_{\alpha \in \{1,2 \dots N_m\}}$ and $\{t_\alpha\}_{\alpha\in \{1, 2 \dots N_m - 1\}}$ which will be useful in the following sections.
\begin{lemma}[Upper bound on $\omega_\alpha, t_\alpha$ (Ref.~\cite{chin2010exact})]
\label{lemma:upper_bound_chain_coeffs}
Given a chain unitary group with $N_m$ modes generated by $v \in L^2(\mathbb{R})$ with $\textnormal{supp}(\hat{v}) \subseteq [-\omega_c, \omega_c]$ for $\omega_c \geq 0$, then
\[
\abs{\omega_\alpha} \leq \omega_c \ \text{and } t_\alpha \leq \omega_c \ \forall \ \alpha \in \{1, 2 \dots N_m\},
\]
where $\{\omega_\alpha \}_{\alpha \in \{1, 2 \dots N_m\}}$ and $\{t_\alpha \}_{\alpha \in \{1, 2 \dots N_m\}}$ are the parameters of the chain unitary group defined in definition \ref{def:chain_unitary}.
\end{lemma}
\begin{lemma}[Ref.~\cite{trivedi2021convergence}]
\label{lemma:truncation_chain_group}
Given $v \in L^2(\mathbb{R}) \cap L^\infty(\mathbb{R})$ with $\textnormal{supp}(f) \in [-\omega_c, \omega_c]$, let $\upnu_t : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be the chain unitary group with $N_m$ modes generated by $v$ (definition \ref{def:chain_unitary}), then $\forall t \geq 0$,
\[
\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 \leq \norm{v}_{L^2}^2 N_m^2 e^{N_m} \bigg(\frac{2\omega_c t}{N_m}\bigg)^{N_m},
\]
where $\uptau_t : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is the translation group $[(\uptau_t f)(x) = f(x + t) \ \forall \ f \in L^2(\mathbb{R})]$.
\end{lemma}
\noindent\emph{Proof}: We define the polynomial $\pi_\alpha$ of degree $\alpha - 1$ via
\[
\pi_\alpha = \frac{\norm{v}_{L^2}}{\norm{p_i \hat{v}}_{L^2}} p_\alpha \ \text{for } \alpha \in \{1, 2 \dots N_m\}.
\]
We will denote by $A \in \mathbb{R}^{N_m \times N_m}$ the matrix
\[
A = \begin{bmatrix}
\omega_1 & t_1 & 0 & 0 &\dots & 0 \\
t_1 & \omega_2 & t_2 & 0 &\dots & 0 \\
0 & t_2 & \omega_3 & t_3 & \dots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \dots & \omega_{N_m}
\end{bmatrix},
\]
We will denote by $\lambda_i \in \mathbb{R}$ and $u^i\in \mathbb{R}^{N_m}, i \in \{1, 2 \dots N_m\}$ the eigenvalues and eigenvectors of the matrix $A$. We note that if $(\lambda \in \mathbb{R}, u \in \mathbb{R}^{N_m})$ is an eigenvalue, eigenvector pair of $A$, then
\begin{align*}
&\big(\omega_1 - \lambda\big) u_1 + t_1 u_{2} = 0, \\
&\big(\omega_i - \lambda\big) u_i + t_{i - 1} u_{i - 1} + t_i u_{i + 1} = 0 \ \text{for }i\in \{2, 3 \dots N_m - 1\}, \\
&\big(\omega_{N_m} - \lambda \big)u_{N_m} + t_{N_m - 1} u_{N_m - 1} = 0.
\end{align*}
Solving these recursions, we obtain that
\[
u_i = u_1 \pi_i(\lambda) \ \text{and } \pi_{N_m + 1}(\Omega) = 0 \ (\text{or } p_{N_m + 1}(\Omega) = 0).
\]
Therefore, the eigenvalues $\lambda_1, \lambda_2 \dots \lambda_{N_m}$ are the roots of the polynomial $p_{N_m + 1}$, and the eigenvectors are given by
\[
u^i_j = \frac{\pi_j(\lambda_i)}{N_i} \ \text{where } N_i = \bigg(\sum_{j = 1}^{N_m}\pi_j^2(\lambda_i) \bigg)^{1/2}
\]
It can be noted that the matrix $A$ is hermitian, and consequently, its eigenvectors for an orthonormal basis for $\mathbb{R}^{N_m}$, which implies that
\begin{align}\label{eq:orthonormality}
\sum_{j = 1}^{N_m} \pi_j(\lambda_i) \pi_j(\lambda_{i'}) = N_i^2 \delta_{i, i'} \text{ and } \sum_{i = 1}^{N_m} \pi_j(\lambda_i) \pi_{j'}(\lambda_i) = N_i^2 \delta_{j, j'}.
\end{align}
We next compute $\upnu_t v$ --- noting that $v \propto \varphi_1$, we obtain that
\[
\upnu_t v = \sum_{\beta = 1}^{N_m} c_\beta(t) \varphi_\beta, \ \text{where } \begin{bmatrix}
c_1(t) \\
c_2(t) \\
\vdots \\
c_{N_m}(t)
\end{bmatrix} = \norm{v}e^{-iAt}
\begin{bmatrix}
1\\
0 \\
\vdots \\
0
\end{bmatrix},
\]
which can be rewritten as
\[
e^{-iAt} \begin{bmatrix}
1 \\
0 \\
\vdots \\
0
\end{bmatrix} = \sum_{i = 1}^{N_m} \frac{\pi_1(\lambda_i)}{N_i}e^{-i\lambda_i t} u^i \implies c_j(t) = \norm{v}_{L^2}\sum_{i = 1}^{N_m} \frac{\pi_1(\lambda_i) \pi_j(\lambda_i)}{N_i^2}e^{-i\lambda_i t}.
\]
We now consider
\begin{align*}
&\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 = \norm{v}_{L^2}^2 - \sum_{j = 1}^{N_m} \text{Re}\bigg(c_j^*(t) \int_{\mathbb{R}} \hat{\varphi}_j^*(\omega) \hat{v}(\omega) e^{-i\omega t} d\omega \bigg) \nonumber\\
&\qquad= \norm{v}_{L^2}^2 - \sum_{i, j = 1}^{N_m} \frac{\pi_1(\lambda_i)\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 \cos((\omega - \lambda_i)t) d\omega.
\end{align*}
We next use the Gauss quadrature theorem --- we note that the polynomials $p_1, p_2 \dots p_{N_m}$ are the polynomials that would be used to approximate the integral of $f(\omega) |\hat{v}(\omega)|^2$ in the interval $[-\omega_c, \omega_c]$ with Gaussian quadrature with $N_m$ interpolating points. In particular, for every $N_m \in \mathbb{Z}_{> 1}$, $\exists w \in [0, \infty)^{N_m}$ with $\norm{w}_{1} = 1$ such that for all polynomials $q$ of degree $\leq 2 N_m - 1$,
\[
\frac{1}{ \norm{v}_{L_2}^2}\int_{-\omega_c}^{\omega_c} q(\omega) |\hat{v}(\omega)|^2d\omega = \sum_{i = 1}^{N_m} w_i q(\lambda_i).
\]
Note that from the Taylor's remainder theorem, it follows that,
\[
\forall \omega \in [-\omega_c, \omega_c], \ \cos(\omega t) = q_{N_m}(\omega) + r_{N_m}(\omega),
\]
where $q_{N_m}$ is a polynomial of degree $N_m$ with $q_{N_m}(0) = 1$, and
\begin{align}\label{eq:estimate_inf_norm_remainder}
\sup_{\omega \in [-2\omega_c, 2\omega_c]} |r_{N_m}(\omega)| \leq (2\omega_c t)^{N_m + 1} / (N_m + 1)!.
\end{align}
We thus obtain that
\begin{align*}
&\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 =\nonumber\\
&\norm{v}_{L^2}^2 - \sum_{i, j = 1}^{N_m} \frac{\pi_1(\lambda_i)\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 q_{N_m}(\omega - \lambda_i) d\omega - \sum_{i, j = 1}^{N_m}\frac{\pi_1(\lambda_i)\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 r_{N_m}(\omega - \lambda_i) d\omega.
\end{align*}
Now, from the Gauss quadrature theorem, it follows that since for $j \in \{1, 2 \dots N_m\}$ degree of $\pi_j(\omega) q_{N_m}(\omega - \lambda_j) \leq 2N_m - 1$
\begin{align*}
\int_{-\omega_c}^{\omega_c} \pi_j(\omega) | \hat{v}(\omega)|^2 q_{N_m}(\omega - \lambda_i) d\omega = \norm{v}_{L_2}^2\sum_{k = 1}^{N_m} w_k \pi_j(\lambda_k) q_{N_m}(\lambda_k - \lambda_i),
\end{align*}
and therefore
\begin{align*}
\sum_{i, j = 1}^{N_m} \frac{\pi_1(\lambda_i)\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 q_{N_m}(\omega - \lambda_i) d\omega = \norm{v}_{L^2}^2 \sum_{i, j, k = 1}^{N_m}w_k \frac{\pi_j(\lambda_i) \pi_j(\lambda_k)}{N_i^2} q_{N_m}(\lambda_k - \lambda_i).
\end{align*}
where we have used that $\pi_1(\omega) = 1 \ \forall \omega\in\mathbb{R}$. Furthermore, using Eq.~\ref{eq:orthonormality}, we obtain that
\[
\sum_{i, j = 1}^{N_m} \frac{\pi_1(\lambda_i)\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 q_{N_m}(\omega - \lambda_i) d\omega = \norm{v}_{L^2}^2 \norm{w}_{1} = \norm{v}_{L_2}^2,
\]
and therefore
\[
\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 = -\sum_{i, j = 1}^{N_m} \frac{\pi_j(\lambda_i)}{N_i^2} \int_{-\omega_c}^{\omega_c} \pi_j(\omega) |\hat{v}(\omega)|^2 r_{N_m}(\omega - \lambda_i) d\omega.
\]
Since for $i \in \{1, 2 \dots N_m\}, \lambda_i \in [-\omega_c, \omega_c]$ and therefore
\[
\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 \leq \sum_{i, j = 1}^{N_m}\bigg|\frac{\pi_j(\lambda_i)}{N_i^2} \bigg| \sup_{\omega\in[-2\omega_c, 2\omega_c]} |r_{N_m}(\omega)| \int_{-\omega_c}^{\omega_c} |\pi_j(\omega)| |\hat{v}(\omega)|^2 d\omega
\]
Note that $\forall i, j \in \{1, 2 \dots N_m\}, |\pi_j(\lambda_i) | \leq N_i$ and $N_i \geq 1$. Using this and the estimate in Eq.~\ref{eq:estimate_inf_norm_remainder}, we obtain that
\[
\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 \leq \frac{(2\omega_c t)^{N_m + 1}}{(N_m + 1)!} \sum_{i, j = 1}^{N_m} \int_{-\omega_c}^{\omega_c} |\pi_j(\omega)| |\hat{v}(\omega)|^2 d\omega \leq \frac{(2\omega_c t)^{N_m + 1}}{(N_m + 1)!} \sum_{i, j = 1}^{N_m} \norm{\hat{v} \pi_j}_{L^2} \norm{v}_{L^2} = \frac{(2\omega_c t)^{N_m + 1}}{(N_m + 1)!} N_m^2 \norm{v}_{L^2}^2.
\]
Finally, using Stirling's approximation to estimate $(N_m + 1)! \geq (N_m + 1)^{N_m + 1} e^{-N_m} \geq N_m^{N_m + 1} e^{-N_m}$, we obtain that
\[
\frac{1}{2}\norm{\uptau_t v - \upnu_t v}_{L^2}^2 \leq \norm{v}_{L^2}^2 N_m^2 \bigg(\frac{2e\omega_c t}{N_m}\bigg)^{N_m},
\]
which proves the lemma statement.
\(\square\)
\begin{replemma}{lemma:star_to_chain}[Star-to-chain transformation]
Consider a non-Markovian model with system Hamiltonian $H_S(t)$, square-integrable coupling functions $\{v_\alpha \in \text{L}^2(\mathbb{R})\}_{\alpha \in \{1, 2 \dots M\}}$ and jump operators $\{L_\alpha\}_{\alpha \in \{1, 2 \dots M\}}$. Furthermore, assume that $\exists\ \omega_c > 0$ such that for $\abs{\omega} \geq \omega_c, v_\alpha(\omega) = 0$. Then there exists a chain dilation of this non-Markovian model with $N_m$ modes and bandwidth $\leq \omega_c$ such that
\begin{align*}
&\norm{\ket{\Psi(t)} - \ket{\hat{\Psi}(t)}} \leq 4\ell t \bigg(1 + \ell^2 t^2 + \mu^{(1)}_{\ket{\Psi_0}}\bigg)^{1/2} N_m \bigg(\frac{2e\omega_c t}{N_m}\bigg)^{N_m/2},
\end{align*}
where $\ket{\Psi(t)}$ and $\ket{\hat{\Psi}(t)}$ are the system-environment states obtained from the model and its chain dilation, $\ket{\Psi(0)} = \ket{\hat{\Psi(0)}} = \ket{\Psi_0}$, $\ell = \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \norm{L_\alpha}$ and $\mu^{(1)}_{\ket{\Psi_0}}$ is the initial expectation value of the particle number operator of the environment.
\end{replemma}
\noindent\emph{Proof}: Let $U(t, s)$ and $\hat{U}(t, s)$ be the propagators corresponding to the exact model and its chain dilation respectively --- we note that both $U(t, s)\ket{\Psi}$ and $\hat{U}(t, s)\ket{\Psi}$ are strongly differentiable with respect to $t$ and $s$ if $\ket{\Psi} \in \mathcal{H}_S \otimes \textnormal{F}_1^M[L^2(\mathbb{R})]$. Consider now,
\[
\norm{\ket{\Psi(t)} - \ket{\hat{\Psi}(t)}} = \norm{\ket{\Psi_0} - \hat{U}(0, t) U(t, 0) \ket{\Psi_0}}.
\]
Now,
\begin{align*}
&\frac{d}{dt}\bigg(\hat{U}(0, t) U(t, 0) \ket{\Psi_0}\bigg) = i\sum_{\alpha = 1}^M \hat{U}(0, t) \big(L_\alpha a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^+ + L_\alpha^\dagger a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^- \big) \ket{\Psi(t)},
\end{align*}
where $\uptau_{\alpha, t} : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ and $\upnu_{\alpha, t} : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ are the time-translation and chain-unitary groups on the $\alpha^\text{th}$ bath respectively. We can thus obtain the estimate,
\[
\norm{\frac{d}{dt} \bigg(\hat{U}(0, t) U(t, 0) \ket{\Psi_0}\bigg)} \leq \sum_{\alpha = 1}^M \norm{L_\alpha} \bigg(\norm{a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^+ \ket{\Psi(t)}} + \norm{a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^- \ket{\Psi(t)}}\bigg).
\]
Moreover,
\begin{align*}
&\norm{a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^+ \ket{\Psi(t)}}^2 \leq \norm{\upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}_{L^2}^2 \sum_{n = 0}^\infty (n + 1)\norm{\Pi_n \ket{\Psi_\uptau(t)}}^2 = \norm{\upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}_{L^2}^2 \big(1 + \mu_{\ket{\Psi(t)}}^{(1)}\big) \ \text{and}\\
&\norm{a_{\alpha, \upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}^- \ket{\Psi(t)}}^2 \leq \norm{\upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}_{L^2}^2 \sum_{n = 0}^\infty n\norm{\Pi_n \ket{\Psi(t)}}^2 = \norm{\upnu_{\alpha, t} v_\alpha - \uptau_{\alpha, t} v_\alpha}_{L^2}^2 \mu_{\ket{\Psi(t)}}^{(1)}
\end{align*}
Finally, using lemma \ref{lemma:particle_num_bound}, we obtain that
\[
\mu^{(1)}_{\ket{\Psi(t)}} \leq 2{\mu^{(1)}_{\ket{\Psi_0}}} +2t^2 \bigg( \sum_{\alpha = 1}^M \norm{L_\alpha} \norm{v_\alpha}_{L^2} \bigg)^2.
\]
From these estimates, we thus obtain
\[
\norm{\ket{\Psi(t)} - \ket{\hat{\Psi}(t)}} \leq 2t \bigg(1 + 2{\mu_{\ket{\Psi_0}}^{(1)}} + 2t^2 \bigg( \sum_{\alpha = 1}^M \norm{L_\alpha} \norm{v_\alpha}_{L^2} \bigg)^2\bigg)^{1/2} \sum_{\alpha = 1}^M \norm{L_\alpha} \sup_{s \in [0,t]}\norm{\upnu_{\alpha, s} v_\alpha - \uptau_{\alpha, s} v_\alpha}_{L^2}
\]
The lemma now follows using this estimate together with lemma~\ref{lemma:truncation_chain_group}. $
\square$
\section{Proof of lemma \ref{lemma:chain_model_bqp} }\label{app:hspace_trunc}
\begin{replemma}{lemma:chain_model_bqp}Problem \ref{prob:chain_model} can be solved on a quantum computer in run time $\textnormal{poly}(n)$.
\end{replemma}
\noindent\emph{Proof}: For notational simplicity, we will denote by $a_{\alpha, j}$ for $\alpha \in \{1, 2 \dots M\}$ and $j \in \{1, 2 \dots N_m\}$ the annihilation operator corresponding to the $j^\text{th}$ chain mode of the $\alpha^\text{th}$ bath, which have the commutation relations $[a_{\alpha, j}, a_{\alpha', j'}^\dagger] = \delta_{\alpha, \alpha'} \delta_{j, j'}$. Problem \ref{prob:chain_model} is then equivalent to the simulation of the Hamiltonian defined on the Hilbert space $\mathcal{H}_S \otimes \textnormal{Fock}[\mathbb{C}^{N_m}]^{\otimes M}$
\begin{align}\label{eq:chain_hamiltonian}
H(t) = H_S(t) + \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \big( L_\alpha a_{\alpha, 1}^\dagger + L_\alpha^\dagger a_{\alpha, 1}\big) + \sum_{\alpha = 1}^M \sum_{j = 1}^{N_m} \omega_{\alpha, j}a_{\alpha, j}^\dagger a_{\alpha, j} + \sum_{\alpha = 1}^M \sum_{j = 1}^{N_m - 1} t_{\alpha, j}\bigg(a_{\alpha, j} a_{\alpha, j + 1}^\dagger + a_{\alpha, j + 1} a_{\alpha, j}^\dagger\bigg),
\end{align}
where $\{\omega_{\alpha, j}\}_{\alpha \in \{1, 2 \dots M\}, \{1, 2 \dots N_m\}}, \{t_{\alpha, j}\}_{\alpha \in \{1, 2 \dots M\}, \{1, 2 \dots N_m\}}$ are the chain parameters corresponding to the unitary group $\upnu_{\alpha, t}$. We first truncate this model into a finite-dimensional model --- to do so, we first derive a bound on the expectation number of particles in the $M$ baths coupling to the system. Denoting by $\mu_\alpha^{(1)} = \langle \sum_{j = 1}^{N_m} a_{\alpha, j}^\dagger a_{\alpha, j}\rangle$ and $\mu_\alpha^{(2)} = \langle \sum_{j = 1}^{N_m} (a_{\alpha, j}^\dagger a_{\alpha, j})^2\rangle$, we obtain from Heisenberg's equations of motion that
\begin{align*}
&\frac{d}{dt} \mu_\alpha^{(1)} = -i\norm{v_\alpha}_{L^2}\bigg( \langle L_\alpha a_{\alpha, 1}^\dagger \rangle - \textnormal{c.c.}\bigg) \leq 2\norm{v_\alpha}_{L^2} \norm{L_\alpha} \sqrt{\langle a_{\alpha,1}^\dagger a_{\alpha, 1}\rangle} \leq 2\norm{v_\alpha}_{L^2} \norm{L_\alpha} \sqrt{\mu_\alpha^{(1)}}, \nonumber\\
&\frac{d}{dt}\mu_\alpha^{(2)} = -i\norm{v_\alpha}_{L^2}\bigg(2\langle L_\alpha \big(a_{\alpha, 1}^\dagger\big)^2 a_{\alpha, 1}\rangle + \langle L_\alpha a_{\alpha, 1}^\dagger \rangle - \textnormal{c.c.}\bigg) \leq 2\norm{v_\alpha}_{L^2}\norm{L_\alpha} \sqrt{\mu_\alpha^{(1)}}\bigg( 2\sqrt{\mu_\alpha^{(2)}} + 1\bigg)
\end{align*}
integrating which yields
\[
\mu_\alpha^{(1)}(t) \leq \bigg(\sqrt{\mu_\alpha^{(1)}(0)} + \norm{v_\alpha}_{L^2} \norm{L_\alpha} t\bigg)^2,
\]
and
\[
\sqrt{\mu_\alpha^{(2)}(t)} - \frac{1}{2}\log\big(1 + 2\sqrt{\mu_\alpha^{(2)}(t)}\big) \leq\sqrt{\mu_\alpha^{(2)}(0)} - \frac{1}{2}\log\big(1 + 2\sqrt{\mu_\alpha^{(2)}(0)}\big) + 2\norm{v_\alpha}_{L^2}\norm{L_\alpha}\bigg(\sqrt{\mu_\alpha^{(1)}(0)} t + \frac{1}{2}\norm{v_\alpha}_{L^2} \norm{L_\alpha} t^2\bigg),
\]
or equivalently
\[
\sqrt{\mu_\alpha^{(2)}(t)} - \bigg(\frac{\mu_\alpha^{(2)}(t)}{4}\bigg)^{1/4} \leq \sqrt{\mu_\alpha^{(2)}(0)}+ 2\norm{v_\alpha}_{L^2}\norm{L_\alpha}\bigg(\sqrt{\mu_\alpha^{(1)}(0)} t + \frac{1}{2}\norm{v_\alpha}_{L^2} \norm{L_\alpha} t^2\bigg),
\]
where we have used that for $x \geq 0$, $0\leq \log(1 + x)\leq \sqrt{x}$. We thus obtain that if $t = \textnormal{poly}(n)$, $\mu_\alpha^{(2)}(t) = \textnormal{poly}(n)$. With this bound, we consider truncating the Hamiltonian --- given $p \in \mathbb{Z}_{>1}$, we consider the projector $\mathcal{P}_p = \textnormal{id}\otimes \Pi_{\leq p}^{\otimes M}$, where $\Pi_{\leq p}$ is a projector onto the space with less than or equal to $p$ particles defined on $\textnormal{Fock}[\mathbb{C}^{N_m}]$. Also, we define $\mathcal{Q}_p = \textnormal{id} - \mathcal{P}_p$. Denoting by $\ket{\Psi(t)}$ the state corresponding to the Hamiltonian under consideration (Eq.~\ref{eq:chain_hamiltonian}) at time $t$ and by $U_{\mathcal{P}}(t, 0)$ the propagator corresponding to the Hamiltonian $\mathcal{P} H(t) \mathcal{P}$, then
\[
\norm{ \ket{\Psi(t)} - U_\mathcal{P}(t, 0) \mathcal{P}\ket{\Psi_0}} \leq \norm{\mathcal{Q}_p\ket{\Psi(t)}} + \int_0^t \norm{\mathcal{P}_p H(s) \mathcal{Q}_p \ket{\Psi(s)}}ds.
\]
Both the terms in the above estimate can be easily bounded from above in terms of $p$ --- note that
\begin{align}\label{eq:projector_q_estimate}
\norm{\mathcal{Q}_p\ket{\Psi(t)}}^2 \leq \sum_{\alpha = 1}^M \bra{\Psi(t)} \textnormal{id}\otimes\big(\textnormal{id}^{\otimes (\alpha - 1)} \otimes \Pi_{> p} \otimes \textnormal{id}^{M- \alpha}\big)\ket{\Psi(t)} \leq \frac{1}{p}\sum_{\alpha = 1}^M \mu_\alpha^{(1)}(t) = \frac{1}{p}\textnormal{poly}(n),
\end{align}
where we have used $M = \textnormal{poly}(n)$. Furthermore, noting that $\mathcal{P}_p, \mathcal{Q}_p$ commute with any system operators, and for $\alpha \in \{1, 2 \dots M\}, i, j \in \{1, 2 \dots N_m\}$, $\mathcal{P}_p a_{\alpha, i}^\dagger a_{\alpha, j} \mathcal{Q}_p = 0$ and $\mathcal{P}_p a_{\alpha, i}^\dagger \mathcal{Q}_{p} = 0$, we obtain that for $s \in (0, t)$,
\[
\norm{\mathcal{P}_p H(s) \mathcal{Q}_p \ket{\Psi(s)}} \leq \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \norm{L_\alpha} \norm{\mathcal{P}_p a_{\alpha, 1} \mathcal{Q}_p \ket{\Psi(s)}} \leq \sum_{\alpha = 1}^M \norm{v_\alpha}_{L^2} \norm{L_\alpha} \norm{a_{\alpha, 1} \mathcal{Q}_p \ket{\Psi(s)}}.
\]
For $\alpha \in \{1, 2 \dots M\}$, we obtain that
\[
\norm{ a_{\alpha, 1} \mathcal{Q}_p \ket{\Psi(s)}}^2 = \bra{{\Psi(s)}} \mathcal{Q}_p a_{\alpha, 1}^\dagger a_{\alpha, 1}\mathcal{Q}_p \ket{\Psi(s)} = \bra{{\Psi(s)}} a_{\alpha, 1}^\dagger a_{\alpha, 1}\mathcal{Q}_p \ket{\Psi(s)} \leq \bra{\Psi(s)}\big(a_{\alpha, 1}^\dagger a_{\alpha, 1}\big)^2\ket{\Psi(s)} \bra{\Psi(s)} \mathcal{Q}_p \ket{\Psi(s)},
\]
and consequently using Eq.~\ref{eq:projector_q_estimate},
\[
\norm{ a_{\alpha, 1} \mathcal{Q}_p \ket{\Psi(s)}}^2 \leq \frac{\mu^{(2)}_\alpha(s)}{p}\sum_{\alpha' = 1}^M \mu^{(1)}_{\alpha'}(s).
\]
Therefore, $\int_0^t \norm{\mathcal{P}_p H(s) \mathcal{Q}_p(s) \ket{\Psi(s)}}ds \leq {\textnormal{poly}(n)}/{\sqrt{p}}$. Thus, we obtain the estimate
\[
\norm{\ket{\Psi(t)} - U_{\mathcal{P}}(t, 0) \mathcal{P}\ket{\Psi_0}} \leq \frac{\textnormal{poly}(n)}{\sqrt{p}}.
\]
Hence, to ensure that the error is within the truncation is below $1 / \textnormal{poly}(n)$, we need to choose $p = \textnormal{poly}(n)$.
Finally, we apply lemma \ref{lemma:hamil_simul} to prove the simulatability of the hamiltonan $\mathcal{P}_p H(t) \mathcal{P}_p$ --- we need to show that on a (suitably chosen) basis, for any choice of computational basis $\ket{b}$, $H(t)\ket{b}$ can be efficiently computed as a sparse vector. We consider the basis set of the form $\mathcal{B} = \mathcal{B}_S \times \mathcal{B}_1 \times \mathcal{B}_2 \dots \times \mathcal{B}_M$, where $\mathcal{B}_S$ is the computational basis for the $n$ qudit system, and for $\alpha \in \{1, 2 \dots M\}$, $\mathcal{B}_\alpha$ is the subset of Fock state basis for the $\alpha^\text{th}$ bath with number of particles less than $p$ i.e.
\[
\mathcal{B}_\alpha = \bigg\{\big(a_{\alpha, 1}^\dagger\big)^{n_1} \big(a_{\alpha, 2}^\dagger\big)^{n_2} \dots \big(a_{\alpha, N_m}^\dagger\big)^{n_{N_m}}\ket{\text{vac}} \bigg| n_1, n_2 \dots n_{N_m} \in \mathbb{Z}_{\geq 0} \text{ with } n_1 + n_2 \dots n_{N_m}\leq p \bigg\}
\]
Consider now the Hamiltonian $\mathcal{P}_p H(t) \mathcal{P}_p$ --- it can be expressed as sum the following terms:
\begin{itemize}
\item $\mathcal{P}_p H_S(t) \mathcal{P}_p$ --- Since by assumption is expressible only as $\textnormal{poly}(n)$ operators that act on at-most $k$ qudits, it immediately follows that $\mathcal{P}_p H_S(t)\mathcal{P}_p \ket{b} = H_S(t)\ket{b}$ can be classically efficiently computed for $\ket{b} \in \mathcal{B}$.
\item $\mathcal{P}_p \omega_{\alpha, j} a_{\alpha, j}^\dagger a_{\alpha, j} \mathcal{P}_p$ for $\alpha \in \{1, 2 \dots M\}, j \in \{1, 2 \dots N_m\}$ --- this term is diagonal in the basis $\mathcal{B}$. Furthermore, since there are only $N_m M = \textnormal{poly}(n)$ such terms, $\mathcal{P}_p\sum_{\alpha = 1}^M \sum_{j = 1}^{N_m} \omega_{\alpha, j} a_{\alpha, j}^\dagger a_{\alpha, j} \mathcal{P}_p \ket{b}$ can be efficiently computed $\forall \ket{b} \in \mathcal{B}$.
\item $\mathcal{P}_p t_{\alpha, j} (a_{\alpha, j} a_{\alpha, j + 1}^\dagger + a_{\alpha, j + 1} a_{\alpha, j}^\dagger) \mathcal{P}_p$ for $\alpha \in \{1, 2 \dots M\}, j \in \{1, 2 \dots N_m - 1\}$ --- applying this term on $\ket{b} \in \mathcal{B}$ produces a vector with at-most two non-zero elements when represented on the same basis. Furthermore, since there are only $(N_m - 1)M = \text{poly}(n)$ such terms, $\mathcal{P}_p \sum_{\alpha = 1}^M \sum_{j = 1}^{N_m - 1}t_{\alpha, j} (a_{\alpha, j} a_{\alpha, j + 1}^\dagger + a_{\alpha, j + 1} a_{\alpha, j}^\dagger) \mathcal{P}_p\ket{b}$ can be efficiently computed $\forall \ket{b}\in \mathcal{B}$.
\item $\mathcal{P}_p \big(L_\alpha a_{\alpha, 1}^\dagger + L_\alpha^\dagger a_{\alpha, 1}\big) \mathcal{P}_p$ for $\alpha \in \{1, 2 \dots M\}$ --- since $L_\alpha$ only acts on at-most $k$ qudits, applying this term on $\ket{b} \in \mathcal{B}$ produces a vector with at-most $2d^k$ non-zero elements when represented on the same basis. Furthermore, since there are only $M =\text{poly}(n)$ such terms, $\mathcal{P}_p \sum_{\alpha = 1}^M \big(L_\alpha a_{\alpha, 1}^\dagger + L_\alpha^\dagger a_{\alpha, 1}\big) \mathcal{P}_p \ket{b}$ can be efficiently computed $\forall \ket{b} \in \mathcal{B}$.
\end{itemize}
It thus follows that $\mathcal{P}_p H(t) \mathcal{P}_p\ket{b}$ can be efficiently computed $\forall \ket{b} \in \mathcal{B}$. Finally, we note from lemma \ref{lemma:upper_bound_chain_coeffs} that $\abs{\omega_{\alpha, j} }, t_{\alpha, j} \leq \omega_c$ for all $\alpha \in \{1, 2 \dots M\}, j \in \{1,2 \dots N_m\}$
\[
\norm{\mathcal{P}_p H(t) \mathcal{P}_p} \leq \norm{H_S(t)} +2\sqrt{p + 1} \sum_{\alpha = 1}^M \norm{L_\alpha} \norm{v_\alpha}_{L^2} + p M N_m \omega_c + 2\omega_c (p + 1) M (N_m - 1),
\]
where we have used the estimates $\norm{\mathcal{P}_p a_{\alpha, j} \mathcal{P}_p}, \norm{\mathcal{P}_p a_{\alpha, j}^\dagger \mathcal{P}_p} \leq \sqrt{p + 1}$, $\norm{\mathcal{P}_p a_{\alpha, j}^\dagger a_{\alpha, j }\mathcal{P}_p} \leq p$. Noting that by assumption $\norm{L_\alpha} \leq 1, \norm{H_S(t)}, p, M, N_m, \omega_c, t = \text{poly}(n)$, we obtain that $\int_0^t \norm{\mathcal{P}_p H(s) \mathcal{P}_p} ds \leq O(\textnormal{poly}(n))$. Thus, from lemma \ref{lemma:hamil_simul}, we can show that there is a circuit with depth $\text{poly}(n)$ that approximates the propagator corresponding to $\mathcal{P}_p H(s) \mathcal{P}_p$ with evolution time $t$ within $1 / \text{poly}(n)$ spectral norm error.Furthermore, the initial state can be efficiently represented on the basis $\mathcal{B}$ since it is efficiently projectable (assumption \ref{assump:initial_state}b), and hence the reduced system state $\rho_S(t)$ can be efficiently simulated on this quantum circuit.
\(\square \)
\end{document} |
\begin{document}
\title[Representations of the quantum toroidal algebra]{Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type $\mathfrak {gl}_N$}
\author{K. Takemura}
\address{Research Institute for Mathematical Sciences, Kyoto University, 606 Kyoto, Japan.}
\email{[email protected]}
\thanks{K.T. is supported by JSPS Research Fellowship for Young Scientists}
\author{D. Uglov}
\address{Research Institute for Mathematical Sciences, Kyoto University, 606 Kyoto, Japan.}
\email{[email protected]}
\subjclass{17B37, 81R50}
\begin{abstract}
A representation of the quantum toroidal algebra of type $\mathfrak {sl}_N$ is constructed on every integrable irreducible highest weight module of the the quantum affine algebra of type $\mathfrak {gl}_N.$
The $q$-version of the level-rank duality giving the reciprocal decomposition of the $q$-Fock space with respect to mutually commutative actions of $\operatorname{U}^{\prime}_q(\widehat{{\mathfrak {gl}}}_N)$ of level $L$ and $\UU_q^{\prime}(\asll_L)$ of level $N$ is described.
\end{abstract}
\maketitle
\section{Introduction}
\noindent In this article we continue our study \cite{STU} of representations of the quantum toroidal algebra of type $\mathfrak {sl}_N$ on irreducible integrable highest weight modules of the quantum affine algebra of type $\mathfrak {gl}_N.$
The quantum toroidal algebra $\ddot{\UU}$ was introduced in \cite{GKV} and \cite{VV1}. The definition of $\ddot{\UU}$ is given in Section \ref{sec:tor}. This algebra is a two-parameter deformation of the enveloping algebra of the universal central extension of the double-loop Lie algebra $\mathfrak {sl}_N[x^{\pm 1}, y^{\pm 1}].$
To our knowledge, no general results on the representation theory of $\ddot{\UU}$ are available at the present.
It therefore appears to be desirable, as a preliminary step towards a development of a general theory, to obtain concrete examples of representations of $\ddot{\UU}.$
The main reason why representations of central extensions of the double-loop Lie algebra, and of their deformations such as $\ddot{\UU},$ are deemed to be a worthwhile topic to study, is that one expects applications to higher-dimensional exactly solvable field theories.
Our motivation to study such representations comes, however, from a different source.
We were led to this topic while trying to understand the meaning of the level 0 action of the quantum affine algebra $\UU_q^{\prime}(\asll_N)$ which was defined in \cite{TU}, based on the earlier work \cite{JKKMP}, on each level 1 irreducible integrable highest weight module of the algebra $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N).$
These level 0 actions appear as the $q$-analogues of the Yangian actions on level 1 irreducible integrable modules of $\widehat{{\mathfrak {sl}}}_N$ discovered in \cite{H,Schoutens}.
Let us recall here, following \cite{STU} and \cite{VV2}, the connection between the level 0 actions and the quantum toroidal algebra $\ddot{\UU}.$ It is known \cite{GKV} (see also Section \ref{sec:tor}) that $\ddot{\UU}$ contains two subalgebras $\operatorname{U}_{h},$ and $\operatorname{U}_v$ such that there are algebra homomorphisms $\UU_q^{\prime}(\asll_N) \rightarrow \operatorname{U}_h,$ and $\UU_q^{\prime}(\asll_N) \rightarrow \operatorname{U}_v.$
As a consequence, every module of $\ddot{\UU}$ admits two actions of $\UU_q^{\prime}(\asll_N):$ the {\em horizontal} action obtained through the first of the above homomorphisms, and the {\em vertical} action obtained through the second one.
It was shown in \cite{STU} and \cite{VV2}, that on each level 1 irreducible integrable highest weight module of $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N)$ there is an action of $\ddot{\UU},$ such that the horizontal action coincides with the standard level 1 action of $\UU_q^{\prime}(\asll_N) \subset \operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N),$ while the vertical action coincides with the level 0 action defined in \cite{TU}.
The aim of the present article is to extend this result to higher level irreducible integrable highest weight modules of $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N).$
The algebra $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N)$ is, by definition, the tensor product of algebras $H\otimes\UU_q(\asll_N),$ where $H$ is the Heisenberg algebra (see Section \ref{sec:bosons}).
Let $\Lambda$ be a level $L$ dominant integral weight of $\UU_q(\asll_N),$ and let $V(\Lambda)$ be the irreducible integrable $\UU_q(\asll_N)$-module of the highest weight $\Lambda.$ As the main result of this article we define an action of $\ddot{\UU}$ on the irreducible $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N)$-module
\begin{equation}
\widetilde{V}(\Lambda) = {\mathbb K\hskip.5pt}[H_-]\otimes V(\Lambda), \label{eq:tv1}
\end{equation}
where ${\mathbb K\hskip.5pt}[H_-]$ is the Fock representation (see Section \ref{sec:decomp}) of $H.$ The corresponding horizontal action of $\UU_q^{\prime}(\asll_N)$ is just the standard, level $L,$ action on the second tensor factor in (\ref{eq:tv1}).
The vertical action of $\UU_q^{\prime}(\asll_N)$ has level zero, this action is a $q$-analogue of the Yangian action constructed recently on each irreducible integrable highest weight module of $\widehat{{\mathfrak {gl}}}_N$ in \cite{U}.
Let us now describe the main elements of our construction of the $\ddot{\UU}$-action on $\widetilde{V}(\Lambda).$
To define the $\ddot{\UU}$-action we introduce a suitable realization of $\widetilde{V}(\Lambda)$ using the $q$-analogue of the classical level-rank duality, due to Frenkel \cite{F1,F2}, between the affine Lie algebras $\widehat{{\mathfrak {sl}}}_N$ and $\widehat{{\mathfrak {sl}}}_L.$
The quantized version of the level-rank duality takes place on the {\em $q$-Fock space} (we call it, simply, the Fock space hereafter). The Fock space is an integrable, level $L,$ module of the algebra $\UU_q^{\prime}(\asll_N).$
The action of this algebra on the Fock space is centralized by a level $N$ action of $\UU_q^{\prime}(\asll_L),$ and the resulting action of $\UU_q^{\prime}(\asll_N)\otimes\UU_q^{\prime}(\asll_L)$ is centralized by an action of the Heisenberg algebra $H.$
We give in the present paper a construction of the Fock space in the spirit of semi-infinite wedges of \cite{S,KMS}. The Fock space defined in \cite{KMS} appears as the special case of our construction when the level $L$ equals $1.$
In Theorem \ref{t:decofF} we describe the irreducible decomposition of the Fock space with respect to the action of $H\otimes\UU_q^{\prime}(\asll_N)\otimes\UU_q^{\prime}(\asll_L).$ This theorem is the $q$-analogue of Theorem 1.6 in \cite{F1}. The decomposition shows that for every level $L$ dominant integral weight $\Lambda$ the corresponding irreducible $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N)$-module $\widetilde{V}(\Lambda)$ is realized as a direct summand of the Fock space, such that the multiplicity space of $\widetilde{V}(\Lambda)$ is a certain level $N$ irreducible integrable highest weight module of $\UU_q^{\prime}(\asll_L).$
To define the action of the quantum toroidal algebra on $\widetilde{V}(\Lambda)$ we proceed very much along the lines of \cite{STU}. The starting point is a representation, due to Cherednik \cite{C3}, of the toroidal Hecke algebra of type $\mathfrak {gl}_n$ on the linear space ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_n^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n}.$ Here ${\mathbb K\hskip.5pt} = \mathbb Q(q^{\frac{1}{2N}}).$
Applying the Varagnolo--Vasserot duality \cite{VV1} between modules of the toroidal Hecke algebra and modules of $\ddot{\UU},$ we obtain a representation of $\ddot{\UU}$ on the $q$-wedge product $\wedge^n V_{\mathrm {aff}},$ where $V_{\mathrm {aff}} = {\mathbb K\hskip.5pt}[z^{\pm1}]\otimes {\mathbb K\hskip.5pt}^N\otimes {\mathbb K\hskip.5pt}^L.$ This $q$-wedge product (we call it, simply, {\em the wedge product} hereafter) is similar to the wedge product of \cite{KMS}, and reduces to the latter when $L=1.$
The Fock space is defined as an inductive limit ($n \rightarrow \infty$) of the wedge product $\wedge^n V_{\mathrm {aff}}.$
We show that the Fock space inherits the $\ddot{\UU}$-action from $\wedge^n V_{\mathrm {aff}}.$ As the final step we demonstrate, that the $\ddot{\UU}$-action on the Fock space can be restricted on $\widetilde{V}(\Lambda)$ provided certain parameters in the $\ddot{\UU}$-action are fixed in an appropriate way.
Let us now comment on two issues which we {\em do not} deal with in the present paper.
The first one is the question of irreducibility of $\widetilde{V}(\Lambda)$ as the $\ddot{\UU}$-module.
Based on analysis of the Yangian limit (see \cite{U}) we expect that $\widetilde{V}(\Lambda)$ is irreducible. However we lack a complete proof of this at the present.
The second issue is the decomposition of $\widetilde{V}(\Lambda)$ with respect to the level 0 vertical action of $\UU_q^{\prime}(\asll_N).$
In the Yangian limit this decomposition was performed in \cite{U} for the vacuum highest weight $\Lambda = L \Lambda_0.$
It is natural to expect, that combinatorially this decomposition will remain unchanged in the $q$-deformed situation.
In particular, the irreducible components are expected to be parameterized by semi-infinite skew Young diagrams, and the $\operatorname{U}_q(\mathfrak {sl}_N)$-characters of these components are expected to be given by the corresponding skew Schur functions. \\
\mbox{} \\
\noindent The paper is organized as follows.
In sections \ref{s:pre} through \ref{s:Fock} we deal with the $q$-analogue of the level-rank duality, and the associated realization of the integrable irreducible modules of $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N).$
Section \ref{s:pre} contains background information on the quantum affine algebras and affine Hecke algebra.
In Section \ref{s:wedgeprod} we introduce the wedge product, and describe the technically important {\em normal ordering rules} for the $q$-wedge vectors. In Section \ref{s:Fock} we define the Fock space, and, on this space, the action of $H\otimes\UU_q^{\prime}(\asll_N)\otimes\UU_q^{\prime}(\asll_L).$
The decomposition of the Fock space as $H\otimes\UU_q^{\prime}(\asll_N)\otimes\UU_q^{\prime}(\asll_L)$-module is given in Theorem \ref{t:decofF}.
In Sections \ref{s:tor} and \ref{s:toract} we deal with the quantum toroidal algebra $\ddot{\UU}$ and its actions. Section \ref{s:tor} contains basic information on the toroidal Hecke algebra and $\ddot{\UU}.$
In Section \ref{s:toract} we define actions of $\ddot{\UU}$ on the Fock space, and on irreducible integrable highest weight modules of $\operatorname{U}_q(\widehat{{\mathfrak {gl}}}_N).$ \\
\section{Preliminaries} \label{s:pre}
\subsection{Preliminaries on the quantum affine algebra}\label{sec:Usl}
For $k,m \in \mathbb Z$ we define the following $q$-integers, factorials, and binomials
$$[k]_q = \frac{q^k-q^{-k}}{q-q^{-1}},\quad [k]_q!=[k]_q[k-1]_q\cdots[1]_q,\quad \text{and} \;\begin{bmatrix}m\\k\end{bmatrix}_q = \frac{[m]_q!}{[m-k]_q![k]_q!}.$$
The quantum affine algebra $\UU_q(\asll_M)$ is the unital associative algebra over ${\mathbb K\hskip.5pt} =\mathbb Q(q)$ generated by the elements $E_i,F_i,K_i,K_i^{-1}, D$ $(0\leqslant i < M)$ subject to the relations:
\begin{gather}
K_i K_j = K_j K_i, \quad DK_i=K_iD,\quad K_i K_i^{-1} = K_i^{-1} K_i = 1, \label{eq:r1}\\
K_i E_j = q^{a_{ij}} E_j K_i, \label{eq:r2}\\
K_i F_j = q^{-a_{ij}} F_j K_i, \label{eq:r3}\\
[D,E_i] = \delta(i=0) E_i, \quad [D,F_i] = -\delta(i=0) F_i,\label{eq:r4} \\
[E_i,F_j] = \delta_{ij} \frac{K_i - K_i^{-1}}{q - q^{-1}},\label{eq:r5}\\
\sum_{k=0}^{1-a_{ij}} (-1)^k \begin{bmatrix} 1- a_{ij} \\ k \end{bmatrix}_q E_i^{1-a_{ij}-k} E_j E_i^k = 0 \quad (i\neq j), \label{eq:r6}\\
\sum_{k=0}^{1-a_{ij}} (-1)^k \begin{bmatrix} 1- a_{ij} \\ k \end{bmatrix}_q F_i^{1-a_{ij}-k} F_j F_i^k = 0 \quad (i\neq j). \label{eq:r7}
\end{gather}
Here $a_{ij} = 2\delta(i=j) - \delta(i=j+1) - \delta(i=j-1),$ and the indices are extended to all integers modulo $M.$ For $P$ a statement, we write $\delta(P) = 1$ if $P$ is true, $\delta(P) = 0$ if otherwise. \\
$\UU_q(\asll_M)$ is a Hopf algebra, in this paper we will use two different coproducts $\Delta^+$ and $\Delta^-$ given by
\begin{alignat}{5}
&\Delta^+(K_i) = K_i\otimes K_i,\quad & &\Delta^-(K_i) = K_i\otimes K_i,& \label{eq:co1}\\
&\Delta^+(E_i) = E_i\otimes K_i + 1\otimes E_i,\quad & &\Delta^-(E_i) = E_i\otimes 1+ K_i\otimes E_i,& \label{eq:co2}\\
&\Delta^+(F_i) = F_i\otimes 1 + K_i^{-1}\otimes F_i,\quad & &\Delta^-(F_i) = F_i\otimes K_i^{-1}+ 1\otimes F_i,& \label{eq:co3} \\
&\Delta^+(D) = D\otimes 1 + 1\otimes D,\quad & &\Delta^-(D) = D\otimes 1 + 1\otimes D.& \label{eq:co4}
\end{alignat}
Denote by $\operatorname{U}_q^{\prime}(\widehat{{\mathfrak {sl}}}_M)$ the subalgebra of $\operatorname{U}_q(\widehat{{\mathfrak {sl}}}_M)$ generated by $E_i,F_i,K_i,K_i^{-1}, 0\leqslant i < M.$ \\
\noindent In our notations concerning weights of $\operatorname{U}_q(\widehat{{\mathfrak {sl}}}_M)$ we will follow \cite{Kac}. Thus we denote by $\Lambda_0,\Lambda_1,\dots,\Lambda_{M-1}$ the fundamental weights, by $\delta$ the null root, and let $\alphapha_i = 2\Lambda_i - \Lambda_{i+1} - \Lambda_{i-1} + \delta_{i,0} \delta$ $(0\leqslant i < M)$ denote the simple roots. The indices are assumed to be cyclically extended to all integers modulo $M.$ Let $P_M = \mathbb Z \delta \oplus \left(\oplus_i \mathbb Z \Lambda_i\right)$ be the set of integral weights. \\
\noindent Let ${\mathbb K\hskip.5pt}^N$ be the $N$-dimensional vector space with basis $\mathfrak v_{1},\mathfrak v_{2},\dots,\mathfrak v_N,$ and let ${\mathbb K\hskip.5pt}^L$ be the $L$-dimensional vector space with basis $\mathfrak e_{1},\mathfrak e_{2},\dots,\mathfrak e_L.$ We set $V_{\mathrm {aff}} = {\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L\otimes {\mathbb K\hskip.5pt}^N.$ $V_{\mathrm {aff}}$ has basis $\{ z^m \mathfrak e_{a} \mathfrak v_{\epsilon} \}$ where $m\in \mathbb Z$ and $1\leqslant a \leqslant L;$ $1\leqslant \epsilon \leqslant N.$ Both algebras $\UU_q(\asll_N)$ and $\UU_q(\asll_L)$ act on $V_{\mathrm {aff}}.$ $\UU_q(\asll_N)$ acts in the following way:
\begin{eqnarray}
K_{i}(z^m\mathfrak e_a\mathfrak v_{\epsilon}) &=& q^{\delta_{\epsilon,i} - \delta_{\epsilon,i+1}}z^m\mathfrak e_a\mathfrak v_{\epsilon}, \\
E_{i}(z^m\mathfrak e_a\mathfrak v_{\epsilon}) &=& \delta_{\epsilon,i+1}z^{m+\delta_{i,0}}\mathfrak e_a\mathfrak v_{\epsilon-1},\\
F_{i}(z^m\mathfrak e_a\mathfrak v_{\epsilon}) &=& \delta_{\epsilon,i}z^{m-\delta_{i,0}} \mathfrak e_a\mathfrak v_{\epsilon+1},\\
D(z^m\mathfrak e_a\mathfrak v_{\epsilon}) &=& mz^m\mathfrak e_a\mathfrak v_{\epsilon};
\end{eqnarray}
where $0 \leqslant i < N,$ and all indices but $a$ should be read modulo $N.$ \\
The action of $\UU_q(\asll_L)$ is given by
\begin{eqnarray}
\dot{K}_{a}(z^m\mathfrak e_b\mathfrak v_{\epsilon}) &=& q^{\delta_{b,L-a+1} - \delta_{b,L-a}}z^m\mathfrak e_b\mathfrak v_{\epsilon}, \label{eq:ul1}\\
\dot{E}_{a}(z^m\mathfrak e_b\mathfrak v_{\epsilon}) &=& \delta_{b,L-a}z^{m+\delta_{a,0}}\mathfrak e_{b+1}\mathfrak v_{\epsilon},\\
\dot{F}_{a}(z^m\mathfrak e_b\mathfrak v_{\epsilon}) &=& \delta_{b,L-a+1}z^{m-\delta_{a,0}}\mathfrak e_{b-1}\mathfrak v_{\epsilon}, \label{eq:ul2}\\
\dot{D}(z^m\mathfrak e_a\mathfrak v_{\epsilon}) &=& mz^m\mathfrak e_a\mathfrak v_{\epsilon}.
\end{eqnarray}
where $0 \leqslant a< N,$ and all indices but $\epsilon$ are to be read modulo $L.$ Above and in what follows we put a dot over the generators of $\UU_q(\asll_L)$ in order to distinguish them from the generators of $\UU_q(\asll_N).$ When both $\UU_q(\asll_N)$ and $\UU_q(\asll_L)$ act on the same linear space and share a vector $v$ as their weight vector, we will understand that $\mathrm {wt}(v)$ is a sum of weights of $\UU_q(\asll_N)$ and $\UU_q(\asll_L).$ Thus
$$ \mathrm {wt}(z^m\mathfrak e_a\mathfrak v_{\epsilon}) = \Lambda_{\epsilon} - \Lambda_{\epsilon-1} + \dot{\Lambda}_{L-a+1} - \dot{\Lambda}_{L-a} + m(\delta + \dot{\delta}).$$
Here, and from now on, we put dots over the fundamental weights, etc. of $\UU_q(\asll_L).$\\
\noindent Iterating the coproduct $\Delta^+$ (cf.(\ref{eq:co1}--\ref{eq:co4})) $n-1$ times we get an action of $\UU_q(\asll_N)$ on the tensor product $V_{\mathrm {aff}}^{\otimes n}.$ Likewise
for $\UU_q(\asll_L),$ but in this case we use the {\em other} coproduct $\Delta^-.$
\subsection{Preliminaries on the affine Hecke algebra} \label{sec:affH}
The affine Hecke algebra of type $\mathfrak {gl}_n,$ $\mathbf {\dot{H}}_n,$ is a unital associative algebra over ${\mathbb K\hskip.5pt}$ generated by elements $T_i^{\pm 1},X_j^{\pm 1}, 1\leqslant i < n, 1\leqslant j \leqslant n.$ These elements satisfy the following relations:
\begin{gather}
T_i T_i^{-1} = T_i^{-1} T_i = 1,\qquad (T_i + 1)(T_i - q^2) = 0, \label{eq:ah1}\\
T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}, \qquad T_i T_j = T_j T_i \quad \text{if $|i-j| > 1,$} \\
X_j X_j^{-1} = X_j^{-1} X_j = 1, \qquad X_i X_j = X_j X_i, \\
T_i X_i T_i = q^2 X_{i+1}, \qquad T_i X_j = X_j T_i \quad \text{if $ j \neq i,i+1.$}\label{eq:ah2}
\end{gather}
The subalgebra $\mathbf {H}_n \subset \mathbf {\dot{H}}_n$ generated by the elements $T_i^{\pm 1}$ alone is known to be isomorphic to the finite Hecke algebra of type $\mathfrak {gl}_n.$\\
\noindent Following \cite{GRV}, \cite{KMS} we introduce a representation of $\mathbf {\dot{H}}_n$ on the linear space $({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}.$ We will identify this space with ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_n^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n}$ by the correspondence
$$ z^{m_1}\mathfrak e_{a_1} \otimes z^{m_2}\mathfrak e_{a_2}\otimes \cdots \otimes z^{m_n}\mathfrak e_{a_n} \mapsto z_1^{m_1}z_2^{m_2}\cdots z_n^{m_n} \otimes (\mathfrak e_{a_1} \otimes \mathfrak e_{a_2}\otimes \cdots \otimes \mathfrak e_{a_n}). $$
Let $E_{a,b} \in \mathrm {End}({\mathbb K\hskip.5pt}^L)$ be the matrix units with respect to the basis $\{\mathfrak e_a\},$ and define the {\em trigonometric R-matrix} as the following operator on $({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes 2} = {\mathbb K\hskip.5pt}[z_1^{\pm 1},z_2^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes 2}:$
\begin{eqnarray*} R(z_1,z_2)& =& (q^2 z_1 - z_2)\sum_{1\leqslant a \leqslant L} E_{a,a}\otimes E_{a,a} + q(z_1 - z_2)\sum_{1\leqslant a \neq b \leqslant L} E_{a,a}\otimes E_{b,b} + \\ & + &z_1(q^2 - 1)\sum_{1\leqslant a < b \leqslant L} E_{a,b}\otimes E_{b,a} + z_2(q^2 - 1)\sum_{1\leqslant b < a \leqslant L} E_{a,b}\otimes E_{b,a}.
\end{eqnarray*}
Let $s$ be the exchange operator of factors in the tensor square $({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes 2},$ and let
\begin{equation} \Tc _{(1,2)}:= \frac{ z_1 - q^2 z_2 }{z_1 - z_2} \cdot \left( 1 - s \cdot \frac{R(z_1,z_2)}{q^2 z_1 - z_2}\right) - 1.
\label{eq:Tc}\end{equation}
The operator $\Tc_{(1,2)}$ is known as {\em the matrix Demazure-Lusztig operator} (cf. \cite{C3}), note that it is an element of $\mathrm {End}\left(({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes 2}\right)$ despite the presence of the denominators in the definition. For $1 \leqslant i < n$ we put
\begin{equation}
\Tc_i := 1^{\otimes (i-1)}\otimes \Tc_{(i,i+1)} \otimes 1^{\otimes (n-i-1)} \quad \in \mathrm {End}\left( ({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}\right).
\label{eq:Tci}\end{equation}
\begin{propos}[\cite{C3}, \cite{GRV}, \cite{KMS}]
The map
\begin{equation}
X_j \mapsto z_j, \qquad T_i \mapsto \Tc_i
\end{equation}
where $z_j$ stands for the multiplication by $z_j,$ extends to a right representation of $\mathbf {\dot{H}}_n$ on $({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}.$
\end{propos}
\noindent Following \cite{J} we define a left action of the finite Hecke algebra $\mathbf {H}_n$ on $({\mathbb K\hskip.5pt}^N)^{\otimes n}$ by
\begin{align}
T_i \mapsto \Ts_i := 1^{\otimes (i-1)}\otimes \Ts \otimes 1^{\otimes (n-i-1)}, \quad \text{where $ \Ts \in \mathrm {End}\left(({\mathbb K\hskip.5pt}^N)^{\otimes 2} \right),$} \label{eq:Tsi}\\
\text{and} \quad \Ts(\mathfrak v_{\epsilon_1}\otimes \mathfrak v_{\epsilon_2}) = \begin{cases} q^2 \mathfrak v_{\epsilon_1}\otimes \mathfrak v_{\epsilon_2} & \text{ if $\epsilon_1 = \epsilon_2,$} \\
q \mathfrak v_{\epsilon_2}\otimes \mathfrak v_{\epsilon_1} & \text{ if $\epsilon_1 < \epsilon_2,$} \\
q \mathfrak v_{\epsilon_2}\otimes \mathfrak v_{\epsilon_1} + (q^2 - 1) \mathfrak v_{\epsilon_1}\otimes \mathfrak v_{\epsilon_2} & \text{ if $\epsilon_1 > \epsilon_2.$} \end{cases} \label{eq:Ts}
\end{align}
\section{The Wedge Product} \label{s:wedgeprod}
\subsection{Definition of the wedge product} \label{sec:wedge}
Identify the tensor product $V_{\mathrm {aff}}^{\otimes n}$ with \\$({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ by the natural isomorphism
$$ z^{m_1}\mathfrak e_{a_1}\mathfrak v_{\epsilon_1} \otimes \cdots \otimes z^{m_n}\mathfrak e_{a_n}\mathfrak v_{\epsilon_n} \mapsto \left(z^{m_1}\mathfrak e_{a_1} \otimes \cdots \otimes z^{m_n}\mathfrak e_{a_n}\right)\otimes\left(\mathfrak v_{\epsilon_1} \otimes \cdots \otimes\mathfrak v_{\epsilon_n}\right).
$$
Then the operators $\Tc_i$ and $\Ts_i$ are extended on $V_{\mathrm {aff}}^{\otimes n}$ as $\Tc_i\otimes 1$ and $1 \otimes \Ts_i$ respectively. In what follows we will keep the same symbol $\Tc_i$ to mean $\Tc_i\otimes 1,$ and likewise for $\Ts_i.$ We define the $n$-fold {\em $q$-wedge product } (or, simply, {\em the wedge product}) $\wedge^n V_{\mathrm {aff}}$ as the following quotient space:
\begin{equation}
\wedge^n V_{\mathrm {aff}} := V_{\mathrm {aff}}^{\otimes n} / \sum_{i=1}^{n-1} \mathrm {Im} (\Tc_i - \Ts_i).
\label{eq:qwp} \end{equation}
Note that under the specialization $q=1$ the operator $\Tc$ (\ref{eq:Tc}) tends to {\em minus} the permutation operator of the tensor square $(\mathbb Q[z^{\pm 1}]\otimes \mathbb Q^L)^{\otimes 2},$ while the operator $\Ts$ (\ref{eq:Ts}) tends to {\em plus} the permutation operator of the tensor square $(\mathbb Q^N)^{\otimes 2},$
so that (\ref{eq:qwp}) is a $q$-analogue of the standard exterior product.
\noindent {\bf Remark.} The wedge product is the dual, in the sense of Chari--Pressley \cite{CP} (see also \cite{C1}) of the $\mathbf {\dot{H}}_n$-module $({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}:$ there is an evident isomorphism of linear spaces
$$ \wedge^nV_{\mathrm {aff}} \cong ({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}\otimes_{\mathbf {H}_n}({\mathbb K\hskip.5pt}^N)^{\otimes n}. $$ \\
\noindent For $ m \in \mathbb Z_{\neq 0}$ define $B^{(n)}_m \in \mathrm {End}(V_{\mathrm {aff}}^{\otimes n})$ as
\begin{equation}
B^{(n)}_m = z_1^m + z_2^m + \cdots + z_n^m.
\end{equation}
In Section \ref{sec:Usl} mutually commutative actions of the quantum affine algebras $\UU_q(\asll_N)$ and $\UU_q(\asll_L)$ were defined on $V_{\mathrm {aff}}^{\otimes n}.$ The operators $B^{(n)}_m$ obviously commute with these actions.\\
\noindent The following proposition is easily deduced from the results of \cite{CP}, \cite{GRV}, \cite{KMS}.
\begin{propos} \label{p:CPD}
For each $i=1,\dots,n-1$ the subspace $\mathrm {Im}(\Tc_i - \Ts_i)\subset V_{\mathrm {aff}}^{\otimes n}$ is invariant with respect to $\UU_q(\asll_N),\UU_q(\asll_L)$ and $B^{(n)}_m$ $(m \in \mathbb Z_{\neq 0}).$ Therefore actions of $\UU_q(\asll_N),\UU_q(\asll_L)$ and $B^{(n)}_m$ are defined on the wedge product $\wedge^n V_{\mathrm {aff}}.$
\end{propos}
\noindent It is clear that the actions of $\UU_q^{\prime}(\asll_N) \subset \UU_q(\asll_N),$ $\UU_q^{\prime}(\asll_L)\subset\UU_q(\asll_L)$ and $B^{(n)}_m$ $(m\in \mathbb Z_{\neq 0})$ on the wedge product are mutually commutative.
\subsection{Wedges and normally ordered wedges} \label{sec:nowedges}
In the following discussion it will be convenient to relabel elements of the basis $\{z^m\mathfrak e_a\mathfrak v_{\epsilon}\}$ of $V_{\mathrm {aff}}$ by single integer. We put $k = \epsilon-N(a+Lm)$ and denote $u_k = z^m\mathfrak e_a\mathfrak v_{\epsilon}.$ Then the set $\{u_k \:|\: k \in \mathbb Z\}$ is a basis of $V_{\mathrm {aff}}.$ Let
\begin{equation} u_{k_1}\wedge u_{k_2} \wedge \cdots \wedge u_{k_n} \label{eq:wedge}
\end{equation}
be the image of the tensor $u_{k_1}\otimes u_{k_2} \otimes \cdots \otimes u_{k_n}$ under the quotient map from $V_{\mathrm {aff}}^{\otimes n}$ to $\wedge^nV_{\mathrm {aff}}.$ We will call a vector of the form (\ref{eq:wedge}) {\em a wedge} and will say that a wedge is {\em normally ordered} if $k_1>k_2>\dots>k_n.$ When $q$ is specialized to $1,$ a wedge is antisymmetric with respect to a permutation of any pair of indices $k_i,k_j,$ and the normally ordered wedges form a basis of $\wedge^nV_{\mathrm {aff}}.$ In the general situation -- when $q$ is a parameter -- the normally ordered wedges still form a basis of $\wedge^nV_{\mathrm {aff}}.$ However the antisymmetry is replaced by a more complicated {\em normal ordering rule} which allows to express any wedge as a linear combination of normally ordered wedges.
\mbox{}
\noindent Let us start with the case of the two-fold wedge product $\wedge^2V_{\mathrm {aff}}.$ The explicit expressions for the operators $\Tc_1$ and $\Ts_1$ lead for all $k \leqslant l$ to the normal ordering rule of the form
\begin{equation}
u_k\wedge u_l = c_{kl}(q) u_l\wedge u_k + (q^2-1)\sum_{i\geqslant 1, l-i > k+i} c_{kl}^{(i)}(q) u_{l-i}\wedge u_{k+i}, \label{eq:norule1}
\end{equation}
were $ c_{kl}(q), c_{kl}^{(i)}(q)$ are Laurent polynomials in $q.$ In particular $c_{kk}(q)=-1,$ and thus $u_k\wedge u_k =0.$
To describe all the coefficients in (\ref{eq:norule1}), we will employ a vector notation. For all $a,a_1,a_2 =1,\dots,L;$ $\epsilon,\epsilon_1,\epsilon_2 $ $=$ $1,\dots,N;$ $m_1,m_2 \in \mathbb Z$ define the following column vectors:
\begin{eqnarray}
X_{a,a}^{\epsilon_1,\epsilon_2}(m_1,m_2) &=&\left(\begin{array}{c} u_{\epsilon_1-N(a+L m_1)}\wedge u_{\epsilon_2-N(a+L m_2)} \\ u_{\epsilon_2-N(a+L m_1)}\wedge u_{\epsilon_1-N(a+L m_2)}\end{array}\right),\\
Y_{a_1,a_2}^{\epsilon,\epsilon}(m_1,m_2) &=& \left(\begin{array}{c} u_{\epsilon-N(a_1+L m_1)}\wedge u_{\epsilon-N(a_2+L m_2)} \\ u_{\epsilon-N(a_2+L m_1)}\wedge u_{\epsilon-N(a_1+L m_2)}\end{array}\right), \\
Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_1,m_2) &=& \left(\begin{array}{c}
u_{\epsilon_1-N(a_1+L m_1)}\wedge u_{\epsilon_2-N(a_2+L m_2)} \\
u_{\epsilon_1-N(a_2+L m_1)}\wedge u_{\epsilon_2-N(a_1+L m_2)} \\
u_{\epsilon_2-N(a_1+L m_1)}\wedge u_{\epsilon_1-N(a_2+L m_2)} \\
u_{\epsilon_2-N(a_2+L m_1)}\wedge u_{\epsilon_1-N(a_1+L m_2)} \end{array}\right).
\end{eqnarray}
Moreover let
\begin{eqnarray}
& X_{a,a}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime}=X_{a,a}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime\prime} = X_{a,a}^{\epsilon_1,\epsilon_2}(m_1,m_2) \quad &\text{if $ m_1\neq m_2,$} \\
& Y_{a_1,a_2}^{\epsilon,\epsilon}(m_1,m_2)^{\prime}=Y_{a_1,a_2}^{\epsilon,\epsilon}(m_1,m_2)^{\prime\prime} = Y_{a_1,a_2}^{\epsilon,\epsilon}(m_1,m_2) \quad &\text{if $ m_1\neq m_2,$} \\
& Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime}=Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime\prime} = Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_1,m_2) \quad &\text{if $ m_1\neq m_2.$}
\end{eqnarray}
And
\begin{eqnarray}
X_{a,a}^{\epsilon_1,\epsilon_2}(m,m)^{\prime} &=&\left(\begin{array}{c}
0 \\
u_{\epsilon_2-N(a+L m)}\wedge u_{\epsilon_1-N(a+L m)}\end{array}\right),\\
Y_{a_1,a_2}^{\epsilon,\epsilon}(m,m)^{\prime} &=& \left(\begin{array}{c}
u_{\epsilon-N(a_1+L m)}\wedge u_{\epsilon-N(a_2+L m)} \\
0 \end{array}\right), \\
Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m,m)^{\prime} &=& \left(\begin{array}{c}
u_{\epsilon_1-N(a_1+L m)}\wedge u_{\epsilon_2-N(a_2+L m)} \\
0 \\
u_{\epsilon_2-N(a_1+L m)}\wedge u_{\epsilon_1-N(a_2+L m)} \\
0
\end{array}\right).
\end{eqnarray}
\begin{eqnarray}
X_{a,a}^{\epsilon_1,\epsilon_2}(m,m)^{\prime\prime} &=&\left(\begin{array}{c}
u_{\epsilon_1-N(a+L m)}\wedge u_{\epsilon_2-N(a+L m)} \\
0
\end{array}\right),\\
Y_{a_1,a_2}^{\epsilon,\epsilon}(m,m)^{\prime\prime} &=& \left(\begin{array}{c}
0 \\
u_{\epsilon-N(a_2+L m)}\wedge u_{\epsilon-N(a_1+L m)}\end{array}\right), \\
Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m,m)^{\prime\prime} &=& \left(\begin{array}{c}
0 \\
u_{\epsilon_1-N(a_2+L m)}\wedge u_{\epsilon_2-N(a_1+L m)} \\
0 \\
u_{\epsilon_2-N(a_2+L m)}\wedge u_{\epsilon_1-N(a_1+L m)} \end{array}\right).
\end{eqnarray}
For $t \in \mathbb Z$ introduce also the matrices:
\begin{alignat}{5}
& M_{X} = \left(\begin{array}{c c} 0 & -q \\ -q & q^2 - 1 \end{array}\right),\quad & &M_{X}(t) = (q^2-1)\left(\begin{array}{c c} q^{2t-2} & -q^{2t-1} \\ -q^{2t-1} & q^{2t} \end{array}\right), & & \\
& M_{Y} = \left(\begin{array}{c c} q^{-2}-1 & -q^{-1} \\ -q^{-1} & 0 \end{array}\right),\quad & & M_{Y}(t) = (q^{-2}-1)\left(\begin{array}{c c} q^{-2t} & -q^{-2t+1} \\ -q^{-2t+1} & q^{-2t+2} \end{array}\right).& &
\end{alignat}
\begin{gather}
M_{Z} = \left(\begin{array}{c c c c}
0 & 0 & -(q-q^{-1}) & -1 \\
0 & 0 & -1 & 0 \\
-(q-q^{-1}) & -1 & (q-q^{-1})^2 & (q-q^{-1}) \\
-1 & 0 & (q-q^{-1}) & 0
\end{array}\right), \\
M_{Z}(t) = \qquad \frac{q^2-1}{q^2+1} \times \\
\left(\begin{array}{c c c c}
q^{2t}-q^{-2t} & q^{2t-1}+q^{-2t+1} & -(q^{2t+1}+q^{-2t-1}) & -(q^{2t}-q^{-2t})\\
q^{2t-1}+q^{-2t+1} & q^{2t-2}-q^{-2t+2} & -(q^{2t}-q^{-2t}) & -(q^{2t-1}+q^{-2t+1}) \\
-(q^{2t+1}+q^{-2t-1}) & -(q^{2t}-q^{-2t}) & q^{2t+2}-q^{-2t-2} & q^{2t+1}+q^{-2t-1} \\
-(q^{2t}-q^{-2t}) & -(q^{2t-1}+q^{-2t+1}) & q^{2t+1}+q^{-2t-1}& q^{2t}-q^{-2t} \end{array}\right). \nonumber
\end{gather}
Note that all entries of the matrix $M_{Z}(t)$ are Laurent polynomials in $q,$ i.e. the numerators are divisible by ${q^2+1}.$
\mbox{}
\noindent Computing $\mathrm {Im}(\Tc - \Ts)$ we get the following lemma:
\begin{lemma}[Normal ordering rules] \label{l:norules}\mbox{}
In $\wedge^2 V_{\mathrm {aff}} $ there are the following relations:
\begin{eqnarray}
& & u_{\epsilon-N(a+Lm_1)}\wedge u_{\epsilon-N(a+Lm_2)} = - u_{\epsilon-N(a+Lm_2)}\wedge u_{\epsilon-N(a+Lm_1)} \quad (m_1 \geqslant m_2), \label{eq:n1}
\end{eqnarray}
\begin{eqnarray}
& &X_{a,a}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime} = M_X\cdot X_{a,a}^{\epsilon_1,\epsilon_2}(m_2,m_1)^{\prime\prime} + \!\!\!\!\sum_{t=1}^{[\frac{m_1-m_2}{2}]} \!\!\!\! M_X(t)\cdot X_{a,a}^{\epsilon_1,\epsilon_2}(m_2+t,m_1-t)^{\prime\prime}\label{eq:n2}\\ & & \quad (m_1\geqslant m_2; \epsilon_1 > \epsilon_2 ), \nonumber\\
& & Y_{a_1,a_2}^{\epsilon,\epsilon}(m_1,m_2)^{\prime} = M_Y\cdot Y_{a_1,a_2}^{\epsilon,\epsilon}(m_2,m_1)^{\prime\prime} + \!\!\!\!\sum_{t=1}^{[\frac{m_1-m_2}{2}]}\!\!\!\! M_Y(t)\cdot Y_{a_1,a_2}^{\epsilon,\epsilon}(m_2+t,m_1-t)^{\prime\prime} \label{eq:n3}\\ & & \quad (m_1\geqslant m_2; a_1 > a_2 ), \nonumber \\
& &Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_1,m_2)^{\prime} = M_Z\cdot Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_2,m_1)^{\prime\prime} + \!\!\!\!\sum_{t=1}^{[\frac{m_1-m_2}{2}]}\!\!\!\! M_Z(t)\cdot Z_{a_1,a_2}^{\epsilon_1,\epsilon_2}(m_2+t,m_1-t)^{\prime\prime} \label{eq:n4} \\ & & \quad (m_1\geqslant m_2; \epsilon_1 > \epsilon_2; a_1 > a_2 ). \nonumber
\end{eqnarray}
\end{lemma}
\noindent The relations (\ref{eq:n1} -- \ref{eq:n4}) indeed have the form (\ref{eq:norule1}), in particular, all wedges $u_{k}\wedge u_{l}$ in the left-hand-sides satisfy $k\leqslant l$ and all wedges in the right-hand-sides are normally ordered. Note moreover, that every wedge $u_{k}\wedge u_{l}$ such that $k\leqslant l$ appears in the left-hand-side of one of the relations. When $L=1$ the normal ordering rules are given by (\ref{eq:n1}) and (\ref{eq:n2}), these relations coincide with the normal ordering rules of \cite[ eq.(43),(45)]{KMS}.
\begin{propos} \label{p:fwbasis}\mbox{} \mbox{}\\
{\em (i)} Any wedge from $\wedge^n V_{\mathrm {aff}}$ is a linear combination of normally ordered wedges with coefficients determined by the normal ordering rules {\em (\ref{eq:n1} -- \ref{eq:n4})} applied in each pair of adjacent factors of $\wedge^n V_{\mathrm {aff}}.$ \\
\noindent {\em (ii)} Normally ordered wedges form a basis of $\wedge^n V_{\mathrm {aff}}.$
\end{propos}
\begin{proof}
(i) follows directly from the definition of $\wedge^n V_{\mathrm {aff}}.$ \\
(ii) In view of (i) it is enough to prove that normally ordered wedges are linearly independent. This is proved by specialization $q=1.$ Let $w_{1},\dots,w_{m}$ be a set of distinct normally ordered wedges in $\wedge^n V_{\mathrm {aff}},$ and let $t_{1},\dots,t_{m} \in V_{\mathrm {aff}}^{\otimes n}$ be the corresponding pure tensors. Assume that
\begin{equation}
\sum c_j(q) w_{j} = 0, \label{eq:l1}
\end{equation}
where $c_1(q),\dots,c_m(q)$ are non-zero Laurent polynomials in $q.$ Then
\begin{equation}
\sum c_j(q) t_{j} \in \sum_{i=1}^{n-1} \mathrm {Im}(\Tc_i-\Ts_i).
\end{equation}
Specializing $q$ to be $1$ this gives
\begin{equation}
\sum c_j(1) t_{j} \in \sum_{i=1}^{n-1} \mathrm {Im}(P_i+1) \subset \otimes_{\mathbb Q}^n {V_{\mathrm {aff}}bar}, \label{eq:l2}
\end{equation}
where ${V_{\mathrm {aff}}bar} = \mathbb Q[z,z^{-1}]\otimes_{\mathbb Q}\mathbb Q^L\otimes_{\mathbb Q}\mathbb Q^N,$ and $P_i$ is the permutation operator for the $i$th and $i+1$th factors in $\otimes_{\mathbb Q}^n {V_{\mathrm {aff}}bar}.$ Since each $t_j$ is a tensor of the form $u_{k_1}\otimes u_{k_2}\otimes \cdots \otimes u_{k_n}$ where $k_1,k_2,\dots,k_n$ is a decreasing sequence, it follows from (\ref{eq:l2}) that $c_j(1)=0$ for all $j.$ Therefore each $c_j(q)$ has the form $(q-1)c_j(q)^{(1)}$ where $c_j(q)^{(1)}$ is a Laurent polynomial in $q.$ Equation (\ref{eq:l1}) gives now
\begin{equation}
\sum c_j(q)^{(1)} w_{j} = 0.
\end{equation}
Repeating the arguments above we conclude that all $c_j(q)$ are divisible by arbitrarily large powers of $(q-1).$ Therefore all $c_j(q)$ vanish.
\end{proof}
\begin{lemma} \label{l:lemma}
Let $l \leqslant m.$ Then the wedges $u_m\wedge u_{m-1} \wedge \cdots \wedge u_{l+1}\wedge u_{l}\wedge u_m$ and $u_l\wedge u_m\wedge u_{m-1} \wedge \cdots \wedge \cdots u_{l+1}\wedge u_{l}$ are equal to zero.
\end{lemma}
\begin{proof}
As particular cases of relations (\ref{eq:n1} -- \ref{eq:n4}) we have for all $k$ and $N\geqslant 2$
$$ u_k\wedge u_k = 0, \quad u_k\wedge u_{k+1} = \begin{cases} -q^{\delta(k\not\equiv 0\bmod N)} u_{k+1}\wedge u_{k} & \text{if $N\geqslant 2,$}\\ -q^{-1}u_{k+1}\wedge u_{k} & \text{if $N=1.$} \end{cases} $$
The lemma follows by induction from (\ref{eq:n1} -- \ref{eq:n4}). \end{proof}
\section{The Fock Space} \label{s:Fock}
\subsection{Definition of the Fock space}
For each integer $M$ we define the Fock space ${\mathcal F}_M$ as the inductive limit $(n \rightarrow \infty)$ of $\wedge^n V_{\mathrm {aff}},$ where maps $\wedge^n V_{\mathrm {aff}} \rightarrow \wedge^{n+1} V_{\mathrm {aff}}$ are given by $v \mapsto v\wedge u_{M-n}.$ For $v \in \wedge^n V_{\mathrm {aff}}$ we denote by $v\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots$ the image of $v$ with respect to the canonical map from $\wedge^n V_{\mathrm {aff}}$ to ${\mathcal F}_M.$ Note that for $v_{(n)} \in \wedge^n V_{\mathrm {aff}},$ $v_{(r)} \in \wedge^r V_{\mathrm {aff}},$ the equality $$ v_{(n)}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots = v_{(r)}\wedge u_{M-r}\wedge u_{M-r-1}\wedge \cdots $$
holds if and only if there is $s \geqslant n,r$ such that
$$ v_{(n)}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots \wedge u_{M-s+1} = v_{(r)}\wedge u_{M-r}\wedge u_{M-r-1}\wedge \cdots \wedge u_{M-s+1} .$$ In particular, $v_{(n)}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots$ vanishes if and only if there is $s \geqslant n$ such that $ v_{(n)}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots \wedge u_{M-s+1} $ is zero. \\
\noindent For a decreasing sequence of integers $(k_1 > k_2 > \cdots )$ such that $k_i = M-i+1$ for $i \gg 1,$ we will call the vector
$ u_{k_1}\wedge u_{k_2} \wedge \cdots \quad \in {\mathcal F}_M$
a (semi-infinite) {\em normally ordered wedge}.
\begin{propos}\label{p:siwbasis}
The normally ordered wedges form a basis of ${\mathcal F}_M.$
\end{propos}
\begin{proof}
For each $w \in {\mathcal F}_M$ there are $n, v\in \wedge^n V_{\mathrm {aff}} $ such that $ w = v \wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots.$ By Proposition \ref{p:fwbasis} the finite normally ordered wedges form a basis of $\wedge^n V_{\mathrm {aff}},$ therefore $w$ is a linear combination of vectors
\begin{equation}
u_{k_1}\wedge u_{k_2}\wedge \cdots \wedge u_{k_n}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots , \quad \text{ where $k_1 > k_2 > \cdots > k_n.$}
\label{eq:swibasis1}
\end{equation}
If $k_n \leqslant M-n,$ then there is $r > n$ such that $u_{k_n}\wedge u_{M-n}\wedge u_{M-n-1}\wedge \cdots \wedge u_{M-r+1}$ vanishes by Lemma \ref{l:lemma}. It follows that (\ref{eq:swibasis1}) is zero if $k_n \leqslant M-n.$ Thus the normally ordered wedges span ${\mathcal F}_M.$ \\
\noindent Suppose $ \sum c_{(k_1,k_2,\dots)} u_{k_1}\wedge u_{k_2}\wedge \cdots = 0,$ where wedges under the sum are normally ordered and $c_{(k_1,k_2,\dots)} \in {\mathbb K\hskip.5pt}.$ Then by definition of the inductive limit there exists $n$ such that $ \sum c_{(k_1,k_2,\dots)} u_{k_1}\wedge u_{k_2}\wedge \cdots \wedge u_{k_n} = 0.$ Thus
linear independence of semi-infinite normally ordered wedges follows from the linear independence of finite normally ordered wedges.
\end{proof}
\subsection{The actions of $\UU_q(\asll_N)$ and $\UU_q(\asll_L)$ on the Fock spaces} \label{sec:UF}
Define the {\em vacuum vector} of ${\mathcal F}_M$ as
$$ |M\rangle = u_M \wedge u_{M-1} \wedge \cdots . $$
Then for
each vector $w$ from ${\mathcal F}_M$ there is a sufficiently large integer $m$ such that $w$ can be represented as
\begin{equation}
w = v\wedge |-NLm\rangle ,\quad \text{ where $v \in \wedge^{M+NLm}V_{\mathrm {aff}}.$} \label{eq:w=vbyvac}
\end{equation}
\noindent For each $M\in \mathbb Z$ we define on ${\mathcal F}_M$ operators $E_i,F_i,K_i^{\pm 1}, D$ $(0\leqslant i < N)$ and $\dot{E}_a,\dot{F}_a,\dot{K}_a^{\pm 1},\dot{D}$ $(0\leqslant a < L)$ and then show, in Theorem \ref{t:UNUL}, that these operators satisfy the defining relations of $\UU_q(\asll_N)$ and $\UU_q(\asll_L)$ respectively.
As the first step we define actions of these operators on vectors of the form $|-NLm\rangle.$ Let $\overline{v} = u_{-NLm}\wedge u_{-NLm -1}\wedge \cdots \wedge u_{-NL(m+1)+1}.$ We set
\begin{alignat}{5}
&D |-NLm\rangle & = & NL\frac{m(1-m)}{2} |-NLm\rangle, \label{eq:dNvac}& \\
&K_i |-NLm\rangle & = & q^{L\delta(i=0)}|-NLm\rangle,\label{eq:KNvac} & \\
&E_i |-NLm\rangle & = & 0, \label{eq:ENvac} & \\
&F_i |-NLm\rangle & = & \begin{cases} 0 & \text{ if $i \neq 0,$} \\
F_0(\overline{v})\wedge |-NL(m+1)\rangle & \text{ if $i = 0.$} \end{cases} & \label{eq:FNvac}
\end{alignat}
And
\begin{alignat}{5}
&\dot{D} |-NLm\rangle & = & NL\frac{m(1-m)}{2} |-NLm\rangle, & \label{eq:dLvac} \\
&\dot{K}_a |-NLm\rangle & = & q^{N\delta(a=0)}|-NLm\rangle,& \label{eq:KLvac} \\
&\dot{E}_a |-NLm\rangle & = & 0, &\label{eq:ELvac} \\
&\dot{F}_a |-NLm\rangle & = & \begin{cases} 0 & \text{ if $a \neq 0,$} \\
q^{-N}\dot{F}_0(\overline{v})\wedge |-NL(m+1)\rangle & \text{ if $a = 0.$} \end{cases}&\label{eq:FLvac}
\end{alignat}
Then the actions on an arbitrary vector $w \in {\mathcal F}_M$ are defined by using the presentation (\ref{eq:w=vbyvac}) and the coproducts (\ref{eq:co1} -- \ref{eq:co4}). Thus for $v \in \wedge^{M+NLm}V_{\mathrm {aff}}$ and $w = v\wedge |-NLm\rangle \in {\mathcal F}_M$ we define
\begin{alignat}{5}
&D(w) & = & D(v)\wedge |-NLm\rangle + v \wedge D |-NLm\rangle , &\label{eq:dN} \\
&K_i(w) &=& K_i(v)\wedge K_i |-NLm\rangle,& \label{eq:KN} \\
&E_i(w) &=& E_i(v)\wedge K_i |-NLm\rangle, & \label{eq:EN} \\
&F_i(w) &=& F_i(v)\wedge |-NLm\rangle + K_i^{-1}(v)\wedge F_i|-NLm\rangle .&\label{eq:FN}
\end{alignat}
And
\begin{alignat}{5}
&\dot{D}(w) & = & \dot{D}(v)\wedge |-NLm\rangle + v \wedge \dot{D} |-NLm\rangle , &\label{eq:dL} \\
&\dot{K}_a(w) &=& \dot{K}_a(v)\wedge \dot{K}_a |-NLm\rangle, & \label{eq:KL} \\
&\dot{E}_a(w) &=& \dot{E}_a(v)\wedge |-NLm\rangle, \label{eq:EL} &\\
&\dot{F}_a(w) &=& \dot{F}_a(v)\wedge \dot{K}_a^{-1}|-NLm\rangle + v\wedge \dot{F}_a|-NLm\rangle .&\label{eq:FL}
\end{alignat}
It follows from Lemma \ref{l:lemma} that the operators $E_i,F_i,K_i^{\pm 1},D$ and $\dot{E}_a,\dot{F}_a,\dot{K}_a^{\pm 1},\dot{D}$ are well-defined, that is do not depend on a particular choice of the presentation (\ref{eq:w=vbyvac}), and for $v \in \wedge^n V_{\mathrm {aff}},$ $u\in {\mathcal F}_{M-n}$ satisfy the following relations, analogous to the coproduct formulas (\ref{eq:co1} -- \ref{eq:co4}):
\begin{alignat}{5}
&D(v\wedge u) & = & D(v)\wedge u + v \wedge D(u) ,& \label{eq:codN} \\
&K_i(v\wedge u) &=& K_i(v)\wedge K_i(u), & \label{eq:coKN} \\
&E_i(v\wedge u) &=& E_i(v)\wedge K_i (u) + v\wedge E_i(u), & \label{eq:coEN} \\
&F_i(v\wedge u) &=& F_i(v)\wedge u + K_i^{-1}(v)\wedge F_i(u) .&\label{eq:coFN}
\end{alignat}
And
\begin{alignat}{5}
&\dot{D}(v\wedge u) & = & \dot{D}(v)\wedge u + v \wedge \dot{D}(u) ,& \label{eq:codL} \\
&\dot{K}_a(v\wedge u) &=& \dot{K}_a(v)\wedge \dot{K}_a(u), & \label{eq:coKL} \\
&\dot{E}_a(v\wedge u) &=& \dot{E}_a(v)\wedge u + \dot{K}_a(v)\wedge \dot{E}_a(u), & \label{eq:coEL} \\
&\dot{F}_a(v\wedge u) &=& \dot{F}_a(v)\wedge \dot{K}_a^{-1}(u) + v\wedge \dot{F}_a(u) .\label{eq:coFL}&
\end{alignat}
Relations ((\ref{eq:dNvac}, \ref{eq:KNvac}),(\ref{eq:dN}, \ref{eq:KN})) and ((\ref{eq:dLvac}, \ref{eq:KLvac}),(\ref{eq:dL}, \ref{eq:KL})) define the weight decomposition of the Fock space ${\mathcal F}_M.$ We have
\begin{equation} \mathrm {wt}( |-NLm\rangle ) = L\Lambda_0 + N\dot{\Lambda}_0 + NL\frac{m(1-m)}{2} (\delta + \dot{\delta}), \label{eq:wt1} \end{equation}
and for $v \in \wedge^{M+NLm} V_{\mathrm {aff}}$
\begin{equation} \mathrm {wt}(v\wedge |-NLm\rangle ) = \mathrm {wt}(v) + \mathrm {wt}( |-NLm\rangle ). \label{eq:wt2} \end{equation}
\begin{theor} \label{t:UNUL} \mbox{} \\
{\em (i)} The operators $E_i,F_i,K_i,D$ $(0\leqslant i <N)$ define on ${\mathcal F}_M$ a structure of an integrable $\UU_q(\asll_N)$-module. And the operators $\dot{E}_a,\dot{F}_a,\dot{K}_a,\dot{D} $ define on ${\mathcal F}_M$ a structure of an integrable $\UU_q(\asll_L)$-module. \\
{\em (ii)} The actions of the subalgebras $\UU_q^{\prime}(\asll_N) \subset \UU_q(\asll_N)$ and $\UU_q^{\prime}(\asll_L) \subset \UU_q(\asll_L)$ on ${\mathcal F}_M$ are mutually commutative.
\end{theor}
\begin{proof}
(i) It is straightforward to verify that the relations (\ref{eq:r1}--\ref{eq:r4}) are satisfied. In particular, the weights of $E_i,F_i$ and $\dot{E}_a,\dot{F}_a$ are $\alphapha_i,-\alphapha_i$ and $\dot{\alphapha}_a,-\dot{\alphapha}_a$ respectively.
To prove the relations
\begin{equation}
[E_i,F_j] = \delta_{ij}\frac{K_i-K_i^{-1}}{q -q^{-1}}, \quad \text{and} \quad [\dot{E}_a,\dot{F}_b] = \delta_{ab}\frac{\dot{K}_a-\dot{K}_a^{-1}}{q -q^{-1}} \label{eq:t1}
\end{equation}
it is enough, by (\ref{eq:KN}--\ref{eq:FN}) and (\ref{eq:KL}--\ref{eq:FL}), to show that these relations hold when applied to a vacuum vector of the form $|-NLm\rangle.$ If $i\neq j ,$ $a\neq b$ we have
$$ [E_i,F_j]|-NLm\rangle = 0,\quad [\dot{E}_a,\dot{F}_b]|-NLm\rangle = 0 $$
because $\alphapha_i-\alphapha_j + \mathrm {wt}(|-NLm\rangle)$ $(i\neq j )$ and $\dot{\alphapha}_a -\dot{\alphapha}_b+\mathrm {wt}(|-NLm\rangle)$ $(a\neq b)$ are not weights of ${\mathcal F}_{-NLm}.$
The relations
\begin{alignat}{4} &[E_i,F_i]|-NLm\rangle &=&\frac{K_i-K_i^{-1}}{q -q^{-1}}|-NLm\rangle \label{eq:t21} \\ &[\dot{E}_a,\dot{F}_a] |-NLm\rangle &= &\frac{\dot{K}_a-K_a^{-1}}{q -q^{-1}}|-NLm\rangle & \label{eq:t22}
\end{alignat}evidently hold by (\ref{eq:KNvac} -- \ref{eq:FNvac}), (\ref{eq:KLvac} -- \ref{eq:FLvac}) when $i \neq 0,$ $a\neq 0.$
Let $a=0.$ We have
\begin{align*} \dot{F}_0 |-NLm\rangle = &q^{-N}\sum_{i =1}^N q^{i} \: u_{N-N(1+Lm)}\wedge u_{N-1-N(1+Lm)}\wedge \cdots \\ & \cdots \wedge u_{i-N(L+L(m-1))} \wedge \cdots \wedge u_{1-N(1+Lm)} \wedge |N-N(2+Lm)\rangle.
\end{align*}
Then by Lemma \ref{l:lemma}
$$ \dot{E}_0 \dot{F}_0 |-NLm\rangle = q^{1-N} \sum_{i=1}^N q^{2(i-1)} |-NLm\rangle = \frac{q^N-q^{-N}}{q -q^{-1}}|-NLm\rangle.$$
This shows the relation (\ref{eq:t22}) for $a=0.$ The relation (\ref{eq:t21}) for $i=0$ is shown in a similar way.
Thus $E_i,F_i,K_i,D$ and $\dot{E}_a,\dot{F}_a,\dot{K}_a,\dot{D} $ satisfy the defining relations (\ref{eq:r1} -- \ref{eq:r5}). Observe that for $i=0,\dots,N-1;$ $a=0,\dots,L-1$ and $\mu \in P_N+P_L,$ $\mu + r \alphapha_i$, $ \mu + n \dot{\alphapha}_a $ are weights of ${\mathcal F}_{M}$ for only a finite number of $r$ and $n.$ Therefore ${\mathcal F}_M$ is an integrable module of $\operatorname{U}_q(\mathfrak {sl}_2)_i = \langle E_i,F_i,K_i^{\pm 1}\rangle $ and $\operatorname{U}_q(\mathfrak {sl}_2)_a = \langle \dot{E}_a,\dot{F}_a,\dot{K}_a^{\pm 1}\rangle .$ By Proposition B.1 of \cite{KMPY} this implies that the Serre relations (\ref{eq:r6}, \ref{eq:r7}) are satisfied.
Eigenspaces of the operator $D$ and eigenspaces of the operator $\dot{D}$ are finite-dimensional. Therefore the integrability with respect to each $\operatorname{U}_q(\mathfrak {sl}_2)_i$ and $\operatorname{U}_q(\mathfrak {sl}_2)_a$ implies the integrability of ${\mathcal F}_M$ as both $\UU_q(\asll_N)$-module and $\UU_q(\asll_L)$-module. \\
\noindent (ii) The Cartan part of $\UU_q^{\prime}(\asll_N)$ evidently commutes with $\UU_q^{\prime}(\asll_L),$ and vice-versa. By (\ref{eq:KN} -- \ref{eq:FN}) and (\ref{eq:KL} -- \ref{eq:FL}) it is enough to prove that commutators between the other generators vanish when applied to a vector of the form $|-NLm\rangle .$
The relation
$$ [E_i,\dot{E}_a] |-NLm\rangle = 0$$
is trivially satisfied by (\ref{eq:ENvac}, \ref{eq:ELvac}). The relations
$$ [F_i,\dot{E}_a] |-NLm\rangle = 0, \quad [E_i,\dot{F}_a] |-NLm\rangle = 0$$
hold because $\dot{\alphapha}_a - \alphapha_i + \mathrm {wt}(|-NLm\rangle)$ and $ \alphapha_i -\dot{\alphapha}_a + \mathrm {wt}(|-NLm\rangle)$ are not weights of ${\mathcal F}_{-NLm}.$
The relations $$[F_i,\dot{F}_a] |-NLm\rangle = 0$$ are trivial by (\ref{eq:FNvac}, \ref{eq:FLvac}) when $i\neq 0,a\neq 0;$ and are verified by using the normal ordering rules (\ref{eq:n1} -- \ref{eq:n4}) and Lemma \ref{l:lemma} in the rest of the cases.
\end{proof}
\subsection{The actions of Bosons} \label{sec:bosons}
We will now define actions of operators $B_n$ $(n\in \mathbb Z_{\neq 0})$ (called {\em bosons}) on ${\mathcal F}_M.$ Let $u_{k_1}\wedge u_{k_2} \wedge \cdots $ ($k_i = M-i+1$ for $i\gg 1$) be a vector of ${\mathcal F}_M.$ By Lemma \ref{l:lemma}, for $n\neq 0$ the sum
\begin{align}
&(z^n u_{k_1})\wedge u_{k_2} \wedge u_{k_3} \wedge \cdots \; + \label{eq:ba}\\
&u_{k_1} \wedge(z^n u_{k_2})\wedge u_{k_3} \wedge \cdots \; + \nonumber\\
&u_{k_1} \wedge u_{k_2} \wedge(z^n u_{k_3})\wedge \cdots \; + \nonumber\\
& \quad + \quad \cdots \quad .\nonumber
\end{align}
contains only a finite number of non-zero terms, and is, therefore, a vector of ${\mathcal F}_M.$ By Proposition \ref{p:CPD} the assignment $u_{k_1}\wedge u_{k_2} \wedge \cdots \mapsto \text{(\ref{eq:ba})}$ defines an operator on ${\mathcal F}_M.$ We denote this operator $B_n.$ By definition we have for $v \in V_{\mathrm {aff}},$ $u \in {\mathcal F}_{M-1}:$
\begin{equation}
B_n(v\wedge u) = (z^nv)\wedge u + v\wedge B_n(u). \label{eq:Bvu}
\end{equation}
\begin{propos}
For all $n\in \mathbb Z_{\neq 0}$ the operator $B_n$ commutes with the actions of $\UU_q^{\prime}(\asll_N)$ and $\UU_q^{\prime}(\asll_L).$
\end{propos}
\begin{proof}
It follows immediately from the definition, that the weight of $B_n$ is $n(\delta + \dot{\delta}).$ Thus $B_n$ commutes with $K_i, \dot{K}_a$ $(0\leqslant i <N, 0\leqslant a < L).$
Let $X$ be any of the operators $E_i,F_i,\dot{E}_a,\dot{F}_a$ $(0\leqslant i <N, 0\leqslant a < L).$ The relations (\ref{eq:coEN}, \ref{eq:coFN}), (\ref{eq:coEL}, \ref{eq:coFL}) and (\ref{eq:Bvu}) imply now that $[B_n , X] = 0$ will follow from $[B_n , X] |-NLm \rangle = 0 $ for an arbitrary integer $m.$
If $n >0,$ we have $[B_n , X] |-NLm \rangle = 0$ because $n(\delta + \dot{\delta})\pm\alphapha_i + \mathrm {wt}(|-NLm \rangle) $ and $n(\delta + \dot{\delta})\pm\dot{\alphapha}_a + \mathrm {wt}(|-NLm \rangle)$ are not weights of ${\mathcal F}_{-NLm}.$
Let $n < 0.$ Consider the expansion
$$
[B_n , X] |-NLm \rangle = \sum_{\nu} c_{\nu} u_{k^{\nu}_1}\wedge u_{k^{\nu}_2}\wedge \cdots
$$
where the wedges in the right-hand-side are normally ordered. Comparing the weights of the both sides, we obtain for all $\nu$ the inequality $k_1^{\nu} > -NLm.$ For $r \geqslant 0$ (\ref{eq:coEN}, \ref{eq:coFN}), (\ref{eq:coEL}, \ref{eq:coFL}) and (\ref{eq:Bvu}) give
\begin{multline}
[B_n , X] |-NLm \rangle = \\ = u_{-NLm}\wedge u_{-NLm-1}\wedge \cdots \wedge u_{-NL(m+r)+1}\wedge [B_n , X] |-NL(m+r) \rangle \label{eq:bx}
\end{multline}
where
$$ [B_n , X] |-NL(m+r) \rangle =
\sum_{\nu} c_{\nu} u_{k^{\nu}_1-NLr}\wedge u_{k^{\nu}_2-NLr}\wedge \cdots .
$$
Now let $r$ be sufficiently large, so that
$$ k_1^{\nu} - NLr \leqslant -NLm$$
holds for all $\nu.$ By Lemma \ref{l:lemma}, the last inequality and $ k_1^{\nu} - NLr > -NL(m+r) $ imply that (\ref{eq:bx}) vanishes.
\end{proof}
\begin{propos}
There are non-zero $\gamma_n(q) \in \mathbb Q[q,q^{-1}]$ (independent on $M$) such that
\begin{equation}
[B_n,B_{n^{\prime}}] = \delta_{n+n^{\prime},0} \gamma_n(q).
\end{equation}
\end{propos}
\begin{proof}
Each vector of ${\mathcal F}_{M^{\prime}}$ $(M^{\prime}\in\mathbb Z)$ is of the form $v\wedge |M\rangle$ where $v \in \wedge^k V_{\mathrm {aff}},$ and $k=M^{\prime}-M$ is sufficiently large. By (\ref{eq:Bvu}) we have
$$ [B_n,B_{n^{\prime}}](v\wedge |M\rangle) = v\wedge[B_n,B_{n^{\prime}}]|M\rangle.$$
The vector $[B_n,B_{n^{\prime}}]|M\rangle$ vanishes if $n+n^{\prime}>0$ because in this case $\mathrm {wt}(|M\rangle) + (n+n^{\prime})(\delta + \dot{\delta})$ is not a weight of ${\mathcal F}_M.$
Let $n+n^{\prime}<0.$ Write $[B_n,B_{n^{\prime}}]|M\rangle$ as the linear combination of normally ordered wedges:
$$ [B_n,B_{n^{\prime}}]|M\rangle = \sum_{\nu} c_{\nu} u_{k_1^{\nu}}\wedge u_{k_2^{\nu}} \wedge \cdots .$$
Since $[B_n,B_{n^{\prime}}]|M\rangle$ is of the weight $\mathrm {wt}(|M\rangle) + (n+n^{\prime})(\delta + \dot{\delta})$ with $n+n^{\prime}<0,$ we necessarily have $k_1^{\nu} > M.$
For any $s > 0$ eq. (\ref{eq:Bvu}) gives
\begin{equation} [B_n,B_{n^{\prime}}]|M\rangle = u_M\wedge u_{M-1} \wedge \cdots \wedge u_{M-NLs+1}\wedge [B_n,B_{n^{\prime}}]|M-NLs\rangle,
\label{eq:pB1}\end{equation}
where
$$
[B_n,B_{n^{\prime}}]|M-NLs\rangle = \sum_{\nu} c_{\nu} u_{k_1^{\nu}-NLs}\wedge u_{k_2^{\nu}-NLs} \wedge \cdots.
$$
Taking $s$ sufficiently large so that $ M-k_1^{\nu} + NLs \geqslant 0$ holds for all $\nu$ above, we have for all $\nu$ the inequalities
$$ k_1^{\nu} - NLs - (M - NLs) > 0, \quad \text{and} \quad M-(k_1^{\nu} - NLs) \geqslant 0. $$
Lemma \ref{l:lemma} now shows that (\ref{eq:pB1}) is zero.
Let now $n+n^{\prime}=0.$ The vector $[B_n,B_{n^{\prime}}]|M\rangle$ has weight $\mathrm {wt}(|M\rangle).$ The weight subspace of this weight is one-dimensional, so we have $ [B_n,B_{-n}]|M\rangle = \gamma_{n,M}(q) |M\rangle $ for $\gamma_{n,M}(q) \in {\mathbb K\hskip.5pt}.$ Since $[B_n,B_{-n}]|M\rangle = u_M \wedge [B_n,B_{-n}]|M-1\rangle,$ $ \gamma_{n,M}(q)$ is independent on $M.$
The coefficients $c_{kl}(q),c_{kl}^{(i)}(q)$ in the normal ordering rules (\ref{eq:norule1}) are Laurent polynomials in $q,$ hence so are $\gamma_{n}(q).$ Specializing to $q=1$ we have $\gamma_{n}(1) = nNL.$ Thus all $\gamma_{n}(q)$ $(n \in \mathbb Z_{\neq 0})$ are non-zero.
\end{proof}
\begin{propos}
If $N=1$ or $L=1$ or $n=1,2$, we have for $\gamma _n(q)$ the following formula:
\begin{equation}
\gamma _n(q) = n \frac{1-q^{2Nn}}{1-q^{2n}} \frac{1-q^{-2Ln}}{1-q^{-2n}}.
\label{Bconst}
\end{equation}
\end{propos}
\begin{proof}
The $L=1$ case is due to \cite{KMS}, and the formula for $N=1$ is obtained from the formula for $L=1$ by comparing the normal ordering rules (\ref{eq:n2}) and (\ref{eq:n3}).
The $n=1,2$ case is shown by a direct but lengthy calculation.
(First act with $B_{-n}$ on the vacuum vector, express all terms as linear combinations of the normally ordered wedges, then act with $B_n$ and, again, rewrite the result in terms of the normally ordered wedges to get the coefficient $\gamma _n(q)$.)
\end{proof}
\begin{conje}
The formula {\em (\ref{Bconst})} is valid for all positive integers $N,L,n$.
\end{conje}
\noindent Let $H$ be the Heisenberg algebra generated by $\{ B_n\}_{n\in\mathbb Z_{\neq 0}}$ with the defining relations $[B_n,B_{n^{\prime}}]=\delta_{n+n^{\prime},0}\gamma_n(q).$ Summarizing this and the previous sections, we have constructed on each Fock space ${\mathcal F}_M$ an action of the algebra $H\otimes \UU_q^{\prime}(\asll_N) \otimes \UU_q^{\prime}(\asll_L).$ Note that the action of $\UU_q^{\prime}(\asll_N)$ has level $L$ and the action of $\UU_q^{\prime}(\asll_L)$ has level $N.$
\subsection{The decomposition of the Fock space} \label{sec:decomp}
Let $P_N^+$ and $P_N^+(L)$ be respectively the set of dominant integral weights of $\UU_q^{\prime}(\asll_N)$ and the subset of dominant integral weights of level $L\in \mathbb N:$
\begin{alignat}{4}
&P_N^+ &= &\{ a_0\Lambda_0 + a_1\Lambda_1 + \cdots + a_{N-1}\Lambda_{N-1} \: | \: a_i \in \mathbb Z_{\geqslant 0} \}, \\
&P_N^+(L) &= &\{ a_0\Lambda_0 + a_1\Lambda_1 + \cdots + a_{N-1}\Lambda_{N-1} \: | \: a_i \in \mathbb Z_{\geqslant 0},\; \sum a_i = L \}.
\end{alignat}
For $\Lambda \in P_N^+$ let $V(\Lambda)$ be the irreducible integrable highest weight module of $\UU_q^{\prime}(\asll_N),$ and let $v_{\Lambda} \in V(\Lambda)$ be the highest weight vector.
Let $\overline{\Lambda}_1,\overline{\Lambda}_2,\dots,\overline{\Lambda}_{N-1}$ be the fundamental weights of $\mathfrak {sl}_N,$ and let $\overline{\alphapha}_i = 2 \overline{\Lambda}_i-\overline{\Lambda}_{i+1}-\overline{\Lambda}_{i-1}$ $ 1 \leqslant i < N$ be the simple roots. Here the indices are cyclically extended to all integers modulo $N,$ and $\overline{\Lambda}_0 :=0.$ Let $\overline{Q}_N = \oplus_{i=1}^{N-1} \mathbb Z \overline{\alphapha}_i $ be the root lattice of $\mathfrak {sl}_N.$ For an $\UU_q^{\prime}(\asll_N)$-weight $\Lambda = \sum_{i=0}^{N-1} a_i \Lambda_i $ we will set $\overline{\Lambda} = \sum_{i=1}^{N-1} a_i \overline{\Lambda}_i.$
A vector $w \in {\mathcal F}_M$ is a {\em highest weight vector} of $H\otimes \UU_q^{\prime}(\asll_N) \otimes \UU_q^{\prime}(\asll_L)$ if it is a highest weight vector with respect to $\UU_q^{\prime}(\asll_N)$ and $\UU_q^{\prime}(\asll_L)$ and is annihilated by $B_n$ with $n > 0.$ We will now describe a family of highest weight vectors.
With every $\Lambda = \sum_{i=0}^{N-1}a_i \Lambda_i \in P_N^+(L),$ such that $\overline{\Lambda} \equiv \overline{\Lambda}_M \bmod \overline{Q}_N,$ we associate $\dot{\Lambda}^{(M)} \in P_L^+(N)$ (i.e. $\dot{\Lambda}^{(M)}$ is a dominant integral weight of $\UU_q^{\prime}(\asll_L)$ of level $N$) as follows. Let $M\equiv s \bmod NL$ $( 0\leqslant s < NL),$ and let $l_1 \geqslant l_2 \geqslant \dots \geqslant l_N $ be the partition defined by the relations:
\begin{align}
&l_i - l_{i+1} = a_i\quad (1 \leqslant i < N), \label{eq:part1}\\
& l_1 + l_2 + \cdots + l_N = s + NL. \label{eq:part2}
\end{align}
Note that all $l_i$ are integers, and that $l_N > 0.$ Then we set
\begin{equation}
\dot{\Lambda}^{(M)} := \dot{\Lambda}_{l_1} + \dot{\Lambda}_{l_2} + \cdots + \dot{\Lambda}_{l_N}. \label{eq:dotLambda}
\end{equation}
Recall that the indices of the fundamental weights are cyclically extended to all integers modulo $L$.
Consider the Young diagram of $l_1 \geqslant l_2 \geqslant \dots \geqslant l_N$ (Fig. 1).We set the coordinates $(x,y)$ of the lowest leftmost square to be $(1,1).$
\begin{center}
\begin{picture}(120,120)(-15,-15)
\multiput(0,0)(10,0){2}{\line(0,1){85}}
\put(0,85){\line(1,0){10}} \put(0,85){\makebox(12,15){{\scriptsize $l_1$}}}
\put(20,0){\line(0,1){60}}
\put(10,60){\line(1,0){10}} \put(10,60){\makebox(12,15){{\scriptsize $l_2$}}}
\put(30,0){\line(0,1){40}}
\put(20,40){\line(1,0){10}} \put(20,40){\makebox(12,15){{\scriptsize $l_3$}}}
\put(40,15){\makebox(30,10){$\cdots$}}
\multiput(80,0)(10,0){2}{\line(0,1){20}}
\put(80,20){\line(1,0){10}} \put(80,20){\makebox(12,15){{\scriptsize $l_N$}}}
\put(0,0){\line(1,0){30}} \put(80,0){\line(1,0){10}}
\put(-10,-10){\vector(1,0){20}} \put(10,-15){\makebox{{\scriptsize $x$}}}
\put(-10,-10){\vector(0,1){20}} \put(-15,10){\makebox{{\scriptsize $y$}}}
\end{picture}
{\large { Fig. 1}} \end{center}
Introduce a numbering of squares of the Young diagram by $1,2,\dots,s+NL$ by requiring that the numbers assigned to squares in the bottom row of a pair of any adjacent rows are greater than the numbers assigned to squares in the top row, and that the numbers increase from right to left within each row (cf. the example below). Letting $(x_i,y_i)$ to be the coordinates of the $i$th square, set $k_i = x_i + N(y_i - L - 1) + M-s.$ Then $k_i > k_{i+1}$ for all $i=1,2,\dots,s+NL-1.$ Now define
\begin{equation}
\psi_{\Lambda} = u_{k_1}\wedge u_{k_2} \wedge \cdots \wedge u_{k_{s+NL}}\wedge |M-s-NL \rangle. \label{eq:hwv}
\end{equation}
Note that $\psi_{\Lambda} \in {\mathcal F}_M,$ and $\psi_{\Lambda}$ is a normally ordered wedge.
\begin{example}
Let $N=3,$ $L=2,$ and $M=0.$ The set $\{ \Lambda \in P_3^+(2) \: | \: \overline{\Lambda} \equiv 0\bmod \overline{Q}_3 \}$ contains the two weights: $2\Lambda_0$ and $\Lambda_1+\Lambda_2$ only. The corresponding weights of $\UU_q^{\prime}(\asll_L)$ and the numbered Young diagrams are shown below.
$$ \begin{picture}(100,60)(0,0)
\put(0,50){$\Lambda = 2\Lambda_0:$}
\put(0,35){$\dot{\Lambda}^{(0)} = 3\dot{\Lambda}_0$}
\multiput(0,0)(0,10){3}{\line(1,0){30}}
\multiput(0,0)(10,0){4}{\line(0,1){20}}
\put(0,0){\makebox(10,10){\scriptsize{$6$}}}
\put(10,0){\makebox(10,10){\scriptsize{$5$}}}
\put(20,0){\makebox(10,10){\scriptsize{$4$}}}
\put(0,10){\makebox(10,10){\scriptsize{$3$}}}
\put(10,10){\makebox(10,10){\scriptsize{$2$}}}
\put(20,10){\makebox(10,10){\scriptsize{$1$}}}
\end{picture}
\begin{picture}(100,60)(0,0)
\put(0,50){$\Lambda = \Lambda_1+\Lambda_2:$}
\put(0,35){$\dot{\Lambda}^{(0)} = \dot{\Lambda}_0 + 2\dot{\Lambda}_1 $}
\multiput(0,0)(0,10){2}{\line(1,0){30}}
\put(0,20){\line(1,0){20}}\put(0,30){\line(1,0){10}}
\multiput(0,0)(10,0){2}{\line(0,1){30}}
\put(20,0){\line(0,1){20}}\put(30,0){\line(0,1){10}}
\put(0,0){\makebox(10,10){\scriptsize{$6$}}}
\put(10,0){\makebox(10,10){\scriptsize{$5$}}}
\put(20,0){\makebox(10,10){\scriptsize{$4$}}}
\put(0,10){\makebox(10,10){\scriptsize{$3$}}}
\put(10,10){\makebox(10,10){\scriptsize{$2$}}}
\put(0,20){\makebox(10,10){\scriptsize{$1$}}}
\end{picture}
$$
\end{example}
\begin{propos}\label{p:hw}
For each $\Lambda \in P_N^+(L)$ such that $\overline{\Lambda} \equiv \overline{\Lambda}_M\bmod \overline{Q}_N,$ $\psi_{\Lambda}$ is a highest weight vector of $H\otimes \UU_q^{\prime}(\asll_N)\otimes \UU_q^{\prime}(\asll_L).$ The $\UU_q^{\prime}(\asll_N)$-weight of $\psi_{\Lambda}$ is $\Lambda,$ and the $\UU_q^{\prime}(\asll_L)$-weight of $\psi_{\Lambda}$ is $\dot{\Lambda}^{(M)}.$
\end{propos}
\begin{proof}
The weights of $\psi_{\Lambda}$ are given by (\ref{eq:wt1}, \ref{eq:wt2}).
To prove that $\psi_{\Lambda}$ is annihilated by $E_i,\dot{E}_a$ and $B_n$ $(n >0)$ we use the following lemma.
\begin{lemma}\label{l:hw}
Keeping $\Lambda$ as in the statement of Proposition \ref{p:hw}, define the decreasing sequence $k_1,k_2,\dots$ from $\psi_{\Lambda} = u_{k_1}\wedge u_{k_2} \wedge \cdots .$
\noindent Then for $l>m$ we have
\begin{equation}
u_{k_l} \wedge u_{k_m} = \sum_{l^{\prime }}c_{\alphapha , k_{l^{\prime }}}u_{\alphapha }\wedge u_{k_{l^{\prime }}} \: \: \mbox{ where } \alphapha > k_{l^{\prime }} \geqslant k_l.
\end{equation}
\end{lemma}
\begin{proof}
Define $\epsilonsilon _{k_i}, a_{k_i}, m_{k_i}$ $(1 \leqslant \epsilonsilon _{k_i}\leqslant N, 1\leqslant a_{k_i} \leqslant L, m_{k_i} \in \mathbb Z)$ by $k_{i} = \epsilonsilon _{k_i}-N(a_{k_i}+L m_{k_i})$.
Using the normal ordering rules, we have
\begin{equation}
u_{k_l} \wedge u_{k_m} = \sum c_{\alphapha ,\beta} u_{\alphapha }\wedge u_{\beta},
\label{albeta}
\end{equation}
where $k_m\geqslant \alphapha > \beta \geqslant k_l$ and $\alphapha = \epsilonsilon _{k_i} -N(a_{k_{i^{\prime}}}+Lm_{\alphapha })$, $\beta = \epsilonsilon _{k_j} -N(a_{k_{j^{\prime}}}+Lm_{\beta })$, $i,j,i^{\prime},j^{\prime} \in \{l,m \} $, $i \neq j$, $i^{\prime} \neq j^{\prime}$, $m_{\alphapha }, m_{\beta } \in \mathbb Z$.
\mbox{} From the explicit expression for $\psi_{\Lambda}$ (cf. \ref{eq:hwv}) it follows that there is at most one integer $\gamma $ such that $\gamma = \epsilonsilon _{k_i} -N(a_{k_{i^{\prime}}}+Lm_{\gamma })$ $(i,i^{\prime } \in \{k,l \}, \; m_{\gamma} \in \mathbb Z)$, $k_l < \gamma < k_m $ and $\gamma \neq k_i.$
Moreover, if the integer $\gamma $ exists, then $a_l \neq a_m$, $\epsilonsilon _l > \epsilonsilon _m$ and $\gamma = e_{k_l}-N(a_{k_l}+L(m_{k_m}+\delta(a_{k_l} <a_{k_m})))$.
Note that $\gamma$ is the maximal element of the set $\{ \gamma^{\prime} | \gamma^{\prime} =\epsilonsilon _{k_i} -N(a_{k_{i^{\prime}}}+Lm_{\gamma^{\prime}}), \; i,i^{\prime } \in \{k,l \}, \; m_{\gamma^{\prime}} \in \mathbb Z, \; k_l < \gamma^{\prime} < k_m \} $.
If the $\gamma $ exists, then $\beta$ in (\ref{albeta}) is distinct from $\gamma $.
Therefore $\beta = k_{l^{\prime }}$ for some $l^{\prime }$ such that $k_{l^{\prime }} \geqslant k_l$, and the lemma follows.
\end{proof}
Now we continue the proof of Proposition \ref{p:hw}.
\mbox{} From the definition of $\psi_{\lambda}$ it follows that $E_i\psi_{\Lambda},\dot{E}_a\psi_{\Lambda}$ and $B_n\psi_{\Lambda}$ $(n >0)$ are linear combinations of vectors of the form
\begin{equation}
u_{k_1} \wedge \dots\wedge u_{k_{i-1}} \wedge u_{k_{j}} \wedge u_{k_{i+1}} \wedge \dots \wedge u_{k_{j}} \wedge\dots .
\label{klwedge}
\end{equation}
Applying Lemma \ref{l:hw} repeatedly, we conclude that vectors (\ref{klwedge}) are all zero.
\end{proof}
\noindent Let ${\mathbb K\hskip.5pt}[H_-]$ be the Fock module of $H.$ That is ${\mathbb K\hskip.5pt}[H_-]$ is the $H$-module generated by the vector $1$ with the defining relations $B_n 1 =0$ for $n >0.$
By Theorem \ref{t:UNUL}, ${\mathcal F}_M$ is an integrable module of $\UU_q^{\prime}(\asll_N)$ and $\UU_q^{\prime}(\asll_L).$ Therefore it is semisimple relative to the algebra $H\otimes \UU_q^{\prime}(\asll_N)\otimes \UU_q^{\prime}(\asll_L).$ Proposition \ref{p:hw} now implies that we have an injective $H\otimes \UU_q^{\prime}(\asll_N)\otimes \UU_q^{\prime}(\asll_L)$ - linear homomorphism
\begin{equation}
\bigoplus_{ \{\Lambda \in P_N^+(L)\: |\: \overline{\Lambda} \equiv \overline{\Lambda}_M\bmod \overline{Q}_N \}} {\mathbb K\hskip.5pt}[H_-]\otimes V(\Lambda)\otimes V(\dot{\Lambda}^{(M)}) \; \rightarrow \; {\mathcal F}_M
\label{eq:hom} \end{equation}
sending $1\otimes v_{\Lambda} \otimes v_{\dot{\Lambda}^{(M)}}$ to $\psi_{\Lambda}.$ It is known (cf. \cite{F1}[Theorem 1.6]) that (\ref{eq:hom}) specializes to an isomorphism when $q=1$.
The characters of ${\mathbb K\hskip.5pt}[H_-],$ $V(\Lambda),$ $V(\dot{\Lambda}^{(M)}),$ and ${\mathcal F}_M$ remain unchanged when $q$ is specialized to $1.$ Therefore (\ref{eq:hom}) is an isomorphism.
Summarizing, we have the following theorem.
\begin{theor} \label{t:decofF}
There is an isomorphism of $H\otimes \UU_q^{\prime}(\asll_N)\otimes \UU_q^{\prime}(\asll_L)$-modules:
\begin{equation}
{\mathcal F}_M \cong \bigoplus_{ \{\Lambda \in P_N^+(L)\: |\: \overline{\Lambda} \equiv \overline{\Lambda}_M\bmod \overline{Q}_N \}} {\mathbb K\hskip.5pt}[H_-]\otimes V(\Lambda)\otimes V(\dot{\Lambda}^{(M)}) .
\end{equation}
\end{theor}
\section{The toroidal Hecke algebra and the quantum toroidal algebra} \label{s:tor}
\subsection{Toroidal Hecke algebra} \label{sec:torHecke}
\mbox{} From now on we will work over the base field $\mathbb Q(q^{\frac{1}{2N}})$ rather than $\mathbb Q(q).$ Until the end of the paper we put ${\mathbb K\hskip.5pt} = \mathbb Q(q^{\frac{1}{2N}}).$ Clearly, all results of the preceding sections hold for this ${\mathbb K\hskip.5pt}.$
The toroidal Hecke algebra of type $\mathfrak {gl}_n$, $\ddot{\mathbf {H}}_n,$ \cite{VV1,VV2} is a unital associative algebra over ${\mathbb K\hskip.5pt}$ with the generators $\mathbf x^{\pm 1},$ $T_i^{\pm 1},X_j^{\pm 1},Y_j^{\pm 1}, 1\leqslant i < n, 1\leqslant j \leqslant n.$ The defining relations involving $T_i^{\pm 1},X_j^{\pm 1}$ are those of the affine Hecke algebra (\ref{eq:ah1} -- \ref{eq:ah2}), and the rest of the relations are as follows:
\begin{eqnarray*}
& \text{ the elements $\mathbf x^{\pm 1}$ are central,} & \qquad \mathbf x \mathbf x^{-1} = \mathbf x^{-1}\mathbf x = 1, \\
& Y_j Y_j^{-1} = Y_j^{-1} Y_j = 1, & \qquad Y_i Y_j = Y_j Y_i, \\
& T_i^{-1} Y_i T_i^{-1} = q^{-2} Y_{i+1}, &\qquad T_i Y_j = Y_j T_i \quad \text{if $ j \neq i,i+1.$} \\
&(X_1 X_2 \cdots X_n)Y_1 = \mathbf x Y_1 (X_1 X_2 \cdots X_n), &\qquad X_2 Y_1^{-1} X_2^{-1} Y_1 = q^{-2} T_1^2.
\end{eqnarray*}
The subalgebras of $\ddot{\mathbf {H}}_n$ generated by $T_i^{\pm 1},X_j^{\pm 1}$ and by $T_i^{\pm 1},Y_j^{\pm 1}$ are both isomorphic to the affine Hecke algebra $\dot{\mathbf {H}}_n$ (cf. \cite{VV1}, \cite{VV2}). \\
\noindent Following \cite{C3} we introduce a representation of the toroidal Hecke algebra on the space $ ({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}$ $=$ $ {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_n^{\pm 1}] \otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n}.$ This representation is an extension of the representation of $\dot{\mathbf {H}}_n = \langle T_i^{\pm 1} , X_j \rangle $ described in Section \ref{sec:affH}.
Let $\nu = \sum_{a=1}^L \nu(a) \epsilon_a,$ where $\epsilon_a = \dot{\overline{\Lambda}}_a - \dot{\overline{\Lambda}}_{a-1},$ be an integral weight of $\mathfrak {sl}_L$ $( \nu(a) \in \mathbb Z).$ Define $q^{\nu^{\vee}} \in \mathrm {End} \left({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L\right)$ as follows:
$$ q^{\nu^{\vee}}( z^m\mathfrak e_a) = q^{\nu(L+1-a)} z^m\mathfrak e_a.$$
Here the basis $\mathfrak e_1,\dots,\mathfrak e_L$ of ${\mathbb K\hskip.5pt}^L$ is the same as in Section \ref{sec:Usl}. For $p \in q^{\mathbb Z}$ define $p^D \in \mathrm {End} \left({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L\right)$ as
$$ p^{D}( z^m\mathfrak e_a) = p^{m} z^m\mathfrak e_a.$$
For $i=1,2,\dots,n-1$ let $s_i$ be the permutation operator of factors $i$ and $i+1$ in $ ({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n},$ and let $\tilde{T}_{i,i+1} = - q (\Tc_i)^{-1}.$ Here $\Tc_i$ is the generator of the finite Hecke algebra defined in (\ref{eq:Tci}). For $X \in \mathrm {End} \left({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L\right)$ let $$(X)_i := 1^{\otimes (i-1)} \otimes X \otimes 1^{\otimes (n-i-1)}\in \mathrm {End} \left({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L\right)^{\otimes n}.$$
For $i=1,2,\dots,n$ define the matrix analogue of the Cherednik-Dunkl operator \cite{C3} as
\begin{equation}
Y_i^{(n)} = \tilde{T}^{-1}_{i,i+1}\cdots \tilde{T}^{-1}_{n-1,n} s_{n-1} s_{n-2} \cdots s_1 (p^D)_1 (q^{\nu^{\vee}})_1 \tilde{T}_{1,2}\cdots \tilde{T}_{i-1,i}.
\label{eq:Y}
\end{equation}
Let $s\in \{0,1,\dots,NL-1\}$ and $m\in \mathbb Z$ be defined from $n = s + NLm.$ Put $\underlineun{n} = Nm.$
\begin{propos}[\cite{C3}] \label{p:torHeckerep}
The map
$$ T_i \mapsto \Tc_i,\quad X_i \mapsto z_i, \quad Y_i \mapsto q^{-{\underlineun{n}}} Y_i^{(n)}, \quad \mathbf x \mapsto p 1 $$
extends to a right representation of $\ddot{\mathbf {H}}_n$ on $({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}.$
\end{propos}
\noindent{\bf Remark.} The normalizing factor $q^{-{\underlineun{n}}}$ in the map $Y_i \mapsto q^{-{\underlineun{n}}} Y_i^{(n)}$ above clearly can be replaced by any coefficient in ${\mathbb K\hskip.5pt}.$ The adopted choice of this factor makes $q^{-{\underlineun{n}}} Y_i^{(n)}$ to behave appropriately (see Proposition \ref{p:inter}) with respect to increments of $n$ by steps of the value $NL.$ \\
\noindent Let $\chi = \sum_{a=1}^L \chi(a) \epsilon_a $ be an integral weight of $\mathfrak {sl}_L.$ Let $\operatorname{U}_q({\mathfrak b}_L)^{\chi}$ be the non-unital subalgebra of $\UU_q^{\prime}(\asll_L)$ generated by the elements
\begin{equation}\dot{F}_0,\dot{F}_1,\dots,\dot{F}_{L-1} \quad \text{and} \quad \dot{K}_a - q^{\chi(a) - \chi(a+1)} 1 \quad (a=1,\dots,L-1).
\label{eq:gen} \end{equation}
We define an action of $\UU_q^{\prime}(\asll_L)$ on ${\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L$ by the obvious restriction of the action on ${\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L \otimes {\mathbb K\hskip.5pt}^N$ defined in (\ref{eq:ul1} -- \ref{eq:ul2}). Iterating the coproduct $\Delta^-$ given in (\ref{eq:co1} -- \ref{eq:co3}) we obtain an action of $\UU_q^{\prime}(\asll_L)$ on $({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}.$
\begin{propos} \label{p:inv1}
Suppose $p = q^{-2L},$ and $ \nu = -\chi - 2\rho,$ where $\rho = \sum_{a=1}^{L-1} \dot{\overline{\Lambda}}_a.$ Then the action of the toroidal Hecke algebra on $({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}$ defined in Proposition \ref{p:torHeckerep} leaves invariant the subspace $\operatorname{U}_q({\mathfrak b}_L)^{\chi} \left(({\mathbb K\hskip.5pt}[z^{\pm 1}] \otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}\right).$
\end{propos}
\begin{proof}
It is clear that the multiplication by $z_i,$ and hence action of $X_i$ commutes with all generators of $\UU_q^{\prime}(\asll_L).$ \mbox{} From the intertwining property of the $R$-matrix it follows that the operators $\Tc_i$ (cf. \ref{eq:Tc}) commute with all generators of $\UU_q^{\prime}(\asll_L)$ as well. With $p = q^{-2L},$ and $ \nu = -\chi - 2\rho,$ a direct computation gives
\begin{align*}
&Y_n^{(n)} \dot{F}_a = \left( (q^{\chi(a)-\chi(a+1)}1 - \dot{K}_a) \dot{K}_a^{-1} (\dot{F}_a)_n (\dot{K}_a)_n + \dot{F}_a (\dot{K}_a)_n \right) Y_n^{(n)} \quad (a=1,\dots,L-1),\\
&Y_n^{(n)} \dot{F}_0 = \left( (q^{\chi(L)-\chi(1)}1 - \dot{K}_0) \dot{K}_0^{-1} (\dot{F}_0)_n (\dot{K}_0)_n + \dot{F}_0 (\dot{K}_0)_n \right) Y_n^{(n)}.
\end{align*}
In view of the relation $\Tc_i Y_{i+1}^{(n)} \Tc_i = q^2 Y_i^{(n)},$ and the commutativity of $\Tc_i$ with the generators of $\UU_q^{\prime}(\asll_L),$ this shows that for all $i$ the operators $Y_i^{(n)}$ leave the image of $\operatorname{U}_q({\mathfrak b}_L)^{\chi} $ invariant.
\end{proof}
\subsection{The quantum toroidal algebra} \label{sec:tor}
Fix an integer $N\geqslant 3.$
The quantum toroidal algebra of type $\mathfrak {sl}_N,$ $\ddot{\UU},$ is an associative unital algebra over ${\mathbb K\hskip.5pt}$ with generators:
$$E_{i,k},\quad F_{i,k},\quad H_{i,l},\quad K_i^{\pm 1}, \quad
q^{\pm\frac12 c}, \quad \mathbf d^{\pm 1},$$
where $k\in {\mathbb Z}$, $l\in {\mathbb Z}\backslash \{0\}$ and $i=0,1,\cdots,N-1$. The generators $q^{\pm\frac12 c}$ and $\mathbf d^{\pm 1}$ are central. The rest of the defining relations are expressed in terms of the formal series
$$E_i(z)=\sum_{k\in {\mathbb Z}}E_{i,k}z^{-k}, \quad
F_i(z)=\sum_{k\in {\mathbb Z}}F_{i,k}z^{-k}, \quad
K_i^{\pm}(z)=K_i^{\pm 1}\exp(\pm (q-q^{-1})\sum_{k\geqslant 1}H_{i,\pm k}
z^{\mp k}),$$
as follows:
\begin{gather}
K_i K_i^{-1}=K_i^{-1}K_i= q^{\frac12 c} q^{-\frac12 c} = q^{-\frac12 c} q^{\frac12 c}= \mathbf d \mathbf d^{-1} = \mathbf d^{-1} \mathbf d = 1, \\
K_i^{\pm}(z)K_j^{\pm}(w)=K_j^{\pm}(w)K_i^{\pm}(z)
\label{rb} \\
\theta_{- a_{ij}}
(q^{-c}\mathbf d^{m_{ij}}\frac{z}{w})K_i^-(z)K_j^+(w)=
\theta_{-a_{ij}}
(q^{c}\mathbf d^{m_{ij}}\frac{z}{w})K_j^+(w)
K_i^-(z)
\label{rc} \\
K_i^{\pm}(z)E_j(w)
=\theta_{\mp a_{ij}}
(q^{-\frac12 c}\mathbf d^{\mp m_{ij}}w^{\pm}z^{\mp})E_j(w)K_i^+(z)
\label{rd} \\
K_i^{\pm}(z)F_j(w)
=\theta_{\pm a_{ij}}
(q^{\frac12 c}\mathbf d^{\mp m_{ij}}w^{\pm}z^{\mp})F_j(w)K_i^+(z) \\
[E_i(z),F_j(w)]=\delta_{i,j}\frac{1}{q-q^{-1}}
\{\delta(q^c\frac{w}{z})K_i^+(q^{\frac12 c}w)-\delta(q^c\frac{z}{w})K_i^
-(q^{\frac12 c}z)\}
\label{re} \\
(\mathbf d^{m_{ij}}z-q^{a_{ij}}w)E_i(z)E_j(w)
=(q^{a_{ij}}\mathbf d^{m_{ij}}z-w)E_j(w)E_i(z)
\label{rf} \\
(\mathbf d^{m_{ij}}z-q^{-a_{ij}}w)F_i(z)F_j(w)
=(q^{-a_{ij}}\mathbf d^{m_{ij}}z-w)F_j(w)F_i(z) \\
\sum_{\sigma\in {\mathfrak S}_m}\sum_{r=0}^m(-1)^r
\begin{bmatrix}m\\r\end{bmatrix}
E_i(z_{\sigma(1)})\cdots E_i(z_{\sigma(r)})E_j(w)E_i(z_{\sigma(r+1)})\cdots
E_i(z_{\sigma(m)})=0
\label{rg} \\
\sum_{\sigma\in {\mathfrak S}_m}\sum_{r=0}^m(-1)^r
\begin{bmatrix}m\\r\end{bmatrix}
{} F_i(z_{\sigma(1)})\cdots F_i(z_{\sigma(r)})F_j(w)F_i(z_{\sigma(r+1)})\cdots
{} F_i(z_{\sigma(m)}) = 0 \label{rg1}
\end{gather}
where in (\ref{rg}) and (\ref{rg1}) $i\ne j$ and $m=1-a_{ij}$.\\
\noindent In these defining relations $\delta(z) = \sum_{n = -\infty}^{\infty} z^n,$ $\theta_m(z) \in {\mathbb K\hskip.5pt}[[z]] $ is the expansion of $\frac{zq^m-1}{z-q^m},$ $a_{ij}$ are the entries of the Cartan matrix of $\widehat{{\mathfrak {sl}}}_N,$ and
$m_{ij}$ are the entries
of the following $N\times N$-matrix
$$
M=\begin{pmatrix}
0 & -1 & 0 & \hdots & 0 & 1\\
1 & 0 & -1 & \hdots & 0 & 0\\
0 & 1 & 0 & \hdots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & 0 & \hdots & 0 & -1\\
-1 & 0 & 0 & \hdots & 1 & 0
\end{pmatrix}.
$$
Let $\operatorname{U}_h$ be the subalgebra of $\ddot{\UU}$ generated by the elements $E_{i,0},F_{i,0},K_i^{\pm 1}$ $(0\leqslant i < N).$ These elements satisfy the defining relations (\ref{eq:r1} -- \ref{eq:r3}) and (\ref{eq:r5} -- \ref{eq:r7}) of $\UU_q^{\prime}(\asll_N).$ Thus the following map extends to a homomorphism of algebras:
\begin{equation}\UU_q^{\prime}(\asll_N) \rightarrow \operatorname{U}_h :\; E_i \mapsto E_{i,0}, \quad F_i \mapsto F_{i,0}, \quad K_i^{\pm 1} \mapsto K_i^{\pm 1}. \label{eq:Uh}
\end{equation}
Let $\operatorname{U}_v$ be the subalgebra of $\ddot{\UU}$ generated by the elements $E_{i,k},F_{i,k},H_{i,l},$ $K_i^{\pm 1}$ $(1\leqslant i < N; k\in \mathbb Z; l\in \mathbb Z_{\neq 0}),$ and $q^{\pm \frac12 c}, \mathbf d^{\pm 1}.$ Recall, that apart from the presentation given in Section \ref{sec:Usl}, the algebra $\UU_q^{\prime}(\asll_N)$ has the ``new presentation'' due to Drinfeld which is similar to that one of $\ddot{\UU}$ above. A proof of the isomorphism between the two presentations is announced in \cite{Drinfeld1} and given in \cite{Beck}. Let $\tilde{E}_{i,k},\tilde{F}_{i,k},\tilde{H}_{i,l},\tilde{K}_{i}^{\pm 1},$ $(1\leqslant i < N; k\in \mathbb Z; l\in \mathbb Z_{\neq 0}),$ and $ q^{\pm \frac12 \tilde{c}}$ be the generators of $\UU_q^{\prime}(\asll_N)$ in the realization of \cite{Drinfeld1}. Comparing this realization of $\UU_q^{\prime}(\asll_N)$ with the defining relations of $\ddot{\UU}$ one easily sees that the map
\begin{align}
\UU_q^{\prime}(\asll_N) \rightarrow \operatorname{U}_v :\; &\tilde{E}_{i,k} \mapsto \mathbf d^{ik}E_{i,k},\quad \tilde{F}_{i,k} \mapsto \mathbf d^{ik}F_{i,k},\quad \tilde{H}_{i,l} \mapsto \mathbf d^{il}H_{i,l},\label{eq:Uv}\\ &\tilde{K}_i^{\pm 1} \mapsto K_i^{\pm 1}, \quad q^{\pm \frac12 \tilde{c}} \mapsto q^{\pm \frac12 c} \nonumber
\end{align}
where $1 \leqslant i < N,$ extends to a homomorphism of algebras.
Thus each module of $\ddot{\UU}$ carries two actions of $\UU_q^{\prime}(\asll_N)$ obtained by pull-backs through the homomorphisms (\ref{eq:Uh}) and (\ref{eq:Uv}). We will say that a module of $\ddot{\UU}$ has {\em level} $(l_v,l_h)$ provided the action of $\UU_q^{\prime}(\asll_N)$ obtained through the homomorphism (\ref{eq:Uh}) has level $l_h,$ and the action of $\UU_q^{\prime}(\asll_N)$ obtained through the homomorphism (\ref{eq:Uv}) has level $l_v.$ On such a module the central elements $q^{\pm \frac12 c}$ act as multiplications by $q^{\pm \frac12 l_v},$ and the element $K_0 K_1 \cdots K_{N-1}$ acts as the multiplication by $q^{l_h}.$
The following proposition, proved in \cite{VV1}, shows that it is sometimes possible to extend a representation of $\UU_q^{\prime}(\asll_N)$ to a representation of $\ddot{\UU}.$
\begin{propos} \label{p:shift1}
Let $W$ be a module of $\UU_q^{\prime}(\asll_N).$ Suppose that there are $a,b \in q^{\mathbb Z},$ and an invertible $\tilde{\psi} \in \mathrm {End}(W)$ such that
\begin{alignat}{5}
&\tilde{\psi}^{-1}\tilde{E}_i(z)\tilde{\psi} = \tilde{E}_{i-1}(az),& &\tilde{\psi}^{-2}\tilde{E}_1(z)\tilde{\psi}^2 = \tilde{E}_{N-1}(bz),& \\
&\tilde{\psi}^{-1}\tilde{F}_i(z)\tilde{\psi} = \tilde{F}_{i-1}(az),& &\tilde{\psi}^{-2}\tilde{F}_1(z)\tilde{\psi}^2 = \tilde{F}_{N-1}(bz),& \\
&\tilde{\psi}^{-1}\tilde{K}_i^{\pm}(z)\tilde{\psi} = \tilde{K}_{i-1}^{\pm}(az),& \qquad&\tilde{\psi}^{-2}\tilde{K}_1^{\pm}(z)\tilde{\psi}^2 = \tilde{K}_{N-1}^{\pm}(bz),&
\end{alignat}
where $2\leqslant i < N.$ Then $W$ is a $\ddot{\UU}$-module with the action given by
\begin{align*}
& X_i(z) = \tilde{X}_i(d^iz) \qquad (1\leqslant i <N), \qquad
X_0(z) = \tilde{\psi}^{-1}\tilde{X}_1(a^{-1}d^{-1}z)\tilde{\psi}, \\
& \mathbf d = d 1, \qquad q^{\frac12 c} = q^{\frac12 \tilde{c}}.
\end{align*}
where $d^N = b/a^2,$ and $X = E,F,K^{\pm}.$
\end{propos}
\subsection{The Varagnolo-Vasserot duality} \label{sec:VVdual} We now briefly review, following \cite{VV1}, the Schur-type duality between the toroidal Hecke algebra $\ddot{\mathbf {H}}_n$ and the quantum toroidal algebra $\ddot{\UU}.$
Let $\operatorname M$ be a right $\ddot{\mathbf {H}}_n $-module, such that the central element $\mathbf x$ of $\ddot{\mathbf {H}}_n$ acts as the multiplication by $x \in q^{\mathbb Z}.$ The algebra $\ddot{\mathbf {H}}_n$ contains two subalgebras: $\dot{\mathbf {H}}_n^h = \langle T_i^{\pm 1} , X_j \rangle ,$ and $\dot{\mathbf {H}}_n^v = \langle T_i^{\pm 1} , Y_j \rangle$ both isomorphic to the affine Hecke algebra $\dot{\mathbf {H}}_n.$ Therefore the duality functor of Chari--Pressley \cite{CP} yields two actions of $\UU_q^{\prime}(\asll_N)$ on the linear space
$ \operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}.$ Here the action of the finite Hecke algebra $\mathbf {H}_n$ on $({\mathbb K\hskip.5pt}^N)^{\otimes n}$ is given by (\ref{eq:Tsi}), and $\mathbf {H}_n$ is embedded into $\ddot{\mathbf {H}}_n$ as the subalgebra generated by $T_i^{\pm 1}.$
For $i,j=1,\dots,N$ let $e_{i,j} \in \mathrm {End}({\mathbb K\hskip.5pt}^N)$ be the matrix units with respect to the basis $\mathfrak v_1,\mathfrak v_2,\dots,\mathfrak v_{N}$ (cf. Section \ref{sec:Usl}). For $i=0,1,\dots,N-1$ let $k_i = q^{e_{i,i} - e_{i+1,i+1}},$ where the indices are cyclically extended modulo $N$. For $X \in \mathrm {End}({\mathbb K\hskip.5pt}^N)$ we put $(X)_i = 1^{\otimes (i-1)}\otimes X \otimes 1^{\otimes (n-i)}.$
The functor of \cite{CP} applied to $\operatorname M$ considered as the $\dot{\mathbf {H}}_n^h$-module gives the following action of $\UU_q^{\prime}(\asll_N)$ on $ \operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}:$
\begin{align}
&E_i(m\otimes v) = \sum_{j=1}^n m X_j^{\delta(i=0)} \otimes (e_{i,i+1})_j (k_i)_{j+1} (k_i)_{j+2}\cdots (k_i)_{n}v, \label{eq:h1}\\
&F_i(m\otimes v) = \sum_{j=1}^n m X_j^{-\delta(i=0)} \otimes (e_{i+1,i})_j (k_i^{-1})_{1} (k_i^{-1})_{2}\cdots (k_i^{-1})_{j-1}v, \\
&K_i(m\otimes v) = m \otimes (k_i)_1(k_i)_2 \cdots (k_i)_n v. \label{eq:h3}
\end{align}
Here $m \in \operatorname M, v \in ({\mathbb K\hskip.5pt}^N)^{\otimes n},$ and the indices are cyclically extended modulo $N.$ Likewise, application of this functor to $\operatorname M$ considered as the $\dot{\mathbf {H}}_n^v$-module gives another action of $\UU_q^{\prime}(\asll_N)$ on $ \operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}:$
\begin{align}
&\hat{E}_i(m\otimes v) = \sum_{j=1}^n m Y_j^{-\delta(i=0)} \otimes (e_{i,i+1})_j (k_i)_{j+1} (k_i)_{j+2}\cdots (k_i)_{n}v, \label{eq:v1}\\
&\hat{F}_i(m\otimes v) = \sum_{j=1}^n m Y_j^{\delta(i=0)} \otimes (e_{i+1,i})_j (k_i^{-1})_{1} (k_i^{-1})_{2}\cdots (k_i^{-1})_{j-1}v, \\
&\hat{K}_i(m\otimes v) = m \otimes (k_i)_1(k_i)_2 \cdots (k_i)_n v. \label{eq:v3}
\end{align}
Here we put hats over the generators in order to distinguish the actions given by (\ref{eq:h1} -- \ref{eq:h3}) and (\ref{eq:v1} -- \ref{eq:v3}).
Varagnolo and Vasserot have proven, in \cite{VV1}, that $ \operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ is a $\ddot{\UU}$-module such that the $\UU_q^{\prime}(\asll_N)$-action (\ref{eq:h1} -- \ref{eq:h3}) is the pull-back through the homomorphism (\ref{eq:Uh}), and the $\UU_q^{\prime}(\asll_N)$-action (\ref{eq:v1} -- \ref{eq:v3}) is the pull-back through the homomorphism (\ref{eq:Uv}). Let us recall here the main element of their proof.
Let $\psi$ be the endomorphism of $ \operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ defined by
\begin{gather}
\psi : m \otimes \mathfrak v_{\epsilon_1}\otimes \mathfrak v_{\epsilon_2}\otimes \cdots \otimes \mathfrak v_{\epsilon_n} \mapsto\label{eq:psi} \\ m X_1^{-\delta_{N,\epsilon_1}}X_2^{-\delta_{N,\epsilon_2}} \cdots X_n^{-\delta_{N,\epsilon_n}} \otimes \mathfrak v_{\epsilon_1+1}\otimes \mathfrak v_{\epsilon_2+1}\otimes \cdots \otimes \mathfrak v_{\epsilon_n+1}, \nonumber
\end{gather}
where $\mathfrak v_{N+1}$ is identified with $\mathfrak v_1.$ Taking into account the defining relations of $\ddot{\mathbf {H}}_n$ one can confirm that $\psi$ is well-defined.
Let $\tilde{E}_{i,k},\tilde{F}_{i,k},\tilde{H}_{i,l},\tilde{K}_{i}^{\pm 1}$ $(k\in \mathbb Z;l\in \mathbb Z_{\neq 0}; 1\leqslant i < N)$ be the generators of the $\UU_q^{\prime}(\asll_N)$-action (\ref{eq:v1} -- \ref{eq:v3}) obtained from $\hat{E}_j,\hat{F}_j,\hat{K}_j^{\pm 1}$ $(0\leqslant j < N)$ by the isomorphism between the two realizations of $\UU_q^{\prime}(\asll_N)$ given in \cite{Beck}. Let $\tilde{E}_i(z), \tilde{F}_i(z), \tilde{K}^{\pm}_i(z)$ be the corresponding generating series.
\begin{propos}[\cite{VV1}] \label{p:fintwist}
The following relations hold in $\operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}:$
\begin{alignat}{5}
&{\psi}^{-1}\tilde{E}_i(z){\psi} = \tilde{E}_{i-1}(q^{-1}z),& &{\psi}^{-2}\tilde{E}_1(z){\psi}^2 = \tilde{E}_{N-1}(x^{-1}q^{N-2}z),& \\
&{\psi}^{-1}\tilde{F}_i(z){\psi} = \tilde{F}_{i-1}(q^{-1}z),& &{\psi}^{-2}\tilde{F}_1(z){\psi}^2 = \tilde{F}_{N-1}(x^{-1}q^{N-2}z),& \\
&{\psi}^{-1}\tilde{K}_i^{\pm}(z){\psi} = \tilde{K}_{i-1}^{\pm}(q^{-1}z),& \qquad&{\psi}^{-2}\tilde{K}_1^{\pm}(z){\psi}^2 = \tilde{K}_{N-1}^{\pm}(x^{-1}q^{N-2}z).&
\end{alignat}
Here $2\leqslant i < N.$
\end{propos}
\noindent Proposition \ref{p:shift1} now implies that $\operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ is a $\ddot{\UU}$-module, in particular, the central element $\mathbf d$ acts as the multiplication by $x^{-1/N}q,$ and the central element $q^{\frac12 c}$ acts as the multiplication by $1.$
\subsection{The action of the quantum toroidal algebra on the wedge product}
In the framework of the preceding section, let $\operatorname M = ({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}$ be the $\ddot{\mathbf {H}}_n$-module with the action given in Proposition \ref{p:torHeckerep}. In view of the remark made in Section \ref{sec:wedge}, the linear space $\operatorname M\otimes_{\mathbf {H}_n} ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ is isomorphic to the wedge product $\wedge^nV_{\mathrm {aff}}.$ Therefore, by the Varagnolo-Vasserot duality, $\wedge^nV_{\mathrm {aff}}$ is a module of $\ddot{\UU}.$ The action of $\UU_q^{\prime}(\asll_N)$ given by (\ref{eq:h1} -- \ref{eq:h3}) coincides with the action of $\UU_q^{\prime}(\asll_N)$ defined on $\wedge^nV_{\mathrm {aff}}$ in Section \ref{sec:wedge}. Following the terminology of \cite{VV2}, we will call this action {\em the horizontal} action of $\UU_q^{\prime}(\asll_N)$ on $\wedge^n V_{\mathrm {aff}}.$ The formulas (\ref{eq:v1} -- \ref{eq:v3}) give another action of $\UU_q^{\prime}(\asll_N)$ on $\wedge^n V_{\mathrm {aff}},$ we will refer to this action as {\em the vertical } action.
Recall, that in Section \ref{sec:wedge} an action of $\UU_q^{\prime}(\asll_L),$ commutative with the horizontal action of $\UU_q^{\prime}(\asll_N),$ was defined on $\wedge^nV_{\mathrm {aff}}.$ Recall, as well, that for each integral weight $\chi$ of $\mathfrak {sl}_L$ we have defined, in Section \ref{sec:torHecke}, the subalgebra $\operatorname{U}_q({\mathfrak b}_L)^{\chi} $ of $\UU_q^{\prime}(\asll_L).$
The $\ddot{\mathbf {H}}_n$-module structure defined in Proposition \ref{p:torHeckerep} depends on two parameters: $\nu$ which is an integral weight of $\mathfrak {sl}_L,$ and $p \in q^{\mathbb Z}.$ The same parameters thus enter into the $\ddot{\UU}$-module structure on $\wedge^n V_{\mathrm {aff}}.$
\begin{propos}\label{p:inv2}
Suppose $p = q^{-2L},$ and $\nu = - \chi - 2\rho$ for an integral $\mathfrak {sl}_L$-weight $\chi.$ Then the action of $\ddot{\UU}$ on $\wedge^n V_{\mathrm {aff}}$ leaves invariant the linear subspace $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left(\wedge^n V_{\mathrm {aff}} \right).$
\end{propos}
\begin{proof}
It is not difficult to see, that the subalgebras $\operatorname{U}_h$ and $\operatorname{U}_v$ generate $\ddot{\UU}$ (cf. Lemma 2 in \cite{STU}). Therefore, to prove the proposition, it is enough to show, that both the horizontal and the vertical actions of $\UU_q^{\prime}(\asll_N)$ on $\wedge^n V_{\mathrm {aff}}$ leave $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left(\wedge^n V_{\mathrm {aff}} \right)$ invariant. However, the horizontal action commutes with the action of $\UU_q^{\prime}(\asll_L),$ while Proposition \ref{p:inv1} implies that the vertical action leaves $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left(\wedge^n V_{\mathrm {aff}} \right)$ invariant.
\end{proof}
\section{The actions of the quantum toroidal algebra on the Fock spaces and on irreducible integrable highest weight modules of $\operatorname{U}_q^{\prime}(\widehat{{\mathfrak {gl}}}_N)$} \label{s:toract}
\subsection{A level 0 action of $\UU_q^{\prime}(\asll_N)$ on the Fock space}
Let $\pi^v_{(n)} : \UU_q^{\prime}(\asll_N) \rightarrow \mathrm {End}(\wedge^n V_{\mathrm {aff}})$ be the map defining the vertical action of $\UU_q^{\prime}(\asll_N)$ on the wedge product $\wedge^nV_{\mathrm {aff}}.$ In accordance with (\ref{eq:v1} -- \ref{eq:v3}), for $f\in ({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}$ and $v \in ({\mathbb K\hskip.5pt}^N)^{\otimes n }$ we have
\begin{align}
&\pi^v_{(n)}(E_i)\cdot \wedge(f\otimes v) = \wedge \sum_{j=1}^n (q^{-\underlineun{n}}Y^{(n)}_j)^{-\delta(i=0)} f \otimes (e_{i,i+1})_j (k_i)_{j+1} (k_i)_{j+2}\cdots (k_i)_{n}v, \label{eq:pi1}\\
&\pi^v_{(n)}(F_i)\cdot \wedge(f\otimes v) = \wedge \sum_{j=1}^n (q^{-\underlineun{n}}Y^{(n)}_j)^{\delta(i=0)} f \otimes (e_{i+1,i})_j (k_i^{-1})_{1} (k_i^{-1})_{2}\cdots (k_i^{-1})_{j-1}v,\\
&\pi^v_{(n)}(K_i)\cdot \wedge(f\otimes v) = \wedge f \otimes (k_i)_1(k_i)_2 \cdots (k_i)_n v, \label{eq:pi3}
\end{align}
where we denote by $\wedge$ the canonical map from $V_{\mathrm {aff}}^{\otimes n}$ $ = $ $
({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes n }$ to $\wedge^n V_{\mathrm {aff}}.$
In this section, for each $M\in \mathbb Z,$ we define a level 0 action of $\UU_q^{\prime}(\asll_N)$ on the Fock space ${\mathcal F}_M.$ Informally, this action arises as the limit $n\rightarrow \infty$ of the vertical action (\ref{eq:pi1} -- \ref{eq:pi3}) on the wedge product. In parallel with the finite case, the Fock space, thus admits two actions of $\UU_q^{\prime}(\asll_N):$ the level $L$ action defined in Section \ref{sec:UF} as the inductive limit of the horizontal action, and an extra action with level zero.
We start by introducing a grading on ${\mathcal F}_M.$ To facilitate this, we adopt the following notational convention. For each integer $k$ we define the unique triple $\overline{k},\dot{k},\underline{k},$ where $\overline{k} \in \{1,2,\dots ,N\},$ $\dot{k} \in \{1,2,\dots ,L\},$ $\underline{k} \in \mathbb Z$ by
$$ k = \overline{k} - N( \dot{k} + L \underline{k}) .$$
Then (cf. Section \ref{sec:nowedges}) we have $u_k = z^{\underline{k}}\mathfrak e_{\dot{k}}\mathfrak v_{\overline{k}}.$ The Fock space ${\mathcal F}_M$ has a basis formed by normally ordered semi-infinite wedges
$ u_{k_1}\wedge u_{k_2} \wedge \cdots $
where the decreasing sequence of momenta $k_1,k_2,\dots$ satisfies the asymptotic condition $k_i = M-i+1$ for $i\gg 1.$ Let $o_1,o_2,\dots$ be the sequence of momenta labeling the vacuum vector $|M\rangle$ of ${\mathcal F}_M,$ i.e.: $o_i = M-i+1$ for all $i \geqslant 1.$ Define the degree of a semi-infinite normally ordered wedge by
\begin{equation}
\deg u_{k_1}\wedge u_{k_2} \wedge \cdots = \sum_{i\geqslant 1} \underline{o_i} - \underline{k_i}.
\label{eq:sideg}\end{equation}
Let ${\mathcal F}_M^d$ be the homogeneous component of ${\mathcal F}_M$ of degree $d.$ Clearly, the asymptotic condition $k_i = M-i+1$ $(i\gg 1)$ implies that
$$ {\mathcal F}_M = \bigoplus_{d=0}^{\infty} {\mathcal F}_M^d.$$
Let $s \in \{0,1,\dots,NL-1\}$ be defined from $M\equiv s\bmod NL.$ For a non-negative integer $l$ we define the linear subspace $V_{M,s+lNL}$ of $\wedge^{s+lNL}V_{\mathrm {aff}}$ by
\begin{equation}
V_{M,s+lNL} = \bigoplus_{\underline{k_{s+lNL}} \leqslant \underline{o_{s+lNL}}} {\mathbb K\hskip.5pt} u_{k_1}\wedge u_{k_2}\wedge \cdots \wedge u_{k_{s+lNL}}, \label{eq:VM}
\end{equation}
where the wedges in the right-hand side are assumed to be normally ordered. For $s=l=0$ we put $V_{M,s+lNL} = {\mathbb K\hskip.5pt}.$ The vector space (\ref{eq:VM}) has a grading similar to that one of the Fock space. Now the degree of a normally ordered wedge is defined as
\begin{equation}
\deg u_{k_1}\wedge u_{k_2} \wedge \cdots\wedge u_{k_{s+lNL}} = \sum_{i=1}^{s+lNL} \underline{o_i} - \underline{k_i}.
\label{eq:fdeg}\end{equation}
Note that this degree is necessarily a non-negative integer since $k_1 > k_2 > \cdots > k_{s+lNL}$ and $\underline{k_{s+lNL}} \leqslant \underline{o_{s+lNL}}$ imply $\underline{k_i} \leqslant \underline{o_i}$ for all $i=1,2,\dots,s+lNl.$ Let $V_{M,s+lNL}^d$ be the homogeneous component of $V_{M,s+lNL}$ of degree $d.$
For non-negative integers $d$ and $l$ introduce the following linear map:
\begin{equation}
\varrho_l^d : V_{M,s+lNL}^d \rightarrow {\mathcal F}_M^d \: : \: w \mapsto w\wedge | M-s-lNL \rangle.
\label{eq:rol}\end{equation}
The proof of the following proposition is straightforward (cf. Proposition 16 in \cite{STU}, or Proposition 3.3 in \cite{U}).
\begin{propos} \label{p:rho}
Suppose $l \geqslant d.$ Then $\varrho_l^d$ is an isomorphism of vector spaces.
\end{propos}
\noindent In view of this proposition, it is clear that for non-negative integers $d,l,m,$ such that $d \leqslant l < m,$ the linear map
\begin{equation}
\varrho_{l,m}^d : V_{M,s+lNL}^d \rightarrow V_{M,s+mNL}^d \: : \: w \mapsto w\wedge u_{M-s-lNL}\wedge u_{M-s-lNL-1}\wedge \cdots \wedge u_{M-s-mNL+1}
\label{eq:rolm}\end{equation}
is an isomorphism of vector spaces as well.
Now let us return to the vertical action $\pi^v_{(n)}$ of $\UU_q^{\prime}(\asll_N)$ on $\wedge^n V_{\mathrm {aff}} $ given by (\ref{eq:pi1} -- \ref{eq:pi3}).
\begin{propos}
For each $d=0,1,\dots$ the subspace $ V_{M,s+lNL}^d \subset \wedge^{s+lNL} V_{\mathrm {aff}} $ is invariant with respect to the action $\pi^v_{(s+lNL)}.$
\end{propos}
\begin{proof}
Let $n=s+lNL,$ and let us identify $V_{\mathrm {aff}}^{\otimes n}$ with ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_n^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes n}$ by the isomorphism
$$ z^{m_1}\mathfrak e_{a_1}\mathfrak v_{\epsilon_1} \otimes \cdots \otimes z^{m_n}\mathfrak e_{a_n}\mathfrak v_{\epsilon_n} \mapsto z_1^{m_1}\cdots z_n^{m_n} \mathfrak e_{a_1} \cdots \mathfrak e_{a_n} \mathfrak v_{\epsilon_n} \cdots \mathfrak v_{\epsilon_n}. $$
Then $V_{M,s+lNL}$ is the image, with respect to the quotient map $\wedge : V_{\mathrm {aff}}^{\otimes n} \rightarrow \wedge^n V_{\mathrm {aff}},$ of the subspace
\begin{equation} (z_1 \cdots z_n)^{\underline{o_n}} {\mathbb K\hskip.5pt}[z_1^{-1},\dots,z_n^{- 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes n} \subset V_{\mathrm {aff}}^{\otimes n},
\label{eq:ss}\end{equation}
while the grading on $V_{M,s+lNL}$ is induced from the grading of (\ref{eq:ss}) by eigenvalues of the operator $D = z_1\frac{\partial}{\partial z_1} + \cdots + z_n\frac{\partial}{\partial z_n}.$
The operators $Y_i^{(n)}$ leave $(z_1 \cdots z_n)^{\underline{o_n}} {\mathbb K\hskip.5pt}[z_1^{-1},\dots,z_n^{- 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes n} $ invariant, and commute with $D.$ Now (\ref{eq:pi1} -- \ref{eq:pi3}) imply the statement of the proposition.
\end{proof}
\begin{propos} \label{p:inter}
Let $0 \leqslant d \leqslant l,$ let $n=s+lNL,$ and let $X$ be any of the generators $E_i,F_i,K_i^{\pm 1}$ $(0\leqslant i < N)$ of $\UU_q^{\prime}(\asll_N).$ Then the following intertwining relation holds for all $w \in V_{M,s+lNL}^d:$
\begin{equation}
\pi^v_{(n+NL)}(X) \cdot \varrho_{l,l+1}^d (w) = \varrho_{l,l+1}^d \left(\pi^v_{(n)}(X)\cdot w \right).
\label{eq:inter}\end{equation}
Consequently, for $0 \leqslant d \leqslant l < m$ the map $\varrho_{l,m}^d$ defined in {\em (\ref{eq:rolm})} is an isomorphism of $\UU_q^{\prime}(\asll_N)$-modules.
\end{propos}
\begin{proof}
The proof is based, in particular, on Lemma \ref{l:ml}, to state which we introduce the following notation. For $\mathbf m = (m_1,m_2,\dots,m_n) \in \mathbb Z^n,$ and $\mathbf a = (a_1,a_2,\dots,a_n) \in \{1,2,\dots,L\}^n$ let
$$ \zeta_i(\mathbf m,\mathbf a) = p^{m_i} q^{\nu(L+1-a_i) + \mu_i(\mathbf m,\mathbf a)} \qquad (i=1,2,\dots,n) $$
where $p, \nu$ are the parameters of the representation of $\ddot{\mathbf {H}}_n$ introduced in Section \ref{sec:torHecke}, and $\mu_i(\mathbf m,\mathbf a) = - \#\{j < i | m_j < m_i, a_j = a_i\} + \#\{j < i | m_j \geqslant m_i, a_j = a_i\} + \#\{j > i | m_j > m_i, a_j = a_i\} - \#\{j > i | m_j \leqslant m_i, a_j = a_i\}.$
\begin{lemma}\label{l:ml}
For $k =1,2,\dots$ consider the following monomial
$$ f = z_1^{m_1} z_2^{m_2} \dots z_{n+k}^{m_{n+k}} \otimes \mathfrak e_{a_1}\mathfrak e_{a_2} \cdots \mathfrak e_{a_{n+k}} \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+k}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+k)}.$$
Assume that $m_1,m_2,\dots,m_n < m_{n+1} = m_{n+2} = \cdots = m_{n+k} =: m,$ and that $a_{n+i} \leqslant a_{n+j}$ for $ 1 \leqslant i < j \leqslant k.$ For $j \in \{ 1,2,\dots,L\}$ put $\overline{n}(j) = \#\{ \: i\: | \: a_{n+i} = j, 1\leqslant i \leqslant k\}.$
Define the linear subspaces $\mathcal K_{n,k}^m,\mathcal L_{n,k}^m \subset {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+k}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+k)}$ as follows:
\begin{align*}
&\mathcal K_{n,k}^m = {\mathbb K\hskip.5pt}\{ z_1^{m_1^{\prime}}\cdots z_{n+k}^{m_{n+k}^{\prime}} \otimes \mathfrak e \: | \: \mathfrak e \in ({\mathbb K\hskip.5pt}^L)^{n+k}; m_1^{\prime},\dots,m_{n+k}^{\prime} \leqslant m; \#\{ m_i^{\prime} | m_i^{\prime} = m \} < k\}, \\
&\mathcal L_{n,k}^m = {\mathbb K\hskip.5pt}\{ z_1^{m_1^{\prime}}\cdots z_{n+k}^{m_{n+k}^{\prime}} \otimes \mathfrak e_{b_1}\cdots \mathfrak e_{b_{n+k}} \: |\: m_1^{\prime},\dots,m_{n}^{\prime} < m ; m_{n+1}^{\prime},\dots,m_{n+k}^{\prime} = m; \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \exists j < a_{n+k} \:\text{{\em s.t.}}\: \#\{\: i \: | b_{n+i} = j, 1 \leqslant i \leqslant k\} > \overline{n}(j)\}.
\end{align*}
Then
\begin{align*}
&\left( Y_i^{(n+k)} \right)^{\pm 1}(f) \equiv \zeta_i(\mathbf m,\mathbf a)^{\pm 1} f \:\bmod \left( \mathcal K_{n,k}^m + \mathcal L_{n,k}^m \right) \quad (i=n+1,n+2,\dots,n+k), \\
&\left( Y_i^{(n+k)} \right)^{\pm 1}(f) \equiv q^{\pm \overline{n}(a_i)} \left( Y_i^{(n)} \right)^{\pm 1}(f) \: \bmod \left( \mathcal K_{n,k}^m + \mathcal L_{n,k}^m \right) \quad (i=1,2,\dots,n).
\end{align*}
Here $\mathbf m = (m_1,\dots,m_{n+k}),$ $\mathbf a = (a_1,\dots,a_{n+k}),$ and in the right-hand side of the last equation $\left( Y_i^{(n)} \right)^{\pm 1}$ act on the first $n$ factors of the monomial $f.$
\end{lemma}
\noindent A proof of the lemma is given in \cite{TU} for $L=1.$ A proof for general $L$ is quite similar and will be omitted here.
Let $w$ be a normally ordered wedge from $V_{M,n}^d,$ and let $\bar{w} = \varrho_{l,l+1}^d(w).$ The vector $\bar{w}$ is a normally ordered wedge from $V_{M,n+NL}^d,$ we have
\begin{equation}
\bar{w} = u_{k_1}\wedge u_{k_2} \wedge \cdots \wedge u_{k_{n+NL}} = \wedge (f \otimes v),
\end{equation}
where
\begin{align}
& f = (z_1^{\underline{k_1}}z_2^{\underline{k_2}} \cdots z_n^{\underline{k_n}})(z_{n+1} \cdots z_{n+NL})^{m}\otimes \label{eq:fff}\\ & \qquad \qquad \qquad \qquad\otimes (\mathfrak e_{\dot{k}_1}\mathfrak e_{\dot{k}_2} \cdots \mathfrak e_{\dot{k}_n})\underlinederbrace{(\mathfrak e_1 \cdots \mathfrak e_1)}_{ N \: {\mathrm {times}}}\underlinederbrace{(\mathfrak e_2 \cdots \mathfrak e_2)}_{ N \:{\mathrm {times}}} \dots \underlinederbrace{(\mathfrak e_L \cdots \mathfrak e_L)}_{ N \:{\mathrm {times}}}, \nonumber \\
& v = (\mathfrak v_{\overline{k_1}}\mathfrak v_{\overline{k_2}} \cdots \mathfrak v_{\overline{k_n}})(\mathfrak v_N \mathfrak v_{N-1} \underlinederbrace{ \cdots \mathfrak v_1 ) \dots (\mathfrak v_N \mathfrak v_{N-1} }_{ L \: {\mathrm {copies}}} \cdots \mathfrak v_1) \in ({\mathbb K\hskip.5pt}^N)^{\otimes (n+NL)},
\end{align}
and $m = \underline{o_{n+1}} = \underline{o_{n+2}} = \cdots =\underline{o_{n+NL}}.$ The monomial $f$ given by (\ref{eq:fff}) satisfies the assumptions of Lemma \ref{l:ml} with $k=NL,$ and $\overline{n}(j) = N$ for all $j \in \{1,2,\dots,L\}.$ Let
$\mathcal K_{n,NL}^m$ and $\mathcal L_{n,NL}^m$ be the corresponding subspaces of ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+NL}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+NL)}.$
\begin{lemma} \label{l:ml1}
Let $y \in ({\mathbb K\hskip.5pt}^N)^{\otimes (n+NL)},$ and let $f_1 \in \mathcal K_{n,NL}^m,$ $f_2 \in \mathcal L_{n,NL}^m. $ Then
\begin{align}
&\wedge( f_1 \otimes y ) \in \oplus_{d^{\prime} > l} V_{M,n+NL}^{d^{\prime}}, \tag{i} \\
&\wedge( f_2 \otimes y ) = 0. \tag{ii}
\end{align}
\end{lemma}
\begin{proof} \mbox{} \\
This lemma is the special case ($b=L$ and $c=N$) of Lemma \ref{l:mlbc}. See the proof of Lemma \ref{l:mlbc}.
\end{proof}
Now we continue the proof of the proposition. \mbox{} From the definitions (\ref{eq:pi1} -- \ref{eq:pi3}) and Lemmas \ref{l:lemma}, \ref{l:ml} and \ref{l:ml1}, it follows that (\ref{eq:inter}) holds modulo $\oplus_{d^{\prime} > d} V_{M,n+NL}^{d^{\prime}}.$ However, the both sides of (\ref{eq:inter}) belong to $V_{M,n+NL}^{d}$ since the action of $\UU_q^{\prime}(\asll_N)$ preserves the degree $d.$ Hence (\ref{eq:inter}) holds exactly.
\end{proof}
Now we are ready to give the definition of the level 0 action of $\UU_q^{\prime}(\asll_N)$ on the Fock space ${\mathcal F}_M.$
\begin{defin} \label{dp:def}
Let $0 \leqslant d \leqslant l.$ We define a $\UU_q^{\prime}(\asll_N)$-action $\pi^v : \UU_q^{\prime}(\asll_N) \mapsto \mathrm {End}({\mathcal F}_M^d)$ as
$$ \pi^v(X) = \varrho_l^d \circ \pi^v_{(s+lNL)}(X) \circ (\varrho_l^d)^{-1} \qquad (X \in \UU_q^{\prime}(\asll_N) ).$$
By Proposition \ref{p:inter} this definition does not depend on the choice of $l$ as long as $l \geqslant d.$
\end{defin}
Thus a $\UU_q^{\prime}(\asll_N)$-action is defined on each homogeneous component ${\mathcal F}_M^d,$ and hence on the entire Fock space ${\mathcal F}_M.$
\subsection{The action of the quantum toroidal algebra on the Fock space}
In Section \ref{sec:UF} we defined a level $L$ action of $\UU_q^{\prime}(\asll_N)$ on ${\mathcal F}_M.$ Let us denote by $\pi^h$ the corresponding map $\UU_q^{\prime}(\asll_N) \rightarrow \mathrm {End}({\mathcal F}_M).$ We refer to $\pi^h$ as the horizontal $\UU_q^{\prime}(\asll_N)$-action on the Fock space. In the preceding section we defined another -- level 0 -- action $ \pi^v : \UU_q^{\prime}(\asll_N) \rightarrow \mathrm {End}({\mathcal F}_M).$ We call $\pi^v$ the vertical $\UU_q^{\prime}(\asll_N)$-action. Note that for $i=1,2,\dots,N-1$ we have
\begin{equation*}
\pi^h(E_i) = \pi^v(E_i), \quad
\pi^h(F_i) = \pi^v(F_i), \quad
\pi^h(K_i) = \pi^v(K_i),
\end{equation*}
i.e. the restrictions of $\pi^h$ and $\pi^v$ on the subalgebra $\operatorname{U}_q(\mathfrak {sl}_N)$ coincide.
In this section we show that $\pi^h$ and $\pi^v$ are extended to an action $\ddot{\pi}$ of the quantum toroidal algebra $\ddot{\UU},$ such that $\pi^h$ is the pull-back of $\ddot{\pi}$ through the homomorphism (\ref{eq:Uh}), and $\pi^v$ is the pull-back of $\ddot{\pi}$ through the homomorphism (\ref{eq:Uv}). The definition of $\ddot{\pi}$ is based on Proposition \ref{p:shift1}.
Let $\psi_n : \wedge^nV_{\mathrm {aff}} \rightarrow \wedge^nV_{\mathrm {aff}}$ be the the map (\ref{eq:psi}) for $\operatorname M = ({\mathbb K\hskip.5pt}[z^{\pm 1}]\otimes {\mathbb K\hskip.5pt}^L)^{\otimes n}.$ That is
\begin{gather}
\psi_n : z^{m_1}\mathfrak e_{a_1}\mathfrak v_{\epsilon_1}\wedge z^{m_2}\mathfrak e_{a_2}\mathfrak v_{\epsilon_2}\wedge \cdots \wedge z^{m_n}\mathfrak e_{a_n}\mathfrak v_{\epsilon_n} \mapsto \label{eq:psin}\\
z^{m_1-\delta_{\epsilon_1,N}}\mathfrak e_{a_1}\mathfrak v_{\epsilon_1+1}\wedge z^{m_2-\delta_{\epsilon_2,N}}\mathfrak e_{a_2}\mathfrak v_{\epsilon_2+1}\wedge \cdots \wedge z^{m_n-\delta_{\epsilon_n,N}}\mathfrak e_{a_n}\mathfrak v_{\epsilon_n+1}, \nonumber
\end{gather}
where $\mathfrak v_{N+1} $ is identified with $\mathfrak v_1.$ Let ${\mathcal F} = \oplus_M {\mathcal F}_M.$ We define a semi-infinite analogue $\psi_{\infty} \in \mathrm {End}({\mathcal F})$ of $\psi_n$ as follows. For $m\in \mathbb Z$ we let
$$ \psi_{\infty} |-mNL \rangle = z^{m-1}\mathfrak e_1\mathfrak v_1\wedge z^{m-1}\mathfrak e_2\mathfrak v_1\wedge \cdots \wedge z^{m-1}\mathfrak e_L\mathfrak v_1 \wedge |-mNL \rangle.$$
Any vector in ${\mathcal F}$ can be presented in the form $v\wedge |-mNL\rangle ,$ where $v \in \wedge^nV_{\mathrm {aff}}$ for suitable $n$ and $m.$ Then we set
$$ \psi_{\infty}( v\wedge |-mNL\rangle ) = \psi_n(v)\wedge \psi_{\infty} |-mNL\rangle.$$
By using the normal ordering rules it is not difficult to verify that $\psi_{\infty}$ is well-defined (does not depend on the choice of $m$). Note that $\psi_{\infty} : {\mathcal F}_M \rightarrow {\mathcal F}_{M+L},$ and that $\psi_{\infty}$ is invertible. Moreover
\begin{equation}
\psi_{\infty}i \pi^h(X_i) \psi_{\infty} = \pi^h(X_{i-1}) \qquad (i=0,1,\dots,N-1), \label{eq:cyc}
\end{equation}
where $X=E,F,K$ and the indices are cyclically extended modulo $N.$
\begin{propos} \label{p:twist}
For each vector $w \in {\mathcal F}_M$ we have
\begin{eqnarray}
{\psi_{\infty}i}\pi^v(\tilde{X}_i(z)){\psi_{\infty}}(w) & = & \pi^v(\tilde{X}_{i-1}(q^{-1}z))(w), \qquad (2\leqslant i \leqslant N-1), \label{eq:tw1} \\
{\psi_{\infty}si}\pi^v(\tilde{X}_1(z)){\psi_{\infty}s}(w) & = & \pi^v(\tilde{X}_{N-1}(p^{-1}q^{N-2}z))(w), \label{eq:tw2}
\end{eqnarray}
where $X = E,F,K^{\pm}.$
\end{propos}
\begin{proof}
To prove the proposition we use the following lemmas.
\begin{lemma}\label{l:mlbc}
Let $0 \leqslant d \leqslant l,$ $n=s+lNL,$ where $M\equiv s\bmod N,$ $s\in \{0,1,\dots,NL-1\}.$ Let $w= z_1^{\underline{k_1}}\mathfrak e_{\dot{k}_1}\mathfrak v_{\overline{k_1}} \wedge z_2^{\underline{k_2}}\mathfrak e_{\dot{k}_2}\mathfrak v_{\overline{k_2}} \wedge \cdots \wedge z_n^{\underline{k_n}}\mathfrak e_{\dot{k}_n}\mathfrak v_{\overline{k_n}}$ be a normally ordered wedge from $V_{M,n}^d,$ let $b,c$ be integers such that $1 \leqslant b \leqslant L,$ $1 \leqslant c \leqslant N$.
We define $f \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+bc}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+bc)}$ as follows.
\begin{align}
& f = (z_1^{\underline{k_1}}z_2^{\underline{k_2}} \cdots z_n^{\underline{k_n}})(z_{n+1} \cdots z_{n+bc})^{m}\otimes \label{eq:ffff}\\ & \qquad \qquad \qquad \qquad\otimes (\mathfrak e_{\dot{k}_1}\mathfrak e_{\dot{k}_2} \cdots \mathfrak e_{\dot{k}_n})\underlinederbrace{(\mathfrak e_1 \cdots \mathfrak e_1)}_{ c \: {\mathrm {times}}}\underlinederbrace{(\mathfrak e_2 \cdots \mathfrak e_2)}_{ c \:{\mathrm {times}}} \dots \underlinederbrace{(\mathfrak e_b \cdots \mathfrak e_b)}_{ c \:{\mathrm {times}}}, \nonumber
\end{align}
where $m = \underline{o_{n+1}} = \underline{o_{n+2}} = \cdots =\underline{o_{n+bc}}.$ The monomial $f$ given by (\ref{eq:ffff}) satisfies the assumptions of Lemma \ref{l:ml} with $k=bc,$ and $\overline{n}(j) = c$ for all $j \in \{1,2,\dots,b\}.$ Let
$\mathcal K_{n,bc}^m$ and $\mathcal L_{n,bc}^m$ be the corresponding subspaces of ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+bc}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+bc)}.$
Let $y= y^{(n)} \otimes ( \mathfrak v_{\epsilon_{1}} \otimes \cdots \otimes \mathfrak v_{\epsilon_{bc}}) \in ({\mathbb K\hskip.5pt}^N)^{\otimes n} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes bc} $ such that $N-c+1 \leqslant \epsilon_{i} \leqslant N$ $(1\leqslant i \leqslant bc)$, and let $f_1 \in \mathcal K_{n,bc}^m,$ $f_2 \in \mathcal L_{n,bc}^m. $ Then
\begin{align}
&\wedge( f_1 \otimes y ) \in \oplus_{d^{\prime} > l} V_{M,n+bc}^{d^{\prime}}, \tag{i} \\
&\wedge( f_2 \otimes y ) = 0. \tag{ii}
\end{align}
\end{lemma}
\begin{proof} \mbox{} \\
(i) The vector $\wedge( f_1 \otimes y )$ is a linear combination of normally ordered wedges
$$ u_{(k_i)} = u_{k_1} \wedge u_{k_2} \wedge \cdots \wedge u_{k_n} \wedge u_{k_{n+1}} \wedge \cdots \wedge u_{k_{n+bc}} $$
such that $\underline{k_{n+1}} < \underline{o_{n+1}}.$ This inequality implies that $\deg u_{(k_i)} \geqslant l+1.$ \\
\noindent (ii)
It is sufficient to show that
\begin{equation}
\wedge( \mathfrak e_{a_1}\mathfrak e_{a_2}\cdots\mathfrak e_{a_{bc}} \otimes \mathfrak v_{\epsilon_{1}} \mathfrak v_{\epsilon_{2}} \cdots \mathfrak v_{\epsilon_{bc}} )\in \wedge^{bc}V_{\mathrm {aff}}
\label{lemw}
\end{equation}
is zero whenever there is $J \in \{1,2,\dots,b\}$ such that $\#\{ i \: | \: 1\leqslant i\leqslant bc ,\; a_i = J \} > c.$
Using the normal ordering rules (\ref{eq:n1} -- \ref{eq:n4}) one can write (\ref{lemw}) as a linear combination of the normally ordered wedges $\mathfrak e_{a^{\prime}_1}\mathfrak v_{\epsilon^{\prime}_1}\wedge \mathfrak e_{a^{\prime}_2}\mathfrak v_{\epsilon^{\prime}_2} \wedge \cdots \wedge \mathfrak e_{a^{\prime}_{bc}}\mathfrak v_{\epsilon^{\prime}_{bc}}$.
The $\operatorname{U}_q(\mathfrak {sl}_N)$ and $\operatorname{U}_q(\mathfrak {sl}_L)$-weights of the both sides in the normal ordering rules are equal. This implies that $\# \{ i \: | \: a_i ^{\prime } =J \} >c $ and $\# \{ j \: | \: \exists i , \; \epsilon^{\prime}_i =j , \; a_i ^{\prime } =J \} \leqslant c$. Therefore, there exists some $i$ such that $a_i ^{\prime }= a_{i+1} ^{\prime }$ and $\epsilon^{\prime}_i= \epsilon^{\prime}_{i+1}$.
On the other hand, we know that $\mathfrak e_{a_i ^{\prime }}\mathfrak v_{\epsilon^{\prime}_i} \wedge \mathfrak e_{a_i ^{\prime }}\mathfrak v_{\epsilon^{\prime}_i} =0$. This implies that $\wedge( f_2 \otimes y ) = 0.$
\end{proof}
\begin{lemma} \label{l:lt}
Suppose $d$ and $l$ are integers such that $0 \leqslant d \leqslant l.$ Let $n= s + lNL,$ where $s \in \{0,1,\dots,NL-1\}$ is defined from $M \equiv s \bmod NL.$ Let $m$ be the integer such that $M-s-lNL = -mNL.$
For $1 \leqslant b \leqslant L$ we put
\begin{align*}
&v_{b,N} = z^m\mathfrak e_1\mathfrak v_N\wedge z^m\mathfrak e_2\mathfrak v_N\wedge \cdots \wedge z^m\mathfrak e_b\mathfrak v_N,\\
&v_{b,N-1} = z^m\mathfrak e_1\mathfrak v_N\wedge z^m\mathfrak e_1\mathfrak v_{N-1}\wedge z^m\mathfrak e_2\mathfrak v_N\wedge z^m\mathfrak e_2\mathfrak v_{N-1}\wedge \cdots \wedge z^m\mathfrak e_b\mathfrak v_N\wedge z^m\mathfrak e_b\mathfrak v_{N-1}.
\end{align*}
Assume $v \in V^d_{M,s+lNL}.$ Then
\begin{align}
&\pi^v_{(n+b)}(\tilde{X}_i(z))(v\wedge v_{b,N}) = \pi^v_{(n)}(\tilde{X}_i(z))(v)\wedge v_{b,N} , \label{eq:lt1}\\
&\pi^v_{(n+2b)}(\tilde{X}_{N-1}(z))(v\wedge v_{b,N-1}) = \pi^v_{(n)}(\tilde{X}_{N-1}(z))(v) \wedge v_{b,N-1} \label{eq:lt2}
\end{align}
Here $1\leqslant i \leqslant N-2.$
\end{lemma}
For the proof, see the appendix.
\noindent Retaining the notations introduced in the statement of the above lemma, we continue the proof of the proposition. We may assume that $w \in {\mathcal F}_M^d.$ Then, by Proposition \ref{p:rho}, $w = v\wedge |-mNL\rangle,$ where $v \in V_{M,s+lNL}^d.$ By Definition \ref{dp:def}, for $ 2 \leqslant i \leqslant N-1$ we have
\begin{equation} \pi^v(\tilde{X}_{i-1}(q^{-1}z))(v\wedge |-mNL\rangle) = \pi_{(n)}^v(\tilde{X}_{i-1}(q^{-1}z))(v )\wedge |-mNL\rangle.
\label{eq:inftwist1}\end{equation}
The definition of $\psi_{\infty}$ yields
$$ |-mNL\rangle = v_{L,N}\wedge \psi_{\infty}i|-mNL\rangle,$$
where $v_{L,N}$ is defined in the statement of Lemma \ref{l:lt}.
Applying (\ref{eq:lt1}) in this lemma, we have
$$\pi^v_{(n+L)}(\tilde{X}_{i-1}(q^{-1}z))(v\wedge v_{L,N}) = \pi^v_{(n)}(\tilde{X}_{i-1}(q^{-1}z))(v)\wedge v_{L,N}.
$$
Taking this, and Proposition \ref{p:fintwist} into account, we find that the right-hand side of (\ref{eq:inftwist1}) equals
$$
\psi_{n+L}^{-1}\pi^v_{(n+L)}(\tilde{X}_i(z))\psi_{n+L}(v\wedge v_{L,N})\wedge \psi_{\infty}i|-mNL\rangle,
$$
which in turn is equal, by definition of $\psi_{\infty},$ to
\begin{equation}
\psi_{\infty}i \left(\pi^v_{(n+L)}(\tilde{X}_i(z))\psi_{n+L}(v\wedge v_{L,N})\wedge |-mNL\rangle \right).
\label{eq:iftwist2}\end{equation}
It is clear, that $\psi_{n+L}(v\wedge v_{L,N}) \in V^{d^{\prime}}_{M+L,n+L}$ for some non-negative integer $d^{\prime}.$ Choosing now $m$ large enough, or, equivalently, $l$ large enough (cf. the statement of Lemma \ref{l:lt}), we have by Definition \ref{dp:def}:
$$ \pi^v_{(n+L)}(\tilde{X}_i(z))\psi_{n+L}(v\wedge v_{L,N})\wedge |-mNL\rangle =
\pi^v(\tilde{X}_i(z))\left(\psi_{n+L}(v\wedge v_{L,N})\wedge |-mNL\rangle \right) .$$
Since $\psi_{\infty}(v\wedge |-mNL\rangle ) = \psi_{n+L}(v\wedge v_{L,N})\wedge |-mNL\rangle,$ we find that (\ref{eq:iftwist2}) equals
$$ \psi_{\infty}i \pi^v(\tilde{X}_i(z))\psi_{\infty}\left(v\wedge|-mNL\rangle \right).$$
Thus (\ref{eq:tw1}) is proved.
A proof of (\ref{eq:tw2}) is similar. Here the essential ingredients are the relation (\ref{eq:lt2}), and those relations of Proposition \ref{p:fintwist} which contain the square of $\psi.$
\end{proof}
Now by Propositions \ref{p:shift1} and \ref{p:twist} we obtain
\begin{theor}
The following map extends to a representation of $\ddot{\UU}$ on ${\mathcal F}_M.$
\begin{alignat}{4}
& \ddot{\pi} : X_i(z) &\quad \mapsto\quad & \pi^v(\tilde{X}_i(d^iz)) \qquad (1\leqslant i <N), \label{eq:tt1}\\
& \ddot{\pi} : X_0(z) &\quad \mapsto \quad & {\psi_{\infty}i}\pi^v(\tilde{X}_1(qd^{-1}z)){\psi_{\infty}}, \label{eq:tt2} \\
& \ddot{\pi} : \mathbf d & \quad \mapsto \quad & d 1, \\
& \ddot{\pi} : q^{\frac12 c} & \quad \mapsto\quad & 1.
\end{alignat}
Here $d = p^{-1/N}q ,$ and $X = E,F,K^{\pm}.$
\end{theor}
\mbox{} From (\ref{eq:tt1}) it follows that the vertical (level $0$) $\UU_q^{\prime}(\asll_N)$-action $\pi^v$ is the pull-back of $\ddot{\pi}$ through the homomorphism (\ref{eq:Uv}). Whereas from (\ref{eq:tt2}) and (\ref{eq:cyc}) it follows that the horizontal (level $L$) $\UU_q^{\prime}(\asll_N)$-action $\pi^h$ the pull-back of $\ddot{\pi}$ through the homomorphism (\ref{eq:Uh}). Thus as an $\ddot{\UU}$-module the Fock space ${\mathcal F}_M$ has level $(0,L)$ (cf. Section \ref{sec:tor}).
\subsection{The actions of the quantum toroidal algebra on irreducible integrable highest weight modules of $\operatorname{U}_q^{\prime}(\widehat{{\mathfrak {gl}}}_N)$ }
Let $\Lambda$ be a level $L$ dominant integral weight of $\UU_q^{\prime}(\asll_N).$ In this section we define an action of the quantum toroidal algebra $\ddot{\UU}$ on the irreducible module
\begin{equation}
\widetilde{V}(\Lambda) = {\mathbb K\hskip.5pt}[H_-]\otimes V(\Lambda)
\end{equation}
of the algebra $\operatorname{U}_q^{\prime}(\widehat{{\mathfrak {gl}}}_N) = H\otimes \UU_q^{\prime}(\asll_N).$ Here (cf. Section \ref{sec:decomp}) ${\mathbb K\hskip.5pt}[H_-]$ is the Fock module of the Heisenberg algebra $H,$ and $V(\Lambda)$ is the irreducible highest weight module of $\UU_q^{\prime}(\asll_N)$ of highest weight $\Lambda.$
In Section \ref{sec:torHecke} we defined, for any integral weight $\chi$ of $\mathfrak {sl}_L,$ the subalgebra $\operatorname{U}_q({\mathfrak b}_L)^{\chi}$ of $\UU_q^{\prime}(\asll_L).$ A level $N$ action of $\UU_q^{\prime}(\asll_L)$ on the Fock space ${\mathcal F}_M$ ($M\in \mathbb Z$) was defined in Section \ref{sec:Usl}, so that there is an action $\operatorname{U}_q({\mathfrak b}_L)^{\chi}$ on ${\mathcal F}_M.$ Recall moreover, that the vertical $\UU_q^{\prime}(\asll_N)$-action $\pi^v$ on ${\mathcal F}_M,$ and, consequently, the action $\ddot{\pi}$ of $\ddot{\UU},$ depend on two parameters: $p \in q^{\mathbb Z},$ and $\nu$ which is an integral weight of $\mathfrak {sl}_L.$
\begin{propos}\label{p:inv3}
Suppose $p = q^{-2L},$ and $\nu = - \chi - 2\rho$ for an integral $\mathfrak {sl}_L$-weight $\chi.$ Then the action $\ddot{\pi}$ of $\ddot{\UU}$ on ${\mathcal F}_M$ leaves invariant the linear subspace $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right).$
\end{propos}
\begin{proof}
It is sufficient to prove that both the horizontal $\UU_q^{\prime}(\asll_N)$-action $\pi^h$ and the vertical $\UU_q^{\prime}(\asll_N)$-action $\pi^v$ leave $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right)$ invariant. The horizontal action commutes with the action of $\UU_q^{\prime}(\asll_L).$ Thus it remains to prove that the vertical action leaves $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right)$ invariant. Let $w \in {\mathcal F}_M^d$ and let $l \geqslant d.$ By Proposition \ref{p:rho} there is a unique $v \in V_{M,s+lNL}^d$ such that
$$ w = v \wedge | M - s - lNL \rangle .$$
Here $s\in \{0,1,\dots,NL-1\},$ $M\equiv s\bmod NL.$ \\
Let $g$ be one of the generators of $\operatorname{U}_q({\mathfrak b}_L)^{\chi}$ (cf. \ref{eq:gen}). For all large enough $l$ we have
\begin{equation}
g(w) = g(v)\wedge | M - s - lNL \rangle c(g),
\label{eq:inv31}\end{equation}
where $c(g) = q^{-N}$ if $g = \dot{F}_0,$ and $c(g) = 1$ if $g = \dot{F}_a, \dot{K}_a - q^{\chi(a) - \chi(a+1)} 1$ $(1\leqslant a < L).$ If $g= \dot{F}_0$ then $g(v) \in V_{M,s+lNL}^{d+1},$ otherwise $g(v) \in V_{M,s+lNL}^{d}.$
Let $X$ be an element of $\UU_q^{\prime}(\asll_N).$ Provided $l$ is sufficiently large, Definition \ref{dp:def} gives
$$ \pi^v(X)g(w) = \pi_{(s+lNL)}^v(X)g(v) \wedge | M - s - lNL \rangle c(g) .$$
By Proposition \ref{p:inv2} the right-hand side of the last equation is a linear combination of vectors
\begin{equation} h (v^{\prime})\wedge | M - s - lNL \rangle ,
\label{eq:inv32}\end{equation}
where $h$ is again one of the generators of $\operatorname{U}_q({\mathfrak b}_L)^{\chi},$ and $v^{\prime}$ belongs to either $V_{M,s+lNL}^{d}$ or $V_{M,s+lNL}^{d+1}.$ Applying (\ref{eq:inv31}) again, the vector (\ref{eq:inv32}) is seen to be proportional to
$$ h(v^{\prime}\wedge | M - s - lNL \rangle ).$$
Thus the vertical action leaves $\operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right)$ invariant.
\end{proof}
Now we use Theorem \ref{t:decofF} to define an action of $\ddot{\UU}$ on $\widetilde{V}(\Lambda).$ Fix the unique $M \in \{0,1,\dots,N-1\}$ such that $\overline{\Lambda} \equiv \overline{\Lambda}_M \bmod \overline{Q}_N.$ Since the dual weights $\dot{\Lambda}^{(M)}$ of $\UU_q^{\prime}(\asll_L)$ are distinct for distinct $\Lambda,$ from Theorem \ref{t:decofF}
we have the isomorphism of $\operatorname{U}_q^{\prime}(\widehat{{\mathfrak {gl}}}_N)$-modules:
\begin{equation} \widetilde{V}(\Lambda) \cong {\mathcal F}_M/ \operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right),\label{eq:is}
\end{equation}
where $\chi $ is the finite part of $\dot{\Lambda}^{(M)}.$ That is for $\dot{\Lambda}^{(M)} = \sum_{a=0}^{L-1} n_a \dot{\Lambda}_a,$ $\chi = \sum_{a=1}^{L-1} n_a \dot{\overline{\Lambda}}_a.$
By Proposition \ref{p:inv2}, the $\ddot{\UU}$-action $\ddot{\pi}$ with $p= q^{-2L},$ $ \nu = - \chi - 2 \rho,$ factors through the quotient map
$$ {\mathcal F}_M \rightarrow {\mathcal F}_M/ \operatorname{U}_q({\mathfrak b}_L)^{\chi}\left({\mathcal F}_M\right),$$
and therefore by (\ref{eq:is}) induces an action of $\ddot{\UU}$ on $\widetilde{V}(\Lambda).$
\appendix
\section{The proof of lemma \ref{l:lt}}
\setcounter{section}{7}
In this appendix we prove Lemma \ref{l:lt}. The idea of the proof is essentially the same as that of the proof of \cite[Lemma 23]{STU}. \\
{\bf Lemma \ref{l:lt}.}
{\em Suppose $d$ and $l$ are integers such that $0 \leqslant d \leqslant l.$ Let $n= s + lNL,$ where $s \in \{0,1,\dots,NL-1\}$ is defined from $M \equiv s \bmod NL.$ Let $m$ be the integer such that $M-s-lNL = -mNL.$
For $1 \leqslant b \leqslant L$ we put
\begin{align*}
&v_{b,N} = z^m\mathfrak e_1\mathfrak v_N\wedge z^m\mathfrak e_2\mathfrak v_N\wedge \cdots \wedge z^m\mathfrak e_b\mathfrak v_N,\\
&v_{b,N-1} = z^m\mathfrak e_1\mathfrak v_N\wedge z^m\mathfrak e_1\mathfrak v_{N-1}\wedge z^m\mathfrak e_2\mathfrak v_N\wedge z^m\mathfrak e_2\mathfrak v_{N-1}\wedge \cdots \wedge z^m\mathfrak e_b\mathfrak v_N\wedge z^m\mathfrak e_b\mathfrak v_{N-1}.
\end{align*}
Assume $v \in V^d_{M,s+lNL}.$ Then
\begin{align}
&\pi^v_{(n+b)}(\tilde{X}_i(z))(v\wedge v_{b,N}) = \pi^v_{(n)}(\tilde{X}_i(z))(v)\wedge v_{b,N} , \label{eq:alt1}\\
&\pi^v_{(n+2b)}(\tilde{X}_{N-1}(z))(v\wedge v_{b,N-1}) = \pi^v_{(n)}(\tilde{X}_{N-1}(z))(v) \wedge v_{b,N-1} \label{eq:alt2}
\end{align}
Here $1\leqslant i \leqslant N-2.$}
\begin{proof}
As is mentioned in the proof of Lemma 22 in \cite{STU}, for each $i$ $(1 \leqslant i \leqslant N-1),$ the subalgebra of $\UU_q^{\prime}(\asll_N)$ generated by $\tilde{E}_{i,l'} , \; \tilde{F}_{i,l'}, \; \tilde{H}_{i,m'}, \tilde{K}^{\pm }_{i}$ $(l' \in \mathbb Z , \; m' \in \mathbb Z \setminus \{ 0 \} )$ is in fact generated by only the elements $\tilde{E}_{i,0}, \tilde{F}_{i,0}, \tilde{K}^{\pm }_{i}, \tilde{F}_{i,1} $ and $\tilde{F}_{i,-1}.$
By the definition of the representation, every generator of the vertical action $\mbox{U}_v$ preserves the degree in the sense of (\ref{eq:fdeg}).
So it is sufficient to show that the actions of $\tilde{E}_{i,0}, \tilde{F}_{i,0}, \tilde{K}^{\pm }_{i}, \tilde{F}_{i,1} $ and $\tilde{F}_{i,-1}$ satisfy the relations (\ref{eq:alt1}, \ref{eq:alt2}).
For $\tilde{E}_{i,0}, \tilde{F}_{i,0}, \tilde{K}^{\pm }_{i}$,
this is shown directly by using the definitions of the actions (\ref{eq:pi1}--\ref{eq:pi3}).
Now we must show that
\begin{align}
&\pi^v_{(n+b)}(\tilde{F}_{i,\pm 1})(v\wedge v_{b,N}) = \pi^v_{(n)}(\tilde{F}_{i,\pm 1})(v)\wedge v_{b,N}, \label{eq:Fi}\\
&\pi^v_{(n+2b)}(\tilde{F}_{N-1,\pm 1})(v\wedge v_{b,N-1}) = \pi^v_{(n)}(\tilde{F}_{N-1,\pm 1})(v)\wedge v_{b,N-1} \label{eq:FN-1}
\end{align}
Here $1\leqslant i \leqslant N-2.$
We will prove (\ref{eq:FN-1}).
For any $ M',M'', M''' $ ($1 \leqslant M', M'', M'''\leqslant N+2b, \; M'\leqslant M'' $), we define an $\UU_q^{\prime}(\asll_N) $--action on the space $ {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b)}$ in terms of the Chevalley generators as follows:
\begin{align}
& E_i( f \otimes \tilde{v} ) = \sum_{j=M'}^{M''} (q^{-\underlineun{M'''}}Y_{j}^{(M''')})^{-\delta (i=0)} f \otimes (e_{i,i+1})_{j} (k_{i})_{j+1} \dots (k_{i})_{M''} \tilde{v}, \label{eM} \\
& F_i ( f \otimes \tilde{v}) = \sum_{j=M'}^{M''} (q^{-\underlineun{M'''}}Y_{j}^{(M''')})^{\delta (i=0)} f \otimes (k_{i}^{-1})_{M'}\dots (k_{i}^{-1})_{j-1}(e_{i+1,i})_{j} \tilde{v}. \label{fM} \\
& K_i ( f \otimes \tilde{v}) = f \otimes (k_i)_{M'} (k_i)_{M'+1} \dots (k_i)_{M''} \tilde{v}. \label{kM}
\end{align}
Here $i= 0, \dots ,N-1$, indices are cyclically extended modulo $N$, $f \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)}$, $ \tilde{v} \in ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b)}$, and the meaning of the notations $(e_{i,i'})_j$, $(k ^{\pm 1}_i)_j$ is the same as in Section \ref{sec:VVdual}. It is understood, that for $M''' < n+2b$ the operators $Y_{i}^{(M''')}$ in (\ref{eM}, \ref{fM}) act non-trivially only on the variables $z_1,z_2,\dots,z_{M'''}$ and on the first $M'''$ factors in ${\mathbb K\hskip.5pt}^{\otimes(n+2b)}.$
Note that the $\UU_q^{\prime}(\asll_N)$-action is well--defined because of the commutativity of $Y_{i}^{(M''')}$ ($i = 1, \dots , M'''$).
The actions of the Drinfeld generators are determined by the actions of the Chevalley generators.
Let $\tilde{X}$ be an element of $\UU_q^{\prime}(\asll_N) $, we denote by $\tilde{X}^{(M',M''), M'''}$ the operator giving the action of $\tilde{X}$ on the space ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b)}$ in accordance with (\ref{eM}--\ref{kM}).
Also, we set $\tilde{X}^{\{ j \}, M''' }$ $=$ $\tilde{X}^{(j,j), M'''}$ $(j=1, \dots , M''').$
With these definitions, for any two elements $\tilde{X}$ and $\tilde{Y}$ from $\UU_q^{\prime}(\asll_N) ,$ the operators $\tilde{X}^{(M',M''), M'''}$ and $\tilde{Y}^{(N',N''), M'''}$ commute if $M''<N'$ or $N''<M'$. Note that for any $\tilde{X} \in \UU_q^{\prime}(\asll_N)$ we have
$$ \pi^v_{(n+2b)}(\tilde{X})\wedge(f \otimes \tilde{v}) = \wedge\left( \tilde{X}^{(1,n+2b),n+2b}(f \otimes \tilde{v})\right).$$
\mbox{}
\noindent Let $UN_+$ and $UN_-^{2}$ be the left ideals in $\UU_q^{\prime}(\asll_N)$ generated respectively by $\{ \tilde{E}_{i,k'} \}$ and $\{ \tilde{F}_{i,k'}\tilde{F}_{j,l'} \}$. Let $UN_+^{(M',M''), M'''},(UN_-^2)^{(M',M''), M'''}$ be the images of these ideals with respect to the map $\UU_q^{\prime}(\asll_N) \rightarrow \mathrm {End}\left({\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b)} \right)$ given by (\ref{eM}--\ref{kM}).
Then the following relations hold:
\begin{align*}
& \pi_{(n+2b)}^v(\tilde{F}_{N-1,1}) \wedge (f \otimes \tilde{v} ) \equiv \wedge( (\tilde{K}_{N-1} ^{(1,n+2b-2), n+2b} \tilde{F}_{N-1,1}^{(n+2b-1,n+2b),n+2b}+ \tilde{F}_{N-1,1}^{(1,n+2b-2), n+2b}) (f \otimes \tilde{v} )) , \\
& \pi_{(n+2b)}^v(\tilde{F}_{N-1,-1}) \wedge (f \otimes \tilde{v} ) \equiv \wedge (((\tilde{K}_{N-1} ^{(1,n+2b-2), n+2b})^{-1} \tilde{F}_{N-1,-1}^{(n+2b-1,n+2b),n+2b}+ \tilde{F}_{N-1,-1}^{(1,n+2b-2), n+2b} \\
& \quad \quad \quad + (q^{-1}-q)(\tilde{K}_{N-1}^{(1,n+2b-2), n+2b})^{-1} \tilde{H}_{N-1,-1}^{(1,n+2b-2), n+2b} \tilde{F}_{N-1,0}^{(n+2b-1,n+2b),n+2b} ) (f \otimes \tilde{v} )) ,\nonumber \\
& \qquad \text{where}\qquad f \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)}, \quad \tilde{v} \in ({\mathbb K\hskip.5pt}^N)^{\otimes n+2b}. \nonumber
\end{align*}
Here the equivalence $\equiv$ is understood to be modulo $$ \wedge (UN_+^{(1,n+2b-2), n+2b}
\cdot (UN_-^2)^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v} )).$$
These relations follow from the the coproduct formulas which have been
obtained in \cite[Proposition 3.2.A]{koyama}:
\begin{eqnarray}
& \Delta^+ (\tilde{F}_{i,1}) \equiv \tilde{K}_{i} \otimes \tilde{F}_{i,1} + \tilde{F}_{i,1} \otimes 1 &
\mbox{mod } UN_+ \otimes UN_-^{2} , \label{cop1} \\
& \Delta^+ (\tilde{F}_{i,-1}) \equiv \tilde{K}_{i}^{-1} \otimes \tilde{F}_{i,-1} + \tilde{F}_{i,-1} \otimes 1 &
\label{cop2} \\
& + (q^{-1}-q)\tilde{K}_i^{-1} \tilde{H}_{i,-1} \otimes \tilde{F}_{i,0} & \mbox{mod } UN_+
\otimes UN_-^{2}. \nonumber
\end{eqnarray}
Recall the definition of $\Delta^+$ given in (\ref{eq:co1} -- \ref{eq:co4}).
Let $w= z_1^{\underline{k_1}}\mathfrak e_{\dot{k}_1}\mathfrak v_{\overline{k_1}} \wedge z_2^{\underline{k_2}}\mathfrak e_{\dot{k}_2}\mathfrak v_{\overline{k_2}} \wedge \cdots \wedge z_n^{\underline{k_n}}\mathfrak e_{\dot{k}_n}\mathfrak v_{\overline{k_n}}$ be a normally ordered wedge from $V_{M,n}^d,$ and define $f \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)}$ and $\tilde{v} \in ({\mathbb K\hskip.5pt}^N)^{\otimes(n+2b)}$ as follows.
\begin{align}
& f = (z_1^{\underline{k_1}}z_2^{\underline{k_2}} \cdots z_n^{\underline{k_n}})(z_{n+1} \cdots z_{n+2b})^{m}\otimes (\mathfrak e_{\dot{k}_1}\mathfrak e_{\dot{k}_2} \cdots \mathfrak e_{\dot{k}_n})(\mathfrak e_1 \mathfrak e_1 \mathfrak e_2 \mathfrak e_2 \cdots \mathfrak e_b \mathfrak e_b), \label{lemf} \\
& \tilde{v} = (\mathfrak v_{\overline{k_1}}\mathfrak v_{\overline{k_2}} \cdots \mathfrak v_{\overline{k_n}})(\mathfrak v_N \underlinederbrace{ \mathfrak v_{N-1} ) (\mathfrak v_N \mathfrak v_{N-1}) \dots (\mathfrak v_N }_{ b\: {\mathrm {copies}}} \mathfrak v_{N-1}), \label{lemv}
\end{align}
where $m = \underline{o_{n+1}} = \underline{o_{n+2}} = \cdots =\underline{o_{n+2b}}.$
Then the monomial $f$ satisfies the assumptions of Lemma \ref{l:ml} with $k=2b,$ and $\overline{n}(j) = 2$ for all $j \in \{1,2,\dots,b\}.$
Now we will show the equality
\begin{equation}
\pi_{(n+2b)}^v(\tilde{F}_{N-1,\pm 1})\wedge (f \otimes \tilde{v} ))
=\wedge(\tilde{F}_{N-1,\pm 1}^{(1,n+2b-2), n+2b} (f \otimes \tilde{v})).
\label{ffstar}
\end{equation}
First let us prove that any element in $UN_+^{(1,n+2b-2), n+2b} \cdot (UN_-^2)^{(n+2b-1,n+2b),n+2b}$
annihilates the vector $f \otimes \tilde{v}$ where $f$ and $\tilde{v}$ are given by (\ref{lemf}) and (\ref{lemv}).
It is enough to show that
\begin{equation}
( \tilde{F}_{i',k'}^{(n+2b-1,n+2b),n+2b}\tilde{F}_{j',l'}^{(n+2b-1,n+2b),n+2b}) ( \bar{v} \otimes (z_{n+2b-1}^m\mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_{N-1}))=0,
\end{equation}
for $\bar{v} \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b-2}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b-2)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b-2)}$.
This follows immediately from the observation that $\mathrm {wt}(\mathfrak v_N)+\mathrm {wt}( \mathfrak v_{N-1})-\overline{\alphapha}_{i'}-\overline{\alphapha}_{j'}$ is not a $\operatorname{U}_q(\mathfrak {sl}_N)$-weight of $ ({\mathbb K\hskip.5pt}^N)^{\otimes 2}$.
Next we will show that $\wedge (\tilde{F}_{N-1,\pm 1}^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v}) )= 0$, (here $f$ and $\tilde{v}$ are given by (\ref{lemf}) and (\ref{lemv})).
By the formulas (\ref{cop1}) and (\ref{cop2}), we have the following identities modulo
$\wedge (UN_+^{\{ n+2b-1 \} ,n+2b } (UN_- ^2) ^{\{ n+2b\},n+2b } ( f \otimes \tilde{v} ))$ (see also \cite{STU}):
\begin{align}
& \wedge ( \tilde{F}^{(n+2b-1,n+2b),n+2b}_{N-1,1} (f \otimes \tilde{v} )) \equiv \wedge ( (\tilde{K}_{N-1} ^{\{ n+2b-1 \},n+2b } \tilde{F}_{N-1,1}^{\{ n+2b \},n+2b }+ \tilde{F}_{N-1,1}^{\{ n+2b-1 \},n+2b }) (f\otimes \tilde{v} )) , \\
& \wedge (\tilde{F}^{(n+2b-1,n+2b),n+2b}_{N-1,-1} (f \otimes \tilde{v} )) \equiv \wedge (((\tilde{K}_{N-1} ^{\{ n+2b-1 \} ,n+2b})^{-1} \tilde{F}_{N-1,-1}^{\{ n+2b \} ,n+2b}+ \tilde{F}_{N-1,-1}^{\{ n+2b-1 \} ,n+2b} \\
& \quad \quad \quad + (q^{-1}-q)[ \tilde{E}_{N-1,0}^{\{ n+2b-1 \} ,n+2b}, \tilde{F}_{N-1,-1}^{\{ n+2b-1 \},n+2b } ] \tilde{F}_{N-1,0}^{\{ n+2b \},n+2b } ) (f \otimes \tilde{v} )) ,\nonumber
\end{align}
The following formula is essentially written in \cite[Proposition 3.2.B]{koyama}:
\begin{align}
& \tilde{F}_{i,\pm 1}^{\{ l\},n+2b } ( f' \otimes (\otimes _{j=1}^{n+2b} \mathfrak v_{\epsilon _j})) \label{ko2} \\
& = (q^{i-\underlineun{n+2b}}(Y_l ^{(n+2b)})^{-1})^{\pm 1} f'\otimes (\otimes _{j=1}^{l-1} \mathfrak v_{\epsilon _j}) \otimes
\delta _{i,\epsilon _l} \mathfrak v_{i+1} \otimes (\otimes _{j=l+1}^{n+2b} \mathfrak v_{\epsilon _j}),
\nonumber
\end{align}
where $f'\in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b)} $ and $\otimes _{j=1}^{n+2b} \mathfrak v_{\epsilon _j} \in ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b)}$.
By (\ref{ko2}) we have $(UN_+^{\{ n+2b-1 \},n+2b } (UN_- ^2)^{\{ n+2b \},n+2b } (f \otimes \tilde{v} )) =0 $, and by (\ref{ko2}) and Lemma \ref{l:ml} we have
\begin{align}
& \tilde{F}_{N-1,\pm 1}^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v} ) \equiv \alphapha _{\pm 1} \bar{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_N \label{ot}\\
& \qquad \qquad \qquad \qquad
\mbox{ mod } (\mathcal K_{n,2b}^m + \mathcal L_{n,2b}^m ) \otimes (\check{v}^{(n)} \otimes
(\mathfrak v_N \underlinederbrace{ \mathfrak v_{N-1} ) \dots (\mathfrak v_N }_{ b-1\: {\mathrm {copies}}} \mathfrak v_{N-1}) (\mathfrak v_N \mathfrak v_N)) , \nonumber
\end{align}
Here $ c _{\pm 1}$ are certain coefficients, $ \bar{v} \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b-2}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b-2)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b-2)}$, $\check{v}^{(n)} \in ({\mathbb K\hskip.5pt}^N)^{\otimes n}$.
Using the normal ordering rules, we have $\wedge (\bar{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_N) =0.$ By Lemma \ref{l:mlbc}, we have
\begin{equation}
\wedge ((\mathcal K_{n,2b}^m + \mathcal L_{n,2b}^m ) \otimes (\check{v}^{(n)} \otimes
(\mathfrak v_N \underlinederbrace{ \mathfrak v_{N-1} ) \dots (\mathfrak v_N }_{ b-1\: {\mathrm {copies}}} \mathfrak v_{N-1}) (\mathfrak v_N \mathfrak v_N))) \in \oplus_{d^{\prime} > l} V_{M,n+2b}^{d^{\prime}}.
\label{ott}
\end{equation}
On the other hand the degree of the wedge (\ref{ott}) is equal to deg$(f \otimes \tilde{v} )=d$. Taking into account that $d \leqslant l,$ we have $\wedge (\tilde{F}_{N-1,\pm 1}^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v} ))=0$.
Now we prove that $\wedge((\tilde{K}^{(1,n+2b-2),n+2b}_{N-1})^{-1} \tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1} \tilde{F}_{N-1,0}^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v}))$ vanishes.
We have
\begin{align}
& \wedge ((\tilde{K}^{(1,n+2b-2),n+2b}_{N-1})^{-1} \tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1} \tilde{F}_{N-1,0}^{(n+2b-1,n+2b),n+2b} \bar{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_{N-1}) \\
& = \wedge(( \tilde{K}^{(1,n+2b-2),n+2b}_{N-1})^{-1} \tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1} \bar{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_N) , \nonumber
\end{align}
here $ \bar{v} \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b-2}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b-2)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b-2)}$.
By (\ref{eM} -- \ref{kM}) the operator $\tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1}$ is a polynomial in the operators $(Y_{j}^{(n+2b)})^{\pm 1}$, $(k_l)^{\pm 1}_j,$ $(e_{l,l'})_j $ where $1 \leqslant j \leqslant n+2b-2 $ and $ 1 \leqslant l,l' \leqslant N$.
By Lemma \ref{l:ml}, we have
\begin{align*}
& (\tilde{K}^{(1,n+2b-2),n+2b}_{N-1})^{-1} \tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1} \bar{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_N \\
& \qquad \qquad \qquad \equiv c(\hat{v} \otimes z_{n+2b-1}^m \mathfrak e_b \mathfrak v_N \otimes z_{n+2b}^m \mathfrak e_b \mathfrak v_N ) \nonumber \\
& \qquad \qquad \qquad \qquad \qquad
\mbox{ mod } (\mathcal K_{n,2b}^m + \mathcal L_{n,2b}^m ) \otimes (\check{v}^{(n)} \otimes
(\mathfrak v_N \underlinederbrace{ \mathfrak v_{N-1} ) \dots (\mathfrak v_N }_{ b-1\: {\mathrm {copies}}} \mathfrak v_{N-1}) (\mathfrak v_N \mathfrak v_N)) , \nonumber
\end{align*}
Here $ c $ is a certain coefficient, $ \bar{v}, \hat{v} \in {\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+2b-2}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+2b-2)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+2b-2)}$, $\check{v}^{(n)}$ is an element in $({\mathbb K\hskip.5pt}^N)^{\otimes n}$.
Repeating the arguments given after the relation (\ref{ott}), we have $\wedge((\tilde{K}^{(1,n+2b-2),n+2b}_{N-1})^{-1} \tilde{H}^{(1,n+2b-2),n+2b}_{N-1,-1} \tilde{F}_{N-1,0}^{(n+2b-1,n+2b),n+2b} (f \otimes \tilde{v}))=0.$
Thus we have shown (\ref{ffstar}).
Repeatedly applying the arguments that led to (\ref{ffstar}), we have
\begin{equation}
\pi_{(n+2b)}^v(\tilde{F}_{N-1, \pm 1})(f \otimes \tilde{v})) = \wedge (\tilde{F}^{(1,n),n+2b}_{N-1, \pm 1}(f \otimes \tilde{v})).
\label{nn2b}
\end{equation}
To prove $\pi_{(n+2b)}^v(\tilde{F}_{N-1, \pm 1})\wedge(f \otimes \tilde{v})) = \wedge (\tilde{F}^{(1,n),n}_{N-1, \pm 1}(f \otimes \tilde{v}))$, we must show that in the right-hand side of (\ref{nn2b}) we can replace $q^{-\underlineun{n+2b}}Y^{(n+2b)}_i$ by $q^{-\underlineun{n}}Y^{(n)}_i$ $(1 \leqslant i \leqslant n).$
Observe that $\tilde{F}^{(1,n),n+2b}_{N-1, \pm 1}$ is a polynomial in the operators
\begin{equation}
(Y_{j}^{(n+2b)})^{\pm 1},\quad (k_l)^{\pm 1}_j,\quad (e_{l,l'})_j \quad\text{where $1 \leqslant j \leqslant n $ and $ 1 \leqslant l,l' \leqslant N.$}
\label{eq:opers}\end{equation}
By Lemma \ref{l:ml} we have
\begin{equation}
(q^{-\underlineun{n+2b}}Y^{(n+2b)}_i)^{\pm 1} (f \otimes \tilde{v}) \equiv (q^{-\underlineun{n}}Y^{(n)}_i)^{\pm 1} (f \otimes \tilde{v}) \mbox{ mod } (\mathcal K ^m _{n,2b} + \mathcal L ^m _{n,2b}) \otimes \tilde{v}.
\end{equation}
For $f^{\prime} \in \mathcal K ^m _{n,2b} + \mathcal L ^m _{n,2b},$ and $\mathcal E_n$ a polynomial in (\ref{eq:opers}), the vector $f^{\prime}\otimes \mathcal E_n \tilde{v}$
satisfies the assumption of Lemma \ref{l:mlbc}. By this lemma, and by the arguments given after (\ref{ott}), we have
\begin{equation}
\wedge ( (\mathcal K ^m _{n,2b} + \mathcal L ^m _{n,2b}) \otimes \mathcal E_n \tilde{v}) =0
\label{lstab}
\end{equation}
Combining (\ref{lstab}), the commutativity of $\mathcal E_n$ and $(Y^{(\tilde{n})}_i)^{\pm 1}$ $(1 \leqslant i \leqslant n, \; \tilde{n} = n $ or $n+2b)$, and the fact that $(Y_i^{(\tilde{n})})^{\pm 1} ( \mathcal K ^m _{n,2b} + \mathcal L ^m _{n,2b}) \subset ( \mathcal K ^m _{n,2b} + \mathcal L ^m _{n,2b}),$ we have $\pi_{(n+2b)}^v(\tilde{F}_{N-1, \pm 1})\wedge (f \otimes \tilde{v})) = \wedge (\tilde{F}^{(1,n),n}_{N-1, \pm 1}(f \otimes \tilde{v}))$. The relation (\ref{eq:FN-1}) follows.
To prove (\ref{eq:Fi}), consider the tensor product ${\mathbb K\hskip.5pt}[z_1^{\pm 1},\dots,z_{n+b}^{\pm 1}]\otimes ({\mathbb K\hskip.5pt}^L)^{\otimes(n+b)} \otimes ({\mathbb K\hskip.5pt}^N)^{\otimes (n+b)}$, use the formulas (\ref{cop1}), (\ref{cop2}) and continue the proof in a way that is completely analogous to the proof of (\ref{eq:FN-1}).
\end{proof}
\newcommand{\BOOK}[6]{\bibitem[{#6}]{#1}{\sc #2}, {\it #3} (#4)#5.}
\newcommand{\JPAPER}[8]{\bibitem[{#8}]{#1}{\sc #2}, `#3',
{\it #4} {\bf #5} (#6) #7.}
\newcommand{\JPAPERS}[9]{\bibitem[{#9}]{#1}{\sc #2}, `#3', {\it #4} #5 #6,
#7 #8.}
\end{document} |
\begin{document}
\title{Key polynomials and pseudo-convergent sequences}
\author{Josnei Novacoski}
\author{Mark Spivakovsky}
\keywords{Key polynomials, Pseudo-convergent sequences, Valuations}
\subjclass[2010]{Primary 13A18}
\begin{abstract}
In this paper we introduce a new concept of key polynomials for a given valuation $\nu$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaqui\'e, for instance, that they are irreducible and that the truncation of $\nu$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation
$\nu$ on $K[x]$ admits a sequence of key polynomials that completely determines $\nu$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.
\end{abstract}
\maketitle
\section{Introduction}
Given a valuation $\nu$ of a field $K$, it is important to understand what are the possible extensions of $\nu$ to $K[x]$. Many different theories have been developed in order to understand such extensions. For instance, in \cite{Mac}, MacLane develops the theory of key polynomials. He proves that given a discrete valuation $\nu$ of $K$, every extention of $\nu$ to $K[x]$ is uniquely determined by a sequence (with order type at most $\omega$) of key polynomials. Recently, M. Vaqui\'e developed a more general theory of key polynomials (see \cite{Vaq}), which extends the results of MacLane for a general valued field (that is, the given valuation of $K$ is no longer assumed to be discrete). At the same time, F.H. Herrera Govantes, W. Mahboub, M.A. Olalla Acosta and M. Spivakovsky developed another definition of key polynomials (see \cite{HOS}). This definition is an adaptation of the concept of generating sequences introduced by Spivakovsky in \cite{Spi1}. A comparison between this two definitions of key polynomials is presented in \cite{Mahboud}.
Roughly speaking, for a given valuation $\mu$ of $K[x]$, a MacLane -- Vaqui\'e key polynomial $\mathbb{P}hi\in K[x]$ for $\mu$ is a polynomial that allows us to obtain a new valuation $\mu_1$ of $K[x]$ with $\mu_1(\mathbb{P}hi)=\gamma_1>\mu(\mathbb{P}hi)$ and $\mu(p)=\mu_1(p)$ for every $p\in K[x]$ with $\deg(p)<\deg(\mathbb{P}hi)$ (in this case we denote $\mu_1$ by $[\mu;\mu_1(\mathbb{P}hi)=\gamma_1]$). Then, for any valuation
$\nu$ of $K[x]$ one tries to obtain a sequence of valuations $\mu_0,\mu_1,\ldots,\mu_n,\ldots$ with $\mu_0$ a monomial valuation and $\mu_{i+1}=[\mu_i;\mu_{i+1}(\mathbb{P}hi_{i+1})=\gamma_{i+1}]$ for a key polynomial $\mathbb{P}hi_{i+1}$ for $\mu_i$, such that
\begin{equation}
\nu=\lim \mu_i\label{eq:nu=lim}
\end{equation}
(in the sense that will be defined precisely below). This process does not work in general, that is, the equality (\ref{eq:nu=lim}) may not hold even after one constructs an infinite sequence $\{\mu_i\}$. This leads one to introduce the concept of ``limit key polynomial".
It is known that valuations which admit limit key polynomials are more difficult to handle. For instance, it was proved by J.-C. San Saturnino (see Theorem 6.5 of \cite{JCSS}), that if a valuation $\nu$ is centered on a noetherian local domain and $\nu$ does not admit limit key polynomials (on any sub-extension $R'\subseteq R'[x]\subseteq R$ with $\dim\ R'=\dim\ R-1$), then it has the local uniformization property (where we assume, inductively, that local uniformization holds for $R'$).
In this paper, we introduce a new concept of key polynomials. Let $K$ be a field and $\nu$ a valuation on $K[x]$. Let $\Gamma$ denote the value group of $K$ and $\Gamma'$ the value group of $K[x]$. For a positive integer $b$, let
$\mathbb{P}artial_b:=\frac{1}{b!}\frac{\mathbb{P}artial^b}{\mathbb{P}artial x^b}$ (this differential operator of order $b$ is sometiems called \textbf{the $b$-th formal derivative}). For a polynomial $f\in K[x]$ let
\[
\epsilon(f)=\max_{b\in \mathbb{N}}\left\{\frac{\nu(f)-\nu(\mathbb{P}artial_bf)}{b}\right\}.
\]
A monic polynomial $Q\in K[x]$ is said to be a \textbf{key polynomial} (of level $\epsilon (Q)$) if for every $f\in K[x]$ if $\epsilon(f)\geq \epsilon(Q)$, then $\deg(f)\geq\deg(Q)$.
This new definition offers many advantages. For instance, it gives a criterion to determine, for a given valuation $\nu$ of $K[x]$, whether any given polynomial is a key polynomial for $\nu$. This has a different meaning than in the approach of MacLane-Vaqui\'e. In their approach, a key polynomial allows us to ``extend the given valuation" and here a key polynomial allows us to ``truncate the given valuation". For instance, our definition of key polynomials treats the limit key polynomials on the same footing as the non-limit ones. Moreover, we present a characterization of key polynomial (Theorem \ref{definofkeypol}) which allows us to determine whether a given key polynomial is a limit key polynomial. A more precise comparison between the concept of key polynomial introduced here and that of MacLane -- Vaqui\'e will be explored in a forthcoming paper by Decoup, Mahboub and Spivakovsky.
Given two polynomials $f,q\in K[x]$ with $q$ monic, we call the \textbf{$q$-standard expansion of $f$} the expression
\[
f(x)=f_0(x)+f_1(x)q(x)+\ldots+f_n(x)q^n(x)
\]
where for each $i$, $0\leq i\leq n$, $f_i=0$ or $\deg(f_i)<\deg(q)$. For a polynomial $q(x)\in K[x]$, the \textbf{$q$-truncation of $\nu$} is defined as
\[
\nu_q(f):=\min_{0\leq i\leq n}\{\nu(f_iq^i)\}
\]
where $f=f_0+f_1q+\ldots+f_nq^n$ is the $q$-standard expansion of $f$. In Section 2, we present an example that shows that $\nu_q$ does need to be a valuation. We also prove (Theorem \ref{proptruncakeypolval}) that if $Q$ is a key polynomial, then $\nu_Q$ is a valuation. A set $\Lambda$ of key polynomials is said to be a \textbf{complete set of key polynomials for $\nu$} if for every $f\in K[x]$, there exists $Q\in \Lambda$ such that $\nu_Q(f)=\nu(f)$. One of the main results of this paper is the following:
\begin{Teo}\label{Theoremexistencecompleteseqkpol}
Every valuation $\nu$ on $K[x]$ admits a complete set of key polynomials.
\end{Teo}
Another way of describing extensions of valuations from $K$ to $K[x]$ is the theory of pseudo-convergent sequences developed by Kaplansky in \cite{Kap}. He uses this theory to determine whether a maximal immediate extension of the valued field $(K,\nu)$ is unique (up to isomorphism). For a valued field $(K,\nu)$, a \textbf{pseudo-convergent sequence} is a well-ordered subset $\{a_{\rho}\}_{\rho<\lambda}$ of $K$, without last element, such that
\[
\nu(a_\sigma-a_\rho)<\nu(a_\tau-a_\sigma)\mbox{ for all }\rho<\sigma<\tau<\lambda.
\]
For a given pseudo-convergent sequence $\{a_{\rho}\}_{\rho<\lambda}$ it is easy to show that either $\nu(a_\rho)<\nu(a_\sigma)$ for all $\rho<\sigma<\lambda$ or there is $\rho<\lambda$ such that $\nu(a_\sigma)=\nu(a_\rho)$ for every $\rho<\sigma<\lambda$. If we set $\gamma_\rho:=\nu(a_{\rho+1}-a_\rho)$, then $\nu(a_\sigma-a_\rho)=\gamma_\rho$ for every $\rho<\sigma<\lambda$. Hence, the sequence $\{\gamma_\rho\}_{\rho<\lambda}$ is an increasing subset of $\Gamma$. An element $a\in K$ is said to be a \textbf{limit} of the pseudo-convergent sequence $\{a_\rho\}_{\rho<\lambda}$ if $\nu(a-a_\rho)=\gamma_\rho$ for every
$\rho<\lambda$.
One can prove that for every polynomial $f(x)\in K[x]$, there exists $\rho_f<\lambda$ such that either
\begin{equation}\label{condforpscstotra}
\nu(f(a_\sigma))=\nu(f(a_{\rho_f}))\mbox{ for every }\rho_f\leq \sigma<\lambda,
\end{equation}
or
\begin{equation}\label{condforpscstoalg}
\nu(f(a_\sigma))>\nu(f(a_{\rho}))\mbox{ for every }\rho_f\leq \rho< \sigma<\lambda.
\end{equation}
If case (\ref{condforpscstotra}) happens, we say that the value of $f$ is fixed by $\{a_\rho\}_{\rho<\lambda}$ (or that $\{a_\rho\}_{\rho<\lambda}$ fixes the value of $f$). A pseudo-convergent sequence $\{a_\rho\}_{\rho<\lambda}$ is said to be of \textbf{transcendental type} if for every polynomial $f(x)\in K[x]$ the condition (\ref{condforpscstotra}) holds.
Otherwise, $\{a_\rho\}_{\rho<\lambda}$ is said to be of \textbf{algebraic type}, i.e., if there exists at least one polynomial for which condition (\ref{condforpscstoalg}) holds.
The concept of key polynomials appears in the approach to local uniformization by Spivakovsky.
On the other hand, the concept of pseudo-convergent sequence plays an important role in the work of Knaf and Kuhlmann (see \cite{KK_1}) on the local uniformization problem. In this paper, we present a comparison between the concepts of key polynomials and pseudo-convergent sequences. More specifically, we prove the following:
\begin{Teo}\label{compthemkppsc}
Let $\nu$ be a valuation on $K[x]$ and let $\{a_\rho\}_{\rho<\lambda}\subset K$ be a pseudo-convergent sequence, without a limit in $K$, for which $x$ is a limit. If $\{a_\rho\}_{\rho<\lambda}$ is of transcendental type, then $\Lambda:=\{x-a_\rho\mid \rho<\lambda\}$ is a complete set of key polynomials for $\nu$. On the other hand, if $\{a_\rho\}_{\rho<\lambda}$ is of algebraic type, then every polynomial $q(x)$ of minimal degree among the polynomials not fixed by $\{a_\rho\}_{\rho<\lambda}$ is a limit key polynomial for $\nu$.
\end{Teo}
\section{Key polynomials}
We will assume throughout this paper that $K$ is a field, $\nu$ a valuation of $K[x]$, non-trivial on $K$ with $\nu(x)\geq 0$. We begin by making some remarks.
\begin{Obs}\label{exlinearnonlinepolarepsk}
\begin{description}
\item[(i)] Every linear polynomial $x-a$ is a key polynomial (of level $\epsilon(x-a)=\nu(x-a)$).
\item[(ii)] Take a polynomial $f(x)\in K[x]$ of degree greater than one and assume that there exists $a\in K$ such that
$\nu(\mathbb{P}artial_bf(a))=\nu(\mathbb{P}artial_bf(x))$ for every $b\in\mathbb{N}$ (note that such an $a$ always exists if the assumptions of Theorem \ref{compthemkppsc} hold and the pseudo-convergent sequence is transcendental or is algebraic and $\deg(f)\leq\deg(q)$). Write
\[
f(x)=f(a)+\sum_{i=1}^n\mathbb{P}artial_if(a)(x-a)^i
\]
and take $h\in\{1,\ldots,n\}$ such that
\[
\nu(\mathbb{P}artial_hf(x))+h\nu(x-a)=\min_{1\leq i\leq n}\{\nu(\mathbb{P}artial_if(x))+i\nu(x-a)\}.
\]
If $\nu(f(a))<\nu(\mathbb{P}artial_hf(x))+h\nu(x-a)$, then $\nu(f(x))=\nu(f(a))$ and hence
\[
\frac{\nu(f(x))-\nu(\mathbb{P}artial_if(x))}{i}<\nu(x-a)
\]
for every $i$, $1\leq i\leq n$. Consequently, $\epsilon(f)<\nu(x-a)=\epsilon(x-a)$ and hence $f$ is not a key polynomial.
On the other hand, if
$$
\nu(\mathbb{P}artial_hf(x))+h\nu(x-a)\leq\nu(f(a)),
$$
then
\begin{equation}\label{eqtaylorexpwithpcs}
\nu(f(x))\geq\nu(\mathbb{P}artial_hf(x))+h\nu(x-a)
\end{equation}
and if the equality holds in (\ref{eqtaylorexpwithpcs}), then
\[
\epsilon(f)=\frac{\nu(f(x))-\nu(\mathbb{P}artial_hf(x))}{h}=\nu(x-a)=\epsilon(x-a)
\]
and hence $f$ is not a key polynomial. In other words, the only situation when $f$ may be a key polynomial is when
$$
f(x)>\min_{1\leq i\leq n}\{f(a),\nu(\mathbb{P}artial_if(x))+i\nu(x-a)\}.
$$
\end{description}
\end{Obs}
\begin{Obs}
We observe that if $Q$ is a key polynomial of level $\epsilon:=\epsilon(Q)$, then for every polynomial $f\in K[x]$ with
$\deg(f)<\deg(Q)$ and every $b\in\mathbb{N}$ we have
\begin{equation}\label{eqpolyndegsmallkeypol}
\nu(\mathbb{P}artial_b(f))>\nu(f)-b\epsilon.
\end{equation}
Indeed, from the definition of key polynomial we have that $\epsilon>\epsilon(f)$. Hence, for every $b\in\mathbb{N}$ we have
\[
\frac{\nu(f)-\nu(\mathbb{P}artial_b(f))}{b}\leq\epsilon(f)<\epsilon
\]
and this implies (\ref{eqpolyndegsmallkeypol}).
\end{Obs}
Let
\[
I(f)=\left\{b\in\mathbb{N}\left|\frac{\nu(f)-\nu(\mathbb{P}artial_bf)}{b}=\epsilon(f)\right.\right\}
\]
and $b(f)=\min I(f)$.
\begin{Lema}
Let $Q$ be a key polynomial and take $f,g\in K[x]$ such that
$$
\deg(f)<\deg(Q)
$$
and
$$
\deg(g)<\deg(Q).
$$
Then for $\epsilon:=\epsilon(Q)$ and any $b\in\mathbb{N}$ we have the following:
\begin{description}\label{lemaonkeypollder}
\item[(i)] $\nu(\mathbb{P}artial_b(fg))>\nu(fg)-b\epsilon$
\item[(ii)] If $\nu_Q(fQ+g)<\nu(fQ+g)$ and $b\in I(Q)$, then $\nu(\mathbb{P}artial_b(fQ+g))=\nu(fQ)-b\epsilon$;
\item[(iii)] If $h_1,\ldots,h_s$ are polynomials such that $\deg(h_i)<\deg(Q)$ for every $i=1,\ldots, s$ and
$\displaystyle\mathbb{P}rod_{i=1}^sh_i=qQ+r$ with $\deg(r)<\deg(Q)$, then
\[
\nu(r)=\nu\left(\mathbb{P}rod_{i=1}^sh_i\right)<\nu(qQ).
\]
\end{description}
\end{Lema}
\begin{proof}
\textbf{(i)} Since $\deg(f)<\deg(Q)$ and $\deg(g)<\deg(Q)$, for each $j\in \mathbb{N}$, we have
\[
\nu(\mathbb{P}artial_jf)>\nu(f)-j\epsilon\mbox{ and }\nu(\mathbb{P}artial_jg)>\nu(g)-j\epsilon.
\]
This, and the fact that
\[
\mathbb{P}artial_b(fg)=\sum_{j=0}^b\mathbb{P}artial_jf\mathbb{P}artial_{b-j}g,
\]
imply that
\[
\nu(\mathbb{P}artial_b(fg))\geq\min_{0\leq j\leq b}\{\nu(\mathbb{P}artial_jf)+\nu(\mathbb{P}artial_{b-j}g)\}>\nu(fg)-b\epsilon.
\]
\textbf{(ii)} If $\nu_Q(fQ+g)<\nu(fQ+g)$, then $\nu(fQ)=\nu(g)$. Hence,
\[
\nu(\mathbb{P}artial_bg)>\nu(g)-b\epsilon=\nu(fQ)-b\epsilon.
\]
Moreover, for every $j\in\mathbb{N}$, we have
\[
\nu(\mathbb{P}artial_j f\mathbb{P}artial_{b-j}Q)=\nu(\mathbb{P}artial_jf)+\nu(\mathbb{P}artial_{b-j}Q)>\nu(f)-j\epsilon+\nu(Q)-(b-j)\epsilon=\nu(fQ)-b\epsilon.
\]
Therefore,
\[
\nu(\mathbb{P}artial_b(fQ+g))=\nu\left(f\mathbb{P}artial_bQ+\sum_{j=1}^b\mathbb{P}artial_jf\mathbb{P}artial_{b-j}Q+\mathbb{P}artial_bg\right)=\nu(fQ)-b\epsilon.
\]
\textbf{(iii)} We proceed by induction on $s$. If $s=1$, then $h_1=qQ+r$ with
$$
\deg(h_1)<\deg(Q),
$$
which implies that $h_1=r$ and $q=0$. Our result follows immediately.
Next, consider the case $s=2$. Take $f,g\in K[x]$ such that $\deg(f)<\deg(Q)$, $\deg(g)<\deg(Q)$ and write $fg=qQ+r$ with
$\deg(r)<\deg(Q)$. Then
$$
\deg(q)<\deg(Q)
$$
and for $b\in I(Q)$ we have
\[
\nu\left(\mathbb{P}artial_b(qQ)\right)=\nu\left(\sum_{j=0}^b\mathbb{P}artial_jq\mathbb{P}artial_{b-j}Q\right)=\nu(qQ)-b\epsilon.
\]
This and part \textbf{(i)} imply that
\begin{displaymath}
\begin{array}{rcl}
\nu(qQ)-b\epsilon &=& \nu\left(\mathbb{P}artial_b(qQ)\right)= \nu(\mathbb{P}artial_b(fg)-\mathbb{P}artial_b(r))\\
&\geq &\min\{\nu\left(\mathbb{P}artial_b(fg)\right),\nu\left(\mathbb{P}artial_b(r)\right)\}\\
&>&\min\{\nu(fg),\nu(r)\}-b\epsilon.
\end{array}
\end{displaymath}
and consequently
\begin{equation}\label{equationwithepsilon}
\nu(r)=\nu(fg)<\nu(qQ).
\end{equation}
Assume now that $s>2$ and define $\displaystyle h:=\mathbb{P}rod_{i=1}^{s-1}h_i$. Write $h=q_1Q+r_1$ with $\deg(r_1)<\deg(Q)$. Then by the induction hypothesis we have
$$
\nu(r_1)=\nu(h)<\nu(q_1Q)
$$
and hence
\[
\nu\left(\mathbb{P}rod_{i=1}^sh_i\right)=\nu(r_1h_s)<\nu(q_1h_sQ).
\]
Write $r_1h_s=q_2Q+r_2$. Then, by equation (\ref{equationwithepsilon}) we have
\[
\nu(r_2)=\nu(r_1h_s)<\nu(q_2Q).
\]
If $\displaystyle \mathbb{P}rod_{i=1}^sh_i=qQ+r$ with $\deg(r)<\deg(Q)$, then
\[
qQ+r=\mathbb{P}rod_{i=1}^sh_i=hh_s=(q_1Q+r_1)h_s=q_1h_sQ+r_1h_s=q_1h_sQ+q_2Q+r_2
\]
and hence $q=q_1h_s+q_2$ and $r=r_2$. Therefore,
\[
\nu(qQ)\geq\min\{\nu(q_1h_sQ),\nu(q_2Q)\}>\nu(r_1h_s)=\nu(r)=\nu\left(\mathbb{P}rod_{i=1}^sh_i\right).
\]
This is what we wanted to prove.
\end{proof}
We denote by $p$ the \textbf{exponent characteristic} of $K$, that is, $p=1$ if $\mathbb{C}h(K)=0$ and $p=\mathbb{C}h(K)$ if $\mathbb{C}h(K)>0$.
\begin{Prop}\label{propaboutpseudkeyool}
Let $Q\in K[x]$ be a key polynomial and set $\epsilon:=\epsilon(Q)$. Then the following hold:
\begin{description}
\item[(i)] Every element in $I(Q)$ is a power of $p$;
\item[(ii)] $Q$ is irreducible.
\end{description}
\end{Prop}
\begin{proof}
\textbf{(i)} Take $b\in I(Q)$ and assume, aiming for contradiction, that $b$ is not a power of $p$. Write $b=p^tr$ where $r>1$ is prime to $p$. Then, by Lemma 6 of \cite{Kap}, $\binom{b}{p^t}$ is prime to $p$ and hence $\nu\binom{b}{p^t}=0$. Since $\binom{b}{p^t}\mathbb{P}artial_b=\mathbb{P}artial_{p^t}\circ\mathbb{P}artial_{b'}$ for $b'=b-p^t$, we have
\[
\nu(\mathbb{P}artial_{b'}Q)-\nu(\mathbb{P}artial_bQ)=\nu(\mathbb{P}artial_{b'}Q)-\nu(\mathbb{P}artial_{p^t}(\mathbb{P}artial_{b'}Q))\leq p^t\epsilon(\mathbb{P}artial_{b'}(Q))<p^t\epsilon
\]
because $\deg(\mathbb{P}artial_{b'}Q)<\deg(Q)$ and $Q$ is a key polynomial. Hence,
\[
b\epsilon=\nu(Q)-\nu(\mathbb{P}artial_bQ)=\nu(Q)-\nu(\mathbb{P}artial_{b'}Q)+\nu(\mathbb{P}artial_{b'}Q)-\nu(\mathbb{P}artial_bQ)<
b'\epsilon+p^t\epsilon=b\epsilon,
\]
which gives the desired contradiction.
\textbf{(ii)} If $Q=gh$ for non-constant polynomials $g,h\in K[x]$, then by Lemma \ref{lemaonkeypollder} \textbf{(i)}, we would have for $b\in I(Q)$ that
\[
\nu(\mathbb{P}artial_bQ)>\nu(Q)-b\epsilon,
\]
which is a contradiction to the definition of $b$ and $\epsilon$.
\end{proof}
We present an example to show that $\nu_q$ does not need to be a valuation for a general polynomial $q(x)\in K[x]$.
\begin{Exa}
Consider a valuation $\nu$ in $K[x]$ such that $\nu(x)=\nu(a)=1$ for some $a\in K$. Take $q(x)=x^2+1$ (which can be irreducible, for instance, if $K=\mathbb{R}$ or $K=\mathbb F_p$ and $-1$ is not a quadratic residue $\mod p$). Since $x^2-a^2=(x^2+1)-(a^2+1)$ we have
\[
\nu_q(x^2-a^2)=\min\{\nu(x^2+1),\nu(a^2+1)\}=0.
\]
On the other hand, $\nu_q(x+a)=\nu(x+a)\geq\min\{\nu(a),\nu(x)\}=1$ (and the same holds for $\nu_q(x-a)$). Hence
\[
\nu_q(x^2-a^2)=0<1+1\leq\nu_q(x-a)+\nu_q(x+a)
\]
which shows that $\nu_q$ is not a valuation.
\end{Exa}
If $f=f_0+f_1q+\ldots+f_nq^n$ is the $q$-standard decomposition of $f$ we set
\[
S_q(f):=\{i\in\{0,\ldots, n\}\mid \nu(f_iq^i)=\nu_q(f)\}\mbox{ and }\delta_q(f)=\max S_q(f).
\]
\begin{Prop}\label{proptruncakeypolval}
If $Q$ is a key polynomial, then $\nu_Q$ is a valuation of $K[x]$.
\end{Prop}
\begin{proof}
One can easily see that $\nu_Q(f+g)\geq\min\{\nu_Q(f),\nu_Q(g)\}$ for every $f,g\in K[x]$. It remains to prove that
$\nu_Q(fg)=\nu_Q(f)+\nu_Q(g)$ for every $f,g\in K[x]$. Assume first that $\deg(f)<\deg(Q)$ and $\deg(g)<\deg(Q)$ and let $fg=aQ+c$ be the $Q$-standard expansion of $fg$. By Lemma \ref{lemaonkeypollder} \textbf{(iii)} we have
\[
\nu(fg)=\nu(c)<\nu(aQ)
\]
and hence
\[
\nu_Q(fg)=\min\{\nu(aQ),\nu(c)\}=\nu(c)=\nu(fg)=\nu(f)+\nu(g)=\nu_Q(f)+\nu_Q(g).
\]
Now assume that $f,g\in K[x]$ are any polynomials and consider the $Q$-expansions
\[
f=f_0+\ldots+f_nQ^n\mbox{ and }g=g_0+\ldots+g_mQ^m
\]
of $f$ and $g$. Then, using the first part of the proof, we obtain
\[
\nu_Q(fg)\geq\min_{i,j}\{\nu_Q(f_ig_jQ^{i+j})\}=\min_{i,j}\{\nu_Q(f_iQ^i)+\nu_Q(g_jQ^j)\}=\nu_Q(f)+\nu_Q(g).
\]
For each $i\in\{0,\ldots,n\}$ and $j\in\{0,\ldots,m\}$, let $f_ig_j=a_{ij}Q+c_{ij}$ be the $Q$-standard expansion of $f_ig_j$. Then, by Lemma \ref{lemaonkeypollder} \textbf{(iii)}, we have
\[
\nu(f_iQ^i)+\nu(g_jQ^j)=\nu(f_ig_j)+\nu(Q^{i+j})=\nu(c_{ij})+\nu(Q^{i+j})=\nu(c_{ij}Q^{i+j}).
\]
Let
\[
i_0=\min\{i\mid\nu_Q(f)=\nu(f_iQ^i)\}\mbox{ and }j_0=\min\{j\mid\nu_Q(g)=\nu(g_jQ^j)\},
\]
and set $k_0:=i_0+j_0$. Then for every $i<i_0$ or $j<j_0$ we have
\begin{equation}\label{eqnatnaksdjs}
\min\left\{\nu(a_{ij}Q^{i+j+1}),\nu(c_{ij}Q^{i+j})\right\}=\nu(f_iQ^i)+\nu(g_jQ^j)>\nu(c_{i_0j_0}Q^{k_0}).
\end{equation}
Let $fg=a_0+a_1Q+\ldots+a_rQ^r$ be the $Q$-standard expansion of $fg$. Then
\[
a_{k_0}=\sum_{i+j+1=k_0}a_{ij}+\sum_{i+j=k_0}c_{ij}.
\]
This and equation (\ref{eqnatnaksdjs}) give us that
\[
\nu(a_{k_0}Q^{k_0})=\nu(c_{i_0j_0}Q^{k_0})=\nu(f_{i_0}Q^{i_0})+\nu(g_{j_0}Q^{j_0})=\nu_Q(f)+\nu_Q(g).
\]
Therefore,
\[
\nu_Q(fg)=\min_{0\leq k\leq r}\{\nu(a_kQ^k)\}\leq \nu_Q(f)+\nu_Q(g),
\]
which completes the proof.
\end{proof}
\begin{Prop}\label{Propdificil}
Let $Q\in K[x]$ be a key polynomial and set $\epsilon:=\epsilon(Q)$. For any $f\in K[x]$ the following hold:
\begin{description}
\item[(i)] For any $b\in\mathbb{N}$ we have
\begin{equation}\label{eqthatcompvalutrunc}
\frac{\nu_Q(f)-\nu_Q(\mathbb{P}artial_bf)}{b}\leq \epsilon;
\end{equation}
\item[(ii)] If $S_Q(f)\neq\{0\}$, then the equality in (\ref{eqthatcompvalutrunc}) holds for some $b\in\mathbb{N}$;
\item[(iii)] If for some $b\in\mathbb{N}$, the equality in (\ref{eqthatcompvalutrunc}) holds and $\nu_Q(\mathbb{P}artial_bf)=\nu(\mathbb{P}artial_bf)$, then
$\epsilon(f)\geq\epsilon$. If in addition, $\nu(f)>\nu_Q(f)$, then $\epsilon(f)>\epsilon$.
\end{description}
\end{Prop}
Fix a key polynomial $Q$ and $h\in K[x]$ with $\deg(h)<\deg(Q)$. Then, for every $b\in\mathbb{N}$ the Leibnitz rule for derivation gives us that
\begin{equation}
\mathbb{P}artial_b(hQ^n)=\sum_{b_0+\ldots+b_r=b}T_b(b_0,\ldots,b_r)
\end{equation}
where
\[
T_b(b_0,\ldots, b_r):=\mathbb{P}artial_{b_0}h\left(\mathbb{P}rod_{i=1}^r\mathbb{P}artial_{b_i}Q\right)Q^{n-r}.
\]
In order to prove Proposition \ref{Propdificil}, we will need the following result:
\begin{Lema}\label{Lemamagic3}
Let $Q$ be a key polynomial, $h\in K[x]$ with $\deg(h)<\deg(Q)$ and set $\epsilon:=\epsilon(Q)$. For any $b\in\mathbb{N}$ we have
\[
\nu_Q(T_b(b_0,\ldots,b_r))\geq \nu(hQ^n)-b\epsilon.
\]
Moreover, if either $b_0>0$ or $b_i\notin I(Q)$ for some $i=1,\ldots, r$, then
\[
\nu_Q(T_b(b_0,\ldots,b_r))> \nu(hQ^n)-b\epsilon.
\]
\end{Lema}
\begin{proof}
Since $\deg(h)<\deg(Q)$ and $Q$ is a key polynomial we have $\epsilon(h)<\epsilon$. Hence, if $b_0>0$ we have
\[
\nu(\mathbb{P}artial_{b_0}h)\geq \nu(h)-b_0\epsilon(h)>\nu(h)-b_0\epsilon.
\]
On the other hand, for every $i=1,\ldots, r$, by definition of $\epsilon$ we have
\[
\nu(\mathbb{P}artial_{b_i}Q)\geq \nu(Q)-b_i\epsilon,
\]
and if $b_i\notin I(Q)$ we have
\[
\nu(\mathbb{P}artial_{b_i}Q)> \nu(Q)-b_i\epsilon.
\]
Since $\nu_Q(\mathbb{P}artial_{b_0}h)=\nu(\mathbb{P}artial_{b_0}h)$ and $\nu_Q(\mathbb{P}artial_{b_i}Q)=\nu(\mathbb{P}artial_{b_i}Q)$, we have
\begin{displaymath}
\begin{array}{rcl}
\nu_Q(T_b(b_0,\ldots,b_r))&=&\displaystyle\nu_Q\left(\mathbb{P}artial_{b_0}h\left(\mathbb{P}rod_{i=1}^r\mathbb{P}artial_{b_i}Q\right)Q^{n-r}\right)\\
&=&\displaystyle\nu_Q(\mathbb{P}artial_{b_0}h)+\sum_{i=1}^r\nu_Q(\mathbb{P}artial_{b_i}Q)+(n-r)\nu_Q(Q)\\
&\geq &\displaystyle\nu(h)-b_0\epsilon+\sum_{i=1}^r\left(\nu(Q)-b_i\epsilon\right)+(n-r)\nu(Q)\\
&\geq &\nu(hQ^n)-b\epsilon.
\end{array}
\end{displaymath}
Moreover, if $b_0>0$ or $b_i\notin I(Q)$ for some $i=1,\ldots, r$, then the inequality above is strict.
\end{proof}
\begin{Cor}\label{Coroaboutderib}
For every $b\in\mathbb{N}$ we have $\nu_Q\left(\mathbb{P}artial_b(aQ^n)\right)\geq\nu(aQ^n)-b\epsilon$.
\end{Cor}
\begin{proof}[Proof of Proposition \ref{Propdificil}]
\textbf{(i)} Take any $f\in K[x]$ and consider its $Q$-standard expansion $f=f_0+f_1Q+\ldots+f_nQ^n$. For each $i=0,\ldots,n$, Corollary \ref{Coroaboutderib} gives us that
\[
\nu_Q\left(\mathbb{P}artial_b(f_iQ^i)\right)\geq \nu (f_iQ^i)-b\epsilon.
\]
Hence,
\[
\nu_Q\left(\mathbb{P}artial_b(f)\right)\geq\min_{0\leq i\leq n}\{\nu_Q(f_iQ^i)\}\geq\min_{0\leq i\leq n}\{\nu(f_iQ^i)-b\epsilon\}=\nu_Q(f)-b\epsilon.
\]
\textbf{(ii)} Assume that $S_Q(f)\neq \{0\}$ and set $j_0=\min S_Q(f)$. Then $j_0=p^er$ for some $e\in\mathbb{N}\cup\{0\}$ and some $r\in\mathbb{N}$ with $(r,p)=1$. We set $b:=p^eb(Q)$ and will prove that $\nu_Q(\mathbb{P}artial_b(f))=\nu_Q(f)-b\epsilon$.
Write
\[
f_{j_0}\left(\mathbb{P}artial_{b(Q)}Q\right)^{p^e}=rQ+h
\]
for some $r,h\in K[x]$ and $\deg(h)<\deg(Q)$ (note that $h\ne0$ because $Q$ is irreducible and $Q\nmid f_{j_0}$ and $Q\nmid \mathbb{P}artial_{b(Q)}Q$). Then Lemma \ref{lemaonkeypollder} \textbf{(iii)} gives us that
\[
\nu(h)=\nu\left(f_{j_0}(\mathbb{P}artial_{b(Q)}Q)^{p^e}\right).
\]
This implies that
\begin{equation}\label{equationboa}
\nu\left(hQ^{j_0-p^e}\right)=\nu_Q(f)-b\epsilon.
\end{equation}
Indeed, we have
\begin{displaymath}
\begin{array}{rcl}
\nu\left(hQ^{j_0-p^e}\right)&=& \nu(h)+\nu\left(Q^{j_0-p^e}\right)=\nu\left(f_{j_0}(\mathbb{P}artial_{b(Q)}Q)^{p^e}\right)+
\nu\left(Q^{j_0-p^e}\right)\\
&=&\nu(f_{j_0})+p^e\nu\left(\mathbb{P}artial_{b(Q)}Q\right)+(j_0-p^e)\nu(Q)\\
&=&\nu(f_{j_0})+p^e\left(\nu\left(Q\right)-b(Q)\epsilon\right)+(j_0-p^e)\nu(Q)\\
&=&\nu(f_{j_0})+j_0\nu(Q)-p^eb(Q)\epsilon\\
&=&\nu(f_{j_0}Q^{j_0})-p^eb(Q)\epsilon=\nu_Q(f)-b\epsilon.\\
\end{array}
\end{displaymath}
Since $f=f_0+f_1Q+\ldots+f_nQ^n$, we have $\mathbb{P}artial_b(f)=\mathbb{P}artial_b(f_0)+\mathbb{P}artial_b(f_1Q)\ldots+\mathbb{P}artial_b(f_nQ^n)$. For each $j=0,\ldots, n$, if $j\notin S_Q(f)$, then
\[
\nu_Q\left(\mathbb{P}artial_b(f_jQ^j)\right)\geq \nu_Q(f_jQ^j)-b\epsilon>\nu_Q(f)-b\epsilon.
\]
We set
\[
h_1=\sum_{j\notin S_Q(f)}f_jQ^j.
\]
Then $\nu_Q(h_1)>\nu_Q(f)-b\epsilon$.
For each $j\in S_Q(f)$ the term $\mathbb{P}artial_b(f_jQ^j)$ can be written as a sum of terms of the form $T_b(b_0,\ldots,b_r)$. For each $T_b(b_0,\ldots,b_r)$ we have the following cases:
\textbf{Case 1:} $b_0>0$ or $b_i\notin I(Q)$ for some $i$.\\
In this case, by Lemma \ref{Lemamagic3} we have $\nu_Q(T_b(b_0,\ldots,b_r))>\nu_Q(f)-b\epsilon$. In particular, if $h_2$ is the sum of all these terms, then $\nu_Q(h_2)>\nu_Q(f)-b\epsilon$.
\textbf{Case 2:} $b_0=0$ and $b_i\in I(Q)$ for every $i=1,\ldots,r$ but $b_{i_0}\neq b(Q)$ for some $i_0=1,\ldots,r$.\\
This implies, in particular, that $j\geq j_0$ and since $b=p^eb(Q)$ we must have $r<p^e$. Hence
\[
T_b(b_0,b_1,\ldots,b_r)=\mathbb{P}artial_{b_0}f_j\left(\mathbb{P}rod_{i=1}^r\mathbb{P}artial_{b_i}Q\right)Q^{j-r}=sQ^{j_0-p^e+1}
\]
for some $s\in K[x]$.
\textbf{Case 3:} $b_0=0$, $j>j_0$ and $b_i=b(Q)$ for every $i=1,\ldots,r$.\\
Since $b=p^eb(Q)$, $b_i=b(Q)$ and $\displaystyle\sum_{i=1}^rb_i=b$ we must have $r=p^e$. Hence
\[
T_b(b_0,b_1,\ldots,b_r)=f_j\left(\mathbb{P}artial_{b(Q)}Q\right)^{p^e}Q^{j-p^e}=s'Q^{j_0-p^e+1}
\]
for some $s'\in K[x]$.\\
\textbf{Case 4:} $b_0=0$, $j=j_0$ and $b_i=b(Q)$ for every $i=1,\ldots,r$.\\
In this case we have
\begin{equation}\label{caseintport}
\begin{array}{rcl}
T_b(b_0,b_1,\ldots,b_r)&=&f_{j_0}\left(\mathbb{P}artial_{b(Q)}Q\right)^{p^e}Q^{j_0-p^e}\\
&=&\left(h-rQ\right)Q^{j_0-p^e}\\
&=&hQ^{j_0-p^e}-rQ^{j_0-p^e+1}.
\end{array}
\end{equation}
Observe that the number of times that the term (\ref{caseintport}) appears in $\mathbb{P}artial_b(f_{j_0}Q^{j_0})$ is $\binom{j_0}{p^e}$, that is, the number of ways that one can choose a subset with $p^e$ elements in a set of $j_0$ elements.
Therefore, we can write
\[
\mathbb{P}artial_b(f)=\binom{j_0}{p^e}hQ^{j_0-p^e}+\left(s+s'-\binom{j_0}{p^e}r\right)Q^{j_0-p^e+1}+h_1+h_2
\]
Since $p\nmid \binom{j_0}{p^e}$ the equation (\ref{equationboa}) gives us that
\[
\nu\left(\binom{j_0}{p^e}hQ^{j_0-p^e}\right)=\nu_Q(f)-b\epsilon.
\]
Then
\[
\nu_Q\left(\binom{j_0}{p^e}hQ^{j_0-p^e}+\left(s+s'-\binom{j_0}{p^e}r\right)Q^{j_0-p^e+1}\right)\leq \nu_Q(f)-b\epsilon.
\]
This and the fact that $\nu_Q(h_1+h_2)>\nu_Q(f)-b\epsilon$ imply that $\nu_Q\left(\mathbb{P}artial_b(f)\right)\leq\nu_Q(f)-b\epsilon$. This concludes the proof of \textbf{(ii)}.
\textbf{(iii)} The assumptions on $b$ give us
\[
\frac{\nu_Q(f)-\nu_Q(\mathbb{P}artial_bf)}{b}= \epsilon
\]
and
\[
\nu_Q(\mathbb{P}artial_bf)=\nu(\mathbb{P}artial_bf).
\]
Consequently,
\[
\epsilon(f)\geq \frac{\nu(f)-\nu(\mathbb{P}artial_bf)}{b}\geq\frac{\nu_Q(f)-\nu_Q(\mathbb{P}artial_bf)}{b}= \epsilon.
\]
In the inequality above, one can see that if $\nu(f)>\nu_Q(f)$, then $\epsilon(f)>\epsilon$.
\end{proof}
\begin{Prop}\label{Propcompkeypol}
For two key polynomials $Q,Q'\in K[x]$ we have the following:
\begin{description}
\item[(i)] If $\deg(Q)<\deg(Q')$, then $\epsilon(Q)<\epsilon(Q')$;
\item[(ii)] If $\epsilon(Q)<\epsilon(Q')$, then $\nu_Q(Q')<\nu(Q')$;
\item[(iii)] If $\deg(Q)=\deg(Q')$, then
\begin{equation}\label{eqwhdegsame}
\nu(Q)<\nu(Q')\Longleftrightarrow \nu_Q(Q')<\nu(Q')\Longleftrightarrow \epsilon(Q)<\epsilon(Q').
\end{equation}
\end{description}
\end{Prop}
\begin{proof}
Item \textbf{(i)} follows immediately from the the definition of key polynomial (in fact, the same holds if we substitute $Q$ for any $f\in K[x]$).
In order to prove \textbf{(ii)} we set $\epsilon:=\epsilon(Q)$ and $b':=b(Q')$. By \textbf{(i)} of Proposition \ref{Propdificil}, we have
\[
\nu_Q(Q')\leq \nu_Q(\mathbb{P}artial_{b'}Q')+b'\epsilon.
\]
Since $\epsilon(Q)<\epsilon(Q')$, we also have
\[
\nu(\mathbb{P}artial_{b'}Q')+b'\epsilon< \nu(\mathbb{P}artial_{b'}Q')+b'\epsilon(Q')=\nu(Q').
\]
This, and the fact that $\nu_Q(\mathbb{P}artial_{b'}Q')\leq \nu(\mathbb{P}artial_{b'}Q')$, imply that $\nu_Q(Q')<\nu(Q')$.
Now assume that $\deg(Q)=\deg(Q')$ and let us prove (\ref{eqwhdegsame}). Since
$$
\deg(Q)=\deg(Q')
$$
and both $Q$ and $Q'$ are monic, the $Q$-standard expansion of $Q'$ is given by
$$
Q'=Q+(Q-Q').
$$
Hence
\[
\nu_Q(Q')=\min\{\nu(Q),\nu(Q-Q')\}.
\]
The first equivalence follows immediately from this. In view of part \textbf{(ii)}, it remains to prove that if $\nu_Q(Q')<\nu(Q')$, then $\epsilon(Q)<\epsilon(Q')$. Since $\nu_Q(Q')<\nu(Q')$ we have $S_Q(Q')\neq \{0\}$. Hence, by Proposition \ref{Propdificil} \textbf{(ii)}, the equality holds in (\ref{eqthatcompvalutrunc}) (for $f=Q'$) for some $b\in\mathbb{N}$. Moreover, since $\deg(Q)=\deg(Q')$, we have
$\deg(\mathbb{P}artial_bQ')<\deg(Q)$ and consequently $\nu_Q(\mathbb{P}artial_bQ')=\nu(\mathbb{P}artial_bQ')$. Then Proposition \ref{Propdificil} \textbf{(iii)} implies that $\epsilon(Q)<\epsilon(Q')$.
\end{proof}
For a key polynomial $Q\in K[x]$, let
\[
\alpha(Q):=\min\{\deg(f)\mid \nu_Q(f)< \nu(f)\}
\]
(if $\nu_Q=\nu$, then set $\alpha(Q)=\infty$) and
\[
\Psi(Q):=\{f\in K[x]\mid f\mbox{ is monic},\nu_Q(f)< \nu(f)\mbox{ and }\alpha(Q)=\deg (f)\}.
\]
\begin{Lema}\label{lemmapsikeypoly}
If $Q$ is a key polynomial, then every element $Q'\in\Psi(Q)$ is also a key polynomial. Moreover, $\epsilon(Q)<\epsilon(Q')$.
\end{Lema}
\begin{proof}
By assumption, we have $\nu_Q(Q')<\nu(Q')$, hence $S_{Q}(Q')\neq \{0\}$. This implies, by Proposition \ref{Propdificil} \textbf{(ii)}, that there exists $b\in\mathbb{N}$ such that
\[
\nu_Q(Q')-\nu_Q(\mathbb{P}artial_b Q')=b\epsilon(Q).
\]
Since $\deg(\mathbb{P}artial_b Q')<\deg(Q')=\alpha(Q)$, we have $\nu_Q(\mathbb{P}artial_b Q')=\nu(\mathbb{P}artial_b Q')$. Consequently, by Proposition \ref{Propdificil} \textbf{(iii)}, $\epsilon(Q)<\epsilon(Q')$.
Now take any polynomial $f\in K[x]$ such that $\deg(f)<\deg(Q')=\alpha(Q)$. In particular, $\nu_Q(f)=\nu(f)$. Moreover, for every
$b\in\mathbb{N}$, $\deg(\mathbb{P}artial_b f)<\deg(Q')= \alpha(Q)$ which implies that $\nu_Q(\mathbb{P}artial_bf)=\nu(\mathbb{P}artial_b f)$. Then, for every $b\in\mathbb{N}$,
\[
\frac{\nu(f)-\nu(\mathbb{P}artial_bf)}{b}=\frac{\nu_Q(f)-\nu_Q(\mathbb{P}artial_bf)}{b}\leq \epsilon(Q)<\epsilon(Q').
\]
This implies that $\epsilon(f)<\epsilon(Q')$, which shows that $Q'$ is a key polynomial.
\end{proof}
\begin{Teo}\label{definofkeypol}
A polynomial $Q$ is a key polynomial if and only if there exists a key polynomial $Q_-\in K[x]$ such that $Q\in \Psi(Q_-)$ or the following conditions hold:
\begin{description}
\item[(K1)] $\alpha(Q_-)=\deg (Q_-)$
\item[(K2)] the set $\{\nu(Q')\mid Q'\in\Psi(Q_-)\}$ does not contain a maximal element
\item[(K3)] $\nu_{Q'}(Q)<\nu(Q)$ for every $Q'\in \Psi(Q_-)$
\item[(K4)] $Q$ has the smallest degree among polynomials satisfying \textbf{(K3)}.
\end{description}
\end{Teo}
\begin{proof}
We will prove first that if such $Q_-$ exists, then $Q$ is a key polynomial. The case when $Q\in \Psi(Q_-)$ follows from Lemma \ref{lemmapsikeypoly}. Assume now that \textbf{(K1) - (K4)} hold. Take $f\in K[x]$ such that $\deg(f)<\deg(Q)$. This implies that $\deg(\mathbb{P}artial_bQ)<\deg(Q)$ and $\deg(\mathbb{P}artial_bf)<\deg(Q)$ for every $b\in\mathbb{N}$. Hence, by \textbf{(K4)}, there exists $Q'\in\Psi(Q_-)$ such that
\[
\nu_{Q'}(f)=\nu(f), \nu_{Q'}(\mathbb{P}artial_bf)=\nu(\mathbb{P}artial_bf) \mbox{ and }\nu_{Q'}(\mathbb{P}artial_bQ)=\nu(\mathbb{P}artial_bQ)\mbox{ for every }b\in\mathbb{N}.
\]
We claim that $\epsilon(Q')<\epsilon(Q)$. If not, by Proposition \ref{Propcompkeypol} \textbf{(i)}, we would have
$\deg(Q)\leq\deg(Q')$. Since $\nu_{Q'}(Q)<\nu(Q)$, this implies that $\deg(Q)=\deg(Q')$. This and Proposition \ref{Propcompkeypol} \textbf{(iii)} give us that $\epsilon(Q')<\epsilon(Q)$ which is a contradiction.
Now,
\[
\epsilon(f)\leq \frac{\nu(f)-\nu(\mathbb{P}artial_bf)}{b}=\frac{\nu_{Q'}(f)-\nu_{Q'}(\mathbb{P}artial_bf)}{b}\leq\epsilon(Q')<\epsilon(Q).
\]
Hence $Q$ is a key polynomial.
For the converse, take a key polynomial $Q\in K[x]$ and consider the set
\[
\mathcal S:=\{Q'\in K[x]\mid Q'\mbox{ is a key polynomial and }\nu_{Q'}(Q)<\nu(Q)\}.
\]
Observe that $\mathcal S\neq\emptyset$. Indeed, if $\deg(Q)>1$, then every key polynomial $x-a\in\mathcal S$. If $Q=x-a$, then there exits $b\in K$ such that $\nu(b)<\min\{\nu(a),\nu(x)\}$. Therefore, $x-b\in \mathcal S$.
If there exists a key polynomial $Q_-\in \mathcal S$ such that $\deg(Q)=\deg(Q_-)$ , then we have $Q\in \Psi(Q_-)$ and we are done. Hence, assume that every polynomial $Q'\in \mathcal S$ has degree smaller that $\deg(Q)$.
Assume that there exists $Q_-\in \mathcal S$ such that for every $Q'\in \mathcal S$ we have
\begin{equation}\label{eqmaxphi}
(\deg(Q_-),\nu(Q_-))\geq ((\deg(Q'),\nu(Q'))
\end{equation}
in the lexicographical ordering. We claim that $Q\in \Psi(Q_-)$. If not, there would exist a key polynomial $Q''$ such that $\nu_{Q_-}(Q'')<\nu(Q'')$ and $\deg(Q'')<\deg(Q)$. Since $\deg(Q'')<\deg(Q)$ Proposition \ref{Propcompkeypol} \textbf{(i)} and \textbf{(ii)} give us that $\nu_{Q''}(Q)<\nu(Q)$. Hence $Q''\in \mathcal S$. The inequality (\ref{eqmaxphi}) gives us that $\deg(Q'')\leq\deg(Q_-)$. On the other hand, since $\nu_{Q_-}(Q'')<\nu(Q'')$ we must have $\deg(Q_-)=\deg(Q'')$. Hence, Proposition \ref{Propcompkeypol} \textbf{(iii)} gives us that $\nu(Q_-)<\nu(Q'')$ and this is a contradiction to the inequality (\ref{eqmaxphi}).
Now assume that for every $Q'\in \mathcal S$, there exists $Q''\in \mathcal S$ such that
\begin{equation}\label{eqmaxphihas}
(\deg(Q'),\nu(Q'))<(\deg(Q''),\nu(Q''))
\end{equation}
in the lexicographical ordering. Take $Q_-\in\mathcal S$ such that $\deg(Q_-)\geq \deg(Q')$ for every $Q'\in\mathcal S$. We will show that the conditions \textbf{(K1) - (K4)} are satisfied. By (\ref{eqmaxphihas}), there exists $Q''\in \mathcal S$ such that
\begin{equation}\label{eqbanolimit}
(\deg(Q_-),\nu(Q_-))<(\deg(Q''),\nu(Q'')).
\end{equation}
In particular, $\deg(Q_-)=\deg(Q'')$ and $\nu(Q_-)<\nu(Q'')$. Proposition \ref{Propcompkeypol} \textbf{(iii)} gives us that
$\nu_{Q_-}(Q'')<\nu(Q'')$. Hence $\alpha(Q_-)=\deg(Q_-)$ and we have proved \textbf{(K1)}. If $Q'\in\Psi(Q_-)$, then
$\deg(Q')=\deg(Q_-)<\deg(Q)$ and hence $\nu_{Q'}(Q)<\nu(Q)$. This implies that $Q'\in\mathcal S$. The equation (\ref{eqbanolimit}) tells us that $\{\nu(Q')\mid Q'\in\Psi(Q_-)\}$ has no maximum, so we have proved \textbf{(K2)}. Now take any element $Q'\in\Psi(Q_-)$. Then $\deg(Q')<\deg(Q)$ and Proposition \ref{Propcompkeypol} \textbf{(i)} and \textbf{(ii)} give us that $\nu_{Q'}(Q)<\nu(Q)$. This proves \textbf{(K3)}. Take a polynomial $\widetilde{Q}$ with $\nu_{Q'}(\widetilde{Q})<\nu(\widetilde{Q})$ for every $Q'\in\Psi(Q_-)$ with minimal degree possible. We want to prove that $\deg(\widetilde Q)=\deg(Q)$. Assume, aiming for a contradiction, that
$\deg(\widetilde{Q})<\deg(Q)$. The first part of the proof gives us that $\widetilde Q$ is a key polynomial. Fix $Q'\in\Psi(Q_-)$. Then
$\nu_{Q'}(\widetilde Q)<\nu(\widetilde Q)$ and consequently $\deg(\widetilde{Q})=\deg(Q')=\deg(Q_-)$. Therefore
$\nu(Q')<\nu(\widetilde Q)$ for every $Q'\in \Psi(Q_-)$, which is a contradiction to (\ref{eqmaxphihas}). This concludes our proof.
\end{proof}
\begin{Def}
When conditions \textbf{(K1) - (K4)} of Theorem \ref{definofkeypol} are satisfied, we say that $Q$ is a \textbf{limit key polynomial}.
\end{Def}
\begin{Obs}
Observe that as a consequence of the proof we obtain that
$$
\epsilon(Q_-)<\epsilon(Q).
$$
\end{Obs}
\begin{proof}[Proof of Theorem \ref{Theoremexistencecompleteseqkpol}]
Consider the set
\[
\Gamma_0:=\{\nu(x-a)\mid a\in K\}.
\]
We have two possibilities:
\begin{itemize}
\item $\Gamma_0$ has a maximal element
\end{itemize}
Set $Q_0:=x-a_0$ where $a_0\in K$ is such that $\nu(x-a_0)$ is a maximum of $\Gamma_0$. If $\nu=\nu_{Q_0}$ we are done, so assume that $\nu\neq\nu_{Q_0}$. If the set
\[
\{\nu(Q)\mid Q\in \Psi(Q_0)\}
\]
has a maximum, choose $Q_1\in \Psi(Q_0)$ such that $\nu(Q_1)$ is this maximum. If not, choose $Q_1$ as any polynomial in $\Psi(Q_0)$. Set $\Lambda_1:=\{Q_0,Q_1\}$ (ordered by $Q_0<Q_1$).
\begin{itemize}
\item $\Gamma_0$ does not have a maximal element
\end{itemize}
For every $\gamma\in \Gamma_0$ set $Q_\gamma:=x-a_\gamma$ for some $a_\gamma\in K$ such that $\nu(x-a_\gamma)=\gamma$. If for every $f\in K[x]$, there exists $\gamma\in \Gamma_0$ such that $\nu(f)=\nu_{Q_\gamma}(f)$ we are done. If not, let $Q$ be a polynomial of minimal degree among all the polynomials for which $\nu_{Q_\gamma}(Q)<\nu(Q)$ for every $\gamma\in \Gamma_0$. If $\alpha(Q)=\deg(Q)$ and the set $\{\nu(Q')\mid Q'\in \Psi(Q)\}$ contains a maximal element, choose $Q_1\in \Psi(Q)$ such that $\nu(Q_1)\geq \nu(Q')$ for every $Q'\in \Psi(Q)$. If not, set $Q_1:=Q$. Set $\Lambda_1:=\{Q_\gamma\mid \gamma\in \Gamma_0\}\cup\{Q_1\}$ (ordered by $Q_1>Q_\gamma$ for every $\gamma\in \Gamma$ and $Q_\gamma>Q_{\gamma'}$ if $\gamma>\gamma'$).
Observe that in either case, $\deg(Q_1)>\deg(Q_0)$ and for $Q,Q'\in \Lambda_1$, $Q<Q'$ if and only if $\epsilon(Q)<\epsilon(Q')$. Moreover, if $\alpha(Q_1)=\deg(Q_1)$, then $\{\nu(Q)\mid Q\in\Psi(Q_1)\}$ does not have a maximum.
Assume that for some $i\in\mathbb{N}$, there exists a totally ordered set $\Lambda_i$ consisting of key polynomials with the following properties:
\begin{description}
\item[(i)] there exist $Q_0,Q_1,\ldots,Q_i\in \Lambda_i$ such that $Q_i$ is the last element of $\Lambda_i$ and $\deg(Q_0)<\deg(Q_1)<\ldots<\deg(Q_i)$.
\item[(ii)] if $\alpha(Q_i)=\deg(Q_i)$, then $\Gamma_i:=\{\nu(Q)\mid Q\in \Psi(Q_i)\}$ does not have a maximum.
\item[(iii)] for $Q,Q'\in \Lambda_i$, $Q<Q'$ if and only if $\epsilon(Q)<\epsilon(Q')$.
\end{description}
If $\nu_{Q_i}\neq \nu$, then we will construct a set $\Lambda_{i+1}$ of key polynomials having the same properties (changing $i$ by
$i+1$).
Since $\nu_{Q_i}\neq \nu$, the set $\Psi(Q_i)$ is not empty. We have two cases:
\begin{itemize}
\item $\alpha(Q_i)>\deg(Q_i)$.
\end{itemize}
If $\Gamma_i$ has a maximum, take $Q_{i+1}\in\Psi(Q_i)$ such that $\nu(Q_{i+1})\geq \Gamma_i$. Otherwise, choose $Q_{i+1}$ to be any element of $\Psi(Q_i)$. Observe that if $\alpha(Q_{i+1})=\deg(Q_{i+1})$, then $\Gamma_{i+1}$ does not have a maximum. Set $\Lambda_{i+1}=\Lambda_i\cup\{Q_{i+1}\}$ with the extension of the order in $\Lambda_i$ obtained by setting $Q_{i+1}>Q$ for every $Q\in \Lambda_i$.
\begin{itemize}
\item $\alpha(Q_i)=\deg(Q_i)$.
\end{itemize}
By assumption, the set $\Gamma_i$ does not have a maximum. For each $\gamma\in\Gamma_i$, choose a polynomial
$Q_\gamma\in \Psi(Q_i)$ such that $\nu(Q_\gamma)=\gamma$. If for every $f\in K[x]$, there exists $\gamma\in \Gamma_i$ such that
$\nu_{Q_\gamma}(f)=\nu(f)$, then we are done. Otherwise, choose a monic polynomial $Q$, of smallest degree possible, such that
$\nu_{Q'}(Q)<\nu(Q)$ for every $Q'\in\Psi(Q_i)$. If $\alpha(Q)=\deg(Q)$ and $\{\nu(Q')\mid Q'\in\Psi(Q)\}$ has a maximum, we choose $Q_{i+1}$ such that $\nu(Q_{i+1})\geq\{\nu(Q')\mid Q'\in\Psi(Q)\}$. Otherwise we set $Q_{i+1}=Q$. Then set
\[
\Lambda_{i+1}:=\Lambda_i\cup\{Q_\gamma\mid \gamma\in\Gamma_i\}\cup\{Q_{i+1}\},
\]
with the extension of the order of $\Lambda_i$ given by
$$
Q_{i+1}>Q'\mbox{ for every }Q'\in \Lambda_{i+1}\setminus\{Q_{i+1}\},
$$
$Q_\gamma> Q'$ for every $\gamma\in \Gamma_i$ and $Q'\in\Lambda_i$ and $Q_\gamma>Q_{\gamma'}$ for $\gamma,\gamma'\in \Gamma_i$ with $\gamma>\gamma'$.
In all cases, the set $\Lambda_{i+1}$ has the properties \textbf{(i)}, \textbf{(ii)} and \textbf{(iii)}.
Assume now that for every $i\in\mathbb{N}$ the sets $\Lambda_i$ and $\Lambda_{i+1}$ can be constructed. Then we can construct a set
\[
\Lambda_\infty:=\bigcup_{i=1}^\infty\Lambda_i
\]
of key polynomials having the property that for $Q,Q'\in \Lambda_\infty$, $Q<Q'$ if and only if $\epsilon(Q)<\epsilon(Q')$ and there are polynomials $Q_0,\ldots,Q_i,\ldots\in \Lambda_\infty$ such that
$$
\deg(Q_{i+1})>\deg(Q_i)
$$
for every $i\in\mathbb{N}$. This means that for every $f\in K[x]$ there exists $i\in\mathbb{N}$ such that $\deg(f)<\deg(Q_i)$, which implies that
$\nu_{Q_i}(f)=\nu(f)$. Therefore, $\Lambda_\infty$ is a complete set of key polynomials for $\nu$.
\end{proof}
Observe that at each stage, the same construction would work if we replaced $\Gamma_i$ by any cofinal set $\Gamma_i'$ of
$\Gamma_i$. Hence, if the rank of $\nu$ is equal to 1, then we can choose $\Gamma_i'$ to have order type at most $\omega$. Then, from the construction of the sets $\Lambda_i$ and $\Lambda_\infty$, we can conclude the following:
\begin{Cor}
If the rank of $\nu$ is equal to one, then there exists a complete sequence of key polynomials of $\nu$ with order type at most $\omega\times\omega$.
\end{Cor}
\section{Pseudo-convergent sequences}
The next two theorems justify the definitions of algebraic and transcendental pseudo-convergent sequences.
\begin{Teo}[Theorem 2 of \cite{Kap}]
If $\{a_\rho\}_{\rho<\lambda}$ is a pseudo-convergent sequence of transcendental type, without a limit in $K$, then there exists an immediate transcendental extension $K(z)$ of $K$ defined by setting $\nu(f(z))$ to be the value $\nu(f(a_{\rho_f}))$ as in condition (\ref{condforpscstotra}). Moreover, for every valuation $\mu$ in some extension $K(u)$ of $K$, if $u$ is a pseudo-limit of $\{a_\rho\}_{\rho<\lambda}$, then there exists a value preserving $K$-isomorphism from $K(u)$ to $K(z)$ taking $u$ to $z$.
\end{Teo}
\begin{Teo}[Theorem 3 of \cite{Kap}]\label{thmonalgimmext}
Let $\{a_\rho\}_{\rho<\lambda}$ be a pseudo-convergent sequence of algebraic type, without a limit in $K$, $q(x)$ a polynomial of smallest degree for which (\ref{condforpscstoalg}) holds and $z$ a root of $q(x)$. Then there exists an immediate algebraic extension of $K$ to $K(z)$ defined as follows: for every polynomial $f(x)\in K[x]$, with $\deg f<\deg q$ we set $\nu(f(z))$ to be the value $\nu(f(a_{\rho_f}))$ as in condition (\ref{condforpscstotra}). Moreover, if $u$ is a root of $q(x)$ and $\mu$ is some extension $K(u)$ of $K$ making $u$ a pseudo-limit of $\{a_\rho\}_{\rho<\lambda}$, then there exists a value preserving $K$-isomorphism from $K(u)$ to $K(z)$ taking $u$ to $z$.
\end{Teo}
For the rest of this paper, let $\{a_\rho\}_{\rho<\lambda}$ be a pseudo-convergent sequence for the valued field $(K,\nu)$, without a limit in $K$. For each $\rho<\lambda$, we denote $\nu_\rho=\nu_{x-a_\rho}$. For a polynomial $f(x)\in K[x]$ and $a\in K$ we consider the Taylor expansion of $f$ at $a$ given by
\[
f(x)=f(a)+\mathbb{P}artial_1f(a)(x-a)+\ldots+\mathbb{P}artial_nf(a)(x-a)^n.
\]
Assume that $\{a_\rho\}_{\rho<\lambda}$ fixes the value of the polynomials $\mathbb{P}artial_if(x)$ for every $1\leq i\leq n$. We denote by $\beta_i$ this fixed value.
\begin{Lema}[Lemma 8 of \cite{Kap}]\label{lemmakaplvalpol}
There is an integer $h$, which is a power of $p$, such that for sufficiently large $\rho$
\[
\beta_i+i\gamma_\rho>\beta_h+h\gamma_\rho\mbox{ whenever }i\ne h\mbox{ and } \nu(f(a_\rho))=\beta_h+h\gamma_\rho.
\]
\end{Lema}
\begin{Cor}\label{correlanurhowithnu}
If $\{a_\rho\}_{\rho<\lambda}$ fixes the value of $f(x)$, then $\nu_\rho(f(x))=\nu(f(x))$. On the other hand, if $\{a_\rho\}_{\rho<\lambda}$ does not fix the value of $f(x)$, then $\nu_\rho(f(x))<\nu(f(x))$ for every $\rho<\lambda$.
\end{Cor}
\begin{proof}
By definition of $\nu_\rho$ we have
\[
\nu_\rho(f(x))=\min_{0\leq i\leq n}\{\nu(\mathbb{P}artial_if(a_\rho)(x-a_\rho)^i)\}=\min_{0\leq i\leq n}\{\beta_i+i\gamma_\rho\},
\]
where $\beta_0:=\nu(f(a_\rho))$. This implies, using the lemma above, that
\[
\nu_\rho(f(x))=\nu(f(a_\rho)).
\]
If $\{a_\rho\}_{\rho<\lambda}$ fixes the value of $f(x)$, then $\nu(f(a_\rho))=\nu(f(x))$ for $\rho$ sufficiently large. Thus $\nu_\rho(f(x))=\nu(f(x))$. On the other hand, if $\{a_\rho\}_{\rho<\lambda}$ does not fix the value of $f(x)$, then $\nu(f(x))>\nu(f(a_\rho))=\nu_\rho(f(x))$ for every $\rho<\lambda$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{compthemkppsc}]
If $\{a_\rho\}_{\rho<\lambda}$ is of transcendental type it fixes, for any polynomial $f(x)\in K[x]$, the values of the polynomials
$\mathbb{P}artial_if(x)$ for every $0\leq i\leq n$ (here $\mathbb{P}artial_0f:=f$). Hence, Corollary \ref{correlanurhowithnu} implies that
$\nu_\rho(f(x))=\nu(f(x))$ for sufficiently large $\rho<\lambda$, which is what we wanted to prove.
Now assume that $\{a_\rho\}_{\rho<\lambda}$ is of algebraic type. Take $\rho<\lambda$ such that
$$
\nu(q(a_\tau))>\nu(q(a_\sigma))
$$
for every $\rho<\sigma<\tau<\lambda$ and set $Q_-=x-a_\rho$. Then
\[
\nu_{Q_-}(x-a_\sigma)=\nu_{Q_-}(x-a_\rho+a_\rho-a_\sigma)=\nu(x-a_\rho)<\nu(x-a_\sigma)
\]
for every $\rho<\sigma<\lambda$. This implies that $\alpha(Q-)=1$ and then $\alpha(Q_-)=\deg (Q_-)$. Consequently, \textbf{(K1)} is satisfied. Moreover,
\[
\Psi(Q_-)=\{x-a\mid \nu_{Q_-}(x-a)<\nu(x-a)\}.
\]
In order to prove \textbf{(K2)} assume, aiming for a contradiction, that $\nu(\Psi(Q_-))$ has a maximum, let us say $\nu(x-a)$. Then, in particular, $\nu(x-a)>\nu(x-a_\sigma)$ for every $\rho<\sigma<\lambda$. This implies that $a\in K$ is a limit of $\{a_\rho\}_{\rho<\lambda}$, which is a contradiction. Condition \textbf{(K3)} and \textbf{(K4)} follow immediately from Corollary \ref{correlanurhowithnu} and the fact that $\{\nu(x-a_\rho)\mid \rho<\lambda\}$ is cofinal in $\nu(\Psi(Q_-))$.
\end{proof}
\end{document} |
\begin{document}
\date{}
\title[Long-time solvability of NSB equations]{Long-time solvability of the Navier-Stokes-Boussinesq equations
with almost periodic initial large data}
\author{Slim Ibrahim and Tsuyoshi Yoneda}
\email{[email protected]} \urladdr{
http://www.math.uvic.ca/~ibrahim/}
\thanks{S. I. is partially supported by NSERC\# 371637-2009 grant and a start up fund from University
of Victoria}
\thanks{T. Y. is partially supported by PIMS Post-doc fellowship at the University
of Victoria, and partially supported by NSERC\# 371637-2009}
\maketitle
\begin{center}
Department of Mathematics and Statistics,
University of Victoria \\
PO Box 3060 STN CSC,
Victoria, BC, Canada, V8W 3R4 \\
\end{center}
\begin{center}
and
\end{center}
\begin{center}
Department of Mathematics, Hokkaido University\\
Sapporo 060-0810, Japan
\end{center}
\vskip0.3cm \noindent {\bf Keywords:}
Navier-Stokes equation, Boussinesq approximation, almost periodic functions
\vskip0.3cm \noindent {\bf Mathematics Subject Classification:}
76D50,42B05
\begin{abstract}
We investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with
spatially almost periodic large data when the density stratification is sufficiently large.
In 1996, Kimura and Herring \cite{KH} examined numerical simulations to show a stabilizing effect due to the stratification.
They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane.
Our result is a mathematical justification of the presence of such two-dimensional pancakes.
To show the existence of solutions for large times,
we use $\ell^1$-norm of amplitudes.
Existence for large times is then proven using techniques of fast singular oscillating limits and
bootstrapping argument from a global-in-time unique solution of the system of limit equations.
\end{abstract}
\section{Introduction}
Large-scale fluids such as atmosphere and ocean are parts of geophysical fluids, and the Coriolis force due to the earth rotation plays a significant role in the large scale flows considered in meteorology and geophysics.\\
Mathematically, it was first investigated by Poincar\'e \cite{Po}. Later on, the problem of strong Coriolis force was extensively studied. Babin, Mahalov and Nicolaenko (BMN) \cite{BMN1,BMN2} studied the incompressible rotating Navier-Stokes and Euler equations in the periodic case while Chemin, Desjardins, Gallagher and Grenier \cite{CDGG} analyzed the case of decaying data and more recently, the second author \cite{Y} considered the almost periodic case. Gallagher in \cite{Ga} studied a more abstract parabolic system. We also refer to Paicu \cite{P} for anisotropic viscous fluids, Benameur, Ibrahim and Majdoub \cite{BIM} for rotating Magneto-Hydro-Dynamic system and to Gallagher and Saint-Raymond \cite{GSR} for inhomogeheous rotating fluid equations.\\
Moreover on the one hand, the case when fluids are governed by both a strong Coriolis force and vertical stratification effects was investigated by BMN in \cite{BMN3} in the periodic setting and Charve in \cite{C} for decaying data. However, their studies do not cover the case when fluid equations are governed by the only effect of stratification.
It is known that a strong Coriolis force has a stabilizing effect (see \cite{BMN1}). However, in
BMN \cite[Section 9.2]{BMN4} the authors observed that for ideal fluids (i.e., with zero viscosity), the only effect of stratification leads to unbalanced dynamics. Moreover, the case of both strong Coriolis and stratification forces in the almost periodic setting seems to remain open. Finally, note that for the almost periodic case, energy type estimates cannot be used, and instead Fujita-Kato's approach has to be used.
On the other hand,
Kimura and Herring \cite{KH}
examined numerical simulations to show a stabilizing effect due to the effect of stratification for viscous fluid. They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane.
Our result can be seen as a mathematical justification of the presence of such two-dimensional pancakes.
More precisely, we study long-time solvability for Navier-Stokes-Boussinesq equation
with stratification effects.
The Navier-Stokes-Boussinesq equations with stratification effects are governed by the following equations.
\begin{equation}\label{eq11}
\begin{cases}
\partial_t u -\nu\Delta u + (u \cdot \nabla) u +\nabla
p= g\rho e_3&x \in \mathbb{R}^3,\quad t>0\\
\partial_t \rho -\kappa\Delta \rho +(u \cdot \nabla) \rho = -\mathcal{N}^2 u_3& x \in \mathbb{R}^3,\quad t>0\\
\nabla \cdot u=0&x\in \mathbb{R}^3,\quad t>0\\
u|_{t=0}=u_0 { \ },\quad \rho|_{t=0}=\rho_0
\end{cases}
\end{equation}
where the unknown functions $u = u(x, t) = (u_1,u_2,u_3)$, $\rho = \rho(x, t)$ and $p = p(x, t)$ are the fluid velocity,
the thermal disturbances and the pressure, respectively. The parameters $\nu > 0$, $\kappa \geq 0$ and $g >0$ are the viscosity,
the thermal diffusivity and the gravity force, respectively. The parameter $\mathcal{N} > 0$ is Brunt-V\"{a}is\"{a}l\"{a} frequency
(stratification-parameter). Recall that $\Delta :=(\partial_1^2+\partial_2^2+\partial_3^2)I_3$, $\nabla :=(\partial_1,\partial_2,\partial_3)$
and $e_3:=(0,0,1)$.
Our method follows the ideas based on BMN. For the limit
equations, we show that it is equivalent
to the 2D-Navier-Stokes equations\footnote{in the sense that there
is a one to one correspondence between solutions of the two
equations}, which is known to have, in the almost periodic setting, a unique global solution, see for example
\cite{GMY}. Then, we show that the global existence for the remainder equations in the limit equations.
Since we handle not only periodic functions, we have to introduce a new analytic functional
setting, which is more suitable for the almost periodic situation as the second author did for the rotating fluid case in \cite{Y}. More
precisely, a straightforward application of an energy inequality is
impossible if the initial data is almost periodic. To overcome this
difficulty, we use $\ell^1$-norm of amplitudes with sum closed
frequency set. We recall the analytic functional setting (see \cite{Y}) as follows:
\begin{defn} (Countable sum closed frequency set.)
\label{countable sum}
A countable set $\Lambda$ in $\mathbb{R}^3$ is called a sum closed frequency set if it satisfies the following properties:
$$
\Lambda=\{a+b:a,b\in\Lambda\}
\quad
and
\quad -\Lambda=\Lambda.
$$
\end{defn}
\begin{remark}\
If $\{e_j\}_{j=1}^3$ is the standard orthogonal basis in
$\mathbb{R}^3$, then the sets $\mathbb{Z}^3$, $\{m_1e_1+\sqrt
2m_2e_2 +m_3e_3:m_1,\cdots,m_3\in\mathbb{Z}\}$ and
$\{m_1e_1+m_2(e_1+e_2\sqrt 2)+m_3(e_2+e_3\sqrt
3):m_1,m_2,m_3\in\mathbb{Z}\}$ are examples of such countable sum
closed frequency sets. Clearly, the case $\mathbb Z^3$ corresponds to the periodic.
Each of the other two cases are dense in $\mathbb{R}^3$ and therefore they correspond to ``purely" almost periodic setting.
\end{remark}
\begin{defn} (An $\ell^1$-type function space)
Let $BUC$ be the space of all bounded uniformly continuous functions defined in $\mathbb{R}^3$ equipped with the $L^\infty$-norm.
For a countable sum closed frequency set $\Lambda\subset\mathbb{R}^3$, let
$$
X^\Lambda(\mathbb{R}^3):=
\{u=\sum_{n\in\Lambda}\hat u_ne^{in\cdot x}\in BUC(\mathbb{R}^3): u_{-n}=u^*_n \quad for\quad n\in\Lambda,\
\|u\|:=\sum_{n\in\Lambda}|u_n|<\infty\},
$$
where $u_n^*$ is the complex conjugate coefficient of $u_n$.
\end{defn}
The second condition in Definition \ref{countable sum} is needed
to include real-valued almost periodic functions in $X^\Lambda$.
\begin{remark}
Note that functions in $\ell^1$ do not necessarily decay as $x\rightarrow\infty$. Also, this almost periodic setting is in general,
different from the periodic case since the frequency set may have accumulation points. The almost periodic setting is somehow between the periodic and the full non-decaying cases.
\end{remark}
Now, we define anisotropic dilation of the frequency set as follows.
\begin{defn}
For $\gamma=(\gamma_1,\gamma_2)\in (0,\infty)^2$, let
\begin{equation}\label{restriction}
\Lambda(\gamma):=\{(\gamma_1 n_1,\gamma_2 n_2,n_3)\in\mathbb{R}^3:(n_1,n_2, n_3)\in\Lambda\}.
\end{equation}
\end{defn}
Now, we specify the following Quasi-Geostrophic equation (a part of limiting system),
and assume (for the moment) that it has a scalar global solution $\theta =\theta (t)=\theta(t,x_1,x_2,x_3)$,
\begin{equation}\label{QG}
\begin{cases}
\partial_t\theta -\Delta_3\theta+(-\Delta_h)^{-1/2}\left[(w\cdot \nabla)\left((-\Delta_h)^{1/2}\theta)\right) \right]=0,\\
w=\left(-\partial_{x_2}(-\Delta_h)^{-1/2}\theta,\partial_{x_1}(-\Delta_h)^{-1/2}\theta\right)\\
\theta(t)|_{t=0}=\theta_0,
\end{cases}
\end{equation}
where $\Delta_h:=\partial_{x_1}^2+\partial_{x_2}^2$ and
$\Delta_3:=\partial_{x_1}^2+\partial_{x_2}^2+\partial_{x_3}^2$.
We will show that the initial value problem
for the QG equation admits a global-in-time unique solution in
$C([0,\infty): X^\Lambda)$ with the initial data $\theta_0=-\partial_{x_2}(-\Delta_h)^{-1/2}u_{0,1}+\partial_{x_1}(-\Delta_h)^{-1/2}u_{0,2}$.
More precisely, we give an explicit one-to-one correspondence between the QG and a 2D type Navier-Stokes equations
(for the existence of the unique global solution to 2D-Navier-Stokes equation with almost periodic initial data, see \cite{GMY}).
Now we state our main result.
\begin{thm}\label{main}
Let $\Lambda$ be a sum closed frequency set. There exists a set of frequencies dilation factors
$\Gamma(\Lambda)\subset (0,\infty)^2$ such that\footnote{the complement set $\Gamma^c$ is at most countable.}:\\
for any $\gamma\in\Gamma$,
for any zero-mean value and divergence free initial vector field $u_0\in X^{\Lambda(\gamma)}$,
initial thermal disturbance $\rho_0\in X^{\Lambda(\gamma)}$,
$\nu>0$, $\kappa\geq 0$
and $T>0$, there exists
$N_0>g$ depending only on $\nu$, $\kappa$, $u_0$, $\rho_0$ such that if
$|N|>N_0$, then there exists a mild solution to the
equation (\ref{eq11}), $u(t) \in
C([0,T]: X^{\Lambda(\gamma)})$ with zero-mean value and divergence free, and $\rho(t) \in
C([0,T]: X^{\Lambda(\gamma)})$ .
\end{thm}
\begin{remark}
For the periodic case, we do not need to restrict the frequency set to $\Gamma$ i.e. we can take $\Gamma(\Lambda)=(0,\infty)^2$.
However, the computation in this case is more complicated and needs a``restricted convolution"
type result in the spirit of \cite{BMN2}.
\end{remark}
\section{Preliminaries}
Before going any further, we first recall the following few facts about the space $X^\Lambda$:
\begin{itemize}
\item $(X^\Lambda, \|\cdot\|)$ is a Banach space, and any almost periodic function $u\in X^\Lambda$ can be decomposed $u(x)=\Sigma_{n\in\Lambda}\hat u_ne^{inx}$,
where each ``Fourier coefficient" $\hat u_n$ is uniquely determined by
$$
\hat u_n=\lim_{|B|\to\infty}\frac1{|B|}\int_{ B}u(x)e^{ix\cdot n}\;dx,
$$
and ${B}$ stands for a ball in $\mathbb R^3$ (see for example \cite{Co}).
\item $X^\Lambda$ is closed subspace of $FM$, the Fourier preimage of the space of all finite Radon measures
proposed by
Giga, Inui, Mahalov and Matsui in 2005 (see \cite{GIMM2,GIMS2,GJMY}).
\item Leray projection on almost periodic functions $\bar P=\{\bar P_{jk}\}_{j,k=1,2,3}$ is defined as
\begin{equation*}
\bar P_{jk}:=
\delta_{jk}+R_j R_k\quad (1\leq j,k \leq3)
\end{equation*}
with $\delta_{jk}$ is Kronecker's delta and $R_j$ is the Riesz transform defined by
\begin{equation*}
R_j=\frac{\partial}{\partial x_j}(-\Delta)^{-\frac{1}{2}} { \ } for { \ } j=1,2,3.
\end{equation*}
The symbol $\sigma(R_j)$ of $R_j$ is $in_j/|n|$, where $i=\sqrt{-1}$ (see \cite{Co}). Let $P$
be the extended Leray projection with Fourier-multiplier $P_n=\{P_{n,ij}\}_{i,j=1,2,3,4}$ given by
\begin{equation*}
P_{n,ij}:=
\begin{cases}
\delta_{ij}-\frac{n_in_j}{|n|^2}\quad (1\leq i,j \leq3),\\
\delta_{ij}\quad (otherwise).
\end{cases}
\end{equation*}
\item Helmotz-Leray decomposition is defined on almost periodic functions in the same way as in the periodic case.
Namely, $u$ is uniquely decomposed as
\begin{equation*}
u=w+\nabla \pi,
\end{equation*}
where $\pi=-(-\Delta)^{-1}\text{div}\ u$ and $w=\bar Pu$.
\end{itemize}
Now we rewrite the system ($\ref{eq11}$) in a more abstract way. Set $N:= \mathcal{N} \sqrt{g}$
and $v \equiv (v_1,v_2,v_3,v_4):=(u_1,u_2,u_3,\frac{\sqrt{g}}{\mathcal{N}}\rho)$.
Then $v$ solves
\begin{equation}\label{eq31}
\begin{cases}
\partial_t v -\tilde\nu\Delta v + NJ v + \nabla_3 p=-(v\cdot\nabla_3) v\\
v|_{t=0}=v_0\\
\nabla_3 \cdot v = 0,\\
\end{cases}
\end{equation}
with $\tilde \nu=\hbox{diag}(\nu,\nu,\nu,\kappa)$, the initial data $v_0=(u_{0,1},u_{0,2},u_{0,3}$,
$\frac{ \sqrt{g}}{\mathcal{N}}\rho_0)$,
$\nabla_3:=(\partial_1,\partial_2,\partial_3,0)$,
\begin{equation*}
J:=
\begin{pmatrix}
0 & 0 & 0 &0\\
0 & 0 & 0 &0\\
0 & 0 & 0 &-1\\
0 & 0 & 1 &0\\
\end{pmatrix},
\end{equation*}
and $(v\cdot \nabla_3) =(v_1 \partial_1 + v_2 \partial_2 + v_3 \partial_3)$.
Observe that under the condition $N>g$ we have $\mathcal N>\sqrt g$
and therefore $\|v_{0,4}\|=\|\frac{\sqrt g}{\mathcal
N}\rho_0\|<\|\rho_0\|$. We will assume this condition throughout the
paper.
Applying the extended Leray projection ${P}$ to ($\ref{eq31}$), we obtain
\begin{equation}\label{eq32}
\begin{cases}
dv/dt +(-\tilde \nu\Delta+NS)v =- {P}( v \cdot \nabla_3 ) v
,\\
v|_{t=0}=P v_0=v_0,\\
\end{cases}
\end{equation}
with $S:= PJP$.
Recall that for $|n|_{h}\neq0$, the matrix $S_n:=P_nJP_n$ has the
following Craya-Herring orthonormal eigen basis
$\{q^1_n,q^{-1}_n,q^0_n ,q^{div}_n\}$ (see \cite{BMN3,EM})
associated to the eigenvalues $\{i\omega_n,-i\omega_n,0,0\}$ with
\begin{equation*}
\omega_n =\frac{|n|_h}{|n|},\ |n|_h=\sqrt{n_1^2+n_2^2}
\end{equation*}
and
\begin{align*}
q^1_n:=(q^1_{1,n}, q^1_{2,n}, q^1_{3,n}, q^1_{4,n} ): = & \frac{1}{\sqrt 2|n|_h^2}
(i\omega_nn_1n_3,i\omega_nn_2n_3,-i|n|_h^2\omega_n,|n|^2_h)=q^{-1*}_n\\
q^{-1}_n:=(q^{-1}_{1,n}, q^{-1}_{2,n}, q^{-1}_{3,n}, q^{-1}_{4,n} ):= & \frac{1}{\sqrt 2|n|^2_h}
(-i\omega_nn_1n_3,-i\omega_nn_2n_3,i|n|_h^2\omega_n,|n|^2_h)
=q^{1*}_n\\
q^0_n := (q^0_{1,n}, q^0_{2,n}, q^0_{3,n}, q^0_{4,n} ):=& \frac{1}{|n|_h}(-n_2,n_1,0,0)=q_n^{0*}\\
q^{div}_n := (q^{div}_{1,n}, q^{div}_{2,n}, q^{div}_{3,n}, q^{div}_{4,n} ):=& \frac{1}{|n|}(n_1,n_2,n_3,0),
\end{align*}
where $q^{1*}_n=(q^1_n)^*$ is the conjugate of $q^1_n$.
The case when $|n|_h=0$ and $n_3\not=0$, we define
\begin{align*}
q^1_n: = & (1/2,1/2,0,1/\sqrt 2)\\
q^{-1}_n:= & (-1/2,-1/2,0,1/\sqrt 2)\\
q^0_n :=& (-1/\sqrt 2,1/\sqrt 2,0,0 )\\
q^{div}_n := & (0,0,1,0).
\end{align*}
In fact, for $|n|_h=0$ and $n_3\not=0$, we have $S_n=P_nJP_n=0$. However, the above choice of the basis is uniquely determined by the conditions
$(\tilde \nu q^1_n\cdot q^{1*}_n)=\left(\frac{\nu+\kappa}{2}\right)$, $(\tilde \nu q^{-1}_n\cdot q^{-1*}_n)=\left(\frac{\nu+\kappa}{2}\right)$
and $(\tilde \nu q^0_n\cdot q^{0*}_n)=\nu$ for \eqref{c}. Moreover, the divergence-free condition requires that $(\hat v_n(t)\cdot q^{div}_n)=0$,
giving $q^{div}_n := (0,0,1,0)$.
Using Craya-Herring basis, one obtains an explicit representation of the solution to the linear version of ($\ref{eq32}$).
For $n\in\Lambda$ and $\hat v_n:=(\hat v_{n,1},\hat v_{n,2}, \hat v_{n,3},\hat v_{n,4})$
such that $\hat v_n\cdot \vec n=0$, we have
\begin{equation*}
e^{tNS_n}\hat v_n
=\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}e^{tNS_n}q_n^{\sigma_0} =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0} e^{i\sigma_0\omega_nNt}q_n^{\sigma_0},
\end{equation*}
with $\vec n:=(n,0)=(n_1,n_2,n_3,0)$,
\begin{equation*}
\hat v_n
=\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0} q_n^{\sigma_0}\quad\hbox{and}\quad a^{\sigma_0}_n:=(\hat v_n \cdot q^{\sigma_0*}_n).
\end{equation*}
Similarly, write a solution $v$ of \eqref{eq32} as
\begin{equation*}
v(t,x)=\sum_{n\in\Lambda}\hat v_n(t)e^{in\cdot x}.
\end{equation*}
From (\ref{eq32}), we derive for $n \in\Lambda$,
\begin{equation}\label{vequation}
\partial_t \hat v_n(t)=-\tilde \nu|n|^2\hat v_n(t)-S_n \hat v_n(t)-i
P_n\sum_{n=k+m}(\hat v_k(t)\cdot \vec m)\hat v_m(t)\quad\text{with}\quad (\vec n\cdot \hat v_n(t))=0.
\end{equation}
In the sequel, we do not distinguish between $\vec n$ and $n$ unless a confusion occurs.
For $n\in \Lambda$ we have
\begin{equation*}
e^{tNS_n}\hat v_n(t)
=\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}(t)e^{tNS_n}q_n^{\sigma_0} =\sum_{\sigma_0\in\{-1,0,1\}}a_n^{\sigma_0}(t) e^{i\sigma_0\omega_nNt}q_n^{\sigma_0},
\end{equation*}
where $a^{\sigma_0}_n(t):=(\hat v_n(t)\cdot q^{\sigma_0*}_n)$. From equation \eqref{vequation}, we get for $\sigma_0=-1,0,1$,
\begin{eqnarray}
\nonumber
\partial_t a^{\sigma_0}_n(t)&=&-a^{\sigma_0}_n(t)((|n|^2 \tilde\nu+NS_n)q^{\sigma_0}_n\cdot q^{\sigma_0*}_n)\\
& -&i\sum_{n=k+m,\;\sigma_1,\sigma_2\in \{-1,0,1\}}c^{\sigma_1}_kc^{\sigma_2}_m
(q^{\sigma_1}_k\cdot m)(P_n q^{\sigma_2}_m\cdot q^{\sigma_0*}_n).
\nonumber
\end{eqnarray}
Note that $P_n$ is self adjoint and $P_nq^{\sigma_0*}_n=q^{\sigma_0*}_n$.
Setting $c^{\sigma_0}_n(t):=e^{-itN \sigma_0\omega_n}a^{\sigma_0}_n(t)$ leads to the following equation
\begin{eqnarray}
\nonumber
\partial_t c^{\sigma_0}_n(t)&=&-c^{\sigma_0}_n(t)|n|^2 (\tilde\nu q^{\sigma_0}_n\cdot q^{\sigma_0*}_n)\\
& -&i\sum_{n=k+m,\;\sigma_1,\sigma_2\in \{-1,0,1\}}e^{iNt\omega^\sigma_{nkm}}c^{\sigma_1}_kc^{\sigma_2}_m
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n).
\nonumber
\end{eqnarray}
where, $\omega^\sigma_{nkm}:=(-\sigma_0\omega_n+\sigma_1\omega_k+\sigma_2\omega_m )$. Now we split the nonlinear part into the ``resonant"
(independent of $N$) and non ``resonant" two parts
defined by
$$
\bar B_n^{\sigma_0}(g^{\sigma_1},h^{\sigma_2}):=-i\sum_{n=k+m,\;\omega^\sigma_{nkm}=0}
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n)
g^{\sigma_1}_kh^{\sigma_2}_m
$$
and
$$
\tilde B_n^{\sigma_0}(Nt, g^{\sigma_1},h^{\sigma_2}):=-i\sum_{n=k+m,\;\omega^\sigma_{nkm}\not=0}
(q^{\sigma_1}_k\cdot m)(q^{\sigma_2}_m\cdot q^{\sigma_0*}_n)
g^{\sigma_1}_kh^{\sigma_2}_m
\exp (i\omega^\sigma_{nkm} Nt),
$$
respectively.
In addition, observe that we have the following estimates:
\begin{equation}\label{desiredestimates}
\begin{cases}
\|e^{-\nu|n|^2t}\tilde B^{\sigma_0}_n(Nt, g^{\sigma_1},h^{\sigma_2})\|\leq \frac{C_{\nu}}{t^{1/2}}\|g^{\sigma_1}\|\|h^{\sigma_2}\|\\
\|e^{-\nu|n|^2t}\bar B^{\sigma_0}_n(g^{\sigma_1},h^{\sigma_2})\|\leq \frac{C_{\nu}}{t^{1/2}}\|g^{\sigma_1}\|\|h^{\sigma_2}\|
\end{cases}
\end{equation}
(for $\sigma_0=-1,0,1$) obtained by estimating the first derivative of the heat kernel as follows
\begin{equation*}
\sup_{n\in\Lambda}\left | |n|e^{- \nu |n|^2 t}\right|\leq \frac{C_{\nu}}{t^{1/2}}.
\end{equation*}
The constant $C_\nu>0$ is independent of $N$.
Then we have the following equations:
\begin{equation}\label{original}
\begin{cases}
\partial_t c_n^0(t)=-\nu |n|^2c_n^0(t)+\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\bar B^0_n(c^{\sigma_1},c^{\sigma_2})
+\tilde B^0_n(Nt, c^{\sigma_1},c^{\sigma_2})\right)\\ \\
\partial_t c^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2} \right)|n|^2c^{\sigma_0}_n(t)
+\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2})
+\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right),
\end{cases}
\end{equation}
for $\sigma_0=\pm1$.
From the condition $\omega^{\sigma}_{nkm}=0$, we easily see that the terms $\bar B_n^{0}(c^{1},c^{1})$, $\bar B_n^{0}(c^{-1},c^{-1})$,
$\bar B_n^{0}(c^{0},c^{\pm 1})$, $\bar B_n^{0}(c^{\pm 1},c^{0})$, $\bar B_n^{\pm 1}(c^{\mp 1},c^{0})$, $\bar B_n^{\pm 1}(c^{0},c^{\mp 1})$
and $\bar B_n^{\pm 1}(c^{0},c^{0})$ disappear.
Now, we define the ``limit equations" by
\begin{equation}\label{c}
\begin{cases}
\partial_t c_n^0(t)=-\nu |n|^2c_n^0(t)+\bar B^0_n(c^0,c^0)+\bar B^0_n(c^1,c^{-1})+\bar B^0_n(c^{-1},c^1),\\ \\
\partial_t c^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2} \right)|n|^2c^{\sigma_0}_n(t)
+\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2\setminus D}\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2}), \quad \sigma_0=\pm1,
\end{cases}
\end{equation}
where $D:=\{(0,0), (-1, 0), (0,-1)\}$ for $\sigma_0=1$ and $D:=\{(0,0), (1, 0), (0,1)\}$ for $\sigma_0=-1$.
Formally, we can get \eqref{c} from \eqref{original} when $N\to \infty$. We will justify this convergence in Lemma \ref{fas}.
Now we show that there is more non trivial cancellation in the limit equations. More precisely,
\begin{lemma}
We have
\begin{equation*}
\bar B^0_n(c^1,c^{-1})+\bar B^0_n(c^{-1},c^1)=0.
\end{equation*}
\end{lemma}
\begin{proof}
To prove the lemma, it suffices to show
\begin{equation}\label{cancellation}
(q^1_k\cdot m)(q^{-1}_m\cdot q^{0*}_n)+(q^{-1}_m\cdot k)(q^1_k\cdot q^{0*}_n)=0\quad\text{for any}\quad n=k+m\quad \text{with}\quad \omega_k=\omega_m.
\end{equation}
First we show that $\omega_k=\omega_m$ if and only if
\begin{equation*}
k,m\in\{n\in\mathbb{Z}^3: |n|_h^2=\lambda n_3^2\}\quad \text{for some}\quad \lambda>0.
\end{equation*}
($\Leftarrow$): This direction is clear. Thus we omit it.
($\mathbb Rightarrow$): Rewrite the identity $\omega_k=\omega_m$ as $F(X)=F(Y)$, where $X:=|k|_h^2/k^2_3$, $Y:=|m|_h^2/ m_2^3$ and
$F(X):=X/(X+1)$. Since the function $F$ is monotone increasing, we see $X=Y$. This means that
\begin{equation*}
k_3=\pm\frac{|k|_h}{\sqrt \lambda}\quad\text{and}\quad m_3=\pm\frac{|m|_h}{\sqrt \lambda}.
\end{equation*}
We only consider the case $k_3=\frac{|k|_h}{\sqrt \lambda}$ and $m_3=\frac{|m|_h}{\sqrt \lambda}$, since the other cases are similar.
A direct calculation shows that
\begin{eqnarray*}
(q^1_k\cdot m)(q^{-1}_m\cdot q^{0*}_n)&=&\frac{1}{\sqrt 2\lambda |m||k||n|_h}\left(\frac{k_h\cdot m_h}{\lambda}-\frac{m_3|k|}{\sqrt{1+\lambda^2}}\right)
(-m_2k_1+m_1k_2),\\
(q^{-1}_m\cdot k)(q^{1}_k\cdot q^{0*}_n)&=&\frac{1}{\sqrt 2\lambda |m||k||n|_h}\left(\frac{k_h\cdot m_h}{\lambda}-\frac{k_3|m|}{\sqrt{1+\lambda^2}}\right)
(-k_2m_1+k_1m_2).\\
\end{eqnarray*}
By $k_3=\frac{|k|_h}{\sqrt \lambda}$ and $m_3=\frac{|m|_h}{\sqrt \lambda}$, we have \eqref{cancellation}.
\end{proof}
Now we show that the function $c^0$ in the limit equations satisfies a quasi geostrophic (QG) equation type
and that this QG equation is equivalent to the 2D type Navier-Stokes equation.
By the following lemma, we can see that the function $c^0$ satisfies the QG equation \eqref{QG}.
\begin{lemma}\label{qg nonlinear}\
Let $|n|_h=\sqrt{n_1^2+n_2^2}$.
The resonant part
$ \bar B_n^0(c^{0},c^{0})$ can be expressed as follows:
\begin{equation*}
\bar B^{0}_n(c^0,c^0)=
-\sum_{n=k+m}
\frac{i(k\times m)|m|_h}{|k|_h|n|_h}
c_{k}^0 c_{m}^0.
\end{equation*}
\end{lemma}
\begin{proof}
Since
$q^0_n := \frac{1}{|n|_h}(-n_2,n_1,0,0)$
and $q^0_n=\frac{1}{|n|_h}(|k|_hq^0_k+|m|_hq^0_m)$ for $n=k+m$,
we have
\begin{eqnarray*}
\bar B_n^0(c^0,c^0)&=&-\sum_{n=k+m}c^0_kc^0_m (q_k^0\cdot im)(q^0_m\cdot q^{0*}_n)\\
&=&-\sum_{n=k+m}\frac{i}{|k|_h|n|_h}(k_2m_1-k_1m_2)\left(q_m^0\cdot (|k|_hq_k^{0*}+|m|_hq_m^{0*})\right)c^0_kc^0_m\\
&=&\sum_{n=k+m}-\frac{i|m|_h}{|k|_h|n|_h}(k_2m_1-k_1m_2)c^0_kc^0_m\\
& &
-
\sum_{n=k+m}\frac{i}{|n|_h}(k_2m_1-k_1m_2)(q^0_m\cdot q_k^{0*})c^0_kc^0_m.
\end{eqnarray*}
Since $k\times m=-(m\times k)$,
we see that
\begin{equation*}
\sum_{n=k+m}\frac{i}{|n|_h}(k_2m_1-k_1m_2)(q^0_m\cdot q_k^{0*})c^0_kc^0_m=0,
\end{equation*}
which leads to the desired formula.
\end{proof}
Now we show that there is a one-to-one correspondence between the QG and a 2D type Navier-Stokes equations.
\begin{lemma}
Let $\Delta_h=\partial_{x_1}^2+\partial_{x_2}^2$ and
\begin{eqnarray*}
w:=(w_1(x_1,x_2,x_3,t),w_2(x_1,x_2,x_3,t)):=
\left(\sum_{n\in\Lambda}\hat w_{1,n}(t)e^{in\cdot x},\sum_{n\in\Lambda}\hat w_{2,n}(t)e^{in\cdot x}\right)
\end{eqnarray*}
and define $\theta=\theta (t,x_1,x_2,x_3):=(-\Delta_h)^{-1/2}\text{rot}_2 w$, with $\text{rot}_2$ is the 2 dimensional curl given by
$$
\text{rot}_2w=\partial_2 w_1-\partial_1w_2.
$$
Then, $w$ solves the following 2D type Navier-Stokes equation
\begin{equation}\label{2DNS}
\begin{cases}
\partial_t w-\Delta w+(w\cdot\nabla_2)w+\nabla_2p=0,\\
\nabla_2\cdot w=0,\ w|_{t=0}=w_0
\end{cases}
\end{equation}
if and only if $\theta$ solves \eqref{QG}.
\end{lemma}
\begin{proof}
Recall that $-\Delta_h=-(\partial_{x_1}^2+\partial_{x_2}^2)$. First observe that for $\theta=(-\Delta_h)^{-1/2}\text{rot}_2 w=\sum_{n\in\Lambda}\hat \theta_n(t)e^{in\cdot x}$, we have by $\nabla_2\cdot w=0$,
$$
w=(i\sum_{n\in\Lambda}\frac{n_2}{|n|_h}\hat \theta_n(t)e^{in\cdot x}, -i\sum_{n\in\Lambda}\frac{n_1}{|n|_h}\hat \theta_n(t)e^{in\cdot x})=
(\partial_2(-\Delta_h)\theta, -\partial_1(-\Delta_h)\theta).
$$
Then, apply rot$_2$ to (\ref{2DNS}), we get
\begin{equation}\label{vortexequation}
\partial_t\text{rot}_2 w-\Delta\text{rot}_2 w+(w\cdot \nabla_2)\text{rot}_2 w=0,
\end{equation}
here, we used the fact that
\begin{equation}\label{commute}
(w\cdot\nabla_2)\text{rot}_2w=\text{rot}_2[(w\cdot\nabla_2)w].
\end{equation}
Finally, apply $(-\Delta_h)^{-1/2}$ to both sides of \eqref{vortexequation}, we see that $\theta=(-\Delta_h)^{-1/2}\text{rot} w$ satisfies the desired QG equation \eqref{QG}.
Conversely, applying $L:=(-(-\Delta_h)^{-1}\partial_{x_2},(-\Delta_h)^{-1}\partial_{x_1})$ (which commutes with $\Delta$) to (\ref{vortexequation}), and by \eqref{commute} we can see that $L\text{rot}_2$ is nothing but the two dimensional Leray projection. Therefore, this implies (\ref{2DNS}) as desired.
\end{proof}
\begin{remark}
We refer to \cite{GMY} for the existence of the unique global solution to 2D type Navier-Stokes equation \eqref{2DNS} with almost periodic initial data.
\end{remark}
In what follows and in order to show the main theorem, we need the following lemma (which is needed only for the almost periodic case)
on the dilation of the frequency set \eqref{restriction}. This kind of restrictions is
technical. However we do not know whether or not such constraints
are removable. This means that the general almost periodic setting seems to remain open.
\begin{lemma}\label{pnkm}
For $n,k,m \in \Lambda$ and $\gamma=(\gamma_1,\gamma_2)\in(0,\infty)^2$, define $\tilde n=(\gamma_1 n_1,\gamma_2 n_2,
n_3)$, $\tilde k=(\gamma_1 k_1,\gamma _2 k_2,k_3)$ and $\tilde m=(\gamma_1 m_1,\gamma_2 m_2,
m_3)$.
Let
$$
P_{ nk m}(\gamma):=| \tilde n|^8| \tilde k|^8|\tilde m|^8\prod_{\sigma\in\{-1,1\}^3}\omega^\sigma_{\tilde n\tilde k\tilde m}.
$$
Given a frequency set $\Lambda$, there is $\Gamma:=\Gamma(\Lambda)\subset(0,\infty)^2$ s.t. for any $\gamma\in\Gamma$, $P_{nkm}(\gamma)\not=0$ for any $n$, $k$, $m\in\Lambda$ such that $(n_h,k_h,m_h)\not=(0,0,0)$.
\end{lemma}
\begin{remark}
If $n_h$, $k_h$, $m_h=0$, then $\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})=0$.
\end{remark}
\begin{proof}
Define $\Gamma$ by
$$
\Gamma:=\{\gamma\in(0,\infty)^2:
P_{n k m}(\gamma)\not=0\quad
for\quad all
\quad n, k, m\in\Lambda \quad with \quad |n|_h,|k|_h, |m|_h\not=0\}.
$$
Nothe that if
$\gamma\in\Gamma$, then $\omega^\sigma_{\tilde n\tilde
k\tilde m}\not=0$.
We show that $\Gamma$ cannot be empty.
By a direct calculation, we have
\begin{eqnarray*}
P_{nkm}(\gamma)&=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8\\
& &
\left( (\omega_{\tilde n}+\omega_{\tilde k}+\omega_{\tilde m})
(-\omega_{\tilde n}+\omega_{\tilde k}+\omega_{\tilde m})
(\omega_{\tilde n}-\omega_{\tilde k}+\omega_{\tilde m})
(\omega_{\tilde n}+\omega_{\tilde k}-\omega_{\tilde m}) \right)^2\\
&=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8
\left( \omega_{\tilde n}^2-(\omega_{\tilde k}+\omega_{\tilde m})^2\right)^2
\left ((\omega_{\tilde n}^2-(\omega_{\tilde k}-\omega_{\tilde m})^2\right)^2\\
&=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8
\left( (\omega_{\tilde n}^2-\omega_{\tilde k}^2-\omega_{\tilde m}^2)^2-4
\omega_{\tilde k}^2\omega_{\tilde m}^2\right)^2\\
&=&
|\tilde n|^8|\tilde k|^8|\tilde m|^8
\left( \omega_{\tilde n}^4+\omega_{\tilde k}^4+\omega_{\tilde m}^4
-2\omega_{\tilde k}^2\omega_{\tilde m}^2
-2\omega_{\tilde m}^2\omega_{\tilde n}^2
-2\omega_{\tilde n}^2\omega_{\tilde k}^2
\right)^2\\
&=&
|\tilde n|_h^4|\tilde k|^4|\tilde m|^4+|\tilde n|^4|\tilde k|_h^4|\tilde m|^4
+|\tilde n|^4|\tilde k|^4|\tilde m|^4_h\\
& &
-2|\tilde n|^2|\tilde n|^2_h|\tilde k|^2|\tilde k|^2_h|\tilde m|^4
-2|\tilde n|^2|\tilde n|^2_h|\tilde k|^4|\tilde m|^2|\tilde m|_h^2
-2|\tilde n|^4|\tilde k|^2|\tilde k|^2_h|\tilde m|^2|\tilde m|_h^2\\
&=&
-3n_1^2k_1^2m_1^2 \gamma_1^6-3n_2^2k_2^2m_2^2 \gamma_2^6\\
& &
-3n_1^2k_1^2m_2^2 \gamma_1^4\gamma_2^2
-3n_1^2k_2^2m_1^2 \gamma_1^4\gamma_2^2
-3n_2^2k_1^2m_1^2 \gamma_1^4\gamma_2^2\\
& &
-3n_2^2k_2^2m_1^2 \gamma_1^2\gamma_2^4
-3n_2^2k_1^2m_2^2 \gamma_1^2\gamma_2^4
-3n_1^2k_2^2m_2^2 \gamma_1^2\gamma_2^4
+\cdots.
\end{eqnarray*}
Since $|n|_h, |k|_h, |m|_h\not=0$, then the highest order terms
never disappear. This means that
$$
|\{\gamma: \cup_{n,k,m}P_{nkm}(\gamma)=0\}|=0.
$$
Thus the complement set of $\Gamma$ is countable (which means that $\Gamma$ is a non-empty set).
\end{proof}
Now we show that the limit equations have a global solution. In the almost periodic case,
the non-resonant part $\bar B^{\pm 1}(c^{\pm 1},c^{\pm 1})$ disappears just by restricting the frequencies set to $\Lambda(\gamma)$.
However, the periodic case is more subtle as we need a lemma on restricted convolution (see \cite{BMN2}).
\begin{lemma}\label{glo}
Let $\Lambda$ be a sum closed frequency set. If $\Lambda=\mathbb{Z}^3$, take $\gamma\in(0,\infty)^2$, otherwise we restrict it to $\gamma\in\Gamma(\Lambda)$.
Then for $\sigma_0=\pm 1$ there exists a global-in-time unique solution $c^{\sigma_0}(t)$ to equations (\ref{c})
such that
$c^{\sigma_0}(t)\in C([0,\infty):\ell^1(\Lambda(\gamma)))$ with $(c^{\sigma_0}_n(t)\cdot n)=0$ for all $n\in\Lambda(\gamma)$ and $c^{\sigma_0}_0(t)=0$.
\end{lemma}
\begin{proof}
Recall that $\tilde n=(\gamma_1 n_1,\gamma_2 n_2,
n_3)$, $\tilde k=(\gamma_1 k_1,\gamma _2 k_2,k_3)$ and $\tilde m=(\gamma_1 m_1,\gamma_2 m_2,
m_3)$.
First we consider the almost periodic case. By restricting $\gamma\in\Gamma$, we can eliminate the worst non-linear
term using Lemma \ref{pnkm}. More precisely, for all $ n\in \Lambda$ and $\sigma=(\sigma_1,\sigma_2,\sigma_3)\in\{-1,1\}^3$, the term
$$
\bar B^{\sigma_0}_{\tilde n}(c^{\sigma_1},c^{\sigma_2})= \sum_{\stackrel{\tilde n=\tilde k+\tilde m}{\omega^\sigma_{\tilde n\tilde k\tilde m}=0}}
(q^{\sigma_1}_{\tilde k}\cdot i\tilde m)(q^{\sigma_2}_{\tilde m}\cdot q^{\sigma_0*}_{\tilde n})c_{\tilde k}^{\sigma_1}c_{\tilde m}^{\sigma_2}\quad\text{disappears}.
$$
Then we have two coupled linear equations for $\{c^{-1}_n\}_n$ and $\{c^1_n\}_n$.
In this case, the global existence will immediately follow from estimates \eqref{desiredestimates}.\\
However, the periodic case requires more details. For $\alpha>0$ and $p\geq 1$, define the weighted $\ell_p^\alpha$ norm as
\begin{equation*}
\|c\|_{\ell^\alpha_p}:=\left(\sum_{n\in\Lambda}|n|^{p\alpha}|c_n|^p\right)^{1/p}.
\end{equation*}
The main step is to show an {\it \`a-priori} bound on $c^{\pm 1}$ in $\ell^1_2$. Since the 3D type Navier-Stokes equation \eqref{c}
is subcritical in the space $\ell^s_2$ with $s>1/2$, then
a bootstrap argument (using the dissipation, see \cite[Proposition 15.1]{Le} for example) enables us to conclude that
\begin{equation*}
c^{\pm 1}(t)\in L^\infty_{loc}([0,\infty):\ell^1_2)\cap L^\infty_{loc}((0,\infty):\ell^s_2)\quad\text{for}\quad 1<s<2,
\end{equation*}
whenever the initial data $c^{\pm 1}(0)\in\ell^1_2$.
More precisely, by \eqref{c} we write the following mild formulation:
\begin{equation}\label{limit mild solution}
c_n^{\sigma_0}(t)=e^{\frac{\nu+\kappa}{2}|n|^2t}c_n^{\sigma_0}(t_0)-\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2\setminus D}\int_{t_0}^t
e^{-\frac{\nu+\kappa}{2}(t-\tau+t_0)|n|^2}\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2})d\tau
\end{equation}
for $0<t_0<t$.
We have a good estimate for the heat kernel with fractional Laplacian,
\begin{equation*}
|n|^se^{-|n|^2t}=t^{-s/2}|t^{1/2}n|^se^{-|t^{1/2}n|^2}\leq
t^{-s/2}\frac{C_\varepsilonilon}{1+|t^{1/2}n|^{1/\varepsilonilon}}\leq t^{-s/2-\varepsilonilon/2}\frac{C_\varepsilonilon}{1+|n|^{1/\varepsilonilon}}
\end{equation*}
for $0<t\leq 1$ and $\varepsilonilon>0$.
Then by H\" older's and Young's inequality for the discrete case, we have the following estimate from \eqref{limit mild solution}:
\begin{eqnarray}\label{regularity estimate}
\|c^{\sigma_0}(t)\|_{\ell^s_2}&\leq &C_{\kappa,\nu,s}\bigg[\|c^{\sigma_0}(t_0)\|_{\ell^1_2}
+
\int_{t_0}^t(t-\tau+t_0)^{-s/2-\varepsilonilon/2}\times\\
& &\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\|c^{\sigma_1}(\tau)\|_{\ell^1_2}\|c^{\sigma_2}(\tau)\|_{\ell^0_2}+
\|c^{\sigma_1}(\tau)\|_{\ell^0_2}\|c^{\sigma_2}(\tau)\|_{\ell^1_2}\right)d\tau\bigg].
\nonumber
\end{eqnarray}
If $1<s<2$, then $\|c^{\sigma_0}(t)\|_{\ell^s_2}$ is finite since the right hand side of \eqref{regularity estimate} is finite.
This means that $c^{\sigma_0}\in L^\infty_{loc}((0,\infty):\ell^s_2)$ for $1<s<2$.
Then, thanks to Bernstein's lemma (which can be applied only for the periodic case), $\sum_n|c_n|\leq \|c\|_{\ell^\alpha_2}$ for $\alpha>3/2$,
we get an {\it \`a-priori} bound of the $\ell^1$-norm.
Now we show an {\it \`a-priori} bound of $c^{\pm 1}$ in $\ell^1_2$. Multiply equation (\ref{c}) by $|n|^2(c_n^{\sigma_0})^*$ and summing, we obtain
\begin{eqnarray*}
\sum_{{\sigma_0}\in\{-1,1\}}\sum_{n\in\mathbb Z^3} \bigg[\frac12\partial_t(|n| c_n^{\sigma_0}(t))^2&+&\left(\frac{\nu+\kappa}2\right)( |n|^2 c_n^{\sigma_0}(t))^2\bigg]\\
&=&\sum_{n\in\mathbb Z^3} \sum_{{\sigma_0}\in\{-1,1\}} \sum_{(\sigma_1,\sigma_2)\in\{-1.0,1\}^2\setminus D}{\bar B}_n^{\sigma_0}(c^{\sigma_1},c^{\sigma_2})|n|^2c_n^{\sigma_0*}(t).
\nonumber
\end{eqnarray*}
Notice that
\begin{eqnarray*}
(c^{\sigma_0}_n)^*&=&
\left[(\hat v_n\cdot q^{\sigma_0*}_n)e^{-itN\sigma_0\omega_n}\right]^*
=
(\hat v^*_n\cdot q^{\sigma_0}_n)e^{itN\sigma_0\omega_n}\\
&=&
(\hat v_{-n}\cdot (q^{-\sigma_0}_{-n})^*)e^{-itN(-\sigma_0)\omega_{-n}}= c_{-n}^{-\sigma_0}.
\end{eqnarray*}
Moreover, from one hand, a direct calculation using H\"older's then young's inequalities shows that
\begin{multline}\label{aprioriestimate2}
\left|\sum_n\bar B^{\pm 1}_n(c^{0},c^{\pm 1})|n|^2c^{\mp 1}_{-n}\right|+
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{0})|n|^2c^{\mp 1}_{-n}\right|\\
\leq
C\|c^{\mp 1}\|_{\ell^2_2}\left(\|c^{\pm 1}\|_{\ell^0_2}\|c^{0}\|_{\ell^1_1}+ \|c^{\pm 1}\|_{\ell^1_2}\|c^{0}\|_{\ell^0_1} \right)
\leq \varepsilonilon\|c^{\mp 1}\|_{\ell_2^2}^2+
C\left(\|c^{\pm 1}\|_{\ell^0_2}^2\|c^{0}\|_{\ell^1_1}^2+ \|c^{\pm 1}\|_{\ell^1_2}^2\|c^{0}\|_{\ell^0_1}^2 \right)
\end{multline}
for sufficiently small $\varepsilonilon>0$.
On the other hand, we show that
\begin{equation}\label{skew1}
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})|n|^2c^{\mp 1}_{-n}\right|
\leq
\sum_n\sum_{-n=k+m}|m||c^{\pm 1}_m||k||c^{\pm 1 }_k||n||c^{\mp 1}_n|.
\end{equation}
Using identities $\alpha\times(\alpha\times \beta)=-|\alpha|^2\beta$ and $(\alpha\cdot (\beta\times\gamma))=((\alpha\times\beta)\cdot\gamma)$, we see that
\begin{eqnarray*}
\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm 1})|n|^2c^{\mp 1}_{-n}
&=&
\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)\left(q^{\pm 1}_m\cdot (in\times (in \times q^{\mp 1}_n))\right)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\
&=&
\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)\left((q^{\pm 1}_m\times in)\cdot( in \times q^{\mp 1}_n)\right)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\
&=&
-\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times ik)( in \times q^{\pm 1}_n)c^{\mp 1}_kc^{\pm 1}_mc^{\mp 1}_{n}\\
& &
-\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\pm 1}_n)c^{\mp 1}_kc^{\pm 1}_mc^{\mp 1}_{n}.
\end{eqnarray*}
By the skew-symmetry, namely,
\begin{eqnarray*}
\sum_n\sum_{-n=k+m}(im\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}&=&\\
-\sum_n\sum_{-n=k+m}(in\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n}&=&\\
-\sum_m\sum_{-m=k+n}(in\cdot q^{\pm 1}_k)(q^{\pm 1}_m\times im)( in \times q^{\mp 1}_n)c^{\pm 1}_kc^{\pm 1}_mc^{\mp 1}_{n},& &\\
\end{eqnarray*}
the second term disappears which obviously leads to (\ref{skew1}).
Now to have the {\it \`a priori} bound of $c^{\pm 1}$ in $\ell^1_2$, we apply a smoothing via time averaging effect due to \cite{BMN2}.
\begin{prop}\label{restconvprop}{Restricted convolution}\cite[Theorem 3.1 and Lemma 3.1]{BMN2}
Assume that the following holds for $|n|_h\neq0$
\begin{equation}\label{restconv}
\sup_n\sum_{k:k+m+n=0,k\in \Sigma_i}\chi(n,k,m)|k|^{-1}\leq C2^{i}
\end{equation}
for every $i=1,2,\cdots$, where
\begin{equation*}
\Sigma_i:=\{k:2^i\leq |k|\leq 2^{i+1}\},
\quad
\chi(n,k,m)=
\begin{cases}
1\quad\text{if}\quad P_{nkm}(1)=0\\
0\quad\text{if}\quad P_{nkm}(1)\not=0.
\end{cases}
\end{equation*}
Then we have
\begin{eqnarray}\label{apriori0}
\left|\sum_n\bar B^{\pm 1}_n(c^{\pm 1},c^{\pm1})|n|^2c^{\mp 1}_{-n}\right|
&\leq&
C\|c^{\pm 1}\|_{\ell^2_2}\|c^{\pm 1}\|_{\ell^1_2}^2\\
\nonumber
&\leq& C\|c^{\pm 1}\|_{\ell_2^1}^4+\varepsilonilon\|c^{\pm 1}\|_{\ell^2_2}^2.
\nonumber
\end{eqnarray}
\end{prop}
Now, combining (\ref{aprioriestimate2}) and (\ref{apriori0}), we obtain the following estimate on $c^{\pm 1}$ in $\ell^1_2$, namely,
\begin{equation}\label{apply Gronwall}
\|c^{\pm 1}(t)\|^2_{\ell^1_2}\leq \|c^{\pm 1}(0)\|^2_{\ell^1_2}+
C\int_0^t\left(\|c^{\pm 1}(s)\|^4_{\ell^1_2}+\|c^{\pm 1}(s)\|^2_{\ell^1_2}\|c^{0}(s)\|^2_{\ell^1_1}\right)ds.
\end{equation}
Moreover we have the following energy inequality:
\begin{eqnarray}\label{energy ineq}
\|c^{\pm 1}(t)||_{\ell^0_2}&+&\left(\frac{\nu+\kappa}{2}\right)\int_0^t\|c^{\pm 1}(s)\|_{\ell^1_2}ds
\nonumber\\
&\leq&
\|c^{\pm 1}(t)||_{\ell^0_2}+\|c^0(t)\|_{\ell^0_2}+\int_0^t\left(\left(\frac{\nu+\kappa}{2}\right)\|c^{\pm 1}(s)\|_{\ell^1_2}+\nu\|c^0(s)\|_{\ell^1_2}\right)ds
\\
&\leq& \|c^{\pm 1}(0)\|_{\ell^0_2}^2+\|c^0(0)\|_{\ell^1_2}^2.
\nonumber
\end{eqnarray}
In fact, multiply the first equation of \eqref{c} by $c^{0*}(t)$ and the second one by $c^{\pm 1*}(t)$,
we have \eqref{energy ineq}.
Since all convection terms disappear due to the skew-symmetry, for example,
\begin{eqnarray*}
\sum_n\left(\bar B_n^1(c^1,c^1)\cdot c^{1*}\right)&=&
-i\sum_n\sum_{-n=k+m,\ -\omega_n^1+\omega^1_k+\omega^1_m=0}(q^1_k\cdot m)(q^1_m\cdot q^{-1}_n)c^1_kc^1_mc^{-1}_n\\
&=&
i\sum_n\sum_{-n=k+m,\ -\omega_n^1+\omega^1_k+\omega^1_m=0}(q^1_k\cdot n)(q^{-1*}_{-m}\cdot q^{-1}_n)c^1_kc^{-1*}_{-m}c^{-1}_n\\
&=&
i\sum_m\sum_{m=k+n,\ \omega_n^{-1}+\omega^1_k-\omega^{-1}_m=0}(q^1_k\cdot n)(q^{-1*}_m\cdot q^{-1}_n)c^1_kc^{-1*}_mc^{-1}_n\\
&=&
-\sum_m\left(\bar B_m^{-1}(c^1,c^{-1})\cdot c^{-1*}\right).
\end{eqnarray*}
To apply Gronwall's inequality, we need the following definition.
Let $C$ be the positive constant appearing in \eqref{apply Gronwall}. Then define $h$ as
\begin{equation*}
h:=\inf\{h'\in [0,\infty):\int _\tau^{\tau+h'}\|c^{\pm
1}(s)\|^2_{\ell_2^1}ds\leq \frac{1}{2C}\quad\text{for any}\quad
\tau>0\}.
\end{equation*}
Note that $h$ is independent of $N$ and can be chosen positive thaks to
\eqref{energy ineq}. From \eqref{apply Gronwall} and
an absorbing argument, we see that
\begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|^2_{\ell^1_2}\leq 2\|c^{\pm
1}(0)\|^2_{\ell^1_2}+ 2C\int_0^t\sup_{0<s''\leq s'}\|c^{\pm
1}(s'')\|^2_{\ell^1_2}\|c^{0}(s')\|^2_{\ell^1_1}ds'\quad\text{for}\quad
t<h.
\end{equation*}
Then by Gronwall's inequality,
\begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|_{\ell_2^1}^2\leq 2\|c^{\pm
1}(0)\|_{\ell_2^1}^2\exp\left(2C\int_0^t
\|c^0(s)\|_{\ell_1^1}^2ds\right)\quad\text{for}\quad t<h.
\end{equation*}
Iterating the same argument, one more time, we obtain
\begin{equation*}
\sup_{0<s\leq t}\|c^{\pm 1}(s)\|_{\ell_2^1}^2\leq 2\|c^{\pm
1}(h)\|_{\ell_2^1}^2\exp\left(2C\int_h^t
\|c^0(s)\|_{\ell_1^1}^2ds\right)\quad\text{for}\quad h\leq t<2h.
\end{equation*}
Note that $\int_0^t\|c^0(s)\|^2_{\ell^1_1}ds$ is always finite for
any fixed $t>0$, since there is a global solution to the 2D
Navier-Stokes equations in $\ell_1^s$-type function spaces. Fixing
$T>0$ and repeating this argument finitely many times, we have an {\it \`a
priori} bound of $c^{\pm 1}$ in $\ell^1_2$ over $t\in [0,T]$.
Finally, to use Proposition \ref{restconvprop} on the restricted
convolution, we need to verify that \eqref{restconv} holds. Observe that
\begin{eqnarray*}
P_{n,k,-n-k}(1)&=&
|n|_h^4| k|^4| n+ k|^4+| n|^4| k|_h^4| n+ k|^4
+| n|^4| k|^4| n+ k|^4_h\\
& &
-2| n|^2| n|^2_h| k|^2| k|^2_h| n+ k|^4
-2| n|^2| n|^2_h| k|^4| n+ k|^2| n+ k|_h^2\\
& &
-2| n|^4| k|^2| k|^2_h| n+ k|^2| n+ k|_h^2\\
&=&
| n|^4_hk^8_3+l.o.t.
\end{eqnarray*}
where $l.o.t.$ stands for lower order terms. Thus, it follows that
$P_{n,k,-n-k}(1)$ is a polynomial of degree eight in $k_3$ with a
nonzero leading coefficient whenever
$|n_1|+|n_2|\not
=0$. Then for fixed $k_1$, $k_2$ and $n$, there are at most eight
$k_3$ satisfying $\chi(n,k,-n-k)=1$. Thus,
\begin{eqnarray*}
\sum_{2^i\leq |k|\leq 2^{i+1}}|k|^{-1}\chi(n,k,-n-k)&\leq& \sum_{0\leq |k|_h\leq 2^{i+1},k_3\in\mathbb{R}}|k|^{-1}_h\chi(n,k,-n-k)\\
&\leq& 8\sum_{j=1}^i\sum_{2^j\leq |k|_h\leq 2^{j+1}}|k|^{-1}_h\leq 8\sum_{j=1}^i2^{2(j+1)}2^{-j}\leq C2^i.
\end{eqnarray*}
\end{proof}
\section{Proof of the main theorem}
Before proving the main theorem, we first mention the local
existence result.
Using estimate (\ref{desiredestimates}), we obtain a local-in-time unique solution to \eqref{original}
in $C([0,T]:\ell^1(\Lambda))$ as stated in the following lemma.
\begin{lemma}\label{local}
Assume that $c(0):=\{c^{\sigma_0}_n(0)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}} \in \ell^1(\Lambda)$
and
$c^{\sigma_0}_0(0)=0$ for $\sigma_0\in\{-1,0,1\}$.
Then there is a local-in-time unique solution
$c(t)\in C([0,T_L]:\ell^1(\Lambda))$
and
$c^{\sigma_0}_0(t)=0$ for $\sigma_0\in\{-1,0,1\}$
satisfying
\begin{equation}\label{T_L}
T_{L}\geq \frac{C}{\|c(0)\|^2},\quad
\sup_{0<t<T_{L}}\|c(t)\|\leq 10\|c(0)\|,
\end{equation}
where $C$ is a positive constant independent of $N$.
\end{lemma}
\begin{proof}
First we recall the mild formulation of \eqref{original}:
\begin{eqnarray*}
c_n^0(t)&=&e^{-\nu |n|^2t}
c^0_n(0)\\
& & +\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}
\int_0^te^{-\nu(t-s)|n|^2} \left(\bar
B^0_n(c^{\sigma_1},c^{\sigma_2})+\tilde B^0_n(Nt,
c^{\sigma_1},c^{\sigma_2})\right)ds \end{eqnarray*}
and
\begin{eqnarray*}
c_n^{\sigma_0}(t)&=&e^{-\frac{\nu+\kappa}{2} |n|^2t}
c^{\sigma_0}_n(0)\\
& &
+\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}
\int_0^te^{-\frac{\nu+\kappa}{2}(t-s)|n|^2}
\left(\bar B^{\sigma_0}_n(c^{\sigma_1},c^{\sigma_2})+\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right)
ds.
\end{eqnarray*}
By \eqref{desiredestimates}, we have the estimates
\begin{equation*}
\|c_n^0(t)\|\leq \|c^0_n(0)\|+C_\nu
t^{1/2}\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\sup_{0\leq
s<t}\|c^{\sigma_1}(s)\|\sup_{0\leq s<t}\|c^{\sigma_2}(s)\|\right)
\end{equation*}
and
\begin{eqnarray*}
\|c_n^{\sigma_0}(t)\|&\leq& \| c^{\sigma_0}_n(0)\|\\
& &+C_{\left(\frac{\nu+\kappa}{2}\right)}t^{1/2}
\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\sup_{0\leq
s<t}\|c^{\sigma_1}(s)\|\sup_{0\leq s<t}\|c^{\sigma_2}(s)\|\right).
\end{eqnarray*}
These {\it \` a-priori} estimates of $\sup_t\|c^0(t)\|$ and
$\sup_t\|c^{\sigma_0}(t)\|$ give us through a standard fixed point
argument the existence of a local-in-time unique solution (for the
detailed computation, see \cite{GIMM2} for example).
\end{proof}
Let $b^{\sigma_0}(t)$ be the solution to the limit equations \eqref{c} and $c^{\sigma_0}(t)$ be the solution to the original equation \eqref{original}.
The point is to control, in the $\ell^1$-norm, the remainder term $r^{\sigma_0}_n(t):=c^{\sigma_0}_n(t)-b^{\sigma_0}_n(t)$ ($\sigma_0=-1,0,1$) by the large
parameter $N$.
More precisely, $r^0$ and $r^{\sigma_0}$ satisfy
\begin{equation*}
\partial_t r_n^0(t)=-\nu |n|^2r_n^0(t)+\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}\left(\bar B^0_n(r^{\sigma_1},c^{\sigma_2})+
\bar B^0_n(b^{\sigma_1},r^{\sigma_2}) +\tilde B^0_n(Nt,
c^{\sigma_1},c^{\sigma_2})\right)
\end{equation*}
and
\begin{eqnarray*}
\partial_t r^{\sigma_0}_n(t)&=&-\left(\frac{\nu+\kappa}{2} \right)|n|^2r^{\sigma_0}_n(t)\\
& &
+\sum_{(\sigma_1,\sigma_2)\in \{-1,0,1\}^2}\left(\bar B^{\sigma_0}_n(r^{\sigma_1},c^{\sigma_2})+
\bar B^{\sigma_0}_n(b^{\sigma_1},r^{\sigma_2})+
\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1},c^{\sigma_2}) \right),
\end{eqnarray*}
respectively.
Once we control the remainder term in the $\ell^1$-norm, we easily
have the main result by a usual bootstrapping argument (see \cite{Y}
for example). Now we show the following lemma concerning the
smallness of the reminder term. Let
$b(t):=\{b^{\sigma_0}_n(t)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}}$
and
$r(t):=\{r^{\sigma_0}_n(t)\}_{n\in\Lambda,\sigma_0\in\{-1,0,1\}}$.
\begin{lemma}\label{fas}
For all $\varepsilonilon>0$, there is $N_0>0$ such that $\|r_n(t)\|\leq\varepsilonilon$ for $0<t<T_L$ and
$|N|>N_0$,
where $T_L$ is the local existence time (see Lemma \ref{local}).
\end{lemma}
\begin{proof}
To simplify the remainder equation, we introduce the following notation.
Let
\begin{eqnarray*}
\bar R^{\sigma_0}_n(r,c,b):&=&\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}
\left(\bar B^{\sigma_0}_n(r^{\sigma_1}, c^{\sigma_2})+\bar B^{\sigma_0}_n(b^{\sigma_1}, r^{\sigma_2})
\right),\\
\tilde R^{\sigma_0}_n(Nt,c):&=&\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}
\tilde B^{\sigma_0}_n(Nt, c^{\sigma_1}, c^{\sigma_2}).\\
\end{eqnarray*}
We rewrite the remainder equations as follows:
\begin{equation}\label{remainder equation}
\begin{cases}
\partial_tr_n^0(t)=-\nu |n|^2 r^0_n(t)+\bar R^0_n(r,c,b)+\tilde R^0_n(Nt,c)\\
\partial_t r^{\sigma_0}_n(t)=-\left(\frac{\nu+\kappa}{2}\right) |n|^2r^{\sigma_0}_n(t)
+\bar R^{\sigma_0}_n(r,c,b)+\tilde R^{\sigma_0}_n(Nt,c)
\quad\text{for}\quad \sigma_0=-1,1.\\
\end{cases}
\end{equation}
To control $r$, the key is to control $\tilde R^{0}_n(Nt,c)$ and $\tilde R^{\sigma_0}_n(Nt,c)$ in \eqref{remainder equation}. To do so,
we need to analyze the following oscillatory
integral of the non-resonant part as follows:
$$
\tilde{\mathcal {B}}^{\sigma_0}_n(N t,g^{\sigma_1},h^{\sigma_2}):=
\sum_{n=k+m, \omega^\sigma_{nkm}\not=0}\frac{1}{iN\omega^\sigma_{nkm}}e^{iN t\omega^\sigma_{nkm}}
(q_k^{\sigma_1}\cdot im)(q^{\sigma_2}_m\cdot q_n^{\sigma_0*})
g^{\sigma_1}_{k}h^{\sigma_2}_{m}
$$
and
\begin{equation}\label{estimate R}
\tilde {\mathcal R}^{\sigma_0}_n(Nt,c):=\sum_{(\sigma_1,\sigma_2)\in\{-1,0,1\}^2}
\tilde {\mathcal B}^{\sigma_0}(c^{\sigma_1}, c^{\sigma_2}).
\end{equation}
Note that we have the following relation between $\tilde B$ and $\tilde{ \mathcal B}$ ($\tilde R$ and $\tilde{\mathcal R}$):
\begin{eqnarray*}
\partial_t\left( \tilde{ \mathcal B}^{\sigma_0}_n(N t,
g^{\sigma_1},h^{\sigma_2})\right)&=&
\tilde B^{\sigma_0}_n(N t, g^{\sigma_1},h^{\sigma_2})+ \tilde {\mathcal B}^{\sigma_0}_n(N t,
\partial_tg^{\sigma_1},h^{\sigma_2})\\
& & +\tilde{\mathcal B}^{\sigma_0}_n(N t, g^{\sigma_1},\partial_th^{\sigma_2})
\end{eqnarray*}
and
\begin{eqnarray}\label{relation}
\partial_t\left( \tilde{ \mathcal R}^{\sigma_0}_n(N t,
c )\right)&=&
\tilde R^{\sigma_0}_n(N t, c)+ \tilde {\mathcal R}^{\sigma_0}_n(N t,
\partial_t c).
\nonumber
\end{eqnarray}
To control $r$, we split \eqref{remainder equation} into two parts: finitely many terms and small (in $\ell^1(\Lambda)$) remainder terms, respectively (cf. \cite[Theorem
6.3]{BMN1}).
For $\eta=1,2,\cdots$, we choose
$\{s_j\}_{j=1}^\infty\subset\mathbb N$ $(s_1<s_2<\cdots )$ in order
to satisfy $\|(I-\mathcal P_\eta)r\|\to 0\quad (\eta\to\infty)$,
where
\begin{eqnarray*}
\mathcal P_\eta r&:=&\bigg\{r_{n_1},r_{n_2},\cdots, r_{n_{s_\eta}}: \\
& &n_1,\cdots,n_{s_\eta}\in\Lambda:
n_k\not=n_\ell\ (k\not=\ell), |n_j|\leq \eta\quad\text{for all}\quad
j=1,\cdots,s_\eta\bigg\}.
\end{eqnarray*}
The choice of $n_1$ $\cdots$ $n_{s_\eta}$ is not uniquely
determined, however this does not matter. Then we can divide $r$ into
two parts: finitely many terms $r_{n_1},\cdots, r_{n_{s_\eta}}$ and
small remainder terms
$\{(I-\mathcal P_\eta)r_n\}_{n\in\Lambda}$.
\begin{remark}\label{beta}
We have the following estimates:
\begin{equation*}
\|\mathcal P_\eta \tilde {\mathcal{B}}^{\sigma_0}_n (\mathcal P_\eta c,\mathcal P_\eta c)\|_0
\leq \frac{\beta(\eta)}{N}(1+\eta^2)^{1/2}\|\mathcal P_\eta c\|_0^2,
\end{equation*}
\begin{equation*}
\|\mathcal P_\eta \bar {R}^{\sigma_0}_n(\mathcal P_\eta y,c,b)\|
\leq (1+\eta^2)^{1/2}\|\mathcal P_\eta y\|(\|c\|+\|b\|)
\end{equation*}
and
\begin{equation*}
\| |n|^2\mathcal P_\eta y\|\leq
(1+\eta^2)\|\mathcal P_\eta y\|_0
\end{equation*}
for $0<t<T_L$ ($T_L$ is a local existence time, see \eqref{T_L}), where
\begin{equation*}
\beta(\eta):=\max\{|\omega^\sigma_{nkm}|^{-1}:k=k_1,\cdots k_{s_\eta},\ n=n_1,\cdots, n_{s_\eta},\ m=n-k\}.
\end{equation*}
Note that $\beta(\eta)$ is always finite, since it only have finite
combinations for the choice of $n$, $k$ and $m$. We can also have
the same type estimate for $\|\partial_t\mathcal P_\eta c\|$ using
\eqref{c}.
\end{remark}
We use a change of variables to control $\tilde R^0$ and $\tilde {R}^{\sigma_0}$.
Let us set $y$ as
\begin{equation*}
y^{0}_n(t):= r^{0}_n(t)- \tilde {\mathcal{R}}^{0}_n(N t,\mathcal P_\eta c)\quad\text{and}\quad
y^{\sigma_0}_n(t):= r^{\sigma_0}_n(t)- \tilde {\mathcal{R}}^{\sigma_0}_n(N t,\mathcal P_\eta c).
\end{equation*}
From \eqref{remainder equation}, we see that
\begin{eqnarray*}
\partial_t\left(y^{0}_n+\tilde{\mathcal R}^{0}_n\right)&=&-\nu|n|^2(y^{0}_n+\tilde{\mathcal R}^{0}_n)
+
\bar R^{0}_n(y^{0}+\tilde {\mathcal R}^{0},c,b)\\
& &
+\tilde R^{0}_n(Nt, \mathcal P_\eta c)+\tilde R^{0}_n(Nt, (I-\mathcal P_\eta )c),\\
\partial_t\left(y^{\sigma_0}_n+\tilde{\mathcal R}^{\sigma_0}_n\right)&=&
-\left(\frac{\nu+\kappa}{2}\right)|n|^2(y^{\sigma_0}_n+\tilde{\mathcal
R}^{\sigma_0}_n) + \bar R^{\sigma_0}_n(y^{\sigma_0}+\tilde {\mathcal
R}^{\sigma_0},c,b)\\
& &
+\tilde R^{\sigma_0}_n(Nt, \mathcal P_\eta c) +\tilde
R^{\sigma_0}_n(Nt, (I-\mathcal P_\eta) c).
\end{eqnarray*}
Now we control $\mathcal P_\eta y^0$ and $\mathcal P_\eta y^{\sigma_0}$ for fixed $\eta$. By \eqref{estimate R},
\begin{eqnarray}\label{heat}
\partial_t\mathcal P_\eta y^{0}_n(t)&=&-\nu|n|^2\mathcal P_\eta y^{0}_n
+\mathcal P_\eta \bar { R}^{0}_n(\mathcal P_\eta y,c,b) +E^{0}_n,\\
\partial_t\mathcal P_\eta y^{\sigma_0}_n(t)&=&-\left(\frac{\nu+\kappa}{2}\right)|n|^2\mathcal P_\eta y^{\sigma_0}_n
+\mathcal P_\eta \bar { R}^{\sigma_0}_n(\mathcal P_\eta y,c,b) +E^{\sigma_0}_n,
\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
E^{0}_n:&=&-\mathcal P_\eta \tilde {\mathcal R}^{0}_n(Nt, \mathcal P_\eta \partial_t c)+
\mathcal P_\eta \bar R^{0}_n(\mathcal P_\eta \tilde {\mathcal R}^{0}_n(Nt,\mathcal P_\eta c),c,b)\\
& &-
\nu|n|^2\mathcal P_\eta \tilde{\mathcal R}^{0}_n(Nt, \mathcal P_\eta c)
+\mathcal P_\eta\tilde R^0_n(Nt, (I-\mathcal P_\eta)c)\\
\end{eqnarray*}
and
\begin{eqnarray*}
E^{\sigma_0}_n:&=&-\mathcal P_\eta \tilde {\mathcal
R}^{\sigma_0}_n(Nt, \mathcal P_\eta \partial_tc)+
\mathcal P_\eta \bar R^{\sigma_0}_n(\mathcal P_\eta \tilde {\mathcal R}^{\sigma_0}_n(Nt,\mathcal P_\eta c),c,b)\\
& &-
\left(\frac{\nu+\kappa}{2}\right)|n|^2\mathcal P_\eta \tilde{\mathcal R}^{\sigma_0}_n(Nt, \mathcal P_\eta c)
+\mathcal P_\eta\tilde R^{\sigma_0}_n(Nt, (I-\mathcal P_\eta)c).
\end{eqnarray*}
Note that \eqref{heat} are linear heat type equations with
external force $E^0$ and $E^{\sigma_0}$. Thus the point is to
control $E^0$ and $E^{\sigma_0}$. By Remark \ref{beta}, we can see
that for any $\varepsilonilon>0$, there is $\eta_0$ and $N_0$ (depending
on $\eta_0$) such that if $N>N_0$ and $\eta>\eta_0$, then
$\|E^{\sigma_0}\|<\varepsilonilon$ and $\|E^{\sigma_0}\|<\varepsilonilon$.
Thus we have from \eqref{heat},
\begin{eqnarray*}
\|\mathcal P_\eta y^{0}_n(t)\|&\leq &\int_0^t
\bigg(C_\nu(1+\eta^2)\|\mathcal P_\eta
y^{0}_n(s)\| \\
& &+(1+\eta^2)^{1/2}\|\mathcal P_\eta y^0(s)\|(\|c(s)\|+\|b(s)\|)
+\varepsilonilon\bigg) ds
\end{eqnarray*}
and
\begin{eqnarray*}
\|\mathcal P_\eta
y^{\sigma_0}_n(t)\|&\leq &\int_0^t \bigg(
C_{\nu,\kappa}(1+\eta^2)\|\mathcal P_\eta
y^{\sigma_0}_n(s)\|\\
& & +(1+\eta^2)^{1/2}\|\mathcal P_\eta
y^{\sigma_0}(s)\|(\|c(s)\|+\|b(s)\|) +\varepsilonilon\bigg) ds.
\nonumber
\end{eqnarray*}
By Gronwall's inequality, we have that for any $\varepsilonilon>0$, there
is $\eta_0$ and $N_0$ (depending on $\eta_0$) such that if $\eta>\eta_0$
and $N>N_0$, then $\|\mathcal P_\eta y^0\|<\varepsilonilon$ and $\|\mathcal
P_\eta y^{\sigma_0}\|<\varepsilonilon$ for $0<t<T_L$. Clearly, we can also control
$(I-\mathcal P_\eta )y$ with sufficiently large $\eta$ (independent
of $N$), and $\mathcal P_\eta \tilde {\mathcal R}^{\sigma_0}_n(Nt,
\mathcal P_\eta c)$ with sufficiently large $N$ for fixed $\eta$.
Thus we can control $r$ for sufficiently large $\eta$ and $N$.
\end{proof}
{\bf Acknowledgments.}
The second author thanks the Pacific Institute for the Mathematical Sciences
for support of his presence there during the academic
year 2010/2011.
This paper developed during a stay of the second author as a PostDoc at the Department of Mathematics and Statistics,
University of Victoria.
\end{document} |
\begin{document}
\tildeitle[Walking within Growing Domains: Recurrence versus Transience ]
{Walking within Growing Domains: \\mathbb{R}ecurrence versus Transience}
\ date{\tildeoday}
\author[A.\ Dembo]{Amir Dembo$^*$}
\author[R.\ Huang]{Ruojun Huang$^\ diamond$}
\author[V.\ Sidoravicius]{Vladas Sidoravicius$^\ dagger$}
\address{$^*$Department of Mathematics, Stanford University, Building 380,
Sloan Hall, Stanford, CA 94305, USA}
\address{$^{*\ diamond}$Department of Statistics, Stanford University,
Sequoia Hall, 390 Serra Mall, Stanford, CA 94305, USA}
\address{$^\ dagger$IMPA, Estrada Dona Castorina 110, Jardim Botanico,
Cep 22460-320, Rio de Janeiro, RJ, Brazil}
\begin{abstract}
For normally reflected Brownian motion and for simple random walk on independently growing
in time $d$-dimensional domains, $d\ge3$, we establish a sharp criterion for
recurrence versus transience in terms of the growth rate.
\end{abstract}
\maketitle
\section{Introduction.}
There has been much interest in studies of random walks in random environment
(see \cite{HMZ}). Of particular challenge are problems in which the walker
affects its environment, as in reinforced random walks. In this context even
the most fundamental question of recurrence versus transience is often open.
For example, the recurrence of two dimensional linearly reinforced random
walk with large enough reinforcement strength has just been recently solved in
\cite{ACK}, \cite{ST} (and for its extensions to other graphs see \cite{ACK}). The
corresponding question by M. Keane for once reinforced random walk remains
open. Moving to $\mathbb{Z}^d$, $d\ge3$, it was conjectured by the last author
that recurrence of once reinforced random walk is sensitive to the strength of the
reinforcement.
In contrast, the motion of walker excited towards the origin on the boundary of its
range is recurrent in any dimension regardless of the strength of the excitation,
see \cite[Section 2]{K1} and \cite{K2}, while as
shown in \cite{BW}, excitation by means
of a drift in $\vec{e}_1$ direction results in transience for any strength of
the drift in any dimension $d \geq 2$ (Cf. \cite{KZ} for results in one dimension,
related excitation models, and open problems).
The case where the walk does not affect the time evolution of its environment
is better understood. For example, time homogeneous, translation invariant
Markovian evolution of the environment is considered in \cite{DKL} and the references
therein.
The quenched \abbr{CLT} for the walk is proved there for stationary initial conditions
subject to suitable locality, ellipticity, spatial and temporal mixing of the
environment.
Our focus is on the recurrence/transience properties of certain
time-varying, highly non-reversible evolutions. Specifically, we
consider the discrete-time simple random walk (\abbr{srw}) $\{Y_t\}$
on non-decreasing connected graphs $\mathbb{G}_t$ of common vertex set.
Namely, having $Y_t = y$, one chooses $Y_{t+1}$ uniformly among
all neighbors of $y$ within $\mathbb{G}_{t+1}$. In this article we propose
three natural general conjectures about the recurrence/transience
of such processes and prove partial results in this direction,
for such \abbr{srw} on subgraphs of $\mathbb{Z}^d$, $d \ge 3$,
which satisfy the following bounded-shape condition.
\begin{assumption}\leftarrowbel{ass-1}
The connected, non-decreasing $t \mapsto \mathbb{D}_t \subseteq \mathbb{Z}^d$,
$d \ge 3$ are such that $f(t)
\mathbb{B}_1 \cap \mathbb{Z}^d \subseteq \mathbb{D}_t \subseteq f(t) \mathbb{B}_c\cap \mathbb{Z}^d$, for some finite
$c$ and non-decreasing, unbounded, strictly positive $f(t)$, $t \ge 0$
(and $\mathbb{B}_c \subset \mathbb{R}^d$ denotes an Euclidean ball of radius
$c$, centered at the origin $0 \in \mathbb{Z}^d$).
\end{assumption}
In this context, we propose the following {\em universality conjecture}
(namely, that only the asymptotic growth rate of $t \mapsto f(t)$ matters
for transience/recurrence of such \abbr{srw}).
\begin{cnj} \leftarrowbel{cj1}
Almost surely,
the \abbr{srw} $\{Y_t \}$ on $\{\mathbb{D}_t\}$
satisfying Assumption \ref{ass-1} and starting at $Y_0=0$,
returns to the origin finitely often iff
\begin{equation}
J_f := \int_0^\infty \fracrac{dt}{f(t)^{d}} < \infty.
\end{equation}
\end{cnj}
\noindent
Indeed, we show in Theorem \ref{rwthm} that under Assumption \ref{ass-1},
having $J_f < \infty$ implies that
$\mathbb{P} (A) = 0$ for $A:= \{ \sum_t \mathbb{I}_{\{y\}} (Y_t) = \infty \}$
and any $y \in \mathbb{Z}^d$. For the more challenging part, namely
\begin{equation}\leftarrowbel{eq:Jf-inf}
J_f = \infty \quad \mathbb{R}ightarrow \quad \mathbb{P} (A) =1,
\end{equation}
we resort to connecting the \abbr{srw} $\{ Y_t \}$ with a normally reflected Brownian motion (in short \abbr{rbm}), via an invariance principle (see Lemma \ref{invariance}). Thus, our approach
yields sample-path recurrence results
for reflected Brownian motion on growing domains in $\mathbb{R}^d$
(in short \abbr{rbmg}, see Definition \ref{rbmg} and
Theorem \ref{bmthm}), which
are of independent interest. This strategy comes however at a cost of imposing certain additional restrictions
on $t \mapsto \mathbb{D}_t$.
Specifically, when proving in part (b) of Theorem \ref{rwthm} the recurrence
of the \abbr{srw} on growing domains $\mathbb{D}_t$ in
$\mathbb{Z}^d$, $d \ge 3$,
we further assume that
$\mathbb{D}_t = f(t) \mathbb{K} \cap \mathbb{Z}^d$ for some $\mathbb{K}$ regular enough,
to which end we recall the following definition.
\begin{defn}
An open connected $\mathbb{K} \subseteq \mathbb{R}^d$ is called a uniform domain
if there exists a constant $C < \infty $ such that for every $x, y \in \mathbb{K}$ there
exists a rectifiable curve $\gamma \subseteq \mathbb{K}$
joining $x$ and $y$, with
$length(\gamma) \leq C|x-y|$ and
$\min \{|x-z|, |z-y| \} \leq C {\rm{dist}} (z, \partial \mathbb{K})$ for all $z \in \gamma$.
\end{defn}
Dealing with a {\em{discrete time}} \abbr{srw},
we may consider without loss of generality only
$t \mapsto f(t)$ {\em{piecewise constant}}, that is,
from the collection
\begin{eqnarray}\leftarrowbel{eq:FF-def}
\mathcal{F}:=\{f(\cdot)\,:f(t)=\sum_{l=1}^\infty a_l\mathbb{I}_{[t_l, t_{l+1})}(t),
\tildeext{ for } t_1=0, \, \{t_l\}\uparrow\infty,0 < a_l\uparrow\infty\}.
\end{eqnarray}
However, as seen in our main result below, for our
proof of \eqref{eq:Jf-inf} we further require
the following separation of scales
\begin{equation}\leftarrowbel{eq:def-Fsar}
\mathcal{F}_\ast:=\{f \in \mathcal{F}: \; (a_l - a_{l-1}) \uparrow \infty,
\quad \sum_{l=1}^\infty a_l^{2-d} \log (1+a_l) <\infty \} \,.
\end{equation}
\begin{thm}
\leftarrowbel{rwthm}
Consider a \abbr{srw} $\{Y_t \}$ on $\{\mathbb{D}_t\}$
satisfying Assumption \ref{ass-1}, with $Y_0=0$.
\newline
(a). Whenever $J_f<\infty$, the \abbr{srw} $\{Y_t \}$ a.s.
visits every $y \in \mathbb{Z}^d$ finitely often.
\newline
(b). Such \abbr{srw} $\{Y_t\}$ a.s. visits every $y \in \mathbb{Z}^d$ infinitely often,
in case $\mathbb{D}_t = f(t) \mathbb{K} \cap \mathbb{Z}^d$ with
$f \in \mathcal{F}_\ast$ such that $J_f=\infty$ and
$\mathbb{K}$ in
\begin{equation}\leftarrowbel{eq:cK-def}
\mathcal{K} := \{\tildeextrm{bounded uniform domain } \; \mathbb{K} \subset \mathbb{R}^d :
\; x \in \mathbb{K} \; \mathbb{R}ightarrow \;
\leftarrowmbda x \in \mathbb{K}
\quad \fracorall \leftarrowmbda\in [0,1]
\}\,.
\end{equation}
\end{thm}
\begin{remark}\leftarrowbel{rmk-sup-lin} Requiring $(a_l - a_{l-1}) \uparrow \infty$
results in $l \mapsto a_l$ super-linear, and hence in
the series $\sum_l a_l^{2-d} \log (1+a_l)$
converging whenever $d \ge 4$ (so the latter
restriction on $f \in \mathcal{F}_\ast$ is relevant only for $d=3$). We need $\mathbb{K}$ to be a uniform domain only
for the invariance principle of Lemma \ref{invariance},
and impose on $\mathbb{K}$ the star-shape condition of
\eqref{eq:cK-def} merely to guarantee that
the corresponding sub-graphs $t \mapsto \mathbb{D}_t$ are non-decreasing.
\end{remark}
One motivating example for our study is the \abbr{srw} $\{Y_t\}$ on the independently
growing Internal Diffusion Limited Aggregation (\abbr{idla})
cluster $\mathbb{D}_t$, formed by particles injected at the origin according
to a Poisson process of bounded away from zero intensity $\leftarrowmbda(t)$,
and independently performing \abbr{srw} with jump-rate $v$.
While the microscopic boundary of such \abbr{idla} cluster $\mathbb{D}_t$
is rather involved, it is well known (see \cite{LBG}),
that $M_t^{-1/d} \mathbb{D}_t \tildeo \mathbb{B}_{\kappa}$, where $M_t$
denotes the number of particles reaching the \abbr{idla} cluster
boundary by time $t$, and the value of $\kappa = \kappa_d$ is chosen
such that $\mathbb{B}_{\kappa}$ has volume one. Consequently, from
part (a) of Theorem \ref{rwthm} we have that
\begin{cor}\leftarrowbel{cor-idla}
The \abbr{srw} on such \abbr{idla} clusters is a.s.
transient when the random variable
$J := \int_1^\infty M_t^{-1}dt$ is finite.
\end{cor}
\noindent
Further, our analysis (i.e. part (b) of Theorem \ref{rwthm}),
suggests the a.s. recurrence of the \abbr{srw} on such
\abbr{idla} clusters, whenever $J = \infty$ (this is
also a special case of Conjecture \ref{cj1}).
\begin{remark} In our \abbr{idla} clusters example,
let $g(t) := \int_0^t \leftarrowmbda(s)ds$ denote the mean of the Poisson
number of particles $N_t$ injected at the origin by time $t$. Then, a.s. $J < \infty$ iff
\begin{equation}
\leftarrowbel{1.1}
\widehat J := \int_1^\infty g(t)^{-1} dt < \infty \,.
\end{equation}
Indeed, clearly $M_t \le N_t$ and for large $t$ the Poisson
variable $N_t$ is concentrated around $g(t)$. Our claim thus
follows immediately when $v \tildeo \infty$, for then
one has further that $M_t \uparrow N_t \sim g(t)$. More
generally, for $v$ finite and $t \gg 1$, the variable $M_t$ is still concentrated,
say around some non-random $u(t)$, which is roughly comparable to
$N_{t- cu(t)^{2/d}}$ for some $c = c(v)$, and thereby also to
$g(t-c u(t)^{2/d})$. Solving for
\begin{equation}
u(t) := \sup \{u: \, g(t-cu^{2/d}) \geq u \} \notag
\end{equation}
it is easy to check that $u(2t) \ge g(t)\wedge(t/c)^{d/2}$, hence for $d\geq 3$
\begin{equation}
\int_1^\infty u(t)^{-1} dt < \infty \quad \tildeext{\it{iff}} \quad \widehat J < \infty. \notag
\end{equation}
\end{remark}
Next, considering part (b) of Theorem \ref{rwthm} for $\mathbb{K} = \mathbb{B}_1$ we see that (at least subject to our conditions
about $\{a_l\}$), Conjecture \ref{cj1} is a consequence
of the more general {\em monotonicity conjecture}:
\begin{cnj}
\leftarrowbel{cj2}
Suppose non-decreasing in $t$ graphs ${\mathbb{G}}_t , \; {\mathbb{G}}'_t$ of uniformly bounded
degrees are such that
${\mathbb{G}}_t \subseteq {\mathbb{G}}'_t$ for all $t$, and the
\abbr{srw} $\{ Y_t \}$ on $\{ \mathbb{G}_t \}$
is transient, i.e. its sample-path a.s. returns
to $Y_0=y_0$ finitely often. Then, the same
holds for the sample path of the \abbr{srw}
$\{ Y'_t \}$ on $\{ {\mathbb{G}}'_t \}$, starting at $Y_0'=y_0$.
\end{cnj}
\begin{remark}
By Rayleigh monotonicity principle, Conjecture \ref{cj2} trivially holds
whenever $\mathbb{G}_t$ and ${\mathbb{G}}'_t$ do not depend on $t$.
However, beware that it may fail when the graphs depend on $t$ and unbounded degrees are allowed. For example,
on $\mathbb{G}_t = \mathbb{Z}^3$ the \abbr{srw} is transient, but
we can force having a.s. infinitely many returns to $0$
by adding to the edges of $\mathbb{Z}^3$,
at times $t_k \uparrow \infty$ fast enough,
edges in $\mathbb{G}'_{t}$, $t \ge t_k$, between $0$
and each vertex in a wide enough annulus
$\mathbb{A}_k := \{x \in \mathbb{Z}^3 : \|x\|_2 \in [r_k,R_k)\}$
(specifically, with $r_k \ll R_k$ suitably chosen
to make sure the \abbr{srw} on $\mathbb{G}'_{t}$ is at
times $t_k$ in $\mathbb{A}_k$ and thereby force at least
one return to zero before exiting $\mathbb{A}_k$,
while $t_{k-1} \ll t_k$ gives separation of scales).
\end{remark}
As shown for example in Theorem \ref{rwthm}, when the \abbr{srw} on
the limiting graph $\mathbb{G}_{\infty}$ is transient,
one may still get recurrence
by imposing slow enough growth on $\mathbb{G}_t$.
In contrast, whenever the \abbr{srw}
on $\mathbb{G}_{\infty}$ is recurrent, we have the following consequence of
the Conjecture \ref{cj2}.
\begin{cnj} \leftarrowbel{cj3}
If \abbr{srw} $\{ Y_t \}$ on a fixed graph $\mathbb{G}_{\infty}$
of uniformly bounded degrees is recurrent, then the
same applies to \abbr{srw} on non-decreasing $\mathbb{G}_t \subseteq \mathbb{G}_{\infty}$,
for {\em any choice} of $\mathbb{G}_t \uparrow \mathbb{G}_{\infty} $.
\end{cnj}
\noindent In particular, Conjecture \ref{cj3}
implies that the \abbr{srw}
on {\em{any}} non-decreasing $\mathbb{D}_t \subseteq \mathbb{Z}^2$ is recurrent. We note in passing
that monotonicity of $t \mapsto \mathbb{D}_t$ is necessary for the latter statement
(hence for Conjectures \ref{cj2} and \ref{cj3}).
Indeed, with $\mathbb{D}_t$ being $\mathbb{Z}^2 $ without edges $(x,y)$ for $|| x||_1 = t$
and $|| y ||_1 = t-1$, we have $\mathbb{D}_t \tildeo \mathbb{Z}^2$ as $t \tildeo \infty$, while
$||Y_t||_1 = t$ for all $t$.
\begin{remark} Conjecture \ref{cj3} was proposed
to us by J. Ding and upon completing this manuscript
we found a more general version of it in \cite{ABGK}. Specifically, \cite{ABGK} conjecture that a
random walk $\{Y_t\}$ on graph $\mathbb{G}_\infty$
with non-decreasing edge conductances
$\{c_t(e)\}$ is recurrent as soon as the walk on $(\mathbb{G}_\infty,\{c_\infty(e)\})$ is recurrent
(Conjecture \ref{cj3} is just its restriction
to $\{0,1\}$-valued conductances). This is
proved for $\mathbb{G}_\infty$ a tree
(by potential theory, see \cite[Theorem 5.1]{ABGK}).
A weaker version of Conjecture \ref{cj2} is also
proposed there (and confirmed in \cite[Theorem 4.2]{ABGK}
for $\mathbb{G}_\infty=\mathbb{Z}_+$), whereby
the transience of the walk on
$(\mathbb{G}_\infty,\{c_t(e)\})$ is
conjectured to hold whenever
the walk on $(\mathbb{G}_\infty,\{c_0(e)\})$ and
the walk on $(\mathbb{G}_\infty,\{c_\infty(e)\})$
are both transient. Finally, we note in passing that the
zero-one law $\mathbb{P}(A) \in \{0,1\}$ in
Conjecture \ref{cj1} is not at all obvious
given \cite[Example 4.5]{ABGK}, where $0<\mathbb{P}(A)<1$
for some random walk on $\mathbb{Z}$ with certain
non-random, non-increasing $c_t(e) \in (0,1]$.
\end{remark}
\begin{remark} Recall \cite{GKZ} that the \abbr{srw}
on the infinite cluster $\mathbb{D}_0$ of Bernoulli bond
percolation on $\mathbb{Z}^d$ is a.s.
recurrent for $d=2$ and transient for any $d \ge 3$.
Hence, by Conjectures
\ref{cj2}
and
\ref{cj3}
the same should apply to the \abbr{srw} on
any independently growing domains
$\mathbb{D}_t \supseteq \mathbb{D}_0$.
Whereas the latter is an open problem, by
\cite[Theorem 1.1]{Ke}
such conclusion trivially holds when $\mathbb{D}_t$
is the set of vertices connected to the origin
by time $t$ in First-Passage Percolation
with finite, non-negative i.i.d. passage
times on $\mathbb{Z}^d$, subject only
to the mild moment condition \cite[(1.6)]{Ke}.
\end{remark}
We consider also Brownian motions on growing domains, as defined next.
\begin{defn}
\leftarrowbel{rbmg}
We call $(W_t,\mathbb{D}_t)$ reflected Brownian motion on growing domains
(\abbr{RBMG}), if the non-random, monotone non-decreasing
$\mathbb{D}_t\subseteq\mathbb{R}^d$ are such that the normally reflected
Brownian motion $W$ on the time-space domain
$\mathcal{D}:=\{(t,x)\in\mathbb{R}^{d+1}:x\in \mathbb{D}_t\}$
is a well-defined strong Markov process solving the corresponding
deterministic Skorohod problem. That is, for any
$(s,x) \in \overline{\mathcal{D}}$ there is a unique pair of continuous
processes $(W, L)$ adapted to the minimal admissible filtration of
Brownian motion $\{U_t\}_{t \geq 0}$, with $L$ non-decreasing, such
that for any $t \geq s$, both $(t,W_t) \in \overline{\mathcal{D}}$ and
\begin{align}
\leftarrowbel{skorohod}
W_t &= x + U_t - U_s + \int_s^t {\bf n}(u, W_u) dL_u\,, \\
L_t &= \int_s^t \mathbb{I}_{\partial \mathcal{D}} (u, W_u) dL_u
\,,
\notag
\end{align}
where $\mathbf{n}(u,y)$ denotes the inward normal unit vector at
$y\in\partial \mathbb{D}_u$.
\end{defn}
As shown in \cite [Theorem 2.1 and 2.5]{BCS}, Definition \ref{rbmg}
applies when $\partial \mathcal{D}$ is $C^3$-smooth with
$\mathbf{\gamma}(t,x) \cdot(0,\mathbf{n}(t,x))$
bounded away from zero uniformly
on compact time intervals, where $\mathbf{\gamma}(t,x)$
denotes the inward
normal unit vector at $(t,x)\in\partial \mathcal{D}$.
Focusing on $\mathbb{D}_t=f(t)\mathbb{K}$
this condition holds whenever both $f(t)$ and
$\partial \mathbb{K}$ are $C^3$-smooth.
Further, this construction easily extends to handle
isolated jumps in $t \mapsto f(t)$.
In the context of $\mathbb{R}^d$-valued stochastic
processes, we define recurrence as follows:
\begin{defn}
\leftarrowbel{rec}
The sample path $x_t$ of a stochastic process
$t\mapsto x_t \in \mathbb{D}_t$ with $x_0=0$, is called recurrent,
if it makes infinitely many excursions to $\mathbb{B}_{\epsilon}$ for any $\epsilon>0$, and
is called transient otherwise. That is, recurrence amounts to the event
$A:=\cap_{\epsilon>0}A_\epsilon$, where
\begin{align*}
\sigma_\epsilon^{(0)}&:=\inf\{t\ge0: \mathbb{D}_t\supseteq \mathbb{B}_\epsilon\},\;\; \\
\tildeau_\epsilon^{(i)}&:=\inf\{t\ge\sigma_\epsilon^{(i-1)}: |x_t|<\epsilon\},\;\; i\ge1\\
\sigma_\epsilon^{(i)}&:=\inf\{t\ge\tildeau_\epsilon^{(i)}: |x_t|>1/2\},\\
A_\epsilon&:=\{\tildeau_\epsilon^{(i)}<\infty, \fracorall i\}.
\end{align*}
\end{defn}
\begin{thm}
\leftarrowbel{bmthm}
Suppose $\mathbb{B}_{f(t)} \subseteq \mathbb{D}_t\subseteq \mathbb{R}^d$,
$d \ge 3$, and $t \mapsto f(t)$ is
positive, non-decreasing.
\newline
(a). The sample path of the \abbr{rbmg} $(W_t, \mathbb{D}_t)$
is a.s. transient whenever $J_f < \infty$.
\newline
(b). The sample path of the \abbr{rbmg} $(W_t, \mathbb{D}_t)$
is a.s. recurrent whenever $J_f=\infty$, provided
$\mathbb{D}_t = f(t)\mathbb{K}$ for
$C^3$-smooth up to isolated
jump points $t \mapsto f(t)$ such that
$\int_0^\infty f'(s)^2 ds$ is finite and
$\mathbb{K} \in \mathcal{K}$ of
$C^3-$smooth boundary $\partial \mathbb{K}$.
\end{thm}
\begin{remark}
In part (a) of Theorem \ref{bmthm} we implicitly assume
that the \abbr{rbmg} $(W_t,\mathbb{D}_t)$ is well defined,
in the sense of Definition \ref{rbmg}. Since $J_f=\infty$
whenever $f(t)$ is bounded, in which case
part (b) trivially holds,
we assume throughout
that $f(t)$ is unbounded.
The condition
$\int_0^\infty f'(s)^2 ds < \infty$ is needed in part (b)
only for $\mathbb{K} \ne \mathbb{B}_r$, and it holds for example whenever
$f(\cdot)$ is piecewise constant, or in case
$f(s)=(c+s)^\alpha$ for some $c>0$ and $\alpha \in [0,1/2)$.
\end{remark}
We prove Theorem \ref{bmthm} in Section \ref{sec2}
and Theorem \ref{rwthm}
in Section \ref{sec3},
whereas in Section \ref{appen} we show that
in the context of Conjecture \ref{cj1},
if recurrence/transience occurs a.s.
with respect to the origin, then the same
applies at any other point.
\section{Proof of Theorem \ref{bmthm}}\leftarrowbel{sec2}
Since the events $A_\epsilon$ are non-decreasing in $\epsilon$, it suffices
for Theorem \ref{bmthm}
to show that
$q_\epsilon:=\mathbb{P}(W\in A_\epsilon)=\mathbb{I}_{\{J_f=\infty\}}$
for each fixed $\epsilon>0$. To this end we require
the following three lemmas.
\begin{lem}\leftarrowbel{coupling}
Suppose $|x_1|\le|x_2|$ and
for some positive $g\uparrow\infty$, and constant $c>1$, one has
\abbr{RBMG}-s $(W_t^{(1)},\mathbb{D}_t^{(1)})$, $(W_t^{(2)},\mathbb{D}_t^{(2)})$, such that
$W_0^{(i)}=x_i$, $i=1,2$,
$\mathbb{D}_t^{(1)}=\mathbb{B}_{g(t)}$, $\mathbb{D}_t^{(2)}\supseteq \mathbb{B}_{cg(t)}$.
\newline
(a). Then, there exists a coupling
$(W_t^{(1)},W_t^{(2)})$ with non-negative
$\psi_t:=|W_t^{(2)}|-|W_t^{(1)}|$.
\newline
(b). Such coupling also exists in case
$\mathbb{D}_t^{(2)}=\mathbb{D}_{t}^{(1)}=\mathbb{B}_{g(t)}$.
\end{lem}
\begin{proof} (a). Given $x, y \in \mathbb{R}^d$
with $|x|=|y|$,
let ${\mathcal{V}} := span\{x,y\}$ and ${\bf{O}} (x,y)$ denote the
unique $d$-dimensional orthogonal matrix acting as the identity
on ${\mathcal{V}}^{\perp}$ and as the rotation such that
${\bf{O}} x =y$ on ${\mathcal{V}}$.
By assumption $\psi_0\ge0$. We run the \abbr{RBMG}-s independently,
until $\eta_1:=\inf\{t\ge0: \psi_t=0\}$, noting that by continuity of
$t\mapsto\psi_t$, the function $\psi_t$ is non-negative on $(0,\eta_1)$.
It thus suffices to consider only $\eta_1<\infty$.
In this case, let $\{W_t^{(1)}, \, t\geq \eta_1\}$, be the solution of
(\ref{skorohod}) driven by Brownian motion
$\{U_t, \, t \geq \eta_1\}$ starting at
$W_{\eta_1}^{(1)}=U_{\eta_1}$. Setting
$\widetilde{U}_t := {\bf{O}} ( W_{\eta_1}^{(1)},W_{\eta_1}^{(2)}) U_t$ let
\begin{eqnarray*}
\tildeau_1:=\inf\{t\ge\eta_1: \widetilde{U}_t \in\partial \mathbb{D}_t^{(2)}\}.
\end{eqnarray*}
Since $|W_{\eta_1}^{(2)}| = |W_{\eta_1}^{(1)}| \leq g(\eta_1) < cg(\eta_1)$,
it follows from the definition of \abbr{RBMG} $(W_t^{(2)}, \mathbb{D}_t^{(2)})$ that
$\{ \widetilde{U}_t\}$ has for $t \in [\eta_1, \tildeau_1]$ the same law as
$\{ W_t^{(2)}\}$.
In particular, a normal reflection at $\partial \mathbb{B}_{g(t)}$ reduces the norm,
hence $|W_t^{(1)}| \leq |U_t| = |\widetilde U_t| = |W_t^{(2)}|$ for such $t$.
That is $\psi_t \geq 0$ on $t \in [\eta_1, \tildeau_1]$.
With $\psi_{\tildeau_1} \geq (c-1)g(\tildeau_1) >0$, clearly
$\eta_2:=\inf\{t\ge\tildeau_1: \psi_t=0\} > \tildeau_1$. In case $\eta_2 < \infty$,
with $W_{\eta_2}^{(2)}\in \mathbb{D}_{\eta_2}^{(1)}$, we repeat the above argument
for $[\eta_2,\tildeau_2]$, then for $[\eta_3,\tildeau_3]$, etc. By construction,
$\eta_n<\tildeau_n<\eta_{n+1}$ for all $n$. Moreover, a.s.
$\tildeau_n \rightarrow\infty$ when $n \tildeo \infty$.
Indeed, assuming without loss of generality that
$\eta_k<\infty$, we have the stopping times
\begin{align*}
\tildeheta_k &:= \inf\{ t \ge \eta_k :
\widetilde{U}_t \in \partial \mathbb{B}_{g(t)} \} \,,\\
\zeta_k &:= \inf\{ t \ge \tildeheta_k :
\widetilde{U}_t \in \partial \mathbb{B}_{cg(\tildeheta_k)} \} \,,
\end{align*}
such that $\tildeheta_k < \zeta_k \le \tildeau_k$ and
conditional on the relevant stopped $\sigma$-algebra
at $\tildeheta_k$,
the random variable $\zeta_k - \tildeheta_k$ has the law
of the time it takes an independent Brownian motion
to get from $\partial \mathbb{B}_{g(\tildeheta_k)}$ to
$\partial \mathbb{B}_{cg(\tildeheta_k)}$. With $g(\tildeheta_k) \ge g(0)$,
by Brownian scaling it follows that the sequence $\{\tildeau_k-\eta_k\}$
stochastically dominates the i.i.d.
$\{\xi_k\}$, each distributed as
$\xi:=g(0)^2 \inf\{t\ge0: |U_t|=c,|U_0|=1\}$.
This induces stochastic domination of the corresponding
partial sums,
$$
\sum_{k=1}^n (\tildeau_k-\eta_k) \succeq \sum_{k=1}^n\xi_k \,.
$$
As $n \tildeo \infty$ the right-hand-side
grows a.s. to infinity and so does the left-hand-side.
\noindent
(b). We follow the construction and reasoning
of part (a), up to time $\eta_1$, setting now
$\widetilde{U}_t := {\bf{O}} ( W_{\eta_1}^{(1)},W_{\eta_1}^{(2)}) W^{(1)}_t$
for all $t \ge \eta_1$. Then, by the
invariance to rotations of $\mathbb{B}_{g(t)}$ and
the fact that only normal reflections are used,
we have that
$t \mapsto W^{(2)}_t 1_{\{t < \eta_1\}} +
\widetilde{U}_t 1_{\{t \ge \eta_1\}}$ is a
realization of the \abbr{rmbg}
$(W_t^{(2)},\mathbb{B}_{g(t)})$, for which $\psi_t$ is
non-negative.
\end{proof}
\begin{lem}
\leftarrowbel{exptail}
Let $\mathbb{P}_x$ denote the law of the
\abbr{rbm}
$Z_t$ on $\mathbb{B}_a$,
starting at $Z_0 = x$. Consider the stopping times
$\tildeau(a):=\inf\{s\ge0: Z_s\in\partial \mathbb{B}_{a}\}$ and
$\sigma(a,r):=\inf\{s\ge0: Z_s\in \mathbb{B}_{r}\}$. Then,
there exists $C = C_d (\ delta) >0$ such that for any $ t, \ delta >0$,
$\fracrac{r}{a} \in [\ delta, 1)$, $d\geq 3$,
\begin{align}
\leftarrowbel{2.1a}
\sup_{x \in \mathbb{B}_a}\mathbb{P}_x(\tildeau(a) >
t a^2)&<C^{-1}e^{-Ct}\,,
\\
\leftarrowbel{2.1b}
\sup_{x \in \mathbb{B}_a \backslash \mathbb{B}_{r}}\mathbb{P}_x(
\sigma(a,r) > ta (a-r))&<C^{-1}e^{-Ct}\,,
\\
\inf_{x \in \mathbb{B}_{a/2}} \mathbb{P}_x(\tildeau(a) > a^2) & > C \,,
\leftarrowbel{2.1c}
\end{align}
\end{lem}
\noindent {\emph{Proof.}}
In case the process starts at $z \in \partial \mathbb{B}_r$
we use $\mathbb{P}_{re_1}$ to indicate probabilities
of events which are invariant under any
rotation of the sample path.
Then, with $U_t$ denoting a
standard Brownian motion, by Brownian scaling
the left-hand side of (\ref{2.1c}) does not depend on $a$
and is merely the
positive probability $\mathbb{P}_{0.5 e_1}(|U_s|<1, \fracorall s \le 1)$.
Further, by the Markov property,
invariance to rotations and Brownian scaling, for
$x \in \mathbb{B}_a$,
\begin{align*}
&\mathbb{P}_{x}(\tildeau(a)>ta^2)=\mathbb{P}_{x}(|Z_s|<a, \fracorall s\le ta^2)
\le \mathbb{B}ig[\sup_{0\le|z|<a}\mathbb{P}_z(|Z_s|<a, \fracorall s\le a^2)\mathbb{B}ig]^{\lfloor{t}\rfloor}\\
&= \big[\mathbb{P}_{0}(|Z_s|<a, \fracorall s\le a^2)\big]^{\lfloor{t}\rfloor}
= \mathbb{P}_0(|U_s|<1, \fracorall s\le 1)^{\lfloor{t}\rfloor} := (1-\eta)^{\lfloor{t}\rfloor},
\end{align*}
with $\eta = \eta_d >0$, out of which we get (\ref{2.1a}).
Proceeding similarly, we have for (\ref{2.1b}) that
\begin{align*}
\mathbb{P}_{x}(\sigma(a,r)& > ta(a-r))=\mathbb{P}_{x}(|Z_s|>r, \fracorall s\le ta(a-r))\\
&\le \big[\sup_{r < |z| \leq a}\mathbb{P}_z(|Z_s|>r, \fracorall s\le a(a-r))\big]^{\lfloor{t}\rfloor}
= \big[\mathbb{P}_{ae_1}(|Z_s|> r, \fracorall s\le a(a-r))\big]^{\lfloor{t}\rfloor} \\
&\leq
\sup_{\ delta \leq \rho < 1}\mathbb{P}_{e_1}(|\widehat{Z}_s| > \rho, \fracorall s\le1-\rho)^{\lfloor{t}\rfloor}
:= (1-\zeta(\ delta))^{\lfloor{t}\rfloor}
\end{align*}
where $\widehat{Z}$ denotes the \abbr{rbm}
on $\mathbb{B}_1$.
Further,
$$
\zeta \ge \inf_{\ delta \leq \rho < 1}
\mathbb{P}_{e_1} (|U_{1-\rho}| \leq \rho) >0
$$
(by the stochastic domination $|U_s| \succeq
|\widehat{Z}_s|$, for example,
due to part (a) of Lemma \ref{coupling}).
\qed
\begin{lem}
\leftarrowbel{estimates}
Let $\mathbb{P}_x$ denote the law of the \abbr{rbm}
$Z_t$ on $\mathbb{B}_a$,
starting at $Z_0 = x$.
Fixing $\epsilon,\ delta \in (0,1/2)$, there exist
finite $M_d(\epsilon,\ delta)$ and
$C=C_d(\epsilon,\ delta)$ such that for all
$M,T,a$ and $r$ with
$M \geq M_d (\epsilon,\ delta)$,
$T \geq M a^2 \log a$ and $a-M \geq r \geq a \ delta$,
\begin{align}
\leftarrowbel{2.3}
\inf_{x \in \mathbb{B}_r}\mathbb{P}_x(\exists s\le T: |Z_s|<\epsilon)& \ge C^{-1} \big[ \fracrac{ T}{a^d}\wedge1\big], \\
\leftarrowbel{2.4}
\sup_{x \in \partial \mathbb{B}_{r}}\mathbb{P}_x(\exists s\le T: |Z_s|<\epsilon)&\le C \big[ \fracrac{ T}{a^d}\wedge1\big]\,.
\end{align}
\end{lem}
\begin{proof}
Starting at $Z_0=x \in \partial \mathbb{B}_r$, let
$\sigma(a,\epsilon) := \inf\{ t \ge 0: |Z_t| \le \epsilon \}$,
and setting $\sigma_0:=0$,
\begin{align}
\tildeau_k&:=\inf\{t\ge\sigma_{k-1}: Z_t\in\partial \mathbb{B}_{a}
\},\quad k\ge1
\leftarrowbel{excursion}\\
\sigma_k&:=\inf\{t\ge\tildeau_k: Z_t\in \mathbb{B}_{a/2}\}\nonumber
\end{align}
we call $Z_\cdot$ restricted to $[\sigma_k,\sigma_{k+1}]$ the $k$-th excursion of $Z$, with $L_k:=\sigma_{k+1}-\sigma_k$ denoting its length.
Obviously, for any $k \in \mathbb{N}$
\begin{equation}\leftarrowbel{eq:basic-bd}
\mathbb{P}(\sigma(a,\ep) \le \sigma_k) -
\mathbb{P}(\sigma_k \ge T) \le \mathbb{P}(\sigma(a,\ep) \le T) \le
\mathbb{P}(\sigma(a,\ep) \le \sigma_k) +
\mathbb{P}(\sigma_k \le T) \,.
\end{equation}
Recall that conditional on their starting and ending
positions, these excursions of the \abbr{rbm} on $\mathbb{B}_a$
are mutually independent. Consequently,
\begin{equation}\leftarrowbel{eq:hit-bd}
\mathbb{P}(\sigma(a,\ep) > \sigma_k) = \mathbb{E} \mathbb{B}ig[
\prod_{i=1}^k \big(1-b_{\epsilon}
(Z_{\sigma_{i-1}},Z_{\sigma_i})\big) \mathbb{B}ig]
\end{equation}
where $b_\epsilon (x,w) :=
\mathbb{P}_x(\inf_{t \le \tildeau_1} |Z_t| \le \epsilon \;|\; Z_{\sigma_1}=w)$
is the probability of entering $\mathbb{B}_\epsilon$ in one
such excursion. Elementary potential theory
(e.g. see \cite[Theorem 3.18]{MP}), yields the formula
\begin{equation}\leftarrowbel{eq:beps}
b_\epsilon (x) =
\fracrac{|x|^{2-d}-a^{2-d}}{\epsilon^{2-d}-a^{2-d}}
\end{equation}
for the unconditional probability
$b_\epsilon (x) := \mathbb{E} [b_\epsilon (x,Z_{\sigma_1})]$.
Hence, applying the strong Markov property of $Z_\cdot$
at the stopping time $\sigma_i$ where
$|Z_{\sigma_i}|=a/2$, going
from $i=k-1$ backwards to $i=1$ we deduce that
$$
q_k := \mathbb{P}(\sigma(a,\ep) \le \sigma_k) =
1-(1-b_\epsilon(re_1))(1-b_\epsilon(\fracrac{a}{2} e_1))^{k-1} \,.
$$
It is easy to check that $b_\epsilon(\fracrac{a}{2} e_1)
= c_0 a^{2-d} (1+o(1/M))$ for
some finite, positive $c_0=c_0(d,\epsilon)$
and all $a \ge M \ge 1/\ delta$, whereas
$b_\epsilon(r e_1) \le c' b_\epsilon(\fracrac{a}{2} e_1)$
for some finite $c' = c'(d,\ delta)$,
and all $r \ge a \ delta \ge 1$.
Thus, setting $k = [T a^{-2} \kappa^{\mp 1}]$ for some
universal $\kappa$ yet to be determined, we see that
\begin{equation}\leftarrowbel{eq:dom-term}
C^{-1} \mathbb{B}ig[\fracrac{T}{a^d} \wedge 1\mathbb{B}ig] \le
q_k \le C \mathbb{B}ig[\fracrac{T}{a^d} \wedge 1\mathbb{B}ig] \,,
\end{equation}
for some finite $C=C(d,\epsilon,\ delta,\kappa)$ and all
$M \ge M_d(\epsilon,\ delta)$ large enough.
Hence, it suffices to show that for some universal
$c=c(d,\epsilon,\ delta)>0$, $\kappa=\kappa(d,\epsilon,\ delta)$
finite and all $a \ge 2M_d$, $k \ge 1$,
$x \in \partial \mathbb{B}_r$
\begin{equation}\leftarrowbel{eq:exc-concent}
\mathbb{P}_x (k^{-1} a^{-2} \sigma_k \ge \kappa) \le e^{-c k}\,,
\qquad
\mathbb{P}_x (k^{-1} a^{-2} \sigma_k \le \kappa^{-1}) \le e^{-c k}\,.
\end{equation}
Indeed, our assumption that $T \ge M a^2 \log a$
translates to $c k \ge c \kappa^{\mp 1} M \log a$, so
that for all large enough $M \ge M_0(\kappa,c,d,C)$
we have that
$$
e^{-c k} \le \fracrac{1}{2 C} \mathbb{B}ig[\kappa^{\pm 1} a^{2-d} \wedge 1 \mathbb{B}ig] \le \fracrac{1}{2} q_k \,,
$$
resulting by (\ref{eq:basic-bd}) and
(\ref{eq:dom-term}) in the claimed bounds.
\noindent
The universal exponential tail bounds of
(\ref{eq:exc-concent}), are a direct consequence
of having control on the log-moment generating
functions
$\mathbb{L}eftarrowmbda_k(\tildeheta):= \log \mathbb{E} [e^{\tildeheta \sigma_k}]$
for large $k$ and small $\widehat{\tildeheta} := \tildeheta a^2$.
Specifically, by Markov's exponential inequality
(also known as Chernoff's bound), we get
(\ref{eq:exc-concent}) as
soon as we show that
\begin{align}\leftarrowbel{eq:upp-tail-mgf}
\kappa_+ & := \liminf_{\widehat{\tildeheta} \ downarrow 0} \;
\limsup_{k \tildeo \infty} \; \widehat{\tildeheta}^{-1} k^{-1}
\sup_{a \ge M} \big\{ \mathbb{L}eftarrowmbda_k(\widehat{\tildeheta} a^{-2})
\big\} < \infty \,,\\
- \kappa_-^{-1} & := \liminf_{\widehat{\tildeheta} \ downarrow 0}
\; \limsup_{k \tildeo \infty} \; \widehat{\tildeheta}^{-1} k^{-1}
\sup_{a \ge M} \big\{
\mathbb{L}eftarrowmbda_k(-\widehat{\tildeheta} a^{-2}) \big\} < 0 \,,
\leftarrowbel{eq:low-tail-mgf}
\end{align}
(provided $\kappa > \kappa_+ \vee \kappa_-$ and
$c=\widehat{\tildeheta}
(\kappa-\kappa_+) \wedge (\kappa_-^{-1} - \kappa^{-1})$
for $\widehat{\tildeheta}>0$ small enough). Turning to control
$\mathbb{L}eftarrowmbda_k(\cdot)$, recall that $\sigma_k = \sum_{i=0}^{k-1} L_i$,
with $\{L_i\}$ mutually independent conditional on the values of
$\{Z_{\sigma_i}\}$. Thus, proceeding in the same manner
as done in (\ref{eq:hit-bd}), we have that for any
$\tildeheta \in \mathbb{R}$ and $k \in \mathbb{N}$,
$$
\mathbb{L}eftarrowmbda_k(\tildeheta) = \log \mathbb{E} \mathbb{B}ig[ \prod_{i=1}^k m(\tildeheta,Z_{\sigma_{i-1}},Z_{\sigma_i}) \mathbb{B}ig] \,,
$$
where $m(\tildeheta,x,w):=\mathbb{E}_x[e^{\tildeheta L_0}|Z_{\sigma_1}=w]$.
By invariance of the joint law of $\{\sigma_k\}$
with respect to rotations of the
\abbr{RBM} sample path $t \mapsto Z_t$,
the unconditional function
$m(\tildeheta,x)=\mathbb{E}[m(\tildeheta,x,Z_{\sigma_1})]=m(\tildeheta,|x|)$ depends
only on $|x|$. Hence, exploiting once more the strong Markov property at the stopping times $\sigma_i$
where $|Z_{\sigma_i}|=a/2$
(first for $i=k-1$, then backwards to $i=1$),
we find that
\begin{equation}\leftarrowbel{eq:lbdk}
\mathbb{L}eftarrowmbda_k(\tildeheta) = (k-1) \log m(\tildeheta,a/2) +
\log m(\tildeheta,r) \,.
\end{equation}
Further, $L_0$
is the sum of two independent variables, having the
laws of $\tildeau(a)$ and $\sigma(a,a/2)$ of Lemma \ref{exptail}. Thus, the universal
upper bounds (\ref{2.1a}) and (\ref{2.1b})
imply that for
any $0 \le \widehat{\tildeheta} < C_d(\ delta)$
(and $C_d(\ delta)>0$ as in Lemma \ref{exptail}),
\begin{equation}\leftarrowbel{eq:bd-mgf-1ex}
\sup_{r \ge a \ delta} \, \{ m(\widehat{\tildeheta} a^{-2},r)
\} \le \big[ 1+\fracrac{\widehat{\tildeheta}}{C_d(C_d-\widehat{\tildeheta})}
\big]^2 \,.
\end{equation}
Combining this with (\ref{eq:lbdk}) we get
(\ref{eq:upp-tail-mgf}) (with $\kappa_+=2 C_d^{-2}$ finite).
Recall that
for any $Y,y \ge 0$,
$$
\log \mathbb{E}[e^{-Y}] \le -(1-\mathbb{E}[e^{-Y}]) \le -(1-e^{-y})\mathbb{P}(Y \ge y)
\,,
$$
hence
$\log m(-\widehat{\tildeheta} a^{-2},r) \le 0$ and
$$
\log m(-\widehat{\tildeheta} a^{-2},a/2) \le - (1-e^{-\widehat{\tildeheta}})
\mathbb{P}( L_1 \ge a^2) \,.
$$
It thus follows from (\ref{2.1c}) and
the stochastic domination $L_1 \succeq \tildeau(a)$
starting at some position $x \in \partial \mathbb{B}_{a/2}$
that $\log m(-\widehat{\tildeheta} a^{-2},a/2)
\le -C_d \widehat{\tildeheta}/e$ for all $a>0$ and $\widehat{\tildeheta} \le 1$,
thereby establishing (\ref{eq:low-tail-mgf})
with $\kappa_-^{-1} = C_d/e$ positive, and completing
the proof of the lemma.
\end{proof}
\noindent {\emph{Proof of Theorem \ref{bmthm}.}}\;
This proof consists of six steps. First, for
$\mathbb{D}_t=\mathbb{B}_{f(t)}$ and $f \in \mathcal{F}_\ast$ of
\eqref{eq:def-Fsar},
we prove in {\tildeextsc{Step I}}
the a.s. recurrence of the
\abbr{rbmg} when
$J_f=\infty$, and in \tildeextsc{Step II}
its a.s. transience when $J_f<\infty$.
Relaxing these conditions, in \tildeextsc{Step III}
we prove part (a), and in \tildeextsc{Step IV}
get part (b) for $\mathbb{K} = \mathbb{B}_1$.
The a.s. sample-path recurrence when $J_f=\infty$ is then
established for $\mathbb{K} \in \mathcal{K}$ of \eqref{eq:cK-def},
when both $\partial \mathbb{K}$ and $t \mapsto f(t)$ are $C^3$-smooth
(see \tildeextsc{Step V}), and further extended to $f(\cdot)$ having
isolated jump points (see \tildeextsc{Step VI}).
\noindent
{\bf Step I.} For $f \in \mathcal{F}_\ast$ we set
$\mathbb{D}elta T_l :=t_{l+1}-t_l$ and
$p_l := a_l^{2-d} \log (1+a_l)$, so that
$\sum_l p_l < \infty$ and
\begin{equation}
\leftarrowbel{eq:jdisc}
J_f=\sum_{l=1}^\infty a_l^{-d} \mathbb{D}elta T_l \,.
\end{equation}
Considering here $\mathbb{D}_t=\mathbb{B}_{f(t)}$ for
$f \in \mathcal{F}_\ast$, we proceed to prove the a.s.
recurrence of the \abbr{rbmg} sample path
in case $J_f=\infty$. To this end,
consider the events
$\mathbb{G}amma_{l}:=\{\exists t \in [t_{l-1}, t_l) : |W_{t}|<\ep\}$,
adapted to the filtration ${\mathcal{G}}_{l}:=\sigma\{W_s, s\le t_{l}\}$. Fixing $\ delta \in (0,1/2)$ we set
$r_l := (a_{l-1}+1) \vee \ delta a_l$ and
further assume that
\begin{equation}
\leftarrowbel{A1}
\mathbb{D}elta T_l
\geq 2 M_d a_l^2 \log (1+a_l) \,.
\end{equation}
Then, since
\begin{equation}
\leftarrowbel{eq:wt}
W_{t_l} \in \overline{\mathbb{B}}_{a_{l-1}} \quad {\rm and} \quad \mathbb{D}_t=\mathbb{B}_{a_l}, \; \fracorall t \in [t_l,t_{l+1}),
\end{equation}
we have by (\ref{2.3}) that
\begin{eqnarray*}
\zeta_l:=\mathbb{P}(\mathbb{G}amma_{l+1}|\mathcal{G}_l)=\mathbb{P}(\mathbb{G}amma_{l+1}|W_{t_l})\ge
\inf_{x\in \overline{\mathbb{B}}_{a_{l-1}}}\mathbb{P}_{x}(\exists s\le\mathbb{D}elta T_l: |Z_{s}|<\ep)\ge C^{-1} \big[\fracrac{\mathbb{D}elta T_l}{a_l^d} \wedge 1\big].
\end{eqnarray*}
Recall that $J_f$ of \eqref{eq:jdisc} is infinite,
hence a.s.
$\{\sum_{l=1}^\infty\zeta_l=\infty \}$, which implies
that $\mathbb{G}amma_l$ occurs infinitely often
(by the conditional version of Borel-Cantelli II, see
\cite[Theorem 5.3.2]{Du}). That is,
\begin{equation}\leftarrowbel{eq:recc}
\exists l_k\uparrow\infty \quad \& \quad
s_k\in[t_{l_k},t_{l_k+1}) \quad
\tildeextrm{ such that } \quad
|W_{s_k}|<\epsilon \,.
\end{equation}
By transience of the $d\geq 3$ dimensional Brownian
motion we can set $k_1 = 1$ and recursively pick $u_j:=\inf\{t> s_{k_j}: |W_t|>1/2\}$,
$k_{j+1} := \inf \{ k: \, s_k > u_j \}$, for $j= 1,2, \ dots, $ thus yielding the event $A_\epsilon$.
To remove the spurious condition (\ref{A1}) set
$\psi_l:=\mathbb{D}elta T_l/(a_l^2 \log (1+a_l))$, so
$\sum_l \psi_l p_l = \sum_l a_l^{-d} \mathbb{D}elta T_l$
diverges by (\ref{eq:jdisc}) whereas $\sum_l p_l$ is finite.
Hence, $\sum_{l}\psi_l p_l \mathbb{I}_{\{\psi_{l} \geq 2 M_d \}} = \infty $, and the
preceding argument is applicable even when restricted to $\{l_k\}\uparrow\infty$ such that
$\psi_{l_k} \geq 2 M_d$.
\noindent
{\bf Step II.} Still considering $\mathbb{D}_t=\mathbb{B}_{f(t)}$ for
$f \in \mathcal{F}_\ast$, we show next that
$\mathbb{P}(A_{\ep})=0$ whenever $J_f$ of
(\ref{eq:jdisc}) is finite. To this end, note that
\begin{equation*}
\tildeau_l:=\inf\{s\ge0:W_s\not\in\overline{\mathbb{B}}_{r_l}\},
\end{equation*}
for $l =1, 2, \ldots , $ are a.s. finite and proceed to show that
\begin{equation}
\leftarrowbel{eq:gsum}
\sum_l \mathbb{P} (\widetilde{\mathbb{G}amma}_{l}) < \infty ,
\end{equation}
where
$
\widetilde{\mathbb{G}amma}_{l}:=\{\exists t \in [\tildeau_{l}, \tildeau_{l+1}) : |W_{t}|<\ep\}.
$
Indeed, in this case by Borel-Cantelli I, a.s. the \abbr{rbmg} does not re-enter $\mathbb{B}_\epsilon$
during $[\tildeau_l, \infty)$, for some $l $ finite. In any finite time, even the
\abbr{rbm}
on $\mathbb{B}_1$ a.s. makes only finitely many excursions between $\mathbb{B}_\epsilon$ and $\mathbb{B}_{1/2}^c$,
hence $\mathbb{P}(A_{\ep})=0$. Turning to prove (\ref{eq:gsum}),
recall that $t_l \leq \tildeau_l$ and $t_{l+1} \leq \tildeau_{l+1}$,
so the interval
$[\tildeau_l, \tildeau_{l+1})$ splits into
$[\tildeau_l, \xi_{l+1})$ and $[\xi_{l+1}, \tildeau_{l+1})$, where $$
\xi_{l+1}:=\inf\{s\ge t_{l+1}: W_s\not\in \overline{\mathbb{B}}_{r_l}\}\,.
$$
Restricted to $t \in [\tildeau_l, \xi_{l+1})$, the process
$\{W_t\}$ has the law of a \abbr{rbm}
on $\mathbb{B}_{a_l}$, and the length of $[\tildeau_l, \xi_{l+1})$ is at most $\mathbb{D}elta T_l$ plus the length of $[t_{l+1}, \xi_{l+1})$. By (\ref{2.1a}), for some constant $C=C_d(\ delta)>0$, any $l$ and all $t$,
\begin{align}
\mathbb{P}(\xi_{l+1}-t_{l+1}>ta_l^2)<C^{-1}e^{-Ct}. \leftarrowbel{length}
\end{align}
Combining (\ref{2.4}) with (\ref{length}) for
$t=M\log a_l$, $M=M_d\vee \fracrac{2}{C}$, we have that
\begin{align}\leftarrowbel{eq:first-bd}
\mathbb{P}(\exists s\in[\tildeau_l, \xi_{l+1}): |W_s|<\ep)\le C[a_l^{-d}\mathbb{D}elta T_l+Mp_l]+C^{-1}a_l^{-2}\,,
\end{align}
with the first term on the right-hand-side
summable in $l$ iff $J_f<\infty$ (the other
two terms are summable for any $f \in \mathcal{F}_\ast$).
Further, restricted to $t \in [\xi_{l+1}, \tildeau_{l+1})$,
the process $\{W_t\}$ has the law of
Brownian motion $\{U_t\}$ (since $r_{l+1}<a_{l+1}$),
hence
\begin{equation}\leftarrowbel{eq:second-bd}
\mathbb{P}(\exists s\in[\xi_{l+1}, \tildeau_{l+1}): |W_s|<\ep)=\fracrac{r_l^{2-d} - r_{l+1}^{2-d}}{\epsilon^{2-d}- r_{l+1}^{2-d}} \leq 2\epsilon^{d-2}(r_l^{2-d} - r_{l+1}^{2-d}).
\end{equation}
Bounding $\mathbb{P}(\widetilde{\mathbb{G}amma}_l)$ by the sum of
the left-hand-sides of
(\ref{eq:first-bd}) and (\ref{eq:second-bd}),
we thus conclude that $\sum_l\mathbb{P}(\widetilde{\mathbb{G}amma}_l)<\infty$ whenever $J_f<\infty$.
\noindent
{\bf Step III}.
Given non-decreasing, unbounded, positive $t \mapsto f(t)$
(which without loss of generality we assume hereafter
to be also right-continuous), let $g \in \mathcal{F}_\ast$ with
$a_l=2^{l-1} f(0)$ and $t_l:=\inf\{t \ge 0: f(t) \geq 2^{l-1} f(0)\}$. Since $g(t) \leq f(t) \leq 2g(t)$ for all $t \ge 0$,
we have by part (a) of Lemma \ref{coupling}, the coupling
$|\widetilde{W}_t| \leq |W'_t|$ for \abbr{RBMG}
$(\widetilde{W}_t,\mathbb{B}_{g(t)/2})$ and $(W'_t,\mathbb{D}_t)$
such that $\mathbb{D}_t \supseteq \mathbb{B}_{f(t)}$.
Further, as $J_{4g} \leq J_{f} \leq J_{\fracrac{1}{2}g}$,
if $J_f<\infty$ then $J_{\fracrac{1}{2}g} = 8^d J_{4g} < \infty$
and in view of Step II, a.s. $\{ \widetilde{W}_t \}$ enters
$\mathbb{B}_\epsilon$ finitely often. Hence,
$\mathbb{P}(A_{\epsilon})=0$, yielding the stated a.s. transience
of the sample path for any such \abbr{rbmg} $(W_t',\mathbb{D}_t)$,
thereby completing the proof of part (a).
\noindent
{\bf Step IV.} Returning to $\mathbb{D}_t = \mathbb{B}_{f(t)} = f(t) \mathbb{B}_1$, now for $t \mapsto f(t)$
which is further $C^3$-smooth up to isolated jump points,
we have by yet another application of part (a) of Lemma \ref{coupling}
that $|W'_t| \leq |W_t^{''}|$ for the \abbr{rbmg}
$(W_t^{''}, \mathbb{B}_{4g(t)})$. Assuming that $J_f=\infty$,
or equivalently that $J_{4g} = \infty$ (with $g \in \mathcal{F}_\ast$
chosen as in Step III), we know from
Step I that for any $u$ fixed,
$\{ W_t^{''}, t \ge u \}$ a.s. makes infinitely many excursions from $\mathbb{B}_\epsilon$ to $\mathbb{B}_{1/2}^c$.
With $|W'_t| \le |W_t^{''}|$ we consequently get \eqref{eq:recc} (for any unbounded $t_l$), which
as we have already seen in Step I of the proof,
implies that $\{W'_t\}$ a.s. makes infinitely many excursions from $\mathbb{B}_\epsilon$ to $\mathbb{B}_{1/2}^c$.
\noindent
{\bf Step V.} We next extend the a.s. recurrence
of the \abbr{rbmg} $(W_t,\mathbb{D}_t)$ sample path to $\mathbb{D}_t = f(t)\mathbb{K}$
with $J_f=\infty$, $\mathbb{K}$ from $\mathcal{K}$ of \eqref{eq:cK-def}, such that
$\partial \mathbb{K}$ and $f(t)$ are both $C^3$-smooth, and
$\int_0^\infty f'(s)^2 ds < \infty$.
To this end, we
assume without loss of generality that
$\mathbb{B}_1 \subseteq \mathbb{K} \subseteq \mathbb{B}_c$ and note that
$t \mapsto \int_0^t \fracrac{1}{f(u)} dL_u =:\widetilde{L}_t$ increases
only when $X_t := \fracrac{1}{f(t)}W_t$
is at $\partial \mathbb{K}$. Hence,
applying Ito's formula to the
$C^{1,2}$-function $v(t,x) = \fracrac{1}{f(t)} x$ (with $v_{xx}=0$),
and the semi-martingale $\{W_t\}$ of (\ref{skorohod}), we get that
$(X,\widetilde{L})$ is the strong Markov
process solving the deterministic Skorohod problem corresponding for
$(s,x) \in \mathbb{R}_+ \tildeimes \mathbb{K}$ to
\begin{align}
\leftarrowbel{skorohoda}
X_t &= x + \int_s^t \fracrac{1}{f(u)} dB_u + \int_s^t {\bf{n}} (X_u) d \widetilde{L}_u\,, \\
\leftarrowbel{skorohodb}
\widetilde{L}_t &= \int_s^t \mathbb{I}_{\partial \mathbb{K}} (X_u) d\widetilde{L}_u \,,
\end{align}
where ${\bf{n}} (x)$ denotes the inward unit normal vector
at $x\in \partial \mathbb{K}$ and
\begin{equation}\leftarrowbel{eq:Bt-def}
B_t = U_t - \int_0^t f'(s) X_s ds, \quad B_0 = 0\,.
\end{equation}
Further, with ${X_t} \in \mathbb{K} \subseteq \mathbb{B}_c$
and $\int_0^\infty f'(s)^2 ds < \infty$, the
quadratic variation
$\leftarrowngle M \rightarrowngle_t = \int_0^t |f'(s) X_s|^2 ds$
of the continuous (local) martingale
$$
M_t =\int_0^t f'(s) X_s d U_s \,,
$$
has uniformly in $t$ bounded exponential moments.
That is, for any $\kappa > 1$,
\begin{equation*}\leftarrowbel{exp-mmt-bd}
\mathbb{E}\mathbb{B}ig[\exp\big\{\kappa \leftarrowngle M \rightarrowngle_\infty \big\}\mathbb{B}ig]
\le\exp\mathbb{B}ig\{c^2 \kappa\int_0^\infty f'(s)^2 ds\mathbb{B}ig\}<\infty\,.
\end{equation*}
By Novikov's criterion,
$Z_t=\exp(M_t - \fracrac{1}{2} \leftarrowngle M \rightarrowngle_t)$
is
a uniformly integrable continuous martingale
(see \cite[Proposition VIII.1.15]{RY}).
The same applies for $Z_t^{-1} = \exp(\widehat{M}_t-\fracrac{1}{2}\leftarrowngle \widehat{M} \rightarrowngle_t)$ and the martingale $\widehat{M}_t = - \int_0^t f'(s) X_s d B_s$ under the measure $\mathbb{Q}$ such
that $\{B_t, t \in [0,\infty)\}$ is a standard
Brownian motion in $\mathbb{R}^d$. Hence, by
Girsanov's theorem, restricted to the completion
of the canonical Brownian filtration, the measure
$\mathbb{Q}$ is equivalent
to $\mathbb{P}$ (see \cite[Proposition VIII.1.1]{RY}).
Moreover, under $\mathbb{Q}$ the process
$\{X_t\}$ is a normally reflected time
changed Brownian motion
(in short \abbr{tcrbm}), on $\mathbb{K}$
for the deterministic time change
\begin{equation}\leftarrowbel{eq:time-change}
\tildeau(t):=\int_0^t f(s)^{-2} ds \,.
\end{equation}
Applying the same procedure for the \abbr{rbmg}
$(W_t',f(t) \mathbb{B}_c)$, such that $W'_0=W_0$,
yields another
probability measure $\mathbb{H}$, likewise equivalent
to $\mathbb{P}$, under which $Y_t := \fracrac{1}{f(t)} W_t'$
is a \abbr{tcrbm} on $\mathbb{B}_{c}$ for the {\em same} time
change $\tildeau(\cdot)$ as in (\ref{eq:time-change}).
Further,
$\{W_\cdot \in A\}$ iff
$\{X_\cdot \in A^{(f)}\}$, and $\{W'_\cdot \in A\}$
iff
$\{Y_\cdot \in A^{(f)}\}$,
where $A^{(f)}:=\bigcap_{\epsilon>0}A^{(f)}_\epsilon$ and
similarly to Definition \ref{rec} we have that
\begin{align*}
\sigma_\epsilon^{(0,f)}&:=0 \\
\tildeau_\epsilon^{(i,f)}&:=\inf\{t\ge\sigma_\epsilon^{(i-1,f)}: |x_t|<\epsilon/f(t)\},\;\; i\ge1\\
\sigma_\epsilon^{(i,f)}&:=\inf\{t\ge\tildeau_\epsilon^{(i,f)}: |x_t|>1/(2f(t))\},\\
A_\epsilon^{(f)}&:=\{\tildeau_\epsilon^{(i,f)}<\infty, \fracorall i\}
\end{align*}
(as $\epsilon<f(0)$ without loss of generality).
For $J_f=\infty$ we saw in Step IV that $\mathbb{P}(W'_\cdot\in A)=1$, hence
$$
\mathbb{P}(Y_\cdot\in A^{(f)})=1
\quad\mathbb{L}eftrightarrow\quad
\mathbb{H}(Y_\cdot\in A^{(f)})=1
\quad\stackrel{(a)}{\mathbb{R}ightarrow}\quad
\mathbb{Q}(X_\cdot\in A^{(f)})=1
\quad \mathbb{L}eftrightarrow\quad
\mathbb{P}(X_\cdot\in A^{(f)})=1
$$
out of which we deduce that $\mathbb{P}(W_\cdot\in A)=1$
as well. The key implication, marked by (a), is a consequence
of the proof of \cite[Theorem 5.4]{Pa}. This theorem is
a comparison result about
Neumann heat kernels over domains $\mathbb{D}_i$, $i=1,2$,
such that $\mathbb{D}_2\subseteq \mathbb{B} \subseteq \mathbb{D}_1\subseteq\mathbb{R}^d$,
$d\ge2$, for bounded domains $\mathbb{D}_1$, $\mathbb{D}_2$ of $C^2$-smooth
boundary, and some ball $\mathbb{B}$ centered at $0$, such that
for any $x \in \mathbb{D}_2$, the line segment
from $0$ to $x$ is in $\mathbb{D}_2$. Its proof in
\cite{Pa} is by constructing a
(mirror) coupling between the \abbr{rbm} $\widehat{X}$ on
$\mathbb{D}_2$ and the \abbr{rbm} $\widehat{Y}$ on $\mathbb{D}_1$, such that
$|\widehat{X}_s| \le |\widehat{Y}_s|$ for all $s \ge 0$
and any common starting point $x \in \mathbb{D}_2$.
We use it here for $\mathbb{D}_2=\mathbb{K} \subseteq \mathbb{B}_c = \mathbb{D}_1$ and note
that the monotonicity of the radial component
under this coupling extends to the \abbr{tcrbm}-s
$X_s$ (under $\mathbb{Q}$), and $Y_s$ (under $\mathbb{H}$),
thereby assuring that $Y \in A^{(f)}$, $\mathbb{H}$-a.s.
implies $X \in A^{(f)}$, $\mathbb{Q}$-a.s.
\noindent
{\bf Step VI.} We proceed to show that the conclusion of Step V holds
in case $t \mapsto f(t)$ has jumps $\mathbb{D}elta_j>0$ at
isolated
jump points
$t_1 <\cdots<t_j<\cdots$.
That is, $f(t)=f_c(t)+f_d(t)$ with a $C^3$-smooth function
$f_c(\cdot)$ and piecewise constant
$f_d(t)=\sum_{j} \mathbb{D}elta_j \mathbb{I}_{t \ge t_j}$.
Setting $t_0=0$ and re-using the notations of Step V, upon applying Ito's formula we get that $X_t$ (and $Y_t$)
solve the corresponding deterministic Skorohod problem
(\ref{skorohoda})-(\ref{skorohodb}) within each interval $[t_{i-1},t_i)$,
and $B_t$ is again defined via (\ref{eq:Bt-def}) except
for $f_c'(t)$ replacing $f'(t)$.
In addition, $X_{t_i}=\eta_i X_{t_i^-}$ and $Y_{t_i}=\eta_i Y_{t_i^-}$
for $i=1,2,\ldots$, where $\eta_i=f(t_i^-)/f(t_i)<1$. Since
$\int_0^\infty f_c'(s)^2 ds$ is finite, as in Step V
we have measures $\mathbb{Q}$ and
$\mathbb{H}$, both equivalent to $\mathbb{P}$, under which
within each interval $[t_{i-1},t_i]$ the processes
$X_t$ and $Y_t$ are \abbr{tcrbm}-s on $\mathbb{K}$ and
$\mathbb{B}_c$, respectively, for the same time change $\tildeau(\cdot)$. With $J_f=\infty$, we already saw in
Step IV that $\mathbb{P}(W'_\cdot \in A)=1$. Following
the argument of Step V this would yield that $\mathbb{P}(W_\cdot \in A)=1$, provided we suitably extend the scope
of the implication (a). That is, suffices to show the
existence of coupling between \abbr{rbm}-s
$\widehat{X}$ on $\mathbb{K}$ and $\widehat{Y}$ on $\mathbb{B}_c$,
such that $|\widehat{X}_s| \le |\widehat{Y}_s|$
for all $s \ge 0$, in the setting where
at a sequence of isolated times $s_i=\tildeau(t_i)$
one applies the common shrinkage by $\eta_i \in (0,1)$
to both $\widehat{X}_{\cdot}$ and
$\widehat{Y}_{\cdot}$. To achieve this, starting at
$\widehat{Y}_0=\widehat{Y}'_0=\widehat{X}_0=x$,
we produce inductively for $i=0,1,\ldots$ another copy
$\{\widehat{Y}'_s : s \in [s_i,s_{i+1})\}$
of the \abbr{rbm} on $\mathbb{B}_c$, with jumps
from $\widehat{Y}'_{s_i^-}$
to $\widehat{Y}'_{s_i}=\widehat{X}_{s_i}$
and a coupling such that $|\widehat{X}_s| \le |\widehat{Y}'_s| \le |\widehat{Y}_s|$ for all $s$.
Indeed, as explained in Step V, employing
\cite[Theorem 5.4]{Pa} separately within each interval $[s_i,s_{i+1})$ yields a (mirror) coupling
of $\widehat{Y}^{'}$ and $\widehat{X}$
that maintains the stated relation
$|\widehat{X}_{s}| \le |\widehat{Y}'_s|$.
Further, applying part (b) of
Lemma \ref{coupling} inductively in $i \ge 0$, we
couple $\widehat{Y}'_s$ and
$\widehat{Y}_s$ within each interval $[s_i,s_{i+1})$,
such that $|\widehat{Y}'_s| \le |\widehat{Y}_s|$
for all $s \ge 0$, provided
$|\widehat{Y}'_{s_i}| \le |\widehat{Y}_{s_i}|$
for all $i \ge 0$. Starting at $\widehat{Y}_0=\widehat{Y}'_0$, we have the latter
inequality at $i=0$. Then, for $i \ge 1$ we have
by induction, upon utilizing our coupling on $[s_{i-1},s_{i})$ that
$|\widehat{X}_{s_i^-}| \le |\widehat{Y}'_{s_i^{-}}|\le|\widehat{Y}_{s_i^{-}}|$. Hence
$|\widehat{Y}'_{s_i}| = |\widehat{X}_{s_i}| \le |\widehat{Y}_{s_i}|$ (after the common shrinkage
by factor $\eta_i$), as needed for concluding the proof.
\qed
\section{Proof of Theorem \ref{rwthm}}\leftarrowbel{sec3}
Hereafter we denote the inner boundary of a discrete set $\mathbb{G}$ by $\partial \mathbb{G}$ and
fix $\mathbb{K}$ from the collection
$\mathcal{K}$ of \eqref{eq:cK-def}, scaled by
a constant factor so as to have
$\mathbb{K} \supseteq \mathbb{B}_2$ and hence
$(\mathbb{B}_a\cap\mathbb{Z}^d) \cap\partial(a\mathbb{K}\cap\mathbb{Z}^d)=\varnothing$
for all $a \ge a_d$ large enough. We then have the
following \abbr{srw} analog of
Lemma \ref{exptail}.
\begin{lem}\leftarrowbel{rwtail}
Let $\mathbb{P}_x$ denote the law of \abbr{srw}
$\{Z_t, t \ge 0\}$
on $a\mathbb{K}\cap\mathbb{Z}^d$, $d \ge 3$,
starting at $Z_0=x \in \mathbb{Z}^d$. Considering
the stopping times
$\tildeau(a):=\inf\{s\ge0: Z_s\in \mathbb{B}_a^c\}$ and $\sigma(a,r):=\inf\{s\ge0: Z_s\in \overline{\mathbb{B}}_r\}$,
there exists $C=C_d(\ delta)>0$ and $a_d=a_d(\ delta)<\infty$ such that for any $t,\ delta>0$, $a \ge a_d$, $\fracrac{r}{a}\in[\ delta,1)$,
\begin{align}
\sup_{x\in \mathbb{B}_a}\mathbb{P}_{x}(\tildeau(a)>ta^2)&<C^{-1}e^{-Ct}\,,
\leftarrowbel{2.6a}\\
\sup_{x\in a\mathbb{K}\backslash \mathbb{B}_r}\mathbb{P}_{x}(\sigma(a,r)>ta^2)&<C^{-1}e^{-Ct}\,,\leftarrowbel{2.6b}\\
\inf_{x \in \overline{\mathbb{B}}_{a/2}}\,
\mathbb{P}_x(\tildeau(a) > a^2) & > C \,. \leftarrowbel{2.6c}
\end{align}
\end{lem}
In proving Lemma \ref{rwtail} we rely on
the following invariance principle in
bounded uniform domains, which allows us to transform hitting probabilities of \abbr{srw} to the corresponding probabilities for an \abbr{rbm}.
\begin{lem}\leftarrowbel{invariance}\cite{BC,CCK}
Fix a bounded uniform domain
$\mathbb{D}\subseteq\mathbb{R}^d$ and
let $Y_t^n:=n^{-1}Y_{\lfloor n^2t\rfloor}$
denote the \abbr{srw}
on $\mathbb{D}\cap(n^{-1}\mathbb{Z})^d$, induced by
the discrete-time \abbr{srw} $\{Y_t\}$
on $n\mathbb{D}\cap\mathbb{Z}^d$.
If $Y_0^n=x_n \rightarrow x\in\overline{\mathbb{D}}$, then
$\{Y_t^n;t\ge0\}$ converges weakly in $D([0,\infty),\overline{\mathbb{D}})$
to $\{W_{\kappa t};t\ge0\}$, where $W$ is the \abbr{rbm}
on $\overline{\mathbb{D}}$ starting from $x$, time changed by constant $\kappa$.
\end{lem}
\noindent
{\emph{Proof:}} Lemma \ref{invariance} merely adapts
facts from \cite[Theorem 3.17 and Section 4.2]{CCK}
to our context (alternatively, it also follows by strengthening \cite[Theorem 3.6]{BC} as
suggested in \cite[Remark 3.7]{BC}).
The original result presented in \cite{CCK}
is for variable-speed and constant-speed random walks (\abbr{vsrw},\abbr{csrw}) on bounded uniform domain with random conductances uniformly bounded up and below. We
are in a special case where all edges in $n\mathbb{D} \cap \mathbb{Z}^d$
are present and have equal non-random conductance.
Hence, here the \abbr{csrw} is merely a continuous-time \abbr{srw} $Z_t$ of unit jump rate on $n \mathbb{D} \cap \mathbb{Z}^d$
and further
the invariance principle holds for
$Z_t^n := n^{-1} Z_{n^2 t}$ and
{\em{any}} choice of $x\in\overline{\mathbb{D}}$.
Indeed, while \abbr{rbm}
$W_t$ constructed via Dirichlet
forms is typically well defined only for
a quasi-everywhere starting point
in $\overline{\mathbb{D}}$, here this can be refined
to every starting point. This is because
in a uniform domain, such \abbr{rbm}
admits a jointly-continuous transition density $p(t,x,y)$ on $\mathbb{R}_{+}\tildeimes\overline{\mathbb{D}}\tildeimes\overline{\mathbb{D}}$
of Aronson's type (see \cite[Theorem 3.10]{GS}),
thereby eliminating the exceptional set in
\cite[Theorem 4.5.4]{FOT}.
It remains only to infer the invariance principle for
the discrete-time \abbr{srw} $\{Y^n_t\}$ out of the
invariance principle for $\{Z^n_t\}$. To this end,
recall the representation $Y^n_t=Z^n_{n^{-2} L(n^2 t)}$
for $L(t):=\inf\{s\ge0: N(s)=\lfloor t\rfloor\}$ and
the independent Poisson process $N(t)$ of
intensity one. Now, fixing $T$ finite,
by the functional strong
law of large numbers for Poisson processes,
$$
\sup_{t \in [0,T]} |n^{-2} L(n^2 t) -t| \stackrel{a.s.}{\rightarrow} 0 \,, \quad \tildeextrm{for} \quad n \tildeo \infty \,.
$$
Further, by \cite[Proposition 3.10 and Section 4.2]{CCK}, for any $r>0$,
$$
\lim_{\ delta\tildeo0}\limsup_{n\tildeo\infty}\mathbb{P}_{nx_n}\big(\sup_{|s_1-s_2|\le\ delta, s_i\le T}|Z_{s_2}^n-Z_{s_1}^n|>r\big)=0.
$$
Hence,
\begin{align*}
\sup_{0\le t\le T} |Y_t^n-Z_t^n|=\sup_{0\le t\le T}|Z_{n^{-2} L(n^2 t)}^n-Z_t^n|\stackrel{p}{\rightarrow} 0 \,.
\end{align*}
and it follows that $(Y_t^n;t\ge0)\stackrel{d}{\rightarrow}(W_{\kappa t};t\ge0)$ as $n\rightarrow\infty$.
\qed
\begin{remark}\leftarrowbel{gen-invariance}
Lemma \ref{invariance} generalizes to
$Y_t^{a_n}\stackrel{d}{\rightarrow}W_{\kappa t}$,
for $Y_t^{a_n}:={a_n}^{-1} Y_{\lfloor {a_n}^2t\rfloor}$
that is induced by the discrete-time \abbr{srw} on
$a_n \mathbb{D} \cap \mathbb{Z}^d$ and
any fixed $a_n \uparrow \infty$ (just note that the conditions
laid out in \cite[first paragraph, Page 13]{CCK} hold
with $a_n$ replacing $n$).
\end{remark}
\noindent
{\emph{Proof of Lemma \ref{rwtail}.}}\,
Consider the \abbr{rbm}
$W_{\cdot}$ on $\overline{\mathbb{K}} \supseteq \mathbb{B}_2$
and the rescaled discrete time \abbr{srw}
$Z_t^a:=a^{-1} Z_{\lfloor a^2 t \rfloor}$.
Starting with the proof of \eqref{2.6a}, for
$a>0$ and $y \in \overline{\mathbb{K}}$, let
\begin{align*}
q^{{\rm \abbr{rw}}} (a,y) &:= \mathbb{P}_y(Z_s^a \in \mathbb{B}_1, \; \fracorall s \le 1),
\qquad
m^{{\rm \abbr{rw}}}(a) := \sup_{y \in \mathbb{B}_1 \cap (a^{-1} \mathbb{Z})^d} \;
q^{{\rm \abbr{rw}}} (a,y) \,, \\
q^{{\rm \abbr{bm}}}(y) &:= \mathbb{P}_y (W_{\kappa s}\in \mathbb{B}_1, \; \fracorall s\le1),
\qquad
m^{{\rm \abbr{bm}}}:=\sup_{y\in\overline{\mathbb{B}}_1} \; q^{{\rm \abbr{bm}}}(y) \,.
\end{align*}
Then,
by the Markov property of the \abbr{srw}, for any
$a,t>0$ and $x \in \mathbb{B}_a \cap \mathbb{Z}^d$,
\begin{align}
\mathbb{P}_{x}(\tildeau(a)>ta^2)
&=
\mathbb{P}_x (Z_s\in \mathbb{B}_{a}, \; \fracorall s\le ta^2)
=\mathbb{P}_{a^{-1}x}(Z_s^{a}\in \mathbb{B}_1, \; \fracorall s\le t)\nonumber\\
&\le
\big[\sup_{y \in \mathbb{B}_1 \cap(a^{-1}\mathbb{Z})^d}q^{{\rm \abbr{rw}}}(a,y)\big]^{\lfloor{t}\rfloor}={m^{{\rm \abbr{rw}}}(a)}^{\lfloor{t}\rfloor} \,.
\leftarrowbel{notation}
\end{align}
An \abbr{rbm}
on uniform domain admits
jointly continuous, positive transition density (\cite[Theorem 3.10]{GS}), and in particular $m^{{\rm \abbr{bm}}}=1-2\eta$ for some $\eta \in (0,1/2)$.
As we show in the sequel, setting $\xi:=\fracrac{1-\eta}{1-2\eta}>1$,
\begin{equation}\leftarrowbel{eq:ad-finite}
a_d := \sup \{ a>0 : m^{{\rm \abbr{rw}}}(a) > \xi m^{{\rm \abbr{bm}}} \} \,,
\end{equation}
is finite. It then follows from
(\ref{notation}) that for some positive $C$, all $a > a_d$ and
$t>0$,
\begin{eqnarray*}
\sup_{x \in \mathbb{B}_a \cap \mathbb{Z}^d} \mathbb{P}_{x}(\tildeau(a)>ta^2)
\le {m^{{\rm \abbr{rw}}}(a)}^{\lfloor{t}\rfloor}
\le(\xi \, m^{{\rm \abbr{bm}}})^{\lfloor t\rfloor}=(1-\eta)^{\lfloor t\rfloor}
\le C^{-1} e^{-Ct} \,.
\end{eqnarray*}
To complete the proof of (\ref{2.6a}), suppose to the contrary
that $a_d=\infty$ in (\ref{eq:ad-finite}), namely
$m^{{\rm \abbr{rw}}}(a_l) > \xi m^{{\rm \abbr{bm}}}$ for some $a_l \uparrow \infty$.
Taking the uniformly bounded
$y_l \in \mathbb{B}_1 \cap (a_l^{-1} \mathbb{Z})^d$ such
that $q^{{\rm \abbr{rw}}}(a_l,y_l)=m^{{\rm \abbr{rw}}}(a_l)$, we pass to a sub-sequence
$\{l_n\}$ along which $y_{l_n} \tildeo x \in \overline{\mathbb{B}}_1$.
Then, considering Remark \ref{gen-invariance} for the sequence
$a_{l_n}$, we deduce that as $n \tildeo \infty$,
$$
m^{{\rm \abbr{rw}}}(a_{l_n}) = q^{{\rm \abbr{rw}}}(a_{l_n},y_{l_n}) \tildeo q^{{\rm \abbr{bm}}} (x) \le m^{{\rm \abbr{bm}}}
\,,
$$
in contradiction with our assumption
that $m^{{\rm \abbr{rw}}}(a_{l_n}) > \xi m^{{\rm \abbr{bm}}}$ for some $\xi>1$ and all $n$.
Likewise, whenever $x \in \overline{B}_{a/2} \cap \mathbb{Z}^d$ we have
that
\begin{align*}
\mathbb{P}_{x}(\tildeau(a)>a^2)=
\mathbb{P}_{a^{-1} x}(Z_s^a \in \mathbb{B}_1, \; \fracorall s \le 1)
\ge
\inf_{y\in \overline{\mathbb{B}}_{1/2} \cap(a^{-1} \mathbb{Z})^d} \; q^{rw}(a,y)
:= m_{{\rm \abbr{rw}}}(a)
\end{align*}
and by the same reasoning as before,
$$
\liminf_{a \tildeo \infty} m_{{\rm \abbr{rw}}}(a) \ge
\inf_{z \in \overline{\mathbb{B}}_{1/2}} \{ q^{{\rm \abbr{bm}}}(z) \} > 0 \,,
$$
yielding the bound (\ref{2.6c}). Next, fixing $\ delta>0$
we turn to the stopping time $\sigma(a,r)$ and set
\begin{align*}
q^{{\rm \abbr{rw}}}_\ast (a,y) &:=
\mathbb{P}_y(Z_s^a \notin \overline{\mathbb{B}}_\ delta, \; \fracorall s \le 1),
\qquad
m^{{\rm \abbr{rw}}}_\ast (a) := \sup_{y \in (\mathbb{K} \backslash \overline{\mathbb{B}}_\ delta) \cap (a^{-1} \mathbb{Z})^d} \;
q^{{\rm \abbr{rw}}}_\ast (a,y) \,, \\
q^{{\rm \abbr{bm}}}_\ast (y) &:= \mathbb{P}_y (W_{\kappa s} \notin \overline{\mathbb{B}}_\ delta, \;
\fracorall s\le 1),
\qquad
m^{{\rm \abbr{bm}}}_\ast :=\sup_{y\in\overline{\mathbb{K}} \backslash \mathbb{B}_\ delta} \; q^{{\rm \abbr{bm}}}_\ast (y) \,,
\end{align*}
getting by Markov property of the \abbr{srw}
that for any $a,t>0$, $r/a \in [\ delta,1)$ and
$x \in (a \mathbb{K} \backslash \mathbb{B}_r) \cap \mathbb{Z}^d$
\begin{align}
\mathbb{P}_x(\sigma(a,r)>ta^2)&=\mathbb{P}_{x}(Z_s\notin
\overline{\mathbb{B}}_r, \; \fracorall s\le ta^2)
\leq
\mathbb{P}_{a^{-1} x}(Z_s^a \notin \overline{\mathbb{B}}_\ delta,
\;\; \fracorall s\le t)\nonumber\\
&\le
\big[\sup_{y\in(\mathbb{K}\backslash \overline{\mathbb{B}}_\ delta)\cap(a^{-1}\mathbb{Z})^d}
q^{{\rm \abbr{rw}}}_\ast (a,y)\big]^{\lfloor{t}\rfloor}={
m^{{\rm \abbr{rw}}}_\ast (a)}^{\lfloor{t}\rfloor}\,.
\leftarrowbel{notation2}
\end{align}
By the same arguments as in case of (\ref{notation}),
again $m^{{\rm \abbr{bm}}}_\ast = 1 - 2\eta$ for some $\eta \in (0,1/2)$, and in view of
Remark \ref{gen-invariance} the corresponding constant
$a_d$ as in (\ref{eq:ad-finite}), is finite, with
(\ref{notation2}) thus yielding (\ref{2.6b}).
\qed
Equipped with Lemma \ref{rwtail}
we can now establish the following
\abbr{srw} analog of Lemma \ref{estimates}.
\begin{lem}\leftarrowbel{rw_estimates}
Let $\mathbb{P}_x$ denotes the law of \abbr{srw} $\{Z_t, t \ge 0\}$
on $a\mathbb{K}\cap\mathbb{Z}^d$, starting at $Z_0 = x \in \mathbb{Z}^d$.
\newline
(a). For $\ delta \in (0,1/2)$, there exist $C = C_d (\ delta)>0$ and $M_d=M_d(\ delta)$ finite, such that for all $M \geq M_d$,
and any $T \geq M a^2 \log a$, $a-M \geq r \geq a \ delta$,
\begin{align}\leftarrowbel{lower_bound}
\inf_{x \in r\mathbb{K}}\mathbb{P}_x(\exists s\le T: |Z_s|=0)& \ge C^{-1} \big[ \fracrac{ T}{a^d}\wedge1\big],\\
\sup_{x \in \partial (r\mathbb{K}\cap\mathbb{Z}^d)}\mathbb{P}_x(\exists s\le T: |Z_s|=0)&\le C \big[ \fracrac{ T}{a^d}\wedge1\big], \leftarrowbel{upper_bound}
\end{align}
\noindent
(b). The uniform bound \eqref{upper_bound} applies for \abbr{srw}
$\{Z_t\}$ on growing domains $\widetilde{\mathbb{D}}_t \supseteq
\mathbb{B}_{a+1} \cap \mathbb{Z}^d$, starting at arbitrary
$Z_0 \in \mathbb{B}_{a \ delta}^c$.
\end{lem}
\begin{proof}
(a). We adapt the proof of Lemma \ref{estimates} to
the current setting of discrete time \abbr{srw}
$Z_t$ on $a \mathbb{K} \cap \mathbb{Z}^d$, by taking throughout
$\epsilon=0$ and re-defining the excursions of length
$L_k:=\sigma_{k+1}-\sigma_k$, $k \ge 0$, to be
determined now by the stopping times $\sigma_0=0$ and
\begin{align*}
\tildeau_k&:=\inf\{t\ge\sigma_{k-1}: Z_t\in \mathbb{B}_{a}^c\},\quad k\ge1
\\
\sigma_k&:=\inf\{t\ge\tildeau_k : Z_t \in \overline{\mathbb{B}}_{a/2}\}\,.\nonumber
\end{align*}
Since the laws of increments of \abbr{srw} are not invariant to rotations, $x \mapsto m(\tildeheta,x)=\mathbb{E}_x[e^{\tildeheta L_0}]$
is not a radial function. However, replacing
Lemma \ref{exptail} (which we used when
bounding $m(\tildeheta,x)$
in case of Brownian motion), by the {\em universal} bounds of Lemma \ref{rwtail}, yields (\ref{eq:upp-tail-mgf}) and
(\ref{eq:low-tail-mgf}) for the \abbr{srw} case
considered here.
Thereby, applying the discrete analogue of (\ref{eq:beps})
\begin{align}
b(x):=\mathbb{P}_x(\inf_{t \le \tildeau_1} |Z_t|=0) = c_d(|x|^{2-d}-a^{2-d})+O(|x|^{1-d}) \,, \leftarrowbel{eq:beps2}
\end{align}
where $0<c_d<\infty$ is a dimensional constant
(see \cite[Proposition 1.5.9]{La}),
at $x \in \partial (r\mathbb{K} \cap \mathbb{Z}^d)$ and $x \in \partial (\overline{\mathbb{B}}_{a/2} \cap \mathbb{Z}^d)$, yields the
\abbr{srw} analog of \eqref{eq:dom-term}, out of which
the stated conclusions follow.
\newline
(b). Let $I_k:=[\sigma_k,\tildeau_{k+1})$, $k \ge 0$.
Our assumptions that $Z_0 \in \mathbb{B}^c_{a \ delta}$
and $\widetilde{\mathbb{D}}_t
\supseteq \mathbb{B}_{a+1} \cap \mathbb{Z}^d$ result in
$\{Z_t, t \in I_k\}$ having for each $k \ge 0$ the same
conditional law given $Z_{\sigma_k}$, as in part (a).
Since the event $|Z_t|=0$ can only occur for
$t \in \cup_k I_k$, the derivation leading to
the \abbr{srw} analog of \eqref{eq:dom-term} applies
here as well. Further, conditional
on $Z_{\sigma_k}=x$, each $L_k$, $k \ge 1$,
stochastically dominates the random variable
$\tildeau(a)$ of Lemma \ref{rwtail} starting at
same point $x$. Consequently
$$
\mathbb{E}_x[e^{-\widehat{\tildeheta} L_k/a^2}]\le\mathbb{E}_x[e^{-
\widehat{\tildeheta} \tildeau(a)/a^2}]\,,
$$
and utilizing the uniform in $x$ and $a$
control on the r.h.s. due to \eqref{2.6a},
establishes yet again the analog of \eqref{eq:low-tail-mgf}.
Examining the proof
of \eqref{2.4} in Lemma \ref{estimates} we see that
this suffices for re-producing the corresponding
uniform upper bound \eqref{upper_bound}.
\end{proof}
\noindent
{\emph{Proof of Theorem \ref{rwthm}}}.
(a). Fix $f(t)$ such that $J_f<\infty$
and consider the \abbr{srw} $\{Y_t\}$
on $\mathbb{D}_t \subseteq \mathbb{Z}^d$, $d \ge 3$
for which Assumption \ref{ass-1} holds.
Similarly to Step II of the proof of Theorem \ref{bmthm},
for $a_l:=(c+1)^l$, $l \ge 1$, define
\begin{align*}
t_l:=&\inf\{s\ge 1: \mathbb{D}_s\cap \mathbb{B}_{a_{l+1}}^c\neq\varnothing\},\\
\tildeau_l:=&\inf\{s\ge0: Y_s\in \mathbb{B}_{a_l}^c\},\\
\widetilde{\mathbb{G}amma}_l:=&\{\exists t\in[\tildeau_{l},\tildeau_{l+1}): Y_t=0\}\,.
\end{align*}
With $f(\cdot)$ unbounded, for any $l$ eventually
$\mathbb{D}_s \supseteq \mathbb{B}_{a_l} \cap \mathbb{Z}^d$ and
by the transience of the \abbr{srw} on $\mathbb{Z}^d$,
necessarily $\tildeau_l$ are a.s. finite.
Thus, by Borel-Cantelli I,
\begin{equation}\leftarrowbel{eq:summable}
\sum_l\mathbb{P}(\widetilde{\mathbb{G}amma}_l)<\infty \quad \mathbb{R}ightarrow \quad
\mathbb{P}_0(Y_t=0\;\; f.o.\,)=1 \,.
\end{equation}
Turning to bound $\mathbb{P}(\widetilde{\mathbb{G}amma}_l)$, note
that $\tildeau_{l+1} \ge t_l$ and
$\mathbb{D}_{t_l} \supseteq \mathbb{B}_{1+a_{l}}\cap\mathbb{Z}^d$
(by Assumption \ref{ass-1} and the choice of $a_l$).
Hence, by (\ref{2.6a}), we have that for some constants
$C_d>0$ and $l_d<\infty$, all $l \ge l_d$ and $t \ge 0$,
\begin{align}
\mathbb{P}(\tildeau_{l+1}-t_l>t a_l^2)<C_d^{-1}e^{-C_d t}. \leftarrowbel{length_control}
\end{align}
Let $\mathbb{D}elta T_l := (t_l-t_{l-1})$ and for
$\ delta=1/(c+1)<1/2$ and $M_d=M_d(\ delta)$
of Lemma \ref{rw_estimates}, set $T_l^\ast := M_d \log a_l$ and $T_l = \mathbb{D}elta T_l + T_l^\ast a_l^2$. Since
$\tildeau_l \ge t_{l-1}$ the length of
$[\tildeau_{l}, \tildeau_{l+1})$ is at
most $\mathbb{D}elta T_l$ plus the length of
$[t_{l}, \tildeau_{l+1})$, which by (\ref{length_control})
is with high probability under $T_l^\ast a_l^2$.
Further, $\mathbb{D}_{\tildeau_l} \supseteq \mathbb{D}_{t_{l-1}} \supseteq \mathbb{B}_{1+a_{l-1}} \cap \mathbb{Z}^d$ and $Y_{\tildeau_l} \in \mathbb{B}_{a_l}^c$,
hence from part (b) of Lemma \ref{rw_estimates} we have that,
\begin{align}
\mathbb{P}(\widetilde{\mathbb{G}amma}_l) &\le
\mathbb{P}(\tildeau_{l+1} - t_l > T_l^\ast a_l^2) +
\mathbb{P} (\exists s\in[\tildeau_l,\tildeau_l+T_l]: Y_s=0) \nonumber \\
&\le C_d^{-1}e^{-C _dT_l^\ast} + C a_{l-1}^{-d} T_l \,.
\leftarrowbel{2.23}
\end{align}
With our choice of $a_l$ growing exponentially in $l$, the
terms $e^{-C_d T_l^\ast}$ and $a_l^{2-d} T_l^\ast$
in the bound (\ref{2.23}) are summable over $l \in \mathbb{N}$.
Hence, the left-hand-side of \eqref{eq:summable} is finite
whenever $\sum_l a_{l-1}^{-d} \mathbb{D}elta T_l$ is finite.
Further, Assumption \ref{ass-1} and our definition of
$t_l$ imply that $f(t_l-1) \le 1+a_{l+1}$. Thus,
$$
J_f \ge \sum_{l \ge 2} f(t_l-1)^{-d} \mathbb{D}elta T_l
\ge (1+c)^{-3d} \sum_{l \ge 2} a_{l-1}^{-d} \mathbb{D}elta T_l \,.
$$
Consequently, finite $J_f$ results in
$\mathbb{P}_0(Y_t=0\;\;$ f.o.$)=1$, which by
Proposition \ref{equiv} extends to $\mathbb{P}_0(Y_t=y\;\;$ f.o.$)=1$
for all $y \in \mathbb{Z}^d$, as claimed.
\noindent
(b). Fix $f \in \mathcal{F}_\ast$ such that $J_f=\infty$
and $\mathbb{K} \in \mathcal{K}$. Since $J_{f/r}=\infty$ for
any $r>0$ and $\mathbb{D}_t = (f(t)/r) (r \mathbb{K}) \cap \mathbb{Z}^d$,
taking $r$ large enough we have with no loss of
generality that $\mathbb{K} \supseteq \mathbb{B}_2$. Then, considering
the \abbr{srw} on $\mathbb{D}_t$,
upon replacing (\ref{2.3}) by
(\ref{lower_bound}), the argument we have used in
Step I of the proof of Theorem \ref{bmthm}
applies here
as well, apart from the obvious notational changes
(of replacing $\mathbb{B}_{a_l}$ and $\overline{\mathbb{B}}_{a_{l-1}}$ in
(\ref{eq:wt}) by $a_l \mathbb{K} \cap \mathbb{Z}^d$
and the collection of all $x \in \mathbb{Z}^d$ within
distance one of $a_{l-1} \mathbb{K}$, respectively).
\qed
\noindent
\section{On recurrence probability independence of target states}\leftarrowbel{appen}
The following, $xy$-recurrence property,
generalizes Definition \ref{rec} to arbitrary
starting and target locations,
$x,y \in \mathbb{R}^d$, respectively.
\begin{defn}\leftarrowbel{xy_rec}
Suppose $\mathbb{D}_t\uparrow\mathbb{R}^d$, $x\in \mathbb{D}_0$, $y\in\mathbb{R}^d$. The sample path $x_t$ of a stochastic process $t\mapsto x_t \in \mathbb{D}_t$ is $xy$-recurrent if
$x_0=x$ and the event $A(y):=\cap_{\epsilon>0}A_\epsilon(y)$ occurs, where
\begin{eqnarray*}
\sigma_\epsilon^{(0)} &:=&\inf\{t\ge0: \mathbb{D}_t\supseteq \mathbb{B}_\epsilon+x\},\;\;\\
\tildeau_\epsilon^{(i)}&:=&\inf\{t\ge\sigma_\epsilon^{(i-1)}: |x_t-y|<\epsilon\},\;\; i\ge1\\ \sigma_\epsilon^{(i)}&:=&\inf\{t\ge\tildeau_\epsilon^{(i)}: |x_t-y|>1/2\},\\
A_\epsilon(y)&:=&\{\tildeau_\epsilon^{(i)}<\infty, \fracorall i\}.
\end{eqnarray*}\end{defn}
\begin{ppn}\leftarrowbel{equiv}
Suppose $\{X_t\}$ is a \abbr{srw}
on $\mathbb{D}_t\uparrow \mathbb{Z}^d$,
or alternatively that $(X_t,\mathbb{D}_t)$
is the \abbr{RBMG} of Definition \ref{rbmg}
with $\mathbb{D}_0$ open connected set and
$\mathbb{D}_t \uparrow \mathbb{R}^d$.
Then, the probability $q_{xy}$ of $xy$-recurrence
does not depend on $y$.
In case of \abbr{rbmg}, if $q_{zy} \in \{0,1\}$
for some $z \in \mathbb{D}_0$ then $q_{xy}=q_{zy}$
for all $x \in \mathbb{D}_0$, whereas in case of \abbr{srw}, if
$q_{zy}=0$ for some $z \in \mathbb{D}_0$ then
$q_{xy}=0$ whenever $\|x-z\|_1$ is even.
\end{ppn}
\begin{remark} Adapting the approach we use for the
\abbr{rbmg}, it is not hard to show that for
{\em{continuous time}} \abbr{srw} (on growing domains
$\mathbb{D}_t \uparrow \mathbb{Z}^d$), having $q_{zz} \in \{0,1\}$
for some $z \in \mathbb{D}_0$ results in $q_{xy}=q_{zz}$
for all $x \in \mathbb{D}_0$ and $y \in \mathbb{Z}^d$. This approach is based
on the equivalence of hitting measures of suitable sets
when starting the process at nearby initial states.
This however does not apply for discrete time \abbr{srw},
hence our limited conclusion in that case.
\end{remark}
\begin{proof} This proof consists of the following
four steps. Starting with the \abbr{SRW} we
show in \tildeextsc{Step I}
that
$q_{xy}$
does not depend on $y$, then for
$x,z\in \mathbb{D}_0$ with $\|x-z\|_1$ even, we prove
in \tildeextsc{Step II}
that $q_{zz}=0$ implies $q_{xx}=0$.
In case of the \abbr{RBMG} we have that
$q^\ep_{xy} := \mathbb{P}_x(A_\ep(y)) \ downarrow q_{xy}$
and deduce the stated claims upon
showing in \tildeextsc{Step III} that if $q_{zz} \in \{0,1\}$
then $q_{xz}=q_{zz}$ for any $x,z \in \mathbb{D}_0$, then conclude
in \tildeextsc{Step IV} that $q^\ep_{xy}=q^\ep_{xx}$
for any fixed $\ep>0$ and all $y \in \mathbb{R}^d$ (even when $0<q_{xx}<1$).
\noindent
{\bf Step I}.
For the
\abbr{SRW} $X_t$ on $\mathbb{D}_t \subseteq \mathbb{Z}^d$
and fixed $s \in \mathbb{N}$ we denote by $\mathbb{P}_x^s(\cdot)$
the law of \abbr{srw} $X_t$ on the shifted-domains
$\mathbb{D}_{t+s}$ starting at $X_0=x$.
Then, for any $x,y \in \mathbb{Z}^d$ and $s \ge 0$,
$$
q_{xy}(s):=\mathbb{P}(X_t=y \; i.o. \; | \; X_s=x)=\mathbb{P}^s_x(X_t=y \; i.o.)\,,
$$
with $q_{xy}:=q_{xy}(0)$. Since $\mathbb{D}_t \uparrow \mathbb{Z}^d$,
clearly any $y,w \in \mathbb{Z}^d$ are also in
$\mathbb{D}_t$ provided $t \ge t_0(y,w)$ is large enough,
with some non-self-intersecting path in
$\mathbb{D}_{t_0}$ connecting $y$ and $w$.
Setting $\mathcal{F}_t^X:=\sigma\{X_s,s\le t\}\uparrow\mathcal{F}_\infty$ and events
$\mathbb{G}amma_{s,t,z,w}:=\{X_s=z, X_u=w$ some $u>t\}$,
we thus have $\eta=\eta(y,w)>0$ such that for any
starting point $x$, all $z,s$ and $t \ge t_0 \vee s$,
$$
\mathbb{P}_x(\mathbb{G}amma_{s,t,z,w}|{\mathcal F}^X_t)
\ge\eta\mathbb{I}_{\{X_s=z, X_t=y\}} \,.
$$
Further as $t \tildeo \infty$ we have that
$$
\mathbb{G}amma_{s,t,z,w} \ downarrow \mathbb{G}amma_{s,z,w} :=
\{X_s=z \tildeextrm{ and } X_u=w \; i.o. \tildeextrm{ in } u\} \,.
$$
Clearly, $\mathbb{G}amma_{s,z,w} \in \mathcal{F}_\infty$ so it follows by
L\'{e}vy's upward theorem (and dominated convergence,
see \cite[Theorem 5.5.9]{Du}),
that for any $x$, a.s.
\begin{eqnarray*}
\mathbb{I}_{\mathbb{G}amma_{s,z,w}}=\mathbb{P}_x(\mathbb{G}amma_{s,z,w}|\mathcal{F}_\infty)= \lim_{t \tildeo \infty} \mathbb{P}_x(\mathbb{G}amma_{s,t,z,w}|\mathcal{F}_t^X)
\ge\eta\limsup_{t\rightarrow\infty}
\mathbb{I}_{\{X_s=z,X_t=y\}}=
\eta\mathbb{I}_{\mathbb{G}amma_{s,z,y}}\,.
\end{eqnarray*}
The same applies with the roles of $y$ and $w$
exchanged and consequently,
a.s. $\mathbb{G}amma_{s,z,y}=\mathbb{G}amma_{s,z,w}$
for all $z,y,w \in \mathbb{Z}^d$ and $s \ge 0$.
In particular, $q_{zy}(s) = \mathbb{P}(\mathbb{G}amma_{s,z,y}| X_s=z)$
is thus independent of $y$, for any $z$ and $s \ge 0$.
\noindent
{\bf Step II}.
Assuming now that $q_{zz}=q_{zz}(0)=0$ for some $z \in \mathbb{D}_0$,
we have from Step I that $q_{zx}=0$. As explained before (in Step I),
$s_0 := \inf\{t : \mathbb{P}_z(X_{t}=x)>0\}$ is a finite integer
and clearly $\mathbb{P}_z(X_{2s+s_0}=x)>0$ for any $s \ge 0$. By
the Markov property at time $2s+s_0$,
$$
0 = q_{zx} \ge \mathbb{P}_z(X_{2s+s_0}=x, X_t=x \;\; i.o.)
= \mathbb{P}_z(X_{2s+s_0}=x) q_{xx}(2s+s_0) \,.
$$
Consequently, for any $s \ge 0$,
$$
\mathbb{P}_x(X_{2s+s_0}=x, X_t=x\;\; i.o.) = \mathbb{P}_x(X_{2s+s_0}=x) q_{xx}(2s+s_0) = 0 \,.
$$
Starting at $X_0=x$, the event $\{X_t=x\}$ is possible only at $t$ even.
Since $\|x-z\|_1$ is even, so is the value of $s_0$ and from the
preceding we know that $\mathbb{P}_x$-a.s. any visit of $x$ at even integer larger than
$s_0$ results in only finitely many visits to $x$. Since there can be
only finitely many visits of $x$ up to time $s_0$, we conclude that
$q_{xx}=0$.
\noindent
{\bf Step III.}
Dealing hereafter with the \abbr{RBMG}, recall that
$A_\ep(y) \ downarrow A(y)$ for
$A_\ep(y)=\{\exists s_k, u_k\uparrow\infty:
|X_{s_k}-y|<\epsilon, |X_{u_k}-y|>1/2, u_k\in(s_k,s_{k+1})\}$.
Let $\mathbb{P}_x^s(\cdot)$ stand for the law of the
\abbr{RBMG} $\{X_t\}$ on shifted-domains $\mathbb{D}_{t+s}$
starting at $X_0=x$, and
$q^\ep_{xy}(s):=\mathbb{P}^s_x(A_\ep(y))$ with $q^\ep_{xy}=q^\ep_{xy}(0)$,
so that $q^\ep_{xy} \ downarrow q^0_{xy} = q_{xy}$ when $\ep \ downarrow 0$.
We first prove that if $q_{zz} \in \{0,1\}$ for some $z \in \mathbb{D}_0$
then $q_{xz}=q_{zz}$ for any $x\in\mathbb{D}_0$ such that
$\fracrac{x+z}{2}+\mathbb{B}_\alpha\subseteq\mathbb{D}_0$
for some $\alpha > |x-z|/2$. Indeed, with
$\mathbb{P}^{x,\alpha}$ denoting the joint law of
$(X_{\tildeau_\alpha},\tildeau_\alpha)$ for the first exit time
$\tildeau_\alpha:=\inf\{s\ge0: X_s\notin \fracrac{x+z}{2}+\mathbb{B}_\alpha\}$
and $X_0=x$, we have that
\begin{align}\leftarrowbel{eq:xz-iden}
q^\ep_{xz} = \int q^\ep_{x'z}(\gamma)d\mathbb{P}^{x,\alpha} (x',\gamma) \,,
\end{align}
for any fixed $\ep >0$. By dominated convergence this
identity extends to $\ep=0$ and considering
it for $x=z$ (and $\ep=0)$, we deduce that
$q_{x'z}(\gamma)=q_{zz} \in \{0,1\}$
for $\mathbb{P}^{z,\alpha}$-a.e. $(x',\gamma)$. By our assumption
about the points $x$ and $z$, the measure
$\mathbb{P}^{z,\alpha}$ is merely the joint law of
exit position and time for $\fracrac{x+z}{2}+ \mathbb{B}_\alpha$
and Brownian motion $X_s$ starting at $z$ and as such
it has a continuous Radon-Nikodym
density with respect to the product of the uniform surface measure
$\omega_{d-1}$ on $\partial(\fracrac{x+z}{2}+\mathbb{B}_\alpha)$
and the Lebesgue measure on $(0,\infty)$ (for example, see
\cite[Theorem 1 and 3]{Hs}). Further, the latter
density is strictly positive due to the continuity of
(killed) Brownian transition kernel. Since the same applies
to the corresponding Radon-Nikodym density
between $\mathbb{P}^{x,\alpha}$ and $d \omega_{d-1} \tildeimes dt$,
we conclude that $\mathbb{P}^{z,\alpha}$
and $\mathbb{P}^{x,\alpha}$ are mutually equivalent measures.
In particular, $q_{x'z}(\gamma)=q_{zz}$ also
for $\mathbb{P}^{x,\alpha}$-a.e. $(x',\gamma)$ and hence
it follows from (\ref{eq:xz-iden}) at $\ep=0$,
that $q_{xz}= q_{zz}$.
Now, since $\mathbb{D}_0$ is an open connected subset of $\mathbb{R}^d$,
any $x,z \in\mathbb{D}_0$ are connected by a continuous path
$w : [0,1] \tildeo \mathbb{D}_0$ such that
dist$(w(\cdot),\mathbb{D}_0^c)>0$. Consequently, there exists a
finite sequence of points $\{w_k\}_{k=0}^K \subseteq \mathbb{D}_0$
with $w_0=z$, $w_K=x$ and $\fracrac{w_{k-1}+w_k}{2}+
\mathbb{B}_{\alpha_k} \subseteq \mathbb{D}_0$, for
$\alpha_k > |w_k-w_{k-1}|/2$ and all $1 \le k \le K$.
Applying iteratively the preceding argument, we conclude that
if $q_{zz} \in \{0,1\}$ then
$q_{zz}=q_{w_1z}=\cdots=q_{w_{K-1}z}=q_{xz}$, as claimed.
\noindent
{\bf Step IV}.
Next, fixing $\ep>0$ and $x \in \mathbb{D}_0$ we proceed to show that
$q^\ep_{xz}=q^\ep_{xy}$ for any $z,y \in \mathbb{R}^d$. To this end,
let $t_0=t_0(y,z)$ be large enough so that $\mathbb{D}_{t_0}$
contains the compact set
$$
\mathbb{K} := \{w : \inf_{\leftarrowmbda \in [0,1]}
|w - \leftarrowmbda z - (1-\leftarrowmbda) y| \le 1 \} \,,
$$
set $\mathcal{F}_t^X:=\sigma\{X_s,s\le t\}\uparrow\mathcal{F}_\infty$
and consider the
$\mathcal{F}^X$-stopping times $\tildeheta_{t,z} \ge \tildeau_{t,z} \ge t$,
given by
$$
\tildeau_{t,z} := \inf \{ u \ge t: |X_u - z| < \ep\}\,, \qquad
\tildeheta_{t,z} := \inf \{ v \ge \tildeau_{t,z} : |X_v - z| \ge 1/2\}
$$
(with $\tildeheta_{t,y} \ge \tildeau_{t,y} \ge t$ defined analogously).
We claim that $\mathbb{P}_x$-a.s. for some non-random
$\eta=\eta(z,y,\ep)>0$ and any $t \ge t_0$,
\begin{equation}\leftarrowbel{eq:bd-zy}
\mathbb{P}_x(\tildeheta_{t,y} < \infty | \mathcal{F}^X_{\tildeheta_{t,z}})
\ge \eta \mathbb{I}_{\{\tildeheta_{t,z}<\infty\}} \,.
\end{equation}
Indeed, assuming without loss of generality that
$\tildeheta=\tildeheta_{t,z}$ is finite, for any given
$w=X_\tildeheta \in z + \partial \mathbb{B}_{1/2}$
let $\psi(\cdot)$ denote the line segment
from $\psi(0)=w$ to $\psi(1)=y$. The event
$\mathbb{G}amma_{w,y} := \sup_{s \in [0,1]} |X_{\tildeheta+s} - \psi(s)| < \ep$
implies that $\tildeau_{t,y} \le \tildeheta + 1$ is finite and thereby
also that $\tildeheta_{t,y} < \infty$. Further,
since $\psi(\cdot) \subseteq \mathbb{K} \subseteq \mathbb{D}_t$
is of distance $1/2 > \ep$ from $\partial \mathbb{K}$,
the probability of $\mathbb{G}amma_{w,y}$ given $\mathcal{F}^X_{\tildeheta}$
is merely
$\ delta(w) := \mathbb{P}(\sup_{s \in [0,1]} |U_s+\psi(0)-\psi(s)| < \ep)$
for a standard $d$-dimensional Brownian motion $\{U_s\}$. Clearly,
$\eta= \inf\{\ delta(w) : |w-z|=1/2 \} > 0$, yielding (\ref{eq:bd-zy}).
Now, considering the conditional expectation of (\ref{eq:bd-zy})
given $\mathcal{F}_t^X$, we find that
$$
\mathbb{P}_x(\tildeheta_{t,y} < \infty | \mathcal{F}^X_t)
\ge \eta
\mathbb{P}_x(\tildeheta_{t,z} < \infty | \mathcal{F}^X_t) \,.
$$
Further, the $\mathcal{F}_\infty$-measurable event $A_\ep(y)$
is the limit of $\{\tildeheta_{t,y} < \infty\}$ as $t \tildeo \infty$
(and the same applies to $A_\ep(z)$), so it follows from
L\'{e}vy's upward theorem (see \cite[Theorem 5.5.9]{Du}),
that $\mathbb{P}_x$-a.s.
\begin{eqnarray*}
\mathbb{I}_{A_\ep(y)}=\mathbb{P}_x( A_\ep(y) |\mathcal{F}_\infty)
= \lim_{t \tildeo \infty} \mathbb{P}_x(\tildeheta_{t,y} < \infty |\mathcal{F}_t^X)
\ge\eta
\lim_{t \tildeo \infty} \mathbb{P}_x(\tildeheta_{t,z} < \infty |\mathcal{F}_t^X)
= \eta\mathbb{I}_{A_\ep(z)}\,.
\end{eqnarray*}
The same applies with the roles of $y$ and $z$ exchanged and
consequently, $\mathbb{P}_x$-a.s. $A_\ep(y)=A_\ep(z)$.
In particular, $q^\ep_{xy}=\mathbb{P}_x(A_\ep(y))=\mathbb{P}_x(A_\ep(z))
=q^\ep_{xz}$, as claimed.
\end{proof}
\vskip 5pt
\noindent
{\bf Acknowledgment}
We thank Z-Q Chen for helpful correspondence, and
G. Ben Arous, J. Ding, H. Duminil-Copin, G. Kozma, T. Kumagai
and O. Zeitouni for fruitful discussions. We are grateful to
the anonymous referees for constructive feedback that
improved the presentation of this work. We also thank the
Courant Institute for hospitality and financial support
of visits (by A.D. and V.S.), during which part of this work
was done. This research was supported in part by NSF
grant DMS-1106627,
by Brazilian CNPq grants 308787/2011-0 and 476756/2012-0,
Faperj grant E-26/102.878/2012-BBP
and by ESF RGLIS Excellence Network.
\end{document} |
\begin{document}
\title{Presentation length and Simon's conjecture}
\author[Ian Agol]{
Ian Agol}
\address{
University of California, Berkeley \\
970 Evans Hall \#3840 \\
Berkeley, CA 94720-3840}
\email{
[email protected]}
\author[Yi Liu]{
Yi Liu}
\address{
University of California, Berkeley \\
970 Evans Hall \#3840 \\
Berkeley, CA 94720-3840}
\email{
[email protected]}
\thanks{Agol and Liu partially supported by NSF grant DMS-0806027}
\subjclass[2010]{57M}
\date{
July 6, 2010}
\begin{abstract}
In this paper, we show that any knot group maps onto at most finitely many knot groups.
This gives an affirmative answer to a conjecture of J. Simon.
We also bound the diameter of a closed hyperbolic 3-manifold
linearly in terms of the presentation length of its fundamental group, improving a result of White.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
In this paper, we prove the following theorem:
\begin{theorem}\label{main} Let $G$ be a finitely generated group with $b_1(G)=1$.
Then there are at most finitely many distinct knot complements $M$ such that there is an epimorphism $\phi:G\twoheadrightarrow\pi_1(M)$.\end{theorem}
As a corollary, we resolve Problem 1.12(D) from Kirby's problem list \cite{Kirby},
conjectured by Jonathan K. Simon in the 1970's. Recall a {\it knot group} is the fundamental group of the complement of a knot in $S^3$.
\begin{cor} Every knot group maps onto at most finitely many knot groups.\end{cor}
Note that this also resolves Problem 1.12(C). Our techniques say nothing about Part (B), and Part (A) is known to be false.
There has been a fair amount of work recently on Simon's conjecture, which partly motivated our work. Daniel S.
Silver and Wilbur Whitten proved that a fibered knot group maps onto at most finitely many knot groups (\cite[the
comment after Conjecture 3.9]{SilverWhitten06}, see also \cite{Leininger09}).
Silver and Whitten also considered a restricted partial order on knot complements where the epimorphism must preserve
peripheral structure, and therefore has a well-defined degree, (\cite{SilverWhitten05, SilverWhitten08}).
Restricting the class of epimorphisms to ones of
non-zero degree, finiteness was shown by Michel Boileau, Hyam Rubinstein and Shicheng Wang \cite{BRW}.
Recently it was proven that a $2$-bridge knot group maps onto at most finitely
many knot groups by Boileau et al. \cite{BBRW}. There
was also some experimental evidence for epimorphsims between prime knot groups of $\leq11$ crossings by Teruaki Kitano, Masaaki Suzuki, et al. (\cite{HKMS, KitanoSuzuki08}).
There are also families of examples of epimorphisms between 2-bridge knot groups, cf. \cite{ORS08, LS, HosteShanahan10}.
Now we give some remarks on the proof of Theorem \ref{main}. The finitely generated case is reduced to the finitely
presented case following a suggestion of Jack Button, so we assume that $G$ is finitely presented.
Given the proof of Simon's conjecture for maps of non-zero degree \cite{BRW}, we need to allow maps of zero degree.
In this case, and allowing finitely presented groups $G$ with $b_1(G)=1$,
techniques such as simplicial volume give no information. Nevertheless, we can show that there is a bound on
the simplicial volume of the image in terms of the \emph{\hyperlink{term-prLen}{presentation length}} of $G$, defined by Daryl Cooper in \cite{Co} and used to bound the volume of a
hyperbolic manifold (Section \ref{Sec-volume}). The \hyperlink{term-prLen}{presentation length} gives a coarse substitute for simplicial volume. Suppose we want to prove Theorem \ref{main}
for epimorphisms to hyperbolic knot groups. Then bounding the volume does not give a finiteness result, since there can
be infinitely many hyperbolic knot complements of bounded volume, such as the twist knots, which are obtained by
Dehn filling on the Whitehead link. A twist knot $k$ with a large number of twists will have a very short geodesic in the hyperbolic metric on its complement,
which contains a very large tubular neighborhood by the Margulis Lemma. What we would like to do is
to factor the epimorphism $G \twoheadrightarrow G_k$ through the fundamental group of the Whitehead link complement obtained by drilling the
short geodesic from the twist knot complement.
Intuitively, we expect that a \hyperlink{term-prLen}{presentation complex} for $G$ mapped into $S^3-k$ should be possible to homotope off
of a deep Margulis tube, since the complex should have bounded area, whereas the meridian disk of the Margulis tube should have large area.
If we could do this, then we obtain a contradiction, since the only finitely generated covers of the Whithead link complement with $b_1=1$ are
elementary covers, and the homomorphism factoring through such a cover could not map onto $G_k$.
However, this factorization cannot be done in general, cf. Example \ref{knotExample}. The substitute for this is to factor through the \emph{\hyperlink{term-extDrilling}{extended drilling}}, which has
enough good properties, such as coherence of the fundamental group (cf. Subsection \ref{Subsec-Scott}),
that we may still obtain a contradiction (Section \ref{Sec-factor}). The general case of non-hyperbolic knots is similar,
but requires some modifications involving the JSJ decomposition of the knot complement and some case-by-case analysis (Sections \ref{Sec-cpnship}, \ref{Sec-number}, \ref{Subsec-SFPieces}, \ref{Sec-clTypes}).
The technique for this factorization is based on a result of Matthew E. White (\cite{Wh}),
which bounds the diameter of a closed hyperbolic $3$-manifold in terms of the length of the presentation of its fundamental group. We improve upon White's result in Section \ref{Sec-diamBound} (to understand the improvement of White's result Theorem \ref{diamBound}(2), you need only read Section \ref{Subsec-DehnExt} and Section \ref{Subsec-factorHyp} ).
{\bf Acknowledgement:} We thank Daniel Groves for helpful conversations. We thank Jack Button, Hongbin Sun, and Shicheng Wang for comments on
a preliminary draft, and the referee for their comments.
\section{Dehn extensions}\label{Sec-eDehn}
In this section and the next, we study factorizations of maps through
\hyperlink{term-extDehnFilling}{extended} \hyperlink{term-extDehnFilling}{Dehn} \hyperlink{term-extDehnFilling}{fillings}.
This is motivated by the na\"ive question: suppose $f_n:K\to M_n$ is a sequence of maps from a finite $2$-complex to orientable aspherical compact $3$-manifolds $M_n$,
which are obtained by Dehn filling on a $3$-manifold $N$ along a sequence of slopes on a torus boundary component of $N$,
does $f_n$ factorize as $i_n\circ f'$ up to homotopy for some $f':K\to N$ for infinitely many $n$,
where $i_n:N\to M_n$ is the Dehn filling inclusion? The answer is negative as it stands, (cf. Subsection \ref{Subsec-example}),
but a modified version will be true in certain natural situations if we allow `\hyperlink{term-extDehnFilling}{extended Dehn fillings}'
(cf. Theorems \ref{factor-hyp}, \ref{factor-SF}).
We introduce and study \hyperlink{term-DehnExt}{Dehn extensions} in Subsections \ref{Subsec-DehnExt}, \ref{Subsec-Scott}.
\subsection{Examples}\label{Subsec-example}
We start with an elementary example. The phenomenon here essentially illustrates why
we should expect the factorization only through the
\hyperlink{term-extDehnFilling}{extended Dehn filling}, and will be used in the construction of Dehn extensions.
\begin{example}\label{elemExample} Let $P={\mathbb Z}\alpha\oplus{\mathbb Z}\beta$ be a rank-$2$ free-abelian group generated by $\alpha,\beta$. We
identify $P$ as the integral lattice of ${\mathbb R}^2$, where $\alpha=(1,0)$, $\beta=(0,1)$. Let
$\omega=a\,\alpha+b\,\beta\in P$ be primitive and $m>1$ be an integer. Define an extended lattice of ${\mathbb R}^2$
by $\tilde{P}=P+{\mathbb Z}\,\frac{\omega}{m}$. Pick any primitive $\zeta\in P$
such that $\omega+\zeta\in m{\mathbb Z}\oplus m{\mathbb Z}$. There are infinitely many such $\zeta$'s,
for example, $\zeta=(my-a)\,\alpha+(-mx-b)\,\beta$ with integers pairs $(x,y)$ such that $ax+by=1$.
Note there are infinitely many such pairs since the general solution to the linear Diophantine equation $ax+by=1$ is $(x,y)=(x^*+bt,y^*-at)$,
where $t\in{\mathbb Z}$ and $(x^*,y^*)$ is a particular solution.
$\zeta$ is primitive because $(-x)(my-a)+(-y)(-mx-b)=1$. Thus there is a well-defined epimorphism, $$\phi_\zeta:\tilde{P}\to P\,/\,{\mathbb Z}\zeta,$$
by requiring $\phi_\zeta|_P$ to be modulo ${\mathbb Z}\zeta$, and $\phi_\zeta(\frac{\omega}{m})=\frac{\omega+\zeta}{m}+{\mathbb Z}\zeta$.
\end{example}
We extend this construction to give a counterexample to the question in the category of $3$-manifolds.
\begin{example}\label{knotExample} Let $k_{p/q}$ be the $p/q$-cable knot ($p,q$ coprime and $q>1$) on a hyperbolic knot $k\subset S^3$. Let $N=S^3-k_{p/q}$, $M=S^3-k$. Identify $P=\pi_1(\partial M)={\mathbb Z}\mu\oplus{\mathbb Z}\lambda$ as a peripheral subgroup of $\pi_1(M)$, where $\mu$, $\lambda$ are the meridian and the longitude. Let $w=\mu^p\lambda^q$, then $\pi_1(N)\cong \pi_1(M)\langle\sqrt[q]{w}\rangle$, by which we mean the group $\pi_1(M)*\langle u\rangle$ modulo $u^q=w$. We may as well write $\pi_1(N)\cong \pi_1(M)*_PQ$, where $Q=P\langle\sqrt[q]{w}\rangle$. Note
the abelianization of $Q$ equals $\tilde{P}=P+{\mathbb Z}\,\frac{\omega}{q}$, where $\omega=p\,\mu+q\,\lambda$, if we identify $P={\mathbb Z}\mu\oplus{\mathbb Z}\lambda$ as the integral lattice of ${\mathbb Q}\oplus{\mathbb Q}$. By Example \ref{elemExample}, there are infinitely many primitive $\zeta$'s such that $\omega+\zeta\in q{\mathbb Z}\oplus q{\mathbb Z}$, and there are epimorphisms $\bar\phi|:\tilde{P}\to P\,/\,{\mathbb Z}\zeta$, which extend as
$\bar\phi:\pi_1(M)*_P\tilde{P}\to\pi_1(M_\zeta)$ in an obvious fashion, where $M_\zeta$ is the Dehn filling along $\zeta\in P$. Hence we obtain epimorphisms $\phi:\pi_1(N)\to\pi_1(M_\zeta)$ via the composition:
$$\pi_1(N)\cong\pi_1(M)*_PQ\to\pi_1(M)*_P\tilde{P}\to\pi_1(M_\zeta).$$
Now there are infinitely many slopes $\zeta\subset\partial M$, such that the corresponding $\phi:\pi_1(N)\to \pi_1(M_\zeta)$ are all surjective. It also is clear that $\phi$ can be realized by a map $$f:N\to M_\zeta,$$ by mapping the companion \hyperlink{term-piece}{piece} of $N$ to $M_\zeta$, and extending over the cable pattern \hyperlink{term-piece}{piece} of $N$.
However, $f$ does not always factor through the Dehn filling $i:M\to M_\zeta$ up to homotopy.
Suppose $f\simeq i\circ h$ for some $h:N\to M$.
Let $\kappa:\tilde{M}\to M$ be the covering corresponding to ${\rm Im}(h_\sharp)$, where $h_\sharp:\pi_1(N)\to\pi_1(M)$ after choosing some base points. Let $j:M\to N$ be the inclusion of $M$ as the companion \hyperlink{term-piece}{piece} of $N$, and let $\kappa':\tilde{M}'\to \tilde{M}$ be the covering corresponding to ${\rm Im}((h\circ j)_\sharp)$.
The homotopy lift $\tilde{h}:M \to\tilde{M}$ is hence $\pi_1$-surjective, and the homotopy lift $\widetilde{h\circ j}: M\to \tilde{M}'$ is also $\pi_1$-surjective. We have the commutative diagram:
$$\begin{CD} M @>j>> N @.\\
@V \widetilde{h\circ j} V V @V \tilde{h} VV \\
\tilde{M}'@>\kappa' >> \tilde{M} @>\kappa>> M.
\end{CD}$$
Note $H_1(N)\cong H_1(M) \cong {\mathbb Z}$ and $\tilde{M}$ and $\tilde{M}'$ are hyperbolic with non-elementary fundamental group, since they map $\pi_1$-surjectively to $M_\zeta$ via the factorization. Clearly $H_1(\tilde{M})$ and $H_1(\tilde{M}')$ are isomorphic to ${\mathbb Z}$.
As we shall see in Theorem \ref{volBound}, the volume of $\tilde{M}$ is at most $\pi\ell(\pi_1(N))$, where $\ell(\pi_1(N))$ is the \hyperlink{term-prLen}{presentation length} of $\pi_1(N)$, and similarly the volume of $\tilde{M}'$ is at most $\pi\ell(\pi_1(M))$. Thus $\tilde{M}$ is a finite covering of $M$ of some degree $d\leq\pi\ell(\pi_1(M))/{\mathrm{Vol}}(M)$. Moreover, $\tilde{h}_*: H_1(N) \to H_1(\tilde{M})$ is an isomorphism as $\tilde{h}$ is $\pi_1$-surjective, and $\kappa_*:H_1(\tilde{M})\to H_1(M)$ is the multiplication by some factor $d_0$ of $d$. Similarly, $(\widetilde{h\circ j})_*: H_1(M) \to H_1(\tilde{M}')$ is an isomorphism, and $\kappa'_*:H_1(\tilde{M}')\to H_1(\tilde{M})$ is multiplication by a factor $q'$, which is the same factor as the multiplication $j_*: H_1(M)\to H_1(N)$. By the cabling construction of knots, the image of $j_*:H_1(M)\to H_1(N)\cong{\mathbb Z}$ is contained in $q{\mathbb Z}$, and thus we see that $q'=q$. Thus:
$$\begin{CD} H_1(M) @>\times q>> H_1(N) @.\\
@V \cong V V @V \cong VV \\
H_1(\tilde{M}')@>\times q>> H_1(\tilde{M}) @>\times d_0>> H_1(M).
\end{CD}$$
However, we have $q= {\mathrm{Vol}}(\tilde{M}')\,/\,{\mathrm{Vol}}(\tilde{M}) \leq \pi\ell(\pi_1(M))\,/\,{\mathrm{Vol}}(M)$. Therefore if
we take $q > \pi \ell(\pi_1(M))\,/\,{\mathrm{Vol}}(M)$, we obtain a contradiction. In other words, for such $q$'s, $f$ does not factor through the Dehn filling $i:M\to M_\zeta$ up to homotopy.
On the other hand, as evidence for Theorem \ref{factor-hyp}, clearly every $\phi:\pi_1(N)\to\pi_1(M_\zeta)$ as above factors through the `extended' Dehn filling epimorphisms $\iota^e:\pi_1(M)*_P(P+{\mathbb Z}\frac{\zeta}{q})\to \pi_1(M_\zeta)$, since $P+{\mathbb Z}\frac{\zeta}{q}=\tilde{P}$.\end{example}
\subsection{Dehn extensions of filling and drilling}\label{Subsec-DehnExt}
In this subsection, we introduce the notion of \hyperlink{term-DehnExt}{Dehn extensions}.
Let $N$ be an aspherical orientable compact $3$-manifold, and $\zeta$ be a slope on an incompressible torus boundary component $T\subset\partial N$. By choosing a base-point of $N$ and a path to $T$, we may identify $P=\pi_1(T)$ as a peripheral subgroup of $\pi_1(N)$. By choosing an orientation of $\zeta$, we identify $\zeta$ as primitive element in $P$. On the other hand, by choosing a basis for $\pi_1(T)$, we may also identify $P\cong{\mathbb Z}\oplus{\mathbb Z}$ as the integral lattice in ${\mathbb Q}\oplus{\mathbb Q}$.
\begin{definition} \raisebox{\baselineskip}[0pt]{\hypertarget{term-DehnExt}}For any integer $m>1$, we define the \emph{Dehn extension} of $\pi_1(N)$
along $\zeta$ with denominator $m$ as the amalgamated product:
$$\pi_1(N)^{e(\zeta,m)}=\pi_1(N)*_P \left(P+{\mathbb Z}\,\frac{\zeta}{m}\right).$$
We often simply abbreviate this $\pi_1(N)^e$ when $m$ and $\zeta$ are clear from the context. There is a natural \emph{extended Dehn filling epimorphism}: $$\iota^e:\pi_1(N)^e\to\pi_1(N_\zeta),$$ defined by quotienting out the normal closure of ${\mathbb Z}\,\frac{\zeta}{m}$, where $N_\zeta$ is the Dehn filling of $N$ along $\zeta$.
We also regard $\pi_1(N)$ as a \emph{trivial} Dehn extension of itself with denominator $m=1$, in which case we do not require $N$ to have any incompressible torus boundary component or non-empty boundary.\end{definition}
From a topological point of view, the \hyperlink{term-DehnExt}{Dehn extension} $\pi_1(N)^e$ may be regarded as the fundamental group of a topological space $N^e=N^{e(\zeta,m)}$, which will be called the \emph{Dehn extension} of $N$ along $\zeta$ with denominator $m$. Certainly $N^e$ could be $N$ itself for the trivial \hyperlink{term-DehnExt}{Dehn extension}. If $m>1$, $N^e$ is obtained from $N$ by gluing $T\subset \partial N$ to the source torus of a mapping cylinder, denoted as $Z$, of the $m$-fold covering map between tori: $${\mathbb R}^2/P\to {\mathbb R}^2/(P+{\mathbb Z}\frac{\zeta}{m}).$$
To visualize this, take the product of the unit interval $I=[0,1]$ with an $m$-pod (i.e. a cone over $m$ points), identify the $0$-slice with the $1$-slice via a primitive cyclic permutation of the legs, and denote the resulting space as $\Psi=\Psi(m)$. \raisebox{\baselineskip}[0pt]{\hypertarget{term-ridge}}The `side' of $\Psi$ is a loop which is homotopic to $m$ wraps along of the `ridge' loop of $\Psi$. The mapping cylinder $Z=Z(m)$ is homeomorphic to $S^1\times \Psi$, and we will refer the product of $S^1$ with the side loop (resp. ridge loop) as the \emph{side torus} (resp. \emph{ridge torus}). Glue the side torus of $Z$ to the boundary $T$ via a homeomorphism, such that the side-loop of $Z$ is identified with $\zeta$, (clearly this determines the resulting space up to homeomorphism), cf. Figure \ref{figDehnExt}. We call $Z$ the \emph{ridge piece}, and the copy of $N$ under the inclusion the \emph{regular part}.
\begin{figure}
\caption{The Dehn extension $N^{e}
\label{figDehnExt}
\end{figure}
The \hyperlink{term-DehnExt}{Dehn extension} $N^e$ may be \textit{ad hoc} characterized as a `ridged manifold'.
Despite the mild singularity near the ridges, $N^e$ behaves like an aspherical compact $3$-manifold in many ways. For instance, $N^e$ is an Eilenberg-MacLane space of $\pi_1(N)^e$, and $H_*(N^e,{\mathbb Q})\cong H_*(N,{\mathbb Q})$.
The \hyperlink{term-DehnExt}{Dehn extension} $N^e$ also admits an analogous JSJ decomposition. \raisebox{\baselineskip}[0pt]{\hypertarget{term-piece}}Recall that the classical Jaco-Shalen-Johanson (JSJ) decomposition says any orientable irreducible compact $3$-manifold can be cut along a minimal finite collection of essential tori into atoroidal and Seifert fibered {\it pieces}, canonical up to isotopy. The analogous JSJ decomposition of $N^e$ consists of the JSJ pieces of $N$ together with a \hyperlink{term-ridge}{ridge piece} $Z$
if $m>1$. These pieces are glued along $\pi_1$-injective tori. This is a graph-of-spaces decomposition, associating to $N^e$ a finite connected graph, called the \emph{JSJ graph}, whose vertices correspond to the JSJ pieces and edges to the JSJ tori.
Moreover, $\pi_1(N^e)=\pi_1(N)^e$ is coherent; indeed, any covering of $N^e$ with finitely
generated fundamental group has a Scott core, as we shall see in Subsection \ref{Subsec-Scott}.
There is also a natural map, called the \raisebox{\baselineskip}[0pt]{\hypertarget{term-extDehnFilling}}\emph{extended Dehn filling} of $N^e$ along $\zeta$,
$$i^e:N^e\to M,$$ defined by collapsing $Z\cong S^1\times \Psi$ to $S^1$. $i^e$ induces
$\iota^e:\pi_1(N)^e\cong\pi_1(N^e)\to\pi_1(N_\zeta)$.
To an essential drilling, we may naturally associate a \hyperlink{term-DehnExt}{Dehn extension}. Specifically, let $M$ be an aspherical orientable compact $3$-manifold, and $\gamma\subset M$ be an essential simple closed curve, i.e. which is not null-homotopic. Then $N=M-\gamma$ is an aspherical orientable compact $3$-manifold, and the boundary component $\partial_\gamma N$ coming from the drilling is an incompressible torus.There is a canonical slope $\zeta\subset\partial_\gamma N$ so that $M=N_\zeta$.
\raisebox{\baselineskip}[0pt]{\hypertarget{term-extDrilling}} For any integer $m>0$, we shall call the \hyperlink{term-DehnExt}{Dehn extension} $N^e(\zeta, m)$ the \emph{extended drilling} of $M$ along $\gamma$ with denominator $m$.
We shall be most interested in \emph{geometric drillings}, namely, when $\gamma$ is a
simple closed geodesic in the interior of some geometric \hyperlink{term-piece}{piece} of $M$. Factorization of
maps through extended geometric drillings will be studied in
Section \ref{Sec-factor}.
\subsection{Coherence and the Scott core} \label{Subsec-Scott}
With notation from Subsection \ref{Subsec-DehnExt}, in this subsection, we show
every \hyperlink{term-DehnExt}{Dehn extension} $N^e$ is \emph{coherent}, in the
sense that every finitely generated subgroup of its fundamental group is finitely
presented. This follows from the existence of a Scott core for any connected finite
generated covering space $\tilde{N}^e$ of $N$ proved in Proposition \ref{ScottCore} below.
The results in this section are preparatory for the arguments in Section \ref{Sec-homeoTypes}.
Recall for a topological space $X$, a \emph{Scott core} of $X$ is a connected compact
subspace $C\subset\tilde{N}^e$, if any, whose inclusion induces a $\pi_1$-isomorphism.
It is named after Peter Scott who first found such cores for $3$-manifolds with finitely generated fundamental group (\cite{Sc}) .
First, we need an auxiliary result. For background on combinatorial group theory, cf. \cite{Cohen}.
\begin{lemma} \label{amalgamation}
Let $G=A\ast_C B$ or $G= A\ast_C$ be a free product with amalgamation or HNN extension. If $G$ and $C$ are finitely generated, then $A$ and $B$ are too.
\end{lemma}
\begin{proof}
We give the argument for the amalgamated product, the HNN case being similar.
Let $g_1,\ldots, g_n$ be generators for $G$, and $c_1,\ldots,c_m$ be generators for $C$.
We may write each element $g_i = a_{i,1} b_{i,1} \cdots a_{i,k(i)} b_{i,k(i)}$, where
$a_{i,j}\in A, b_{i,j}\in B$, $i=1,\ldots, n$. Let $A' = \langle a_{i,j}, c_l \rangle, B' =\langle a_{i,j}, c_l\rangle$, so
that $C< A', C<B'$.
Then we have $G'= A'\ast_C B' \to \langle A',B'\rangle \leq A\ast_C B = G$ injects by \cite[Proposition 28]{Cohen}. But clearly also $G\leq G'$,
so $G=G', A'=A, B'=B$, and we see that $A$ and $B$ are finitely generated.
\end{proof}
\begin{cor} \label{coherent}
If $G$ is a finitely generated group which is a finite graph of groups with finitely generated edge groups,
then the vertex groups are also finitely generated.
\end{cor}
\begin{proof}
Use Lemma \ref{amalgamation} together with induction on the number of vertices of the graph of groups.
\end{proof}
\begin{prop}\label{ScottCore} Let $N$ be an orientable aspherical compact $3$-manifold, and $N^e$ be the \hyperlink{term-DehnExt}{Dehn extension}
of $N$ with denominator $m>0$ along some slope on an incompressible torus boundary component.
Then any connected covering space $\kappa:\tilde{N}^e\to N^e$ with finitely generated fundamental
group has an aspherical Scott core $C\subset\tilde{N}^e$. Furthermore, for any component of the preimage
of a JSJ torus of $N^e$, $C$ either meets it in a Scott core of the component or misses it. For
any component of the preimage of a JSJ \hyperlink{term-piece}{piece} of $N^e$,
$C$ either meets it in an aspherical Scott core of the component or misses it.
\end{prop}
\begin{remark} The construction here includes the case when $N^e$ is the
trivial \hyperlink{term-DehnExt}{Dehn} \hyperlink{term-DehnExt}{extension}, and so will Lemmas \ref{trivialCk} and \ref{simplifyCore}.\end{remark}
\begin{proof}
Let $\kappa:\tilde{N}^e\to N^e$ be the covering map, and let ${\mathcal{T}}$ be the union of the JSJ tori of $N^e$.
Pick a finite bouquet of circles $L=S^1\vee\cdots\vee S^1$ mapping $\pi_1$-surjectively into $\tilde{N}^e$,
say $f:L\to\tilde{N}^e$. There are only finitely many components of the preimage of JSJ tori meeting $f(L)$,
and there are only finitely many components of the preimage of \hyperlink{term-piece}{pieces} meeting $f(L)$.
Let $T\subset{\mathcal{T}}$ be a JSJ torus. For any component $\tilde{T}$ of $\kappa^{-1}(T)$ meeting $f(L)$,
clearly $\tilde{T}$ is either a torus, or a cylinder, or a plane, so we may take the torus, or an essential annulus,
or a disk in $\tilde{T}$ containing $\tilde{T}\cap f(L)$, respectively, and denote as $C_{\tilde T}$.
Let $C_T$ be the union of $C_{\tilde{T}}$ for all such $\tilde{T}$'s, and let $C_{{\mathcal{T}}}$ be the union of all $C_T$'s.
For a regular \hyperlink{term-piece}{piece} $J$, first observe any component $\tilde{J}$ of $\kappa^{-1}(J)$ has finitely generated fundamental group.
To see this, we may assume $\tilde{J}$ meets $f(L)$. Then there are at most finitely many components of $\tilde{N}^e-\tilde{J}$
which meet $f(L)$, say $U_1,\cdots,U_t$, and $\tilde{J}\cup U_1\cup\cdots\cup U_t\subset \tilde{N}^e$ is
$\pi_1$-surjective as it contains $f(L)$. Clearly it is also $\pi_1$-injective, so $\pi_1(\tilde{N}^e)\cong\pi_1(\tilde{J}\cup U_1\cup\cdots\cup U_t)$.
Note $\tilde{J}$ and $U_i$'s are glued along $\pi_1$-injective coverings of a JSJ torus, namely planes, cylinders or tori,
so we have a graph-of-groups decomposition of $\pi_1(\tilde{N}^e)$,
whose vertex groups are $\pi_1(\tilde{J}),\pi_1(U_1),\cdots,\pi_1(U_t)$ and whose edge groups are finitely generated.
By Corollary \ref{coherent}, it follows the vertex groups must all be finitely generated as $\pi_1(\tilde{N}^e)$ is finitely generated.
In particular, $\pi_1(\tilde{J})$ is finitely generated.
Thus, as $\tilde{J}\cap C_{{\mathcal{T}}} \subset \partial\tilde{J}$ is a (possibly empty) compact sub-manifold,
by a theorem of Darryl McCullough \cite[Theorem 2]{McC86}, there is a Scott core of $\tilde{J}$ which meets
$\partial\tilde{J}$ exactly in $\tilde{J}\cap C_{{\mathcal{T}}}$. Moreover, since $\tilde{J}$ is aspherical,
we may also make the core aspherical by adding to the core bounded complementary components
whose inclusion into $\tilde{J}$ is $\pi_1$-trivial. The result is denoted as $C_{\tilde{J}}$.
Let $C_J$ be the union of all such $\tilde{J}$'s.
Let $Z$ be the ridge \hyperlink{term-piece}{piece} of $N^e$.
\raisebox{\baselineskip}[0pt]{\hypertarget{term-ridgeCovering}} For each component $\tilde{Z}$ of $\kappa^{-1}(Z)$ meeting $f(L)$,
the preimage of the \hyperlink{term-ridge}{ridge torus} is either a torus, or a cylinder, or a plane,
so we may take the torus, or an essential annulus, or a disk in the image, respectively,
and thicken it up to a regular neighborhood in $\tilde{Z}$ which meets the preimage of the
\hyperlink{term-ridge}{side torus} exactly in $\tilde{Z}\cap C_{{\mathcal{T}}}$.
This is a Scott core $C_{\tilde{Z}}$ of $\tilde{Z}$. Let $C_Z$ be the union of all such $\tilde{Z}$'s.
Let $C$ be the union of all the $C_J$'s and $C_Z$ constructed above. Then $C$ is an aspherical Scott core of $\tilde{N}^e$ as required.
\end{proof}
\raisebox{\baselineskip}[0pt]{\hypertarget{term-chunk}}In this paper, we often refer to any connected union $Q$ of a few components from $C_Z$ and $C_J$'s as a \emph{chunk} of $C$, following the notation in the proof of Proposition \ref{ScottCore}. It is a \emph{ridge} chunk (resp. a \emph{hyperbolic} chunk, or a \emph{Seifert fibered} chunk) if it is a single component from $C_Z$ (resp. some $C_J$ where $J$ is a hyperbolic \hyperlink{term-piece}{piece}, or a Seifert fibered \hyperlink{term-piece}{piece}). It is a \emph{regular} chunk if it is contained in $C-C_Z$. Note if the denominator of the \hyperlink{term-DehnExt}{Dehn extension} $N^e$ is $2$, there could be some ridge chunk which is a manifold, homeomorphic to either $I\times D^2$, or $S^1\times A$, or $I\times A$, or $S^1\times R$, or $I\times R$, where $A$ is an annulus and $R$ is a M\"obius strip, but we do not regard such as a regular chunk. In particular, regular chunks are all orientable.
The \raisebox{\baselineskip}[0pt]{\hypertarget{term-cutBdry}} \emph{cut boundary} $\partial_{{\mathcal{T}}}Q$ of a chunk $Q$ is the union of the components of $C_{{\mathcal{T}}}$ which are contained in the frontier of $Q$ in $\tilde{N}^e$. For example, if $Q$ is regular, $\partial_{{\mathcal{T}}}Q$ is a (possibly disconnected, with boundary) essential compact subsurface of $\partial Q$.
The preimage of the JSJ tori ${\mathcal{T}}$ of $N^e$ cuts $C$ into minimal \hyperlink{term-chunk}{chunks} along $C_{\mathcal{T}}=C\cap\kappa^{-1}({\mathcal{T}})$. This induces a graph-of-spaces decomposition of $C$ over a finite connected simplical graph $\Lambda$. The vertices of $\Lambda$ correspond to the components of $C-C_{\mathcal{T}}$, and whenever two components are adjacent to each other, there are edges joining them, each corresponding to a distinct component of $C_{\mathcal{T}}$ along which they are adjacent. Thus a \hyperlink{term-chunk}{chunk} $Q$ may be regarded as the subspace of $C$ associated to a connected complete subgraph of $\Lambda$. The graph-of-spaces decomposition also induces a graph-of-groups decomposition of $\pi_1(C)$ along free abelian edge groups of rank at most $2$.
In the rest of this subsection, we discuss how to get rid of unnecessary \hyperlink{term-chunk}{chunks} when $b_1(\tilde{N}^e)=1$.
\begin{lemma}\label{trivialCk} A \hyperlink{term-chunk}{chunk} $Q\subset C$ is contractible if and only if $b_1(Q)=0$.\end{lemma}
\begin{proof}
It suffices to prove the `if' direction. This is clear if $Q$ is a ridge \hyperlink{term-chunk}{chunk}. Also, if $Q$ a regular \hyperlink{term-chunk}{chunk}, $\partial Q$ must be a union of spheres so $Q$ is contractible as it is aspherical. To see the general case, first observe any component of $C_{{\mathcal{T}}}\cap Q$ is separating in $Q$ as $b_1(Q)=0$. If there were a ridge \hyperlink{term-chunk}{subchunk} $S\subset Q$ with a \hyperlink{term-ridgeCovering}{ridge annulus or torus}, then any regular \hyperlink{term-chunk}{chunk} $R$ adjacent to $S$ would have at least one component of $\partial R$ which is not a sphere since it contains an essential loop, so $b_1(S)\geq1$. The union $Q'$ of $S$ and all its adjacent maximal regular \hyperlink{term-chunk}{chunks} has $b_1(Q')\geq 1$ by an easy Mayer-Vietoris argument. Furthermore, if $S'$ is a ridge \hyperlink{term-chunk}{chunk} adjacent to $Q'$, let $Q''$ is the union of $Q'$, $S'$ and all other maximal regular \hyperlink{term-chunk}{chunks} adjacent to $S'$. No matter $S'$ has a \hyperlink{term-ridgeCovering}{ridge disk, annulus or torus}, a Mayer-Vietoris argument again shows $b_1(Q'')\geq 1$. Keep going in this way, then in the end we would find $b_1(Q)\geq 1$.
Since $b_1(Q)=0$ by assumption, we see every ridge \hyperlink{term-chunk}{chunk} has a \hyperlink{term-ridgeCovering}{disk ridge}, hence is contractible. However, in this case, $\partial_{{\mathcal{T}}}S$ is a union of disks for any ridge \hyperlink{term-chunk}{chunk} $S$, so for any maximal regular \hyperlink{term-piece}{piece} $R$ we must have $b_1(R)=0$ under the assumption $b_1(Q)=0$. This implies $Q$ is also contractible as we mentioned at the beginning. Therefore, $Q$ must be a contractible \hyperlink{term-chunk}{chunk}.
\end{proof}
For any component $\tilde{J}\subset \tilde{N}^e$ of the preimage of a JSJ \hyperlink{term-piece}{piece} $J\subset N^e$, the corresponding \hyperlink{term-chunk}{chunk} $C_{\tilde{J}}=C\cap \tilde{J}$ is called \emph{elementary} if $\pi_1(\tilde{J})$ is abelian, and it is called \emph{central} if $\pi_1(\tilde{J})$ is a subgroup of the center of $\pi_1(J)$. A central hyperbolic \hyperlink{term-chunk}{chunk} is always contractible, and a central Seifert fibered \hyperlink{term-chunk}{chunk} is either contractible or homeomorphic to a trivial $S^1$-bundle over a disk, and every ridge \hyperlink{term-chunk}{chunk} is central.
\begin{lemma}\label{simplifyCore} If $\pi_1(\tilde{N}^e)$ is non-abelian and $b_1(\tilde{N}^e)=1$, one may assume $C$ has no contractible \hyperlink{term-chunk}{chunk}, no non-central elementary \hyperlink{term-chunk}{chunk}, and $C_{{\mathcal{T}}}=C\cap\kappa^{-1}({\mathcal{T}})$ has no disk component.\end{lemma}
\begin{proof} We show this by modifying $C$. If there is a contractible \hyperlink{term-chunk}{chunk} $Q\subset C$,
the \hyperlink{term-cutBdry}{cut} \hyperlink{term-cutBdry}{boundary} $\partial_{{\mathcal{T}}}Q$ is a disjoint union of disks.
If $D\subset \partial_{{\mathcal{T}}}Q$ were non-separating in $C$, $C'=C-D$ will have $b_1(C')=0$,
then by Lemma \ref{trivialCk}, $C'$ is contractible, so $C$ is homotopy equivalent to a circle,
which is impossible as $\pi_1(C)=\pi_1(\tilde{N}^e)$ is non-abelian by assumption.
Thus $D\subset \partial_{{\mathcal{T}}}Q$ is separating in $C$, so $C-D=C'\sqcup C''$,
and $b_1(C')=0$ or $b_1(C'')=0$. Say $b_1(C')=0$, by Lemma \ref{trivialCk},
$C'$ is contractible. We may discard a small regular neighborhood of $C'$ from $C$,
and the rest of $C''$ is still a Scott core.
As there are at most finitely many \hyperlink{term-chunk}{chunks} in $C$, we may discard all the contractible ones, and obtain a Scott core, still denoted as $C$, with no contractible \hyperlink{term-chunk}{chunk}.
A similar argument shows one may assume $C_{{\mathcal{T}}}$ has no disk component.
Now for any non-central elementary \hyperlink{term-chunk}{chunk} $Q$,
$Q\neq C$ since $\pi_1(C)$ is non-abelian. As $C$ has no contractible \hyperlink{term-chunk}{chunk},
at least one component $U$ of $\partial_{{\mathcal{T}}}Q$ is a torus or an annulus.
Note when $Q$ is a hyperbolic \hyperlink{term-chunk}{chunk}, it is homeomorphic to $U\times I$,
and when $Q$ is a Seifert fibered \hyperlink{term-chunk}{chunk},
it is either a trivial $S^1$-bundle over an annulus if $U$ is a torus,
or an $I$-bundle over an annulus if $U$ is an annulus, since $Q$ is non-central.
Let $\tilde{J}$ be the component of the preimage of a regular JSJ \hyperlink{term-piece}{piece} $J$
such that $U$ is carried by a component $\tilde{T}\subset\partial\tilde{J}$.
Suppose there were some other boundary component $U'$ carried by some
other component $\tilde{T}'\subset\partial\tilde{J}$. As $U$,
$U'$ are both essential annuli or tori by the construction of Proposition \ref{ScottCore},
$T$, $T'$ cannot be disjoint cylinders or tori, otherwise $\tilde{J}$ would not be elementary.
We conclude $T=T'$, hence $U=U'$ by the construction.
It is also clear that $\pi_1(U)\cong\pi_1(Q)\cong\pi_1(\tilde{J})$ induced by the inclusion.
We may discard a small regular neighborhood of $Q$ from $C$,
and the rest is still connected and is a Scott core.
Thus we discard all non-central elementary \hyperlink{term-chunk}{chunks} in this fashion.
In the end, we may assume $C$ to have no non-central elementary \hyperlink{term-chunk}{chunk}.
Note also the modifications here do not affect the properties of $C$ described in Proposition \ref{ScottCore},
so we obtain a Scott core $C$ as required.
\end{proof}
\section{Factorization through extended geometric drilling} \label{Sec-factor}
In this section, we show that a map from a given finite $2$-complex to an orientable aspherical compact $3$-manifold factorizes up to homotopy through \hyperlink{term-extDrilling}{extended} \hyperlink{term-extDrilling}{drilling} along sufficiently short geodesics in the hyperbolic \hyperlink{term-piece}{pieces} (Theorem \ref{factor-hyp}) and through \hyperlink{term-extDrilling}{extended} \hyperlink{term-extDrilling}{drillings} along a sufficiently sharp cone-fiber in the Seifert fibered \hyperlink{term-piece}{pieces} (Theorem \ref{factor-SF}). We need the notion of \hyperlink{term-prLen}{presentation length} for the statement and the proof of these results, so we introduce it in Subsection \ref{Subsec-prlen}.
\subsection{Presentation length}\label{Subsec-prlen}
For a finitely presented group $G$, the presentation length $\ell(G)$ of $G$ turns out to be useful in $3$-manifold topology, especially for the study of hyperbolic $3$-manifolds. For example, Daryl Cooper proved the volume of an orientable closed hyperbolic $3$-manifold $M$ is at most $\pi\ell(\pi_1(M))$ (\cite{Co}). Matthew E. White also gave an upper bound on the diameter of an orientable closed hyperbolic $3$-manifold $M$ in terms of $\ell(\pi_1(M))$ in a preprint (\cite{Wh}).
\begin{definition}\label{prlen} \raisebox{\baselineskip}[0pt]{\hypertarget{term-prLen}}Suppose $G$ is a nontrivial finitely presented group. For any presentation $\mathcal{P}=(x_1,\ldots,x_n;r_1,\ldots,r_m)$ of $G$ with the word-length of each relator
$|r_j|\geq 2$, define the \emph{length} of $\mathcal{P}$ as
$$\ell(\mathcal{P})=\sum_{j=1}^m (|r_j|-2),$$
and the \emph{presentation length} $\ell(G)$ of $G$ as the minimum of $\ell(\mathcal{P})$ among all such presentations. \end{definition}
Note by adding finitely many generators and discarding single-letter relators, $\ell(G)$ can always be realized by a \emph{triangular presentation}, namely of which the word-length of every relator equals $2$ or $3$. The length of a triangular presentation equals the number of relators of length $3$.
Associated to any finite presentation $\mathcal{P}$ of $G$, there is a \emph{presentation $2$-complex} $K$,
which consists of a single $0$-cell $*$, and $1$-cells corresponding to the generators attached to $*$,
and $2$-cells corresponding to the relators attached to the $1$-skeleton with respect to $\mathcal{P}$.
The $2$-cells are all $2$-simplices or bigons if $\mathcal{P}$ is triangular.
\subsection{Drilling a short geodesic in a hyperbolic piece}\label{Subsec-factorHyp}
In this subsection, we show that maps factorize through the \hyperlink{term-extDrilling}{extended drilling}
of a short simple closed geodesic in a hyperbolic \hyperlink{term-piece}{piece}.
The precise statement is as follows.
\begin{theorem}\label{factor-hyp} Let $G$ be a finitely presented group, and $M$ be an orientable aspherical compact $3$-manifold. Suppose there is a simple closed geodesic $\gamma$ in the interior of
a hyperbolic \hyperlink{term-piece}{piece} $J\subset M$ with length sufficiently small, with respect to some complete hyperbolic metric on the interior of $J$, but depending only on the \hyperlink{term-prLen}{presentation length} $\ell(G)$. Then any homomorphism $\phi:G\to\pi_1(M)$ factors through the \hyperlink{term-extDehnFilling}{extended Dehn filling} epimorphism $\iota^e:\pi_1(N^{e(\zeta,m)})\to \pi_1(M)$ of some denominator $0<m\leq T(\ell(G))$, where $N=M-\gamma$, $\zeta$ is the meridian of the Margulis tube about $\gamma$, and $T(n)=2\cdot 3^n$. Namely, $\phi=\iota^e\circ\phi^e$ for some $\phi^e:G\to\pi_1(N^e)$.
\end{theorem}
\begin{remark} By Mostow Rigidity, the complete hyperbolic metric on the interior of $J$ is unique, of finite volume, if $\partial J$ consists of tori or is empty.\end{remark}
We prove Theorem \ref{factor-hyp} in the rest of this subsection. The approach here is inspired by a paper
of Matthew E. White (\cite{Wh}).
To begin with, take a presentation $2$-complex $K$ of a triangular presentation $\mathcal{P}$ achieving $\ell(G)$, and a PL map $f:K\to M$ realizing $\phi:G\to\pi_1(M)$. We may assume $f(K)$ intersects the JSJ tori ${\mathcal{T}}$ in general position and the image of the $0$-simplex $*\in K$ misses $J$. Let $\epsilon_3>0$ be the Margulis constant for ${\mathbb H}^3$, and $J_{\mathtt{geo}}=
(\mathring{J},\rho)$ be the interior of $J$ with the complete hyperbolic metric $\rho$ as assumed. By picking a sufficiently small $\epsilon<\epsilon_3$, we may endow $M$ with a complete Riemannian metric so that $J$ is isometric to $J_{\mathtt{geo}}$ with the open $\epsilon$-thin horocusps corresponding to $\partial J$ removed.
We may homotope $f$ so that $f(K)\cap J$ is totally geodesic on each $2$-simplex of $K$, and the total area of $f(K)\cap J$ is at most $\pi\ell(\mathcal{P})$. In fact, we may first homotope $f$ so that $f(K)$ meets the JSJ tori in minimal normal position, (i.e. the number of $K^{(1)}\cap f^{-1}({\mathcal{T}})$ is minimal), then homotope rel $f^{-1}({\mathcal{T}})$ so that $f(K)\cap J$ becomes totally geodesic. Roughly speaking, noting that $K\cap f^{-1}(J)$ is a union of $1$-handles (bands) and monkey-handles (hexagons) by the normal position assumption, the total area of $f(K)\cap J$ is approximately the total area of the monkey-handles. The area of each monkey-handle is bounded by $\pi$, which is the area of an ideal hyperbolic triangle, and there are at most $\ell(\mathcal{P})$ monkey-handles, as $\ell(\mathcal{P})$ equals the number of $2$-simplices of $K$. It is not hard to make a rigorous argument of the estimation using elementary hyperbolic geometry, so the total area of $f(K)\cap J$ can be bounded by $\pi\ell(\mathcal{P})$.
A theorem of Chun Cao, Frederick W. Gehring and Gavin J. Martin \cite{CGM} says that if $\gamma$ has length $l<\frac{\sqrt{3}}{2\pi}(\sqrt{2}-1)$, then there is an embedded tube $V\subset M$ of radius $r$ with the core geodesic $\gamma$, such that: $$\sinh^2(r)=\frac{\sqrt{1-(4\pi l/\sqrt{3})}}{4\pi l/\sqrt{3}}-\frac{1}{2},$$ \cite{CGM}. This means if $\gamma$ is very short, it lies in a very deep tube $V$. In particular, any meridian disk of $V$ will have very large area.
Up to a small adjustment of the radius of $V$, we may assume $f(K)$ intersects $\partial V$ in general position,
and the $0$-simplex $*\in K$ misses $V$. Denote $K_V=f^{-1}(V)$, and $K_{\partial{V}}=f^{-1}(\partial V)$.
Let $\zeta\subset\partial V$ be an oriented simple closed curve bounding a meridian disk in $V$.
Topologically, let $i:N\to M$ be the Dehn filling inclusion identifying $N$ as $M\setminus\mathring{V}$.
Remember $N^e=N^{e(\zeta,m)}=N\cup Z(m)$, where $Z(m)$ is the \hyperlink{term-ridge}{ridge piece} of some denominator $m>0$.
We must show if $\gamma$ is sufficiently short, then for some denominator $m$, there is a map $$f^e:K\to N^e,$$ such that $f\simeq i^e\circ f^e$.
The lemma below is an easy criterion.
\begin{lemma}\label{factorCriterion}
Suppose $f_*:H_2(K_V,K_{\partial V};{\mathbb Q})\to H_2(V,\partial V;{\mathbb Q})$ vanishes, and
let $m >0$ be the maximal order of torsion elements of $H_1(K_V,K_{\partial V};{\mathbb Z})$.
Then $f^e|_{K_V}$, and hence $f^e$, exists for $N^{e(\zeta,m)}$.
\end{lemma}
\begin{proof} It suffices to show that there exists a lift $f^e|_{K_V}:K_V\to Z(m)$ commuting with the diagram up to homotopy:
$$\begin{CD} K_{\partial V} @>f_{\partial V}^e>> Z(m)\\
@V j V\cap V @V i^e VV\\
K_V@>f_V >> V,
\end{CD} $$
where $f_V, f_{\partial V}$ are the restrictions of $f:K\to M$ to $K_V, K_{\partial V}$ respectively, and $f_{\partial V}^e$ is $f_{\partial V}$ post-composed with $\partial V\subset Z(m)$. This is a relative homotopy extension problem which can be resolved by obstruction theory, but we give a manual proof here for the reader's reference. Because $\pi_1(Z(m))$ and $\pi_1(V)$ are abelian, this is the same as finding a lifing $\psi:H_1(K_V)\to H_1(Z(m))$ commuting with the diagram above on homology. If $f_*:H_2(K_V,K_{\partial V};{\mathbb Q})\to H_2(V,\partial V;{\mathbb Q})$ vanishes, so does $f_*:H_2(K_V,K_{\partial V})\to H_2(V,\partial V)$ since $H_2(V,\partial V)\cong{\mathbb Z}$. Note that since $K_V$ is homotopy equivalent to a graph, $H_2(K_V)=0$ in the relative homology sequence:
$$\cdots\to H_2(K_V)\to H_2(K_V,K_{\partial V})\to H_1(K_{\partial V})\to H_1(K_V)\to\cdots.$$
Thus $H_2(K_V,K_{\partial V})\leq H_1(K_{\partial V})$ is the kernel of $H_1(K_{\partial V})\to H_1(K_V)$.
From the commutative diagram:
$$\begin{CD} H_2(K_V,K_{\partial V}) @>0>> H_2(V,\partial V)\\
@VVV @VVV\\
H_1(K_{\partial V})@>>> H_1(\partial V)@>\subset>> H_1(Z(m)),
\end{CD} $$
we conclude the kernel of $H_1(K_{\partial V})\to H_1(K_V)$ is contained
in the kernel of $H_1(K_{\partial V})\to H_1(Z(m))$. Denote $A=H_1(K_V)$, $B={\rm Im}\{H_1(K_{\partial V})\to H_1(K_V)\}$. Since $A$ is a finitely generated abelian group,
$A=\bar{B}\oplus {\mathbb Z} [u_1]\oplus\cdots\oplus{\mathbb Z} [u_t]$ where $\bar{B}/B$ is torsion and $[u_i]\in A$, for $1\leq i\leq t$. Take $m>0$ to be the least common multiple of the orders of elements in $\bar{B}/B$ (equivalently, the maximal order of torsion elements in $H_1(K_V,K_{\partial V};{\mathbb Z})$). Then $\psi:A\to H_1(Z(m))$ can be constructed as follows.
Let $\eta\in \partial V$ be a slope intersecting the filling slope $\zeta\subset\partial V$ in one point, such that $i^e_*[\eta]=[\gamma]$. For any $[u_i]$, $1\leq i\leq t$, define $\psi([u_i])=[\tilde{u}_i]\in H_1(Z(m))$ such that $i^e_*[\tilde{u}_i]=f_{V*}[u_i]$. For any $[v]\in \bar{B}$ with order $s>0$ in $\bar{B}/B$, let $[w]\in H_1(K_{\partial{V}})$ be such that $j_*(w)=s\,[v]$. Note $f_{V*}(s\,[v])=s\,f_{V*}[v]=sb\,[\gamma]$, for some integer $b$. Then $i^e_*f_{\partial V*}[w]=i^e_*(sb\,[v])$. This means $f_{\partial V*}[w]=sb\,[\eta]+c\,[\zeta]$ for some integer $c$. Since $s$ divides $m$ by the choice of $m$, $b[\eta]+\frac{c}{s}[\zeta]\in H_1(\partial V)+{\mathbb Z}\frac{[\zeta]}{m}\cong H_1(Z(m))$. We may define $\psi([v])=b\,[\eta]+\frac{c}{s}\,[\zeta]$. It is straightforward to check $\psi$ is a well-defined homomorphism as required.
\end{proof}
For any ${\mathbb R}$-coefficient chain of PL singular $2$-simplices into $M$, its area is known as the sum of the unsigned pull-back areas of the $2$-simplices, weighted by the absolute values of the coefficients. Because any PL singular relative ${\mathbb Z}$-cycle which represents a nontrivial element of $H_2(V,\partial V;{\mathbb Z})\cong{\mathbb Z}$ will have arbitrarily large area if $V$ has sufficiently large radius, to apply the criterion in Lemma \ref{factorCriterion}, it suffices to show that $H_2(K_V,K_{\partial V};{\mathbb Q})$ has a generating set whose elements are represented by relative ${\mathbb Z}$-cycles each with area bounded in terms of $\ell(\mathcal{P})$.
\begin{lemma}\label{smallGen} There is a generating set of $H_2(K_V,K_{\partial V};{\mathbb Q})$ whose elements are represented by relative ${\mathbb Z}$-cycles each with area bounded by $A(\ell(\mathcal{P}))$, where $A(n)=27^n(9n^2+4n)\pi$.\end{lemma}
\begin{proof} Without loss of generality, we may assume $K_V$ does not contain the $0$-simplex $*$ of $K$. Because $V$ is convex and each $2$-simplex of $K$ has convex image within the hyperbolic \hyperlink{term-piece}{piece}, $K_V$ is a finite union of \emph{$0$-handles} (half-disks), \emph{$1$-handles} (bands), \emph{monkey-handles} (hexagons), and possibly a few \emph{isolated disks} (disks whose boundary do not meet the $1$-skeleton of $K$), cf. Figure \ref{figKV}. It is clear that the number of monkey-handles is at most the number of $2$-simplices, hence bounded by $\ell(\mathcal{P})$, and the union of $1$-handles in $K_V$ is an $I$-bundle over a (possible disconnected) graph. By fixing an orientation for each of them, the handles and the isolated disks give a CW-complex structure on $K_V$ in an obvious fashion. Let $\mathcal{C}_*(K_V,K_{\partial V})$, $\mathcal{Z}_*(K_V,K_{\partial V})$, $\mathcal{B}_*(K_V,K_{\partial V})$ denote the free ${\mathbb Z}$-modules of cellular relative chains, cycles and boundaries, respectively. $\mathcal{C}_2(K_V,K_{\partial V})$ has a natural basis consisting of the handles and the isolated disks.
\begin{figure}
\caption{$K_V$ and $K_{\partial V}
\label{figKV}
\end{figure}
To prove the lemma, it suffices to find a generating set for $\mathcal{Z}_2(K_V,K_{\partial V};{\mathbb Q})$ whose elements are in $\mathcal{Z}_2(K_V,K_{\partial V})\leq\mathcal{C}_2(K_V,K_{\partial V})$ with bounded coefficients over the natural basis. Decompose $K_V$ as: $$K_V=S_V\sqcup E_V\sqcup K'_V,$$ where $S_V$ is the union of the isolated disk components, $E_V$ is the union of the components that contain no monkey-handles, and $K'_V$ is the union of the components that contain at least one monkey-handle. Let $S_{\partial V}$, $E_{\partial V}$, $K'_{\partial V}$ be the intersection of $S_V$, $E_V$, $K'_V$ with $K_{\partial V}$, respectively. $$\mathcal{Z}_2(K_V,K_{\partial V};{\mathbb Q})=\mathcal{Z}_2(S_V,S_{\partial V};{\mathbb Q})\oplus\mathcal{Z}_2(E_V,E_{\partial V};{\mathbb Q})\oplus\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q}).$$
It suffices to find bounded generating relative ${\mathbb Z}$-cycles for the direct-summands separately.
First, consider $\mathcal{Z}_2(S_V,S_{\partial V};{\mathbb Q})$. Clearly, it has a generating set whose elements are the isolated disks. Hence absolute value of the coefficients over the natural basis are bounded $\leq 1$ for every element of the generating set.
Secondly, consider $\mathcal{Z}_2(E_V,E_{\partial V};{\mathbb Q})$. We show that it has a generating set whose elements have coefficients bounded $\leq2$ in absolute value over the natural basis. To see this, note $E=E_I\cup D_1\cup\cdots \cup D_s$ is a union of an $I$-bundle $E_I$ over a (possibly disconnected) graph $\Gamma_I$ together with $0$-handles $D_j$, $1\leq i\leq s$. $K_{\partial V}\cap E_I$ is an embedded $\partial I$-bundle $E_{\partial I}$. Now $\mathcal{Z}_2(E_I,E_{\partial I};{\mathbb Q})$ can be generated by all the relative ${\mathbb Z}$-cycles, in fact finitely many, of the following forms: (i) $A_I\in\mathcal{Z}_2(E_I,E_{\partial I})$, where $(A_I,A_{\partial I})\subset (E_I,E_{\partial I})$ is a sub-$I$-bundle which is an embedded annulus; or (ii) $R_I+2B_I+R'_I\in\mathcal{Z}_2(E_I,E_{\partial I})$, where $(R_I,R_{\partial I}),(R_I',R_{\partial I}')\subset (E_I,E_{\partial I})$ are sub-$I$-bundles which are embedded M\"obius strips, and $(B_I,B_{\partial I})\subset (E_I,E_{\partial I})$ is a sub-$I$-bundle which is an embedded band joining $R_I$ and $R'_I$. Moreover, $\mathcal{Z}_2(E_V,E_{\partial V};{\mathbb Q})\,/\,\mathcal{Z}_2(E_I,E_{\partial I};{\mathbb Q})$ can be generated by the residual classes represented by all the relative ${\mathbb Z}$-cycles, in fact finitely many, of the following forms: (i) $D_j+B_I\pm D_{j'}\in\mathcal{Z}_2(E_V,E_{\partial V})$, where $D_j$, $D_{j'}$ are distinct $0$-handles, and $(B_I,B_{\partial I})\subset(E_I,E_{\partial I})$ is a sub-$I$-bundle which is an embedded band joining $D_j$ and $D_{j'}$; or (ii) $2D_j+2B_I+R_I\in\mathcal{Z}_2(E_V,E_{\partial V})$, where $D_j$ is a $0$-handle, and $(R_I,R_{\partial I})\subset (E_I,E_{\partial I})$ is a sub-$I$-bundle which is an embedded M\"obius strip, and $(B_I,B_{\partial I})\subset (E_I,E_{\partial I})$ is a sub-$I$-bundle which is an embedded band joining $D_j$ and $R_I$.
All these relative ${\mathbb Z}$-cycles together generate $\mathcal{Z}_2(E_V,E_{\partial V};{\mathbb Q})$, and each of them has coefficients bounded $\leq2$ in absolute value over the natural basis.
Finally, consider $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$. We show that it has a generating set whose elements have coefficients bounded $\leq27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})+4)$ in absolute value over the natural basis. To see this, note $K'_V=K'_I\cup D_1\cup\cdots\cup D_s\cup F_1\cdots F_t$ is a union of an $I$-bundle $K'_I$ over a (possibly disconnected) graph, and $0$-handles $D_j$, $1\leq j\leq s$, and monkey-handles $F_k$, $1\leq k\leq t$. Note $t\leq\ell(\mathcal{P})$. Moreover, $K'_I=B_1\cup\cdots\cup B_r$ is a union of $1$-handles $B_i$, $1\leq i\leq r$, and it is also a disjoint union of components $K'_{I,1},\cdots, K'_{I,p}$, where $p\leq 3t\leq 3\ell(\mathcal{P})$.
Let $\bar\partial:\mathcal{C}_2(K'_V,K'_{\partial V})\to \mathcal{C}_1(K'_V,K'_{\partial V})$ be the relative boundary operator. Then $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$ is by definition the solution space of: $$\bar\partial U=0,$$
for $U\in\mathcal{C}_2(K'_V,K'_{\partial V};{\mathbb Q})$.
We shall first solve the residual equation $\bar\partial U=0$ modulo $\mathcal{B}_1(K'_I,K'_{\partial I})$, then lift a set of fundamental solutions to solutions of $\bar\partial U=0$ by adding chains from $\mathcal{C}_2(K'_I,K'_{\partial I})$. This set of solutions together with a generating set of $\mathcal{Z}_2(K'_I,K'_{\partial I};{\mathbb Q})$ will be a generating set of $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$.
To solve $\bar\partial U=0$ modulo $\mathcal{B}_1(K'_I,K'_{\partial I})$, we write:
$$U=\sum_{i=1}^r x_i\,B_i+\sum_{j=1}^s y_j\,D_j+\sum_{k=1}^t z_k\,F_k.$$
The topological interpretation of $\bar\partial U$ modulo $\mathcal{B}_1(K'_I,K'_{\partial I})$ is the total `contribution' of the base elements $B_i$, $D_j$, $F_k$'s to the fiber of each component of $K'_I$.
To make sense of this, on each component $K'_{I,l}$ of $K'_I$, we pick an oriented fiber $\varphi_l$, $1\leq l\leq p$. Note $\mathcal{C}_1(K'_V,K'_{\partial V})=\mathcal{C}_1(K'_I,K'_{\partial I})=\mathcal{C}_1(K'_{I,1},K'_{\partial I,1})\oplus\cdots\oplus \mathcal{C}_1(K'_{I,p},K'_{\partial I,p})$, and:
$$\mathcal{C}_1(K'_I,K'_{\partial I})\,/\,\mathcal{B}_1(K'_I,K'_{\partial I})\cong{\mathbb Z}^{\oplus p},$$ generated by $\varphi_1,\cdots, \varphi_p\bmod\mathcal{B}_1(K'_I,K'_{\partial I})$.
The contribution of $B_i$, $D_j$, $F_k$ on $\phi_l$ is formally the value of $\bar\partial B_i$, $\bar\partial D_j$, $\bar\partial F_k$ modulo $\mathcal{B}_1(K'_I,K'_{\partial I}))$ on the $l$-th direct-summands. In other words, we count algebraically how many components of $\bar\partial B_i$ is parallel to $\varphi_l$ in $K'_{I,l}$, and similarly for $\bar\partial D_j$, $\bar\partial F_k$. In this sense, on any $\varphi_l$, each $B_i$ contributes $0$ or $\pm 2$, each $D_j$ contributes $0$ or $\pm 1$, and each $F_k$ contributes $0$, $\pm 1$, $\pm 2$ or $\pm 3$. Let $\vec{u}$ be the column vector of coordinates $(x_1,\cdots,x_r,y_1,\cdots, y_s,z_1,\cdots,z_t)^T$, and $q=r+s+t$. Let $a_{lm}$ be the contribution of the $m$-th basis vector (corresponding to some $B_i$, $D_j$ or $F_k$) on $\varphi_l$. Thus, $a_{lm}$ are integers satifying $|a_{lm}|\leq 3$, for $1\leq l\leq p$, $1\leq m\leq q$, and $\sum_{l=1}^p |a_{lm}|\leq 3$, for $1\leq m\leq q$. The residual equation $\bar\partial U=0\bmod B_1(K'_I,K'_{\partial I})$ becomes a linear system of equations: $$A\vec{u}=\vec{0},$$ where $A=(a_{lm})$ is a $p\times q$ integral matrix. Every column of $A$ has at most $3$ nonzero entries, and the sum of their absolute values is at most $3$. Our aim is to find a set of fundamental solutions over ${\mathbb Q}$ with bounded integral entries.
Picking out a maximal independent collection of equations if necessary, we may assume $p$ equals the rank of $A$ over ${\mathbb Q}$. We may also re-order the coordinates and assume the first $p$ columns of $A$ are linearly independent over ${\mathbb Q}$. Let $A=(P,Q)$ where $P$ consists of the first $p$ columns and $Q$ of the rest $q-p$ columns. Let $\vec{u}=\left(\begin{array}{c}\vec{v}\\ \vec{w}\end{array}\right)$ be the corresponding decomposition of coordinates. Then the linear system becomes $P\vec{v}+Q\vec{w}=\vec{0}$. Basic linear algebra shows that a set of fundamental solutions
is $\vec{v}_n=-P^{-1}Q\,\vec{e}_n$, $\vec{w}_n=\vec{e}_n$, where $1\leq n\leq q-p$ and $(\vec{e}_1,\cdots,\vec{e}_{q-p})$ is the natural basis of ${\mathbb R}^{q-p}$. We clear the denominator by letting ${\vec{v}_n}^*=-P^*Q\,\vec{e}_n$, ${\vec{w}_n}^*=\det(P)\,\vec{e}_n$, where $P^*$ is the adjugate matrix of $P$. The corresponding $\vec{u}^*_1,\cdots, \vec{u}^*_{q-p}$ is a set of fundamental solutions over ${\mathbb Q}$ of the linear system $A\vec{u}=\vec{0}$ with integral entries.
For each $1\leq n\leq q-p$, $\vec{u}^*_n$ has at most $p+1$ non-zero entries, and the absolute value of the entries are all bounded $3^p$. Indeed, $\vec{u}^*$ has at most $p+1$ non-zero entries by the way we picked $\vec{v}^*_n$ and $\vec{w}^*_n$. To bound the absolute value of entries, note each column of $P$ has at most $3$ nonzero entries whose absolute value sum $\leq 3$.
It is easy to see $|\det(P)|\leq 3^p$ by an induction on $p$ using column expansions. Similarly, the absolute value of each entry of $P^*$ is at most $3^{p-1}$, and each column of $Q$ has at most $3$ nonzero entries whose absolute value sum $\leq 3$, so the absolute value of any entry of $-P^*Q$ is also $\leq 3^p$.
Let $U^*_1,\cdots, U^*_{q-p}\in\mathcal{C}_2(K'_V,K'_{\partial V})$ be the relative $2$-chains corresponding to the fundamental solutions $\vec{u}^*_1,\cdots,\vec{u}^*_{q-p}$ respectively as obtained above. Then the $U^*_n$'s form
a set of fundamental solutions to $\bar\partial U=0\bmod \mathcal{B}_1(K'_I,K'_{\partial I})$. To lift $U^*_n$
to a solution of $\bar\partial U=0$, note $\bar\partial U^*_n$ is the ${\mathbb Z}$-algebraic sum of $1$-simplices each parallel to a fiber $\varphi_l$. For a $1$-simplex $\sigma$ parallel to $\varphi_l$ coming from $\bar\partial U^*_n$, we pick a sub-$I$-bundle of $K'_{I,l}$ which is an embedded band joining $\sigma$ and $\varphi_l$, and let $L_n\in \mathcal{C}_2(K'_I,K'_{\partial I})$ be the relative ${\mathbb Z}$-chain which is the algebraic sum of all such sub-$I$-bundles.
Since each sub-$I$-bundle as a relative ${\mathbb Z}$-chain has coefficient bounded by $1$ in absolute value over the natural basis, the absolute values of coefficients of $L_n$ are bounded $\leq 3\cdot3^p(p+1)=3^{p+1}(p+1)$.
Let $\hat{U}_n=U^*_n-L_n$, $1\leq n\leq q-p$, then $\bar\partial \hat{U}_n=0$, with coefficients bounded $\leq 3^{p+1}(p+1)+3^p=3^p(3p+4)$ in absolute value.
In other words, $\hat{U}_n\in\mathcal{Z}_2(K'_V,K'_{\partial V})$, $1\leq n\leq q-p$. Moreover, $\hat{U}_n$'s together with a generating set of $\mathcal{Z}_2(K'_I,K'_{\partial I};{\mathbb Q})$ generate $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$. Note $K'_I$ has no monkey-handle, the no-monkey-handle case implies that $\mathcal{Z}_2(K'_I,K'_{\partial I};{\mathbb Q})$ has a generating set of relative ${\mathbb Z}$-cycles with coefficients bounded by $2$ in absolute value. Therefore, $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$ has a generating set of relative ${\mathbb Z}$-cycles, consisting of $\hat{U}_n$'s and the generating set of $\mathcal{Z}_2(K'_I,K'_{\partial I};{\mathbb Q})$ as above, with coefficients bounded by $3^p(3p+4)$ in absolute value. Remember $p\leq 3t\leq 3\ell(\mathcal{P})$, the absolute values of coefficients are bounded $\leq 3^{3\ell(\mathcal{P})}(3\cdot 3\ell(\mathcal{P})+4)=27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})+4)$.
To sum up, putting the generating sets of $\mathcal{Z}_2(S_V,S_{\partial V};{\mathbb Q})$, $\mathcal{Z}_2(E_V,E_{\partial V};{\mathbb Q})$, $\mathcal{Z}_2(K'_V,K'_{\partial V};{\mathbb Q})$ together, we obtain a generating set of $\mathcal{Z}_2(K_V,K_{\partial V};{\mathbb Q})$ of relative ${\mathbb Z}$-cycles with coefficients bounded by $27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})+4)$ over the natural basis. In particular, they represent homology classes that generate $H_2(K_V,K_{\partial V};{\mathbb Q})$. Remember the natural basis of $\mathcal{C}_2(K_V,K_{\partial V})$ consists of handles and isolated disks, whose total area is bounded by $\pi\ell(\mathcal{P})$. Therefore, the generating set consists of relative ${\mathbb Z}$-cycles with area bounded $\leq27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})+4)\cdot{\mathrm{Area}}(K_V)\leq A(\ell(\mathcal{P}))$, where $A(n)=27^n(9n^2+4n)\pi$.
\end{proof}
The following lemma bounds the torsion orders of $H_1(K_V,K_{\partial V};{\mathbb Z})$:
\begin{lemma}\label{smallTor} The maximal order of torsion elements of $H_1(K_V,K_{\partial V};{\mathbb Z})$ is bounded by
$T(\ell(\mathcal{P}))$, where $T(n)=2\cdot 3^n$.\end{lemma}
\begin{proof} It suffices to show that for any component $C_V$ of $K_V$, the order of torsion
elements of $H_1(C_V,C_{\partial V};{\mathbb Z})$ is at most $T(\ell(\mathcal{P}))$, where $C_{\partial V}
=K_{\partial V}\cap C_V$. If $C_V$ is an isolated disk, $H_1(C_V,C_{\partial V};{\mathbb Z})$ is trivial.
Thus we may assume $C_V$ is a union of $0$-handles, $1$-handles and monkey-handles.
Let $E_V$ be a maximal union of $1$-handles in $C_V$ which forms a trivial $I$-bundle over
a (possibly disconnected) finite graph (we also include in $E_V$ isolated edges which are not contained
in any such trivial $I$-bundle, which may be thought of as trivial $I$-bundles over isolated vertices
of the finite graph).
Suppose $E_V=E^1_V\sqcup\cdots\sqcup E^s_V$, where
each $E^j_V$ is a connected component, and let $e^j$ be a (directed) fiber of $E^j_V$.
Clearly, $H_1(E_V,E_{\partial V};{\mathbb Z})$, where $E_{\partial V}=E_V\cap K_{\partial V}$,
is torsion-free, rank-$s$, spanned by:
$$[e^1],\cdots,[e^s].$$
Moreover, $H_1(C_V,C_{\partial V};{\mathbb Z})$ is generated by these $[e^j]$'s as well.
Suppose $\sigma^1,\cdots,\sigma^r$ are the rest of the handles of $C_V$, {\it i.e.} which are not
in $E_V$. The boundary of each $\sigma^i$ gives a linear combination:
$$a_{i1}\,[e^1]+\cdots +a_{is}\,[e^s]\in H_1(E_V,E_{\partial V};{\mathbb Z}).$$
Moreover, if $\sigma^i$ is a $0$-handle or $1$-handle, there is only one
non-zero coefficient which is $\pm 1$ or $\pm 2$, respectively, (the $1$-handle
case follows from the maximality of $E_V$). If $\sigma^i$ is a monkey-handle,
the absolute value of coefficients sum up
to $3$, so in particular, at most three entries are non-zero. Thus we obtain an integral $r\times s$-matrix
$A=(a_{ij})$, which is a presentation matrix of the module $H_1(C_V,C_{\partial V};{\mathbb Z})$,
so that at most $t$ rows have more than one non-zero entry, where $t\leq\ell(\mathcal{P})$.
We may suppose these are the first $t$ rows, forming a $t\times s$-submatrix
$A'$, and the rest of the $(r-t)$ rows form a $(r-t)\times s$-submatrix $A''$.
Note the entries of $A'$ have absolute value at most $3$, so the order of any torsion elements
of ${\rm Coker}(A')$ is bounded by the greatest common divisor of the
minors of $A'$ of square submatrices of size $\mathrm{rank}(A')$, and hence is bounded by $3^t$. As $H_1(C_V,C_{\partial V};{\mathbb Z})$
is the quotient of $A'$ by further killing relators given by rows of $A''$, which at most doubles
the order of the torsion, we conclude
that the orders of torsion elements of $H_1(C_V,C_{\partial V};{\mathbb Z})$ is
at most $2\cdot 3^t$, where $t\leq \ell(\mathcal{P})$. This completes the proof.
\end{proof}
To finish the proof of Theorem \ref{factor-hyp}, if the area of the meridian disk of $V$ is larger than $A(\ell(P))$ as in Lemma \ref{smallGen}, then $f_*:H_2(K_V,K_{\partial V})\to H_2(V,\partial V)$ vanishes.
This amounts to requiring the radius $r$ of $V$ satisfy: $$\pi\sinh^2(r)>A(\ell(\mathcal{P})).$$
If $\gamma$ is so short that this inequality holds, by Lemma \ref{factorCriterion}, we may factorize any $f:K\to M$ up to homotopy, and hence any $\phi:G\to\pi_1(M)$, through the \hyperlink{term-extDehnFilling}{extended Dehn filling}. The denominator of the drilling is bounded by the order of the torsion of $H_1(K_V,K_{\partial V})$ by Lemma \ref{factorCriterion}. By Lemma \ref{smallTor}, the order of the torsion is bounded by $T(\ell(G))$. This completes the proof of Theorem \ref{factor-hyp}.
\subsection{Drilling a sharp cone-fiber in a Seifert fibered piece}
In this subsection, we show a similar result to Theorem \ref{factor-hyp} for Seifert fibered \hyperlink{term-piece}{pieces}, that maps factorize through the \hyperlink{term-extDrilling}{extended drilling} of an exceptional fiber at a sharp cone point in a Seifert fibered \hyperlink{term-piece}{piece}. To make this precise, we need recall some facts about Seifert fibered spaces.
Let $J$ be an orientable compact Seifert fibered space. The interior of $J$ may be regarded as an $S^1$-bundle over a finitely generated $2$-orbifold $\mathcal{O}$. In general, $\mathcal{O}$ is isomorphic to a surface with cone points and/or punctures $F(q_1,\cdots,q_s)$, where $F$ is a closed (possibly non-orientable) surface, and each integer $1<q_i\leq\infty$ ($1\leq i\leq s$) corresponds to either a cone point on $F$ with the cone angle $\frac{2\pi}{q_i}$ if $q_i<\infty$, or a puncture if $q_i=\infty$. $\mathcal{O}$ can be endowed with a complete hyperbolic structure of finite area if and only if the orbifold Euler characteristic $\chi(\mathcal{O})=\chi(F)-\sum_{i=1}^s(1-\frac{1}{q_i})$ is negative, where $\frac{1}{\infty}=0$ by convention. In other words, in this case $J$ is either ${\mathbb E}\times{\mathbb H}^2$-geometric or $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$-geometric. In an orientable aspherical compact $3$-manifold $M$, any Seifert fibered \hyperlink{term-piece}{piece} $J$ whose base $2$-orbifold has a sufficiently sharp cone point (i.e. the cone angle is sufficiently small) is ${\mathbb E}\times{\mathbb H}^2$-geometric, unless $M$ is itself $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$-geometric. In fact, if $M$ is neither $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$-geometric nor virtually solvable, any Seifert \hyperlink{term-piece}{piece} $J$ is either ${\mathbb E}\times{\mathbb H}^2$-geometric or homeomorphic to the nontrivial $S^1$-bundle over a M\"obius strip.
\begin{theorem}\label{factor-SF} Let $G$ be a finitely presented group, and $M$ be an orientable aspherical compact $3$-manifold. Suppose there is a sufficiently sharp cone point in the base $2$-orbifold of
a Seifert fibered \hyperlink{term-piece}{piece} $J\subset M$, depending only on the \hyperlink{term-prLen}{presentation length} $\ell(G)$. Let $\gamma\subset J$ be the corresponding exceptional fiber, and $N=M-\gamma$ be the drilling along $\gamma$. Then any homomorphism $\phi:G\to\pi_1(M)$ factors through the \hyperlink{term-extDehnFilling}{extended Dehn filling} epimorphism $\iota^e:\pi_1(N^e)\to \pi_1(M)$ of some denominator $m\leq T(\ell(G))$. Namely, $\phi=\iota^e\circ\phi^e$ for some $\phi^e:G\to\pi_1(N^e)$.\end{theorem}
The proof is almost the same as the hyperbolic case, so we only give a sketch highlighting necessary modifications.
We may assume $J$ is either ${\mathbb E}\times{\mathbb H}^2$-geometric or $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$-geometric, and let $J_{\mathtt{geo}}=(\mathring{J},\rho)$ be the interior of $J$ with a complete Riemanianian metric $\rho$ of finite volume, induced by a complete hyperbolic structure on its base $2$-orbifold $\mathcal{O}$ requiring the length of any ordinary fiber to be $1$. Let $x\in\mathcal{O}$ be the cone point as assumed with cone angle $\frac{2\pi}{q}$. A result of Gaven J. Martin implies that for any complete hyperbolic $2$-orbifold $\mathcal{O}$ with a cone point of angle $\frac{2\pi}{q}$, there is an embedded cone centered at the point with radius $r$ satisfying:
$$\cosh(r)=\frac{1}{2\sin\frac{\pi}{q}},$$
which is optimal in $S^2(2,3,q)$, (cf. \cite[Theorem 2.2]{Martin96}). Applying to $x$ as above, the preimage of the embedded cone in $J_{\mathtt{geo}}$ is a tube, which will have very large radius if the cone is very sharp.
There is a natural notion of the horizontal area of a PL singular $2$-complex in $J_{\mathtt{geo}}$, heuristically the area of its projection on the base $2$-orbifold. Formally, let $\tilde{\omega}$ be the pull-back of the area form of ${\mathbb H}^2$ via the natural projection ${\mathbb E}\times{\mathbb H}^2\to{\mathbb H}^2$ or $\widetilde{{\mathrm{SL}}}_2({\mathbb R})\to{\mathbb H}^2$, which is invariant under isometry. As it is invariant under the holonomy action of $\pi_1(J_{\mathtt{geo}})$, $\tilde{\omega}$ descends to a $2$-form $\omega$ on $J_{\mathtt{geo}}$. For any PL singular $2$-simplex $j:\Delta\to J_{\mathtt{geo}}$, we define the \emph{horizontal area} to be: $${\mathrm{Area}}^{\tt h}(j(\Delta))=\left|\,\int_\Delta j^*\omega\,\right|,$$ and define the horizontal area of a ${\mathbb R}$-coefficient PL singular $2$-chain in $J_{\mathtt{geo}}$ to be the sum of the horizontal areas of its simplices, weighted by the absolute values of coefficients.
Because for the ${\mathbb E}\times{\mathbb H}^2$-geometry, resp. for the $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$-geometry, any path in $J_{\mathtt{geo}}$ can be pulled straight, namely, homotoped rel end-points to a unique geodesic segment. Moreover, any immersed $2$-simplex in $J_{\mathtt{geo}}$ can be homotoped rel vertices to a ruled $2$-simplex with geodesic sides. In fact, let $\Delta=[0,1]\times[0,1]/\sim$ where $(t,0)\sim (t',0)$ for any ($0\leq t,t'\leq 1$), and $j:\Delta\to J_{\mathtt{geo}}$ be an immersion in its interior. One may first pull straight the sides by homotopy, then simultaneously homotope so that $j(\seq{t}\times[0,1])$ becomes geodesic for every $0\leq t\leq 1$. Note every geodesic in ${\mathbb E}\times{\mathbb H}^2$, resp. in $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$, projects to a geodesic in ${\mathbb H}^2$, it is clear that any ruled $2$-simplex (lifted) in ${\mathbb E}\times{\mathbb H}^2$, resp. in $\widetilde{{\mathrm{SL}}}_2({\mathbb R})$, projects to a totally geodesic triangle in ${\mathbb H}^2$. This implies any ruled $2$-simplex in $J_{\mathtt{geo}}$ has horizontal area at most $\pi$. More generally, ruled triangular $2$-complexes in $J_{\mathtt{geo}}$ with $m$ $2$-simplices have horizontal area at most $m\pi$.
To prove Theorem \ref{factor-SF}, pick a presentation $2$-complex $K$ of a triangular presentation $\mathcal{P}$ achieving $\ell(G)$, and a PL map $f:K\to M$ realizing $\phi$.
By picking a sufficiently small $\epsilon<\epsilon_2$, we may endow $M$ with a complete Riemannian metric such that $J$ is isometric to $J_{\mathtt{geo}}$ with the $\epsilon$-thin tubes corresponding to $\partial J$ removed. We pull the part of $f^{-1}(J)$ straight, namely, homotope it rel $f^{-1}(\partial J)$ to a ruled $2$-complex. If $\epsilon$ is sufficiently small, we may assume the horizontal area of $f^{-1}(J)$ to be at most $\pi\ell(\mathcal{P})$ by the discussion above.
Suppose $\gamma$ is the singular fiber in $J$ with sufficiently small cone angle, then there is an embedded tube $V\subset J$ containing $\gamma$ with sufficiently large radius. Since $V$ is convex and the simplices meeting $V$ are ruled, it is easy to see that $K_V=f^{-1}(V)$ is a finite union of $0$-handles, $1$-handles and monkey-handles and possible a few isolated disks. The number of monkey-handles is at most the number of simplices $\ell(P)$. Let $K_{\partial V}=f^{-1}(\partial V)$. Now the horizontal area of $K_V$ is at most $\pi\ell(\mathcal{P})$.
The factorization criterion in Lemma \ref{factorCriterion} is a general fact which also applies here.
By the same argument as Lemma \ref{smallGen}, $H_2(K_V,K_{\partial V};{\mathbb Q})$ has a generating set whose elements are represented by relative ${\mathbb Z}$-cycles with horizontal area bounded by $27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})+4)\cdot{\mathrm{Area}}^{\tt h}(K_V)\leq
27^{\ell(\mathcal{P})}(9\ell(\mathcal{P})^2+4\ell(\mathcal{P}))\pi$. Thus, if $\gamma$ is an exceptional fiber with the corresponding cone angle sufficiently small such that it has a tubular neighborhood $V\subset M$ of radius $r$ satisfying: $$\pi\sinh^2(r)>A(\ell(\mathcal{P})),$$ where
$A(n)=27^n(9n^2+4n)\pi$, $f_*:H_2(K_V,K_{\partial V})\to H_2(V,\partial V)$ vanishes. This implies $f:K\to M$ factors through the \hyperlink{term-extDehnFilling}{extended Dehn filling} $i^e:N^{e(\zeta,m)}\to M$ up to homotopy, where denominator $m$ of the drilling is bounded by the order of the torsion of $H_1(K_V,K_{\partial V})$ by Lemma \ref{factorCriterion}. By Lemma \ref{smallTor}, the order of the torsion is bounded by $T(\ell(G))$.
This completes the proof of Theorem \ref{factor-SF}.
\section{A bound of the simplicial volume}\label{Sec-volume}
In this section, we give an upper-bound of the volume of $M$ in terms of $G$, under the assumptions of Theorem \ref{main}. This gives some restrictions to the geometry of the hyperbolic \hyperlink{term-piece}{pieces} of $M$, which will be useful in Section \ref{Sec-homeoTypes}. For the purpose of certain independent interest, we prefer to prove a slightly more general result, allowing $M$ to be an compact orientable aspherical $3$-manifold with tori boundary.
For any compact orientable manifold $M$ with tori boundary, we denote the simplicial volume of $M$ as $v_3\norm{M}$, where $v_3\approx 1.01494$ is the volume of an ideal regular hyperbolic tetrahedron and $\norm{M}$ stands for the Gromov norm. We prove the following theorem.
\begin{theorem}\label{volBound} Suppose $G$ is a finitely presented group with $b_1(G)=1$, and $M$ is an orientable compact aspherical $3$-manifold with (possibly empty) tori boundary. If $G$ maps onto $\pi_1(M)$, then: $$v_3\norm{N}\leq\pi\ell(G).$$
\end{theorem}
More generally, one may assume that $G$ is only finitely
generated with $b_1(G)=1$ in this theorem, since any such group is the quotient of a finitely presented group $G' \twoheadrightarrow G$ with $b_1(G')=1$.
We shall prove Theorem \ref{volBound} in the rest of this section.
The idea is as follows. First take a finite $2$-complex $K$ realizing a triangular presentation $\mathcal{P}$ which achieves $\ell(G)$.
Take a PL map $f:K\to M$ realizing an epimorphism $\phi:G\twoheadrightarrow\pi_1(M)$. We first show that $M-f(K)$ consists of elementary components, in the sense that the inclusion of any such component induces a homomorphism on $\pi_1$ with abelian image. By `pulling straight' $f$ in hyperbolic \hyperlink{term-piece}{pieces} of $M$ via homotopy, we may apply an isoperimetric inequality to bound the sum of their volumes by $\pi\ell(\mathcal{P})$. Then the theorem follows because $v_3\norm{M}$ is equal to the sum of the volume of hyperbolic \hyperlink{term-piece}{pieces}.
We first show $M-f(K)$ consists of elementary components. The approach we are taking here is a `drilling argument' which will also be used to prove Theorem \ref{main}.
\begin{prop}\label{elemCplment} Let $K$ be a finite $2$-complex with $b_1(K)=1$, and $M$ be an orientable compact aspherical $3$-manifold. Suppose $f:K\to M$ is a PL map (with respect to any PL structures of $K$, $M$) which induces an epimorphism on the fundamental group. Then $M-f(K)$ consists of elementary components, i.e. whose inclusion into $M$ has abelian $\pi_1$-image. \end{prop}
\begin{proof}
This is trivial if $M$ is itself elementary. We shall assume $M$ to be non-elementary without loss of generality.
To argue by contradiction, suppose there is a non-elementary component $U$ of $M-f(K)$. We may take an embedded finite connected simplicial graph $\Gamma\subset U$ such that $\Gamma$ is non-elementary in $M$, i.e. $\pi_1(\Gamma)\to \pi_1(M)$ has non-abelian image. Let $N=M-\Gamma$. Observe that $N$ is aspherical, because if there is an embedded sphere in $N$, it bounds a ball in $M$. This ball cannot contain $\Gamma$ as $\Gamma$ is non-elementary, and hence the ball is contained in $N$. Thus, $N$ is irreducible, and therefore aspherical by the Sphere Theorem \cite{Papa}. Denote the induced map:
$$f':K\to N.$$ By picking base points of $K$ and $N$, there is an induced homomorphism $f'_\sharp:\pi_1(K)\to\pi_1(N)$. Note ${\rm Im}(f'_\sharp)\leq\pi_1(N)$ is in general of infinite index as $b_1(K)=1$ and $b_1(N)>1$. We consider the covering space $\tilde{N}$ of $N$ corresponding to ${\rm Im}(f'_\sharp)$, with the covering map:
$$\kappa:\tilde{N}\to N.$$
Assume we can prove $\chi(\tilde{N})<0$, and hence $b_1(\tilde{N})>1$ at this point, then we obtain a contradiction because $b_1(K)=1$ and $\pi_1(\tilde{N})\cong{\rm Im}(f'_\sharp)$. We shall show $\chi(\tilde{N})<0$ in a separate lemma, (Lemma \ref{notTiny}), and with that done, the proof is completed.
\end{proof}
\begin{lemma}\label{notTiny} With the assumptions in the proof of Proposition \ref{elemCplment}, $$\chi(\tilde{N})<0.$$\end{lemma}
\begin{proof} Let ${\mathcal{T}}$ be the JSJ tori of $N$. Note $\partial M$ is at most a torus under the assumption that $f:K\to M$ is $\pi_1$-surjective and $b_1(K)=1$. Consider the JSJ decomposition of $N$. Then the \hyperlink{term-piece}{piece} $Y$ containing the component of $\partial N$ coming from drilling $\Gamma$ is necessarily hyperbolic, and $\chi(Y)<0$. Let $Y_\Gamma\subset M$ be the union of $Y$ with the component of $M-Y$ that contains $\Gamma$. To argue by contradiction, suppose $b_1(\tilde{N})=1$.
By Proposition \ref{ScottCore} $\tilde{N}$ has an aspherical Scott core $C$
such that $C\cap\kappa^{-1}({\mathcal{T}})$ are essential annuli and/or tori. Moreover, by Lemma \ref{simplifyCore}, $C$ has no non-central elementary \hyperlink{term-chunk}{chunk}, in particular, no elementary hyperbolic \hyperlink{term-chunk}{chunk}.
We claim that $C$ contains a hyperbolic \hyperlink{term-chunk}{chunk} $Q$ mapping to $Y$ under $C\subset\tilde{N}\stackrel{\kappa}\to N$.
To see this, note $f':K\to N$ factorizes as: $$K\stackrel{\tilde{f}'}{\longrightarrow} C\stackrel{\subset}{\longrightarrow} \tilde{N}
\stackrel{\kappa}{\longrightarrow} N,$$
up to homotopy. If $C$ had no hyperbolic \hyperlink{term-chunk}{chunk} mapping to $Y$, $f'$ would miss the interior of $Y_\Gamma$ up to homotopy.
Then $f:K\to M$ may be homotoped to $g:K\to M$ within $N$ such that $g(K)$ misses the interior of $Y_\Gamma$. Clearly $\partial Y_\Gamma$ has some component which is not parallel to $\partial M$, because otherwise either $\Gamma$ or $g(K)$ is contained in a collar neighborhood of $\partial M$. This either contradicts $\Gamma$ being non-elementary, or contradicts $f\simeq g$ being $\pi_1$-surjective as $M$ is assumed to be non-elementary. Let $T$ be such a component of $\partial Y_\Gamma$. $T$ cannot be incompressible in $M$, otherwise $g(K)$ is not surjective by the Van Kampen theorem or the HNN extension.
If $T$ is compressible, let $D\subset M$ be a compressing disk of $T$. One component of $\partial W$, where $W$ is a regular neighborhood of $D\cup T$, is a sphere $S\subset M$, which must bound a ball $B\subset M$. There are four cases:
if $D\subset Y_\Gamma$ and $B\subset Y_\Gamma$, then $Y_\Gamma$ is a solid torus containing $\Gamma$, which contradicts $\Gamma$ being non-elementary; if $D\subset Y_\Gamma$ and $B\subset (M\setminus\mathring{Y}_\Gamma)\cup W$, then $B$ contains $g(K)$, which contradicts $g$ being $\pi_1$-surjective; if $D\subset M\setminus\mathring{Y}_\Gamma$ and $B\subset M\setminus\mathring{Y}_\Gamma$, then $M\setminus\mathring{Y}_\Gamma$ is a solid torus containing $g(K)$, which contradicts the assumption that $M$ is non-elementary; if $D\subset M\setminus\mathring{Y}_\Gamma$ and $B\subset W\cup Y_\Gamma$, then $B$ contains $\Gamma$, which contradicts $\Gamma$ being non-elementary. This means $T$ cannot be compressible either. This contradiction proves the claim that $C$ must have some hyperbolic \hyperlink{term-chunk}{chunk} $Q$.
Now $Q$ is a non-elementary hyperbolic \hyperlink{term-chunk}{chunk} of $C$ by Lemma \ref{simplifyCore}. Let $\tilde{Y}$ be the component of $\kappa^{-1}(Y)$ containing $Q$. We have $\pi_1(Q)\cong\pi_1(\tilde{Y})\leq\pi_1(Y)$ is a non-elementary subgroup of $\pi_1(Y)$. Since $\chi(Y)<0$, we conclude $\chi(Q)<0$. Because $C$ cuts along annuli and tori into non-contractible \hyperlink{term-chunk}{chunks}, $$\chi(\tilde{N})=\chi(C)\leq\chi(Q)<0.$$ This implies $b_1(\tilde{N})>1$ as $H_n(\tilde{N})=0$ for $n>2$, a contradiction to the assumption that $b_1(\tilde{N})=1$.
\end{proof}
Let $J_1,\cdots, J_s$ ($s\geq0$) be the hyperbolic \hyperlink{term-piece}{pieces} in the JSJ decomposition of $M$ as assumed in Theorem \ref{volBound}. As before, we write $J_{i,\,{\mathtt{geo}}}=(\mathring{J}_i,\rho_i)$ for the interior of $J_i$ with the complete hyperbolic metric of finite volume. It is a well-known fact that only hyperbolic \hyperlink{term-piece}{pieces} contribute to the simplicial volume, namely, $$v_3\norm{M}=\sum_{i=1}^s{\mathrm{Vol}}(J_{i,\,{\mathtt{geo}}}),$$
cf. \cite[Theorem 1]{So}. Therefore, to prove Theorem \ref{volBound}, it suffices to bound the volume of hyperbolic \hyperlink{term-piece}{pieces}
of $M$, assuming $s>0$.
By picking a positive $\epsilon<\epsilon_3$, where $\epsilon_3$ is the Margulis constant for ${\mathbb H}^3$, we may endow $M$ with a complete Riemannian metric so that $J_i$ is isometric to $J_{i,\,{\mathtt{geo}}}$ with the $\epsilon$-thin horocusps corresponding to $\partial J_i$ removed, (remember the JSJ \hyperlink{term-piece}{pieces} are the components of $M$ with an open regular neighborhood of the JSJ tori removed).
Remember $K$ is a finite $2$-complex with a single base point $*$ and $2$-simplices corresponding to the relators of the triangular presentation $\mathcal{P}$. If $\epsilon>0$ is sufficiently small, we may homotope $f$ so that $*$ is not in any hyperbolic \hyperlink{term-piece}{piece}, and that $f(K)\cap J_i$ is totally geodesic on each $2$-simplex of $K$. As $K_{J_i}=f^{-1}(J_i)$ is a union of $1$-handles (bands) and at most $\ell({\mathcal{P}})$ monkey-handles (hexagons), we may bound the area of $K_{J_i}$ by $\pi m_i$, where $m_i$ is the number of monkey-handles in $K_{J_i}$, if $\epsilon>0$ is sufficiently small. Note $m_1+\cdots+m_s\leq\ell(\mathcal{P})$. Moreover, the area of $\partial J_i$ is bounded by $\frac{\epsilon^2A_i}{\epsilon_3^2}$, where $A_i$ is total area of the $\epsilon_3$-horocusp boundaries of $J_{i,\,{\mathtt{geo}}}$ corresponding to $\partial J_i$.
We need an isoperimetric inequality as below at this point (cf. \cite[Lemma 3.2]{Rafalski07}).
\begin{lemma}\label{isopIneq} Let $Y$ be a hyperbolic $3$-manifold, and $R\subset Y$ be a connected compact PL sub-$3$-manifold. If $R$ is elementary in $Y$, then $${\mathrm{Vol}}(R)\leq \frac{1}{2}{\mathrm{Area}}(\partial R).$$ \end{lemma}
\begin{proof} Pass to the covering of $X$ of $Y$ corresponding to the image of $\pi_1(R)\to\pi_1(Y)$, then there is a copy of $R$ in $X$ lifted from $R\subset Y$. As $R$ is elementary in $Y$, $\pi_1(X)$ is free abelian of rank $\leq2$, so we pick a $\pi_1$-injective map to a torus $f:X\to T^2$. Let $W\subset T^2$ be the union of two generator slopes on $T^2$ meeting in one point. We may assume $f$ and $f|_{\partial R}$ are transversal to both circles in $W$, then $\Sigma=f^{-1}(W)$ is a $2$-sub-complex in $X$ with finite area (since compact and measurable) such that the universal covering space $\tilde{X}$, isometric to ${\mathbb H}^3$, can be constructed by gluing copies of $C_g=X\setminus\Sigma$ indexed by $g\in\pi_1(X)$. Let $\kappa:\tilde{X}\to X$, then any connected component $\tilde{R}$ of $\kappa^{-1}(R)$ is a universal covering of $R$. To illustrate, consider $\pi_1(X)\cong{\mathbb Z}\oplus{\mathbb Z}$ for instance, and let $\alpha,\beta$ be two generators such that $C_\alpha\cap C_0\neq\emptyset$, $C_\beta\cap C_0\neq\emptyset$. For any $m>0$, let $\tilde{R}_n$ be the union of all the $\tilde{R}\cap C_{i\,\alpha+j\,\beta}$, where $-m\leq i,j\leq m$. It is clear ${\mathrm{Vol}}(\tilde{R}_m)=4m^2\,{\mathrm{Vol}}(R)$, and ${\mathrm{Area}}(\partial\tilde{R}_n)=4m^2\,{\mathrm{Area}}(\partial R)+2m\,{\mathrm{Area}}(\Sigma)$. Using the isoperimetric inequality in ${\mathbb H}^3$, we have $${\mathrm{Vol}}(\tilde{R}_m)\leq \frac{1}{2}{\mathrm{Area}}(\partial\tilde{R}_m).$$
This implies ${\mathrm{Vol}}(R)\leq\frac{1}{2}{\mathrm{Area}}(\partial R)$ as $m\to+\infty$. When $\pi_1(X)$ is isomorphic to ${\mathbb Z}$ or trivial, the argument is similar.
\end{proof}
To finish the proof of Theorem \ref{volBound}, by Proposition \ref{elemCplment}, the compactification of each components of $J_i\setminus f(K)$ is also elementary. Thus, by Lemma \ref{isopIneq},
${\mathrm{Vol}}(J_i)\leq \pi m_i+{\epsilon^2A_i}/{\epsilon_3^2}$.
We obtain:
$$\sum_{i=0}^s{\mathrm{Vol}}(J_i)\leq\pi\sum_{i=1}^s m_i+\frac{\epsilon^2}{\epsilon_3^2}\sum_{i=1}^sA_i\leq \pi\ell(\mathcal{P})+\frac{\epsilon^2A}{\epsilon_3^2},$$
where $A=A_1+\cdots+A_s$ is a constant independent of $\epsilon>0$. As $\epsilon\to0$, the left-hand side goes to
$v_3\norm{M}=\sum_{i=1}^s{\mathrm{Vol}}(J_{i,\,{\mathtt{geo}}})$, and the right-hand side goes to $\pi\ell(\mathcal{P})$. We conclude:
$$v_3\norm{M}\leq\pi\ell(\mathcal{P}).$$
This completes the proof of Theorem \ref{volBound}.
\section{The JSJ decomposition of knot complements}\label{Sec-cpnship}
In this section, we review the JSJ decomposition of knot complements following \cite{Bu}, and provide an equivalent data-structural description of a knot complement as a \hyperlink{term-rootedTree}{rooted tree} with vertices decorated by compatible geometric \hyperlink{term-node}{nodes}, (Proposition \ref{knotData}). This is in preparation of the proof of Theorem \ref{main}.
Let $k$ be a knot in $S^3$. For the knot complement $M=S^3-k$, i.e. $S^3$ with an open regular neighborhood of $k$ removed, the JSJ graph $\Lambda$ is a finite tree as every embedded torus in $S^3$ is separating. Moreover, $\Lambda$ has a natural rooted tree structure. \raisebox{\baselineskip}[0pt]{\hypertarget{term-rootedTree}} Recall that a finite tree is \emph{rooted} if it has a specified vertex, called the \emph{root}. The edges are naturally directed toward the root, thus every non-root vertex has a unique \emph{parent} adjacent to it. The adjacent vertices of a vertex except its parent are called its \emph{children}. Every vertex is contained in a unique \emph{complete rooted subtree}, namely the maximal subtree with the induced edge directions in which the vertex becomes the root.
\begin{definition}\label{rootedJSJ} \raisebox{\baselineskip}[0pt]{\hypertarget{term-rootedJSJTree}} For a knot complement $M$, the associated \emph{rooted JSJ tree} $\vec\Lambda$ is a rooted tree isomorphic to the JSJ tree $\Lambda$ of $M$ with the root corresponding to the unique JSJ \hyperlink{term-piece}{piece} containing $\partial M$.
\end{definition}
The \hyperlink{term-rootedJSJTree}{rooted JSJ tree} is related to the satellite constructions of knots. In fact, for any \hyperlink{term-rootedTree}{complete rooted subtree} $\vec\Lambda_{\mathtt{c}}\subset\vec\Lambda$, the subspace $M_{\mathtt{c}}\subset M$ over $\vec\Lambda_{\mathtt{c}}$ is homeomorphic to the complement of a knot $k_{\mathtt{c}}$ in $S^3$, and the subspace $N\subset M$ over $\vec\Lambda\setminus\vec\Lambda_{\mathtt{c}}$ is homeomorphic to the complement of a knot $k_{\mathtt{p}}$ in a solid torus $S^1\times D^2$ with the natural product structure. Thus $k$ is the satellite knot of $k_{\mathtt{c}}\subset S^3$ and $k_{\mathtt{p}}\subset S^1\times D^2$.
To give a more precise description of the JSJ \hyperlink{term-piece}{pieces} and how they are glued together, \cite{Bu} introduced the notion of \hyperlink{term-KGL}{KGLs}.
\begin{definition}[{\cite[Definition 4.4]{Bu}}]\label{KGL} \raisebox{\baselineskip}[0pt]{\hypertarget{term-KGL}}A \emph{knot-generating link} (KGL) is an oriented link $L=k_{\mathtt{p}}\sqcup k_{{\mathtt{c}}_1}\sqcup\cdots\sqcup k_{{\mathtt{c}}_r}\subset S^3$, ($r\geq 0$), such that
$k_{{\mathtt{c}}_1}\sqcup\cdots\sqcup k_{{\mathtt{c}}_r}$ is an oriented unlink.
\end{definition}
\begin{example}\label{KGL-example} \raisebox{\baselineskip}[0pt]{\hypertarget{term-KGLSF}} Figure \ref{figKGL} exhibits three families of \hyperlink{term-KGL}{KGLs}, namely, (right-handed) \emph{$r$-key-chain links} ($r>1$), \emph{$p/q$-torus knots} ($p,q$ coprime, $|p|>1$, $q>1$), and \emph{$p/q$-cable links} ($p,q$ coprime, $q>1$). Their complements are all Seifert fibered. \raisebox{\baselineskip}[0pt]{\hypertarget{term-KGLhyp}} There are also \emph{hyperbolic} KGLs, namely whose complements are hyperbolic, such as the Borromean rings with suitable assignments of components.
\begin{figure}
\caption{Three families of Seifert fibered KGLs.}
\label{figKGL}
\end{figure}
\end{example}
According to \cite{Bu}, the JSJ decomposition of knot complements may be described as below.
\begin{theorem}[{Cf. \cite[Theorem 4.18]{Bu}}]\label{BudneyThm}
Suppose $k$ is a nontrivial knot in $S^3$. Let $\vec\Lambda$ be the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} of $M=S^3-k$. Then:
\noindent(1) Every vertex $v\in\vec\Lambda$ is associated to a \hyperlink{term-KGL}{KGL}, $L_v=k_{\mathtt{p}}\sqcup k_{{\mathtt{c}}_1}\sqcup\cdots\sqcup k_{{\mathtt{c}}_r}\subset S^3$, satisfying the following requirements: the JSJ \hyperlink{term-piece}{piece} $J_v$ corresponding to $v$ is homeomorphic to $S^3-L_v$; $\partial J_v=\partial_{\mathtt{p}} J_v\sqcup\partial_{{\mathtt{c}}_1}J_v\sqcup\cdots\sqcup\partial_{{\mathtt{c}}_r}J_v$, $r\geq0$, where $\partial_{\mathtt{p}} J_v$ is the torus adjacent to the parent \hyperlink{term-piece}{piece} of $J_v$, or is $\partial M$ if $v$ is the root, and each $\partial_{{\mathtt{c}}_i}J_v$ is adjacent to a distinct child vertex of $v$; and if $v'$ is a child of $v$, and let $k_{{\mathtt{c}}'}\subset L_v$, $k'_{\mathtt{p}}\subset L_{v'}$ be the components so that $\partial_{{\mathtt{c}}'} J_v$ is glued to $\partial_{\mathtt{p}} J_{v'}$, then the meridian of $k_{{\mathtt{c}}'}$ is glued to the longitude of $k'_{\mathtt{p}}\subset S^3$ preserving orientations. Note the longitude and the meridian of a component of an oriented link are naturally oriented.
\noindent(2) There are only four possible families of \hyperlink{term-KGL}{KGLs} that could be associated to vertices of $\vec\Lambda$, namely, $r$-key-chain links ($r>1$), $p/q$-torus knots ($p,q$ coprime, $|p|>1$, $q>1$), $p/q$-cable links ($p,q$ coprime, $q>1$), and hyperbolic \hyperlink{term-KGL}{KGLs}. Furthermore, no key-chain-link vertex has a key-chain-link child.
\noindent(3) The associated \hyperlink{term-KGL}{KGLs} are canonical
up to unoriented isotopies of $L_v$'s with respect to the requirements.
Moreover, any \hyperlink{term-rootedJSJTree}{rooted tree} $\vec\Lambda$ with an
assignment of vertices to \hyperlink{term-KGL}{KGLs} satisfying the
properties above realizes a unique nontrivial knot $k$ in $S^3$ up to isotopy.
\end{theorem}
In view of the satellite construction, the minimal complete rooted subtree of $\vec\Lambda$
containing a cable-link vertex corresponds to the complement of a cable knot,
and the minimal \hyperlink{term-rootedTree}{complete rooted subtree} of $\vec\Lambda$
containing a key-chain-link vertex corresponds to the complement of a connected sum of knots.
For our purpose of use, we prefer to encode a \hyperlink{term-KGL}{KGL} by its complement, forgetting the embedding into $S^3$,
but remembering the children longitudes:
\begin{definition}\label{node} \raisebox{\baselineskip}[0pt]{\hypertarget{term-node}}
A \emph{node} is a triple $(J,\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}})$, $r\geq0$,
where $J$ is an oriented compact $3$-manifold homeomorphic to an irreducible $(r+1)$-component
\hyperlink{term-KGL}{KGL} complement with incompressible boundary, $\mu_{\mathtt{p}}$
and $\lambda_{{\mathtt{c}}_i}$'s are slopes on distinct components of $\partial J$,
i.e. oriented simple closed curves up to isotopy on $\partial J$,
such that the Dehn filling of $J$ along $\mu_{\mathtt{p}}$ yields an $r$-component unlink complement
with meridian slopes $\seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}}$ .
We call $\mu_{\mathtt{p}}$ the \emph{parent meridian} and the $\lambda_{{\mathtt{c}}_i}$'s the \emph{children longitudes}.
It is \emph{compatible} with a vertex $v$ in a rooted tree $\vec\Lambda$ if $r$
equals the number of the children of $v$. Two nodes
$(J,\mu_{{\mathtt{p}}},\seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}})$, $(J',\mu'_{{\mathtt{p}}},\seq{\lambda'_{{\mathtt{c}}_1},\cdots,\lambda'_{{\mathtt{c}}_{r'}}})$
are \emph{isomorphic} if there is an orientation-preserving homeomorphism
between the pairs $(J,\mu_{{\mathtt{p}}}\sqcup\lambda_{{\mathtt{c}}_1}\sqcup\cdots\sqcup\lambda_{{\mathtt{c}}_r})$
and $(J',\mu'_{{\mathtt{p}}}\sqcup \lambda'_{{\mathtt{c}}_1}\sqcup\cdots\sqcup\lambda'_{{\mathtt{c}}_{r'}})$, and in particular, $r=r'$.
\end{definition}
\begin{remark}
For a node $(J,\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}})$,
$\partial J$ is the disjoint union of components $\partial_{\mathtt{p}} J\sqcup \partial_{{\mathtt{c}}_1}J\sqcup\cdots\sqcup\partial_{{\mathtt{c}}_r}J$,
such that $\mu_{\mathtt{p}}\subset \partial_{\mathtt{p}} J$ and $\lambda_{{\mathtt{c}}_i}\subset \partial_{{\mathtt{c}}_i}J$ for $1\leq i\leq r$.
Each $\partial_{{\mathtt{c}}_i}J$ is called a \emph{child boundary}, and $\partial_{\mathtt{p}} J$ is called the \emph{parent boundary}.
The \emph{parent meridian} $\mu_{\mathtt{p}}$ is determined up to finitely many possibilities by $\seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}}$, and up to orientation.
If $J$ is a key-chain, then $\mu_{\mathtt{p}}$ is the boundary slope induced by the Seifert fibering.
If $J$ is a torus knot, then $\mu_{\mathtt{p}}$ is determined by the unique meridian which makes it a knot complement.
If $J$ is a cable link, then $\mu_{\mathtt{p}}$ must intersect the fiber slope once if Dehn filling is to give the unknot.
All of these possible meridians intersecting the fiber slope once
are related by Dehn twists along the annulus connecting the two boundary components,
and therefore $\mu_{\mathtt{p}}$ is uniquely determined by $\lambda_{{\mathtt{c}}_1}$. Otherwise, if $J$ is hyperbolic,
then there are at most $3$ possibilities for $\mu_{\mathtt{p}}\subset \partial_{\mathtt{p}} J$ by \cite[Theorem 2.4.4]{CGLS}.
There are naturally defined \emph{children meridians}, up to isotopy on $\partial J$,
which are the oriented simple closed curves $\mu_{{\mathtt{c}}_i}\subset\partial_{{\mathtt{c}}_i}J$
such that $\mu_{{\mathtt{c}}_i}$ is null-homotopic in the Dehn filling of $J$ along $\mu_{\mathtt{p}}$,
and that the orientation induced by $(\lambda_{{\mathtt{c}}_i},-\mu_{{\mathtt{c}}_i})$ coincides with that of $\partial_{{\mathtt{c}}_i}J$.
There is also a naturally defined \emph{parent longitude} $\lambda_{\mathtt{p}}\subset\partial_{\mathtt{p}} J$, up to isotopy on $\partial J$,
which is the oriented simple closed curve such that $\lambda_{\mathtt{p}}$ is null-homological in the Dehn filling of $J$ along
$\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}$, and that the orientation induced by $(\lambda_{\mathtt{p}},\mu_{\mathtt{p}})$ coincides with that of $\partial_{\mathtt{p}} J$.
\end{remark}
We say a \hyperlink{term-node}{node} is \emph{geometric} if $J$ is either Seifert fibered or hyperbolic. More specifically, we say \emph{key-chain nodes}, \emph{torus-knot nodes}, \emph{cable nodes} and \emph{hyperbolic nodes}, according to their defining \hyperlink{term-KGL}{KGLs}. The first three families are also called \emph{Seifert fibered nodes}.
Now Theorem \ref{BudneyThm} may be rephrased as follows.
\begin{prop}\label{knotData} Every nontrivial knot complement $M$ is completely characterized by the following data: (i) the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} $\vec\Lambda$; and (ii) the assignment of the vertices of $\vec\Lambda$ to compatible geometric \hyperlink{term-node}{nodes}, each of which is either a key-chain node, or a torus-knot node, or a cable node, or a hyperbolic node.\end{prop}
Provided Proposition \ref{knotData}, in order to prove Theorem \ref{main}, we must bound the number of allowable isomorphism types of the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} and the number of allowable \hyperlink{term-node}{node} types, under the assumption that $G$ maps onto the fundamental group of the knot complement $M$. This amounts to bounding the number of JSJ \hyperlink{term-piece}{pieces}, the
homeomorphism types of the JSJ \hyperlink{term-piece}{pieces}, as well as the number of allowable assignments of \hyperlink{term-node}{children longitudes}.
\section{Isomorphism types of the rooted JSJ tree}\label{Sec-number}
We start to prove Theorem \ref{main} in this section. Suppose $G$ is a finitely presented group with $b_1(G)=1$, and $M$ is a nontrivial knot complement such that there is an epimorphism $\phi:G\twoheadrightarrow\pi_1(M)$. In this section, we show that there are at most finitely many allowable isomorphism types of the \hyperlink{term-rootedJSJTree}{rooted JSJ trees} of $M$. In Section \ref{Sec-homeoTypes}, we shall show that there are at most finitely many homeomorphism types of geometric \hyperlink{term-piece}{pieces} that are allowed to be a geometric \hyperlink{term-piece}{piece} of $M$. In Section \ref{Sec-clTypes}, we shall show that there are at most finitely many allowable assignments of \hyperlink{term-node}{children longitudes} for any such \hyperlink{term-piece}{piece} to make it a \hyperlink{term-node}{node} decorating the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} of $M$. By Proposition \ref{knotData}, this will complete the proof of Theorem \ref{main}.
\begin{lemma}\label{pieceNumber} Suppose $G$ is a finitely generated group of rank $n$, and $M$ is a knot complement such that $G$ maps onto $\pi_1(M)$. Then $M$ has at most $4n-3$ \hyperlink{term-piece}{pieces} in its JSJ decomposition. Hence there are at most finitely many allowable isomorphism types of \hyperlink{term-rootedJSJTree}{rooted JSJ trees}.\end{lemma}
\begin{proof}
The upper-bound of the number of geometric \hyperlink{term-piece}{pieces} is a quick consequence from a theorem of Richard Weidmann. In \cite[Theorem 2]{We}, he proved that if $G$ is a non-cyclic freely-indecomposable $n$-generated group with a minimal $k$-acylindrical action on a simplicial tree, then the graph-of-groups decomposition induced by the action has at most $1+2k(n-1)$ vertices. Recall that for a group $G$, a $G$-action on a simplicial tree $T$ is called minimal if there is no proper subtree which is $G$-invariant, and is called $k$-acylindrical if no nontrivial element of $G$ fixes a segment of length $>k$.
Note there is a $\pi_1(M)$-action on the Bass-Serre tree $T$ associated to the JSJ decomposition of $M$.
Precisely, $T$ is the simplicial tree constructed as follows.
Let $\tilde{M}$ be the universal covering of $M$,
then the preimage of any geometric \hyperlink{term-piece}{piece} is a collection of component.
A vertex of $T$ is a component of a geometric \hyperlink{term-piece}{piece} of $M$;
two vertices are joined by an edge if and only if they are adjacent to each other.
By specifying a base-point of $M$, there is a natural $\pi_1(M)$-action on $T$
induced by the covering transformation.
Because there is no geometric \hyperlink{term-piece}{piece} homeomorphic to the nontrivial $S^1$-bundle
over a M\"obius strip as $M$ is a knot complement, it is known that $\pi_1(M)$-action on $T$ is minimal and $2$-acylindrical, cf. \cite[p. 298]{BW}. Therefore, the induced $\phi(G)$-action on $T$
is also minimal and $2$-acylindrical. Since $\phi(G)$ is finitely generated as is $G$,
we may apply Weidmann's theorem to obtain an upper-bound of the number
of geometric \hyperlink{term-piece}{pieces} by $1+4(n-1)=4n-3$, where $n$ is the rank of $G$.
The `hence' part follows since there are only finitely many isomorphism types of \hyperlink{term-rootedTree}{rooted} \hyperlink{term-rootedTree}{trees} with at most $4n-3$ vertices.
\end{proof}
\section{Homeomorphism types of geometric pieces}\label{Sec-homeoTypes}
In this section, we show there are at most finitely many allowable homeomorphism types of geometric \hyperlink{term-piece}{pieces} under the assumption of Theorem \ref{main}. We consider the hyperbolic case and the Seifert fibered case in Subsections \ref{Subsec-hypPieces} and \ref{Subsec-SFPieces}, respectively.
\subsection{Homeomorphism types of hyperbolic pieces}\label{Subsec-hypPieces}
In this subsection, we show there are at most finitely many allowable homeomorphism types of hyperbolic \hyperlink{term-piece}{pieces}:
\begin{prop}\label{hypPieces} Let $G$ be a finitely presented group with $b_1(G)=1$, then there are at most finitely homeomorphism types of hyperbolic \hyperlink{term-piece}{pieces} $J$ such that $J$ is a hyperbolic \hyperlink{term-piece}{piece} of
some knot complement $M$ such that $G$ maps onto $\pi_1(M)$.
\end{prop}
We prove Proposition \ref{hypPieces} in the rest of this subsection.
Let $K$ be a finite $2$-complex $K$ of a presentation $\mathcal{P}$ of $G$ achieving $\ell(G)$.
To argue by contradiction, suppose $G$ maps onto infinitely
many knot groups $\pi_1(M_n)$ such that there are infinitely many homeomorphically distinct hyperbolic \hyperlink{term-piece}{pieces} showing up.
By Theorem \ref{volBound}, ${\mathrm{Vol}}(M_n)\leq\pi\ell(G)$. The J{\o}rgensen-Thurston
theorem (\cite[Theorem 5.12.1]{Th}) implies infinitely many of these \hyperlink{term-piece}{pieces} are distinct hyperbolic Dehn fillings
of some hyperbolic $3$-manifold of finite volume. In particular, for any $\delta>0$
infinitely many of these \hyperlink{term-piece}{pieces} contain closed geodesics of length $<\delta$.
Let $M$ be a knot complement with a hyperbolic \hyperlink{term-piece}{piece} $J$ containing a sufficiently
short closed geodesic $\gamma$ so that Theorem \ref{factor-hyp} holds.
By assumption, there is a
$\pi_1$-surjective map $f:K\to M$. By Theorem \ref{factor-hyp},
it factorizes through the \hyperlink{term-extDehnFilling}{extended Dehn filling} $i^e:N^e\to M$ of some denominator $m>0$, up to homotopy,
namely $f\simeq i^e\circ f^e$, where $f^e:K\to N^e$. Remember $N=M-\gamma$ and $N^e=N\cup Z$.
Note that $b_1(K)=1$ and $b_1(N^e)=2$, so $f^e$ is not $\pi_1$-surjective. We consider the covering space
$\kappa:\tilde{N}^e\to N^e$ corresponding to ${\rm Im}(\pi_1(f^e))$, after choosing some base-points.
Let ${\mathcal{T}}$ be union of the JSJ tori of $N^e$.
Note ${\mathcal{T}}$ is the union of the JSJ tori of $M$ together with the drilling boundary $\partial_\gamma N$,
and the \hyperlink{term-ridge}{ridge piece} $Z$ of $N^e$ is only adjacent to the hyperbolic \hyperlink{term-piece}{piece} $Y=J-\gamma$.
By Proposition \ref{ScottCore},
there is an aspherical Scott core $C$ of $\tilde{N}^e$ such
that $C\cap\kappa^{-1}({\mathcal{T}})$ are essential annuli and/or tori. Moreover, because $b_1(G)=1$ implies $b_1(\tilde{N}^e)=1$, by Lemma \ref{simplifyCore},
$C$ contains no contractible \hyperlink{term-chunk}{chunk} or non-central elementary \hyperlink{term-chunk}{chunk},
in particular, no elementary hyperbolic \hyperlink{term-chunk}{chunk}.
Also, $b_0(C)=1, b_3(C)=0$ and $\chi(C)\leq 0$ (since each chunk of $C$ has $\chi\leq 0$, and the chunks are glued along tori and annuli by Lemma \ref{simplifyCore}). Therefore, since $b_1(C)=1$, we also have $b_2(C)=0$ and $\chi(C)=0$.
However, there must be some hyperbolic \hyperlink{term-chunk}{chunk} $Q$ covering $Y$,
because otherwise $\kappa|:C\to N^e$ would miss the interior of $Y$ up to homotopy,
so would map into $Z$ or $N-Y$. Either case contradicts $f:K\to M$ being $\pi_1$-surjective by the Van Kampen theorem.
Thus $Q$ is a non-elementary hyperbolic \hyperlink{term-chunk}{chunk} of $C$.
Suppose ${\rm Im}(\pi_1(Q)\to \pi_1(Y))$ has finite index in $\pi_1(Y)$. Then there is a torus boundary component
$\tilde{T}\subset \partial Q\cap \kappa^{-1}(\mathcal{T})$ which is adjacent to a
ridge piece. If so, then $\tilde{T}$ covers the \hyperlink{term-ridge}{side torus} $T=\partial_\gamma Y \subset N^e$. Since $0\neq [T]\subset H_2(N^e;{\mathbb Q})$,
we conclude that $0\neq [\tilde{T}] \in H_2(C;{\mathbb Q})$. Thus, $b_2(C) >0$, which gives a contradiction.
We conclude $\pi_1(Q)$ is isomorphic to a non-elementary Kleinian group with infinite covolume, so $\chi(Q)<0$. Note $C$ is cut along tori/annuli into non-contractible \hyperlink{term-chunk}{chunks} with nonpositive Euler characteristics. We conclude $\chi(\tilde{N}^e)=\chi(C)\leq\chi(Q)<0$, a contradiction.
\subsection{Homeomorphism types of Seifert fibered pieces}\label{Subsec-SFPieces}
In this subsection, we show there are at most finitely many homeomorphism types of Seifert fibered \hyperlink{term-piece}{pieces}.
\begin{prop}\label{SFPieces} Let $G$ be a finitely presented group with $b_1(G)=1$. Then there are at most finitely many homeomorphism types of Seifert fibered \hyperlink{term-piece}{pieces} $J$ such that $J$ is a Seifert fibered \hyperlink{term-piece}{piece} of
some knot complement $M$ such that $G$ maps onto $\pi_1(M)$. In fact, there are at most finitely many allowable values of $q$ for a $p/q$-cable \hyperlink{term-piece}{piece}, where $p\neq 0, q>1$ and $p,q$ are coprime integers, and there are at most finitely many allowable values of $p,q$ for a $p/q$-torus-knot \hyperlink{term-piece}{piece}, where $|p|>1,q>1$ and $p,q$ are coprime integers.
\end{prop}
We prove Proposition \ref{SFPieces} in the rest of this subsection.
We first explain why the `in fact' part implies the first statement. Remember from Section \ref{Sec-cpnship} that there are only three families of Seifert fibered \hyperlink{term-piece}{pieces} that could be a JSJ \hyperlink{term-piece}{piece} of a nontrivial knot complement, namely the key-chain link complements, cable-link complements, and torus-knot complements, (cf. Example \ref{KGL-example}). For an $(r+1)$-component key-chain link, the homeomorphism type of its complement is determined by $r$, indeed, it is homeomorphic to $F_{0,r+1}\times S^1$ where $F_{0,r+1}$ is $S^2$ with $(r+1)$ open disks removed. Thus the allowed values of $r$ are bounded by the number of JSJ \hyperlink{term-piece}{pieces} of $M$, which is bounded in terms of $G$ by Lemma \ref{pieceNumber}. For a $p/q$-cable-link, the homeomorphism type of its complement is determined by $q$ together with the residual class of $p\bmod q$, (possibly with some redundancy). For a $p/q$-torus knot, the homeomorphism type of its complement is determined by the value $p/q\in{\mathbb Q}$, (possibly with some redudancy). Therefore, the `in fact' part and Lemma \ref{pieceNumber} implies that there are at most finitely many allowable homeomorphism types of Seifert fibered \hyperlink{term-piece}{pieces}.
It now suffices to prove the `in fact' part. The arguments for the $p/q$-cable case and the $p/q$-torus-knot case are essentially the same, but we treat them as two cases for convenience.
\noindent\textbf{Case 1}. Homeomorphism types of cable \hyperlink{term-piece}{pieces}.
Let $K$ be a finite $2$-complex $K$ of a presentation $\mathcal{P}$ of $G$ achieving $\ell(G)$.
To argue by contradiction, suppose $G$ maps onto infinitely
many knot groups $\pi_1(M_n)$ such that there are infinitely many homeomorphically
distinct $p_n/q_n$-cable \hyperlink{term-piece}{pieces} $J_n\subset M_n$ arising. Then for infinitely many $n$, $q_n>1$ are sufficiently large. Let $M=M_n$ be such a knot complement, $q=q_n$, $J=J_n$, and $f:K\to M$ be a $\pi_1$-surjective map as assumed.
By Theorem \ref{factor-SF}, $f$ factors through the \hyperlink{term-extDehnFilling}{extended Dehn filling} $f^e:K\to N^e$ of some denominator $m \leq T(\ell(G))$ up to homotopy, where $N=M-\gamma$ is the drilling along the corresponding exceptional fiber $\gamma\subset J$ with boundary $T$, and $N^e=N\cup_T Z$ is the \hyperlink{term-DehnExt}{Dehn extension}.
Consider the covering space $\kappa:\tilde{N}^e\to N^e$ corresponding to ${\rm Im}(\pi_1(f^e))$, after choosing some base-points. Then there is a homotopy lift $\tilde{f}^e:K\to \tilde{N}^e$, such that $\tilde{f}^e$ is $\pi_1$-surjective and $f^e\simeq\kappa\circ\tilde{f}^e$. Therefore $b_1(\tilde{N}^e)=1$ by the assumption $b_1(K)=1$.
We wish to show, however, $b_1(\tilde{N}^e)>1$ in order to reach a contradiction.
Let ${\mathcal{T}}$ be the union of the JSJ tori of $N^e$. Let $Y=J-\gamma$ be the regular cable \hyperlink{term-piece}{piece} of $N^e$, and let $Z$ be the \hyperlink{term-ridge}{ridge piece} of $N^e$ adjacent to $Y$. By Proposition \ref{ScottCore}, there is an aspherical Scott core $C$ of $\tilde{N}^e$ such that $C_{{\mathcal{T}}}=C\cap\kappa^{-1}({\mathcal{T}})$ are essential annuli and/or tori. Moreover, because $b_1(G)=1$ implies $b_1(\tilde{N}^e)=1$, by Lemma \ref{simplifyCore} $C$ has no contractible \hyperlink{term-chunk}{chunk} and no non-central elementary \hyperlink{term-chunk}{chunk}, in particular, no Seifert fibered \hyperlink{term-chunk}{chunk} which is an $I$-bundle over an annulus.
Also, $b_0(C)=1, b_3(C)=0$ and $\chi(C)= 0$. Therefore, since $b_1(C)=1$, we also have $b_2(C)=0$.
Note $Y$ is homeomorphic to a trivial $S^1$-bundle over a pair of pants $\Sigma$, and $\partial Y$ consists of three components, namely, the parent boundary ${\partial_{\mathtt{p}}}Y$ (the `pattern boundary'), the child boundary ${\partial_{\mathtt{c}}}Y$ (the `companion boundary'), and the drilling boundary $T=\partial_\gamma Y$. We also write $\partial\Sigma={\partial_{\mathtt{p}}}\Sigma\sqcup{\partial_{\mathtt{c}}}\Sigma\sqcup\partial_\gamma\Sigma$ correspondingly. Let $\tilde{Y}$ be a component of $\kappa^{-1}(Y)$ and $Q=C\cap\tilde{Y}$ be a cable \hyperlink{term-chunk}{chunk}. Then $\tilde{Y}$ is homeomorphic to either a trivial $S^1$-bundle or a trivial ${\mathbb R}$-bundle over a finitely generated covering $\bar{\kappa}:\tilde{\Sigma}\to\Sigma$, and $Q$ is homeomorphic to either a trivial $S^1$-bundle or a trivial $I$-bundle over a Scott core $W$ of $\tilde\Sigma$, which is an orientable compact surface. Moreover, the \hyperlink{term-cutBdry}{cut boundary} $\partial_{{\mathcal{T}}}Q=Q\cap\kappa^{-1}({\mathcal{T}})\subset\partial Q$ is a union of annuli and/or tori. It is a sub-bundle over a corresponding union of arcs and/or loops $\partial_{\mathcal T}W\subset\partial W$, and can be decomposed as a disjoint union $${\partial_{\mathtt{p}}}Q\sqcup{\partial_{\mathtt{c}}}Q\sqcup\partial_\gamma Q,$$
according to the image under $\kappa|:\partial Q\to\partial Y$. We also write $\partial_{\mathcal{T}} W={\partial_{\mathtt{p}}} W\sqcup{\partial_{\mathtt{c}}} W\sqcup\partial_\gamma W$ correspondingly. We will use the same notation for a Seifert fibered \hyperlink{term-chunk}{chunk} which is not $Y$, except that $\partial_\gamma$ will be empty in such a case, and $W$ may be an orbifold instead of a surface.
The following lemma rules out the case that $Q$ is an $I$-bundle.
Remembering that there is no disk component in $C_{\mathcal{T}}$ by Lemma \ref{simplifyCore}, the lemma below says any component of $C_{\mathcal{T}}$ is a torus unless it is adjacent to a ridge chunk.
\begin{lemma}\label{annulusRidge} If $A$ is an annulus component of $C_{\mathcal{T}}$, then $A$ is adjacent to a ridge \hyperlink{term-chunk}{chunk}, and its core is a fiber in $Q$.\end{lemma}
\begin{proof} Suppose $A$ were adjacent to regular \hyperlink{term-chunk}{chunks} on both sides. If $A$ is adjacent to a
hyperbolic \hyperlink{term-chunk}{chunk} $Q$, then by the argument of Proposition \ref{hypPieces},
$\chi(Q)<0$, and hence $\chi(C)<0$, so $b_1(C)>1$, a contradiction.
If $A$ is adjacent to Seifert fibered \hyperlink{term-chunk}{chunks} on both sides,
then the core loop of $A$ can only cover a fiber in one of the corresponding
\hyperlink{term-piece}{pieces} of $N$ under $\kappa$. Then the other \hyperlink{term-chunk}{chunk}
must be a Seifert fibered \hyperlink{term-chunk}{chunk} which is an $I$-bundle over an orientable compact surface $W$.
Then $\partial_{\mathcal T} W$ cannot be arcs of $\partial W$ as $\partial_{{\mathcal{T}}}Q$ has no disk component.
Hence $\partial_{\mathcal T}W$ are a few components of $\partial W$.
Note $W$ is a compact orientable surface, which cannot be a disk since $Q$ is not contractible.
If $\chi(W)=0$, $W$ is an annulus, so $Q$ is an $I$-bundle over an annulus,
which has been ruled out by our simplification of the Scott core $C$, (Lemma \ref{simplifyCore}).
We conclude $\chi(W)<0$. Note $C$ is cut into \hyperlink{term-chunk}{chunks} along annuli and tori,
$\chi(C)\leq\chi(Q)=\chi(W)<0$, so in this case $b_1(\tilde{N}^e)=b_1(C)>1$. This contradicts $b_1(C)=1$.
If the core of $A$ is not a fiber in the cable \hyperlink{term-chunk}{chunk} $Q$, then again we conclude that
$\chi(Q)<0$, a contradiction.
\end{proof}
Thus we may assume that every Seifert fibered \hyperlink{term-chunk}{chunk} is an $S^1$-bundle over an orientable compact surface orbifold.
\begin{lemma} \label{ridge torus}
There is no torus $\tilde{T}$ of $C_\mathcal{T}\subset C$ adjacent to a ridge \hyperlink{term-chunk}{chunk}.
\end{lemma}
\begin{proof}
If so, then $\tilde{T}$ covers the \hyperlink{term-ridge}{side torus} $T=\partial_\gamma Y \subset N^e$. Since $0\neq [T]\subset H_2(N^e;{\mathbb Q})$,
we conclude that $0\neq [\tilde{T}] \in H_2(C;{\mathbb Q})$. Thus, $b_2(C) >0$, a contradiction.
\end{proof}
The following lemma rules out non-separating components of $C_{\mathcal{T}}$.
\begin{lemma}\label{separating} Any component of $C_{{\mathcal{T}}}$ is separating in $C$.\end{lemma}
\begin{proof} To argue by contradiction, suppose there were some non-separating component of $C_{{\mathcal{T}}}$. Remember $C_{\mathcal{T}}$ induces a graph-of-spaces decomposition of $C$ over a finite connected simplicial graph $\Lambda$. Because $b_1(C)=1$ and there is a non-separating edge, it is clear that the graph $\Lambda$ has a unique embedded loop.
Let $C^\sigma\subset C$ be the union of the \hyperlink{term-chunk}{chunk} over the embedded loop together with maximal regular \hyperlink{term-chunk}{chunks} that are adjacent to this \hyperlink{term-chunk}{chunk} at its regular \hyperlink{term-chunk}{subchunks}, and let $\sigma\subset\Lambda$ be the underlying subgraph of $C^\sigma$. Remember we do not regard ridge \hyperlink{term-chunk}{subchunks} which are homeomorphic to a $3$-manifold as regular, and hence regular \hyperlink{term-chunk}{subchunks} are all orientable. Thus the vertices of $\sigma$ corresponding to ridge \hyperlink{term-chunk}{subchunks} all lie on the embedded loop with valence $2$ in $\sigma$.
Consider the compact $3$-manifold $\hat{C}^\sigma$ obtained by replacing every ridge \hyperlink{term-chunk}{subchunk} $S\subset C^\sigma$ by a thickened annulus $A^2\times [0,1]$. This is possible because the \hyperlink{term-ridge}{ridge piece} has easily understood coverings. There is a natural `resolution' map: $$\varrho:\hat{C}^\sigma\to C^\sigma,$$ such that $A^2\times\seq{\frac{1}{2}}$ covers the ridge of $S$, and $\varrho$ induces an isomorphism on the rational homology. In particular, $H_1(\hat{C}^\sigma;{\mathbb Q})\cong H_1(C^\sigma;{\mathbb Q})$. Note that $\hat{C}^\sigma$ may be non-orientable, but $\hat{C}^\sigma$ cut along any non-separating component of $C_{\mathcal{T}}$ is always orientable.
However, we claim $b_1(\hat{C}^\sigma)>1$, and hence $b_1(C^\sigma)>1$.
To see this, first note that $\hat{C}^\sigma$ is irreducible, and has no sphere boundary components by Lemma \ref{trivialCk}.
Thus, $\chi(\hat{C}^\sigma) \leq 0$.
Suppose some non-separating component of $C_{{\mathcal{T}}}$ is a torus $T$.
Then $T\subset C^\sigma$, and correspondingly $T$ lies in the interior of $\hat{C}^\sigma$.
However, $\partial \hat{C}^\sigma$ must be non-empty, since otherwise any maximal regular
\hyperlink{term-chunk}{subchunk} $R$ of $C^\sigma$ would only have tori boundary which
covers $\partial_\gamma N$ under $\kappa$, but this is clearly impossible as there is another
component $\partial M\subset\partial N$. Thus, $b_3(\hat{C}^\sigma)=0$.
Also, $0\neq [T] \in H_2(\hat{C}^\sigma;{\mathbb Q})$ since $T$ is $2$-sided, orientable,
and non-separating, so $b_2(\hat{C}^\sigma)\geq 1$. Thus,
$0\geq \chi(\hat{C}^\sigma) = b_0(\hat{C}^\sigma)-b_1(\hat{C}^\sigma)+b_2(\hat{C}^\sigma) \geq 2-b_1(\hat{C}^\sigma)$, which implies $b_1(\hat{C}^\sigma)>1$.
Now suppose every non-separating component of $C_{\mathcal{T}}$ is an annulus.
If $A$ is such an annulus, then it must be adjacent to a ridge \hyperlink{term-chunk}{subchunk} $S\subset C^\sigma$ by Lemma \ref{annulusRidge}, so $A$ is also a component of $\partial_\gamma Q$ for the cable \hyperlink{term-chunk}{subchunk} $Q\subset C^\sigma$ adjacent to $S$ along $A$. As $\partial_\gamma Q$ is a trivial $S^1$-bundle over $\partial_\gamma W$, where $W$ is the compact orientable surface as described before,
the core loop $\xi$ of $A$ covers an ordinary fiber in $Y$ under $\kappa$, so $\xi$ is sent to a cover of an ordinary fiber of the cable \hyperlink{term-piece}{piece} $J\subset M$ under the composition: $$\hat{C}^\sigma\stackrel{\varrho}\longrightarrow C^\sigma\stackrel{\subset}\longrightarrow \tilde{N}^e\stackrel{\kappa}\longrightarrow N^e\stackrel{i^e}\longrightarrow M.$$ Note $\partial A\subset\partial\hat{C}^\sigma$, and $A$ is $2$-sided.
If $\chi(\hat{C}^\sigma) <0$, as before we conclude that $b_1(\hat{C}^\sigma) >1$, so we may assume that $\chi(\partial \hat{C}^\sigma) = 2\chi(\hat{C}^\sigma)=0$.
If a component of $\partial \hat{C}^\sigma$ containing a component of $\partial A$ is a Klein bottle, then
the fiber $\xi$ either lies in the boundary of a M\"obius strip, or is freely homotopic
to its orientation-reversal. The former case is impossible because otherwise $\hat{C}^\sigma-A$ would have at least one non-orientable boundary component, no matter the components of $\partial A$ lie on $1$ or $2$ boundary components of $\hat{C}^\sigma$, which contradicts $\hat{C}^\sigma-A$ being orientable; the latter case is impossible because $\xi$ covers an ordinary fiber of a cable \hyperlink{term-piece}{piece} of $M$, which cannot be freely homotopic to the orientation-reversal due to the $2$-acylindricity of the JSJ decomposition of $M$, cf. \cite[p. 298]{BW}. Thus $\partial A$ lies in (possibly the same) tori components of $\partial \hat{C}^\sigma$.
If $\hat{C}^\sigma$ is non-orientable with a torus boundary component, then $b_2(\hat{C}^\sigma)>0$ from the exact
sequence:
$$0=H_3(\hat{C}^\sigma,T^2;{\mathbb Q}) \to H_2(T^2;{\mathbb Q}) \to H_2(\hat{C}^\sigma;{\mathbb Q}),$$ so $b_1(\hat{C}^\sigma)>1$. So
we may assume that $\hat{C}^\sigma$ is orientable. First suppose $\partial A$ lies on a single component of $\partial\hat{C}^\sigma$.
In this case, $\partial A$ must be separating in $\partial \hat{C}^\sigma$, because otherwise
the union of $A$ and a component of $\partial\hat{C}^\sigma-\partial A$ would be a Klein bottle,
thus each component of $\partial A$ with the induced orientation would be freely homotopic to its
orientation-reversal via the Klein bottle, but this is impossible since the components of
$\partial A$ cover ordinary fibers in $Y\subset M$, which cannot be freely homotopic to
their orientation-reversal in $M$ as before. Therefore, the union of $A$ and a component of $\partial\hat{C}^\sigma-\partial A$ is parallel to a non-separating $2$-sided torus in the interior of $\hat{C}^\sigma$, which implies $b_1(\hat{C}^\sigma)>1$ as before.
Now suppose $\partial A$ lies on two different components of $\partial\hat{C}^\sigma$, then $b_1(\hat{C}^\sigma)>1$ as $\hat{C}^\sigma$ is orientable with at least two tori boundary components. This proves the claim.
To finish the proof of this lemma, we successively add adjacent \hyperlink{term-chunk}{chunks} to $C^\sigma$.
Let $C'$ be the union of $C^\sigma$ with all the adjacent ridge \hyperlink{term-chunk}{chunks},
then $H_1(C';{\mathbb Q})\cong H_1(C^\sigma;{\mathbb Q})$ so $b_1(C')>1$.
For any maximal regular \hyperlink{term-chunk}{chunk} $R\subset C$ adjacent to $C'$,
they are adjacent along a \hyperlink{term-ridge}{side annulus} of a ridge \hyperlink{term-chunk}{chunk} $S\subset C'$.
If they are adjacent along an annulus $A$,
$\partial R$ is non-empty, so $b_1(R\cup C')\geq b_1(R)+b_1(C')-b_1(A)\geq 1+2-1=2$.
Thus, let $C''$ be the union of $C'$ with all the adjacent maximal regular \hyperlink{term-chunk}{chunks},
$b_1(C'')>1$. Continuing in this fashion by induction, we see $b_1(C)>1$.
This is a contradiction since we have said $b_1(C)=1$.
\end{proof}
The last part of the proof of Lemma \ref{separating} is a useful argument, so we extract it as below.
\begin{lemma}\label{MVarg} In Case 1, assume all the components of $C_{\mathcal{T}}$ are separating. If $C'\subset C$ is a \hyperlink{term-chunk}{chunk} with $b_1(C')>1$, and all \hyperlink{term-chunk}{chunks} adjacent to $C'$ are ridge \hyperlink{term-chunk}{chunks}, then $b_1(C)>1$.\end{lemma}
\begin{proof} Similar to the last paragraph in the proof of Lemma \ref{separating}, successively enlarge $C'$ by attaching all the adjacent ridge \hyperlink{term-chunk}{chunks}, and then all the adjacent maximal regular \hyperlink{term-chunk}{chunks}, and continue alternately in this fashion.\end{proof}
Lemmas \ref{separating} and \ref{MVarg} allow us to use Mayer-Vietoris arguments based on the homology of cable \hyperlink{term-chunk}{chunks}. This is carried out in the lemma below. Remember we have assumed any cable \hyperlink{term-chunk}{chunk} $Q$ is a trivial $S^1$-bundle over a compact orientable surface $W$.
\begin{lemma}\label{CblCkBdry} With notation as above, for any cable \hyperlink{term-chunk}{chunk} $Q\subset C$, the base surface $W$ is planar, and ${\partial_{\mathtt{p}}}W\sqcup\partial_\gamma W$ are arcs contained in a single component of $\partial W$. Hence $W$ is homeomorphic to a regular neighborhood of the union of $\partial_{\mathcal{T}} W$ together with an embedded simplicial tree of which each end-point lies on a distinct component of $\partial_{\mathcal{T}} W$, connecting all the components of $\partial_{\mathcal{T}} W$.\end{lemma}
\begin{proof}
Suppose $W$ were not planar, there is an embedded non-separating torus $T\subset Q$ which is the sub-$S^1$-bundle over a non-separating simple closed curve of $W$. Let $C'$ be the maximal regular \hyperlink{term-chunk}{chunk} containing $W$. First suppose $\partial_{\mathcal{T}} C'$ is empty, then $C=C'$ but $\partial C$ is non-empty with no sphere component. Thus $b_1(C)>1$ since it has non-empty aspherical boundary and a non-separating embedded torus, contrary to $b_1(\tilde{N}^e)=1$. Now suppose $\partial_{\mathcal{T}} C'$ is non-empty, for the same reason as above, $b_1(C')>1$. By Lemmas \ref{separating}, \ref{MVarg}, $b_1(\tilde{N}^e)=b_1(C)>1$, contrary to $b_1(\tilde{N}^e)=1$.
To see the second part, we first show ${\partial_{\mathtt{p}}}W\sqcup\partial_\gamma W$ is contained in a single component of $\partial W$. Suppose on the contrary that ${\partial_{\mathtt{p}}}W\sqcup\partial_\gamma W$ meets at least $2$ components of $\partial W$, then $b_1(Q)>1+k$ where $k\geq 0$ is the number of torus components of ${\partial_{\mathtt{c}}} Q$. Let $R\subset C$ be any maximal regular \hyperlink{term-chunk}{chunk} adjacent to $Q$ that misses the interior of $C\cap\kappa^{-1}Y$. Note if $R$ is adjacent to $Q$ along any torus component of ${\partial_{\mathtt{p}}} Q\sqcup\partial_\gamma Q$, then $R$ has some boundary component other than this torus. Thus, let $Q'$ be $Q$ together with all such \hyperlink{term-chunk}{chunks}. A Mayer-Vietoris argument shows $b_1(Q')>1+k'$ where $k'$ is the number of torus components of ${\partial_{\mathtt{c}}} Q'$. Continuing in this fashion, in the end, we have $b_1(C')>1$, where $C'\subset C$ is the maximal regular \hyperlink{term-chunk}{chunk} containing $Q$, which must have no torus components of ${\partial_{\mathtt{c}}} C'$, (indeed, ${\partial_{\mathtt{p}}}C'\sqcup{\partial_{\mathtt{c}}}C'=\emptyset$). By Lemmas \ref{separating}, \ref{MVarg}, again we obtain a contradiction.
Now we show $\partial_{\mathtt{p}} W\sqcup\partial_\gamma W$ are disjoint arcs rather than a single loop. Otherwise there would be two cases. If $\partial_{\mathtt{p}} W\sqcup\partial_\gamma W=\partial_{\mathtt{p}} W$ were a single loop, then $\partial_{\mathtt{p}} Q$ would contain a torus mapping to the homologically non-trivial torus $\partial_{\mathtt{p}} Y$. Thus $b_2(C)>0$, a contradiction. If $\partial_{\mathtt{p}} W\sqcup\partial_\gamma W=\partial_\gamma W$ were a single loop, then $\partial_\gamma Q$ would contain a torus, contradicting Lemma \ref{ridge torus}.
The `hence' part is an immediate consequence from the first part.
\end{proof}
To finish the proof of Case 1, we observe the lemma below. Remember $J$ is a $p/q$-cable pattern \hyperlink{term-piece}{piece}, where $q>1$, and \textit{a priori} $M=M_{\mathtt{p}}\cup J\cup M_{\mathtt{c}}$, where $M_{\mathtt{p}}$ and $M_{\mathtt{c}}$ are the pattern component and the companion component of $M-J$, respectively. Recall $m$ is the denominator of the Dehn extension $N^e$. Let $q'=q/\gcd(q,m)$.
\begin{lemma}\label{notOnto} In the current situation, $M=J\cup M_{\mathtt{c}}$, and the image of $(i^e\circ\kappa)_*:H_1(\tilde{N}^e)\to H_1(M)\cong{\mathbb Z}$ is contained in $q'{\mathbb Z}$.\end{lemma}
\begin{proof} We first show $M=J\cup M_{\mathtt{c}}$, namely $\partial_{\mathtt{p}} J$ is parallel to $\partial M$. Suppose this is not the case, then let
$S=\partial_{\mathtt{p}} J$ be the pattern boundary of $J$, then $S\subset N^e$.
Notice that $0\neq [S]\in H_2(N^e;{\mathbb Q})$. If $\kappa^{-1}(S) = \emptyset$, then
$f:K \to N$ misses $S$ up to homotopy, and we conclude that $\pi_1(K) \ntwoheadrightarrow \pi_1(M)$ unless
$J$ is the \hyperlink{term-rootedJSJTree}{root} of the JSJ tree.
So $\kappa^{-1}(S)$ is non-empty, and by Lemma \ref{annulusRidge}, each component $\tilde{S}$ of
$\kappa^{-1}(S)$ is a torus. But then $0\neq [\tilde{S}]\in H_2(\tilde{N}^e;{\mathbb Q})$, since $\kappa_|: H_2(\tilde{S};{\mathbb Q}) \to H_2(S;{\mathbb Q})$
is non-zero. This implies that $b_2(\tilde{N}^e) >0$, a contradiction.
Next, we show the image of $H_1(\tilde{N}^e)\to H_1(M)\cong{\mathbb Z}$ is contained in $q'{\mathbb Z}$.
It suffices to show for $H_1(C)\to H_1(M)$ as $C\subset\tilde{N}^e$ is a homotopy equivalence.
Now $C$ is the union of cable \hyperlink{term-chunk}{chunks},
ridge \hyperlink{term-chunk}{chunks} and regular $M_{\mathtt{c}}$-\hyperlink{term-chunk}{chunks},
namely the components of $C\cap\kappa^{-1}(M_{\mathtt{c}})$. By the cabling construction of knots,
the image of $H_1(M_{\mathtt{c}})\to H_1(M)\cong{\mathbb Z}$ is contained in $q{\mathbb Z}$,
so every $M_{\mathtt{c}}$-\hyperlink{term-chunk}{chunk} maps into $q{\mathbb Z}$ on homology.
Note the ordinary fiber of $J$ also lies in $q{\mathbb Z}$, by Lemma \ref{CblCkBdry},
every cable \hyperlink{term-chunk}{chunk} also maps into $q{\mathbb Z}$ on homology.
Note every ridge \hyperlink{term-chunk}{chunk} is adjacent to at least one cable \hyperlink{term-chunk}{chunk},
Lemma \ref{CblCkBdry} implies any ridge \hyperlink{term-chunk}{chunk} $S$ is homeomorphic to $\mathcal{M}_u\times I$,
where $u:S^1\to S^1$ is a finite cyclic cover (from an ordinary Seifert fiber in $T$), and $\mathcal{M}_u$ is the mapping cylinder $u$.
To bound the degree of
$u$, we take a basis for the homology of the torus $H_1(T)\cong {\mathbb Z}\oplus{\mathbb Z}$ with meridian $(1,0)$ and longitude $(0,1)$. Then the fiber slope is $(p,q)$.
When we take the \hyperlink{term-DehnExt}{Dehn extension}, we embed ${\mathbb Z} + {\mathbb Z} \subset \frac{1}{m} {\mathbb Z} + {\mathbb Z}$. Letting $m'=\gcd(m,q)$, we see that the maximal root of the fiber
slope $(p,q)$ in $\frac{1}{m} {\mathbb Z} + {\mathbb Z}$ is $(p/m', q/m')= (p/m', q')$. Thus, the degree of the map $u$ must divide $m'$ since $\mathcal{M}_u$ is a core for a cyclic cover of the ridge chunk $Z$.
This implies every ridge \hyperlink{term-chunk}{chunk} maps into $q'{\mathbb Z}$ on homology. Because $C\cap\kappa^{-1}(\partial Y)$, where $Y=J-\gamma$, has no non-separating component by Lemma \ref{separating}, the Mayer-Vietoris sequence implies $C$ maps into $q'{\mathbb Z}$ on homology. This completes the proof of the second part.
\end{proof}
When $q>m$, then $q'>1$, so Lemma \ref{notOnto} gives a contradiction to the assumption that $f:K\to M$ is $\pi_1$-surjective as $f$ is homotopic to the composition: $$K\stackrel{\tilde{f}^e}\longrightarrow \tilde{N}^e\stackrel{\kappa}\longrightarrow N^e\stackrel{i^e}\longrightarrow M.$$ Thus, $q\leq m\leq T(\ell(G))$ by Thoerem \ref{factor-SF}. This completes the proof of Case 1.
\noindent\textbf{Case 2}. Homeomorphism types of torus-knot \hyperlink{term-piece}{pieces}.
We'll use the same notation as the beginning of the proof of Case 1.
In fact, the $p/q$-torus-knot complement $J$ is a Seifert fibered space over the base $2$-orbifold $S^2(p,q,\infty)$.
If it is the $q$-exceptional fiber $\gamma$ (i.e. the fiber over the cone point correpsonding to $q$)
that has been drilled out for the \hyperlink{term-DehnExt}{Dehn extension}, let $N^e= N \cup_T Z$
as in Case 1. Let $S=\partial_{\mathtt{p}} J$ be the pattern boundary of $J$, then $S\subset N^e$.
Notice that $0\neq [S]\in H_2(N^e;{\mathbb Q})$. If $\kappa^{-1}(S) = \emptyset$, then
$f:K \to N$ misses $S$ up to homotopy, and we conclude that $\pi_1(K) \ntwoheadrightarrow \pi_1(M)$ unless
$J$ is the \hyperlink{term-rootedJSJTree}{root} of the JSJ tree.
So $\kappa^{-1}(S)$ is non-empty, and by Lemma \ref{annulusRidge}, each component $\tilde{S}$ of
$\kappa^{-1}(S)$ is a torus. But then $0\neq [\tilde{S}]\in H_2(\tilde{N}^e;{\mathbb Q})$, since $\kappa_|: H_2(\tilde{S};{\mathbb Q}) \to H_2(S;{\mathbb Q})$
is non-zero. This implies that $b_2(\tilde{N}^e) >0$, a contradiction.
The only possibility if $J$ is the \hyperlink{term-rootedJSJTree}{root} of the JSJ tree of $M$ is that $J=M$, i.e. $M$ is a torus knot complement.
In this case, we obtain a contradiction by showing that covers $\tilde{N}^e\to N^e$ with $b_1(\tilde{N}^e)=1$
are elementary as in the proof of Lemma \ref{CblCkBdry}.
Cases 1 and 2 together completes the proof of Proposition \ref{SFPieces}.
\section{Choices of parent meridians and children longitudes}\label{Sec-clTypes}
We shall finish the proof of Theorem \ref{main} in this section.
Up to now, it remains to bound the allowable choices of \hyperlink{term-node}{children longitudes}
on an allowable JSJ \hyperlink{term-piece}{piece}. Provided Propositions \ref{hypPieces}, \ref{SFPieces},
this will bound the allowable isomorphism types of \hyperlink{term-node}{nodes}.
\begin{lemma}\label{hypCl} Let $G$ be a finitely presented group with $b_1(G)=1$.
If $J$ is an orientable compact $3$-manifold homeomorphic to a hyperbolic
\hyperlink{term-piece}{piece} of some knot complement $M$ such that $G$ maps onto $\pi_1(M)$,
then there are at most finitely many choices of slopes $\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}}$
on $\partial J$, depending only on $G$ and $J$, so that $(J,\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}})$
is a hyperbolic \hyperlink{term-node}{node} decorating a vertex of the \hyperlink{term-rootedJSJTree}{rooted JSJ tree}
for some such $M$.\end{lemma}
\begin{proof}
Up to finitely many choices, we may assume the \hyperlink{term-node}{children boundaries} $\partial_{{\mathtt{c}}_1} J\sqcup\cdots\sqcup\partial_{{\mathtt{c}}_r} J$ and the \hyperlink{term-node}{parent boundary} ${\partial_{\mathtt{p}}} J$ are assigned, (cf. the remark of Definition \ref{node}). Let $K$ be a finite presentation $2$-complex of $G$ as before. We need only show that there are finitely many possible choices for the children longitudes, since for each such choice, there are only finitely many possible \hyperlink{term-node}{parent meridian} choices.
To argue by contradiction, suppose there are infinitely many $\pi_1$-surjective maps
$f_n:K\to M_n$ such that $J$ is a hyperbolic \hyperlink{term-piece}{piece} of a knot complement $M_n$ with the \hyperlink{term-node}{children} \hyperlink{term-node}{and parent} \hyperlink{term-node}{boundaries} compatible with the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} of $M_n$, and that the sets of \hyperlink{term-node}{children} \hyperlink{term-node}{longitudes} $\seq{\lambda_{{\mathtt{c}}_1,n},\cdots,\lambda_{{\mathtt{c}}_r,n}}$ are distinct up to isotopy on $\partial J$ for different $n$'s.
After passing to a subsequence and re-indexing the \hyperlink{term-node}{children boundaries} if necessary, we may assume that $\lambda_{{\mathtt{c}}_1,n}\subset\partial_{{\mathtt{c}}_1} J$ are distinct slopes for different $n$'s, without loss of generality .
For every $n$, we may write $M_n=M_{{\mathtt{p}},n}\cup J\cup M_{{\mathtt{c}}_1,n}\cup\cdots\cup M_{{\mathtt{c}}_r,n}$, where $M_{{\mathtt{p}},n}$ and $M_{{\mathtt{c}}_i,n}$'s are the components of $M_n-J$ adjacent to $J$ along $\partial_{\mathtt{p}} J$ and $\partial_{{\mathtt{c}}_i}J$'s, respectively. ($M_{{\mathtt{p}},n}$ is possibly empty if $\partial_{\mathtt{p}} J$ equals $\partial M_n$). Let $J'_n$ be the Dehn filling of $J$ along $\lambda_{{\mathtt{c}}_1,n}$, and $M'_n=M_{{\mathtt{p}},n}\cup J'_n\cup M_{{\mathtt{c}}_2,n}\cup\cdots\cup M_{{\mathtt{c}}_r,n}$. Then $M_n'$ is still a knot complement and there is a `de-satellitation' map:
$$\alpha_n:M_n\to M'_n,$$
namely, such that $\alpha_n|:M_{{\mathtt{c}}_1,n}\to S^1\times D^2\cong J'_n-J$ is the degree-one map induced by the abelianization $\pi_1(M_{{\mathtt{c}}_1,n})\to{\mathbb Z}$, and that $\alpha_n$ is the identity restricted to the rest of $M_n$. Since $\alpha_n$ is degree-one and hence $\pi_1$-surjective, $f'_n=\alpha_n\circ f_n:K\to M'_n$ is $\pi_1$-surjective for any $n$. However, by the J{\o}rgensen-Thurston theorem on hyperbolic Dehn fillings (\cite[Theorem 5.12.1]{Th}), for all but finitely many slopes $\lambda_{{\mathtt{c}}_1,n}$, $J'_n$ are hyperbolic and mutually distinct. Thus $J'_n$'s are homeomorphically distinct hyperbolic \hyperlink{term-piece}{pieces} of $M'_n$'s, but this contradicts Proposition \ref{hypPieces} that there are only finitely many allowable homeomorphism types of hyperbolic \hyperlink{term-piece}{pieces}.
\end{proof}
\begin{lemma}\label{SFCl}
Let $G$ be a finitely presented group with $b_1(G)=1$. If $J$ is an orientable compact $3$-manifold
homeomorphic to a Seifert fibered \hyperlink{term-piece}{piece} of some knot complement $M$
such that $G$ maps onto $\pi_1(M)$, then there are at most finitely many choices of slopes
$\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}}$ on $\partial J$, depending only on $G$ and $J$,
so that $(J,\mu_{\mathtt{p}}, \seq{\lambda_{{\mathtt{c}}_1},\cdots,\lambda_{{\mathtt{c}}_r}})$ is a Seifert fibered \hyperlink{term-node}{node}
decorating a vertex of the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} for some such $M$.
\end{lemma}
\begin{proof} Among the three possible families of Seifert fibered \hyperlink{term-node}{nodes} (cf. Proposition \ref{knotData}),
the choice of \hyperlink{term-node}{children longitude} is uniquely determined up to orientation for key-chain \hyperlink{term-node}{nodes},
and there is no \hyperlink{term-node}{children longitude} for a torus-knot \hyperlink{term-node}{node}.
It remains to bound the allowable choices of \hyperlink{term-node}{children longitudes} for an allowable cable \hyperlink{term-piece}{piece}.
As before, there will be only finitely many possible choices of \hyperlink{term-node}{parent meridians} for a give choice of \hyperlink{term-node}{children longitudes}.
Note the homeomorphism type of a $p/q$-cable \hyperlink{term-piece}{piece} ($p,q$ coprime, $q>1$)
is determined by $q$ and the residual class of $p\bmod q$, and the \hyperlink{term-node}{children longitude}
is determined by the integer $p$ provided $q$.
Thus it suffices to bound the allowable values of $p$ as the allowable values of $q$ are already bounded by Proposition \ref{SFPieces}.
To see this, let $G$ is a finitely presented group with $b_1(G)=1$, represented by a finite presentation $2$-complex $K$.
Let $J$ be an orientable compact $3$-manifold homeomorphic to a cable \hyperlink{term-piece}{piece} of some knot complement $M$ as assumed. There is a unique choice of the parent (i.e. pattern) component and the child (i.e. companion) component, $\partial J=\partial_{\mathtt{c}} J\sqcup\partial_{\mathtt{p}} J$. Let $M=M_{\mathtt{p}}\cup J\cup M_{\mathtt{c}}$ where $M_{\mathtt{p}}$, $M_{\mathtt{c}}$ are the components of $M-J$ adjacent to $J$ along $\partial_{\mathtt{p}} J$, $\partial_{\mathtt{c}} J$, respectively. If $J$ is realized as a $p/q$-cable in $M$, let $\lambda_{\mathtt{c}}\subset\partial_{\mathtt{c}} J$ be the \hyperlink{term-node}{child longitude} realized in $M$, and $J'$ be the Dehn filling of $J$ along $\lambda_{\mathtt{c}} $, and $M'=M_{\mathtt{p}}\cup J'$. Then $M'$ is still a knot complement and there is a `de-satellitation' map:
$$\alpha:M\to M',$$
induced by the abelianization $\pi_1(M_{\mathtt{c}})\to{\mathbb Z}$. Because $\alpha$ is degree-one and hence $\pi_1$-surjective, $f'=\alpha\circ f:K\to M'$ is also $\pi_1$-surjective. However, now $J'$ is a $p/q$-torus-knot complement, so there are at most finitely many allowable values of $p$ by Proposition \ref{SFPieces}.
\end{proof}
\begin{proof}[{Proof of Theorem \ref{main}}]
First, we reduce from the case that $G$ is a finitely generated group with $b_1(G)=1$ to the case that
$G$ is finitely presented. For any presentation $\mathcal{P} = (x_1,\ldots,x_n;r_1,\ldots,r_m, \ldots)$ of $G$, we may choose a finite
collection of relators $\{r_1,\ldots, r_k\}$ such that the group $G' = \langle x_1, \ldots, x_n; r_1, \ldots, r_k\rangle$ has $b_1(G')=1$.
If $G'$ has only finitely many homomorphisms to knot groups, then so does $G$, since we have an epimorphism $G'\to G$.
We thank Jack Button for pointing out this observation to us. Thus, we may assume that $G$ is finitely presented.
By Lemma \ref{pieceNumber}, there are at most finitely many allowable isomorphism types of the \hyperlink{term-rootedJSJTree}{rooted JSJ tree} of $M$ (Definition \ref{rootedJSJ}) under the assumption in the statement. By Propositions \ref{hypPieces}, \ref{SFPieces}, there are at most finitely many allowable homeomorphism types of JSJ \hyperlink{term-piece}{pieces}, and by Lemmas \ref{hypCl}, \ref{SFCl}, each of them allows at most finitely many choices of \hyperlink{term-node}{children longitudes} and \hyperlink{term-node}{parent meridian}. Hence there are at most finitely many allowable isomorphism types of compatible geometric \hyperlink{term-node}{nodes}. By Proposition \ref{knotData}, we conclude that there are at most finitely many allowable homeomorphisms types of knot complements $M$ as assumed.
\end{proof}
\section{A diameter bound for closed hyperbolic 3-manifolds} \label{Sec-diamBound}
In this section, we generalize and improve the diameter bound for closed hyperbolic $3$-manifolds obtained in \cite{Wh}.
\begin{theorem}\label{diamBound} There exists a universal constant $C>0$,
such that for any orientable closed hyperbolic 3-manifold $M$, the following statements are true.
\noindent(1) The diameter of $M$ is bounded by $C\,\ell(G)$ for any finitely presented group $G$ with $b_1(G)=0$ if $G$ maps onto $\pi_1(M)$.
\noindent(2) The diameter of $M$ is also bounded by $C\,\ell(G)$ for $G=\pi_1(M)$.\end{theorem}
\begin{proof} (1) Let $\epsilon=\epsilon_3>0$ be the Margulis constant of ${\mathbb H}^3$.
Let $M=M_\epsilon\cup V_1\cup\cdots\cup V_s$ where $M_\epsilon$ is the $\epsilon$-thick part,
and $V_i$'s are the components of the $\epsilon$-thin parts, which are homeomorphic to solid tori.
To bound the diameter of $M_\epsilon$, pick a maximal collection of pairwise disjoint balls of radii $\frac{\epsilon}{2}$.
Then the centers of the balls form an $\epsilon$-net of $M_\epsilon$.
In particular, ${\mathrm{diam}}(M_\epsilon)$ is bounded by $2\epsilon$ times the number of balls.
On the other hand, writing $\omega$ for $\pi(\sinh(\epsilon)-\epsilon)$,
i.e. the volume of a hyperbolic ball of radius $\frac{\epsilon}{2}$,
the number of balls is at most $\frac{{\mathrm{Vol}}(M_\epsilon)}{\omega}\leq\frac{{\mathrm{Vol}}(M)}{\omega}$,
which is bounded by $\frac{\pi\ell(G)}{\omega}$ by Theorem \ref{volBound}.
We have:
$${\mathrm{diam}}(M_\epsilon)\leq \frac{2\pi\epsilon\ell(G)}{\omega}=C_1\,\ell(G),$$
where $C_1=\frac{2\pi\epsilon}{\omega}$.
To bound the diameter of the thin tubes, let $V=V_i$ be a component.
The core loop of $V$ is a simple closed geodesic $\gamma$.
If the length of $\gamma$ were so short that the tube radius of $V$ satisfies $\pi\sinh^2(r)>A(\ell(G))$,
where $A(n)=27^n(9n^2+4n)\pi$, (cf. Lemma \ref{smallGen}),
then by Theorem \ref{factor-hyp}, the assumed epimorphism $\phi:G\to \pi_1(M)$
would factorize through some \hyperlink{term-extDrilling}{extended drilling} $\pi_1(N^e)$
where $N=M-\gamma$, as $\phi=\iota^e\circ\phi^e$,
where $\iota^e:\pi_1(N^e)\to M$ is the \hyperlink{term-extDehnFilling}{extended Dehn filling} epimorphism and $\phi^e:G\to\pi_1(N^e)$.
Consider the covering $\kappa:\tilde{N}^e\to N^e$ corresponding to ${\rm Im}(\phi^e)$,
and an argument similar to Proposition \ref{hypPieces} would give a contradiction.
Note assuming $b_1(G)=0$ is necessary here since after drilling one can only conclude $b_1(N^e)>0$.
Indeed, assuming $b_1(G)=1$ would not work, for example,
any one-cusped hyperbolic $3$-manifold maps $\pi_1$-onto infinitely many Dehn fillings whose diameters can be arbitrarily large.
The contradiction implies the tube radius $r$ satisfies:
\begin{eqnarray*}
r&\leq& {\rm arcsinh} \sqrt{27^{\ell(G)}(9\ell(G)^2+4\ell(G))}\\
&<&\ln \left[(1+\sqrt{2})\cdot\sqrt{27^{\ell(G)}(9\ell(G)^2+4\ell(G))}\right]\\
&<&C_2\,\ell(G),
\end{eqnarray*}
for some constant $C_2>0$. We have ${\mathrm{diam}}(V)<{\rm Length}(\gamma)+2r<\epsilon+2C_2\,\ell(G)$.
Combining the bounds for the different parts, we have
$${\mathrm{diam}}(M)\leq{\mathrm{diam}}(M_\epsilon)+2\,\max_{1\leq i\leq s}\,{\mathrm{diam}}(V_i)<2\epsilon+(C_1+2C_2)\ell(G)<C\,\ell(G),$$
for some sufficiently large universal constant $C>0$.
(2) The proof of (1) works for this situation after two modifications.
First, in bounding the diameter of the thick part, we use \cite[Theorem 0.1]{Co} instead of Theorem \ref{volBound} to conclude ${\mathrm{Vol}}(M)\leq\pi\ell(G)$.
Secondly, in bounding the diameter of the thin tubes, after the factorization $\phi=\iota^e\circ\phi^e$,
we consider the sequence of homomorphisms:
$$G\stackrel{\tilde{\phi}^e}\longrightarrow\pi_1(\tilde{N}^e)\stackrel{\kappa_\sharp}\longrightarrow\pi_1(N^e)\stackrel{\iota^e}\longrightarrow\pi_1(M).$$
The composition is $\pi_1$-isomorphic, so should that on $H_3(-;{\mathbb Q})$. Note $H_3(G;{\mathbb Q})\cong H_3(M;{\mathbb Q})\cong{\mathbb Q}$,
but $H_3(N^e;{\mathbb Q})\cong H_3(N;{\mathbb Q})=0$, cf. Subsection \ref{Subsec-DehnExt}.
This is a contradiction, which in turn implies the tube radius $r$ satisfies $\pi\sinh^2(r)\leq A(\ell(G))$. The rest of the proof proceeds the same way as in (1).
\end{proof}
\section{Conclusion}
We believe that some of the techniques in this paper may have applications to questions regarding
homomorphisms from groups to 3-manifold groups.
\begin{question} \label{homomorphism}
For a finitely generated group $G$, is there a uniform description of all homomorphisms from $G$ to
all 3-manifold groups?
\end{question}
If $G$ is a free group of rank $n$, then this question is asking for a description of all 3-manifolds which have
a subgroup of rank $n$, which is probably too difficult to carry out in general. So to make progress on this
question, one would likely have to make certain restrictions on the group $G$, such as the hypothesis
in this paper of $b_1(G)=1$. The answer will likely involve factorizations through \hyperlink{term-DehnExt}{Dehn extensions}, and
may be analogous to the theory of limit groups and Makanin-Razborov diagrams \cite{Sela01}.
More specifically, we ask for an effective version of Theorem \ref{main}:
\begin{question}
For a finitely presented group $G$ with $b_1(G)=1$, is there an algorithmic description
of all knot complements $M$
for which there is an epimorphism $G\twoheadrightarrow \pi_1(M)$, and for each
such $M$, an algorithmic description of the epimorphisms?
\end{question}
As an aspect of Question \ref{homomorphism}, we ask:
\begin{question}
If $G$ is finitely generated (but infinitely presented), is there a finitely presented group $\hat{G}$ and an epimorphism $e: \hat{G}\to G$
such that it induces a bijection $e^*:{\rm Hom}(G,\Gamma)\to {\rm Hom}(\hat{G},\Gamma)$, for every $3$-manifold group $\Gamma=\pi_1(M)$?
\end{question}
This is true if we restrict $M$ to be hyperbolic.
The proof of Theorem \ref{main} holds if we restrict $M$ to be a hyperbolic knot complement in a rational homology sphere.
The place that we used that $M$ is a knot complement in $S^3$ is in the JSJ decomposition in
Section \ref{Subsec-SFPieces} and in bounding the companions in Section \ref{Sec-clTypes}.
\begin{question}
If $G$ is finitely generated with $b_1(G)=1$, are there only finitely many $M$ a knot complement in a rational homology sphere
for which there is an epimorphism $G \twoheadrightarrow \pi_1(M)$?
\end{question}
\end{document} |
\begin{document}
\title{Teleportation scheme implementing contextually the Universal Optimal Quantum
Cloning Machine and the Universal Not Gate. Complete experimental realization }
\author{M. Ricci,\ F. Sciarrino,\ C. Sias, and F. De Martini}
\address{Dipartimento di Fisica and \\
Istituto Nazionale per la Fisica della Materia\\
Universit\`{a} di Roma ''La Sapienza'', Roma, 00185 - Italy}
\maketitle
\begin{abstract}
By a significant modification of the standard protocol of quantum state
Teleportation two processes ''forbidden'' by quantum mechanics in their
exact form, the Universal NOT gate and the Universal Optimal Quantum Cloning
Machine, have been implemented {\it contextually} and {\it optimally} by a
fully {\it linear }method{\it .} In particular, the first experimental
demonstration of the \ {\it Tele-UNOT}\ {\it Gate}, a novel quantum
information protocol has been reported (cfr. quant-ph/0304070). A complete
experimental realization of the protocol is presented here.
\end{abstract}
\pacs{03.67.Dd, 03.65.Ud}
Classical information is encoded in bits, viz. dichotomic variables that can
assume the values $0$ or $1.$ For such variables there are no theoretical
limitations as far as the ''cloning'' and/or ''spin-flipping'' processes are
concerned. However manipulations on the quantum analogue of a bit, a qubit,
have strong limitations due to fundamental requirements by quantum
mechanics. For instance, it has been shown that an arbitrary unknown qubit
cannot be perfectly cloned: $\left| \Psi \right\rangle \rightarrow \left|
\Psi \right\rangle \left| \Psi \right\rangle $, a consequence of the
so-called ``no cloning theorem'' \cite{1}. Another ''impossible'' device is
the quantum NOT gate, the transformation that maps any qubit into the
orthogonal one $\left| \Psi \right\rangle \rightarrow \left| \Psi ^{\perp
}\right\rangle $ \cite{2}. In the last years a great deal of theoretical
investigation has been devoted to finding the best approximation allowed by
quantum mechanics to these processes and to establish the corresponding
''optimal'' values of the ''fidelity'' $F<1$.\ This problem has been solved
in the general case \cite{3,4}. In particular, it was found that a
one-to-two universal optimal quantum cloning machine (UOQCM), i.e. able to
clone one qubit\ into two qubits ($1\rightarrow 2$), can be realized with a
fidelity $F_{CLON}=\frac{5}{6}$. The UOQCM has been experimentally realized
by a quantum optical ''amplification'' method, i.e. by associating the
cloning effect with a QED\ ''stimulated emission'' process \cite{5,6}. Very
recently it has been argued by \cite{6} that when the cloning process is
realized in a subspace $H$ of a larger nonseparable Hilbert space $H\otimes K
$ which is acted upon by a physical apparatus, the same apparatus performs
{\it contextually} in the space $K$ the ''flipping'' of the input injected
qubit, then realizing a $(1\rightarrow 1)$ Universal-NOT gate (U-NOT) with a
fidelity $F_{NOT}=\frac{2}{3}$ \cite{4}. As an example, a UOQCM can be
realized on one output mode of a non-degenerate ''quantum-injected'' optical
parametric amplifier (QI-OPA), while the U-NOT transformation is realized on
the other mode \cite{7}.
In the present work this relevant, somewhat intriguing result is
investigated under a new perspective implying a {\it modified} Quantum State
Teleportation (QST)\ protocol, according to the following scheme. The QST\
protocol implies that an unknown input qubit $\left| \phi \right\rangle
_{S}=\alpha \left| 0\right\rangle _{S}+\beta \left| 1\right\rangle _{S}\ $is
destroyed at a sending place (Alice:\ ${\cal A}$) while its perfect replica
appears at a remote place (Bob:\ ${\cal B}$) via dual {\it quantum} and {\it
classical} channels \cite{8}. Let us assume that Alice and Bob share the
entangled ''singlet''\ state $\left| \Psi ^{-}\right\rangle _{AB}=2^{-
{\frac12}
}\left( \left| 0\right\rangle _{A}\left| 1\right\rangle _{B}-\left|
1\right\rangle _{A}\left| 0\right\rangle _{B}\right) $, and that we want to
teleport the generic qubit $\left| \phi \right\rangle _{S}\equiv \left| \phi
\right\rangle $. The \ {\it singlet} state is adopted hereafter because its
well known invariance under SU(2) transformations will ensure the
''universality'' of the cloning and U-NOT processes, as we shall see \cite
{5,6,7}. The overall state of the system is then $\left| \Omega
\right\rangle _{SAB}$= $\left| \phi \right\rangle _{S}\left| \Psi
^{-}\right\rangle _{AB}$. Alice performs a Bell measurement by projecting
the joint state of the qubits $S$ and $A$ into the four Bell states $\left\{
\left| \Psi ^{-}\right\rangle _{SA},\left| \Psi ^{+}\right\rangle
_{SA},\left| \Phi ^{-}\right\rangle _{SA},\left| \Phi ^{+}\right\rangle
_{SA}\right\} $ spanning the 4-dimensional Hilbert space $H\equiv $ $
H_{A}\otimes H_{S}$ and then sends the result to Bob by means of 2 bits of
classical information. In order to obtain $\left| \phi \right\rangle _{B}$,
Bob applies to the received state the appropriate unitary transformation $
U_{B}$ according to the following protocol: $\left| \Psi ^{-}\right\rangle
_{SA}\rightarrow U_{B}={\Bbb I}$, $\left| \Psi ^{+}\right\rangle
_{SA}\rightarrow U_{B}=\sigma _{Z}$, $\left| \Phi ^{-}\right\rangle
_{SA}\rightarrow U_{B}=\sigma _{X}$, $\left| \Phi ^{+}\right\rangle
_{SA}\rightarrow U_{B}=\sigma _{Y}$ where the kets express the received
corresponding information,$\ I,\sigma _{Z},\sigma _{X}$ are respectively the
identity, phase flip, spin flip operators and $\sigma _{Y}=-i\sigma
_{Z}\sigma _{X}$. At last, the QST channel acts on the input state $\rho
_{S}\equiv \left| \phi \right\rangle \left\langle \phi \right| $ as the
identity operator: $E_{QST}(\rho _{S})={\Bbb \rho }_{B}$. In absence of the
{\it classical} channel, i.e. of the appropriate $U_{B}$ transformation, the
apparatus realizes the map $E_{B}(\rho _{S})=
{\frac12}
{\Bbb I}_{B}$, corresponding to the {\it depolarization} channel $E_{DEP}$.
This is the worst possible case because any information about the initial
state $\rho _{S}$ is lost.
In order to implement the UOQCM and U-NOT at Alice's and Bob's sites, in the
present work we modify the QST\ protocol by performing a different \
measurement on the system $S$ $+$ $A$. This leads to a different content of
information to be transferred by the {\it classical} channel from ${\cal A}$
to ${\cal B}$. Precisely, the ''{\it Bell measurement}'', able to
discriminate between the 4 Bell states, is replaced here by a dichotomic ''
{\it projective} {\it measurement}'' able to identify $\left| \Psi
^{-}\right\rangle _{SA}$ , i.e. the {\it anti-symmetric} subspace of $
H\equiv $ $H_{A}\otimes H_{S}$, and its complementary {\it symmetric}
subspace. Let us analyze the outcomes of such strategy, schematically
represented by Figure 1. With a probability $p=\frac{1}{4}$ the $\left| \Psi
^{-}\right\rangle _{SA}$ is detected by ${\cal A}$. In this case the correct
QST channel $E_{QST}$ is realized. However, if this is not the case, with
probability $p=\frac{3}{4}$, Bob cannot apply any unitary transformation to
the set of the non identified Bell states $\left\{ \left| \Psi
^{+}\right\rangle _{SA},\left| \Phi ^{-}\right\rangle _{SA},\left| \Phi
^{+}\right\rangle _{SA}\right\} $, and then the QST channel implements the
statistical map $E\left( \rho \right) $ = $\frac{1}{3}\left[ \sigma _{Z}\rho
\sigma _{Z}+\sigma _{X}\rho \sigma _{X}+\sigma _{Y}\rho \sigma _{Y}\right] $
. As we shall see, this map coincides with the map $E_{UNOT}\left( \rho
_{S}\right) $ which realizes the Universal Optimal-NOT gate, i.e. the one
that approximates {\it optimally} the flipping of one qubit $\left| \phi
\right\rangle $ into the orthogonal qubit $\left| \phi ^{\perp
}\right\rangle $, i.e. $\rho _{S}$ into $\rho _{S}^{\perp }\equiv \left|
\phi ^{\perp }\right\rangle \left\langle \phi ^{\perp }\right| $. Bob
identifies the two different maps realized at his site by reading the
information (1 bit) received by Alice on the classical channel. For example,
such {\it bit} can assume the value $0$ if Alice identifies the Bell state $
\left| \Psi ^{-}\right\rangle _{SA}$ and $1$ if she does not. We name this
process $Tele-UNOT$ since it consists in the Teleportation of an {\it optimal
} antiunitary map acting on any input qubit, the U-NOT Gate \cite{7,9}.
As already stated, it has been shown that in a bipartite entangled system
the {\it optimal} U-NOT Gate is generally realized {\it contextually}
together with an {\it optimal} quantum cloning process \cite{6,7}. Therefore
it is worth analyzing what happens when the overall state $\left| \Omega
\right\rangle _{SAB}$\ is projected onto the subspace orthogonal to $\left|
\Psi ^{-}\right\rangle _{SA}\left\langle \Psi ^{-}\right| _{SA}\otimes H_{B}$
by the projector:
\begin{equation}
P_{SAB}=({\Bbb I}_{SA}-\left| \Psi ^{-}\right\rangle _{SA}\left\langle \Psi
^{-}\right| _{SA})\otimes {\Bbb I}_{B}. \label{proiettore}
\end{equation}
This procedure generates the normalized state $\left| \widetilde{\Omega }
\right\rangle \equiv P_{SAB}\left| \Omega \right\rangle _{SAB}=\sqrt{\frac{2
}{3}}[\left| \xi _{1}\right\rangle _{SA}\otimes \left| 1\right\rangle
_{B}-\left| \xi _{0}\right\rangle _{SA}\otimes \left| 0\right\rangle _{B}]$
where: $\left| \xi _{1}\right\rangle _{SA}$= $\alpha \left| 0\right\rangle
_{S}\left| 0\right\rangle _{A}$ + $
{\frac12}
\beta (\left| 1\right\rangle _{S}\left| 0\right\rangle _{A}+\left|
0\right\rangle _{S}\left| 1\right\rangle _{A})$ and $\left| \xi
_{0}\right\rangle _{SA}$= $\beta \left| 1\right\rangle _{S}\left|
1\right\rangle _{A}$+ $
{\frac12}
\alpha (\left| 1\right\rangle _{S}\left| 0\right\rangle _{A}+\left|
0\right\rangle _{S}\left| 1\right\rangle _{A})$. By tracing this state over
the $SA$ and $B$ manifolds we get: $\rho _{SA}\equiv Tr_{B}\left| \widetilde{
\Omega }\right\rangle \left\langle \widetilde{\Omega }\right| $ = $\frac{2}{3
}\left| \phi \right\rangle \left| \phi \right\rangle _{SA}\left\langle \phi
\right| \left\langle \phi \right| _{SA}+\frac{1}{3}\left| \left\{ \phi ,\phi
^{\perp }\right\} \right\rangle _{SA}\left\langle \left\{ \phi ,\phi ^{\perp
}\right\} \right| _{SA}$ and $\rho _{B}^{out}\equiv Tr_{SA}\left| \widetilde{
\Omega }\right\rangle \left\langle \widetilde{\Omega }\right| =\frac{1}{3}
(2\rho _{S}^{\perp }+\rho _{S})$ , where: $\left| \left\{ \phi ,\phi ^{\perp
}\right\} \right\rangle _{SA}$= $2^{-
{\frac12}
}\left( \left| \phi ^{\perp }\right\rangle _{S}\left| \phi \right\rangle
_{A}+\left| \phi \right\rangle _{S}\left| \phi ^{\perp }\right\rangle
_{A}\right) $. We further project on the spaces $A$ and $B$ and obtain the
output states, which are found mutually equal: $\rho _{S}^{out}\equiv
Tr_{A}\rho _{SA}$ = $\frac{1}{6}(5\rho _{S}+\rho _{S}^{\perp })$ = $\rho
_{A}^{out}\equiv Tr_{S}\rho _{SA}$. At last, by these results the expected
{\it optimal} values for the fidelities of the two ''forbidden''\ processes
are obtained: $F_{CLON}=Tr[\rho _{S}^{out}\rho _{S}]=Tr[\rho _{A}^{out}\rho
_{S}]=\frac{5}{6}$ and $F_{UNOT}=Tr[\rho _{B}^{out}\rho _{S}^{\perp }]=\frac{
2}{3}\ $\cite{6,7}.
In the last years many experimental realizations of quantum state
Teleportation have been achieved \cite{10}. Recently the first ''active'',
i.e. {\it complete} version of the QST\ protocol was realized by our
Laboratory by physically implementing the Bob's unitary operation \cite{11}.
In the present experiment the input qubit was codified as the {\it
polarization} state of a single photon belonging to the input mode $k_{S}$: $
\left| \phi \right\rangle _{S}=\alpha \left| H\right\rangle _{S}+\beta
\left| V\right\rangle _{S}$, $\left| \alpha \right| ^{2}+\left| \beta
\right| ^{2}=1$: Figure 2. Here $\left| H\right\rangle $ and $\left|
V\right\rangle $ correspond to the horizontal and vertical polarizations,
respectively. In addition, an entangled pair of photons, $A$ and $B$ were
generated on the modes $k_{A}$ and $k_{B}$ by Spontaneous Parametric Down
Conversion (SPDC) in the {\it singlet} state: $\left| \Psi ^{-}\right\rangle
_{AB}$= $2^{-1/2}\left( \left| H\right\rangle _{A}\left| V\right\rangle
_{B}-\left| V\right\rangle _{A}\left| H\right\rangle _{B}\right) $. The {\it
projective measurement} in the space $H=$ $H_{A}\otimes H_{S}$ was realized
by linear superposition of the modes $k_{S}$ and $k_{A}$ on a $50:50$ {\it
beam-splitter, }$BS_{A}.$ Consider the overall output state realized on the
two output modes $k_{1}\,$and $k_{2}$ of $BS_{A}$ and expressed by a linear
superposition of the states: $\left\{ \left| \Psi ^{-}\right\rangle
_{SA},\left| \Psi ^{+}\right\rangle _{SA},\left| \Phi ^{-}\right\rangle
_{SA},\left| \Phi ^{+}\right\rangle _{SA}\right\} $. As it is well known,
the realization of the singlet $\left| \Psi _{SA}^{-}\right\rangle $ is
identified by the emission of \ one photon on each output mode of $BS_{A}$
while the realization of the set of the other three Bell states implies the
emission of 2 photons either on mode $k_{1}\,$or on mode $k_{2}$. The
realization of the last process, sometimes dubbed as ''Bose Mode
Coalescence'' (BMC) or ''Mode-occupation enhancement'' \cite{12} \ was
experimentally identified by the simultaneous clicking of the detectors $
D_{2}$ and $D_{2}^{\ast }$ coupled to the output mode $k_{2}$ by the $50:50$
beam-splitter $BS_{2}$. The identical effect expected on mode $k_{1}$ was
not exploited, for simplicity. As shown by the theoretical analysis \ above,
the last condition implied the simultaneous realization in our experiment of
the U-NOT and UOQCM processes, here detected by a {\it post-selection}
technique. Interestingly enough, the {\it symmetry} of the projected
subspace of $H$ identified by BMC is implied by the intrinsic {\it Bose
symmetry} of the 2 photon Fock state realized at the output of $BS_{A}$.
The source of the SPDC process was a Ti:Sa mode-locked pulsed laser with
wavelength (wl) $\lambda =795nm$ and repetition rate $76MHz$. A weak beam,
deflected from the laser beam by mirror $M$, strongly attenuated by grey
filters $(At)$, delayed by $Z${\bf \ }$=2c\Delta t${\bf \ }via{\bf \ }a
micrometrically adjustable optical ''trombone'' was the source of \ the {\it
quasi} single-photon state injected into $BS_{A}$ over the mode $k_{s}$. The
average number of injected photons was $\overline{n}$ $\simeq $ $0.1$ and
the probability of a spurious 2 photon injection was evaluated to be a
factor $0.05$ lower than for single photon injection. Different qubit states
$\left| \phi \right\rangle _{S}$ were prepared via the optical Wave-Plate
(wp) $WP_{S}$, either a $\lambda /2$ or $\lambda /4$ wp. The main UV\ laser
beam with wl $\lambda _{p}=397.5nm$. generated by Second Harmonic
Generation, focused into a $1.5mm$ thick nonlinear (NL) crystal of $\beta $
-barium borate (BBO) cut for Type II phase-matching, excited the SPDC\
source of the {\it singlet} $\left| \Psi ^{-}\right\rangle _{AB}$. The
photons $A$ and $B$ of each entangled pair were emitted over the modes $
k_{A} $ and $k_{B}$ with equal wls $\lambda =795nm$. All adopted
photodetectors $(D)$ were equal SPCM-AQR14 single photon counters. One
interference filter with bandwidth $\Delta \lambda =3nm$ was placed in front
of each $D$ and determined the coherence time of \ the optical pulses: $\tau
_{coh}\simeq 350f\sec $.
In order to realize the $Tele-UNOT$ protocol, the BMC\ process on the output
mode $k_{2}$ of $BS_{A}$ was detected by a coincidence technique involving $
D_{2}$ and $D_{2}^{\ast }$ [Fig.2, inset:\ (a)]. In order to enhance the ''
{\it visibility}'' of the Ou-Mandel interference at the output of $BS_{A}$,
a $\pi -$preserving mode-selector $(MS)$ was inserted on mode $k_{2}$. On
the Bob's site, the polarization $\pi -state$ on the mode $k_{B}$ was
analyzed by the combination of the wp $WP_{B}$ and of the {\it polarization
beam splitter} $PBS_{B}$. For each input $\pi -state$ $\left| \phi
\right\rangle _{B}$, $WP_{B}$ was set in order to make the $PBS_{B}$ to
transmit $\left| \phi \right\rangle _{B}$ and to reflect $\left| \phi
^{\perp }\right\rangle _{B}$, by then exciting $D_{B}$ and $D_{B}^{\ast }$
correspondingly. First consider the Teleportation (QST)\ turned off, by
setting the optical delay $\left| Z\right| \gg c\tau _{coh\text{\ }}$i.e. by
spoiling the interference of photons $S$ and $A$ in $BS_{A}$. In this case,
since the states $\left| \phi \right\rangle _{B}$ and $\left| \phi ^{\perp
}\right\rangle _{B}$ were realized with the same probability on mode $k_{B}$
, the rate of coincidences detected by the $D$-sets $[D_{B},D_{2},D_{2}^{
\ast }]$ and $[D_{B}^{\ast },D_{2},D_{2}^{\ast }]\ $were expected to be
equal. By turning on the QST, i.e. by setting $\left| Z\right| <<c\tau
_{coh} $, the output state $\rho _{B}^{out}=(2\rho _{S}^{\perp }+\rho
_{S})/3 $ was realized implying a factor $R=2$ {\it enhancement} of the
counting rate $[D_{B}^{\ast },D_{2},D_{2}^{\ast }]$ and {\it no enhancement}
of $[D_{B},D_{2},D_{2}^{\ast }]$. The actual measurement of $R$ was carried
out,\ and the {\it universality} of the $Tele-UNOT$ process demonstrated, by
the experimental results shown in Figure 3. These 3-coincidence results,
involving the sets $[D_{B},D_{2},D_{2}^{\ast }]$ and $[D_{B}^{\ast
},D_{2},D_{2}^{\ast }]$ correspond to the injection of three different input
states: $\left| \phi \right\rangle _{S}=\left| H\right\rangle $, $\left|
\phi \right\rangle _{S}=2^{-1/2}(\left| H\right\rangle +\left|
V\right\rangle )$, $\left| \phi \right\rangle _{S}=2^{-1/2}(\left|
H\right\rangle +i\left| V\right\rangle )$. In Figure 3 the square and
triangular markers refer respectively to the $[D_{B}^{\ast
},D_{2},D_{2}^{\ast }]$ and $[D_{B},D_{2},D_{2}^{\ast }]$ coincidences
versus the delay $Z=2c\Delta t.$ We may check that the $Tele-UNOT$ process
only affects the $\left| \phi ^{\perp }\right\rangle _{B}$ component, as
expected. The {\it signal-to-noise} $(S/N)$\ ratio $R\;$was determined as
the ratio between the peak values, i.e. for $Z\simeq 0$, and the no BMC
enhancement values, i.e. for $\left| Z\right| \gg c\tau _{coh\text{\ }}$.
The experimental values of the $UNOT$ {\it fidelity\ }$F=R(R+1)^{-1}$ are: $
F_{H}=0.641\pm 0.005$;$\ F_{H+V}=0.632\pm 0.006$; $F_{H+iV}=0.619\pm 0.006$,
for the three injection states $\left| \phi \right\rangle _{S}$. These
results, to be compared with the {\it optimal} value $F_{th}=2/3\approx
0.666 $ corresponding to the {\it optimal}$\;R=2$, have been evaluated by
taking into account the reduction, by a factor $\xi =0.7$, of the
coincidence rate due to the spurious simultaneous injection of two photons
on the mode $k_{S}$ and to the simultaneous emission of two SPDC\ pairs. The
factor $\xi $ was carefully evaluated by a side experiment involving the
detectors $D_{2}$and $D_{2}^{\ast }$. Note that the experimental Tele-UNot
peaks shown in Figure 3 indeed demonstrate the simultaneous, {\it contextual}
realization of the Quantum Cloning and UNot Processes on Alice' s and Bob's
sites, respectively.
For the sake of completeness, we wanted to gain insight in the linear {\it
probabilistic} UOQCM\ process by investigating whether at the output of $
BS_{A}$ the photons $S$ and $A$ were indeed left in the state $\rho
_{S}^{out}=(5\rho _{S}+\rho _{S}^{\perp })/6$ = $\rho _{A}^{out}$ after the
state projection. The cloning analysis was realized on the $BS_{A}$ output
mode $k_{2}$ by replacing the measurement set (a) with the (b) one, shown in
the inset of Figure 2. The polarization state on mode $k_{2}$ was analyzed
by the combination of the wp $WP_{C}$ and of the polarizer beam splitters $
PBS$. For each input $\pi $-state $\left| \phi \right\rangle _{S}$, $WP_{C}$
was set in order to make $PBS^{\prime }$s to transmit $\left| \phi
\right\rangle _{S}$ and to reflect $\left| \phi ^{\perp }\right\rangle _{S}$
. The ''cloned'' state $\left| \phi \phi \right\rangle _{S}$ was detected on
mode $k_{2}$ by a coincidence between the detectors $D_{C}$, $D_{C}^{\prime
} $. The generation of an entangled pair was assured by detecting one photon
on the mode $k_{B}$; in this case $PBS_{B}$ was removed and the field of
mode $k_{B}$ was coupled directly to $D_{B}$. Any coincidence detected by
the sets $[D_{C},D_{C}^{\prime },D_{B}]$ and $[D_{C},D_{C}^{\ast },D_{B}]$
implied the realization of\ the states $\left| \phi \phi \right\rangle _{S}$
and $\left| \phi \phi ^{\perp }\right\rangle _{S}$, respectively. In analogy
with the previous experiment, when $\left| Z\right| >>c\tau _{coh\text{\ }}$
the rate of coincidences detected by $[D_{C},D_{C}^{\prime },D_{B}]$ and $
2\times \lbrack D_{C}^{\ast },D_{C}^{\prime },D_{B}]$ were expected to be
equal. By turning on the cloning machine, $\left| Z\right| <<c\tau _{coh}$,
an {\it enhancement }by a factor $R=2$ of the counting rate by $
[D_{C},D_{C}^{\prime },D_{B}]$ and {\it no enhancement} by $
[D_{C},D_{C}^{\ast },D_{B}]$ were expected. The experimental results of the $
S/N$ ratio $R$, carried out by coincidence measurements involving $
[D_{C},D_{C}^{\prime },D_{B}]$ and $[D_{C},D_{C}^{\ast },D_{B}]$ are
reported\ in the lower plots of Figure 3, again for the three different
input states: $\left| \phi \right\rangle _{S}=\left| H\right\rangle $, $
\left| \phi \right\rangle _{S}=2^{-1/2}(\left| H\right\rangle +\left|
V\right\rangle )$, $\left| \phi \right\rangle _{S}=2^{-1/2}(\left|
H\right\rangle +i\left| V\right\rangle )$. The square and triangular markers
there refer respectively to the $[D_{C},D_{C}^{\prime },D_{B}]$ and $
[D_{C},D_{C}^{\ast },D_{B}]$ coincidence plots Vs the delay $Z$. The
following values of the {\it cloning fidelity }$F=(2R+1)(2R+2)^{-1}$ were
found: $F_{H}=0.821\pm 0.003$;$\ F_{H+V}=0.813\pm 0.003$; $F_{H+iV}=0.812\pm
0.003$, to be compared with the {\it optimal} $F_{th}=5/6\approx 0.833$
corresponding to the limit $S/N$ value: $R=2$. As for the $Tele-UNOT$ gate
experiment, the factor $\xi =0.7$ has been corrected during the evaluation
of the fidelities. Finally note that, in the present context the entangled
{\it singlet} state $\left| \Psi ^{-}\right\rangle _{AB}\ $was not strictly
necessary for the solely implementation of quantum cloning as we could model
the {\it local} effect of the {\it singlet} on the input mode $k_{A}$ by a
{\it fully mixed }state $\rho _{A}=
{\frac12}
{\Bbb I}_{A}$ spanning a $2$ dimensional space. This has been realized
successfully in another experiment \cite{13}.
In summary, two relevant quantum information processes, forbidden by quantum
mechanics in their exact form are found to be connected {\it contextually}
by a modified quantum state teleportation scheme and can be ''optimally''
realized. \ At \ variance with previous experiments, the complete
implementation of the new protocol has been successfully performed by a
fully $linear$ optical setup. The results are found in full agreement with
theory.
This work has been supported by the FET European Network on Quantum
Information and Communication (Contract IST-2000-29681: ATESIT), by I.N.F.M.
(PRA\ ''CLON'')\ and by M.I.U.R.(COFIN 2002).
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\centerline{\bf Figure Captions}
\end{references}
\vskip 8mm
\parindent=0pt
\parskip=3mm
Figure.1. (COLOR\ ONLINE) General scheme for the simultaneous realization of
the Tele-UNOT Gate and of the {\it probabilistic} Universal Quantum Cloning
Machine (UOQCM).
Figure.2. (COLOR\ ONLINE) Setup for the optical implementation of the {\it
Tele-UNOT Gate} and the {\it probabilistic} UOQCM. The measurement setup
used for the verification of the cloning experiment is reported in the INSET
(b).
Figure.3.\ (COLOR\ ONLINE) Experimental results of the {\it Tele-UNOT Gate}
and the UOQCM for three input qubits. {\it Filled squares:} plots
corresponding to the ''correct'' polarization; {\it Open triangles}: plots
corresponding to the ''wrong'' polarization. The solid line represents the
best gaussian fit expressing the {\it correct} polarization.
\end{document} |
\begin{document}
\title{Effective sketching methods for value function approximation}
\begin{abstract}
High-dimensional representations, such as radial basis function networks or tile coding, are common choices for policy evaluation in reinforcement learning. Learning with such high-dimensional representations, however, can be expensive, particularly for matrix methods, such as least-squares temporal difference learning or quasi-Newton methods that approximate matrix step-sizes. In this work, we explore the utility of sketching for these two classes of algorithms. We highlight issues with sketching the high-dimensional features directly, which can incur significant bias. As a remedy, we demonstrate how to use sketching more sparingly, with only a left-sided sketch, that can still enable significant computational gains and the use of these matrix-based learning algorithms that are less sensitive to parameters. We empirically investigate these algorithms, in four domains with a variety of representations. Our aim is to provide insights into effective use of sketching in practice.
{\mathop{\mathrm{te}}xtrm{e-}}nd{abstract}
\section{INTRODUCTION}
A common strategy for function approximation in reinforcement learning is
to overparametrize: generate a large number of features to provide
a sufficiently complex function space.
For example, one typical representation is a radial basis function network,
where the centers for each radial basis function are chosen to exhaustively cover the
observation space.
Because the environment is unknown---particularly for the incremental learning setting---such an overparameterized representation
is more robust to this uncertainty because a reasonable representation is guaranteed for any part
of the space that might be visited.
Once interacting with the environment, however, it is likely not all features
will become active, and that a lower-dimensional subspace will be visited.
A complementary approach for this high-dimensional representation expansion in reinforcement learning,
therefore, is to use projections. In this way, we can overparameterize for robustness,
but then use a projection to a lower-dimensional space to make learning feasible.
For an effectively chosen projection, we can avoid discarding important information,
and benefit from the fact that the agent only visits a lower-dimensional subspace of the environment
in the feature space.
Towards this aim, we investigate the utility of sketching: projecting with a random matrix.
Sketching has been extensively used for efficient communication
and solving large linear systems,
with a solid theoretical foundation and a variety of different sketches \perp\!\!\!\perptep{woodruff2014sketching}.
Sketching has been previously used in reinforcement learning, specifically to reduce the dimension of the features.
\perp\!\!\!\perptet{bellemare2012sketch} replaced the standard biased hashing function used for tile coding \perp\!\!\!\perpte{sutton1996generalization}, instead using count-sketch.\footnote{They called the sketch the tug-of-war sketch, but it is more standard to call it count-sketch.}
\perp\!\!\!\perptet{ghavamzadeh2010lstd} investigated sketching features to reduce the dimensionality
and make it feasible to run least-squares temporal difference learning (LSTD) for policy evaluation.
In LSTD, the value function is estimated by incrementally computing a $d\timesd$ matrix $\mathbf{A}$, where $d$ is the number of features, and an $d$-dimensional vector $\mathbf{b}$, where the parameters are estimated as the solution to this linear system.
Because $d$ can be large, they randomly project the features to reduce the matrix size to $k\timesk$, with $k \ll d$.
For both of these previous uses of sketching, however, the resulting value function estimates
are biased. This bias, as we show in this work, can be quite significant, resulting in significant
estimation error in the value function for a given policy. As a result, any gains from using
LSTD methods---over stochastic temporal difference (TD) methods---are largely overcome by this bias.
A natural question is if we can benefit from sketching,
with minimal bias or without incurring any bias at all.
In this work, we propose to instead sketch the linear system
in LSTD. The key idea
is to only sketch the constraints of the system (the left-side of $\mathbf{A}$) rather than the variables
(the right-side of $\mathbf{A}$).
Sketching features, on the other hand, by design, sketches both constraints and variables.
We show that even with a straightforward linear system solution,
the left-sided sketch can significantly reduce bias.
We further show how to use this left-sided sketch within a quasi-Newton algorithm,
providing an unbiased policy evaluation algorithm that can still benefit from the computational improvements of sketching.
The key novelty in this work is designing such system-sketching algorithms
when also incrementally computing the linear system solution.
There is a wealth of literature on sketching linear systems, to reduce computation.
In general, however, many sketching approaches cannot be applied
to the incremental policy evaluation problem, because
the approaches are designed for a static linear system.
For example, \perp\!\!\!\perptet{gower2015randomized} provide a host of possible
solutions for solving large linear systems. However, they assume access to $\mathbf{A}$ upfront,
so the algorithm design, in memory and computation, is not suitable
for the incremental setting.
Some popular sketching approaches, such as Frequent Directions \perp\!\!\!\perptep{ghashami2014improved},
has been successfully used for the online setting, for quasi-Newton algorithms \perp\!\!\!\perptep{luo2016efficient};
however, they sketch symmetric matrices, that are growing with number of samples.
This paper is organized as follows.
We first introduce the policy evaluation problem---learning a value function for a fixed policy---and provide background
on sketching methods. We then illustrate issues with only sketching features, in terms of quality of the value function approximation.
We then introduce the idea of using asymmetric sketching for policy evaluation with LSTD,
and provide an efficient incremental algorithm that is $O(d k)$ on each step.
We finally highlight settings where we expect sketching to perform particularly well in practice,
and investigate the properties of our algorithm on four domains, and with a variety of representation properties.
\section{PROBLEM FORMULATION}
We address the policy evaluation problem within reinforcement learning,
where the goal is to estimate the value function for a given policy\footnote{To focus the investigation on sketching, we consider
the simpler on-policy setting in this work. Many of the results, however, generalize to the off-policy setting,
where data is generated according to a behavior policy different than the given target policy we wish to evaluate.}.
As is standard, the agent-environment interaction is formulated as a Markov decision process $(\mathcal{S}, \mathcal{A}, \mathrm{Pr}, r)$,
where
$\mathcal{S}$ is the set of states,
$\mathcal{A}$ is the set of actions, and
$\mathrm{Pr}: \mathcal{S} \times \mathcal{A} \times \mathcal{S} \mathop{\mathrm{ri}}ghtarrow [0,\infty)$ is the one-step state transition dynamics.
On each time step $t = 1,2,3,...$, the agent selects an action according to its policy $\pi$, $A_t\sim\pi(S_t, \cdot)$,
with $\pi: \mathcal{S} \times \mathcal{A} \mathop{\mathrm{ri}}ghtarrow [0, \infty)$ and transitions into a new state $S_{t+1}\sim\mathrm{Pr}(S_t, A_t, \cdot)$ and obtains scalar reward $R_{t+1} \defeq r(S_t,A_t,S_{t+1})$.
For policy evaluation, the goal is to estimate the value function, $v_\pi: \mathcal{S} \mathop{\mathrm{ri}}ghtarrow \mathbb{R}$, which corresponds to the expected return when following policy $\pi$
\begin{equation*}
v_\pi(s) \defeq \mathbb{E}_\pi[G_t | S_t = s],
{\mathop{\mathrm{te}}xtrm{e-}}nd{equation*}
where $\mathbb{E}_\pi$ is the expectation over future states when selecting actions according to $\pi$. The return, $G_t \in \mathbb{R}$ is the discounted sum of future rewards given actions are selected according to $\pi$:
\begin{align}
G_t &\defeq R_{t+1} + \gamma_{t+1} R_{t+2} + \gamma_{t+1}\gamma_{t+2} R_{t+3} + ... \\
&= R_{t+1} + \gamma_{t+1} G_{t+1} \nonumber
{\mathop{\mathrm{te}}xtrm{e-}}nd{align}
where $\gamma_{t+1}\in[0,1]$ is a scalar that depends on $S_t, A_t, S_{t+1}$ and discounts the contribution of future rewards exponentially with time. A common setting, for example, is a constant discount. This recent generalization to state-dependent discount \perp\!\!\!\perptep{sutton2011horde,white2016unifying} enables either episodic or continuing problems, and so we adopt this more general formalism here.
We consider linear function approximation to estimate the value function. In this setting,
the observations are expanded to a higher-dimensional space, such as through tile-coding, radial basis functions or Fourier basis.
Given this nonlinear encoding $x: \mathcal{S} \mathop{\mathrm{ri}}ghtarrow \mathbb{R}^d$, the value is approximated as
$v_\pi(S_t) \approx \mathbf{w}^\top \mathbf{x}_t$ for $\mathbf{w}\in\mathbb{R}^d$ and $\mathbf{x}_t \defeq x(S_t)$.
One algorithm for estimating $\mathbf{w}$ is least-squares temporal difference learning (LSTD).
The goal in LSTD($\lambda$) \perp\!\!\!\perptep{boyan1999least} is to minimize the mean-squared projected Bellman error, which can be represented
as solving the following linear system
\begin{align*}
\mathbf{A} &\defeq \mathbb{E}_\pi[{\mathop{\mathrm{te}}xtrm{e-}}vec_t (\mathbf{x}_t-\gamma_{t+1}\mathbf{x}_{t+1})^\top]\\
\mathbf{b} &\defeq \mathbb{E}_\pi[R_{t+1}{\mathop{\mathrm{te}}xtrm{e-}}vec_t]
.
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
where ${\mathop{\mathrm{te}}xtrm{e-}}vec_t \defeq \gamma_{t+1} \lambda {\mathop{\mathrm{te}}xtrm{e-}}vec_{t-1} + \mathbf{x}_t$ is called the eligibility trace
for trace parameter $\lambda \in [0,1]$.
To obtain $\mathbf{w}$, the system $\mathbf{A}$ and $\mathbf{b}$ are incrementally estimated, to
solve $\mathbf{A} \mathbf{w} = \mathbf{b}$.
For a trajectory $\{(S_t,A_t,S_{t+1},R_{t+1})\}_{t=0}^{T-1}$, let $\mathbf{d}_t \defeq \mathbf{x}_t - \gamma_{t+1} \mathbf{x}_{t+1}$, then the above two expected terms are usually computed via sample average
that can be recursively computed, in a numerically stable way, as
\begin{align*}
\mathbf{A}_{t+1} &= \mathbf{A}_{t} + \frac{1}{t+1} \left( {\mathop{\mathrm{te}}xtrm{e-}}vec_{t} \mathbf{d}_t^\top - \mathbf{A}_{t}\mathop{\mathrm{ri}}ght)\\
\mathbf{b}_{t+1} &= \mathbf{b}_{t} + \frac{1}{t+1} \left({\mathop{\mathrm{te}}xtrm{e-}}vec_{t} R_{t+1} - \mathbf{b}_{t} \mathop{\mathrm{ri}}ght)
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
with $\mathbf{A}_0 = \vec{0}$ and $\mathbf{b}_0 = \vec{0}$. The incremental estimates $\mathbf{A}_t$ and $\mathbf{b}_t$
converge to $\mathbf{A}$ and $\mathbf{b}$. A naive algorithm, where $\mathbf{w} = \mathbf{A}_t^\inv \mathbf{b}_t$ is recomputed on each step, would result in $\mathcal{O}(d^3)$ computation to compute the inverse $\mathbf{A}_t^\inv$.
Instead, $\mathbf{A}_t^\inv$ is incrementally updated using the Sherman-Morrison formula, with $\mathbf{A}_0^\inv = \xi$ for a small $\xi>0$
\begin{align*}
\mathbf{A}_t^\inv &= \left(\frac{t-1}{t} \mathbf{A}_{t-1} + \frac{1}{t} {\mathop{\mathrm{te}}xtrm{e-}}vec_{t} \mathbf{d}_t^\top\mathop{\mathrm{ri}}ght)^\inv \\
&= \frac{t}{t-1}\left(\mathbf{A}_{t-1}^\inv + \frac{ \mathbf{A}_{t-1}^\inv {\mathop{\mathrm{te}}xtrm{e-}}vec_{t} \mathbf{d}_t^\top\mathbf{A}_{t-1}^\inv}{t-1+ \mathbf{d}_t^\top\mathbf{A}_{t-1}^\inv {\mathop{\mathrm{te}}xtrm{e-}}vec_{t} }\mathop{\mathrm{ri}}ght)
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
requiring
$\mathcal{O}(d^2)$ storage and computation per step.
Unfortunately, this quadratic cost is prohibitive for many incremental learning settings. In our experiments, even $d=10,000$ was prohibitive, since $d^2 = 100$ million.
A natural approach to improve computation to solve for $\mathbf{w}$ is to use stochastic methods,
such as TD($\lambda$) \perp\!\!\!\perptep{sutton1988learning}. This algorithm incrementally updates with $\mathbf{w}_{t+1} = \mathbf{w}_t + \alpha \delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t$ for stepsize $\alpha > 0$ and TD-error $\delta_t = R_{t+1} + (\gamma_{t+1} \mathbf{x}_{t+1} - \mathbf{x}_t)^\top \mathbf{w}_t$. The expectation of this update is $\mathbb{E}_\pi[\delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t] = \mathbf{b} - \mathbf{A} \mathbf{w}_t$; the fixed-point solutions are the same for both LSTD and TD, but LSTD corresponds to a batch solution whereas TD corresponds to a stochastic update.
Though more expensive than TD---which is only $O(d)$---LSTD does have several advantages.
Because LSTD is a batch method, it summarizes all samples (within $\mathbf{A}$), and so can be more
sample efficient. Additionally, LSTD has no step-size parameter, using a closed-form solution for $\mathbf{w}$.
Recently, there has been some progress in better balancing between TD and LSTD.
\perp\!\!\!\perptet{pan2017accelerated} derived a quasi-Newton algorithm, called accelerated gradient TD (ATD),
giving an unbiased algorithm that has some of
the benefits of LSTD, but with significantly reduced computation because they only maintain a low-rank approximation to $\mathbf{A}$.
The key idea is that $\mathbf{A}$ provides curvature information, and so can significantly improve step-size selection for TD and
so improve the convergence rate. The approximate $\mathbf{A}$ can still provide useful curvature information, but can be significantly cheaper to compute.
We use the ATD update to similarly obtain an unbiased algorithm, but use sketching approximations instead of low-rank approximations.
First, however, we investigate some of the properties of sketching.
\section{ISSUES WITH SKETCHING THE FEATURES}\label{sec_issues}
One approach to make LSTD more feasible is to project---sketch---the features. Sketching involves
sampling a random matrix $\mathbf{S}: \mathbb{R}^{k \times d}$ from a family of matrices $\mathcal{S}$, to project a given $d$-dimensional vector $\mathbf{x}$ to a (much smaller) $k$-dimensional vector $\mathbf{S} \mathbf{x}$. The goal in defining this class of sketching matrices
is to maintain certain properties of the original vector. The following is a standard definition for such a family.
\begin{definition}[Sketching]\label{defSketch} Let $d$ and $k$ be positive integers, $\delta\in (0,1)$, and ${\mathop{\mathrm{te}}xtrm{e-}}psilon \in \mathbb{R}^+$. Then, $\mathcal{S}\subset \mathbb{R}^{k\times d}$ is called a family of sketching matrices with parameters $({\mathop{\mathrm{te}}xtrm{e-}}psilon,\delta)$, if for a random matrix, $\mathbf{S}$, chosen uniformly at random from this family, we have that $\forall \mathbf{x}\in \mathbb{R}^d$
\begin{equation*}
\mathbb{P}\Big[ (1-{\mathop{\mathrm{te}}xtrm{e-}}psilon) \| \mathbf{x} \|_2^2 \leq \| \mathbf{S} \mathbf{x} \|_2^2 \leq (1+{\mathop{\mathrm{te}}xtrm{e-}}psilon) \| \mathbf{x} \|_2^2\Big] \geq 1-\delta.
{\mathop{\mathrm{te}}xtrm{e-}}nd{equation*}
where the probability is w.r.t. the distribution over $\mathbf{S}$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{definition}
We will explore the utility of sketching the features with several
common sketches.
These sketches all require $k = \Omega({\mathop{\mathrm{te}}xtrm{e-}}psilon^{-2} \ln (1/\delta) \ln d)$.
\begin{figure*}[]
\subfigure[Mountain Car, tile coding]{
\includegraphics[width=0.33\mathop{\mathrm{te}}xtwidth]{figures/mcartile_50r_compskts.pdf}\label{fig:mcar_compskts_tile50r}}
\subfigure[Mountain Car, RBF]{
\includegraphics[width=0.33\mathop{\mathrm{te}}xtwidth]{figures/mcarrbf_50r_compskts.pdf}\label{fig:mcar_compskts_rbf50r}} \\
\subfigure[Puddle World, tile coding]{
\includegraphics[width=0.33\mathop{\mathrm{te}}xtwidth]{figures/pdtile_50r_compskts.pdf}\label{fig:pd_compskts_tile50r}}
\subfigure[Puddle World, RBF]{
\includegraphics[width=0.33\mathop{\mathrm{te}}xtwidth]{figures/pdrbf_50r_compskts.pdf}\label{fig:pd_compskts_rbf50r}}
\begin{minipage}{0.1cm}
{\mathop{\mathrm{te}}xtrm{e-}}nd{minipage}
\begin{minipage}{0.32\mathop{\mathrm{te}}xtwidth}
\caption{
Efficacy of different sketches for sketching the features for LSTD, with $k = 50$. The RMSE is w.r.t. the optimal value function, computed using rollouts.
LSTD($\lambda$) is included as the baseline, with $\mathbf{w} = \mathbf{A}^\inv \mathbf{b}$, with the other curves corresponding to different sketches of the features, to give $\mathbf{w} = (\mathbf{S} \mathbf{A} \mathbf{S}^\top)^\inv \mathbf{S} \mathbf{b}$ as used for the random projections LSTD algorithm. The RBF width in Mountain Car is $\sigma = 0.12$ times the range of the state space and in Puddle World is $\sigma = \sqrt{0.0072}$. The $1024$ centers for RBFs are chosen to uniformly cover the 2-d space in a grid. For tile coding, we discretize each dimension by $10$, giving $10\times10$ grids, use $10$ tilings, and set the memory size as $1024$. The bias is high for tile coding features, and much better for RBF features, though still quite large. The different sketches perform similarly.
}\label{fig:compare_skts50r}
{\mathop{\mathrm{te}}xtrm{e-}}nd{minipage}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\mathop{\mathrm{te}}xtbf{Gaussian random projections}, also known as the JL-Transform \perp\!\!\!\perptep{johnson1984extensions}, has each entry in $\mathbf{S}$ i.i.d. sampled from a Gaussian, $\mathcal{N}(0, \frac{1}{k})$.
\mathop{\mathrm{te}}xtbf{Count sketch} selects exactly one uniformly picked non-zero entry in each column, and sets that entry to either $1$ or $-1$ with equal probability \perp\!\!\!\perptep{charikar2002finding,gilbert2010sparse}.
The Tug-of-War sketch \perp\!\!\!\perptep{alon1996space} performs very similarly to Count sketch in our experiments, and so we omit it.
\mathop{\mathrm{te}}xtbf{Combined sketch} is the product of a count sketch matrix and a Gaussian projection matrix \perp\!\!\!\perptep{wang2015apractical,chris2015pca}.
\mathop{\mathrm{te}}xtbf{Hadamard sketch}---the Subsampled Randomized Hadamard Transform---is computed as $\mathbf{S} = \frac{1}{\sqrt{k d}}\mathbf{D} \mathbf{H}_d \mathbf{P}$, where $\mathbf{D} \in \mathbb{R}^{d \times d}$ is a diagonal matrix with each diagonal element uniformly sampled from $\{1, -1\}$, $\mathbf{H}_d \in \mathbb{R}^{d \times d}$ is a Hadamard matrix and $\mathbf{P} \in \mathbb{R}^{d \times k}$ is a column sampling matrix \perp\!\!\!\perptep{nir2006fastjlt}.
Sketching provides a low-error between the recovery $\mathbf{S}^\top \mathbf{S} \mathbf{x}$ and the original $\mathbf{x}$, with high probability.
For the above families, the entries in $\mathbf{S}$ are zero-mean i.i.d. with variance 1,
giving $\mathbb{E}[\mathbf{S}^\top \mathbf{S}] = \mathbf{I}$ over all possible $\mathbf{S}$. Consequently,
in expectation, the recovery $\mathbf{S}^\top \mathbf{S} \mathbf{x}$ is equal to $\mathbf{x}$. For a stronger result,
a Chernoff bound can be used to bound the deviation of $\mathbf{S}^\top \mathbf{S}$ from this expected value: for the parameters $({\mathop{\mathrm{te}}xtrm{e-}}psilon,\delta)$ of the matrix family, we get that $\mathbb{P}\Big[ (1-{\mathop{\mathrm{te}}xtrm{e-}}psilon)I\prec \mathbf{S}^\top \mathbf{S}\prec (1+{\mathop{\mathrm{te}}xtrm{e-}}psilon)I\Big] \geq 1-\delta$.
These properties suggest that using sketching for the feature vectors should provide effective approximations. \perp\!\!\!\perpte{bellemare2012sketch} showed that they could use these projections for tile coding, rather than the biased hashing function that is typically used, to improve learning performance for the control setting. The efficacy, however, of sketching given features, versus using the unsketched features, is less well-understood.
We investigate the properties of sketching the features, shown in Figure~\ref{fig:compare_skts50r} with a variety of sketches in two benchmark domains for RBF and tile-coding representations (see \perp\!\!\!\perpte[Chapter 8]{sutton1998reinforcement} for an overview of these representations). For both domains, the observations space is 2-dimensional, with expansion to $d = 1024$ and $k = 50$. The results are averaged over 50 runs, with $\xi,\lambda$ swept over 13 values, with ranges listed in Appendix \ref{app_experiments}. We see that sketching the features can incur significant bias, particularly for tile coding, even with a reasonably large $k = 50$ to give $O(d k)$ runtimes. This bias reduces with $k$, but remains quite high and so is likely too unreliable for practical use.
\section{SKETCHING THE LINEAR SYSTEM}
All of the work on sketching within reinforcement learning has investigated
sketching the features; however, we can instead consider sketching the linear system,
$\mathbf{A} \mathbf{w} = \mathbf{b}$. For such a setting, we can sketch the left and right subspaces
of $\mathbf{A}$ with different sketching matrices, $\mathbf{S}_L \in \mathbb{R}^{k_L \times d}$ and
$\mathbf{S}_R \in \mathbb{R}^{k_R \times d}$. Depending on the choices of $k_L$ and $k_R$,
we can then solve the smaller system $\mathbf{S}_L \mathbf{A} \mathbf{S}_R^\top \mathbf{S}_R \mathbf{w} = \mathbf{S}_L \mathbf{b}$ efficiently.
The goal is to better take advantage of the properties for the different sides of an asymmetric matrix $\mathbf{A}$.
One such natural improvement should be in one-sided sketching.
By only sketching from the left, for example, and setting $\mathbf{S}_R = \mathbf{I}$,
we do not project $\mathbf{w}$. Rather, we only project the constraints to the linear
system $\mathbf{A} \mathbf{w} = \mathbf{b}$. Importantly, this does not introduce bias: the original solution
$\mathbf{w}$ to $\mathbf{A} \mathbf{w} = \mathbf{b}$ is also a solution to $\mathbf{S}\mathbf{A} \mathbf{w} = \mathbf{S}\mathbf{b}$ for any sketching
matrix $\mathbf{S}$. The projection, however, removes uniqueness in terms of the solutions $\mathbf{w}$,
since the system is under-constrained.
Conversely, by only sketching from the right, and setting $\mathbf{S}_L = \mathbf{I}$,
we constrain the space of solutions to a unique set, and do not remove any constraints.
For this setting, however, it is unlikely that $\mathbf{w}$ with $\mathbf{A} \mathbf{w} = \mathbf{b}$ satisfies $\mathbf{A} \mathbf{S}^\top \mathbf{w} = \mathbf{b}$.
The conclusion from many initial experiments is that the key benefit from asymmetric sketching is
when only sketching from the left.
We experimented with all pairwise combinations of Gaussian random projections, Count sketch, Tug-of-War sketch and Hadamard sketch for $\mathbf{S}_L$ and $\mathbf{S}_R$. We additionally experimented with only sketching from the right, setting $\mathbf{S}_L = \mathbf{I}$.
In all of these experiments, we found asymmetric
sketching provided little to no benefit over using $\mathbf{S}_L = \mathbf{S}_R$ and that sketching only
from the right also performed similarly to using $\mathbf{S}_L = \mathbf{S}_R$. We further investigated
column and row selection sketches (see \perp\!\!\!\perpte{wang2015apractical} for a thorough overview), but also found
these to be ineffective. We therefore proceed with an investigation into effectively using left-side sketching.
In the next section, we provide an efficient $\mathcal{O}(d k)$ algorithm to compute $(\mathbf{S}_L \mathbf{A})^\pinv$,
to enable computation of $\mathbf{w} = (\mathbf{S}_L \mathbf{A})^\pinv \mathbf{S}_L \mathbf{b}$ and for use
within an unbiased quasi-Newton algorithm.
We conclude this section with an interesting connection to a data-dependent projection
method that has been used for policy evaluation, that further motivates the utility of sketching
only from the left.
This algorithm, called truncated LSTD (tLSTD) \perp\!\!\!\perptep{gehring2016incremental},
incrementally maintains a rank $k$ approximation of $\mathbf{A}$ matrix, using
an incremental singular value decomposition.
We show below that this approach corresponds to projecting $\mathbf{A}$ from the left with the top $k$ left singular vectors.
This is called a data-dependent projection, because the projection depends on the observed data,
as opposed to the data-independent projection---the sketching matrices---which is randomly sampled independently of the data.
\begin{proposition}
Let $\mathbf{A}=\mathbf{U}\boldsymbol{\Sigma} \mathbf{V}^\top$ be singular value decomposition of the true $\mathbf{A}$.
Assume the singular values are in decreasing order and let $\boldsymbol{\Sigma}_k$
be the top $k$ singular values, with corresponding $k$ left singular vectors $\mathbf{U}_k$ and $k$ right singular vectors $\mathbf{V}_k$.
Then the solution $\mathbf{w} = \mathbf{V}_k \boldsymbol{\Sigma}_k^\pinv \mathbf{U}_k^\top \mathbf{b}$ (used for tLSTD) corresponds
to LSTD using asymmetric sketching with $\mathbf{S}_L=\mathbf{U}_k^\top$ and $\mathbf{S}_R=\mathbf{I}$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{proposition}
\begin{proof}
We know $\mathbf{U} = [\mathbf{u}_1, \ldots, \mathbf{u}_d]$ for singular vectors $\mathbf{u}_i \in \mathbb{R}^d$ with $\mathbf{u}_i^\top \mathbf{u}_i = 1$ and $\mathbf{u}_i^\top \mathbf{u}_j = 0$
for $i\neq j$. Since $\mathbf{U}_k = [\mathbf{u}_1, \ldots, \mathbf{u}_k]$, we get that
$\mathbf{U}_k^\top \mathbf{U} = [\mathbf{I}_{k} \ \ \vec{0}_{d-k}] \in \mathbb{R}^{k \times d}$ for $k$-dimensional identity matrix $\mathbf{I}_k$ and zero matrix $\vec{0}_{d-k} \in \mathbb{R}^{k \times (d-k)}$. Then we get that
$\mathbf{S}_L \mathbf{b} = \mathbf{U}_k^\top \mathbf{b}$ and
$\mathbf{S}_L \mathbf{A}
= [\mathbf{I}_{k} \ \ \vec{0}_{d-k}] \boldsymbol{\Sigma} \mathbf{V}^\top
= \boldsymbol{\Sigma}_k \mathbf{V}^\top
= \boldsymbol{\Sigma}_k \mathbf{V}_k^\top
.$
Therefore, $\mathbf{w} = (\mathbf{S}_L \mathbf{A})^\pinv \mathbf{S}_L \mathbf{b} = \mathbf{V}_k \boldsymbol{\Sigma}_k^\pinv \mathbf{U}_k^\top \mathbf{b}$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{proof}
\section{LEFT-SIDED SKETCHING ALGORITHM}\label{sec_left}
In this section, we develop an efficient approach to use the smaller, sketched matrix $\mathbf{S} \mathbf{A}$
for incremental policy evaluation. The most straightforward way to use $\mathbf{S}\mathbf{A}$ is to
incrementally compute $\mathbf{S}\mathbf{A}$, and periodically solve $\mathbf{w} = (\mathbf{S}\mathbf{A})^\pinv \mathbf{S}\mathbf{b}$. This
costs $O(d k)$ per step, and $O(d^2 k)$ every time the solution is recomputed. To maintain
$O(d k)$ computation per-step, this full solution could only be computed every $d$ steps,
which is too infrequent to provide a practical incremental policy evaluation approach. Further, because it is an underconstrained
system, there are likely to be infinitely many solutions to
$\mathbf{S}\mathbf{A} \mathbf{w} = \mathbf{S}\mathbf{b}$; amongst those solutions, we would like to sub-select amongst the unbiased solutions
to $\mathbf{A} \mathbf{w} = \mathbf{b}$.
We first discuss how to efficiently maintain $(\mathbf{S}\mathbf{A})^\pinv$, and then describe
how to use that matrix to obtain an unbiased algorithm. Let $\tilde{\mathbf{A}} \defeq \mathbf{S}\mathbf{A} \in \mathbb{R}^{k \times d}$.
For this underconstrained system with $\tilde{\mathbf{b}} \defeq \mathbf{S} \mathbf{b}$, the minimum norm solution to $\tilde{\mathbf{A}} \mathbf{w} = \tilde{\mathbf{b}}$ is\footnote{We show in Proposition \ref{lem_rowrank}, Appendix \ref{app_theory}, that $\tilde{\mathbf{A}}$ is full row rank with high probability. This property is required to ensure that the inverse of $\tilde{\mathbf{A}} \tilde{\mathbf{A}}^\top$ exists. In practice, this is less of a concern, because we initialize
the matrix $\tilde{\mathbf{A}}_0 \tilde{\mathbf{A}}_0^\top$ with a small positive value, ensuring invertibility for $\tilde{\mathbf{A}}_t \tilde{\mathbf{A}}_t^\top$ for finite $t$.}
$\mathbf{w} = \tilde{\mathbf{A}}^\top (\tilde{\mathbf{A}} \tilde{\mathbf{A}}^\top)^\inv \tilde{\mathbf{b}}$ and $\tilde{\mathbf{A}}^\pinv = \tilde{\mathbf{A}}^\top (\tilde{\mathbf{A}} \tilde{\mathbf{A}}^\top)^\inv \in \mathbb{R}^{d \times k}$.
To maintain $\tilde{\mathbf{A}}_t^\pinv$ incrementally, therefore, we simply need to maintain $\tilde{\mathbf{A}}_t$ incrementally
and the $k\timesk$-matrix $(\tilde{\mathbf{A}}_t \tilde{\mathbf{A}}_t^\top)^\inv$ incrementally.
Let $\tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t \defeq \mathbf{S}{\mathop{\mathrm{te}}xtrm{e-}}vec_t$, $\mathbf{d}_t \defeq \mathbf{d}_t$
and $\mathbf{h}_t \defeq \tilde{\mathbf{A}}_{t}\mathbf{d}_t$. We can update the sketched system in $O(d k)$ time and space
\begin{align*}
\tilde{\mathbf{A}}_{t+1} &= \tilde{\mathbf{A}}_{t} + \tfrac{1}{t+1}\left(\tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t} \mathbf{d}_{t}^\top -\tilde{\mathbf{A}}_{t}\mathop{\mathrm{ri}}ght)\\
\tilde{\mathbf{b}}_{t+1} &= \tilde{\mathbf{b}}_{t} + \tfrac{1}{t+1}\left(\tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t} R_{t+1} -\tilde{\mathbf{b}}_{t}\mathop{\mathrm{ri}}ght)
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
To maintain $(\tilde{\mathbf{A}}_t \tilde{\mathbf{A}}_t^\top)^\inv$ incrementally, notice that the unnormalized update is
\begin{align*}
\tilde{\mathbf{A}}_{t+1} \tilde{\mathbf{A}}_{t+1}^\top
&= (\tilde{\mathbf{A}}_{t} + \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t \mathbf{d}_t^\top)(\tilde{\mathbf{A}}_{t} +\tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t \mathbf{d}_t^\top) \\
&= \tilde{\mathbf{A}}_{t} \tilde{\mathbf{A}}_{t}^\top + \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t \mathbf{h}_t^\top + \mathbf{h}_t \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t ^\top + ||\mathbf{d}_t||_2^2 \| \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_t ^\top
.
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
Hence, $(\tilde{\mathbf{A}}_{t+1} \tilde{\mathbf{A}}_{t+1}^\top)^\inv$ can be updated from $(\tilde{\mathbf{A}}_{t} \tilde{\mathbf{A}}_{t}^\top)^\inv$ by applying
the Sherman-Morrison update three times.
For a normalized update, based on samples, the update is
\begin{align*}
\tilde{\mathbf{A}}_{t+1} \tilde{\mathbf{A}}_{t+1}^\top
&= \left(\tfrac{t}{t+1}\mathop{\mathrm{ri}}ght)^2\tilde{\mathbf{A}}_{t} \tilde{\mathbf{A}}_{t}^\top
+ \tfrac{t}{(t+1)^2}\left(\tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t} \mathbf{h}_{t}^\top + \mathbf{h}_{t} \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t} ^\top\mathop{\mathrm{ri}}ght) \\
&+ \tfrac{1}{(t+1)^2}||\mathbf{d}_{t}||_2^2 \| \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t} \tilde{{\mathop{\mathrm{te}}xtrm{e-}}vec}_{t}^\top
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
We can then compute $\mathbf{w}_t = \tilde{\mathbf{A}}_t (\tilde{\mathbf{A}}_t \tilde{\mathbf{A}}_t^\top)^\pinv \tilde{\mathbf{b}}_t$ on each step.
This solution, however, will provide the minimum norm solution, rather than the unbiased solution,
even though the unbiased solution is feasible for the underconstrained system. To instead push the preference
towards this unbiased solution, we use the stochastic approximation algorithm, called ATD \perp\!\!\!\perptep{pan2017accelerated}.
This method is a quasi-second order method, that relies on a low-rank approximation $\hat{\mathbf{A}}_t$ to $\mathbf{A}_t$;
using this approximation, the update is
$\mathbf{w}_{t+1} = \mathbf{w}_t + (\alpha_t\hat{\mathbf{A}}^\pinv_t + {\mathop{\mathrm{te}}xtrm{e-}}ta \mathbf{I}) \delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t$.
Instead of being used to explicitly solve for $\mathbf{w}$, the approximation matrix is used to provide
curvature information. The inclusion of ${\mathop{\mathrm{te}}xtrm{e-}}ta$ constitutes a small regularization component, that pushes
the solution towards the unbiased solution.
We show in the next proposition that for our alternative approximation, we still obtain unbiased solutions.
We use results for iterative methods for singular linear systems \perp\!\!\!\perptep{shi2011convergence,wang2013ontheconvergence}, since
$\mathbf{A}$ may be singular. $\mathbf{A}$ has been shown to be positive semi-definite under standard assumptions on the MDP \perp\!\!\!\perptep{yu2015onconvergence}; for simplicity, we assume $\mathbf{A}$ is positive semi-definite, instead of providing these MDP assumptions.
\begin{assumption}
For $\mathbf{S} \in \mathbb{R}^{k \times d}$ and
$\mathbf{B} = \alpha (\mathbf{S} \mathbf{A})^\pinv \mathbf{S} + {\mathop{\mathrm{te}}xtrm{e-}}ta \mathbf{I}$ with $\mathbf{B} \in \mathbb{R}^{d \times d}$, the matrix $\mathbf{B} \mathbf{A}$
is diagonalizable.
{\mathop{\mathrm{te}}xtrm{e-}}nd{assumption}
\begin{assumption}
$\mathbf{A}$ is positive semi-definite.
{\mathop{\mathrm{te}}xtrm{e-}}nd{assumption}
\begin{assumption}
$\alpha \in (0,\tfrac{1}{2})$ and $0 < \eta \le \tfrac{1}{2{\mathop{\mathrm{te}}xtrm{e-}}igmax(\mathbf{A})}$ where ${\mathop{\mathrm{te}}xtrm{e-}}igmax(\mathbf{A})$ is the maximum eigenvalue of $\mathbf{A}$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{assumption}
\begin{theorem} \label{thm_main}
Under Assumptions 1-3,
the expected updating rule
$\mathbf{w}_{t+1} = \mathbf{w}_t + \mathbb{E}_\pi[\mathbf{B} \delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t]$
converges to a fixed-point $\mathbf{w}^\star = \mathbf{A}^\pinv \mathbf{b}$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{theorem}
\begin{proof}
The expected updating rule is $\mathbb{E}_\pi[\mathbf{B} \delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t] = \mathbf{B}(\mathbf{b} - \mathbf{A} \mathbf{w}_t)$.
As in the proof of convergence for ATD \perp\!\!\!\perpte[Theorem 1]{pan2017accelerated},
we similarly verify the conditions from \perp\!\!\!\perptep[Theorem 1.1]{shi2011convergence}.
Notice first that \ \ \ \
$\mathbf{B} \mathbf{A} = \alpha (\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A} + \eta \mathbf{A} $.
For singular value decomposition, $\mathbf{S} \mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^\top$, we have that
$(\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A} = \mathbf{V} \boldsymbol{\Sigma}^\pinv \mathbf{U}^\top \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^\top
= \mathbf{V} [\mathbf{I}_{\tilde{k}} \ \vec{0}_{d -k}] \mathbf{V}^\top$, where $\tilde{k} \le k$ is the rank of $\mathbf{S}\mathbf{A}$. The maximum eigenvalue of
$(\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A}$ is therefore $1$.
Because $(\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A} $ and
$\mathbf{A}$ are both positive semidefinite, $\mathbf{B} \mathbf{A}$ is positive semi-definite.
By Weyl's inequalities,
\begin{equation*}
{\mathop{\mathrm{te}}xtrm{e-}}igmax(\mathbf{B} \mathbf{A}) \le \alpha{\mathop{\mathrm{te}}xtrm{e-}}igmax( (\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A}) + \eta {\mathop{\mathrm{te}}xtrm{e-}}igmax(\mathbf{A})
.
{\mathop{\mathrm{te}}xtrm{e-}}nd{equation*}
Therefore, the eigenvalues of $\mathbf{I} - \mathbf{B} \mathbf{A}$ have absolute value strictly less than 1,
because $\eta \le (2{\mathop{\mathrm{te}}xtrm{e-}}igmax(\mathbf{A}))^\inv$ and $\alpha < 1/2 = (2{\mathop{\mathrm{te}}xtrm{e-}}igmax( (\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A}))^\inv $
by assumption.
For the second condition, since $\mathbf{B}\mathbf{A}$ is PSD and diagonalizable, we
can write $\mathbf{B}\mathbf{A} = \mathbf{Q} \boldsymbol{\Lambda} \mathbf{Q}^\inv$ for some matrices $\mathbf{Q}$ and diagonal matrix $\boldsymbol{\Lambda}$
with eigenvalues greater than or equal to zero. Then $(\mathbf{B}\mathbf{A})^2 = \mathbf{Q} \boldsymbol{\Lambda} \mathbf{Q}^\inv\mathbf{Q} \boldsymbol{\Lambda} \mathbf{Q}^\inv
= \mathbf{Q} \boldsymbol{\Lambda}^2 \mathbf{Q}^\inv$ has the same rank.
\newcommand{\mathop{\mathrm{null}}pace}{\mathop{\mathrm{te}}xt{nullspace}}
For the third condition, because $\mathbf{B} \mathbf{A}$ is the sum of two positive semi-definite matrices,
the nullspace of $\mathbf{B}\mathbf{A}$ is a subset of the nullspace of each of those matrices individually:
$\mathop{\mathrm{null}}pace(\mathbf{B}\mathbf{A}) = \mathop{\mathrm{null}}pace(\alpha (\mathbf{S} \mathbf{A})^\pinv \mathbf{S} \mathbf{A} + \eta \mathbf{A} ) \subseteq \eta \mathop{\mathrm{null}}pace( \eta \mathbf{A} ) = \mathop{\mathrm{null}}pace(\mathbf{A})$. In the other direction, for all $\mathbf{w}$ such that $\mathbf{A}\mathbf{w} = \vec{0}$, its clear
that $\mathbf{B} \mathbf{A} \mathbf{w} = \vec{0}$, and so $\mathop{\mathrm{null}}pace(\mathbf{A}) \subseteq \mathop{\mathrm{null}}pace(\mathbf{B}\mathbf{A})$.
Therefore, $\mathop{\mathrm{null}}pace(\mathbf{A}) = \mathop{\mathrm{null}}pace(\mathbf{B}\mathbf{A})$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{proof}
\begin{figure*}[t!]
\subfigure[Mountain Car, RBF]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_rbf50r.pdf} \label{fig:mcar_gau_rbf50r}}
\subfigure[Mountain Car, Tile coding]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_tile50r.pdf} \label{fig:mcar_gau_tile50r}}\\
\subfigure[Mountain Car, RBF, Sensitivity]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_rbf50r_sensi.pdf}\label{fig:mcar_gau_rbf50rsensi}}
\subfigure[Mountain Car, Tile coding, Sensitivity]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_tile50r_sensi.pdf}\label{fig:mcar_gau_tile50rsensi}}
\begin{minipage}{0.32\mathop{\mathrm{te}}xtwidth}
\caption{
\mathop{\mathrm{te}}xtbf{(a)} and \mathop{\mathrm{te}}xtbf{(b)} are learning curves on Mountain Car with $k = 50$, and \mathop{\mathrm{te}}xtbf{(c)} and \mathop{\mathrm{te}}xtbf{(d)} are their corresponding parameter-sensitivity plots. The sensitivity plots report average RMSE over the entire learning curve, for the best $\lambda$ for each parameter. The stepsize $\alpha$ is reported for TD, the initialization parameter $\xi$ for the LSTD methods
and the regularization parameter ${\mathop{\mathrm{te}}xtrm{e-}}ta$ for the ATD methods. The initialization for the matrices in the ATD methods is fixed to the identity. The range for the regularization term ${\mathop{\mathrm{te}}xtrm{e-}}ta$ is $0.1$ times the range for $\alpha$. As before, the sketching approaches with RBFs perform better than with tile coding. The sensitivity of the left-side projection methods is significantly lower than the TD methods. ATD-L also seems to be less sensitive than ATD-SVD, and incurs less bias than LSTD-L.
}\label{fig:mcar_lc}
{\mathop{\mathrm{te}}xtrm{e-}}nd{minipage}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\section{\!\!WHEN SHOULD SKETCHING HELP?}
To investigate the properties of these sketching approaches, we need to understand
when we expect sketching to have the most benefit.
Despite the wealth of literature on sketching and strong theoretical results, there seems to be fewer empirical investigations
into when sketching has most benefit. In this section, we elucidate some hypotheses
about when sketching should be most effective, which we then explore in our experiments.
In the experiments for sketching the features in Section \ref{sec_issues},
it was clear that sketching the RBF features was much more effective than sketching the tile coding features.
A natural investigation, therefore, is into the properties of representations that
are more amenable to sketching. The key differences between these two representations is in
terms of smoothness, density and overlap. The tile coding representation has non-smooth 0,1 features,
which do not overlap in each grid. Rather, the overlap for tile coding results from
overlapping tilings.
This differs from RBF overlap, where centers are arranged in a grid and only edges of the RBF features overlap. The density of RBF features is significantly higher,
since more RBFs are active for each input. Theoretical work in sketching for regression \perp\!\!\!\perptep{maillard2012linear}, however, does not require features to be smooth.
We empirically investigate these three properties---smoothness, density and overlap.
There are also some theoretical results that suggest sketching could be more amenable for more distinct features---less overlap or potentially less tilings.
\perp\!\!\!\perptet{balcan2006kernels} showed a worst-case setting where data-independent sketching results in poor performance.
They propose a two-stage projection, to maintain separability in classification.
The first stage uses a data-dependent projection, to ensure features are not highly correlated, and the second uses a data-independent projection (a sketch)
to further reduce the dimensionality after the orthogonal projection. The implied conclusion from this result
is that, if the features are not highly correlated, then the first step can be avoided and the data independent sketch
should similarly maintain classification accuracy.
This result suggests that sketching for feature expansions with less redundancy should
perform better.
We might also expect sketching to be more robust to the condition number of the matrix.
For sketching in regression, \perp\!\!\!\perpte{fard2012compressed} found a bias-variance trade-off when increasing
$k$, where for large $k$, estimation error from a larger number of parameters became
a factor. Similarly, in our experiments above, LSTD using an incremental Sherman-Morrison update
has periodic spikes in the learning curve, indicating some instability.
The smallest eigenvalue of the
sketched matrix should be larger than that of the original matrix;
this improvement in condition number compensates for the loss in information.
Similarly, we might expect that maintaining an incremental singular value decomposition,
for ATD, could be less robust than ATD with left-side sketching.
\begin{figure*}[htp!]
\centering
\subfigure[Puddle World, RBF, $k = 25$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_rbf25r.pdf} \label{fig:pd_gau_rbf25r}}
\subfigure[Puddle World, RBF, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_rbf50r.pdf} \label{fig:pd_gau_rbf50r}}
\subfigure[Puddle World, RBF, $k = 75$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_rbf75r.pdf}\label{fig:pd_gau_rbf75r}}
\caption{
Change in performance when increasing $k$, from $25$ to $75$. Two-sided projection (i.e., projecting the features) significantly improves with larger $k$, but is strictly dominated by left-side projection. At $k = 50$, the left-side projection methods are outperforming TD
and are less variant. ATD-SVD seems to gain less with increasing $k$, though in general we found ATD-SVD to perform more poorly than ATD-P particularly for RBF representations.
}\label{fig:pd_rank_lc}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\begin{figure*}[htp!]
\subfigure[RBF, $k = 50$]{
\includegraphics[width=0.235\mathop{\mathrm{te}}xtwidth]{figures/pd_overlap_rbf50r.pdf} \label{fig:pd_dense_tile50r}}
\subfigure[Spline, $k = 50$]{
\includegraphics[width=0.235\mathop{\mathrm{te}}xtwidth]{figures/pd_overlap_spline50r.pdf} \label{fig:pd_dense_rbf50r}}
\subfigure[Tile coding, $k = 50$]{
\includegraphics[width=0.235\mathop{\mathrm{te}}xtwidth]{figures/pd_redund_tile50r.pdf} \label{fig:pd_tile_redund}}
\subfigure[RBFs with tilings, $k = 50$]{
\includegraphics[width=0.235\mathop{\mathrm{te}}xtwidth]{figures/pd_redund_tilerbf50r.pdf} \label{fig:pd_tilerbf_redund}}
\caption{
The effect of varying the representation properties, in Puddle World with $d = 1024$. In \mathop{\mathrm{te}}xtbf{(a)} and \mathop{\mathrm{te}}xtbf{(b)}, we examine the impact of varying the overlap, for both smooth features (RBFs) and 0-1 features (Spline). For spline, the feature is 1 if $||\mathbf{x} - \mathbf{c}_i|| < \sigma$ and otherwise 0. The spline feature represents a bin, like for tile coding, but here we adjust the widths of the bins so that they can overlap and do not use tilings.
The x-axis has four width values, to give a corresponding feature vector norm of about $20, 40, 80, 120$.
In \mathop{\mathrm{te}}xtbf{(c)} and \mathop{\mathrm{te}}xtbf{(d)}, we vary the redundancy, where number of tilings is increased and the total number of features kept constant. We generate tilings for RBFs like for tile coding, but for each grid cell use an RBF similarity rather than a spline similarity. We used $4 \times 16 \times 16, 16 \times 8 \times 8$ and $64 \times 4 \times 4$.
}\label{fig:pd_overlap_redund}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\section{EXPERIMENTS}
In this section, we test the efficacy of sketching for LSTD and ATD in four domains: Mountain Car, Puddle World, Acrobot and Energy Allocation. We set $k = 50$, unless otherwise specified, average all results over 50 runs and sweep parameters for each algorithm. Detailed
experimental settings, such as parameter ranges, are in Appendix \ref{app_experiments}. To distinguish projections, we add -P for two-sided and -L for left-sided to the algorithm name.
We conclude that 1) two-sided projection---projecting the features---generally does much worse than only projecting the left-side of $\mathbf{A}$, 2) higher feature density is more amenable to sketching, particularly for two-sided sketching, 3) smoothness of features only seems to impact two-sided sketching, 4) ATD with sketching decreases bias relative to its LSTD variant and 5) ATD with left-sided sketching typically performs as well as ATD-SVD, but is significantly faster.
\mathop{\mathrm{te}}xtbf{Performance and parameter sensitivity for RBFs and Tile coding.} We first more exhaustively compare the algorithms in Mountain Car and Puddle World, in Figures \ref{fig:mcar_lc} and \ref{fig:pd_rank_lc} with additional such results in the appendix.
As has been previously observed, TD with a well-chosen stepsize can perform almost as well as LSTD in terms of sample efficiency,
but is quite sensitive to the stepsize. Here, therefore, we explore if our matrix-based learning algorithms can reduce this parameter sensitivity. In Figure \ref{fig:mcar_lc}, we can indeed see that this is the case. The LSTD algorithms look a bit more sensitive, because we sweep over small initialization values for completeness. For tile coding, the range is a bit more narrow, but for RBFs, in the slightly larger range, the LSTD algorithms are quite insensitive for RBFs. Interestingly, LSTD-L seems to be more robust.
We hypothesize that the reason for this is that LSTD-L only has to initialize a smaller $k\timesk$ symmetric matrix, $(\mathbf{S} \mathbf{A} (\mathbf{S}\mathbf{A})^\top)^\inv = {\mathop{\mathrm{te}}xtrm{e-}}ta \mathbf{I}$, and so is much more robust to this initialization. In fact, across settings, we found initializing to $\mathbf{I}$ was effective. Similarly, ATD-L benefits from this robustness, since it needs to initialize the same matrix, and then further overcomes bias using the approximation to $\mathbf{A}$ only for curvature information.
\mathop{\mathrm{te}}xtbf{Impact of the feature properties.}
We explored the feature properties---smoothness, density, overlap and redundancy---where we hypothesized sketching should help, shown in Figure~\ref{fig:pd_overlap_redund}.
The general conclusions are 1) the two-side sketching methods improve---relative to LSTD---with increasing density (i.e., increasing overlap and increasing redundancy), 2) the smoothness of the features (RBF versus spline) seems to affect the two-side projection methods much more, 3) the shape of the left-side projection methods follows that of LSTD and 4) ATD-SVD appears to follow the shape of TD more. Increased density generally seemed to degrade TD, and so ATD-SVD similarly suffered more in these settings. In general, the ATD methods had less gain over their corresponding LSTD variants, with increasing density.
\mathop{\mathrm{te}}xtbf{Experiments on high dimensional domains.} We finally apply our sketching techniques on two high dimensional domains to illustrate practical usability: Acrobot and Energy allocation. The Acrobot domain \perp\!\!\!\perptep{sutton1998reinforcement} is a four dimensional episodic task, where the goal is to raise an arm to a certain height.
The Energy allocation domain \perp\!\!\!\perpte{salas2013benchmarking} is a five-dimensional continuing task, where the goal is to store and allocate energy to maximize profit. For Acrobot, we used $14,400$ uniformly-spaced centers and for Energy allocation, we used the same tile coding of $8192$ features as \perp\!\!\!\perptet{pan2017accelerated}. We summarize the results in the caption of Figure \ref{fig:high_domain}, with the overall conclusion that ATD-L provides an attractive way to reduce parameter sensitivity of TD, and benefit from sketching to reduce computation.
\begin{figure*}[htp!]
\centering
\subfigure[Acrobot, RBF]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/acro_gau_rbf50r.pdf}\label{fig:acro_rbf50r}}
\subfigure[Acrobot, RMSE vs Time]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/acro_time_50r.pdf}}
\subfigure[Energy allocation, Tile coding]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/energy_gau_tile50r.pdf}\label{fig:energy_tile50r}}
\caption{
Results in domains with high-dimensional features, using $k = 50$ and with results averaged over $30$ runs.
For Acrobot, the (left-side) sketching methods perform well and are much less sensitive to parameters than TD. For runtime, we show RMSE versus time allowing the algorithms to process up to $25$ samples per second, to simulate a real-time setting learning; slow algorithms cannot process all $25$ within a second. With computation taken into account, ATD-L has a bigger win over ATD-SVD, and does not lose relative to TD. Total runtime in seconds for one run for each algorithm is labeled in the plot. ATD-SVD is much slower, because of the incremental SVD.
For the Energy Allocation domain, the two-side projection methods (LSTD-P, ATD-P) are significantly worse than other algorithms. Interestingly, here ATD-SVD has a bigger advantage, likely because sketching the tile coding features is less effective.
}\label{fig:high_domain}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\section{CONCLUSION AND DISCUSSION}
In this work, we investigated how to benefit from sketching approaches
for incremental policy evaluation. We first showed that sketching features can have significant bias issues,
and proposed to instead sketch the linear system, enabling better control over how much information is lost.
We highlighted that sketching for radial basis features seems to be much more effective, than for tile coding,
and further that a variety of natural asymmetric sketching approaches for sketching the linear system are not effective.
We then showed that more carefully using sketching---particularly with left-side sketching within a quasi-Newton update---enables us to obtain an unbiased approach that can improve sample efficiency without incurring significant computation.
Our goal in this work was to provide practical methods that can benefit from sketching,
and start a focus on empirically investigating settings in which sketching is effective.
Sketching has been used for quasi-Newton updates in online learning; a natural
question is if those methods are applicable for policy evaluation.
\perp\!\!\!\perptet{luo2016efficient} consider sketching approaches for an online Newton-update, for general functions
rather than just the linear function approximation case we consider here.
They similarly have to consider updates amenable to incrementally approximating a matrix (a Hessian in their case).
In general, however, porting these quasi-Newton updates to policy evaluation for reinforcement learning is problematic
for two reasons. First, the objective function for temporal difference learning is the mean-squared projected Bellman error,
which is the product of three expectations. It is not straightforward to obtain an unbiased sample of this gradient, which is why \perp\!\!\!\perptet{pan2017accelerated} propose a slightly different quasi-Newton update that uses $\mathbf{A}$ as a preconditioner. Consequently, it is
not straightforward to apply quasi-Newton online algorithms that assume access to unbiased gradients.
Second, the Hessian can be nicely approximated in terms of gradients, and is symmetric; both are exploited when deriving the sketched online Newton-update \perp\!\!\!\perptep{luo2016efficient}. We, on the other hand, have an asymmetric matrix $\mathbf{A}$.
In the other direction, we could consider if our approach could be beneficial for the online regression setting.
For linear regression, with $\gamma =0$, the matrix $\mathbf{A}$ actually corresponds to the Hessian.
In contrast to previous approaches that sketched the features \perp\!\!\!\perptep{maillard2012linear,fard2012compressed,luo2016efficient},
therefore, one could instead sketch the system
and maintain $(\mathbf{S} \mathbf{A})^\pinv$. Since the second-order update is $\mathbf{A}^\inv \mathbf{g}_t$ for gradient $\mathbf{g}_t$
on iteration $t$, an approximate second-order update could be computed as $((\mathbf{S}\mathbf{A})^\pinv\mathbf{S} + {\mathop{\mathrm{te}}xtrm{e-}}ta \mathbf{I}) \mathbf{g}_t$.
In our experiments, we found sketching both sides of $\mathbf{A}$ to be less effective and found little benefit from modifying the chosen
sketch; however, these empirical conclusions warrant further investigation. With more understanding into
the properties of $\mathbf{A}$, it could be possible to benefit from this variety. For example,
sketching the left-side of $\mathbf{A}$ could be seen as sketching the eligibility trace, and the right-side as sketching
the difference between successive features. For some settings, there could be properties of either of these vectors
that are particularly suited to a certain sketch. As another example, the key benefit of many of the sketches over Gaussian random projections is in enabling the dimension $k$ to be larger, by using (sparse) sketching matrices where dot product are efficient. We could not easily benefit from these properties, because $\mathbf{S} \mathbf{A}$ could be dense and computing matrix-vector products and incremental inverses would be expensive for larger $k$. For sparse $\mathbf{A}$, or when $\mathbf{S}\mathbf{A}$ has specialized properties,
it could be more possible to benefit from different sketches.
Finally, the idea of sketching fits well into a larger theme of random representations
within reinforcement learning. A seminal paper on random representations \perp\!\!\!\perptep{sutton1993online} demonstrates
the utility of random threshold units, as opposed to more carefully learned units. Though end-to-end
training has become more popular in recent years, there is evidence that random representations
can be quite powerful \perp\!\!\!\perptep{aubry2002random,rahimi2007random,rahimi2008uniform,maillard2012linear}, or even combined with descent strategies \perp\!\!\!\perptep{mahmood2013representation}.
For reinforcement learning,
this learning paradigm is particularly suitable, because data cannot be observed upfront.
Data-independent representations, such as random representations and sketching approaches,
are therefore particularly appealing and warrant further investigation for the incremental
learning setting within reinforcement learning.
}
\appendix
\section{Row-rank properties of $\mathbf{S}\mathbf{A}$}\label{app_theory}
To ensure the right pseudo-inverse is well-defined in Section \ref{sec_left}, we show that the projected matrix $\mathbf{S}\mathbf{A}$ is full row-rank with high probability, if $\mathbf{A}$ has sufficiently high rank.
We know that the probability measure of row-rank deficient matrices for $\mathbf{S}$ has zero mass.
However in the following, we prove a stronger and practically more useful claim that $\mathbf{S}\mathbf{A}$ is far from being row-rank deficient. Formally, we define a matrix to be {\mathop{\mathrm{te}}xtrm{e-}}mph{$\delta$-full row-rank} if there is no row that can be replaced by another row with distance at most $\delta$ to make that matrix row-rank deficient.
\begin{proposition}\label{lem_rowrank}
Let $\mathbf{S} \in \mathbb{R}^{k \times d}$ be any Gaussian matrix with $0$ mean and unit variance. For $r_A=rank(\mathbf{A})$ and
for any $\delta > 0$, $\mathbf{S}\mathbf{A}$ is $\delta$-full row-rank with probability at least $1-{\mathop{\mathrm{te}}xtrm{e-}}xp(-2\frac{(r_A(1-0.8\delta)-k)^2}{r_A})$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{proposition}
\begin{proof}
Let $\mathbf{A}=\mathbf{U}\boldsymbol{\Sigma} \mathbf{V}^\top$ be the SVD for $\mathbf{A}$. Since $\mathbf{U}$ is an orthonormal matrix, $\mathbf{S}'=\mathbf{S}\mathbf{U}$ has the same distribution as $\mathbf{S}$ and the rank of $\mathbf{S} \mathbf{A}$ is the same as $\mathbf{S}' \boldsymbol{\Sigma}$. Moreover notice that the last $d-r_A$ columns of $\mathbf{S}'$ get multiplied by all-zero rows of $\boldsymbol{\Sigma}$.
Therefore, in what follows, we assume we draw a random matrix $\mathbf{S}' \in \mathbb{R}^{k \times r_A}$(similar to how $\mathbf{S}$ is drawn), and that $\boldsymbol{\Sigma} \in \mathbb{R}^{r_A \times r_A}$ is a full rank diagonal matrix. We study the rank of $\mathbf{S}' \boldsymbol{\Sigma}$.
Consider iterating over the rows of $\mathbf{S}'$, the probability that any new row is $\delta$-far from being a linear combination of the previous ones is at least $1-0.8\delta$. To see why,
assume that you currently have $i$ rows and sample another vector $\mathbf{v}$ with entries sampled i.i.d. from a standard Gaussian
as the candidate for the next row in $\mathbf{S}'$.
The length corresponding to the projection of any row $\mathbf{S}_{j:}'$ onto $\mathbf{v}$, i.e., $\mathbf{S}_{j:}' \mathbf{v} \in \mathbb{R}$, is a Gaussian random variable.
Thus, the probability of the $\mathbf{S}_{j:}' \mathbf{v}$ being within $\delta$
is at most $0.8\delta$.
This follows from the fact that the area under probability density function of a standard Gaussian random variable over $[0,x]$ is at most $0.4x$, for any $x>0$.
This stochastic process is a Bernoulli trial with success probability of at least $1-0.8\delta$. The trial stops when there are $k$ successes or when the number of iterations reaches $r_A$.
The Hoeffding inequality bounds the probability of failure by ${\mathop{\mathrm{te}}xtrm{e-}}xp(-2\frac{(r_A(1-0.8\delta)-k)^2}{r_A})$.
{\mathop{\mathrm{te}}xtrm{e-}}nd{proof}
\section{Alternative iterative updates}
In addition to the proposed iterative algorithm using a left-sided sketch of $\mathbf{A}$,
we experimented with a variety of alternative updates that proved ineffective.
We list them here for completeness.
We experimented with a variety of iterative updates. For a linear system, $\mathbf{A} \mathbf{w} = \mathbf{b}$,
one can iteratively update using $\mathbf{w}_{t+1} = \mathbf{w}_t + \alpha (\mathbf{b} - \mathbf{A} \mathbf{w}_t)$
and $\mathbf{w}_t$ will converge to a solution of the system (under some conditions). We tested the following ways to use sketched linear systems.
\mathop{\mathrm{te}}xtbf{First}, for the two-sided sketched $\mathbf{A}$, we want to solve for $\mathbf{S}_L \mathbf{A} \mathbf{S}_R^\top \mathbf{w} = \mathbf{S}_L \mathbf{b}$.
If $\tilde{\mathbf{A}}_t = \mathbf{S}_L \mathbf{A}_t \mathbf{S}_R^\top$ is square, we can use the iterative update
\begin{align*}
\tilde{\mathbf{A}}_{t+1} &= \tilde{\mathbf{A}}_t + \frac{1}{t+1}\left(\mathbf{S}_L{\mathop{\mathrm{te}}xtrm{e-}}vec_t(\mathbf{S}_R \mathbf{d}_t)^\top - \tilde{\mathbf{A}}_t \mathop{\mathrm{ri}}ght) \\
\tilde{\mathbf{b}}_{t+1} &= \tilde{\mathbf{b}}_t + \frac{1}{t+1}\left(r_{t+1} \mathbf{S}_L {\mathop{\mathrm{te}}xtrm{e-}}vec_t - \tilde{\mathbf{b}}_t \mathop{\mathrm{ri}}ght)\\
\mathbf{w}_{t+1} &= \mathbf{w}_t + \alpha_t (\tilde{\mathbf{b}}_{t+1} - \tilde{\mathbf{A}}_{t+1} \mathbf{w}_t)\\
&= \mathbf{w}_t + \alpha_t (\mathbf{S}_L\mathbf{b}_{t+1} - \mathbf{S}_L \mathbf{A}_{t+1} \mathbf{S}_R^\top \mathbf{w}_t)
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
and use $\mathbf{w}$ for prediction on the sketched features.
Another option is to maintain the inverse incrementally, using Sherman-Morrison
\begin{align*}
\mathbf{a}_d &= \mathbf{d}_t^\top \mathbf{S}_R ^\top \tilde{\mathbf{A}}_t^\inv \\
\mathbf{a}_u &= \tilde{\mathbf{A}}_t^\inv \mathbf{S}_L {\mathop{\mathrm{te}}xtrm{e-}}vec_t\\
\tilde{\mathbf{A}}_t^\inv &= \tilde{\mathbf{A}}_t^\inv - \frac{\mathbf{a}_u \mathbf{a}_d}{1+ \mathbf{d}_t^\top \mathbf{a}_u} \\
\tilde{\mathbf{b}}_t &= \tilde{\mathbf{b}}_t + \frac{r_{t+1} \mathbf{S}_L {\mathop{\mathrm{te}}xtrm{e-}}vec_t - \tilde{\mathbf{b}}_t }{t}\\
\mathbf{w} &= \tilde{\mathbf{A}}_t^\inv \tilde{\mathbf{b}}_t
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
If $\mathbf{S}_L \mathbf{A} \mathbf{S}_R^\top$ is not square (e.g., $\mathbf{S}_R = \mathbf{I}$), we instead
solve for $\mathbf{S}_L^\top\mathbf{S}_L \mathbf{A} \mathbf{S}_R^\top\mathbf{S}_R \mathbf{w} = \mathbf{S}_L^\top\mathbf{S}_L \mathbf{b}$, where applying $\mathbf{S}_L^\top$ provides the recovery from the left and $\mathbf{S}_R$ the recovery from the right.
\mathop{\mathrm{te}}xtbf{Second}, with the same sketching, we also experimented with $\mathbf{S}_L^\pinv$, instead of $\mathbf{S}_L^\top$ for the recovery, and similarly for $\mathbf{S}_R$, but this
provided no improvement.
For this square system, the iterative update is
\begin{align*}
\mathbf{w}_{t+1} &= \mathbf{w}_t + \alpha_t \mathbf{S}_L^\pinv (\tilde{\mathbf{b}}_{t+1} - \tilde{\mathbf{A}}_{t+1} \mathbf{S}_R \mathbf{w}_t)\\
&= \mathbf{w}_{t+1} + \alpha_t \mathbf{S}_L^\pinv (\mathbf{S}_L \mathbf{b}_{t+1} - \mathbf{S}_L \mathbf{A}_{t+1} \mathbf{S}_R^\top \mathbf{S}_R \mathbf{w}_t)
{\mathop{\mathrm{te}}xtrm{e-}}nd{align*}
for the same $\tilde{\mathbf{b}}_t$ and $\tilde{\mathbf{A}}_t$ which can be efficiently kept incrementally, while the pseudoinverse of $\mathbf{S}_L$ only needs to be computed once at the beginning.
\mathop{\mathrm{te}}xtbf{Third}, we tried to solve the system $\mathbf{S}_L^\top\mathbf{S}_L \mathbf{A} \mathbf{w} = \mathbf{b}$, using the updating rule $\mathbf{w}_{t+1} = \mathbf{w}_t + \alpha_t (\mathbf{b}_{t+1} - \mathbf{S}_L^\top \mathbf{S}_L \mathbf{A}_{t+1} \mathbf{w}_t)$, where the matrix $\mathbf{S}_L \mathbf{A}_{t+1}$ can be incrementally maintained at each step by using a simple rank-one update.
\mathop{\mathrm{te}}xtbf{Fourth}, we tried to explicitly regularize these iterative updates by adding a small step in the direction of $\delta_t {\mathop{\mathrm{te}}xtrm{e-}}vec_t$.
In general, none of these iterative methods performed well. We hypothesize this may be due to difficulties in choosing stepsize parameters. Ultimately, we found the sketched updated within ATD to be the most effective.
\section{Experimental details}\label{app_experiments}
Mountain Car is a classical episodic task with the goal of driving the car to the top of mountain. The state is 2-dimensional, consisting of the (position, velocity) of the car. We used the specification from \perp\!\!\!\perptep{sutton1998reinforcement}. We compute the true values of $2000$ states, where each testing state is sampled from a trajectory generated by the given policy. From each test state, we estimate the value---the expected return---by computing the average over 1000 returns, generated by rollouts. The policy for Mountain Car is the energy pumping policy with $20\%$ randomness starting from slightly random initial states. The discount rate is 1.0, and is 0 at the end of the episode, and the reward is always $-1$.
Puddle World \perp\!\!\!\perpte{boyan1995generalization} is an episodic task, where the goal is for a robot in a continuous gridworld to reach a goal state within as fewest steps as possible. The state is 2-dimensional, consisting of ($x,y$) positions. We use the same setting as described in \perp\!\!\!\perptep{sutton1998reinforcement}, with a discount of 1.0 and -1 per step, except when going through a puddle that gives higher magnitude negative reward. We compute the true values from $2000$ states in the same way as Mountain Car.
A simple heuristic policy choosing the action leading to shortest Euclidean distance with $10\%$ randomness is used.
Acrobot is a four-dimensional episodic task, where the goal is to raise an arm to certain level. The reward is $-1$ for non-terminal states and $0$ for goal state, again with discount set to 1.0. We use the same tile coding as described in \perp\!\!\!\perptep{sutton1998reinforcement}, except that we use memory size $2^{15} = 32,768$. To get a reasonable policy, we used true-online Sarsa($\lambda$) to go through $15000$ episodes with stepsize $\alpha = 0.1/48$ and bootstrap parameter $\lambda = 0.9$. Each episode starts with a slight randomness. The policy is ${\mathop{\mathrm{te}}xtrm{e-}}psilon-$greedy with respect to state value and ${\mathop{\mathrm{te}}xtrm{e-}}psilon = 0.05$. The way we compute true values and generate training trajectories are the same as we described for the above two domains.
Energy allocation \perp\!\!\!\perptep{salas2013benchmarking} is a continuing task with a five-dimensional state, where we use the same settings as in \perp\!\!\!\perpte{pan2017accelerated}.
The matrix $\mathbf{A}$ was shown to have a low-rank structure \perp\!\!\!\perptep{pan2017accelerated} and hence matrix approximation methods are expected to perform well.
For radial basis functions, we used format $k(\mathbf{x}, \mathbf{c}) = {\mathop{\mathrm{te}}xtrm{e-}}xp(-\frac{||\mathbf{x} - \mathbf{c}||_2^2}{2\sigma^2})$ where $\sigma$ is called RBF width and $\mathbf{c}$ is a feature. On Mountain Car, because the position and velocity have different ranges, we set the bandwidth separately for each feature using $k(\mathbf{x}, \mathbf{c}) = {\mathop{\mathrm{te}}xtrm{e-}}xp(-((\frac{\mathbf{x}_1 - \mathbf{c}_1}{0.12r_1})^2 + (\frac{\mathbf{x}_2 - \mathbf{c}_2}{0.12r_2})^2))$, where $r_1$ is the range of the first state variable and $r_2$ is the range of second state variable.
In Figure~\ref{fig:pd_overlap_redund}, we used a relatively rarely used representation which we call spline feature. For sample $\mathbf{x}$, the $i$th spline feature is set to 1 if $||\mathbf{x} - \mathbf{c}_i|| < \delta$ and otherwise set as 0.
The centers are selected in exactly the same way as for the RBFs.
\paragraph{Parameter optimization.} We swept the following ranges for stepsize ($\alpha$), bootstrap parameter ($\lambda$), regularization parameter (${\mathop{\mathrm{te}}xtrm{e-}}ta_t$), and initialization parameter $\xi$ for all domains:
\begin{enumerate}[nolistsep]
\item $\alpha \in \{0.1 \times 2.0^j | j = -7,-6, ...,4,5\}$ divided by $l_1$ norm of feature representation, 13 values in total.
\item $\lambda \in \{0.0, 0.1, ..., 0.9, 0.93,0.95,0.97,0.99, 1.0\}$, 15 values in total.
\item ${\mathop{\mathrm{te}}xtrm{e-}}ta \in \{0.01 \times 2.0^j | j = -7,-6, ...,4,5\}$ divided by $l_1$ norm of feature representation, 13 values in total.
\item $\xi \in \{10^j | j = -5, -4.25, -3.5, ..., 2.5, 3.25, 4.0\}$, 13 values in total.
{\mathop{\mathrm{te}}xtrm{e-}}nd{enumerate}
To choose the best parameter setting for each algorithm, we used the sum of RMSE across all steps for all the domains Energy allocation. For this domain, optimizing based on the whole range causes TD to pick an aggressive step-size to improve early learning at the expense of later learning. Therefore, for Energy allocation, we instead select the best parameters based on the sum of the RMSE for the second half of the steps.
For the ATD algorithms, as done in the original paper, we set $\alpha_t = \frac{1}{t}$ and only swept the regularization parameter ${\mathop{\mathrm{te}}xtrm{e-}}ta$,
which can also be thought of a (smaller) final step-size. For this reason, the range of ${\mathop{\mathrm{te}}xtrm{e-}}ta$ is set to $0.1$ times the range of $\alpha$,
to adjust this final stepsize range to an order of magnitude lower.
\paragraph{Additional experimental results.}
In the main paper, we demonstrated a subset of the results to highlight conclusions. For example, we showed the learning curves and parameter sensitivity in Mountain Car, for RBFs and tile coding. Due to space, we did not show the corresponding results for Puddle World in the main paper; we include these experiments here. Similarly, we only showed Acrobot with RBFs in the main text, and include results with tile coding here.
\begin{figure*}[htp!]
\subfigure[Puddle World, tile coding, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_tile50r.pdf} \label{fig:pd_gau_tile50r}}
\subfigure[Puddle World, RBF, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_rbf50r.pdf}} \\
\subfigure[Puddle World, tile coding, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_tile50r_sensi.pdf}\label{fig:pd_gau_tile50rsensi}}
\subfigure[Puddle World, RBF, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/pd_gau_rbf50r_sensi.pdf}\label{fig:pd_gau_rbf50rsensi}}
\begin{minipage}{0.32\mathop{\mathrm{te}}xtwidth}
\caption{
The two sensitivity figures are corresponding to the above two learning curves on Puddle World domain. Note that we sweep initialization for LSTD-P, but keep initialization parameter fixed across all other settings. The one-side projection is almost insensitive to initialization and the corresponding ATD version is insensitive to regularization. Though ATD-SVD also shows insensitivity, performance of ATD-SVD is much worse than sketching methods for the RBF representation. And, one should note that ATD-SVD is also much slower as shown in the below figures.
}\label{fig_pd_mcar_sensi}
{\mathop{\mathrm{te}}xtrm{e-}}nd{minipage}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\begin{figure*}[htp!]
\centering
\subfigure[Mountain Car, Tile coding, $k = 25$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_tile25r.pdf} \label{fig:mcar_gau_tile25r}}
\subfigure[Mountain Car, Tile coding, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_tile50r.pdf}
}
\subfigure[Mountain Car, Tile coding, $k = 75$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/mcar_gau_tile75r.pdf}\label{fig:mcar_gau_tile75r}}
\caption{
Change in performance when increasing $k$, from $25$ to $75$. We can draw similar conclusions to the same experiments in Puddle World in the main text. Here, the unbiased of ATD-L is even more evident; even with as low a dimension as 25, it performs similarly to LSTD.
}\label{fig:mcar_tile_rank_lc}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
\begin{figure*}[htp!]
\centering
\subfigure[Acrobot, tile coding, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/acro_gau_tile50r.pdf} \label{fig:acro_gau_tile50r}}
\subfigure[Acrobot, tile coding, $k = 50$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/acro_gau_tile50r_sensi.pdf}
}
\subfigure[Acrobot, RBF, $k = 75$]{
\includegraphics[width=0.32\mathop{\mathrm{te}}xtwidth]{figures/acro_gau_rbf50r_sensi.pdf}\label{fig:acro_gau_rbf75r}}
\caption{
Additional experiments in Acrobot, for tile coding with $k = 50$ and for RBFs with $k = 75$.
}\label{fig:mcar_rank_lc}
{\mathop{\mathrm{te}}xtrm{e-}}nd{figure*}
{\mathop{\mathrm{te}}xtrm{e-}}nd{document} |
\begin{document}
\begin{abstract}
We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$.
Combining this with the result of Stibitz and Zhuang \cite{SZ19} on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$.
\end{abstract}
\maketitle
\tableofcontents
\chapter{Introduction} \label{sec:intro}
Throughout the article, the ground field is assumed to be the complex number field $\mathbb{C}$.
\section{K-stability, birational superrigidity, and a conjecture}
\label{sec:KstBSRConj}
The notion of K-stability was introduced by Tian \cite{Tia97} as an attempt to characterize the existence of K\"ahler--Einstein metrics (KE metrics, for short) on Fano manifolds.
Later K-stability was extended and reformulated by Donaldson \cite{Don02} in algebraic terms.
The notion of K-stability emerged in the study of KE metrics (see \cite{Don02}, \cite{Tia97}), and it gives a characterization of the existence of a KE metric for smooth Fano manifolds (see \cite{CDS}, \cite{Tia15}).
Birational (super)rigidity means the uniqueness of a Mori fiber space in the birational equivalence class (see Definition \ref{def:BSR}), and this notion has its origin in the rationality problem of Fano varieties.
Specifically it grew out of the study of birational self-maps of smooth quartic 3-folds by Iskovskikh and Manin \cite{IM71} (see \cite{Puk13} and \cite{Che05} for surveys).
K-stability and birational superrigidity have completely different origins and we are unable to find a similarity in their definitions.
However, both of them are closely related to some mildness of singularities of pluri-anticanonical divisors (or linear systems).
For example, it is proved by Odaka and Sano \cite{OS12} (see also \cite{Tia87}) that a Fano variety $X$ of dimension $n$ is K-stable if $\alpha (X) > n/(n+1)$.
Here
\[
\alpha (X) = \sup \{\, c \in \mathbb{Q}_{> 0} \mid \text{$(X, c D)$ is log canonical for any $D \in \left| - K_X \right|_{\mathbb{Q}}$} \,\}
\] is called the {\it alpha invariant} of $X$ and it measures singularities of pluri-anticanonical divisors.
We refer readers to \cite{Fuj19b}, \cite{Li17}, \cite{FO18} and \cite{BJ20} for criteria for K-stability in terms of beta and delta invariants which are more or less related to singularities of pluri-anticanonical divisors.
On the other hand, it is known that a Fano variety of Picard number one is birationally superrigid if and only if the pair $(X, \lambda \mathcal{M})$ is canonical for any $\lambda \in \mathbb{Q}_{> 0}$ and any movable linear system $\mathcal{M}$ such that $\lambda \mathcal{M} \sim_{\mathbb{Q}} - K_X$ (see Theorem \ref{thm:charactbsr}).
With these relations in mind, one may expect a positive answer to the following.
\begin{Conj} \label{conj:BSRKst}
A birationally superrigid Fano variety is K-stable.
\end{Conj}
Actually we expect stronger conjectures to hold (see \S \ref{section:BRKst} for discussions).
The main aim of this article is to verify Conjecture \ref{conj:BSRKst} for quasi-smooth Fano 3-fold weighted hypersurfaces.
\section{Evidences for the conjecture}
\subsection{Smooth Fano manifolds}
\label{sec:introsmFano}
Smooth quartic $3$-folds and double covers of $\mathbb{P}^3$ branched along a smooth hypersurface of degree $6$ (or equivalently smooth weighted hypersurfaces of degree $6$ in $\mathbb{P} (1, 1, 1, 1, 3)$) are the only smooth Fano 3-fold that are birationally superrigid (see \cite{IM71}, \cite{Isk80}, \cite{Che05}).
K-stability (and hence the existence of a KE metric) is proved for smooth quartic 3-folds (\cite[Corollary 1.4]{Fuj19a}) and for smooth weighted hypersurfaces of degree $6$ in $\mathbb{P} (1, 1, 1, 1, 3)$ (\cite[Corollary 3.4]{CPW14}).
We have evidences in arbitrary dimension $n \ge 3$.
After the results established in low dimensional cases in \cite{IM71}, \cite{Puk87} and \cite{dFEM03}, it is finally proved by de Fernex \cite{dF13} that any smooth hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ is birationally superrigid for $n \ge 3$.
On the other hand, it is proved by Fujita \cite{Fuj19a} that any such hypersurface is K-stable (hence admits a KE metric).
It is also proved in \cite{Zhu20b} that a smooth Fano complete intersection $X \subset \mathbb{P}^{n+r}$ of Fano index $1$, codimension $r$ and dimension $n \ge 10 r$ is birationally superrigid and K-stable.
\subsection{Fano 3-fold weighted hypersurfaces}
\label{sec:introFanoWH}
By a {\it quasi-smooth Fano $3$-fold weighted hypersurface}, we mean a Fano 3-fold (with only terminal singularities) embedded as a quasi-smooth hypersurface in a well-formed weighted projective 4-space $\mathbb{P} (a_0, \dots, a_4)$ (see \S \ref{sec:wfqsm} for quasi-smoothness and well-formedness).
Let $X = X_d \subset \mathbb{P} (a_0, \dots, a_4)$ be a quasi-smooth Fano 3-fold weighted hypersurface of degree $d$.
Then, the class group $\operatorname{Cl} (X)$ is isomorphic to $\mathbb{Z}$ and is generated by $\mathcal{O}_X (1)$ (see for example \cite[Remark 4.2]{Oka19}).
By adjunction, we have $\mathcal{O}_X (-K_X) \cong \mathcal{O}_X (\iota_X)$, where
\[
\iota_X := \sum_{i=0}^4 a_i - d \in \mathbb{Z}_{> 0}.
\]
We call $\iota_X$ the {\it Fano index} (or simply {\it index}) of $X$.
By \cite{IF00} and \cite{CCC11}, quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$ are classified and they consist of 95 families.
Among them, quartic 3-folds and weighted hypersurfaces of degree $6$ in $\mathbb{P} (1, 1, 1, 1, 3)$ are smooth and the remaining 93 families consist of singular Fano 3-folds (with terminal quotient singularities).
The descriptions of these 93 families are given in Table \ref{table:main}.
\begin{Thm}[\cite{CP17}, \cite{CPR00}] \label{thm:BRWH}
Any quasi-smooth Fano $3$-fold weighted hypersurface of index $1$ is birationally rigid.
\end{Thm}
Among the 95 families, any quasi-smooth member of each of specific 50 families is birationally superrigid.
The 50 families consist of 48 families in Table \ref{table:main} which do not admit singularity with ``QI" or ``EI" in the 4th column plus the 2 families of smooth Fano weighted hypersurfaces.
For each of the remaining 45 families, a general quasi-smooth member is strictly birationally rigid (meaning that it is not birationally superrigid) but some special quasi-smooth members are birationally superrigid (see \S \ref{sec:95fam} for details).
\begin{Thm}[{\cite[Corollary 1.45]{Che09}}] \label{thm:KstWHgen}
A general quasi-smooth member of each of the 95 families is K-stable and admits a KE metric.
\end{Thm}
The generality assumption is crucial in Theorem \ref{thm:KstWHgen}.
In particular, it is highly likely that birationally superrigid special members of each of the above mentioned 45 families are not treated in Theorem \ref{thm:KstWHgen}.
Although Theorems \ref{thm:BRWH} and \ref{thm:KstWHgen} give a strong evidence for Conjecture \ref{conj:BSRKst}, it is very important to consider special (quasi-smooth) members for Conjecture \ref{conj:BSRKst}.
\subsection{Conceptual evidences}
Apart from evidences by concrete examples given in \S \ref{sec:introsmFano} and \S \ref{sec:introFanoWH}, we have conceptual results supporting Conjecture \ref{conj:BSRKst}.
The notion of slope stability for polarized varieties was introduced by Ross and Thomas (cf.\ \cite{RT07}).
For a Fano variety $X$, slope stability of $(X, -K_X)$ is a weaker version of K-stability.
\begin{Thm}[{\cite[Theorem 1.1]{OO13}}]
Let $X$ be a birationally superrigid Fano manifold of Fano index $1$.
If $\left| -K_X \right|$ is base point free, then $(X, -K_X)$ is slope stable.
\end{Thm}
As it is explained in \S \ref{sec:KstBSRConj}, K-stability of a Fano variety $X$ of dimension $n$ follows from the inequality $\alpha (X) > n/(n+1)$.
In practice, the computations of alpha invariants are very difficult and hence it is not easy to prove the inequality $\alpha (X) > n/(n+1)$.
\begin{Rem}
In fact, our results show that there exists a birationally superrigid Fano 3-fold $X$ such that $\alpha (X) < 3/4$ (see Example \ref{ex:No46degQI}).
\end{Rem}
Recently Stibitz and Zhuang relaxed the assumption on the alpha invariants significantly under the assumption of birationally superrigidity, and obtained the following.
\begin{Thm}[{\cite[Theorem 1.2, Corollary 3.1]{SZ19}}] \label{thm:SZ}
Let $X$ be a birationally superrigid Fano variety.
If $\alpha (X) \ge 1/2$, then $X$ is K-stable.
\end{Thm}
Note that the assumption on the alpha invariant is $\alpha (X) > 1/2$ in \cite[Theorem 1.2]{SZ19}, but the equality is allowed by \cite[Corollary 3.1]{SZ19}.
It is informed by C. Xu and Z. Zhuang that one can even conclude the uniform K-stability of $X$ in Theorem \ref{thm:SZ} under the same assumption.
The notion of uniform K-stability is originally introduced in \cite{Der16b} and \cite{BHJ17} (see also \cite{Fuj19b} and \cite{BJ20}) and it is stronger than K-stability.
Moreover it is very important to mention that uniform K-stability implies the existence of a KE metric (\cite{LTW19}).
Combining these results, we have the following.
\begin{Thm}[{\cite[Theorem 9.6]{Xu20}, \cite{SZ19}, \cite{LTW19}}] \label{thm:SXZ}
Let $X$ be a birationally superrigid Fano variety and assume that $\alpha (X) \ge 1/2$.
Then $X$ is uniformly K-stable.
In particular, $X$ is K-stable and it admits a KE metric.
\end{Thm}
\section{Main results}
\label{sec:mainresults}
We state Main Theorem of this article.
\begin{Thm}[Main Theorem] \label{mainthm}
Let $X$ be a quasi-smooth Fano $3$-fold weighted hypersurface of index $1$.
Then $\alpha (X) \ge 1/2$.
\end{Thm}
The following is a direct consequence of Theorems \ref{mainthm} and \ref{thm:SXZ}.
\begin{Cor} \label{maincor:Kst}
Any birationally superrigid quasi-smooth Fano $3$-fold weighted hypersurface of index $1$ is K-stable and admits a KE metric.
\end{Cor}
By \cite[Corollary 1.3]{ACP20}, a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurface necessarily has Fano index $1$.
Thus we obtain the following.
\begin{Cor}
Conjecture \ref{conj:BSRKst} is true for quasi-smooth Fano 3-fold weighted hypersurfaces.
\end{Cor}
It is natural to consider a generalization of Conjecture \ref{conj:BSRKst} by relaxing the assumption of birational superrigidity to birational rigidity (see \S \ref{section:BRKst}), or to expect that the conclusion of Corollary \ref{maincor:Kst} holds without the assumption of birational superrigidity.
We are unable to relax the assumption of birational superrigidity to birational rigidity in Theorem \ref{thm:SZ} or \ref{thm:SXZ}, and thus we cannot conclude K-stability for strictly birationally rigid members as a direct consequence of Theorem \ref{mainthm}.
By the arguments delivered in this article, we are able to prove $\alpha (X) > 3/4$ for any quasi-smooth member $X$ of suitable families.
As a consequence, we have the following (see \S \ref{sec:KEmetric} for details).
\begin{Thm}[= Theorem \ref{thm:qsmWHKE}] \label{thm:introKE}
Let $X$ be any quasi-smooth member of a family which is given ``KE" in the right-most column of Table \ref{table:main}.
Then $X$ is K-stable and admits a KE metric.
\end{Thm}
We can also prove K-stability for any quasi-smooth member (which is not necessarily birationally superrigid) of suitable families.
\begin{Thm}[= Corollary \ref{cor:Kstqsm}] \label{thm:introKst}
Let $X$ be any quasi-smooth member of a family which is given ``K" or ``KE" in the right-most column of Table \ref{table:main}.
Then $X$ is K-stable.
\end{Thm}
We explain the organization of this article.
In Chapter \ref{chap:prelim}, we recall definitions and basic properties of relevant notions such as birational (super)rigidity, log canonical thresholds, alpha invariants, and weighted projective varieties.
In Chapter \ref{chap:methods}, we explain methods of computing log canonical thresholds and alpha invariants.
By applying these methods, we compute local alpha invariants $\alpha_{\mathsf{p}} (X)$ for any point $\mathsf{p}$ on a quasi-smooth Fano 3-fold weighted hypersurface $X$ of index $1$.
In Chapters \ref{chap:smpt} and \ref{chap:singpt}, we compute local alpha invariants at smooth and singular points, respectively.
At this stage, Theorem \ref{mainthm} is proved except for specific 7 families.
These exceptional families are families No. 2, 4, 5, 6, 8, 10 and 14, and we need extra arguments to prove $\alpha (X) \ge 1/2$, which will be done in Chapter \ref{chap:exc}.
In Chapter \ref{chap:discuss}, we will consider and prove further results such as Theorems \ref{thm:introKE} and \ref{thm:introKst}.
We will also discuss related problems that arise naturally through the experience of huge amount of computations.
Finally, in Chapter \ref{chap:table}, various information on the families of quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$ are summarized.
\end{Ack}
\chapter{Preliminaries} \label{chap:prelim}
\section{Basic definitions and properties} \label{sec:basicdefs}
We refer readers' to \cite{KM98} for standard notions of birational geometry which are not explained in this article.
\begin{Def}
By a {\it Fano variety}, we mean a normal projective $\mathbb{Q}$-factorial variety with at most terminal singularities whose anticanonical divisor is ample.
\end{Def}
For a variety $X$, we denote by $\operatorname{Sm} (X)$ the smooth locus of $X$ and $\operatorname{Sing} (X) = X \setminus \operatorname{Sm} (X)$ the singular locus of $X$.
For a subset $\Gamma \subset X$, we define $\operatorname{Sing}_{\Gamma} (X) := \operatorname{Sing} (X) \cap \Gamma$.
Let $X$ be a normal variety and $D$ a Weil divisor (class) on $X$.
We denote by $|D|_{\mathbb{Q}}$ the set of effective $\mathbb{Q}$-divisors on $X$ which are $\mathbb{Q}$-linearly equivalent to $D$.
For a smooth point $\mathsf{p} \in X$, we define $|\mathcal{I}_{\mathsf{p}} (D)|$ to be the linear subspace of $|D|$ consisting of members of $|D|$ passing through $\mathsf{p}$.
\subsection{Birational (super)rigidity of Fano varieties}
Let $X$ be a normal $\mathbb{Q}$-factorial variety, $D$ a $\mathbb{Q}$-divisor on $X$ and $\mathcal{M}$ a movable linear system on $X$.
For a prime divisor $E$ over $X$, we define $\operatorname{ord}_E (D)$ to be the coefficient of $E$ in $\varphi^*D$, where $\varphi \colon Y \to X$ be a birational morphism such that $E \subset Y$, and we set $m_E (\mathcal{M}) := \operatorname{ord}_E (M)$, where $M$ is a general member of $\mathcal{M}$.
For a positive rational number $\lambda$, we say that a pair $(X, \lambda \mathcal{M})$ is {\it canonical} if
\[
\lambda m_E (\mathcal{M}) > a_E (K_X)
\]
for any exceptional prime divisor $E$ over $X$.
Let $X$ be a Fano variety of Picard number one.
Note that we can view $X$ (or more precisely the structure morphism $X \to \operatorname{Spec} \mathbb{C}$) as a Mori fiber space.
\begin{Def} \label{def:BSR}
We say that $X$ is {\it birationally rigid} if the existence of a Mori fiber space $Y \to T$ such that $Y$ is birational to $X$ implies that $Y$ is isomorphic to $X$ (and $T = \operatorname{Spec} \mathbb{C}$).
We say that $X$ is {\it birationally superrigid} if $X$ is birationally rigid and $\operatorname{Bir} (X) = \operatorname{Aut} (X)$.
\end{Def}
\begin{Def}
A closed subvariety $\Gamma \subset X$ is called a {\it maximal center} if there exists a movable linear system $\mathcal{M} \sim_{\mathbb{Q}} - n K_X$ and an exceptional prime divisor $E$ over $X$ such that $m_E (\mathcal{M}) > n a_E (K_X)$.
\end{Def}
\begin{Thm}[{\cite[Theorem 1.26]{CS}}] \label{thm:charactbsr}
A Fano variety $X$ of Picard number $1$ is birationally superrigid if and only if the pair $(X, \frac{1}{n} \mathcal{M})$ is canonical for any movable linear system $\mathcal{M}$ on $X$, where $n \in \mathbb{Q}_{> 0}$ is such that $\mathcal{M} \sim_{\mathbb{Q}} - n K_X$, or equivalently if and only if there is no maximal center on $X$.
\end{Thm}
\subsection{Log canonical thresholds and alpha invariants}
\begin{Def} \label{def:glct}
Let $(X, \Delta)$ be a pair, $D$ an effective $\mathbb{Q}$-divisor on $X$, and let $\mathsf{p} \in X$ be a point.
Assume that $(X, \Delta)$ has at most log canonical singularities.
We define the {\it log canonical threshold} (abbreviated as LCT) of $(X, \Delta; D)$ {\it at} $\mathsf{p}$ and the {\it log canonical threshold} of $(X, \Delta; D)$ to be the numbers
\[
\begin{split}
\operatorname{lct}_{\mathsf{p}} (X, \Delta; D) &= \sup \{\, c \in \mathbb{Q}_{\ge 0} \mid \text{$(X, \Delta + c D)$ is log canonical at $\mathsf{p}$} \,\}, \\
\operatorname{lct} (X, \Delta; D) &= \sup \{\, c \in \mathbb{Q}_{\ge 0} \mid \text{$(X, \Delta + c D)$ is log canonical} \,\},
\end{split}
\]
respectively.
We set $\operatorname{lct}_{\mathsf{p}} (X; D) = \operatorname{lct}_{\mathsf{p}} (X, 0; D)$ and $\operatorname{lct}_{\mathsf{p}} (X; D) = \operatorname{lct} (X, \Delta; D)$ when $\Delta = 0$.
Assume that $\left| -K_X \right|_{\mathbb{Q}} \ne \emptyset$.
Then we define the {\it alpha invariant of $X$ at $\mathsf{p}$} and the {\it alpha invariant of $X$} to be the numbers
\[
\begin{split}
\alpha_{\mathsf{p}} (X) &= \inf \{\, \operatorname{lct}_{\mathsf{p}} (X, D) \mid D \in \left| -K_X \right|_{\mathbb{Q}} \,\}, \\
\alpha (X) &= \inf \{\, \alpha_{\mathsf{p}} (X) \mid \mathsf{p} \in X \, \},
\end{split}
\]
respectively.
\end{Def}
The following fact is frequently used.
\begin{Rem} \label{rem:covex}
Let $\mathsf{p}$ be a point on $X$, and let $D_1, D_2$ be effective $\mathbb{Q}$-divisors on $X$.
If both $(X, D_1)$ and $(X, D_2)$ are log canonical at $\mathsf{p}$, then the pair
\[
(X, \lambda D_1 + (1-\lambda) D_2)
\]
is log canonical at $\mathsf{p}$ for any $\lambda \in \mathbb{Q}$ such that $0 \le \lambda \le 1$.
In particular, if $\alpha_{\mathsf{p}} (X) < c$ for some number $c > 0$, then there exists an irreducible $\mathbb{Q}$-divisor $D \in \left|-K_X \right|_{\mathbb{Q}}$ such that $(X, c D)$ is not log canonical at $\mathsf{p}$.
Here a $\mathbb{Q}$-divisor is {\it irreducible} if its support $\operatorname{Supp} (D)$ is irreducible.
\end{Rem}
\subsection{Cyclic quotient singularities and orbifold multiplicities}
\begin{Def}
Let $r > 0$ and $a_1, \dots, a_n$ be integers.
Suppose that the cyclic group $\boldsymbol{\mu}_r$ of $r$th roots of unity in $\mathbb{C}$ acts on the affine $n$-space $\mathbb{A}^n$ with affine coordinates $x_1, \dots, x_n$ via
\[
(x_1, \dots, x_n) \mapsto (\zeta^{a_1} x_1, \dots, \zeta^{a_n} x_n),
\]
where $\zeta \in \boldsymbol{\mu}_r$ is a fixed primitive $r$th root of unity.
We denote by $\bar{o} \in \mathbb{A}^n/\boldsymbol{\mu}_r$ the image of the origin $o \in \mathbb{A}^n$ under the quotient morphism $\mathbb{A}^n \to \mathbb{A}^n/\boldsymbol{\mu}_r$.
A singularity $\mathsf{p} \in X$ is a {\it cyclic quotient singularity} of type $\frac{1}{r} (a_1, \dots, a_n)$ if $\mathsf{p} \in X$ is analytically isomorphic to an analytic germ $\bar{o} \in \mathbb{A}^n/\boldsymbol{\mu}_r$.
In this case $r$ is called the {\it index} of the cyclic quotient singularity $\mathsf{p} \in X$.
\end{Def}
\begin{Rem}
Let $\mathsf{p} \in X$ be an $n$-dimensional cyclic quotient singular point.
Then we have a suitable action of $\boldsymbol{\mu}_r$ on $\mathbb{A}^n$ such that there is an analytic isomorphism $\bar{o} \in \mathbb{A}^n/\boldsymbol{\mu}_r \cong \mathsf{p} \in X$ of (analytic) germs.
In the following, the germ $o \in \mathbb{A}^n$ is often denoted by $\check{\mathsf{p}} \in \check{X}$.
By identifying $\mathsf{p} \in X \cong \bar{o} \in \mathbb{A}^n/\boldsymbol{\mu}_r$, the quotient morphism $o \in \mathbb{A}^n \to \bar{o} \in \mathbb{A}^n/\boldsymbol{\mu}_r$ is denoted by $q_{\mathsf{p}} \colon \check{X} \to X$ and is called the {\it quotient morphism} of $\mathsf{p} \in X$.
\end{Rem}
Note that, by convention, the case $r = 1$ is allowed in the definition of cyclic quotient singularity.
A cyclic quotient singularity $\mathsf{p} \in X$ of index $1$ is nothing but a smooth point $\mathsf{p} \in X$ and in that case the quotient morphism $q_{\mathsf{p}} \colon \check{X} \to X$ is simply an isomorphism.
\begin{Def}
Let $\mathsf{p} \in X$ be a cyclic quotient singularity and let $q_{\mathsf{p}} \colon \check{X} \to X$ be its quotient morphism with $\check{\mathsf{p}} \in \check{X}$ the preimage of $\mathsf{p}$.
For an effective $\mathbb{Q}$-divisor $D$ on $X$, we define
\[
\operatorname{omult}_{\mathsf{p}} (D) := \operatorname{mult}_{\check{\mathsf{p}}} (q_{\mathsf{p}}^*D)
\]
and call it the {\it orbifold multiplicity} of $D$ at $\mathsf{p}$.
By convention, we set $\operatorname{omult}_{\mathsf{p}} (D) = \operatorname{mult}_{\mathsf{p}} (D)$ when $\mathsf{p} \in X$ is a smooth point.
\end{Def}
\subsection{Kawamata blowup}
Let $\mathsf{p} \in V$ be a $3$-dimensional terminal quotient singularity.
Then it is of type $\frac{1}{r} (1, a, r-a)$, where $r$ and $a$ are coprime positive integers with $r > a$ (see \cite{MS84}).
Let $\varphi \colon W \to V$ be the weighted blowup of $V$ at $\mathsf{p}$ with weight $\frac{1}{r} (1,a,r-a)$.
By \cite{Kawamata}, $\varphi$ is the unique divisorial contraction centered at $\mathsf{p}$ and we call $\varphi$ the {\it Kawamata blowup} of $V$ at $\mathsf{p}$.
If we denote by $E$ the $\varphi$-exceptional divisor, then $E \cong \mathbb{P} (1,a,r-a)$ and we have
\[
K_W = \varphi^*K_V + \frac{1}{r} E,
\]
and
\[
(E^3) = \frac{r^2}{a (r-a)}.
\]
\section{Weighted projective varieties}
We recall basic definitions of various notions concerning weighted projective spaces and their subvarieties.
We refer readers to \cite{IF00} for details.
\subsection{Weighted projective space}
Let $N$ be a positive integer.
For positive integers $a_0, \dots, a_N$, let
\[
R (a_0, \dots, a_N) := \mathbb{C} [x_0, \dots, x_N]
\]
be the graded ring whose grading is given by $\deg x_i = a_i$.
We define
\[
\mathbb{P} (a_0,\dots, a_N) := \operatorname{Proj} R (a_0, \dots, a_N),
\]
and call it the {\it weighted projective space} with homogeneous coordinates $x_0, \dots, x_N$ (of degree $\deg x_i = a_i$).
We sometimes denote
\[
\mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N}
\]
in order to make it clear the homogeneous coordinates $x_0, \dots, x_N$.
For $i = 0 \dots, N$, we denote by
\begin{equation} \label{eq:wpvvertex}
\mathsf{p}_{x_i} = (0\!:\!\cdots\!:\!1\!:\!\cdots\!:\!0) \in \mathbb{P} (a_0, \dots, a_N)
\end{equation}
the coordinate point at which only the coordinate $x_i$ does not vanish.
Let $f \in R := R (a_0, \dots, a_N) = \mathbb{C} [x_0, \dots, x_N]$ be a polynomial.
We say that $f$ is {\it quasi-homogeneous} (resp.\ {\it homogeneous}) if it is homogeneous with respect to the grading $\deg x_i = a_i$ (resp.\ $\deg x_i = 1$) for $i = 0, 1, \dots, N$.
For a polynomial $f \in \mathbb{C} [x_0, \dots, x_N]$ and a monomial $M = x_0^{m_0} \cdots x_N^{m_N}$, we denote by
\[
\operatorname{coeff}_f (M) \in \mathbb{C}
\]
the coefficient of $M$ in $f$, and, by a slight abuse of notation, we write $M \in f$ if $\operatorname{coeff}_f (M) \ne 0$.
For quasi-homogeneous polynomials $f_1, \dots, f_k \in R$, we denote by
\[
(f_1 = \cdots = f_k = 0) \subset \mathbb{P} (a_0, \dots, a_N)
\]
the closed subscheme defined by the quasi-homogeneous ideal $(f_1, \dots, f_k) \subset R$.
Moreover, for a closed subscheme $X \subset \mathbb{P} (a_0, \dots, a_N)$ and quasi-homogeneous polynomials $g_1, \dots, g_l \in R$, we define
\[
(g_1 = \cdots = g_l = 0)_X := (g_1 = \cdots = g_l = 0) \cap X,
\]
which is a closed subscheme of $X$.
For $i = 0, \dots, N$, we define
\begin{equation} \label{eq:wpvHU}
\begin{split}
\mathcal{H}_{x_i} &:= (x_i = 0) \subset \mathbb{P} (a_0, \dots, a_N), \\
\mathcal{U}_{x_i} &:= \mathbb{P} (a_0, \dots, a_N) \setminus \mathcal{H}_{x_i}.
\end{split}
\end{equation}
\begin{Rem}
The weighted projective space $\mathbb{P} (a, b, c, d, e)$ with homogeneous coordinates $x$, $y$, $z$, $t$, $w$ of degrees $a$, $b$, $c$, $d$, $e$, respectively, is sometimes denoted by
\[
\mathbb{P} (a, b, c, d, e)_{x, y, z, t, w}
\]
in order to emphasize the homogeneous coordinates.
For a coordinate $v \in \{x, \dots, w\}$, the point $\mathsf{p}_v \in \mathbb{P} (a, b, c, d, e)$, the quasi-hyperplane $\mathcal{H}_v = (v = 0) \subset \mathbb{P} (a, b, c, d, e)$ and the open set $\mathcal{U}_v = \mathbb{P} (a, b, c, d, e) \setminus \mathcal{H}_v$ are similarly defined as in \eqref{eq:wpvvertex} and \eqref{eq:wpvHU}.
\end{Rem}
\subsection{Well-formedness and quasi-smoothness}
\label{sec:wfqsm}
\begin{Def}
We say that a weighted projective space $\mathbb{P} (a_0, \dots, a_N)$ is {\it well-formed} if
\[
\gcd \{a_0, \dots, \hat{a}_i, \dots, a_N\} = 1
\]
for any $i = 0, 1, \dots, N$.
\end{Def}
\begin{Def}
Let $\mathbb{P} (a_0, \dots, a_N)$ be a weighted projective space such that $\gcd \{a_0, \dots, a_N\} = 1$.
For $j = 0, 1, \dots, N$, we set
\[
\begin{split}
l_j &:= \gcd \{ a_0, a_1, \dots, \hat{a}_j, \dots, a_N\}, \\
m_j &:= l_0 l_1 \cdots \hat{l}_j \cdots l_N, \\
b_j &:= \frac{a_j}{m_j}.
\end{split}
\]
We then define
\[
\mathbb{P} (a_0, \dots, a_N)^{\operatorname{wf}} := \mathbb{P} (b_0, \dots, b_N)
\]
and call it the {\it well-formed model} of $\mathbb{P} (a_0, \dots, a_N)$.
\end{Def}
\begin{Rem}
Any weighted projective space is isomorphic to a well-formed one (see e.g.\ \cite[Lemma 5.7]{IF00}).
More precisely, for a weighted projective space $\mathbb{P} = \mathbb{P} (a_0, \dots, a_N)$ with $\gcd \{a_0, \dots, a_N\} = 1$, there exists an isomorphism
\[
\phi \colon \mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N} \to \mathbb{P}^{\operatorname{wf}} = \mathbb{P} (b_0, \dots, b_N)_{y_0, \dots, y_N},
\]
such that $\phi^* \mathcal{H}_{y_i} = m_i \mathcal{H}_{x_i}$ for $i = 0, 1, \dots, N$, where $\mathcal{H}_{x_i} = (x_i = 0) \subset \mathbb{P}$ and $\mathcal{H}_{y_i} = (y_i = 0) \subset \mathbb{P}^{\operatorname{wf}}$.
\end{Rem}
In the following, we set $\mathbb{P} := \mathbb{P} (a_0, \dots, a_N)$ and we denote by
\[
\Pi \colon \mathbb{A}^{N+1} \setminus \{o\} \to \mathbb{P}, \quad
(\alpha_0, \dots, \alpha_N) \mapsto (\alpha_0\!:\!\cdots\!:\!\alpha_N),
\]
the canonical projection.
Let $X \subset \mathbb{P}$ be a closed subscheme.
We set $C_X^* := \Pi^{-1} (X)$ and call it the {\it punctured affine quasi-cone} over $X$.
The {\it affine quasi-cone} $C_X$ over $X$ is the closure of $C_X^*$ in $\mathbb{A}^{N+1}$.
We set $\pi := \Pi|_{C_X^*} \colon C_X^* \to X$.
\begin{Def}
We say that a closed subscheme $X \subset \mathbb{P}$ is {\it well-formed} if $\mathbb{P}$ is well-formed and $\operatorname{codim}_X (X \cap \operatorname{Sing} (\mathbb{P})) \ge 2$.
\end{Def}
\begin{Def}
Let $X \subset \mathbb{P}$ be a closed subscheme as above.
We define the {\it quasi-smooth locus} of $X$ as
\[
\mathrm{QI}Sm (X) := \pi (\operatorname{Sm} (C^*_X)) \subset X.
\]
Let $S$ be a subset of $X$.
We say that $X$ is {\it quasi-smooth along} $S$ if $S \subset \mathrm{QI}Sm (X)$.
We simply say that $X$ is {\it quasi-smooth} when $X = \mathrm{QI}Sm (X)$.
\end{Def}
\subsection{Orbifold charts}
Let $\mathcal{U}_{x_i}$ be the open subset of $\mathbb{P} = \mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N}$ as in \eqref{eq:wpvHU}, where $i \in \{0, 1, \dots, N\}$.
We call $\mathcal{U}_{x_i}$ the {\it standard affine open subset} of $\mathbb{P}$ containing $\mathsf{p}_{x_i}$.
We denote by $\breve{\mathcal{U}}_{x_i}$ the affine $N$-space $\mathbb{A}^N$ with affine coordinates $\breve{x}_0, \dots, \widehat{\breve{x}_i}, \dots, \breve{x}_N$.
Consider the $\boldsymbol{\mu}_{a_i}$-action on $\breve{\mathcal{U}}_i$ defined by
\[
\breve{x}_j \mapsto \zeta^{a_j} \breve{x}_j, \quad \text{for $j = 0, \dots, \hat{i}, \dots, N$},
\]
where $\zeta \in \boldsymbol{\mu}_{a_i}$ is a primitive $a_i$th root of unity.
Then the open set $\mathcal{U}_i$ can be naturally identified with the quotient $\breve{\mathcal{U}}_{x_i}/\boldsymbol{\mu}_{a_i}$.
In fact, this can be seen by the identification
\[
\breve{x}_j = \frac{x_j}{x_i^{a_j/a_i}}, \quad \text{for $j = 0, \dots, \hat{i}, \dots, N$}.
\]
The quotient morphism $\breve{\mathcal{U}}_{x_i} \to \breve{\mathcal{U}}_{x_i}/\boldsymbol{\mu}_{a_i} = \mathcal{U}_{x_i}$ is denoted by
\[
\rho_{x_i} \colon \breve{\mathcal{U}}_{x_i} \to \mathcal{U}_{x_i}
\]
and is called the {\it orbifold chart} of $\mathbb{P}$ containing $\mathsf{p}_{x_i}$.
Let $X \subset \mathbb{P}$ be a subscheme.
Usually, we denote by $U_{x_i} \subset X$ the open set $\mathcal{U}_{x_i} \cap X$, and we call $U_{x_i}$ the {\it standard affine open subset} of $X$ containing $\mathsf{p}_{x_i}$.
In this case, we set $\breve{U}_{x_i} = \rho_{x_i}^{-1} (U_{x_i}) \subset \breve{\mathcal{U}}_{x_i}$.
By a slight abuse of notation, the morphism $\rho_{x_i}|_{U_i} \colon \breve{U}_{x_i} \to U_{x_i}$ is also denote by
\[
\rho_{x_i} \colon \breve{U}_{x_i} \to U_{x_i}
\]
and is called the {\it orbifold chart} of $X$ containing $\mathsf{p}_{x_i}$.
When we are using the notation $\mathsf{p} = \mathsf{p}_{x_i}$, the morphism $\rho_{x_i}$ is sometimes denoted by $\rho_{\mathsf{p}}$.
Note that $\breve{U}_{x_i}$ is not necessary smooth in general.
Suppose that $X \subset \mathbb{P}$ is a closed subvariety containing the point $\mathsf{p} = \mathsf{p}_{x_i}$.
The preimage $\breve{\mathsf{p}}$ of $\mathsf{p}$ is the origin of $\breve{U}_{x_i} \subset \breve{\mathcal{U}}_{x_i} = \mathbb{A}^N$.
It is straightforward to see that $X$ is quasi-smooth at $\mathsf{p}$ if and only if $\breve{U}_{x_i}$ is smooth at $\breve{\mathsf{p}}$.
Suppose that $X$ is quasi-smooth at $\mathsf{p}$.
A system of local coordinates of $U_{x_i}$ at $\breve{\mathsf{p}}$ is called a system of {\it local orbifold coordinates} of $X$ at $\mathsf{p}$.
In this case, $\mathsf{p} \in X$ is a cyclic quotient singularity of index $a_i$ and $\rho_{x_i} \colon \breve{U}_{x_i} \to U_{x_i}$ can be identified with (or analytically equivalent to) the quotient morphism $q_{\mathsf{p}}$ of $\mathsf{p} \in X$ after shrinking $U_{x_i}$ and then $\breve{U}_{x_i}$.
Moreover, if $X$ is quasi-smooth, then $\breve{U}_i$ is smooth for any $i$.
\begin{Rem}
When we work with $\mathbb{P} = \mathbb{P} (a, b, c, d, e)_{x, y, z, t, w}$ and its closed subscheme $X \subset \mathbb{P}$, then $\mathcal{U}_{v} = \mathbb{A}^5_{\breve{x}, \dots, \hat{\breve{v}}, \dots, \breve{w}}$, $\rho_v \colon \breve{\mathcal{U}}_v \to \mathcal{U}_v$, $\breve{U}_{v} = \rho_v^{-1} (U_v) \subset \mathcal{U}_v$ and $\rho_v \colon \breve{U}_v \to U_v$ are similarly defined.
\end{Rem}
\subsection{Weighted hypersurfaces and quasi-tangent divisors}
As in the previous subsections, we work with $\mathbb{P} = \mathbb{P} (a_0, \dots, a_N)_{x_0, \dots, x_N}$.
\begin{Def}
A {\it quasi-linear polynomial} (or a {\it quasi-linear form}) in variables $x_0, \dots, x_{n+1}$ is a quasi-homogeneous polynomial $f = f (x_0,\dots, x_{n+1})$ such that $x_i \in f$ for some $i = 0, \dots, n+1$.
\end{Def}
\begin{Def}
We say that a subvariety $S \subset \mathbb{P}$ is a {\it quasi-linear subspace} of $\mathbb{P}$ if it is a complete intersection in $\mathbb{P}$ defined by quasi-linear equations of the form
\[
\ell_1 + f_1 = \ell_2 + f_2 = \cdots = \ell_k + f_k = 0,
\]
where $\ell_1, \ell_2, \dots, \ell_k$ are linearly independent linear forms in variables $x_0, \dots, x_{n+1}$ and $f_1, \dots, f_k \in \mathbb{C} [x_0, \dots, x_{n+1}]$ are quasi-homogeneous polynomials which are not quasi-linear.
A quasi-linear subspace of $\mathbb{P}$ of codimension $1$ (resp.\ dimension $1$) is called a {\it quasi-hyperplane} (resp.\ {\it quasi-line}) of $\mathbb{P}$.
\end{Def}
It is clear that a quasi-linear subspace of $\mathbb{P}$ is isomorphic to a weighted projective space.
In particular, a quasi-line is isomorphic to $\mathbb{P}^1$.
Let $X$ be a hypersurface in $\mathbb{P} = \mathbb{P} (a_0, \dots, a_N)$ defined by a quasi-homogeneous polynomial of degree $d$.
We often denote it as $X = X_d \subset \mathbb{P} (a_0, \dots, a_N)$.
Suppose that $X$ is quasi-smooth at a point $\mathsf{p} = \mathsf{p}_{x_i}$.
Then the defining polynomial $F = F (x_0, \dots, x_N)$ of $X$ can be written as
\begin{equation}
\label{eq:qtangpoly}
F = x_i^m f + x_i^{m-1} g_{m-1} + \cdots + x_i g_1 + g_0,
\end{equation}
where $m \ge 0$, $f = f (x_0, \dots, x_N)$ is a quasi-homogeneous polynomial of degree $d - m a_i$ which is quasi-linear and $g_k = g_k (x_0, \dots,\hat{x}_i, \dots, x_N)$ is a quasi-homogeneous polynomial of degree $d - k a_i$ which is not quasi-linear for $0 \le k \le m - 1$.
Note that the expression \eqref{eq:qtangpoly} is uniquely determined once the homogeneous coordinates of $\mathbb{P}$ are fixed.
\begin{Def} \label{def:qtangdiv}
Under the notation and assumptions as above, we call $f$ the {\it quasi-tangent polynomial} of $X$ at $\mathsf{p}$ and the divisor $(f = 0)_X$ on $X$ is called the {\it quasi-tangent divisor} of $X$ at $\mathsf{p}$.
When $f = x_j$ for some $j$, then we also call $x_j$ as the {\it quasi-tangent coordinate} of $X$ at $\mathsf{p}$.
\end{Def}
\begin{Rem}
Let $X = X_7 \subset \mathbb{P} (1, 1, 1, 2, 3)_{x, y, z, t, w}$ be a weighted hypersurface of degree $7$.
Suppose that its defining polynomial is of the form
\[
F = t^3 x + t^2 w + t g_5 + g_7,
\]
where $g_5, g_7 \in \mathbb{C} [x, y, z, w]$ are quasi-homogeneous polynomials of degree $5, 7$, respectively.
In this case $X$ is quasi-smooth at $\mathsf{p} = \mathsf{p}_t$.
The quasi-tangent polynomial of $X$ at $\mathsf{p}$ is $t x + w$.
Note that $x$ is not a quasi-tangent coordinate of $X$ at $\mathsf{p}$ because of the presence of $t^2 w \in F$.
\end{Rem}
\begin{Lem}
Let $X \subset \mathbb{P}$ be a weighted hypersurface of degree $d$.
Assume that $X$ is quasi-smooth at a point $\mathsf{p} = \mathsf{p}_{x_i}$ for some $i = 0, 1, \dots, N$ and let $x_j$ be a homogeneous coordinate such that $x_j \in f$, where $f$ is the quasi-tangent polynomial of $X$ at $\mathsf{p}$.
Then, after a suitable choice of homogeneous coordinates $x_0, \dots, x_N$, the defining polynomial $F$ of $X$ can be written as
\[
F = x_i^m x_j + x_i^{m-1} g_{m - 1} + \cdots + x_i g_1 + g_0,
\]
where $g_k = g_k (x_0, \dots, \hat{x}_i, \dots, x_N)$ is a quasi-homogeneous polynomial of degree $d - k a_i$ which is not quasi-linear.
\end{Lem}
\begin{proof}
We can write $F = x_i^m f + g$, where $m \ge 0$, $f = f (x_0, \dots, x_N) \ni x_j$ is the quasi-tangent polynomial and $g$ is a quasi-homogeneous polynomial of degree $d$ which does not involve a monomial divisible by $x_i^m$ and which is contained in the ideal $(x_0, \dots, \hat{x}_i, \dots, x_N)^2$.
We write $g = x_j^e h_{e} + \cdots + x_j h_1 + h_0$, where $e \ge 0$ and $h_k$ is a quasi-homogeneous polynomial of degree $d - k a_j$ which does not involve the variables $x_j$.
By rescaling $x_j$, we may assume $f = x_j - \tilde{f}$, where $x_j \notin \tilde{f}$, and we write $\tilde{f} = x_i^n \tilde{f}_n + \cdots + x_i \tilde{f}_1 + \tilde{f}_0$, where $\tilde{f}_k$ is a quasi-homogeneous polynomial of degree $d - (m + k) a_i$ which does not involve the variable $x_i$.
We consider the coordinate change $x_j \mapsto x_j + \tilde{f}$.
Then the new defining polynomial can be written as
\[
\begin{split}
F &= x_i^m x_j + (x_j + x_i^n \tilde{f}_n + \cdots)^e h_{e} + \cdots + (x_j + x_i^n \tilde{f}_n + \cdots) h_1 + h_0 \\
&= (x_i^{n e - m} \tilde{f}_n^e + \cdots + x_j) x_i^m + \cdots
\end{split}
\]
It follows that the new quasi-tangent polynomial is $x_i^{n e - m} \tilde{f}_n^e + \cdots + x_j$.
We claim that $n e - m< n$.
We have $d = m a_i + a_j$, $d = e a_j + \deg h_e \ge e a_j$ and $a_j = n a_i + \deg \tilde{f}_n > n a_i$, which implies $n e - m < n$.
Thus, repeating the above coordinate change, we can drop the degree of the quasi-tangent coordinate with respect to $x_i$, and we may assume $F = x_i^m f + x_i^{m-1} g_{m-1} + \cdots + x_i g + g_0$, where $f \ni x_j$ and $g_k$ are quasi-homogeneous polynomials of degree $a_j$ and $d - k a_i$, respectively, which do not involve the variable $x_i$.
Moreover $g_k$ is not quasi-linear for $0 \le k \le m-1$.
Finally, replacing $x_j$, we may assume $f = x_j$ and this completes the proof.
\end{proof}
\begin{Rem}
Suppose that a weighted hypersurface $X \subset \mathbb{P}$ is quasi-smooth at $\mathsf{p} = \mathsf{p}_{x_i}$.
Then $\operatorname{omult}_{\mathsf{p}} ((f = 0)_X) > 1$ for the quasi-tangent polynomial $f$ of $X$ at $\mathsf{p}$.
Moreover, $x_j$ is a quasi-tangent coordinate of $X$ at $\mathsf{p}$ if and only if $\operatorname{omult}_{\mathsf{p}} (H_{x_j}) > 1$.
\end{Rem}
\section{The 95 families}
\label{sec:95fam}
\subsection{Definition of the families}
As it is explained in \S \ref{sec:introFanoWH}, quasi-smooth Fano 3-fold weighted hypersurfaces of index 1 are classified and they form 95 families.
According to the classification, the minimum of the weights of an ambient space is $1$.
Hence a family is determined by a quadruple $(a_1, a_2, a_3, a_4)$, which means that the family corresponding to a quadruple $(a_1, a_2, a_3, a_4)$ is the family of weighted hypersurfaces of degree $d = a_1 + a_2 + a_3 + a_4$ in $\mathbb{P} (1, a_1, a_2, a_3, a_4)$.
The 95 families are numbered in the lexicographical order on $(d, a_1, a_2, a_3, a_4)$, and each family is referred to as family No.~$\mathsf{i}$ for $\mathsf{i} \in \{1, 2, \dots, 95\}$.
Families No.~$1$ and $3$ are the families consisting of quartic 3-folds and degree $6$ hypersurfaces in $\mathbb{P} (1, 1, 1, 1, 3)$, respectively, and for any smooth member of these 2 families, K-stability (and hence the existence of KE metrics) is known.
\begin{Def}
We set
\[
\mathsf{I} := \{1, 2, \dots, 95\} \setminus \{1, 3\},
\]
and, for $\mathsf{i} \in \mathsf{I}$, we denote by $\mathcal{F}_{\mathsf{i}}$ the family consisting of the quasi-smooth members of family No.~$\mathsf{i}$.
\end{Def}
The main objects of this article is thus the members of $\mathcal{F}_{\mathsf{i}}$ for $\mathsf{i} \in \mathsf{I}$.
We set
\[
\mathsf{I}_1 := \{2, 4, 5, 6, 8, 10, 14\}.
\]
The set $\mathsf{I}_1$ is characterized as follows: let $X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)$, $a_1 \le a_2 \le a_3 \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
Then $\mathsf{i} \in \mathsf{I}_1$ if and only if $a_2 = 1$.
The computations of alpha invariants will be done in a relatively systematic way for families $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ (see Chapters \ref{chap:smpt} and \ref{chap:singpt}), while the computations will be done separately for families $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}_1$ (see Chapter \ref{chap:exc}).
We explain notation and conventions concerning the main objects of this article.
Let $X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4) =: \mathbb{P}$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
\begin{itemize}
\item Unless otherwise specified, we assume $a_1 \le a_2 \le a_3 \le a_4$.
\item In many situations (especially when we treat a specific family), we denote by $x, y, z, t, w$ the homogeneous coordinates of $\mathbb{P}$ of degree respectively $1, a_1, a_2, a_3, a_4$.
\item We denote by $F$ the polynomial defining $X$ in $\mathbb{P}$, which is quasi-homogeneous of degree $d = a_1 + a_2 + a_3 + a_4$.
\item We set $A = -K_X$, which is the positive generator of of $\operatorname{Cl} (X) \cong \mathbb{Z}$.
Note that we have
\[
(-K_X)^3 = (A^3) = \frac{d}{a_1 a_2 a_3 a_4} = \frac{a_1 + a_2 + a_3 + a_4}{a_1 a_2 a_3 a_4}.
\]
\end{itemize}
\subsection{Definitions of QI and EI centers, and birational (super)rigidity}
\label{sec:defBI}
In this subsection, let
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4, a_5)_{x, y, z, t, w}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$, where $a_1 \le a_2 \le a_3 \le a_4$.
We give definitions of QI and EI centers, which are particular singular points on $X$ and are important for understanding birational (super)rigidity of $X$.
For EI centers, we only give an ad hoc definition (see \cite[Section 4.10]{CPR00} and \cite[Section 4.2]{CP17} for more detailed treatments).
\begin{Def} \label{def:BIcenter}
Let $\mathsf{p} \in X$ be a singular point.
We say that $\mathsf{p} \in X$ is an {\it EI center} if the upper script $\mathrm{EI}$ is given in the 4th column of Table \ref{table:main}, or equivalently if $\mathsf{i}$ and $\mathsf{p}$ belong to one of the following.
\begin{itemize}
\item $\mathsf{i} = 7$ and $\mathsf{p}$ is of type $\frac{1}{2} (1, 1, 1)$.
\item $\mathsf{i} \in \{23, 40, 44, 61, 76\}$ and $\mathsf{p} = \mathsf{p}_t$.
\item $\mathsf{i} \in \{20, 36\}$ and $\mathsf{p} = \mathsf{p}_z$.
\end{itemize}
We say that $\mathsf{p} \in X$ is a {\it QI center} if there are distinct $j$ and $k$ such that $d = 2 a_k + a_j$ and the index of the cyclic quotient singularity $\mathsf{p} \in X$ coincides with $a_k$.
We say that $\mathsf{p} \in X$ is a {\it BI center} if it is either an EI center or a QI center.
\end{Def}
\begin{Rem} \label{rem:maxcent}
Let $X$ be a member of $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
Then the following are proved in \cite{CP17}.
\begin{enumerate}
\item No smooth point on $X$ is a maximal center.
\item A singular point $\mathsf{p} \in X$ is a maximal center only if $\mathsf{p}$ is a BI center.
\end{enumerate}
Note that a BI center $\mathsf{p} \in X$ is not always a maximal center (see \S \ref{sec:eqQI}, especially Remark \ref{rem:QImaxcent}, for the complete analysis for QI centers).
\end{Rem}
\begin{Def}
We define the subset $\mathsf{I}_{\mathrm{BSR}} \subset \mathsf{I}$ as follows: $\mathsf{i} \in \mathsf{I}_{\mathrm{BSR}}$ if and only if a member $X$ of $\mathcal{F}_{\mathsf{i}}$ does not admit a BI center.
We then define $\mathsf{I}_{\mathrm{BR}} = \mathsf{I} \setminus \mathsf{I}_{\mathrm{BSR}}$.
\end{Def}
Note that $|\mathsf{I}_{\mathrm{BSR}}| = 48$ and $|\mathsf{I}_{\mathrm{BR}}| = 45$.
The following is a more precise version of Theorem \ref{thm:BRWH}.
\begin{Thm}[{\cite{CP17}}]
Let $X$ be a member of $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
\begin{enumerate}
\item If $\mathsf{i} \in \mathsf{I}_{\mathrm{BSR}}$, then any member of $\mathcal{F}_{\mathsf{i}}$ is birationally superrigid.
\item If $\mathsf{i} \in \mathsf{I}_{\mathrm{BR}}$, then any member of $\mathcal{F}_{\mathsf{i}}$ is birationally rigid while its general member is not birationally superrigid.
\end{enumerate}
\end{Thm}
We emphasize that a family $\mathcal{F}_{\mathsf{i}}$, where $\mathsf{i} \in \mathsf{I}_{\mathrm{BR}}$, can contain (in fact does contain for most of $\mathsf{i} \in \mathsf{I}_{\mathrm{BR}}$) birationally superrigid Fano 3-folds as special members.
\subsection{Numerics on weights and degrees} \label{sec:smptnum}
Let
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_{\mathsf{i}}$.
Throughout the subsection, we assume that $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ and that $a_1 \le a_2 \le a_3 \le a_4$.
We collect some elementary numerical results on weights $a_1, \dots, a_4$, the degree $d = a_1 + a_2 + a_3 + a_4$ of the defining polynomial $F = F (x, y, z, t, w)$ of $X$, and the anticanonical degree $(A^3)$ of $X$ which will be repeatedly used in the rest of this article.
\begin{Lem} \label{lem:smptHLdegwt}
One of the following happens.
\begin{enumerate}
\item $d = 2 a_4$.
\item $d = 3 a_4$.
\item $d = 2 a_4 + a_j$ for some $j \in \{1, 2, 3\}$.
\end{enumerate}
\end{Lem}
\begin{proof}
We see that either $w^n \in F$ for some $n \ge 2$ or $x^n v \in F$ for some $n \ge 1$ and $v \in \{x, y, z, t\}$ by the quasi-smoothness of $X$.
Suppose $w^n \in F$ for some $n \ge 2$.
Then we have
\[
d = n a_4 = a_1 + a_2 + a_3 + a_4 < 4 a_4.
\]
Hence $n = 2, 3$ and we are in case (1) or (2).
Suppose $w^n v \in F$ for some $n \ge 1$ and $v \in \{y, z, t\}$.
Then we have $d = n a_4 + a_j$ and moreover we have
\[
a_4 + a_j < d = a_1 + a_2 + a_3 + a_4 < 3 a_4 + a_j.
\]
This shows $n = 2$, that is, $d = 2 a_4 + a_j$.
If $a_1 = 1$, then the proof is completed.
It remains to show that the case $d = 2 a_4 + 1$ does not take place assuming $a_1 \ge 2$.
Suppose $d = 2 a_4 + 1$ and $a_1 \ge 2$.
Then $w^2 x \in F$ and the singularity of $\mathsf{p}_w \in X$ is of type $\frac{1}{a_4} (a_1, a_2, a_3)$.
There exist distinct $i, j \in \{1,2,3\}$ such that $a_i + a_j$ is divisible by $a_4$ since $\mathsf{p}_w \in X$ is terminal.
We have $a_i + a_j = a_4$ since $0 < a_i + a_j < 2 a_4$.
Let $k \in \{1,2,3\}$ be such that $\{i, j, k\} = \{1,2,3\}$.
Then
\[
d = a_1 + a_2 + a_3 + a_4 = a_k + 2 a_4.
\]
Combining this with $d = 2 a_4 + 1$, we have $a_k = 1$.
This is a contradiction since $a_k \ge a_1 \ge 2$.
\end{proof}
\begin{Lem} \label{lem:wtnumerics}
\begin{enumerate}
\item We have $\mathsf{i} \in \{9, 17\}$ if and only if $d = 3 a_4$ and $a_1 = 1$.
\item We have $a_1 a_2 a_3 (A^3) \le 3$ and the equality holding if and only if $d = 3 a_4$.
\item If $a_1 < a_2$, then we have $a_1 (A^3) < 1$.
\item If $1 < a_1 < a_2$, then $a_1 a_3 (A^3) \le 1$.
\item If $a_1 < a_2$ and $d > 2 a_4$, then $a_1 a_4 (A^3) \le 2$.
\item If $d$ is divisible by $a_4$ and $\mathsf{i} \notin \{9, 17\}$, then $a_2 a_3 (A^3) \le 2$.
\item If $d$ is not divisible by $a_4$ and $a_1 \ge 2$, then $a_2 a_4 (A^3) \le 2$.
\end{enumerate}
\end{Lem}
\begin{proof}
We prove (1).
The ``only if" part is obvious.
Suppose $d = 3 a_4$ and $a_1 = 1$.
Then we have $2 a_4 = 1 + a_2 + a_3$.
This implies $a_2 = a_4 - 1$ and $a_3 = a_4$ since $a_2 \le a_3 \le a_4$.
Then, by setting $a = a_2 \ge 2$, $X$ is a weighted hypersurface in $\mathbb{P} (1, 1, a, a+1, a+1)$ of degree $3 (a+1)$.
Suppose $\mathsf{p}_z \notin X$.
Then some power of $z$ is contained in $F$ and this implies that $3 (a+1)$ is divisible by $a$.
In particular, we have $a = 3$ and this case corresponds to $\mathsf{i} = 17$.
Suppose $\mathsf{p}_z \notin X$, then either $3 (a+1) \equiv 1 \pmod{a}$ or $3 (a+1) \equiv a+1 \pmod{a}$ by the quasi-smoothness of $X$.
In both cases, we have $a = 2$, ad hence $\mathsf{i} = 9$.
Thus (1) is proved.
The assertion (2) follows immediately since we have
\[
a_1 a_2 a_3 (A^3) = \frac{d}{a_4} \le 3
\]
and $d \le 3 a_4$ by Lemma \ref{lem:smptHLdegwt}.
We prove (3).
Note that $2 \le a_2 \le a_3 \le a_4$.
Note also that $a_4 > a_2$ because otherwise $X$ has non-isolated singularity along $L_{xy}$ which is impossible.
In particular, we have $a_1 + \cdots + a_4 < 4 a_4$ and $a_2 a_3 \ge 4$, and we have
\[
a_1 (A^3) = \frac{a_1 (a_1 + a_2 + a_3 + a_4)}{a_1 a_2 a_3 a_4}
< \frac{4}{a_2 a_3}
\le 1,
\]
which proves (3).
We prove (4).
We have $a_2 \ge 3$ since $a_2 > a_1 > 1$ and thus
\[
a_1 a_3 (A^3) = \frac{d}{a_2 a_4} \le \frac{3}{a_2} \le 1.
\]
We prove (5).
We have $d > 2 a_4$ by assumption.
Then, by Lemma \ref{lem:smptHLdegwt}, we have $d = 2 a_4 + a_j$ for some $j \in \{1,2,3,4\}$, and combining this with $d = a_1 + a_2 + a_3 + a_4$, we have
\[
a_4 = a_1 + a_2 + a_3 - a_j \le a_2 + a_3.
\]
If $a_1 > 1$, then we have $a_2, a_3 \ge 3$ and thus
\[
a_1 a_4 (A^3) = \frac{a_1 + a_2 + a_3 + a_4}{a_2 a_3} < \frac{3 a_2 + 2 a_3}{a_2 a_3} = \frac{3}{a_3} + \frac{2}{a_2} \le \frac{5}{3}.
\]
Suppose $a_1 = 1$.
In this case $2 \le a_2 \le a_3$.
If $a_3 \ge 3$, then
\[
a_4 (A^3) = \frac{1 + a_2 + a_3 + a_4}{a_2 a_3} \le \frac{1 + 2 a_2 + 2 a_3}{a_2 a_3} = \frac{1}{a_2 a_3} + \frac{2}{a_3} + \frac{2}{a_2} \le \frac{11}{6}.
\]
Suppose $a_3 = 2$, that is, $a_2 = a_3 = 2$.
Then we have $a_4 = 3$ and $d = 8$ since $d = 5 + a_4 > 2 a_4$ and $a_4$ is odd.
In this case we have $a_4 (A^3) = 2$.
This proves (5).
We prove (6).
By Lemma \ref{lem:smptHLdegwt} and (1), either $d = 2 a_4$ or $d = 3 a_4$ and $a_1 \ge 2$.
If $d = 2 a_4$ (resp.\ $d = 3 a_4$ and $a_1 \ge 2$), then
\[
a_2 a_3 (A^3) = \frac{2}{a_1} \le 2 \quad (\text{resp.\ } a_2 a_3 (A^3) = \frac{3}{a_1} \le 2).
\]
This proves (6).
We prove (7).
By Lemma \ref{lem:smptHLdegwt}, we have $d = 2 a_4 + a_j$ for some $j \in \{1, 2, 3\}$.
Then we have $a_4 = a_1 + a_2 + a_3 - a_j \le a_2 + a_3$.
If $a_1 \ge 3$, then
\[
a_2 a_4 (A^3) = \frac{a_1 + a_2 + a_3 + a_4}{a_1 a_3} \le \frac{a_1 + 4 a_3}{a_1 a_3} = \frac{1}{a_3} + \frac{4}{a_1} \le \frac{5}{3}.
\]
We continue the proof assuming $a_1 = 2$.
If in addition $a_2 < a_3$, then
\[
a_2 a_4 (A^3) = \frac{2 + a_2 + a_3 + a_4}{2 a_3} \le \frac{2 + 2 a_2 + 2 a_3}{2 a_3} \le \frac{4 a_3}{2 a_3} = 2.
\]
We continue the proof assuming $a_1 = 2$ and $a_2 = a_3$.
In this case, by setting $a = a_2 = a_3$ and $b = a_4$, $X$ is a weighted hypersurface of degree $d$ in $\mathbb{P} (1, 2, a, a, b)$ and either $d = 2 b + 2$ or $d = 2 b + a$.
If $d = 2 b + 2$, then $b = 2 a$ but this is impossible since $X$ has only terminal singularities.
Hence $d = 2 b + a$.
In this case $b = a + 2$ and $d = 3 a + 4$.
By the quasi-smoothness of $X$, we see that $d = 3 a + 4$ is divisible by $a$.
This implies that $a \in \{2, 4\}$.
This is impossible since $X$ has only terminal singularities.
Therefore (7) is proved.
\end{proof}
\subsection{How to compute alpha invariants?}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
For the proof of Theorem \ref{mainthm}, it is necessary to show $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any point $\mathsf{p} \in X$.
Let $\mathsf{p} \in X$ be a point.
We briefly explain the most typical method of bounding $\alpha_{\mathsf{p}} (X)$ from below, which goes as follows.
\begin{enumerate}
\item Choose and fix a divisor $S$ on $X$ which vanishes at $\mathsf{p}$ to a relatively large (orbifold) multiplicity $m = \operatorname{omult}_{\mathsf{p}} (S) > 0$.
In some cases $S = H_x$ (when $\mathsf{p} \in H_x$) and in other cases $S$ is the quasi-tangent divisor of $X$ at $\mathsf{p}$.
Let $a$ be the positive integer such that $S \sim a A$.
\item Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{a} S$.
Then $D \cdot S$ is an effective $1$-cycle on $X$.
\item Find a $\mathbb{Q}$-divisor $T \in |e A|_{\mathbb{Q}}$ for some $e \in \mathbb{Z}_{> 0}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot S$.
We will find such a $\mathbb{Q}$-divisor $T$ by considering $\mathsf{p}$-isolating set or class which will be explained in \S \ref{sec:isol}.
\item Let $q = q_{\mathsf{p}}$ be the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ be the preimage of $\mathsf{p}$ via $q$.
By the above choices, $\operatorname{Supp} (q^*D) \cap \operatorname{Supp} (q^*S) \cap \operatorname{Supp} (q^*T)$ is a finite set of points including $\check{\mathsf{p}}$, and hence the local intersection number $(q^*D \cdot q^*S \cdot q^*T)_{\check{\mathsf{p}}}$ is defined (see \S \ref{sec:intnumber}).
Then, we have the inequalities
\[
m \operatorname{omult}_{\mathsf{p}} (D) \le (q_{\mathsf{p}}^*D \cdot q_{\mathsf{p}}^*S \cdot q_{\mathsf{p}}^*T)_{\check{\mathsf{p}}} \le r (D \cdot S \cdot T) = r a e (A^3),
\]
where $r$ is the index of the cyclic quotient singularity $\mathsf{p} \in X$. Note that $q$ is the identity morpphism and $r = 1$ when $\mathsf{p} \in X$ is a smooth point.
By Lemma \ref{lem:multlct} which will be explained below, we have
\[
\operatorname{lct}_{\mathsf{p}} (X;D) \ge \frac{r a e (A^3)}{m}.
\]
for any $D$ as in (2).
\item As a conclusion, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X; S), \ \frac{r a e (A^3)}{m} \right\}.
\]
\item It remains to bound $\operatorname{lct}_{\mathsf{p}} (X;S)$ from below.
This is easy when $S$ is quasi-smooth at $\mathsf{p}$ because in that case we have $\operatorname{lct}_{\mathsf{p}} (X;S) = 1$.
The computation gets involved when $S$ is the quasi-tangent divisor, but will be done by considering suitable weighted blowups which will be explained in \S \ref{sec:compwbl}.
\end{enumerate}
We need to consider variants of the above explained method, or other methods especially for points in special positions.
These will be explained in Chapter \ref{chap:methods}.
\chapter{Methods of computing log canonical thresholds}
\label{chap:methods}
\section{Auxiliary results}
\subsection{Some results on multiplicities and log canonicity}
\label{sec:intnumber}
Let $V$ be an $n$-dimensional variety.
For effective Cartier divisors $D_1, \dots, D_n$ on $V$ and a point $\mathsf{p} \in V$ which is an isolated component of $\operatorname{Supp} (D)_1 \cap \cdots \cap \operatorname{Supp} (D_n)$, the {\it intersection multiplicity}
\[
i (\mathsf{p}, D_1 \cdots D_n;V)
\]
is defined (see \cite[Example 7.1.10]{Fulton}).
Suppose that $V$ is $\mathbb{Q}$-factorial.
Then this definition is naturally generalized to effective $\mathbb{Q}$-divisors $D_1, \dots, D_n$ as follows:
\[
i (\mathsf{p}, D_1, \cdots, D_n;V) := \frac{1}{d^n} i (\mathsf{p}, d D_1, \cdots, d D_n;V),
\]
where $d$ is a positive integer such that $d D_i$ is a Cartier divisor for any $i$.
In this paper, we set
\[
(D_1 \cdots D_n)_{\mathsf{p}} := i (\mathsf{p}, D_1 \cdots D_n;V)
\]
and call it the {\it local intersection number} of $D_1, \dots, D_n$ at $\mathsf{p}$.
\begin{Rem}
If $\mathsf{p} \in V$ is a smooth point, $D_1, \dots, D_n$ are effective divisors defined by $f_1, \dots, f_n \in \mathcal{O}_{V, \mathsf{p}}$ around $\mathsf{p}$, and $\mathsf{p}$ is an isolated component of $\operatorname{Supp} (D_1) \cap \cdots \cap \operatorname{Supp} (D_n)$, then
\[
(D_1 \cdots D_n)_{\mathsf{p}} = \dim_{\mathbb{C}} \mathcal{O}_{V, P}/(f_1, \dots, f_n).
\]
If $X \subset \mathbb{P} (a_0, \dots, a_N)$ is an $n$-dimensional subvariety which is quasi-smooth at $\mathsf{p} = \mathsf{p}_{x_i} \in V$, $D_1 = (G_1 = 0)_X, \dots, D_n = (G_n = 0)_X$ are effective Weil divisors such that $\mathsf{p}$ is an isolated component of $D_1 \cap \cdots \cap D_n$, where $G_i = G_i (x_0, \dots, x_N)$ is a quasi-homogeneous polynomial of degree $d_i$, then
\[
(D_1 \cdots D_n)_{\mathsf{p}}
= \frac{1}{a_i} (\rho^*D_1 \cdots \rho^* D_n)_{\breve{\mathsf{p}}}
= \frac{1}{a_i} \dim_{\mathbb{C}} \mathcal{O}_{\breve{U}_{\mathsf{p}}, \breve{\mathsf{p}}}/(g_1, \dots, g_n),
\]
where $\rho = \rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}} := X \cap \mathcal{U}_{\mathsf{p}}$ is the orbifold chart with $\breve{\mathsf{p}} \in \breve{U}_{\mathsf{p}}$ the preimage of $\mathsf{p}$ and $g_i = G (\breve{x}_0, \dots, 1, \dots, \breve{x}_N)$ with $\breve{x}_j = x_j/x_i^{a_j/a_i}$ for $j \ne i$.
\end{Rem}
We will frequently use the following property of local intersection numbers.
Let $D_1, \dots, D_n$ be effective $\mathbb{Q}$-divisors on $X$ and $\mathsf{p} \in X$ be a smooth point.
If $\mathsf{p}$ is an isolated component of $\operatorname{Supp} (D_1) \cap \cdots \cap \operatorname{Supp} (D_n)$, then
\[
(D_1 \cdot \ldots \cdot D_n)_{\mathsf{p}} \ge \prod_{i=1}^n \operatorname{mult}_{\mathsf{p}} (D_i).
\]
We refer readers to \cite[Corollary 12.4]{Fulton} for a proof.
Although the following results are well known to experts, we include their proofs for readers' convenience.
\begin{Lem} \label{lem:multlct}
Let $\mathsf{p} \in X$ be either a germ of a smooth variety or a germ of a cyclic quotient singular point and let $D$ be an effective $\mathbb{Q}$-divisor on $X$.
Then the inequality
\[
\frac{1}{\operatorname{omult}_{\mathsf{p}} (D)} \le \operatorname{lct}_{\mathsf{p}} (X,D)
\]
holds.
\end{Lem}
\begin{proof}
Let $q = q_{\mathsf{p}} \colon \check{X} \to X$ be the quotient morphism of $\mathsf{p} \in X$, which is \'{e}tale in codimension $1$, and let $\check{\mathsf{p}} \in \check{X}$ be the preimage of $\mathsf{p}$.
By \cite[20.4 Corollary]{FA}, we have $\operatorname{lct}_{\mathsf{p}} (X;D) = \operatorname{lct}_{\check{\mathsf{p}}} (\check{X};q^*D)$.
Note that $\check{\mathsf{p}} \in \check{X}$ is smooth.
Hence, by \cite[8.10 Lemma]{Kol}, we have
\[
\frac{1}{\operatorname{omult}_{\mathsf{p}} (D)} = \frac{1}{\operatorname{mult}_{\check{\mathsf{p}}} (q^*D)} \le \operatorname{lct}_{\check{\mathsf{p}}} (\check{X};q^*D),
\]
and the proof is completed.
\end{proof}
\begin{Lem}[{$2n$-inequality, cf.\ \cite[Corollary 3.5]{Corti}}] \label{lem:2nineq}
Let $\mathsf{p} \in X$ be a germ of a smooth $3$-fold, $D$ an effective $\mathbb{Q}$-divisor on $X$, $n > 0$ a rational number and let $\varphi \colon Y \to X$ be the blowup of $X$ at $\mathsf{p}$ with exceptional divisor $E$.
If $(X, \frac{1}{n} D)$ is not canonical at $\mathsf{p}$, then there exists a line $L \subset E \cong \mathbb{P}^2$ with the following property.
\begin{itemize}
\item For any prime divisor $T$ on $X$ such that $T$ is smooth at $\mathsf{p}$ and that its proper transform $\tilde{T}$ contains $L$, we have $\operatorname{mult}_{\mathsf{p}} (D|_T) > 2 n$.
\end{itemize}
\end{Lem}
\begin{proof}
We set $m = \operatorname{mult}_{\mathsf{p}} (D)$.
By \cite[Corollary 3.5]{Corti}, one of the following holds.
\begin{enumerate}
\item $m > 2 n$.
\item There is a line $L \subset E$ such that the pair
\[
\left(Y, \left(\frac{m}{n} - 1\right)E + \frac{1}{n} \tilde{D} \right)
\]
is not log canonical at the generic point of $L$.
\end{enumerate}
Note that in \cite[Corollary 3.5]{Corti} the boundary is a movable linear system $\mathcal{H}$ but the same argument applies if we replace $\mathcal{H}$ by an effective $\mathbb{Q}$-divisor $D$.
We may assume $m \le 2n$ because otherwise $\operatorname{mult}_{\mathsf{p}} (D|_T) > 2 n$ for any prime divisor $T$ which is smooth at $\mathsf{p}$ and the assertion follows by choosing any line on $E$.
Thus the option (2) takes place.
Let $T$ be a prime divisor on $X$ such that $T$ is smooth at $\mathsf{p}$ and $\tilde{T} \supset L$.
We have
\[
K_Y + \left( \frac{m}{n} - 1 \right) E + \frac{1}{n} \tilde{D} + \tilde{T} = \varphi^* \left( K_X + \frac{1}{n} D + T \right).
\]
Note that $E|_{\tilde{T}} = L$ and we can write $\tilde{D}|_{\tilde{T}} = \alpha L + G$, where $\alpha \ge 0$ is a rational number and $G$ is an effective $\mathbb{Q}$-divisor on $\tilde{T}$.
Thus, by restricting the above equation to $\tilde{T}$, we have
\[
K_{\tilde{T}} + \left( \frac{m}{n} - 1 + \alpha \right) L + G = \varphi^* \left(K_T + \frac{1}{n} D|_T \right),
\]
and the pair
\[
\left( \tilde{T}, \left( \frac{m}{n} - 1 + \alpha \right) L + G \right)
\]
is not log canonical at the generic point of $L$.
This implies $\frac{m}{n} - 1 + \alpha > 1$ and we have
\[
\frac{1}{n} \operatorname{mult}_{\mathsf{p}} (D|_T) = \left( \frac{m}{n} - 1 + \alpha \right) + 1 > 2.
\]
Thus $\operatorname{mult}_{\mathsf{p}} (D|_T) > 2 n$ and the proof is completed.
\end{proof}
\begin{Lem} \label{lem:lctP2cubic}
Let $D \in \left| \mathcal{O}_{\mathbb{P}^2} (3) \right|$ be a divisor on $\mathbb{P}^2$ which is not a triple line.
Then $\operatorname{lct} (\mathbb{P}^2; D) \ge 1/2$.
\end{Lem}
\begin{proof}
We have the following possibilities for $D$.
\begin{enumerate}
\item $D$ is irreducible and reduced.
\item $D = Q + L$, where $Q$ is an irreducible conic and $L$ is a line.
\item $D = L_1 + 2 L_2$, where $L_1, L_2$ are distinct lines.
\item $D = L_1 + L_2 + L_3$, where $L_1, L_2, L_3$ are mutually distinct lines.
\end{enumerate}
If we are in one of the cases (1), (2) and (3), then $\operatorname{mult}_{\mathsf{p}} (D) \le 2$ for any point $\mathsf{p} \in D$ and thus $(\mathbb{P}^2, \frac{1}{2} D)$ is log canonical.
If we are in case (3), then it is obvious that the pair $(\mathbb{P}^2, \frac{1}{2} D) = (\mathbb{P}^2, \frac{1}{2} L_1 + L_2)$ is log canonical.
\end{proof}
\begin{Lem} \label{lem:singnoncanbd}
Let $X$ be a Fano $3$-fold of Picard number one and let $\mathsf{p} \in X$ be a cyclic quotient terminal singular point (which is not a smooth point).
If $\mathsf{p} \in X$ is not a maximal center, then there is at most one irreducible $\mathbb{Q}$-divisor $D \in \left| -K_X \right|_{\mathbb{Q}}$ such that $(X, D)$ is not canonical at $\mathsf{p}$.
\end{Lem}
\begin{proof}
Suppose that there are two distinct irreducible $\mathbb{Q}$-divisors $D_i \sim_{\mathbb{Q}} - K_X$ such that $(X, D_i)$ is not canonical at $\mathsf{p}$ for $i = 1, 2$.
Let $r > 1$ be the index of the singularity $\mathsf{p} \in X$ and let $\varphi \colon Y \to X$ be the Kawamata blowup at $\mathsf{p}$ with exceptional divisor $E$.
By \cite{Kawamata}, we have $\operatorname{ord}_E (D_i) > 1/r$.
Take a positive integer $n$ such that $n D_1, n D_2$ are both integral and $n D_1 \sim n D_2$.
Then the pencil $\mathcal{M} \sim - n K_X$ generated by $n D_1$ and $n D_2$ is a movable linear system and we have $\operatorname{ord}_E (\mathcal{M}) \ge n/r$.
It follows that the pair $(X, \frac{1}{n} \mathcal{M})$ is not canonical at $\mathsf{p}$.
This is a contradiction since $\mathsf{p} \in X$ is not a maximal center.
\end{proof}
\begin{Lem} \label{lem:qtangdivncan}
Let
\[
X = X_d \subset \mathbb{P} (1, b_1, b_2, b_3, b_4)_{x, y_1, y_2, y_3, y_4}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
Let $i \in \{1, 2, 3, 4\}$ be such that $b_i > 1$ and $\mathsf{p} := \mathsf{p}_{y_i} \in X$.
If $H_x$ is the quasi-tangent divisor of $X$ at $\mathsf{p}$, then the pair $(X, H_x)$ is not canonical at $\mathsf{p}$.
\end{Lem}
\begin{proof}
Note that the point $\mathsf{p} \in X$ is of type $\frac{1}{b_i} (b_j, b_k, b_l)$, where $\{i, j, k, l\} = \{1, 2, 3, 4\}$, and it is a terminal singularity.
Let $\varphi \colon Y \to X$ be the Kawamata blowup with exceptional divisor $E$.
Since $H_x$ is the quasi-tangent divisor of $X$ at $\mathsf{p}$, we have
\[
\operatorname{ord}_E (H_x) > \frac{1}{b_i}.
\]
Combining this with
\[
K_Y = \varphi^*K_X + \frac{1}{b_i} E,
\]
we see that the discrepancy of the pair $(X, H_x)$ along $E$ is negative.
This completes the proof.
\end{proof}
\subsection{Some results on singularities of weighted hypersurfaces}
\begin{Lem} \label{lem:normalqhyp}
Let $X$ be a quasi-smooth weighted hypersurface in $\mathbb{P} (b_0, \dots, b_4)$.
Assume that $\mathbb{P} (b_0, \dots, b_4)$ is well-formed and $X$ has at most isolated singularities.
Then any quasi-hyperplane section on $X$ is a normal surface.
\end{Lem}
\begin{proof}
Let $x_0, \dots, x_4$ be the homogeneous coordinates of $\mathbb{P} = \mathbb{P} (b_0, \dots, b_4)$ of degree $b_0, \dots, b_4$, respectively.
Let $F = F (x_0,\dots,x_4)$ be the defining polynomial of $X$ and let $S$ be a quasi-hyperplane section on $X$.
After replacing homogeneous coordinates, we may assume $S = (x_4 = 0)_X = (x_4 = F = 0) \subset \mathbb{P}$.
It is enough to show that the singular locus $\operatorname{Sing} (S)$ of $S$ is a finite set of points.
We write $F = x_4 G + \bar{F}$, where $G = G (x_0, \dots, x_4)$ and $\bar{F} = \bar{F} (x_0, \dots, x_3)$ are quasi-homogeneous polynomials.
We set $\bar{\mathbb{P}} = \mathbb{P} (b_0, \dots, b_3)$.
We claim that $\bar{\mathbb{P}}$ is well-formed.
Suppose it is not.
Then $\operatorname{Sing} (\mathbb{P})$ contains a $2$-dimensional stratum.
We have $\operatorname{Sing} (X) = \operatorname{Sing} (\mathbb{P}) \cap X$ since a quasi-smooth weighted hypersurface is well-formed (\cite[Theorem 6.17]{IF00}).
It follows that $\operatorname{Sing} (X)$ cannot be a finite set of points.
This is a contradiction and the claim is proved.
The surface $S$ is identified with the hypersurface $(\bar{F} = 0) \subset \bar{\mathbb{P}}$ and we have
\[
\operatorname{Sing} (S) = (S \setminus \mathrm{QI}Sm (S)) \cup (\operatorname{Sing} (\bar{\mathbb{P}}) \cap S).
\]
We claim that $\operatorname{Sing} (\bar{\mathbb{P}}) \cap S$ is a finite set of points.
Suppose not.
Then $\operatorname{Sing} (\bar{\mathbb{P}}) \cap S$ contains a curve and so does $\operatorname{Sing} (\mathbb{P}) \cap S$.
In particular $\operatorname{Sing} (X) = \operatorname{Sing} (\mathbb{P}) \cap X$ contains a curve.
This is impossible since $\operatorname{Sing} (X)$ is a finite set of points, and the claim is proved.
It remains to show that the closed subset $\Sigma := S \setminus \mathrm{QI}Sm (S) \subset \mathbb{P}$ is a finite set of points.
Let $\Pi \colon \mathbb{A}^5 \setminus \{o\} \to \mathbb{P}$ be the natural quotient morphism.
Then $\Sigma = \Pi (\operatorname{Sing} (C^*_S))$, where $C^*_S \subset \mathbb{A}^5 \setminus \{o\}$ is the punctured affine quasi-cone of $S$.
We have $\operatorname{Sing} (C_X) \cap C_S = \operatorname{Sing} (C_S) \cap (G = 0)$.
By the quasi-smoothness of $X$, we have $\operatorname{Sing} (C_X) = \{o\} \subset \mathbb{A}^5$.
This implies $\Sigma \cap (G = 0) = \emptyset$.
Since $(G = 0)$ is an ample divisor on $\mathbb{P}$, we see that $\Sigma$ is a finite set of points.
This completes the proof.
\end{proof}
\begin{Lem} \label{lem:qsminvhypsec}
Let $S$ be a weighted hypersurface in $\mathbb{P} (b_0, b_1, b_2, b_3)$ and let $T \subset \mathbb{P} (b_0, b_1, b_2, b_3)$ be a quasi-hyperplane.
If the scheme-theoretic intersection $S \cap T$ is quasi-smooth at a point $\mathsf{p}$, then $S$ is quasi-smooth at $\mathsf{p}$.
\end{Lem}
\begin{proof}
Let $x_0, x_1, x_2, x_3$ be the homogeneous coordinates of $\mathbb{P} = \mathbb{P} (b_0, b_1, b_2, b_3)$ of degree $b_0, b_1, b_2, b_3$, respectively, and let $F= F (x_0, x_1, x_2, x_3)$ be the defining polynomial of $S$.
We may assume $T = H_{x_3} \subset \mathbb{P}$ and we write $F = x_3 G + \bar{F}$, where $G = G (x_0, x_1, x_2, x_3)$ and $\bar{F} = \bar{F} (x_0, x_1, x_2)$ are quasi-homogeneous polynomials.
Then $S \cap T$ is the closed subscheme in $\mathbb{P} (b_0,b_1,b_2,b_3)$ defined by $x_3 = \bar{F} = 0$.
By the quasi-smoothness of $S \cap T$ at $\mathsf{p}$, there exists $i \in \{0, 1, 2\}$ such that
\[
\frac{\partial \bar{F}}{\partial x_i} (\mathsf{p}) \ne 0.
\]
It follows that
\[
\frac{\partial F}{\partial x_i} (\mathsf{p}) = \frac{\partial \bar{F}}{\partial x_i} (\mathsf{p}) \ne 0
\]
since $\mathsf{p} \in H_{x_3}$.
Thus $S$ is quasi-smooth at $\mathsf{p}$.
\end{proof}
\begin{Lem} \label{lem:pltsurfpair}
Let $S$ be a normal weighted hypersurface in a well formed weighted projective $3$-space $\mathbb{P} (b_0,\dots, b_3)$ and $T \subset \mathbb{P} (b_0,\dots, b_3)$ a quasi-hyperplane such that $T \ne S$.
Let $\Gamma$ be an irreducible component of $S \cap T$ and we assume that
\[
T|_S = \Gamma + \Delta,
\]
where $\Delta$ is an effective divisor on $S$ such that $\Gamma \not\subset \operatorname{Supp} (\Delta)$.
If $\Gamma$ is a smooth weighted complete intersection curve and $S$ is quasi-smooth at each point of $\Gamma \cap \operatorname{Supp} (\Delta)$, then $S$ is quasi-smooth along $\Gamma$ and the pair $(S, \Gamma)$ is plt along $\Gamma$.
\end{Lem}
\begin{proof}
We set $\Xi = \Gamma \cap \operatorname{Supp} (\Delta)$.
By \cite[Theorem 12.1]{IF00}, $\Gamma$ is quasi-smooth.
We have $(S \cap T) \setminus \Xi = \Gamma \setminus \Xi$.
It follows that $S \cap T$ is quasi-smooth along $\Gamma \setminus \Xi$.
By Lemma \ref{lem:qsminvhypsec}, $S$ is quasi-smooth along $\Gamma \setminus \Xi$.
Therefore $S$ is quasi-smooth along $\Gamma$.
For $i = 0,1,2,3$, let $S_i = (x_i \ne 0) \cap S$ be the standard open set of $S$ and let $\rho_i \colon \breve{S}_i \to S_i$ be the orbifold chart.
Note that $\rho_i$ is a finite surjective morphism of degree $b_i$ which is \'etale in codimension $1$.
By the quasi-smoothness of $S$, the affine varieties $\breve{S}_i$ and $\rho_i^*(\Gamma \cap S_i)$ are smooth.
Hence the pair $(\breve{S}_i, \rho_i^*(\Gamma \cap S_i))$ is plt along $\rho_i^*(\Gamma \cap S_i)$.
By \cite[Corollary 20.4]{FA}, the pair $(S_i, \Gamma \cap S_i)$ is plt along $\Gamma \cap S_i$.
This completes the proof.
\end{proof}
\begin{Rem} \label{rem:compselfint}
Let $S$, $T$ and $\Gamma$ be as in Lemma \ref{lem:pltsurfpair}.
We assume in addition that $\Gamma$ is rational, i.e.\ $\Gamma \cong \mathbb{P}^1$.
Let $\operatorname{Sing}_{\Gamma} (S) = \{\mathsf{p}_1,\dots,\mathsf{p}_n\}$ be the set of singular points of $S$ along $\Gamma$ and let $m_i$ be the index of the quotient singular point $\mathsf{p}_i \in S$.
Then, since the pair $(S, \Gamma)$ is plt along $\Gamma$, we can apply \cite[Proposition 16.6]{FA} and we have
\[
(K_S + \Gamma)|_{\Gamma} = K_{\Gamma} + \sum_{i=1}^n \frac{m_i-1}{m_i} \mathsf{p}_i.
\]
Thus we have
\[
(\Gamma^2)_S = - (K_S \cdot \Gamma)_S -2 + \sum_{i=1}^n \frac{m_i-1}{m_i}.
\]
\end{Rem}
\subsection{Isolating set and class}
\label{sec:isol}
We recall the definitions of isolating set and class which are introduced by Corti, Pukhlikov and Reid \cite{CPR00} as well as their basic properties.
Let $V$ be a normal projective variety embedded in a weighted projective space $\mathbb{P} = \mathbb{P} (a_0, \dots, a_N)$ with homogeneous coordinates $x_0, \dots, x_N$ with $\deg x_i = a_i$, and let $A$ be a Weil divisor on $V$ such that $\mathcal{O}_V (A) \cong \mathcal{O}_V (1)$.
We do not assume that $a_0 \le \cdots \le a_N$.
\begin{Def}
Let $\mathsf{p} \in V$ be a point.
We say that a set $\{g_1, \dots, g_m\}$ of quasi-homogeneous polynomials $g_1, \dots, g_m \in \mathbb{C} [x_0, \dots, x_N]$ {\it isolates} $\mathsf{p}$ or is a $\mathsf{p}$-{\it isolating set} if $\mathsf{p}$ is an isolated component of the set
\[
(g_1 = \cdots = g_m = 0) \cap V.
\]
\end{Def}
\begin{Def}
Let $\mathsf{p} \in V$ be a smooth point and let $L$ be a Weil divisor class on $V$.
For positive integers $k$ and $l$, we define $|\mathcal{I}_{\mathsf{p}}^k (l L)|$ to be the linear subsystem of $|l L|$ consisting of divisors vanishing at $\mathsf{p}$ with multiplicity at least $k$.
We say that $L$ {\it isolates} $\mathsf{p}$ or is a $\mathsf{p}$-{\it isolating class} if $\mathsf{p}$ is an isolated component of the base locus of $|\mathcal{I}^k_{\mathsf{p}} (kL)|$.
\end{Def}
\begin{Lem}[{\cite[Lemma 5.6.4]{CPR00}}]
Let $\mathsf{p} \in V$ be a smooth point.
If $\{g_1, \dots, g_m\}$ is a $\mathsf{p}$-isolating class, then $l A$ is a $\mathsf{p}$-isolating class, where
\[
l = \max \{\, \deg g_i \mid i = 1, 2, \dots, m \,\}.
\]
\end{Lem}
\begin{Lem} \label{lem:isolexT}
Let $\mathsf{p} \in V$ be a point, $Z_1, \dots, Z_k$ irreducible closed subsets of $V$ such that $\dim Z_i > 0$ for any $i$, and let $g_1, \dots, g_n \in \mathbb{C} [x_0, \dots, x_N]$ be quasi-homogeneous polynomials.
Suppose that $V$ is quasi-smooth at $\mathsf{p}$ and that $\{g_1, \dots, g_n\}$ isolates $\mathsf{p}$.
We set $G_i = (g_i = 0)_V$ and we set
\[
\mu := \min \left\{\, \frac{\operatorname{omult}_{\mathsf{p}} (G_i)}{\deg g_i} \; \middle| \; i = 1, \dots, n \,\right\}.
\]
Then there exists an effective $\mathbb{Q}$-divisor $T \sim_{\mathbb{Q}} A$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge \mu$ and $\operatorname{Supp} (T)$ does not contain any $Z_i$.
\end{Lem}
\begin{proof}
Let $d$ be the least common multiple of $\deg g_1, \dots, \deg g_n$ and we set $e_i = d/\deg g_i$.
Consider the linear system $\Lambda \subset |d A|$ on $V$ generated by $g_1^{e_1}, \dots, g_n^{e_n}$.
We see that $\mathsf{p}$ is an isolating component of $\operatorname{Bs} \Lambda$ since $\{g_1, \dots, g_n\}$ isolates $\mathsf{p}$.
Hence a general $D \in \Lambda$ does not contain any $Z_i$ in its support.
Moreover, for any $D \in \Lambda$, we have
\[
\operatorname{omult}_{\mathsf{p}} (D) \ge \min \{ \, e_i \operatorname{omult}_{\mathsf{p}} (D_i) \mid i = 1, \dots, n \,\} = d \mu.
\]
Thus the assertion follows by setting $T = \frac{1}{d} D \sim_{\mathbb{Q}} A$ for a general $D \in \Lambda$.
\end{proof}
\begin{Rem} \label{rem:isolT}
Lemma \ref{lem:isolexT} will be frequently applied in the following way:
under the same notation and assumptions as in Lemma \ref{lem:isolexT}, there exists an effective $\mathbb{Q}$-divisor $T \sim_{\mathbb{Q}} e A$, where
\[
e = \max \{ \, \deg g_i \mid i = 1, \dots, n \,\},
\]
such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any $Z_i$.
\end{Rem}
\begin{Lem} \label{lem:isolclass}
Let $X = X_d \subset \mathbb{P} (1, a_1, \dots, a_4)_{x, y, z, t, w}$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$, where we assume that $a_1 \le a_2 \le a_3 \le a_4$, and let $\mathsf{p} \in H_x \setminus L_{xy}$.
Then $a_1 a_4 A$ isolates $\mathsf{p}$.
If $w^k$ appears in the defining polynomial of $X$ with nonzero coefficient, then $a_1 a_3 A$ isolates $\mathsf{p}$.
\end{Lem}
\begin{proof}
We can write $\mathsf{p} = (0\!:\!1\!:\!\alpha_2\!:\!\alpha_3\!:\!\alpha_4)$ for some $\alpha_2, \alpha_3, \alpha_4 \in \mathbb{C}$.
Then it is to see that the set
\[
\{ x, z^{a_1} - \alpha_2^{a_1} y^{a_2}, t^{a_1} - \alpha_3^{a_3} y^{a_3}, w^{a_1} - \alpha_4^{a_4} y^{a_4} \}
\]
the isolates $\mathsf{p}$, and thus $a_1 a_4 A$ isolates $\mathsf{p}$.
Suppose that $w^k$ appears in the defining polynomial of $X$.
Then the natural projection $\mathbb{P} (1, a_1, \dots, a_4) \dashrightarrow \mathbb{P} (1, a_1, a_2, a_3)$ restricts to a finite morphism $\pi \colon X \to \mathbb{P} (1, a_1, a_2, a_3)$.
The common zero locus (in $X$) of the sections contained in the set
\[
\{ x, z^{a_1} - \alpha_2^{a_1} y^{a_2}, t^{a_1} - \alpha_3^{a_3} y^{a_3} \}
\]
coincides with the set $\pi^{-1} (\mathsf{q})$, where $\mathsf{q} = (0\!:\!1\!:\!\alpha_2\!:\!\alpha_3) \in \mathbb{P} (1, a_1, a_2, a_3)$.
It follows that the above set isolates $\mathsf{p}$ since $\pi^{-1} (\mathsf{q})$ is a finite set containing $\mathsf{p}$.
Thus $a_1 a_3 A$ isolates $\mathsf{p}$.
\end{proof}
\section{Methods}
\subsection{Computations by intersecting two divisors}
We recall methods of computing LCTs and consider their generalizations for some of them.
\begin{Lem}[{cf.\ \cite[Lemma 2.5]{KOW18}}] \label{lem:exclL}
Let $X$ be a normal projective $\mathbb{Q}$-factorial $3$-fold with nef and big anticanonical divisor, and let $\mathsf{p} \in X$ be either a smooth point or a terminal quotient singular point of index $r$ (Below we set $r = 1$ when $\mathsf{p} \in X$ is a smooth point).
Suppose that there are prime divisors $S \sim_{\mathbb{Q}} - a K_X$ and $T \sim_{\mathbb{Q}} - b K_X$ with $a, b \in \mathbb{Q}$ such that $S \cap T$ is irreducible and $q^* S \cdot q^* T = m \check{\Gamma}$, where $q = q_{\mathsf{p}} \colon \check{U} \to U$ is the quotient morphism of an analytic neighborhood $\mathsf{p} \in U$ of $\mathsf{p} \in X$, $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q$, $m$ is a positive integer and $\check{\Gamma}$ is an irreducible and reduced curve on $\check{U}$.
Then, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X; \tfrac{1}{a} S), \ \frac{b}{m \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma})}, \ \frac{1}{r a b (-K_X)^3} \right\}.
\]
\end{Lem}
\begin{proof}
We set
\[
c := \min \left\{ \operatorname{lct}_{\mathsf{p}} (X; \tfrac{1}{a} S), \ \frac{b}{m \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma})}, \ \frac{1}{r a b (-K_X)^3} \right\}.
\]
We will derive a contradiction assuming $\alpha_{\mathsf{p}} (X) < c$.
By the assumption, there is an irreducible $\mathbb{Q}$-divisor $D \in \left| - K_X \right|_{\mathbb{Q}}$ such that $(X, c D)$ is not log canonical at $\mathsf{p}$.
Then the pair $(\check{U}, c \rho^*D)$ is not log canonical at $\check{\mathsf{p}}$ and we have
\begin{equation} \label{eq:exclL-1}
\operatorname{mult}_{\check{\mathsf{p}}} (q^*D) > \frac{1}{c}.
\end{equation}
Since $q^*S \cdot q^*T = m \check{\Gamma}$ and $S \cap T$ is irreducible, we have $S \cdot T = m \Gamma$, where $\Gamma$ is an irreducible and reduced curve such that $\check{\Gamma} = q^*\Gamma$.
We have
\begin{equation} \label{eq:exclL-2}
(-K_X \cdot \Gamma) = \frac{1}{m} (-K_X \cdot S \cdot T) = \frac{a b (-K_X)^3}{m}.
\end{equation}
This in particular implies
\begin{equation} \label{eq:exclL-3}
(T \cdot \Gamma) = b (-K_X \cdot \Gamma) = \frac{a b^2 (-K_X)^3}{m}.
\end{equation}
We have $\operatorname{Supp} (D) \ne S$ since $\operatorname{lct}_{\mathsf{p}} (X;S) \ge c$, and thus $q^*D \cdot q^*S$ is an effective $1$-cycle on $\check{U}$.
We write $q^*D \cdot q^*S = \gamma \check{\Gamma} + \check{\Delta}$, where $\gamma \ge 0$ and $\check{\Delta}$ is an effective $1$-cycle on $\check{U}$ such that $\check{\Gamma} \not\subset \operatorname{Supp} (\check{\Delta})$.
Then $D \cdot S = \gamma \Gamma + \Delta + \Xi$, where $\Delta = \frac{1}{r} q_*\check{\Delta}$ and $\Xi$ is an effective $1$-cycle such that $\Gamma \not\subset \operatorname{Supp} (\Xi)$.
By \eqref{eq:exclL-2}, we have
\[
a (-K_X)^3 = (-K_X \cdot D \cdot S) \ge \gamma (-K_X \cdot \Gamma) = \frac{ab (-K_X)^3}{m} \gamma,
\]
where the inequality follows since $-K_X$ is nef.
Note that $(-K_X)^3 > 0$ since $-K_X$ is nef and big.
Hence
\begin{equation} \label{eq:exclL-4}
\gamma \le \frac{m}{b}.
\end{equation}
By \eqref{eq:exclL-1} and \eqref{eq:exclL-3}, we have
\[
\begin{split}
r \left(a b (-K_X)^3 - \frac{a b^2 (-K_X)^3}{m} \gamma\right)
&= r (T \cdot (D \cdot S - \gamma \Gamma)) \\
&= r (T \cdot (\Delta + \Xi)) \ge r (T \cdot \Delta) \\
&\ge r (T \cdot \Delta)_{\mathsf{p}} = (q^*T \cdot \check{\Delta})_{\check{\mathsf{p}}} \\
&\ge \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Delta}) \\
&> \frac{1}{c} - \gamma \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma}),
\end{split}
\]
where $(\ \cdot \ )_{\mathsf{p}}$ and $(\ \cdot \ )_{\check{\mathsf{p}}}$ denote the local intersection numbers at $\mathsf{p}$ and $\check{\mathsf{p}}$ respectively.
It follows that
\begin{equation} \label{eq:exclL-5}
\left(\operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma}) - \frac{r a b^2 (-K_X)^3}{m}\right) \gamma > \frac{1}{c} - r a b (-K_X)^3.
\end{equation}
We have $\operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma}) - r a b^2 (-K_X)^3/m > 0$ since $1/c - r a b (-K_X)^3 \ge 0$ by the definition of $c$.
Combining \eqref{eq:exclL-4} and \eqref{eq:exclL-5}, we have
\[
c > \frac{b}{m \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma})}.
\]
This contradicts the definition of $c$ and the proof is completed.
\end{proof}
Lemma \ref{lem:exclL} is very useful in computing alpha invariants but works only when $S \cap T$ is irreducible.
We consider its generalization that can be applied when $S \cap T$ is reducible.
\begin{Def}
Let $M = (a_{ij})$ be an $n \times n$ matrix with entries in $\mathbb{R}$, where $n \ge 2$.
For a non-empty subset $I \subset \{1,2,\dots,n\}$, we denote by $M_I$ the submatrix of $M$ consisting of $i$th rows and columns for $i \in I$.
We say that $M$ satisfies the {\it condition $(\star)$} if the following are satisfied.
\begin{itemize}
\item $(-1)^{|I|} \det M_I \ge 0$ for any non-empty proper subset $I \subset \{1,2,\dots,n\}$.
\item $(-1)^{n-1} \det M > 0$.
\item $a_{ij} > 0$ for any $i, j$ with $i \ne j$.
\end{itemize}
For $v = {}^t (v_1,\dots,v_n), w = {}^t (w_1,\dots,w_n) \in \mathbb{R}^n$, the expression $v \le w$ means $v_i \le w_i$ for any $i$.
\end{Def}
\begin{Lem} \label{lem:matrix}
Let $M = (a_{i j})$ be an $n \times n$ matrix with entries in $\mathbb{R}$ satisfying the condition $(\star)$, and let $v, w \in \mathbb{R}^n$.
Then $M v \le M w$ implies $v \le w$.
\end{Lem}
\begin{proof}
It is enough to show that $v \le 0$ assuming $M v \le 0$ for $v \in \mathbb{R}^n$.
We prove this assertion by induction on $n \ge 2$.
The case $n = 2$ is easily done and we omit it.
Assume $n \ge 3$.
Suppose that there is a diagonal entry $a_{kk}$ such that $a_{kk} = 0$.
Then, we have $\det M_{\{k,l\}} < 0$ since $a_{k l}, a_{l k} > 0$.
By the condition $(\star)$, this is impossible since $n \ge 3$.
In the following we may assume that $a_{ii} \ne 0$ for any $i$.
By the condition $(\star)$, we have $a_{ii} = \det M_{\{i\}} \le 0$ and hence $a_{ii} < 0$ for any $i$.
Let $M'$ be the matrix obtained by adding the $1$st row multiplied by the positive integer $-a_{i1}/a_{11}$ to the $i$th row, for $i = 2,\dots,n$.
Then we obtain the inequality $M' v \le 0$ and we can write
\[
M' =
\arraycolsep5pt
\left(
\begin{array}{@{\,}cccc@{\,}}
a_{11} & a_{12} & \cdots &a_{1n}\\
0&&&\\
0&\operatorname{mult}icolumn{3}{c}{\raisebox{-5pt}[0pt][0pt]{\Huge $N'$}}\\
0&&&\\
\end{array}
\right),
\]
where $N'$ is an $(n-1) \times (n-1)$ matrix.
It is straightforward to check that $N$ satisfies the condition $(\star)$.
Since $N' {}^t (v_2 \ \dots \ v_n) \le 0$, we have $v_2, \dots, v_n \le 0$ by induction hypothesis.
Next, let $M''$ be the matrix obtained by adding the $n$th row multiplied by the positive integer $- a_{i n}/a_{nn}$ to the $i$th row, for $i = 1, 2, \dots, n-1$.
Then we have $M'' v \le 0$ and, by repeating the similar argument as above, we conclude $v_1, \dots, v_{n-1} \le 0$ by inuction.
This completes the proof.
\end{proof}
\begin{Def}
Let $S$ be a normal projective surface and let $\Gamma_1, \dots, \Gamma_k$ be irreducible and reduced curves on $S$.
Then the $k \times k$ matrix
\[
M (\Gamma_1, \dots, \Gamma_k) = ((\Gamma_i \cdot \Gamma_j)_S)_{1 \le i, j \le k}
\]
is called the {\it intersection matrix} of curves $\Gamma_1, \dots, \Gamma_k$ on $S$.
\end{Def}
\begin{Lem} \label{lem:mtdLred}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
Let $S \in \left| - a K_X \right|$ be a normal surface on $X$, $T \in \left| - b K_X \right|$ an effective divisor and $\mathsf{p} \in S$ a point, where $a, b > 0$.
We set $r = 1$ when $\mathsf{p} \in X$ is a smooth point, and otherwise we denote by $r$ the index of the cyclic quotient singularity $\mathsf{p} \in X$.
Suppose that
\[
T|_S = m_1 \Gamma_1 + m_2 \Gamma_2 + \cdots + m_k \Gamma_k,
\]
where $\Gamma_1, \dots, \Gamma_k$ are distinct irreducible and reduced curves on $S$ and $m_1, \dots, m_k$ are positive integers, and the following properties are satisfied.
\begin{itemize}
\item $r b \deg \Gamma_1 \le m_1$.
\item $\mathsf{p} \in \Gamma_1 \setminus (\cup_{i \ge 2} \Gamma_i)$, and $S, \Gamma_1$ are both quasi-smooth at $\mathsf{p}$.
\item The intersection matrix $M (\Gamma_1, \dots, \Gamma_k)$ satisfies the condition $(\star)$.
\end{itemize}
Then we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ a, \ \frac{m_1}{r a b (-K_X)^3 + \frac{m_1^2}{b} - r m_1 \deg \Gamma_1} \right\}.
\]
\end{Lem}
\begin{proof}
Let $D \in \left| - K_X \right|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor.
If $\operatorname{Supp} (D) = S$, then $D = \frac{1}{a} S$ and we have $\operatorname{lct}_{\mathsf{p}} (X, D) \ge a$ since $S$ is quasi-smooth at $\mathsf{p}$.
We assume $\operatorname{Supp} (D) \ne S$.
It is enough to prove the inequality
\begin{equation}
\label{eq:mtdLred1}
\operatorname{lct}_{\mathsf{p}} (X; D) \ge \frac{m_1}{r a b (-K_X)^3 + \frac{m_1^2}{b} - r m_1 \deg \Gamma_1}.
\end{equation}
We can write
\[
D|_S = \gamma_1 \Gamma_1 + \cdots + \gamma_k \Gamma_k + \Delta,
\]
where $\gamma_1, \dots, \gamma_k \ge 0$ and $\Delta$ is an effective $\mathbb{Q}$-divisor on $S$ such that $\Gamma_i \not\subset \operatorname{Supp} (\Delta)$ for $i = 1, \dots, k$.
We set $\sigma_i = (\Gamma_i^2)_S$ and $\chi_{i,j} = (\Gamma_i \cdot \Gamma_j)_S$.
For $i = 1, \dots, k$, we have
\begin{equation}
\label{eq:mtdLred2}
\begin{split}
b \deg \Gamma_i &= (T|_S \cdot \Gamma_i)_S \\
&= m_1 \chi_{1,i} + \cdots + m_{i-1} \chi_{i - 1, i} + m_i \sigma_i + m_{i+1} \chi_{i+1, i} + \cdots + m_k \chi_{k,i},
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:mtdLred3}
\begin{split}
\deg \Gamma_i &= (D|_S \cdot \Gamma_i)_S \\
&\ge \gamma_1 \chi_{1, i} + \cdots + \gamma_{i-1} \chi_{i-1} + \gamma_i \sigma_i + \gamma_{i+1} \chi_{i+1,i} + \cdots + \gamma_k \chi_{k,i+1}.
\end{split}
\end{equation}
We set $M = M (\Gamma_1, \dots, \Gamma_k)$.
Combining inequalities \eqref{eq:mtdLred2} and \eqref{eq:mtdLred3}, we have
\[
M \begin{pmatrix} b \gamma_1 \\ \vdots \\ b \gamma_k \end{pmatrix} \le M \begin{pmatrix} m_1 \\ \vdots \\ m_k \end{pmatrix}.
\]
By Lemma \ref{lem:matrix}, this implies $\gamma_i \le m_i/b$ for any $i$.
When $\mathsf{p}$ is a singular point, then we set $\rho = \rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$, which is the orbifold chart of $X$ containing $\mathsf{p}$.
When $\mathsf{p}$ is a smooth point of $X$, then we set $\breve{U} = U = X$ and $\rho \colon \breve{U} \to U$ is assumed to be the identity morphism.
Moreover we set $\breve{S} := q^*(S \cap U)$ and $\rho_S = \rho|_{\breve{S}} \colon \breve{S} \to S \cap U$.
We see that $\breve{S}$ is smooth at the preimage $\breve{\mathsf{p}}$ of $\mathsf{p}$ since $S$ is quasi-smooth at $\mathsf{p}$, and
\[
\rho^*D|_{\breve{S}} = \gamma_1 \rho_S^*\Gamma_1 + \cdots \gamma_k \rho_S^*\Gamma_k + \rho_S^*\Delta.
\]
This implies
\[
\operatorname{mult}_{\breve{\mathsf{p}}} (\rho_S^*\Delta) \ge \operatorname{omult}_{\mathsf{p}} (D) - \gamma_1
\]
since $\rho_S^*\Gamma_i$ does not pass through $\breve{\mathsf{p}}$ for $i \ge 2$ and $\rho_S^*\Gamma_1$ is smooth at $\breve{\mathsf{p}}$ by the quasi-smoothness of $\Gamma_1$ at $\mathsf{p}$.
We have
\[
\begin{split}
r (ab (-K_X)^3 - b \gamma_1 \deg \Gamma_1) &\ge
r (T|_S \cdot (D|_S - \gamma_1 \Gamma_1 - \cdots - \gamma_k \Gamma_k))_S \\
&= r (T|_S \cdot \Delta)_S \\
&\ge m_1 r (\Gamma_1 \cdot \Delta)_S \\
&\ge m_1 (\rho_S^*\Gamma_1 \cdot \rho_S^*\Delta)_{\breve{\mathsf{p}}} \\
&\ge m_1 \operatorname{mult}_{\breve{\mathsf{p}}} (\rho_S^*\Delta) \\
&\ge m_1 (\operatorname{omult}_{\mathsf{p}} (D) - \gamma_1).
\end{split}
\]
Since $m_1 - r b \deg \Gamma_1 \ge 0$ and $\gamma_1 \le m_1/b$, we have
\[
\begin{split}
\operatorname{omult}_{\mathsf{p}} (D) &\le \frac{1}{m_1} (r a b (-K_X)^3 + (m_1 - r b \deg \Gamma_1) \gamma_1) \\
& \le \frac{1}{m_1} (r a b (-K_X)^3 + \frac{m_1^2}{b}- r m_1 \deg \Gamma_1).
\end{split}.
\]
This implies \eqref{eq:mtdLred1} and the proof is completed.
\end{proof}
The following is a version of Lemma \ref{lem:mtdLred}, which may be effective when $S$ is singular at $\mathsf{p}$.
\begin{Lem} \label{lem:mtdLredSsing}
Let $X$ be a normal projective $\mathbb{Q}$-factorial $3$-fold.
Let $S \sim_{\mathbb{Q}} - a K_X$ be a normal surface on $X$, $T \sim_{\mathbb{Q}} - b K_X$ an effective divisor and $\mathsf{p} \in S$ a point, where $a, b$ are positive rational numbers.
Suppose that
\[
T|_S = m_1 \Gamma_1 + m_2 \Gamma_2 + \cdots + m_k \Gamma_k,
\]
where $\Gamma_1, \dots, \Gamma_k$ are distinct irreducible and reduced curves on $S$ and $m_1, \dots, m_k$ are positive integers, and the following properties are satisfied.
\begin{itemize}
\item $b \deg \Gamma_1 \le \operatorname{mult}_{\mathsf{p}} (\Gamma_1)$.
\item $\mathsf{p} \in \Gamma_1 \setminus (\cup_{i \ge 2} \Gamma_i)$, and $X$ is smooth at $\mathsf{p}$.
\item The intersection matrix $M (\Gamma_1, \dots, \Gamma_k)$ satisfies the condition $(\star)$.
\end{itemize}
Then we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \frac{a}{\operatorname{mult}_{\mathsf{p}} (S)}, \ \frac{\operatorname{mult}_{\mathsf{p}} (S)}{a b (-K_X)^3 + \frac{m_1}{b} \operatorname{mult}_{\mathsf{p}} (\Gamma_1) - m_1 \deg \Gamma_1} \right\}.
\]
\end{Lem}
\begin{proof}
Let $D \in \left| - K_X \right|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor.
If $\operatorname{Supp} (D) = S$, then $D = \frac{1}{a} S$ and we have $\operatorname{lct}_{\mathsf{p}} (X, D) \ge a/\operatorname{mult}_{\mathsf{p}} (S)$.
We assume $\operatorname{Supp} (D) \ne S$.
It is enough to show that
\begin{equation}
\label{eq:mtdLredSsing1}
\operatorname{lct}_{\mathsf{p}} (X; D) \ge \frac{\operatorname{mult}_{\mathsf{p}} (S)}{a b (-K_X)^3 + \frac{m_1}{b} \operatorname{mult}_{\mathsf{p}} (\Gamma_1) - m_1 \deg \Gamma_1}.
\end{equation}
We write
\[
D|_S = \gamma_1 \Gamma_1 + \cdots + \gamma_k \Gamma_k + \Delta,
\]
where $\gamma_1, \dots, \gamma_k \ge 0$ and $\Delta$ is an effective divisor on $S$ such that $\Gamma_i \not\subset \operatorname{Supp} (\Delta)$ for $i = 1, \dots, k$.
By the same argument as in the proof of Lemma \ref{lem:mtdLred}, we have $\gamma_i \le m_i/b$ for any $i$.
We consider the $1$-cycle $D \cdot S = \gamma_1 \Gamma_1 + \cdots + \gamma_k \Gamma_k$ on $X$ and we have
\[
\begin{split}
ab (-K_X)^3 - b \gamma_1 \deg \Gamma_1 &\ge
(T \cdot (D \cdot S - \gamma_1 \Gamma_1 - \cdots - \gamma_k \Gamma_k))_X \\
& = (T \cdot \Delta)_X \\
&\ge \operatorname{mult}_{\mathsf{p}} (\Delta) \\
&\ge (\operatorname{mult}_{\mathsf{p}} (S)) (\operatorname{mult}_{\mathsf{p}} (D)) - \gamma_1 \operatorname{mult}_{\mathsf{p}} (\Gamma_1).
\end{split}
\]
Since $\operatorname{mult}_{\mathsf{p}} (\Gamma_1) - b \deg \Gamma_1 \ge 0$ and $\gamma_1 \le m_1/b$, we have
\[
\begin{split}
\operatorname{mult}_{\mathsf{p}} (D) &\le \frac{1}{\operatorname{mult}_{\mathsf{p}} (S)} (ab (-K_X)^3 + (\operatorname{mult}_{\mathsf{p}} (\Gamma_1) - b \deg \Gamma_1) \gamma_1) \\
& \le \frac{1}{\operatorname{mult}_{\mathsf{p}} (S)} \left( ab (-K_X)^3 + \frac{m_1}{b} \operatorname{mult}_{\mathsf{p}} (\Gamma_1) - m_1 \deg \Gamma_1 \right).
\end{split}.
\]
This implies \eqref{eq:mtdLredSsing1} and the proof is completed.
\end{proof}
\begin{Lem} \label{lem:mtdLintpt}
Let $X$ be a normal projective $\mathbb{Q}$-factorial $3$-fold.
Let $S \sim_{\mathbb{Q}} - a K_X$ be a normal surface on $X$, $T \sim_{\mathbb{Q}} - b K_X$ an effective divisor and $\mathsf{p} \in X$ a point, where $a, b$ be positive rational numbers.
Suppose that
\[
T|_S = \Gamma_1 + \Gamma_2,
\]
where $\Gamma_1, \Gamma_2$ are distinct irreducible and reduced curves on $S$, and the following properties are satisfied.
\begin{itemize}
\item $\deg \Gamma_i \le 2/b$ for $i = 1, 2$.
\item $\mathsf{p} \in \Gamma_1 \cap \Gamma_2$ and all the $X$, $S$, $\Gamma_1$ and $\Gamma_2$ are smooth at $\mathsf{p}$.
\item The intersection matrix $M (\Gamma_1, \Gamma_2)$ satisfies the condition $(\star)$.
\end{itemize}
Then we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ a, \ \frac{b}{2} \right\}.
\]
\end{Lem}
\begin{proof}
We have $\operatorname{lct}_{\mathsf{p}} (X, \frac{1}{a} S) \ge a$ since $S$ is smooth at $\mathsf{p}$ by assumption.
Let $D \in \left| - K_X \right|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ such that $\operatorname{Supp} (D) \ne S$.
It is enough to prove the inequality $\operatorname{lct}_{\mathsf{p}} (X;D) \ge b/2$.
We write
\[
D|_S = \gamma_1 \Gamma_1 + \gamma_2 \Gamma_2 + \Delta,
\]
where $\gamma_1, \gamma_2 \ge 0$ and $\Delta$ is an effective divisor on $S$ with $\Gamma_1, \Gamma_2 \not\subset \operatorname{Supp} (\Delta)$.
By the proof of Lemma \ref{lem:mtdLred}, we have $\gamma_1, \gamma_2 \le 1/b$.
We have
\begin{equation} \label{eq:lem:methodLintpt1}
a b (-K_X)^3 = (-K_X|_S \cdot T|_S)_S = \deg \Gamma_1 + \deg \Gamma_2.
\end{equation}
Since $\operatorname{mult}_{\mathsf{p}} (T|_S) = 2$ and $\operatorname{mult}_{\mathsf{p}} (\Delta) \ge \operatorname{mult}_{\mathsf{p}} (D) - \gamma_1 - \gamma_2$, we have
\[
\begin{split}
ab (-K_X)^3 - b \gamma_1 \deg \Gamma_1 - b \gamma_2 \deg \Gamma_2 &=
(T|_S \cdot (D|_S - \gamma_1 \Gamma_1 - \gamma_2 \Gamma_2))_S \\
&= (T|_S \cdot \Delta)_S \\
& \ge 2 (\operatorname{mult}_{\mathsf{p}} (D) - \gamma_1 - \gamma_2).
\end{split}
\]
By \eqref{eq:lem:methodLintpt1}, the assumption $\deg \Gamma_1, \deg \Gamma_2 \le 2/b$ and $\gamma_1, \gamma_2 \le 1/b$, we have
\[
\begin{split}
\operatorname{mult}_{\mathsf{p}} (D) &\le\frac{1}{2} (ab (-K_X)^3 + (2 - b \deg \Gamma_1) \gamma_1 + (2- b \deg \Gamma_2) \gamma_2) \\
& \le \frac{2}{b}.
\end{split}
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X; D) \ge b/2$ and thus $\alpha_{\mathsf{p}} (X) \ge \min \{a, b/2\}$.
\end{proof}
\subsection{Computations by weighted blowups}
\label{sec:compwbl}
We explain methods of computing LCTs via suitable weighted blowups.
Let $\mathsf{p} \in X$ be a germ of a smooth variety of dimension $n$ with a system of local coordinates $\{x_1, \dots, x_n\}$ at $\mathsf{p}$, and let $D$ be an effective $\mathbb{Q}$-divisor on $X$.
Let $\varphi \colon Y \to X$ be the weighted blowup at $\mathsf{p}$ with weight $\operatorname{wt} (x_1, \dots, x_n) = (c_1, \dots, c_n)$, where $\underline{c} = (c_1, \dots, c_n)$ is a tuple of positive integers such that $\gcd \{c_1, \dots, c_n\} = 1$.
Let $E \cong \mathbb{P} (\underline{c}) = \mathbb{P} (c_1, \dots, c_n)$ be the exceptional divisor of $\varphi$.
Note that $Y$ can be singular along a divisor on $E$ (see Remark \ref{rem:wbldiff} below) so that we cannot expect the usual adjunction $(K_Y + E)|_E = K_E$.
In general we need a correction term and we have
\[
(K_Y + E)|_E = K_E + \operatorname{Diff}
\]
where the correction term $\operatorname{Diff}$ is a $\mathbb{Q}$-divisor on $E$ which is called the {\it different} (see \cite[Chapter 16]{FA}).
\begin{Rem} \label{rem:wbldiff}
We give a concrete description of $\operatorname{Diff}$.
Let $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ be the well-formed model of $\mathbb{P} (\underline{c})$ and we identify $E$ with $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
For $i = 0, 1, \dots, n$, let
\[
H^{\operatorname{wf}}_i = (\tilde{x}_i = 0) \subset E \cong \mathbb{P} (\underline{c})^{\operatorname{wf}}
\]
be the quasi-hyperplane of $\mathbb{P} (\underline{c})^{\operatorname{wf}}_{\tilde{x}_1, \dots, \tilde{x}_n}$, and we set $m_i = \gcd \{c_0, \dots, \hat{c}_i, \dots, c_n\}$.
We see that $Y$ is singular at the generic point of $H^{\operatorname{wf}}_i$ if and only if $m_i > 1$, and if this is the case, then the singularity of $Y$ along $H^{\operatorname{wf}}_i$ is a cyclic quotient singularity of index $m_i$.
It follows from \cite[Proposition 16.6]{FA} that
\[
\operatorname{Diff} = \sum_{i=1}^n \frac{m_i-1}{m_i} H^{\operatorname{wf}}_i
\]
under the identification $E \cong \mathbb{P} (\underline{c})^{\operatorname{wf}}$.
\end{Rem}
\begin{Lem} \label{lem:lctwbl}
Let the notation and assumption as above.
Then we have
\begin{equation} \label{eq:lctwbl-1}
\operatorname{lct}_{\mathsf{p}} (X;D) \ge \min \left\{ \frac{c_1 + \cdots + c_n}{\operatorname{ord}_E (D)}, \ \operatorname{lct} (E, \operatorname{Diff} ; \tilde{D}|_E) \right\},
\end{equation}
where $\tilde{D}$ is the proper transform of $D$.
If in addition the inequality
\begin{equation} \label{eq:lctwbl-2}
\frac{c_1 + \cdots + c_n}{\operatorname{ord}_E (D)} \le \operatorname{lct} (E, \operatorname{Diff}_E; \tilde{D}|_E)
\end{equation}
holds, then we have
\begin{equation} \label{eq:lctwbl-3}
\operatorname{lct}_{\mathsf{p}} (X; D) = \frac{c_1 + \cdots + c_n}{\operatorname{ord}_E (D)}.
\end{equation}
\end{Lem}
\begin{proof}
We set $c = c_1 + \cdots + c_n$ and let $\lambda$ be any rational number such that
\[
0 < \lambda \le \min \left\{ \frac{c}{\operatorname{ord}_E (D)}, \ \operatorname{lct} (E; \tilde{D}|_E) \right\}.
\]
We will show that the pair $(X, \lambda D)$ is log canonical at $\mathsf{p}$, which will prove the inequality \eqref{eq:lctwbl-1}.
We assume that the pair $(X, \lambda D)$ is not log canonical at $\mathsf{p}$.
We have
\begin{equation} \label{eq:lctwbl-4}
K_Y + \lambda \tilde{D} + (\lambda \operatorname{ord}_E (D) - c + 1) E = \varphi^* (K_X + \lambda D),
\end{equation}
and the pair $(Y, \lambda \tilde{D} + (\lambda \operatorname{ord}_E (D) - c + 1)E)$ is not log canonical along $E$.
Since $\lambda \le c/\operatorname{ord}_E (D)$, we have
\[
\lambda \operatorname{ord}_E (D) - c + 1 \le 1,
\]
which implies that the pair $(Y, \lambda \tilde{D} + E)$ is not log canonical along $E$.
Thus the pair $(E, \operatorname{Diff} + \lambda \tilde{D}|_E)$ is not log canonical.
This is impossible since $\lambda \le \operatorname{lct} (E, \operatorname{Diff};\tilde{D}|_E)$.
Therefore the pair $(X, \lambda D)$ is log canonical at $\mathsf{p}$, and the inequality \eqref{eq:lctwbl-1} is proved.
By considering the coefficient of $E$ in \eqref{eq:lctwbl-4}, it is easy to see that
\[
\operatorname{lct}_{\mathsf{p}} (X; D) \le \frac{c}{\operatorname{ord}_E (D)}.
\]
Under the assumption \eqref{eq:lctwbl-2}, this shows the equality \eqref{eq:lctwbl-3}.
\end{proof}
We consider Lemma \ref{lem:lctwbl} in more details in a concrete setting.
\begin{Def}
Let $\underline{c} = (c_1, \dots, c_n)$ be an $n$-tuple of positive integers such that $\gcd \{c_1, \dots, c_n\} = 1$ and we set
\[
m_i = \gcd \{c_1, \dots, \hat{c}_i, \dots, c_n\}
\]
for $i = 1, \dots, n$.
Let $f = f (x_1, \dots, x_n)$ be a polynomial which is quasi-homogeneous with respect to $\operatorname{wt} (x_1, \dots, x_n) = \underline{c}$.
If $f$ is irreducible and $f \ne x_i$ for $i = 1, \dots, n$, then there exists an irreducible polynomial $f^{\operatorname{wf}} = f^{\operatorname{wf}} (\tilde{x}_1, \dots, \tilde{x}_n)$ (in new variables $\tilde{x}_1, \dots, \tilde{x}_n$) such that
\[
f^{\operatorname{wf}} (x_1^{m_1}, \dots, x_n^{m_n}) = f (x_1, \dots, x_n).
\]
We call $f^{\operatorname{wf}}$ the {\it well-formed model} of $f$ (with respect to the weight $\operatorname{wt} (x_1, \dots, x_n) = \underline{c}$).
In general we have a decomposition
\[
f = x_1^{\lambda_1} \cdots x_n^{\lambda_n} f_1^{\mu_1} \cdots f_k^{\mu_k},
\]
where $k, \lambda_1, \dots, \lambda_n, \mu_1, \dots, \mu_k$ are nonnegative integers and $f_1, \dots, f_k$ are irreducible polynomials in variables $x_1, \dots, x_n$ which are quasi-homogeneous with respect to $\operatorname{wt} (x_1, \dots, x_n) = \underline{c}$ and which are not $x_i$ for any $i$.
We define
\[
f^{\operatorname{wf}} := f (\tilde{x}_1^{1/m_1}, \dots, \tilde{x}_n^{1/m_n})
= \tilde{x}^{\lambda_1/m_1} \cdots \tilde{x}_n^{\lambda_n/m_n} (f_1^{\operatorname{wf}})^{\mu_1} \cdots (f_k^{\operatorname{wf}})^{\mu_k}
\]
and call it the {\it well-formed model} of $f$.
Note that $f^{\operatorname{wf}}$ is in general not a polynomial since $\lambda_i/m_i$ need not be an integer.
In this case the effective $\mathbb{Q}$-divisor
\[
\mathcal{D}^{\operatorname{wf}}_f := \sum_{i=1}^n \frac{\lambda_i}{m_i} H^{\operatorname{wf}}_i + \sum_{j=1}^k \mu_j (f_j^{\operatorname{wf}} = 0)
\]
on the well-formed model $\mathbb{P} (\underline{c})^{\operatorname{wf}}_{\tilde{x}_1, \dots, \tilde{x}_n}$ of $\mathbb{P} (\underline{c})$ is called the {\it effective $\mathbb{Q}$-divisor on $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ associated to} $f$, where $H_i^{\operatorname{wf}}$ is the quasi-hyperplane on $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ defined by $\tilde{x}_i = 0$.
\end{Def}
\begin{Lem} \label{lem:lctwblwh}
Let $\mathbb{P} (\underline{b}) := \mathbb{P} (b_0, \dots, b_{n+1})_{x_0, \dots, x_{n+1}}$ be a well-formed weighted projective space and let $X \subset \mathbb{P} (\underline{b})$ be a normal weighted hypersurface with defining polynomial $F = F (x_0, \dots, x_{n+1})$.
Let $\underline{c} = (c_1, \dots, c_n)$ be a tuple of positive integers such that $\gcd \{c_1, \dots, c_n\} = 1$.
Assume that
\[
F = x_0^e x_{n+1} + \sum_{i=1}^e x_0^{e-i} f_i,
\]
where $e \in \mathbb{Z}_{> 0}$ and $f_i = f_i (x_1, \dots, x_n, x_{n+1})$ is a quasi-homogeneous polynomial of degree $i b_0 + b_{n+1}$.
Let $G = G (x_1, \dots x_n)$ be the lowest weight part of $\bar{F} := F (1, x_1, \dots, x_n,0)$ with respect to $\operatorname{wt} (x_1, \dots, x_n) = \underline{c}$
Then, for the point $\mathsf{p} = \mathsf{p}_{x_0} = (1\!:\!0\!:\!\cdots\!:\!0) \in X$, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_{x_{n+1}})
\ge \min \left\{ \frac{c_1 + \cdots + c_n}{\operatorname{wt}_{\underline{c}} (\bar{F})}, \ \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}^{\operatorname{wf}}_G) \right\},
\]
where $\operatorname{wt}_{\underline{c}} (\bar{F})$, $\operatorname{Diff}$ and $\mathcal{D}^{\operatorname{wf}}_G$ are as follows.
\begin{itemize}
\item $\operatorname{wt}_{\underline{c}} (\bar{F})$ is the weight of $\bar{F}$ with respect to $\operatorname{wt} (x_1, \dots, x_n) = \underline{c}$.
\item $\operatorname{Diff} = \sum_{i=1}^n \frac{m_i-1}{m_i} H^{\operatorname{wf}}_i$, where $H^{\operatorname{wf}}_i = (\tilde{x}_i = 0)$ is the $i$th coordinate quasi-hyperplane of $\mathbb{P} (\underline{c})^{\operatorname{wf}}_{\tilde{x}_1, \dots, \tilde{x}_n}$ and $m_i = \gcd \{c_1, \dots, \hat{c}_i, \dots, c_n\}$ for $i = 1, \dots, n$.
\item $\mathcal{D}^{\operatorname{wf}}_G$ is the effective $\mathbb{Q}$-divisor on $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ associated to $G$.
\end{itemize}
If in addition the inequality
\[
\frac{c_1 + \cdots + c_n}{\operatorname{wt} (\bar{F})} \le \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}^{\operatorname{wf}}_G)
\]
holds, then we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_{x_{n+1}}) = \frac{c_1 + \cdots + c_n}{\operatorname{wt} (\bar{F})}.
\]
\end{Lem}
\begin{proof}
Let $\rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}} \subset X$, where $U_{\mathsf{p}} = U_{x_0}$, be the orbifold chart containing $\mathsf{p}$ and we set $\rho = \rho_{\mathsf{p}}, \breve{U} = \breve{U}_{\mathsf{p}}$ and $U = U_{\mathsf{p}}$.
We set $H = H_{x_{n+1}}$ and $\breve{H} = \rho^*H$.
We have $\operatorname{lct}_{\mathsf{p}} (X; H) = \operatorname{lct}_{\check{\mathsf{p}}} (\check{U}; \check{H})$.
The variety $\breve{U}$ is the hypersurface in $\mathcal{U}_{x_0} = \mathbb{A}^{n+1}_{\breve{x}_1, \dots, \breve{x}_{n+1}}$ defined by the equation
\begin{equation} \label{eq:lctwblwh-1}
F (1, \breve{x}_1, \dots, \breve{x}_{n+1}) = \breve{x}_{n+1} + \sum_{i=1}^e \breve{f}_i = 0,
\end{equation}
where $\breve{f}_i = f_i (\breve{x}_1, \dots, \breve{x}_{n+1})$, and $\breve{\mathsf{p}}$ corresponds to the origin.
We see that $\{\breve{x}_1, \dots, \breve{x}_n\}$ is a system of local coordinates of $\breve{U}$ at $\breve{\mathsf{p}}$.
Let $\varphi \colon Y \to \breve{U}$ be the weighted blowup at $\breve{\mathsf{p}}$ with $\operatorname{wt} (\breve{x}_1, \dots, \breve{x}_n) = (c_1, \dots, c_n)$.
We can identify the $\varphi$-exceptional divisor $E$ with $\mathbb{P} (\underline{c})_{\breve{x}_1, \dots, \breve{x}_n}$.
Filtering off terms divisible by $\breve{x}_{n+1}$ in \eqref{eq:lctwblwh-1}, we have
\[
(-1 + \cdots ) \breve{x}_{n+1} = F (1, \breve{x}_1, \dots, \breve{x}_n, 0)
\]
on $\breve{U}$, where the omitted term in the left-hand side is a polynomial vanishing at $\breve{\mathsf{p}}$.
Since $\breve{H}$ is the divisor on $\breve{U}$ defined by $\breve{x}_{n+1} = 0$, we see that $\operatorname{ord}_E (\breve{H}) = \operatorname{wt}_{\underline{c}} (\bar{F})$ and the divisor $\tilde{H}|_E$ corresponds to the divisor $\mathcal{D}^{\operatorname{wf}}_G$ on $E \cong \mathbb{P} (\underline{c})^{\operatorname{wf}}$, where $\tilde{H}$ is the proper transform of $\breve{H}$ on $Y$.
Therefore the proof is completed by Lemma \ref{lem:lctwbl} and Remark \ref{rem:wbldiff}.
\end{proof}
\begin{Lem} \label{lem:lcttangcube}
Let $X \subset \mathbb{P} (a, b_1, b_2, b_3, r)_{x, y_1, y_2, y_3, z}$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$ with defining polynomial $F = F (x, y_1, y_2, y_3, z)$.
Assume that $F$ can be written as
\[
F = z^k x + z^{k-1} f_{r + a} + z^{k-2} f_{2 r + a} + \cdots + f_{k r + a},
\]
where $f_i \in \mathbb{C} [x, y_1, y_2, y_3]$ is a quasi-homogeneous polynomial of degree $i$, and we set
\[
\bar{F} := F (0,y_1, y_2, y_2, 1) \in \mathbb{C} [y_1, y_2, y_3].
\]
If either $\bar{F} \in (y_1, y_2, y_3)^2 \setminus (y_1, y_2, y_3)^3$ or $\bar{F} \in (y_1, y_2, y_3)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form in $y_1, y_2, y_3$, then for the point $\mathsf{p} := \mathsf{p}_z \in X$, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge \frac{1}{2}.
\]
If in addition $a = 1$, $r > 1$ and $\mathsf{p} \in X$ is not a maximal center, then
\[
\alpha_{\mathsf{p}} (X) = \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}.
\]
\end{Lem}
\begin{proof}
Let $\rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$ be the orbifold chart of $X$ containing $\mathsf{p}$.
We see that Let $\breve{U}_{\mathsf{p}}$ be the hypersurface in $\breve{\mathcal{U}}_{\mathsf{p}} = \mathbb{A}^4_{\breve{x}, \breve{y}_1, \breve{y}_2, \breve{y}_3}$ defined by the equation
\[
F (\breve{x}, \breve{y}_1, \breve{y}_2, \breve{y}_3, 1) = 0.
\]
We see that $\breve{U}_{\mathsf{p}}$ is smooth and the morphism $\rho_{\mathsf{p}}$ can be identified with the quotient morphism of the singularity $\mathsf{p} \in X$ over a suitable analytic neighborhood of $\mathsf{p}$.
We denote by $\breve{\mathsf{p}} \in \breve{U}$ the origin of $\breve{\mathcal{U}}_{\mathsf{p}} = \mathbb{A}^4$ which is the preimage of $\mathsf{p}$ via $\rho_{\mathsf{p}}$.
Filtering off terms divisible by $x$ in $F (x, y_1, y_2, y_3, 1)$, we have
\[
(-1 + \cdots) \breve{x} = F (0, \breve{y}_1, \breve{y}_2, \breve{y}_3, 1) = \bar{F} (\breve{y}_1, \breve{y}_2, \breve{y}_3) =: \breve{F}
\]
on $\breve{U}_{\mathsf{p}}$.
Note that we can choose $\{\breve{y}_1, \breve{y}_2, \breve{y}_3\}$ as a system of local coordinates of $\breve{U}_{\mathsf{p}}$ at $\breve{\mathsf{p}}$.
If $\bar{F} \in (y_1, y_2, y_3)^2 \setminus (y_1, y_2, y_3)^3$, then $\operatorname{omult}_{\mathsf{p}} (H_x) = \operatorname{mult}_{\breve{\mathsf{p}}} (\rho_{\mathsf{p}}^*H_x) = 2$ and hence $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$.
Suppose that $\bar{F} \in (y_1, y_2, y_3)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form in $y_1, y_2, y_3$.
Let $\varphi \colon V \to \breve{U}$ be the blowup of $\breve{U}$ at $\breve{\mathsf{p}}$ with exceptional divisor $E \cong \mathbb{P}^2$.
We set $D = \rho_{\mathsf{p}}^*H_x$.
Since $\operatorname{mult}_{\breve{\mathsf{p}}} (D) = 3$, we have
\[
K_V + \frac{1}{2} \tilde{D} = \varphi^* \left(K_{\check{U}} + \frac{1}{2} D \right) + \frac{1}{2} E,
\]
where $\tilde{D}$ is the proper transform of $D$ on $V$.
The divisor $\tilde{D}|_E$ on $E$ is isomorphic to the hypersurface in $\mathbb{P}^2_{\breve{y}_1, \breve{y}_2, \breve{y}_3}$ defined by the cubic part of $\bar{F} (\breve{y}_1, \breve{y}_2, \breve{y}_3)$, and the pair $(E, \frac{1}{2} \tilde{D}|_E)$ is log canonical by Lemma \ref{lem:lctP2cubic}.
It then follows that the pair $(V, \frac{1}{2} \tilde{D})$ is log canonical along $E$.
This shows that the pair $(\breve{U}, \frac{1}{2} D)$ is log canonical at $\breve{\mathsf{p}}$, and hence $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ as desired.
Suppose in addition that $r > 1$ and $\mathsf{p} \in X$ is not a maximal center.
By Lemma \ref{lem:qtangdivncan}, the pair $(X, H_x)$ is not canonical at $\mathsf{p} = \mathsf{p}_z$ and thus we have $\alpha_{\mathsf{p}} (X) = \min \{1, \operatorname{lct}_{\mathsf{p}}; (X;H_x)\}$ by Lemma \ref{lem:singnoncanbd}.
This proves the latter assertion.
\end{proof}
\subsection{Computations by $2 n$-inequality}
\begin{Lem} \label{lem:complctsingtang}
Let $X \subset \mathbb{P} (b_1, b_2, b_3, c, r)_{x_1, x_2, x_3, y, z}$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$ with defining polynomial $F = F (x_1, x_2, x_3, y, z)$ and suppose $\mathsf{p} :=\mathsf{p}_z \in X$.
We assume that $b_1 \le b_2 \le b_3$ and that we can choose $y$ as a quasi-tangent coordinate of $X$ at $\mathsf{p}$.
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{r b_2 b_3 (A^3)}.
\]
In particular, if $r b_2 b_3 (A^3) \le 4$, then $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\end{Lem}
\begin{proof}
Let $\rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$ be the orbifold chart of $X$ containing $\mathsf{p} = \mathsf{p}_z$.
We set $\rho = \rho_{\mathsf{p}}$, $\breve{U} = \breve{U}_{\mathsf{p}}$ and $U = U_{\mathsf{p}}$.
We see that $\breve{U}$ is the hypersurface in $\breve{\mathcal{U}}_{\mathsf{p}} = \mathbb{A}^4_{\breve{x}_1, \breve{x}_2, \breve{x}_3, \breve{y})}$ defined by the equation
\[
F (\breve{x}_1, \breve{x}_2, \breve{x}_3, \breve{y}, 1) = 0.
\]
We see that $\breve{U}$ is smooth and the morphism $\rho$ can be identified with the quotient morphism of $\mathsf{p} \in X$ over a suitable analytic neighborhood of $\mathsf{p}$.
We denote by $\breve{\mathsf{p}} \in \breve{U}$ the origin which is the preimage of $\mathsf{p}$ via $\rho$.
By the assumption, we can choose $\breve{x}_1, \breve{x}_2, \breve{x}_3$ as a system of local coordinates of $\breve{U}$ at $\breve{\mathsf{p}}$.
We set
\[
\lambda := \frac{2}{r b_2 b_3 (A^3)}
\]
and assume that $\alpha_{\mathsf{p}} (X) < \lambda$.
Then there exists an irreducible $\mathbb{Q}$-divisor $D \sim_{\mathbb{Q}} A$ such that the pair $(X, \lambda D)$ is not log canonical at $\mathsf{p}$.
In particular the pair $(\breve{U}, \lambda \rho^*D)$ is not log canonical at $\breve{\mathsf{p}}$.
Let $\varphi \colon V \to \breve{U}$ be the blowup of $\breve{U}$ at $\breve{\mathsf{p}}$ with exceptional divisor $E \cong \mathbb{P}^2$.
By Lemma \ref{lem:2nineq}, there exists a line $L \subset E$ with the property that for any prime divisor $T$ on $\breve{U}$ such that $T$ is smooth at $\breve{\mathsf{p}}$ and that its proper transform $\tilde{T}$ contains $L$, we have $\operatorname{mult}_{\breve{\mathsf{p}}} (D|_T) > 2/\lambda$.
By a slight abuse of notation, we have an isomorphism $E \cong \mathbb{P}^2_{\breve{x}_1, \breve{x}_2, \breve{x}_3}$.
The line $L \subset E$ is isomorphic to $(\alpha_1 \breve{x}_1 + \alpha_2 \breve{x}_2 + \alpha_3 \breve{x}_3 = 0) \subset \mathbb{P}^2$, for some $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{C}$ with $(\alpha_1, \alpha_2, \alpha_3) \ne (0, 0, 0)$.
We set
\[
\breve{T} := (\alpha_1 \breve{x}_1 + \alpha_2 \breve{x}_2 + \alpha_3 \breve{x}_3 = 0) \subset \breve{U}.
\]
Then $\breve{T}$ is smooth at $\breve{\mathsf{p}}$ and its proper transform on $V$ contains $L$.
It follows that $\operatorname{mult}_{\breve{\mathsf{p}}} (\rho^*D|_{\breve{T}}) > 2/\lambda$.
Set $k := \max \{\, i \mid \alpha_i \ne 0 \, \}$.
We have $r \breve{T} = \rho^*G$ for some effective Weil divisor $G \sim r b_k A$.
Let $j \in \{1, 2, 3\}$ be such that
\[
b_j = \max \{\, b_i \mid 1 \le i \le 3, i \ne k \, \}.
\]
Then, since $\{x_1, x_2, x_3\}$ isolates $\mathsf{p}$, we can take an effective $\mathbb{Q}$-divisor $S \sim_{\mathbb{Q}} A$ such that $\operatorname{omult}_{\mathsf{p}} (S) \ge 1/b_j$ and $\rho^*S$ does not contain any component of $\rho^*D|_{\breve{T}}$.
Hence we have
\[
r b_k (A^3) = (D \cdot G \cdot S) \ge (\rho^*D \cdot \breve{T} \cdot \rho^*S)_{\breve{\mathsf{p}}} > \frac{2}{b_j \lambda} = \frac{r b_2 b_3 (A^3)}{b_j}.
\]
This is a contradiction since $b_j b_k \le b_2 b_3$, and the proof is completed.
\end{proof}
\subsection{Computations by $\operatorname{\overline{NE}}$}
Let $X$ be a quasi-smooth Fano 3-fold weighted hypersurface of index $1$.
Let $\mathsf{p} \in X$ be a singular point and we denote by $\varphi \colon Y \to X$ the Kawamata blowup at $\mathsf{p}$.
In \cite{CP17}, the assertion $(-K_Y)^2 \notin \operatorname{Int} \operatorname{\overline{NE}} (Y)$ is verified in many cases, where $\operatorname{\overline{NE}} (Y)$ is the cone of effective curves on $Y$.
Thus the following result is very useful.
\begin{Lem}[{\cite[Lemma 2.8]{KOW18}}] \label{lem:singptNE}
Let $\mathsf{p} \in X$ be a terminal quotient singular point and $\varphi \colon Y \to X$ the Kawamata blowup at $\mathsf{p}$.
Suppose that $(-K_Y)^2 \notin \operatorname{Int} \operatorname{\overline{NE}} (Y)$ and there exists a prime divisor $S$ on $X$ such that $\tilde{S} \sim_{\mathbb{Q}} - m K_Y$ for some $m > 0$, where $\tilde{S}$ is the proper transform of $S$ on $Y$.
Then $\alpha_{\mathsf{p}} (X) \ge 1$.
\end{Lem}
\chapter{Smooth points} \label{chap:smpt}
The aim of this chapter is to prove the following.
\begin{Thm} \label{thm:smpt}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any smooth point $\mathsf{p} \in X$.
\end{Thm}
We explain the organization of this chapter.
Throughout the present chapter, let
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$, where we assume $a_1 \le \cdots \le a_4$.
Note that $a_2 \ge 2$ since $\mathsf{i} \notin \mathsf{I}_1$.
Recall that we denote by $F = F (x, y, z, t, w)$ the defining polynomial of $X$ with $\deg F = d$ and we set $A := -K_X$.
We set
\[
\begin{split}
U_1 &:= \bigcup_{v \in \{x,y,z,t,w\}, \deg v = 1} (v \ne 0) \cap X, \\
L_{xy} &:= H_x \cap H_y = (x = y = 0) \cap X.
\end{split}
\]
Note that $U_1$ is an open subset of $X$ contained in $\operatorname{Sm} (X)$, and $L_{xy}$ is a $1$-dimensional closed subset of $X$.
The proof of the inequality $\alpha_{\mathsf{p}} (X) \ge 1/2$ for $\mathsf{p} \in U_1$ will be done in \S \ref{sec:smptU1}.
The proof for the other smooth points will be done as follows.
\begin{itemize}
\item If $1 < a_1 < a_2$, then $\operatorname{Sm} (X) \subset U_1 \sqcup (H_x \setminus L_{xy}) \sqcup L_{xy}$.
In this case, the proof of $\alpha_{\mathsf{p}} (X) \ge 1/2$ for smooth point $\mathsf{p}$ of $X$ contained in $H_x \setminus L_{xy}$ (resp.\ $L_{xy}$) will be done in \S \ref{sec:smptHminusL} (resp.\ \S \ref{sec:smptL1} and \S \ref{sec:smptL2}), respectively.
\item If $1 = a_1 < a_2$, then $\operatorname{Sm} (X) \subset U_1 \sqcup L_{xy}$.
In this case, the proof of $\alpha_{\mathsf{p}} (X) \ge 1/2$ for smooth point $\mathsf{p}$ of $X$ contained in $L_{xy}$ will be done in \S \ref{sec:smptL1} and \S \ref{sec:smptL2}.
\item If $1 < a_1 = a_2$, then $\operatorname{Sm} (X) \subset U_1 \sqcup H_x$.
In this case, the proof of $\alpha_{\mathsf{p}} (X) \ge 1/2$ for smooth point $\mathsf{p}$ of $X$ contained in $H_x$ will be done in \S \ref{sec:smptH}.
\end{itemize}
Therefore Theorem \ref{prop:smptU1} will follow from Propositions \ref{prop:smptU1}, \ref{prop:smptHminusL}, \ref{prop:smptL1}, \ref{prop:smptL2} and \ref{prop:smptH}, which are the main results of \S \ref{sec:smptU1}, \S \ref{sec:smptHminusL}, \S \ref{sec:smptL1}, \S \ref{sec:smptL2} and \S \ref{sec:smptH}, respectively.
\section{Smooth points on $U_1$ for families indexed by $\mathsf{I} \setminus \mathsf{I}_1$} \label{sec:smptU1}
\begin{Lem} \label{lem:nsptU1-1}
We have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{a_2 a_4 (A^3)}
\]
for any point $\mathsf{p} \in U_1$.
\end{Lem}
\begin{proof}
Let $\mathsf{p} \in U_1$ be a point.
We may assume $\mathsf{p} = \mathsf{p}_x$ by a change of coordinates.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor.
The linear system $|\mathcal{I}_{\mathsf{p}} (a_2 A)|$ is movable, and let $S \in |\mathcal{I}_{\mathsf{p}} (a_2 A)|$ be a general member so that $\operatorname{Supp} (S) \ne \operatorname{Supp} (D)$.
The set $\{y, z, t, w\}$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ (see Remark \ref{rem:isolT}).
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = a_2 a_4 (A^4).
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/a_2 a_4 (A^3)$ and the proof is completed.
\end{proof}
\begin{Lem} \label{lem:nsptU1-2}
Suppose that $d$ is divisible by $a_4$.
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{a_2 a_3 (A^3)}
\]
for any point $\mathsf{p} \in U_1$.
\end{Lem}
\begin{proof}
Let $\mathsf{p} \in U_1$ be a point.
We may assume $\mathsf{p} = \mathsf{p}_x$.
We can choose coordinates so that $w^{d/a_4} \in F$.
Indeed, if $a_4 > a_3$, then $w^{d/a_4} \in F$ by the quasi-smoothness of $X$.
If $a_4 = a_3$, then there is a monomial of degree $d$ consisting of $t, w$ by the quasi-smoothness of $X$ and we can choose coordinates $t, w$ so that $w^{d/a_4} \in F$.
Under the above choice of coordinates, we see that $\{y, z, t\}$ isolates $\mathsf{p}$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor.
Let $S$ be a general member of the movable linear system $|\mathcal{I}_{\mathsf{p}} (a_2 A)|$, so that $\operatorname{Supp} (S)$ does not contain $\operatorname{Supp} (D)$.
We can take a $\mathbb{Q}$-divisor $T \in |a_3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ since $\{y, z, t\}$ isolates $\mathsf{p}$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = a_2 a_3 (A^3).
\]
This shows $\alpha_{\mathsf{p}} (D) \ge 1/a_2 a_3 (A^3)$ and the proof is completed.
\end{proof}
\begin{Rem} \label{rem:lemsmpttrue}
The objects of Chapter \ref{chap:smpt} are members of families $\mathcal{F}_{\mathsf{i}}$ for $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ and the inequality $a_2 \ge 2$ is assumed throughout the present chapter.
It is however noted that in Lemmas \ref{lem:nsptU1-1} and \ref{lem:nsptU1-2}, the assumption $a_2 \ge 2$ is not required and the statement holds for members of $\mathcal{F}_{\mathsf{i}}$ for any $\mathsf{i} \in \mathsf{I}$.
\end{Rem}
\begin{Lem} \label{lem:nsptU1-3}
Suppose that $d$ is not divisible by $a_4$ and $a_1 = 1$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X;S_{\mathsf{p}}), \ \frac{1}{a_4 (A^3)} \right\} \ge \frac{1}{2}
\]
for any $\mathsf{p} \in U_1$, where $S_{\mathsf{p}}$ is the unique member of $|\mathcal{I}_{\mathsf{p}} (A)|$.
\end{Lem}
\begin{proof}
Let $\mathsf{p} \in U_1$ be a point.
We may assume $\mathsf{p} = \mathsf{p}_x$.
Note that we have $a_2 > 1$ and thus the linear system $|\mathcal{I}_{\mathsf{p}} (A)|$ indeed consists of a unique member $S_{\mathsf{p}}$.
In this case $S_{\mathsf{p}} = H_y$.
We first prove $\operatorname{lct}_{\mathsf{p}} (X;S_{\mathsf{p}}) \ge 1/2$, that is, $(X, \frac{1}{2} H_y)$ is log canonical at $\mathsf{p}$.
Assume to the contrary that $(X, \frac{1}{2} H_y)$ is not log canonical at $\mathsf{p}$.
Then $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 3$.
Suppose $\operatorname{mult}_{\mathsf{p}} (H_y) = 3$.
Then, by Lemma \ref{lem:lctP2cubic}, the degree $3$ part of $F (1,0,z,t,w)$ with respect to $\deg (z,t,w) = (1,1,1)$ is a cube of a linear form, that is, it can be written as $(\alpha z + \beta t + \gamma w)^3$ for some $\alpha, \beta, \gamma \in \mathbb{C}$.
By Lemma \ref{lem:smptHLdegwt}, we have $d < 3 a_4$ since $d$ is not divisible by $a_4$.
From this we deduce $\gamma = 0$.
Then we can write
\[
F = x^{d-1} y + x^{d-3 a_3} (\alpha x^{a_3-a_2} z + \beta t)^3 + g + y h,
\]
where $g = g (x,z,t,w) \in (z,t,w)^4 \subset \mathbb{C} [x,z,t,w]$ and $h = h (x,y,z,t,w)$.
By the inequality $d < 3 a_4$, any monomial in $g$ cannot be divisible by $w^3$, so that $g \in (z,t)^2$.
But then $X$ is not quasi-smooth along the non-empty subset
\[
(y = x^{d-1} + h = z = t = 0) \subset X.
\]
This is impossible and we have $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 4$.
By the same argument as above, we can write
\[
F = x^{d-1} y + g + y h,
\]
where $g \in (z,t)^2$, which implies that $X$ is not quasi-smooth.
This is a contradiction and thus $\operatorname{lct}_{\mathsf{p}} (X; S_{\mathsf{p}}) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $S_{\mathsf{p}} = H_y$.
Note that $D \cdot H_y$ is an effective $1$-cycle.
The set $\{y, z, t, w\}$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_y$.
We have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_y \cdot T) \le a_4 (A^3) \le 2,
\]
where the last inequality follows from (5) of Lemma \ref{lem:wtnumerics} since $a_1 = 1$.
Thus $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/a_4 (A^3) \ge 1/2$ and the proof is completed.
\end{proof}
\begin{Lem} \label{lem:nsptU1-4}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{9,17\}$.
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any point $\mathsf{p} \in U_1$.
\end{Lem}
\begin{proof}
In this case,
\[
X = X_{3 a + 3} \subset \mathbb{P} (1, 1, a, a+1, a+1)_{x, y, z, t, w},
\]
where $a = 2, 3$ if $\mathsf{i} = 9, 17$, respectively.
Let $\mathsf{p} \in U_1$ be a point.
We may assume $\mathsf{p} = \mathsf{p}_x$.
We show that $(X, \frac{1}{2} H_y)$ is log canonical at $\mathsf{p}$.
Assume to the contrary that it is not.
Then $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 3$ and, by Lemma \ref{lem:lctP2cubic}, we can write
\[
F = x^{3 a + 2} y + x^3 (\alpha z x + \beta t + \gamma w)^3 + g + y h,
\]
where $g = g (x,z,t,w) \in (z,t,w)^4 \subset \mathbb{C} [x,y,t,w]$ and $h = h (x,y,z,t,w)$.
By degree reason, any monomial in $g \in (z,t,w)^4$ is divisible by $z^3$.
It follows that $X$ is not quasi-smooth along the non-empty subset
\[
(y = x^{3a+2} + h = z = \beta t + \gamma w = 0) \subset X.
\]
This is a contradiction and the pair $(X, \frac{1}{2} H_y)$ is log canonical at $\mathsf{p}$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_y$.
The set $\{y, z, t, w\}$ clearly isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |(a+1) A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_y$.
Then, we have
\[
2 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_y \cdot T)_{\mathsf{p}} \le (D \cdot H_y \cdot T) = \frac{3}{a} \le \frac{3}{2},
\]
since $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 2$.
This shows $\operatorname{lct}_{\mathsf{p}} (X, D) \ge 4/3$ and the proof is completed.
\end{proof}
\begin{Prop} \label{prop:smptU1}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge 1
\]
for any point $\mathsf{p} \in U_1$.
\end{Prop}
\begin{proof}
Let $X = X_d \subset \mathbb{P} (1,a_1,a_2,a_3,a_4)$ be a member of $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$, where we assume $a_1 \le \cdots \le a_4$.
\begin{itemize}
\item If $d$ is not divisible by $a_4$ and $a_1 \ge 2$, then $a_2 a_4 (A^3) \le 2$ and the assertion follows from Lemma \ref{lem:nsptU1-1}.
\item If $d$ is not divisible by $a_4$ and $a_1 = 1$, then the assertion follows from Lemma \ref{lem:nsptU1-3}.
\item If $d$ is divisible by $a_4$ and $\mathsf{i} \notin \{9, 17\}$, then $a_2 a_3 (A^3) \le 2$ by Lemma \ref{lem:wtnumerics} and the assertion follows from Lemma \ref{lem:nsptU1-2}.
\item If $\mathsf{i} \in \{9, 17\}$, then the assertion follows from Lemma \ref{lem:nsptU1-4}.
\end{itemize}
This completes the proof.
\end{proof}
\section{Smooth points on $H_x \setminus L_{xy}$ for families with $1 < a_1 < a_2$} \label{sec:smptHminusL}
Let
\[
X = X_d \subset \mathbb{P} (1, a_1, \dots, a_4)_{x, y, z, t, w}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfying $1 < a_1 < a_2 \le a_3 \le a_4$.
In this section, we set $\bar{F} = F (0,y,z,t,w)$.
Then $H_x$ is isomorphic to the weighted hypersurface in $\mathbb{P} (a_1, \dots, a_4)$ defined by $\bar{F} = 0$.
We note that if a smooth point $\mathsf{p} \in X$ contained in $H_x$ satisfies $\operatorname{mult}_{\mathsf{p}} (H_x) > 2$, then $\mathsf{p}$ belongs to the subset
\[
\bigcap_{v_1, v_2 \in \{y,z,t,w\}} \left( \frac{\partial^2 \bar{F}}{\partial v_1 \partial v_2} = 0 \right) \cap X.
\]
The following is the main result of this section.
\begin{Prop} \label{prop:smptHminusL}
Let $X = X_d \subset \mathbb{P} (1, a_1, \dots, a_4)$, $a_1 \le \dots \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfying $1 < a_1 < a_2$.
Then
\[
\alpha_{\mathsf{p}} (X) \ge 1
\]
for any smooth point $\mathsf{p}$ of $X$ contained in $H_x \setminus L_{xy}$.
\end{Prop}
The rest of this section is entirely devoted to the proof of Proposition \ref{prop:smptHminusL} which will be done by division into several cases.
\subsection{Case: $1 < a_1 < a_2$ and $d = 2 a_4$}
We will prove Proposition \ref{prop:smptHminusL} under the assumption of $1 < a_1 < a_2$ and $d = 2 a_4$.
Let $\mathsf{p} \in X$ be a smooth point contained in $H_x \setminus L_{xy}$.
We have $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$ since $w^2 \in F$, and thus $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
By Lemma \ref{lem:isolclass}, $a_1 a_3 A$ isolates $\mathsf{p}$, and hence
we can take a $\mathbb{Q}$-divisor $T \in |a_1 a_3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = a_1 a_3 (A^3) \le 1,
\]
where the last inequality follows from Lemma \ref{lem:wtnumerics}.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ and the proof is completed.
\subsection{Case: $1 < a_1 < a_2$ and $d = 2 a_4 + a_1$}
We will prove Proposition \ref{prop:smptHminusL} under the assumption of $1 < a_1 < a_2$ and $d = 2 a_4 + a_1$.
Let $\mathsf{p} \in X$ be a smooth point contained in $H_x \setminus L_{xy}$.
We can write
\[
F = w^2 y + w f + g,
\]
where $f, g \in \mathbb{C} [x, y, z, t]$ are quasi-homogeneous polynomials of degree $d-a_4$ and $d$ respectively.
Since $\partial^2 \bar{F}/\partial w^2 = y$ and $y$ does not vanish at $\mathsf{p} \in H_x \setminus L_{xy}$, we have $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
By Lemma \ref{lem:isolclass}, $a_1 a_4 A$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_1 a_4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = a_1 a_4 (A^3) \le 2,
\]
where the last inequality from Lemma \ref{lem:wtnumerics}.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/2$ and the proof is completed.
\subsection{Case: $1 < a_1 < a_2$ and $d = 2 a_4 + a_2$} \label{sec:smptHminusLc}
We will prove Proposition \ref{prop:smptHminusL} under the assumption of $1 < a_1 < a_2$ and $d = 2 a_4 + a_2$.
Let $\mathsf{p} \in X$ be a smooth point contained in $H_x \setminus L_{xy}$.
We can write
\[
F = w^2 z + w (z f + g) + h,
\]
where $f, h \in \mathbb{C} [x,y,z,t]$ and $g \in \mathbb{C} [x,y,t]$ are quasi-homogeneous polynomials of degrees $a_4, a_2 + a_4$ and $d$, respectively.
\begin{Claim} \label{clm:smptHminusL1}
$\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:smptHminusL1}]
We prove $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$.
Assume to the contrary that $\operatorname{mult}_{\mathsf{p}} (H_x) > 2$.
Since
\[
\frac{\partial^2 \bar{F}}{\partial w^2} = z, \quad \frac{\partial^2 \bar{F}}{\partial w \partial z} = 2 w + f,
\]
the point $\mathsf{p}$ is contained in $H_x \cap H_z \cap (2 w + f = 0)$.
Suppose in addition that $\mathsf{p} \in H_t$.
Note that we have $a_4 = a_1 + a_3$ since $d = a_1 + a_2 + a_3 + a_4$ and $d = 2 a_4 + a_2$.
We see that $a_1$ does not divide $a_4 = a_1 + a_3$ because otherwise $\gcd \{a_1, a_3, a_4\} > 1$ and $X$ has a non-isolated singularity which is clearly worse than terminal, a contradiction.
It follows that $f$ does not contain a power of $y$, i.e.\ $f (0,y,0) = 0$, and
\[
\mathsf{p} \in H_x \cap H_z \cap H_t \cap (2 w + f = 0) = H_x \cap H_z \cap H_t \cap H_w = \{\mathsf{p}_y\}.
\]
This is impossible since $\mathsf{p}_y$ is a singular point of $X$.
Thus $\mathsf{p} \notin H_t$.
Since $a_4 = a_1 + a_3$, we may assume that $\mathsf{p} \in H_w$ after replacing $w$ by $w - \xi y t$ for some $\xi \in \mathbb{C}$.
We can write $\mathsf{p} = (0\!:\!1\!:\!0\!:\!\lambda\!:\!0)$ for some non-zero $\lambda \in \mathbb{C}$.
The set $\{x, z, w, t^{a_1} - \lambda^{a_1} y^{a_3}\}$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_1 a_3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $H_x \cdot H_z$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (H_x) \le (H_x \cdot H_z \cdot T)_{\mathsf{p}} \le (H_x \cdot H_z \cdot T) = a_1 a_2 a_3 (A^3) < 3,
\]
where the last inequality follows from Lemma \ref{lem:wtnumerics}.
This shows $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$, and thus $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$.
\end{proof}
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
By Lemma \ref{lem:isolclass}, $a_1 a_4 A$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_1 a_4 A|_{mbQ}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$.
We have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = a_1 a_4 (A^3) < 2,
\]
where the last inequality follows from Lemma \ref{lem:wtnumerics}.
Thus $\operatorname{lct}_{\mathsf{p}} (X; D) > 1/2$ and we conclude $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsection{Case: $1 < a_1 < a_2$, $d = 2 a_4 + a_3$ and $a_3 \ne a_4$}
The proof of Proposition \ref{prop:smptHminusL} under the assumption of $1 < a_1 < a_2$ and $d = 2 a_4 + a_3$ is completely parallel to the one given in \S \ref{sec:smptHminusLc}.
Indeed, the same proof applies after interchanging the role of $z$ and $t$ (and hence $a_2$ and $a_3$).
Thus we omit the proof.
\subsection{Case: $1 < a_1 < a_2$ and $d = 3 a_4$}
We will prove Proposition \ref{prop:smptHminusL} under the assumption of $1 < a_1 < a_2$ and $d = 3 a_4$.
Let $\mathsf{p} \in X$ be a smooth point contained in $H_x \setminus L_{xy}$.
We first prove $\alpha_{\mathsf{p}} (X) \ge 1/2$ assuming that the inequality $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$ holds.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
By Lemma \ref{lem:isolclass}, $a_1 a_4 A$ isolates $\mathsf{p}$, and hence we can take a $\mathbb{Q}$-divisor $T \in |a_1 a_4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = a_1 a_4 (A^3) \le 2,
\]
where the last inequality follows from Lemma \ref{lem:wtnumerics}.
This shows $\operatorname{mult}_{\mathsf{p}} (D) \le 2$ and we have $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/2$.
Therefore the proof of Proposition \ref{prop:smptHminusL} under the assumption of $1< a_1 < a_2$ and $d = 3 a_4$ is reduced to the following.
\begin{Claim} \label{cl:smHcase3e}
We have $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$ for any smooth point $\mathsf{p} \in X$ contained in $H_x \setminus L_{xy}$.
\end{Claim}
The rest of this subsection is devoted to the proof of Claim \ref{cl:smHcase3e}, which will be done by considering each family individually.
The families satisfying $1 < a_1 < a_2$ and $d = 3 a_4$ are families $\mathcal{F}_{\mathsf{i}}$, where
\[
\mathsf{i} \in \{19, 27, 39, 49, 59, 66, 84\}.
\]
In the following, for a polynomial $f (x,y,z,\dots)$ in variables $x, y, z, \dots$, we set $\bar{f} = \bar{f} (y,z,\dots) := f (0,y,z,\dots)$.
We first consider the family $\mathcal{F}_{27}$, which is the unique family satisfying $d = 3 a_3 = 3 a_4$
We then consider the rest of the families which satisfy $d = 3 a_4 > 3 a_3$.
\subsubsection{The family $\mathcal{F}_{27}$}
We can choose $w$ and $t$ so that
\[
F = w^2 t + w t^2 + w t b_5 + w c_{10} + t d_{10} + e_{15},
\]
where $b_5, c_{10}, d_{10}, e_{15} \in \mathbb{C} [x,y,z]$ are quasi-homogeneous polynomials of indicated degrees.
Let $\mathsf{p} \in H_x \setminus L_{xy}$ be a smooth point of $X$ and we assume $\operatorname{mult}_{\mathsf{p}} (H_x) > 2$.
Since
\[
\frac{\partial^2 \bar{F}}{\partial w^2} = 2 t, \quad
\frac{\partial^2 \bar{F}}{\partial t^2} = 2 w,
\]
we have $\mathsf{p} \in H_t \cap H_w$.
Then we can write $\mathsf{p} = (0\!:\!1\!:\!\lambda\!:\!0\!:\!0)$ for some non-zero $\lambda \in \mathbb{C}$ since $\mathsf{p} \notin H_y$ and $\mathsf{p} \ne \mathsf{p}_y$.
We can write $\bar{e}_{15} = z (z^2 - \lambda^2 y^3)(z^2 - \mu y^3)$ for some $\mu \in \mathbb{C}$.
We have
\[
\begin{split}
\frac{\partial^2 \bar{F}}{\partial z^2} (\mathsf{p}) &= 2 \lambda (7 \lambda^2 - 3 \mu) = 0, \\
\frac{\partial^2 \bar{F}}{\partial z \partial y} (\mathsf{p}) &= - 3 \lambda^2 (3 \lambda^2 + \mu) = 0,
\end{split}
\]
which implies $\lambda = 0$.
This is a contradiction and Claim \ref{cl:smHcase3e} is proved for the family $\mathcal{F}_{27}$.
We consider the rest of the families, which satisfies $d = 3 a_4 > 3 a_3$.
Replacing $w$ if necessary, we can write
\[
F = w^3 + w g_{2 a_4} + h_{3 a_4},
\]
where $g_{2 a_4}, h_{3 a_4} \in \mathbb{C} [x,y,z,t]$ are quasi-homogeneous polynomials of degree $2 a_4, 3 a_4$, respectively.
Let $\mathsf{p} \in H_x \setminus L_{xy}$ be a smooth point of $X$ and we assume $\operatorname{mult}_{\mathsf{p}} (H_x) > 2$.
Since $\partial^2 \bar{F}/\partial w^2 = 6 w$, we have $\mathsf{p} \in H_w$, so that $\mathsf{p} \in H_x \cap H_w$ and $\mathsf{p} \notin H_y$.
In the following we will derive a contradiction by considering each family separately.
\subsubsection{The family $\mathcal{F}_{19}$}
Replacing $t \mapsto t - \xi z$ for a suitable $\xi \in \mathbb{C}$, we may assume $\mathsf{p} \in H_t$.
Since $\mathsf{p} \in H_x \cap H_t \cap H_w$, $\mathsf{p} \notin H_y$ and $\mathsf{p} \ne \mathsf{p}_y$, we have $\mathsf{p} = (0\!:\!1\!:\!\lambda\!:\!0\!:\!0)$ for a non-zero $\lambda \in \mathbb{C}$.
We can write $\bar{h}_{12} = (z^2 - \lambda^2 y^3)(z^2 - \mu y^3) + t e_9$ for some $\mu \in \mathbb{C}$ and $e_9 = e_9 (y,z,t)$.
It is then straightforward to check that
\[
\frac{\partial^2 \bar{F}}{\partial z^2} (\mathsf{p}) = \frac{\partial^2 \bar{F}}{\partial z \partial y} (\mathsf{p}) = 0
\]
is impossible, and this is a contradiction.
\subsubsection{The family $\mathcal{F}_{39}$}
We have $g_{2 a_4} = g_{12}$, $h_{3 a_4} = h_{18}$ and we can write
\[
g_{12} (0,y,z,t) = \alpha t z y + \lambda z^3 + \mu y^3,
\]
where $\alpha, \lambda, \mu \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\lambda \ne 0$ and $\mu \ne 0$.
We have
\[
\frac{\partial^2 \bar{F}}{\partial w \partial t} = \alpha z y, \quad
\frac{\partial^2 \bar{F}}{\partial w \partial z} = \alpha t y + 3 \lambda z^2, \quad
\frac{\partial^2 \bar{F}}{\partial w \partial y} = \alpha t z + 3 \mu y^2.
\]
Suppose that $\alpha \ne 0$, then, since both $\alpha z y$ and $\alpha t y + 3 \lambda z^2$ vanish at $\mathsf{p}$ and $y$ does not vanish at $\mathsf{p}$, we have $\mathsf{p} \in H_z \cap H_t$.
It follows that $\mathsf{p} = \mathsf{p}_y$.
This is impossible since $\mathsf{p}_y$ is a singular point of $X$.
Thus $\alpha = 0$.
Then, both $3 \lambda z^2$ and $\alpha t z + 3 \mu y^2$ vanish at $\mathsf{p}$, which implies that $y$ vanishes at $\mathsf{p}$.
This is a contradiction.
\subsubsection{The family $\mathcal{F}_{49}$}
We have $g_{2 a_4} = g_{14}, h_{3 a_4} = h_{21}$ and we can write
\[
h_{24} (0,y,z,t) = \lambda t^3 y + \alpha t^2 y^3 + \beta t z^3 + \gamma t y^5 + \delta z^3 y^2 + \varepsilon y^7,
\]
where $\lambda, \alpha, \beta, \dots, \varepsilon \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\lambda \ne 0$, and by replacing $t$, we can assume that $\alpha = 0$.
Since $\mathsf{p} \in H_w$, $\mathsf{p} \notin H_y$ and
\[
\frac{\partial^2 \bar{F}}{\partial t^2} = w \frac{\partial^2 g_{14} (0, y, z, t)}{\partial t^2} + 6 \lambda t y,
\]
we have $\mathsf{p} \in H_t$.
Then $\mathsf{p} \notin H_z$ because otherwise $\mathsf{p} = \mathsf{p}_y$ is a singular point and this is impossible.
We have
\[
\frac{\partial^2 \bar{F}}{\partial z \partial y} = w \frac{\partial g_{14} (0,y,z,t)}{\partial z \partial y} + 6 \delta z^2 y,
\]
which implies $\delta = 0$.
Then, by the quasi-smoothness of $X$, we have $\varepsilon \ne 0$.
But the polynomial
\[
\frac{\partial^2 \bar{F}}{\partial y^2} = w \frac{\partial^2 g_{14} (0,y,z,t)}{\partial y^2} + 20 \gamma t y^3 + 42 \varepsilon y^5
\]
does not vanish at $\mathsf{p}$.
This is a contradiction.
\subsubsection{The family $\mathcal{F}_{59}$}
We have $g_{2 a_4} = g_{16}, h_{3 a_4} = h_{24}$ and we can write
\[
h_{24} (0,y,z,t) = \lambda t^3 y + \mu z^4 + \alpha z^3 y^2 + \beta z^2 y^4 + \gamma z y^6 + \delta y^8,
\]
where $\lambda, \mu, \alpha, \beta, \gamma, \delta \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\lambda \ne 0$ and $\mu \ne 0$.
Since $\mathsf{p} \notin H_y$, $\lambda \ne 0$ and
\[
\frac{\partial^2 \bar{F}}{\partial t^2} = w \frac{\partial^2 g_{16} (0,y,z,t)}{\partial t^2} + 6 \lambda t y,
\]
we have $\mathsf{p} \in H_t$.
This is a contradiction since $\mathsf{p} \in H_x \cap H_t \cap H_w$ but $H_x \cap H_t \cap H_w$ consists of singular points.
\subsubsection{The family $\mathcal{F}_{66}$}
We have $h_{3 a_4} = h_{27}$ and we can write
\[
h_{27} (0,y,z,t) = \lambda t^3 z + \mu t y^4 + \alpha z^2 y^3,
\]
where $\alpha, \lambda, \mu \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\lambda \ne 0$ and $\mu \ne 0$.
We have
\[
\frac{\partial^2 \bar{F}}{\partial t \partial y} = w \frac{\partial^2 g_{18} (0,y,z,t)}{\partial t \partial y} + 4 \mu y^3.
\]
This is a contradiction since $\mathsf{p} \in H_w$, $\mathsf{p} \notin H_y$ and $\mu \ne 0$.
\subsubsection{The family $\mathcal{F}_{84}$}
We have
\[
g_{24} (0,y,z,t) = \alpha t z y + \lambda z^3, \quad
h_{36} (0,y,z,t) = \mu t^4 + \beta z y^4,
\]
where $\alpha, \beta, \lambda, \mu \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\lambda \ne 0$ and $\mu \ne 0$.
We have
\[
\frac{\partial^2 \bar{F}}{\partial t^2} = w \frac{\partial^2 g_{24} (0,y,z,t)}{\partial t^2} + 12 \mu t^2,
\]
which implies $\mathsf{p} \in H_t$ since $\mathsf{p} \in H_w$ and $\mu \ne 0$.
We have
\[
\frac{\partial^2 \bar{F}}{\partial w \partial z} = \frac{g_{24} (0,y,z,t)}{\partial z} = \alpha t y + 3 \lambda z^2,
\]
which implies $\mathsf{p} \in H_z$ since $\mathsf{p} \in H_t$ and $\lambda \ne 0$.
This shows $\mathsf{p} = \mathsf{p}_y$, and this is a contradiction since $\mathsf{p}_y$ is a singular point.
Therefore we derive a contradiction for all families and the proof of Claim \ref{cl:smHcase3e} is completed.
\section{Smooth points on $L_{xy}$ for families with $a_1 < a_2$, Part 1} \label{sec:smptL1}
In this and next sections, we consider a member
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, , t, w}
\]
of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfying $a_1 < a_2 \le a_3 \le a_4$ and prove $\alpha_{\mathsf{p}} (X) \ge 1/2$ for a smooth point $\mathsf{p}$ of $X$ contained in $L_{xy}$.
The objects of this section are members $X$ such that the $1$-dimensional scheme $L_{xy} = (x = y = 0) \cap X$ is irreducible and reduced.
\begin{Lem} \label{lem:Lirred}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$ which satisfies $a_1 < a_2$ and which is listed in \emph{Table \ref{table:Lsmooth}} $($resp.\ \emph{Table \ref{table:Lsing}}$)$.
Then, $L_{xy}$ is an irreducible smooth curve $($resp.\ irreducible and reduced curve which is smooth along $L_{xy} \cap \operatorname{Sm} (X)$$)$.
\end{Lem}
\begin{proof}
Let $F$ be the defining polynomial of $X$ and set $d = \deg F$.
The scheme $L_{xy}$ is isomorphic to the hypersurface in $\mathbb{P} (a_2,a_3,a_4)_{z, t, w}$ defined by the polynomial $f := F (0,0,z,t,w)$.
We explain that $f$ can be transformed into the polynomial given in Tables \ref{table:Lsmooth} and \ref{table:Lsing} by a suitable change of homogeneous coordinates.
Suppose $\mathsf{i} \notin \{11, 15, 16, 17, 21, 27, 34\}$.
Then, there are only a few monomials of degree $d$ in variables $z, t, w$.
We first simply express $f$ as a linear combination of those monomials, and then consider the following coordinate change.
\begin{itemize}
\item Suppose $d = 2 a_4$.
In this case, $f$ is quadratic with respect to $w$ and we eliminate the term of the form $w g (z, t)$ by replacing $w$ suitably.
Then, we may assume $f = w^2 + h (z, t)$ for some quasi-homogeneous polynomial $h = h (z, t)$ of degree $d$.
\item Suppose $d = 3 a_4$.
In this case, $f$ is cubic with respect to $w$ and we eliminate the term of the form $w^2 g (z, t)$ by replacing $w$ suitably.
Then, we may assume $f = w^3 + w h_1 (z, t) + h_2 (z,t)$ for some quasi-homogeneous polynomials $h_1 = h_1 (z, t), h = h_2 (z, t)$ of degrees $d - a_4 = 2 a_4$ and $d = 3 a_4$, respectively.
\end{itemize}
After the above coordinate change, we observe that $f$ is a linear combination of at most $3$ distinct monomials and it is possible to make those coefficients $1$ by rescaling $z, t, w$.
The resulting polynomial is the one given in Tables \ref{table:Lsmooth} and \ref{table:Lsing}.
Once an explicit form of the defining polynomial is given, it is then easy to show that $L_{xy}$ is entirely smooth or is smooth along $L_{xy} \cap \operatorname{Sm} (X)$.
For $\mathsf{i} = \{11, 15, 16, 17, 21, 27, 34\}$, the description of $f$ is explained as follows.
\begin{itemize}
\item Suppose $\mathsf{i} = 11$.
In this case, $f = w^2 + h (z,t)$, where $h$ is a quintic form in $z, t$.
The solutions of the equation $h = 0$ correspond to the $5$ singular points of type $\frac{1}{2} (1,1,1)$.
Thus $h$ does not have a multiple component and in particular $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 15$.
In this case, $f = w^2 + \alpha t^4 + \beta t^2 z^3 + \gamma z^6$ for some $\alpha, \beta, \gamma \in \mathbb{C}$.
We have $\alpha \ne 0$ (resp.\ $\gamma \ne 0$) because otherwise $X$ cannot be quasi-smooth at $\mathsf{p}_t$ (resp.\ $\mathsf{p}_z$).
Replacing $t$ and rescaling $z$, we may assume $\alpha = 1$, $\beta = 0$ and $\gamma = 1$, and we obtain the desired form $f = w^2 + t^4 + z^6$.
It is easy to see that $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 16$.
In this case, $f = w^2 z + h (z, t)$, where $h = \alpha t^3 + \beta t^2 z^2 + \gamma t z^4 + \delta z^6$ for some $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $\alpha \ne 0$.
Moreover the solutions of $h (z, t) = 0$ correspond to $3$ singular points of type $\frac{1}{2} (1,1,1)$.
Thus $h (z, t)$ does not have a multiple component and in particular $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 17$.
In this case, $f = c (t, w) + \alpha z^4$, where $\alpha \in \mathbb{C}$ and $c (t, w)$ is a cubic form in $t, w$.
By the quasi-smoothness of $X$, we have $\alpha \ne 0$ and we may assume $\alpha = 1$ by rescaling $z$.
Moreover the solutions of $c (t, w) = 0$ correspond to $3$ singular points of type $\frac{1}{4} (1,1,3)$.
Thus $c (t, w)$ does not have a multiple component and in particular $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 21$.
In this case, $f = w^2 + h (z, t)$, where $h = \alpha t^3 z + \beta t^2 z^3 + \gamma t z^5 + \delta z^7$ for some $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
By the quasi-smoothness of $X$ at $\mathsf{p}_t$, we have $t^3 z \in F$, that is, $\alpha \ne 0$.
Moreover the solutions of $\alpha t^3 + \beta t^2 z^2 + \gamma t z^4 + \delta z^6 = 0$ correspond to the $3$ singular points of type $\frac{1}{2} (1,1,1)$.
Thus $h$ does not have a multiple component and in particular $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 27$.
In this case, $f = c (t, w) + \alpha z^5$, where $\alpha \in \mathbb{C}$ and $c (t, w)$ is a cubic form in $t, w$.
By the same arguments as in the case of $\mathsf{i} = 17$, $c (t, w)$ does not have a multiple component and we may assume $\alpha =1$.
Thus $L_{xy}$ is smooth.
\item Suppose $\mathsf{i} = 34$.
In this case, $f = w^2 + h (z, t)$, where $h = \alpha t^3 + \beta t^2 z^3 + \gamma t z^6 + \delta z^9$ for some $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
By the quasi-smoothness of $X$, we have $t^3 \in F$, that is, $\alpha \ne 0$.
Moreover the solutions of $h = 0$ correspond to $3$ singular points of type $\frac{1}{2} (1,1,1)$.
Thus $h$ does not have a multiple component and in particular $L_{xy}$ is smooth.
\end{itemize}
This completes the proof.
\end{proof}
\begin{table}[h]
\renewcommand{1.35}{1.1}
\begin{center}
\caption{$L_{xy}$: Irreducible and smooth case}
\label{table:Lsmooth}
\begin{tabular}{cc|cc}
No. & Equation & No. & Equation \\
\hline
11 & $w^2 + h (z,t)$ & 55 & $w^2 + t^3 z + z^8$ \\
15 & $w^2 + t^4 + z^6$ & 57 & $w^2 + t^4 z + z^6$ \\
16 & $w^2 z + h (z, t)$, $t^3 \in h$ & 66 & $w^3 + w z^3 + t^3 z$ \\
17 & $c (t,w) + z^4$ & 68 & $w^2 + t^4 + z^7$ \\
19 & $w^3 + h (z,t)$ & 70 & $w^2 + t^3 + t z^5$ \\
21 & $w^2 + h (z, t)$ & 71 & $w^2 + t^3 z + z^5$ \\
26 & $w^2 z + z^5 + t^3$ & 72 & $w^2 + t^3 + z^{10}$ \\
27 & $c (t,w) + z^5$ & 74 & $w^2 z + t^3 + t z^5$ \\
34 & $w^2 + h (z, t)$ & 75 & $w^2 + t^5 + z^6$ \\
35 & $w^2 + t^3 z + z^6$ & 76 & $w^2 t + t^3 z + z^5$ \\
36 & $w^2 z + t^3 + t z^3$ & 80 & $w^2 + t^3 z + t z^6$ \\
41 & $w^2 + t^4 + z^5$ & 84 & $w^3 + w z^3 + t^4$ \\
45 & $w^2 z + t^4 + z^5$ & 86 & $w^2 + t^4 z + t z^5$ \\
48 & $w^2 z + t^3 + z^7$ & 88 & $w^2 + t^3 + z^7$ \\
51 & $w^2 + t^3 z + t z^4$ & 90 & $w^2 + t^3 + t z^7$ \\
53 & $w^2 + t^3 + z^8$ & 93 & $w^2 + t^5 + t z^5$ \\
54 & $w^2 z + t^3 + z^4$ & 95 & $w^2 + t^3 + z^{11}$
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\caption{$L_{xy}$: Irreducible and singular case}
\label{table:Lsing}
\begin{tabular}{ccc|ccc}
No. & Equation & Sing. & No. & Equation & Sing. \\
\hline
43 & $t^4 + z^5$ & $\mathsf{p}_w$ & 77 & $w^2 + t^3 z$ & $\mathsf{p}_z$ \\
44 & $w^2 t + z^4$ & $\mathsf{p}_t$ & 78 & $w^2 + t z^5$ & $\mathsf{p}_t$ \\
46 & $t^3 + z^7$ & $\mathsf{p}_w$ & 79 & $w^2 z + t^3$ & $\mathsf{p}_{z}$ \\
47 & $w^2 z + t^3$ & $\mathsf{p}_z$ & 81 & $w^2 + t^4 z$ & $\mathsf{p}_z$ \\
56 & $t^3 + z^8$ & $\mathsf{p}_w$ & 82 & $w^2 + t^3$ & $\mathsf{p}_z$ \\
59 & $w^3 + z^4$ & $\mathsf{p}_t$ & 83 & $w^2 + z^9$ & $\mathsf{p}_t$ \\
61 & $w^2 t + z^5$ & $\mathsf{p}_t$ & 85 & $w^2 + t^3 z$ & $\mathsf{p}_z$ \\
62 & $w^2 + t^3 z$ & $\mathsf{p}_z$ & 87 & $w^2 + t^5$ & $\mathsf{p}_z$ \\
65 & $w^2 z + t^3$ & $\mathsf{p}_z$ & 89 & $w^2 + t^3$ & $\mathsf{p}_z$ \\
67 & $w^2 + z^7$ & $\mathsf{p}_t$ & 91 & $w^2 + t^3 z$ & $\mathsf{p}_z$ \\
69 & $w^2 z + t^4$ & $\mathsf{p}_z$ & 92 & $w^2 + t^3$ & $\mathsf{p}_z$ \\
73 & $w^2 + z^5$ & $\mathsf{p}_t$ & 94 & $w^2 + t^3$ & $\mathsf{p}_z$
\end{tabular}
\end{center}
\end{table}
\begin{Prop} \label{prop:smptL1}
Let $X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)$, $a_1 \le a_2 \le a_3 \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ which satisfies $a_1 < a_2$ and which is listed in Tables \ref{table:Lsmooth} and \ref{table:Lsing}.
Then
\[
\alpha_{\mathsf{p}} (X) \ge 1
\]
for any smooth point $\mathsf{p}$ of $X$ contained in $L_{xy}$.
\end{Prop}
\begin{proof}
Take a point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
Let $S \in |A|$ and $T \in |a_1 A|$ be general members.
By Lemma \ref{lem:Lirred}, $L_{xy}$ is an irreducible and reduced curve, and we have $S \cdot T = L_{xy}$.
Note that $L_{xy}$ is quasi-smooth at $\mathsf{p}$ and we have $\operatorname{mult}_{\mathsf{p}} (L_{xy}) = 1$.
By Lemma \ref{lem:qsminvhypsec}, $S$ is quasi-smooth at $\mathsf{p}$.
It follows that $\operatorname{lct}_{\mathsf{p}} (X;S) = 1$.
By Lemma \ref{lem:exclL}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X;S), \ \frac{a_1}{\operatorname{mult}_{\mathsf{p}} (L_{xy})}, \ \frac{1}{a_1 (A^3)} \right\} = 1
\]
since $1/a_1 (A^3) > 1$ by Lemma \ref{lem:wtnumerics}.
\end{proof}
\section{Smooth points on $L_{xy}$ for families with $a_1 < a_2$, Part 2} \label{sec:smptL2}
In this section, we consider the case where $L_{xy}$ is not necessarily irreducible or reduced.
Specifically the aim of this section is to prove the following.
\begin{Prop} \label{prop:smptL2}
Let $X = X_d \subset \mathbb{P} (1,a_1,a_2,a_3,a_4)$, $a_1 \le a_2 \le a_3 \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ which satisfies $a_1 < a_2$ and which is not listed in Tables \ref{table:Lsmooth} and \ref{table:Lsing}.
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any smooth point $\mathsf{p}$ of $X$ contained in $L_{xy}$.
\end{Prop}
Note that a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfies the assumption of Proposition \ref{prop:smptL2} if and only if
\[
\begin{split}
\mathsf{i} \in \{ & 7, 9, 12, 13, 15, 20, 23, 24, 25, 29, 30, 31, 32, \\
& 33, 37, 38, 39, 40, 42, 49, 50, 52, 58, 60, 63, 64\}.
\end{split}
\]
The proof of Proposition \ref{prop:smptL2} will be completed in \S \ref{sec:proofsmptL} by considering each family separately and by case-by-case arguments.
Those arguments form several patterns and we describe them in \S \ref{sec:smptLgen}.
\subsection{General arguments} \label{sec:smptLgen}
In this subsection, let
\[
X = X_d \subset \mathbb{P} (1, a, b_1, b_2, b_3)_{x, y, z_1, z_2, z_3}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$.
Throughout this subsection, we assume that $a < b_i$ for $i = 1, 2, 3$.
Note that we do not assume $b_1 \le b_2 \le b_3$.
As before, we denote by $F = F (x, y, z_1, z_2, z_3)$ the defining polynomial of $X$, and we set $A := -K_X$.
The following very elementary lemma will be used several times.
\begin{Lem} \label{lem:numelem}
Let $a, e_1, e_2, e_3$ be positive integers such that $a < e_i$ for $i = 1, 2, 3$ and $\gcd \{e_1, e_2, e_3\} = 1$, and let $\lambda \ge 1$ be a number.
Then the following inequalities hold.
\begin{enumerate}
\item $\frac{1 + e_2 + e_3}{e_1 e_2 e_3} \le \frac{1}{2}$.
\item $\frac{a + e_2 + e_3}{e_1 e_2 e_3} + \frac{\lambda}{a} \le \frac{1}{2} + \lambda$.
\item $a (a + e_2 + e_3) < e_1 e_2 e_3$.
\end{enumerate}
\end{Lem}
\begin{proof}
In view of the assumption $\gcd \{e_1, e_2, e_3\} = 1$, it is easy to see that
\[
\frac{1 + e_2 + e_3}{e_1 e_2 e_3} = \frac{1}{e_1 e_2 e_3} + \frac{1}{e_1 e_3} + \frac{1}{e_1 e_2}
\]
attains its maximum when $(e_1, e_2, e_3) = (2, 2, 3)$, which implies (1).
It is also easy to see that
\[
\frac{a + e_2 + e_3}{e_1 e_2 e_3} + \frac{\lambda}{a},
\]
viewed as a function of $a$, attains its maximum when $a = 1$ since $1 \le a \le \sqrt{\lambda e_1 e_2 e_3}$.
Combining this with (1), the inequality (2) follows.
By the assumption, we have $e_1 e_2 \ge (a+1)^2$.
Hence we have
\[
\begin{split}
e_1 e_2 e_3 - a (a + e_2 + e_3)
&= e_3 (e_1 e_2 - a) - a^2 - a e_2 \\
& \ge e_3 (a^2 + a + 1) - a^2 - a e_2 \\
&= a^2 (e_3 - 1) + a (e_3 - e_2) + e_3 \\
&> 0.
\end{split}
\]
This proves (3).
\end{proof}
\begin{Lem} \label{lem:Linteg}
Suppose that $L_{xy} := (x = y = 0)_X$ is an irreducible and reduced curve which is smooth along $L_{xy} \cap \operatorname{Sm} (X)$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge 1
\]
for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Lem}
\begin{proof}
Let $S \in |A|$ and $T \in |a A|$ be general members.
Then we have $S \cap T = L_{xy}$.
Take any point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
By Lemma \ref{lem:qsminvhypsec}, $S$ is smooth at $\mathsf{p}$.
It follows that $\operatorname{mult}_{\mathsf{p}} (L_{xy}) = 1$ and $\operatorname{lct}_{\mathsf{p}} (X;S) = 1$.
By Lemma \ref{lem:exclL}, we have
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X;S), \ \frac{1}{\operatorname{mult}_{\mathsf{p}} (L_{xy})}, \ \frac{1}{a (A^3)} \right\} = 1,
\]
since $1/a (A^3) > 1$ by Lemma \ref{lem:wtnumerics}.
\end{proof}
\begin{Lem} \label{lem:Lredcp1}
Let $S \in |A|$ and $T \in |aA|$ be general members.
Suppose that the following assertions are satisfied.
\begin{enumerate}
\item $T|_S = \Gamma + \Delta$, where $\Gamma = (x = y = z_1 = 0)$ is a quasi-line and $\Delta$ is an irreducible and reduced curve which is quasi-smooth along $\Delta \cap \operatorname{Sm} (X)$.
\item $S$ is quasi-smooth along $\Gamma \cap \Delta$.
\item $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_{z_2}, \mathsf{p}_{z_3}\}$.
\end{enumerate}
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Lem}
\begin{proof}
By assumptions (1), (2) and Lemma \ref{lem:pltsurfpair}, $S$ is quasi-smooth along $\Gamma$.
\begin{Claim} \label{cl:Lredcp1-1}
The intersection matrix $M = M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:Lredcp1-1}]
By the assumption (3) and the quasi-smoothness of $S$, we see that $\operatorname{Sing}_{\Gamma} (S) = \{\mathsf{p}_{z_2}, \mathsf{p}_{z_3}\}$ and $\mathsf{p}_{z_i} \in S$ is a cyclic quotient singularity of index $b_i$ for $i = 2, 3$.
By Remark \ref{rem:compselfint}, we have
\[
(\Gamma^2)_S = -2 + \frac{b_2-1}{b_2} + \frac{b_3-1}{b_3} = - \frac{b_2 + b_3}{b_2 b_3} < 0.
\]
By taking the intersection number of $T|_S = \Gamma + \Delta$ and $\Gamma$, we have
\[
(\Gamma \cdot \Delta)_S
= - (\Gamma^2)_S + (T \cdot \Gamma)
= \frac{a + b_2 + b_3}{b_2 b_3} > 0.
\]
Note that we have
\[
(T \cdot \Delta)
= (T^2 \cdot S) - (T \cdot \Gamma)
= a^2 (A^3) - \frac{a}{b_2 b_3}
= \frac{a (a + b_2 + b_3)}{b_1 b_2 b_3},
\]
and then by taking the intersection number of $T|_S = \Gamma + \Delta$ and $\Delta$, we have
\[
(\Delta^2)_S = (T \cdot \Delta) - (\Gamma \cdot \Delta)_S
= - \frac{(b_1-a)(a + b_2 + b_3)}{b_1 b_2 b_3} < 0.
\]
Finally we have
\[
\begin{split}
\det M &= \frac{b_2 + b_3}{b_2 b_3} \cdot \frac{(b_1-a)(a + b_2 + b_3)}{b_1 b_2 b_3} - \frac{(a + b_2 + b_3)^2}{b_2^2 b_3^2} \\
&= - \frac{a (a + b_2 + b_3)(b_1 + b_2 + b_3)}{b_1 b_2^2 b_3^2} < 0.
\end{split}
\]
It follows that $M$ satisfies the condition $(\star)$.
\end{proof}
Let $\mathsf{p} \in (\Gamma \setminus \Delta) \cap \operatorname{Sm} (X)$ be a point.
By Lemma \ref{lem:normalqhyp}, $S$ is a normal surface.
It is easy to check that $a \deg \Gamma = a/(b_2 b_3) \le 1$, and that $X$, $S$ and $\Gamma$ are smooth at $\mathsf{p}$.
Thus, we can apply Lemma \ref{lem:mtdLred} and we conclude
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ 1, \ \frac{1}{a (A^3) + \frac{1}{a} - \deg \Gamma} \right\}
= \min \left\{ 1, \ \frac{1}{\frac{a + b_2 + b_3}{b_1 b_2 b_3} + \frac{1}{a}} \right\}
\ge \frac{2}{3},
\]
where the last inequality follows from Lemma \ref{lem:numelem}.
Let $\mathsf{p} \in (\Delta \setminus \Gamma) \cap \operatorname{Sm} (X)$ be a point.
Note that $\Delta$ is smooth at $\mathsf{p}$ since it is quasi-smooth at $\mathsf{p}$ by the assumption (1).
We have
\[
a \deg \Delta = a \left( a (A^3) - \frac{1}{b_2 b_3} \right) = \frac{a (a + b_2 + b_3)}{b_1 b_2 b_3} < 1
\]
by Lemma \ref{lem:numelem}.
Note that we have
\[
a (A^3) + \frac{1}{a} - \deg \Delta = \frac{1}{a} + \frac{1}{b_2 b_3} \le 1 + \frac{1}{4} = \frac{5}{4}
\]
since $1 \le a < b_i$.
Thus, we can apply Lemma \ref{lem:mtdLred} and conclude
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ 1, \ \frac{1}{a (A^3) + \frac{1}{a} - \deg \Delta} \right\} \ge \frac{4}{5}.
\]
Finally let $\mathsf{p} \in (\Gamma \cap \Delta) \cap \operatorname{Sm} (X)$ be a point.
Note that $S$ is smooth at $\mathsf{p}$ by the assumption (2), and we have
\[
\deg (\Gamma) = \frac{1}{b_2 b_3} < \frac{2}{a}, \quad
\deg (\Delta) = \frac{a + b_2 + b_3}{b_1 b_2 b_3} < \frac{2}{a},
\]
where the former inequality is obvious and the latter follows from Lemma \ref{lem:numelem}.
Thus we can apply Lemma \ref{lem:mtdLintpt} and conclude that
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{1, \frac{a}{2} \right\} \ge \frac{1}{2}.
\]
Therefore the proof is completed.
\end{proof}
\begin{Rem} \label{rem:Lredcp1}
Let the notation and assumption as in Lemma \ref{lem:Lredcp1}.
Assume in addition that $a \ge 2$ and $\Gamma \cap \Delta \subset \operatorname{Sing} (X)$.
Then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{43}{54} > \frac{3}{4}
\]
for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
Indeed, since $\Gamma \cap \Delta \subset \operatorname{Sing} (X)$, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{1, \ \frac{1}{\frac{a + b_2 + b_3}{b_1 b_2 b_3} + \frac{1}{a}}, \ \frac{4}{5} \right\}
\]
by the proof of Lemma \ref{lem:Lredcp1}.
Since $2 \le a < \sqrt{b_1 b_2 b_3}$ and $a < b_i$ for $i = 1, 2, 3$, we have
\[
\frac{a + b_2 + b_3}{b_1 b_2 b_3} + \frac{1}{a}
\le \frac{3 a + 1}{(a+1)^3} + \frac{1}{a}
\le \frac{43}{54}.
\]
This proves the desired inequality.
\end{Rem}
\begin{Lem} \label{lem:Lredcp2}
Suppose that $b_1, b_2, b_3$ are mutually coprime and $a \in \{1, 2\}$.
Suppose in addition that $F$ can be written as
\[
F = f_1 (z_1, z_2) x + f_2 (z_1, z_2) y + z_3^m z_2 + g (x, y, z_1, z_2, z_3),
\]
where $m \in \{2, 3\}$ and $f_1, f_2 \in \mathbb{C} [z_1, z_2], g \in [x, y, z_1, z_2, z_3]$ are quasi-homogeneous polynomials satisfying the following properties.
\begin{enumerate}
\item $\deg F = b_1 b_2 + a$.
\item $g$ is contained in the ideal $(x, y) \cap (x, y, z_3)^2 \subset \mathbb{C} [x, y, z_1, z_2, z_3]$.
\end{enumerate}
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Lem}
\begin{proof}
We have $d = m b_3 + b_2$ since $z_3^m z_2 \in F$, and combining this with $d = a + b_1 + b_2 + b_3$, we have
\begin{equation} \label{eq:Lredcp2-1}
a + b_1 + b_3 = m b_3.
\end{equation}
\begin{Claim} \label{clm:Lredcp2-1}
We can assume
\begin{equation} \label{eq:Lredcp2-2}
F =
\begin{cases}
z_2^{b_1} x - z_1^{b_2} y + z_3^m z_2 + g, & \text{if $a = 1$}, \\
z_1^k z_2^l x + (z_1^{b_2} - z_2^{b_1}) y + z_3^m z_2 + g, & \text{if $a = 2$}, \\
\end{cases}
\end{equation}
after replacing $x$ and $y$ suitably, where $k$ and $l$ are non-negative integers.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:Lredcp2-1}]
Suppose $a = 1$.
Then, since $\deg f_1 = \deg f_2 = b_1 b_2$, we can write
\[
f_1 (z_1,z_2) x + f_2 (z_1,z_2) y = z_2^{b_1} \ell_1 (x, y) + z_1^{b_2} \ell_2 (x, y),
\]
where $\ell_1, \ell_2$ are linear forms in $x, y$.
We see that $\ell_1, \ell_2$ are linearly independent because otherwise we can write $\ell_2 = \alpha \ell_1$ for some nonzero $\alpha \in \mathbb{C}$ and $X$ is not quasi-smooth along
\[
(x = y = w = \ell_1 = z_2^{b_2} + \alpha z_1^{b_1} = 0) \subset X.
\]
This is a contradiction.
Thus $\ell_1, \ell_2$ are linearly independent and we may assume $\ell_1 = x$ and $\ell_2 = y$, as desired.
Suppose $a = 2$.
We have $f_2 = \alpha z_1^{b_2} + \beta z_2^{b_1}$ for some $\alpha, \beta \in \mathbb{C}$ since $\deg f_2 = b_1 b_2$.
By the quasi-smooth of $X$ at $\mathsf{p}_z, \mathsf{p}_t$, we have $\alpha, \beta \ne 0$, and thus we may assume $\alpha =1$, $\beta = -1$ by rescaling $z_1, z_2$.
We see that the equation $f_1 (z_1, z_2) = f_2 (z_1, z_2) = 0$ on variables $z_1, z_2$ has only trivial solution because otherwise $X$ is not quasi-smooth along the nonempty set
\[
(x = y = w = f_1 (z_1, z_2) = f_2 (z_1, z_2) = 0) \subset X,
\]
which is impossible.
This implies that $f_1 \ne 0$ as a polynomial, and there exists a monomial $z_1^k z_2^l$ of degree $b_1 b_2 + 1$.
Since $b_1$ is coprime to $b_2$, $z_1^k z_2^l$ is the unique monomial of degree $b_1 b_2 + 1$ in variables $z_1, z_2$.
Thus we have $f_2 = \gamma z_1^k z_2^l$ for some nonzero $\gamma \in \mathbb{C}$.
Rescaling $x$, we may assume $\gamma = 1$, and the claim is proved.
\end{proof}
We continue the proof of Lemma \ref{lem:Lredcp2}.
Let $S \in |A|$ and $T \in |a A|$ be general members.
We have
\[
T|_S = \Gamma + m \Delta,
\]
where
\[
\Gamma = (x = y = z_2 = 0), \quad
\Delta = (x = y = z_3 = 0),
\]
since $F (0,0,z,t,w) = z_3^m z_2$.
We see that $\Gamma$ and $\Delta$ are quasi-lines of degree $1/b_1 b_3$ and $1/b_1 b_2$, respectively, and $\Gamma \cap \Delta = \{\mathsf{p}_{z_1}\} \subset \operatorname{Sing} (X)$.
We see that $S$ is quasi-smooth at $\mathsf{p}_{z_1}$ since $z_1^{b_2} y \in F$.
By Lemma \ref{lem:pltsurfpair}, $S$ is quasi-smooth along $\Gamma$ and the pair $(S, \Gamma)$ is plt.
\begin{Claim} \label{clm:Lredcp2-2}
The intersection matrix $M = M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:Lredcp2-2}]
We see that $d$ is not divisible by $b_1$ or $b_3$ since $d = b_1 b_2 + a = m b_3 + b_2$, $a < b_1$ and $b_2$ is coprime to $b_3$.
It follows that $\operatorname{Sing}_{\Gamma} (S) = \{\mathsf{p}_{z_1}, \mathsf{p}_{z_3}\}$, and $\mathsf{p}_{z_i} \in S$ is a cyclic quotient singularity of index $b_i$ for $i = 1, 3$.
By Remark \ref{rem:compselfint}, we have
\[
(\Gamma^2)_S
= -2 + \frac{b_1 - 1}{b_1} + \frac{b_3 - 1}{b_3}
= - \frac{b_1 + b_3}{b_1 b_3} < 0.
\]
By taking intersection number of $T|_S = \Gamma + m \Delta$ and $\Gamma$, we obtain
\[
(\Gamma \cdot \Delta)_S
= \frac{1}{m} (a \deg \Gamma - (\Gamma^2)_S)
= \frac{a + b_1 + b_3}{m b_1 b_3} = \frac{1}{b_1} > 0.
\]
Similarly, by taking intersection number of $T|_S$ and $\Delta$, we obtain
\[
(\Delta^2)_S
= \frac{1}{m} (a \deg \Delta - (\Gamma \cdot \Delta)_S)
= - \frac{b_2 - a}{m b_1 b_2} < 0,
\]
where the second equality follows from \eqref{eq:Lredcp2-1}.
Finally, we have
\[
\det M = \frac{b_1 + b_3}{b_1 b_3} \cdot \frac{b_2 - a}{m b_1 b_2} - \frac{1}{b_1^2} = - \frac{a (b_1 + b_2 + b_3)}{b_1^2 b_2 b_3} < 0,
\]
where the second equality follows from \eqref{eq:Lredcp2-1}.
It follows that $M$ satisfies the condition $(\star)$.
\end{proof}
Let $\mathsf{p} \in (\Gamma \setminus \Delta) \cap \operatorname{Sm} (X)$.
We see that $X, S$ and $\Gamma$ are smooth at $\mathsf{p}$, and it is easy to see $a \deg \Gamma = a/(b_1 b_3) < 1$.
Hence we can apply Lemma \ref{lem:mtdLred} and we have
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ 1, \ \frac{1}{a (A^3) + \frac{1}{a} - \deg \Gamma} \right\}
= \min \left\{ 1, \ \frac{1}{\frac{a + b_1 + b_3}{b_1 b_2 b_3} + \frac{1}{a}} \right\}
\ge \frac{2}{3},
\]
where the last inequality follows from Lemma \ref{lem:numelem}.
It remains to consider $\mathsf{p} \in (\Delta \setminus \Gamma) \cap \operatorname{Sm} (X)$ since $\Gamma \cap \Delta = \{\mathsf{p}_{z_1}\} \subset \operatorname{Sing} (X)$.
We first consider the case when $a = 2$.
In this case $S = H_x$ is quasi-smooth along $\Delta \setminus \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!1\!:\!1\!:\!0) \in (\Delta \setminus \Gamma) \cap \operatorname{Sm} (X)$, and $S$ has a double point at $\mathsf{q}$.
We have $\operatorname{mult}_{\mathsf{p}} (\Delta) = 1$, and $a \deg \Delta = a/(b_1 b_2) < 1$.
Thus we can apply Lemma \ref{lem:mtdLredSsing} and conclude
\[
\begin{split}
\alpha_{\mathsf{p}} (X)
&\ge \min \left\{ \frac{2}{\operatorname{mult}_{\mathsf{p}} (S)}, \ \frac{\operatorname{mult}_{\mathsf{p}} (S)}{2 (A^3) + \frac{m}{2} - m \deg \Delta} \right\} \\
&= \min \left\{ \frac{2}{\operatorname{mult}_{\mathsf{p}} (S)}, \ \frac{\operatorname{mult}_{\mathsf{p}} (S)}{\frac{1}{b_1 b_3} + \frac{m}{2}} \right\} \\
&\ge \min \left\{ 1, \ \frac{1}{\frac{1}{12} + \frac{3}{2}} \right\} \\
&= \frac{12}{19},
\end{split}
\]
since $1/(b_1 b_3) \le 1/12$, $m \in \{2, 3\}$ and $\operatorname{mult}_{\mathsf{p}} (S) \in \{1, 2\}$.
Suppose $a = 1$.
We set $S' = (z_3 = 0) \cap X \in |b_3 A|$.
For $\lambda \in \mathbb{C}$, we set $T'_{\lambda} = (y - \lambda x = 0) \cap X \in |A|$.
We can write $g (x, \lambda x, z_1, z_2, 0) = x^2 h_{\lambda}$ for some $h_{\lambda} = h_{\lambda} (x,z_1,z_2)$ since $g \in (x,y,z_3)^2$.
In view of \eqref{eq:Lredcp2-2}, we have
\[
F (x, \lambda x, z_1, z_2, 0) = x \phi_{\lambda} (x, z_1, z_2),
\]
where
\[
\phi_{\lambda} (x, z_1, z_2) =
z_1^{b_2} - \lambda z_2^{b_1} + x h_{\lambda}
\]
The polynomial $\phi_{\lambda}$ is irreducible for any nonzero $\lambda \in \mathbb{C}$.
We have
\[
T'_{\lambda}|_{S'} = \Delta + \Xi_{\lambda},
\]
where
\[
\Xi_{\lambda} = (y - \lambda x = z_3 = \phi_{\lambda} = 0)
\]
is an irreducible and reduced curve.
We have $\Delta \cap \Xi_{\lambda} = \{\mathsf{q}_{\lambda}\} \subset \operatorname{Sm} (X)$, where $\mathsf{q}_{\lambda} = (0\!:\!0\!:\!\sqrt[b_2]{\lambda}\!:\!1\!:\!0)$.
It is easy to see that $S'$ is quasi-smooth at $\mathsf{q}_{\lambda}$.
Hence, $S'$ is quasi-smooth along $\Delta$ by Lemma \ref{lem:pltsurfpair}.
\begin{Claim} \label{clm:Lredcp2-3}
The intersection matrix $M' = M (\Delta, \Xi_{\lambda})$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:Lredcp2-3}]
By Remark \ref{rem:compselfint}, we have
\[
(\Delta^2)_S = - \frac{b_3 -1}{b_1 b_2} - 2 + \frac{b_1 - 1}{b_1} + \frac{b_2 - 1}{b_2} = - \frac{b_1 + b_2 + b_3 - 1}{b_1 b_2} < 0.
\]
By taking intersection number of $T'_{\lambda}|_{S'} = \Delta + \Xi_{\lambda}$ and $\Delta$, we obtain
\[
(\Delta \cdot \Xi_{\lambda})_S = \frac{b_1 + b_2 + b_3}{b_1 b_2} > 0.
\]
By taking intersection number of $T'_{\lambda}|_{S'}$ and $\Xi_{\lambda}$, we obtain
\[
(\Xi_{\lambda}^2)_S = 0.
\]
It is then obvious that $\det M' < 0$ and the proof is completed.
\end{proof}
Now we take any point $\mathsf{p} \in (\Delta \setminus \Gamma) \cap \operatorname{Sm} (X)$, and then we can choose a nonzero $\lambda \in \mathbb{C}$ so that $\mathsf{p} \ne \mathsf{q}_{\lambda}$.
By Lemma \ref{lem:qsminvhypsec}, $S'$ is smooth at $\mathsf{p}$ since $S' \cap T'_{\lambda}$ is smooth at $\mathsf{p}$.
It is easy to see that $\deg \Delta = 1/(b_1 b_2) < 1$.
Thus we can apply Lemma \ref{lem:mtdLred} and conclude
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ b_3, \ \frac{1}{b_3 (A^3) + 1 - \deg \Delta} \right\}
= \min \left\{ b_3, \ \frac{1}{2} \right\} = \frac{1}{2},
\]
where the first equality follows since
\[
b_3 (A^3) + 1 - \deg \Delta = \frac{d - 1}{b_1 b_2} + 1 = 2.
\]
This completes the proof.
\end{proof}
\begin{Lem} \label{lem:Lnonredu2}
Suppose that $b_1, b_2, b_3$ are mutually coprime and $a \in \{1, 2, 3\}$.
Suppose in addition that $F$ can be written as
\[
F = z_3^m + z_1^{e_1} y - z_2^{e_2} x + g (x, y, z_1, z_2, z_3),
\]
where $m \ge 2, e_1, e_2$ are positive integers and $g \in \mathbb{C} [x, y, z_1 z_2, z_3]$ is a homogeneous polynomial satisfying the following properties.
\begin{enumerate}
\item If $a \ge 2$, then $m \le 2 a$.
\item If $a = 1$, then $e_1 \le b_2$.
\item $g$ is a homogeneous polynomial contained in the ideal $(x, y) \cap (x, y, z_3)^2 \subset \mathbb{C} [x, y, z_1, z_2, z_3]$.
\end{enumerate}
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Lem}
\begin{proof}
We first consider the case where $a \ge 2$.
Let $S \in |A|$ and $T \in |a A|$ be general members.
We have
\[
S \cdot T = m \Gamma,
\]
where
\[
\Gamma = (x = y = z_3 = 0)
\]
is a quasi-line of degree $1/(b_1 b_2)$.
Let $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
It is straightforward to check that $S$ is smooth at $\mathsf{p}$, which implies $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{a} S) = a$.
By Lemma \ref{lem:exclL}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ a, \ \frac{a}{m}, \ \frac{1}{a (A^3)} \right\} \ge \frac{1}{2},
\]
since $a/m \ge 1/2$ and $1/(a (A^3)) > 1$ by the assumption (1) and Lemma \ref{lem:wtnumerics}, respectively.
In the following we assume $a = 1$.
We set $S' = (z_3 = 0) \cap X \in |b_3 A|$ and $\Gamma = (x = y = z_3 = 0) \subset S'$.
We have $L_{xy} = \Gamma$ set-theoretically.
For $\lambda \in \mathbb{C}$, we set $T'_{\lambda} = (y - \lambda x = 0) \cap X \in |aA|$.
We can write
\[
g (x, \lambda x, z_1, z_2, 0) = x^2 h_{\lambda} (x, z_1, z_2),
\]
where $h_{\lambda}$ is a quasi-homogeneous polynomial in variables $x, z_1, z_2$ since $g \in (x, y, z_3)^2$.
We have
\[
F (x, \lambda x, z_1, z_2, 0) = x (\lambda z_1^{e_1} - z_2^{e_2} + x h_{\lambda}).
\]
\begin{Claim} \label{clm:Lnonredu2-1}
The quasi-homogeneous polynomial
\[
\phi_{\lambda} := \lambda z_1^{e_1} - z_2^{e_2} + x h_{\lambda} \in \mathbb{C} [x, z_1, z_2]
\]
is irreducible for any $\lambda \in \mathbb{C} \setminus \{0\}$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:Lnonredu2-1}]
We assume $\lambda \ne 0$.
If $\phi_{\lambda}$ is a reducible polynomial, then we can write
\[
\phi_{\lambda} = - (z_2^{c_2} + \cdots + \alpha z_1^{c_1} + \cdots)(z_2^{e_2-c_2} + \cdots + \beta z_1^{e_1-c_1} + \cdots)
\]
for some $c_1, c_2 \in \mathbb{Z}_{\ge 0}$ with $c_1 \le e_1$ and $0 < c_2 < e_2$, and nonzero $\alpha, \beta \in \mathbb{C}$ such that $\alpha \beta = \lambda$.
We have $c_2 b_2 = c_1 b_1$.
Since $b_1$ is coprime to $b_2$, we see that $c_1$ is divisible by $b_2$.
This implies $c_1 = e_1 = b_2$ since $c_1 \le e_1 \le b_2$.
By the equality $e_2 b_2 = e_1 b_1$, we have $c_2 = e_2 = b_1$.
This is a contradiction since $c_2 < e_2$.
Therefore $\phi_{\lambda}$ is irreducible for $\lambda \ne 0$.
\end{proof}
We continue the proof of Lemma \ref{lem:Lnonredu2}.
By Claim \ref{clm:Lnonredu2-1}, we have
\[
T'_{\lambda}|_{S'} = \Gamma + \Delta_{\lambda},
\]
where
\[
\Delta_{\lambda} = (y - \lambda x = z_3 = \phi_{\lambda} = 0)
\]
is an irreducible and reduced curve for any $\lambda \in \mathbb{C} \setminus \{0\}$.
We have $\Gamma \cap \Delta_{\lambda} = \{\mathsf{q}_{\lambda}\}$, where
\[
\mathsf{q}_{\lambda} =
(0\!:\!0\!:\!1\!:\!\sqrt[e_2]{\lambda}\!:\!0).
\]
It is easy to check that $S'$ is quasi-smooth at $\mathsf{q}_{\lambda}$.
By Lemma \ref{lem:pltsurfpair}, $S'$ is quasi-smooth along $\Gamma$ and the pair $(S', \Gamma)$ is plt.
\begin{Claim} \label{clm:Lnonredu2-2}
The intersection matrix $M' = M (\Gamma, \Xi'_{\lambda})$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:Lnonredu2-2}]
We see that $\operatorname{Sing}_{\Gamma} (S') = \{\mathsf{p}_{z_1}, \mathsf{p}_{z_2}\}$ and $\mathsf{p}_{z_i} \in S'$ is a cyclic quotient singular point of index $b_i$ for $i = 1, 2$.
By the same computation as in Claim \ref{clm:Lredcp2-3}, we have
\[
\begin{split}
(\Gamma^2)_{S'} &= - \frac{b_1 + b_2 + b_3 -1}{b_1 b_2} < 0, \\
(\Gamma \cdot \Delta_{\lambda})_{S'} &= \frac{b_1 + b_2 + b_3}{b_1 b_2} > 0, \\
(\Delta_{\lambda}^2)_{S'} &= 0.
\end{split}
\]
It is then easy to see that $\det M' < 0$, which shows that $M'$ satisfies $(\star)$.
\end{proof}
Now take a point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X) = \Gamma \cap \operatorname{Sm} (X)$.
We choose and fix a general $\lambda \in \mathbb{C}$ so that $\Delta_{\lambda}$ is irreducible and $\mathsf{q}_{\lambda} \ne \mathsf{p}$.
Then $\mathsf{p} \in (\Gamma \setminus \Xi_{\lambda}) \cap \operatorname{Sm} (X)$.
We see that $X$, $S'$ and $\Gamma$ are smooth at $\mathsf{p}$, and $\deg \Gamma = 1/(b_1 b_2) < 1$.
Thus we can apply Lemma \ref{lem:mtdLred} and conclude
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ b_3, \ \frac{1}{b_3 (A^3) + 1 - \deg \Gamma} \right\}
= \min \left\{ b_3, \ \frac{1}{\frac{e_1}{b_2} + 1} \right\}
\ge \frac{1}{2},
\]
where the last inequality follows from the assumption (2).
This completes the proof.
\end{proof}
\begin{Lem} \label{lem:nsptLpuredouble}
Let $S \in |A|$ and $T \in |a A|$ be general members.
Suppose that
\[
S \cdot T = 2 \Gamma,
\]
where $\Gamma = (x = y = z_3 = 0)$.
Then
\[
\alpha_{\mathsf{p}} (X)
\ge \min \left\{ \operatorname{lct}_{\mathsf{p}} (X; S), \ \frac{a}{2}, \ \frac{1}{a (A^3)} \right\}
\ge \frac{1}{2}
\]
for any point $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Lem}
\begin{proof}
Let $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
We have $\operatorname{mult}_{\mathsf{p}} (\Gamma) = 1$ and the first inequality follows from Lemma \ref{lem:exclL}.
We have $\operatorname{mult}_{\mathsf{p}} (S) \le \operatorname{mult}_{\mathsf{p}} (S \cdot T) = 2$, which implies $\operatorname{lct}_{\mathsf{p}} (X; S) \ge 1/2$.
Thus the second inequality in the statement follows since $1/(a (A^3)) > 1$ by Lemma \ref{lem:wtnumerics}.
\end{proof}
\subsection{Proof of Proposition \ref{prop:smptL2}} \label{sec:proofsmptL}
This subsection is entirely devoted to the proof of Proposition \ref{prop:smptL2}.
Let $X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)$, $a_1 \le \cdots \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfying $a_1 < a_2$.
Let $S \in |A|$ and $T \in |a_1 A|$ are general members so that their scheme-theoretic intersection $S \cap T$ coincides with $L_{xy}$.
Note that $S$ is a normal surface by Lemma \ref{lem:normalqhyp} and $T$ is a quasi-hyperplane section on $X$.
We set
\[
f := F (0, 0, z, t, w),
\]
so that $L_{xy}$ is isomorphic to the hypersurface in $\mathbb{P} (a_2, a_3, a_4)_{z, t, w}$ defined by $f = 0$.
\subsubsection{The family $\mathcal{F}_7$}
\label{sec:smptL2-7}
We have
\[
f = w^2 \ell (z,t) + h (z,t),
\]
where $\ell, h$ are linear and quadratic forms in $z, t$, respectively.
By the quasi-smoothness of $X$, we have $\ell (z, t) \ne 0$, and $h (z,t)$ does not have a multiple component.
\begin{itemize}
\item Case (i): $h$ is not divisible by $\ell$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $h$ is divisible by $\ell$.
Replacing $z$ and $t$, we may assume $\ell (z,t) = z$.
We can write $h = z c (z,t)$, where $c (z,t)$ is a cubic form in $z, t$.
Note that $c (z,t)$ is not divisible by $z$ since $h (z,t) = z c (z,t)$ does not have a multiple component, and we can assume $c (0,t) = - t^3$ by re-scaling $t$.
In this case $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + c (z,t) = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve since $c (z, t)$ does not have a multiple component.
We have $\Gamma \cap \Delta = \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!0\!:\!1\!:\!1) \in \operatorname{Sm} (X)$.
We claim that $S$ is quasi-smooth (and hence smooth) at $\mathsf{q}$.
We have $(\partial F/\partial z) (\mathsf{q}) = (\partial F/\partial t)(\mathsf{q}) = (\partial F/\partial w) (\mathsf{q}) = 0$.
Hence at least one of $(\partial F/\partial x)(\mathsf{q})$ and $(\partial F/\partial y)(\mathsf{q})$ is nonzero by the quasi-smoothness of $X$.
By choosing $x$ and $y$, we may assume that $S = H_x$ and $(\partial F/\partial y)(\mathsf{q}) \ne 0$.
It then follows that $S$ is quasi-smooth at $\mathsf{q}$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t, \mathsf{p}_w\}$.
Thus the assumptions of Lemma \ref{lem:Lredcp1} are satisfied and we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_9$}
\label{sec:smptL2-9}
We have
\[
f = c (t,w) + z^3 \ell (t,w),
\]
where $\ell = \ell (t,w)$ and $c = c (t,w)$ are linear and cubic forms in $t, w$ respectively.
By the quasi-smoothness of $X$, $c (t, w)$ does not have a multiple component.
\begin{itemize}
\item Case (i): $\ell (t,w) \ne 0$ and $c (t,w)$ is not divisible by $\ell (t,w)$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\ell (t, w) \ne 0$ and $c (t, w)$ is divisible by $\ell (t,w)$.
We write $c (t,w) = \ell (t, w) q (t, w)$, where $q (t,w)$ is a quadratic form in $t, w$ which is not divisible by $\ell (t,w)$.
Replacing $t$ and $w$, we may assume $\ell = t$, that is, $f = w (q (t,w) + z^3)$.
We may also assume $q (0,w) = - w^2$ since $c (t,w)$ does not have a multiple component.
In this case $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = q + z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve since $q (t,w)$ is not a square of a linear form.
We have $\Gamma \cap \Delta = \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!1\!:\!0\!:\!1) \in \operatorname{Sm} (X)$.
By the similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we can conclude that $S$ is quasi-smooth at $\mathsf{q}$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\ell (t,w) = 0$.
In this case $f = \ell_1 \ell_2 \ell_3$, where $\ell_1, \ell_2, \ell_3$ are linear forms in $t, w$ which are not mutually proportional, and $T|_S = \Gamma_1 + \Gamma_2 + \Gamma_3$, where $\Gamma_1, \Gamma_2, \Gamma_3$ are as follows.
\begin{itemize}
\item For $i = 1, 2, 3$, $\Gamma_i = (x = y = \ell_i = 0)$ is a quasi-line and $\operatorname{Sing}_{\Gamma_i} = \{1 \times \frac{1}{2} (1, 1), 1 \times \frac{1}{3} (1, 2)\}$.
\item $\Gamma_i \cap \Gamma_j = \{\mathsf{p}_z\} \subset \operatorname{Sing} (X)$ for $i \ne j$.
Moreover, $S$ is quasi-smooth at $\mathsf{p}_z$ since $S \in |A|$ is general.
\end{itemize}
We can compute $(\Gamma_i^2)_S = -5/6$ by the method explained in Remark \ref{rem:compselfint} and then we have $(\Gamma_i \cdot \Gamma_j)_S = 1/2$ for $i \ne j$ by considering $(\Gamma_l \cdot T|_S)_S$ for $l = 1, 2, 3$.
Thus the intersection matrix of $\Gamma_1, \Gamma_2, \Gamma_3$ is given by
\[
((\Gamma_i \cdot \Gamma_j)_S) =
\begin{pmatrix}
- \frac{5}{6} & \frac{1}{2} & \frac{1}{2} \\[1mm]
\frac{1}{2} & - \frac{5}{6} & \frac{1}{2} \\[1mm]
\frac{1}{2} & \frac{1}{2} & - \frac{5}{6}
\end{pmatrix}
\]
and it satisfies the condition $(\star)$.
By Lemma \ref{lem:mtdLred}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{1, \frac{3}{4} \right\} = \frac{3}{4}
\]
for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{12}$}
We have $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
Hence rescaling $w$, we can write $f = w^2 z + \alpha w t^2 + \lambda w z^3 + \beta t^2 z^2 + \mu z^5$, where $\alpha, \beta, \lambda, \mu \in \mathbb{C}$.
We can eliminate the monomial $z^5$ by replacing $w$ and hence we assume $\mu = 0$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_z$, we have $\lambda \ne 0$ and we may assume $\lambda = -1$ by rescaling $z$.
Thus we can write
\[
f = w^2 z + \alpha w t^2 - w z^3 + \beta t^2 z^2.
\]
\begin{itemize}
\item Case (i): $\alpha \ne 0$ and $\beta \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha \ne 0$ and either $\beta = 0$ or $\beta = \alpha$.
When $\beta = \alpha$, we replace $w \mapsto w - z^2$ and $z \mapsto - z$.
After this replacement, we may assume $\beta = 0$.
Moreover we may assume $\alpha = 1$ by re-scaling $t$.
Then we have $f = w (w z + t^2 - z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w z + t^2 - z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve.
We have $\Gamma \cap \Delta = \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!1\!:\!1\!:\!0) \in \operatorname{Sm} (X)$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{q}$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = 0$ and $\beta \ne 0$.
Re-scaling $t$, we may assume $\beta = 1$.
Then we have $f = z (w^2 + w z^2 + t^2 z)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + w z^2 + t^2 z = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$.
By the similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{p}_t$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iv): $\alpha = \beta = 0$.
In this case $f = z w (w + z^2)$ and $T|_S = \Gamma_1 + \Gamma_2 + \Gamma_3$, where $\Gamma_1, \Gamma_2, \Gamma_3$ are as follows.
\begin{itemize}
\item $\Gamma_1 = (x = y = z = 0)$ is a quasi-line of degree $1/12$ and $\operatorname{Sing}_{\Gamma_1} (S) = \{1 \times \frac{1}{3} (1,3), 1 \times \frac{1}{4} (1,3)\}$.
\item $\Gamma_2 = (x = y = w = 0)$ and $\Gamma_3 = (x = y = w + z^2 = 0)$ are quasi-lines of degree $1/6$ and $\operatorname{Sing}_{\Gamma_i} (S) = \{1 \times \frac{1}{2} (1,1), \frac{1}{3} (1,2)\}$ for $i = 2, 3$.
\item For $1 \le i < j \le 3$, we have $\Gamma_i \cap \Gamma_j = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$.
Moreover, $S$ is quasi-smooth at $\mathsf{p}_t$ since $S \in |A|$ is general.
\end{itemize}
By the similar computation as in Case (iii) of \S \ref{sec:smptL2-9}, the intersection matrix of $\Gamma_1, \Gamma_2, \Gamma_3$ is given by
\[
\begin{pmatrix}
- \frac{7}{12} & \frac{1}{3} & \frac{1}{3} \\[1mm]
\frac{1}{3} & - \frac{5}{6} & \frac{2}{3} \\[1mm]
\frac{1}{3} & \frac{2}{3} & - \frac{5}{6}
\end{pmatrix}
\]
and it satisfies the condition $(\star)$.
By Lemma \ref{lem:mtdLred}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
3/4, & \text{if $\mathsf{p} \in \Gamma_1 \cap \operatorname{Sm} (X)$}, \\
4/5, & \text{if $\mathsf{p} \in \Gamma_i \cap \operatorname{Sm} (X)$ for $i = 2, 3$}
\end{cases}
\]
Thus, $\alpha_{\mathsf{p}} (X) \ge 3/4$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{13}$}
We have
\[
f = \alpha w t^2 + \beta w z^3 + \gamma t^3 z + \delta t z^4,
\]
where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
Note that $(\alpha,\gamma) \ne (0,0)$ since $X$ is quasi-smooth at $\mathsf{p}_t$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$, $(\beta, \delta) \ne (0,0)$ and $(\alpha,\gamma)$ is not proportional to $(\beta,\delta)$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_w \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha \ne 0$, $(\beta, \delta) \ne (0,0)$ and $(\alpha,\gamma)$ is proportional to $(\beta,\delta)$.
In this case $f = (\alpha w + \gamma t z)(t^2 + \varepsilon z^3)$, where $\varepsilon := \beta/\alpha \in \mathbb{C}$ is non-zero.
Replacing $w$ and $z$, we may assume $f = w (t^2 - z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad \Delta = (x = y = t^2 - z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!1\!:\!1\!:\!0) \in \operatorname{Sm} (X)$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{q}$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha \ne 0$ and $(\beta,\delta) = (0,0)$.
In this case $f = t^2 (\alpha w + \gamma t z)$ and we may assume $f = t^2 w$ by replacing $w$.
We can write
\[
F = f_1 (z,w) x + f_2 (z,w) y + t^2 w + g (x,y,z,t,w),
\]
where $f_1, f_2 \in \mathbb{C} [z,w]$ and $g \in \mathbb{C} [x,y,z,t,w]$ are homogeneous polynomials such that $g \in (x, y) \cap (x,y,t)^2$.
By Lemma \ref{lem:Lredcp2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iv): $\alpha = 0$ and $\beta \ne 0$.
Note that $\gamma \ne 0$.
In this case $f = z (\beta w z^2 + \gamma t^3 + \delta t z^3)$.
Then $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = \beta w z^2 + \gamma t^3 + \delta t z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\} \subset \operatorname{Sing} (X)$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{p}_w$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t, \mathsf{p}_w\}$.
Then, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (v): $\alpha = \beta = 0$ and $\delta \ne 0$.
Note that $\gamma \ne 0$.
In this case $f = z t (\gamma t^2 + \delta z^3)$ and $T|_S = \Gamma_1 + \Gamma_2 + \Gamma_3$, where $\Gamma_1, \Gamma_2, \Gamma_3$ are as follows.
\begin{itemize}
\item $\Gamma_1 = (x = y = z = 0)$ is a quasi-line of degree $1/15$ and $\operatorname{Sing}_{\Gamma_1} (S) = \{1 \times \frac{1}{3} (1,2), 1 \times \frac{1}{5} (2,3)\}$.
\item $\Gamma_2 = (x = y = t = 0)$ is a quasi-line of degree $1/10$ and $\operatorname{Sing}_{\Gamma_2} (S) = \{1 \times \frac{1}{2} (1,1), 1 \times \frac{1}{5} (2,3)\}$.
\item $\Gamma_3 = (x = y = \gamma t^2 + \delta z^3 = 0)$ is an irreducible smooth curve of degree $1/5$.
\item For $ 1 \le i < j \le 3$, we have $\Gamma_i \cap \Gamma_j = \{\mathsf{p}_w\} \subset \operatorname{Sing} (X)$.
Moreover $S$ is quasi-smooth at $\mathsf{p}_w$.
\end{itemize}
We compute $(\Gamma_1^2)_S = - 8/15$ and $(\Gamma_2^2)_S = - 15/19$ by the method explained in Remark \ref{rem:compselfint}.
We can choose $z, t$ as orbifold coordinates of $S$ at $\mathsf{p}_w$.
It follows that $\Gamma_1, \Gamma_2$ intersect transversally at the point over $\mathsf{p}_w$ on the orbifold chart of $S$ at $\mathsf{p}_w$ and we have $(\Gamma_1 \cdot \Gamma_2)_S = 1/5$.
Then, by taking intersections with $T|_S = \Gamma_1 + \Gamma_2 + \Gamma_3$ with $\Gamma_i$ for $i = 1,2,3$, we see that the intersection matrix of $\Gamma_1, \Gamma_2, \Gamma_3$ is given by
\[
\begin{pmatrix}
- \frac{8}{15} & \frac{1}{5} & \frac{2}{5} \\[1mm]
\frac{1}{5} & - \frac{7}{10} & \frac{3}{5} \\[1mm]
\frac{2}{5} & \frac{3}{5} & - 1
\end{pmatrix}
\]
and it satisfies the condition $(\star)$.
By Lemma \ref{lem:mtdLred},
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
10/13, & \text{if $\mathsf{p} \in \Gamma_1 \cap \operatorname{Sm} (X)$}, \\
15/19, & \text{if $\mathsf{p} \in \Gamma_2 \cap \operatorname{Sm} (X)$}, \\
6/7, & \text{if $\mathsf{p} \in \Gamma_3 \cap \operatorname{Sm}(X)$}.
\end{cases}
\]
Thus, we have $\alpha_{\mathsf{p}} (X) \ge 10/13$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (vi): $\alpha = \beta = \delta = 0$.
In this case $\gamma \ne 0$ and we may assume that $f = t^3 z$ by re-scaling $z$.
We can write
\[
F = f_1 (z,w) x + f_2 (z,w) y + t^3 z + g (x,y,z,t,w),
\]
where $f_1, f_2 \in \mathbb{C} [z,w]$ and $g \in \mathbb{C} [x,y,z,t,w]$ are homogeneous polynomials such that $g \in (x,y) \cap (x,y,t)^2$.
By Lemma \ref{lem:Lredcp2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{15}$}
We have $w^2 \in f$.
Replacing $w$, we may assume that the coefficients of $z^6$ and $t^4$ are both $0$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_z, \mathsf{p}_t \in X$, we have $t^2 w, z^3 w \in F$.
Hence, by rescaling $w, t$ and $z$, we can write
\[
f = w^2 + (t^2 - z^3)w + \alpha t^2 z^3
\]
for some $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
Replacing $w$ and rescaling $z$, we may assume $f = w(w + t^2 - z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w + t^2 - z^3 = 0).
\]
We see that $\Gamma$ and $\Delta$ are both quasi-lines.
We have $\Gamma \cap \Delta = \{\mathsf{q}\}$, where $\mathsf{q} = (0\!:\!0\!:\!1\!:\!1\!:\!0) \in \operatorname{Sm} (X)$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{q}$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{20}$}
We have $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
Hence we can write
\[
f = w^2 z + \alpha w t^2 + \beta t z^3,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$ and $\beta \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha \ne 0$ and $\beta = 0$.
We have $f = w (w z + \alpha t^2)$ and thus $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w z + \alpha t^2 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\} \subset \operatorname{Sing} (X)$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{p}_z$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = 0$ and $\beta \ne 0$.
We have $f = z (w^2 + \beta t z^2)$ and thus $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + \beta t z^2 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\}$.
By a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}, we conclude that $S$ is quasi-smooth at $\mathsf{p}_t$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iv): $\alpha = \beta = 0$.
In this case $f = w^2 z$ and we can write
\[
F = f_1 (z,t) x + f_2 (z,t) y + w^2 z + g (x,y,z,t,w),
\]
where $f_1, f_2 \in \mathbb{C} [z,t]$ and $g \in \mathbb{C} [x,y,z,t,w]$ are homogeneous polynomials such that $g \in (x,y) \cap (x,y,w)^2$.
By Lemma \ref{lem:Lredcp2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{23}$}
We have $w^2 t \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
Hence we can write
\[
f = w^2 t + \alpha w z^3 + \beta t^2 z^2,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$ and $\beta \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha \ne 0$ and $\beta = 0$.
We have $f = w (w t + \alpha z^3)$ and thus $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w t + \alpha z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_t$ since $t^3 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = 0$ and $\beta \ne 0$.
We have $f = t (w^2 + \beta t z^2)$ and thus $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = w^2 + \beta t z^2 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\}$, and $S$ is quasi-smooth at $\mathsf{p}_z$ since $z^4 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iv): $\alpha = \beta = 0$.
In this case $f = w^2 t$ and we can write
\[
F = f_1 (z,t) x + f_2 (z,t) + w^2 t + g (x,y,z,t,w),
\]
where $f_1, f_2 \in \mathbb{C} [z,t]$ and $g \in \mathbb{C} [x,y,z,t,w]$ are homogeneous polynomials such that $g \in (x,y) \cap (x,y,w)^2$.
By Lemma \ref{lem:Lredcp2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{24}$}
We have $t^3 \in F$ and, by rescaling $t$, we can write
\[
f = \alpha w z^4 + t^3 + \beta t z^5,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_w \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$ and $\beta \ne 0$.
By rescaling $z$, we may assume $f = t (t^2 + z^5)$.
Then $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = t^2 + z^5 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\}$, and $S$ is quasi-smooth at $\mathsf{p}_w$ by a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = \beta = 0$.
In this case, $f = t^3$ and the defining polynomial $F$ of $X$ can be written as
\[
F = w^2 \ell_1 (x,y) + t^3 + z^7 \ell_2 (x,y) + w h_8 (x,y,z,t) + h_{15} (x,y,z,t),
\]
where $h_8, h_{15} \in \mathbb{C} [x,y,z,t]$ are homogeneous polynomials of degrees $8, 15$, respectively, such that $z^4 \notin h_8$ and $t^3, z^7 x, z^7 y \notin h_{15}$, and $\ell_1, \ell_2$ are linear forms in $x, y$.
Note that $h_8, h_{15} \in (x,y) \cap (x,y,t)^2$.
By the quasi-smoothness of $X$, we see that $\ell_1$ and $\ell_2$ are linearly independent.
Replacing $x, y$, we can assume that
\[
F = w^2 x + t^3 - z^7 y + g,
\]
where $h = w h_8 + h_{15} \in (x, y) \cap (x, y, t)^2$.
Thus, by Lemma \ref{lem:Lnonredu2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{25}$}
We have $z^5 \in F$ and, by rescaling $z$, we can write
\[
f = \alpha w t^2 + \beta t^3 z + z^5,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_w \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
By the quasi-smoothness of $X$ at $\mathsf{p}_t$, we have $\beta \ne 0$, and hence we may assume $\beta = 1$.
Then $f = z ( t^3 + z^5)$ and we have $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = t^3 + z^5 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\}$, and $S$ is quasi-smooth at $\mathsf{p}_w$ by a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{29}$}
We have $w^2 \in F$ and, by rescaling $w$, we can write $f = w^2 + \lambda w z^4 + \alpha t^2 z^3 + \mu z^8$, where $\alpha, \lambda, \mu \in \mathbb{C}$.
By replacing $w$, we can eliminate the term $\mu z^8$, i.e.\ we may assume $\mu = 0$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_z$, we have $w z^4 \in F$, i.e.\ $\lambda \ne 0$.
Thus we can write
\[
f = w^2 + w z^4 + \alpha t^2 z^3.
\]
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
Then we have $f = w(w+z^4)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w + z^4).
\]
We see that $\Gamma$ and $\Delta$ are both quasi-lines.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\}$, and $S$ is quasi-smooth at $\mathsf{p}_t$ by a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{30}$}
We have $w^2 \in F$ and, by rescaling $w$, we can write $f = w^2 + \lambda w t^2 + \mu t^4 + \alpha t z^4$, where $\alpha, \lambda, \mu \in \mathbb{C}$.
We may assume $\mu = 0$ by replacing $w$, and then we have $w t^2 \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_t$.
Thus we can write
\[
f = w^2 + w t^2 + \alpha t z^4.
\]
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
Then we have $f = w(w+t^2)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w + t^2 = 0).
\]
We see that $\Gamma$ and $\Delta$ are both quasi-lines.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\}$, and $S$ is quasi-smooth at $\mathsf{p}_z$ by a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{31}$}
We have $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$, and we have $z^4 \in F$.
Rescaling $w$ and $z$, we can write
\[
f = w^2 z + \alpha w t^2 - z^4,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = z (w^2 - z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 - z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_t$ by a similar argument as in Case (ii) of \S \ref{sec:smptL2-7}.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{32}$}
We have $t^4 \in F$, and we can write
\[
f = \alpha w z^3 + t^4 + \beta t z^4,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_w \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$ and $\beta \ne 0$.
Re-scaling $z$, we may assume $\beta = 1$ and $f = t (t^3 + z^4)$.
Then $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = t^3 + z^4 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\}$, and $S$ is quasi-smooth at $\mathsf{p}_w$ since $w^2 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} = \{\mathsf{p}_z, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = \beta = 0$.
In this case $f = t^4$.
Since $w z^3, t z^4 \notin F$, we have $z^5 x \in F$, and we can write
\[
F = w^2 y + t^4 + z^5 x + w h_9 (x,y,z,t) + h_{16} (x,y,z,t),
\]
where $g_i \in \mathbb{C} [x,y,z,t]$ is a homogeneous polynomial of degree $i$ such that $z^3 \notin h_9$ and $t^4, t z^4, z^5 x \notin h_{16}$.
Note that $h_9, h_{16} \in (x, y) \cap (x, y, t)^2$.
Thus, by Lemma \ref{lem:Lnonredu2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{33}$}
We have $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$, and we can write
\[
f = w^2 z + \alpha w t^2 + \beta t z^4,
\]
where $\alpha, \beta \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$ and $\beta \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha \ne 0$ and $\beta = 0$.
In this case $f = w (w z + \alpha z^2)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w z + \alpha z^2 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible quasi-smooth curve.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_z$ since $z^5 t \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iii): $\alpha = 0$ and $\beta \ne 0$.
In this case $f = z (w^2 + \beta t z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + \beta t z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_t$ since $t^3 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (iv): $\alpha = \beta = 0$.
In this case $f = w^2 z$ and we can write
\[
F = f_1 (z,t) x + f_2 (z,t) y + w^2 z + g (x,y,z,t,w),
\]
where $f_1, f_2 \in ^mbC [z,t]$ and $g \in \mathbb{C} [x,y,z,t,w]$ are homogeneous polynomials such that $g \in (x,y) \cap (x,y,w)^2$.
By Lemma \ref{lem:Lredcp2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{37}$}
\label{sec:smptL2-37}
We have $w^2 \in F$ and we can write $f = w^2 + \lambda w z^3 + \alpha t^3 z^2 + \mu z^6$, where $\alpha, \lambda, \mu \in \mathbb{C}$.
Replacing $w$, we may assume $\mu = 0$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_z$, we have $\lambda \ne 0$.
Re-scaling $z$, we can write
\[
f = w^2 + w z^3 + \alpha t^3 z^2.
\]
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = w(w+z^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w + z^3 = 0),
\]
are both quasi-lines.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\}$, and $S$ is quasi-smooth at $\mathsf{p}_t$ since $t^4 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{38}$}
We have $z^6 \in F$, and we can write
\[
f = \alpha w t^2 + \beta t^3 z + z^6,
\]
where $\alpha, \beta \in \mathbb{C}$.
Note that we have $(\alpha,\beta) \ne (0,0)$ by the quasi-smoothness of $X$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_w \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap X_{\operatorname{sm}}$.
\item Case (ii): $\alpha = 0$.
Note that $\beta \ne 0$.
In this case $f = z (\beta t^3 + z^5)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = \beta t^3 + z^5 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_w\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_w$ since $w^2 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{39}$}
\label{sec:smptL2-39}
We have $w^3 \in F$ and $w z^3 \in F$ by the quasi-smoothness of $X$.
Rescaling $w$ and $z$, we can write
\[
f = w^3 + w z^3 + \alpha t^2 z^2,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = z (w^2 + z^3)$ and we have $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + z^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} = \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\}$, and $S$ is quasi-smooth at $\mathsf{p}_t$ since $t^3 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{40}$}
\label{sec:smptL2-40}
We have $w^2 t \in F$ and $t^3 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$ and $\mathsf{p}_t$.
Rescaling $w$ and $z$, we can write
\[
f = w^2 t + \alpha w z^3 + t^3 z,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = t (w^2 + t^2 z)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = w^2 + t^2 z = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_z\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\}$, and $S$bis quasi-smooth at $\mathsf{p}_z$ since $z^4 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{42}$}
\label{sec:smptL2-42}
We have $w^2 \in F$ and we can write $f = w^2 + \lambda w t^2 + \mu t^4 + \alpha t z^5$, where $\alpha, \lambda, \mu \in \mathbb{C}$.
Replacing $w$, we may assume $\mu = 0$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_t$, we have $\lambda \ne 0$.
Rescaling $t$, we can write
\[
f = w^2 + w t^2 + \alpha t z^5.
\]
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = w(w+t^2)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = w = 0), \quad
\Delta = (x = y = w + t^2 = 0).
\]
We see that $\Gamma$ and $\Delta$ are both quasi-lines.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\} \subset \operatorname{Sing} (X)$, and $S$ is quasi-smooth at $\mathsf{p}_z$ since $z^6 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z, \mathsf{p}_t\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{49}$}
We have $w^3 \in F$, and we can write
\[
f = w^3 + \alpha t z^3,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = w^3$ and, after replacing $x, y$ suitably, the defining polynomial $F$ of $X$ can be written as
\[
F = w^3 + t^3 y - z^3 x + g_{21} (x,y,z,t,w),
\]
where $g_{21} \in \mathbb{C} [x,y,z,t,w]$ is a homogeneous polynomial of degree $21$ such that $g \in (x,y) \cap (x,y,w)^2$.
By Lemma \ref{lem:Lnonredu2}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{50}$}
We have $w^2 \in F$, and we can write
\[
f = w^2 + \alpha t z^5,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
We have $S \cdot T = 2 \Gamma$, where $\Gamma = (x = y = w = 0)$.
By Lemma \ref{lem:nsptLpuredouble}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{52}$}
We have $w^2 \in F$, and we can write
\[
f = w^2 + \alpha t^2 z^3,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\{\mathsf{p}_z, \mathsf{p}_t\} \subset \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
We have $S \cdot T = 2 \Gamma$, where $\Gamma = (x = y = w = 0)$.
By Lemma \ref{lem:nsptLpuredouble}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{58}$}
We have $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
Also we have $z^6 \in F$.
Rescaling $w$ and $z$, we can write
\[
f = w^2 z + \alpha w t^2 + z^6,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = z (w^2 + z^5)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = z = 0), \quad
\Delta = (x = y = w^2 + z^5 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \{\mathsf{p}_t\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_t\}$, and $S$ is quasi-smooth at $\mathsf{p}_t$ since $t^3 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_t,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{60}$}
We have $w^2 t \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
Also we have $t^4 \in F$.
Rescaling $w$ and $t$, we can write
\[
f = w^2 t + \alpha w z^3 + t^4,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and smooth.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
In this case $f = t (w^2 + t^3)$ and $T|_S = \Gamma + \Delta$, where
\[
\Gamma = (x = y = t = 0), \quad
\Delta = (x = y = w^2 + t^3 = 0).
\]
We see that $\Gamma$ is a quasi-line and $\Delta$ is an irreducible curve which is quasi-smooth along $\Delta \setminus \mathsf{p}_z\} \supset \Delta \cap \operatorname{Sm} (X)$.
We have $\Gamma \cap \Delta = \{\mathsf{p}_z\}$, and $S$ is quasi-smooth at $\mathsf{p}_z$ since $z^4 y \in F$ and $S = H_x$.
Finally we have $\operatorname{Sing}_{\Gamma} (X) = \{\mathsf{p}_z,\mathsf{p}_w\}$.
Thus, by Lemma \ref{lem:Lredcp1}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{63}$}
\label{sec:smptL2-63}
We have $w^2 \in F$, and we can write
\[
f = w^2 + \alpha t z^6,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $
\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
We have $S \cdot T = 2 \Gamma$, where $\Gamma = (x = y = w = 0)$.
By Lemma \ref{lem:nsptLpuredouble}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\subsubsection{The family $\mathcal{F}_{64}$}
We have $w^2 \in F$, and we can write
\[
f = w^2 + \alpha t z^4,
\]
where $\alpha \in \mathbb{C}$.
\begin{itemize}
\item Case (i): $\alpha \ne 0$.
In this case $S \cdot T = L_{xy}$ is irreducible and is smooth outside $\mathsf{p}_t \in \operatorname{Sing} (X)$.
By Lemma \ref{lem:Linteg}, we have $\alpha_{\mathsf{p}} (X) = 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\item Case (ii): $\alpha = 0$.
We have $S \cdot T = 2 \Gamma$, where $\Gamma = (x = y = w = 0)$.
By Lemma \ref{lem:nsptLpuredouble}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{itemize}
\section{Smooth points on $H_x$ for families with $1 < a_1 = a_2$} \label{sec:smptH}
The aim of this section is to prove the following.
\begin{Prop} \label{prop:smptH}
Let $X = X_d \subset \mathbb{P} (1, a_1, \dots, a_4)$, $a_1 \le \dots \le a_4$, be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ such that $1 < a_1 = a_2$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any smooth point $\mathsf{p} \in X$ contained in $H_x$.
\end{Prop}
Note that a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ satisfies the assumption of Proposition \ref{prop:smptH} if and only if
\[
\mathsf{i} \in \{18, 22, 28\}.
\]
\subsection{The family $\mathcal{F}_{18}$}
This subsection is devoted to the proof of Proposition \ref{prop:smptH} for the family $\mathcal{F}_{18}$.
Let $X = X_{12} \subset \mathbb{P} (1, 2, 2, 3, 5)$ be a member of $\mathcal{F}_{18}$.
By the quasi-smoothness of $X$, We have $t^4 \in F$ and we may assume $\operatorname{coeff}_F (t^4) = 1$ by rescaling $t$.
We have $(x = y = z = 0) \cap X = \{\mathsf{p}_w\} \subset \operatorname{Sing} (X)$.
Hence we may assume $\mathsf{p} \in H_y$ and $\mathsf{p} \notin H_z$ after possibly replacing $y$ and $z$, and we can write $\mathsf{p} = (0\!:\!0\!:\!1\!:\!\lambda\!:\!\mu)$ for some $\lambda, \mu \in \mathbb{C}$.
We can write
\[
F (0, 0, z, t, w) = \alpha w^2 z + \beta w t z^2 + t^4 + \gamma t^2 z^3 + \delta z^6,
\]
where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
We will derive a contradiction by assuming $\alpha_{\mathsf{p}} (X) < 1/2$.
By the assumption, there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ such that $\operatorname{lct}_{\mathsf{p}} (X;D) < 1/2$.
Suppose $\lambda \ne 0$.
Then, by replacing $w$ by $\lambda w - \mu z t$, we may assume $\mathsf{p} = (0\!:\!0\!:\!1\!:\!\lambda\!:\!0)$.
Let $S$ be a general member of the pencil $|\mathcal{I}_{\mathsf{p}} (2 A)|$ so that $\frac{1}{2} S \ne D$.
We can take a $\mathbb{Q}$-divisor $T \in |5 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ since $\{x, y, w\}$ isolates $\mathsf{p}$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = 2 \cdot 5 \cdot (A^3) = 2,
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/2$.
This is impossible and we have $\lambda = 0$.
By rescaling $w$, we may assume $\mathsf{p} = (0\!:\!0\!:\!1\!:\!0\!:\!1)$.
Suppose $\alpha \ne 0$.
Then we have $\delta = - \alpha$ since $F (\mathsf{p}) = 0$.
In this case $H_x$ is smooth at $\mathsf{p}$ since $(\partial F/\partial z) (\mathsf{p}) = - 5 \alpha \ne 0$.
In particular $H_x \ne D$.
We can take a $\mathbb{Q}$-divisor $T \in |3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$ since $\{x, y, t\}$ isolates $\mathsf{p}$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = 3 (A^3) = \frac{3}{5},
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 5/3$.
This is impossible and we have $\alpha = 0$.
Note that $\delta = 0$ since $F (\mathsf{p}) = 0$, and we have
\[
F (0,0,z,t,w) = t (\beta w z^2 + t^3 + \gamma t z^3).
\]
We claim $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$.
We set $\zeta := \operatorname{coeff}_F (w^2 y)$ and $\eta := \operatorname{coeff}_F (z^5 y)$.
By the quasi-smoothness of $X$, we see $\zeta, \eta \ne 0$ since $w^2 y, z^6 \notin F$.
We have
\[
\frac{\partial F}{\partial y} (\mathsf{p}) = \zeta + \eta, \quad
\frac{\partial F}{\partial t} (\mathsf{p}) = \beta.
\]
If either $\zeta + \eta \ne 0$ or $\beta \ne 0$, then we have $\operatorname{mult}_{\mathsf{p}} (H_x) = 1$.
If $\zeta + \eta = \beta = 0$, then we have $\operatorname{mult}_{\mathsf{p}} (H_x) = 2$ since the term $\zeta y (w^2 - z^5)$ appears in $F$.
Thus the claim is proved.
By the claim, we have $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ and in particular $D \ne H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |10 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$ since $\{x, y, t, w^2 - z^5\}$ isolates $\mathsf{p}$.
Then we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = 10 (A^3) = 2,
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/2$.
This is a contradiction and the proof is completed.
\subsection{The family $\mathcal{F}_{22}$}
This subsection is devoted to the proof of Proposition \ref{prop:smptH} for the family $\mathcal{F}_{22}$.
Let $X = X_{14} \subset \mathbb{P} (1, 2, 2, 3, 7)$ be a member of $\mathcal{F}_{22}$.
By the quasi-smoothness of $X$, we have $w^2 \in F$ and we may assume $\operatorname{coeff}_F (w^2) = 1$ by rescaling $w$.
We see that $(x = y = z = 0) \cap X = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$.
Hence we may assume $\mathsf{p} = (0\!:\!0\!:\!1\!:\!\lambda\!:\!\mu)$ for some $\lambda, \mu \in \mathbb{C}$ after possibly replacing $y$ and $z$.
We can write
\[
F (0,0,z,t,w) = w^2 + \alpha w t z^2 + \beta t^4 z + \gamma t^2 z^4 + \delta z^7,
\]
where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$.
We will derive a contradiction by assuming $\alpha_{\mathsf{p}} (X) < 1/2$.
By the assumption, there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ such that $\operatorname{lct}_{\mathsf{p}} (X;D) < 1/2$.
Let $S$ be a general member of the pencil $|\mathcal{I}_{\mathsf{p}} (2 A)|$ so that $\frac{1}{2} S \ne D$.
Suppose $\lambda = 0$.
In this case, we can take a $\mathbb{Q}$-divisor $T \in |3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ since $\{x, y, t\}$ isolates $\mathsf{p}$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = 2 \cdot 3 \cdot (A^3) = 1,
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$.
This is impossible and we have $\lambda \ne 0$.
Replacing $w$ by $\lambda w - \mu t z^2$, we may assume $\mu = 0$, that is, $\mathsf{p} = (0\!:\!0\!:\!1\!:\!\lambda\!:\!0)$.
We see that the set $\{x, y, t^2 - \lambda^2 z^3\}$ isolates $\mathsf{p}$.
It follows that we can take a $\mathbb{Q}$-divisor $T \in |6 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ which does not contain any component of $D \cdot S$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = 2 \cdot 6 \cdot (A^3) = 2,
\]
which implies $\alpha_{\mathsf{p}} (X) \ge 1/2$.
This is a contradiction and the proof is completed.
\subsection{The family $\mathcal{F}_{28}$}
This subsection is devoted to the proof of Proposition \ref{prop:smptH} for the family $\mathcal{F}_{28}$.
Let $X = X_{15} \subset \mathbb{P} (1, 3, 3, 4, 5)$ be a member of $\mathcal{F}_{28}$.
By the quasi-smoothness of $X$, we have $w^3 \in F$ and we may assume $\operatorname{coeff}_F (w^3) = 1$ by rescaling $w$.
We see that $(x = y = z = 0) \cap X = \{\mathsf{p}_t\} \subset \operatorname{Sing} (X)$.
Hence we may assume $\mathsf{p} = (0\!:\!0\!:\!1\!:\!\lambda\!:\!\mu)$ for some $\lambda, \mu \in \mathbb{C}$ after possibly replacing $y$ and $z$.
We can write
\[
F (0,0,z,t,w) = w^3 + \alpha w t z^2 + \beta t^3 z + \gamma z^5,
\]
where $\alpha, \beta, \gamma \in \mathbb{C}$.
We will derive a contradiction by assuming $\alpha_{\mathsf{p}} (X) < 1/2$.
By the assumption, there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ such that $\operatorname{lct}_{\mathsf{p}} (X;D) < 1/2$.
Let $S$ be a general member of the pencil $|\mathcal{I}_{\mathsf{p}} (3 A)|$ so that $\operatorname{Supp} (S) \ne \operatorname{Supp} (D)$.
Suppose $\lambda \ne 0$ and $\mu \ne 0$.
In this case the set $\{x, y, \mu t^2 - \lambda^2 w z\}$ isolates $\mathsf{p}$, and we can take a $\mathbb{Q}$-divisor $T \in |8 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = 3 \cdot 8 \cdot (A^3) = 2,
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1/2$.
This is impossible and we have either $\lambda = 0$ or $\mu = 0$.
Suppose $\lambda = 0$.
In this case we can take a $\mathbb{Q}$-divisor $T \in |4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ since $\{x, y, t\}$ isolates $\mathsf{p}$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = 3 \cdot 4 \cdot (A^3) = 1,
\]
which implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$.
This is impossible.
We have $\lambda \ne 0$ and $\mu = 0$.
In this case we may assume $\lambda = 1$ by rescaling $t$, that is, we may assume $\mathsf{p} = (0\!:\!0\!:\!1\!:\!1\!:\!0)$.
We claim $\operatorname{mult}_{\mathsf{p}} (H_x) \le 2$.
We set $\zeta := \operatorname{coeff}_F (t^3 y)$ and $\eta := \operatorname{coeff}_F (z^4 y)$.
We have $\beta + \gamma = 0$ since $F (\mathsf{p}) = 0$.
Then,
\[
\frac{\partial F}{\partial z} (\mathsf{p}) = \beta + 5 \gamma = 4 \gamma, \quad
\frac{\partial F}{\partial y} (\mathsf{p}) = \zeta + \eta.
\]
If either $\gamma \ne 0$ or $\zeta + \eta \ne 0$, then we have $\operatorname{mult}_{\mathsf{p}} (H_x) = 1$.
It remains to consider the case where $\gamma = \zeta + \eta = 0$.
Note that we have $\beta = 0$ since $\beta + \gamma = 0$.
By the quasi-smoothness of $X$ at $\mathsf{p}_t$, we have $\zeta \ne 0$.
Then, we see that $\operatorname{mult}_{\mathsf{p}} (H_x) = 2$ since the term $\zeta y (t^3 - z^4)$ appears in $F$.
Thus the claim is proved.
By the claim, we have $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ and in particular $D \ne H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |12 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$ since $\{x, y, w, t^3 - z^4\}$ isolates $\mathsf{p}$.
Then, we have
\[
\operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_x \cdot T)_{\mathsf{p}} \le (D \cdot H_x \cdot T) = 12 (A^3) = 1,
\]
which implies $\alpha_{\mathsf{p}} (X) \ge 1$.
This is a contradiction and the proof is completed.
\chapter{Singular points}
\label{chap:singpt}
The aim of this chapter is to prove the following.
\begin{Thm} \label{thm:singpt}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}
\]
for any singular point $\mathsf{p} \in X$.
\end{Thm}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$.
Then the inequality $\alpha_{\mathsf{p}} (X) \ge 1/2$ will follow from Propositions \ref{prop:singptCP}, \ref{prop:lctsingptL}, \ref{prop:singptrem} for singular points $\mathsf{p} \in X$ which are not BI centers, from Proposition \ref{prop:alphaEI} for EI centers, and from Propositions \ref{prop:lctexcQI}, \ref{prop:lctdegQI} and \ref{prop:ndQIcent} for QI centers.
This will complete the proof of Theorem \ref{thm:singpt}.
\section{Non-BI centers}
Throughout the present section, let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
\subsection{Computation by $\operatorname{\overline{NE}} (Y)$}
\begin{Prop} \label{prop:singptCP}
Let $\mathsf{p} \in X$ be a singular point with subscript $\heartsuit$ in the $5$th column of Table \ref{table:main}, and let $\varphi \colon Y \to X$ be the Kawamata blowup at $\mathsf{p}$.
Then $(-K_Y)^2 \notin \operatorname{Int} \operatorname{\overline{NE}} (Y)$ and $\tilde{D} \sim -K_Y$ for the proper transform of a general member $D \in |A|$.
In particular, we have
\[
\alpha_{\mathsf{p}} (X) \ge 1.
\]
\end{Prop}
\begin{proof}
Let $r > 1$ be the index of the quotient singular point $\mathsf{p} \in X$.
For every instance, we have either $a_1 = 1$ or $d - 1$ is not divisible by $r$.
This means that we can take $x$ as a part of local orbifold coordinates of $X$ at $\mathsf{p}$, and hence $\tilde{D} \sim - K_Y$ for a general $D \in |A|$.
The point $\mathsf{p}$ is excluded as a maximal center by either Lemma 3.2.2 or 3.2.4 of \cite{CP17}.
We set $S := \tilde{D} \sim - K_Y$, where $D \in |A|$ is a general member.
If $\mathsf{p}$ is excluded by \cite[Lemma 3.2.2]{CP17}, then it follows from its proof that $(- K_Y)^2 = (-K_Y) \cdot S \notin \operatorname{Int} \operatorname{\overline{NE}} (Y)$.
If $\mathsf{p}$ is excluded by \cite[Lemma 3.2.4]{CP17}, then there exists a nef divisor $T$ on $Y$ such that $(T \cdot S \cdot - K_Y) \le 0$, which implies $(-K_Y)^2 = (-K_Y) \cdot S \notin \operatorname{Int} \operatorname{\overline{NE}} (Y)$.
The latter assertion follows from Lemma \ref{lem:singptNE}.
\end{proof}
\subsection{Computation by $L_{xy}$}
\begin{Prop} \label{prop:lctsingptL}
Let $\mathsf{p} \in X$ be a singular point with the subscript $\diamondsuit$ or $\diamondsuit'$ in the $5$th column of Table \ref{table:main}, and let $q = q_{\mathsf{p}}$ be the quotient morphism of $\mathsf{p} \in X$.
We denote by $r$ the index of the cyclic quotient singularity $\mathsf{p} \in X$.
Let $S \in |A|$ and $T \in |a_1 A|$ be general members.
Then the following assertions hold.
\begin{enumerate}
\item The pair $(X, S)$ is log canonical at $\mathsf{p}$.
\item The intersection $S \cap T$ is irreducible and we have $q^*S \cdot q^*T = \check{\Gamma}$, where $\check{\Gamma}$ is an irreducible and reduced curve such that
\[
0 < \operatorname{mult}_{\check{\mathsf{p}}} (\check{\Gamma}) \le a_1.
\]
\item We have
\[
r a_1 (A^3) \le
\begin{cases}
1, & \text{if the subscript of $\mathsf{p}$ is $\diamondsuit$}, \\
\frac{3}{2}, & \text{if the subscript of $\mathsf{p}$ is $\diamondsuit'$}.
\end{cases}
\]
\end{enumerate}
In particular,
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
1, & \text{if the subscript of $\mathsf{p}$ is $\diamondsuit$}, \\
\frac{3}{2}, & \text{if the subscript of $\mathsf{p}$ is $\diamondsuit'$}.
\end{cases}
\]
\end{Prop}
\begin{proof}
Let $\mathsf{p} \in X$ be as in the statement.
The assertion (3) can be checked individually and it remains to consider (1) and (2).
It is straightforward to check that $X$ is a member of a family $\mathcal{F}_{\mathsf{i}}$ which is listed in one of the Tables \ref{table:Lsmooth} and \ref{table:Lsing}.
It follows that $S \cap T = L_{xy}$ is irreducible.
It is easy to check that we may assume $\mathsf{p} = \mathsf{p}_{v}$ for some $v \in \{z, t, w\}$ after replacing coordinates.
Let $\rho = \rho_v \colon \breve{U}_v \to U_v$ be the orbifold chart.
We set $\breve{\Gamma} = (\breve{x} = \breve{y} = 0) \subset U_v$.
We see that $\breve{\Gamma}$ is an irreducible and reduced curve since so is $L_{xy}$, and that $\rho^*S \cdot \rho^*T = \breve{\Gamma}$.
Note that $q$ can be identified with $\rho$ over a suitable analytic neighborhood of $\mathsf{p} \in U_v$, and hence it is enough to prove the inequality $\operatorname{mult}_{\breve{\mathsf{p}}} (\breve{\Gamma}) \le a_1$ for the proof of (2).
If $X$ is listed in Table \ref{table:Lsmooth}, then $\breve{\Gamma}$ is irreducible and smooth by Lemma \ref{lem:Lirred}.
In this case $S$ is quasi-smooth at $\mathsf{p}$ and thus both (1) and (2) are clearly satisfied.
Suppose that $X$ is listed in Table \ref{table:Lsing}.
Then $X$ is a member of a family $\mathcal{F}_{\mathsf{i}}$, where
\[
\mathsf{i} \in \{44, 47, 61, 62, 65, 69, 77, 79, 83, 85\}.
\]
If $\mathsf{p}$ is not the unique singular point of $L_{xy}$ which is described in Table \ref{table:Lsing}, then (1) and (2) follow immediately.
Suppose that $\mathsf{p}$ is the unique singular point of $L_{xy}$.
Then we have $\mathsf{i} \in \{44, 61, 83\}$ and $\mathsf{p} = \mathsf{p}_t$.
By the equation given in Table \ref{table:Lsing}, we compute $\operatorname{mult}_{\breve{\mathsf{p}}} (\breve{\Gamma}) = 2$.
This shows (2) since $a_1 \ge 2$.
We see that $r = a_3$ does not divide $d-1$, which implies that $S$ is quasi-smooth at $\mathsf{p}$ and hence (1) follows.
Therefore (1), (2) and (3) are verified and the assertion on $\alpha_{\mathsf{p}} (X)$ follows from Lemma \ref{lem:exclL}.
\end{proof}
\subsection{Computation by isolating class}
\begin{Prop} \label{prop:singptic}
Let $\mathsf{p} \in X$ be a singular point with subscript $\clubsuit$ in the 5th column of Table \ref{table:main} which is also listed in Table \ref{table:isolsingpt}.
Then the set of coordinates given in the 5th column of Table \ref{table:isolsingpt} isolates $\mathsf{p}$.
In particular, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, c\} \ge \frac{1}{2},
\]
where $c$ is the number given in the 7th column of Table \ref{table:isolsingpt}.
\end{Prop}
\begin{proof}
Let $\mathcal{C}$ be the set of homogeneous coordinates given in the 5th column of Table \ref{table:isolsingpt}.
It is straightforward to check that
\[
\bigcap_{v \in \mathcal{C}} (v = 0) \cap X
\]
is a finite set of points including $\mathsf{p}$, which shows that $\mathcal{C}$ isolates $\mathsf{p}$.
Let $c$ be the number listed in the 7th column of Table \ref{table:isolsingpt} and assume that $\alpha_{\mathsf{p}} (X) < \min \{1, c\}$.
Then there exists an irreducible $\mathbb{Q}$-divisor $D \sim_{\mathbb{Q}} A$ such that $(X, c D)$ is not log canonical at $\mathsf{p}$.
In particular we have $\operatorname{omult}_{\mathsf{p}} (D) > 1/c$.
If $H_x$ (resp.\ $|n A|$ for some $n > 0$) is given in the 4th column of Table \ref{table:isolsingpt}, then we set $S := H_x$ (resp. we define $S$ to be a general member of $|n A|$).
We set $n = 1$ if $S = H_x$ so that $S \sim n A$ in any case.
Let $r$ be the index of the cyclic quotient singularity $\mathsf{p} \in X$.
We claim that $\operatorname{Supp} (D)$ is not contained in $S$.
This is clear when $S \in |nA|$ is a general member.
Suppose that $S = H_x$.
Then we see that $d-1$ is not divisible by $r$, which implies that $S = H_x$ is quasi-smooth at $\mathsf{p}$.
Hence $(X, S)$ is log canonical at $\mathsf{p}$ and we have $D \ne H_x$ as desired.
By the claim, $D \cdot S$ is an effective $1$-cycle on $X$.
Let $e$ be the integer given in the 6th column of Table \ref{table:isolsingpt}.
Note that $e = \max \{\, \deg v \mid v \in \mathcal{C} \,\}$ and
\[
r n e_{\max} (A^3) = \frac{1}{c}.
\]
There exists an irreducible $\mathbb{Q}$-divisor $T \sim_{\mathbb{Q}} e A$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot S$ since $\mathcal{C}$ isolates $\mathsf{p}$.
It follows that
\[
\frac{1}{c} < \operatorname{omult}_{\mathsf{p}} (D)
\le (q_{\mathsf{p}}^*D \cdot q_{\mathsf{p}}^*S \cdot q_{\mathsf{p}}^*T)_{\check{\mathsf{p}}}
\le r (D \cdot S \cdot T)
= r n e (A^3)
= \frac{1}{c},
\]
where $q = q_{\mathsf{p}}$ is the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q$.
This is a contradiction and the inequality $\alpha_{\mathsf{p}} (X) \ge \min \{1, c\}$ is proved.
\end{proof}
\begin{table}[h]
\renewcommand{1.35}{1.15}
\begin{center}
\caption{Isolating set}
\label{table:isolsingpt}
\begin{tabular}{ccccccc}
No. & Pt. & Type & $S$ & Isol.\ set & $e_{\max}$ & $c$\\
\hline
10 & $\mathsf{p}_t$ & $\frac{1}{3} (1,1,2)$ & $|A|$ & $\{x,y,z\}$ & $1$ & $1/2$ \\
23 & $\mathsf{p}_{yt}$ & $\frac{1}{2} (1,1,1)$ & $H_x$ & $\{x,z,w\}$ & $5$ & $6/7$ \\
23 & $\mathsf{p}_z$ & $\frac{1}{3} (1,1,2)$ & $H_x$ & $\{x,y,t,w\}$ & $5$ & $4/7$ \\
23 & $\mathsf{p}_t$ & $\frac{1}{4} (1,1,3)$ & $H_x$ & $\{x,y,z\}$ & $3$ & $5/7$ \\
29 & $\mathsf{p}_{zw}$ & $\frac{1}{2} (1,1,1)$ & $|A|$ & $\{x,y,t\}$ & $5$ & $1/2$ \\
29 & $\mathsf{p}_t$ & $\frac{1}{5} (1,2,3)$ & $|A|$ & $\{x,y,z\}$ & $2$ & $1/2$ \\
31 & $\mathsf{p}_{zw}$ & $\frac{1}{2} (1,1,1)$ & $|A|$ & $\{x,y,t\}$ & $5$ & $3/4$ \\
33 & $\mathsf{p}_z$ & $\frac{1}{3} (1,1,2)$ & $H_x$ & $\{x,y,t,w\}$ & $7$ & $10/17$ \\
37 & $\mathsf{p}_{zw}$ & $\frac{1}{3} (1,1,2)$ & $H_x$ & $\{x,y,t\}$ & $4$ & $1$ \\
39 & $\mathsf{p}_{yw}$ & $\frac{1}{3} (1,1,2)$ & $H_x$ & $\{x,z,t\}$ & $5$ & $1$ \\
39 & $\mathsf{p}_z$ & $\frac{1}{4} (1,1,3)$ & $H_x$ & $\{x,y,t\}$ & $5$ & $1$ \\
40 & $\mathsf{p}_z$ & $\frac{1}{4} (1,1,3)$ & $H_x$ & $\{x,y,t,w\}$ & $7$ & $15/19$ \\
40 & $\mathsf{p}_t$ & $\frac{1}{5} (1,2,3)$ & $H_x$ & $\{x,y,z,w\}$ & $4$ & $1$ \\
50 & $\mathsf{p}_t$ & $\frac{1}{7} (1,3,4)$ & $|A|$ & $\{x,y,z\}$ & $3$ & $1/2$ \\
61 & $\mathsf{p}_y$ & $\frac{1}{4} (1, 1, 3)$ & $H_x$ & $\{x, z, t, w\}$ & $9$ & $7/5$ \\
63 & $\mathsf{p}_t$ & $\frac{1}{8} (1,3,5)$ & $H_x$ & $\{x, y, z\}$ & $3$ & $1$ \\
64 & $\mathsf{p}_z$ & $\frac{1}{5} (1,2,3)$ & $|2 A|$ & $\{x, y, t\}$ & $6$ & $1/2$ \\
66 & $\mathsf{p}_y$ & $\frac{1}{5} (1,1,4)$ & $H_x$ & $\{x,z,t\}$ & $7$ & $1$ \\
68 & $\mathsf{p}_y$ & $\frac{1}{3} (1,1,2)$ & $|4 A|$ & $\{x,z,t\}$ & $7$ & $1/2$ \\
80 & $\mathsf{p}_y$ & $\frac{1}{3} (1,1,2)$ & $|4A|$ & $\{x,z,t\}$ & $10$ & $1/2$ \\
93 & $\mathsf{p}_y$ & $\frac{1}{7} (1,3,4)$ & $|8A|$ & $\{x,z,t\}$ & $10$ & $1/2$ \\
95 & $\mathsf{p}_y$ & $\frac{1}{5} (1,2,3)$ & $|6A|$ & $\{x,z,t\}$ & $22$ & $1/2$
\end{tabular}
\end{center}
\end{table}
\subsection{Remaining non-BI centers}
\begin{Prop} \label{prop:singptrem}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$ and let $\mathsf{p} \in X$ be a singular point with subscript $\spadesuit$ in the 5th column of Table \ref{table:main} which is also listed below.
\begin{itemize}
\item $\mathsf{i} = 12$ and singular points of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 13$ and the singular point of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 24$ and the singular point of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 27$ and the singular point of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 32$ and the singular point of type $\frac{1}{3} (1,1,2)$.
\item $\mathsf{i} = 33$ and the singular point of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 40$ and the singular point of type $\frac{1}{3} (1,1,2)$.
\item $\mathsf{i} = 47$ and the singular point of type $\frac{1}{5} (1, 2, 3)$.
\item $\mathsf{i} = 48$ and the singular point of type $\frac{1}{2} (1,1,1)$.
\item $\mathsf{i} = 49$ and the singular point of type $\frac{1}{5} (1, 2, 3)$.
\item $\mathsf{i} = 62$ and the singular point of type $\frac{1}{5} (1, 2, 3)$.
\item $\mathsf{i} = 65$ and the singular point of type $\frac{1}{2} (1, 1, 1)$.
\item $\mathsf{i} = 67$ and the singular point of type $\frac{1}{9} (1, 4, 5)$.
\item $\mathsf{i} = 82$ and the singular point of type $\frac{1}{5} (1, 2, 3)$.
\item $\mathsf{i} = 84$ and the singular point of type $\frac{1}{7} (1, 2, 5)$.
\end{itemize}
Then, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}.
\]
\end{Prop}
The rest of this subsection is to prove Proposition \ref{prop:singptrem} which will be separately for each family.
\subsubsection{The family $\mathcal{F}_{12}$, points of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{10} \subset \mathbb{P} (1, 1, 2, 3, 4)$ be a member of $\mathcal{F}_{12}$ and $\mathsf{p}$ a singular point of type $\frac{1}{2} (1,1,1)$.
We may assume $\mathsf{p} = \mathsf{p}_z$ after replacing $w$.
Then we have $z^3 w \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{2 \cdot 1 \cdot 3 \cdot (A^3)} = \frac{4}{5},
\]
and the proof is completed in this case.
\subsubsection{The family $\mathcal{F}_{13}$, the singular point of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{11} \subset \mathbb{P} (1, 1, 2, 3, 5)$ be a member of $\mathcal{F}_{13}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{2} (1,1,1)$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
\frac{2}{2 \cdot 1 \cdot 3 \cdot (A^3)} = \frac{10}{11}, & \text{if $z^3 w \in F$}, \\
\frac{2}{2 \cdot 1 \cdot 5 \cdot (A^3)} = \frac{6}{11}, & \text{if $z^3 w \notin F$ and $z^4 t \in F$}.
\end{cases}
\]
It remains to consider the case where $z^3 w, z^4 t \notin F$.
Then, by choosing $x, y$ suitably, we can write
\[
F = z^5 x + z^4 f_3 + z^3 f_5 + z^2 f_7 + z f_9 + f_{11},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_3$ and $w \notin f_5$.
We claim $w^2 y \in F$.
Assume $w^2 y \notin F$.
Then, by the quasi-smoothness of $X$ at $\mathsf{p}_w$, we have $w^2 x \in F$ and we may assume $\operatorname{coeff}_F (w^2 x) = -1$.
We can write $F = (z^5 - w^2) x + f'$, where $f' = z^4 f_3 + z^3 f_5 + z^2 f_7 + z f_9 + f_{11} + w^2 x$.
It is straightforward to check that $f' \in (x, y, t)^2$ and thus $X$ is not quasi-smooth at the point $(0\!:\!0\!:\!1\!:\!0\!:\!1) \in X$, which is a contradiction.
Thus $w^2 y \in F$.
We see that $\bar{F} := F (0, y, 1, t, w) \in (y, t, w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^2 y \in \bar{F}$ and $w^3 \notin \bar{F}$.
Thus we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ by Lemma \ref{lem:lcttangcube}.
\subsubsection{The family $\mathcal{F}_{24}$, the point of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{15} \subset \mathbb{P} (1, 1, 2, 5, 7)$ be a member of $\mathcal{F}_{24}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{2} (1,1,1)$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
\frac{2}{2 \cdot 1 \cdot 5 \cdot (A^3)} = \frac{14}{15}, & \text{if $z^4 w \in F$}, \\
\frac{2}{2 \cdot 1 \cdot 7 \cdot (A^3)} = \frac{2}{3}, & \text{if $z^4 w \notin F$ and $z^5 t \in F$}.
\end{cases}
\]
Suppose $z^4 w, z^5 t \notin F$.
Then we can write
\[
F = z^7 x + z^6 f_3 + z^5 f_5 + z^4 f_7 + z^3 f_9 + z^2 f_{11} + z f_{13} + f_{15},
\]
where $f_i = f_i (x, y, t, w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_5$ and $w \notin f_7$.
We claim $w^2 y \in F$.
Assume to the contrary $w^2 y \notin F$.
Then we can write $F = (z^7 + g) x + h$, where $g \in \mathbb{C} [x,y,z,t,w]$ and $h \in \mathbb{C} [y,z,t,w]$ are quasi-homogeneous polynomials such that $h \in (y,t)^2$.
But then $X$ is not quasi-smooth along the non-empty subset
\[
(x = y = t = z^7 + g = 0) \subset X.
\]
This is a contradiction and the claim is proved.
We see that $\bar{F} := F (0, y, 1, t, w) \in (y,t,w)^3$, $w^2 y \in \bar{F}$ and $w^3 \notin \bar{F}$.
In particular, $\bar{F}$ cannot be a cube of a linear form and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$ by Lemma \ref{lem:lcttangcube}.
\subsubsection{The family $\mathcal{F}_{27}$, the point of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{15} \subset \mathbb{P} (1, 2, 3, 5, 5)$ be a member of $\mathcal{F}_{27}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{2} (1,1,1)$.
If either $y^5 w \in F$ or $y^5 t \in F$, then we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{2 \cdot 3 \cdot 5 \cdot (A^3)} = \frac{2}{3}
\]
by Lemma \ref{lem:complctsingtang}.
Suppose $y^5 w, y^5 t \notin F$ and $y^6 z \in F$.
Then we can write
\[
F = y^6 z + y^5 f_5 + y^4 f_7 + y^3 f_9 + y^2 f_{11} + y f_{13} + f_{15},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $w, t \notin f_5$.
We see that $\operatorname{omult}_{\mathsf{p}} (H_z) = 3$ and thus $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{3} H_z) \ge 1$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{3} H_z$.
Then, we can take a $\mathbb{Q}$-divisor $T \in |5 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$ since $\{x, z, t, w\}$ isolates $\mathsf{p}$.
Since $\operatorname{omult}_{\mathsf{p}} (H_z) = 3$, we have
\[
3 \operatorname{omult}_{\mathsf{p}} (D) \le 3 (q^*D \cdot q^*H_z \cdot q^*T)_{\check{\mathsf{p}}} \le 2 (D \cdot H_z \cdot T) = 3,
\]
where $q = q_{\mathsf{p}}$ is the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q$.
This shows $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 1$ and thus $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $y^5 w, y^5 t, y^6 z \notin F$.
Then we can write
\[
F = y^7 x + y^6 f_3 + y^5 f_5 + y^4 f_7 + y^3 f_9 + y^2 f_{11} + y f_{13} + f_{15},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $z \notin f_3$ and $w, t \notin f_5$.
We see that $\bar{F} = F (0,1,z,t,w) \in (z,t,w)^3$ and the cubic part of $\bar{F}$ cannot be cube of a linear form since $F (0,1,0,t,w) = F (0,0,0,t,w)$ is a product of three linearly independent linear forms in $t, w$ by the quasi-smoothness of $X$.
By Lemma \ref{lem:lcttangcube}, we have $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{32}$, the point of type $\frac{1}{3} (1,1,2)$}
Let $X = X_{16} \subset \mathbb{P} (1, 2, 3, 4, 7)$ be a member of $\mathcal{F}_{32}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{3} (1,1,2)$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
\frac{2}{3 \cdot 2 \cdot 4 \cdot (A^3)} = \frac{7}{8}, & \text{if $z^3 w \in F$}, \\
\frac{2}{3 \cdot 2 \cdot 7 \cdot (A^3)} = \frac{1}{2}, & \text{if $z^3 w \notin F$ and $z^4 t \in F$}.
\end{cases}
\]
Suppose $z^3 w, z^4 t \notin F$.
Then $z^5 x \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}$ and we can write
\[
F = z^5 x + z^4 f_4 + z^3 f_7 + z^2 f_{10} + z f_{13} + f_{16},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $w^2 y \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
It follows that either $\bar{F} = F (0,y,1,t,w) \in (y,t,w)^2 \setminus (y,t,w)^3$ or $\bar{F} \in (y,t,w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^3 \notin \bar{F}$.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{33}$, the point of type $\frac{1}{2} (1, 1, 1)$}
Let $X = X_{17} \subset \mathbb{P} (1, 2, 3, 5, 7)$ be a member of $\mathcal{F}_{33}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{2} (1, 1, 1)$.
Suppose that at least one of $y^5 w$, $y^6 t$ and $y^7 z$ appear in $F$ with nonzero coefficient.
In this case, $H_x$ is quasi-smooth at $\mathsf{p}$ and we have $\operatorname{lct}_{\mathsf{p}} (X; H_x) = 1$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |7 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of of the effective $1$-cycle $D \cdot H_x$ since the set $\{x, z, t, w\}$ isolates $\mathsf{p}$.
It follows that
\[
\operatorname{omult}_{\mathsf{p}} (D) \le (q^*_{\mathsf{p}} D \cdot q^*_{\mathsf{p}} H_x \cdot q^*_{\mathsf{p}} T)_{\check{\mathsf{p}}} \le 2 (D \cdot H_x \cdot T) = \frac{17}{15}.
\]
Thus $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 15/17$ and we have $\alpha_{\mathsf{p}} (X) \ge 15/17$.
Suppose $y^5 w, y^6 t, y^7 z \notin F$.
Then $y^8 x \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}$ and we can write
\[
F = y^8 x + y^7 f_3 + y^6 f_5 + y^5 f_7 + y^4 f_9 + y^3 f_{11} + y^2 f_{13} + y f_{15} + f_{17},
\]
where $f_i = f_i (x, z, t, w)$ is a quasi-homogeneous polynomial of degree $i$.
We see that $\bar{F} := F (0, 1, z, t, w) \in (z, t, w)^3$, $w^3 \notin \bar{F}$ and $w^2 z \in \bar{F}$.
It follows that $\bar{F} \in (z, t, w)^3$ and it cannot be a cube of a linear form.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{40}$, the point of type $\frac{1}{3} (1, 1, 2)$}
Let $X = X_{19} \subset \mathbb{P} (1, 3, 4, 5, 7)$ be a member of $\mathcal{F}_{40}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{3} (1, 1, 2)$.
Suppose that either $y^4 w \in F$ or $y^5 z \in F$.
In this case $\operatorname{lct}_{\mathsf{p}} (X; H_x) = 1$ since $H_x$ is quasi-smooth at $\mathsf{p}$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |7 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of of the effective $1$-cycle $D \cdot H_x$ since the set $\{x, z, t, w\}$ isolates $\mathsf{p}$.
It follows that
\[
\operatorname{omult}_{\mathsf{p}} (D) \le (q^*_{\mathsf{p}} D \cdot q^*_{\mathsf{p}} H_x \cdot q^*_{\mathsf{p}} T)_{\check{\mathsf{p}}} \le 3 (D \cdot H_x \cdot T) = \frac{19}{20}.
\]
Thus $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 20/19$ and we have $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $y^4 w, y^5 z \notin F$.
Then $y^6 x \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}$ and we can write
\[
F = y^6 x + y^5 f_4 + y^4 f_7 + y^3 f_{10} + y^2 f_{13} + y f_{16} + f_{19},
\]
where $f_i = f_i (x, z, t, w)$ is a quasi-homogeneous polynomial of degree $i$.
We set $\bar{F} := F (0, 1, z, t, w) \in (z, t, w)^3$.
It is easy to see that $w^3 \notin \bar{F}$ and $w^2 z \in \bar{F}$.
It follows that either $\bar{F} \in (z, t, w)^2$ or $\bar{F} \in (z, t, w)^3$ and it cannot be a cube of a linear form.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{47}$, the point of type $\frac{1}{5} (1, 2, 3)$}
Let $X = X_{21} \subset \mathbb{P} (1, 1, 5, 7, 8)$ be a member of $\mathcal{F}_{47}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{5} (1,2,3)$.
We can write
\[
F = z^4 x + z^3 f_6 + z^2 f_{11} + z f_{16} + f_{21},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$.
By the quasi-smoothness of $X$ at $\mathsf{p}_w$, we have $w^2 \in f_{16}$, which implies $\bar{F} = F (0, y, 1, t, w) \in (y, t, w)^2 \setminus (y, t, w)^3$.
Thus, by Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{48}$, the point of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{21} \subset \mathbb{P} (1, 2, 3, 7, 9)$ be a member of $\mathcal{F}_{48}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{2} (1,1,1)$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
\frac{2}{2 \cdot 3 \cdot 7 \cdot (A^3)} = \frac{6}{7}, & \text{if $y^6 w \in F$}, \\
\frac{2}{2 \cdot 3 \cdot 9 \cdot (A^3)} = \frac{2}{3}, & \text{if $y^6 w \notin F$ and $y^7 t \in F$}.
\end{cases}
\]
Suppose $y^6 w, y^7 t \notin F$ and $y^9 z \in F$.
We can write
\[
F = y^9 z + y^8 f_5 + y^7 f_7 + y^6 f_9 + \cdots + f_{21},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_7$ and $w \notin f_9$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 1$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) = 1$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |9 A|_{\mathbb{Q}}$ with such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$ since $\{x, z, t, w\}$ isolates $\mathsf{p}$.
Then we have
\[
\operatorname{omult}_{\mathsf{p}} (D) \le (q^*D \cdot q^*H_x \cdot q^*T)_{\check{\mathsf{p}}} \le 2 (D \cdot H_x \cdot T) = 2 \cdot 1 \cdot 9 \cdot (A^3) = 1,
\]
where $q = q_{\mathsf{p}}$ is the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ and thus $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $y^6 w, y^7 t, y^9 z \notin F$.
Then $y^{10} x \in F$ and we can write
\[
y^{10} x + y^9 f_3 + y^8 f_5 + y^7 f_7 + y^6 f_9 + \cdots + f_{21},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $z \notin f_3$, $t \notin f_7$ and $w \notin f_9$.
We see that $\bar{F} := F (0,1,z,t,w) \in (z,t,w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^2 z \in \bar{F}$ and $w^3 \notin \bar{F}$.
Thus, by Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{49}$, the point of type $\frac{1}{5} (1,2,3)$}
Let $X = X_{21} \subset \mathbb{P} (1, 3, 5, 6, 7)$ be a member of $\mathcal{F}_{49}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{5} (1,2,3)$.
If $z^3 t \in F$, then
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{5 \cdot 3 \cdot 7 \cdot (A^3)} = \frac{4}{7}
\]
by Lemma \ref{lem:complctsingtang}.
Suppose $z^3 t \notin F$.
Then $z^4 x \in F$ and we can write
\[
F = z^4 x + z^3 f_6 + z^2 f_{11} + z f_{16} + f_{21},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_6$.
We have $w^3, t^3 y \in F$ and we may assume $\operatorname{coeff}_F (w^3) = \operatorname{coeff}_F (t^3 y) = 1$.
We claim $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
If $y^2 \in f_6$, then $\operatorname{omult}_{\mathsf{p}} (H_x) = 2$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
We assume $y^2 \notin f_6$.
Then we can write
\[
F (0,y,z,t,w) = z (\alpha w t y + \beta w y^3) + w^3 + y (t^3 + \gamma t^2 y^2 + \delta t y^4 + \varepsilon y^6),
\]
where $\alpha, \beta, \gamma, \delta, \varepsilon \in \mathbb{C}$.
We set $\bar{F} := F (0, y, 1, t, w) \in \mathbb{C} [y, t, w]$.
\begin{itemize}
\item Suppose $\alpha \ne 0$.
Then, $\bar{F} \in (y,t,w)^3$ and its cubic part is $\alpha w t y + w^3$.
By Lemma \ref{lem:lcttangcube}, we have $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ in this case.
\item Suppose $\alpha = 0$ and $\beta \ne 0$.
Then the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (y,t,w) = (6,7,9)$ is $\beta w y^3 + w^3 + t^3 y$.
By Lemma \ref{lem:lctwblwh}, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge \min \left\{ \frac{22}{27}, \operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D} ) \right\},
\]
where
\begin{itemize}
\item $\tilde{\mathbb{P}} = \mathbb{P} (2, 7, 9)_{\tilde{y}, \tilde{t}, \tilde{w}} = \mathbb{P} (6, 7, 9)^{\operatorname{wf}}$,
\item $\operatorname{Diff} = \frac{2}{3} H_{\tilde{t}}$ with $H_{\tilde{t}} = (\tilde{t} = 0) \subset \tilde{\mathbb{P}}$, and
\item $\mathcal{D}$ is the prime divisor $(\beta \tilde{w} \tilde{y}^3 + \tilde{w}^3 + \tilde{t} \tilde{y} = 0)$ on $\tilde{\mathbb{P}}$.
\end{itemize}
We see that $\mathcal{D}$ is quasi-smooth and it intersects $H_{\tilde{t}}$ transversally.
It follows that $\operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D}) = 1$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 22/27$.
\item Suppose $\alpha = \beta = 0$.
Then the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (y,z,t) = (3,6,7)$ is $w^3 + t^3 y + \gamma t^2 y^3 + \delta t y^5 + \varepsilon y^7$.
By Lemma \ref{lem:lctwblwh}, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge \min \left\{ \frac{16}{21}, \operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D} ) \right\},
\]
where
\begin{itemize}
\item $\tilde{\mathbb{P}} = \mathbb{P} (1, 2, 9)_{\tilde{y}, \tilde{t}, \tilde{w}} = \mathbb{P} (3, 6, 7)^{\operatorname{wf}}$,
\item $\operatorname{Diff} = \frac{2}{3} H_{\tilde{w}}$ with $H_{\tilde{w}} = (\tilde{w} = 0) \subset \tilde{\mathbb{P}}$, and
\item $\mathcal{D}$ is the prime divisor $(\tilde{w} + \tilde{t}^3 \tilde{y} + \gamma \tilde{t}^2 \tilde{y}^3 + \delta \tilde{t} \tilde{y}^5 + \varepsilon \tilde{y}^7 = 0)$ on $\tilde{\mathbb{P}}$.
\end{itemize}
We see that $\mathcal{D}$ is quasi-smooth.
The solutions of the equation $\tilde{t}^3 \tilde{y} + \gamma \tilde{t}^2 \tilde{y}^3 + \delta \tilde{t} \tilde{y}^5 + \varepsilon \tilde{y}^7 = 0$ corresponds to the 3 points of type $\frac{1}{3} (1, 1, 2)$ on $X$.
In particular the equation has 3 distinct solutions.
It follows that $\mathcal{D}$ intersects $H_{\tilde{w}}$ transversally and we have $\operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D}) = 1$.
Thus $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 16/21$.
\end{itemize}
Thus the claim is proved.
The point $\mathsf{p}$ is not a maximal center and the pair $(X, H_x)$ is not canonical by Lemma \ref{lem:qtangdivncan}.
Thus,
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x) \} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
\subsubsection{The family $\mathcal{F}_{62}$, the point of $\frac{1}{5} (1,2,3)$}
Let $X = X_{26} \subset \mathbb{P} (1, 1, 5, 7, 13)$ be a member of $\mathcal{F}_{62}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{5} (1,2,3)$.
Replacing $x$ and $y$, we can write
\[
F = z^5 x + z^4 f_6 + z^3 f_{11} + z^2 f_{16} + z f_{21} + f_{26},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 2$ since $w^2 \in F$.
Hence $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
The point $\mathsf{p}$ is not a maximal center and the pair $(X, H_x)$ is not canonical at $\mathsf{p}$ by Lemma \ref{lem:qtangdivncan}.
Thus
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
\subsubsection{The family $\mathcal{F}_{65}$, the point of type $\frac{1}{2} (1,1,1)$}
Let $X = X_{27} \subset \mathbb{P} (1, 2, 5, 9, 11)$ be a member of $\mathcal{F}_{65}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{2} (1,1,1)$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases}
\frac{2}{2 \cdot 5 \cdot 9 \cdot (A^3)} = \frac{22}{27}, & \text{if $y^8 w \in F$}, \\
\frac{2}{2 \cdot 5 \cdot 11 \cdot (A^3)} = \frac{2}{3}, & \text{if $y^8 w \notin F$ and $y^9 t \in F$}.
\end{cases}
\]
Suppose $y^8 w, y^9 t \notin F$ and $y^{11} z \in F$.
Then we can write
\[
F = y^{11} z + y^{10} f_7 + \cdots + y f_{25} + f_{27},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_9$ and $w \notin f_{11}$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 1$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) = 1$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
We see that $\{x, z, t, w\}$ isolates $\mathsf{p}$, hence we can take a $\mathbb{Q}$-divisor $T \in |11 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$.
Then we have
\[
\operatorname{omult}_{\mathsf{p}} (D)
\le (q_{\mathsf{p}}^*D \cdot q_{\mathsf{p}}^*H_x \cdot q_{\mathsf{p}}
^*T)_{\check{\mathsf{p}}}
\le 2 (D \cdot H_x \cdot T)
= 2 \cdot 1 \cdot 11 \cdot (A^3) = \frac{3}{5},
\]
where $q_{\mathsf{p}}$ is the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q_{\mathsf{p}}$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ and thus $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose that $y^8 w, y^9 t, y^{11} z \notin F$.
Then $y^{13} x \in F$ and we can write
\[
F = y^{13} x + y^{12} f_3 + \cdots + y f_{25} + f_{27},
\]
where $f_i = f_i (x,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $z \notin f_5$, $t \notin f_9$ and $w \notin f_{11}$.
We see that $\bar{F} := F (0,1,z,t,w) \in (z,t,w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^2 z \in \bar{F}$ and $w^3 \notin \bar{F}$.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{The family $\mathcal{F}_{67}$, the point of $\frac{1}{9} (1,4,5)$}
Let $X = X_{28} \subset \mathbb{P} (1, 1, 4, 9, 14)$ be a member of $\mathcal{F}_{67}$ and $\mathsf{p} = \mathsf{p}_t$ the singular point of type $\frac{1}{9} (1,4,5)$.
Replacing $x$ and $y$, we can write
\[
F = t^3 x + t^2 f_{10} + t f_{19} + f_{28},
\]
where $f_i = f_i (x,y,z,w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 2$ since $w^2 \in F$.
Hence $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ by Lemma \ref{lem:qtangdivncan}.
Thus,
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
\subsubsection{The family $\mathcal{F}_{82}$, the point of $\frac{1}{5} (1,2,3)$}
Let $X = X_{36} \subset \mathbb{P} (1, 1, 5, 12, 18)$ be a member of $\mathcal{F}_{82}$ and $\mathsf{p} = \mathsf{p}_z$ the singular point of type $\frac{1}{5} (1,2,3)$.
Replacing $x$ and $y$, we can write
\[
F = z^7 x + z^6 f_6 + z^5 f_{11} + \cdots + f_{36},
\]
where $f_i = f_i (x, y, t, w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 2$ since $w^2 \in F$.
Hence $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$ by Lemma \ref{lem:qtangdivncan}.
Thus,
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
\subsubsection{The family $\mathcal{F}_{84}$, the point of type $\frac{1}{7} (1,2,5)$}
Let $X = X_{36} \subset \mathbb{P} (1, 7, 8, 9, 12)$ be a member of $\mathcal{F}_{84}$ and $\mathsf{p} = \mathsf{p}_y$ the singular point of type $\frac{1}{7} (1,2,5)$.
By the quasi-smoothness of $X$, either $y^4 z \in F$ or $y^5 x \in F$.
Moreover, we have $w^3, t^4, z^3 w \in F$ and we assume $\operatorname{coeff}_F (w^3) = \operatorname{coeff}_F (t^4) = \operatorname{coeff}_F (z^3 w) = 1$ by rescaling $z, t, w$.
Suppose $y^4 z \in F$.
Let $\rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$ the orbifold chart of $X$ containing $\mathsf{p}$.
Then we have $\rho_{\mathsf{p}}^* H_x \cdot \rho^*_{\mathsf{p}} H_z = \breve{\Gamma}$, where
\[
\breve{\Gamma} = (\breve{x} = \breve{z} = \breve{w}^3 + \breve{t}^4 = 0) \subset \breve{U}_{\mathsf{p}}
\]
is an irreducible and reduced curve with $\operatorname{mult}_{\breve{\mathsf{p}}} (\breve{\Gamma}) = 3$.
We see that $(X, H_x)$ is log canonical at $\mathsf{p}$ since $H_x$ is quasi-smooth at $\mathsf{p}$.
Thus, by Lemma \ref{lem:exclL}, we have $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $y^4 z \notin F$.
Then $y^5 x \in F$ and we can write
\[
F = y^5 x + y^4 f_8 + y^3 f_{15} + y^2 f_{22} + y f_{29} + f_{36},
\]
where $f_i = f_i (x, z, t, w)$ is a quasi-homogeneous polynomial of degree $i$ with $z \notin f_8$.
By setting $\alpha := \operatorname{coeff}_F (y w t z)$, we have
\[
\bar{F} := F (0,1,z,t,w) = \alpha w t z + w^3 + w z^3 + t^4.
\]
\begin{itemize}
\item If $\alpha \ne 0$, then $\bar{F} \in (z,t,w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form.
Hence $\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge 1/2$ by Lemma \ref{lem:lcttangcube}.
\item If $\alpha = 0$, then the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (z, t, w) = (8,9,12)$ is $\bar{F} = w^3 + w z^3 + t^4$.
By Lemma \ref{lem:lctwblwh}, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_x) \ge \min \left\{ \frac{29}{36}, \operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D} ) \right\},
\]
where
\begin{itemize}
\item $\tilde{\mathbb{P}} = \mathbb{P} (2, 3, 1)_{\tilde{z}, \tilde{t}, \tilde{w}} = \mathbb{P} (8, 9, 12)^{\operatorname{wf}}$,
\item $\operatorname{Diff} = \frac{2}{3} H_{\tilde{z}} + \frac{3}{4} H_{\tilde{t}}$ with $H_{\tilde{z}} = (\tilde{z} = 0) \subset \tilde{\mathbb{P}}$, $H_{\tilde{t}} = (\tilde{t} = 0) \subset \tilde{\mathbb{P}}$, and
\item $\mathcal{D}$ is the prime divisor $(\tilde{w}^3 + \tilde{w} \tilde{z} + \tilde{t} = 0)$ on $\tilde{\mathbb{P}}$.
\end{itemize}
We see that $\mathcal{D}$ is quasi-smooth, $\mathcal{D} \cap H_{\tilde{z}} \cap H_{\tilde{t}} = \emptyset$ and any two of $\mathcal{D}, H_{\tilde{z}}, H_{\tilde{t}}$ intersect transversally.
It follows that $\operatorname{lct} (\tilde{\mathbb{P}}, \operatorname{Diff}; \mathcal{D}) = 1$ and thus $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 29/36$.
\end{itemize}
Note that $\mathsf{p} \in X$ is not a maximal center and the pair $(X, H_x)$ is not canonical at $\mathsf{p}$ by Lemma \ref{lem:qtangdivncan}.
Thus
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
\section{EI centers}
\begin{Prop} \label{prop:alphaEI}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$ and $\mathsf{p} \in X$ an EI center.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}.
\]
\end{Prop}
\begin{proof}
We have $\alpha_{\mathsf{p}} (X) \ge 1$ by Proposition \ref{prop:lctsingptL} for a member $X$ of $\mathcal{F}_{\mathsf{i}}$ and $\mathsf{p} \in X$, where
\begin{itemize}
\item $\mathsf{i} = 36$ and $\mathsf{p}$ is of type $\frac{1}{4} (1, 1, 3)$.
\item $\mathsf{i} = 44$ and $\mathsf{p}$ is of type $\frac{1}{6} (1, 1, 5)$.
\item $\mathsf{i} = 61$ and $\mathsf{p}$ is of type $\frac{1}{7} (1, 2, 5)$.
\item $\mathsf{i} = 76$ and $\mathsf{p}$ is of type $\frac{1}{8} (1, 3, 5)$.
\end{itemize}
We have $\alpha_{\mathsf{p}} (X) \ge 1/2$ by Proposition \ref{prop:singptic} for a member $X$ of $\mathcal{F}_{\mathsf{i}}$ and $\mathsf{p} \in X$, where
\begin{itemize}
\item $\mathsf{i} = 23$ and $\mathsf{p}$ is of type $\frac{1}{4} (1, 1, 3)$.
\item $\mathsf{i} = 40$ and $\mathsf{p}$ is of type $\frac{1}{5} (1, 2, 3)$.
\end{itemize}
It remains to consider members of families $\mathcal{F}_7$ and $\mathcal{F}_{20}$, and singular points of types $\frac{1}{2} (1, 1, 1)$ and $\frac{1}{3} (1, 1, 2)$, respectively.
Let $X = X_8 \subset \mathbb{P} (1, 1, 2, 2, 3)$ be a member of $\mathcal{F}_7$ and $\mathsf{p}$ a singular point of type $\frac{1}{2} (1, 1, 1)$.
Replacing homogeneous coordinates, we may assume $\mathsf{p} = \mathsf{p}_t$ and we can write
\[
F = t^3 z + t^2 f_4 + t f_6 + f_8,
\]
where $f_i = f_i (x, y, z, w)$ is a quasi-homogeneous polynomial of degree $i$.
Hence, by Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{2 \cdot 1 \cdot 3 \cdot (A^3)} = \frac{1}{2}.
\]
Let $X = X_{13} \subset \mathbb{P} (1, 1, 3, 4, 5)$ be a member of $\mathcal{F}_{20}$ and $\mathsf{p} = \mathsf{p}_z$ be the singular point of type $\frac{1}{3} (1, 1, 2)$.
If $z^3 t \in F$, then we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{3 \cdot 1 \cdot 5 \cdot (A^3)} = \frac{8}{13}
\]
by Lemma \ref{lem:complctsingtang}.
Suppose $z^3 t \notin F$.
Then we can write that
\[
F = z^4 x + z^3 f_4 + z^2 f_7 + z f_{10} + f_{13},
\]
where $f_i = f_i (x,y,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_4$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 2$ since $w^2 z \in F$ by the quasi-smoothness of $X$ at $\mathsf{p}_w$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
The point $\mathsf{p}$ is not a maximal singularity and the pair $(X, H_x)$ is not canonical at $\mathsf{p}$ by Lemma \ref{lem:qtangdivncan}.
Thus,
\[
\alpha_{\mathsf{p}} (X) \ge \min \{1, \operatorname{lct}_{\mathsf{p}} (X;H_x)\} \ge \frac{1}{2}
\]
by Lemma \ref{lem:singnoncanbd}.
This completes the proof.
\end{proof}
\section{Equations for QI centers}
\label{sec:eqQI}
Let
\[
X = X_d \subset \mathbb{P} (1,a_1,\dots, a_4)_{x_0, \dots, x_4} =: \mathbb{P}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$.
We set $a_0 = 1$ and let $F = F (x_0, \dots, x_4)$ be the defining polynomial of $X$.
\begin{Def}
Let $\mathsf{p} \in X$ be a QI center and let $j, k$ be such that $j \ne k$, $d = 2 a_k + a_j$ and the index of $\mathsf{p} \in X$ coincides with $a_k$.
Then we can choose coordinates so that $\mathsf{p} = \mathsf{p}_{x_k}$.
We say that $\mathsf{p}$ is {\it an exceptional QI center} if $x_k^2 x_l \notin F$ for any $l \in \{0, \dots, 4\}$.
\end{Def}
\begin{Lem} \label{lem:QIcoord}
Let $\mathsf{p} \in X$ be a non-exceptional QI center.
Then we can choose homogeneous coordinates $x_{i_1}, x_{i_2}, x_{i_3}, x_j, x_k$ of $\mathbb{P}$, where $\{i_1, i_2, i_3, j, k\} = \{0, 1, 2, 3, 4\}$, such that $a_{i_1}, a_{i_2}, a_{i_3} < a_k$, $\mathsf{p} = \mathsf{p}_{x_k}$ and
\begin{equation} \label{eq:QIcoord}
F = x_k^2 x_j + x_k f (x_{i_1}, x_{i_2}, x_{i_3},x_j) + g (x_{i_1},x_{i_2},x_{i_3},x_j)
\end{equation}
for some quasi-homogeneous polynomials $f, g \in \mathbb{C} [x_{i_1}, x_{i_2}, x_{i_3}, x_j]$ of degree $d - a_k$, $d$, respectively.
\end{Lem}
\begin{proof}
Basically this follows by looking at Table \ref{table:main}.
See also \cite[Theorem 4.9]{CPR00}.
\end{proof}
Let $\mathsf{p} \in X$ be a non-exceptional QI center and we choose and fix homogeneous coordinates $x_{i_1}, x_{i_2}, x_{i_3}, x_j, x_k$ of $\mathbb{P}$ as in Lemma \ref{lem:QIcoord}.
\begin{Def}
We say that $\mathsf{p}$ is a {\it degenerate QI center} if $f (x_{i_1}, x_{i_2}, x_{i_3},0) = 0$ as a polynomial, otherwise we call $\mathsf{p}$ a {\it non-degenerate QI center}.
\end{Def}
\begin{Rem} \label{rem:QImaxcent}
It is proved in \cite[Section 4.1]{CP17} that a QI center $\mathsf{p} \in X$ is a maximal center if and only if it is non-degenerate.
\end{Rem}
\begin{Lem} \label{lem:eqdegenQI}
Let $\mathsf{p}$ be a degenerate QI center.
Then we can choose homogeneous coordinates $x_{i_1}, x_{i_2}, x_{i_3}, x_j, x_k$ of $\mathbb{P}$ such that $a_{i_1}, a_{i_2}, a_{i_3} < a_k$, $\mathsf{p} = \mathsf{p}_{x_k}$ and
\begin{equation} \label{eq:eqdegenQI}
F = x_k^2 x_j + g (x_{i_1},x_{i_2},x_{i_3},x_j)
\end{equation}
for some quasi-homogeneous polynomial $g \in \mathbb{C} [x_{i_1},x_{i_2},x_{i_3},x_j]$ of degree $d$.
Moreover the hypersurface
\[
(g (x_{i_1}, x_{i_2}, x_{i_3}, 0) = 0) \subset \mathbb{P} (a_{i_1}, a_{i_2}, a_{i_3})_{x_{i_1}, x_{i_2}, x_{i_3}}
\]
is quasi-smooth.
\end{Lem}
\begin{proof}
We have $f = x_j f'$ for some $f' \in \mathbb{C} [x_{i_1}, x_{i_2}, x_{i_3}, x_j]$ since $\mathsf{p}$ is degenerate.
Filtering off terms divisible by $x_j$ in \eqref{eq:QIcoord}, we have
\[
F = x_j (x_k^2 + x_k f') + g.
\]
We can eliminate the term $x_k x_j f'$ by replacing $x_k \mapsto x_k - f'/2$.
This shows the first assertion.
We choose and fix homogeneous coordinates so that $F$ is of the form \eqref{eq:eqdegenQI}.
We set $\bar{g} = g (x_{i_1},x_{i_2},x_{i_3},0)$.
Then we can write $g = \bar{g} + x_j h$, where $h = h (x_{i_1},x_{i_2},x_{i_3},x_j)$.
Suppose to the contrary that $(\bar{g} = 0) \subset \mathbb{P} (a_{i_1},a_{i_2},a_{i_3})$ is not quasi-smooth at a point $(\alpha_1\!:\!\alpha_2\!:\!\alpha_3)$.
We choose and fix $\beta \in \mathbb{C}$ such that $\beta^2 + h (\alpha_1,\alpha_2,\alpha_3,0) = 0$, and set
\[
\mathsf{q} := (\alpha_1\!:\!\alpha_2\!:\!\alpha_3\!:\!0\!:\!\beta) \in \mathbb{P} (a_{i_1},a_{i_2},a_{i_3},a_j,a_k) = \mathbb{P}.
\]
It is easy to see that $(\partial F/\partial v) (\mathsf{q}) = 0$ for any $v \in \{x_{i_1},x_{i_2},x_{i_3},x_j,x_k\}$.
This is impossible since $X$ is quasi-smooth.
Therefore $(\bar{g} = 0) \subset \mathbb{P} (a_{i_1},a_{i_2},a_{i_3})$ is quasi-smooth.
\end{proof}
\begin{Lem} \label{lem:QIeqtypes}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \{2, 8\}$.
Suppose that $X$ has a QI center.
Then one of the following holds.
\begin{enumerate}
\item $X$ has a unique QI center.
In this case, by a choice of homogeneous coordinates, we have
\[
X = X_{2 r + c} \subset \mathbb{P} (1, a, b, c, r)_{x, s, u, v, w},
\]
where $a$ is coprime to $b$, $a < b$, $a + b = r$, $c < r$, and the unique QI center is the point $\mathsf{p} = \mathsf{p}_w$, which is of type $\frac{1}{r} (1, a, b)$.
\item $X$ has exactly $3$ distinct QI centers.
In this case, by a choice of homogeneous coordinates, we have
\[
X = X_{3 r} \subset \mathbb{P} (1, a, b, r, r)_{x, y, z, t, w},
\]
where $a$ is coprime to $b$, $a \le b$ and $a + b = r$.
The $3$ QI centers are the $3$ points in $(x = y = z = 0) \cap X$ and they are all of type $\frac{1}{r} (1, a, b)$.
\item $X$ has exactly $2$ distinct QI centers and their singularity types are equal.
In this case, by a choice of homogeneous coordinates, we have
\[
X = X_{4 r} \subset \mathbb{P} (1, a, b, r, 2 r)_{x, y, z, t, w},
\]
where $a$ is coprime to $b$, $a \le b$ and $a + b = r$.
The QI centers are the 2 points in $(x = y = z = 0) \cap X$ and they are both of type $\frac{1}{r} (1, a, b)$.
\item $X$ has exactly $2$ distinct QI centers and their singularity types are distinct.
In this case, by a choice of homogeneous coordinates, we have
\[
X = X_{4 a + 3 b} \subset \mathbb{P} (1, a, b, r_1, r_2)_{x, u, v, t, w},
\]
where $a$ is coprime to $b$, $a + b = r_1$ and $2 a + b = r_2$.
The QI centers are $\mathsf{p}_t$ and $\mathsf{p}_w$ which are of types $\frac{1}{r_1} (1, a, b)$ and $\frac{1}{r_2} (1, a, a + b)$, respectively.
\end{enumerate}
\end{Lem}
\begin{proof}
Let
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x_0, x_1, x_2, x_3, x_4}
\]
be a member of $\mathcal{F}_{\mathsf{i}}$ with$\mathsf{i} \in \mathsf{I} \setminus \{2, 8\}$.
We set $a_0 = 1$ and assume $a_1 \le \dots \le a_4$.
We assume that $X$ has at least one QI center.
Suppose $d = 3 a_4$.
Let $\mathsf{p} \in X$ be a QI center.
Then, after replacing homogeneous coordinates, we may assume $\mathsf{p} = \mathsf{p}_{x_i}$ and $x_i^2 x_j \in F$ for some $i \in \{0, 1, 2, 3, 4\}$ and $j \in \{0, 1, 2, 3, 4\} \setminus \{i\}$.
In particular we have $d = 3 a_4 = 2 a_i + a_j$, which is possible if and only if $a_i = a_j = a_4$.
Thus we have $a_3 = a_4$ and we may assume $i = 4, j = 3$.
We see that $\mathsf{p} \in X$ is of type $\frac{1}{a_4} (1, a_1, a_2)$ and $\mathsf{p} \in X$ is terminal.
It follows that $a_1 + a_2 = a_4$ and $a_l$ is coprime to $a_4$ for $l = 1, 2$.
By setting $a := a_1$, $b := a_2$ and $r := a_3 = a_4$, this case corresponds to (2).
In the following we assume $d < 3 a_4$.
Suppose $d = 2 a_4$.
We have $a_4 = a_1 + a_2 + a_3$ since $d = a_1 + a_2 + a_3 + a_4 = 2 a_4$.
Let $\mathsf{p} \in X$ be a QI center.
Then we may assume $\mathsf{p} = \mathsf{p}_{x_i}$ and $x_i^2 x_j \in F$ for some $i \in \{0, 1, 2, 3\}$ and $j \in \{0, 1, 2, 3, 4\} \setminus \{i\}$.
In particular we have $d = 2 a_i + a_j$, and hence
\[
2 a_i + a_j = a_1 + a_2 + a_3 + a_4 = 2 (a_1 + a_2 + a_3),
\]
which is only possible when $j = 4$ and $i = 3$.
Hence $i = 3$, $j = 4$, and we have $a_4 = 2 a_3$ since $d = 2 a_3 + a_4 = 2 a_4$.
We see that $\mathsf{p} \in X$ is of type $\frac{1}{a_3} (1, a_1, a_2)$ and $\mathsf{p} \in X$ is terminal.
It follows that $a_3 = a_1 + a_2$ and $a_l$ is coprime to $a_3$ for $l = 1, 2$.
By setting $a:= a_1$, $b := a_2$, $r := a_3$, this case corresponds to (3).
Suppose $d = 2 a_4 + a_3$.
We have $a_4 = a_1 + a_2$ since $d = a_1 + a_2 + a_3 + a_4$.
We see that $\mathsf{p}_4 \in X$ is of type $\frac{1}{a_4} (1, a_1, a_2)$ and it is a QI center.
It follows that $a_4$ is coprime to $a_l$ for $l = 1, 2$ since $\mathsf{p}_4 \in X$ is a terminal singularity.
If $X$ admits a QI center other than $\mathsf{p}_4$, then we have $d = 2 a_i + a_j$, where $i \in \{1, 2, 3\}$ and $j \in \{0, 1, 2, 3, 4\} \setminus \{
i\}$ which is impossible.
Thus $\mathsf{p}_4 \in X$ is a unique QI center, and we are in case (1) by setting $a := a_1$, $b := a_2$, $c := a_3$ and $r := a_4$.
Note that we have $a < b$ because otherwise we have $a_1 = a_2 = 1$ and $a_4 = 2$ and $X$ belongs to a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{2, 8\}$ which is impossible.
Suppose $d = 2 a_4 + a_2$.
Then $a_4 = a_1 + a_3$.
We see that $\mathsf{p}_4 \in X$ is of type $\frac{1}{a_4} (1, a_1, a_3)$ and it is a QI center.
If $\mathsf{p}_4$ is a unique QI center, then we are in case (1) by setting $a := a_1$, $b := a_3$, $c := a_2$ and $r := a_4$.
We assume that $X$ admits a QI center $\mathsf{p} \in X$ other than $\mathsf{p}_4$.
We may assume $\mathsf{p} = \mathsf{p}_i$ after replacing homogeneous coordinates and we have $d = 2 a_i + a_j$ for some $i \in \{1, 2, 3\}$ and $j \in \{0, 1, 2, 3, 4\} \setminus \{i\}$.
Then we have $a_j = a_4$ and $a_i = a_3$.
Thus $i = 3$, $j = 4$ and we have $a_3 = a_1 + a_2$.
The singularity of $\mathsf{p} = \mathsf{p}_i \in X$ is of type $\frac{1}{a_3} (1, a_1, a_2)$ and it is terminal.
It follows that $a_1$ is coprime to $a_2$.
Thus we are in case (4) by setting $a := a_1$, $b := a_2$, $r_1 := a_3$ and $r_2 = a_4$.
Suppose $d = 2 a_4 + a_1$.
Then, by interchanging the role of $a_1$ and $a_2$ in the privious arguments, we conclude that this case corresponds to either (1) or (4).
This completes the proof.
\end{proof}
\begin{Lem} \label{lem:uniqQItypes}
Let
\[
X = X_{2 r + c} \subset \mathbb{P} (1, a, b, c, r)_{x, s, u, v, w}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \{2, 8\}$ with a unique QI center, where $a$ is coprime to $b$, $a < b$, $r = a + b$ and $c < r$.
Then the following assertions hold.
\begin{enumerate}
\item If $c = 1$, then $X$ belongs to a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{24, 46\}$.
\item If $2 r + c$ is not divisible by $b$, then $b = a + 1$, $c = a + 2$, $r = 2 a + 1$ and $a \in \{2, 3, 4\}$.
\end{enumerate}
\end{Lem}
\begin{proof}
This follows from Table \ref{table:main}.
\end{proof}
\begin{Lem} \label{lem:QIexceq}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ and let $\mathsf{p} \in X$ be an exceptional QI center.
Then we are in Case (4) of Lemma \ref{lem:QIeqtypes} and $\mathsf{p} = \mathsf{p}_t$.
Moreover we can write
\begin{equation} \label{eq:QIexceq}
F = t^3 u + t^2 f_{2 a + b} + t f_{3 a + 2 b} + f_{4 a + 3 b},
\end{equation}
where $f_i \in \mathbb{C} [x, u, v, w]$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_{2 a + b}$.
\end{Lem}
\begin{proof}
We are in (1), (2), (3) and (4) of Lemma \ref{lem:QIeqtypes}.
Suppose that we are in (1).
Then $\mathsf{p} = \mathsf{p}_w$.
Since $\mathsf{p} \in X$ is exceptional and $X$ is quasi-smooth at $\mathsf{p}$, we have $w^m q \in F$ for some $m \ge 3$ and a homogeneous coordinate $q \in \{x, s, u, v\}$.
This implies
\[
2 r + c = d = m r + \deg{q} \ge 3 r + 1,
\]
which is impossible since $c < r$.
By similar arguments we can show that (2) and (3) are both impossible.
It follows that we are in Case (4).
In this case either $\mathsf{p} = \mathsf{p}_t$ or $\mathsf{p} = \mathsf{p}_w$.
The latter is impossible since $d = 4 a + 3 b < 3 r_2$.
Hence $\mathsf{p} = \mathsf{p}_t$.
We have $t^m q \in F$ for some integer $m \ge 3$ and a homogeneous coordinate $q \in \{x, u, v, w\}$.
It is easy to see that this is possible if and only if $m = 3$ and $\deg q = a$.
Possibly replacing coordinates we may assume $q = u$.
Then it is straightforward to see that $F$ can be written as \eqref{eq:QIexceq}.
\end{proof}
\section{QI centers: exceptional case}
The aim of this section is to prove the following.
\begin{Prop} \label{prop:lctexcQI}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \mathsf{I}_1$ and let $\mathsf{p} \in X$ be an exceptional QI center.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}.
\]
\end{Prop}
Let $X$ be a member of $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \{2, 8\}$ which admits an exceptional QI center,
Then, by Lemma \ref{lem:QIexceq} and Table \ref{table:main}, we have
\[
\mathsf{i} \in \{12, 13, 20, 25, 31, 33, 38, 58\}.
\]
The rest of this section is devoted to the proof of Proposition \ref{prop:lctexcQI} which will be done by division into cases.
By Lemma \ref{lem:QIexceq}, we can choose coordinates $x, u, v, t, w$ of $\mathbb{P} = \mathbb{P} (1, a, b, r_1, r_2)$ as in Case (4) of Lemma \ref{lem:QIeqtypes} with $\mathsf{p} = \mathsf{p}_t$ and the defining polynomial $F$ is as in \eqref{eq:QIexceq}.
\subsection{Case: $a \ge 2$ and $4 a \le 3 b$}
This case corresponds to families $\mathcal{F}_{33}$ and $\mathcal{F}_{58}$.
We have $w^2 v \in f_{4 a + 3 b}$ since $X$ is quasi-smooth at $\mathsf{p}_w$.
Moreover, we see that no quadratic monomial in variables $x, v, w$ appear in $f_{2 a + b}, f_{3 a + 2 b}, f _{4 a + 3 b}$.
This implies $\operatorname{omult}_{\mathsf{p}} (H_u) = 3$, and we have
\[
\alpha_{\mathsf{p}} \left( X; \frac{1}{a} H_u \right) \ge \frac{a}{3} \ge \frac{2}{3}.
\]
Let $D \in |A|_{\mathbb{Q}}$ be an effective $\mathbb{Q}$-divisor other than $\frac{1}{a} H_u$.
We can take a $\mathbb{Q}$-divisor $T \in |r_2 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_u$ since $\{x, u, v, w\}$ isolates $\mathsf{p}$.
Let $q = q_{\mathsf{p}}$ be the quotient morphism of $\mathsf{p} \in X$ and let $\check{\mathsf{p}}$ be the preimage of $\mathsf{p}$ via $q$.
Then we have
\[
3 \operatorname{omult}_{\mathsf{p}} (D)
\le (q^*D \cdot q^*H_u \cdot q^*T)_{\check{\mathsf{p}}}
\le r_1 (D \cdot H_u \cdot T)
= \frac{4 a + 3 b}{b} \le 6
\]
since $4 a \le 3 b$.
Thus $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 1/2$ and the inequality $\alpha_{\mathsf{p}} (X) \ge 1/2$ is proved.
\subsection{Case: $a = 1$}
This case corresponds to families $\mathcal{F}_{12}$, $\mathcal{F}_{20}$ and $\mathcal{F}_{31}$.
We have either $\bar{F} = F (x, 0, v, 1, w) \in (x, v, w)^2 \setminus (x, v, w)^3$ or $\bar{F} \in (x, v, w) \in (x, v, w)^3$ and its cubic part is not a cube of a linear form since $w^2 v \in F$ and $w^3 \notin F$.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ since $a = 1$.
\subsection{Case: $X$ is a member of the family $\mathcal{F}_{13}$}
Let
\[
X = X_{11} \subset \mathbb{P} (1, 1, 2, 3, 5)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_{13}$ and $\mathsf{p} \in X$ an exceptional QI center.
Then we have
\[
F = t^3 z + t^2 f_5 + t f_8 + f_{11},
\]
where $f_i \in \mathbb{C} [x, y, z, w]$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_5$, and $\mathsf{p} = \mathsf{p}_t$.
Let $S, T \in |A|$ be general members.
We have
\[
F (0, 0, z, t, w) = t^3 z + \alpha t z^4 + \beta w z^3 = z (t^3 + \alpha t z^3 + \beta w z^2),
\]
where $\alpha, \beta \in \mathbb{C}$.
We set $\Gamma = (x = y = z = 0)$, which is a quasi-line of degree $1/15$.
If $\beta \ne 0$, then we set
\[
\Delta = (x = y = t^3 + \alpha t z^3 + \beta w z^2 = 0),
\]
which is clearly an irreducible and reduced curve of degree $3/10$ and does not pass through $\mathsf{p}$.
Moreover, we have
\[
T|_S = \Gamma + \Delta.
\]
\begin{Claim} \label{clm:lctexcQINo13-1}
If $\beta \ne 0$, then the intersection matrix $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:lctexcQINo13-1}]
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\}$ and it is easy to see that $S$ is quasi-smooth at $\mathsf{p}_w$ since $S \in |A|$ is general.
By Lemma \ref{lem:pltsurfpair}, $S$ is quasi-smooth along $\Gamma$ and we have $\operatorname{Sing}_{\Gamma} (S) = \{\mathsf{p}_t, \mathsf{p}_w\}$, where $\mathsf{p}_t, \mathsf{p}_w \in S$ are of types $\frac{1}{3} (1, 2)$, $\frac{1}{5} (2, 3)$, respectively.
By Remark \ref{rem:compselfint}, we have
\[
(\Gamma^2)_S = -2 + \frac{2}{3} + \frac{4}{5} = - \frac{8}{15}.
\]
By taking intersection numbers of $T|_S = \Gamma + \Delta$ and $\Gamma, \Delta$, we have
\[
(\Gamma \cdot \Delta)_S = \frac{3}{5}, \quad
(\Delta^2)_S = - \frac{3}{10}.
\]
Thus $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{proof}
Suppose $\beta = 0$ and $\alpha \ne 0$.
We set
\[
\Xi = (x = y = t = 0), \quad
\Theta = (x = y = t^2 + \alpha z^3 = 0),
\]
which are irreducible and reduced curves of degrees $1/10$, $1/5$, respectively, which do not pass through $\mathsf{p}$.
We have
\[
T|_S = \Gamma + \Xi + \Theta.
\]
\begin{Claim} \label{clm:lctexcQINo13-2}
If $\beta = 0$ and $\alpha \ne 0$, then the intersection matrix $M (\Gamma, \Xi, \Theta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:lctexcQINo13-2}]
We have $(\Gamma^2)_S = - 8/15$ by the proof of Claim \ref{clm:lctexcQINo13-1}.
We see that $\Xi \cap (\Gamma \cup \Theta) = \{\mathsf{p}_w\}$ and $S$ is quasi-smooth at $\mathsf{p}_w$.
By Lemma \ref{lem:pltsurfpair}, $S$ is quasi-smooth along $\Xi$ and we have $\operatorname{Sing}_{\Xi} (S) = \{\mathsf{p}_z, \mathsf{p}_w\}$, where $\mathsf{p}_z, \mathsf{p}_w \in S$ are of types $\frac{1}{2} (1, 1)$, $\frac{1}{5} (2, 3)$, respectively.
By Remark \ref{rem:compselfint}, we have
\[
(\Xi^2)_S = -2 + \frac{1}{2} + \frac{4}{5} = - \frac{7}{10}.
\]
We compute the intersection number $(\Gamma \cdot \Xi)_S$.
We have $\Gamma \cap \Xi = \{\mathsf{p}_w\}$ and the germ $\mathsf{p}_w \in S$ is analytically isomorphic to $\bar{o} \in \mathbb{A}^2_{z, t}/\boldsymbol{\mu}_5$, where the $\boldsymbol{\mu}_5$-action on $\mathbb{A}^2_{z, t}$ is given by
\[
(z, t) \mapsto (\zeta^2 z, \zeta^3 t),
\]
and $\bar{o}$ is the image of the origin $o \in \mathbb{A}^2_{z, t}$.
Under the isomorphism, $\Gamma$ and $\Xi$ corresponds to the quotient of $(z = 0)$ and $(t = 0)$.
It follows that
\[
(\Gamma \cdot \Xi)_S = (\Gamma \cdot \Xi)_{\mathsf{p}_w} = \frac{1}{5}.
\]
Then, by taking intersection numbers of $T|_S = \Gamma + \Xi + \Theta$ and $\Gamma, \Xi, \Theta$, we have
\[
(\Gamma \cdot \Theta)_S = \frac{2}{5}, \quad
(\Xi \cdot \Theta)_S = \frac{3}{5}, \quad
(\Theta^2)_S = - \frac{4}{5}.
\]
Thus $M (\Gamma, \Xi, \Theta)$ satisfies the condition $(\star)$.
\end{proof}
Suppose $\beta = \alpha = 0$.
Then
\[
T|_S = \Gamma + 3 \Xi,
\]
where $\Xi = (x = y = t = 0)$.
\begin{Claim} \label{clm:lctexcQINo13-3}
If $\beta = \alpha = 0$, then the intersection matrix $M (\Gamma, \Xi)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:lctexcQINo13-3}]
We have $(\Gamma^2)_S = - 8/15$ by the proof of Claim \ref{clm:lctexcQINo13-1}.
By taking intersection numbers of $T|_S = \Gamma + 3 \Xi$ and $\Gamma, \Xi$, we have
\[
(\Gamma \cdot \Xi)_S = \frac{1}{5}, \quad
(\Xi^2)_S = - \frac{1}{30}.
\]
Thus $M (\Gamma, \Xi)$ satisfies the condition $(\star)$.
\end{proof}
By Claims \ref{clm:lctexcQINo13-1}, \ref{clm:lctexcQINo13-2}, \ref{clm:lctexcQINo13-3} and Lemma \ref{lem:mtdLred}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ 1, \ \frac{1}{3 (A^3) + 1 - 3 \deg \Gamma} \right\} = \frac{10}{19}.
\]
\subsection{Case: $X$ is a member of the family $\mathcal{F}_{25}$}
Let
\[
X = X_{15} \subset \mathbb{P} (1, 1, 3, 4, 7)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_{25}$ and let $\mathsf{p}$ be an exceptional QI center.
Then we have
\[
F = t^3 z + t^2 f_7 + t f_{11} + f_{15},
\]
where $f_i \in \mathbb{C} [x, y, z, w]$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_7$, and we have $\mathsf{p} = \mathsf{p}_t$.
By the quasi-smoothness we have $z^5 \in f_{15}$ and we may assume $\operatorname{coeff}_{f_{15}} (z^5) = 1$.
Then we have
\[
F (0, 0, z, t, w) = t^3 z + z^5 = z (t^3 + z^4).
\]
Let $S, T \in |A|$ be general members.
Then we have
\[
T|_S = \Gamma + \Delta,
\]
where $\Gamma = (x = y = z = 0)$ is a quasi-line of degree $1/28$ and $\Delta = (x = y = t^3 + z^4)$ is an irreducible and reduced curve of degree $1/7$ that does not pass through $\mathsf{p}$.
\begin{Claim} \label{clm:lctexcQINo25-1}
The intersection matrix $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:lctexcQINo25-1}]
We have $\Gamma \cap \Delta = \{\mathsf{p}_w\}$ and $S$ is quasi-smooth at $\mathsf{p}_w$.
Hence $S$ is quasi-smooth along $\Gamma$ by Lemma \ref{lem:pltsurfpair}, and we have $\operatorname{Sing}_{\Gamma} (S) = \{ \mathsf{p}_t, \mathsf{p}_w\}$, where $\mathsf{p}_t, \mathsf{p}_w \in S$ are of types $\frac{1}{4} (1, 3), \frac{1}{7} (3, 4)$, respectively.
By Remark \ref{rem:compselfint}, we have
\[
(\Gamma^2)_S = -2 + \frac{3}{4} + \frac{6}{7} = - \frac{11}{28}.
\]
By taking intersection numbers of $T|_S = \Gamma + \Delta$ and $\Gamma, \Delta$, we have
\[
(\Gamma \cdot \Delta)_S = \frac{3}{7}, \quad
(\Delta^2)_S = - \frac{2}{7}.
\]
It follows that $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{proof}
By Claim \ref{clm:lctexcQINo25-1} and Lemma \ref{lem:mtdLred}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ 1, \ \frac{1}{4 (A^3) + 1 - 4 \deg \Gamma} \right\} = \frac{7}{11}.
\]
\subsection{Case: $X$ is a member of the family $\mathcal{F}_{38}$}
Let
\[
X = X_{18} \subset \mathbb{P} (1, 2, 3, 5, 8)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_{38}$ and $\mathsf{p} \in X$ an exceptional QI center.
Then we have
\[
F = t^3 z + t^2 f_8 + t f_{13} + f_{18},
\]
where $f_i \in \mathbb{C} [x, y, z, w]$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_8$, and $\mathsf{p} = \mathsf{p}_t$.
By the quasi-smoothness of $X$, we have $z^6 \in f_{18}$ and we may assume $\operatorname{coeff}_{f_{18}} (z^6) = 1$.
Then we have
\[
F (0, 0, z, t, w) = t^3 z + z^6 = z (t^3 + z^5).
\]
We set $S = H_x$ and $T = H_y$.
We have
\[
T|_S = \Gamma + \Delta,
\]
where $\Gamma = (x = y = z = 0)$ is a quasi-line of degree $1/40$ and $\Delta = (x = y = t^3 + z^5 = 0)$ is an irreducible and reduced curve of degree $1/8$ that does not pass through $\mathsf{p}$.
\begin{Claim} \label{clm:lctexcQINo38-1}
The intersection matrix $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:lctexcQINo38-1}]
By similar arguments as in the proof of Claim \ref{clm:lctexcQINo25-1}, we have
\[
(\Gamma^2)_S = -2 + \frac{4}{5} + \frac{7}{8} = - \frac{13}{40},
\]
and
\[
(\Gamma \cdot \Delta)_S = \frac{3}{8}, \quad
(\Delta^2)_S = - \frac{1}{8}.
\]
Thus $M (\Gamma, \Delta)$ satisfies the condition $(\star)$.
\end{proof}
By Claim \ref{clm:lctexcQINo38-1} and Lemma \ref{lem:mtdLred}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ 1, \ \frac{1}{5 \cdot 2 \cdot (A^3) + \frac{1}{2} - 5 \deg \Gamma} \right\} = \frac{8}{9}.
\]
This completes the proof of Proposition \ref{prop:lctexcQI}.
\section{QI centers: degenerate case}
The aim of this section is to prove the following, which gives the exact value of $\alpha_{\mathsf{p}} (X)$ for a degenerate QI center $\mathsf{p} \in X$.
Let
\[
X = X_d \subset \mathbb{P} (a_0, a_1, \dots, a_4)_{x_0, x_1, x_2, x_3, x_4}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$, where $1 = a_0 \le a_1 \le \cdots \le a_4$, and let $\mathsf{p} \in X$ be a degenerate QI center.
We choose homogeneous coordinates as in Lemma \ref{lem:eqdegenQI}.
\begin{Prop} \label{prop:lctdegQI}
Let the notation as above and let $\mathsf{p} = \mathsf{p}_{x_k} \in X$ be a degenerate QI center.
Then,
\[
\alpha_{\mathsf{p}} (X) =
\begin{cases}
\frac{a_k + 1}{2 a_k + 1}, & \text{if $a_j = 1$}, \\
1, & \text{otherwise}.
\end{cases}
\]
In particular, we have $\alpha_{\mathsf{p}} (X) > \frac{1}{2}$.
\end{Prop}
\begin{proof}
Let $\varphi \colon Y \to X$ be the Kawamata blowup at $\mathsf{p}$ with exceptional divisor $E$.
Note that we can choose $x_{i_1}, x_{i_2}, x_{i_3}$ as a system of orbifold coordinates at $\mathsf{p}$ and $\varphi$ is the weighted blowup with weight $\operatorname{wt} (x_{i_1}, x_{i_2}, x_{i_3}) = \frac{1}{a_k} (a_{i_1}, a_{i_2}, a_{i_3})$.
Filtering off terms divisible by $x_j$ in \eqref{eq:eqdegenQI}, we have
\[
x_j (x_k^2 + \cdots) = g (x_{i_1}, x_{i_2}, x_{i_3}, 0) =: \bar{g}.
\]
Since the polynomial $x_k^2 + \cdots$ does not vanish at $\mathsf{p}$, the vanishing order of $x_j$ along $E$ coincides with that of $\bar{g}$, which is clearly $d/a_k$.
Hence we have
\begin{equation} \label{eq:complctdegQI}
K_Y + \frac{1}{a_j} \tilde{H}_{x_j} + \frac{2}{a_j} E
= \varphi^* \left( K_X + \frac{1}{a_j} H_{x_j} \right),
\end{equation}
where $\tilde{H}_{x_j}$ is the proper transform of $H_{x_j}$ on $Y$.
In particular $(X, \frac{1}{a_j} H_{x_j})$ is not canonical at $\mathsf{p}$.
By Lemma \ref{lem:singnoncanbd}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ if $(X, \frac{1}{a_j} H_{x_j})$ is log canonical at $\mathsf{p}$, and otherwise $\alpha_{\mathsf{p}} (X) = \operatorname{lct}_{\mathsf{p}} (X;\frac{1}{a_j} H_{x_j})$.
Suppose $a_j > 1$.
The pair $(E, \frac{1}{a_j} \tilde{H}_{x_j}|_E)$ is log canonical since $\tilde{H}_{x_j}|_E$ is isomorphic to
\[
(g (x_{i_1},x_{i_2},x_{i_3},0) = 0) \subset \mathbb{P} (a_{i_1},a_{i_2},a_{i_3}) \cong E,
\]
which is quasi-smooth by Lemma \ref{lem:eqdegenQI}.
By the inversion of adjunction, the pair $(Y, \frac{1}{a_j} \tilde{H}_{x_j} + E)$ is log canonical along $E$, and so is the pair $(Y, \frac{1}{a_j} \tilde{H}_{x_j} + \frac{2}{a_j} E)$ since $2/a_j \le 1$.
By \eqref{eq:complctdegQI}, the pair $(X, \frac{1}{a_j} H_{x_j})$ is log canonical at $\mathsf{p}$.
Thus $\alpha_{\mathsf{p}} (X) \ge 1$.
The existence of the prime divisor $H_{x_0} \in A$ passing through $\mathsf{p}$ shows $\alpha_{\mathsf{p}} (X) \le 1$, and we conclude $\alpha_{\mathsf{p}} (X) = 1$ in this case.
Suppose $a_j = 1$.
We set
\[
\theta = \frac{a_k+1}{2 a_k + 1},
\]
and prove $\operatorname{lct}_{\mathsf{p}} (X;\frac{1}{a_j} H_{x_j}) = \theta$.
For a rational number $c \ge 0$, it is easy to see that the discrepancy of the pair $(X, \frac{c}{a_j} H_{x_j})$ along $E$ is
\[
\frac{1}{a_k} - \frac{c d}{a_j a_k}
\]
and it is at least $-1$ if and only if $c \le \theta$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;\frac{1}{a_j} H_{x_j}) \le \theta$.
Moreover, since
\[
K_Y + \frac{\theta}{a_j} \tilde{H}_{x_j} + E = \varphi^* \left( K_X + \frac{\theta}{a_j} H_{x_j} \right)
\]
and the pair $(Y, \frac{\theta}{a_j} \tilde{H}_{x_j} + E)$ is log canonical along $E$, the pair $(X, \frac{\theta}{a_j} H_{x_j})$ is log canonical at $\mathsf{p}$.
This shows $\alpha_{\mathsf{p}} (X) = \theta$ and the proof is completed.
\end{proof}
\begin{Ex} \label{ex:No46degQI}
Let $X = X_{21} \subset \mathbb{P} (1,1,3,7,10)$ be a member of the family $\mathcal{F}_{46}$ and $\mathsf{p} = \mathsf{p}_w$ the $\frac{1}{10} (1,3,7)$ point, which is the center of a quadratic involution.
Assume that $\mathsf{p}$ is degenerate, which is equivalent to $X$ being birationally superrigid.
Then, by Proposition \ref{prop:lctdegQI}, we have
\[
\alpha (X) \le \alpha_{\mathsf{p}} (X) = \frac{11}{21}.
\]
\end{Ex}
\section{QI centers: non-degenerate case}
The aim of this section is to prove the following.
\begin{Prop} \label{prop:ndQIcent}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I} \setminus \{2, 5, 8\}$ and $\mathsf{p} \in X$ be a nondegenerate QI center.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}.
\]
\end{Prop}
The rest of this section is entirely devoted to the proof of Proposition \ref{prop:ndQIcent}, which will be done by dividing into several cases.
\subsection{Case: $X$ has a unique QI center}
By Lemma \ref{lem:QIeqtypes}, we can choose homogeneous coordinates so that
\[
X = X_{2 r + c} \subset \mathbb{P} (1, a, b, c, r)_{x, s, u, v, w},
\]
where $a$ is corprime to $b$, $a < b$, $r = a + b$ and $c < r$.
Let $\mathsf{p} \in X$ be the QI center.
Then $\mathsf{p} = \mathsf{p}_w$ and the defining polynomial $F$ of $X$ can be written as
\[
F = w^2 v + w f_{r + c} + f _{2 r + c},
\]
where $f_{r + c} = f_{r + c} (x, s, u)$ and $f_{2 r + c} = f_{2 r + c} (x, s, u, v)$ are quasi-homogeneous polynomials of the indicated degree.
We will show that $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{c} H_v) \ge 1/2$.
\begin{Claim} \label{clm:ndQIcent-1}
Suppose that $c \ge 2$ and $2 r + c$ is divisible by $b$.
Then
\[
\operatorname{lct}_{\mathsf{p}} \left(X; \frac{1}{c} H_v \right) \ge \frac{1}{2}.
\]
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-1}]
We first show that $\mathsf{p}_u \notin X$ unless $X$ belongs to the family $\mathcal{F}_7$.
Suppose $\mathsf{p}_u \in X$.
By the quasi-smoothness of $X$ at $\mathsf{p}_u$, we have $d = m b + e$, where $m \in \mathbb{Z}_{> 0}$ and $e \in \{1, a, c, r\}$.
Since $d$ is divisible by $b$, we see that $e$ is divisible by $b$.
This is possible only when $e = c$ since $r = a + b$ and $a$ are both coprime to $b$.
Thus we can write $c = k b$ for some $k \in \mathbb{Z}_{> 0}$.
Take any point $\mathsf{q} \in (x = s = w = 0) \cap X$.
The singularity $\mathsf{q} \in X$ is of type $\frac{1}{b} (1, a, r) = \frac{1}{b} (1, a, a)$.
It follows that $a = 1$ and $b = 2$ since $\mathsf{q} \in X$ is terminal.
We have $r = a + b = 3$ and $c = 2$ since $c = k b = 2 k < r = 3$.
Thus $X = X_8 \subset \mathbb{P} (1, 1, 2, 2, 3)$ and this belongs to $\mathcal{F}_7$.
We first consider the case where $\mathsf{p}_u \notin X$.
This means that $u^m \in F$ for some $m \in \mathbb{Z}_{> 0}$.
We have $\operatorname{omult}_{\mathsf{p}} (H_v) \le m = (2 r + c)/b$ and hence
\[
\operatorname{lct}_{\mathsf{p}} \left( X; \frac{1}{c} H_v \right) \ge \frac{b c}{2 r + c}.
\]
It remains to prove the inequality $bc/(2r + c) \ge 1/2$, which is equivalent to $(2 b - 1) c \ge 2 r$.
We have
\[
(2 b - 1) c \ge 2 (2 b - 1) = 2 (b-1) + 2 b \ge 2 a + 2 b = 2 r,
\]
since $c \ge 2$ and $b > a$.
This shows $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{c} H_v) \ge 1/2$.
We next consider the case where $\mathsf{p}_u \in X$.
In this case $X$ belongs to the family $\mathcal{F}_7$ and $X = X_8 \subset \mathbb{P} (1, 1, 2, 2, 3)_{x, s, u, v, w}$ with defining polynomial
\[
F = w^2 v + w f_5 (x, s, u) + f_8 (x, s, u, v).
\]
We have $u^4 \notin F$ since $\mathsf{p}_u \in X$.
We show that $f_5 (x, s, u)$ contains a monomial involving $u$.
Suppose to the contrary that $f_5 = f_5 (x, s)$ is a polynomial in variables $x$ and $s$.
We can write $f_8 = u^3 g_2 + u^2 g_4 + u g_6 + g_8 + v h_6$, where $g_i = g_i (x, s)$ and $h_6 = h_6 (x, s, u, v)$ are quasi-homogeneous polynomials of indicated degree.
Then we have $F = v (w^2 + h_6) + g$, where $g = w f_5 + u^3 g_2 + u^2 g_4 + u g_6 + g_8 \in (x, s)^2$, and we see that $X$ is not quasi-smooth at any point in the nonempty set
\[
(t = w^2 + h_6 = x = s = 0) \subset \mathbb{P} (1, 1, 2, 2, 3).
\]
This is a contradiction.
It follows that there is a monomial involving $u$ which appears in $f_5$ with nonzero coefficient.
This implies $\operatorname{omult}_{\mathsf{p}} (H_v) \le 4$ and we have $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{2} H_v) \ge 1/2$ as desired.
\end{proof}
\begin{Claim} \label{clm:ndQIcent-2}
Suppose that $c \ge 2$ and $2 r + c$ is not divisible by $b$.
Then
\[
\operatorname{lct}_{\mathsf{p}} \left(X; \frac{1}{c} H_v \right) \ge \frac{c}{5} \ge \frac{4}{5}.
\]
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-2}]
By Lemma \ref{lem:uniqQItypes}, we have
\[
X = X_{5 a + 4} \subset \mathbb{P} (1, a, a + 1, a + 2, 2 a + 1)_{x, s, u, v, w}
\]
with $a \in \{2, 3, 4\}$.
Moreover $\mathsf{p} = \mathsf{p}_w$ and the defining polynomial of $X$ can be written as
\[
F = w^2 v + w f_{3 a + 3} (x, s, u) + f_{5 a + 4} (x, s, u, v).
\]
By the quasi-smoothness of $X$ at $\mathsf{p}_z$, we have either $u^3 \in f_{3 a + 3}$ or $u^4 s \in f_{5 a + 4}$.
This implies $\operatorname{omult}_{\mathsf{p}} (H_v) \le 5$ and we have
\[
\operatorname{lct}_{\mathsf{p}} \left(X; \frac{1}{c} H_v \right) \ge \frac{c}{5} = \frac{a+2}{5} \ge \frac{4}{5}.
\]
This proves the claim.
\end{proof}
It remains to consider the case where $c = 1$.
By Lemma \ref{lem:uniqQItypes}, $X$ belongs to a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{24, 46\}$.
\begin{Claim} \label{clm:ndQIcent-3}
Suppose $X$ is a member of the family $\mathcal{F}_{24}$.
Then $\operatorname{lct}_{\mathsf{p}} (X;H_v) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-3}]
We have
\[
X = X_{15} \subset \mathbb{P} (1, 2, 5, 1, 7)_{x, s, u, v, w}
\]
and $\mathsf{p} = \mathsf{p}_w$ is of type $\frac{1}{7} (1, 2, 5)$.
We can write
\[
F = w^2 v + w f_8 (x, s, u) + f_{15} (x, s, u, v),
\]
where $f_8 = f_8 (x, s, u) \ne 0$ and $f_{15} = f_{15} (x, s, u, v)$ are quasi-homogeneous polynomials of degree $8$ and $15$, respectively.
We have $u^3 \in f_{15}$ and we may assume $\operatorname{coeff}_{f_{15}} (u^3) = 1$.
We set $\bar{F} := F (x, s, u, 0, 1)$.
For a given $\underline{c} = (c_1, c_2, c_3) \in (\mathbb{Z}_{ > 0})^3$, we denote by $G_{\underline{c}} \in \mathbb{C} [x, s, u]$ the lowest weight part of $\bar{F}$ with respect to the weight $\operatorname{wt} (x, s, u) = \underline{c}$ and let
\[
\mathcal{D}_{\underline{c}} = \mathcal{D}^{\operatorname{wf}}_{G_{\underline{c}}}
\]
be the effective $\mathbb{Q}$-divisor on $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ associated to $G_{\underline{c}}$.
By Lemma \ref{lem:lctwblwh}, we have
\begin{equation} \label{eq:ndBIuniclaim}
\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge \min \left\{ \frac{c_1 + c_2 + c_3}{\deg G_{\underline{c}}}, \ \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) \right\},
\end{equation}
where $\deg G_{\underline{c}}$ is the degree with respect to the weight $\operatorname{wt} (x, s, u) = \underline{c}$.
Suppose $u s x \in f_8$.
In this case, we may assume $\operatorname{coeff}_{f_8} (u s x) = 1$ and we have $G_{\underline{c}} = u s x + u^3$ for $\underline{c} = (1, 1, 1)$.
In this case $\mathbb{P} (\underline{c})^{\operatorname{wf}} = \mathbb{P}^2$, $\operatorname{Diff} = 0$ and $\mathcal{D}_{\underline{c}}$ is the sum of a line and a conic intersection at distinct 2 points.
It is straightforward to check
\[
\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})
= \operatorname{lct} (\mathbb{P}^2; \mathcal{D}_{\underline{c}})
= 1
\] and we have $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge 1$ in this case.
Suppose that $u s x \notin f_8$ and $s^4 \in f_8$.
In this case, we may assume $\operatorname{coeff}_{f_8} (s^4) = 1$ and $\operatorname{coeff}_{f_{15}} (u s^5) = 0$ by replacing $s$ and $w$.
Hence we have
\[
F (0, s, u, 0, 1) = s^4 + u^3,
\]
and thus $\operatorname{lct}_{\mathsf{p}} (X;H_v) \ge \operatorname{lct}_{\mathsf{p}} (H_x, H_v|_{H_x}) = 7/12$, where the equality follows from \cite[8.21 Proposition]{Kol} (or by Lemma \ref{lem:lctwblwh} with $\operatorname{wt} (s, u) = (3, 4)$).
Suppose $u x^3 \in f_8$.
In this case we may assume $\operatorname{coeff}_{f_8} (u s x) = 1$ by rescaling $x$.
We consider a weight $\underline{c} = (2, e, 3)$, where $e$ is a sufficiently large integer which is corprime to $6$.
Then $G_{\underline{c}} = u (x^3 + u^2)$, $\mathbb{P} (\underline{c})^{\operatorname{wf}} = \mathbb{P} (2, e, 3)$, $\operatorname{Diff} = 0$ and $\mathcal{D}_{\underline{c}}$ is the union of $2$ quasi-smooth curves $(u = 0)$ and $(x^3 + u^2 = 0)$.
We have
\[
\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) = \operatorname{lct} (\mathbb{P} (2, e, 3); \mathcal{D}_{\underline{c}})
= \operatorname{lct}_{(0:1:0)} (\mathbb{P} (2, e, 3); \mathcal{D}_{\underline{c}}) = \frac{5}{9},
\]
and thus $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge 5/9$ in this case.
In the following we assume $u s x, s^4, u x^3 \notin f_8$.
We can write
\[
\bar{F} = (\alpha_1 s^3 x^2 + \alpha_2 s^2 x^4 + \alpha_3 s x^6 + \alpha_4 x^8) + (u^3 + \beta u s^5 + \gamma s^7 x + g_{15}),
\]
where $\alpha_1, \dots, \alpha_4, \beta, \gamma \in \mathbb{C}$ and $g_{15} = g_{15} (x, s, u) \not\ni u^3$ is a quasi-homogeneous polynomial of degree $15$ which is contained in the ideal $(x, u)^2 \subset \mathbb{C} [x, s, u]$.
Note that at least one of $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ is nonzero since $f_8 (x, s, u) \ne 0$.
Note also that $\lambda u s^5$ and $\mu s^7 x$ are the only terms in $\bar{F}$ which is not contained in $(x, u)^2$.
It follows from the quasi-smoothness of $X$ that $(\lambda, \mu) \ne (0,0)$.
Suppose $\beta \ne 0$.
Replacing $u$, we may assume $\beta = 1$ and $\gamma = 0$.
There exists $j \in \{1, 2, 3, 4\}$ such that $\alpha_j \ne 0$ since $f_8 \ne 0$ as a polynomial, and thus we set $i = \min \{\, j \mid \alpha_j \ne 0 \,\} \in \{1, 2, 3, 4\}$.
We may assume $\alpha_i = 1$ by rescaling $x$.
We set $\underline{c} = (2 i + 7, 4 i, 10 i)$.
We have $G_{\underline{c}} = s^{4-i} x^{2 i} + u^3 + u s^5$ for $1 \le i \le 4$.
Moreover we see that
\[
\mathbb{P} (\underline{c})^{\operatorname{wf}} =
\begin{cases}
\mathbb{P} (2 i + 7, 2, 5)_{\tilde{x}, \tilde{s}, \tilde{u}}, & \text{if $1 \le i \le 3$}, \\
\mathbb{P} (3, 2, 1)_{\tilde{x}, \tilde{s}, \tilde{u}}, & \text{if $i = 4$},
\end{cases}
\]
and
\[
(\operatorname{Diff}, \mathcal{D}_{\underline{c}}) =
\begin{cases}
(\frac{2 i -1}{2 i} H_{\tilde{x}}, D_i), & \text{if $1 \le i \le 3$}, \\
(\frac{7}{8} H_{\tilde{x}} + \frac{4}{5} H_{\tilde{s}}, D'), & \text{if $i = 4$},
\end{cases}
\]
where
\[
D_i = (\tilde{s}^{4-i} \tilde{x} + \tilde{u}^3 + \tilde{u} \tilde{s}^5 = 0), \quad
D' = (\tilde{x} + \tilde{u}^3 + \tilde{u} \tilde{s} = 0)
\]
are prime divisors on $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
We first consider the case where $1 \le i \le 3$.
We see that $H_{\tilde{x}}$ is quasi-smooth, and $D_i$ is quasi-smooth outside $\{\mathsf{q}\}$, where $\mathsf{q} = (1\!:\!0\!:\!0) \in \mathbb{P} (\underline{c})^{\operatorname{wf}}$.
Moreover they intersect at two points $(0\!:\!1\!:\!0)$ and $(0\!:\!-1\!:\!1)$ transversally.
It follows that $\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) = \min \{1, \ \operatorname{lct}_{\mathsf{q}} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \mathcal{D}_{\underline{c}})\}$ since $H_{\tilde{x}}$ does not pass through $\mathsf{q}$.
If $i = 3$, then $D_3$ is also quasi-smooth at $\mathsf{q}$, which implies $\operatorname{lct} (\mathbb{P} (\underline{c}), \operatorname{Diff}; \mathcal{D}_{\underline{c}}) = 1$.
If $i = 1, 2$, then we have
\[
\begin{split}
\operatorname{lct}_{\mathsf{q}} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \mathcal{D}_{\underline{c}})
&= \operatorname{lct}_{(0,0)} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, (\tilde{s}^{4 - i} + \tilde{u}^3 + \tilde{u} \tilde{s}^5 = 0)) \\
&= \operatorname{lct}_{(0,0)} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, (\tilde{s}^{4-i} + \tilde{u}^3 = 0)) \\
&= \frac{3 + (4-i)}{3 (4-i)} = \frac{7 - i}{3 (4-i)}
\end{split}
\]
since $(\tilde{s}^{4 - i} + \tilde{u}^3 + \tilde{u} \tilde{s}^5 = 0)$ is analytically equivalent to $(\tilde{s}^{4-i} + \tilde{u}^3 = 0)$.
Thus, by Lemma \ref{lem:lctwblwh}, we have
\[
\begin{split}
\operatorname{lct}_{\mathsf{p}} (X; H_v)
&\ge \min \left\{ \frac{(2 i + 7) + 4 i + 10 i}{30 i}, \ \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) \right\} \\
&= \begin{cases}
\frac{2}{3}, & \text{if $i = 1$}, \\
\frac{13}{20}, & \text{if $i = 2$}, \\
\frac{11}{18}, & \text{if $i = 3$}.
\end{cases}
\end{split}
\]
Suppose $\beta = 0$.
In this case we have $\gamma \ne 0$.
We set $i = \min \{\, j \mid \alpha_j \ne 0 \, \} \in \{1, 2, 3, 4\}$.
We may assume $\gamma = \alpha_i = 1$ by rescaling $x$ and $s$ appropriately.
We set
\[
\underline{c} =
\begin{cases}
(3 (i + 3), 3 (2 i - 1), 15 i - 4), & \text{if $1 \le i \le 3$}, \\
(3, 3, 8), & \text{if $i = 4$}.
\end{cases}
\]
We have $G_{\underline{c}} = s^{4-i} x^{2 i} + u^3 + s^7 x$ for $1 \le i \le 4$.
Moreover we see that
\[
\mathbb{P} (\underline{c})^{\operatorname{wf}} =
\begin{cases}
\mathbb{P} (i+3, 2 i - 1, 15 i - 4)_{\tilde{x}, \tilde{s}, \tilde{u}}, & \text{if $1 \le i \le 3$}, \\
\mathbb{P} (1, 1, 8)_{\tilde{x}, \tilde{s}, \tilde{u}}, & \text{if $i = 4$},
\end{cases}
\]
and
\[
\operatorname{Diff} = \frac{2}{3} H_{\tilde{u}}, \quad
\mathcal{D}_{\underline{c}} = (\tilde{s}^{4-i} \tilde{x}^{2 i} + \tilde{u} + \tilde{s}^7 \tilde{x} = 0).
\]
We see that $H_{\tilde{u}}$ and $\mathcal{D}_{\underline{c}}$ are both quasi-smooth.
If $i = 3, 4$, then $H_{\tilde{u}}$ and $\mathcal{D}_{\underline{c}}$ intersect transversally and thus we have $\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) = 1$.
Suppose $i = 1, 2$.
Then $H_{\tilde{u}}$ and $\mathcal{D}_{\underline{c}}$ intersect transversally except at $\mathsf{p}_{\tilde{x}} = (1\!:\!0\!:\!0) \in \mathbb{P} (\underline{c})^{\operatorname{wf}}$, and we have
\[
\begin{split}
\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})
&= \operatorname{lct}_{\mathsf{p}_{\tilde{x}}} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) \\
&= \operatorname{lct} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, \tfrac{2}{3} (\tilde{u} = 0); (\tilde{s}^{4-i} + \tilde{u} + \tilde{s}^7 = 0)) \\
&= \operatorname{lct} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, \tfrac{2}{3} (\tilde{u} = 0); (\tilde{s}^{4-i} + \tilde{u} = 0)) \\
&=
\begin{cases}
\frac{2}{3}, & \text{if $i = 1$}, \\
1, & \text{if $i = 2$}.
\end{cases}
\end{split}
\]
Thus, by Lemma \ref{lem:lctwblwh}, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge
\begin{cases}
\frac{2}{3}, & \text{if $i = 1$}, \\
\frac{25}{39}, & \text{if $i = 2$}, \\
\frac{74}{123}, & \text{if $i = 3$}, \\
\frac{7}{12}, & \text{if $i = 4$}.
\end{cases}
\]
This proves the claim.
\end{proof}
\begin{Claim} \label{clm:ndQIcent-4}
Suppose $X$ is a member of the family $\mathcal{F}_{46}$.
Then $\operatorname{lct}_{\mathsf{p}} (X;H_v) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-4}]
We have
\[
X = X_{21} \subset \mathbb{P} (1, 3, 7, 1, 10)_{x, s, u, v, w}
\]
and $\mathsf{p} = \mathsf{p}_w$ is of type $\frac{1}{10} (1, 3, 7)$.
We can write
\[
F = w^2 v + w f_{11} (x, s, u) + f_{21} (x, s, u, v),
\]
where $f_{11} = f_{11} (x, s, u) \ne 0$ and $f_{21} = f_{21} (x, s, u, v)$ are quasi-homogeneous polynomials of degree $11$ and $21$, respectively.
We have $u^3, s^7 \in F$ by the quasi-smoothness of $X$, and we may assume $\operatorname{coeff}_F (u^3) = \operatorname{coeff}_F (s^7) = 1$.
We set $\bar{F} = F (x, s, u, 0, 1)$, which can be written as
\[
\bar{F} = (\alpha u s x + \beta s^3 x^2 + \gamma u x^4 + \delta s^2 x^5 + \varepsilon s x^8 + \zeta x^{11}) + (u^3 + s^7),
\]
where $\alpha, \beta, \dots, \zeta \in \mathbb{C}$.
We introduce various $3$-tuples $(\underline{c} = (c_1, c_2, c_3)$ of positive integers according to the following division into cases.
We denote by $G_{\underline{c}}$ the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (x, s, u) = \underline{c}$.
\begin{itemize}
\item[(i)] $\alpha \ne 0$.
In this case we may assume $\alpha = 1$.
We set $\underline{c} = (1, 1, 1)$.
Then we have $G_{\underline{c}} = u s x + u^3$.
\item[(ii)] $\alpha = 0$ and $\beta \ne 0$.
In this case we may assume $\beta = 1$.
We set $\underline{c} = (6, 3, 7)$.
Then we have $G_{\underline{c}} = s^3 x^2 + u^3 + s^7$.
\item[(iii)] $\alpha = \beta = 0$ and $\gamma \ne 0$.
In this case we may assume $\gamma = 1$.
We set $\underline{c} = (7, 6, 14)$.
Then we have $G_{\underline{c}} = u x^4 + u^3 + s^7$.
\item[(iv)] $\alpha = \beta = \gamma = 0$ and $\delta \ne 0$.
In this case we may assume $\delta = 1$.
We set $\underline{c} = (3, 3, 7)$.
In this case we have $G_{\underline{c}} = s^2 x^5 + u^3 + s^7$.
\item[(v)] $\alpha = \beta = \gamma = \delta = 0$ and $\varepsilon \ne 0$.
In this case we may assume $\varepsilon = 1$.
We set $\underline{c} = (9, 12, 28)$.
In this case we have $G_{\underline{c}} = s x^8 + u^3 + s^7$.
\item[(vi)] $\alpha = \beta = \gamma = \delta = \varepsilon = 0$.
In this case we may assume $\delta = 1$.
We set $\underline{c} = (21, 33, 77)$.
In this case we have $x^{11} + u^3 + s^7$.
\end{itemize}
The descriptions of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$, $\operatorname{Diff}$ and $G_{\underline{c}} ^{\operatorname{wf}}$ are given in Table \ref{table:No46lwp}, where we choose $\tilde{x}, \tilde{s}, \tilde{u}$ as homogeneous coordinates of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
\begin{table}[h]
\renewcommand{1.35}{1.15}
\begin{center}
\caption{Family $\mathcal{F}_{46}$: Weights and LCT}
\label{table:No46lwp}
\begin{tabular}{ccccccc}
Case & $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ & $\operatorname{Diff}$ & $G_{\underline{c}}^{\operatorname{wf}}$ & $\eta$ & $\theta$ \\
\hline
(i) & $\mathbb{P} (1, 1, 1)$ & $0$ & $\tilde{u} (\tilde{s} \tilde{x} + \tilde{u}^2)$ & $1$ & $1$ \\
(ii) & $\mathbb{P} (2, 1, 7)$ & $\frac{2}{3} H_{\tilde{u}}$ & $\tilde{s}^3 \tilde{x}^2 + \tilde{u} + \tilde{s}^7$ & $2/3$ & $2/3$ \\
(iii) & $\mathbb{P} (1, 3, 1)$ & $\frac{1}{2} H_{\tilde{x}} + \frac{6}{7} H_{\tilde{s}}$ & $\tilde{u} \tilde{x}^2 + \tilde{u}^3 + \tilde{s}$ & $1$ & $9/14$ \\
(iv) & $\mathbb{P} (1, 1, 7)$ & $\frac{2}{3} H_{\tilde{u}}$ & $\tilde{s}^2 \tilde{x}^5 + \tilde{u} + \tilde{s}^7$ & $5/6$ & $13/21$ \\
(v) & $\mathbb{P} (3, 1, 7)$ & $\frac{3}{4} H_{\tilde{x}} + \frac{2}{3} H_{\tilde{u}}$ & $\tilde{s} \tilde{x}^2 + \tilde{u} + \tilde{s}^7$ & $1$ & $7/12$ \\
(vi) & $\mathbb{P} (1, 1, 1)$ & $\frac{10}{11} H_{\tilde{x}} + \frac{6}{7} H_{\tilde{s}} + \frac{2}{3} H_{\tilde{u}}$ & $\tilde{x} + \tilde{u} + \tilde{s}$ & $1$ & $131/231$
\end{tabular}
\end{center}
\end{table}
We set $\mathcal{D}_{\underline{c}} = \mathcal{D}^{\operatorname{wf}}_{G_{\underline{c}}}$.
We explain the computation of $\eta := \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})$ whose value is given in the 5th column of Table \ref{table:No46lwp}.
The computation $\eta = 1$ is straightforward when we are in case (i) since $\mathcal{D}_{\underline{c}}$ is the union of of a line and a conic on $\mathbb{P}^2$ intersecting at $2$ points.
In the other cases, $\mathcal{D}_{\underline{c}}$ is the divisor defined by $G_{\underline{c}}^{\operatorname{wf}} = 0$ which is a quasi-line in $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
If we are in one of the cases (iii), (v) and (vi), then any 2 of the components of $\operatorname{Diff} + \mathcal{D}_{\underline{c}}$ intersect transversally, which implies $\eta = 1$.
If we are in case (ii) or (iv), then $H_{\tilde{u}}$ and $\mathcal{D}_{\underline{c}}$ intersect transversally except at $\mathsf{q} = (1\!:\!0\!:\!0) \in \mathbb{P} (\underline{c})^{\operatorname{wf}}$.
We set $e = 3, 2$ if we are in case (ii), (iv), respectively.
Then we have
\[
\begin{split}
\operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})
&= \operatorname{lct}_{\mathsf{q}} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}}) \\
&= \operatorname{lct}_{(0,0)} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, \tfrac{2}{3} (\tilde{u} = 0); (\tilde{s}^e + \tilde{u} + \tilde{s}^7 = 0)) \\
&= \operatorname{lct}_{(0,0)} (\mathbb{A}^2_{\tilde{s}, \tilde{u}}, \tfrac{2}{3} (\tilde{u} = 0); (\tilde{s}^e + \tilde{u} = 0)).
\end{split}
\]
This completes the explanations of the computations of $\eta$.
We set
\[
\theta = \left\{\frac{c_1 + c_2 + c_3}{\deg_{\underline{c}} (G_{\underline{c}}^{\operatorname{wf}})}, \ \eta \right\}
\]
which is described in the 6th column of Table \ref{table:No46lwp}.
By Lemma \ref{lem:lctwblwh}, we have $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge \theta \ge 1/2$ and the claim is proved.
\end{proof}
By Claims \ref{clm:ndQIcent-1}, \ref{clm:ndQIcent-2}, \ref{clm:ndQIcent-3} and \ref{clm:ndQIcent-4}, we have $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{c} H_v) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{c} H_v$.
We set $\lambda = (r + c)/(2 r + c)$ and we will show that $\operatorname{lct}_{\mathsf{p}} (X; D) \ge \lambda$.
Suppose not, that is, $(X, \lambda D)$ is not log canonical at $\mathsf{p}$.
Let $\varphi \colon Y \to X$ be the Kawamata blowup of $\mathsf{p} \in X$.
Then, for the proper transforms $\tilde{H}_t$ and $\tilde{D}$ of $H_t$ and $D$, respectively, we have
\[
\begin{split}
\tilde{H}_t &\sim c \varphi^*A - \frac{r + c}{r} E, \\
\tilde{D} & \sim_{\mathbb{Q}} \varphi^*A - \frac{e}{r} E,
\end{split}
\]
where $e \in \mathbb{Q}_{\ge 0}$.
By \cite{Kawamata}, the discrepancy of the pair $(X, \lambda D)$ along $E$ is negative, and thus we have $e > 1/\lambda$.
By \cite[Theorem 4.9]{CPR00}, $-K_Y \sim_{\mathbb{Q}} \varphi^* A - \frac{1}{r} E$ is nef (more precisely, $- m K_Y$ defines the flopping contraction for a sufficiently divisible $m > 0$).
Hence $(-K_Y \cdot \tilde{H}_t \cdot \tilde{D}) \ge 0$ and we have
\[
\begin{split}
0 \le (-K_Y \cdot \tilde{H}_t \cdot \tilde{D}) &= c (A^3) - \frac{e (r+c)}{r^3} (E^3) \\
&= \frac{2 r + c}{a b r} - \frac{e (r+c)}{a b r} < \frac{2 r + c}{a b r} - \frac{r+c}{\lambda a b r} = 0.
\end{split}
\]
This is a contradiction.
Therefore $\operatorname{lct}_{\mathsf{p}} (X;D) \ge \lambda$ and thus
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ \operatorname{lct}_{\mathsf{p}} \left(X; \frac{1}{c} H_v \right), \ \frac{r + c}{2 r + c} \right\} \ge \frac{1}{2}.
\]
This completes the proof of Proposition \ref{prop:ndQIcent} when $X$ has a unique QI center.
\subsection{Case: $X$ has exactly $3$ distinct QI centers}
\label{sec:ndQI3ditinct}
By Lemma \ref{lem:QIeqtypes}, we can choose homogeneous coordinates so that
\[
X = X_{3 r} \subset \mathbb{P} (1, a, b, r, r)_{x, y, z, t, w},
\]
where $a$ is coprime to $b$, $a \le b$ and $a + b = r$.
Let $\mathsf{p} \in X$ be a QI center.
Then we may assume $\mathsf{p} = \mathsf{p}_w$ by replacing $t$ and $w$ suitably.
Then the defining polynomial $F$ of $X$ can be written as
\[
F = w^2 t + w f_{2 r} + f_{3r},
\]
where $f_{2 r} (x, y, z)$ and $f_{3 r} (x, y, z, t)$ are quasi-homogeneous polynomials of degrees $2 r$ and $3 r$, respectively.
We have $(A^3) = 3 r/a b r^2 = 3 / a b r$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{r a b (A^3)} = \frac{2}{3},
\]
and Proposition \ref{prop:ndQIcent} is proved when $X$ has exactly $3$ distinct QI centers.
\subsection{Case: $X$ has exactly $2$ distinct QI centers and their singularity types are equal}
\label{sec:ndQI2eqtypes}
By Lemma \ref{lem:QIeqtypes}, we can choose homogeneous coordinates so that
\[
X = X_{4 r} \subset \mathbb{P} (1, a, b, r, 2 r)_{x, y, z, t, w},
\]
where $a$ is coprime to $b$, $a \le b$ and $a + b = r$.
Let $\mathsf{p} \in X$ be a QI center.
We may assume $\mathsf{p} = \mathsf{p}_t$ by replacing $w$ suitably.
Then the defining polynomial $F$ of $X$ can be written as
\[
F = t^2 w + t f_{3 r} + f_{4 r},
\]
where $f_{3 r} (x, y, z)$ and $f_{4 r} (x, y, z, w)$ are quasi-homogeneous polynomials of degrees $3 r$ and $4 r$, respectively.
Note that $(A^3) = 4 r/2 a b r^2 = 2/a b r$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{r a b (A^3)} = 1,
\]
and Proposition \ref{prop:ndQIcent} is proved in this case.
\subsection{Case: $X$ has exactly $2$ distinct QI centers and their singularity types are distinct}
By Lemma \ref{lem:QIeqtypes}, we have
\[
X = X_{4 a + 3 b} \subset \mathbb{P} (1, a, b, r_1, r_2)_{x, u, v, t, w},
\]
where $a$ is coprime to $b$, $r_1 = a + b$ and $r_2 = 2 a + b$.
We first consider the QI center $\mathsf{p} = \mathsf{p}_t \in X$ of type $\frac{1}{r_1} (1, a, b)$.
The defining polynomial $F$ of $X$ can be written as
\[
F = t^2 w + t f_{3 a + 2 b} + f_{4 a + 3 b},
\]
where $f_{3 a + 2 b} = f_{3 a + 2 b} (x, u, v)$ and $f_{4 a + 3 b} = f_{4 a + 3 b} (x, u, v, w)$ are quasi-homogeneous polynomials of the indicated degrees.
Note that $(A^3) = (4 a + 3 b)/a b r_1 r_2$.
By Lemma \ref{lem:QIeqtypes}, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{r_1 a b (A^3)} = \frac{2 r_2}{4 a + 3b} = \frac{4 a + 2 b}{4 a + 3 b} > \frac{2}{3}.
\]
We next consider the QI center $\mathsf{p} = \mathsf{p}_w \in X$ of type $\frac{1}{r_2} (1, a, a + b)$.
Then the defining polynomial $F$ of $X$ can be written as
\[
F = w^2 v + w f_{2 a + 2 b} + f_{4 a + 3 b},
\]
where $f_{2 a + 2 b} = f_{2 a + 2 b} (x, u, t)$ and $f_{4 a + 3 b} = f_{4 a + 3 b} (x, u, v, t)$ are quasi-homogeneous polynomials of the indicated degree.
Suppose $t^2 w \in F$, that is, $t^2 \in f_{2 a + 2 b}$.
Then $\operatorname{omult}_{\mathsf{p}} (H_v) = 2$ and we have $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{b} H_v) \ge b/2 \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{b} H_v$.
We see that the set $\{x, u, v\}$ isolates $\mathsf{p}$ since $t^2 w \in F$.
In particular, a general member $T \in |a A|$ does not contain any component of the effective $1$-cycle $D \cdot H_z$.
Then we have
\[
\begin{split}
2 \operatorname{omult}_{\mathsf{p}} (D)
&\le (\rho_{\mathsf{p}}^*D \cdot \rho_{\mathsf{p}}^*H_v \cdot \rho_{\mathsf{p}}^*T)_{\check{\mathsf{p}}} \le r_2 (D \cdot H_v \cdot T) \\
&= r_2 b a (A^3) = \frac{4 a + 3 b}{a + b}.
\end{split}
\]
This implies
\[
\operatorname{lct}_{\mathsf{p}} (X;D) \ge \frac{2 a + 2 b}{4 a + 3 b} > \frac{1}{2}.
\]
Therefore we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
In the following, we consider the case where $t^2 w \notin F$.
\begin{Claim} \label{clm:ndQIcent-5}
If $b \ge 2$, then $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{b} H_v) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-5}]
By the quasi-smoothness of $X$ at $\mathsf{p}_t$, we have $t^3 u \in f_{4 a + 3 b}$ since $t^2 w \notin F$ by assumption.
Hence we have $\operatorname{omult}_{\mathsf{p}} (H_v) \le 4$ and this shows $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{b} H_v) \ge 1/2$.
\end{proof}
If $b = 1$, then $X$ is a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{13, 25\}$.
\begin{Claim} \label{clm:ndQIcent-6}
If $X$ is a member of the family $\mathcal{F}_{13}$, then $\operatorname{lct}_{\mathsf{p}} (X;H_v) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-6}]
We have
\[
F = w^2 v + w f_6 (x, u, v, t) + f_{11} (x, u, v, t).
\]
Note that $f_6 (x, u, 0, t) \ne 0$ as a polynomial since $\mathsf{p} \in X$ is non-degenerate.
We set $\bar{F} = F (x, u, 0, t, 1) \in \mathbb{C} [x, u, t]$.
We have $t^3 u \in F$ and we may assume $\operatorname{coeff}_F (t^3 u) = 1$.
If $t u x \in f_6$, then the cubic part of $\bar{F}$ is not a cube of a linear form and thus we have $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge 1/2$ by Lemma \ref{lem:lcttangcube}.
In the following, we assume $t u x \notin f_6$.
Then we can write
\[
\bar{F} = (\alpha u^3 + \beta u^2 x^2 + \gamma t x^3 + \delta u x^4 + \varepsilon x^6) + (t^3 u + \lambda t u^4 + x g_{10}),
\]
where $\alpha, \beta, \dots, \varepsilon, \lambda \in \mathbb{C}$ and $g_{10} = g_{10} (x, u, t)$ is a quasi-homogeneous polynomial of degree $10$.
We introduce $3$-tuples $\underline{c} = (c_1, c_2, c_3)$ of positive integers according to the following division into cases.
We denote by $G_{\underline{c}}$ the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (x, u, t) = \underline{c}$.
\begin{enumerate}
\item[(i)] $\alpha \ne 0$.
In this case we may assume $\alpha = 1$.
We choose and fix a sufficiently large integer $e$ which is coprime to $2$ and $3$, and we set $\underline{c} = (e, 3, 2)$.
Then we have $G_{\underline{c}} = u^3 + t^3 u$.
\item[(ii)] $\alpha = 0$ and $\beta \ne 0$.
In this case we may assume $\beta = 1$.
We set $\underline{c} = (7, 4, 6)$.
Then we have $G_{\underline{c}} = u^2 x^2 + t^3 u + \lambda t u^4$.
\item[(iii)] $\alpha = \beta = 0$ and $\gamma \ne 0$.
In this case we may assume $\gamma = 1$.
We set $\underline{c} = (8, 6, 9)$.
Then $G_{\underline{c}} = t x^3 + t^3 u + \lambda t u^4$.
\item[(iv)] $\alpha = \beta = \gamma = 0$ and $\delta \ne 0$.
In this case we may assume $\delta = 1$.
We set $\underline{c} = (9, 8, 12)$.
Then $G_{\underline{c}} = u x^4 + t^3 u + \lambda t u^4$.
\item[(v)] $\alpha = \beta = \gamma = \delta = 0$.
In this case we may assume $\varepsilon = 1$.
We set $\underline{c} = (11, 12, 18)$.
Then we have $x^6 + t^3 u + \lambda t u^4$.
\end{enumerate}
\begin{table}[h]
\renewcommand{1.35}{1.15}
\begin{center}
\caption{Family $\mathcal{F}_{13}$: Weights and LCT}
\label{table:No13lwp}
\begin{tabular}{ccccccc}
Case & $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ & $\operatorname{Diff}$ & $G_{\underline{c}}^{\operatorname{wf}}$ & $\eta$ & $\theta$ \\
\hline
(i) & $\mathbb{P} (e, 3, 2)$ & $0$ & $\tilde{u} (\tilde{u}^2 + \tilde{t}^3)$ & $5/9$ & $5/9$ \\
(ii) & $\mathbb{P} (7, 2, 3)$ & $\frac{1}{2} H_{\tilde{x}}$ & $\tilde{u} (\tilde{u} \tilde{x} + \tilde{t}^3 + \lambda \tilde{t} \tilde{u}^3)$ & $2/3$ & $2/3$ \\
(iii) & $\mathbb{P} (4, 1, 3)$ & $\frac{2}{3} H_{\tilde{x}} + \frac{1}{2} H_{\tilde{t}}$ & $\tilde{t}^{1/2} (\tilde{x} + \tilde{t} \tilde{u} + \lambda \tilde{u}^4)$ & $\ge 5/9$ & $\ge 5/9$ \\
(iv) & $\mathbb{P} (3, 2, 1)$ & $\frac{3}{4} H_{\tilde{x}} + \frac{2}{3} H_{\tilde{u}}$ & $\tilde{u}^{1/3} (\tilde{x} + \tilde{t}^3 + \lambda \tilde{t} \tilde{u})$ & $\ge 7/12$ & $\ge 7/12$ \\
(v) & $\mathbb{P} (11, 2, 3)$ & $\frac{5}{6} H_{\tilde{x}}$ & $\tilde{x} + \tilde{t}^3 \tilde{u} + \lambda \tilde{t} \tilde{u}^4$ & $\ge 1/2$ & $\ge 1/2$
\end{tabular}
\end{center}
\end{table}
The descriptions of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$, $\operatorname{Diff}$ and $G_{\underline{c}}^{\operatorname{wf}}$ are given in Table \ref{table:No13lwp}, where
we choose $\tilde{x}, \tilde{u}, \tilde{t}$ as homogeneous coordinates of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
We set $\mathcal{D}_{\underline{c}} = \mathcal{D}_{G_{\underline{c}}}^{\operatorname{wf}}$.
We explain the computation of $\eta := \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})$ whose value (or lower bound) is given in the 5th column of Table \ref{table:No13lwp}.
\end{proof}
\begin{Claim} \label{clm:ndQIcent-7}
If $X$ is a member of the family $\mathcal{F}_{25}$, then $\operatorname{lct}_{\mathsf{p}} (X;H_v) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:ndQIcent-7}]
We have
\[
F = w^2 v + w f_8 (x, u, v, t) + f_{15} (x, u, v, t).
\]
Note that $f_8 (x, u, 0, t) \ne 0$ as a polynomial since $\mathsf{p} \in X$ is non-degenerate.
We set $\bar{F} = F (x, u, 0, t, 1) \in \mathbb{C} [x, u, t]$.
We have $t^3 u, u^5 \in F$ and we may assume $\operatorname{coeff}_F (t^3 u) = \operatorname{coeff}_F (u^5) = 1$.
If $t u x \in f_8$, then the cubic part of $\bar{F}$ is not a cube of a linear form and thus we have $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge 1/2$ by Lemma \ref{lem:lcttangcube}.
In the following, we assume $t u x \notin f_8$.
Then we can write
\[
\bar{F} = (\alpha u^2 x^2 + \beta t x^4 + \gamma u x^5 + \delta x^8) + (t^3 u + u^5 + x g_{14}),
\]
where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ and $g_{14} = g_{14} (x, u, t)$ is a quasi-homogeneous polynomial of degree $14$.
We introduce $3$-tuples $\underline{c} = (c_1, c_2, c_3)$ of positive integers according to the following division into cases.
We denote by $G_{\underline{c}}$ the lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (x, u, t) = \underline{c}$.
\begin{enumerate}
\item[(i)] $\alpha \ne 0$.
In this case we may assume $\alpha = 1$.
We set $\underline{c} = (9, 6, 8)$.
Then we have $G_{\underline{c}} = u^2 x^2 + t^3 u + u^5$.
\item[(ii)] $\alpha = 0$ and $\beta \ne 0$.
In this case we may assume $\beta = 1$.
We set $\underline{c} = (11, 12, 16)$.
Then we have $G_{\underline{c}} = t x^4 + t^3 u + u^5$.
\item[(iii)] $\alpha = \beta = 0$ and $\gamma \ne 0$.
In this case we may assume $\gamma = 1$.
We set $\underline{c} = (12, 15, 20)$.
Then $G_{\underline{c}} = u x^5 + t^3 u + u^5$.
\item[(iv)] $\alpha = \beta = \gamma = 0$ and $\delta \ne 0$.
In this case we may assume $\delta = 1$.
We set $\underline{c} = (15, 24, 32)$.
Then $G_{\underline{c}} = x^8 + t^3 u + u^5$.
\end{enumerate}
\begin{table}[h]
\renewcommand{1.35}{1.15}
\begin{center}
\caption{Family $\mathcal{F}_{25}$: Weights and LCT}
\label{table:No25lwp}
\begin{tabular}{ccccccc}
Case & $\mathbb{P} (\underline{c})^{\operatorname{wf}}$ & $\operatorname{Diff}$ & $G_{\underline{c}}^{\operatorname{wf}}$ & $\eta$ & $\theta$ \\
\hline
(i) & $\mathbb{P} (3, 1, 4)$ & $\frac{1}{2} H_{\tilde{x}} + \frac{2}{3} H_{\tilde{t}}$ & $\tilde{u} (\tilde{u} \tilde{x} + \tilde{t} + \tilde{u}^4)$ & $1$ & $23/30$ \\
(ii) & $\mathbb{P} (11, 3, 4)$ & $\frac{3}{4} H_{\tilde{x}}$ & $\tilde{t} \tilde{x} + \tilde{t}^3 \tilde{u} + \tilde{u}^5$ & $1$ & $13/20$ \\
(iii) & $\mathbb{P} (1, 1, 1)$ & $\frac{4}{5} H_{\tilde{x}} + \frac{3}{4} H_{\tilde{u}} + \frac{2}{3} H_{\tilde{t}}$ & $\tilde{u}^{1/4} (\tilde{x} + \tilde{t} + \tilde{u})$ & $1$ & $47/75$ \\
(iv) & $\mathbb{P} (5, 1, 4)$ & $\frac{7}{8} H_{\tilde{x}} + \frac{3}{4} H_{\tilde{t}}$ & $\tilde{x} + \tilde{t} \tilde{u} + \tilde{u}^5$ & $1$ & $71/120$ \\
\end{tabular}
\end{center}
\end{table}
The descriptions of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$, $\operatorname{Diff}$ and $G_{\underline{c}}^{\operatorname{wf}}$ are given in Table \ref{table:No25lwp}, where
we choose $\tilde{x}, \tilde{u}, \tilde{t}$ as homogeneous coordinates of $\mathbb{P} (\underline{c})^{\operatorname{wf}}$.
We set $\mathcal{D}_{\underline{c}} = \mathcal{D}_{G_{\underline{c}}}^{\operatorname{wf}}$.
We explain the computation of $\eta := \operatorname{lct} (\mathbb{P} (\underline{c})^{\operatorname{wf}}, \operatorname{Diff}; \mathcal{D}_{\underline{c}})$ whose value is given in the 5th column of Table \ref{table:No25lwp}.
Suppose that we are in case (ii) or (iv).
Then $\mathcal{D}_{\underline{c}}$ is a prime divisor which is quasi-smooth and intersects any component of $\operatorname{Diff}$ transversally.
This shows $\eta = 1$.
Suppose that we are in case (i).
Then $\mathcal{D}_{\underline{c}} = H_{\tilde{u}} + \Gamma$, where $\Gamma = (\tilde{u} \tilde{x} + \tilde{t} + \tilde{u}^4 = 0)$ is a quasi-line.
We see that any two of $H_{\tilde{x}}, H_{\tilde{u}}, H_{\tilde{t}}, \Gamma$ intersect transversally, and thus $\eta = 1$.
Suppose that we are in case (iii).
Then $\mathcal{D}_{\underline{c}} = \frac{1}{4} H_{\tilde{u}} + \Gamma$, where $\Gamma = (\tilde{x} + \tilde{t} \tilde{u} + \tilde{u}^5 = 0)$ is a quasi-line.
We see that any two of $H_{\tilde{x}}, H_{\tilde{u}}, H_{\tilde{t}}$ and $\Gamma$ intersect transversally, and thus $\eta = 1$.
We set
\[
\theta := \min \left\{\ \frac{c_1 + c_2 + c_3}{\operatorname{wt}_{\underline{c}} (\bar{F})}, \eta \right\},
\]
which is listed in the 6th column of Table \ref{table:No25lwp}.
By Lemma \ref{lem:lctwblwh}, we have $\operatorname{lct}_{\mathsf{p}} (X; H_v) \ge \theta \ge 1/2$ and the claim is proved.
\end{proof}
By Claims \ref{clm:ndQIcent-5}, \ref{clm:ndQIcent-6} and \ref{clm:ndQIcent-7}, we have $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{b} H_v) \ge 1/2$.
Suppose $\alpha_{\mathsf{p}} (X) < 1/2$.
Then there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ other than $\frac{1}{b} H_v$ such that $(X, \frac{1}{2} D)$ is not log canonical at $\mathsf{p}$.
Let $\varphi \colon Y \to X$ be the Kawamata blowup at $\mathsf{p}$ with exceptional divisor $E$.
We set $\lambda = \operatorname{ord}_E (D)$.
Since the pair $(X, \frac{1}{2} D)$ is not canonical at $\mathsf{p}$, the discrepancy of $(X, \frac{1}{2} D)$ along $E$ is negative, which implies
\[
\lambda > \frac{2}{r_2}.
\]
By \cite[Theorem 4.9]{CPR00}, the divisor $- K_Y \sim_{\mathbb{Q}} \varphi^*A - \frac{1}{r_2} E$ is nef.
We see that $\tilde{D} \cdot \tilde{H}_v$ is an effective $1$-cycle on $Y$, where $\tilde{D}$ and $\tilde{H}_v$ are proper transforms of $D$ and $H_v$, respectively.
It follows that
\[
\begin{split}
0 &\le (- K_Y \cdot \tilde{D} \cdot \tilde{H}_v)
= b (A^3) - \frac{(2 a + 2 b) \lambda}{r_2^2} (E^3) \\
&= \frac{(4 a + 3 b) - (2 a + 2 b) r_2 \lambda}{a r_1 r_2}
< - \frac{b}{a r_1 r_2} < 0.
\end{split}
\]
This is a contradiction and we have $\alpha_{\mathsf{p}} (X) \ge 1/2$.
Therefore, the proof of Proposition \ref{prop:ndQIcent} is completed.
\chapter{Families $\mathcal{F}_2$, $\mathcal{F}_{4}$, $\mathcal{F}_5$, $\mathcal{F}_6$, $\mathcal{F}_8$, $\mathcal{F}_{10}$ and $\mathcal{F}_{14}$}
\label{chap:exc}
This chapter is devoted to the proof of the following theorem.
\begin{Thm} \label{thm:famI1}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}_1$.
Then,
\[
\alpha (X) \ge \frac{1}{2}.
\]
\end{Thm}
\section{Families $\mathcal{F}_6$, $\mathcal{F}_{10}$ and $\mathcal{F}_{14}$}
In this section, we prove Theorem \ref{thm:famI1} for families $\mathcal{F}_6$, $\mathcal{F}_{10}$ and $\mathcal{F}_{14}$ whose member is a weighted hypersurface
\[
X = X_{2 (a+2)} \subset \mathbb{P} (1, 1, 1, a, a+ 2)_{x, y, z, t, w},
\]
where $a = 2, 3, 4$, respectively.
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{6, 10, 14\}$.
Let $\mathsf{p} \in X$ be a smooth point.
We may assume $\mathsf{p} = \mathsf{p}_x$ by a suitable choice of coordinates.
By Lemma \ref{lem:nsptU1-2} (see also Remark \ref{rem:lemsmpttrue}), we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{1}{1 \cdot a \cdot (A^3)} = \frac{1}{2}.
\]
Let $\mathsf{p} \in X$ be a singular point.
If $\mathsf{i} = 14$, then $\mathsf{p} \in X$ is of type $\frac{1}{2} (1, 1, 1)$ and we have $\alpha_{\mathsf{p}} (X) \ge 1$ by Proposition \ref{prop:singptCP}.
If $\mathsf{i} = 6, 10$, then $\mathsf{p} \in X$ is of type $\frac{1}{2} (1, 1, 1)$, $\frac{1}{3} (1, 1, 2)$, respectively, and in both cases we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ by Proposition \ref{prop:singptic}.
Thus the proof of Theorem \ref{thm:famI1} for families $\mathcal{F}_6$, $\mathcal{F}_{10}$ and $\mathcal{F}_{14}$ is completed.
\section{The family $\mathcal{F}_2$}
This section is devoted to the proof Theorem \ref{thm:famI1} for the family $\mathcal{F}_2$.
In the following, let
\[
X = X_5 \subset \mathbb{P} (1,1,1,1,2)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_2$ with defining polynomial $F = F (x,y,z,t,w)$.
\subsection{Smooth points}
Let $\mathsf{p} \in X$ be a smooth point.
In this subsection, we will prove $\alpha_{\mathsf{p}} (X) \ge 1/2$.
We may assume $\mathsf{p} = \mathsf{p}_x$ by a choice of coordinates.
The proof will be done by division into cases.
\subsubsection{Case: $x^3 w \in F$}
In this case, we can write
\[
F = x^3 w + x^2 f_3 + x f_4 + f_5,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $\operatorname{mult}_{\mathsf{p}} (H_w) \ge 3$.
\begin{Claim} \label{cl:No2-1}
$\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{2} H_w) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:No2-1}]
This is obvious when $\operatorname{mult}_{\mathsf{p}} (H_w) \le 4$, hence we assume $\operatorname{mult}_{\mathsf{p}} (H_w) \ge 5$.
Then we can write
\[
F = x^3 w + x^2 w a_1 + x (\alpha w^2 + w b_2) + w^2 c_1 + w d_3 + e_5,
\]
where $\alpha \in \mathbb{C}$ and $a_1, b_2, c_1, d_3, e_5 \in \mathbb{C} [y,z,t]$ are quasi-homogeneous polynomials of indicated degrees.
We show that $(e_5 = 0) \subset \mathbb{P} (1,1,1)_{y, z, t}$ is smooth.
Indeed, if it has a singular point at $(y\!:\!z\!:\!t) = (\lambda\!:\!\mu\!:\!\nu)$, then, by setting $\theta \in \mathbb{C}$ to be a solution of the equation
\[
x^3 + x^2 a_1 (\lambda,\mu,\nu) + x b_2 (\lambda,\mu,\nu) + d_3 (\lambda,\mu,\nu) = 0,
\]
we see that $X$ is not quasi-smooth at the point $(\theta\!:\!\lambda\!:\!\mu\!:\!\nu\!:\!0)$ and this is a contradiction.
The lowest weight part of $F (1, y, z, t, 0) = e_5$ with respect to $\operatorname{wt} (y, z, t) = (1, 1, 1)$ is $e_5$ which defines a smooth hypersurface in $\mathbb{P}^2$.
By Lemma \ref{lem:lctwblwh}, we have $\operatorname{lct}_{\mathsf{p}} (X, H_w) \ge 3/5$.
Thus $\operatorname{lct}_{\mathsf{p}} (X;\frac{1}{2} H_w) \ge 6/5$ in this case and the claim is proved.
\end{proof}
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{2} H_w$.
We can take a $\mathbb{Q}$-divisor $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_w$ since $\{y, z, t\}$ isolates $\mathsf{p}$.
It follows that
\[
3 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_w \cdot T)_{\mathsf{p}} \le (D \cdot H_w \cdot T) = 5.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 3/5$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{Case: $x^3 w \notin F$}
By a choice of coordinates, we can write
\[
F = x^4 t + x^3 f_2 + x^2 f_3 + x f_4 + f_5,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_2$.
Suppose $w^2 \in f_4$.
In this case, $\operatorname{mult}_{\mathsf{p}} (H_t) = 2$ and hence $\operatorname{lct}_{\mathsf{p}} (X;H_t) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_t$.
We can take a $\mathbb{Q}$-divisor $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_t$ since $\{y, z, t\}$ isolates $\mathsf{p}$, so that
\[
2 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_t \cdot T)_{\mathsf{p}} \le (D \cdot H_t \cdot T) = \frac{5}{2}.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 4/5$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$ in this case.
Suppose $w^2 \notin f_4$.
We have $\operatorname{Bs} |\mathcal{I}_{\mathsf{p}} (A)| = \Gamma$, where $\Gamma = (y = z = t = 0) \subset X$ is a quasi-line.
We assume $\alpha_{\mathsf{p}} (X) < 1/2$.
Then there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ such that $(X, \frac{1}{2} D)$ is not log canonical at $\mathsf{p}$.
Let $S \in |\mathcal{I}_{\mathsf{p}} (A)|$ be a general member so that $S \ne \operatorname{Supp} (D)$.
Then $S$ is a normal surface by Lemma \ref{lem:normalqhyp} and it is quasi-smooth along $\Gamma$.
Moreover, for another general $T \in |\mathcal{I}_{\mathsf{p}} (A)|$, the multiplicity of $T|_S$ along $\Gamma$ is $1$, that is, we can write
\[
T|_S = \Gamma + \Delta,
\]
where $\Delta$ is an effective divisor on $S$ such that $\Gamma \not\subset \operatorname{Supp} (\Delta)$.
We see that $\Gamma$ is a quasi-line, $S$ is quasi-smooth at $\mathsf{p}_w$, $\Gamma$ passes through the $\frac{1}{2} (1,1)$ point $\mathsf{p}_w$ of $S$ and $(K_S \cdot \Gamma) = 0$.
It follows that
\[
(\Gamma^2)_S = - 2 + \frac{1}{2} = - \frac{3}{2},
\]
by Remark \ref{rem:compselfint}.
Hence
\[
(\Delta \cdot \Gamma)_S = (T|_S \cdot \Gamma)_S - (\Gamma^2)_S = 2.
\]
The divisor $D|_S$ on $S$ is effective and we write $\frac{1}{2} D|_S = \gamma \Gamma + \Xi$, where $\gamma \ge 0$ and $\Xi$ is an effective divisor on $S$ such that $\Gamma \not\subset \operatorname{Supp} (\Xi)$.
Since $\operatorname{Bs} |\mathcal{I}_{\mathsf{p}} (A)| = \Gamma$ and $S$ is general, we may assume that $\operatorname{Supp} (\Xi)$ does not contain any component of $\operatorname{Supp} (\Delta)$.
In particular $(\Xi \cdot \Delta)_S \ge 0$.
Note also that
\[
(D|_S \cdot \Delta)_S = (T|_S \cdot \Delta)_S = ((A^3) - (T \cdot \Gamma)_S) = 2.
\]
It follows that
\[
2 = (D|_S \cdot \Delta)_S \ge 2 \gamma (\Gamma \cdot \Delta)_S = 4 \gamma,
\]
which implies $\gamma \le \frac{1}{2}$.
We see that $(X, \frac{1}{2} D|_S)$ is not log canonical at $\mathsf{p}$, and hence $(S, \Gamma + \Xi) = (S, \frac{1}{2} D|_S + (1-\gamma) \Gamma)$ is not log canonical at $\mathsf{p}$.
By the inversion of adjunction, we have
\[
1 \ge \frac{1}{4} + \frac{3}{2} \gamma = ((\frac{1}{2} D|_S - \gamma \Gamma) \cdot \Gamma)_S = (\Delta \cdot \Gamma)_S \ge \operatorname{mult}_{\mathsf{p}} (\Delta|_{\Gamma}) > 1.
\]
This is a contradiction and the inequality $\alpha_{\mathsf{p}} (X) \ge 1/2$ is proved.
\subsection{The singular point of type $\frac{1}{2} (1,1,1)$}
Let $\mathsf{p} = \mathsf{p}_w$ be the singular point of type $\frac{1}{2} (1,1,1)$.
Note that the point $\mathsf{p} \in X$ is a QI center.
\subsubsection{Case: $\mathsf{p}$ is non-degenerate}
By a choice of coordinates, we can write
\[
F = w^2 t + w f_3 (x,y,z) + g_5 (x,y,z,t),
\]
where $f_3, g_5$ are non-zero homogeneous polynomials such that $f_3 \ne 0$ as a polynomial.
Let $\varphi \colon Y \to X$ be the Kawamata blowup at $\mathsf{p}$ with exceptional divisor $E$.
\begin{Claim} \label{cl:No2-2}
$\operatorname{lct}_{\mathsf{p}} (X, H_t) \ge \frac{1}{2}$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:No2-2}]
The lowest weight part of $F (x,y,z,0,1)$ with respect to $\operatorname{wt} (x,y,z) = (1,1,1)$ is $f_3$.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ unless $f_3$ is a cube of a linear form.
Hence it remains to prove the claim assuming that $f_3$ is a cube of a linear form.
By a choice of coordinates, we may assume $f_3 = z^3$.
Let $S$ be the divisor on $X$ defined by $x - \lambda y = 0$ for a general $\lambda \in \mathbb{C}$.
By the quasi-smoothness of $X$, the polynomial $F$ cannot be contained in the ideal $(z, t) \subset \mathbb{C} [x, y, z, t, w]$.
This implies $g_5 (x, y, 0, 0) \ne 0$, and hence $g_5 (\lambda y, y,0,0) \ne 0$.
By eliminating $x$, the surface $S$ is isomorphic to the hypersurface in $\mathbb{P} (1, 1, 1, 2)_{y, z, t, w}$ defined by
\[
G := w^2 t+ w z^3 + \alpha y^5 + z a_4 + t b_4 = 0,
\]
where $a_4 = a_4 (y, z), b_4 = b_4 (y, z, t)$ are homogeneous polynomials of degree $4$ and $\alpha \ne 0$ is a constant.
The lowest weight part of $G (x,z,0,1)$ with respect to $\operatorname{wt} (y, z) = (3, 5)$ is $z^3 + \alpha y^5$ which defines a smooth point of $\mathbb{P} (3, 5)_{y, z}$.
By Lemma \ref{lem:lctwblwh}, $\operatorname{lct}_{\mathsf{p}} (S; H_t|_S) \ge 8/15$, and hence $\operatorname{lct}_{\mathsf{p}} (X;H_t) \ge 8/15$.
Thus the claim is proved.
\end{proof}
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_t$.
We can take $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_t$ since $\{x, y, z, t\}$ isolates $\mathsf{p}$.
Then
\[
3 \operatorname{omult}_{\mathsf{p}} (D) < 2 (D \cdot H_t \cdot T) = 5
\]
since $\operatorname{omult}_{\mathsf{p}} (H_t) = 3$.
This shows $\operatorname{lct}_{\mathsf{p}} (X; D) \ge \frac{3}{5}$ and thus $\alpha_{\mathsf{p}} (X) \ge \frac{1}{2}$.
\subsubsection{Case: $\mathsf{p}$ is degenerate}
In this case we have $\alpha_{\mathsf{p}} (X) = 3/5$ by Proposition \ref{prop:lctdegQI}.
Therefore the proof of Theorem \ref{thm:famI1} for the family $\mathcal{F}_2$ is completed.
\section{The family $\mathcal{F}_4$}
This subsection is devoted to the proof of Theorem \ref{thm:famI1} for the family $\mathcal{F}_4$.
In the following, let
\[
X = X_6 \subset \mathbb{P} (1,1,1,2,2)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_4$ with defining polynomial $F = F (x, y, z, t, w)$.
\subsection{Smooth points}
Let $\mathsf{p}$ be a smooth point of $X$.
We will prove $\alpha_{\mathsf{p}} (X) \ge 1/2$.
We may assume $\mathsf{p} = \mathsf{p}_x$ by a choice of coordinates.
The proof will be done by division into cases.
\subsubsection{Case: Either $x^4 w \in F$ or $x^4 t \in F$}
In this case we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{1 \cdot 1 \cdot 2 \cdot (A^3)} = \frac{2}{3}.
\]
by Lemma \ref{lem:complctsingtang}.
\subsubsection{Case: $x^4 w, x^4 t \notin F$}
We can write
\[
F = x^5 y + x^4 f_2 + x^3 f_3 + x^2 f_4 + x f_5 + f_6,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t, w \notin f_2$.
We claim $\operatorname{lct}_{\mathsf{p}} (X;H_y) \ge 1/2$.
This is obvious when $\operatorname{mult}_{\mathsf{p}} (H_y) \le 2$ and hence we assume $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 3$.
Then we can write
\[
\bar{F} := F (1, 0, z, t, w)
= \sum_{i=2}^6 f_i (0, z, t, w)
= \alpha z^3 + \beta t z^2 + \gamma w z^2 + c (t, w) + h,
\]
where $c (t, w) = f_6 (0, 0, z, t)$ and $h = h (y, t, w)$ is in the ideal $(y, t, w)^4$.
By the quasi-smoothness of $X$, $c$ cannot be a cube of a linear form.
This implies that the cubic part of $\bar{F}$ is not a cube of a linear form.
Thus $\operatorname{lct}_{\mathsf{p}} (X;H_y) \ge 1/2$ by Lemma \ref{lem:lcttangcube} and the claim is proved.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_y$.
We can take $T \in |2 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_y$ since $\{y, z, t, w\}$ isolates $\mathsf{p}$.
Then we have
\[
2 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_y \cdot T) = 2 (A^3) = 3
\]
since $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 2$.
This implies $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 2/3$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsection{Singular points of type $\frac{1}{2} (1,1,1)$}
Let $\mathsf{p}$ be a singular point of type $\frac{1}{2} (1,1,1)$.
Then we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ by Proposition \ref{prop:ndQIcent} (actually we have $\alpha_{\mathsf{p}} (X) \ge 2/3$ by the argument in \S \ref{sec:ndQI3ditinct}).
Therefore the proof of Theorem \ref{thm:famI1} for the family $\mathcal{F}_4$ is completed.
\section{The family $\mathcal{F}_5$}
This subsection is devoted to the proof Theorem \ref{thm:famI1} for the family $\mathcal{F}_5$.
In the following, let
\[
X = X_7 \subset \mathbb{P} (1,1,1,2,3)_{x, y, z, t, w}
\]
be a member of family $\mathcal{F}_5$ with defining polynomial $F = F (x, y, z, t, w)$.
\subsection{Smooth points}
Let $\mathsf{p}$ be a smooth point of $X$.
We will prove $\alpha_{\mathsf{p}} (X) \ge 1/2$.
The proof will be done by division into cases.
\subsubsection{Case: $\mathsf{p} \in U_x \cup U_y \cup U_z$}
By a choice of coordinates $x, y, z$, we may assume $\mathsf{p} = \mathsf{p}_x$.
By Lemma \ref{lem:complctsingtang}, we have
\[
\alpha_{\mathsf{p}} (X) \ge
\begin{cases} \frac{2}{1 \cdot 1 \cdot 2 \cdot (A^3)} = \frac{6}{7}, & \text{if $x^4w \in F$}, \\
\frac{2}{1 \cdot 1 \cdot 3 \cdot (A^3)} = \frac{4}{7}, & \text{if $x^4 w \notin F$ and $x^5 t \in F$}.
\end{cases}
\]
It remains to consider the case where $x^4 w, x^5 t \notin F$.
In this case we can write
\[
F = x^6 y + x^5 f_2 + x^4 f_3 + x^3 f_4 + x^2 f_5 + x f_6 + f_7,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $t \notin f_2$ and $w \notin f_3$.
\begin{Claim} \label{cl:No5-1}
$\operatorname{lct}_{\mathsf{p}} (X;H_y) \ge 1/2$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:No5-1}]
This is obvious when $\operatorname{mult}_{\mathsf{p}} (H_y) = 2$, and we assume $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 3$.
It follows that each monomial appearing in $F$ is contained in $(y) \cup (z, t, w)^3$.
A monomial of degree $d \in \{2, 3, 4, 5, 6, 7\}$ in variables $y, z, t, w$ which is contained in $(y) \cup (z, t, w)^3$ is contained in $(y) \cup (z, t)^2$ except for the monomial $w^2 z$ of degree $7$.
Hence we can write
\[
F = x^6 y + y g + h + \alpha w^2 z,
\]
where $g = g (x, y, z, t, w) \in \mathbb{C} [x, y, z, t, w]$ and $h = h (x, z, t, w) \in (z, t)^2$.
If $\alpha = 0$, then $X$ is not quasi-smooth at any point of the nonempty set
\[
(y = x^6 + g = z = t = 0) \subset \mathbb{P} (1,1,1,2,3).
\]
Thus $w^2 z \in F$ and we see that $\bar{F} = F (1,0,z,t,w) \in (z,t,w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^2 z \in \bar{F}$ and $w^3 \notin \bar{F}$.
By Lemma \ref{lem:lcttangcube}, we have $\operatorname{lct}_{\mathsf{p}} (X;H_y) \ge 1/2$ and the claim is proved.
\end{proof}
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_y$.
We can take a $\mathbb{Q}$-divisor $T \in |3 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_y$ since $\{y, z, t, w\}$ isolates $\mathsf{p}$.
Then
\[
2 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_y \cdot T)_{\mathsf{p}} \le (D \cdot H_y \cdot T) = 3 (A^3) = \frac{7}{2}
\]
since $\operatorname{mult}_{\mathsf{p}} (H_y) \ge 2$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 4/7$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{Case $\mathsf{p} \notin U_x \cup U_y \cup U_z$}
If $w t^2 \in F$, then $X \setminus (U_x \cup U_y \cup U_z)$ consists of singular points.
Hence we have $w t^2 \notin F$ in this case, and $\mathsf{p}$ is contained in the quasi-line $\Gamma := (x = y = z = 0) \subset X$.
We will show $\alpha_{\mathsf{p}} (X) \ge 1$.
Assume to the contrary that $\alpha_{\mathsf{p}} (X) < 1$.
Then there exists an irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ such that the pair $(X, D)$ is not log canonical at $\mathsf{p}$.
Let $S \in |A|$ be a general member and write $D|_S = \gamma \Gamma + \Delta$, where $\gamma \ge 0$ is a rational number and $\Delta$ is an effective $1$-cycle on $S$ such that $\Gamma \not\subset \operatorname{Supp} (\Delta)$.
\begin{Claim} \label{cl:No5-2}
$(\Gamma^2)_S = - 5/6$ and $\gamma \le 1$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:No5-2}]
We see that $S$ has singular points of type $\frac{1}{2} (1,1)$ and $\frac{1}{3} (1,2)$ at $\mathsf{p}_t$ and $\mathsf{p}_w$, respectively, and smooth elsewhere since $S \in |A|$ is general.
Since $\Gamma$ is a quasi-line on $S$ passing through $\mathsf{p}_t, \mathsf{p}_w$ and $K_S = (K_X + S)|_S \sim 0$ by adjunction, we have
\[
(\Gamma^2)_S = -2 + \frac{1}{2} + \frac{2}{3} = - \frac{5}{6}.
\]
We choose a general member $T \in |A|$ which does not contain any component of $\Delta$.
This is possible since $\operatorname{Bs} |A| = \Gamma$.
We write $T|_S = \Gamma + \Xi$, where $\Xi$ is an effective divisor on $S$ such that $\Gamma \not\subset \operatorname{Supp} (\Xi)$.
We have
\[
\begin{split}
(D|_S \cdot \Xi)_S &= (D|_S \cdot (T|_S - \Gamma))_S = \frac{7}{6} - \frac{1}{6} = 1, \\
(\Gamma \cdot \Xi)_S &= (\Gamma \cdot (T|_S - \Gamma))_S = \frac{1}{6} + \frac{5}{6} = 1.
\end{split}
\]
Note that $\Xi$ does not contain any component of $\Delta$ by our choice of $T$, and hence
\[
1 = (D|_S \cdot \Xi)_S = ((\gamma \Gamma + \Delta) \cdot \Xi)_S \ge \gamma (\Gamma \cdot \Xi)_S = \gamma,
\]
as desired.
\end{proof}
The pair $(S, D|_S) = (S, \gamma \Gamma + \Delta)$ is not log canonical at $\mathsf{p}$.
Hence the pair $(S, \Gamma + \Delta)$ is not log canonical at $\mathsf{p}$ since $\gamma \le 1$.
By the inversion of adjunction, we have $\operatorname{mult}_{\mathsf{p}} (\Delta|_{\Gamma}) > 1$ and thus
\[
1 < \operatorname{mult}_{\mathsf{p}} (\Delta|_{\Gamma})
\le (\Delta \cdot \Gamma)_S
= ((D|_S - \gamma \Gamma) \cdot \Gamma)_S
= \frac{1}{6} + \frac{5}{6} \gamma
\le 1.
\]
This is a contradiction and we have $\alpha_{\mathsf{p}} (X) \ge 1$.
\subsection{The singular point of type $\frac{1}{2} (1,1,1)$}
Let $\mathsf{p} = \mathsf{p}_t$ be the singular point of type $\frac{1}{2} (1,1,1)$.
\subsubsection{Case: $t^2 w \in F$}
In this case, we have
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{2 \cdot 1 \cdot 1 \cdot (A^3)} = \frac{6}{7}
\]
by Lemma \ref{lem:complctsingtang}.
\subsubsection{Case: $t^2 w \notin F$}
\label{sec:No5singpt2Case2}
Replacing $x, y, z$, we can write
\[
F = t^3 x + t^2 f_3 + t f_5 + f_7,
\]
where $f_i = f_i (x, y, z, w)$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_3$.
\begin{Claim} \label{cl:No5-3}
If $\operatorname{mult}_{\mathsf{p}} (H_x) \ge 3$, then either $w^2 y \in F$ or $w^2 z \in F$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{cl:No5-3}]
Suppose $w^2 y, w^2 z \notin F$.
Then $h := F (0, y, z, t, w)$ is contained in the ideal $(y, z)^2 \subset \mathbb{C} [y, z, t, w]$, and we can write $F = x g + h$, where $g = g (x, y, z, t, w)$.
We see that $X$ is not quasi-smooth at any point in the nonempty subset
\[
(x = y = z = g = 0) \subset \mathbb{P} (1, 1, 1, 2, 3).
\]
This is a contradiction and the claim is proved.
\end{proof}
We set $\bar{F} := F (0, y, z, 1, w)$.
By Claim \ref{cl:No5-3}, either $\bar{F} \in (y, z, w)^2 \setminus (y, z, w)^3$ or $\bar{F} \in (y, z, w)^3$ and the cubic part of $\bar{F}$ is not a cube of a linear form since $w^3 \notin \bar{F}$.
By Lemma \ref{lem:lcttangcube}, we have $\alpha_{\mathsf{p}} (X) \ge 1/2$ since $\mathsf{p} \in X$ is not a maximal center.
\subsection{Singular point of type $\frac{1}{3} (1,1,2)$}
Let $\mathsf{p} = \mathsf{p}_w$ be the singular point of type $\frac{1}{3} (1,1,2)$.
We can write
\[
F = w^2 x + w (\alpha t^2 + t a_2 (y, z) + b_4 (y, z)) + f_7 (x, y, z, t),
\]
where $\alpha \in \mathbb{C}$ and $a_2 = a_2 (x, y), b_4 = b_4 (y, z), f_7 = f_7 (x, y, z, t)$ are quasi-homogeneous polynomials of degree $2, 4, 7$, respectively.
Let $q = q_{\mathsf{p}}$ be the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ be the preimage of $\mathsf{p}$.
\subsubsection{Case: $\alpha \ne 0$}
We have $\operatorname{mult}_{\mathsf{p}} (H_x) = 2$ and $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$ since $\{x, y, z\}$ isolates $\mathsf{p}$.
Then
\[
2 \operatorname{omult}_{\mathsf{p}} (D) \le (q^*D \cdot q^*H_x \cdot q^*T)_{\check{\mathsf{p}}} \le 3 (D \cdot H_x \cdot T) = \frac{7}{2}.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 4/7$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{Case $\alpha = 0$ and $a_2 \ne 0$}
The cubic part of $F (0, y, z, t, 1)$ is $t a_2$ and, by Lemma \ref{lem:lcttangcube}, we have $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 1/2$.
Let $D \sim_{\mathbb{Q}} A$ be an irreducible $\mathbb{Q}$-divisor on $X$ other than $H_x$.
Then we can take a general $T \in |\mathcal{I}_{\mathsf{p}} (2 A)| = |2 A|$ which does not contain any component of $D \cap H_x$ since $\operatorname{Bs} |2 A| =\mathsf{p}$.
We see that $T$ is defined by $t - q (x, y, z) = 0$ on $X$, where $q \in \mathbb{C} [x,y,z]$ is a general quadratic form.
Let $\rho = \rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$ be the orbifold chart of $X$ containing $\mathsf{p}$ and let $\breve{\mathsf{p}}$ be the preimage of $\mathsf{p}$.
It is then easy to see that the effective $1$-cycle $\rho^*H_x \cdot \rho^*T$ on $\breve{U}_{\mathsf{p}}$ has multiplicity $4$ at $\breve{\mathsf{p}}$.
Then we have
\[
4 \operatorname{omult}_{\mathsf{p}} (D) \le
(\rho^*D \cdot \rho^*H_x \cdot \rho^*T)_{\breve{\mathsf{p}}}
\le 3 (D \cdot H_x \cdot T) = 7
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 4/7$ and thus $\alpha_{\mathsf{p}} (X) \ge 4/7$.
\subsubsection{Case: $\alpha = a_2 = 0$ and $b_4 \ne 0$}
By similar arguments as in the proof of Claim \ref{cl:No5-3}, we see that either $t^3 y \in f_7$ or $t^3 z \in f_7$.
We choose $z$ and $t$ so that $b_4 (0,z) = z^4$ and $\operatorname{coeff}_{f_7} (t^3 z) = 1$.
Then we have
\[
F (0,0,z,t,1) = z^4 + t^3 z + \beta t^2 z^3 + \gamma t z^5 + \delta z^7,
\]
where $\beta, \gamma, \delta \in \mathbb{C}$.
The lowest weight part of $F (0,0,z,t,1)$ with respect to $\operatorname{wt} (z,t) = (1,1)$ is $z^4 + t^3 z$ which defines $4$ distinct points of $\mathbb{P}^1_{z, t}$.
Hence we have
\[
\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge \operatorname{lct}_{\mathsf{p}} (H_y;H_x|_{H_y}) \ge \frac{1}{2}
\]
by Lemma \ref{lem:lctwblwh}.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_x$.
We can take a $\mathbb{Q}$-divisor $T \in |2 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_x$ since $\{x, y, z, t\}$ isolates $\mathsf{p}$.
Then
\[
4 \operatorname{omult}_{\mathsf{p}} (D) \le
(q^*D \cdot q^*H_x \cdot q^*T)_{\check{\mathsf{p}}}
\le 3 (D \cdot H_x \cdot T) = 7.
\]
since $\operatorname{omult}_{\mathsf{p}} (H_x) = 4$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 4/7$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
\subsubsection{Case: $\alpha = a_2 = b_4 = 0$}
In this case the point $\mathsf{p} \in X$ is a degenerate QI center and we have $\alpha_{\mathsf{p}} (X) = 4/7$ by Proposition \ref{prop:lctdegQI}.
\section{The family $\mathcal{F}_8$}
This subsection is devoted to the proof of Theorem \ref{thm:famI1} for the family $\mathcal{F}_8$.
In the following, let
\[
X = X_9 \subset \mathbb{P} (1,1,1,3,4)_{x, y, z, t, w}
\]
be a member of $\mathcal{F}_8$ with defining polynomial $F = F (x, y, z, t, w)$.
\subsection{Smooth points}
Let $\mathsf{p} \in X$ be a smooth point.
We will prove $\alpha_{\mathsf{p}} (X) \ge 1/2$.
We may assume $\mathsf{p} = \mathsf{p}_x$.
The proof will be done by division into cases.
\subsubsection{Case: $x^5 w \in F$}
We can write
\[
F = x^5 w + x^4 f_5 + x^3 f_6 + x^2 f_7 + x f_8 + f_9,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$.
We have $\operatorname{mult}_{\mathsf{p}} (H_w) = 3$ since $t^3 \in f_9$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor on $X$.
Let $S \in |\mathcal{I}_{\mathsf{p}} (A)|$ be a general member so that $\operatorname{Supp} (D) \ne S$.
Since $\{y, z, w\}$ isolates $\mathsf{p}$, we can take a $\mathbb{Q}$-divisor $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot S$ and $\operatorname{mult}_{\mathsf{p}} (T) \ge 3/4$ (Note that $T$ is one of $H_y, H_z$ and $\frac{1}{4} H_w$).
Then we have
\[
\frac{3}{4} \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot S \cdot T)_{\mathsf{p}} \le (D \cdot S \cdot T) = \frac{3}{4}.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ and thus $\alpha_{\mathsf{p}} (X) \ge 1$.
\subsubsection{Case: $x^5 w \notin F$ and $x^6 t \in F$}
We can write
\[
F = x^6 t + x^5 f_4 + x^4 f_5 + x^3 f_6 + x^2 f_7 + x f_8 + f_9,
\]
where $f_i = f_i (y,z,t,w)$ is a quasi-homogeneous polynomial of degree $i$ with $w \notin f_4$.
Let $S, T \in |\mathcal{I}_{\mathsf{p}} (A)|$ be general members.
Note that $S$ is smooth at $\mathsf{p}$.
The intersection $S \cap T$ is isomorphic to the subscheme in $\mathbb{P} (1_x, 3_t, 4_w)$ defined by the equation $F (x,0,0,t,w) = 0$ and we can write
\[
F (x,0,0,t,w) = x^6 t + \alpha x^3 t^2 + \beta x^2 w t + \gamma x w^2 + t^3,
\]
where $\alpha, \beta, \gamma \in \mathbb{C}$.
\begin{Claim} \label{clm:No8smpt2-1}
If $\gamma \ne 0$, then $S \cdot T = \Gamma$, where $\Gamma$ is an irreducible and reduced curve of degree $3/4$ that is smooth at $\mathsf{p}$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:No8smpt2-1}]
Suppose $\gamma \ne 0$.
Then it is easy to see that the polynomial $F (x, 0, 0, t, w)$ is irreducible.
Hence the curve
\[
\Gamma = (y = z = F (x,0,0,t,w) = 0) \subset \mathbb{P} (1,1,1,3,4).
\]
is irreducible and reduced.
It is also obvious that $\deg \Gamma = 3/4$ and $\Gamma$ is smooth at $\mathsf{p}$.
\end{proof}
If $\gamma \ne 0$, then we have $\alpha_{\mathsf{p}} (X) \ge 1$ by Claim \ref{clm:No8smpt2-1} and Lemma \ref{lem:exclL}.
In the following we consider the case where $\gamma = 0$.
We set
\[
\Delta = (y = z = t = 0) \subset \mathbb{P} (1, 1, 1, 3, 4),
\]
which is a quasi-line of degree $1/4$ passing through $\mathsf{p}$.
Note that $\Delta$ is smooth at $\mathsf{p}$.
\begin{Claim} \label{clm:No8smpt2-2}
If $\gamma = 0$ and $\beta \ne 0$, then $T|_S = \Delta + \Xi$, where $\Xi$ is an irreducible and reduced curve which does not pass through $\mathsf{p}$.
Moreover the intersection matrix $M (\Delta, \Xi)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:No8smpt2-2}]
We have
\[
F (x, 0, 0, t, w) = t (x^6 + \alpha x^3 t + \beta x^2 w + t^2),
\]
and the polynomial $x^6 + \alpha x^3 t + \beta x^2 w + t^2$ is irreducible since $\beta \ne 0$.
It follows that $T|_S = \Delta + \Xi$, where
\[
\Xi = (y = z = x^6 + \alpha x^3 t + \beta x^2 w + t^2 = 0) \subset \mathbb{P} (1, 1, 1, 3, 4)
\]
is an irreducible and reduced curve of degree $1/2$ that does not pass through $\mathsf{p}$.
We have $\Delta \cap \Xi = \{\mathsf{p}_w, \mathsf{q}\}$, where $\mathsf{q} = (1\!:\!0\!:\!0\!:\!0\!:\!-1/\beta)$.
It is easy to see that $S$ is quasi-smooth at $\mathsf{p}_w$ and $\mathsf{q}$, hence $S$ is quasi-smooth along $\Delta$ by Lemma \ref{lem:pltsurfpair}.
We have $\operatorname{Sing}_{\Gamma} (S) = \{\mathsf{p}_w\}$ and $\mathsf{p}_w \in S$ is of type $\frac{1}{4} (1, 3)$.
By Remark \ref{rem:compselfint}, we have
\[
(\Delta^2)_S = -2 + \frac{3}{4} = - \frac{5}{4}.
\]
By taking intersection number of $T|_S = \Delta + \Xi$ and $\Delta$, and then $T|_S$ and $\Xi$, we have
\[
(\Delta \cdot \Xi) = \frac{3}{2}, \quad
(\Xi^2)_S = -1.
\]
It follows that the intersection matrix $M (\Delta, \Xi)$ satisfies the condition $(\star)$.
\end{proof}
\begin{Claim} \label{clm:No8smpt2-3}
If $\gamma = \beta = 0$ and $\alpha \ne \pm 2$, then $T|_S = \Delta + \Theta_1 + \Theta_2$, where $\Theta_1$ and $\Theta_2$ are distinct quasi-lines which does not pass through $\mathsf{p}$.
Moreover the intersection matrix $M (\Delta, \Theta_1, \Theta_2)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:No8smpt2-3}]
We have
\[
F (x, 0, 0, t, w) = t (x^6 + \alpha x^3 t + t^2) = t (t - \lambda x^3)(t - \lambda^{-1} x^3),
\]
where $\lambda \ne 0, 1$ is a complex number such that $\alpha = \lambda + \lambda^{-1}$.
Hence we have
\[
T|_S = \Delta + \Theta_1 + \Theta_2,
\]
where
\[
\Xi_1 = (y = z = t - \lambda x^3 = 0), \quad
\Xi_2 = (y = z = t - \lambda^{-1} x^3 = 0)
\]
are both quasi-lines of degree $1/4$ that do not pass through $\mathsf{p}$.
We have $\Delta \cap (\Theta_1 \cup \Theta_2) = \{\mathsf{p}_w\}$ and $S$ is clearly quasi-smooth at $\mathsf{p}_w$.
It follows that $S$ is quasi-smooth along $\Gamma$ by Lemma \ref{lem:pltsurfpair}, and $\operatorname{Sing}_{\Delta} (S) = \{\mathsf{p}_w\}$, where $\mathsf{p}_w \in S$ is of type $\frac{1}{4} (1, 3)$.
Thus we have
\[
(\Delta^2)_S = -\frac{5}{4}.
\]
By similar arguments, we see that $S$ is quasi-smooth along $\Theta_i$ and $\operatorname{Sing}_{\Theta} (S) = \{\mathsf{p}_w\}$ for $i = 1, 2$, and hence
\[
(\Theta_i^2)_S = - \frac{5}{4}.
\]
By taking intersection number of $T|_S = \Delta + \Theta_1 + \Theta_2$ and $\Delta, \Theta_1, \Theta_2$, we conclude
\[
(\Delta \cdot \Theta_1)_S = (\Delta \cdot \Theta_2)_S = (\Theta_1 \cdot \Theta_2)_S = \frac{3}{4}.
\]
It is then straightforward to see that $M (\Delta, \Theta_1, \Theta_2)$ satisfies the condition $(\star)$.
\end{proof}
\begin{Claim} \label{clm:No8smpt2-4}
If $\gamma = \beta = 0$ and $\alpha = \pm 2$, then $T|_S = \Delta + 2 \Theta$, where $\Theta$ is an irreducible and reduced curve which does not pass through $\mathsf{p}$.
Moreover the intersection matrix $M (\Delta, \Theta)$ satisfies the condition $(\star)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:No8smpt2-4}]
Without loss of generality, we may assume $\alpha = -2$.
We have
\[
F (x, 0, 0, t, w) = t (t - x^3)^2,
\]
and hence
\[
T|_S = \Delta + 2 \Theta,
\]
where
\[
\Theta = (y = z = t - x^3 = 0) \subset \mathbb{P} (1, 1, 1, 3, 4)
\]
is a quasi-line of degree $1/4$ that does not pass through $\mathsf{p}$.
By the same arguments as in Claim \ref{clm:No8smpt2-3}, we have
\[
(\Delta^2)_S = - \frac{5}{4}.
\]
Then, by taking intersection number of $T|_S = \Delta + 2 \Theta$ and $\Delta, \Theta$, we have
\[
(\Delta \cdot \Theta)_S = \frac{3}{4}, \quad
(\Theta^2)_S = - \frac{1}{4}.
\]
Thus the matrix $M (\Delta, \Theta)$ satisfies the condition $(\star)$.
\end{proof}
By Claims \ref{clm:No8smpt2-2}, \ref{clm:No8smpt2-3}, \ref{clm:No8smpt2-4} and Lemma \ref{lem:mtdLred}, we conclude
\[
\alpha_{\mathsf{p}} (X) \ge \min \left\{ 1, \ \frac{1}{(A^3) + 1 - \deg \Delta} \right\} = \frac{2}{3}.
\]
\subsubsection{Case: $x^5 w, x^6 t \notin F$}
Replacing $y$ and $z$, we can write
\[
F = x^8 y + x^7 f_2 + x^6 f_3 + x^5 f_4 + x^4 f_5 + x^3 f_6 + x^2 f_7 + x f_8 + f_9,
\]
where $f_i = f_i (y,z,t,w)$ is a homogeneous polynomial of degree $i$ with $w \notin f_4$ and $t \notin f_3$.
Note that we have $2 \le \operatorname{mult}_{\mathsf{p}} (H_y) \le 3$ since $t^3 \in F$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_y$.
We can take a $\mathbb{Q}$-divisor $T \in |4 A|_{\mathbb{Q}}$ such that $\operatorname{mult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_y$ since $\{y, z, t, w\}$ isolates $\mathsf{p}$.
Then
\[
2 \operatorname{mult}_{\mathsf{p}} (D) \le (D \cdot H_y \cdot T)_{\mathsf{p}} \le (D \cdot H_y \cdot T) = 4 (A^3) = 3.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 2/3$ and thus it remains to show that $\operatorname{lct} (X; H_y) \ge 1/2$.
Suppose that either $\operatorname{mult}_{\mathsf{p}} (H_y) = 2$ or $\operatorname{mult}_{\mathsf{p}} (H_y) = 3$ and the cubic part of $\bar{F} := F (1,0,z,t,w)$ is a cube of a linear form.
Then $\operatorname{lct}_{\mathsf{p}} (X;H_y) \ge 1/2$ by Lemma \ref{lem:lcttangcube}, and we are done.
In the following, we assume that $\operatorname{mult}_{\mathsf{p}} (H_y) = 3$ and the cubic part of $\bar{F}$ is a cube of a linear form.
Since $t^3 \in \bar{F}$ and $w^3 \notin \bar{F}$, we may assume that the cubic part of $\bar{F}$ is $t^3$ after replacing $t$.
We claim $w^2 z \in F$.
We see that a monomial other than $w^2 z$ which appear in $\bar{F}$ with nonzero coefficient is contained in the ideal $(z, t)^2 \subset \mathbb{C} [z, t, w]$.
We can write $F = y G + F (x, 0, z, t, w)$ for some homogeneous polynomial $G (x, y, z, t, w)$.
If $w^2 z \notin F$, then $F (x, 0, z, t, w) \in (z, t)^2$ and $X$ is not quasi-smooth at any point contained in the nonempty set
\[
(y = z = t = G = 0) \subset \mathbb{P} (1,1,1,3,4).
\]
This is a contradiction and the claim is proved.
Then, we may assume $\operatorname{coeff}_F (w^2 z) = 1$ and, by replacing $t$ and $w$, we can write
\[
\begin{split}
\bar{F} = \alpha_4 z^4 & + \alpha_5 z^5 + (\beta t z^3 + \alpha_6 z^6) + (\gamma w z^3 + \delta t z^4 + \alpha_7 z^7) + \\
+ & (\varepsilon w z^4 + \zeta t^2 z^2 + \eta t z^5 + \alpha_8 z^8)
+ (w^2 z + t^3 + \theta t^2 z^3 + \lambda t z^6 + \alpha_9 z^9),
\end{split}
\]
where $\alpha_4, \dots, \alpha_9, \beta, \gamma, \dots, \lambda \in \mathbb{C}$.
The lowest weight part of $\bar{F}$ with respect to $\operatorname{wt} (z,t,w) = (6,8,9)$ is $G := \alpha_4 z^4 + w^2 z + t^3$.
We set $\mathbb{P} = \mathbb{P} (6, 8, 9)$.
Then $\mathbb{P}^{\operatorname{wf}} = \mathbb{P} (1, 4, 3)_{\tilde{z}, \tilde{t}, \tilde{w}}$ and, by Lemma \ref{lem:lctwblwh}, we have
\[
\operatorname{lct}_{\mathsf{p}} (X; H_y) \ge \min \left\{ \frac{23}{24}, \ \operatorname{lct} (\mathbb{P}^{\operatorname{wf}}, \operatorname{Diff}; \Gamma) \right\},
\]
where
\[
\begin{split}
\operatorname{Diff} &= \frac{2}{3} H^{\operatorname{wf}}_{\tilde{t}} + \frac{1}{2} H^{\operatorname{wf}}_{\tilde{w}}, \\
\Gamma &= \mathcal{D}^{\operatorname{wf}}_G = (\alpha_4 \tilde{z}^4 + \tilde{w} \tilde{z} + \tilde{t} = 0) \subset \mathbb{P} (1, 4, 3),
\end{split}
\]
are $(\mathbb{Q}$-)divisors on $\mathbb{P}^{\operatorname{wf}}$ with $H^{\operatorname{wf}}_{\tilde{t}} = (\tilde{t} = 0)$ and $H^{\operatorname{wf}}_{\tilde{w}} = (\tilde{w} = 0)$.
It is easy to see that any pair of curves $H^{\operatorname{wf}}_{\tilde{t}}, H^{\operatorname{wf}}_{\tilde{w}}$ and $\Gamma$ intersect transversally.
If $\alpha_4 \ne 0$, then $H^{\operatorname{wf}}_{\tilde{t}} \cap H^{\operatorname{wf}}_{\tilde{w}} \cap \Gamma = \emptyset$, and thus $\operatorname{lct} (\mathbb{P}^{\operatorname{wf}}, \operatorname{Diff}; \Gamma) = 1$.
If $\alpha_4 = 0$, then $H^{\operatorname{wf}}_{\tilde{t}} \cap H^{\operatorname{wf}}_{\tilde{w}} \cap \Gamma = \{\mathsf{p}_{\tilde{z}}\}$.
In this case, by consider the the blowup at $\mathsf{p}_{\tilde{z}}$, we can confirm the equality $\operatorname{lct} (\mathbb{P}^{\operatorname{wf}}, \operatorname{Diff}; \Gamma) = 5/6$.
Thus, we have $\operatorname{lct}_{\mathsf{p}} (X; H_y) \ge 5/6$, and the proof is completed.
\subsection{The singular point of type $\frac{1}{4} (1,1,3)$}
Let $\mathsf{p} = \mathsf{p}_w$ be the singular point of type $\frac{1}{4} (1,1,3)$.
We can write
\[
F = w^2 x + w (t a_2 (y, z) + b_5 (y, z)) + f_9 (x, y, z, t),
\]
where $a_2 = a_2 (y, z)$, $b_5 = b_5 (y, z)$ and $f_9 =f_9 (x, y, z, t)$ are homogeneous polynomials of degrees $2$, $5$ and $9$, respectively.
Suppose that $a_2 \ne 0$ as a polynomial.
Then $\bar{F} := F (0, y, z, t, 1) \in (y, z, t)^3$ and its cubic part $t a_2 + t^3$ is not a cube of a linear form.
It follows that $\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge 2/3$ by Lemma \ref{lem:lcttangcube}.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $H_x$.
Since the set $\{x, y, z\}$ isolates $\mathsf{p}$, we can take a $\mathbb{Q}$-divisor $T \in |A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of $D \cdot H_x$.
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 3$.
It follows that
\[
3 \operatorname{omult}_{\mathsf{p}} (D) \le (q^*D \cdot q^*H_x \cdot q^*T)_{\check{\mathsf{p}}} \le 4 (D \cdot H_x \cdot T) = 3,
\]
where $q = q_{\mathsf{p}}$ is the quotient morphism of $\mathsf{p} \in X$ and $\check{\mathsf{p}}$ is the preimage of $\mathsf{p}$ via $q$.
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ and thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
Finally, suppose that $a_2 = 0$ and $b_5 \ne 0$.
Replacing $y$ and $z$, we may assume $z^5 \in b_5$ and $\operatorname{coeff}_{b_5} (z^5) = 1$.
We may also assume that $\operatorname{coeff}_{f_9} (t^3) = 1$ by rescaling $t$.
Then we have
\[
F (0,0,z,t,1) = z^5 + f_9 (0,0,z,t).
\]
The lowest weight part with respect to the weight $\operatorname{wt} (z,t) = (3,5)$ is $z^5 + t^3$ and thus
\[
\operatorname{lct}_{\mathsf{p}} (X;H_x) \ge \operatorname{lct}_{\mathsf{p}} (H_y;H_x|_{H_y}) = \frac{8}{15}.
\]
We have $\operatorname{omult}_{\mathsf{p}} (H_x) = 3$ and the set $\{x, y, z\}$ isolates $\mathsf{p}$.
Hence by the same argument as in the the case $a_2 \ne 0$, we have $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$ for any irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ other than $H_x$.
Thus $\alpha_{\mathsf{p}} (X) \ge 1/2$.
Suppose that $a_2 = b_5 = 0$.
Then we have $\alpha_{\mathsf{p}} (X) = 5/9$ by Proposition \ref{prop:lctdegQI}, and the proof is completed.
\begin{Rem} \label{rem:QIcent-8}
The singular point $\mathsf{p} \in X$ of type $\frac{1}{4} (1, 1, 3)$ is a QI center.
When $\mathsf{p}$ is non-degenerate, the above proof shows that $\operatorname{lct}_{\mathsf{p}} (X; D) \ge 1$ for any irreducible $\mathbb{Q}$-divisor $D \in |A|_{\mathbb{Q}}$ other than $H_x$ and $\operatorname{lct}_{\mathsf{p}} (X; H_x) > 1/2$.
\end{Rem}
\chapter{Further results and discussion on related problems}
\label{chap:discuss}
\section{Birationally superrigid Fano $3$-folds of higher codimensions}
We can embed a Fano 3-fold into a weighted projective space by choosing (minimal) generators of the anticanonical graded ring.
We consider embedded Fano 3-folds.
We have satisfactory results on the classification of Fano 3-folds of low codimensions (\cite{IF00}, \cite{CCC11}, \cite{ABR02}) and the following are known for their birational (super)rigidity.
\begin{itemize}
\item Fano 3-folds of codimension $2$ are all weighted complete intersections and they consist of 85 families.
Among them, there are exactly $19$ families whose members are birationally rigid (\cite{Oka14}, \cite{AZ16}).
\item Fano 3-folds of codimension $3$ consist of 69 families of so called Pfaffian Fano 3-folds and 1 family of complete intersections of three quadrics in $\mathbb{P}^6$.
Among them, there are exactly $3$ families whose members are birationally rigid (\cite{AO18}).
\item Constructions of many families of Fano 3-folds of codimension 4 has been known (see e.g.\ \cite{BKR12}, \cite{CD18}), but their classification is not completed.
There are at least $2$ families of birationally superrigid Fano 3-folds of codimension 4 (\cite{Oka20a}).
\end{itemize}
For birationally rigid Fano 3-folds of codimension 2 and 3, K-stability and existence of KE metrics are known under some generality assumptions.
\begin{Thm}[{\cite{KOW18}}] \label{thm:KOW18}
Let $X$ be a general quasi-smooth Fano 3-folds of codimension $c \in \{2, 3\}$ which is birationally rigid.
We assume that $X$ is a complete intersection of a quadric and cubic in $\mathbb{P}^5$ when $c = 2$.
Then $\alpha (X) \ge 1$, $X$ is K-stable and admits a KE metric.
\end{Thm}
\begin{Thm}[{\cite[Theorem 1.3]{Zhu20b}}] \label{thm:Zhu20}
Let $X$ be a smooth complete intersection of a quadric and cubic in $\mathbb{P}^5$.
Then $X$ is K-stable and admits a KE metric.
\end{Thm}
\begin{Question}
Can we conclude K-stability for any quasi-smooth Fano 3-fold of codimension 2 and 3 which is birationally (super)rigid?
How about for Fano 3-folds of codimension 4 or higher?.
\end{Question}
\section{Lower bound of alpha invariants}
In the context of Theorem \ref{thm:SZ}, the following is a very natural question to ask.
\begin{Question}
Is it true that $\alpha (X) \ge 1/2$ (or $\alpha (X) > 1/2$) for any birationally superrigid Fano variety?
If yes, can we find a lower bound better than $1/2$?
\end{Question}
The following example suggests that the number $1/2$ is optimal (or the lower bound can be even smaller).
\begin{Ex} \label{ex:bsrlcthalf}
For an integer $a \ge 2$, let $X_a$ be a weighted hypersurfaces of degree $2 a + 1$ in $\mathbb{P} (1^{a+2}, a) = \operatorname{Proj} \mathbb{C} [x_1,\dots,x_{a+2},y]$, given by the equation
\[
y^2 x_1 + f (x_1,\dots,x_{a+2}) = 0,
\]
where $f$ is a general homogeneous polynomial of degree $2 a + 1$.
Then $X_a$ is a quasi-smooth Fano weighted hypersurface of dimension $a+1$ and Picard number $1$ with the unique singular point $\mathsf{p}$ of type
\[
\frac{1}{a} (\overbrace{1,\dots,1}^{a+1}).
\]
The singularity $\mathsf{p} \in X$ is terminal.
By the same argument as in the proof of Proposition \ref{prop:lctdegQI}, we obtain
\[
\alpha (X) \le \alpha_{\mathsf{p}} (X) = \operatorname{lct}_{\mathsf{p}} (X; H_{x_1}) = \frac{a+1}{2 a + 1}.
\]
When $a = 2$, $X_a = X_2$ is a member of the family $\mathcal{F}_2$ and it is birationally superrigid.
We expect that $X_a$ is birationally superrigid although this is not proved at all when $a \ge 3$.
If $X_a$ is birationally superrigid for $a \gg 0$, then it follows that there exists a sequence of birationally superrigid Fano varieties whose alpha invariants are arbitrary close to (or less than) $\frac{1}{2}$.
\end{Ex}
\begin{Question}
Let $X_a$ be as in Example \ref{ex:bsrlcthalf}.
Is $X_a$ birationally superrigid for $a \ge 3$?
\end{Question}
\section{Existence of K\"ahler--Einstein metrics}
\label{sec:KEmetric}
For a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ which is strictly birationally rigid, we are unable to conclude the existence of a K\"{a}hler--Einstein metric as a direct consequence of Theorem \ref{mainthm}.
However, for a Fano variety $X$ of dimension $n$ with only quotient singularities, the implication
\[
\alpha (X) > \frac{n}{n + 1} \ \Longrightarrow \ \text{existence of a KE metric on $X$}
\]
is proved in \cite[Section 6]{DK}.
The aim of this section is to prove the existence of KE metrics on quasi-smooth members of suitable families.
We set
\[
\mathsf{I}'_{\mathrm{KE}} = \{42, 44, 45, 61, 69, 74, 76, 79 \} \subset \mathsf{I}_{\mathrm{BR}},
\]
and
\[
\mathsf{I}_{\mathrm{KE}} = \mathsf{I}_{\mathrm{BSR}} \sqcup \mathsf{I}'_{\mathrm{KE}}.
\]
Note that $|\mathsf{I}_{\mathrm{KE}}| = 56$.
For a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}$, the mark ``KE" is given in the right-most column of Table \ref{table:main} if and only if $\mathsf{i} \in \mathsf{I}_{\mathrm{KE}}$.
\begin{Thm} \label{thm:qsmWHKE}
For a member $X$ of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}'_{\mathrm{KE}}$, we have
\[
\alpha (X) > \frac{3}{4}.
\]
In particular, any member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{I}_{\mathrm{KE}}$ admits a KE metric and is K-stable.
\end{Thm}
\begin{proof}
By Corollary \ref{maincor:Kst} and the above arguments, it is enough to prove the first assertion.
Let
\[
X = X_d \subset \mathbb{P} (1, a_1, a_2, a_3, a_4)_{x, y, z, t, w}
\]
be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}'_{\mathrm{KE}}$, where we assume $a_1 \le a_2 \le a_3 \le a_4$.
Note that $1 < a_1 < a_2$.
\begin{Claim} \label{clm:KEU1}
$\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in U_1 \cap \operatorname{Sm} (X)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:KEU1}]
Let $\mathsf{p}$ be a smooth point of $X$ contaied in $U_1$.
Suppose $\mathsf{i} = 42$.
Then $d = 20$ is divisible by $a_4 = 10$ and $a_2 a_3 (A^3) = 1$.
By Lemma \ref{lem:nsptU1-2}, we have $\alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $\mathsf{i} \in \{ 69, 74, 76, 79 \}$.
Then $a_2 a_4 (A^3) \le 1$.
By Lemma \ref{lem:nsptU1-1}, we have $\alpha_{\mathsf{p}} (X) \ge 1$ in this case.
Suppose $\mathsf{i} \in \{44, 45, 61\}$.
Then $a_3 a_4 (A^3) \le 2$.
We may assume $\mathsf{p} = \mathsf{p}_x$.
Then we have $\alpha_{\mathsf{p}} (X) \ge 2/a_3 a_4 (A^3) \ge 1$ by Lemma \ref{lem:complctsingtang}.
This completes the proof.
\end{proof}
\begin{Claim} \label{clm:KEH}
$\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in (H_x \setminus L_{xy}) \cap \operatorname{Sm} (X)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:KEH}]
This follows immediately from Proposition \ref{prop:smptHminusL}.
\end{proof}
\begin{Claim} \label{clm:KEL}
$\alpha_{\mathsf{p}} (X) \ge 43/54 > 3/4$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:KEL}]
Let $\mathsf{p}$ be a smooth point of $X$ contained in $L_{xy}$.
Suppose that $X$ is a member of one of the families listed in Tables \ref{table:Lsmooth} or \ref{table:Lsing}, that is, $X$ is a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \{44, 45, 61, 69, 74, 76, 79\}$.
Then the claim follows immediately from Proposition \ref{prop:smptL1}.
Suppose $\mathsf{i} = 42$.
Then, by the proof of Proposition \ref{prop:smptL2} (see \S \ref{sec:smptL2-42}), either $\alpha_{\mathsf{p}} (X) \ge 1$ for any $\mathsf{p} \in L_{xy} \cap \operatorname{Sm} (X)$ or $X$ satisfies the assumption of Lemma \ref{lem:Lredcp1}.
In the latter case we have $\alpha_{\mathsf{p}} (X) \ge 43/54$ by Remark \ref{rem:Lredcp1}.
This completes the proof.
\end{proof}
By Claims \ref{clm:KEU1}, \ref{clm:KEH} and \ref{clm:KEL}, we have $\alpha_{\mathsf{p}} (X) \ge 3/4$ for any smooth point $\mathsf{p} \in X$.
It remains to consider singular points.
\begin{Claim} \label{clm:KEsing}
$\alpha_{\mathsf{p}} (X) > 3/4$ for any $\mathsf{p} \in \operatorname{Sing} (X)$.
\end{Claim}
\begin{proof}[Proof of Claim \ref{clm:KEsing}]
Let $\mathsf{p} \in X$ be a singular point.
If the subscript $\heartsuit$ (resp.\ $\diamondsuit$) is given in Table \ref{table:main}, then $\alpha_{\mathsf{p}} (X) \ge 1$ by Proposition \ref{prop:singptCP} (resp.\ Proposition \ref{prop:lctsingptL}).
It remains to consider the case where $\mathsf{i} = 42$ and $\mathsf{p}$ is of type $\frac{1}{5} (1, 2, 3)$.
In this case we have $\alpha_{\mathsf{p}} (X) \ge 1$ by the proof of Proposition \ref{prop:ndQIcent} (see \S \ref{sec:ndQI2eqtypes}).
\end{proof}
This completes the proof of Theorem \ref{thm:qsmWHKE}.
\end{proof}
\section{Birational rigidity and K-stability} \label{section:BRKst}
\subsection{Generalizations of the conjecture}
Birational superrigidity is a very strong property.
It is natural to relax the assumption of birational superrigidity to birational rigidity in Conjecture \ref{conj:BSRKst}, and we still expect a positive answer to the following.
\begin{Conj}[{\cite[Conjecture 1.9]{KOW18}}] \label{conj:BRKst}
A birationally rigid Fano variety is K-stable.
\end{Conj}
We explain the situation for smooth Fano 3-folds.
There are exactly 2 families of smooth Fano 3-folds which is strictly birationally rigid: one is the family of complete intersections of a quadric and cubic in $\mathbb{P}^6$ (\cite{IP96}), and another is the family of double covers $V$ of a smooth quadric $Q$ of dimension 3 branched along a smooth surface degree $8$ on $Q$ (\cite{Isk80}).
Former Fano 3-folds are K-stable and admit KE metrics (\cite{Zhu20b}), and so are the latter Fano 3-folds (this follows from \cite{Der16a} since $Q$ is K-semistable).
Some more evidences are already provided by Theorems \ref{thm:BRWH} and \ref{thm:KOW18}, and we will provide further evidences in the next subsection (see Corollary \ref{cor:Kstqsm}).
It may be interesting to consider further generalization of Conjecture \ref{conj:BRKst}.
According to systematic studies of Fano 3-folds of codimension 2 \cite{Oka14, Oka18, Oka20b}, existence of many birationally bi-rigid Fano 3-folds are verified.
Here a Fano variety $X$ of Picard number $1$ is {\it birationally bi-rigid} if there exists a Fano variety $X'$ of Picard number $1$ which is birational but not isomorphic to $X$, and up to isomorphism $\{X, X'\}$ are all the Mori fiber space in the birational equivalence class of $X$.
Extending bi-rigidity, tri-rigidity and so on, notion of solid Fano variety in introduced in \cite{AO18}: a Fano variety of Picard number $1$ is {\it solid} if any Mori fiber space in the birational equivalence class is a Fano variety of Picard number $1$.
Solid Fano varieties are expected to behave nicely in moduli (\cite{Zhu20a}).
Only a few evidences are known (\cite{KOW19}) for the following question.
\begin{Question}
Is it true that any solid Fano variety is K-stable?
\end{Question}
\subsection{On K-stability for 95 families}
\label{sec:Kstqsm}
For strictly birationally rigid members of the 95 families, we are unable to conclude K-stability by Theorem \ref{mainthm}, except for those treated in Theorem \ref{thm:qsmWHKE}.
The aim of this subsection is to prove K-stability for all the quasi-smooth members of suitable families indexed by $\mathsf{I}_{\mathrm{BR}}$.
This will be done by combining the inequality $\alpha \ge 1/2$ obtained by Theorem \ref{mainthm} and an additional information on local movable alpha invariants which are introduced below.
\begin{Def}
Let $X$ be a Fano variety of Picard number $1$ and $\mathsf{p} \in X$ a point.
For a non-empty linear system $\mathcal{M}$ on $X$, we define $\lambda_{\mathcal{M}} \in \mathbb{Q}_{> 0}$ to be the rational number such that $\mathcal{M} \sim_{\mathbb{Q}} - \lambda_{\mathcal{M}} K_X$.
For a movable linear system $\mathcal{M}$ on $X$ and a positive rational number $\mu$, we define the {\it movable log canonical threshold} of $(X, \mu \mathcal{M})$ at $\mathsf{p}$ to be the number
\[
\operatorname{lct}^{\operatorname{mov}}_{\mathsf{p}} (X; \mu \mathcal{M}) = \sup \{\, c \in \mathbb{Q}_{\ge 0} \mid \text{$(X, c \mu \mathcal{M})$ is log canonical at $\mathsf{p}$} \, \},
\]
and then we define the {\it movable alpha invariant of $X$ at} $\mathsf{p}$ as
\[
\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) = \inf \{\, \operatorname{lct}_{\mathsf{p}}^{\operatorname{mov}} (X, \lambda_{\mathcal{M}}^{-1} \mathcal{M}) \mid \text{$\mathcal{M}$ is a movable linear system on $X$} \}.
\]
\end{Def}
\begin{Prop}[{cf.\ \cite[Corollary 3.1]{SZ19}}] \label{prop:Kstmovalpha}
Let $X$ be a quasi-smooth Fano $3$-fold weighted hypersurface of index $1$.
Assume that, for any maximal center $\mathsf{p} \in X$, we have
\[
\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge 1 \quad \text{and} \quad
(\alpha^{\operatorname{mov}}_{\mathsf{p}} (X), \alpha_{\mathsf{p}} (X)) \ne (1, 1/2).
\]
Then $X$ is K-stable.
\end{Prop}
\begin{proof}
By the main result of \cite{CP17} (cf.\ Remark \ref{rem:maxcent}), we have $\alpha^{\operatorname{mov}}_{\mathsf{q}} (X) \ge 1$ for any point $\mathsf{q} \in X$ which is not a maximal center.
It follows that the pair $(X, \lambda_{\mathcal{M}}^{-1} \mathcal{M})$ is log canonical for any movable linear system $\mathcal{M}$ on $X$.
Combining this with the inequality $\alpha (X) \ge 1/2$ obtained by Theorem \ref{mainthm}, we see that $X$ is K-semistable by \cite[Theorem 1.2]{SZ19}.
Suppose that $X$ is not K-stable.
Then, by \cite[Corollary 3.1]{SZ19}, there exists a prime divisor $E$ over $X$, a movable linear system $\mathcal{M} \sim_{\mathbb{Q}} - n K_X$ and an effective $\mathbb{Q}$-divisor $D \sim_{\mathbb{Q}} - K_X$ such that $E$ is a log canonical place of $(X, \frac{1}{n} \mathcal{M})$ and $(X, \frac{1}{2} D)$.
Note that the center $\Gamma$ of $E$ on $X$ is necessarily a maximal center, and a maximal center on $X$ is a BI center.
Thus $\Gamma = \mathsf{p}$ is a BI center, and this implies $(\alpha_{\mathsf{p}}^{\operatorname{mov}} (X), \alpha_{\mathsf{p}} (X)) = (1, 1/2)$.
This is impossible by the assumption.
Therefore $X$ is K-stable.
\end{proof}
We define
\[
\mathsf{I}'_{\mathrm{K}} = \{6, 8, 15, 16, 17, 26, 27, 30, 36, 41, 47, 48, 54, 56, 60, 65, 68\} \subset \mathsf{I}_{\mathrm{BR}}.
\]
\begin{Thm} \label{thm:qsmWHKst}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}'_{\mathrm{K}}$.
Then, for any BI center $\mathsf{p} \in X$, we have
\begin{equation} \label{eq:qsmWHKst}
\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge 1 \quad \text{and} \quad \alpha_{\mathsf{p}} (X) > \frac{1}{2}.
\end{equation}
In particular, $X$ is K-stable.
\end{Thm}
\begin{proof}
Let $X$ be a member of $\mathcal{F}_{\mathsf{i}}$, where $\mathsf{i} \in \mathsf{I}'_{\mathrm{K}}$.
We first show that the inequalities \eqref{thm:qsmWHKst} are satisfied.
Suppose $\mathsf{i} \in \{16, 17, 26, 27, 36, 47, 48, 54, 65\}$.
Then, the subscript $\diamondsuit$ is given in the 4th column of Table \ref{table:main} for any BI center on $X$.
By Proposition \ref{prop:lctsingptL}, we have $\alpha_{\mathsf{p}} (X) \ge 1$, and hence $\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge 1$, for any BI center $\mathsf{p} \in X$.
Suppose $\mathsf{i} \in \{6, 15, 30, 41, 68\}$.
In this case, $X$ admits two QI centers of equal singularity type, and does not admit any other BI center.
By the proof of Proposition \ref{prop:ndQIcent} (see \S \ref{sec:ndQI2eqtypes}), we have $\alpha_{\mathsf{p}} (X) \ge 1$ for any QI center $\mathsf{p} \in X$.
In particular, we have $\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge \alpha_{\mathsf{p}} (X) \ge 1$.
Suppose $\mathsf{i} \in \{8, 56, 60\}$.
In this case $X$ admits a unique BI center and it is a QI center.
The inequalities \eqref{eq:qsmWHKst} follows from Remark \ref{rem:QIcent-8} and Propositions \ref{prop:No56QIex}, \ref{prop:No60QIex}.
This completes the verifications for the inequalities \eqref{eq:qsmWHKst}.
The K-stability of $X$ follows from the inequalities \eqref{eq:qsmWHKst}, Theorem \ref{mainthm}, and Proposition \ref{prop:Kstmovalpha}.
\end{proof}
We define
\[
\mathsf{I}_{\mathrm{K}} := \mathsf{I}'_{\mathrm{K}} \sqcup \mathsf{I}_{\mathrm{KE}}.
\]
Note that $|\mathsf{I}_{\mathrm{K}}| = 73$.
Combining Theorems \ref{thm:qsmWHKE}, \ref{thm:qsmWHKst} and Corollary \ref{maincor:Kst}, we obtain the K-stability of arbitrary quasi-smooth member for families indexed by $\mathsf{I}_{\mathrm{K}}$.
\begin{Cor} \label{cor:Kstqsm}
Let $X$ be a member of a family $\mathcal{F}_{\mathsf{i}}$ with $\mathsf{i} \in \mathsf{I}_{\mathrm{K}}$.
Then $X$ is K-stable.
\end{Cor}
\section{Further computations of alpha invariants}
In this section, we compute local alpha invariants for a few families in order to give better lower bounds.
The results obtained in this section are used only in the proof of Theorem \ref{thm:qsmWHKst}.
\begin{Prop} \label{prop:No56QIex}
Let $X$ be a member of the family $\mathcal{F}_{56}$ and $\mathsf{p} = \mathsf{p}_w \in X$ be the singular point of type $\frac{1}{11} (1, 3, 8)$.
Then,
\[
\alpha_{\mathsf{p}} (X) \ge \frac{2}{3} \quad \text{and} \quad
\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge 1.
\]
\end{Prop}
\begin{proof}
We set $\mathsf{p} = \mathsf{p}_w$.
We can write the defining polynomial of $X$ as
\[
F = w^2 y + f_{13} + f_{24},
\]
where $f_i = f_i (x, y, z, t)$ is a quasi-homogeneous polynomial of degree $i$.
By the quasi-smoothness of $X$, we have $t^3 \in F$.
It is easy to see that $F (x, 0, z, t, 1) \in (x, z, t)^3$.
It follows that $\operatorname{omult}_{\mathsf{p}} (H_y) = 3$, which in particular implies $\operatorname{lct}_{\mathsf{p}} (X; \frac{1}{2} H_y) \ge 2/3$.
Let $D \in |A|_{\mathbb{Q}}$ be an irreducible $\mathbb{Q}$-divisor other than $\frac{1}{2} H_y$.
We can take a $\mathbb{Q}$-divisor $T \in |3 A|_{\mathbb{Q}}$ such that $\operatorname{omult}_{\mathsf{p}} (T) \ge 1$ and $\operatorname{Supp} (T)$ does not contain any component of the effective $1$-cycle $D \cdot H_y$.
We have
\[
3 \operatorname{omult}_{\mathsf{p}} (D) \le 11 (D \cdot H_y \cdot T) = 3.
\]
This shows $\operatorname{lct}_{\mathsf{p}} (X;D) \ge 1$.
Therefore $\alpha^{\operatorname{mov}}_{\mathsf{p}} (X) \ge 1$ and $\alpha_{\mathsf{p}} (X) \ge 2/3$.
\end{proof}
\begin{Prop} \label{prop:No60QIex}
Let $X$ be a member of the family $\mathcal{F}_{60}$ and let $\mathsf{p} = \mathsf{p}_w$ be the singular point of type $\frac{1}{9} (1, 4, 5)$.
Then
\[
\alpha_{\mathsf{p}} (X) = 1.
\]
\end{Prop}
\begin{proof}
We set $S = H_x \sim A$, $T = H_z$ and $\Gamma = S \cap T = (x = z = 0)_X$.
Let $\rho = \rho_{\mathsf{p}} \colon \breve{U}_{\mathsf{p}} \to U_{\mathsf{p}}$ be the orbifold chart of $\mathsf{p} \in X$, and we set $\breve{\Gamma} = (\breve{x} = \breve{z} = 0) \subset \breve{U}_{\mathsf{p}}$.
We can write the defining polynomial of $X$ as
\[
F = w^2 t + w f_{15} + f_{24},
\]
where $f_i = f_i (x, y, z, t)$ is a quasi-homogeneous polynomial of degree $i$.
By the quasi-smoothness of $X$, we have $t^4, y^6 \in F$, and we may assume $\operatorname{coeff}_F (t^4) = \operatorname{coeff}_F (y^6) = 1$ by rescaling $y$ and $t$.
We set $\lambda = \operatorname{coeff}_F (t^2 y^3) \in \mathbb{C}$.
Then,
\[
\begin{split}
\Gamma &\cong (w^2 t + t^4 + \lambda t^2 y^3 + y^6 = 0) \subset \mathbb{P} (3, 10, 17)_{y, t, w}, \\
\breve{\Gamma} & \cong (\breve{w}^2 \breve{t} + \breve{t}^4 + \lambda \breve{t}^{\hspace{0.3mm} 2} \breve{y}^3 + \breve{y}^6 = 0) \subset \mathbb{A}^3_{\breve{y}, \breve{t}, \breve{w}}.
\end{split}
\]
It is easy to see that $\Gamma$ is an irreducible and reduced curve, and $\operatorname{mult}_{\breve{\mathsf{p}}} (\breve{\Gamma}) = 1$, where $\breve{\mathsf{p}} = o \in \mathbb{A}^3$ is the preimage of $\mathsf{p}$ via $\rho$.
We see that $H_x$ is quasi-smooth at $\mathsf{p}$, and hence $\operatorname{lct}_{\mathsf{p}} (X; H_x) = 1$.
Therefore, we have $\alpha_{\mathsf{p}} (X) \ge 1$ by Lemma \ref{lem:exclL}.
\end{proof}
\chapter{The table}
\label{chap:table}
The list of the 93 families together with their basic information are summarized in Table \ref{table:main}, and we explain the contents.
The first two columns indicate basic information of each family and the anticanonical degree $(A^3) = (-K_X)^3$ is indicated in the 3rd column.
In the 4th column, the number and the singularities of $X$ are described.
The symbol $\frac{1}{r} [a, r-a]$ stands for the cyclic quotient singularity of type $\frac{1}{r} (1, a, r-a)$, where $1 \le a \le r/2$.
Moreover the symbols $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$ stand for singularities of types $\frac{1}{2} (1, 1, 1)$, $\frac{1}{3} (1, 1, 2)$ and $\frac{1}{4} (1, 1, 3)$.
The superscript $\mathrm{QI}$ and $\mathrm{EI}$ indicates that the corresponding singular point $\mathsf{p}$ is a QI center and EI center, respectively (see \S \ref{sec:defBI} for definitions).
The meaning of the subscripts is explained as follows.
\begin{itemize}
\item The subscript $\heartsuit$ indicates that $\alpha_{\mathsf{p}} (X) \ge 1$ is proved by Proposition \ref{prop:singptCP}.
\item The subscript $\diamondsuit$ (resp.\ $\diamondsuit'$) indicates that $\alpha_{\mathsf{p}} (X) \ge 1$ (resp.\ $\alpha_{|msp} (X) \ge 2/3$) is proved by Proposition \ref{prop:lctsingptL}.
\item The subscript $\clubsuit$ indicates that $\alpha_{\mathsf{p}} (X) \ge 1/2$ is proved by Proposition \ref{prop:singptic}.
\item The subscript $\spadesuit$ indicates that $\alpha_{\mathsf{p}} (X) \ge 1/2$ is proved by Proposition \ref{prop:singptrem}.
\end{itemize}
In Theorem \ref{mainthm}, any birational superrigid member of each of the 95 families is proved.
Apart from this main result, we have results on the existence of KE metrics or K-stability for any quasi-smooth member of suitable families.
In the right-most column the mark ``KE" and ``K" are given and their meanings are as follows.
\begin{itemize}
\item The mark ``KE" in the right-most column means that any quasi-smooth member admits a KE metric and is K-stable (see \S \ref{sec:KEmetric}).
\item The mark ``K" in the right-most column means that any quasi-smooth member is K-stable (see \S \ref{sec:Kstqsm}).
\end{itemize}
\begingroup
\renewcommand{1.35}{1.35}
\begin{longtable}{clclc}
\caption{The 93 families}
\label{table:main}
\\
\hline\hline
No.\ & $X_d \subset \mathbb{P} (1,a_1,a_2,a_3,a_4)$ & $(A^3)$ & Singular points & \\
\hline\hline
\endfirsthead
\hline\hline
No.\ & $X_d \subset \mathbb{P} (1,a_1,a_2,a_3,a_4)$ & $(A^3)$ & Singular points & \\
\hline\hline
\endhead
\rowcolor{lightgray}
2 & $X_5 \subset \mathbb{P} (1,1,1,1,2)$ & $\frac{5}{2}$ & $\frac{1}{2}^{\mathrm{QI}}$ & \\
4 & $X_6 \subset \mathbb{P} (1,1,1,2,2)$ & $\frac{3}{2}$ & $3 \! \times \! \frac{1}{2}^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
5 & $X_7 \subset \mathbb{P} (1,1,1,2,3)$ & $\frac{7}{6}$ & $\frac{1}{2}^{\mathrm{QI}}, \ \frac{1}{3}^{\mathrm{QI}}$ & \\
6 & $X_8 \subset \mathbb{P} (1,1,1,2,4)$ & $1$ & $2 \! \times \! \frac{1}{2}^{\mathrm{QI}}$ & K \\
\rowcolor{lightgray}
7 & $X_8 \subset \mathbb{P} (1,1,2,2,3)$ & $\frac{2}{3}$ & $4 \! \times \! \frac{1}{2}^{\mathrm{EI}}, \ \frac{1}{3}^{\mathrm{QI}}$ & \\
8 & $X_9 \subset \mathbb{P} (1,1,1,3,4)$ & $\frac{3}{4}$ & $\frac{1}{4}^{\mathrm{QI}}$ & K \\
\rowcolor{lightgray}
9 & $X_9 \subset \mathbb{P} (1,1,2,3,3)$ & $\frac{1}{2}$ & $\frac{1}{2}_{\heartsuit}, \ 3 \! \times \! \frac{1}{3}^{\mathrm{QI}}$ & \\
10 & $X_{10} \subset \mathbb{P} (1,1,1,3,5)$ & $\frac{2}{3}$ & $\frac{1}{3}_{\clubsuit}$ & KE \\
\rowcolor{lightgray}
11 & $X_{10} \subset \mathbb{P} (1,1,2,2,5)$ & $\frac{1}{2}$ & $5 \! \times \! \frac{1}{2}_{\heartsuit}$ & KE \\
12 & $X_{10} \subset \mathbb{P} (1,1,2,3,4)$ & $\frac{5}{12}$ & $2 \! \times \! \frac{1}{2}_{\spadesuit}, \ \frac{1}{3}^{\mathrm{QI}}, \ \frac{1}{4}^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
13 & $X_{11} \subset \mathbb{P} (1,1,2,3,5)$ & $\frac{11}{30}$ & $\frac{1}{2}_{\spadesuit}, \ \frac{1}{3}^{\mathrm{QI}}, \ \frac{1}{5} [2, 3]^{\mathrm{QI}}$ & \\
14 & $X_{12} \subset \mathbb{P} (1,1,1,4,6)$ & $\frac{1}{2}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
15 & $X_{12} \subset \mathbb{P} (1,1,2,3,6)$ & $\frac{1}{3}$ & $2 \! \times \! \frac{1}{2}_{\diamondsuit}, \ 2 \! \times \! \frac{1}{3}_{\diamondsuit}^{\mathrm{QI}}$ & K \\
16 & $X_{12} \subset \mathbb{P} (1,1,2,4,5)$ & $\frac{3}{10}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [1, 4]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
17 & $X_{12} \subset \mathbb{P} (1,1,3,4,4)$ & $\frac{1}{4}$ & $3 \! \times \! \frac{1}{4}^{\mathrm{QI}}_{\diamondsuit}$ & K \\
18 & $X_{12} \subset \mathbb{P} (1,2,2,3,5)$ & $\frac{1}{5}$ & $6 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [2, 3]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
19 & $X_{12} \subset \mathbb{P} (1,2,3,3,4)$ & $\frac{1}{6}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ 4 \! \times \! \frac{1}{3}_{\heartsuit}$ & KE \\
20 & $X_{13} \subset \mathbb{P} (1,1,3,4,5)$ & $\frac{13}{60}$ & $\frac{1}{3}^{\mathrm{EI}}, \ \frac{1}{4}^{\mathrm{QI}}, \ \frac{1}{5} [1, 4]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
21 & $X_{14} \subset \mathbb{P} (1,1,2,4,7)$ & $\frac{1}{4}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\diamondsuit}$ & KE \\
22 & $X_{14} \subset \mathbb{P} (1,2,2,3,7)$ & $\frac{1}{6}$ & $7 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
23 & $X_{14} \subset \mathbb{P} (1,2,3,4,5)$ & $\frac{7}{60}$ & $3 \! \times \! \frac{1}{2}_{\clubsuit}, \ \frac{1}{3}_{\clubsuit}, \ \frac{1}{4}^{\mathrm{EI}}_{\clubsuit}, \ \frac{1}{5} [2, 3]^{\mathrm{QI}}$ & \\
24 & $X_{15} \subset \mathbb{P} (1,1,2,5,7)$ & $\frac{3}{14}$ & $\frac{1}{2}_{\spadesuit}, \ \frac{1}{7} [2, 5]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
25 & $X_{15} \subset \mathbb{P} (1,1,3,4,7)$ & $\frac{5}{28}$ & $\frac{1}{4}^{\mathrm{QI}}, \ \frac{1}{7} [3, 4]^{\mathrm{QI}}$ & \\
26 & $X_{15} \subset \mathbb{P} (1,1,3,5,6)$ & $\frac{1}{6}$ & $2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{6} [1, 5]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
27 & $X_{15} \subset \mathbb{P} (1,2,3,5,5)$ & $\frac{1}{10}$ & $\frac{1}{2}_{\spadesuit}, 3 \! \times \! \frac{1}{5} [2, 3]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
28 & $X_{15} \subset \mathbb{P} (1,3,3,4,5)$ & $\frac{1}{12}$ & $5 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
29 & $X_{16} \subset \mathbb{P} (1,1,2,5,8)$ & $\frac{1}{5}$ & $2 \! \times \! \frac{1}{2}_{\clubsuit}, \ \frac{1}{5} [2, 3]_{\clubsuit}$ & KE \\
30 & $X_{16} \subset \mathbb{P} (1,1,3,4,8)$ & $\frac{1}{6}$ & $\frac{1}{3}_{\heartsuit}, \ 2 \! \times \! \frac{1}{4}^{\mathrm{QI}}$ & K \\
\rowcolor{lightgray}
31 & $X_{16} \subset \mathbb{P} (1,1,4,5,6)$ & $\frac{2}{15}$ & $\frac{1}{2}_{\clubsuit}, \ \frac{1}{5} [1, 4]^{\mathrm{QI}}, \ \frac{1}{6} [1, 5]^{\mathrm{QI}}$ & \\
32 & $X_{16} \subset \mathbb{P} (1,2,3,4,7)$ & $\frac{2}{21}$ & $4 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\spadesuit}, \ \frac{1}{7} [3, 4]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
33 & $X_{17} \subset \mathbb{P} (1,2,3,5,7)$ & $\frac{17}{210}$ & $\frac{1}{2}_{\spadesuit}, \ \frac{1}{3}_{\clubsuit}, \ \frac{1}{5} [2, 3]^{\mathrm{QI}}, \ \frac{1}{7} [2, 5]^{\mathrm{QI}}$ & \\
34 & $X_{18} \subset \mathbb{P} (1,1,2,6,9)$ & $\frac{1}{6}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
35 & $X_{18} \subset \mathbb{P} (1,1,3,5,9)$ & $\frac{2}{15}$ & $2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\diamondsuit}$ & KE \\
36 & $X_{18} \subset \mathbb{P} (1,1,4,6,7)$ & $\frac{3}{28}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{4}^{\mathrm{EI}}_{\diamondsuit}, \ \frac{1}{7} [1, 6]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
37 & $X_{18} \subset \mathbb{P} (1,2,3,4,9)$ & $\frac{1}{12}$ & $4 \! \times \! \frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{3}_{\clubsuit}, \ \frac{1}{4}_{\heartsuit}$ & KE \\
38 & $X_{18} \subset \mathbb{P} (1,2,3,5,8)$ & $\frac{3}{40}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [2, 5]^{\mathrm{QI}}, \frac{1}{8} [3, 5]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
39 & $X_{18} \subset \mathbb{P} (1,3,4,5,6)$ & $\frac{1}{20}$ & $ \frac{1}{2}_{\heartsuit}, \ 3 \! \times \! \frac{1}{3}_{\clubsuit}, \ \frac{1}{4}_{\clubsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}$ & KE \\
40 & $X_{19} \subset \mathbb{P} (1,3,4,5,7)$ & $\frac{19}{420}$ & $\frac{1}{3}_{\spadesuit}, \ \frac{1}{4}_{\clubsuit}, \ \frac{1}{5} [2, 3]^{\mathrm{EI}}_{\clubsuit}, \ \frac{1}{7} [3, 4]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
41 & $X_{20} \subset \mathbb{P} (1,1,4,5,10)$ & $\frac{1}{10}$ & $\frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{5} [1, 4]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
42 & $X_{20} \subset \mathbb{P} (1,2,3,5,10)$ & $\frac{1}{15}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ 2 \! \times \! \frac{1}{5} [2, 3]^{\mathrm{QI}}$ & KE \\
\rowcolor{lightgray}
43 & $X_{20} \subset \mathbb{P} (1,2,4,5,9)$ & $\frac{1}{18}$ & $5 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{9} [4, 5]^{\mathrm{QI}}$ & \\
44 & $X_{20} \subset \mathbb{P} (1,2,5,6,7)$ & $\frac{1}{21}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{6} [1, 5]^{\mathrm{EI}}_{\diamondsuit}, \ \frac{1}{7} [2, 5]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
\rowcolor{lightgray}
45 & $X_{20} \subset \mathbb{P} (1,3,4,5,8)$ & $\frac{1}{24}$ & $\frac{1}{3}_{\heartsuit}, \ 2 \! \times \! \frac{1}{4}_{\heartsuit}, \ \frac{1}{8} [3, 5]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
46 & $X_{21} \subset \mathbb{P} (1,1,3,7,10)$ & $\frac{1}{10}$ & $\frac{1}{10} [3, 7]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
47 & $X_{21} \subset \mathbb{P} (1,1,5,7,8)$ & $\frac{3}{40}$ & $\frac{1}{5} [2, 3]_{\spadesuit}, \ \frac{1}{8} [1, 7]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
48 & $X_{21} \subset \mathbb{P} (1,2,3,7,9)$ & $\frac{1}{18}$ & $\frac{1}{2}_{\spadesuit}, \ 2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{9} [2, 7]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
49 & $X_{21} \subset \mathbb{P} (1,3,5,6,7)$ & $\frac{1}{30}$ & $3 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\spadesuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}$ & KE \\
50 & $X_{22} \subset \mathbb{P} (1,1,3,7,11)$ & $\frac{2}{21}$ & $\frac{1}{3}_{\heartsuit}, \ \frac{1}{7} [3, 4]_{\clubsuit}$ & KE \\
\rowcolor{lightgray}
51 & $X_{22} \subset \mathbb{P} (1,1,4,6,11)$ & $\frac{1}{12}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{6} [1, 5]_{\diamondsuit}$ & KE \\
52 & $X_{22} \subset \mathbb{P} (1,2,4,5,11)$ & $\frac{1}{20}$ & $5 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
53 & $X_{24} \subset \mathbb{P} (1,1,3,8,12)$ & $\frac{1}{12}$ & $2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}$ & KE \\
54 & $X_{24} \subset \mathbb{P} (1,1,6,8,9)$ & $\frac{1}{18}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{9} [1, 8]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
55 & $X_{24} \subset \mathbb{P} (1,2,3,7,12)$ & $\frac{1}{21}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{7} [2, 5]_{\diamondsuit}$ & KE \\
56 & $X_{24} \subset \mathbb{P} (1,2,3,8,11)$ & $\frac{1}{22}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{11} [3, 8]^{\mathrm{QI}}$ & K \\
\rowcolor{lightgray}
57 & $X_{24} \subset \mathbb{P} (1,3,4,5,12)$ & $\frac{1}{30}$ & $2 \! \times \! \frac{1}{3}_{\heartsuit}, \ 2 \! \times \! \frac{1}{4}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\heartsuit}$ & KE \\
58 & $X_{24} \subset \mathbb{P} (1,3,4,7,10)$ & $\frac{1}{35}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{7} [3, 4]^{\mathrm{QI}}, \ \frac{1}{10} [3, 7]^{\mathrm{QI}}$ & \\
\rowcolor{lightgray}
59 & $X_{24} \subset \mathbb{P} (1,3,6,7,8)$ & $\frac{1}{42}$ & $\frac{1}{2}_{\heartsuit}, \ 4 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{7} [1, 6]_{\heartsuit}$ & KE \\
60 & $X_{24} \subset \mathbb{P} (1,4,5,6,9)$ & $\frac{1}{45}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{9} [4, 5]^{\mathrm{QI}}$ & K \\
\rowcolor{lightgray}
61 & $X_{25} \subset \mathbb{P} (1,4,5,7,9)$ & $\frac{5}{252}$ & $\frac{1}{4}_{\clubsuit}, \ \frac{1}{7} [2, 5]^{\mathrm{EI}}_{\diamondsuit}, \ \frac{1}{9} [4, 5]_{\diamondsuit}$ & KE \\
62 & $X_{26} \subset \mathbb{P} (1,1,5,7,13)$ & $\frac{2}{35}$ & $\frac{1}{5} [2, 3]_{\spadesuit}, \ \frac{1}{7} [1, 6]_{\diamondsuit}$ & KE \\
\rowcolor{lightgray}
63 & $X_{26} \subset \mathbb{P} (1,2,3,8,13)$ & $\frac{1}{24}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{8} [3, 5]_{\clubsuit}$ & KE \\
64 & $X_{26} \subset \mathbb{P} (1,2,5,6,13)$ & $\frac{1}{30}$ & $4 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\clubsuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
65 & $X_{27} \subset \mathbb{P} (1,2,5,9,11)$ & $\frac{3}{110}$ & $\frac{1}{2}_{\spadesuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{11} [2, 9]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
66 & $X_{27} \subset \mathbb{P} (1,5,6,7,9)$ & $\frac{1}{70}$ & $\frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\clubsuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}, \ \frac{1}{7} [2, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
67 & $X_{28} \subset \mathbb{P} (1,1,4,9,14)$ & $\frac{1}{18}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{9} [4, 5]_{\spadesuit}$ & KE \\
68 & $X_{28} \subset \mathbb{P} (1,3,4,7,14)$ & $\frac{1}{42}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\clubsuit}, \ 2 \! \times \! \frac{1}{7} [3, 4]^{\mathrm{QI}}_{\diamondsuit}$ & K \\
\rowcolor{lightgray}
69 & $X_{28} \subset \mathbb{P} (1,4,6,7,11)$ & $\frac{1}{66}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}, \ \frac{1}{11} [4, 7]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
70 & $X_{30} \subset \mathbb{P} (1,1,4,10,15)$ & $\frac{1}{20}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
71 & $X_{30} \subset \mathbb{P} (1,1,6,8,15)$ & $\frac{1}{24}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{8} [1, 7]_{\diamondsuit}$ & KE \\
72 & $X_{30} \subset \mathbb{P} (1,2,3,10,15)$ & $\frac{1}{30}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
73 & $X_{30} \subset \mathbb{P} (1,2,6,7,15)$ & $\frac{1}{42}$ & $5 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{7} [1, 6]_{\heartsuit}$ & KE \\
74 & $X_{30} \subset \mathbb{P} (1,3,4,10,13)$ & $\frac{1}{52}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{13} [3, 10]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
\rowcolor{lightgray}
75 & $X_{30} \subset \mathbb{P} (1,4,5,6,15)$ & $\frac{1}{60}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ 2 \! \times \! \frac{1}{5} [1, 4]_{\heartsuit}$ & KE \\
76 & $X_{30} \subset \mathbb{P} (1,5,6,8,11)$ & $\frac{1}{88}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{8} [3, 5]^{\mathrm{EI}}_{\diamondsuit}, \ \frac{1}{11} [5, 6]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
\rowcolor{lightgray}
77 & $X_{32} \subset \mathbb{P} (1,2,5,9,16)$ & $\frac{1}{45}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{9} [2, 7]_{\diamondsuit}$ & KE \\
78 & $X_{32} \subset \mathbb{P} (1,4,5,7,16)$ & $\frac{1}{70}$ & $2 \! \times \! \frac{1}{4}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{7} [2, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
79 & $X_{33} \subset \mathbb{P} (1,3,5,11,14)$ & $\frac{1}{70}$ & $\frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{14} [3, 11]^{\mathrm{QI}}_{\diamondsuit}$ & KE \\
80 & $X_{34} \subset \mathbb{P} (1,3,4,10,17)$ & $\frac{1}{60}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\clubsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{10} [3, 7]_{\diamondsuit}$ & KE \\
\rowcolor{lightgray}
81 & $X_{34} \subset \mathbb{P} (1,4,6,7,17)$ & $\frac{1}{84}$ & $2 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}, \ \frac{1}{7} [3, 4]_{\heartsuit}$ & KE \\
82 & $X_{36} \subset \mathbb{P} (1,1,5,12,18)$ & $\frac{1}{30}$ & $\frac{1}{5} [2, 3]_{\spadesuit}, \frac{1}{6} [1, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
83 & $X_{36} \subset \mathbb{P} (1,3,4,11,18)$ & $\frac{1}{66}$ & $\frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{11} [4, 7]_{\diamondsuit}$ & KE \\
84 & $X_{36} \subset \mathbb{P} (1,7,8,9,12)$ & $\frac{1}{168}$ & $\frac{1}{3}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{7} [2, 5]_{\spadesuit}, \ \frac{1}{8} [1, 7]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
85 & $X_{38} \subset \mathbb{P} (1,3,5,11,19)$ & $\frac{2}{165}$ & $\frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{11} [3, 8]_{\diamondsuit}$ & KE \\
86 & $X_{38} \subset \mathbb{P} (1,5,6,8,19)$ & $\frac{1}{120}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{6} [1, 5]_{\heartsuit}, \ \frac{1}{8} [3, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
87 & $X_{40} \subset \mathbb{P} (1,5,7,8,20)$ & $\frac{1}{140}$ & $\frac{1}{4}_{\heartsuit}, \ 2 \! \times \! \frac{1}{5} [2, 3]_{\heartsuit}, \ \frac{1}{7} [1, 6]_{\heartsuit}$ & KE \\
88 & $X_{42} \subset \mathbb{P} (1,1,6,14,21)$ & $\frac{1}{42}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{7} [1, 6]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
89 & $X_{42} \subset \mathbb{P} (1,2,5,14,21)$ & $\frac{1}{70}$ & $3 \! \times \! \frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{7} [2, 5]_{\heartsuit}$ & KE \\
90 & $X_{42} \subset \mathbb{P} (1,3,4,14,21)$ & $\frac{1}{84}$ & $\frac{1}{2}_{\heartsuit}, \ 2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{7} [3, 4]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
91 & $X_{44} \subset \mathbb{P} (1,4,5,13,22)$ & $\frac{1}{130}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\heartsuit}, \ \frac{1}{13} [4, 9]_{\diamondsuit}$ & KE \\
92 & $X_{48} \subset \mathbb{P} (1,3,5,16,24)$ & $\frac{1}{120}$ & $2 \! \times \! \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [1, 4]_{\heartsuit}, \ \frac{1}{8} [3, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
93 & $X_{50} \subset \mathbb{P} (1,7,8,10,25)$ & $\frac{1}{280}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\heartsuit}, \ \frac{1}{7} [3, 4]_{\clubsuit}, \ \frac{1}{8} [1, 7]_{\heartsuit}$ & KE \\
94 & $X_{54} \subset \mathbb{P} (1,4,5,18,27)$ & $\frac{1}{180}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{4}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\heartsuit}, \ \frac{1}{9} [4, 5]_{\heartsuit}$ & KE \\
\rowcolor{lightgray}
95 & $X_{66} \subset \mathbb{P} (1,5,6,22,33)$ & $\frac{1}{330}$ & $\frac{1}{2}_{\heartsuit}, \ \frac{1}{3}_{\heartsuit}, \ \frac{1}{5} [2, 3]_{\clubsuit}, \ \frac{1}{11} [5, 6]_{\heartsuit}$ & KE
\end{longtable}
\endgroup
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Capacity of the range in dimension $5$}
\runtitle{Capacity of the range in dimension $5$}
\author{\fnms{Bruno} \snm{Schapira}\ead[label=e1]{[email protected]}}
\address{Aix-Marseille Universit\'e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France\\ \printead{e1}}
\runauthor{Bruno Schapira}
\begin{abstract}
We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb Z^5$,
under only a moment condition on the step distribution. The result is analogous to
the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance.
The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent random walks intersect.
The latter are then used for computing covariances of some intersection events, at the leading order.
\end{abstract}
\begin{keyword}[class=MSC]
\kwd{60F05; 60G50; 60J45}
\end{keyword}
\begin{keyword}
\kwd{Random Walk}
\kwd{Range}
\kwd{Capacity}
\kwd{Central Limit Theorem}
\kwd{Intersection of random walk ranges}
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec:intro}
Consider a random walk $(S_n)_{n\ge 0}$ on $\mathbb{Z}^d$, that is a process of the form $S_n=S_0+X_1+\dots + X_n$, where the $(X_i)_{i\ge 1}$ are independent and identically distributed. A general question is to understand the geometric properties of its range, that is the random set $\mathbb{R}R_n:=\{S_0,\dots,S_n\}$, and more specifically to analyze its large scale limiting behavior as the time $n$ is growing.
In their pioneering work, Dvoretzky and Erd\'os \cite{DE51} proved a strong law of large numbers for the number of distinct sites in $\mathbb{R}R_n$, in any dimension $d\ge 1$.
Later a central limit theorem was obtained first by Jain and Orey \cite{JO69} in dimensions $d\ge 5$, then by Jain and Pruitt \cite{JP71} in dimension $3$ and higher, and finally by Le Gall \cite{LG86} in dimension $2$, under fairly general hypotheses on the common law of the $(X_i)_{i\ge 1}$. Furthermore, a lot of activity has been focused on analyzing the large and moderate deviations, which we will not discuss here.
More recently some papers were concerned with other functionals of the range, including its entropy \cite{BKYY}, and its boundary \cite{AS17, BKYY, BY, DGK, Ok16}.
Here we will be interested in another
natural way to measure the size of the range, which also captures some properties of its shape.
Namely we will consider its Newtonian capacity, defined for a finite subset $A\subset \mathbb{Z}^d$, as
\begin{equation}\label{cap.def}
\mathrm{Cap}(A) :=\sum_{x\in A} \mathbb{P}_x[H_A^+ = \infty],
\end{equation}
where $\mathbb{P}_x$ is the law of the walk starting from $x$, and $H_A^+$ denotes the first return time to $A$ (see \eqref{HAHA+} below).
Actually the first study of the capacity of the range goes back to the earlier work by Jain and Orey \cite{JO69},
who proved a law of large numbers in any dimension $d\ge 3$; and more precisely that almost surely, as $n\to \infty$,
\begin{equation}\label{LLN.cap}
\frac 1n \mathrm{Cap}(\mathbb{R}R_n)\to \gamma_d,
\end{equation}
for some constant $\gamma_d$, which is nonzero if and only if $d\ge 5$ -- the latter observation being actually directly related to the fact that it is only in dimension $5$
and higher that two independent ranges have a positive probability not to intersect each other. However, until
very recently to our knowledge there were no other work on the capacity of the range, even though the results of Lawler on the intersection of random walks
incidentally gave a sharp asymptotic behavior of the mean in dimension four, see \cite{Law91}.
In a series of recent papers \cite{Chang, ASS18, ASS19}, the central limit theorem has been established for the simple random walk in any dimension $d\ge 3$,
except for the case of dimension $5$, which remained unsolved so far. The main goal of this paper is to fill this gap, but in the mean time we obtain general results on
the probability that the ranges of two independent walks intersect, which might be of independent interest.
We furthermore obtain estimates for the covariances between such events, which is arguably one of the main novelty of our work; but we shall come back on this point a bit later.
Our hypotheses on the random walk are quite general: we only require that the distribution of the $(X_i)_{i\ge 1}$ is a symmetric and irreducible probability measure on $\mathbb{Z}^d$, which has a finite $d$-th moment.
Under these hypotheses our first result is the following.
\begin{theoremA}\label{theoremA}
Assume $d=5$. There exists a constant $\sigma>0$, such that as $n\to \infty$,
$$\operatorname{Var}(\mathrm{Cap}(\mathbb{R}R_n)) \sim \sigma^2 \, n\log n.$$
\end{theoremA}
We then deduce a central limit theorem.
\begin{theoremB}\label{theoremB}
Assume $d=5$. Then,
$$\frac{\mathrm{Cap}(\mathbb{R}R_n) - \gamma_5 n}{\sigma \sqrt{n\log n}}\quad \stackrel{(\mathcal L)}{\underset{n\to \infty}{\Longrightarrow}} \quad \mathcal N(0,1).$$
\end{theoremB}
As already mentioned, along the proof we also obtain a precise asymptotic estimate for the probability that the ranges of two independent walks starting from far away intersect.
Previously to our knowledge only the order of magnitude up to multiplicative constants had been established, see \cite{Law91}.
Since our proof works the same in any dimension $d\ge 5$, we state our result in this general setting. Recall that to each random walk one can associate a norm (see below for a formal definition), which we denote here by $\mathcal J(\cdot)$ (in particular in the case of the simple random walk it coincides with the Euclidean norm).
\begin{theoremC}\label{theoremC}
Assume $d\ge 5$. Let $S$ and $\widetilde S$ be two independent random walks starting from the origin (with the same distribution).
There exists a constant $c>0$, such that as $\|x\|\to \infty$,
$$\mathbb{P}\left[\mathbb{R}R_\infty \cap (x+\widetilde \mathbb{R}R_\infty)\neq \varnothing\right]\sim \frac{c}{\mathcal J(x)^{d-4}}.$$
\end{theoremC}
In fact we obtain a stronger and more general result. Indeed, first we get some control on the second order term, and show that it is $\mathcal{O} (\|x\|^{4-d-\nu})$, for some constant $\nu>0$. Moreover, we also consider some functionals of the position of one of the two walks at its hitting time of the other range. More precisely, we obtain asymptotic estimates for quantities of the form $\mathbb{E}[F(S_\tau){\text{\Large $\mathfrak 1$}}\{\tau<\infty\}]$,
with $\tau$ the hitting time of the range $x+\widetilde \mathbb{R}R_\infty$,
for functions $F$ satisfying some regularity property, see \eqref{cond.F}. In particular, it applies to functions of the form $F(x)=1/\mathcal J(x)^\alpha$, for any $\alpha\in [0,1]$, for which we obtain that for some constants $\nu>0$, and $c>0$,
$$ \mathbb{E}\left[\frac{ {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} }{1+\mathcal J(S_\tau)^\alpha}\right] = \frac{c}{\mathcal J(x)^{d-4+\alpha}} + \mathcal{O}\left(\|x\|^{4-\alpha-d- \nu}\right).$$
Moreover, the same kind of estimates is obtained when one considers rather $\tau$ as the hitting time of $x+\widetilde \mathbb{R}R[0,\ell]$, with $\ell$ a finite integer. These results are then
used to derive asymptotic estimates for covariances of hitting events in the following four situations: let $S$, $S^1$, $S^2$, and $S^3$, be four independent random walks on $\mathbb{Z}^5$, all starting from the origin and consider either
\begin{itemize}
\item[$(i)$] $A=\{\mathbb{R}R_\infty^1 \cap \mathbb{R}R[k,\infty)\neq \varnothing\}, \ \text{and}\ B= \{\mathbb{R}R_\infty^2 \cap (S_k + \mathbb{R}R_\infty^3 )\neq \varnothing\}$,
\item[$(ii)$] $A=\{\mathbb{R}R_\infty^1 \cap \mathbb{R}R[k,\infty)\neq \varnothing\}, \text{ and } B= \{(S_k+\mathbb{R}R_\infty^2) \cap \mathbb{R}R[k+1,\infty)\neq \varnothing\}$,
\item[$(iii)$] $A=\{\mathbb{R}R_\infty^1 \cap \mathbb{R}R[k,\infty)\neq \varnothing\}, \ \text{and}\ B= \{(S_k+\mathbb{R}R_\infty^2) \cap \mathbb{R}R[0,k-1] \neq \varnothing\}$,
\item[$(iv)$] $A=\{\mathbb{R}R_\infty^1 \cap \mathbb{R}R[1,k] \neq \varnothing\}, \ \text{and}\ B= \{(S_k+\mathbb{R}R_\infty^2) \cap \mathbb{R}R[0,k-1] \neq \varnothing\}$.
\end{itemize}
In all these cases, we show that for some constant $c>0$, as $k\to \infty$,
$$\operatorname{Cov}(A,B) \sim \frac{c}{k}. $$
Case $(i)$ is the easiest, and follows directly from Theorem C, since actually one can see that in this case both $\mathbb{P}[A\cap B]$ and $\mathbb{P}[A]\cdot \mathbb{P}[B]$ are asymptotically equivalent to a constant times the inverse of $k$. However, the other cases are more intricate, partly due to some cancellations that occur between the two terms,
which, if estimated separately, are both of order $1/\sqrt{k}$ in cases $(ii)$ and $(iii)$, or even of order $1$ in case $(iv)$. In these cases, we rely on the extensions of Theorem C, that we just mentioned above.
More precisely in case $(ii)$ we rely on the general result applied
with the functions $F(x)=1/\|x\|$, and its convolution with the distribution of $S_k$, while in cases $(iii)$ and $(iv)$ we use the extension to hitting times of
finite windows of the range. We stress also that showing the positivity of the constants $c$ here is a delicate part of the proof, especially in case $(iv)$, where it relies
on the following inequality:
$$\int_{0\le s \le t\le 1}\left( \mathbb{E}\left[\frac{1}{\|\beta_s-\beta_1\|^3\cdot \|\beta_t\|^3}\right] -\mathbb{E}\left[\frac{1}{\|\beta_s-\beta_1\|^3}\right] \mathbb{E}\left[\frac{1}{\|\beta_t\|^3}\right]\right) \, ds\, dt>0,$$
with $(\beta_u)_{u\ge 0}$ a standard Brownian motion in $\mathbb{R}^5$.
The paper is organized as follows. The next section is devoted to preliminaries, in particular we fix the main notation, recall known results on the transition kernel
and the Green's function, and derive some basic estimates. In Section 3 we give the plan of the proof of Theorem A, which is cut into a number of intermediate results:
Propositions \ref{prop.error}--\ref{prop.phipsi.2}. Propositions \ref{prop.error}--\ref{prop.phi0} are then proved in Sections $4$--$6$.
The last one, which is also the most delicate one, requires
Theorem C and its extensions. Its proof is therefore postponed to Section 8, while we first prove our general results on the intersection of two independent ranges in Section $7$, which is written in the general setting of random walks on $\mathbb{Z}^d$, for any $d\ge 5$, and can be read independently of the rest of the paper.
Finally Section 9 is devoted to the proof
of Theorem B, which is done by following a relatively
well-established general scheme, based on the Lindeberg-Feller theorem for triangular arrays.
\section{Preliminaries}
\subsection{Notation}
We recall that we assume the law of the $(X_i)_{i\ge 1}$ to be a symmetric and irreducible probability measure\footnote{symmetric means that for all $x\in \mathbb{Z}^d$, $\mathbb{P}[X_1=x]=\mathbb{P}[X_1=-x]$, and irreducible means that for all $x$,
$\mathbb{P}[S_n=x]>0$, for some $n\ge 1$.} on $\mathbb{Z}^d$, $d\ge 5$, with a finite $d$-th moment\footnote{this means that $\mathbb{E}[\|X_1\|^d]<\infty$, with $\|\cdot \|$ the Euclidean norm.}.
The walk is called aperiodic if the probability to be at the origin at time $n$ is nonzero for all $n$ large enough, and it is called bipartite if this probability is nonzero only when $n$ is even.
Note that only these two cases may appear for a symmetric random walk.
Recall also that for $x\in \mathbb{Z}^d$, we denote by $\mathbb{P}_x$ the law of the walk starting from $S_0=x$. When $x=0$, we simply write it as $\mathbb{P}$.
We denote its total range as $\mathbb{R}R_\infty :=\{S_k\}_{k\ge 0}$, and for $0\le k\le n\le +\infty$, set $\mathbb{R}R[k,n]:=\{S_k,\dots,S_n\}$.
For an integer $k\ge 2$, the law of $k$ independent random walks (with the same step distribution) starting from some $x_1,\dots, x_k\in \mathbb{Z}^5$, is denoted by
$\mathbb{P}_{x_1,\dots,x_k}$, or simply by $\mathbb{P}$ when they all start from the origin.
We define
\begin{equation}\label{HAHA+}
H_A:=\inf\{n\ge 0\ : \ S_n\in A\},\quad \text{and} \quad H_A^+ :=\inf\{n\ge 1\ :\ S_n\in A\},
\end{equation}
respectively for the hitting time and first return time to a subset $A\subset \mathbb{Z}^d$, that we abbreviate respectively as $H_x$ and $H_x^+$ when $A$ is a singleton $\{x\}$.
We let $\|x\|$ be the Euclidean norm of $x\in \mathbb{Z}^d$.
If $X_1$ has covariance matrix $\Gamma= \Lambda \Lambda^t$, we define its associated norm as
$$\mathcal J^*(x) := |x\cdot \Gamma^{-1} x|^{1/2} = \|\Lambda^{-1} x\|,$$
and set $\mathcal J(x)= d^{-1/2}\mathcal J^*(x)$ (see \cite{LL} p.4 for more details).
For $a$ and $b$ some nonnegative reals, we let $a\wedge b:=\min(a,b)$ and $a\vee b:= \max(a,b)$.
We use the letters $c$ and $C$ to denote constants (which could depend on the covariance matrix of the walk), whose values might change from line to line. We also use standard notation for the comparison of functions: we write $f=\mathcal{O}(g)$, or sometimes $f\lesssim g$, if there exists a constant $C>0$, such that $f(x) \le Cg(x)$, for all $x$. Likewise, $f=o(g)$ means that $f/g \to 0$, and $f\sim g$ means that $f$ and $g$ are equivalent, that is if $|f-g| = o(f)$. Finally we write $f\asymp g$, when both $f=\mathcal{O}(g)$, and $g=\mathcal{O}(f)$.
\subsection{Transition kernel and Green's function}
We denote by $p_n(x)$ the probability that a random walk starting from the origin ends up at position $x\in \mathbb{Z}^d$ after $n$ steps, that is $p_n(x):=\mathbb{P}[S_n=x]$, and note that for any $x,y\in \mathbb{Z}^d$, one has $\mathbb{P}_x[S_n=y] = p_n(y-x)$. Recall the definitions of $\Gamma$ and $\mathcal J^*$ from the previous subsection, and define
\begin{equation}\label{pnbar}
\overline p_n(x) := \frac {1}{(2\pi n)^{d/2} \sqrt{\det \Gamma}} \cdot e^{-\frac{\mathcal J^*(x)^2}{2n}}.
\end{equation}
The first tool we shall need is a local central limit theorem, roughly saying that $p_n(x)$ is well approximated by $\overline p_n(x)$, under appropriate hypotheses.
Such result has a long history, see in particular the standard books by Feller \cite{Feller} and Spitzer \cite{Spitzer}. We refer here to the more recent book of
Lawler and Limic \cite{LL}, and more precisely to their Theorem 2.3.5 in the case of an
aperiodic random walk, and to (the proof of) their Theorem 2.1.3 in the case of bipartite walks, which provide the result we need under minimal hypotheses (in particular it only requires a finite fourth-moment for $\|X_1\|$).
\begin{theorem}[{\bf Local Central Limit Theorem}]\label{LCLT}
There exists a constant $C>0$, such that for all $n\ge 1$, and all $x\in \mathbb{Z}^d$,
\begin{eqnarray*}
|p_n(x)-\overline p_n(x)| \le \frac{C}{n^{(d+2)/2}},
\end{eqnarray*}
in the case of an aperiodic walk, and for bipartite walks,
$$|p_n(x)+p_{n+1}(x)-2\overline p_n(x)| \le \frac{C}{n^{(d+2)/2}}.$$
\end{theorem}
In addition, under our hypotheses (in particular assuming $\mathbb{E}[\|X_1\|^d]<\infty$), there exists a constant $C>0$, such that for any $n\ge 1$ and any
$x\in \mathbb{Z}^d$ (see Proposition 2.4.6 in \cite{LL}),
\begin{equation}\label{pn.largex}
p_n(x)\le C\cdot \left\{
\begin{array}{ll}
n^{-d/2} & \text{if }\|x\|\le \sqrt n,\\
\|x\|^{-d} & \text{if }\|x\|>\sqrt n.
\end{array}
\right.
\end{equation}
It is also known (see the proof of Proposition 2.4.6 in \cite{LL}) that
\begin{equation}\label{norm.Sn}
\mathbb{E}[\|S_n\|^d] =\mathcal{O}(n^{d/2}).
\end{equation}
Together with the reflection principle (see Proposition 1.6.2 in \cite{LL}), and Markov's inequality, this gives that for any $n\ge 1$ and $r\ge 1$,
\begin{equation}\label{Sn.large}
\mathbb{P}\left[\max_{0\le k\le n} \|S_k\|\ge r\right] \le C \cdot \left(\frac{\sqrt n}{r}\right)^{d}.
\end{equation}
Now we define for $\ell \ge 0$, $G_\ell(x) := \sum_{n\ge \ell} p_n(x)$.
The {\bf Green's function} is the function $G:=G_0$. A union bound gives
\begin{equation}\label{Green.hit}
\mathbb{P}[x\in \mathbb{R}R[\ell,\infty)] \le G_\ell(x).
\end{equation}
By \eqref{pn.largex} there exists a constant $C>0$, such that for any $x\in \mathbb{Z}^d$, and $\ell \ge 0$,
\begin{equation}\label{Green}
G_\ell(x) \le \frac{C}{\|x\|^{d-2} + \ell^{\frac{d-2}{2}} + 1}.
\end{equation}
It follows from this bound (together with the corresponding lower bound $G(x)\ge c\|x\|^{2-d}$, which can be deduced from Theorem \ref{LCLT}), and the fact that $G$ is
harmonic on $\mathbb{Z}^d\setminus\{0\}$, that the hitting probability of a ball is bounded as follows (see the proof of \cite[Proposition 6.4.2]{LL}):
\begin{equation}\label{hit.ball}
\mathbb{P}_x\left[\eta_r<\infty \right] =\mathcal{O}\left(\frac{r^{d-2}}{1+\|x\|^{d-2}}\right), \quad \text{with}\quad \eta_r:=\inf\{n\ge 0\ :\ \|S_n\|\le r\}.
\end{equation}
We shall need as well some control on the overshoot. We state the result we need as a lemma and provide a short proof for the sake of completeness.
\begin{lemma}[{\bf Overshoot Lemma}]\label{hit.ball.overshoot}
There exists a constant $C>0$, such that for all $r\ge 1$, and all $x\in \mathbb{Z}^d$, with $\|x\|\ge r$,
\begin{equation*}
\mathbb{P}_x[\eta_r<\infty,\, \|S_{\eta_r}\| \le r/2] \le \frac{C}{1+\|x\|^{d-2}}.
\end{equation*}
\end{lemma}
\begin{proof}
We closely follow the proof of Lemma 5.1.9 in \cite{LL}. Note first that one can alway assume that $r$ is large enough, for otherwise the result follows from \eqref{hit.ball}. Then define for $k\ge 0$,
$$Y_k:= \sum_{n= 0}^{\eta_r} {\text{\Large $\mathfrak 1$}}\{r+k \le \|S_n\|< r+(k+1)\}.$$
Let
$$g(x,k) = \mathbb{E}_x[Y_k] = \sum_{n=0}^\infty \mathbb{P}_x[r+k \le \|S_n\|\le r+k+1,\, n< \eta_r].$$
One has
\begin{align*}
& \mathbb{P}_x[\eta_r<\infty, \, \|S_{\eta_r}\| \le r/2] = \sum_{n=0}^\infty \mathbb{P}_x[\eta_r=n+1, \, \|S_{\eta_r}\| \le r/2] \\
& = \sum_{n=0}^\infty \sum_{k=0}^\infty
\mathbb{P}_x[\eta_r=n+1, \, \|S_{\eta_r}\| \le r/2,\, r+k \le \|S_n\|< r+k+1]\\
& \le \sum_{k=0}^\infty \sum_{n=0}^\infty \mathbb{P}_x\left[\eta_r>n,\, r+k \le \|S_n\|\le r+k+1,\, \|S_{n+1}-S_n\| \ge \frac r2 + k\right]\\
& = \sum_{k=0}^\infty g(x,k) \mathbb{P}\left[\|X_1\| \ge \frac r2 + k \right] = \sum_{k=0}^\infty g(x,k) \sum_{\ell = k}^\infty \mathbb{P}\left[\frac r2 +\ell \le \|X_1\|< \frac r2 + \ell+1 \right]\\
& = \sum_{\ell = 0}^\infty \mathbb{P}\left[\frac r2 +\ell \le \|X_1\|< \frac r2 + \ell+1 \right]\sum_{k=0}^\ell g(x,k).
\end{align*}
Now Theorem \ref{LCLT} shows that one has $\mathbb{P}_z[\|S_{\ell^2}\| \le r]\ge \rho$, for some constant $\rho>0$, uniformly in $r$ (large enough), $\ell\ge 1$, and $r\le \|z\|\le r+\ell$. It follows, exactly as in the
proof of Lemma 5.1.9 from \cite{LL}, that for any $\ell\ge 1$,
$$\max_{\|z\|\le r+\ell} \sum_{0\le k< \ell} g(z,k) \le \frac{\ell^2}{\rho}.$$
Using in addition \eqref{hit.ball}, we get with the Markov property,
$$\sum_{0\le k< \ell} g(x,k) \le C \frac{(r+\ell)^{d-2}}{1+\|x\|^{d-2}} \cdot \ell^2,$$
for some constant $C>0$.
As a consequence one has
\begin{align*}
& \mathbb{P}_x[\eta_r<\infty, \, \|S_{\eta_r}\| \le r/2] \\
& \le \frac C{1+\|x\|^{d-2}} \sum_{\ell = 0}^\infty \mathbb{P}\left[\frac r2+ \ell \le \|X_1\|< \frac r2 +\ell + 1\right](r+\ell)^{d-2}(\ell+1) ^2\\
& \le \frac C{1+\|x\|^{d-2}} \mathbb{E}\left[\|X_1\|^{d-2}(\|X_1\| - r/2)^2{\text{\Large $\mathfrak 1$}}\{\|X_1\|\ge r/2\}\right] \le \frac{C}{1+\|x\|^{d-2}},
\end{align*}
since by hypothesis, the $d$-th moment of $X_1$ is finite.
\end{proof}
\subsection{Basic tools}
We prove here some elementary facts, which will be needed throughout the paper, and which are immediate consequences of the results from the previous subsection.
\begin{lemma}\label{lem.upconvolG}
There exists $C>0$, such that for all $x\in \mathbb{Z}^d$, and $\ell \ge 0$,
$$ \sum_{z\in \mathbb{Z}^d} G_\ell(z) G(z-x) \le \frac{C}{\|x\|^{d-4}+\ell^{\frac{d-4}{2}} + 1}.$$
\end{lemma}
\begin{proof}
Assume first that $\ell =0$. Then by \eqref{Green},
\begin{align*}
\sum_{z\in \mathbb{Z}^d} G(z) G(z-x) & \lesssim \frac{1}{1+\|x\|^{d-2}} \left(\sum_{\|z\|\le 2\|x\|} \frac{1}{1+\|z\|^{d-2}} +\sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{1}{1+\|z-x\|^{d-2}} \right)\\
& \quad + \sum_{\|z\|\ge 2\|x\|}\frac{1}{1+\|z\|^{2(d-2)}} \lesssim \frac{1}{1+\|x\|^{d-4}}.
\end{align*}
Assume next that $\ell\ge 1$. We distinguish two cases: if $\|x\|\le \sqrt \ell$, then by using \eqref{Green} again we deduce,
$$\sum_{z\in \mathbb{Z}^d} G_\ell(z) G(z-x) \lesssim \frac{1}{\ell^{d/2}}\cdot \sum_{\|z\|\le 2\sqrt \ell} \frac{1}{1+ \|z-x\|^{d-2}} + \sum_{\|z\|\ge 2\sqrt \ell} \frac 1{\|z\|^{2(d-2)}}
\lesssim \frac{1}{ \ell^{\frac{d-4}2}}.$$
When $\|x\| >\sqrt \ell$, the result follows from case $\ell =0$, since $G_\ell(z) \le G(z)$.
\end{proof}
\begin{lemma}\label{lem.sumG}
One has,
\begin{equation}\label{exp.Green}
\sup_{x\in \mathbb{Z}^d} \, \mathbb{E}[G(S_n-x)] = \mathcal{O}\left(\frac 1{n^{\frac{d-2}{2}}}\right),
\end{equation}
and for any $\alpha \in [0,d)$,
\begin{equation}\label{exp.Green.x}
\sup_{n\ge 0} \, \mathbb{E}\left[\frac 1{1+\|S_n-x\|^\alpha } \right] = \mathcal{O}\left(\frac 1{1+\|x\|^\alpha}\right).
\end{equation}
Moreover, when $d=5$,
\begin{equation} \label{sumG2}
\mathbb{E}\left[\Big(\sum_{n\ge k} G(S_n)\Big)^2\right] = \mathcal{O}\left(\frac 1k\right).
\end{equation}
\end{lemma}
\begin{proof}
For \eqref{exp.Green}, we proceed similarly as in the proof of Lemma \ref{lem.upconvolG}. If $\|x\| \le \sqrt{n}$, one has using \eqref{pn.largex} and \eqref{Green},
\begin{align*}
& \mathbb{E}[G(S_n-x)] = \sum_{z\in \mathbb{Z}^d}p_n(z) G(z-x) \\
& \lesssim \frac 1{n^{d/2}} \sum_{\|z\|\le 2\sqrt n} \frac{1}{1+\|z-x\|^{d-2}} + \sum_{\|z\|>2\sqrt n} \frac {1}{\|z\|^{2d-2}} \lesssim n^{\frac{2-d}{2}},
\end{align*}
while if $\|x\|>\sqrt{n}$, we get as well
$$\mathbb{E}[G(S_n-x)]\lesssim \frac 1{n^{d/2}} \sum_{\|z\|\le \sqrt n/2} \frac{1}{\|x\|^{d-2}} + \sum_{\|z\|>\sqrt n/2} \frac {1}{\|z\|^d(1+\|z-x\|)^{d-2}} \lesssim n^{\frac{2-d}{2}}.$$
Considering now \eqref{exp.Green.x}, we write
\begin{align*}
& \mathbb{E}\left[\frac 1{1+\|S_n-x\|^\alpha} \right] \le \frac{C}{1+\|x\|^\alpha} + \sum_{\|z-x\|\le \|x\|/2} \frac{p_n(z)}{1+\|z-x\|^\alpha} \\
& \stackrel{\eqref{pn.largex}}{\lesssim} \frac{1}{1+\|x\|^\alpha} + \frac{1}{1+\|x\|^d} \sum_{\|z-x\|\le \|x\|/2} \frac{1}{1+\|z-x\|^\alpha} \lesssim \frac{1}{1+\|x\|^\alpha}.
\end{align*}
Finally for \eqref{sumG2}, one has using the Markov property at the second line,
\begin{align*}
&\mathbb{E}\left[\Big(\sum_{n\ge k} G(S_n)\Big)^2\right] = \sum_{x,y} G(x)G(y)\mathbb{E}\left[\sum_{n,m\ge k}{\text{\Large $\mathfrak 1$}}\{S_n=x,S_m=y\}\right]\\
&\le 2\sum_{x,y} G(x)G(y)\sum_{n\ge k}\sum_{\ell \ge 0} p_n(x)p_\ell(y-x)= 2\sum_{x,y} G(x)G(y)G_k(x)G(y-x)\\
& \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_x \frac{1}{\|x\|^4}G_k(x) \stackrel{\eqref{Green}}{\lesssim} \frac 1k.
\end{align*}
\end{proof}
The next result deals with the probability that two independent ranges intersect. Despite its proof is a rather straightforward
consequence of the previous results, it already provides upper bounds of the right order (only off by a multiplicative constant).
\begin{lemma}\label{lem.simplehit}
Let $S$ and $\widetilde S$ be two independent walks starting respectively from the origin and some $x\in \mathbb{Z}^d$. Let also $\ell$ and $m$ be two given nonnegative integers (possibly infinite for $m$). Define
$$\tau:=\inf\{n\ge 0\ :\ \widetilde S_n\in \mathbb{R}R[\ell,\ell+m]\}.$$
Then, for any function $F:\mathbb{Z}^d\to \mathbb{R}_+$,
\begin{equation}\label{lem.hit.1}
\mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{\tau <\infty\}F(\widetilde S_\tau)] \le \sum_{i=\ell}^{\ell + m} \mathbb{E}[G(S_i-x)F(S_i)].
\end{equation}
In particular, uniformly in $\ell$ and $m$,
\begin{equation} \label{lem.hit.2}
\mathbb{P}_{0,x}[\tau<\infty] = \mathcal{O}\left(\frac{1}{1+\|x\|^{d-4} }\right).
\end{equation}
Moreover, uniformly in $x\in \mathbb{Z}^d$,
\begin{equation}\label{lem.hit.3}
\mathbb{P}_{0,x}[\tau<\infty] = \left\{
\begin{array}{ll}
\mathcal{O}\left(m\cdot \ell^{\frac{2-d}2}\right) & \text{if }m<\infty \\
\mathcal{O}\left(\ell^{\frac{4-d}2} \right) & \text{if }m=\infty.
\end{array}
\right.
\end{equation}
\end{lemma}
\begin{proof}
The first statement follows from \eqref{Green.hit}. Indeed using this, and the independence between $S$ and $\widetilde S$, we deduce that
\begin{align*}
\mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{\tau <\infty\}F(\widetilde S_\tau)] & \le \sum_{i=\ell}^{\ell + m} \mathbb{E}_{0,x}[{\text{\Large $\mathfrak 1$}}\{S_i\in \widetilde \mathbb{R}R_\infty \} F(S_i)] \stackrel{\eqref{Green.hit}}{\le} \sum_{i=\ell}^{\ell +m} \mathbb{E}[G(S_i-x)F(S_i)].
\end{align*}
For \eqref{lem.hit.2}, note first that it suffices to consider the case when $\ell=0$ and $m=\infty$, as otherwise the probability is just smaller. Taking now $F\equiv 1$ in \eqref{lem.hit.1}, and using Lemma \ref{lem.upconvolG} gives the result. Similarly \eqref{lem.hit.3} directly follows from \eqref{lem.hit.1} and \eqref{exp.Green}.
\end{proof}
\section{Scheme of proof of Theorem A}
\subsection{A last passage decomposition for the capacity of the range}
We provide here a last passage decomposition for the capacity of the range, in the same fashion as the well-known decomposition for the size of the range, which goes
back to the seminal paper by
Dvoretzky and Erd\'os \cite{DE51}, and which was also used by Jain and Pruitt \cite{JP71} for their proof of the central limit theorem. We note that Jain and Orey \cite{JO69} used as well a similar decomposition in their analysis of the capacity of the range (in fact they used instead a first passage decomposition).
So let $(S_n)_{n\ge 0}$ be some random walk starting from the origin, and set
$$\operatorname{Var}phi_k^n:= \mathbb{P}_{S_k}[H_{\mathbb{R}R_n}^+=\infty\mid \mathbb{R}R_n], \text{ and } Z_k^n := {\text{\Large $\mathfrak 1$}}\{S_\ell \neq S_k,\ \text{for all } \ell = k+1,\dots, n\},$$
for all $0\le k\le n$,
By definition of the capacity \eqref{cap.def}, one can write by recording the sites of $\mathbb{R}R_n$ according to their last visit,
$$\mathrm{Cap}(\mathbb{R}R_n)=\sum_{k=0}^n Z_k^n\cdot \operatorname{Var}phi_k^n.$$
A first simplification is to remove the dependance in $n$ in each of the terms in the sum. To do this, we need some additional notation: we
consider $(S_n)_{n\in \mathbb{Z}}$ a two-sided random walk starting from the origin (that is $(S_n)_{n\ge 0}$ and $(S_{-n})_{n\ge 0}$ are two independent walks starting from the origin), and denote its total range
by $\overline \mathbb{R}R_\infty :=\{S_n\}_{n\in \mathbb{Z}}$. Then for $k\ge 0$, let
$$\operatorname{Var}phi(k):=\mathbb{P}_{S_k}[H_{\overline \mathbb{R}R_\infty}^+ = \infty \mid (S_n)_{n\in \mathbb{Z}}], \text{ and } Z(k):= {\text{\Large $\mathfrak 1$}}\{S_\ell \neq S_k,\text{ for all }\ell \ge k+1 \}.$$
We note that $\varphi(k)$ can be zero with nonzero probability, but that $\mathbb{E}[\varphi(k)]\neq 0$ (see the proof of Theorem 6.5.10 in \cite{LL}). We then define
$$\mathcal C_n : = \sum_{k=0}^n Z(k)\operatorname{Var}phi(k),\quad \text{ and }\quad W_n:=\mathrm{Cap}(\mathbb{R}R_n) - \mathcal C_n.$$
We will prove in a moment the following estimate.
\begin{lemma} \label{lem.Wn} One has
$$\mathbb{E}[W_n^2] = \mathcal O(n).$$
\end{lemma}
Given this result, Theorem A reduces to an estimate of the variance of $\mathcal C_n$.
To this end, we first observe that
$$\operatorname{Var}(\mathcal C_n) = 2 \sum_{0\le \ell < k\le n} \operatorname{Cov}( Z(\ell)\operatorname{Var}phi(\ell), Z(k)\operatorname{Var}phi(k)) + \mathcal{O}(n).$$
Furthermore, by translation invariance, for any $\ell < k$,
$$\operatorname{Cov}( Z(\ell)\operatorname{Var}phi(\ell), Z(k)\operatorname{Var}phi(k)) = \operatorname{Cov}(Z(0)\operatorname{Var}phi(0), Z(k-\ell)\operatorname{Var}phi(k-\ell)),$$
so that in fact
$$\operatorname{Var}(\mathcal C_n) = 2\sum_{\ell = 1}^n \sum_{k=1}^{\ell} \operatorname{Cov}( Z(0)\operatorname{Var}phi(0), Z(k)\operatorname{Var}phi(k)) + \mathcal{O}(n).$$
Thus Theorem A is a direct consequence of the following theorem.
\begin{theorem}\label{prop.cov}
There exists a constant $\sigma>0$, such that
\begin{equation*}
\operatorname{Cov}( Z(0)\operatorname{Var}phi(0), Z(k)\operatorname{Var}phi(k)) \sim \frac{\sigma^2}{2k}.
\end{equation*}
\end{theorem}
This result is the core of the paper, and uses in particular Theorem C (in fact some more general statement, see Theorem \ref{thm.asymptotic}).
More details about its proof will be given in the next subsection, but first we
show that $W_n$ is negligible by giving the proof of Lemma \ref{lem.Wn}.
\begin{proof}[Proof of Lemma \ref{lem.Wn}]
Note that $W_n=W_{n,1} + W_{n,2}$, with
$$W_{n,1} = \sum_{k=0}^n (Z_k^n - Z(k))\operatorname{Var}phi_k^n, \quad \text{and}\quad W_{n,2} = \sum_{k=0}^n(\operatorname{Var}phi_k^n- \operatorname{Var}phi(k)) Z(k).$$
Consider first the term $W_{n,1}$ which is easier. Observe that $Z_k^n-Z(k)$ is nonnegative and bounded by the indicator function of the event
$\{S_k\in \mathbb{R}R[n+1,\infty)\}$. Bounding also $\operatorname{Var}phi_k^n$ by one, we get
\begin{align*}
\mathbb{E}[W_{n,1}^2] & \le \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{E}[(Z_\ell^n-Z(\ell))(Z_k^n -Z(k))] \\
&\le \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{P}\left[S_\ell \in \mathbb{R}R[n+1,\infty), \, S_k\in \mathbb{R}R[n+1,\infty)\right].
\end{align*}
Then noting that $(S_{n+1-k}-S_{n+1})_{k\ge 0}$ and $(S_{n+1+k}-S_{n+1})_{k\ge 0}$ are two independent random walks starting from the origin, we obtain
\begin{align*}
\mathbb{E}[W_{n,1}^2] & \le \sum_{\ell =1}^{n+1} \sum_{k=1}^{n+1} \mathbb{P}[H_{S_\ell} <\infty, \, H_{S_k}<\infty]\le 2 \sum_{\ell = 1}^{n+1} \sum_{k=1}^{n+1} \mathbb{P}[H_{S_\ell} \le H_{S_k} <\infty] \\
&\le 2 \sum_{1\le \ell \le k\le n+1} \mathbb{P}[H_{S_\ell} \le H_{S_k} <\infty] + \mathbb{P}[H_{S_k} \le H_{S_\ell} <\infty].
\end{align*}
Using next the Markov property and \eqref{Green.hit}, we get with $S$ and $\widetilde S$ two independent random walks starting from the origin,
\begin{align*}
\mathbb{E}[W_{n,1}^2] & \le 2 \sum_{1\le \ell \le k\le n+1} \mathbb{E}[G(S_\ell) G(S_k-S_\ell)] + \mathbb{E}[G(S_k) G(S_k- S_\ell)]\\
&\le 2 \sum_{\ell = 1}^{n+1} \sum_{k=0}^n \mathbb{E}[G(S_\ell)] \cdot \mathbb{E}[G(S_k)] + \mathbb{E}[G(S_\ell + \widetilde S_k) G(\widetilde S_k)]\\
&\le 4 \left(\sup_{x\in \mathbb{Z}^5} \sum_{\ell \ge 0} \mathbb{E}[G(x+S_\ell)]\right)^2 \stackrel{\eqref{exp.Green}}{=} \mathcal O(1).
\end{align*}
We proceed similarly with $W_{n,2}$. Observe first that for any $k\ge 0$,
$$0 \le \operatorname{Var}phi_k^n - \operatorname{Var}phi(k) \le \mathbb{P}_{S_k}[H_{\mathbb{R}R(-\infty,0]}<\infty\mid S] + \mathbb{P}_{S_k}[H_{\mathbb{R}R[n,\infty)}<\infty\mid S].$$
Furthermore, for any $0\le \ell \le k\le n$, the two terms $\mathbb{P}_{S_\ell}[H_{\mathbb{R}R(-\infty,0]}<\infty\mid S]$ and $\mathbb{P}_{S_k}[H_{\mathbb{R}R[n,\infty)}<\infty\mid S]$ are independent.
Therefore,
\begin{align}\label{Wn2}
\nonumber \mathbb{E}[W_{n,2}^2] \le & \sum_{\ell = 0}^n \sum_{k=0}^n \mathbb{E}[(\operatorname{Var}phi_\ell^n-\operatorname{Var}phi(\ell))(\operatorname{Var}phi_k^n-\operatorname{Var}phi(k))] \le 2\left(\sum_{\ell = 0}^n \mathbb{P}\left[ H_{\mathbb{R}R[\ell, \infty)}<\infty\right]\right)^2 \\
& + 4 \sum_{0\le \ell \le k\le n} \mathbb{P}\left[\mathbb{R}R^3_\infty \cap (S_\ell + \mathbb{R}R^1_\infty) \neq \varnothing, \, \mathbb{R}R^3_\infty \cap (S_k+\mathbb{R}R_\infty^2)\neq \varnothing \right],
\end{align}
where in the last term $\mathbb{R}R^1_\infty$, $\mathbb{R}R^2_\infty$ and $\mathbb{R}R^3_\infty$ are the ranges of three (one-sided) independent walks, independent of $(S_n)_{n\ge 0}$, starting from the origin (denoting here $(S_{-n})_{n\ge 0}$ as another walk $(S^3_n)_{n\ge 0}$).
Now \eqref{lem.hit.3} already shows that the first term on the right hand side of \eqref{Wn2} is $\mathcal{O}(n)$.
For the second one, note that for any $0\le \ell \le k\le n$, one has
\begin{align*}
&\mathbb{P}\left[\mathbb{R}R^3_\infty \cap (S_\ell + \mathbb{R}R^1_\infty) \neq \varnothing, \, \mathbb{R}R^3_\infty \cap (S_k+\mathbb{R}R_\infty^2)\neq \varnothing \right]\\
\le & \
\mathbb{E}\left[|\mathbb{R}R^3_\infty \cap (S_\ell + \mathbb{R}R^1_\infty)| \cdot | \mathbb{R}R^3_\infty \cap (S_k+\mathbb{R}R_\infty^2)| \right]\\
= &\ \mathbb{E}\left[\mathbb{E}[|\mathbb{R}R^3_\infty \cap (S_\ell + \mathbb{R}R^1_\infty)|\mid S,\, S^3] \cdot \mathbb{E}[| \mathbb{R}R^3_\infty \cap (S_k+\mathbb{R}R_\infty^2)|\mid S,\, S^3] \right] \\
\stackrel{\eqref{Green.hit}}{\le} & \mathbb{E}\left[ \Big(\sum_{m\ge 0} G(S^3_m - S_\ell) \Big) \Big(\sum_{m\ge 0} G(S^3_m - S_k) \Big) \right] = \mathbb{E}\left[ \Big(\sum_{m\ge k} G(S_m - S_{k-\ell}) \Big) \Big(\sum_{m\ge k} G(S_m) \Big) \right]\\
\le & \ \mathbb{E}\left[ \Big(\sum_{m\ge \ell} G(S_m) \Big)^2\right]^{1/2} \cdot \mathbb{E}\left[ \Big(\sum_{m\ge k} G(S_m)\Big)^2\right]^{1/2} \stackrel{\eqref{sumG2}}{=} \mathcal{O}\left(\frac{1}{1+\sqrt{k\ell}}\right),
\end{align*}
using invariance by time reversal at the penultimate line, and Cauchy-Schwarz at the last one.
This concludes the proof of the lemma.
\end{proof}
\subsection{Scheme of proof of Theorem \ref{prop.cov}}
We provide here some decomposition of $\varphi(0)$ and $\varphi(k)$ into a sum of terms involving intersection and non-intersection probabilities of different parts of the path $(S_n)_{n\in \mathbb{Z}}$.
For this, we consider some sequence of integers $(\operatorname{Var}epsilon_k)_{k\ge 1}$ satisfying $k>2\varepsilon_k$, for all $k\ge 3$, and whose value will be fixed later.
A first step in our analysis is to reduce the influence of the random variables $Z(0)$ and $Z(k)$, which play a
very minor role in the whole proof.
Thus we define
$$Z_0:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq 0,\, \forall \ell=1,\dots,\operatorname{Var}epsilon_k\}, \text{ and }
Z_k:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq S_k, \, \forall \ell =k+1,\dots,k+\operatorname{Var}epsilon_k\}.$$
Note that these notation are slightly misleading (as in fact $Z_0$ and $Z_k$ depend on $\varepsilon_k$, but this shall hopefully not cause any confusion).
One has
$$\mathbb{E}[|Z(0)- Z_0|]= \mathbb{P}[0\in \mathbb{R}R[\operatorname{Var}epsilon_k+1,\infty)] \stackrel{\eqref{Green.hit}}{\le} G_{\operatorname{Var}epsilon_k}(0) \stackrel{\eqref{Green}}{=} \mathcal{O}(\operatorname{Var}epsilon_k^{-3/2}),$$
and the same estimate holds for $\mathbb{E}[|Z(k)-Z_k|]$, by the Markov property. Therefore,
$$\operatorname{Cov}(Z(0)\operatorname{Var}phi(0),Z(k)\operatorname{Var}phi(k)) = \operatorname{Cov}(Z_0\operatorname{Var}phi(0),Z_k\operatorname{Var}phi(k)) + \mathcal{O}(\operatorname{Var}epsilon_k^{-3/2}).$$
Then recall that we consider a two-sided walk $(S_n)_{n\in \mathbb{Z}}$, and that $\operatorname{Var}phi(0) = \mathbb{P}[H_{\mathbb{R}R(-\infty, \infty)}^+=\infty\mid S]$. Thus one can decompose $\operatorname{Var}phi(0)$ as follows:
$$\operatorname{Var}phi(0) =\operatorname{Var}phi_0- \operatorname{Var}phi_1- \operatorname{Var}phi_2-\operatorname{Var}phi_3 + \operatorname{Var}phi_{1,2} +\operatorname{Var}phi_{1,3} + \operatorname{Var}phi_{2,3} - \operatorname{Var}phi_{1,2,3},$$
with
$$\operatorname{Var}phi_0:=\mathbb{P}[H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty\mid S],\quad \operatorname{Var}phi_1: = \mathbb{P}[H^+_{\mathbb{R}R(-\infty,-\operatorname{Var}epsilon_k-1]}<\infty, \, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S], $$
$$\operatorname{Var}phi_2 := \mathbb{P}[H^+_{\mathbb{R}R[\operatorname{Var}epsilon_k+1,k]}<\infty, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S], \, \operatorname{Var}phi_3 := \mathbb{P}[H^+_{\mathbb{R}R[k+1,\infty)}<\infty , H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty\mid S],$$
$$\operatorname{Var}phi_{1,2}: = \mathbb{P}[H^+_{\mathbb{R}R(-\infty,-\varepsilon_k-1]}<\infty , \, H^+_{\mathbb{R}R[\varepsilon_k+1,k]}<\infty , \, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S], $$
$$\operatorname{Var}phi_{1,3} := \mathbb{P}[H^+_{\mathbb{R}R(-\infty,-\varepsilon_k-1]}<\infty , \, H^+_{\mathbb{R}R[k+1,\infty)}<\infty, \, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S],$$
$$\operatorname{Var}phi_{2,3} := \mathbb{P}[H^+_{\mathbb{R}R[\varepsilon_k+1,k]}<\infty,\, H^+_{\mathbb{R}R[k+1,\infty)}<\infty, \, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S],$$
$$\operatorname{Var}phi_{1,2,3} := \mathbb{P}[H^+_{\mathbb{R}R(-\infty, -\operatorname{Var}epsilon_k-1]}<\infty, H^+_{\mathbb{R}R[\operatorname{Var}epsilon_k+1,k]}<\infty, H^+_{\mathbb{R}R[k+1,\infty)}<\infty, H_{\mathbb{R}R[-\operatorname{Var}epsilon_k,\operatorname{Var}epsilon_k]}^+=\infty \mid S].$$
We decompose similarly
$$\operatorname{Var}phi(k)=\psi_0 - \psi_1 - \psi_2 - \psi_3 + \psi_{1,2} + \psi_{1,3} + \psi_{2,3} - \psi_{1,2,3},$$
where index $0$ refers to the event of avoiding $\mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k]$, index $1$ to the event of hitting $\mathbb{R}R(-\infty,-1]$, index $2$ to the event of hitting $\mathbb{R}R[0,k-\varepsilon_k-1]$ and index $3$ to the event of hitting $\mathbb{R}R[k+\varepsilon_k+1,\infty)$ (for a walk starting from $S_k$ this time).
Note that $\operatorname{Var}phi_0$ and $\psi_0$ are independent. Then write
\begin{align}\label{main.dec}
& \operatorname{Cov}(Z_0\operatorname{Var}phi(0),Z_k\operatorname{Var}phi(k)) = -\sum_{i=1}^3 \left(\operatorname{Cov}(Z_0\operatorname{Var}phi_i,Z_k\psi_0)+ \operatorname{Cov}(Z_0\operatorname{Var}phi_0,Z_k\psi_i)\right) \\
\nonumber & + \sum_{i,j=1}^3 \operatorname{Cov}(Z_0\operatorname{Var}phi_i , Z_k\psi_j) +\sum_{1\le i <j\le 3} \left(\operatorname{Cov}(Z_0\varphi_{i,j},Z_k\psi_0) + \operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{i,j})\right)+ R_{0,k},
\end{align}
where $R_{0,k}$ is an error term.
Our first task will be to show that it is negligible.
\begin{proposition}\label{prop.error}
One has $|R_{0,k}| = \mathcal{O}\left(\varepsilon_k^{-3/2}\right)$.
\end{proposition}
The second step is the following.
\begin{proposition}\label{prop.ij0}
One has
\begin{itemize}
\item[(i)]
$|\operatorname{Cov}(Z_0\varphi_{1,2},Z_k\psi_0)|+|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{2,3})| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right)$,
\item[(ii)] $|\operatorname{Cov}(Z_0\varphi_{1,3},Z_k\psi_0)| + |\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{1,3})| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}) + \frac{1}{\varepsilon_k^{3/4} \sqrt k}\right)$,
\item[(iii)]
$|\operatorname{Cov}(Z_0\varphi_{2,3},Z_k\psi_0)| +|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_{1,2})|= \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k})+ \frac{1}{\varepsilon_k^{3/4} \sqrt k}\right)$.
\end{itemize}
\end{proposition}
In the same fashion as Part (i) of the previous proposition, we show:
\begin{proposition}\label{prop.phipsi.1}
For any $1\le i<j\le 3$,
$$|\operatorname{Cov}(Z_0\varphi_i,Z_k\psi_j)| =\mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right), \quad |\operatorname{Cov}(Z_0\varphi_j,Z_k\psi_i)| =\mathcal{O}\left(\frac{1}{\varepsilon_k}\right).$$
\end{proposition}
The next step deals with the first sum in the right-hand side of \eqref{main.dec}.
\begin{proposition}\label{prop.phi0}
There exists a constant $\alpha\in (0,1)$, such that
$$\operatorname{Cov}(Z_0\operatorname{Var}phi_1, Z_k\psi_0) = \operatorname{Cov}(Z_0\operatorname{Var}phi_0,Z_k\psi_3) = 0,$$
$$|\operatorname{Cov}(Z_0\operatorname{Var}phi_2,Z_k\psi_0)|+ |\operatorname{Cov}(Z_0\operatorname{Var}phi_0,Z_k\psi_2)| = \mathcal{O}\left(\frac{\sqrt{\operatorname{Var}epsilon_k}}{k^{3/2}} \right),$$
$$|\operatorname{Cov}(Z_0\operatorname{Var}phi_3,Z_k\psi_0)|+ |\operatorname{Cov}(Z_0\operatorname{Var}phi_0,Z_k\psi_1)| = \mathcal{O}\left(\frac{ \operatorname{Var}epsilon_k^\alpha}{k^{1+\alpha}} \right).$$
\end{proposition}
At this point one can already deduce the bound $\operatorname{Var}(\mathrm{Cap}(\mathbb{R}R_n))= \mathcal{O}(n \log n)$, just applying the previous propositions with say $\varepsilon_k:=\lfloor k/4\rfloor$.
In order to obtain the finer asymptotic result stated in Theorem \ref{prop.cov}, it remains to identify the leading terms in \eqref{main.dec},
which is the most delicate part. The result reads as follows.
\begin{proposition}\label{prop.phipsi.2}
There exists $\delta>0$, such that if $\varepsilon_k\ge k^{1-\delta}$ and $\varepsilon_k = o(k)$, then for some positive constants $(\sigma_{i,j})_{1\le i\le j\le 3}$,
$$\operatorname{Cov}(Z_0\varphi_j,Z_k\psi_i)\sim \operatorname{Cov}(Z_0\varphi_{4-i},Z_k\psi_{4-j}) \sim \frac{\sigma_{i,j}}{k}.$$
\end{proposition}
Note that Theorem \ref{prop.cov} is a direct consequence of \eqref{main.dec} and Propositions \ref{prop.error}--\ref{prop.phipsi.2}, which we prove now in the following sections.
\section{Proof of Proposition \ref{prop.error}}
We divide the proof into two lemmas.
\begin{lemma}\label{lem.123}
One has
$$\mathbb{E}[\operatorname{Var}phi_{1,2,3}] = \mathcal{O}\left(\frac{1}{\operatorname{Var}epsilon_k \sqrt k}\right),\quad \text{and}\quad \mathbb{E}[\psi_{1,2,3}] = \mathcal{O}\left(\frac{1}{\varepsilon_k\sqrt k}\right).$$
\end{lemma}
\begin{lemma}\label{lem.ijl}
For any $1\le i<j\le 3$, and any $1\le \ell\le 3$,
$$ \mathbb{E}[\operatorname{Var}phi_{i,j}\psi_\ell] =\mathcal{O}\left(\varepsilon_k^{-3/2} \right), \quad \text{and} \quad \mathbb{E}[\operatorname{Var}phi_{i,j}] \cdot \mathbb{E}[\psi_\ell] =\mathcal{O}\left(\varepsilon_k^{-3/2} \right) .$$
\end{lemma}
Observe that the $(\operatorname{Var}phi_{i,j})_{i,j}$ and $(\psi_{i,j})_{i,j}$ have the same law (up to reordering), and similarly for the $(\operatorname{Var}phi_i)_i$ and $(\psi_i)_{i}$.
Furthermore, $\operatorname{Var}phi_{i,j}\le \operatorname{Var}phi_i$ for any $i,j$.
Therefore by definition of $R_{0,k}$ the proof of Proposition \ref{prop.error} readily follows from these two lemmas.
For their proofs, we will use the following fact.
\begin{lemma}\label{lem.prep.123}
There exists $C>0$, such that for any $x,y\in \mathbb{Z}^5$, $0\le \ell \le m$,
\begin{equation*}
\sum_{i=\ell}^m \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \le
\frac{C}{(1+\|x\|+\sqrt m)^5} \left(\frac 1{1+\|y-x\|} + \frac{1}{1+\sqrt{\ell}+\|y\|}\right).
\end{equation*}
\end{lemma}
\begin{proof}
Consider first the case $\|x\|\le \sqrt m$. By \eqref{pn.largex} and Lemma \ref{lem.upconvolG},
$$
\sum_{i=\ell}^{\lfloor m/2\rfloor } \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{1+m^{5/2}} \sum_{z\in \mathbb{Z}^5} G_{\ell}(z) G(z-y)
\lesssim \frac{(1+ m)^{-5/2}}{1+\sqrt{\ell}+\|y\|}, $$
with the convention that the first sum is zero when $m<2\ell$, and
$$\sum_{i=\lfloor m/2\rfloor }^m \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{1+m^{5/2}} \sum_{z\in \mathbb{Z}^5} G(z-y) G(z-x)
\lesssim \frac{(1+m)^{-5/2}} {1+\|y-x\|}. $$
Likewise, when $\|x\|>\sqrt m$, applying again \eqref{pn.largex} and Lemma \ref{lem.upconvolG}, we get
\begin{align*}
& \sum_{i=\ell}^{m} \sum_{\|z-x\| \ge \frac{\|x\|}{2}} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{\|x\|^5} \sum_{z\in \mathbb{Z}^5} G_{\ell}(z)G(z-y) \lesssim \frac{\|x\|^{-5}}{ 1+\sqrt{\ell}+\|y\|},\\
& \sum_{i=\ell}^{m} \sum_{\|z-x\| \le \frac{\|x\|}{2}} p_i(z) G(z-y) p_{m-i}(z-x) \lesssim \frac{1}{\|x\|^5} \sum_{z\in \mathbb{Z}^5} G(z-y)G(z-x) \lesssim \frac{\|x\|^{-5}}{1+\|y-x\|},
\end{align*}
which concludes the proof of the lemma.
\end{proof}
One can now give the proof of Lemma \ref{lem.123}.
\begin{proof}[Proof of Lemma \ref{lem.123}]
Since $\operatorname{Var}phi_{1,2,3}$ and $\psi_{1,2,3}$ have the same law, it suffices to prove the result for $\operatorname{Var}phi_{1,2,3}$. Let $(S_n)_{n\in \mathbb{Z}}$ and $(\widetilde S_n)_{n\ge 0}$ be two independent random walks starting from the origin. Define
$$\tau_1:=\inf\{n\ge 1\, :\, \widetilde S_n \in \mathbb{R}R(-\infty,-\operatorname{Var}epsilon_k-1]\},\ \tau_2:=\inf\{n\ge 1\, :\, \widetilde S_n \in \mathbb{R}R[\operatorname{Var}epsilon_k+1,k]\},$$
and
$$\tau_3:= \inf\{n\ge 1\, :\, \widetilde S_n\in \mathbb{R}R[k+1,\infty)\}.$$
One has
\begin{equation}\label{phi123.tauij}
\mathbb{E}[\varphi_{1,2,3}] \le \sum_{i_1\neq i_2 \neq i_3} \mathbb{P}[\tau_{i_1}\le \tau_{i_2}\le \tau_{i_3}].
\end{equation}
We first consider the term corresponding to $i_1=1$, $i_2=2$, and $i_3=3$.
One has by the Markov property,
\begin{align*}
\mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2<\infty\}}{1+\|\widetilde S_{\tau_2} - S_k\|}\right]\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{i=\operatorname{Var}epsilon_k}^k \mathbb{E}\left[ \frac{G(S_i-\widetilde S_{\tau_1}){\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\| S_i - S_k\|}\right].
\end{align*}
Now define $\mathcal{G}_i:=\sigma((S_j)_{j\le i})\vee \sigma((\widetilde S_n)_{n\ge 0})$, and note that $\tau_1$ is $\mathcal{G}_i$-measurable for any $i\ge 0$.
Moreover, the Markov property and \eqref{pn.largex} show that
$$\mathbb{E}\left[\frac{1}{1+\|S_i - S_k\|}\mid \mathcal{G}_i\right] \lesssim \frac{1}{\sqrt{k-i}}.$$
Therefore,
\begin{align*}
& \mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \lesssim \sum_{i=\operatorname{Var}epsilon_k}^k \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\cdot \frac{G(S_i-\widetilde S_{\tau_1})}{1+\sqrt{k-i}}\right]\\
& \lesssim \sum_{z\in \mathbb{Z}^5}\mathbb{P}[\tau_1<\infty, \, \widetilde S_{\tau_1}=z] \cdot\left( \sum_{i=\varepsilon_k}^{k/2} \frac{\mathbb{E}[G(S_i-z)]}{\sqrt k} + \sum_{i=k/2}^k \frac{\mathbb{E}[G(S_i-z)]}{1+\sqrt{k-i}}\right)\\
& \stackrel{\eqref{exp.Green}}{\lesssim} \frac{1}{\sqrt{k\varepsilon_k}}\cdot \mathbb{P}[\tau_1<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim}\frac 1{\varepsilon_k\sqrt k}.
\end{align*}
We consider next the term corresponding to $i_1=1$, $i_2=3$ and $i_3=2$, whose analysis slightly differs from the previous one.
First Lemma \ref{lem.prep.123} gives
\begin{align}\label{tau132}
& \mathbb{P}[\tau_1\le \tau_3\le \tau_2<\infty] =\sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty, \widetilde S_{\tau_3}=y, S_k=x\} \sum_{i=\varepsilon_k}^k G(S_i-y) \right]\\
\nonumber = & \sum_{x,y\in \mathbb{Z}^5} \left(\sum_{i=\varepsilon_k}^k \sum_{z\in \mathbb{Z}^5} p_i(z)G(z-y) p_{k-i}(x-z)\right) \mathbb{P}\left[\tau_1\le \tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x\right]\\
\nonumber \lesssim & \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\| + \sqrt k)^5}\left(\frac{\mathbb{P}[\tau_1\le \tau_3 <\infty\mid S_k=x]}{\sqrt{\varepsilon_k}} +
\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right] \right).
\end{align}
We then have
\begin{align*}
& \mathbb{P}[\tau_1\le \tau_3<\infty\mid S_k=x] \stackrel{\eqref{lem.hit.2}}{\lesssim}
\mathbb{E}\left[ \frac { {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\|\widetilde S_{\tau_1}-x\|}\right] \\
&\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{y\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \frac{1}{(1+\|x\|) \sqrt \varepsilon_k} + \sum_{\|y-x\|\le \frac{\|x\|}{2}}\frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} .
\end{align*}
Moreover, when $\|x\|\ge \sqrt {\operatorname{Var}epsilon_k}$, one has
\begin{align*}
\sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\eqref{Green}}{\lesssim} \frac{1}{\|x\|^6} \sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{1}{1+\|y-x\|}
\lesssim \frac{1}{\|x\|^2},
\end{align*}
while, when $\|x\|\le \sqrt {\operatorname{Var}epsilon_k}$,
$$\sum_{\|y-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|} \stackrel{\eqref{Green}}{\lesssim} (1+\|x\|) \varepsilon_k^{-3/2} \lesssim \frac {1}{\operatorname{Var}epsilon_k}.$$
Therefore, it holds for any $x$,
\begin{equation}\label{tau132.1}
\mathbb{P}[\tau_1\le \tau_3<\infty\mid S_k=x] \lesssim \frac{1}{(1+\|x\|)\sqrt \operatorname{Var}epsilon_k}.
\end{equation}
Similarly, one has
\begin{align}\label{tau132.2}
\nonumber & \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right] \le \mathbb{E}\left[ \sum_{y\in \mathbb{Z}^5} \frac{G(y-\widetilde S_{\tau_1}) G(y-x)}{1+\|y-x\|}{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\right]\\
& \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+\|\widetilde S_{\tau_1} - x\|^2}\right] \le \sum_{y\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(y) G(y)}{1+\|y-x\|^2}
\lesssim \frac{1}{(1+\|x\|^2)\sqrt{\varepsilon_k}}.
\end{align}
Injecting \eqref{tau132.1} and \eqref{tau132.2} into \eqref{tau132} finally gives
$$\mathbb{P}[\tau_1\le \tau_2\le \tau_3<\infty] \lesssim \frac{1}{\varepsilon_k\sqrt k}.$$
The other terms in \eqref{phi123.tauij} are entirely similar, so this concludes the proof of the lemma.
\end{proof}
For the proof of Lemma \ref{lem.ijl}, one needs some additional estimates that we state as two separate lemmas.
\begin{lemma}\label{lem.prep.ijl}
There exists a constant $C>0$, such that for any $x,y\in \mathbb{Z}^5$,
\begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k}\sum_{z\in \mathbb{Z}^5} & \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|} + \frac 1{\sqrt{k-i}}\right) \\
& \le C\cdot \left\{
\begin{array}{ll}
\frac{1}{k^{5/2}}\left( \frac 1{1+\|x\|^2} + \frac{1}{\operatorname{Var}epsilon_k}\right) + \frac{1}{k^{3/2}\varepsilon_k^{3/2}(1+\|y-x\|)}&\quad \text{if }\|x\|\le \sqrt k\\
\frac{1}{\|x\|^5\varepsilon_k}\left(1+\frac{k}{\sqrt{\varepsilon_k}(1+\|y-x\|)} \right) &\quad \text{if }\|x\|>\sqrt k.
\end{array}
\right.
\end{align*}
\end{lemma}
\begin{proof}
We proceed similarly as for the proof of Lemma \ref{lem.prep.123}. Assume first that $\|x\|\le \sqrt k$. On one hand, using Lemma \ref{lem.upconvolG}, we get
$$\sum_{i=\varepsilon_k}^{k/2} \frac{1}{\sqrt{k-i}} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\lesssim \frac{1}{k^3} \sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) G(z-y)
\lesssim \frac{1}{k^{5/2}\sqrt{k\varepsilon_k}},$$
and,
\begin{align*}
&\sum_{i=\varepsilon_k}^{k/2} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5(1+\|z-x\|)} \lesssim \frac{1}{k^{5/2}}\sum_{z\in \mathbb{Z}^5} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|}\\
&\lesssim \frac{1}{k^{5/2}} \left(\sum_{\|z-x\|\ge \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|} + \sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{G_{\varepsilon_k}(z)G(z-y)}{1+\|z-x\|}\right) \\
& \lesssim \frac{1}{k^{5/2}} \left(\frac{1}{(1+\|x\|)\sqrt{\varepsilon_k}} + \frac{1}{1+\|x\|^2}\right) \lesssim \frac{1}{k^{5/2}} \left(\frac{1}{1+\|x\|^2}+ \frac 1{\varepsilon_k}\right).
\end{align*}
On the other hand, by \eqref{pn.largex}
\begin{align*}
\sum_{i=k/2}^{k-\varepsilon_k} \sum_{\|z\|>2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|}+ \frac{1}{\sqrt{k-i}} \right)
\lesssim \frac{1}{k^2}\sum_{\|z\|>2\sqrt k} \frac{G(z-y)}{\|z\|^5} \lesssim k^{-\frac 72}.
\end{align*}
Furthermore,
\begin{align*}
\sum_{i=k/2}^{k-\varepsilon_k} \frac 1{\sqrt{k-i}} \sum_{\|z\|\le 2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}
\lesssim \frac{1}{k^2\varepsilon_k}\sum_{\|z\|\le 2\sqrt k} \frac{G(z-y)}{1+\|z-x\|^3} \lesssim \frac{(k^2\varepsilon_k)^{-1}}{1+\|y-x\|},
\end{align*}
and
\begin{align*}
\sum_{i=\frac k2}^{k-\varepsilon_k} \sum_{\|z\|\le 2\sqrt k} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\frac{1}{1+\|z-x\|} & \lesssim \frac{1}{k^{3/2}\varepsilon_k^{3/2}}\sum_{\|z\|\le 2\sqrt k} \frac{G(z-y)}{1+\|z-x\|^3} \\
& \lesssim \frac{1}{k^{3/2}\varepsilon_k^{3/2}} \frac{1}{1+\|y-x\|}.
\end{align*}
Assume now that $\|x\|>\sqrt k$. One has on one hand, using Lemma \ref{lem.upconvolG},
\begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{\|z-x\|\ge \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5}\left(\frac{1}{1+\|z-x\|}+ \frac{1}{\sqrt{k-i}} \right)
\lesssim \frac{1}{\|x\|^5 \varepsilon_k}.
\end{align*}
On the other hand,
\begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{\|z-x\|\le \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} \frac{1}{1+\|z-x\|} & \lesssim \frac{k}{\|x\|^5 \varepsilon_k^{3/2}} \sum_{z\in \mathbb{Z}^5} \frac{G(z-y)}{1+\|z-x\|^3} \\
& \lesssim \frac{k}{\|x\|^5 \varepsilon_k^{3/2}(1+\|y-x\|)},
\end{align*}
and
\begin{align*}
\sum_{i=\varepsilon_k}^{k-\varepsilon_k}\frac 1{\sqrt {k-i}} \sum_{\|z-x\|\le \frac{\|x\|}2} \frac{p_i(z) G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} & \lesssim \frac{\sqrt k}{\|x\|^5 \varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{G(z-y)}{1+\|y-x\|^3} \\
& \lesssim \frac{\sqrt k}{\|x\|^5 \varepsilon_k(1+\|y-x\|)},
\end{align*}
concluding the proof of the lemma.
\end{proof}
\begin{lemma}\label{lem.prep.ijl2}
There exists a constant $C>0$, such that for any $x,y\in \mathbb{Z}^5$,
\begin{align*}
& \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v\|+\sqrt k)^5} \left(\frac{1}{1+\|x-v\|} + \frac{1}{ 1+\|x\|}\right)\frac 1{(\|x-v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right)\\
& \le C\cdot \left\{
\begin{array}{ll}
\frac{1}{k^2\varepsilon_k} \left( \frac 1{\sqrt {\varepsilon_k}} + \frac{1}{1+\|x\|} +\frac 1{1+\|y-x\|}+\frac{\sqrt{\varepsilon_k}}{(1+\|x\|) (1+\|y-x\|)}\right) & \quad \text{if }\|x\|\le \sqrt k\\
\frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5\sqrt{\varepsilon_k}} \left(\frac{1}{1+\|y-x\|} + \frac 1{\sqrt k} \right) & \quad \text{if }\|x\|>\sqrt k.
\end{array}
\right.
\end{align*}
\end{lemma}
\begin{proof}
Assume first that $\|x\| \le \sqrt k$. In this case it suffices to notice that on one hand, for any $\alpha\in \{3,4\}$, one has
$$\sum_{\|v\|\le 2 \sqrt k} \frac{1}{(1+\|x-v\|^\alpha)(1+\|y-v\|^{4-\alpha})} = \mathcal{O}(\sqrt k),$$
and on the other hand, for any $\alpha, \beta \in \{0,1\}$,
$$\sum_{\|v\|>2\sqrt k} \frac{1}{\|v\|^{10+\alpha} (1+\|y-v\|)^\beta} = \mathcal{O}(k^{-5/2 - \alpha - \beta}). $$
Assume next that $\|x\|>\sqrt k$. In this case it is enough to observe that
$$\sum_{\|v\|\le \frac{\sqrt k}{2}} \left(\frac{1}{1+\|x-v\|} + \frac{1}{\|x\|}\right) \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right) \lesssim \frac {k^2}{(1+\|y-x\|)},$$
$$\sum_{\|v\|\ge \frac{\sqrt k}{2}} \frac{1}{\|v\|^5 (\sqrt{\varepsilon_k}+\|x-v\|)^5} \lesssim \frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5} ,$$
$$\sum_{\|v\|\ge \frac{\sqrt k}{2}} \frac{1}{\|v\|^5 (\sqrt{\varepsilon_k}+\|x-v\|)^5(1+\|y-v\|)} \lesssim \frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5}\left(\frac 1{\sqrt k} + \frac 1{1+\|y-x\|}\right). $$
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem.ijl}]
First note that for any $\ell$, one has $\mathbb{E}[\psi_\ell] = \mathcal{O}(\varepsilon_k^{-1/2})$, by \eqref{lem.hit.3}. Using also similar arguments as in the proof of Lemma \ref{lem.123}, that we will not reproduce here, one can see that $\mathbb{E}[\varphi_{i,j}] = \mathcal{O}(\varepsilon_k^{-1})$, for any $i\neq j$. Thus only the terms of the form $\mathbb{E}[\varphi_{i,j}\psi_\ell]$ are at stake.
Let $(S_n)_{n\in \mathbb{Z}}$, $(\widetilde S_n)_{n\ge 0}$ and $(\widehat S_n)_{n\ge 0}$ be three independent walks starting from the origin.
Recall the definition of $\tau_1$, $\tau_2$ and $\tau_3$ from the proof of Lemma \ref{lem.123}, and define analogously
$$\widehat \tau_1:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathbb{R}R(-\infty,-1]\}, \, \widehat \tau_2:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathbb{R}R[0,k-\varepsilon_k-1]\},$$
and
$$\widehat \tau_3:=\inf\{n\ge 1 : S_k+\widehat S_n \in \mathbb{R}R[k+\varepsilon_k+1,\infty)\}.$$
When $\ell\neq i,j$, one can take advantage of the independence between the different parts of the range of $S$, at least once we condition on the value of $S_k$. This allows for instance to write
$$\mathbb{E}[\varphi_{1,2}\psi_3] \le \mathbb{P}[\tau_1<\infty,\, \tau_2<\infty,\, \widehat \tau_3<\infty] =\mathbb{P}[\tau_1<\infty,\, \tau_2<\infty] \mathbb{P}[\widehat \tau_3<\infty] \lesssim \varepsilon_k^{-3/2},$$
using independence for the second equality and our previous estimates for the last one. Similarly,
\begin{align*}
& \mathbb{E}[\varphi_{1,3} \psi_2] \le \sum_{x\in \mathbb{Z}} \mathbb{P}[\tau_1<\infty, \, \tau_3<\infty \mid S_k=x] \times \mathbb{P}[\widehat \tau_2<\infty,\, S_k=x] \\
&\lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(1+\|x\|)\sqrt \varepsilon_k}\cdot \frac 1{(1+\|x\| + \sqrt k)^5}\left(\frac{1}{1+\|x\|}+\frac 1{\sqrt{\varepsilon_k}}\right) \lesssim \frac{1}{\varepsilon_k \sqrt k},
\end{align*}
using \eqref{tau132.1} and Lemma \ref{lem.prep.123} for the second inequality. The term $\mathbb{E}[\varphi_{2,3} \psi_1]$ is handled similarly.
We consider now the other cases. One has
\begin{equation}\label{phi233}
\mathbb{E}[\varphi_{2,3} \psi_3] \le \mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] + \mathbb{P}[\tau_3\le \tau_2<\infty,\, \widehat \tau_3<\infty].
\end{equation}
By using the Markov property at time $\tau_2$, one can write
\begin{align*}
& \mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] \\
& \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=0}^\infty G(S_i-y+x)\right)\left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right) \right] \mathbb{P}[\tau_2<\infty, \widetilde S_{\tau_2}=y, S_k=x].
\end{align*}
Then applying Lemmas \ref{lem.upconvolG} and \ref{lem.prep.123}, we get
\begin{align}\label{tau33}
\nonumber & \mathbb{E}\left[ \left(\sum_{i=0}^{\varepsilon_k} G(S_i-y+x)\right) \left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right)\right]\\
& \nonumber = \sum_{v\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right){\text{\Large $\mathfrak 1$}}\{S_{\varepsilon_k} = v\} \right] \mathbb{E}\left[\left(\sum_{j=0}^\infty G(S_j+v) \right)\right]\\
\nonumber & \lesssim \sum_{v\in \mathbb{Z}^5} \frac{1}{1+\|v\|}\cdot \left(\sum_{i=0}^{\varepsilon_k} p_i(z) G(z-y+x) p_{\varepsilon_k-i}(v-z) \right)\\
& \lesssim \sum_{v\in \mathbb{Z}^5} \frac{1}{1+\|v\|} \frac{1}{(\|v\|+ \sqrt{\varepsilon_k})^5} \left(\frac{1}{1+\|v-y+x\|} + \frac{1}{1+\|y-x\|}\right) \lesssim \frac{\varepsilon_k^{-1/2}}{ 1+\|y-x\|}.
\end{align}
Likewise,
\begin{align}\label{tau33bis}
\nonumber \mathbb{E}\left[ \left(\sum_{i=\varepsilon_k}^\infty G(S_i-y+x)\right)\left(\sum_{j=\varepsilon_k}^\infty G(S_j) \right)\right] &\le \sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) \left(\frac{G(z-y+x)}{1+\|z\|} + \frac{G(z)}{1+\|z-y+x\|}\right)\\
&\lesssim \frac{1}{\sqrt{\varepsilon_k} (1+\|y-x\|)}.
\end{align}
Recall now that by \eqref{lem.hit.3}, one has $\mathbb{P}[\tau_2<\infty] \lesssim \varepsilon_k^{-1/2}$. Moreover, from the proof of Lemma \ref{lem.123}, one can deduce that
$$\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}}{\|\widetilde S_{\tau_2}-S_k\|}\right] \lesssim \frac{1}{\sqrt {k \varepsilon_k}}.$$
Combining all these estimates we conclude that
$$\mathbb{P}[\tau_2\le \tau_3<\infty,\, \widehat \tau_3<\infty] \lesssim \frac 1{\varepsilon_k\sqrt k}.$$
We deal next with the second term in the right-hand side of \eqref{phi233}. Applying the Markov property at time $\tau_3$, and then Lemma \ref{lem.prep.123}, we obtain
\begin{align}\label{tau323.start}
\nonumber & \mathbb{P}[\tau_3\le \tau_2<\infty,\, \widehat \tau_3<\infty] \\
&\nonumber \le \sum_{x,y\in \mathbb{Z}^5}\left(\sum_{i=\varepsilon_k}^k \mathbb{E}[G(S_i-y){\text{\Large $\mathfrak 1$}}\{S_k=x\}]\right) \mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
\nonumber & \lesssim \sum_{x,y\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5} \left(\frac{1}{1+\|y-x\|}+\frac{1}{\sqrt{\varepsilon_k}}\right) \mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
\nonumber &\lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5}\left( \frac{\mathbb{P}[\tau_3<\infty, \widehat \tau_3<\infty\mid S_k=x]}{\sqrt{\varepsilon_k}} +
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right]\right)\\
& \lesssim \sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5}\left( \frac{1}{\varepsilon_k(1+\|x\|)} + \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big| \ S_k=x\right]\right),
\end{align}
using also \eqref{tau33} and \eqref{tau33bis} (with $y=0$) for the last inequality.
We use now \eqref{hit.ball} and Lemma \ref{hit.ball.overshoot} to remove the denominator in the last expectation above.
Define for $r\ge 0$, and $x\in \mathbb{Z}^5$,
$$\eta_r(x):=\inf\{n\ge 0\ :\ \|\widetilde S_n -x\|\le r\}.$$
On the event when $r/2\le \|\widetilde S_{\eta_r(x)} -x\| \le r$, one applies the Markov property at time $\eta_r(x)$, and we deduce from \eqref{hit.ball} and Lemma
\ref{hit.ball.overshoot} that
\begin{align*}
&\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big|\ S_k=x\right] \le \frac{\mathbb{P}[\tau_3<\infty, \, \widehat \tau_3<\infty\mid S_k=x]}{1+\|x\| } \\
&\qquad + \sum_{i=0}^{\log_2\|x\|}
\frac{\mathbb{P}\left[\tau_3<\infty, \, \widehat \tau_3<\infty,\, 2^i \le \|\widetilde S_{\tau_3} -x\| \le 2^{i+1}\mid S_k=x\right]}{2^i} \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|^2)} + \sum_{i=0}^{\log_2\|x\|}
\frac{\mathbb{P}\left[\eta_{2^{i+1}}(x)\le \tau_3<\infty, \, \widehat \tau_3<\infty\mid S_k=x\right]}{2^i} \\
& \lesssim \frac{\varepsilon_k^{-1/2}}{1+\|x\|^2} + \frac{\mathbb{P}[\widehat \tau_3<\infty]}{1+\|x\|^3} + \sum_{i=0}^{\log_2\|x\|} \frac{2^{2i}}{1+\|x\|^3} \max_{\|z\|\ge 2^i} \mathbb{P}_{0,0,z}\left[H_{\mathbb{R}R[\varepsilon_k,\infty)}<\infty, \widetilde H_{\mathbb{R}R_\infty}<\infty \right],
\end{align*}
where in the last probability, $H$ and $\widetilde H$ refer to hitting times by
two independent walks, independent of $S$, starting respectively from the origin and from $z$.
Then it follows from \eqref{tau33} and \eqref{tau33bis} that
\begin{equation}\label{remove.denominator}
\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty, \widehat \tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} - x\|}\ \Big|\ S_k=x\right] \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|^2)}.
\end{equation}
Combining this with \eqref{tau323.start}, it yields that
\begin{align*}
\mathbb{P}[\tau_2\le \tau_3<\infty, \widehat \tau_3 <\infty] \lesssim \frac 1{\varepsilon_k\sqrt k}.
\end{align*}
The terms $\mathbb{E}[\varphi_{1,3} \psi_3]$ and $\mathbb{E}[\varphi_{1,3}\psi_1]$ are entirely similar, and we omit repeating the proof. Thus it only remains to consider the terms $\mathbb{E}[\varphi_{2,3} \psi_2]$ and $\mathbb{E}[\varphi_{1,2} \psi_2]$. Since they are also similar we only give the details for the former.
We start again by writing
\begin{equation}\label{tau232}
\mathbb{E}[\varphi_{2,3} \psi_2]\le \mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] + \mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2<\infty].
\end{equation}
Then one has
\begin{align}\label{Sigmai}
&\mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2 <\infty] \\
\nonumber \le & \sum_{x,y\in \mathbb{Z}^5}\mathbb{E}\left[\left(\sum_{i=\varepsilon_k}^k G(S_i-y)\right) \left( \sum_{j=0}^{k-\varepsilon_k} G(S_j-x)\right) {\text{\Large $\mathfrak 1$}}\{S_k=x\} \right]
\mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x]\\
\nonumber \le & \sum_{x,y\in \mathbb{Z}^5} \left(\sum_{i=\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k}\sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\right) \\
\nonumber & \qquad \times \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x].
\end{align}
Now for any $x,y\in \mathbb{Z}^5$,
\begin{align*}
& \mathcal{S}igma_1(x,y):= \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{j=\varepsilon_k}^{k-\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z,\, S_j=w,\, S_k=x] G(z-y)G(w-x)\\
& \le 2\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}p_i(z) G(z-y) \left(\sum_{j=i}^{k-\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_{j-i}(w-z)G(w-x) p_{k-j}(x-w) \right)\\
& = 2\sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}p_i(z) G(z-y) \left(\sum_{j=\varepsilon_k}^k \sum_{w\in \mathbb{Z}^5} p_j(w) G(w) p_{k-i-j}(w+x-z) \right)\\
& \stackrel{\text{Lemma } \ref{lem.prep.123}}{\lesssim} \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{p_i(z)G(z-y)}{(\|z-x\| + \sqrt{k-i})^5} \left(\frac 1{1+\|z-x\|} + \frac{1}{\sqrt{k-i}}\right)\\
& \stackrel{\text{Lemma } \ref{lem.prep.ijl}}{\lesssim} \left\{
\begin{array}{ll}
\frac{1}{k^{5/2}}\left( \frac 1{1+\|x\|^2} + \frac{1}{\operatorname{Var}epsilon_k}\right) + \frac{1}{k^{3/2}\varepsilon_k^{3/2}(1+\|y-x\|)} & \text{if }\|x\|\le \sqrt k\\
\frac{1}{\|x\|^5\varepsilon_k}\left(1+ \frac{k}{\sqrt{\varepsilon_k}(1+\|y-x\|)}\right) &\text{if }\|x\|>\sqrt k.
\end{array}
\right.
\end{align*}
We also have
\begin{align*}
& \mathcal{S}igma_2(x,y) := \sum_{i=k-\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\\
=& \sum_{i=k-\varepsilon_k}^k \sum_{j=0}^{k-\varepsilon_k} \sum_{z,v,w\in \mathbb{Z}^5} \mathbb{P}[S_j= w, S_{k-\varepsilon_k} = v, S_i=z, S_k=x] G(z-y)G(w-x)\\
=& \sum_{v\in \mathbb{Z}^5} \left(\sum_{j=0}^{k-\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_j(w) p_{k-\varepsilon_k-j}(v-w) G(w-x)\right) \left(\sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z-v)
p_{\varepsilon_k-i}(x-z)G(z-y)\right),
\end{align*}
and applying then Lemmas \ref{lem.prep.123} and \ref{lem.prep.ijl2}, gives
\begin{align*}
& \mathcal{S}igma_2(x,y) \\
\lesssim & \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v\|+\sqrt k)^5}\left(\frac{1}{1+\|x-v\|} + \frac{1}{ 1+\|x\|}\right)\frac 1{(\|x-v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|y-x\|}+\frac 1{1+\|y-v\|}\right)\\
\lesssim & \left\{
\begin{array}{ll}
\frac{1}{k^2\varepsilon_k} \left( \frac 1{\sqrt {\varepsilon_k}} + \frac{1}{1+\|x\|} +\frac 1{1+\|y-x\|}+\frac{\sqrt{\varepsilon_k}}{(1+\|x\|) (1+\|y-x\|)}\right) & \quad \text{if }\|x\|\le \sqrt k\\
\frac{\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})}{\|x\|^5\sqrt{\varepsilon_k}} \left(\frac{1}{1+\|y-x\|} + \frac 1{\sqrt k} \right) &\quad \text{if }\|x\|>\sqrt k.
\end{array}
\right.
\end{align*}
Likewise, by reversing time, one has
\begin{align*}
& \mathcal{S}igma_3(x,y):= \sum_{i=\varepsilon_k}^k \sum_{j=0}^{\varepsilon_k} \sum_{z,w\in \mathbb{Z}^5} \mathbb{P}[S_i= z, S_j=w, S_k=x] G(z-y)G(w-x)\\
=& \sum_{i=0}^{k-\varepsilon_k} \sum_{j=k-\varepsilon_k}^k \sum_{z,v,w\in \mathbb{Z}^5} \mathbb{P}[S_i = z-x, S_{k-\varepsilon_k} = v-x, S_j= w-x, S_k=-x] G(z-y)G(w-x)\\
=& \sum_{v\in \mathbb{Z}^5} \left(\sum_{i=0}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z-x) p_{k-\varepsilon_k-i}(v-z) G(z-y)\right) \left(\sum_{j=0}^{\varepsilon_k} \sum_{w\in \mathbb{Z}^5} p_j(w-v)
p_{\varepsilon_k-j}(w)G(w-x)\right) \\
\lesssim & \sum_{v\in \mathbb{Z}^5} \frac{1}{(\|v-x\|+\sqrt k)^5}\left(\frac{1}{1+\|y-v\|} + \frac{1}{ 1+\|y-x\|}\right)\frac 1{(\|v\|+ \sqrt{\varepsilon_k} )^5} \left(\frac{1}{1+\|x\|}+\frac 1{1+\|x-v\|}\right),
\end{align*}
and then a similar argument as in the proof of Lemma \ref{lem.prep.ijl2} gives the same bound for $\mathcal{S}igma_3(x,y)$ as for $\mathcal{S}igma_2(x,y)$.
Now recall that \eqref{Sigmai} yields
$$\mathbb{P}[\tau_3\le \tau_2<\infty, \widehat \tau_2 <\infty] \le \sum_{x,y\in \mathbb{Z}^5} \left(\mathcal{S}igma_1(x,y) + \mathcal{S}igma_2(x,y) + \mathcal{S}igma_3(x,y)\right) \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3}=y\mid S_k=x]. $$
Recall also that by \eqref{lem.hit.2},
$$\mathbb{P}[\tau_3<\infty\mid S_k=x] \lesssim \frac{1}{1+\|x\|},$$
and
\begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}}{1+\|\widetilde S_{\tau_3} -x \| }\, \Big|\, S_k=x\right]
\le \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)}{1+\|y-x\|}
\lesssim \frac{1}{1+\|x\|^2}.
\end{align*}
Furthermore, for any $\alpha\in\{1,2,3\}$, and any $\beta\ge 6$,
$$\sum_{\|x\|\le \sqrt k} \frac{1}{1+\|x\|^\alpha} \lesssim k^{\frac{5-\alpha}{2}}, \quad \sum_{\|x\|\ge \sqrt k} \frac {\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})} {\|x\|^{\beta}} \le \sum_{\|x\|\ge \sqrt{\varepsilon_k}} \frac {\log (\frac{\|x\|}{\sqrt{\varepsilon_k}})} {\|x\|^{\beta}}\lesssim \varepsilon_k^{\frac{5-\beta}{2}}.$$
Putting all these pieces together we conclude that
$$\mathbb{P}[\tau_3\le \tau_2<\infty, \, \widehat \tau_2 <\infty] \lesssim \varepsilon_k^{-3/2}. $$
We deal now with the other term in \eqref{tau232}. As previously, we first write using the Markov property, and then using \eqref{lem.hit.1} and Lemma \ref{lem.upconvolG},
\begin{align*}
\mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty, \, \widehat \tau_2<\infty\}}{1+\|\widetilde S_{\tau_2} - S_k\|}\right].
\end{align*}
Then using \eqref{hit.ball} and Lemma \ref{hit.ball.overshoot} one can handle the denominator in the last expectation, the same way as for \eqref{remove.denominator},
and we conclude similarly that
\begin{align*}
\mathbb{P}[\tau_2\le \tau_3<\infty, \, \widehat \tau_2 <\infty] \lesssim \varepsilon_k^{-3/2}.
\end{align*}
This finishes the proof of Lemma \ref{lem.ijl}.
\end{proof}
\section{Proof of Propositions \ref{prop.ij0} and \ref{prop.phipsi.1}}
For the proof of these propositions we shall need the following estimate.
\begin{lemma}\label{lem.prep.0ij}
One has for all $x,y\in \mathbb{Z}^5$,
\begin{align*}
\sum_{i=k-\varepsilon_k}^k & \mathbb{E}\left[G(S_i-y) {\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \\
& \lesssim \varepsilon_k \left(\frac{\log (2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{(\|x\|+\sqrt k)^5(\|y-x\| + \sqrt{\varepsilon_k})^3}
+ \frac{\log (2+\frac{\|y\|}{\sqrt{k}})}{(\|x\|+\sqrt{\varepsilon_k})^5(\|y\| + \sqrt{k})^3}\right).
\end{align*}
\end{lemma}
\begin{proof}
One has using \eqref{pn.largex} and \eqref{Green},
\begin{align*}
&\sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[G(S_i-y){\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] = \sum_{i=k-\varepsilon_k}^k \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y) p_{k-i}(x-z)\\
& \lesssim \sum_{z\in \mathbb{Z}^5} \frac{\varepsilon_k}{(\|z\| + \sqrt k)^5(1+\|z-y\|^3) (\|x-z\| +\sqrt{\varepsilon_k})^5} \\
& \lesssim \frac{1}{\varepsilon_k^{3/2}(\|x\|+\sqrt k)^5} \sum_{\|z-x\|\le \sqrt{\varepsilon_k}}\frac 1{1+\|z-y\|^3} \\
& \quad + \frac{\varepsilon_k}{(\|x\|+\sqrt k)^5} \sum_{\sqrt{\varepsilon_k}\le \|z-x\|\le \frac{\|x\|}{2}} \frac{1}{(1+\|z-y\|^3)(1+\|z-x\|^5)} \\
&\quad + \frac{\varepsilon_k}{(\|x\|+\sqrt {\varepsilon_k})^5} \sum_{\|z-x\|\ge \frac{\|x\|}{2}}\frac{1}{(\|z\|+\sqrt k)^5(1+\|z-y\|^3)}.
\end{align*}
Then it suffices to observe that
$$ \sum_{\|z-x\|\le \sqrt{\varepsilon_k}}\frac 1{1+\|z-y\|^3} \lesssim \frac{\varepsilon_k^{5/2}}{(\|y-x\| + \sqrt{\varepsilon_k})^3},$$
$$ \sum_{\sqrt{\varepsilon_k}\le \|z-x\|\le \frac{\|x\|}{2}} \frac{1}{(1+\|z-y\|^3)(1+\|z-x\|^5)} \lesssim
\frac{\log(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{(\|y-x\| + \sqrt{\varepsilon_k})^3}, $$
$$ \sum_{z\in \mathbb{Z}^5}\frac{1}{(\|z\|+\sqrt k)^5(1+\|z-y\|^3)} \lesssim \frac{\log (2+\frac{\|y\|}{\sqrt{k}})}{(\|y\| + \sqrt{k})^3}. $$
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.ij0} (i)]
This part is the easiest: it suffices to observe that $\varphi_{1,2}$
is a sum of one term which is independent of $Z_k\psi_0$ and another one, whose expectation is negligible. To be more precise, define
$$\varphi_{1,2}^0 := \mathbb{P}\left[H_{\mathbb{R}R[-\varepsilon_k,\varepsilon_k]}^+ = \infty,\, H^+_{\mathbb{R}R(-\infty,-\varepsilon_k-1]}<\infty, \, H^+_{\mathbb{R}R[\varepsilon_k+1,k-\varepsilon_k-1]}<\infty \mid S\right],$$
and note that $Z_0\varphi_{1,2}^0$ is independent of $Z_k\psi_0$. It follows that
$$|\operatorname{Cov}(Z_0\varphi_{1,2},Z_k\psi_0)| =|\operatorname{Cov}(Z_0(\varphi_{1,2}-\varphi_{1,2}^0),Z_k\psi_0)|\le \mathbb{P}\left[\tau_1<\infty, \, \tau_*<\infty\right],$$
with $\tau_1$ and $\tau_*$ the hitting times respectively of $\mathbb{R}R(-\infty,-\varepsilon_k]$ and $\mathbb{R}R[k-\varepsilon_k,k]$ by another walk $\widetilde S$ starting from the origin, independent of $S$.
Now, using \eqref{pn.largex}, we get
\begin{align*}
\mathbb{P}[\tau_1\le \tau_*<\infty] & \le \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty \} \left(\sum_{i=k-\varepsilon_k}^k G(S_i-\widetilde S_{\tau_1})\right)\right] \\
& \le \sum_{y\in \mathbb{Z}^5} \left(\sum_{z\in \mathbb{Z}^5} \sum_{i=k-\varepsilon_k}^k p_i(z) G(z-y) \right)\mathbb{P}[\tau_1<\infty, \, \widetilde S_{\tau_1} = y]\\
& \lesssim \frac{\varepsilon_k}{k^{3/2}} \, \mathbb{P}[\tau_1<\infty] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
Likewise, using now Lemma \ref{lem.upconvolG},
\begin{align*}
\mathbb{P}[\tau_*\le \tau_1<\infty] & \le \mathbb{E}\left[{\text{\Large $\mathfrak 1$}}\{\tau_*<\infty \} \left(\sum_{i=\varepsilon_k}^\infty G(S_{-i}-\widetilde S_{\tau_*})\right)\right] \\
& \le \sum_{y\in \mathbb{Z}^5} \left(\sum_{z\in \mathbb{Z}^5} G_{\varepsilon_k}(z) G(z-y) \right)\mathbb{P}[\tau_*<\infty, \, \widetilde S_{\tau_*} = y]\\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}}\, \mathbb{P}[\tau_*<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},
\end{align*}
and the first part of (i) follows. But since $Z_0$ and $Z_k$ have played no role here, the same computation gives the result for the covariance between $Z_0\varphi_0$ and $Z_k \psi_{2,3}$ as well.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.ij0} (ii)-(iii)]
These parts are more involved. Since they are entirely similar, we only prove (iii), and as for (i) we only give the details for the covariance between $Z_0 \varphi_{2,3}$ and $Z_k\psi_0$, since $Z_0$ and $Z_k$ will not play any role here. We define similarly as in the proof of (i),
$$\varphi_{2,3}^0 := \mathbb{P}\left[H_{\mathbb{R}R[-\varepsilon_k,\varepsilon_k]}^+ = \infty,\, H^+_{\mathbb{R}R[\varepsilon_k,k-\varepsilon_k]}<\infty, \, H^+_{\mathbb{R}R[k+\varepsilon_k,\infty)}<\infty \mid S\right],$$
but observe that this time, the term $\varphi_{2,3}^0$ is no more independent of $\psi_0$. This entails some additional difficulty, on which we shall come back later,
but first we show that one can indeed replace $\varphi_{2,3}$ by $\varphi_{2,3}^0$ in the computation of the covariance. For this, denote
respectively by $\tau_2$, $\tau_3$, $\tau_*$ and $\tau_{**}$ the hitting times of $\mathbb{R}R[\varepsilon_k,k]$, $\mathbb{R}R[k,\infty)$, $\mathbb{R}R[k-\varepsilon_k,k]$, and $\mathbb{R}R[k,k+\varepsilon_k]$ by $\widetilde S$. One has
\begin{align*}
\mathbb{E}[|\varphi_{2,3} - \varphi_{2,3}^0|]& \le \mathbb{P}[\tau_2<\infty, \, \tau_{**}<\infty] +\mathbb{P}[\tau_3<\infty, \, \tau_*<\infty].
\end{align*}
Using \eqref{pn.largex}, \eqref{lem.hit.1} and Lemma \ref{lem.upconvolG}, we get
\begin{align*}
\mathbb{P}[\tau_*\le \tau_3<\infty] & \le \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\tau_*<\infty\}}{1+\|\widetilde S_{\tau_*} - S_k\|}\right] \le \sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[ \frac{G(S_i)}{1+\|S_i-S_k\|}\right] \\
&\lesssim \sum_{i=k-\varepsilon_k}^k \mathbb{E}\left[ \frac{G(S_i)}{1+\sqrt{k-i} }\right] \lesssim \sum_{z\in \mathbb{Z}^5} \sum_{i=k-\varepsilon_k}^k \frac{p_i(z) G(z)}{1+\sqrt{k-i}}\\
& \lesssim \sqrt{\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\frac 1{(\|z\| + \sqrt k)^5} G(z) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
Next, applying Lemma \ref{lem.prep.0ij}, we get
\begin{align*}
& \mathbb{P}[\tau_3\le \tau_*<\infty] \\
& \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\left(\sum_{i=k-\varepsilon_k}^k G(S_i-y)\right){\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \mathbb{P}[\tau_3<\infty, \widetilde S_{\tau_3} = y\mid S_k=x]\\
& \lesssim \varepsilon_k \sum_{x\in \mathbb{Z}^5} \left( \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log (2+\frac{\|\widetilde S_{\tau_3}-x\|}{\sqrt{\varepsilon_k}})}{(\|x\|+\sqrt k)^5(\sqrt{\varepsilon_k}+\|\widetilde S_{\tau_3} - x\|)^3}\, \Big|\, S_k=x \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log(2+\frac{\|\widetilde S_{\tau_3}\|}{\sqrt k})}{(\|x\|+\sqrt{\varepsilon_k})^5(\sqrt{k}+\|\widetilde S_{\tau_3}\|)^3}\, \Big|\, S_k=x \right] \right).
\end{align*}
Moreover,
\begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log (2+\frac{\|\widetilde S_{\tau_3}-x\|}{\sqrt{\varepsilon_k}})}{(\sqrt{\varepsilon_k}+\|\widetilde S_{\tau_3} - x\|)^3}\, \Big|\, S_k=x \right]
& \stackrel{\eqref{lem.hit.1}}{\le} \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)
\log (2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}})}{ (\sqrt{\varepsilon_k}+\|y - x\|)^3} \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}(1+\|x\|)^3},
\end{align*}
and
\begin{align*}
\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_3<\infty\}\log(2+\frac{\|\widetilde S_{\tau_3}\|}{\sqrt k})}{(\sqrt{k}+\|\widetilde S_{\tau_3}\|)^3}\, \Big|\, S_k=x \right] & \stackrel{\eqref{lem.hit.1}}{\le} \sum_{y\in \mathbb{Z}^5} \frac{G(y) G(y-x)\log(2+\frac{\|y\|}{\sqrt k})}{ (\sqrt{k}+\|y \|)^3} \\
& \lesssim \frac{1}{\sqrt{k}(1+\|x\|)(\sqrt k + \|x\|)^2}.
\end{align*}
Furthermore, it holds
$$\sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt k)^5(1+\|x\|)^3} \lesssim \frac{1}{k^{3/2}},$$
$$\sum_{x\in \mathbb{Z}^5} \frac{1}{(\|x\|+\sqrt{\varepsilon_k})^5(1+\|x\|)(\sqrt k + \|x\|)^2} \lesssim \frac{1}{\sqrt{k\varepsilon_k}},$$
which altogether proves that
$$ \mathbb{P}[\tau_3\le \tau_*<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
Likewise,
\begin{align*}
\mathbb{P}[\tau_2\le \tau_{**}<\infty] \le \sum_{x,y\in \mathbb{Z}^5} \mathbb{E}\left[\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right] \mathbb{P}[\tau_2<\infty, \widetilde S_{\tau_2} = y, S_k=x],
\end{align*}
and using \eqref{Green}, we get
\begin{align*}
& \mathbb{E}\left[\sum_{i=0}^{\varepsilon_k} G(S_i-y+x) \right] = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z) G(z-y+x)\\
& \lesssim \sum_{\|z\|\le \sqrt{\varepsilon_k}} G(z) G(z-y+x) + \varepsilon_k\, \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{G(z-y+x)}{\|z\|^5} \\
&\lesssim \frac{\varepsilon_k}{(\|y-x\| +\sqrt{\varepsilon_k})^2(1+\|y-x\|)} + \varepsilon_k \, \frac{\log\left(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}}\right) }{(\|y-x\| + \sqrt{\varepsilon_k})^3}\\
& \lesssim \varepsilon_k \, \frac{\log\left(2+\frac{\|y-x\|}{\sqrt{\varepsilon_k}}\right) }{(\|y-x\| + \sqrt{\varepsilon_k})^2(1+\|y-x\|)}.
\end{align*}
Therefore, using the Markov property,
\begin{align*}
&\mathbb{P}[\tau_2\le \tau_{**}<\infty] \lesssim \varepsilon_k \cdot \mathbb{E}\left[ \frac{\log\left(2+\frac{\|\widetilde S_{\tau_2}-S_k\|}{\sqrt{\varepsilon_k}}\right) \cdot {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}}{(\|\widetilde S_{\tau_2}-S_k\| + \sqrt{\varepsilon_k})^2(1+\|\widetilde S_{\tau_2} - S_k\|)} \right]\\
&\lesssim \varepsilon_k \sum_{i=\varepsilon_k}^k \mathbb{E}[G(S_i)] \cdot \mathbb{E}\left[\frac{\log\left(2+\frac{\|S_{k-i}\|}{\sqrt{\varepsilon_k}}\right) }{(\|S_{k-i}\| + \sqrt{\varepsilon_k})^2(1+\| S_{k-i}\|)} \right].
\end{align*}
Furthermore, using \eqref{pn.largex} we obtain after straightforward computations,
$$\mathbb{E}\left[\frac{\log\left(2+\frac{\|S_{k-i}\|}{\sqrt{\varepsilon_k}}\right) }{(\|S_{k-i}\| + \sqrt{\varepsilon_k})^2(1+\| S_{k-i}\|)} \right] \lesssim \frac{\log\left(2+\frac{k-i}{\varepsilon_k}\right) }{\sqrt{k-i}(\varepsilon_k + k-i)},$$
and using in addition \eqref{exp.Green}, we conclude that
$$\mathbb{P}[\tau_2\le \tau_{**}<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$
Similarly, using Lemma \ref{lem.prep.123} we get
\begin{align*}
&\mathbb{P}[\tau_{**} \le \tau_2<\infty] \\
& = \sum_{x,y\in \mathbb{Z}^5} \mathbb{P}[\tau_{**}<\infty,\, \widetilde S_{\tau_{**}} = y \mid S_k=x] \cdot \mathbb{E}\left[\sum_{i=\varepsilon_k}^k G(S_i-y) {\text{\Large $\mathfrak 1$}}\{S_k=x\}\right] \\
& \lesssim \sum_{x\in\mathbb{Z}^5} \frac{1}{(\|x\| + \sqrt{k})^5} \left(\mathbb{E}\left[\frac{1\{\tau_{**}<\infty\}}{ 1+\|\widetilde S_{\tau_{**}}-x\|}\, \Big|\, S_k=x\right] + \frac{\mathbb{P}[\tau_{**}<\infty\mid S_k=x]}{\sqrt{\varepsilon_k}}\right).
\end{align*}
Moreover, one has
\begin{align*}
\mathbb{P}[\tau_{**}<\infty\mid S_k=x] &\le \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_i+x)]\lesssim \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{1}{(1+\|z\| + \sqrt i)^5(1+\|z+x\|^3)} \\
&\lesssim \sum_{\|z\| \le \sqrt{\varepsilon_k}} \frac{1}{(1+\|z\|^3)(1+\|z+x\|^3)} + \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{\varepsilon_k}{\|z\|^5(1+\|z+x\|^3)}\\
&\lesssim \frac{\varepsilon_k \log(2+\frac{\|x\|}{\sqrt{\varepsilon_k}})}{(\sqrt{\varepsilon_k}+\|x\|)^2(1+\|x\|)},
\end{align*}
and likewise
\begin{align*}
\mathbb{E}\left[\frac{1\{\tau_{**}<\infty\}}{ 1+\|\widetilde S_{\tau_{**}}-x\|}\, \Big|\, S_k=x\right] &\le \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \frac{1}{(1+\|z\| + \sqrt i)^5(1+\|z-x\|^3)(1+\|z\|)} \\
&\lesssim \sum_{\|z\| \le \sqrt{\varepsilon_k}} \frac{1}{(1+\|z\|^4)(1+\|z-x\|^3)} + \sum_{\|z\|\ge \sqrt{\varepsilon_k}} \frac{\varepsilon_k}{\|z\|^6(1+\|z-x\|^3)}\\
&\lesssim \frac{\sqrt{\varepsilon_k} }{(\|x\|+\sqrt{\varepsilon_k})(1+\|x\|^2)}.
\end{align*}
Then it follows as above that
$$\mathbb{P}[\tau_{**} \le \tau_2<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$
In other words we have proved that
$$\mathbb{E}[|\varphi_{2,3} - \varphi_{2,3}^0|]\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}).$$
We then have to deal with the fact that $Z_0\varphi_{2,3}^0$ is not really independent of $Z_k\psi_0$. Therefore, we introduce the new random variables
$$\widetilde Z_k := {\text{\Large $\mathfrak 1$}}\{S_i \neq S_k \ \forall i=k+1,\dots,\varepsilon'_k\}, \ \widetilde \psi_0:=\mathbb{P}_{S_k}\left[H^+_{\mathbb{R}R[k-\varepsilon'_k,k+\varepsilon'_k]}=\infty\mid S\right],$$
where $(\varepsilon'_k)_{k\ge 0}$ is another sequence of integers, whose value will be fixed later. For the moment we only assume that it satisfies $\varepsilon'_k\le \varepsilon_k/4$, for all $k$.
One has by \eqref{Green} and \eqref{lem.hit.3},
\begin{equation}\label{tildepsi0}
\mathbb{E}[|Z_k\psi_0 - \widetilde Z_k\widetilde \psi_0|] \lesssim \frac{1}{\sqrt{\varepsilon'_k}}.
\end{equation}
Furthermore, for any $y\in \mathbb{Z}^5$,
\begin{align}\label{cov.230}
& \mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right] = \sum_{x\in \mathbb{Z}^5} \mathbb{E}\left[\varphi_{2,3}^0 {\text{\Large $\mathfrak 1$}}\{S_{k-\varepsilon_k}=x\}\mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right]\\
\nonumber & \le \sum_{x\in \mathbb{Z}^5}\mathbb{P}\left[\widetilde \mathbb{R}R_\infty \cap \mathbb{R}R[\varepsilon_k,k-\varepsilon_k]\neq \varnothing,\, \widetilde \mathbb{R}R_\infty \cap (x+y+\widehat \mathbb{R}R_\infty)\neq \varnothing,\, S_{k-\varepsilon_k}=x\right],
\end{align}
where in the last probability, $\widetilde \mathbb{R}R_\infty$ and $\widehat \mathbb{R}R_\infty$ are the ranges of two independent walks, independent of $S$, starting from the origin.
Now $x$ and $y$ being fixed, define
$$\tau_1:=\inf\{n\ge 0 : \widetilde S_n\in \mathbb{R}R[\varepsilon_k,k-\varepsilon_k]\}, \ \tau_2:= \inf\{n\ge 0 : \widetilde S_n\in (x+y + \widehat \mathbb{R}R_\infty)\}.$$
Applying \eqref{lem.hit.1} and the Markov property we get
\begin{align*}
&\mathbb{P}[\tau_1\le \tau_2<\infty,\, S_{k-\varepsilon_k} = x] \le \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty,\, S_{k-\varepsilon_k}=x\}}{1+\|\widetilde S_{\tau_1} - (x+y)\|}\right] \\
& \le \sum_{i=\varepsilon_k}^{k-\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\frac{p_i(z) G(z) p_{k-\varepsilon_k-i}(x-z)}{1+\|z-(x+y)\|} \\
&\lesssim \frac{1}{(\|x\|+\sqrt k)^5}\left(\frac{1}{\sqrt{\varepsilon_k} (1+\|x+y\|)} + \frac{1}{1+\|x\|^2}\right),
\end{align*}
using also similar computations as in the proof of Lemma \ref{lem.prep.123} for the last inequality.
It follows that for some constant $C>0$, independent of $y$,
$$\sum_{x\in \mathbb{Z}^5} \mathbb{P}[\tau_1\le \tau_2<\infty,\, S_{k-\varepsilon_k} = x] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$
On the other hand, by Lemmas \ref{lem.prep.123} and \ref{lem.simplehit},
\begin{align*}
\mathbb{P}[\tau_2\le \tau_1<\infty,\, S_{k-\varepsilon_k} = x] & \lesssim \frac{1}{( \| x \| +\sqrt k)^5} \left(\mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\} }{1+\|\widetilde S_{\tau_2} - x\|}\right] + \frac{\mathbb{P}[\tau_2<\infty]}{\sqrt{\varepsilon_k} }\right) \\
& \lesssim \frac{1}{(\|x\|+\sqrt k)^5}\left(\frac{1}{\sqrt{\varepsilon_k} (1+\|x+y\|)} + \frac{1}{1+\|x\|^2}\right),
\end{align*}
and it follows as well that
$$\sum_{x\in \mathbb{Z}^5} \mathbb{P}[\tau_2\le \tau_1<\infty, S_{k-\varepsilon_k} = x] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$
Coming back to \eqref{cov.230}, we deduce that
\begin{equation}\label{phi230.cond}
\mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right] \lesssim \frac{1}{\sqrt{k\varepsilon_k}},
\end{equation}
with an implicit constant independent of $y$.
Together with \eqref{tildepsi0}, this gives
\begin{align*}
& \mathbb{E}\left[\varphi_{2,3}^0|Z_k\psi_0-\widetilde Z_k\widetilde \psi_0|\right] \\
& = \sum_{y\in \mathbb{Z}^5} \mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} -S_{k-\varepsilon_k}= y\right]\cdot \mathbb{E}\left[|Z_k\psi_0-\widetilde Z_k\widetilde \psi_0|{\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k}-S_{k-\varepsilon_k} = y\}\right] \\
&\lesssim \frac{1}{\sqrt{k\varepsilon_k\varepsilon'_k}}.
\end{align*}
Thus at this point we have shown that
$$ \operatorname{Cov}(Z_0\varphi_{2,3},Z_k\psi_0) = \operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) +\mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\cdot \log(\frac k{\varepsilon_k}) + \frac{1}{\sqrt{k\varepsilon_k\varepsilon'_k}}\right).$$
Note next that
\begin{align*}
\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) & = \sum_{y,z\in \mathbb{Z}^5} \mathbb{E}\left[Z_0\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} - S_{k-\varepsilon'_k} =y\right] \\
& \times \mathbb{E}\left[\widetilde Z_k \widetilde \psi_0{\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon'_k}-S_{k-\varepsilon'_k}=z\}\right] \left(p_{\varepsilon_k-\varepsilon'_k}(y-z) - p_{\varepsilon_k+\varepsilon'_k}(y)\right).
\end{align*}
Moreover, one can show exactly as \eqref{phi230.cond} that uniformly in $y$,
$$\mathbb{E}\left[\varphi_{2,3}^0 \mid S_{k+\varepsilon_k} - S_{k-\varepsilon'_k} =y\right] \lesssim \frac{1}{\sqrt{k\varepsilon_k}}.$$
Therefore by using also \eqref{Sn.large} and Theorem \ref{LCLT}, we see that
\begin{align*}
& |\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) | \\
& \lesssim \frac{1}{\sqrt{k\varepsilon_k}} \sum_{\|y\| \le \varepsilon_k^{\frac 6{10}}} \sum_{\|z\|\le \varepsilon_k^{\frac 1{10}}\cdot \sqrt{\varepsilon'_k} } p_{2\varepsilon'_k}(z)\, |\overline p_{\varepsilon_k-\varepsilon'_k}(y-z) - \overline p_{\varepsilon_k+\varepsilon'_k}(y)| + \frac 1{\varepsilon_k \sqrt{k} }.
\end{align*}
Now straightforward computations show that for $y$ and $z$ as in the two sums above, one has for some constant $c>0$,
$$|\overline p_{\varepsilon_k-\varepsilon'_k}(y-z) - \overline p_{\varepsilon_k+\varepsilon'_k}(y)| \lesssim \left(\frac{\|z\|}{\sqrt{\varepsilon_k}} + \frac{\varepsilon'_k}{\varepsilon_k}\right)\overline p_{\varepsilon_k-\varepsilon'_k}(cy),$$
at least when $\varepsilon'_k\le \sqrt{\varepsilon_k}$, as will be assumed in a moment.
Using also that $\sum_z \|z\|p_{2\varepsilon'_k}(z) \lesssim \sqrt{\varepsilon'_k}$, we deduce that
$$ |\operatorname{Cov}(Z_0\varphi_{2,3}^0,\widetilde Z_k\widetilde \psi_0) | = \mathcal{O}\left(\frac{\sqrt{\varepsilon'_k}}{\varepsilon_k \sqrt k}\right).$$
This concludes the proof as we choose $\varepsilon'_k = \lfloor \sqrt{\varepsilon_k} \rfloor$.
\end{proof}
We can now quickly give the proof of Proposition \ref{prop.phipsi.1}.
\begin{proof}[Proof of Proposition \ref{prop.phipsi.1}]
\underline{Case $1\le i<j\le 3$.} First note that $Z_0\varphi_1$ and $Z_k\psi_3$ are independent, so only the cases $i=1$ and $j=2$, or $i=2$ and $j=3$ are at stake. Let us only consider the case $i=2$ and $j=3$, since the other one is entirely similar. Define, in the same fashion as in the proof of Proposition \ref{prop.ij0},
$$\varphi_2^0:= \mathbb{P}\left[H^+_{\mathbb{R}R[-\varepsilon_k,\varepsilon_k]}=\infty,\, H^+_{\mathbb{R}R[\varepsilon_k+1,k-\varepsilon_k]}<\infty\mid S\right]. $$
One has by using independence and translation invariance,
$$\mathbb{E}[|\varphi_2-\varphi_2^0| \psi_3] \le \mathbb{P}[H_{\mathbb{R}R[k-\varepsilon_k,k]}<\infty]\cdot \mathbb{P}[H_{\mathbb{R}R[\varepsilon_k,\infty)}<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},$$
which entails
$$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_3) = \operatorname{Cov}(Z_0\varphi_2^0,Z_k\psi_3) + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},$$
since $Z_0\varphi_2^0$ and $Z_k\psi_3$ are independent.
\underline{Case $1\le j\le i\le 3$}. Here one can use entirely similar arguments as those from the proof of Lemma \ref{lem.ijl}, and we therefore omit the details.
\end{proof}
\section{Proof of Proposition \ref{prop.phi0}}
We need to estimate here the covariances $\operatorname{Cov}(Z_0\varphi_i, Z_k\psi_0)$ and $\operatorname{Cov}(Z_0\varphi_0, Z_k\psi_{4-i})$, for all $1\le i \le 3$.
\underline{Case $i=1$.} It suffices to observe that
$Z_0\varphi_1$ and $Z_k\psi_0$ are independent, as are $Z_0\varphi_0$ and $Z_k\psi_3$. Thus their covariances are equal to zero.
\underline{Case $i=2$.} We first consider the covariance between $Z_0\varphi_2$ and $Z_k\psi_0$, which is easier to handle. Define
$$\widetilde \varphi_2:=\mathbb{P}\left[H^+_{\mathbb{R}R[-\varepsilon_k,k-\varepsilon_k-1]}=\infty,\, H^+_{\mathbb{R}R[k-\varepsilon_k,k]}<\infty\mid S\right],$$
and note that $Z_0(\varphi_2 - \widetilde \varphi_2)$ is independent of $Z_k\psi_0$. Therefore
$$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) = \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0).$$
Then we decompose $\psi_0$ as $\psi_0=\psi_0^1-\psi_0^2$, where
$$\psi_0^1:=\mathbb{P}_{S_k}[H^+_{\mathbb{R}R[k,k+\varepsilon_k]}=\infty\mid S],\ \psi_0^2:=\mathbb{P}_{S_k}[H^+_{\mathbb{R}R[k,k+\varepsilon_k]}=\infty, H^+_{\mathbb{R}R[k-\varepsilon_k,k-1]}<\infty \mid S].$$
Using now that $Z_k\psi_0^1$ is independent of $Z_0\widetilde \varphi_2$ we get
$$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) =- \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2).$$
Let $(\widetilde S_n)_{n\ge 0}$ and $(\widehat S_n)_{n\ge 0}$ be two independent walks starting from the origin, and define
$$\tau_1:=\inf \{n\ge 0 : S_{k-n}\in \widetilde \mathbb{R}R[1,\infty)\},\ \tau_2:=\inf\{n\ge 0 : S_{k-n}\in (S_k + \widehat \mathbb{R}R[1,\infty))\}.$$
We decompose
\begin{align*}
& \operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2)\\
& = \mathbb{E}\left[Z_0\widetilde \varphi_2Z_k\psi_0^2{\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2\}\right] + \mathbb{E}\left[Z_0\widetilde \varphi_2Z_k\psi_0^2{\text{\Large $\mathfrak 1$}}\{\tau_1> \tau_2\}\right] - \mathbb{E}[Z_0\widetilde \varphi_2] \mathbb{E}[Z_k\psi_0^2].
\end{align*}
We bound the first term on the right-hand side simply by the probability of the event $\{\tau_1\le \tau_2\le \varepsilon_k\}$, which we treat later, and for the difference between the last two terms, we use that
$$\left| {\text{\Large $\mathfrak 1$}}\{\tau_2<\tau_1\le \varepsilon_k\} - \sum_{i=0}^{\varepsilon_k} {\text{\Large $\mathfrak 1$}}\left\{\tau_2=i,\, H^+_{\mathbb{R}R[k-\varepsilon_k,k-i-1]}<\infty\right\}\right| \le {\text{\Large $\mathfrak 1$}}\{\tau_1\le \tau_2\le \varepsilon_k\}.$$
Using also that the event $\{\tau_2=i\}$ is independent of $(S_n)_{n\le k-i}$, we deduce that
\begin{align*}
& |\operatorname{Cov}(Z_0\widetilde \varphi_2, Z_k\psi_0^2)| \\
&\le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2=i] \left|\mathbb{P}\left[H^+_{\mathbb{R}R[k-\varepsilon_k,k-i]}<\infty \right] - \mathbb{P}\left[H^+_{\mathbb{R}R[k-\varepsilon_k,k]}<\infty \right] \right|\\
& \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2=i] \cdot \mathbb{P}\left[H^+_{\mathbb{R}R[k-i,k]}<\infty \right] \\
& \stackrel{\eqref{lem.hit.3}}{\le} 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} i \mathbb{P}[\tau_2=i] \\
& \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} \mathbb{P}[\tau_2\ge i]\\
& \stackrel{\eqref{lem.hit.3}}{\le} 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C}{k^{3/2}}\sum_{i=0}^{\varepsilon_k} \frac{1}{\sqrt i} \le 2\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] + \frac{C\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
Then it amounts to bound the probability of $\tau_1$ being smaller than $\tau_2$:
\begin{align*}
&\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] =\sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\tau_1=i, i\le \tau_2\le \varepsilon_k, S_k=x,\, S_{k-i} = x+y\right]\\
\le & \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\tau_1=i, S_{k-i}=x+y, (x+\widehat \mathbb{R}R_\infty) \cap \mathbb{R}R[k-\varepsilon_k,k-i]\neq \varnothing, S_k=x\right]\\
\le & \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \mathbb{P}\left[\widetilde \mathbb{R}R_\infty \cap (x+\mathbb{R}R[0,i-1])=\varnothing, S_i=y,\, x+y\in \widetilde \mathbb{R}R_\infty\right]\\
& \qquad \times \mathbb{P}\left[ \widehat \mathbb{R}R_\infty \cap (y+ \mathbb{R}R[0,\varepsilon_k-i])\neq \varnothing, S_{k-i}=-x-y\right],
\end{align*}
using invariance by time reversal of $S$, and where we stress the fact that in the first probability in the last line, $\mathbb{R}R$ and $\widetilde \mathbb{R}R$ are two independent ranges starting from the origin.
Now the last probability can be bounded using \eqref{Green.hit} and Lemma \ref{lem.prep.123}, which give
\begin{align*}
&\mathbb{P}\left[ \widehat \mathbb{R}R_\infty \cap (y+ \mathbb{R}R[0,\varepsilon_k-i])\neq \varnothing, S_{k-i}=-x-y\right] \le \sum_{j=0}^{\varepsilon_k-i} \mathbb{E}\left[G(S_j+y){\text{\Large $\mathfrak 1$}}\{S_{k-i} = -x-y\}\right]\\
=& \sum_{j=0}^{\varepsilon_k-i} \sum_{z\in \mathbb{Z}^5} p_j(z) G(z+y)p_{k-i-j}(z+x+y) =\sum_{j=k-\varepsilon_k}^{k-i} \sum_{z\in \mathbb{Z}^5} p_j(z) G(z-x)p_{k- i-j}(z-x-y) \\
\lesssim & \frac{1}{(\|x+y\|+\sqrt k)^5}\left(\frac 1{1+\|y\|} + \frac 1{\sqrt k + \|x\|}\right).
\end{align*}
It follows that
\begin{align*}
\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] \lesssim \sum_{x,y\in \mathbb{Z}^5} \sum_{i=0}^{\varepsilon_k} \frac{G(x+y) p_i(y)}{(\|x+y\| + \sqrt k)^5}\left(\frac 1{1+\|y\|} + \frac 1{\sqrt k+\|x\|}\right),
\end{align*}
and then standard computations show that
\begin{equation}\label{tau12eps}
\mathbb{P}[\tau_1\le \tau_2\le \varepsilon_k] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{equation}
Taking all these estimates together proves that
$$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_0) \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
We consider now the covariance between $Z_0\varphi_0$ and $Z_k\psi_2$. Here a new problem arises due to the random variable $Z_0$, which does not play the same role as $Z_k$,
but one can use similar arguments. In particular the previous proof gives
$$\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_2) = -\operatorname{Cov}((1-Z_0)\varphi_0,Z_k\psi_2) + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right).$$
Then we decompose as well $\varphi_0=\varphi_0^1 - \varphi_0^2$, with
$$\varphi_0^1:=\mathbb{P}[H^+_{\mathbb{R}R[k-\varepsilon_k,k]}=\infty\mid S],\ \varphi_0^2:=\mathbb{P}[H^+_{\mathbb{R}R[k-\varepsilon_k,k]}=\infty, H^+_{\mathbb{R}R[k+1,k+\varepsilon_k]}<\infty \mid S].$$
Using independence we get
$$\operatorname{Cov}((1-Z_0)\varphi_0^1,Z_k\psi_2) = \mathbb{E}[\varphi_0^1]\cdot \operatorname{Cov}((1-Z_0),Z_k\psi_2).$$
Then we define in the same fashion as above,
$$\widetilde \tau_0:= \inf\{n\ge 1 : S_n=0\}, \ \widetilde \tau_2:= \inf\{n\ge 0 : S_n\in (S_k+\widehat \mathbb{R}R[1,\infty))\},$$
with $\widehat \mathbb{R}R$ the range of an independent walk starting from the origin. Recall that by definition $1-Z_0 = {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 \le \varepsilon_k\}$.
Thus one can write
$$\operatorname{Cov}((1-Z_0),Z_k\psi_2) = \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_2 \le \widetilde \tau_0\le \varepsilon_k\}] + \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 < \widetilde \tau_2\}] -
\mathbb{P}[\widetilde \tau_0\le \varepsilon_k] \mathbb{E}[Z_k\psi_2].$$
On one hand, using \eqref{Green.hit}, the Markov property, and \eqref{exp.Green},
\begin{align*}
& \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_2 \le \widetilde \tau_0\le \varepsilon_k\}] \le \mathbb{P}[\widetilde \tau_2 \le \widetilde \tau_0 \le \varepsilon_k]\le \sum_{y\in \mathbb{Z}^5} \mathbb{P}[\widetilde \tau_2\le \varepsilon_k,\, S_{\widetilde \tau_2} =y] \cdot G(y)\\
& \le \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[G(S_i-S_k)G(S_i)\right] \le \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_{k-i})]\cdot \mathbb{E}[G(S_i)] \lesssim \frac{1}{k^{3/2}} \sum_{i=0}^{\varepsilon_k} \frac{1}{1+i^{3/2}} \lesssim \frac{1}{k^{3/2}}.
\end{align*}
On the other hand, similarly as above,
\begin{align}\label{tau20eps}
\nonumber & \mathbb{E}[Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{\widetilde \tau_0 < \widetilde \tau_2\}] - \mathbb{P}[\widetilde \tau_0\le \varepsilon_k]\cdot \mathbb{E}[Z_k\psi_2] \\
\nonumber & \le \mathbb{P}[\widetilde \tau_2 \le \widetilde \tau_0 \le \varepsilon_k] + \sum_{i=1}^{\varepsilon_k}\mathbb{P}[\widetilde \tau_0=i]
\left(\mathbb{P}\left[(S_k+\widehat \mathbb{R}R[1,\infty))\cap \mathbb{R}R[i+1,\varepsilon_k]\neq \varnothing\right] - \mathbb{P}[\widetilde \tau_2\le \varepsilon_k]\right)\\
\nonumber &\lesssim \frac{1}{k^{3/2}} + \sum_{i=1}^{\varepsilon_k}\mathbb{P}[\widetilde \tau_0=i] \mathbb{P}[\widetilde \tau_2\le i] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{k^{3/2}} +
\frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} i \mathbb{P}[\widetilde \tau_0=i] \\
& \lesssim \frac{1}{k^{3/2}} + \frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} \mathbb{P}[\widetilde \tau_0\ge i] \stackrel{\eqref{Green.hit}, \eqref{Green}}{\lesssim} \frac{1}{k^{3/2}}+ \frac{1}{k^{3/2}}\sum_{i=1}^{\varepsilon_k} \frac{1}{1+i^{3/2}} \lesssim \frac{1}{k^{3/2}}.
\end{align}
In other terms, we have already shown that
$$|\operatorname{Cov}((1-Z_0)\varphi_0^1,Z_k\psi_2) | \lesssim \frac{1}{k^{3/2}}.$$
The case when $\varphi_0^1$ is replaced by $\varphi_0^2$ is entirely similar. Indeed, we define
$$\widetilde \tau_1:=\inf\{n\ge 0 : S_n\in \widetilde \mathbb{R}R[1,\infty)\},$$
with $\widetilde \mathbb{R}R$ the range of a random walk starting from the origin, independent of $S$ and $\widehat \mathbb{R}R$. Then we set $\widetilde \tau_{0,1} := \max(\widetilde \tau_0,\widetilde \tau_1)$, and
exactly as for \eqref{tau12eps} and \eqref{tau20eps}, one has
\begin{equation*}
\mathbb{P}[\widetilde \tau_2\le \widetilde \tau_{0,1} \le \varepsilon_k] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}},
\end{equation*}
and
\begin{align*}
& \mathbb{E} \left[(1-Z_0) \varphi_0^2 Z_k\psi_2 {\text{\Large $\mathfrak 1$}}\{ \widetilde \tau_{0,1}<\widetilde \tau_2 \} \right] - \mathbb{E}[(1-Z_0) \varphi_0^2]\cdot \mathbb{E}[ Z_k\psi_2] \\
& \le \mathbb{P}[\widetilde \tau_2\le \widetilde \tau_{0,1} \le \varepsilon_k] +\sum_{i=0}^{\varepsilon_k} \mathbb{P}[\widetilde \tau_{0,1}= i] \cdot \mathbb{P}[\widetilde \tau_2\le i]
\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
Altogether, this gives
$$|\operatorname{Cov}(Z_0\varphi_0,Z_k\psi_2)|\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
\underline{Case $i=3$.} We only need to treat the case of the covariance between $Z_0\varphi_3$ and $Z_k\psi_0$, as the other one is entirely similar here. Define
$$\widetilde \varphi_3:=\mathbb{P}\left[H^+_{\mathbb{R}R[-\varepsilon_k,\varepsilon_k]\cup \mathbb{R}R[k+\varepsilon_k+1,\infty)}=\infty,\, H^+_{\mathbb{R}R[k,k+\varepsilon_k]}<\infty\mid S\right].$$
The proof of the case $i=2$, already shows that
$$|\operatorname{Cov}(Z_0\widetilde \varphi_3,Z_k\psi_0)|\lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
Define next
$$h_3:=\varphi_3-\widetilde \varphi_3=\mathbb{P}\left[H^+_{\mathbb{R}R[-\varepsilon_k,\varepsilon_k]}=\infty,\, H^+_{\mathbb{R}R[k+\varepsilon_k+1,\infty)}<\infty\mid S\right].$$
Assume for a moment that $\varepsilon_k\ge k^{\frac{9}{20}}$. We will see later another argument when this condition is not satisfied.
Then define $\varepsilon_k':= \lfloor \varepsilon_k^{10/9}/k^{1/9}\rfloor$, and note that one has $\varepsilon'_k\le \varepsilon_k$. Write $\psi_0=\psi'_0+h_0$, with
$$\psi_0':= \mathbb{P}\left[H^+_{\mathbb{R}R[k-\varepsilon'_k+1,k+\varepsilon'_k-1]}=\infty \mid S\right], $$
and
$$h_0:= \mathbb{P}\left[H^+_{\mathbb{R}R[k-\varepsilon'_k+1,k+\varepsilon'_k-1]}=\infty, \, H^+_{\mathbb{R}R[k-\varepsilon_k,k-\varepsilon'_k]\cup\mathbb{R}R[k+\varepsilon'_k,k+\varepsilon_k] } <\infty \mid S\right]. $$
Define also
$$Z'_k:={\text{\Large $\mathfrak 1$}}\{S_\ell\neq S_k,\text{ for all }\ell=k+1,\dots,k+\varepsilon'_k-1\}.$$
One has
$$\operatorname{Cov}(Z_0h_3,Z_k\psi_0) = \operatorname{Cov}(Z_0h_3,Z'_k\psi'_0) + \operatorname{Cov}(Z_0h_3,Z'_kh_0) +\operatorname{Cov}(Z_0h_3,(Z_k-Z'_k)\psi_0) .$$
For the last of the three terms, one can simply notice that, using the Markov property at the first return time to $S_k$ (for the walk $S$), and then \eqref{Green.hit}, \eqref{Green}, and \eqref{lem.hit.3}, we get
\begin{align*}
\mathbb{E}[h_3(Z_k-Z'_k)] & \le \mathbb{E}[Z_k-Z'_k] \times \mathbb{P}[\widetilde \mathbb{R}R_\infty \cap \mathbb{R}R\left[k,\infty)\neq \varnothing\right] \\
& \lesssim \frac{1}{(\varepsilon'_k)^{3/2}\sqrt k} \lesssim \frac 1{\varepsilon_k^{5/3}k^{1/3} }\lesssim \frac{1}{k^{\frac{13}{12}}},
\end{align*}
using our hypothesis on $\varepsilon_k$ for the last equality.
As a consequence, it also holds
$$|\operatorname{Cov}(Z_0h_3,(Z_k-Z'_k)\psi_0)| \lesssim k^{-\frac{13}{12}}. $$
Next we write
\begin{equation}\label{cov.i3}
\operatorname{Cov}(Z_0h_3,Z'_kh_0) = \sum_{x,y\in \mathbb{Z}^5} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x),
\end{equation}
where
$$H_1(y) := \mathbb{E}\left[Z'_kh_0 {\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k} - S_{k-\varepsilon_k} = y\}\right], \
H_2(x) := \mathbb{E}\left[Z_0h_3 \mid S_{k+\varepsilon_k} - S_{\varepsilon_k} = x\right].$$
Define $r_k:=(k/\varepsilon'_k)^{1/8}$. By using symmetry and translation invariance,
\begin{align*}
&\sum_{\|y\|\ge \sqrt{\varepsilon_k} r_k} H_1(y) \le \mathbb{P}\left[H_{\mathbb{R}R[-\varepsilon_k,-\varepsilon'_k]\cup\mathbb{R}R[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} - S_{-\varepsilon_k}\|\ge \sqrt{\varepsilon_k} r_k\right]\\
&\le 2 \mathbb{P}\left[H_{\mathbb{R}R[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}{2}\right] + 2\mathbb{P}\left[H_{\mathbb{R}R[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{-\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}{2} \right] \\
& \stackrel{\eqref{lem.hit.3}, \, \eqref{Sn.large}}{\le} 2 \mathbb{P}\left[H_{\mathbb{R}R[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}2\right] + \frac{C}{\sqrt{\varepsilon'_k} r_k^5}.
\end{align*}
Considering the first probability on the right-hand side, define $\tau$ as the first hitting time (for $S$), after time $\varepsilon'_k$, of another independent walk $\widetilde S$ (starting from the origin). One has
\begin{align*}
& \mathbb{P}\left[H_{\mathbb{R}R[\varepsilon'_k,\varepsilon_k]}<\infty, \, \|S_{\varepsilon_k} \|\ge \sqrt{\varepsilon_k} \frac{r_k}2\right] \\
& \le \mathbb{P}[\|S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] + \mathbb{P}[\|S_{\varepsilon_k} - S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] .
\end{align*}
Using then the Markov property at time $\tau$, we deduce with \eqref{lem.hit.3} and \eqref{Sn.large},
$$\mathbb{P}[\|S_{\varepsilon_k} - S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] \lesssim \frac{1}{\sqrt{\varepsilon'_k} r_k^5}.$$
Likewise, using the Markov property at the first time when the walk exit the ball of radius $\sqrt{\varepsilon_k} r_k/4$, and applying then \eqref{Sn.large} and \eqref{lem.hit.2}, we get as well
$$ \mathbb{P}[\|S_\tau\| \ge \sqrt{\varepsilon_k} \frac{r_k}4,\, \tau\le \varepsilon_k] \lesssim \frac{1}{\sqrt{\varepsilon_k} r_k^6}.$$
Furthermore, for any $y$, one has
$$\sum_{x\in \mathbb{Z}^5} p_{k-2\varepsilon_k}(x-y) H_2(x) \stackrel{\eqref{pn.largex}, \eqref{lem.hit.2}}{\lesssim} \sum_{x\in \mathbb{Z}^5} \frac{1}{(1+\|x+y\|)(\|x\|+\sqrt{k})^5}
\lesssim \frac{1}{\sqrt k},$$
with an implicit constant, which is uniform in $y$ (and the same holds with $p_k(x)$ instead of $p_{k-2\varepsilon_k}(x-y)$).
Similarly, define $r'_k:=(k/\varepsilon'_k)^{\frac 1{10}}$. One has for any $y$, with $\|y\|\le \sqrt{\varepsilon_k}r_k$,
$$\sum_{\|x\|\ge \sqrt{k}r_k'} p_{k-2\varepsilon_k}(x-y) H_2(x) \stackrel{\eqref{Sn.large}, \eqref{lem.hit.2}}{\lesssim} \frac{1}{\sqrt{k}(r'_k)^6}.$$
Therefore coming back to \eqref{cov.i3}, and using that by \eqref{lem.hit.2}, $\sum_y H_1(y)\lesssim 1/\sqrt{\varepsilon'_k}$, we get
\begin{align*}
& \operatorname{Cov}(Z_0h_3,Z'_kh_0) \\
& = \sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x) + \mathcal{O}\left(\frac{1}{\sqrt{k\varepsilon'_k}(r'_k)^6} + \frac{1}{\sqrt{k\varepsilon'_k} r_k^5}\right)\\
& = \sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) - p_k(x)) H_1(y)H_2(x) + \mathcal{O}\left(\frac{(\varepsilon'_k)^{\frac{1}{10} } }{k^{\frac{11}{10}}}\right).
\end{align*}
Now we use the fact $H_1(y) = H_1(-y)$. Thus the last sum is equal to half of the following:
\begin{align*}
&\sum_{\|x\|\le \sqrt{k}r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (p_{k-2\varepsilon_k}(x-y) + p_{k-2\varepsilon_k}(x+y) - 2p_k(x)) H_1(y)H_2(x) \\
&\stackrel{\text{Theorem }\ref{LCLT},\eqref{lem.hit.2}}{\le} \sum_{\|x\|\le \sqrt{k} r'_k} \sum_{\|y\|\le \sqrt{\varepsilon_k}r_k} (\overline p_{k-2\varepsilon_k}(x-y) + \overline p_{k-2\varepsilon_k}(x+y) - 2\overline p_k(x)) H_1(y)H_2(x) \\
& \qquad + \mathcal{O}\left(\frac{(r'_k)^4}{k^{3/2}\sqrt{\varepsilon'_k}} \right),
\end{align*}
(with an additional factor $2$ in front in case of a bipartite walk). Note that the error term above is $\mathcal{O}(k^{-11/10})$, by definition of $r'_k$.
Moreover, straightforward computations show that for any $x$ and $y$ as in the sum above,
$$|\overline p_{k-2\varepsilon_k}(x-y) + \overline p_{k-2\varepsilon_k}(x+y) - 2\overline p_k(x)| \lesssim \left(\frac{\|y\|^2+\varepsilon_k}{k} \right) \overline p_k(cx). $$
In addition one has (with the notation as above for $\tau$),
\begin{align*}
&\sum_{y\in \mathbb{Z}^5} \|y\|^2\, H_1(y) \le 2 \mathbb{E}\left[\|S_{\varepsilon_k} - S_{-\varepsilon_k}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \le 4 \mathbb{E}[\|S_{\varepsilon_k}\|^2] \mathbb{P}[\tau\le \varepsilon_k] + 4 \mathbb{E}\left[\|S_{\varepsilon_k}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \stackrel{\eqref{Sn.large}, \eqref{lem.hit.3}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}} + \mathbb{E}\left[\|S_{\tau}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] + \mathbb{E}\left[\|S_{\varepsilon_k}-S_{\tau}\|^2 {\text{\Large $\mathfrak 1$}}\{\tau\le \varepsilon_k\}\right] \\
& \stackrel{\eqref{Sn.large}, \eqref{lem.hit.3}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}} + \sum_{r\ge \sqrt{\varepsilon_k}} r \mathbb{P}\left[\|S_{\tau}\|\ge r, \, \tau\le \varepsilon_k\right] \stackrel{\eqref{Sn.large}, \eqref{lem.hit.2}}{\lesssim} \frac{ \varepsilon_k}{\sqrt{\varepsilon'_k}},
\end{align*}
using also the Markov property in the last two inequalities (at time $\tau$ for the first one, and at the exit time of the ball of radius $r$ for the second one).
Altogether, this gives
$$|\operatorname{Cov}(Z_0h_3,Z'_kh_0)| \lesssim \frac{\varepsilon_k }{k^{3/2}\sqrt{\varepsilon'_k}} + \frac{(\varepsilon'_k)^{\frac{1}{10} } }{k^{\frac{11}{10}}}
\lesssim \frac{(\varepsilon_k)^{\frac{1}{9} } }{k^{ \frac{10}{9} }}. $$
In other words, for any sequence $(\varepsilon_k)_{k\ge 1}$, such that $\varepsilon_k\ge k^{9/20}$, one has
$$\operatorname{Cov}(Z_0h_3,Z_k\psi_0) = \operatorname{Cov}(Z_0h_3,Z'_k\psi'_0) + \mathcal{O}\left(\frac{(\varepsilon_k)^{\frac{1}{9} } }{k^{ \frac{10}{9} }} + \frac{1}{k^{\frac{13}{12}} }\right).$$
One can then iterate the argument with the sequence $(\varepsilon'_k)$ in place of $(\varepsilon_k)$, and (after at most a logarithmic number of steps), we are left to consider a sequence $(\varepsilon_k)$, satisfying $\varepsilon_k\le k^{9/20}$. In this case, we use similar arguments as above. Define $\widetilde H_1(y)$ as $H_1(y)$, but with $Z_k\psi_0$ instead of $Z'_kh_0$ in the expectation, and choose $r_k:= \sqrt{k/\varepsilon_k}$, and $r'_k=k^{\frac 1{10}}$. Then we obtain exactly as above,
\begin{align*}
&\operatorname{Cov}(Z_0h_3,Z_k\psi_0) \\
& = \sum_{\|x\|\le \sqrt k r'_k} \sum_{\|y\|\le \sqrt{k}} ( p_{k-2\varepsilon_k}(x-y) - p_k(x)) \widetilde H_1(y) H_2(x) + \mathcal{O}\left(\frac{1}{r_k^5\sqrt k} + \frac 1{(r'_k)^6 \sqrt k}\right)\\
& = \sum_{\|x\|\le \sqrt k r'_k} \sum_{\|y\|\le \sqrt{k}} ( \overline p_{k-2\varepsilon_k}(x-y) - \overline p_k(x)) \widetilde H_1(y) H_2(x) + \mathcal{O}\left(\frac{1}{k^{\frac {11}{10}}} \right)\\
& \lesssim \frac{\varepsilon_k}{k^{3/2}} + \frac{1}{k^{\frac {11}{10}}} \lesssim \frac{1}{k^{\frac {21}{20}}},
\end{align*}
which concludes the proof of the proposition.
\section{Intersection of two random walks and proof of Theorem C} \label{sec.thmC}
In this section we prove a general result, which will be needed for proving Proposition \ref{prop.phipsi.2}, and which also gives Theorem C as a corollary.
First we introduce some general condition for a function $F:\mathbb{Z}^d\to \mathbb{R}$, namely:
\begin{equation}\label{cond.F}
\begin{array}{c}
\text{there exists a constant $C_F>0$, such that }\\
|F(y) - F(x)|\le C_F\, \frac{\|y-x\|}{1+\|y\|} \cdot |F(x)|, \quad \text{for all }x,y\in \mathbb{Z}^d.
\end{array}
\end{equation}
Note that any function satisfying \eqref{cond.F} is automatically bounded.
Observe also that this condition is satisfied by functions which are equivalent to $c/\mathcal J(x)^\alpha$, for some constants $\alpha\in [0,1]$, and $c>0$.
On the other hand, it is not satisfied by functions which are $o(1/\| x\|)$, as $\|x\|\to \infty$.
However, this is fine, since the only two cases that will be of interest for us here are when either $F$ is constant, or when $F(x)$ is of order $1/\|x\|$.
Now for a general function $F:\mathbb{Z}^d\to \mathbb{R}$, we define for $r>0$,
$$\overline F(r) := \sup_{r\le \|x\|\le r+1} |F(x)|.$$
Then, set
$$I_F(r):= \frac {\log (2+r)}{r^{d-2}}\int_0^r s\cdot \overline F(s)\, ds + \int_r^\infty \frac{\overline F(s)\log(2+s)}{s^{d-3} }\, ds,$$
and, with $\chi_d(r):= 1+(\log (2+r)){\text{\Large $\mathfrak 1$}}_{\{d=5\}}$,
$$ J_F(r):= \frac{\chi_d(r)}{r^{d-2}} \int_0^r \overline F(s)\, ds + \int_r^\infty \frac{\overline F(s)\chi_d(s)}{s^{d-2}}\, ds.$$
\begin{theorem}\label{thm.asymptotic}
Let $(S_n)_{n\ge 0}$ and $(\widetilde S_n)_{n\ge 0}$ be two independent random walks on $\mathbb{Z}^d$, $d\ge 5$, starting respectively from the origin and some $x\in \mathbb{Z}^d$. Let $\ell \in \mathbb{N}\cup \{\infty\}$, and define
$$\tau:=\inf\{n\ge 0\, : \, \widetilde S_n \in \mathbb{R}R[0,\ell] \}.$$
There exists $\nu\in (0,1)$, such that for any $F:\mathbb{Z}^d\to \mathbb{R}$, satisfying \eqref{cond.F},
\begin{align}\label{thm.asymp.formula}
\mathbb{E}_{0,x}\left[F(\widetilde S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\}\right] = &\ \frac {\gamma_d}{\kappa}\cdot \mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right] \\
\nonumber & + \mathcal{O}\left(\frac{I_F(\|x\|)}{(\ell\wedge \|x\|)^\nu} + (\ell\wedge \|x\|)^\nu J_F(\|x\|)\right),
\end{align}
where $\gamma_d$ is as in \eqref{LLN.cap}, and $\kappa$ is some positive constant given by
$$\kappa:=\mathbb{E}\left[\Big(\sum_{n\in \mathbb{Z}} G(S_n)\Big)\cdot \mathbb{P}\left[H^+_{\overline \mathbb{R}R_\infty}=+\infty \mid \overline \mathbb{R}R_\infty \right]\cdot {\text{\Large $\mathfrak 1$}}\{S_n\neq 0,\, \forall n\ge 1\}\right],$$
with $(S_n)_{n\in\mathbb{Z}}$ a two-sided walk starting from the origin and $\overline \mathbb{R}R_\infty := \{S_n\}_{n\in \mathbb{Z}}$.
\end{theorem}
\begin{remark}\emph{Note that when $F(x) \sim c/\mathcal J(x)^{\alpha}$, for some constants $\alpha \in [0,1]$ and $c>0$, then $I_F(r)$ and $J_F(r)$ are respectively of order $1/r^{d-4+ \alpha}$, and $1/r^{d-3+\alpha}$ (up to logarithmic factors), while
one could show that
$$\mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right] \sim \frac{c'}{\mathcal J(x)^{d-4+\alpha}}, \quad \text{as }\|x\|\to \infty\text{ and }\ell/\|x\|^2\to \infty,$$
for some other constant $c'>0$ (see below for a proof at least when $\ell = \infty$ and $\alpha=0$). Therefore in these cases Theorem \ref{thm.asymptotic}
provides a true equivalent for the term on the left-hand side of \eqref{thm.asymp.formula}.
}
\end{remark}
\begin{remark}
\emph{This theorem strengthens Theorem C in two aspects: on one hand it allows to consider functionals of the position of one of the two walks at its hitting time of the other path, and on the other hand it also allows to consider only a finite time horizon for one of the two walks (not mentioning the fact that it gives additionally some bound on the error term). Both these aspects will be needed later (the first one in the proof of Lemma \ref{lem.var.2}, and the second one in the proofs of Lemmas \ref{lem.var.3} and \ref{lem.var.4}). }
\end{remark}
Given this result one obtains Theorem C as a corollary. To see this, we first recall an asymptotic result on the Green's function:
in any dimension $d\ge 5$, under our hypotheses on $\mu$, there exists a constant $c_d>0$, such that as $\|x\|\to \infty$,
\begin{equation}\label{Green.asymp}
G(x)= \frac{c_d}{\mathcal J(x)^{d-2}} + \mathcal{O}(\|x\|^{1-d}).
\end{equation}
This result is proved in \cite{Uchiyama98} under only the hypothesis that $X_1$ has a finite $(d-1)$-th moment (we refer also to Theorem 4.3.5 in \cite{LL}, for a proof under the stronger hypothesis that $X_1$ has a finite $(d+1)$-th moment).
One also needs the following elementary fact:
\begin{lemma}\label{Green.convolution}
There exists a positive constant $c$, such that as $\|x\|\to \infty$,
$$\sum_{y\in \mathbb{Z}^d\setminus\{0,x\}} \frac{1}{\mathcal J(y)^{d-2}\cdot \mathcal J(y-x)^{d-2}} = \frac{c}{\mathcal J(x)^{d-4}} + \mathcal{O}\left(\frac 1{\|x\|^{d-3}}\right).$$
\end{lemma}
\begin{proof}
The proof follows by first an approximation by an integral, and then a change of variables. More precisely, letting $u:=x/\mathcal J(x)$, one has
\begin{align*}
& \sum_{y\in \mathbb{Z}^d\setminus\{0,x\} } \frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-x)^{d-2}} = \ \int_{\mathbb{R}^d} \frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-x)^{d-2}} \, dy +
\mathcal{O}(\|x\|^{3-d}) \\
& = \frac 1{\mathcal J(x)^{d-4}} \int_{\mathbb{R}^5}\frac{1}{\mathcal J(y)^{d-2} \mathcal J(y-u)^{d-2} }\, dy + \mathcal{O}(\|x\|^{3-d}),
\end{align*}
and it suffices to observe that by rotational invariance, the last integral is independent of $x$.
\end{proof}
\begin{proof}[Proof of Theorem C]
The result follows from Theorem \ref{thm.asymptotic}, by taking $F\equiv 1$ and $\ell = \infty$, and then by using \eqref{Green.asymp} together with Lemma \ref{Green.convolution}.
\end{proof}
It amounts now to prove Theorem \ref{thm.asymptotic}.
For this, we need some technical estimates that we gather in Lemma \ref{lem.thm.asymptotic} below. Since we believe this is not the most interesting part, we defer its proof to the end of this section.
\begin{lemma}\label{lem.thm.asymptotic}
Assume that $F$ satisfies \eqref{cond.F}. Then
\begin{enumerate}
\item There exists a constant $C>0$, such that for any $x\in \mathbb{Z}^d$,
\begin{equation}\label{lem.thm.asymp.1}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{j=0}^\infty G(S_j-S_i)\frac{\|S_j-S_i\|}{1+\|S_j\|}\right) \cdot |F(S_i)| G(S_i-x)\right] \le C J_F(\|x\|).
\end{equation}
\item There exists $C>0$, such that for any $R>0$, and any $x\in \mathbb{Z}^d$,
\begin{equation}\label{lem.thm.asymp.2}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{|j-i|\ge R}G(S_j-S_i)\right) |F(S_i)| G(S_i-x)\right] \le \frac{C}{R^{\frac{d-4}2}}\cdot I_F(\|x\|),
\end{equation}
\begin{equation}\label{lem.thm.asymp.2bis}
\sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{|j-i|\ge R}G(S_j-S_i) |F(S_j)|\right) G(S_i-x)\right] \le \frac{C}{R^{\frac{d-4}2} }\cdot I_F(\|x\|) .
\end{equation}
\end{enumerate}
\end{lemma}
One also need some standard results from (discrete) potential theory. If $\Lambda$ is a nonempty finite subset of $\mathbb{Z}^d$, containing the origin, we define
$$\text{rad}(\Lambda):=1+\sup_{x\in \Lambda} \|x\|,$$
and also consider for $x\in \Lambda$,
$$e_\Lambda(x):=\mathbb{P}_x[H_\Lambda^+=\infty], \quad \text{and} \quad \overline e_\Lambda(x):=\frac{e_\Lambda(x)}{\mathrm{Cap}(\Lambda)}.$$
The measure $\overline e_\Lambda$ is sometimes called the harmonic measure of $\Lambda$ from infinity, due to the next result.
\begin{lemma}\label{lem.potential}
There exists a constant $C>0$, such that for any finite subset $\Lambda\subseteq \mathbb{Z}^d$ containing the origin, and any $y\in \mathbb{Z}^d$, with $\| y\|>2\text{rad}(\Lambda)$,
\begin{eqnarray}\label{cap.hitting}
\mathbb{P}_y[H_\Lambda<\infty] \le C\cdot \frac{\mathrm{Cap}(\Lambda)}{1+\|y\|^{d-2}}.
\end{eqnarray}
Furthermore, for any $x\in \Lambda$, and any $y\in \mathbb{Z}^d$,
\begin{eqnarray}\label{harm.hit}
\Big| \mathbb{P}_y[S_{H_\Lambda}=x\mid H_\Lambda<\infty] - \overline e_\Lambda(x)\Big| \le C\cdot \frac{\text{rad}(\Lambda)}{1+\|y\|}.
\end{eqnarray}
\end{lemma}
This lemma is proved in \cite{LL} for finite range random walks. The proof extends to our setting, but some little care is needed, so
we shall give some details at the end of this section.
Assuming this, one can now give the proof of our main result.
\begin{proof}[Proof of Theorem \ref{thm.asymptotic}]
The proof consists in computing the quantity
\begin{equation}\label{eq.A}
A:= \mathbb{E}_{0,x}\left[\sum_{i=0}^\ell \sum_{j=0}^\infty {\text{\Large $\mathfrak 1$}}\{S_i = \widetilde S_j\}F(S_i)\right],
\end{equation}
in two different ways\footnote{This idea goes back to the seminal paper of Erd\'os and Taylor \cite{ET60}, even though it was not used properly there and was corrected only a few years later by Lawler, see \cite{Law91}.}. On one hand, by integrating with respect to the law of $\widetilde S$ first, we obtain
\begin{equation}\label{A.first}
A= \mathbb{E}\left[\sum_{i=0}^\ell G(S_i-x)F(S_i)\right].
\end{equation}
On the other hand, the double sum in \eqref{eq.A} is nonzero only when $\tau$ is finite. Therefore, using also the Markov property at time $\tau$, we get
\begin{align*}
A &= \mathbb{E}_{0,x}\left[\left(\sum_{i=0}^\ell \sum_{j=0}^\infty {\text{\Large $\mathfrak 1$}}\{S_i = \widetilde S_j\}F(S_i)\right) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right]\\
& = \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i) F(S_j)\right)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],
\end{align*}
where we recall that $Z_i^\ell = {\text{\Large $\mathfrak 1$}}\{S_j \neq S_i,\, \forall j=i+1,\dots,\ell\}$. The computation of this last expression is divided in a few steps.
\underline{Step 1.} Set
$$B:= \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i) \right)F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$
and note that,
\begin{align*}
& |A-B| \stackrel{\eqref{cond.F}}{\le} C_F\, \sum_{i=0}^\ell \mathbb{E}_{0,x}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i)\frac{\|S_j-S_i\|}{(1+\|S_j\|)} \right)|F(S_i)| {\text{\Large $\mathfrak 1$}}\{S_i\in \widetilde \mathbb{R}R_\infty\} \right]\\
&\stackrel{\eqref{Green.hit}}{\le} C_F\, \sum_{i=0}^\ell \mathbb{E}\left[ \left( \sum_{j=0}^\ell G(S_j-S_i)\frac{\|S_j-S_i\|}{(1+\|S_j\|)} \right)|F(S_i)| G(S_i-x)\right] \stackrel{\eqref{lem.thm.asymp.1}}{=} \mathcal{O}\left(J_F(\|x\|) \right).
\end{align*}
\underline{Step 2.} Consider now some positive integer $R$, and define
$$D_R:= \sum_{i=0}^{\ell} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R,\ell} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$
with $\mathcal G_{i,R,\ell}:= \sum_{j=(i-R)\vee 0}^{(i+R)\wedge \ell} G(S_j-S_i)$.
One has
$$|B-D_R| \stackrel{\eqref{Green.hit}}{\le} \sum_{i=0}^{\ell} \mathbb{E}\left[ \left(\sum_{|j-i|>R} G(S_j-S_i)\right) |F(S_i)|G(S_i-x)\right]
\stackrel{\eqref{lem.thm.asymp.2}}{\lesssim} \frac{ I_F(\|x\|)}{R^{\frac{d-4}2}}.$$
\underline{Step 3.} Let $R$ be an integer larger than $2$, and such that $\ell\wedge \|x\|^2 \ge R^6$.
Let $M:=\lfloor \ell / R^5\rfloor -1$, and define for $0\le m\le M$,
$$I_m:=\{mR^5+R^3,\dots, (m+1)R^5-R^3\}, \text{ and } J_m:=\{mR^5,\dots, (m+1)R^5-1\}.$$
Define further
$$E_R := \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau<\infty, \widetilde S_\tau = S_i\} \right],$$
with $\mathcal G_{i,R} := \sum_{j=i-R}^{i+R} G(S_j-S_i)$.
One has, bounding $\mathcal G_{i,R}$ by $(2R+1)G(0)$,
\begin{align*}
|D_R - E_R| & \le (2R+1) G(0)\\
& \times \left\{ \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] + \sum_{i=(M+1) R^5}^\ell \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \right\},
\end{align*}
with the convention that the last sum is zero when $\ell$ is infinite.
Using $\ell \ge R^6$, we get
\begin{align*}
&\sum_{i=(M+1) R^5}^\ell \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \le \sum_{z\in \mathbb{Z}^d} |F(z)| G(z-x) \sum_{i=(M+1)R^5}^{(M+2)R^5} p_i(z) \\
&\stackrel{\eqref{pn.largex},\, \eqref{Green}}{\lesssim} \frac{R^5}{\ell} \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|}{(1+ \|z-x\|^{d-2})(1+\|z\|^{d-2})} \lesssim \frac{R^5}{ \ell} \cdot I_F(\|x\|).
\end{align*}
Likewise, since $\|x\|^2\ge R^6$,
\begin{align}\label{final.step3}
\nonumber & \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \mathbb{E} \left[|F(S_i)| G(S_i-x) \right] \le \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} p_i(z)\\
\nonumber &\stackrel{\eqref{Green}}{\lesssim} \frac{1}{1+\|x\|^{d-2}} \sum_{\|z\|^2\le R^5} \frac{1}{1+\|z\|^{d-2}} \\
\nonumber & \qquad + \sum_{\|z\|^2 \ge R^5} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \sum_{m=0}^M \sum_{i\in J_m\setminus I_m} \left(\frac{{\text{\Large $\mathfrak 1$}}\{i\le \|z\|^2\}}{1+\|z\|^d} + \frac{{\text{\Large $\mathfrak 1$}}\{i\ge \|z\|^2\}}{i^{d/2}}\right) \\
& \lesssim \frac{R^5}{1+\|x\|^{d-2}} + \frac{1}{R^2} \cdot I_F(\|x\|),
\end{align}
using for the last inequality that the proportion of indices $i$ which are not in one of the $I_m$'s, is of order $1/R^2$.
\underline{Step 4.}
For $0\le m \le M+1$, set
$$\mathbb{R}R^{(m)}:=\mathbb{R}R[mR^5,(m+1)R^5-1], \quad \text{and}\quad \tau_m:= \inf\{ n \ge 0 \, :\, \widetilde S_n \in \mathbb{R}R^{(m)}\}.$$
Then let
$$F_R := \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R} F(S_i)Z_i^\ell \cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty, \widetilde S_{\tau_m} = S_i\} \right].$$
Since by definition $\tau\le \tau_m$, for any $m$, one has for any $i\in I_m$,
\begin{align*}
& |\mathbb{P}_{0,x}[\tau<\infty, \widetilde S_{\tau} = S_i\mid S] - \mathbb{P}_{0,x}[\tau_m<\infty, \widetilde S_{\tau_m} = S_i\mid S] | \\
& \le \mathbb{P}_{0,x}[\tau<\tau_m<\infty, \widetilde S_{\tau_m}=S_i\mid S]
\le \sum_{j\notin J_m} \mathbb{P}_{0,x}[\tau<\tau_m<\infty, \widetilde S_{\tau} = S_j, \widetilde S_{\tau_m} = S_i\mid S]\\
&\stackrel{\eqref{Green.hit}}{\le} \sum_{j\notin J_m} G(S_j-x) G(S_i-S_j).
\end{align*}
Therefore, bounding again $\mathcal G_{i,R}$ by $(2R+1)G(0)$, we get
\begin{align*}
|E_R-F_R| & \lesssim R \, \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}\left[\left(\sum_{j\notin J_m} G(S_i-S_j) G(S_j-x)\right)\cdot |F(S_i)| \right]\\
&\lesssim R \, \sum_{i=0}^\infty \mathbb{E}\left[\left(\sum_{j\, :\, |j-i|\ge R^3} G(S_i-S_j) G(S_j-x)\right)\cdot |F(S_i)| \right] \\
& \stackrel{\eqref{lem.thm.asymp.2bis}}{\lesssim} \frac{1}{ R^{3\frac{d-4}{2}-1}} \cdot I_F(\|x\|)\lesssim \frac{1}{\sqrt R} \cdot I_F(\|x\|).
\end{align*}
\underline{Step 5.} For $m\ge 0$ and $i\in I_m$, define
$$e_i^m := \mathbb{P}_{S_i} \left[H_{\mathbb{R}R^{(m)}}^+=\infty\mid S\right], \quad \text{and}\quad \overline e_i^m:= \frac{e_i^m}{\mathrm{Cap}(\mathbb{R}R^{(m)}) }.$$
Then let
$$H_R: = \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \mathcal G_{i,R}F(S_i)Z_i^\ell \overline e_i^m \cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right].$$
Applying \eqref{harm.hit} to the sets $\Lambda_m:=\mathbb{R}R^{(m)}-S_{i_m}$, we get for any $m\ge 0$, and any $i\in I_m$,
\begin{eqnarray}\label{harm.application}
\left| \mathbb{P}_{0,x}[\widetilde S_{\tau_m} = S_i\mid \tau_m<\infty, S] - \overline e_i^m \right| \le C\, \frac{\text{rad}(\Lambda_m)}{1+\|x-S_{i_m}\|}.
\end{eqnarray}
By \eqref{cap.hitting}, it also holds
\begin{align}\label{hit.cap.application}
\nonumber \mathbb{P}_{0,x}[\tau_m<\infty \mid S ] & \le \frac{ CR^5}{1+\|x-S_{i_m}\|^{d-2}} + {\text{\Large $\mathfrak 1$}}\{\|x-S_{i_m} \| \le 2\text{rad}(\Lambda_m)\}\\
& \lesssim \frac{ R^5+\text{rad}(\Lambda_m)^{d-2}}{1+\|x-S_{i_m}\|^{d-2}},
\end{align}
using that $\mathrm{Cap}(\Lambda_m)\le |\Lambda_m| \le R^5$. Note also that by \eqref{norm.Sn} and Doob's $L^p$-inequality (see Theorem 4.3.3 in \cite{Dur}), one has for any $1< p\le d$,
\begin{equation}\label{Doob}
\mathbb{E}[\text{rad}(\Lambda_m)^p] = \mathcal{O}(R^{\frac{5p}{2}}).
\end{equation}
Therefore,
\begin{align*}
& |F_R - H_R| \stackrel{\eqref{harm.application}}{\lesssim} R
\sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[ \frac{|F(S_i)|\cdot \text{rad}(\Lambda_m)}{1+\|x-S_{i_m}\|} {\text{\Large $\mathfrak 1$}}\{ \tau_m<\infty\}\right] \\
&\stackrel{\eqref{cond.F}}{\lesssim} R^6
\sum_{m=0}^M \mathbb{E}_{0,x}\left[ \frac{|F(S_{i_m})|\cdot \text{rad}(\Lambda_m)^2}{1+\|x-S_{i_m}\|} {\text{\Large $\mathfrak 1$}}\{ \tau_m<\infty\}\right] \\
& \stackrel{\eqref{hit.cap.application}, \eqref{Doob}}{\lesssim} R^{6+\frac{5d}{2}} \sum_{m=0}^M \mathbb{E}\left[ \frac{|F(S_{i_m})|}{1+\|x-S_{i_m}\|^{d-1}}\right] \lesssim R^{6+\frac{5d}{2}}\, \sum_{z\in \mathbb{Z}^d} \frac{|F(z)|G(z)}{1+\|x-z\|^{d-1}} \\
& \stackrel{\eqref{Green}}{\lesssim} \frac{R^{6+\frac{5d}{2}}}{1+\|x\|} \cdot I_F(\|x\|).
\end{align*}
\underline{Step 6.}
Let
$$K_R:= \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] \cdot \mathbb{E}\left[F(S_{i_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right].$$
One has, using the Markov property and a similar argument as in the previous step,
\begin{align*}
& |K_R-H_R| \stackrel{\eqref{cond.F}}{\lesssim}
R \sum_{m=0}^M \sum_{i\in I_m} \mathbb{E}_{0,x}\left[\frac{|F(S_{i_m})|\cdot (1+\|S_i-S_{i_m}\|^2)}{1+\|S_{i_m}\|}\cdot {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] \\
& \stackrel{\eqref{hit.cap.application}, \eqref{Sn.large}}{\lesssim} R^{6+\frac{5d}{2}} \sum_{m=0}^M \mathbb{E}\left[ \frac{|F(S_{i_m})|}{(1+\|S_{i_m}\|)(1+\|x-S_{i_m}\|^{d-2})}\right] \lesssim R^{6+\frac{5d}{2}}\cdot J_F(\|x\|).
\end{align*}
\underline{Step 7.}
Finally we define
$$\widetilde A:= \frac{\kappa}{\gamma_d} \cdot \mathbb{E}_{0,x}\left[F(\widetilde S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right].$$
We recall that one has (see Lemmas 2.1 and 2.2 in \cite{AS19}),
\begin{eqnarray}\label{easy.variance}
\mathbb{E}\left[\left(\mathrm{Cap}(\mathbb{R}R_n) - \gamma_d n\right)^2\right] = \mathcal{O}(n(\log n)^2).
\end{eqnarray}
It also holds for any nonempty subset $\Lambda\subseteq \mathbb{Z}^d$,
\begin{eqnarray}\label{min.cap}
\mathrm{Cap}(\Lambda) \ge c|\Lambda|^{1-\frac 2d}\ge c|\Lambda|^3,
\end{eqnarray}
using $d\ge 5$ for the second inequality (while the first inequality follows from \cite[Proposition 6.5.5]{LL} applied to the constant function equal to $c/|\Lambda|^{2/d}$, with $c>0$ small enough). As a consequence, for any $m\ge 0$ and any $i\in I_m$,
\begin{align*}
& \left|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] - \frac{\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell e_i^m\right]}{\gamma_dR^5} \right| \lesssim
\frac{1}{R^4} \mathbb{E}\left[ \frac {|\mathrm{Cap}(\mathbb{R}R^{(m)}) - \gamma_dR^5|}{\mathrm{Cap}(\mathbb{R}R^{(m)})}\right] \\
\stackrel{\eqref{easy.variance}}{\lesssim} & \frac{\log R}{R^{3/2}} \mathbb{E}\left[\frac 1{\mathrm{Cap}(\mathbb{R}R^{(m)})^2}\right]^{1/2}
\stackrel{\eqref{min.cap}}{\lesssim} \frac{\log R}{R^{3/2}} \left(\frac{\mathbb{P}[\mathrm{Cap}(\mathbb{R}R^{(m)}) \le \gamma_d R^5/2]}{R^6} + \frac{1}{R^{10}}\right)^{1/2} \\
\stackrel{\eqref{easy.variance}}{\lesssim} & \frac{\log R}{R^{3/2}} \left(\frac{(\log R)^2}{R^{11}} + \frac{1}{R^{10}}\right)^{1/2} \lesssim \frac{1}{R^6}.
\end{align*}
Next, recall that $Z(i)={\text{\Large $\mathfrak 1$}}\{S_j\neq S_i,\, \forall j>i\}$, and note that
$$|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell e_i^m\right] - \mathbb{E} \left[ \mathcal G_{i,R} Z(i) e_i^m\right]| \stackrel{\eqref{Green.hit},\, \eqref{Green}}{\lesssim} \frac{1}{R^{7/2}}.$$
Moreover, letting $e_i:=\mathbb{P}_{S_i}[H^+_{\overline \mathbb{R}R_\infty} = \infty\mid \overline \mathbb{R}R_\infty]$ (where we recall $\overline \mathbb{R}R_\infty$ is the range of a two-sided random walk), one has
$$ |\mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i^m\right] - \mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i \right]| \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{\sqrt R}, $$
$$| \mathbb{E} \left[ \mathcal G_{i,R} Z_i e_i \right] - \kappa| \le 2\, \mathbb{E}\left[\sum_{j>R} G(S_j)\right] \stackrel{\eqref{exp.Green}}{\lesssim}\frac 1{\sqrt R}.$$
Altogether this gives for any $m\ge 0$ and any $i\in I_m$,
$$\left|\mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] - \frac{\kappa}{\gamma_dR^5}\right|\lesssim \frac{1}{R^{5+\frac 12}},$$
and thus for any $m\ge 0$,
$$\left| \left(\sum_{i\in I_m} \mathbb{E} \left[ \mathcal G_{i,R} Z_i^\ell \overline e_i^m\right] \right) - \frac{\kappa}{\gamma_d}\right|\lesssim \frac{1}{\sqrt R}.$$
Now, a similar argument as in Step 6 shows that
$$\sum_{m=0}^M \left|\mathbb{E}_{0,x}\left[F(S_{i_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] - \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right]\right| \lesssim R^{\frac{5d}2} J_F(\|x\|). $$
Furthermore, using that
\begin{align*}
F(\widetilde S_\tau){\text{\Large $\mathfrak 1$}}\{\tau<\infty\} & = \sum_{m=0}^{M+1} F(\widetilde S_{\tau_m}){\text{\Large $\mathfrak 1$}}\{\tau=\tau_m<\infty\}\\
& =\sum_{m=0}^{M+1} F(\widetilde S_{\tau_m})({\text{\Large $\mathfrak 1$}} \{\tau_m<\infty\} - {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}),
\end{align*}
(with the convention that the term corresponding to index $M+1$ is zero when $\ell =\infty$) we get,
\begin{align*}
& \left| \sum_{m=0}^M \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau_m}) {\text{\Large $\mathfrak 1$}}\{\tau_m<\infty\} \right] - \mathbb{E}_{0,x}\left[F(\widetilde S_{\tau}) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right]\right| \\
& \lesssim \mathbb{P}_{0,x}[\tau_{M+1}<\infty] + \sum_{m=0}^M \mathbb{E}_{0,x}\left[|F(\widetilde S_{\tau_m})| {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}\right].
\end{align*}
Using \eqref{hit.cap.application}, \eqref{Doob} and \eqref{exp.Green.x}, we get
$$\mathbb{P}_{0,x}[\tau_{M+1}<\infty] \lesssim \frac{R^{\frac{5(d-2)}2}}{1+\|x\|^{d-2} }.$$
On the other hand, for any $m\ge 0$,
\begin{align*}
&\mathbb{E}\left[|F(\widetilde S_{\tau_m})| {\text{\Large $\mathfrak 1$}}\{\tau<\tau_m<\infty\}\right] \le \sum_{j\in J_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right]\\
& \le \sum_{j\in I_m} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \\
& \qquad + \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right].
\end{align*}
The first sum is handled as in Step 4. Namely,
\begin{align*}
& \sum_{m=0}^M \sum_{j\in I_m} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \\
& \le \sum_{j\ge 0} \sum_{|j-i|>R^3} \mathbb{E}\left[|F(S_j)| G(S_i-S_j)G(S_i-x)\right] \stackrel{\eqref{lem.thm.asymp.2bis}}{\lesssim} \frac{I_F(\|x\|)}{R^{3/2}} .
\end{align*}
Similarly, defining $\widetilde J_m:=\{mR^5,\dots, mR^5+R\} \cup \{(m+1)R^5-R,\dots,(m+1)R^5-1\}$, one has,
\begin{align*}
&\sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \le \sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{|i-j|>R} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \quad +\sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m,\, |i-j|\le R} \mathbb{E}\left[|F(S_j)|G(S_i-S_j)G(S_i-x)\right] \\
& \stackrel{\eqref{lem.thm.asymp.2bis}, \eqref{cond.F}}{\lesssim} \frac{I_F(\|x\|)}{\sqrt R} + \sum_{m=0}^M \sum_{j\in J_m\setminus I_m} \sum_{i\notin J_m,\, |i-j|\le R} \mathbb{E}\left[|F(S_i)|G(S_i-x)\right]\\
& \lesssim \frac{ I_F(\|x\|)}{\sqrt R} + R \sum_{m= 0}^M \sum_{i \in \widetilde J_m} \mathbb{E}\left[|F(S_i)|G(S_i-x)\right] \lesssim \frac{I_F(\|x\|)}{\sqrt R} + \frac{R^5}{1+\|x\|^{d-2}},
\end{align*}
using for the last inequality the same argument as in \eqref{final.step3}.
Note also that
$$\mathbb{E}[|F(\widetilde S_\tau)|{\text{\Large $\mathfrak 1$}}\{\tau<\infty\}] \stackrel{\eqref{lem.hit.1}}{\le} \sum_{i\ge 0} \mathbb{E}[|F(S_i)|G(S_i-x)]\lesssim I_F(\|x\|). $$
Therefore, putting all pieces together yields
$$|K_R - \widetilde A| \lesssim \frac{I_F(\|x\|)}{\sqrt{R}} + R^{\frac{5d}2}\cdot J_F(\| x\|) + \frac{R^{\frac{5(d-2)}{2}}}{1+\|x\|^{d-2}}.$$
\underline{Step 8.}
Altogether the previous steps show that for any $R$ large enough, any $\ell \ge 1$, and any $x\in\mathbb{Z}^d$, satisfying $\ell\wedge \|x\|^2 \ge R^6$,
$$|A-\widetilde A| \lesssim \left(\frac{1}{\sqrt{R}} + \frac{R^{6+\frac{5d}{2}}}{1+\|x\|}\right) \cdot I_F(\|x\|) + \frac{R^{\frac{5(d-2)}{2}}}{1+\|x\|^{d-2}} + R^{6+\frac{5d}{2}}\cdot J_F(\|x\|). $$
The proof of the theorem follows by taking for $R$ a sufficiently small power of $\|x\|\wedge \ell$, and observing that
for any function $F$ satisfying \eqref{cond.F}, one has $\liminf_{\|z\|\to \infty} |F(z)|/\|z\|>0$, and thus also $I_F(\|x\|) \ge \frac{c}{1+\|x\|^{d-3}}$.
\end{proof}
It amounts now to give the proofs of Lemmas \ref{lem.thm.asymptotic} and \ref{lem.potential}.
\begin{proof}[Proof of Lemma \ref{lem.thm.asymptotic}]
We start with the proof of \eqref{lem.thm.asymp.1}. Recall the definition of $\chi_d$ given just above Theorem \ref{thm.asymptotic}. One has for any $i\ge 0$,
\begin{align*}
&\mathbb{E}\left[\sum_{j= i+1}^\infty G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} \mid S_i \right] \stackrel{\eqref{Green}}{\lesssim}
\mathbb{E}\left[\sum_{j= i+1}^\infty \frac{1}{(1+\|S_j - S_i\|^{d-3})(1+\|S_j\|)} \mid S_i \right] \\
& \lesssim \sum_{z\in \mathbb{Z}^d} G(z) \frac{1}{(1+\|z\|^{d-3})(1+\|S_i+z\|)}\stackrel{\eqref{Green}}{\lesssim} \frac{\chi_d(\|S_i\|)}{1+\|S_i\|},
\end{align*}
and moreover,
\begin{align}\label{lem.FG}
\nonumber &\sum_{i=0}^\infty \mathbb{E}\left[\frac{|F(S_i)|\chi_d(\|S_i\|)}{1+\|S_i\|}G(S_i-x)\right] = \sum_{z\in \mathbb{Z}^d} G(z) \frac{|F(z)|\chi_d(\|z\|)}{1+\|z\|} G(z-x) \\
\nonumber \stackrel{\eqref{Green}}{\lesssim}& \frac{ \chi_d(\|x\|)}{1+\|x\|^{d-2}} \sum_{\|z\|\le \frac{\|x\|}{2}} \frac{|F(z)|}{1+\|z\|^{d-1}} + \sum_{\|z\|\ge \frac{\|x\|}{2}} \frac{|F(z)|\chi_d(\|z\|)}{1+\|z\|^{2d-3}} \\
\nonumber & \qquad + \frac{\chi_d(\|x\|)}{1+\|x\|^{d-1}} \sum_{\|z-x\|\le \frac{\|x\|}{2}} \frac{|F(z)|}{1+\|z-x\|^{d-2}} \\
\stackrel{\eqref{cond.F}}{\lesssim} & J_F(\|x\|/2) + \frac{|F(x)|\chi_d(\|x\|)}{1+\|x\|^{d-3}}\lesssim J_F(\|x\|),
\end{align}
where the last inequality follows from the fact that by \eqref{cond.F},
$$ \int_{\|x\|/2}^{\|x\|} \frac{\overline F(s)\chi_d(s)}{s^{d-2}} \, ds\, \asymp\, \frac{|F(x)|\chi_d(\|x\|)}{1+\|x\|^{d-3}}\, \asymp\, \frac{\chi_d(\|x\|)}{1+\|x\|^{d-2}} \int_{\|x\|/2}^{\|x\|} \overline F(s)\, ds.$$
Thus
$$\sum_{i=0}^\infty \sum_{j=i+1}^\infty \mathbb{E}\left[G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} |F(S_i)| G(S_i-x) \right] = \mathcal{O}(J_F(\|x\|)).$$
On the other hand, for any $j\ge 0$,
\begin{align}\label{lem.FG.2}
\nonumber &\mathbb{E}\left[\sum_{i= j+1}^\infty G(S_j-S_i) \|S_j-S_i\| \cdot |F(S_i)|G(S_i-x) \mid S_j \right] \\
\nonumber \stackrel{\eqref{Green}}{\lesssim} &
\sum_{i=j+1}^\infty \mathbb{E}\left[ \frac{|F(S_i)|G(S_i-x)}{1+\|S_j - S_i\|^{d-3}} \mid S_j \right]
\stackrel{\eqref{Green}}{\lesssim} \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j+z)| G(S_j + z-x)}{1+\|z\|^{2d-5}}\\
\nonumber \stackrel{\eqref{cond.F}, \eqref{Green}}{\lesssim} & \sum_{z\in \mathbb{Z}^d}
\frac{|F(S_j)|}{(1+\|z\|^{2d-5})(1+\|S_j+z-x\|^{d-2})} + \frac{1}{1+\|S_j\|^{2d-5}}\sum_{\|u\|\le \|S_j\|}\frac{|F(u)| }{1+\|u-x\|^{d-2}} \\
\nonumber \stackrel{\eqref{cond.F}}{\lesssim} & \frac{|F(S_j)|\chi_d(\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le \|x\|/2\}\cdot |F(S_j)|}{(1+\|x\|^{d-2})(1+\|S_j\|^{d-5})} \\
\nonumber & \qquad + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \|x\|/2\}}{1+\|S_j\|^{2d-5}}\left(|F(x)|(1+\| x\|^2) + |F(S_j)|(1+\|S_j\|^2)\right) \\
\lesssim & \frac{|F(S_j)|\chi_d(\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le \|x\|\} |F(S_j)|}{1+\|x\|^{d-2}} + \frac{{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \|x\| \} |F(S_j)|}{1+\|S_j\|^{d-2}},
\end{align}
where for the last two inequalities we used that by \eqref{cond.F}, if $\|u\|\le \|v\|$, then $|F(u)|\lesssim | F(v)| (1+\|v\|)/(1+\|u\|)$, and also that $d\ge 5$ for the last one.
Moreover, for any $r\ge 0$
$$\sum_{j=0}^\infty \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|S_j\|\le r\} \cdot |F(S_j)| }{1+\|S_j\|} \right] = \sum_{\|z\| \le r} \frac{G(z)|F(z)|}{1+\|z\|} \stackrel{\eqref{Green}}{=}\mathcal{O} \left(\int_0^{r} \overline F(s)\, ds\right),$$
$$\sum_{j=0}^\infty \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge r\} \cdot |F(S_j)| }{1+\|S_j\|^{d-1}} \right] = \sum_{\|z\| \ge r} \frac{G(z)|F(z)|}{1+\|z\|^{d-1}} \stackrel{\eqref{Green}}{=}\mathcal{O} \left(\int_{r}^\infty \frac{\overline F(s)}{s^{d-2}}\, ds\right).$$
Using also similar computations as in \eqref{lem.FG} to handle the first term in \eqref{lem.FG.2}, we conclude that
$$\sum_{j=0}^\infty \sum_{i=j+1}^\infty \mathbb{E}\left[G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} |F(S_i)| G(S_i-x) \right] = \mathcal{O}(J_F(\|x\|)),$$
which finishes the proof of \eqref{lem.thm.asymp.1}.
We then move to the proof of \eqref{lem.thm.asymp.2}. First note that for any $i\ge 0$,
$$\mathbb{E}\left[\sum_{j\ge i+R} G(S_j-S_i)\mid S_i \right] = \mathbb{E}\left[\sum_{j\ge R} G(S_j) \right] \stackrel{\eqref{exp.Green}}{=} \mathcal{O}\left(R^{\frac{4-d}{2}}\right),$$
and furthermore,
\begin{equation}\label{lem.FG.3}
\sum_{i=0}^\infty \mathbb{E}[|F(S_i)|G(S_i-x)] = \sum_{z\in \mathbb{Z}^d} |F(z)| G(z-x)G(z) \stackrel{ \eqref{cond.F},\, \eqref{Green}}{=} \mathcal{O}(I_F(\|x\|)),
\end{equation}
which together give the desired upper bound for the sum on the set $\{0\le i\le j-R\}$.
On the other hand, for any $j\ge 0$, we get as for \eqref{lem.FG.2},
\begin{align*}
&\mathbb{E}\left[\sum_{i\ge j+R} G(S_j-S_i)|F(S_i)|G(S_i-x) \mid S_j \right] = \sum_{z\in \mathbb{Z}^d} G(z)|F(S_j+z)| G(S_j+z-x) G_R(z) \\
& \stackrel{\eqref{Green}}{\lesssim} \frac{1}{R^{\frac{d-4}{2}}} \cdot \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j+z)|}{ (1+\|z\|^d)(1+\|S_j+z-x\|^{d-2})}\\
& \stackrel{ \eqref{cond.F}}{\lesssim} \frac{1}{R^{\frac{d-4}{2}}} \left\{ \sum_{z\in \mathbb{Z}^d} \frac{|F(S_j)|}{ (1+\|z\|^d)(1+\|S_j+z-x\|^{d-2})} + \frac 1{1+\|S_j\|^d} \sum_{\|u\|\le \|S_j\|} \frac{|F(u)|}{1+\|u-x\|^{d-2}}\right\}\\
&\lesssim \frac{1}{R^{\frac{d-4}{2}}} \left\{ \frac{|F(S_j)|\log (2+\|S_j-x\|)}{1+\|S_j-x\|^{d-2}} + \frac{|F(S_j)|}{1+\|x\|^{d-2}+\|S_j\|^{d-2}} \right\}.
\end{align*}
Then similar computation as above, see e.g. \eqref{lem.FG.3}, give
\begin{equation}\label{FSj}
\sum_{j\ge 0} \mathbb{E}\left[\frac{|F(S_j)|\log (2+\|S_j-x\|)}{1+\|S_j-x\|^{d-2}}\right] = \mathcal{O}(I_F(\|x\|)),
\end{equation}
\begin{equation*}
\sum_{j\ge 0} \mathbb{E}\left[\frac{|F(S_j)|}{1+\|x\|^{d-2} + \|S_j\|^{d-2}}\right] = \mathcal{O}(I_F(\|x\|)),
\end{equation*}
which altogether proves \eqref{lem.thm.asymp.2}.
The proof of \eqref{lem.thm.asymp.2bis} is entirely similar: on one hand, for any $i\ge 0$,
\begin{align*}
& \mathbb{E}\left[\sum_{j= i+R}^\infty G(S_j-S_i) |F(S_j)| \mid S_i \right] \stackrel{\eqref{cond.F}}{\lesssim} \mathbb{E}\left[\sum_{j= i+R}^\infty G(S_j-S_i) \frac{\|S_j-S_i\|}{1+\|S_j\|} \mid S_i \right] |F(S_i)| \\
& \lesssim \sum_{z\in \mathbb{Z}^d} G_R(z) \frac{|F(S_i)|}{(1+\|z\|^{d-3})(1+\|S_i+z\|)} \\
& \lesssim \sum_{z\in \mathbb{Z}^d} \frac{|F(S_i)|}{(R^{\frac{d-2}{2}} + \|z\|^{d-2})(1+\|z\|^{d-3})(1+\|S_i+z\|)} \lesssim \frac{|F(S_i)|}{R^{\frac{d-4}{2}}},
\end{align*}
and together with \eqref{FSj}, this yields
$$\sum_{i=0}^\infty \sum_{j=i+R}^\infty \mathbb{E}\left[G(S_j-S_i) |F(S_j)| G(S_i-x) \right]\lesssim \frac{I_F(\|x\|)}{R^{\frac{d-4}{2}} }.$$
On the other hand, for any $j\ge 0$, using \eqref{Green},
\begin{align*}
\mathbb{E}\left[\sum_{i\ge j+R} G(S_j-S_i)G(S_i-x) \mid S_j \right] \lesssim \sum_{z\in \mathbb{Z}^d} \frac{G(S_j + z-x)}{R^{\frac{d-4}{2}}(1+\|z\|^d)}
\lesssim \frac{\log (2+ \|S_j-x\|)}{R^{\frac{d-4}{2}}(1+\|S_j-x\|^{d-2})},
\end{align*}
and we conclude the proof of \eqref{lem.thm.asymp.2bis} using \eqref{FSj} again.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem.potential}] The first statement follows directly from \eqref{Green.asymp} and the last-exit decomposition (see Proposition 4.6.4 (c) in \cite{LL}):
$$\mathbb{P}_y[H_\Lambda<\infty] = \sum_{x\in \Lambda} G(y-x) e_\Lambda(x).$$
Indeed if $\|y\|>2 \text{rad}(\Lambda)$, using \eqref{Green} we get $G(y-x)\le C\|y\|^{2-d}$, for some constant $C>0$ independent of $x\in \Lambda$, which gives well \eqref{cap.hitting}, since by definition $\sum_{x\in \Lambda} e_\Lambda(x) = \mathrm{Cap}(\Lambda)$.
The second statement is more involved. Note that one can always assume $\mathcal J(y)>C\text{rad}(\Lambda)$, for some constant $C>0$, for otherwise the result is trivial. We use similar notation as in \cite{LL}. In particular $G_A(x,y)$ denotes the Green's function restricted to a subset $A\subseteq \mathbb{Z}^d$, that is the expected number of visits to $y$ before exiting $A$ for a random walk starting from $x$, and $H_A(x,y)=\mathbb{P}_x[H_{A^c}=y]$. We also let $\mathbb{C}C_n$ denote the (discrete) ball of radius $n$ for the norm $\mathcal J(\cdot)$. Then exactly as in \cite{LL} (see Lemma 6.3.3 and Proposition 6.3.5 thereof), one can see using \eqref{Green.asymp} that for all $n\ge 1$,
\begin{equation}\label{GA}
\left| G_{\mathbb{C}C_{n}}(x,w) - G_{\mathbb{C}C_{n}}(0,w) \right| \le C \frac{\|x\|}{1+\|w\|} \, G_{\mathbb{C}C_{n}}(0,w),
\end{equation}
for all $x\in \mathbb{C}C_{n/4}$, and all $w$ satisfying $2\mathcal J(x) \le \mathcal J(w) \le n/2$. One can then derive an analogous estimate for the (discrete) derivative of $H_{\mathbb{C}C_n}$. Define $A_n=\mathbb{C}C_n\setminus \mathbb{C}C_{n/2}$, and $\rho = H^+_{A_n^c}$. By the last-exit decomposition (see \cite[Lemma 6.3.6]{LL}), one has for $x\in \mathbb{C}C_{n/8}$ and $z\notin \mathbb{C}C_n$,
\begin{align*}
\nonumber & |H_{\mathbb{C}C_n}(x,z) - H_{\mathbb{C}C_n}(0,z)|\le \sum_{w\in \mathbb{C}C_{n/2}}|G_{\mathbb{C}C_n}(x,w) - G_{\mathbb{C}C_n}(0,w)|\cdot \mathbb{P}_w[S_\rho = z]\\
\nonumber & \stackrel{\eqref{GA}, \eqref{Green}}{\lesssim} \frac{\|x\|}{n}\cdot H_{\mathbb{C}C_n}(0,z) + \sum_{2\mathcal J(x)\le \mathcal J(w)\le \frac n4} \frac{\|x\|}{\|w\|^{d-1}} \mathbb{P}_w[S_\rho = z] \\
& \qquad + \sum_{\mathcal J(w) \le 2\mathcal J(x)} \left(\frac 1{1+\|w-x\|^{d-2}} + \frac 1{1+\|w\|^{d-2}}\right)\mathbb{P}_w[S_\rho = z].
\end{align*}
Now, observe that for any $y\notin \mathbb{C}C_n$, any $w\in \mathbb{C}C_{n/4}$, and any $A\subseteq \mathbb{Z}^d$,
\begin{align*}
\sum_{z\notin \mathbb{C}C_n} G_{A} (y,z) \mathbb{P}_w[S_\rho = z] \lesssim \sum_{z\notin \mathbb{C}C_n} \mathbb{P}_w[S_\rho = z] \lesssim \mathbb{P}_w[\mathcal J(S_1)>\frac n2] \lesssim \mathbb{P}[\mathcal J(X_1)> \frac n4] \lesssim n^{-d},
\end{align*}
using that by hypothesis $\mathcal J(X_1)$ has a finite $d$-th moment.
It follows from the last two displays that
\begin{align}\label{potentiel.3}
\sum_{z\notin \mathbb{C}C_n} G_{A} (y,z) H_{\mathbb{C}C_n}(x,z) = \left(\sum_{z\notin \mathbb{C}C_n} G_{A} (y,z) H_{\mathbb{C}C_n}(0,z)\right)\left(1+\mathcal{O}\Big(\frac{\|x\|}{n}\Big)\right)+ \mathcal{O}\left( \frac{\|x\|}{n^{d-1}} \right).
\end{align}
Now let $\Lambda$ be some finite subset of $\mathbb{Z}^d$ containing the origin, and let $m:=\sup\{\mathcal J(u)\, :\, \|u\|\le 2\text{rad}(\Lambda)\}$. Note that $m=\mathcal{O}(\text{rad}(\Lambda))$, and thus one can assume $\mathcal J(y)>16m$. Set $n:=\mathcal J(y)-1$.
Using again the last-exit decomposition and symmetry of the step distribution, we get for any $x\in \Lambda$,
\begin{align}\label{potentiel.4}
\mathbb{P}_y[S_{H_\Lambda} =x,\, H_\Lambda<\infty] = \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z) \mathbb{P}_x[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+],
\end{align}
with $\tau_n:= H_{\mathbb{C}C_n^c}$.
We then write, using the Markov property,
\begin{align}\label{potentiel.5}
\nonumber \mathbb{P}_x[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+]& =\sum_{x'\in \mathbb{C}C_{n/8}\setminus \mathbb{C}C_m}\mathbb{P}_x[\tau_m<H_\Lambda^+,\, S_{\tau_m} =x']\cdot \mathbb{P}_{x'}[S_{\tau_n} = z,\, \tau_n<H_\Lambda^+] \\
& \qquad + \mathbb{P}_x\left[\mathcal J(S_{\tau_m}) >\frac n8,\, S_{\tau_n}=z\right],
\end{align}
with $\tau_m:=H_{\mathbb{C}C_m^c}$. Concerning the last term we note that
\begin{align}\label{potentiel.5.bis}
\nonumber & \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z) \mathbb{P}_x\left[\mathcal J(S_{\tau_m}) >\frac n8,\, S_{\tau_n}=z\right] \\
\nonumber & \stackrel{\eqref{Green.hit}}{\le} \sum_{z\notin \mathbb{C}C_n} G(z-y) \left\{\mathbb{P}_x[S_{\tau_m}=z] + \sum_{u\in \mathbb{C}C_n\setminus \mathbb{C}C_{n/8}} \mathbb{P}_x[S_{\tau_m} =u]G(z-u)\right\}\\
\nonumber & \stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_{z\notin \mathbb{C}C_n} G(z-y) \mathbb{P}_x[S_{\tau_m}=z] + \sum_{u\in \mathbb{C}C_n\setminus \mathbb{C}C_{n/8}} \frac{\mathbb{P}_x[S_{\tau_m} =u]}{\|y-u\|^{d-4}} \\
\nonumber & \lesssim \mathbb{P}_x[\mathcal J(S_{\tau_m})>n/8] \lesssim \sum_{u\in \mathbb{C}C_m} G_{\mathbb{C}C_m}(x,u) \mathbb{P}[J(X_1)>\frac n8 - m] \\
& \stackrel{\eqref{Green}}{=}\mathcal{O}\left(\frac{m^2}{n^d}\right) = \mathcal{O}\left(\frac{m}{n^{d-1}}\right),
\end{align}
applying once more the last-exit decomposition at the penultimate line, and the hypothesis that $\mathcal J(X_1)$ has a finite $d$-th moment at the end.
We handle next the sum in the right-hand side of \eqref{potentiel.5}. First note that \eqref{potentiel.3} gives for any $x'\in \mathbb{C}C_{n/8}$,
\begin{align}\label{potentiel.6}
\nonumber \sum_{z\notin \mathbb{C}C_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z] = \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(x', z)\\
& = \left(\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right) + \mathcal{O}\left( \frac{\|x'\|}{n^{d-1}} \right).
\end{align}
Observe then two facts. On one hand, by the last exit-decomposition and symmetry of the step distribution,
\begin{equation}\label{pot.1}
\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z) \le \sum_{z\notin \mathbb{C}C_n} G_{\mathbb{Z}^d\setminus \{0\}}(y,z)H_{\mathbb{C}C_n}(0, z) = \mathbb{P}[H_y<\infty] \stackrel{\eqref{Green.hit}, \eqref{Green}}{\lesssim} n^{2-d},
\end{equation}
and on the other hand by Proposition 4.6.2 in \cite{LL},
\begin{align}\label{pot.2}
& \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z) H_{\mathbb{C}C_n}(0, z) \\
\nonumber & = \sum_{z\notin \mathbb{C}C_n} G_{\mathbb{Z}^d\setminus \{0\}}(y,z)H_{\mathbb{C}C_n}(0, z) + \sum_{z\notin \mathbb{C}C_n} \left(G_{\Lambda^c}(y,z) - G_{\mathbb{Z}^d\setminus \{0\}}(y,z)\right) H_{\mathbb{C}C_n}(0, z) \\
\nonumber & \ge \mathbb{P}[H_y<\infty] - \mathcal{O}\left(\mathbb{P}_y[H_\Lambda<\infty]\sum_{z\notin \mathbb{C}C_n} G(z) H_{\mathbb{C}C_n}(0, z)\right) \\
\nonumber & \stackrel{\eqref{hit.ball}}{\ge} \mathbb{P}[H_y<\infty] - \mathcal{O}\left(n^{2-d}\sum_{z\notin \mathbb{C}C_n} G(z)^2\right)
\stackrel{\eqref{Green.asymp}}{\ge} \frac{c}{n^{d-2}}.
\end{align}
This last fact, combined with \eqref{potentiel.6} gives therefore, for any $x'\in \mathbb{C}C_{n/8}$,
\begin{align}\label{potentiel.6.bis}
\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z] = \left(\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right).
\end{align}
By the Markov property, we get as well
\begin{align*}
\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z\mid H_\Lambda<\tau_n ] = \left(\sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}n\Big)\right),
\end{align*}
since by definition $\Lambda\subseteq \mathbb{C}C_m\subset \mathbb{C}C_{n/8}$, and thus
\begin{align*}
\sum_{z\notin \mathbb{C}C_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z,\, H_\Lambda<\tau_n ] \\
& = \mathbb{P}_{x'}[H_\Lambda<\tau_n]\left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}n\Big)\right).
\end{align*}
Subtracting this from \eqref{potentiel.6.bis}, we get for $x'\in \mathbb{C}C_{n/8}\setminus \mathbb{C}C_m$,
\begin{align*}
\sum_{z\notin \mathbb{C}C_n} & G_{\Lambda^c}(y,z)\mathbb{P}_{x'}[S_{\tau_n} = z,\, \tau_n<H_\Lambda ] \\
& = \mathbb{P}_{x'}[\tau_n<H_\Lambda]\left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{\|x'\|}n\Big)\right),
\end{align*}
since by \eqref{hit.ball}, one has $\mathbb{P}_{x'}[\tau_n<H_\Lambda]>c$, for some constant $c>0$, for any $x'\notin \mathbb{C}C_m$ (note that the stopping time theorem gives in fact
$\mathbb{P}_{x'}[H_\Lambda<\infty] \le G(x') / \inf_{\|u\|\le \text{rad}(\Lambda)} G(u)$, and thus by using \eqref{Green.asymp}, one can ensure $\mathbb{P}_{x'}[H_\Lambda<\infty]\le 1-c$, by taking $\|x'\|$ large enough, which is always possible).
Combining this with \eqref{potentiel.4}, \eqref{potentiel.5} and \eqref{potentiel.5.bis}, and using as well \eqref{pot.1} and \eqref{pot.2}, we get
\begin{align*}
& \mathbb{P}_y[S_{H_\Lambda} =x,\, H_\Lambda<\infty] = \mathbb{P}_x[\tau_n<H_\Lambda]\left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right) \\
& \qquad + \mathcal{O}\left(\frac 1{n^{d-1}}\sum_{x'\in\mathbb{C}C_{n/8} \setminus \mathbb{C}C_m} \mathbb{P}_x[S_{\tau_m}=x'] \cdot \|x'\| \right) + \mathcal{O}\left(\frac{m}{n^{d-1}}\right)\\
\stackrel{\eqref{hit.ball}}{=} & e_\Lambda(x)\left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right)+ \mathcal{O}\left(\frac 1{n^{d-1}}\sum_{r=2m}^{n/8} \frac{m^2}{r^{d-1}} \right) + \mathcal{O}\left(\frac{m}{n^{d-1}}\right)\\
= & \ e_\Lambda(x)\left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right),
\end{align*}
using the same argument as in \eqref{potentiel.5.bis} for bounding $\mathbb{P}_{x'}[\mathcal J(S_{\tau_m})\ge r]$, when $r\ge 2m$.
Summing over $x\in \Lambda$ gives
\begin{align*}
\mathbb{P}_y[H_\Lambda<\infty]= \mathrm{Cap}(\Lambda) \left( \sum_{z\notin \mathbb{C}C_n} G_{\Lambda^c}(y,z)H_{\mathbb{C}C_n}(0, z)\right)\left(1+\mathcal{O}\Big(\frac{m}{n}\Big)\right),
\end{align*}
and the proof of the lemma follows from the last two displays.
\end{proof}
\section{Proof of Proposition \ref{prop.phipsi.2}}
The proof is divided in four steps, corresponding to the next four lemmas.
\begin{lemma} \label{lem.var.1}
Assume that $\varepsilon_k\to \infty$, and $\varepsilon_k/k\to 0$. There exists a constant $\sigma_{1,3}>0$, such that
$$\operatorname{Cov}(Z_0\varphi_3, Z_k\psi_1) \sim \frac{\sigma_{1,3}}{k}.$$
\end{lemma}
\begin{lemma} \label{lem.var.2}
There exist positive constants $\delta$ and $\sigma_{1,1}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$,
$$ \operatorname{Cov}(Z_0\varphi_1,Z_k\psi_1) \sim \operatorname{Cov}(Z_0\varphi_3, Z_k\psi_3) \sim \frac{\sigma_{1,1}}{k}.$$
\end{lemma}
\begin{lemma}\label{lem.var.3}
There exist positive constants $\delta$ and $\sigma_{1,2}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$,
$$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_1) \sim \operatorname{Cov}(Z_0\varphi_3, Z_k\psi_2) \sim \frac{\sigma_{1,2}}{k}.$$
\end{lemma}
\begin{lemma}\label{lem.var.4}
There exist positive constants $\delta$ and $\sigma_{2,2}$, such that when $\varepsilon_k\ge k^{1-\delta}$, and $\varepsilon_k/k\to 0$,
$$ \operatorname{Cov}(Z_0\varphi_2,Z_k\psi_2) \sim \frac{\sigma_{2,2}}{k}.$$
\end{lemma}
\subsection{Proof of Lemma \ref{lem.var.1}}
We assume now to simplify notation that the distribution $\mu$ is aperiodic, but it should be clear from the proof that the case of a bipartite walk could be handled similarly.
The first step is to show that
\begin{equation}\label{var.1.1}
\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \rho^2 \left\{\sum_{x\in \mathbb{Z}^5} p_k(x) \operatorname{Var}phi_x^2 - \left(\sum_{x\in \mathbb{Z}^5} p_k(x) \operatorname{Var}phi_x\right)^2\right\} + o\left(\frac 1k\right),
\end{equation}
where $\rho$ and $\operatorname{Var}phi_x$ are defined respectively as
\begin{eqnarray}\label{def.rho}
\rho:= \mathbb{E}\left[ \mathbb{P}\left[H^+_{\overline \mathbb{R}R_\infty } = \infty \mid (S_n)_{n\in \mathbb{Z}} \right] \cdot {\text{\Large $\mathfrak 1$}}\{S_\ell \neq 0,\, \forall \ell \ge 1\}\right] ,
\end{eqnarray}
and
$$\operatorname{Var}phi_x : = \mathbb{P}_{0,x} [\mathbb{R}R_\infty \cap \widetilde \mathbb{R}R_\infty \neq \varnothing].$$
To see this, one needs to dissociate $Z_0$ and $Z_k$, as well as the events of avoiding $\mathbb{R}R[-\varepsilon_k,\varepsilon_k]$ and $\mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k]$ by two independent walks starting respectively from the origin and from $S_k$,
which are local events (in the sense that they only concern small parts of the different paths), from the events of hitting $\mathbb{R}R[k+1,\infty)$ and $\mathbb{R}R(-\infty,-1]$ by these two walks, which involve different parts of the trajectories.
To be more precise, consider $(S_n^1)_{n\ge 0}$ and $(S_n^2)_{n\ge 0}$, two independent random walks starting from the origin, and independent of $(S_n)_{n\in \mathbb{Z}}$. Then define
$$\tau_1:= \inf\{n\ge \varepsilon_k : S_n^1 \in \mathbb{R}R[k+\varepsilon_k,\infty) \}, \ \tau_2:= \inf \{n\ge \varepsilon_k : S_k + S_n^2 \in \mathbb{R}R(-\infty,-\varepsilon_k]\}.$$
We first consider the term $\mathbb{E}[Z_0\varphi_3]$. Let
$$\tau_{0,1}:=\inf \left\{n\ge \varepsilon_k \, :\, S_n^1 \in \mathbb{R}R[-\varepsilon_k,\varepsilon_k]\right\},$$
and
$$\Delta_{0,3}:= \mathbb{E}\left[Z_0\cdot {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k]\cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] =\varnothing\}\cdot {\text{\Large $\mathfrak 1$}}\{ \tau_1 <\infty\}\right].$$
One has,
\begin{align*}
\left| \mathbb{E}[Z_0\varphi_3] - \Delta_{0,3}\right| &\le \mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty \right] + \mathbb{P}\left[ \mathbb{R}R^1[0,\varepsilon_k] \cap \mathbb{R}R[k,\infty)\neq \varnothing \right] \\
& \qquad +\mathbb{P}[\mathbb{R}R^1_\infty \cap \mathbb{R}R[k,k+\varepsilon_k]\neq \varnothing] \\
& \stackrel{\eqref{lem.hit.3}}{\le} \mathbb{P}[\tau_1\le \tau_{0,1}<\infty] + \mathbb{P}[\tau_{0,1}\le \tau_1<\infty ] + \mathcal{O}\left(\frac{\varepsilon_k}{k^{3/2}} \right).
\end{align*}
Next, conditioning on $\mathbb{R}R[-\varepsilon_k,\varepsilon_k]$ and using the Markov property at time $\tau_{0,1}$, we get with $X=S_{\varepsilon_k} - S^1_{\tau_{0,1}}$,
\begin{align*}
\mathbb{P}[\tau_{0,1}\le \tau_1<\infty ] &\le \mathbb{E}\left[\mathbb{P}_{0,X}[\mathbb{R}R[k,\infty)\cap \widetilde \mathbb{R}R_\infty \neq \varnothing] \cdot {\text{\Large $\mathfrak 1$}}\{\tau_{0,1}<\infty\}\right] \\
& \stackrel{\eqref{lem.hit.3}}{=}
\mathcal{O}\left(\frac{\mathbb{P}[\tau_{0,1}<\infty] }{\sqrt{k}}\right) \stackrel{\eqref{lem.hit.3}}{=} \mathcal{O}\left(\frac 1{\sqrt {k \varepsilon_k}} \right).
\end{align*}
Likewise, using the Markov property at time $\tau_1$, we get
\begin{align*}
& \mathbb{P}[\tau_1\le \tau_{0,1}<\infty ] \stackrel{\eqref{lem.hit.1}}{\le} \mathbb{E}\left[\left(\sum_{j=-\varepsilon_k}^{\varepsilon_k} G(S_j- S_{\tau_1}^1)\right) {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}\right] \\
& \stackrel{\eqref{lem.hit.1}}{\le}\sum_{i=k+\varepsilon_k}^\infty \sum_{j=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_j- S_i)G(S_i-S^1_{\varepsilon_k})\right]\\
& \le (2\varepsilon_k+1) \sup_{x\in \mathbb{Z}^5} \sum_{i=k}^\infty \mathbb{E}\left[G(S_i)G(S_i-x)\right] \\
& \le (2\varepsilon_k+1) \sup_{x\in \mathbb{Z}^5} \sum_{z\in \mathbb{Z}^5} G(z) G(z-x) G_k(z) \stackrel{\eqref{Green}, \, \text{Lemma }\ref{lem.upconvolG}}{=}\mathcal{O}\left(\frac{\varepsilon_k}{k^{3/2}}\right).
\end{align*}
Now define for any $y_1,y_2\in \mathbb{Z}^5$,
\begin{equation}\label{Hy1y2}
H(y_1,y_2):= \mathbb{E}\left[Z_0 {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k]\cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] =\varnothing, S_{\varepsilon_k }= y_1, S^1_{\varepsilon_k} = y_2 \}\right].
\end{equation}
One has by the Markov property
\begin{align*}
\Delta_{0,3} =\sum_{x\in \mathbb{Z}^5} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1) \operatorname{Var}phi_x.
\end{align*}
Observe that typically $\|y_1\|$ and $\|y_2\|$ are much smaller than $\|x\|$, and thus $p_k(x+y_2-y_1)$ should be also typically close to $p_k(x)$.
To make this precise, consider $(\chi_k)_{k\ge 1}$ some sequence of positive integers, such that $\varepsilon_k \chi_k^3 \le k$, for all $k\ge 1$, and $\chi_k\to \infty$, as $k\to \infty$.
One has using Cauchy-Schwarz at the third line,
\begin{align*}
& \sum_{\|x\|^2\le k/\chi_k} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1)\operatorname{Var}phi_{x} \\
& \le \sum_{\|x\|^2\le k/\chi_k } \sum_{y_2\in \mathbb{Z}^5} p_{\varepsilon_k}(y_2) p_{k+\varepsilon_k}(x) \operatorname{Var}phi_{x-y_2}
\stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|S_{k+\varepsilon_k}\|^2 \le k/\chi_k\}}{1+\|S_{k+\varepsilon_k} - S^1_{\varepsilon_k}\|}\right]\\
& \lesssim \mathbb{E}\left[\frac 1{1+\|S_{k+2\varepsilon_k}\|^2}\right]^{1/2}\cdot \mathbb{P}\left[\|S_{k+\varepsilon_k}\|^2\le k/\chi_k\right]^{1/2}\stackrel{\eqref{pn.largex}}{\lesssim} \frac 1{\sqrt{k}\cdot \chi_k^{5/4}}.
\end{align*}
Likewise, using just \eqref{Sn.large} at the end instead of \eqref{pn.largex}, we get
$$\sum_{\|x\|^2 \ge k\chi_k} \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) p_k(x+y_2-y_1)\operatorname{Var}phi_{x} \lesssim \frac 1{\sqrt{k}\cdot \chi_k^{5/4}},$$
and one can handle the sums on the sets $\{\|y_1\|^2\ge \varepsilon_k\chi_k\}$ and $\{\|y_2\|^2\ge \varepsilon_k\chi_k\}$ similarly.
Therefore, it holds
$$\Delta_{0,3} =\sum_{k/\chi_k \le \|x\|^2 \le k \chi_k } \sum_{\|y_1\|^2\le \varepsilon_k \chi_k} \sum_{\|y_2\|^2\le \varepsilon_k \chi_k} H(y_1,y_2) p_k(x+y_2-y_1) \operatorname{Var}phi_x + \mathcal{O}\left(\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} \right).$$
Moreover, Theorem \ref{LCLT} shows that for any $x,y_1,y_2$ as in the three sums above, one has
$$|p_k(x+y_2-y_1)-p_k(x)| = \mathcal{O}\left(\frac{\sqrt{\varepsilon_k} \cdot \chi_k}{\sqrt k}\cdot p_k(x) + \frac 1{k^{7/2}}\right).$$
Note also that by \eqref{lem.hit.2}, one has
\begin{equation}\label{sum.pkxphix}
\sum_{x,y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2)p_k(x) \varphi_x\le \sum_{x\in \mathbb{Z}^5} p_k(x) \operatorname{Var}phi_x = \mathcal{O}\left(\frac 1{\sqrt{k}}\right).
\end{equation}
Using as well that $\sqrt{\varepsilon_k}\chi_k \le \sqrt {k/\chi_k}$, and $\sum_{\|x\|^2 \le k\chi_k} \varphi_x = \mathcal{O}(k^2\chi_k^2)$, we get
\begin{align*}
\Delta_{0,3} =\rho_k \sum_{x\in \mathbb{Z}^5 } p_k(x) \operatorname{Var}phi_x + \mathcal{O}\left(\frac{1}{\sqrt{k\cdot \chi_k}} + \frac{\chi_k^2}{k^{3/2}}\right),
\end{align*}
with
$$\rho_k:= \sum_{y_1,y_2\in \mathbb{Z}^5} H(y_1,y_2) = \mathbb{E}\left[Z_0\cdot {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k] \cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] = \varnothing\}\right].$$
Note furthermore that one can always take $\chi_k$ such that $\chi_k=o( \sqrt k)$, and that by \eqref{Green.hit}, \eqref{Green} and \eqref{lem.hit.3}, one has
$|\rho_k - \rho| \lesssim \varepsilon_k^{-1/2}$.
This gives
\begin{eqnarray}\label{Z0phi3.final}
\mathbb{E}[Z_0\varphi_3] = \rho \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x + o\left(\frac{1}{\sqrt{k}}\right).
\end{eqnarray}
By symmetry the same estimate holds for $\mathbb{E}[Z_k\psi_1]$, and thus using again \eqref{sum.pkxphix}, it entails
$$\mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_1] = \rho^2\left( \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x\right)^2 + o\left(\frac{1}{k}\right).$$
The estimate of $\mathbb{E}[Z_0\varphi_3Z_k\psi_1]$ is done along the same line, but is a bit more involved. Indeed, let
\begin{align*}
\Delta_{1,3}:= \mathbb{E}\left[\right. &Z_0Z_k {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k]\cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] =\varnothing\} \\
& \left. \times {\text{\Large $\mathfrak 1$}}\{(S_k+\mathbb{R}R^2[1,\varepsilon_k])\cap \mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k] =\varnothing, \tau_1 <\infty, \tau_2<\infty\}\right].
\end{align*}
The difference between $\mathbb{E}[Z_0\varphi_3Z_k\psi_1]$ and $\Delta_{1,3}$ can be controlled roughly as above, but one needs additionally to handle the probability of $\tau_2$ being finite.
Namely one has using symmetry,
\begin{align}\label{Delta13}
& |\mathbb{E}[Z_0\varphi_3Z_k\psi_1] - \Delta_{1,3} | \le 2\left(\mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \overline \tau_2<\infty \right] \right. \\
\nonumber + & \mathbb{P}\left[ \mathbb{R}R^1[0,\varepsilon_k] \cap \mathbb{R}R[k,\infty)\neq \varnothing,\, \overline \tau_2<\infty \right] + \left. \mathbb{P}[\mathbb{R}R^1_\infty \cap \mathbb{R}R[k,k+\varepsilon_k]\neq \varnothing, \, \overline \tau_2<\infty]\right),
\end{align}
with
$$\overline \tau_2:= \inf\{n\ge 0 \, : \, S_k+S^2_n \in \mathbb{R}R(-\infty,0]\}.$$
The last term in \eqref{Delta13} is handled as follows:
\begin{align*}
&\mathbb{P}\left[\mathbb{R}R^1_\infty \cap \mathbb{R}R[k,k+\varepsilon_k]\neq \varnothing, \overline \tau_2<\infty\right] =\sum_{x\in \mathbb{Z}^5}
\mathbb{P}[\mathbb{R}R^1_\infty \cap \mathbb{R}R\left[k,k+\varepsilon_k]\neq \varnothing, \overline \tau_2<\infty, S_k=x\right] \\
& \stackrel{\eqref{lem.hit.1}}{\le} \sum_{x\in \mathbb{Z}^5} p_k(x)\varphi_x \sum_{i=0}^{\varepsilon_k} \mathbb{E}[G(S_i+x)]
\stackrel{\eqref{Green}, \eqref{lem.hit.2}, \eqref{exp.Green.x}}{\lesssim} \varepsilon_k \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|^4} \stackrel{\eqref{pn.largex}}{\lesssim} \frac{\varepsilon_k}{k^2}.
\end{align*}
The same arguments give as well
$$\mathbb{P}\left[ \mathbb{R}R^1[0,\varepsilon_k] \cap \mathbb{R}R[k,\infty)\neq \varnothing,\, \overline \tau_2<\infty \right] \lesssim \frac{\varepsilon_k}{k^2},$$
$$\mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \overline \tau_2<\infty \right] = \mathbb{P}\left[\tau_{0,1}<\infty,\, \tau_1<\infty, \, \tau_2<\infty \right] + \mathcal{O}\left(\frac{\varepsilon_k}{k^2}\right).$$
Then we can write,
\begin{align*}
&\mathbb{P}\left[\tau_{0,1}\le \tau_1<\infty, \tau_2<\infty \right] = \mathbb{E} \left[ \mathbb{P}_{0,S_{k+\varepsilon_k}-S_{\tau_{0,1}}}[\mathbb{R}R_\infty \cap \widetilde \mathbb{R}R_\infty \neq \varnothing]
{\text{\Large $\mathfrak 1$}}\{\tau_{0,1}<\infty, \tau_2<\infty\}\right] \\
& \stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac {1}{1+\|S_{k+\varepsilon_k}-S_i\|}\cdot \frac{G(S_i-S^1_{\varepsilon_k})}{1+\|S_k-S_{-\varepsilon_k}\|}\right]\\
& \stackrel{\eqref{exp.Green}}{\lesssim}\frac{1}{\varepsilon_k^{3/2}} \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac {1}{1+\|S_k-S_i\|}\cdot \frac{1}{1+\|S_k-S_{-\varepsilon_k}\|}\right] \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}} \max_{k-\varepsilon_k\le j \le k+\varepsilon_k}\, \sup_{u\in\mathbb{Z}^d}\, \mathbb{E}\left[\frac {1}{1+\|S_j\|}\cdot \frac{1}{1+\|S_j+u\|}\right]
\lesssim \frac{1}{k\sqrt{\varepsilon_k}},
\end{align*}
where the last equality follows from straightforward computations, using \eqref{pn.largex}. On the other hand,
\begin{align*}
&\mathbb{P}\left[\tau_1\le \tau_{0,1}<\infty, \tau_2<\infty \right] \stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=k+\varepsilon_k}^\infty \sum_{j=-\varepsilon_k}^{\varepsilon_k}
\mathbb{E}\left[\frac{G(S_j-S_i)G(S_i-S_{\varepsilon_k}^1)}{1+\|S_k-S_{-\varepsilon_k}\|}\right] \\
& \stackrel{\eqref{Green}, \eqref{exp.Green.x}}{\lesssim} \sum_{j=-\varepsilon_k}^{\varepsilon_k} \sum_{i=k+\varepsilon_k}^\infty
\mathbb{E}\left[\frac{G(S_j-S_i)}{(1+\|S_i\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\right] \\
& \lesssim \sum_{j=-\varepsilon_k}^{\varepsilon_k}\sum_{z\in \mathbb{Z}^d} G_{\varepsilon_k}(z) \mathbb{E}\left[\frac{G(z+ S_k-S_j)}{(1+\|z+S_k\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\right].
\end{align*}
Note now that for $x,y\in \mathbb{Z}^5$, by \eqref{Green} and Lemma \ref{lem.upconvolG},
\begin{align*}
\sum_{z\in \mathbb{Z}^d} \frac{G_{\varepsilon_k}(z)}{(1+\|z-x\|^3)(1+\|z-y\|^3)} \lesssim \frac{1}{1+\|x\|^3} \left(\frac 1{\sqrt{\varepsilon_k}} + \frac{1}{1+\|y-x\|}\right) .
\end{align*}
It follows that
\begin{align*}
&\mathbb{P}\left[\tau_1\le \tau_{0,1}<\infty, \tau_2<\infty \right] \lesssim \sum_{j=-\varepsilon_k}^{\varepsilon_k}
\mathbb{E}\left[\frac{1}{(1+\|S_k\|^3)(1+\|S_k-S_{-\varepsilon_k}\|)}\left(\frac 1{\sqrt{\varepsilon_k}} + \frac 1{1+\|S_j\|}\right) \right]\\
& \stackrel{\eqref{exp.Green.x}}{\lesssim} \mathbb{E}\left[\frac {\sqrt{\varepsilon_k}}{1+\|S_k\|^4}\right] + \sum_{j=-\varepsilon_k}^0 \mathbb{E}\left[\frac 1{(1+\|S_k\|^3)(1+\|S_k-S_j\|)(1+\|S_j\|)}\right] \\
& \qquad + \sum_{j=1}^{\varepsilon_k} \mathbb{E}\left[\frac 1{(1+\|S_k\|^4)(1+\|S_j\|)}\right]\\
& \lesssim \frac{1}{k^2} \left(\sqrt{\varepsilon_k} + \sum_{j=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[\frac 1{1+\|S_j\|}\right] \right)\lesssim \frac{\sqrt{\varepsilon_k}}{k^2},
\end{align*}
using for the third inequality that by \eqref{pn.largex}, it holds uniformly in $x\in \mathbb{Z}^5$ and $j\le \varepsilon_k$,
$$\mathbb{E}\left[\frac 1{1+\|S_k-S_j+x\|^4}\right] \lesssim k^{-2}, \quad \mathbb{E}\left[\frac 1{(1+\|S_k\|^3) (1+\|S_k+x\|)}\right] \lesssim k^{-2}.$$
Now we are left with computing $\Delta_{1.3}$. This step is essentially the same as above, so we omit to give all the details. We first define for $y_1,y_2,y_3 \in \mathbb{Z}^5$,
$$H(y_1,y_2,y_3):= \mathbb{E}\left [Z_0 {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k]\cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] =\varnothing, S_{\varepsilon_k} =y_1, S^1_{\varepsilon_k} = y_2, S_{-\varepsilon_k} = y_3\}\right],$$
and note that
$$\Delta_{1,3}= \sum_{\substack{y_1,y_2,y_3\in \mathbb{Z}^5 \\ z_1,z_2,z_3\in \mathbb{Z}^5 \\ x\in \mathbb{Z}^5}} H(y_1,y_2,y_3) H(z_1,z_2,z_3) p_{k-2\varepsilon_k}(x-y_1+z_3) \operatorname{Var}phi_{x+z_1-y_2} \operatorname{Var}phi_{x + z_2-y_3}.$$
Observe here that by Theorem C, $\varphi_{x+z_1-y_2}$ is equivalent to $\varphi_x$, when $\|z_1\|$ and $\|y_2\|$ are small when compared to $\|x\|$, and similarly for $\varphi_{x+z_2-y_3}$. Thus using similar arguments as above, and in particular that by \eqref{pn.largex} and \eqref{lem.hit.2},
\begin{equation}\label{sum.pkxphix.2}
\sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x^2 = \mathcal{O}\left(\frac 1k\right),
\end{equation}
we obtain
$$\Delta_{1,3} = \rho^2 \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x^2 + o\left(\frac 1k\right).$$
Putting all pieces together gives \eqref{var.1.1}.
Using in addition \eqref{pn.largex}, \eqref{lem.hit.2} and Theorem \ref{LCLT}, we deduce that
$$\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \rho^2 \left\{\sum_{x\in \mathbb{Z}^5} \overline p_k(x) \operatorname{Var}phi_x^2 - \left(\sum_{x\in \mathbb{Z}^5} \overline p_k(x) \operatorname{Var}phi_x\right)^2\right\} + o\left(\frac 1k\right).$$
Then Theorem C, together with \eqref{sum.pkxphix} and \eqref{sum.pkxphix.2} show that
$$\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) = \sigma \left\{\sum_{x\in \mathbb{Z}^5} \frac{\overline p_k(x)}{1+\mathcal J(x)^2} - \left(\sum_{x\in \mathbb{Z}^5} \frac{\overline p_k(x)}{1+\mathcal J(x)} \right)^2\right\} +o\left(\frac 1k\right), $$
for some constant $\sigma>0$. Finally an approximation of the series with an integral and a change of variables gives, with $c_0:=(2\pi)^{-5/2} (\det \Gamma)^{-1/2}$,
\begin{align*}
\operatorname{Cov}(Z_0\varphi_3, Z_k \psi_1) & = \frac{\sigma c_0}{k} \left\{\int_{\mathbb{R}^5} \frac{e^{- 5 \mathcal J(x)^2/2}}{\mathcal J(x)^2} \, dx - c_0\left(\int_{\mathbb{R}^5} \frac{e^{- 5 \mathcal J(x)^2/2}}{\mathcal J(x)} \, dx \right)^2\right\} +o\left(\frac 1k\right).
\end{align*}
The last step of the proof is to observe that the difference between the two terms in the curly bracket is well a positive real. This follows simply by Cauchy-Schwarz, once we observe that $c_0\int_{\mathbb{R}^5} e^{-5 \mathcal J(x)^2/2} \, dx = 1$, which itself
can be deduced for instance from the fact that $1= \sum_{x\in \mathbb{Z}^5} p_k(x) \sim c_0\int_{\mathbb{R}^5}
e^{- 5 \mathcal J(x)^2/2} \, dx$, by the above arguments. This concludes the proof of Lemma \ref{lem.var.1}.
$\square$
\subsection{Proof of Lemma \ref{lem.var.2}}
Let us concentrate on the term $\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_3)$, the estimate of $\operatorname{Cov}(Z_0\varphi_1,Z_k\psi_1)$ being entirely similar. We also assume to simplify notation that
the walk is aperiodic.
We consider as in the proof of the previous lemma $(S_n^1)_{n\ge 0}$ and $(S_n^2)_{n\ge 0}$ two independent random walks starting from the origin, independent of $(S_n)_{n\in \mathbb{Z}}$, and define this time
$$\tau_1:= \inf\{n\ge k+\varepsilon_k : S_n \in \mathbb{R}R^1[\varepsilon_k,\infty) \}, \ \tau_2:= \inf \{n\ge k+\varepsilon_k : S_n \in S_k+\mathbb{R}R^2[\sqrt \varepsilon_k,\infty)\}.$$
Define as well
$$\overline \tau_1:= \inf\{n\ge k+\varepsilon_k : S_n \in \mathbb{R}R^1_\infty \}, \ \overline \tau_2:= \inf \{n\ge k+\varepsilon_k : S_n \in S_k+\mathbb{R}R^2_\infty\}.$$
\underline{Step $1$.} Our first task is to show that
\begin{equation}\label{cov.33.first}
\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_3) = \rho^2 \cdot \operatorname{Cov}\left({\text{\Large $\mathfrak 1$}}\{\overline \tau_1<\infty\} ,\, {\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\}\right) + o\left(\frac 1k\right),
\end{equation}
with $\rho$ as defined in \eqref{def.rho}. This step is essentially the same as in the proof of Lemma \ref{lem.var.1}, but with some additional technical difficulties, so let us give some details.
First, the proof of Lemma \ref{lem.var.1} shows that (using the same notation),
$$\mathbb{E}[Z_0\varphi_3] = \Delta_{0,3} + \mathcal{O}\left(\frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}}\right),$$
and that for any sequence $(\chi_k)_{k\ge 1}$ going to infinity with $\varepsilon_k \chi_k^{2+\frac 14} \le k$,
$$\Delta_{0,3} = \sum_{k/\chi_k \le \|x\|^2 \le k \chi_k } \sum_{\substack{\|y_1\|^2\le \varepsilon_k \chi_k \\ \|y_2\|^2\le \varepsilon_k \chi_k}} H(y_1,y_2) p_k(x+y_2-y_1) \operatorname{Var}phi_x + \mathcal{O}\left(\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} \right).$$
Observe moreover, that by symmetry $H(y_1,y_2) = H(-y_1,-y_2)$, and that by Theorem \ref{LCLT}, for any $x$, $y_1$, and $y_2$ as above, for some constant $c>0$,
$$\left|p_k(x+y_2-y_1) + p_k(x+y_1-y_2) - p_k(x) \right| = \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k} \overline p_k(cx) + \frac 1{k^{7/2}}\right),$$
It follows that one can improve the bound \eqref{Z0phi3.final} into
\begin{align}\label{Z0phi3.bis}
\nonumber \mathbb{E}[Z_0\varphi_3] & = \rho \sum_{x\in \mathbb{Z}^5} p_k(x) \varphi_x + \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k^{3/2}} + \frac{\chi_k^2}{k^{3/2}} +\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} + \frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}} \right)\\
& = \rho \, \mathbb{P}[\overline\tau_1<\infty] + \mathcal{O}\left(\frac{\varepsilon_k\chi_k}{k^{3/2}} + \frac{\chi_k^2}{k^{3/2}} +\frac 1{\sqrt{k}\cdot \chi_k^{5/4}} + \frac{1}{\sqrt{k\varepsilon_k}} + \frac{\varepsilon_k}{k^{3/2}} \right).
\end{align}
Since by \eqref{lem.hit.3} one has
$$\mathbb{E}[Z_k\psi_3] \le \mathbb{E}[\psi_3] = \mathcal{O}\left(\frac 1{\sqrt{\varepsilon_k}}\right),$$
this yields by taking $\chi_k^{2+1/4} := k/\varepsilon_k$, and $\varepsilon_k\ge k^{2/3}$ (but still $\varepsilon_k= o(k)$),
\begin{align}\label{Z03.1}
\mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_3] = \rho\, \mathbb{P}[\overline\tau_1<\infty]\cdot \mathbb{E}[Z_k\psi_3] + o\left(\frac 1k \right).
\end{align}
We next seek an analogous estimate for $\mathbb{E}[Z_k\psi_3]$.
Define $Z'_k:=1\{S_{k+i}\neq S_k,\, \forall i=1,\dots,\varepsilon_k^{3/4}\}$, and
$$\Delta_0:= \mathbb{E}\left[Z'_k\cdot {\text{\Large $\mathfrak 1$}}\left\{\mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k^{3/4}] \cap (S_k+\mathbb{R}R^2[1,\sqrt{\varepsilon_k}])=\varnothing, \, \tau_2<\infty \right\}\right].$$
Note that (with $\mathbb{R}R$ and $\widetilde \mathbb{R}R$ two independent walks),
\begin{align*}
\left| \mathbb{E}[Z_k\psi_3] - \Delta_0\right| & \le \mathbb{P}\left[0\in \mathbb{R}R[\varepsilon_k^{3/4},\varepsilon_k]\right] + \mathbb{P}\left[\widetilde \mathbb{R}R[0,\sqrt {\varepsilon_k}]\cap \mathbb{R}R[\varepsilon_k,\infty)\neq \varnothing\right] \\
& + \mathbb{P}\left[\widetilde \mathbb{R}R_\infty\cap \mathbb{R}R[\varepsilon_k^{3/4},\varepsilon_k] \neq \varnothing, \widetilde \mathbb{R}R_\infty\cap \mathbb{R}R[\varepsilon_k,\infty) \neq \varnothing \right] \\
& + \mathbb{P}\left[\widetilde \mathbb{R}R[\sqrt{\varepsilon_k},\infty) \cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k]\neq \varnothing, \widetilde \mathbb{R}R[\sqrt{\varepsilon_k},\infty) \cap \mathbb{R}R[\varepsilon_k,\infty)\neq \varnothing \right].
\end{align*}
Moreover,
\begin{equation}\label{ZkZ'k}
\mathbb{P}\left[0\in \mathbb{R}R[\varepsilon_k^{3/4},\varepsilon_k]\right] \stackrel{\eqref{Green.hit}, \eqref{exp.Green}}{\lesssim} \varepsilon_k^{-9/8},\quad \mathbb{P}\left[\widetilde \mathbb{R}R[0,\sqrt {\varepsilon_k}]\cap \mathbb{R}R[\varepsilon_k,\infty)\neq \varnothing\right] \stackrel{\eqref{lem.hit.3}}{\lesssim} \varepsilon_k^{- 1}.
\end{equation}
Using also the same computation as in the proof of Lemma \ref{lem.123}, we get
\begin{equation*}
\mathbb{P}\left[\widetilde \mathbb{R}R_\infty\cap \mathbb{R}R[\varepsilon_k^{3/4},\varepsilon_k] \neq \varnothing,\, \widetilde \mathbb{R}R_\infty\cap \mathbb{R}R[\varepsilon_k,\infty) \neq \varnothing \right] \lesssim \varepsilon_k^{-\frac 38 - \frac 12},
\end{equation*}
\begin{equation} \label{tau01tau2}
\mathbb{P}\left[\widetilde \mathbb{R}R[\sqrt{\varepsilon_k},\infty) \cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k]\neq \varnothing, \widetilde \mathbb{R}R[\sqrt{\varepsilon_k},\infty) \cap \mathbb{R}R[\varepsilon_k,\infty)\neq \varnothing \right] \lesssim \varepsilon_k^{-\frac 14 - \frac 12}.
\end{equation}
As a consequence
\begin{align}\label{Zk3.1}
\mathbb{E}[Z_k\psi_3] = \Delta_0 + \mathcal{O}\left(\varepsilon_k^{-3/4}\right).
\end{align}
Introduce now
\begin{align*}
\widetilde H(y_1,y_2) := \mathbb{E}\left[Z'_k\cdot \right. & {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k^{3/4}]\cap (S_k+ \mathbb{R}R^2[1,\sqrt{\varepsilon_k}])=\varnothing\} \\
& \times \left. {\text{\Large $\mathfrak 1$}}\{S_{k+\varepsilon_k^{3/4}}-S_k = y_1, S^2_{\sqrt{\varepsilon_k}} = y_2\}\right],
\end{align*}
and note that
$$\Delta_0 = \sum_{x\in \mathbb{Z}^d} \sum_{y_1,y_2\in \mathbb{Z}^d} \widetilde H(y_1,y_2) p_{\varepsilon_k-\varepsilon_k^{3/4}}(x+y_2-y_1) \varphi_x.$$
Let $\chi_k:= \varepsilon_k^{1/8}$. As above, we can see that
\begin{align*}
\Delta_0 & = \sum_{\substack{\varepsilon_k/\chi_k\le \|x\|^2\le \varepsilon_k\chi_k \\ \|y_1\|^2\le \varepsilon_k^{3/4}\chi_k \\ \|y_2\|^2 \le \sqrt{\varepsilon_k}\chi_k}} \widetilde H(y_1,y_2) p_{\varepsilon_k-\varepsilon_k^{3/4}}(x+y_2-y_1) \varphi_x + \mathcal{O}\left(\frac{1}{\sqrt{\varepsilon_k} \chi_k^{5/4}}\right)\\
&= \left(\sum_{y_1,y_2\in \mathbb{Z}^d} \widetilde H(y_1,y_2)\right) \left(\sum_{x\in \mathbb{Z}^d} p_{\varepsilon_k}(x)\varphi_x\right) + \mathcal{O}\left( \frac{\chi_k}{\varepsilon_k^{3/4}} + \frac{\chi_k^2}{\varepsilon_k^{3/2}} + \frac{1}{\sqrt{\varepsilon_k} \chi_k^{5/4}}\right) \\
& = \rho\cdot \mathbb{P}[\overline \tau_2<\infty] + \mathcal{O}(\varepsilon_k^{-5/8}).
\end{align*}
Then by taking $\varepsilon_k\ge k^{5/6}$, and recalling \eqref{Z03.1} and \eqref{Zk3.1}, we obtain
\begin{align}\label{Z03.2}
\mathbb{E}[Z_0\varphi_3] \cdot \mathbb{E}[Z_k\psi_3] = \rho^2\cdot \mathbb{P}[\overline\tau_1<\infty]\cdot \mathbb{P}[\overline \tau_2<\infty]+ o\left(\frac 1k \right).
\end{align}
Finally, let
\begin{align*}
\Delta_{3,3}:= & \mathbb{E} [Z_0Z'_k {\text{\Large $\mathfrak 1$}}\{\mathbb{R}R^1[1,\varepsilon_k]\cap \mathbb{R}R[-\varepsilon_k,\varepsilon_k] =\varnothing\}\\
& \times {\text{\Large $\mathfrak 1$}}\{ (S_k+\mathbb{R}R^2[1,\sqrt{\varepsilon_k}])\cap \mathbb{R}R[k-\varepsilon_k^{\frac 34},k+\varepsilon_k^{\frac 34}] =\varnothing, \tau_1 <\infty, \tau_2<\infty \}].
\end{align*}
It amounts to estimate the difference between $\Delta_{3,3}$ and $\mathbb{E}[Z_0Z_k\varphi_3\psi_3]$.
Define
$$\widetilde \tau_1:=\inf\{n\ge k+\varepsilon_k : S_n\in \mathbb{R}R^1[0,\varepsilon_k]\}, \ \widetilde \tau_2:=\inf\{n\ge k+\varepsilon_k : S_n\in S_k+\mathbb{R}R^2[0,\sqrt{\varepsilon_k}]\}.$$
Observe first that
\begin{align}\label{tilde1bar2}
\nonumber & \mathbb{P}[\widetilde \tau_1\le \overline \tau_2<\infty] \stackrel{\eqref{lem.hit.2}}{\lesssim} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\widetilde \tau_1<\infty\}}{1+\|S_{\widetilde \tau_1}-S_k\|} \right]
\stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[\frac{G(S_i^1-S_{k+\varepsilon_k})}{1+\|S_i^1-S_k\|} \right]\\
\nonumber & \lesssim \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} p_i(z) \mathbb{E}\left[\frac{G(z-S_{k+\varepsilon_k})}{1+\|z-S_k\|} \right]\stackrel{\eqref{pn.largex}}{\lesssim}
\sum_{z\in \mathbb{Z}^5} \frac {\sqrt{\varepsilon_k}}{1+\|z\|^4}\, \mathbb{E}\left[\frac{G(z-S_{k+\varepsilon_k})}{1+\|z-S_k\|} \right] \\
&\stackrel{\eqref{Green}}{\lesssim} \mathbb{E}\left[\frac{\sqrt{\varepsilon_k}}{(1+\|S_{k+\varepsilon_k}\|^2)(1+\|S_k\|)} \right]
\stackrel{\eqref{exp.Green.x}}{\lesssim} \mathbb{E}\left[\frac{\sqrt{\varepsilon_k}}{1+\|S_k\|^3} \right] \stackrel{\eqref{exp.Green}}{\lesssim} \frac{\sqrt{\varepsilon_k}}{k^{3/2}},
\end{align}
and likewise,
\begin{align*}
& \mathbb{P}[\overline \tau_1\le \widetilde \tau_2<\infty] \stackrel{\eqref{lem.hit.1}}{\le} \sum_{j\ge 0}\sum_{i=0}^{\varepsilon_k}\mathbb{E}\left[G(S_k + S_i^2 - S_j^1) G(S_j^1- S_{k+\varepsilon_k})\right] \\
& = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5}\mathbb{E}\left[G(z) G(S_k + S_i^2 - z) G(z- S_{k+\varepsilon_k})\right] \\
& \le C\sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[\frac 1{1+\|S_k + S_i^2\|^3}\left(\frac 1{1+\|S_{k+\varepsilon_k}\|} +\frac 1{1+ \|S_{k+\varepsilon_k}-S_k-S_i^2\|}\right)\right]\\
& \stackrel{\eqref{exp.Green},\, \eqref{exp.Green.x}}{\le} C\mathbb{E}\left[\frac {\varepsilon_k}{1+\|S_k\|^4}\right] +C \mathbb{E}\left[\frac {\sqrt{\varepsilon_k}}{1+\|S_k\|^3} \right]= \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right).
\end{align*}
Additionally, it follows directly from \eqref{lem.hit.3} that
$$\mathbb{P}[\overline \tau_2 \le \widetilde \tau_1<\infty] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}, \quad \text{and} \quad \mathbb{P}[\widetilde \tau_2 \le \overline \tau_1<\infty] \lesssim \frac{1}{\varepsilon_k \sqrt k},$$
which altogether yields
$$|\mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty] - \mathbb{P}[\tau_1<\infty,\, \tau_2<\infty]| \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}} + \frac{1}{\varepsilon_k \sqrt k}.$$
Similar computations give also
\begin{equation}\label{tau1tau2}
\mathbb{P}[\overline \tau_1<\infty, \, \overline \tau_2<\infty] \lesssim \frac 1{\sqrt{k \varepsilon_k}}.
\end{equation}
Next, using \eqref{ZkZ'k} and the Markov property, we get
$$\mathbb{E}[|Z_k-Z'_k|{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}] \lesssim \frac 1{\varepsilon_k^{9/8}\sqrt k}.$$
Thus, for $\varepsilon_k \ge k^{5/6}$,
\begin{align*}
\left|\mathbb{E}[Z_0Z_k\varphi_3\psi_3] - \Delta_{3,3}\right| & \le \mathbb{P}[\tau_{0,1}<\infty, \tau_1<\infty, \tau_2<\infty] +\mathbb{P}[\tau_{0,2}<\infty, \tau_1<\infty, \tau_2<\infty] \\
& \qquad + \mathbb{P}[\widetilde \tau_{0,2}<\infty, \tau_1<\infty, \tau_2<\infty] + o\left(\frac 1k\right),
\end{align*}
where $\tau_{0,1}$ is as defined in the proof of Lemma \ref{lem.var.1},
$$\tau_{0,2} : =\inf\{n\ge \sqrt{\varepsilon_k} : S_k + S_n^2 \in \mathbb{R}R[k-\varepsilon_k,k+\varepsilon_k]\},$$
and
$$\widetilde \tau_{0,2} : =\inf\{n\le \sqrt{\varepsilon_k} : S_k + S_n^2 \in \mathbb{R}R[k-\varepsilon_k,k-\varepsilon_k^{3/4}]\cup \mathbb{R}R[k+\varepsilon_k^{3/4},k+\varepsilon_k]\}.$$
Applying \eqref{lem.hit.3} twice already shows that
$$\mathbb{P}[\widetilde \tau_{0,2}<\infty,\, \tau_1<\infty] \lesssim \frac{1}{\sqrt k} \cdot \mathbb{P}[\widetilde \tau_{0,2}<\infty] \lesssim \frac{1}{\sqrt{k}\varepsilon_k^{5/8}} = o\left(\frac 1k\right). $$
Then, notice that \eqref{tilde1bar2} entails
$$\mathbb{P}[\mathbb{R}R[k+\varepsilon_k,\infty) \cap \mathbb{R}R^1[0,\tau_{0,1}]\neq \varnothing, \, S^1_{\tau_{0,1}} \in \mathbb{R}R[-\varepsilon_k,0]] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.$$
On the other hand,
\begin{align*}
&\mathbb{P}[\mathbb{R}R[k+\varepsilon_k,\infty) \cap \mathbb{R}R^1[0,\tau_{0,1}]\neq \varnothing, S^1_{\tau_{0,1}} \in \mathbb{R}R[0,\varepsilon_k]] \\
\stackrel{\eqref{lem.hit.1}}{\le} & \sum_{i=0}^{\varepsilon_k} \sum_{j=k+\varepsilon_k}^\infty
\mathbb{E}[G(S_i-S_{k+j})G(S_{k+j} - S_k)] = \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5}
\mathbb{E}[G(S_i-S_k + z)G(z)G_{\varepsilon_k}(z)] \\
\stackrel{\eqref{exp.Green}}{\lesssim} & \frac{\varepsilon_k}{k^{3/2}} \sum_{z\in \mathbb{Z}^5} G(z)G_{\varepsilon_k}(z)\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim}
\frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
By \eqref{lem.hit.1} and \eqref{exp.Green}, one has with $\widetilde \mathbb{R}R_\infty$ an independent copy of $\mathbb{R}R_\infty$,
\begin{align*}
& \mathbb{P}[\tau_{0,1}<\infty, \tau_2<\infty, \mathbb{R}R[k+\varepsilon_k,\infty) \cap \mathbb{R}R^1[\tau_{0,1},\infty)\neq \varnothing ] \\
\lesssim & \frac{1}{\sqrt{\varepsilon_k}}
\max_{-\varepsilon_k\le i\le \varepsilon_k}\mathbb{P}[\tau_2<\infty,\, \mathbb{R}R[k+\varepsilon_k,\infty) \cap (S_i+ \widetilde \mathbb{R}R_\infty) \neq \varnothing ] \lesssim \frac{1}{\varepsilon_k\sqrt{k}},
\end{align*}
where the last equality follows from \eqref{tau1tau2}. Thus
$$\mathbb{P}[\tau_{0,1}<\infty, \tau_1<\infty, \tau_2<\infty] =o\left(\frac 1k\right).$$
In a similar fashion, one has
$$\mathbb{P}[\tau_{0,2}<\infty, \tau_2\le \tau_1<\infty] \stackrel{\eqref{lem.hit.3}}{\lesssim} \frac{1}{\sqrt k} \mathbb{P}[\tau_{0,2}<\infty, \tau_2<\infty]
\stackrel{\eqref{tau01tau2}}{\lesssim} \frac{1}{\varepsilon_k^{3/4}\sqrt{k}},$$
as well as,
\begin{align*}
& \mathbb{P}\left[\tau_{0,2}<\infty, \, \tau_1\le \tau_2<\infty,\, S_{\tau_2}\in (S_k+\mathbb{R}R^2[0, \tau_{0,2}])\right] \\
\stackrel{\eqref{lem.hit.1}}{\le} & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{j\ge 0} \sum_{\ell \ge 0} \mathbb{E}[G(S_i-\widetilde S_j - S^1_\ell) G(\widetilde S_j + S^1_\ell-S_k) G(S^1_\ell - S_{k+\varepsilon_k})] \\
\le & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{\ell \ge 0} \sum_{z\in\mathbb{Z}^5} \mathbb{E}[G(z) G(S_i- S^1_\ell - z) G(z + S^1_\ell-S_k) G(S^1_\ell - S_{k+\varepsilon_k})]\\
\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} & \sum_{i=k-\varepsilon_k}^{k+\varepsilon_k} \sum_{\ell \ge 0}
\mathbb{E}\left[\frac{G(S^1_\ell - S_{k+\varepsilon_k})}{1+\|S^1_\ell - S_k\|^3} \left(\frac 1{1+\|S_\ell^1- S_i\|} + \frac 1{1+\|S_i-S_k\|}\right)\right]\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \sum_{i=0}^{\varepsilon_k} \sum_{\ell \ge 0}
\left\{\mathbb{E}\left[\frac{\varepsilon_k^{-3/2}}{1+\|S^1_\ell - S_k\|^3} \left(\frac 1{1+\|S_\ell^1- S_{k-i}\|} + \frac 1{1+\|S_{k-i}-S_k\|}\right)\right] \right. \\
+& \left. \mathbb{E}\left[\frac{1}{(1+\|S^1_\ell - S_{k+i}\|^3)(1+\|S^1_\ell - S_k\|^3)} \left(\frac 1{1+\|S_\ell^1- S_{k+i}\|} + \frac 1{1+\|S_{k+i}-S_k\|}\right)\right] \right\}\\
\stackrel{\eqref{pn.largex}, \eqref{exp.Green.x}}{\lesssim} & \sum_{i=0}^{\varepsilon_k} \sum_{\ell \ge 0}
\left\{\mathbb{E}\left[\frac{\varepsilon_k^{-3/2}}{1+\|S^1_\ell - S_{k-i}\|^3} \left(\frac 1{1+\|S_\ell^1- S_{k-i}\|} + \frac 1{1+\sqrt{i}}\right)\right] \right. \\
& \qquad + \left. \mathbb{E}\left[\frac{(1+i)^{-1/2}}{1+\|S^1_\ell - S_k\|^6} \right] \right\} \\
\lesssim &\ \frac {\sqrt{\varepsilon_k}}{k^{3/2}} ,
\end{align*}
and
\begin{align*}
& \mathbb{P}\left[\tau_{0,2}<\infty, \, \tau_1\le \tau_2<\infty,\, S_{\tau_2}\in (S_k+\mathbb{R}R^2[\tau_{0,2},\infty))\right] \\
\stackrel{\eqref{Green.hit}}{\le} &
\sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) {\text{\Large $\mathfrak 1$}}\{\tau_1<\infty,\, \mathbb{R}R[\tau_1,\infty) \cap (S_{k+i}+ \widetilde \mathbb{R}R_\infty) \neq \varnothing\}\right]\\\
\stackrel{\eqref{lem.hit.2}}{\lesssim} & \sum_{i=-\varepsilon_k}^{\varepsilon_k} \mathbb{E}\left[G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}})
\frac{{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}}{1+ \|S_{\tau_1} - S_{k+i}\|} \right] \\
\stackrel{\eqref{lem.hit.1}}{\lesssim} & \sum_{i=-\varepsilon_k}^{\varepsilon_k}\sum_{j\ge k+\varepsilon_k} \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_j)}{1+ \|S_j - S_{k+i}\|} \right] \\
\lesssim & \sum_{i=0}^{\varepsilon_k} \sum_{z\in \mathbb{Z}^5} \left\{\mathbb{E}\left[\frac{G(S_{k-i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_k+z)G(z)}{1+ \|z + S_k - S_{k-i}\|} \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) G(S_{k+i}+z)G(z)}{1+ \|z\|} \right] \right\}\\
\lesssim & \sum_{i=0}^{\varepsilon_k} \left\{\mathbb{E}\left[\frac{G(S_{k-i}-S_k - S^2_{\sqrt{\varepsilon_k}}) }{1+ \|S_k\|^2} \right]
+ \mathbb{E}\left[\frac{G(S_{k+i}-S_k - S^2_{\sqrt{\varepsilon_k}}) }{1+ \|S_{k+i}\|^2} \right] \right\}\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \frac{1}{\varepsilon_k^{3/4}}\sum_{i=0}^{\sqrt{\varepsilon_k}} \mathbb{E}\left[\frac{1}{1+ \|S_k\|^2} +\frac{1}{1+ \|S_{k+i}\|^2}\right] + \sum_{i = \sqrt{\varepsilon_k}}^{\varepsilon_k} \mathbb{E}\left[\frac{G(S_{k-i}-S_k)}{1+ \|S_k\|^2} +\frac{G(S_{k+i}-S_k)}{1+ \|S_{k+i}\|^2}\right]\\
\stackrel{\eqref{pn.largex}, \eqref{exp.Green}}{\lesssim} & \frac{1}{\varepsilon_k^{1/4}k} + \sum_{i = \sqrt{\varepsilon_k}}^{\varepsilon_k} \frac 1{i^{3/2}}\cdot \mathbb{E}\left[\frac{1}{1+ \|S_{k-i}\|^2} +\frac{1}{1+ \|S_k\|^2}\right] \lesssim \frac 1{\varepsilon_k^{1/4} k}.
\end{align*}
Thus at this point we have shown that
\begin{eqnarray}\label{approx.Delta3}
\left|\mathbb{E}[Z_0Z_k\varphi_3\psi_3] - \Delta_{3,3}\right| = o\left(\frac 1k\right).
\end{eqnarray}
Now define
\begin{align*}
\widetilde H(z_1,z_2,z_3) := \mathbb{P}\left[0\notin \mathbb{R}R[1,\varepsilon_k^{3/4}], \right. & \widetilde \mathbb{R}R[1,\sqrt{\varepsilon_k}]\cap \mathbb{R}R[-\varepsilon_k^{ 3/4},\varepsilon_k^{ 3/4}]=\varnothing, \\
& \left. S_{\varepsilon_k^{3/4}}=z_1, S_{-\varepsilon_k^{3/4}}=z_3, \widetilde S_{\sqrt{\varepsilon_k}} = z_3\right],
\end{align*}
and recall also the definition of $H(y_1,y_2)$ given in \eqref{Hy1y2}.
One has
$$\Delta_{3,3} = \sum H(y_1,y_2) \widetilde H(z_1,z_2,z_3)
p_{k-\varepsilon_k - \varepsilon_k^{3/4}}(x - y_1+y_2+z_3-z_2) p_{\varepsilon_k- \varepsilon_k^{3/4}}(u-z_1+z_2) \operatorname{Var}phi_{x,u},$$
where the sum runs over all $x,u,y_1,y_2,z_1,z_2,z_3\in \mathbb{Z}^5$, and
\begin{align*}
\varphi_{x,u} := \mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty\mid S_k = x,\, S_{k+\varepsilon_k} = x+u].
\end{align*}
Note that the same argument as for \eqref{tau1tau2} gives also
\begin{eqnarray}\label{phixu}
\varphi_{x,u} \lesssim \frac 1{1+\|u\|}\left(\frac 1{1+\|x+u\|} +\frac 1{1+\|x\|}\right).
\end{eqnarray}
Using this it is possible to see that in the expression of $\Delta_{3,3}$ given just above, one can restrict the sum to typical values of the parameters.
Indeed, consider for instance the sum on atypically large values of $x$. More precisely, take $\chi_k$, such that $\varepsilon_k \chi_k^{2+1/4} =k$, and note that by \eqref{phixu},
\begin{align*}
& \sum_{\substack{\|x\|^2\ge k\chi_k \\ u,y_1,y_2,z_1,z_2,z_3}} H(y_1,y_2) \widetilde H(z_1,z_2,z_3)
p_{k-\varepsilon_k - \varepsilon_k^{3/4}}(x - y_1+y_2+z_3-z_2) p_{\varepsilon_k- \varepsilon_k^{3/4}}(u-z_1+z_2) \operatorname{Var}phi_{x,u}\\
& \le \mathbb{P}\left[ \|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}, \tau_1<\infty, \tau_2<\infty\right]\le \mathbb{P}\left[ \|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}, \tau_1<\infty, \overline \tau_2<\infty\right] \\
& \lesssim \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|S_k-S_{\varepsilon_k}^1\|\ge \sqrt{k\chi_k}\}}{1+\|S_{k+\varepsilon_k}-S_k\|} \left(\frac {1}{1+\|S_k-S^1_{\varepsilon_k}\|} + \frac 1{1+\|S_{k+\varepsilon_k}-S^1_{\varepsilon_k}\|}\right)\right] \\
& \lesssim \frac{1}{\chi_k^{5/4}\sqrt{k\varepsilon_k} },
\end{align*}
where the last equality follows by applying Cauchy-Schwarz inequality and \eqref{Sn.large}. The other cases are entirely similar. Thus
$\Delta_{3,3}$ is well approximated by the sums on typical values of the parameters (similarly as for $\Delta_0$ for instance), and then we can deduce with Theorem \ref{LCLT} and \eqref{phixu} that
$$\Delta_{3,3} = \rho^2\cdot \mathbb{P}[\overline \tau_1<\infty,\, \overline \tau_2<\infty]+ o\left(\frac 1k\right).$$
Together with \eqref{approx.Delta3} and \eqref{Z03.2} this proves \eqref{cov.33.first}.
\underline{Step $2$.}
For a (possibly random) time $T$, set
$$\overline \tau_1\circ T := \inf \{n\ge T\vee \varepsilon_k : S_n \in \mathbb{R}R^1_\infty\},\ \overline \tau_2\circ T := \inf \{n\ge T\vee \varepsilon_k : S_n \in (S_k+\mathbb{R}R^2_\infty)\}. $$
Observe that
\begin{equation}\label{main.tau.1}
\mathbb{P}[\overline \tau_1\le \overline \tau_2<\infty] = \mathbb{P}[\overline \tau_1\le \overline \tau_2\circ \overline \tau_1<\infty] - \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2\le \overline \tau_2\circ \overline \tau_1\circ \overline \tau_2<\infty],
\end{equation}
and symmetrically,
\begin{equation}\label{main.tau.2}
\mathbb{P}[\overline \tau_2\le \overline \tau_1<\infty] = \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] - \mathbb{P}[\overline \tau_1\le\overline \tau_2\circ \overline \tau_1\le \overline \tau_1\circ \overline \tau_2\circ \overline \tau_1<\infty].
\end{equation}
Our aim here is to show that the two error terms appearing in \eqref{main.tau.1} and \eqref{main.tau.2} are negligible.
Applying repeatedly \eqref{lem.hit.1} gives
\begin{align*}
E_1& := \mathbb{P}[\overline \tau_1\le\overline \tau_2\circ \overline \tau_1\le \overline \tau_1\circ \overline \tau_2\circ \overline \tau_1<\infty] \\
&\lesssim \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^1 - S_k - S_\ell^2) G(S_k + S_\ell^2 - S_m^1) G(S_m^1 - S_{k+\varepsilon_k})\right]\\
& \stackrel{\eqref{exp.Green.x}}{\lesssim} \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^1 - S_k - S_\ell^2) G(S_k + S_\ell^2 - S_m^1) G(S_m^1 - S_k)\right]\\
& \lesssim \sum_{j\ge 0} \sum_{m\ge 0} G(z) \mathbb{E}\left[ G(S_j^1 - S_k - z) G(S_k + z - S_m^1) G(S_m^1 - S_k)\right].
\end{align*}
Note also that by using Lemma \ref{lem.upconvolG} and \eqref{Green}, we get
$$\sum_{z\in \mathbb{Z}^5}G(z-x) G(z-y) G(z) \lesssim \frac 1{1+\|x\|^3} \left(\frac 1{1+\|y\|} + \frac {1}{1+ \|y-x\|} \right).$$
Thus, distinguishing also the two cases $j\le m$ and $m\le j$, we obtain
\begin{align*}
E_1& \lesssim \sum_{j\ge 0} \sum_{m\ge 0}\mathbb{E}\left[\frac {G(S_m^1 - S_k)}{1+\|S_j^1-S_k\|^3} \left( \frac 1{1+\|S_m^1-S_k\|} + \frac 1{1+\|S_m^1- S_j^1\|}\right) \right] \\
& \lesssim \sum_{j\ge 0} \sum_{z\in \mathbb{Z}^5} G(z)\left\{ \mathbb{E}\left[\frac {G(z+S_j^1 - S_k)}{1+\|S_j^1-S_k\|^3} \left( \frac 1{1+\|z+S_j^1-S_k\|} + \frac 1{1+\|z\|}\right) \right] \right. \\
& \qquad \left. + \mathbb{E}\left[\frac {G(S_j^1 - S_k)}{1+\|z+S_j^1-S_k\|^3} \left( \frac 1{1+\|S_j^1-S_k\|} + \frac 1{1+\|z\|}\right) \right]\right\}\\
& \lesssim \sum_{j\ge 0} \mathbb{E}\left[\frac {1}{1+\|S_j^1-S_k\|^5} \right] \lesssim \mathbb{E}\left[\frac {\log (1+\|S_k\|)}{1+\|S_k\|^3}\right] \lesssim \frac{\log k}{k^{3/2}}.
\end{align*}
Similarly,
\begin{align*}
& \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2\le \overline \tau_2\circ \overline \tau_1\circ \overline \tau_2<\infty] \\
&\lesssim \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0} \mathbb{E}\left[ G(S_j^2 + S_k - S_\ell^1) G(S_\ell^1 - S_k- S_m^2) G(S_m^2 + S_k - S_{k+\varepsilon_k})\right]\\
& \stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} \frac {1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0} \sum_{\ell \ge 0} \sum_{m\ge 0}\mathbb{E}\left[ \frac{G(S_j^2 + S_k - S_\ell^1) G(S_\ell^1 - S_k- S_m^2) }{1+\|S_m^2\|^2} \right]\\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0} \sum_{m\ge 0}\mathbb{E}\left[\frac 1{(1+\|S_m^2\|^2)(1+\|S_j^2+S_k\|^3)} \left( \frac 1{1+\|S_m^2+S_k\|} + \frac 1{1+\|S_m^2- S_j^2\|}\right) \right] \\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}} \sum_{j\ge 0}\mathbb{E}\left[\frac 1{(1+\|S_j^2\|)(1+\|S_j^2+S_k\|^3)} +\frac 1{(1+\|S_j^2\|^2)(1+\|S_j^2+S_k\|^2)} \right] \\
&\lesssim \frac{1}{\sqrt{\varepsilon_k}}\cdot \mathbb{E}\left[\frac {\log (1+\|S_k\|)}{1+\|S_k\|^2}\right] \lesssim \frac{\log k}{k \sqrt{\varepsilon_k}}.
\end{align*}
\underline{Step $3$.} We now come to the estimate of the two main terms in \eqref{main.tau.1} and \eqref{main.tau.2}.
In fact it will be convenient to replace $\overline \tau_1$ in the first one by
$$\widehat \tau_1:= \inf \{n\ge k : S_n\in \mathbb{R}R_\infty^1\}.$$
The error made by doing this is bounded as follows: by shifting the origin to $S_k$, and using symmetry of the step distribution, we can write
\begin{align*}
& \left| \mathbb{P}[\overline \tau_1\le \overline \tau_2\circ \overline \tau_1<\infty] - \mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty]\right| \le \mathbb{P}\left[\mathbb{R}R_\infty^1\cap \mathbb{R}R[k,k+\varepsilon_k] \neq \varnothing, \overline \tau_2<\infty\right] \\
& \stackrel{\eqref{Green.hit}}{\le} \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i- \widetilde S_k)\right) \left(\sum_{j=\varepsilon_k}^\infty G(S_j)\right)\right] \\
& = \mathbb{E}\left[\left(\sum_{i=0}^{\varepsilon_k} G(S_i- \widetilde S_k)\right) \left(\sum_{z\in \mathbb{Z}^5} G(z) G(z+S_{\varepsilon_k})\right)\right] \\
&\stackrel{\text{Lemma }\ref{lem.upconvolG}}{\lesssim} \sum_{i=0}^{\varepsilon_k} \mathbb{E}\left[ \frac{G(S_i- \widetilde S_k)}{1+\|S_{\varepsilon_k}\|}\right] \stackrel{\eqref{exp.Green}}{\lesssim} \frac {\varepsilon_k}{k^{3/2}}\cdot \mathbb{E}\left[ \frac 1{1+\|S_{\varepsilon_k}\|}\right] \lesssim \frac{\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
Moreover, using Theorem C, the Markov property and symmetry of the step distribution, we get for some constant $c>0$,
\begin{align*}
& \mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty]
= c \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1} - S_k) }\right] + o\left(\frac 1k\right) \\
& = c \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1}) }\right] + o\left(\frac 1k\right)
= c \sum_{x\in \mathbb{Z}^5}p_k(x) \, \mathbb{E}_{0,x} \left[F(S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\}\right] + o\left(\frac 1k\right),
\end{align*}
with $\tau$ the hitting time of two independent walks starting respectively from the origin and from $x$, and $F(z) := 1/(1+ \mathcal{J}(z))$.
Note that the bound $o(1/k)$ on the error term in the last display comes from the fact that
$$\mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\widehat \tau_1<\infty\}}{1+\mathcal{J}(S_{\widehat \tau_1})} \right] \stackrel{\eqref{lem.hit.1}}{\lesssim} \sum_{j\ge 0} \mathbb{E}\left[\frac{G(\widetilde S_j-S_k)}{1+\|\widetilde S_j\|}\right] \lesssim \sum_{z\in \mathbb{Z}^5}
\mathbb{E}\left[\frac{G(z)G(z-S_k)}{1+\| z\|}\right] \lesssim \frac 1k. $$
Then by applying Theorem \ref{thm.asymptotic}, we get
\begin{equation}\label{main.tau.1.2}
\mathbb{P}[\widehat \tau_1\le \overline \tau_2\circ \widehat \tau_1<\infty] = c_0 \sum_{x\in \mathbb{Z}^5} p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-x)}{1+\mathcal J(z)} + o\left(\frac 1k\right),
\end{equation}
for some constant $c_0>0$.
Likewise, by Theorem \ref{thm.asymptotic} one has for some constant $\nu\in (0,1)$,
\begin{align*}
\mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c\, \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})}\right] +
\mathcal{O}\left(\mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})^{1+\nu}} \right] \right).
\end{align*}
Furthermore,
\begin{align*}
& \mathbb{E}\left[ \frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})^{1+\nu}} \right] \lesssim \sum_{j\ge 0} \mathbb{E}\left[ \frac {G(S_j^2+S_k-S_{k+\varepsilon_k}) }{1+ \| S_j^2+S_k\|^{1+\nu}} \right] \\
& \stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} \frac{1}{\sqrt{\varepsilon_k}}\sum_{j\ge 0} \mathbb{E}\left[ \frac {1 }{(1+\|S_j^2\|^2)(1+ \| S_j^2+S_k\|^{1+\nu})} \right] \\
& \lesssim \frac{1}{\sqrt{\varepsilon_k}} \mathbb{E}\left[\frac{\log (1+\|S_k\|)}{1+\|S_k\|^{1+\nu}}\right] \lesssim \frac {\log k}{ k^{(1+ \nu)/2}\sqrt{\varepsilon_k}}.
\end{align*}
Therefore, taking $\varepsilon_k\ge k^{1-\nu/2}$, we get
\begin{align}\label{main.tau.2.1}
\nonumber \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c\, \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\overline \tau_2<\infty\} }{1+ \mathcal{J}(S_{\overline \tau_2})}\right] + o\left(\frac 1k\right) \\
\nonumber & = c \sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \mathbb{E}_{0,u} \left[\frac{{\text{\Large $\mathfrak 1$}}\{\tau<\infty\} }{1+ \mathcal{J}(S_\tau - S_k)}\right] + o\left(\frac 1k\right) \\
& = c \sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \mathbb{E}_{0,u} \left[\widetilde F(S_\tau) {\text{\Large $\mathfrak 1$}}\{\tau<\infty\} \right] + o\left(\frac 1k\right),
\end{align}
with $\tau$ the hitting time of two independent walks starting respectively from the origin and from $u$, and
$$\widetilde F(z):= \mathbb{E}\left[\frac 1{1+ \mathcal{J}(z-S_k)} \right]. $$
We claim that this function $\widetilde F$ satisfies \eqref{cond.F}, for some constant $C_{\widetilde F}$ which is independent of $k$.
Indeed, first notice that
$$\widetilde F(z) \asymp \frac 1{1+\|z\| + \sqrt{k}},\quad \text{and}\quad \mathbb{E}\left[ \frac 1{1+\mathcal{J}(z-S_k)^2}\right] \asymp \frac 1{1+\|z\|^2 + k},$$
which can be seen by using Theorem \ref{LCLT}. Moreover, by triangle inequality, and Cauchy-Schwarz,
\begin{align*}
|\widetilde F(y) - \widetilde F(z) | & \lesssim \mathbb{E}\left[\frac {\|y-z\|}{(1+\|y-S_k\|)(1+\|z-S_k\|)}\right] \\
& \lesssim \|y-z\| \, \mathbb{E}\left[\frac 1{1+\|y-S_k\|^2}\right]^{\frac 12} \mathbb{E}\left[\frac 1{1+\|z-S_k\|^2}\right]^{\frac 12} \\
& \lesssim \frac{\|y-z\|}{(1+\|y\|+\sqrt k)(1+\|z\| +\sqrt{k})} \lesssim \frac{\|y-z\|}{1+\|y\|}\cdot \widetilde F(z),
\end{align*}
which is the desired condition \eqref{cond.F}. Therefore, coming back to \eqref{main.tau.2.1} and applying Theorem \ref{thm.asymptotic} once more gives,
\begin{align}\label{main.tau.2.1.bis}
\nonumber \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] & = c_0\sum_{u\in \mathbb{Z}^5} p_{\varepsilon_k}(u) \sum_{z\in \mathbb{Z}^5} G(z)G(z-u)\widetilde F(z) + o\left(\frac 1k\right) \\
& = c_0\sum_{u\in \mathbb{Z}^5} \sum_{x\in \mathbb{Z}^5}p_{\varepsilon_k}(u) p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-u)}{1+\mathcal J(z-x)} + o\left(\frac 1k\right).
\end{align}
Similarly, one has
\begin{align}\label{main.tau.product}
\nonumber &\mathbb{P}[\overline \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] =\mathbb{P}[\widehat \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] + \mathcal{O}\left(\frac{\sqrt{\varepsilon_k}}{k^{3/2}}\right) \\
& = c_0 \sum_{u\in \mathbb{Z}^5} \sum_{x\in \mathbb{Z}^5} p_{\varepsilon_k}(u)p_k(x) \sum_{z\in \mathbb{Z}^5} \frac{G(z)G(z-u)}{1+\mathcal J(x)} + o\left(\frac 1k\right).
\end{align}
Note in particular that the constant $c_0$ that appears here is the same as in \eqref{main.tau.1.2} and \eqref{main.tau.2.1.bis}.
\underline{Step $4$.} We claim now that when one takes the difference between the two expressions in \eqref{main.tau.2.1.bis} and \eqref{main.tau.product},
one can remove the parameter $u$ from the factor $G(z-u)$ (and then absorb the sum over $u$).
Indeed, note that for any $z$ with $\mathcal{J}(z)\le \mathcal{J}(x)/2$, one has
$$\left| \frac 1{1+\mathcal{J}(z+x)} + \frac 1{1+\mathcal{J}(z-x)} - \frac 2{1+\mathcal{J}(x)}\right| \lesssim \frac{\|z\|^2}{1+\|x\|^3}.$$
It follows that, for any $\chi_k\ge 2$,
\begin{align*}
& \sum_{\substack{u,x\in \mathbb{Z}^5 \\ \mathcal{J}(z) \le \frac{\mathcal{J}(x)}{\chi_k}}} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left|\frac 1{1+\mathcal{J}(z-x)}+\frac 1{1+\mathcal{J}(z+x)} - \frac 2{1+\mathcal{J}(x)}\right| \\
&\lesssim \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|^3} \sum_{\mathcal{J}(z) \le \mathcal{J}(x)/\chi_k } \frac{\mathbb{E}[G(z-S_{\varepsilon_k})]}{1+\|z\|} \stackrel{\eqref{exp.Green.x}}{\lesssim}
\frac{1}{k\chi_k}.
\end{align*}
In the same way, for any $z$ with $\mathcal{J}(z) \ge 2\mathcal{J}(u)$, one has
$$|G(z-u) -G(z)| \lesssim \frac{\|u\|}{1+\|z\|^4},$$
$$\left|\frac 1{1+\mathcal{J}(z-x)} - \frac 1{1+\mathcal{J}(x)}\right| \lesssim \frac{\|z\|}{(1+\|x\|)(1+\|z-x\|)}.$$
Therefore, for any $\chi_k\ge 2$,
\begin{align*}
& \sum_{\substack{u,x\in \mathbb{Z}^5 \\ \mathcal{J}(z) \ge (\mathcal{J}(u)\chi_k)\vee \frac{\mathcal{J}(x)}{\chi_k}}} p_{\varepsilon_k}(u) p_k(x) G(z) |G(z-u)-G(z)| \left|\frac 1{1+\mathcal{J}(z-x)} - \frac 1{1+\mathcal{J}(x)}\right| \\
&\lesssim \sqrt{\varepsilon_k} \sum_{x\in \mathbb{Z}^5} \frac{p_k(x)}{1+\|x\|} \sum_{\mathcal{J}(z) \ge \mathcal{J}(x)/\chi_k } \frac{1}{\|z\|^6(1+\|z-x\|)}
\stackrel{\eqref{exp.Green.x}}{\lesssim} \frac{\chi_k^2\sqrt{\varepsilon_k}}{k^{3/2}}.
\end{align*}
On the other hand by taking $\chi_k = (k/\varepsilon_k)^{1/6}$, we get using \eqref{pn.largex} and \eqref{Sn.large},
\begin{align*}
\sum_{\substack{x,z\in \mathbb{Z}^5 \\ \mathcal{J}(u)\ge \sqrt{\varepsilon_k}\chi_k }} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left(\frac 1{1+\mathcal{J}(z-x)} + \frac 1{1+\mathcal{J}(x)}\right) \lesssim \frac{1}{\chi_k^5\sqrt{k\varepsilon_k}} = o \left(\frac 1k\right),
\end{align*}
\begin{align*}
& \sum_{\substack{u,z\in \mathbb{Z}^5 \\ \mathcal{J}(x)\le \sqrt{k}/\chi_k }} p_{\varepsilon_k}(u) p_k(x) G(z) G(z-u) \left(\frac 1{1+\mathcal{J}(z-x)} + \frac 1{1+\mathcal{J}(x)}\right) = o \left(\frac 1k\right).
\end{align*}
As a consequence, since $\mathcal{J}(u)\le \sqrt{\varepsilon_k} \chi_k$ and $\mathcal{J}(x)\ge \sqrt{k}/\chi_k$, implies $\mathcal{J}(u)\le \mathcal{J}(x)/\chi_k$, with our choice of $\chi_k$,
we get as wanted (using also symmetry of the step distribution) that
\begin{align}\label{main.tau.combined.1}
& \mathbb{P}[\overline \tau_2\le \overline \tau_1\circ \overline \tau_2<\infty] - \mathbb{P}[\overline \tau_1<\infty] \cdot \mathbb{P}[\overline \tau_2<\infty] \\
\nonumber & = c_0 \sum_{x,z\in \mathbb{Z}^5} p_k(x) G(z)^2 \left(\frac 1{1+\mathcal{J}(z-x)} - \frac{1}{1+\mathcal J(x)}\right) + o \left(\frac 1k\right)\\
\nonumber &= \frac{c_0}{2} \sum_{x,z\in \mathbb{Z}^5} p_k(x) G(z)^2 \left(\frac 1{1+\mathcal{J}(z-x)} +\frac 1{1+\mathcal{J}(z+x)}- \frac{2}{1+\mathcal J(x)}\right) + o \left(\frac 1k\right).
\end{align}
\underline{Step 5.} The previous steps show that
$$\operatorname{Cov}\left(\{\overline \tau_1<\infty\} , \{\overline \tau_2<\infty\}\right) = c_0 \sum_{x,z\in \mathbb{Z}^5} p_k(x) \left(\frac{G(z)G(z-x)}{1+\mathcal{J}(z)} + \frac{G(z)^2}{1+\mathcal{J}(z-x)} - \frac{G(z)^2}{1+\mathcal{J}(x)}\right).$$
Now by approximating the series with an integral (recall \eqref{Green.asymp}), and doing a change of variables, we get with $u:=x/\mathcal{J}(x)$ and $v:=\Lambda^{-1} u$, and for some constant $c>0$ (that might change from line to line),
\begin{align}\label{last.lem.integral}
\nonumber & \sum_{z\in \mathbb{Z}^5}\left(\frac{G(z)G(z-x)}{1+\mathcal{J}(z)} + \frac{G(z)^2}{1+\mathcal{J}(z-x)} - \frac{G(z)^2}{1+\mathcal{J}(x)}\right) \\
\nonumber & \sim c \int_{\mathbb{R}^5} \left\{\frac{1}{\mathcal{J}(z)^4\cdot \mathcal{J}(z-x)^3} + \frac{1}{\mathcal{J}(z)^6} \left(\frac 1{\mathcal{J}(z-x)} -\frac 1{\mathcal{J}(x)}\right)\right\}\, dz \\
\nonumber & = \frac{c}{\mathcal{J}(x)^2} \int_{\mathbb{R}^5}\left\{\frac{1}{\mathcal{J}(z)^4\cdot \mathcal{J}(z-u)^3} + \frac{1}{\mathcal{J}(z)^6} \left(\frac 1{\mathcal{J}(z-u)} -1\right)\right\}\, dz \\
& = \frac{c}{\mathcal{J}(x)^2} \int_{\mathbb{R}^5} \left\{\frac{1}{\|z\|^4\cdot \|z-v\|^3} + \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right)\right\} \, dz.
\end{align}
Note that the last integral is convergent and independent of $v$ (and thus of $x$ as well) by rotational invariance. Therefore, since $\sum_{x\in \mathbb{Z}^5} p_k(x) / \mathcal{J}(x)^2\sim \sigma/k$, for some constant $\sigma>0$ (for instance by applying Theorem \ref{LCLT}), it only remains to show that the integral above is positive.
To see this, we use that the map $z\mapsto \|z\|^{-3}$ is harmonic outside the origin, and thus satisfies the mean value property on $\mathbb{R}^5\setminus \{0\}$.
In particular, using also the rotational invariance, this shows (with $\mathcal{B}_1$ the unit Euclidean ball and $\partial \mathcal{B}_1$ the unit sphere),
\begin{align}\label{last.lem2.a}
\int_{\mathcal{B}_1^c} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz &= \frac 1{|\partial \mathcal{B}_1|} \int_{\partial \mathcal{B}_1} \, dv \int_{\mathcal{B}_1^c} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz\\
\nonumber & = \int_{\mathcal{B}_1^c} \frac 1{\|z\|^7} \, dz = c_1 \int_1^\infty \frac 1{r^3}\, dr = \frac {c_1}2,
\end{align}
for some constant $c_1>0$.
Likewise,
\begin{equation}\label{last.lem2.b}
\int_{\mathcal{B}_1} \frac{1}{\|z\|^4\cdot \|z-v\|^3}\, dz = \frac{c_1}{|\partial \mathcal{B}_1|} \int_0^1 \, dr \int_{\partial \mathcal{B}_1} \frac{du}{\|ru - v\|^3} = c_1,
\end{equation}
with the same constant $c_1$ as in the previous display.
On the other hand
\begin{equation}\label{last.lem2.c}
\int_{\mathcal{B}_1^c} \frac 1{\|z\|^6}\, dz = c_1 \int_1^\infty \frac 1{r^2} \, dr = c_1.
\end{equation}
Furthermore, using again the rotational invariance,
\begin{align}\label{last.lem.2}
& \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right) \, dz = \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{2\|z-v\|} + \frac 1{2\|z+v\|} -1\right) \, dz \\
\nonumber & = \frac{c_1}{|\partial \mathcal{B}_1|} \int_0^1 \frac {dr}{r^2} \int_{\partial \mathcal{B}_1}\left(\frac 1{2\|v-ru\|} + \frac 1{2\|v+ru\|} -1\right)\, du.
\end{align}
Now we claim that for any $u,v\in \partial \mathcal{B}_1$, and any $r\in (0,1)$,
\begin{equation}\label{claim.geom}
\frac 12\left(\frac 1{\|v-ru\|} + \frac 1{\|v+ru\|}\right) \ge \frac{1}{\sqrt{1+r^2}}.
\end{equation}
Before we prove this claim, let us see how we can conclude the proof.
It suffices to notice that, if $f(s) = (1+s^2)^{-1/2}$, then $f'(s) \ge - s$, for all $s\in (0,1)$, and thus
\begin{equation}\label{lower.sqrt}
\frac 1{\sqrt{1+r^2}} - 1 = f(r) - f(0) \ge - \int_0^r s\, ds \ge -r^2/2.
\end{equation}
Inserting this and \eqref{claim.geom} in \eqref{last.lem.2} gives
$$ \int_{\mathcal{B}_1} \frac{1}{\|z\|^6}\left(\frac 1{\|z-v\|} -1\right) \, dz \ge -\frac {c_1}{2}. $$
Together with \eqref{last.lem2.a}, \eqref{last.lem2.b}, \eqref{last.lem2.c}, this shows that the integral in \eqref{last.lem.integral} is well positive. Thus all that remains to do is proving the claim \eqref{claim.geom}. Since the origin, $v$, $v+ru$, and $v-ru$ all lie in a common two-dimensional plane, one can
always work in the complex plane, and assume for simplicity that $v= 1$, and $u=e^{i\theta}$, for some $\theta\in [0,\pi/2]$. In this case, the claim is equivalent to showing that
$$\frac 12 \left( \frac 1{\sqrt{1+ r^2 + 2r\cos \theta} }+ \frac 1{\sqrt{1+ r^2 - 2r\cos \theta}}\right) \ge \frac 1{\sqrt{1+r^2}},$$
which is easily obtained using that the left hand side is a decreasing function of $\theta$.
This concludes the proof of Lemma \ref{lem.var.2}.
$\square$
\begin{remark}\emph{Note that the estimate of the covariance mentioned in the introduction, in case $(ii)$, can now be done as well. Indeed, denoting by
$$\widehat \tau_2:= \inf\{n\ge k+1\, :\, S_n\in S_k + \mathbb{R}R_\infty^2\},$$
it only remains to show that
$$\left|\mathbb{P}[\widehat \tau_2\le k+\varepsilon_k, \, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \cdot \mathbb{P}[\overline \tau_1<\infty]\right| = o\left(\frac 1k\right).$$
Using similar estimates as above we get, with $\chi_k = (k/\varepsilon_k)^{4/5}$,
\begin{align*}
& \left|\mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \cdot \mathbb{P}[\overline \tau_1<\infty]\right| \\
\stackrel{\eqref{Sn.large}}{=} & \left|\mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k, \|S_{\widehat \tau_2} - S_k\|\le \sqrt{\varepsilon_k\chi_k}, \overline \tau_1<\infty ] - \mathbb{P}[\widehat \tau_2\le k+ \varepsilon_k] \mathbb{P}[\overline \tau_1<\infty]\right| + \mathcal{O}\left(\frac 1{\sqrt k \chi_k^{\frac 52}}\right)\\
=& \sum_{\substack{x\in \mathbb{Z}^5 \\ \|y\|\le \sqrt{\varepsilon_k \chi_k}} } \left|\frac{p_k(x+y) + p_k(x-y)}{2}-p_k(x)\right| \mathbb{P}[\widehat \tau_2\le k+\varepsilon_k, S_{\widehat \tau_2} -S_k= y] \operatorname{Var}phi_x + \mathcal{O}\left(\frac 1{\sqrt k \chi_k^{\frac 52}}\right)\\
\lesssim & \frac 1{k^{\frac 32}} \mathbb{E}\left[\|S_{\widehat \tau_2}-S_k\|^2 {\text{\Large $\mathfrak 1$}}\{\|S_{\widehat \tau_2}-S_k\|\le \sqrt{\varepsilon_k\chi_k} \}\right] + \frac 1{\sqrt k \chi_k^{\frac 52}}\lesssim \frac 1{\sqrt k \chi_k^{\frac 52}}+ \frac {\sqrt{\varepsilon_k\chi_k}}{k^{\frac 32}},
\end{align*}
using that by \eqref{lem.hit.2} and the Markov property, one has $\mathbb{P}[\|S_{\widehat \tau_2}-S_k\|\ge t] \lesssim \frac 1t $.
}
\end{remark}
\subsection{Proof of Lemma \ref{lem.var.3}}
We consider only the case of $\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_1)$, the other one being entirely similar. Define
$$\tau_1:= \inf\{n\ge 0 : S_n^1\in \mathbb{R}R[\varepsilon_k,k]\},\ \tau_2:=\inf\{n\ge 0 : S_k+S_n^2\in \mathbb{R}R(-\infty, 0] \},$$
with $S^1$ and $S^2$ two independent walks, independent of $S$.
The first step is to see that
$$\operatorname{Cov}(Z_0\varphi_3,Z_k\psi_2)= \rho^2\cdot \operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\},{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) +o\left(\frac 1k\right),$$
with $\rho$ as in \eqref{def.rho}. Since the proof of this fact has exactly the same flavor as in the two previous lemmas, we omit the details and directly move to the next step.
Let $\eta\in (0,1/2)$ be some fixed constant (which will be sent to zero later). Notice first that
\begin{align}\label{eta.tau1}
& \nonumber \mathbb{P}\left[S^1_{\tau_1} \in \mathbb{R}R[(1-\eta) k,k],\, \tau_2<\infty\right] \stackrel{\eqref{Green.hit}, \eqref{lem.hit.2}}{\lesssim} \sum_{i=\lfloor (1-\eta)k\rfloor }^k
\mathbb{E}\left[\frac{G(S_i)}{1+\|S_k\|}\right] \\
& \stackrel{\eqref{exp.Green}}{\lesssim} \sum_{i=\lfloor (1-\eta)k\rfloor }^k
\frac{\mathbb{E}\left[G(S_i)\right] }{1+\sqrt{k-i}}
\stackrel{\eqref{exp.Green}}{\lesssim}\frac{\sqrt{\eta}}{k}.
\end{align}
Next, fix another constant $\delta\in (0,1/4)$ (which will be soon chosen small enough).
Then let $N: = \lfloor (1-\eta)k/\varepsilon_k^{1-\delta}\rfloor$, and for $i=1,\dots,N$, define
$$\tau_1^i:= \inf\{n\ge 0 \, :\, S_n^1 \in \mathbb{R}R[k_i,k_{i+1}]\},\quad \text{with}\quad k_i:= \varepsilon_k + i\lfloor \varepsilon_k^{1-\delta}\rfloor .$$
We claim that with sufficiently high probability, at most one of these hitting times is finite. Indeed, for $i\le N$,
set $I_i := \{k_i,\dots,k_{i+1}\}$, and notice that
\begin{align*}
&\sum_{1\le i< j\le N} \mathbb{P}[\tau_1^i <\infty, \, \tau_1^j<\infty,\, \tau_2<\infty] \\
\le & \sum_{1\le i< j\le N} \left(\mathbb{P}[\tau_1^i \le \tau_1^j<\infty,\, \tau_2<\infty] + \mathbb{P}[\tau_1^j \le \tau_1^i<\infty,\, \tau_2<\infty]\right) \\
\stackrel{\eqref{lem.hit.1}, \eqref{lem.hit.2}}{\lesssim} & \sum_{\substack{i=1,\dots,N, j\neq i \\ \ell \in I_i, m\in I_j}} \mathbb{E}\left[\frac{G(S_\ell - S_m) G(S_m)}{1+\|S_k\|} \right] \lesssim \frac{1}{\sqrt{k}} \sum_{\substack{i=1,\dots,N, j\neq i \\ \ell \in I_i, m\in I_j}} \mathbb{E}\left[G(S_\ell - S_m) G(S_m)\right]\\
\stackrel{\eqref{exp.Green}, \eqref{exp.Green.x}}{\lesssim} & \frac{1}{\sqrt{k}} \sum_{\substack{i=1,\dots, N, j\neq i \\ \ell \in I_i, m\in I_j}} \frac{1}{(1+ |m-\ell |^{3/2}) (m\wedge \ell)^{3/2}} \lesssim \frac {N\varepsilon_k^{(1-\delta)/2}}{\varepsilon_k^{3/2}\sqrt k} = o\left(\frac 1k\right),
\end{align*}
where the last equality follows by assuming $\varepsilon_k\ge k^{1-c}$, with $c>0$ small enough.
Therefore, as claimed
$$\mathbb{P}[\tau_1<\infty,\, \tau_2<\infty ] = \sum_{i=1}^{N} \mathbb{P}[\tau_1^i <\infty, \, \tau_2<\infty] + o\left(\frac 1k\right), $$
and one can show as well that,
$$\mathbb{P}[\tau_1<\infty]\cdot \mathbb{P}[ \tau_2<\infty ] = \sum_{i=1}^{N-2} \mathbb{P}[\tau_1^i <\infty] \cdot \mathbb{P}[\tau_2<\infty] +o\left(\frac 1k\right).$$
Next, observe that for any $i\le N$, using H\"older's inequality at the third line,
\begin{align*}
&\mathbb{P}\left[\tau_1^i <\infty, \tau_2<\infty, \|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\right] \stackrel{\eqref{Green.hit}, \eqref{lem.hit.2}}{\lesssim} \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ \frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\} }{1+\|S_k\|}\right]\\
& \stackrel{\eqref{exp.Green}}{\lesssim} \frac {1}{\sqrt k} \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\} \right]\\
& \lesssim \frac {1}{\sqrt k}\left( \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[\frac 1{1+\|S_j\|^4}\right]^{3/4}\right) \cdot \mathbb{P}\left[\|S_{k_{i+1}} - S_{k_i}\|^2\ge \varepsilon_k^{1-\delta/2}\right]^{1/4} \\
& \stackrel{\eqref{Sn.large}}{\lesssim} \frac {\varepsilon_k^{1-\delta}}{ k_i^{3/2}\sqrt{k}} \cdot \frac 1{\varepsilon_k^{5\delta/16}}= o\left(\frac 1{Nk}\right),
\end{align*}
by choosing again $\varepsilon_k\ge k^{1-c}$, with $c$ small enough.
Similarly, one has using Cauchy-Schwarz,
\begin{align*}
& \mathbb{P}\left[\tau_1^i <\infty, \tau_2<\infty, \|S_k - S_{k_{i+1}}\|^2\ge k \varepsilon_k^{\delta/2}\right] \lesssim \sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ \frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_k - S_{k_{i+1}}\|^2 \ge k \varepsilon_k^{\delta/2}\} }{1+\|S_k\|}\right]\\
& \lesssim \frac{1}{\varepsilon_k^{5\delta/8}}\sum_{j=k_i}^{k_{i+1}} \mathbb{E}\left[ G(S_j) \mathbb{E}\left[\frac 1{1+\|S_k\|^2} \mid S_j \right]^{1/2}\right]\lesssim \frac {\varepsilon_k^{1-\delta}}{ k_i^{3/2}\sqrt{k}} \cdot \frac 1{\varepsilon_k^{5\delta/8}}= o\left(\frac 1{Nk}\right).
\end{align*}
As a consequence, using also Theorem \ref{LCLT}, one has for $i\le N$, and with $\ell := k_{i+1}-k_i$,
\begin{align*}
& \mathbb{P}[\tau_1^i <\infty, \, \tau_2<\infty] \\
& =\sum_{x\in \mathbb{Z}^5} \sum_{\substack{ \|z\|^2 \le k \varepsilon_k^{\delta/2} \\ \|y\|^2 \le \varepsilon_k^{1-\delta/2} }} p_{k_i}(x)\mathbb{P}_{0,x}\left[ \mathbb{R}R_\infty \cap \widetilde \mathbb{R}R[0,\ell] \neq\varnothing, \widetilde S_{\ell} = y\right] p_{k-k_{i+1}}(z-y) \varphi_{x+z} + o\left(\frac 1{Nk}\right)\\
& = \sum_{x\in \mathbb{Z}^5} \sum_{\underset{\|z\|^2 \le k \varepsilon_k^{\delta/2} }{\|y\|^2 \le \varepsilon_k^{1-\delta/2}} } p_{k_i}(x) \mathbb{P}_{0,x}
\left[ \mathbb{R}R_\infty \cap \widetilde \mathbb{R}R[0,\ell] \neq\varnothing, \widetilde S_\ell = y\right] p_{k-k_i}(z) \varphi_{x+z} + o\left(\frac 1{Nk}\right)\\
& = \sum_{x,z\in \mathbb{Z}^5} p_{k_i}(x) \mathbb{P}_{0,x} \left[ \mathbb{R}R_\infty \cap \widetilde \mathbb{R}R[0,\ell] \neq\varnothing \right] p_{k-k_i}(z) \varphi_{x+z} + o\left(\frac 1{Nk}\right).
\end{align*}
Moreover, Theorem \ref{thm.asymptotic} yields for any nonzero $x\in \mathbb{Z}^5$, and some $\nu>0$,
\begin{align}\label{RtildeRell}
\mathbb{P}_{0,x} \left[ \mathbb{R}R_\infty \cap \widetilde \mathbb{R}R[0,\ell] \neq\varnothing \right] = \frac{\gamma_5}{\kappa}\cdot \mathbb{E}\left[ \sum_{j=0}^{\ell} G(x+\widetilde S_j) \right] + \mathcal{O}\left(\frac {\log(1+ \|x\|)}{\|x\| (\|x\|\wedge \ell)^{\nu}}\right).
\end{align}
Note also that for any $\varepsilon \in [0,1]$,
\begin{align*}
\sum_{x,z\in \mathbb{Z}^5} \frac{p_{k_i}(x)}{1+\|x\|^{1+\varepsilon}} p_{k-k_i}(z) \varphi_{x+z} = \mathbb{E}\left[\frac 1{(1+\|S_{k_i}\|^{1+\varepsilon})(1+\|S_k\|)}\right]
\lesssim \frac 1{\sqrt{k_i}^{1+\varepsilon} \sqrt k},
\end{align*}
and thus
$$\sum_{i=1}^N \sum_{x,z\in \mathbb{Z}^5} \frac{p_{k_i}(x)}{1+\|x\|^{1+\varepsilon}} p_{k-k_i}(z) \varphi_{x+z} = \mathcal{O}\left(\frac 1{\ell k^{\varepsilon}}\right).$$
In particular, the error term in \eqref{RtildeRell} can be neglected, as we take for instance $\delta=\nu/2$, and $\varepsilon_k\ge k^{1-c}$, with $c$ small enough.
It amounts now to estimate the other term in \eqref{RtildeRell}. By \eqref{pn.largex}, for any $x\in \mathbb{Z}^5$ and $j\ge 0$,
\begin{equation*}
\mathbb{E}[G(x+S_j)] = G_j(x) = G(x)-\mathcal O(\frac{j}{1+ \|x\|^d}).
\end{equation*}
As will become clear the error term can be neglected here.
Furthermore, similar computations as above show that for any $j\in \{k_i,\dots,k_{i+1}\}$,
$$\sum_{x,z\in \mathbb{Z}^5} p_{k_i}(x) G(x) p_{k-k_i}(z) \varphi_{x+z} = \sum_{x,z\in \mathbb{Z}^5} p_j(x) G(x) p_{k-j}(z) \varphi_{x+z}+ o\left(\frac 1{Nk}\right),$$
Altogether, and applying once more Theorem \ref{thm.asymptotic}, this gives for some $c_0>0$,
\begin{equation}\label{cov.3.sum}
\sum_{i=1}^N \mathbb{P}[\tau_1^i <\infty, \tau_2<\infty] = \sum_{j=\varepsilon_k}^{(1-\eta) k} \mathbb{E}[G(S_j)\varphi_{S_k}]+ o\left(\frac 1k\right) =c_0 \sum_{j=\varepsilon_k}^{\lfloor (1-\eta) k\rfloor} \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right]+ o\left(\frac 1k\right).
\end{equation}
We treat the first terms of the sum separately. Concerning the other ones notice that by \eqref{Green.asymp} and Donsker's invariance principle, one has
\begin{align*}
\sum_{j=\lfloor \eta k\rfloor}^{\lfloor (1-\eta)k\rfloor} \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right] & = \frac 1k \int_\eta^{1-\eta} \mathbb{E}\left[\frac{G(\Lambda \beta_s)}{\mathcal{J}(\Lambda \beta_1)}\right] \, ds+ o\left(\frac 1k\right) \\
& = \frac {c_5}k \int_\eta^{1-\eta} \mathbb{E}\left[\frac{1}{\|\beta_s\|^3 \cdot \|\beta_1\|}\right] \, ds+ o\left(\frac 1k\right),
\end{align*}
with $(\beta_s)_{s\ge 0}$ a standard Brownian motion, and $c_5>0$ the constant that appears in \eqref{Green.asymp}.
In the same way, one has
\begin{align*}
\sum_{i=1}^N \mathbb{P}[\tau_1^i <\infty] \cdot \mathbb{P}[\tau_2<\infty] & = c_0 \sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \mathbb{E}[G(S_j)]\mathbb{E}\left[\frac{1}{1+\mathcal{J}(S_k)}\right] \\
& \quad + \frac{c_0c_5}{k} \int_\eta^{1-\eta}\mathbb{E}\left[\frac{1}{\|\beta_s\|^3}\right] \mathbb{E}\left[ \frac 1{\|\beta_1\|}\right] \, ds+ o\left(\frac 1k\right),
\end{align*}
with the same constant $c_0$, as in \eqref{cov.3.sum}.
We next handle the sum of the first terms in \eqref{cov.3.sum} and show that its difference with the sum from the previous display is negligible.
Indeed, observe already that with $\chi_k := k/(\eta\varepsilon_k)$,
$$\sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \mathbb{E}\left[\frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_j\|\ge \eta^{1/4} \sqrt k\}}{1+\mathcal{J}(S_k)}\right]+ \mathbb{E}\left[\frac{G(S_j){\text{\Large $\mathfrak 1$}}\{\|S_k\|\ge \sqrt{k\chi_k}\}}{1+\mathcal{J}(S_k)}\right] \lesssim \frac{\eta^{1/4}}{k}. $$
Thus one has, using Theorem \ref{LCLT},
\begin{align}\label{eta.tau1.bis}
&\sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor}\left| \mathbb{E}\left[\frac{G(S_j)}{1+\mathcal{J}(S_k)}\right] - \mathbb{E}[G(S_j)]\cdot \mathbb{E}\left[\frac{1}{1+\mathcal{J}(S_k)}\right]\right| \\
\nonumber & \lesssim \sum_{j=\varepsilon_k}^{\lfloor \eta k\rfloor} \sum_{\underset{\|x\|\le \eta^{1/4}\sqrt{k}}{\|z\|\le \sqrt{k\chi_k}} } \frac{p_j(x)G(x)}{1+\|z\|} \left|\overline p_{k-j}(z-x) +\overline p_{k-j}(z+x) -2 \overline p_k(z)\right| + \frac {\eta^{1/4}}k \lesssim \frac {\eta^{1/4}}k.
\end{align}
Define now for $s\in (0,1]$,
$$ H_s: = \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_1\|}\right] - \mathbb{E}\left[\frac 1{\|\beta_s\|^3}\right] \cdot \mathbb{E}\left[\frac 1{\|\beta_1\|}\right].$$
Let $f_s(\cdot)$ be the density of $\beta_s$ and notice that as $s\to 0$,
\begin{align*}
& H_s=\int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac { f_s(x) f_{1-s}(y)}{\|x\|^3 \|x+y\|} \, dx\, dy - \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac { f_s(x) f_1(y)}{\|x\|^3 \|y\|} \, dx\, dy \\
& = \frac {1}{s^{3/2}} \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac{f_1(x)f_1(y)}{\|x\|^3} \left(
\frac {1}{\|y\sqrt{1-s} +x \sqrt s \| } -\frac {1}{\|y\|}\right) \, dx\, dy\\
& = \frac {1}{s^{3/2}} \int_{\mathbb{R}^5} \int_{\mathbb{R}^5} \frac{f_1(x)f_1(y)}{\|x\|^3\|y\|}
\left\{\left(\frac 12 + \frac{\|x\|^2}{2\|y\|^2} + \frac{\langle x,y\rangle^2}{\|y\|^4} \right)s + \mathcal{O}(s^{3/2}) \right\}\, dx\, dy =\frac c{\sqrt{s}} + \mathcal{O}(1),
\end{align*}
with $c>0$.
Thus the map $s\mapsto H_s$ is integrable at $0$, and since it is also continuous on $(0,1]$, its integral on this interval is well defined. Since $\eta$ can be taken arbitrarily small
in \eqref{eta.tau1} and \eqref{eta.tau1.bis}, in order to finish the proof it just remains to show that the integral of $H_s$ on $(0,1]$ is positive.
To this end, note first that $\widetilde \beta_{1-s}:= \beta_1 - \beta_s$ is independent of $\beta_s$. We use then \eqref{claim.geom}, which implies,
with $q=\mathbb{E}[1/\|\beta_1\|^3]$,
\begin{align*}
& \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_1\|}\right] = \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \|\beta_s + \widetilde \beta_{1-s}\|}\right]
\ge \mathbb{E}\left[\frac 1{\|\beta_s\|^3 \sqrt{\|\beta_s\|^2 + \|\widetilde \beta_{1-s}\|^2}}\right]\\
& = \frac{(5q)^2}{s^{3/2}} \int_0^\infty \int_0^\infty \frac{r e^{-\frac 52 r^2} u^4 e^{-\frac 52u^2}}{\sqrt{sr^2 + (1-s)u^2}} \, dr\, du\\
& = \frac{q^2}{5s^{3/2}} \int_0^\infty \int_0^\infty \frac{r e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2}}{\sqrt{sr^2 + (1-s)u^2}} \, dr\, du.
\end{align*}
We split the double integral in two parts, one on the set $\{sr^2\le (1-s)u^2\}$, and the other one on the complementary set $\{sr^2\ge (1-s)u^2\}$.
Call respectively $I_s^1$ and $I_s^2$ the integrals on these two sets. For $I_s^1$, \eqref{lower.sqrt} gives
\begin{align*}
I_s^1 & \ge \frac 1{\sqrt{1-s}} \int_0^\infty u^3 e^{-\frac{u^2}2} \int_0^{\sqrt{\frac{1-s}{s}}u} r e^{-\frac{r^2}2}\, dr \, du\\
& \qquad - \frac{s}{2(1-s)^{3/2}} \int_0^\infty u e^{-\frac{u^2}2} \int_0^{\sqrt{\frac{1-s}{s}}u} r^3 e^{-\frac{r^2}2}\, dr \, du\\
& = \frac{2(1-s^2)}{\sqrt{1-s}} + \frac{s^2}{\sqrt{1-s}} - \frac{s}{\sqrt{1-s}}= \frac{2-s - s^2}{\sqrt{1-s}}.
\end{align*}
For $I_s^2$ we simply use the rough bound:
$$I_s^2 \ge \frac 1{\sqrt{2s}} \int_0^\infty \int_0^\infty e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2}{\text{\Large $\mathfrak 1$}}\{sr^2\ge (1-s)u^2\} \, dr \, du, $$
which entails
\begin{align*}
& \int_0^1 \frac{I_s^2}{s^{3/2}}\, ds \ge \frac 1{\sqrt 2} \int_0^\infty \int_0^\infty e^{-\frac{r^2}2} u^4 e^{-\frac{u^2}2}
\left(\int_{\frac{u^2}{u^2+r^2}}^1 \frac 1{s^2}\, ds\right)\, dr \, du \\
& = \frac 1{\sqrt 2} \left(\int_0^\infty r^2 e^{-\frac{r^2}2} \, dr\right)^2= \frac 1{\sqrt 2} \left(\int_0^\infty e^{-\frac{r^2}2}\, dr\right)^2 =\frac{\pi}{2\sqrt 2}>1,
\end{align*}
where for the last inequality we use $\sqrt{2}<3/2$.
Note now that
$$\mathbb{E}\left[\frac 1{\|\beta_s\|^3}\right] \cdot \mathbb{E}\left[\frac 1 {\|\beta_1\|}\right] = \frac {2q^2}{5s^{3/2}} , $$
and
\begin{align*}
& \int_0^1 \frac{I_s^1 - 2}{s^{3/2}}\, ds \ge \int_0^1 s^{-3/2} \left\{(2- s- s^2)(1+\frac s2 + \frac{3s^2}{8}) - 2\right\} \, ds \\
& = - \int_0^1 (\frac 34 \sqrt s + \frac 78 s^{3/2} +\frac {3}{8} s^{5/2} ) \, ds = - (\frac 12 + \frac 7{20} + \frac{3}{28}) = - \frac{134}{140} > -1.
\end{align*}
Altogether this shows that the integral of $H_s$ on $(0,1]$ is well positive as wanted. This concludes the proof of the lemma.
$\square$
\begin{remark}\emph{The value of $H_1$ can be computed explicitely and one can check that it is positive. Similarly, by computing the leading order term in $I_s^2$, we could show that $H_s$ is also positive in a neighborhood of the origin, but it would be interesting to know whether $H_s$ is positive for all $s\in (0,1)$. }
\end{remark}
\subsection{Proof of Lemma \ref{lem.var.4}}
We define here
$$\tau_1:= \inf\{n\ge 0 : S_n^1\in \mathbb{R}R[\varepsilon_k,k-\varepsilon_k]\},\ \tau_2:=\inf\{n\ge 0 : S_k+S_n^2\in \mathbb{R}R[\varepsilon_k,k-\varepsilon_k]\},$$
with $S^1$ and $S^2$ two independent walks, independent of $S$. As in the previous lemma, we omit the details of the fact that
$$\operatorname{Cov}(Z_0\varphi_2,Z_k\psi_2)= \rho^2\cdot \operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\},{\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) +o\left(\frac 1k\right).$$
Then we define $N:=\lfloor (k-3\varepsilon_k)/\varepsilon_k\rfloor$ and let $(\tau_1^i)_{i=1,\dots,N}$ be as in the proof of Lemma \ref{lem.var.3}.
Define also $(\tau_2^i)_{i=1,\dots,N}$ analogously. Similarly as before one can see that
\begin{eqnarray}\label{tau1i.tau2j}
\mathbb{P}[\tau_1<\infty, \tau_2<\infty ]= \sum_{i=1}^{N} \sum_{j=1}^{N} \mathbb{P}[\tau_1^i <\infty, \tau_2^j <\infty ] + o\left(\frac 1k\right).
\end{eqnarray}
Note also that for any $i$ and $j$, with $|i-j| \le 1$, by \eqref{Green.hit} and \eqref{exp.Green},
$$\mathbb{P}[\tau_1^i<\infty, \, \tau_2^j<\infty] = \mathcal{O}\left(\frac {\varepsilon_k^{2(1-\delta)}}{k_i^{3/2} (k-k_i)^{3/2} }\right), $$
so that in \eqref{tau1i.tau2j}, one can consider only the sum on the indices $i$ and $j$ satisfying $|i-j|\ge 2$.
Furthermore, when $i<j$, the events $\{\tau_1^i<\infty\}$ and $\{\tau_2^j<\infty\}$ are independent. Thus altogether this gives
\begin{align*}
\operatorname{Cov}( &{\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) \\
& = \sum_{i = 1}^{N-2} \sum_{j=i+2}^N \left( \mathbb{P}[ \tau_1^j <\infty, \tau_2^i<\infty] - \mathbb{P}[\tau_1^j<\infty] \mathbb{P}[\tau_2^i<\infty] \right) + o\left(\frac 1k\right).
\end{align*}
Then by following carefully the same steps as in the proof of the previous lemma we arrive at
$$\operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, \, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\}) = \frac {c}{k} \int_0^1 \widetilde H_t\, dt + o\left(\frac 1k\right), $$
with $c>0$ some positive constant and,
$$\widetilde H_t := \int_0^t \left(\mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] - \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3}\right] \cdot\mathbb{E}\left[\frac 1{ \|\beta_t\|^3} \right] \right) \, ds,$$
at least provided we show first that $\widetilde H_t$ it is well defined and that its integral over $[0,1]$ is convergent.
However, observe that for any $t\in (0,1)$, one has with $q=\mathbb{E}[\|\beta_1\|^{-3}]$,
$$\int_0^t \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3}\right] \cdot\mathbb{E}\left[\frac 1{ \|\beta_t\|^3} \right] =\frac{q^2}{t^{3/2}} \int_0^t \frac 1{(1-s)^{3/2}}\, ds = \frac{2q^2(1-\sqrt{1-t})}{t^{3/2}\sqrt{1-t}},$$
and therefore this part is integrable on $[0,1]$. This implies in fact that the other part in the definition of $\widetilde H_t$ is also well defined and integrable, since we already know that
$\operatorname{Cov}({\text{\Large $\mathfrak 1$}}\{\tau_1<\infty\}, \, {\text{\Large $\mathfrak 1$}}\{\tau_2<\infty\})=\mathcal{O}(1/k)$. Thus it only remains to show that the integral of $\widetilde H_t$ on $[0,1]$ is positive.
To this end, we write $\beta_t = \beta_s + \gamma_{t-s}$, and $\beta_1 = \beta_s + \gamma_{t-s} + \delta_{1-t}$, with $(\gamma_u)_{u\ge 0}$ and $(\delta_u)_{u\ge 0}$
two independent Brownian motions, independent of $\beta$. Furthermore, using that the map $z\mapsto 1/\|z\|^3$ is harmonic outside the origin, we can compute:
\begin{align*}
& I_1:= \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge \|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] = \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge \|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\gamma_{t-s} + \delta_{1-t}\|^3 \cdot \|\beta_s\|^3} \right] \\
= & \frac{5q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \} }{\|\gamma_{t-s} + \delta_{1-t}\|^3} \int_{\frac{\|\gamma_{t-s}\|}{\sqrt s}}^\infty re^{-\frac 52 r^2} \, dr\right] = \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \}}{\|\gamma_{t-s} + \delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] \\
=& \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\ge \|\delta_{1-t}\| \}}{\|\gamma_{t-s}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] = \frac{5q^2}{s^{3/2}(t-s)^{3/2}} \mathbb{E}\left[\int_{\frac {\|\delta_{1-t}\|}{\sqrt{t-s}}}^\infty r e^{-\frac 52 r^2 (1+ \frac {t-s}{s})}\, dr \right] \\
=& \frac{q^2}{\sqrt{s}(t-s)^{3/2}t}\mathbb{E}\left[ e^{- \frac{\|\delta_{1-t}\|^2t}{s(t-s)}}\right] = \frac{5q^3}{\sqrt{s}(t-s)^{3/2}t} \int_0^\infty r^4 e^{-\frac 52r^2(1+ \frac{t(1-t)}{s(t-s)})}\, dr = \frac{q^2s^2(t-s)}{t\, \Delta^{5/2}},
\end{align*}
with
$$\Delta := t(1-t) + s(t-s) = (1-t)(t-s) + s(1-s).$$
Likewise,
\begin{align*}
I_2&:= \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\ge\|\gamma_{t-s}\|,\, \|\delta_{1-t}\| \ge \|\gamma_{t-s}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] = \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\le \|\delta_{1-t}\| \}}{\|\gamma_{t-s} + \delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right] \\
=& \frac{q}{s^{3/2}} \mathbb{E}\left[\frac{{\text{\Large $\mathfrak 1$}}\{\|\gamma_{t-s}\|\le \|\delta_{1-t}\| \}}{\|\delta_{1-t}\|^3} e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \right]
= \frac{5q^2}{s^{3/2}(1-t)^{3/2}} \mathbb{E}\left[ e^{-\frac 5{2s} \|\gamma_{t-s}\|^2} \int_{\frac {\|\gamma_{t-s}\|}{\sqrt{1-t}}}^\infty r e^{-\frac 52 r^2}\, dr \right] \\
=& \frac{q^2}{s^{3/2}(1-t)^{3/2}} \mathbb{E}\left[ e^{-\frac 5{2} \|\gamma_{t-s}\|^2(\frac 1s + \frac {1}{1-t})}\right] = \frac{q^2s(1-t)}{\Delta^{5/2}}.
\end{align*}
Define as well
\begin{align*}
I_3 & := \mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\le \|\gamma_{t-s}\|\le \|\delta_{1-t}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right],
\end{align*}
\begin{align*}
I_4:=\mathbb{E}\left[\frac {{\text{\Large $\mathfrak 1$}}\{ \|\delta_{1-t}\|\le\|\beta_s\|\le \|\gamma_{t-s}\| \} }{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right], \ I_5:= \mathbb{E}\left[\frac { {\text{\Large $\mathfrak 1$}}\{\|\beta_s\|\le \|\delta_{1-t}\|\le \|\gamma_{t-s}\| \} }{ \|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right].
\end{align*}
Note that by symmetry one has
$$\int_{0\le s\le t \le 1} I_1 \, ds \, dt = \int_{0\le s\le t \le 1} I_3 \, ds \, dt,\text{ and } \int_{0\le s\le t \le 1} I_4 \, ds \, dt = \int_{0\le s\le t \le 1} I_5 \, ds \, dt. $$
Observe also that,
$$I_1+I_2 = \frac{q^2s}{t \Delta^{3/2}}. $$
Moreover, using symmetry again, we can see that
$$\int_0^t \frac {s-t/2}{ \Delta^{3/2}} \, ds = 0,$$
and thus
$$\int_0^t (I_1+I_2)\, ds = \frac {q^2}{2} \int_0^t \frac{1}{ \Delta^{3/2}}\, ds. $$
Likewise,
\begin{align*}
& \int_{0\le s\le t\le 1} I_1\, ds \, dt = \int_{0\le s\le t\le 1} \frac{q^2s(t-s)^2}{t\Delta^{5/2}}\, ds \, dt = \frac 12\int_{0\le s\le t\le 1} \frac{q^2s(t-s)}{\Delta^{5/2}}\, ds \, dt \\
=& \int_{0\le s\le t\le 1} \frac{q^2(1-t)(t-s)}{2\Delta^{5/2}}\, ds \, dt = \int_{0\le s\le t\le 1} \frac {q^2t(1-t)}{4\Delta^{5/2}}\, ds\, dt = \int_{0\le s\le t\le 1} \frac {q^2}{6\Delta^{3/2}}\, ds\, dt.
\end{align*}
It follows that
$$\int_{0\le s\le t\le 1} (I_1+I_2+I_3) \, ds\, dt = \frac {2q^2}3 \int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\, dt.$$
We consider now the term $I_4$, which is a bit more complicated to compute, thus we only give a lower bound on a suitable interval. To be more precise, we first define for $r\ge 0$ and $\lambda\ge 0$,
$$F(r):=\int_0^r s^4e^{-5s^2/2}\, ds,\quad \text{and}\quad F_2(\lambda, r):=\int_0^r F(\lambda s) s^4e^{-5s^2/2}\, ds,$$
and then we write,
\begin{align*}
& I_4= \mathbb{E}\left[ \frac { {\text{\Large $\mathfrak 1$}}\{ \|\delta_{1-t}\|\le\|\beta_s\|\le \|\gamma_{t-s}\| \} }{ \|\gamma_{t-s}\|^6} \right]
= 5q\cdot \mathbb{E}\left[\frac { {\text{\Large $\mathfrak 1$}}\{ \|\beta_s\|\le \|\gamma_{t-s}\| \} }{ \|\gamma_{t-s}\|^6} F\left(\frac{\|\beta_s\|}{\sqrt{1-t}}\right)\right]\\
=& \mathbb{E}\left[\frac {(5q)^2}{ \|\gamma_{t-s}\|^6} F_2\left(\frac{\sqrt s}{\sqrt{1-t}},\frac{\|\gamma_{t-s}\|}{\sqrt s}\right)\right]
= \frac{(5q)^3}{(t-s)^3} \int_0^\infty \frac{e^{-\frac{5r^2}{2}}}{r^2} F_2\left(\frac{\sqrt s}{\sqrt{1-t}},r\frac{\sqrt{t-s}}{\sqrt s}\right)\, dr \\
=& \frac{(5q)^3}{(t-s)^3} \left\{ \frac{\sqrt{t-s}}{\sqrt s} \int_0^\infty F\left(r\frac{\sqrt{t-s}}{\sqrt{1-t}}\right) r^3e^{-\frac{5r^2}{2}}\, dr -5\int_0^\infty F_2\left(\frac{\sqrt s}{\sqrt{1-t}},r\frac{\sqrt{t-s}}{\sqrt s}\right) e^{-\frac{5r^2}{2}}\, dr \right\}\\
\ge & \frac{(5q)^3}{(t-s)^3} \left\{ \frac{(t-s)^{\frac 32}}{s^{3/2}} \int_0^\infty F\left( r\frac{\sqrt{t-s}}{\sqrt {1-t}}\right) r^3 e^{-\frac {5r^2t}{2s}}\, dr +\frac{(2s-t)\sqrt{t-s}}{s^{3/2}} \int_0^\infty F\left(r\frac{\sqrt{t-s}}{\sqrt{1-t}}\right) r^3e^{-\frac {5r^2}{2}}\, dr \right\},
\end{align*}
using that
$$F_2(\lambda,r)\le \frac 15 r^3F(\lambda r)(1-e^{-5r^2/2}).$$
Therefore, if $t/2\le s\le t$,
\begin{align*}
I_4& \ge \frac{(5q)^3}{[s(t-s)]^{3/2}} \int_0^\infty r^3 F\left( r\frac{\sqrt{t-s}}{\sqrt {1-t}}\right) e^{-\frac {5r^2t}{2s}}\, dr \\
& = \frac{(5q)^3\sqrt s}{t^2(t-s)^{3/2}} \int_0^\infty r^3 F\left( r\frac{\sqrt{s(t-s)}}{\sqrt {t(1-t)}}\right) e^{-5r^2/2}\, dr\\
& \ge \frac{2\cdot 5^2q^3\sqrt s}{t^2(t-s)^{3/2}}\int_0^\infty F\left( r\frac{\sqrt{s(t-s)}}{\sqrt {t(1-t)}}\right) re^{-\frac{5r^2}{2}}\, dr
=\frac{2\cdot 5q^3s^3(t-s)}{t^2[t(1-t)]^{5/2}}\int_0^\infty r^4 e^{-\frac{5r^2\Delta}{2t(1-t)}}\, dr\\
& = \frac{2 q^2s^3(t-s)}{t^2\Delta^{5/2}}\ge \frac{q^2s(t-s)}{2\Delta^{5/2}},
\end{align*}
and as a consequence,
\begin{align*}
\int_{0\le s\le t\le 1} I_4\, ds\, dt & \ge \int_{t/2\le s\le t\le 1}I_4\, ds\, dt \ge \frac {q^2}2 \int_{t/2\le s\le t\le 1}\, \frac{s(t-s)}{\Delta^{5/2}}\, ds\, dt \\
&=\frac {q^2}4 \int_{0\le s\le t\le 1}\frac{s(t-s)}{\Delta^{5/2}}\, ds\, dt = \frac {q^2}{12} \int_{0\le s\le t\le 1}\Delta^{-3/2}\, ds\, dt.
\end{align*}
Putting all these estimates together yields
$$\int_{0\le s\le t\le 1} \mathbb{E}\left[\frac 1{\|\beta_s-\beta_1\|^3 \cdot \|\beta_t\|^3} \right] \, ds\, dt = \sum_{k=1}^5 \int_{0\le s\le t\le 1} I_k\, ds\, dt \ge \frac 56 \int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\,dt.$$
Thus it just remains to show that
\begin{eqnarray}\label{finalgoal}
\int_{0\le s\le t\le 1} \Delta^{-3/2}\, ds\,dt \ge \frac 65 \int_{0\le s\le t\le 1} \widetilde \Delta^{-3/2}\, ds\,dt,
\end{eqnarray}
where $\widetilde \Delta := t(1-s)$.
Note that $\Delta = \widetilde \Delta +(t-s)^2$.
Recall also that for any $\alpha \in \mathbb{R}$, and any $x\in (-1,1)$,
\begin{eqnarray}\label{DL1+x}
(1+x)^\alpha = 1+ \sum_{i\ge 1}\frac{\alpha (\alpha-1)\dots(\alpha-i+1)}{i!} x^i.
\end{eqnarray}
Thus
$$\frac 1{\Delta^{3/2}} = \frac 1{\widetilde \Delta^{3/2}} \left(1+\sum_{k\ge 1} \frac{(3/2)(5/2)\dots (k+1/2)}{k!} \cdot \frac{(t-s)^{2k}}{\widetilde \Delta^k}\right).$$
One needs now to compute the coefficients $C_k$ defined by
$$C_k := \frac{(3/2)(5/2)\dots (k+1/2)}{k!} \int_{0\le s\le t\le 1} \frac{(t-s)^{2k}}{\widetilde \Delta^{k+3/2}}\, ds\,dt.$$
We claim that one has for any $k\ge 0$,
\begin{eqnarray}\label{Ckformula}
C_k= \frac{2^{2k+2}}{2k+1}(-1)^k \mathcal{S}igma_k,
\end{eqnarray}
with $\mathcal{S}igma_0=1$, and for $k\ge 1$,
$$\mathcal{S}igma_k = 1+ \sum_{i=1}^{2k}(-1)^i \frac{(k+1/2)(k-1/2)\dots(k-i +3/2)}{i!}.$$
We will prove this formula in a moment, but let us conclude the proof of the lemma first, assuming it is true.
Straightforward computations show by \eqref{Ckformula} that
$$C_0 = 4,\quad C_1= \frac 23, \quad \text{and}\quad C_2= \frac {3}{10},$$
and $C_0+C_1+C_2\ge 6C_0/5$, gives \eqref{finalgoal} as wanted.
So let us prove \eqref{Ckformula} now. Note that one can assume $k\ge 1$, as the result for $k=0$ is immediate. By \eqref{DL1+x}, one has
$$(1-s)^{-k-3/2}= 1+ \sum_{i\ge 1} \frac{(k+3/2)(k+5/2)\dots (k+ i +1/2)}{i!} s^i.$$
Thus by integrating by parts, we get
$$\int_0^t \frac{(t-s)^{2k}}{(1-s)^{k+3/2}} \, ds = (2k)! \sum_{i\ge 0} \frac{(k+3/2)\dots(k+i+1/2)}{(2k + i +1)!}\cdot t^{2k+i+1},$$
and then
$$\int_0^1 \int_0^t \frac{(t-s)^{2k}}{t^{k+3/2}(1-s)^{k+3/2}} \, ds\, dt = (2k)! \sum_{i\ge 0} \frac{(k+3/2)\dots(k+i-1/2)}{(2k + i +1)!}.$$
As a consequence,
\begin{align*}
C_k& = \frac{(2k)!}{k!} \sum_{i\ge 0} \frac{(3/2)(5/2)\dots(k+i-1/2)}{(2k+i+1)!} \\
& = \frac{(2k)!}{(k+1/2)(k-1/2)\dots(3/2)(1/2)^2\cdot k!} \sum_{i\ge 0} \frac{|(k+1/2)(k-1/2)\dots(-k-i+1/2)|}{(2k + i +1)!} \\
& = \frac{2^{2k+2}}{2k+1} \sum_{i\ge 0} \frac{|(k+1/2)(k-1/2)\dots(-k-i+1/2)|}{(2k + i +1)!},
\end{align*}
and it just remains to observe that the last sum is well equal to $\mathcal{S}igma_k$. The latter is obtained by taking the limit as $t$ goes to $1$ in the
formula \eqref{DL1+x} for $(1-t)^{k+1/2}$. This concludes the proof of Lemma \ref{lem.var.4}.
$\square$
\begin{remark}
\emph{It would be interesting to show that the covariance between $1/\|\beta_s-\beta_1\|^3$ and $1/\|\beta_t\|^3$ itself is positive for all $0\le s\le t\le 1$, and not just its integral, as we have just shown. }
\end{remark}
\section{Proof of Theorem B}
The proof of Theorem B is based on the Lindeberg-Feller theorem for triangular arrays, that we recall for convenience (see Theorem 3.4.5 in \cite{Dur}):
\begin{theorem}[Lindeberg-Feller]\label{thm:lind}
For each $n$ let $(X_{n,i}: \, 1\leq i\leq n)$ be a collection of independent random variables with zero mean. Suppose that the following two conditions are satisfied
\newline
{\rm{(i)}} $\sum_{i=1}^{n}\mathbb{E}[X_{n,i}^2] \to \sigma^2>0$ as $n\to \infty$, and
\newline
{\rm{(ii)}} $\sum_{i=1}^{n}\mathbb{E}\left[(X_{n,i})^2{\text{\Large $\mathfrak 1$}}\{|X_{n,i}|>\varepsilon\}\right] \to 0$, as $n\to \infty$, for all $\varepsilon>0$.
\newline
Then, $S_n=X_{n,1}+\ldots + X_{n,n} \Longrightarrow \mathbb{N}N(0,\sigma^2)$, as $n\to \infty$.
\end{theorem}
In order to apply this result, one needs three ingredients. The first one is an asymptotic estimate for the variance of the capacity of the range,
which is given by our Theorem A.
The second ingredient is a decomposition of the capacity of two sets as a sum of the capacities of the two sets minus some error term, in the spirit of the inclusion-exclusion
formula for the cardinality of a set, which allows to decompose the capacity of the range up to time $n$ into a sum of independent pieces having the law of the capacity of the range up to a smaller time index, and finally the last ingredient is a sufficiently good bound on the centered fourth moment.
This strategy has been already employed successfully for the capacity of the range in dimension six and more in \cite{ASS18} (and for the size of the range as well, see \cite{JO69, JP71}).
In this case the asymptotic of the variance followed simply from a sub-additivity argument, but the last two ingredients are entirely similar in dimension $5$ and in higher dimension. In particular one has the following decomposition (see Proposition 1.6 in \cite{ASS19}): for any two subsets $A,B\subset \mathbb{Z}^d$, $d\ge 3$,
\begin{eqnarray}\label{cap.decomp}
\mathrm{Cap}(A\cup B) = \mathrm{Cap}(A) + \mathrm{Cap}(B) - \chi(A,B),
\end{eqnarray}
where $\chi(A,B)$ is some error term. Its precise expression is not so important here. All one needs to know is that
$$|\chi(A,B)| \le 3\sum_{x\in A}\sum_{y\in B} G(x,y),$$
so that by \cite[Lemma 3.2]{ASS18}, if $\mathbb{R}R_n$ and $\widetilde \mathbb{R}R_n$ are the ranges of two independent walks in $\mathbb{Z}^5$, then
\begin{eqnarray}\label{bound.chin}
\mathbb{E}[\chi(\mathbb{R}R_n,\widetilde \mathbb{R}R_n)^4] = \mathcal{O}(n^2).
\end{eqnarray}
We note that the result is shown for the simple random walk only in \cite{ASS18}, but the proof applies as well to our setting (in particular Lemma 3.1 thereof also follows from \eqref{exp.Green}).
Now as noticed already by Le Gall in his paper \cite{LG86} (see his remark (iii) p.503), a good bound on the centered fourth moment follows from \eqref{cap.decomp} and \eqref{bound.chin}, and the triangle inequality in $L^4$. More precisely in dimension $5$, one obtains (see for instance the proof of Lemma 4.2 in \cite{ASS18} for some more details):
\begin{eqnarray}\label{cap.fourth}
\mathbb{E}\left[\left(\mathrm{Cap}(\mathbb{R}R_n)-\mathbb{E}[\mathrm{Cap}(\mathbb{R}R_n)]\right)^4\right] = \mathcal{O}(n^2(\log n)^4).
\end{eqnarray}
Actually we would even obtain the slightly better bound $\mathcal{O}(n^2(\log n)^2)$, using our new bound on the variance $\operatorname{Var}(\mathrm{Cap}(\mathbb{R}R_n))=\mathcal{O}(n\log n)$, but this is not needed here.
Using next a dyadic decomposition of $n$, one can write with $T:=\lfloor n/(\log n)^4\rfloor$,
\begin{eqnarray}\label{cpRn}
\mathrm{Cap}(\mathbb{R}R_n) = \sum_{i=0}^{\lfloor n/T\rfloor} \mathrm{Cap}(\mathbb{R}R^{(i)}_T) - R_n,
\end{eqnarray}
where the $(\mathbb{R}R^{(i)}_T)_{i=0,\dots,n/T}$ are independent pieces of the range of length either $T$ or $T+1$, and
$$
R_n= \sum_{\ell =1}^L \sum_{i=0}^{2^{\ell-1}} \chi(\mathbb{R}R^{(2i)}_{n/2^\ell},\mathbb{R}R^{(2i+1)}_{n/2^\ell}),
$$
is a triangular array of error terms (with $L=\log_2(\log n)^4$). Then it follows from \eqref{bound.chin}, that
\begin{align*}
\operatorname{Var}(R_n) &\le L \sum_{\ell=1}^L \operatorname{Var}\left(\sum_{i=1}^{2^{\ell-1}} \chi(\mathbb{R}R^{(2i)}_{n/2^\ell},\mathbb{R}R^{(2i+1)}_{n/2^\ell})\right)\le L\sum_{\ell=1}^L \sum_{i=1}^{2^{\ell-1}} \operatorname{Var}\left(\chi(\mathbb{R}R^{(2i)}_{n/2^\ell},\mathbb{R}R^{(2i+1)}_{n/2^\ell})\right)\\
& = \mathcal{O}(L^2 n)=\mathcal{O}(n(\log \log n)^2).
\end{align*}
In particular $(R_n-\mathbb{E}[R_n])/\sqrt{n\log n}$ converges in probability to $0$. Thus one is just led to show the convergence in law of the remaining sum in \eqref{cpRn}.
For this, one can apply Theorem \ref{thm:lind}, with
$$X_{n,i}:=\frac{\mathrm{Cap}(\mathbb{R}R^{(i)}_T)-\mathbb{E}\left[\mathrm{Cap}(\mathbb{R}R^{(i)}_T)\right]}{\sqrt{n\log n}}.$$
Indeed, Condition (i) of the theorem follows from Theorem A, and Condition (ii) follows
from \eqref{cap.fourth} and Markov's inequality (more details can be found in \cite{ASS18}). This concludes the proof of Theorem B.
$\square$
\section*{Acknowledgments} We thank Fran{\c c}oise P\`ene for enlightening discussions at an early stage of this project, and Pierre Tarrago for Reference \cite{Uchiyama98}. We also warmly thank Amine Asselah and Perla Sousi for our many discussions related to the subject of this work, which grew out of it. The author was supported by the ANR SWIWS (ANR-17-CE40-0032) and MALIN (ANR-16-CE93-0003).
\end{document} |
\begin{document}
\title{Entanglement breaking channels and entanglement sudden death}
\author{Laura T. Knoll}
\affiliation{DEILAP, CITEDEF \& CONICET, J.B. de La Salle 4397, 1603 Villa Martelli, Buenos Aires, Argentina}
\author{Christian T. Schmiegelow}
\affiliation{Laboratorio de Iones y \'Atomos Frí\'{\i}os, Departamento de F\'{\i}sica, FCEN, UBA \& IFIBA, CONICET, Pabell\'on 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina}
\author{Osvaldo~Jim\'enez~Far\'{\i}as}
\affiliation{ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain”}
\affiliation{Centro Brasileiro de Pesquisas F\'isicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, 22290-180 Rio de Janeiro, Brazil}
\author{Stephen P. Walborn}
\affiliation{Instituto de F\'{\i}sica, Universidade Federal do Rio de
Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil}
\author{Miguel A. Larotonda}
\affiliation{DEILAP, CITEDEF \& CONICET, J.B. de La Salle 4397, 1603 Villa Martelli, Buenos Aires, Argentina}
\date{\today}
\begin{abstract}
The occurrence of entanglement sudden death in the evolution of a bipartite system depends on both the initial state and the channel responsible for the evolution. An extreme case is that of entanglement braking channels, which are channels that acting on only one of the subsystems drives them to full disentanglement regardless of the initial state. In general, one can find certain combinations of initial states and channels acting on one or both subsystems that can result in entanglement sudden death or not. Neither the channel nor the initial state, but their combination, is responsible for this effect, but their combination. In this work we show that, in all cases, when entanglement sudden death occurs, the evolution can be mapped to that of an effective entanglement breaking channel on a modified initial state. Our results allow to anticipate which states will suffer entanglement sudden death or not for a given evolution. An experiment with polarization entangled photons demonstrates the utility of this result in a variety of cases.
\end{abstract}
\pacs{}
\maketitle
\section{Introduction}
Quantum entanglement is a property of physical systems composed of two or more parts and is a consequence of the superposition principle on bipartite or multipartite systems. It takes the form of correlations of measurement results that cannot be reproduced by any classical mechanism. For this reason, entanglement has become a physical concept of central importance for the foundations of Quantum Mechanics \cite{schrodinger1935discussion}.
Apart from its conceptual relevance, entanglement is also a resource that can be used to accomplish informational tasks like teleportation \cite{bennett1993teleporting}, quantum key distribution and super dense coding \cite{horodecki2009quantum,gisin2002quantum,agrawal2006perfect}.
The interaction of an entangled system with its environment results in an irreversible distribution of the entanglement between the system and the environment \cite{aguilar2014experimental,zurek2003decoherence,aguilar2014flow,dur2000three,xu2009experimental,aolitarev}. Interestingly, through the so-called Choi-Jamiolkowski relation \cite{jamiolkowski1972linear,choi1975completely}, the representation of a quantum channel and a state are equivalent within the theory. In this report we explore this connection between environments and states and its implications for the dynamics of entanglement. We perform experiments where we produce pairs of polarization entangled photons, initially in a variety of entangled mixed states. Then, one of the photons is exposed to a local environment. For a given environment $E$, some states of the system $S$ suffer Entanglement Sudden Death (ESD), while for other states the entanglement will disappear only asymptotically, as the interaction time tends to infinity \cite{almeida2007environment,laurat2007heralded,eberly2007end,yu2009sudden,aolitarev,drumond2009asymptotic,cunha2007geometry}. Theoretical considerations allow us to show that ESD occurs if and only if there exists a local effective Entanglement-Breaking Channel (EBC). This effect arises as the interplay between an initial state and a channel that annihilates any entanglement that one could try to establish through it \cite{horodecki2003entanglement,ruskai2003qubit}. The effective EBC is not the channel $E$ nor the state of the system $S$, rather it is a combination of the properties of both of them that we are able to measure experimentally. We show that this allows one to anticipate which states will suffer ESD or not for a given channel. This represents our main result.
Section \ref{chanstate} is devoted to review and define notation on the duality between quantum channels and states, for a general $d$-dimension bipartite system. In section \ref{qubitevo} we address the problem of the entanglement evolution on a two-qubit system, and we explicitly show the relation between entanglement breaking channels and the sudden death of entanglement. Finally we apply these results in section \ref{exper} to two experimental situations using polarization-entangled photon pairs, where one of the qubits evolves through a noisy channel, by interacting with a controlled environment implemented using it's path internal degree of freedom.
\section{The relation between channels and states}
\label{chanstate}
The evolution of a system $S$ of dimension $d$ due to an environment $E$, can be represented by at most $d^2$ Kraus operators, $\oper{K}_i$ by
\begin{eqnarray}
\oper{\mathcal{E}}[\rho_S]=\sum_{i=1}^{d^2}\oper{K}_i\rho_S\oper{K}_i^{\dagger},
\end{eqnarray}
where $\rho_S$ is the the state of the system. To conserve probabilities, we choose to work with trace preserving maps. Such assumption implies that the Kraus operators must satisfy
\begin{eqnarray}
\sum_{i=1}^{d^2}\oper{K}_i^{\dagger}\oper{K}_i=\oper{\mathcal{I}}.
\end{eqnarray}
Time dependence of the Kraus operators $\oper{K}_i(t)$ is omitted from this notation, for simplicity.
A mathematical relation can be established between channels and states; this is the aforementioned Jamiolkowski-Choi (J-CH) isomorphism \cite{jamiolkowski1972linear,choi1975completely}. More than just a theoretical tool the so-called \textit{Channel-State duality} has many practical implications and can be stated as follows:
The set of channels $\{\oper{\mathcal{E}}\}$ acting on $\mathcal{C}^d$ is isomorphic to the set of bipartite states $\{\rho^* \}$ in $\mathcal{C}^d\otimes \mathcal{C}^d$, satisfying ${\rm Tr}_{2}[\rho^*]=\frac{1}{d}$, where $\mathcal{C}^d$ is a $d$-dimensional Hilbert space and ${\rm Tr}_{2}[\rho^*]$ is the partial trace over the second system \cite{horodecki1999general}.
The isomorphism can be established using a maximally entangled state $\ket{\phi_+}=\frac{1}{\sqrt{d}}\sum_i \ket{i}\ket{i}$ as illustrated in fig \ref{fig:chanstate}a). For two qubits one then has that $\rho^*=(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\ket{\phi_+}\bra{\phi_+}$ satisfies the isomorphism.
\begin{figure}
\caption{\label{fig:chanstate}
\label{fig:chanstate}
\end{figure}
The relation between channels and states can be extended for arbitrary states in $\mathcal{C}^d\otimes\mathcal{C}^d$ beyond the J-CH isomorphism, removing the condition ${\rm Tr}_{2}[\rho^*]=\frac{1}{d}$: it was shown by Werner \cite{werner2001quantum} that there always exists a channel $\oper{\Gamma}$ and a pure state $\sigma$ such that any bipartite density matrix $\rho$ can be written as $\rho=(\oper{\mathcal{I}}\otimes\oper{\Gamma})\sigma$, where $\sigma$ is a pure-state density matrix.
This result associates two physical objects to a mixed entangled state: A pure state $\sigma$, and a channel $\oper{\Gamma}$. In contrast to the J-CH isomorphism where there is a one to one correspondence between the target state and a unilateral channel, here the combination of a channel and a pure state is not unique: an arbitrary mixed state $\rho$ can be written as:
\begin{eqnarray}
\label{decomp}
\rho= (\oper{\mathcal{I}}\otimes\oper{\Gamma})\sigma= (\oper{\Lambda}\otimes\oper{\mathcal{I}})\sigma'
\end{eqnarray}
with $\oper{\Gamma} \neq \oper{\Lambda}$, and $\sigma \neq \sigma'$ are two distinct pure states. This represents two ways of preparing the bipartite state $\rho$ as illustrated in Fig \ref{fig:chanstate}b).
\section{Qubit to qubit entanglement and its evolution}
\label{qubitevo}
Consider a quantum operation $\oper{\mathcal{E}}$ acting in one of the qubits, so that the bipartite state evolves through $\oper{\mathcal{I}}\otimes \oper{\mathcal{E}} $. Konrad \emph{et al.}~\cite{konrad2008evolution} demonstrated that the concurrence~\cite{wootters2001entanglement} of pure bipartite states, when exposed to one qubit channels, evolves according to
\begin{eqnarray}
C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\ket{\chi}\bra{\chi}]= C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\ket{\phi_+}\bra{\phi_+}]C[\ket{\chi}].
\label{factpuro}
\end{eqnarray}
This equation shows that, for pure states, the entanglement evolution depends on the state only by means of the initial concurrence $C[\ket{\chi}]$; all the information about the evolution of the entanglement is contained in the dual state of the channel $\oper{\mathcal{E}}$.
We can also derive an evolution equation for the concurrence of mixed states. According to Eq.~\eqref{decomp} one can consider a mixed state as the combination of a pure state and a unilateral channel. Then, under the action of the channel $\oper{\mathcal{E}} $, a mixed state $\rho$ can be written as $C\left[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\rho\right]=C\left[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})(\oper{\mathcal{I}}\otimes\oper{\Gamma})\sigma\right]$, where $\sigma$ a pure state. It follows from Eq. \eqref{factpuro} that
\begin{equation}
\label{general}
C\left[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\rho\right]=C\left[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}\oper{\Gamma})|\phi_+\rangle\langle\phi_+|\right]C[\sigma].
\end{equation}
This is the extended version of Eq.~\eqref{factpuro}, which was reported and experimentally tested in Ref. \cite{farias2009determining}. It shows that, even if the input state is mixed, the final concurrence is given by the product of two factors: the concurrence of a Bell state evolving under the action of the product of two channels $\oper{\mathcal{E}}\oper{\Gamma}$, acting on the second qubit, and the concurrence of a pure state, $C[\sigma]$. Both the channel $\oper{\Gamma}$ and the factor $C[\sigma]$ depend only on the initial mixed state.
\par
With these tools, we now study the situation where entanglement in the qubits vanishes in finite time, a phenomenon better known as Entanglement Sudden Death (ESD).
\subsection{Unilateral Channels - Pure States}
Let's consider first the situation of a pure initial state $\ket{\chi}$ subject to a unilateral channel, $\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}$. In this case the evolution equation \eqref{factpuro} says that the concurrence depends, up to a constant factor, only on the channel through $C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}})\ket{\phi_+}\bra{\phi_+}]$. No matter what the initial entangled pure state is, if the entanglement vanishes there is only one cause for this: the channel $\oper{\mathcal{E}}$. This is the definition of an Entanglement Breaking Channel (EBC) rephrased in terms of the evolution equation. Many channels such as Amplitude Damping (ADC) or Phase Damping (PDC) are EBCs only asymptotically, and so they do not produce ESD alone. An example of an EBC for qubits is the Depolarizing channel \cite{nielsen2010quantum}.
\subsection{Unilateral Channels - Mixed States}
Now we consider a \emph{mixed} entangled state $\rho$ evolving under the action of $\oper{\mathcal{I}}\otimes \oper{\mathcal{E}} $. In this situation the evolution depends on the initial state $\rho$, and Eq.\eqref{general} tells us exactly how. If the channel $\oper{\mathcal{E}}$ is an EBC for finite times, then it is clear that the composition will be also an EBC and ESD will occur. On the other hand, if the channel $\oper{\mathcal{E}}$ is not an EBC, ESD may still occur because of the mixture of the state $\rho$. In both cases there is an effective EBC $\oper{\mathcal{E}} \oper{\Gamma}$ acting on a maximally entangled state, that is responsible for the ESD. Indeed, an initially mixed state defines a whole family of EBC's through the equation
\begin{equation}
C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}\oper{\Gamma})\ket{\phi_+}\bra{\phi_+}]=0.
\end{equation}
Here we see the usefulness of the generalized evolution equation, as it separates correctly the characteristics of the mixed states that lead to a decrease or loss of entanglement.
\subsection{Bilateral Channels - Pure and Mixed States}
Consider now two different pure bipartite entangled states $\ket{\Phi_1}$ and $\ket{\Phi_2}$ under the action of two channels $ \mathcal{E}_1$ and $\mathcal{E}_2$, such that none of these channels produce ESD by itself; i.e.
\begin{align*}
C[(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{I}})\ket{\Phi_i}\bra{\Phi_i}]&\neq0,\\ C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_2)\ket{\Phi_i}\bra{\Phi_i}]&\neq0,
\end{align*}
$i=1,2$. We may consider the action of both channels acting on $\ket{\Phi_1}$ and $\ket{\Phi_2}$, such that
\begin{eqnarray}
C[(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{E}}_2)\ket{\Phi_1}\bra{\Phi_1}]=0,
\label{sudden2}
\end{eqnarray}
while
\begin{eqnarray}
C[(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{E}}_2)\ket{\Phi_2}\bra{\Phi_2}]\neq0. \label{sudden}
\end{eqnarray}
This condition was first observed in \cite{almeida2007environment}. In view of Eq. (\ref{sudden2}) we notice that by use of Eq. (\ref{decomp}) one can find a channel $\oper{\Gamma}$ and a pure state $\Sigma$ such that
\begin{align*}
(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{E}}_2)\ket{\Phi_1}\bra{\Phi_1}&= (\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_2)(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{I}})\ket{\Phi_1}\bra{\Phi_1}\\
&= (\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_2)(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}'}_1)\Sigma\\
&= (\oper{\mathcal{I}}\otimes\oper{\Gamma})\Sigma.
\end{align*}
By applying Eq. (\ref{factpuro}), the concurrence depends on the action of $\oper{\Gamma}$ on a maximally entangled state and on the concurrence of the state $\Sigma$; satisfying Eq. (\ref{sudden2}) then implies
\begin{eqnarray}
C[(\oper{\mathcal{I}}\otimes\oper{\Gamma})\Sigma]= C[(\oper{\mathcal{I}}\otimes\oper{\Gamma})\ket{\phi_+}\bra{\phi_+}]C[\Sigma]=0.
\label{macelo1}
\end{eqnarray}
Since $\oper{\mathcal{E}}_1$ does not produce ESD by itself as stated above, we obtain
\begin{align*}
0 &\neq C[(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{I}})\ket{\Phi_1}\bra{\Phi_1}]=C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}'}_1)\ket{\phi_+}\bra{\phi_+}]C[\Sigma],
\end{align*}
and therefore we know that $C[\Sigma]\neq 0$. Hence, according to Eq. (\ref{macelo1}), ESD is caused by the action of the effective channel $\oper{\Gamma}$ on a maximally entangled state, which is indeed an EBC.
In the same way, the channel-state combination of Eq. (\ref{sudden}) can be written as a channel $\oper{\Lambda}$ acting on another pure state $\Sigma'$:
\begin{eqnarray}
(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{E}}_2)\ket{\Phi_2}\bra{\Phi_2}=(\oper{\mathcal{I}}\otimes \oper{\Lambda})\Sigma'
\end{eqnarray}
where, by following the same steps that lead to Eq. (\ref{macelo1}) we find that $\oper{\Lambda}$ is not an EBC.
In summary, this shows our claim that for every Entanglement Sudden Death, one can associate an effective Entanglement Breaking Channel acting on only one of the subsystems.
\section{Experimental Investigation of Entanglement Dynamics}
\label{exper}
\begin{figure}
\caption{\label{fig:setup}
\label{fig:setup}
\end{figure}
In this section we describe an experiment where we observe the evolution of the entanglement of a pair of polarization qubits in two different scenarios and prove the use of the new interpretation presented in the previous sections.
The experimental set-up for the investigation of the entanglement dynamics is sketched in Fig. \ref{fig:setup}. Polarization entangled photon pairs are produced via spontaneous parametric down-conversion (SPDC) using a CW diode laser to pump a nonlinear crystal arrangement. A more detailed description of the entangled pair source can be found in \cite{knoll2014remote,knoll2014noisy}. After implementation of the specific quantum channels -which are described below- the photons from both paths, after passing through specific quantum gates implemented for the experiments, are directed to polarization analysis schemes and then detected in coincidence with single-photon detectors, with a temporal coincidence window of 8ns. Full tomography of the two photon polarizations state was implemented.
With this setup we are able to prepare the following quantum state
\begin{align}
\ket{\alpha}= \alpha\ket{HH}+\sqrt{1-|\alpha|^2}\ket{VV}.
\label{eq:spdc}
\end{align}
The real amplitude $\alpha$ can be controlled by rotating the angle $\theta$ of a half-wave plate (H3 in figure \ref{fig:setup}) in the pump laser beam~\cite{kwiat1999ultrabright}.
Rotating H3 to $45^{\circ}$ produces the state $\ket{HH}\bra{HH}$, and adding another HWP at $45^{\circ}$ on Bob's path can further transform the downconverted pair into the state $\ket{HV}\bra{HV}$.
Starting from state \eqref{eq:spdc} we were able to produce several families of initially mixed states by weighted averages of different pure states. In the next section we explain in detail the families of mixed entangled states that were created.
Open system dynamics can be induced on one of the polarization-encoded qubits by using its path internal degree of freedom. The polarization qubit is coupled to the path, or linear momentum qubit in a controlled manner using a displaced Sagnac interferometer, and the information on this path qubit is traced out before the polarization analysis. This interferometer has been shown to implement a few characteristic quantum channels, including the amplitude damping channel and the phase damping channel~\cite{salles2008experimental}:
Bob's photon enters a Sagnac interferometer based on a polarization beamsplitter (PBS), where the $H$ and $V$ polarization components are routed in different directions. If $\phi=0^{\circ}$ both polarization components are coherently recombined in the PBS and exit the interferometer in mode $a$. For any other angle of $\phi$ the $V$ component is transformed into an $H$ polarized photon with probability $g=\sin^{2}(2\phi)$ and exits the interferometer in mode $b$. Both modes are later incoherently recombined using a half waveplate (HWP) oriented at $45^{\circ}$ on mode $b$ (H2) and another polarizing beam splitter, which corresponds to a partial tracing operation over the environment.
\subsection{Amplitude damping channel acting on a mixed state}
We use quantum process tomography to characterize the effect of the Sagnac interferometer, set to implement an amplitude damping channel of strength $g$. By doing so we obtain the operator representation of the noisy channel $\oper{\mathcal{E}}_g$. The reconstructed Kraus operators are shown on Fig \ref{fig:kraus}a), for a damping parameter $g=0.6$. We want to observe the evolution of mixed states through the channel.
We are able to prepare a family of mixed entangled states characterized by the parameter $p$,
\begin{equation}
\rho=(1-p)\rho_1+p\rho_2\\
\label{simuA}
\end{equation}
where $\rho_{1}=\ket{\alpha}\bra{\alpha}$ given by \eqref{eq:spdc}
and $\rho_2=\ket{HV}\bra{HV}$.
In this way, the noise on Alice's qubit is simulated as a weighted average of different experimentally obtained pure states, as described in \cite{knoll2014noisy}.
Once again, by using Eq. (\ref{decomp}), we can find a channel $\mathcal{E}_1$ acting on a pure state $\ket{\chi}$, such that
\begin{center}
$\rho=(\oper{\mathcal{E}}_1\otimes\oper{\mathcal{I}})\ket{\chi}\bra{\chi}$.
\end{center}
$\mathcal{E}_1$ may describe the situation where an amplitude damping channel with damping parameter $h$ acts on the first qubit of the pure state $\ket{\chi}=\sqrt{\omega}\ket{HH}+\sqrt{1-\omega}\ket{VV}$. In this way, $\rho$ is now characterized by the two parameters $h$ and $\omega$, such that $p=h(1-\omega)$.
Next we start the evolution through the amplitude damping channel (ADC) $\oper{\mathcal{I}} \otimes \oper{\mathcal{E}}_g$ on Bob's side and calculate the concurrence of the output state $\rho_{out}=(\oper{\mathcal{I}}\otimes \oper{\mathcal{E}}_g)\rho=(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_g\oper{\Gamma})\sigma$. From the state tomography of initial state $\rho$, Eq.~\eqref{simuA} we can extract the map $\oper{\Gamma}$. Process tomography allows to obtain the Kraus operators corresponding to the ADC and combining those two, the effective channel $\oper{\mathcal{E}}_g\oper{\Gamma}$. A convenient representation of this channel is expressed through the Kraus operators
\begin{equation*}
K_1=\begin{pmatrix}
\sqrt{\frac{\omega+gh(1-\omega)}{\omega+h(1-\omega)}} & 0\\
0 & \sqrt{\frac{\omega(1-g)}{\omega+gh(1-\omega)}}
\end{pmatrix}\; \; \;
K_3=\begin{pmatrix}
0 & \sqrt{g}\\
0 & 0
\end{pmatrix}
\end{equation*}
\begin{equation}
K_2=\begin{pmatrix}
0 & 0\\
\sqrt{\frac{h(1-g)(1-\omega)}{\omega+h(1-\omega)}} & 0
\end{pmatrix}\;\;\;
K_4=\begin{pmatrix}
0 & 0\\
0 & \sqrt{\frac{(1-g)hg(1-\omega)}{\omega+gh(1-\omega)}}
\end{pmatrix}.
\label{adcadckraus}
\end{equation}
Figure \ref{fig:kraus}b) shows the measured Kraus operators for the effective channel, for the particular choice $\omega=0.25$ and $h=0.5$. These operators are measured for a damping parameter $g=0.6$.
\begin{figure}
\caption{\label{fig:kraus}
\label{fig:kraus}
\end{figure}
The concurrence of the family of initial states studied is
\begin{equation}
C(\sigma)=2\sqrt{(1-h)(1-\omega)[\omega+h(1-\omega)]},
\label{cdesigma}
\end{equation}
which is a positive function in the $(0,1)\times(0,1)$ interval. Therefore, the vanishing points of $C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_g)\rho]$ are also the vanishing points of the concurrence of the map applied to a maximally entangled state, $C[(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_g\oper{\Gamma})|\phi_+\rangle\langle\phi_+|]$. Entanglement breaking characteristics of the map can be studied with this scheme. With this in mind, we can monitor the evolution of the concurrence in the parameter space, where we remind that $g$ parametrizes the damping parameter of the noisy channel.
\begin{figure*}
\caption{\label{fig:maps}
\label{fig:maps}
\end{figure*}
Figure \ref{fig:maps}a) shows gray-scale maps representing the concurrence values as a function of the degree of mixture $h$ and the noise parameter on Bob's qubit $g$, for states with initial degree of coherent superposition $\omega$=0.12. The colorbar on the right indicates the value of the concurrence. Contour lines are also plotted as a reference. The color within the circles represent the actual measured values of the concurrence. A symbol and its background with similar gray level means a good agreement between measurements and theory.
In figures \ref{fig:maps}a) and c), horizontal lines correspond to the evolution of the concurrence of a given initial state.
As an example, the entanglement dynamics for $h=0.21$ and $\omega=0.12$ is represented in the inset at the top of figure \ref{fig:maps}a), where we can see that ESD occurs for $g\approx0.65$. The contour line for $\mathcal{C}=0$ shows whether the map applied to a specific initial state generates ESD or not, at some instance in the evolution through the damping channel. For $\omega\sim0.5$ there is no sudden death, as the initial input state becomes the maximally entangled state $\ket{\Phi_+}$ \cite{almeida2007environment}.
Thanks to Eq. \eqref{general}, we can find regions in the parameter space $\omega, h$ (initial coherent superposition and degree of mixture) for which the channel is EBC, i.e. where the concurrence vanishes for some value of $g<1$. This is depicted in figure \ref{fig:maps}b); for values of $\omega<$ 0.5, a certain initial value of $h$ converts the channel from non-EB to EB. From the quantum state point of view, figure \ref{fig:maps}b) simply shows which are the initial mixed states \eqref{simuA} that suffer ESD under the action of an AD channel. More interestingly, from the quantum channel point of view, this figure shows the entanglement-breaking capacity of the map $(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}_g\oper{\Gamma})$ specified in \eqref{adcadckraus}, with decay probability $g$: maps with parameters $h,\omega$ that falls on the left side of the boundary curve are EBC's, i.e. the concurrence vanishes for a finite evolution time ($g<1$), while maps with parameters that lie on the right side of the
figure
are non-EBC. This feature was also experimentally verified: states with different values of $\omega,h$ were prepared, and the concurrence for increasing values of the interaction parameter $g$ was obtained by performing quantum state tomography. Initial conditions that lead to an experimental observation of the sudden death of entanglement are depicted with blue symbols, while conditions in which ESD was not observed for any evolution time are marked in red, showing a very good agreement with the theoretical prediction.
\subsection{Phase Damping Channel acting on X-states}
In a second set of measurements, we prepare the family of pure states $\rho_X=h\ket{\phi_+}\bra{\phi_+}+(1-h)\ket{\psi_+}\bra{\psi_+}$; both $\ket{\phi_+}$ and $\ket{\psi_+}$ are accessible experimentally. We can generate the state $\rho_X$ by averaging the results of the experiments using these two Bell states with their respective statistical weights, and calculate the concurrence for different values of $h$ and noise parameters. Again, $h$ characterizes the initial degree of mixture of the state.
We study the dynamics by letting the state $\rho_X$ evolve through the phase damping channel (PDC), $\oper{\mathcal{E}}$ on Bob's side, obtaining $\rho^{PDC}=(\oper{\mathcal{I}}\otimes\oper{\mathcal{E}}'_g)\rho_X$.
We can once again find a channel $\oper{\Lambda}$ and a pure state $\sigma^\prime$ such that $\rho_X=(\oper{\mathcal{I}}\otimes\oper{\Lambda})\sigma^\prime$. In this case, the channel $\oper{\Lambda}$ corresponds to a bit-flip channel with probability $h$, and $\sigma^\prime=\ket{\phi_+}\bra{\phi_+}$.
Recalling \eqref{general}, we see that the concurrence of the output state, is just the concurrence of a Bell state evolved through the channel $\oper{\mathcal{E}}'_g\oper{\Lambda}_{h}$, since the concurrence of a Bell state $C[\ket{\phi_+}\bra{\phi_+}]=1$.
Following the evolution of an X-state through local phase damping channels can therefore give information about the entanglement breaking capacity of the channel $\oper{\mathcal{E}}'_g\oper{\Lambda}_{h}$.
As in the previous experiment, the noise on Bob's side is implemented through the interaction with a path qubit in a polarization sensitive Sagnac interferometer, with a slight modification that allows for the implementation of a phase decay map \cite{almeida2007environment}.
Figure \ref{fig:maps}c) shows a gray-scale map representing the concurrence values
of the Bell state $\ket{\phi_+}$ evolved through the channel $\oper{\mathcal{E}}'_g\oper{\Lambda}_{h}$,
as a function of both $h$ and $g$. The measured values of the concurrence for the output state $\rho^{PDC}$ are plotted on top, with the same gray-scale coded values as the theoretical map.
X-states with $h$ close to 1/2 have a large degree of mixture. Evolution of the bit-flip channel can be observed by following the concurrence through vertical lines, while temporal evolution of the phase damping channel is obtained by tracing horizontal lines in the figure.
Accordingly, the concurrence of states with $h=1/2$ is 0 for all values of $g$.
On the other hand, for $h=0,1$ we obtain the states $\ket{\psi_+}, \ket{\phi_+}$ respectively, for which the concurrence drops to zero only when $g=1$. The complete map is EBC except for these limits.
As opposed to the ADC map, the phase damping has a symmetric behavior on the initial state populations. As the noise increases, the zero concurrence region becomes larger symmetrically with respect to $h$, due to the fact that there is no change in the populations.
\section{Conclusion}
We have presented experimental results for the dynamics of entanglement for mixed states, monitoring the evolution of the concurrence through noisy environments given by the amplitude damping and phase damping channels acting on different initial two-qubit states. Using the connection between channels and bipartite states, we were able to express the concurrence of the output state as the product of the concurrence of a Bell state evolved through an effective channel acting on a single qubit, and the concurrence of a pure state. In doing so, we could study the entanglement-breaking capacity of different effective channels, and to establish conditions on the map parameters that produce an EBC. We experimentally tested these conclusions by observing the evolution of an entangled state through different local damping channels. In particular, we studied the action of two local amplitude damping channels on a pure state and we related its dynamics to the evolution of a particular family of mixed states through an
amplitude damping channel. The state-channel duality was also used to study the action of phase damping channels on X-states. An \emph{a priori} knowledge of the initial mixed state and/or the quantum channel that will affect the entangled resource could allow one to choose the optimal set of parameters for efficient quantum information processing.
\begin{acknowledgments}
We acknowledge financial support from the Brazilian funding
agencies CNPq, CAPES, and FAPERJ, and the Argentine funding agencies CONICET and ANPCyT. This work was performed as part of the Brazilian National Institute of Science and Technology for Quantum Information. O.J.F. was supported by the Beatriu de Pin\'os fellowship (nº 2014 BP-B 0219) and Spanish MINECO (Severo Ochoa grant SEV-2015-0522). We thank Corey O'Meara for useful discussions.
\end{acknowledgments}
\end{document} |
\begin{document}
\title{Online network change point detection with missing values and temporal dependence}
\begin{abstract}
In this paper we study online change point detection in dynamic networks with time-heterogeneous missing pattern within networks and dependence across the time course. The missingness probabilities, the entrywise sparsity of networks, the rank of networks and the jump size in terms of the Frobenius norm, are all allowed to vary as functions of the pre-change sample size. On top of a thorough handling of all the model parameters, we notably allow the edges and missingness to be dependent. To the best of our knowledge, such general framework has not been rigorously nor systematically studied before in the literature. We propose a polynomial-time change point detection algorithm, with a version of soft-impute algorithm \citep[e.g.][]{mazumder2010spectral, klopp2015matrix} as the imputation sub-routine. Piecing up these standard sub-routines algorithms, we are able to solve a brand new problem with sharp detection delay subject to an overall Type-I error control. Extensive numerical experiments are conducted demonstrating the outstanding performances of our proposed method in practice.
\end{abstract}
\section{Introduction}
In recent years, statistical analysis on dynamic networks proliferates, due to the soaring interest from application areas, including neuroscience \citep[e.g.][]{bassett2017network}, biology \citep[e.g.][]{kim2014inference}, climatology \citep[e.g.][]{cross201215}, finance \citep[e.g.][]{schuenemann2020japanese}, economics \citep[e.g.][]{wu2020dependency} and cybersecurity \citep[e.g.][]{basaras2013detecting}, to name a few. To be specific, by dynamic networks, we mean a collection of relational observations (nodes) and their relation (edges), all of which may evolve along a certain linear ordering, referred to as time.
Based on the form of available data and different sets of assumptions, there have been different types of research on dynamic networks. When data are random observations associated with nodes, dynamic Gaussian graphical models and/or Markov random fields models are usually deployed \citep[see e.g.][]{keshavarz2018sequential, roy2017change}. When observations are regarding edges, Erd\H{o}s--R\'enyi models, stochastic block models and random dot product models are often summoned \citep[see e.g.][]{chi2009evolutionary, zhou2010time, athreya2017statistical, arroyo2021inference}. In terms of dynamic mechanisms, there are three main types of assumptions: the underlying distributions (1) stay unchanged during the time course \citep[e.g.][]{bhattacharyya2018spectral}, (2) slowly evolving along time allowing for small changes \citep[e.g.][]{zhou2010time, danaher2014joint}, and (3) abruptly change at certain unknown locations or otherwise unchanged \citep[e.g.][]{liu2018global, yu2021optimal}.
When assuming the situation (3) above, the problems are referred to as change point analysis. Dynamic network change point detection problems have been studied from different angles in \cite{keshavarz2018sequential}, \cite{liu2018global}, \cite{cribben2017estimating}, \cite{wang2021optimal} and \cite{yu2021optimal}, among others. Change point analysis on its own is an area with rich literature, and has been studied in univariate, multivariate, matrix, functional, manifold types of data, from both \emph{online} (detecting change points while collecting data) and \emph{offline} (estimating change points retrospectively with all data already available) perspectives, concerning testing and estimating goals. We refer to \cite{yu2020review} and \cite{aminikhanghahi2017survey} for recent reviews.
Data collected in practice often contain missing values, especially for large-scale network-type data. The statistical research on missing data enjoys a vast body of literature, including the renowned expectation-maximisation algorithm \citep[e.g.][]{dempster1977maximum, balakrishnan2017statistical}. Due to the fact that the network data concerned in this paper are in the form of random matrices, we focus on the matrix completion literature, where \cite{candes2010matrix}, \cite{candes2010power}, \cite{candes2009exact} and others, are among the first line of attack introducing nuclear-norm penalisation estimators, based on low-rank assumptions. Since then, different structural assumptions, missingness patterns and penalisation methods have been introduced \citep[e.g.][]{keshavan2010matrix, cai2016matrix, wang2015orthogonal}. Theoretical guarantees of different estimators are also investigated, ranging from estimation error bounds to minimax optimality, for both one-step and iterative estimators \cite[e.g.][]{koltchinskii2011nuclear, klopp2014noisy, klopp2015matrix, klopp2011rank, bhaskar2016probabilistic, carpentier2018adaptive}.
Missingness has also been previously considered in the change point analysis literature. For high-dimensional vector-valued data with missing entries, \cite{xie2012change} devised a generalised likelihood ratio based statistic to study an online change point detection problem; \cite{foll:21} was concerned with the localisation error control in an offline perspective. For Gaussian graphical models with potential missingness, \cite{londschien2021change} studied an offline change point detection and focused only on the computational aspects. Other more application-oriented work includes \cite{muniz2011random} and \cite{yang2020changepoint}, among others. \cite{enikeeva2021change}, arguably, is the closest-related to us in the literature. \cite{enikeeva2021change} was concerned with testing the existing of and estimating the location of one change point in an offline setting, on dynamic networks with missingness. Optimal offline detection and localisation rates were investigated in \cite{enikeeva2021change}.
On top of the missingness, startling different from the aforementioned literature, we also consider temporal dependence in both (1) the network generating mechanism and (2) the missingness patterns, as well as the dependence between (1) and (2). This generality and flexibility is the first of its kind. Having said this, modelling only the dynamic network processes with temporal dependence has been considered in literature, such as Markov chain-based dynamic network models \citep[e.g.][]{snijders2005models, ludkin2018dynamic, jiang2020autoregressive}. We focus on modelling dynamic networks using temporally dependent latent positions, under $\phi$-mixing conditions, with an example in \Cref{sec:example}. The introduction of latent positions allows for edge-dependence within networks, in addition to temporal dependence.
In this paper, we assume that the data are a sequence of random dot product networks (\Cref{def-rdpg}), with a heterogeneous Bernoulli sampling missingness pattern (\Cref{def-sample}). Both the dynamic networks and the missingness are driven by a sequence of temporally dependent latent positions associated with temporally fixed nodes. The unknown underlying distribution of the networks is assumed to change at a certain time point, namely the change point. Our goal is to detect this change point as soon as it occurs, while controlling the probability of false alarms.
The contributions of this paper are summarised as follows. Firstly, this is, to the best of our knowledge, the first paper studying online network change point detection possessing missing values with rigorous theoretical justifications. In a rather general framework, we study a change point estimator based on the soft-impute matrix completion algorithm \citep[e.g.][]{mazumder2010spectral} and derive a nearly-optimal upper bound on the detection delay, while controlling the probability of false alarms. Secondly, we allow for the $\phi$-mixing in the latent position sequence as well as the missing patterns. Under such generality, we present a multiple-copy version of the soft-impute algorithm and verify the theoretical guarantees of its output, allowing for different copies possessing different missingness patterns. Lastly, extensive numerical analysis is conducted to support our theoretical findings.
\textbf{Notation.} We do not distinguish a network and its adjacency matrix. Let $\|\cdot\|_{\infty}$, $\|\cdot\|_{\mathrm{F}}$, $\|\cdot\|_*$ and $\|\cdot\|_{\mathrm{op}}$ be the entrywise-maximum, Frobenius, nuclear and operator norms of a matrix, respectively. Let $\|\cdot\|$ be the $\ell_2$-norm of a vector. For any matrix $A \in \mathbb{R}^{n \times n}$ and a 0/1-matrix $B \in \{0, 1\}^{n \times n}$, let $A_B$ and $A_{\overline{B}}$ be the sub-matrices of $A$ indexed by the one and zero entries of $B$, respectively. For any matrix $M \in \mathbb{R}^{n \times m}$, let $M_i$ be $M$'s $i$th row, $i = 1, \ldots, n$. For two deterministic or random $\mathbb R$-valued sequences $a_n, b_n > 0$, write $a_n \gg b_n$ if $a_n/b_n \to \infty$ as n diverges. Write $a_n \lesssim b_n$ if $a_n/b_n \leq C$, and write $a_n \asymp b_n$ if $c \leq a_n/b_n \leq C$, for some absolute constants $c, C > 0$ for all $n \geq 1$. Denote $\mathbb N$ and $\mathbb N^*$ the set of non-negative integers and the set of natural numbers, respectively. Let $\odot$ be the matrix Hadamard product operator. For any two matrices $A, B$ of the same dimensions, where $B$ is a $0/1$ matrix, let $A_B = A \odot B$ denote $A_{ij}$ is observed if and only if $B_{ij} = 1$.
\section{Problem setup}
This paper contains four key ingredients: dynamic networks, missing data analysis, temporal dependence and online change point detection. We unfold the framework concerned in this paper.
We start by considering the marginal network models. Let $n \in \mathbb N^*$ be the number of nodes. At each time, we assume that the adjacency matrix $Y \in \{0,1\}^{n \times n}$ is marginally generated from a random dot product graph (RDPG) model. The latent positions $\{X_i\}_{i = 1}^n \subset \mathbb R^d$ are assumed to be independent and identically distributed (i.i.d.)~and follow the inner product distribution defined below. Note that the inner product distribution and RDPG refer to the marginal distributions of the latent positions and the adjacency matrix, respectively.
\begin{definition}[Inner product distribution] \label{def-inner-product-dist}
Let $F$ be a probability distribution whose support is given by $\mathcal{X}_F \subset \mathbb{R}^d$ for some $d \in \mathbb N^*$. We say that $F$ is an $d$-dimensional inner product distribution on $\mathbb{R}^d$ if for all $x, y \in \mathcal{X}_F$, it holds that $x^{\top}y \in [0, 1]$.
\end{definition}
\begin{definition}[RDPG]
\label{def-rdpg}
Let $Y \in \{0, 1\}^{n \times n}$ be the adjacency matrix of an RDPG with latent positions $\{X_i\}_{i = 1}^n$, if $Y_{ij}|\{X_i, X_j\} \overset{\mbox{ind.}}{\sim} \mathrm{Bernoulli}(P_{ij})$, with $P_{ij} = X_i^{\top}X_j$, $i < j$. Denote further the associated graphon matrix as $\Theta = \{\mathbb E(P_{ij})\}_{(i,j) \in \{1,\dots,n\}^{\otimes 2}} \in [0, 1]^{n \times n}$.
\end{definition}
This modelling approach based on Definitions \ref{def-inner-product-dist} and \ref{def-rdpg} is general, including stochastic block models as special cases. We refer readers to \cite{athreya2017statistical} for a comprehensive review. For brevity, we focus on undirected networks in this paper.
In practice, large networks often contain missingness. Formulating networks into adjacency matrices naturally leads to the territory of matrix completion analysis. In the existing literature, different assumptions have been proposed based on missingness patterns. Besides structural missingness patterns (for instance, \citealp{cai2016structured} assumed observing a few full rows and columns), two main types of sampling assumptions have been used in the literature: the Bernoulli sampling model (BSM, \Cref{def-sample}) and the uniform sampling at random (USR). Given the matrix basis $\mathcal{B} = \{E_{kl} = e_k e_l^{\top}, k, l = 1, \ldots, n\} \subset \mathbb{R}^{n \times n}$, where $e_k \in \mathbb{R}^n$ with $(e_k)_j = \mathbbm{1}\{k =j\}$, the USR assumes that observations are i.i.d.~from~$\mathcal{B}$. Since under the USR, the probability of observing one entry multiple times is non-zero, we resort to the BSM detailed in \Cref{def-sample} to suit the purpose of this paper. \Cref{def-sample}, with the latent positions $\{X_i\}_{i = 1}^n$ as the inputs, is reminiscent to the graph associated sampling scheme considered in \cite{bhojanapalli2014universal} and \cite{bhaskar2016probabilistic}.
\begin{definition}[Bernoulli sampling model] \label{def-sample}
Let $\Omega \in \{0, 1\}^{n \times n}$ be the Bernoulli sampling matrix of an RDPG with latent positions $\{X_i\}_{i=1}^n$ and link function $q: \mathbb R^d \times \mathbb R^d \to [0,1]$, if $\Omega_{ij}|\{X_i, X_j\} \overset{\mbox{ind.}}{\sim} \mathrm{Bernoulli}(Q_{ij})$, with $Q_{ij} = q(X_i,X_j)$, $i < j$. Denote the associated missingness probability matrix as $\Pi = \{\mathbb E(Q_{ij})\}_{(i,j) \in \{1, \ldots, n\}^{\otimes 2}} \in [0, 1]^{n \times n}$.
\end{definition}
In our framework, the adjacency matrix $Y$ and the missingness $\Omega$ are dependent through the same latent positions. This is a more challenging case but this can be easily extended to the scenarios where they are defined through different latent positions, by letting $P_{ij} = X_{1,i}^{\top}X_{1,j}$, $Q_{ij} = q(X^{\top}_{2, i}X_{2, j})$ and $X_i = (X_{1, i}^{\top}, X_{2, i}^{\top})^{\top}$.
Moving on from marginal modelling of a static network to dynamic networks, we let $\{Y(t)\}_{t \in \mathbb N^*}$ and $\{\Omega(t)\}_{t \in \mathbb N^*}\subset \mathbb \{0,1\}^{n \times n}$ be sequences of adjacency and missingness matrices, respectively, where the number of nodes $n$ is assumed to be fixed across $t$. Let $\{X_i(t)\}_{i = 1,\, t \in \mathbb N^*}^n$ be a sequence of latent positions. For each $i \in \{1, \ldots, n\}$, we allow for temporal dependence in the sequence $\{X_i(t)\}_{t \in \mathbb N^*}$, measured by its $\phi$-mixing coefficients defined below.
\begin{definition}[$\phi$-mixing coefficients]\label{def-alpha-mixing}
For a sequence of random objects $\{Z(t)\}_{t \in \mathbb{N}^*}$ and two integers $k \geq j$, let $\mathcal{A}_{j}^k$ be the $\sigma$-algebra generated by $\{Z(t)\}_{t = j}^k$. The $\phi$-mixing coefficients for any $\ell \in \mathbb{N}^*$ are defined as
\[
\phi_Z(\ell) = \sup_{j \in \mathbb{N}^*}\sup_{A \in \mathcal{A}_1^j,\, B \in \mathcal{A}_{j+\ell}^{\infty}}\big|\mathbb{P}(B|A) - \mathbb{P}(B)\big|.
\]
\end{definition}
The $\phi$-mixing coefficients are just one of the many ways to measure temporal dependence, other examples including $\alpha$-, $\beta$-mixing coefficients \citep{bradley2005basic} and the functional dependence measure \citep{wu2005nonlinear}. We adopt the $\phi$-mixing condition in this paper owing to its relevance to the ergodicity of Markov chains, which is a more natural choice for the dependent dynamic models that we will elaborate later. For each $i \in \{1, \ldots, n\}$, denote $\{\phi_{X_i}(\ell)\}_{\ell \in \mathbb{N}^*}$ as $\{\phi_i(\ell)\}_{\ell \in \mathbb{N}^*}$.
With the aforementioned dynamic networks, sampling models and $\phi$-mixing coefficients, we are now ready to unveil the full framework.
\begin{assumption}[Model] \label{assume-model-new}
Let the process $\{Y_{\Omega}(t), \Omega(t)\}_{t \in \mathbb{N}^*} \subset \mathbb{R}^{n \times n}$ satisfy the following conditions.
\begin{enumerate}[leftmargin=*,align=left, nolistsep, topsep=0pt]
\item[a.] For each $t \in \mathbb{N}^*$, $n$ latent positions $\{X_i(t)\}_{i = 1}^n \overset{i.i.d.}{\sim} F$, where $F$ is an inner product distribution.
\item[b.] For any $\ell \in \mathbb{N}$, assume that there exists $\rho \in (0,1)$ satisfying that $\phi(\ell) = \max_{i = 1}^n \phi_i(\ell) \leq C\rho^\ell$, where $C > 0$ is an absolute constant.
\item[c.] For each $t \in \mathbb{N}^*$, let $Y(t)$ be the adjacency matrix of an RDPG with latent positions $\{X_i(t)\}_{i = 1}^n$,
denoting its graphon matrix by $\Theta(t)$.
Assume that $0 < \vartheta_1 \leq \inf_{t \in \mathbb{N}^*}\inf_{1 \leq i\leq j \leq n} \Theta_{ij}(t) \leq \sup_{t \in \mathbb{N}^*}\sup_{1 \leq i\leq j \leq n} \Theta_{ij}(t) \leq \vartheta_2 < 1$ and $\sup_{t \in \mathbb N^*}\mathrm{rank}\{\Theta(t)\} \leq r$.
\item[d.] For each $t \in \mathbb{N}^*$, let $\Omega(t)$ be the Bernoulli sampling matrix of an RDPG with latent positions $\{X_i(t)\}_{i = 1}^n$ and link function $q_t$.
Assume that
$q_t$ satisfies $0 < q_1 \leq \inf_{t \in \mathbb{N}^*}\inf_{x,y \in \mathcal{X}_F} q_{t}(x,y) \leq \sup_{t \in \mathbb{N}^*}\sup_{x,y \in \mathcal{X}_F} q_{t}(x,y) \leq q_2 \leq 1$.
\item[e.] Sequences $\{Y(t)\}_{t \in \mathbb{N}^*}$ and $\{\Omega(t)\}_{t \in \mathbb{N}^*}$ are conditional independent, given $\{X_i(t)\}_{i = 1, \, t \in \mathbb N^*}^n$.
\end{enumerate}
\end{assumption}
\Cref{assume-model-new}b.~formalises the exponential decay of the $\phi$-mixing coefficients of the latent positions. The temporal dependence of $\{X_i(t)\}_{i = 1, \, t \in \mathbb{N}^*}^n$ cause the temporal dependence of $\{Y(t)\}_{t \in \mathbb N^*}$ and $\{\Omega(t)\}_{t \in \mathbb N^*}$. This statement is formalised in the next lemma.
\begin{lemma}\label{theorem:markov-phi}
Under \Cref{assume-model-new}, for any $(i,j) \in \{1, \ldots, n\}^{\otimes 2}$, let $\{\phi_Y^{ij}(\ell)\}_{\ell \in \mathbb{N}}$ and $\{\phi_{\Omega}^{ij}(\ell)\}_{\ell \in \mathbb{N}}$ be the $\phi$-mixing coefficient sequences of $\{Y_{ij}(t)\}_{t \in \mathbb{N}^*}$ and $\{\Omega_{ij}(t)\}_{t \in \mathbb{N}^*}$, respectively. It holds that $\max\{\phi_Y^{ij}(\ell), \, \phi_{\Omega}^{ij}(\ell)\} \leq 2C \rho^{\ell}$,
with the $\rho$ and $C$ in \Cref{assume-model-new}b.
\end{lemma}
The proof of \Cref{theorem:markov-phi}, along with all the other technical details of this paper, is collected in Appendix. To accommodate the change points, we introduce the following two assumptions focusing on two scenarios.
\begin{assumption}[No change point] \label{assume-no-change}
Assume that $\Theta(1) = \Theta(2) = \cdots$.
\end{assumption}
\begin{assumption}[One change point] \label{assume-one-change}
Assume that there exists a positive integer $\Delta \geq 1$ such that $\Theta(1) = \cdots \Theta(\Delta) \neq \Theta(\Delta + 1) = \cdots$.
\end{assumption}
In view of Assumptions~\ref{assume-model-new}, \ref{assume-no-change} and \ref{assume-one-change}, we are concerned with detecting change points in the underlying network distributions, but not the missingness mechanisms. In fact, we allow the missingness probability matrix $\Pi(t) = \mathbb E[Q(t)]$ defined in \Cref{def-sample} to be distinct at every time point. The heterogeneity of the missingness probability matrices, undoubtedly, increases the difficulty in detecting the changes in the graphon changes.
From the online change point detection point of view, our task is to seek $\widehat{\Delta}$, an estimator of $\Delta$, such that the following hold. The quantities $\mathbb{P}_{\infty}(\cdot)$ and $\mathbb{E}_{\infty}(\cdot)$ indicate the probability and expectation under \Cref{assume-no-change}, with $\mathbb{P}_{\Delta}(\cdot)$ and $\mathbb{E}_{\Delta}(\cdot)$ their counterparts under \Cref{assume-one-change}.
\begin{enumerate}[leftmargin=*,align=left, nolistsep, topsep=0pt]
\item [$\bullet$] The overall false alarm control. With a pre-specified $\alpha > 0$, it holds $\mathbb{P}_{\infty}\{\exists t \in \mathbb{N}^*: \, \widehat{\Delta} \leq t\} < \alpha$.
\item [$\bullet$] The detection delay control. If $\Delta < \infty$, then $\mathbb{P}_{\Delta}\{\Delta \leq \widehat{\Delta} \leq \Delta + d\} > 1 - \alpha$, where $d$ is referred to as detection delay and is to be minimised.
\end{enumerate}
We remark that in the online change point detection literature, there are two types of control. In addition to the false alarm controls, it is also popular to lower bound $\mathbb{E}_{\infty}[\widehat{\Delta}]$ - the expectation of the change point location estimator under \Cref{assume-no-change}, see e.g.~\cite{lorden1971procedures}, \cite{lai1995sequential} and \cite{lai1998information}. These two types of control share many similarities but with some subtle differences: controlling the overall false alarm probability is more preferable in theoretical analysis and controlling the average run length is handier in practice. We refer readers to \cite{yu2020note} for detailed discussions on this matter and stick with controlling the overall false alarm probability in this paper.
\section{Optimal network change point detection with missingness and dependence}
Given the model described in \Cref{assume-model-new}, for the task of online change point detection, in this section, we propose a soft-impute-based change point estimator, with nearly-optimal theoretical guarantees. The sub-routine soft-imputation is presented in \Cref{sec-soft-impute}, the change point analysis is conducted in \Cref{sec-change-point}, the fundamental limits are discussed in \Cref{sec-minimax} and a special example on a Markov chain-type dynamic network is investigated in \Cref{sec:example}.
\subsection{The soft-impute algorithm}\label{sec-soft-impute}
As for the task of matrix completion, a rich collection of estimators have been proposed, most of which are penalisation-based methods working under some form of sparsity assumptions. The soft-impute algorithm \citep[e.g.][]{mazumder2010spectral} iteratively solves a nuclear norm penalised least squares estimator. We present a multiple-copy version of the soft-impute algorithm, which is a variant of the soft-impute algorithm studied in \cite{klopp2015matrix}.
Given any $s, e \in \mathbb{N}$ with $s < e$, let
\begin{align}\label{eq-obj-function-hatM}
\widetilde{\Theta}_{s:e} \in \argmin_{\Theta \in \mathbb{R}^{n \times n}}f\left(\Theta; \{Y_{\Omega}(i), \Omega(i)\}_{i = s+1}^e, \widehat{\Theta}, \lambda_{s, e}\right),
\end{align}
where $f(\Theta; \{Y_{\Omega}(i), \Omega(i)\}_{i = s+1}^e, \widehat{\Theta}, \lambda_{s, e}) = \{2(e-s)\}^{-1} \sum_{i = s+1}^e \big\|Y_{\Omega}(i) + \widehat{\Theta}_{\overline{\Omega(i)}} - \Theta\big\|^2_{\mathrm{F}} + \lambda_{s, e} \|\Theta\|_*$, $\lambda_{s, e} > 0$ is a tuning parameter and $\widehat{\Theta} \in \mathbb{R}^{n \times n}$ is an optimiser obtained from the previous step or the initialiser of the algorithm. In fact, $\widehat{\Theta}$ can be any dimensional-compatible matrix. Then, $\widehat{\Theta}_{s:e}$ is obtained by truncated each entry of $\widetilde{\Theta}_{s:e}$ by some tuning paramter $a \in (0,1)$.
\begin{algorithm}
\begin{algorithmic}
\INPUT $\{Y_{\Omega}(t), \Omega(t)\}_{t = s+1, \ldots, e}$, $\lambda_{s, e}, a > 0$, $\widehat{\Theta} \in \{0,1\}^{n \times n}$
\State $\mathrm{FLAG} \leftarrow 0$
\While{$\mathrm{FLAG} = 0$}
\State{
\begin{equation}\label{eq-soft-impute}
\widetilde{\Theta} \leftarrow S_{\lambda_{s, e}} \bigg\{(e-s)^{-1}\sum_{t = s+1}^e \left[Y_{\Omega}(t) + \widehat{\Theta}_{\overline{\Omega(t)}})\right]\bigg\}
\end{equation}}
\If{$\|(e-s)^{-1} \sum_{t = s+1}^e (\widetilde{\Theta} - \widehat{\Theta})_{\overline{\Omega(t)}}\|_{\mathrm{op}} < \lambda_{s, e}/3$ and $\|\widetilde{\Theta} - \widehat{\Theta}\|_{\infty} < a$}
\State $\mathrm{FLAG} \leftarrow 1$
\EndIf
\State $\widehat{\Theta}_{ij} \leftarrow \mathrm{sign}(\widetilde{\Theta}_{ij}) \min\{|\widetilde{\Theta}_{ij}|, \, a\}$
\EndWhile
\OUTPUT $\widehat{\Theta}$
\caption{Soft-Impute\label{alg-main-klopp}}
\end{algorithmic}
\end{algorithm}
The soft-thresholding estimator $S_{\lambda}(\cdot)$ in \eqref{eq-soft-impute} is defined as follows. For any matrix $W \in \mathbb{R}^{n \times n}$, let $S_{\lambda}(W) = UD_{\lambda} V^{\top}$, where $W = UDV^{\top}$ is a singular value decomposition of $W$, $D = \mathrm{diag}\{d_i, i = 1, \ldots, n\}$ is a diagonal matrix containing all singular values of $W$ including zero and $D_{\lambda} = \mathrm{diag}\{(d_i - \lambda)_+, i = 1, \ldots, n\}$.
For any natural numbers pair $s < e$, we solve \eqref{eq-obj-function-hatM} using \Cref{alg-main-klopp}, with pre-specified $\lambda, a >0$. \Cref{alg-main-klopp} terminates when the convergence criteria are met. When $e-s = 1$, it is well-understood that \eqref{eq-soft-impute} is the minimiser of \eqref{eq-obj-function-hatM} and the iterative algorithm converges \citep{mazumder2010spectral, klopp2015matrix}. As for the multiple-copy version that we exploit in this paper, we provide a sanity check in \Cref{lem-soft-impute-property}. We would like to emphasise that \Cref{lem-soft-impute-property} is a deterministic result, working with any given input matrices.
\begin{lemma}\label{lem-soft-impute-property}
For any $m \in \mathbb{N}^*$, any sequence of matrices $\{R(t)\}_{t = 1}^m$ and $\lambda > 0$, the solution to the optimisation problem $\min_Z \left\{(2m)^{-1} \sum_{t = 1}^m \|R(t) - Z\|^2_{\mathrm{F}} + \lambda \|Z\|_*\right\}$
is given by $\widehat{Z} = S_{\lambda} \{m^{-1} \sum_{t = 1}^m R(t) \}$.
Let $\{M_k\}_{k \in \mathbb{N}}$ be the sequence of solutions produced by \Cref{alg-main-klopp}, with input indexed by $t \in \{1, \ldots, m\}$. We have that $\max\{\|M_{k+1} - M_k\|_{\mathrm{F}}$, $\|M_{k+1} - M_k\|_{\infty}\} \to 0$ as $k \to \infty$. For any $t \in \{1, \ldots, m\}$, $\|(M_{k+1} - M_k)_{\overline{\Omega(t)}}\|_{\mathrm{op}} \to 0$ as $k \to \infty$.
\end{lemma}
\subsection{The change point detection algorithm}\label{sec-change-point}
With the soft-impute estimator in hand, we define the change point estimator as
\begin{equation}\label{eq-widehat-delt-defi}
\widehat{\Delta} = \min\left\{t \geq 2: \, \max_{s = 1}^{t-1} \widehat{D}_{s, t} \geq \varepsilon_{s, t}\right\},
\end{equation}
where $\widehat{D}_{s, t} = \|\widehat{\Theta}_{0:s} - \widehat{\Theta}_{s:t}\|_{\mathrm{F}}$ for any pair of positive integers $s, t$, $t \geq 2$ and $\{\varepsilon_{s, t}\}$ is a sequence of pre-specified tuning parameters. The search of the change point is terminated once a change point is detected, or there is no more new data point observed. To understand the theoretical performances of our change point estimator $\widehat{\Delta}$, we require the following signal-to-noise ratio condition.
\begin{assumption}[Signal-to-noise ratio] \label{assume-snr}
Under \Cref{assume-one-change}, let $\kappa = \|\Theta(\Delta) - \Theta(\Delta+1)\|_{\mathrm{F}}$. For a pre-specified $\alpha \in (0, 1)$, it satisfies that $\kappa^2 \Delta \geq C_{\mathrm{SNR}} rnq_2 \max\{\vartheta_2^2, (1-\vartheta_1)^2\}q_1^{-2}\log^2(\Delta/\alpha)$,
where $C_{\mathrm{SNR}} > 0$ is a large enough absolute constant.
\end{assumption}
We refer to \Cref{assume-snr} as the signal-to-noise ratio condition, in the sense that the signal strength is completely characterised by the jump size $\kappa$ and the pre-change sample size $\Delta$, and the noise level is quantified by the network size $n$, network entrywise-sparsity parameters $\vartheta_1$ and $\vartheta_2$, network low-rank sparsity $r$ and the strength of missingness $q_2/q_1^2$. When $q_1 = q_2 =1$, i.e.~without missingness, up to logarithmic factors, \Cref{assume-snr} has the same form as what is required for a polynomial-time algorithm to yield a nearly-optimal detection delay, in an online network change point detection problem without missing entries \citep{yu2021optimal}.
\begin{theorem}\label{thm-upper-bound}
Consider the model described in \Cref{assume-model-new}. Let $\alpha \in (0, 1)$ and $\widehat{\Delta}$ be defined in \eqref{eq-widehat-delt-defi}, with the inputs of \Cref{alg-main-klopp} being $a = \vartheta_{2}$, for any $s, t \in \mathbb N^*$, $s < t$, the penalisation levels being
\begin{align*}
\lambda_{s, t} = C_{\lambda}q_2\max\{\vartheta_2, 1-\vartheta_1\}\sqrt{(t-s)^{-1}\{n+\log(1/\alpha)\}}\max\left\{\sqrt{\log(t-s)},\sqrt{\log(n/\alpha)}\right\},
\end{align*}
and with the change point thresholds being
\begin{align*}
\varepsilon_{s, t} = C_{\varepsilon} \sqrt{rnq_2 q^{-2}_1\max\{\vartheta_2^2, (1-\vartheta_1)^2\}}\left(\sqrt{s^{-1}\log(s/\alpha)} + \sqrt{(t-s)^{-1}\log(t/\alpha)}\right)\sqrt{\log(nt/\alpha)},
\end{align*}
where $C_{\lambda}, C_{\varepsilon} > 0$ are sufficiently large absolute constants.
If \Cref{assume-no-change} holds, then we have that $\mathbb{P}\{\exists t \in \mathbb{N}^*: \, \widehat{\Delta} \leq t\} \leq \alpha$.
If Assumptions~\ref{assume-one-change} and \ref{assume-snr} hold, then we have that $\mathbb{P}\left\{\Delta \leq \widehat{\Delta} \leq \Delta + d \right\} \geq 1 - \alpha$, where $d = C_d rnq_2q_1^{-2} \kappa^{-2}\max\{\vartheta_2^2, (1-\vartheta_1)^2\} \log^2(\Delta/\alpha)$
and $C_d > 0$ is a sufficiently large absolute constant.
\end{theorem}
It is shown in \Cref{thm-upper-bound} that, with the overall false alarm probability upper bounded by pre-specified $\alpha$, if a change point does occur, our procedure will, with probability at least $1 - \alpha$, detect the change point with a detection delay upper bounded by
\begin{equation}\label{eq-detection-delay-rate}
C_d rnq_2 q_1^{-2}\kappa^{-2}\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log^2(\Delta/\alpha).
\end{equation}
The derivation of \Cref{thm-upper-bound} relies on two key ingredients: an estimation error bound on imputing a low-rank matrix and an online change point analysis procedure. To further understand \Cref{thm-upper-bound}, we compare our result with \cite{klopp2015matrix} and \cite{yu2021optimal}. Before delving into details, we highlight the dependence we allow. Compared to \cite{klopp2015matrix}, we allow for the dependence within network edges and missingess, as well as the dependence between the presences and missingness of edges. This is materalised by the interconnection via random latent positions. Compared to \cite{yu2021optimal}, we allow for the temporal dependence across time and this is formalised by the $\phi$-mixing coefficients.
\begin{proposition} \label{cor-main-klopp-cor-3}
Assume that Assumptions~\ref{assume-model-new} and \ref{assume-no-change} hold. Denote $\Theta(t) = \Theta_0$, $t \in \mathbb{N}^*$, with $\mathrm{rank}(\Theta_0) \leq r$. For any integer pair $1 \leq s < e$, let $\widehat{\Theta}_{s:e}$ be the output of \Cref{alg-main-klopp} with $a = \vartheta_{2}$ and
\begin{align*}
\lambda_{s,e} = C_{\lambda}q_2\max\{\vartheta_2, 1-\vartheta_1\}\sqrt{(e-s)^{-1}\{n+\log(1/\delta)\}}\max\left\{\sqrt{\log(e-s)}, \sqrt{\log(n/\delta)}\right\}.
\end{align*}
Then with probability at least $1 - 3\delta$, it holds that
\begin{align}\label{eq-estimation-error-bounds}
\|\widehat{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2 \leq \frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/\delta)}{q_1^2(e-s)}\max\left\{\log(e-s), \log(n/\delta)\right\},
\end{align}
where $C_{\mathrm{noise}} > 0$ is an absolute constant.
\end{proposition}
\begin{remark}
When $e-s \geq n/\delta$, \Cref{cor-main-klopp-cor-3} claims that with probability at least $1 - 3\delta$, the estimation error is upper bounded that
\begin{align*}
\|\widehat{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2 \leq C_{\mathrm{noise}}r(e-s)^{-1}nq_1^{-2}q_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/\delta) \log(e-s).
\end{align*}
The interesting term is $\log(e-s)$, which would not appear if we impose temporal independence assumption and which is the cost of the temporal dependence. Compared to \cite{klopp2015matrix}, since we deal with multiple copies of input matrices, the term $(e-s)^{-1}$ incurrs.
\end{remark}
\textbf{Online change point analysis.} \cite{yu2021optimal} studies an optimal online network change point detection without missingness. To compare with \cite{yu2021optimal}, we note that when $q_1 = q_2 = 1$, \eqref{eq-detection-delay-rate} corresponds to the detection delay rate without missingness that
\begin{equation}\label{eq-detection-delay-without-missingness}
rn \max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log^2(\Delta/\alpha) \kappa^{-2}.
\end{equation}
It is shown in \cite{yu2021optimal} that, without missingness, a minimax lower bound on the detection delay is of order $\log(1/\alpha) \max\{r^2/n, \, 1\} n \vartheta_2 \kappa^{-2}$, which can be obtained, off by a logarithmic factor, by an NP-hard algorithm. As for polynomial-time algorithms, \cite{yu2021optimal} shows that without missingness, the detection delay is exactly that in \eqref{eq-detection-delay-without-missingness}.
\textbf{Choices of tuning parameters.} The tuning parameters involved in the procedure are $\lambda_{s, e}$ - the penalisation level and the convergence indicator when calculating $\widehat{\Theta}_{s, e}$ using \Cref{alg-main-klopp}, $a$ - the truncation parameter used in \Cref{alg-main-klopp} tailored for the sparsity of networks and $\{\varepsilon_{s, t}\}$ - the sequence of change point thresholds. To guarantee the theoretical results, all these tuning parameters involve unknown model parameters. The practical guidance is left to \Cref{sec-numerical}. We are to explain their theoretical choices here.
As for $\lambda_{s, e}$, it is chosen to be $\lambda_{s,e} \asymp \|\frac{1}{e-s} \sum_{t = s+1}^e \Omega_{ij}(t)\{Y_{ij}(t) - \Theta_{ij}(t)\} E_{ij}\|_{\mathrm{op}}$,
where $E_{ij} = e_ie_j^{\top}$ is a matrix basis.
That is to say, $\lambda_{s, e}$ is chosen to overcome the noise matrix with observed entries. We do not have guarantees if there is underestimation on $\lambda_{s,e}$, but an over-estimated $\lambda_1 \geq \lambda_{s, e}$ still leads to sufficient guarantees by inflating estimation error bound \eqref{eq-estimation-error-bounds} that
\[
\|\widehat{\Theta}_{s:e} - \Theta_0 \|^2_{\mathrm{F}} \lesssim \frac{\lambda^2_1 \vee \left\{rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\} \log(1/\delta)\max\left\{\log(e-s), \log(n/\delta)\right\}\right\}}{q_1^{2} (e-s)},
\]
which, with a correspondingly adjusted threshold, directly implies the same false alarm guarantees and an inflated detection delay. The truncating parameter $a$ is chosen to be the same as the entrywise-sparsity level of the underlying networks. We do not have theoretical guarantees when $a$ is under-estimated, but when it is over-estimated (in fact, $a \leq 1$ always holds), with a correspondingly adjusted threshold, directly implies the same false alarm guarantees and an inflated detection delay. The change point thresholds are set to be the minimum order to cast $\max_{s, t} \|\widehat{\Theta}_{0:s} - \widehat{\Theta}_{s:t}\|_{\mathrm{F}}$. Under- and over-estimation of the change point thresholds leads to a distortion on the overall false alarm probability and detection delay controls, respectively.
\subsection{The fundamental limits}\label{sec-minimax}
It is studied in \cite{yu2021optimal} that without missingness or any form of dependence, a minimax lower bound on the detection delay is of the form
\begin{align}\label{eq-minimiax-lower-bound}
\log(\alpha^{-1}) \max\{r^2/n, \, 1\} n \vartheta_2 \kappa^{-2} = \log(\alpha^{-1}) \max\{r^2/n, \, 1\}\kappa^{-2}_0 n^{-1} \vartheta_2^{-1},
\end{align}
where $\kappa_0 = \kappa n^{-1}\vartheta_2^{-1}$ is a scale-free normalised jump size. The lower bound \eqref{eq-minimiax-lower-bound}, up to a logarithmic factor, can be matched by an NP-hard algorithm with the upper bound $\log(\Delta/\alpha) \kappa_0^{-2}n^{-1}\vartheta_2^{-1} \max\{r^2/n, \, \log(r)\}$.
With the presence of missingness, it is equivalent to consider the variance of each entry is inflated from $\vartheta_2$ to $\vartheta_2 q$, where we use $q = q_2/q_1^2 \geq 1$ to denote the strength of missingness and where we assume a homogeneous missingness pattern for simplicity. This is exactly the same treatment when constructing the minimax lower bound in the matrix completion literature \citep[e.g.][]{koltchinskii2011nuclear}. We immediately have a minimax lower bound on the detection delay based on \eqref{eq-minimiax-lower-bound} and replacing $\vartheta_2$ with $\vartheta_2 q$, i.e.
\begin{equation}\label{eq-lb-rate}
\log(1/\alpha) \max\{r^2/n, \, 1\} \kappa^{-2}_0 (n \vartheta_2 q)^{-1}.
\end{equation}
Under the homogeneous missingness assumption, the detection delay in \Cref{thm-upper-bound}, which is based on a polynomial-time algorithm, is of order
\begin{equation}\label{eq-ub-rate}
\log^2(\Delta/\alpha) r\kappa_0^{-2} (n\vartheta_2 q)^{-1}.
\end{equation}
Comparing \eqref{eq-lb-rate} and \eqref{eq-ub-rate}, we see that when $r \asymp 1$, i.e.~in a low-rank regime, up to a logarithmic factor, the detection delay in \Cref{thm-upper-bound} is optimal.
Note that compared to the upper bound, the discussions in this section is under temporal independence.
\subsection{An example: A Markov chain-based RDPG model}\label{sec:example}
We conclude this section with some further justification of the $\phi$-mixing conditions we imposed in \Cref{assume-model-new}. The ``sticky'' dynamic network process studied in \cite{padilla2019change} is defined based on RDPG, with latent position $X_i(t) \in \mathbb{R}^d$ satisfying that
\begin{equation}\label{eq:markov_latentpostion}
X_i(t) \begin{cases}
= X_i(t-1), &\quad\text{with probability $\rho$},\\
\overset{ind.}{\sim} F, &\quad\text{with probability $1-\rho$},
\end{cases}
\end{equation}
where $\rho \in [0,1)$ and $F$ is an inner product distribution on $\mathbb{R}^d$ defined in \Cref{def-inner-product-dist}. The latent positions process given in \eqref{eq:markov_latentpostion} is in fact a Markov chain and satisfied \Cref{assume-model-new}b, as suggested below.
\begin{lemma}\label{lemma:ergodic}
For any $i \in \{1, \ldots, n\}$, the latent position sequence $\{X_{i}(t)\}_{t \in \mathbb{N}^*}$, given in \eqref{eq:markov_latentpostion} with $\rho \in [0,1)$, is a geometric ergodic Markov chain, and its $\phi$-mixing coefficients satisfy $\phi_{X}(\ell) \leq \rho^{\ell}$, i.e.~$\phi_{X}(\ell)$ decays exponentially in $\ell$.
\end{lemma}
With \Cref{lemma:ergodic} at hand, we see that the model studied in \cite{padilla2019change} is a special case of the model we considered in this paper.
\section{Numerical experiments} \label{sec-numerical}
We now illustrate the numerical efficacy of our proposed online change point detection estimator. More numerical details, simulation scenarios and a real data example can be found in the Supplementary Materials. Our proposed methods are implemented in the R package \texttt{changepoints} \citep{changepoints_R}.
\noindent \textbf{Competitors.} Three competitors are considered. \textbf{1}. Instead of using the soft-impute algorithm for matrix completion, we consider the Rank-Constrained Maximum Likelihood Estimator (RC) \citep{cai:2013,bhaskar2016probabilistic}.
\textbf{2.} Instead of using the $\widehat{D}_{s,t}$-type scan-statistics to detect change points, we deploy the $k$-nearest neighbour ($k$-NN) based procedure \citep{chu:18,chen:19}, and still call \Cref{alg-main-klopp} for matrix completion. Three types of statistics are considered in the R package \texttt{gStream} \citep{chen2019package}: the original edge-count scan statistic (ORI), the weighted edge-count scan statistic (W) and the generalised edge-count scan statistic (G).
\textbf{3.} We consider an online adaptation of the MissInspect change point estimator \citep{foll:21}, which is applied to the vectorised networks with missing elements.
\noindent \textbf{Evaluation measurements.} Two metrics are considered: the average detection delay $\mathrm{Delay} = \big(\sum_{j = 1}^N\mathbbm{1}\{\tilde{\Delta} \geq \Delta\}\big)^{-1} {\sum_{j = 1}^N (\tilde{\Delta} - \Delta)\mathbbm{1}\{\tilde{\Delta} \geq \Delta\}}$
and the proportion of false alarms $\mathrm{PFA} = N^{-1} \sum_{j = 1}^N\mathbbm{1}\{\tilde{\Delta} < \Delta\}$,
where $N = 100$ represents the number of repetitions we conduct in each setting, $\tilde{\Delta}$ is the minimum of the total time length and the estimated change points to account for the situations when no change point is detected.
\noindent \textbf{Change points.} We generate a sequence of network data $\{Y_{\Omega}(t), \Omega(t)\}_{t = 1}^T$ each time, with a single change point at $\Delta = 150$ and the total length $T = 300$. If no change point is detected before $T$, then we set $\widehat{\Delta}=\infty$ and $\tilde{\Delta}=\min\{T, \widehat{\Delta}\}$.
\noindent \textbf{Setting.} We only present a setting on the RDPG with temporal dependence here. More simulation results can be found in the Supplementary Materials. Let $n = 100$, we generate $n$ latent positions $X_i(1) \sim \mathcal{N}(0, I_5)$, and for $t \in \{2, \dots, \Delta\}$,
\begin{align*}
X_i(t) \begin{cases}
= X_i(t-1), & \text{with probability } 0.5,\\
\overset{ind.}{\sim} \mathcal{N}(0, I_5), & \text{with probability } 0.5.
\end{cases}
\end{align*}
Let
\begin{align*}
P_{ij}(t) = \frac{\exp\{X_i(t)^{\top}X_i(t)\}}{1+\exp\{X_i(t)^{\top}X_i(t)\}} \;\; \text{and} \;\; Q_{ij}(t) = \max\left\{q_1, \frac{\exp\{X_i(t)^{\top}X_i(t)\}}{1+\exp\{X_i(t)^{\top}X_i(t)\}}\right\},
\end{align*}
where $q_1 \in \{0.7, 0.8\}$.
After the change point, we generate $n$ latent positions $X_i(\Delta+1) \sim \mathrm{Unif}[0,1]^{\otimes 5}$, and for $t \in \{\Delta+2, \dots, T\}$
\begin{align*}
X_i(t) \begin{cases}
= X_i(t-1), & \text{with probability } 0.5,\\
\overset{ind.}{\sim} \mathrm{Unif}[0,1]^{\otimes 5}, & \text{with probability } 0.5.
\end{cases}
\end{align*}
Let
\begin{align*}
P_{ij}(t) = \frac{X_i^{\top}X_j}{\Vert X_i \Vert \Vert X_j \Vert} \;\; \text{and} \;\; Q_{ij}(t) = \max\left\{q_1, \frac{X_i^{\top}X_j}{\Vert X_i \Vert \Vert X_j \Vert}\right\}.
\end{align*}
For $i < j$, we generate independently
\begin{align*}
Y_{ij}(t)|\{X_i(t), X_j(t)\} \overset{ind.}{\sim} \mathrm{Bernoulli}\big(P_{ij}(t)\big) \, \mbox{and} \, \Omega_{ij}(t)|\{X_i(t), X_j(t)\} \overset{ind.}{\sim} \mathrm{Bernoulli}\big(Q_{ij}(t)\big).
\end{align*}
We consider symmetric adjacency matrices without self-loop, which does not guarantee a low-rank graphon. However, the adjacency and missingness matrices are generated from latent positions based on a Markov chain-based RDPG model. As a result, these processes possess temporal dependence.
In \Cref{tab: compare2}, we exclude the $k$-NN based approaches, which perform very poorly. We observe that MI does not perform well in this setting. Although RC provides the shortest detection delay, it does not guarantee the false alarm rate, particularly when the missingness is high (i.e.~$q_1 = 0.7$). Our method performs well in this scenario, as demonstrated in \Cref{tab: compare2}. We conjecture that a blockwise permutation for selecting $\{\epsilon_{s,t}\}$ may further improve our performance, given the temporal dependence in the data.
\begin{table}
\caption{Average detection delays and false alarms rates over 100 repetitions of RDPG considered in Scenario 2, with the missingness probabilities $q_1 \in \{0.7, 0.8\}$ and $q_2 = 1$. SI: our proposed methods; RC: rank-constrained maximum likelihood estimator; MI: MissInspect; $\alpha$, the false alarm controls. }
\label{tab: compare2}
\centering
\begin{tabular}{lllllll}
\toprule
$\alpha$ & SI & RC & MI & SI & RC & MI \\
\midrule
& \multicolumn{3}{c}{Delay,\; $q_1 = 0.7$} & \multicolumn{3}{c}{Delay,\; $q_1 = 0.8$}\\
\cmidrule(r){2-4} \cmidrule(r){5-7}
0.05 & 5.17 & 1.88 & 23.07 & 4.60 & 1.91 & 22.33 \\
0.01 & 5.47 & 2.04 & 25.48 & 5.10 & 2.34 & 23.52 \\ [3pt]
& \multicolumn{3}{c}{PFA,\; $q_1 = 0.7$} & \multicolumn{3}{c}{PFA,\; $q_1 = 0.8$} \\
\cmidrule(r){2-4} \cmidrule(r){5-7}
0.05 & 0.02 & 0.15 & 0.28 & 0.06 & 0.08 & 0.20 \\
0.01 & 0 & 0.07 & 0.08 & 0 & 0 & 0.10\\
\bottomrule
\end{tabular}
\end{table}
\section{Conclusion}
In this paper, we study an online change point detection in a dynamic networks with heterogenius missingness. A highlight of our framework is that we allow for dependence: edge dependence and missingness dependence within a network, temporal dependence across time and the dependence between the edge pattern and missingness pattern. Despite the technical condition on the $\phi$-mixing coefficients, we have demonstrated its practicality through concrete examples. Having such a general framework, we are still able to show a near-optimal detection delay upper bound, subject to an overall Type-I error control.
\appendix
More discussions on the theoretical results derived, additional numerical experiments results and all technical details are collected in this document.
\section{More discussions of the theoretical results}
\textbf{The hierarchy of constants.} When there exists a change point, i.e.~under Assumption~\ref{assume-one-change}, we can see from \Cref{thm-upper-bound} that there are four absolute constants $C_{\mathrm{SNR}}$, $C_{\lambda}$, $C_{\varepsilon}$ and $C_d$. We do not claim optimality for these constants, but only show here that the feasible choices of these constants do not form an empty set. One first needs to establish an absolute constant $C_{\mathrm{noise}} > 0$ appearing in \Cref{cor-main-klopp-cor-3}. The choice of $C_{\lambda}$ solely depends on $C_{\mathrm{noise}}$, $C_{\varepsilon}$ only depends on $C_{\lambda}$, and then $C_d$ only depends on $C_{\varepsilon}$. Once $C_{\varepsilon}$ and $C_d$ are chosen, one only needs to ensure that $C_{\mathrm{SNR}}$ is large enough -- larger than some increasing functions of $C_{\varepsilon}$ and $C_d$.
\textbf{Estimation error analysis.} Our estimation error analysis is built upon the analysis in \cite{klopp2015matrix}, which presents an estimation error bound of a soft-impute estimator based on only one incomplete matrix. To deal with the dynamic networks scenario, we derive a multiple-copy version of Corollary~3 in \cite{klopp2015matrix} in \Cref{cor-main-klopp-cor-3}, under temporal dependence and network dependence.
\textbf{Computational complexity.} To obtain our change point estimator $\widehat{\Delta}$, we need a sequence of matrix estimators $\{\widehat{\Theta}_{s:t}\}$, output by \Cref{alg-main-klopp}. The computational cost includes that of conducting a truncated singular value decomposition and reconstructing a low-rank matrix, and depends on the iterations, warm-start choices, etc. We omit the details here and refer readers to Section~5 in \cite{mazumder2010spectral}. On top of the matrix estimation, we also have the change point analysis procedure. In \eqref{eq-widehat-delt-defi}, for notational simplicity, we present a rather expensive procedure exploiting all integer pairs $s < t$. In fact, without any loss of statistical accuracy, one can instead exploit a dyadic grid $s \in \{(t - 2^j) \vee 1\}_{j \in \mathbb{N}^*}$, for any $t \geq 2$, see \cite{yu2020note} for detailed discussions. This means, for every newly collected time point $t \geq 2$, the additional cost of estimating the change point is of order $O\{\log(t) \mathcal{Q}\}$, where $\mathcal{Q}$ is the cost of \Cref{alg-main-klopp}. We comment that the logarithmic cost of online change point detection cannot be improved, without knowing the exact pre-change distribution.
\section{Simulation experiments}
\noindent \textbf{More details on the competitors.}
\textbf{1}. For the Rank-Constrained Maximum Likelihood Estimator (RC) \citep{cai:2013,bhaskar2016probabilistic}. to be specific, we replace $\widehat{D}_{s,t}$ in \eqref{eq-widehat-delt-defi} with $\widehat{D}_{s, t}^{\mathrm{MLE}} = \|\widehat{\Theta}_{0:s}^{\mathrm{MLE}} - \widehat{\Theta}_{s:t}^{\mathrm{MLE}}\|_{\mathrm{F}}$, where for $e > s$, $\widehat{\Theta}_{s:e}^{\mathrm{MLE}} = \argmin_{\|\Theta\|_{\infty} \leq a, \mathrm{rank}(\Theta) \leq r} F_{s:e}(\Theta)$, $a > 0$, $r \in \mathbb{N}$ and $F_{s:e}(\Theta)$ is the negative log-likelihood of $\{Y_{\Omega}(t), \Omega(t)\}_{s+1}^e$. We compute $\widehat{\Theta}_{s:e}^{\mathrm{MLE}}$ using the approximate projected gradient method \citep[see Algorithm 1 in ][]{bhaskar2016probabilistic}, with the rank constraint $r$ the dimension of the matrix as done in \cite{bhaskar2016probabilistic} and the sup-norm constraint $a = 0.99$. The step size $\tau$ of the approximate projected gradient method is selected via a backtracking line search.
\textbf{2.} For the $k$-nearest neighbour ($k$-NN) based procedure \citep{chu:18,chen:19}, for fair comparisons, we calibrate the thresholds thereof on training data so that the false alarm rate is controlled at the level $\alpha$ instead of the average run length as proposed in \cite{chen:19}. In the simulations we set $k=1$.
\noindent \textbf{Tuning parameters.} To choose $\{\lambda_{s,t}, \varepsilon_{s, t}\}$, each experiment comprises of a calibration step carried out on training data $\{Y_{\Omega}(t), \Omega(t)\}_{t = 1}^{T_{\mathrm{train}}}$, $T_{\mathrm{train}}=200$. The training data do not possess change points.
To be specific, for $\{\lambda_{s,t}\}$ we set the sparsity parameter $\vartheta_2$ ($\vartheta_1$) of the adjacency matrices as the $95\%$ ($5\%$) quantile of $\{\widehat{\Theta}_{0:T_{\mathrm{train}}, ij}\}_{i,j}$, set $a = 1$ in \Cref{alg-main-klopp}, set the maximum (minimum) missingness probability $q_2$ ($q_1$) as the $95\%$ ($5\%$) quantile of $\{T^{-1}_{\mathrm{train}}\sum_{t = 1}^{T_{\mathrm{train}}}\Omega(t)\}_{i,j}$, set the rank $r$ as the rank of $\widehat{M}_{0:T_{\mathrm{train}}}$, and fix $C_\lambda=2/3$ in all numerical experiments. The robustness of the choice of $C_\lambda$ is investigated in an additional simulation provided in \Cref{sec-addition}.
As for $\{\varepsilon_{s,t}\}$, we randomly permute the training data $K = 100$ times with respective to $t$, and compute the $K$ replicates of the CUSUM statistic in \eqref{eq-widehat-delt-defi}, where for each $s$ and $t$, $\Vert\widehat{D}_{s,t}^{(k)}\Vert_{\mathrm{F}}$ involves only the $k$th permutation, $k \in \{1, \ldots, K\}$. We choose $C_{\varepsilon}$ such that the proportion of $\{\Vert\widehat{D}_{s,t}^{(k)}\Vert_{\mathrm{F}}\}_{k=1}^K$ which cross the detection threshold is capped at $\alpha$.
The detection thresholds for the competitors are also calibrated in a similar manner using the training data.
We consider two additional scenarios in this section.
\subsection{Scenario 1: SBM} \label{sec: scenario1}
Let the data be from SBMs with $n = 100$ and three equally-sized communities. Let $z_i \in \{1, 2, 3\}$ denote the community membership label of node $i$, $i \in \{1, 2, \dots, n\}$. Let the $(i,j)$th entry of the underlying graphon matrix $\Theta(t)$ be $\Theta_{ij}(t) = \vartheta_2 B_{z_iz_j}(t)$, $i,j \in \{1,2, \dots, n\}$, where $\vartheta_2 = 0.5$ and $B(t)$'s are
\[
B(t) =
\begin{cases}
\begin{pmatrix}
0.6 & 1.0 & 0.6\\
1.0 & 0.6 & 0.5\\
0.6 & 0.5 & 0.6
\end{pmatrix}, & t \in \{1,2, \dots, \Delta\}, \\
\begin{pmatrix}
0.6 & 0.5 & 0.6\\
0.5 & 0.6 & 1.0\\
0.6 & 1.0 & 0.6
\end{pmatrix}, &
t \in \{\Delta+1, \dots, T\}.
\end{cases}
\]
For $t \in \{1, 2, \dots, T\}$, we first generate the unobserved, symmetric adjacency matrices $\{Y(t)\}_{t = 1}^T$ as
\begin{align*}
Y_{ij}(t) \overset{ind.}{\sim} \mathrm{Bernoulli}(\Theta_{ij}(t)), \quad i \leq j,
\end{align*}
followed by the missingness pattern matrices $\{\Omega(t)\}_{t = 1}^T$ as
\begin{align*}
\Omega_{ij}(t) \overset{ind.}{\sim} \mathrm{Bernoulli}(\pi), \quad i \leq j,
\end{align*}
where $\pi = q_1 = q_2$ controls the overall proportion of missing entries.
We finally let the observed, symmetric adjacency matrices with missing values $\{Y_{\Omega}(t)\}_{t=1}^T$ be
\begin{align*}
Y_{\Omega,\, ij}(t) = \Omega_{ij}(t) Y_{ij}(t), \quad i \leq j.
\end{align*}
In this scenario, we consider undirected graphs with self-loops, which implies low-rank of the graphon matrices $\{\Theta(t)\}_{t = 1}^T$, i.e.~$r = 3$ across all $t$.
Tables \ref{tab:delay-1} and \ref{tab:delay} show that our approach outperforms the competitors by having the smallest average detection delay for varying degrees of missingness, while maintaining exceptional control over proportion of false alarms. While the performance of RC is close, it fails to keep the desired level of false alarms in some instances.
\iffalse
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|cccccc}
\hline
\multicolumn{1}{l|}{} & \multicolumn{6}{c}{$\{\mathrm{PNM}_t\}_{t = 1}^{300}$} \\
$\pi_{\mathrm{LB}}$ & min & 1st Qu. & Median & Mean & 3rd Qu. & Max \\ \hline
0.7 & 0.8375 & 0.8458 & 0.8488 & 0.8490 & 0.8525 & 0.8616 \\
0.8 & 0.8864 & 0.8960 & 0.8986 & 0.8986 & 0.9012 & 0.9093 \\
0.9 & 0.9428 & 0.9483 & 0.9507 & 0.9503 & 0.9525 & 0.9588 \\ \hline
\end{tabular}
}
\caption{Summary of proportion of non-missing entries by $\pi_{LB}$.}
\label{tab:pnm_t}
\end{table}
\fi
\begin{table}[ht]
\centering
\begin{tabular}{ccccccc}
\hline
$\alpha$ & SI & RC & ORI & W & G & MI \\ \hline\hline
\multicolumn{7}{c}{$\pi = 0.7$}\\
0.05 & \textbf{3.12} & 3.82 & 16.09 & 15.06 & 16.05 & 36.38 \\
0.01 & \textbf{3.37} & 3.86 & 19.68 & 17.11 & 19.42 & 37.25 \\
\multicolumn{7}{c}{$\pi = 0.8$}\\
0.05 & \textbf{2.98} & 3.69 & 15.28 & 15.00 & 16.28 & 34.57 \\
0.01 & \textbf{3.08} & 3.97 & 16.60 & 15.80 & 17.79 & 35.23 \\
\multicolumn{7}{c}{$\pi = 0.9$}\\
0.05 & \textbf{2.87} & 3.50 & 16.12 & 15.36 & 17.26 & 47.03 \\
0.01 & \textbf{2.90} & 3.60 & 17.68 & 16.52 & 18.63 & 48.22 \\
\multicolumn{7}{c}{$\pi = 0.95$}\\
0.05 & \textbf{2.69} & 3.42 & 15.15 & 14.95 & 16.85 & 51.62 \\
0.01 & \textbf{2.76} & 3.58 & 15.95 & 18.18 & 18.99 & 52.36 \\ \hline
\end{tabular}
\caption{Average detection delays over 100 repetitions of SBM considered in Scenario 1. SI: our proposed methods; RC: rank-constrained maximum likelihood estimator; ORI, W and G: three different statistics based on $k$-NN; MI: MissInspect; $\pi$, the missing probability; $\alpha$, the false alarm controls.}
\label{tab:delay-1}
\end{table}
\begin{table}[ht]
\centering
\begin{tabular}{ccccccc}
\hline
$\alpha$ & SI & RC & ORI & W & G & MI \\ \hline\hline
\multicolumn{7}{c}{$\pi = 0.7$}\\
0.05 & {0} & 0.06 & 0.03 & 0.02 & 0.04 & 0.07 \\
0.01 & {0} & 0.04 & {0} &{ 0} & {0} & 0.04 \\
\multicolumn{7}{c}{$\pi = 0.8$}\\
0.05 & {0} & 0.05 & 0.04 & 0.03 & 0.06 & {0} \\
0.01 & {0} & 0.01 & 0.02 & 0.02 & 0.01 & {0} \\
\multicolumn{7}{c}{$\pi = 0.9$}\\
0.05 & {0} & 0.1 & 0.11 & 0.05 & 0.05 & 0.04 \\
0.01 & {0} & 0.04 & 0.01 & 0.03 & 0.01 & 0.02 \\
\multicolumn{7}{c}{$\pi = 0.95$}\\
0.05 & {0} & 0.08 & 0.09 & 0.08 & 0.06 & 0.03 \\
0.01 & {0} & 0.05 & 0.06 & 0.02 & 0.03 & 0.01 \\ \hline
\end{tabular}
\caption{False alarms rates over 100 repetitions of SBM considered in Scenario 1. SI: our proposed methods; RC: rank-constrained maximum likelihood estimator; ORI, W and G: three different statistics based on $k$-NN; MI: MissInspect; $\pi$, the missing probability; $\alpha$, the false alarm controls.}
\label{tab:delay}
\end{table}
\subsection{Scenario 2: RDPG} \label{sec: scenario2}
To generate data from RDPGs, we let $n = 100$ and the latent positions be $X$ for $t \in \{1, \ldots, \Delta\}$ and $\widetilde X$ for $\{\Delta+1, \ldots, T\}$. Suppose
$X, X^{\prime} \in \mathbb{R}^{n \times 5}$ are generated from $X_{ij}, X^{\prime}_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathrm{Unif}[0,1]$, $i \in \{1, 2, \dots, n\}$, $j \in \{1, 2, \dots, 5\}$. Then
for $i \in \{1, 2, \dots, n\}$,
\begin{align*}
\widetilde{X}_i = \begin{cases}
X^{\prime}_i, & i \leq \lfloor n/4 \rfloor,\\
X_i, & \mbox{otherwise}.
\end{cases}
\end{align*}
The latent positions $X$ and $\widetilde{X}$ are fixed throughout.
For $i < j$, the $(i,j)$th entries of the unobserved, symmetric adjacency matrices $\{Y_{ij}(t)\}_{t=1}^T$ are generated independently as
\begin{align*}
Y_{ij}(t) \sim \begin{cases}
\mathrm{Bernoulli}\bigg(\frac{X_i^{\top}X_j}{\Vert X_i \Vert \Vert X_j \Vert}\bigg), & t \in \{1, \ldots, \Delta\},\\
\mathrm{Bernoulli}\bigg(\frac{\widetilde X_i^{\top}\widetilde X_j}{\Vert \widetilde X_i \Vert \Vert \widetilde X_j \Vert}\bigg), & t \in \{\Delta+1, \ldots, T\}.
\end{cases}
\end{align*}
The matrices $\{\Omega(t)\}_{t = 1}^T$ are generated in the same way as in \Cref{sec: scenario1} and we fix $\pi = q_1 = q_2 = 0.9$. Different from \Cref{sec: scenario1}, we consider symmetric adjacency matrices without self-loop, a more challenging setting as this does not guarantee a low-rank graphon.
The observed, symmetric adjacency matrices with missing values $\{Y_{\Omega}(t)\}_{t=1}^T$ are generated as $Y_{\Omega, \,ij}(t) = \Omega_{ij}(t) A_{ij}(t)$, $i < j$.
\Cref{tab: compare} shows that our method outperforms the $k$-NN graph based approach and MissInspect in the RDPG setting. The RC approach performs best across the board. We conjecture that better choices of constants, e.g.~$C_{\lambda}$, in our algorithm may improve our performances further.
\begin{table}[ht]
\centering
\begin{tabular}{ccccccc}
\hline
$\alpha$ & SI & RC & ORI & W & G & MI \\ \hline\hline
\multicolumn{7}{c}{Delay}\\
0.05 & 9.67 & \textbf{4.00} & 23.53 & 24.26 & 24.85 & 19.53 \\
0.01 & 9.62 & \textbf{4.04} & 26.58 & 25.85 & 26.88 & 21.60 \\
\multicolumn{7}{c}{PFA}\\
0.05 & {0} & {0} & 0.04 & 0.04 & 0.01 & 0.11 \\
0.01 & {0} & {0} & {0} & 0.01 & 0.01 & {0}\\
\hline
\end{tabular}
\caption{Average detection delays and false alarms rates over 100 repetitions of RDPG considered in Scenario 2, with the missing probability $\pi = q_1 = q_2 =0.9$. SI: our proposed methods; RC: rank-constrained maximum likelihood estimator; ORI, W and G: three different statistics based on $k$-NN; MI: MissInspect; $\alpha$, the false alarm controls. }
\label{tab: compare}
\end{table}
\section{Real data analysis} \label{sec-data}
In this section, we apply the proposed online change point detection method and competitors discussed in \Cref{sec: scenario2} to networks obtained from the weekly log-returns of $29$ companies, based on the Dow Jones Industrial Average index from April 1990 to January 2012 \citep[see the R package \texttt{ecp},][]{james2013ecp} and are processed as follows. We first obtain sequences of outer-product matrices of the companies’ weekly log-returns at each week and then threshold the entries of matrices by setting the entries with values above the $95\%$ quantile as $1$ and zero otherwise. Following this we apply the missing scheme described in \Cref{sec: scenario1} with $\pi= 0.9$. Finally, the network sequence, with missing values, together with their associated missingness pattern matrices are treated as the observed data. We consider two time periods: period 1 is from 2-Apr-1990 to 31-May-2004 and period 2 is from 31-May-2004 to 1-Mar-2010.
\begin{table}[ht]
\centering
\begin{tabular}{cccc}
\hline
$\alpha$ & SI & RC & ORI \\ \hline\hline
\multicolumn{4}{c}{Period 1} \\
0.05 & 2001-10-22 & Inf & Inf \\
0.01 & 2001-12-17 & Inf & Inf \\
\multicolumn{4}{c}{Period 2} \\
0.05 & 2008-04-14 & 2008-06-23 & 2009-08-03 \\
0.01 & 2008-06-16 & 2009-06-01 & 2009-08-03 \\ \hline
$\alpha$ & W & G & MI \\ \hline\hline
\multicolumn{4}{c}{Period 1} \\
0.05 & 2002-12-30 & Inf & 2001-07-23 \\
0.01 & Inf & Inf & Inf \\
\multicolumn{4}{c}{Period 2} \\
0.05 & 2009-04-20 & 2009-08-03 & 2008-12-22 \\
0.01 &2009-08-03 & 2009-08-03 & Inf \\ \hline
\end{tabular}
\caption{Stock exchange networks: Estimated change points with the missing probability $\pi=0.9$. SI: our proposed methods; RC: rank-constrained maximum likelihood estimator; ORI, W and G: three different statistics based on $k$-NN; MI: MissInspect; $\alpha$, the false alarm controls.}
\label{tab:data}
\end{table}
For period 1, we let the data dated from 2-Apr-1990 to 4-Jan-1999 be the training set, used to calibrate the tuning parameters, and detect change points in the rest in an online fashion. To be specific, instead of letting all data available at once, we pretend that the data are obtained sequentially. Note that there was dramatic financial turbulence right after the terrorist attacks on 11-Sept-2001. Our proposed method successfully detects a change point at 22-Oct-2001 (17-Dec-2001) with $\alpha = 0.05$ (0.01), indicating a small detection delay. As for the competitors, the weighted edge count statistic (W) detects a much later change point and MissInspect (MI) detects the date 23-July-2001, which appears more likely to be a false alarm.
For period 2, we let the data dated from 31-May-2004 to 15-Jan-2007 be the training set and detect change points in the rest in an online fashion. Note that there was a financial crisis in 2007-2008. We detect a change point in April (June) 2008 with $\alpha = 0.05$ (0.01), with a significantly smaller detection delay than the competitors.
\section{Additional simulation results}\label{sec-addition}
In this section, we provide additional simulation results, demonstrating the effect of the low-rank assumption and the robustness of the choices of $C_{\lambda}$ in our simulations.
Let the network size $n = 100$ and let $b$ be the number of equally-sized communities, each of which has size around $n/b$. Let $z_i \in \{1, \ldots, b\}$ be the community membership of node $i$, $i \in \{1, \ldots, n\}$. Let the $(i,j)$th entry of the underlying graphon matrix be $\Theta_{ij}(t) = \vartheta_2 B_{z_iz_j}(t)$, where $\vartheta_2 = 0.5$ and the matrices $B(t)$'s are
\begin{align*}
B(t) = \begin{cases}
0.2 \times \mathbf{1}_{b} + 0.7 \times \mathbf{I}_b, & t \in \{1,2, \dots, \Delta\}, \\
0.1 \times \mathbf{1}_b + 0.4 \times \mathbf{I}_b, & t \in \{\Delta+1, \Delta+2, \dots, T\},
\end{cases}
\end{align*}
where $\mathbf{1}_b$ is a $b \times b$ matrix of ones and $\mathbf{I}_b$ is an identity matrix of size $b$.
For $t \in \{1, 2, \dots, T\}$, we first generate the unobserved, symmetric adjacency matrices $\{Y(t)\}_{t = 1}^T$ as
\begin{align*}
Y_{ij}(t) \overset{\mathrm{ind.}}{\sim} \mathrm{Bernoulli}(\Theta_{ij}(t)), \quad i \leq j,
\end{align*}
followed by the missingness matrices $\{\Omega(t)\}_{t = 1}^T$ as
\begin{align*}
\Omega_{ij}(t) \overset{\mathrm{ind.}}{\sim} \mathrm{Bernoulli}(\pi), \quad i \leq j,
\end{align*}
where $\pi = q_1 = q_2 = 0.9$. Note that we consider symmetric adjacency matrices with self-loop, which ensures the symmetrized graphon matrix of rank $b$. We consider $b \in \{3, 5, 10, 25, 50\}$.
\textbf{Varying rank.} Fixing $C_{\lambda} = 0.67$, the performance of our method is reported in \Cref{tab:delay-s1}, implying the trend that the detection delay increases as the rank $b$ increases. This is as expected and is indicated in \Cref{thm-upper-bound}.
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
\hline
$\alpha$ & $b = 3$ & $b = 5$ & $b = 10$ & $b = 25$ & $b = 50$ \\ \hline\hline
\multicolumn{6}{c}{Delay}\\
0.05 & 3.33 & 3.90 & 6.67 & 8.81 & 8.50 \\
0.01 & 3.62 & 4.29 & 6.78 & 8.21 & 8.33 \\
\multicolumn{6}{c}{PFA}\\
0.05 & 0 & 0 & 0 & 0 & 0 \\
0.01 & 0 & 0 & 0 & 0 & 0 \\ \hline
\end{tabular}
\caption{Average detection delays and false alarms
rates of our proposed method over 100 repetitions of SBM, with $q_1 = q_2 = 0.9$. $\alpha$, the false alarm controls; b, the number of communities of underlying graphon.}
\label{tab:delay-s1}
\end{table}
\textbf{Robustness of the choices of $C_{\lambda}$.} We let $C_{\lambda} \in \{0.25, 0.50, 0.67, 1.00, 1.25\}$ and collect results in \Cref{tab:delay-s2}. It can be seen that the $C_{\lambda}$, which produces the smallest detection delay among candidates, varies in different settings.
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
\hline
$\alpha$ & $C_{\lambda} = 0.25$ & $C_{\lambda} = 0.50$ & $C_{\lambda} = 0.67$ & $C_\lambda = 1.00$ & $C_{\lambda} = 1.25$ \\ \hline\hline
\multicolumn{6}{c}{b = 3}\\
0.05 & 6.15 (0.02) & \textbf{2.72} (0.02) & 3.33 (0) & 3.21 (0) & 3.76 (0) \\
0.01 & 6.15 (0.01) & \textbf{2.97} (0) & 3.62 (0) & 3.43 (0) & 3.85 (0) \\
\multicolumn{6}{c}{b = 5}\\
0.05 & 6.64 (0) & 4.10 (0) & \textbf{3.90} (0) & 4.95 (0) & 5.98 (0) \\
0.01 & 6.36 (0) & 4.57 (0) & \textbf{4.29} (0) & 5.29 (0) & 6.31 (0) \\
\multicolumn{6}{c}{b = 10}\\
0.05 & 8.38 (0) & \textbf{6.01} (0) & 6.67 (0) & 9.68 (0) & 13.29 (0) \\
0.01 & 8.00 (0) & \textbf{5.91} (0) & 6.78 (0) & 9.67 (0) & 13.27 (0) \\
\multicolumn{6}{c}{b = 25}\\
0.05 & 15.90 (0) & 9.78 (0) & \textbf{8.81} (0) & 9.18 (0) & 9.25 (0) \\
0.01 & 14.60 (0) & 9.27 (0) & \textbf{8.21} (0) & 8.79 (0) & 9.01 (0) \\
\multicolumn{6}{c}{b = 50}\\
0.05 & 16.89 (0) & 10.73 (0) & 8.50 (0) & 5.28 (0) & \textbf{4.60} (0) \\
0.01 & 15.57 (0) & 10.65 (0) & 8.33 (0) & 5.08 (0) & \textbf{4.52} (0) \\
\hline
\end{tabular}
\caption{Average detection delays and false alarms
rates (in parenthesis) of our proposed method over 100 repetitions of SBM, with $q_1 = q_2 = 0.9$. $\alpha$, the false alarm controls; b, the number of communities.}
\label{tab:delay-s2}
\end{table}
\section{Auxiliary lemmas}
This section collects some existing results which are used in our proofs. We use $\phi$-mixing throughout our paper. Howeve, there are other variants of mixing coefficients, such as the $\alpha$ and $\beta$-mixing coefficients \cite[e.g.][]{bradley2005basic,doukhan2012mixing}. Note that the conditions based on the $\alpha$-mixing are weaker than that of $\beta$-mixing, which are further weaker than that of $\phi$-mixing. More precisely, it follows that $2\alpha(\ell) \leq \beta(\ell) \leq \phi(\ell)$ for any $\ell \in \mathbb{N}$ \cite[e.g.][]{bradley2005basic}.
\iffalse
\begin{lemma}[Theorem 3.4 in \cite{bradley2005basic}]\label{thm:phi-mixing}
Suppose $\{X(t)\}_{t \in \mathbb{N}^*}$ is a strictly stationary Markov chain which is irreducible and aperiodic. If $\phi(\ell) < 1$ for some $\ell \geq 1$, then $\phi(\ell) \to 0$ at least exponentially fast as $\ell \to \infty$.
\end{lemma}
\Cref{thm:phi-mixing} considers Markov chains whose state space not necessary be finite or countable. Note that the $\phi$-mixing coefficients satisfy that $0 \leq \phi(\ell) \leq 1$ and $2\alpha(\ell) \leq \beta(\ell) \leq \phi(\ell)$ for any $\ell \in \mathbb{N}$. \Cref{thm:phi-mixing} states that for a strictly stationary irreducible and aperiodic Markov chain, excluding $\phi(\ell) = 1$ for some $\ell \geq 1$ guarantees $\phi(\ell)$ decays to zero exponentially as $\ell \to \infty$.
\fi
\begin{lemma}[Theorem 5.2 in \citealt{bradley2005basic}]\label{lemma:mixing_indep_comp}
Denote $X^{(k)} = \{X^{(k)}(t)\}_{t \in \mathbb{N}^*}$, for $k \in \mathbb N^*$. Suppose these sequences, $X^{(k)}$, are mutually independent across $k$. Suppose that for each $t \in \mathbb{N}^*$, $h_t: \mathbb{R}\times \mathbb{R} \times \mathbb{R} \times \dots \mapsto \mathbb{R}$ is a Borel function. Define the sequence $X = \{X(t)\}_{t \in \mathbb{N}^*}$ of random variables with $X(t) = h_t\big(X^{(1)}(t),X^{(2)}(t),X^{(3)}(t),\dots\big)$, for $t \in \mathbb{N}^*$. Then for each $\ell \geq 1$, it follows that $\phi_X(\ell) \leq 1-\prod_{k=1}^{\infty}\big(1-\phi_{X^{(k)}}(\ell)\big) \leq \sum_{k = 1}^{\infty}\phi_{X^{(k)}}(\ell)$.
\end{lemma}
\iffalse
\begin{lemma}[Theorem 1 in \citealt{merlevede2011bernstein}]\label{lemma:bernstein}
Let $\{X_i\}_{i \geq 1}$ be a sequence of centered $\mathbb{R}$-valued random variables with $\alpha$-mixing coefficients $\alpha(\ell)$. Suppose there exist constants $c, \gamma_1, \gamma_2 > 0$ and $1/\gamma = 1/\gamma_1 + 1/\gamma_2 > 1$, such that for any $\ell \geq 1$
\begin{equation*}
\alpha(\ell) \leq \exp(-c\ell^{\gamma_1}),
\end{equation*}
and for any $u > 0$
\begin{equation}\label{eq:marginal_tail}
\sup_{i \geq 1}\mathbb{P}(|X_i| > u) \leq \exp(1-u^{\gamma_2}).
\end{equation}
Then for any $n \geq 4$, there exist positive absolute constants $C_1$ and $C_2$ depending only on $c$, $\gamma_1$ and $\gamma_2$ such that for any $x \geq 1$
\begin{equation*}
\mathbb{P}\bigg(n^{-1/2}\bigg|\sum_{i = 1}^nX_i\bigg| \geq x\bigg) \leq n\exp\bigg(-\frac{x^{\gamma}n^{\gamma/2}}{C_1}\bigg) + \exp\bigg(-\frac{x^2n}{C_2(1+nV^2)} \bigg),
\end{equation*}
where
\begin{equation*}
V^2 = \sup_{i \geq 1}\bigg\{\var(X_i) + 2\sum_{j > i}|\cov(X_i, X_j)|\bigg\}.
\end{equation*}
\end{lemma}
\fi
\iffalse
{\color{orange}
REMOVE?
\begin{lemma}[Theorem 1 in \cite{banna2016bernstein}]\label{lemma:matrix_bernstein}
Let $\{M_i\}_{i \in \mathbb{N}^*}$ is a family of self-adjoint random matrices of size $d$. Assume that there exists a constant $c > 0$ such that for any $\ell \geq 1$, $\beta_M(\ell) \leq \exp(1-c\ell)$, and there exist a positive constant $D$ such that for any $i \in \mathbb{N}^*$,
\begin{align*}
\mathbb{E}[M_i] = 0 \quad \text{and} \quad \lambda_{\max}(M_i) \leq D \quad \text{almost surely}.
\end{align*}
Then there exists an absolute constant $C$ such that for any $x > 0$ and any integer $n \geq 2$,
\[
\mathbb{P}\left(\lambda_{\max}\left(\sum_{i=1}^nM_i\right) \geq x\right) \leq d\exp\left(-\frac{Cx^2}{\nu^2n + c^{-1}D^2 + xD\gamma(c,n)}\right),
\]
where
\[
\nu^2 = \sup_{\mathcal{K} \subseteq\{1, \dots, n\}}\frac{1}{|\mathcal{K}|}\lambda_{\max}\left(\mathbb{E}\left(\sum_{i \in \mathcal{K}}M_i\right)^2\right)
\]
and
\[
\gamma(c,n) = \frac{\log n}{\log 2}\max\left\{2, \frac{32\log n}{c\log 2}\right\}.
\]
\end{lemma}
{\color{red}
By the Gershgorin circle theorem, we have that
\[
|\lambda_{\max}(A)| \leq \max_{j \in \{1,2,\dots,d\}}\sum_{k = 1}^d|A_{jk}|
\]
}
Define the spectral-wise truncated version of $M_i$ as
\[
M_i^{\delta} = \frac{\delta \vee \|M_i\|_{\mathrm{op}}}{\|M_i\|_{\mathrm{op}}}M_i.
\]
\begin{align*}
&\mathbb{P}\left(\frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega(t) - \mathbb{E}[\Omega(t)]\big)\right\|_{\mathrm{op}} \geq x\right)\\
\leq&\mathbb{P}\left(\frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega(t) - \Omega^{\delta}(t)+ \Omega^{\delta}(t)- \mathbb{E}[\Omega^{\delta}(t)] + \mathbb{E}[\Omega^{\delta}(t)] -\mathbb{E}[\Omega(t)]\big)\right\|_{\mathrm{op}} \geq x\right)\\
\leq&\mathbb{P}\left(\frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega(t) - \Omega^{\delta}(t))\right\|_{\mathrm{op}} + \frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega^{\delta}(t)- \mathbb{E}[\Omega^{\delta}(t)] + \mathbb{E}[\Omega^{\delta}(t)] -\mathbb{E}[\Omega(t)]\big)\right\|_{\mathrm{op}} \geq x\right)\\
\leq&\mathbb{P}\left(\left\|\sum_{t=1}^T\big(\Omega(t) - \Omega^{\delta}(t))\right\|_{\mathrm{op}} >0\right) + \mathbb{P}\left(\frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega^{\delta}(t)- \mathbb{E}[\Omega^{\delta}(t)] + \mathbb{E}[\Omega^{\delta}(t)] -\mathbb{E}[\Omega(t)]\big)\right\|_{\mathrm{op}} \geq x\right)\\
\leq&\sum_{t=1}^T\mathbb{P}\left(\Omega(t) \neq \Omega^{\delta}(t)\right) + \mathbb{P}\left(\left\|\sum_{t=1}^T\big(\Omega^{\delta}(t)- \mathbb{E}[\Omega^{\delta}(t)]\big)\right\|_{\mathrm{op}} \geq Tx - \sum_{t=1}^T\big\|\mathbb{E}[\Omega^{\delta}(t)] -\mathbb{E}[\Omega(t)]\big\|_{\mathrm{op}}\right)\\
\end{align*}
Set $\delta = C\max_{ij}\|\Omega_{ij}(t)\|_{\psi_2}\big(\sqrt{\log(2T/\epsilon)} \vee \sqrt{\log(4Tnq_2/\epsilon)} + 2\sqrt{n}\big)$, we have that
\begin{align*}
\sum_{t = 1}^T\mathbb{P}\left(\Omega(t) \neq \sum_{t=1}^T\Omega^{\delta}(t)\right) =& \mathbb{P}\left(\|\Omega(t)\|_{\mathrm{op}} > \delta\right) \leq 2T\exp\left(-\left(\frac{\delta}{C\max_{ij}\|\Omega_{ij}(t)\|_{\psi_2}}-2\sqrt{n}\right)^2\right)\\
\leq& \epsilon,
\end{align*}
where the inequality follows from Theorem 4.4.5 in \cite{vershynin2018high}.
We also have that
\begin{align*}
\big\|\mathbb{E}[\Omega^{\delta}(t)] -\mathbb{E}[\Omega(t)]\big\|_{\mathrm{op}} =& \left\|\mathbb{E}\left[\left(1-\frac{\delta}{\|\Omega(t)\|_{\mathrm{op}}}\right)\Omega(t)\mathbbm{1}\{\|\Omega(t)\|_{\mathrm{op}}>\delta\}\right]\right\|_{\mathrm{op}}\\
\leq& \sup_{u \in \mathbb{S}^{n-1}}\mathbb{E}\left[u^{\top}\Omega(t)u\mathbbm{1}\{\|\Omega(t)\|_{\mathrm{op}}>\delta\}\right]\\
\leq& \sup_{u \in \mathbb{S}^{n-1}}\sqrt{u^{\top}\mathbb{E}[\Omega^2(t)]u}\sqrt{\mathbb{P}(\{\|\Omega(t)\|_{\mathrm{op}}>\delta\}}\\
\leq& \sqrt{2\max_{i}\left\{\sum_{j = 1}^n\mathbb{E}[\Omega_{ij}^2(t)]\right\}}\exp\left(-\frac{1}{2}\left(\frac{\delta}{C\max_{ij}\|\Omega_{ij}(t)\|_{\psi_2}}-2\sqrt{n}\right)^2\right)\\
\leq& \sqrt{2nq_2}\exp\left(-\frac{1}{2}\left(\frac{\delta}{C\max_{ij}\|\Omega_{ij}(t)\|_{\psi_2}}-2\sqrt{n}\right)^2\right)\\
\leq& \sqrt{\frac{\epsilon}{T}}
\end{align*}
To obtain the tail probability bound of $\|\Omega^{\delta}(t) - \mathbb{E}[\Omega^{\delta}(t)]\|_{\mathrm{op}}$, we use \Cref{lemma:matrix_bernstein} with $M_i = \Omega^{\delta}(t) - \mathbb{E}[\Omega^{\delta}(t)]$.
Note that for any $\delta > 0$
\[
0 < \sup_{t \in \mathbb{N}^*}\frac{\delta \vee \|\Omega(t)\|_{\mathrm{op}}}{\|\Omega(t)\|_{\mathrm{op}}} \leq 1,
\]
and we have that
\[
\sup_{\mathcal{K} \subseteq\{1, \dots, n\}}\frac{1}{|\mathcal{K}|}\lambda_{\max}\left(\mathbb{E}\left(\sum_{t \in \mathcal{K}}\Omega^{\delta}(t)\right)^2\right) \leq \sup_{\mathcal{K} \subseteq\{1, \dots, n\}}\frac{1}{|\mathcal{K}|}\lambda_{\max}\left(\mathbb{E}\left(\sum_{t \in \mathcal{K}}\Omega(t)\right)^2\right).
\]
Note that due to the independence across entries of $\Omega(t)$ for any $t \in \mathbb{N}^*$, we have that for any $\ell \in \mathbb{N}^*$
\begin{align*}
\mathbb{E}[(\Omega(t) - \mathbb{E}[\Omega(t)])(\Omega(t+\ell) - \mathbb{E}[\Omega(t+\ell)])] =& \diag\left\{\sum_{j = 1}^n\cov\big(\Omega_{ij}(t),\Omega_{ij}(t+\ell)\big)\right\}_{i = 1}^n\\
\preceq& 2\alpha_{\Omega}(\ell)n\max\{q_2^2, (1-q_2)^2\}\bm{I}_n.
\end{align*}
Thus, we have that
\begin{align*}
\nu^2 \leq \left(1+4\sum_{\ell}\alpha_{\Omega}(\ell)\right)n\max\{q_2^2, (1-q_2)^2\} = C_{\Omega}n\max\{q_2^2, (1-q_2)^2\},
\end{align*}
where $C_{\Omega} = 1+4\sum_{\ell}\alpha_{\Omega}(\ell) \in (0,\infty)$.
\begin{align*}
\mathbb{P}\left(\left\|\frac{1}{T}\sum_{i=1}^T\big(\Omega^{\delta}(t) - \mathbb{E}[\Omega^{\delta}(t)]\big)\right\|_{\mathrm{op}} \geq x\right) \leq& 2n\exp\left(-\frac{CT^2x^2}{C_{\Omega}n\max\{q_2^2, (1-q_2)^2\}T + c^{-1}\delta^2 + x\delta\gamma(c,T)}\right)\\
\leq& 2n\exp\left(-\left(\frac{CT^2x^2}{C_{\Omega}n\max\{q_2^2, (1-q_2)^2\}T + c^{-1}\delta^2} \bigg\wedge \frac{CTx}{\delta\gamma(c,T)}\right) \right).
\end{align*}
\begin{align*}
&\mathbb{P}\left(\frac{1}{T}\left\|\sum_{t=1}^T\big(\Omega(t) - \mathbb{E}[\Omega(t)]\big)\right\|_{\mathrm{op}} \geq x+\sqrt{\frac{\epsilon}{T}}\right)\\
\leq&\sum_{t=1}^T\mathbb{P}\left(\Omega(t) \neq \Omega^{\delta}(t)\right) + \mathbb{P}\left(\left\|\frac{1}{T}\sum_{t=1}^T\big(\Omega^{\delta}(t)- \mathbb{E}[\Omega^{\delta}(t)]\big)\right\|_{\mathrm{op}} \geq x\right)\\
\leq& 2n\exp\left(-\left(\frac{CT^2x^2}{C_{\Omega}n\max\{q_2^2, (1-q_2)^2\}T + c^{-1}\delta^2} \bigg\wedge \frac{CTx}{\delta\gamma(c,T)}\right) \right) + \epsilon
\end{align*}
}
\fi
\iffalse
\begin{lemma}[Proposition 2.5(i) in \citealt{fan2003nonlinear}]\label{lemma:cov_inequ_mixing}
Let $X$ and $Y$ be two random variables, and $\alpha$ be the $\alpha$-mixing coefficient of $\{X, Y\}$. If $\|X\|_p + \|Y\|_q < \infty$ for some $p,q \geq 1$ and $1/p + 1/q < 1$, it holds that
\begin{equation*}
|\cov(X,Y)| \leq 8\alpha^{1/r}\|X\|_p\|Y\|_q,
\end{equation*}
where $r = 1-1/p-1/q$.
\end{lemma}
\fi
\iffalse
\begin{lemma}[Theorem 1.1 in \cite{rio1993covariance}]\label{lemma:cov_inequ}
Let $X$ and $Y$ be two integrable real-valued random variables, and $\alpha$ be the $\alpha$-mixing coefficient of $\{X, Y\}$. Denote the quantile function of $|X|$ as
\[
Q_X(u) = \inf\left\{t: \mathbb{P}(|X| > t) \leq u\right\}
\]
Assume furthermore that $Q_XQ_Y$ is integrable on $[0,1]$. Then
\begin{equation*}
|\cov(X,Y)| \leq 2\int_{0}^{2\alpha}Q_X(u)Q_Y(u)du.
\end{equation*}
\end{lemma}
\begin{remark}\label{remark:quantile_fct_ber}
Consider a random variable $X \sim \mathrm{Bernoulli}(p)$. The quantile function of $|X-p|$ can be written as
\[
Q_{X-p}(u) = \begin{cases}
(1-p)\vee p, \quad &\text{if} \;\; 0 \leq u < (1-p) \wedge p,\\
(1-p)\wedge p, \quad &\text{if} \;\; (1-p) \wedge p \leq u < 1,\\
0, \quad &\text{if} \;\; u = 1.
\end{cases}
\]
\end{remark}
\begin{corollary}\label{coro:cov_inequ_ber}
Let $X \sim \mathrm{Bernoulli}(p_X)$ and $Y \sim \mathrm{Bernoulli}(p_Y)$. With the same notation as that in \Cref{lemma:cov_inequ}, we have that
\begin{equation*}
|\cov(X,Y)| \leq 4\alpha\max\{(1-p_X)(1-p_Y), (1-p_X)p_Y, p_X(1-p_Y), p_Xp_Y\}.
\end{equation*}
\end{corollary}
\begin{proof}
Note that $\cov(X,Y) = \cov(X-p_X,Y-p_Y)$. Since $0 \leq 2\alpha \leq 1/2$, the proof directly follows from \Cref{lemma:cov_inequ} and \Cref{remark:quantile_fct_ber}.
\end{proof}
\fi
\begin{lemma}[Proposition 2.6.1 in \citealt{vershynin2018high}]\label{lemma:subGaussian_ind_sum}
Let $X_1, \dots, X_n$ be independent, mean zero, sub-Gaussian random variables. Then $\sum_{i=1}^n X_i$ is also a sub-Gaussian random variable, and
\[
\Big\|\sum_{i=1}^n X_i \Big\|_{\psi_2}^2 \leq C\sum_{i = 1}^n\|X_i\|_{\psi_2}^2,
\]
where $C > 0$ is a universal constant.
\end{lemma}
\begin{theorem}[Theorem 4.4.5 \& Corollary 4.4.8 in \citealt{vershynin2018high}]\label{thm:op_norm_tail}
(a) Let $X \in \mathbb{R}^{m \times n}$ be a random matrix whose entries $X_{ij}$ are independent, centred and sub-Gaussian. Then, for any $t > 0$, we have that
\[
\|X\|_{\mathrm{op}} \leq CK(\sqrt{m} + \sqrt{n} + t),
\]
with probability at least $1 - \exp(-t^2)$, and $K = \max_{i,j}\|X_{ij}\|_{\psi_2}$.
(b) Let $X \in \mathbb{R}^{n \times n}$ be a symmetric random matrix whose upper diagonal entries $X_{ij}$ are independent, centred and sub-Gaussian. Then, for any $t > 0$, we have that
\[
\|X\|_{\mathrm{op}} \leq CK(\sqrt{m} + \sqrt{n} + t),
\]
with probability at least $1 - \exp(-t^2)$, and $K = \max_{i,j}\|X_{ij}\|_{\psi_2}$.
\end{theorem}
\iffalse
\begin{lemma}[Exercise 4.4.3 in \citealt{vershynin2018high}]\label{lemma:epsilon_net}
Let $A \in \mathbb{R}^{m \times n}$ be a matrix and $\epsilon \in [0,1/2)$, for any $\epsilon$-net $\mathcal{N}$ of the sphere $\mathbb{S}^{n-1}$ and $\mathcal{M}$ of $\mathbb{S}^{m-1}$, we have that
\begin{align*}
\mathbb{P}\big(\|A\|_{\mathrm{op}} \geq u\big) \leq 9^{n+m}\max_{x \in \mathcal{N}, y \in \mathcal{M}}\mathbb{P}\big(\big|\langle Ax, y \rangle\big| \geq (1-2\epsilon)u\big)
\end{align*}
\end{lemma}
\begin{proof}
It follows from the definition of the $\epsilon$-net that for any $x \in \mathbb{S}^{n-1}$, there exists $x_0 \in \mathcal{N}$ such that
\begin{align*}
|x - x_0|_2 \leq \epsilon.
\end{align*}
For any $x\in \mathbb{S}^{n-1}$ and $y \in \mathbb{S}^{m-1}$, we have that
\begin{align*}
\langle Ax, y \rangle = \langle Ax_0, y_0 \rangle + \langle A(x - x_0), y_0 \rangle + \langle Ax, y - y_0 \rangle,
\end{align*}
and
\begin{align*}
\big|\langle Ax, y \rangle\big| \leq \big|\langle Ax_0, y_0 \rangle\big| + 2\epsilon \| A \|_{\mathrm{op}}.
\end{align*}
Taking sup on the left-hand side of the above inequality leads to
\begin{align*}
\|A\|_{\mathrm{op}} \leq \frac{1}{1-2\epsilon}\max_{x \in \mathcal{N}, y \in \mathcal{M}}\big|\langle Ax, y \rangle\big|.
\end{align*}
It follows Corollary 4.2.13 in \cite{vershynin2018high} that
\begin{align*}
|\mathcal{N}| \leq 9^n \quad \text{and} \quad |\mathcal{M}| \leq 9^m.
\end{align*}
Furthermore, for any $u > 0$, we have that
\begin{align*}
\mathbb{P}\big(\|A\|_{\mathrm{op}} \geq u\big) \leq \mathbb{P}\bigg(\frac{1}{1-2\epsilon}\max_{x \in \mathcal{N}, y \in \mathcal{M}}\big|\langle Ax, y \rangle\big| \geq u\bigg) \leq 9^{n+m}\max_{x \in \mathcal{N}, y \in \mathcal{M}}\mathbb{P}\big(\big|\langle Ax, y \rangle\big| \geq (1-2\epsilon)u\big)
\end{align*}
\end{proof}
\begin{lemma}[Theorem 1 in \citealt{doukhan2012mixing}]\label{lemma:coupling}
Let $E$ and $F$ be two polish spaces and $(X, Y)$ some $E\times F $-valued random variables. Denote by $\beta(\sigma(X), \sigma(Y))$ the $\beta$-mixing coefficients relative to the $\sigma$-algebras generated by $X$ and $Y$. A random variable $Y^*$ can be defined with the same probability distribution as $Y$, independent of $X$ and such that
$$\mathbb{P}(Y\not = Y^*) = \beta(\sigma(X), \sigma(Y)).$$
For some measurable function $f$ on $E\times F \times [0,1]$, and some uniform random variable $U$ on the interval $[0,1]$, $Y^*$
takes the form $Y^* = f(X, Y, U)$.
\end{lemma}
In the following, we describe the blocking technique introduced by \cite{yu1994rates} to analyze a stationary $\beta$-mixing process $\{X_t\}_{t = 1}^T$.
spliting the sample into $2M$ blocks of equal size $N$. For simplity, assume $T = 2MN$. For $j \in \{1, \dots, 2M\}$, denote the $j$th block by
\[
B_j = \{X_t\}_{t = 1+(j-1)N}^{jN}.
\]
Denote the set of odd blocks and the set even blocks respectively by
\[
B_{o} = \{B_1, B_3, \ldots, B_{2m-1}\}, \quad \text{and} \quad B_{e} = \{B_2, B_4, \ldots, B_{2m}\}.
\]
Consider also the set
\[
B^*_{o} = \{B^*_1, B^*_3, \ldots, B^*_{2m-1}\} \quad \text{and} \quad B^*_j = \{\widetilde X_t\}_{t = 1+(j-1)a}^{ja}.
\]
where the blocks $B^*_1, B^*_3, \ldots, B^*_{2m-1}$ are mutually independent and following the same distributions corresponding to that of $B_1, B_3, \ldots, B_{2m-1}$. The sequence of independent blocks $B^*_{o}$ can be constructed using \Cref{lemma:coupling}. Similarly, we also define $B^*_{e} = \{B^*_2, B^*_4, \ldots, B^*_{2m}\}$. The sequence $\{\widetilde X_t\}_{t = 1}^T$ is implied by such construction, and for fixed $\ell \in \{1, \dots, N\}$, $\widetilde X_{1+(k-1)N+\ell}$ are mutually independent across $k \in \{1, 2, \dots, 2M\}$.
\\
\\
Using the notations above, we now state the key result of the blocking technique.
\begin{lemma}[Corollary 2.7 in \citealt{yu1994rates}]\label{lemma:blocking}
Let $h$ be a measurable function, such that $|h| \leq H$ for some $M > 0$, then
\[
\left| \mathbb{E}[h(B_{o})] - \mathbb{E}[h(B_{o}^*)] \right| \leq 2H(M-1)\beta(N).
\]
The same result can be obtained for $\left| \mathbb{E}[h(B_{e})] - \mathbb{E}[h(B_{e}^*)] \right|$.
\end{lemma}
\begin{proof}
The proof follows directly from \Cref{lemma:coupling}.
\end{proof}
\fi
The next theorem gives a variant of Talagrand's concentration inequality under $\phi$-mixing condition.
\begin{theorem}[Corollary 4 in \citealt{samson2000concentration}]\label{thm-talagrand_dep}
Suppose that $f: \, [-1, 1]^n \to \mathbb{R}$ is convex and $L$-Lipschitz continus with respect to the $l_2$-norm. Let $\epsilon_1, \ldots, \epsilon_n$ be a sequence of random variables taking values in $[-1, 1]$. Let $Z = f(\epsilon_1, \ldots, \epsilon_n)$. Then for any $t \geq 0$, it holds that
\[
\mathbb{P}\left\{|Z - \mathbb{E}(Z)| \geq t\right\} \leq 2\exp\left\{-\frac{t^2}{2L^2\|\Gamma\|^2_{\mathrm{op}}}\right\},
\]
where $\Gamma = (\gamma_{ij})_{1\leq i,j \leq n}$ is a lower triangular matrix having $1$s along its diagonal. For $1 \leq i < j \leq n$, let $\epsilon_i^j$ represent the vector $(\epsilon_i, \dots, \epsilon_j)$ and $\mathcal{L}(\epsilon_{j}^n|\epsilon_1^i = y_1^i)$ denote the law of $\epsilon_j^n$ conditionally to $\epsilon_1^i = y_1^i$. Then set
\begin{align*}
(\gamma_{ij})^2 = 2\esssup_{y_1^i \in \mathbb R^i, \mathcal{L}(\epsilon_1^i)} \left\|\mathcal{L}\left(\epsilon_{j}^n|\epsilon_1^i = y_1^i\right) - \mathcal{L}\left(\epsilon_j^n\right)\right\|_{\mathrm{TV}},
\end{align*}
where $\esssup_{y_1^i \in \mathbb R^i, \mathcal{L}(\epsilon_1^i)}$ is the essential supremum with respect to the measure $\mathcal{L}(\epsilon_1^i)$ and $\|\cdot\|_{\mathrm{TV}}$ is the total variation of a signed measure.
Further assume that the $\phi$-mixing coefficients of $\epsilon_1, \ldots, \epsilon_n$ decay exponentially, i.e.~there exists some $C > 0$ and $\rho \in (0,1)$ such that $\phi(\ell) \leq C\rho^\ell$, for any $\ell \in \mathbb{N}$, then it follows that $\|\Gamma\|_{\mathrm{op}} \leq \sum_{\ell = 1}^{\infty}\sqrt{\phi(\ell)} < \infty$.
\end{theorem}
\begin{remark}
Note that the above theorem is stated under the $\phi$-mixing conditions, which are stronger than the $\alpha$ or $\beta$-mixing conditions with the same decay rate. This theorem is the main reason for which we impose the $\phi$-mixing conditions. To the best of our knowledge, it is still unknown if the Talagrand's concentration inequality could be generalised under the more general $\alpha$ or $\beta$-mixing conditions.
\end{remark}
\iffalse
\begin{theorem}[Corollary 3.3 in \citealt{bandeira2016sharp}]\label{thm:expectation_op}
(a) Let $X \in \mathbb{R}^{n\times m}$ be a random matrix with $X_{ij}$, $1\leq i \leq n, 1\leq j \leq m$, be independent centered bounded random variables. Then
\[
\mathbb{E}\left[\|X\|_{\mathrm{op}}\right] \leq C\max\left\{\max_{i}\sqrt{\sum_{j}\mathbb E[X_{ij}^2]},\; \max_{j}\sqrt{\sum_{i}\mathbb E[X_{ij}^2]}, \;\max_{ij}|X_{ij}|\sqrt{\log(n \wedge m)} \right\},
\]
for some absolute constant $C > 0$.
(b) Let $X \in \mathbb{R}^{n\times n}$ be a symmetric random matrix with $X_{ij}$, $1\leq i \leq j \leq n$, be independent centered bounded random variables. Then
\[
\mathbb{E}\left[\|X\|_{\mathrm{op}}\right] \leq C\max\left\{\max_{i}\sqrt{\sum_{j}\mathbb E[X_{ij}^2]},\; \max_{ij}|X_{ij}|\sqrt{\log n} \right\},
\]
for some absolute constant $C > 0$.
\end{theorem}
\begin{theorem}[Corollary 3.12 in \citealt{bandeira2016sharp}; Proposition 13 in \citealt{klopp2015matrix}]\label{thm:tailprop_op}
(a) Let $X \in \mathbb{R}^{n\times m}$ be a random matrix whose entries $X_{ij}$ are independent centered bounded random variables. Then for any $0 < \epsilon \leq 1/2$, there exists an absolute constant $C_{\epsilon}$, such that for any $t \geq 0$
\[
\mathbb{P}\left(\|X\|_{\mathrm{op}} \geq (1+\epsilon)2\sqrt{2}(\sigma_1 \vee \sigma_2) + t \right) \leq (n \wedge m)\exp\left(-\frac{t^2}{C_{\epsilon}\sigma_*^2}\right),
\]
where
\[
\sigma_1 = \max_i\sqrt{\sum_j\mathbb{E}[X_{ij}^2]}, \quad \sigma_2 = \max_j\sqrt{\sum_i\mathbb{E}[X_{ij}^2]}, \quad \text{and} \quad \sigma_* = \max_{ij}|X_{ij}|.
\]
(b) Let $X \in \mathbb{R}^{n\times n}$ be a symmetric random matrix with $X_{ij}$, $1 \leq i \leq j \leq n$, be independent centered bounded random variables. Then for any $0 < \epsilon \leq 1/2$, there exists an absolute constant $C_{\epsilon}$, such that for any $t \geq 0$
\[
\mathbb{P}\left(\|X\|_{\mathrm{op}} \geq (1+\epsilon)2\sigma_1 + t \right) \leq n\exp\left(-\frac{t^2}{C_{\epsilon}\sigma_*^2}\right).
\]
\end{theorem}
\begin{lemma}\label{lem1-mix}
Let $\{X_t\}_{t \in \mathbb Z}$ be a centered possibly nonstationary sequence. Denote the corresponding $\alpha$-mixing coefficients by $\alpha(\ell)$ for $\ell \in \mathbb N$. Suppose that there exist $\delta, \Delta>0$, $2s \leq \delta \leq 2(s+1)$, $s = 0,1,2,\dots$, such that
$$
\mathbb{E}\left[|X_t|^{2+\delta+\Delta}\right] \leq D_1, \;\; \text{for all} \;\; t \in \mathbb Z,
$$
and
$$
\sum_{\ell=0}^{\infty}(\ell+1)^{2s + 2} \alpha(\ell)^{\Delta /(2s+4+\Delta)} \leq D_2(\delta, \Delta) .
$$
Then
$$
\mathbb{E}\Big[\max _{k=1, \ldots, n}\Big|\sum_{t=1}^k X_t\Big|^{2+\delta}\Big] \leq D n^{(2+\delta) / 2},
$$
where $D$ only depends on $\delta$ and the joint distribution of $\{X_t\}_{t \in \mathbb Z}$.
\end{lemma}
\begin{proof}
This is Lemma B.8. of \cite{kirch2006resampling}.
\end{proof}
\begin{lemma}\label{lem2-mix}
Under the conditions of \Cref{lem1-mix}, it holds that for any $d \in \mathbb N^*,0<\nu<1$ and $x>0$,
$$
\mathbb{P}\left(\max_{k \in[\lceil \nu d \rceil, d]} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq x\right) \leq C x^{-2-\delta},
$$
where $C$ is some constant.
\end{lemma}
\begin{proof}
Let
$$
S_d^*=\max _{k=1, \ldots, d}\Big|\sum_{t=1}^k X_t\Big| .
$$
Then \Cref{lem1-mix} implies that
$$
\|S_d^*\|_{L_{2+\delta}} \leq C_1 d^{1 / 2}
$$
Therefore it holds that
$$
\mathbb{P}\Big(\Big|\frac{S_d^*}{\sqrt{d}}\Big| \geq x\Big)=\mathbb{P}\Big(\Big|\frac{S_d^*}{\sqrt{d}}\Big|^{2+\delta} \geq x^{2+\delta}\Big) \leq C_1 x^{-2-\delta} .
$$
Observe that
$$
\frac{|S_d^*|}{\sqrt{d}}=\max _{k=1, \ldots, d} \frac{|\sum_{t=1}^k X_t|}{\sqrt{d}} \geq \max _{k \in[\lceil \nu d \rceil, d]} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{d}} \geq \max _{k \in[\lceil \nu d \rceil, d]} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k / \nu}}
$$
Therefore
$$
\mathbb{P}\left(\max _{k \in[\lceil\nu d\rceil, d]} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq x / \sqrt{\nu}\right) \leq \mathbb{P}\left(\Big|\frac{S_d^*}{\sqrt{d}}\Big| \geq x\right) \leq C_1 x^{-2-\delta},
$$
which gives
$$
\mathbb{P}\left(\max _{k \in[\lceil\nu d\rceil, d]} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq x\right) \leq C_2 x^{-2-\delta} .
$$
\end{proof}
\begin{theorem}\label{thm3-mix}
Under the conditions of \Cref{lem1-mix}, for any $0<a, \nu <1$ it holds that
$$
\mathbb{P}\left(\Big|\sum_{t=1}^d X_t\Big| \leq \frac{C}{a} \sqrt{d}\{\log (d \nu)+1\} \;\; \text{for all} \;\; d \geq 1 / \nu \;\;\text{and}\;\; d \in \mathbb N^*\right) \geq 1-a^2,
$$
where $C > 0$ is some absolute constant.
\end{theorem}
\begin{proof}
Let $s \in \mathbb{N}^{*}$ and $\mathcal{T}_s=\big[2^s / \nu, 2^{s+1} / \nu\big]$. By \Cref{lem2-mix}, for all $x \geq 1$,
$$
\mathbb{P}\left(\sup _{k \in \mathcal{T}_s \cap \mathbb N^*} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq x\right) \leq C_1 x^{-2-\delta} \leq C_1 x^{-2} .
$$
Therefore by a union bound, for any $0<a<1$,
$$
\mathbb{P}\left(\exists s \in \mathbb{N}^{*}: \sup _{k \in \mathcal{T}_s \cap \mathbb N^*} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq \frac{\sqrt{C_1}}{a}(s+1)\right) \leq \sum_{s=0}^{\infty} \frac{a^2}{(s+1)^2}=a^2 \pi^2 / 6 .
$$
For any $k \in\big[2^s / \nu, 2^{s+1} / \nu\big], s \leq \log (k \nu) / \log (2)$, and therefore
$$
\mathbb{P}\left(\exists s \in \mathbb{N}^{*}: \sup _{k \in \mathcal{T}_s\cap \mathbb N^*} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq \frac{\sqrt{C_1}}{a}\Big\{\frac{\log (k \nu)}{\log (2)}+1\Big\}\right) \leq a^2 \pi^2 / 6,
$$
which directly gives
$$
\mathbb{P}\left(\sup _{k \in \mathcal{T}_s \cap \mathbb N^*} \frac{\big|\sum_{t=1}^k X_t\big|}{\sqrt{k}} \geq \frac{C}{a}\{\log (k \nu)+1\}\right) \leq a^2 .
$$
\end{proof}
\fi
\section[]{Proof of \Cref{theorem:markov-phi}}
\begin{proof}[Proof of \Cref{theorem:markov-phi}]
It suffice to consider a fixed $(i,j) \in \{1, \ldots, n\}^{\otimes 2}$.
Let $\{U_{ij}(t)\}_{t \in \mathbb{N}^*}$ be a sequence of i.i.d. $\textrm{Unif}~[0,1]$ random variables, which are independent of $\{X_i(t), X_j(t)\}_{t \in \mathbb{N}^*}$. We have equivalently that
\begin{equation*}
Y_{ij}(t) = \mathbbm{1}\{X_{i}(t)^{\top}X_j(t) + U_{ij}(t) \geq 1\}.
\end{equation*}
Since $\mathbbm{1}\{\cdot \geq 1\}: [0,2] \mapsto \{0, 1\}$ is a measurable function, for any $t \in \mathbb{N}^*$, we have that the $\sigma$-algebra satisfies
\begin{align*}
\sigma\big(\mathbbm{1}\{X_{i}(t)^{\top}X_j(t) + U_{ij}(t) \geq 1\}
\subset \sigma\big(X_{i}(t),X_{j}(t), U_{ij}(t)\big).
\end{align*}
Due to the temporal independence, the $\phi$-mixing coefficient of $\{U_{ij}(t)\}_{t \in \mathbb{N}^*}$ is zero for any $\ell \geq 1$. By \Cref{lemma:mixing_indep_comp}, we have that for any $\ell \in \mathbb{N}$,
\begin{equation*}
\phi_{Y}^{ij}(\ell) \leq 2\phi(\ell) \leq 2C\rho^{\ell},
\end{equation*}
which completes the first part of the proof. The proof for $\Omega_{ij}(t)$ follows similarly and thus omitted.
\end{proof}
\section[]{Proof of \Cref{lem-soft-impute-property}}
\begin{proof}[Proof of \Cref{lem-soft-impute-property}]
The results have two parts and we prove them separately. The first part is an adaptation of the proof Lemma~1 in \cite{mazumder2010spectral} and the second part is an adaptation of the proof Lemma~1 in \cite{klopp2015matrix}.
\noindent \textbf{The form of $\widehat{Z}$.}
For any matrix $Z \in \mathbb{R}^{n \times n}$, let $Z = \widetilde{U} \widetilde{D} \widetilde{V}^{\top}$ be a singular value decomposition of $Z$. Note that
\begin{align}
& \frac{1}{2m_1} \sum_{t = 1}^{m_1} \|R(t) - Z\|_{\mathrm{F}}^2 + \lambda\|Z\|_* = \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\{\left\|R(t)\right\|_{\mathrm{F}}^2 - 2\sum_{i = 1}^n \tilde{d}_i \tilde{u}_i^{\top} R(t) \tilde{v}_i + \sum_{i = 1}^n \tilde{d}_i^2\right\} + \lambda \sum_{i = 1}^n \tilde{d}_i \nonumber \\
= & \frac{1}{2} \left\{\frac{1}{m_1}\sum_{t = 1}^{m_1}\|R(t)\|_{\mathrm{F}}^2 - 2 \sum_{i = 1}^n \tilde{d}_i \tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i + \sum_{i = 1}^n \tilde{d}_i^2\right\} + \lambda \sum_{i = 1}^n \tilde{d}_i, \label{eq-proof-lem1-1}
\end{align}
where
\[
\widetilde{D} = \mathrm{diag}\{\tilde{d}_1, \ldots, \tilde{d}_n\}, \quad \widetilde{U} = (\tilde{u}_1, \ldots, \tilde{u}_n) \quad \mbox{and} \quad \widetilde{V} = (\tilde{v}_1, \ldots, \tilde{v}_n),
\]
and the first identity follows from
\[
\mathrm{tr}\{Z^{\top}R(t)\} = \mathrm{tr}\{\widetilde{V} \widetilde{D} \widetilde{U}^{\top} R(t)\} = \sum_{i = 1}^n \mathrm{tr} \{\tilde{d}_i \tilde{v}_i \tilde{u}_i^{\top} R(t)\} = \sum_{i = 1}^n \tilde{d}_i \tilde{u}_i^{\top} R(t)\tilde{v}_i.
\]
Minimising \eqref{eq-proof-lem1-1} is equivalent to minimising
\[
- 2 \sum_{i = 1}^n \tilde{d}_i \tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i + \sum_{i = 1}^n \tilde{d}_i^2 + 2\lambda \sum_{i = 1}^n \tilde{d}_i,
\]
with respect to $(\tilde{u}_i, \tilde{v}_i, \tilde{d}_i)$, $i = 1, \ldots, n$, under the constraints $\widetilde{U}^{\top}\widetilde{U} = \widetilde{V}^{\top}\widetilde{V} = I_n$ and $\tilde{d}_i \geq 0$.
Observe that the above is equivalent to minimising (with respect to $\widetilde{U}$ and $\widetilde{V}$) the function $g(\widetilde{U}, \widetilde{V})$,
\begin{equation}\label{eq-proof-lem1-2}
g(\widetilde{U}, \widetilde{V}) = \min_{\widetilde{D} \geq 0} \left\{- 2 \sum_{i = 1}^n \tilde{d}_i \tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i + \sum_{i = 1}^n \tilde{d}_i^2 + 2\lambda \sum_{i = 1}^n \tilde{d}_i\right\}.
\end{equation}
Since the objective \eqref{eq-proof-lem1-2} to be minimised with respect to $\widetilde{D}$, is separable in $\tilde{d}_i$, $i = 1, \ldots, n$. It suffices to minimise it with respect to each $\tilde{d}_i$ separately.
The problem
\[
\min_{\tilde{d}_i \geq 0} \left\{- 2 \tilde{d}_i \tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i + \tilde{d}_i^2 + 2\lambda \tilde{d}_i\right\}
\]
can be solved by the solution
\[
S_{\lambda}\left(\tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i\right) = \left(\tilde{u}_i^{\top} \left[\frac{1}{m_1} \sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i - \lambda\right)_+.
\]
Plugging this into \eqref{eq-proof-lem1-2} leads to
\begin{align*}
g(\widetilde{U}, \widetilde{V}) & = \frac{1}{2} \Bigg\{\frac{1}{m_1}\sum_{t = 1}^{m_1} \|R(t)\|_{\mathrm{F}}^2 - 2\sum_{i = 1}^n \left(\tilde{u}_i^{\top} \left[\frac{1}{m_1}\sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i - \lambda\right)_+\left(\tilde{u}_i^{\top} \left[\frac{1}{m_1}\sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i - \lambda\right) \\
& \hspace{1cm} + \sum_{i = 1}^n \left(\tilde{u}_i^{\top} \left[\frac{1}{m_1}\sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i - \lambda\right)_+^2\Bigg\}.
\end{align*}
Minimising $g(\widetilde{U}, \widetilde{V})$ with respect to $(\widetilde{U}, \widetilde{V})$ is equivalent to maximising
\begin{equation}\label{eq-proof-lemma1-singular-values}
\sum_{\tilde{u}_i^{\top} \left[\frac{1}{m_1}\sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i > \lambda} \left(\tilde{u}_i^{\top} \left[\frac{1}{m_1}\sum_{t = 1}^{m_1} R(t)\right] \tilde{v}_i - \lambda\right)_+^2.
\end{equation}
Due to the definitions of singular values, we see that \eqref{eq-proof-lemma1-singular-values} is solved by the left and right singular vectors of the matrix $m_1^{-1}\sum_{t = 1}^{m_1} R(t)$. The final result then follows.
\noindent \textbf{The convergence.}
Since for any $t \in \{1, \ldots, m_1\}$ and $k \in \mathbb{N}$, it holds that
\[
\|(M_{k+1} - M_k)_{\overline{\Omega(t)}}\|_{\mathrm{op}} \leq \|(M_{k+1} - M_k)_{\overline{\Omega(t)}}\|_{\mathrm{F}} \leq \|M_{k+1} - M_k\|_{\mathrm{F}}
\]
and
\[
\|M_{k+1} - M_k\|_{\infty} \leq \|M_{k+1} - M_k\|_{\mathrm{F}},
\]
it suffices to show that $\|M_{k+1} - M_k\|_{\mathrm{F}} \to 0$, as $k \to \infty$.
Denote by $M_k^*$ the $k$th step solution before truncating.
For any $k \in \mathbb{N}^*$, we have that
\begin{align}
& \|M_{k+1} - M_k\|_{\mathrm{F}} \leq \|M^*_{k+1} - M^*_k\|_{\mathrm{F}} \nonumber \\
= & \left\|S_{\lambda} \left\{m_1^{-1} \sum_{t = 1}^{m_1} \left[(Y(t))_{\Omega(t)} + (M_k)_{\overline{\Omega(t)}}\right]\right\} - S_{\lambda} \left\{m_1^{-1} \sum_{t = 1}^{m_1} \left[(Y(t))_{\Omega(t)} + (M_{k-1})_{\overline{\Omega(t)}}\right]\right\}\right\|_{\mathrm{F}} \nonumber \\
\leq & \left\|\left\{m_1^{-1} \sum_{t = 1}^{m_1} \left[(Y(t))_{\Omega(t)} + (M_k)_{\overline{\Omega(t)}}\right]\right\} - \left\{m_1^{-1} \sum_{t = 1}^{m_1} \left[(Y(t))_{\Omega(t)} + (M_{k-1})_{\overline{\Omega(t)}}\right]\right\}\right\|_{\mathrm{F}} \nonumber \\
= & \left\|m_1^{-1} \sum_{t = 1}^{m_1} (M_k - M_{k-1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}} \leq \|M_k - M_{k-1}\|_{\mathrm{F}}. \label{eq-pf-lem-1-klopp-1}
\end{align}
This implies that the sequence $\{\|M_k - M_{k-1}\|_{\mathrm{F}}\}_{k \in \mathbb{N}^*}$ converges. It remains to show that this sequence converges to zero.
Note that \eqref{eq-pf-lem-1-klopp-1} imply that
\[
\|M_k - M_{k-1}\|_{\mathrm{F}}^2 - \left\|m_1^{-1} \sum_{t = 1}^{m_1} (M_k - M_{k-1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 = \left\|m_1^{-1} \sum_{t = 1}^{m_1} (M_k - M_{k-1})_{\Omega(t)}\right\|_{\mathrm{F}}^2 \to 0,
\]
as $k \to \infty$. We therefore only need to show that
\begin{equation}\label{eq-lemma1-proof-sufficient-cond}
\left\|m_1^{-1} \sum_{t = 1}^{m_1} (M_k - M_{k-1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}} \to 0, \quad \mbox{as } k \to \infty.
\end{equation}
Define
\[
h(A, B) = \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(Y(t) - B)_{\Omega(t)}\right\|_{\mathrm{F}}^2 + \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(A - B)_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 + \lambda \|B\|_*.
\]
Due to the definition of $M^*_{k+1}$, we have that
\begin{align*}
& h(M_k, M^*_k) = \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(Y(t) - M^*_k)_{\Omega(t)}\right\|_{\mathrm{F}}^2 + \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(M_k - M^*_k)_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 + \lambda \|M^*_k\|_* \\
\geq & \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(Y(t) - M^*_{k+1})_{\Omega(t)}\right\|_{\mathrm{F}}^2 + \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(M_k - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 + \lambda \|M^*_{k+1}\|_* \\
= & h(M_k, M^*_{k+1}) \geq \frac{1}{2m_1} \sum_{t = 1}^{m_1} \left\|(Y(t) - M^*_{k+1})_{\Omega(t)}\right\|_{\mathrm{F}}^2 + \frac{1}{2m_1}\sum_{t = 1}^{m_1}\left\|(M_{k+1} - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 + \lambda \|M^*_{k+1}\|_* \\
= & h(M_{k+1}, M^*_{k+1}),
\end{align*}
which shows that the sequence $\{h(M_k, M^*_k)\}_{k \geq 1}$ converges. Therefore,
\begin{align*}
& h(M_k, M^*_{k+1}) - h(M_{k+1}, M^*_{k+1}) \\
= & \frac{1}{2m_1}\sum_{t = 1}^{m_1}\left\|(M_k - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 - \frac{1}{2m_1}\sum_{t = 1}^{m_1}\left\|(M_{k+1} - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 \to 0.
\end{align*}
Since
\begin{align*}
& \frac{1}{2m_1}\sum_{t = 1}^{m_1}\left\|(M_k - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 - \frac{1}{2m_1}\sum_{t = 1}^{m_1}\left\|(M_{k+1} - M^*_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2 \geq \frac{1}{2m_1}\sum_{t = 1}^{m_1} \left\|(M_k - M_{k+1})_{\overline{\Omega(t)}}\right\|_{\mathrm{F}}^2,
\end{align*}
combining with \eqref{eq-lemma1-proof-sufficient-cond}, we complete the proof.
\end{proof}
\section[]{Proof of \Cref{thm-upper-bound}}
In this section, we provide the proof of \Cref{thm-upper-bound}, starting with introducing extra notation needed for the proofs.
\subsection{Notation}
For notational simplicity, in the proof, we adopt an equivalence notation system. Let the observations be $\{Y_{\Omega}(t), \Omega(t)\}_{t \in \mathbb{N}^*}$, and for each $(i, j) \in \{1, \ldots, n\}^{\otimes 2}$
\[
Y_{\Omega, ij}(t) = \Omega_{ij}(t)\{P_{ij}(t) + \xi_{ij}(t)\},
\]
where $P_{ij}(t) = \{X_i(t)\}^{\top}\{X_j(t)\}$ and the noise $\xi_{ij}(t)$ satisfies that
\begin{align*}
\mathbb{P}\big(\xi_{ij}(t) = 1 - P_{ij}(t)|\{X_i(t), X_j(t)\}\big) = 1 - \mathbb{P}\big(\xi_{ij}(t) = -P_{ij}(t)|\{X_i(t), X_j(t)\}\big) = P_{ij}(t).
\end{align*}
For any $(i, j) \in \{1, \ldots, n\}^{\otimes 2}$, let $E_{ij} = e_i e_j^{\top}$ be a matrix basis and
\begin{equation} \label{eq-Sigma-mat-def}
\Sigma(t) = \sum_{(i, j)} \Omega_{ij}(t)\big\{P_{ij}(t) - \mathbb E[P_{ij}(t)] + \xi_{ij}(t)\big\} E_{ij}.
\end{equation}
and
\begin{equation} \label{eq-Sigma-prime-maat-def}
\Sigma_{R}(t) = \sum_{(i,j)}\big\{\Omega_{ij}(t) - \mathbb{E}[\Omega_{ij}(t)]\big\}E_{ij}.
\end{equation}
Note that under Assumption~\ref{assume-model-new}, we have that conditional on $\{X_i(t)\}_{i \in \{1, \ldots, n\}, \, t \in \mathbb N^*}$, the sequence $\{\Omega(t)\}_{t \in \mathbb N^*}$ is independent of the sequence $\{\xi(t)\}_{t \in \mathbb N^*}$. Thus, we have that
\[
\mathbb E[\Sigma(t)] = \mathbb E\left[\mathbb E_{|\{X_i(t)\}_{i \in \{1, \ldots, n\}, \, t \in \mathbb N^*}}[\Sigma(t)]\right] = 0_{n \times n}.
\]
For any linear vector subspace $S \subset \mathbb{R}^n$, let $P_S$ be the projector onto $S$ and $S^{\perp}$ be the orthogonal complement of $S$. For any matrix $A$, let $\{u_j(A)\}$ and $\{v_j(A)\}$ be the left and right orthonormal singular vectors of $A$, respectively. Let $S_1(A) = \mathrm{span}\{u_j(A)\}$ and $S_2(A) = \mathrm{span}\{v_j(A)\}$. For any matrix $B$, let
\begin{equation}\label{eq-mathbfP-def}
\mathbf{P}_A(B)^{\perp} = P_{S_1^{\perp}(A)} BP_{S_2^{\perp}(A)} \quad \mbox{and} \quad \mathbf{P}_A(B) = B - \mathbf{P}_A(B)^{\perp}.
\end{equation}
For any matrix $A \in \mathbb{R}^{n \times n}$, define the weighted Frobenius norm of $A$ as
\begin{equation}\label{eq-weightedFrob-def}
\|A\|^2_{L_2(\Pi)} = \sum_{(i, j)} \Pi_{ij} A_{ij}^2,
\end{equation}
where the weights are given by the determinestic matrix $\Pi = (\Pi_{ij})_{(i,j) \in \{1, \ldots, n\}^{\otimes 2}}$.
For the latent positions each $t \in \mathbb N^*$, we denote $X_1^n(t) = \{X_i(t)\}_{i = 1}^n$ for $t \in \mathbb N^*$ for brevity.
\subsection{Estimation bounds}
We collect results on the estimation error bounds on the soft-impute estimators in this subsection. The results here are based on their counterparts in \cite{klopp2015matrix}. Note that \cite{klopp2015matrix} is only concerned with one matrix observation, while we deal with a multiple-copy version and allow for temporal and cross-sectional dependence.
\iffalse
{\color{blue}
\begin{proposition}
Assume that Assumptions~\ref{assume-model-new} and \ref{assume-no-change} hold. Denote $\Theta(t) = \Theta_0$, $t \in \mathbb{N}^*$. For any integer pair $0 \leq s < e$, define $\Sigma_{s:e} = (e-s)^{-1} \sum_{t = s+1}^e \Sigma(t)$, where $\Sigma(t)$ is defined in \eqref{eq-Sigma-mat-def}. Let $\widehat{\Theta}_{s:e}$ be the output of \Cref{alg-main-klopp} with
\begin{equation}\label{eq-prop-3-lambda-cond}
\lambda \geq 3\|\Sigma_{s:e}\|_{\mathrm{op}},
\end{equation}
initialiser $\widetilde{M}$ and \textcolor{red}{$a = \rho$}.
With probability at least $1 - \delta$, it holds that
\begin{align*}
\|\widehat{M}_{s:e} - M_0\|_{\mathrm{F}}^2 \leq \frac{C}{p^2} \Bigg\{\mathrm{rank}(M_0)\left[\lambda^2 + \rho^2 \left\{\mathbb{E}(\|\Sigma_{R, s:e}\|_{\mathrm{op}})\right\}^2\right] + \frac{\rho^2}{e-s} + \frac{\log(1/\delta)}{e-s}\Bigg\},
\end{align*}
where $\Sigma_{R, s:e} = (e-s)^{-1} \sum_{t = s+1}^e \Sigma_R(t)$ and $C > 0$ is an absolute constant.
\end{proposition}
}
\fi
\begin{proposition} \label{thm-main-klopp-cor-3}
Assume that Assumptions~\ref{assume-model-new} and \ref{assume-no-change} hold. Denote $\Theta(t) = \Theta_0$ for $t \in \mathbb{N}^*$. For any integer pair $0 \leq s < e$, let $\Sigma_{s:e} = (e-s)^{-1} \sum_{t = s+1}^e \Sigma(t)$, where $\Sigma(t)$ is defined in \eqref{eq-Sigma-mat-def}. Let $\widehat{\Theta}_{s:e}$ be the output of \Cref{alg-main-klopp} with
\begin{equation}\label{eq-prop-3-lambda-cond}
\lambda \geq 3\|\Sigma_{s:e}\|_{\mathrm{op}},
\end{equation}
$a = \vartheta_2$ given in Assumption~\ref{assume-model-new} and a generic initialiser $\widehat{\Theta} \in \{0,1\}^{n \times n}$. Then, with probability at least $1 - \delta$, it holds that
\begin{align*}
\|\widehat{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2 \leq \frac{C}{q_1^2} \Bigg\{\mathrm{rank}(\Theta_0)\left(\lambda^2 + \vartheta_{2}^2 \left\{\mathbb{E}\left[\|\Sigma_{R, s:e}\|_{\mathrm{op}}\right]\right\}^2\right) + \frac{\log(1/\delta)}{e-s}\Bigg\},
\end{align*}
where $\Sigma_{R,s:e} = (e-s)^{-1}\sum_{t = s+1}^e\Sigma_R(t)$ and $\Sigma_R(t)$ is defined in \eqref{eq-Sigma-prime-maat-def}, and $C > 0$ is an absolute constant.
\end{proposition}
\begin{proof}[Proof of \Cref{thm-main-klopp-cor-3}]
Since $\|\widehat{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2 \leq \|\widetilde{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2$, it suffice to show that
\begin{align*}
\|\widetilde{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}}^2 \leq \frac{C}{q_1^2} \Bigg\{\mathrm{rank}(\Theta_0)\left(\lambda^2 + \vartheta_{2}^2 \left\{\mathbb{E}\left[\|\Sigma_{R, s:e}\|_{\mathrm{op}}\right]\right\}^2\right) + \frac{\log(1/\delta)}{e-s}\Bigg\}.
\end{align*}
Recall that
\begin{align*}
\widetilde{\Theta}_{s:e} & \in \argmin_{\Theta \in \mathbb{R}^{n \times n}} f_{\lambda, s:e} = \argmin_{\Theta \in \mathbb{R}^{n \times n}} f(\Theta; \{Y_{\Omega}(t), \Omega(t)\}_{t = s+1}^e, \widehat{\Theta}, \lambda) \\
& = \argmin_{\Theta \in \mathbb{R}^{n \times n}} \left\{\frac{1}{2(e-s)} \sum_{t = s+1}^e \left\|Y_{\Omega}(t) + \widehat{\Theta}_{\overline{\Omega(t)}} - \Theta\right\|_{\mathrm{F}}^2 + \lambda\|\Theta\|_*\right\}.
\end{align*}
\noindent \textbf{Step 1.} It follows from \Cref{lem-soft-impute-property} that $\widetilde{\Theta}_{s:e}$ minimises $f_{\lambda, s:e}(\Theta)$. Using the sub-gradient stationary conditions we have that
\begin{equation}\label{eq-proof-prop3-step1-1}
- \Bigg\langle \frac{1}{e-s} \sum_{t = s+1}^e \left\{Y_{\Omega}(t) + \widehat{\Theta}_{\overline{\Omega(t)}}\right\} - \widetilde{\Theta}_{s:e}, \, \widetilde{\Theta}_{s:e} - \Theta_0\Bigg\rangle + \lambda \left\langle \widehat{V}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \leq 0,
\end{equation}
where $\widehat{V} \in \partial \left\|\widehat{\Theta}_{s:e}\right\|_*$. We have that
\begin{align}
& \sum_{t = s+1}^e \left\langle \left(Y(t) - \Theta_0\right)_{\Omega(t)}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle + \sum_{t = s+1}^e \left\langle \left(\widehat{\Theta} - \widetilde{\Theta}_{s:e}\right)_{\overline{\Omega(t)}}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \nonumber\\
&+ \lambda (e-s) \left\langle \widehat{V}, \, \Theta_0 - \widetilde{\Theta}_{s:e}\right\rangle \nonumber \\
= & \sum_{t = s+1}^e \left\langle \left(Y(t) - \Theta_0\right)_{\Omega(t)} + \left(\widehat{\Theta} - \widetilde{\Theta}_{s:e}\right)_{\overline{\Omega(t)}}, \, \widetilde{\Theta}_{s:e} - \Theta_0 \right\rangle - \lambda (e-s) \left\langle \widehat{V}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \nonumber \\
\geq & \sum_{t = s+1}^e \left\langle \left(Y(t) - \Theta_0\right)_{\Omega(t)} + \left(\widehat{\Theta} - \widetilde{\Theta}_{s:e}\right)_{\overline{\Omega(t)}}, \, \widetilde{\Theta}_{s:e} - \Theta_0 \right\rangle \nonumber \\
& \hspace{1cm} - \sum_{t = s+1}^e \left\langle \left\{Y_{\Omega}(t) + \widehat{\Theta}_{\overline{\Omega(t)}}\right\} - \widetilde{\Theta}_{s:e}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \nonumber \\
= & \sum_{t = s+1}^e \left\langle\left (\widetilde{\Theta}_{s:e} - \Theta_0\right)_{\Omega(t)}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle = \sum_{t = s+1}^e \left\|\left(\widetilde{\Theta}_{s:e} - \Theta_0\right)_{\Omega(t)}\right\|_{\mathrm{F}}^2, \label{eq-proof-prop3-step1-2}
\end{align}
where the inequality follows from \eqref{eq-proof-prop3-step1-1} and the second identity follows from the fact that $Y_{\Omega}(t)$'s are only observed on $\Omega(t)$'s.
\\
Equation~\eqref{eq-proof-prop3-step1-2} implies that
\begin{align}
& \frac{1}{e-s} \sum_{t = s+1}^e \left\|\left(\widetilde{\Theta}_{s:e} - \Theta_0\right)_{\Omega(t)}\right\|_{\mathrm{F}}^2 \nonumber \\
\leq & \left|\frac{1}{e-s} \sum_{t = s+1}^e \left\langle (Y(t) - \Theta_0)_{\Omega(t)}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \right|\label{eq-decomp-1} \\
& + \left|\frac{1}{e-s} \sum_{t = s+1}^e \left\langle \left(\widehat{\Theta} - \widetilde{\Theta}_{s:e}\right)_{\overline{\Omega(t)}}, \, \widetilde{\Theta}_{s:e} - \Theta_0\right\rangle \right|\label{eq-decomp-2} \\
& + \lambda \left\langle \widehat{V}, \, \Theta_0 - \widetilde{\Theta}_{s:e}\right\rangle. \label{eq-decomp-3}
\end{align}
\\
For term \eqref{eq-decomp-1}, recalling the definition of $\Sigma(t)$ in \eqref{eq-Sigma-mat-def}, we have that
\begin{align}
\eqref{eq-decomp-1} =& \left|\Bigg\langle \frac{1}{e-s} \sum_{t = s+1}^e \Sigma(t), \, \widetilde{\Theta}_{s:e} - \Theta_0 \Bigg\rangle\right| \leq \left\|\frac{1}{e-s} \sum_{t = s+1}^e \Sigma(t)\right\|_{\mathrm{op}} \left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_* \nonumber \\
\leq & \frac{\lambda}{3} \left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_*, \label{eq-decomp-1-2}
\end{align}
due to the condition on $\lambda$ in \eqref{eq-prop-3-lambda-cond}.
\\
For term \eqref{eq-decomp-2}, we have that
\begin{align} \label{eq-decomp-2-2}
\eqref{eq-decomp-2} \leq \left\|\frac{1}{e-s} \sum_{t = s+1}^e \left(\widehat{\Theta} - \widetilde{\Theta}_{s:e}\right)_{\overline{\Omega(t)}}\right\|_{\mathrm{op}} \left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_* \leq \frac{\lambda}{3} \left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_*,
\end{align}
due to the stopping rule of \Cref{alg-main-klopp}.
\\
For term \eqref{eq-decomp-3}, due to the monotonicity of sub-differentials of convex functions, we have that for any $V \in \partial \|\Theta_0\|_*$,
\[
\left\langle \widehat{V}, \Theta_0 - \widetilde{\Theta}_{s:e} \right\rangle \leq \left\langle V, \Theta_0 - \widetilde{\Theta}_{s:e} \right\rangle.
\]
We then have that
\begin{align} \label{eq-decomp-3-2}
\eqref{eq-decomp-3} \leq \lambda \left(\left\|\mathbf{P}_{\Theta_0} (\Theta_0 - \widetilde{\Theta}_{s:e})\right\|_* - \left\|\mathbf{P}_{\Theta_0} (\widetilde{\Theta}_{s:e})^{\perp}\right\|_*\right).
\end{align}
\\
Combining \eqref{eq-decomp-1-2}, \eqref{eq-decomp-2-2} and \eqref{eq-decomp-3-2}, recalling the notation introduced in \eqref{eq-mathbfP-def}, we have that
\begin{align}\label{eq-frobenius-nuclear}
\frac{1}{e-s} \sum_{t = s+1}^e \left\|\left(\widetilde{\Theta}_{s:e} - \Theta_0\right)_{\Omega(t)}\right\|_{\mathrm{F}}^2 \leq \frac{5\lambda}{3} \left\|\mathbf{P}_{\Theta_0}(\Theta_0 - \widetilde{\Theta}_{s:e})\right\|_* \leq \frac{5\lambda}{3} \sqrt{2\mathrm{rank}(\Theta_0)} \left\|\Theta_0 - \widetilde{\Theta}_{s:e}\right\|_{\mathrm{F}}
\end{align}
and
\[
\frac{\lambda}{3}\left\|\mathbf{P}_{\Theta_0}(\widetilde{\Theta}_{s:e})^{\perp}\right\|_* \leq \frac{5\lambda}{3} \left\|\mathbf{P}_{\Theta_0}(\Theta_0 - \widetilde{\Theta}_{s:e})\right\|_*.
\]
We therefore have that
\[
\left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_* \leq 6\left\|\mathbf{P}_{\Theta_0}(\Theta_0 - \widetilde{\Theta}_{s:e})\right\|_* \leq \sqrt{72 \mathrm{rank}(\Theta_0)} \left\|\widetilde{\Theta}_{s:e} - \Theta_0\right\|_{\mathrm{F}}.
\]
\noindent \textbf{Step 2.} For any matrix $A \in \mathbb{R}^{n \times n}$, recall the notation given in \eqref{eq-weightedFrob-def}
\[
\|A\|^2_{L_2(\Pi), s:e} = \frac{1}{e-s} \sum_{t = s+1}^e \sum_{(i, j)} \Pi_{ij}(t) A_{ij}^2 \geq q_1 \|A\|^2_{\mathrm{F}}.
\]
For absolute constants $C, c_1 > 0$, let
\[
Q_{s:e} = \frac{C\log(1/\delta)}{c_1q_1 (e-s) \log(6/5)}.
\]
Note that $\|\widetilde{\Theta}_{s:e} - \widehat{\Theta}\|_{\infty} \leq \vartheta_{2}$ and $\|\widehat{\Theta}\|_{\infty} \leq \vartheta_{2}$, which implies that $\|\widetilde{\Theta}_{s:e} - \Theta_0\|_{\infty} \leq 3\vartheta_{2}$. We now consider two cases.
\noindent \textbf{Step 2.1.} If $\|\widetilde{\Theta}_{s:e} - \Theta_0\|^2_{L_2(\Pi), s:e} < Q_{s:e}$, then we complete the result.
\noindent \textbf{Step 2.2.} If $\|\widetilde{\Theta}_{s:e} - \Theta_0\|^2_{L_2(\Pi), s:e} \geq Q_{s:e}$, then we consider the following set
\[
\mathcal{C}(r, e-s) = \Big\{A \in \mathbb{R}^{n \times n}: \, \|A\|_{\infty} = 1, \, \|A\|_{L_2(\Pi), s:e}^2 \geq \frac{C\log(1/\delta)}{c_1q_1 (e-s) \log(6/5)}, \, \|A\|_* \leq \sqrt{r}\|A\|_{\mathrm{F}}\Big\}.
\]
We then have that
\[
(3\vartheta_{2})^{-1}(\widetilde{\Theta}_{s:e} - \Theta_0) \in \mathcal{C}(72\mathrm{rank}(\Theta_0), e-s).
\]
Due to \Cref{lem-9-klopp_dep}, with probability at least $1 - \delta$, one has that
\begin{align*}
& \|\widetilde{\Theta}_{s:e} - \Theta_0\|_{L_2(\Pi), s:e}^2 \leq \frac{2}{e-s} \sum_{t = s+1}^e \|(\widetilde{\Theta}_{s:e} - \Theta_0)_{\Omega(t)}\|_{\mathrm{F}}^2 + C\frac{\vartheta_2^2}{q_1}\mathrm{rank}(\Theta_0)\left\{\mathbb{E}(\|\Sigma_{R, s:e}\|_{\mathrm{op}})\right\}^2\\
\leq & \frac{10\lambda}{3} \sqrt{2\mathrm{rank}(\Theta_0)} \|\Theta_0 - \widetilde{\Theta}_{s:e}\|_{\mathrm{F}} + C\frac{\vartheta_{2}^2}{q_1}\mathrm{rank}(\Theta_0)\left\{\mathbb{E}(\|\Sigma_{R, s:e}\|_{\mathrm{op}})\right\}^2\\
\leq & \frac{C\lambda^2}{q_1} \mathrm{rank}(\Theta_0) + Cq_1 \|\Theta_0 - \widetilde{\Theta}_{s:e}\|_{\mathrm{F}}^2 + C\frac{\vartheta_{2}^2}{q_1}\mathrm{rank}(\Theta_0)\left\{\mathbb{E}(\|\Sigma_{R,s:e}\|_{\mathrm{op}})\right\}^2,
\end{align*}
where the second inequality follows from \eqref{eq-frobenius-nuclear}.
\\
Therefore we have that
\[
\|\widetilde{\Theta}_{s:e} - \Theta_0\|_{L_2(\Pi)}^2 \leq \frac{C}{q_1} \mathrm{rank}(\Theta_0)\left[\lambda^2 + \vartheta_{2}^2 \left\{\mathbb{E}(\|\Sigma_{R, s:e}\|_{\mathrm{op}})\right\}^2\right].
\]
The final result holds due to the fact that $\Pi_{ij}(t) \geq q_1$, for any $(i, j) \in \{1, \ldots, n\}^{\otimes 2}$ and $t \in \mathbb{N}^*$.
\end{proof}
\subsection[]{Proof of \Cref{cor-main-klopp-cor-3}}
\begin{proof}[Proof of \Cref{cor-main-klopp-cor-3}]
This is an immediate consequence of \Cref{thm-main-klopp-cor-3} and \Cref{lem-lem4-klopp_dep}.
\end{proof}
\subsection{Change point analysis}
\begin{lemma}\label{lem-large-prob-event-2}
Under all the conditions and notation in \Cref{cor-main-klopp-cor-3}, define
\begin{equation}\label{eq-event-G-def}
\mathcal{E} = \left\{\forall s, t \in \mathbb{N}^*, \, t > s, \, t \geq 2: \, \|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} < \zeta_{s, t}\right\},
\end{equation}
where
\begin{align*}
\zeta_{s, t} = \sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/{\delta}_t)}{q_1^2(t-s)}}\sqrt{\max\left\{\log(t-s),\log(n/\delta_t)\right\}},
\end{align*}
with $\alpha \in (0, 1)$,
\begin{equation}\label{eq-lemma-4-delta-t-def}
\delta_t = \frac{\alpha}{24 t^2}
\end{equation}
and $C_{\mathrm{noise}} > 0$ being a sufficiently large absolute constant. It holds that
\[
\mathbb{P}\{\mathcal{E}\} \geq 1 - \alpha.
\]
\end{lemma}
\begin{proof}[Proof of \Cref{lem-large-prob-event-2}]
Let $\mathcal{E}^c$ be the complement of $\mathcal{E}$. We thus have that
\begin{align}
\mathbb{P}\{\mathcal{E}^c\} & = \mathbb{P}\left\{\exists s, t \in \mathbb{N}, \, t > s, \, t \geq 2 : \, \|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} \geq \zeta_{s, t}\right\} \nonumber \\
& \leq \sum_{j = 1}^{\infty} \mathbb{P}\left\{\max_{2^j \leq t < 2^{j+1}} \max_{0 \leq s < t} \left\{\|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} - \zeta_{s, t}\right\} \geq 0\right\} \nonumber \\
& \leq \sum_{j = 1}^{\infty} 2^j \max_{2^j \leq t < 2^{j+1}} \mathbb{P}\left\{ \max_{0 \leq s < t} \left\{\|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} - \zeta_{s, t} \right\} \geq 0 \right\} \nonumber \\
& \leq \sum_{j = 1}^{\infty} 2^{2j+1} \max_{2^j \leq t < 2^{j+1}} \max_{0 \leq s < t} \mathbb{P}\left\{\|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} \geq \zeta_{s, t}\right\}. \label{eq-lemma-s4-pf-1}
\end{align}
Due to \Cref{cor-main-klopp-cor-3}, we have that for any integer pair $1 \leq s < t$ and any $\delta > 0$,
\begin{align*}
&\mathbb{P}\Bigg(\|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} \geq \sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/\delta)}{q_1^2(t-s)}}\sqrt{\max\left\{\log(t-s),\log(n/\delta)\right\}}\Bigg)\\
&< 3\delta
\end{align*}
Choosing
\begin{align*}
\widetilde{\zeta}_{s, t} = \sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/\widetilde{\delta}_t)}{q_1^2(t-s)}}\sqrt{\max\left\{\log(t-s),\log(n/\widetilde\delta_t)\right\}},
\end{align*}
and
\[\widetilde{\delta}_t = \frac{\alpha \log^2(2)}{6 \{\log(t) + \log(2)\}^2 t^2},
\]
due to \eqref{eq-lemma-s4-pf-1}, we have that
\begin{align*}
\mathbb{P}\{\mathcal{E}^c\} & \leq \sum_{j = 1}^{\infty} 2^{2j+1} \max_{2^j \leq t < 2^{j+1}} \max_{0 \leq s < t} \mathbb{P}\left\{\|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} \geq \widetilde{\zeta}_{s, t}\right\}\\
& \leq \sum_{j = 1}^{\infty} 2^{2j+1} \max_{2^j \leq t < 2^{j+1}} \max_{0 \leq s < t} \frac{\alpha \log^2(2)}{2 \{\log(t) + \log(2)\}^2 t^2} \\
& \leq \alpha \sum_{j = 1}^{\infty} 2^{2j+1} \frac{1}{2\left\{\frac{\log(2^j)}{\log(2)} + 1\right\}^2 2^{2j}} \leq \alpha \sum_{j = 1}^{\infty} \frac{1}{(j+1)^2} \leq \alpha \sum_{j = 1}^{\infty} \frac{1}{j(j+1)} = \alpha,
\end{align*}
where the first inequality holds since $\delta_t \leq \widetilde{\delta}_t$, with $t \geq 2$, $\delta_t$ defined in \eqref{eq-lemma-4-delta-t-def}.
\end{proof}
\subsection[]{Proof of \Cref{thm-upper-bound}}
\begin{proof}[Proof of \Cref{thm-upper-bound}]
The proof is conducted in the event $\mathcal{E}$, with the event $\mathcal{E}$ defined as
\[
\mathcal{E} = \left\{\forall s, t \in \mathbb{N}^*, \, t > s, \, t \geq 2: \, \|\widehat{\Theta}_{s:t} - \Theta_0\|_{\mathrm{F}} < \zeta_{s, t}\right\}.
\]
It follows from \Cref{lem-large-prob-event-2} that $\mathbb{P}\{\mathcal{E}\} > 1 - \alpha$.
\begin{align*}
\|\widehat{\Theta}_{s:e} - \Theta_0\|_{\mathrm{F}} \leq \sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(1/\delta)}{q_1^2(e-s)}}\sqrt{\max\left\{\log(e-s),\log(n/\delta)\right\}},
\end{align*}
For any $t \leq \Delta$, we have that
\begin{align*}
& \|\widehat{\Theta}_{0:s} - \widehat{\Theta}_{s:t}\|_{\mathrm{F}} \leq \|\widehat{\Theta}_{0:s} - \Theta_{\Delta}\|_{\mathrm{F}} + \|\widehat{\Theta}_{s:t} - \Theta_{\Delta}\|_{\mathrm{F}} \\
\leq & \left(\sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(32s^2/\alpha)}{q_1^2s}} + \sqrt{\frac{C_{\mathrm{noise}}rnq_2\max\{\vartheta_2^2, (1-\vartheta_1)^2\}\log(32t^2/\alpha)}{q_1^2(t-s)}}\right)\\
&\times \sqrt{\max\big\{\log s, \log(t-s), \log(32nt^2/\alpha)\big\}}\\
\leq & C_{\varepsilon} \frac{\sqrt{ rnq_2}\max\{\vartheta_2, 1-\vartheta_1\}}{q_1}\left(\sqrt{\frac{\log(s/\alpha)}{s}} + \sqrt{\frac{\log(t/\alpha)}{t-s}}\right)\sqrt{\log(nt/\alpha)}\\
=& \varepsilon_{s, t},
\end{align*}
where $C_{\varepsilon} > 0$ is a large enough absolute constant depending only on $C_{\mathrm{noise}}$. We thus have that $\widehat{\Delta} > t$ and consequently $\widehat{\Delta} > \Delta$.
Now we consider $t > \Delta$. Let
\[
\widetilde{\Delta} = \min\{t > \Delta: \, \|\widehat{\Theta}_{0: \Delta} - \widehat{\Theta}_{\Delta: t}\|_{\mathrm{F}}^2 \geq \varepsilon_{\Delta, t}\}.
\]
From the definition of $\widehat{\Delta}$, it holds that $\widetilde{\Delta} \geq \widehat{\Delta}$. It now suffices to upper bound $(\widetilde{\Delta} - \Delta)_+$.
On the event $\mathcal{E}$, we have that
\begin{align}
& \|\widehat{\Theta}_{0:\Delta} - \widehat{\Theta}_{\Delta: \widetilde{\Delta}}\|_{\mathrm{F}} = \|\widehat{\Theta}_{0:\Delta} - \Theta_{\Delta} + \Theta_{\Delta} - \Theta_{\Delta + 1} - \Theta_{\Delta+1} - \widehat{\Theta}_{\Delta: \widetilde{\Delta}}\|_{\mathrm{F}} \nonumber \\
\geq & \|\Theta_{\Delta} - \Theta_{\Delta+1}\|_{\mathrm{F}} - \left(\|\widehat{\Theta}_{0:\Delta} - \Theta_{\Delta}\|_{\mathrm{F}} + \|\widehat{\Theta}_{\Delta: \widetilde{\Delta}} - \Theta_{\Delta+1}\|_{\mathrm{F}} \right) \nonumber \\
\geq & \kappa - C_{\varepsilon} \frac{\sqrt{r nq_2}\max\{\vartheta_2, 1-\vartheta_1\}}{q_1}\Bigg(\sqrt{\frac{\log(\Delta/\alpha)}{\Delta}} + \sqrt{\frac{\log\{(d+\Delta)/\alpha\}}{d}}\Bigg)\sqrt{\log(n(d+\Delta)/\alpha)}, \label{eq-proof-main-11}
\end{align}
where we let $d = \widetilde{\Delta} - \Delta$ in the last inequality.
It now suffices to show that with the choice of
\begin{align*}
d = C_d \frac{rnq_2 \max\{\vartheta_2^2, (1-\vartheta_1)^2\}}{q_1^{2}\kappa^2}\big(\log(\Delta/\alpha)\big)^{2},
\end{align*}
where $C_d > 0$ is a large enough absolute constant depending only on $C_{\varepsilon}$, it holds that $\eqref{eq-proof-main-11} \geq \varepsilon_{\Delta, \Delta + d}$, which is equivalent to
\[
\kappa \geq 2\varepsilon_{\Delta, \Delta + d}.
\]
Due to Assumption~\ref{assume-snr}, i.e.~\begin{align*}
\kappa^2 \Delta \geq C_{\mathrm{SNR}}\frac{rnq_2 \max\{\vartheta_2^2, (1-\vartheta_1)^2\}}{q_1^{2}}\big(\log(\Delta/\alpha)\big)^{2},
\end{align*}
where $C_{\mathrm{SNR}} \geq C_d$ is a large enough absolute constant, we have that $\Delta > d$ and $\Delta > n$, which leads to
\begin{align*}
&2\varepsilon_{\Delta, \Delta+d}\\
=& 2C_{\varepsilon}\frac{\sqrt{rnq_2}\max\{\vartheta_2, 1-\vartheta_1\}}{q_1} \left(\sqrt{\frac{\log(\Delta/\alpha)}{\Delta}} + \sqrt{\frac{\log(\Delta+d/\alpha)}{d}}\right)\sqrt{\log(n(d+\Delta)/\alpha)}\\
\leq& 2C_{\varepsilon}^{\prime}\frac{\sqrt{rnq_2}\max\{\vartheta_2, 1-\vartheta_1\}}{q_1} \left(\sqrt{\frac{\log(\Delta/\alpha)}{\Delta}} + \sqrt{\frac{\log(\Delta+d/\alpha)}{d}}\right)\sqrt{\log (\Delta /\alpha)}\\
\leq& \kappa.
\end{align*}
\end{proof}
\section[]{Proof of \Cref{lemma:ergodic}}
\begin{proof}[Proof of \Cref{lemma:ergodic}]
Note that for any $i \in \{1, \ldots, n\}$, $\{X_i(t)\}_{t \in \mathbb{N}^*}$ is a Markov chain with state space $\mathcal{X}$. By Theorem 3.7 in \cite{bradley2005basic}, it suffices to verify Doeblin’s condition \citep[e.g.][]{rosenthal1995convergence}, which holds on the transition probability $\mathbb P(\cdot, \cdot)$, if there exists an $\epsilon > 0$ and a probability measure $\mu(\cdot)$, such that for all $x \in \mathcal{X}$ and measurable subsets $A \subseteq \mathcal{X}$,
$\mathbb P(x, A) \geq \epsilon\mu(A)$.
We write \eqref{eq:markov_latentpostion} equivalently as
\[
X_i(t) = (1-U)\cdot X_i(t-1) + U\cdot Z,
\]
where $U \sim \mathrm{Bernoulli}(1-\rho)$, $Z \sim F$, and $U$ is independent of $X_i(t-1)$ and $Z$. We have for any $A \subseteq \mathcal{X}$
\begin{align*}
\mathbb P(x, A) =& \mathbb P(X_i(t) \in A| X_i(t-1) = x)\\
=& \mathbb P(X_i(t) \in A, U = 1| X_i(t-1) = x)\\
&+ \mathbb P(X_i(t) \in A, U = 0| X_i(t-1) = x)\\
\geq& \mathbb P(X_i(t) \in A, U = 1| X_i(t-1) = x)\\
=& \mathbb P(Z \in A, U = 1)\\
=& (1-\rho)\mathbb P(Z \in A).
\end{align*}
Thus, we have $\epsilon = 1-\rho > 0$, which concludes the proof.
\end{proof}
\section{Auxiliary results}
For $r \in \mathbb{N}^*$ and integer pair $1 \leq s < e$, define the set
\begin{align}\label{eq-def-c-r-es}
\mathcal{C}(r, e-s) = \Big\{A \in \mathbb{R}^{n \times n}: \, \|A\|_{\infty} = 1, \, \|A\|_{L_2(\Pi), s:e}^2 \geq \frac{C\log(1/\delta)}{c_1q_1 (e-s) \log(6/5)}, \, \|A\|_* \leq \sqrt{r}\|A\|_{\mathrm{F}}\Big\},
\end{align}
where for any matrix $A \in \mathbb{R}^{n \times n}$, the norm $\|A\|_{L_2(\Pi), s:e}$ is defined in \eqref{eq-weightedFrob-def}
and $\{\Pi(t)\}_{t \in \mathbb N^*}$ is a sequence of missingness probability matrices. Recall that
\[
\Sigma_{R,s:e} = \frac{1}{e-s}\sum_{t = s+1}^e\Sigma_R(t),
\]
where $\Sigma_R(t)$ is defined in \eqref{eq-Sigma-prime-maat-def}.
\begin{lemma}\label{lem-9-klopp_dep}
For any integer pair $1 \leq s < e$ and for any $A \in \mathcal{C}(r, e-s)$ as defined in \eqref{eq-def-c-r-es}, it holds with probability at least $1 - \delta$ that,
\[
\frac{1}{2}\|A\|^2_{L_2(\Pi), s:e} - \frac{54r}{q_1} \left\{\mathbb{E}[\|\Sigma_{R, s:e}\|_{\mathrm{op}}]\right\}^2 \leq \frac{1}{e-s} \sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2.
\]
\end{lemma}
\begin{proof}[Proof of \Cref{lem-9-klopp_dep}]
Let
\[
R = \frac{54r}{q_1}\left\{\mathbb{E}[\|\Sigma_{R, s:e}\|_{\mathrm{op}}]\right\}^2,
\]
and the event
\[
\mathcal{E} = \left\{\exists A \in \mathcal{C}(r, e-s) \,\mbox{ s.t. } \left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right| > \frac{1}{2} \|A\|^2_{L_2(\Pi), s:e} + R\right\}.
\]
We are to show that $\mathcal{E}$ holds with small probability.
In order to estimate the probability of $\mathcal{E}$, we use a peeling argument. Let $\alpha = 4.5$ and $v > 0$ to be specified. For $l \in \mathbb{N}^*$, set
\[
\mathcal{S}_l = \left\{A \in \mathcal{C}(r, e-s): \, \alpha^{l-1} v \leq \|A\|^2_{L_2(\Pi), s:e} \leq \alpha^l v\right\}.
\]
If the event $\mathcal{E}$ holds for some matrix $A \in \mathcal{C}(r, e-s)$, then there exists $l \in \mathbb{N}$ such that $A \in \mathcal{S}_l$ and
\begin{align*}
\left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right| > \frac{1}{2} \|A\|^2_{L_2(\Pi), s:e} + R \geq \frac{1}{2} \alpha^{l-1} v + R = \frac{1}{9} \alpha^l v + R.
\end{align*}
For $W > v$, consider the following set of matrices
\[
\mathcal{C}(r, e-s, W) = \left\{A \in \mathcal{C}(r, e-s): \, \|A\|^2_{L_2(\Pi), s:e} \leq W\right\}
\]
and the following event
\[
\mathcal{E}_l = \left\{\exists A \in \mathcal{C}(r, e-s, \alpha^l v):\, \left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right| > \frac{1}{9} \alpha^l v + R\right\}.
\]
Note that $A \in \mathcal{S}_l$ implies that $A \in \mathcal{C}(r, e-s, \alpha^l v)$, and consequently that $\mathcal{E} \subset \cup_{l \in \mathbb{N}^*} \mathcal{E}_l$. This means that it suffices to estimate the probability of $\mathcal{E}_l$ and the apply a union bound argument.
Let
\[
Z_{W, s:e} = \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right|.
\]
Due to \Cref{lem-lem10-klopp_dep}, we have that
\[
\mathbb{P}(\mathcal{E}_l) \leq 2\exp\left(-C_1q_1 (e-s) \alpha^l v\right).
\]
Using a union bound argument, we have that
\begin{align*}
\mathbb{P}(\mathcal{E}) & \leq \sum_{l = 1}^{\infty} \mathbb{P}(\mathcal{E}_l) \leq 2\sum_{l = 1}^{\infty}\exp\{-C_1q_1 (e-s) \alpha^l v\} \leq 2\sum_{l = 1}^{\infty}\exp\{-C_1q_1(e-s) \log(\alpha) l v\} \\
& \leq \frac{2\exp\{-C_1q_1(e-s) \log(\alpha) v\}}{1 - \exp\{-C_1q_1(e-s) \log(\alpha)v\}}.
\end{align*}
Letting
\[
v = \frac{\log(4/\delta)}{C_1q_1(e-s) \log(\alpha)},
\]
we have that
\[
\mathbb{P}(\mathcal{E}) \leq \delta.
\]
\end{proof}
\begin{lemma}\label{lem-lem10-klopp_dep}
Let the sequence of missingness matrices $\{\Omega(t)\}_{t \in \mathbb{N}} \subset \mathbb{R}^{n \times n}$ and the sequence of missingness probability matrices $\{\Pi(t)\}_{t \in \mathbb{N}} \subset \mathbb{R}^{n \times n}$ satisfy Assumption~\ref{assume-model-new}. For any integer pair $1 \leq s < e$ and any $W > 0$, let
\[
Z_{W, s:e} = \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right|
\]
and
\[
\mathcal{C}(r, e-s, W) = \left\{A \in \mathcal{C}(r, e-s): \, \|A\|^2_{L_2(\Pi), s:e} \leq W\right\}.
\]
We have that
\[
\mathbb{P}\left(Z_{W, e:s} \geq \frac{W}{9} + \frac{54r}{q_1}\left\{\mathbb{E}[\|\Sigma_{R, s:e}\|_{\mathrm{op}}]\right\}^2\right) \leq 2 \exp\left(-C q_1(e-s)W\right).
\]
where $C > 0$ is some absolute constant.
\end{lemma}
\begin{proof}[Proof of \Cref{lem-lem10-klopp_dep}]
\
\\
\noindent\textbf{Step 1.} We will start by showing that $Z_{W, s:e}$ concentrates around its expectation and then upper bound the expectation. Recall that
\begin{align*}
Z_{W, s:e} & = \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\frac{1}{e-s}\sum_{t = s+1}^e \|A_{\Omega(t)}\|_{\mathrm{F}}^2 - \|A\|^2_{L_2(\Pi), s:e}\right| \\
& = \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\frac{1}{e-s} \sum_{t = s+1}^e \sum_{(i, j)} \Omega_{ij}(t)A^2_{ij} - \frac{1}{e-s} \sum_{t = s+1}^e \sum_{(i, j)} \mathbb{E}[\Omega_{ij, t}] A_{ij}^2\right| \\
& = \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\frac{1}{e-s} \sum_{t = s+1}^e \sum_{(i, j)} \left\{\Omega_{ij}(t) - \mathbb{E} [\Omega_{ij}(t)]\right\}A^2_{ij} \right|.
\end{align*}
We are to apply \Cref{thm-talagrand_dep}. Let
\[
f(x_{ij,t};\; i, j = 1, \ldots, n;\; t = s+1, \ldots, e) = \frac{1}{e-s}\sup_{A \in \mathcal{C}(r, e-s, W)} \left|\sum_{t = s+1}^e \sum_{(i, j)} \{x_{ij, t} - \Pi_{ij}(t)\} A_{ij}^2\right|.
\]
We first show that $f(\cdot)$ is a Lipschitz function with the constant $\sqrt{W/\{(e-s)q_1\}}$. It follows that
\begin{align*}
& |f(x_{ij, t}) - f(z_{ij, t})| \\
= & \frac{1}{e-s} \left|\sup_{A \in \mathcal{C}(r, e-s, W)} \left|\sum_{t = s+1}^e \sum_{(i, j)} \{x_{ij, t} - \Pi_{ij}(t)\} A_{ij}^2\right| - \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\sum_{t = s+1}^e \sum_{(i, j)} \{z_{ij, t} - \Pi_{ij}(t)\} A_{ij}^2\right|\right| \\
\leq & \frac{1}{e-s} \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\left|\sum_{t = s+1}^e \sum_{(i, j)} \{x_{ij, t} - \Pi_{ij}(t)\} A_{ij}^2\right| - \left|\sum_{t = s+1}^e \sum_{(i, j)} \{z_{ij, t} - \Pi_{ij}(t)\} A_{ij}^2\right|\right| \\
\leq & \frac{1}{e-s} \sup_{A \in \mathcal{C}(r, e-s, W)} \left|\sum_{t = s+1}^e \sum_{(i, j)} \{x_{ij, t} - z_{ij}(t)\} A_{ij}^2\right| \\
\leq & \frac{1}{\sqrt{e-s}} \sup_{A \in \mathcal{C}(r, e-s, W)} \sqrt{\frac{1}{e-s}\sum_{t = s+1}^e \sum_{(i, j)} \Pi_{ij}(t) A_{ij}^4} \sqrt{\sum_{t = s+1}^e \sum_{(i, j)} (\Pi_{ij}(t))^{-1} (x_{ij, t} - z_{ij, t})^2} \\
\leq & \frac{1}{\sqrt{(e-s)q_1}}\sup_{A \in \mathcal{C}(r, e-s, W)} \sqrt{\frac{1}{e-s}\sum_{t = s+1}^e \sum_{(i, j)} \Pi_{ij}(t) A_{ij}^2} \sqrt{\sum_{t = s+1}^e \sum_{(i, j)} (x_{ij, t} - z_{ij, t})^2} \\
\leq & \sqrt{\frac{W}{(e-s)q_1}} \sqrt{\sum_{t = s+1}^e \sum_{(i, j)} (x_{ij, t} - z_{ij, t})^2}.
\end{align*}
According to \Cref{theorem:markov-phi}, the $\phi$-mixing coefficients of $\{\Omega_{ij}(t)\}_{t \in \mathbb N^*}$ satisfies that $\phi_{\Omega}^{ij}(\ell) \leq 2\rho^{\ell}$ for any $\ell \in \mathbb N$ and some $\rho \in (0,1)$. Then \Cref{thm-talagrand_dep} leads to that
\[
\mathbb{P}\left(Z_{W, e:s} \geq \mathbb{E}[Z_{W, e:s}] + t\right) \leq 2\exp\left(-\frac{t^2 (e-s)q_1}{2\|\Gamma\|^2_{\mathrm{op}}W}\right),
\]
where $\|\Gamma\|_{\mathrm{op}} < \infty$ under Assumption~\ref{assume-model-new}.
Taking $t = W/27$, we have that
\[
\mathbb{P}\left(Z_{W, e:s} \geq \mathbb{E}[Z_{W, e:s}] + \frac{W}{27}\right) \leq 2 \exp(-C q_1(e-s)W),
\]
with some absolute constant $C > 0$.
\noindent\textbf{Step 2.} Next, we bound the expectation $\mathbb{E}[Z_W]$. It follows that
\begin{align*}
\mathbb{E}[Z_{W, e:s}] & = \frac{1}{e-s}\mathbb{E}\left[\sup_{A \in \mathcal{C}(r, s:e, W)} \left|\sum_{t = s+1}^e \sum_{(i, j)}\left\{\Omega_{ij}(t) - \Pi_{ij}(t)\right\} A^2_{ij}\right|\right] \\
&= \mathbb{E}\left[\sup_{A \in \mathcal{C}(r, s:e, W)} \left|\langle \Sigma_{R,s:e}, A\rangle\right|\right],
\end{align*}
where the second identity follows from the definition
\[
\Sigma_{R,s:e} = \frac{1}{e-s}\sum_{t = s+1}^e\sum_{(i,j)}\left\{\Omega_{ij}(t) - \mathbb{E}[\Omega_{ij}(t)]\right\}E_{ij}.
\]
For $A \in \mathcal{C}(r, e-s, W)$, we have that
\[
\|A\|_* \leq \sqrt{r} \|A\|_{\mathrm{F}} \leq \sqrt{\frac{r}{q_1}} \|A\|_{L_2(\Pi), s:e} \leq \sqrt{\frac{rW}{q_1}}.
\]
Therefore, we have that
\[
\mathbb{E}[Z_{W, e:s}] \leq \sqrt{\frac{rW}{q_1}}\mathbb{E}\left[\|\Sigma_{R, s:e}\|_{\mathrm{op}}\right].
\]
\noindent\textbf{Step 3.} Finally, using
\[
\mathbb{E}[Z_{W, e:s}] + \frac{W}{27} \leq \frac{W}{9} + \frac{54r}{q_1}\left(\mathbb{E}\left[\|\Sigma_{R, s:e}\|_{\mathrm{op}}\right]\right)^2,
\]
we have that
\[
\mathbb{P}\left(Z_{W, e:s} \geq \frac{W}{9} + \frac{54r}{q_1}\left(\mathbb{E}\left[\|\Sigma_{R, s:e}\|_{\mathrm{op}}\right]\right)^2\right) \leq 2 \exp(-C q_1(e-s)W).
\]
\end{proof}
\iffalse
\begin{theorem}[Talagrand's concentration inequality]\label{thm-talagrand}
Suppose that $f: \, [-1, 1]^N \to \mathbb{R}$ is a convex Lipschitz function with Lipschitz constant $L$. Let $\epsilon_1, \ldots, \epsilon_n$ be independent random variables taking values in $[-1, 1]$. Let $Z = f(\epsilon_1, \ldots, \epsilon_N)$. Then for any $t \geq 0$, it holds that
\[
\mathbb{P}\left\{|Z - \mathbb{E}(Z)| \geq 16L + t\right\} \leq 4\exp\left\{-\frac{t^2}{2L^2}\right\}.
\]
\end{theorem}
\fi
\iffalse
\begin{theorem}[Proposition 13 in \cite{klopp2015matrix}]\label{thm:tailprop_op}
Let $X \in \mathbb{R}^{n\times m}$ be a random matrix whose entries $X_{ij}$ are independent centered bounded random variables. Then for any $0 < \epsilon \leq 1/2$, there exists an absolute constant $C_{\epsilon}$, such that for any $t \geq 0$
\[
\mathbb{P}\left(\|X\|_{\mathrm{op}} \geq (1+\epsilon)2\sqrt{2}(\sigma_1 \vee \sigma_2) + t \right) \leq (n \wedge m)\exp\left(-\frac{t^2}{C_{\epsilon}\sigma_*^2}\right),
\]
where
\[
\sigma_1 = \max_i\sqrt{\sum_j\mathbb{E}[X_{ij}^2]}, \quad \sigma_2 = \max_j\sqrt{\sum_i\mathbb{E}[X_{ij}^2]}, \quad \text{and} \quad \sigma_* = \max_{ij}|X_{ij}|.
\]
\end{theorem}
\fi
\begin{lemma}\label{lem-lem4-klopp_dep}
Suppose Assumption~\ref{assume-model-new} holds.\\
(i) For any integer pair $1 \leq s < e$, it holds that
\begin{align}
\mathbb{E}\left[\| \Sigma_{R,s:e}\|_{\mathrm{op}} \right]\leq& C\sqrt{\frac{q_2n}{e-s}}.
\end{align}\label{eq-lem-lem4-klopp-r}
where $C > 0$ is an absolute constant.
\\
(ii) For any integer pair $1 \leq s < e$ and $\delta > 0$, with probability at least $1 - 3\delta$, it holds that
\begin{align} \label{eq-lem-lem4-klopp}
\|\Sigma_{s:e}\|_{\mathrm{op}} \leq& Cq_2\max\{\vartheta_2, 1-\vartheta_1\}\sqrt{\frac{n+\log(1/\delta)}{e-s}}\max\left\{\sqrt{\log(e-s)}, \sqrt{\log n+\log(1/\delta)}\right\}.
\end{align}
\end{lemma}
\begin{proof}
(i) \textbf{Upper bound of $\mathbb{E}[\|\Sigma_{R, s:e}\|_{\mathrm{op}}]$}.
\\
Recall that
\[
\Sigma_{R,s:e} = \frac{1}{e-s}\sum_{(i,j)}E_{ij}\sum_{t = s+1}^e\big\{\Omega_{ij}(t) - \mathbb{E}[\Omega_{ij}(t)]\big\}.
\]
Note that under Assumption~\ref{assume-model-new}, conditional on the latent positions $\{X_1^n(t)\}_{t = s+1}^e$, $\{\Omega(t)\}_{t = s+1}^e$ are independent across $t$, and for each $t$, the upper diagonal entries of $\Omega(t)$ are independent. Let $\{\epsilon_{ij,t}\}_{t \in \mathbb Z,\, 1\leq i,j \leq n}$ be a sequence of i.i.d. Rademacher random variables independent of $\{\Omega(t)\}_{t = s+1}^e$. We have that
\begin{align*}
\mathbb E[\|\Sigma_{R,s:e}\|_{\mathrm{op}}] =& \mathbb E_{\{X_1^n(t)\}_{t = s+1}^e}\left[\mathbb E_{|\{X_1^n(t)\}_{t = s+1}^e}\big[\|\Sigma_{R,s:e}\|_{\mathrm{op}}\big]\right]\\
=& \frac{1}{e-s}\mathbb E_{\{X_1^n(t)\}_{t = s+1}^e}\left[\mathbb E_{|\{X_1^n(t)\}_{t = s+1}^e}\left\{\left\|\sum_{(i,j)}E_{ij}\sum_{t = s+1}^e\big(\Omega_{ij}(t) - \mathbb{E}[\Omega_{ij}(t)]\big)\right\|_{\mathrm{op}}\right\}\right]\\
\leq& \frac{2}{e-s}\mathbb E_{\{X_1^n(t)\}_{t = s+1}^e}\left[\mathbb E_{ \epsilon, \,|\{X_1^n(t)\}_{t = s+1}^e}\left\{\left\|\sum_{(i,j)}E_{ij}\sum_{t = s+1}^e \epsilon_{ij,t}\Omega_{ij}(t)\right\|_{\mathrm{op}}\right\}\right],
\end{align*}
where the inequality follows from the symmetrization argument. We further bound the above conditional expectation by applying \Cref{thm:op_norm_tail} and taking an integral of the tail probability bound.
Note that for each $(i,j)$ and $t = s+1, \dots, e$, conditional on $\{X_{1}^n(t)\}_{t = s+1}^e$
\[ \epsilon_{ij,t}\Omega_{ij}(t) = \begin{cases}
-1 \quad &\text{w.p.} \;\; Q_{ij}(t)/2\\
0 \quad &\text{w.p.} \;\; 1 - Q_{ij}(t)\\
1 \quad &\text{w.p.} \;\; Q_{ij}(t)/2
\end{cases}.
\]
If follows that
\[
\mathbb{E}_{\epsilon, \,|\{X_{1}^n(t)\}_{t = s+1}^e}\big[\epsilon_{ij,t}\Omega_{ij}(t) \big] = 0,
\]
and conditonal on $\{X_{1}^n(t)\}_{t = s+1}^e$, the sub-Gaussian norm (e.g.~Definition 2.5.6 in \citealt{vershynin2018high}) satisfies
\[
\big\|\epsilon_{ij,t} \Omega_{ij}(t)\big\|_{\psi_2} \leq \frac{1}{\sqrt{\log\big(1+1/Q_{ij}(t)\big)}} \leq \sqrt{q_2+1} = C_1\sqrt{q_2}.
\]
By \Cref{lemma:subGaussian_ind_sum}, we have that conditonal on $\{X_{1}^n(t)\}_{t = s+1}^e$, the sub-Gaussian norm
\[
\max_{1 \leq i,j \leq n}\left\|\sum_{t = s+1}^e\epsilon_{ij,t}\Omega_{ij}(t)\right\|_{\psi_2} \leq C_2\sqrt{q_2(e-s)}.
\]
By \Cref{thm:op_norm_tail}, for any $t > 0$ that
\begin{align*}
\mathbb{P}_{\epsilon,\,|\{X_{1}^n(t)\}_{t=s+1}^e}\left(\frac{C_3}{\sqrt{q_2(e-s)}}\left\|\sum_{(ij)}E_{ij}\sum_{t = s+1}^e\epsilon_{ij,t} \Omega_{ij}(t)\right\|_{\mathrm{op}} -\sqrt{n} \geq t \right)\leq 2\exp(-t^2),
\end{align*}
which leads to
\begin{align*}
\mathbb{E}_{\epsilon,\,|\{ X_{1}^n(t)\}_{t =s+1}^e}\left[\left\|\sum_{(ij)}E_{ij} \sum_{t = s+1}^e\epsilon_{ij,t} \Omega_{ij}(t)\right\|_{\mathrm{op}} \right]\leq C_4\sqrt{q_2(e-s)n}.
\end{align*}
Thus,
\begin{align*}
\mathbb E[\|\Sigma_{R,s:e}\|_{\mathrm{op}}] \leq C_5\sqrt{\frac{q_2n}{e-s}}.
\end{align*}
(ii) \textbf{Tail probability bound of $\|\Sigma_{s:e}\|_{\mathrm{op}}$}.
\\
Recall the definition that
\begin{align*}
\Sigma_{s:e} = \frac{1}{e-s} \sum_{(i, j)}E_{ij} \sum_{t = s+1}^e \Omega_{ij}(t)\big\{P_{ij}(t) - \mathbb E[P_{ij}(t)] + \xi_{ij}(t)\big\}.
\end{align*}
Note that under Assumption~\ref{assume-model-new}, conditional on the latent positions $\{X_1^n(t)\}_{t = s+1}^e$, $\{\Omega(t)\}_{t = s+1}^e$ and $\{\xi(t)\}_{t = s+1}^e$ are mutually independent and are both independent across $t$. Moreover, for each $t$, the upper diagonal entries of $\Omega(t)$ and $\xi(t)$ are independent.
For any $t > 0$, it holds that
\begin{align*}
\mathbb P\left(\|\Sigma_{s:e}\|_{\mathrm{op}} \geq t \right) = \mathbb E_{\{X_1^n(t)\}_{t = s+1}^e}\left[\mathbb P_{|\{X_{1}^n(t)\}_{t = s+1}^e}\big(\|\Sigma_{s:e}\|_{\mathrm{op}} \geq t\big)\right].
\end{align*}
Next, we will upper bound the conditional tail probabilities using \Cref{thm:op_norm_tail}.
Note that for each $(i,j)$ and $t = s+1, \dots, e$, conditional on $\{X_{1}^n(t)\}_{t = s+1}^e$
\[ \Omega_{ij}(t)\big\{P_{ij}(t) - \mathbb E[P_{ij}(t)] + \xi_{ij}(t)\big\} = \Omega_{ij}(t)\xi_{ij}(t), \quad \mathbb E_{|\{X_{1}^n(t)\}_{t = s+1}^e}[\Omega_{ij}(t)\xi_{ij}(t)] = 0,
\]
and by \Cref{lemma:subGaussian_ind_sum}, we have that conditonal on $\{X_{1}^n(t)\}_{t = s+1}^e$, the sub-Gaussian norms
\[
\left\|\frac{1}{e-s}\sum_{t = s+1}^e\Omega_{ij}(t)\xi_{ij}(t)\right\|_{\psi_2} \leq \frac{Q_{ij}(t)P_{ij}(t)}{\sqrt{e-s}} \leq \frac{q_2P_{ij}(t)}{\sqrt{e-s}}.
\]
By \Cref{thm:op_norm_tail}, for any $\delta > 0$ and conditional on $\{X_{1}^n(t)\}_{t = s+1}^e$, we have that
\begin{align*}
&\mathbb P_{|\{X_{1}^n(t)\}_{t = s+1}^e}\left(\|\Sigma_{s:e}\|_{\mathrm{op}} \geq C_1q_2\max_{ij,t}P_{ij}(t)\sqrt{\frac{n+\log(1/\delta)}{e-s}} \right) \leq \delta.
\end{align*}
Next, we bound $\max_{ij,t}P_{ij}(t)$ with high probability. For any $u > 0$, we have that
\begin{align*}
\mathbb P\left(\max_{1\leq i,j \leq n,\, s+1 \leq t \leq e}P_{ij}(t) \geq u\right) \leq \frac{n(n+1)(e-s)}{2}\max_{1\leq i,j \leq n,\, s+1 \leq t \leq e}\mathbb P\left(P_{ij}(t) \geq u\right)\\
\leq \frac{n(n+1)(e-s)}{2}\max_{1\leq i,j \leq n,\, s+1 \leq t \leq e}\exp\left(-C_2\frac{\big(u - \mathbb E[X^{\top}_i(t)X_j(t)]\big)^2}{\max\big\{\vartheta_2^2, (1-\vartheta_1)^2\big\}}\right)\\
\leq \max_{1\leq i,j \leq n,\, s+1 \leq t \leq e}\exp\left(-C_2\frac{\big(u - \mathbb E[X^{\top}_i(t)X_j(t)]\big)^2}{\max\big\{\vartheta_2^2, (1-\vartheta_1)^2\big\}} + 2\log n + \log(e-s)\right).
\end{align*}
It follows that for any $\delta > 0$, with probability at least $1-\delta$
\begin{align*}
\max_{1\leq i,j \leq n,\, s+1 \leq t \leq e}P_{ij}(t) \leq C_3\max\{\vartheta_2, 1-\vartheta_1\}\left(\sqrt{\log(e-s)} + \sqrt{\log n + \log(1/\delta)}\right).
\end{align*}
Therefore, we have for any $\delta > 0$, with probability at least $1-2\delta$
\begin{align*}
\|\Sigma_{s:e}\|_{\mathrm{op}} \leq& C_4q_2\max\{\vartheta_2, 1-\vartheta_1\}\sqrt{\frac{n+\log(1/\delta)}{e-s}}\max\left\{\sqrt{\log(e-s)}, \sqrt{\log n+\log(1/\delta)}\right\}.
\end{align*}
\end{proof}
\end{document} |
\begin{equation}gin{document}
\title{Long-range quantum entanglement in noisy cluster states}
\author{Robert Raussendorf, Sergey Bravyi and Jim Harrington}
\affiliation{California Institute of Technology,\\
Institute for Quantum Information, Pasadena, CA 91125, USA}
\date{\today}
\begin{equation}gin{abstract}
We describe a phase transition for long-range entanglement in a
three-dimensional cluster state affected by noise. The partially decohered
state is modeled by the thermal state of a short-range
translation-invariant Hamiltonian. We find
that the temperature at which the entanglement length changes from infinite
to finite is nonzero. We give an upper and lower bound to this transition
temperature.
\end{abstract}
\pacs{3.67.Lx, 3.67.-a}
\maketitle
\section{Introduction}
\langlebel{intro}
Nonlocality is an essential feature of quantum mechanics, put to the
test by the famous Bell inequalities \cite{Bell} and verified in a
series of experiments, see e.g. \cite{Aspect}. Entanglement
\cite{Schr} is an embodiment of this nonlocality which has become a
central notion in quantum information theory.
In realistic physical systems, decoherence represents a formidable but
surmountable obstacle to the creation of entanglement among far
distant particles. Devices such as quantum repeaters \cite{Rep} and
fault-tolerant quantum computers are being envisioned in which the
entanglement length \cite{LE,Aha} is infinite, provided the
noise is below a critical level. Here we are interested in the
question of whether an infinite entanglement length can also be found
in spin chains with a short-range interaction that are subjected to noise.
A prerequisite for
our investigation is the existence of systems with infinite
entanglement length at zero temperature. An example of such behavior
has been discovered by Verstraete, Mart\'in-Delgado, and
Cirac~\cite{Verstraete_Delgado_Cirac_04} with spin-1 chains in the
AKLT-model \cite{AKLT}, and by Pachos and Plenio with cluster Hamiltonians
\cite{PP}; see also \cite{Kay}. In this paper, we study the case of finite
temperature. We present a short-range, translation-invariant Hamiltonian
for which the entanglement
length remains infinite until a critical temperature $T_c$ is
reached. The system we consider is a thermal cluster state in three
dimensions. We show that the transition from infinite to finite
entanglement length occurs in the interval $0.30\, \Delta \leq T_c
\leq 1.15\,\Delta$, with $\Delta$ being the energy gap of the
Hamiltonian.
We consider a simple 3D cubic lattice ${\cal{C}}$
with one spin-$1/2$ particle (qubit) living
at each vertex of the lattice.
Let $X_u$, $Y_u$, and $Z_u$ be the Pauli operators acting
on the spin at a vertex $u\in {\cal C }$. The model Hamiltonian is
\begin{equation}gin{equation}\langlebel{Hmltn}
H= - \frac{\Delta}{2}\, \,
\sum_{u\in {\cal{C}}} K_u,
\quad
K_u = X_u \!\!\!\!\prod_{v\in \nb{(u)}}
\!\!Z_v.
\end{equation}
Here $\nb{(u)}$ is a set of nearest neighbors of
vertex $u$. The ground state of $H$ obeys eigenvalue equations
$K_u |\phi\rangle_{{\cal C }} = |\phi\rangle_{{\cal C }}$
and coincides with a cluster state~\cite{BR}.
We define a thermal cluster state at a temperature $T$ as
\begin{equation}
\rhosub{CS} = \frac1{{\cal Z }} \exp{( -\begin{equation}ta H )},
\end{equation}
where ${\cal Z }=\trace{ e^{-\begin{equation}ta H}}$ is a partition function and $\begin{equation}ta\equiv T^{-1}$.
Since all terms in $H$
commute, one can easily get
\begin{equation}gin{equation}\langlebel{CS}
\rhosub{CS} = \frac{1}{2^{|{\cal{C}}|}}
\prod_{u\in {\cal{C}}}
\left( I + \tanh\left(\begin{equation}ta\Delta/2\right)\, K_u \right).
\end{equation}
Let $A, B \subset {\cal C }$ be two distant regions on the lattice.
Our goal is to create as much entanglement between $A$ and $B$ as possible by
doing local measurements on all spins not belonging to $A\cup B$.
Denote $\alpha$ as the list of all outcomes obtained in these measurements
and $\rhosubsup{\alpha}{AB}$ as the state of $A$ and $B$ conditioned on the
outcomes $\alpha$. Let $E[\rho]$ be some measure of bipartite entanglement.
Following~\cite{LE} we define
the localizable entanglement between $A$ and $B$ as
\begin{equation}gin{equation}\langlebel{LE}
E(A,B)= \max \sum_\alpha p_\alpha\, E[ \rhosubsup{\alpha}{AB} ],
\end{equation}
where $p_\alpha$ is a probability to observe the outcome $\alpha$ and the
maximum is taken over all possible patterns of local measurements.
To specify the entanglement measure $E[\rho]$ it is useful to regard
$\rhosubsup{\alpha}{AB}$ as an encoded two-qubit
state with the first logical qubit residing in $A$ and the second in $B$.
We choose $E[\rho]$ as the maximum amount of two-qubit entanglement
(as measured by entanglement of formation) contained in $\rho$.
Thus $0\le E(A,B)\le 1$ and an equality $E(A,B)=1$ implies that
a perfect Bell pair can be created between $A$ and $B$. Conversely,
$E(A,B)=0$ implies that any choice of a measurement pattern produces a
separable state.
In this paper we consider a finite 3D cluster
\[
{\cal C } = \{ u=(u_1,u_2,u_3) \, : \, 1\le u_1,u_2+1 \le l; \, \,
1\le u_3 \le d\}\]
and choose a pair of opposite 2D faces as $A$ and $B$:
\[
A = \{ u \in {\cal C } \, : \, u_3=1\}, \quad
B = \{ u \in {\cal C } \, : \, u_3=d\},
\]
so that the separation between the two regions is $d-1$.
In Section~\ref{sec:LB} we show
that~\footnote{Refs. \cite{WHP,PT_Nishimori} consider a lattice with proportions of a cube, corresponding to $l=d$. However, numerical simulations
indicate that $\lim_{l,d\to \infty} E(A,B)=1$ even if
$l=C\ln{(d)}$; see remarks to Section~\ref{sec:LB}.}
\[
\lim_{l,d\to \infty} E(A,B)=1 \quad \mbox{for} \quad T < 0.30\, \Delta.
\]
Further, we show in Section~\ref{sec:UB} that
if $T > 1.15\,\Delta$ then
$E(A,B)=0$ for $d\geq2$
and arbitrarily large $l$.
\section{Lower bound}
\langlebel{sec:LB}
We relate the lower bound on the transition temperature to quantum
error correction. From Eq.~(\ref{CS}) it follows that
$\rhosub{CS}$ can be prepared from the perfect cluster state $|\phi\rangle_{{\cal C }}$
by applying the Pauli operator $Z_u$ to each spin $u\in {\cal C }$
with a probability
\begin{equation}
\langlebel{Eprob}
p=\frac{1}{1+\exp(\begin{equation}ta\Delta)}.
\end{equation}
Thus, thermal
fluctuations are equivalent to independent local $Z$-errors with an error rate
$p$.
We use a
single copy of $\rhosub{CS}$ and apply a specific pattern of local
measurements which creates an
encoded Bell state among sets of particles in $A$ and $B$. For encoding we use
the planar
code, which belongs to the family of surface codes introduced by Kitaev.
The 3D cluster state has, as opposed to its 1D counterpart \cite{BR}, an
intrinsic error correction capability which we use in the measurement pattern described below. Therein, the measurement outcomes are individually random but not independent; parity constraints exist among them. The violation of any of these indicates an error. Given sufficiently many such constraints, the measurement outcomes specify a syndrome from which typical errors can be reliably identified. The optimal error correction given this syndrome breaks down at a certain error rate (temperature), and the Bell correlations can no longer be mediated. This temperature is a lower bound to $T_c$, because in principle there may exist a more effective measurement pattern.
To describe the measurement pattern we use, let us introduce two
cubic sublattices $T_e, T_o\subset {\cal C }$ with a double spacing.
Each qubit $u\in {\cal C }$ becomes either a vertex or an edge in one of
the sublattices $T_e$ and $T_o$. The sets of vertices $V(T_e)$ and $V(T_o)$
are defined as
\[
\begin{array}{rcl}
V(T_e) &=& \{ u=(e,e,e) \in {\cal C } \},\\
V(T_o) &=& \{ u=(o,o,o) \in {\cal C } \},\\
\end{array}
\]
where $e$ and $o$ stand for even and odd coordinates.
The sets of edges $E(T_e)$ and $E(T_o)$ are defined as
\[
\begin{array}{rcl}
E(T_e) &=& \{ u=(e,e,o),(e,o,e),(o,e,e) \in {\cal C }\},\\
E(T_o) &=& \{ u=(o,o,e),(o,e,o),(e,o,o) \in {\cal C }\}.\\
\end{array}
\]
The lattices $T_e$, $T_o$ play an important role in
the identification of error correction on the cluster state with a
$\mathbb{Z}_2$ gauge model \cite{TCDS}. They are displayed in
Fig.~\ref{measpatt}
Let us assume that the lengths $l$ and $d$ are odd~\footnote{There is no
loss of generality here since one can decrease the size of the lattice
by measuring all of the qubits on some of the 2D faces in the $Z$-basis.}.
The Bell pair to be
created between $A$ and $B$ will be encoded into subsets of qubits
\[
\begin{array}{rcl}
L &=& \{ u=(o,e,1),(e,o,1) \in {\cal C }\} \subset A,\\
R &=& \{ u=(o,e,d),(e,o,d) \in {\cal C }\} \subset B.\\
\end{array}
\]
Each qubit $u\in {\cal C }$ is measured either in the $Z$- or
$X$-basis unless it belongs to $L$ or $R$. Denoting $M_X$ and $M_Z$ local
$X$- and $Z$-measurements, we can now present the measurement pattern:
\begin{equation}gin{equation}
\langlebel{M1}
\begin{equation}gin{array}{lr}
M_Z: & \;\;\;\; \forall u \in V(T_e) \cup V(T_o),\\
M_X: & \;\;\;\; \forall u \in E(T_e) \cup E(T_o)\begin{array}ckslash (L \cup R),\\
\end{array}
\end{equation}
We denote the measurement outcome $\pm1$ at vertex $u$ by $z_u$ or
$x_u$, respectively.
A graphic illustration of the measurement patterns for the individual slices is
given in Fig.~\ref{OneSlice}.
\begin{equation}gin{figure}
\begin{equation}gin{center}
\epsfig{file=TwoSlices.eps, width=7cm}
\caption{\langlebel{OneSlice}(color online)
(a) Measurement pattern on the first
and last slice of
${\cal{C}}$, for $l=5$. The resulting state is in the
code space of the planar
code. (The unmeasured qubits are displayed as shaded circles.) (b)
Lattice for the planar code. (c)/(d) Measurement pattern for
even and odd inner slices.}
\end{center}
\end{figure}
Before we consider errors, let us discuss the effect of this
measurement pattern on a perfect cluster state. Consider some
fixed outcomes $\{x_u\}$, $\{z_u\}$ of local measurements and
let $|\psi\rangle_{LR}$ be the reduced state of
the unmeasured qubits $L$ and $R$. We will now show that $|\psi\rangle_{LR}$
is, modulo local unitaries, an encoded Bell pair, with each qubit encoded
by the planar code \cite{Kit2}, the planar
counterpart of the toric code \cite{Kit1}. The initial cluster state
obeys eigenvalue equations $K_u|\phi\ranglengle_{\cal{C}} =
|\phi\ranglengle_{\cal{C}}$. This implies for the
reduced state
\begin{equation}gin{equation}
\langlebel{plaquette}
Z_{P,u}
|\psi\rangle_{LR} =
\langlembda_{P,u} |\psi\rangle_{LR},\;\;\forall \,\, u = (e,e,1),
\end{equation}
where $Z_{P,u} = \bigotimes_{v \in \nb(u) \cap L}Z_v$ is a
plaquette ($z$-type) stabilizer operator for the
planar code \cite{Kit2}. The eigenvalue $\langlembda_{P,u}$
depends upon the measurements outcomes as
$\langlembda_{P,u}= x_u z_{(u_1,u_2,2)}$.
Note that in the planar code the qubits live on
the edges of a lattice rather than on its vertices. The planar code
lattice is distinct from the cluster lattice
${\cal{C}}$; see Figs.~\ref{OneSlice},\ref{measpatt}.
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\epsfig{file=cube1.eps, width=8.5cm}\\
\epsfig{file=cube2.eps, width=8.5cm}
\caption{\langlebel{measpatt}(color online) The measurement pattern on the
cluster
${\cal{C}}$. The sublattices $T_e$ and $T_o$ are displayed (thick lines).
For reference, the cluster lattice is also shown (thin lines) and
the axis labeling shows the cluster coordinates.
Cluster qubits measured in the $Z$-basis (on the sites of $T_o$ and $T_e$)
are displayed in black, and qubits measured in the $X$-basis (on the edges
of $T_o$ and $T_e$) are displayed in gray (red). The large circles to left
and to
the right denote the unmeasured qubits which form the encoded Bell pair. The
measurement pattern has a bcc symmetry.}
\end{center}
\end{figure}
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\epsfig{file=cube3.eps, width=8.5cm}
\caption{\langlebel{cycl}(color online) A homologically nontrivial and a
homologically trivial error cycle on the lattice $T_o$. The nontrivial error
cycle stretches from one rough face to the opposite one while the trivial
error has both ends on the same face. Only the qubits belonging to $T_o$ are
shown and the qubits important for establishing the
$\overline{X}_A\overline{X}_B$-correlation are displayed enlarged.}
\end{center}
\end{figure}
From the equation $\prod_{v \in \nb(u)} K_v\, |\phi\ranglengle_{\cal{C}}=
|\phi\ranglengle_{\cal{C}}$, for $u=(o,o,1)$, we obtain
\begin{equation}gin{equation}
\langlebel{site}
X_{S,u}|\psi\rangle_{LR} = \langlembda_{S,u}
|\psi\rangle_{LR},\;\;\forall \,\, u = (o,o,1),
\end{equation}
where $X_{S,u}=\bigotimes_{v \in \nb(u) \cap L}X_v$ coincides with a site
($x$-type) stabilizer operator for the planar code
\cite{Kit2}, and $\langlembda_{S,u}= x_{(u_1,u_2,2)} z_u\prod_{v\in \nb_o(u)} z_v$,
where $\nb_o$ refers to a neighborhood relation on the sublattice $T_o$.
The code stabilizer operators in
Eq.~(\ref{plaquette}) and (\ref{site}) are algebraically
independent. There are $(l^2-1)/2$ code stabilizer generators for
$(l^2+1)/2$ unmeasured qubits, such that there exists one encoded
qubit on $L$. By direct analogy, there is also one encoded qubit
located on $R$.
Next, we show that $|\psi\rangle_{LR}$ is an eigenstate of
$\ovl{X}_L \ovl{X}_R$ and $\ovl{Z}_L \ovl{Z}_R$, where $\ovl{X}$ and
$\ovl{Z}$ are the encoded Pauli operators $X$ and $Z$, respectively,
i.e. $|\psi\rangle_{LR}$ is an encoded Bell pair. The encoded Pauli
operators \cite{Kit2} on $L$ and $R$ are $\ovl{X}_{L[R]} =
\bigotimes\limits_{u_1\,\, \text{odd}} X_{(u_1,u_2,1 [d])}$ for any
even $u_2$, and $\ovl{Z}_{L [R]} =
\bigotimes\limits_{u_2\,\,\text{even}} Z_{(u_1,u_2,1 [d])}$ for any
odd $u_1$. To derive the Bell-correlations of $|\psi\rangle_{LR}$
let us introduce 2D slices
\[
\begin{array}{rcl}
\Tsubsup{XX}{u_2} &=& \{ u = (o,u_2,o) \in {\cal C } \} \subset T_o,\\
\Tsubsup{ZZ}{u_1} &=& \{ u = (u_1,e,e) \in {\cal C } \} \subset T_e.\\
\end{array}
\]
The eigenvalue equation $\prod_{v \in
\Tsubsup{XX}{u_2}} K_v\, |\phi\ranglengle_{\cal{C}}= |\phi\ranglengle_{\cal{C}}$
with even $u_2$ implies for the reduced state
\begin{equation}gin{equation}
\langlebel{XXcorr}
\ovl{X}_L\ovl{X}_R\, |\psi\rangle_{LR} = \langlembda_{XX}\,
|\psi\rangle_{LR},
\end{equation}
with
$
\langlembda_{XX}=\prod_{v \in \Tsubsup{XX}{u_2+1}
\cup \Tsubsup{XX}{u_2-1} } z_v
\prod_{v \in \Tsubsup{XX}{u_2}\begin{array}ckslash(L \cup R)} x_v.
$
Here and thereafter it is
understood that $x_u=z_u=1$ for all
$u \not \in {\cal{C}}$.
Similarly, from $|\phi\ranglengle_{\cal{C}}=\prod_{v \in \Tsubsup{ZZ}{u_1}}
K_v\, |\phi\ranglengle_{\cal{C}}$, for $u_1$ odd, we obtain for the reduced state
\begin{equation}gin{equation}
\langlebel{ZZcorr}
\ovl{Z}_L\ovl{Z}_R\, |\psi\rangle_{LR} = \langlembda_{ZZ}\,
|\psi\rangle_{LR},
\end{equation}
with $ \langlembda_{ZZ}=\prod_{v \in \Tsubsup{ZZ}{u_1+1} \cup
\Tsubsup{ZZ}{u_1-1}} z_v \prod_{v \in \Tsubsup{ZZ}{u_1}} x_v$. Thus the
eigenvalue Eqs.~(\ref{plaquette}-\ref{ZZcorr}) show that the
measurement pattern of Eq.~(\ref{M1}) projects the initial perfect
cluster state into a state equivalent under local unitaries to the Bell pair,
with each qubit encoded by the planar code.
It is crucial that the measurement outcomes $\{z_u\}$ and $\{x_v\}$
are not completely independent. Indeed, for any vertex
$u\in T_o$ with $1<u_3<d$ the eigenvalue equation
$\prod_{v \in \nb(u)} K_v\, |\phi\ranglengle_{\cal{C}} = |\phi\ranglengle_{{\cal C }}$
implies the constraint
\begin{equation}gin{equation}
\langlebel{Sy1}
\prod_{v \in \nb(u)} x_v \cdot \prod_{w\in \nb_o(u)} z_w =1 .
\end{equation}
Analogously, for any vertex $u\in T_e$ one has a constraint
\begin{equation}gin{equation}
\langlebel{Sy2}
\prod_{v \in \nb(u)} x_v \cdot \prod_{w\in \nb_e(u)} z_w = 1,
\end{equation}
where $\nb_e$ refers to a neighborhood relation on the lattice $T_e$.
Thus there exists one syndrome bit for each vertex of $T_e$ and $T_o$,
(with exception for the vertices of $T_o$ with $u_3=1$ or $u_3=d$).
What are the errors detected by these syndrome bits?
Since we have only $Z$-errors (for generalization, see remark 1),
only the $X$-measurements are affected by
them. Each $X$-measured qubit is either on an edge of $T_o$ or
$T_e$. Thus, we can identify the locations of the elementary errors
with $E(T_o)$ and $E(T_e)$. From the equations Eq.~(\ref{Sy1},\ref{Sy2}),
each error located on an edge creates a syndrome at its end vertices.
Let us briefly compare with \cite{TCDS}. Therein, independent local $X$-and $Z$-errors were considered for storage whose correction runs completely independently. The $X$-errors in this model correspond to our $Z$-errors on qubits in $E(T_e)$, and the $Z$-storage errors to our $Z$-errors on qubits in $E(T_o)$, if the $X-$ and $Z-$error correction phases in \cite{TCDS} are pictured as alternating in time.
The syndrome information provided by Eqs.~(\ref{Sy1},\ref{Sy2}) is not
yet complete. There are two important issues to be
addressed: (i) There are no syndrome bits at the vertices of $T_o$ with
$u_3=1$ or $u_3=d$; (ii) Edges of $T_e$ with $u_3=1$ or $u_3=d$ have only
one end vertex, so errors that occur on these edges create only one
syndrome bit. Concerning (i), to get the missing syndrome bits we will measure
eigenvalues $\langlembda_{P,u}$ and $\langlembda_{S,u}$ for
the plaquette and the site stabilizer
operators living on the faces $A$ and $B$,
see Eqs.~(\ref{plaquette},\ref{site}).
Such measurements are local operations within $A$ or within $B$, so they
can not increase entanglement between $A$ and $B$.
For any $u=(o,o,1)$ or $u=(o,o,d)$ it follows from Eq.~(\ref{site}) that
\begin{equation}
\langlebel{Sy3}
\begin{equation}gin{array}{rclr}
\ds{\langlembda_{S,u}\, x_{(u_1,u_2,2)}\, z_u \prod_{v\in \nb_o(u)}z_v} &=& 1,&
\mbox{for } u_3=1,\\
\ds{\langlembda_{S,u}\,x_{(u_1,u_2,d-1)}\, z_u \prod_{v\in \nb_o(u)}z_v} &=& 1, &
\mbox{for } u_3=d.
\end{array}
\end{equation}
For any vertex $u=(o,o,1)$ or $u=(o,o,d)$ there are several
edges of the lattice $T_o$ incident to $u$.
It is easy to see that a single $Z$-error that occurs on any of these edges
changes a sign in Eqs.~(\ref{Sy3}).
Thus, these two constraints yield the syndrome bits living at the vertices
$u=(o,o,1)$ and $u=(o,o,d)$, so the issue~(i) is addressed.
Concerning (ii), we make use of Eq.~(\ref{plaquette}) and obtain
\begin{equation}
\langlebel{Sy5}
\begin{equation}gin{array}{rclr}
\langlembda_{P,u}\, x_u z_{(u_1,u_2,2)} &=& 1,& \mbox{for any }
u=(e,e,1),
\\
\langlembda_{P,u}\, x_u z_{(u_1,u_2,d-1)} &=& 1,& \mbox{for any }
u=(e,e,d).
\end{array}
\end{equation}
Since we have only $Z$-errors, the eigenvalues $\langlembda_{P,u}$ and
the outcomes $z_{(u_1,u_2,2)}$, $z_{(u_1,u_2,d-1)}$ are not affected by
errors. Thus the syndrome bits Eqs.~(\ref{Sy5}) are equal to $-1$
iff an error has occurred on the edge $u=(e,e,1)$ or $u=(e,e,d)$ of the
lattice $T_e$. Since each of these errors shows itself in a corresponding
syndrome bit which is not affected by any other error, we can reliably
identify
these errors. This is equivalent to actively correcting them with unit
success probability. We can therefore assume in the subsequent analysis
that no errors occur on the edges $(e,e,1)$ and $(e,e,d)$, which concludes
the discussion of the issue (ii).
As in \cite{Kit1}, we
define an error chain ${\cal{E}}$ as a collection of edges where an
elementary error has occurred. Each of the two lattices $T_e$ and $T_o$ has its own error chain. An error chain ${\cal{E}}$ shows a syndrome only at its
boundary $\partial({\cal{E}})$, and errors with the same boundary thus
have the same syndrome. One may identify an error ${\cal{E}}$ only
modulo a cycle $D$, ${\cal{E}}^\prime = {\cal{E}} + D$, with
$\partial(D)=0$.
There are homologically trivial and nontrivial cycles. A cycle $D$ is
trivial if it is a closed loop in $T_{o}$ ($T_e$), and homologically
nontrivial if it stretches from one rough face in $T_{o}$ ($T_{e}$) to
another. A rough face here is the 2D analogue of a rough edge on a
planar code \cite{Kit2}. The rough faces of $T_o$
are on the upper and lower side of ${\cal{C}}$, and the rough faces of
$T_e$ are on the front and back of ${\cal{C}}$ (recall that no errors
occur on the left and right rough faces of $T_e$).
Let us now study the effect of error cycles on the identification of
the state $|\psi\rangle_{LR}$ from the measurement outcomes. We only discuss
the error chains on $T_o$ here, which potentially affect the
eigenvalue Eq.~(\ref{XXcorr}). The discussion of the error chains in
$T_e$---which disturb the $\ovl{Z}_L\ovl{Z}_R$-correlations---is
analogous. An individual qubit error on $v\in {\cal{C}}$ will modify
the $\ovl{X}_L\ovl{X}_R$ correlation of $|\psi\rangle_{LR}$ if it either
affects $\ovl{X}_L$, $\ovl{X}_R$ or $\langlembda_{XX}$. That happens if $v
\in T_{XX}^{(u_2)}$. Now, the vertices in $T_{XX}^{(u_2)}$ correspond
to edges in $T_o$. If an error cycle $D$ in $T_o$ is homologically
trivial, it intersects $T_{XX}^{(u_2)}$ in an even number of
vertices; see Fig.~\ref{cycl}. This has no effect on the eigenvalue
Eq.~(\ref{XXcorr}). However, if the cycle is homologically nontrivial,
i.e. if it stretches between the upper and lower face of ${\cal{C}}$,
then it intersects $T_{XX}^{(u_2)}$ in an odd number of vertices. This
does modify the eigenvalue Eq.~(\ref{XXcorr}) by a sign factor of
$(-1)$ on the l.h.s., which leads to a logical error.
Therefore, for large system size, we require the probability
of misinterpreting the syndrome by a nontrivial cycle to be negligible
\cite{TCDS}:
\begin{equation}gin{equation}
\sum_{\cal{E}} \text{prob}({\cal{E}}) \sum_{D\; \text{nontrivial}}
\text{prob}({\cal{E}}+D|\,{\cal{E}})\, \approx \,0.
\end{equation}
We have now traced back the problem of reconstructing an encoded Bell pair
$|\psi\rangle_{LR}$ to the same setting that was found in \cite{TCDS}
to describe fault-tolerant data storage with the toric code. Via the
measurement pattern Eq.~(\ref{M1}), we may introduce two lattices
$T_o$, $T_e$ such that 1) Syndrome bits are located on the vertices of
these lattices, 2) Independent errors live on the edges and show a
syndrome on their boundary, 3) Only the homologically nontrivial
cycles give rise to a logical error. This error model can be
mapped onto a random plaquette $Z_2$-gauge field theory in 3 dimensions
\cite{TCDS,WHP} which undergoes a phase transition between an
ordered low temperature and a disordered high temperature phase. In the limit of $l,d \longrightarrow \infty$, full error-correction is possible in the low temperature phase.
In our setting, the error probabilities for all edges are equal to
$p$. For this case the critical error probability has been computed numerically in a lattice simulation \cite{PT_Nishimori}, $p_c=0.033\pm 0.001$. This value corresponds, via Eq.~(\ref{Eprob}), to $T_c=(0.296\pm 0.003)\Delta$.
{\em{Remarks:}} 1) The error model equivalent to Eq. (\ref{CS}), i.e.
$Z$-errors only, is very restricted. We have a physical motivation for this model, but we would like to point out that the very strong assumptions we have made about the noise are not crucial to our result of the threshold error rate being non-zero. One may, for example, generalize the error model from a dephasing channel to a depolarizing channel, with $p_x=p_y=p_z=p^\prime/3$.
Then, two changes need to be addressed, those in the bulk and those on the faces $L$ and $R$. Concerning the faces, the errors on the rough faces to the left and right of $T_e$ can no longer be unambiguously identified by measurements of the code stabilizer (\ref{Sy5}), which raises the question of whether---for depolarizing errors---it may be these surface errors that set the threshold for long-range entanglement. This is not the case. To see this, note that two slices of 2D cluster states may be attached to the left and right of ${\cal{C}}$, at $u_3=0,-1$ and $u_3=d+1,d+2$. The required operations are assumed to be perfect. They do not change the localizable entanglement between the left and right side of the cluster ${\cal{C}}$ because they act locally on the slices $-1 .. 1$ and $d .. d+2$, respectively. The subsets $A$ and $B$ of spins are re-located to the slices $-1$ and $d+2$, with the corresponding changes in the measurement pattern. The effect of this procedure is that the leftmost and rightmost slice of the enlarged cluster are error-free \footnote{ The following operations are required to attach a slice: (I) $\Lambda(Z)$-gates within the slice, (II) $\Lambda(Z)$-gates between the slice and its next neighboring slice, (III) $X$- and $Z$-measurements within the slice, see Fig.~1. All these operations are assumed to be perfect, and the errors on slices $1$ and $d$ are not propagated to slices $-1$ and $d+2$ by the $\Lambda(Z)$-gates (II).}, and only the bulk errors matter.
Concerning the bulk, note that the cluster qubits measured in $Z$-basis
serve no purpose and may be left out from
the beginning. Then, the considered lattice for the initial cluster
state has a bcc symmetry and double spacing. The lattices $T_o$, $T_e$
remain unchanged. Further, $X$-errors are absorbed in the $X$-measurements
and $Y$-errors act like $Z$-errors, such that we still map to the original
$\mathbb{Z}_2$ gauge model \cite{TCDS} at the Nishimori line. The threshold
for local depolarizing
channels applied to this configuration is thus $p^{\prime}_c=3/2 \,p_c=
4.9\%$. In addition, numerical simulations performed for
the initial simple cubic cluster and depolarizing channel
yield an estimate of the critical error probability of
$p^{\prime\prime}_c=1.4\%$. \\ 2)
Finite size effects. We carried out numerical simulations of error
correction on an $l \times l \times d$ lattice with periodic boundary
conditions (as opposed to the open boundary conditions of the planar
codes within the cluster state). For differing error rates below the
threshold value of $2.9\%$ \cite{WHP}, we found good
agreement for the fidelity $F$ between the perfect and the error-corrected encoded Bell state with the model $F \sim \exp( -d k_1 \exp( -l
k_2))$. Some data is shown in Fig. \ref{phaseErrorsFigure}
corresponding to a $Z$-error rate of $1.0\%$. Provided that planar
codes and toric codes have similar behavior away from threshold, our
simulations suggest that, in order to achieve constant fidelity, the
length $l$ specifying the surface code need only scale
\emph{logarithmically} with the distance $d$.\\ 3) For even $d$,
the construction presented above
can be used to mediate an encoded conditional $Z$-gate on
distant encoded qubits located on slices $1$ and $d$.
\begin{equation}gin{figure}
\begin{equation}gin{center}
\epsfig{file=phaseErrorsFigureBW.eps,width=8cm}
\caption{\langlebel{phaseErrorsFigure}This figure plots data for simulations
of error correction on an $L
\times L \times d_\text{toric}$ lattice, with periodic boundary
conditions in the first two directions, for various $L$
and $d_\text{toric}$ ($d=2d_\text{toric}+1$, $l=2L$).
The error rate is $p=0.01$. The logs are
base $e$. Two standard deviations
above and below the computed values (as given by statistical noise due
to the sample sizes) are shown by the error bars. The solid lines
each have slope one, and they are spaced equally apart. This lends
good support to the model of fidelity $F \sim \exp( -d k_1 \exp( -l
k_2))$ for error rates below threshold.}
\end{center}
\end{figure}
\section{Upper bound}
\langlebel{sec:UB}
In this section we analyze the high-temperature behavior of thermal cluster
states and find an upper bound on the critical temperature $T_c$.
Our analysis is based on the isomorphism between cluster states and the
so-called Valence Bond Solids (VBS) pointed out by Verstraete and Cirac
in~\cite{Verstraete_Cirac_VBS} which can easily be generalized to a finite
temperature.
\begin{equation}gin{figure}[ht]
\begin{equation}gin{center}
\epsfig{file=ValenceBond.eps, width=8cm}
\caption{\langlebel{VBpict}(color online) (a) Correspondence between physical
and virtual qubits. Domains are shown by dashed lines.
(b) A bipartite cut of a cubic lattice. The regions $A$ and $B$ are
highlighted.}
\end{center}
\langlebel{fig:VBS}
\end{figure}
With each physical qubit $u\in {\cal C }$ we associate
a domain $u.*$ of $d(u)$ virtual qubits,
where $d(u)=|\nb(u)|$ is the number of nearest neighbors of $u$
(see Fig.~\ref{VBpict}~(a)).
Let us label virtual qubits from a domain $u.*$ as $u.v$, $v\in \nb(u)$.
Denote $E$ to be the set of edges of the lattice ${\cal C }$ and define a thermal
VBS state $\rhosub{V\!BS}$ as
\begin{equation}gin{equation}\langlebel{VBS}
\rhosub{V\!BS}= \prod_{e=(u,v)\in E}
\frac14 \left( I + \omega_e X_{u.v}Z_{v.u}\right)
\left( I + \omega_e Z_{u.v}X_{v.u}\right).
\end{equation}
Here $\{\omega_e\}$ are arbitrary weights such that $0\le \omega_e \le 1$.
It should be emphasized that $\rhosub{V\!BS}$ is a state of virtual qubits
rather than physical ones.
Our goal is to convert $\rhosub{V\!BS}$ into $\rhosub{CS}$
by local transformations mapping a domain $u.*$ into a single qubit $u\in {\cal C }$.
The following theorem is a straightforward generalization of the
Verstraete and Cirac construction (here we put $\Delta/2=1$).
\begin{equation}gin{theorem}\langlebel{theorem1}
Let $\rhosub{CS}$ be a thermal cluster state on the 3D cubic lattice ${\cal C }$
at a temperature $T\equiv \begin{equation}ta^{-1}$.
Consider a thermal VBS state $\rhosub{V\!BS}$
as in Eq.~(\ref{VBS}) such that the weights $\omega_e$ satisfy
\begin{equation}gin{equation}\langlebel{div}
\prod_{v\in \nb{(u)}}
\omega_{(u,v)}\, \ge \tanh{(\begin{equation}ta)} \quad \mbox{for each} \quad u\in {\cal C }.
\end{equation}
Then $\rhosub{V\!BS}$ can be converted into $\rhosub{CS}$ by
applying a completely positive transformation
${\cal{W}}_u$ to each domain $u.*$,
\begin{equation}gin{equation}\langlebel{CS&VBS}
\rhosub{CS} = {\cal{W}}( \rhosub{V\!BS} ), \quad {\cal{W}}=\bigotimes_{u\in {\cal C }} {\cal{W}}_u.
\end{equation}
\end{theorem}
Let us first discuss the consequences of this theorem. Note that each
edge $e\in E$ of $\rhosub{V\!BS}$ carries a two-qubit state
\begin{equation}gin{equation}\langlebel{bond state}
\rho_e =\frac14 (I + \omega_e X_1 Z_2)(I + \omega_e Z_1 X_2).
\end{equation}
The Peres-Horodecki partial transpose criterion~\cite{Peres_PPT,Horodecki_PPT}
tells us that $\rho_e$ is separable if and only if $\omega_e \le \sqrt{2}-1$.
Consider a bipartite cut of the lattice by a hyperplane of
codimension 1 (see Fig.~\ref{VBpict}~(b)).
We can satisfy
Eq.~(\ref{div}) by setting $\omega_e=\tanh{(\begin{equation}ta)}$ for all edges crossing
the cut and setting $\omega_e=1$ for all other edges.
Clearly, the state $\rhosub{V\!BS}$ is bi-separable whenever
$\tanh{(\begin{equation}ta)}\le \sqrt{2}-1$. But bi-separability of $\rhosub{V\!BS}$
implies bi-separability of $\rhosub{CS}$. We conclude that the localizable
entanglement between the regions $A$ and $B$ is zero whenever $\tanh{(\begin{equation}ta)}\le \sqrt{2}-1$,
which yields the upper bound on $T_c$ presented earlier.
{\it Remarks:} We can also satisfy Eq.~(\ref{div}) by setting $\omega_e=\omega$
for all $e\in E$, with $\omega^6 = \tanh{(\begin{equation}ta)}$.
This choice demonstrates that
$\rhosub{CS}$ is completely separable for $\tanh{(\begin{equation}ta)}<(\sqrt{2}-1)^6$
(that is $T\approx 200$).
It reproduces the upper bound~\cite{MacE} of D\"ur and Briegel
on the separability threshold error rate for cluster states.
In the remainder of this section we prove Theorem~\ref{theorem1}.
Consider an algebra ${\cal A }_u$ of operators acting on some particular
domain $u.*$. It is generated by the Pauli operators
$Z_{u.v}$ and $X_{u.v}$ with $v\in \nb{(u)}$.
The transformation ${\cal{W}}_u$ maps
${\cal A }_u$ into the one-qubit algebra generated
by the Pauli operators $Z_u$ and $X_u$. First, we choose
\[
{\cal{W}}_u(\eta)= W_u^\dag\, \eta \, W_u, \quad
W_u = |0^{\otimes d(u)}\rangle\langle 0| + |1^{\otimes d(u)}\rangle\langle 1|.
\]
One can easily check that
\begin{equation}gin{equation}\langlebel{conj2}
W_u^\dag Z_{u.v} = Z_u W_u^\dag \quad \mbox{and} \quad
Z_{u.v} W_u = W_u Z_u,
\end{equation}
for any $v\in \nb{(u)}$. As for commutation relations between
$W_u$ and $X_{u.v}$ one has
\begin{equation}gin{equation}\langlebel{conj3}
\begin{equation}gin{array}{rcl}
W_u^\dag \, \left( \prod_{v\in \nb{(u)}} X_{u.v}\right)
\, W_u &=& X_u,\\
W_u^\dag \, \left( \prod_{v\in S} X_{u.v}\right) \, W_u
&=&0,
\end{array}
\end{equation}
for any non-empty proper subset $S\subset \nb{(u)}$.
Taking ${\cal{W}}=\bigotimes_{u\in {\cal C }} {\cal{W}}_u$ and using
Eqs.~(\ref{conj2}), (\ref{conj3}) one can easily get
\begin{equation}gin{equation}\langlebel{quasy_CS}
{\cal{W}}(\rhosub{V\!BS}) = \frac{1}{4^{|E|}} \prod_{u\in {\cal C }}
\left( I + \eta_u K_u \right), \quad
\eta_u = \prod_{v\in \nb{(u)}} \omega_{(u,v)}.
\end{equation}
We can regard the state in Eq.~(\ref{quasy_CS})
as a thermal cluster state with a local temperature
$\tanh{(\begin{equation}ta_u)}\equiv \eta_u$ depending upon $u$.
The inequality of Eq.~(\ref{div}) implies that $\begin{equation}ta_u\ge \begin{equation}ta$ for all $u$.
To achieve a uniform temperature distribution $\begin{equation}ta_u=\begin{equation}ta$ one can
intentionally apply local $Z$-errors with properly chosen
probabilities.
\section{Conclusion}
Thermal cluster states in three dimensions exhibit a transition from
infinite to finite entanglement length at a non-zero transition
temperature $T_c$. We have given a lower and an upper bound to $T_c$,
$0.3\,\Delta\leq T_c\leq 1.15\,\Delta$ ($\Delta=$ energy gap of the
Hamiltonian). The reason for $T_c$ being non-zero is an intrinsic
error-correction capability of 3D cluster states. We have devised an
explicit measurement pattern that establishes a connection between
cluster states and surface codes. Using this, we have described how to
create a Bell state of far separated encoded qubits in the
low-temperature regime $T<0.3\, \Delta$, making the entanglement
contained in the initial thermal state accessible for quantum
communication and computation.
\begin{equation}gin{acknowledgments}
We would like to thank Hans Briegel and Frank Verstraete for bringing to our
attention the problem of entanglement localization in thermal
cluster states.
This work was supported by the National Science Foundation
under grant number EIA-0086038.
\end{acknowledgments}
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\end{document} |
\begin{document}
\title{Quantum-Space Attacks}
\author{Ran Gelles \hskip 2cm Tal Mor \\
\\
Technion - Israel Institute of Technology \\
Computer Science Department \\
{\tt \{gelles, talmo\}@cs.technion.ac.il}
}
\maketitle
\begin{abstract}
Theoretical quantum key distribution (QKD) protocols
commonly rely on the use of qubits (quantum bits).
In reality, however, due to practical limitations,
the legitimate users are forced to employ a larger quantum
(Hilbert) space,
say a quhexit (quantum six-dimensional) space, or even
a much larger quantum Hilbert space.
Various specific attacks exploit
of these limitations. Although security can still be proved in some
very special cases,
a general framework that
considers such realistic QKD protocols,
{\em as well as} attacks on such protocols,
is still missing.
We describe a general method of attacking realistic QKD
protocols, which we call the `quantum-space attack'.
The description is based on assessing
the enlarged quantum space actually used by a protocol,
the `quantum space of the protocol'.
We demonstrate these new methods by
classifying various (known) recent
attacks against several QKD schemes,
and by analyzing a novel attack
on interferometry-based QKD.
\end{abstract}
\section{Introduction}\label{sec:Introduction}
Quantum cryptography has brought us new ways of
exchanging a secret key between two
users (known as Alice and Bob).
The security of such Quantum Key Distribution (QKD) methods
is based on a very basic rule of nature and quantum mechanics---the
``no-cloning'' principle.
The first QKD protocol was suggested in a seminal paper by
Bennett and Brassard~\cite{BB84} in 1984, and is now known
as BB84.
During recent years many security analyses were published
\cite{Y95, Mayers01, BBBMR-STOC, ShorPres00, BBBMR, Gisin05}
which proved the information-theoretical security of
the BB84 scheme
against the most general attack by an unlimited adversary (known as Eve),
who has full control over the quantum channel\footnote{All QKD protocols
assume that Alice and Bob also use an insecure, yet unjammable, classical channel.}.
Those security proofs are
limited as they always
consider a theoretical QKD that uses perfect qubits.
Although these security proofs do take errors into account,
and the protocols use error correction and privacy amplification (to
compensate for these errors and for reducing any partial knowledge that Eve
might have),
in general, they
avoid security issues that arise from the implementation
of qubits in the {\em real world}.
A pivotal paper by Brassard,
L\"utkenhaus, Mor, and Sanders~\cite{BLMS-ec00,BLMS00}
presented the ``Photon Number Splitting (PNS) attack'' and
exposed a security flaw in experimental and practical QKD:
One must take into account the fact that Alice
does not generate perfect qubits (2 basis-states of a single photon),
but, instead, generates states that reside in an enlarged
Hilbert space (we call it ``quantum space'' here), of six dimensions.
The reason for that discrepancy in the size of the used quantum
space is that each
electromagnetic pulse that Alice generates
contains (in addition to the two dimensions spanned by the single-photon states)
also a vacuum state and three 2-photon states,
and these are extremely useful to the eavesdropper.
That paper proved that, in contrast to what was assumed in
previous papers,
Eve can make use of the enlarged space,
and get a lot of
information on the secret key, sometimes even full information,
without inducing any noise.
Many attacks on the practical protocols then
followed (e.g.,~\cite{H03,HLP04,GLLP04,NSG05,HM05,GFKZR06}),
based on
extensions of the quantum spaces,
exploring various additional security flaws; other papers~\cite{H03,SARG04,W05}
suggested possible ways to overcome such attacks.
On the one hand,
several security proofs, considering specific imperfections,
were given for the BB84 protocol \cite{GLLP04, ILM2007}.
Yet on the other hand, it is
generally impossible now to prove the security of
a practical protocol, since
{\em a general framework} that
considers such realistic QKD protocols,
{\em and} the possible attacks on such protocols,
is still missing.
We show that
the PNS attack, and actually all attacks directed at
the channel, are various special cases
of a general attack that we define here,
the {\em Quantum-Space Attack} (QSA).
The QSA generalizes existing attacks and also offers novel attacks.
The QSA is based on the fact that the ``qubits'' manipulated in
the QKD protocol actually reside in a larger Hilbert space, and this enlarged
space {\em can be assessed}. Although this enlarged space is
not fully accessible to the legitimate users,
they can still analyze it, and learn what a fully powerful eavesdropper
can do.
We believe that this assessment of the
enlarged ``quantum space of the protocol''
is a vital step on the way to proving
or disproving the unconditional security of practical QKD schemes.
We focus on schemes in which the quantum communication is
uni-directional, namely, from Alice's laboratory (lab) to Bob's lab.
We consider
an adversary that can attack all the quantum states
that come out of Alice's lab,
and all the quantum states that go into Bob's lab.
The paper is organized as follows:
Definitions of the
quantum spaces involved in the
realization of a protocol, and of the
``quantum space of the protocol'',
are presented
and discussed in Section~\ref{sec:QSoP}.
The ``quantum-space attack'' is defined and discussed in Section~\ref{sec:QSA}.
Using the general framework when the information carriers are
photons is discussed in
Section~\ref{sec:QSAphotonicWorld}.
Next, in Section~\ref{sec:knownQSA} we show
that the best known attacks on practical QKD
are special cases of the QSA.
Section~\ref{sec:InterferoBB84} demonstrates and analyzes
a novel QSA on an interferometric implementation of the BB84
and the six-state QKD protocols.
Last, we discuss a few subtleties and open problems for future research
in Section~\ref{sec:conclusion}.
We would like to emphasize that our (crypt)analysis presents
the difficulty of proving unconditional
security for practical QKD setups,
yet also provides an important (probably even vital)
step in that direction.
\section{The Quantum Space of the Protocol}
\label{sec:QSoP}
The Quantum Space Attack (QSA) is the most general attack
on the quantum channel that connects Alice to Bob. It
can be applied to any realistic
QKD protocol, yet here we focus on uni-directional schemes
and on implementations of the BB84 protocol and the six-state protocol.
We need to have a proper model of the protocol
in order to understand the Hilbert space
that an unlimited Eve can attack.
This space has never been analyzed before except for specific cases.
Our main finding is a proper description of this space,
which allows, for the first time,
defining the most general eavesdropping attack on the channel.
We start with a model of a practical ``qubit'', continue with understanding
the spaces used by Alice and Bob, and end by defining the relevant space,
the {\em Quantum Space of the Protocol}
(QSoP), used by Eve to attack the protocol.
The attacks on the QSoP are what we call {\em Quantum-Space Attacks}.
\subsection{Alice's realistic space}
\label{sec:HA}
In most QKD protocols, Alice sends Bob qubits, namely,
states of
2 dimensional quantum spaces ($H_2$).
A realistic view should take into account
any deviation from theory, caused by Alice's equipment.
For example, Alice might encode the qubit via a polarized photon:
$\ket{0_z}$ via a photon polarized horizontally, and
$\ket{1_z}$ polarized vertically. This can be written using
Fock notation\footnote{States written using the Fock notation
$\fet{\cdot}$ are called Fock states, see Section~\ref{sec:QSAphotonicWorld}.}
as $\fet{n_{h},n_v}$ where $n_h$ ($n_v$) represents
the number of horizontal (vertical) photons; then
$\ket{0_z} \equiv \fet{1,0}$ and $\ket{1_z} \equiv \fet{0,1}$.
When Alice's photon is lost within her equipment
(or during the transmission), Bob gets the
state $\fet{0,0}$,
so that Alice's realistic space becomes $H_3$. Alice might send
multiple photons and then $H^A$ is of higher dimension, see
Section~\ref{sec:AliceRealPhotonic}.
\begin{definition}
\label{def:HA}
{\bf Alice's realistic space, $H^A$,}
is the minimal space containing the actual quantum states
sent by Alice to Bob during the QKD protocol.
\end{definition}
In the BB84 protocol, Alice sends qubits in two\footnote{The six-state scheme
uses the three conjugate bases of the qubit space;
namely, also
$\ket{0_y}=\left (\ket{0_z}+i\ket{1_z}\right)/\sqrt{2}$, etc.}
fixed conjugate bases.
Theoretically, Alice randomly chooses a basis and a bit value
and sends the chosen bit encoded in the appropriate chosen basis as
a state in $H_2$ (e.g.\ $\ket{0_z}$,$\ket{1_z}$,
$\ket{0_x}=\left (\ket{0_z}+\ket{1_z}\right)/\sqrt{2}$, and
$\ket{1_x}= \left (\ket{0_z}-\ket{1_z}\right)/\sqrt{2}$).
To a better approximation, the states sent by Alice
are four different states
$\ket{\psi_i}_A$ ($i= 1,2,3,4$)
in her realistic space $H^A$, spanned by these
four states. This space $H^A$ is
of dimension $|H^A|$, commonly between 2 and 4,
depending on the specific implementation.
As practical instruments often diverse from theory,
Alice might send quite different states. As an extreme example,
see the {\em tagging attack} (Section~\ref{sec:TagAsQSA}),
which is based on the fact that Alice's space could
contain more than just these four theoretical states, so that
$|H^A| > 4$ is possible.
\subsection{Extension of Alice's space}
\label{sec:ExtendAliceSpace}
Bob commonly receives one of several possible states
$\ket{\psi_i}_{A}$ sent by Alice, and measures it. The most
general measurement Bob can perform is to add an ancilla, perform
a unitary transformation on the joint system, perform a
complete measurement,
and potentially
``forget''\footnote{By the term
``forget'' we mean that Bob's detection is unable
to distinguish between several measured states.}
some of the outcomes\footnote{This entire process can be described
in a compact way by using a POVM~\cite{Peres93}.}. However, once
Alice's space is larger than $H_2$, the extra dimensions provided
by Alice could be used by Bob for his measurement,
{\em instead of} adding an ancilla.
Interestingly, by his measurement Bob might be {\em extending}
the space vulnerable to Eve's attack well beyond $H^A$.
This is possible since in many cases the realistic space, $H^A$,
is embedded inside a larger space $M$.
\begin{definition}
\label{def:M}
The space $M$ is the space in which
$H^A$ is embedded, $H^A \subseteq M$.
The space $M$ is the actual space available for Alice and an Eavesdropper.
\end{definition}
Due to the presence of an eavesdropper,
Bob's choice whether to add an ancilla or
to use the extended space $M$
is vital for security analysis.
In the first case the ancilla
is added by Bob, inside his lab,
while in the second it is controlled by Alice,
transferred through the quantum channel
and exposed to Eve's deeds.
Eve might attack the extended space $M$,
and thus have a different
effect on Bob, considering his measurement method.
For example,
suppose Alice sends two non-orthogonal states of a qubit,
$\theta_0 = {\cos\theta \choose \sin\theta}$
and $\theta_1 = {\cos\theta \choose -\sin\theta}$, with
a fixed and known angle $0 \ge \theta \ge 45^{\circ}$.
Bob would like to distinguish between them,
while allowing inconclusive results
sometimes, but no errors~\cite{Peres88}.
Bob can add the ancilla $\ket{0}_{{ Anc}} \equiv {1 \choose 0}_{Anc}$
and perform the following
transformation~${\cal U}$:
\begin{multline}
\ket{0}_{{ Anc}} \otimes {\cos\theta \choose \pm \sin\theta}
=
\left ( \begin{array}{c} \cos\theta \\ \pm\sin\theta \\0 \\0
\end{array} \right )
\mapm{{\cal U}}
\left ( \begin{array}{c} \sin\theta \\ \pm\sin\theta \\
\sqrt{\cos 2\theta} \\ 0 \end{array} \right )
\\
= \sqrt{2}\sin\theta
\ket{0}_{{ Anc}} \otimes {1/\sqrt{2} \choose \pm 1/\sqrt{2}} +
\sqrt{\cos 2\theta} \ket{1}_{{ Anc}} \otimes {1 \choose 0}
\end{multline}
where
$\ket{1}_{{ Anc}} \equiv {0 \choose 1}_{Anc}$.
This operation leads to a conclusive result with probability
$2\sin^2\theta$ (when the measured ancilla is $\ket{0}_{{ Anc}}$),
and inconclusive result otherwise. It is simple to see that the
same measurement can be done, {\em without the use of an ancilla},
if the states $\theta_0$ and $\theta_1$ are embedded at
Alice's lab in a larger
space $M$, e.g.\ $M = H_3$, using Bob's transformation
\begin{equation}\label{eqn:3dimMeasurmetn}
\left ( \begin{array}{c} \cos\theta \\ \pm\sin\theta \\ 0 \end{array} \right )
\mapm{{\cal U}}
\left ( \begin{array}{c} \sin\theta \\ \pm\sin\theta \\
\sqrt{\cos 2\theta} \end{array} \right ) \text{.}
\end{equation}
In the general case, the space $M$ might be very large,
even infinite. Bob might use only
parts of it, for his measurements.
A complication in performing security analysis is due to
Bob's option to {\em both} use an ancilla and extend
the space used by Alice.
Our analysis in the following sections starts with the
space extension only (Sections \ref{sec:HB}--\ref{sec:HP}),
and later on deals with the general case (Sections
\ref{sec:HB+anc}--\ref{sec:HP+anc}).
\subsection{Bob's space, without an ancilla}
\label{sec:HB}
Let us formulate the spaces involved in the protocol, as described
above. Assume Alice uses the space $H^A$ according to
Definition~\ref{def:HA},
which is embedded in a
(potentially larger) space $M$.
Ideally, in the BB84 protocol,
Bob would like to measure just the states in $H^A$, but
in practice he usually can not do so.
Each one of Alice's states $\ket{\psi_i}_{ A}$ is transformed
by Bob's equipment into some pure\footnote{The case in which
Bob transposes the state into a mixed state is a special case
of the analysis done in Section~\ref{sec:HB+anc}.
For the notion of mixed states or quantum mixture see~\cite{NC00,Peres93}.}
state $\ket{\psi_i}_{ M} \in M$. The
space which is spanned by those states contains all the
information about Alice's states $\{ \ket{\psi_i}_{ A} \}$.
More important, Bob might be measuring un-needed
subspaces of $M$ which Alice's states do not span.
For instance, examine the case where Bob uses detectors to measure
the Fock states $\fet{1,0}$ and $\fet{0,1}$. Bob is usually able
to distinguish a loss (the state $\fet{0,0}$) or an error (e.g.\ $\fet{1,1}$,
one horizontal photon and one vertical photon),
from the two desired states,
but he cannot distinguish between other states containing multiple photons.
This means that Bob measures a much larger subspace of
the entire space $M$, but (inevitably) interprets
outcomes outside $H^A$ as legitimate states;
e.g.\ the states $\fet{2,0}$, $\fet{3,0}$, etc. are (mistakenly)
interpreted as $\fet{1,0}$.
See further discussion in Section~\ref{sec:PhotonicExtension}.
We denote Bob's setup
(beam splitters, phase shifters, etc.) by the unitary
operation ${\cal U}_B$, followed by a measurement; all these
operations are operating on the space $M$ (or parts of it).
Bob might have several different setups
(e.g.\ a different setup for the $z$-basis and for the $x$-basis).
Let $\mathbf{U}$ be the set of unitary transformations
in all Bob's setups.
\begin{definition} \label{def:HB}
{\bf [This definition is Temporary.]}
Given a specific setup-transformation ${\cal U}_j \in \mathbf{U}$, let
$H^{B_{j}} \subseteq M$ be the subsystem actually measured by Bob,
having $K$ basis states $\{ \ket{\phi_k}_{B_j} \}_{k= 0 \ldots K-1}$.
The set of {\bf Bob's Measured Spaces}
is the set $\{ H^{B_{j}} \}_{j=0 \ldots J-1}$
of $J= |\mathbf{U}|$ spaces.
\end{definition}
We have already seen that Bob might be measuring un-needed dimensions.
On the other hand he might not measure
certain subspaces of $M$, even when Alice's state might reach there.
In either case, the deviation
is commonly due to limitations of Bob's equipment.
\subsection{The quantum space of the protocol, without an ancilla}
\label{sec:HP}
The ``quantum space of the protocol'' (QSoP) is in fact Alice's {\em extended}
space, taking into consideration its {\em extensions}
due to Bob's measurements.
The security analysis of a protocol
depends on the space $H^{B^{-1}}$ defined below.
\begin{definition}\label{def:HB-1}
{\bf [This definition is Temporary.]}
{\bf The reversed space $H^{B^{-1}}$} is the Hilbert space
spanned by the states ${\cal U}_j^{-1} (\ket{\phi_k}_{B_j})$,
for each possible setup ${\cal U}_j \in \mathbf U$,
and for each basis state
$\ket{\phi_k}_{B_j}$
of the appropriate $H^{B_j} \subseteq M$.
\end{definition}
The Space $H^{B^{-1}}$
usually resides in a larger space than $H^A$.
For instance, using photons, the ideal space $H^A$ consists of
two modes with 2 basis states,
see Section~\ref{sec:QSAphotonicWorld}.
Now $H^{B^{-1}}$ could have an infinite space in each mode,
but also could have more modes.
In order to derive the quantum space of the protocol we need to
define the way Alice's space is extended according to $H^{B^{-1}}$,
for this simple case where Bob does not add an ancilla.
In this case, the space $H^{B^{-1}}$ simply extends Alice's space to
yield the QSoP via $H^P = H^A + H^{B^{-1}}$.
Formally speaking
\begin{definition}\label{def:HP}
{\bf [This definition is Temporary.]}
{\bf The Quantum Space of the Protocol}, $H^P$,
is the space spanned by the basis states of the space $H^A$
and the basis states of the space $H^{B^{-1}}$.
\end{definition}
If Alice's realistic space
is fully measured by Bob's detection process,
then $H^A$ is a subspace of $H^{B^{-1}}$, hence
$H^P = H^{B^{-1}}$.
\subsection{Bob's space (general case)}
\label{sec:HB+anc}
In the general case, one must consider Bob's option to add an ancilla
during his measurement process.
This addition causes a considerable
difficulty in analyzing a protocol, however
it is often an inherent part of the protocol,
and can not be avoided.
We denote the added ancilla as the state $\ket{0}_{B'}$
that resides in the space $H^{B'}$.
\begin{definition}\label{def:M+anc}
$M'$ is the space that includes the physical space
used by Alice as defined in Definition~\ref{def:M},
in addition to Bob's ancilla, $M' = M \otimes H^{B'}$.
\end{definition}
Bob measures a subspace of the space $M'$, so the
(permanent)
definitions of his measured spaces $H^{B_j}$ and the reversed space
$H^{B^{-1}}$ should be modified accordingly.
\begin{definition} \label{def:HB+anc}
Given a specific setup-transformation ${\cal U}_j \in \mathbf{U}$ let
$H^{B_{j}} \subseteq M'$ be the subsystem actually measured by Bob,
having $K$ basis states $\{ \ket{\phi_k}_{B_j} \}_{k= 0 \ldots K-1}$.
The set of {\bf Bob's Measured Spaces},
is the set $\{ H^{B_{j}} \}_{j=0 \ldots J-1}$
of $J= |\mathbf{U}|$ spaces.
\end{definition}
\subsection{The quantum space of the protocol (general case)}
\label{sec:HP+anc}
The quantum space of the protocol is still Alice's {\em extended}
space, while considering its {\em extensions}
due to Bob's measurements. Yet, the added ancilla makes things
much more complex.
The security analysis of a protocol
depends now {\em not} on the space $H^{B^{-1}}$ defined below,
but on a (potentially {\em much larger})
space obtained from it by tracing-out Bob's ancilla.
As before, we first define the reversed space.
\begin{definition}\label{def:HB-1+anc}
{\bf The reversed space $H^{B^{-1}}$} is the Hilbert space
spanned by the states ${\cal U}_j^{-1} (\ket{\phi_k}_{B_j})$,
for each possible setup ${\cal U}_j \in \mathbf U$,
and for each basis state
$\ket{\phi_k}_{B_j}$
of the appropriate $H^{B_j} \subseteq M'$.
\end{definition}
Once a basis state of one of
Bob's measured spaces $\ket{\phi_k}_{B_j}$
is reversed by ${\cal U}_j^{-1}$ we result with a state that
might, partially, reside in Bob's ancillary space $H^{B'}$.
Since Eve has no access to this space\footnote
{Giving this space to Eve
(for getting an upper bound on her information),
might be easier to analyze, but
is usually not possible since it would
give her too much power, making the protocol insecure.}
it must be traced-out (separated out), for deriving the QSoP.
Let us redefine the QSoP given the addition of the ancilla:
\begin{definition}\label{def:HP+anc}
{\bf The Quantum Space of the Protocol, $H^P$}, is the space
spanned by {\bf (a)} the basis states of the space $H^A$; and {\bf (b)}
the states $\mathrm{Tr}_{Bob}[ {\cal U}_j^{-1} (\ket{\phi_k}_{B_j})]$,
(namely, after tracing out Bob),
for each possible setup ${\cal U}_j \in \mathbf U$, and
for each basis state $\ket{\phi_k}_{B_j}$ of the appropriate
space $H^{B_j}$.
\end{definition}
Whenever ${\cal U}_B$ entangles Bob's ancilla with the system sent from Alice,
tracing out Bob's ancilla after performing ${\cal U}_B^{-1}$
might cause an increase of the QSoP to the dimension of
Bob's ancillary space.
For instance, assume Alice's state is embedded in an $n$-qubit space
to which Bob adds an ancilla of $n$-qubits and performs a unitary
transformation ${\cal U}$, such that for one state measured
by Bob, $\ket{\Psi}_{B} \mapm{{\cal U}^{-1}}
\frac{1}{2^{n/2}}\sum_{k=0}^{2^n-1}
\ket{k}_{P}\ket{k}_{B'}$.
Tracing out Bob from this state yields the
maximally mixed state
$\rho_{P} = \frac{1}{2^n}\sum_{k=0}^{2^n-1} \ket{k}\bra{k}$,
so that in this example
the whole $n$-qubits space is spanned.
\section{The Quantum Space Attack}
\label{sec:QSA}
\subsection{Eavesdropping on qubits}
When Alice and Bob use qubits, in theoretical QKD, Eve can attack
the protocol in many
ways. In her simplest attack, the so-called ``measure-resend attack'',
Eve performs any measurement (of her choice) on the qubit,
and accordingly decides what to send to Bob.
A generalization of that attack is the ``translucent attack'',
in which Eve attaches an ancilla, in an initial state $\ket{0}_E$ (and in any
dimension she likes), and entangles the ancilla and Alice's qubit, using
$\ket{0}_E \ket{i}_A \rightarrow \sum_{j=0}^{1} \ket{E_{ij}}_E \ket{j}_A$
where $\ket{i}_A$ is a basis for Alice's qubit, and Eve's states after
the unitary transformation are $\ket{E_{ij}}_E$.
Using this transformation one can define the most general
``individual-particle attack''~\cite{EHPP94,FGGNP97},
and also the most general
``collective attack''~\cite{BM97a,BBBGM02}.
In the individual-particle attack
Eve delays the measurement of her ancilla till after learning
anything she can about the qubit
(e.g., its basis), while in the collective attack Eve delays her measurements
further till she learns anything she can
about {\em all} the qubits (e.g., how the final key is generated from the
obtained string of shared bits), so she attacks directly the {\em final key}.
The most general attack that Eve could perform on the channel
is to attack all those qubits
transmitted from Alice to Bob, using {\em one} large ancilla.
This is the ``joint attack''.
Security, in case Eve tries to learn a maximal information on the final key,
was proven in~\cite{Y95,Mayers01,BBBMR-STOC,ShorPres00,BBBMR} via various methods.
The attack's unitary transformation is written as before, but with
$i$ a binary string of $n$ bits, and so is $j$,
$ \ket{0}_E\ket{i}_A\rightarrow \sum_{j=0}^{2^n-1}
\ket{E_{ij}}_E\ket{j}$.
\subsection{Eavesdropping on the quantum space of the protocol}
By replacing the qubit space $H_2$ by Alice's realistic ``qubit'' in the space
$H^A$, and by defining Eve's attack on the entire space of the protocol
$H^P$, we can generalize each of the known attacks on theoretical QKD
to a ``quantum space attack'' (QSA).
We can easily define now Eve's most general
{\it individual-transmission QSA}
on a realistic ``qubit'',
which generalizes the individual-particle attack earlier described.
Eve prepares an ancilla in a state $\ket{0}_{E}$,
and attaches it to Alice's state, but actually her ancilla is
now attached to the entire QSoP.
Eve performs a unitary
transformation ${\cal U}_E$ on the joint state.
If Eve's attack is only on $H^A$, we write
the resulting transformation on any basis state
of $H^A$,
$\ket{i}_A$, as
$\ket{0}_E\ket{i}_A \rightarrow \sum_j \ket{E_{ij}}_{E}\ket{j}_{A}$,
where the sum is over the dimension of $H^A$.
The Photon-Number-Splitting attack (see Section~\ref{sec:PNSasQSA})
is an example for such an attack.
The most general
individual-transmission QSA is based on a translucent
QSA on the QSoP,
\begin{equation}
\label{eqn:TranslucentQSA}
\ket{0}_E\ket{i}_P \rightarrow \sum_j \ket{E_{ij}}_{E}\ket{j}_P\text{,}
\end{equation}
where the sum is over the dimension of $H^P$.
The subsystem in $H^P$ is then sent to Bob
while the rest (the subsystem $H^E$) is kept by Eve.
We write
the transformation on any basis state
of $H^P$, $\ket{i}_P$, but
note that it is sufficient to define the transformation on the different
states in $H^A$, namely
for all states of the form $\ket{i}_A$, since other states of the QSoP are
never sent by Alice (any other additional subsystem of the QSoP
is necessarily at a known state when it enters Eve's transformation).
Attacks that are more general than the {\it individual
transmission QSA}, the
{\it collective QSA} and the {\it joint QSA},
can now be defined
accordingly.
In the most general collective QSA, Eve performs the above
translucent QSA
on many (say, $n$) realistic ``qubits''
(potentially a different attack on each one, if she
likes), waits till she gets all data regarding the generation
of the final key,
and she then measures all the ancillas together, to obtain the optimal
information on the final key or the final secret.
The most general attack that Eve could perform on the channel
is to attack all those realistic ``qubits''
transmitted from Alice to Bob, using
{\em one} large ancilla. This is the
``joint QSA''.
The attack's unitary transformation is written as before, but with
$i$ a string of $n$ digits rather than a single digit
(digits of the relevant dimension of $H^P$),
and so is $j$,
\begin{equation}
\label{eqn:JointQSA}
\ket{0}_E\ket{i}_{P^{\otimes n}} \rightarrow \sum_{j=0}^{{|H^P|}^n-1}
\ket{E_{ij}}_E\ket{j}_{P^{\otimes n}}\text{.}
\end{equation}
Eve measures the ancilla, after learning all classical information,
to obtain the optimal
information on the final key or the final secret.
As before, it is sufficient to define the transformation on
the different input states from $(H^A)^{\otimes n}$.
We would like to emphasize several issues:
1.-- When analyzing specific attacks,
or when trying to obtain a limited
security result,
it is always legitimate to restrict the analysis to
the relevant (smaller) subspace of the QSoP, for simplicity,
e.g., to $H^A$, or to $H^{B^{-1}}$, etc.
2.-- Any bi-directional protocol will have a much more complicated QSoP,
thus it might be extremely difficult to analyze any type of
QSA (even the simplest ones) on such protocols.
This remark is especially important since
bi-directional protocols play a very important
role in QKD, since they appear in many
interesting protocols such as the
plug-and-play~\cite{MHHTG97},
the ping-pong~\cite{BF02}, and the classical Bob~\cite{BKM07} protocols.
Specifically they provided (via the plug-and-play)
the only commerical QKD so far~\cite{idQ, magiQ}.
3.-- It is well known that the collective or joint
attack is only finished after Eve gets all
quantum and classical information, since she delays her measurements
till then~\cite{BM97a, BBBGM02, BBBMR-STOC, Mayers01, BBBMR}; if she expects
more information, she better wait and attack the final secret rather
than the final key; it is important to notice that if the key will be used
to encode quantum information (say, qubits) then the quantum-space of the
protocol will require a modification, potentially a major one;
It is interesting to study if this new notion of QSoP has an influence on
analysis of such usage of the key as done (for the ideal qubits)
in~\cite{BHLMO05}.
\section{Photonic Quantum Space Attacks}
\label{sec:QSAphotonicWorld}
\subsection{Photons as quantum-information carriers}
\label{sec:PhotonAsQuantumSystem}
Since most of the practical QKD experiments and products are done
using photons, in this section we demonstrate
our QSoP and QSA definitions and methods via photons.
Our analysis uses the Fock-Space\footnote{A description of the
Fock space and Fock notations can be found in
various quantum optic books, e.g.\ \cite{SZ97}.} notations
for describing photonic quantum spaces.
For clarity, states written using the Fock notation are denoted
with the superscript `{\tiny F}', e.g.\ $\fet{0}$,
$\fet{3}$, and
$\fet{0,3,1}$.
A photon can not be treated as a quantum system
in a straightforward way. For instance, unlike dust particles or grains
of sand,
photons are indistinguishable particles, meaning that when a couple of
photons are interacting, one cannot define the evolution of the specific
particle, but rather describe the whole system.
Let us examine a cavity, for instance.
It can contain photons of specific wavelengthes
($\lambda_1$, $\lambda_2$, etc.) and
the energy of a photon of wavelength $\lambda$ is
directly proportional to $1/\lambda$.
While one cannot distinguish between photons of the same wavelength, one can
distinguish between photons of different wavelengths.
Therefore, it is convenient to define distinguishable
``photonic modes'', such that each wavelength corresponds to a specific mode
(so a mode inside a cavity can be denoted by its wavelength),
and then count the number of photons in each mode.
If a single photon in a specific mode carries some unit of energy, then $n$
such photons of the same wavelength carry $n$ times that energy.
If the cavity is at its ground (minimal) energy level,
we say that there are
``no photons'' in the cavity and denote the state as $\fet{0}$---the vacuum
state.
The convention is to denote only those modes that
are potentially populated, so
if we can find $n$ photons in one mode, and no photons in any other mode,
we write, $\fet{n}$.
If two modes are populated by $n_a$ and $n_b$ photons, and all
other modes are surely empty,
we write $\fet{n_a,n_b}$ (or $\fet{m,n}_{ab}$).
When there is no danger of confusion, and the number
of photons per mode is small (smaller than ten), we just write
$\fet{mn}$ for $m$ photons in one mode and $n$ in the other.
In addition to its wavelength, a photon also has a property called
polarization, and a basis for
that property is, for instance, the horizontal and vertical polarizations
mentioned earlier. Thus, two modes (in a cavity) can also
have the same energy,
but different polarizations.
Outside a cavity photons travel with the speed of light,
say from Alice to Bob,
yet modes can still be described, e.g., by using ``pulses'' of light~\cite{BLPS90}.
The modes can then be distinguished by different directions of the light
beams (or by different paths),
or by the timing of pulses (these modes are denoted by non-overlapping
time-bins), or by orthogonal
polarizations.
A proper description of a photonic qubit is
commonly based on using two modes `$a$' and `$b$' which are
populated by exactly a single photon,
namely, a photon in mode $a$, so the state is $\fet{10}_{ab}$,
or a photon in mode $b$,
so the state is $\fet{01}_{ab}$.
However, a quantum space that consists of a
single given photonic mode `$a$' is not restricted to a single photon,
and can be populated by any number of photons.
A basis for this space is $\{\fet{n}_a\}$ with $n \ge 0$, so that
the quantum space is infinitely large, $H_\infty$.
Theoretically, a general state in this space is can be written
as the superposition $\sum_{n=0}^\infty c_{n}\fet{n}_a$,
with $\sum_n|c_n|^2=1$, $c_n \in \mathbb{C}$.
Similarly, a quantum space that consists of two photonic modes
has the basis states $\fet{n_a,n_b}$, for $n_a , n_b \ge 0$ and
a general state is of the form
$\sum_{n_a , n_b = 0}^\infty c_{n_a,n_b}\fet{n_a,n_b}$
with $\sum_{n_a , n_b = 0}^\infty |c_{n_a,n_b}|^2 = 1$,
$c_{n_a,n_b} \in \mathbb{C}$.
This quantum space is described as a tensor product of two ``systems''
$H_\infty \otimes H_\infty$.
Using {\em exactly} two photons in two different
(and orthogonal) modes assists in clarifying
the difference between photons and dust particles (or grains of sand):
Due to the indistiguishability of photons, only 3 different
states can exist (instead of 4): $\fet{20}_{ab}$, $\fet{02}_{ab}$ and
$\fet{11}_{ab}$.
The last state has one photon in mode `$a$' and another photon
in `$b$', however, exchanging the photons is meaningless
since one can never tell one photon from another.
A realistic model
of a photon source (in a specific mode)
is of a coherent pulse (a Poissonian distribution)
\[
\ket{\alpha} = e^{- \frac{|\alpha|^2}{2}}
\sum^{\infty}_{n=0}\frac{\alpha^n}{\sqrt{n!}}\ket{n}
\]
including terms
that describe the possibility of emitting any number $n$ of photons.
As the number of photons increases beyond some number,
the probability decreases, so it
is common to neglect the higher orders.
In QKD, experimentalists commonly use a ``weak'' coherent state (such that
$|\alpha|\ll 1$) and then terms with $n\ge3$ can usually be neglected.
There is also a lot of research about
sources that emit (to a good approximation) single photons,
and then, again,
terms with $n\ge3$ can usually be neglected.
\subsection{Alice's realistic photonic space}
\label{sec:AliceRealPhotonic}
While the theoretical qubit lives in $H_2$,
a realistic view defines the space actually used by Alice
to be much larger.
The possibility to emit empty pulses
increases Alice's realistic space into $H_3$,
due to the vacuum state $\fet{00}_{ab}$.
When Alice sends a qubit using two modes,
using a weak coherent state (or a ``single-photon'' source),
her realistic space, $H^A$,
is embedded in $H_{\infty}\otimes H_{\infty}$.
Terms containing more than two photons can be neglected,
so these are excluded from Alice's space $H^A$.
The appropriate realistic quantum space of Alice, $H^A$,
is now a quhexit: the six-dimensional space
spanned by
$\chi^6 =\{\fet{00}$, $\fet{10}$,
$\fet{00}$, $\fet{11}$, $\fet{20}$, $\fet{02} \}$.
The PNS attack
demonstrated in Section~\ref{sec:PNSasQSA},
is based on attacking
this 6 dimensional space $H^A$.
Note also that terms with more than two photons still appear in $M$,
and thus could potentially appear in the QSoP (and then used by Eve).
At times, Alice's realistic space is even larger, due to extra modes
that are sent through the channel, and are not meant to be
a part of the protocol.
These extra modes might severely compromise the security of the protocol,
since they might carry some vital information about the protocol.
A specific QSA based on that flaw is the
``tagging attack'' (Section~\ref{sec:TagAsQSA}).
Note that even if Alice uses exactly two modes, the quantum space
$M$ where $H^A$ is embedded, certainly contains other modes as well.
\subsection{Extensions of the photonic space; the QSoP}
\label{sec:PhotonicExtension}
Let us discuss Bob's measurement of photonic spaces.
There are (mainly) two types of detectors that can be used.
The common detector can not distinguish
a single photon from more than one photon
(these kind of detectors are known as {\em threshold detectors}).
The Hilbert space where Bob's measurement
is defined is infinite\footnote{
In practice, that space is as large as Eve
might wish it to be.
We can ignore the case where Eve uses too
many photons so that the detector
could burn due to the high energy,
since it is not in Eve's interest.
Thus, in some of the analyses below we replace
$\infty$ by some large number $L$.},
since a click in the detector tells Bob that the
number of photons occupying the mode is ``not zero'' i.e.\
the detector clicks when $\fet{n}$ is detected,
for $n \ge 1$.
This means that Bob
measures the state $\fet{0}$, or he measures
$\fet{1}$, $\fet{2}$, $\ldots$ but then ``forgets''
how many photons were detected.
Bob might severely compromise the security,
since he inevitably interprets a
measurement of a state containing multiple photons
as the ``legal'' state that contains only a single photon.
An attack based on a similar limitation is the
``Trojan-Pony'' attack
described below, in Section~\ref{subsec:Trojan}.
In order to avoid false interpretations of the photon number
reaching the detector, Bob could use an enhanced type of detector
known as the {\em photon-number resolving detector}
or a {\em counter} (which is still under development).
This device distinguishes a single photon from $n \ge 2$ photons, hence any
eavesdropping attempt that generates multi-photon states can
potentially be noticed by Bob.
A much enhanced security
can be achieved now,
although the QSoP is infinite also in this case,
due to identifying correctly the legitimate state $\fet{1}$, from
various legitimate states.
The number of modes in the QSoP
depends on Bob's detectors as well.
Bob commonly increases the number of measured
modes by ``opening''
his detector for more time-bin modes or more frequency modes.
For instance, suppose Bob is using a detector
whose detection time-window is quite larger than the width of the
pulse used in the protocol, since he does not know when exactly Alice's
pulse might arrive.
The result is an extension of the space used by Alice,
so that the QSoP includes the subspace
of $M$ that contains all these measured modes.
When a single detector is used to measure
more than one mode {\em without distinguishing them}, the
impact on the security might be severe, see the
``Fake state'' attack (Section~\ref{sec:FakeAsQSA}).
In addition to the known attacks described in the following subsection,
a new QSA is analyzed in Section~\ref{sec:InterferoBB84}, where
we examine the more general case of QSA, in which
Bob adds an ancilla during the process.
\section{Known Attacks as Quantum-Space Attacks}\label{sec:knownQSA}
All known attacks can be considered as special cases of the
Quantum-Space Attack. In this section we show a description of
several such attacks using QSA terms. For each and every attack we
briefly describe the specific protocol used, the quantum space of
the protocol, and a realization of the attack as a QSA.
\subsection{The photon number splitting attack~\cite{BLMS00}}
\label{sec:PNSasQSA}
{\bf The Protocol.} Consider a BB84 protocol, where Alice uses a
``weak pulse'' laser to send photons in two modes corresponding
to the vertical and horizontal polarizations when using the $z$
basis (the diagonal polarizations then relate to using the $x$
basis). Bob uses a device called a Pockel cell to rotate the
polarization (by $45^\circ$) for measuring the $x$ basis, or
performs no rotation if measuring the $z$ basis. The measurement
of the state is then done using two detectors and a ``polarization
beam splitter'' that passes the first mode to one detector and the
second mode to the other detector (for a survey of
polarization-based QKD experiments, see~\cite{GRTZ02, DLH06}).
{\bf The Quantum Space of the Protocol.} Every pulse sent by Alice
is in one of four states, each in a superposition of the 6
orthogonal states $\chi^6 = \{\fet{00}$, $\fet{10}$,
$\fet{01}$,$\fet{11}$,$\fet{20}$, $\fet{02}\}$, where the space
used by Alice is $H^A = H_6$. Bob uses two setups, ${\cal U}_{B_z}=
I$ for the $z$ basis, and ${{\cal U}_{B_x}}$ for the $x$ basis, which
is more complex and described in Appendix~\ref{app:polU}.
The detectors used by Bob cannot distinguish between modes having
single photon and multiple photons. Each one of his two detectors
measures the basis elements $\{ \fet{n} \}$ for $n\ge0$ (of the
specific mode directed to that specific detector), where Bob
interprets the states $\{ \fet{n} \}$ with $n > 1$ as measuring
the state $\fet{1}$ of the same mode. Bob's measured space $H^{B}$
is thus infinite and spanned by the states $\{ \fet{mn} \}$ for
$m,n \ge 0$. The QSoP $H^P$ is equal to $H^{B_z}$ ($=H^{B_x}$)
since performing ${\cal U}^{-1}$ does not change the dimensionality of
the spanned space (in both setups).
{\bf The Attack.}
Eve measures the number of photons in the pulse, using
non-demolition measurement. If she finds that the number of
photons is $\ge 1$, she blocks the pulse and generates a loss. In
the case she finds that the pulse consists of 2 photons, she
splits one photon out of the pulse and sends it to Bob, keeping
the other photon until the bases are revealed, thus getting full
information of the key-bit. Eve sends the eavesdropped qubits to
Bob via a lossless channel so that Bob will not notice the
enhanced loss-rate. As is common in experimental QKD, Bob is
willing to accept a high loss-rate (he does not count losses as
errors), since most of Alice's pulses are empty. See the precise
mathematical description of this attack in
Appendix~\ref{App:MathPNS}.
\subsection{The tagging attack (based on~\cite{GLLP04})}\label{sec:TagAsQSA}
{\bf The Protocol.} Consider a BB84 QKD protocol in which Alice
sends an enlarged state rather than a qubit. This state contains,
besides the information qubit, a {\em tag} giving Eve some
information about the bit. The tag can, for example, tell Eve the
basis being used by Alice. For a potentially realistic example,
let the tag be an additional qutrit indicating if Alice used the
$x$-basis, or the $z$-basis, or whether the basis is {\em
unknown}: whenever Alice switches basis, a single photon comes out
of her lab prior to the qubit-carrying pulse, telling the basis,
say using the states $\fet{10}_{\textit{tag}}$ and
$\fet{01}_{\textit{tag}}$, and when there is no change of basis,
what comes out prior to the qubit is just the vacuum
$\fet{00}_{\textit{tag}}$.
{\bf The Quantum Space of the Protocol.} In this example, Alice is
using the space $H^A = H_2 \otimes H_{\textit{tag}} = H_2 \otimes
H_3$. Bob, unaware of the enlarged space used by Alice, expects
and receives only the subspace $H_2$. We assume that Bob
ideally measures this space with a single setup ${\cal U}_B = I$,
therefore $H^{B} = H_2$. Since Bob's setup does not change the
space, $H^{B^{-1}} = H_2$ as well. However, the tag is of a much
use to Eve, and indeed the QSoP following Definition~\ref{def:HP},
defined to be $H^P = H_2 \otimes H_{\textit{tag}}$.
{\bf The Attack.} Eve uses the tag in order to
retrieve information about the qubit without inducing error (e.g.\
via cloning the qubit in the proper basis).
The attack is then an intercept-resend QSA. We mention that this
attack is very similar to a side-channel cryptanalysis of classic
cryptosystems.
\vskip 12 pt
{\bf A Short Summery.} It can be seen that the PNS attack
described above is actually a special case of the tagging attack,
where the {\it tag} in that case is in fact another copy of the
transmitted qubit. This copy is kept by Eve until the bases are
revealed, then it can be measured so the the key-bit value is
exposed with certainty. Both those QSA attacks are based on the
fact that Alice (realistic) space is larger than the theoretical
one. Although in the PNS example, the QSoP is further extended due
to Bob's measurement, the attack is not based on that extension but
on the fact that $H^A$ is larger than $H_2$. In the following
attacks Bob's measurements cause the enlargement of the QSoP,
allowing Eve to exploit the larger QSoP for her attack.
\subsection{The Trojan-pony attack \cite{GLLP04, HLP04}}
\label{subsec:Trojan}
In Trojan-pony attacks Eve modifies the state sent to Bob in a way
that gives her information. In contrast to a ``Trojan-horse'' that
goes in-and-out of Bob's lab, the ``pony'' only goes in,
therefore, it is not considered an attack on the lab, but only on
the channel. We present here an interesting example~\cite{GLLP04}.
{\bf The Protocol.} Assume a polarization-encoded BB84 protocol,
in which Alice is ideal, namely, sending perfect qubits
($H^A=H_2$). However, Bob uses realistic threshold detectors that
suffer from losses and dark counts, and that cannot distinguish
between one photon and $k$ photons for $1<k<L$.
In order to be able to ``prove'' security, for a longer distance
of transmission Bob wants to keep the error-rate low although the
increase of dark counts' impact with the
distance~\cite{BLMS00}. Therefore, Bob assumes that Eve has no
control over dark counts, and whenever both detectors click, Alice
and Bob agree to consider it as {\em a loss} since it is outside
of Eve's control (i.e.\ the QSoP is falsely considered to be
$H_2$). Namely, they assume that {\em an error} occurs only when
Bob measures in the right basis, and only one detector clicks,
(which is the detector corresponding to the wrong bit-value).
{\bf The Quantum Space of the Protocol.} Same as in
Section~\ref{sec:PNSasQSA}, Bob's measured spaces $H^{B_z}$,
$H^{B_x}$, the reversed space $H^{B^{-1}}$ as well as the QSoP
$H^P$, are merely the spaces describing two modes (with up to $L$
photons), $H_L \otimes H_L$.
Bob's detectors cannot distinguish between receiving a
single-photon pulse from a multi-photon pulse, so his measurement
is properly described as a projection of the received state onto
the space containing $\{\fet{ij}\}$ followed by ``forgetting'' the
exact result, and keeping only one of three results:
``$\{10\}\equiv$ detector-1 clicks'', ``$\{01\}\equiv$ detector-2
clicks'', and else it is $\{00\}$, a ``loss''. In formal, {\em
generalized-measurements} language (called POVM,
see~\cite{Peres93,NC00}) these three possible results are written
as: $\{10\}\equiv \sum_{k=1}^{L-1} \fet{k0}\fra{k0}$,
$\{01\}\equiv \sum_{k=1}^{L-1} \fet{0k}\fra{0k}$, $\{00\}\equiv
\fet{00}\fra{00} + \sum_{k_1,k_2\ =1}^{L-1}
\fet{k_1k_2}\fra{k_1k_2}$, and their sum is the identity matrix.
{\bf The Attack.} Eve's attack is the following: (a) Randomly
choose a basis (b) Measure the arriving qubit in that specific
chosen basis (c) Send Bob $m$-photons identical to the measured
qubit, where $m \gg 1$. Obviously, when Eve chooses the same basis
as Alice and Bob then Bob measures the exact value sent by Alice,
and Eve gets full information. Otherwise, both of his detectors
click, implying a ``loss'', except for a negligible probability,
$\approx 2^{(-m+1)}$, thus Eve induces no errors.
The main observation of this measure-resend QSA is that treating a
count of more than a single photon as a loss, rather than as an
error, is usually not justified. A second conclusion is that
letting Bob use counters instead of threshold detectors (to
distinguish a single photon from multiple photons), together with
treating any count of more than one photon as an error, could be
vital for proving security against QSA. The price is that dark
counts put severe restrictions on the distance to which
communication can still be considered secure, as suggested already
by~\cite{BLMS00}.
\subsection{The fake-state attack (based on~\cite{HM05,MAS06})}
\label{sec:FakeAsQSA}
{\bf The Protocol.} In this example, we examine a polarization
encoded BB84 protocol, and an ideal Alice ($H^A = H_2$). This time
Bob's detectors are imperfect so that their detection windows do
not fully overlap, meaning that there exist times in which one
detector is blocked (or it has a low efficiency), while the other
detector is still regularly active. Thus, if Eve can control the
precise timing of the pulse, she can control whether the photon
will be detected or lost. The setup is built four detectors and a
rotating mirror (since Bob does not want to spend money on a
Pockel cell (polarization rotator), he actually uses 2 fixed
different setups). Using the rotating mirror Bob sends the photon
into a detection setup for basis $z$ or a detection setup for
basis $x$. Suppose the two detection setups use slightly different
detectors, or slightly different delay lines, or slightly
different shutters, and Eve is aware of this (or had learnt it
during her past attacks on the system).
For simplicity, we model the non-overlapping detection windows, as
additional two modes, one slightly prior to Alice's intended mode
(the pulse), and one right after it.
{\bf The Quantum Space of the Protocol.} The original qubit is
sent in a specific time-bin $t_0$ (namely, $H^A=H_2$). The setup
${\cal U}_Z$ is a set of two detectors and a polarized beam splitter,
separating the horizontal and the vertical modes to the detectors,
where ${\cal U}_x$ separate the diagonal modes into a set of two
(different) detectors. Let the detectors for one basis, say $z$,
be able to measure a pulse arriving at $t_0$ or $t_1$, while the
detectors for the other basis ($x$) measure pulses arriving at
$t_{-1}$ or $t_0$.
For simplicity, we degenerate the space to contain one or less
photons\footnote {As mentioned above, this is used for
non-security proof, and is not legitimate assumption for proving
unconditional security, where the three time-modes
should be considered as $H_L \otimes H_L \otimes H_L$.},
so that $H^{B_z}$ is $H_5$, i.e. two possible time-bins consisting
each of two (polarization) modes of one or less photons. The
measured space of the $x$-setup has two possible time-bins and two
possible polarization modes, thus $H^{B_x}=H_5$ as well, however,
the two time-bins for this setup are $t_0$ and $t_1$. Following
Definition~\ref{def:HB-1} we get that that the reversed space
$H^{B^{-1}}$ contains three time-bins ($t_{-1}$, $t_0$ and $t_1$)
with two polarization modes in each, therefore $H^{B^{-1}} = H_7$,
under the single-photon assumption. The QSoP, following
Definition~\ref{def:HP} equals $H^{B^{-1}}$ since $H^A \subset
H^{B^{-1}}$.
{\bf The Attack.} Eve exploit the larger space by sending ``fake''
states using the external time bins ($t_{-1}$ and $t_1$). Eve
randomly chooses a basis, measures the qubit sent by Alice, and
sends Bob the same polarization state she found, but at $t_{-1}$
if she have used the $x$ basis, or at $t_{1}$ if she have used the
$z$ basis. Since no ancilla is kept by Eve, this is an
intercept-resend QSA.
Bob will get the same result as Eve if he uses the same basis,
or {\em a loss} otherwise.
The mathematical description of the
attack is as follows: Eve can generate superpositions of states of
the form $\fet{{V_{t_{-1}}}{H_{t_{-1}}}
{V_{t_0}}{H_{t_0}}{V_{t_1}}{H_{t_1}}}$, where the index $\{H, V
\}$ denotes this mode has Vertical or Horizontal polarization, and
its subscript denotes the time-bin of the mode. Eve's
measure-resend attack is described as measuring Alice's qubit in the
$x$ basis, creating a new copy of the measured qubit, and
performing the transformation $(\fet{001000} \rightarrow
\fet{100000})$; $(\fet{000100} \rightarrow \fet{010000})$ or as
performing a measurement in the $z$ basis, and performing the
transformation $(\fet{001000} \rightarrow \fet{000010})$;
$(\fet{000100} \rightarrow \fet{000001})$ on the generated copy.
\vskip 12 pt
{\bf A short summery}
We see that Eve can ``force'' a desired value (or a loss) on Bob,
thus gaining all the information while inducing no errors (but
increasing the loss rate). Bob can use a shutter to block the
irrelevant time-bins but such a shutter could generate a similar
problem in the frequency domain. This attack is actually a special
case of the Trojan-pony attack, in which the imperfections of
Bob's detectors allow Eve to send states that will be un-noticed
unless the measured basis equals to Eve's chosen basis.
\section{Interferometric BB84 and 6-state Protocols}
\label{sec:InterferoBB84}
In order to demonstrate the power of QSA,
and to see its advantages, this section presents
a partial security analysis
of some interferometric BB84 and 6-state schemes.
Interferometric schemes are more common than any other type
of implementation in QKD
experiments~\cite{T94,MT95,GRTZ02,BBN03,DLH06,MHHTG97}
and products~\cite{idQ,magiQ}.
In this section we define the specific equipment used by Bob,
and we formulate ${\cal U}_B$ and Bob's measurements. We then find
the spaces $H^A$, $H^{B_j}$,
$H^{B^{-1}}$ and the QSoP, $H^P$.
Finally, we demonstrate a novel attack which is found to be very
successful
against a specific variant of the BB84 interferometric scheme;
this specific QSA, which we call
the ``reversed-space attack'', is designed using the
tools
developed in Sections~\ref{sec:QSoP} and~\ref{sec:QSA}.
\subsection{Bob's equipment}\label{sec:xy-setup}
We begin with a description of interferometric
(BB84 and six-state) schemes,
which is based on sending phase-encoded qubits
arriving in two time-separated modes~\cite{T94, MT95}.
Alice encodes her qubit using two time-bins $t'_0$ and $t'_1$,
where a photon in the first mode, $\fet{10}_{t'_0t'_1}$,
represents the state $\ket{0_z}$, and a photon in the other mode,
$\fet{01}_{t'_0t'_1}$, represents $\ket{1_z}$.
The BB84 protocol of~\cite{T94, MT95} (and many others)
uses the $x$ and $y$ bases, meaning that Alice (ideally) sends
one of the following four states:
$\ket{0_x} = (\fet{10}_{t'_0t'_1}+\fet{01}_{t'_0t'_1} ) /\sqrt{2}$;
$\ket{1_x} = (\fet{10}_{t'_0t'_1}-\fet{01}_{t'_0t'_1} ) /\sqrt{2}$;
$\ket{0_y} = (\fet{10}_{t'_0t'_1}+i\fet{01}_{t'_0t'_1} ) /\sqrt{2}$; and
$\ket{1_y} = (\fet{10}_{t'_0t'_1}-i\fet{01}_{t'_0t'_1} ) /\sqrt{2}$.
Bob uses an interferometer built from two beam
splitters with one short path and one long path (Figure~\ref{fig:lab-xy}).
A pulse of light travels through the short arm of the interferometer
in $T_{\rm short}$ seconds, and through the long arm in $T_{\rm long}
= T_{\rm short} +\Delta T$
seconds, where $\Delta T$ is also {\em precisely} the time
separation between the two arriving modes of the qubit, $\Delta T =
t'_1 - t'_0$.
A controlled phase shifter
$P_\phi$, is placed in the long arm of the interferometer.
It performs a phase
shift by a given phase $\phi$,
i.e. $P_\phi(\ket{\psi}) = e^{i\phi}\ket{\psi}$.
The phase shifter is set to $\phi=0$ ($\phi = \pi /2 $)
when Bob measures the $x$ ($y$) basis.
\begin{figure}\label{fig:lab-xy}
\end{figure}
Each beam splitter interferes
two input arms (modes 1, 2) into two output arms (modes 3, 4),
in the following way (for a single photon):
$\fet{10}_{1,2} \mapsto
\frac{1}{\sqrt{2}}\fet{10}_{3,4}+\frac{i}{\sqrt{2}}\fet{01}_{3,4}$, and
$\fet{01}_{1,2} \mapsto
\frac{i}{\sqrt{2}}\fet{10}_{3,4}+\frac{1}{\sqrt{2}}\fet{01}_{3,4}$.
The photon is
transmitted/reflected with a probability of $50\%$;
The transmitted part keeps the same phase as
the incoming photon, while the reflected
part gets an extra phase of $e^{i\pi/2}$, if it carries a single photon.
When a single mode, carrying at least a single photon,
enters a beam splitter
from one arm, and nothing enters the other input arm, we must
consider the other entry to be an additional mode (an ancilla)
in a vacuum state.
When a single mode (carrying one or more photons) enters the interferometer
at time $t'_0$,
see Figure~\ref{fig:lab-xy}, it yields
two modes at time $t_0$ due to traveling through the short arm,
and two modes at time $t_1$ due to traveling through the long arm.
Those four output modes are:
times $t_0$, $t_1$ in the `$s$' (straight) arm of the interferometer,
and times $t_0$, $t_1$ in the `$d$' (down) arm.
A basis state in this Fock space is then
$\fet{n_{s_0}, n_{s_1}, n_{d_0}, n_{d_1}}$.
In the case of having that single mode carrying exactly a single photon,
the transformation, which requires three additional empty ancillas\footnote
{See a brief description in Appendix~\ref{app:interferometer}.},
is $\fet{1}_{t'_0}\fet{000} \mapsto
(\fet{1000}-\fet{0100}+i\fet{0010} +i\fet{0001}) \thickspace / 2$.
Note that
a pulse which is sent at
a different time (say, $t'_x$)
results in the same output state, but with the appropriate
delays, i.e.\
\begin{equation}
\label{eqn:pulse_in_inerferometer}
\fet{1}_{t'_x}\fet{000} \mapsto
(\fet{1000}-\fet{0100}+i\fet{0010} +i\fet{0001}) \thickspace / 2\text{,}
\end{equation}
where the resulting state is defined in the Fock space
whose basis states are $\ket{n_{s_x}, n_{s_{x+1}}, n_{d_x}, n_{d_{x+1}}}$.
Let us now examine any superposition of two modes ($t'_0$ and $t'_1$)
that enter the interferometer one after the other,
with exactly the same time difference $\Delta T$ as the
difference lengths of the arms.
The state evolves in the following way (see Appendix~\ref{app:modesEvo}):
\begin{multline}
\cos\theta\fet{10}_{t'_0t'_1}\fet{0000} +
\sin\theta e^{i\varphi}\fet{01}_{t'_0t'_1}\fet{0000}
\mapsto
\\
\Bigl (
\cos\theta\fet{100000}_B +
(-\cos\theta e^{i\phi} + \sin\theta e^{i\varphi})\fet{010000}_B -
\sin\theta e^{i(\varphi+\phi)}\fet{001000}_B
\\ +
i\cos\theta\fet{000100}_B +
i(\cos\theta e^{i\phi}+\sin\theta e^{i\varphi})\fet{000010}_B +
i\sin\theta e^{i(\varphi +\phi)}\fet{000001}_B \Bigr ) /2
\label{eqn:interf_evu}
\end{multline}
describing the evolution
for any possible BB84 state sent by Alice ($\ket{0_x}$, $\ket{1_x}$,
$\ket{0_y}$, $\ket{1_y}$ determined by the value of $\varphi = 0$,
$\pi$, $\frac{\pi}{2}$, $\frac{3\pi}{2}$ respectively,
when $\theta=\frac{\pi}{4}$).
As a result of this precise timing, these two modes are transformed into
a superposition of 6 possible modes (and not 8 modes) at the outputs,
due to interference at the second beam splitter.
Only four vacuum-states ancillas (and not six)
are required for that process.
The resulting 6 modes are $t_0$, $t_1$, $t_2$ in the `$s$' arm
and in the `$d$' arm of the interferometer.
Denote this Fock space as $H^B$, with basis elements
$\fet{n_{s_0}, n_{s_1}, n_{s_2}, n_{d_0}, n_{d_1}, n_{d_2}}_B$.
The measurement is performed as follows:
Bob opens his detectors at time $t_1$ in both
output arms of the interferometer.
A click in the ``down'' direction
means measuring the bit-value $0$, while
a click in the ``straight'' direction means $1$.
The other modes are commonly considered as a
loss (they are not measured)
since they give an inconclusive result
regarding the original qubit.
We refer this BB84 variant as ``$xy$-BB84''.
One might want to use the $z$ basis in his QKD protocol
(using $\varphi = 0$, and $\theta=0$ or $\theta = \frac{\pi}{2}$),
for instance, in order to avoid the need for a controlled
phase shifter or for another equipment-related reason,
or in order to perform
``QKD with classical Bob''~\cite{BKM07}.
A potentially more important reason might be to perform
the 6-state QKD~\cite{Brus98, BG99,L01} protocol, due to its
improved immunity against errors (27.4\% errors versus only 20\% in
BB84~\cite{C02}).
A possible and easy to implement variant for realizing a measurement
in the $z$ basis is the following:
Bob uses the setup ${\cal U}_{B_x}$
(i.e.\ he sets $P_\phi$ to $\phi = 0$),
and opens his detectors at times $t_0$ and $t_2$,
corresponding to the bit-values $0$ and $1$ respectively
(See Equation~(\ref{eqn:interf_evu})).
Unfortunately, technological limitations,
e.g.\ of telecommunication
wavelength (IR) detectors,
might make it difficult for Bob
to open his detectors for more than
a single detection window per pulse.
Bob could perform a measurement of {\em just} the states
$\{\fet{000100}_B,\fet{001000}_B \}$,
opening the $d$ arm detector at time $t_0$
(to measure $\ket{0_z}$) and the $s$ arm detector
at time $t_2$ (to measure $\ket{1_z}$).
We refer this variant as ``$xyz$-six-state''.
\subsection{The quantum space of the interferometric protocols}
\label{sec:ibb84-QSOP}
We assume Alice to be almost ideal, having the realistic space $H^A=H_3$
(a qubit or a vacuum state), using two time-bin modes.
As we have seen,
four ancillary modes in vacuum states are added to each transmission.
Therefore, the interferometer setups ${\cal U}_{B_x}$ and ${\cal U}_{B_y}$
transform the 2-mode states of $H^A$ into a subspace that
resides in the 6 modes space $H^B$.
For simplicity, we assume that Eve does not generate $n$-photon states,
with $n \ge 2$,
so we can ignore high photon numbers in the $H^B$ space\footnote
{As mentioned in Section~\ref{sec:QSoP}, this assumption is not legitimate
when proving unconditional security of a protocol.}.
Therefore, we redefine $H^B = H_7$, the space spanned by the vacuum,
and the six single-photon terms in each of the above modes.
Using the $x$ and $y$ bases,
Bob measures only time-bin $t_1$, so his actual measured spaces
consist of two modes: time-bin $t_1$ in the `$s$' arm and the `$d$' arm.
In that case, the measured spaces are $H^{B_x} = H^{B_y} = H_3$,
spanned by the states
$\{\fet{000000}_B$, $\fet{010000}_B$, $\fet{000010}_B \}$.
When Bob uses the $z$ basis,
he measures two different modes, so $H^{B_z}$ is spanned
by the states $\{\fet{000000}_B$, $\fet{000100}_B$, $\fet{001000}_B \}$.
Let us define the appropriate space
$H^{B^{-1}}$ for the 6-state protocol,
according to Definition~\ref{def:HB-1+anc}.
The space $H^{B^{-1}}$ is spanned by the states given by performing
${\cal U} \in \{ {\cal U}_{B_x}, {\cal U}_{B_y} \}$
on $\{\fet{000000}_{B}$, $\fet{010000}_{B}$, $\fet{000010}_{B} \}$,
as well as the states given by performing ${\cal U}_{B_z}$ on
$\{\fet{000000}_{B}$, $\fet{000100}_{B}$ , $\fet{001000}_{B} \}$.
Interestingly, once applying ${\cal U}^{-1}$, the resulting states
are embedded in an 8-mode space
defined by the two
incoming arms of the interferometer, `$a$' (from Alice) and `$b$' (from Bob),
at time bins $t'_{-1}$, $t'_{0}$, $t'_{1}$, and $t'_{2}$. The basis states
of $H^{B^{-1}}$ are listed in Appendix~\ref{app:interf_HB-1_basis}.
Following Definition~\ref{def:HP+anc},
the QSoP $H^P$ of this implementation for the 6-state protocol,
is the subsystem of $H^{B^{-1}}$
which is {\em controlled} by Eve.
It is spanned by the 8-mode states spanning
$H^{B^{-1}}$ after tracing out Bob.
The space that contains those ``traced-out'' states
has only four modes that are controlled by Eve, specifically, input `$a$'
of the interferometer at times $t'_{-1}$ to $t'_2$,
having a basis state of the form
$\fet{a_{t'_{-1}}a_{t'_0}a_{t'_1}a_{t'_2}}_P$.
Given the single-photon restriction, we get
$H^P = H_5$, namely,
the space spanned by the vacuum state,
and a single photon in each of the four modes, i.e.\
$\{ \fet{0000}_{P}$, $\fet{1000}_{P}$,
$\fet{0100}_P$, $\fet{0010}_P$ , $\fet{0001}_P \}$.
This same result is obtained also if Bob measures all the
six modes in $H^B$.
Bob might want to see how the basis states of the
4-mode QSoP, $H^P$,
evolve through the interferometer
in order to place detectors on the resulting modes, which
will be used to identify Eve's attack.
It is interesting to note, that those basis states result in
{\em 10 different non-empty modes (!)}.
If Bob measures all these modes, he {\em increases} the QSoP,
and maybe allows Eve to attack a larger space,
and so on and so forth.
Therefore, in order to perform a security analysis,
one must first fix the scheme and only then assess the QSoP.
Otherwise, a ``ping-pong'' effect might
increase the spaces' dimensions to infinity.
A similar, yet reversed logic, hints that it could actually be better
for Bob, in terms of the simplicity of the analysis for
the ``$xy$-BB84'' scheme, to measure
{\em just} the two modes at $t_1$
(i.e.\ the space spanned by $\fet{0,n_{s_1},0,0,n_{d_1},0}_B$),
thus reducing the QSoP to a 2-mode space, $H^P=H^A$, see
Appendix~\ref{app:QSoP2ModesIntf}.
Although Eve is allowed to attack a larger
space than this two-mode $H^P$, she has no advantage in doing so:
pulses that enter the interferometer on different modes
(i.e.\ other time-bins than $t'_0$ and $t'_1$), never interfere
with the output pulses of time-bin $t_1$ measured by Bob. Therefore,
state occupying different modes can not
be distinguished from the states in which those modes are empty.
\subsection{The ``Reversed-Space'' attack on interferometric protocols}
\label{sec:RSA}
Consider a BB84 variant
in which Bob uses only the $x$ and the $z$ bases,
using a single interferometer, where the $z$-basis measurement
is performed according to the description in the last few lines
of Section~\ref{sec:xy-setup}. We refer this variant as
``$xz$-BB84''. The QSoP of this scheme, $H^P$ is the space
described above for the ``$xyz$-six-state'' protocol.
The following attack
\begin{align}
\ket{0}_{E}\ket{0100}_{P} \mapm{{\cal U}_{E}} &
{\frac{1}{2}}\ket{E_0}_{ E}\mathbf{i}gl (\fet{1000}_{P} + \fet{0100}_{P} \mathbf{i}gr )
{}+ {\frac{1}{2}}\ket{E_1}_{ E}\mathbf{i}gl (\fet{0010}_{P} + \fet{0001}_{P} \mathbf{i}gr )
\\
\ket{0}_{E}\ket{0010}_{P} \mapm{{\cal U}_{E}} &
{\frac{1}{2}}\ket{E_1}_{ E}\mathbf{i}gl (-\fet{1000}_{P} + \fet{0100}_{P} \mathbf{i}gr )
{}+{\frac{1}{2}}\ket{E_0}_{ E}\mathbf{i}gl (\fet{0010}_{P} - \fet{0001}_{P} \mathbf{i}gr )
\end{align}
which we call ``the Reversed-Space Attack'', allows Eve
to acquire information about the
transmitted qubits, without inducing {\em any} errors.
The states $\ket{\cdot}_E$ denote Eve's ancilla
which is not necessarily a photonic system.
The state $\ket{0_z}_{A} \equiv \fet{0100}_P$ and
$\ket{1_z}_{A} \equiv \fet{0010}_P$ are the regular states
send by Alice, where we added the relevant extension
of $H^A$ in $H^P$.
When $\ket{0_z}_{A}$ is sent by Alice,
the attacked state ${\cal U}_{E} \ket{0}_{E}\ket{0_z}_{A}$
reaches Bob's interferometer, and interferes
in a way such that it can never reach Bob's detector at time $t_2$,
i.e. $\fra{001000}_B {\cal U}_{B_x} \left(
\left ({\cal U}_{E}\ket{0}_{E}\ket{0_z}_{A} \right )\fet{0000}_{B'} \right )=0$.
Although the attacked state ${\cal U}_{E}\ket{0}_{E}\ket{0_z}_{A} $
reaches modes that Alice's original state $\ket{0_z}_{A}$ can never
reach, Bob never measures those modes, and cannot notice the attack.
A similar argument applies when Alice sends $\ket{1_z}_{A}$.
As for the $x$ basis\footnote{For simplicity
we use the shorter notation
$\ket{0_x}\equiv(\fet{0100}_P+\fet{0010})/\sqrt{2}$, etc.},
this attack satisfies
\begin{multline}
\ket{0}_{E}\ket{0_x}_{A} \mapsto
\\
\frac{1}{\sqrt{8}}
(\ket{E_0}_{ E} + \ket{E_1}_{ E})
(\fet{0100}_P + \fet{0010}_P) +
\frac{1}{\sqrt{8}}
(\ket{E_0}_{ E} - \ket{E_1}_{ E})
(\fet{1000}_P - \fet{0001}_P) \phantom{1\text{.}}
\end{multline}
\begin{multline}
\ket{0}_{E}\ket{1_x}_{A} \mapsto
\\
\frac{1}{\sqrt{8}}
(\ket{E_0}_{ E} - \ket{E_1}_{ E})
(\fet{0100}_P - \fet{0010}_P) +
\frac{1}{\sqrt{8}}
(\ket{E_0}_{ E} + \ket{E_1}_{ E})
(\fet{1000}_P + \fet{0001}_P) \text{.}
\end{multline}
The first element in the sum results in the desired interference in Bob's lab,
while the second is not measured by Bob's detectors at time $t_1$.
By letting Eve's probes $\ket{E_0}_{E}$ and $\ket{E_1}_{E}$ be orthogonal states,
Eve gets a lot of information while inducing no errors at all.
Yet, we find that Eve is increasing the loss rate by this attack to
87.5\%, but a very high loss rate is anyhow expected by Bob
(as explained in the analysis of the PNS~\cite{BLMS00}
and the tagging~\cite{GLLP04} attacks).
In conclusion,
this attack demonstrates the risk of using various
setups without giving full security analysis for the {\em specific} setup.
We are not familiar
with any other security analysis that takes into account the enlarged
space generated by the inverse-transformation of
Bob's space.
\section{Conclusion}
\label{sec:conclusion}
In this paper we have defined the QSA,
a novel attack that generalizes all currently
known attacks on the channel.
This new attack brings a new
method for performing security analysis of protocols.
The attack is based on a realistic view of
the quantum spaces involved, and in particular, the spaces
that become larger than the theoretical ones,
due to practical considerations.
Although this paper is explicitly
focused on the case of uni-directional
implementations of a few schemes, its main observations
and methods apply to any uni-directional
QKD protocol, to bi-directional QKD protocols,
and maybe also to any realistic
quantum cryptography scheme beyond QKD.
The main conclusion of this research is that the quantum space which is
attacked by Eve can be assessed, given a proper understanding of the
experimental limitations. This assessment requires a novel cryptanalysis
formalism --- analyzing the states generated in Alice's lab, as well as
the states that are to be measured by
Bob (assessing them as if they go backwards in time from Bob's lab);
this type of analysis resembles the two-time formalism in quantum
theory~\cite{AAD85,VAA87}.
Open problems for further theoretical research include:
1.-- Generalization of the QSA to other conventional
protocols (such as the two-state
protocol, EPR-based protocols, d-level protocols, etc.); such a generalization
should be rather straightforward.
2.-- Proving unconditional security (or more limited security results such as
``robustness''~\cite{BKM07}) against various QSAs. This is especially important for the
interferometric setup, where the QSoP is much larger than Alice's
six-dimensional space (the one spanned by $\chi^6$).
3.-- Describing the QSA for more complex protocols, such
as two-way protocols \cite{MHHTG97,BF02, BKM07}
in which the quantum communication is bi-directional,
and protocols which use a larger set of
states such as data-rejected protocols \cite{BHP93}
or decoy-state protocols \cite{H03,W05,LMC05, YSS07}.
4.-- Extend the analysis and results to composable QKD~\cite{BHLMO05}.
5(a).-- In some cases, if Bob uses ``counters''
and treats various measurement outcomes as errors,
the effective QSoP relevant for proving security is potentially
{\em much smaller} than the QSoP defined here.
5(b).-- Adding counters on more modes
increases the QSoP defined here, but might
allow analysis of a smaller ``attack's QSoP'',
if those counters are used
to identify Eve's attack.
More generally,
the connection between the way Bob interprets
his measured outcomes, and the
``attack's QSoP'' is yet to be further analyzed.
\paragraph{{\bf Acknowledgments.}}
We thank Michel Boyer, Dan Kenigsberg and Hoi-Kwong Lo
for helpful remarks.
\begin{appendix}
\section*{Appendix}
\section{Mathematical Description of the PNS attack}
\label{App:MathPNS}
The PNS attack can be realized using (an infinite set of)
polarization independent beams splitters.
Eve uses a beam splitter to split photons from Alice's state.
Using a non-demolition measurement Eve measures the number of photons
in one output of the beam splitter, and repeat the splitting
until she acquires exactly one photon.
Formally ${\cal U}_E$ is defined:
\begin{align*}
\fet{00}_E\fet{02}_A &\mapsto \fet{01}_E\fet{01}_P &
\fet{00}_E\fet{10}_A &\mapsto \fet{10}_E\fet{00}_P \\
\fet{00}_E\fet{20}_A &\mapsto \fet{10}_E\fet{10}_P &
\fet{00}_E\fet{01}_A &\mapsto \fet{01}_E\fet{00}_P \\
\fet{00}_E\fet{11}_A &\mapsto (\fet{01}_E\fet{10}_P +
\fet{10}_E\fet{01}_A) / \sqrt{2}\text{.}
\end{align*}
Whenever Alice sends a pulse with two photons of
the same polarization, Eve and Bob end up, each, with
having a single photon of the original polarization.
\begin{proposition}
Eve's PNS attack for a pulse of 2 photons,
gives Eve full information while inducing no errors.
\end{proposition}
\begin{proof}
According to its definition
it is trivial to verify the
attack for the horizontal and
vertical polarizations $\ket{0_z}^{(2)}$ and $\ket{1_z}^{(2)}$
(where $\ket{P}^{(k)}$ means $k$ photons having polarization $P$).
Using the standard creation and annihilation operators ($a^\dagger$ and $a$)\footnote{
See any quantum optics book, e.g.\ \cite{SZ97}},
we can write the state of two photons in the diagonal polarization ($x$ basis):
$\ket{0_x}^{(2)} =
\left (\frac{1}{\sqrt{2}} (a^\dagger_1 + a^\dagger_2) \right )^2 \fet{00} =
\frac{1}{2}\mathbf{i}gl (\fet{20} + \sqrt{2}\fet{11} + \fet{02}\mathbf{i}gr ) $,
similarly $\ket{1_x}^{(2)} = \frac{1}{2}\mathbf{i}gl
(\fet{20} - \sqrt{2}\fet{11} + \fet{02}\mathbf{i}gr ) $.
\begin{eqnarray*}
\fet{00}_E\ket{0_x}^{(2)}_{ P} &\equiv & \frac{1}{2}
\fet{00}_E \mathbf{i}gl (\fet{20} +
\sqrt{2}\fet{11} + \fet{02}\mathbf{i}gr )_P \\
&\mapm{{\cal U}_E} & \frac{1}{2} \mathbf{i}gl (\fet{10}_E\fet{10}_P +
\fet{01}_E\fet{10}_P + \fet{10}_E\fet{01}_P +
\fet{01}_E\fet{01}_P \mathbf{i}gr ) \\
& = & \frac{1}{2} \mathbf{i}gl ( (\fet{10}_E+\fet{01}_E )\fet{10}_P
+ (\fet{10}_E+\fet{01}_E )\fet{01}_P \mathbf{i}gr ) \\
& = & \frac{1}{2} (\fet{10}_E+\fet{01}_E)(\fet{10}_P+\fet{01}_P) \\
&\equiv & \ket{0_x}_{ E}\ket{0_x}^{(1)}_{ P}
\end{eqnarray*}
\begin{eqnarray*}
\fet{00}_E\ket{1_x}^{(2)}_{ P} &\equiv &
\frac{1}{2} \fet{00}_E \mathbf{i}gl(\fet{20} - \sqrt{2}\fet{11} + \fet{02}\mathbf{i}gr )_P \\
&\mapm{{\cal U}_E}& \frac{1}{2} \mathbf{i}gl (\fet{10}_E\fet{10}_P -
\fet{01}_E\fet{10}_P - \fet{10}_E\fet{01}_P +
\fet{01}_E\fet{01}_P \mathbf{i}gr ) \\
& = & \frac{1}{2} \mathbf{i}gl ( (\fet{10}_E-\fet{01}_E )\fet{10}_P
- (\fet{10}_E-\fet{01}_E )\fet{01}_P \mathbf{i}gr ) \\
& = & \frac{1}{2} (\fet{10}_E-\fet{01}_E)(\fet{10}_P-\fet{01}_P) \\
&\equiv & \ket{1_x}_{ E}\ket{1_x}^{(1)}_{ P}
\end{eqnarray*}
Which completes the proof.
\end{proof}
\subsection{Polarization change}
\label{app:polU}
A Polarization based QKD protocol makes a use of a Pockel cell (${\cal U}_{B_x}$),
rotating the polarization of the photons going through it.
For a single photon, its action is trivial,
\begin{align}
\nonumber
\fet{10} &\mapm{{\cal U}_{B_x}}
\frac{1}{\sqrt{2}} \left (\fet{10}+\fet{01} \right )\text{, and} \\
\nonumber
\fet{01} &\mapm{{\cal U}_{B_x}}
\frac{1}{\sqrt{2}} \left (\fet{10}-\fet{01} \right )\text{.} \\
\end{align}
For a state that contains multiple photons, the transformation is not
intuitive, and most simply defined using the creation and annihilation operators.
In a somewhat simplified way, the Pokcel cell
can be considered as performing
$a^\dagger_1 \mapsto \left (\frac{1}{\sqrt{2}} (a^\dagger_1 + a^\dagger_2) \right )$
and
$a^\dagger_2 \mapsto \left (\frac{1}{\sqrt{2}} (a^\dagger_1 - a^\dagger_2) \right )$,
so that a state is transformed in the following way
\begin{align}
\fet{nm} = {\left (a^\dagger_1 \right)}^n {\left (a^\dagger_2 \right )}^m
\fet{00} \mapm{{\cal U}_{B_x}}
{\left (\frac{1}{\sqrt{2}} (a^\dagger_1 + a^\dagger_2) \right )}^n
{\left (\frac{1}{\sqrt{2}} (a^\dagger_1 - a^\dagger_2) \right )}^m\fet{00}
\text{.}
\end{align}
\section {QSoP of the Interferometeric Scheme: Supplementary Information}
\subsection{A (brief) graphical description of pulses evolution through interferometer}
\label{app:interferometer}
See Figure~\ref{fig:1evo} for evolution of a single occupied mode through
the interferometer, and Figure~\ref{fig:2evo} for evolution of
two superpositioned modes.
\begin{figure}
\caption{Evolution in time of a single photon pulse through an interferometer
satisfying $\ket{1000}
\label{fig:1evo-t0}
\label{fig:1evo-t1}
\label{fig:1evo-t2}
\label{fig:1evo-t3}
\label{fig:1evo}
\end{figure}
\begin{figure}
\caption{Evolution in time of two modes through an interferometer satisfying
$ (\alpha\ket{1}
\label{fig:2evo-0}
\label{fig:2evo-1}
\label{fig:2evo-2}
\label{fig:2evo-3}
\label{fig:2evo-4}
\label{fig:2evo}
\end{figure}
\subsection{Evolution of modes through the interferometer}
\label{app:modesEvo}
In order to simplify the analysis
(a simplification that is not allowed when proving the full security of
a scheme) we look at the
ideal case in which exactly one photon (or none)
is sent by Alice.
The basis states are then the vacuum $\fet{000000}_B \equiv \fet{V}_B$,
and the six states (that we denote for simplicity by)
$\fet{100000}_B \equiv \fet{s_0}_B$;
$\fet{010000}_B \equiv \fet{s_1}_B$;
$\fet{001000}_B \equiv \fet{s_2}_B$;
$\fet{000100}_B \equiv \fet{d_0}_B$;
$\fet{000010}_B \equiv \fet{d_1}_B$ and
$\fet{000001}_B \equiv \fet{d_2}_B$.
The full transformation of a single photon pulse through the
interferometer is given by Equation~\eqref{eqn:pulse_in_inerferometer}.
Alice sends photons at time bins $t'_0$ and $t'_1$ only, so
the interferometer transformation on Alice's basis states is
$\fet{00}_A\fet{0000}_{\hat B} \mapsto \fet{V}_{B}$, and
\begin{eqnarray}\label{eqnB+-On01}
\begin{array}{l}
\fet{10}_A\fet{0000}_{\hat B} \mapsto
(\fet{s_0}_{B}-e^{i\phi}\fet{s_1}_{B}+i\fet{d_0}_{B}
+ie^{i\phi}\fet{d_1}_{B}) \thickspace / 2 \\
\fet{01}_A\ket{0000}_{\hat B} \mapsto
(\fet{s_1}_{B}-e^{i\phi}\fet{s_2}_{B}+i\fet{d_1}_{B}
+ie^{i\phi}\fet{d_2}_{B})\thickspace / 2
\ ,
\end{array}
\end{eqnarray}
where $\ket{0000}_{\hat B}$ denotes ancilla added
during the process\footnote{Those ancillas (the space $H^{\hat B}$) are originated
by Alice extended space $H^P$ and by Bob ($H^{B'}$).
Performing ${\cal U}^{-1}$ reveals the exact origin of those ancillas.}.
Equation~\ref{eqnB+-On01} can be used to describe the interferometer effect on a
general qubit, shown in Equation~\eqref{eqn:interf_evu}.
The states sent by Alice during the ``$xy$-BB84'' protocol
evolve in the interferometer as follows:
\begin{eqnarray}
\begin{array}{rcl}
\label{eqnB+-On+-}
\ket{0_x}_{A} &\mapm{\phi=0} &
(\fet{s_0}_{B} \phantom{{}-2\fet{s_1}_B} -\fet{s_2}_{B}
+i\fet{d_0}_{B} +2i \fet{d_1}_{B} +i\fet{d_2}_{B}) \thickspace / \sqrt{8}\\
\ket{1_x}_{A} &\mapm{\phi=0} &
(\fet{s_0}_{B} -2 \fet{s_1}_{B} +\fet{s_2}_{B}
+i\fet{d_0}_{B} \phantom{{}+2i\fet{d_1}_B} -i\ket{d_2}_{B}) \thickspace / \sqrt{8} \\
\ket{0_y}_{A} &\mapm{\phi=\pi/2}&
(\fet{s_0}_{B} \phantom{{}+2i\fet{s_1}_B} +\fet{s_2}_{B}
+i\fet{d_0}_{B} -2 \fet{d_1}_{B} -i\fet{d_2}_{B}) \; /\sqrt{8}\\
\ket{1_y}_{A} & \mapm{\phi=\pi/2} &
(\fet{s_0}_{B} {}-2i \fet{s_1}_{B} -\fet{s_2}_{B}
+i\fet{d_0}_{B} \phantom{{}+2\fet{d_1}_B} +i\fet{d_2}_{B}) \; /\sqrt{8}
\end{array}
\end{eqnarray}
Bob can distinguish the computation basis elements
of bases $x$ and $y$, measuring time-bin $t_1$, i.e.\ the states $\fet{d_1}$ for $\ket{0}$
and $\fet{s_1}$ for $\ket{1}$ in the measured basis.
Other states give Bob no information
about the state sent by Alice.
\subsection{$H^{B^{-1}}$ of the ``$xyz$-six-state'' scheme}
\label{app:interf_HB-1_basis}
Let Bob be using interferometric
setups ${\cal U}_{B_x}$ and measuring 6 modes
(corresponding the space with a basis
state $\fet{n_{s_0}n_{s_1}n_{s_2}n_{d_0}n_{d_1}n_{d_2}}_B$) with
one or less photons.
Following Definition~\ref{def:HB-1+anc},
the states spanning the space $H^{B^{-1}}$ can be derived
using Equation~\eqref{eqn:interf_evu} (adjusted to the appropriate space):
\begin{align}
\nonumber
\fet{000000}_B &\mapm{{\cal U}_{B_x}^{-1} } \fet{00000000}_{PB'}
\\
\nonumber
\fet{010000}_B &\mapm{{\cal U}_{B_x}^{-1} }
\frac{1}{2} \left (-\fet{01000000}_{PB'} + \fet{00100000}_{PB'}
-i \fet{00000100}_{PB'} -i\fet{00000010}_{PB'} \right )
\\
\nonumber
\fet{000010}_B &\mapm{{\cal U}_{B_x}^{-1}}
\frac{1}{2} \left (-i\fet{01000000}_{PB'} -i \fet{00100000}_{PB'}
+ \fet{00000100}_{PB'} -\fet{00000010}_{PB'} \right )
\\
\nonumber
\fet{000000}_B &\mapm{{\cal U}_{B_z}^{-1} } \fet{00000000}_{PB'}
\\
\nonumber
\fet{001000}_B &\mapm{{\cal U}_{B_z}^{-1}}
\frac{1}{2} \left (-\fet{00100000}_{PB'} + \fet{00010000}_{PB'}
-i \fet{00000010}_{PB'} -i\fet{00000001}_{PB'} \right )
\\
\nonumber
\fet{000100}_B &\mapm{{\cal U}_{B_z}^{-1}}
\frac{1}{2} \left (-i\fet{10000000}_{PB'} -i \fet{01000000}_{PB'}
+ \fet{00001000}_{PB'} -\fet{00000100}_{PB'} \right )
\\
\nonumber
\fet{000000}_B &\mapm{{\cal U}_{B_y}^{-1} } \fet{00000000}_{PB'}
\\
\nonumber
\fet{010000}_B &\mapm{{\cal U}_{B_y}^{-1} }
\frac{1}{2} \left (i\fet{01000000}_{PB'} + \fet{00100000}_{PB'}
- \fet{00000100}_{PB'} -i\fet{00000010}_{PB'} \right )
\\
\fet{000010}_B &\mapm{{\cal U}_{B_y}^{-1}}
\frac{1}{2} \left (-\fet{01000000}_{PB'} -i \fet{00100000}_{PB'}
-i \fet{00000100}_{PB'} -\fet{00000010}_{PB'} \right )
\end{align}
defined over the space $H^P \otimes H^{B'}$ with basis state
$\fet{a_{t'_{-1}}a_{t'_0}a_{t'_1}a_{t'_2}b_{t'_{-1}}b_{t'_0}b_{t'_1}b_{t'_2}}_{PB'}$.
Note that performing ${\cal U}^{-1}$ requires an additional ancilla,
since the modes number increases from six to eight.
\subsection{QSoP of the ``$xy$-BB84'' scheme}
\label{app:QSoP2ModesIntf}
Assume Bob measures only time-bin $t_1$ in both output arms of the
interferometer, i.e.\ the measured space is $H^B$ subspace
spanned by $\fet{0,n_{s_1},0,0,n_{d_1},0}_B$.
Assuming a single-photon restriction,
the reversed space, of that measured space that is spanned by:
\begin{align}
\nonumber
\fet{000000}_{B} &\mapm{{\cal U}_{B_x}^{-1}} \fet{0000}_{PB'}
\\
\nonumber
\fet{010000}_{B} &\mapm{{\cal U}_{B_x}^{-1}}
\frac{1}{2} \left (-\fet{1000}_{PB'} + \fet{0100}_{PB'}
-i \fet{0010}_{PB'} -i\fet{0001}_{PB'} \right )
\\
\nonumber
\fet{000010}_{B} &\mapm{{\cal U}_{B_x}^{-1}}
\frac{1}{2} \left (-i\fet{1000}_{PB'} -i \fet{0100}_{PB'}
+ \fet{0010}_{PB'} -\fet{0001}_{PB'} \right )
\\
\nonumber
\fet{000000}_{B} &\mapm{{\cal U}_{B_y}^{-1}} \fet{0000}_{PB'}
\\
\nonumber
\fet{010000}_{B} &\mapm{{\cal U}_{B_y}^{-1}}
\frac{1}{2} \left (i\ket{1000}_{PB'} + \fet{0100}_{PB'}
- \fet{0010}_{PB'} -i\fet{0001}_{PB'} \right )
\\
\fet{000010}_{B} &\mapm{{\cal U}_{B_y}^{-1}}
\frac{1}{2} \left (-\fet{1000}_{PB'} -i \fet{0100}_{PB'}
-i \fet{0010}_{PB'} -\fet{0001}_{PB'} \right )
\label{eqn:x4bb84_HB-1_basis}
\end{align}
as can be verified using Equation~\eqref{eqn:interf_evu}.
The space $H^{B^{-1}}$ is embedded in a 4-mode space $H^P\otimes H^{B'}$,
having the basis
element $\fet{a_{t'_0}a_{t'_1}b_{t'_0}b_{t'_1}}_{PB'}$, i.e.\
Alice modes at times $t'_0$ and $t'_1$ and Bob's added ancillary modes at times
$t'_0$ and $t'_1$ respectively.
The resulting six states \eqref{eqn:x4bb84_HB-1_basis}
span a 4-dimensional space, i.e.\ $H^{B^{-1}}=H_4$. The QSoP in this special case
is $H^P = H_3$, spanned by $\fet{a_{t'_0}a_{t'_1}}$ with one or less photons.
\end{appendix}
\end{document} |
\begin{document}
\title{Sampling of globally depolarized random quantum circuit}
\begin{flushright}
YITP-19-90
\end{flushright}
\author{Tomoyuki Morimae}
\email{[email protected]}
\affiliation{Yukawa Institute for Theoretical Physics,
Kyoto University, Kitashirakawa Oiwakecho, Sakyoku, Kyoto
606-8502, Japan}
\affiliation{JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama,
332-0012, Japan}
\author{Yuki Takeuchi}
\email{[email protected]}
\affiliation{NTT Communication Science Laboratories,
NTT Corporation, 3-1 Morinosato Wakamiya,
Atsugi, Kanagawa 243-0198, Japan}
\author{Seiichiro Tani}
\email{[email protected]}
\affiliation{NTT Communication Science Laboratories,
NTT Corporation, 3-1 Morinosato Wakamiya,
Atsugi, Kanagawa 243-0198, Japan}
\date{\today}
\begin{abstract}
The recent
paper [F. Arute et al. Nature {\bf 574}, 505 (2019)]
considered exact classical sampling of
the output probability distribution of
the globally depolarized
random quantum circuit.
In this paper, we show three results.
First, we consider the case
when the fidelity $F$ is constant.
We show that if the distribution is
classically sampled in polynomial time within a constant
multiplicative error, then
${\rm BQP}\subseteq{\rm SBP}$,
which means that BQP is in the second level of the
polynomial-time hierarchy.
We next show that for any $F\le1/2$,
the distribution is classically trivially sampled by the uniform
distribution
within the multiplicative error $F2^{n+2}$,
where $n$ is the number of qubits.
We finally show that for any $F$,
the distribution is classically trivially sampled by the uniform
distribution
within the additive error $2F$.
These last two results show that if we consider
realistic cases, both $F\sim2^{-m}$ and $m\gg n$,
or at least $F\sim2^{-m}$,
where $m$ is the number of gates,
quantum supremacy does not exist
for approximate sampling even with the exponentially-small errors.
We also argue that if $F\sim2^{-m}$ and $m\gg n$,
the standard approach will not work
to show quantum supremacy even for
exact sampling.
\end{abstract}
\maketitle
\section{Introduction}
Several sub-universal quantum computing
models are shown to be hard to classically simulate.
For example, output probability distributions of
the depth-four model~\cite{TD},
the Boson Sampling model~\cite{BS},
the IQP model~\cite{IQP1,IQP2},
the one-clean qubit
model~\cite{KL,MFF,M,Kobayashi,KobayashiICALP},
and the random circuit model~\cite{random1,random2}
cannot be classically sampled in polynomial time unless some conjectures
in classical complexity theory (such as the infiniteness of the polynomial-time
hierarchy) are refuted.
Impossibilities of exponential-time classical simulations of sub-universal
quantum computing models
have also been shown recently based on classical fine-grained complexity
theory~\cite{Dalzell,DalzellPhD,Huang,Huang2,MorimaeTamaki}.
Let $p_z$ be the probability that an $n$-qubit ideal
random quantum circuit outputs the $n$-bit string $z\in\{0,1\}^n$.
Ref.~\cite{Google}
considered the globally depolarized version
where
the probability $p_z'$ that the output is
$z\in\{0,1\}^n$ is written as
\begin{eqnarray*}
p_z'=Fp_z+\frac{1-F}{2^n},
\end{eqnarray*}
where $0< F<1$ is the fidelity.
In this paper, we show the following three results:
\begin{theorem}
\label{theorem:result}
Assume that $F$ is constant. Then,
if the probability distribution $\{p_z'\}_{z\in\{0,1\}^n}$
is sampled in classical $poly(n)$ time within a constant multiplicative
error $\epsilon<1$,
then ${\rm BQP}\subseteq{\rm SBP}$.
\end{theorem}
\begin{theorem}
\label{theorem:result2}
For any $F\le\frac{1}{2}$,
$\{p_z'\}_{z\in\{0,1\}^n}$ is classically sampled
by the uniform distribution within the
multiplicative error $F2^{n+2}$.
\end{theorem}
\begin{theorem}
\label{theorem:result3}
For any $F$,
$\{p_z'\}_{z\in\{0,1\}^n}$ is classically sampled
by the uniform distribution within the additive error $2F$.
\end{theorem}
Proofs are given in the later sections.
In the rest of this section, we provide several remarks.
First, the class SBP~\cite{SBP} is defined as follows.
\begin{definition}
A language $L$ is in SBP if and only if there exist a polynomial
$s$
and a classical polynomial-time probabilistic algorithm
such that if
$x\in L$ then $p_{acc}\ge 2^{-s(|x|)}$, and
if $x\notin L$ then $p_{acc}\le 2^{-s(|x|)-1}$.
Here, $p_{acc}$ is the acceptance probability.
\end{definition}
Note that the class SBP remains unchanged even when the
two thresholds, $2^{-s(|x|)}$ and $2^{-s(|x|)-1}$, are replaced with
$\alpha 2^{-s(|x|)}$ and $\beta2^{-s(|x|)}$, respectively, for any
constants
$\alpha$ and $\beta$ satisfying $0\le\beta<\alpha\le1$.
It is known that SBP is in AM, and therefore ${\rm BQP}\subseteq{\rm SBP}$
means that BQP is in the second level of the polynomial-time hierarchy.
The containment of BQP in the polynomial-time hierarchy is not
believed. (For example, there is an oracle separation~\cite{Raz}.)
Second,
note that
quantum supremacy for the globally depolarized circuits
was previously studied
in Ref.~\cite{nonclean}
for the one-clean qubit model.
Third, Theorem~\ref{theorem:result} holds for a broader class
of quantum circuits than the
globally depolarized random circuit.
In particular, we can replace our random gate application
with the coherent one.
In this paper, however, we concentrate on the
globally depolarized random circuit
for the simplicity.
Theorem~\ref{theorem:result2} and
Theorem~\ref{theorem:result3} hold for the output probability
distribution $\{p_z\}_z$ of any quantum circuit.
Finally,
in Ref.~\cite{Google}, it was claimed that if the exact polynomial-time
classical sampling
of $\{p_z'\}_z$ is possible, then estimating
$|\langle0^n|U|0^n\rangle|^2$ for an $n$-qubit unitary $U$
can be done
by an Arthur-Merlin protocol with the $F^{-1}$-time Arthur.
However, if we consider the realistic case,
$F\sim2^{-m}$ and $m\gg n$, where $m$ is the number of gates,
the time-complexity of Arthur is $\sim2^m$.
(On the other hand, the exact computation of $|\langle0^n|U|0^n\rangle|^2$
can be done in time $\sim2^n$.)
Moreover, although Ref.~\cite{Google} considered exact sampling of
$\{p_z'\}_z$,
what a realistic
quantum computer can do is approximately sampling
$\{p_z'\}_z$.
Theorem~\ref{theorem:result2}
shows that
if we consider the realistic case, $F\sim2^{-m}$ and $m\gg n$,
quantum supremacy does not exist
for approximate sampling of $\{p_z'\}_z$
even with the exponentially-small multiplicative
error $\sim 2^{-(m-n)}$.
Theorem~\ref{theorem:result3}
shows that
if we consider the realistic case, $F\sim2^{-m}$,
quantum supremacy does not exist
for
approximate sampling of
$\{p_z'\}_z$ even with the exponentially-small additive
error $\sim2^{-m}$.
\section{Discussion}
Our theorems show that if $F\sim2^{-m}$ and $m\gg n$,
or at least $F\sim2^{-m}$,
quantum supremacy does not exist for
approximate sampling of $\{p_z'\}_z$.
In this section, we argue that if $F\sim2^{-m}$ and $m\gg n$,
the standard approach will not work to show
quantum supremacy for exact sampling of $\{p_z'\}_z$.
In the standard proof of quantum
supremacy~\cite{TD,BS,IQP1,MFF,Kobayashi,KobayashiICALP},
we first consider the following
promise problem: given the classical description of an $n$-qubit
$m$-size quantum circuit $U$, and parameters $a$ and $b$, decide
$p_{acc}\ge a$ or $p_{acc}\le b$,
where $p_{acc}$ is the acceptance probability.
In the standard proof of quantum supremacy, we take the promise problem
as the complete problem of a ``strong" quantum class,
such as postBQP, SBQP, or NQP.
We next assume that $p_{acc}'\equiv Fp_{acc}+\frac{1-F}{2^n}$
is exactly classically sampled. It means that there exists a
polynomial-time classical probabilistic algorithm that accepts with
probability $q_{acc}$ such that
$q_{acc}=p_{acc}'$.
If the answer of the promise problem is yes, then
$q_{acc}\ge Fa+\frac{1-F}{2^n}\equiv\alpha$.
If the answer of the promise problem is no, then
$q_{acc}\le Fb+\frac{1-F}{2^n}\equiv\beta$.
In the standard proof of quantum supremacy, we then conclude
that the promise problem is in a ``weaker" class
(such as postBPP, SBP, or NP) that leads to an unlikely
consequence in complexity theory,
such as ${\rm postBQP}\subseteq{\rm postBPP}$,
${\rm SBQP}\subseteq{\rm SBP}$, or
${\rm NQP}\subseteq{\rm NP}$.
However,
deciding
$q_{acc}\ge \alpha$ or $q_{acc}\le \beta$
seems to be ``more difficult" than the original promise problem:
the original promise problem can be solved in time $\sim2^n$,
while deciding $q_{acc}\ge\alpha$ or $q_{acc}\le\beta$
will not be solved in that time
because $\alpha-\beta=F(a-b)=O(2^{-m})$, and $m\gg n$.
Therefore we will not have any unlikely consequence in
this approach.
Although the above argument does not exclude the existence of
a completely new supremacy proof for the exact sampling
of $\{p_z'\}_z$ that works even when $F\sim2^{-m}$ and $m\gg n$,
we can also argue that even if the realistic quantum
computer exactly samples
$\{p_z'\}_z$, it is ``effectively" classically samplable
by the uniform distribution when $F\sim2^{-m}$
unless we can access exponentially many samples.
To see this,
let us consider the task of distinguishing
$\rho_0\equiv\frac{I^{\otimes n}}{2^n}$
and
$\rho_1\equiv F\rho+(1-F)\frac{I^{\otimes n}}{2^n}$,
where $\rho$ is any $n$-qubit state.
Assume that we can measure $k$ copies of $\rho_0$ or $\rho_1$.
Let $\Pi_0$ $(\Pi_1)$ be the POVM element that we conclude
that the actual state is $\rho_0^{\otimes k}$
($\rho_1^{\otimes k}$), where
$\Pi_0+\Pi_1=I^{\otimes nk}$.
The probability $p_{correct}$ that we make the correct decision is
\begin{eqnarray*}
p_{correct}&\equiv&
\frac{1}{2}{\rm Tr}(\Pi_0\rho_0^{\otimes k})
+\frac{1}{2}{\rm Tr}(\Pi_1\rho_1^{\otimes k})\\
&=&
\frac{1}{2}+\frac{1}{2}\Big[
{\rm Tr}(\Pi_0\rho_0^{\otimes k})
-{\rm Tr}(\Pi_0\rho_1^{\otimes k})\Big]\\
&\le&
\frac{1}{2}+
\frac{1}{4}\big\|\rho_0^{\otimes k}
-\rho_1^{\otimes k}\big\|_1\\
&\le&
\frac{1}{2}
+\frac{k}{4}
\big\|\rho_0-\rho_1\big\|_1\\
&=&
\frac{1}{2}+
\frac{k}{4}\Big\|\frac{I^{\otimes n}}{2^n}
-\Big[F\rho+(1-F)\frac{I^{\otimes n}}{2^n}\Big]\Big\|_1\\
&=&
\frac{1}{2}+
\frac{kF}{4}\Big\|
\rho-\frac{I^{\otimes n}}{2^n}\Big\|_1\\
&\le&\frac{1}{2}+\frac{kF}{2}.
\end{eqnarray*}
If $F\sim2^{-m}$ and $k=o(2^m)$,
$p_{correct}\to \frac{1}{2}$.
\section{Proof of Theorem~\ref{theorem:result}}
Assume that a language $L$ is in BQP. Then
for any polynomial $r$,
there exists a polynomial-time uniformly generated
family $\{V_x\}_x$ of quantum circuits such that
if $x\in L$ then $p_{acc}\ge 1-2^{-r(|x|)}$,
and
if $x\notin L$ then $p_{acc}\le 2^{-r(|x|)}$.
Here
\begin{eqnarray*}
p_{acc}\equiv\langle0^w|V_x|0^w\rangle
\end{eqnarray*}
with $w=poly(|x|)$ is the acceptance probability.
Let $m$ be the number of elementary gates in $V_x$, i.e.,
$V_x=w_mw_{m-1}...w_2w_1$, where each $w_j$ is an elementary gate
(such as $H$, $CNOT$, and $T$, etc.).
Let us consider the following random quantum circuit
on $n\equiv w+m$ qubits:
\begin{itemize}
\item[1.]
The initial state is $|0^w\rangle\otimes|0^m\rangle$, where we call
the first $w$-qubit register the main register,
and the second $m$-qubit register the ancilla register.
\item[2.]
For each $j=1,2,...,m$, apply $w_j\otimes I$ or
$\eta_j\otimes X$ with probability
1/2, where $\eta_j$ is any elementary gate,
$w_j$ and $\eta_j$ act on the main register,
and $I$ and $X$ act on the $j$th qubit of the ancilla register.
Thus obtained the final state is
\begin{eqnarray}
\frac{1}{2^m}\sum_{\alpha\in\{0,1\}^m}
\xi_m^{\alpha_m}...\xi_1^{\alpha_1}|0^w\rangle\langle0^w|
(\xi_1^{\alpha_1})^\dagger...(\xi_m^{\alpha_m})^\dagger
\otimes|\alpha\rangle\langle\alpha|,
\label{final}
\end{eqnarray}
where $\alpha\equiv(\alpha_1,...,\alpha_m)\in\{0,1\}^m$ is an $m$-bit string,
$\xi_j^0=w_j$,
and $\xi_j^1=\eta_j$.
\item[3.]
Measure all $n$ qubits in the computational basis.
If all results are 0, accept. Otherwise, reject.
\end{itemize}
If we consider the globally depolarized version,
the state of Eq.~(\ref{final}) is replaced with
\begin{eqnarray*}
\frac{F}{2^m}\sum_{\alpha\in\{0,1\}^m}
\xi_m^{\alpha_m}...\xi_1^{\alpha_1}|0^w\rangle\langle0^w|
(\xi_1^{\alpha_1})^\dagger...(\xi_m^{\alpha_m})^\dagger
\otimes|\alpha\rangle\langle\alpha|
+(1-F)\frac{I^{\otimes n}}{2^n}.
\end{eqnarray*}
The acceptance probability
$p_{acc}'$
is
\begin{eqnarray*}
p_{acc}'=\frac{Fp_{acc}^2}{2^m}+\frac{1-F}{2^n}.
\end{eqnarray*}
Assume that there exists a classical $poly(n)$-time probabilistic algorithm
that accepts with probability $q_{acc}$ such that
$|p_{acc}'-q_{acc}|\le \epsilon p_{acc}'$,
where $\epsilon<1$ is a constant.
Then, if $x\in L$,
\begin{eqnarray*}
q_{acc}&\ge&(1-\epsilon)p_{acc}'\\
&=&(1-\epsilon)\Big(\frac{Fp_{acc}^2}{2^m}+\frac{1-F}{2^n}\Big)\\
&\ge&(1-\epsilon)F2^{-m}(1-2^{-r})^2,
\end{eqnarray*}
and if $x\notin L$,
\begin{eqnarray*}
q_{acc}&\le&(1+\epsilon)p_{acc}'\\
&=&(1+\epsilon)\Big(\frac{Fp_{acc}^2}{2^m}+\frac{1-F}{2^n}\Big)\\
&\le&(1+\epsilon)\Big(F2^{-2r-m}+\frac{1-F}{2^n}\Big)\\
&=&2^{-m}(1+\epsilon)F\Big(2^{-2r}+\frac{1-F}{F2^w}\Big).
\end{eqnarray*}
If $r$ and $w$ are sufficiently large, $L$ is in SBP.
\fbox
Note that although here we have considered constant $F$,
the same result also holds for other ``not so small" $F$
such as $F=\frac{1}{poly(m)}$.
\section{Proof of Theorem~\ref{theorem:result2}}
Let us take $\epsilon=F2^{n+2}$.
For any $z\in\{0,1\}^n$,
\begin{eqnarray*}
\Big|p_z'-\frac{1}{2^n}\Big|
=\Big|\Big(Fp_z+\frac{1-F}{2^n}\Big)
-\frac{1}{2^n}\Big|\le F\Big(1+\frac{1}{2^n}\Big)< \epsilon p_z',
\end{eqnarray*}
where in the last inequality, we have used
\begin{eqnarray*}
\epsilon p_z'-F\Big(1+\frac{1}{2^n}\Big)
&=&
\epsilon\Big(Fp_z+\frac{1-F}{2^n}\Big)-F\Big(1+\frac{1}{2^n}\Big)\\
&\ge&\frac{\epsilon(1-F)}{2^n}-F\Big(1+\frac{1}{2^n}\Big)\\
&=&\frac{F2^{n+2}(1-F)}{2^n}-F\Big(1+\frac{1}{2^n}\Big)\\
&=&4F(1-F)-F\Big(1+\frac{1}{2^n}\Big)\\
&>0&.
\end{eqnarray*}
\fbox
\section{Proof of Theorem~\ref{theorem:result3}}
\begin{eqnarray*}
\sum_{z\in\{0,1\}^n}\Big|p_z'-\frac{1}{2^n}\Big|
= F\sum_{z\in\{0,1\}^n}\Big|p_z-\frac{1}{2^n}\Big|
\le 2F.
\end{eqnarray*}
\fbox
\acknowledgements
TM is supported by
MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067394,
JST PRESTO No.JPMJPR176A,
and the Grant-in-Aid for Young Scientists (B) No.JP17K12637 of JSPS.
YT is supported by
MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067394.
\end{document} |
\begin{document}
\title[On the Brauer group of a generic Godeaux surface]{On the Brauer group of a generic Godeaux surface}
\author{Theodosis Alexandrou}
\setstretch{1.16}
\keywords{}
\subjclass[2010]{}
\makeatletter
\hypersetup{
pdfauthor=,
pdfsubject=\@subjclass,
pdfkeywords=\@keywords
}
\makeatother
\maketitle
\begin{abstract} Let $X$ be a Godeaux surface and $q_{X}\colon Y\to X$ be its universal cover. We show that the pullback map $q_{X}^{*}\colon\br(X)\to\br(Y)$ is injective if $\rho(Y)=9$. Our arguments rely on a degeneration technique that also applies to other examples.\end{abstract}
\section{Introduction}\label{sec:1}The Brauer group $\br(X)$ of a smooth projective variety $X$ over a field $k$ is the abelian group $H^{2}(X_{\text{\'et}},\mathbb{G}_{m})$. It can be also regarded as the group of equivalence classes of \'etale locally trivial $\mathbb{P}^{n}$ bundles over $X$ modulo Zariski locally trivial ones. This is always a torsion group and an important birational invariant of $X$.\par Let $q\colon Y\to X$ be a finite \'etale covering. This defines a natural pull-back map $q^{*}\colon \br(X)\to\br(Y)$ on Brauer groups. Since $q_{*}q^{*}=\deg(q)\cdot\id$, the kernel of $q^{*}$ is killed by the positive integer $d\coloneqq\deg(q)$. Therefore, detecting whether a given $d$-torsion class $a\in\br(X)[d]$ pulls back to zero is a subtle problem.\par Beauville studied this question in the case of complex Enriques surfaces, which was first stated in \cite{hsk}. Namely, an Enriques surface $X$ admits a natural double cover $Y$, which is a $K3$ surface. Beauville \cite{bea} showed that the kernel of the map $\mathbb{Z}/2=\br(X)\to\br(Y)=(\mathbb{Q}/\mathbb{Z})^{22-\rho},\ 1\leq\rho\leq 20,$ depends on $X$. More specifically, he described lattice theoretically the surfaces $X$ in the coarse moduli space of Enriques for which the kernel of $q^{*}_{X}$ is non-trivial and showed that this locus is a countable, infinite union of non-empty algebraic hypersurfaces. His method fails to give a concrete example but among others Garbagnati and Sch\"utt \cite{gm} constructed examples of Enriques surfaces $X$ over $\mathbb{Q}$ with $q^{*}_{X}$ being injective or trivial.\par Recent work of Bergstr\"om, Ferrari, Tirabassi and Vodrup \cite{bie} uses Beauvilles' method to answer similar questions for a Bi-elliptic surface $X$ over $\mathbb{C}$.\par In \cite{talk}, Beauville asked the above question for Godeaux surfaces. A Godeaux surface $X$ is a classical example of a surface of general type with $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2$ and $\NS(X)_{\tor}\neq 0$($\cong\mathbb{Z}/5$). They usually appear as $\mathbb{Z}/5$-quotients of smooth quintics $Y\subset\mathbb{P}^{3}_{\mathbb{C}}$ that are invariant and fixed point free by the automorphism \[\varphi\colon (x_{0}:x_{1}:x_{2}:x_{3})\mapsto (x_{0}:\zeta x_{1}:\zeta^{2}x_{2}:\zeta^{3}x_{3}),\tag{1.1}\label{1.1}\]where $\zeta$ is a primitive $5$-th root of unity (see \cite[$\S$2]{lw}).\par The main result of this paper answers the question for $\rho(Y)=9$, which is the generic case (see \cite[Example 3]{BoM}).\begin{theorem}\label{thm:1.1}Let $X$ be a Godeaux surface over $\mathbb{C}$ and $q_{X}\colon Y\to X$ be its universal cover. If the Picard number of $Y$ is $9$, then the pull-back map $q^{*}_{X}\colon\br(X)\to\br(Y)$ is injective.\end{theorem} As an immediate consequence, we also obtain:\begin{corollary}\label{cor:1.2}For the very general Godeaux surface $X$ over $\mathbb{C}$, the pull-back map $q^{*}_{X}$ is injective.\end{corollary}Theorem \ref{thm:1.1} will be deduced from the following result.\begin{theorem}\label{thm:1.3} Let $k$ be an algebraically closed field of characteristic $0$. Pick a generic pencil $\mathcal{Y}\to\mathbb{P}^{1}_{k}\subset|\mathcal{O}_{\mathbb{P}^{3}_{k}}(5)|^{\mathbb{Z}/5}$ of $\mathbb{Z}/5$-invariant quintics, with some fibre $\mathcal{Y}_{t_{0}},\ t_{0}\in\mathbb{P}^{1}_{k}(k)$ not passing through any of the 4 fixed points of the action \eqref{1.1} and having 5 triple points as its only singularities. Let $q\colon\mathcal{Y}\to\mathcal{X}$ be the quotient map. Then the pull-back map $q^{*}_{\bar{\eta}}\colon\br(\mathcal{X}_{\bar{\eta}})\to\br(\mathcal{Y}_{\bar{\eta}})$ is injective.\end{theorem} A pencil with the properties mentioned in Theorem \ref{thm:1.3} exists. This relies on the fact that the invariant quintics $|\mathcal{O}_{\mathbb{P}^{3}_{k}}(5)|^{\mathbb{Z}/5}$ form a 12-dimensional linear space and for a given point $p\in\mathbb{P}^{3}_{k}(k)$ not in any coordinate hyperplane, we can always find a quintic surface $Y$ invariant and fixed point free under the $\mathbb{Z}/5$-action \eqref{1.1} with 5 triple points at the orbit of $p$ and no other singularities (see \cite[Appendix 5, Lemma 1]{persson}).\par It remains unclear whether there are actually examples of Godeaux surfaces $X$, such that $q^{*}_{X}=0$.\par Our method is adjustable and can be also applied to other surfaces. In section \ref{sec:6} we show that complex Enriques surfaces $X$ with $q^{*}_{X}=0$ do not admit a type II degeneration (see Theorem \ref{thm:6.2}). By comparing the latter with Beauvilles' result \cite[Corollary 6.5]{bea}, one can determine hypersurfaces in the coarse moduli space of Enriques, with the property that any of their members cannot be degenerated to a type II (see Corollary \ref{cor:6.5}). In the last section we also study the case of cyclic quotients of products of curves (see Theorem \ref{thm:7.1}).\subsection{A degeneration technique.}\label{subsec:1.1} Theorem \ref{thm:1.3} relies on the following, which is the crucial technical result of the paper.\begin{theorem}\label{thm:1.4} Let $q\colon\mathcal{Y}\to\mathcal{X}$ be a finite \'etale Galois covering of strictly semi-stable $R$-schemes, such that the \'etale cover $q_{0}$ is trivial over $\mathcal{X}^{\sing}_{0}\cap\mathcal{X}_{0,i}$ for every component $\mathcal{X}_{0,i}$ of the special fibre $\mathcal{X}_{0}$. Assume that the degree $d\coloneqq\deg(q)$ is invertible in $R$. Then \eqref{1'''} implies \eqref{2'''}:\begin{enumerate}
\item \label{1'''} The pull-back $q_{\bar{\eta}}^{*}\colon\br(\mathcal{X}_{\bar{\eta}})[d]\to\br(\mathcal{Y}_{\bar{\eta}})[d]$ is the zero map.\item\label{2'''} Up to some finite ramified base change of discrete valuation rings $R\subset\tilde{R}$ followed by a resolution of $\mathcal{X}_{\tilde{R}},$ the restriction $\br(\mathcal{X})[d]\to\br(\mathcal{X}_{\bar{\eta}})[d]$ is surjective.
\end{enumerate}\end{theorem} When performing a ramified base change $R\subset\tilde{R},$ the model $\mathcal{X}_{\tilde{R}}$ becomes singular, but it follows from \cite[Proposition 2.2]{Har}, that the family $\mathcal{X}_{\tilde{R}}\to\Spec\tilde{R}$ can be made again into a strictly semi-stable by repeatedly blowing up the non-Cartier divisors of the special fibre.\par Theorem \ref{thm:1.4} establishes a link between the following two properties (see also Theorem \ref{thm:3.1}):\begin{enumerate}[label=(\roman*)]
\item\label{i} A class $\alpha\in\br(\mathcal{X}_{\bar{\eta}})$ pulls back to zero, i.e. $q_{\bar{\eta}}^{*}\alpha=0\in\br(\mathcal{Y}_{\bar{\eta}})$.\item\label{ii} Up to some finite base change of discrete valuation rings $R\subset\tilde{R}$ followed by a resolution of $\mathcal{X}_{\tilde{R}}$, $\alpha\in\br(\mathcal{X}_{\bar{\eta}})$ lifts to an honest class in $\br(\mathcal{X})$.\end{enumerate}
The advantage of relating these properties is that in many cases it is not hard to check whether a given class $\alpha\in\br(\mathcal{X}_{\bar{\eta}})$ lifts to the total space $\mathcal{X}$. In all applications presented in this paper, the restriction map $\br(\mathcal{X})\to\br(\mathcal{X}_{\bar{\eta}})$ happens to be trivial, either because $\br(\mathcal{X})=0$ or $\br(\mathcal{X})$ is a divisible group and $\br(\mathcal{X}_{\bar{\eta}})$ is finite (see Lemmas \ref{lem:5.2}, \ref{lem:6.3} and \ref{lem:7.2}). An example with $\br(\mathcal{X})\to\br(\mathcal{X}_{\bar{\eta}})$ being surjective appears in \cite[Theorem 7.1]{Stef}.\par The property \ref{ii} had been also studied in \cite{Stef}, where a relation to algebraic torsion classes had been found.\subsection{Outline of the proofs.}\label{subsec:1.2} The proof of Theorem \ref{thm:1.4} relies on the following observations. Pick a Brauer class $\alpha\in\br(\mathcal{X}_{\eta})\{\ell\}$ with $q^{*}_{\eta}\alpha=0$. Up to a suitable finite ramified base change, it is possible to achieve vanishing of the residues $\partial_{i}\alpha\in H^{1}_{\text{\'et}}(\mathcal{X}_{0,i},\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ for all irreducible components $\mathcal{X}_{0,i}$ that have not been introduced by resolving the singularities that appeared due to the base change. Using our assumption for $q_{0},$ we see that over the new components the covering map $q$ is trivial. Hence, the usual pull-push argument shows that the remaining residues of $\alpha$ coincide with the ones of its pull-back $0=q^{*}_{\eta}\alpha\in\br(\mathcal{Y}_{\eta})\{\ell\}$ and so, all vanish. This will conclude the proof of Theorem \ref{thm:1.4}.\par Let $\mathcal{Y}\to\mathbb{P}^{1}_{k}\subset|\mathcal{O}_{\mathbb{P}^{1}_{k}}(5)|^{\mathbb{Z}/5}$ be as in Theorem \ref{thm:1.3} and perform a base change with respect to the local ring at $t_{0}\in\mathbb{P}^{1}_{k}$. By applying semi-stable reduction to $\mathcal{Y}$ and its $\mathbb{Z}/5$-quotient $\mathcal{X}$, we arrive at a $5:1$ \'etale cover $q\colon\mathcal{Y}\to\mathcal{X}$, that is trivial over $\mathcal{X}^{\sing}_{0}\cap\mathcal{X}_{0,i}$ for all components $\mathcal{X}_{0,i}$ of the special fibre $\mathcal{X}_{0}$. Replacing $R\coloneqq\mathcal{O}_{\mathbb{P}^{1}_{k},t_{0}}$ with its completion $\hat{R}\cong k[[t]]$, the proper base change theorem can be used to show that the restriction map $\br(\mathcal{X})\to\br(\mathcal{X}_{0})_{\tor}$ is an isomorphism (see Lemma \ref{Lem:4.1}). Then an explicit calculation using the Mayer-Vietoris exact sequence gives $\br(\mathcal{X}_{0})=0$ (see Lemma \ref{lem:5.2}). Hence, $\br(\mathcal{X})=0$ and so a Brauer class $0\neq\alpha\in\br(X)\cong\mathbb{Z}/5$ never lifts to the total space $\mathcal{X}$. We thus obtain Theorem \ref{thm:1.3} by applying Theorem \ref{thm:1.4}.\par Theorem \ref{thm:1.3} can be used to produce a Godeaux surface $X$ with universal cover $Y$ of Picard number $9$, such that $q^{*}_{X}$ is injective. As a final step, a specialization argument comes in hand, which shows that the map in question is in fact injective for every $Y$ with $\rho(Y)=9$ (see Proposition \ref{prp:4.4}). This completes the proof of Theorem \ref{thm:1.1}.\section{Preliminaries}\label{sec:2}\subsection{Conventions and notations.}\label{subsec:2.1} Let $A$ be an abelian group and let $\ell$ be a prime number. We denote by $A[\ell^{r}]$ the subgroup of $\ell^{r}$-torsion elements. We write $A\{\ell\}$ for the $\ell$-primary subgroup of $A$. If $n\in\mathbb{N}$, then $[n]\colon A\to A$ is the map that sends $x$ to $nx$. Also $A_{\text{div}}$ denotes the maximal divisible subgroup of $A$.\par All schemes are assumed to be seperated. A variety is a geometrically integral scheme of finite type over a field. The regular locus of $X$ will be denoted by $X^{\reg}$ and its singular by $X^{\sing}$. For an $S$-scheme $X\to S$ and any morphism $S'\to S$, we denote by $X_{S'}\coloneqq X\times_{S}S'$ the base change.\par We denote by $k^{sep}$ and $\bar{k}$ the seperable and algebraic closure of a field $k$, respectively. We use the notation $R^{\text{h}}$ for the henselization of a local ring $R$.\par Let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$. For any $R$-scheme $\mathcal{X}\to\Spec R$, we write $\mathcal{X}_{0}\coloneqq\mathcal{X}\times_{R}k$ (resp. $\mathcal{X}_{\bar{0}}\coloneqq\mathcal{X}\times_{R}\bar{k}$) for the special (resp. geometric special) fibre and $\mathcal{X}_{\eta}\coloneqq\mathcal{X}\times_{R}K$ (resp. $\mathcal{X}_{\bar{\eta}}\coloneqq\mathcal{X}\times_{R}\bar{K}$) for the generic (resp. geometric generic) fibre.\par A projective flat $R$-scheme $\mathcal{X}\to\Spec R$ is called \textit{strictly semi-stable}, if $\mathcal{X}$ is an integral regular scheme, the generic fibre $\mathcal{X}_{\eta}$ is smooth and the special fibre $\mathcal{X}_{0}$ is a geometrically reduced simple normal crossing divisor on $\mathcal{X}$, i.e. the irreducible components $\mathcal{X}_{0}$ are all smooth varieties and the scheme-theoretic intersection of $n$ distinct components is either empty or smooth and equi-dimensional of codimension $n$ in $\mathcal{X}$. A strictly semi-stable $R$-scheme $\mathcal{X}\to\Spec R$ is called \textit{triple-point free} if the intersection of any three pairwise distinct components in the special fibre is empty.\subsection{\'Etale Cohomology and Brauer Group.}\label{subsec:2.2} We collect some results from \'etale cohomology that will be used through out this paper. The results that we discuss can be found in \cite{mil} and \cite{cts}. For a scheme $X$, we denote by $X_{\text{\'et}}$ the small \'etale site of $X$. If $\mathcal{F}\in Sh(X_{\text{\'et}})$ is a sheaf of abelian groups, then we write $H^{i}(X,\mathcal{F})\coloneqq H^{i}(X_{\text{\'et}},\mathcal{F})$ for the \'etale cohomology groups with coefficients in $\mathcal{F}$. If $X=\Spec A$ for some ring $A$, we prefer to write $H^{i}(A,F)$, instead. For a positive integer $n$, we denote by $\mu_{n}$ the subsheaf of $n$-th roots of unity of the multiplicative sheaf $\mathbb{G}_{m}$ on $X_{\text{\'et}}$.\par Let $X$ be a scheme and $n$ a positive integer that is invertible on $X$. We shall frequently use the following description of the $n$-torsion subgroup of the Brauer group of $X$, which is induced from the Kummer sequence \[\coker(c_{1}\colon\Pic(X)\to H^{2}(X,\mu_{n}))\cong \br(X)[n].\tag{2.1}\label{2.6}\]\begin{lemma}\label{lem:2.1}Let $X$ be an integral, regular scheme of finite type over a field $k$ (resp. a dvr $R$) and let $Z\subset X$ be a closed subset of codimension one in $X$. Let $U\subset X$ be the complement of $Z$ in $X$. Let $\ell$ be a prime invertible on $X$. If $Z_{1}, Z_{2},\ldots,Z_{t}$ are the components of $Z$ with codimension one in $X$, then we have the following exact sequences:\[0\to\br(X)[\ell^{r}]\to\br(U)[\ell^{r}]\overset{\{\partial_{i}\}}\to\bigoplus_{i=1}^{t}H^{1}(k(Z_{i}),\mathbb{Z}/\ell^{r}),\tag{2.2}\label{2.1}\]\[0\to\br(X)\{\ell\}\to\br(U)\{\ell\}\overset{\{\partial_{i}\}}\to\bigoplus_{i=1}^{t}H^{1}(k(Z_{i}),\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}).\tag{2.3}\label{2.2}\]\end{lemma}\begin{proof} See \cite[Section\ 3.7]{cts} for a detailed proof.\end{proof}\begin{lemma}\label{lem:2.2} Let $f\colon X'\to X$ be a flat morphism of relative dimension $n\coloneqq\dim X'-\dim X$, between integral, regular schemes of finite type over a field $k$ (resp. a dvr $R$). For a prime divisor $Z\subset X$, we write $f^{*}Z=\sum_{i=1}^{t}\lambda_{i}Z'_{i}$ for its pull-back, where $Z'_{i}$ are the prime divisors on $X'$ supported on $f^{-1}(Z)$ and $\lambda_{i}$ denote their multiplicities. Set $U\coloneqq X\setminus Z$ and $U'\coloneqq f^{-1}(U)\subset X'$. Then for any prime $\ell$ invertible on $X$, the following diagram commutes:\[\begin{tikzcd}
{\br(U)[\ell^{r}]} \arrow[r, "\partial"] \arrow[d, "f^{*}"'] & {H^{1}(k(Z),\mathbb{Z}/\ell^{r})} \arrow[d, "{\{\lambda_{i}f|_{Z'_{i}}^{*}\}_{i}}"'] \\
{\br(U')[\ell^{r}]} \arrow[r, "\{\partial_{i}\}"] & {{\bigoplus_{i=1}^{t}H^{1}(k(Z'_{i}),\mathbb{Z}/\ell^{r})}.}
\end{tikzcd}\tag{2.4}\label{2.3}\]\end{lemma}
\begin{proof} This follows from \cite[Theorem 3.7.5]{cts}.\end{proof}
\begin{lemma}\label{lem:2.3} Let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$. Let $\mathcal{X}\to\Spec R$ be a triple-point free strictly semi-stable $R$-family. Then we have the following Mayer-Vietoris exact sequence:\[\dots\to\bigoplus_{i}\Pic(\mathcal{X}_{0,i})\overset{r_{1}}\to\bigoplus_{i<j}\Pic(\mathcal{X}_{0,i}\cap\mathcal{X}_{0,j})\to\br(\mathcal{X}_{0})\overset{r_{2}}\to\bigoplus_{i}\br(\mathcal{X}_{0,i})\to \dots,\tag{2.5}\label{2.4}\] where the map $r_{1}$ sends $(\mathcal{L}_{i})$ to $(\mathcal{L}_{i}|_{\mathcal{X}_{0,i}\cap\mathcal{X}_{0,j}}-\mathcal{L}_{j}|_{\mathcal{X}_{0,i}\cap\mathcal{X}_{0,j}})$ and $r_{2}$ takes $\alpha$ to $(\alpha|_{\mathcal{X}_{0,i}})$.\end{lemma}\begin{proof} Consider the morphisms $p\colon Y\coloneqq\bigsqcup_{i}\mathcal{X}_{0,i}\to\mathcal{X}_{0}$ and $\rho_{i,j}\colon X_{i,j}\coloneqq\mathcal{X}_{0,i}\cap\mathcal{X}_{0,j}\hookrightarrow\mathcal{X}_{0}$ for $i\neq j$. The triple point free assumption implies exactness of the following sequence of sheaves for the \'etale topology on $\mathcal{X}_{0}$:\[1\to\mathbb{G}_{m,\mathcal{X}_{0}}\to p_{*}\mathbb{G}_{m,Y}\to \bigoplus_{i<j}{\rho_{i,j}}_{*}\mathbb{G}_{m,X_{i,j}}\to 1. \tag{2.6}\label{2.5}\] Since the morphisms $p$ and $\rho_{i,j}$ are finite, the push-forwards $p_{*}$ and ${\rho_{i,j}}_{*}$ are both exact functors in the \'etale topology (see \cite[Corollary II.3.6]{mil}). Hence, $R^{s}p_{*}=R^{s}{\rho_{i,j}}_{*}=0$ for any $s\geq 1$ and the Leray spectral sequence gives rise to the isomorphisms $H^{s}(\mathcal{X}_{0},p_{*}\mathbb{G}_{m})\cong H^{s}(Y,\mathbb{G}_{m})$ and $H^{s}(\mathcal{X}_{0},{\rho_{i,j}}_{*}\mathbb{G}_{m})\cong H^{s}(X_{i,j},\mathbb{G}_{m})$. Finally, the associated long exact sequence of \eqref{2.5} together with the use of the above isomorphisms lead to the sequence \eqref{2.4}.\end{proof}\section{Main Technical Result}\label{sec:3} Theorem \ref{thm:1.4} is an immediate corollary of the following.\begin{theorem}\label{thm:3.1} Let $q\colon\mathcal{Y}\to\mathcal{X}$ be a finite \'etale Galois covering of strictly semi-stable $R$-schemes, such that the \'etale cover $q_{0}$ is trivial over $\mathcal{X}^{\sing}_{0}\cap\mathcal{X}_{0,i}$ for every irreducible component $\mathcal{X}_{0,i}$ of $\mathcal{X}_{0}$. Fix a prime number $\ell$, that is invertible in $R$. Pick a class $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\{\ell\}$. Then the following are equivalent:\begin{enumerate}\item\label{1} The class $\alpha$ pulls-back to zero, i.e. $q_{\bar{\eta}}^{*}\alpha=0$.\item\label{2} Up to some finite ramified base change of discrete valuation rings $R\subset\tilde{R}$ followed by a resolution of $\mathcal{X}_{\tilde{R}},$ the class $\alpha$ lifts to a class $\tilde{\alpha}$ in $\br(\mathcal{X})\{\ell\}$ with $q^{*}(\tilde{\alpha})=0$.\end{enumerate}\end{theorem}\begin{proof} The implication \eqref{2} $\implies$ \eqref{1} is clear by functoriality of pullbacks.\par We prove the implication \eqref{1} $\implies$ \eqref{2}. We may assume that $\alpha\in\br(\mathcal{X}_{\bar{\eta}})[\ell^{r}]$ for some $r\geq 1$, such that $\ell^{r}$ divides $d\coloneqq\deg(q)$. By \cite[Section 2.2.2]{cts}, we have \[\br(\mathcal{X}_{\bar{\eta}})=\lim_{\underset{K\subset L}\longrightarrow}\br(\mathcal{X}_{L}),\] where $\mathcal{X}_{L}=\mathcal{X}_{\eta}\times_{K} L$ and the co-limit is taken over all finite seperable field extensions $L$ of $K$. Hence, we can find a finite seperable field extension $L$ of $K$, such that $\alpha\in\im(\br(\mathcal{X}_{L})[\ell^{r}]\to\br(\mathcal{X}_{\bar{\eta}})[\ell^{r}])$. We pick a lift of $\alpha$ in $\br(\mathcal{X}_{L})[\ell^{r}]$ and denote it again by the same symbol. Since $q^{*}_{\bar{\eta}}\alpha=0,$ the same reasoning as above gives $q^{*}_{L}\alpha=0$ (up to replacing $L$ if neccessary with some finite seperable field extension). We let $\tilde{R}\subset L$ be the normalization of $R$ in $L$. Then $\tilde{R}$ is a Dedekind domain finite over $R$ (see \cite[Proposition 12.53]{gw}). Choose a maximal ideal $\mathfrak{m}\subset R$ and consider the discrete valuation ring $R'\coloneqq \tilde{R}_{\mathfrak{m}}$. We perform a base change corresponding to the extension of discrete valuation rings $R'$ of $R$ and note that the models $\mathcal{X}$ and $\mathcal{Y}$ may become singular. By \cite[Proposition 2.2]{Har}, both $\mathcal{X}_{R'}$ and $\mathcal{Y}_{R'}$ can be made into strictly semi-stable $R'$-schemes. Specifically, by repeatedly blowing up all non-Cartier components of the special fibre of $\mathcal{Y}\to\Spec R'$, we can arrive at a strictly semi-stable model $\mathcal{Y}'\to\Spec R'$. As we blow up along $G$-invariant centers, the action on $\mathcal{Y}$ given by the Galois group $G\coloneqq\Aut(q)$, can be naturally lifted to a fixed point free action on $\mathcal{Y}'$. Passing to the quotient $\mathcal{X}'\coloneqq\mathcal{Y}'/G$, we obtain a semi-stable model for $\mathcal{X}_{R'}$. In particular, according to the above, we may assume that $\alpha\in\br(\mathcal{X}_{\eta})[\ell^{r}]$ and $q_{\eta}^{*}\alpha=0\in\br(\mathcal{Y}_{\eta})[\ell^{r}]$.\par Fix a uniformizer $\pi\in R$ and perform the $\ell^{r}:1$ base change $\pi\mapsto\pi^{\ell^{r}}$ followed by resolving the singularities, as above. We keep the notations $\tilde{\mathcal{X}},\tilde{\mathcal{Y}}$ and $\tilde{q}\colon\tilde{\mathcal{Y}}\to\tilde{\mathcal{X}}$ for the resulting models and covering over $R$. Write $\mathcal{X}_{0}=\sum_{t\in T}P_{t}$ for the special fibre of $\mathcal{X}\to R$ and denote by $\tilde{P}_{t}\subset\tilde{\mathcal{X}_{0}}$ the irreducible component that maps isomorphically onto $P_{t}$ via the composition $f\colon\tilde{\mathcal{X}}\to\mathcal{X}_{R}\to\mathcal{X}$ for every $t\in T$. By Lemma \ref{lem:2.2}, we have the commutative diagram:\begin{center}\begin{tikzcd}
{\br(\mathcal{X}_{\eta})[\ell^{r}]} \arrow[r, "\{\partial_{t}\}"] \arrow[d, "f^{*}"] & {\oplus_{t\in T}H^{1}(k(P_{t}),\mathbb{Z}/\ell^{r})} \arrow[d, "\{\ell^{r}f^{*}\}=0"] \\
{\br(\tilde{\mathcal{X}}_{\eta})[\ell^{r}]} \arrow[r, "\{\partial_{t}\}"] & {\oplus_{t\in T}H^{1}(k(\tilde{P}_{t}),\mathbb{Z}/\ell^{r})},
\end{tikzcd}\end{center}from which one obtains $\partial_{t}f^{*}\alpha=0$ for all $t\in T$. It remains to check that $\tilde{\alpha}\coloneqq f^{*}\alpha\in\br(\tilde{\mathcal{X}}_{\eta})[\ell^{r}]$ has trivial residue along each irreducible component that has been introduced by resolving the singularities that appeared due to the base change. Over a new component the restriction of $\tilde{q}$ is trivial. It follows that for such a component $V\subset\tilde{\mathcal{X}}_{0}$, if we write $\tilde{q}^{-1}(V)\cong\bigsqcup_{g\in G} \{g\}\times V$ and let $\tilde{q_{g}}\colon\{g\}\times V\to V $ be the restriction of $\tilde{q}$ to $\{g\}\times V$, then $0=(\tilde{q_{g}})_{*}(\partial_{g}\tilde{q_{\eta}}^{*}\tilde{\alpha})=(\tilde{q_{g}})_{*}\tilde{q_{g}}^{*}(\partial_{V}\tilde{\alpha})=\partial_{V}\tilde{\alpha}\in H^{1}(k(V),\mathbb{Z}/\ell^{r})$. Here, the first equality holds, because $\tilde{q_{\eta}}^{*}\tilde{\alpha}=0$, whereas for the last one, we use that $(\tilde{q_{g}})_{*}\tilde{q_{g}}^{*}=\id$ on $V$. Finally, the lift $\tilde{\alpha}\in\br(\tilde{\mathcal{X}})\{\ell\}$ satisfies $\tilde{q}^{*}\tilde{\alpha}=0\in\br(\tilde{\mathcal{Y}})$, as the restriction of $\tilde{q}^{*}\tilde{\alpha}$ to $\tilde{\mathcal{Y}_{\eta}}$ is zero and the natural pull-back map $\br(\tilde{\mathcal{Y}})\to\br(\mathcal{Y}_{\eta})$ is injective. The proof now is complete.\end{proof}\section{Brauer Group of the Total space and a specialization argument}\label{sec:4}Let $p\colon\mathcal{X}\to\Spec R$ be a strictly semi-stable scheme over a henselian discrete valuation ring $R$ with seperably closed residue field $k$. The main aim of this section is the study of the restriction map $\br(\mathcal{X})\to\br(\mathcal{X}_{0})_{\tor}$.\begin{lemma}\label{Lem:4.1} Let $p\colon\mathcal{X}\to\Spec R$ be a strictly semi-stable scheme over a henselian discrete valuation ring $R$ with seperably closed residue field $k$. Assume that $H^{1}(X,\mathcal{O}_{X})=0$ and $b_{2}=\rho$, where $b_{2}$ is the second $\ell$-adic betti number of $X\coloneqq\mathcal{X}_{\bar{\eta}}$ for some prime $\ell$ invertible in $R$ and $\rho$ is the rank of the N\'eron severi group of $X$. Then for any prime $\ell$ invertible in $R$, restriction yields an isomorphism $\br(\mathcal{X})\{\ell\}\to\br(\mathcal{X}_{0})\{\ell\}.$\end{lemma}\begin{proof} We follow the proof in \cite[Proposition 7.2]{Stef}. In what follows, $H^{i}_{cont}$ stands for Jannsen's continuous \'etale cohomology \cite{Jan}. The proper base change theorem (see \cite[Corollary VI.2.7]{mil}) gives rise to the natural isomorphisms\[H^{i}(\mathcal{X},\mu_{\ell^{r}})\cong H^{i}(\mathcal{X}_{0},\mu_{\ell^{r}}) \tag{4.1}\label{4.1}\]\[H_{cont}^{i}(\mathcal{X},\mathbb{Z}_{\ell}(1))\cong H^{i}(\mathcal{X}_{0},\mathbb{Z}_{\ell}(1)).\tag{4.2}\label{4.2}\] Note that \eqref{4.2} follows from \eqref{4.1}. Indeed, \eqref{4.1} implies that the groups $H^{i}(\mathcal{X},\mu_{\ell^{r}})$ are finite (see \cite[Corollary VI.2.8]{mil}) and thus, in this case Jannsen's continuous \'etale cohomology coincides with the usual $\ell$-adic cohomology (see \cite[(0.2)]{Jan}).\par We prove that the $\ell$-adic cycle class map $c_{\ell,1}\colon\Pic(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to H^{2}(\mathcal{X},\mathbb{Z}_{\ell}(1))$ is surjective. To this end, we compute $H^{2}_{cont}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1))$, using the Hochschild-Serre spectral sequence (see \cite[Corollary 3.4]{Jan})\[E^{p,q}_{2}\coloneqq H^{p}_{cont}(G,H^{q}(X,\mathbb{Z}_{\ell}(1)))\implies H^{p+q}_{cont}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1)), \tag{4.3}\label{4.3}\] where $G\coloneqq\Gal(K^{sep}/K)$ is the absolute Galois group of the fraction field $K$ of $R$ and $X\coloneqq\mathcal{X}_{\bar{\eta}}$. We observe that \[H^{1}(X,\mathbb{Z}_{\ell}(1))\coloneqq\lim_{\underset{r}\longleftarrow} H^{1}(X,\mu_{\ell^{r}})\cong\lim_{\underset{r}\longleftarrow}\Pic(X)[\ell^{r}]\eqqcolon T_{\ell}\Pic(X),\tag{4.4}\label{4.4}\]where the isomorphism is canonical and is induced from the Kummer sequence (see \cite[Corollary 4.18]{mil}). Since $H^{1}(X,\mathcal{O}_{X})=0$, the identity component of the Picard scheme $\Pic^{0}_{X/\bar{K}}$ is just a point (see \cite[Corollary 5.13]{kl}) and so $\Pic(X)=\NS(X)$. The N\'eron Severi group of a proper variety over an algebraically closed field is finitely generated (see \cite[Theorem V.3.25]{mil}). Hence, the torsion subgroup of $\Pic(X)$ is finite and thus $T_{\ell}\Pic(X)=0$. Therefore, \eqref{4.4} gives $E^{p,1}_{2}=0$ for every $p\geq 0$.\par Next, we calculate $E^{2,0}_{2}=H^{2}_{cont}(G,\mathbb{Z}_{\ell}(1))$. Consider the Gysin sequence
\[H^{1}(R,\mu_{\ell^{r}})\to H^{1}(G,\mu_{\ell^{r}})\overset{\partial}\to H^{0}(k,\mathbb{Z}/\ell^{r})\to H^{2}(R,\mu_{\ell^{r}}).\tag{4.5}\label{4.5}\] Proper base change (see \cite[Corollary VI.2.7]{mil}) together with $k^{sep}=k$, gives $H^{1}(R,\mu_{\ell^{r}})\cong H^{1}(k,\mu_{\ell^{r}})\cong k^{*}/(k^{*})^{\ell^{r}}=0$ and $H^{2}(R,\mu_{\ell^{r}})\cong H^{2}(k,\mu_{\ell^{r}})=\br(k)[\ell^{r}]=0$. Hence, \eqref{4.5} yields an isomorphism $\partial\colon H^{1}(G,\mu_{\ell^{r}})\overset{\cong}\to \mathbb{Z}/\ell^{r}$. The latter shows that the groups $H^{1}(G,\mu_{\ell^{r}})$ are finite for all $r$ and so we obtain $H^{2}_{cont}(G,\mathbb{Z}_{\ell}(1))=\lim H^{2}(G,\mu_{\ell^{r}})$ (see \cite[(2.1)]{Jan}). By \cite[Proposition 1.4.5]{cts}, we have that $cd_{\ell}(K)\leq 1$ and thus, $H^{2}_{cont}(G,\mathbb{Z}_{\ell}(1))=0$.\par From the above calculations, we find that $E^{p,1}_{\infty}=E^{p,1}_{2}=0$ and $E^{2,0}_{\infty}=E^{2,0}_{2}=0$. In addition, we note that $E^{0,2}_{\infty}=E^{0,2}_{2}$. To see this, we need to show that the term $E^{3,0}_{\infty}=E^{3,0}_{2}$ also vanishes. But this follows again from \cite[(2.1)]{Jan}, using that $cd_{\ell}(K)\leq 1$. Consequently, \eqref{4.3} yields an isomorphism \[H^{2}_{cont}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1))\cong H^{2}(X,\mathbb{Z}_{\ell}(1))^{G}.\tag{4.6}\label{4.6}\]\par We consider the following short exact sequence induced from the Kummer exact sequence\[0\to\Pic(X)\otimes\mathbb{Z}/\ell^{r}\to H^{2}(X,\mu_{\ell^{r}})\to\br(X)[\ell^{r}]\to 0\tag{4.7}\label{4.7}.\] Taking the inverse limit of \eqref{4.7} over $r$, we obtain the short exact sequence \[0\to\Pic(X)\otimes\mathbb{Z}_{\ell}\overset{c_{\ell,1}}\to H^{2}(X,\mathbb{Z}_{\ell}(1))\to T_{\ell}\br(X)\to0\tag{4.8}\label{4.8}.\] The last term of \eqref{4.8} is a free $\mathbb{Z}_{\ell}$-module of finite rank. Tensoring \eqref{4.8} with $\mathbb{Q}_{\ell},$ we see that the rank of $T_{\ell}\br(X)$ is $b_{2}-\rho$ and so vanishes. As clearly $\mathcal{X}^{\sm}_{0}(k)\neq\emptyset$, the henselian property of $R$ implies that the $R$-scheme $\mathcal{X}\to\Spec R$ admits a section (see \cite[Theorem I.4.2]{mil}). From this follows that the relative Picard functor $\Pic_{\mathcal{X}_{\eta}/K}$ is representable by a $K$-scheme (see \cite[Theorem 2.5]{kl} and \cite[Theorem 4.8]{kl}) and hence, $\Pic(X)^{G}={\Pic_{\mathcal{X}_{\eta}/K}(K^{sep})}^{G}=\Pic_{\mathcal{X}_{\eta}/K}(K)=\Pic(\mathcal{X}_{\eta})$. Passing to $G$-invariants the $\ell$-adic cycle class map induces by \eqref{4.6} an isomorphism $c_{\ell,1}\colon\Pic(\mathcal{X}_{\eta})\otimes\mathbb{Z}_{\ell}\to H_{cont}^{2}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1))$.\par The Gysin sequence for Jannsen's continuous \'etale cohomology \cite[\S 6]{stefanos}, yields an exact sequence\[\oplus^{n}_{i=1}\mathbb{Z}_{\ell}[\mathcal{X}_{0,i}]\to H^{2}(\mathcal{X},\mathbb{Z}_{\ell}(1))\to H_{cont}^{2}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1)).\tag{4.9}\label{4.9}\] Since we know now that $H_{cont}^{2}(\mathcal{X}_{\eta},\mathbb{Z}_{\ell}(1))$ is algebraic, taking closures of divisors on $\mathcal{X}_{\eta}$ in $\mathcal{X}$ shows via \eqref{4.9} that $H^{2}(\mathcal{X},\mathbb{Z}_{\ell}(1))$ is algebraic, too. By \eqref{4.2} and the functoriality with respect to pullbucks of the $\ell$-adic cycle class map, we obtain similarly that $c_{\ell,1}\colon\Pic(\mathcal{X}_{0})\otimes\mathbb{Z}_{\ell}\to H^{2}(\mathcal{X}_{0},\mathbb{Z}_{\ell}(1))$ is surjective. In particular, \eqref{2.6} implies \[\coker(H^{2}(\mathcal{X},\mathbb{Z}_{\ell}(1))\to H^{2}(\mathcal{X},\mu_{\ell^{r}}))\cong \br(\mathcal{X})[\ell^{r}]\]\[\coker(H^{2}(\mathcal{X}_{0},\mathbb{Z}_{\ell}(1))\to H^{2}(\mathcal{X}_{0},\mu_{\ell^{r}}))\cong \br(\mathcal{X}_{0})[\ell^{r}].\] The natural isomorphism $\br(\mathcal{X})[\ell^{r}]\cong\br(\mathcal{X}_{0})[\ell^{r}]$ then follows from \eqref{4.1} and \eqref{4.2}.\end{proof}\begin{remark}\label{rem:4.2} Regarding the characteristic zero case, the conditions in the above Lemma are satisfied, when $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2$.\end{remark} In general, $\rho\leq b_{2}$ and one can study the restriction map $\br(\mathcal{X})\to\br(\mathcal{X}_{0})_{\tor}$ via the following diagram with exact rows:
\begin{center}\begin{tikzcd}
(4.10) & 0 \arrow[r] & \Pic(\mathcal{X})\otimes \mathbb{Z}/\ell^{r} \arrow[r, "c_{1}"] \arrow[d, hook] & {H^{2}(\mathcal{X},\mu_{\ell^{r}})} \arrow[r] \arrow[d, "\cong"] & {\br(\mathcal{X})[\ell^{r}]} \arrow[r] \arrow[d, two heads] & 0 \\
& 0 \arrow[r] & \Pic(\mathcal{X}_{0})\otimes \mathbb{Z}/\ell^{r} \arrow[r, "c_{1}"] & {H^{2}(\mathcal{X}_{0},\mu_{\ell^{r}})} \arrow[r] & {\br(\mathcal{X}_{0})[\ell^{r}]} \arrow[r] & 0,
\end{tikzcd}
\end{center}
where $\ell$ is any prime invertible in $R$ and the middle restriction map is isomorphism by the proper base change theorem (see \cite[Corollary VI.2.7]{mil}). Applying the snake Lemma to the above diagram, we deduce the isomorphism\[\ker(\br(\mathcal{X})\to\br(\mathcal{X}_{0}))[\ell^{r}]\cong \coker(\Pic(\mathcal{X})\to\Pic(\mathcal{X}_{0}))\otimes\mathbb{Z}/\ell^{r}.\tag{4.11}\label{4.13}\]Passing to the colimit over $r$, we find that the map $\br(\mathcal{X})\{\ell\}\to\br(\mathcal{X}_{0})\{\ell\}$ is isomorphism if and only if \[\frac{\Pic(\mathcal{X}_{0})}{\Pic(\mathcal{X})}\otimes\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}=0.\tag{4.12}\label{4.14}\]Elementary deformation theory thus gives the following result.\begin{lemma}\label{Lem:4.3}Let $p\colon\mathcal{X}\to\Spec R$ be a strictly semi-stable scheme over a complete discrete valuation ring $R$. If $H^{2}(\mathcal{X}_{0},\mathcal{O}_{\mathcal{X}_{0}})=0$, then for any prime $\ell$ invertible in $R$, the map $\br(\mathcal{X})\{\ell\}\to\br(\mathcal{X}_{0})\{\ell\}$ is an isomorphism.\end{lemma}\begin{proof}The obstruction to extending a line bundle lies in $H^{2}(\mathcal{X}_{0},\mathcal{O}_{\mathcal{X}_{0}})$. Since this is zero, we may lift any line bundle $\mathcal{L}_{0}$ on $\mathcal{X}_{0}$ to the total space $\mathcal{X}$. Thus, the result follows immediately from \eqref{4.14}.\end{proof}\begin{lemma}\label{lem:4.4} Let $k$ be a field and set $p=\Char(k)$ if $\Char(k)>0$ and $p=1,$ otherwise. Let $\mathcal{Z}\to B$ be a smooth proper family of $k$-varieties with geometrically irreducible fibres. Then for $t\in B$ and a prime $\ell\neq p,$ we have a specialization map on Brauer groups $sp_{\bar{\eta},\bar{t}}\colon \br(\mathcal{Z}_{\bar{\eta}})\{\ell\}\to \br(\mathcal{Z}_{\bar{t}})\{\ell\},$ which is functorial with respect to pullbacks. In addition, this map is always surjective and is an isomorphism if and only if $\rho(\mathcal{Z}_{\bar{\eta}})=\rho(\mathcal{Z}_{\bar{t}})$.\end{lemma}\begin{proof} Fix a point $t\in B$ and a prime $\ell\neq p$. Recall that we have specialization maps $sp_{\bar{\eta},\bar{t}}\colon \NS(\mathcal{Z}_{\bar{\eta}})\to\NS(\mathcal{Z}_{\bar{t}})$ (see \cite[\S 20.3]{fw}) and $sp_{\bar{\eta},\bar{t}}\colon H^{2}(\mathcal{Z}_{\bar{\eta}},\mu_{\ell^{r}})\to H^{2}(\mathcal{Z}_{\bar{t}},\mu_{\ell^{r}}),$ that are both functorial with respect to pullbacks. The second one is an isomorphism and is induced by the smooth proper base change theorem (see \cite[Corollary VI.4.2]{mil}). As specialization commutes with the cycle class maps (see \cite[App. X]{gr}), \eqref{2.6} yields the homomorphism $sp_{\bar{\eta},\bar{t}}\colon \br(\mathcal{Z}_{\bar{\eta}})[\ell^{r}]\to \br(\mathcal{Z}_{\bar{t}})[\ell^{r}]$. The functoriality of this homomorphism with respect to pullbacks is clear.\par We have the following diagram with exact rows\[\begin{tikzcd}
0 \arrow[r] & \NS(\mathcal{Z}_{\bar{\eta}})\otimes\mathbb{Z}/\ell^{r} \arrow[r] \arrow[d, "{sp_{\bar{\eta},\bar{t}}}"] & {H^{2}(\mathcal{Z}_{\bar{\eta}},\mu_{\ell^{r}})} \arrow[r] \arrow[d, "{sp_{\bar{\eta},\bar{t}}}"] & {\br(\mathcal{Z}_{\bar{\eta}})[\ell^{r}]} \arrow[r] \arrow[d, dashed, "{sp_{\bar{\eta},\bar{t}}}"] & 0 \\
0 \arrow[r] & \NS(\mathcal{Z}_{\bar{t}})\otimes\mathbb{Z}/\ell^{r} \arrow[r] & {H^{2}(\mathcal{Z}_{\bar{t}},\mu_{\ell^{r}})} \arrow[r] & {\br(\mathcal{Z}_{\bar{t}})[\ell^{r}]} \arrow[r] & 0.\tag{4.13}\label{4.15}\end{tikzcd}\] Since the middle vertical map is an isomorphism, the first vertical map is always injective and the last one is surjective.\par Next, we assume $\rho(\mathcal{Z}_{\bar{\eta}})=\rho(\mathcal{Z}_{\bar{t}})$ and want to show that $sp_{\bar{\eta},\bar{t}}\colon \br(\mathcal{Z}_{\bar{\eta}})\{\ell\}\to \br(\mathcal{Z}_{\bar{t}})\{\ell\}$ is an isomorphism. All the terms in \eqref{4.15} are finite, and so taking inverse limits over $r$ is exact. Thus, the last vertical map in \eqref{4.15} induces a surjection between the associated $\ell$-adic Tate modules. As both $T_{\ell}\br(\mathcal{Z}_{\bar{\eta}})$ and $T_{\ell}\br(\mathcal{Z}_{\bar{t}})$ are free $\mathbb{Z}_{\ell}$-modules of the same rank $b_{2}-\rho$, the map $T_{\ell}\br(\mathcal{Z}_{\bar{\eta}})\to T_{\ell}\br(\mathcal{Z}_{\bar{t}})$ is in fact an isomorphism. Applying the snake Lemma to the inverse limit of \eqref{4.15}, the latter also yields that the specialization map $sp_{\bar{\eta},\bar{t}}\colon \NS(\mathcal{Z}_{\bar{\eta}})\otimes\mathbb{Z}_{\ell}\to\NS(\mathcal{Z}_{\bar{t}})\otimes\mathbb{Z}_{\ell}$ is an isomorphism for every $\ell\neq p$. The faithfully flat property of $\bigsqcup_{\ell\neq p}\Spec\mathbb{Z}_{\ell}\to\Spec\mathbb{Z}[1/p],$ implies $sp_{\bar{\eta},\bar{t}}\colon \NS(\mathcal{Z}_{\bar{\eta}})\otimes\mathbb{Z}[1/p]\to\NS(\mathcal{Z}_{\bar{t}})\otimes\mathbb{Z}[1/p]$ is an isomorphism. As a consequence, we also obtain $sp_{\bar{\eta},\bar{t}}\colon\br(\mathcal{Z}_{\bar{\eta}})\{\ell\}\to\br(\mathcal{Z}_{\bar{t}})\{\ell\}$ is an isomorphism for all $\ell\neq p$, since this holds for the first two vertical maps in \eqref{4.15}.\par The converse is true as well. Namely, if $sp_{\bar{\eta},\bar{t}}\colon\br(\mathcal{Z}_{\bar{\eta}})\{\ell\}\to\br(\mathcal{Z}_{\bar{t}})\{\ell\}$ is an isomorphism, then applying the snake Lemma to \eqref{4.15}, one gets $\rho(\mathcal{Z}_{\bar{\eta}})=\rho(\mathcal{Z}_{\bar{t}})$.\end{proof}
\begin{proposition}\label{prp:4.4} Let $k$ be a field and set $p=\Char(k)$ if $\Char(k)>0$ and $p=1,$ otherwise. Let $f\colon\mathcal{X}\to B$ and $g\colon\mathcal{Y}\to B$ be smooth proper morphisms of $k$-varieties with geometrically irreducible fibres. Let $q\colon\mathcal{Y}\to\mathcal{X}$ be a morphism of $B$-schemes. Assume that $\rho(\mathcal{X}_{\bar{\eta}})=b_{2},$ where $b_{2}$ is the second $\ell$-adic betti number of $\mathcal{X}_{\bar{\eta}}$ for some prime $\ell\neq p$. For $t\in B$, consider the pull-back map $q_{\bar{t}}^{*}\colon\br(\mathcal{X}_{\bar{t}})\to\br(\mathcal{Y}_{\bar{t}})$. Then for every prime $\ell\neq p$ and $t\in B$, such that $\rho(\mathcal{Y}_{\bar{t}})=\rho(\mathcal{Y}_{\bar{\eta}}),$ we have $\ker(q_{\bar{t}}^{*})\{\ell\}\cong \ker (q_{\bar{\eta}}^{*})\{\ell\}$.\end{proposition}\begin{proof}We have $\rho(\mathcal{X}_{\bar{\eta}})\leq \rho(\mathcal{X}_{\bar{t}})\leq b_{2}$ for all $t\in B$. The inequalities follow from the injectivity of $sp_{\bar{\eta},\bar{\tau}}\colon \NS(\mathcal{X}_{\bar{\eta}})\otimes\mathbb{Z}[1/p]\to\NS(\mathcal{X}_{\bar{t}})\otimes\mathbb{Z}[1/p]$ and of the $\ell$-adic cycle class map $c_{\ell,1}$ for a prime $\ell\neq p$. The assumption $\rho(\mathcal{X}_{\bar{\eta}})=b_{2}$ implies that the Picard number in the family $\mathcal{X}\to B$ is constant and equals $b_{2}$. By Lemma \ref{lem:4.4}, we deduce that for $t\in B,$ with $\rho(\mathcal{Y}_{\bar{t}})=\rho(\mathcal{Y}_{\bar{\eta}})$, both $sp_{\bar{\eta},\bar{t}}\colon\br(\mathcal{X}_{\bar{\eta}})\{\ell\}\to\br(\mathcal{X}_{\bar{t}})\{\ell\}$ and $sp_{\bar{\eta},\bar{t}}\colon\br(\mathcal{Y}_{\bar{\eta}})\{\ell\}\to\br(\mathcal{Y}_{\bar{t}})\{\ell\}$ are isomorphisms. The result follows as $sp_{\bar{\eta},\bar{t}}\circ q^{*}_{\bar{\eta}}=q_{\bar{t}}^{*}\circ sp_{\bar{\eta},\bar{t}}$.\end{proof}\section{Main Application: Godeaux Surface}\label{sec:5} Let $k$ be an algebraically closed field of characteristic $0$. Any quintic surface $Y,$ which is invariant and fixed point free under the $\mathbb{Z}/5$-action \eqref{1.1} can be realised as a hypersurface in $\mathbb{P}_{k}^{3}$ defined by an equation of the form\begin{center}$G_{\alpha_{1},\ldots,\alpha_{8}}(T_{0},\ldots,T_{3})= T^{5}_{0}+T^{5}_{1}+T^{5}_{2}+T^{5}_{3}+\alpha_{1}T^{3}_{0}T_{2}T_{3}+\alpha_{2}T_{0}T^{3}_{1}T_{2}+\alpha_{3}T_{1}T^{3}_{2}T_{3}+\alpha_{4}T_{0}T_{1}T^{3}_{3}+\alpha_{5}T_{0}T^{2}_{2}T^{2}_{3}+\alpha_{6}T^{2}_{0}T_{1}T^{2}_{2}+\alpha_{7}T^{2}_{1}T_{2}T_{3}^{2}+\alpha_{8}T^{2}_{0}T^{2}_{1}T_{3},$\end{center}where $\alpha\coloneqq(\alpha_{1},\alpha_{2},\ldots,\alpha_{8})\in\mathbb{A}_{k}^{8}(k)$. Given a point $\alpha\in\mathbb{A}_{k}^{8}(k)$, we set $Y_{\alpha}\coloneqq\mathbb{V}(G_{\alpha})\subset\mathbb{P}^{3}_{k}$.\begin{proposition}[\cite{persson}]\label{prp:5.1}Let $k$ be an algebraically closed field of characteristic $0$. There exists a triple-point free semi-stable degeneration $\mathcal{X}\to\Spec k[[t]]$ of a Godeaux surface, such that the dual graph $\Gamma$ of the special fibre $\mathcal{X}_{0}$ is a chain. If we number the components $\mathcal{X}_{0,i}$ of $\mathcal{X}_{0}$, so that $\mathcal{X}_{0,i}\cap\mathcal{X}_{0,i+1}\neq\emptyset$ for every $i$, then all $\mathcal{X}_{0,i}$ are elliptic ruled surfaces except the last one $\mathcal{X}_{0,n}$, which is a cubic surface. The components $\mathcal{X}_{0,i}$ and $\mathcal{X}_{0,i+1}$ are glued along smooth elliptic curves $C_{i,i+1}$. In addition, if $C_{\mathcal{X}_{0,i}}\coloneqq\mathcal{X}_{0,i}\cap\mathcal{X}^{\sing}_{0},$ then $C_{\mathcal{X}_{0,1}}$ sits as a $5$-fold cover onto the base of the ruled surface $\mathcal{X}_{0,1}$, $C_{\mathcal{X}_{0,i}}$ consists of two disjoint sections of the ruled surface $\mathcal{X}_{0,i}$ for $i=2,\ldots,n-1$ and $C_{\mathcal{X}_{0,n}}+K_{\mathcal{X}_{0,n}}=0\in\Pic(\mathcal{X}_{0,n})$.\end{proposition}\begin{proof}We briefly sketch Perssons' construction (see \cite[Appendix 3]{persson}). Pick $p\in\mathbb{P}^{3}_{k}(k)$ not in any coordinate hyperplane. Then there exists $\alpha_{0}\in\mathbb{A}^{8}_{k}(k),$ such that $Y_{0}\coloneqq Y_{\alpha_{0}}$ has $5$ triple points at the $\mathbb{Z}/5$-orbit of $p$ and no other singularities (see \cite[Appendix 3, Lemma 1]{persson}). The resolution $\nu\colon\tilde{Y}_{0}\to Y_{0}$ obtained by blowing up the $5$ triple points is an elliptic ruled surface and the curves $E_{i}\coloneqq\nu^{-1}(p_{i})$, where $p_{i}$ is a singular point of $Y_{0},$ are smooth elliptic and constitute sections in $\tilde{Y}_{0}$ (see \cite[Appendix 3, Lemma 2]{persson}). Since we blow up with respect to a $\mathbb{Z}/5$-invariant center, the $\mathbb{Z}/5$-action on $Y_{0}$ lifts naturally to $\tilde{Y}_{0}$. This action has no fixed points on $\tilde{Y}_{0}$ and permutes the curves $E_{i}$. Hence, passing to the quotient by the $\mathbb{Z}/5$-action on $\tilde{Y}_{0},$ we obtain an elliptic ruled surface $\tilde{X}_{0}$ and the curves $E_{i}$ become a $5$-fold cover onto the base of $\tilde{X}_{0}$ (see \cite[Appendix 3, Lemma 2]{persson}).\par Choose $\alpha_{1}\in\mathbb{A}^{8}_{k}(k),$ such that $Y_{1}\coloneqq Y_{\alpha_{1}}$ is smooth and $p\notin Y_{1}$. Consider the one-parameter family of $\mathbb{Z}/5$-invariant quintics $\mathcal{Y}\coloneqq\mathbb{V}(G_{\alpha_{0}}+t(G_{\alpha_{1}}-G_{\alpha_{0}}))\subset\mathbb{P}^{3}_{k}\times_{k}\mathbb{A}^{1}_{k}$. The condition $p\notin Y_{1}$ implies that $\mathcal{Y}$ is smooth along $Y_{0}$. We perform a base change with respect to the local ring at $0\in\mathbb{A}^{1}_{k}(k)$ and note that the total space of $\mathcal{Y}\to B$, $B\coloneqq\Spec\mathcal{O}_{\mathbb{A}^{1}_{k},0},$ remains regular. Let $\tilde{\mathcal{Y}}\to\mathcal{Y}$ be the blow up at the $5$ triple points of $Y_{0}$. Then the special fibre of the degeneration $\tilde{\mathcal{Y}}\to B$ is given by $\tilde{Y}_{0}$ and along each elliptic curve $E_{i}$, there is glued a $P_{i}\cong\mathbb{P}^{2}$ with multiplicity $3$. Consequently, $\mathcal{O}_{\tilde{\mathcal{Y}}}(\tilde{Y}_{0})\cong \mathcal{O}_{\tilde{\mathcal{Y}}}(-\sum_{i}P_{i})^{\otimes 3}$. We take the triple cyclic covering $\mathcal{Y}^{'}\to\tilde{\mathcal{Y}}$ branched along $\tilde{Y}_{0}$. Then $\mathcal{Y}'$ is regular and the composite $\mathcal{Y}^{'}\to B$ factors through $ B\to B, t\mapsto t^{3},$ giving rise to a degeneration $\mathcal{Y}'\to B$ of a $\mathbb{Z}/5$-invariant quintic. The special fibre of the resulting degeneration consists of $\tilde{Y}_{0}$ and along each $E_{i}$, there is glued a cubic surface $R_{i}$ with multiplicity $1$. Finally, forming the $\mathbb{Z}/5$-quotient, we deduce a semi-stable degeneration $\mathcal{X}\to B$ of a Godeaux surface. The special fibre of this family consists of two components, the elliptic ruled surface $\tilde{X}_{0}$ and a cubic surface $R$. They intersect along a smooth elliptic curve $C,$ which sits as a $5$-fold section in $\tilde{X}_{0}$ and satisfies $K_{\tilde{X}_{0}}+C=0\in\Pic(\tilde{X}_{0})$.\end{proof}
\begin{proposition}\label{prp:5.2} Let $k$ be an algebraically closed field of characteristic $0$. Let $\mathcal{X}\to\Spec k[[t]]$ be a semi-stable degeneration of a Godeaux surface with special fibre as in Proposition \ref{prp:5.1}. Then $\mathcal{X}\to\Spec k[[t]]$ admits a natural $5:1$ \'etale cover $\mathcal{Y}\to\Spec k[[t]]$, which is a triple-point free semi-stable degeneration of a smooth quintic surface. The dual graph $\Gamma$ of the special fibre $\mathcal{Y}_{0}$ is a flower pot. In particular, $\mathcal{Y}_{0}$ is given by an elliptic ruled surface $Y_{0}$ with five disjoint sections $E_{i},$ such that along each $E_{i},$ there is glued a chain of elliptic ruled surfaces $Y_{i,j}$ with a cubic surface $Y_{i,n}$ as an end-component. These chains are isomorphic to each other and they are permuted by the $\mathbb{Z}/5$-action of the covering, whereas $Y_{0}$ is invariant under this action.\end{proposition}
\begin{proof}The Mayer-Vietoris sequence \eqref{2.4} gives rise to the following exact sequence\[0\to\Pic(\mathcal{X}_{0})\to\bigoplus_{i=1}^{n}\Pic(\mathcal{X}_{0,i})\overset{r}\to\bigoplus_{i=1}^{n-1}\Pic(C_{i,i+1})\tag{5.1}\label{5.1}.\]\par We aim to compute the torsion group $\Pic(\mathcal{X}_{0})_{\tor}$. Since $\mathcal{X}_{0,n}$ is a rational surface, we have $\Pic(\mathcal{X}_{0,n})_{\tor}=0$. Thus, the restriction $r_{\tor}$ of $r$ to $\bigoplus_{i=1}^{n}\Pic(\mathcal{X}_{0,i})_{\tor},$ takes $(\mathcal{L}_{i})_{i}$ to $(\mathcal{L}_{1}|_{C_{1,2}}-\mathcal{L}_{2}|_{C_{1,2}},\ldots,\mathcal{L}_{n-1}|_{C_{n-1,n}})$. Recall that for $i=2,\ldots,n-1,$ the components $\mathcal{X}_{0,i}$ are ruled and both curves $C_{i-1,i}$ and $C_{i,i+1}$ constitute sections in $\mathcal{X}_{0,i}$. Hence, restriction of line bundles yields isomorphisms $\Pic(\mathcal{X}_{0,i})_{\tor}\cong\Pic(C_{i-1,i})_{\tor}$ (resp. $ \Pic(C_{i,i+1})_{\tor}$). It follows that the kernel of $r_{\tor}$ coincides with the one of $\Pic(\mathcal{X}_{0,1})_{\tor}\to\Pic(C_{1,2})_{\tor},\mathcal{L}\mapsto\mathcal{L}|_{C_{1,2}}$. The last map can be identified with ${(\psi|_{C_{1,2}})}^{*}\colon\Pic(E)_{\tor}\cong\Pic(\mathcal{X}_{0,1})_{\tor}\to\Pic(C_{1,2})_{\tor},$ where $\psi\colon\mathcal{X}_{0,1}\to E$ is the fibering of the ruled surface $\mathcal{X}_{0,1}$. Since $C_{1,2}$ is a $5$-fold cover onto the base curve $E$, the kernel of the pullback map ${(\psi|_{C_{1,2}})}^{*}$ is the Cartier dual of $\mathbb{Z}/5$, which is $\mu_{5}\cong\mathbb{Z}/5$. Consequently, $\Pic(\mathcal{X}_{0})_{\tor}\cong\mathbb{Z}/5$.\par To conclude, we need to show that a non-trivial $5$-torsion line bundle on $\mathcal{X}_{0}$ lifts to a $5$-torsion line bundle on the total space $\mathcal{X}$. But torsion line bundles always extend to torsion ones (see \cite[Proposition 22.5]{hr}). Lastly, the desired form of the special fibre for the associated cover $\mathcal{Y}$ is justified by the fact that the restriction of $0\neq\mathcal{L}\in\Pic(\mathcal{X}_{0})_{\tor}\cong\mathbb{Z}/5$ to $\cup_{i>1}\mathcal{X}_{0,i}$ is trivial, whereas $\mathcal{L}|_{\mathcal{X}_{0,1}}\neq 0$.\end{proof}\begin{lemma}\label{lem:5.2} Let $k$ be an algebraically closed field of characteristic $0$. Let $\mathcal{X}\to\Spec k[[t]]$ be a semi-stable degeneration of a Godeaux surface with special fibre $\mathcal{X}_{0},$ as in Proposition \ref{prp:5.1}. Then $\br(\mathcal{X})=0$.\end{lemma}\begin{proof} A Godeaux surface $X$ over $\overline{k((t))}$ satisfies $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2$. Hence, we may apply Lemma \ref{4.1} and obtain the isomorphism $\br(\mathcal{X})\to\br(\mathcal{X}_{0})_{\tor}$.\par We compute $\br(\mathcal{X}_{0})$ with the help of the Mayer-Vietoris exact sequence \eqref{2.4}. As the components of the special fibre $\mathcal{X}_{0}$ are all ruled or rational and the ground field is algebraically closed, we find $\br(\mathcal{X}_{0,i})=0$ for all $i$. Thus, \eqref{2.4} gives rise to the exact sequence \[\bigoplus^{n}_{i=1}\Pic(\mathcal{X}_{0,i})\overset{r}\to\bigoplus^{n-1}_{i=1}\Pic(C_{i,i+1})\to \br(\mathcal{X}_{0})\to 0,\tag{5.2}\label{5.2}\] and we need to show that $r$ is surjective. Consider the fibrations $p_{i}\colon\mathcal{X}_{0,i}\to E_{i}$ and let $s_{i,j}\colon E_{i}\to C_{j,j+1}\subset\mathcal{X}_{0,i}$ be the corresponding sections, where $j=i-1,i$ and $i=2,\ldots,n-1$. For a given point $x\in E_{i}$, we set $F^{i}_{x}\coloneqq p^{-1}_{i}(x)$ for the fibre of $p_{i}$ over $x$ and note that, \[r(0,\ldots,[F^{i}_{x}],\ldots,0)=(0,\ldots,-[s_{i,i-1}(x)],[s_{i,i}(x)],\ldots,0).\tag{5.3}\label{5.3}\] As the double curves $C_{i,i+1}$ are isomorphic to each other, \eqref{5.3} implies that it is enough to show $([x],0,\ldots,0)\in\im(r)$ for all points $x\in C_{1,2}$. In fact, it suffices to find a single point $x\in C_{1,2}$ for which the latter holds. Indeed, this follows from $\Pic(C_{1,2})=\mathbb{Z}[x]\oplus\Pic^{0}(C_{1,2})$ and that the composite $\Pic^{0}(E_{1})\subset\Pic^{0}(\mathcal{X}_{0,1})\to\Pic^{0}(C_{1,2})$ is surjective.\par Since $\mathcal{X}_{0,n}$ is a cubic surface, we may pick a $(-1)$-rational curve $D\subset\mathcal{X}_{0,n}$. Recall that $K_{\mathcal{X}_{0,n}}=- C_{n-1,n}$. Thus, the adjunction formula $(K_{\mathcal{X}_{0,n}}+D).D=-2$ yields $C_{n-1,n}.D=1$ and so $r(0,\ldots,0,-[D])=(0,\ldots,[x])$ for some point $x\in C_{n-1,n}$. Finally, by \eqref{5.3} we get a point $x\in C_{1,2}$, such that $([x],0,\ldots,0)\in\im(r)$, finishing the proof.\end{proof} We are in the position to prove the main result of this paper.\begin{proof}[Proof of Theorem \ref{thm:1.3}.] Pick a generic pencil $\mathcal{Y}\to\mathbb{P}^{1}_{k}\subset|\mathcal{O}_{\mathbb{P}^{3}_{k}}(5)|^{\mathbb{Z}/5}$ of $\mathbb{Z}/5$-invariant quintics, such that some fibre $\mathcal{Y}_{t_{0}},$ $t_{0}\in\mathbb{P}^{1}_{k}(k)$ is not passing through any of the $4$ fixed points of the action \eqref{1.1} and has $5$ triple points as its only singularities. Since the pencil is generic, we may assume that the $5$ triple points are not base points and so the total space $\mathcal{Y}$ is smooth along $\mathcal{Y}_{t_{0}}$. We perform a base change with respect to the completion of the local ring of $\mathbb{P}^{1}_{k}$ at $t_{0}$. Applying semi-stable reduction to the family $\mathcal{Y}$ and its $\mathbb{Z}/5$-quotient $\mathcal{X}$, we arrive at the $5:1$ \'etale cover $q\colon\tilde{\mathcal{Y}}\to\tilde{\mathcal{X}},$ described in Proposition \ref{prp:5.2} (see proof of Proposition \ref{prp:5.1}). We want to show that the pullback map $q^{*}_{\bar{\eta}}$ is injective. If $q^{*}_{\bar{\eta}}=0$, then Theorem \ref{thm:1.4} implies that up to some finite base change of discrete valuation rings followed by a resolution of the total space (see \cite{Har}), the restriction map $\br(\tilde{\mathcal{X}})\to\br(\mathcal{X}_{\bar{\eta}})$ is surjective. The latter contradicts Lemma \ref{lem:5.2}.\end{proof}\begin{remark}\label{rem:5.4}In general the universal cover $Y$ of a Godeaux surface $X$ is not a quintic, but rather its canonical model $\nu\colon Y\to Y_{can}$ that is obtained by contracting the $(-2)$-curves (see \cite[Theorem 2]{lw}). In particular, $Y_{can}\subset\mathbb{P}^{3}_{\mathbb{C}}$ is invariant, fixed point free under the $\mathbb{Z}/5$-action \eqref{1.1} and $X_{can}=Y_{can}/(\mathbb{Z}/5)$ (see \cite[$\S 2$]{lw}).\par We note that if $\nu\colon Y\to Y_{can}$ is not an isomorphism, then $\rho(Y)\geq 13$. Especially, $\rho(Y)=9$ implies that $Y$ is a quintic surface.\par Indeed, let $n$ denote the number of dinstinct $\mathbb{Z}/5$-orbits of rational double points in $Y_{can}$. Pick a representative $p_{i}\in Y_{can}$ for each $\mathbb{Z}/5$-orbit and let $n_{i}$ be the number of components of the fibre $E_{i}\coloneqq \nu^{-1}(p_{i})$, where $i=1,\ldots,n$. Then we have the relations \[\rho(Y)=\rho(Y_{can})+\sum^{n}_{i=1}5n_{i}\ \text{and}\ 9=\rho(X)=\rho(X_{can})+\sum^{n}_{i=1}n_{i}.\]Since $\rho(Y_{can})\geq\rho(X_{can}),$ we find $\rho(Y)\geq 9 +\sum^{n}_{i=1}4n_{i}\geq 13$, as claimed.\end{remark}\begin{proof}[Proof of Theorem \ref{thm:1.1}.]The generic Picard number in the family of smooth, $\mathbb{Z}/5$-invariant and fixed point free quintics $Y$ over $\mathbb{C}$ is $9$ (see \cite[Example 3]{BoM}). By Remark \ref{rem:5.4} we know that if the universal cover of a Godeaux surface has Picard number $9$, then it is a quintic. Hence, Proposition \ref{prp:4.4} shows that it suffices to find a single smooth, $\mathbb{Z}/5$-invariant and fixed point free quintic $Y$ with $\rho(Y)=9$, so that if $X\coloneqq Y/(\mathbb{Z}/5)$, then $q^{*}_{X}\colon\br(X)\to\br(Y)$ is injective. But this follows immediately from Theorem \ref{thm:1.3}, by considering a generic pencil of $\mathbb{Z}/5$-invariant quintics, such that some member has $5$ triple points as its only singularities (see \cite[Appendix 3, Lemma 1]{persson}).\end{proof}\section{Application II: Enriques Surface}\label{sec:6} Let $k$ be an algebraically closed field of characteristic $0$ and let $R\coloneqq k[[t]]$. By \cite[Corollary 6.2]{mor}, up to birational equivalence, there are exactly three types of strictly semi-stable degenerations of Enriques $\mathcal{X}\to\Spec R$, such that the canonical divisor $K_{\mathcal{X}}$ is $2$-torsion. We refer to them as $(\text{i}a)$ $\mathcal{X}_{0}$ is a smooth Enriques' surface, $(\text{ii}a)$ $\mathcal{X}_{0}$ is an elliptic chain with one rational component and $(\text{iii}a)$ $\mathcal{X}_{0}$ is rationally polygonal and its dual graph $\Gamma$ is a triangulation of $\mathbb{R}\mathbb{P}^{2}$ (see \cite[Corollary 6.2]{mor}).\par The corresponding double cover $\mathcal{Y}\to\Spec R$ is a strictly semi-stable degeneration of a $K3$ surface, with $(\text{i}a)$ $\mathcal{Y}_{0}$ is a $K3$ surface, $(\text{ii}a)$ $\mathcal{Y}_{0}$ is an elliptic chain with two rational components and $(\text{iii}a)$ $\mathcal{Y}_{0}$ is rationally polygonal and its dual graph $\Gamma$ is a triangulation of $S^{2}$ (see \cite[Theorem 6.1]{mor}).\par Let $\mathcal{M}$ denote the coarse moduli space of Enriques surfaces. Beauville showed that Enriques surfaces $X$ with $q^{*}_{X}=0$ form an infinite, countable union of non-empty hypersurfaces in $\mathcal{M}$ (see \cite[Corollary 6.5]{bea}). These hypersurfaces have been explicitly described in \cite{bea}, using lattice theory (see \cite[Proposition 6.2]{bea}). In what follows, $\mathcal{H}\subset\mathcal{M}$ denotes the union of these hypersurfaces. To the author's knowledge the following result is new:\begin{theorem}\label{thm:6.2} Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $\neq2$. Let $\mathcal{X}\to\Spec R$ be a type $(\text{ii}a)$ degeneration of an Enriques surface. Consider its canonical cover $q\colon\mathcal{Y}\to\mathcal{X}$. Then $q^{*}_{\bar{\eta}}\colon\br(\mathcal{X}_{\bar{\eta}})[2]\to\br(\mathcal{Y}_{\bar{\eta}})[2]$ is non-zero.\end{theorem}\begin{proof} It is clear that a type $(\text{ii}a)$ degeneration of an Enriques surface and its double covering $q\colon\mathcal{Y}\to\mathcal{X}$ satisfy the conditions in Theorem \ref{thm:1.4}. Hence, it suffices to show that the generator $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\cong\mathbb{Z}/2$ cannot be lifted to $\br(\mathcal{X})\{2\}$. This holds true due to Lemma \ref{lem:6.3} (cf. \cite[Remark 7.4]{Stef}).\end{proof}\begin{lemma}\label{lem:6.3} Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $\neq2$. Let $\mathcal{X}\to\Spec R$ be a type $(\text{ii}a)$ degeneration of an Enriques surface. Then the map $\br(\mathcal{X})\to\br(\mathcal{X}_{\bar{\eta}})$ is trivial.\end{lemma}\begin{proof} The proof is more or less the same as in Lemma \ref{lem:5.2}. We may replace $R$ by its henselization $R^{\text{h}}$. We number the components $\mathcal{X}_{0,i},$ such that $C_{i,i+1}\coloneqq\mathcal{X}_{0,i}\cap\mathcal{X}_{0,i+1}\neq\emptyset$ for all $i$. We also set $C_{\mathcal{X}_{0,i}}\coloneqq\mathcal{X}_{0,i}\cap\mathcal{X}^{\sing}_{0}$.\par An Enriques surface has invariants $p_{q}=q=0$. Hence, Lemma \ref{Lem:4.1} yields the isomorphism $\br(\mathcal{X})\cong\br(\mathcal{X}_{0})_{\tor}$ and it suffices to prove $\br(\mathcal{X}_{0})=0$. The exact sequence \eqref{5.2} holds here as well and so we only need to show that the map $r$ is surjective. Recall that the special fibre $\mathcal{X}_{0}$ is an elliptic chain with rational end-component $\mathcal{X}_{0,n},$ and thus the relation \eqref{5.3} is also true. It is therefore enough to prove that $([x],0,\ldots,0)\in\im(r)$ for a single point $x\in C_{1,2}$, as $C_{1,2}$ is an \'etale double cover onto the base of the elliptic ruled component $\mathcal{X}_{0,1}$ (cf. the argument in Lemma \ref{lem:5.2}).\par We claim that there is at least one component $\mathcal{X}_{0,i}$, which is not a minimal surface. The canonical bundle formula gives $K_{\mathcal{X}_{0,i}}=K_{\mathcal{X}}\arrowvert_{\mathcal{X}_{0,i}}-C_{\mathcal{X}_{0,i}}$. Since $K_{\mathcal{X}}$ is $2$-torsion in $\Pic(\mathcal{X})$, we obtain $K_{\mathcal{X}_{0,i}}+C_{\mathcal{X}_{0,i}}\equiv0$. For a contradiction, assume now that all components of the special fibre $\mathcal{X}_{0}$ are minimal. As $\mathcal{X}_{0,i}$ are ruled for $i=1,\ldots,n-1$, we get $K^{2}_{\mathcal{X}_{0,i}}=C^{2}_{\mathcal{X}_{0,i}}=0$. Recall that $\mathcal{X}_{0,n}$ is a rational surface and so $K^{2}_{\mathcal{X}_{0,n}}=C^{2}_{n-1,n}=8$ or $9$. The first relation yields $C^{2}_{n-1,n}=0$, which contradicts the last one.\par Pick a component $\mathcal{X}_{0,i}$, which is not minimal. For a $(-1)$-rational curve $D\subset\mathcal{X}_{0,i}$, the adjunction formula $(K_{\mathcal{X}_{0,i}}+D).D=-2$ yields $D.C_{\mathcal{X}_{0,i}}=1$. Therefore, $r(0,\ldots,[D],0,\dots,0)=(0,\ldots,[x],\ldots,0)$ for some point $x\in C_{j-1,j}$, where $j=i$ or $i+1$. By \eqref{5.3} we also deduce a point $x\in C_{1,2}$, such that $([x],\ldots,0)\in\im(r),$ finishing the proof.\end{proof}
As an immediate consequence of Theorem \ref{thm:6.2}, we obtain:
\begin{corollary}\label{cor:6.5} Any member $X$ in $\mathcal{H}$ cannot be degenerated to a singular Enriques surface, such that after applying semi-stable reduction the special fibre becomes of type $(\text{ii}a)$.\end{corollary}\section{Application III: Quotients of Products of Curves} As a last application of Theorem \ref{thm:1.4}, we prove the following.\begin{theorem}\label{thm:7.1}Let $E$ be a smooth elliptic curve and $C$ a smooth projective curve over $\mathbb{C}$. Let $\psi$ be an automorphism of $C$ of finite order $d,$ with $\Fix(\psi)\neq\emptyset$. Assume $\Hom(E,\Jac(C))=\{0\}$. Then there exists $0\neq\tau\in E[d]$ with the following property: If $Y\coloneqq E\times C$ and $q_{\tau}\colon Y\to X\coloneqq E\times C/(\mathbb{Z}/d)$ denotes the quotient by the diagonal action \[(x,y)\mapsto (x+\tau,\psi(y)),\tag{7.1}\label{7.1}\] then for any $\alpha\in\br(X)\setminus\br(X)_{\text{div}},$ the pullback $q^{*}_{\tau}\alpha\in\br(Y)$ is non-zero.\end{theorem} We follow the same strategy as in the previous sections. We first review degenerations of elliptic curves.\subsection{Cycle degenerations}\label{subsec:7.1}Let $R$ be a discrete valuation ring with residue field $k$ of any characteristic and fraction field $K$. Let $E$ be an elliptic curve over $K$. It is well-known that we can always find a finite extension $R\subset\tilde{R}$ of discrete valuation rings, such that the fraction field $\tilde{K}$ of $\tilde{R}$ is a finite seperable field extension of $K$ and the minimal model $\mathcal{E}\to\Spec\tilde{R}$ of the base change $E_{\tilde{K}}$ is semi-stable over $\tilde{R}$ (see \cite[\href{https://stacks.math.columbia.edu/tag/0CDM}{Tag 0CDM}]{stacks-project}). There are two particular possibilities for the special fibre $E_{0}\coloneqq\mathcal{E}_{0}$:\begin{enumerate}
\item [$(I_{0})$] $E_{0}$ is a smooth elliptic curve.
\item [$(I_{\nu})$] $E_{0}$ is a N\'eron $\nu$-gon for some integer $\nu\geq 2$, i.e. $E_{0}$ is isomorphic to the quotient of $\mathbb{P}^{1}_{\tilde{k}}\times(\mathbb{Z}/\nu)$ obtained by identifying the $\infty$-section of the $i$-th copy of $\mathbb{P}^{1}$ with the $0$-section of $(i+1)$-st.
\end{enumerate}\par The smooth locus $\mathcal{E}^{\sm}$ is a commutative smooth group scheme over $\tilde{R}$, which is the N\'eron model of $E_{\tilde{K}}$. In case $E_{0}$ is a N\'eron $\nu$-gon, the smooth locus of the special fibre is the affine group scheme $\mathbb{G}_{m,\tilde{k}}\times\mathbb{Z}/\nu$. If furthermore, $\nu$ is invertible in $\tilde{R}$, then the group scheme $\mathcal{E}^{\sm}[\nu]\coloneqq\ker(\mathcal{E}^{\sm}\overset{\times\nu}\to\mathcal{E}^{\sm})$ is \'etale locally isomorphic to $(\mathbb{Z}/\nu)^{2}$ (see \cite[Proposition 20.7]{ab}) and thus, after an unramified base change of discrete valuation rings, we may assume that \[\mathcal{E}^{\sm}[\nu]\cong(\mathbb{Z}/\nu)^{2}.\tag{7.2}\label{7.2}\]\par The natural action of $\mathcal{E}^{\sm}[\nu]$ on $\mathcal{E}^{\sm}$ extends to $\mathcal{E}$ (see \cite[Proposition 9.3.13]{lq}). The action on the special fibre $E_{0}$ can be described as follows: The first direct summand of $E^{\sm}_{0}[\nu]\cong\mu_{\nu}\times\mathbb{Z}/\nu$ acts on each component of the special fibre via multiplication by $\nu$-th roots of unity and fixes the two vertices $0$ and $\infty$, while the second summand rotates the components.\par\subsection{Proof of Theorem \ref{thm:7.1}}\label{subsec:7.2} Fix a positive integer $d>1$. Let $\mathcal{E}\to\Spec R$ be a semi-stable model of an elliptic curve $E_{K}$, whose special fibre is of type $I_{\kappa d}$ for some $\kappa\geq2$ and $\nu\coloneqq \kappa d$ is invertible in $R$. Up to some unramified base change, we may assume that \eqref{7.2} holds. There are exactly $n_{d}\coloneqq d\varphi(d)$ points $0\neq\tau\in\mathcal{E}^{\sm}[d]$, such that the quotient map $E_{0}\to E_{0}/\tau$ is trivial over each component of $E_{0}/\tau$. Via the isomorphism $E^{\sm}_{0}[\nu]\cong\mu_{\nu}\times\mathbb{Z}/\nu$, they correspond to tuples ($\zeta^{\kappa},i\kappa$), where $\zeta$ is a $d$-th root of unity and $1\leq i\leq\nu$ is an integer prime to $d$.\par Pick any $0\neq\tau\in\mathcal{E}^{\sm}[d]$ from these $n_{d}$ points. Consider a smooth proper family of curves $\mathcal{C}\to\Spec R$ and an $R$-automorphism $\psi$ of order $d$ that fixes a section of this family. We let $\mathbb{Z}/d$ act diagonally on the product $\mathcal{Y}\coloneqq\mathcal{E}\times_{R}\mathcal{C}$ as in \eqref{7.1} and form the quotient \[q_{\tau}\colon\mathcal{Y}\to\mathcal{X}\coloneqq(\mathcal{E}\times_{R}\mathcal{C})/(\mathbb{Z}/d) \label{7.6}. \tag{7.3}\]\par In contrast to the previous two applications, the Brauer group of the special fibre of a cycle degeneration has infinitely many torsion elements.\begin{lemma}\label{lem:7.2} Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $N_{0}$ be a smooth projective variety over $k$. Consider a triple-point free semi-stable degeneration $\mathcal{N}\to\Spec R$, such that the dual graph $\Gamma$ of $\mathcal{N}_{0}$ is a cycle with components $\mathcal{N}_{0,i}\cong\mathbb{P}^{1}\times N_{0}$ and each double intersection $\mathcal{N}_{0,i}\cap\mathcal{N}_{0,i+1}$ is isomorphic to a fibre of $\mathbb{P}^{1}\times N_{0}\to\mathbb{P}^{1}$. Then there is a short exact sequence \[0\to\Pic(N_{0})\to\br(\mathcal{N}_{0})\to\br(N_{0})\to0.\tag{7.4}\label{7.3}\]\end{lemma}\begin{proof} The Mayer-Vietoris sequence \eqref{2.4} gives rise to the following exact sequence \[\bigoplus^{n}_{i=1}\Pic (\mathbb{P}^{1}\times N_{0})\overset{\alpha}\longrightarrow\bigoplus^{n}_{i=1}\Pic(N_{0})\longrightarrow\br(\mathcal{N}_{0})\longrightarrow\bigoplus^{n}_{i=1}\br(\mathbb{P}^{1}\times N_{0})\overset{\beta}\longrightarrow\bigoplus^{n}_{i=1}\br(N_{0}).\label{7.7}\tag{7.5}\] We note that $\Pic(\mathbb{P}^{1}\times N_{0})\cong\Pic(\mathbb{P}^{1})\times\Pic(N_{0})\cong\mathbb{Z}[N_{0}]\times \Pic(N_{0})$. Since the restriction of $\mathcal{O}_{\mathbb{P}^{1}\times N_{0}}([N_{0}])$ to $N_{0}$ is the trivial line bundle, we deduce that the cokernel of $\alpha$ coincides with the one of the map $\bigoplus^{n}_{i=1}\Pic(N_{0})\to\bigoplus^{n}_{i=1}\Pic(N_{0}),\ (D_{1},\ldots,D_{n})\mapsto (D_{1}-D_{2},\ldots,D_{n}-D_{1}).$ It is readily checked that the summation map $\bigoplus^{n}_{i=1}\Pic(N_{0})\to\Pic(N_{0}),(D_{i})_{i}\mapsto \Sigma_{i}D_{i}$ yields an isomorphism $\coker(\alpha)\cong \Pic(N_{0})$. On the other hand, recall that the Brauer group of smooth projective varieties is a stably birational invariant (see \cite[Proposition 6.2.9]{cts}). Hence, we have a natural isomorphism $\br(N_{0})\cong\br(N_{0}\times\mathbb{P}^{1})$ (see \cite[Corollary 6.2.11]{cts}). Under the above identification the map $\beta$ takes the form $(\alpha_{1},\ldots,\alpha_{n})\mapsto(\alpha_{1}-\alpha_{2},\ldots,\alpha_{n}-\alpha_{1})$ and so its kernel is isomorphic to $\br(N_{0})$. The exact sequence \eqref{7.3} thus follows from \eqref{7.7}.\end{proof}\begin{lemma}\label{lem:7.3} Let $R$ be a strictly henselian discrete valuation ring. Let $q_{\tau}\colon\mathcal{Y}\to\mathcal{X}\coloneqq(\mathcal{E}\times_{R}\mathcal{C})/(\mathbb{Z}/d)$ be the quotient map in \eqref{7.6}. For any prime $\ell$ invertible in $R$, restriction yields isomorphisms \[\br(\mathcal{Y})\{\ell\}\overset{\cong}\to\br(\mathcal{Y}_{0})\{\ell\}\tag{7.6}\label{7.4}\]\[\br(\mathcal{X})\{\ell\}\overset{\cong}\to\br(\mathcal{X}_{0})\{\ell\}.\tag{7.7}\label{7.5}\]\end{lemma}\begin{proof} Note that the sufficient conditions of Lemmas \ref{Lem:4.1} and \ref{Lem:4.3} are not true here and so we cannot get \eqref{7.4}, \eqref{7.5} for free. Instead, we proceed as follows. The Mayer-Vietoris sequence \eqref{2.4} gives rise to the following short exact sequences:\[1\to k^{*}\to\Pic (\mathcal{Y}_{0})\to (\bigoplus^{\nu}_{i=1}\mathbb{Z}[C_{0}])\times \Pic(C_{0})\to 1\]\[1\to k^{*}\to\Pic (\mathcal{X}_{0})\to (\bigoplus^{\kappa}_{i=1}\mathbb{Z}[C_{0}])\times\Pic(C_{0})\to 1.\] Here, we think of $\Pic(C_{0})$ as a subgroup of $\bigoplus\Pic(\mathbb{P}^{1}\times C_{0})$ via the diagonal. Since $R$ is strictly henselian, the residue field $k$ is seperably closed and tensoring with $\mathbb{Z}/\ell^{r}$ gives isomorphisms \[\Pic (\mathcal{Y}_{0})\otimes\mathbb{Z}/\ell^{r}\cong (\bigoplus^{\nu}_{i=1}(\mathbb{Z}/\ell^{r})[C_{0}])\times (\Pic(C_{0})\otimes\mathbb{Z}/\ell^{r})\] \[\Pic (\mathcal{X}_{0})\otimes\mathbb{Z}/\ell^{r}\cong (\bigoplus^{\kappa}_{i=1}(\mathbb{Z}/\ell^{r})[C_{0}])\times (\Pic(C_{0})\otimes\mathbb{Z}/\ell^{r}).\]\par By \eqref{4.13} it suffices to prove \[\frac{\Pic(\mathcal{Y}_{0})}{\Pic(\mathcal{Y})}\otimes\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}=0\ \text{and}\ \frac{\Pic(\mathcal{X}_{0})}{\Pic(\mathcal{X})}\otimes\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}=0.\]\par Using the group scheme structure of $\mathcal{E}^{\sm}$, we find $\nu$ sections $s_{i}\colon \Spec R\to\mathcal{E}^{\sm}$, one for each component of the special fibre. We set $D_{i}\coloneqq\im(s_{i}\times\id_{\mathcal{C}})\subset\mathcal{Y}$ and $F_{i}\coloneqq q(D_{i})\subset\mathcal{X}$ for all $i$. Clearly, the restrictions $D_{i}|_{\mathcal{Y}_{0}}\in\Pic(\mathcal{Y}_{0})$ (resp. $F_{i}|_{\mathcal{X}_{0}}\in\Pic(\mathcal{X}_{0})$) are generators in $\bigoplus^{\nu}_{i=1}(\mathbb{Z}/\ell^{r})[C_{0}]$ (resp. $\bigoplus^{\kappa}_{i=1}(\mathbb{Z}/\ell^{r})[C_{0}]$).\par Recall that $\Pic(C_{0})\otimes\mathbb{Z}/\ell^{r}\cong\mathbb{Z}/\ell^{r}$ is generated by any closed point. Pick a fixed point $x\in\mathcal{C}(R)$ of $\psi$ and consider the prime divisors $D\coloneqq\im(\id_{\mathcal{E}}\times x)\subset \mathcal{Y}$ and the quotient $F\coloneqq D/(\mathbb{Z}/d)\subset \mathcal{X}$. Then the restrictions $D|_{\mathcal{Y}_{0}}\in\Pic(\mathcal{Y}_{0})$ and $F|_{\mathcal{X}_{0}}\in\Pic(\mathcal{X}_{0})$ yield generators of $\Pic(C_{0})\otimes\mathbb{Z}/\ell^{r}$, finishing the proof.\end{proof}\begin{proposition}\label{prp:7.4}Let $R$ be a discrete valuation ring with fraction field $K$ and algebraically closed residue field $k$. Let $q_{\tau}\colon\mathcal{Y}\to\mathcal{X}\coloneqq(\mathcal{E}\times_{R}\mathcal{C})/(\mathbb{Z}/d)$ be the quotient map in \eqref{7.6}. Let $\ell$ be a prime invertible in $R$. For any class $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\{\ell\}\setminus\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}\{\ell\},$ the pull-back to $\mathcal{Y}_{\bar{\eta}}$ is non-zero: $0\neq q^{*}_{\bar{\eta}}\alpha\in\br(\mathcal{Y}_{\bar{\eta}})$.\end{proposition}\begin{proof} We may assume that $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\{\ell\}\setminus\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}\{\ell\},$ for some prime factor $\ell$ of $d$. A sufficient condition for the pull-back $q^{*}_{\bar{\eta}}\alpha$ to be non-zero is to show that up to every finite base change of discrete valuation rings followed by a resolution of the total space (see \cite{Har}), $\alpha$ cannot be lifted to a class in $\br(\mathcal{X})\{\ell\}$ (see Theorem \ref{thm:3.1}).\par As the Brauer group of a curve over an algebraically closed field is zero (see \cite[Theorem 5.6.1]{cts}), Lemma \ref{lem:7.2} implies $\br(\mathcal{X}_{0})\{\ell\}\cong\Pic(C_{0})\{\ell\}$. By passing to the henselization of the base, Lemma \ref{lem:7.3} yields $\br(\mathcal{X})\{\ell\}\cong\Pic(C_{0})\{\ell\}$. The group $\Pic(C_{0})\{\ell\}\cong(\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{2g}$ is divisible, whereas $\br(\mathcal{X}_{\bar{\eta}})\{\ell\}/\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}\{\ell\}\cong H^{3}(\mathcal{X}_{\bar{\eta}},\mathbb{Z}_{\ell}(1))_{\tor}$ is finite (see \cite[Proposition 5.2.9]{cts}). These observations certainly imply that $\br(\mathcal{X})\{\ell\}\to\br(\mathcal{X}_{\bar{\eta}})\{\ell\}/\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}\{\ell\}$ is always the zero map, as claimed.\end{proof} Finally, with the help of the above preparation, we are able to prove Theorem \ref{thm:7.1}.\begin{proof}[Proof of Theorem \ref{thm:7.1}] Let $C$ be a smooth complex projective curve and let $\psi\in\Aut(C)$ be an automorphism of order $d$, with $\Fix(\psi)\neq\emptyset$. We claim that Proposition \ref{prp:7.4} yields an example of an elliptic curve $E$ over $\mathbb{C}$ with $\Hom(E,\Jac(C))=\{0\}$ and a point $0\neq\tau\in E[d]$, for which the conclusion of Theorem \ref{thm:7.1} holds true.\par To see this, we may choose a countable algebraically closed field $k\subset\mathbb{C}$, such that $C=C_{k}\times_{k}\mathbb{C}$. Let $\mathcal{E}$ be a cycle degeneration over the local ring of a smooth pointed $k$-curve $(B,0)$ with special fibre of type $I_{\kappa d}$, $\kappa\geq 2$. We may assume that \eqref{7.2} holds. Set $\mathcal{C}\coloneqq C_{k}\times_{k}\mathcal{O}_{B,0}$ and consider the quotient map $q_{\tau}\colon\mathcal{Y}\to\mathcal{X}\coloneqq(\mathcal{C}\times_{\mathcal{O}_{B,0}}\mathcal{E})/(\mathbb{Z}/d)$ (from \eqref{7.6}) for some point $0\neq\tau\in\mathcal{E}^{\sm}[d]$ that restricts via the isomorphism $E^{\sm}_{0}[\nu]\cong \mu_{\nu}\times\mathbb{Z}/\nu$ to a tuple of the form $(\zeta^{\kappa},i\kappa)$, where the integer $1\leq i\leq \nu\coloneqq\kappa d$ is prime to $d$ and $\zeta$ is any $d$-th root of unity.\par Proposition \ref{prp:7.4} implies that $q^{*}_{\bar{\eta}}\alpha\neq0$ for all $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\setminus\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}$. We pick an embedding $k(B)\subset\mathbb{C}$ that respects the given one $k\subset\mathbb{C}$ and perform the base change. This yields an example $q_{\tau}\colon E\times C\to(E\times C)/(\mathbb{Z}/d)$ defined over $\mathbb{C}$ that satisfies the conclusion of Theorem \ref{thm:7.1}. Note that $\Hom(E,\Jac(C))=\{0\}$, because $\mathcal{E}_{\eta}$ has multiplicative reduction at $0\in B$, whereas $\Jac(C_{\eta})$ has abelian reduction everywhere in $B$.\par Next, we proceed as in the proof of Theorem \ref{thm:1.1}. We consider the Legendre family of elliptic curves $p\colon\mathcal{F}\subset\mathbb{P}_{\mathbb{C}}^{2}\times U\to U$, $U\coloneqq\mathbb{A}^{1}_{\mathbb{C}}-\{0,1\}$, whose fibres $\mathcal{F}_{\lambda}$ are defined by the affine equation \[y^{2}=x(x-1)(x-\lambda)\] and recall that every elliptic curve is isomorphic to some fibre of this family. We regard $\mathcal{F}\to U$ as an abelian scheme, with the identity section given by the point $(0:1:0)$.\par Pick $\lambda_{0}\in U$, such that $E=\mathcal{F}_{\lambda_{0}}$. Up to some finite base change, we may assume that $\tau\in E[d]$ lifts to a section of the family $\mathcal{F}\to U$. We let the group $\mathbb{Z}/d$ act on $\mathcal{F}$ via the translation $\tau$ and on $C$ via the automorphism $\psi$ and consider the quotient map \[q_{\mathcal{F}}\colon \mathcal{F}\times C\to(\mathcal{F}\times C)/(\mathbb{Z}/d).\]For $\lambda\in U$, we set $q_{\lambda}\coloneqq q_{\mathcal{F}_{\lambda}}$, $Y_{\lambda}\coloneqq \mathcal{F}_{\lambda}\times C$ and $X_{\lambda}\coloneqq (\mathcal{F}_{\lambda}\times C)/(\mathbb{Z}/d)$.\par Choose $\lambda_{1}\in U$, such that $\Hom(\mathcal{F}_{\lambda_{1}},\Jac(C))=\{0\}$. Then the maps $sp_{\bar{\eta},\lambda_{i}}\colon\br(Y_{\bar{\eta}})\to\br(Y_{\lambda_{i}})$ are isomorphisms for $i=0,1$ (see Lemma \ref{lem:4.4}). By the smooth proper base change theorem, the quotient of the Brauer group of a smooth projective variety over an algebraically closed field of characteristic $0$ by its maximal divisible subgroup is invariant in smooth proper families (see \cite[Proposition 5.2.9]{cts}). Thus, the surjectivity of $sp_{\bar{\eta},\lambda_{i}}\colon\br(X_{\bar{\eta}})\rightarrow\hspace*{-.14in}\rightarrow\br(X_{\lambda_{i}})$ yields an isomorphism \[\br(X_{\bar{\eta}})/\br(X_{\bar{\eta}})_{\text{div}}\cong\br(X_{\lambda_{i}})/\br(X_{\lambda_{i}})_{\text{div}}.\]Since specialization is compatible with pullbacks (see Lemma \ref{lem:4.4}) the claim follows by comparing the three pull-backs $q^{*}_{\bar{\eta}}$, $q^{*}_{\lambda_{0}}$ and $q^{*}_{\lambda_{1}}$ via the specialization maps. The proof of Theorem \ref{thm:7.1} is complete.\end{proof}\begin{remark}\label{rem:5.9} Assume that the pair $(C,\psi)$ satisfies $C/\psi\cong\mathbb{P}^{1}$. Then $\br((E\times C)/(\mathbb{Z}/d))_{\text{div}}=0$ and so Theorem \ref{thm:7.1} says that the pull-back map $q_{\tau}^{*}\colon\br((E\times C)/(\mathbb{Z}/d))\to\br(E\times C)$ is injective for some $0\neq\tau\in E[d]$, if $\Hom(E,\Jac(C))=\{0\}$.\par The conclusion of Theorem \ref{thm:7.1} is known for Bi-elliptic surfaces and their canonical covers, without any restriction on the choice of the torsion point $\tau\in E[d]$ (see \cite[Theorem B]{bie}).\end{remark}\begin{example} It is readily checked that the datum $(C,\psi)$ can be replaced by any pair $(V,\psi)$, where $V$ is a smooth complex projective variety and $\psi$ is an automorphism of $V$ of order $d$ satisfying the following two properties:\begin{enumerate}
\item\label{1'} For all prime factors $\ell$ of $d,$ the group $\NS(V)\otimes\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}$ is generated by prime divisors that are invariant under $\psi$.
\item\label{2'} $\bigoplus_{\ell|d}\Pic(V)\{\ell\}=\bigoplus_{\ell|d}\Pic^{0}(V)\{\ell\}$.
\end{enumerate} Item \eqref{1'} is needed to ensure $\bigoplus_{\ell|d}\br(\mathcal{X})\{\ell\}\cong\bigoplus_{\ell|d}\br(\mathcal{X}_{0})\{\ell\}$, where $\mathcal{X}=(\mathcal{E}\times V)/(\mathbb{Z}/d)$ (cf. Proof of Lemma \ref{lem:7.3}). In the proof of Proposition \ref{prp:7.4} item \eqref{2'} was the key to show that classes $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\setminus\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}$ do not lift to the total space $\mathcal{X}$. However, the group $\br(\mathcal{X}_{0})_{\tor}$ may not be divisible (see \eqref{7.3}) and so the same argument might not work here. We prove Proposition \ref{prp:7.4} for the quotient map $q_{\tau}\colon\mathcal{Y}\coloneqq \mathcal{E}\times V\to\mathcal{X}\coloneqq(\mathcal{E}\times V)/(\mathbb{Z}/d)$, where $0\neq \tau\in\mathcal{E}[d]$ is chosen so that $E_{0}\to E_{0}/\tau$ is generically trivial: For a contradiction, assume that there exists a non-zero $d$-torsion class $\alpha\in\br(\mathcal{X}_{\bar{\eta}})\setminus\br(\mathcal{X}_{\bar{\eta}})_{\text{div}}$, such that $q^{*}_{\bar{\eta}}(\alpha)=0\in \br(\mathcal{Y}_{\bar{\eta}})$. Then after a finite base change of discrete valuation rings folllowed by a resolution of the total space, we may assume that $\alpha$ lifts to a class $\tilde{\alpha}\in\br(\mathcal{X})[d]$ and $q^{*}\tilde{\alpha}=0\in\br(\mathcal{Y})[d]$ (see Theorem \ref{thm:3.1}). But by item \eqref{2'} via the last homomorphism in \eqref{7.3} the restriction $0\neq\tilde{\alpha}|_{\mathcal{X}_{0}}$ must map to a non-zero element in $\br(V)$. Finally, the commutativity of the following diagram\[\begin{tikzcd}
\br(\mathcal{X}) \arrow[r,] \arrow[d, "q^{*}"] & \br(\mathcal{X}_{0}) \arrow[r,] \arrow[d, "q_{0}^{*}"] & \br(V) \arrow[d, "\{(\psi^{i})^{*}\}^{d}_{i=1}"] \\
\br(\mathcal{Y}) \arrow[r] & \br(\mathcal{Y}_{0}) \arrow[r] & \bigoplus^{d}_{i=1}\br(V),
\end{tikzcd}\]yields $q^{*}\tilde{\alpha}\neq 0$, which contradicts $q^{*}\tilde{\alpha}=0$.\end{example}
\end{document} |
\begin{document}
\thispagestyle{empty}
\begin{center}
{\Huge From Schr\"odinger spectra
to orthogonal polynomials,
via a functional equation}
{\Large Arieh Iserles\footnote{Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, England.}}
\parbox[t]{130mm}{{\bf Abstract} The main difference between certain
spectral problems for linear Schr\"odinger operators, e.g. the almost
Mathieu equation, and three-term recurrence relations for orthogonal
polynomials is that in the former the index ranges across $\ZZ$ and in
the latter only across $\Zp$. We present a technique that, by a
mixture of Dirichlet and Taylor expansions, translates the almost
Mathieu equation and its generalizations to three term recurrence
relations. This opens up the possibility of exploiting the full power
of the theory of orthogonal polynomials in the analysis of
Schr\"odinger spectra.
Aforementioned three-term recurrence relations share the property
that their coefficients are almost periodic. We generalize a method of
proof, due originally to Jeff Geronimo and Walter van Assche, to
investigate essential support of the Borel measure of associated
orthogonal polynomials, thereby deriving information on the underlying
absolutely continuous spectra of Schr\"odinger operators.}
\end{center}
\eject
\section{The almost Mathieu equation and orthogonal polynomials}
\label{sec1}
The point of departure of our analysis is the {\em almost Mathieu
equation\/} (also known as the {\em Harper equation\/}). We seek
$\lambda\in\RR$ and $\{a_n\}_{n\in\ZZ}\in\ell_1[\ZZ]$ that satisfy
\begin{equation}\label{1.1}
a_{n-1}-2\kappa\cos(\alpha n+\beta)a_n+a_{n+1}=\lambda a_n,\qquad n\in\ZZ.
\end{equation}
$\alpha,\beta$ and $\kappa\neq0$ being given real constants.
The almost Mathieu equation features in a number of applications
\cite{Hofstadter} and has been already extensively studied
\cite{Bellissard,Cycon,Last,Simon}. The purpose of our analysis is not to
reveal new features of the spectrum of \R{1.1} {\em per se\/}, since
the latter is quite comprehensively known. Instead, we intend to
demonstrate that the almost Mathieu equation exhibits an intriguing
connection with orthogonal polynomials, a connection that lends itself
to far reaching generalizations.
Let $\omega={\rm e}^{\I\alpha}$, $b_1=\kappa{\rm e}^{\I\beta}$ and
$b_2=\kappa{\rm e}^{-\I\beta}=\bar{b}_1$. We rewrite \R{1.1} as
\begin{equation}\label{1.2}
(b_1\omega^{2n}+b_2)a_n=\omega^n(a_{n-1}-\lambda a_n+a_{n+1}),\qquad n\in\ZZ
\end{equation}
and consider the {\em Dirichlet expansion\/}
$$y(z)=\sum_{n=-\infty}^\infty a_n \exp\left\{b_1^{\frac12}\omega^n
z\right\},\qquad z\in\CC.$$
The choice of a specific branch of the square root of $b_1$ is
arbitrary. Note that, since $|\omega|=1$, it is easy to demonstrate that
$\{a_n\}_{n\in\ZZ}\in\ell_1[\ZZ]$ implies convergence of the series
for all $z\in\CC$ \cite{Hardy,Iserles1}.
We multiply \R{1.2} by $\exp \left\{ b_1^{\frac12}\omega^n z\right\}$
and sum for $n\in\ZZ$. Since
$$y'(z)=b_1^{\frac12}\sum_{n=-\infty}^\infty a_n\omega^n
{\rm e}^{b_1^{\frac12}\omega^n z},\qquad y''(z)=b_1\sum_{n=-\infty}^\infty a_n
\omega^{2n} {\rm e}^{b_1^{\frac12}\omega^n z},$$
we readily deduce that $y$ obeys the functional differential equation
\begin{equation}\label{1.3}
y''(z)+b_2y(z)=b_1^{-\frac12} \left\{ \omega^{-1}
y'(\omega^{-1}z) -\lambda y'(z)+\omega y'(\omega z)\right\}.
\end{equation}
The solution of \R{1.3} is determined uniquely by the values of $y(0)$
and $y'(0)$.
Dirichlet expansions have been employed by Gregor\u{\i} Defel
and Stanislav Molchanov \cite{Derfel1} to investigate the simplified
spectral problem
$$a_{n-1}-2\kappa q^n a_n+a_{n-1}=\lambda a_n,\qquad n\in\ZZ,$$
where $\kappa\in\RR\setminus\{0\}$ and $q>0$, and they have
demonstrated that it obeys another functional differential equation.
Both the Derfel--Molchanov equation and \R{1.3} are a generalization
of the {\em pantograph equation\/}, that has been extensively analysed
in \cite{Kato} and \cite{Iserles1}. However, it is important to
emphasize that \R{1.3} has an important feature that sets it apart from
other functional equations of the pantograph type, namely that, unless
$\omega\in\RR$, its evolution makes sense only for complex $z$ and
proceeds along circles of constant $|z|$, emanating from the origin.
Inasmuch as the equation \R{1.3} can be analysed directly, our next
step entails expanding it in Taylor series. Thus, letting
$$y(z)=\sum_{m=0}^\infty \frac{y_m}{m!} z^m,$$
substitution in \R{1.3} readily yields
\begin{equation}\label{1.4}
y_{m+1}=\sqrt{b_1}^{-\frac12} (2\cos\alpha m-\lambda)y_m-b_2
y_{m-1},\qquad m=1,2,\ldots.
\end{equation}
It is beneficial to treat $y_m$ as a function of the spectral
parameter $\lambda$ and to define
$$\tilde{y}_m(t):=b_2^{-m/2} y_m(-(b_1b_2)^{1/2}t),\qquad
m=0,1,\ldots.$$
Brief manipulation affirms that \R{1.4} is equivalent to
\begin{equation}\label{1.5}
\tilde{y}_{m+1}(t)=\left(t+\frac{2}{\kappa}\cos\alpha m\right)
\tilde{y}_m(t)-\tilde{y}_{m-1}(t),\qquad m=0,1,\ldots.
\end{equation}
To specify the solution of \R{1.5} in a unique fashion we need to
choose $\tilde{y}_0$ and $\tilde{y}_1$, which, of course, corresponds
to equipping \R{1.3} with requisite initial conditions. We note in
passing that $t=-\kappa\lambda$, hence real initial values in
\R{1.3} correspond to $\tilde{y}_0,\tilde{y}_1\in\RR$.
Each solution of \R{1.5} is a linear combination of two linearly
independent solutions. Setting $\sigma=2/\kappa$, we let
\begin{eqnarray}
r_{-1}(t)&\equiv&0,\nonumber\\
r_0(t)&\equiv&1,\nonumber\\
r_{m+1}(t)&=&(t+\sigma\cos\alpha m)r_m(t)-r_{m-1}(t),\qquad m=0,1,\ldots,
\label{1.6}
\end{eqnarray}
and
\begin{eqnarray}
s_{-1}(t)&\equiv&0\nonumber\\
s_0(t)&\equiv&1\nonumber\\
s_{m+1}(t)&=&(t+\sigma\cos\alpha(m+1)) s_m(t)-s_{m-1}(t),\qquad
m=0,1,\ldots. \label{1.7}
\end{eqnarray}
It is trivial to verify that $\{r_m\}_{m\in\Zp}$ and
$\{s_{m+1}\}_{m\in\Zp}$ are linearly independent, hence they span all
solutions of \R{1.5}. Moreover, we observe that each $r_m(t)$ and
$s_m(t)$ is an $m$th degree monic polynomial in $t$.
This is a crucial observation, by virtue of the Favard theorem
\cite{Chihara}: Given any three-term recurrence relation of the form
\begin{eqnarray*}
p_{-1}(t)&\equiv&0,\\
p_0(t)&\equiv&1,\\
p_{m+1}(t)&=&(t+c_m)p_m(t)-d_m p_{m-1}(t),\qquad m=0,1,\ldots,
\end{eqnarray*}
the monic polynomial sequence $\{p_m\}_{m\in\Zp}$ is orthogonal with
respect to some Borel measure $\D\varphi$, i.e.\
$$\int_{\RR} p_m(t)p_n(t)\D\varphi(t)=0,\qquad m\neq n.$$
The essential support $\Xi$ of $\D\varphi$ is of central importance to the
matter in hand, since it is an easy consequence of, for example, the
$n$th root asymptotics of orthogonal polynomials \cite{Stahl} that, as
long as $\Xi$ is bounded, the sum $\sum_{m=0}^\infty p_m(t)z^m/m!$
converges for $z\in\CC$, whereas convergence fails for $z\neq0$ for
$t\not\in\Xi$. Thus, travelling all the way back from orthogonal
polynomials to functional equations, and hence to the almost Mathieu
equation, we deduce that, subject to the linear transformation
$\lambda=-t/\kappa$, essential supports of the Borel measures
corresponding to \R{1.6} and \R{1.7} result in the essential spectrum
of \R{1.1}.
We note that, in principle, the Favard theorem falls short of
producing a unique measure and it is entirely possible that there
exist many Borel measures that produce an identical set of monic
orthogonal polynomials. This, however, is ruled out by the determinacy
of the underlying Hamburger moment problem and the latter can be
affirmed for both \R{1.6} and \R{1.7} by the Carleman criterion
\cite{Chihara}.
The remainder of this paper is devoted toward the determination of
$\Xi$. In Section \ref{sec2} we demonstrate, by extending a technique
due to Jeff Geronimo and Walter van Assche, that the essential support
-- and, indeed, the underlying Borel measure -- can be specified
explicitly when $\alpha/\pi$ is rational. We prove in essence that
$\D\varphi$ is a linear combination of Chebyshev measures of the
second kind, supported on a set of disjoint intervals.
Various results on irrational $\alpha/\pi$ are reported in Section
\ref{sec3}. Inasmuch as the general form of $\D\varphi$ is currently a
matter for conjecture, we derive a number of results that, beside
being of interest on their own merit, are fully consistent with known
results about the almost Mathieu operator, in particular with the
theorem that the absolutely continuous spectrum of \R{1.1} is (for
irrational $\alpha/\pi$) a Cantor set.
The equation \R{1.1} has been extensively studied in the past and,
inasmuch as there are few outstanding conjectures, our knowledge of
the spectrum of the almost Mathieu operator is quite comprehensive.
Although the approach of this paper introduces a new perspective,
there is no claim that Sections \ref{sec2}--\ref{sec3} add to the
current state of knowledge of Schr\"odinger spectra. This state of
affairs is remedied in Section \ref{sec4}, where the framework of our
discussion undergoes a far reaching generalization. Firstly, we
demonstrate that general periodic potentials with a finite number of
Fourier harmonics lend themselves to similar analysis, except that,
instead of orthogonal polynomials, the outcome is a generalized
eigenvalue problem for a certain matrix pencil. Secondly, we prove
that a multivariate extension of the almost Mathieu equation can be
`transformed' by our techniques to a problem in (univariate)
orthogonal polynomials.
A possible application of our analysis, and in particular of Section
\ref{sec4}, is numerical computation of the essential spectrum of
\R{1.1} and of its generalizations. We do not pursue this further in
the present paper.
The observation that \R{1.1} is `almost' a three-term recurrence
relation -- only the index range is wrong -- hence that the almost
Mathieu equation might be connected with orthogonal polynomials, is
not new. Most prominently, it had been made in \cite{Ismail}, where it
motivated a very interesting generalization of Chebyshev polynomials.
The most important innovation in the present paper is in a technique
that reduces the index range from $\ZZ$ to $\Zp$ by passing from
Dirichlet to Taylor expansions and which can be extended to cater for
a substantially more general problem.
\section{Orthogonal polynomials with periodic recurrence coefficients}
\label{sec2}
The focus of our attention in this section is the three-term
recurrence
\begin{eqnarray}
p_{-1}(t)&\equiv&0,\nonumber\\
p_0(t)&\equiv&1,\nonumber\\
p_{m+1}(t)&=&(t-\alpha_m)p_m(t)-p_{m-1}(t),\qquad m=0,1,\ldots, \label{2.1}
\end{eqnarray}
where the sequence $\{\alpha_m\}_{m\in\Zp}$ is $K$-periodic,
\begin{equation}\label{2.2}
\alpha_{m+K}=\alpha_m,\qquad m=0,1,\ldots.
\end{equation}
Note that both \R{1.6} and \R{1.7} assume this form when
$\alpha/\pi$ is rational. Our objective is to determine the Borel
measure that renders $\{p_m\}_{m\in\Zp}$ into an orthogonal polynomial
system (OPS).
In \cite{vanAssche} the authors consider the following problem. Let
$\{Q_m\}_{m\in\Zp}$ be an OPS whose measure has an essential support
$\Xi_0\subseteq[-1,1]$ and let $T$ be a given $N$th degree polynomial.
Setting $\Xi=T^{-1}(\Xi_0)$ (the latter set is, generically, a union
of $\leq N$ disjoint intervals), they derive a new OPS, whose Borel measure
is supported in $\Xi$, explicitly in terms of $\{Q_m\}_{m\in\Zp}$.
This construction is intimately related to the discussion of this
section, except that we need, in a manner of speech, to travel in the
opposite direction. As it turns out, the Borel measure associated with
\R{2.1} inhabits a sets of disjoint intervals and we identify it by
choosing an appropriate polynomial transformation $T$.
Let
$$q_n(t):=p_{(n+1)K-1}(t),\qquad n=0,1,\ldots.$$
Note that $q_{-1}\equiv0$ and, moreover, \R{2.1} and \R{2.2} imply
\begin{equation}\label{2.3}
p_{nK}(t)=(t-\alpha_0)q_{n-1}-p_{nK-2}(t),\qquad n=1,2,\ldots.
\end{equation}
We seek polynomials $\alpha_\ell,\beta_\ell$, $\ell=0,1,\ldots,K-1$,
such that
$$p_{nK+\ell}(t)=a_\ell(t)q_{n-1}(t)-b_\ell(t)p_{nK-2}(t),\qquad
\ell=0,1,\ldots,K-1,\quad n=1,2,\ldots.$$
Because of \R{2.3} and the definition of $q_n$, we have
\begin{equation}\label{2.4}
\begin{array}{rclcrcl}
a_{-1}(t) & \equiv & 1, & \qquad\qquad & b_{-1}(t) & \equiv & 0,\\
a_0(t) & = & t-\alpha_0, & \qquad\qquad & b_0(t) & \equiv & 1.
\end{array}
\end{equation}
We next substitute in the recurrence relation \R{2.1} and, by virtue
of \R{2.2}, obtain
\begin{eqnarray*}
p_{nK+\ell+1}(t)&=&(t-\alpha_{\ell+1})p_{nK+\ell}(t)-p_{nK+\ell-1}(t)\\
&=&(t-\alpha_{\ell+1})\{ a_\ell(t)q_{n-1}(t)-b_\ell(t)p_{nK-2}(t)\}\\
&&\quad\mbox{}-\{a_{\ell-1}(t)q_{n-1}(t)-b_{\ell-1}(t)p_{nK-2}(t)\}.
\end{eqnarray*}
Thus, comparing coefficients, we derive the recurrences
\begin{eqnarray}
a_{\ell+1}(t)&=&(t-\alpha_{\ell+1})a_\ell(t)-a_{\ell-1}(t),
\label{2.5}\\
b_{\ell+1}(t)&=&(t-\alpha_{\ell+1})b_\ell(t)-b_{\ell-1}(t),\qquad
\ell=0,1,\ldots,K-2, \label{2.6}
\end{eqnarray}
which, in tandem with \R{2.4}, determine $\{a_\ell,b_\ell\}_{\ell=0}
^{K-1}$.
Let $\ell=K-1$, then
$$p_{nK-2}(t)=\frac{a_{K-1}(t)q_{n-1}(t)-q_n(t)}{b_{K-1}(t)}$$
and, shifting the index,
$$p_{(n+1)K-2}(t)=\frac{a_{K-1}(t)q_n(t)-q_{n+1}(t)}{b_{K-1}(t)}$$
Substituting both expressions into
$$p_{(n+1)K-2}(t)=a_{K-2}(t)q_{n-1}(t)-b_{K-2}(t)p_{nK-2}(t)$$
yields the recurrence relation
\begin{equation}\label{2.7}
q_{n+1}(t)=(a_{K-1}(t)-b_{K-2}(t))q_n(t)-\Delta_{n-2}(t)q_{n-1}(t),
\end{equation}
where
$$\Delta_\ell(t)=\det\left[\begin{array}{ll}
a_\ell(t) & a_{\ell+1}(t)\\ b_\ell(t) & b_{\ell+1}(t)
\end{array}\right], \qquad \ell=0,1,\ldots,K-1.$$
We multiply \R{2.6} by $a_\ell(t)$, \R{2.5} by $b_\ell(t)$ and
subtract from each other. This readily affirms by induction that
$$\Delta_\ell(t)=\Delta_{\ell-1}(t)=\cdots=1$$
and \R{2.7} simplifies into
\begin{equation}\label{2.8}
q_{n+1}(t)=(a_{K-1}(t)-b_{K-2}(t))q_n(t)-q_{n-1}(t),\qquad
n=0,1,\ldots.
\end{equation}
Note that \R{2.8} is consistent with $n=0$, since $q_{-1}\equiv0$. To
further simplify the recurrence, we observe that
$q_0(t)=a_{K-1}(t)-b_{K-2}(t)$, hence, letting
$$\tilde{q}_n(x):=\frac{q_n(t)}{q_0(t)},\qquad n=-1,0,\ldots,$$
where $x=q_0(t)$, we obtain the three-term recurrence
\begin{eqnarray*}
\tilde{q}_{-1}(x)&\equiv&0,\\
\tilde{q}_0(x)&\equiv&1,\\
\tilde{q}_{n+1}(x)&=&x\tilde{q}_n(x)-\tilde{q}_{n-1}(x),\qquad n=0,1,\ldots.
\end{eqnarray*}
Thus, each $\tilde{q}_n$ is an $n$th degree monic polynomial and, by
virtue of the Favard theorem, $\{\tilde{q}_n\}_{n\in\Zp}$ is an OPS.
It can be easily identified as a shifted and scaled {\em Chebyshev
polynomial of the second kind\/},
$$\tilde{q}_n(x)=2^n U_n\left(\Frac12 x\right),\qquad
n=0,1,\ldots.$$
We thus deduce that
\begin{equation}\label{2.9}
q_n(t)=2^nq_0(t)U_n\left(\Frac12 q_0(t)\right),\qquad n=0,1,\ldots.
\end{equation}
Before we identify the underlying Borel measure, let us `fill in' the
remaining values of $p_m$. By definition, $p_{nK-1}=q_{n-1}$,
$p_{n(K+1)-1}=q_n$, hence the recurrence \R{2.1} gives
\begin{eqnarray*}
(x-\alpha_1)p_{nK}(t)-p_{nK+1}(t)&=&q_{n-1}(t),\\
-p_{nK+\ell-1}(t)+(t-\alpha_{\ell+1})p_{nK+\ell}(t)-p_{nK+\ell+1}(t)&=&0,
\qquad \ell=1,2,\ldots,K-3,\\
-p_{(n+1)K-3}(t)+(t-\alpha_{K-1})p_{(n+1)K-2}(t)&=&q_n(t).
\end{eqnarray*}
This is a linear system of equations, which we write as
\begin{equation}\label{2.10}
A_{K-1}{\bf p}_n={\bf q}_n,
\end{equation}
where
$$A_m=\left[\begin{array}{cccccc}
t-\alpha_1 & -1 \\
-1 & t-\alpha_2 & -1\\
& -1 & t-\alpha_3 & -1\\
& & \ddots & \ddots & \ddots\\
& & & -1 & t-\alpha_{m-1} & -1\\
& & & & -1 & t-\alpha_m
\end{array}\right],\qquad m=1,2,\ldots,K-1.,$$
$${\bf p}_n=\left[\begin{array}{c}
p_{nK}(t)\\p_{nK+1}(t)\\\vdots\\p_{(n+1)K-3}(t)\\p_{(n+1)K-2}(t)
\end{array}\right] \qquad\mbox{and}\qquad {\bf q}_n =\left[
\begin{array}{c}q_{n-1}(t)\\0\\\vdots\\0\\q_n(t)\end{array}\right].$$
We expand the determinant of $A_m$ in its bottom row and rightmost
column. This results in a three-term recurrence relation and
comparison with \R{2.4} and \R{2.6} affirms that $\det A_m=b_m(t)$.
Hence, solving \R{2.10} with Cramer's rule, we deduce that there exist
$(K-2)$-degree polynomials $\tilde{a}_\ell$ and $\tilde{b}_\ell$,
$\ell=0,1,\ldots,K-2$, such that
\begin{equation}\label{2.11}
p_{nK+\ell}(t)=\frac{\tilde{a}_\ell(t)
q_{n-1}(t)+\tilde{b}_\ell(t)q_n(t)}{b_{K-1}(t)},\qquad
\ell=0,1,\ldots,K-2.
\end{equation}
Bearing in mind the definition of $a_\ell$ and $b_\ell$,
$$p_{nK+\ell}(t)=a_\ell(t)q_{n-1}(t)-b_\ell(t)p_{nK-2}(t),$$
we obtain from \R{2.11} the identity
$$\frac{\tilde{a}_\ell(t)q_{n-1}(t)+\tilde{b}_\ell(t)q_n(t)}
{b_{K-1}(t)}=a_\ell(t)q_{n-1}(t)-\frac{b_\ell(t)(\tilde{a}_{K-2}(t)
q_{n-2}(t)+\tilde{b}_{K-2}(t)q_{n-1}(t))}{b_{K-1}(t)}.$$
We next substitute
$$q_{n-2}(t)=(a_{K-1}(t)-b_{K-2}(t))q_{n-1}(t)-q_n(t)$$
({\em pace\/} \R{2.8}) and rearrange terms, whereby
\begin{eqnarray*}
&&\{\tilde{b}_\ell(t)-b_\ell(t)\tilde{a}_{K-2}(t)\}q_n(t)\\
&=&\{-\tilde{a}_\ell+a_\ell(t)b_{K-1}(t)-(a_{K-1}(t)b_\ell(t)-b_\ell(t)
b_{K-2}(t)) \tilde{a}_{K-2}(t)-b_\ell(t)\tilde{b}_{K-2}(t)\} q_{n-1}(t).
\end{eqnarray*}
However, consecutive orthogonal polynomials $q_{n-1}$ and $q_n$ cannot
share zeros \cite{Chihara}, therefore both sides of the last equality
identically vanish and we derive the explicit expressions
\begin{eqnarray}
\tilde{a}_\ell(t)&=&a_\ell(t)b_{K-1}(t)-(a_{K-1}(t)b_\ell(t)-b_\ell(t)
b_{K-2}(t))
\tilde{a}_{K-2}(t)-b_\ell(t)\tilde{b}_{K-2}(t),\qquad\label{2.12}\\
\tilde{b}_\ell(t)&=&b_\ell(t)\tilde{a}_{K-2}(t). \label{2.13}
\end{eqnarray}
Letting $\ell=K-2$ in \R{2.13} gives
$\tilde{b}_{K-2}(t)=b_{K-2}(t)\tilde{a}_{K-2}(t)$ and we substitute
this into \R{2.12}. The outcome is
\begin{equation}\label{2.14}
\tilde{a}_\ell(t)=a_\ell(t)b_{K-1}(t)-a_{K-1}(t)b_\ell(t)
\tilde{a}_{K-2}(t).
\end{equation}
In particular, $\ell=K-2$ and the definition of $\Delta_m$ result in
\begin{eqnarray*}
(1+a_{K-1}(t)b_{K-2}(t))\tilde{a}_{K-2}(t)&=&a_{K-2}(t)b_{K-1}(t)
=a_{K-1}(t)b_{K-2}(t)+\Delta_{K-2}(t)\\
&=&1+a_{K-1}(t)b_{K-2}(t).
\end{eqnarray*}
Since $a_{K-1}b_{K-2}\not\equiv-1$, we conclude that
$\tilde{a}_{K-2}\equiv1$ and substitution in \R{2.13} and \R{2.14}
yields the explicit formulae
$$\tilde{a}_\ell=\det\left[\begin{array}{cc}
a_\ell(t) & a_{K-1}(t) \\ b_\ell(t) & b_{K-1}(t)
\end{array}\right],\qquad \tilde{b}_\ell(t)=b_\ell(t),\qquad
\ell=0,1,\ldots,K-2.$$
\noindent {\bf Theorem 1} The OPS $\{p_m\}_{m\in\Zp}$ has an explicit
representation in the form
\begin{equation}\label{2.15}
p_{nK+\ell}(t)=\frac{1}{b_{K-1}(t)} \left\{
\det\left[\begin{array}{cc} a_\ell(t) & a_{K-1}(t) \\ b_\ell(t) &
b_{K-1}(t) \end{array}\right] q_{n-1}(t)+b_\ell(t)q_n(t)\right\},
\end{equation}
where $n=0,1,\ldots$, $\ell=0,1,\ldots,K-1$ and the OPS
$\{q_n\}_{n\in\Zp}$ satisfies the three-term recurrence \R{2.8}. \QED
Note that letting $\ell=K-1$ or $\ell=-1$ in \R{2.15}, in tandem with
\R{2.4}, results in $p_{(n+1)K-1}=q_n$ and $p_{nK-1}=q_{n-1}$
respectively, as required.
Let $t\in\Xi$. Then, by the discussion preceding the representation
\R{2.9}, we know that $\frac12 q_0(t)\in[-1,1]$, and there exists
$\theta\in[-\pi,\pi]$ such that $\frac12 q_0(t)=\cos\theta$. Since, by
the definition of Chebyshev polynomials of the second kind,
$$U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin\theta},$$
\R{2.9} implies that
$$q_n(t)=2^n q_0(t)\frac{\sin(n+1)\theta}{\sin\theta}.$$
Substitution into \R{2.15} results in
$$p_{nK+\ell}(t)=\frac{2^nq_0(t)}{b_{K-1}(t)\sin\theta} \left\{
\det\left[\begin{array}{cc} a_\ell(t) & a_{K-1}(t) \\ b_\ell(t) &
b_{K-1}(t) \end{array}\right] \sin n\theta+b_\ell(t)\sin(n+1)\theta
\right\}.$$
We next proceed to determine the essential support $\Xi$ of the Borel
measure $\D\varphi$ which corresponds to the OPS $\{p_n\}_{n\in\Zp}$.
As we have already mentioned, this is very similar to the construction
of Geronimo and van Assche in \cite{vanAssche}, with
$\{q_n\}_{n\in\Zp}$, $\frac12 q_0$ and $[-1,1]$ plying the role of
$\{Q_n\}_{n\in\Zp}$, $T$ and $\Xi_0$ respectively. This similarity
notwithstanding, there are some important differences -- not least
that our argument advances in an opposite direction to that of
Geronimo and van Assche -- and we present here a complete derivation
of $\Xi$. We commence by observing that, by virtue of Theorem~1,
everything depends on the support of $\tilde{q}_n(x(t))=2^n
U_n\left(\frac12 q_0(t)\right)$, $n=0,1,\ldots$. Thus, we seek a Borel
measure $\D\psi$ such that
$$I_{n,m}=\int_{-\infty}^\infty \tilde{q}_n(x(t))\tilde{q}_m(x(t))
\D\psi(t)=0,\qquad n,m=0,1,\ldots, \quad n\neq m.$$
It follows from the defintion of $\tilde{q}_n$ that
$$I_{n,m}=2^{n+k}\int_{-\infty}^\infty U_n\left(\Frac12 q_0(t)\right)
U_m\left(\Frac12 q_0(t)\right) \D\psi(t).$$
Similarly to \cite{vanAssche}, we seek the inverse function to
$x=\frac12 q_0(t)$. Let $\xi_1<\xi_2<\cdots<\xi_s$ be all the minima
and maxima of $q_0$ in $\RR$ (of course, $s\leq K-2$) and define
$\xi_0=-\infty$, $\xi_{s+1}=\infty$. In each interval
$[\xi_j,\xi_{j+1}]$, $j=0,1,\ldots,s$, the function $\frac12 q_0(t)$
is monotone, hence it possesses there a well-defined inverse. We
denote it by $X_j(x)$, hence $\frac12 q_0(X_j(x))=x$. Changing the
integration variable, we have
\begin{eqnarray}
I_{n,m}&=&2^{n+m}\sum_{j=0}^s \int_{\xi_j}^{\xi_{j+1}}
U_n\left(\Frac12 q_0(t)\right) U_m\left(\Frac12 q_0(t)\right)
\D\psi(t) \nonumber\\
&=&2^{n+m} \sum_{j=0}^s \int_{\frac12 q_0(\xi_j)}^{\frac12
q_0(\xi_{j+1})} U_n(x)U_m(x)\D\psi(X_j(x)). \label{2.16}
\end{eqnarray}
We recall that $\{U_n\}_{n\in\Zp}$ is an OPS with respect to the Borel
measure $(1-x^2)^{\frac12}\D x$, supported by $x\in[-1,1]$. Thus, for
every $j=0,1,\ldots,s$ we distinguish among the following cases:
\noindent{\bf Case 1:} $q_0(\xi_j)\leq-2$ and $2\leq
q_0(\xi_{j+1})$.\\
We stipulate that $\D\psi(X_j(x))$ vanishes for all
$$x\in\left[\Frac12 q_0(\xi_j),\Frac12 q_0(\xi_{j+1})\right]
\setminus[-1,1].$$
Since $X_j$ increases monotonically in $[\xi_j,\xi_{j+1}]$, the
contribution of this interval to \R{2.16} is
\begin{equation}\label{2.17}
2^{n+m}\int_{-1}^1 U_n(x)U_m(x)\D\psi(|X_j(x)|).
\end{equation}
\noindent{\bf Case 2:} $q_0(\xi_{j+1})\leq-2$ and $2\leq
q_0(\xi_j)$.\\
Likewise, we require that the support of $\D\psi(X_j)$ is restricted
to $[-1,1]$. $q_0$ decreases monotonically within $[\xi_j,\xi_{j+1}]$
and straightforward manipulation affirms that \R{2.17} represents the
contribution of this interval to \R{2.16}.
\noindent{\bf Case 3:} $\min\{q_0(\xi_j),q_0(\xi_{j+1})\}>-2$ or
$\max\{q_0(\xi_j),q_0(\xi_{j+1})\}<2$.\\
In that case we cannot fit $[-1,1]$ into $[\xi_j,\xi_{j+1}]$, hence we
stipulate that $\D\psi(X_j)$ is not supported in $[\xi_j,\xi_{j+1}]$.
Let $\nu_1<\nu_2<\cdots<\nu_r$ be all the indices in
$\{0,1,\ldots,s\}$ such that either Case 1 or Case 2 holds. We require
that $r\geq1$. Then \R{2.16} reduces to
$$I_{n,m}=\int_{-1}^1 U_n(x)U_m(x)\sum_{\ell=1}^r
\D\psi(|X_{\nu_\ell}(x)|).$$
Since the Hamburger moment problem for the Chebyshev measure of the
second kind is determinate, it follows that necessarily
$$\sum_{\ell=1}^r \D\psi(|X_{\nu_\ell}(x)|)=(1-x^2)^{\frac12}\D
x,\qquad x\in[-1,1].$$
\noindent {\bf Theorem 2} The orthogonality measure corresponding to
the OPS $\{p_m\}_{m\in\Zp}$ is supported by
$$\Xi={\cal I}_1\cup{\cal I}_2\cup\cdots\cup{\cal I}_r,$$
where for each $\ell=1,2,\ldots,r$ ${\cal I}_\ell\subseteq
[\xi_{\nu_\ell}, \xi_{\nu_{\ell+1}}]$ is the unique interval such that
$|q_0(t)|=2$ at its endpoints.
{\bf Proof.} Follows at once from our construction. \QED
Figure 1 displays two examples of the present construction, for
different cases of $q_0$. In each case $\Xi$ is the union of the
`thick' intervals.
Harking back to \R{1.6} and \R{1.7}, we let
$\alpha_m=-\sigma\cos\alpha m$ and $\alpha_m=-\sigma\cos(m+1)\alpha$,
$m=0,1,\ldots$, respectively, where $\alpha=2\pi L/K$. Thus,
$\{\alpha_m\}_{m\in\Zp}$ is indeed $K$-periodic. Unsurprisingly, the
outcome of our analysis are the familiar spectral bounds
\cite{Bellissard,Cycon}. The merits of our approach are, however, not
just in providing an alternative proof of known results but also in
extending the framework to the multivariate case in Section \ref{sec4}.
We mention in passing that the analysis of this section can be easily
extended to recurrences of the form
\begin{eqnarray*}
p_{-1}(t)&\equiv&0,\\
p_0(t)&\equiv&1,\nonumber\\
p_{m+1}(t)&=&(t-\alpha_m)p_m(t)-\beta_m p_{m-1}(t),\qquad
m=0,1,\ldots,
\end{eqnarray*}
where both $\{\alpha_m\}_{m\in\Zp}$ and $\{\beta_m\}_{m\in\Zp}$ are
$K$-periodic. This, however, is of little relevance to the theme of
this paper.
\eject
\special{psfile="inv1.ps" voffset=-420 hoffset=-70 hscale=90
vscale=65}
\special{psfile="inv2.ps" voffset=-710 hoffset=-70 hscale=90
vscale=65}
\rule{0pt}{500pt}
\begin{figure}
\caption{The sets $\Xi$ for two different polynomials $q_0$.}
\end{figure}
\eject
\section{Orthogonal polynomials with almost periodic recurrence
coefficients} \label{sec3}
The three-term recurrence relations \R{1.6} and \R{1.7} assume,
$\alpha/\pi$ being irrational, almost-periodic recurrence coefficients
and this state of affairs is even more important in a multivariate
generalization of \R{1.1} in Section~\ref{sec4}. Unfortunately, no
general theory exists to cater for orthogonal polynomials with almost
periodic recurrence coefficients. The theme of the present section is
a preliminary and -- in the nature of things -- incomplete
investigation of the case when an irrational $\alpha/\pi$ is
approximated by rationals. In other words, we commence with the
$K$-periodic recurrence \R{2.1}, except that we will allow the period
$K$ to become unbounded.
The motivation for our analysis is an observation which is interesting
on its own merit. Denote by $\sigma_K$ the value of
$q_{K-1}(0)$ for $\alpha_\ell=-\cos\frac{2\pi\ell}{K}$,
$\ell=0,1,\ldots,K-1$, i.e.\
$$\sigma_K=\det\left[\begin{array}{ccccc}
\cos\frac{2\pi}{K} & 1\\
1 & \cos\frac{4\pi}{K} & 1\\
& \ddots & \ddots\\
& & 1 & \cos\frac{2(K-2)\pi}{K} & 1\\
& & & 1 & \cos\frac{2(K-1)\pi}{K} \end{array}\right],\qquad K=1,2,\ldots.$$
Computation indicates that
\begin{equation}\label{3.1}
\sigma_{4L}\equiv0,\qquad
\lim_{L\rightarrow\infty}\sigma_{4L+2}=0,\qquad
\lim_{L\rightarrow\infty} \sigma_{4L+1}=-\lim_{L\rightarrow\infty}
\sigma_{4L+3}=\frac{2\sqrt{3}}{3}.
\end{equation}
It is easy to prove that $\sigma_{4L}=0$ for all $L\geq1$. Thus, let
$K=4L$ and set $\ell=L$ in
\begin{equation}\label{3.2}
\nu_\ell=\cos\frac{2\pi\ell}{K}\nu_{\ell-1}-\nu_{\ell-2}, \qquad
\ell=1,2,\ldots,4L-1
\end{equation}
(note that \R{3.2}, in tandem with $\nu_{-1}=0$, $\nu_0=1$, yields
$\sigma_K=\nu_{K-1}$). This yields $\nu_L+\nu_{L-2}=0$ and we claim
that, in general,
\begin{equation}\label{3.3}
\nu_{L+k}+(-1)^k \nu_{L-k}=0,\qquad k=-1,0,\ldots,L-1.
\end{equation}
We have already proved \R{3.3} for $k=0$ and it is trivially true for
$k=-1$. We continue by induction and assume that \R{3.3} is true for
$k=-1,0,\ldots,s-1$. Letting $\ell=L\pm s$ in \R{3.2}, we have
\begin{eqnarray*}
\nu_{L+s}&=&-\sin\Frac{\pi s}{2L}\nu_{L+s-1}-\nu_{L+s-2},\\
\nu_{L-s}&=&+\sin\Frac{\pi s}{2L}\nu_{L-s-1}-\nu_{L-s-2}.
\end{eqnarray*}
We multiply the second equation by $(-1)^s$ and add to the first, thus
$$(\nu_{L+s}+(-1)^s\nu_{L-s})=-\sin\Frac{\pi s}{2L}
(\nu_{L+s-1}+(-1)^{s-1} \nu_{L-s+1})-(\nu_{L+s-2}+(-1)^{s-2}
\nu_{L-s+2})$$
and \R{3.3} follows at once.
Hence, letting $k=L-1$ in \R{3.3} and recalling that $\nu_{-1}=0$, we
obtain $\nu_{2L-1}=0$. Moreover, similarly to \R{3.3}, we can prove
that
$$\nu_{3L+k}+(-1)^k\nu_{3L-k}=0,\qquad k=-1,0,\ldots,L-1$$
and $k=L-1$ gives
$$\sigma_{4L}=\nu_{4L-1}=(-1)^L\nu_{2L-1}=0.$$
This completes the proof of the lemma. \QED
Other observations in \R{3.1} are also true and in the sequel we prove
them in a generalized setting.
Given $\sigma\in(-1,1)$, we define
\begin{eqnarray}
A_{-1}(t)&\equiv&0,\qquad\qquad A_0(t)\equiv1, \nonumber\\
A_n(t)&=&t\xi_n A_{n-1}(t)-A_{n-2}(t),\qquad n=1,2,\ldots, \label{3.4}
\end{eqnarray}
where $\{\xi_n\}_{n=1}^\infty$ is a given real sequence. To emphasize
the dependence on parameters, we write, as and when necessary,
$A_n(\,\cdot\,)=A_n(\,\cdot\,;\xi_1,\xi_2,\ldots,\xi_n)$.
An alternative representation of $A_n$ is
$$A_n(t)=\det\left[\begin{array}{cccc}
t\xi_1 & 1\\ 1 & t\xi_2 & \ddots\\ & \ddots & \ddots & 1\\ & & 1 &
t\xi_n \end{array}\right],\qquad n=1,2,\ldots,$$
hence $A_n$ is an $n$th degree polynomial. We observe that
$$A_n(0)=\left\{\begin{array}{lcl}(-1)^s & \qquad &:n=2s,\\0 & \qquad
& :n=2s+1. \end{array}\right.$$
Moreover, differentiating with respect to $t$, we obtain
$$A_n'(t)=\sum_{k=1}^n \det \left[ \begin{array}{ccccccccc}
t\xi_1 & 1 \\
1 & \ddots & \ddots\\
& \ddots & \ddots & \ddots\\
& & 1 & t\xi_{k-1} & 1\\
& & & 0 & \xi_k & 0\\
& & & & 1 & t\xi_{k+1} & 1\\
& & & & & \ddots & \ddots & \ddots\\
& & & & & & \ddots & \ddots & 1\\
& & & & & & & 1 & t\xi_n \end{array}\right],$$
therefore, expanding in the $k$th row, we derive the identity
\begin{equation}\label{3.5}
A_n'(t;\xi_1,\ldots,\xi_n) =\sum_{k=1}^n \xi_k
A_{k-1}(t;\xi_1,\ldots,\xi_{k-1}) A_{n-k}(t;\xi_{k+1},\ldots,x_n).
\end{equation}
\noindent {\bf Proposition 3} $A_n^{(r)}(0)=0$ whenever $n+r$ is odd,
hence the polynomial $A_n$ has the same parity as $n$.
{\bf Proof.} By induction on $r$. The assertion is true for $r=0$.
Moreover, repeatedly differentiating \R{3.5} with the Leibnitz rule
and letting $t=0$, we obtain
$$A_{2n}^{(2r+1)}(0;\xi_1,\ldots,\xi_{2n})=\sum_{\ell=0}^{2r}
{{2r}\choose \ell} \sum_{k=1}^{2n} \xi_k A_{k-1}^{(\ell)}(0;\xi_1,
\ldots,\xi_{k-1}) A_{2n-k}^{(2r-\ell)}(0;\xi_{k+1},\ldots,\xi_{2n}).$$
But
$$(k-1)+\ell \quad \mbox{is even} \qquad\Longleftrightarrow\qquad
(2n-k)+(2r-\ell) \quad \mbox{is odd},$$
therefore for all $\ell=0,1,\ldots,2r$ and $k=1,2,\ldots,2n$ the
induction hypothesis affirms that at least one of the terms in the
product vanishes. Similar argument demonstrates that
$A_{2n+1}^{(2r)}(0)=0$. \QED
We therefore let for all $n=0,1,\ldots$, $s=1,2,\ldots$,
\begin{eqnarray*}
A_{2n}(t;\xi_s,\ldots,\xi_{2n+s-1})&=&\sum_{r=0}^n
B_{2n,s}^{(2r)}t^{2r}\\
A_{2n+1}(t;\xi_s,\ldots,\xi_{2n+s})&=&\sum_{r=0}^n B_{2n+1,s}^{(2r+1)}
t^{2r+1}.
\end{eqnarray*}
Substitution into \R{3.4} (where we replace $\xi_n$ by $\xi_{n+s-1}$)
results in the recurrences
\begin{eqnarray}
B_{2n,s}^{(2r)}&=&\xi_{s+2n-1}
B_{2n-1,s}^{(2r-1)}-B_{2n-2,s}^{(2r)}, \label{3.6}\\
B_{2n+1,s}^{(2r+1)}&=&\xi_{s+2n}B_{2n,s}^{(2r)}-B_{2n-1,s}^{(2r+1)}.
\label{3.7}
\end{eqnarray}
Given $q\in\CC$, we recall that the {\em $q$-factorial symbol\/} is
defined as
$$(z;q)_n=\prod_{k=0}^{n-1}(1-q^k z),\qquad z\in\CC,\quad
n\in\ZZ\cup\{\infty\},$$
whereas the {\em $q$-binomial\/} reads
$$\qbin{n}{m}:=\frac{(q;q)_n}{(q;q)_m(q;q)_{n-m}},\qquad 0\leq m\leq
n.$$
\noindent {\bf Lemma 4} Let $\xi_s=q^{\frac12 s}+q^{-\frac12 s}$,
$s=1,2,\ldots$, where $q\in\CC$ is given. Then, for every
$n=0,1,\ldots$, $s=1,2,\ldots$ and $r=0,1,\ldots,n$,
\begin{eqnarray}
B_{2n,s}^{(2r)}&=&(-1)^{n+r}q^{-r\left(2n+s-r-\frac12\right)}
\qbin{n+r}{2r} (-q^{n+s-r};q)_{2r}, \label{3.8}\\
B_{2n+1,s}^{(2r+1)}&=&(-1)^{n+r} q^{-\left(r+\frac12\right) (2n+s-r)}
\qbin{n+r+1}{2r+1} (-q^{n+s-r};q)_{2r+1}.\label{3.9}
\end{eqnarray}
{\bf Proof.} By induction on $n$, using \R{3.8} and \R{3.9}.
Obviously, the assertion of the lemma is true for $n=0$. Otherwise,
for even values,
\begin{eqnarray*}
&&\left(q^{n+\frac{s-1}{2}}+q^{-n-\frac{s-1}{2}}\right)
B_{2n-1}^{(2r-1)}-B_{2n-2,s}^{(2r)}\\
&=&(-1)^{n+r}q^{-\left(r-\frac12\right)(2n+s-r-1)}
\left(q^{n+\frac{s-1}{2}}+q^{-n-\frac{s-1}{2}}\right)
\qbin{n+r-1}{2r-1} (-q^{n+s-r};q)_{2r-1}\\
&&\quad\mbox{}-(-1)^{n+r-1} q^{-r\left(2n+s-r-\frac52\right)}
\qbin{n+r-1}{2r} (-q^{n+s-r-1};q)_{2r}\\
&=&(-1)^{n+r}q^{-r\left(2n+s-r-\frac12\right)} \frac{(q;q)_{n+r-1}}
{(q;q)_{2r}(q;q)_{n-r}} (-q^{n+s-r};q)_{2r-1}\\
&&\quad\mbox{}\times \left\{(1+q^{2n+s-1})(1-q^{2r})
+q^{2r}(1-q^{n-r})(1+q^{n+s-r-1})\right\}\\
&=&(-1)^{n+r}q^{-r\left(2n+s-r-\frac12\right)} \qbin{n+r}{2r}
(-q^{n+s-r};q)_{2r}.
\end{eqnarray*}
This accomplishes a single inductive step for \R{3.8}. We prove
\R{3.9} in an identical manner, by considering odd values of $n$.
\QED
Recall that our interest in the polynomials $A_n$ has been sparked by
the observation \R{3.1}. Thus, we require to recover cosine terms, and
to this end we choose $q$ of unit modulus.
\noindent {\bf Proposition 5} Suppose that $q^{n+1}=1$ and $q^m\neq1$ for
$m=1,2,\ldots,n$. Then $B_{2n+1,1}^{(2r+1)}=0$ for all $0\leq r\leq
\frac{n}{2}$.
{\bf Proof.} Let $2r\leq n$. We have from \R{3.9} that
$$B_{2n+1,1}^{(2r+1)}=(-1)^{n+r}q^{\left(r+\frac12\right)(r+1)}
\frac{(q;q)_{n+r+1}}{(q;q)_{2r+1}(q;q)_{n-r}} (-q^{-r};q)_{2r+1}.$$
However,
$$\frac{(q;q)_{n+r+1}}{(q;q)_{n-r}}=\prod_{\ell=-r}^r (1-q^\ell)=0,$$
whereas, because of our restriction on $r$,
$$(q;q)_{2r+1}=\prod_{\ell=1}^{2r+1}(1-q^\ell)\neq0,$$
since $q$ is a root of unity of {\em minimal\/} degree $n+1$. The
proposition follows. \QED
\noindent {\bf Corollary} Let $q=\exp\frac{2\pi\I m}{n+1}$, where $m$
and $n$ are relatively prime. Then it is true that
\begin{equation}\label{3.10}
\lim_{n\rightarrow\infty} A_{2n+1}(t)=0
\end{equation}
for every $t\in(-1,1)$.
{\bf Proof.} Straightforward, since $A_{2n+1}(t)=\O{t^{\frac12 n}}$.
\QED
\noindent {\bf Proposition 6} Suppose that $\omega=q^{\frac12}$ is a root
of unity of minimal degree $2n+1$. Then, for all $r=0,1,\ldots,n$ it
is true that
\begin{equation}\label{3.11}
(-1)^n B_{2n,1}^{(2r)}=\prod_{\ell=1}^r \frac{\sin (2\ell-1)\phi}{\sin
2\ell\phi},
\end{equation}
where $\phi=\arg\omega$.
{\bf Proof.} Since $q^{n+\frac12}=1$, it follows from \R{3.8} that
$$(-1)^n B_{2n,1}^{(2r)}=(-1)^r q^{r\left(r+\frac12\right)}
\frac{(q;q)_{n+r}}{(q;q)_{2r}(q;q)_{n-r}} (-q^{-r+\frac12};q)_{2r}.$$
But
$$\frac{(q;q)_{n+r}(q^{-r+\frac12};q)_{2r}}{(q;q)_{n-r}}=
\prod_{\ell=-r}^{r-1}(1-q^{2\ell+1}).$$
Moreover, $(q;q)_{2r}\neq0$ for $r=0,1,\ldots,n$, since $2n+1$ is the
least nontrivial degree of the root of unity $q$, and we deduce that
$$(-1)^n B_{2n,1}^{(2r)}=(-1)^r q^{r\left(r+\frac12\right)}
\frac{\prod_{\ell=-r}^{r-1}(1-q^{2\ell+1})}{\prod_{\ell=1}^{2r}
(1-q^\ell)}.$$
But
$$\prod_{\ell=1}^{2r}(1-q^\ell)=\prod_{\ell=1}^{2r}\omega^\ell
(\omega^\ell-\omega^{-\ell})=q^{r\left(r+\frac12\right)}
\prod_{\ell=1}^{2r} (\omega^\ell-\omega^{-\ell})$$
and, likewise,
$$\prod_{\ell=-r}^{r-1}(1-q^{2\ell+1})=\prod_{\ell=-r}^{r-1}
\omega^{2\ell+1} (\omega^{2\ell+1}-\omega^{-2\ell-1})=(-1)^r
\prod_{\ell=0}^{r-1} (\omega^{2\ell+1}-\omega^{-2\ell-1})^2.$$
Consequently,
$$(-1)^n B_{2n,1}^{(2r)}=\frac{\prod_{\ell=0}^{r-1} (\omega^{2\ell+1}
-\omega^{-2\ell-1})^2}{\prod_{\ell=1}^{2r}
(\omega^\ell-\omega^{-\ell})}.$$
This is precisely the identity \R{3.11}. \QED
Next, we consider progression to a limit as $n\rightarrow\infty$ --
\R{3.1} is a special case. Thus, suppose that we have a sequence
$\Phi=\{\phi_n\}_{n\in{\cal I}}$, where $\phi_n=2\pi m_n/(n+1)$,
$m_n\in\Zp$, ${\cal I}\subseteq\Zp$ is a set of infinite cardinality
and
$$\lim_{\stackrel{\scriptstyle n\rightarrow\infty}{n\in{\cal I}}}
\phi_n=\phi\in[0,2\pi).$$
Set
$$C(t,\Phi)=\lim_{\stackrel{\scriptstyle n\rightarrow\infty}{n\in{\cal
I}}} (-1)^{[n/2]}A_n(t;\xi_1^{(n)},\xi_2^{(n)},\ldots,\xi_n^{(n)}),$$
where $\xi_\ell^{(n)}=2\cos\ell\phi_n$, $\ell=1,2,\ldots,n$,
$n=1,2,\ldots$. Thus, $\xi_\ell^{(n)}=q_n^\ell+q_n^{-\ell}$, where
$q_n=\exp\frac{4\pi\I}{n+1}$. Consequently, according to Proposition 5, if
$\cal I$ consists of only odd indices, necessarily $C(t,\Phi)\equiv0$.
This proves, icidentally, that $\sigma_{2L}\rightarrow0$ in \R{3.1}.
\noindent {\bf Lemma 7} Suppose that ${\cal I}\subseteq 2\Zp$ and that
$\phi=0$. Then, provided that $m_n=o(n^{\frac23})$, it is true that
$C(t,\Phi)=(1-t^2)^{-\frac12}$.
{\bf Proof.} Since
$$\frac{\sin(2\ell-1)\phi_n}{\sin2\ell\phi_n}=\frac{2\ell-1}{2\ell}+
\O{\phi_n^3},$$
we deduce from \R{3.11} that
$$(-1)^n B_{2n,1}^{(2r)}=r^{-r}{{2r}\choose r}+\O{n\phi_n^3}.$$
According to the assumption, $\O{n\phi_n^3}=o(1)$ and the lemma
follows from
$$\sum_{r=0}^\infty {{2r}\choose
r}\frac{t^{2r}}{4^r}=\frac{1}{\sqrt{1-t^2}}.$$
$\Box$
Letting $t=\frac12$ affirms the remaining part of \R{3.1}.
Similarly to the last proposition, it is possible to derive an
explicit expression for $C(t,\Phi)$, provided that $\phi/\pi$ is
rational and that $n(\phi-\phi_n)^3=o(1)$ as $n\rightarrow\infty$. The
derivation is long and it will be published elsewhere. It suffices to
mention here the remarkable sensitivity of $C(t,\Phi)$ to both the
choice of $\cal I$ and to the specific nature of $\phi$.
What happens when $\phi/\pi$ is irrational? This is, as things stand,
an open problem. It is possible to show that, formally,
$$C(t,\Phi)=\sum_{\ell=0}^\infty t^{2\ell} \prod_{k=1}^\ell
\frac{\sin(2k-1)\phi}{2k\phi}=
\sum_{\ell=0}^\infty t^{2\ell}q^{\frac14 \ell}
\frac{(q^{\frac12};q)_\ell}{(q;q)_\ell},$$
where $q={\rm e}^{4\I\phi}$. The latter series can be summed up by means of
the Gau\ss--Heine theorem \cite{Gasper} for $|q|<1$ and, after simple
manipulation, for $|q|>1$. Unfortunately, because of a breakdown in
H\"older-continuity across $|q|=1$, it is impossible to deduce its
value on the unit circle from the values within and without by means,
for example, of the Sokhotsky formula \cite{Henrici}.
Clearly, there is much to be done to understand better the behaviour
of the $p_n$'s when the period $K$ becomes infinite. In this section we
have established few results with regard to the values at the origin.
They should be regarded as a preliminary foray into an interesting
problem in orthogonal polynomial theory {\em cum\/} linear algebra and
we hope to return to this theme in the future.
\section{Generalizations} \label{sec4}
There are two natural ways of generalizing an almost Mathieu equation
\R{1.1}, by either specifying a more general periodic potential
or replacing the index by a multi-index. Remarkably, the basic
framework of this paper -- replacing a doubly-infinite recurrence by a
functional equation which, in turn, is replaced by a singly-infinite
recurrence -- survives both generalizations! In the present section we
describe briefly this state of affairs.
Firstly, suppose that the cosine term in \R{1.1} is replaced by a more
general harmonic term and we consider the spectral problem
\begin{equation}\label{4.1}
a_{n-1}-2\left\{\sum_{\ell=1}^m \kappa_\ell \cos(n\ell\theta+\psi_\ell)
\right\} a_n +a_{n+1}=\lambda a_n,\qquad n\in\ZZ.
\end{equation}
We assume that $\kappa_1,\kappa_2,\ldots,\kappa_m\in\RR$ and, without
loss of generality, that $\kappa_m\neq0$. Letting $q={\rm e}^{\I\theta}$,
we set
$$\kappa_\ell^*={\rm e}^{\I\psi_\ell}\kappa_\ell,\quad
\kappa_{-\ell}^*={\rm e}^{-\I\psi_\ell} \kappa_\ell,\qquad
\ell=1,2,\ldots,m,$$
and $\kappa_0^*=0$. Therefore \R{4.1} assumes the form
\begin{equation}\label{4.2}
q^{mn}a_{n-1}-\left\{\sum_{\ell=0}^{2m}
\kappa^*_{\ell-m}q^{n\ell}\right\} a_n+q^{mn}a_{n+1}=\lambda
q^{mn}a_n,\qquad n\in\ZZ.
\end{equation}
Let $c\in\CC\setminus\{0\}$ and consider the Dirichlet series
$y(t)=\sum_{n=-\infty}^\infty a_n \exp\{cq^n t\}$. Since, formally,
$$y^{(\ell)}(t)=c^\ell \sum_{n=-\infty}^\infty q^{n\ell}a_n
{\rm e}^{cq^nt},\qquad \ell=0,1,\ldots,$$
we obtain from \R{4.2} the functional differential equation
\begin{equation}\label{4.3}
\sum_{\ell=0}^{2m}\kappa^*_{\ell-m}c^{m-\ell} y^{(\ell)}(t)= q^m
y^{(m)}(qt) -\lambda y^{(m)}(t)+q^{-m}y^{(m)}(q^{-1}t).
\end{equation}
The derivation is identical to that of \R{1.3} and is left to the
reader.
In line with Section \ref{sec1}, we next expand the solution of
\R{4.3} in Taylor series, $y(t)=\sum_{n=0}^\infty p_n t^n/n!$. This
readily yields
$$\sum_{\ell=0}^{2m}\kappa^*_{\ell-m} c^{m-\ell}
p_{n+\ell}=(q^{n+m}+q^{-n-m}-\lambda) p_{n+m},\qquad n=0,1,\ldots,$$
hence, replacing $n+m$ by $n$,
$$\sum_{\ell=-m}^m \kappa_\ell^* c^{-\ell} p_{n+\ell}=(2\cos
n\theta-\lambda) p_n,\qquad \ell=m,m+1,\ldots.$$
Finally, we choose $c=\exp\{\I\psi_m/m\}$, hence
$\kappa_m^*c^{-m}=\kappa^*_{-m}= \kappa_m\in\RR\setminus\{0\}$. We
thus define $\alpha_\ell=c^{-\ell}\kappa_\ell^*/\kappa_m$, $|\ell|\leq
m$ and replace $\lambda$ by $-\lambda \kappa_m$. This results in the
recurrence
\begin{equation}\label{4.4}
\sum_{\ell=-m}^m \alpha_\ell p_{n+\ell}=(\lambda-\beta_n)p_n,\qquad
n=m,m+1,\ldots,
\end{equation}
where
$$\beta_n=-\frac{\cos n\theta}{\kappa_m},\qquad n=m,m+1,\ldots.$$
Note that, inasmuch as the $\alpha_\ell$s may be complex, we have
$\alpha_{-\ell}= \bar{\alpha}_\ell$, $\ell=1,2,\ldots,m$,
$\alpha_0=0$.
The recurrence \R{4.4} is spanned by $2m$ linearly independent
solutions. However, unless $m=1$, it is no longer true that, for
appropriate choice of $p_0,p_1,\ldots,p_{2m-1}$, each $p_n$ is a
polynomial of degree $n+k$ for some $k$, independent of $n$.
Indeed, it is easy to verify that the degree of $p_n$ increases
roughly as $[n/m]$. Hence, orthogonality is lost. Fortunately, an
important feature of orthogonal polynomials, namely that their zeros
are eigenvalues of a truncated Jacobi matrix \cite{Chihara}, can be
generalized to the present framework. It is possible to show that the
zeros of $p_n$ are generalized eigenvalues of a specific pencil of
`truncated' matrices and this provides a handle on their location. We
expect to address ourselves to this issue in a future publication.
Another generalization of \R{1.1} allows the index $n$ to be replaced
by a multi-index ${\bf n}=(n_1,n_2,\ldots,n_d)\in\ZZ^d$. Thus, let
${\bf e}_\ell\in\ZZ^d$ be the $\ell$th unit vector, $\ell=1,2,\ldots,d$,
and consider the spectral problem
\begin{equation}\label{4.5}
\sum_{\ell=1}^d (a_{{\bf n}+{\bf e}_\ell}+a_{{\bf n}-{\bf e}_\ell})
-2\kappa \cos\left(\sum_{\ell=1}^d \alpha_\ell
n_\ell+\beta\right)a_{\bf n}=\lambda a_{\bf n},\qquad {\bf n}\in\ZZ^d.
\end{equation}
In line with Section \ref{sec1}, we let
$$b_1=b{\rm e}^{\I\beta},\quad b_2={\rm e}^{-\I\beta},\qquad
q_\ell={\rm e}^{\I\alpha_\ell},\quad \ell=1,2,\ldots,d,$$
whereupon \R{4.5} becomes
\begin{equation}\label{4.6}
\sum_{\ell=1}^d {\bf q}^{\bf n}(a_{{\bf n}+{\bf e}_\ell} +a_{{\bf
n}-{\bf e}_\ell})-(b_1{\bf q}^{2{\bf n}}+b_2)a_{\bf n}=\lambda {\bf
q}^{\bf n}a_{\bf n},\qquad {\bf n}\in\ZZ^d.
\end{equation}
The last formula employs standard multi-index notation, e.g.\ ${\bf
q}^{\bf n}=q_1^{n_1}q_2^{n_2}\cdots q_d^{n_d}$.
We let formally
$$y(t)=\sum_{{\bf n}\in\ZZ^d} a_{\bf n}\exp\left\{ b_1^{\frac12}{\bf
q}^{\bf n}t\right\}$$
and note that
\begin{eqnarray*}
y'(t)&=&b_1^{\frac12} \sum_{{\bf n}\in\ZZ^d} a_{\bf n}{\bf q}^{\bf n}
\exp\left\{ b_1^{\frac12}{\bf q}^{\bf n}t\right\},\\
y''(t)&=&b_1^{\frac12} \sum_{{\bf n}\in\ZZ^d} a_{\bf n}{\bf q}^{2\bf n}
\exp\left\{ b_1^{\frac12}{\bf q}^{\bf n}t\right\}.
\end{eqnarray*}
Therefore, multiplying \R{4.6} by $\exp \left\{ b_1^{\frac12}{\bf
q}^{\bf n}t\right\}$ and summing up for ${\bf n}\in\ZZ^d$ yields,
after brief manipulation, the complex functional differential equation
\begin{equation}\label{4.7}
y''(t)+b_2y(t)=b_1^{-\frac12} \left\{ \sum_{\ell=1}^d
\left(q_\ell^{-1} y'(q_\ell^{-1}t)+q_\ell y'(q_\ell t)\right) -\lambda
y'(t)\right\}.
\end{equation}
Equation \R{4.7} is of independent interest, being a special case of
the equation
$$y''(t)+c_1y'(t)+c_2 y(t)=\int_0^{2\pi}
y({\rm e}^{\I\theta}t)\D\mu(\theta),$$
where $\D\mu$ is a complex-valued Borel measure. This, in turn, is
similar to the functional integro-differential equations of the form
$$y''(t)+c_1y'(t)+c_2 y(t)=\int_0^1 y(qt)\D\eta(q),$$
say, where $\D\eta$ is, again, a complex-valued Borel measure.
Equations of this kind have been considered by the present
author, jointly with Yunkang Liu \cite{Iserles2}, with an emphasis on
their dynamics and asymptotic behaviour. In the present
paper, however, we are interested in the spectral problem for \R{4.7},
and to this end we again expand $y$ in Taylor series,
$y(t)=\sum_{m=0}^\infty y_m t^m/m!$. It is easy to affirm by
substitution into \R{4.7} the three-term recurrence relation
\begin{equation}\label{4.8}
y_{m+1}=b_1^{-\frac12}\left(2\sum_{\ell=1}^d \cos\alpha_\ell
m-\lambda\right) y_m-b_2y_{m-1},\qquad m=1,2,\ldots.
\end{equation}
Note a most remarkable phenomenon -- although \R{4.5} is
$d$-dimensional, the index in \R{4.8} lives in $\Zp$! In other words,
the dimensionality of the resultant three-term recurrence is
independent of $d$ -- it is, instead, expressed as the number of
harmonics in the recurrence coefficient. Moreover, inasmuch as \R{4.8}
is more complicated for $d\geq2$ then its one-dimensional counterpart
\R{1.4}, both recurrences display similar qualitative characteristics.
In particular, we can use the theory of Section~\ref{sec2} to cater for
the case of $\alpha_1/\pi,\alpha_2/\pi,\ldots,\alpha_d/\pi$ being all
rational.
In line with the analysis of Section \ref{sec1}, we let
$\tilde{y}_m(t)=b_2^{\frac12 m}y_m(-(b_1b_2)^{\frac12} t)$, $m\in\Zp$,
whereupon \R{4.8} becomes
\begin{equation}\label{4.9}
\tilde{y}_{m+1}(t)=\left(t+\frac{2}{\kappa} \sum_{\ell=1}^d
\cos\alpha_\ell m\right) \tilde{y}_m(t)-\tilde{y}_{m-1}(t),\qquad
m=1,2,\ldots.
\end{equation}
To recover all solutions of \R{4.9} we need to consider a linearly
independent two-dimensional set of solutions. Letting $r_{-1}=0$,
$r_0=1$ and $s_0=0$, $s_1=1$, we recover, similarly to $(1.6$--$7)$,
two sequences $\{r_m\}_{m\in\Zp}$ and $\{s_m\}_{m\in\Zp}$ that span
all solutions of \R{4.9} and such that $\deg r_m=m$, $\deg s_m=m-1$.
In other words, by the Favard theorem both $\{r_m\}_{m\in\Zp}$ and
$\{s_{m+1}\}_{m\in\Zp}$ are OPS and, in line with our analysis of the
one-dimensional almost Mathieu equation \R{1.1}, we are in position
to exploit the theory of orthogonal polynomials.
\end{document} |
\begin{document}
\author{Mario Alberto Castagnino.}
\address{I.A.F.E. (Univ. de Bs. As.)}
\author{Adolfo Ram\'{o}n Ord\'{o}\~{n}ez.}
\address{I.F.I.R. / Fac. de Ciencias Exactas, Ing. y Agrim. (Univ. Nac. de Rosario)}
\author{Daniela Beatriz Emmanuele.}
\address{Fac. Cs. Exactas, Ing.y Agrim.U.N.R.}
\title{A general mathematical structure for the time-reversal operator.}
\date{December 27th., 2000}
\maketitle
\begin{abstract}
The aim of this work is the mathematical analysis of the physical
time-reversal operator and its definition as a geometrical structure{\bf , }
in such a way that it could be generalized to the purely mathematical realm.
Rigorously, only having such a ``time-reversal structure'' it can be decided
whether a dynamical system is time-symmetric or not.{\it \ }The
``time-reversal structures'' of several important physical and mathematical
examples are presented, showing that there are some mathematical categories
whose objects are the (classical or abstract) ``time-reversal systems'' and
whose morphisms generalize the Wigner transformation.
\end{abstract}
\section{Introduction.}
The dynamics and the thermodynamics of both, classical and quantum physical
systems, are modelized by the mathematical theory of classical and abstract
dynamical systems. It is obvious that the {\it physical} notion of
``time-symmetric (or asymmetric) dynamical systems'' requires the definition
of a ``time-reversal operator'', $K$ \cite{7}$.$ In fact, every known model
of a physical dynamical system {\it has} some $K$ operator. E.g., the
dynamic of classical physical systems is described in the cotangent fiber
bundle $T^{*}(N)$ of its configuration manifold $N$, and therefore the
action of $K:T^{*}(N)\rightarrow T^{*}(N)$ is defined as
\begin{equation}
p_{q}\mapsto -p_{q} \label{0.1}
\end{equation}
for any linear functional $p$ on $q\in N,$ or in coordinate's language:
\begin{equation}
(q^{i},p_{i})\mapsto (q^{i},-p_{i}) \label{0.2}
\end{equation}
in a particular $(q^{i},p_{i})$ coordinate system.
In Quantum Mechanics there is the well known Wigner antiunitary operator $K$
defined through the complex conjugation in the position (wave functions)
representation:
\begin{equation}
\psi (x,t)\mapsto \psi (x,-t)^{*} \label{0.3}
\end{equation}
In the last few years it was demonstrated that ordinary Quantum Mechanics
(with no superselection sectors) can be naturally included in the
Hamiltonian formalism of its real K\"{a}hlerian differentiable manifold of
quantum states \cite{Cire0} \cite{Cire1} \cite{Cire2}. The latter one is the
real but infinite dimensional manifold of the associated projective space $
{\bf P}({\cal H})$ of its Hilbert states space ${\cal H}$ \footnote{
We should remember the fact that ordinary pure quantum states are not {\it
vectors} $\psi $ (normalized or not) of a Hilbert space ${\cal H}$, but rays
or {\it projective equivalence classes of vectors} $[\psi ]\in {\bf P}({\cal
H})$.}. From this point of view, $K:{\bf P}({\cal H})\rightarrow {\bf P}(
{\cal H})$ acts as the cannonical projection to the quotient of (\ref{0.3})
\begin{equation}
\lbrack \psi (x,t)]\mapsto \left[ \psi (x,-t)^{*}\right] \label{0.4}
\end{equation}
Moreover, this result has been generalized to more general quantum systems
through its characteristic C*-algebra $A$, and its pure quantum states space
$\partial K(A)$ turns out to be a projective K\"{a}hler bundle over the
spectrum $\widehat{A},$ whose fiber over the class of a state $\psi ,$ is
isomorphic to ${\bf P}({\cal H}_{\psi }),$ being ${\cal H}_{\psi }$ its GNS
(Gelfand-Naimark-Segall) representation's space \cite{Cire3}. More recently
these authors have relaxed the K\"{a}hlerian structure, showing ''minimal''
mathematical structures involved in the quantum principles of superposition
and uncertainty, with the aim of considering non linear extensions of
quantum mechanics \cite{Cire4}.
But, what happens in more general dynamical systems? Some of them -such as
the Bernouilli systems and certain Kolmogorov-systems \cite{2}- are purely
mathematical. Nevertheless, the notion of time-symmetry seems to make sense
also for them. So, it would be interesting to know what kind of mathematical
structures are involved in these systems.
Our aim is to show that:
1.-{\it The mere existence of the time reversal operator is a mathematical
structure }consisting of a non trivial involution $K$ of the states space of
a general system (with holonomic constraints) $M,$ which splits into a $K$
-invariant submanifold (coordinatized by ${\it q}^{i}$) and whose
complementary set (the field of the effective action of $K,$ coordinatized
by ${\it p}_{i}$) is a manifold with the same dimension of $M.$ This
structure is logically independent of the symplectic one \cite{1}, which
doesn't require such a splitting at all. In fact, the essence of symplectic
geometry, as its own etymology shows, is the ''common enveloping'' of $q$
and $p$, loosing any ''privilege'' between them. Actually, {\it this }$K$
{\it -structure is defined by the action of that part of the complete
Galilei group -including the time-reversal- which is allowed to act on the
phase space manifold }$M$ {\it by the constraints. }In fact, only on the
homogeneous Euclidean phase space $M^{\prime }={\Bbb R}^{6n},$ it is
possible to have the transitive action \cite{4} of the complete Galilei group
{\it . }
2.-{\it It is possible to define generalized and purely mathematical
``time-reversals'' }allowing a generalization of the notion of
``time-symmetry'' for a wider class of dynamical systems, including all
Bernouilli systems. In fact there are mathematical categories whose objects
are the time-reversal (classical or abstract) systems $(M,K)$ and whose
morphisms generalize the Wigner transformation \cite{Wigner}{\it . }
3.-When the states space has additional structures, {\it there is a
possibility of getting a richer time-reversal compatible with these
structures. }For example, in Classical Mechanics the canonical $K$ is a
symplectomorphism of phase space, and the Wigner quantum operator is
compatible with the K\"{a}hlerian structure of{\it \ }${\bf P}({\cal H}).$
At first sight (\ref{0.2}) is quite similar to (\ref{0.3}) and it seems to
be some kind of ''complex conjugation'' (and the even dimensionality of
phase space reinforces this idea). We will prove this fact, namely, the
existence of an almost complex structure $J$ with respect to which $K$ is an
almost complex time-reversal. This increases the analogy with Quantum
Mechanics, where the strong version of the Heisemberg Uncertainty Principle,
\cite{Cire1} \cite{Cire2} is equivalent to the existence of a complex
structure $J,$ by means of which the Wigner time-reversal is defined.
4.-{\it It is possible to make a definition of time-reversal systems so
general }that it includes among its examples the real line, the Minkowski
space-time, the cotangent fiber bundles, the quantum systems, the classical
densities function space, the quantum densities operator space, the
Bernouilli systems, etc.
The paper is organized as follows: In section 2, the general theory of{\it \
}{\bf reversals }and{\bf \ time-reversal systems} is developed. In section
3, the theory of {\bf abstract} {\bf reversals }and {\bf abstract\
time-reversal systems} is given. In section 4, many important examples of
time-reversal systems are given{\it .} In section 5, we give the abstract
time-reversal of Bernouilli systems and we show explicitly the geometrical
meaning of our definitions for the Baker's transformation.
\section{Reversals and time-reversal systems}
{\bf Definition:} Let $M$ be a real paracompact, connected, finite or
infinite-dimensional differentiable manifold, and let $K:M\rightarrow M$ be
a diffeomorphism. We will say that $K$ is a {\bf reversal} on $M,$ and that $
(M,K)$ is a {\bf reversal system} if the following conditions are satisfied:
(r.1) $K$ is an involution, i.e. $K^2=I_M$
(r.2) The set $N$ of all fixed points of $K$ is an immersed submanifold of $
M,$ such that $M-N$ is a connected or disconnected submanifold of the same
dimension of $M$. (In particular, this implies that $M$ is a non trivial
involution, i.e. $K\neq I_{M},$ the identity function on $M$)
We will say that $N$ is the {\bf invariant} {\bf submanifold} of the
reversal system.
{\bf Definition:} Let $(M,K)$ be a reversal system. We will say that $M$ is $
K${\bf -orientable} if $M-N$ is composed of two diffeomorphic connected
components $M_{+}$ y $M_{-}$, and if
\begin{equation}
K(M_{+})=M_{-}\;,\;\text{and }K(M_{-})=M_{+} \label{1.0}
\end{equation}
$M$ is $K${\bf -oriented} when -conventionally or arbitrarily- one of these
components is selected as {\bf ``positively oriented''}. In that case, $K$
changes the $K${\bf -}orientation of $M.$
If there is a complex (or almost complex) structure $J$ on $M$ (and
therefore $J^{\prime }=-J$ is another one) and if, in addition, $K$
satisfies:
(c.r.) $K$ is complex (or almost complex) as a map from $(M,J)$ to $
(M,J^{\prime })=(M,-J)$, i.e. :
\begin{equation}
K_{*}\circ J=-J\circ K_{*} \label{1.1}
\end{equation}
\noindent we will say that $K$ is a {\bf complex (or almost complex)
reversal, or a conjugation. }
Similarly, if a symplectic (or almost symplectic) 2-form $\omega $ is given
on $M$ \footnote{
In the infinite-dimensional case we require $\omega $ to be {\it strongly
non-degenerate }\cite{Cire4} in the sense that the map $X\mapsto \omega
(X,.) $ is a toplinear isomorphism.} (and therefore $\omega ^{\prime
}=-\omega $ is another one) and if, in addition to (r.1) y (r.2), $K$
satisfies:
(s.r.) $K$ is a symplectomorphism from $(M,\omega )$ to $(M,\omega ^{\prime
})=(M,-\omega )$, i.e. :
\begin{equation}
K^{*}\omega =-\omega \label{1.2}
\end{equation}
\noindent we will say that $K$ is a {\bf symplectic (or almost symplectic)
reversal. }If we have a symplectic (or almost symplectic) reversal system $
(M,\omega ,K),$ then for every
\[
m\in M:K_{*}:T_{m}(M)\rightarrow T_{m}(M)
\]
is a (toplinear) isomorphism, and if
\[
i:N\rightarrow M\text{ is the immersion}:i(q)=m
\]
and $i_{*}(X_{q})=X_{i(q)}$ is the induced isomorphism, we can define an
almost complex structure $J$:
\begin{eqnarray}
J\left( X_{i(q)}\right) &:&=Y_{m}\Leftrightarrow \omega \left(
X_{i(q)},Y_{m}\right) =1 \nonumber \\
J\left( Y_{m}\right) &:&=-X_{i(q)} \label{1.2a}
\end{eqnarray}
that is to say, by defining the pairs of ``conjugate'' vectors (and
extending by linearity). Then
\[
K_{*}\left( X_{i(q)}\right) =X_{i(q)}\;,\;K_{*}\left( Y_{m}\right) =-Y_{m}
\]
When $(M,\omega ,J,g)$ is a K\"{a}hler (or almost K\"{a}hler) manifold, and $
K$ satisfies the properties (r.1), (r.2), (c.r.) and (s.r.), we will say
that $K$ is a {\bf K\"{a}hlerian (or almost K\"{a}hlerian) reversal. }In
that case, $K$ is also an isometry
\begin{equation}
K^{*}g=g \label{1.3}
\end{equation}
with respect to the K\"{a}hler metric $g$ defined by:
\begin{equation}
g(X,Y)=-\omega (X,JY)\text{ for all vector fields }X\text{ and }Y.
\label{1.4}
\end{equation}
{\bf Definition:} Let $(M,K)$ be a reversal system such that there is a
class ${\cal F}$ of flows $\left( S_t\right) _{t\in {\Bbb R}}$ or / and
cascades $\left( S_t=S^t\right) _{t\in {\Bbb Z}}$ on $M$ such that, for any $
m\in M,$ and any $t$ in ${\Bbb R}$ (or in ${\Bbb Z}$) satisfies:
\begin{equation}
(K\circ S_t\circ K)(m)=S_{-t}(m) \label{1.5}
\end{equation}
Then we will say that $K$ is a {\bf time-reversal }for ${\cal F}${\bf \ }on $
M.$ (In Physics we can take ${\cal F}$ as a class of physical interest. For
example, in Classical Mechanics we can take the class of all Hamiltonian
flows over a fixed phase space and in Quantum Mechanics the class of
solutions of the Schr\"{o}dinger equation in a fixed states space, etc.)
Only having a time-reversal on $M,$ {\bf time-symmetric} {\bf (or asymmetric)
} dynamical systems $\left( S_{t}\right) $ (flows $\left( S_{t}\right)
_{t\in {\Bbb R}}$ ; or cascades $\left( S_{t}=S^{t}\right) _{t\in {\Bbb Z}}$
) can be defined. In fact, $\left( S_{t}\right) $ will be considered as{\bf
\ symmetric with respect to the time-reversal }$K${\bf ,} if it fulfills $
\forall m\in M$ the condition (\ref{1.5}) (or {\bf asymmetric }if it
doesn't).
When $M$ is orientable (oriented) with respect to a time-reversal $K$, we
will say that it is {\bf time-orientable (oriented). }
{\bf Definition: }By a {\bf morphism} of the reversal system $(M,K)$ into $
(M^{\prime },K^{\prime }),$ we mean a differentiable map $f$ of $M$ into $
M^{\prime }$ such that
\begin{equation}
f\circ K=K^{\prime }\circ f \label{1.6}
\end{equation}
As the composition of two morphisms is a morphism and the identity $I_{M}$
is a morphism, we get a {\bf category of reversal systems}, whose objects
are the reversal systems and whose morphisms are the morphisms of reversal
systems. Also we have the subcategories of symplectic, almost complex,
K\"{a}hlerian, etc. reversal systems.
\section{Abstract reversals and abstract time-reversal systems}
{\bf Definition:} Let $(M,\mu )$, be a measure space, and let $
K:M\rightarrow M$ be an isomorphism (mod 0) \cite{2}. We will say that $K$
is an {\bf abstract} {\bf reversal} on $(M,\mu )$ and that $(M,\mu ,K)$ is
an {\bf abstract} {\bf reversal system} if the following conditions are
satisfied:
(a.r.1) $K$ is an involution, i.e. $K^2=I_M$ (mod 0)
(a.r.2) The set $N$ of all fixed points of $K$ is a measurable subset of
null measure of $M,$ and so $\mu [M-N]=\mu [M]$ (In particular, this implies
that $M$ is a non trivial involution, i.e. $K\neq I_{M},$ the identity
function on $M$)
We will say that $N$ is the {\bf invariant} {\bf subset} of the abstract
reversal system.
{\bf Definition:} Let $(M,\mu ,K)$ be an abstract reversal system such that
there is a class ${\cal F}$ of measure preserving flows $\left( S_{t}\right)
_{t\in {\Bbb R}}$ or / and cascades $\left( S_{t}=S^{t}\right) _{t\in {\Bbb Z
}}$ on $M$ such that, for all $m\in M,$ and all $t$ in ${\Bbb R}$ (or in $
{\Bbb Z}$) it satisfies (\ref{1.5}). Then, we will say that $K$ is a {\bf
time reversal }for ${\cal F}${\bf \ }on $(M,\mu )$.
Only having an abstract time-reversal on $M,$ {\bf time-symmetric} {\bf (or
asymmetric)} abstract dynamical systems $\left( S_{t}\right) $ (flows $
\left( S_{t}\right) _{t\in {\Bbb R}}$ ; or cascades $\left(
S_{t}=S^{t}\right) _{t\in {\Bbb Z}}$ ) can be defined. In fact, $\left(
S_{t}\right) $ will be regarded as{\bf \ symmetric with respect to the
time-reversal }$K${\bf ,} if it fulfills $\forall m\in M$ the condition (\ref
{1.5}) (or {\bf asymmetric }if it doesn't)
{\bf Definition: }By a {\bf morphism} of the abstract reversal system $
(M,\mu ,K)$ into $(M^{\prime },\mu ^{\prime },K^{\prime }),$ we mean a
measurable map $f$ of $(M,\mu )$ into $(M^{\prime },\mu ^{\prime })$ such
that, $\forall A^{\prime }\subset M^{\prime }$ measurable:
\begin{equation}
\mu \left( f^{-1}(A^{\prime })\right) =\mu ^{\prime }\left( A^{\prime
}\right) \text{ mod 0} \label{1.7}
\end{equation}
and
\begin{equation}
f\circ K=K^{\prime }\circ f \label{1.8}
\end{equation}
As the composition of two morphisms is a morphism and the identity $I_{M}$
is a morphism, we get a {\bf category of abstract reversal systems}, whose
objects are the abstract reversal systems and whose morphisms are the
morphisms of abstract reversal systems.
\section{Examples of time-reversal systems}
We will see how the mathematical structure just described can be implemented
in all the classical and quantum physical systems, and also generalized to
more abstract purely mathematical dynamical systems, as the Bernouilli
systems.
\subsection{The real line}
Let us consider in the real line ${\Bbb R}$, the mapping $K:{\Bbb R}
\rightarrow {\Bbb R}$ defined by:
\begin{equation}
K(t)=-t \label{2.1}
\end{equation}
Clearly, ${\Bbb R}$ is $K$-orientable, because if $N=\{0\},$ then ${\Bbb R}
-\{0\}={\Bbb R}_{+}\cup {\Bbb R}_{-},$ and $K$ is a canonical time-reversal
for the family of translations: for $a\in {\Bbb R}$ fixed, and $t\in {\Bbb R}
$,
\begin{equation}
\text{ }S_{t}^{a}(x):=x+ta \label{2.2}
\end{equation}
\subsection{The Minkowski space-time}
Let $({\Bbb R}^4,\eta )$ be the Minkowski space-time, with $\eta =$ diag $
(1,-1,-1,-1)$. The invariant submanifold is the spacelike hyperplane
\[
N=\left\{ (0,x,y,z):x,y,z\in {\Bbb R}\right\}
\]
Clearly, fixing $M_{+}$ as the halfspace containing the ''forward'' light
cone
\[
\left\{ (ct,x,y,z):c^{2}t^{2}-x^{2}-y^{2}-z^{2}>0\text{ and }t>0\right\}
\]
and $M_{-}$ as the halfspace containing the ''backward'' light cone
\[
\left\{ (ct,x,y,z):c^{2}t^{2}-x^{2}-y^{2}-z^{2}>0\text{ and }t<0\right\}
\]
and defining $K:{\Bbb R}^{4}\rightarrow {\Bbb R}^{4}$ by:
\begin{equation}
K(ct,x,y,z)=(-ct,x,y,z) \label{2.3}
\end{equation}
we get a canonical $K$-orientation, equivalent to the ussual
time-orientation. $K$ is a time-reversal with respect to the temporal
translations
\begin{equation}
S_{t}^{A}(X):=X+tA=(x^{0}+ta^{0},x^{1},x^{2},x^{3}) \label{2.4}
\end{equation}
for $A=(a^{0},0,0,0)\in {\Bbb R}^{4}:a^{0}\neq 0$ fixed, and $t\in {\Bbb R}$.
{\bf Remark: }As an effect of curvature, not every general 4-dimensional
Lorentzian manifold $(M,g),$ will be time-orientable \cite{Licner}.
Nevertheless, a time-orientable general space-time is necessary if we are
searching for a model of a universe with an arrow of time \cite{cosmic arrow}
\cite{Casta}. In fact, if our universe were represented by a
non-time-orientable manifold, it would be impossible to define past and
future in a global sense, in contradiction with all our present cosmological
observations. Precisely, we know that there are no parts of the universe
where the local arrow of time points differently from our own arrow.
\subsection{The cotangent fiber bundle. Classical Mechanics.}
A general cotangent fiber bundle needs not to be $K$-orientable.
Nevertheless, we have the following result:
{\bf Theorem}: The cotangent fiber bundle (of a finite dimensional
differentiable manifold) $\left( T^{*}(N),\pi ,N\right) $ \cite{1} has a
canonical almost K\"{a}hlerian time-reversal (for the Hamiltonians flows on
it).
{\bf Proof:} Let $M$ be the cotangent fiber bundle $T^{*}(N)$ of a real
n-dimensional manifold $N.$ In this case $N$ is an embedded submanifold of $
M $, being the embedding $i:N\rightarrow T^{*}(N)$ such that:
\[
\text{ }i(q)=0_{q}\text{ (the null functional at }q\text{)}
\]
Let's define
\begin{eqnarray}
K &:&T^{*}(N)\rightarrow T^{*}(N) \nonumber \label{4.1} \\
\forall p_q &\in &T_q^{*}(N):K(p_q)=-p_q \label{2.5}
\end{eqnarray}
Because of its definition, this map is obviously an involution, and by its
linearity is differentiable and its differential or tangent map
\[
K_{*}:T\left( T^{*}(N)\right) \rightarrow T\left( T^{*}(N)\right)
\]
verifies:
\begin{equation}
K_{*}(X_{p_{q}})=
{X_{p_{q}}\text{ if }X_{p_{q}}\in i_{*}\left( T_{q}(N)\right) \atopwithdelims\{. -X_{p_{q}}\text{ if }X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) }
\label{2.6}
\end{equation}
$X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) $ means that it is
``vertical'' or tangent to the point $p_{q}$ of the fibre in $q$. It must be
taken into account that by joining a vertical base with the image of a base
in $N$ by the isomorphism $i_{*}$ $,$ we get a base of $T_{p_{q}}\left(
T^{*}(N)\right) .$
Let $\omega $ be the canonical symplectic 2-form of the cotangent fiber
bundle. As $\omega $ is antisymmetric, in order to evaluate $K^{*}\omega ,$
it is sufficient to consider only three possibilities:
\begin{eqnarray*}
1)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\;,Y_{p_q}\in i_{*}\left(
T_q(N)\right) \\
2)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\;,Y_{p_q}\in T_{p_q}\left( \pi
^{-1}(q)\right) \\
3)\;\left( X_{p_q},Y_{p_q}\right) &:&X_{p_q}\in i_{*}\left( T_q(N)\right)
\text{ but }Y_{p_q}\in T_{p_q}\left( \pi ^{-1}(q)\right)
\end{eqnarray*}
In case 1)
\begin{equation}
\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =\omega \left(
X_{p_q},Y_{p_q}\right) =0 \label{2.7}
\end{equation}
In case 2)
\begin{equation}
\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =\omega \left(
-X_{p_q},-Y_{p_q}\right) =\omega \left( X_{p_q},Y_{p_q}\right) =0
\label{2.8}
\end{equation}
In case 3)
\begin{eqnarray}
\omega \left( K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) &=&\omega \left(
X_{p_q},-Y_{p_q}\right) \nonumber \\
&=&-\omega \left( X_{p_q},Y_{p_q}\right) \label{2.9}
\end{eqnarray}
Thus, in any case
\begin{equation}
\left( K^{*}\omega \right) \left( X_{p_q},Y_{p_q}\right) =\omega \left(
K_{*}(X_{p_q}),K_{*}(Y_{p_q})\right) =-\omega \left( X_{p_q},Y_{p_q}\right)
\label{2.10}
\end{equation}
\noindent which proves that $K^{*}\omega =-\omega ,$ the (s.r.) property$.$
Now, let us define
\begin{eqnarray}
J &:&T\left( T^{*}(N)\right) \rightarrow T\left( T^{*}(N)\right) \nonumber
\\
J(X_{p_q}) &=&Y_{p_q}\Leftrightarrow \omega \left( X_{p_q},Y_{p_q}\right) =1
\label{2.10.1}
\end{eqnarray}
that is to say, $J(X_{p_q})$ is the canonical conjugate of $X_{p_q}.$
Then, by the antisymmetry of $\omega ,$ clearly $J^{2}=-I.$ In addition
\begin{eqnarray}
\left( K_{*}\circ J\right) (X_{p_{q}}) &=&K_{*}\left( J(X_{p_{q}})\right)
=-J\left( X_{p_{q}}\right) \nonumber \\
&=&-J\left( K_{*}\left( X_{p_{q}}\right) \right) \nonumber \\
&=&\left( -J\circ K_{*}\right) (X_{p_{q}}) \label{2.10.2}
\end{eqnarray}
if $X_{p_{q}}\in i_{*}\left( T_{q}(N)\right) $ and
\begin{eqnarray}
\left( K_{*}\circ J\right) (X_{p_{q}}) &=&K_{*}\left( J(X_{p_{q}})\right)
=J(X_{p_{q}}) \nonumber \\
&=&J\left( -K_{*}\left( X_{p_{q}}\right) \right) \nonumber \\
&=&\left( -J\circ K_{*}\right) (X_{p_{q}}) \label{2.10.3}
\end{eqnarray}
if $X_{p_{q}}\in T_{p_{q}}\left( \pi ^{-1}(q)\right) .$ So, $K$ preserves
both $\omega $ and $J$, and therefore is almost K\"{a}hlerian. $\Box $
As it is well known, the phase space $M$ of a classical system with a finite
number ($n$) of degrees of freedom and holonomic constraints has the
particular form $T^{*}(N),$ where $N$ denotes the configuration space of the
system. This implies the existence of a privileged submanifold $N$ of $M$.
We may enquire: why is this so? The answer is: because every law of
Classical Mechanics is invariant with respect to the Galilei group {\it
which contains all the spatial translations} (and it is itself a contraction
of the inhomogeneous Lorentz group \cite{Hermann}). This forces the
configuration space to be a submanifold of some {\it homogeneuos} ${\Bbb R}
^{d}$ space. Now, in this submanifold we also have a privileged system of
coordinates: the spatial position coordinates $q_{1}=x_{1},...,q_{d}=x_{d}$,
with respect to which the action of the Galilei group has its simplest
affine expression. Nevertheless, in general this action will take us away
from the configuration manifold $N,$ because it doesn't fit with the
constraints (think for example in the configuration space of a double
pendulum -with two united threads- which is a 2-torus, contained in ${\Bbb R}
^{3}$).
\subsection{Classical Statistical Mechanics}
Let us consider the phase space of a classical system $\left(
T^{*}(N),\omega \right) $ and take the real Banach space $V=L_{{\Bbb R}
}^1\left( T^{*}(N),\sigma \right) $ containing the probability densities
over the phase space, where $\sigma =\omega \wedge ...\wedge \omega $ ($n$
times) is the Liouville measure. $V$ is a real infinite dimensional
differentiable manifold modelled by itself. Then, the above defined almost
K\"{a}hlerian time-reversal $K$ on $T^{*}(N)$ induces $\widetilde{K}
:V\rightarrow V$ by:
\begin{equation}
\rho \mapsto \widetilde{K}(\rho ):\left( \widetilde{K}(\rho )\right)
(m):=\rho \left( K(m)\right) \label{2.11}
\end{equation}
Clearly, $\widetilde{K}$ is a toplinear involution. Now, let us consider the
set $P$ of all (``almost everywhere'' equivalent classes of) ``$\widetilde{K}
$-even'' integrable functions
\begin{equation}
P=\left\{ \rho \in V:\rho \left( K(m)\right) =\rho (m)\right\} \label{2.12}
\end{equation}
and the set $I$ of all (``almost everywhere'' equivalent classes of) the ``$
\widetilde{K}$-odd'' integrable functions
\begin{equation}
I=\left\{ \rho \in V:\rho \left( K(m)\right) =-\rho (m)\right\} \label{2.13}
\end{equation}
Trivially, $V=P\oplus I$ , and there are two toplinear projectors mapping
any $\rho \in V$ into its ``$\widetilde{K}$-even'' and ``$\widetilde{K}$
-odd'' parts. $P$ is the invariant subspace of $\widetilde{K}.$ Its
complement is the (infinite dimensional) open submanifold of (``almost
everywhere'' equivalent classes of) integrable functions whose ``$\widetilde{
K}$-odd'' projection doesn't vanish$.$
Now, every dynamical system $(S_t)$ (in particular those of the class ${\cal
F}$ of $K$) on $T^{*}(N)$ induces another dynamical system $(U_t)$ on $V:$
\begin{equation}
\left( U_t(\rho )\right) (m):=\rho \left( S_{-t}(m)\right) \label{2.14}
\end{equation}
Considering the class $\widetilde{{\cal F}},$ induced by ${\cal F},$ we
conclude that $\widetilde{K}$ is a time-reversal.
So we have another example lacking time orientability but having a
time-reversal structure .
\subsection{Complex Banach spaces}
A {\bf complex structure} on a {\it real} finite (or infinite) Banach space $
V$ is a linear (toplinear) transformation $J$ of $V$ such that $J^2=-I $ ,
where $I$ stands for the identity transformation of $V$. \cite{4}
In the case of a {\it complex} Banach space $V_{{\Bbb C}}$ , we can consider
the associated real vector space (its ``realification'') $V=V_{{\Bbb R}}$
composed of the same set of vectors, but with ${\Bbb R}$ instead of ${\Bbb C}
$, as the field of its scalars. Then, $J=iI$ is the canonical complex
structure of $V_{{\Bbb R}}.$
If $J$ is a complex structure on a finite dimensional real vector space, its
dimension must be even. In any case, there exist elements $X_1$, $X_2$,..., $
X_n,...$ of $V$ such that
\[
\left\{ X_1,...,X_n,...,JX_1,...JX_n,...\right\}
\]
is a basis for $V$ \cite{4}.
Let us define $K:V\rightarrow V$ as the ``conjugation'', i.e. extending by
linearity the assignment
\begin{equation}
\forall i=1,2,...:K(X_{i})=X_{i}\;;\;K(JX_{i})=-JX_{i} \label{2.15}
\end{equation}
Then the real subspace generated by $\{X_{1},X_{2},...\},$ is the subspace
of fixed points of $K,$ $N$. So, $(V,J,K)$ is a complex time-reversal system
for the class of ''non real translations'' $\left( S_{t}^{A}\right) _{t\in
{\Bbb R}}$ ($A$ being a linear combination of $JX_{1},...JX_{n},...)$:
\begin{equation}
S_{t}^{A}(X)=X+tA\;,\;t\in {\Bbb R} \label{2.16}
\end{equation}
In fact,
\begin{eqnarray}
\left( K\circ S_{t}^{A}\circ K\right) (X) &=&K\left( K(X)+tA\right) =X-tA
\nonumber \\
&=&S_{-t}^{A}(X) \label{2.17}
\end{eqnarray}
\subsection{Ordinary quantum mechanical systems}
As a particular case of the previous example, let us consider a classical
system whose phase space is ${\Bbb R}^{6n},$ and take $V={\cal H}
=L^{2}\left( {\Bbb R}^{3n},\sigma \right) $ (actually, its realification).
This choice is motivated by the fact that we want to have a
Galilei-invariant Quantum Mechanics, and so {\it we must quantify the
spatial position coordinates}. There is no cannonical or symplectic symmetry
here. Only acting on the wave functions of the position coordinates the
Wigner time-reversal operator will be expressed as the complex conjugation.
So, we get a complex time-reversal structure.
Now, following \cite{Cire1}, let us consider the real but infinite
dimensional K\"{a}hlerian manifold $\left( {\bf P}({\cal H}),\widetilde{J},
\widetilde{\omega },g\right) $ of the associated projective space ${\bf P}(
{\cal H})$ of the Hilbert states space ${\cal H}$ of an ordinary quantum
mechanical system. $J$ is the complex structure of ${\cal H}$, and it is the
local expresion of
\[
\widetilde{J}:T\left( {\bf P}({\cal H})\right) \rightarrow T\left( {\bf P}(
{\cal H})\right)
\]
$\left( {\bf P}({\cal H}),\widetilde{J},\widetilde{\omega },g\right) $ has a
canonical K\"{a}hlerian time-reversal structure. In fact, we define $K:{\cal
H}\rightarrow {\cal H}$ as in the previous example, and take:
\[
\widetilde{K}:{\bf P}({\cal H})\rightarrow {\bf P}({\cal H})\text{ by: }
\widetilde{K}\left[ \psi \right] =\left[ K(\psi )\right]
\]
then, all the desired properties follows easily.
\subsection{Quantum Statistical Mechanics}
Let $V=L^{1}({\cal H})$ denote the complex Banach space generated by all
nuclear operators on ${\cal H}$ with the trace norm. This set contains the
density operators of Quantum Statistical Mechanics. $V$ is a real infinite
dimensional differentiable manifold modelled by itself. Then, the above
defined complex time-reversal $K$ on ${\cal H}$ induces $\widehat{K}
:V\rightarrow V$ by:
\begin{equation}
\widehat{\rho }\mapsto \widehat{K}(\widehat{\rho }):\left( \widehat{K}(
\widehat{\rho })\right) (\psi ):=\widehat{\rho }\left( K(\psi )\right)
\label{2.18}
\end{equation}
Clearly, $\widehat{K}$ is a toplinear involution. Now, let us consider the
set $R$ of all ``$\widehat{K}$-real'' densities
\begin{equation}
R=\left\{ \widehat{\rho }\in V:\widehat{\rho }\left( K(\psi )\right) =
\widehat{\rho }(\psi )\right\} \label{2.19}
\end{equation}
and the set $I$ of all the ``$\widehat{K}$-imaginary'' densities
\begin{equation}
I=\left\{ \widehat{\rho }\in V:\widehat{\rho }\left( K(\psi )\right) =-
\widehat{\rho }(\psi )\right\} \label{2.20}
\end{equation}
Trivially, $V=R\oplus I$ , and there are two toplinear projectors mapping
any $\rho \in V$ into its ``$\widehat{K}$-real'' and ``$\widehat{K}$
-imaginary'' parts. $R$ is the invariant subspace of $\widehat{K}.$ Its
complement is the (infinite dimensional) open submanifold of (``almost
everywhere'' equivalent classes of) integrable functions whose ``$\widehat{K}
$-imaginary'' projection is not null$.$
Now, every dynamical system $(S_{t})$ (in particular those of the class $
{\cal F}$ of $K$) on ${\cal H}$ induces another dynamical system $(U_{t})$
on $V$ by setting$:$
\begin{equation}
\left( U_{t}(\widehat{\rho })\right) (\psi ):=\widehat{\rho }\left(
S_{-t}(\psi )\right) \label{2.21}
\end{equation}
Considering the class $\widehat{{\cal F}},$ induced by ${\cal F},$ we
conclude that $\widehat{K}$ is a complex time-reversal. So we have another
example lacking time orientability but having a time-reversal structure.
In both Classical and Quantum Statistical Mechanics, we have used the same
criterium to choose $N$ and ${\cal H}$ respectively. The densities of the
two theories are related by the Wigner integral $W$, which is an essential
ingredient in the theory of the classical limit \cite{Casta-Laura}. In the
one dimensional case, it is the mapping $\widehat{\rho }\mapsto \rho
=W\left( \widehat{\rho }\right) $ given by:
\begin{equation}
\rho (q,p)=\frac{1}{\pi }\int\limits_{-\infty }^{+\infty }\widehat{\rho }
(q-\lambda ,q+\lambda )\,e^{2ip\lambda }\,d\lambda \label{2.21a}
\end{equation}
where $q$ is the spatial position coordinate \footnote{
We want to emphasize the necessity of having an homogeneous configuration
space (${\Bbb R}$ in the one dimensional case) in order to have the
translations $q\mapsto q\pm \lambda $ in $W$ integral.}, $p$ its conjugate
momentum, $\rho (q,p)$ a classical density function, and
\begin{eqnarray}
\widehat{\rho }(x,x^{\prime }) &=&\left( \sum\limits_{j=1}^{\infty }\rho
_{j}\,\overline{\psi _{j}}\otimes \psi _{j}\right) (x,x^{\prime }) \nonumber
\\
&=&\sum\limits_{j=1}^{\infty }\rho _{j}\,\overline{\psi _{j}}(x)\psi
_{j}(x^{\prime }) \label{2.21b}
\end{eqnarray}
is a generic matrix element of a quantum density, being $\left\{ \psi
_{j}\right\} _{j=1}^{\infty }$ an orthonormal base of ${\cal H}$, $\rho
_{j}\geqslant 0$ and $\sum\limits_{j=1}^{\infty }\rho _{j}=1$.
As it is obvious by a simple change of variables,
\begin{equation}
W\left[ \widehat{K}\left( \widehat{\rho }\right) \right] \left( q,p\right)
=\rho (q,-p)=\rho \left( K(q,p)\right) =\widetilde{K}\left[ W\left( \widehat{
\rho }\right) \right] \left( q,p\right) \label{2.21c}
\end{equation}
and therefore, $W$ is a morphism between $\left( L^{1}({\cal H}),\widehat{K}
\right) $ and $\left( L_{{\Bbb R}}^{1}\left( T^{*}(N)\right) ,\widetilde{K}
\right) .$
\subsection{Koopman treatment of Kolmogorov-Systems}
With the definition of time-reversal in the physical examples above, we now
face the same definition in purely mathematical dynamical systems.
Let $(M,\mu ,S_t)$ be a Kolmogorov system (cascade or flow). As it is well
known, this implies that the induced unitary evolution $U_t$ in ${\cal H}
=[1]^{\bot }$ the orthogonal complement of the one dimensional subspace of
the classes a. e. of the constant functions in the Hilbert space $L^2(M,\mu
),$ has uniform Lebesgue spectrum of numerable constant multiplicity. This,
in turn, implies the existence of a system of imprimitivity $(E_s)_{s\in
{\Bbb G}}$ based on ${\Bbb G}$ for the group $(U_t)_{t\in {\Bbb G}}$ , where
${\Bbb G}$ is ${\Bbb Z}$ or ${\Bbb R}$:
\begin{equation}
E_{s+t}=U_tE_sU_t^{-1} \label{2.22}
\end{equation}
Following Misra \cite{Misra}, we define the ``Aging'' operator
\begin{equation}
T=\int\limits_{{\Bbb G}}s\,dE_{s}=
{\int\limits_{{\Bbb R}}s\,dE_{s}\text{ for fluxes} \atopwithdelims\{. \sum\limits_{s\in {\Bbb Z}}sE_{s}\text{ for cascades}}
\label{2.23}
\end{equation}
Then
\begin{equation}
U_{-t}TU_t=T+t \label{2.24}
\end{equation}
$T$ is selfadjoint in the discrete case, and essentially selfadjoint in the
continuous case, and there are eigenvectors in the discrete case, and
generalized eigenvectors (antifunctionals) in certain riggings of ${\cal H}$
by a nuclear space $\Phi $ ($\Phi \prec {\cal H}\prec \Phi ^{\times }$) in
the continuous case, $\left( \left| \tau ,n\right\rangle \right) _{\tau \in
{\Bbb G}}$ , such that:
\begin{eqnarray}
T\left| \tau ,n\right\rangle &=&\tau \left| \tau ,n\right\rangle
\label{2.25} \\
U_{t}\left| \tau ,n\right\rangle &=&(\tau +t)\left| \tau ,n\right\rangle
\label{2.26}
\end{eqnarray}
Defining
\begin{equation}
K\left| \tau ,n\right\rangle =-\left| \tau ,n\right\rangle \label{2.27}
\end{equation}
It follows easilly that $K$ restricted to ${\cal H}$ is a time-reversal for $
{\cal F}=\{(U_t)\}$ with respect to which $U_t$ is symmetric.
\section{Examples of abstract time-reversals}
\subsection{Bernouilli schemes}
Let $M$ be the set $\Sigma ^{{\Bbb Z}}$ of all bilateral sequences (of
``bets'')
\begin{equation}
m=(a_{j})_{j\in {\Bbb Z}}=(...a_{-2},a_{-1},a_{0},a_{1},a_{2},...)
\label{3.1}
\end{equation}
on a finite set $\Sigma $ with $n$ elements (a ``dice'' with $n$ faces). Let
${\frak X}$ be the $\sigma $-algebra on $M$ generated by all the subset of
the form
\begin{equation}
A_{j}^{s}=\{m:a_{j}=s\in \Sigma \} \label{3.2}
\end{equation}
Clearly,
\begin{equation}
M=\bigcup\limits_{s\in \Sigma }A_j^s=\bigcup\limits_{k=1}^nA_j^{s_k}
\label{3.3}
\end{equation}
Let's define a normalized measure $\mu $ on $M$ by choosing $n$ ordered
positive real numbers $p_{1},...,p_{n}$ whose sum is equal to one ($p_{k}$
is the ``probability'' of getting $s_{k}$ when the ``dice'' is thrown), and
setting:
\begin{equation}
\forall k:k=1,...,n\;:p_{k}=\mu (A_{j}^{s_{k}}) \label{3.4}
\end{equation}
\begin{equation}
\mu \left( A_{j_1}^{s_1}\cap ...\cap A_{j_k}^{s_k}\right) =\mu
(A_{j_1}^{s_1})...\mu (A_{j_k}^{s_k}) \label{3.5}
\end{equation}
where $j_1,...,j_k$ are all different.
Let the dynamical authomorphism $S$ be the shift to the right:
\begin{eqnarray}
S\left( (a_{j})_{j\in {\Bbb Z}}\right) &=&(a_{j}^{\prime })_{j\in {\Bbb Z}}
\nonumber \\
\text{where: } &&a_{j}^{\prime }:=a_{j-1} \label{3.6}
\end{eqnarray}
The shift preserves $\mu $ because
\begin{equation}
\mu \left( S(A_j^{s_k})\right) =\mu (A_{j+1}^{s_k})=p_k \label{3.7}
\end{equation}
The above abstract dynamical scheme is called a Bernouilli scheme and
denoted $B(p_1,...,p_n).$
Let's define a ``cannonical'' abstract reversal by:
\begin{eqnarray}
K\left( (a_{j})_{j\in {\Bbb Z}}\right) &=&(a_{j}^{\prime })_{j\in {\Bbb Z}}
\nonumber \\
a_{j}^{\prime } &=&a_{-j+1} \label{3.8}
\end{eqnarray}
Clearly, $K$ is an isomorphism, and its invariant set
\begin{equation}
N=\bigcap\limits_{j\in {\Bbb Z}}\left\{ \bigcup\limits_{s\in \Sigma }\left(
A_j^s\cap A_{-j}^s\right) \right\} \label{3.9}
\end{equation}
has $\mu $-measure $0.$ In addition $K$ is a time-reversal for the class $
{\cal F}$ of all Bernouilli schemes, because
\begin{equation}
K\circ S\circ K=S^{-1} \label{3.10}
\end{equation}
being $S^{-1}$ the shift to the left.
\subsection{The Baker's transformation}
We will show the geometrical meaning of the last two time-reversals for the
Baker's transformation.
The measure space is the torus
\[
M=\left[ 0,1\right] \times \left[ 0,1\right] \;/\sim \;=\left\{ (x,y)
\mathop{\rm mod}
1=[x,y]:x,y\in \left[ 0,1\right] \right\}
\]
that is to say, $\sim $ is the equivalence relation that identifies the
following boundary points:
\[
(0,x)\sim (1,x)\;\text{and}\;(x,0)\sim (x,1)
\]
with its Lebesgue measure. The automorphism $S$ acts as follows:
\begin{equation}
S(x,y)=
{(2x,\frac 12y)\;\;\;\;\;\;\;\;\;\;if\;0\leq x\leq \frac 12\;,\;0\leq y\leq 1 \atopwithdelims\{. (2x-1,\frac 12y+\frac 12)\;if\;\frac 12\leq x\leq 1\;,\;0\leq y\leq 1}
\label{5.1}
\end{equation}
It's clear that $S$ is a non-continuous{\bf \ }but measure preserving
transformation which involves a contraction in the $y$ direction and a
dilatation in the $x$ direction: the contracting and dilating directions at
every point $m\in M$ (that is, the vertical and the horizontal lines through
each $m$).
The torus is a compact, connected Lie group and we can define an involutive
automorphism $K$ on $M$ by putting:
\begin{equation}
K[x,y]=[y,x]\;;\;x,y\in I=\left[ 0,1\right] \label{5.2}
\end{equation}
The fixed points of $K$ constitute a submanifold of the torus: the
projection of the diagonal $\Delta $ of the unit square $I\times I$
\[
N=\Delta \;/\sim =\left\{ [x,x]:x\in I\right\}
\]
Then:
\begin{equation}
K\circ S^t\circ K=S^{-t},\;\forall t\in {\Bbb Z} \label{5.3}
\end{equation}
In fact, the first application of $K$ to the generating partition of $S$,
rotates the unit square, interchanging the $x$ fibers with the $y$ ones.
Then by applicating $S^{t}$ (that is $t$ times $S$) we get a striped pattern
of horizontal lines, which is rotated and yields a striped pattern of
vertical lines when $K$ is applicated again. The same pattern would be
obtained if $S^{-t}$ was used.
As it is well known \cite{2}, the Baker transformation is isomorphic to $B(
\frac{1}{2},\frac{1}{2})$. In fact, the map
\begin{equation}
(x,y)\mapsto m=(a_{j})_{j\in {\Bbb Z}}\Leftrightarrow
x=\sum\limits_{j=0}^{\infty }\frac{a_{-j}}{2^{j+1}}\;\text{and }
y=\sum\limits_{j=1}^{\infty }\frac{a_{j}}{2^{j}} \label{5.4}
\end{equation}
is an isomorphism (mod 0). Moreover it is an isomorphism of abstract
time-reversal systems, because it sends the time-reversal of (\ref{5.2}) in
the time-reversal of (\ref{3.8}). In particular, this implies that the
Baker's map is a Kolmogorov system, and therefore having the corresponding
time-reversal for its Koopman treatment. These three reversals are related.
Let $\{A,B\}$ be the partition of the unit square into its left and right
halves. As it is well known this partition is both independent and
generating for the Baker's map. Let's define
\begin{equation}
\theta _{0}=1-\chi _{A}=
{\text{\ }1\text{ in }A \atopwithdelims\{. -1\text{ in }B}
\label{5.5}
\end{equation}
where $\chi _{A}$ is the characteristic function of the set $A,$ as well as
\begin{equation}
\theta _{n}=U^{n}(\theta _{0})=\theta _{0}\circ S^{-n}=
{\text{\ }1\text{ in }S^{n}(A) \atopwithdelims\{. -1\text{ in }S^{n}(B)}
\label{5.6}
\end{equation}
and for any finite set $F=\{n_{1},...,n_{F}\}\subset {\Bbb Z}$, put
\begin{equation}
\theta _{F}=\theta _{n_{1}}...\theta _{n_{F}}\;\text{(ordinary product of
functions)} \label{5.7}
\end{equation}
Then, all the eigenvectors of the Aging operator $T$ of $U$ are of the form
\cite{Prigo}:
\begin{equation}
T\theta _{F}=n_{m}\theta _{F} \label{5.8}
\end{equation}
where $n_{m}=\max F.$ Geometrically speaking, $\rho =\theta _{0}$ -that we
can identify with $\{A,B\}$- is an eigenvector of age $0,$ and if $U$ acts $
n $ times on it we get an eigenvector of age $n$: $\theta _{n}$ -which can
be identified with a set of horizontal fringes-$.$ On the other hand if $
U^{-1}$ acts $n$ times on it we get an eigenvector of age $-n$: $\theta
_{-n} $ -which can be identified with a set of vertical fringes-$.$ As
expected, the induced action of $K$ sends the ``future'' horizontal
eigenstates of $T$ to the ``past'' vertical ones, and reciprocally.
\begin{center}
ACKNOWLEDGMENT
\end{center}
The authors wish to express their gratitude to Dr. Sebastiano Sonego for
providing an initial and fruitful discussion on the subject of this paper.
This work was partially supported by grant PIP 4410 of CONICET (Argentine
National Research Council)
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\end{document} |
\begin{document}
\begin{abstract}
We survey the development of the notion of Lazarsfeld-Mukai bundles
together with various applications, from the classification of Mukai manifolds
to Brill-Noether theory and syzygies of $K3$ sections.
To see these techniques at work, we present a short
proof of a result of M. Reid on the existence of elliptic pencils.
\end{abstract}
\maketitle
\section*{Introduction}
Lazarsfeld--Mukai bundles appeared naturally in connection with two
completely different important problems in algebraic geometry from the
1980s. The first problem, solved by Lazarsfeld, was to find explicit examples
of smooth curves which are generic in the sense of Brill-Noether-Petri
\cite{LazarsfeldJDG}. The second problem was the classification
of prime Fano manifolds of coindex 3 \cite{MukaiPNAS}. More recently,
Lazarsfeld--Mukai bundles have found
applications to syzygies and higher-rank Brill--Noether theory.
The common
feature of all these research topics is the central role played by
$K3$ surfaces and their hyperplane sections. For the Brill--Noether--Petri genericity, Lazarsfeld
proves that a general curve in a linear system that generates
the Picard group of a $K3$ surface satisfies this condition.
For the classification of prime Fano manifolds of coindex 3,
after having proved the existence of smooth fundamental divisors,
one uses the geometry of a two-dimensional linear section which is a very general
$K3$ surface.
The idea behind this definition is that the Brill--Noether theory
of smooth curves~on a $K3$ surface, also called \textit{$K3$ sections},
is governed by higher-rank vector bundles on the surface. To be
more precise, consider $S$ a $K3$ surface (considered always to be
smooth, complex, projective), $C$ a smooth curve on $S$ of genus $\ge 2$,
and $|A|$ a base-point-free pencil on $C$.
If we attempt to lift the linear system
$|A|$ to the surface $S$, in~most cases, we will fail.
For instance, $|A|$ cannot
lift to a pencil on $S$ if $C$ generates
$\mathrm{Pic}(S)$ or if $S$ does not contain any elliptic curve at all.
However, interpreting a general divisor in $|A|$
as a zero-dimensional subscheme of $S$, it is
natural to try and find a rank-two bundle $E$ on $S$ and
a global section of $E$ whose scheme of zeros coincides with
the divisor in question. Varying the divisor, one~should exhibit
in fact a two-dimensional space of global sections of $E$.
The effective construction of $E$ is realized through
elementary modifications, see Sect.\,\ref{section: Def}, and this
is precisely a Lazarsfeld--Mukai bundle of rank two.
The passage to higher ranks is natural, if we start
with a complete, higher-dimensional, base-point-free linear
system on $C$. At the end, we obtain vector bundles
with unusually high number of global sections, which
provide us with a rich geometric environment.
The structure of this chapter is as follows.
In the first section, we recall the definition of Lazarsfeld--Mukai
bundles and its first properties.
We note equivalent conditions for a bundle to be Lazarsfeld--Mukai in
Sect.\,\ref{subsection: Definition LM},
and we discuss simplicity in the rank-two case in
Sect.\,\ref{subsection: Simple LM}.
The relation with the Petri conjecture and the
classification of Mukai manifolds, the original
motivating problems for the~definition, are considered in
Sects.\,\ref{subsection: BNP} and \ref{subsection: Mukai manifolds}, respectively.
In Sect.\,\ref{section: Constancy}
we treat the problem of constancy of invariants in
a given linear system. For small gonalities, Saint-Donat
and Reid proved that minimal pencils on $K3$ sections
are induced from elliptic pencils on the $K3$ surface;
we present a short proof using Lazarsfeld--Mukai bundles
in Sect.\,\ref{subsection Gonality I}.
Harris and Mumford conjectured that the gonality
should always be constant. We discuss the evolution
of this conjecture, from Donagi--Morrison's counterexample,
Sect.\,\ref{subsection Gonality I},
to Green--Lazarsfeld's reformulation in terms
of Clifford index, Sect.\,\ref{subsection: Cliff} and to Ciliberto--Pareschi's results on the subject, Sect.\,\ref{subsection: Gonality II}.
The works around this problem emphasized the importance
of parameter spaces of Lazarsfeld--Mukai bundles.
We conclude the section with a discussion of
dimension calculations of these spaces, Sect.\,\ref{subsection: Parameter}, which are
applied afterwards to Green's conjecture.
Sect.\,\ref{section: Green} is devoted to Koszul cohomology
and notably to Green's conjecture for $K3$ sections.
After recalling the definition and the motivations that
led to the definition, we discuss the statement of Green's
conjecture, and we sketch the proof for $K3$ sections.
Voisin's approach using punctual Hilbert schemes, which is
an essential ingredient, is examined in Sect.\,\ref{subsection: Hilbert}. Lazarsfeld--Mukai bundles
are fundamental objects in this topic, and their
role is outlined in Sect.\,\ref{subsection: Role of LM}.
The final step in the solution of Green's conjecture for
$K3$ sections is tackled in Sect.\,\ref{subsection: Green for K3}.
We conclude this chapter with a short discussion on
Farkas--Ortega's new applications of Lazarsfeld--Mukai bundles to
Mercat's conjecture (which belongs to the rapidly developing
higher-dimensional Brill--Noether theory),
Sect.\,\ref{section: Higher BN}.
\noindent{\it Notation.} The additive and the multiplicative notation for divisors and line bundles
will be mixed sometimes.
If $E$ is a vector bundle on $X$ and $L\in \mbox{Pic}(X)$, we set $E(-L):=E\otimes L^*$;
this notation will be used especially when $E$ is
replaced by the canonical bundle $K_C$ of a curve $C$.
\section{Definition, Properties, the First Applications}
\label{section: Def}
\subsection{Definition and First Properties}
\label{subsection: Definition LM}
We fix $S$ a smooth, complex, projective $K3$ surface and $L$ a globally
generated line bundle on $S$ with
$L^2=2g-2$. Let $C\in|L|$ be a
smooth curve and $A$ be a base-point-free line bundle
in $W^r_d(C)\setminus W^{r+1}_d(C)$.
As mentioned in the Introduction, the definition of Lazarsfeld--Mukai
bundles emerged from the attempt to lift
the linear system
$A$ to the surface $S$. Since it is virtually impossible to
lift it to another linear system, a higher-rank vector bundle
is constructed such that $H^0(C,A)$ corresponds to an $(r+1)$-dimensional
space of global sections. Hence $|A|$ lifts to a higher-rank
analogue of a linear system.
The kernel of the evaluation of sections of $A$\vspace*{-3pt}
\begin{equation}
\label{eqn: F}
0\to F_{C,A}\to H^0(C,A)\otimes \mathcal{O}_S \buildrel{\mathrm{ev}}\over{\to} A\to 0\vspace*{-3pt}
\end{equation}
is a vector bundle of rank $(r+1)$.
\begin{defn}[Lazarsfeld \cite{LazarsfeldJDG}, Mukai \cite{MukaiPNAS}]
The \textit{Lazarsfeld--Mukai bundle} $E_{C,A}$ associated to the pair
$(C,A)$ is the dual of $F_{C,A}$.
\end{defn}
By dualizing the sequence~(\ref{eqn: F}) we obtain the short exact sequence\vspace*{-3pt}
\begin{equation}
\label{eqn: E}
0\to H^0(C,A)^*\otimes \mathcal{O}_S \to E_{C,A}\to K_C(-A)\to 0,\vspace*{-3pt}
\end{equation}
and hence $E_{C,A}$ is obtained
from the trivial bundle by modifying it along the curve $C$
and comes equipped with a natural $(r+1)$-dimensional
space of global sections as planned.
We note here the first properties of $E_{C,A}$:
\begin{prop}[Lazarsfeld]
\label{prop: E_{C,A}}
The invariants of $E$ are the following:
\begin{itemize}
\item[(1)] $\det (E_{C,A})= L$.
\item[(2)] $c_2(E_{C,A})=d$.
\item[(3)] $h^0 (S,E_{C,A})= h^0(C,A)+h^1(C,A)= 2r-d+1+g$.
\item[(4)] $h^1(S,E_{C,A})=h^2(S,E_{C,A})=0$.
\item[(5)] $\chi(S,E_{C,A}\otimes F_{C,A})=2(1-\rho(g,r,d))$.
where $\rho(g,r,d)=g-(r+1)(g-d+r)$.
\item[(6)] $E_{C,A}$ is globally generated off the base locus of $K_C(-A)$;
in particular, $E_{C,A}$ is globally generated if $K_C(-A)$
is globally generated.
\end{itemize}
\end{prop}
It is natural to ask conversely if given $E$ a vector
bundle on $S$ with $\mbox{rk}(E)=r+1$, $h^1(S,E)=h^2(S,E)=0$, and
$\det (E)= L$, $E$ is the Lazarsfeld--Mukai bundle associated to a pair
$(C,A)$.
To this end, note that there is a rational map\vspace*{-3pt}
\[
h_E : G(r+1,H^0(S,E)) \dashrightarrow |L|\vspace*{-3pt}
\]
defined in the following way. A general subspace $\varLambda\in G(r+1,H^0(S,E))$
is mapped to the degeneracy locus of the evaluation map:
$
\mathrm{ev} _{\varLambda} : \varLambda \otimes \mathcal{O}_S \to E.
$
If~the image $h_E(\varLambda)$ is a smooth curve $C\in |L|$, we set
$\mbox{Coker}(\mathrm{ev}_{\varLambda}):=K_C(-A)$,
where $A\in \mbox{Pic}(C)$ and $\mbox{deg}(A)=c_2(E)$, and
observe that $E=E_{C,A}$. Indeed, since $h^1(S,E)=0$, $A$
is globally generated, and from $h^2(S,E)=0$ it follows
that $\varLambda\cong H^0(C,A)^*$.
The conclusion is that:
\begin{prop}
\label{prop: caracterizare LM}
A rank-$(r+1)$ vector bundle $E$ on $S$ is a Lazarsfeld--Mukai bundle if
and only if $H^1(S,E)=H^2(S,E)=0$ and there exists an $(r+1)$-dimensional subspace
of sections $\varLambda\subset H^0(S,E)$, such that the degeneracy locus of
the morphism $\mathrm{ev}_{\varLambda}$ is a smooth curve. In particular, being a
Lazarsfeld--Mukai vector bundle is an \textit{open condition}.
\end{prop}
Note that
there might be different pairs with the same Lazarsfeld--Mukai
bundles, the difference being given by the corresponding spaces
of global sections.
\subsection{Simple and Non-simple Lazarsfeld--Mukai Bundles}
\label{subsection: Simple LM}
We keep the notation from the previous subsection.
In the original situation, the bundles used by Lazarsfeld \cite{LazarsfeldJDG}
and Mukai \cite{MukaiPNAS} are simple.
The non-simple Lazarsfeld--Mukai bundles are, however, equally useful
\cite{Aprodu-FarkasCOMP,Ciliberto-PareschiCRELLE}.
For instance, Lazarsfeld's argument is partly based on an analysis of the
non-simple bundles.
Proposition~\ref{prop: E_{C,A}} already shows that for $\rho(g,r,d)<0$
the associated Lazarsfeld--Mukai bundle cannot be simple.
The necessity of making a distinction between simple and non-simple bundles
for nonnegative $\rho$ will become more evident in the next sections.
In the rank-two case, one can give a precise description \cite{Donagi-MorrisonJDG} of non-simple
Lazarsfeld--Mukai bundles, see also \cite{Ciliberto-PareschiCRELLE} Lemma 2.1:
\begin{lem}[Donagi--Morrison]
\label{lemma: DM}
Let $E_{C,A}$ be a non-simple Lazarsfeld--Mukai bundle.
Then there exist line bundles $M,N\in \mathrm{Pic}(S)$ such that
$h^0(S,M)$, $h^0(S,N)\ge 2$,
$N$ is globally generated, and there exists a
locally complete intersection subscheme $\xi$ of $S$,
either of dimension zero or the empty set, such that
$E_{C,A}$ is expressed as an extension
\begin{equation}
\label{eq: DM}
0\to M\to E_{C,A}\to N\otimes I_{\xi} \to 0.
\end{equation}
Moreover, if $h^0(S,M\otimes N^*)=0$, then $\xi=\emptyset$ and the extension splits.
\end{lem}
One can prove furthermore that $h^1(S,N)=0$, \cite{Aprodu-FarkasCOMP} Remark 3.6.
We say that~(\ref{eq: DM}) is the \emph{Donagi--Morrison extension}
associated to $E_{C, A}$. This notion makes perfect sense as
this extension is uniquely determined by the vector bundle,
if it is indecomposable \cite{Aprodu-FarkasCOMP}.
Actually, a \textit{decomposable} Lazarsfeld--Mukai bundle $E$ cannot
be expressed as an extension~(\ref{eq: DM}) with $\xi\ne\emptyset$, and
hence a Donagi--Morrison extension is always
unique, up to a permutation of factors in the decomposable case.
Moreover, a Lazarsfeld--Mukai bundle is decomposable
if and only if the corresponding Donagi--Morrison extension is trivial.
In the higher-rank case, we do not have such a precise description.\footnote{In fact,
we do have a Harder--Narasimhan filtration,
but we cannot control all the factors.} However, a similar
sufficiently strong statement is still valid \cite{LazarsfeldJDG,LazarsfeldICTP,PareschiJAG}.
\begin{prop}[Lazarsfeld]
Notation as above. If $E_{C,A}$ is not simple, then
the linear system $|L|$ contains a reducible or a multiple
curve.
\end{prop}
In the rank-two case, this statement comes from the decomposition
$L\cong M\otimes N$.
\subsection{The Petri Conjecture Without Degenerations}
\label{subsection: BNP}
A smooth curve of genus $g$ is said to satisfy \textit{Petri's condition},
or to be \textit{Brill--Noether--Petri generic},
if the multiplication map (the Petri map)
\[
\mu_{0,A}: H^0(C, A) \otimes H^0(C, K_C(-A))
\to H^0(C, K_C),
\]
is injective for any line bundle $A$ on $C$.
One consequence of this condition is that
all the Brill--Noether loci $W^r_d(C)$ have the expected
dimension and are smooth away from $W^{r+1}_d(C)$;
recall that the tangent space at the point $[A]$
to $W^r_d(C)$ is naturally isomorphic to the dual
of $\mathrm{Coker}(\mu_{0,A})$ \cite{Arbarello-Cornalba-Griffiths-Harris}.
The Petri conjecture, proved by degenerations by Gieseker,
states that a general curve satisfies Petri's condition.
Lazarsfeld \cite{LazarsfeldJDG} found a simpler and
elegant proof without degenerations by analyzing curves
on very general $K3$ surfaces.
Lazarsfeld's idea is to relate the Petri maps
to the Lazarsfeld--Mukai bundles; this relation is valid
in general and has many other
applications. Suppose, as in the previous subsections,
that $S$ is a $K3$ surface and $L$ is a globally generated
line bundle on $S$. For the moment, we do not need to assume
that $L$ generates the Picard group.
E. Arbarello and M. Cornalba constructed
a scheme $\mathcal{W}^r_d(|L|)$
parameterizing pairs $(C,A)$ with $C\in |L|$ smooth and
$A\in W^r_d(C)$ and a morphism\vspace*{-3pt}
\[
\pi_S:\mathcal{W}^r_d(|L|)\to |L|.\vspace*{-3pt}
\]
Assume that $A\in W^r_d(C)\setminus W^{r+1}_d(C)$ is globally generated,
and consider $M_A$ the vector bundle of rank $r$
on $C$ defined as the kernel of the evaluation map
\begin{equation}\label{MA}
0\to M_A \to H^0(C,A)\otimes \mathcal{O}_C \buildrel{\mathrm{ev}}\over{\to} A\to 0.
\end{equation}
Twisting~(\ref{MA}) with $K_C\otimes A^*$, we obtain the
following description of the kernel of the Petri map:\footnote{This ingenious procedure is an efficient replacement
of the base-point-free pencil trick;
``it has killed the base-point-free pencil trick,'' to quote Enrico Arbarello.}\vspace*{4pt}
\[
\mbox{Ker}(\mu_{0,A})= H^0(C, M_A\otimes K_C\otimes A^*).
\]
There is another exact sequence on $C$\vspace*{-4pt}
\[
0\to \mathcal{O}_C\to F_{C,A}|_C\otimes K_C\otimes A^*\to M_A\otimes K_C\otimes A^*\to 0,\vspace*{-4pt}
\]
and from the defining sequence of $E_{C,A}$ one obtains the exact sequence on $S$\vspace*{-4pt}
\[
0\to H^0(C, A)^*\otimes F_{C,A}\to E_{C,A}\otimes F_{C,A}\to
F_{C, A}|_C\otimes K_C\otimes A^*\to 0.\vspace*{-4pt}
\]
From the vanishing of $h^0(C,F_{C,A})$
and of $h^1(C, F_{C, A})$, we obtain\vspace*{-4pt}
\[
H^0(C,E_{C,A}\otimes F_{C,A})=H^0(C,F_{C,A}|_C\otimes K_C\otimes A^*).\vspace*{-4pt}
\]
Suppose that $\mathcal{W}\subset \mathcal{W}^r_d(|L|)$ is a dominating component and
$(C,A)\in\mathcal{W}$ is an element such that $A$ is
globally generated and $h^0(C,A)=r+1$. A deformation-theoretic
argument shows that if the Lazarsfeld--Mukai
bundle $E_{C,A}$ is simple, then the coboundary map
$H^0(C,M_A\otimes K_C\otimes A^*)\to H^1(C,\mathcal{O}_C)$
is zero \cite{PareschiJAG}, which eventually implies the
injectivity of $\mu_{0,A}$.
By reduction to complete base-point-free bundles on the curve \cite{LazarsfeldJDG,PareschiJAG} this analysis yields:
\begin{thm}[Lazarsfeld]
\label{thm: Lazarsfeld}
Let $C$ be a smooth curve of genus $g\ge 2$ on a $K3$
surface~$S$, and assume that any divisor in the linear
system $|C|$ is reduced and irreducible. Then a
generic element in the linear system $|C|$
is Brill--Noether--Petri generic.
\end{thm}
A particularly interesting case is when the Picard group of $S$ is
generated by $L$ and $\rho(g,r,d)=0$. Obviously, the condition
$\rho=0$ can be realized only for composite genera, as
$g=(r+1)(g-d+r)$, for example, $r=1$ and $g$ even.
Under these assumptions, there is a unique
Lazarsfeld--Mukai bundle $E$ with $c_1(E)=L$ and $c_2(E)=d$,
and different pairs $(C,A)$ correspond to different
\hbox{$\varLambda\in G(r+1,$} $H^0(S,E))$; in other words
the natural rational map $G(r+1,H^0(S,E))\dashrightarrow
\mathcal{W}^r_d(|L|)$ is dominating.
Note that $E$ must be stable and globally generated.
\subsection{Mukai Manifolds of Picard Number One}
\label{subsection: Mukai manifolds}
A Fano manifold $X$ of dimension $n\ge 3$
and index $n-2$ (i.e., of coindex 3) is called
a \textit{Mukai manifold}.\footnote{Some authors consider
that Mukai manifolds have dimension four or more.}
In the classification, special attention is given to prime Fano manifolds:
note that if $n\ge 7$, $X$ is automatically prime as shown by Wisniewski;
see, for example, \cite{Iskovskih-Prokhorov}.
Assume that the Picard group of $X$ is generated
by an ample line bundle $L$, and let the sectional genus
$g$ be the integer $(L^n)/2+1$. Mukai and Gushel
used vector bundle techniques to obtain a complete
classification of these manifolds. A~first major obstacle
is to prove that the fundamental linear system contains indeed
a smooth element, aspect which is settled by
Shokurov and Mella; see, for example, \cite{Iskovskih-Prokhorov}.
Then the $(g+n-2)$-dimensional linear system
$|L|$ is base-point-free, and
a general linear section
with respect to the generator of the Picard group
is a $K3$ surface. More precisely, if $\mathrm{Pic}(X)
=\mathbb{Z}\cdot L$, then for $H_1,\cdots,H_{n-2}$
general elements in the fundamental linear system
$|L|$, $S:=H_1\cap\cdots\cap H_{n-2}$ is
scheme-theoretically a $K3$ surface. Note that if $n\ge 4$
and $i\ge 3$, the intersection $H_1\cap\cdots\cap H_{n-i}$
is again a Fano manifold of coindex 3.
Mukai noticed that the fundamental linear system either
is very ample, and the image of $X$ is projectively
normal or is associated to a double covering of $\mathbb{P}^n$
($g=2$) or of the hyper-quadric $Q^n\subset\mathbb{P}^{n+1}$
($g=3$). The difficulty of the problem is thus
to classify all the possible cases where
$|L|$ is normally generated, called \textit{of the first
species}. Taking linear
sections one reduces (not quite immediately) to the case $n=3$
\cite{Iskovskih-Prokhorov} p.110.
For simplicity, let us assume that
$X$ is a prime Fano 3-fold of index 1. If
$g=4$ and $g=5$, $X$ is a complete intersection; hence
the hard cases begin with genus $6$.
A hyperplane section $S$ is a $K3$ surface, and,
by a result of Moishezon, $\mathrm{Pic}(S)$ is generated
by $L|_S$.
Let us denote by $\mathcal{F}_g$ the moduli
space of polarized $K3$ surfaces of degree $2g-2$, by $\mathcal{P}_g$
the moduli space of pairs $(K3\ \mbox{surface},\mbox{curve})$ and
$\mathcal{M}_g$ the moduli space of genus-$g$ curves.
There are two nice facts in Mukai's proof involving
these two moduli spaces. His first observation is
that if there exists a prime Fano 3-fold $X$
of the first species of genus $g\ge 6$ and index $1$,
the rational map $\phi_g:\mathcal{P}_g\dashrightarrow \mathcal{M}_g$
is \textit{not} generically finite \cite{MukaiLMS}. The second nice
fact is that $\phi_g$ is generically finite
if and only if $g=11$ or $g\ge 13$ \cite{MukaiLMS}.\footnote{In genus $11$, it
is actually birational \cite{MukaiLNPAM}.} Hence, one
is reduced to study the genera $6\le g\le 12$
with $g\ne 11$. At this point, Lazarsfeld--Mukai
bundles are employed. By the discussion from Sect.\,\ref{subsection: BNP},
for any decomposition $g=(r+1)(g-d+r)$, with
$r\ge 1$, $d\le g-1$, there exists a unique
Lazarsfeld--Mukai bundle $E$ of rank $(r+1)$.
It has already been noticed that the bundle $E$ is stable and globally generated.
Moreover, the determinant map\vspace*{-4pt}
\[
\mathrm{det}:\wedge^{r+1}H^0(S,E)\to H^0(S,L)\vspace*{-4pt}
\]
is surjective \cite{MukaiPNAS}, and hence it induces
a linear embedding\vspace*{-4pt}
\[
\mathbb{P}H^0(S,L)^*\hookrightarrow \mathbb{P}(\wedge^{r+1}H^0(S,E)^*).\vspace*{-4pt}
\]
Following \cite{MukaiPNAS}, we have a commutative diagram
\begin{center}
\mbox{\xymatrix{S \ar[r]^{\phi_E}\ar@{^(->}[d]^{\phi_{|L|}} & G \ar@{^(->}[d]^{\mathrm{Pluecker}}\\
\mathbb{P}H^0(L)^*\ar@{^(->}[r] & \mathbb{P}(\wedge^{r+1}H^0(E)^*)}}
\end{center}
where $G:=G(r+1,H^0(S,E)^*)$ and $\phi_E$ is given by $E$.
This diagram shows that $S$ is embedded in a suitable linear section
of the Grassmannian $G$. Moreover, this diagram
extends over $X$: by a result of Fujita, $E$
extends to a stable vector bundle on $X$, and
the diagram over $X$ is obtained for similar reasons.
Hence $X$ is a linear section of a Grassmannian.
By induction on the dimension, $X$ is contained in
a \textit{maximal} Mukai manifold, which is also
a linear section of the Grassmannian.
A complete list of maximal Mukai manifolds
is given in \cite{MukaiPNAS}. Notice that
in genus $12$, the maximal Mukai manifolds are
threefold already.
\setcounter{thm}{0}
\section{Constancy of Invariants of $K3$ Sections}
\label{section: Constancy}
\subsection{Constancy of the Gonality. I}
\label{subsection Gonality I}
In his analysis of linear systems on $K3$ surfaces Saint--Donat
\cite{Saint-DonatAJM}
shows that any smooth curve which is linearly equivalent
to a hyperelliptic or trigonal curve is also hyperelliptic, respectively trigonal.
The idea was to prove that the
minimal pencils are induced by elliptic pencils
defined on the surface. This result was sensibly extended by Reid
\cite{ReidJLM} who proved the following existence result:
\begin{thm}[Reid]
Let $C$ be a smooth curve of genus $g$ on a $K3$ surface $S$
and $A$ be a complete, base-point-free $g^1_d$ on
$C$. If\vspace*{-3pt}
\[
\frac{d^2}{4}+d+2<g,\vspace*{-3pt}
\]
then $A$ is the restriction of an elliptic pencil on $S$.
\end{thm}
It is a good occasion to present here, as a direct application of techniques involving
Lazarsfeld--Mukai bundles, an alternate shorter proof of Reid's theorem.
\begin{proof}
We use the notation of previous sections.
By the hypothesis, the Lazarsfeld--Mukai bundle $E$ is not simple, and
hence we have a unique Donagi--Morrison extension\vspace*{3pt}
\[
0\to M\to E\to N\otimes I_{\xi}\to 0,\vspace*{2pt}
\]
with $\xi$ of length $\ell$. Note that $M\cdot N=d-\ell\le d$.
By the Hodge index theorem, we have
$(M^2)\cdot(N^2)\le (M\cdot N)^2\le d^2$, whereas from
$M+N=C$ we obtain $(M^2)=2(g-1-d)-(N^2)$, hence\vspace*{-4pt}
\[
(N^2)\le\frac{d^2}{2(g-1-d)-(N^2)}.\vspace*{-4pt}
\]
Therefore, the even integer $x:=(N^2)$ satisfies the following
inequality
$x^2-2x(g-1-d)+d^2\ge 0.$
The hypothesis shows that the above inequality fails for $x\ge 2$,
and hence $N$ must be an elliptic pencil.
\end{proof}
In conclusion, for small values, the gonality\footnote{The gonality
$\mathrm{gon}(C)$ of a curve $C$ is the minimal degree of a morphism
from $C$ to the projective~line.}
is constant in the linear system. Motivated
by these facts, Harris and Mumford conjectured
that \textit{the gonality of \hbox{$K3$-sections}
should always be constant} \cite{Harris-MumfordINVENTIONES}.
This conjecture is unfortunately wrong as stated: Donagi and Morrison
\cite{Donagi-MorrisonJDG} gave the following counterexample:
\begin{ex}
\label{ex: DM}
Let $S\to \mathbb{P}^2$ be a double cover branched
along a smooth sextic and $L$ be the pull-back of
$\mathcal{O}_{\mathbb{P}^2}(3)$. The curves in $|L|$
have all genus $10$. The general curve $C\in|L|$
is isomorphic to a smooth plane sextic, and hence
it is pentagonal. On the other hand, the pull-back of
a general smooth plane cubic $\varGamma$ is a double
cover of $\varGamma$, and thus it is tetragonal.
\end{ex}
\subsection{Constancy of the Clifford Index}
\label{subsection: Cliff}
Building on his work on Koszul cohomology and its relations
with geometry, M. Green proposed a reformulation of the
Harris-Mumford conjecture replacing the gonality by
the Clifford index.
Recall that the \textit{Clifford index} of
a nonempty linear system $|A|$ on a smooth curve $C$
is the codimension of the image
of the natural addition map $|A|\times|K_C(-A)|\to |K_C|$.
This definition is nontrivial only for relevant linear
systems $|A|$, i.e., such that both $|A|$ and $|K_C(-A)|$
are at least one-dimensional; such an $A$ is said
to \textit{contribute to the Clifford index}.
The \textit{Clifford index of $C$} is the minimum of all
the Clifford indices taken over the linear
systems that contribute to the Clifford index and
is denoted by $\mathrm{Cliff}(C)$.
The Clifford index is related to the gonality
by the following inequalities\vspace*{3pt}
\[
\mathrm{gon}(C)-3\le \mathrm{Cliff}(C)\le \mathrm{gon}(C)-2,\vspace*{3pt}
\]
and curves with $\mathrm{gon}(C)-3 = \mathrm{Cliff}(C)$
are very rare: typical examples are plane curves and
Eisenbud--Lange--Martens--Schreyer curves \cite{ELMS,KnutsenIJM}.\footnote{It
is conjectured that the only other examples should
be some half-canonical curves of even genus
and maximal gonality \cite{ELMS}; however,
this conjecture seems to be very difficult.}
From the Brill--Noether theory, we obtain the
bound $\mathrm{Cliff}(C)\le\left[(g-1)/2\right]$
(and, likewise, $\mathrm{gon}(C)\le\left[(g+3)/2\right]$),
and it is known that the equality is achieved for general curves.
The Clifford index is in fact a measure of how special
a curve is in the moduli space.
The precise statement obtained by Green and Lazarsfeld
is the following \cite{Green-LazarsfeldINVENTIONES}:
\begin{thm}[Green--Lazarsfeld]
\label{thm: GL Cliff}
Let $S$ be a $K3$ surface and $C\subset S$ be a smooth
irreducible curve of genus $g\ge 2$. Then $\mathrm{Cliff}(C')
=\mathrm{Cliff}(C)$ for every smooth curve $C'\in|C|$. Furthermore,
if $\mathrm{Cliff}(C)$ is strictly less than the generic value
$ \left[(g-1)/2\right]$, then there exists a line bundle
$M$ on $S$ whose restriction to any smooth curve $C'\in|C|$ computes
the Clifford index of~$C'$.
\end{thm}
The proof strategy is based on a reduction method of the
associated Lazarsfeld--Mukai bundles. The bundle $M$
is obtained from the properties of the reductions; we
refer to \cite{Green-LazarsfeldINVENTIONES} for
details.
From the Clifford index viewpoint, Donagi--Morrison's
example is not different from the other cases. Indeed, all
smooth curves in $|L|$ have Clifford index $2$.
We shall see in the next subsection that
Donagi--Morrison's example
is truly an isolated exception for the constancy of the gonality.
\subsection{Constancy of the Gonality. II}
\label{subsection: Gonality II}
As discussed above, the Green--Lazarsfeld proof of the
constancy of the Clifford index was mainly based on
the analysis of Lazarsfeld--Mukai bundles.
It is natural to try and explain the peculiarity of
Donagi--Morrison's example from this point of view.
This was done in \cite{Ciliberto-PareschiCRELLE}.
The surprising answer found by Ciliberto and
Pareschi \cite{Ciliberto-PareschiCRELLE}
(see also \cite{Donagi-MorrisonJDG})
is the following:
\begin{thm}[Ciliberto--Pareschi]
\label{thm: Ciliberto-Pareschi}
Let $S$ be a $K3$ surface and $L$ be an ample line bundle on $S$.
If the gonality of the smooth curves in $|L|$ is not constant,
then $S$ and $L$ are as in Donagi--Morrison's example.
\end{thm}
Theorem~\ref{thm: Ciliberto-Pareschi} was refined
by Knutsen \cite{KnutsenIJM} who replaced ampleness by the more
general condition that $L$ be globally generated.
The extended setup covers also the case of exceptional curves, as introduced
by Eisenbud, Lange, Martens, and Schreyer \cite{ELMS}.
The proof of Theorem~\ref{thm: Ciliberto-Pareschi}
consists of a thorough analysis of the
loci $\mathcal{W}^1_d(|L|)$, where $d$ is the minimal
gonality of smooth curves in $|L|$, through the
associated Lazarsfeld--Mukai bundles. The authors identify
Donagi--Morrison's example in the following way:
\begin{thm}[Ciliberto--Pareschi]
\label{thm: Ciliberto-Pareschi2}
Let $S$ be a $K3$ surface and $L$ be an ample line bundle on $S$.
If the gonality of smooth curves in $|L|$ is not constant and if
there is a pair $(C,A)\in\mathcal{W}^1_d(|L|)$ such that
$h^1(S,E_{C,A}\otimes F_{C,A})=0$, then $S$ and
$L$ are as in Donagi--Morrison's example.
\end{thm}
To conclude the proof of Theorem~\ref{thm: Ciliberto-Pareschi}, Ciliberto and Pareschi prove that
non-constancy of the gonality implies the existence
of a pair $(C,A)$ with $h^1(S,E_{C,A}\otimes F_{C,A})=0$;
see \cite{Ciliberto-PareschiCRELLE} Proposition 2.4.
It is worth to notice that, in Example~\ref{ex: DM}, if $C$ is the
inverse image of a plane cubic and $A$ is a $g^1_4$
(the pull-back of an involution), then
$E_{C,A}$ is the pull-back of
$\mathcal{O}_{\mathbb{P}^2}(1)\oplus\mathcal{O}_{\mathbb{P}^2}(2)$
\cite{Ciliberto-PareschiCRELLE}, and hence
the vanishing of $h^1(S,E_{C,A}\otimes F_{C,A})$ is guaranteed
in this case.
\subsection{Parameter Spaces of Lazarsfeld--Mukai
Bundles and~Dimension of Brill--Noether Loci}
\label{subsection: Parameter}
We have already seen that the Brill--Noether loci
are smooth of expected dimension at pairs corresponding
to simple Lazarsfeld--Mukai bundles. It
is interesting to know what is the dimension of
these loci at other points as well. Precisely,
we look for a uniform bound on the dimension
of Brill--Noether loci of general curves in a linear
system.
A first step was made by Ciliberto and Pareschi
\cite{Ciliberto-PareschiCRELLE} who proved,
as a necessary step in Theorem~\ref{thm: Ciliberto-Pareschi},
that an ample curve of gonality strictly less
than the generic value, general in its linear system,
carries finitely many minimal pencils. This result
was extended to other Brill--Noether loci
\cite{Aprodu-FarkasCOMP},
proving a phenomenon of \textit{linear growth} with the degree; see below.
Let us mention that, for the moment, the only results
in this direction are known to hold for pencils
\cite{Aprodu-FarkasCOMP} and nets \cite{Lelli-Chiesa}.
As before, we consider $S$ a $K3$ surface and $L$
a globally generated line bundle on $S$.
In order to parameterize all pairs $(C, A)$
with non-simple Lazarsfeld--Mukai bundles, we need a global construction.
We fix a nontrivial globally generated line bundle
$N$ on $S$ with $H^0(L(-2N))\neq 0$
and an integer $\ell\ge 0$. We set $M:=L(-N)$
and $g:=1+L^2/2$.
Define $\widetilde{\mathcal{P}}_{N,\ell}$ to be the family of \textit{vector bundles} of rank $2$ on
$S$ given by nontrivial extensions\vspace*{-3pt}
\begin{equation}
\label{eq: extension}
0\to M\to E\to N\otimes I_{\xi}\to 0,\vspace*{-3pt}
\end{equation}
where $\xi$ is a zero-dimensional locally complete intersection subscheme
(or the empty set) of $S$ of length $\ell$, and set
\[
\mathcal{P}_{N,\ell}:=\{[E]\in\widetilde{\mathcal{P}}_{N, \ell}:\ h^1(S,E)=h^2(S,E)=0\}.
\]
Equivalently (by Riemann--Roch), $[E]\in \mathcal{P}_{N, \ell}$ if and only if
$h^0(S,E)=$\break $g-c_2(E)+3$ and $h^1(S,E)=0$. Note that any non-simple
Lazarsfeld--Mukai bundle on $S$ with
determinant $L$ belongs to some family $\mathcal{P}_{N,\ell}$,
from Lemma~\ref{lemma: DM}.
The family $\mathcal{P}_{N,\ell}$,
which, a priori, might be the empty set, is an open Zariski subset
of a projective bundle of the Hilbert scheme~$S^{[\ell]}$.
Assuming that $\mathcal{P}_{N,\ell}\ne \emptyset$,
we consider the Grassmann bundle $\mathcal{G}_{N,\ell}$ over $\mathcal{P}_{N,\ell}$
classifying pairs $(E,\varLambda)$ with $[E]\in\mathcal{P}_{N,\ell}$ and
$\varLambda\in \mathrm{G}(2,H^0(S,E))$. If $d:=c_2(E)$ we define the rational map $h_{N, \ell}: \mathcal{G}_{N, \ell}
\dashrightarrow \mathcal{W}^1_d(|L|)$, by setting $h_{N, \ell}(E, \varLambda):=(C_{\varLambda}, A_{\varLambda})$, where $A_{\varLambda}\in \mbox{Pic}^d(C_{\varLambda})$ is such that the following exact sequence on $S$ holds:\vspace*{4pt}
\[
0\to \varLambda\otimes \mathcal{O}_S\stackrel{\mathrm{ev}_{\varLambda}}\to E\to K_{C_{\varLambda}}\otimes A_{\varLambda}^*\to 0.\vspace*{3pt}
\]
One computes
$\dim\ \mathcal{G}_{N,\ell}=g+\ell+h^0(S,M\otimes N^*)$.
If we assume furthermore that $\mathcal{P}_{N,\ell}$ contains a Lazarsfeld--Mukai vector bundle $E$ on $S$ with $c_2(E)=d$
and consider $\mathcal{W}\subset \mathcal{W}^1_d(|L|)$ the closure of the
image of the rational map $h_{N,\ell}:\mathcal{G}_{N,\ell}\dashrightarrow \mathcal{W}^1_d(|L|)$,
then we find $\dim\ \mathcal{W} =g+d-M\cdot N=g+\ell$.
On the other hand, if $C\in|L|$ has Clifford
dimension one and $A$ is a globally generated line
bundle on $C$ with $h^0(C,A)=2$ and $[E_{C,A}]\in\mathcal{P}_{N,\ell}$,
then $M\cdot N\ge \mathrm{gon}(C)$.
These considerations on the indecomposable
case, together with a simpler analysis of
decomposable bundles, yield finally \cite{Aprodu-FarkasCOMP}:
\begin{thm}
\label{thm: Green Cliffdim 1}
Let $S$ be a $K3$ surface and $L$ a globally generated line bundle on $S$,
such that general curves in $|L|$ are of Clifford dimension one. Suppose that
$\rho(g,1,k)\leq 0$, where $L^2=2g-2$ and $k$ is the (constant) gonality of
all smooth curves in $|L|$. Then for a
general curve $C\in|L|$, we have
\begin{equation}
\label{lgc}
\mathrm{dim}\ W^1_{k+d}(C)=d\mbox{ for all }
0\le d\le g-2k+2.
\end{equation}
\end{thm}
The condition~(\ref{lgc}) is called the \textit{linear growth condition}.
It is equivalent to
\[
\mathrm{dim}\ W^1_{g-k+2}(C)=\rho(g,1,g-k+2)=g-2k+2.
\]
Note that the condition that $C$ carry finitely many minimal
pencils, which is a part of~(\ref{lgc}), appears explicitly in~\cite{Ciliberto-PareschiCRELLE}.
It is directly related to the constancy of the gonality
discussed before.
\setcounter{thm}{0}
\section{Green's Conjecture for Curves on $K3$ Surfaces}
\label{section: Green}
\subsection{Koszul Cohomology}\label{ch1:sec3.1}
Let $X$ be a (not necessarily smooth) complex, irreducible,
projective variety and $L\in\mathrm{Pic}(X)$ globally generated.
The Euler sequence on the projective space $\mathbb{P}(H^0(X,L)^*)$
pulls back to a short exact sequence of vector bundles
on~$X$
\begin{equation}
\label{eqn: Euler}
0 \to M_L\to H^0(X,L)\otimes \mathcal{O}_X\to L\to 0.
\end{equation}
After taking exterior powers in the sequence~(\ref{eqn: Euler}),
twisting with multiples of $L$ and going to global sections,
we obtain an exact sequence for any nonnegative $p$ and $q$:\vspace*{-5pt}
\begin{align}
\label{eqn: WedgeEuler}
0\to H^0(\wedge^{p+1}M_L\otimes L^{q-1})
\to \wedge^{p+1}H^0(L)\otimes H^0(L^{q-1})
\stackrel{\delta}{\to} H^0(\wedge^pM_L\otimes L^q).\nonumber\\ \vspace*{-5pt}
\end{align}
The finite-dimensional vector space $K_{p,q}(X,L):=\mathrm{Coker}(\delta)$
is called the \textit{Koszul cohomology space}\footnote{The
indices $p$ and $q$ are usually forgotten when defining Koszul cohomology.}
of $X$ with values in $L$
\cite{LazarsfeldICTP,GreenJDG,GreenICTP}.
Observe that $K_{p,q}$ can be defined alternatively as:
\[
K_{p,q}(X,L)=\mathrm{Ker}\left(H^1(\wedge^{p+1}M_L\otimes L^{q-1})
\to \wedge^{p+1}H^0(L)\otimes H^1(L^{q-1})\right),\vspace*{-3pt}
\]
description which is particularly useful when $X$ is a curve.
Several versions are used in practice, for example, replace
$H^0(L)$ in~(\ref{eqn: Euler}) by a subspace that generates $L$
or twist~(\ref{eqn: WedgeEuler}) by $\mathcal{F}\otimes L^{q-1}$
where $\mathcal{F}$ is a coherent sheaf. For our presentation,
however, we do not need to discuss these natural generalizations.
Composing the maps\vspace*{-3pt}
\[
\wedge^{p+1}H^0(L)\otimes H^0(L^{q-1})
\stackrel{\delta}{\to} H^0(\wedge^pM_L\otimes L^q)
\hookrightarrow \wedge^pH^0(L)\otimes H^0(L^q)\vspace*{-3pt}
\]
we obtain, by iteration, a complex\vspace*{-3pt}
\[
\wedge^{p+1}H^0(L)\otimes H^0(L^{q-1})\to
\wedge^pH^0(L)\otimes H^0(L^q)\to
\wedge^{p-1}H^0(L)\otimes H^0(L^{q+1})\vspace*{-3pt}
\]
whose cohomology at the middle is
$K_{p,q}(X,L)$, and this is the definition given by Green
\cite{GreenJDG}.
An important property of Koszul cohomology is upper-semicontinuity
in flat families with constant cohomology; in particular,
vanishing of Koszul cohomology is an open
property in such families. For curves, constancy
of $h^1$ is a consequence of flatness and of constancy of $h^0$, as
shown by the Riemann--Roch theorem.
The original motivation for studying Koszul cohomology
spaces was given by the relation with minimal resolutions
over the polynomial ring. More precisely, if $L$ is very
ample, then the Koszul cohomology computes the minimal resolution
of the graded module\vspace*{4pt}
\[
R(X,L):=\bigoplus_qH^0(X,L^q)\vspace*{3pt}
\]
over the polynomial ring
\cite{GreenJDG,GreenICTP}; see also \cite{EisenbudBOOK,Aprodu-NagelULECT}, in the sense that any graded piece
that appears in the minimal resolution is (non-canonically) isomorphic to a
$K_{p,q}$. If the image of $X$ is projectively normal,
this module coincides with the homogeneous coordinate
ring of $X$. The projective normality of $X$ can also
be read off Koszul cohomology, being
characterized by the vanishing condition $K_{0,q}(X,L)=0$ for all
$q\ge 2$. Furthermore, for a projectively normal $X$,
the homogeneous ideal is generated by quadrics if and
only if $K_{1,q}(X,L)=0$ for all $q\ge 2$.\footnote{The
dimension of $K_{1,q}$ indicates the number of generators
of degree $(q+1)$ in the homogeneous~ideal.} The phenomenon
continues as follows: if $X$ is projectively normal and the homogeneous ideal
is generated by quadrics,
then the relations between the generators are linear if and only if
$K_{2,q}(X,L)=0$ for all $q\ge 2$, whence the relation with
syzygies ~\cite{GreenJDG}.
Other notable application of Koszul cohomology is the description
of Castelnuovo--Mumford regularity, which coincides
with, \cite{GreenJDG,Aprodu-NagelULECT}
\[
\mathop{\mathrm{min}}_q\{K_{p,q}(X,L)=0,\mbox{ for all }p\}.
\]
Perhaps the most striking property of Koszul cohomology,
discovered by Green and Lazarsfeld \cite[Appendix]{GreenJDG},
is a consequence of a nonvanishing result:
\begin{thm}[Green--Lazarsfeld]
\label{thm: GL nonvan}
Suppose $X$ is smooth and $L=L_1\otimes L_2$ with $r_i:=h^0(X,L_i)-1\ge 1$.
Then $K_{r_1+r_2-1,1}(X,L)\ne 0$.
\end{thm}
Note that the spaces $K_{p,1}$ have the following particular
attribute: if $K_{p,1}\ne 0$ for some $p\ge 1$ then
$K_{p',1}\ne 0$ for all $1\le p'\le p$. This is obviously
false for $K_{p,q}$ with $q\ge 2$.
Theorem~\ref{thm: GL nonvan} shows that the existence of nontrivial decompositions
of $L$ reflects onto the existence of nontrivial
Koszul classes in some space $K_{p,1}$.
Its most important applications are for curves, in
particular for canonical curves, case which is discussed in the next subsection.
In the higher-dimensional cases, for surfaces, for instance, the
meaning of Theorem~\ref{thm: GL nonvan} becomes more
transparent if it is accompanied by a restriction theorem which
compares the Koszul cohomology of $X$
with the Koszul cohomology of the linear sections
\cite{GreenJDG}:
\begin{thm}[Green]
\label{thm: Lefschetz}
Suppose $X$ is smooth and $h^1(X,L^q)=0$ for all $q\ge 1$.
Then for any connected reduced divisor $Y\in|L|$,
the restriction map induces an isomorphism\vspace*{4pt}
\[
K_{p,q}(X,L)\stackrel{\sim}{\to}K_{p,q}(Y,L|_Y),\vspace*{3pt}
\]
for all $p$ and $q$.
\end{thm}
The vanishing of $h^1(X,\mathcal{O}_X)$ suffices
to prove that the restriction is an isomorphism between
the spaces $K_{p,1}$ \cite{Aprodu-NagelULECT}.
In the next subsections, we shall apply Theorem~\ref{thm: Lefschetz} for $K3$ sections.
\begin{cor}
\label{cor: Lefschetz K3}
Let $C$ be a smooth connected curve on a $K3$
surface $S$. Then\vspace*{-4pt}
\[
K_{p,q}(S,\mathcal{O}_S(C))\cong
K_{p,q}(C,K_C)\vspace*{-4pt}
\]
for all $p$ and $q$.
\end{cor}
One direct consequence is a duality theorem for
Koszul cohomology of $K3$ surfaces.\footnote{Duality
for Koszul cohomology of curves follows from Serre's duality. For higher-dimensional
manifolds, some supplementary vanishing conditions are required \cite{GreenJDG,GreenICTP}.} It shows the
symmetry of the table containing the dimensions
of the spaces $K_{p,q}$, called \textit{the Betti
table}.
\vspace*{-4pt}
\subsection{Statement of Green's Conjecture}\label{ch1:sec3.2}
Let us particularize Theorem~\ref{thm: GL nonvan} for a
canonical curve. Consider $C$ a smooth curve and
choose a decomposition $K_C=A\otimes K_C(-A)$.
Theorem~\ref{thm: GL nonvan} applies only if
$h^0(C,A)\ge 2$ and $h^1(C,A)\ge 2$, i.e., if
$A$ contributes to the Clifford index.
The~quantity $r_1+r_2-1$ which appears in the statement
equals $g-\mathrm{Cliff}(A)-2$, and hence, if $A$
\textit{computes} the Clifford index, we obtain
the following:
\begin{thm}[Green--Lazarsfeld]
For any smooth curve $C$ of genus $g$ Clifford index $c$
we have $K_{g-c-2,1}(C,K_C)\ne 0$.
\end{thm}
It is natural to determine whether or not this result is sharp, question
which is addressed in the statement Green's conjecture:
\begin{con}[Green]
Let $C$ be a smooth curve. For all $p\ge g-c-1$,
we have $K_{p,1}(C,K_C) = 0$.
\end{con}
For the moment, Green's conjecture remains a hard open problem.
At the same time, strong evidence has been discovered.
For instance, it is known to hold for general curves \cite{VoisinJEMS,VoisinCOMP}, for curves of odd genus and maximal
Clifford index \cite{VoisinCOMP,Hirschowitz-RamananAENS},
for general curves of given gonality
\cite{VoisinJEMS,TeixidorDUKE},\footnote{Voisin's and Teixidor's cases complete each other quite remarkably.}
\cite{SchreyerLNM}, for curves with small Brill--Noether loci \cite{AproduMRL},
for plane curves \cite{LooseMANUSCRIPTA},
for curves on $K3$ surfaces \cite{VoisinJEMS,VoisinCOMP,Aprodu-FarkasCOMP},~etc.;
see also \cite{Aprodu-NagelULECT}
for a discussion.
We shall consider in the sequel the case of curves on
$K3$ surfaces with emphasis on Voisin's approach to the problem
and the role played by Lazarsfeld--Mukai bundles.
It is interesting to notice that Green's conjecture for $K3$ sections can be formulated directly
in the $K3$ setup, as a vanishing result on the moduli space
$\mathcal{F}_g$ of polarized $K3$ surfaces. However, in the proof of this statement,
as it usually happens in mathematics, we have to exit the $K3$ world, prove
a more general result in the extended setup, and return to $K3$ surfaces.
The steps we have to take, ordered logically and not chronologically, are
the following.
In the first, most elaborated step, one finds an example for
odd genus \cite{VoisinCOMP,VoisinJEMS}. At this stage,
we are placed in the moduli space $\mathcal{F}_{2k+1}$.
Secondly, we exit the $K3$ world, land in $\mathcal{M}_{2k+1}$,
and prove the equality of two divisors \cite{Hirschowitz-RamananAENS,VoisinJEMS}.
The first step is used, and the identification of~the divisors extends
to their closure over the component $\varDelta_0$
of the boundary \cite{AproduMRL}.
In~the third step, we jump from a gonality stratum
$\mathcal{M}^1_{g,d}$ in a moduli space $\mathcal{M}_g$
to the~boundary of another moduli space of stable curves $\overline{\mathcal{M}}_{2k+1}$,
where $k=g-d+1$ \cite{AproduMRL}. The second step reflects into a vanishing result on
an explicit open subset of $\mathcal{M}^1_{g,d}$. Finally one goes back to
$K3$ surfaces and applies
the latter vanishing result \cite{Aprodu-FarkasCOMP} on $\mathcal{F}_g$.
In the steps
concerned with $K3$ surfaces (first and last), the Lazarsfeld--Mukai bundles are central objects.
\subsection{Voisin's Approach}
\label{subsection: Hilbert}
The proof of the generic Green conjecture was achieved by Voisin
in two papers \cite{VoisinJEMS,VoisinCOMP}, using a completely different
approach to Koszul cohomology via Hilbert scheme of points.
Let $X$ be
a complex connected projective manifold and $L$ a line
bundle on $X$. It is obvious that any global
section $\sigma$ is uniquely determined by the
collection $\{\sigma(x)\}_x$, where $\sigma(x)\in L|_x\cong\mathbb{C}$
and $x$ belongs to a nonempty open subset of $X$.
One tries to find a similar fact for
multisections in $\wedge^nH^0(X,L)$.
Let $\sigma_1\wedge\cdots\wedge\sigma_n$ be
a decomposable element in $\wedge^nH^0(X,L)$ with $n\ge 1$.
By analogy with the
case $n=1$, we have to look at the restriction
$\sigma_1|_{\xi}\wedge\cdots\wedge\sigma_n|_{\xi}\in \wedge^nL|_{\xi}$ where
$\xi$ is now a zero-dimensional subscheme, and it is clear
that we need $n$ points for otherwise this restriction would be zero.
Note that a zero-dimensional subscheme of length $n$ defines
a point in the punctual Hilbert scheme $X^{[n]}$. For technical
reasons, we shall restrict to curvilinear subschemes\footnote{A curvilinear
subscheme is defined locally, in the classical topology, by $x_1=\cdots=x_{s-1}=x_s^k=0$;
equivalently, it is locally embedded in a smooth curve.}
which form a large open subset $X^{[n]}_c$ in a connected component
of the Hilbert scheme.\footnote{The connectedness of $X^{[n]}_c$
follows from the observation that a curvilinear subscheme is a deformation
of a reduced subscheme.}
Varying $\xi\in X^{[n]}_c$, the collection
$\{\sigma_1|_{\xi}\wedge\cdots\wedge\sigma_n|_{\xi}\}_{\xi}$
represents a section in a line bundle described as follows.
Put $\varXi_n\subset X^{[n]}_c\times X$ the incidence
variety and denote by $q$ and $p$ the projections on
the two factors; note that $q$ is finite of degree $n$.
Then $L^{[n]}:=q_*p^*(L)$ is a vector bundle of rank $n$
on $X^{[n]}_c$, and the fibre at a point $\xi\in X^{[n]}_c$
is $L^{[n]}|_{\xi}\cong L|_{\xi}$. In conclusion,
the collection $\{\sigma|_{\xi}\wedge\cdots\wedge\sigma|_{\xi}\}_{\xi}$
defines a section in the line bundle
$\mathrm{det}(L^{[n]})$.
The map we are looking at $\wedge^nH^0(L)\to H^0(\mathrm{det}(L^{[n]}))$
is deduced from the evaluation map
$\mathrm{ev}_n:H^0(L)\otimes \mathcal{O}_{X^{[n]}_c}\to L^{[n]}$,
taking $\wedge^n\mathrm{ev}_n$ and applying $H^0$.
It is remarkable that \cite{VoisinJEMS,VoisinCOMP,Ellingsrud-Goettsche-LehnJAG}:
\begin{thm}[Voisin, Ellingsrud--G\"ottsche--Lehn]
The map\vspace*{-3pt}
\[
H^0(\wedge^n\mathrm{ev}_n):\wedge^nH^0(X,L)\to
H^0\left(X^{[n]}_c,\mathrm{det}(L^{[n]})\right)\vspace*{-3pt}
\]
is an isomorphism.
\end{thm}
Since the exterior powers of $H^0(L)$ are building blocks
for Koszul cohomology, it is natural to believe that the
isomorphism above yields a relation between
the Koszul cohomology and the Hilbert scheme. To this
end, the Koszul differentials must be reinterpreted in
the new context.
There is a natural birational morphism\footnote{We see
one advantage of working on $X^{[n]}_c$: subtraction makes sense only for curvilinear subschemes.}\vspace*{-3pt}
\[
\tau:\varXi_{n+1}\to X^{[n]}_c\times X,\
(\xi,x)\mapsto(\xi-x,x)\vspace*{-3pt}
\]
presenting $\varXi_{n+1}$ as the blowup of $X^{[n]}_c\times X$ along $\varXi_n$.
If we denote by $D_{\tau}$ the exceptional locus, we obtain
an inclusion \cite{VoisinJEMS}
\[
q^*\mathrm{det}(L^{[n+1]})\cong\tau^*(\mathrm{det}(L^{[n]})\boxtimes
L)(-D_{\tau})\hookrightarrow \tau^*(\mathrm{det}(L^{[n]})\boxtimes
L)
\]
whence
\[
H^0\left(X^{[n+1]}_c,\mathrm{det}(L^{[n+1]})\right)\hookrightarrow
H^0(X^{[n]}_c\times X,\mathrm{det}(L^{[n]})\boxtimes L),
\]
identifying the left-hand member with the kernel of a Koszul
differential \cite{VoisinJEMS}. A~version of this identification
leads us to \cite{VoisinJEMS,VoisinCOMP}:
\begin{thm}[Voisin]
\label{thm: Hilbert scheme}
For any integers $m$ and $n$, $K_{n,m}(X,L)$ is isomorphic to the
cokernel of the restriction map:
\[
H^0\left(X^{[n+1]}_c\times X,\mathrm{det}(L^{[n+1]})\boxtimes L^{m-1}\right)\to
H^0\left(\varXi_{n+1},\mathrm{det}(L^{[n+1]})\boxtimes L^{m-1}|_{\varXi_{n+1}}\right).
\]
\end{thm}
The vanishing of Koszul cohomology is thus reduced to proving
surjectivity of the restriction map above. In general, it
is very hard to prove surjectivity directly, and one
has to make a suitable base-change \cite{VoisinJEMS}.
\subsection{The Role of Lazarsfeld--Mukai Bundles in the Generic
Green Conjecture and Consequences}
\label{subsection: Role of LM}
In order to prove Green's conjecture for general curves,
it suffices to exhibit one example of a curve of
maximal Clifford index, which verifies the predicted
vanishing. Afterwards, the vanishing of
Koszul cohomology propagates by semicontinuity.
Even so, finding one
single example is a task of major difficulty.
The curves used by Voisin in \cite{VoisinJEMS,VoisinCOMP}
are $K3$ sections, and the setups change slightly, according to the parity
of the genus. For even genus, we have \cite{VoisinJEMS}:
\begin{thm}[Voisin]
\label{thm: Voisin even}
Suppose that $g=2k$. Consider $S$ a $K3$ surface
with $\mathrm{Pic}(S)\cong \mathbb{Z}\cdot L$, $L^2=2g-2$, and
$C\in|L|$ a smooth curve. Then $K_{k,1}(C,K_C)=0$.
\end{thm}
For odd genus, the result is \cite{VoisinCOMP}:
\begin{thm}[Voisin]
\label{thm: Voisin odd}
Suppose that $g=2k+1$. Consider $S$ a $K3$ surface
with $\mathrm{Pic}(S)\cong \mathbb{Z}\cdot L\oplus \mathbb{Z}\cdot \varGamma$, $L^2=2g-2$,
$\varGamma$ a smooth rational curve. $L\cdot \varGamma =2$ and
$C\in|L|$ a smooth curve. Then $K_{k,1}(C,K_C)=0$.
\end{thm}
Note that the generic value for the Clifford index
in genus $g$ is $[(g-1)/2]$, and hence, in both cases, the prediction made by Green's
conjecture for general curve $C$ is precisely $K_{k,1}(C,K_C)=0$.
There are several reasons for making these choices:
the curves have maximal Clifford index, by
Theorem~\ref{thm: GL Cliff} (and the Clifford
dimension is one), the Lazarsfeld--Mukai
bundles associated to minimal pencils are $L$-stable,
the hyperplane section theorem applies, etc.
We outline here the role played by Lazarsfeld--Mukai bundles in
Voisin's proof and, for simplicity, we restrict to the even-genus case.
By the \hbox{hyperplane} section Theorem~\ref{thm: Lefschetz}, the required
vanishing on the curve is equivalent to \hbox{$K_{k,1}(S,L)=0$.}
From the description of Koszul cohomology in terms of Hilbert
schemes, Theorem \ref{thm: Hilbert scheme}, adapting the notation from the previous subsection, one has to prove the surjectivity of the map
\[
q^*: H^0\left(S_c^{[n+1]},\mathrm{det}(L^{[n+1]})\right)\to
H^0\left(\varXi_{n+1},q^*\mathrm{det}(L^{[n+1]})|_{\varXi_{n+1}}\right).
\]
The surjectivity is proved after performing a suitable base-change.
We are in the case $\rho(g,1,k+1)=0$; hence there is a unique
Lazarsfeld--Mukai bundle $E$ on $S$ associated to all $g^1_{k+1}$
on curves in $|L|$. The uniqueness yields an alternate description
of $E$ as extension
\[
0\to \mathcal{O}_S\to E\to L\otimes I_{\xi}\to 0,
\]
where $\xi$ varies in $S_c^{[k+1]}$.
There exists a morphism
$\mathbb{P} H^0(S,E)\rightarrow S^{[k+1]}$ that sends a global section $s\in H^0(S,E)$
to its zero set $Z(s)$. By restriction to an open subset
$\mathbb{P}\subset\mathbb{P}
H^0(S,E)$, we obtain a morphism $\mathbb{P}\rightarrow S_c^{[k+1]}$,
inducing a commutative diagram
\begin{center}
\mbox{\xymatrix{
\mathbb{P}^{\widetilde{\mathcal{R}}ime} = \mathbb{P}\times_{S_c^{[k+1]}}\varXi_{k+1}\ar[r]\ar[d]^{q^{\widetilde{\mathcal{R}}ime}} &\varXi_{k+1} \ar[d]^q \\
\mathbb{P} \ar[r] & S_c^{[k+1]}.
}}
\end{center}
Set-theoretically
\[
\mathbb{P}^{\widetilde{\mathcal{R}}ime} = \{(Z(s),x)|s\in H^0(S,E),x\in Z(s)\}.
\]
Unfortunately, this very natural base-change does not satisfy the
necessary conditions that imply the surjectivity
of $q^*$, \cite{VoisinJEMS}. Voisin modifies slightly this construction
and replaces $\mathbb{P}$ with another variety related to $\mathbb{P}$ which
parameterizes zero-cycles of the form
$Z(s)-x+y$
with $[s]\in\mathbb{P}$, $x\in\mathrm{Supp}(Z(s))$ and $y\in S$.
It turns out, after numerous elaborated calculations using the
rich geometric framework provided by the Lazarsfeld--Mukai
bundle, that the
new base-change is suitable and the surjectivity of $q^*$ follows
from vanishing results on the Grassmannian \cite{VoisinJEMS}.
In the odd-genus case, Voisin proves first Green's conjecture
for smooth curves in $|L+\varGamma|$, which are easily seen to
be of maximal Clifford index. The situation on $|L+\varGamma|$
is somewhat close to the setup of Theorem~\ref{thm: Voisin even},
and the proof is similar. The next hard part is to descend
from the vanishing of $K_{k+1,1}(S,L\otimes\mathcal{O}_S(\varGamma))$
to the vanishing of $K_{k,1}(S,L)$. This step uses again
intensively the unique Lazarsfeld--Mukai bundle associated
to any $g^1_{k+2}$ on curves in $|L+\varGamma|$.
The odd-genus case is of maximal interest:
mixed with Hirschowitz-Ramanan result
\cite{Hirschowitz-RamananAENS}, Theorem~\ref{thm: Voisin odd}
gives a solution to Green's conjecture
for \textit{any} curve of odd genus and maximal Clifford index:
\begin{thm}[Hirschowitz--Ramanan, Voisin]
\label{thm: HRV}
Let $C$ be a smooth curve of odd genus $2k+1\ge 5$
and Clifford index $k$. Then $K_{k,1}(C,K_C)=0$.
\end{thm}
Note that Theorem~\ref{thm: HRV} implies
the following statement:
\begin{cor}
A smooth curve of odd genus
and maximal Clifford index
has Clifford dimension one.
\end{cor}
The proof of Theorem~\ref{thm: HRV}
relies on the comparison of two effective
divisors on the moduli space of curves $\mathcal{M}_{2k+1}$,
one given by the condition $\mathrm{gon}(C)\le k+1$, which
is known to be a divisor from \cite{Harris-MumfordINVENTIONES},
and the second given by $K_{k,1}(C,K_C)\ne 0$.
By duality $K_{k,1}(C,K_C)\cong K_{k-2,2}(C,K_C)$.
Note that $K_{k-2,2}(C,K_C)$ is isomorphic~to\vspace*{-3pt}
\[
\mathrm{Coker}
\left(\wedge^kH^0(K_C)\otimes H^0(K_C)/\wedge^{k+1}H^0(K_C)
\to H^0(\wedge^{k-1}M_{K_C}\otimes K_C^2)\right)\vspace*{-3pt}
\]
and the two members have the same dimension. The locus of curves
with $K_{k,1}\ne 0$ can be described as the degeneracy locus
of a morphism between vector bundles of the same dimension, and hence it
is a virtual divisor. Theorem~\ref{thm: Voisin odd} implies that this locus
is not the whole space, and in conclusion it must be an
effective divisor. \hbox{Theorem \ref{thm: GL nonvan}}
already gives an inclusion between the supports of
two divisors in question, and the
set-theoretic equality is obtained from a divisor class
calculation~\cite{Hirschowitz-RamananAENS}.
\vspace*{-6pt}
\subsection{Green's Conjecture for Curves on $K3$ Surfaces}
\label{subsection: Green for K3}
We have already seen that general $K3$ sections have
a mild behavior from the Brill--Noether theory viewpoint. In some
sense, they behave like general curves in any
gonality stratum of the moduli space of curves.
As in the previous subsections, fix a $K3$ surface $S$ and a globally
generated line bundle $L$ with $L^2=2g-2$ on $S$, and denote by $k$ the
gonality of a general smooth curve in the linear system $|L|$.
Suppose that $\rho(g, 1, k)\le 0$ to exclude the case $g=2k-3$
(when $\rho(g, 1, k)=1$).
If in addition the curves in $|L|$ have Clifford dimension one,
Theorem~\ref{thm: Green Cliffdim 1}
shows that\vspace*{-3pt}
\[
\mathrm{dim}\ W^1_{g-k+2}(C)=\rho(g,1,g-k+2)=g-2k+2,\vspace*{-3pt}
\]
property which was called the \textit{linear growth condition}.
This property appears in connection with Green's conjecture~\cite{AproduMRL}
for a much larger class of curves:
\begin{thm}
\label{thm: Aprodu}
If $C$ is any smooth curve of genus $g\ge 6$ and gonality $3\le k<[g/2]+2$
with $\mathrm{dim}\ W^1_{g-k+2}(C)=\rho(g,1,g-k+2)$,
then $K_{g-k+1,1}(C,K_C)=0$.
\end{thm}
One effect of Theorems~\ref{thm: Aprodu} and~\ref{thm: GL nonvan} is that an arbitrary curve that satisfies the
linear growth condition is automatically of Clifford
dimension one and verifies Green's conjecture.
Theorem~\ref{thm: Aprodu} is a consequence of Theorem~\ref{thm: HRV} extended over the boundary of the moduli space.
Starting from a $k$-gonal smooth curve $[C]\in \mathcal{M}_g$, by identifying pairs of general points
$\{x_i,y_i\}\subset C$ for $i=0, \dots, g-2k+2$ we produce a stable
irreducible curve
\[
\left[X:=C/(x_0\sim y_0, \ldots, x_{g-2k+2}\sim y_{g-2k+2})\right]
\in \overline{\mathcal{M}}_{2(g-k+1)+1},
\]
and the Koszul cohomology of $C$ and of $X$
are related by the inclusion $K_{p,1}(C,K_C)$ $\subset K_{p,1}(X,\omega_X)$
for all $p\ge 1$, \cite{VoisinJEMS}.
If $C$ satisfies the linear growth condition then $X$
has maximal gonality\footnote{The gonality for
a singular stable curve is defined in terms of admissible
covers \cite{Harris-MumfordINVENTIONES}.} $\mathrm{gon}(X)=g-k+3$, i.e., $X$ lies outside the
closure of the divisor $\mathcal{M}_{2(g-k+1)+1, g-k+2}^1$
consisting of curves with a pencil
$g^1_{g-k+2}$. The class of the failure locus of
Green's conjecture on $\overline{\mathcal{M}}_{2(g-k+1)+1}$
is a multiple of the divisor $\overline{\mathcal{M}}_{2(g-k+1)+1, g-k+2}^1$;
hence Theorem~\ref{thm: HRV} extends to irreducible stable curves
of genus $2(g-k+1)+1$ of maximal gonality $(g-k+3)$.
In particular, $K_{g-k+1,1}(X,\omega_X)=0$, implying
$K_{g-k+1,1}(C,K_C)=0$.
Coming back to the original situation, we conclude from
Theorems~\ref{thm: Aprodu} and~\ref{thm: Green Cliffdim 1}
and Corollary~\ref{cor: Lefschetz K3}
that Green's conjecture holds for a $K3$ section $C$
having Clifford dimension one.
If $\mbox{Cliff}(C)=\mathrm{gon}(C)-3$,
either $C$ is a smooth plane curve or else there
exist smooth curves $D, \varGamma \subset S$, with
$\varGamma^2=-2, \varGamma\cdot D=1$ and $D^2\geq 2$, such that $C\equiv 2D+
\varGamma$ and $\mbox{Cliff}(C)=\mbox{Cliff}(\mathcal{O}_C(D))$
\cite{Ciliberto-PareschiCRELLE,KnutsenIJM}. The
linear growth condition is no longer satisfied,
and this case is treated differently, by
degeneration to a reduced curve
with two irreducible components \cite{Aprodu-FarkasCOMP}.
The outcome of this analysis of the Brill--Noether loci
is the following \cite{VoisinJEMS,VoisinCOMP,Aprodu-FarkasCOMP}:
\begin{thm}
\label{thm: Green on K3}
Green's conjecture is valid for any smooth curve on a $K3$ surface.
\end{thm}
Applying Theorem~\ref{thm: Green on K3},
Theorem~\ref{thm: Lefschetz}, and the duality,
we obtain a full description of the situations when Koszul
cohomology of a $K3$ surface is zero \cite{Aprodu-FarkasCOMP}:
\begin{thm}
\label{thm: K3}
Let $S$ be a $K3$ surface and $L$ a globally generated line bundle
with $L^2=2g-2\ge 2$. The Koszul cohomology group $K_{p,q}(S,L)$ is nonzero if and only if one of the following cases occurs:
\begin{itemize}
\item[(1)] $q=0$ and $p=0$, or
\item[(2)] $q=1$, $1\le p\le g-c-2$, or
\item[(3)] $q=2$ and $c\le p\le g-1$, or
\item[(4)] $q=3$ and $p=g-2$.
\end{itemize}
\end{thm}
The moral is that the shape of the Betti table, i.e., the distribution of
zeros in the table, of a polarized $K3$ surface is completely
determined by the geometry of hyperplane sections;
this is one of the many situations where algebra and geometry are intricately related.
\section{Counterexamples to Mercat's Conjecture in Rank Two}
\label{section: Higher BN}
Starting from Mukai's works, experts tried
to generalize the classical Brill--Noether
theory to higher-rank vector bundles on curves.
Within these extended theories,\footnote{Higher-rank
Brill--Noether theory is a major, rapidly growing research field,
and it deserves a separate dedicated survey.}
we note the attempt
to find a proper generalization of the Clifford index.
H. Lange and P. Newstead proposed the following
definition. Let $E$ be a semistable vector
bundle of rank $n$ of degree $d$ on a smooth curve $C$.
Put
\[
\gamma(E):=\mu(E)-2\frac{h^0(E)}{n}+2.
\]
\begin{defn}[Lange--Newstead]
The \textit{Clifford index} of rank $n$ of $C$ is
\[
\mathrm{Cliff}_n(C):=\mathrm{min}\{\gamma(E):\ \mu(E)\le g-1,\ h^0(E)\ge 2n\}.
\]
\end{defn}
From the definition, it is clear that $\mathrm{Cliff}_1(C)=\mathrm{Cliff}(C)$
and $\mathrm{Cliff}_n(C)\le\mathrm{Cliff}(C)$ for all $n$.\footnote{For
any line bundle $A$, we have $\gamma(A^{\oplus n})=\mathrm{Cliff}(A)$.}
Mercat conjectured \cite{MercatIJM} that $\mathrm{Cliff}_n(C)=\mathrm{Cliff}(C)$.
In rank two, the conjecture is known to hold in a number of cases: for general curves of
small gonality, i.e., corresponding to a general point in a gonality
stratum $\mathcal{M}^1_{g,k}$ for small $k$ (Lange-Newstead), for plane curves
(Lange--Newstead), for general curves of genus $\le 16$
(Farkas--Ortega), etc.
However, even in rank two, the conjecture is false. It is remarkable
that counterexamples are found for curves of maximal Clifford index~\cite{Farkas-OrtegaIJM}:
\begin{thm}[Farkas--Ortega]
Fix $p\ge 1$, $a\ge 2p+3$. Then there exists a smooth curve
of genus $2a+1$ of maximal Clifford index lying on a smooth $K3$
surface $S$ with $\mathrm{Pic}(S)=\mathbb{Z}\cdot C\oplus \mathbb{Z}\cdot H$,
$H^2=2p+2$, $C^2=2g-2$, $C\cdot H=2a+2p+1$, and there exists a
stable rank-two vector bundle $E$ with $\mathrm{det}(E)=\mathcal{O}_S(H)$
with $h^0(E)=p+3$, $\gamma(E)=a-\frac{1}{2}<a=\mathrm{Cliff}(A)$, and
hence Mercat's conjecture in rank two fails for~$C$.
\end{thm}
The proof uses restriction of Lazarsfeld--Mukai bundles.
However, it is interesting that the bundles are not
restricted to the same curves to which they
are associated. More precisely, the genus of
$H$ is $2p+2$ and $H$ has maximal gonality $p+2$.
Consider $A$ a minimal pencil on $H$, and take
$E=E_{H,A}$ the associated Lazarsfeld--Mukai bundle.
The restriction of $E$ to $C$ is stable and verifies
all the required properties.
A particularly interesting case is $g=11$.
In this case, as shown by Mukai \cite{MukaiLNPAM},
a general curve $C$ lies on a unique $K3$
surface $S$ such that $C$ generates $\mathrm{Pic}(S)$.\vadjust{\pagebreak}
It~is~remarkable that the failure locus of Mercat's conjecture in rank
two \textit{coincides} with the Noether-Lefschetz divisor
\[
\mathcal{NL}^4_{11,13}:=\left\{[C]\in \mathcal{M}_{11}:
\begin{array}{l}
C\mbox{ lies on a } K3 \mbox{ surface } S, \
\mathrm{Pic}(S)\supset \mathbb{Z}\cdot C\oplus \mathbb{Z}\cdot H,\\
H\in \mathrm{Pic}(S) \mbox{ is nef},
H^2=6, \ C\cdot H=13, \ C^2=20 \end{array}
\right\}
\]
inside the moduli space $\mathcal{M}_{11}$.
We refer to \cite{Farkas-OrtegaIJM} for details.
\end{document} |
\mathfrak{b}egin{document}
\mathfrak{b}egin{abstract}
We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(\frac 12,f\times g)$ twisted by $\lambda_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$.
The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
\end{abstract}
\maketitle
\section{Introduction}
Let $p,q$ be distinct primes.
Fix a holomorphic cusp form $g$ of weight $\kappa$ on $\mathcal SL_2(\mathbb{Z})$.
In this paper we establish a reciprocity relation between the twisted second moment of the central values of Rankin-Selberg $L$-functions:
\[
\sum_{f \text{ level } q} \omega_f \lambda_f(p) L(\tfrac{1}{2},f\times g)^2 \leadsto \sum_{f \text{ level } p} \omega_f \lambda_f(q) L(\tfrac{1}{2}, f\times g)^2.
\]
Here $\omega_f = \pm 1$ is the eigenvalue of $f$ under the Fricke involution and $\lambda_f(p)$ is the $p$\textsuperscript{th} Hecke eigenvalue of $f$. The sum on each side should be understood as a complete sum/integral over the full spectrum of level $q$ or $p$ modular forms, including the holomorphic, discrete, and continuous spectra. The exact formulation is given below in Theorem \ref{thm:MainTheorem}.
Our work is motivated by the case when $g$ is an Eisenstein series and the sum is over Hecke cusp forms $f$ of a given weight. In that case preliminary calculations with the Petersson trace formula and transforms on sums of Kloosterman sums lead to a formula of rough shape
\mathfrak{b}egin{equation}\label{eq:twisted-fourth-moment}
\sum_{f \text{ level } q} \lambda_f(p) L(\tfrac12,f)^4 \leadsto \sum_{f \text{ level } p} \lambda_f(q) L(\tfrac12,f)^4.
\end{equation}
Note that when $f$ is a holomorphic modular form, $\omega_f = -1$ implies $L(\frac 12,f) = 0$, and therefore the identity \eqref{eq:twisted-fourth-moment} does not include $\omega_f$.
One may use the amplification method in conjunction with such an identity to obtain a subconvexity result for $L(\frac12, f)$ in the level aspect.
We were led to consider such a reciprocity relation after the works of Conrey~\mathfrak{c}ite{conrey2007mean}, Young~\mathfrak{c}ite{YoungReciprocity}, and Bettin~\mathfrak{c}ite{BettinReciprocity},
who discovered and elaborated upon an identity relating $\mathcal M(a,q)$ to $\mathcal M(-q,a)$, where $\mathcal M(a,q)$ is the second moment of Dirichlet $L$-functions $L(\frac 12,\mathfrak{c}hi)$ modulo $q$, twisted by $\mathfrak{c}hi(a)$.
Our method is structurally similar to Motohashi's proof of a beautiful formula discovered by Kuznetsov and then fully proven in \mathfrak{c}ite{MotohashiFourthMomentFE}.
Similar to Motohashi we apply the Bruggeman/Kuznetsov trace formula, followed by the
${\rm GL}_2$-Voronoi fomula twice, giving us again a sum over Kloosterman sums.
We then apply the Bruggeman/Kuznetsov trace formula again in order to obtain the reciprocal moment.
Our work is distinct in at least three ways from that of Motohashi.
First, we are working in the congruence subgroup $\Gamma_0(q)$ and twisting by the Fourier coefficient $\lambda_f(p)$.
This allows us to see the reciprocity relation exchanging the level and the twist parameters.
Second, we twist our moments further by $\omega_f$, so that when we apply the Bruggeman/Kuznetsov trace formula we are working with the cusp-pair $0\infty$.
The Kloosterman sums associated to the cusp-pair $0\infty$ feature $p$ and $q$ in more symmetric roles, and it becomes conceptually clear how the reciprocity occurs (see Theorem~\ref{thm:S-reciprocity} below).
As can be seen in \mathfrak{c}ite{KiralYoungFifthMoment} and \mathfrak{c}ite{BlomerKhanReciprocity} the trick of moving to the $0\infty$ cusp-pair may be avoided in the fourth moment case by inserting an arithmetic reciprocity relation between the Voronoi formulas, but this trick does not work in the Rankin-Selberg second moment case (more on \mathfrak{c}ite{BlomerKhanReciprocity} below).
Third, our formula relates the twisted Rankin-Selberg second moment rather than the fourth moment.
In principle, we could obtain the fourth moment if $g$ were chosen as an Eisenstein series.
Practically, this corresponds to replacing every instance of $\lambda_g(m)$ in this paper by $\tau_w(m) = \sum_{ab = n}\left(\tfrac{a}{b}\right)^w$, which introduces {main terms} at various points.
Motohashi has produced other beautiful formulas relating different moments of $L$-functions. For example, in \mathfrak{c}ite{MotohashiFourthMomentZetaThirdMomentMaass} he gives an exact identity relating the weighted fourth moment of the Riemann zeta function on the critical line to third moments of central values of Maass forms of level $1$.
Later with Ivic \mathfrak{c}ite{IvicMotohashiFourthMomentZetaError} they use this exact formula to give an asymptotic for the fourth moment of the Riemann zeta function with an error term of size $O(T^{2/3}(\log T)^c)$.
Several authors have discovered and applied identites between moments of $L$-functions either exact or approximate; see \mathfrak{c}ite{PetrowTwistedMotohashi,YoungFourthMoment} and the references therein.
Recently Blomer, Li and Miller \mathfrak{c}ite{BLMSpectralReciprocity} announced an identity involving the first moment of $L(\frac 12,\mathbb{P}i \times u_j)$ where $\mathbb{P}i$ is a self-dual cusp form on ${\rm GL}_4$ and $u_j$ runs over ${\rm GL}_2$ Maass forms.
Notice that Motohashi's formula on the fourth moment could be interpreted as the case where $\mathbb{P}i$ is a $4 = 1 + 1 + 1 + 1$ isobaric sum.
During the preparation of this manuscript, Blomer and Khan posted their preprint \mathfrak{c}ite{BlomerKhanReciprocity}, in which they addressed the twisted fourth moment problem \eqref{eq:twisted-fourth-moment} and realized independently that one obtains a kind of reciprocity relation exchanging $p$ and $q$.
We assume $p$ and $q$ to be prime for simplicity; {additionally, we include the Fricke eigenvalue in the moment, which allows us make use of arithmetic features coming from the $0\infty$ cusp-pair Kloosterman sums in the Kuznetsov formulas.}
As an application, Blomer and Khan sum over the twist variable and reconstruct the subconvexity-implying fifth moment bound in \mathfrak{c}ite{KiralYoungFifthMoment}.
They start by considering the moment $\sum_{\pi \text{ level } q} L(1/2,F \times \pi) L(1/2,\pi) \lambda_\pi(\ell)$ where $F$ is a ${\rm GL}_3$ automorphic form. This is the $4 = 3 + 1$ decomposition {in the framework of \mathfrak{c}ite{BLMSpectralReciprocity}}.
When $F$ is an Eisenstein series one obtains the twisted fourth moment.
In the framework above, our result corresponds to the $4 = 2 + 2$ setup.
This difference is the reason why we use the ${\rm GL}_2$ instead of the ${\rm GL}_3$ Voronoi summation formula.
\section{Statement of Results} \label{sec:preliminaries}
We begin by fixing notation, which we mostly borrow from \mathfrak{c}ite{IwaniecSpectralBook}.
Let $\Gamma = \Gamma_0(N)$ for some squarefree integer $N$ and let $k\geq 0$ be an even integer.
The weight $k$ Petersson inner product is defined as
\[
\langle h_1,h_2\rangle = \iint_{\Gamma\mathfrak{b}ackslash \mathcal{H}} h_1(z) \overline{h_2(z)} y^k \, \frac{dx dy}{y^2}.
\]
Here $z=x+iy$ and $h_1,h_2$ are holomorphic cusp forms of weight $k$ or Maass cusp forms (in the latter casse $k=0$).
Let $S_k(N)$ denote the space of holomorphic cusp forms of weight $k$ on $\Gamma_0(N)$, and let $\mathcal B_k(N)$ denote an orthonormal basis of $S_k(N)$.
We will always use $f$ or $g$ to denote an element of $\mathcal B_k(N)$.
The Fourier expansion of such an $f$ at a cusp $\mathfrak{a}$ of $\Gamma$ is given by
\[
j(\sigma_\mathfrak{a},z)^{-k} f(\sigma_\mathfrak{a} z) = \sum_{n=1}^\infty \rho_{\mathfrak{a} f}(n) e(nz),
\]
where $j(\pmatrix abcd, z) = cz+d$ and $\sigma_\mathfrak{a}$ is a scaling matrix (see Section~\ref{sec:kloo-b/k}).
When $\mathfrak{a}=\infty$ we will often drop the dependence on $\mathfrak{a}$ from the notation.
We adopt the standard notation $e(x):= e^{2\pi ix}$.
We normalize the coefficients $\rho_{\mathfrak{a} f}(n)$ by setting
\[
\nu_{\mathfrak{a} f}(n) = \left(\frac{\pi^{-k}\Gamma(k) }{(4n)^{k-1}}\right)^{\frac12} \rho_{\mathfrak{a} f}(n).
\]
Without loss of generality we may assume that the elements of $\mathcal B_k(N)$ are eigenforms of the Hecke operators $T_n$ for $(n,N)=1$ with eigenvalues $\lambda_f(n)$, and that they satisfy
\mathfrak{b}egin{equation} \label{eq:fricke}
N^{-\frac k2} z^{-k} f(-1/Nz) = \omega_f f(z)
\end{equation}
where $\omega_f=\pm 1$ is the eigenvalue of the Fricke involution.
Since the Fricke involution swaps the cusps $\infty$ and $0$, we have the relation
\mathfrak{b}egin{equation}
\rho_{0 f}(n) = \omega_f \, \rho_{\infty f}(n).
\end{equation}
Similarly, let $\mathcal U(N) = \{u_j\}$ denote a complete orthonormal system of Maass cusp forms with Fourier expansions
\mathfrak{b}egin{equation}
u_j(\sigma_\mathfrak{a} z) = \sqrt y \sum_{n \neq 0} \rho_{\mathfrak{a} j}(n) K_{i t_j}(2\pi |n| y) e(nx),
\end{equation}
where $\frac 14+it_j$ is the Laplace eigenvalue and $K_\nu(x)$ is the $K$-Bessel function.
We normalize the coefficients $\rho_{\mathfrak{a} j}(n)$ by
\mathfrak{b}egin{equation}
\nu_{\mathfrak{a} j}(n) = \left(\frac{\pi}{\mathfrak{c}osh(\pi t_j)}\right)^{\frac12} \rho_{\mathfrak{a} j}(n).
\end{equation}
We may also assume, as above, that the $u_j$ are eigenforms for the Hecke operators $T_n$ for $(n,N)=1$ and for the Fricke involution (i.e. that $u_j$ satisfies \eqref{eq:fricke} with $k=0$).
We write their eigenvalues $\lambda_j(n)$ and $\omega_j$, respectively.
Let $(\nu,\lambda)$ denote one of the pairs $(\nu_{f},\lambda_f)$ or $(\nu_{j},\lambda_j)$.
As long as $(n,N) = 1$ we have the relation
\mathfrak{b}egin{equation}\label{eq:hecke}
\nu(m)\lambda(n) = \sum_{d | (m,n) } \nu\left(\frac{mn}{d^2}\right)
\end{equation}
see \mathfrak{c}ite[(8.37)]{IwaniecSpectralBook}.
This implies implies that
\[
\nu(n) = \nu(1)\lambda(n)
\]
as long as $(n,N) = 1$.
Both sides are zero when $(\nu,\lambda)$ does not correspond to a newform.
For each cusp $\mathfrak{c}$ of $\Gamma$ (see Section~\ref{sec:kloo-b/k}) and for $\re(u)>1$ let
\[
E_\mathfrak{c}(z,u) := \sum_{\gamma \in \Gamma_\infty\mathfrak{b}ackslash \Gamma} \im(\sigma_\mathfrak{c}^{-1} \gamma z)^u
\]
denote the Eisenstein series associated to $\mathfrak{c}$.
This has Fourier expansion
\mathfrak{b}egin{equation}\label{eq:EisensteinFourierExpansion}
E_\mathfrak{c}(\sigma_\mathfrak{a} z,u)
= \delta_{\mathfrak{a}\mathfrak{c}} y^{u} + \rho_{\mathfrak{a}\mathfrak{c}}(0,u) y^{1 - u}
+ \sqrt y \sum_{n \neq 0} \rho_{\mathfrak{a}\mathfrak{c}}(n,u) K_{u-\frac 12}(2\pi |n|y) e(nx)
\end{equation}
which has a meromorphic continuation to $u\in \mathbb{C}$.
On the line $\re(u) = \frac 12$ we normalize the coefficients by
\mathfrak{b}egin{equation}
\nu_{\mathfrak{a}\mathfrak{c}}(n,t) = \left(\frac{\pi}{\mathfrak{c}osh \pi t}\right)^{\frac12} \rho_{\mathfrak{a}\mathfrak{c}} (n,\tfrac 12+it).
\end{equation}
For the remainder of the paper, fix a
holomorphic newform $g\in S_\kappa(1)$.
For $s=\sigma+it$ with $\sigma$ sufficiently large,
let us call
\mathfrak{b}egin{equation}\label{eq:RawLFunction}
\widetildede L(s, h \times g) = \zeta_N(2s) \sum_{n=1}^\infty \frac{\lambda_g(n)\nu_h(n)}{n^s},
\end{equation}
where $\zeta_N(s) = \prod_{p \nmid N} (1 - p^{-s})^{-1}$.
This is the ``raw $L$-function'' involving {$\nu_h$ as opposed to $\lambda_h$}; it has an analytic continuation and a functional equation.
We choose this notation {in order to simultaneously cover oldforms.}
The coefficients $\nu_h(n)$ come up in applications of the Bruggeman/Kuznetsov trace formula and hence the Dirichlet series $\widetildede L(s, g \times h)$ naturally appears.
If $h$ is a newform (holomorphic or Maass), then {we} simply {have} $\widetildede L(s,h) = \nu_h(1) L(s,h)$, where $L(s,h)$ is the usual $L$-function of $h$.
Let $\varphi$ be a smooth function defined on the nonnegative reals such that
\mathfrak{b}egin{equation}\label{eq:phiKuznetsovdGrowth}
\varphi(0) = 0 \quad \text{ and } \quad \varphi^{(j)}(x) \ll (1+x)^{-2-\vepsilon} \quad \text{ for }j=0,1,2,
\end{equation}
and let $\varphi_h$ and $\varphi_+$ denote the integral transforms in \eqref{eq:KuznetsovTransformDefinitions} below.
Define
\mathfrak{b}egin{equation}\label{eq:Ngdisc}
\mathcal N_g^{d}(p,q;s;\varphi) = \sum_{u_j \in \mathcal U(q)} \omega_j \, \varphi_+(t_j) \widetildede L(s, g \times u_j)^2\lambda_j(p),
\end{equation}
\mathfrak{b}egin{equation}\label{eq:Nghol}
\mathcal N_g^{h}(p,q;s;\varphi) = \sum_{\substack{\ell \text{ even}}} i^{\ell} \varphi_h(\ell) \sum_{f \in \mathcal B_\ell(q)} \omega_f \, \widetildede L(s,g \times f)^2 \lambda_f(p),
\end{equation}
and
\mathfrak{b}egin{equation}\label{eq:Ngcts}
\mathcal N_g^{c}(p,q;s;\varphi) = \frac{1}{4\pi}\sum_{\mathfrak{c}} \int_{-\infty}^\infty \varphi_+(t) \widetildede L\left(s, g\times E_\mathfrak{c}(\sigma_0*, \tfrac12 + it)\right) \widetildede L\left(s, g\times E_\mathfrak{c}(*,\tfrac12 - it)\right) \tau_{it}(p) \, dt,
\end{equation}
where $\sum_\mathfrak{c}$ is over a set of inequivalent cusps of $\Gamma$.
Call
\mathfrak{b}egin{equation}\label{eq:NgDefinition}
\mathcal N_g(p,q;s;\varphi) = \mathcal N_g^d(p,q;s;\varphi) + \mathcal N_g^c(p,q;s;\varphi) + \mathcal N_g^h(p,q;s;\varphi).
\end{equation}
With $\varphi$ as above, the function $\mathcal N_g(p,q;s;\varphi)$ is holomorphic at $s=\frac 12$, which is the point we are most interested in, and we let $\mathcal N_g(p,q;\varphi) := \mathcal N_g(p,q;\frac 12;\varphi)$.
We are now ready to state our main theorem.
\mathfrak{b}egin{theorem}\label{thm:MainTheorem}
Let $p$ and $q$ be distinct primes and let $\phi$ be a function on $[0,\infty)$ satisfying the conditions in the beginning of Theorem \ref{thm:S-reciprocity}.
With the notation above, define
\mathfrak{b}egin{equation}
\mathcal M_g(p,q;\varphi) := (1-p^{-2}) \, \mathcal N_g(p,q;\varphi) - \frac{2\lambda_g(p)}{\sqrt p}(1-p^{-1}) \, \mathcal N_g(1,q;\varphi) + \frac{1}{\sqrt p} \, \mathcal N_g(1,pq;\varphi).
\end{equation}
Then
\mathfrak{b}egin{equation}
\sqrt q \, \mathcal M_g(p,q;\phi) = \sqrt p \, \mathcal M_g(q,p;\mathbb{P}hi),
\end{equation}
where $\mathbb{P}hi$ is an integral transform of $\phi$ given in \eqref{eq:Phi-def}.
\end{theorem}
The various steps of the proof are detailed in the remaining sections, but we give a high-level outline here.
\mathfrak{b}egin{proof}[Proof of Theorem~\ref{thm:MainTheorem}]
Using equation \eqref{eq:Ng-S} { with $\mathcal{S}$ defined as in \eqref{eq:S-def}} below we have that
\[
\mathcal{S}(p,q;s;\phi) = \frac{1- p^{-4s}}{\zeta_{pq}(2s)^2} \mathcal N_g(p,q;s;\phi) - \frac{2\lambda_g(p)}{p^s}\frac{\left(1 - p^{-2s}\right)}{\zeta_{pq}(2s)^2}\mathcal N_g(1,q;s;\phi) + \frac{\mathcal N_g(1,pq;s;\phi)}{\sqrt{p}\zeta_{pq}(2s)^2}.
\]
Specialization of $\zeta_{pq}(2s)^2 \mathcal S(p,q;s;\phi)$ at $s = \tfrac12$ yields $\mathcal{M}_g(p,q;\phi)$. Now we apply the reciprocity relation for $\mathcal S$ in Theorem \ref{thm:S-reciprocity} below, which gives
\[
\zeta_{pq}(2s)^2\mathcal S(p,q;s,\phi) = \zeta_{pq}(2s)^2 \left(\frac pq\right)^{2s - 1} \frac{\sqrt{p}}{\sqrt{q}} \mathcal S(q,p;s;\mathbb{P}hi)
\]
with $\mathbb{P}hi$ as in \eqref{eq:Phi-def}. Specializing to $s = \tfrac12$ yields the result.
\end{proof}
\section{The Bruggeman/Kuznetsov trace formula} \label{sec:kloo-b/k}
The purpose of this section is to state the Bruggeman/Kuznetsov trace formula associated to the $0\infty$ cusp pair, which will allow us to relate the moments in Theorem~\ref{thm:MainTheorem} to sums of Kloosterman sums.
Let $\Gamma=\Gamma_0(N)$ and let $\mathfrak{a},\mathfrak{b} \in \mathbb{Q} \mathfrak{c}up \{\infty\}$ denote two cusps.
A scaling matrix $\sigma_\mathfrak{a} \in \mathcal SL_2(\mathbb{R})$ for the cusp $\mathfrak{a}$ satisfies the properties
\mathfrak{b}egin{equation} \label{eq:ScalingMatrixProperties}
\sigma_\mathfrak{a} \infty = \mathfrak{a}, \quad \text{ and } \quad \sigma_\mathfrak{a}^{-1} \Gamma_\mathfrak{a} \sigma_\mathfrak{a} = \left\{ \pm \pmatrix 1n01 : n \in \mathbb{Z}\right\}
\end{equation}
where $\Gamma_\mathfrak{a}$ is the stabilizer of $\mathfrak{a}$ in $\Gamma$.
We are primarily interested in the cases $\mathfrak{a}=\infty$ and $\mathfrak{a}=0$, for which we have
\mathfrak{b}egin{equation}
\sigma_\infty = \pMatrix 1001 \quad \text{ and } \quad \sigma_0 = \pMatrix 0{-1/\sqrt{N}}{\sqrt N}0.
\end{equation}
Given a pair of cusps $\mathfrak{a}, \mathfrak{b}$ and associated scaling matrices $\sigma_\mathfrak{a}, \sigma_\mathfrak{b}$
a set of allowed moduli for the Kloosterman sums is given by
\[
\mathcal{C}_{\mathfrak{a},\mathfrak{b}} = \{\gamma >0: \left(\mathfrak{b}egin{smallmatrix} *&*\\\gamma&*\end{smallmatrix}\right) \in \sigma_\mathfrak{a}^{-1} \Gamma\sigma_\mathfrak{b}\}.
\]
The Kloosterman sum for a modulus $\gamma \in \mathcal{C}_{\mathfrak{a},\mathfrak{b}}$ is defined as
\mathfrak{b}egin{equation} \label{eq:Kloo-cusps}
S^\Gamma_{\mathfrak{a}\mathfrak{b}}(m,n;\gamma) = \sum_{\pmatrix ab\gamma d \in \Gamma_\infty \mathfrak{b}ackslash \sigma_\mathfrak{a}^{-1} \Gamma\sigma_\mathfrak{b}/\Gamma_\infty} e\left(\frac{am + dn}{\gamma}\right).
\end{equation}
When $\mathfrak{a}=\mathfrak{b}=\infty$ we obtain the usual Kloosterman sum (for $c\equiv 0\pmod N$)
\mathfrak{b}egin{equation}
S^\Gamma_{\infty\infty}(m,n;c) = S(m,n;c) := \sum_{\substack{d\mathfrak{b}mod c \\ (d,c)=1}} e\pfrac{\mathfrak{b}ar dm+d n}{c}.
\end{equation}
For these sums we have Weil's bound
\mathfrak{b}egin{equation}\label{eq:WeilBound}
|S(m, n;c)| \leq \tau_0(c) (m,n,c)^{\frac 12} c^{\frac12}
\end{equation}
where $\tau_0(c)$ is the number of divisors of $c$.
For the $\infty0$ cusp-pair, and for the choices above, the set of allowed moduli is $\mathcal C_{\infty,0}=\{\gamma = c\sqrt{N} : (c,N)=1\}$.
Then \eqref{eq:Kloo-cusps} can be expressed in terms of $S(m,n;c)$ via
\[
S_{\infty0}^{\Gamma}(m,n;c\sqrt{N}) = S(\mathfrak{b}ar N m, n;c).
\]
Kloosterman sums appear in the Fourier expansion of the Eisenstein series.
For $n\neq 0$ and $\re (u)>1$ we have (see \mathfrak{c}ite[Theorem~3.4]{IwaniecSpectralBook})
\mathfrak{b}egin{equation}
\rho_{\mathfrak{a}\mathfrak{c}}(n,u) = \frac{2\pi^u}{\Gamma(u)} |n|^{u-\frac 12} \sum_{c\in \mathcal C_{\mathfrak{c},\mathfrak{a}}} \frac{S_{\mathfrak{c}\mathfrak{a}}(0,n;c)}{c^{2u}}.
\end{equation}
The Kloosterman sum $S(0,n;c)$ is equal to the Ramanujan sum $\sum_{d\mid (n,c)} \mu(c/d) d$ and is multiplicative as a function of $c$.
It follows that, for $q$ prime,
\mathfrak{b}egin{equation}
\rho_{\infty 0}(n,u) = \frac{2\pi^u\tau^{(q)}_{u-\frac 12}(n)}{\Gamma(u)\zeta_q(2u)} \quad \text{ and } \quad \rho_{\infty\infty}(n,u) = \frac{2\pi^u}{\Gamma(u)} \left( \frac{\tau_{u-\frac 12}(n)}{\zeta(2u)} - \frac{\tau^{(q)}_{u-\frac 12}(n)}{\zeta_q(2u)} \right),
\end{equation}
where $\tau_w(n) = \sum_{ab=|n|}(a/b)^w$ and $\tau_w^{(q)}(n) = \tau_w(n/(n,q))$ (cf. \mathfrak{c}ite[(2.27)]{IwaniecSpectralBook}).
From this we can derive a Hecke relation for the pair $(\nu_{\mathfrak{a}\mathfrak{c}}(n,t),\tau_{it}(p))$, where $p \neq q$ is a prime.
{Suppose that $(n,q)=1$ and that $(\mathfrak{a},\mathfrak{c})=(\infty,\infty)$ or $(\infty,0)$. Then since}
\mathfrak{b}egin{equation}
\tau_w^{(q)}(m) \tau_w^{(q)}(n) = \sum_{d\mid (m,n)} \tau_w^{(q)} \pfrac{mn}{d^2}
\end{equation}
{we have} the relation
\mathfrak{b}egin{equation}
\nu_{\mathfrak{a}\mathfrak{c}}(n,t) \tau_{it}(p) = \nu_{\mathfrak{a}\mathfrak{c}}(np, t) + \delta_{p\mid n} \nu_{\mathfrak{a}\mathfrak{c}}(n/p,t).
\end{equation}
The Bruggeman/Kuznetsov trace formula relates sums of Kloosterman sums to coefficients of cusp forms and Eisenstein series.
Let $\varphi$ be a smooth function defined on the nonnegative reals satisfying
\mathfrak{b}egin{equation}\label{eq:phiKuznetsovdGrowth}
\varphi(0) = 0 \quad \text{ and } \quad \varphi^{(j)}(x) \ll (1+x)^{-2-\vepsilon} \quad \text{ for }j=0,1,2,
\end{equation}
and define
\mathfrak{b}egin{align} \label{eq:KuznetsovTransformDefinitions}
\varphi_h(\ell) &= \int_0^\infty J_{\ell-1} (x) \varphi(x) \, \frac{dx}{x}, \notag \\
\varphi_+(t) &= \frac{i}{2\sinh\pi t}\int_{0}^{\infty} \left( J_{2it}(x) - J_{-2it}(x) \right) \varphi(x) \, \frac{dx}{x}.
\end{align}
The following can be found in \mathfrak{c}ite[Theorem~9.8]{IwaniecSpectralBook}.
\mathfrak{b}egin{proposition}[Bruggeman/Kuznetsov] \label{prop:0inftyKuznetsov}
Suppose that $m,n\geq 1$.
Let $\varphi$ be a smooth function defined on $[0,\infty)$ which satisfies \eqref{eq:phiKuznetsovdGrowth}.
Define
\[
\mathcal K(m,n,N;\varphi) = \sum_{(c,N) = 1} \frac{S(\overline{N}m,n;c)}{c\sqrt{N}} \varphi\left(\frac{4\pi \sqrt{mn}}{c\sqrt{N}}\right).
\]
Then
\[
\mathcal K = \mathcal K^d + \mathcal K^c + \mathcal K^h,
\]
where
\mathfrak{b}egin{align}
\mathcal K^d(m,n,N;\varphi) &= \sum_{u_j \in \mathcal U(N)} \varphi_+(t_j) \overline{\nu_{0j}}(m) \nu_{\infty j}(n), \\
\mathcal K^c (m,n,N;\varphi) &= \sums_{\mathfrak c} \frac{1}{4\pi} \int_{-\infty}^\infty \varphi_+(t) \overline{\nu_{0\mathfrak c}}(m,t) \nu_{\infty \mathfrak c}(n,t) \, dt, \\
\mathcal K^h(m,n,N;\varphi) &= \sum_{\ell \equiv 0(2)} i^\ell \varphi_h(\ell) \sum_{f\in \mathcal B_\ell(N)} \overline{\nu_{0f}}(m) \nu_{\infty f}(n).
\end{align}
\end{proposition}
\section{Calculations with the Hecke relations and Kloosterman sums}
Let $\phi$ be as in Theorem~\ref{thm:MainTheorem}.
The purpose of this section is to prove the identity
\mathfrak{b}egin{equation}\label{eq:Ng-S}
\mathcal N_g(p,q;s;\phi) = \frac{2\lambda_g(p)}{p^s(1 + p^{-2s})}
\mathcal N_g(1,q;s;\phi) - \frac{1}{\sqrt{p}} \frac{\mathcal N_g(1,pq;s;\phi)}{\left(1 - p^{-4s}\right)} + \zeta_{pq}(2s)^2 \frac{ \mathcal S(p,q;s,\phi)}{\left(1 - p^{-4s}\right) },
\end{equation}
where $\mathcal S$ is defined in \eqref{eq:S-def} below.
The following section shows that $\mathcal S$ is the quantity that satisfies a natural reciprocity relation.
We begin with explicit computations in the case of the holomorphic spectrum $\mathcal N_g^h(p,q;s;\phi)$.
Suppose that $\sigma>1$.
Opening the $L$-functions {as Dirichlet series} we find that
\mathfrak{b}egin{equation} \label{eq:N-g-h-1}
\mathcal N_g^h(p,q;s;\phi) = \zeta_q^2(2s) \sum_{m\geq 1} \frac{\lambda_g(m)}{m^s} \sum_{\ell \equiv 0(2)} i^\ell \phi_h(\ell) \sum_{f\in \mathcal{H}_\ell(q)} \omega_f \nu_f(m) \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu_f(n) \lambda_f(p).
\end{equation}
We apply the Hecke relations \eqref{eq:hecke} in order to absorb the $\lambda_f(p)$ factor.
\mathfrak{b}egin{lemma}\label{lem:Lsfg-moment-in-Dirichlet-2}
Let $p$ be a prime number and let $g\in S_\kappa(1)$.
Suppose that $(\nu,\lambda)$ is a pair of arithmetic functions satisfying the relation
\mathfrak{b}egin{equation} \label{eq:nu-hecke}
\nu(n)\lambda(p) = \nu(np) + \delta_{p\mid n} \nu(n/p)
\end{equation}
and the bound $\nu(n) \ll n^{\mathfrak{a}lpha}$ for some $\mathfrak{a}lpha>0$.
Then for $\sigma>1+\mathfrak{a}lpha$ we have
\mathfrak{b}egin{equation} \label{eq:Mg-S1}
\sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(n)\lambda(p)
= \frac{\lambda_g(p)}{p^s} \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(n) + \left(1-p^{-2s}\right) \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(np).
\end{equation}
\end{lemma}
\mathfrak{b}egin{proof}
Using the relation \eqref{eq:nu-hecke}
we find that
\mathfrak{b}egin{equation}
\sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(n)\lambda(p) = \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(np) + \sum_{n\geq 1} \frac{\lambda_g(np)}{(np)^s} \nu(n).
\end{equation}
Since $\lambda_g$ also satisfies the Hecke relations \eqref{eq:nu-hecke}
we see that
\mathfrak{b}egin{equation}
\sum_{n\geq 1} \frac{\lambda_g(np)}{(np)^s} \nu(n) = \frac{\lambda_g(p)}{p^s} \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(n) - \frac{1}{p^{2s}} \sum_{n\geq 1} \frac{\lambda_g(n)}{n^s} \nu(np).
\end{equation}
The lemma follows.
\end{proof}
Applying Lemma~\ref{lem:Lsfg-moment-in-Dirichlet-2} to \eqref{eq:N-g-h-1}, we find that
\mathfrak{b}egin{equation} \label{eq:N-g-h-2}
\mathcal N_g^h(p,q;s;\phi) = \frac{\lambda_g(p)}{p^s} \mathcal N_g^h(1,q;s;\phi) + (1-p^{-2s}) \zeta_q(2s)^2 \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \mathcal K^h(m,pn,q;\phi).
\end{equation}
A similar computation confirms that \eqref{eq:N-g-h-2} holds for the discrete and continuous spectra, simply replacing $h$ by $d$ or $c$ above.
After applying Proposition~\ref{prop:0inftyKuznetsov}, it follows that
\mathfrak{b}egin{equation} \label{N-g-1}
\mathcal N_g(p,q;s;\phi) = \frac{\lambda_g(p)}{p^s} \mathcal N_g(1,q;s;\phi) + (1-p^{-2s}) \zeta_q(2s)^2 \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \mathcal K(m,pn,q;\phi).
\end{equation}
Our next aim is to relate the sum on the right-hand side of \eqref{N-g-1}
to the sum
\mathfrak{b}egin{equation} \label{eq:S-def}
\mathcal S(p,q;s,\phi) := \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \sum_{(c,pq)=1} \frac{S(m\overline{q},np;c)}{c\sqrt{q}} \phi\left(\frac{4\pi \sqrt{mnp}}{c\sqrt{q}}\right).
\end{equation}
This involves a sieving process which leaves us with a sum over $c$ relatively prime to $pq$.
\mathfrak{b}egin{proposition}\label{prop:Mg-S}
Let $p$ be a prime number and let $g\in S_\kappa(1)$. For $\sigma>\frac 54$ we have
\mathfrak{b}egin{multline} \label{eq:Mg-S}
(1+p^{-2s}) \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \mathcal K(m,pn,q;\phi)
\\ = \mathcal S(p,q;s,\phi) - \frac{1}{\sqrt p} \zeta_{pq}(2s)^{-2} \mathcal N_g(1,pq;s;\phi) + \frac{\lambda_g(p)}{p^s} \zeta_q(2s)^{-2} \mathcal N_g(1,q;s;\phi).
\end{multline}
\end{proposition}
In the proof of Proposition~\ref{prop:Mg-S} we will need the following facts about Kloosterman sums, all of which follow from standard exponential sum manipulations.
\mathfrak{b}egin{lemma} \label{lem:kloo-props}
Suppose that $m,n,c \in \mathbb{Z}$ with $c>0$ and that $p$ is prime.
\mathfrak{b}egin{enumerate}
\item If $p\mid\mid c$ then
\mathfrak{b}egin{equation} \label{eq:kloo-p-1}
S(m,pn;c) = S(\mathfrak{b}ar p m, n; c/p) \times
\mathfrak{b}egin{cases}
-1 & \text{ if } p\nmid m, \\
p-1 & \text{ if }p\mid m,
\end{cases}
\end{equation}
where $\mathfrak{b}ar pp\equiv 1\pmod{c/p}$.
\item If $p^2 \mid c$ then
\mathfrak{b}egin{equation} \label{eq:kloo-p-2}
S(m,pn;c)=0 \quad \text{ unless } p\mid m.
\end{equation}
\item We have
\mathfrak{b}egin{equation} \label{eq:kloo-p-3}
S(pm,pn,p^2c) = p S(m,n,pc).
\end{equation}
\end{enumerate}
\end{lemma}
\mathfrak{b}egin{proof}[Proof of Proposition~\ref{prop:Mg-S}]
Let
\mathfrak{b}egin{align}
\mathcal R :=& \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \mathcal K(m,pn,q;\phi) \\
=& \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \sum_{(c,q)=1} \frac{S(\mathfrak{b}ar qm,pn;c)}{c\sqrt q} \phi \pfrac{4\pi\sqrt{mnp}}{c\sqrt q},
\end{align}
and write $\mathcal R = \mathcal S + \mathcal T$,
where $\mathcal S=\mathcal S(p,q;s,\phi)$ is defined in \eqref{eq:S-def} and $\mathcal T$ comprises those terms of $\mathcal R$ with $p\mid c$.
If furthermore $p^2 \mid c$, we apply \eqref{eq:kloo-p-2}, and if $p\mid\mid c$, we apply \eqref{eq:kloo-p-1}, obtaining
\mathfrak{b}egin{multline}
\mathcal T =
\sum_{\substack{m,n\geq 1}} \frac{\lambda_g(pm)\lambda_g(n)}{(pm)^s n^s} \sum_{\substack{(c,q)=1 \\ p\mid c}} \frac{S(\mathfrak{b}ar qpm,pn;pc)}{pc\sqrt q} \phi \pfrac{4\pi \sqrt{mn}}{c\sqrt{q}} \\
- \sum_{\substack{m,n\geq 1\\p\nmid m}} \frac{\lambda_g(m)\lambda_g(n)}{m^s n^s} \sum_{\substack{(c,pq)=1}} \frac{S(\mathfrak{b}ar {qp}m,n;c)}{pc\sqrt q} \phi \pfrac{4\pi \sqrt{mn}}{c\sqrt{pq}} \\
+(p-1) \sum_{\substack{m,n\geq 1\\p\mid m}} \frac{\lambda_g(m)\lambda_g(n)}{m^s n^s} \sum_{\substack{(c,pq)=1}} \frac{S(\mathfrak{b}ar {qp}m,n;c)}{pc\sqrt q} \phi \pfrac{4\pi \sqrt{mn}}{c\sqrt{pq}}.
\end{multline}
Now we apply \eqref{eq:kloo-p-3} to the terms of the first sum.
In the last line we separate $p-1$, combining the $p$ term with the first line, and the $-1$ term with the second. We find that
\mathfrak{b}egin{multline}
\mathcal T = \sum_{\substack{m,n\geq 1}} \frac{\lambda_g(pm)\lambda_g(n)}{(pm)^s n^s} \sum_{\substack{(c,q)=1}} \frac{S(\mathfrak{b}ar qm,n;c)}{c\sqrt q} \phi \pfrac{4\pi\sqrt{mn}}{c\sqrt{q}} \\
- \frac{1}{\sqrt p} \sum_{\substack{m,n\geq 1}} \frac{\lambda_g(m)\lambda_g(n)}{m^s n^s} \sum_{\substack{(c,pq)=1}} \frac{S(\mathfrak{b}ar {qp}m,n;c)}{c\sqrt {pq}} \phi \pfrac{4\pi\sqrt{mn}}{c\sqrt{pq}}.
\end{multline}
Finally, we apply the Hecke relation again to $\lambda_g(pm)$ in the first term, and conclude that
\mathfrak{b}egin{equation}
\mathcal T = \frac{\lambda_g(p)}{p^s} \zeta_q(2s)^{-2} \mathcal N_g(1,q;s;\phi) - \frac{1}{p^{2s}} \mathcal R - \frac{1}{\sqrt p} \zeta_{pq}(2s)^{-2} \mathcal N_g(1,pq;s;\phi)
\end{equation}
since
\mathfrak{b}egin{equation}
S(\mathfrak{b}ar q pm,n;c) = S(pm,\mathfrak{b}ar q n;c) = S(\mathfrak{b}ar q n, pm;c).
\end{equation}
On the other hand, $\mathcal T = \mathcal R - \mathcal S$, so we have
\mathfrak{b}egin{equation}
(1+p^{-2s})\mathcal R = \mathcal S - \frac{1}{\sqrt p} \zeta_{pq}(2s)^{-2} \mathcal N_g(1,pq;s;\phi) + \frac{\lambda_g(p)}{p^s} \zeta_q(2s)^{-2} \mathcal N_g(1,q;s;\phi),
\end{equation}
as desired.
\end{proof}
Now let us combine the last two results. Combining \eqref{N-g-1} and Proposition \ref{prop:Mg-S} we obtain
\mathfrak{b}egin{multline}
\mathcal N_g(p,q;s;\phi) = \frac{\lambda_g(p)}{p^s}\mathcal N(1,q,s,\phi) + \frac{\zeta_{pq}(2s)^2 \left(1 + p^{-2s}\right)}{\left(1 - p^{-2s}\right)\left(1 + p^{-2s}\right)} \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n) }{(mn)^s} K(m,pn;s;\phi)\\
= \left(\frac{\lambda_g(p)}{p^s} + \frac{\lambda_g(p)}{p^s}\frac{\left(1 - p^{-2s}\right) }{\left(1 + p^{-2s}\right)}\right) \mathcal N_g(1,q;s;\phi) - \frac{1}{\sqrt{p}} \frac{\mathcal N_g(1,pq;s;\phi)}{\left(1 - p^{-4s}\right)} + \zeta_{pq}(2s)^2 \frac{ S(p,q;s,\phi)}{\left(1 - p^{-4s}\right)} .
\end{multline}
Rearanging the terms in the first parentheses we obtain \eqref{eq:Ng-S}.
\section{Reciprocity for \texorpdfstring{$\mathcal S(p,q;s,\phi)$}{S(p,q;s,phi)}}
In this section we will prove the following reciprocity relation for $\mathcal S(p,q;s,\phi)$.
\mathfrak{b}egin{theorem} \label{thm:S-reciprocity}
Let $\phi$ be a smooth test function satisfying $\phi^{(j)}(0) = 0$ and $\phi^{(j)}(x) \ll (1 + x)^{-A}$ for $0\leq j\leq 12$ and for some $A>12$.
Suppose that $\operatorname{Re}(s) = \sigma>\frac 54$ and that $p,q$ are distinct primes.
Then
\mathfrak{b}egin{equation}
\sqrt q \, \mathcal S(p,q;s,\phi) = \pfrac{p}{q}^{2s-1} \sqrt p \, \mathcal S(q,p;s,\mathbb{P}hi),
\end{equation}
where, for any $\xi$ satisfying $0<\xi+12<A$, $\mathbb{P}hi(x)$ is defined as
\mathfrak{b}egin{equation} \label{eq:Phi-def}
\mathbb{P}hi(x) = \mathbb{P}hi_{\kappa,s}(x) := \frac{1}{2\pi i} \int_{(\xi)} \widetildede\phi(u) 2^{-u} \left[\frac{\Gamma(\frac{\kappa+1}{2}-s-\frac{u}{2})}{\Gamma(\frac{\kappa-1}{2}+s+\frac{u}{2})}\right]^2 \left(\frac{x}{2}\right)^{u+4s-2} \, du
\end{equation}
and $\widetildede\phi$ is the Mellin transform
\mathfrak{b}egin{equation}
\widetildede\phi(u) = \int_0^\infty t^u \phi(t) \, \mfrac{dt}{t}.
\end{equation}
Furthermore, the function $\mathbb{P}hi_{\kappa, s}$ can be analytically continued to $\frac12 \leq \sigma \leq \frac54 + \vepsilon$ and it satisfies \eqref{eq:phiKuznetsovdGrowth} in that region.
\end{theorem}
Suppose that $\phi^{(j)}(0)=0$ and $\phi^{(j)}(x) \ll (1+x)^{-A}$ for all $j\in \{0,1,2,\ldots, J\}$, {for some integer $J$}.
{We will later see that we may take $J = 12$.}
By the decay of $\phi$ and its derivatives at $0$ and $\infty$, we may apply integration by parts $j$ times and obtain that
\mathfrak{b}egin{equation}\label{eq:MellinPhiBound}
\widetilde{\phi}(u) = (-1)^j \int_{0}^\infty \phi^{(j)}(x) \mfrac{x^{u + j}}{u (u + 1) \mathfrak{c}dots (u + j-1)}\, \mfrac{d x}{x} \ll (1 + |u|)^{-j}
\end{equation}
as long as the integral converges.
Near $0$ we have $\phi^{(j)}(x)\ll x^{J-j}$, so the integrand is majorized by $x^{J+\xi-1}$ {(with $\xi = \operatorname{Re}(u)$)} as $x\to 0$.
For large $x$ the integrand is majorized by $x^{-A+\xi+j-1}$.
It follows that the integral is absolutely convergent (and thus $\eqref{eq:MellinPhiBound}$ holds with $j=J$) as long as $-J<\xi<A-J$.
Starting with \eqref{eq:S-def}, we write $\phi$ as the inverse Mellin transform of $\widetilde{\phi}$ via
\mathfrak{b}egin{equation}
\phi(x) = \frac{1}{2\pi i} \int_{(\xi)} \widetildede\phi(u) x^{-u} \, du.
\end{equation}
We then interchange integral and summation to obtain
\mathfrak{b}egin{align}\label{eq:S-integral-series}
\sqrt q \, \mathcal S(p,q;s,\phi) &= \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \sum_{(c,pq)=1} \frac{S(m\overline{q},np;c)}{c} \frac{1}{2\pi i}
\int_{(\xi)} \widetilde{\phi}(u)\left(\frac{4\pi \sqrt{mnp}}{c\sqrt{q}}\right)^{-u} \, du \notag\\
&= \frac{1}{2\pi i} \int_{(\xi)} \widetildede\phi(u) \pfrac{4\pi \sqrt p}{\sqrt q}^{-u} \sum_{(c,pq)=1} \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)S(\mathfrak{b}ar qm, pn; c)}{c^{1-u}(mn)^{s+\frac u2}} \, du.
\end{align}
As the following proposition shows, the innermost sum above satisfies a functional equation which exchanges the roles of $p$ and $q$.
\mathfrak{b}egin{proposition} \label{prop:D-props}
Let $g\in S_k(1)$ be a newform.
Suppose that $a$, $b$, and $c$ are integers with $c$ positive and $(a,c)=(b,c)=1$.
For $\sigma>1$, define
\mathfrak{b}egin{equation} \label{eq:H-def}
D_g(a,b,c;s) := \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)S(am, bn; c)}{(mn)^s}.
\end{equation}
Then $D_g(a,b,c;s)$ extends to an entire function of $s$ which satisfies the functional equation
\mathfrak{b}egin{equation} \label{eq:H-func-eq}
D_g(a,b,c;s) = \pfrac{2\pi}{c}^{4s-2} \left[\frac{\Gamma(\frac{\kappa+1}{2}-s)}{\Gamma(\frac{\kappa-1}{2}+s)}\right]^2 D_g(\mathfrak{b}ar a, \mathfrak{b}ar b, c;1-s),
\end{equation}
where $a\mathfrak{b}ar a \equiv b\mathfrak{b}ar b \equiv 1 \pmod{c}$.
Furthermore, for any $\vepsilon>0$, as $|t|\to\infty$ with $\sigma$ bounded we have
\mathfrak{b}egin{equation} \label{eq:H-bound}
D_g(a,b,c;s) \ll
\mathfrak{b}egin{cases}
c^{\frac 12+\vepsilon} & \text{ if } \sigma \geq 1+\vepsilon, \\
c^{\frac 52-2\sigma+\vepsilon}|t|^{2-2\sigma+\vepsilon} & \text{ if } -\vepsilon \leq \sigma \leq 1+\vepsilon, \\
c^{\frac 52-4\sigma+\vepsilon} |t|^{2-4\sigma} & \text{ if } \sigma\leq -\vepsilon.
\end{cases}
\end{equation}
\end{proposition}
\mathfrak{b}egin{proof}
Suppose that $\sigma>1$.
Opening up the Kloosterman sum, we find that
\mathfrak{b}egin{equation}\label{eq:Dg-in-L}
D_g(a,b,c;s) = \sums_{d\mathfrak{b}mod c} L(s,g,\tfrac{a\mathfrak{b}ar d}{c}) L(s,g,\tfrac{b\mathfrak{b}ar d}{c}),
\end{equation}
where
\mathfrak{b}egin{equation}\label{eq:LgAdditiveTwist}
L(s,g,x) := \sum_{m=1}^\infty \frac{\lambda_g(m)e(mx)}{m^s}.
\end{equation}
The $L$-series $L(s,g,x)$ extends to an entire function of $s$ which satisfies, for $x=\frac dc$ with $(d,c)=1$, the functional equation
\mathfrak{b}egin{equation}\label{eq:LgAdditiveTwistFE}
L(s,g,\tfrac dc) = (-1)^{\frac \kappa2} \pfrac{2\pi}{c}^{2s-1} \frac{\Gamma(\frac{\kappa+1}2-s)}{\Gamma(\frac{\kappa-1}2+s)} L(1-s,g,-\tfrac{\mathfrak{b}ar d}{c}).
\end{equation}
Thus $D_g(a,b,c;s)$ inherits the functional equation \eqref{eq:H-func-eq} after using that $S(-a,-b;c)=S(a,b;c)$.
To prove the estimates \eqref{eq:H-bound} we apply the Phragmen-Lindel\"of principle (see, e.g. IK, Theorem 5.53).
The estimate in the range $\sigma \geq 1+\vepsilon$ follows from a standard computation involving the Weil bound \eqref{eq:WeilBound}.
In the range $\sigma \leq -\vepsilon$ we apply the functional equation \eqref{eq:H-func-eq}, together with Stirling's formula and the bound in the $\sigma\geq 1+\vepsilon$ range.
Then Phragmen-Lindel\"of yields the bound in the middle range $-\vepsilon\leq \sigma \leq 1+\vepsilon$.
\end{proof}
The innermost sum in \eqref{eq:S-integral-series} equals $c^{u-1}D_g(\overline{q}, p, c;s + \tfrac{u}{2})$.
We would like to apply the functional equation \eqref{eq:H-func-eq} and write the result as a sum of Kloosterman sums.
To do this, we will need to work in the region of absolute convergence of the function $D_g(q,\mathfrak{b}ar p,c;1-\frac u2-s)$, so we move the line of integration in $u$ to $\xi=-2\sigma-\vepsilon$.
By the estimates \eqref{eq:H-bound} and \eqref{eq:MellinPhiBound}, the resulting integral converges as long as $J\geq 4$. {Note that $\sigma > \tfrac 54$ and hence the sum over $c$ converges.}
The function is entire, as can be seen from \eqref{eq:Dg-in-L} and we do not pick up any poles in the process\footnote{Note that if we were to replace $\lambda_g(n)$ with $\tau_w(n)$, the resulting function $D$ would be a finite sum of products of two Estermann zeta functions, and we would have picked up poles which in turn would constitute the main terms one encounters in the reciprocity formula of the fourth moment.}.
\mathfrak{b}egin{proof}[Proof of Theorem~\ref{thm:S-reciprocity}]
Starting with the \eqref{eq:S-integral-series} represenation of $\sqrt q \, \mathcal S(p,q;s)$ with $\xi$ satisfying $\xi = - 2\sigma-\vepsilon$, we apply the functional equation \eqref{eq:H-func-eq} to obtain
\mathfrak{b}egin{multline}
\sqrt {q/p} \, \mathcal S(p,q;s;\phi)
= \left(\frac{p}{q}\right)^{2s-1} \frac{1}{2\pi i} \int_{(\xi)} \widetilde{\phi}(u) \left[\frac{\Gamma(\frac{\kappa+1}{2}-s-\frac{u}{2})}{\Gamma(\frac{\kappa-1}{2}+s+\frac{u}{2})}\right]^2 2^{-4s - 2u + 2}\\
\times \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^{s}} \sum_{(c,pq)=1} \frac{S(qm,\mathfrak{b}ar pn;c)}{c\sqrt p} \pfrac{4\pi \sqrt{mnq}}{c\sqrt p}^{4s + u - 2} \, du.
\end{multline}
The integral and sums are absolutely convergent for $\sigma>\frac 54$, so we can interchange them to obtain
\mathfrak{b}egin{align}
\sqrt {q/p} \, \mathcal S(p,q;s;\phi) &= \left(\frac{p}{q}\right)^{2s-1} \sum_{m,n\geq 1} \frac{\lambda_g(m)\lambda_g(n)}{(mn)^s} \sum_{(c,pq) = 1} \frac{S(qm, \overline{p}n;c)}{c\sqrt{p}} \mathbb{P}hi\left(\frac{4\pi \sqrt{mnq}}{c\sqrt{p}}\right).
\end{align}
The sums on the right hand side are exactly $\mathcal S(q,p;s;\mathbb{P}hi)$.
Let us finish the proof by specifying the conditions for $\phi$ so that $\mathbb{P}hi$ satisfies \eqref{eq:phiKuznetsovdGrowth}.
This is necessary in order for us to apply the Kuznetsov formula to the sum of Kloosterman sums with the weight function $\mathbb{P}hi$.
So far we have required that $J\geq 4$ and that $-J<\xi<A-J$.
By Stirling's formula the integrand in the definition of $\mathbb{P}hi$ is majorized by $( 1 + |\Im(u)|)^{-J + 2 - 4\sigma - 2\xi}$, so the integral converges as long as
$2\xi+4\sigma > 3 - J$.
Suppose that $\sigma \geq \frac 12$.
Then $\mathbb{P}hi(0)=0$ as long as we can take $\xi>0$ (note that we can easily stay away from the poles of the gamma factor since $\kappa\geq 12$).
This requires $A>J$.
On the other hand, to estimate $\mathbb{P}hi^{(\ell)}(x)$ for large $x$ and for $\ell=0,1,2$, we want to move $\xi$ as far to the left as possible.
Differentiating under the integral, we find that
\mathfrak{b}egin{align}
\mathbb{P}hi^{(\ell)} (x) \ll \frac{1}{2\pi i} \int_{(\xi)}|\widetildede\phi(u)| \left|\frac{\Gamma(\frac{\kappa+1}{2}-s-\frac{u}{2})}{\Gamma(\frac{\kappa-1}{2}+s+\frac{u}{2})}\right|^2 x^{4\sigma-2+\xi- \ell} ( 1 + | u|)^{\ell} \, du \ll (1 + x)^{4\sigma - 2 + \xi - \ell},
\end{align}
assuming the integral converges.
To ensure convergence, we need that $2\xi+4\sigma>\ell+3-J$, and for $\mathbb{P}hi^{(\ell)} (1+x) \ll x^{-2-\vepsilon}$ to hold we need that $4\sigma + \xi <\ell$.
Suppose that $\frac 12\leq \sigma\leq \frac 54+\vepsilon$.
Then we need
\mathfrak{b}egin{equation}
\ell-J+1 < 2\xi < 2\ell-10 \qquad \text{ for }\ell=0,1,2.
\end{equation}
For this interval to be nonempty, it is enough to take $J\geq 12$.
\end{proof}
\mathfrak{b}ibliographystyle{plain}
\mathfrak{b}ibliography{../biblio}
\end{document} |
\begin{document}
\title{Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy}
\author[1]{Geir-Arne Fuglstad\thanks{Corresponding author, [email protected]}}
\author[2]{Finn Lindgren}
\author[1]{Daniel Simpson}
\author[1]{HÃ¥vard Rue}
\affil[1]{Department of Mathematical Sciences, NTNU, Norway}
\affil[2]{Department of Mathematical Sciences, University of Bath, UK}
\date{April 25, 2014}
\maketitle
\begin{abstract}
Gaussian random fields (GRFs) constitute an important part of spatial
modelling, but can be computationally infeasible for general
covariance structures. An efficient approach is to specify GRFs
via stochastic partial differential equations (SPDEs) and
derive Gaussian Markov random field (GMRF) approximations of the
solutions. We consider the construction of a class of non-stationary
GRFs with varying local anisotropy, where the
local anisotropy is introduced by allowing
the coefficients in the SPDE to vary with position. This is done by
using a form of diffusion equation driven by Gaussian white
noise with a spatially varying diffusion matrix. This allows
for the introduction of parameters that control the GRF by
parametrizing the diffusion matrix. These parameters and the GRF
may be considered to be part of a hierarchical model and the
parameters estimated in a Bayesian framework. The results show that the
use of an SPDE with non-constant coefficients is a promising way of
creating non-stationary spatial GMRFs that allow for physical
interpretability of the parameters, although there are several remaining
challenges that would need to be solved before these models can be put
to general practical use.
\noindent \textbf{Keywords:} Non-stationary, Spatial, Gaussian random fields,
Gaussian Markov random fields, Anisotropy, Bayesian
\end{abstract}
\section{Introduction}
Many spatial models for continuously indexed phenomena, such as temperature,
precipitation and air pollution, are based on Gaussian random fields (GRFs).
This is mainly due to the fact that their theoretical properties are
well understood and that their distributions can be fully
described by mean and covariance functions. In principle, it is enough to specify the mean at each location and the covariance between any two locations.
However, specifying covariance functions is hard and specifying covariance
functions that can be controlled by parameters in useful ways is even harder.
This is the reason why the covariance function usually is selected from a class of
known covariance functions such as the exponential covariance function, the
Gaussian covariance function or the Mat\'{e}rn covariance function.
But even when the covariance function is selected
from one of these classes, the feasible
problem sizes are severely limited by a cubic increase in computation time
as a function of the number of observations and a quadratic
increase in computation time as a function of the number of prediction locations.
This computational challenge is usually tackled
either by reducing the dimensionality of the
problem~\cites{Cressie2008, Banerjee2008}, by introducing sparsity in the
precision matrix~\cite{Rue2005} or the covariance matrix~\cite{Furrer2006},
or by using an approximate likelihood~\cites{Stein2004, Fuentes2007}. \ocite{Sun2012}
offers comparisons of the advantages and challenges associated with the usual
approaches to large spatial datasets.
The main goal of this paper is to explore a new class of
non-stationary GRFs that provide both an easy way to specify the parameters
and allows for fast computations. The main computational tool used is Gaussian
Markov random fields ({GMRFs})~\cite{Rue2005} with a spatial Markovian
structure where each position is conditionally dependent only on positions
close to itself. The strong connection between the Markovian structure
and the precision matrix results in sparse precision matrices that
can be exploited in computations.
The main problem associated with such an approach is that GMRFs must be
constructed through conditional distributions, which presents a challenge
as it is generally not easy to determine whether a set of conditional
distributions gives a valid joint distribution. Additionally, the
conditional distributions have to be controlled by useful parameters in
such a way that not only the joint distribution is valid, but also such that
the effect of the
parameters is understood. Lastly, it is desirable that the GMRF is
a consistent approximation of a GRF
in the sense that when the distances between the positions decrease, the
GMRF ``approaches'' a continuous GRF. These issues are even more challenging
for non-stationary GMRFs. It is is extremely hard to specify the
non-stationarity directly through conditional distributions.
There is no generally accepted way to handle non-stationary GRFs, but many
approaches have been suggested. There is a large literature on methods based on
the deformation method of~\ocite{Sampson1992}, where a stationary process is
made non-stationary by deforming the space on which it is defined. Several
Bayesian extensions of the method have been
proposed~\cites{Damian2001, Damian2003, Schmidt2003, Schmidt2011}, but all these methods require replicated realizations which might not be
available. There has been some development
towards an approach for a single realization, but with a ``densely''
observed realization~\cite{Anderes2008}. Other approaches use
kernels which are convolved with Gaussian
white noise~\cites{Higdon1998, Paciorek2006},
weighted sums of stationary processes~\cite{Fuentes2001} and expansions
into a basis such as a wavelet basis~\cite{Nychka2002}.
Conceptually simpler methods have been made with
``stationary windows''~\cites{Haas1990a, Haas1990b} and with piecewise
stationary Gaussian processes~\cite{Kim2005}. There has also been some progress
with methods based on the spectrum of the
processes~\cites{Fuentes2001, Fuentes2002b, Fuentes2002a}.
Recently, a new type of method based on a connection between stochastic
partial differential equations ({SPDEs}) and some classes
of GRFs was proposed by~\ocite{Lindgren2011}. They use an
SPDE to model the GRF and construct a GMRF approximation to the
GRF for computations. An application of a non-stationary model
of this type to ozone data can be found in~\ocite{Bolin2011} and
an application to precipitation data can be found in~\ocite{Rikke2013}.
This paper extends on the work of~\ocite{Lindgren2011} and explores the possibility
of constructing a non-stationary GRF by varying the local anisotropy.
The interest lies both in considering the different types of
structures that can be achieved, and how to parametrize the GRF and
estimate the parameters in a Bayesian setting. The construction of the GRF
is based on an SPDE which describes the
GRF as the result of a linear filter applied to Gaussian white noise. Basically,
the SPDE expresses how the smoothing of the Gaussian white noise varies at
different locations. This construction bears some
resemblance to the deformation method of~\ocite{Sampson1992} in the sense that
parts of the spatial variation of the linear filter can be understood as
a local deformation of the space, only with an associated spatially varying
variance for the Gaussian white noise.
The main idea for
computations is that since this filter works locally, it implies a Markovian
structure on the GRF. This Markovian structure can be transferred to a GMRF which
approximates the GRF, and in turn fast computations can be done with sparse matrices.
This paper presents a first look into a new type of model and the main goal is to
explore what can be achieved in terms of models and inference with the model.
Section~\ref{sec:Preliminaries} contains the motivation and introduction to the
class of non-stationary GRFs that is studied in the other sections. The form
of the SPDE that generates the class is given and it is related to more
standard constructions of GMRFs. In Section~\ref{sec:Examples} illustrative
examples are given on both stationary and non-stationary constructions. This
includes some discussion on how to control the non-stationarity of the GRF. Then
Section~\ref{sec:Inference} explores parameter estimation for these types of models
through different examples with simulated data. The paper ends with discussion of
extensions in Section~\ref{sec:Extensions} and general discussion
and concluding remarks in Section~\ref{sec:Discussion}.
\section{New class of non-stationary GRFs}
\label{sec:Preliminaries}
A GMRF \(\boldsymbol{u}\) is usually parametrized through a mean
\(\boldsymbol{\mu}\) and a precision matrix \(\mathbf{Q}\) such that
\(\boldsymbol{u}\sim\mathcal{N}(\boldsymbol{\mu}, \mathbf{Q}^{-1})\). The main
advantage of this formulation compared to the usual parametrization of
multivariate Gaussian distributions through
the covariance matrix is that the Markovian structure is represented in the non-zero
structure of the precision matrix \(\mathbf{Q}\)~\cite{Rue2005}.
Off-diagonal entries are non-zero if and only if the corresponding elements of
\(\boldsymbol{u}\) are conditionally independent. This can be seen
from the conditional properties of a GMRF,
\[
\mathrm{E}(u_i | \boldsymbol{u}_{-i}) = \mu_i-\frac{1}{Q_{i,i}}\sum_{j\neq i} Q_{i,j}(u_j-\mu_j)
\]
and
\[
\mathrm{Var}(u_i | \boldsymbol{u}_{-i}) = \frac{1}{Q_{i,i}},
\]
where \(\boldsymbol{u}_{-i}\) denotes the vector \(\boldsymbol{u}\) with
element \(i\) deleted. For a spatial GMRF the non-zeros of
\(\mathbf{Q}\) can correspond to grid-cells that are close to
each other in a grid, neighbouring regions in a Besag model and so on.
However, even when this non-zero structure is determined it is not clear what
values should be given to the non-zero elements of the precision matrix.
This is the framework of the
conditionally auto-regressive (CAR) models, whose conception predates
the advances in modern computational statistics~\cites{Whittle1954, Besag1974}.
In the multivariate Gaussian case it is clear that the requirement for
a valid joint distribution is that \(\mathbf{Q}\) is positive definite, which
is not an easy condition to check.
Specification of a GMRF through the conditional properties given above is
usually done in a somewhat ad-hoc manner. For regular grids, a process such as random
walk can be constructed and the only major issue is to get the conditional
variance correct as a function of step-length. For irregular grids the situation
is not as clear because each of the conditional means and variances must
depend on the varying step-lengths. In~\ocite{Lindgren2008} it is demonstrated that
some such constructions for second-order random walk can lead to
inconsistencies as new grid points are added, and they offer a surprisingly
simple construction for second-order random walk based on the SPDE
\[
-\frac{\partial^2}{\partial x^2} u(x) = \sigma \mathcal{W}(x),
\]
where \(\sigma>0\) and \(\mathcal{W}\) is standard Gaussian white noise.
If the precision matrix is chosen according to their scheme one does not
have to worry about scaling as the grid is refined, as it automatically
approaches the continuous second-order random walk. There is an
automatic procedure to select the form of the conditional means
and variances.
A one-dimensional second-order random walk is a relatively simple example
of a process
with the same behaviour everywhere. To approximate a two-dimensional,
non-stationary GRF, a scheme would require (possibly)
different anisotropy and correct
conditional variance at each location. To select the precision matrix in this
situation poses a large problem and there is abundant use of simple models
such as a spatial moving average
\[
\mathrm{E}(u_{i,j}|\boldsymbol{u}_{-\{(i,j)\}}) = \frac{1}{4}(u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1})
\]
with a constant conditional variance \(1/\alpha\). There are ad-hoc ways to extend
such a scheme to a situation with varying step-lengths in each direction,
but little theory for more irregular choices of locations.
This is why the choice was made to start with the close connection between
SPDEs and some classes of GRFs that was presented in~\ocite{Lindgren2011},
which is not plagued by the issues above.
From~\ocite{Whittle1954} it is known that the SPDE
\begin{equation}
(\kappa^2-\Delta)u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in\mathbb{R}^2,
\label{eq:LindSPDE}
\end{equation}
where \(\kappa^2>0\) and \(\Delta = \frac{\partial^2}{\partial s_1^2}+\frac{\partial^2}{\partial s_2^2}\) is the Laplacian, gives rise to a GRF \(u\) with the
Mat\'{e}rn covariance function
\[
r(\boldsymbol{s}) = \frac{1}{4\pi \kappa^2}(\kappa\lvert\lvert\boldsymbol{s}\rvert\rvert)K_1(\kappa\lvert\lvert\boldsymbol{s}\rvert\rvert),
\]
where \(K_1\) is the modified Bessel function of the second kind of order 1.
Equation~\eqref{eq:LindSPDE} can be extended to fractional operator orders
in order to obtain other smoothness parameters in the Mat\'ern covariance
function. However, for practical applications, the true smoothness of
the field is very hard to estimate from data, in particular when the
model is used in combination with an observation noise model.
Restricting the development to smoothness 1 in the Mat\'ern family is
therefore unlikely to be a major practical serious limitation.
However, for practical computations the model will be discretised
using methods similar to \ocite{Lindgren2011}, which does
permit other operator orders. Integer orders are easiest, but for
stationary models, fractional orders are also achievable
\cite{Lindgren2011}*{Authors' discussion response}. For
non-stationary models, techiques similar to \ocite{Bolin2013}*{Section 4.2}
would be possible to use. This means that even though we here
will restrict the model development to the special case in
Equation~\eqref{eq:LindSPDE}, other smoothnesses, e.g.\ exponential covariances, will
be reachable by combining the different approximation techniques.
The intriguing part, that~\ocite{Lindgren2011} expanded upon in
Equation~\eqref{eq:LindSPDE}, is that
\((\kappa^2-\Delta)\) can be interpreted as a linear filter acting locally. This
means that if the continuously indexed process \(u\) were instead represented
by a GMRF \(\boldsymbol{u}\) on a grid or a triangulation, with appropriate
boundary conditions, one
could replace this operator with
a matrix, say \(\mathbf{B}(\kappa^2)\), only involving neighbours of
each location such that
Equation~\eqref{eq:LindSPDE} becomes approximately
\begin{equation}
\mathbf{B}(\kappa^2)\boldsymbol{u} \sim \mathcal{N}(0,\mathbf{I}).
\label{eq:MatrixEquation}
\end{equation}
The matrix \(\mathbf{B}(\kappa^2)\) depends on the chosen grid, but after the
relationship is derived, the calculation of \(\mathbf{B}(\kappa^2)\)
is straightforward for any \(\kappa^2\).
Since \(\mathbf{B}(\kappa^2)\) is sparse, the resulting precision matrix
\(\mathbf{Q}(\kappa^2) = \mathbf{B}(\kappa^2)^\mathrm{T}\mathbf{B}(\kappa^2)\) for
\(\boldsymbol{u}\)
is also sparse. This means that by
correctly discretizing the operator (or linear filter), it
is possible to devise a GMRF with approximately the same distribution as the
continuously indexed GRF. And because it comes from a continuous equation one
does not have to worry about changing behaviour as the grid is refined.
The class of models that are studied in this paper is the one that can be constructed
from Equation~\eqref{eq:LindSPDE}, but with anisotropy added to the \(\Delta\)
operator. A function \(\mathbf{H}\), that gives \(2\times 2\) symmetric
positive definite matrices at each position, is introduced and the operator
is changed to
\begin{align*}
\nabla\cdot\mathbf{H}(\boldsymbol{s})\nabla &=
\frac{\partial}{\partial s_1}\left(h_{11}(\boldsymbol{s})\frac{\partial}{\partial s_1} \right)+
\frac{\partial}{\partial s_1}\left(h_{12}(\boldsymbol{s})\frac{\partial}{\partial s_2} \right) \\
&\phantom{=}+\frac{\partial}{\partial s_2}\left(h_{21}(\boldsymbol{s})\frac{\partial}{\partial s_1} \right)+
\frac{\partial}{\partial s_2}\left(h_{22}(\boldsymbol{s})\frac{\partial}{\partial s_2} \right).
\end{align*}
This induces different strength of local dependence in different directions,
which results in a range that varies with direction at all locations. Further, it is necessary for the discretization procedure to restrict the SPDE to a bounded domain.
The chosen SPDE is
\begin{equation}
(\kappa^2-\nabla\cdot\mathbf{H}(\boldsymbol{s})\nabla)u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s} \in \mathcal{D}=[A_1, B_1]\times[A_2, B_2]\subset\mathbb{R}^2,
\label{eq:fullSPDE}
\end{equation}
where the rectangular domain makes it possible to use periodic
boundary conditions. Neither the rectangular shape of the domain nor
the periodic boundary conditions are essential restrictions for the
model, but are merely the practical restrictions we choose to work
with in this paper, in order to focus on the non-stationarity itself.
When using periodic boundary conditions when approximating the
likelihood of a stationary process on an unbounded domain, the
parameter estimates will be biased, e.g.\ when using the Whittle
likelihood in the two-dimensional case \cite{Dahlhaus1987edge}.
However, as \ocite{Lindgren2011}*{Appendix A.4} notes for the
case with Neumann boundary conditions (i.e.\ normal derivatives set to
zero), the effect of the boundary conditions is limited to a region in
the vicinity of the boundary. At a distance greater than twice the
correlation range away from the boundary the bounded domain model is
nearly indistinguishable from the model on an unbounded domain.
Therefore, the bias due to boundary effects can be eliminated by
embedding the domain of interest into a larger region, in effect
moving the boundary away from where it would influence the likelihood
function. For non-stationary models, defining appropriate boundary
conditions becomes part of the practical model formulation itself.
For simplicity we will therefore ignore this issue here, leaving
boundary specification for future development, but provide some
additional practical comments in Section~\ref{sec:Extensions}.
Both for interpretation and for the practical use of Equation~\eqref{eq:fullSPDE}
it is useful to decompose \(\mathbf{H}\) into scalar functions. The anisotropy
due to \(\mathbf{H}\) is decomposed as
\[
\mathbf{H}(\boldsymbol{s}) = \gamma\mathbf{I}_2 + \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
where \(\gamma\) specifies the isotropic, baseline effect and the vector field
\(\boldsymbol{v}(\boldsymbol{s}) = [v_x(\boldsymbol{s}), v_y(\boldsymbol{s})]^\mathrm{T}\)
specifies the direction and magnitude of the local, extra anisotropic effect
at each location. In this way, one can, loosely speaking, think of different Mat\'{e}rn
like fields locally each with its own anisotropy that are
combined into a full process.
An example of an extreme case of a process with a strong local anisotropic effect
is shown in Example~\ref{exmp:nonStat}. The example
shows that there is a close connection between the vector field and the resulting
covariance structure of the GRF.
The main computational challenge is to determine the appropriate discretization
of the SPDE in Equation~\eqref{eq:fullSPDE}, that is how to derive a
matrix \(\mathbf{B}\) such as in Equation~\eqref{eq:MatrixEquation}.
The idea is to look to the field of numerics for discretization methods
for differential equations. Then combine these with properties of Gaussian
white noise. Namely, that for a Lebesgue measurable
subset $A$ of $\mathbb{R}^n$, for some $n > 0$,
\[
\int_A \! \mathcal{W}(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} \sim \mathcal{N}(0,|A|),
\]
where $|A|$ is the Lebesgue measure of $A$, and that for two disjoint Lebesgue
measurable subsets $A$ and $B$ of $\mathbb{R}^n$ the integral over $A$ and the
integral over $B$ are independent~\cite{Adler2007}*{pp.~24--25}.
A matrix equation such as Equation~\eqref{eq:MatrixEquation}
was derived for the SPDE in Equation~\eqref{eq:fullSPDE} with a finite volume
method. The derivations are quite involved and technical and are in
Appendix~\ref{app:DerivationPrecisionMatrix}.
However, when the form of the discretized SPDE
has been derived as an expression of the coefficients in the SPDE and the
grid, the conversion from SPDE to GMRF is automatic for any choice of coefficients
and rectangular domain.
\section{Examples of models}
\label{sec:Examples}
The simplest case of Equation~\eqref{eq:fullSPDE} is with constant
coefficients. In this case one has an isotropic model (up to boundary
effects) if $\mathbf{H}$ is a constant times the identity matrix
or a stationary anisotropic model (up to boundary effects) if this
is not the case. In both cases it is possible to calculate
an exact expression for the covariance function and the marginal variance
for the corresponding SPDE solved over $\mathbb{R}^2$.
For this purpose write
\[
\mathbf{H} = \begin{bmatrix} H_1 & H_2 \\ H_2 & H_3 \end{bmatrix},
\]
where $H_1$, $H_2$ and $H_3$ are constants. This gives the SPDE
\begin{equation}
\label{eq:genHomSPDE}
\left[\kappa^2-H_1 \frac{\partial^2}{\partial x^2}-2H_2 \frac{\partial^2}{\partial x \partial y}-H_3 \frac{\partial^2}{\partial y^2}\right]u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in\mathbb{R}^2.
\end{equation}
But if $\lambda_1$ and $\lambda_2$ are the eigenvalues of $\mathbf{H}$, then
the solution of the SPDE is actually only a rotated version of the solutions
of
\begin{equation}
\label{eq:rotHomSPDE}
\left[\kappa^2-\lambda_1 \frac{\partial^2}{\partial \tilde{x}^2}-\lambda_2 \frac{\partial^2}{\partial \tilde{y}^2}\right]u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in\mathbb{R}^2.
\end{equation}
Here the new $x$-axis is parallel to the eigenvector of $\mathbf{H}$
corresponding to $\lambda_1$ in the old coordinate system and
the new $y$-axis is parallel to the eigenvector of $\mathbf{H}$
corresponding to $\lambda_2$ in the old coordinate system.
From Proposition~\ref{prop:margVar} one can see that the marginal
variance of $u$ is
\[
\sigma_m^2 = \frac{1}{4\pi\kappa^2\sqrt{\mathrm{det}(\mathbf{H})}} = \frac{1}{4\pi\kappa^2\sqrt{\lambda_1\lambda_2}}.
\]
One can think of the eigenvectors of $\mathbf{H}$ as the two principal
directions and $\lambda_1$ and $\lambda_2$ as a measure of the ``strength''
of the diffusion in these principal directions. Additionally, if
$\lambda_1 = \lambda_2$, which is equivalent to $\mathbf{H}$ being
equal to a constant times the identity matrix, the SPDE is rotation
and translation invariant and the solution is isotropic. If
$\lambda_1 \neq \lambda_2$, the SPDE is still translation invariant, but
not rotation invariant, and the solutions are stationary, but not
isotropic.
In our case the domain is not $\mathbb{R}^2$, but $[0,A]\times[0,B]$ with
periodic boundary conditions. This means that a boundary effect is introduced
and the above results are only approximately true.
\subsection{Stationary models}
For a constant \(\mathbf{H}\) the SPDE in Equation~\eqref{eq:fullSPDE} becomes
\[
[\kappa^2-\nabla\cdot\mathbf{H}\nabla] u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in[0,A]\times[0,B].
\]
This SPDE can be rewritten as
\begin{equation}
[1-\nabla\cdot\hat{\mathbf{H}}\nabla] u(\boldsymbol{s}) = \sigma\mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in[0,A]\times[0,B],
\label{eq:ReparSPDE}
\end{equation}
where \(\hat{\mathbf{H}} = \mathbf{H}/\kappa^2\) and \(\sigma = 1/\kappa^2\).
From this form it is clear that \(\sigma\) is only a scale parameter and that it is
enough to solve for \(\sigma=1\) and then multiply the solution with the desired
value of \(\sigma\). Therefore, it is the effect of \(\hat{\mathbf{H}}\) that
is most interesting to study.
It is useful to parametrize \(\hat{\mathbf{H}}\) as
\[
\hat{\mathbf{H}} = \gamma \mathbf{I}_2 + \beta \boldsymbol{v}(\theta)\boldsymbol{v}(\theta)^\mathrm{T},
\]
where $\boldsymbol{v}(\theta) = [\cos(\theta),\sin(\theta)]^\mathrm{T}$,
\(\gamma>0\) and \(\beta>0\). In this
parametrization one can think of $\gamma$ as the coefficient of
the second order derivative in the direction orthogonal to
$\boldsymbol{v}(\theta)$ and $\gamma+\beta$ as the coefficient
of the second order derivative in the direction $\boldsymbol{v}(\theta)$.
Ignoring boundary effects, $\gamma$ and $\gamma+\beta$
are the coefficients of the second order derivatives in
Equation~\eqref{eq:rotHomSPDE} and $\theta$ is how much
the coordinate system has been rotated in positive direction.
\begin{exmp}[Stationary GMRF]
\label{exmp:statAni}
The purpose of this example is to consider the effects of using a constant
\(\hat{\mathbf{H}}\). Use the SPDE in Equation~\eqref{eq:ReparSPDE} with
domain \([0,20]\times[0,20]\) and periodic boundary conditions, and
discretize with a regular \(200\times 200\) grid. Two different values of
\(\hat{\mathbf{H}}\) are used, an isotropic case with
\(\hat{\mathbf{H}} = \mathbf{I}_2\) and an anisotropic case with
\(\gamma = 1\), \(\beta=8\) and \(\theta = \pi/4\). The anisotropic case
corresponds to a coefficient 9 in the \(x\)-direction and a coefficient
1 in the \(y\)-direction, and then a rotation of \(\pi/4\) in the
positive direction. The isotropic GMRF has marginal variances \(0.0802\)
and the anisotropic GMRF has marginal variances \(0.0263\). For
comparison Proposition~\ref{prop:margVar} gives \(0.0796\) and
\(0.0263\).
Figure~\ref{fig:statAni-obs} shows one realization for each of the
cases. Comparing Figure~\ref{fig:statAni-1-obs} and
Figure~\ref{fig:statAni-9-obs} it seems that the direction with the
higher coefficient for the second-order derivative has longer
range and more regular behaviour. Compared to the corresponding
partial differential equation (PDE) without the white noise,
this is what one would expect since
large values of the coefficient penalize large values of the second
order derivatives. One should expect that the correlation range
increases when the coefficient is increased.
\begin{figure}
\caption{\subref{fig:statAni-1-obs}
\label{fig:statAni-1-obs}
\label{fig:statAni-9-obs}
\label{fig:statAni-obs}
\end{figure}
This is in fact what happens. Figure~\ref{fig:statAni-cov} shows the
correlation of the variable at \((9.95, 9.95)\) with every other
point in the grid for the isotropic and the anisotropic case. This is
sufficient to describe all the correlations since the solutions are
stationary. One can immediately note that the iso-correlation curves
are close to ellipses with semi-axes along \(\boldsymbol{v}(\theta)\)
and the direction orthogonal to \(\boldsymbol{v}(\theta)\). One can see that the
correlation decreases most slowly and most quickly in the directions
used to specify \(\hat{\mathbf{H}}\), with slowest decrease along
\(\boldsymbol{v}(\theta)\). It is interesting to see
that both the isotropic case and the non-isotropic case has approximately
the same length for the minor semi-axis of the iso-correlation curves, and
that the major semi-axis is longer for the anisotropic case. This is
due to the fact that the lengths of the semi-axes are connected with
\(\sqrt{\gamma}\) and \(\sqrt{\gamma+\beta}\).
\begin{figure}
\caption{\subref{fig:statAni-1-cov}
\label{fig:statAni-1-cov}
\label{fig:statAni-9-cov}
\label{fig:statAni-cov}
\end{figure}
\end{exmp}
From the example above one can see that the use of 3 parameters allow for
the creation of GMRFs which are more regular in one direction than the other.
One can use the parameters \(\gamma\), \(\beta\) and \(\theta\) to control
the form of the correlation function and \(\sigma\) to get
the desired marginal variance.
\subsection{Non-stationary models}
To make the solution of the SPDE in Equation~\eqref{eq:fullSPDE}
non-stationary, either $\kappa^2$ or $\mathbf{H}$ has to be a non-constant
function. One way to achieve non-stationarity is by choosing
\[
\mathbf{H}(\boldsymbol{s}) = \gamma \mathbf{I}_2 + \beta \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
where $\boldsymbol{v}$ is a non-constant vector field on $[0,A]\times[0,B]$
which satisfy the periodic boundary conditions and $\gamma > 0$ and
$\beta > 0$ are constants.
\begin{exmp}[Non-stationary GMRF]
\label{exmp:nonStat}
Use the domain $[0,20]^2$ with a $200\times 200$ grid and
periodic boundary conditions for the SPDE in
Equation~\eqref{eq:fullSPDE}. Let $\kappa^2$ be equal to
$1$ and let ${\sf \bf H}$ be given as
\[
\mathbf{H}(\boldsymbol{s}) = \gamma \mathbf{I}_2 + \beta \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
where $\boldsymbol{v}$ is a $2$-dimensional vector field on $[0,20]^2$
which satisfies the periodic boundary conditions and $\gamma > 0$ and
$\beta > 0$ are constants.
To create an interesting vector field, start with the function
$f:[0,20]^2\rightarrow\mathbb{R}$ defined by
\[
f(x,y) = \left(\frac{10}{\pi}\right) \left(\frac{3}{4}\sin(2\pi x/20)+\frac{1}{4}\sin(2\pi y/20)\right).
\]
Then calculate the gradient $\nabla f$ and let
$\boldsymbol{v}:[0,20]^2\rightarrow\mathbb{R}^2$ be the gradient
rotated $90^{\circ}$ counter-clockwise at each point.
Figure~\ref{fig:nonStat-fvalues} shows the values of the function $f$
and Figure~\ref{fig:nonStat-ffield} shows the resulting vector field
$\boldsymbol{v}$. The vector field is calculated on a $400 \times 400$
regular grid, because the values between neighbouring cells in the
discretization are needed.
\begin{figure}
\caption{The gradient of the function illustrated
in~\subref{fig:nonStat-fvalues}
\label{fig:nonStat-fvalues}
\label{fig:nonStat-ffield}
\label{fig:nonStat-fill}
\end{figure}
Figure~\ref{fig:nonStat-obs} shows one realization from the resulting
GMRF with $\gamma = 0.1$ and $\beta =25$. A much higher value for
$\beta$ than $\gamma$ is chosen to illustrate the connection between
the vector field and the resulting covariance structure. From the
realization it is clear that there is stronger dependence along the
directions of the vector field shown in
Figure~\ref{fig:nonStat-ffield} at each point than in the other
directions. In addition, from Figure~\ref{fig:nonStat-mvar} it seems
that positions with large values for the norm of the vector field has
smaller marginal variance than positions with small values and vice
versa. This feature introduces an undesired connection between
anisotropy and marginal variances. It is possible to reduce this interaction
between the vector field and the marginal variances by
reformulating the controlling SPDE as discussed briefly
in Section~\ref{sec:Extensions}.
\begin{figure}
\caption{One observation and the marginal variances of the
solution of the SPDE in Equation~\eqref{eq:fullSPDE}
\label{fig:nonStat-obs}
\label{fig:nonStat-mvar}
\label{fig:nonStat-fobs}
\end{figure}
From Figure~\ref{fig:nonStat-covmm} and Figure~\ref{fig:nonStat-covs}
one can see that the correlations depend on the direction and norm
of the vector field, and that there is clearly non-stationarity.
Figure~\ref{fig:nonStat-covld} and Figure~\ref{fig:nonStat-covlu}
show that the correlations with the positions $(4.95,1.95)$
$(4.95,7.95)$ tend to follow the vector field around the point
$(5,5)$, whereas Figure~\ref{fig:nonStat-covrd} and
Figure~\ref{fig:nonStat-covru} show that the correlations with
the positions $(14.95,1.95)$ and $(14.95,7.95)$ tend to
follow the vector field away from the point $(15,5)$.
Figure~\ref{fig:nonStat-covlm} shows that the correlations with
position $(4.95, 4.95)$ and every other point is not isotropic, but
concentrated close to the point itself, and Figure~\ref{fig:nonStat-covrm}
shows that the correlations with position $(14.95,4.95)$ have
high correlation along four directions which extends out from the
point. Figure~\ref{fig:nonStat-covmm} shows that the
correlations with position \((9.95, 9.95)\) ``follow'' the vector field
with high correlations in the vertical direction.
\begin{figure}
\caption{Correlations with position $(9.95,9.95)$ and all
other points for the solution of the SPDE in
Example~\ref{exmp:nonStat}
\label{fig:nonStat-covmm}
\end{figure}
\begin{figure}
\caption{Correlations for different points with all other points for
the solution of the SPDE in Example~\ref{exmp:nonStat}
\label{fig:nonStat-covld}
\label{fig:nonStat-covrd}
\label{fig:nonStat-covlu}
\label{fig:nonStat-covru}
\label{fig:nonStat-covlm}
\label{fig:nonStat-covrm}
\label{fig:nonStat-covs}
\end{figure}
\end{exmp}
From this example one can see that allowing $\mathbf{H}$ to be non-constant
means that one can vary the dependence structure in more interesting ways than
the stationary anisotropic fields. Secondly, using a vector field to control
how $\mathbf{H}$ varies means that the resulting correlation structure can be
partially visualized from the vector field. Thirdly, when $\gamma >0$ this
construction guarantees that $\mathbf{H}$ is everywhere positive definite.
\section{Inference}
\label{sec:Inference}
This section begins with a discussion of the parametrization of the model
and the derivation of the posterior distribution. Then the properties
of the inference scheme are discussed through some examples with simulated data.
\subsection{Posterior distribution and parametrization}
The first step for inference is to introduce parameters that control the
behaviour of the coefficients in Equation~\eqref{eq:fullSPDE} and in turn
the behaviour of the GMRF. The way this is done is by expanding each of
the functions in a basis and use a linear combination of the basis functions
weighted by parameters. For \(\kappa^2\) only one parameter, say
\(\theta_1\), is needed as it is assumed constant, but for the function
\(\mathbf{H}\) a vector of parameters \(\boldsymbol{\theta}_2\) is needed.
Set \(\boldsymbol{\theta} = (\theta_1, \boldsymbol{\theta}_2^\mathrm{T})\)
and give it a prior \(\boldsymbol{\theta}\sim \pi(\boldsymbol{\theta})\).
Then for each value of \(\boldsymbol{\theta}\), the discretization
in Appendix~\ref{sec:discScheme} is used to construct the GMRF
\(\boldsymbol{u}|\boldsymbol{\theta} \sim \mathcal{N}(\boldsymbol{0}, \mathbf{Q}(\boldsymbol{\theta})^{-1})\).
Combine the prior of $\boldsymbol{\theta}$ with this conditional distribution
to find the joint distribution of the parameters and $\boldsymbol{u}$.
Together with a model for how an observation \(\mathbf{y}\) is made from the
underlying GMRF this forms a hierarchical spatial model. The relationship
between \(\boldsymbol{y}\) and \(\boldsymbol{u}\) is chosen to be particularly
simple, namely that linear combinations of \(\mathbf{u}\) are observed
with Gaussian noise,
\[
\boldsymbol{y}|\boldsymbol{u} \sim \mathcal{N}(\mathbf{A}\boldsymbol{u},\mathbf{Q}_\mathrm{N}^{-1}),
\]
where $\mathbf{Q}_\mathrm{N}$ is a known precision matrix.
The purpose of the hierarchical model is to do inference
on $\boldsymbol{\theta}$ based on an observation of $\boldsymbol{y}$. With a
Gaussian latent model it is possible to integrate out the
latent field \(\boldsymbol{u}\) exactly and this leads to the log-posterior
\begin{eqnarray}
\lefteqn{\log(\pi(\boldsymbol{\theta}|\boldsymbol{y})) =} \notag\\
& & \mathrm{Const}+\log(\pi(\boldsymbol{\theta}))+\frac{1}{2}\log(|\mathbf{Q}(\boldsymbol{\theta})|) \notag \\
& & { } -\frac{1}{2}\log(|\mathbf{Q}_\mathrm{C}(\boldsymbol{\theta})|)+\frac{1}{2}\boldsymbol{\mu}_C(\boldsymbol{\theta})^\mathrm{T}\mathbf{Q}_\mathrm{C}(\boldsymbol{\theta})\boldsymbol{\mu}_C(\boldsymbol{\theta}),
\label{eq:postty}
\end{eqnarray}
where $\mathbf{Q}_\mathrm{C}(\boldsymbol{\theta}) = \mathbf{Q}(\boldsymbol{\theta})+\mathbf{A}^\mathrm{T}\mathbf{Q}_\mathrm{N} \mathbf{A}$ and $\boldsymbol{\mu}_\mathrm{C}(\boldsymbol{\theta}) = \mathbf{Q}_\mathrm{C}(\boldsymbol{\theta})^{-1}\mathbf{A}^\mathrm{T}\mathbf{Q}_\mathrm{N} \boldsymbol{y}$.
From the above expression one can see that the posterior distribution of
$\boldsymbol{\theta}$ contains terms which are hard to handle analytically.
It is hard to say anything about both the determinants and the quadratic term
as functions of $\boldsymbol{\theta}$. Therefore, the inference is done
numerically. The model is on a form which could be handled by the INLA
methodology~\cite{Rue2009}, but at the time of writing the R-INLA
software\footnote{\url{www.r-inla.org}} does not have the model implemented. Instead
the parameters are estimated with maximum a posteriori estimates
based on the posterior density given in Equation~\eqref{eq:postty}. In
addition, the standard deviations are estimated from the square roots
of the diagonal elements of the observed information matrix.
The parametrization of \(\mathbf{H}\) introduced in the previous section employs
a pre-defined vector field and a parameter \(\beta\) that controls the magnitude
of the anisotropy due to this vector field. This is a useful representation for
achieving a desired dependence structure, but in a inference setting there
may not be any pre-defined vector field to input. Therefore, the vector field
itself must be estimated. In this context the decomposition of \(\mathbf{H}\)
introduced in Section~\ref{sec:Preliminaries},
\[
\mathbf{H}(\boldsymbol{s}) = \gamma \mathbf{I}_2 + \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
is more useful. For inference it is necessary to control the vector
field by a finite number of parameters. The simple case of a
constant matrix requires 3
parameters. Use parameters \(\gamma\), \(v_1\) and \(v_2\) and write
\[
\mathbf{H}(\boldsymbol{s}) \equiv \gamma \mathbf{I}_2 + \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\begin{bmatrix}v_1 & v_2 \end{bmatrix}.
\]
If $\mathbf{H}$ is not constant, it is necessary to parametrize the
vector field $\boldsymbol{v}$ in some manner. Any vector field is
possible for $\boldsymbol{v}$, so a basis which can generate any vector
field is desirable. The Fourier basis possesses this
property, but is only one of many possible choices.
Let the domain be $[0,A]\times[0,B]$ and assume that $\boldsymbol{v}$ is a differentiable, periodic
vector field on the domain. Then each component of the vector field can be written as a Fourier
series of the form
\[
\sum_{(k,l)\in \mathbb{Z}^2}C_{k,l}\exp\left[2\pi i\left(\frac{k}{A}x+\frac{l}{B}y\right)\right],
\]
where $i$ is the imaginary unit. But since the components are real-valued,
each of them can also be written as a real $2$-dimensional Fourier series of the form
\[
A_{0,0}+\sum_{(k,l)\in E} \left[A_{k,l}\cos\left[2\pi\left(\frac{k}{A}x+\frac{l}{B}y\right)\right] + B_{k,l} \sin\left[2\pi\left(\frac{k}{A}x+\frac{l}{B}y\right)\right]\right],
\]
where the set $E\subset\mathbb{Z}^2$ is given by
\[
E = (\mathbb{N}\times\mathbb{Z})\cup (\{0\}\times\mathbb{N}).
\]
Putting these Fourier series together gives
\begin{eqnarray}
\lefteqn{\boldsymbol{v}(\boldsymbol{s}) =} \notag \\
& & \begin{bmatrix}A_{0,0}^{(1)} \\ A_{0,0}^{(2)} \end{bmatrix}+\sum_{(k,l)\in E} \begin{bmatrix} A_{k,l}^{(1)} \\ A_{k,l}^{(2)} \end{bmatrix}\cos\left[2\pi\left(\frac{k}{A}x+\frac{l}{B}y\right)\right] + \notag\\
& & \sum_{(k,l)\in E} \begin{bmatrix} B_{k,l}^{(1)} \\ B_{k,l}^{(2)}\end{bmatrix} \sin\left[2\pi\left(\frac{k}{A}x+\frac{l}{B}y\right)\right],
\label{chap3:eq:fullFour}
\end{eqnarray}
where $A_{k,l}^{(1)}$ and $B_{k,l}^{(1)}$ are the coefficients for the first component of $\boldsymbol{v}$ and
$A_{k,l}^{(2)}$ and $B_{k,l}^{(2)}$ are the coefficients of the second component.
This gives 2 coefficients when only the zero-frequency is included, then 18 parameters
when the $(0,1)$, $(1,-1)$, $(1,0)$ and $(1,1)$ frequencies are included. When
the number of frequencies used in each direction doubles, the number
of required parameters quadruples.
\subsection{Inference on simulated data}
In this section we consider data generated from a known set of parameters.
The prior used is an improper prior that disallows illegal parameter values.
It is uniform on \((0,\infty)\) for \(\gamma\) and uniform on
\(\mathbb{R}\) for the rest of the parameters in \(\mathbf{H}\).
The first issue to investigate is whether it is possible to estimate the
stationary model with exactly observed data and whether the approximate
estimation scheme performs well.
\begin{exmp}
\label{exmp:aniInf}
Use the SPDE
\begin{equation}
\label{eq:aniInf}
u(\boldsymbol{s}) - \nabla\cdot\mathbf{H}\nabla u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s}\in[0,20]\times[0,20],
\end{equation}
where $\mathcal{W}$ is a standard Gaussian white noise process and
$\mathbf{H}$ is a $2 \times 2$ matrix, with periodic boundary conditions. Let
\[
\mathbf{H} = 3 \mathbf{I}_2 + 2\boldsymbol{v}\boldsymbol{v}^\mathrm{T},
\]
with $\boldsymbol{v} = (1,\sqrt{3})/2$. This means that $\mathbf{H}$ has eigenvector
$\boldsymbol{v}$ with eigenvalue $5$ and an eigenvector orthogonal to $\boldsymbol{v}$ with
eigenvalue $3$. Construct the GMRF on a $100 \times 100$ grid.
One observation of the solution is shown in Figure~\ref{fig:aniInf}. Assume
that the fact that $\mathbf{H}$ is constant is known, but that its value is not. Then using
the decomposition from the previous sections one can write
\[
\mathbf{H} = \gamma \mathbf{I}_2 + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}\begin{bmatrix}v_1 & v_2 \end{bmatrix},
\]
where $\gamma$, $v_1$ and $v_2$ are the parameters. Since the process is assumed to
be exactly observed, we can use the distribution of
\(\boldsymbol{\theta}|\boldsymbol{u}\). This gives the posterior estimates shown in
Table~\ref{tab:aniInfConst}. From the table one can see
that all the estimates are accurate to one digit, and within one standard deviation of
the true value. Actually, this decomposition of $\mathbf{H}$ is
invariant to changing $\boldsymbol{v}$ with $-\boldsymbol{v}$, so
there are two choices of parameters that means the same.
\begin{figure}
\caption{One realization of the solution of the SPDE in Example~\ref{exmp:aniInf}
\label{fig:aniInf}
\end{figure}
\begin{table}
\centering
\caption{\sf Parameter estimates for Example~\ref{exmp:aniInf}.}
\begin{tabular}{llll}
\textbf{Parameter} & \textbf{True value} & \textbf{Estimate} & \textbf{Std.dev.} \\
$\gamma$ & 3 & 2.965 & 0.070 \\
$v_1$ & 0.707 & 0.726 & 0.049 \\
$v_2$ & 1.225 & 1.231 & 0.039
\end{tabular}
\label{tab:aniInfConst}
\end{table}
The biases in the estimates were evaluated by generating 10000 datasets
from the true model and estimating the parameters for each dataset. The estimated bias was less than or equal to
\(0.1\%\) of the true value for each parameter. Additionally, the
sample standard deviations based on the estimation of the parameters
for each of the
10000 datasets were \(0.070\), \(0.050\) and \(0.039\) for \(\gamma\), \(v_1\) and \(v_2\), respectively. Each one corresponds well to the
corresponding approximate standard deviation, computed
via the observed
information matrix as described in the previous section, that is shown
in Table~\ref{tab:aniInfConst}.
\end{exmp}
In the above example it is possible to estimate the model,
but this is under the assumption that
it is known beforehand that the model is stationary. In general, it is
not reasonable to be able to know this beforehand. Therefore, the estimation
is repeated for the more complex model developed in the previous sections
which allows for significant non-stationarity controlled through a vector field.
The intention is to evaluate whether the more complex model
is able to detect that the true model is a stationary model and if there
are identifiability issues.
\begin{exmp}
\label{exmp:15par}
Use the same SPDE and observation as in Example~\ref{exmp:aniInf}, but
assume that it is not known that $\mathbf{H}$ is constant. Add the
terms in the Fourier series corresponding to the next
frequencies, $(k,l) = (0,1)$, $(k,l)=(1,-1)$, $(k,l) = (1,0)$
and $(k,l)=(1,1)$. The observation is still assumed to be exact, but
there are 16 additional parameters, 4 additional
parameters for each frequency.
First two arbitrary starting positions are chosen for the optimization.
The first is $\gamma = 3.0$ and all other parameters at $0.1$. And the
second is $\gamma = 3.0$, $A_{0,0}^{(1)} = 0.1$, $A_{0,0}^{(2)} = 0.1$ and
all other parameters equal to $0$. For both of these starting points the
optimization converges to non-global maximums. Parameter estimates and
approximate standard deviations are not show, but Figure~\ref{fig:aniInf-vec}
shows the two different vector fields found.
A third optimization is done with starting values close to the correct
parameter values. This gives a vector field close to the actual one, with
estimates for $\gamma$, $A_{0,0}^{(1)}$ and $A_{0,0}^{(2)}$ that agree with
the ones in Example~\ref{exmp:aniInf} to two digits. The other frequencies
all had coefficients close to zero, with the largest having an absolute value
of $0.058$.
\begin{figure}
\caption{Two different local maxima found for the vector field. The vector field in \subref{fig:aniInf-vec1}
\label{fig:aniInf-vec1}
\label{fig:aniInf-vec2}
\label{fig:aniInf-vec}
\end{figure}
The results illustrate a difficulty with estimation caused by the
the inherent non-identifiability of the sign of the vector field.
The true vector field is constant and in Figure~\ref{fig:aniInf-vec}
one can see that each vector field has large parts
which has the correct appearance if one only considers the lines
defined by the arrows and not in which of the two possible directions
that the arrow points. The positions where the vector field
is wrong are smaller areas where the vector field flips its direction.
The problem is that it is difficult to reverse this flipping as it
requires moving through states with smaller likelihood. Thus creating
undesirable local maximums. One approach to improving the
situation would be to force an apriori preference for vector
fields without abrupt changes. That is to introduce a prior
which forces higher frequencies
of the Fourier basis to be less desirable. This is an issue that
needs to be addressed for an application and is briefly discussed
in Section~\ref{sec:Extensions}.
By acknowledging the issue and starting close to the true value, one
can do repeated simulations of datasets and prediction of parameters
to evaluate how well the non-stationary model captures the fact
that the true model is
stationary and see if there is any consistent bias. 1000 datasets
were simulated and the estimation of the parameters was done for each
dataset with a starting value close to the true value. This gives
the result summarized in the boxplot in
Figure~\ref{fig:biasInVectorField}. There does not appear to be
any significant bias and the parameters that give non-stationarity
are all close to zero.
\begin{figure}
\caption{Boxplot of estimated parameters for 1000 simulated
datasets in Example~\ref{exmp:15par}
\label{fig:biasInVectorField}
\end{figure}
\end{exmp}
The example shows that there are issues in estimating the anisotropy in the
non-stationary model due to the non-identifiability of the sign of the
vector field, but that if one avoids the local maximums the estimated model
is close to the true stationary model in this case. In addition,
there is a significant
increase in computation time when increasing the parameter space from
\(3\) to \(19\) parameters. The computation time required is increased
by a factor of approximately $10$.
A high-dimensional model increases the flexibility, but as seen above also
adds additional difficulties. In situations where there is a physical
explanation of the additional dependence in one direction, it would be
desirable to do a simpler model with one parameter for the baseline
isotropic effect and one parameter specifying the degree of anisotropy
caused by a pre-defined vector field such as in Example~\ref{exmp:nonStat}.
This presents a simplification from the previous inference examples because
the vector field itself does not need to be estimated.
\begin{exmp}
\label{exmp:finf}
Use a $100 \times 100$ grid of $[0,20]^2$ and
periodic boundary conditions for the SPDE in Equation~\eqref{eq:fullSPDE}.
Let $\kappa^2$ be equal to $1$ and let $\mathbf{H}$ be parametrized
as
\[
\mathbf{H}(\boldsymbol{s}) = \gamma \mathbf{I}_2 + \beta \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
where $\boldsymbol{v}$ is the vector field from Example~\ref{exmp:nonStat}.
Figure~\ref{fig:inf-fobs} shows one observation of the solution with
$\gamma = 0.5$ and $\beta = 5$. In this case one expects that it is
possible to make accurate estimates about $\gamma$ and $\beta$ as the
situation is simpler than in the previous example.
\begin{figure}
\caption{An observation of the SPDE in Equation~\eqref{eq:fullSPDE}
\label{fig:inf-fobs}
\end{figure}
The estimated parameters are shown in Table~\ref{tab:finf}.
From the table one can see that the estimates for both $\gamma$ and $\beta$
are quite accurate, which is reflected both in the actual value of the estimates and the approximated standard deviations. The estimates for both
$\gamma$ and $\beta$ are accurate to $2$ digits. In a similar way as
in the previous example, the bias is estimated to be less than \(0.02\%\)
for each each parameter, and the sample standard deviation from
estimation over many datasets is 0.008 and 0.08 for \(\gamma\) and
\(\beta\) respectively.
\begin{table}
\centering
\caption{\sf Posterior inference on parameters in Example~\ref{exmp:finf}.}
\begin{tabular}{llll}
{\bf Parameter} & {\bf True value} & {\bf Estimate} & {\bf Std.dev.} \\
$\gamma$ & 0.5 & 0.5012 & 0.0081 \\
$\beta$ & 5 & 5.014 & 0.084
\end{tabular}
\label{tab:finf}
\end{table}
\end{exmp}
The above example does not have the same issues as Example~\ref{exmp:15par} where
the vector field itself must be estimated. The example shows that when using
only the $\gamma \mathbf{I}_2$ term and fixed vector field where only the
magnitude of the effect is controlled by a parameter \(\beta\), the estimates
of the parameters are quite accurate. The accuracy of the estimates
will of course depend on the vector field used.
In a more realistic situation the actual basis needed for the vector field
is not known and there is observation noise. In the following example the
estimation is compared when all required frequencies are included and
when only a subset of the required frequencies of the Fourier basis is included.
\begin{exmp}
\label{exmp:noisy-inf}
Use a $100 \times 100$ grid of $[0,20]^2$ and
periodic boundary conditions for the SPDE in Equation~\eqref{eq:fullSPDE}.
Let $\kappa^2$ be equal to $1$ and let $\mathbf{H}$ be given
as
\[
\mathbf{H}(\boldsymbol{s}) = \mathbf{I}_2 + \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T},
\]
where $\boldsymbol{v}$ is the vector field
\[
\boldsymbol{v}(x,y) = \begin{bmatrix} 2+\cos\left(\frac{\pi}{10}x\right) \\ 3+2\sin\left(\frac{\pi}{10}y\right)+\sin\left(\frac{\pi}{10}(x+y)\right)\end{bmatrix}.
\]
One observation with i.i.d.\ Gaussian noise with precision $400$ is shown
in Figure~\ref{fig:noisy-inf-obs}.
Based on this realization it is desired to estimate the correct value of
$\gamma$ and the correct vector field
$\boldsymbol{v}$ in the parametrization
\[
\mathbf{H}(\boldsymbol{s}) = \gamma \mathbf{I}_2 + \boldsymbol{v}(\boldsymbol{s})\boldsymbol{v}(\boldsymbol{s})^\mathrm{T}.
\]
First use only one extra frequency in each direction, that is only the frequencies $(0,0)$, $(0,1)$ and $(1,0)$.
This gives the estimated vector field shown in Figure~\ref{fig:noisy-vec-some}. Then add the missing frequency
and use the frequencies $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. This gives the estimated vector field shown in
Figure~\ref{fig:noisy-vec-all}. The true vector field is shown in Figure~\ref{fig:noisy-vec-real}.
\begin{figure}
\caption{An observation of the SPDE in Example~\ref{exmp:noisy-inf}
\label{fig:noisy-inf-obs}
\end{figure}
Both estimated vector fields are quite similar to the true vector field, and
the $\gamma$ parameter was estimated to $1.14$ in the first case and $1.09$ in the latter case. There is a clear bias in the estimate of \(\gamma\), but
this must be expected as there is a need to compensate for the lacking
frequencies. All parameter values were estimated, but are not shown.
For the first case many parameters is more than two standard deviations
from their correct values and in the second case this only
happens for one parameter. For each case the difference between the
true \(\mathbf{H}\) and the estimated \(\hat{\mathbf{H}}\) is calculated
through
\[
\frac{1}{100}\sqrt{\sum_{i=1}^{100}\sum_{j=1}^{100} \left\lvert\left\lvert \mathbf{H}(\boldsymbol{s}_{i,j})-\hat{\mathbf{H}}(\boldsymbol{s}_{i,j})\right\rvert\right\rvert_2^2},
\]
where \(\boldsymbol{s}_{i,j}\) are the centres of the cells in the
grid and \(||\cdot||_2\) denotes the 2-norm.
The case with frequencies \((0,0)\), \((0,1)\) and \((1,0)\) gives
7.9 and the case with frequencies \((0,0)\), \((0,1)\), \((1,0)\) and
\((1,1)\) gives 1.5.
\begin{figure}
\caption{True vector field and inferred vector fields in
Example~\ref{exmp:noisy-inf}
\label{fig:noisy-vec-some}
\label{fig:noisy-vec-all}
\label{fig:noisy-vec-real}
\label{fig:noisy-vec}
\end{figure}
\end{exmp}
These examples focus on simple cases where specific issues can be highlighted.
The inherent challenges in estimating a spatially varying direction and strength
are equally important in the more general setting where also
\(\kappa\) and the baseline effect \(\gamma\) is allowed to vary. The estimation
of the vector field presents an important component that must be dealt with
in any inference strategy for the more general case.
\section{Extensions}
\label{sec:Extensions}
The class of models discussed in the previous sections offers a
flexible way to introduce and control directional dependence at each
location using a vector field. This is an important step towards a full
flexible non-stationary
model for practical applications, but still leaves something to be desired.
To make the model applicable to real-world datasets it is necessary to also
make the parameters \(\kappa\) and \(\gamma\) spatially varying functions.
This results in some control also over the marginal variance and the strength
of the local baseline component of the anisotropy at each location. A varying
\(\kappa\) is discussed briefly in Section~3.2 in~\ocite{Lindgren2011}.
However, this comes at the cost of two more functions that must be
inferred together with the vector field \(\boldsymbol{v}\). Which in turn means
two more functions that need to be expanded into bases. This could be done
in a similar way as for the vector field with a Fourier basis, but the Fourier
basis does not constitute the only possible choice, and any basis which
respects the boundary condition could in principle be used. But the amount
of freedom available by having four spatially varying functions comes at a
price, and it would be necessary to introduce some apriori restrictions on
the behaviour of the functions.
In Example~\ref{exmp:15par} the challenge with the non-identifiability
of the sign of the vector field is demonstrated. It would be possible to make
the situation less problematic by enforcing more structure in the
estimated vector field. For example, through spline penalties which adds a
preference for components without abrupt changes. Such apriori restrictions
make sense both from a modelling perspective, in the sense that the properties
should not change to quickly, and from a computational perspective, in
the sense that it is desirable to avoid situations as the one encountered
in the previous section where the direction of the vector field flips.
The full model could be used in a real-world application through a
three step approach. First, choose an appropriate basis to use for each
function and select an appropriate prior. This means deciding how many
basis elements one is willing to use, from a computational point of view,
and how strong the apriori penalties
needs to be. Second, find the maximum aposteriori estimate of the
functions \(\kappa\), \(\gamma\), \(v_1\) and \(v_2\). Third, assume the
maximum aposteriori estimates are the true functions and calculate the
predicted values and prediction variances. The full details of such
an approach is beyond the scope of this paper and is being studied
in current work on an application to annual rainfall data in the conterminous
US~\cite{Fuglstad2013b}.
Another way forward deals with the interactions of
the functions \(\kappa\), \(\gamma\), \(v_1\) and \(v_2\).
The functions interact in difficult ways to control marginal
variance and to control anisotropy. As seen in Example~\ref{exmp:nonStat} the
vector field that controls the anisotropic behaviour is also linked to the
marginal variances of the field. It would be desirable to try to separate
the functions that are allowed to affect the marginal variances and the
functions that are allowed to affect the correlation structure. This
may present a useful feature in applications, both for interpretability
and for constructing priors.
One promising way to greatly reduce this interaction is to extend on the
ideas presented in Section~3.4 in~\ocite{Lindgren2011}. The section links
the use of an anisotropic Laplacian to the deformation method
of~\ocite{Sampson1992}. The link presented is in itself
too restrictive, but the last comments about the connection to metric tensors
leads to a useful way to rewrite the SPDE in Equation~\eqref{eq:fullSPDE}.
This is work in progress and involves interpreting the simple SPDE
\begin{equation}
[1-\Delta]u = \mathcal{W}
\label{eq:SIMPLE}
\end{equation}
as an SPDE on a Riemannian manifold with an inverse metric tensor
defined through the strength of dependence in different directions in a similar way
as the spatially varying matrix \(\mathbf{H}\). This leads to a slightly
different SPDE, where a separate function, which does not affect
correlation structure, can be used to control marginal standard deviations.
However, the separation is not perfect since the varying metric tensor
gives a curved space and thus affects the marginal variances of the solution
of the above SPDE. But the effect
of the metric tensor on marginal standard deviations appears small, and it
appears to be a promising way forward.
Another issue which is not addressed in the previous sections is how
to define relevant boundary conditions. For rectangular domains,
periodic boundary conditions as used here are simple to implement, but
a naive use of such conditions will typically not inappropriate in
practical applications due to the resulting spurious dependence
between physically distant locations. This problem can be partly
rectified by embedding the region of interest into a larger covering
domain, so that the boundary effects are moved away from the region
that directly influences the likelihood function. It is also possible
to apply Neumann type boundary conditions similar to the ones used by
\ocite{Lindgren2011}. These are easier to adapt to more general
domains, but they still require a domain extension in order to remove
the influence of the boundary condition on the likelihood. A more
theoretically appealing, and computationally potentially less
expensive, solution would be to directly define the behaviour of the
field along the boundary so that the models would contain stationary
fields as a neutral case. Work is underway to design stochastic
boundary conditions to accomplish this, and some of the solutions show
potential for extension to non-stationary models.
\section{Discussion}
\label{sec:Discussion}
The paper explores different aspects of a new class of non-stationary
GRFs based on local anisotropy. The benefit of the formulation presented is that
it allows for flexible models with few requirements on the parameters.
Since the GRF is based on an SPDE, there is no need to worry about how to
change the discretized model in a consistent manner when the grid is refined.
In other words, one does not need to worry
about how the precision matrix must be changed to give a similar
covariance structure when the number of grid points is increased. This is
one of the more attractive features of the SPDE-based modelling.
The focus of the examples has been the matrix \(\mathbf{H}\) introduced
in the Laplace-operator. The examples show that a variety of different
effects can be achieved by using different types of spatially varying
matrices. Constant matrices of the form \(\gamma \mathbf{I}_2\) give
isotropic random fields and constant matrices of other forms give anisotropic,
stationary random fields.
As shown in Section~\ref{sec:Examples} the anisotropic fields have
anisotropic Mat\'{e}rn-like covariance functions, through stretching and rotating
the domain, and can be controlled by four parameters. It is possible to
control the marginal variances, the principal directions and the range
in each of the principal directions. A spatially varying \(\mathbf{H}\)
gives non-stationary random fields. And by using a vector field to
specify the strength and direction of extra spatial dependence in
each location, there is a clear connection between the vector field
and the resulting covariance structure. The covariance structure
can be partially visualized from the vector field.
From the examples in Section~\ref{sec:Inference} one can see that sensible
values for the parameters are estimated both with and without noise, except
for problems with multimodality in Example~\ref{exmp:15par}, which uses
a more flexible construction for the vector field than the other examples.
Additionally, the examples show no significant biases in the estimate.
The last example presents the most challenging case, where the true model
cannot be represented by the model estimated, and is perhaps closest to a
real scenario. In the example good results are achieved when estimating
the vector field with only a subset of the frequencies required to fully
describe it.
There are many avenues that are not explored in this paper due to
the fact that it is a first look into a new type of model. The chief
motivation is to explore the class of models both in the sense of
what can be achieved and associated challenges for inference with the
model. In this paper it is shown that a vector field constitutes a useful way
to control local anisotropy in the SPDE-model of~\ocite{Lindgren2011}.
What remains for a fully flexible spatial model is to allow
also $\kappa$ and $\gamma$ to be spatially varying functions. However,
this is a simpler task than the anisotropy component since they do
not require vector fields. For this more complex model there will
be \(4\) spatially varying functions to estimate and an expansion
of each of these functions into a basis will lead to many parameters.
This means it is necessary to explore ways of dealing with
high-dimensional estimation problems. Additionally, it remains to
investigate appropriate choices of priors for use in applications.
This question is connected with the discussion in
Section~\ref{sec:Extensions} on an alternative construction of the
model which separates the functions that are allowed to affect
marginal variances and the functions that allowed to affect
correlation structure.
\section{Derivation of precision matrix}
\label{app:DerivationPrecisionMatrix}
\subsection{Formal equation}
The SPDE is
\begin{equation}
(\kappa^2(\boldsymbol{s}) -\nabla\cdot\mathbf{H}(\boldsymbol{s}))\nabla u(\boldsymbol{s}) = \mathcal{W}(\boldsymbol{s}), \qquad \boldsymbol{s} \in [0,A]\times [0,B],
\label{eq:mainSPDE}
\end{equation}
where $A$ and $B$ are strictly positive constants, $\kappa^2$ is a scalar
function, ${\sf \bf H}$ is a $2\times 2$ matrix-valued function,
$\nabla=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$
and $\mathcal{W}$ is a standard Gaussian white noise process. In addition,
$\kappa^2$ is assumed to be a continuous, strictly positive function and
$\mathbf{H}$ is assumed to be a continuously differentiable function which
gives a positive definite matrix $\mathbf{H}(\boldsymbol{s})$ for each
$\boldsymbol{s}\in[0,A]\times[0,B]$.
Further, periodic boundary conditions are used, which means that opposite sides
of the rectangle $[0,A]\times[0,B]$ are identified. This gives
additional requirements for $\kappa^2$ and $\mathbf{H}$. The values of
$\kappa^2$ must agree on opposite edges and the values of $\mathbf{H}$ and
its first order derivatives must agree on opposite edges.
The periodic boundary conditions are not essential to the methodology presented
in what follows, but were chosen to avoid the issue of appropriate boundary
conditions.
\subsection{Finite volume methods}
\label{sec:finVolMeth}
In the discretization of the SPDE in Equation~\eqref{eq:mainSPDE} a finite
volume method is employed. The finite volume methods are useful for
creating discretizations of conservation laws of the form
\[
\nabla\cdot \boldsymbol{F}(\boldsymbol{x},t) = f(\boldsymbol{x},t),
\]
where $\nabla\cdot$ is the spatial divergence operator. This equation relates
the spatial divergence of the flux $\boldsymbol{F}$ and the sink-/source-term $f$.
The main tool in these methods is the use of the divergence theorem
\begin{equation}
\int_E \! \nabla\cdot\boldsymbol{F} \, \mathrm{d}V = \oint_{\partial E} \! \boldsymbol{F}\cdot \boldsymbol{ n} \,\mathrm{d}\sigma,
\label{eq:divThm}
\end{equation}
where $\boldsymbol{n}$ is the outer normal vector of the surface $\partial E$
relative to $E$.
The main idea is to divide the domain of the SPDE in
Equation~\eqref{eq:mainSPDE} into smaller parts and consider the resulting
``flow'' between the different parts. A lengthy treatment of finite volume
methods is not given, but a comprehensive treatment of the method for
deterministic differential equations can be found in~\ocite{eymard2000finite}.
\subsection{Derivation}
\label{sec:discScheme}
To keep the calculations simple the domain is divided into a regular grid of
rectangular cells. Use $M$ cells in the $x$-direction and $N$ cells in the
$y$-direction. Then for each cell the sides parallel to the $x$-axis have
length $h_x=A/M$ and the sides parallel to the $y$-axis have length
$h_y = B/N$. Number the cells by $(i,j)$, where $i$ is the column of the cell
(along the $x$-axis) and $j$ is the row of the cell (along the $y$-axis). Call
the lowest row $0$ and the leftmost column $0$, then cell $(i,j)$ is
\[
E_{i,j} = [i h_x,(i+1) h_x] \times [j h_y, (j+1) h_y].
\]
Using this notation the set of cells, $\mathcal{I}$, is given by
\[
\mathcal{I} = \{E_{i,j} : i = 0,1,\ldots,M-1, j = 0,1,\ldots, N-1 \}.
\]
Figure~\ref{fig:discGrid} shows an illustration of the discretization of
$[0,A]\times[0,B]$ into the cells $\mathcal{I}$.
\begin{figure}
\caption{Illustration of the division of $[0,A]\times[0,B]$ into a
regular $M\times N$ grid of rectangular cells.}
\label{fig:discGrid}
\end{figure}
Each cell has four faces, two parallel to the $x$-axis (top and bottom) and
two parallel to the $y$-axis (left and right). Let the right face, top face,
left face and bottom face of cell $E_{i,j}$ be denoted $\sigma_{i,j}^{\mathrm{R}}$,
$\sigma_{i,j}^{\mathrm{T}}$, $\sigma_{i,j}^{\mathrm{L}}$ and $\sigma_{i,j}^{\mathrm{B}}$,
respectively. Additionally, denote by $\sigma(E_{i,j})$ the set of faces
of cell $E_{i,j}$.
For each cell $E_{i,j}$, $\boldsymbol{s}_{i,j}$
gives the centroid of the cell, and $\boldsymbol{s}_{i+1/2,j}$,
$\boldsymbol{s}_{i,j+1/2}$, $\boldsymbol{s}_{i-1/2,j}$
and $\boldsymbol{s}_{i,j-1/2}$ give the centres of
the faces of the cell. Due to the periodic boundary conditions, the $i$-index
and $j$-index in $\boldsymbol{s}_{i,j}$ are modulo $M$ and modulo $N$,
respectively. Figure~\ref{fig:oneCell} shows one cell $E_{i,j}$ with the
centroid and the faces marked on the figure.
Further, let $u_{i,j} = u(\boldsymbol{s}_{i,j})$ for each cell and denote the
area of $E_{i,j}$ by $V_{i,j}$. Since the grid is regular, all $V_{i,j}$ are
equal to $V = h_x h_y$.
\begin{figure}
\caption{One cell, $E_{i,j}
\label{fig:oneCell}
\end{figure}
To derive the finite volume scheme, begin by integrating
Equation~\eqref{eq:mainSPDE} over a cell, $E_{i,j}$. This gives
\begin{equation}
\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s})u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} - \int_{E_{i,j}} \! \nabla\cdot \mathbf{H}(\boldsymbol{s})\nabla u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} = \int_{E_{i,j}} \! \mathcal{W}(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s},
\label{eq:fInt}
\end{equation}
where $\mathrm{d}\boldsymbol{s}$ is an area element.
The integral on the right hand side is
distributed as a Gaussian variable with mean $0$ and variance $V$ for each
$(i,j)$~\cite{Adler2007}*{pp.~24--25}.
Further, the integral on the right hand side is independent for
different cells, because two different cells can at most share a common
face. Thus Equation~\eqref{eq:fInt} can be written as
\[
\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s})u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} - \int_{E_{i,j}} \! \nabla\cdot \mathbf{H}(\boldsymbol{s})\nabla u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} = \sqrt{V} z_{i,j},
\]
where $z_{i,j}$ is a standard Gaussian variable for each $(i,j)$ and the
Gaussian variables are independent.
By the divergence theorem in Equation~\eqref{eq:divThm}, the second integral
on the left hand side can be written as an integral over the boundary of the
cell. This results in
\begin{equation}
\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s})u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} - \oint_{\partial E_{i,j}} \! (\mathbf{H}(\boldsymbol{s}) \nabla u(\boldsymbol{s}))^\mathrm{T} \boldsymbol{n}(\boldsymbol{s}) \, \mathrm{d}\sigma = \sqrt{V}z_{i,j},
\label{eq:divApplied}
\end{equation}
where $\boldsymbol{n}$ is the exterior normal vector of $\partial E_{i,j}$
with respect to $E_{i,j}$ and $\mathrm{d}\sigma$ is a line element. It is
useful to divide the integral over the boundary in
Equation~\eqref{eq:divApplied} into integrals over each face,
\begin{equation}
\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s})u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} - \left(W_{i,j}^R+W_{i,j}^T+W_{i,j}^L+W_{i,j}^B\right) = \sqrt{V}z_{i,j},
\label{eq:flux}
\end{equation}
where $W_{i,j}^{\mathrm{dir}} = \int_{\sigma_{i,j}^{\mathrm{dir}}} (\mathbf{H}(\boldsymbol{s})\nabla u(\boldsymbol{s}))^\mathrm{T} \boldsymbol{n}(\boldsymbol{s}) \, \mathrm{d}\sigma$.
The first integral on the left hand side of Equation~\eqref{eq:flux} is
approximated by
\begin{equation}
\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s}) u(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s} = V \kappa_{i,j}^2 u(\boldsymbol{s}_{i,j}) = V \kappa_{i,j}^2 u_{i,j},
\label{eq:fApp}
\end{equation}
where $\kappa_{i,j}^2 = \frac{1}{V}\int_{E_{i,j}} \! \kappa^2(\boldsymbol{s}) \, \mathrm{d}\boldsymbol{s}$.
The function $\kappa^2$ is assumed to be continuous and $\kappa_{i,j}^2$ is
approximated by $\kappa^2(\boldsymbol{s}_{i,j})$.
The second part of Equation~\eqref{eq:flux} requires the approximation of the
surface integral over each face of a given cell. The values of $\mathbf{H}$ are
in general not diagonal, so it is necessary to estimate both components of the
gradient on each face of the cell. For simplicity, it is assumed that the
gradient is constant on each face and that it is identically equal to the value
at the centre of the face. On a face parallel to the $y$-axis the estimate of
the partial derivative with respect to $x$ is simple since the centroid of each
of the cells which share the face have the same $y$-coordinate. The problem is
the estimate of the partial derivative with respect to $y$. The reverse is true
for the top and bottom face of the cell.
It is important to use a scheme which gives the same estimate of the gradient
for a given face no matter which of the two neighbouring cells are chosen. For
the right face of $E_{i,j}$, that is $\sigma_{i,j}^{\mathrm{R}}$, the approximation
used is
\[
\frac{\partial}{\partial y} u(\boldsymbol{s}_{i+1/2,j}) \approx \frac{1}{h_y}(u(\boldsymbol{s}_{i+1/2,j+1/2})-u(\boldsymbol{s}_{i+1/2,j-1/2})).
\]
where the values of $u$ at $\boldsymbol{s}_{i+1/2,j+1/2}$ and
$\boldsymbol{s}_{i+1/2,j-1/2}$ are linearly interpolated from the values at
the four closest cells. More precisely, because of the regularity of the grid
the mean of the four closest cells are used. This gives
\begin{equation}
\frac{\partial}{\partial y}u(\boldsymbol{s}_{i+1/2,j}) \approx \frac{1}{4h_y}(u_{i+1,j+1}+u_{i,j+1}-u_{i,j-1}-u_{i+1,j-1}).
\label{eq:ygrady}
\end{equation}
Note that this formula can be used for the partial derivative with respect to
$y$ on any face parallel to the $y$-axis by suitably changing the $i$ and $j$
indices. The partial derivative with respect to $x$ on a face parallel to the
$y$-axis can be approximated directly by
\begin{equation}
\frac{\partial}{\partial x}u(\boldsymbol{s}_{i+1/2,j}) \approx \frac{1}{h_x} (u_{i+1,j}-u_{i,j}).
\label{eq:ygradx}
\end{equation}
In more or less exactly the same way the two components of the gradient on the
top face of cell $E_{i,j}$ can be approximated by
\[
\frac{\partial}{\partial x}u(\boldsymbol{s}_{i,j+1/2}) \approx \frac{1}{4h_x}(u_{i+1,j+1}+u_{i+1,j}-u_{i-1,j}-u_{i-1,j+1})
\]
and
\[
\frac{\partial}{\partial y}u(\boldsymbol{s}_{i,j+1/2}) \approx \frac{1}{h_y}(u_{i,j+1}-u_{i,j}).
\]
These approximations can be used on any side parallel to the $x$-axis by
changing the indices appropriately.
\begin{table}
\centering
\caption{\sf Finite difference schemes for the partial derivative with
respect to $x$ and $y$ at the different faces of cell
$E_{i,j}$.}
\begin{tabular}{lcc}
{\bf Face} & $\frac{\partial}{\partial x}u(s)$ & $\frac{\partial}{\partial y}u(s)$ \\[0.1cm]
\hline \\[-0.25cm]
$\sigma_{i,j}^{\mathrm{R}}$ & $\frac{u_{i+1,j}-u_{i,j}}{h_x}$ & $\frac{u_{i,j+1}+u_{i+1,j+1}-u_{i,j-1}-u_{i+1,j-1}}{4h_y}$ \\[0.2cm]
$\sigma_{i,j}^{\mathrm{T}}$ & $\frac{u_{i+1,j}+u_{i+1,j+1}-u_{i-1,j}-u_{i-1,j+1}}{4h_x}$ & $\frac{u_{i,j+1}-u_{i,j}}{h_y}$ \\[0.2cm]
$\sigma_{i,j}^{\mathrm{L}}$ & $\frac{u_{i,j}-u_{i-1,j}}{h_x}$ & $\frac{u_{i-1,j+1}+u_{i,j+1}-u_{i-1,j-1}-u_{i,j-1}}{4h_y}$ \\[0.2cm]
$\sigma_{i,j}^{\mathrm{B}}$ & $\frac{u_{i+1,j}+u_{i+1,j-1}-u_{i-1,j-1}-u_{i-1,j}}{4h_x}$ & $\frac{u_{i,j}-u_{i,j-1}}{h_y}$ \\
\end{tabular}
\label{table:findiff}
\end{table}
The approximations for the partial derivatives on each face are collected in
Table~\ref{table:findiff}. Using this table one can find the approximations
needed for the second part of Equation~\eqref{eq:flux}.
It is helpful to write
\[
W_{i,j}^{\mathrm{dir}} = \int_{\sigma_{i,j}^{\mathrm{dir}}} \! (\mathbf{H}(\boldsymbol{s})\nabla u(\boldsymbol{s}))^\text{T} \boldsymbol{n}(\boldsymbol{s}) \, \mathrm{d}\sigma = \int_{\sigma_{i,j}^{\mathrm{dir}}} \! (\nabla u(\boldsymbol{s}))^\text{T} (\mathbf{H}(\boldsymbol{s}) \boldsymbol{n}(\boldsymbol{s})) \, \mathrm{d}\sigma,
\]
where the symmetry of $\mathbf{H}$ is used to avoid transposing the matrix.
Assuming that the gradient is identically equal to the value at the centre of
the face, one finds
\[
W_{i,j}^{\mathrm{dir}} \approx \left(\nabla u(\boldsymbol{c}_{i,j}^{\mathrm{dir}})\right)^\text{T} \int_{\sigma_{i,j}^{\mathrm{dir}}} \! \mathbf{H}(\boldsymbol{s})\boldsymbol{n}(\boldsymbol{s}) \, \mathrm{d}\sigma,
\]
where $\boldsymbol{c}_{i,j}^{\mathrm{dir}}$ is the centre of face $\sigma_{i,j}^{\mathrm{dir}}$.
Since the cells form a regular grid, $\boldsymbol{n}$ is constant on each face.
Let $\mathbf{H}$ be approximated by its value at the centre of the face, then
\begin{equation}
W_{i,j}^{\mathrm{dir}} \approx m(\sigma_{i,j}^{\mathrm{dir}})\left(\nabla u(\boldsymbol{c}_{i,j}^{\mathrm{dir}})\right)^{\mathrm{T}}\left(\mathbf{H}(\boldsymbol{c}_{i,j}^{\mathrm{dir}})\boldsymbol{n}(\boldsymbol{c}_{i,j}^{\mathrm{dir}})\right),
\label{eq:Wapp}
\end{equation}
where $m(\sigma_{i,j}^{\mathrm{dir}})$ is the length of the face. Note that
the length of the face is either $h_x$ or $h_y$ and that the normal vector is
parallel to the $x$-axis or the $y$-axis.
Let
\[
\mathbf{H}(\boldsymbol{s}) = \begin{bmatrix}
H^{11}(\boldsymbol{s}) & H^{12}(\boldsymbol{s}) \\
H^{21}(\boldsymbol{s}) & H^{22}(\boldsymbol{s})
\end{bmatrix},
\]
then using Table~\ref{table:findiff} one finds the approximations
\begin{eqnarray*}
\lefteqn{\hat{W}_{i,j}^{\mathrm{R}} =} \\
& & h_y \left[H^{11}(\boldsymbol{s}_{i+1/2,j})\frac{u_{i+1,j}-u_{i,j}}{h_x}\right]+ \\
& & h_y\left[H^{21}(\boldsymbol{s}_{i+1/2,j})\frac{u_{i,j+1}+u_{i+1,j+1}-u_{i,j-1}-u_{i+1,j-1}}{4 h_y}\right],
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{\hat{W}_{i,j}^{\mathrm{T}} =} \\
& & h_x \left[H^{12}(\boldsymbol{s}_{i,j+1/2})\frac{u_{i+1,j+1}+u_{i+1,j}-u_{i-1,j+1}-u_{i-1,j}}{4 h_x}\right]+ \\
& & h_x\left[H^{22}(\boldsymbol{s}_{i,j+1/2})\frac{u_{i,j+1}-u_{i,j}}{h_y}\right],
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{\hat{W}_{i,j}^{\mathrm{L}} =} \\
& & h_y \left[H^{11}(\boldsymbol{s}_{i-1/2,j})\frac{u_{i-1,j}-u_{i,j}}{h_x}\right]+ \\
& & h_y\left[H^{21}(\boldsymbol{s}_{i-1/2,j})\frac{u_{i,j-1}+u_{i-1,j-1}-u_{i-1,j+1}-u_{i,j+1}}{4 h_y}\right]
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{\hat{W}_{i,j}^{\mathrm{B}} =} \\
& & h_x \left[H^{12}(\boldsymbol{s}_{i,j-1/2})\frac{u_{i-1,j}+u_{i-1,j-1}-u_{i+1,j}-u_{i+1,j-1}}{4 h_x}\right]+ \\
& & h_x\left[H^{22}(\boldsymbol{s}_{i,j-1/2})\frac{u_{i,j-1}-u_{i,j}}{h_y}\right].
\end{eqnarray*}
These approximations can be combined with the approximations in
Equation~\eqref{eq:fApp} and inserted into Equation~\eqref{eq:flux} to give
\[
V \kappa_{i,j}^2 u_{i,j} - \left(\hat{W}_{i,j}^\mathrm{R}+\hat{W}_{i,j}^\mathrm{T}+\hat{W}_{i,j}^\mathrm{L}+\hat{W}_{i,j}^\mathrm{B}\right) = \sqrt{V} z_{i,j}.
\]
Stacking the variables $u_{i,j}$ row-wise in a vector $\boldsymbol{u}$, that
is first row $0$, then row $1$ and so on, gives the linear system of equations,
\begin{equation}
\mathbf{D}_V \mathbf{D}_{\kappa^2}\boldsymbol{u}-\mathbf{A}_{\mathbf{H}} \boldsymbol{u} = \mathbf{D}_V^{1/2}\boldsymbol{z},
\label{eq:mateq}
\end{equation}
where $\mathbf{D}_V = V \mathbf{I}_{MN}$,
$\mathbf{D}_{\kappa^2} = \mathrm{diag}(\kappa_{0,0}^2,\ldots,\kappa_{M-1,0}^2,\kappa_{0,1}^2,\ldots,\kappa_{M-1,N-1}^2)$, $\boldsymbol{z} \sim \mathcal{N}_{MN}(\boldsymbol{0}, \mathbf{I}_{MN})$ and $\mathbf{A}_\mathbf{H}$
is considered more closely in what follows.
The construction of the matrix $\mathbf{A}_\mathbf{H}$, which depends on the function $\mathbf{H}$, requires
only that one writes out the sum
\[
\hat{W}_{i,j}^{\mathrm{R}}+\hat{W}_{i,j}^\mathrm{T}+\hat{W}_{i,j}^\mathrm{L}+\hat{W}_{i,j}^\mathrm{B}
\]
and collects the coefficients of the different $u_{a,b}$ terms.
This is not difficult, but requires many lines of equations.
Therefore, only the resulting coefficients are
given. Fix $(i,j)$ and consider the equation for cell $E_{i,j}$. For convenience,
let $i_p$ and $i_n$ be the column left and right of the current column respectively and
let $j_n$ and $j_p$ be the row above and below the current row respectively. These
rows and columns are $0$-indexed and due to the periodic boundary conditions
one has, for example, that column $0$ is to the right of column $M-1$.
Further, number the rows and columns of the matrix $\mathbf{A}_\mathbf{H}$ from $0$ to $MN-1$.
For row $jM+i$ the coefficient of $u_{i,j}$ itself is given by
\begin{eqnarray*}
\lefteqn{(\mathbf{A}_\mathbf{H})_{jM+i,jM+i} =} \\
& & { } -\frac{h_y}{h_x}\left[H^{11}(\boldsymbol{s}_{i+1/2,j})+H^{11}(\boldsymbol{s}_{i-1/2,j})\right] \\
& & { } -\frac{h_x}{h_y}\left[H^{22}(\boldsymbol{s}_{i,j+1/2})+H^{22}(\boldsymbol{s}_{i,j-1/2})\right].
\end{eqnarray*}
The four closest neighbours have coefficients
\[
\begin{aligned}
(\mathbf{A}_\mathbf{H})_{jM+i,jM+i_p} &= \frac{h_y}{h_x}H^{11}(s_{i-1/2,j})-\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j+1/2})-H^{12}(\boldsymbol{s}_{i,j-1/2})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,jM+i_n} &= \frac{h_y}{h_x}H^{11}(\boldsymbol{s}_{i+1/2,j})+\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j+1/2})-H^{12}(\boldsymbol{s}_{i,j-1/2})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,j_nM+i} &= \frac{h_x}{h_y}H^{22}(\boldsymbol{s}_{i,j+1/2})+\frac{1}{4}\left[H^{21}(\boldsymbol{s}_{i+1/2,j})-H^{21}(\boldsymbol{s}_{i-1/2,j})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,j_pM+i} &= \frac{h_x}{h_y}H^{22}(\boldsymbol{s}_{i,j-1/2})-\frac{1}{4}\left[H^{21}(\boldsymbol{s}_{i+1/2,j})-H^{21}(\boldsymbol{s}_{i-1/2,j})\right].
\end{aligned}
\]
Lastly, the four diagonally closest neighbours have coefficients
\[
\begin{aligned}
(\mathbf{A}_\mathbf{H})_{jM+i,j_pM+i_p} &= +\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j-1/2})+H^{21}(\boldsymbol{s}_{i-1/2,j})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,j_pM+i_n} &= -\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j-1/2})+H^{21}(\boldsymbol{s}_{i+1/2,j})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,j_nM+i_p} &= -\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j+1/2})+H^{21}(\boldsymbol{s}_{i-1/2,j})\right], \\
(\mathbf{A}_\mathbf{H})_{jM+i,j_nM+i_n} &= +\frac{1}{4}\left[H^{12}(\boldsymbol{s}_{i,j+1/2})+H^{21}(\boldsymbol{s}_{i+1/2,j})\right].
\end{aligned}
\]
The rest of the elements of row $jM+i$ are $0$.
Based on Equation~\eqref{eq:mateq} one can write
\[
\boldsymbol{z} = \mathbf{D}_V^{-1/2}\mathbf{A}\boldsymbol{u},
\]
where $\mathbf{A} = \mathbf{D}_V\mathbf{D}_{\kappa^2}-\mathbf{A}_\mathbf{H}$.
This gives the joint distribution of $\boldsymbol{u}$,
\begin{align}
\nonumber \pi(\boldsymbol{u}) &\propto \pi(\boldsymbol{z}) \propto \exp\left(-\frac{1}{2}\boldsymbol{z}^{\mathrm{T}}\boldsymbol{z}\right) \\
\nonumber \pi(\boldsymbol{u}) &\propto \exp\left(-\frac{1}{2}\boldsymbol{u}^{\text{T}}\mathbf{A}^{\mathrm{T}}\mathbf{D}_V^{-1}\mathbf{A}\boldsymbol{u}\right) \\
\nonumber \pi(\boldsymbol{u}) &\propto \exp\left(-\frac{1}{2}\boldsymbol{u}^{\text{T}}\mathbf{Q}\boldsymbol{u}\right),
\end{align}
where $\mathbf{Q} = \mathbf{A}^{\mathrm{T}}\mathbf{D}_V^{-1}\mathbf{A}$. This
is a sparse matrix with a maximum of $25$ non-zero elements on each row,
corresponding to the point itself, its 8 closest neighbours and the 8 closest
neighbours of each of the 8 closest neighbours.
\section{Marginal variances with constant coefficients}
\begin{prop}
\label{prop:margVar}
Let $u$ be a stationary solution of the SPDE
\begin{equation}
\kappa^2 u(x,y) - \nabla\cdot \mathbf{H} \nabla u(x,y) = \mathcal{W}(x,y), \qquad (x,y)\in\mathbb{R}^2,
\label{chap2:eq:propR2}
\end{equation}
where $\mathcal{W}$ is a standard Gaussian white noise process,
$\kappa^2 > 0$ is a constant, $\mathbf{H}$ is a positive definite
$2\times 2$ matrix and
$\nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$.
Then $u$ has marginal variance
\[
\sigma_m^2 = \frac{1}{4\pi \kappa^2 \sqrt{\mathrm{det}(\mathrm{H})}}.
\]
\begin{proof}
Since the solution is stationary, Gaussian white noise is
stationary and the SPDE has constant coefficients, the SPDE
is acting as a linear filter. Thus one can use spectral theory
to find the marginal variance. The transfer function of the
SPDE is
\[
g(\boldsymbol{w}) = \frac{1}{\kappa^2+\boldsymbol{w}^\mathrm{T}\mathbf{H}\boldsymbol{w}}.
\]
Further, the spectral density of a standard Gaussian white
noise process on $\mathbb{R}^2$ is identically equal to
$1/(2\pi)^2$. It follows that the spectral density of the
solution is
\[
f_S(\boldsymbol{w}) = \left(\frac{1}{2\pi}\right)^2 \frac{1}{(\kappa^2+\boldsymbol{w}^\mathrm{T}\mathbf{H}\boldsymbol{w})^2}.
\]
From the spectral density it is only a matter of integrating
the density over $\mathbb{R}^2$,
\[
\sigma_m^2 = \int_{\mathbb{R}^2} \! f_S(\boldsymbol{w}) \, \mathrm{d}\boldsymbol{w}.
\]
The matrix $\mathbf{H}$ is (symmetric) positive definite
and, therefore, has a (symmetric) positive definite square
root, say $\mathbf{H}^{1/2}$. Use the change of variables
$\boldsymbol{w} = \kappa\mathbf{H}^{-1/2}\boldsymbol{z}$
to find
\begin{align*}
\sigma_m^2 &= \frac{1}{4\pi^2} \int_{\mathbb{R}^2} \! \frac{1}{(\kappa^2+\kappa^2\boldsymbol{z}^\mathrm{T}\boldsymbol{z})^2} \mathrm{det}(\kappa\mathbf{H}^{-1/2}) \, \mathrm{d}\boldsymbol{z} \\
&= \frac{1}{4\pi^2 \kappa^2 \sqrt{\mathrm{det}(\mathbf{H})}}\int_{\mathbb{R}^2} \! \frac{1}{(1+\boldsymbol{z}^\mathrm{T}\boldsymbol{z})^2} \, \mathrm{d}\boldsymbol{z} \\
&= \frac{1}{4\pi \kappa^2\sqrt{\mathrm{det}(\mathbf{H})}}.
\end{align*}
\end{proof}
\end{prop}
\input{NewClassNonstationary.bbl}
\end{document} |
\begin{document}
\title{Two-party Models and the No-go Theorems}
\date{\today}
\author{Minh-Dung Dang}
\affiliation{GET-ENST \& LTCI-UMR 5141 CNRS, 46 rue
Barrault, 75634 Paris Cedex 13, France}
\begin{abstract}
In this paper, we reconsider the communication model used in the no-go
theorems on the impossibility of quantum bit commitment and oblivious
transfer. We state that a macroscopic classical channel may not be
replaced with a quantum channel which is used in the reduced model
proving the no-go theorems. We show that in some restricted cases, the
reduced model is insecure while the original model with a classical
channel is secure.
\end{abstract}
\pacs{03.67.Dd, 03.67.Hk}
\maketitle
\section{\label{sec:introduction}Introduction}
The theorems on the \emph{security} of quantum key
distribution~\cite{LC99,SP00}, following the first protocol of Bennett
and Brassard~\cite{BB84}, and the theorems on the \emph{insecurity} of
two-party quantum bit commitment~\cite{LC97,May97,Bub01}, oblivious
transfer and general two-party quantum secure computation~\cite{Lo97}
are among the most interesting subjects in the field of quantum
cryptography.
In a bit commitment protocol, Alice sends the commitment information
about a secret bit to Bob, who cannot discover the bit, and when Alice
is supposed to reveal the bit, Bob can detect if Alice changes the
value of the bit. Oblivious transfer is a computation protocol where
Alice enters two secret bits, Bob enters a choice to gain only one of
these while Alice cannot know Bob's choice. Coin flipping is a
protocol for Alice and Bob sharing a fairly random bits, i.e. none of
the parties can affect the probability distribution of the outcome.
These three primitives are used to construct secure computation for
generic two-party functions~\cite{Gol04book}. Classically, bit
commitment implements coin flipping and is implemented by oblivious
transfer~\cite{Kil88}, while oblivious transfer can be built from bit
commitment by transmitting quantum information~\cite{Cre94,Yao95}.
With the introduction of quantum information~\cite{Wie83},
cryptographers were willing to build unconditionally secure bit
commitment~\cite{BB84,BCJ+93}. But the first no-go theorem eliminated
these proposals~\cite{LC97,May97}. This knock alerted people about the
provable security of other two-party protocols, including key
distribution. While quantum key distribution is proved to be
secure~\cite{LC99,SP00}, two other no-go theorems were issued: quantum
secure two-party computation and so oblivious transfer are
impossible~\cite{Lo97}; quantum coin flipping with arbitrarily small
bias is impossible~\cite{Kit02}. We can add to the list another
no-go result: quantum bit commitment cannot be built from coin
flipping~\cite{Ken99}.
These no-go theorems execute the protocols on a quantum two-party
model. This model is proposed first by Yao, and consists of
a quantum machine on Alice side interacting with a quantum machine on
Bob side via a quantum machine~\cite{Yao95,Kre95}. The global states,
or \emph{images}, of a protocol are then described in a joint space of
Alice side $\hilbert{A}$, Bob side $\hilbert{B}$ and the quantum
channel $\hilbert{C}$. The states of the execution can be mixed
states, but all local random choice and computation made by Alice and
Bob can be purified and kept
at the quantum level~\cite{May97,LC97,Lo97}. The global state is then
described by a pure state in a larger space $\hilbert{A'}\otimes
\hilbert{C}\otimes \hilbert{B'}$ where $\hilbert{A'},\hilbert{B'}$ are
extended from $\hilbert{A},\hilbert{B}$ to purify the random choices
and measurements. Adopting the option that parts of the channel are
controlled by one of the two participants, the no-go theorems converts
the state of the joint computation (at a moment) into a bipartite
state of joint space $\hilbert{A''}\otimes\hilbert{B''} =
(\hilbert{A'}\otimes\hilbert{CA})\otimes(
\hilbert{CB}\otimes\hilbert{B'})$ where $\hilbert{CA},\hilbert{CB}$
are the channel parts respectively controlled by Alice and Bob. In
such a bipartite space, bit commitment and oblivious transfer are
impossible.
In this paper, I try to criticize the application of the no-go
theorems to all possible two-party protocols, including
quantum-classical mixed protocols. In fact, a specified protocol with
classical messages would rather be implemented on a real physical
model with a classical channel. We state that a classical channel
can be macroscopic and cannot be controlled by any participants in
terms of purification. It should be viewed as a measurement which is
trusted by both Alice and Bob. I give some restricted cases where the
no-go theorems are valid with quantum communications but not with
quantum-classical mixed communications.
This critique is not to say that quantum bit commitment is
possible. Indeed, another reconsideration of bit commitment on a
complete model was made recently~\cite{AKS+06}, and we should wait for
reviews to confirm the results.
First, in Section \ref{sec:revision}, we do a revision on the no-go
theorems of Mayers, Lo \& Chau and the quantum model used in their
proofs. With the same arguments, in Section~\ref{sec:ext-nogo}, we
extend the no-go theorems with presence of a particular quantum
trusted third-party. By this extension of the no-go theorems for a
particular quantum trusted third-party model, I show a result similar
to~\cite{Ken99}: bit commitment and oblivious transfer cannot be built
upon coin flipping with purely quantum communications.
Then, in Section~\ref{sec:reduction}, we
re-question about the arguments of Mayers, Lo \& Chau in reducing
general protocols from the mixed model with a classical channel to the
purely quantum model. In that section, we propose a generalized
model to adopt a real classical channel, and we give a particular
case-study where the no-go theorems are valid with purely quantum
communications but not with mixed communications.
In the following, we will use shortly ``protocols'' for all protocols
with quantum-classical mixed communications while ``quantum
protocols'' for protocols with purely quantum communications.
\section{\label{sec:revision}Quantum model and no-go theorems}
\subsection{\label{sec:quantum-model}Quantum model}
In~\cite{Yao95}, for proving the security of bit-commitment-based
quantum oblivious transfer, Yao defined a quantum two-party protocol
as a pair of quantum machines interacting through a quantum
channel.
The protocol is then executed on a joint system consisting of Alice's
machine $\hilbert{A}$, Bob's machine $\hilbert{B}$, and the quantum
channel $\hilbert{C}$. Initially, each participant prepares a state
for its private system and the channel is in state $\ket{0}$.
The execution is alternating rounds of one-way
communications. For each round of one
participant $D \in \{A,B\}$, this performs a computation on the joint
space of his private system $\hilbert{D}$ and the messages
$\hilbert{C}$. The messages will be taken to the location of the
other for the next round.
\begin{figure}
\caption{\label{fig:quantum-model}
\label{fig:quantum-model}
\end{figure}
This model is general for all quantum protocols, and has been widely
used for analyzing quantum protocols, e.g. the complexity of quantum
communication~\cite{Kre95,Wol02} and quantum interactive
proofs~\cite{Wat99}.
\subsection{\label{sec:nogo-theorems}No-go theorems}
The no-go theorems used the quantum model of Yao to prove the
insecurity of two-party protocols.
For protocols with quantum-classical mixed communications, Yao said
``Although the above description is general enough to incorporate
classical computations and transmissions of classical information, it
is useful to separate out the classical parts in describing
protocols''~\cite{Yao95}. But his model was then widely used for all
quantum-classical mixed protocols. The arguments is that a classical
bit can be transmitted as a qubit. Mayers explicitly explained this
kind of \emph{quantum} communication for classical
information~\cite{May97}.
Because all random choices are made by either Alice or Bob, these
two participant can keep the random choices at the quantum level by
holding the entanglement with the quantum coins. Indeed, the fact that
a participant purifies or not his random variables does not change the
correctness of the protocol and the cheating strategy of
the other. All of local computations can be also kept at the quantum
level by delaying measurements to the final step. Thus, the global
\emph{images} of the the computation
can be described by a pure state lying on a larger space purifying all
local random variables and measurements:
$\hilbert{A'}\otimes\hilbert{C}\otimes\hilbert{B'}$. The execution of
the protocol is then a well specified unitary transformation
of the input state into the output state, and surprisingly
\emph{deterministic}.
In the quantum model, at each moment between two rounds
of communication, any quantum part of the channel must lie on either
Alice or Bob side. Thus, the global image at any moment can be
considered as a state lying in a bipartite space
$\hilbert{A''}\otimes\hilbert{B''} =
\hilbert{A'}\otimes\hilbert{CA}\otimes\hilbert{CB}\otimes\hilbert{B'}$
where $\hilbert{CD}$ is for the channel part held by participant $D
\in \{A,B\}$.
Then, at the considered moment, the partial images of the computations
on Alice and Bob sides are
$$
\rho^{A''} = tr_{B''}(\rho), \quad \rho^{B''} = tr_{A''}(\rho),
$$
where $\rho$ is the global image which is a pure state in the global
space $\hilbert{A''}\otimes\hilbert{B''}$.
For the security of bit commitment on Bob side, the partial images on
Bob side at the moment before the opening phase must identical for the
commitments of $0$ and $1$:
$$
tr_{A''}(\rho_0) = tr_{A''}(\rho_1)\cdot
$$
The no-go theorem on bit commitment is issued by a theorem that states
that, in such a case, there exists a local unitary transformation on
$\hilbert{A''}$ that transforms $\rho_0$ to $\rho_1$. Alice can then
switch the computation before opening the secret bit~\cite{May97,LC97}.
For the security of one-sided computation, Lo discovered that if the
protocol is secure against Alice, then Bob has a local
unitary transformation, independent from Alice's input, that helps Bob
to learn the results computed from all of possible local
inputs~\cite{Lo97}.
We revise here Lo's theorem for one-sided computation. In fact, to
compute $f(i,j)$, Alice and Bob run together a unitary $U$
transformation on Alice's input $\ket{i}:i \in \{i_1,..,i_m\}$ joint
with Bob's input $\ket{j}:j \in \{j_1,..,j_n\}$. Other known local
variables can be omitted without generalization. At the end, Bob can
learn the result from the output state $\ket{v_{ij}} =
U(\ket{i}_A\otimes\ket{j}_B)$. But Alice can entangle her input $A$
with a private quantum dice $P$, i.e. prepares the initial state
$\frac{1}{\sqrt{n}}\sum_i \ket{i}_P\otimes\ket{i_A}$.
If Bob inputs $j_1$ then the initial state for the protocol is
\begin{equation}\label{equa:lo-initial}
\ket{u'}_{in} = \frac{1}{\sqrt{n}}\sum_i
\ket{i}_P\otimes\ket{i_A}\otimes\ket{j_1}_B,
\end{equation}
and at the end, the output state is
$$
\ket{v_{j_1}} = \frac{1}{\sqrt{n}}\sum_i
\ket{i}_P\otimes U (\ket{i_A}\otimes\ket{j_1}_B)\cdot
$$
Similarly, if Bob inputs $j_2$ then the output state is
$$
\ket{v_{j_2}} = \frac{1}{\sqrt{n}}\sum_i
\ket{i}_P\otimes U (\ket{i_A}\otimes\ket{j_2}_B) \cdot
$$
For the security on Alice side, the partial images must be identical,
i.e.
$$
tr_B(\projection{v_{j_1}}{v_{j_1}}) =
tr_B(\projection{v_{j_2}}{v_{j_2}})
$$
and then, there exists a local unitary transformation $U^{j_1,j_2}$ on
Bob local system such that
$$
\ket{v_{j_2}} = U^{j_1,j_2}\ket{v_{j_1}}\cdot
$$
Therefore, because $_P\bracket{i}{v_j} =
\frac{1}{\sqrt{n}}\ket{v_{ij}}$, the transformation $U^{j_1,j_2}$ is
universal for all Alice input $i$:
$$
\ket{v_{ij_2}} = U^{j_1,j_2}\ket{v_{ij_1}}\cdot
$$
Bob can enter $\ket{j_1}$, computes $\ket{v_{ij_1}}$ and measures it
to learn $f(i,j_1)$. However, to enable Bob to \emph{unambiguously}
get the result, $\ket{v_{ij_1}}$ must not be perturbed by his
measurement. Bob can transform it to $\ket{v_{ij_2}}$ by
$U^{j_1,j_2}$, measures to learn $f(i,j_2)$, and so on.
More strongly, imperfect protocols are also banned from reaching an
arbitrarily high security. There exist always a trade-off between the
security on one side and the insecurity on the other
side~\cite{SR01b}.
\section{\label{sec:ext-nogo}Extensions of no-go theorems for the
quantum model} In this section, we suppose an honest third-party that
helps Alice and Bob to do some computations. We define a quantum
trusted third-party as a quantum device that can help Alice
and Bob to do any required computation. The third-party can used some
local pure variables for the computations. The local variables of the
trusted party are initialized to $\ket{0}$. At the end of the required
computation, the third-party splits all of the outputs, included the
local variables, into two parts, redirects one part to Alice, and one
part to Bob, cf. figure~\ref{fig:trusted-computation}. The execution
time of the computation done by the third-party is a elementary unit,
and we can consider as it immediately returns the results to the
participants.
\begin{figure}
\caption{\label{fig:trusted-computation}
\label{fig:trusted-computation}
\end{figure}
With such a trusted third-party, we can extend the results of
\cite{May97,LC97,Lo97} for the quantum model:
\begin{theorem}[Extension of no-go theorem on bit commitment]
\label{thm:ext-bc-nogo}
Quantum bit commitment is insecure even with help of the specified quantum
trusted third-party.
\end{theorem}
\begin{theorem}[Extension of no-go theorem on secure computations]
\label{thm:ext-ot-nogo}
Quantum two-party secure computations are insecure even with help of
the specified quantum trusted third-party.
\end{theorem}
In fact, when the third-party uses only pure states as local input,
and immediately, splits and sends all of the qubits which participate
to the computations to Alice and Bob, the global state at any considered
moment is in some known pure state, according to the algorithm, in a
bipartite space relating only Alice and Bob sides. Therefore, the
no-go theorems remain valid.
For example, to prove the later theorem for one-sided secure
computation. We start with equation
\eqref{equa:lo-initial}. Attaching a pure state $\ket{0}_{A'B'}$,
locally prepared by the third-party, the initial state is
$$
\ket{u'}_{in} = \frac{1}{\sqrt{n}}\sum_i \ket{i}_P \otimes \ket{i}_A
\otimes \ket{j_1}_B \otimes \ket{0}_{A'B'}\cdot
$$
At the end of the computation, with help of the third-party, the
combined system is in state
$$
\ket{v_{j1}} = \frac{1}{\sqrt{n}}\sum_i \ket{i}_P \otimes U(\ket{i}_A
\otimes \ket{j_1}_B \otimes \ket{0}_{A'B'})
$$
where system $A'$ is set to $A$, system $B'$ is set to $B$ after the
split. Therefore, the remaining arguments of Lo's proofs can be
followed, cf. Section \ref{sec:nogo-theorems}.
\section{\label{sec:cf-based}Is Coin Flipping weaker than Bit Commitment?}
As a corollary of the extensions of the no-go theorems, cf. Section
\ref{sec:ext-nogo}, we conclude that
\begin{corollary}
In the quantum model, Coin Flipping is weaker than Bit Commitment and
Oblivious Transfer.
\end{corollary}
In \cite{Ken99}, Kent shown a similar result. In his paper, he
established a relativist model to implement coin flipping. With an
assumed quantum trusted party, we made the model more
comprehensible from a non-relativist point of view.
It's because we can suppose a trusted third-party that creates a
pair of qubits in Bell state $\ket{\Phi+}$ and sends each part to an
user. In such a model, coin flipping is realizable while bit commitment
and oblivious transfer are not with quantum communication, as shown by
Theorems \ref{thm:ext-bc-nogo}, \ref{thm:ext-ot-nogo}.
\section{\label{sec:reduction}Quantum model is not to prove the
insecurity} \subsection{\label{sec:mixed-model}Quantum-classical mixed
model} What is the difference between the two channel? A quantum
channel is a medium that we can used to transmit directly a quantum
state without disturbing it while a classical channel permits only one of
two discrete values for a classical bit. For example a macroscopic
electrical wire with tension $+5V$ for $0$ and $-5V$ for $1$. It is
natural that in reality, a classical channel is well coupled with the
environment, and the decoherence is so strong that only $\ket{0}$ or
$\ket{1}$ is accepted, in terms of quantum information.
Imagine that in the specification of a protocol, at a certain
moment, a party has to measure some results of its quantum computation
and send the resulting classical messages to the other party. With a
macroscopic classical channel, the measurement must be carried
out. The sender can also memorize the emitted message. It is
convenient to see the classical channel as a trusted measurement
machine: the sender sends the qubits to the machine that measures,
doubles the output state, which is an eigenstate and cloneable, sends
one copy to the receiver and one copy back to the sender for
memorizing it.
Yao's model should be generalized for two-party protocols as a pair of
quantum machines interacting through a quantum channel \emph{and
necessarily a classical channel}. The model is also alternating rounds
of one-way communications. In each round, a participant $D \in
\{A,B\}$ performs a computation on the joint space of his private
system $\hilbert{D}$, the quantum messages $\hilbert{C}$ the classical
messages $\hilbert{R_M,D}$ received from the trusted measurement
machine $M$, and the messages $\hilbert{S_M,D}$ to be sent to the
measurement machine for producing classical messages.
For simplifying, the measurement machine should not copy the
output. This task can be carried out by the sender. In fact, measuring
a state $a\ket{0} + b\ket{1}$, the above machine produces $\ket{00}$
or $\ket{11}$. Instead, the sender can create $a\ket{00} + b\ket{11}$,
sends the first qubit to the machine that measures it and sends the
output state to the receiver. By this way, the sender keep a copy of
the measurement.
Therefore, the model is simplified, and consists of two machines
$\hilbert{A},\hilbert{B}$, a quantum channel $\hilbert{C}$ for both
quantum and classical messages and a trusted measurement machine $M$
with ancillas $\hilbert{M}$. The measurement is in fact a CNOT-like
gate whose controlling inputs are in the space of the sender's
``classical messages'' and targets are ancillas in the macroscopic
environment space $\hilbert{M}$, cf. figure~\ref{fig:mixed-model}.
In each communication round, a participant $D \in \{A,B\}$ does an
unitary computation on $\hilbert{D'}\otimes\hilbert{C}$ where
$\hilbert{D'}$ is extended from $\hilbert{D}$ to purify local
variables and measurements; the trusted machine applies the CNOT gate
to the ``classical messages'' in $\hilbert{C}$ and the environment of
the classical channel $\hilbert{M}$. The quantum messages and
``classical messages'' in $\hilbert{C}$ of the round are taken to the
other location for the next round. $\hilbert{M}$ is not controlled by
any participant.
\begin{figure}
\caption{\label{fig:mixed-model}
\label{fig:mixed-model}
\end{figure}
\subsection{\label{cogent-reduction}Is the reduction cogent?}
It is obvious that all protocols can be \emph{correctly} implemented
on the quantum model by replacing classical bits by qubits. By this
way, a protocol is secure if its simulating quantum protocol is
secure. It's reasonable to use simulating quantum protocols \emph{to
prove the security of original protocols}~\cite{Yao95}. But,
vice-versa, it is not evident. Unfortunately, the no-go theorems used
the simulating quantum protocols \emph{to prove the insecurity of
original protocols}.
\begin{figure*}
\caption{\label{fig:quantum-vs-mixed}
\label{fig:quantum-vs-mixed}
\end{figure*}
For example, in the specification of a mixed protocol, a
participant makes the measurement of its computation to produce a
classical message, sends the messages via a the classical channel
which reproduces a corresponding state in the computation basis for
the other participant. In the model of the specification with a
macroscopic channel for sending classical message, the receiver really
receives one of the eigen-states of the measurement made by the sender.
While, by simulating with a quantum channel, the sender
equivalently creates a quantum mixed state as the sum of the above
measurement eigen-states weighted by the corresponding
probabilities. However, the sender can prepare this mixed state as
part of a bipartite pure entangled state. We see this by citing to an
early simple example: at a certain moment, a participant gets
$a\ket{0} + b\ket{1}$, measures it and sends the result, either
$\ket{0}$ with probability $a^2$ or $\ket{1}$ with probability
$b^2$. This classical message is described in terms of quantum
information as $a^2\projection{0}{0}+b^2\projection{0}{0}$. With a
quantum channel, the sender can send $(a\ket{0} +
b\ket{1})\otimes\ket{0}_C$ through a CNOT gate to make
$(a\ket{0}\otimes\ket{0}_C + b\ket{1}\otimes\ket{1}_C)$ and send
qubit $C$ to the receiver.
The above quantum communication of classical messages gave to the
participants an extra entanglement that does not exist in the
specification of the protocol with mixed communications. Indeed, this
entanglement could be used as a powerful attack. We will explicitly
expose this with a case-study where, if the receiver used the received
message to do some quantum computation and sends back the result, the
sender could learn more information with entanglement attack by the
effect of \emph{super-dense coding}~\cite{BW92}.
Such an bipartite entanglement should be destroyed by the measurement
of the classical channel. In our model for the classical channel, the
sender has to send qubit $C$ to the measurement machine of the
classical channel that makes instead the state
$(a\ket{0}\otimes\ket{0}_M\otimes\ket{0}_C +
b\ket{1}\otimes\ket{1}_M\otimes\ket{1}_C)$ and sends qubit $C$ to the
receiver.
From a global view, one can see that if the two parties follow a
protocol using a defined classical channel, the
global system should lie in a tripartite spaces joining
$\hilbert{A}\otimes\hilbert{C,A}$, $\hilbert{B}\otimes\hilbert{C,B}$
and $\hilbert{M}$. The crucial argument that the whole system is
described in a bipartite space, used in the proofs of the no-go
theorems~\cite{May97,LC97,Lo97}, is not valid in the mixed
model, cf. figure~\ref{fig:quantum-vs-mixed}.
All of the proposed quantum bit commitment protocols fall into the
quantum model~\cite{BB84,BCJ+93,Yue00}, where all communications are
realized with quantum messages except the final step to open the
committed bit. Then, they are directly attacked by the no-go
theorems~\cite{May97,LC97,Bub01}.
Maybe, unconditionally secure bit commitment is also impossible in the
mixed model, as concluded by a recent study in a preprint paper of
Mauro d'Ariano et al.~\cite{AKS+06}, but the arguments used in Mayer
and Lo \& Chau proofs for all possible protocols are not evident.
We think that, the insecurity of a protocol should be considered when
implementing it on a quantum-classical mixed model that match better
the real world. Normally, a macroscopic classical channel is coupled
with a trusted environment and measures the quantum messages sent
through it. We expect that, to be more convincing, no-go theorems
should be proved for the mixed model we proposed in the previous
section.
\subsection{\label{sec:honest-oot-gate} Case-study: an honest
third-party O-OT gate}
Let verify a quantum oblivious transfer protocol with a pure trusted
party, cf. \ref{sec:ext-nogo}. In our protocol, the pure trusted
party uses tree local qubits. For simplifying, we initialize the first
and the second qubit to be entangled and in state $\ket{\Phi+}$. In
fact, the trusted party can prepare this state from $\ket{00}$ by a
doing $\pi/2$ rotation on the first qubit and sending the two qubit
through CNOT gate whose target is the second qubit. The third qubit is
initialized to $\ket{0}$.
Inspired from Bennett et al.~\cite{BDS+96}, we use the notations:
\begin{align*}
\widetilde{00} = \ket{\Phi+} = (\ket{00} + \ket{11})/\sqrt{2},\\
\widetilde{01} = \ket{\Phi-} = (\ket{00} - \ket{11})/\sqrt{2},\\
\widetilde{10} = \ket{\Psi+} = (\ket{01} + \ket{10})/\sqrt{2},\\
\widetilde{11} = \ket{\Psi-} = (\ket{01} - \ket{10})/\sqrt{2}\cdot
\end{align*}
Let $b_0,b_1$ be the two bits that Alice want to send and $c$ be Bob's
choice. The trusted party does a controlled $\pi$ rotation $R_{b_0b_1}$
on the first qubit, according to $b_0,b_1$:
$$
R_{00} = I, R_{01} = \sigma_z, R_{10} = \sigma_x, R_{11} = \sigma_y\cdot
$$
The first and second qubits are obtained in state
$\widetilde{b_0b_1}$. Next, the trusted party applies the bilateral
$\pi/2$ rotation $B_y$ to the first and second qubits in case
$c=1$~\cite{BDS+96}:
\begin{align*}
\widetilde{00} \rightarrow_{B_y} \widetilde{00},\\
\widetilde{01} \rightarrow_{B_y} \widetilde{10},\\
\widetilde{10} \rightarrow_{B_y} \widetilde{01},\\
\widetilde{11} \rightarrow_{B_y} \widetilde{11}\cdot
\end{align*}
The trusted party applies then the CNOT gate whose controlling input is
the first qubit and the target is the third qubit. Finally, the
trusted party splits the outputs, sends back Alice's qubits
with his first local qubit to Alice, and sends back Bob's
qubit with its second and third local qubits to Bob,
cf. figure~\ref{fig:oot-gate}.
\begin{figure}
\caption{\label{fig:oot-gate}
\label{fig:oot-gate}
\end{figure}
In case Alice and Bob communicate with the trusted party via quantum
channels, they can send directly quantum inputs. The computation of
the trusted party is a quantum circuit acting on
$6$ qubits: two for Alice's inputs, tree for the local qubits, one for
Bob's input. Alice can prepare a superposition
$$
\frac{1}{2}(\ket{00} + \ket{01} + \ket{10} + \ket{11})\cdot
$$
The global input state is then
$$
\ket{in} = \frac{1}{2}(\ket{00}_A + \ket{01}_A + \ket{10}_A +
\ket{11}_A)\ket{\Phi+}_T\ket{0}_T\ket{c}_B\cdot
$$
If Bob sends $\ket{c} = \ket{0}$ then the computation is
\begin{align*}
\ket{in} & \rightarrow_{R_{b_0b_1}} & \frac{1}{2} [ &
\ket{00}_A\widetilde{00}_T + \ket{01}_A\widetilde{01}_T\\
&&& + \ket{10}_A\widetilde{10}_T +
\ket{11}_A\widetilde{11}_T]\ket{0}_T\ket{0}_B\\
&\rightarrow_{B_y} & \frac{1}{2\sqrt{2}}[ &
\ket{00}_A(\ket{00}_T+\ket{11}_T)\\
&&& + \ket{01}_A(\ket{00}_T - \ket{11}_T)\\
&&& + \ket{10}_A(\ket{01}_T+\ket{10}_T) +\\
&&&\ket{11}_A(\ket{01}_T - \ket{10}_T)]\ket{0}_T\ket{0}_B\\
&\rightarrow_{CNOT} & \frac{1}{2\sqrt{2}}[ &
\ket{00}_A(\ket{000}_T+\ket{111}_T)\\
&&&+ \ket{01}_A(\ket{000}_T - \ket{111}_T)\\
&&& + \ket{10}_A(\ket{010}_T+\ket{101}_T)\\
&&& + \ket{11}_A(\ket{010}_T - \ket{101}_T)]\ket{0}_B\\
&\rightarrow_{split} & \frac{1}{2\sqrt{2}}[ & (\ket{000}_A +
\ket{010}_A)\ket{000}_B\\
&&& + (\ket{001}_A - \ket{011}_A)\ket{110}_B\\
&&& + (\ket{100}_A + \ket{110}_A)\ket{100}_B\\
&&& + (\ket{101}_A - \ket{111}_A)\ket{010}_B]\cdot
\end{align*}
If Bob sends $\ket{c}=\ket{1}$ then the computation is
\begin{align*}
\ket{in} & \rightarrow_{R_{b_0b_1}} & \frac{1}{2}[ &
\ket{00}_A\widetilde{00}_T + \ket{01}_A\widetilde{01}_T +\\
&&& \ket{10}_A\widetilde{10}_T +
\ket{11}_A\widetilde{11}_T]\ket{0}_T\ket{1}_B\\
&\rightarrow_{B_y} & \frac{1}{2\sqrt{2}}[ &
\ket{00}_A(\ket{00}_T+\ket{11}_T)\\
&&&+ \ket{01}_A(\ket{01}_T + \ket{10}_T)\\
&&& + \ket{10}_A(\ket{00}_T-\ket{11}_T)\\
&&& + \ket{11}_A(\ket{01}_T - \ket{10}_T)]\ket{0}_T\ket{1}_B\\
&\rightarrow_{CNOT} & \frac{1}{2\sqrt{2}}[ &
\ket{00}_A(\ket{000}_T+\ket{111}_T)\\
&&& + \ket{01}_A(\ket{010}_T + \ket{101}_T)\\
&&& + \ket{10}_A(\ket{000}_T - \ket{111}_T)\\
&&& +\ket{11}_A(\ket{010}_T - \ket{101}_T)]\ket{1}_B\\
&\rightarrow_{split} & \frac{1}{2\sqrt{2}}[ & (\ket{000}_A +
\ket{100}_A)\ket{001}_B\\
&&& + (\ket{001}_A - \ket{101}_A)\ket{111}_B\\
&&& + (\ket{010}_A + \ket{110}_A)\ket{100}_B\\
&&& + (\ket{011}_A - \ket{111}_A)\ket{011}_B]\cdot
\end{align*}
We see that the reduced density matrices at Alice's location are
different for the two cases, and so $c$ is not secure against Alice.
\begin{figure}
\caption{\label{fig:measure-oot-gate}
\label{fig:measure-oot-gate}
\end{figure}
However, if Alice and Bob are subjected to send $b_0,b_1,c$ to $T$ via
classical channels, cf. figure \ref{fig:measure-oot-gate}. The inputs
will be measured and projected onto the computation basis. For example
on Alice side, simply speaking, Alice inputs $\ket{b_0b_1}$ can only
take one of the $4$ values $\ket{00}, \ket{01}, \ket{10}, \ket{11}$
and for any of these cases, Alice cannot discover $c$. Using the
defined model for the classical channel, Alice sends her inputs through
CNOT gates whose targets are in the measurement machine $M$ of the
classical channel. The output is entangled with $M$. The final states
of the computations for $c = 0$ and $c=1$ are
\begin{align*}
\ket{out_0} &= \frac{1}{2\sqrt{2}}[ & (\ket{000}_A\ket{00}_M +
\ket{010}_A\ket{01}_M)\ket{000}_B\\
&& + (\ket{001}_A\ket{00}_M - \ket{011}_A\ket{01}_M)\ket{110}_B\\
&& + (\ket{100}_A\ket{10}_M + \ket{110}_A\ket{11}_M)\ket{100}_B\\
&& + (\ket{101}_A\ket{10}_M - \ket{111}_A\ket{11}_M)\ket{010}_B],\\
\ket{out_1} &= \frac{1}{2\sqrt{2}} [ & (\ket{000}_A\ket{00}_M +
\ket{100}_A\ket{10}_M)\ket{001}_B\\
&& + (\ket{001}_A\ket{00}_M - \ket{101}_A\ket{10}_M)\ket{111}_B\\
&& + (\ket{010}_A\ket{01}_M + \ket{110}_A\ket{11}_M)\ket{100}_B\\
&& + (\ket{011}_A\ket{01}_M - \ket{111}_A\ket{11}_M)\ket{011}_B]\cdot
\end{align*}
The reduced matrices of tree qubits at Alice location are gained by
tracing out $M$ part and $B$ part, and become $I/8$ for both two
values of $c$.
Similar analyses on Bob side shows that the protocol is also
secure. Therefore, with help of classical communications, the protocol
becomes secure on both sides.
\section{\label{conclusions}Summary}
Our arguments were based on the difference between communicating
classical information in a classical manner and in a quantum manner.
\begin{figure}
\caption{\label{fig:public-cat}
\label{fig:public-cat}
\end{figure}
The discussions take us back to a similar problem of Schrodinger's
Cat~\cite{Gri04}. Imagine that Alice owns a Schrodinger Box, and at
a certain moment, has to tell Bob whether the cat is dead or alive.
If Alice and Bob live in a same public environment, e.g. in a same
room, Alice does this via a classical channel, e.g. the acoustic
channel: Alice has to open the box and sound what she sees. It is
equivalent to as though they open the box together,
cf. figure~\ref{fig:public-cat}. In another way, Alice can give the
box to Bob and let him open it. However, Bob can open the box in a
private environment. We can say that Alice and Bob live in two
separate quantum worlds. Imagine that Alice and Bob live in two
isolated rooms. Alice puts the observable hole of her box through the
wall into Bob's room, and the two rooms remain always
isolated, cf. figure~\ref{fig:private-cat}. It is as though Bob's
measurement devices are thrown to a private quantum space.
\begin{figure}
\caption{\label{fig:private-cat}
\label{fig:private-cat}
\end{figure}
We see that, within the classical concepts, a classical message
transmitted from Alice to Bob must be ``comprehensible'' by both
parties in a same reference frame. It is due to a classical channel as
a common environment that both Alice and Bob refer to. Classical
information can be viewed as quantum information, but measured by this
reference environment. All quantum measurement devices making classical
messages in a protocol between Alice and Bob must be thrown to this
trusted Hilbert space. In this paper, we have used these concepts of
communication of classical information, as in the former case of
telling the cat's state.
Nevertheless, in the purely quantum model used by Mayers and Lo \&
Chau, there is no such a common space, and the measurement devices of
each party for making classical information are thrown to a private
Hilbert space of that party~\cite{May97}.
Another example is the difference between the result of the
tossing of a classical random bit and the splitting of the EPR pair
$\ket{\Phi+}$. We see that the result of tossing of a random bit is a
pair of $(0_A,0_B)$ or $(1_A,1_B)$ with equal probability $1/2$. In
terms of quantum information, the random bits are described by the
density matrix:
$$
r_{AB} = \frac{1}{2}\projection{0_A0_B}{0_A0_B} +
\frac{1}{2}\projection{1_A1_B}{1_A1_B}\cdot
$$
On the other hand, the EPR pair is
$$
\ket{\Phi+} = \frac{1}{\sqrt{2}}(\ket{0_A0_B} + \ket{1_A1_B})
$$
The two states are indeed different. The EPR pair really implements
the tossing only when Alice and Bob have a common reference, for
example measurement devices coupled with a common environment that
project each qubit to a same basis $\{\ket{0},\ket{1}\}$. This
measurement is trusted by both party, and used as a Hilbert space
$\hilbert{M}$ of reference, and the measurement devices can be thrown
to it. The measurement of the EPR pair gives
$$
\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{0_A0_B0_M} + \ket{1_A1_B0_M}),
$$
and Alice and Bob get exactly a pair of random bits $r_{AB}$:
$$
tr_M(\projection{\psi}{\psi}) = \frac{1}{2}\projection{0_A0_B}{0_A0_B}
+ \frac{1}{2}\projection{1_A1_B}{1_A1_B}\cdot
$$
With the above concepts of communicating classical information, we
summarize that
\begin{itemize}
\item A general protocol, specified with classical and quantum
messages can be correctly implemented in the quantum model with only a
quantum channel. We can say that the original protocol is secure if
the simulating protocol in the quantum model is secure. However, we
have no right to use the simulating protocol in the quantum model to
prove the insecurity of the original protocol. We should consider its
insecurity in a model that would match better the real world with
quantum and classical channels.
\item We supposed that a classical channel is normally well coupled
with the environment and may not be controlled by neither Alice nor
Bob. It is convenient to see it as a trusted measurement which sends
back the classical outcomes to Alice and Bob.
\item We shown that in some special cases, the original protocol is
secure in presence of a classical channel while its simulating
protocol in the purely quantum model is insecure.
\end{itemize}
\end{document} |
\begin{document}
\title[Supercentralizers for deformations of the Pin osp dual pair ]{Supercentralizers for deformations of the Pin osp dual pair}
\author{Roy Oste}
\address{Department of Applied Mathematics, Computer Science and Statistics, Faculty of Sciences, Ghent University, Krijgslaan 281, Building S9, 9000 Gent, Belgium}
\email{[email protected]}
\thanks{
The author was supported by a postdoctoral fellowship, fundamental research, of the Research Foundation -- Flanders (FWO), number 12Z9920N}
\subjclass[2010]{16S80,
17B10,
20F55
}
\keywords{}
\begin{abstract}
In recent work, we examined the algebraic structure underlying a class of elements supercommuting with a realization of the Lie superalgebra $\fr{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra.
This generalization contained in particular the deformation by means of Dunkl operators associated with a real reflection group, yielding a rational Cherednik algebra instead of the Weyl algebra.
The aim of this work is to show that this is the full supercentralizer, give a (minimal) set of generators, and to describe the relation with the $(\mathrm{Pin}(d),\fr{osp}(2m+1|2n))$ Howe dual pair.
\end{abstract}
\maketitle
\section{Introduction}
In recent work~\cite{DOV}, we started the investigation of the algebraic structure underlying a class of elements supercommuting with a realization of the Lie superalgebra $\fr{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra.
This generalization contained in particular the deformation by means of Dunkl operators associated with a real reflection group, yielding a rational Cherednik algebra instead of the Weyl algebra.
The aim of this work is to show that this is the full supercentralizer inside the tensor product of rational Cherednik $H_\kappa$ and a Clifford algebra, give a (minimal) set of generators, and to describe the relation with the $(\mathrm{Pin}(d),\fr{osp}(2m+1|2n))$ Howe dual pair.
An explicit realisation of $H_\kappa$ is given by means of Dunkl operators (for the elements of $V$ )
and coordinate variables (the elements of $V^*$), which gives a natural (faithful) action on the polynomial space $S(V^*)$.
We will work over $\mathbb{C}$, the field of complex numbers with $i^2=-1$.
Throughout, $[\cdot,\cdot]$ will denote the skew-supersymmetric operation on a Lie superalgebra or the supercommutator~\eqref{e:supercomm}.
The notation $\{\cdot,\cdot\}$ will denote the antisupercommutator~\eqref{e:ascom}.
Moreover, a sign above the comma will sometimes, mostly in Section~\ref{s:rels}, be used to indicate the actual sign used in a(n anti)supercommutator. For instance, if $a$ and $b$ are odd then
$\scom{a}{b} = ab + ba $, so we will write $\scom[+]{a}{b} $, while if $a$ or $b$ is even, we have $\scom[-]{a}{b} = ab - ba $.
Tensor products are assumed to be $\mathbb{Z}_2$-graded, unless stated otherwise. The notation $ \odot $ will be used for the supersymmetric tensor product~\eqref{e:supersymtensor}.
Notations are not final.
$O_u = \tilde\sigma_u$ for $u\in V^*$,
\section{Lie superalgebras}\label{s:LSA}
\subsection{Preliminaries}
Denote $\bar 0$ and $\bar 1$ the elements of $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, the residue class ring mod 2.
The term superspace is used to refer to a $\mathbb{Z}_2$-graded vector space $\ca V =\ca V_{\bar 0 } \oplus\ca V_{\bar 1}$, and superalgebra for a $\mathbb{Z}_2$-graded algebra. The parity or ($\mathbb{Z}_2$-)degree of a homogeneous element $a \in \ca V_j$ is denoted by $|a| = j \in \mathbb{Z}_2$.
The parity reversing functor $\Pi$ sends a superspace $\ca V = \ca V_{\bar 0 } \oplus\ca V_{\bar 1}$ to the superspace $\Pi(\ca V)$ with the opposite $\mathbb{Z}_2$-grading:
$\Pi(\ca V_j) = \ca V_{j+\bar 1}$ for $j\in\mathbb{Z}_2$.
A Lie superalgebra $\mathfrak{g}= \mathfrak{g}_{\bar 0} \oplus \mathfrak{g}_{\bar 1}$ is a superalgebra whose product $[\cdot,\cdot]$ satisfies, for homogeneous elements $x,y\in \mathfrak{g}_{\bar 0} \cup \mathfrak{g}_{\bar 1}$ and $z\in \mathfrak{g}$,
\begin{align}
[x,y] & = -(-1)^{|x||y|}[y,x] &&\text{(super skew-symmetry)} \\*
[x,[y,z]] & =[[x,y],z] +(-1)^{|x||y|}[y,[x,z]] && \text{(super Jacobi identity).}\label{e:Jacobi}
\end{align}
A bilinear form $b$ on a Lie superalgebra $\mathfrak{g}$ is called invariant if $b([x,y],z) = b(x,[y,z])$ for all elements $x,y,z\in\mathfrak{g}$.
For a superspace $\ca V$, the general linear Lie superalgebra $\fr{gl}(\ca V)$ is formed by equipping
the associative superalgebra $A = \End(\ca V)$ with the supercommutator
\begin{equation}
\label{e:supercomm}
[a,b] = ab - (-1)^{|a||b|} ba\p,
\end{equation}
for $a,b\in A$ homogeneous elements for the $\mathbb{Z}_2$-grading on $ A$.
Denote $\mathbb{C}^{m|n}$ for the superspace $\ca V =\ca V_{\bar 0 } \oplus\ca V_{\bar 1}$ with $\ca V_{\bar 0 } = \mathbb{C}^m$ and $\ca V_{\bar 1} = \mathbb{C}^n$. In this case, the notation $\mathfrak{gl}(m|n)$ is used instead of $\mathfrak{gl}(\ca V)$.
A bilinear form $b$ on a superspace $\ca V = \ca V_{\bar 0 } \oplus \ca V_{\bar 1}$
is called consistent or even, if for $i,j\in\mathbb{Z}_2$, one has $b(\ca V_i ,\ca V_j ) = 0$ unless $i+ j =\bar{0}$. A consistent bilinear form $b$ is said to be supersymmetric (resp.\ skew-supersymmetric), if $b|_{\ca V_{\bar 0 }\times \ca V_{\bar 0 }}$
is
symmetric (resp.\ skew-symmetric) and $b|_{\ca V_{\bar 1}\times \ca V_{\bar 1}}$
is skew-symmetric (resp.\ symmetric). The subspace where $b$ restricts to a skew-symmetric form necessarily has even dimension.
If $b$ is a non-degenerate, consistent, supersymmetric or skew-supersymmetric, bilinear form on $\ca V$, the
orthosymplectic Lie superalgebra $\mathfrak{osp}(b)=\mathfrak{osp}(\ca V,b)$ is the
subalgebra of $\mathfrak{gl}(\ca V)$ that
preserves $b$.
As Lie superalgebras $\mathfrak{osp}(\ca V,b)\cong\mathfrak{osp}(\Pi(\ca V),\Pi (b))$, where $\Pi (b)$ is the form on $\Pi(\ca V)$ induced by means of the parity reversing functor $\Pi$. If $b$ is supersymmetric, then $\Pi (b)$ is skew-supersymmetric and vice versa.
For $\ca V = \mathbb{C}^{M|2n}$ with the standard supersymmetric form, the associated orthosymplectic Lie superalgebra is denoted by $\mathfrak{osp}(M|2n)$ or $\mathfrak{osp}^{sy}(M|2n)$.
For $\ca V = \mathbb{C}^{2n|M}$ with a skew-supersymmetric form, the associated orthosymplectic (or symplectico-orthogonal) Lie superalgebra is
$\mathfrak{spo}(2n|M)$ or $\mathfrak{osp}^{sk}(2n|M)$.
Given the isomorphism between them via $\Pi$, the notation $\mathfrak{osp}(M|2n)$ is sometimes used to refer to either of them.
\subsection{\texorpdfstring{$\fr{osp}(1|2)$}{osp(1|2)}}
The Lie superalgebra $\mathfrak{osp}(1|2)$ has a one-dimensional Cartan subalgebra $\mathfrak{h} = \langle h \rangle$ and root system $\Phi = \{\pm 2 \delta \} \cup \{ \pm \delta \}$, where $\delta\in \mathfrak{h}^*$ is the dual of $h$.
The even subalgebra is $\mathfrak{osp}(1|2)_{\bar 0} \cong \fr{sp}(2)\cong \fr{sl}(2)$ with root system $\{\pm 2 \delta \}$.
The odd root vectors $e_{\delta},e_{-\delta}$ satisfy the relations
\begin{equation}\label{e:osp12r}
[ e_{\delta} , e_{-\delta} ] = ( e_{\delta},e_{-\delta}) h_{\delta} \p,\qquad [h, e_{\pm\delta} ] = \pm e_{\pm\delta}\p,
\end{equation}
where $h_{\delta} = (\delta,\delta)h$ is the coroot of $\delta$. Here, $(\cdot,\cdot)$ denotes the unique (up to a constant factor)
non-degenerate consistent invariant supersymmetric bilinear form on $\mathfrak{g}$, and also the (symmetric) non-degenerate restriction of the form to $\mathfrak{h}$, and the induced form on $\mathfrak{h}^*$.
With the following normalization for the root vectors
\begin{equation}\label{e:osp12n}
F^{\pm} \colonequals e_{\pm\delta}/\sqrt{(e_{\delta},e_{-\delta})(\delta,\delta)}\p, \qquad
E^{\pm} \colonequals \pm [e_{\pm\delta}, e_{\pm\delta} ] / (2( e_{\delta},e_{-\delta})(\delta,\delta) )\p,
\end{equation}
and denoting also $H=h$, we have $\mathfrak{osp}(1|2) = \Span\{H,F^\pm,E^\pm\}$ with the relations that have nonzero right-hand side given by
\begin{equation}\label{e:osp12re}
\begin{aligned}[]
[ F^{+} , F^{-} ] & = H \p,\quad
& [ H , F^{\pm} ] &= \pm F^{\pm}\p,
&
[ F^{\pm} , F^{\pm} ] & = \pm 2\,E^{\pm}\p,
\\
[ E^{+} , E^{-} ] & = H\p,
& [ H , E^{\pm} ] & = \pm2\,E^{\pm}\p, \quad &
[ F^{\pm},E^{\mp} ] & = F^{\mp}\p.
\end{aligned}
\end{equation}
The even subalgebra is $ \Span\{H,E^\pm\} \cong \fr{sl}(2)$.
\subsection{Realization and centralizer}
Let $A = A_{\bar0} \oplus A_{\bar1} $ be an associative unital superalgebra.
Using the supercommutator~\eqref{e:supercomm}, if there are elements $e_{\delta},e_{-\delta}\in A_{\bar1}$ satisfying
\begin{equation}\label{e:osp12}
[ [ e_{\delta} , e_{-\delta} ] , e_{\pm\delta} ] = \pm C\, e_{\pm\delta}\p,
\end{equation}
for a non-zero constant $C$, then there is a realization of $\mathfrak{osp}(1|2)$ in $A$. The constant $C$ is related to the bilinear forms on $\mathfrak{osp}(1|2)$ and on $\mathfrak{h}^*$
by $C=( e_{\delta},e_{-\delta})(\delta,\delta)$. The elements $e_{\delta},e_{-\delta}\in A_{\bar1}$ can be rescaled as in~\eqref{e:osp12n} to have the commutation relations~\eqref{e:osp12re}.
Now, assume that we have a realization $\pi \colon \mathfrak{osp}(1|2) \to A$ of $\mathfrak{osp}(1|2)$ in $A$, with the product given by the supercommutator~\eqref{e:supercomm} in $A$.
For a subspace $ B \subset A$, the supercentralizer in $ A$ is
\begin{equation}
\label{e:centralizer}
\Cent_{ A}( B) = \{\, a \in A \mid [a,b] = 0 \text{ for all } b\in B \,\} \rlap{\,.}
\end{equation}
The idea is to describe the supercentralizer of $\mathfrak{osp}(1|2)$ in $A$.
Hereto, we consider an adjoint action of $\mathfrak{osp}(1|2) $ on $ A$:
\begin{equation}\label{e:ad}
\mathfrak{osp}(1|2) \times A \to A \colon g \mapsto
(g,a) \mapsto [\pi(g),a] \p.
\end{equation}
For the action~\eqref{e:ad} of $\mathfrak{osp}(1|2) $, in the representation space $ A$, every element of the centralizer $\Cent_{ A}(\mathfrak{osp}(1|2))$ is a copy of the one-dimensional, trivial module. Conversely, the isotypic component of the trivial module is precisely $\Cent_{ A}(\mathfrak{osp}(1|2))$.
The following proposition gives a way to determine $\Cent_{ A}(\mathfrak{osp}(1|2))$ from the centralizer of the even subalgebra $\Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})$.
To this end, we consider the following (even) elements in the universal enveloping algebra
$ U(\fr{osp}(1|2))$:
\begin{equation}\label{e:Pdelta}
P_+ \colonequals 1 - F^-F^+ \quad\text{ and }\quad P_{-} \colonequals 1 + F^+F^-\p,
\end{equation}
where we use the normalization~\eqref{e:osp12n}.
\begin{propo}\label{p:osp12}
For $\mathfrak{osp}(1|2)$, realized in an associative unital superalgebra $ A$, with even subalgebra $ \mathfrak{osp}(1|2)_{\bar0} \cong\fr{sl}(2)$,
one has
\[
\Cent_{ A}(\mathfrak{osp}(1|2)) = P_{+} \Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0}) = P_{-} \Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})\p,
\]
with $P_\pm$ acting on $ A$ as in~\eqref{e:ad}, that is
\[
P_{\pm} \colon A \to A \colon a \mapsto P_{\pm}(a)=a\mp [ F^{\mp} , [ F^{\pm}, a]] \p.
\]
\end{propo}
\begin{proof}
First, note that, by means of the relations~\eqref{e:osp12re},
\begin{equation}\label{e:P-dPd}
P_{-} = 1 + F^+F^- = 1 + (-F^-F^+ + H) = P_+ + H \p,
\end{equation}
so when $[H,\cdot]$ acts by zero, as is the case on $\Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})$, the actions of $P_{-}$ and $P_+$ coincide.
By definition, we have
$
\Cent_{ A}(\mathfrak{osp}(1|2)) \subset \Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})
$
and $
P_+ a = a
$ for $a \in \Cent_{ A}(\mathfrak{osp}(1|2))$,
hence
$
\Cent_{ A}(\mathfrak{osp}(1|2)) \subset P_+\Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})
$.
To prove the other inclusion, let $a \in \Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})$, we show that $P_+(a) \in \Cent_{ A}(\mathfrak{osp}(1|2))$. We have, using the super Jacobi identity~\eqref{e:Jacobi},
\begin{align*}
[F^+, P_+(a)]
& = [F^+,a] - [F^+,[F^-,[F^+,a]]]\\
& = [F^+,a] - [[F^+,F^-],[F^+,a]]+ [F^-,[F^+,[F^+,a]]]\\
& = [F^+,a] - [H,[F^+,a]]+ [F^-,[E^+,a]]\\
& = [F^+,a] - [[H,F^+],a]- [F^+,[H,a]]\\
& = [F^+,a] - [F^+,a]\p.
\end{align*}
Meanwhile, for $a \in \Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})$, we also have
\begin{align*}
[F^-, P_+(a)]
& = [F^-,a] - [F^-,[F^-,[F^+,a]]]\\
& = [F^-,a] + [E^-,[F^+,a]]\\
& = [F^-,a] + [[E^-,F^+],a]+ [F^+,[E^-,a]]\\
& = [F^-,a] - [F^-,a]\p.
\end{align*}
As $F^\pm$ generate $\mathfrak{osp}(1|2)$, this proves the other inclusion $ P_+\Cent_{ A}(\mathfrak{osp}(1|2)_{\bar 0})\subset
\Cent_{ A}(\mathfrak{osp}(1|2))
$.
\end{proof}
The next result gives a way to obtain, under certain conditions, the centralizer $\Cent_{ A}(\mathfrak{osp}(1|2)_{\bar{0}})$, which can then be used to describe the supercentralizer $\Cent_{ A}(\mathfrak{osp}(1|2))$ by the previous result.
\begin{propo}\label{p:sl2}
Let $\mathfrak{sl}(2)$ be realized in a unital associative algebra $ A$
by the elements $e_{\alpha},e_{-\alpha},h_{\alpha}$ satisfying the commutation relations
\[
[e_{\alpha},e_{-\alpha}] = h_{\alpha}\p, \qquad [ h_{\alpha},e_{\pm\alpha} ]=\pm(\alpha,\alpha) e_{\pm\alpha}\p.
\]
If $\Cent_{ A}(\mathfrak{h})$ decomposes into only finite-dimensional irreducible $\mathfrak{sl}(2)$-modules for the adjoint action of $\mathfrak{sl}(2)$ on $ A$, then
\[
\Cent_{ A}(\mathfrak{sl}(2)) = P_{\alpha} \Cent_{ A}(\mathfrak{h}) = P_{-\alpha} \Cent_{ A}(\mathfrak{h}) \p.
\]
where
\begin{equation}\label{e:Palpha}
P_{\alpha} = \sum_{k \geq 0} \frac{(-2)^k e_{-\alpha}^k e_{\alpha}^k}{(\alpha,\alpha)^k k!(k+1)!} \p,
\end{equation}
with the elements of $\mathfrak{osp}(1|2)$ acting on $\ca{A}$ via the action~\eqref{e:ad}.
\end{propo}
\begin{proof}
The elements of $\Cent_{ A}(\mathfrak{h})$ form the space of weight zero for the action of $\mathfrak{sl}(2)$ given by~\eqref{e:ad}.
Assume $\Cent_{ A}(\mathfrak{h})$ decomposes into only finite-dimensional irreducible $\mathfrak{sl}(2)$-modules.
For $v\in \Cent_{ A}(\mathfrak{h})$,
the sum in~\eqref{e:Palpha} reduces to a finite one when acting on $v$, and as
$e_{\alpha} P_{\alpha}v = 0 $, which follows via
\begin{align*}
e_{\alpha} e_{-\alpha}^k e_{\alpha}^k =&\ e_{-\alpha}^k e_{\alpha}^{k+1} + k\, e_{-\alpha}^{k-1}(h_{\alpha}-(\alpha,\alpha)(k-1)/2) e_{\alpha}^k \\
=&\ e_{-\alpha}^k e_{\alpha}^{k+1} + k\, e_{-\alpha}^{k-1} e_{\alpha}^k(h_{\alpha}+(\alpha,\alpha)(k+1)/2) \p,
\end{align*}
so $P_{\alpha} v$ is a highest weight vector of a finite-dimensional $\mathfrak{sl}(2)$-module with weight zero, hence a trivial module.
\end{proof}
\begin{remar}\label{r:proj}
The formulas~\eqref{e:Pdelta} and~\eqref{e:Palpha} correspond to a so-called ``extremal projector'', see for instance~\cite{AshSmiTol79,Tolstoi1985,Zhelobenko}, when acting on a space of weight zero.
For $\fr{g}$ a basic classical Lie (super)algebra (or generalization thereof), the extremal projector $P$ is an element of an extension of the universal enveloping algebra $U(\fr{g})$ containing formal power series with coefficients in the field of fractions of $U(\mathfrak{h})$, also called a transvector algebra or Mickelsson-Zhelobenko algebra (localization with respect to $\mathfrak{h}$) \cite{Zhelobenko,RE}. The element $P$ is the unique nonzero solution to the equations
\begin{equation}\label{e:exproj}
P^2 = P, \qquad e_{\alpha} P = 0 = P e_{-\alpha} \p,\quad \text{for all positive roots }\alpha\text{ of } \fr{g}\p.
\end{equation}
Hence, when acting on a representation space it would project onto highest weight vectors.
However, for Lie superalgebras with isotropic roots, when acting on a space of weight zero the formula for the extremal projector $P$ can result in denominators becoming zero.
\end{remar}
We now consider a lemma with some properties of $P_\pm$ that will be used in Section~\ref{s:osp12}. Similar properties hold for $P_{\alpha}$.
\begin{lemma}\label{l:Pdelta} For a realization of $\mathfrak{osp}(1|2)$ in $ A$, and $P_{\pm}$ given by~\eqref{e:Pdelta}, let $a,b,c\in A$, then
\[
P_{\pm} (a+b) = P_{\pm} (a)+ P_{\pm} ( b) \rlap{\,.}
\]
If $a\in\Cent_{ A}(\mathfrak{osp}(1|2))$, then
\[
P_{\pm} (a) = a\rlap{\,.}
\]
If $a\in\Cent_{ A}(\mathfrak{osp}(1|2))$ or $b\in\Cent_{ A}(\mathfrak{osp}(1|2))$, then
\[
P_{\pm} (ab) = P_{\pm} (a)P_{\pm} (b)\rlap{\,.}
\]
If $a,c\in\Cent_{ A}(\mathfrak{osp}(1|2))$, then
\[
P_{\pm} (abc) = aP_{\pm} (b)c\rlap{\,.}
\]
\end{lemma}
\begin{proof}
The first relation follows immediately from the bilinearity of the supercommutator and the second relation from the
definition of $\Cent_{ A}(\mathfrak{osp}(1|2))$.
For the third result, assume $a$ is a homogeneous element for the $\mathbb{Z}_2$-grading on $A$. With the action of $\mathfrak{osp}(1|2)$ on $A$ given by~\eqref{e:ad}, we have
\[
P_{+}(ab)
= ab - F^-F^+(ab)
= ab - F^-(F^+(a)b + (-1)^{|a|}aF^+(b))
\p.
\]
On the one hand, if $a\in\Cent_{A}(\mathfrak{osp}(1|2))$, then $F^{\pm}(a)=0$, so this becomes
\[
P_{+}(ab)
= ab - (-1)^{|a|} (F^-(a)F^+(b) + (-1)^{|a|}a (F^-F^+b))
= a P_{+}(b)
\p.
\]
On the other hand, if $b\in\Cent_{ A}(\mathfrak{osp}(1|2))$, so $F^{\pm}(b)=0$, this becomes
\begin{align*}
P_{+}(ab)
& = ab - (F^-F^+a)b = P_+(a) b\p.
\end{align*}
The final relation follows in the same manner.
\end{proof}
\begin{examp}
In the universal enveloping algebra $ U(\fr{osp}(1|2))$, the
centralizer of the even subalgebra is generated by the
element $F^+F^--F^-F^+ $ and the constants.
This follows by classical invariant theory.
Indeed, using~\eqref{e:osp12re}, we have
\begin{align*}
[E^\pm, F^+F^--F^-F^+ ] = \mp (F^\pm)^2 \pm (F^\pm)^2= 0 \p.
\end{align*}
Now, applying $P_+$ and using~\eqref{e:osp12re}, we find
\begin{align*}
P_+(F^+F^- )
& = F^+ F^- - [F^-,[F^+,F^+ F^- ]] \\
& = F^+ F^-
- [F^-,2E^+F^-
- F^+ H]\\
& = F^+ F^-
- 2F^+F^-
+4E^+E^-
+H^2
-F^+ F^- \\
& = H^2
+4E^+E^-
-2F^+ F^-
\p,
\end{align*}
with a similar expression for
\begin{align*}
P_+(F^-F^+ )
& = F^- F^+ - [F^-,[F^+,F^- F^+ ]] \\
& = F^- F^+
- [F^-,HF^+
- 2F^- E^+]\\
& = F^- F^+
-F^-F^+ - H^2 -4E^-E^+ - 2F^-F^+
\\
& =
- H^2 -4E^-E^+ -2F^- F^+
\p.
\end{align*}
Combining the two yields that $P_+(F^+ F^- - F^-F^+ )$ is proportional to the quadratic Casimir element of $ U(\fr{osp}(1|2))$:
\begin{equation}\label{e:Casiosp}
\Omega_{\fr{osp}} = H^2
+2(E^+E^-+E^-E^+)
-(F^+ F^- -F^-F^+) \p,
\end{equation}
where $\Omega_{\fr{sl}(2)} = H^2
+2(E^+E^-+E^-E^+)$ is the quadratic Casimir element of $U(\fr{sl}(2))$.
Note that $F^+ F^- - F^-F^+$ is related to the $\fr{osp}(1|2)$ Scasimir element~\cite{Casi}:
\begin{equation}\label{e:SCasiosp}
S = F^+F^- -F^-F^+ + 1/2 \p,
\end{equation}
which squares to $S^2 = \Omega_{\fr{osp}} + 1/4$.
The Scasimir element $S$
commutes with the even elements and anticommutes with the odd elements, so it antisupercommutes with $ U(\fr{osp}(1|2))$ since its parity is even.
By the above we also have
\begin{equation}\label{e:SCasiosprel}
P_\pm(S) = 2\Omega_{\fr{osp}} + 1/2 = 2S^2\p.
\end{equation}
\end{examp}
\subsection{Generalized symmetries}\label{s:gensyms}
Let $A$ be an associative unital (super)algebra.
We say that an element $a\in A$ is a generalized symmetry of $F\in A$ if there exists $b\in A$ such that $F\, a = b\, F$. Note that $a$ preserves the kernel of $F$.
Extremal projectors and transvector algebras can be used to construct generalized symmetries of either the positive or negative root vectors of a Lie (super)algebra realized in $A$, see also \cite{Zhelobenko,RE}.
Note that the extremal projectors can contain
fractions of $U(\mathfrak{h})$, so depending on the algebra $A$ one works in, a multiplication by elements of $U(\mathfrak{h})$ can be required to cancel denominators.
We now consider the case of $\fr{osp}(1|2)$, which we will use in Section~\ref{s:gensyms2} for an explicit realization.
Define
\begin{equation}\label{e:Q}
Q^{\pm} \colon \ca A \to \ca A \colon a \mapsto Q^{\pm}(a)=(H\pm1)a\mp F^{\mp} [ F^{\pm}, a] \p.
\end{equation}
\begin{propo}\label{p:gensym}
Let $a \in A$ be such that $[E^-,a] = b F^-$ for some $b \in A$, then
\[
Q^-(a) = (H-1)a - F^+ [F^-,a]
\]
is a generalized symmetry of $F^-$.
\end{propo}
\begin{proof}
Assume $a$ is homogeneous for the $\mathbb{Z}_2$-grading
\begin{align*}
F^-Q^-(a) & = F^-(H-1)a - F^-F^+ [F^-,a] \\*
& = HF^-a - (H-F^+F^- ) [F^-,a] \\*
& =(-1)^{|a|} H a F^- + F^+F^- (F^-a - (-1)^{|a|} a F^- ) \\*
& = (-1)^{|a|} H a F^- - F^+E^- a - F^+(-1)^{|a|} a F^- \\*
& = (-1)^{|a|} H a F^- - F^+ ( a E^- + bF^-)
- (-1)^{|a|} F^+ F^- aF^- \\*
& = (-1)^{|a|}( H a + (-1)^{|a|} F^+ a F^- -(-1)^{|a|} F^+b
- F^+ F^- a)F^- \\*
& = (-1)^{|a|}( Q^-(a) + a -(-1)^{|a|} F^+b )F^-
\end{align*}
which shows that $Q^-(a) $ is a generalized symmetry of $F^-$.
\end{proof}
A similar result holds for $F^+$ using $Q^+$.
More generally, we can take $a\in A$ such that for some $b\in A$
\[
F^- [F^-,a] = F^-(F^-a - (-1)^{|a|} a F^- ) = b F^-\p,
\]
which is the case if for some $c\in A$
\[
F^-F^-a = c F^-\p.
\]
\section{Dunkl realization} \label{s:Dd}
We consider a complex vector space $ V\cong \mathbb{C}^{d}$, for a positive integer $d$.
The ring of polynomial functions on $V$ is the symmetric algebra $S(V^*)$ of the dual space $V^*$.
In the next sections our focus will be on the dual space $V^*$.
The notation $\langle \cdot,\cdot \rangle$ will denote the natural bilinear pairing between a space and its dual.
\subsection{Bilinear form}\label{s:bilform}
Let $B$ denote a non-degenerate symmetric bilinear form on $V$.
The orthogonal group $\mathrm{O} \colonequals\mathrm{O}(V,B) \cong\mathrm{O}(d,\bb C)$ is the group of invertible linear transformations of $V$ that preserve the form $B$. The action of $\mathrm{O}$ on $V$ is naturally extended to $V^*$ as the contragradient action, and in turn also to tensor products of those spaces.
There is an isomorphism $\beta \colon V \to V^*\colon v \mapsto \beta(v)
$ given by
\begin{equation}\label{e:beta}
\langle \beta(u_1) , u_2 \rangle
= B(u_1,u_2)\p, \quad\text{for } u_1,u_2\in V\p,
\end{equation}
which commutes with the action of $\mathrm{O}$.
Hence, the spaces $V$ and $V^*$ are isomorphic as $\mathrm{O}$-modules.
We will use $\beta$ also to denote its inverse so $\beta$ becomes an involution of $V\oplus V^*$. Moreover, we will denote by $B$ also the induced bilinear form on $V^*$ given by $B(u,v) = B(\beta(u),\beta(v))$ for $u,v\in V^*$.
\subsection{Bases}\label{s:bases}
If no specific properties are needed, $v_1^*,\dotsc,v_d^*$ will denote a basis of $V^*$, dual to a basis $v_1,\dotsc,v_d$ of $V$, so $ \langle v^*_p,v_k\rangle = \delta_{p,q}$ for $p,q\in \{1,\dots,d\}$.
In terms of these bases, for $u \in V^*$ and $v\in V$ we have
\begin{equation}\label{e:B}
\beta(v) = \sum_{p=1}^d B( v, v_p) v^*_p\p, \qquad
\beta(u) = \sum_{p=1}^d B( u, v^*_p) v_p\p,
\end{equation}
and
\begin{equation}\label{e:BB}
v = \sum_{p=1}^d B( v, v_p) \beta(v^*_p)
\p, \qquad
u = \sum_{p=1}^d B( u, v^*_p) \beta(v_p)
\p.
\end{equation}
so
\begin{equation}\label{e:BBB}
v = \sum_{p,q=1}^d B(v,v_p) B(v_p^*,v_q^*) v_q
\p, \qquad
u =\sum_{p,q=1}^d B(u,v_p^*) B(v_p,v_q) v_q^*
\p.
\end{equation}
For brevity, we will sometimes denote $B_{pq} = B(v_p,v_q)$.
When needed, $y_1,\dotsc,y_d$ will denote a basis of $V$ and $x_1,\dotsc,x_d\in V^*$ its dual basis, such that $\delta_{j,k} =\langle x_j,y_k\rangle = B(x_j,x_k) = B(y_j,y_k)$.
Note that the involution $\beta$ sends $y_j$ to $x_j$ and vice versa.
Let $V^*=V^+\oplus V^0 \oplus V^-$ be a Witt decomposition of $V^*$ where $V^+$ and $V^-$ are complementary maximal $B$-isotropic subspaces of $V^*$ of dimension $\ell = \lfloor d/2 \rfloor$, and $V^0 = \emptyset$ for $d$ even while $V^0$ is anisotropic and one-dimensional for $d$ odd.
Let $z_1^+,\dotsc,z_\ell^+$ denote a basis of $V^+$ and $z_1^-,\dotsc,z_\ell^-$ a basis of $V^-$ such that $B(z_j^+,z_k^-) = \delta_{j,k}/2$.
For $d$ odd, denote $z_0$ an element of $V^0$ satisfying $B(z_0,z_0) = 1$.
The dual basis of $V$ is given by $w_j^{\pm} \colonequals 2\beta(z_j^\pm)$, satisfying $\langle z_j^\pm , w_j^\mp \rangle = \delta_{j,k}$.
\subsection{Clifford algebra}
As our focus will be on $V^*$, we will construct the Clifford algebra associated with $V^*$ and $B$. Note that this results in the same Clifford algebra as when using $V$ and $B$.
The Clifford algebra $\mathcal{C} \colonequals \mathcal{C}(V^*,B)$ is
the quotient of the tensor algebra $T(V^*)$ by the ideal generated by all elements of the form $v\otimes v - B(v,v)1 $ for $v\in V^*$.
The quotient map from the embedding $V^* \to T(V^*)$ gives a canonical map $\gamma \colon V^* \to \mathcal{C}$.
We have that $\mathcal{C}$ is the associative algebra with 1 generated by $\gamma(V^*)$, subject to the anticommutation relations
\begin{equation}\label{e:Clifcom}
\gamma(u)\gamma(v)+\gamma(v)\gamma(u) = 2 B(u,v)\,1\qquad\text{for }u,v\in V^*\p.
\end{equation}
The Clifford algebra inherits the structure of a filtered super algebra from $T(V^*)$, with the generators $\gamma(V^*)$ being odd and having filtration degree 1.
For $u\in V^*$, we will denote $\gamma_u \colonequals \gamma(u) \in \mathcal{C}$, and for $u_1,\dotsc,u_k\in V^*$ we let $\gamma_{u_1\dotsm u_k} \colonequals \gamma_{u_1}\dotsm \gamma_{u_k} \in \mathcal{C}$.
Moreover, we denote $\gamma_{u_1\dotsm \widehat{u}_j\dotsm u_n}\colonequals \gamma_{u_1} \dotsm \widehat{\gamma}_{u_j} \dotsm \gamma_{u_n}$ where the notation $ \widehat{\gamma}_{u_j}$ indicates that the factor $ \gamma_{u_j}$ is omitted in the product.
For the chosen bases of $V^*$ we denote $e_j = \gamma(x_j)$ for $j\in\{1,\dotsc,d\}$, and
$\theta^\pm_k = \gamma(z^\pm_k)$ for $k\in \{1,\dotsc,\ell\} $ with $\theta_0 = \gamma(z_0)$ for $d$ odd.
In $\ca C$, they satisfy the relations
\begin{equation}
e_je_k+e_ke_j = 2\delta_{j,k} \p, \quad \theta_j^\pm\theta_k^\pm + \theta_k^\pm\theta_j^\pm = 0\p, \quad \theta_j^+\theta_k^- + \theta_k^+\theta_j^- = \delta_{j,k}\p,
\end{equation}
and when $d$ is odd, also $\theta_0^2 = 1$ and $\theta_0\theta_k^\pm + \theta_k^\pm\theta_0 = 0$.
For a subset $A \subset \{1,\dotsc,d\}$, with elements $A= \{ a_{1},a_{2},\dotsc,a_{n}\} $ such that $1\leq a_{1}<a_{2}<\cdots <a_{n}\leq d$, we denote
$
e_A = e_{a_{1}}e_{a_
{2}}\cdots e_{a_{n}}
$.
Let $e_{\emptyset} = 1$, then a basis for $\ca C$ as a vector space is given by $\{e_A \mid A \subset \{1,\dotsc,d\} \}$.
We denote the chirality element of the Clifford algebra as
\begin{equation}\label{e:Gamma}
\Gamma \colonequals i^{d(d-1)/2} e_1 \dotsm e_d \in \ca C;
\end{equation}
it satisfies $\Gamma^2 = 1$ and $\Gamma \gamma_u = (-1)^{d-1} \gamma_u \Gamma$ for $u\in V^*$.
Let $\ca A$ denote the anti-symmetrization operator, which has the following action on a multilinear expression in $n$ indices
\begin{equation}\label{e:asym}
\ca A( f_{u_1u_2\dotsm u_n})= \frac{1}{n!} \sum_{s \in \rm{S}_n} \sgn (s) f_{u_{s(1)} \dotsm u_{s(n)}} \p,
\end{equation}
where $\rm{S}_n$ is the symmetric group of degree $n$.
We have $\ca A \ca A = \ca A$ and
\begin{equation}\label{e:asym2}
\ca A( f_{u_1u_2\dotsm u_n})= \frac{1}{n} \sum_{j=1}^n (-1)^{j-1} f_{u_j} \ca A( {}_{u_1\dotsm \widehat{u}_j \dotsm u_n}) \p.
\end{equation}
The symbol map and its inverse, the quantization map
\begin{equation}\label{e:quant}
q \colon \bigwedge( V^*) \to \ca C \colon u_1 \wedge u_2 \wedge \dotsb \wedge u_k \mapsto \cA{u_1\dotsm u_k}\p,
\end{equation}
are isomorphisms of $\mathrm{O}$-modules and filtered super vector spaces~\cite[Section 2.2.5]{Meinrenken}.
For $u,v,w,x\in V^*$, we have
\begin{align}
\ca A (\gamma_{uv}) & =\gamma_{uv}-B(u,v)\label{e:Auv}\\
\ca A (\gamma_{uvw}) &=\gamma_{uvw}-B(u,v)\gamma_w +B(u,w)\gamma_v -B(v,w)\gamma_u \label{e:Auvw}\\
\ca A (\gamma_{uvwx})& =\gamma_{uvwx} -B(u,v)\gamma_w\gamma_x +B(u,w)\gamma_v\gamma_x -B(v,w)\gamma_u \gamma_x\label{e:Auvwx}\\
& \quad
-B(u,x)\gamma_v\gamma_w +B(v,x)\gamma_u\gamma_w -B(w,x)\gamma_u \gamma_v\notag\\
& \quad
+B(u,v)B(w,x) -B(u,w)B(w,x) +B(v,w)B(w,x) \notag\p.
\end{align}
Note that if $u_1,\dotsc,u_n\in V^*$ are $B$-orthogonal, then
\[
\ca A(\gamma_{u_1u_2\dotsm u_n})= \gamma_{u_1u_2\dotsm u_n}\p.
\]
With the commutator, the space $q(\wedge^2(V^*))$ in $\ca C$ forms a realization of the Lie algebra $\fr{so}(V^*,B) \cong \fr{so}(V,B) \cong \fr{so}(d,\mathbb{C})$. We have the following adjoint action on $\gamma(V^*)$:
\begin{equation}\label{e:adso}
[\gamma_{uv}/2,\gamma_w] = \gamma_u[\gamma_v,\gamma_w]/2 - [\gamma_u,\gamma_w]\gamma_v/2 = B(v,w) \gamma_u- B(u,w)\gamma_v \p.
\end{equation}
\subsection{Spinor space}
When $d$ is odd, there is a unique isomorphism class of irreducible $\mathbb{Z}_2$-graded $\ca C$-modules, and there are two isomorphism classes of irreducible ungraded $\ca C$-modules, see \cite[Theorem~3.10]{Meinrenken}.
When $d$ is even, there are two isomorphism classes of irreducible $\mathbb{Z}_2$-graded $\ca C$-modules, and there is a unique isomorphism class of irreducible ungraded $\ca C$-modules.
A model for an irreducible $\mathbb{Z}_2$-graded $\ca C$-module $\bb S$ is given by the exterior algebra $ \bigwedge (V^+ \oplus V^0)$, where $V^*=V^+\oplus V^0 \oplus V^-$ is a Witt decomposition of $V^*$.
On the space $\bb S$ the elements of $\gamma(V^+)$ act by exterior multiplication and those of $\gamma(V^-)$ by interior multiplication or contraction.
In terms of a $B$-isotropic basis of $V^*$ as defined in Section~\ref{s:bases},
$\theta^-_{j}\in \gamma(V^-)$ acts as the odd differential operator corresponding to the odd variable $\theta^+_{j}\in \gamma(V^+)$.
When $d$ is even, $V^0 = \emptyset$ and this gives the complete action of $\ca C$ on $\bb S$. The parity-reversed space $\Pi ( \bb S ) $ is another irreducible $\mathbb{Z}_2$-graded $\ca C$-module, they are isomorphic as ungraded $\ca C$-modules.
The spinor space $\bb S$ can be realized explicitly inside the Clifford algebra as
\begin{equation}
\bb S =\bigwedge (V^+ ) \prod_{j} (\theta_{j}^- \theta_{j}^+) \p,
\end{equation}
with the action of $\ca C$ given by Clifford algebra multiplication, since the product $\prod\theta_{j}^- \theta_{j}^+$ is annihilated by all $\theta^-_{k}$.
When $d$ is odd, recall $\theta_0\in \gamma(V^0)$ with $B(\theta_0,\theta_0)=1$, we can write $\bb S = \bb S_+ \oplus \bb S_-$ where
$\bb S_\pm = \bigwedge (V^+) (1\pm\theta_0)/2 $ are non-isomorphic irreducible ungraded $\ca C$-modules.
The action of $\theta_{0}$ on $S_\pm$ is given by
\begin{equation}\label{e:acttheta0}
\theta_{0} \cdot \theta = \pm (-1)^k \theta \p,\quad \text{ for } \theta \in \textstyle\bigwedge^k (V^+)(1\pm\theta_0)/2\subset \bb S_\pm\p,
\end{equation}
and extending by linearity. The sign in the action~\eqref{e:acttheta0} of $\theta_{0}$ distinguishes between the spaces $\bb S_{\pm}$.
The spinor space $\bb S$ can be realized explicitly inside the Clifford algebra by letting
\begin{equation}
\bb S_{\pm} = \bigwedge (V^+ )(1\pm\theta_0)/2 \prod_{j} (\theta_{j}^- \theta_{j}^+) \p,
\end{equation}
with the action of $\ca C$ given by Clifford algebra multiplication, since $\theta_0(1\pm\theta_0)/2 = \pm (1\pm\theta_0)/2 $.
\subsection{Reflection group}
We fix a finite real reflection group $G\subset \mathrm{O}$.
Denote by $\mathcal S$ the set of reflections of $G$.
For each $s\in \mathcal{S}$, fix $\alpha_s\in V^*$ to be a $-1$ eigenvector for the action of $s$.
By definition, $\beta(\alpha_s)\in V$ is a $-1$ eigenvector for the action of $s$ on $V$.
Denote $ \alpha_s^{\vee}
\colonequals 2\beta(\alpha_s) / B(\alpha_s,\alpha_s) $, then
for $v\in V$ and $u\in V^*$, the reflection $s\in \mathcal{S}$ acts as
\begin{equation}\label{e:s}
s( v) = v - \langle\alpha_s, v \rangle \alpha_s^{\vee}\p,\qquad
s(u) = u - \alpha_s\langle\alpha_s^{\vee}, u \rangle \p.
\end{equation}
Define $T(V \oplus V^*) \rtimes G$ to be the quotient of $T(V \oplus V^*) \otimes \mathbb{C}[G]$ by the relations
\begin{equation}\label{e:crossedproduct}
(u,g)(v,h) = (u\,g(v),gh) \qquad \text{for } u,v\in T(V \oplus V^*) \text{ and } g,h\in G\p,
\end{equation}
so in $T(V \oplus V^*) \rtimes G$, we have $gug^{-1} = g(u)$ for $g\in G$ and $u\in T(V \oplus V^*)$.
We fix a map $\kappa \colon \mathcal S \to \mathbb{C} $ that is $G$-invariant (for the conjugation action), so that the elements of an orbit all have the same image.
\begin{defin}
Define $H_\kappa = H_\kappa(G,V)$ to be the quotient of $T(V \oplus V^*) \rtimes G$
by the relations
\begin{equation}
\label{e:RC}
\begin{aligned}{}
[x,u] &= 0 = [y,v]\p, \qquad\text{for } y,v \in V \text{ and } x,u\in V^*\\
[y,x] &= \langle y, x\rangle + \sum_{s\in\mathcal S} \langle y, \alpha_s\rangle\langle \alpha_s^{\vee}, x \rangle \kappa(s) s \p, \quad\text{for } y \in V,x\in V^*\p.
\end{aligned}
\end{equation}
\end{defin}
When $\kappa$ is the zero map, the relations~\eqref{e:RC} reduce to the canonical commutation relations of the Weyl algebra $\ca W = \ca W(V)$, so $ H_0(G,V) = \ca W(V) \rtimes G $.
\begin{remar}
The algebra $H_\kappa(G,V)$ is called a rational Cherednik algebra, and is
a rational degeneration of a double affine Hecke algebra \cite{EtGi,GGOR}.
More generally, a complex reflection group $G\subset GL(V)$ can be used.
Also, there can be an extra parameter $t$ accompanying $\langle y, x\rangle$ in \eqref{e:RC}, which we have taken $t=1$ here. See \cite{Kieran} for the case where $t\in \mathbb{C}^\times$.
A realization of $H_\kappa$ is given by means of Dunkl operators~\cite{Du}
\begin{equation}\label{e:dunkl}
\ca{D}_{y} = \frac{\partial}{\partial y} + \sum_{s\in\mathcal S} \kappa(s) \frac{\langle y, \alpha_s\rangle}{\alpha_s} (1-s)\p,\qquad \text{for } y\in V\p,
\end{equation}
and coordinate variables (for the elements of $V^*$), which gives a natural, faithful action on the polynomial space $S(V^*)$.
In the context of a rational Cherednik algebra, the parameter function is usually chosen the opposite sign compared to the one used for Dunkl operators, corresponding to the substitution $\kappa = -c$.
\end{remar}
\begin{lemma}\label{l:Buv}
For $u,v\in V^*$ or $u,v\in V$, in $H_\kappa$ we have $[\beta(u),v]= [\beta(v), u]$.
\end{lemma}
\begin{proof}
Let $u,v\in V^*$, then we have
\begin{equation}\label{e:combuv}
[\beta(u),v]= B(u,v) + 2\sum_{s\in\mathcal S} \frac{B( \alpha_s,u)B( v,\alpha_s ) }{B(\alpha_s,\alpha_s)} \kappa(s) s = [\beta(v), u]\p,
\end{equation}
where the last equality follows from $B$ being symmetric.
\end{proof}
\subsection{Superalgebra}
We consider the superspace $\ca V = \mathbb{C}^{2|1}$ equipped with a non-degenerate, skew-supersymmetric, consistent, bilinear form $b$.
Denote by $\omega$ the skew-symmetric bilinear form on $\ca V$, and also its restriction to $\ca V_{\bar 0}=\mathbb{C}^{2}$, that equals $b$ on $\ca V_{\bar 0} $ and is zero on $(\ca V \times \ca V ) \setminus (\ca V_{\bar 0} \times \ca V_{\bar 0} )$.
The tensor product $\ca U = V^*\otimes \ca V$ is again a superspace, inheriting the $\mathbb{Z}_2$-grading from $\ca V$, so $\ca U_{\bar 0} = V^*\otimes \ca V_{\bar 0}$ and $\ca U_{\bar 1} = V^*\otimes \ca V_{\bar 1}$.
There is a natural action of $\mathrm{O}$ on $\ca U = V^*\otimes \ca V$ as
$\mathrm{O}\otimes \Id_{\ca V}$ where $\Id_{\ca V}$ denotes the identity on $\ca V$. For $G \subset\mathrm{O}$
we consider also another action on $\ca U$:
\begin{equation}\label{e:actO}
a_0 \colon G \times \ca U \to \ca U \colon
\begin{cases}
a_0(g,u\otimes v) = (g\cdot u)\otimes v & \text{ for }u\in V^*, v \in \ca V_{\bar 0} \\
a_0(g,u\otimes v) =u\otimes v & \text{ for }u\in V^*, v \in \ca V_{\bar 1}\p,
\end{cases}
\end{equation}
where $g\cdot u$ denotes the action of $g\in G$ on $u\in V^*$.
The ``missing'' interaction of $G$ and $\ca U_{\bar 1}$ will be provided by means of the $\mathrm{Pin}$-group inside the Clifford algebra, see~\eqref{e:rho}.
The action \eqref{e:actO} is extended naturally to the tensor superalgebra $T(\ca U) = \bigoplus_n \ca U^{\otimes n}$, which uses $\mathbb{Z}_2$-graded tensor products. Using~\eqref{e:actO}, define $T(\ca U) \rtimes G$ to be the quotient of $T(\ca U) \otimes \mathbb{C}[G]$ by the relations
\begin{equation}\label{e:crossproduct2}
(u,g)(v,h) = (u\,a_0(g,v),gh) \qquad \text{for } u,v\in T(\ca U), g,h\in G\p.
\end{equation}
Now, we consider the $G$-invariant symmetric bilinear map
$\psi_\kappa^B(\cdot,\cdot) \colon V^*\times V^* \to \mathbb{C}[G] $
defined as
\begin{equation}\label{e:psiB}
\psi_\kappa^B(u,v) =
2\sum_{s\in\mathcal S}
\frac{B( \alpha_s,u)B( v,\alpha_s )
}{B(\alpha_s,\alpha_s)} \kappa(s) s \p, \qquad\text{for }u,v\in V^*\p.
\end{equation}
\begin{defin}
Define the superalgebra $A_\kappa$ to be the quotient of
$(T(\ca U) \rtimes G) $ by the relations
\begin{equation}\label{e:comre}
u v - (-1)^{|u||v|} vu = b_{\ca{U}}(u,v)\,1+ \psi_\kappa(u,v)\,1 \qquad\text{for }u,v\in \ca U_{\bar0} \cup \ca{U}_{\bar1}\p,
\end{equation}
where the right-hand side is defined for $u \otimes w, v\otimes z \in V^*\otimes \ca V = \ca U $ as
\[
b_{\ca{U}}(u \otimes w,v\otimes z) = B(u,v)b(w,z) \p,
\qquad
\psi_\kappa(u \otimes w,v\otimes z) =\psi_\kappa^B(u,v) \omega(w, z)\p.
\]
\end{defin}
The superalgebra $A_\kappa$ is generated, as an algebra, by $\ca U$ and $G$.
We now show that $A_\kappa$ is the tensor product of the rational Cherednik algebra $H_\kappa$ and the Clifford algebra $\ca C$.
Fix $x^+,x^-\in \ca V _{\bar 0}$ and $\gamma\in \ca V _{\bar 1}$ to be a basis of $\ca V = \mathbb{C}^{2|1}$ satisfying $b(x^-,x^+) = 1 = -b(x^+,x^-)$ and $b(\gamma,\gamma)=2$, with $b$ zero for all other combinations.
For the elements of
the form $ u \otimes \gamma \in \ca U$,
we see that the relations~\eqref{e:comre} correspond precisely to the Clifford algebra relations~\eqref{e:Clifcom}. For $u\in V^*$, we will identify $\gamma_u= u \otimes \gamma$.
Identifying $u\in V^*$ with $u \otimes x^+ \in \ca U$, and $u \otimes x^- \in \ca U$ with $\beta(u) \in V$,
the relations~\eqref{e:comre} then correspond precisely to~\eqref{e:RC}.
An element $v \in V$ then corresponds to $\beta(v)\otimes x^- \subset \ca U$.
\subsection{Double cover}
The group $\mathrm{Pin} \colonequals \mathrm{Pin}(V,B)$ is a double cover $p\colon\mathrm{Pin}\to\mathrm{O}$ and is realized in the Clifford algebra $\ca C$
as the set of products $\gamma_{u_1\dotsm u_k}$ where $u_j\in V^*$ with $B(u_j,u_j) =1$~\cite{Meinrenken}. The subgroup $\mathrm{Spin}(V,B)$ consists of similar products with $k$ even.
For a reflection $s\in \ca S \subset \mathrm{O}$, denote $\tilde s \colonequals \gamma(\alpha_s)/\sqrt{B( \alpha_s,\alpha_s)} \in \mathrm{Pin}\subset \ca C$, then $p(\widetilde s) = s$ and $p^{-1}(s) =\{ \pm\tilde s\}\subset \ca C$.
The preimage of the identity $\Id \in \mathrm{O}$ is $p^{-1}(\Id) =\{ \pm1\}\subset \ca C$.
Define the pin double cover of $G\subset \mathrm{O}$ as $\widetilde G \colonequals p^{-1}(G) \subset \mathrm{Pin}$. The conditions for $\widetilde G$ to be a non-trivial central extension of $g$ are in~\cite{Morris}.
See~\cite{Morris} also for a presentation in terms of generators and relations for $G$ and $\widetilde G$.
\begin{remar}
Note that this is the version of the $\mathrm{Pin}$-group, and thus of $\widetilde G$, where the preimages (for the covering map $p$) of a reflection in $\mathrm{O}$ have order two (and not four).
The (non-isomorphic) other version can be obtained by using elements $u_j\in V^*$ with $B(u_j,u_j) =-1$, or by adding a minus sign to the defining relations of the Clifford algebra.
We refer to~\cite[Section~3.7.2]{Meinrenken} for the definition and the distinction with the group $\mathrm{Pin}_c$.
\end{remar}
In the superalgebra $A_\kappa \cong H_\kappa \otimes \ca C$, there is a copy of the group $G\subset H_\kappa$ and also of the group $ \widetilde{G} \subset \ca C$. We use these to define a group homomorphism
\begin{equation}\label{e:rho}
\rho \colon \tilde G \to A \colon \tilde s \mapsto p(\tilde s) \, \tilde s\p,
\end{equation}
which is extended linearly to a map on the group algebra $\mathbb{C}[\widetilde G]$.
We note that $\rho(\mathbb{C}[\widetilde G])$ is a quotient of the group algebra $\mathbb{C}[\widetilde G]$, since the central element (the non-trivial preimage of the identity) is given by the scalar $-1\in\mathbb{C}$
for the realization of $\widetilde{G}$ in $\mathrm{Pin} \subset\ca C$, see~\cite{Kieran}.
Recall that, as an algebra, the superalgebra $A_\kappa$ is generated by $\ca U = V^*\otimes \ca V$ and $G$.
\begin{defin}\label{d:actG}
For $g\in G$ and $u\in A_\kappa$, we denote by $g\cdot u$ the action $G \times A_\kappa \to A_\kappa$ that is the extension of the natural action of $G\subset \mathrm{O}$ on $V^*$, acting as $G \otimes \Id_{\ca V}$ on $V^* \otimes \ca V$, and of the action by conjugation on the copy $G\subset H_\kappa$.
\end{defin}
This action of $G$ is related to the action of $\rho(\widetilde G)$ inside $A_\kappa$ as follows.
\begin{lemma}\label{l:rhoG}
For $\tilde g \in \widetilde G$ and $u\in A_\kappa$ a homogeneous element for the $\mathbb{Z}_2$-grading, in $A_\kappa$ we have
\begin{equation}
\rho(\tilde g) u \rho(\tilde g^{-1}) = (-1)^{|\tilde g||u|}p(\tilde g) \cdot u\p.
\end{equation}
\end{lemma}
\begin{proof}
Use~\eqref{e:actO}, \eqref{e:rho} and the properties of the $\mathrm{Pin}$-group.
\end{proof}
Finally, for $u\in V^*$, we define the following elements in $ \rho(\widetilde G)$:
\begin{equation}\label{e:Ov}
\tilde\sigma_u \colonequals \frac12\sum_{s\in\mathcal S} \alpha_s^{\vee}(u ) \, \kappa(s) \,s \, \gamma_{\alpha_s}
= \sum_{s\in\mathcal S} \frac{B(\alpha_s,u)}{\sqrt{ B(\alpha_s,\alpha_s)}} \, \kappa(s) \rho(\tilde s ) \p.
\end{equation}
By~\eqref{e:comre} and~\eqref{e:combuv}, we have that for $u,v\in V^*$
\begin{equation}\label{e:Ogamma}
\scom[+]{ \gamma_u}{\tilde\sigma_v} = \scom[-]{\beta(u)}{v} - B(u,v) = \scom[+]{ \gamma_v}{\tilde\sigma_u} \p,
\end{equation}
where the last equality follows by Lemma~\ref{l:Buv}.
Expanding the anticommutators in~\eqref{e:Ogamma} gives rise to the following result (cfr.~\cite[Lemma 3.10]{DOV}).
\begin{lemma}\label{l:Ogammas}
Let $n \in \{1,2,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\begin{equation}\label{e:Ogammas}
\ca A (\tilde\sigma_{u_1} \gamma_{u_2 \dotsm u_n})
= \ca A (\gamma_{u_1} \tilde\sigma_{u_2} \gamma_{u_3 \dotsm u_n})
= \dotsb
= \ca A (\gamma_{u_1 \dotsm u_{n-1}}\tilde\sigma_{u_n})\p.
\end{equation}
\end{lemma}
\begin{proof}
The first non-trivial case, for $n=2$, follows immediately from~\eqref{e:Ogamma}:
\begin{equation}\label{e:gammaO}
\tilde\sigma_u \gamma_v - \tilde\sigma_v \gamma_u = \gamma_u \tilde\sigma_v - \gamma_v \tilde\sigma_u
\p.
\end{equation}
We can then use this to find for general $n\in \{3,\dotsc,d\}$
\begin{align*}
\ca A ( \tilde\sigma_{u_1} \gamma_{u_2 \dotsm u_n})
&=
\frac{1}{n}\sum_{j=1}^n(-1)^{j-1} \tilde\sigma_{u_j} \ca A (\gamma_{u_1 \dotsm\widehat{u}_j \dotsm u_n}) \\
& =
\frac{1}{n(n-1)}\sum_{1\leq j < k \leq n}(-1)^{j+k-1} (\tilde\sigma_{u_j}\gamma_{u_k}-\tilde\sigma_{u_k}\gamma_{u_j}) \ca A ( \gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n}) \\
& =
\frac{1}{n(n-1)}\sum_{1\leq j < k \leq n}(-1)^{j+k-1} (\gamma_{u_j}\tilde\sigma_{u_k}-\gamma_{u_k}\tilde\sigma_{u_j}) \ca A (\gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n}) \\
& = \ca A (\gamma_{u_1} \tilde\sigma_{u_2} \gamma_{u_3 \dotsm u_n})\p,
\end{align*}
and the other equalities follow by repeated application of the same steps.
\end{proof}
\subsection{Lie (super)algebra realizations}
The bilinear form $B$ on $V$ naturally corresponds to an element of $(V\otimes V)^*$, a linear map on $V\otimes V$. Since $V$ is finite-dimensional and $B$ is symmetric, we have $B \in S^2(V^*) \subset V^* \otimes V^*$. As above, let $v_1^*,\dotsc,v_d^*$ denote a basis of $V^*$, dual to a basis $v_1,\dotsc,v_d$ of $V$, then
\begin{equation}
B = \sum_{p,q = 1}^d B(v_p,v_q) v_p^* \otimes v_q^* = \sum_{p = 1}^d v_p^* \otimes \beta(v_p) = \sum_{p = 1}^d \beta(v_p) \otimes v_p^* \p,
\end{equation}
which is, by definition, invariant for the action of the group $\mathrm{O}(V,B)$.
Every element of $\ca{V}$ corresponds to a copy of $V^*$ in $V^*\otimes \ca V$.
We can use this to map $B\in S^2(V^*)$ in $S^2( V^*\otimes \ca V)$, by viewing $\nu \in \ca V$ as the map $\nu \colon V^* \to V^* \otimes \ca V\colon v \mapsto v \otimes \nu$.
For $w,z\in\ca{V}$ homogeneous elements for the $\mathbb{Z}_2$-grading,
the supersymmetric tensor product is given by
\begin{equation}
\label{e:supersymtensor}
w \odot z = (w \otimes z + (-1)^{|w||z|} z \otimes w)/2\p,
\end{equation}
and we then consider the following elements of $S^2( V^*\otimes \ca V)$:
\begin{equation}\label{e:Bwz}
(w \odot z)(B) = \frac12\sum_{p,q=1}^d B(v_p,v_q)((v_p^* \otimes w) (v_q^* \otimes z) + (-1)^{|w||z|} (v_p^* \otimes z )( v_q^* \otimes w ) ) \p.
\end{equation}
Under the quotient by the relations~\eqref{e:comre}, in the superalgebra $A_\kappa$ we have:
\begin{equation}\begin{aligned}\label{e:Bwz2}
(w \odot z)(B) & =
\sum_{p,q=1}^d B(v_p,v_q)((v_p^* \otimes w)( v_q^* \otimes z) - [v_p^* \otimes w , v_q^* \otimes z]/2 )\\
& = \sum_{p,q=1}^d v_p^* \otimes w \,B(v_p,v_q) v_q^* \otimes z -b( w , z)\,d/2-\omega( w , z) \Omega_\kappa \p,
\end{aligned}\end{equation}
where we used~\eqref{e:BB} and denote
\begin{equation}\label{e:Omega}
\Omega_\kappa = \sum_{s\in\ca S} \kappa(s) s\p,
\end{equation}
which is a central element in the group algebra $\mathbb{C}[G] $.
In the tensor product of a Weyl and Clifford algebra $\ca W \otimes \ca C$, the space of invariants for the action (as in Definition~\ref{d:actG}) of $\mathrm{O}$ is generated by the elements of the form $(w \odot z)(B) $~\cite[Proposition~5.11]{CW}.
\begin{lemma}\label{l:Bg}
For $w,z\in\ca{V}$, in $A_\kappa$ we have $[(w \odot z)(B) ,\rho(\widetilde G)]=0$.
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{l:rhoG} and that $G\subset \mathrm{O}$ preserves $B$.
\end{proof}
Next, we consider the adjoint action of elements of the form $(w \odot z)(B) $ on the space $\ca U =V^* \otimes \ca V$ in $A_\kappa$.
Recall that $\gamma \in \ca V_{\bar 1}$ satisfies $b(\gamma,\gamma) =2$.
\begin{lemma} \label{l:lemma3}
Let $u\in V^*$, while $\xi_1,\xi_2\in \ca V_{\bar0}$ and $\eta\in \ca V$.
In $A_\kappa$, we have
\begin{align}\label{e:3}
[(\xi_1 \odot \xi_2)(B),u\otimes \eta]
& = b(\xi_2, \eta ) u \otimes \xi_1 +b(\xi_1, \eta ) u\otimes \xi_2 \p,
\\
\label{e:1}
[({\xi_1 \odot \gamma})(B),u\otimes \eta] & = b(\gamma, \eta )u\otimes \xi_1 + b(\xi_1, \eta ) (u\otimes \gamma +2 \tilde\sigma_u)
\p,
\end{align}
where $ \tilde\sigma_u$ is given by~\eqref{e:Ov}.
\end{lemma}
\begin{proof}
Let $\xi,\eta \in \ca V$ be homogeneous elements for the $\mathbb{Z}_2$-grading and $\xi_1\in \ca V_{\bar0}$, then by~\eqref{e:Bwz2}
\[
[(\xi_1 \odot \xi)(B),u\otimes \eta] = \sum_{p,q=1}^d B_{pq} [ (v_p^*\otimes \xi_1)( v_q^*\otimes \xi) , u \otimes \eta] - \omega(\xi_1,\xi) \sum_{s\in\ca S} \kappa(s) [s,u\otimes \eta] \p.
\]
First, note that $\omega(\xi_1,\xi) =0$ if $\xi \in \ca V_{\bar1}$, and $ [s,u\otimes \eta]=0$ for $\eta \in \ca V_{\bar1}$.
For $\eta \in \ca V_{\bar0}$, via~\eqref{e:actO} and~\eqref{e:s} we have
\[
[s,u\otimes \eta] = ((s\cdot u - u )\otimes \eta )\, s= - \alpha_s^{\vee}( u ) \, (\alpha_s\otimes \eta)\, s \p.
\]
By means of~\eqref{e:comre}, we find
\begin{align*}
[ (v^*_p\otimes \xi_1)( v_q^*\otimes \xi ), u \otimes \eta]
= \ & v_p^* \otimes \xi_1 [ v_q^*\otimes \xi , u \otimes \eta]+ (-1)^{|\xi||\eta|} [ v_p^* \otimes \xi_1 , u\otimes \eta ]v_q^*\otimes \xi \\
= \ & v_p^* \otimes \xi_1 (B( v_q^*, u )b(\xi, \eta )+\psi_\kappa^B( v_q^*, u )\omega(\xi, \eta )) \\*
& + (-1)^{|\xi||\eta|} (B( v_p^*, u )b(\xi_1, \eta ) +\psi_\kappa^B( v_p^*, u )\omega(\xi_1, \eta ) ) v_q^*\otimes \xi \p.
\end{align*}
For $\eta = \gamma$, we have $\omega(\cdot,\gamma)=0$ and the cases where $\eta = \gamma$ now follow by~\eqref{e:BBB}.
Next, we consider $\eta \in \ca V_{\bar0}$. Let $\xi = \xi_2 \in \ca V_{\bar0}$. We find using~\eqref{e:BB} and~\eqref{e:psiB}
\begin{align*}
\sum_{p,q=1}^d B_{pq} v_p^*\otimes \xi_1\psi_\kappa^B( v_q^*, u )
&= \sum_{s\in\mathcal S}\alpha_s^{\vee}( u ) \, \kappa(s) \alpha_s \otimes \xi_1 s \p,
\\
\sum_{p,q=1}^d \psi_\kappa^B( v_p^*, u ) B_{pq} v_q^*\otimes \xi_2
&= \sum_{s\in\mathcal S} \alpha_s^{\vee}( u ) \, \kappa(s)\, s \, \alpha_s \otimes \xi_2
\p.
\end{align*}
Now, collecting the appropriate terms, and using that $ \alpha_s$ is a $-1$ eigenvector of $s\in\mathcal S$, we have
\[
\sum_{s\in\mathcal S} \alpha_s^{\vee}( u ) \, \kappa(s) \alpha_s \otimes (\omega(\xi_1, \xi_2 )\eta + \omega(\xi_2, \eta ) \xi_1 - \omega(\xi_1, \eta ) \xi_2 ) s \p,
\]
where $\omega(\xi_1, \xi_2 )\eta + \omega(\xi_2, \eta ) \xi_1- \omega(\xi_1, \eta ) \xi_2 =0$ for all $\xi_1,\xi_2,\eta\in \ca V_{\bar0} =\mathbb{C}^2 $.
Finally, we consider the case $\xi = \gamma \in \ca V_{\bar1}$ and $\eta = \xi_2 \in \ca V_{\bar0}$. Here, the remaining terms are
\begin{align*}
[(\xi_1 \odot \gamma)(B),u\otimes \xi_2] &= \sum_{p,q=1}^d B_{pq} (B( v_p^*, u )b(\xi_1, \xi_2 ) +\psi_\kappa^B( v_p^*, u )\omega(\xi_1, \xi_2 ) ) v_q^*\otimes \gamma \\
& = b(\xi_1, \xi_2 )u\otimes \gamma + \omega(\xi_1, \xi_2 ) \sum_{s\in\mathcal S} \alpha_s^{\vee}( u ) \, \kappa(s)\, s \, \alpha_s \otimes \gamma
\p.
\end{align*}
The desired result now follows by~\eqref{e:Ov}.
\end{proof}
When restricted to $ \ca V_{\bar0}$, in terms of the basis $x^+,x^-$, the relations of Lemma~\ref{l:lemma3} can be written as follows.
For $v^-\in V$ and $v^+\in V^*$, in $H_\kappa \subset A_\kappa$, we have
\begin{align}\label{e:lemma1a}
[(x^-\odot x^-)(B),v^+] &= 2\beta(v^+)\p, & [(x^+\odot x^+)(B),v^-]& = -2\beta(v^-) \p,
\\\label{e:lemma1b}
[(x^+\odot x^-)(B),v^+]& = v^+\p,& [(x^+\odot x^-)(B),v^-] &= -v^- \p.
\end{align}
Rewriting~\eqref{e:1} for $\xi_1=x^-$ and $\xi_2=x^+$, we get for $u\in V^*$
\begin{equation}\label{e:Ov2}
\tilde\sigma_u = \frac12\big([(x^-\odot \gamma)(B),u] - \gamma_u\big) = \frac12\Big(\sum_{p=1}^d [v_{p},u]\gamma_{v^{*}_{p}} - \gamma_u\Big)\p,
\end{equation}
where $v_1^*,\dotsc,v_d^*$ denotes a basis of $V^*$ dual to a basis $v_1,\dotsc,v_d$ of $V$. Using~\eqref{e:comre}, the expression~\eqref{e:Ov2} reduces to~\eqref{e:Ov}.
We can also consider the map $\tilde\sigma \colon V^* \to \rho(\widetilde G) \colon u \mapsto \tilde\sigma_u$, where $\tilde\sigma_u$ is given by~\eqref{e:Ov}.
We then have the following result for
\begin{equation}
(\tilde\sigma \otimes \gamma)(B) =\sum_{p,q=1}^d \tilde\sigma_{v_p^*} \,B(v_p,v_q) \gamma_{v_q^*} = \sum_{p=1}^d \tilde\sigma_{x_p} e_p\p.
\end{equation}
\begin{lemma}\label{l:Oug} In $A_\kappa$, one has
\[
(\tilde\sigma \otimes \gamma)(B) = \Omega_\kappa = (\gamma \otimes \tilde\sigma)(B)\p.
\]
\end{lemma}
\begin{proof}
Using~\eqref{e:Ov} and \eqref{e:BBB}, we have
\[
(\tilde\sigma \otimes \gamma)(B)
= \sum_{p,q=1}^d B(v_p,v_q)\frac12\sum_{s\in\mathcal S} \alpha_s^{\vee}(v^*_p ) \, \kappa(s) \,s \, \gamma_{\alpha_s} \, \gamma_{v_q^*}
= \sum_{s\in\mathcal S} \frac{ \kappa(s) \,s \, \gamma_{\alpha_s} \, \gamma_{\alpha_s}}{B(\alpha_s,\alpha_s)}
\p.\qedhere
\]
\end{proof}
\begin{propo}\label{p:BB}
For
$z_1,z_2,z_3,z_4\in \ca{V}$ homogeneous for the $\mathbb{Z}_2$-grading, denoting $ w_1 = b(z_2,z_3)z_1 + (-1)^{|z_2||z_3|}b(z_1, z_3)z_2$ and $w_2 =b( z_2 , z_4)z_1 + (-1)^{|z_2||z_4|}b( z_1 , z_4)z_2$,
in $A_\kappa$ one has
\begin{equation}\label{e:BB1}
\begin{aligned}~
& [ (z_1\odot z_2)(B),(z_3\odot z_4)(B)]= (w_1\odot z_4)(B)
+(-1)^{(|z_1|+|z_2|)|z_3|} (z_3\odot w_2 )(B) \p.
\end{aligned}
\end{equation}
\end{propo}
\begin{proof}
Follows by direct computation using Lemma~\ref{l:lemma3} and the fact that $\dim_{\mathbb{C}}(\ca{V}_{\bar 1}) = 1$.
For instance, using Lemma~\ref{l:lemma3},~\eqref{e:comre}, \eqref{e:BB} and Lemma~\ref{l:Oug}, we have
\begin{align*}
& \quad [ (\xi_1\odot \gamma)(B),(\xi_2\odot \gamma)(B)]\\ & = \sum_{p,q=1}^d B(v_p,v_q) \Big( [ (\xi_1\odot \gamma)(B), v_p^* \otimes \xi_2 ] v_q^* \otimes \gamma
+ v_p^* \otimes \xi_2 [ (\xi_1\odot \gamma)(B), v_q^* \otimes \gamma ] \Big) \\
& = \sum_{p,q=1}^d B(v_p,v_q) \Big( b(\xi_1,\xi_2)(v_p^* \otimes\gamma + 2 \tilde\sigma_{v^*_p}) v_q^* \otimes \gamma
+ (v_p^* \otimes \xi_2) 2( v_q^* \otimes \xi_1) \Big)\\
& = 2 (\xi_2\odot \xi_1)(B) \p. \qedhere
\end{align*}
\end{proof}
Proposition~\ref{p:BB} shows that the elements $(w \odot z)(B) $ for $w,z\in \ca V $ form a realization of the Lie superalgebra $\fr{osp}(\ca V,b) \cong \fr{osp}(1|2)$ in $A_\kappa$.
The elements $(w \odot z)(B) $ for $w,z\in \ca V_{\bar{0}}=\mathbb{C}^2 $ form a realization of the even subalgebra $\fr{sp}(\ca V_{\bar{0}},\omega) \cong \fr{sl}(2)$ in $H_\kappa$.
In particular, using the basis $x^-,x^+,\gamma$ of $\ca V$, we have that the elements
\begin{equation}
\label{e:osp}
\begin{aligned}
F^+ & \colonequals \frac{1}{\sqrt2} (x^+ \odot \gamma)(B)
= \frac{1}{\sqrt2}\sum_{p,q=1}^d v^{*}_{p}B_{pq}\gamma_{v^{*}_{q}}\\
F^- & \colonequals \frac{1}{\sqrt2}(x^- \odot \gamma)(B)
= \frac{1}{\sqrt2}\sum_{p,q=1}^d \beta(v^{*}_{p})B_{pq}\gamma_{v^{*}_{q}}
= \frac{1}{\sqrt2}\sum_{p=1}^d v_{p}\gamma_{v^{*}_{p}}\\
H &\colonequals (x^+ \odot x^-)(B)
=\sum_{p,q=1}^d v^{*}_{p}B_{pq}\beta(v^{*}_{q}) +\frac{d}{2}+ \Omega_\kappa
=\sum_{p=1}^d v^{*}_{p}v_{p} +\frac{d}{2}+ \Omega_\kappa,\\
E^+& \colonequals \frac12 (x^+ \odot x^+)(B)
= \frac12\sum_{p,q=1}^d v^{*}_{p}B_{pq}v^{*}_{q},\\
E^-& \colonequals -\frac12 (x^- \odot x^-)(B)
= -\frac12\sum_{p,q=1}^d \beta(v^{*}_{p})B_{pq}\beta(v^{*}_{q})
= -\frac12\sum_{p,q=1}^d v_{p}B_{pq}v_{q}\p,
\end{aligned}
\end{equation}
satisfy the commutation relations~\eqref{e:osp12re}.
\section{Supercentralizers}\label{s:supercent}
To describe the supercentralizer of the realization of $\mathfrak{osp}(1|2)$ in $A_\kappa$ given by~\eqref{e:osp}, we first look at the centralizer of its even subalgebra $\mathfrak{sl}(2)$.
\subsection{Centralizer of \texorpdfstring{$\mathfrak{sl}(2)$}{sl(2)}}\label{s:sl2}
In $H_\kappa\subset A_\kappa$, we define $M_{uv} = M(u,v) \colonequals u\beta(v) - v\beta(u)$ for $u,v\in V^*$.
Similar to $(w \odot z)(B) $ in~\eqref{e:Bwz}, every element of $V^*$ corresponds to a copy of $\ca V$ in $V^* \otimes \ca V$, or a copy of $\ca V_{\bar0}$ in $V^* \otimes \ca V_{\bar0}$.
Using now the skew-symmetric form $\omega$, we have for $u,v\in V^*$,
\begin{equation}\label{e:DAMO}
\begin{aligned}
(u\wedge v)(\omega) &= \frac12( u\beta(v) - \beta(u)v - v\beta(u) + \beta(v) u) \\
& = u\beta(v) - v\beta(u) + \frac12( [\beta(v), u]-[\beta(u),v] ) \\
& = u\beta(v) - v\beta(u) \\
& = \beta(v)u - \beta(u)v \p,
\end{aligned}
\end{equation}
where we used Lemma~\ref{l:Buv}.
In the Dunkl operator realization, $M(u,v)$ becomes an angular momentum operator where the partial derivative is replaced by a Dunkl operator, see also~\cite{Feigin,Calvert}.
By means of the relations~(\ref{e:lemma1a}--\ref{e:lemma1b}),
it is easily verified that the elements of the form $M(u,v)$, for $u,v\in V^*$, commute with $(w \odot z)(B) $, for $w,z\in\ca V_{\bar 0}$.
In~\cite[Theorem 6.5]{CDM}, the authors proved that the centralizer of the $\fr{sl}(2)$ realization inside $H_\kappa$ is the associative subalgebra of $H_\kappa$ generated by the group $G$ and the Dunkl angular momentum operators, that is
\begin{equation}\label{e:Centsl2}
\Cent_{H_\kappa}(\fr{sl}(2)) = \langle M_{uv} \mid u,v \in V^*\rangle \rtimes G\p,
\end{equation}
where the action of $g\in G$ on $M_{uv}$ is given by $ M_{g\cdot u\,g\cdot v}$ for $u,v \in V^*$ and $g\in G$.
The elements $M(u,v)$ for $u,v\in V^*$ generate a deformation of (the associative algebra generated by) the orthogonal Lie algebra $\fr{so}(V,B)\cong\fr{so}(d)$. The proof proceeds in the same way as the one for \cite[Theorem~2.5]{DOV} or \cite[Proposition~6.7]{CDM}.
\begin{propo}\label{p:bbH}
For
$u,v,x,y\in V^*$, in $H_\kappa$ one has
\begin{equation}\label{e:bbH}
\begin{aligned}~
[ M(u,v),M(x,y)]=\ & M(v,x)B_\kappa(u, y) -M(u, x)B_\kappa(v, y) \\
& -M( v , y)B_\kappa(u, x) + M( u , y)B_\kappa(v, x) \p,
\end{aligned}
\end{equation}
where $B_\kappa = B +\psi_\kappa^B$, with the latter given in~\eqref{e:psiB}.
\end{propo}
\begin{proof}Use $ M(u,v) = u\beta(v) - v\beta(u)$, $ M(x,y) = x\beta(y) - y\beta(x)$
\begin{align*}
& [ M(u,v),M(x,y)]=
u [\beta(v),x]\beta(y) - v [\beta(u),x]\beta(y)
-u [\beta(v),y]\beta(x) + v [\beta(u),y]\beta(x) \\
& \qquad\qquad\qquad\quad- x [\beta(y),u]\beta(v) + x [\beta(y),v]\beta(u)
+y [\beta(x),u]\beta(v) - y [\beta(x),v]\beta(u)
\end{align*}
which, using~\eqref{e:combuv}, equals
\begin{align*}
= \ &
M(u,y) [\beta(x),v] - M(v,y) [\beta(u),x]
-M(u,x) [\beta(y),v] + M(v,x) [\beta(u),y] \\
&+ u ([[\beta(x),v],\beta(y) ]-[[\beta(y),v],\beta(x)])
- v ([[\beta(x),u],\beta(y)]- [[\beta(y),u],\beta(x)]) \\
&
- x ([[\beta(u),y],\beta(v)] - [[\beta(v),y],\beta(u)])
+y ([[\beta(u),x],\beta(v)] - [[\beta(v),x],\beta(u)])\p.
\end{align*}
The last two lines vanish by Lemma~\ref{l:uvx}, which we prove next. Hence, the desired result follows by~\eqref{e:comre}.
\end{proof}
\begin{lemma}\label{l:uvx}
For $u,v \in V$ and $x^*,y^* \in V^*$, in $H_\kappa$
\[
[[x^*,u],v] = [[x^*,v],u] \p,\qquad [[x^*,v],y^*] = [[y^*,v],x^*] \p.
\]
\end{lemma}
\begin{proof}
Writing out $[[x^*,u],v] - [[x^*,v],u]$ we have
\[
(x^*u- ux^*)v-v(x^*u- ux^*) -(x^*v-vx^*)u + u(x^*v-vx^*) \p,
\]
where all terms cancel using~\eqref{e:RC}. The other identity follows in the same way.
\end{proof}
\subsection{Supercentralizer of \texorpdfstring{$\mathfrak{osp}(1|2)$}{osp(1|2)} }\label{s:osp12}
Recall that $P_\pm$ is given by~\eqref{e:Pdelta}, the anti-symmetrization operator by~\eqref{e:asym}, and the quantization map by~\eqref{e:quant}.
We now define the following elements, which, by Proposition~\ref{p:osp12}, are in the supercentralizer of $\mathfrak{osp}(1|2)$ in $ A_\kappa = H_\kappa \otimes \ca C$. An explicit expression is given in Lemma~\ref{l:Oun}.
\begin{defin}\label{d:OA}
For a positive integer $n$ and $u_1,\dotsc,u_n\in V^*$, we define
\begin{equation}\label{e:O}
O_{u_1\dotsm u_n}
\colonequals - P_{\pm} ( q (u_1 \wedge \dotsm \wedge u_n))/2 = - P_{\pm} ( \ca A (\gamma_{u_1 \dotsm u_n}))/2 \in A_\kappa \p,
\end{equation}
which is skew-symmetric multilinear in its indices.
\end{defin}
Note that the factor $-1/2$ is chosen to coincide with the definition in~\cite{DOV}, and to have a coefficient of 1 for $M_{uv}$ in~\eqref{e:Ouv2}.
\begin{lemma}\label{l:groupaction}
The group $\rho(\widetilde{G})$ interacts with the elements~\eqref{e:O} as follows
\begin{equation}
\rho(\tilde g ) O_{u_1 \dotsb u_n} = (-1)^{|\tilde g|n} O_{p(\tilde g)\cdot u_1 \dotsb p(\tilde g)\cdot u_n} \rho(\tilde g ) \p,
\end{equation}
for $\tilde g \in \widetilde G$ and $u_1,\dotsc,u_n\in V^*$.
\end{lemma}
\begin{proof}
This follows from $P_{\pm} $ being an even element of $U(\fr{osp}(1|2)$ and thus commuting with $\rho(\tilde g)$ by
Lemma~\ref{l:Bg}, that the quantization map~\eqref{e:quant} is a $G$-module isomorphism and Lemma~\ref{l:rhoG}.
\end{proof}
Next, we want to give an explicit expression for the elements~\eqref{e:O}. Recall that $\tilde\sigma_{u}$ for $u\in V^*$ is given by~\eqref{e:Ov}. The case $n=1$ of the next result, shows that $O_u = \tilde\sigma_u$ for $u\in V^*$, see also~\eqref{e:Pgammav}.
\begin{lemma}\label{l:Pdeltagamma}
Let $n \in \{1,2,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\begin{align*}
P_\pm (\gamma_{u_1} \dotsc \gamma_{u_n}) &= (1-n)\gamma_{u_1} \dotsm \gamma_{u_n} -2 \sum_{j=1}^n \gamma_{u_1} \dotsm \tilde\sigma_{u_j}\dotsm \gamma_{u_n}
\\
& \quad -2\sum_{1\leq j<k \leq n}(-1)^{j+k-1}(u_j\beta(u_k) - \beta(u_j)u_k) \gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n} \p.
\end{align*}
\end{lemma}
\begin{proof}
For $v\in V^*$, using~\eqref{e:Pdelta}, \eqref{e:osp}, and~\eqref{e:1}, we have
\begin{equation}\label{e:Pgammav}
P_\pm (\gamma_v) = \gamma_v - [F^-,[F^+,\gamma_v ]]
= \gamma_v - [(x^-\odot \gamma)(B),v ]
= -2 \tilde\sigma_v\p.
\end{equation}
Now, for $n\in \{2,\dotsc,d\}$ and
$u_1,\dotsc,u_n\in V$,
we have
\begin{align*}
P_\pm (\gamma_{u_1} \dotsm \gamma_{u_n}) = \ & \gamma_{u_1} \dotsm \gamma_{u_n} - \left[ F^-,[ F^+,\gamma_{u_1} ]\gamma_{u_2}\dotsm \gamma_{u_n}-\gamma_{u_1}[F^+,\gamma_{u_2}\dotsm \gamma_{u_n} ]\right]\\
= \ & \gamma_{u_1} \dotsm \gamma_{u_n} - [ F^-,[ F^+,\gamma_{u_1} ]]\gamma_{u_2}\dotsm \gamma_{u_n}- [F^+,\gamma_{u_1}][F^-,\gamma_{u_2}\dotsm \gamma_{u_n} ]
\\
& +[F^-,\gamma_{u_1}][F^+,\gamma_{u_2}\dotsm \gamma_{u_n} ]
-\gamma_{u_1}[F^-,[F^+,\gamma_{u_2}\dotsc \gamma_{u_n} ]]\p.
\end{align*}
By definition~\eqref{e:Pdelta} and relation~\eqref{e:1}, this becomes
\begin{align*}
P_\pm (\gamma_{u_1} \dotsm \gamma_{u_n}) = \ & P_\pm (\gamma_{u_1})\gamma_{u_2}\dotsm \gamma_{u_n}
+ \gamma_{u_1}P_\pm (\gamma_{u_2}\dotsm \gamma_{u_n}) - \gamma_{u_1} \dotsm \gamma_{u_n}
\\
& - 2\sum_{j=2}^n(-1)^{j-2}(u_1\beta(u_j) - \beta(u_1)u_j) \gamma_{u_2}\dotsm \widehat{\gamma}_{u_j}\dotsm \gamma_{u_n} \p.
\end{align*}
Applying this formula recursively yields the desired result.
\end{proof}
\begin{lemma}\label{l:Oun}
Let $n \in \{1,2,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\begin{align*}
&\quad O_{u_1\dotsm u_n} = \frac{n-1}{2} \ca A (\gamma_{u_1\dotsm u_n})
+ n\, \ca A (\tilde\sigma_{u_1}\gamma_{u_2\dotsm u_n})
+ \frac{n(n-1)}{2} \ca A ( M_{u_1u_2} \gamma_{u_3\dotsm u_n})
\\
& = -\frac{(n-1)(n-2)}{4} \ca A (\gamma_{u_1\dotsm u_n})
- n(n-2) \ca A (\tilde\sigma_{u_1}\gamma_{u_2\dotsm u_n} )
+ \frac{n(n-1)}{2} \ca A (O_{u_1u_2} \gamma_{u_3\dotsm u_n})
\p.
\end{align*}
\end{lemma}
Note that the antisymmetrization in these expressions expands to
\begin{align*}
O_{u_1\dotsm u_n}
&= \frac{n-1}{2} \ca A (\gamma_{u_1\dotsm u_n})
+ \sum_{j=1}^n (-1)^{j-1} O_{u_j} \ca A (\gamma_{u_1 \dotsm \widehat{u}_j\dotsm u_n})\nonumber \\*
& \quad +\sum_{1\leq j<k \leq n}(-1)^{j+k-1}M(u_j,u_k) \ca A (\gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n}) \nonumber
\\
& =-\frac{(n-1)(n-2)}{4} \ca A (\gamma_{u_1\dotsm u_n})
- (n-2)\sum_{j=1}^n (-1)^{j-1} O_{u_j} \ca A (\gamma_{u_1 \dotsm \widehat{u}_j\dotsm u_n}) \\*
& \quad +\sum_{1\leq j<k \leq n}(-1)^{j+k-1}O_{u_ju_k} \ca A ( \gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n})
\p.
\end{align*}
\begin{proof}
The first expression follows by antisymmetrizing the result of Lemma~\ref{l:Pdeltagamma} (multiplied by $-1/2$), using
Lemma~\ref{l:Ogammas} and noting that, by~\eqref{e:combuv},
\[
\ca A (u_j\beta(u_k) - \beta(u_j)u_k) = (u_j\beta(u_k) - \beta(u_j)u_k - u_k\beta(u_j) + \beta(u_k)u_j)/2 = M (u_j,u_k) \p.
\]
For $u,v\in V^*$, the first expression gives
\begin{equation}\label{e:Ouv2}
O_{uv} = \ca A (\gamma_{uv})/2 + 2 \ca A (O_{u}\gamma_{v} )+ M_{uv} \p.
\end{equation}
For general $u_1,\dotsc,u_n\in V^*$, we can use~\eqref{e:Ouv2}
to replace $M(u_j,u_k)$ in the lemma's first expression for $O_{u_1\dotsm u_n} $ to find the second.
\end{proof}
\begin{lemma}\label{l:O2gammas}
Let $n \in \{2,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\begin{equation}\label{e:O2gammas}
\ca A (O_{u_1u_2} \gamma_{u_3 \dotsm u_n})
= \ca A (\gamma_{u_1} O_{u_2u_3} \gamma_{u_4 \dotsm u_n})
= \dotsb
= \ca A (\gamma_{u_1 \dotsm u_{n-2}}O_{u_{n-1}u_n})\p.
\end{equation}
\end{lemma}
\begin{proof}
This follows by means of~\eqref{e:Ouv2} and then using Lemma~\ref{l:Pdeltagamma} and the fact that
$M(u_j,u_k)$ commutes with all factors of $ \ca A (\gamma_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n}) $.
\end{proof}
\begin{lemma}\label{l:Oun2}
Let $n \in \{3,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\[
(n-3) O_{u_1\dotsm u_n} = - 4 (n-2)\ca A ( O_{u_1} O_{u_2 \dotsm u_n}) + 2(n-1) \ca A (O_{u_1u_2} O_{u_3\dotsm u_n})
\p.
\]
\end{lemma}
Note that expanding the antisymmetrization gives the following expressions
\begin{align*}
\frac{n(n-3)}{4} O_{u_1\dotsm u_n} &= - (n-2)\sum_{j=1}^n (-1)^{j-1} O_{u_j} O_{u_1 \dotsm \widehat{u}_j\dotsm u_n} \\ &\quad +\sum_{1\leq j<k \leq n}(-1)^{j+k-1}O_{u_ju_k} O_{u_1 \dotsm \widehat{u}_j \dotsm \widehat{u}_k\dotsm u_n}\p.
\end{align*}
\begin{proof}
The result follows by applying $-P_\pm/2 $ to the second expression of Lemma~\ref{l:Oun} and using Lemma~\ref{l:Pdelta}.
\end{proof}
The previous result shows that all elements of the form~\eqref{e:O} can be written in terms of those having one, two or three indices.
In particular, for $u,v\in V^*$, by~\eqref{e:Auv}, \eqref{e:Ouv2} becomes
\begin{equation}\label{e:Ouv}
\begin{aligned}[]
O_{uv} &= u\beta(v) - \beta(u)v + (\gamma_u\gamma_v+B(u,v))/2 + \tilde\sigma_u\gamma_v + \gamma_u \tilde\sigma_v\\
&= u\beta(v) - v\beta(u) + (\gamma_u\gamma_v-B(u,v))/2 + \tilde\sigma_u\gamma_v - \tilde\sigma_v\gamma_u
\p.
\end{aligned}
\end{equation}
Note also that
\begin{equation}
\label{e:POuv}
- P_\delta (\gamma_u\gamma_v) /2= O_{uv} - B(u,v)/2\p.
\end{equation}
For $u,v,w\in V^*$, by~\eqref{e:Auvw} we have
\begin{equation}\label{e:Ouvw}
\begin{aligned}[]
O_{uvw} =\ & \ca A (\gamma_{uvw})+ M(v,w)\gamma_u -M(u,w)\gamma_v + M(u,v)\gamma_w \\ &
+ \tilde\sigma_u \ca A (\gamma_{vw}) - \tilde\sigma_v \ca A (\gamma_{uw}) + \tilde\sigma_w \ca A (\gamma_{uv})
\p.
\end{aligned}
\end{equation}
\begin{propo}\label{p:ospcent}
For $\mathfrak{osp}(1|2)$ realized in $A_\kappa = H_\kappa \otimes \ca C$ by the elements~\eqref{e:osp},
its supercentralizer $\Cent_{A_\kappa}(\mathfrak{osp}(1|2))$ is generated by $\rho(\widetilde G)$ and the elements $O_{uv}$ and $O_{uvw}$ for $u,v,w\in V^*$.
\end{propo}
\begin{proof}
By Proposition~\ref{p:osp12}, we obtain the centralizer of $\mathfrak{osp}(1|2)$ by applying the operator $P_\pm$ to $\Cent_{A_\kappa}(\mathfrak{osp}(1|2)_{\bar 0})$.
In $H_\kappa$, the centralizer $\Cent_{H_\kappa}(\mathfrak{osp}(1|2)_{\bar 0})$ of the even subalgebra $\mathfrak{osp}(1|2)_{\bar 0} \cong \fr{sl}(2)$ is generated by $\{M_{uv}\mid u,v \in V^*\}$ and the group $G$, see~\ref{e:Centsl2}.
As $H_\kappa$ does not interact with the Clifford algebra part of $A_\kappa$, we have
\begin{equation}\label{e:Centsl2c}
\Cent_{A_\kappa}(\mathfrak{osp}(1|2)_{\bar 0}) = \Cent_{H_\kappa}(\mathfrak{osp}(1|2)_{\bar 0}) \otimes \ca C\p.
\end{equation}
First, we note that this means the elements $O_{u_1\dotsm u_n}$ for $u_1,\dotsc,u_n\in V^*$ are in $\Cent_{A_\kappa}(\mathfrak{osp}(1|2))$ by Definition~\ref{d:OA}.
Also, $\rho(\mathbb{C}[\widetilde G]) \subset \Cent_{A_\kappa}(\mathfrak{osp}(1|2))$ by Lemma~\ref{l:Bg}.
Now, a general element of $\Cent_{A_\kappa}(\mathfrak{osp}(1|2)_{\bar 0})$ can be written as a sum of terms of the form
$
m\,c\, g
$
where $m$ is a product of elements of $\{M_{uv}\mid u,v \in V^*\}$, $c \in \ca C$ and $g\in G$.
Assume $m$ is a non-zero product of at least one element of $\{M_{uv} \mid u,v \in V^*\}$. Then, there are $u,v\in V^*$ such that $m = M_{uv} m'$ with $m'$ a product of elements of $\{M_{uv} \mid u,v \in V^*\}$, one fewer than $m$.
By~\eqref{e:Ouv} and using Lemma~\ref{l:Pdelta}, we can write
\begin{align*}
P_\pm(m\,c\, g) =\ & P_\pm(M_{uv}m'\,c\, g)\\
=\ & P_\pm((O_{uv} -(\gamma_{uv}-B(u,v))/2 - \tilde\sigma_u\gamma_v + \tilde\sigma_v\gamma_u )m'\,c\, g) \\
=\ & (O_{uv}+B(u,v)/2)P_\pm(m'\,c\, g) -P_\pm(m'\,\gamma_{uv} c\, g)/2\\
&- \tilde\sigma_uP_\pm(m'\,\gamma_v c\, g)+ \tilde\sigma_vP_\pm( m'\,\gamma_uc\, g)\p,
\end{align*}
since $O_{uv},\tilde\sigma_u,\tilde\sigma_v$ are in $ \Cent_{A_\kappa}(\mathfrak{osp}(1|2))$.
Hence, by proving the property for the case where $m$ is a constant, the result follows by induction.
Note that
for the case $m\,c\, g = M(u,v)$ we find in this way
\begin{equation}\label{e:Pomegauv}
P_\pm(M_{uv} )
=
2 O_{uv}+2 \tilde\sigma_u\tilde\sigma_v-2 \tilde\sigma_v\tilde\sigma_u\p.
\end{equation}
An element $g\in G$ can be written as a product of reflections in $\ca S$. For each $s\in \ca S$, since $\gamma_{\alpha_s}^2 = B(\alpha_s,\alpha_s) \neq 0$ in $\ca C$, we can write
\[
s =\gamma_{\alpha_s}^2 s /B(\alpha_s,\alpha_s) = \gamma_{\alpha_s}\rho(\tilde s) /\sqrt{B(\alpha_s,\alpha_s)}\p.
\]
In this way, we can write $g = c_g\, \tilde g$ where $c_g\in \ca C$ and $\tilde g\in \tilde G$.
By Lemma~\ref{l:Pdelta}, we have
\[
P_\pm( c\,g) = P_\pm( c\,c_g \,\tilde g) = P_\pm( c\,c_g )\,\tilde g\p.
\]
In particular, for $s\in\ca S$ this becomes
\begin{equation}\label{e:Ps}
P_\pm(s) = -2 \,\tilde\sigma_{\alpha_s} \rho(\tilde s) /\sqrt{B(\alpha_s,\alpha_s)} \p.
\end{equation}
The elements of $P_\pm(\Cent_{A_\kappa}(\mathfrak{osp}(1|2)_{\bar 0}))$ are thus given by products of elements in $\rho(\widetilde G)$ and elements of the form $O_{u_1\dotsm u_n}$ for $u_1,\dotsc,u_n\in V^*$
Lemma~\ref{l:Oun2} then shows that the latter can be generated by means of those having one, two or three indices, which completes the proof.
\end{proof}
Given the interaction with
the group $\rho(\widetilde{G})$ in Lemma~\ref{l:groupaction}, a minimal set of generators for $\Cent_{A_\kappa}(\mathfrak{osp}(1|2))$ requires only a subset of the elements $O_{uv}$ and $O_{uvw}$ for $u,v,w\in V^*$.
\subsection{Supercentralizer algebra relations}\label{s:rels}
We can apply $P_\pm $ to both sides of an equality to determine relations for elements of the centralizer $O_{\kappa}$ in this realization.
We recall that throughout the $\mathbb{Z}_2$-graded commutator is used $[a,b] = ab - (-1)^{|a||b|} ba$, where $a,b$ are homogeneous elements for the $\mathbb{Z}_2$-grading, and we also denote
\begin{equation}\label{e:ascom}
\ascom{a}{b} = ab + (-1)^{|a||b|} ba\p.
\end{equation}
The following result includes a generalization of \cite[Theorem 3.13]{DOV}.
\begin{propo}\label{p:OujOun}
Let $n \in \{2,\dotsc,d\}$, and
$u_1,\dotsc,u_n\in V^*$, then
\begin{equation*}\label{e:Ogammas2}
\ca A( O_{u_1}O_{u_2\dotsm u_{n}} ) = \ca A( O_{u_1\dotsm u_{n-1}}O_{u_{n}} )
\quad\text{ or }\quad
\ca A( \scom{O_{u_1}}{O_{u_2\dotsm u_{n}} } )
= 0\p,
\end{equation*}
and
\begin{equation*}\label{e:O2gammas2}
\ca A(O_{u_1u_2} O_{u_3 \dotsm u_n})
= \ca A(O_{u_1 \dotsm u_{n-2}}O_{u_{n-1}u_n})
\quad\text{ or }\quad
\ca A( \scom{O_{u_1u_2}}{ O_{u_3 \dotsm u_n}})
=0\p.
\end{equation*}
\end{propo}
\begin{proof}
By Lemma~\ref{l:Ogammas}, we have $ \ca A( O_{u_1} \gamma_{u_2\dotsm u_n})= \ca A(\gamma_{u_1\dotsm u_{n-1}} O_{u_n})$.
The first result follows by applying $-P_\pm/2 $ and using Lemma~\ref{l:Pdelta}. To see that this implies $\ca A( [O_{u_1},O_{u_2\dotsm u_{n}} ] ) = 0$, we expand the antisymmetrization $\ca A( O_{u_1}O_{u_2\dotsm u_{n}} ) = \ca A( O_{u_1\dotsm u_{n-1}}O_{u_{n}} )$ as
\begin{equation*}
\frac{1}{n} \sum_{j=1}^n (-1)^{j-1} O_{u_j} O_{u_1\dotsm \widehat{u}_j \dotsm u_n}
= \frac{1}{n}\sum_{j=1}^n (-1)^{n-j} O_{u_1\dotsm \widehat{u}_j \dotsm u_n} O_{u_j}\p.
\end{equation*}
Similarly, by Lemma~\ref{l:O2gammas} we have
$ \ca A(O_{u_1u_2} \gamma_{u_3 \dotsm u_n})
= \ca A(\gamma_{u_1 \dotsm u_{n-2}}O_{u_{n-1}u_n})$
and again the result follows by applying $-P_\pm/2 $.
\end{proof}
Specific cases of the previous proposition are, for $u,v,w,z\in V^*$,
\begin{align}
& \scom[-]{ O_{uv} }{ O_{w} } - \scom[-]{ O_{uw} }{ O_v } + \scom[-]{ O_{vw} }{ O_u }= 0\\
& \scom[+]{O_{uvw} }{ O_z } - \scom[+]{O_{uvz} }{ O_w } + \scom[+]{O_{uwz} }{ O_v } - \scom[+]{O_{vwz} }{ O_u } = 0\p, \label{e:OuvwOx}
\end{align}
Note also that by Proposition~\ref{p:OujOun}, one can rewrite antisymmetrized products using~\eqref{e:ascom}, for instance $\ca A (O_{u_1u_2} O_{u_3 u_4}) = \ca A (\ascom[+]{O_{u_1u_2}}{ O_{u_3 u_4}})/2$.
\begin{lemma}\label{l:O45}
For
$u_1,\dotsc,u_5\in V^*$,
\begin{align*}
O_{u_1\dotsm u_4}
& = 6 \, \ca A (O_{u_1u_2} O_{u_3 u_4})- 8\, \ca A ( O_{u_1u_2 u_3 } O_{u_4})
\end{align*}
and
\begin{align*}
O_{u_1\dotsm u_5} &= 4 \ca A (O_{u_1u_2u_3} O_{u_4 u_5})
+48 \ca A ( O_{u_1u_2 u_3 } O_{u_4} O_{u_5})
- 36\ca A ( O_{u_1u_2} O_{u_3 u_4}O_{u_5})
\p.
\end{align*}
\end{lemma}
\begin{proof}
By Lemma~\ref{l:Oun2} and Proposition~\ref{p:OujOun}, we have
\[
(n-3) O_{u_1\dotsm u_n} =2(n-1) \ca A (O_{u_1u_2} O_{u_3\dotsm u_n}) - 4 (n-2)\ca A ( O_{u_1 \dotsm u_{n-1}} O_{u_n})
\p.
\]
For $n=4$, this gives the desired result. For $n=5$, we have
\[
2 O_{u_1\dotsm u_5} = 8 \ca A (O_{u_1u_2} O_{u_3u_4 u_5})- 12\ca A ( O_{u_1 \dotsm u_{4}} O_{u_5})
\p,
\]
where we can use the $n=4$ result to obtain the desired result.
\end{proof}
\begin{propo}
For $a,b,u,v\in V^*$, we have
\begin{align*}
\scom[-]{O_{ab}}{O_{uv}} =\ & B(b,u)( O_{av} + \GG{a}{v} )
- B(a,u)(O_{bv}+\GG{b}{v})\\ &
- B(b,v)( O_{au} + \GG{a}{u} )
+ B(a,v)(O_{bu}+\GG{b}{u})\\ &
+(\scom[+]{O_a}{O_{buv}}
- \scom[+]{O_b}{O_{auv}} + \scom[+]{O_{abu}}{O_v} -\scom[+]{O_{abv}}{O_u} )/2
\p,
\end{align*}
or, denoting
$\hat u = B(b,u)a - B(a,u)b$ and $\hat v = B(b,v)a - B(a,v)b$,
\begin{align*}
\scom[-]{O_{ab}}{O_{uv}} &= O_{\hat u v} + \ascom[-]{O_{\hat u}}{O_v} + \scom[+]{O_{abu}}{O_v} \\ & \quad+ O_{u\hat v} + \ascom[-]{O_u}{O_{\hat v}}
- \scom[+]{O_{abv}}{O_u}\p.
\end{align*}
\end{propo}
\begin{proof}
Using the definition~\eqref{e:Ouv} together with~\eqref{e:gammaO} and~\eqref{e:adso}
\begin{align*}
[O_{ab},\ca A(\gamma_{uv})]
= \ & [ \gamma_{ab}/2 , \gamma_{uv}]
+[ O_a\gamma_b - O_b\gamma_a,\gamma_{uv} ] \\
=\ & [\gamma_a\gamma_b, \gamma_u]\gamma_v/2
+
\gamma_u[\gamma_a \gamma_b, \gamma_v]/2 \\ &
+ (O_a\gamma_b
- O_b\gamma_a)\gamma_u\gamma_v
- \gamma_u \gamma_v ( \gamma_a O_b
- \gamma_b O_a) \\
=\ & (B(b,u) \gamma_a
- B(a,u)\gamma_b)\gamma_v
+
\gamma_u(B(b,v)\gamma_a
-
B(a,v) \gamma_b) \\ &
+ O_a\gamma_b\gamma_u \gamma_v
- O_b\gamma_a\gamma_u\gamma_v
- \gamma_u \gamma_v \gamma_a O_b
+ \gamma_u \gamma_v \gamma_b O_a \p.
\end{align*}
By~\eqref{e:O} and \eqref{e:POuv},
applying $-P_\pm/2 $ to the previous computation gives
\begin{align*}
[O_{ab},O_{uv}]
=\ & B(b,u) (O_{av} - B(a,v)/2 ) - B(a,u)(O_{bv} -B(b,v)/2)\\
&
+
B(b,v)(O_{ua} -B(u,a)/2)
-
B(a,v) (O_{ub} -B(u,b)/2)\\ &
+ O_a(O_{buv} +B(u,v)O_b-B(b,v)O_u+B(b,u)O_v)\\ &
- O_b(O_{auv} +B(u,v)O_a-B(a,v)O_u+B(a,u)O_v)\\ &
-(O_{uva} +B(v,a)O_u-B(u,a)O_v+B(u,v)O_a) O_b \\ &
+ (O_{uvb} +B(v,b)O_u-B(u,b)O_v+B(u,v)O_b)O_a \p.
\end{align*}
Collecting the appropriate terms and using~\eqref{e:OuvwOx} gives the desired results.
\end{proof}
The next two propositions contain expressions with four or five indices, which are given in Lemma~\ref{l:O45}.
\begin{propo}\label{p:O2O34}
Let $a,b,c,u,v,w\in V^*$, and denote $\hat x = B(b,x) a - B(a,x)b$ for $x \in \{c,u,v,w\}$. We have
\begin{align*}
\scom[-]{ O_{ab} }{ O_{uvw} } &= O_{\hat u vw} + O_{u\hat vw} + O_{ uv\hat w} \\*
& \quad +
\ascom[+]{ O_{\hat u}}{ O_{vw}} -
\ascom[+]{ O_{\hat v}}{ O_{uw}} +
\ascom[+]{ O_{\hat w}}{ O_{uv}}\\* & \quad + \scom[-]{O_a}{O_{buvw} } - \scom[-]{O_b}{ O_{auvw} }\p,
\end{align*}
and
\begin{align*}
\scom[-]{O_{ab}}{O_{cuvw}} &=
O_{\hat cuvw} + O_{c\hat uvw} + O_{cu\hat vw} + O_{cuv\hat w}\\*
& \quad + \ascom[-]{O_{\hat c}}{O_{uvw}} + \ascom[-]{O_{\hat u}}{O_{cvw}} + \ascom[-]{O_{\hat v}}{O_{cuw}} + \ascom[-]{O_{\hat w}}{O_{cuv}}
\\* & \quad
\scom[+]{O_a}{O_{bcuvw} } -\scom[+]{O_b}{O_{acuvw}}
\p.
\end{align*}
\end{propo}
\begin{proof}
For the first relation, using~\eqref{e:Ouv} and~\eqref{e:gammaO}
\[
[O_{ab},\ca A(\gamma_{xyz}) ] =
[\gamma_{uv}/2,\ca A(\gamma_{xyz}) ] + [O_u\gamma_v - O_v\gamma_u,\ca A(\gamma_{xyz}) ] \p.
\]
On the one hand, by~\eqref{e:adso} we have
\begin{align*}
\scom{\gamma_{uv}/2}{\ca A(\gamma_{xyz}) } & = \gamma_{\hat x yz} + \gamma_{x\hat yz} + \gamma_{ x y\hat z} \\
& \quad - B(y,z)\gamma_{\hat x}+ B(x,z)\gamma_{\hat y}- B(x,y)\gamma_{\hat z} \\
& = \ca A(\gamma_{\hat x yz} )+ \ca A(\gamma_{x\hat yz}) + \ca A(\gamma_{ x y\hat z} )
\end{align*}
where we used that
\[
B(\hat x , z ) = B(v,x) B(u , z ) - B(u,x)B(v, z ) = - B(x,\hat z)\p.
\]
On the other hand,
\begin{align*}
& \quad \ \scom{O_u\gamma_v - O_v\gamma_u}{\ca A(\gamma_{xyz}) } \\
& = (O_u\gamma_v - O_v\gamma_u)\ca A(\gamma_{xyz}) - \ca A(\gamma_{xyz}) (\gamma_uO_v - \gamma_vO_u)\\
& =\scom{O_u}{\ca A(\gamma_{vxyz}) } - \scom{O_v}{\ca A(\gamma_{uxyz}) } \\
& \quad +
\ascom{ O_u }{ B(v,x)\ca A(\gamma_{yz}) - B(v,y)\ca A(\gamma_{xz}) + B(v,z)\ca A(\gamma_{xy}) }\\
& \quad -
\ascom{ O_v}{ B(u,x)\ca A(\gamma_{yz}) - B(u,y)\ca A(\gamma_{xz})+ B(u,z)\ca A(\gamma_{xy}) }\\
& =
\scom{O_u}{\ca A(\gamma_{vxyz}) } - \scom{O_v}{\ca A(\gamma_{uxyz}) } \\
& \quad +
\ascom{ B(v,x)O_u - B(u,x)O_v}{ \ca A(\gamma_{yz})} \\
& \quad -
\ascom{ B(v,y)O_u - B(u,y)O_v}{ \ca A(\gamma_{xz})} \\
& \quad +
\ascom{ B(v,z)O_u - B(u,z)O_v}{ \ca A(\gamma_{xy})}\p.
\end{align*}
The result follows after applying $-P_\pm/2 $.
Similarly, for the second relation, we have
\[
\scom[-]{O_{ab}}{\ca A ( \g{cuvw})} =
[\cA{ab}/2 + O_a \g{b} - O_b\g{a} , \cA{cuvw} ]
\]
where
\[
[\cA{ab}/2 , \cA{cuvw} ]
= \cA{\hat cuvw} + \cA{c\hat uvw} +\cA{cu\hat vw} +\cA{cuv\hat w}
\p,
\]
and
\begin{align*}
&\quad\ (O_b \g{c} - O_c\g{b})\cA{auvw} - \cA{auvw} (\g{b}O_c-\g{c} O_b )\\
& = \scom[+]{O_b}{ \ca A ( \g{cauvw})}
- B(c,w) \ascom[-]{O_b}{\ca A ( \g{auv})} \\ & \quad
-\scom[+]{O_c}{\ca A ( \g{bauvw} )} -B(b,v) \ascom[-]{O_c}{\ca A ( \g{auw})} \p,
\end{align*}
while
\[
\scom[-]{O_{ab}}{\ca A ( \g{cuvw})} =
[\cA{ab}/2 + O_a \g{b} - O_b\g{a} , \cA{cuvw} ]
\]
where
\[
[\cA{ab}/2 , \cA{cuvw} ]
=B(a,u)\cA{bcvw} + B(b,c) \cA{auvw} + B(b,v)\cA{acuw}
\p,
\]
and
\begin{align*}
&\quad\ (O_a \g{b} - O_b\g{a})\cA{cuvw}
- \cA{cuvw} (\gO{a}{b})\\
& = \scom[+]{O_a}{\ca A ( \g{bcuvw} )} -B(b,c) \ascom[-]{O_a}{\cA{uvw}} -B(b,v) \ascom[-]{O_a}{\ca A ( \g{cuw})}
\\ & \quad
-\scom[+]{O_b}{ \ca A ( \g{acuvw})}
- B(a,u) \ascom[-]{O_b}{\ca A ( \g{cvw})}
\p.
\end{align*}
The results follow after applying $-P_\pm/2 $.
\end{proof}
Moreover, for brevity assuming also $B(a,c) = 0 = B(b,c)$, we have
\begin{propo}\label{p:O3O3}
Let $a,b,c,u,v,w\in V^*$ be such that the only $B$-pairings between $\{a,b,c\}$ and $\{u,v,w\}$ that can be non-zero are $B(a,u),B(b,v),B(c,w)$. We have
\begin{align*}
\scom[+]{O_{abc}}{O_{uvw}} &=
B(a,u) (O_{bcvw} + \ascom[+]{O_{bc}}{ O_{vw} } )
+ \scom[+]{O_a}{O_{bcuvw} }
\\ & \quad
+ B(b,v) (O_{acuw} + \ascom[+]{O_{ac}}{ O_{uw} } )
- \scom[+]{ O_b }{O_{acuvw} }
\\ & \quad
+ B(c,w) (O_{abuv} +\ascom[+]{O_{ab} }{ O_{uv} } )
+ \scom[+]{ O_c }{O_{abuvw} }
\\ & \quad
+ B(b,v)B(c,w) \scom[+]{O_a}{O_u}
+ B(a,u)B(c,w)\scom[+]{O_b}{O_v}
\\
& \quad
+B(a,u)B(b,v)\scom[+]{O_c}{ O_w}
- B(a,u) B(b,v)B(c,w)/2
\end{align*}
where expressions for elements of the form $O_{bcvw}$ and $O_{bcuvw}$ are given in Lemma~\ref{l:O45}.
\end{propo}
\begin{proof}
We have
\begin{align*}
\scom[+]{O_{abc} }{ \ca A ( \gamma_{uvw} ) } & =
\scom[+]{ -\ca A ( \gamma_{abc} )/2 - 3\ca A (O_{a}\gamma_{bc}) + 3\ca A (O_{ab}\gamma_c ) }{ \ca A ( \gamma_{uvw} ) } \p.
\end{align*}
Now, using~\eqref{e:Auvw} and~\eqref{e:Auvwx}
\begin{align*}
\scom[+]{ -\ca A ( \gamma_{abc} )/2 }{ \ca A ( \gamma_{uvw} ) }
& =
-B(a,u) \ca A ( \gamma_{bcvw} ) - B(b,v) \ca A ( \gamma_{acuw} ) - B(c,w) \ca A ( \gamma_{abuv} ) \\ & \quad + B(a,u) B(b,v)B(c,w)
\p,
\end{align*}
while $\scom[+]{ - 3\ca A (O_{a}\gamma_{bc}) }{ \ca A ( \gamma_{uvw} ) } $ becomes
\begin{align*}
&
-\scom[+]{O_a}{ \ca A ( \gamma_{bcuvw} )}
- \ascom[-]{O_a}{ B(b,v) \ca A ( \gamma_{cuw} ) + B(c,w) \ca A ( \gamma_{buv} )}
+ \scom[+]{O_a}{ B(b,v)B(c,w)\gamma_u}
\\ &
+ \scom[+]{ O_b }{\ca A ( \gamma_{acuvw} )}
- \ascom[-]{O_b}{ B(a,u) \ca A ( \gamma_{cvw} ) + B(c,w) \ca A ( \gamma_{auv} ) }
+ \scom[+]{O_b}{ B(a,u)B(c,w) \gamma_v}
\\ &
- \scom[+]{ O_c }{\ca A ( \gamma_{abuvw} )}
+ \ascom[-]{O_c}{ B(a,u) \ca A ( \gamma_{bvw} ) + B(b,v) \ca A ( \gamma_{auw} )}
+ \scom[+]{O_c}{ B(a,u)B(b,v) \gamma_w}
\end{align*}
and
\begin{align*}
\scom[+]{ 3\ca A (O_{ab}\gamma_{c}) }{ \ca A ( \gamma_{uvw} ) }
& =
\scom[-]{O_{ab}}{ \ca A ( \gamma_{cuvw} ) }+ B(c,w)\ascom[+]{O_{ab} }{ \ca A ( \gamma_{uv} ) }
\\ & \quad
- \scom[-]{O_{ac} }{\ca A ( \gamma_{buvw} ) } + B(b,v) \ascom[+]{O_{ac}}{ \ca A ( \gamma_{uw} )}
\\ & \quad
+ \scom[-]{ O_{bc}}{ \ca A ( \gamma_{auvw} )} +B(a,u) \ascom[+]{O_{bc}}{ \ca A ( \gamma_{vw} ) }
\p.
\end{align*}
The result follows after applying $-P_\pm/2 $ and using Proposition~\ref{p:O2O34}.
\end{proof}
\subsubsection{Special bases}
We will use the following notational conventions when a vector of the chosen bases of Section~\ref{s:bases} appears as a subscript index in an element of the form~\eqref{e:O}:
An element of the set $\{1,\dotsc,d\}$ will be used to refer to the corresponding element of $\{x_p\}_{p=1}^d$. For instance, $O_{12} = O_{x_1x_2}$.
For $\ell=\lfloor d/2\rfloor$, an element of the set $\{1,\dotsc,\ell\}$ with a $+$ or $-$ above it, will be used refer to the corresponding element of $\{z_p^\pm\}_{p=1}^\ell$, which is (part of) the $B$-isotropic basis of $V^*$.
If $d$ is odd, an index $0$ will be used to refer to $z_0$.
For instance, $O_{\ib{+}{1}\ib{-}{1}0} = O_{z_1^+z_1^-z_0^{~}} $
The relations determined above, reduce to the following for elements of the basis $\{x_p\}_{p=1}^d$. The proofs are also easier in this case, so they are included for completeness.
\begin{corol}
For $i,j,k,l,m,n$ distinct elements of the set $\{1,\dotsc,d\}$,
\begin{align}
\scom[-]{O_{ij}}{O_{ki}}
&= O_{jk}+ \GG{i}{j}
+[O_{ijk},O_i]
\\
[O_{ij},O_{kl}]
&= \frac12( [O_i,O_{jkl}]
-[O_j,O_{ikl}]-[O_{ijl},O_k]
+[O_{ijk},O_l]
)\\
&= [O_i,O_{jkl}]
-[O_j,O_{ikl}]
\p,
\end{align}
where the last equality follows by~\eqref{e:OuvwOx}.
\begin{align}
[O_{jk},O_{lmn} ] & = [O_j, O_{klmn}] - [O_k,O_{jlmn}]\\
[O_{jk},O_{jlm} ] & = -O_{klm} - \ascom{O_k}{O_{lm}} - [O_j,O_{jklm}]\\
[O_{jk},O_{jkl} ] & = -\ascom{O_j}{O_{jl}} - \ascom{O_k}{O_{kl}} \p.
\end{align}
\begin{align}\label{e:24}
\scom[+]{O_{ijk}}{O_{ijk}} & = 2\left((O_{i})^2+ (O_{j})^2 +
(O_{k})^2 + (O_{ij})^2 + (O_{ik})^2 + (O_{jk})^2\right) -\frac12\\
\scom[+]{O_{ijk}}{O_{ijl}} & = [O_k,O_l]
+ \ascom{O_{ik}}{O_{il}}+ \ascom{O_{jk}}{O_{jl}}\\
\scom[+]{O_{ijk}}{O_{imn}} & = O_{jkmn} + \ascom{O_{jk}}{O_{mn}}
+\scom{O_i}{O_{ijkmn}}\\
\scom[+]{O_{ijk}}{O_{lmn}}
& =
\scom{O_i}{ O_{jklmn}}
- \scom{O_j }{ O_{iklmn}}
+\scom{O_k }{ O_{ijlmn}} \label{e:27}
\p.
\end{align}
\begin{align}
\scom{O_{jk}}{ O_{jklm}}
& = - \ascom{O_j}{ O_{jlm}} - \ascom{O_k}{O_{klm}} \\
\scom{O_{jk}}{ O_{jlmn}}
& = - O_{klmn}- \ascom{O_k}{O_{lmn}} - \scom{O_j}{ O_{jklmn}} \\
\scom{O_{ij}}{ O_{klmn}} & = \scom{O_i}{ O_{jklmn}} - \scom{O_j}{ O_{iklmn}} \p.
\end{align}
\end{corol}
\begin{proof}
For a 3-element set $\{a,b,c\} \subset \{1,\dotsc,d\}$, Proposition~\ref{p:OA2} below gives
\begin{equation}\label{e:Ouvw2}
(O_{abc})^2 = -\frac14 + (O_{a})^2+ (O_{b})^2 + (O_{c})^2 + (O_{ab})^2 + (O_{ac})^2 + (O_{bc})^2\p.
\end{equation}
For the second relation, we compute
\begin{align*}
[O_{jkl},e_{jkm}]
& = [ - e_{jkl}/2 - 3\ca A (O_{j}e_{kl}) + 3\ca A (O_{jk}e_{l} ),e_{jkm}] \\
& =
-(O_j e_{kl} - O_k e_{jl}+O_l e_{jk} )e_{jkm} - e_{jkm}(e_{jk} O_l -e_{jl} O_k + e_{kl} O_j) \\* &\quad
+(O_{jk} e_{l} -O_{jl} e_{k} + O_{kl} e_j)e_{jkm} + e_{jkm}(e_{j} O_{kl} - e_{k} O_{jl}+e_{l} O_{jk} )
\\
& = O_je_{jlm} -e_{jlm}O_j + O_ke_{klm} -e_{klm}O_k +O_le_m +e_mO_l \\*
& \quad + O_{jk} e_{jklm} - e_{jklm}O_{jk} + O_{jl} e_{jm} + e_{jm}O_{jl}+ O_{kl} e_{km} + e_{km}O_{kl}\p.
\end{align*}
The result follows after applying $-P_\pm/2 $ to both sides and using Proposition~\ref{p:O2O34}.
Similarly, the final two relations follow from
\begin{align*}
& \quad [O_{jkl},e_{jmn}]
= [ - e_{jkl}/2 - 3\ca A (O_{j}e_{kl}) + 3\ca A (O_{jk}e_{l} ) ,e_{jmn}] \\
& = - e_{klmn}
-(O_j e_{kl} - O_k e_{jl}+O_l e_{jk} )e_{jmn}
- e_{jmn}(e_{jk} O_l -e_{jl} O_k + e_{kl} O_j) \\* &\quad
+(O_{jk} e_{l} -O_{jl} e_{k} + O_{kl} e_j)e_{jmn}
+ e_{jmn}(e_{j} O_{kl} - e_{k} O_{jl}+e_{l} O_{jk} )
\\
& = - e_{klmn}
-O_je_{jklmn} -e_{jklmn}O_j
- O_ke_{lmn} + e_{lmn}O_k
+O_le_{kmn} -e_{kmn}O_l \\*
& \quad
- O_{jk} e_{jlmn} + e_{jlmn}O_{jk}
+ O_{jl} e_{jkmn} - e_{jkmn}O_{jl}
+ O_{kl} e_{mn} + e_{mn}O_{kl}
\p,
\end{align*}
and
\begin{align*}
[O_{ijk},e_{lmn}]
& = [ - e_{ijk}/2 - 3\ca A (O_{i}e_{jk}) + 3\ca A (O_{ij}e_{k} ),e_{lmn}] \\
& =
-(O_i e_{jk} - O_j e_{ik}+O_k e_{ij} )e_{lmn} - e_{lmn}(e_{ij} O_k -e_{ik} O_j + e_{jk} O_i) \\* &\quad
+(O_{ij} e_k -O_{ik} e_j + O_{jk} e_i)e_{lmn}
+ e_{lmn}(e_i O_{jk} - e_j O_{ik}+e_k O_{ij} )\p,
\end{align*}
after applying $-P_\pm/2 $ and using Proposition~\ref{p:O2O34}.
\end{proof}
A slight modification of the previous proof also yields the following relations
\begin{align*}
\ascom[-]{O_{jkl}}{O_{jkm}} & = -O_{lm} - \ascom{O_l}{O_m}
+ \ascom{O_{jk} }{ O_{jklm}}\\
\ascom[-]{O_{jkl}}{O_{jmn}} & = - \ascom{O_j}{O_{jklmn}}
+ \ascom{O_{jk} }{ O_{jlmn}} + \ascom{O_{jl}}{O_{jkmn}}
\p.
\end{align*}
Together with the relations~(\ref{e:24}--\ref{e:27}) this shows that in $A_\kappa$ a product of elements of the form $O_{uvw}$ for $u,v,w\in V^*$ can be reduced to terms containing at most a single 3-index element.
\begin{propo}\label{p:OA2}
For $n \in \{1,\dotsc,d\}$ and $A = \{ a_{1},a_{2},\dotsc,a_{n}\}\subset \{1,\dotsc,d\}$
\begin{align*}
(O_A)^2
= (-1)^{n(n-1)/2}\bigg( \frac{(n-1)(n-2)}{8} - (n-2)\sum_{a\in A} (O_{a})^2 -\sum_{\{a,b\}\subset A } ({O}_{ab})^2 \bigg)\p.
\end{align*}
\end{propo}
\begin{proof}
Note that $(e_A)^2=(-1)^{n(n-1)/2}$ if $|A|=n$.
By the last expression of Lemma~\ref{l:Oun}
\begin{align*}
O_A e_A & = -\frac{(n-1)(n-2)}{4} (e_A)^2
- (n-2)\sum_{a\in A} O_{a} e_{a}(e_A)^2
-\sum_{\{a,b\}\subset A } {O}_{ab}
e_ae_b(e_A)^2 \p.
\p.
\end{align*}
Applying $-P_\pm/2 $ to both sides and using Lemma~\ref{l:Pdelta} yields the desired result.
\end{proof}
\begin{propo}
For $u,v,w\in V^*$, we have
\begin{equation}\label{e:OD}
\ascom[(-)^d]{ O_{1\dotsm d} }{O_u} = 0\p,\qquad
\scom[-]{ O_{1\dotsm d} }{O_{uv}} = 0\p,\qquad
\ascom[(-)^d]{ O_{1\dotsm d} }{O_{uvw}} = 0\p.
\end{equation}
Moreover, the expression
\begin{equation}\label{e:central}
\Omega = (d-2)\sum_{j=1}^d (O_{j})^2 +\sum_{1\leq j<k \leq d } ({O}_{jk})^2
\end{equation}
is central in $\Cent(\fr{osp}(1|2))$.
\end{propo}
Note that for $d$ odd, the relations~\eqref{e:OD} imply that $O_{1\dotsm d}$ commutes (but not supercommutes as it has odd $\mathbb{Z}_2$-degree) with everything in $\Cent(\fr{osp}(1|2))$.
\begin{proof}
The element $e_{1\dotsm d}$ antisupercommutes with the generators of $\ca C$.
In this way, for $u,v,w\in V^*$, using the expressions~\eqref{e:Ov}, \eqref{e:Ouv}, \eqref{e:Ouvw}, we find
\[
\ascom[(-)^d]{ e_{1\dotsm d} }{O_u} = 0\p,\qquad
\scom[-]{ e_{1\dotsm d} }{O_{uv}} = 0\p,\qquad
\ascom[(-)^d]{ e_{1\dotsm d} }{O_{uvw}} = 0\p,
\]
so the relations~\eqref{e:OD} follow after applying $-P_\pm/2 $.
By Proposition~\ref{p:OA2}, we have
\begin{align*}
(O_{1\dotsm d})^2 = (-1)^{d(d-1)/2}\bigg( \frac{(d-1)(d-2)}{8} - (d-2)\sum_{j=1}^d (O_{j})^2 -\sum_{1\leq j<k \leq d } ({O}_{jk})^2 \bigg)\p.
\end{align*}
Using again the properties of $e_{1\dotsm d}$, the element $O_{1\dotsm d}$ equals
\[
O_{1\dotsm d} = -\frac12P_\delta(e_{1\dotsm d}) = -\frac12(e_{1\dotsm d} - [F^-,[F^+,e_{1\dotsm d}]]) = (F^-F^+ - F^+F^- - 1/2)e_{1\dotsm d}
\]
so, up to the sign $(-1)^{d(d-1)/2}$, the expression $O_{1\dotsm d} e_{1\dotsm d}$ equals the $\fr{osp}(1|2)$
Scasimir element \eqref{e:SCasiosp}.
Since $-P_\pm( O_{1\dotsm d} e_{1\dotsm d})/2 = O_{1\dotsm d}^2 $, we find by~\eqref{e:SCasiosprel} that, up to constants, $\Omega$ equals the $\fr{osp}(1|2)$
Casimir element \eqref{e:Casiosp} in $ U(\fr{osp}(1|2))$ and thus is central in $\Cent(\fr{osp}(1|2))$.
\end{proof}
Using~\eqref{e:Ouvw2}, we have
\[
\sum_{i<j<k}^d (O_{ijk})^2 = -\frac{d(d-1)(d-2)}{24} + \frac{(d-1)(d-2)}{2}\sum_{j=1}^d (O_{j})^2 + (d-2)\sum_{ j<k }^d ({O}_{jk})^2 \p,
\]
so the following combination is also central
\[
\frac{(d-1)(d-2)}{2}\sum_{j=1}^d (O_{j})^2 +\sum_{1\leq j<k \leq d } ({O}_{jk})^2 + \sum_{i<j<k}^d (O_{ijk})^2 \p.
\]
\begin{lemma}\label{l:OO} In $A_\kappa$, one has
\[
(\tilde\sigma \otimes \gamma)(B) = \Omega_\kappa = (\gamma \otimes \tilde\sigma)(B)\p.
\]
\end{lemma}
\begin{proof}
Using~\eqref{e:Ov} and \eqref{e:BBB}, we have
\begin{align*}
(\tilde\sigma \otimes \gamma)(B) & =\sum_{p,q=1}^d \tilde\sigma_{v_p^*} \,B(v_p,v_q) \gamma_{v_q^*} \\
& = \sum_{p,q=1}^d B(v_p,v_q)\frac12\sum_{s\in\mathcal S} \alpha_s^{\vee}(v^*_p ) \, \kappa(s) \,s \, \gamma_{\alpha_s} \, \gamma_{v_q^*}\\
& = \sum_{s\in\mathcal S} \, \kappa(s) \,s \, \gamma_{\alpha_s} \, \gamma_{\alpha_s}/B(\alpha_s,\alpha_s)
\p.\qedhere
\end{align*}
\end{proof}
\begin{corol}
For $i,j,k,r,s,t$ distinct elements of the set $\{1,\dotsc,d\}$,
\begin{align*}
[O_{ij},O_{ki}]
&= O_{jk}+ \GG{i}{j}
+[O_{ijk},O_i]
\\
[O_{ij},O_{kl}]
&= \frac12( [O_i,O_{jkl}]
-[O_j,O_{ikl}]-[O_{ijl},O_k]
+[O_{ijk},O_l]
)\\
&= [O_i,O_{jkl}]
-[O_j,O_{ikl}]
\p,
\end{align*}
where the last equality follows by~\eqref{e:OuvwOx}.
\begin{align*}
[O_{jk},O_{lmn} ] & = [O_j, O_{klmn}] - [O_k,O_{jlmn}]\\
[O_{jk},O_{jlm} ] & = -O_{klm} - \ascom{O_k}{O_{lm}} - [O_j,O_{jklm}]\\
[O_{jk},O_{jkl} ] & = -\ascom{O_j}{O_{jl}} - \ascom{O_k}{O_{kl}} \p.
\end{align*}
\begin{align*}
\scom[+]{O_{ijk}}{O_{ijk}} & = 2\left((O_{i})^2+ (O_{j})^2 +
(O_{k})^2 + (O_{ij})^2 + (O_{ik})^2 + (O_{jk})^2\right) -\frac12\\
\scom[+]{O_{ijk}}{O_{ijt}} & = [O_c,O_t]
+ \ascom{O_{ik}}{O_{it}}+ \ascom{O_{jc}}{O_{jt}}\\
\scom[+]{O_{ijk}}{O_{ist}} & = O_{jkst} + \ascom{O_{jk}}{O_{st}}
+\scom{O_i}{O_{ijkst}}\\
\scom[+]{O_{ijk}}{O_{rst}}
& =
\scom{O_i}{ O_{jkrst}}
- \scom{O_j }{ O_{ikrst}}
+\scom{O_j }{ O_{ijrst}}
\p.
\end{align*}
\begin{align*}
\scom{O_{jk}}{ O_{jklm}}
& = - \ascom{O_j}{ O_{jlm}} - \ascom{O_k}{O_{klm}} \\
\scom{O_{jk}}{ O_{jlmn}}
& = - O_{klmn}- \ascom{O_k}{O_{lmn}} - \scom{O_j}{ O_{jklmn}} \\
\scom{O_{ij}}{ O_{klmn}} & = \scom{O_i}{ O_{jklmn}} - \scom{O_j}{ O_{iklmn}} \p.
\end{align*}
\end{corol}
For the $B$-isotropic basis we have the following. Fix $j,k \in \{1\dotsc,\ell\}$ with $j\neq k$ and let $u\in X$ such that $B(z^\pm_j,u) = 0 = B(z^\pm_k,u)$, then
\begin{align*}
[O_{\ib{+}{j}z_j^-},O_{z_j^\pm u}]
&= \pm 2 (O_{z_j^\pm u}+ \GG{z_j^\pm}{u})
-[O_{z_j^+z_j^-u},O_{z_j^\pm}]
\\
[O_{z_j^\pm z_k^\pm},O_{z_j^\mp u}]
&= \pm 2 (O_{z_j^\pm u}+ \GG{z_j^\pm}{u})
-[O_{z_j^+z_j^-u},O_{z_j^\pm}]
\\
[O_{ij},O_{kl}]
&= \frac12( [O_i,O_{jkl}]
-[O_j,O_{ikl}]-[O_{ijl},O_k]
+[O_{ijk},O_l]
)\\
&= [O_i,O_{jkl}]
-[O_j,O_{ikl}]
\p,
\end{align*}
\subsection{Generalized symmetries} \label{s:gensyms2}
Recall \eqref{e:Q}, given in Section~\ref{s:gensyms}, and the $\fr{osp}(1|2)$-elements~\eqref{e:osp}.
Denote $X \colonequals (x^+ \odot \gamma)(B) = \sum_{j=1}^d x_je_j$ and $D \colonequals (x^- \odot \gamma)(B) = \sum_{j=1}^d y_je_j$.
\begin{propo}\label{p:gensym2}
For $u \in V^*$, the element
\begin{equation}\label{e:gensym}
R_{u} \colonequals Q^-(\gamma_u) = (H-1)\gamma_u +
X \, \beta(u)
\end{equation}
is a generalized symmetry of $D$.
\end{propo}
\begin{proof}
Follows by Proposition~\ref{p:gensym}, using~\eqref{e:1} of Lemma~\ref{l:lemma3} to work out
\begin{equation}
Q^{-}(\gamma_u)=(H-1)\gamma_u+ F^{+} [ F^{-}, \gamma_u] \p.
\end{equation}
\end{proof}
\subsection{Representations and Howe correspondence}
\marginpar{TODO: put in TeX}
\section*{Appendix}
\end{document} |
\begin{document}
\begin{abstract}
We introduce a hierarchy of split principles and show that it parallels the hierarchy of large cardinals. In the typical case, a cardinal being large is equivalent to the corresponding split principle failing. As examples, we show how inaccessibility, weak compactness, subtlety, almost ineffability and ineffability can be characterized. We also consider two-cardinal versions of these principles. Some natural notions in the split hierarchy correspond to apparently new large cardinal notions. Such split principles come with certain ideals, and one of the split principles characterizing a version of $\kappa$ being $\lambda$-Shelah gives rise to a normal ideal on $\ensuremath{\mathcal P_\kappa \lambda}$. We also investigate the splitting numbers and the ideals induced by these split principles, and the relationship to partition relations.
\end{abstract}
\title{Split Principles, large cardinals, splitting families, and split ideals}
\section{Introduction}
A version of the split principle was first considered by Fuchs, Gitman, and Hamkins in the course of their work on \mathfrak{c}ite{FGH:IncomparableModels}, the intended use being the construction of ultrafilters with certain properties. Later, the first author observed that the split principle is an ``anti-large-cardinal axiom'' which characterizes the failure of a regular cardinal to be weakly compact. In the present paper, we consider several versions of the principle that provide simple combinatorial characterizations of the failure of various large cardinal properties.
The split principles for $\kappa$ say that there is a sequence $\vec d = \seq{d_\alpha}{\alpha<\kappa}$ that ``splits" every subset $A$ of $\kappa$ that's large into two subsets of $A$ that are also large, meaning that there is one ordinal $\beta$ such that for many $\alpha\in A$, we have that $\beta$ belongs to $d_\alpha$, but also, for many $\alpha\in A$, it is the case that $\beta$ belongs to $\alpha\setminus d_\alpha$. By varying the meanings of ``large", we obtain a host of natural split principles. Interestingly, in the two-cardinal version, some very natural split principles give rise to what appear to be entirely new large cardinal notions.
In Section \ref{sec:SplittingSubsetsOfKappa}, we introduce the orignial split principle in detail, and we show that if the notions of largeness used are reasonable (namely, the largeness of the sets split into is determined by a tail of the sets), then the failure of the split principle at $\kappa$ says that the corresponding splitting number is larger than $\kappa$. The notion of splitting number and splitting family here is the obvious generalization to arbitrary $\kappa$ of the well-known concepts at $\omega$.
We then show that the nonexistence of a $\kappa$-list that splits unbounded subsets of a regular cardinal $\kappa$ into unbounded sets is equivalent to the weak compactness of $\kappa$, or equivalently, that the corresponding splitting number is greater than $\kappa$.
We then show that the nonexistence of a sequence that splits stationary subsets of a regular cardinal into various classes (anything between the class of nonempty sets and the class of stationary subsets) characterizes its ineffability, and a regular cardinal $\kappa$ is almost ineffable if and only if there is no sequence that splits unbounded subsets into nonempty subsets. This latter characterization does not correspond to a statement about splitting numbers in any obvious way. We also obtain a characterization of subtle cardinals.
In Section \ref{sec:SplittingSubsetsOfPkl}, we consider split principles asserting the existence of lists that split subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. The nonexistence of a sequence that splits unbounded subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ into unbounded is a large cardinal concept which we call wild ineffability and which is situated somewhere between mild and almost ineffability. Interestingly, we don't know where it lies - it may be equivalent to one of those large cardinal notions. We characterize wild ineffability in terms of delayed coherence properties of $\ensuremath{\mathcal P_\kappa \lambda}$-lists, showing that the concept is a natural one, and we show that it is implied by the partition property $\Part{\kappa,\lambda}^3_<$. We give split principle characterizations of the two cardinal versions of almost ineffability and ineffability as before. Versions of the split principle which postulate that functions $F:\lambda\longrightarrow\lambda$ can be split allow us to characterize versions of $\kappa$ being $\lambda$-Shelah. We call the one corresponding to the failure of the functional split principle splitting unbounded sets into unbounded sets wild Shelahness. Again this seems to give rise to an entirely new large cardinal notion.
Finally, in Section \ref{sec:SplitIdeals}, we introduce the $\ensuremath{\mathcal P_\kappa \lambda}$-ideals of sets on which the split principles hold (and thus, where the corresponding large cardinal properties fail), and show that the ideal associated to the functional split principle is strongly normal. A lot of our analysis of $\ensuremath{\mathcal P_\kappa \lambda}$ split principles and the ideals they give rise to is closely related to prior work of Donna Carr (\mathfrak{c}ite{Carr:Diss,Carr1987:PklPartitionRelations}) as well as DiPrisco and Zwicker (\mathfrak{c}ite{DiPriscoZwicker:FlippingAndSC}).
\section{Splitting subsets of $\kappa$}
\label{sec:SplittingSubsetsOfKappa}
Let $\kappa$ be a cardinal.
We shall use the terminology of \mathfrak{c}ite{Weiss:Diss}, and refer to a sequence of the form $\seq{ d_\alpha }{ \alpha < \kappa }$ as a \emph{$\kappa$-list} if for all $\alpha<\kappa$, $d_\alpha \subseteqseteq \alpha$.
\begin{definition}
\label{definition:KappaSplitPrinciple}
Let $\mathcal{A}$ and $\mathcal{B}$ be families of subsets of $\kappa$. The principle $\SP{\kappa}{\mathcal{A},\mathcal{B}}$ says that there is a $\kappa$-list $\vec{d}$ that \emph{splits $\mathcal{A}$ into $\mathcal{B}$,} meaning that for every $A\in\mathcal{A}$, there is a $\beta<\kappa$ such that both $A^+_{\beta,\vec{d}}=A^+_\beta=\{\alpha\in A\; | \;\beta\in d_\alpha\}$ and $A^-_{\beta,\vec{d}}=A^-_\beta=\{\alpha\in A\; | \;\beta\in\alpha\setminus d_\alpha\}$ are in $\mathcal{B}$. In this case, we also say that $\beta$ splits $A$ into $\mathcal{B}$ with respect to $\vec{d}$.
We abbreviate $\SP{\kappa}{\mathcal{A},\mathcal{A}}$ by $\SP{\kappa}{\mathcal{A}}$ .
The collections of all unbounded, all stationary and all $\kappa$-sized subsets of $\kappa$ are denoted by $\mathcal{U}nbounded$, $\ensuremath{\mathsf{stationary}}$ and $\ensuremath{[\kappa]^\kappa}$ respectively.
\end{definition}
The idea is that $\mathcal{A}$ and $\mathcal{B}$ are collections of ``large'' sets, and $\SP{\kappa}{\mathcal{A},\mathcal{B}}$ says that there is one $\kappa$-list that can split any set that's large in the sense of $\mathcal{A}$ into two disjoint sets that are large in the sense of $\mathcal{B}$, in the uniform way described in the definition. There is a very close connection to the concept of splitting families, which can be made explicit after considering a wider range of split principles, in which we drop the assumption that the sequence is a $\kappa$-list.
\begin{definition}
\label{definition:GeneralNonsenseOnKappaSplitting}
Let $\kappa$ and $\tau$ be cardinals, and let $\mathcal{A}$ and $\mathcal{B}$ be families of subsets of $\kappa$. The principle $\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ says: there is a sequence $\vec{d}=\seq{d_\alpha}{\alpha<\kappa}$ of subsets of $\tau$ such that for every $A\in\mathcal{A}$, there is an ordinal $\beta<\tau$ such that the sets ${\tilde{A}}^+_\beta={\tilde{A}}^+_{\beta,\vec{d}}=\{\alpha\in A\; | \;\beta\in d_\alpha\}$ and ${\tilde{A}}^-_\beta={\tilde{A}}^-_{\beta,\vec{d}}=\{\alpha\in A\; | \;\beta\notin d_\alpha\}$ belong to $\mathcal{B}$. Such a sequence is called a \emph{$\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ sequence.}
If $\mathcal{A}=\mathcal{B}$, we don't mention $\mathcal{B}$ in the notation. Thus, $\SP{\kappa,\tau}{\mathcal{A},\mathcal{A}}$ is $\SP{\kappa,\tau}{\mathcal{A}}$.
In this context, a family $\mathcal{F}\subseteq\power{\kappa}$ is an \emph{$(\mathcal{A},\mathcal{B})$-splitting family for $\kappa$} if for every $A\in\mathcal{A}$, there is an $S\in\mathcal{F}$ such that both $A\mathfrak{c}ap S$ and $A\setminus S$ belong to $\mathcal{B}$. The $(\mathcal{A},\mathcal{B})$-\emph{splitting number, denoted $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\kappa)$} is the least size of an $(\mathcal{A},\mathcal{B})$-splitting family. We write $\mathfrak{s}_{\mathcal{A}}(\kappa)$ for $\mathfrak{s}_{\mathcal{A},\mathcal{A}}(\kappa)$. $\mathfrak{s}(\kappa)$ stands for $\mathfrak{s}_{\ensuremath{[\kappa]^\kappa}}(\kappa)$.
\end{definition}
Of course, $\mathfrak{s}=\mathfrak{s}(\omega)$ is a well-known cardinal characteristic of the continuum. Several authors have considered the generalization $\mathfrak{s}(\kappa)$, for uncountable $\kappa$, see \mathfrak{c}ite{Suzuki:OnSplittingNumbers}, \mathfrak{c}ite{Zapletal:SplittingNumberAtUncountableCards}. Note that for regular $\kappa$, $\mathcal{U}nbounded=\ensuremath{[\kappa]^\kappa}$. We'll only use $\ensuremath{[\kappa]^\kappa}$ when $\kappa$ is singular.
We will first explore the relation between the two types of split principles introduced so far.
\begin{observation} \label{observation:IndependentOfInitialSegmentsGivesSplitEquivalentToGeneralSplit}
Let $\kappa$ be a cardinal, and let $\mathcal{A}$ and $\mathcal{B}$ be collections of subsets of $\kappa$ such that for all $B\subseteq\kappa$ and all $\beta<\kappa$, $B\in\mathcal{B}$ iff $B\setminus\beta\in\mathcal{B}$ (``$\mathcal{B}$ is independent of initial segments''). Then $\SP{\kappa,\kappa}{\mathcal{A},\mathcal{B}}$ is equivalent to
$\SP{\kappa}{\mathcal{A},\mathcal{B}}$.
\end{observation}
\begin{proof} If $\vec e$ is a $\SP{\kappa}{\mathcal{A},\mathcal{B}}$-sequence, clearly it is also a $\SP{\kappa,\kappa}{\mathcal{A},\mathcal{B}}$-sequence.
For the other direction, let $\vec{d}$ be a $\SP{\kappa,\kappa}{\mathcal{A},\mathcal{B}}$ sequence. Define the $\kappa$-list $\vec{e}$ by $e_\alpha=d_\alpha\mathfrak{c}ap\alpha$. It follows that $\vec{e}$ is a $\SP{\kappa}{\mathcal{A},\mathcal{B}}$ sequence, because if $A\in\mathcal{A}$ and $\beta<\kappa$ is such that ${\tilde{A}}^+_{\beta,\vec{d}}$ and ${\tilde{A}}^-_{\beta,\vec{d}}$ are in $\mathcal{B}$ by $\SP{\kappa,\kappa}{\mathcal{A},\mathcal{B}}$, then $A^+_{\beta,\vec{e}}={\tilde{A}}^+_{\beta,\vec{d}}\setminus(\beta+1)$, and similarly, $A^-_{\beta,\vec{e}}={\tilde{A}}^-_{\beta,\vec{d}}\setminus(\beta+1)$. It follows from our assumption on $\mathcal{B}$ that $A^+_{\beta,\vec{e}}$ and $A^-_{\beta,\vec{e}}$ are in $\mathcal{B}$. \end{proof}
Note that $\mathcal{U}nbounded$, $\ensuremath{\mathsf{stationary}}$ and $\ensuremath{[\kappa]^\kappa}$ all are independent of initial segments. The following lemma says that the split principles can be viewed as statements about the sizes of the corresponding splitting numbers.
\begin{lemma}
\label{lemma:SNleLambdaEquivalentToKappaLambdaSplit-General}
Let $\kappa$ and $\tau$ be cardinals, and let $\mathcal{A}$, $\mathcal{B}$ be families of subsets of $\kappa$. Then $\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ holds iff $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\kappa)\le\tau$.
\end{lemma}
\noindent\emph{Note:} In other words, $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\kappa)$ is the least $\tau$ such that $\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ holds.
\begin{proof} For the direction from right to left, if $\mathcal{S}=\{x_\alpha\; | \;\alpha<\tau\}$ is an $(\mathcal{A},\mathcal{B})$-splitting family for $\kappa$, then we can define a sequence $\kla{d_\alpha\; | \;\alpha<\kappa}$ of subsets of $\tau$ by setting
\[d_\alpha=\{\gamma<\tau\; | \;\alpha\in x_\gamma\}.\]
Then $\vec{d}$ is a $\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ sequence, because if $A\in\mathcal{A}$, then there is a $\beta<\tau$ such that both $A\mathfrak{c}ap x_\beta$ and $A\setminus x_\beta$ belong to $\mathcal{B}$, but $A\mathfrak{c}ap x_\beta={\tilde{A}}^+_{\beta,\vec{d}} \ $ and $A\setminus x_\beta={\tilde{A}}^-_{\beta,\vec{d}} \ $ so we are done.
Conversely, if $\vec{d}$ is a $\SP{\kappa,\tau}{\mathcal{A},\mathcal{B}}$ sequence, then for each $\gamma<\tau$, we define a subset $x_\gamma$ of $\kappa$ by
\[x_\gamma=\{\alpha<\kappa\; | \;\gamma\in d_\alpha\}.\]
Then $\mathcal{S}=\{x_\gamma\; | \;\gamma<\tau\}$ is an $(\mathcal{A},\mathcal{B})$-splitting family for $\kappa$, because if $A\in\mathcal{A}$, then there is a $\beta<\tau$ such that both ${\tilde{A}}^+_{\beta,\vec{d}}$ and ${\tilde{A}}^-_{\beta,\vec{d}}$ are in $\mathcal{B}$, but as before, ${\tilde{A}}^+_{\beta,\vec{d}}=A\mathfrak{c}ap x_\beta$ and ${\tilde{A}}^-_{\beta, \vec d}=A\setminus x_\beta$. \end{proof}
What this proof shows is that if $X\subseteq\kappa\times\lambda$ is a set and we let, for $\beta<\lambda$, be $X^\beta=\{\alpha<\kappa\; | \;\kla{\alpha,\beta}\in X\}$ be the horizontal section at height $\beta$, and for $\alpha<\kappa$, $X_\alpha=\{\beta<\lambda\; | \;\kla{\alpha,\beta}\in X\}$ be the vertical section at $\alpha$, then $\seq{X_\alpha}{\alpha<\kappa}$ is a \SP{\kappa,\lambda}{\mathcal{A},\mathcal{B}} sequence iff $\seq{X^\beta}{\beta<\lambda}$ is an $(\mathcal{A},\mathcal{B})$-splitting family.
\begin{corollary}
If $\mathcal{B}$ is independent of initial segments, then $\SP{\kappa}{\mathcal{A},\mathcal{B}}$ holds iff $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\kappa)\le\kappa$.
\end{corollary}
\begin{observation}
\label{observation:TrivialInequality}
Let $\kappa$ be a cardinal, and let $\mathcal{A},\mathcal{A}',\mathcal{B},\mathcal{B}'$ be collections of subsets of $\kappa$. If $\mathcal{A}\subseteq\mathcal{A}'$ and $\mathcal{B}'\subseteq\mathcal{B}$, then $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\kappa)\le\mathfrak{s}_{\mathcal{A}',\mathcal{B}'}(\kappa)$.
\end{observation}
\begin{proof}
Under the assumptions stated, every $(\mathcal{A}',\mathcal{B}')$-splitting family is also $(\mathcal{A},\mathcal{B})$-splitting.
\end{proof}
We are now ready to characterize inaccessible cardinals by split principles. The equivalence \ref{item:Inaccessible}$\iff$\ref{item:Unbounded} follows from work of Motoyoshi.
\begin{lemma}
\label{lemma:CharacterizationOfInaccessibility}
Let $\kappa$ be an uncountable regular cardinal. The following are equivalent:
\begin{enumerate}
\item
\label{item:Inaccessible}
$\kappa$ is inaccessible.
\item
\label{item:StationaryNonempty}
$\SP{\kappa,\tau}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ fails for every $\tau<\kappa$. Equivalently, $\mathfrak{s}_{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}(\kappa)\ge\kappa$.
\item
\label{item:Stationary}
$\SP{\kappa,\tau}{\ensuremath{\mathsf{stationary}}}$ fails for every $\tau<\kappa$. Equivalently, $\mathfrak{s}_{\ensuremath{\mathsf{stationary}}}(\kappa)\ge\kappa$.
\item
\label{item:UnboundedNonempty}
$\SP{\kappa,\tau}{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}$ fails for every $\tau<\kappa$. Equivalently, $\mathfrak{s}_{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}(\kappa)\ge\kappa$.
\item
\label{item:Unbounded}
$\SP{\kappa,\tau}{\mathcal{U}nbounded}$ fails for every $\tau<\kappa$. Equivalently, $\mathfrak{s}_{\mathcal{U}nbounded}(\kappa)\ge\kappa$.
\end{enumerate}
It follows that for any collection $\mathcal{B}$ with $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$, these conditions are equivalent to the failure of $\SP{\kappa,\tau}{\ensuremath{\mathsf{stationary}},\mathcal{B}}$, and similarly for any $\mathcal{B}$ with $\mathcal{U}nbounded\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$, they are equivalent to the failure of $\SP{\kappa,\tau}{\mathcal{U}nbounded,\mathcal{B}}$.
Moreover, (\ref{item:StationaryNonempty}), (\ref{item:UnboundedNonempty}) are equivalent to $\kappa$ being inaccessible even if $\kappa$ is not assumed to be regular (not even to be a cardinal).
\end{lemma}
\begin{proof} The equivalence (\ref{item:Inaccessible})$\iff$(\ref{item:Unbounded}) follows from previously known results as follows. According to \mathfrak{c}ite{Suzuki:OnSplittingNumbers}, it was shown in \mathfrak{c}ite{Motoyoshi:Masters} that for an uncountable regular cardinal $\kappa$, $\kappa$ is inaccessible iff $\mathfrak{s}(\kappa)\ge\kappa$ (see \mathfrak{c}ite[Lemma 3]{Zapletal:SplittingNumberAtUncountableCards} for a proof). By Lemma \ref{lemma:SNleLambdaEquivalentToKappaLambdaSplit-General}, this is equivalent to saying that for no $\tau<\kappa$ does $\SP{\kappa,\tau}{\mathcal{U}nbounded}$ hold.
However, we will give a self-contained proof here.
The implications (\ref{item:StationaryNonempty})$\implies$(\ref{item:Stationary}) and (\ref{item:StationaryNonempty})$\implies$(\ref{item:UnboundedNonempty})$\implies$(\ref{item:Unbounded}) are trivial.
To prove (\ref{item:Inaccessible})$\implies$(\ref{item:StationaryNonempty}), let $\kappa$ be inaccessible, and let $\tau<\kappa$. Let $\vec{d}=\seq{d_\alpha}{\alpha<\kappa}$ be a sequence of subsets of $\tau$. We will show that $\vec d$ does not split stationary subsets of $\kappa$ into nonempty sets. Since $2^\tau<\kappa$, it follows that there is a stationary set $S\subseteq\kappa$ and a set $e\subseteq\tau$ such that for all $\alpha\in S$, $d_\alpha=e$. Let $\beta<\tau$. Then $\tilde{S}^+_\beta=\{\alpha\in S\; | \;\beta\in e\}$ and $\tilde{S}^-_\beta=\{\alpha\in S\; | \; \beta\notin e\}$, so one of these is $S$ and the other is $\emptyset$. So $\beta$ does not split $S$ into nonempty sets.
To show (\ref{item:Unbounded})$\implies$(\ref{item:Inaccessible}), suppose $\kappa$ is not inaccessible. Since $\kappa$ is assumed to be regular, it follows that there is a $\tau<\kappa$ such that $2^\tau\ge\kappa$. Let $\vec{d}=\seq{d_\alpha}{\alpha<\kappa}$ be a sequence of distinct subsets of $\tau$. By \ref{item:Unbounded}, $\vec{d}$ is not a $\SP{\kappa,\tau}{\mathcal{U}nbounded}$ sequence. Hence, there is an unbounded set $U\subseteq\kappa$ such that no $\beta<\tau$ splits $U$ into unbounded sets. So, for every $\beta<\tau$, exactly one of $U^+_\beta$ and $U^-_\beta$ is bounded. Let $\xi_\beta<\kappa$ be such that the bounded one is contained in $\xi_\beta$. Let $\xi=\sup_{\beta<\tau}\xi_\beta$. Then $\xi<\kappa$, since $\kappa$ is regular, and we claim that for $\xi<\gamma<\delta$ with $\gamma,\delta\in U$, it follows that $d_\gamma=d_\delta$. For if $\beta<\tau$ and $U^-_\beta$ is bounded, then, since $\gamma\in U\setminus\xi_\beta$, it follows that $\gamma\in U^+_\beta$, which means that $\beta\in d_\gamma$, and for the same reason, $\beta\in d_\delta$. And if $U^+_\beta$ is bounded, then it follows that $\beta\notin d_\gamma$ and $\beta\notin d_\delta$. So $d_\gamma=d_\delta$, a contradiction.
A similar argument shows the final implication, (\ref{item:Stationary})$\implies$(\ref{item:Inaccessible}). Assume $\kappa$ were not inaccessible. Let $\tau<\kappa$ be such that $2^\tau\ge\kappa$. Let $\vec{d}=\seq{d_\alpha}{\alpha<\kappa}$ be a sequence of distinct subsets of $\tau$. By \ref{item:Stationary}, $\vec{d}$ is not a $\SP{\kappa,\tau}{\ensuremath{\mathsf{stationary}}}$ sequence. Hence, there is a stationary set $S\subseteq\kappa$ such that no $\beta<\lambda$ splits $S$ into stationary sets. So, for every $\beta<\tau$, exactly one of $S^+_\beta$ and $S^-_\beta$ is nonstationary. Let $C_\beta$ be a club in $\kappa$, disjoint from the nonstationary one of the two. Let $T=S\mathfrak{c}ap\bigcap_{\beta<\tau}C_\beta$. This is a stationary set, and we claim that for $\gamma<\delta$ are both in $T$, it follows that $d_\gamma=d_\delta$. For if $\beta<\tau$ and $S^-_\beta$ is nonstationary, then, since $\gamma\in S\mathfrak{c}ap C_\beta$, it follows that $\gamma\in S^+_\beta$, which means that $\beta\in d_\gamma$, and for the same reason, $\beta\in d_\delta$. And if $S^+_\beta$ is nonstationary, then it follows that $\beta\notin d_\gamma$ and $\beta\notin d_\delta$. So $d_\gamma=d_\delta$, a contradiction.
The claim about families $\mathcal{B}$ with $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$ follows from the equivalence of \ref{item:StationaryNonempty}.~and \ref{item:Stationary}, and the claim about $\mathcal{B}$ with $\mathcal{U}nbounded\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$ follows from the equivalence of \ref{item:UnboundedNonempty}. and \ref{item:Unbounded}.
For the last claim, it suffices to show that \ref{item:UnboundedNonempty}.~implies that $\kappa$ is regular. But this is obvious, since if $\seq{\xi_\alpha}{\alpha<\mathrm{cf}(\kappa)}$ is cofinal in $\kappa$, then $\{\xi_\alpha\; | \;\alpha<\mathrm{cf}(\kappa)\}$ (viewed as a collection of subsets of $\kappa$) is an $(\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}})$-splitting family. So $\mathfrak{s}_{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}(\kappa)\le\mathrm{cf}(\kappa)$, so $\SP{\kappa,\mathrm{cf}(\kappa)}{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}$ holds. So it has to be the case that $\mathrm{cf}(\kappa)=\kappa$.
\end{proof}
Recall that a regular cardinal $\kappa$ is weakly compact if $\kappa$ is inaccessible and the tree property $\TP(\kappa)$ holds at $\kappa$, which states that every $\kappa$-tree has a cofinal branch, where a $\kappa$-tree is a tree of height $\kappa$ all of whose levels have size less than $\kappa$.
We will show that a regular cardinal $\kappa$ is weakly compact if and only if $\SP{\kappa}{\mathcal{U}nbounded}$ fails, for example. Toward this end, we will first define the analogue of the tree property for $\kappa$-lists.
\begin{definition} A $\kappa$-list $\vec d = \seq{ d_\alpha }{ \alpha < \kappa }$ has a \emph{cofinal branch}, or has a $\kappa$-branch, so long as there is
a $b \subseteqseteq \kappa$ such that for all $\gamma < \kappa$ there is an $\alpha \geq \gamma$ such that
$d_\alpha \mathfrak{c}ap \gamma = b \mathfrak{c}ap \gamma.$
We say that the branch property $\BP(\kappa)$ holds if every $\kappa$-list has a cofinal branch.
Given a $\kappa$-list $\vec d = \seq{ d_\alpha }{ \alpha < \kappa }$, for each $\alpha < \kappa$ let $d^c_\alpha: \alpha \longrightarrow 2$ denote the characteristic function of $d_\alpha$. The sequential tree corresponding to the $\kappa$-list is given by
$T_{\vec d} = \set{ d^c_\alpha \mathbin{\upharpoonright} \beta }{\beta\le\alpha<\kappa}$,
and the tree ordering is set inclusion. As is customary, we refer to a function
$b: \kappa \longrightarrow 2$ as a (cofinal) branch through $T_{\vec d}$ if for all $\gamma < \kappa$, $b\mathbin{\upharpoonright}\gamma\in T_{\vec{d}}$, which means that for every $\gamma<\kappa$, there is an $\alpha\ge\gamma$ such that
$b \mathbin{\upharpoonright} \gamma = d^c_\alpha \mathbin{\upharpoonright} \gamma.$
\end{definition}
\begin{observation}
\label{observation:ListBranchCorrespondsToTreeBranch}
A $\kappa$-list $\vec d$ has a cofinal branch if and only if the corresponding tree $T_{\vec{d}}$ has a cofinal branch.
\end{observation}
It turns out that for regular $\kappa$, the properties of a $\kappa$-list of splitting unbounded sets and having a cofinal branch are complementary.
\begin{theorem}
\label{theorem:ListSplitsIffItHasNoBranch}
Let $\kappa$ be regular, and let $\vec{d}$ be a $\kappa$-list. The following are equivalent:
\begin{enumerate}
\item $\vec{d}$ is a $\SP{\kappa}{\mathcal{U}nbounded}$ sequence.
\item $\vec{d}$ does \emph{not} have a cofinal branch.
\end{enumerate}
\end{theorem}
\begin{proof}
(1)$\implies$(2): Assume towards a contradiction that $\vec{d}$ has a cofinal branch $b \subseteqseteq \kappa$, and that $\vec d$ splits unbounded sets. We will first define a function $f:\kappa \longrightarrow \kappa$ as follows:
\[f(\gamma) =\text{the least $\alpha \geq \gamma$ such that $b\mathfrak{c}ap\gamma = d_\alpha \mathfrak{c}ap \gamma.$}\]
Note that $f$ is weakly increasing, thus letting $A = f``\kappa$, $A$ is unbounded in $\kappa$. So there is $\beta < \kappa$ which splits $A$ (with respect to $\vec{d}$), i.e., both of the sets
$A_\beta^+ = \set{ f(\gamma) \in A }{ \beta \in d_{f(\gamma)} }$ and $A_\beta^- = \set{ f(\gamma) \in A }{ \beta \in f(\gamma) \setminus d_{f(\gamma)} }$
are unbounded in $\kappa$. There are two cases.
\emph{Case 1: $\beta \notin b$.} Since $A_\beta^+$ is unbounded, we may choose $f(\gamma) \in A_\beta^+$ satisfying $f(\gamma)>f(\beta)$. By the weak monotonicity of $f$, it follows that $\gamma>\beta$. Then $\beta \in d_{f(\gamma)}$ by the definition of $A_\beta^+$ and it follows that
$\beta \in d_{f(\gamma)} \mathfrak{c}ap \gamma= b \mathfrak{c}ap \gamma,$
contradicting the assumption that $\beta \notin b$.
\emph{Case 2: $\beta \in b$.} We may run a similar argument to the previous case in order to obtain a contradiction. In this case, we use that $A_\beta^-$ is unbounded to choose $f(\gamma) \in A_\beta^-$ satisfying $f(\gamma) > f(\beta)$, so that $\gamma>\beta$, and get the contradiction that $\beta\notin d_{f(\gamma)}$ while $\beta \in b\mathfrak{c}ap\gamma=d_{f(\gamma)} \mathfrak{c}ap \gamma$.
$\neg(1)$$\implies$$\neg(2)$: Assume that $\vec{d}$ does not split unbounded sets. We will show that then $\vec{d}$ has a cofinal branch. Let $A \subseteqseteq \kappa$ be unbounded such that no $\beta < \kappa$ splits $A$ (with respect to $\vec{d}$). Thus, for each $\beta < \kappa$, exactly one of $A^+_\beta$ or $A^-_\beta$ is bounded in $\kappa$, since $A = A_\beta^+ \mathfrak{c}up A_\beta^-$ (so since $A$ is unbounded, it can't be that both $A^+_\beta$ and $A^-_\beta$ are bounded). Now we may define our branch $b\subseteqseteq\kappa$ as follows:
\[\beta \in b \iff A_\beta^- \text{ is bounded in $\kappa$ } (\iff A_\beta^+ \text{ is unbounded in $\kappa$.})\]
To see that this works, note that for each $\beta < \kappa$ there is a least $\xi_\beta < \kappa$ such that either:
$$\text{ $\beta \in d_\delta$ for all $\delta \in A \setminus \xi_\beta$ or $\beta \notin d_\delta$ for all $\delta \in A \setminus \xi_\beta$, } $$
Letting $\gamma < \kappa$ be arbitrary, using the fact that $\kappa$ is regular, there is an $\alpha \in A$ such that
$\alpha > \sup_{\beta < \gamma} \xi_\beta.$
It follows that $b \mathfrak{c}ap \gamma = d_\alpha \mathfrak{c}ap \gamma$.
To see this, let $\beta < \gamma$. We have two cases.
\emph{Case 1: $\beta \in b$.} Then $A^-_\beta$ is bounded, so for all $\delta\in A\setminus\xi_\beta$, $\beta\in d_\delta$, so $\beta\in d_\alpha$, since $\alpha\in A\setminus\xi_\beta$.
\emph{Case 2: $\beta \notin b$.} Then $A^+_\beta$ is bounded, so $\beta \notin d_\alpha$, since $\alpha \in A \setminus \xi_\beta$.
So we have reached the desired contradiction that $b$ is a cofinal branch.
\end{proof}
\begin{corollary}
\label{corollary:TreeCharacterizationIfKappaIsRegular}
Let $\kappa$ be regular. Then $\SP{\kappa}{\mathcal{U}nbounded}$ holds iff there is a sequential tree $T\subseteq{}^{{<}\kappa}2$ of height $\kappa$ that has no cofinal branch.
\end{corollary}
Note that a sequential tree $T\subseteq{}^{{<}\kappa}2$ of height $\kappa$ without a cofinal branch is not necessarily an Aronszajn tree, as it doesn't even have to be a $\kappa$-tree -- $T$ may have levels of size $\kappa$. But if $\kappa$ is inaccessible, then such a $T$ is Aronszajn.
\proof
For the direction from right to left, let $T$ be a sequential tree $T\subseteq{}^{{<}\kappa}2$ of height $\kappa$ that has no cofinal branch. For each $\alpha<\kappa$, let $s_\alpha$ be a node at the $\alpha$-th level of $T$, i.e., a sequence $s_\alpha:\alpha\longrightarrow 2$ with $s_\alpha\in T$. Let $d_\alpha$ be the sequence $s_\alpha$, viewed as a subset of $\alpha$, i.e., $d_\alpha=\{\gamma<\alpha\; | \; s_\alpha(\gamma)=1\}$. In other words, $s_\alpha=d^c_\alpha$. Then $T_{\vec{d}}\subseteq T$, and so, since $T$ does not have a cofinal branch, $T_{\vec{d}}$ has no cofinal branch, which means, by Observation \ref{observation:ListBranchCorrespondsToTreeBranch}, that $\vec{d}$ has no cofinal branch, and this is equivalent to saying that $\vec{d}$ splits unbounded sets, by Theorem \ref{theorem:ListSplitsIffItHasNoBranch}. So $\SP{\kappa}{\mathcal{U}nbounded}$ holds.
For the converse, let $\vec{d}$ be a $\SP{\kappa}{\mathcal{U}nbounded}$-sequence. By Theorem \ref{theorem:ListSplitsIffItHasNoBranch}, $\vec{d}$ has no cofinal branch. So by Observation \ref{observation:ListBranchCorrespondsToTreeBranch}, $T_{\vec{d}}$ does not have a cofinal branch, so $T_{\vec{d}}$ is as wished. \qed
\subseteqsection{Weakly compact cardinals}
\begin{definition} An uncountable cardinal $\kappa$ is weakly compact so long as $\kappa$ is inaccessible and $\kappa$ has the tree property, meaning that every $\kappa$-tree has a branch of size $\kappa$. \end{definition}
We shall use the previous result to show that the split principle can be used to characterize weakly compact cardinals.
\begin{corollary}
\label{corollary:SplitKappaIffNotWC}
Let $\kappa$ be a regular cardinal. Then $\SP{\kappa}{\mathcal{U}nbounded}$ if and only if $\kappa$ is not weakly compact. (This is true for $\kappa=\omega$ as well, if we consider $\omega$ to be weakly compact, which is not standard.)
\end{corollary}
\begin{proof}
We shall show both directions of the equivalence separately.
$\Longrightarrow$: Assuming $\SP{\kappa}{\mathcal{U}nbounded}$, we have to show that $\kappa$ is not weakly compact. If $\kappa$ is not inaccessible, then we are done, so let's assume it is. By Corollary \ref{corollary:TreeCharacterizationIfKappaIsRegular}, there is a sequential tree $T\subseteq{}^{{<}\kappa}2$ of height $\kappa$ with no cofinal branch. Since $\kappa$ is inaccessible, $T$ is a $\kappa$-tree, and thus, $\kappa$ does not have the tree property, so $\kappa$ is not weakly compact.
$\Longleftarrow$: Let $\kappa$ fail to be weakly compact. We split into two cases.
\emph{Case 1: $\kappa$ is not inaccessible.} Then by \ref{lemma:CharacterizationOfInaccessibility}, there is a $\tau<\kappa$ such that $\SP{\kappa,\tau}{\mathcal{U}nbounded}$ holds. This clearly implies that $\SP{\kappa,\kappa}{\mathcal{U}nbounded}$ holds, and, since $\mathcal{U}nbounded$ is independent of initial segments, this is equivalent to $\SP{\kappa}{\mathcal{U}nbounded}$.
\emph{Case 2: $\kappa$ is inaccessible but not weakly compact.} Then $\TP(\kappa)$ fails, and this is witnessed by a sequential tree $T$ on ${}^{{<}\kappa}2$ that has no cofinal branch.
Thus, $\Split{\kappa}$ holds, by Corollary \ref{corollary:TreeCharacterizationIfKappaIsRegular}.
\end{proof}
So a regular cardinal $\kappa$ is weakly compact iff $\SP{\kappa}{\mathcal{U}nbounded}$ fails. Since this is equivalent to saying that $\SP{\kappa,\kappa}{\mathcal{U}nbounded}$ fails, this can be equivalently expressed by saying that $\mathfrak{s}(\kappa)>\kappa$, by Lemma \ref{lemma:SNleLambdaEquivalentToKappaLambdaSplit-General}. This latter characterization of weak compactness was shown in \mathfrak{c}ite{Suzuki:OnSplittingNumbers}.
It should be noted, however, before moving on to larger large cardinals, that what we really showed with our proof of Theorem \ref{theorem:ListSplitsIffItHasNoBranch} is that $\Split{\kappa}$ is equivalent to the failure of something seemingly stronger than every $\kappa$-list having a cofinal branch, namely the failure of the \textit{strong} branch property, which we define below.
\begin{definition}
\label{definition:StrongBranch}
Let $\vec d = \seq{d_\alpha}{\alpha < \kappa}$ be a $\kappa$-list. A cofinal branch $b\subseteqseteq\kappa$ is a \emph{strong branch} of $\vec{d}$ so long as there are is an unbounded $U \subseteqseteq \kappa$ such that for each $\gamma < \kappa$, there is $\alpha > \gamma$ such that for all $\delta \in U$ with $\delta>\alpha$ we have that
$d_\delta \mathfrak{c}ap \gamma = b \mathfrak{c}ap \gamma.$ In this case we say that the unbounded set $U$ \textit{guides} the cofinal branch $b$.
If every $\kappa$-list has a strong branch, we say $\SBP(\kappa)$ holds.
\end{definition}
Indeed the argument in the beginning of the proof of Theorem \ref{theorem:ListSplitsIffItHasNoBranch} shows that if a $\kappa$-list has a cofinal branch, then that branch is a strong branch (and it is not necessary to assume that $\kappa$ is regular here). The unbounded set verifying strongness is the range of the function $f$ in that proof. Trivially, every strong branch is a branch, and so, these concepts are equivalent for $\kappa$-lists. The situation will turn out to be less clear when dealing with $\ensuremath{\mathcal P_\kappa \lambda}$-lists, as we will do in Section \ref{sec:SplittingSubsetsOfPkl}.
\subseteqsection{Ineffable cardinals}
\label{subsec:Ineffables}
\begin{definition}
Let $\mathcal{A}$ be a family of subsets of $\kappa$. A $\kappa$-list $\vec d = \seq{ d_\alpha }{\alpha< \kappa}$ has an $\mathcal{A}$ branch iff there is a set $b \subseteqseteq \kappa$ and a set $A\in\mathcal{A}$ such that for all $\alpha \in A$, we have that
$d_\alpha = b \mathfrak{c}ap \alpha$. Keeping with tradition, a $\ensuremath{\mathsf{stationary}}$ branch is called an \emph{ineffable} branch, and an $\mathcal{U}nbounded$ branch is called an \emph{almost ineffable} branch. $\kappa$ is $\mathcal{A}$-ineffable if $\kappa$ is regular and every $\kappa$-list has an $\mathcal{A}$ branch.
Using this language, it is clear that by our definition, \ensuremath{\mathsf{stationary}}-ineffable cardinals are exactly \textit{ineffable cardinals}, and \mathcal{U}nbounded-ineffable ones are called \textit{almost ineffable cardinals}.
We say that $\kappa$ has the ineffable branch property, or $\IBP(\kappa)$ holds, if every $\kappa$-list has an ineffable branch.
\end{definition}
We have the following string of implications: $\IBP(\kappa) \implies \SBP(\kappa) \implies \BP(\kappa)$. In particular, ineffable cardinals are weakly compact.
We start by giving a general and uniform characterization of $\mathcal{A}$-ineffability.
\begin{theorem}
\label{theorem:GeneralABranchesForKappaListsAndGeneralIneffability}
Let $\kappa$ be a cardinal, $\mathcal{A}$ a family of subsets of $\kappa$ and $\vec{d}$ a $\kappa$-list. Then the following are equivalent:
\begin{enumerate}
\item $\vec{d}$ is a \SP{\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}-sequence.
\item $\vec{d}$ has no $\mathcal{A}$ branch.
\end{enumerate}
Thus, a regular, uncountable cardinal $\kappa$ is $\mathcal{A}$-ineffable if and only if $\SP{\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$ fails. In particular, $\kappa$ is ineffable if and only if $\SP{\kappa}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ fails, and $\kappa$ is almost ineffable if and only if $\SP{\kappa}{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}$ fails.
\end{theorem}
\begin{proof}
(1)$\implies$(2): Suppose $B$ is an $\mathcal{A}$ branch for $\vec{d}$. Let $A\in\mathcal{A}$ be such that for all $\alpha\in A$, $d_\alpha=B\mathfrak{c}ap\alpha$. Let $\beta$ be such that both $A^+_\beta$ and $A^-_\beta$ are nonempty. Let $\gamma\in A^+_\beta$ and $\delta\in A^-_\beta$. Then $\beta\in\gamma$, $\gamma\in A$, $\beta\in d_\gamma$ and $d_\gamma=B\mathfrak{c}ap\gamma$, so $\beta\in B$. On the other hand, $\beta\in\delta$, $\delta\in A$, $\beta\notin d_\delta$ and $d_\delta=B\mathfrak{c}ap\delta$, so $\beta\notin B$. This is a contradiction.
(2)$\implies$(1): We show the contrapositive. So assuming $\vec{d}$ is not a \SP{\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}-sequence, we have to show that it has an $\mathcal{A}$ branch. Let $A\in\mathcal{A}$ be such that no $\beta$ splits $A$ into nonempty sets. So for every $\beta<\kappa$, it's not the case that both $A^+_\beta$ and $A^-_\beta$ are nonempty. Set
\[B=\{\beta<\kappa\; | \; A^+_\beta\neq\emptyset\}.\]
It follows that for every $\alpha\in A$, $d_\alpha=B\mathfrak{c}ap\alpha$ (and hence that $B$ is an $\mathcal{A}$ branch). To see this, let $\alpha\in A$, and let $\beta<\alpha$. If $\beta\in B$, then $A^+_\beta\neq\emptyset$, so $A^-_\beta=\emptyset$. It follows that $\beta\in d_\alpha$ (because if we had $\beta\notin d_\alpha$, it would follow that $\alpha\in A^-_\beta$). And if $\beta\notin B$, then $A^+_\beta=\emptyset$. Since $\beta\in\alpha$ and $\alpha\in A$, it follows that $\beta\notin d_\alpha$, because if we had $\beta\in d_\alpha$, then it would follow that $\alpha\in A^+_\beta=\emptyset$. This shows that $d_\alpha=B\mathfrak{c}ap\alpha$, as claimed.
\end{proof}
This general theorem on the failure of splitting into nonempty sets allows us to characterize subtle cardinals as well, after introducing an additional concept.
\begin{definition}
\label{definition:SplitIdealsOnKappa}
Let $\mathcal{A},\mathcal{B}\subseteq\power{\kappa}$ be families of subsets of $\kappa$, and let $A\subseteq\kappa$ be a fixed subset of $\kappa$. Then we write $\mathcal{A}\mathbin{\upharpoonright} A$ for $\mathcal{A}\mathfrak{c}ap\power{A}$, i.e., for the family of sets in $\mathcal{A}$ that are contained in $A$.
We let
\[\mathcal{I}(\SP{\kappa}{\mathcal{A},\mathcal{B}})=\{A\subseteq\kappa\; | \; \SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} A,\mathcal{B}} \ \text{holds}\}\]
\end{definition}
Note that if a $\kappa$-list $\vec{d}$ witnesses that $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} A,\mathcal{B}}$ holds, then the values of $d_\alpha$ for $\alpha\in\kappa\setminus A$ are irrelevant, and hence it makes sense to restrict $\vec{d}$ to $A$, call it an $A$-list, and view $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} A,\mathcal{B}}$ as postulating the existence of a splitting $A$-list. Note also that if $A\subseteq B\subseteq\kappa$ and $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} B,\mathcal{B}}$ holds, then so does $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} A,\mathcal{B}}$, since $\mathcal{A}\mathbin{\upharpoonright} A\subseteq\mathcal{A}\mathbin{\upharpoonright} B$. If $I\subseteq\power{X}$ is an ideal, then we write $I^+$ for the collection of $I$-positive sets, i.e., $\power{X}\setminus I$. We write $I^*$ for the dual filter associated to $I$, which consists of the complements of sets in $I$.
\begin{observation}
\label{observation:WhenSplitIdealsAreIdealsOnKappa}
Let $\mathcal{A},\mathcal{B}\subseteq\power{\kappa}$ be families of subsets of $\kappa$. Suppose that $\mathcal{A}=I^+$, for some ideal $I$ on $\kappa$, and that $\mathcal{B}$ is closed under supersets. Then $\mathcal{I}(\SP{\kappa}{\mathcal{A},\mathcal{B}})$ is an ideal.
\end{observation}
\begin{proof} We have already pointed out that $\mathcal{I}(\SP{\kappa}{\mathcal{A},\mathcal{B}})$ is closed under subsets. Now suppose $X,Y\in\mathcal{I}(\SP{\kappa}{\mathcal{A},\mathcal{B}})$. Let $\vec{d}$, $\vec{e}$ be $X$,$Y$-lists witnessing that $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} X,\mathcal{B}}$, $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} Y,\mathcal{B}}$ holds, respectively. Let ${\vec{f}}=\vec{d}\mathfrak{c}up\vec{e}\mathbin{\upharpoonright}(Y\setminus X)$. Then ${\vec{f}}$ is a $\SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright}(X \mathfrak{c}up Y),\mathcal{B}}$-list: let $Z\in\mathcal{A}$, $Z\subseteq X\mathfrak{c}up Y$. Then it must be the case that $Z\mathfrak{c}ap X$ or $Z\mathfrak{c}ap Y$ is in $I^+$, because otherwise $Z=(Z\mathfrak{c}ap X)\mathfrak{c}up(Z\mathfrak{c}ap Y)$ would be the union of two members of $I$. If $Z\mathfrak{c}ap X\in I^+$, then $Z\mathfrak{c}ap X\in\mathcal{A}\mathbin{\upharpoonright} X$, and so, there is a $\beta<\kappa$ such that both $(Z\mathfrak{c}ap X)^+_{\beta,\vec{d}}$ and $(Z\mathfrak{c}ap X)^-_{\beta,\vec{d}}$ are in $\mathcal{B}$. Since ${\vec{f}}\mathbin{\upharpoonright} X=\vec{d}$, it follows that $(Z\mathfrak{c}ap X)^+_{\vec{d},\beta}\subseteq Z^+_{\beta,\vec{e}}$ and $(Z\mathfrak{c}ap X)^-_{\beta,\vec{d}}\subseteq Z^-_{\beta,\vec{e}}$, so, since $\mathcal{B}$ is closed under supersets, $Z^+_{\beta,\vec{e}}$ and $Z^-_{\beta,\vec{e}}$ are in $\mathcal{B}$. If $Z\mathfrak{c}ap X$ is in $I$, then $Z\mathfrak{c}ap(Y\setminus X)\in I^+$, then we can argue similarly, replacing $Z\mathfrak{c}ap X$ with $Z\mathfrak{c}ap(Y\setminus X)$. This means that $I$ is closed under unions and is thus an ideal as desired. \end{proof}
We will explore split ideals more in the $\ensuremath{\mathcal P_\kappa \lambda}$-context, in Section \ref{sec:SplitIdeals}.
Let's now characterize when $\kappa$ is subtle. Recall that by definition, a regular cardinal $\kappa$ is subtle iff for every $\kappa$-list $\vec{d}$ and every club $C\subseteq\kappa$, there are $\alpha<\beta$ in $C$ such that $d_\alpha=d_\beta\mathfrak{c}ap\alpha$.
\begin{lemma}
\label{lemma:CharacterizingSubtlety}
A regular cardinal is subtle iff $\mathcal{I}(\SP{\kappa}{[\kappa]^{2},\ensuremath{\mathsf{nonempty}}})$ contains no club, i.e., iff for every club $C\subseteq\kappa$, $\SP{\kappa}{[\kappa]^2 \mathbin{\upharpoonright} C,\ensuremath{\mathsf{nonempty}}}$ fails.
\end{lemma}
\begin{proof}
Let $\kappa$ be subtle, and suppose $\SP{\kappa}{[\kappa]^2\mathbin{\upharpoonright} C,\ensuremath{\mathsf{nonempty}}}$ held for some club $C\subseteq\kappa$. Let $\vec{d}$ be a $C$-list witnessing this. The proof of Theorem \ref{theorem:GeneralABranchesForKappaListsAndGeneralIneffability} then relativizes to $C$ and shows that $\vec{d}$ has no $[\kappa]^2\mathbin{\upharpoonright} C$ branch. This means that there is no two-element subset of $C$ on which $\vec{d}$ coheres, contradicting our assumption that $\kappa$ is subtle.
Vice versa, if $\kappa$ is not subtle, then there is a club $C$ and a $C$-list $\vec{d}$ such that $\vec{d}$ does not cohere on any two-element subset of $C$, i.e., $\vec{d}$ has no $[\kappa]^2\mathbin{\upharpoonright} C$-branch, and again, by (a relativized version of) Theorem \ref{theorem:GeneralABranchesForKappaListsAndGeneralIneffability},
this is equivalent to $\SP{\kappa}{[\kappa]^2 \mathbin{\upharpoonright} C,\ensuremath{\mathsf{nonempty}}}$.
\end{proof}
Note that the family of nonempty subsets of $\kappa$ is not independent of initial segments, and so Observation \ref{observation:IndependentOfInitialSegmentsGivesSplitEquivalentToGeneralSplit} does not apply, and we do not know that $\SP{\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$ is equivalent to $\SP{\kappa,\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$. The latter is equivalent to $\mathfrak{s}_{\mathcal{A},\ensuremath{\mathsf{nonempty}}}(\kappa)>\kappa$. If every $A\in\mathcal{A}$ has at least two elements, then $\SP{\kappa,\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$ holds, as witnessed by the $(\mathcal{A},\ensuremath{\mathsf{nonempty}})$-splitting family $\{\{\alpha\}\; | \;\alpha\in\kappa\}$, while $\SP{\kappa}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$ characterizes the failure of $\kappa$ being $\mathcal{A}$-ineffable. The following theorem will allow us to characterize ineffability by split principles that can be expressed as statements about splitting numbers.
\begin{theorem}
\label{theorem:CharacterizationOfIneffability}
Let $\kappa$ be a regular cardinal. The following are equivalent.
\begin{enumerate}
\item $\kappa$ is ineffable.
\item $\SP{\kappa}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ fails.
\item $\SP{\kappa}{\ensuremath{\mathsf{stationary}},\mathcal{U}nbounded}$ fails. Equivalently, $\mathfrak{s}_{\ensuremath{\mathsf{stationary}},\mathcal{U}nbounded}(\kappa)>\kappa$.
\item $\SP{\kappa}{\ensuremath{\mathsf{stationary}}}$ fails. Equivalently, $\mathfrak{s}_\ensuremath{\mathsf{stationary}}(\kappa)>\kappa$.
\end{enumerate}
It follows that for any family $\mathcal{B}\subseteq\power{\kappa}$ satisfying $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$, the failure of $\SP{\kappa}{\ensuremath{\mathsf{stationary}},\mathcal{B}}$ characterizes ineffability.
\end{theorem}
\begin{proof}
(1)$\iff$(2) follows from Theorem \ref{theorem:GeneralABranchesForKappaListsAndGeneralIneffability}.
(2)$\implies$(3)$\implies$(4) is trivial
For (4)$\implies$(1), let $\vec{d}$ be a $\kappa$-list. We have to find an ineffable branch. Since $\vec{d}$ is not a $\SP{\kappa}{\ensuremath{\mathsf{stationary}}}$ sequence, it follows that there is a stationary set $S\subseteq\kappa$ such that for no $\beta<\kappa$ do we have that both $S^+_\beta=S^+_{\beta,\vec{d}}$ and $S^-_\beta=S^-_{\beta,\vec{d}}$ are stationary in $\kappa$. But clearly, one of them is. For each $\beta< \kappa$ let $C_\beta$ be club in $\kappa$ and disjoint from the nonstationary one of $S^+_\beta$ and $S^-_\beta$. Let $C=\dintersection_{\beta<\kappa}C_\beta$. Let $T=S\mathfrak{c}ap C$. Then $\vec{d}$ coheres on the stationary set $T$, because if $\gamma<\delta$ both are members of $T$, then for $\xi<\gamma$, it follows that $\gamma,\delta\in C_\xi$, so if $S^-_\xi$ is nonstationary, then $\gamma,\delta\in S^+_\xi$, which means that $\xi\in d_\gamma$ and $\xi\in d_\delta$. And if $S^+_\xi$ is nonstationary, then $\gamma,\delta\in S^-_\xi$ and it follows that $\xi\notin d_\gamma$ and $\xi\notin d_\delta$. So $d_\gamma=d_\delta\mathfrak{c}ap\gamma$. Thus, $b=\bigcup_{\alpha\in T}d_\alpha$ is an ineffable branch.
The statements about the splitting numbers follow because the families we are splitting into are independent of initial segments.
\end{proof}
\section{Splitting subsets of $\ensuremath{\mathcal P_\kappa \lambda}$}
\label{sec:SplittingSubsetsOfPkl}
We may generalize the split principles to the context of $\ensuremath{\mathcal P_\kappa \lambda}$-lists, and assume for the present section that $\kappa$ is regular and that $\lambda>\kappa$ is a cardinal. $\ensuremath{\mathcal P_\kappa \lambda}$-lists are sequences of the form $\seq{ d_x }{ x \in \ensuremath{\mathcal P_\kappa \lambda} }$ satisfying $d_x \subseteqseteq x$ for each $x \in \ensuremath{\mathcal P_\kappa \lambda}$.
We use Jech's approach to stationary subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Thus, a set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ is called \emph{unbounded} if for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, there is a $y\in U$ with $x\subseteq y$. A set $C\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ is \textit{club} if it is unbounded and closed under unions of increasing chains of length less than $\kappa$, and a set $S\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ is \textit{stationary} iff it intersects every club subset of $\ensuremath{\mathcal P_\kappa \lambda}$, see \mathfrak{c}ite[Def.~8.21]{ST3}. If $f:[\lambda]^n\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$, for some $n<\omega$, or $f:[\lambda]^{{<}\omega}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$, then we write $\mathcal{C}_f$ for the set $\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\forall a\in [x]^{<\omega}\mathfrak{c}ap\dom(f)\quad f(a)\subseteq x\}$. It was shown by Menas~\mathfrak{c}ite[Thm.~1.5]{Menas1974:StrongAndSuper} that for every club subset $C$ of $\ensuremath{\mathcal P_\kappa \lambda}$, there is a function $f:[\lambda]^{2}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that $\mathcal{C}_f\setminus\{\emptyset\}\subseteq C$. Since $\mathcal{C}_f$ is itself club, the club filter on $\ensuremath{\mathcal P_\kappa \lambda}$ is generated by the sets of the form $\mathcal{C}_f$, and as a result, a subset $S$ of $\ensuremath{\mathcal P_\kappa \lambda}$ is stationary iff it intersects every set of the form $\mathcal{C}_f$.
\begin{definition}
\label{definition:PklSplitPrinciple}
Let $\kappa$ be regular and $\lambda>\kappa$ be a cardinal. Let $\mathcal{A}$ and $\mathcal{B}$ be families of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. The principle $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$ says that there is a $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec{d}=\seq{d_x}{x\in\ensuremath{\mathcal P_\kappa \lambda}}$ that splits $\mathcal{A}$ into $\mathcal{B}$, meaning that for every $A\in\mathcal{A}$, there is a $\beta<\lambda$ such that both $A^+_{\beta, \vec d} = A^+_\beta =\{x\in A\; | \;\beta\in d_x\}$ (note that $\beta\in d_x\implies \beta\in x$) and $A^-_{\beta, \vec d}=A^-_\beta=\{x\in A\; | \;\beta\in x\setminus d_x\}$ are in $\mathcal{B}$. We write $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A}}$ for $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{A}}$.
Following Carr's notation, for $x\in\ensuremath{\mathcal P_\kappa \lambda}$, let $\widehat{x}=\{y\in\ensuremath{\mathcal P_\kappa \lambda}\; | \; x\subseteq y\}$. \end{definition}
We will also use generalized two-cardinal versions of the split principles, where we do not insist that the sequences are $\ensuremath{\mathcal P_\kappa \lambda}$-lists, as follows. As with the original split principle, these have a close connection to splitting families and splitting numbers.
\begin{definition}
\label{definition:GeneralNonsenseOnPklSplitting}
Let $\kappa$ be regular, $\lambda>\kappa$ a cardinal, and $\tau$ a cardinal. Let $\mathcal{A}$ and $\mathcal{B}$ be families of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Define $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ as before, i.e., it says that there is a sequence $\vec{d}=\seq{d_x}{x\in\ensuremath{\mathcal P_\kappa \lambda}}$ of subsets of $\tau$ that splits $\mathcal{A}$ into $\mathcal{B}$, meaning that for every $A\in\mathcal{A}$, there is a $\beta<\tau$ such that both ${\tilde{A}}^+_\beta = {\tilde{A}}^+_{\beta,\vec d} =\{x\in A\; | \;\beta\in d_x\}$ and ${\tilde{A}}^-_\beta= {\tilde{A}}^-_{\beta, \vec d} =\{x\in A\; | \;\beta\notin d_x\}$ are in $\mathcal{B}$.
Similarly, define that a collection $\mathcal{S}$ of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ is an $(\mathcal{A},\mathcal{B})$-\emph{splitting family} for $\ensuremath{\mathcal P_\kappa \lambda}$, if for every $A\in\mathcal{A}$ there is an $S\in\mathcal{S}$ such that $A\mathfrak{c}ap S$ and $A\setminus S$ are in $\mathcal{B}$ ($\mathcal{S}$ splits $\mathcal{A}$ into $\mathcal{B}$). The splitting number $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\ensuremath{\mathcal P_\kappa \lambda})$ is the smallest cardinality of an $(\mathcal{A},\mathcal{B})$-splitting family.
\end{definition}
The following lemma says that the generalized split principles on $\ensuremath{\mathcal P_\kappa \lambda}$ can be viewed as statements about the sizes of the corresponding splitting numbers.
\begin{lemma}
\label{lemma:SNleLambdaEquivalentToPKappaLambdaSplit-General}
Let $\kappa<\lambda$ be cardinals, $\tau$ a cardinal, and let $\mathcal{A}$, $\mathcal{B}$ be families of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Then $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ holds iff $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\ensuremath{\mathcal P_\kappa \lambda})\le\tau$.
\end{lemma}
\noindent\emph{Note:} In other words, $\mathfrak{s}_{\mathcal{A},\mathcal{B}}(\ensuremath{\mathcal P_\kappa \lambda})$ is the least $\tau$ such that $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ holds.
\begin{proof} For the direction from right to left, if $\mathcal{S}=\{S_\alpha\; | \;\alpha<\tau\}$ is an $(\mathcal{A},\mathcal{B})$-splitting family for $\ensuremath{\mathcal P_\kappa \lambda}$, then we can define a sequence $\kla{d_x\; | \; x\in\ensuremath{\mathcal P_\kappa \lambda}}$ of subsets of $\tau$ by setting
\[d_x=\{\gamma<\tau\; | \; x\in S_\gamma\}.\]
Then $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ sequence, because if $A\in\mathcal{A}$, then there is a $\beta<\tau$ such that both $A\mathfrak{c}ap S_\beta$ and $A\setminus S_\beta$ belong to $\mathcal{B}$, but $A\mathfrak{c}ap S_\beta={\tilde{A}}^+_{\beta,\vec{d}}$ and $A\setminus S_\beta={\tilde{A}}^-_{\beta,\vec{d}}$, so we are done.
Conversely, if $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ sequence, then for each $\gamma<\tau$, we define a subset $S_\gamma$ of $\ensuremath{\mathcal P_\kappa \lambda}$ by
\[S_\gamma=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\gamma\in d_x\}.\]
Then $\mathcal{S}=\{S_\gamma\; | \;\gamma<\tau\}$ is an $(\mathcal{A},\mathcal{B})$-splitting family for $\ensuremath{\mathcal P_\kappa \lambda}$, because if $A\in\mathcal{A}$, then there is a $\beta<\tau$ such that both ${\tilde{A}}^+_{\beta,\vec{d}}$ and ${\tilde{A}}^-_{\beta,\vec{d}}$ are in $\mathcal{B}$, but as before, ${\tilde{A}}^+_{\beta,\vec{d}}=A\mathfrak{c}ap S_\beta$ and ${\tilde{A}}^-_\beta=A\setminus S_\beta$. \end{proof}
The following are the families of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ we will mostly be working with as our collections $\mathcal{A}$ and $\mathcal{B}$ with split principles and splitting numbers.
\begin{definition}
\label{definition:FamiliesOfSubsetsOfPkl} \mbox{}
\begin{itemize}
\item $\mathcal{U}nbounded$ is the set of unbounded subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ (note that ``unbounded'' is not the same as ``not bounded'' - the more correct term would be ``cofinal'', but ``unbounded'' is the commonly accepted term. So $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ is unbounded iff for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$ there is a $y\in\ensuremath{\mathcal P_\kappa \lambda}$ with $x\subseteq y$.)
\item $\ensuremath{\mathsf{covering}}$ is the set of $A\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that $\mathfrak{c}up A=\lambda$, i.e., for every $\xi<\lambda$, there is an $x\in A$ with $\xi\in x$ (note that in the space $\kappa$ rather than $\ensuremath{\mathcal P_\kappa \lambda}$, not bounded, unbounded and covering are equivalent).
\item $\ensuremath{\mathsf{stationary}}$ is the collection of stationary subsets of $\ensuremath{\mathcal P_\kappa \lambda}$.
\item $\ensuremath{\mathsf{nonempty}}$ is the collection of nonempty subsets of $\ensuremath{\mathcal P_\kappa \lambda}$.
\end{itemize}
\end{definition}
\begin{observation}
\label{observation:WhenWeCanUsePklLists}
Let $\tau\le\kappa\le\lambda$ be cardinals, and let $\mathcal{A}$ and $\mathcal{B}$ be collections of subsets of $\kappa$ such that for all $B\subseteq\kappa$ and all $\beta<\tau$, $B\in\mathcal{B}$ iff $B\mathfrak{c}ap\widehat{\{\beta\}}\in\mathcal{B}$ (``$\mathcal{B}$ is independent of initial segments''). Then $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$ is equivalent to the existence of a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$-sequence $\vec{d}$ of subsets of $\tau$.
\end{observation}
\begin{proof} If $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\tau}{\mathcal{A},\mathcal{B}}$-sequence, then the $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec{e}$ defined by $e_x=d_x\mathfrak{c}ap x$ is a sequence of subsets of $\tau$ that is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$-sequence.
This is because if $A\in\mathcal{A}$ and $\beta<\tau$ is such that ${\tilde{A}}^+_{\beta,\vec{d}}$ and ${\tilde{A}}^-_{\beta,\vec{d}}$ are in $\mathcal{B}$, then $A^+_{\beta,\vec{e}}\mathfrak{c}ap\widehat{\{\beta\}}={\tilde{A}}^+_{\beta,\vec{d}}\mathfrak{c}ap\widehat{\{\beta\}}$, and similarly, $A^-_{\beta,\vec{e}}\mathfrak{c}ap\widehat{\{\beta\}}={\tilde{A}}^-_{\beta,\vec{d}}\mathfrak{c}ap\widehat{\{\beta\}}$. It follows from our assumption on $\mathcal{B}$ that $A^+_{\beta,\vec{e}}$ and $A^-_{\beta,\vec{e}}$ are in $\mathcal{B}$. \end{proof}
Note that $\mathcal{U}nbounded$ and $\ensuremath{\mathsf{stationary}}$ are independent of initial segments, while $\ensuremath{\mathsf{covering}}$ and $\ensuremath{\mathsf{nonempty}}$ are not.
\begin{observation}
Let $\mathcal{A},\mathcal{B}\subseteq\power{\kappa}$, and let $\mathcal{B}$ be closed under supersets (i.e., if $B\in\mathcal{B}$ and $B\subseteq C\subseteq\ensuremath{\mathcal P_\kappa \lambda}$, then $C\in\mathcal{B}$). Then every $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$-sequence is a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\lambda}{\mathcal{A},\mathcal{B}}$-sequence.
\end{observation}
\begin{proof} Let $\vec{d}$ be a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$-sequence. For any $A\in\mathcal{A}$ and any $\beta<\lambda$, $A^+_{\vec{d},\beta}={\tilde{A}}^+_{\vec{d},\beta}$ and $A^-_{\vec{d},\beta}\subseteq{\tilde{A}}^-_{\vec{d},\beta}$. So since $\mathcal{B}$ is closed under supersets, it follows that $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\lambda}{\mathcal{A},\mathcal{B}}$-sequence. \end{proof}
\begin{observation}
\label{observation:FromCoveringToNonempty}
Let $\mathcal{A},\mathcal{B}\subseteq\power{\kappa}$ such that $\mathcal{B}\subseteq\ensuremath{\mathsf{covering}}$. Then the principle $\SP{\ensuremath{\mathcal P_\kappa \lambda},\lambda}{\mathcal{A},\mathcal{B}}$ implies $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$.
\end{observation}
\begin{proof}
Let $\vec{d}$ be a $\SP{\ensuremath{\mathcal P_\kappa \lambda},\lambda}{\mathcal{A},\mathcal{B}}$ sequence. Let $e_x=d_x\mathfrak{c}ap x$, for $x\in\ensuremath{\mathcal P_\kappa \lambda}$. We show that $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}$ sequence: let $A\in\mathcal{A}$. Let $\beta<\lambda$ be such that ${\tilde{A}}^+_\beta$ and ${\tilde{A}}^-_\beta$ both are in $\mathcal{B}$. In particular, both of these sets are covering. So let $x\in{\tilde{A}}^+_\beta$ and let $y\in{\tilde{A}}^-_\beta$. Then $x\in A$, and $\beta\in d_x\mathfrak{c}ap x=e_x$, so $x\in A^+_{\beta,\vec{e}}$, and $y\in A$, and $\beta\in y\setminus d_y=y\setminus e_y$, so $y\in A^-_{\beta,\vec{e}}$.
\end{proof}
The terminology introduced in the next definition follows \mathfrak{c}ite{Weiss:Diss}. It is the concept corresponding to sequential binary trees from the previous section.
\begin{definition}
A \emph{forest} on $\ensuremath{\mathcal P_\kappa \lambda}$ is a set $\mathcal{F}\subseteq\set{{}^{x}2}{x\in\ensuremath{\mathcal P_\kappa \lambda}}$ such that for every $f\in\mathcal{F}$, if $x\subseteq\dom(f)$, then $f\mathbin{\upharpoonright} x\in\mathcal{F}$, and such that for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, there is an $f\in\mathcal{F}$ such that $x=\dom(f)$. A cofinal branch through $\mathcal{F}$ is a function $B:\lambda\longrightarrow 2$ such that for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, $B\mathbin{\upharpoonright} x\in\mathcal{F}$.
\end{definition}
In \mathfrak{c}ite{CombProblems}, forests were referred to as \emph{binary $(\kappa,\lambda)$-messes}, and cofinal branches through forests were called \emph{solutions to binary messes}. Clearly, there is an obvious way to assign forests to lists.
\begin{definition}
Given a $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec d = \seq{ d_x }{ x \in \ensuremath{\mathcal P_\kappa \lambda} }$, for each $x \in \ensuremath{\mathcal P_\kappa \lambda}$ let $d_x^c : x \to 2$ denote the characteristic function of $d_x$.
By closing these characteristic functions downward, we may consider the forest $\mathcal F_{\vec d}$ corresponding to the $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec d$, defined by
$\mathcal F_{\vec d} = \set{ d_x^c \mathbin{\upharpoonright} y }{ y \subseteqseteq x \in \ensuremath{\mathcal P_\kappa \lambda} }.$
\end{definition}
Below we define several types of branches for $\ensuremath{\mathcal P_\kappa \lambda}$-lists, similar to our treatment of $\kappa$-lists. Some of our terminology is inspired by work of DiPrisco, Zwicker and Carr. We introduce the notion of wild ineffability here, which appears to be new.
\begin{definition}
\label{definition:BranchesOfPklLists}
Let $\vec d = \seq{ d_x }{ x \in \ensuremath{\mathcal P_\kappa \lambda} }$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list.
A set $B \subseteqseteq \lambda$ is a \emph{cofinal branch} through $\vec{d}$ so long as for all $x \in \ensuremath{\mathcal P_\kappa \lambda}$, there is some $y \in \ensuremath{\mathcal P_\kappa \lambda}$ with $y\supseteq x$ such that
$d_y \mathfrak{c}ap x=B \mathfrak{c}ap x.$
The \emph{branch property} $\BP(\kappa, \lambda)$ holds iff every $\ensuremath{\mathcal P_\kappa \lambda}$-list has a cofinal branch, and $\kappa$ is \emph{mildly $\lambda$-ineffable} iff $\BP(\kappa,\lambda)$ holds (the origin of mild ineffability is \mathfrak{c}ite{DiPriscoZwicker:FlippingAndSC}).
A cofinal branch $B$ is $\emph{guided}$ by a set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ if for all $x \in \ensuremath{\mathcal P_\kappa \lambda}$ there is a $y \supseteq x$ such that for all $z \supseteq y$ with $z \in U$,
$d_z \mathfrak{c}ap x = B \mathfrak{c}ap x.$
A cofinal branch $B$ is \emph{strong} if it is guided by an unbounded set.
If every $\ensuremath{\mathcal P_\kappa \lambda}$-list has a strong branch, then the \emph{strong branch property} $\SBP(\kappa,\lambda)$ holds, and we say that $\kappa$ is \emph{wildly} $\lambda$-ineffable.
An \emph{almost ineffable branch} through $\vec{d}$ is a subset $B\subseteq\lambda$ such that there is an unbounded set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x\in U$, $d_x=B\mathfrak{c}ap x.$ If every $\ensuremath{\mathcal P_\kappa \lambda}$-list has an almost ineffable branch, then $\AIBP(\kappa,\lambda)$ holds, and $\kappa$ is \emph{almost $\lambda$-ineffable.}
An \emph{ineffable branch} through $\vec{d}$ is a subset $B \subseteqseteq \lambda$ which comes with a stationary set $S \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x \in S$,
$d_x = B \mathfrak{c}ap x.$
If every $\ensuremath{\mathcal P_\kappa \lambda}$-list has an ineffable branch, then $\IBP(\kappa, \lambda)$ holds, and $\kappa$ is \emph{$\lambda$-ineffable.}
In general, if $\mathcal{A}$ is a family of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ and $B\subseteq\lambda$, then we say that $B$ is an $\mathcal{A}$ branch of $\vec{d}$ if there is a set $A\in\mathcal{A}$ such that for all $x\in A$, $d_x=B\mathfrak{c}ap x$.
If $\mathcal{F}=\mathcal{F}_{\vec{d}}$ is the forest corresponding to $\vec{d}$, then we refer to a function $B:\lambda\longrightarrow 2$ as a (cofinal, strong, almost ineffable, ineffable) branch of $\mathcal{F}$ if the set $\{\alpha<\lambda\; | \; B(\alpha)=1\}$ is a (cofinal, strong, almost ineffable, ineffable) branch through $\vec{d}$.
\end{definition}
\begin{observation}
\label{observation:RelationshipsBetweenBranches}
Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list, and let $B\subseteq\lambda$. If $B$ is an ineffable branch then it is an almost ineffable branch. If it is an almost ineffable branch, then it is a strong branch. If it is a strong branch, then it is a cofinal branch. So we have the following string of implications: $\IBP(\kappa, \lambda) \implies \AIBP(\kappa,\lambda)\implies\SBP(\kappa, \lambda) \implies \BP(\kappa, \lambda)$.
\end{observation}
\subseteqsection{Split characterizations of two cardinal versions of ineffability}
The next goal is to establish characterizations of the classical two cardinal versions of ineffability and almost ineffability. First, let us state a very general theorem.
\begin{theorem}
\label{theorem:GeneralABranchesForPklLists}
Let $\kappa\le\lambda$ be cardinals, $\mathcal{A}$ a family of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ and $\vec{d}$ a $\ensuremath{\mathcal P_\kappa \lambda}$-list. Then the following are equivalent:
\begin{enumerate}
\item $\vec{d}$ is a \SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}-sequence.
\item $\vec{d}$ has no $\mathcal{A}$ branch.
\end{enumerate}
\end{theorem}
\begin{proof}
(1)$\implies$(2): Suppose $B$ is an $\mathcal{A}$ branch for $\vec{d}$. Let $A\in\mathcal{A}$ be such that for all $x\in A$, $d_x=B\mathfrak{c}ap x$. Let $\beta<\lambda$ be such that both $A^+_\beta$ and $A^-_\beta$ are nonempty. Let $x\in A^+_\beta$ and $y\in A^-_\beta$. Then $\beta\in x$, $x\in A$, $\beta\in d_x$ and $d_x=B\mathfrak{c}ap x$, so $\beta\in B$. On the other hand, $\beta\in y$, $y\in A$, $\beta\notin d_y$ and $d_y=B\mathfrak{c}ap y$, so $\beta\notin B$. This is a contradiction.
(2)$\implies$(1): We show the contrapositive. So assuming $\vec{d}$ is not a \SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\ensuremath{\mathsf{nonempty}}}-sequence, we have to show that it has an $\mathcal{A}$ branch. Let $A\in\mathcal{A}$ be such that no $\beta<\lambda$ splits $A$ into nonempty sets. So for every $\beta<\lambda$, it is not the case that both $A^+_\beta$ and $A^-_\beta$ are nonempty. Set
\[B=\{\beta<\lambda\; | \; A^+_\beta\neq\emptyset\}\]
It follows that for every $x\in A$, $d_x=B\mathfrak{c}ap x$ (and hence that $B$ is an $\mathcal{A}$ branch). To see this, let $x\in A$, and let $\beta\in x$. If $\beta\in B$, then $A^+_\beta\neq\emptyset$, so $A^-_\beta=\emptyset$. It follows that $\beta\in d_x$ (because if we had $\beta\notin d_x$, it would follow that $x\in A^-_\beta$). And if $\beta\notin B$, then $A^+_\beta=\emptyset$. Since $\beta\in x$ and $x\in A$, it follows that $\beta\notin d_x$, because if we had $\beta\in d_x$, then it would follow that $x\in A^+_\beta=\emptyset$. This shows that $d_x=B\mathfrak{c}ap x$, as claimed.
\end{proof}
The following lemma is an immediate consequence.
\begin{lemma}
Let $\kappa\le\lambda$.
\begin{enumerate}
\item $\kappa$ is almost $\lambda$-ineffable iff $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded,\ensuremath{\mathsf{nonempty}}}$ fails.
\item $\kappa$ is $\lambda$-ineffable iff $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ fails.
\end{enumerate}
\end{lemma}
It turns out that the characterization of ineffability by the failure of split principles of the form $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\mathcal{B}}$ is very robust.
\begin{theorem}
\label{theorem:StationaryPklSplitIffNoIneffableBranch}
Let $\kappa$ be regular and uncountable, and $\lambda\ge\kappa$ be a cardinal.
Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list. The following are equivalent:
\begin{enumerate}
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}}}$ sequence.
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\mathcal{U}nbounded}$ sequence.
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{covering}}}$ sequence.
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ sequence.
\item $\vec{d}$ has no ineffable branch.
\end{enumerate}
In general, these are equivalent to saying that $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\mathcal{B}}$ sequence whenever $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$. \end{theorem}
\begin{proof}
(1)$\implies$(2)$\implies$(3)$\implies$(4)~is immediate, because every stationary set is unbounded, every unbounded set is covering, and every covering set is nonempty.
(4)$\implies$(5)~follows from Theorem \ref{theorem:GeneralABranchesForPklLists}.
For (5)$\implies$(1),
we prove the contrapositive, i.e., assuming $\vec{d}$ does not split stationary sets into stationary sets, we show that $\vec{d}$ has an ineffable branch.
Let $S \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$ be a stationary set such that for each $\beta < \lambda$, not both $S^+_\beta$ and $S^-_\beta$ are stationary in $\ensuremath{\mathcal P_\kappa \lambda}$. Since $S\mathfrak{c}ap\widehat{\{\beta\}}= S^+_\beta \mathfrak{c}up S^-_\beta$, this means that exactly one of them is stationary (for each $\beta$). Define
\[B=\{\beta<\lambda\; | \; S^+_\beta\ \text{is stationary}\}.\]
Let $C_\beta$ be club in $\ensuremath{\mathcal P_\kappa \lambda}$ and disjoint from the nonstationary one of $S^+_\beta$ and $S^-_\beta$. Let $D=\dintersection_{\beta<\lambda}C_\beta=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\forall\beta\in x\quad x\in C_\beta\}$. Then $D$ is club, and so, $E=S\mathfrak{c}ap D$ is stationary. But $\vec{d}$ coheres with $B$ on $E$: let $x\in E$. We have to show that $B\mathfrak{c}ap x=d_x$. So let $\beta\in x$. Then $x\in C_\beta$. If $\beta\in B$, then $S^+_\beta$ is stationary, so $C_\beta\mathfrak{c}ap S^-_\beta=\emptyset$, and so, $x\in S^+_\beta$ (since $\beta\in x$), which means that $\beta\in d_x$. And if $\beta\notin B$, then $S^-_\beta$ is stationary, and it follows that $x\in S^-_\beta$, so $\beta\notin d_x$. This shows that $B$ is an ineffable branch.
\end{proof}
As an immediate consequence, we get:
\begin{lemma}
If $\kappa$ is regular and uncountable, and $\lambda\ge\kappa$ is a cardinal, then the following are equivalent:
\begin{enumerate}
\item $\kappa$ is $\lambda$-ineffable.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}}}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\mathcal{U}nbounded}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{covering}}}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\ensuremath{\mathsf{nonempty}}}$ fails.
\end{enumerate}
In general, these are equivalent to saying that $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{stationary}},\mathcal{B}}$ fails whenever $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$.
\end{lemma}
The previous two facts go through in more generality:
\begin{theorem}
\label{theorem:NormalIdeal+PklSplitIffNoIneffableBranch}
Let $\kappa$ be regular and uncountable, and $\lambda\ge\kappa$ be a cardinal.
Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list, and let $\mathcal{I}$ be a normal ideal on $\ensuremath{\mathcal P_\kappa \lambda}$. The following are equivalent:
\begin{enumerate}
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{I}^+}$ sequence.
\item $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{I}^+,\ensuremath{\mathsf{nonempty}}}$ sequence.
\item $\vec{d}$ has no $\mathcal{I}^+$ branch.
\end{enumerate}
In general, these are equivalent to saying that $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{I}^+,\mathcal{B}}$ sequence whenever $\mathcal{I}^+\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$. \end{theorem}
\begin{proof}
(1)$\implies$(2) is immediate, and (2)$\implies$(3)~follows from Theorem \ref{theorem:GeneralABranchesForPklLists}.
For (3)$\implies$(1),
we prove the contrapositive, i.e., assuming $\vec{d}$ does not split $\mathcal{I}$-positive sets into sets in $\mathcal{I}^+$, we show that $\vec{d}$ has an $\mathcal{I}^+$ branch.
Let $S\in\mathcal{I}^+$ be such that for each $\beta < \lambda$, not both $S^+_\beta$ and $S^-_\beta$ are in $\mathcal{I}^+$.
Note that $\widehat{\{\beta\}}\in\mathcal{I}^*$: this is because $\widehat{\{\beta\}}$ is club, and the club filter is the minimal normal filter (see \mathfrak{c}ite{Carr:Diss}), and $\mathcal{I}^*$ is a normal filter.
It follows that for each $\beta<\kappa$, $S\mathfrak{c}ap\widehat{\{\beta\}}\in\mathcal{I}^+$ (this is equivalent to saying that $S\mathfrak{c}ap\widehat{\{\beta\}}\notin\mathcal{I}$). To see this, suppose instead we had $S\mathfrak{c}ap\widehat{\{\beta\}}\in\mathcal{I}$. As $\ensuremath{\mathcal P_\kappa \lambda}\setminus\widehat{\{\beta\}}\in\mathcal{I}$ also $(S\mathfrak{c}ap\widehat{\{\beta\}})\mathfrak{c}up(\ensuremath{\mathcal P_\kappa \lambda}\setminus\widehat{\{\beta\}})\in\mathcal{I}$, but $S$ is a subset of that, so $S\in \mathcal{I}$, a contradiction.
Since $S\mathfrak{c}ap\widehat{\{\beta\}}= S^+_\beta \mathfrak{c}up S^-_\beta$, this means that exactly one of $S^+_\beta$, $S^-_\beta$ is in $\mathcal{I}^+$ (for each $\beta$) - we know that not both of them are in $\mathcal{I}^+$. If neither of them were in $\mathcal{I}^+$, then both would be in $\mathcal{I}$, but then their union would also be in $\mathcal{I}$.
Define
\[B=\{\beta<\lambda\; | \; S^+_\beta\ \text{is in $\mathcal{I}^+$}\}.\]
We show that $B$ is an $\mathcal{I}$ branch as follows. Let $C_\beta$ be in $\mathcal{I}^*$ and disjoint from the one of $S^+_\beta$ and $S^-_\beta$ that's in $\mathcal{I}$. By normality, $D=\dintersection_{\beta<\lambda}C_\beta=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\forall\beta\in x\quad x\in C_\beta\}$ is in $\mathcal{I}^*$, and so, $E=S\mathfrak{c}ap D$ is in $\mathcal{I}^+$. But $\vec{d}$ coheres with $B$ on $E$: let $x\in E$. We have to show that $B\mathfrak{c}ap x=d_x$. So let $\beta\in x$. Then $x\in C_\beta$. If $\beta\in B$, then $S^+_\beta$ is in $\mathcal{I}^+$, $C_\beta\mathfrak{c}ap S^-_\beta=\emptyset$, and so, $x\in S^+_\beta$ (since $\beta\in x$), which means that $\beta\in d_x$. And if $\beta\notin B$, then $S^-_\beta$ is in $\mathcal{I}^+$, and it follows that $x\in S^-_\beta$, so $\beta\notin d_x$. This shows that $B$ is an $\mathcal{I}^+$ branch.
\end{proof}
It turns out that the failure of split principles of the form $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded,\mathcal{B}}$ is less robust. We have seen that if $\mathcal{B}=\ensuremath{\mathsf{nonempty}}$, the principle characterizes almost ineffability. The following theorem explores the other extreme, $\mathcal{B}=\mathcal{U}nbounded$.
\begin{theorem}
\label{theorem:PklSplitIffNotSBP}
Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list. Then $\vec{d}$ is a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ sequence iff $\vec{d}$ does not have a strong branch.
\end{theorem}
\begin{proof} We will show each direction of the implication separately.
$\Longrightarrow$:
Towards a contradiction, assume $B$ is a strong branch through $\vec{d}$, guided by the unbounded set $U \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$. Now define a function $f$ so that for all $x \in \ensuremath{\mathcal P_\kappa \lambda}$, $f(x)\supseteq x$, $f(x)\in U$ and for all $z \supseteq f(x)$, if $z \in U$ then $d_z \mathfrak{c}ap x = B \mathfrak{c}ap x$.
Consider the unbounded set $A = f``\,\ensuremath{\mathcal P_\kappa \lambda}\subseteq U$, and let $\beta$ split $A$ into unbounded sets, with respect to $\vec{d}$.
Suppose that $\beta\in B$. Now we may choose $y \in A^-_\beta$ such that $y \supseteq f(\{ \beta \})$, since $ A^-_\beta$ is unbounded. Then $y\in U$, and by the definition of $f(\{ \beta \})$ we have that $d_y \mathfrak{c}ap \{ \beta \} = B \mathfrak{c}ap \{ \beta \}$. Since $\beta\in B$, this means that $\beta\in d_y$, but since $y\in A^-_\beta$, $\beta\notin d_y$, a contradiction. The case $\beta\notin B$ works similarly by picking some $y \supseteq f(\{ \beta \})$ with $y \in A^+_\beta$.
$\Longleftarrow$: We will show the contrapositive.
So suppose $\vec d = \seq{ d_x }{ x \in \ensuremath{\mathcal P_\kappa \lambda} }$
does not split unbounded sets.
We claim that $\vec{d}$ has a strong branch. Since $\vec{d}$ does not split unbounded sets, there is an unbounded $A \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$ such that for each $\beta < \lambda$, exactly one of $A^+_\beta$ and $A^-_\beta$ is unbounded in $\ensuremath{\mathcal P_\kappa \lambda}$. Define a strong branch $B\subseteq\lambda$ by setting
\[\beta\in B\iff A^+_\beta\ \text{is unbounded in $\ensuremath{\mathcal P_\kappa \lambda}$}.\]
We claim that $B$ is a strong branch, guided by the unbounded set $A$. To see this, fix $x \in \ensuremath{\mathcal P_\kappa \lambda}$. For each element $\beta < \lambda$ of $x$, note that there has to be a $y_\beta \in \ensuremath{\mathcal P_\kappa \lambda}$ such that either for all $z \supseteq y_\beta$, if $z \in A$ then $\beta \in d_z$; or for all $z \supseteq y_\beta$, if $z \in A$ then $\beta \notin d_z$. The former holds if $A^-_\beta$ is not unbounded, and the latter holds if $A^+_\beta$ is not unbounded. Since one of those statements has to be true, $y_\beta$ is defined for each $\beta < \lambda$. Let $y=\bigcup_{\beta \in x} y_\beta$. Since $\kappa$ is regular, $y \in \ensuremath{\mathcal P_\kappa \lambda}$. Pick $z \supseteq y$ such that $z \in A$. Now $B \mathfrak{c}ap x = d_z \mathfrak{c}ap x$ as desired.
\end{proof}
\begin{corollary}
\label{corollary:NotPklSplitIffForestsHaveBranches}
The following are equivalent:
\begin{enumerate}
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ fails.
\item $\kappa$ is wildly $\lambda$-ineffable.
\item Every forest on $\ensuremath{\mathcal P_\kappa \lambda}$ has a strong branch.
\end{enumerate}
\end{corollary}
\begin{proof} (1)~and (2)~are obviously equivalent.
For (3)$\implies$(1), assume the contrary, and let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list that splits unbounded sets into unbounded sets. Then its forest $\mathcal{F}_{\vec{d}}$ does not have a strong branch, by Theorem \ref{theorem:PklSplitIffNotSBP}. This contradicts the assumption that every forest on $\ensuremath{\mathcal P_\kappa \lambda}$ has a strong branch.
For (1)$\implies$(3), given a forest $\mathcal{F}$, for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$ choose a function $f_x\in\mathcal{F}$ with $\dom(f_x)=x$, and let $d_x=\{\gamma\in x\; | \; f_x(\gamma)=1\}$. Then $\vec{d}$ is a $\ensuremath{\mathcal P_\kappa \lambda}$-list, and since $\Split{\ensuremath{\mathcal P_\kappa \lambda}}$ fails, $\vec{d}$ does not split unbounded sets, so that by Theorem \ref{theorem:PklSplitIffNotSBP}, $\vec{d}$ has a strong branch. Letting $B$ be the characteristic function of this strong branch, it follows that $B$ is a strong branch of $\mathcal{F}_{\vec{d}}$, and since $\mathcal{F}_{\vec{d}}\subseteq\mathcal{F}$, $B$ is also a strong branch of $\mathcal{F}$. \end{proof}
However, the exact relationship between the existence of a strong branch and a cofinal branch for $\ensuremath{\mathcal P_\kappa \lambda}$-lists remains somewhat unclear.
\begin{question} \label{question:PklListWithCofNoStrongBranch}
Can there be a $\ensuremath{\mathcal P_\kappa \lambda}$-list that has a cofinal branch but no strong branch?
\end{question}
Still, in light of the previous corollary, the concept of a strong branch comes up naturally in the context of split principles. Since its exact relationship to the concept of a cofinal branch is somewhat mysterious, we want to take some time to elaborate on it. First, the existence of strong branches through $\ensuremath{\mathcal P_\kappa \lambda}$-lists can be formulated as coherence properties.
\begin{observation}
\label{observation:BranchExistenceAsListCoherence}
Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list.
\begin{enumerate}
\item $\vec{d}$ has an ineffable branch if there is a stationary set $S\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x,y\in S$ with $x\subseteq y$, $d_x=d_y\mathfrak{c}ap x$.
\item $\vec{d}$ has an almost ineffable branch if there is an unbounded set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x,y\in U$ with $x\subseteq y$, $d_x=d_y\mathfrak{c}ap x$.
\item $\vec{d}$ has a strong branch if there is an unbounded set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x\in\ensuremath{\mathcal P_\kappa \lambda}$, there is a $y\supseteq x$ such that for all $z_0,z_1\supseteq y$ with $z_0,z_1\in U$, $d_{z_0}\mathfrak{c}ap x=d_{z_1}\mathfrak{c}ap x$.
\end{enumerate}
\end{observation}
We may now see that wild ineffability can be viewed as expressing a delayed coherence property of $\ensuremath{\mathcal P_\kappa \lambda}$-lists, removing all mention of the existence of strong branches.
\begin{definition}
\label{definition:DelayedCoherence}
Let's call a function $f:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ a \emph{delay function} if for all $x\in\ensuremath{\mathcal P_\kappa \lambda}$ we have that $x\subseteq f(x)$. Let's say that a $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec{d}$ \emph{coheres on a set $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ with delay function $f$} if for all $x$ and all $z_0,z_1\supseteq f(x)$ with $z_0,z_1\in U$ we have that $d_{z_0}\mathfrak{c}ap x=d_{z_1}\mathfrak{c}ap x$. If a $\ensuremath{\mathcal P_\kappa \lambda}$-list coheres on $U$ with delay function $\textup{\ensuremath{\text{id}}}$, then let's say that it coheres immediately on $U$. A \emph{continuous delay function} is a delay function $f$ such that for some function $g:\lambda\longrightarrow\lambda$, $f(x)=x\mathfrak{c}up g``x$.
\end{definition}
Using this vocabulary, $\kappa$ is almost $\lambda$-ineffable if every $\ensuremath{\mathcal P_\kappa \lambda}$-list coheres immediately on an unbounded set, it is $\lambda$-ineffable if every $\ensuremath{\mathcal P_\kappa \lambda}$-list coheres immediately on a stationary set, and it is wildly $\lambda$-ineffable if it coheres on an unbounded set with some delay function.
Actually, an analysis of the proof of Theorem \ref{theorem:PklSplitIffNotSBP} shows that $\kappa$ is $\lambda$-wildly ineffable, then every $\ensuremath{\mathcal P_\kappa \lambda}$-list coheres on an unbounded set with a delay function of the form $f(x)=x\mathfrak{c}up\bigcup_{\alpha\in x}g(\alpha)$, where $g:\lambda\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$. Observe that if $\vec{d}$ coheres on a stationary set $S \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$ with such a delay function, then it has an ineffable branch, because the set $C_g=\{x\; | \;\forall\alpha\in x\quad g(\alpha)\subseteq x\}$ is club, and so, $S\mathfrak{c}ap C_g$ is stationary, but if $x\subseteq y$ with $x,y\in S\mathfrak{c}ap C_g$, then $f(x)\subseteq x\subseteq y$, so $d_x=d_x\mathfrak{c}ap x=d_{y}\mathfrak{c}ap x$.
We have explored $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded,\mathcal{B}}$ for $\mathcal{B}=\mathcal{U}nbounded$ and $\mathcal{B}=\ensuremath{\mathsf{nonempty}}$. It turns out that the case $\mathcal{B}=\ensuremath{\mathsf{covering}}$ characterizes continuously delayed coherence.
\begin{lemma}
Let $\kappa\le\lambda$ be cardinals, and let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list. The following are equivalent:
\begin{enumerate}
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded,\ensuremath{\mathsf{covering}}}$.
\item $\vec{d}$ does not cohere on an unbounded set with a continuous delay function.
\end{enumerate}
\end{lemma}
\begin{proof}
(1)$\implies$(2): Suppose there were a set $B\subseteq\lambda$ such that for some unbounded $U$ and some $f:\lambda\longrightarrow\lambda$, we'd have that for every $x$ and every $y\in U$ with $x\mathfrak{c}up f``x\subseteq y$, $B\mathfrak{c}ap x=d_y\mathfrak{c}ap x$ - this is equivalent to continuously delayed coherence on an unbounded set. Let $\beta$ split $U$ into covering sets. Let $x\in U^+_\beta$, $y\in U^-_\beta$ be such that $f(\beta)\in x$, $f(\beta)\in y$. Let $a=\{\beta\}$. Then $b:=a\mathfrak{c}up f``a=\{\beta,f(\beta)\}\subseteq x\in U$, so $B\mathfrak{c}ap a =d_y\mathfrak{c}ap a$, and since $x\in U^+_\beta$, it follows that $\beta\in d_x$, so $\beta\in B$. But also, $b\subseteq y\in U$, and since $y\in U^-_\beta$, it follows that $\beta\notin d_y$, and $B\mathfrak{c}ap a=d_y\mathfrak{c}ap a$, i.e., $\beta\notin B$, a contradiction.
(2)$\implies$(1): We show the contrapositive. So assuming $\vec{d}$ doesn't split unbounded sets into covering sets, we have to prove coherence with continuous delay. Let $U$ be an unbounded set that is not split into covering sets by any $\beta<\lambda$, wrt.~$\vec{d}$. Then for each $\beta<\lambda$, it's not the case that both $U^+_\beta$ and $U^-_\beta$ cover $\lambda$. So there is an $f(\beta)<\lambda$ such that
\begin{enumerate}
\item[(a)] for every $x\in U$ with $f(\beta)\in x$, $x\notin U^+_\beta$, OR
\item[(b)] for every $x\in U$ with $f(\beta)\in x$, $x\notin U^-_\beta$.
\end{enumerate}
Note that these two cases are mutually exclusive, because there is an $x\in U$ with $\{\beta,f(\beta)\}\subseteq x$, and if $x\notin U^+_\beta$, since $\beta\in x$, it follows that $\beta\notin d_x$, and so, $x\in U^-_\beta$. So exactly one of the two holds. Let
\[B=\{\beta<\lambda\; | \;\forall x\in U\ f(\beta)\in x \implies x\notin U^-_\beta\}\]
We claim that for every $x$, and every $y\in U$ with $x\mathfrak{c}up f``x\subseteq y$, it follows that $B\mathfrak{c}ap x=d_y\mathfrak{c}ap x$ (and this implies that $\vec{d}$ coheres on $U$ with delay function $x\mapsto x\mathfrak{c}up f``x$). To see this, let $\beta\in x$. If $\beta\in B$, then it follows from the definition of $B$ that $y\notin U^-_\beta$. But since $\beta\in x\subseteq y$ and $y\in U$, this implies that $\beta\in d_y$. On the other hand, if $\beta\notin B$, then we're not in case (b) above, so we're in case (a). So $y\in U^+_\beta$, so $\beta\in d_y$.
\end{proof}
\begin{definition}
Let $f:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$, and let $\vec U = \seq{U_x}{x\in\ensuremath{\mathcal P_\kappa \lambda}}$ be a sequence of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Define the \emph{$f$-diagonal intersection of $\vec{U}$} as
\[\dintersection^f\vec{U}=\dintersection^f_{x\in\ensuremath{\mathcal P_\kappa \lambda}}U_x=\{z\; | \;\forall x \ (f(x)\subseteq z \implies z\in U_x)\}.
\]
\end{definition}
\begin{observation}
Suppose $B$ is a cofinal branch for the $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec{d}$. For $x\in\ensuremath{\mathcal P_\kappa \lambda}$, let
\[U_x=\{y\supseteq x\; | \; B\mathfrak{c}ap x=d_y\mathfrak{c}ap x\}.\]
Then $B$ is a strong branch iff there is a delay function $f$ such that $U=\dintersection^f_{x\in\ensuremath{\mathcal P_\kappa \lambda}}U_x$ is unbounded.
\end{observation}
\begin{proof} From right to left, if $f$ and $U$ are as stated, then $U$ is an unbounded set that guides $B$, because for any $x$, if $z\supseteq f(x)$ is in $U$, then $z\in U_x$, and so, $B\mathfrak{c}ap x=d_z\mathfrak{c}ap x$.
Vice versa, if $B$ is strong and $U'$ is an unbounded set that guides $B$, then we can define a delay function $f:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, and for every $z\supseteq f(x)$ with $z\in U'$, $d_z\mathfrak{c}ap x=B\mathfrak{c}ap x$. It follows that $U'\subseteq\dintersection^f_{x\in\ensuremath{\mathcal P_\kappa \lambda}}U_x$, because if $z\in U'$ and $x$ is such that $f(x)\subseteq z$, then by the property of $f$, $d_z\mathfrak{c}ap x=B\mathfrak{c}ap x$, that is, $z\in U_x$. So since $U'$ is unbounded, so is $\dintersection^f\vec{U}$. \end{proof}
Digressing briefly, we want to explore a connection to $\ensuremath{\mathcal P_\kappa \lambda}$-partition properties, see \mathfrak{c}ite[p.~346]{THI2} for an overview.
For a natural number $n$ and a subset $X\subseteq\ensuremath{\mathcal P_\kappa \lambda}$, write $[X]^n_{\subseteq}$ for the set
$$\set{\{a_0,a_1,\ldots,a_{n-1}\} \subseteqseteq X}{ a_0\subseteqsetneq a_1\subseteqsetneq \ldots \subseteqsetneq a_{n-1}},$$
and say that $\Part{\kappa,\lambda}^n$ holds if for every partition function $F:[\ensuremath{\mathcal P_\kappa \lambda}]^n_{\subseteq}\longrightarrow 2$, there is an unbounded set $H\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ that's homogeneous for $F$, meaning that $F\mathbin{\upharpoonright}[H]^n_{\subseteq}$ is constant. $\Part{\kappa,\lambda}$ is just $\Part{\kappa,\lambda}^2$.
We give some relevant known results on $\Part{\kappa,\lambda}$ below.
\begin{fact}[\mathfrak{c}ite{Matet:StrongCompactnessAndPartitionProperty}] \mbox{}\begin{enumerate}
\item If $\kappa$ is almost $\lambda^{{<}\kappa}$-ineffable, then $\Part{\kappa,\lambda}$ holds.
\item If $\kappa$ is mildly $\lambda^{{<}\kappa}$-ineffable and $\text{cov}(M_{\kappa,\lambda})>\lambda^{{<}\kappa}$, then $\Part{\kappa,\lambda}$ holds.
\item If $\Part{\kappa,2^{2^{\lambda^{{<}\kappa}}}}$ holds, then $\kappa$ is $\lambda$-compact.
\end{enumerate}\end{fact}
The following has a precursor in \mathfrak{c}ite[Thm.~2.2]{Carr:Diss}, which shows that if $\Part{\kappa, \lambda}^3$ holds, then $\kappa$ is mildly $\lambda$-ineffable.
\begin{theorem}
$\Part{\kappa,\lambda}^3$ implies that every $\ensuremath{\mathcal P_\kappa \lambda}$-list has a strong branch (i.e., that $\kappa$ is wildly $\lambda$-ineffable, or $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ fails).
\end{theorem}
\proof The proof of \mathfrak{c}ite[Thm.~2.2]{Carr:Diss}, in which it was pointed out that the assumption implies that $\kappa$ is inaccessible, works here as well. Let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list, and define a partition $F:[\ensuremath{\mathcal P_\kappa \lambda}]^3_{\subseteq}\longrightarrow 2$ by setting, for $x\subseteqsetneq y\subseteqsetneq z$,
\[F(x,y,z)=\begin{cases}
0 & \text{if}\ d_y\mathfrak{c}ap x=d_z\mathfrak{c}ap x\\
1 & \text{otherwise.}
\end{cases}\]
Let $H$ be an unbounded subset of $\ensuremath{\mathcal P_\kappa \lambda}$ that is homogeneous for $F$.
We first show that $H$ cannot be 1-homogeneous. Suppose it were. Fix $x\in H$. Then, for any $y_0,y_1\in H$ with $x\subseteqsetneq y_0\subseteqsetneq y_1$, $d_{y_0}\mathfrak{c}ap x\neq d_{y_1}\mathfrak{c}ap x$. Let ${\bar{\kappa}}$ be the cardinality of $\power{x}$. Then ${\bar{\kappa}}<\kappa$, since $\kappa$ is inaccessible. But there is a sequence $\seq{y_\alpha}{\alpha<{\bar{\kappa}}^+}$ with $x\subseteqsetneq y_0$ and $y_\alpha\subseteqsetneq y_\beta$ for all $\alpha<\beta<{\bar{\kappa}}^+$, because ${\bar{\kappa}}^+<\kappa$ and $\kappa$ is regular. This is a contradiction, because for all such $\alpha,\beta$, we would have that $d_{y_\alpha}\mathfrak{c}ap x\neq d_{y_{\beta}}\mathfrak{c}ap x$, giving us ${\bar{\kappa}}^+$ distinct subsets of $x$.
Thus, $H$ is 0-homogeneous. Set
\[B=\bigcup\{d_y\mathfrak{c}ap x\; | \; x,y\in H\ \text{and}\ x\subseteqsetneq y\}.\]
It follows that:
\begin{equation}
\label{eqn:Coherence}
\text{If $x\subseteqsetneq y$ with $x,y\in H$ then $B\mathfrak{c}ap x=d_y\mathfrak{c}ap x$.}
\end{equation}
\prooff{(\ref{eqn:Coherence})} The inclusion from right to left is clear, by definition of $B$. For the converse suppose $\alpha\in B\mathfrak{c}ap x$. Let $x'\subseteqsetneq y'$, $x',y'\in H$, with $\alpha\in d_{y'}\mathfrak{c}ap x'$. Pick $z\in H$ with $y\mathfrak{c}up y'\subseteqsetneq z$. Then $\alpha\in d_{y'}\mathfrak{c}ap x'=d_z\mathfrak{c}ap x'$, so $\alpha\in d_z\mathfrak{c}ap x$, since $\alpha\in x$. So $\alpha\in d_z\mathfrak{c}ap x=d_y\mathfrak{c}ap x$, as claimed. \qedd{(\ref{eqn:Coherence})}
It follows that $B$ is a strong branch, as verified by $H$. To see this, let $x\in\ensuremath{\mathcal P_\kappa \lambda}$ be given. Find $x'\in H$ with $x\subseteq x'$, and let $x\subseteqsetneq t$, $t\in H$. We claim that for every $u\in H$ with $t\subseteq u$, $B\mathfrak{c}ap x=d_u\mathfrak{c}ap x$. But this is immediate, since $x'\subseteqsetneq u$ and $x',u\in H$, so by (\ref{eqn:Coherence}), $B\mathfrak{c}ap x'=d_u\mathfrak{c}ap x'$, which implies that $B\mathfrak{c}ap x=d_u\mathfrak{c}ap x$, since $x\subseteq x'$. \qed
One approach to generalizing the theory of ideals on $\kappa$ to $\ensuremath{\mathcal P_\kappa \lambda}$ involved working with the ordering
\[x<y\iff x\subseteq y \ \land\ |x|<|\kappa\mathfrak{c}ap y|\]
instead of set inclusion - this was spearheaded by Donna Carr (see \mathfrak{c}ite{THI2} for the history). This leads to a natural weakening of the partition properties, namely, for $n\in\omega$ and a subset $X\subseteq\ensuremath{\mathcal P_\kappa \lambda}$, write $[X]^n_{<}$ for the set
$\{\{a_0,a_1,\ldots,a_{n-1}\}\; | \; a_0 < a_1 < \ldots < a_{n-1}\in X\},$
and say that $\Part{\kappa,\lambda}^n_<$ holds if for every function $F:[\ensuremath{\mathcal P_\kappa \lambda}]^n_<\longrightarrow 2$, there is an unbounded set $H\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ that's homogeneous for $F$, meaning that $F\mathbin{\upharpoonright}[H]^n$ is constant. It is easy to see that the proof of the previous theorem actually shows the following corollary; one just has to replace every instance of ``$\subseteqsetneq$'' in the proof with ``$<$''.
\begin{corollary}
$\Part{\kappa,\lambda}^3_<$ implies that every $\ensuremath{\mathcal P_\kappa \lambda}$-list has a strong branch (i.e., that $\kappa$ is wildly $\lambda$-ineffable, or $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ fails).
\end{corollary}
Still we are left wondering about the status of wild ineffability as a new large cardinal notion; a negative answer to
Question \ref{question:PklListWithCofNoStrongBranch} would imply that wild ineffability and mild ineffability are the same.
Finally, towards characterizing mild ineffability and strong compactness, we need a slight variation of the split principle.
\begin{definition}
\label{definition:SplitPrincipleForSplittingFunctions}
Let $\mathcal{F}$ be a set of functions from $\ensuremath{\mathcal P_\kappa \lambda}$ to $\ensuremath{\mathcal P_\kappa \lambda}$, and let $\mathcal{B}$ be a family of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Then $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{F},\mathcal{B}}$ is the principle saying that there is a $\ensuremath{\mathcal P_\kappa \lambda}$-list $\vec{d}$ such that for every function $f\in\mathcal{F}$, there is a $\beta<\lambda$ such that both $f^+_\beta=\{x\; | \;\beta\in x\land\beta\in d_{f(x)}\}$ and $f^-_\beta=\{x\; | \;\beta\in x\land\beta \notin d_{f(x)}\}$ belong to $\mathcal{B}$.
Let $\ensuremath{\mathsf{delay\text{-}functions}}$ be the set of delay functions from $\ensuremath{\mathcal P_\kappa \lambda}$ to $\ensuremath{\mathcal P_\kappa \lambda}$, i.e., the set of functions $f:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x\in\ensuremath{\mathcal P_\kappa \lambda}$, $x\subseteq f(x)$.
\end{definition}
\begin{theorem}
\label{theorem:CharacterizationsOfMildIneffability}
Let $\kappa$ be regular and $\lambda\ge\kappa$ be a cardinal. Then the following are equivalent:
\begin{enumerate}
\item $\kappa$ is mildly $\lambda$-ineffable.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{nonempty}}}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{covering}}}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\mathcal{U}nbounded}$ fails.
\item $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{stationary}}}$ fails.
\end{enumerate}
So $\kappa$ is mildly ineffable iff $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\mathcal{B}}$ fails, for some (equivalently, all) $\mathcal{B}$ with $\ensuremath{\mathsf{stationary}}\subseteq\mathcal{B}\subseteq\ensuremath{\mathsf{nonempty}}$. It follows that $\kappa$ is strongly compact iff these equivalent conditions hold for every $\lambda$.
\end{theorem}
\begin{proof}
(1)$\implies$(2): Suppose $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{nonempty}}}$ held. Let $\vec{d}$ witness this. Since $\kappa$ is mildly $\lambda$-ineffable, $\vec{d}$ has a cofinal branch $B\subseteq\lambda$. There is then a delay function $f:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, $B\mathfrak{c}ap x=d_{f(x)}\mathfrak{c}ap x$. By $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{nonempty}}}$, let $\beta$ be such that both $f^+_\beta$ and $f^-_\beta$ are nonempty. Let $x_0,x_1$ be such that $x_0\in f^+_\beta$ and $x_1\in f^-_\beta$. This means that $\beta\in d_{f(x_0)}$ and $\beta\notin d_{f(x_1)}$. Note that by definition, $\beta\in x_0,x_1$. But $d_{f(x_0)}=B\mathfrak{c}ap x_0$, so $\beta\in B$, while on the other hand, $d_{f(x_1)}=B\mathfrak{c}ap x_1$, so $\beta\notin B$, a contradiction.
(2)$\implies$(3)$\implies$(4)$\implies$(5)~is trivial.
(5)$\implies$(1): To prove that $\kappa$ is mildly $\lambda$-ineffable, let $\vec{d}$ be a $\ensuremath{\mathcal P_\kappa \lambda}$-list. We have to find a cofinal branch. Since $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{stationary}}}$ fails, $\vec{d}$ is not a $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{stationary}}}$ sequence. This means that there is a delay function $f$ on $\ensuremath{\mathcal P_\kappa \lambda}$ that is not split into stationary sets by any $\beta<\lambda$ with respect to $\vec{d}$. As in previous arguments, this means that exactly one of $f^+_\beta$ and $f^-_\beta$ is stationary (note that $\widehat{\{\beta\}}$ is the disjoint union of $f^+_\beta\mathfrak{c}up f^-_\beta$). So for every $\beta<\lambda$, there is a club set $C_\beta$ in $\ensuremath{\mathcal P_\kappa \lambda}$ that's disjoint from the nonstationary one of $f^+_\beta$ and $f^-_\beta$. Let
\[B=\{\beta<\lambda\; | \; f^+_\beta\ \text{is stationary}\}\]
Let $D=\dintersection_{\beta<\lambda}C_\beta$. To see that $B$ is a cofinal branch, let $x\in\ensuremath{\mathcal P_\kappa \lambda}$ be given. Let $x\subseteq x'\in D$. We claim that $B\mathfrak{c}ap x=d_{f(x')}\mathfrak{c}ap x$. This completes the proof, since $x\subseteq x'\subseteq f(x')$. So let $\beta\in x$. Then $\beta\in x'$, and so, $x'\in C_\beta$. If $\beta\in B$, then $f^+_\beta$ is stationary, so $C_\beta\mathfrak{c}ap f^-_\beta=\emptyset$. But again, since $\beta\in x'$, it follows that $x'\in f^+_\beta$, so that $\beta\in d_{f(x')}$. If $\beta\notin B$, then $f^+_\beta\mathfrak{c}ap C_\beta=\emptyset$, and since $\beta\in x'$, it follows that $x'\in f^-_\beta$, so $\beta\notin d_{f(x')}$.
The point about the failure of $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\mathcal{B}}$ follows now, because (2)~implies the failure of $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\mathcal{B}}$, and this implies (5).
The claim about $\kappa$ being strongly compact now follows because it is easy to see that $\kappa$ is mildly $\lambda$-ineffable iff every binary $(\kappa,\lambda)$-mess is solvable (see \mathfrak{c}ite[Thm.~1.4, p.~35]{Carr:Diss}), and Jech showed in \mathfrak{c}ite[2.2, p.~174]{CombProblems} that every binary $(\kappa,\lambda)$-mess is solvable iff $\kappa$ is $\lambda$-compact.
\end{proof}
\subseteqsection{Split characterizations of two cardinal versions of Shelah cardinals}
We will now introduce versions of the split principle whose failure can capture variants of the notion of $\kappa$ being $\lambda$-Shelah. This large cardinal notion was introduced by Carr (see \mathfrak{c}ite{Carr:Diss}).
\begin{definition}
\label{definition:FunctionalVersions}
Let $A\subseteq\ensuremath{\mathcal P_\kappa \lambda}$, $\mathcal{A},\mathcal{B}\subseteq\power{\ensuremath{\mathcal P_\kappa \lambda}}$. Then we write $\mathcal{A}\mathbin{\upharpoonright}riction A:=\mathcal{A}\mathfrak{c}ap\power{A}$.
A \emph{functional} $A$-\emph{list} is a sequence $\vec f = \seq{f_x}{x\in A}$ of functions $f_x:x\longrightarrow x$. A functional $A$-list \emph{splits} a set $X \subseteqseteq \ensuremath{\mathcal P_\kappa \lambda}$ into $\mathcal{B}\mathbin{\upharpoonright}riction A$ if there is a pair $\kla{\beta,\delta}\in\lambda\times\lambda$ such that the sets
\[X^+_{\beta,\delta}=\{x\in X\mathfrak{c}ap A\; | \; \beta\in x \land f_x(\beta)=\delta\}\ \text{and}\ X^-_{\beta,\delta}=\{x\in X\mathfrak{c}ap A\; | \; \beta,\delta\in x \land f_x(\beta)\neq\delta\}\]
are in $\mathcal{B}\mathbin{\upharpoonright}riction B$. The principle $A$-$\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}}$ says that there is an $A$-list that splits every $X\in\mathcal{A}\mathbin{\upharpoonright}riction A$ into $\mathcal{B}\mathbin{\upharpoonright}riction A$.
Given a functional $A$-list $\vec{f}$, a function $F:\lambda\longrightarrow\lambda$ is a \begin{itemize}
\item \emph{cofinal branch} for $\vec{f}$ if for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, there is a $y\in\ensuremath{\mathcal P_\kappa \lambda}$ with $x\subseteq y$, such that $f_y\mathbin{\upharpoonright} x=F\mathbin{\upharpoonright} x.$
\item \emph{strong branch} for $\vec f$ if there is an unbounded set $U\subseteq A$ such that for every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, there is a $y\in\ensuremath{\mathcal P_\kappa \lambda}$ with $x\subseteq y$, such that for every $z\in U$ with $y\subseteq z$, $f_z\mathbin{\upharpoonright} x=F\mathbin{\upharpoonright} x,$ and in this case, we say that $F$ is \emph{guided} by $U$.
\item \emph{almost ineffable branch} for $\vec f$ if for unboundedly many $x$, $f_x=F\mathbin{\upharpoonright} x.$
\item \emph{ineffable branch} for $\vec f$ if for stationarily many $x$, $f_x=F\mathbin{\upharpoonright} x.$
\end{itemize}
$A$ is $\lambda$-\emph{Shelah} if every functional $A$-list has a cofinal branch, and $\kappa$ is called $\lambda$-Shelah if $\ensuremath{\mathcal P_\kappa \lambda}$ is $\lambda$-Shelah. $A$ is \emph{wildly $\lambda$-Shelah} if every functional $A$-list has a strong branch, and $\kappa$ is \emph{wildly $\lambda$-Shelah} if $\ensuremath{\mathcal P_\kappa \lambda}$ is wildly $\lambda$-Shelah.
\end{definition}
Note that $f_x(\beta)=\delta$ can be equivalently expressed by saying that $\kla{\beta,\delta}\in f_x$, so the concept of a splitting functional list is a direct generalization of a splitting list.
It was shown in \mathfrak{c}ite{Carr:Diss} that $\kappa$ is (almost) $\lambda$-ineffable iff every functional $\ensuremath{\mathcal P_\kappa \lambda}$-list has an (almost) ineffable branch. So moving from $\ensuremath{\mathcal P_\kappa \lambda}$-lists to functional $\ensuremath{\mathcal P_\kappa \lambda}$-lists does not make a difference for these concepts. Obviously, the logical relationship between these various types of branches for functional $\ensuremath{\mathcal P_\kappa \lambda}$-lists is as for regular $\ensuremath{\mathcal P_\kappa \lambda}$-lists, see Observation \ref{observation:RelationshipsBetweenBranches}. As a consequence, if $\kappa$ is almost $\lambda$-ineffable, then it is wildly $\lambda$-Shelah, and this implies $\lambda$-Shelahness and wild $\lambda$-ineffability.
It was shown in \mathfrak{c}ite{Carr:Diss} that the ideal corresponding to the failure of the Shelah property is normal, while the non-mildly ineffable ideal is equal to the ideal of the non-unbounded sets (assuming the corresponding large cardinal property).
\begin{theorem}
\label{theorem:FunctionalListSplitsIffNoStrongBranch}
Let $A\subseteq\ensuremath{\mathcal P_\kappa \lambda}$.
A functional $A$-list $\vec{f}$ splits all unbounded subsets of $A$ iff it does not have a strong branch.
\end{theorem}
\begin{proof}
We show both implications separately.
$\Longrightarrow$:
Towards a contradiction, assume $F$ is a strong branch through the $A$-list $\vec{f}$, guided by the unbounded set $U \subseteqseteq A$. Let $g:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow U$ be so that for all $x \in \ensuremath{\mathcal P_\kappa \lambda}$, $g(x)\supseteq x$ and for all $z \supseteq g(x)$ with $z\in U$, $f_z\mathbin{\upharpoonright} x=F\mathbin{\upharpoonright} x$. Let $\kla{\beta,\delta}$ split $\tilde{U}=\ran(g)$ with respect to $\vec f$.
If $F(\beta)=\delta$, then we choose $y \in\tilde{U}^-_{\beta,\delta}$ with $y\supseteq g(\{\beta\})$. Then $f_y(\beta)=F(\beta)=\delta$, but since $y\in\tilde{U}^-_{\beta,\delta}$, $f_y(\beta)\neq\delta$, a contradiction. If $F(\beta)\neq\delta$, then we instead choose $y\in\tilde{U}^+_{\beta,\delta}$ with $y\supseteq g(\{\beta\})$ and get the same contradiction.
$\Longleftarrow$: We will show the contrapositive.
So suppose $\vec{f}$ is a functional $A$-list that does not split unbounded sets.
There is then an unbounded $U\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ such that for each $\kla{\beta,\delta}\in\lambda\times\lambda$, exactly one of $U^+_\kla{\beta,\delta}$ and $U^-_\kla{\beta,\delta}$ is unbounded in $\ensuremath{\mathcal P_\kappa \lambda}$. Define $F\subseteq\lambda\times\lambda$ by setting
\[\kla{\beta,\delta}\in F\iff U^+_{\beta,\delta}\ \text{is unbounded in $\ensuremath{\mathcal P_\kappa \lambda}$}.\]
We claim that $U$ guides $F$,
making it a strong branch.
To see this, fix $x \in \ensuremath{\mathcal P_\kappa \lambda}$. For each pair $\kla{\beta,\delta}\in x\times x$, there has is a $y_\kla{\beta,\delta}\in\ensuremath{\mathcal P_\kappa \lambda}$ such that either for all $z\supseteq y_\kla{\beta,\delta}$ with $z\in U$ we have that $f_z(\beta)=\delta$, or for all $z\supseteq y_\kla{\beta,\delta}$ with $z\in U$, $f_z(\beta)\neq\delta$. The former holds if $U^-_\beta$ is not unbounded, and the latter holds if $U^+_\beta$ is not unbounded. Let $y=\bigcup_{\kla{\beta,\delta}\in x\times x} y_\kla{\beta,\delta}$, and pick $z\supseteq y$ with $z\in U$.
Now $F\mathfrak{c}ap(x\times x)=f_z\mathfrak{c}ap(x\times x)$: From right to left, suppose $\kla{\beta,\delta}\in x\times x$ and $f_z(\beta)=\delta$. Then for all $z'\supseteq z$ with $z'\in U$, $f_{z'}(\beta)=\delta$, because $z'\supseteq z\supseteq y_{\kla{\beta,\delta}}$. So $U^+_\kla{\beta,\delta}$ is unbounded, and hence, $\kla{\beta,\delta}\in F$. Vice versa, if $\kla{\beta,\delta}\in F\mathfrak{c}ap(x\times x)$, then $U^+_{\beta,\delta}$ is unbounded, and hence, for all $z'\supseteq y_\kla{\beta,\delta}$ with $z'\in U$, $f_{z'}(\beta)=\delta$ (because the alternative would be that for all $z'\supseteq y_\kla{\beta,\delta}$ with $z'\in U$, $f_{z'}(\beta)\neq\delta$, but that would mean that $U^+_\kla{\beta,\delta}$ is not unbounded). So since $z\supseteq y_\kla{\beta,\delta}$ and $z\in U$, $f_z(\beta)=\delta$, as claimed.
This implies that $F$ is a function, and hence that it is a strong branch in the functional sense.
\end{proof}
\begin{lemma}
Let $\kappa$ be regular. $\kappa$ is wildly $\lambda$-Shelah iff $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ fails. It follows that $\kappa$ is supercompact iff $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ fails, for all $\lambda$.
\end{lemma}
\begin{proof}
It follows by \mathfrak{c}ite{Carr:Diss} and \mathfrak{c}ite{Magidor74:CombCharactSupercompactness} that $\kappa$ is supercompact iff $\kappa$ is $\lambda$-Shelah for all $\lambda$ iff $\kappa$ is almost $\lambda$-ineffable for all $\lambda$ iff $\kappa$ is $\lambda$-ineffable for all $\lambda$. Moreover, \mathfrak{c}ite[p.~52, Cor.~1.4]{Carr:Diss} shows that if $\kappa$ is almost $\lambda$-ineffable, then every functional $\ensuremath{\mathcal P_\kappa \lambda}$-list has an almost ineffable branch, and if $\kappa$ is $\lambda$-ineffable, then every functional $\ensuremath{\mathcal P_\kappa \lambda}$-list has an ineffable branch. In particular, if $\kappa$ is almost $\lambda$-ineffable, then it is wildly $\lambda$-Shelah. It follows from all of this that the failure of $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$ for every $\lambda$ characterizes the supercompactness of $\kappa$.
\end{proof}
In order to characterize when $\kappa$ is $\lambda$-Shelah, we need a modification of the functional split principles similar to the modification that was needed in order to characterize mild ineffability.
\begin{definition}
Let $\mathcal{F}$ be a set of functions from $\ensuremath{\mathcal P_\kappa \lambda}$ to $\ensuremath{\mathcal P_\kappa \lambda}$, and let $\mathcal{B}$ be a family of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$. Then $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{F},\mathcal{B}}$ is the principle saying that there is a functional list ${\vec{f}}$ such that for every function $g\in\mathcal{F}$, there is a pair $\kla{\beta,\delta}\in\lambda\times\lambda$ such that both $g^+_{\kla{\beta,\delta}}=\{x\; | \;\beta\in x\land f_{g(x)}(\beta)=\delta\}$ and $g^-_{\kla{\beta,\delta}}=\{x\; | \;\beta\in x\land f_{g(x)}(\beta)\neq\delta\}$ belong to $\mathcal{B}$.
\end{definition}
\begin{lemma}
Let $\kappa$ be regular and $\lambda\ge\kappa$ be a cardinal. The following are equivalent:
\begin{enumerate}
\item $\kappa$ is $\lambda$-Shelah
\item $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{nonempty}}}$ fails.
\item $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{covering}}}$ fails.
\item $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\mathcal{U}nbounded}$ fails.
\item $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\ensuremath{\mathsf{delay\text{-}functions}},\ensuremath{\mathsf{stationary}}}$ fails.
\end{enumerate}
It follows that $\kappa$ is supercompact iff these conditions hold for arbitrarily large $\lambda$.
\end{lemma}
\begin{proof}
The proof of \ref{theorem:CharacterizationsOfMildIneffability} goes through with minor modifications. By \mathfrak{c}ite[p.~63, Cor.~2.1]{Carr:Diss}, $\kappa$ is supercompact iff $\kappa$ is $\lambda$-Shelah, for every $\lambda$.
\end{proof}
\section{Split ideals}
\label{sec:SplitIdeals}
In the study of $\ensuremath{\mathcal P_\kappa \lambda}$-combinatorics, it has proven fruitful to investigate ideals associated to various large cardinal properties. This was done, for example, for the ideal $\mathsf{NIn}_{\kappa,\lambda}$ of non-ineffable subsets of $\ensuremath{\mathcal P_\kappa \lambda}$, the ideal $\mathsf{NAIn}_{\kappa,\lambda}$ of non-almost-ineffable subsets and the ideal $\mathsf{NMI}_{\kappa,\lambda}$ of non-mildly-ineffable subsets of $\ensuremath{\mathcal P_\kappa \lambda}$ in \mathfrak{c}ite{Carr:Diss}. It was shown there that if $\kappa$ is mildly $\lambda$-ineffable, then $\mathsf{NMI}_{\kappa,\lambda}$ is equal to the ideal $\mathsf{I}_{\kappa,\lambda}$ of non-unbounded subsets of $\ensuremath{\mathcal P_\kappa \lambda}$, and that the other ideals are normal ideals if $\kappa$ has the corresponding large cardinal property.
Since the split principles characterize the failure of a large cardinal property, they allow us to define such ideals in a natural way.
Since some of the large cardinal properties sprouting from our investigation of split principles, such as wild ineffability, appear to be new, it seems worthwhile to investigate these ideals.
\begin{definition}
\label{definition:SplitIdeal}
Let $\mathcal{A},\mathcal{B}\subseteq\power{\ensuremath{\mathcal P_\kappa \lambda}}$ be families of subsets of $\ensuremath{\mathcal P_\kappa \lambda}$, and let $A\subseteq\ensuremath{\mathcal P_\kappa \lambda}$. Then we write $\mathcal{A}\mathbin{\upharpoonright} A$ for $\mathcal{A}\mathfrak{c}ap\power{A}$, i.e., for the family of sets in $\mathcal{A}$ that are contained in $A$.
We let
\[\mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}})=\{A\subseteq\ensuremath{\mathcal P_\kappa \lambda}\; | \; \SP{\kappa}{\mathcal{A} \mathbin{\upharpoonright} A,\mathcal{B}}\ \text{holds}\}\]
\end{definition}
In the more natural cases, $\mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{A},\mathcal{B}})$ is an ideal - this is the case if $\mathcal{A}=I^+$, for an ideal $I$ on $\ensuremath{\mathcal P_\kappa \lambda}$. The proof of Observation \ref{observation:WhenSplitIdealsAreIdealsOnKappa} goes through.
We will first work with the principle $\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded}$, whose failure characterizes the wildly ineffable cardinals.
\begin{lemma}
\label{lemma:SplitIdealIsKappaComplete}
$\mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$ is a $\kappa$-complete ideal containing $\mathsf I_{\kappa,\lambda}$.
\end{lemma}
\proof Clearly, $\mathsf I_{\kappa,\lambda}\subseteq \mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$, since a non-unbounded set has no unbounded subsets. To see that it is an ideal, first observe that if $Y\in \mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$ and $X\subseteq Y$, then $X\in \mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$, because if $\vec{d}$ is a $Y$-list that splits every unbounded subset of $Y$, then $\vec{d}\mathbin{\upharpoonright} X$ is an $X$-list that splits every unbounded subset of $X$. Second, we show $\kappa$-completeness. Thus, let ${\bar{\kappa}}<\kappa$ and let $\seq{X_\gamma}{\gamma<{\bar{\kappa}}}$ be a sequence of sets in $\mathcal{I}(\SP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$. For each $\gamma<{\bar{\kappa}}$, let $\seq{d^\gamma_x}{x\in X_\gamma}$ be a sequence which splits every unbounded subset of $X_\gamma$. Let $X=\bigcup_{\gamma<{\bar{\kappa}}}X_\gamma$. We have to show that there is an $X$-list that splits every unbounded subset of $X$, and to achieve this, we amalgamate the lists $\vec{d}^\gamma$ by letting, for $x\in X$, $\nu(x)$ be least such that $x\in X_{\nu(x)}$, and by setting
\[d_x=d^{\nu(x)}_x\]
for $x\in X$. We claim that $\vec{d}$ splits every unbounded subset of $X$. To see this, let $Y\subseteq X$ be unbounded. Then for some $\gamma<{\bar{\kappa}}$,
the set $\bar{Y}=\{x\in Y\; | \;\nu(x)=\gamma\}$ is unbounded, because the ideal of non-unbounded sets is $\kappa$-complete. Since $\vec{d}^\gamma$ splits every unbounded subset of $X_\gamma$, and since $\bar{Y}$ is an unbounded subset of $X_\gamma$, there is a $\beta$ that splits $\bar{Y}$ with respect to $\vec{d}^\gamma$. But $\vec{d}^\gamma\mathbin{\upharpoonright}\bar{Y}=\vec{d}\mathbin{\upharpoonright}\bar{Y}$, so $\beta$ splits $\bar{Y}$ with respect to $\vec{d}$. \qed
Our lack of knowledge about the relationship between mild ineffability and wild ineffability is reflected by some open questions about the split ideal. It was shown in \mathfrak{c}ite{Carr:Diss} that if $\kappa$ is mildly $\lambda$-ineffable, then $\mathsf{NMI}_{\kappa,\lambda}=\mathsf I_{\kappa,\lambda}$. We do not know whether this is true of the split ideal, and we do not know whether the split ideal is normal, assuming that $\kappa$ is wildly $\lambda$-ineffable.
The ideal corresponding to wild $\lambda$-Shelahness, on the other hand, is normal, like the one corresponding to $\lambda$-Shelahness. The latter was shown by Carr, and her proof generalizes very directly.
Recall that
an ideal $I$ on $\ensuremath{\mathcal P_\kappa \lambda}$ is \emph{normal} if for every sequence $\seq{X_\nu}{\nu<\lambda}$ of members of $I$, the diagonal union
\[\dunion_{\nu<\lambda}X_\nu=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\exists\nu\in x\quad x\in X_\nu\}\]
belongs to $I$.
\begin{theorem}
\label{theorem:FunctionalSplitIdealIsNormal}
$\kappa$ is wildly $\lambda$-Shelah iff $I:=\mathcal{I}(\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$ is a normal proper ideal on $\ensuremath{\mathcal P_\kappa \lambda}$.
\end{theorem}
\proof The direction from right to left is trivial, since if $I$ is a proper ideal, then \fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded} fails, which implies that $\kappa$ is wildly $\lambda$-Shelah, by Theorem \ref{theorem:FunctionalListSplitsIffNoStrongBranch}.
For the substantial forward direction, assume that $\kappa$ is wildly Shelah. According to \mathfrak{c}ite[Lemma 2.2]{Carr:Diss}, to show that $I$ is a normal proper ideal on $\ensuremath{\mathcal P_\kappa \lambda}$ it suffices to show that (0) $I$ is a proper ideal, (1) $I$ is closed under subsets, (2) if $X\in I$ and $Y\in I_{\kappa,\lambda}$, then $X\mathfrak{c}up Y\in I$ and (3) $I$ is closed under diagonal unions.
(0) is clear by our assumption that $\kappa$ is wildly $\lambda$-Shelah. (1) is obvious, as in Lemma \ref{lemma:SplitIdealIsKappaComplete}. (2) is clear because if $X\in I$ and $Y$ is not unbounded, then we can let $\vec{f}$ be a functional $X$-list splitting every unbounded subset of $X$, and extend it arbitrarily to a functional $X\mathfrak{c}up Y$-list $\vec{f'}$. If $A\subseteq X\mathfrak{c}up Y$ is unbounded, then $A=(A\mathfrak{c}ap X) \mathfrak{c}up (A\mathfrak{c}ap Y)$, so one of $A\mathfrak{c}ap X$ and $A\mathfrak{c}ap Y$ is unbounded, as the non-unbounded sets form an ideal. Clearly then, $A\mathfrak{c}ap X$ is unbounded, so split by $\vec{f}$, and hence, $A$ is split by $\vec{f'}$.
The crucial point is (3), the closure of $I$ under diagonal unions. So let $\seq{X_\nu}{\nu<\lambda}$ be a sequence with $X_\nu\in I$ for all $\nu<\lambda$. Fix, for every such $\nu$, a functional $X_\nu$-list $\seq{f^\nu_x}{x\in X_\nu}$ that splits every unbounded subset of $X_\nu$, and let $X'=\dunion_{\nu<\lambda}X_\nu$. That is, for $x\in\ensuremath{\mathcal P_\kappa \lambda}$, $x\in X$ iff there is $\nu\in x$ such that $x\in X_\nu$. For $x\in X'$, let $\gamma(x)\in x$ such that $x\in X_{\gamma(x)}$. We follow the proof of \mathfrak{c}ite[Thm.~2.3]{Carr:Diss} closely here.
Let $\widehat{\{0\}}=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \; 0\in x\}$. Let $X=X'\mathfrak{c}ap\widehat{\{0\}}=\{x\in X\; | \; 0\in x\}$. It suffices to show that $X\in I$, since then it follows by (2) that $X'=X\mathfrak{c}up (X\setminus\widehat{\{0\}})\in I$, as $X\setminus\widehat{\{0\}}\in {\mathsf I}_{\kappa,\lambda}$.
For every $x\in\ensuremath{\mathcal P_\kappa \lambda}$, let $\seq{\alpha^x_\xi}{\xi<\otp(x)}$ be the monotone enumeration of $x$. Since for every $x\in X$ we have $0\in x$, it follows that $\alpha^x_0=0$. We amalgamate the functional lists $\vec{f}^\nu$ into one functional $X$-list $g$ by defining $g_x:x\longrightarrow x$, for $x\in X$, as follows.
\[g_x(\alpha^x_\xi)=
\begin{cases}
\gamma(x) & \text{if}\ \xi=0\ \text{or}\ \xi\ \text{is a limit ordinal},\\
f^{\gamma(x)}_x(\alpha^x_{\xi-1}) & \text{if}\ \xi\ \text{is a successor ordinal}
\end{cases}\]
for $\xi<\otp(x)$. Assuming that $X$ is not in $I$, the functional split ideal on $\ensuremath{\mathcal P_\kappa \lambda}$, no functional $X$-list splits all unbounded subsets of $X$, so Theorem \ref{theorem:FunctionalListSplitsIffNoStrongBranch} implies that every functional $X$-list has a strong branch. Let $G:\lambda\longrightarrow\lambda$ be a strong branch for $\vec{g}$, guided by the unbounded set $U\subseteq X$. Let $\gamma=G(0)$, and define $F:\lambda\longrightarrow\lambda$ by $F(\xi)=G(\xi+1)$.
We claim that
$F$ is a strong branch for $\vec{f^\gamma}$, guided by $U\mathfrak{c}ap X_\gamma$.
To see this, let $x\in\ensuremath{\mathcal P_\kappa \lambda}$. Set $x'=x\mathfrak{c}up\{0\}\mathfrak{c}up\{\xi+1\; | \;\xi\in x\}$. Let $y\in\ensuremath{\mathcal P_\kappa \lambda}$ with $x'\subseteq y$ be such that for all $z\in U$ with $y\subseteq z$, $G\mathbin{\upharpoonright} x'=g_z\mathbin{\upharpoonright} x'$. Since $0\in x'$, it follows that $g_z(0)=G(0)=\gamma$. So for every $\xi\in x$, we have that $\xi,\xi+1\in x'$, so $F(\xi)=G(\xi+1)=g_z(\xi+1)=f_z^\gamma(\xi)$. Note that since $U$ is unbounded, there are such $z$ (meaning $z\in U$ with $y\subseteq z$), and for every such $z$, since $g_z(0)=G(0)=\gamma=\gamma(z)$, it follows that $z\in X_\gamma$. So $U\mathfrak{c}ap X_\gamma$ is unbounded.
We have reached a contradiction, since we assumed that $\vec{f^\gamma}$ splits all unbounded subsets of $X_\gamma$, which implies, by Theorem \ref{theorem:FunctionalListSplitsIffNoStrongBranch}, that it does not have a strong branch. \qed
\begin{definition}
Let $I$ be an ideal on $\ensuremath{\mathcal P_\kappa \lambda}$.
$I$ is \emph{strongly normal} iff every function $f:X\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that $X\in I^+$ and for every $x\in X$, $f(x)<x$, it follows that there is a $y$ such that $f^{-1}``\{y\}\in I^+$.
\end{definition}
Using methods from \mathfrak{c}ite{Carr1987:PklPartitionRelations}, it is not hard to improve the previous theorem as follows, assuming $\lambda^{{<}\kappa}=\lambda$.
\begin{theorem}
Suppose $\kappa$ is wildly $\lambda$-Shelah, where $\lambda^{{<}\kappa}=\lambda$. Then $I:=\mathcal{I}(\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded})$ is a strongly normal ideal on $\ensuremath{\mathcal P_\kappa \lambda}$.
\end{theorem}
\noindent Note: It was shown in \mathfrak{c}ite{Johnson1990:PartRelForIdealsOnPkl} that if $\kappa$ is $\lambda$-Shelah and $\mathrm{cf}(\lambda)\ge\kappa$, then $\lambda^{{<}\kappa}=\kappa$.
\proof First, let's write $\NSh{\kappa,\lambda}$ for the ideal of subsets $X$ of $\ensuremath{\mathcal P_\kappa \lambda}$ that are not $\lambda$-Shelah. Clearly then, $\NSh{\kappa,\lambda}\subseteq I$, since if $X\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ is not $\lambda$-Shelah, then $\fSP{\ensuremath{\mathcal P_\kappa \lambda}}{\mathcal{U}nbounded \mathbin{\upharpoonright} X}$ holds, or else, $X$ would be wildly $\lambda$-Shelah, and hence $\lambda$-Shelah. As a result, the same relation holds between the dual filters associated with these ideals: $\NShStar{\kappa,\lambda}\subseteq I^*$.
Fix a bijection $\varphi:\ensuremath{\mathcal P_\kappa \lambda}\longrightarrow\lambda$. It was shown in \mathfrak{c}ite[Prop.~3.4]{Carr1987:PklPartitionRelations} that if $\kappa$ is $\lambda$-Shelah and $\lambda=\lambda^{{<}\kappa}$ it follows that the sets $A=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \; x\mathfrak{c}ap\kappa\ \text{is an inaccessible cardinal}\}$ and
$B=\{x\in\ensuremath{\mathcal P_\kappa \lambda}\; | \;\varphi``(\mathcal{P}_{\kappa_x}(x))=x\}$ (where $\kappa_x=|x\mathfrak{c}ap\kappa|$) belong to $\NShStar{\kappa,\lambda}$. It follows that they belong to $I^*$.
Using these facts, the proof of \mathfrak{c}ite[Thm.~3.5]{Carr1987:PklPartitionRelations} goes through, to show the claim. Namely, given $X\in I^+$ (i.e., a set $X\subseteq\ensuremath{\mathcal P_\kappa \lambda}$ that is wildly $\lambda$-ineffable) and a function $f:X\longrightarrow\ensuremath{\mathcal P_\kappa \lambda}$ such that for all $x\in X$, $f(x)<x$, we have to show that there is a $y\in\ensuremath{\mathcal P_\kappa \lambda}$ such that $f^{-1}``\{y\}\in I^+$. Let $X_1=X\mathfrak{c}ap B$, and note that $X_1\in I^+$. Define $g:X_1\longrightarrow\lambda$ by $g(x)=\varphi(f(x))$. Then $g(x)\in x$, and hence $g$ is regressive. Since $I$ is normal, by Theorem \ref{theorem:FunctionalSplitIdealIsNormal}, there is an $\alpha<\lambda$ such that $g^{-1}``\{\alpha\}$ is in $I^+$. But $g^{-1}``\{\alpha\}=f^{-1}``\{\varphi^{-1}(\alpha)\}$, so we are done. \qed
\end{document} |
\begin{document}
\title[The Bochner-Riesz means for Fourier-Bessel expansions]
{The Bochner-Riesz means for Fourier-Bessel expansions: norm inequalities
for the maximal operator and almost everywhere convergence}
\author[\'O. Ciaurri and L. Roncal]{\'Oscar Ciaurri and Luz Roncal}
\address{Departamento de Matem\'aticas y Computaci\'on\\
Universidad de La Rioja\\ 26004 Logro\~no, Spain}
\email{[email protected], [email protected]}
\thanks{Research supported by the grant MTM2009-12740-C03-03 from Spanish Government.}
\keywords{Fourier-Bessel expansions, Bochner-Riesz means, almost
everywhere convergence, maximal operators, weighted inequalities}
\subjclass[2010]{Primary: 42C10, Secondary: 42C20, 42A45}
\begin{abstract}
In this paper, we develop a thorough analysis of the boundedness properties
of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel
expansions. For this operator, we study weighted and unweighted inequalities in the
spaces $L^p((0,1),x^{2\nu+1}\, dx)$. Moreover, weak and restricted weak type
inequalities are obtained for the critical values of~$p$. As a consequence,
we deduce the almost everywhere pointwise convergence of these means.
\end{abstract}
\maketitle
\section{Introduction and main results}
Let $J_{\nu}$ be the Bessel function of order $\nu$. For $\nu>-1$ we
have that
\[
\int_0^1 J_{\nu}(s_jx)J_{\nu}(s_kx)x\,dx= \frac12
(J_{\nu+1}(s_j))^2\delta_{j,k},\quad j,k=1,2,\dots
\]
where $\{s_j\}_{j\ge 1}$ denotes the sequence of successive
positive zeros of $J_{\nu}$. From the previous identity we can check
that the system of functions
\begin{equation}
\label{eq:FBesselSystemI}
\psi_j(x)=\frac{\sqrt{2}}{|J_{\nu+1}(s_j)|}x^{-\nu}J_{\nu}(s_jx),\quad
j=1,2,\dots
\end{equation}
is orthonormal and complete in $L^2((0,1),d\mu_\nu)$, with
$d\mu_\nu(x)=x^{2\nu+1}\, dx$ (for the completeness,
see~\cite{Hochstadt}). Given a function $f$ on $(0,1)$, its
Fourier series associated with this system, named as
Fourier-Bessel series, is defined by
\begin{equation}\label{SeriesCoeficientes}
f\sim \sum_{j=1}^\infty a_j(f)\psi_j,\qquad\text{with}\qquad
a_j(f)=\int_0^1 f(y)\psi_j(y)\,d\mu_{\nu}(y),
\end{equation}
provided the integral exists. When $\nu=n/2-1$, for $n\in
\mathbb{N}$ and $n\ge 2$, the functions $\psi_j$ are the
eigenfunctions of the radial Laplacian in the multidimensional
ball $B^n$. The eigenvalues are the elements of the sequence
$\{s_j^2\}_{j\ge 1}$. The Fourier-Bessel series corresponds with
the radial case of the multidimensional Fourier-Bessel expansions
analyzed in~\cite{Bal-Cor}.
For each $\delta>0$, we define the Bochner-Riesz means for
Fourier-Bessel series as
\begin{equation*}
\mathcal{B}_R^{\delta}(f,x)=\sum_{j\ge 1}
\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}a_j(f)\psi_j(x),
\end{equation*}
where $R>0$ and $(1-s^2)_+=\max\{1-s^2,0\}$. Bochner-Riesz means
are a regular summation method used oftenly in harmonic analysis.
It is very common to analyze regular summation methods for Fourier
series when the convergence of the partial sum fails. Ces\`{a}ro
means are other of the most usual summation methods. B.
Muckenhoupt and D. W. Webb \cite{Mu-We} give inequalities for
Ces\`{a}ro means of Laguerre polynomial series and for the
supremum of these means with certain parameters and $1<p\leq
\infty$. For $p=1$, they prove a weak type result. They also
obtain similar estimates for Ces\`{a}ro means of Hermite
polynomial series and for the supremum of those means in
\cite{Mu-We-Her}. An almost everywhere convergence result is
obtained as a corollary in \cite{Mu-We} and \cite{Mu-We-Her}. The
result about Laguerre polynomials is an extension of a previous
result in \cite{Stem}. This kind of matters has been also studied
by the first author and J. L. Varona in \cite{Ciau-Var} for the
Ces\`{a}ro means of generalized Hermite expansions. The Ces\`{a}ro
means for Jacobi polynomials were analyzed by S. Chanillo and B.
Muckenhoupt in \cite{Ch-Muc}. The Bochner-Riesz means themselves
have been analyzed for the Fourier transform and their boundedness
properties in $L^p(\mathbb{R}^n)$ is an important unsolved problem
for $n>2$ (the case $n=2$ is well understood, see \cite{Car-Sjo}).
The target of this paper is twofold. First we will analyze the
almost everywhere (a. e.) convergence, for functions in
$L^p((0,1),d\mu_\nu)$, of the Bochner-Riesz means for
Fourier-Bessel expansions. By the general theory \cite[Ch.
2]{Duoa}, to obtain this result we need to estimate the maximal
operator
\[
\mathcal{B}^{\delta}(f,x)=\sup_{R>0}\left|\mathcal{B}_R^{\delta}(f,x)\right|,
\]
in the $L^p((0,1),d\mu_\nu)$ spaces. A deep analysis of the
boundedness properties of this operator will be the second goal of
our paper. This part of our work is strongly inspired by the
results given in \cite{Ch-Muc} for the Fourier-Jacobi expansions.
Before giving our results we introduce some notation. Being
$p_0=\frac{4(\nu+1)}{2\nu+3+2\delta}$ and
$p_1=\frac{4(\nu+1)}{2\nu+1-2\delta}$, we define
\begin{align}
\label{eq:endpoints0} p_0(\delta)&=\begin{cases}
1,& \delta> \nu+1/2\text{ or } -1<\nu\le -1/2,\\
p_0,& \delta\le \nu+1/2\text{ and } \nu>-1/2,
\end{cases}\\
\notag p_1(\delta)&=\begin{cases}
\infty,& \delta> \nu+1/2\text{ or } -1<\nu\le -1/2,\\
p_1,& \delta\le \nu+1/2\text{ and } \nu>-1/2.
\end{cases}
\end{align}
Concerning to the a. e. convergence of the Bochner-Riesz means,
our result reads as follows
\begin{Theo}
\label{th:fin} Let $\nu>-1$, $\delta>0$, and $1\le p<\infty$.
Then,
\begin{equation*}
\mathcal{B}_R^\delta(f,x)\to f(x)\quad \text{a.
e., for $f\in L^p((0,1),d\mu_\nu)$}
\end{equation*}
if and only if $p_0(\delta)\le p$, where $p_0(\delta)$ is as in
\eqref{eq:endpoints0}.
\end{Theo}
Proof of Theorem \ref{th:fin} is contained in Section
\ref{sec:Proof Th1} and is based on the following arguments. On
one hand, to prove the necessity part, we will show the existence
of functions in $L^p((0,1),d\mu_{\nu})$ for $p<p_0(\delta)$ such
that $\mathcal{B}^\delta_R$ diverges for them. In order to do
this, we will use a reasoning similar to the one given by C.
Meaney in \cite{Meaney} that we describe in Section \ref{sec:Proof
Th1}. On the other hand, for the sufficiency, observe that the
convergence result follows from the study of the maximal operator
$\mathcal{B}^\delta f$. Indeed, it is sufficient to get
$(p_0(\delta),p_0(\delta))$-weak type estimates for this operator
and this will be the content of Theorem
\ref{th:AcDebilMaxRedonda}.
Regarding the boundedness properties of $\mathcal{B}^\delta f$ we
have the following facts. First, a result containing the
$(p,p)$-strong type inequality.
\begin{Theo}
\label{th:max} Let $\nu>-1$, $\delta>0$, and $1< p\le\infty$.
Then,
\begin{equation*}
\left\|\mathcal{B}^{\delta}f\right\|_{L^p((0,1),d\mu_{\nu})}\le C
\|f\|_{L^p((0,1),d\mu_{\nu})}
\end{equation*}
if and only if
\[
\begin{cases}
1<p\le \infty, &\text{for $-1<\nu\le -1/2$ or $\delta>\nu+1/2$},\\
p_0<p<p_1, &\text{for $\delta\le \nu+1/2$ and $\nu>-1/2$.}
\end{cases}
\]
\end{Theo}
In the lower critical value of $p_0(\delta)$ we can prove a
$(p_0(\delta),p_0(\delta))$-weak type estimate.
\begin{Theo}
\label{th:AcDebilMaxRedonda} Let $\nu>-1$, $\delta>0$, and
$p_0(\delta)$ be the number in \eqref{eq:endpoints0}. Then,
\[
\left\|\mathcal{B}^{\delta}f\right\|_{L^{p_0(\delta),\infty}((0,1),d\mu_{\nu})}\le
C \|f\|_{L^{p_0(\delta)}((0,1),d\mu_{\nu})},
\]
with $C$ independent of $f$.
\end{Theo}
Finally, for the upper critical value, when $0<\delta<\nu+1/2$ and
$\nu>-1/2$, it is possible to obtain a $(p_1,p_1)$-restricted weak
type estimate.
\begin{Theo}
\label{th:AcDebilRestMaxRedonda} Let $\nu>-1/2$ and
$0<\delta<\nu+1/2$. Then,
\[
\left\|\mathcal{B}^{\delta}\chi_E\right\|_{L^{p_1,\infty}((0,1),d\mu_{\nu})}\le
C \|\chi_E\|_{L^{p_1}((0,1),d\mu_{\nu})},
\]
for all measurable subsets $E$ of $(0,1)$ and $C$ independent of
$E$.
\end{Theo}
The previous results about norm inequalities are summarized in Figure 1 (case $-1<\nu \le -1/2$) and Figure 2 (case $\nu>-1/2$).
\begin{center}
\begin{tikzpicture}[scale=2.45]
\fill[black!10!white] (0.1,0.1) -- (2.1,0.1) -- (2.1,2.1) -- (0.1,2.1) -- cycle;
\draw[thick, dashed] (2.1,0.1) -- (2.1,2.1);
\draw[very thick] (0.1,2.1) -- (0.1,0.1);
\draw[very thin] (0.1,0.1) -- (2.15,0.1);
\node at (0.1,0) {$0$};
\node at (2.1,0) {$1$};
\node at (2.2,0.1) {$\frac{1}{p}$};
\node at (0.1,2.175) {$\delta$};
\draw (0,1.4) node [rotate=90] {\tiny{$(p,p)$-strong}};
\draw (2.2,1.4) node [rotate=90] {\tiny{$(p,p)$-weak}};
\draw (1.1,-0.2) node {Figure 1: case $-1<\nu\le-\tfrac{1}{2}$.};
\fill[black!10!white] (2.7,0.75) -- (3.35,0.1) -- (4.05,0.1) -- (4.7,0.75) -- (4.7,2.1) -- (2.7,2.1) -- cycle;
\draw[thick, dashed] (4.7,0.75) -- (4.7,2.1);
\draw[very thick] (2.7,2.1) -- (2.7,0.78);
\filldraw[fill=white] (2.7,0.75) circle (0.8pt);
\draw[thick, dotted] (2.72,0.73) -- (3.35,0.1);
\draw[thick, dashed] (4.05,0.1) -- (4.7,0.75);
\draw[very thin] (2.7,0.1) -- (4.75,0.1);
\draw[very thin] (2.7,0.1) -- (2.7,0.72);
\draw[very thin] (4.7,0.1) -- (4.7,0.13);
\node at (2.7,0) {$0$};
\node at (4.7,0) {$1$};
\node at (4.8,0.1) {$\frac{1}{p}$};
\node at (2.7,2.175) {$\delta$};
\draw (2.43,0.75) node {\tiny{$\delta=\nu+\tfrac{1}{2}$}};
\draw (3.35,0) node {\tiny{$\tfrac{2\nu+1}{4(\nu+1)}$}};
\draw (4.05,0) node {\tiny{$\tfrac{2\nu+3}{4(\nu+1)}$}};
\draw (2.6,1.4) node [rotate=90] {\tiny{$(p,p)$-strong}};
\draw (3.1,0.45) node [rotate=-45] {\tiny{$(p,p)$-restric. weak}};
\draw (4.3,0.45) node [rotate=45] {\tiny{$(p,p)$-weak}};
\draw (4.8,1.4) node [rotate=90] {\tiny{$(p,p)$-weak}};
\draw (3.7,-0.2) node {Figure 2: case $\nu>-\tfrac{1}{2}$.};
\end{tikzpicture}
\end{center}
At this point, a comment is in order. Note that J. E.
Gilbert \cite{Gi} also proves weak type norm inequalities for
maximal operators associated with orthogonal expansions. The method used
cannot be applied in our case, and the reason is the same as can
be read in \cite{Ch-Muc}, at the end of Sections 15 and 16
therein. Following the technique in \cite{Gi} we have to analyze
some weak type inequalities for Hardy operator and its adjoint with weights and these
inequalities do not hold for $p=p_0$ and $p=p_1$.
The proof of the sufficiency in Theorem \ref{th:max} will be
deduced from a more general result in which we analyze the
boundedness of the operator $\mathcal{B}^\delta f$ with potential
weights. Before stating it, we need a previous definition. We say
that the parameters $(b,B,\nu,\delta)$ satisfy the $C_p$
conditions if
\begin{align}
b& > \frac{-2(\nu+1)}{p} \,\,\, (\ge \text{ if }p=\infty), \label{ec:con1B}\\
B& < 2(\nu+1)\left(1-\frac1p\right) \,\,\, (\le \text{ if }
p=1), \label{ec:con2B}\\
b& > 2(\nu+1)\left(\frac12-\frac1p\right)-\delta-\frac12\,\,\,
(\ge \text{ if }p=\infty), \label{ec:con3B}\\
B& \le 2(\nu+1)\left(\frac12-\frac1p\right)+\delta+\frac12, \label{ec:con4B}\\
B &\le b \label{ec:con5B},
\end{align}
and in at least one of each of the following pairs the inequality
is strict: \eqref{ec:con2B} and \eqref{ec:con5B}, \eqref{ec:con3B}
and \eqref{ec:con5B}, and \eqref{ec:con4B} and \eqref{ec:con5B}
except for $p=\infty$. The result concerning inequalities with
potential weights is the following.
\begin{Theo}
\label{th:AcFuerteMaxRedonda} Let $\nu>-1$, $\delta>0$, and $1<
p\le\infty$. If $(b,B,\nu,\delta)$ satisfy the $C_p$ conditions,
then
\begin{equation*}
\left\|x^b\mathcal{B}^{\delta}f\right\|_{L^p((0,1),d\mu_{\nu})}\le
C \|x^Bf\|_{L^p((0,1),d\mu_{\nu})},
\end{equation*}
with $C$ independent of $f$.
\end{Theo}
A result similar to Theorem \ref{th:AcFuerteMaxRedonda} for the
partial sum operator was proved in \cite [Theorem 1]{GuPeRuVa}. It
followed from a weighted version of a general Gilbert's maximal
transference theorem, see \cite[Theorem 1]{Gi}. The weighted
extension of Gilbert's result given in \cite{GuPeRuVa} depended
heavily on the $A_p$ theory and it can not be used in our case
because it did not capture all the information relative to the
weights. On the other hand, it is also remarkable the paper by K.
Stempak \cite{Stem2} in which maximal inequalities for the partial
sum operator of Fourier-Bessel expansions and divergence and
convergence results are discussed.
The necessity in Theorem \ref{th:max} will follow by showing that
the operator $\mathcal{B}^\delta f$ is neither $(p_1,p_1)$-weak
nor $(p_0,p_0)$-strong for $\nu>-1/2$ and $0<\delta\le \nu+1/2$.
This is the content of the next theorems.
\begin{Theo}
\label{th:noweak} Let $\nu>-1/2$. Then
\[
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{p_1,\infty}((0,1),d\mu_\nu)}\ge
C (\log R)^{1/p_0},
\]
if $0<\delta<\nu+1/2$; and
\[
\sup_{\|f\|_{L^\infty((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{\infty}((0,1),d\mu_\nu)}\ge C
\log R,
\]
if $\delta= \nu+1/2$.
\end{Theo}
\begin{Theo}
\label{th:nostrong} Let $\nu>-1/2$. Then
\[
\sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\ge
C (\log R)^{1/p_0},
\]
if $0<\delta<\nu+1/2$; and
\[
\sup_{\|f\|_{L^1((0,1),d\mu_\nu)}=1}
\|\mathcal{B}^{\delta}_{R}f\|_{L^{1}((0,1),d\mu_\nu)}\ge C \log R,
\]
if $\delta= \nu+1/2$.
\end{Theo}
The paper is organized as follows. In the next section, we give
the proof of Theorem \ref{th:fin}. In Section
\ref{sec:proofAcFuerte} we first relate the Bochner-Riesz means
$\mathcal{B}_R^{\delta}$ to the Bochner-Riesz means operator
associated with the Fourier-Bessel system in the Lebesgue measure
setting. Then, we prove weighted inequalities for the supremum of
this new operator. With the connection between these means and the
operator $\mathcal{B}_R^{\delta}$, we obtain Theorem
\ref{th:AcFuerteMaxRedonda} and, as a consequence, the sufficiency
of Theorem \ref{th:max}. Sections
\ref{sec:ProofThAcDebilMaxRedonda} and \ref{sec:acdelrest} will be
devoted to the proofs of Theorems \ref{th:AcDebilMaxRedonda} and
\ref{th:AcDebilRestMaxRedonda}, respectively. The proofs of
Theorems \ref{th:noweak} and \ref{th:nostrong} are contained in
Section \ref{sec:negativeths}. One of the main ingredients in the
proofs of Theorems \ref{th:noweak} and \ref{th:nostrong} will be
Lemma \ref{lem:pol}, this lemma is rather technical and it will be
proved in the Section \ref{sec:techlemma}.
Throughout the paper, we will use the following notation: for each
$p\in[1,\infty]$, we will denote by $p'$ the conjugate of $p$,
that is, $\tfrac{1}{p}+\tfrac{1}{p'}=1$. We shall write $X\simeq Y$
when simultaneously $X\le C Y$ and $Y \le C X$.
\section{Proof of Theorem \ref{th:fin}}\label{sec:Proof Th1}
The proof of the sufficiency follows from Theorem
\ref{th:AcDebilMaxRedonda} and standard arguments.
In order to prove the necessity, let us see that, for
$0<\delta<\nu+1/2$ and $\nu>-1/2$, there exists a function $f\in
L^{p}((0,1),d\mu_{\nu})$, $p\in [1,p_0)$, for which
$\mathcal{B}_R^{\delta}(f,x)$ diverges. We follow some ideas
contained in \cite{Meaney} and \cite{Stem2}.
First, we need a few more ingredients. Recall the well-known
asymptotics for the Bessel functions (see \cite[Chapter 7]{Wat})
\begin{equation}\label{zero}
J_\nu(z) = \frac{z^\nu}{2^\nu \Gamma(\nu+1)} + O(z^{\nu+2}),
\quad |z|<1,\quad |\arg(z)|\leq\pi,
\end{equation}
and
\begin{equation}
\label{infty} J_\nu(z)=\sqrt{\frac{2}{\pi z}}\left[
\cos\left(z-\frac{\nu\pi}2
- \frac\pi4 \right) + O(e^{\mathop{\rm Im}(z)}z^{-1}) \right], \quad |z|
\ge 1,\quad |\arg(z)|\leq\pi-\theta,
\end{equation}
where $D_{\nu}=-(\nu\pi/2+\pi/4)$. It will also be useful the fact
that (cf. \cite[(2.6)]{OScon})
\begin{equation}
\label{eq:zerosCons} s_{j}=O(j).
\end{equation}
For our purposes, we need estimates for the $L^p$ norms of the
functions $\psi_j$. These estimates are contained in the following
lemma, whose proof can be read in \cite[Lemma 2.1]{Ci-RoWave}.
\begin{Lem}
\label{Lem:NormaFunc} Let $1\le p\le\infty$ and $\nu>-1$. Then,
for $\nu>-1/2$,
\begin{equation*}
\|\psi_j\|_{L^p((0,1),d\mu_{\nu})}\simeq
\begin{cases} j^{(\nu+1/2)-\frac{2(\nu+1)}{p}}, &
\text{if $p>\frac{2(\nu+1)}{\nu+1/2}$},\\
(\log j)^{1/p}, & \text{if $p=\frac{2(\nu+1)}{\nu+1/2}$},\\
1, & \text{if $p<\frac{2(\nu+1)}{\nu+1/2}$},
\end{cases}
\end{equation*}
and, for $-1<\nu\le-1/2$,
\begin{equation*}
\|\psi_j\|_{L^p((0,1),d\mu_{\nu})}\simeq
\begin{cases}
1, & \text{if $p<\infty$},\\
j^{\nu+1/2},& \text{if $p=\infty$}.
\end{cases}
\end{equation*}
\end{Lem}
We will also use a slight modification of a result by G. H. Hardy
and M. Riesz for the Riesz means of order $\delta$, that is
contained in \cite[Theorem 21]{HaRi}. We present here this
result, adapted to the Bochner-Riesz means. We denote by $S_R(f,x)$
the partial sum associated to the Fourier-Bessel expansion, namely
\[
S_R(f,x)=\sum_{0<s_j\le R} a_j(f)\psi_j(x).
\]
The result reads as follows.
\begin{Lem}
\label{lem:HardyRiesz} Suppose that $f$ can be expressed as a
Fourier-Bessel expansion and for some $\delta>0$ and $x\in(0,1)$
its Bochner-Riesz means $\mathcal{B}_R^\delta(f,x)$ converges to
$c$ as $R\rightarrow \infty$. Then, for $s_n\le R < s_{n+1}$,
\[
|S_R(f,x)-c|\le A_{\delta}n^{\delta}\sup_{0<t\le
s_{n+1}}|\mathcal{B}_{t}^{\delta}(f,x)|.
\]
\end{Lem}
By using this lemma, we can write
\begin{equation}
\label{ConsecuenciaLemaHardyRiesz}
|a_j(f)\psi_j(x)|=|(S_{s_{j}}(f,x)-c)-(S_{s_{j-1}}(f,x)-c)|\le
A_{\delta}j^\delta\sup_{0<t\le
s_{j+1}}|\mathcal{B}_{t}^{\delta}(f,x)|.
\end{equation}
Let us proceed with the proof of the necessity. Let $1\le p<p_0$.
Note that $p_0'=p_1$. Therefore,
$p'>p_0'>\tfrac{2(\nu+1)}{\nu+1/2}$, and
$\delta<\nu+1/2-\frac{2(\nu+1)}{p'}:=\lambda$. By Lemma
\ref{Lem:NormaFunc}, $\|\psi_j\|_{L^{p'}((0,1),d\mu_{\nu})}\ge C
j^{\lambda}$. Then, we have that the mapping $f\mapsto a_j(f)$,
where $a_j(f)$ was given in \eqref{SeriesCoeficientes}, is a
bounded linear functional on $L^{p}((0,1),d\mu_{\nu})$ with norm
bounded below by a constant multiple of $j^\lambda$. By uniform
boundedness principle, for $p$ conjugate to $p'$ and each $0\le
\varepsilon<\lambda$, there is a function $f_0\in L^p((0,1),
d\mu_{\nu})$ so that $a_j(f_0)j^{-\varepsilon}\rightarrow \infty$
as $j\rightarrow \infty$. By taking $\varepsilon=\delta$, we have
that
\begin{equation}\label{coeficienteInfinito}
a_j(f_0)j^{-\delta}\rightarrow \infty \quad \textrm{ as } \quad
j\rightarrow \infty.
\end{equation}
Suppose now that $B_R^\delta(f_0,x)$ converges. Then, by Egoroff's
theorem, it converges on a subset $E$ of positive measure in
$(0,1)$ and, clearly, we can think that $E\subset (\eta, 1)$ for
some fixed $\eta>0$. For each $x\in E$, we can consider $j$ such
that $s_j x\ge 1$ and, by \eqref{infty},
\begin{align*}
|a_j(f_0)\psi_j(x)|&=\big|a_j(f_0)
\Big(\frac{\sqrt2}{|J_{\nu+1}(s_j)|}
x^{-\nu}J_{\nu}(s_jx)\\
&-\frac{\sqrt2}{|J_{\nu+1}(s_j)|}x^{-\nu}
\Big(\frac{2}{\pi s_jx}\Big)^{1/2}
\cos(s_jx+D_{\nu})\Big)\\
&+a_j(f_0) \frac{\sqrt2}{|J_{\nu+1}(s_j)|}x^{-\nu}
\Big(\frac{2}{\pi s_jx}\Big)^{1/2}
\cos(s_jx+D_{\nu})\big|\\
&=Cs_j^{-1/2}\frac{\sqrt 2}{|J_{\nu+1}(s_j)|}|a_j(f_0)x^{-\nu-1/2}
\big(O((s_jx)^{-1})+\cos(s_jx+D_{\nu})\big)|\\
&\simeq |a_j(f_0)x^{-\nu-1/2}(\cos(s_jx+D_{\nu})+O((s_jx)^{-1}))|.
\end{align*}
By \eqref{ConsecuenciaLemaHardyRiesz} on this set $E$,
\[
|a_j(f_0)x^{-\nu-1/2}(\cos(s_jx+D_{\nu})+O((j)^{-1}))|\le
A_{\delta}j^{\delta}\sup_{0<t\le
s_{j+1}}|\mathcal{B}_{t}^{\delta}(f_0,x)|\le K_Ej^{\delta},
\]
uniformly on $x\in E$. We also used \eqref{eq:zerosCons} in the
latter. The inequality above is equivalent to
$$
|a_j(f_0)(\cos(s_jx+D_{\nu})+O(j^{-1}))|\le K_E
x^{\nu+1/2}j^{\delta}\le K_{E}j^{\delta}.
$$
Therefore,
\begin{equation}
\label{ec:boundFj}
|a_j(f_0)j^{-\delta}(\cos(s_jx+D_{\nu})+O((j)^{-1}))|\le K_E.
\end{equation}
Now, taking the functions
\[
F_j(x)=a_j(f_0)j^{-\delta}(\cos(s_jx+D_{\nu})+O(j^{-1})), \qquad
x\in E,
\]
and using an argument based on the Cantor-Lebesgue and
Riemann-Lebesgue theorems, see \cite[Section 1.5]{Meaney} and
\cite[Section IX.1]{Zyg}, we obtain that
\[
\int_E |F_j(x)|^2\, dx\ge C |a_j(f_0)j^{-\delta}|^2|E|,
\]
where, as usual, $|E|$ denotes the Lebesgue measure of the set
$E$. On the other hand, by \eqref{ec:boundFj},
\[
\int_E |F_j(x)|^2\, dx\le K_E^2 |E|.
\]
Then, from the previous estimates, it follows that
$|a_j(f_0)j^{-\delta}|\le C$, which contradicts
\eqref{coeficienteInfinito}.
\section{Bochner-Riesz means for Fourier-Bessel expansions in the Lebesgue
measure setting. Proof of Theorem
\ref{th:AcFuerteMaxRedonda}}\label{sec:proofAcFuerte}
For our convenience, we are going to introduce a new orthonormal
system. We will take the functions
\[
\phi_j(x)=\frac{\sqrt{2x}J_{\nu}(s_jx)}{|J_{\nu+1}(s_j)|},\quad
j=1,2,\dots.
\]
These functions are a slight modification of the functions
\eqref{eq:FBesselSystemI}; in fact,
\begin{equation}
\label{eq:Relation}
\phi_j(x)=x^{\nu+1/2}\psi_j(x).
\end{equation}
The system $\{\phi_j(x)\}_{j\ge1}$ is a complete orthonormal basis
of $L^2((0,1),dx)$.
In this case, the corresponding Fourier-Bessel expansion of a
function $f$ is
\[
f\sim\sum_{j=1}^{\infty}b_j(f) \phi_j(x), \qquad \text{with}
\qquad b_j(f)=\left(\int_0^1 f(y)\phi_j(y)\, dy\right)
\]
provided the integral exists, and for $\delta>0$ the Bochner-Riesz
means of this expansion are
\[
B_R^{\delta}(f,x)=\sum_{j\ge 1}
\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}b_j(f)\phi_j(x),
\]
where $R>0$ and $(1-s^2)_+=\max\{1-s^2,0\}$. It follows that
\[
B_R^{\delta}(f,x)=\int_0^1 f(y)K_R^\delta(x,y)\, dy
\]
where
\begin{equation}
\label{ec:kern} K_R^\delta(x,y)=\sum_{j\ge
1}\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}\phi_j(x)\phi_j(y).
\end{equation}
Our next target is the proof of Theorem
\ref{th:AcFuerteMaxRedonda}. Taking into account that
\[
\mathcal{B}_R^\delta f(x)=\int_0^1 f(y)\mathcal{K}_R^\delta(x,y)
\, d\mu_\nu(y),
\]
where
\[
\mathcal{K}_R^\delta(x,y)=\sum_{j\ge
1}\left(1-\frac{s_j^2}{R^2}\right)_+^{\delta}\psi_j(x)\psi_j(y),
\]
it is clear, from \eqref{eq:Relation}, that
$\mathcal{K}_R^\delta(x,y)=(xy)^{-(\nu+1/2)}K_R^{\delta}(x,y)$.
Then, it is verified that the inequality
\[
\|x^b\mathcal{B}^{\delta}(f,x)\|_{L^p((0,1),d\mu_{\nu})}\le
C\|x^Bf(x)\|_{L^p((0,1),d\mu_{\nu})}
\]
is equivalent to
\begin{equation*}
\|x^{b+(\nu+1/2)(2/p-1)}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le
C\|x^{B+(\nu+1/2)(2/p-1)}f(x)\|_{L^p((0,1),dx)},
\end{equation*}
that is, we can focus on the study of a weighted inequality for
the operator $B_R^{\delta}(f,x)$. The first results about
convergence of this operator can be found in \cite{Ci-Ro}.
We are going to prove an inequality of the form
\begin{equation*}
\|x^{a}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le C
\|x^{A}f(x)\|_{L^p((0,1),dx)}
\end{equation*}
for $\delta>0$, $1< p\leq \infty$, under certain conditions for
$a, A,\nu$ and $\delta$. Besides, a weighted weak type result for
$\sup_{R>0}|B_R^{\delta}(f,x)|$ will be proved for $p=1$. The
abovementioned conditions are the following. Let $\nu>-1$,
$\delta>0$ and $1\leq p\leq \infty$; parameters $(a,A,\nu,\delta)$
will be said to satisfy the $c_p$ conditions provided
\begin{align}
a & > -1/p-(\nu+1/2) \,\,\, (\ge \text{ if } p=\infty), \label{ec:con1}\\
A & < 1-1/p+(\nu+1/2)\,\,\, (\le \text{ if } p=1),\label{ec:con2}\\
a &> -\delta-1/p\,\,\, (\ge \text{ if }p=\infty),\label{ec:con3}\\
A &\le 1+\delta-1/p, \label{ec:con4}\\
A &\le a\label{ec:con5}
\end{align}
and in at least one of each of the following pairs the inequality
is strict: \eqref{ec:con2} and \eqref{ec:con5}, \eqref{ec:con3}
and \eqref{ec:con5}, and \eqref{ec:con4} and \eqref{ec:con5}
except for $p=\infty$.
The main results in this section are the following:
\begin{Theo}
\label{th:main1} Let $\nu>-1$, $\delta>0$ and $1< p\le \infty$. If
$(a, A, \nu, \delta)$ satisfy the $c_p$ conditions, then
\[
\|x^{a}B^{\delta}(f,x)\|_{L^p((0,1),dx)}\le C
\|x^{A}f(x)\|_{L^p((0,1),dx)},
\]
with $C$ independent of $f$.
\end{Theo}
\begin{Theo}
\label{th:main2} Let $\nu>-1$ and $\delta>0$. If $(a, A, \nu,
\delta)$ satisfy the $c_1$ conditions and
\[
E_{\lambda}=\left\{x\in (0,1)\colon x^{a}
\sup_{R>0}\left(|B_R^{\delta}(f,x)|\right)>\lambda \right\},
\]
then
\[
|E_{\lambda}|\leq C \frac{\|x^{A}
f(x)\|_{L^1((0,1),dx)}}{\lambda},
\]
with $C$ independent of $f$ and $\lambda$.
\end{Theo}
Note that, taking $a=b+(\nu+1/2)(2/p-1)$ and
$A=B+(\nu+1/2)(2/p-1)$, Theorem \ref{th:AcFuerteMaxRedonda}
follows from Theorem \ref{th:main1}.
The proofs of Theorem \ref{th:main1} and Theorem \ref{th:main2}
will be achieved by decomposing the square $(0,1)\times (0,1)$
into five regions and obtaining the estimates therein. The regions
will be:
\begin{align}
\label{regions}
\notag A_1&=\{(x,y):0 < x, y\leq 4/R\},\\
\notag A_2&=\{(x,y):4/R<\max\{x,y\}<1,\, |x-y|\le 2/R \},\\
A_3&=\{(x,y): 4/R \leq x < 1,\, 0 < y\leq x/2\},\\
\notag A_4&=\{(x,y):0 < x \leq y/2,\, 4/R \leq y< 1\}, \\
\notag A_5&=\{(x,y): 4/R < x < 1, \, x/2 < y< x- 2/R\}\\
\notag &\kern25pt\cup
\{(x,y): y/2 < x \leq y-2/R,\, 4/R \leq y<1\}.
\end{align}
Theorem~\ref{th:main1} and Theorem~\ref{th:main2} will follow by
showing that, if $1\leq p\leq \infty$, then
\begin{equation}
\label{eq:des_1} \left\|\sup_{R> 0}\int_0^1
y^{-A}x^a|K_R^\delta(x,y)||f(y)|\chi_{A_j}\,dy\right\|_{L^p((0,1),dx)}
\leq C\|f(x)\|_{L^p((0,1),dx)}
\end{equation}
holds for $j=1,3,4$ and that
\begin{equation}
\label{eq:des_2} \int_0^1
y^{-A}x^a|K_R^\delta(x,y)||f(y)|\chi_{A_j}\,dy \leq C M (f,x),
\end{equation}
for $j=2,5$, where $M$ is the Hardy-Littlewood maximal function of
$f$, and $C$ is independent of $R, x$ and $f$. These results and
the fact that $M$ is $(1,1)$-weak and $(p,p)$-strong if $1<p\leq
\infty$ complete the proofs.
To get \eqref{eq:des_1} and \eqref{eq:des_2} we will use a very
precise pointwise estimate for the kernel $K_R^\delta(x,y)$,
obtained in \cite{Ci-Ro}; there, it was shown that
\begin{equation}
\label{ec:kernel}
|K_R^\delta(x,y)|\le C \begin{cases} (xy)^{\nu+1/2}R^{2(\nu+1)}, &
(x,y) \in A_1,\\ R, & (x,y) \in A_2\\
\frac{\Phi_\nu(Rx)\Phi_{\nu}(Ry)}{R^{\delta}|x-y|^{\delta+1}}, &
(x,y) \in A_3\cup A_4 \cup A_5, \end{cases}
\end{equation}
with
\begin{equation}
\label{ec:aux}
\Phi_\nu(t)=\begin{cases}t^{\nu+1/2}, & \text{ if $0<t<2$},\\
1,& \text{ if $t\ge 2$}.\end{cases}
\end{equation}
The proof of \eqref{eq:des_2} follows from the given estimate for
the kernel $K_R^\delta(x,y)$ and $y^{-A}x^a\simeq C$ in $A_2\cup
A_5$ because $A\le a$. In the case of $A_2$, from
$|K_R^\delta(x,y)|\le C R$ we deduce easily the required
inequality. For $A_5$ the result is a consequence of
$\Phi_{\nu}(Rx)\Phi_\nu(Ry)\le C$ and of a decomposition of the
region in strips such that $R|x-y|\simeq 2^{k}$, with $k=0,\dots,
[\log_2 R]-1$; this can be seen in \cite[p. 109]{Ci-Ro}
In this manner, to complete the proofs of Theorem~\ref{th:main1}
and Theorem~\ref{th:main2} we only have to show \eqref{eq:des_1}
for $j=1,3,4$ in the conditions $c_p$ for $1\le p \le \infty$, and
this is the content of Corollary~\ref{cor:corolario2} in
Subsection~\ref{subsec:reg1}. In its turn,
Corollary~\ref{cor:corolario2} follows from Lemmas~\ref{lem:lema7}
and~\ref{lem:lema8} in the same subsection. Previously, Subsection
\ref{subsec:lemmas} contains some technical lemmas that will be
used in the proofs of Lemmas~\ref{lem:lema7} and~\ref{lem:lema8}.
\subsection{Technical Lemmas}
\label{subsec:lemmas}
To prove \eqref{eq:des_1} for $j=1,3,4$ we will use an
interpolation argument based on six lemmas. These are stated
below. They are small modifications of the six lemmas contained in
Section 3 of \cite{Mu-We} where a sketch of their proofs can be
found.
\begin{Lem}
\label{lem:lema1} Let $\xi_0>0$, if $r<-1$, $r+t\leq-1$ and
$r+s+t\leq-1$, then for $p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi_0\leq\xi\leq x}\xi^s
\int_\xi^x y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq
C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $r\leq0$, $r+t\leq-1$ and
$r+s+t\leq-1$ with equality holding in at most one of the first
two inequalities, then this holds for $p=\infty$.
\end{Lem}
\begin{Lem}
\label{lem:lema2} Let $\xi_0>0$, if $t\leq0$, $r+t\leq-1$ and
$r+s+t\leq-1$, with strict inequality in the last two in case of
equality in the first, then for $p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi_0\leq\xi\leq x}\xi^s
\int_x^\infty y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq
C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $t<-1$, $r+t\leq-1$ and
$r+s+t\leq-1$, then this holds for $p=\infty$.
\end{Lem}
\begin{Lem}
\label{lem:lema3} If $s<0$, $s+t\leq0$ and $r+s+t\leq-1$,with
equality holding in at most one of the last two inequalities, then
for $p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s \int_x^\xi
y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $s<0$, $s+t\leq-1$ and
$r+s+t\leq-1$ this holds for $p=\infty$.
\end{Lem}
\begin{Lem}
\label{lem:lema4} If $t\leq0$, $s+t\leq0$ and $r+s+t\leq-1$,with
strict inequality holding in the first two in case the third is an
equality, then for $p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s
\int_\xi^\infty y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq
C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $t<-1$, $s+t\leq-1$ and
$r+s+t\leq-1$ then this holds for $p=\infty$.
\end{Lem}
\begin{Lem}
\label{lem:lema5} If $s<0$, $r+s<-1$ and $r+s+t\leq-1$, then for
$p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{\xi\geq x}\xi^s \int_1^x
y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $s<0$, $r+s\leq 0$ and
$r+s+t\leq-1$, with equality holding in at most one of the last
two inequalities, this holds for $p=\infty$.
\end{Lem}
\begin{Lem}
\label{lem:lema6} If $r<-1$, $r+s<-1$ and $r+s+t\leq-1$, then for
$p=1$
\[
\left\|x^r\chi_{[1,\infty)}(x) \sup_{1\leq\xi\leq x}\xi^s
\int_1^{\xi} y^t|f(y)|\,dy\right\|_{L^p((0,\infty),dx)}\leq
C\|f(x)\|_{L^p((0,\infty),dx)}
\]
with $C$ independent of $f$. If $r\leq0$, $r+s\leq0$ and
$r+s+t\leq-1$, with equality in at most one of the last two
inequalities, this holds for $p=\infty$.
\end{Lem}
\subsection{Proofs of Theorem~\ref{th:main1} and Theorem~\ref{th:main2}
for regions $A_1$, $A_3$ and $A_4$}
\label{subsec:reg1}
This section contains the proofs of the inequality
\eqref{eq:des_1} for regions $A_1$, $A_3$ and $A_4$. The results
we will prove are included in the following
\begin{Lem}
\label{lem:lema7} If $\nu>-1$, $\delta>0$, $R>0$, $j=1, 3, 4$ and
$(a, A, \nu, \delta)$ satisfy the $c_1$ conditions, then
\eqref{eq:des_1} holds for $p=1$ with $C$ independent of $f$.
\end{Lem}
\begin{Lem}
\label{lem:lema8} If $\nu>-1$, $\delta>0$, $R>0$, $j=1, 3, 4$ and
$(a, A, \nu, \delta)$ satisfy the $c_{\infty}$ conditions, then
\eqref{eq:des_1} holds for $p=\infty$ with $C$ independent of $f$.
\end{Lem}
\begin{Cor}
\label{cor:corolario2} If $1\leq p\leq\infty$, $\nu>-1$,
$\delta>0$, $R>0$, $(a, A, \nu, \delta)$ satisfy the $c_p$
conditions and $j=1,3,4$, then \eqref{eq:des_1} holds with $C$
independent of $f$.
\end{Cor}
\textbf{Proof of Corollary~\ref{cor:corolario2}}. It is enough to
observe that if $1< p<\infty$ and $(a, A, \nu, \delta)$ satisfy
the $c_p$ conditions, then $(a-1+1/p, A-1+1/p, \nu, \delta)$
satisfy the $c_1$ conditions. So, by Lemma~\ref{lem:lema7}
\begin{multline*}
\left\|\sup_{R\geq 0}\int_0^1
y^{-A+1-1/p}x^{a-1+1/p}|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\,
dy \right\|_{L^1((0,1),dx)}\\\leq C\|f(x)\|_{L^1((0,1),dx)},
\end{multline*}
and this is equivalent to
\[
\int_0^1 x^{a+1/p}\left(\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right)
\frac{dx}{x}\leq C\int_0^1 x^{A+1/p}|f(x)|\frac{dx}{x},
\]
where $j=1,3,4$. Similarly, if $(a,A,\nu,\delta)$ verify the $c_p$
conditions, then $(a+1/p, A+1/p, \nu, \delta )$ satisfy the
$c_{\infty}$ conditions. Hence, by Lemma~\ref{lem:lema8}
\begin{multline*}
\left\|x^{a+1/p}\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right\|_{L^{\infty}((0,1),dx)}
\\\leq C\|x^{A+1/p}f(x)\|_{L^{\infty}((0,1),dx)}.
\end{multline*}
Now, we can use the Marcinkiewicz interpolation theorem to obtain the inequality
\begin{multline*}
\int_0^1 \left(x^{a+1/p} \left(\sup_{R\geq 0}\int_0^1
|K_R^{\delta}(x,y)|\chi_{A_j}(x,y)|f(y)|\, dy\right)\right)^p
\frac{dx}{x}\\
\leq
C\int_0^1 \left(x^{A+1/p}|f(x)|\right)^p\frac{dx}{x},
\end{multline*}
for $1<p<\infty$ and the proof is finished.
Finally, we will prove Lemmas~\ref{lem:lema7} and~\ref{lem:lema8}
for $A_j$, $j=1, 3$ and $4$, separately.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8}
for $A_1$}. First of all, we have to note that $B_R^\delta
(f,x)=0$ when $0<R<s_1$, being $s_1$ the first positive zero of
$J_\nu$. Using the estimate~\eqref{ec:kernel}, the left side of
\eqref{eq:des_1} in this case is bounded by
\[
C\left\|x^{a+\nu+1/2}\chi_{[0,1]}(x)\sup_{s_1<R\leq
4/x}R^{2(\nu+1)} \int_0^{4/R}y^{-A+\nu+1/2}|f(y)|\,
dy\right\|_{L^p((0,1),dx)}.
\]
Making the change of variables $x=4/u$ and $y=4/v$, we have
\[
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_R^\infty
v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)},
\]
where $\|\cdot\|_{L^p((0,\infty),du)}$ denotes the $L^p$ norm in
the variable $u$, and
\[
g(v)=v^{-2/p}|f(4v^{-1})|.
\]
Note that function $g(v)$ is supported in $(1,\infty)$ and
$\|g\|_{L^p((0,\infty),du)}=\|f\|_{L^p((0,1),dx)}$. The function
$g$ will be used through the subsection, but the value $4$ may be
changed by another one, at some points, without comment. Now,
splitting the inner integral at $u$, we obtain the sum of
\begin{equation}
\label{ec:pa11}
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_R^u
v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)}
\end{equation}
and
\begin{equation}
\label{ec:pa12}
C\left\|u^{-a-\nu-\frac12-\frac2p}\chi_{[4,\infty)}(u)
\sup_{s_1\leq R \leq u}R^{2(\nu+1)}\int_u^\infty
v^{A-(\nu+\frac12)-2+\frac2p}g(v)\,dv\right\|_{L^p((0,\infty),du)}.
\end{equation}
From Lemma~\ref{lem:lema1} we get the required estimate for
\eqref{ec:pa11}, using conditions \eqref{ec:con1} and
\eqref{ec:con5}; Lemma~\ref{lem:lema2} is applied to inequality
\eqref{ec:pa12}, there we need conditions \eqref{ec:con2} and
\eqref{ec:con5} and the restriction on them. This completes the
proof of Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=1$.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8}
for $A_3$}. Clearly, the left side of \eqref{eq:des_1} is bounded
by
\[
C\left\|x^a \chi_{[4/R,1]}(x)\sup_{4/x\leq R}\int_0^{x/2}y^{-A}
|K_R^{\delta}(x,y)||f(y)|\, dy\right\|_{L^p((0,1),dx)}.
\]
Splitting the inner integral at $2/R$, using the bound for the
kernel given in \eqref{ec:kernel} and the definition of
$\Phi_\nu$, we have this expression majorized by the sum of
\begin{equation}
\label{ec:pa21} \left\|x^a \chi_{[0,1]}(x)\sup_{4/x\leq
R}\int_0^{2/R}|f(y)|\frac{(Ry)^{\nu+1/2}y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\,
dy\right\|_{L^p((0,1),dx)}
\end{equation}
and
\begin{equation}
\label{ec:pa22} \left\|x^a \chi_{[0,1]}(x)\sup_{4/x\leq
R}\int_{2/R}^{x/2}\frac{|f(y)|y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy
\right\|_{L^p((0,1),dx)}.
\end{equation}
For \eqref{ec:pa21}, taking into account that $|x-y|\simeq x$ in
$A_3$, the changes of variables $x=4/u$, $y=2/v$ give us
\[
\left\|u^{-a+(\delta+1)-\frac 2p}\chi_{[4,\infty)}(u)\sup_{u\leq
R} R^{-\delta+(\nu+1/2)}\int_R^{\infty}v^{-(\nu+1/2)+A+\frac
2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}.
\]
Lemma~\ref{lem:lema4} can be used here. The required conditions
for $p=1$ are \eqref{ec:con2}, \eqref{ec:con4} and \eqref{ec:con5}
with the restriction in the pairs therein. For $p=\infty$ the same
inequalities are needed.
On the other hand, in \eqref{ec:pa22}, using again that $|x-y|\simeq
x$, by changing of variables $x=4/u$ and $y=2/v$ we have
\begin{multline*}
C\left\|u^{-a+(\delta+1)-\frac
2p}\chi_{[4,\infty)}(u)\sup_{u\leq R}R^{-\delta}\int_{2u}^R
v^{A+\frac 2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}\\
\leq C\left\|u^{-a+(\delta+1)-\frac
2p}\chi_{[4,\infty)}(u)\sup_{u\leq R}R^{-\delta}\int_u^R
v^{A+\frac 2p-2}g(v)\, dv\right\|_{L^p((0,\infty),du)}.
\end{multline*}
Lemma~\ref{lem:lema3} can then be applied. For $p=1$, we need
$\delta>0$, which is an hypothesis, and \eqref{ec:con4} and
\eqref{ec:con5} with its corresponding restriction. For $p=\infty$
the inequalities are the same, with the requirement that
\eqref{ec:con4} is strict. This completes the proof of
Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=3$.
\textbf{Proof of Lemma~\ref{lem:lema7} and Lemma~\ref{lem:lema8}
for $A_4$}. In this case, the left hand side of \eqref{eq:des_1}
is estimated by
\[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{R>4}\int_{\max(4/R,2x)}^1
y^{-A}|K_R^{\delta}(x,y)||f(y)|\, dy\right\|_{L^p((0,1),dx)}.
\]
To majorize this, we decompose the $R$-range in two regions:
$4<R\leq 2/x$ and $R\geq 2/x$. In this manner, with the bound for
the kernel given in \eqref{ec:kernel} and the definition of
$\Phi_\nu$, the previous norm is controlled by the sum of
\[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{4<R\leq 2/x}
\int_{4/R}^1 |f(y)|\frac{(Rx)^{\nu+1/2}y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy
\right\|_{L^p((0,1),dx)}
\]
and
\[
C\left\|x^a \chi_{[0,1/2]}(x)\sup_{R\geq 2/x}\int_{2x}^1
\frac{|f(y)|y^{-A}}{R^{\delta}|x-y|^{\delta+1}}\, dy\right\|_{L^p((0,1),dx)}.
\]
Next, using that $|x-y|\simeq y$ in $A_4$, with the changes of
variables $x=2/u$ and $y=1/v$ the previous norms are controlled by
\begin{equation}
\label{ec:pa31} C\left\|u^{-a-\frac 2p-(\nu+\frac
12)}\chi_{[4,\infty)}(u)\sup_{4<R\leq u}R^{-\delta+(\nu +\frac
12)} \int_1^{R/4}v^{A+\frac 2p
-2+(\delta+1)}g(v)\,dv\right\|_{L^p((0,\infty),du)}
\end{equation}
and
\begin{equation}
\label{ec:pa32} C\left\|u^{-a-\frac
2p}\chi_{[4,\infty)}(u)\sup_{R\geq u}R^{-\delta}\int_1^{u/4}
v^{A+\frac 2p -2+(\delta+1)}g(v)\,dv\right\|_{L^p((0,\infty),du)}.
\end{equation}
In \eqref{ec:pa31}, we use Lemma~\ref{lem:lema6}; for $p=1$,
conditions \eqref{ec:con1}, \eqref{ec:con3} and \eqref{ec:con5}
are needed; we need the same for $p=\infty$. For \eqref{ec:pa32},
Lemma~\ref{lem:lema5} requires the hypothesis $\delta>0$ and
conditions \eqref{ec:con3} and \eqref{ec:con5} for $p=1$ and the
same for $p=\infty$ with the restrictions in the pairs therein.
This proves Lemmas~\ref{lem:lema7} and~\ref{lem:lema8} for $j=4$.
\section{Proof of Theorem \ref{th:AcDebilMaxRedonda}}
\label{sec:ProofThAcDebilMaxRedonda}
Now we shall prove Theorem \ref{th:AcDebilMaxRedonda}. First note
that, by \eqref{eq:Relation}, we can write
\begin{equation*}
\mathcal{B}_R^{\delta}(f,x)
=\int_0^1f(y)\left(\frac{y}{x}\right)^{\nu+1/2}K_R^{\delta}(x,y)\,dy,
\end{equation*}
where $K_R^\delta$ is the kernel in \eqref{ec:kern}. By taking
$g(y)=f(y)y^{\nu+1/2}$, to prove the result it is enough to check
that
\[
\int_{E}\,d\mu_\nu(x)\le \frac{C}{\lambda^{p}}
\int_0^1|g(x)|^{p}x^{(\nu+1/2)(2-p)}\,dx,
\]
where $E=\left\{x\in(0,1):
\sup_{R>0}x^{-(\nu+1/2)}\int_0^1|g(y)||K_R^{\delta}(x,y)|\,dy>\lambda\right\}$
and $p=p_0(\delta)$. We decompose $E$ into four regions, such that
$E=\bigcup_{i=1}^{4}J_i$, where
\begin{equation*}
J_i=\left\{x\in(0,1): \sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}|g(y)|
\chi_{B_i}(x,y)|K_R^{\delta}(x,y)|\,dy>\lambda\right\}
\end{equation*}
for $i=1,\dots,4$, with $B_1=A_1$, $B_2=A_2\cup A_5$, $B_3=A_3$,
and $B_4=A_4$ where the sets $A_i$ were defined in
\eqref{regions}. Note also that
$\int_{E}\,d\mu_\nu(x)\le\sum_{i=1}^4\int_{J_i}\,d\mu_\nu(x)$,
then we need to prove that
\begin{equation}
\label{ec:boundweak}
\int_{J_i}\,d\mu_\nu(x)\le \frac{C}{\lambda^{p}}
\int_0^1|g(x)|^{p}x^{(\nu+1/2)(2-p)}\,dx,
\end{equation}
for $i=1,\dots,4$ and $p=p_0(\delta)$. At some points along the
proof we will use the notation
\begin{equation}
\label{eq:integral}
I_p:=\int_0^{1}|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy.
\end{equation}
In $J_1$, by applying \eqref{ec:kernel} and H\"older inequality
with $p=p_0$, we have
\begin{multline*}
x^{-(\nu+1/2)}\int_0^{1}|g(y)|\chi_{B_1}(x,y)
|K_R^{\delta}(x,y)|\,dy\\
\begin{aligned}
&\le Cx^{-(\nu+1/2)}\int_0^{4/R}
|g(y)|(xy)^{\nu+1/2}R^{2(\nu+1)}\,dy\\
&\le C R^{2(\nu+1)}\left(\int_0^{4/R}
|g(y)|^{p_0}y^{(\nu+1/2)(2-p_0)}\,dy\right)^{1/p_0}
\left(\int_0^{4/R}y^{(2\nu+1)}\,dy\right)^{1/p'_0}\\
&=C R^{\frac{2(\nu+1)}{p_0}}
\left(\int_0^{4/R}|g(y)|^{p_0}y^{(\nu+1/2)(2-p_0)}\,dy\right)^{1/p_0}
\le C R^{\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}.
\end{aligned}
\end{multline*}
Therefore,
\begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy
&\le C\sup_{R>0}\chi_{[0,4/R]}(x)R^{\frac{2(\nu+1)}{p_0}}
I_{p_0}^{1/p_0}\\&\le C
x^{-\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}.
\end{align*}
In the case $p=1$, it is clear that
\[
x^{-(\nu+1/2)}\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy\le
C R^{2(\nu+1)}I_1
\]
and
\[
\sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_1}(x,y)|K_R^{\delta}(x,y)|\,dy\le C
x^{-2(\nu+1)}I_1.
\]
Hence, for $p=p_0(\delta)$,
\[
J_1\subseteq \{x\in(0,1): C x^{-\frac{2(\nu+1)}{p}}I_{p}^{1/p} >\lambda\},
\]
and this gives \eqref{ec:boundweak} for $i=1$.
In $J_3$, note first that
\begin{multline*}
\sup_{R>0}x^{-(\nu+1/2)}
\int_0^{1}|g(y)|\chi_{B_3}(x,y)|K_R^{\delta}(x,y)|\,dy\\
=\sup_{R>0}x^{-(\nu+1/2)}\chi_{[4/R,1]}(x)
\left(\int_0^{2/R}|g(y)||K_R^{\delta}(x,y)|\,dy+
\int_{2/R}^{x/2}|g(y)||K_R^{\delta}(x,y)|\,dy\right)\\:=R_1+R_2.
\end{multline*}
For $R_1$, using \eqref{ec:kernel}, the inequality $x/2<x-y$,
which holds in $B_3$, and H\"older inequality with $p=p_0$,
\begin{align*}
R_1
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)
\int_0^{2/R}R^{\nu+1/2-\delta}y^{\nu+1/2}|g(y)|\,dy\\
&\le\sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)
R^{\nu+1/2-\delta}R^{-\frac{2(\nu+1)}{p'_0}}I_{p_0}^{1/p_0}
\le C x^{-\frac{2(\nu+1)}{p_0}}I_{p_0}^{1/p_0},
\end{align*}
where $I_{p_0}$ is the same as in \eqref{eq:integral}. In the case
$p=1$, the estimate $R_1\le C x^{-2(\nu+1)}I_{1}$ can be obtained
easily.
On the other hand, for $R_2$, by using \eqref{ec:kernel} and
H\"older inequality with $p=p_0$ again,
\begin{align*}
R_2
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}
\chi_{[4/R,1]}(x)I_{p_0}^{1/p_0}R^{-\delta}
\left(\int_{2/R}^{x/2}y^{-(\nu+1/2)\frac{(2-p_0)p'_0}{p_0}}\,dy\right)^{1/p'_0}\\
&\le \sup_{R>0}x^{-(\nu+3/2+\delta)}
\chi_{[4/R,1]}(x)I_{p_0}^{1/p_0}R^{-\delta}
\left(\int_{2/R}^{x/2}y^{(\nu+1/2)\frac{2-p_0}{1-p_0}}\,dy\right)^{1/p'_0}.
\end{align*}
Using that $(\nu+1/2)\frac{2-p_0}{1-p_0}<-1$ and $4/R<x<1$, we
have that
\begin{align*}
R^{-\delta}\left(\int_{2/R}^{x/2}
y^{(\nu+1/2)\frac{2-p_0}{1-p_0}}\,dy\right)^{1/p'_0}
\le C\left(R^{-(\nu+1/2)\frac{2-p_0}{1-p_0}-1}\right)^{1/p'_0}
R^{-\delta}= C
\end{align*}
and the last inequality is true because the exponent of $R$ is
zero. Then
\[
R_2\le C x^{\frac{-2(\nu+1)}{p_0}}I_{p_0}^{1/p_0}.
\]
In the case $p=1$ applying H\"older inequality, then
\begin{equation*}
R_2\le \sup_{R>0}x^{-(\nu+3/2+\delta)}\chi_{[4/R,1]}(x)I_1
\,R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}.
\end{equation*}
Now, if $\nu+1/2>0$ and $\nu+1/2<\delta$,
\begin{multline*}
\sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}\\
=C\sup_{R>0}\chi_{[4/R,1]}(x)R^{\nu+1/2-\delta}\le
Cx^{-\nu-1/2+\delta};
\end{multline*}
and if $\nu+1/2\le0$,
\begin{multline*}
\sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}\sup_{y\in[2/R,x/2]}y^{-(\nu+1/2)}\\
=C\sup_{R>0}\chi_{[4/R,1]}(x)R^{-\delta}x^{-(\nu+1/2)}\le
Cx^{-\nu-1/2+\delta}.
\end{multline*}
In this manner
\[
R_2\le C x^{-2(\nu+1)}I_{1}.
\]
Therefore, collecting the estimates for $R_1$ and $R_2$ for
$p=p_0$ and $p=1$, we have shown that
\[
J_3\subseteq \{x\in(0,1): C x^{\frac{-2(\nu+1)}{p}}(x)I^{1/p}
>\lambda\},
\]
hence we can deduce \eqref{ec:boundweak} for $i=3$.
For the region $J_4$, we proceed as follows
\begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}&
\int_{0}^1|g(y)|\chi_{B_4}(x,y)|K_R^\delta(x,y)|\,dy\\
&\le \sup_{R>0}x^{-(\nu+1/2)}\chi_{[0,2/R]}(x)
\int_{4/R}^1|g(y)||K_R^\delta(x,y)|\,dy\\
&\kern20pt+\sup_{R>0}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)
\int_{2x}^1|g(y)||K_R^\delta(x,y)|\,dy\\
&\le C\sup_{R>0}x^{-(\nu+1/2)}\chi_{[0,2/R]}(x)(Rx)^{\nu+1/2}
\int_{4/R}^1\frac{|g(y)|}{R^{\delta}|x-y|^{\delta+1}}\,dy\\
&\kern20pt+ C\sup_{R>0}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)
\int_{2x}^1\frac{|g(y)|}{R^{\delta}|x-y|^{\delta+1}}\,dy:=S_1+S_2.
\end{align*}
We first deal with $S_1$, we use that $y-x>y/2$, then
\begin{align*}
S_1\le &C\sup_{R>0}\chi_{[0,2/R]}(x) R^{\nu+1/2-\delta}
\int_{4/R}^1\frac{|g(y)|}{y^{\delta+1}}\,dy\\
&\le C\sup_{R>0}\chi_{[0,2/R]}(x) R^{\nu+1}\int_{4/R}^1\frac{|g(y)|}{\sqrt{y}}\,dy
\le C x^{-(\nu+1)}\int_x^1\frac{|g(y)|}{\sqrt{y}}\,dy.
\end{align*}
Now for $p=p_0$ or $p=1$, we have that $2\nu+1-p(\nu+1)>-1$ and
Hardy's inequality \cite[Lemma 3.14, p. 196]{SteinWeiss} is
applied in the following estimate
\begin{align*}
\int_0^1|S_1(x)|^{p}x^{2\nu+1}\,dx& \le C
\int_0^1\left(\int_x^1\frac{|g(y)|}{\sqrt{y}}\,dy\right)^{p}
x^{2\nu+1-p(\nu+1)}\,dx\\
&\le C \int_0^1\left|\frac{g(y)}{\sqrt y}\right|^{p}y^{2\nu+1-p\nu}\,dy
=C\int_0^1|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy.
\end{align*}
Concerning $S_2$, observe that
$\sup_{R>0}\chi_{[2/R,1]}(x)R^{-\delta}\le Cx^{\delta}$, thus
\[
S_2\le C
x^{-\nu-1/2+\delta}\int_x^1\frac{|g(y)|}{y^{\delta+1}}\,dy.
\]
Since for $p=p_0$ or $p=1$ we have that
$2\nu+1-p(\nu+1/2-\delta)>-1$, we can use again Hardy's inequality
to complete the required estimate. Indeed,
\begin{align*}
\int_0^1|S_2(x)|^{p}x^{2\nu+1}\,dx&
\le C\int_0^1\left(\int_x^1\frac{|g(y)|}{y^{\delta+1}}\,dy\right)^{p}
x^{2\nu+1-p(\nu+1/2-\delta)}\,dx\\
&\le C\int_0^1\left|\frac{g(y)}{y^{\delta+1}}\right|^{p}
y^{2\nu+1-p(\nu+1/2-\delta)+p}\,dy\\&
=C\int_0^1|g(y)|^{p}y^{(\nu+1/2)(2-p)}\,dy.
\end{align*}
With the inequalities for $S_1$ and $S_2$, we can conclude
\eqref{ec:boundweak} for $i=4$.
To prove \eqref{ec:boundweak} for $i=2$ we define, for $k$ a
nonnegative integer, the intervals
\[
I_k=[2^{-k-1},2^{-k}], \qquad N_k=[2^{-k-3},2^{-k+2}]
\]
and the function $g_k(y)=|g(y)|\chi_{I_k}(y)$. By using
\eqref{ec:kernel} for $x/2<y<2x$, with $x\in (0,1)$, we have the
bound
\[
|K_R^\delta (x,y)|\le \frac{C}{R^{\delta}(|x-y|+2/R)^{\delta+1}}.
\]
Then
\[
J_{2}\subset \left\{x\in (0,1): \sup_{R>0}\sum_{k=0}^\infty
\int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy> C \lambda
x^{\nu+1/2}\right\}.
\]
Since at most three of these integrals are not zero for each $x\in
(0,1)$
\begin{align*}
J_2&\subset \bigcup_{k=0}^\infty \left\{x\in (0,1):
3\sup_{R>0}\int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy> C \lambda
x^{\nu+1/2}\right\}\\
&\subset \bigcup_{k=0}^\infty \left\{x\in N_k : M(g_k,x)> C
\lambda x^{\nu+1/2}\right\}
\end{align*}
where in the las step we have used that
\[
\sup_{R>0}\int_{x/2}^{\min{\{2x,1\}}}
\frac{g_k(t)}{R^{\delta}(|x-y|+2/R)^{\delta+1}}\, dy\le C
M(g_k,x).
\]
By using the estimate $x\simeq 2^{-k}$ for $x\in N_k$, we can
check easily that
\[
J_2\subset \bigcup_{k=1}^\infty \left\{x \in N_k : M(g_k,x)> C
\lambda 2^{-k(\nu+1/2)}\right\}.
\]
Finally by using again that $x\simeq 2^{-k}$ for $x\in I_k, N_k$ and the
weak type norm inequality for the Hardy-Littlewood maximal
function we have
\begin{align*}
\int_{J_2}x^{2\nu+1}\, dx &\le C \sum_{k=0}^\infty 2^{-k(2\nu+1)}
\int_{\left\{x\in N_k : M(g_k,x)> C \lambda
2^{-k(\nu+1/2)}\right\}}\, dx\\&\le C \sum_{k=0}^\infty
\frac{2^{pk(\nu+1/2)-k(2\nu+1)}}{\lambda^p}\int_{I_k}|g(y)|^p\,
dy\\&\le \frac{C}{\lambda^p}\int_0^1 |g(y)|^p y^{(\nu+1/2)(2-p)}\,
dy
\end{align*}
and the proof is complete.
\section{Proof of Theorem \ref{th:AcDebilRestMaxRedonda}}
\label{sec:acdelrest}
To conclude the result we have to prove \eqref{ec:boundweak} with
$g(x)=\chi_E(x)$ and $p=p_1$. For $J_1$ and $J_2$ the result
follows by using the steps given in the proof of Theorem
\ref{th:AcDebilMaxRedonda} for the same intervals. To analyze
$J_3$ we proceed as we did for $J_4$ in the proof of Theorem
\ref{th:AcDebilMaxRedonda}. In this case we obtain that
\begin{multline*}
\sup_{R>0}x^{-(\nu+1/2)}\int_0^1
|g(y)|\chi_{B_3}(x,y)|K_r^\delta(x,y)|\\\le
C\left(x^{-(\nu+1)}\int_0^x \frac{|g(y)|}{\sqrt{y}}\, dy+
x^{-(\nu+3/2+\delta)}\int_0^x |g(y)| y^\delta\, dy\right).
\end{multline*}
Now taking into account that for $p=p_1$ we have
$2\nu+1-p(\nu+1)<-1$ and $2\nu+1-p(\nu+3/2+\delta)<-1$ we can
apply Hardy's inequalities to obtain that
\[
\int_0^1\left(x^{-(\nu+1)}\int_0^x \frac{|g(y)|}{\sqrt{y}}\,
dy\right)^{p}x^{2\nu+1}\, dx\le C \int_0^1 |g(y)|^p
y^{(\nu+1/2)(2-p)}\, dy
\]
and
\[
\int_0^1\left(x^{-(\nu+3/2+\delta)}\int_0^x |g(y)| y^\delta\,
dy\right)^{p}x^{2\nu+1}\, dx\le C \int_0^1 |g(y)|^p
y^{(\nu+1/2)(2-p)}\, dy,
\]
with these two inequalities we can deduce that
\eqref{ec:boundweak} holds for $J_3$ with $p=p_1$ in this case.
The main difference with the previous proof appears in the
analysis of $J_4$. To deal with this case, we have to use the
following lemma \cite[Lemma 16.5]{Ch-Muc}
\begin{Lem}
\label{lem:Muck} If $1<p<\infty$, $a>-1$, and $E\subset
[0,\infty)$, then
\[
\left(\int_{E}x^a\, dx\right)^p\le
2^p(a+1)^{1-p}\int_{E}x^{(a+1)p-1}\, dx.
\]
\end{Lem}
In this case, it is enough to prove that
\[
\int_{\mathcal{J}}\, d\mu_\nu(x)\le \frac{C}{\lambda^p}\int_{0}^1
\chi_E(y)\,d\mu_\nu(y),
\]
where
\begin{equation*}
\mathcal{J}=\left\{x\in(0,1):
\sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)
\chi_{B_4}(x,y)y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy>\lambda\right\},
\end{equation*}
and this can be deduced immediately by using the inclusion
\begin{equation}
\label{ec:final}
\mathcal{J}\subseteq [0,\min\{1,H\}]
\end{equation}
with
\[
H^{2(\nu+1)}=\frac{C}{\lambda^p}\int_{0}^1 \chi_E(y)\,d\mu_\nu(y).
\]
Let's prove \eqref{ec:final}. By using \eqref{ec:kern} and the
estimate $y-x>y/2$, we have
\begin{multline*}
\sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)
\chi_{B_4}(x,y)y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy
\\
\le C\sup_{R>0}
R^{-\delta+\nu+1/2}\chi_{[0,2/R]}(x)\int_{4/R}^1\chi_E(y)
y^{-\delta+\nu-1/2}\, dy\\
+ C\sup_{R>0}
R^{-\delta}x^{-(\nu+1/2)}\chi_{[2/R,1]}(x)\int_{2x}^1 \chi_E(y)
y^{-\delta+\nu-1/2}\, dy.
\end{multline*}
In the first summand we can use that $R^{-\delta+\nu+1/2}\le C
x^{\delta-\nu-1/2}$ and in the second one that $R^{-\delta}\le
x^{\delta}$. Moreover observing that with $p=p_1$ it holds
$-\delta+\nu+1/2=2(\nu+1)/p$ we obtain that
\begin{align*}
\sup_{R>0}x^{-(\nu+1/2)}\int_0^{1}\chi_E(y)\chi_{B_4}y^{\nu+1/2}|K_R^{\delta}(x,y)|\,dy&\le
C x^{-2(\nu+1)/p}\int_{E}y^{-1+2(\nu+1)/p}\, dy\\
&\le C x^{-2(\nu+1)/p}\int_{E}\,d\mu_\nu(y),
\end{align*}
where in the last step we have used Lemma \ref{lem:Muck}, and this
is enough to deduce the inclusion in \eqref{ec:final}.
\section{Proofs of Theorem \ref{th:noweak} and Theorem \ref{th:nostrong}}
\label{sec:negativeths}
This section will be devoted to the proofs of Theorem
\ref{th:noweak} and Theorem \ref{th:nostrong}. To this end we need
a suitable identity for the kernel and in order to do that we have
to introduce some notation. $H_{\nu}^{(1)}$ will denote the Hankel
function of the first kind, and it is defined as follows
\[
H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z),
\]
where $Y_{\nu}$ denotes the Weber's function, given by
\begin{equation*}
Y_{\nu}(z)=\frac{J_{\nu}(z)\cos \nu \pi-J_{-\nu}(z)}{\sin \nu
\pi},\,\, \nu\notin \mathbb{Z}, \text{ and } Y_n(z)=\lim_{\nu\to
n}\frac{J_{\nu}(z)\cos \nu \pi-J_{-\nu}(z)}{\sin \nu \pi}.
\end{equation*}
From these definitions, we have
\begin{equation*}
H_{\nu}^{(1)}(z)=\frac{J_{-\nu}(z)-e^{-\nu \pi i}J_{\nu}(z)}{i\sin
\nu
\pi}, \,\, \nu\notin \mathbb{Z},\\
\text{ and } H_n^{(1)}(z)=\lim_{\nu\to n}\frac{J_{-\nu}(z)-e^{-\nu
\pi i}J_{\nu}(z)}{i\sin \nu \pi}.
\end{equation*}
For the function $H_\nu^{(1)}$, the asymptotic
\begin{equation}
\label{inftyH} H_{\nu}^{(1)}(z)=\sqrt{\frac{2}{\pi
z}}e^{i(z-\nu\pi/2-\pi/4)}[A+O(z^{-1})], \quad |z|>1,\quad -\pi <
\arg(z)<2\pi,
\end{equation}
holds for some constant $A$.
In \cite[Lemma 1]{Ci-Ro} the following lemma was proved
\begin{Lem}
\label{lem:expresnucleo} For $R>0$ the following holds:
\[K_R^\delta(x,y)=I_{R,1}^\delta(x,y)+I_{R,2}^\delta(x,y)\]
with
\[
I_{R,1}^{\delta}(x,y)=(xy)^{1/2}\int_0^{R}z\multiJ_{\nu}(zx)J_{\nu}(zy)\,
dz
\]
and
\[ I_{R,2}^{\delta}(x,y)=\lim_{\varepsilon\to 0}
\frac{(xy)^{1/2}}{2}\int_{\mathbf{S_\varepsilon}}
\multi \frac{z H^{(1)}_{\nu}(z)J_{\nu}(zx)J_{\nu}(zy)}{J_{\nu}(z)}\,dz,
\]
where, for each $\varepsilon>0$, $\mathbf{S_\varepsilon}$ is the
path of integration given by the interval
$R+i[\varepsilon,\infty)$ in the direction of increasing imaginary
part and the interval $-R+i[\varepsilon,\infty)$ in the opposite
direction.
\end{Lem}
Then, by Lemma \ref{lem:expresnucleo} we have
\[
\mathcal{K}_R^\delta(x,y)=\mathcal{I}_{R,1}^\delta(x,y)+\mathcal{I}_{R,2}^\delta(x,y)
\]
where
$\mathcal{I}_{R,j}^\delta(x,y)=(xy)^{-(\nu+1/2)}I_{R,j}^{\delta}(x,y)$
for $j=1,2$. The main tool to deduce our negative results will be
the following lemma
\begin{Lem}
\label{lem:zero} For $\nu>-1/2$, $\delta>0$, and $R>0$ it is
verified that
\[
\mathcal{K}_R^\delta(0,y)=\frac{2^{\delta-\nu}\Gamma(\delta+1)}{\Gamma(\nu+1)}R^{2(\nu+1)}
\frac{J_{\nu+\delta+1}(yR)}{(yR)^{\nu+\delta+1}}+\mathcal{I}_{R,2}^\delta(0,y),
\]
where
\begin{equation}
\label{ec:boundI} \left|\mathcal{I}_{R,2}^\delta(0,y)\right|\le
C\begin{cases} R^{2\nu-\delta+1}, &
yR\le 1,\\
R^{\nu-\delta+1/2}y^{-(\nu+1/2)}, & yR>1.
\end{cases}
\end{equation}
\end{Lem}
\begin{proof}
From \eqref{zero}, it is clear that
\[
\mathcal{I}_{R,1}^\delta(0,y)=\frac{y^{-\nu}}{2^\nu\Gamma(\nu+1)}\int_0^R
z^{\nu+1}\multiJ_{\nu}(zy)\, dz.
\]
Now, by using Sonine's identity \cite[Ch. 12, 12.11, p. 373]{Wat}
\[
\int_0^1 s^{\nu+1}\left(1-s^2\right)^\deltaJ_{\nu}(sy)\,
ds=2^{\delta}\Gamma(\delta+1)\frac{J_{\nu+\delta+1}(y)}{y^{\delta+1}},
\qquad \nu,\delta>-1,
\]
we deduce the leading term of the expression for
$\mathcal{K}_{R}^\delta(0,y)$.
To control the term
\[
\mathcal{I}_{R,2}^\delta(0,y)=\lim_{\varepsilon\to
0}\frac{y^{-(\nu+1/2)}}{2}\int_{\mathbf{S_\varepsilon}} \multi
\frac{z^{\nu+1/2}
H^{(1)}_{\nu}(z)(zy)^{1/2}J_{\nu}(zy)}{J_{\nu}(z)}\,dz,
\]
we start by using the asymptotic expansions given in \eqref{inftyH}
and \eqref{infty} for $H_{\nu}^{(1)}(z)$ and $J_{\nu}(z)$. We see
that on $\mathbf{S_\varepsilon}$, the path of integration
described in Lemma \ref{lem:expresnucleo}, for $t=\mathop{\rm Im}(z)$ the
estimate
\[
\left|\frac{H_{\nu}(z)}{J_{\nu}(z)}\right|\leq C e^{-2t},
\]
holds for $t>0$. Now, from \eqref{zero} and \eqref{infty}, it is
clear that for $z=\pm R+it$
\[
|\sqrt{zy}J_{\nu}(zy)|\le Ce^{yt}\Phi_\nu((R+t)y)
\]
where $\Phi_\nu$ is the function in \eqref{ec:aux}. Then
\[
|\mathcal{I}_{R,2}^\delta(0,y)|\le C
y^{-(\nu+1/2)}R^{-2\delta}\int_0^\infty
t^{\delta}(R+t)^{\nu+\delta+1/2}\Phi_\nu((R+t)y)e^{-(2-y)t}\, dt.
\]
If $y>1/R$ we have the inequality $\Phi_\nu((R+t)y)\le C$, then
\begin{align*}
|\mathcal{I}_{R,2}^\delta(0,y)| &\le C y^{-(\nu+1/2)} R^{-2\delta}
\int_0^\infty t^{\delta}(R+t)^{\nu+\delta+1/2}
e^{-(2-y)t}\, dt\\
&\le C y^{-(\nu+1/2)}R^{-\delta}(R^{\nu+1/2}+R^{-\delta})\le C
R^{\nu-\delta+1/2}y^{-(\nu+1/2)}
\end{align*}
and \eqref{ec:boundI} follows in this case. If $y\le 1/R$ we
obtain the bound in \eqref{ec:boundI} with the estimate
$\Phi_\nu((R+t)y)\le C (\Phi_\nu(yR)+(yt)^{\nu+1/2})$. Indeed,
\begin{multline*}
|\mathcal{I}_{R,2}^\delta(0,y)| \le C y^{-(\nu+1/2)} R^{-2\delta}
\Phi_\nu(yR) \int_0^\infty t^{\delta}(R+t)^{\nu+\delta+1/2}
e^{-(2-y)t}\, dt\\+ C R^{-2\delta} \int_0^\infty
t^{\nu+\delta+1/2}(R+t)^{\nu+\delta+1/2} e^{-(2-y)t}\, dt\\\le C
(R^{2\nu-\delta+1}+R^{\nu-2\delta+1/2}+R^{\nu-\delta+1/2}+R^{-2\delta})\le
R^{2\nu-\delta+1}.
\end{multline*}
\end{proof}
\begin{Lem}
\label{lem:cota0} For $\nu>-1/2$ and $0<\delta\le \nu+1/2$, the
estimate
\[
\|\mathcal{K}_R^\delta(0,y)\|_{L^{p_0}((0,1),d\mu_\nu)}\ge C
R^{\nu-\delta+1/2}(\log R)^{1/p_0}
\]
holds.
\end{Lem}
\begin{proof}
We will use the decomposition in Lemma \ref{lem:zero}. By using
\eqref{zero} and \eqref{infty} as was done in \cite[Lemma
2.1]{Ci-RoWave} we obtain that
\[
\left\|R^{2(\nu+1)}
\frac{J_{\nu+\delta+1}(yR)}{(yR)^{\nu+\delta+1}}\right\|_{L^{p_0}((0,1),d\mu_\nu)}\ge
C R^{\nu-\delta+1/2}(\log R)^{1/p_0}.
\]
With the bound \eqref{ec:boundI} it can be deduced that
\[
\left\|\mathcal{I}_{R,2}^\delta(0,y)\right\|_{L^{p_0}((0,1),d\mu_\nu)}\le
C R^{\nu-\delta+1/2}.
\]
With the previous estimates the proof is completed.
\end{proof}
Finally, the last element that we need to prove Theorems
\ref{th:noweak} and \ref{th:nostrong} is the norm inequality for
finite linear combinations of the functions $\{\psi_j\}_{j\ge 1}$
contained in the next lemma. Its proof is long and technical and
it will be done in the last section.
\begin{Lem}
\label{lem:pol} For $\nu>-1/2$, $R>0$, $1<p<\infty$ and $f$ a
linear combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$
with $N(R)$ a positive integer such that $N(R)\simeq R$, the inequality
\[
\|f\|_{L^\infty ((0,1),d\mu_\nu)}\le C
R^{2(\nu+1)/p}\|f\|_{L^{p,\infty} ((0,1),d\mu_\nu)}
\]
holds.
\end{Lem}
\begin{proof}[Proof of Theorem \ref{th:noweak}]
With the bound in Lemma \ref{lem:cota0} we have
\begin{align*}
(\log R)^{1/p_0}&\le C R^{-2(\nu+1)/p_1}
\left\|\mathcal{K}_{R}^\delta(0,y)\right\|_{L^{p_0}((0,1),d\mu_\nu)}\\
& = C R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\int_0^1
\mathcal{K}_{R}^\delta(0,y) f(y)\, d\mu_\nu\right|\\
& = C R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\mathcal{B}_{R}^\delta
f(0)\right|.
\end{align*}
From the previous estimate the result for
$\delta=\nu+1/2$ follows. In the case $\delta<\nu+1/2$ it is
obtained by using Lemma \ref{lem:pol} because
\begin{multline*}
R^{-2(\nu+1)/p_1}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left|\mathcal{B}_{R}^\delta
f(0)\right|\\\le C
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}\left\|\mathcal{B}_{R}^\delta
f(x)\right\|_{L^{p_1,\infty}((0,1),d\mu_\nu)}
\end{multline*}
since $\mathcal{B}_{R}^\delta f(x)$ is a linear combination of the
functions $\{\psi_j\}_{1\le j\le N(R)}$ with $N(R)\simeq R$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:nostrong}]
In the case $\delta <\nu+1/2$, the result follows from Theorem
\ref{th:noweak} by using a duality argument. Indeed, it is clear
that
\begin{align}
\sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}} &=\sup_{E\subset (0,1)}
\sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1}
\frac{\left|\int_0^1f(y)\mathcal{B}^{\delta}_{R}\chi_E(y)\,
d\mu_\nu\right|}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\notag\\
&= \sup_{\|f\|_{L^{p_1}((0,1),d\mu_\nu)}=1} \sup_{E\subset (0,1)}
\frac{\left|\int_0^1\chi_E(y)\mathcal{B}^{\delta}_{R}f(y)\,
d\mu_\nu\right|}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}\label{ec:lambdacero}.
\end{align}
By Theorem \ref{th:noweak} it is possible to choose a function $g$
such that $\|g\|_{L^{p_1}((0,1),d\mu_\nu)}=1$ and
\[
\|\mathcal{B}_{R}^\delta
g(x)\|_{L^{p_1,\infty}((0,1),d\mu_\nu)}\ge C (\log R)^{1/p_0}.
\]
Then, with the notation
\[
\mu_\nu(E)=\int_{E}\, d\mu_\nu,
\]
we have
\begin{equation}
\label{ec:lambda} \lambda^{p_1}\mu_\nu(A)\ge C (\log R)^{p_1/p_0},
\end{equation}
for some positive $\lambda$ and $A=\{x\in (0,1): |B_{R}^\delta
g(x)|>\lambda\}$. Now, we consider the subsets of $A$
\[
A_1=\{x\in (0,1): B_{R}^\delta g(x)>\lambda\} \qquad\text{ and }
\qquad A_2=\{x\in (0,1): B_{R}^\delta g(x)<-\lambda\}
\]
and we define $D=A_1$ if $\mu_\nu(A_1)\ge \mu_\nu(A)/2$ and
$D=A_2$ otherwise. Then, by \eqref{ec:lambda}, we deduce that
\begin{equation}
\label{ec:lambda2} \lambda \ge C \frac{(\log
R)^{1/p_0}}{\mu_\nu(D)^{1/p_1}}.
\end{equation}
Taking $f=g$ and $E=D$ in \eqref{ec:lambdacero} and using
\eqref{ec:lambda2}, we see that
\[
\sup_{E\subset
(0,1)}\frac{\|\mathcal{B}^{\delta}_{R}\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}}
{\|\chi_E\|_{L^{p_0}((0,1),d\mu_\nu)}} \ge C
\lambda\frac{\mu_\nu(D)}{\|\chi_D\|_{L^{p_0}((0,1),d\mu_\nu)}} \ge
C (\log R)^{1/p_0}
\]
and the proof is complete in this case. For $\delta=\nu+1/2$ the
result follows from Theorem \ref{th:noweak} with a standard
duality argument.
\end{proof}
\section{Proof of Lemma \ref{lem:pol}}
\label{sec:techlemma}
To proceed with the proof of Lemma \ref{lem:pol} we need some
auxiliary results that are included in this section.
We start by defining a new operator. For each non-negative integer
$r$, we consider the vector of coefficients
$\alpha=(\alpha_1,\dots,\alpha_{r+1})$ and we define
\[
T_{r,R,\alpha}f(x)=\sum_{\ell=1}^{r+1}\alpha_\ell
\mathcal{B}_{\ell R}^{r}f(x).
\]
This new operator is an analogous of the \textit{generalized
delayed means} considered in \cite{SteinDuke}. In \cite{SteinDuke}
the operator is defined in terms of the Ces\`{a}ro means instead
of the Bochner-Riesz means. The properties of $T_{r,R,\alpha}$
that we need are summarized in the next lemma
\begin{Lem}
\label{lem:delay} For each non-negative integer $r$ and $\nu\ge
-1/2$, the following statements hold
\begin{enumerate}
\item[a)] $T_{r,R,\alpha}f$ is a linear combination of the
functions $\{\psi_j\}_{1\le j\le N((r+1)R)}$, where $N((r+1)R)$ is
a non-negative integer such that $N((r+1)R)\simeq (r+1)R$;
\item[b)] there exists a vector of coefficients $\alpha$,
verifying that $|\alpha_\ell|\le A$, for $\ell=1,\dots, r+1$, with
$A$ independent of $R$ and such that $T_{r,R,\alpha}f(x)=f(x)$ for
each linear combination of the functions $\{\psi_j\}_{1\le j\le
N(R)}$ where $N(R)$ is a positive integer. Moreover, in this case,
for $r>\nu+1/2$,
\[
\|Tf_{r,R,\alpha}\|_{L^1 ((0,1),d\mu_\nu)}\le
C\|f\|_{L^1((0,1),d\mu_\nu)}\] and
\[\|T_{r,R,\alpha}f\|_{L^\infty ((0,1),d\mu_\nu)}\le C
\|f\|_{L^\infty((0,1),d\mu_\nu)},
\]
with $C$ independent of $R$ and $f$.
\end{enumerate}
\end{Lem}
\begin{proof}
Part a) is a consequence of the definition of $T_{r,R,\alpha}$ and
the fact that the $m$-th zero of the Bessel function $J_\nu$, with
$\nu \ge-1/2$, is contained in the interval
$(m\pi+\nu\pi/2+\pi/2,m\pi+\nu\pi/2+3\pi/4)$.
To prove b) we consider $f(x)=\sum_{j=1}^{N(R)} a_j \psi_j(x)$. In
order to obtain the vector of coefficients such that
$T_{r,R,\alpha}f(x)=f(x)$ the equations
\[
\sum_{\ell=1}^{r+1}\alpha_\ell \left(1-\frac{s_{k}^2}{(\ell
R)^2}\right)^r=1,
\]
for all $k=1,\dots,N(R)$, should be verified. After some elementary manipulations each
one of the previous equations can be written as
\[
\sum_{j=0}^r
s_k^{2j}\binom{r}{j}\frac{(-1)^j}{R^{2j}}\sum_{\ell=1}^{r+1}
\frac{\alpha_\ell}{\ell^{2j}}=1
\]
and this can be considered as a polynomial in $s_k^2$ which must
be equal $1$, therefore we have the system of equations
\[
\sum_{\ell=1}^{r+1}\frac{\alpha_\ell}{\ell^{2j}}=\delta_{j,0},
\qquad j=0,\dots,r.
\]
This system has an unique solution because the determinant of the
matrix of coefficients is a Vandermonde's one. Of course for each
$\ell=1,\dots,r+1$, it is verified that $|\alpha_\ell|\le A$, with
$A$ a constant depending on $r$ but not on $N(R)$.
The norm estimates are consequence of the uniform boundedness
\[
\|\mathcal{B}_R^\delta f\|_{L^p((0,1),d\mu_\nu)}\le C
\|f\|_{L^p((0,1),d\mu_\nu)},
\]
for $p=1$ and $p=\infty$ when $\delta > \nu+1/2$ (see
\cite{Ci-Ro}).
\end{proof}
In the next lemma we will control the $L^\infty$-norm of a finite linear
combination of the functions $\{\psi_j\}_{j\ge 1}$ by its
$L^1$-norm.
\begin{Lem}
\label{lem:infty1} If $\nu>-1/2$ and $f(x)$ is a linear
combination of the functions $\{\psi_j\}_{1\le j\le N(R)}$ with $N(R)$ a positive integer such that
$N(R)\simeq R$, the inequality
\[
\|f\|_{L^\infty ((0,1),d\mu_\nu)}\le C
R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)}
\]
holds.
\end{Lem}
\begin{proof}
It is clear that
\[
f(x)=\sum_{j=1}^{N(R)} \psi_j(x)\int_0^1 f(y) \psi_j(y)\,
d\mu_\nu(y).
\]
Now, using H\"{o}lder inequality and Lemma \ref{Lem:NormaFunc} we
have
\begin{align*}
\|f\|_{L^\infty ((0,1),d\mu_\nu)}&\le C\sum_{j=1}^{N(R)}
\|\psi_j\|_{L^\infty
((0,1),d\mu_\nu)}^2\|f\|_{L^{1}((0,1),d\mu_\nu)}\\&\le C
\|f\|_{L^{1}((0,1),d\mu_\nu)} \sum_{j=1}^{N(R)}j^{2\nu+1}\le C
R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)}.
\end{align*}
\end{proof}
The following lemma is a version in the space $((0,1),d\mu_\nu)$
of Lemma 19.1 in \cite{Ch-Muc}. The proof can be done in the same
way, with the appropriate changes, so we omit it.
\begin{Lem}
\label{lem:fuerdeb} Let $\nu>-1$, $1<p<\infty$ and $T$ be a linear
operator defined for functions in $L^1((0,1),d\mu_\nu)$ and such
that
\[
\|Tf\|_{L^\infty ((0,1),d\mu_\nu)}\le A
\|f\|_{L^1((0,1),d\mu_\nu)} \,\text{ and }\,\|Tf\|_{L^\infty
((0,1),d\mu_\nu)}\le B \|f\|_{L^\infty((0,1),d\mu_\nu)},
\]
then
\[
\|Tf\|_{L^\infty ((0,1),d\mu_\nu)}\le C
A^{1/p}B^{1/p'}\|f\|_{L^{p,\infty}((0,1),d\mu_\nu)}.
\]
\end{Lem}
Now, we are prepared to conclude the proof of Lemma \ref{lem:pol}.
\begin{proof}[Proof of Lemma \ref{lem:pol}]
We consider the operator $T_{r,R,\alpha}f$ given in Lemma
\ref{lem:delay} b) with $r>\nu+1/2$. By Lemma \ref{lem:delay} and
Lemma \ref{lem:infty1} we have
\begin{align*}
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}&\le C
((r+1)R)^{2(\nu+1)}
\|T_{r,R,\alpha}f\|_{L^1((0,1),d\mu_\nu)}\\&\le C
R^{2(\nu+1)}\|f\|_{L^1((0,1),d\mu_\nu)}.
\end{align*}
From b) in Lemma \ref{lem:delay} we obtain the estimate
\[
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}\le C
\|f\|_{L^\infty((0,1),d\mu_\nu)}.
\]
So, by using Lemma \ref{lem:fuerdeb}, we obtain the inequality
\[
\|T_{r,R,\alpha}f\|_{L^\infty((0,1),d\mu_\nu)}\le C
R^{2(\nu+1)/p}\|f\|_{L^{p,\infty}((0,1),d\mu_\nu)}
\]
for any $f\in L^1((0,1),d\mu_{\nu})$. Now, since
$T_{r,R,\alpha}f(x)=f(x)$ for a linear combination of the
functions $\{\psi_j\}_{1\le j\le N(R)}$, the proof is complete.
\end{proof}
\end{document} |
\begin{document}
\title{An alternative framework for quantifying coherence of quantum channels}
\author{Shi-Yun Kong$^1$}
\author{Ya-Juan Wu$^2$}
\author{Qiao-Qiao Lv$^1$}
\author{Zhi-Xi Wang$^1$}
\author{Shao-Ming Fei$^{1,3}$}
\affiliation{
{\footnotesize $^1$School of Mathematical Sciences, Capital Normal University, Beijing 100048, China}\\
{\footnotesize $^2$School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China}\\
{\footnotesize $^3$Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany}
}
\begin{abstract}
We present an alternative framework for quantifying the coherence of quantum channels, which contains three conditions: the faithfulness, nonincreasing under sets of all the incoherent superchannels and the additivity. Based on the characteristics of the coherence of quantum channels and the additivity of independent channels, our framework provides an easier way to certify whether a function is a bona fide coherence measure of quantum channels. Detailed example is given to illustrate the effectiveness and advantages of our framework.
\end{abstract}
\maketitle
\section{Introduction}
As one of the characteristic features that marks the departure of quantum mechanics from the classical realm, the quantum coherence plays a central role in quantum optics \cite{glauber,scully,gyongyosi}, thermodynamics \cite{brandao,gour,narasimhachar,aberg2,lostaglio1,lostaglio2,gentaro,mayuhan,xushengzhi}, nanoscale physics \cite{plenio,rebentrost,licheming,huelga} and quantum measurements \cite{napoli,mateng,longguilu}. Quantum coherence is also the origin of many quantum phenomena such as entanglement and multiparticle interference. Recently the coherence of quantum states, concerning the quantifications \cite{baumgratz,girolami}, interconversion \cite{bromley} and applications \cite{luo,aberg,monras}, has been extensively investigated.
Similar to the resource theory of quantum entanglement, Baumgratz, Cramer and Plenio presented a rigorous framework (BCP framework) for quantifying the coherence of quantum states and introduced several measures of coherence including the relative entropy of coherence, the $l_1-$norm of coherence and fidelity \cite{baumgratz}. The BCP framework is widely used in quantifying coherence. Inspired by the BCP framework, Yu $et.~al.$ put forward another equivalent framework \cite{yuxiaodong}, which can be more conveniently used in some cases and is applicable to various physical contexts.
The quantum state transfer depends on quantum channels. The coherence of quantum channels characterizes the ability to optimize the coherence of all output states of the channels \cite{xueyuanhu,bendana,korzekawa,theurer,chandan}. Similar to the resource theory of coherence for quantum states, the resource theory of coherence for quantum channels has attracted much attention. With respect to the BCP framework for coherence of quantum states, Xu established a framework for quantifying the coherence of quantum channels \cite{xujianwei}.
In this paper, similar to the alternative framework for quantum states given in \cite{yuxiaodong} which simplifies the BCP framework in particular cases, we establish a new framework for quantifying the coherence of quantum channels, which improves the applications of the previous framework given in \cite{xujianwei}. Detailed examples are presented to illustrate the advantages of our framework.
\maketitle
\section{An alternative framework for coherence of quantum channels}
Let $H_A$ and $H_B$ be Hilbert spaces with dimensions ${\rm dim} H_A=|A|$ and ${\rm dim} H_B=|B|$, and $\{|j\rangle\}_j,~\{|k\rangle\}_k$ and $\{|\alpha\rangle\}_\alpha,~\{|\beta\rangle\}_\beta$ be the fixed orthonormal bases of $H_A$ and $H_B$, respectively. Denote $\mathcal{D}_A$ ($\mathcal{D}_B$) the set of all density operators on $H_A$ ($H_B$). Let $\mathcal{C}_{AB}$ be the set of all channels from $\mathcal{D}_A$ to $\mathcal{D}_B$.
A quantum channel $\phi\in \mathcal{C}_{AB}$ is a linear completely positive and trace preserving (CPTP) map with Kraus operators $\{K_n\}_n$ satisfying $\sum_n K_n^\dagger K_n=I$ such that $\phi(\rho)=\sum_n K_n\rho K_n^\dagger$. The corresponding Choi matrix with respect to the channel $\phi\in\mathcal{C}_{AB}$ has the following form,
\begin{equation}\langlebel{W1}
J_{\phi}=\sum_{jk}|j\rangle\langle k|\otimes\phi(|j\rangle\langle k|)=\sum_{jk,\alpha\beta}\phi_{jk\alpha\beta}|j\rangle\langle k|\otimes|\alpha\rangle\langle\beta|,
\end{equation}
where $\phi_{jk\alpha\beta}=\langle\alpha|\phi(|j\rangle\langle k|)|\beta\rangle$ are complex numbers and $\sum_{\alpha}\phi_{jk,\alpha\alpha}=\delta_{jk}$.
If $C$ is a coherence measure of quantum states, then $C(\phi)=C(\frac{J_\phi}{|A|})$ is a corresponding coherence measure for quantum channels \cite{xujianwei}, which quantifies the coherence of the channel by the coherence of the state $\frac{J_\phi}{|A|}$. The Choi matrix of a channel gives a relation between the coherence measures for quantum channels and for that quantum states.
Let $\mathcal{I}$ be the set of all incoherent states whose density matrices are diagonal in the given basis. An incoherent state $\rho\in H_A$ satisfies $\Delta_A(\rho)=\rho$ with $\Delta_A(\rho)=\sum_j\langle j|\rho|j\rangle|j\rangle\langle j|$ being a completely dephasing channel. $\phi\in\mathcal{C}_{AB}$ is called an incoherent quantum channel if $\Upsilon(\phi)=\phi$, where $\Upsilon(\phi)=\Delta_B \phi \Delta_A$, $\Delta_A$ and $\Delta_B$ are resource destroying maps\cite{liuziwen}. We denote $\mathcal{IC}$ the set of all the incoherent channels.
Let $\mathcal{SC}_{ABA'B'}$ be the superchannels that are linear maps from $\mathcal{C}_{AB}$ to $\mathcal{C}_{A'B'}$, we define the Choi matrix of the superchannel $\Theta\in\mathcal{SC}_{ABA'B'}$ as $J_\Theta=\sum_{jk\alpha\beta}|j\alpha\rangle\langle k\beta|\otimes\Theta(|j\alpha\rangle\langle k\beta|)$. Any superchannel $\Theta$ is an incoherent superchannel ($\mathcal{ISC}$) if there exists an expression of Kraus operators $\Theta=\{M_m\}_m$ such that for each $m$, $M_m=\sum_{j\alpha}M_{mj\alpha}|f(j\alpha)\rangle\langle j\alpha|$ with $f(j\alpha)=f(j,\alpha)\in \{(j',\alpha')|_{j'=1}^{|A'|},_{\alpha'=1}^{|B'|}\}$.
In \cite{xujianwei}, the author presented a framework for quantifying the coherence of quantum channels. A proper measure $C$ of the coherence for quantum channels must satisfy
the following conditions:
\begin{enumerate}
\item[$\mathrm{(B1)}$] For any quantum channel $C(\phi)\geqslant0$, $C(\phi)=0$ if and only if $\phi\in\mathcal{IC}_{AB}$;
\item[$\mathrm{(B2)}$] $C(\phi)\geqslant C[\Theta(\phi)]$ for any incoherent superchannel $\Theta$;
\item[$\mathrm{(B3)}$] $C(\phi)\geqslant\sum_m p_mC(\phi_m)$ for any incoherent superchannel $\Theta$, with $\{M_m\}_m$ an incoherent Kraus operator of $\Theta$, $p_m=\frac{\mathrm{tr}(M_m J_\phi M_m^\dagger)}{|A'|}$, and $J_{\phi_m}=|A'|\frac{M_m J_\phi M_m^\dagger}{\mathrm{tr}(M_m J_\phi M_m^\dagger)}$;
\item[$\mathrm{(B4)}$] $C(\sum_m p_m \phi_m)\leqslant\sum_m p_m C(\phi_m)$ for any set of quantum channels $\{\phi_m\}$ and any probability distribution $\{p_m\}_m$.
\end{enumerate}
The items (B1)-(B4) give necessary conditions for a bona fide measure of coherence for quantum channels, from which the quantification of the coherence for quantum channels has been further
investigated.
Nevertheless, similar to the case of coherence measures for quantum states, the last two conditions (B3) and (B4) are rather difficult to be verified for a given measure of coherence
for quantum channels. In the following we present an alternative framework consisting of three conditions, which is equivalent to the above framework but can be easily applied.
A function $C$ is a well defined coherence measure of quantum channels if it satisfies the following three conditions:
\begin{enumerate}
\item[$\mathrm{(C1)}$] $\mathit{Faithfulness}$. $C(\phi)\geqslant0$ for any $\phi\in \mathcal{C}_{AB}$, and $C(\phi)=0$ if and only if $\phi\in\mathcal{IC}_{AB}$;
\item[$\mathrm{(C2)}$] $\mathit{Nonincreasing\ under\ ISCs}.$ $C(\phi)\geqslant C[\Theta(\phi)]$ for any $\Theta\in\mathcal{ISC}_{ABA'B'}$;
\item[$\mathrm{(C3)}$] $\mathit{Additivity}$. $C(\Phi)=p_1C(\phi_1)+p_2C(\phi_2)$ for $p_1+p_2=1$, $\phi_1\in \mathcal{C}_{AB_1}$ and $\phi_2\in \mathcal{C}_{AB_2}$, where $\Phi(|j\rangle\langle k|)=p_1\phi_1(|j\rangle\langle k|)\oplus p_2\phi_2(|j\rangle\langle k|)$, $\Phi\in \mathcal{C}_{AB}$,\ and $|B|=|B_1|+|B_2|$.
\end{enumerate}
In the following, we prove that the framework given by (B1)-(B4) is equivalent to the one given by (C1)-(C3). We first prove that (B1)-(B4) give rise to (C1)-(C3), namely, (B3)(B4) give rise to (C3) since (C1) and (C2) are the same as (B1) and (B2).
Consider a CPTP map $\Theta_1\in\mathcal{ISC}_{ABAB}$, $\Theta_1(\cdot)=Q_1\cdot Q_1^\dagger+Q_2\cdot Q_2^\dagger$, where
\begin{equation}
\begin{aligned}
Q_1=&|0\rangle\langle0|+\cdots+||B_1|-1\rangle\langle |B_1|-1|+||B_1|+|B_2|\rangle\langle |B_1|+|B_2||+\cdots+|2|B_1|+|B_2|-1\rangle\\
&\langle 2|B_1|+|B_2|-1|+\cdots+|(|A|-1)(|B_1|+|B_2|)\rangle\langle(|A|-1)(|B_1|+|B_2|)|+\cdots\\
&+|(|A|-1)(|B_1|+|B_2|)+|B_1|-1\rangle\langle(|A|-1)(|B_1|+|B_2|)+|B_1|-1|
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
Q_2=&||B_1|\rangle\langle |B_1||+\cdots+||B_1|+|B_2|-1\rangle\langle |B_1|+|B_2|-1|+|2|B_1|+|B_2|\rangle\langle2|B_1|+|B_2||+\cdots\\
&+|2(|B_1|+|B_2|)-1\rangle\langle2(|B_1|+|B_2|)-1|+\cdots+||A||B_1|+(|A|-1)|B_2|\rangle\\
&\langle |A||B_1|+(|A|-1)|B_2||+\cdots+||A|(|B_1|+|B_2|)-1\rangle\langle |A|(|B_1|+|B_2|)-1|.
\end{aligned}
\end{equation}
Note that $Q_1$ and $Q_2$ are just projectors onto $\mathcal{C}_{AB}$. Obviously, one sees that $Q_i\mathcal{IC}Q_i^\dagger\subset\mathcal{IC}$. The Choi matrix of $\Phi$ in (C3) is given by \begin{equation}\langlebel{W3}
J_\Phi=\sum_{j,k}|j\rangle\langle k|\otimes[p_1\phi_1(|j\rangle\langle k|)\oplus p_2\phi_2(|j\rangle\langle k|)].
\end{equation}
Then
\begin{equation}
\Theta_1(\Phi)=Q_1J_\Phi Q_1^\dagger+Q_2J_\Phi Q_2^\dagger=p_1J_{\tilde{\phi}_1}+p_2J_{\tilde{\phi}_2},
\end{equation}
where $p_1=\frac{\mathrm{tr}(Q_1J_\Phi Q_1^\dagger)}{|A|}$, $p_2=\frac{\mathrm{tr}(Q_2J_\Phi Q_2^\dagger)}{|A|}$ and $J_{\tilde{\phi}_1}=|A|\frac{Q_1J_\Phi Q_1^\dagger}{\mathrm{tr}(Q_1J_\Phi Q_1^\dagger)}$, $J_{\tilde{\phi}_2}=|A|\frac{Q_2J_\Phi Q_2^\dagger}{\mathrm{tr}(Q_2J_\Phi Q_2^\dagger)}$. From (B2) and (B3) we have
\begin{equation}\langlebel{W4}
C(\Phi)\geqslant p_1C(\tilde{\phi_1})+p_2C(\tilde{\phi_2}),
\end{equation}
where $\tilde{\phi}_1,\,\tilde{\phi}_2\in\mathcal{C}_{AB}$, $\tilde{\phi}_1(|j\rangle\langle k|)=\phi_1(|j\rangle\langle k|)\oplus{\bf0}(|j\rangle\langle k|)$ and $\tilde{\phi}_2(|j\rangle\langle k|)={\bf0}(|j\rangle\langle k|)\oplus\phi_2(|j\rangle\langle k|)$ with ${\bf0}$ a zero map.
From (B4) we have
\begin{equation}\langlebel{W5}
C(\Phi)\leqslant p_1C(\tilde{\phi_1})+p_2C(\tilde{\phi_2}).
\end{equation}
Combining \eqref{W4} with \eqref{W5}, we get
\begin{equation}\langlebel{W6}
C(\Phi)=p_1C(\tilde{\phi_1})+p_2C(\tilde{\phi_2}).
\end{equation}
To obtain (C3), we need to certify $C(\tilde{\phi_1})=C(\phi_1)$ further.
Under an incoherent superchannel, any $\phi$ and $\overline{\Theta}(\phi)=\sum P_nJ_\phi P_n^\dagger$ can be transformed into each other, where the Kraus operators $\{P_n\}_n$ are permutation matrices. By (B2) we have $C(\overline{\Theta}(\phi))=C(\phi)$. Based on this fact, we define an incoherent superchannel $\overline{\Theta}_1\in\mathcal{ISC}_{ABAB}$ with Kraus operators $\{P_{nl}^{(1)}\}_{nl}$ as the permutation matrices,
\begin{equation}
P_{nl}^{(1)}(i,j)=\left\{
\begin{aligned}
&1,~{\rm if}~(i, j)=(i_{nl}, j_{nl}) ~{\rm or} ~(i, j)=(j_{nl}, i_{nl}),\\
&1,~{\rm if}~i=j,\\
&0,~\rm{other wise},
\end{aligned}\right.
\end{equation}
where $i_{nl}=(n-1)(|B_1|+|B_2|)+l$, $j_{nl}=(n-1)|B_1|+l$, $i,j=1,\cdots,|A||B|$, $n=1,\cdots,|A|$ and $l=1,\cdots,|B_1|$.
Then, \begin{equation}\overline{\Theta}_1(\tilde{\phi}_1)=\sum_{n,l}P_{nl}^{(1)}J_{\tilde{\phi}_1}P_{nl}^{(1)\dagger}=J_{\phi_1}\oplus O_{AB_2},\end{equation}
where $O_{AB_2}$ is a $|A||B_2|\times|A||B_2|$ null matrix. It is easily seen that
\begin{equation}\langlebel{W7}
C(\tilde{\phi}_1)=C(\overline{\Theta}_1(\tilde{\phi}_1)).
\end{equation}
Next, we need to prove $C(\overline{\Theta}_1(\tilde{\phi}_1))=C(\phi_1)$. For this, we define two incoherent superchannels: $\Theta_2\in\mathcal{ISC}_{AB_1AB}$ with Kraus operator $M_0$ satisfying $\langle j|M_0|k\rangle=\delta_{jk}$ and $\Theta_3\in\mathcal{ISC}_{ABAB_1}$ with Kraus operators $\{M_n\}_{n=0}^{\lceil\frac{|B_2|}{|B_1|}\rceil}$ satisfying $\langle j|M_n|k\rangle=\delta_{j,k-n|B_1|}$. Then we get, \begin{equation}~\Theta_2(\phi_1)=M_0J_{\phi_1}M_0^\dagger=J_{\phi_1}\oplus O_{AB_2}\end{equation}
and
\begin{equation}
\Theta_3[(\overline{\Theta}_1(\tilde{\phi}_1)]
=\sum_{n=0}^{\lceil\frac{|B_2|}{|B_1|}\rceil}M_n(J_{\phi_1}\oplus O_{AB_2})M_n^\dagger=J_{\phi_1}.
\end{equation}
From (B2) we obtain
\begin{equation}\langlebel{6jia}
C(\overline{\Theta}_1(\tilde{\phi}_1))=C(\phi_1).
\end{equation}
Combining \eqref{6jia} with \eqref{W6} and \eqref{W7}, we get the condition (C3).
We have shown that any $C$ satisfying (B1)-(B4) also satisfies (C1)-(C3). Next, we prove that any $C$ satisfying (C1)-(C3) must satisfy (B3) and (B4).
First, we prove (B3), i.e., $C$ is convex. Define $\Phi_1\in\mathcal{C}_{AB'}$ as,
\begin{equation}\Phi_1(|j\rangle\langle k|)=\phi(|j\rangle\langle k|)\oplus {\bf 0}(|j\rangle\langle k|)\oplus\cdots\oplus {\bf 0}(|j\rangle\langle k|),\end{equation}
where $\phi\in\mathcal{C}_{AB},\ H_{B'}=\underbrace{H_B\otimes H_B\otimes\cdots\otimes H_B}_{M}.$
From (C3), we have
\begin{equation}\langlebel{W8}
C(\Phi_1)=C(\phi).
\end{equation}
Consider $\overline{\Theta}_2\in\mathcal{ISC}_{AB'AB'}$, with its Kraus operators $\{P_n^{(2)}\}_n$ being the permutation matrices, such that
$\overline{\Theta}_2(\Phi_1)=\sum_n P_n^{(2)}J_{\Phi_1}P_n^{(2)\dagger}=J_\phi\oplus \underbrace{O_{AB}\oplus\cdots\oplus O_{AB}}_{M-1}$. Apply an incoherent superchannel $\Theta_4\in \mathcal{SC}_{ABA_1B'_{1}}$ with Kraus operators $\{U_m\otimes M_m\}_m$ such that
\begin{equation}
\Theta_4[\overline{\Theta}_2(\Phi_1)]=\sum_{m=0}^{M-1}(U_m\otimes M_m)(\sum_nP_n^{(2)}J_{\Phi_1}P_n^{(2)\dagger})(U_m\otimes M_m)^\dagger,
\end{equation}
where $U_m=\sum_{k=0}^{M-1}|(k+m)~\mathrm{mod}~M\rangle\langle k|,~H_{B'_1}=\underbrace{H_{B_1}\otimes H_{B_1}\otimes\cdots\otimes H_{B_1}}_{M}$, and $\{M_m\}$ are incoherent Kraus operators of the superchannel in $\mathcal{ISC}_{ABA_1B_1}$. One can easily see that $(U_m\otimes M_m)\mathcal{IC}(U_m\otimes M_m)^\dagger \subset\mathcal{IC}$,
\begin{equation}
\Theta_4[\overline{\Theta}_2(\Phi_1)]=\sum_{m=0}^{M-1}p_m |m\rangle\langle m|\otimes J_{\phi_m},
\end{equation}
where $p_m=\frac{\rm{tr}(M_mJ_\phi M_m^\dagger)}{|A_1|}$ and $J_{\phi_m}=|A_1|\frac{M_mJ_\phi M_m^\dagger}{\rm{tr}(M_mJ_\phi M_m^\dagger)}$.
Similarly, there exists $\overline{\Theta}_3\in\mathcal{ISC}_{A_1B_1'A_1B_1'}$, with its Kraus operators $\{P_n^{(3)}\}_n$ being the permutation matrices, such that $\overline{\Theta}_3[\Theta_4(\overline{\Theta}_2(\Phi_1))]=\sum P_n^{(3)}(\sum_{m=0}^{M-1}p_m |m\rangle\langle m|\otimes J_{\phi_m})P_n^{(3)\dagger}=\sum_{i,j=0}^{M-1}|i\rangle\langle j|\otimes[p_0\phi_0(|i\rangle\langle j|)\oplus\cdots\oplus p_{M-1}\phi_{M-1}(|i\rangle\langle j|)]$. $\overline{\Theta}_3[\Theta_4(\overline{\Theta}_2(\Phi_1))]$ matches to a channel $\Phi_2\in C_{A_1{B_1}'},~\Phi_2(|i\rangle\langle j|)=p_0\phi_0(|i\rangle\langle j|)\oplus\cdots\oplus p_{M-1}\phi_{M-1}(|i\rangle\langle j|)$.
\\Following (C3), we have
\begin{equation}\langlebel{W9}
C(\Phi_2)=\sum_{m=0}^{M-1}p_m C(\phi_m).
\end{equation}
By (C2), \eqref{W8} and \eqref{W9} we can have\begin{equation}C(\phi)\geqslant\sum_{m=0}^{M-1}p_m C(\phi_m),
\end{equation}
which proves that (B3) holds.
We now prove (B4). We first define an initial channel $\Phi_3\in C_{AB'}$ satisfying \begin{equation}\Phi_3(|j\rangle\langle k|)=\oplus_{m=0}^{M-1}p_m\phi_m(|j\rangle\langle k|),\end{equation}
where $\phi_m\in\mathcal{C}_{AB},~\{p_m\}$ are the probability distribution of $\{\phi_m\}$ and $\sum_{m=0}^{M-1}p_m=1$.
According to (C3), one has
\begin{equation}\langlebel{W10}
C(\Phi_3)=\sum_{m=0}^{M-1}p_m C(\phi_m).
\end{equation}
Apply $\overline{\Theta}_4\in\mathcal{ISC}_{AB'AB'}$, with its Kraus operators being the permutation matrices $\{P_n^{(4)}\}$, such that $\overline{\Theta}_4(\Phi_3)=\sum_n P_n^{(4)} J_{\Phi_3} P_n^{(4)\dagger}=\oplus_{m=0}^{M-1}p_m J_{\phi_m}$.
Let $\Theta_5\in\mathcal{ISC}_{AB'AB'}$ be an incoherent super channel such that
\begin{equation}\begin{aligned}
\Theta_5\overline{\Theta}_4(\Phi_3)&=\sum_{m=0}^{M-1}(|0\rangle\langle m|\otimes I)(\sum_n P_n^{(4)} J_{\Phi_3} P_n^{(4)\dagger})(|0\rangle\langle m|\otimes I)^\dagger\\
&=\sum_{m=0}^{M-1}p_mJ_{\phi_m}\oplus O_{A_1B_1}\oplus\cdots\oplus O_{A_1B_1}.\end{aligned}
\end{equation}
Apply $\overline{\Theta}_5\in\mathcal{ISC}_{AB'AB'}$, with Kraus operators $\{P_n^{(5)}\}$ as permutation matrices, such that
\begin{equation}\begin{aligned}
\overline{\Theta}_5[\Theta_5\overline{\Theta}_4(\Phi_3)]&=\sum_n P_n^{(5)}(\sum_{m=0}^{M-1}p_mJ_{\phi_m}\oplus O_{A_1B_1}\oplus\cdots\oplus O_{A_1B_1})P_n^{(5)\dagger}\\
&=\sum_{j,k=0}^{|A|-1}|j\rangle\langle k|\otimes[\sum_{m=0}^{M-1}p_m\phi_m(|j\rangle\langle k|)\oplus{\bf 0}(|j\rangle\langle k|)\oplus\cdots\oplus{\bf0}(|j\rangle\langle k|)].
\end{aligned}\end{equation}
Thus, $\overline{\Theta}_5[\Theta_5\overline{\Theta}_4(\Phi_3)]$ corresponds to $\Phi_4\in\mathcal{C}_{AB'}$ with $\Phi_4(|j\rangle\langle k|)=\sum_{m=0}^{M-1}p_m\phi_m(|j\rangle\langle k|)\oplus{\bf0}(|j\rangle\langle k|)\oplus\cdots\oplus{\bf0}(|j\rangle\langle k|)$. From (C3) we have
\begin{equation}\langlebel{W11}
C(\Phi_4)=C(\sum_{m=0}^{M-1}p_m\phi_m).
\end{equation}
Combining (C2) with \eqref{W10} and \eqref{W11}, we get \begin{equation}
\sum_{m=0}^{M-1}p_mC(\phi_m)\geqslant C(\sum_{m=0}^{M-1}p_m\phi_m),
\end{equation}
namely, (B4) holds.
We usually get coherence measures for quantum channels from corresponding coherence measures for quantum states. For instance, the $l_1-$norm of coherence $C_{l_1}(\rho)=\sum_{i\ne j}|\rho_{i,j}|$\cite{baumgratz} and the relative entropy of coherence $C_{\rm rel.}(\rho)=S(\rho_{\rm diag})-S(\rho),$ where $S$ is the von Neumann entropy and $\rho_{\rm diag}$ denotes the state obtained from $\rho$ by deleting all off-diagonal elements\cite{baumgratz}, are coherence measures for quantum states, on this basis, $C_{l_1}(\phi)=\sum_{i\ne j}|\frac{J_\phi}{|A|}|$ and $C_{\rm rel.}(\phi)=S(\phi_{\rm diag})-S(\phi)=S(\frac{J_{\phi_{\rm diag}}}{|A|})-S(\frac{J_\phi}{|A|})$ both are coherence measures for quantum channels\cite{xujianwei}.
The above proof shows that our new framework is equivalent to the framework given by (B1)-(B4) for quantum channels. In determining whether a function $C$ can be used as a coherence measure for channels, in some cases, it is not easy to verify whether $C$ satisfies (B3). The condition (C3) in our framework provides a new way to solve the problem. We give an example to show the efficiency of our framework.
{\bf Example}
The trace distance measure of coherence defined by $C_{\rm tr}(\rho):=\min_{\delta \in \cal{I}} \|\rho-\delta\|_{\rm tr}=\min_{\delta \in \cal{I}}{\rm tr}|\rho-\delta|$ is not a well defined coherence measure for quantum states \cite{yuxiaodong}. Let us check whether $C_{\rm tr}(\phi)=C_{\rm tr}(\frac{J_\phi}{|A|})$ is a bona fide coherence measure for quantum channels or not.
Here we define
\begin{equation}
C_{\rm tr}(\phi):=\min_{{\tilde{\phi}}\in \cal{IC}}\|\phi-\tilde{\phi}\|_{\rm tr},
\end{equation}
where $\|\phi-\tilde{\phi}\|_{\rm tr}=\|\frac{J_\phi}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}\|_{\rm tr}={\rm tr}|\frac{J_\phi}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}|$ is the trace norm between $\phi$ and $\tilde{\phi}$ with $\phi,\,\tilde{\phi}\in \mathcal{C}_{AB}$.
We need to verify that $C_{\rm tr}(\phi)$ satisfies either (B1)-(B4) or (C1)-(C3). It has been already proved in previous works \cite{baumgratz,bromley} that $C_{\rm tr}$ satisfies (B1), (B2) and (B4). However, the verification of (B3) is rather difficult. The inequality can only be fulfilled for qubit and $X$ quantum states \cite{shaolianhe,Rana}.
We use condition (C3) to verify the validity of $C_{\rm tr}(\phi)$.
For the isometry channel $\phi_{\rm max}\in \mathcal{C}_{AB}$ \cite{xujianwei},
\begin{equation}\phi_{\rm max}(|j\rangle\langle k|)=\frac{1}{|B|}\sum_{\alpha, \beta=0}^{|B|-1}e^{i(\theta_{j\alpha}-\theta_{k\beta})}|\alpha\rangle\langle\beta|,\end{equation}
we have
\begin{equation}C_{\rm tr}(\phi_{\rm max})=\min_{\tilde{\phi}\in\mathcal{IC}}\|\frac{J_{\phi_{\rm max}}}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}\|_{\rm tr},\end{equation}
where
\begin{equation}\frac{J_{\phi_{\rm max}}}{|A|}=\frac{1}{|A||B|}\sum_{j,k=0}^{|A|-1}|j\rangle\langle k|\otimes(\sum_{\alpha,~\beta=0}^{|B|-1}e^{i(\theta_{j\alpha}-\theta_{k\beta})}|\alpha\rangle\langle\beta|)=|\psi\rangle\langle\psi|,
\end{equation}
with $|\psi\rangle=\frac{1}{\sqrt{|A||B|}}\sum_{j=0}^{|A|-1}\sum_{\alpha=0}^{|B|-1}e^{i\theta_{j\alpha}}|j\alpha\rangle$.
Set $U_n=\sum_{k=0}^{|A||B|-1}e^{i(\theta_{(k+n)~\rm mod~|A||B|}-\theta_{k})}|(k+n)~{\rm mod}~|A||B|\rangle\langle k|$. Then we have $U_n|\psi\rangle=|\psi\rangle$.
Since $\|A\|_{\rm tr}+\|B\|_{\rm tr}\geqslant\|A+B\|_{\rm tr}$ and $\|U_n|\psi\rangle\|_{\rm tr}=~\||\psi\rangle\|_{\rm tr}$ for the unitary operation $U_n$, we obtain
\begin{equation}
\begin{array}{rcl}
\|\frac{J_{\phi_{\rm max}}}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}\|_{\rm tr}&=&\frac{1}{|A||B|}\sum_{n=0}^{|A||B|-1}\|U_n(\frac{J_{\phi_{\rm max}}}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}){U_n}^\dagger\|_{\rm tr}\\[1mm]
&\geqslant&\frac{1}{|A||B|}\|\sum_{n=0}^{|A||B|-1}(U_n(\frac{J_{\phi_{\rm max}}}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}){U_n}^\dagger)\|_{\rm tr}.
\end{array}
\end{equation}
As
\begin{equation}
U_n\frac{J_{\phi_{\rm max}}}{|A|}{U_n}^\dagger=U_n|\psi\rangle\langle\psi|{U_n}^\dagger=\frac{J_{\phi_{\rm max}}}{|A|}
\end{equation}
and
\begin{equation}\sum_{n=0}^{|A||B|-1}U_n\frac{J_{\tilde{\phi}}}{|A|}{U_n}^\dagger=I_{|A||B|},
\end{equation}
we have
\begin{equation}
\|\frac{J_\phi}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}\|_{\rm tr}\geqslant\|\frac{J_\phi}{|A|}-\frac{1}{|A||B|}I_{|A||B|}\|_{\rm tr}.
\end{equation}
Therefore,
\begin{equation}\langlebel{12}
\min_{\tilde{\phi}\in\mathcal{IC}}\|\frac{J_{\phi_{\rm max}}}{|A|}-\frac{J_{\tilde{\phi}}}{|A|}\|_{\rm tr}=\|\frac{J_{\phi_{\rm max}}}{|A|}-\frac{1}{|A||B|}I_{|A||B|}\|_{\rm tr}=\frac{2(|A||B|-1)}{|A||B|}.
\end{equation}
Next, we consider a specific channel $\phi\in \mathcal{C}_{AB}$,
\begin{equation}
\phi(|j\rangle\langle k|)=\frac{1}{2}\phi_1(|j\rangle\langle k|)\oplus\frac{1}{2}\phi_2(|j\rangle\langle k|),
\end{equation}
where
\begin{equation}
\phi_1(|j\rangle)=\frac{1}{\sqrt{2}}\sum_{\alpha=0}^{1}e^{i\theta_{j\alpha}}|\alpha\rangle,
~~~\phi_2(|j\rangle)=\frac{1}{\sqrt{3}}\sum_{\beta=0}^{2}e^{i\theta_{j\beta}}|\beta\rangle,
\end{equation}
with $\phi_1\in\mathcal{C}_{AB_1}$ and $\phi_2\in\mathcal{C}_{AB_2}$ the isometry channels, $|A|=2,~|B_1|=2,~|B_2|=3$ and $|B|=5$.
In particular, we take the incoherent channel $\phi_0\in C_{AB},~\phi_0(|j\rangle\langle k|)=\frac{1}{4}\delta_{jk}(|j\rangle\langle k|)\oplus \mathbf{0}(|j\rangle\langle k|)$. Then
\begin{equation}
C_{\rm tr}(\phi)=\displaystyle\min_{\tilde{\phi}\in\mathcal{IC}}\|\frac{J_\phi}{2}-\frac{J_{\tilde{\phi}}}{2}\|_{\rm tr}\leqslant\|\frac{J_\phi}{2}-\frac{J_{\phi_0}}{2}\|_{\rm tr}.
\end{equation}
From (10) we get $\frac{1}{2}C_{\rm tr}(\phi_1)+\frac{1}{2}C_{\rm tr}(\phi_2)=\frac{19}{12}$. However, $C_{\rm tr}(\phi)\leqslant\|\frac{J_\phi}{2}-\frac{J_{\phi_0}}{2}\|_{\rm tr}=1$. Obviously, $C_{\rm tr}(\phi)\ne\frac{1}{2}C_{\rm tr}(\phi_1)+\frac{1}{2}C_{\rm tr}(\phi_2)$. Therefore, the trace norm of coherence is not a well defined coherence measure of quantum channels. In other words, it also violates (B3). Here, inspired by the definition of trace norm, one may propose a similar trace norm function
${C_{\rm tr}}'(\phi)=\displaystyle\min_{\langlembda\geqslant0,
~\tilde{\phi}\in\mathcal{IC}}\|\phi-\langlembda\tilde{\phi}\|_{\rm tr}$, which can be shown to be a legal coherent measure for quantum channels \cite{yuxiaodong,xujianwei}.
We have studied the coherence of quantum channels based on the corresponding Choi matrices to the quantum channels. Note that $p_1J_{\phi_1}\oplus p_2J_{\phi_2}$ is not necessarily a Choi matrix for arbitrary channels $\phi_1\in \mathcal{C}_{AB_1}$ and $\phi_2\in \mathcal{C}_{AB_2}$. From (1) the Choi matrix corresponding to a channel $\phi\in \mathcal{C}_{AB}$ is a $|A||B|\times|A||B|$ positive definite matrix where each $\phi(|j\rangle\langle k|)$ is a $|B|\times|B|$ block matrix and ${\rm tr}(\phi(|j\rangle\langle j|))=1$. Assuming that there is a channel $\Phi\in \mathcal{C}_{AB}$ such that $J_{\Phi}=p_1J_{\phi_1}\oplus p_2J_{\phi_2}$ with $|B|=|B_1|+|B_2|$. With respect to the matrix $p_iJ_{\phi_i}$, each $p_i\phi_i(|j\rangle\langle k|)$ is a $|B_i|\times|B_i|$ block matrix and ${\rm tr}[p_i\phi_i(|j\rangle\langle j|)]=p_i$, $i=1,2$. One can see that the trace of the $|B|\times|B|$ block matrix on all diagonals cannot always be 1 for arbitrary probability $p_1$ and $p_2$. In other words, $p_1J_{\phi_1}\oplus p_2J_{\phi_2}$ is not necessarily a Choi matrix, as it does not satisfy the structure of the Choi matrix corresponding to the channels.
\section{Conclusions}
We have presented an alternative framework to quantify the coherence of quantum channels. Our framework and the framework given by (B1)-(B4) for quantum channels are equivalent. We have used this framework to certify the validity of the trace norm coherence measure for quantum channels. Similar to the case for the coherence measure of quantum states \cite{yuxiaodong}, our framework has the similar unique superiorities and may significantly simplify the quantification for coherence of quantum channels. Our results may highlight further investigations on the resource theory of quantum channels.
\noindent{\bf Acknowledgments}\, \, This work is supported by NSFC (Grant Nos. 12075159 and 12171044), Beijing Natural Science Foundation (Z190005), Academy for Multidisciplinary Studies, Capital Normal University, the Academician Innovation Platform of Hainan Province, and Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (No. SIQSE202001); Academician Innovation Platform of Hainan Province.
\end{document} |
\begin{document}
\title{Global Navier-Stokes flows for non-decaying initial data with slowly decaying oscillation}
\author{Hyunju Kwon \and Tai-Peng Tsai}
\date{}
\maketitle
\begin{abstract}
Consider the Cauchy problem of incompressible Navier-Stokes equations in $\mathbb{R}^3$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants, or when the velocity gradients are in $L^p$ with finite $p$. In this paper, we construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.
{\it Keywords}: incompressible Navier-Stokes equations, non-decaying initial data, oscillation decay, global existence, local energy solution
{\it Mathematics Subject Classification (2010)}: 35Q30, 76D05, 35D30
\end{abstract}
\section{Introduction}
In this paper, we consider the incompressible Navier-Stokes equations
\begin{equation}\label{NS}\tag{NS}
\begin{cases}
\partial_t v -\De v + (v\cdot \nabla )v + \nabla p = 0 \\
\mathop{\rm div} v =0 \\
v|_{t=0}=v_0
\end{cases}
\end{equation}
in $\mathbb{R}^3\times (0,T)$ for $0 <T\le \infty$. These equations describe the flow of incompressible viscous fluids, so the solution $v:\mathbb{R}^3\times (0,T)\to \mathbb{R}^3$ and $p:\mathbb{R}^3\times (0,T)\to \mathbb{R}$ represent the flow velocity and the pressure, respectively.
For an initial datum with finite kinetic energy, $v_0\in L^2(\mathbb{R}^3)$, the existence of a time-global weak solution dates back to Leray \cite{leray}. This solution has a finite global energy, i.e, it satisfies the energy inequality:
\EQ{\label{energy.ineq}
\norm{v(\cdot, t)}_{L^2(\mathbb{R}^3)}^2 + 2\norm{\nabla v}_{L^2(0,t;L^2(\mathbb{R}^3))}^2
\leq \norm{v_0}_{L^2(\mathbb{R}^3)}^2, \quad \forall t>0.
}
In Hopf \cite{Hopf}, this result is extended to smooth bounded domains with the Dirichlet boundary condition. We say $v$ is \textit{a Leray-Hopf weak solution} to \eqref{NS} in $\Omega \times (0,T)$ for a domain $\Om\subset \mathbb{R}^3$, if
\[
v\in L^\infty(0,T;L^2_{\si}(\Om))\cap L^2(0,T;H_{0,\si}^1(\Om))\cap C_{wk}([0,T);L^2_{\si}(\Om))
\]
satisfies the weak form of \eqref{NS} and the energy inequality \eqref{energy.ineq}.
However, when a fluid fills an unbounded domain, it is possible to have finite local energy but
infinite global energy. One such example is a fluid with constant velocity. There are also many interesting non-decaying infinite energy flows like time-dependent spatially periodic flows (flows on torus) and \emph{two-and-a-half dimensional flows}; see \cite[Section 2.3.1]{MaBe} and \cite{Gallagher}. Can we get global existence for such data? To analyze the motion of such fluids, one may consider the class $ L^2_{\mathrm{uloc}}$ for velocity field $v_0$ in $\mathbb{R}^3$ whose kinetic energy is uniformly locally bounded. Here, for $1\le q \le \infty$, we denote by $L^q_{\mathrm{uloc}}$ the space of functions in $\mathbb{R}^3$ with
\[
\norm{v_0}_{L^q_{\mathrm{uloc}}} := \sup_{x_0 \in \mathbb{R}^3} \norm{v_0}_{L^q(B(x_0,1))}
<\infty.
\]
We also denote its subsapce with spatial decay
\[
E^q = \big\{v_0 \in L^q_{{\mathrm{uloc}}}: \, \lim_{|x_0|\to \infty}\norm{v_0}_{L^q(B(x_0,1))} =0\big\}.
\]
In \cite{LR}, Lemari\'{e}-Rieusset introduced the class of \textit{local energy solutions} for initial data $v_0 \in L^2_{\mathrm{uloc}}$ (see Section \ref{loc.ex.sec} for details). He proved the short time existence for initial data in $L^2_{\mathrm{uloc}}$, and the global in time existence for $v_0\in E^2$, those initial data in $L^2_{\mathrm{uloc}}$ which further satisfy the
spatial decay condition
\EQ{\label{ini.E2}
\lim_{|x_0|\to \infty}\int_{B(x_0,1)} |v_0|^2 dx=0.
}
Then, Kikuchi-Seregin \cite{KS} added more details to the results in \cite{LR}, especially the careful treatment of the pressure. They also allowed a force term $g$ in \eqref{NS} which satisfies $\mathop{\rm div} g=0$ and
\[
\lim_{|x_0|\to \infty}\int_0^T\!\int_{B(x_0,1)}|g(x,t)|^2 dxdt =0, \quad \forall T>0.
\]
Recently, Maekawa-Miura-Prange \cite{MaMiPr} generalized this result to the half-space $\mathbb{R}^3_+$. The treatment of the pressure in \cite{MaMiPr} is even more complicated.
One key difficulty in the study of infinite energy solutions is the estimates of the pressure. While finite energy solutions have enough decay at spatial infinity and one may often get the pressure from the equation $p = (-\De)^{-1}\partial_i\partial_j(v_iv_j)$, this is not applicable to infinite energy solutions because of their slow (or no) spatial decay.
To estimate the pressure, the definition of a local energy solution in \cite{KS} includes a locally-defined pressure decomposition near each point in $\mathbb{R}^3$, see condition (v) in Definition \ref{les}. (It is already in \cite{LR} but not part of the definition.) In \cite{JiaSverak-minimal}-\cite{JiaSverak}, on the other hand, Jia and \v Sver\'ak use a slightly different definition by replacing the decomposition condition by the spatial decay of the velocity
\begin{equation}
\label{decay.condi.parR}
\lim_{|x_0|\to \infty}\int_0^{R^2}\int_{B(x_0,R)}|v(x,t)|^2 dxdt =0, \quad\forall R>0.
\end{equation}
Under the decay assumption \eqref{ini.E2} on initial data, these two definitions can be shown to be equivalent; see \cite{MaMiPr, KMT}. However, for general non-decaying initial data, the decay condition $\eqref{decay.condi.parR}$ is not expected, while the decomposition condition still works. For this reason, we follow the definition of Kikuchi-Seregin \cite{KS} in this paper.
A new feature in the study of infinite energy solutions with non-decaying initial data is the abundance of \emph{parasitic solutions},
\[
v(x,t) = f(t),\quad p(x,t) = -f'(t)\cdot x
\]
for a smooth vector function $f(t)$. They solve the Navier-Stokes equations with initial data $f(0)$. If we choose $f_1(t)\not = f_2(t)$ with $f_1(0)=f_2(0)$, the corresponding parasitic solutions give two different local energy solutions with the same initial data. Such solutions have non-decaying initial data, and can be shown to fail the pressure decomposition condition. More generally, if $(v,p)$ is a solution to \eqref{NS}, then the following \emph{parasitic transform}
\EQ{
u(x,t) = v(y,t)+q'(t), \quad \pi(x,t)= p(y,t) - q''(t) \cdot y, \quad
y=x-q(t)
}
gives another solution $(u,\pi)$ to \eqref{NS} with the same initial data $v_0$ for any vector function $q(t)$ satisfying $q(0)=q'(0)=0$.
We now summarize the known existence results in $\mathbb{R}^3$.
In addition to the weak solution approach based on the a priori bound \eqref{energy.ineq} following Leray and Hopf, another fruitful approach is the theory of \emph{mild solutions}, treating the nonlinear term as a source term of the nonhomogeneous Stokes system. In the framework of $L^q(\mathbb{R}^3)$, there exist short time mild solutions in $L^q(\mathbb{R}^3)$
when $3 \le q \le \infty$ (\cite{FJR,Kato84,GIM}).
When $q=3$, these solutions exist for all time for sufficiently small initial data in $L^3(\mathbb{R}^3)$; see \cite{Kato84}. Similar small data global existence results hold for many other spaces of similar scaling property, such as $L^3_{\text{weak}}$, Morrey spaces $M_{p,3-p}$, negative Besov spaces $\dot B^{3/q-1}_{q,\infty}$, $3<q<\infty$, and the Koch-Tataru space BMO$^{-1}$; See e.g.~\cite{GiMi,Kato,KoYa,Barraza,CP,BCD,Koch-Tataru}.
For any data $v_0 \in L^q(\mathbb{R}^3)$, $2<q<3$,
Calder\'on \cite{Calderon} constructed a global solution. His strategy is to first decompose $v_0 = a_0+b_0$ with small $a_0 \in L^3(\mathbb{R}^3)$ and large $b_0\in L^2(\mathbb{R}^3)$. A solution is then obtained as $v=a+b$, where $a$ is a global small mild solution of \eqref{NS} in $L^3(\mathbb{R}^3)$ with $a(0)=a_0$, and $b$ is a global weak solution of the $a$-perturbed Navier-Stokes equations in the energy class
with $b(0)=b_0$.
This idea is then used by Lemari\'{e}-Rieusset \cite{LR} to construct global local energy solutions for $v_0 \in E^2$; also see Kikuchi-Seregin \cite{KS}.
We now summarize the known existence results for non-decaying initial data. For the local existence, many mild solution existence theorems mentioned earlier allow non-decaying data. The most relevant to us are Giga-Inui-Matsui \cite{GIM} for initial data in $L^\infty(\mathbb{R}^3)$ and $BUC(\mathbb{R}^3)$, and Maekawa-Terasawa \cite{MT} for initial data in the closure of ${\bigcup_{p>3}L^p_{\mathrm{uloc}}}$ in $L^3_{\mathrm{uloc}}$-norm, and any small initial data in $L^3_{\mathrm{uloc}}$. Smallness is needed for $L^3_{\mathrm{uloc}}$ data even for short time existence.
When it comes to the global existence for non-decaynig data, a solution theory for perturbations of constant vectors seems straightforward. Lemari\'{e}-Rieusset \cite[Theorem 1(C)]{LR-Morrey} constructed global weak solutions for $u_0$ in Morrey space $M^{2,1}$, which contains non-decaying functions, e.g.
\[
v_0(x) = \sum_{k \in \mathbb{N}} \zeta(x-x_k)
\]
with $|x_k|\to \infty$ rapidly as $k \to \infty$. Here $\zeta$ is any smooth divergence free vector field with compact support.
The only other result we are aware of is the recent paper Maremonti-Shimizu \cite{MaSe}, which proved the global existence of weak solutions for initial data $v_0$ in $L^\infty(\mathbb{R}^3)\cap \overline{C_0(\mathbb{R}^3)}^{\dot{W}^{1,q}}$, $3<q<\infty$. In particular, they assume $\nablabla v_0 \in L^q(\mathbb{R}^3)$. Their strategy is to decompose the solution $v = U + w$, $U=\sum_{k=1}^n v^k$, where $v^1$ solves the Stokes equations with the given initial data, and $v^{k+1}$, $k\geq 1$, solves the linearized Navier-Stokes equations with the force $f^k=-v^{k}\cdot\nabla v^k$ and homogeneous initial data. The force $f^1 \in L^q(0,T;L^q(\mathbb{R}^3))$ thanks to the assumption on $v_0$. In each iteration, we get an additional decay of the force $f^k$. The perturbation $w$ is then solved in the framework of weak solutions. The paper \cite{MaSe} motivated this paper.
We now state our main theorem.
Denote the average of a function $v$ in a set $O\subset \mathbb{R}^3$ by $(v)_O = \frac 1{|O|}\int_O v(x)\, dx$.
We denote $w\in E^2_\si$ if $w\in E^2$ and $\mathop{\rm div} w=0$.
\begin{theorem}\label{global.ex}
For any vector field $v_0\in E^2_\si + L^3_{\mathrm{uloc}}$ satisfying $\mathop{\rm div} v_0=0$ and
\EQ{\label{ini.decay}
\lim_{|x_0|\to \infty}\int_{B(x_0,1)}| v_0- (v_0)_{B(x_0,1)}| dx =0,
}
we can find a time-global local energy solution $(v,p)$ to the Navier-Stokes equations \eqref{NS} in $\mathbb{R}^3 \times (0,\infty)$, in the sense of Definition \ref{les}.
\end{theorem}
Our main assumption is the ``\emph{oscillation decay}'' condition \eqref{ini.decay}. Note that all $v_0\in L^2_{\mathrm{uloc}}$ satisfying \eqref{ini.E2} also satisfy \eqref{ini.decay}. Furthermore, for $v_0\in L^2_{\mathrm{uloc}}$, either $v_0 \in E^1$ or $\nabla v_0\in E^{1}$ implies the condition \eqref{ini.decay}. Recall $E^q$ for $1 \le q \le \infty$ is the space of functions in $L^q_{\mathrm{uloc}}$ whose $L^q$-norm in a ball $B_1(x_0)$ goes to zero as $|x_0|$ goes to infinity. In particular, our result generalizes the global existence for decaying initial data $v_0\in E^2$ in \cite{LR} and \cite{KS}. It also extends \cite{MaSe} for $v_0 \in L^\infty$ and $\nablabla v_0 \in L^q$.
\begin{example}
Consider
\[
v_0=v_1+v_2, \quad
v_1 = \frac {(-x_2, \ x_1, \ 0)}{\sqrt{|x|^2+1}},\quad
v_2=\frac {(-x_2, \ x_1, \ 0)}{|x|^2+1} \sin \bke{(x^2+1)^{100}}.
\]
We have
$\mathop{\rm div} v_1=\mathop{\rm div} v_2 = 0$, $v_0 \not \in E^2$,
$v_0$ satisfies the oscillation decay condition, and
\[
\limsup_{|x_0|\to \infty} \int_{B_1(x_0)} |\nablabla v_0| =\infty.
\]
In particular, $v_0 \in L^\infty$ but $\nablabla v_0 \not \in L^q$ for any $q \le \infty$.
Moreover, $v_0$ is not a perturbation of constant, although it converges to a constant along each direction.
\end{example}
The condition $v_0 \in E^2_\si+L^3_{\mathrm{uloc}}$ gives us more regularity on the nondecaying part of $v_0$. We do not know if it is necessary for the global existence, but it is essential for our proof, and enables us to prove that for small $t>0$,
\EQ{\label{eq1.5}
\norm{w(t)\chi_R}_{L^2_{\mathrm{uloc}}} \lesssim (t^\frac 1{20} + \norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}),
}
where $\chi_R(x)$ is a cut-off function supported in $|x|>R$, we decompose $v_0=w_0+u_0$ with $w_0\in E^2_\si$ and $u_0 \in L^3_{{\mathrm{uloc}}}$, and $w(t) = v(t) -e^{t\De}u_0$ with $w(0)=w_0$. This estimate shows that $\norm{w(t)\chi_R}_{L^2_{\mathrm{uloc}}}$ vanishes as $ t\to 0_+$ and $R\to \infty$.
The idea of our proof is as follows. First, we construct a local energy solution in a short time. For $v_0 \in L^2_{\mathrm{uloc}}$, this is done in \cite{LR} but not in \cite{KS}. However, we use a slightly revised approximation scheme to make all statements about the pressure easy to verify. In our scheme, we not only mollify the non-linear term as in \cite{leray} and \cite{LR}, but also insert a cut-off function, so that the non-linear term $(v\cdot \nabla)v$ is replaced by $(\mathcal{J}_\ep(v)\cdot \nabla)(v\Phi_{\ep})$, where $\mathcal{J}_\ep$ is a mollification of scale $\ep$ and $\Phi_\ep$ is a radial bump function supported in the ball $B(0, 2\ep^{-1})$.
Once we have a local-in-time local energy solution, we need some smallness to extend the solution globally in time. To this end, we decompose the solution as $v=V+w$ where $V(t)=e^{t\De}u_0$ solves the heat equation. The main effort is to show that $w(t) \in E^2$ for all $t$ and $w(t) \in E^6$ for almost all $t$. The proof is similar to the decay estimates in \cite{LR,KS} and we try to do local energy estimate for $w \chi_R$. The background $V$ has no spatial decay, but we can show the decay of $\nablabla V(x,t)$ in $L^\infty(B_R^c\times (t_0,\infty))$ as $R\to \infty$ for any $t_0>0$. This decay is not uniform up to $t_0=0$ as $u_0$ is rather rough. We need a new decomposition formula of the pressure, so that in the intermediate regions we can show the decay of the pressure using the decay of $\nablabla V$. Because the decay of $\nablabla V$ is not up to $t_0=0$, we need to do the local energy estimate in the time interval $[t_0,T)$, $0<t_0\ll1$. This forces us to prove the estimate \eqref{eq1.5}, and the \emph{strong local energy inequality} for $w$ away from $t=0$.
Once we have shown $w(t) \in E^6$ for almost all $t<T$, we can extend the solution as in \cite{LR} and \cite{KS}. However, we avoid using the strong-weak uniqueness as in \cite{LR,KS}, and choose to verify the definition of local energy solutions directly as in \cite{MaMiPr}.
The rest of the paper consists of the following sections. In Section \ref{pre}, we discuss the properties of the heat flow $e^{t\De}u_0$, especially the decay of its gradient at spatial infinity assuming \eqref{ini.decay}. In Section \ref{loc.ex.sec}, we recall the definition of local energy solutions as in \cite{KS} and use our revised approximation scheme to find a local energy solution local-in-time. In Section \ref{decay.est.sec}, we find a new pressure decomposition formula suitable of using the decay of $\nablabla V$, prove the estimate \eqref{eq1.5} and the strong local energy inequality, and then do the local energy estimate of $w\chi_R$, which implies $w(t) \in E^6$ for almost all $t$. In Section \ref{global.sec}, we construct the desired time-global local energy solution. In Section \ref{sec6}, by a similar and easier proof, we additionally obtain perturbations of time-global solutions with no spatial oscillation decay.
\section{Notations and preliminaries}\label{pre}
\subsection{Notation}
Given two comparable quantities $X$ and $Y$, the inequality $X\lesssim Y$ stands for $X\leq C Y$ for some positive constant $C$. In a similar way, $\gtrsim$ denotes $\geq C$ for some $C>0$. We write $ X \sim Y$ if $X\lesssim Y$ and $Y\lesssim X$. Furthermore, in the case that a constant $C$ in $X\leq C Y$ depends on some quantities $Z_1$, $\cdots$, $Z_n$, we write $X\lesssim_{Z_1,\cdots,Z_n}Y$. The notations $\gtrsim_{Z_1,\cdots,Z_n}$ and $\sim_{Z_1,\cdots,Z_n}$ are similarly defined.
For a point $x\in \mathbb{R}^3$ and a positive real number $r$, $B(x,r)$ is the Euclidean ball in $\mathbb{R}^3$ centered at $x$ with a radius $r$,
\[
B(x,r) =B_r(x)= \{y\in \mathbb{R}^3: |y-x|<r\}.
\]
When $x=0$, we denote $B_r = B(0,r)$. For a point $x\in \mathbb{R}^3$ and $r>0$, we denote the open cube centered at $x$ with a side length $2r$ as
\[
Q(x,r)=Q_r(x) = \bket{ y \in \mathbb{R}^3: \max_{i=1,2,3} |y_i -x_{i}| < r}.
\]
We denote the mollification $\mathcal{J}_\ep(v) = v\ast \eta_\ep$, $\ep>0$, where the mollifier is $\eta_\ep(x) = \e^{-3} \eta\left(\frac x{\ep}\right)$ and $\eta$ is a fixed nonnegative radial bump function in $C_c^\infty(\mathbb{R}^3)$ supported in $B(0,1)$ satisfying
$\int\eta\, dx =1$.
Various test functions in this paper are defined by rescaling and translating a non-negative radially decreasing bump function $\Phi$ satisfying $\Phi = 1$ on $B(0,1)$ and $\supp(\Phi)\subset B(0, \frac 32)$.
For $k \in \mathbb{N} \cup \{0,\infty\}$, let
$C^k_c(\mathbb{R}^3)$ be the subset of functions in $C^k(\mathbb{R}^3)$ with compact supports, and
\[
C^{k}_{c,\si}(\mathbb{R}^3) = \bket{ u \in C^{k}_{c}(\mathbb{R}^3;\mathbb{R}^3):\ \mathop{\rm div} u =0}.
\]
\subsection{Uniformly locally integrable spaces}
To consider infinite energy flows, we work in the spaces $L^q_{\mathrm{uloc}}$, $1\leq q\leq \infty$, and $U^{s,p}(t_0,t)$ for $1\leq s, p\leq \infty$ and $0\leq t_0<t\le \infty$, defined by
\[
L^q_{\mathrm{uloc}} =
\bket{u\in L^1_{{\mathrm{loc}}}(\mathbb{R}^3): \norm{u}_{L^q_{\mathrm{uloc}}} = \sup_{x_0\in \mathbb{R}^3} \norm{u}_{L^q(B_1(x_0))} <+\infty
}
\]
and
\[
U^{s,p}(t_0,t) =
\bket{u \in L^1_{{\mathrm{loc}}}(\mathbb{R}^3\times(t_0,t)): \norm{u}_{U^{s,p}(t_0,t)}= \sup_{x_0\in \mathbb{R}^3} \norm{u}_{L^s(t_0,t;L^p(B_1(x_0)))}<+\infty }.
\]
When $t_0=0$, we simply use $U^{s,p}_T = U^{s,p}(0,T)$. Note that $U^{\infty,p}(t_0,t)=L^\infty(t_0,t;L^p_{\mathrm{uloc}})$, $1\leq p \leq \infty$,
but for general $1\leq s<\infty$ and $1\leq p \le \infty$, $U^{s,p}(t_0,t)$ and $L^s(t_0,t;L^p_{\mathrm{uloc}})$ are not equivalent norms. Indeed, we can only guarantee that
\EQ{\label{Usp.le.LsUp}
\norm{u}_{U^{s,p}(t_0,t)} \leq \norm{u}_{L^s(t_0,t;L^p_{\mathrm{uloc}})},
}
but not the inequality of the other direction.
\begin{example}
Fix $1\le s <\infty$ and $p \in [1,\infty]$. Let $x_k$ be a sequence in $\mathbb{R}^3$ with disjoint $B_1(x_k)$, $k \in \mathbb{N}$, and let $t_k = t_0+2^{-k}$. Define a function $u$ by $u(x,\tau)=2^{k/s}$ on $B_1(x_k) \times (t_0,t_k)$, $k \in \mathbb{N}$, and $u(x,\tau)=0$ otherwise. It is defined independently of $p$. We have $u \in U^{s,p}(t_0,t)$, but
\[
\int _{t_0}^{t_1} \norm{u(\cdot,\tau)}_{L^p_{\mathrm{uloc}}}^s d\tau = \sum_{k=1}^\infty \int_{t_{k+1}}^{t_k} c_p2^k d\tau = \sum_{k=1}^\infty \frac 12 c_p=\infty,
\]
and hence $u \not \in L^s(t_0,t;L^p_{\mathrm{uloc}})$.
\qed
\end{example}
We define a local energy space $\mathcal{E}(t_0,t)$ by
\EQ{\label{cE.def}
\mathcal{E}(t_0,t) = \bket{u\in L^2_{\mathrm{loc}}([t_0,t]\times \mathbb{R}^3;\mathbb{R}^3) \ : \ \mathop{\rm div} u =0, \
\norm{u}_{\mathcal{E}(t_0,t)} <+\infty},
}
where
\[
\norm{u}_{\mathcal{E}(t_0,t)} := \norm{u}_{U^{\infty,2}(t_0,t)} + \norm{\nabla u}_{U^{2,2}(t_0,t)}.
\]
When $t_0=0$, we use the abbreviation $\mathcal{E}_T = \mathcal{E}(0,T)$.
The spaces $E^p$ and $G^p(t_0,t)$, $1\leq p\leq \infty$, are defined by an additional decay condition at infinity,
\[
E^p:= \{f\in L^p_{\mathrm{uloc}}: \norm{f}_{L^p(B(x_0,1))} \to 0, \quad\text{as }|x_0|\to \infty \},
\]
and
\[
G^p(t_0,t) :=\{u \in U^{p,p}(t_0,t): \ \norm{u}_{L^p([t_0,t]\times B(x_0,1))} \to 0, \quad\text{as } |x_0|\to \infty
\}.
\]
We let $L^p_{{\mathrm{uloc}},\si}$, $E^p_\si$ and $G^p_\si(t_0,t)$ denote divergence-free vector fields with components in $L^p_{{\mathrm{uloc}}}$, $E^p$ and $G^p(t_0,t)$, respectively.
The space $E^p$, $1\leq p<\infty$, can be characterized as $\overline{C_{c}^\infty(\mathbb{R}^3)}^{L^p_{\mathrm{uloc}}}$.
The analogous statement for $E^p_\si$ is true.
\begin{lemma}\textup{(\cite[Appendix]{KS})}\label{decomp.Ep}
Suppose that $f\in E^p_\si$ for some $1\leq p<\infty$. Then, for any $\ep>0$, we can find $f^\ep \in C_{c,\si}^\infty(\mathbb{R}^3)$ such that
\[
\norm{f-f^\ep}_{L^p_{\mathrm{uloc}}}<\ep.
\]
\end{lemma}
\subsection{Heat and Oseen kernels on $L^q_{\mathrm{uloc}}$}
Now, we study the operators $e^{t\De}$ and $e^{t\De}\mathbb{P}\nabla \cdot$ on $L^q_{\mathrm{uloc}}$. Here $\mathbb{P}$ denotes the Helmholtz projection in $\mathbb{R}^3$. Both are defined as convolution operators
\[
e^{t\De} f = H_t \ast f, \quad\text{and}\quad
e^{t\De}\mathbb{P}_{ij}\partial_kF_{jk} = \partial_k S_{ij}\ast F_{jk},
\]
where $H_t$ and $S_{ij}$ are the heat kernel and the Oseen tensor, respectively,
\EQN{
H_t(x) = \frac 1{\sqrt{4\pi t}^3} \exp\left(-\frac {|x|^2}{4t}\right),
}
and
\[
S_{ij}(x,t) = H_t(x)\de_{ij} + \frac 1{4\pi} \frac{\partial^2}{\partial_{x_i}\partial_{x_j}} \int_{\mathbb{R}^3} \frac{H_t(y)}{|x-y|}dy.
\]
In this note, we use $(\mathop{\rm div} F)_i = (\nabla \cdot F)_i = \partial_j F_{ji}$. Note that the Oseen tensor satisfies the following pointwise estimates
\EQ{\label{pt.est.oseen}
|\nabla_{x}^l \partial_t^k S(x,t)| \leq C_{k,l} (|x|+\sqrt{t})^{-3-l-2k}.
}
We have the following estimates.
\begin{lemma}[Remark 3.2 in \cite{MT}]\label{lemma23}
For $1\leq q\leq p\leq \infty$, the following holds. For any vector field $f$ and any 2-tensor $F$ in $\mathbb{R}^3$,
\[
\norm{\partial^{\al}_t \partial^{\be}_x e^{t\De}f}_{L^p_{\mathrm{uloc}}}
\lesssim \frac 1{t^{|\al|+\frac{|\be|}{2}}}\left(1+ \frac 1{t^{\frac 32\left(\frac 1q-\frac 1p\right)}}\right)\norm{f}_{L^q_{\mathrm{uloc}}},
\]
\[
\norm{\partial^{\al}_t \partial^{\be}_x e^{t\De}\mathbb{P}\nabla \cdot F}_{L^p_{\mathrm{uloc}}}
\lesssim \frac 1{t^{|\al|+\frac{|\be|}{2}+\frac 12}}\left(1+ \frac 1{t^{\frac 32\left(\frac 1q-\frac 1p\right)}}\right)\norm{F}_{L^q_{\mathrm{uloc}}}.
\]
\end{lemma}
Note $p=\infty$ is allowed, with $L^\infty_{\mathrm{uloc}}= L^\infty$.
\begin{lemma}\label{lemma24}
For any $T>0$, if $f\in L^2_{\mathrm{uloc}}$ and $F\in U^{2,2}_T$, then we have
\EQN{
\norm{e^{t\De}f}_{\mathcal{E}_T}
\lesssim (1+T^\frac 12)\norm{f}_{L^2_{\mathrm{uloc}}},\\
\norm{\int_0^t e^{(t-s)\De}\mathbb{P}\nabla \cdot F(s)\, ds}_{\mathcal{E}_T} \lesssim (1+T)\norm{F}_{U^{2,2}_T}.
}
\end{lemma}
Recall $\norm{u}_{\mathcal{E}_T} = \norm{u}_{U^{\infty,2}_T} + \norm{\nabla u}_{U^{2,2}_T}$. Similar estimates can be found in the proof
of \cite[Theorem 14.1]{LR2}.
We give a slightly revised proof here for completeness.
\begin{proof} Fix $x_0\in \mathbb{R}^3$ and let $\phi_{x_0}(x) = \Phi\left(\frac{x-x_0}2\right)$. We decompose $f$ and $F$ as
\[
f = f\phi_{x_0}+ f(1-\phi_{x_0}) = f_1 + f_2
\]
and
\[
F = F\phi_{x_0}+ F(1-\phi_{x_0}) = F_1 + F_2.
\]
Since $f_1\in L^2(\mathbb{R}^3)$ and $F_1\in L^2(0,T;L^2(\mathbb{R}^3))$, by the usual energy estimates for the heat equation and the Stokes system, we get
\EQ{\label{0708-1}
\norm{e^{t\De}f_1}_{\mathcal{E}_T} \lesssim \norm{f_1}_{2} \lesssim \norm{f}_{L^2_{\mathrm{uloc}}}
}
and
\EQ{\label{0709-1}
\norm{\int_0^t e^{(t-s)\De}\mathbb{P}\nabla \cdot F_1(s) ds}_{\mathcal{E}_T} \lesssim
\norm{F_1}_{L^2(0,T;L^2(\mathbb{R}^3))} \lesssim \norm{F}_{U^{2,2}_T}.
}
On the other hand, by Lemma \ref{lemma23},
\[
\norm{e^{t\De}f_2}_{U^{\infty,2}_T}=\norm{e^{t\De}f_2}_{L^\infty(0,T;L^2_{\mathrm{uloc}})} \lesssim \norm{f_2}_{L^2_{\mathrm{uloc}}} \lesssim \norm{f}_{L^2_{\mathrm{uloc}}}.
\]
Together with \eqref{0708-1}, we get
\EQ{\label{0708-2}
\norm{e^{t\De}f}_{U^{\infty,2}_T} \lesssim \norm{f}_{L^2_{\mathrm{uloc}}}.
}
(This also follows from Lemma \ref{lemma23}.)
By the heat kernel estimates,
\EQN{
\norm{\nabla e^{t\De}f_2}_{L^2((0,T)\times B(x_0,1))}
& \lesssim T^{\frac 12}\norm{\nabla e^{t\De}f_2}_{L^\infty((0,T)\times B(x_0,1))} \\
& \lesssim T^{\frac 12}\int_{B(x_0,2)^c}\frac 1{|x_0-y|^4} |f_2(y)|dy\\
&\le T^{\frac 12}\sum_{k=1}^\infty \int_{B(x_0,2^{k+1})\setminus B(x_0,2^{k})}\frac 1{|x_0-y|^4} |f(y)|dy\\
& \lesssim T^{\frac 12}\sum_{k=1}^\infty \frac 1{2^{4k}} \int_{B(x_0,2^{k+1})} |f(y)|dy.
}
We may cover $B(x_0,2^{k+1})$ by $\bigcup_{j=1}^{J_k} B(x_j^k,1)$ with
$J_k$ bounded by $C_0 2^{3k}$ for some constant $C_0>0$. Then
\[
\norm{\nabla e^{t\De}f_2}_{L^2((0,T)\times B(x_0,1))} \lesssim
T^{\frac 12}\sum_{k=1}^\infty \frac 1{2^{4k}} \sum_{j=1}^{J_k} \int_{B(x_j^k,1)} |f(y)|dy\\
\lesssim T^{\frac 12} \norm{f}_{L^2_{\mathrm{uloc}}} .
\]
Together with \eqref{0708-1}, we get
\[
\norm{\nabla e^{t\De}f}_{L^2((0,T)\times B(x_0,1))} \lesssim (1+T^{\frac 12}) \norm{f}_{L^2_{\mathrm{uloc}}}.
\]
Taking supremum in $x_0$, we obtain
\[
\norm{\nabla e^{t\De}f}_{U^{2,2}_T} \lesssim (1+T^{\frac 12}) \norm{f}_{L^2_{\mathrm{uloc}}}.
\]
This and \eqref{0708-2} show the first bound of the lemma,
$\norm{e^{t\De}f}_{\mathcal{E}_T} \lesssim (1+T^\frac 12)\norm{f}_{L^2_{\mathrm{uloc}}}$.
Denote $\Psi F (t) = \int_0^t e^{(t-s)\De}\mathbb{P}\nabla \cdot F(s) ds$. By the pointwise estimates \eqref{pt.est.oseen} for Oseen tensor, we have
\EQN{
\norm{\Psi F_2}_{L^\infty(0,T;L^2( B(x_0,1)))}
& \lesssim \int_0^t\! \int_{B(x_0,2)^c} \frac 1{|x_0-y|^4} |F_2(y,s)|dy ds\\
&\leq \sum_{k=1}^\infty \frac 1{2^{4k}}\int_0^t\! \int_{B(x_0,2^{k+1})} |F(y,s)|dy ds\\
&\leq \sum_{k=1}^\infty \frac 1{2^{4k}}\sum_{j=1}^{J_k}\int_0^t\! \int_{B(x_j^k,1)} |F(y,s)|dy ds\\
& \lesssim \norm{F}_{U^{1,1}_T} \lesssim T^\frac 12\norm{F}_{U^{2,2}_T}
}
and
\EQN{
\norm{\nabla \Psi F_2}_{L^2((0,T)\times B(x_0,1))}
& \lesssim T^\frac 12 \norm{\nabla \Psi F_2}_{{L^\infty((0,T)\times B(x_0,1))}} \\
& \lesssim T^\frac 12\int_0^t\! \int_{B(x_0,2)^c} \frac 1{|x_0-y|^5} |F_2(y,s)|dy ds\\
&\leq T^\frac 12\sum_{k=1}^\infty \frac 1{2^{5k}}\int_0^t\! \int_{B(x_0,2^{k+1})} |F(y,s)|dy ds\\
& \lesssim T\norm{F}_{U^{2,2}_T}.
}
Combined with \eqref{0709-1}, we have
\[
\norm{\Psi F}_{L^\infty(0,T;L^2( B(x_0,1)))} \lesssim (1+T^\frac 12)\norm{F}_{U^{2,2}_T}
\]
and
\[
\norm{\nabla \Psi F}_{L^2((0,T)\times B(x_0,1))}
\lesssim (1+T)\norm{F}_{U^{2,2}_T}.
\]
Finally, we take suprema in $x_0$ to get
\[
\norm{\int_0^t e^{(t-s)\De}\mathbb{P}\nabla \cdot F(s) ds}_{\mathcal{E}_T}
\lesssim (1+T) \norm{F}_{U^{2,2}_T}.
\]
This is the second bound of the lemma.
\end{proof}
\subsection{Heat kernel on $L^1_{\mathrm{uloc}}$ with decaying oscillation}
In this subsection, we investigate how the decaying oscillation assumption \eqref{ini.decay} on initial data affects the heat flow. Recall
\[
(u)_{Q_r(x)} = \fint_{Q_r(x)} u(y)\, dy = \frac 1{|Q_r(x)|} \int_{Q_r(x)} u(y)\, dy.
\]
\begin{lemma}\label{from.ini}
Suppose that $u\in L^1_{{\mathrm{uloc}}}(\mathbb{R}^3)$ satisfies
\EQ{\label{0505-1}
\lim_{|x_0|\to \infty} \int_{Q_1(x_0)} |u - (u)_{Q_1(x_0)}|dx =0.
}
Then, for any $r>0$, we have
\EQ{\label{0505-2}
\lim_{|x_0|\to \infty} \int_{Q_r(x_0)} |u - (u)_{Q_r(x_0)}| dx=0,
}
and
\EQ{\label{0505-3}
\lim_{|x_0|\to \infty} \sup_{y\in \overline{Q_{2r}(x_0)}} |(u)_{Q_r(y)} - (u)_{Q_r(x_0)}| =0.
}
\end{lemma}
\begin{proof}
First note that $(u)_{Q_r(x)}$ is finite for any $x\in \mathbb{R}^3$ and $r>0$. Indeed,
\[
|(u)_{Q_r(x)}| \le C_r\norm{u}_{L^1_{\mathrm{uloc}}}
\]
for a constant $C_r$ independent of $x$, $C_r < C$ for $r>1$, and $C_r \sim r^{-3}$ for $r \ll 1$.
Fix $x_0\in \mathbb{R}^3$ and $r>0$. For any constant $c\in \mathbb{R}$, we get
\EQN{
\fint_{Q_r(x_0)} |u - (u)_{Q_r(x_0)}| dx
& \le \fint_{Q_r(x_0)} |u - c| + |(u)_{Q_{r}(x_0)} - c| dx
\\
& = \fint_{Q_r(x_0)} |u - c| dx + \abs{ \fint_{Q_r(x_0)} \bke{u - c} dx}
\\
& \le 2\fint_{Q_r(x_0)} |u - c| dx .
}
Then, for $Q_r =Q_r(x_1)\subset Q_R(x_0)$, $R>r$, we get
\EQ{\label{0505-4}
\fint_{Q_r} |u - (u)_{Q_r}| dx \le 2\fint_{Q_r} |u - (u)_{Q_R(x_0)}|dx \le \frac{2R^3}{r^3} \fint_{Q_R(x_0)} |u - (u)_{Q_R(x_0)}| dx .
}
With $x_0=x_1$ and $R=1$ in \eqref{0505-4}, \eqref{0505-1} implies \eqref{0505-2} for all $r \in (0,1)$.
If $y \in \overline{Q_{2r}(x_0)}$, then
\[
Q_r(x_0) \cup Q_r(y) \subset Q_R(x_1), \quad x_1 = \frac 12(x_0+y), \quad R \ge2r.
\]
Thus,
\EQ{\label{0505-7}
\abs{(u)_{Q_r(x_0)} - (u)_{Q_r(y)}}
&\le \abs{ \fint_{Q_r(x_0)} u -(u)_{Q_R(x_1)} dx} +\abs{ \fint_{Q_r(y)} u -(u)_{Q_R(x_1)} dx}
\\
&\le \fint_{Q_r(x_0)} |u - (u)_{Q_R(x_1)}| dx + \fint_{Q_r(y)} |u - (u)_{Q_R(x_1)}| dx
\\
& \le \frac{2R^3}{r^3} \fint_{Q_R(x_1)} |u - (u)_{Q_R(x_1)}| dx .
}
With $R=1$, this and \eqref{0505-1} imply \eqref{0505-3} for all $r \in (0,\frac12]$.
Now, for any $Q_r(x_0)$ with $r>1$, choose the smallest integer $N>2r$ and let $\rho = r/N< \frac 12$. We can find a set $S=S_{x_0,r}$ of $N^3$ points such that $\{ Q_\rho(z): z \in S \}$ are disjoint and
\[
\overline{ Q_r(x_0)} = \bigcup_{z \in S} \overline{Q_\rho(z)}.
\]
For any $z,z' \in S$, we can connect them by points $z_j$ in $S$, $j=0,1,\dots, N$, such that
$z_0=z$, $z_N=z'$, and
$z_{j} \in \overline{Q_{2\rho}(z_{j-1})}$, $j=1, \ldots , N$. We allow $z_{j+1}=z_j$ for some $j$. Thus
\[
|(u)_{Q_\rho(z)} - (u)_{Q_\rho(z')}| \le \sum_{j=1}^{N} |(u)_{Q_\rho(z_{j})} - (u)_{Q_\rho(z_{j-1})}|,
\]
and hence
\EQ{\label{0505-5}
\max_{z,z'\in S_{x_0,r}} |(u)_{Q_\rho(z)} - (u)_{Q_\rho(z')}| =o(1)\quad \text{as } |x_0| \to \infty
}
by \eqref{0505-3} as $\rho \in (0,\frac12)$.
We have
\EQN{
\fint_{Q_r(x_0)}& |u - (u)_{Q_r(x_0)}| dx
\\
&= \sum_{z \in S} N^{-3} \fint_{Q_\rho(z)} |u - (u)_{Q_r(x_0)}| dx
\\
&\le \sum_{z \in S} N^{-3}\bke{ \fint_{Q_\rho(z)} |u - (u)_{Q_\rho(z)}| + |(u)_{Q_r(x_0)} - (u)_{Q_\rho(z)}| dx }
\\
&\le \bke{\sum_{z \in S} N^{-3} \fint_{Q_\rho(z)} |u - (u)_{Q_\rho(z)}| dx} + \max_{z,z'\in S} |(u)_{Q_\rho(z)} - (u)_{Q_\rho(z')}|
\\
&=o(1)\qquad \text{as } |x_0| \to \infty
}
by \eqref{0505-2} and \eqref{0505-5} for $\rho \in (0,\frac12)$. This shows
\eqref{0505-2} for all $r>1$.
Finally, \eqref{0505-3} for $r>1/2$ follows from \eqref{0505-2} and \eqref{0505-7}.
\end{proof}
The following lemma says that decaying oscillation over \emph{cubes} is equivalent with decaying oscillation over \emph{balls}.
\begin{lemma}
Suppose $u \in L^1_{\mathrm{uloc}}$. Then $u$ satisfies \eqref{0505-1} if and only if
\EQ{\label{0607-1}
\lim_{|x_0|\to \infty} \int_{B_1(x_0)} |u - (u)_{B_1(x_0)}|dx =0.
}
\end{lemma}
\begin{proof}
Let $\rho = 3^{-1/2}$. We have $Q_\rho(x_0) \subset B_1(x_0) \subset Q_1(x_0)$. Similar to the proof of \eqref{0505-4}, we have
\[
\int_{B_1(x_0)} |u - (u)_{B_1(x_0)}|dx \le C\int_{Q_1(x_0)} |u - (u)_{Q_1(x_0)}|dx
\]
and hence \eqref{0607-1} follows from \eqref{0505-1}. Similarly, we also have
\[
\int_{Q_\rho(x_0)} |u - (u)_{Q_\rho(x_0)}|dx \le C
\int_{B_1(x_0)} |u - (u)_{B_1(x_0)}|dx
\]
and hence \eqref{0505-2} for $r = \rho$ follows from \eqref{0607-1}.
Then $v(x)=u(\rho x)$ satisfies \eqref{0505-1}.
By Lemma \ref{from.ini}, $v$ satisfies \eqref{0505-2} for any $r>0$, and we get \eqref{0505-1} for $u$.
\end{proof}
\begin{lemma}\label{decay.na.U}
Suppose $v_0\in L^1_{\mathrm{uloc}}$ and
\[
\int_{Q(x_0,1)}|v_0-(v_0)_{Q(x_0,1)}|\to 0, \quad\text{ as } |x_0|\to \infty.
\]
Let $V=e^{t\De}v_0$.
Then $(\nabla V)(t_0)\in C_0(\mathbb{R}^3)$ for every $t_0>0$. Furthermore, for any $t_0>0$, we have
\EQ{\label{decay.naV}
\sup_{t>t_0}\norm{\nabla V(\cdot, t)}_{L^\infty(B(x_0,1))} \to 0, \quad\text{ as } |x_0|\to \infty.
}
\end{lemma}
\begin{proof}
For $k \in \mathbb{Z}^3$, let $\Si_k$ denote the set of its neighbor integer points,
\[
\Si_k = \mathbb{Z}^3 \cap Q(k,1.01)\setminus \{ k\}.
\]
Let
\[
a_k = (v_0)_{Q_1(k)}, \quad b_k = \max _{ k' \in \Si_k} |a_{k'}-a_k|,
\quad
c_k = \int_{ Q_1(k)} |v_0(x)-a_k| dx.
\]
By the assumption, $c_k\to 0$ as $|k|\to \infty$ and by Lemma \ref{from.ini}, $b_k\to 0$ as $|k|\to \infty$.
Choose a nonnegative $\phi \in C^\infty_c(\mathbb{R}^3)$ with $\supp \phi \subset Q_1(0)$ and
\[
\sum_{k \in \mathbb{Z}^3} \phi_k (x)=1 \quad \forall x \in \mathbb{R}^3, \quad \phi_k(x) = \phi(x-k).
\]
Define
\[
v_1(x) = \sum_{k \in \mathbb{Z}^3} a_k \phi_k(x).
\]
Since $|a_k| \lesssim \norm{v_0}_{L^1_{\mathrm{uloc}}}$, $v_1$ is in $L^\infty(\mathbb{R}^3)$. For $x\in Q_1(k)$, it can be written as
\[
v_1(x) = a_k + \sum_{k' \in \Si_k} (a_{k'} -a_k)\phi_{k'}(x).
\]
Thus
\EQ{\label{diff.v01}
\int_{ Q_1(k)} |v_0(x)-v_1(x)| dx &\le \int_{ Q_1(k)} |v_0(x)-a_k| dx + \sum_{k' \in \Si_k} \int_{ Q_1(k)} |a_k -a_{k'}|\phi_{k'}(x) dx
\\
& \le c_k +C b_k ,
}
and
\EQ{\label{nb.v1}
\sup_{x \in Q_1(k)} |\nablabla v_1(x)|
\le \sup_{x \in Q_1(k)} \sum_{k' \in \Si_k} |a_{k'} -a_k| \cdot |\nablabla \phi_{k'}(x)|
\le Cb_{k} .
}
Let $\psi_R(x) = \Phi\left(\frac{x}{R}\right)$.
We decompose
\EQN{
\nablabla V(x,t) &= \int \nablabla H_t(x-y) v_0(y) (1-\psi_R(x-y))dy
\\
&\quad+ \int \nablabla H_t(x-y) [v_0(y)-v_1(y)] \psi_R(x-y) dy
\\
&\quad+ \int \nablabla H_t(x-y) v_1(y) \psi_R(x-y) dy = I_1+ I_2 +I_3.
}
By integration by parts, we can rewrite $I_3$,
\[
I_3 = \int H_t(x-y) \nablabla v_1(y) \psi_R(x-y) dy
-\int H_t(x-y) v_1(y) (\nablabla\psi_R)(x-y) dy = I_{31} + I_{32}.
\]
Fix $\ep>0$ and consider $t>t_0>0$.
Since for any $t>0$ and $x\in \mathbb{R}^3$, we have
\EQN{
|I_1| & \lesssim \int_{B(x,R)^c} \frac {|x-y|^5}{t^{\frac 52}}e^{-\frac {|x-y|^2}{4t}} \frac 1{|x-y|^4}|v_0(y)| dy \\
& \lesssim \int_{B(x,R)^c} \frac 1{|x-y|^4}|v_0(y)| dy \lesssim \frac 1R\norm{v_0}_{L^1_{\mathrm{uloc}}},
}
and
\EQN{
|I_{32}| \lesssim \norm{H_t}_1\norm{v_1}_\infty \norm{\nabla \psi_R}_\infty
\lesssim \frac 1R\norm{v_0}_{L^1_{\mathrm{uloc}}},
}
we can choose sufficiently large $R>0$ such that
\[
|I_1, I_{32}|<\ep.
\]
The integrands of both $I_2$ and $I_{31}$ are supported in $|y-x|\leq 2R$. If $|x|>2\rho$ with $\rho >2R$ and
$|y-x|\leq 2R$, then
$|y|\geq |x|-|x-y| > \rho$. Let
\[
1_{>\rho}(y) = 1\quad \text{for} \quad |y|>\rho,\quad \text{and}\quad 1_{>\rho}(y) = 0\quad \text{for} \quad |y|\leq \rho.
\]
We have
\EQN{
|I_2| \leq \norm{ |\nabla H_t|\ast |v_0-v_1|1_{>\rho}}_{L^\infty(\mathbb{R}^3)}
\lesssim {t_0^{-\frac 12}}\left(1+{t_0^{-\frac 32}}\right)\norm{|v_0-v_1|1_{>\rho}}_{L^1_{\mathrm{uloc}}}
}
by Lemma \ref{lemma23}, and
\[
|I_{31}| \leq \norm{e^{t\De}(|\nabla v_1|1_{>\rho})}_{L^\infty(\mathbb{R}^3)}
\lesssim \norm{|\nabla v_1|1_{>\rho}}_{L^\infty(\mathbb{R}^3)} .
\]
If we take $\rho$ sufficiently large,
by \eqref{diff.v01} and \eqref{nb.v1}, we have $|I_2|+ |I_{31}| \leq 2 \ep$.
Since for any $t>t_0$ and $\ep>0$, we can choose $\rho>0$ such that
\[
\sup_{t>t_0}\norm{\nabla V(\cdot,t)}_{L^\infty(B(0, 2\rho)^c)}<4\ep,
\]
we get \eqref{decay.naV}.
\end{proof}
\section{Local existence}\label{loc.ex.sec}
In this section, we recall the definition of local energy solutions and prove their \emph{time-local} existence using a revised approximation scheme. Note that we do not assume spatial decay of initial data for the time-local existence.
As mentioned in the introduction, we follow the definition in Kikuchi-Seregin \cite{KS}.
\begin{defn}[local energy solution]\label{les}
Let $v_0 \in L^2_{\mathrm{uloc}}$ with $\mathop{\rm div} v_0=0$.
A pair $(v,p)$ of functions is a local energy solution to the Navier-Stokes equations \eqref{NS} with initial data $v_0$
in $\mathbb{R}^3\times (0,T)$, $0 <T< \infty$, if it satisfies the followings.
\begin{enumerate}[(i)]
\item $v\in \mathcal{E}_{T}$, defined in \eqref{cE.def}, and $p \in L^\frac 32_{\mathrm{loc}}([0,T)\times \mathbb{R}^3)$.
\item $(v,p)$ solves the Navier-Stokes equations \eqref{NS} in the distributional sense.
\item For any compactly supported function $\ph\in L^2(\mathbb{R}^3)$, the function $\int_{\mathbb{R}^3} v(x,t)\cdot \ph(x)\, dx$ of time is continuous on $[0,T]$. Furthermore, for any compact set $K\subset \mathbb{R}^3$,
\[
\norm{v(\cdot,t)-v_0}_{L^2(K)} \rightarrow 0, \quad \text{as }t\to 0^+.
\]
\item $(v,p)$ satisfies the local energy inequality (LEI) for any $t\in(0,T)$:
\EQ{\label{LEI}
\int_{\mathbb{R}^3} & |v|^2\xi (x,t)dx + 2\int_0^t\! \int_{\mathbb{R}^3}|\nabla v|^2\xi \,dxds\\
&\quad \leq \int_0^t\! \int_{\mathbb{R}^3} |v|^2(\partial_s\xi + \De \xi) + (|v|^2+2p)(v\cdot \nabla)\xi \,dxds,
}
for all non-negative smooth functions $\xi\in C^\infty_c((0,T)\times\mathbb{R}^3)$.
\item For each $x_0\in \mathbb{R}^3$, we can find $c_{x_0}\in L^\frac 32(0,T)$ such that
\EQ{\label{pressure.decomp}
p(x,t)=\widehat{p}_{x_0}(x,t)+c_{x_0}(t),
\qquad\text{in } L^{\frac 32}(B(x_0, \tfrac32)\times(0,T)),
}
where
\EQ{\label{hp.def}
\widehat{p}_{x_0}(x,t) =& -\frac 13 |v(x,t)|^2 + \pv \int_{B(x_0,2)} K_{ij}(x-y)v_iv_j(y,t)dy\\
&+\int_{B(x_0,2)^c} (K_{ij}(x-y)-K_{ij}(x_0-y))v_iv_j(y,t)dy
}
for $K(x)=\frac 1{4\pi|x|}$ and $K_{ij} = \partial_{ij}K$.
\end{enumerate}
We say the pair $(v,p)$ is a local energy solution to \eqref{NS} in $\mathbb{R}^3\times (0,\infty)$ if it is a local energy solution to \eqref{NS} in $\mathbb{R}^3\times (0,T)$ for all $0 <T< \infty$.\qed
\end{defn}
For an initial data $v_0\in L^2_{\mathrm{uloc}}$ whose local kinetic energy is uniformly bounded, we reprove the local existence of a local energy solution of \cite[Chapt 32]{LR}.
\begin{theorem}[Local existence]\label{loc.ex}
Let $v_0\in L^2_{\mathrm{uloc}}$ with $\mathop{\rm div} v_0 =0$. If
\[
T\le \frac{\e_1}{1+\norm{v_0}_{L^2_{\mathrm{uloc}}}^4}
\]
for some small constant $\e_1>0$,
we can find a local energy solution $(v,p)$ on $\mathbb{R}^3\times (0,T)$ to Navier-Stokes equations \eqref{NS} for the initial data $v_0$, satisfying $\norm{v}_{\mathcal{E}_T} \le C \norm{v_0}_{L^2_{\mathrm{uloc}}}$.
\end{theorem}
Note that we do not assume $v_0 \in E^2$, i.e., we do not assume spatial decay of $v_0$. Although the local existence theorem is proved in \cite[Chapt 32]{LR}, a few details are missing there, in particular those related to the pressure. These details are given in \cite{KS} for the case $v_0 \in E^2$. Here we treat the general case $v_0 \in L^2_{\mathrm{uloc}}$.
Recall the definitions of $\mathcal{J}_\ep(\cdot)$ and $\Phi$ in Section \ref{pre} and let $\Phi_\ep(x)= \Phi (\ep x )$, $\ep>0$. To prove Theorem \ref{loc.ex}, we consider approximate solutions $(v^\ep, p^\ep)$ to the localized-mollified Navier-Stokes equations
\EQ{\label{reg.NS}
\begin{cases}
\partial_t v^\ep - \De v^\ep + (\mathcal{J}_\ep(v^\ep)\cdot\nabla) (v^\ep\Phi_\ep) +\nabla p^\ep =0\\
\mathop{\rm div} v^\ep =0 \\
v^\ep|_{t=0} = v_0
\end{cases}
}
in $\mathbb{R}^3 \times (0,T)$.
Since $v_0\in L^2_{\mathrm{uloc}}$ has no decay, it cannot be approximated by $L^2$-functions, as was done in \cite{KS} when $v_0 \in E^2$. Hence
the approximation solution $v^\ep$ cannot be constructed in the energy class $L^\infty(0,T; L^2(\mathbb{R}^3)) \cap L^2(0,T; \dot H^1(\mathbb{R}^3))$, and has to be constructed in $\mathcal{E}_T$ directly.
Compared to \cite{LR,KS}, our mollified nonlinearity has an additional localization factor $\Phi_\ep$. It makes the decay of the Duhamel term apparent when the approximation solutions have no decay.
We first construct a mild solution $v^\ep$ of \eqref{reg.NS} in $\mathcal{E}_T$.
\begin{lemma}\label{mild.sol}
For each $0<\ep<1$ and $v_0$ with $\norm{v_0}_{L^2_{\mathrm{uloc}}}\leq B$, if $0<T< \min(1,c\ep^3B^{-2})$, we can find a unique solution $v=v^\ep$ to the integral form of \eqref{reg.NS}
\EQ{\label{op.rNS}
v(t) = e^{t\De}v_0 - \int_0^t e^{(t-s)\De} \mathbb{P} \nabla \cdot (\mathcal{J}_\ep(v) \otimes v \Phi_\ep)(s) ds
}
satisfying
\[
\norm{v}_{\mathcal{E}_T} \leq 2C_0B,
\]
where $c>0$ and $C_0>1$ are absolute constants and $(a\otimes b)_{jk} = a_jb_k$.
\end{lemma}
\begin{proof}
Let $\Psi (v)$ be the map defined by the right side of \eqref{op.rNS} for $v \in \mathcal{E}_T$. By Lemma \ref{lemma24} and $T\le 1$,
\EQN{
\norm{\Psi (v)}_{\mathcal{E}_T}
& \lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}+ \norm{\mathcal{J}_\ep(v)\otimes v\Phi_\ep}_{U^{2,2}_T}
\\
& \lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}+ \norm{\mathcal{J}_\ep(v)}_{L^\infty(0,T;L^\infty(\mathbb{R}^3))}\norm{v}_{U^{2,2}_T}
\\
& \lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}+ \ep^{-\frac 32} \sqrt{T}\norm{v}_{U^{\infty,2}_T}^2.
}
Thus
\[
\norm{\Psi (v)}_{\mathcal{E}_T} \leq C_0\norm{v_0}_{L^2_{\mathrm{uloc}}} + C_1\ep^{-\frac 32}\sqrt{T} \norm{v}_{\mathcal{E}_T}^2,
\]
for some constants $C_0,C_1>0$.
Similarly, for $v,u \in \mathcal{E}_T$,
\[
\norm{\Psi (v)-\Psi (u)}_{\mathcal{E}_T} \le
C_1\ep^{-\frac 32}\sqrt{T} \bke{\norm{v}_{\mathcal{E}_T}+ \norm{u}_{\mathcal{E}_T}}
\norm{v-u}_{\mathcal{E}_T}.
\]
By the Picard contraction theorem, if $T$ satisfies
\[
T< \frac {\ep^3}{64(C_0C_1B)^2} = c\ep^3B^{-2},
\]
then we can always find a unique fixed point $v\in \mathcal{E}_T$ of $v = \Psi(v)$, i.e., \eqref{op.rNS},
satisfying
\[
\norm{v}_{\mathcal{E}_T} \leq 2C_0B.\qedhere
\]
\end{proof}
\begin{lemma}\label{sol.rNS} Let $v_0\in L^2_{\mathrm{uloc}}$ with $\mathop{\rm div} v_0 =0$.
For each $\ep\in (0,1)$,
we can find $v^\ep$ in $\mathcal{E}_T$ and $p^\ep$ in $L^\infty(0,T;L^2(\mathbb{R}^3))$ for some positive $T=T(\ep, \norm{v_0}_{L^2_{\mathrm{uloc}}})$ which solves the localized-mollified Navier-Stokes equations \eqref{reg.NS} in the sense of distributions, and $\lim _{t \to 0_+} \norm{v^\ep(t)-v_0}_{L^2(E)}=0$ for any compact subset $E$ of $\mathbb{R}^3$.
\end{lemma}
\begin{proof} By Lemma \ref{mild.sol}, there is a mild solution $v^\ep \in \mathcal{E}_T$ of \eqref{op.rNS} for some $T=T(\ep, \norm{v_0}_{L^2_{\mathrm{uloc}}})$. Apparently,
\EQN{
\norm{v^\ep-e^{t\De}v_0}_{U^{\infty,2}_t}
&= \norm{\int_0^t e^{(t-s)\De} \mathbb{P} \nabla \cdot (\mathcal{J}_\ep(v) \otimes v \Phi_\ep)(s) ds}_{U^{\infty,2}_t} \\
& \lesssim \norm{\mathcal{J}_\ep(v) \otimes v \Phi_\ep}_{U^{2,2}_t}
\lesssim \ep^{-\frac 32}\sqrt{t}\norm{v}_{U^{\infty,2}_T}^2.
}
Also, for any compact subset $E$ of $\mathbb{R}^3$, we have $\norm{e^{t\De}v_0-v_0}_{L^2(E)}\to 0$ as $t$ goes to $0$; by Lebesgue's convergence theorem
\[
\norm{e^{t\De}v_0-v_0}_{L^2(E)} \leq \frac 1{(4\pi)^\frac32}\int e^{-\frac{|z|^2}{4}} \norm{v_0(\cdot -\sqrt{t}z) - v_0}_{_{L^2(E)}} dz \to 0,
\]
as $t \to 0+$. Then, it follows that
$\lim _{t \to 0_+} \norm{v^\ep(t)-v_0}_{L^2(E)}=0$ for any compact subset $E$ of $\mathbb{R}^3$.
Note that $ e^{t\De}v_0$ with $v_0\in L^2_{\mathrm{uloc}}$ solves the heat equation in the distributional sense. Also, using $\mathop{\rm div} v_0 =0$, we can easily see that $\mathop{\rm div} e^{t\De}v_0 =0$.
On the other hand, $\mathcal{J}_\ep(v^\ep)\in L^\infty(\mathbb{R}^3\times [0,T])$ and $v^\ep\in \mathcal{E}_{T}$ imply
\[
\mathcal{J}_\ep(v^\ep) \otimes v^\ep \Phi_\ep\in L^\infty(0,T;L^2(\mathbb{R}^3))
\]
and hence by the classical theory, $w^\ep = v^\ep-V$ and $p^\ep$ defined by
\EQ{\label{pep.def}
p^\ep = (-\De)^{-1}\partial_i\partial_j (\mathcal{J}_\ep(v_i^\ep) v_j^\ep \Phi_\ep) \in L^\infty(0,T;L^2(\mathbb{R}^3)).
}
solves Stokes system with the source term $\nabla \cdot (\mathcal{J}_\ep(v^\ep) \otimes v^\ep \Phi_\ep)$ in the distribution sense.
By adding the heat equation for $V$ with $\mathop{\rm div} V =0$ and the Stokes system for $(w^\ep,p^\ep)$, $v^\ep = V+w^\ep $ satisfies
\[
\partial_t v^\ep - \De v^\ep + (\mathcal{J}_\ep(v^\ep)\cdot \nabla) (v^\ep \Phi_\ep) + \nabla p^\ep =0
\]
in the sense of distribution.
\end{proof}
To extract a limit solution from the family $(v^\ep, p^\ep)$ of approximation solutions, we need a uniform bound of $(v^\ep, p^\ep)$ on a uniform time interval $[0,T]$, $T>0$.
\begin{lemma}\label{uni.est.ep}
For each $\ep \in (0,1)$, let $(v^\ep, p^\ep)$ be the solution on $\mathbb{R}^3 \times [0,T_\ep]$, for some $T_\ep>0$, to the localized-mollified Navier-Stokes equations \eqref{reg.NS} constructed in Lemma \ref{sol.rNS}. There is a small constant $\e_1>0$, independent of $\e$ and $\norm{v_0}_{L^2_{\mathrm{uloc}}}^2$, such that, if $T_\ep \leq T_0= {\e_1}(1+\norm{v_0}_{L^2_{\mathrm{uloc}}}^4)^{-1}$, then $v^\ep$ is uniformly bounded
\EQ{\label{uni.est.ep.li}
\norm{v^\ep}_{\mathcal{E}_{T_\ep}} \leq C\norm{v_0}_{L^2_{\mathrm{uloc}}},
}
where the constant $C$ on the right hand side is independent of $\ep$ and $T_\ep$.
\end{lemma}
\begin{proof}
Let $\phi_{x_0} = \Phi(\cdot-x_0)$ be a smooth cut-off function supported around $x_0$. For the convenience, we drop the index $x_0$. Starting from $v^\ep \in \mathcal{E}_{T_\ep}$ and $p^\ep \in L^\infty_{T_\ep} L^2$, and using the interior regularity theory for perturbed Stokes system with smooth coefficients, we have
\[
\norm{v^\ep, \partial_t v^\ep, \nabla v^\ep, \De v^\ep }_{L^\infty((\de,T_\ep)\times \mathbb{R}^3)} <+\infty
\]
for any $\de\in (0,T_\ep)$. Using $2v^\ep \psi$ with $\psi \in C^\infty_c((0,T_\ep)\times \mathbb{R}^3)$ as a test function in \eqref{reg.NS}, we get
\EQN{
2\int_0^T\!\! \int |\nabla v^\ep|^2\psi dxds
=&
\int_0^T\!\!\int |v^\ep|^2(\partial_s \psi +\De \psi) dxds
+\int_0^T\!\!\int |v^\ep|^2\Phi_\ep ({\cal J}_\ep(v^\ep) \cdot \nabla)\psi dxds\\
&+ 2\int_0^T\!\! \int p^\ep v^\ep \cdot \nabla \psi dxds
-\int_0^T\!\!\int |v^\ep|^2\psi ({\cal J}_\ep(v^\ep)\cdot \nabla)\Phi_\ep dxds.
}
Using $\lim_{t\to 0_+}\norm{v^\ep(t)-v_0}_{L^2(B_n)} =0$ for any $n\in\mathbb{N}$ (Lemma \ref{sol.rNS}), we can show
\EQ{\label{LEI.vep00}
\int |v^\ep|^2 & \psi(x,t) dx+ 2\int_0^t\!\! \int |\nabla v^\ep|^2\psi dxds
= \int |v_0|^2 \psi(\cdot,0) dx
\\
&+\int_0^t\!\!\int |v^\ep|^2(\partial_s \psi +\De \psi) dxds
+\int_0^t\!\!\int |v^\ep|^2\Phi_\ep ({\cal J}_\ep(v^\ep) \cdot \nabla)\psi dxds\\
&+ 2\int_0^t\!\! \int p^\ep v^\ep \cdot \nabla \psi dxds
-\int_0^t\!\!\int |v^\ep|^2\psi ({\cal J}_\ep(v^\ep)\cdot \nabla)\Phi_\ep dxds
}
for any $\psi \in C^\infty_c([0,T_\ep)\times \mathbb{R}^3)$ and $0<t<T_\ep$.
We suppress the index $\ep$ in $v^\ep$ and $p^\ep$, and take $\psi(x,s)= \phi(x) \th(s)$ where $\th(s) \in C^\infty_c([0,T_\ep))$ and $\th(s)=1$ on $[0,t]$ to get
\EQ{\label{pre.lei.locex}
\norm{v(t) \phi}_2^2 &+ 2\norm{|\nabla v|\phi}_{L^2([0,t]\times \mathbb{R}^3)}^2\\
\lesssim & \norm{v_0}_{L^2_{\mathrm{uloc}}}^2
+ \left|\int_0^t\!\int |v|^2|\De \phi^2|dxds \right|
+\left|\int_0^t\!\int |v|^2\phi^2 (\mathcal{J}_\ep(v)\cdot \nabla)\Phi_\ep dxds\right|\\
&+\left|\int_0^t\!\int |v|^2\Phi_\ep (\mathcal{J}_\ep(v)\cdot \nabla)\phi^2 dxds\right|
+\left|\int_0^t\!\int 2\widehat{p} (v\cdot \nabla)\phi^2 dxds\right| \\
=& \norm{v_0}_{L^2_{\mathrm{uloc}}}^2 + I_1 +I_2+ I_3 +I_4,
}
where $\widehat{p} = \widehat{p}^\ep_{x_0}$ will be defined later in \eqref{phat.def} as a function satisfying $\nabla(p - \widehat{p})=0 $ on $B(x_0, \frac 32)\times (0,T)$.
The bounds of $I_1$, $I_2$ and $I_3$ can be easily obtained by H\"{o}lder inequalities,
\EQ{\label{I.123}
I_1 \lesssim \norm{v}_{U^{2,2}_t}^2, \quad \text{and} \quad
I_2, I_3 \lesssim \norm{v}_{U^{3,3}_t}^3.
}
Here we have used $|\nabla \Phi_\ep| \lesssim \ep \leq 1$.
On the other hand, $I_4$ can be estimated as
\[
I_4 \lesssim \norm{\widehat{p}}_{L^\frac32([0,t]\times B(x_0,\frac 32))}\norm{v}_{U^{3,3}_t}.
\]
Now, we define $\widehat{p}^\ep$ on $B(x_0,\frac 32)\times [0,T]$ by
\EQ{\label{phat.def}
\widehat{p}^\ep(x,t)
=&-\frac 13 \mathcal{J}_\ep(v^\ep)\cdot v^\ep\Phi_\ep(x,t)
+ \pv \int_{B(x_0,2)} K_{ij}(x-y) \mathcal{J}_\ep(v^\ep_i) v^\ep_j(y,t) \Phi_\ep(y) dy\\
&+ \int_{B(x_0,2)^c} (K_{ij}(x-y)-K_{ij}(x_0-y)) \mathcal{J}_\ep(v^\ep_i) v^\ep_j(y,t) \Phi_\ep(y) dy \\
=& \ \widehat{p}^1 + \widehat{p}^2 + \widehat{p}^3.
}
Comparing the above with \eqref{pep.def} for $p^\ep$, which has the singular integral form
\EQN{
p^\ep(x,t) = -\frac 13 \mathcal{J}_\ep(v^\ep)\cdot v^\ep(x,t)\Phi_\ep(x) + \pv \int K_{ij}(x-y) \mathcal{J}_\ep(v_i^\ep) v_j^\ep(y,t) \Phi_\ep(y) dy,
}
we see that $p-\widehat{p}$ depends only on $t$, and hence $\nabla \widehat{p} =\nabla p$ on $B(x_0,\frac 32)\times [0,T]$.
Then, we take $L^\frac32([0,t]\times B(x_0,\frac 32))$-norm for each term to get
\[
\norm{\widehat{p}^1}_{L^\frac32([0,t]\times B(x_0,\frac 32))}
\lesssim \norm{v}_{U^{3,3}_t}^2,
\]
and
\[
\norm{\widehat{p}^2}_{L^\frac32([0,t]\times B(x_0,\frac 32))}
\leq \norm{\widehat{p}^2}_{L^{\frac 32}([0,t]\times \mathbb{R}^3)}
\lesssim \norm{\mathcal{J}_\ep(v_i) v_j \Phi_\ep}_{L^\frac32([0,t]\times B(x_0,2))}
\lesssim \norm{v}_{U^{3,3}_t}^2.
\]
The second inequality for $\widehat{p}^2$ follows from Calderon-Zygmund theorem. Finally, using
\[
|K_{ij}(x-y)-K_{ij}(x_0-y)| \lesssim \frac {|x-x_0|}{|x_0-y|^4}
\]
for $x\in B(x_0,\frac 32)$ and $y\in B(x_0,2)^c$, we have
\EQN{
\norm{\widehat{p}^3}_{L^\frac32([0,t]\times B(x_0,\frac 32))}
& \lesssim \norm{\int_{B(x_0,2)^c} \frac 1{|x_0-y|^4}\mathcal{J}_\ep(v_i) v_j(y,s) \Phi_\ep(y)dy}_{L^\frac32(0,t)}\\
& \lesssim \norm{\sum_{k=1}^\infty \frac {1}{2^{4k}} \int_{B(x_0,2^{k+1})} |\mathcal{J}_\ep(v_i) v_j| (y,s) dy}_{L^\frac32(0,t)}\\
&\le \sum_{k=1}^\infty \frac {1}{2^{4k}} \norm{\sum_{j=1}^{J_k} \int_{B(x^k_j,1)} |\mathcal{J}_\ep(v_i) v_j| (y,s) dy}_{L^\frac32(0,t)}\\
& \lesssim \sum_{k=1}^\infty \frac {J_k}{2^{4k}}\norm{\mathcal{J}_\ep(v_i) v_j}_{U^{\frac 32,\frac 32}_t}
\lesssim \norm{v}_{U^{3,3}_t}^2.
}
Above we have taken $B(x_0,2^{k+1}) \subset \cup_{j=1}^{J_k} B(x^k_j,1)$ with $J_k \lesssim 2^{3k}$.
Therefore, we get
\EQ{\label{est.hp}
\norm{\widehat{p}}_{L^\frac32([0,t]\times B(x_0,\frac 32))} \lesssim \norm{v}_{U^{3,3}_t}^2
}
and
\[
I_4 \lesssim \norm{v}_{U^{3,3}_t}^3.
\]
Combining this with \eqref{I.123} and taking supremum on \eqref{pre.lei.locex} over $\{x_0\in \mathbb{R}^3\}$, we have
\[
\norm{v(t)}_{L^2_{\mathrm{uloc}}}^2 + 2\norm{\nabla v}_{U^{2,2}_t}^2
\lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}^2 + \int_0^t \norm{v(s)}_{L^2_{\mathrm{uloc}}}^2 ds
+ \norm{v}_{U^{3.3}_t}^3.
\]
Then, using the interpolation inequality and Young's inequality,
\EQN{
\norm{v}_{U^{3,3}_t}^3
& \lesssim \norm{v}_{U^{6,2}_t}^{3/2} \norm{v}_{U^{2,6}_t}^{3/2}
\\
& \lesssim \norm{v}_{L^6(0,t; L^2_{\mathrm{uloc}} )}^6 + \norm{v}_{L^2(0,t; L^2_{\mathrm{uloc}} )}^2 +\norm{\nabla v}_{U^{2,2}_t}^2,
}
we get
\EQ{\label{pre.uni.bdd}
\norm{v(t)}_{L^2_{\mathrm{uloc}}}^2 + \norm{\nabla v}_{U^{2,2}_t}^2
\lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}^2 + \int_0^t \norm{v(s)}_{L^2_{\mathrm{uloc}}}^2 ds
+ \int_0^t \norm{v(s)}_{L^2_{\mathrm{uloc}}}^6 ds.
}
Finally, we apply the Gr\"{o}nwall inequality, so that there is a small $\e_1>0$ such that, if $v^\ep$ exists on $[0,T]$ for
$T\le T_0$, $T_0= {\e_1}\bke{1+\norm{v_0}_{L^2_{\mathrm{uloc}}}^4}^{-1}$,
then we have
\[
\sup_{0<t<T} \norm{v^\ep(t)}_{L^2_{\mathrm{uloc}}} \lesssim \norm{v_0}_{L^2_{\mathrm{uloc}}}\left(1- \frac{Ct \norm{v_0}_{L^2_{\mathrm{uloc}}}^4}{\min(1,\norm{v_0}_{L^2_{\mathrm{uloc}}})^4}\right)^{-\frac 14}
\le \norm{v_0}_{L^2_{\mathrm{uloc}}}(1- {C\e_1} )^{-\frac 14}.
\]
Together with \eqref{pre.uni.bdd}, this completes the proof.
\end{proof}
\begin{lemma} \label{vep.uniform.interval}
The distributional solutions $\{(v^\ep,p^\ep)\}_{0<\ep<1}$ of \eqref{reg.NS} constructed in Lemma \ref{sol.rNS} can be extended
to the uniform time interval $[0,T_0]$, where $T_0$ is as in Lemma \ref{uni.est.ep}.
\end{lemma}
\begin{proof}
We will prove it by iteration. For the convenience, we fix $0<\ep<1$ and drop the index $\ep$ in $v^\ep$ and $p^\ep$. Denote the uniform bound in Lemma \ref{uni.est.ep} by
\[
B=C(\norm{v_0}_{L^2_{\mathrm{uloc}}}), \quad B\geq \norm{v_0}_{L^2_{\mathrm{uloc}}}.
\]
If an initial data $v(t_0)$ satisfies $\norm{v(\cdot,t_0)}_{L^2_{\mathrm{uloc}}}\leq B$, by Lemma \ref{sol.rNS}, we get $S=S(\ep,B)>0$ and a unique solution $v(x,t+t_0)$ on $\mathbb{R}^3\times [0,S]$ to \eqref{op.rNS} satisfying
\[
\norm{v(t+t_0)}_{\mathcal{E}_{S}} \leq 2C_0 B.
\]
Now, we start the iteration scheme. Since $\norm{v_0}_{L^2_{\mathrm{uloc}}}\leq B$, a unique solution $v$ exists in $ \mathcal{E}_{S}$ to \eqref{op.rNS}. By Lemma \ref{sol.rNS} and Lemma \ref{uni.est.ep}, $v$ satisfies
\[
\norm{v}_{\mathcal{E}_{S}} \leq B.
\]
Then, we choose $\tau \in (\frac 34S,S)$, so that $\norm{v(\tau)}_{L^2_{\mathrm{uloc}}}\leq B$, and hence we obtain a solution $\tilde {v} \in \mathcal{E}(\tau,\tau+S)$ to
\[
\tilde {v}(t) = e^{(t-\tau)\De}v|_{t=\tau} + \int_{\tau}^{t} e^{(t-s)\De}\mathbb{P}\nabla \cdot N^\ep(\tilde v) (s) ds,
\]
where we denote $N^\ep(v) = \mathcal{J}_\ep(v) \otimes v \Phi_\ep$.
Denote the glued solution by $u(x,t) = v(x,t) 1_{[0,\tau]}(t) + \tilde{v}(x,t) 1_{(\tau,\tau+S]}(t)$, where $1_E$ is a characteristic function of a set $E\subset [0,\infty)$. We claim that it solves \eqref{op.rNS} in $(0,\tau+S)$; it is obvious for $t\in (0,\tau]$, and for $t\in (\tau,\tau+S]$,
\EQN{
u(t) =& \ \tilde v(t)
\\
=& e^{(t-\tau)\De}\left(e^{\tau\De}v_0 + \int_0^{\tau} e^{(\tau-s)\De}\mathbb{P}\nabla \cdot N^\ep(v)(s) ds\right)
+ \int_{\tau}^{t} e^{(t-s)\De}\mathbb{P}\nabla \cdot N^\ep(\tilde v)(s) ds\\
=& \ e^{t\De} v_0 + \int_0^{\tau} e^{(t-s)\De}\mathbb{P}\nabla \cdot N^\ep(v)(s) ds
+ \int_{\tau}^{t} e^{(t-s)\De}\mathbb{P}\nabla \cdot N^\ep(\tilde v)(s) ds\\
=& \ e^{t\De} v_0 + \int_0^t e^{(t-s)\De}\mathbb{P}\nabla \cdot N^\ep(u)(s) ds.
}
By Lemma \ref{uni.est.ep} again, it satisfies
\[
\norm{u}_{\mathcal{E}(0,\tau+S)} \leq B.
\]
By uniqueness, we get $u= v$ for $0\le t \le S$. In other words, $u$ is an extension of $v$.
Repeat this until the extended solution exists on $[0,T_0]$. Since at each iteration, we can extend the time interval by at least $\frac 34 S$, in finite numbers of iterations, we have a distributional solution $(v^\ep,p^\ep)$ of \eqref{reg.NS} on $\mathbb{R}^3\times [0,T_0]$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{loc.ex}]
For $0< \ep \ll 1$, let $(v^\ep,\bar{p}^\ep)$ be the distributional solution to the localized-mollified Navier-Stokes equations \eqref{reg.NS} on $\mathbb{R}^3 \times [0,T]$ constructed in Lemmas \ref{sol.rNS} and \ref{vep.uniform.interval}, where $T=T(\norm{v_0}_{L^2_{\mathrm{uloc}}})$ is independent of $\ep$. By Lemma \ref{uni.est.ep},
\[
\norm{v^\ep}_{\mathcal{E}_T}\le C(\norm{v_0}_{L^2_{\mathrm{uloc}}}).
\]
We then define $p^\ep \in L^{\frac 32}_{\mathrm{loc}}( [0,T]\times \mathbb{R}^3)$ by
\EQ{\label{p.ep}
p^\ep (x,t)
&=-\frac 13 \mathcal{J}_\ep(v^\ep)\cdot v^\ep(x,t)\Phi_\ep(x)
+ \pv \int_{B_2} K_{ij}(x-y) N^\ep_{ij}(y,t) dy\\
&\quad + \pv \int_{B_2^c} (K_{ij}(x-y)-K_{ij}(-y)) N^\ep_{ij}(y,t) dy,
\\
N^\ep_{ij}(y,t) & = \mathcal{J}_\ep(v_i^\ep) v_j^\ep(y,t) \Phi_\ep(y).
}
Because $N^\ep_{ij} \in L^\infty(0,T; L^2(\mathbb{R}^3))$, the right side of \eqref{p.ep} is defined in $L^\infty(0,T; L^2(\mathbb{R}^3)) + L^\infty(0,T)$.
Note that $\nabla(\bar{p}^\ep -p^\ep) =0$ because
\[
(\bar{p}^\ep -p^\ep)(t) = \int_{B_2^c} K_{ij}(-y) \mathcal{J}_\ep(v_i^\ep) v_j^\ep(y,t) \Phi_\ep(y) dy \in L^{\frac 32}(0,T).
\]
Therefore, $(v^\ep, p^\ep)$ is another distributional solution to the localized-mollified equations \eqref{reg.NS}. We will show that for each $n\in \mathbb{N}$, $p^\ep$ has a bound independent of $\ep$ in $L^\frac 32([0,T]\times B_{2^n})$. We drop the index $\ep$ in $v^\ep$ and $p^\ep$ for a moment.
For $n\in \mathbb{N}$,
we rewrite \eqref{p.ep} for $x\in B_{2^{n}}$ as follows.
\EQN{
p(x,t) =&-\frac 13 \mathcal{J}_\ep(v)\cdot v(x,t)\Phi_\ep(x)
+ \pv \int_{B_2} K_{ij}(x-y) N_{ij}^{\ep}(y,t) dy\\
&+ \left(\pv \int_{B_{2^{n+1}}\setminus B_2}+\pv \int_{B_{2^{n+1}}^c}\right) (K_{ij}(x-y)-K_{ij}(-y)) N_{ij}^{\ep}(y,t) dy\\
=& \ p_1 + p_2 + p_3 +p_4.
}
All $p_i$ are defined in $L^\infty(0,T; L^2)+L^\infty(0,T)$.
By Lemma \ref{uni.est.ep}, we have
\EQ{\label{est.Nij}
\norm{N_{ij}^\ep}_{U^{\frac 32,\frac32}_T} \lesssim \norm{\mathcal{J}_\ep(v)}_{U^{3,3}_T}\norm{v}_{U^{3,3}_T}
\leq C(\norm{v_0}_{L^2_{\mathrm{uloc}}}),
}
and
\EQ{\label{est.Nij2}
\norm{{N}_{ij}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2^n})}
\lesssim 2^{2n}\norm{\mathcal{J}_\ep(v)}_{U^{3,3}_T}\norm{v}_{U^{3,3}_T}
\leq C(n, \norm{v_0}_{L^2_{\mathrm{uloc}}}), \quad\forall n\in \mathbb{N}.
}
Then, the bound of $p_1$ can be obtained since
\[
\norm{p_1}_{L^{\frac 32}([0,T]\times B_{2^n})} \lesssim \sum_{i=1}^3\norm{{N}_{ii}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2^n})}.
\]
Using Calderon-Zygmund theorem, we get
\[
\norm{p_2}_{L^{\frac 32}([0,T]\times B_{2^n})}
\lesssim \norm{{N}_{ij}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2})},
\]
and
\[
\norm{p_{31}}_{L^{\frac 32}([0,T]\times B_{2^n})}
\lesssim \norm{{N}_{ij}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2^{n+1}})},
\]
where
\[
p_{31}(x,t) = \pv \int_{B_{2^{n+1}}\setminus B_2} K_{ij}(x-y) \mathcal{J}_\ep(v_i)v_j(y,t)\Phi_\ep(y)dy.
\]
On the other hand, $p_{32}= p_3-p_{31}$ satisfies
\EQN{
\norm{p_{32}}_{L^{\frac 32}([0,T]\times B_{2^n})}
& \lesssim 2^{2n} \norm{\frac 1{|y|^3}}_{L^3(B_{2^{n+1}}\setminus B_2)}
\norm{{N}_{ij}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2^{n+1}})}
\\
& \lesssim 2^{2n} \norm{{N}_{ij}^{\ep}}_{L^{\frac 32}([0,T]\times B_{2^{n+1}})}.
}
Since for $x\in B_{2^{n}}$ and $y\in B_{2^{n+1}}^c$, we have
\[
|K_{ij}(x-y)-K_{ij}(-y)| \lesssim \frac{|x|}{|y|^4} \lesssim \frac{2^n}{|y|^4},
\]
the bound of $p_4$ can be obtained as
\EQN{
\norm{p_4}_{L^\frac 32([0,T]\times B_{2^n})}
& \lesssim 2^{2n} \norm{p_4}_{L^\frac 32(0,T; L^\infty( B_{2^n}))}
\lesssim 2^{3n}\norm{\int_{B_{2^{n+1}}^c} \frac 1{|y|^4} |N_{ij}^\ep|(y,t) dy}_{L^\frac 32(0,T)}\\
& \lesssim 2^{3n} \sum_{k=n+1}^\infty \frac 1{2^{4k}}\norm{N_{ij}^\ep}_{L^\frac 32(0,T;L^1(B_{2^{k+1}}))}
\lesssim _n \norm{N_{ij}^\ep}_{U^{\frac 32,1}_T}.
}
Adding the estimates and using \eqref{est.Nij}-\eqref{est.Nij2}, we get for each $n\in \mathbb{N}$,
\EQ{\label{uni.bdd.p}
\norm{p^\ep}_{L^\frac 32([0,T]\times B_{2^n})} \leq C(n,\norm{v_0}_{L^2_{\mathrm{uloc}}}).
}
Now, we find a limit solution of $(v^\ep, p^\ep)$ up to subsequence on each $[0,T]\times B_{2^n}$, $n\in \mathbb{N}$.
First, construct the solution $v$ on the compact set $[0,T]\times B_2$. By uniform bounds on $v^\ep$ and the compactness argument, we can extract a sequence $v^{1,k}$ from $\{v^\ep\}$ satisfying
\EQN{
&v^{1,k} \stackrel{\ast}{\rightharpoonup} v^1 \qquad\qquad \text{in } L^\infty(0,T;L^2(B_2)),\\
&v^{1,k} \rightharpoonup v^1 \qquad\qquad\text{in }L^2(0,T;H^1(B_2)),\\
&v^{1,k} \rightarrow v^1 \qquad\qquad\text{in }L^3(0,T;L^3(B_2)),\\
&{\cal J}_{1,k}(v^{1,k}) \rightarrow v^1 \quad\text{ in }L^3(0,T;L^3(B_{2^-})),
}
as $k\to \infty$. Let $v=v^1$ on $[0,T]\times B_2$.
Then, we extend $v$ to $[0,T]\times B_4$ as follows. In a similar way to getting $v^1$, we can find a subsequence $\{(v^{2,k}, p^{2,k})\}_{k\in \mathbb{N}}$ of $\{(v^{1,k}, p^{1,k})\}_{k\in \mathbb{N}}$ which satisfies the following convergence:
\EQN{
&v^{2,k} \stackrel{\ast}{\rightharpoonup} v^2 \qquad\qquad \text{in } L^\infty(0,T;L^2(B_4)),\\
&v^{2,k} \rightharpoonup v^2 \qquad\qquad\text{in }L^2(0,T;H^1(B_4)),\\
&v^{2,k} \rightarrow v^2 \qquad\qquad\text{in }L^3(0,T;L^3(B_4)),\\
&{\cal J}_{2,k}(v^{2,k}) \rightarrow v^2\quad\text{ in }L^3(0,T;L^3(B_{4^-})),
}
as $k\to \infty$. Here, we can easily check that $v^2=v^1$ on $[0,T]\times B_2$, so that $v=v^2$ is the desired extension. By repeating this argument, we can construct a sequence $\{v^{n,k}\}$ and its limit $v$. Indeed, by the diagonal argument, $v$ can be approximated by
\[
v^{(k)}= \begin{cases}
v^{k,k} & [0,T]\times B_{2^k},\\
0& \text{otherwise}
\end{cases}, \quad \forall k\in \mathbb{N}
\]
More precisely, on each $[0,T]\times B_{2^n}$, $\{v^{(k)}\}_{k=n}^\infty$ enjoys the same convergence properties as above. This follows from that $\{v^{m,j}\}_{j\in \mathbb{N}}$, $m\geq n$ is a subsequence of $\{v^{n,j}\}_{j\in \mathbb{N}}$. Indeed, for each $v^{k,k}$, $k\geq n$, we can find $j_k\geq k$ such that
\[
v^{k,k}= v^{n,j_k}.
\]
Then, by its construction, for each $n\in \mathbb{N}$, $\{v^{(k)}\}_{k=n}^\infty$ satisfies
\begin{align}
&v^{(k)} \stackrel{\ast}{\rightharpoonup} v \qquad\qquad \text{in } L^\infty(0,T;L^2(B_{2^n})), \label{conv.weak.star} \\
&v^{(k)} \rightharpoonup v \qquad\qquad\text{in }L^2(0,T;H^1(B_{2^n})),\label{conv.weak}\\
&v^{(k)} \rightarrow v \qquad\qquad\text{in }L^3(0,T;L^3(B_{2^n})), \label{conv.strong}\\
&{\cal J}_{(k)}(v^{(k)}) \rightarrow v \quad\text{ in }L^3(0,T;L^3(B_{{2^n}^-})) \label{conv.moli}
\end{align}
as $k\to \infty$.
Furthermore, since $v^\ep$ are uniformly bounded in $\mathcal{E}_T$, we can easily see that $v\in \mathcal{E}_T$ and $v\in U^{3,3}_T$,
\[
\norm{v}_{\mathcal{E}_T} + \norm{v}_{U^{3,3}_T} \le C(\norm{v_0}_{L^2_{\mathrm{uloc}}}).
\]
Now, we construct a pressure $p$ corresponding to $v$. Using \eqref{p.ep}, we define $p^{(k)}$ by
\EQ{\label{pk.def}
p^{(k)}(x,t)
=&-\frac 13 {\cal J}_{(k)}(v^{(k)})\cdot v^{(k)}(x,t)\Phi_{(k)}(x) \\
&+ \pv \int_{B_2} K_{ij}(x-y) {\cal J}_{(k)}(v_i^{(k)}) v_j^{(k)}(y,t) \Phi_{(k)}(y) dy\\
&+ \pv \int_{B_2^c} (K_{ij}(x-y)-K_{ij}(-y)) {\cal J}_{(k)}(v^{(k)}_i) v^{(k)}_j(y,t) \Phi_{(k)}(y) dy.
}
where $\Phi_{(k)} = \Phi_{\ep_k}$ for $\ep_k$ satisfying $v^{k,k} = v^{\ep_k}$. Also define
\EQ{\label{p.def}
p(x,t) = \lim_{n \to \infty} \bar p^n(x,t)
}
where $\bar p^n(x,t)$ is defined for $|x|<2^{n}$ by
\EQ{\label{p34.dec}
\bar p^n(x,t)
=&-\frac 13 |v(x,t)|^2
+\pv \int_{B_2} K_{ij}(x-y) v_i v_j(y,t) dy + \bar p^n_3+\bar p^n_4,
}
with
\EQN{
\bar p^n_3(x,t) &=\pv \int_{B_{2^{n+1}}\setminus B_2} (K_{ij}(x-y)-K_{ij}(-y)) v_iv_j(y,t)\, dy ,
\\
\bar p^n_4(x,t) &=\int_{B_{2^{n+1}}^c} (K_{ij}(x-y)-K_{ij}(-y)) v_iv_j(y,t) \, dy .
}
The first two terms in $\bar p^n$ are defined in $U^{\frac32, \frac 32}_T$ and independent of $n$. Among the last two terms, $\bar p^n_4$ converges absolutely but $\bar p^n_3$ only in $U^{\frac32, \frac 32}_T$. By estimates similar to those for $p^\ep$, we get $\bar p^n_3, \bar p^n_4 \in L^{3/2}((0,T)\times B_{2^n})$ and
\[
\bar p^n_3+\bar p^n_4 = \bar p^{n+1}_3+\bar p^{n+1}_4 , \quad \text{in}\quad L^{3/2}((0,T)\times B_{2^n})
\]
Thus $\bar p^n(x,t)$ is independent of $n$ for $n> \log_2 |x|$.
Our goal is to show that the strong convergences \eqref{conv.strong}-\eqref{conv.moli} of $\{v^{(k)}\}$ gives
\EQ{\label{conv.p.st}
p^{(k)} \to {p} \quad \text{ in }L^{\frac 32}([0,T]\times B_{2^{n}}), \quad \text{ for each }n\in \mathbb{N},
}
Let $N_{ij}^{(k)} = {\cal J}_{(k)}(v^{(k)}_i) v^{(k)}_j\Phi_{(k)}$ and $N_{ij} = v_iv_j$. For any fixed $R>0$, we have
\EQ{\label{conv.in.pres}
&\norm{N_{ij}^{(k)}-N_{ij}}_{L^{\frac 32}([0,T]\times B_R)}
\\
&\leq \norm{\left({\cal J}_{(k)}(v_i^{(k)})-v_i\right) v_j^{(k)}\Phi_{(k)}}_{L^{\frac 32}([0,T]\times B_R)}
+ \norm{v_i (v_j^{(k)} - v_j)\Phi_{(k)}}_{L^{\frac 32}([0,T]\times B_R)}\\
&\quad +\norm{v_i v_j(1-\Phi_{(k)})}_{L^{\frac 32}([0,T]\times B_R)}\\
& \lesssim \norm{{\cal J}_{(k)}(v^{(k)})-v}_{L^3([0,T]\times B_R)}\norm{v^{(k)}}_{L^3([0,T]\times B_R)}\\
&\quad+\norm{v^{(k)}-v}_{L^3([0,T]\times B_R)}\norm{v}_{L^3([0,T]\times B_R)}
+\norm{|v|^2(1-\Phi_{(k)})}_{L^{\frac 32}([0,T]\times B_R)}
\longrightarrow 0
}
by \eqref{conv.strong}, \eqref{conv.moli}, and Lebesgue dominated convergence theorem. Then, it provides the convergence of $p^{(k)}$ to $p$: On $[0,T]\times B_{2^{n}}$, for $m >n$,
\EQN{
p^{(k)}-p
=&-\frac 13 \operatorname{tr}(N^{(k)}-N)
+ \pv \int_{B_2} K_{ij}(\cdot-y) (N_{ij}^{(k)}-N_{ij})(y) dy\\
&+ \left[\pv \int_{B_{2^{n+1}}\setminus B_2}+\int_{B_{2^{m}}\setminus B_{2^{n+1}}} +\int_{B_{2^{m}}^c}\right] (K_{ij}(\cdot-y)-K_{ij}(-y)) (N_{ij}^{(k)}-N_{ij})(y) dy\\
=& \ q_1 + q_2 + q_3 + q_4 +q_5.
}
In a similar way to getting \eqref{uni.bdd.p}, we have
\[
\norm{q_1,q_2, q_3}_{L^\frac 32([0,T]\times B_{2^{n}})} \lesssim _n \norm{N^{(k)}-N }_{L^\frac 32([0,T]\times B_{2^{n+1}})},
\]
and
\[
\norm{q_4}_{L^\frac 32([0,T]\times B_{2^{n}})} \lesssim \norm{N^{(k)}-N }_{L^\frac 32([0,T]\times B_{2^{m}})},
\]
On the other hand, using
\[
|K_{ij}(x-y)-K_{ij}(-y)| \lesssim \frac {|x|}{|y|^4}
\]
we obtain
\EQN{
\norm{q_5}_{L^\frac 32([0,T]\times B_{2^{n}})}
\lesssim \frac {2^{3n}}{2^m} \left(\norm{v}_{U^{3,3}_T}^2 +\norm{{\cal J}_{(k)} (v^{(k)}) }_{U^{3,3}_T}\norm{v^{(k)}}_{U^{3,3}_T}\right) \leq C(n,\norm{v_0}_{L^2_{\mathrm{uloc}}},T)\frac 1{2^m}.
}
Therefore, for fixed $n$, if we choose sufficiently large $m$, we can make $q_5$ very small in $L^\frac 32([0,T]\times B_{2^{n}})$ and then for sufficiently large $k$, $q_1$, $q_2$, $q_3$, and $q_4$ also become very small in $L^\frac 32([0,T]\times B_{2^{n}})$ because of \eqref{conv.in.pres}. This gives the desired convergence \eqref{conv.p.st} of $p^{(k)}$ to $p$.
Now, we check that $(v,p)$ is a local energy solution. It is easy to prove that $(v,p)$ solves the Navier-Stokes equation in distributional sense by using the distributional form
of \eqref{reg.NS} for $(v^{(k)},p^{(k)})$ and the convergence \eqref{conv.weak.star}-\eqref{conv.moli} and \eqref{conv.p.st}-\eqref{conv.in.pres}. For example,
for any $\xi \in C_c^\infty((0,T)\times \mathbb{R}^3;\mathbb{R}^3)$,
\[
\left.
\begin{split}
\int_0^T\int v^{(k)}\cdot \partialrtial_t \xi dxdt &\to \int_0^T\int v\cdot \partialrtial_t \xi dxdt
\\
\int_0^T\int {\cal J}_{(k)}(v^{(k)})(v^{(k)}\Phi_{(k)}) : \nablabla \xi dxdt
&\to \int_0^T\int v \otimes v: \nablabla \xi dxdt
\end{split}
\right\}
\quad \text{as}\quad k \to \infty.
\]
Since we have
\EQN{
\int_0^t\!\int &(\De v- (v \cdot \nabla )v - \nabla p )\cdot \phi dxdt\\
\leq& \left| \int_0^t\! \int \nabla v \cdot \nabla \phi dx dt\right|
+ \left|\int_0^t\! \int v(v\cdot \nabla ) \phi dxdt\right|
+ \left|\int_0^t\! \int p \mathop{\rm div} \phi dxdt\right|\\
\lesssim & \norm{\nabla v}_{L^2(0,T;L^2(B_{2^n}))} \norm{\nabla \phi}_{L^2(0,T;L^2(\mathbb{R}^3))}\\
&+ \bke{\norm{v}_{L^3(0,T;L^3(B_{2^n}))}^2+ \norm{p}_{L^\frac 32(0,T;L^{\frac 32}(B_{2^n}))}} \norm{\nabla \phi}_{L^3(0,T;L^3(\mathbb{R}^3))}\\
\leq &\ C(n,T,\norm{v_0}_{L^2_{\mathrm{uloc}}})\norm{\nabla \phi}_{L^3(0,T;L^3(\mathbb{R}^3))},
}
for any $\phi\in C_c^\infty([0,T]\times B_{2^n})$, $n\in \mathbb{N}$, it follows that
\EQN{
\partial_t v = \De v - (v \cdot \nabla )v - \nabla p \in X_n
}
for any $n\in \mathbb{N}$, where $X_n$ is the dual space of $L^3(0,T;{W}^{1,3}_0(B_{2^n}))$.
With this bound of $\partialrtial_t v$, for each $n \in \mathbb{N}$, we may redefine $v(t)$ on a measure-zero subset $\Si_n$ of $[0,T]$ such that the function
\EQ{\label{weak.conti}
t \longmapsto \int_{\mathbb{R}^3} v(x,t)\cdot \zeta(x)\, dx
}
is continuous for any vector $\zeta\in C_c^\infty(B_{2^n})$. Redefine $v(t)$ recursively for all $n$ so that \eqref{weak.conti} is true for any $\zeta\in C_c^\infty(\mathbb{R}^3)$. It is then true for any $\zeta\in L^2(\mathbb{R}^3)$ with a compact support using $v\in L^\infty(0,T;L^2_{\mathrm{uloc}})$.
Furthermore, consider the local energy equality \eqref{LEI.vep00} for $(v^{(k)},p^{(k)})$ on the time interval $(0,T)$ for a non-negative $\psi\in C_c^\infty([0,T)\times \mathbb{R}^3)$. The first term $\int |v^{(k)}|^2 \psi(x,T) dx$ vanishes.
Taking limit infimum as $k$ goes to infinity, and using the weak convergence \eqref{conv.weak} and
the strong convergence \eqref{conv.strong}-\eqref{conv.moli} and \eqref{conv.p.st}-\eqref{conv.in.pres},
we get
\EQ{\label{LEI.v.nt}
2\int_0^T\!\!\int & |\nabla v|^2\psi\, dxds
\leq \int|v_0|^2\psi(\cdot,0) dx \\
&+\int_0^T\!\!\int |v|^2(\partial_s\psi +\De\psi) +(|v|^2+2\widehat{p}) (v\cdot \nabla)\psi\, dxds,
}
for any non-negative $\psi\in C_c^\infty([0,T)\times \mathbb{R}^3)$.
Then, for any $t\in (0,T)$ and non-negative $\ph\in C_c^\infty([0,T)\times \mathbb{R}^3)$, take $\psi(x,s) = \ph(x,s) \th_\e(s)$, $\ep \ll 1$, where $\th_\e(s) =\th\left(\frac{s -t}{\ep}\right)$ for some $\th\in C^\infty(\mathbb{R})$ such that $\th(s)=1$ for $s\le 0$ and $\th(s) = 0$ for $s\ge 1$, and $\th'(s)\le 0$ for all $s$. Note that $\th_\e(s) = 1$ for $s \le t$ and $\th_\e(s) =0$ for $s \ge t+\ep$. Sending $\e \to 0$ and using
\[
\int |v(t)|^2 \ph\, dx \leq \liminf_{\ep\to 0} \int_0^t \int |v|^2\ph (-\th_\e') \,dx\,ds
\]
due to the weak local $L^2$-continuity \eqref{weak.conti},
we get
\EQ{\label{pre.lei}
\int & |v(t)|^2 \ph\, dx + 2\int_0^t\! \int |\nabla v|^2\ph\, dxds\\
&\leq \int |v_0|^2 \ph(\cdot,0) dx + \int_0^t\! \int \bket{ |v|^2(\partialrtial_s \ph+ \De \ph)
+(|v|^2+2\widehat{p}) (v\cdot \nabla)\ph} dxds
}
for any $t\in (0,T)$ and non-negative $\ph\in C_c^\infty([0,T)\times \mathbb{R}^3)$. The local energy inequality \eqref{LEI} is a special case of
\eqref{pre.lei} for test functions vanishing at $t=0$.
Sending $t\to 0_+$ in \eqref{pre.lei} we get $\limsup_{t \to 0_+} \int |v(t)|^2 \ph\, dx\le \int |v_0|^2 \ph(\cdot,0) dx$ for any non-negative $\ph\in C_c^\infty$. Together with the weak continuity \eqref{weak.conti}, we get $\lim_{t \to 0_+} \int_{B_n} |v(x,t)-v_0(x)|^2 dx =0$ for any $n \in \mathbb{N}$.
Finally, we consider the decomposition of the pressure. Recall that the pressure $p$ is defined recursively by \eqref{p.def}-\eqref{p34.dec}. For any $x_0\in \mathbb{R}^3$ define $\widehat{p}_{x_0}\in L^{\frac 32}([0,T]\times B(x_0, \frac 32))$ by \eqref{hp.def}, i.e.,
\EQN{
\widehat{p}_{x_0}(x,t) =& -\frac 13 |v|^2 (x,t)
+\pv \int_{B(x_0,2)} K_{ij}(x-y) v_i v_j(y,t) dy\\
&+ \int_{B(x_0,2)^c} (K_{ij}(x-y)-K_{ij}(x_0-y)) v_iv_j(y,t) dy.
}
Let $c_{x_0} = p-\widehat{p}_{x_0}$. If $ B(x_0,\frac 32)\subset B_{2^n}$, then
\EQ{\label{cx0.def}
c_{x_0}(t)
=& \int_{B_{2^{n+1}}\setminus B(x_0,2)} K_{ij}(x_0-y) v_iv_j(y,t) dy\\
&- \int_{B_{2^{n+1}}\setminus B_2} K_{ij}(-y) v_iv_j(y,t) dy\\
&+ \int_{B_{2^{n+1}}^c} (K_{ij}(x_0-y)-K_{ij}(-y)) v_iv_j(y,t) dy .
}
Note that $c_{x_0} \in L^{3/2}(0,T)$, and $c_{x_0}(t)$ is independent of $x\in B(x_0,\frac 32)$, $n$, and $T$. Therefore, we get the desired decomposition \eqref{pressure.decomp} of the pressure.
\end{proof}
\begin{remark}
Our approach in this section is similar to that in Kikuchi-Seregin \cite{KS}. However, there are two significant differences:
\begin{enumerate}
\item Since we include initial data $v_0$ not in $E^2$, we add an additional localization factor $\Phi_{(k)}$ to the nonlinearity in the localized-mollified equations \eqref{reg.NS}. Our approximation solutions $v^\ep$ live in $L^2_{\mathrm{uloc}}$ and are no longer in the usual energy class.
\item The pressure $p$ and $c_{x_0}$ are implicit in \cite{KS}, but are explicit in this paper. We first specify the formula \eqref{p.def} of the pressure and then justify the strong convergence and decomposition. In particular, our $c_{x_0}(t)$ is given by \eqref{cx0.def} and independent of $T$.
\end{enumerate}
\end{remark}
\begin{remark}
Estimate \eqref{est.hp} and its proof for $\widehat{p}^{\,\e}_{x_0}$ are not limited to our approximation solutions. They are in fact also valid for any local energy solution $(v,p)$ in $(0,T)$ with local pressure $\widehat{p}_{x_0}$ given by \eqref{hp.def}, that is,
\EQ{\label{est.hp2}
\norm{\widehat{p}_{x_0}}_{L^\frac32([0,t]\times B(x_0,\frac 32))} \le C \norm{v}_{U^{3,3}_t}^2, \quad \forall t<T,
}
with a constant $C$ independent of $t,T$.
\end{remark}
\section{Spatial decay estimates}\label{decay.est.sec}
Recall that our initial data $v_0\in E^2_{\si} + L^3_{{\mathrm{uloc}},\si}$. In Sections \ref{decay.est.sec} and \ref{global.sec}, we decompose
\EQ{\label{v0.dec}
v_0 = w_0+u_0, \quad w_0\in E^2_\si, \quad u_0\in L^3_{{\mathrm{uloc}},\si}.
}
Our goal in this section is to show that, although the solution $v$ has no spatial decay, its difference from the linear flow, $w= v-V$, $V(t)=e^{t\De}u_0$, does decay due to the decay of the oscillation of $u_0$. Here, the oscillation decay of $u_0$ follows from that of $v_0$ and $w_0\in E^2$. The main task is to show that the contribution from the nonlinear source term
\[
(V \cdot \nablabla) V = \nablabla \cdot (V \otimes V)
\]
has decay, although $V$ itself does not.
On the other hand, we also need the decay of the pressure. However, $\widehat{p}_{x_0}$ given by \eqref{hp.def}
does not decay. Thus
we need a different decomposition of the pressure $p$ near each point $x_0\in \mathbb{R}^3$.
\begin{lemma}[New pressure decomposition]\label{new.decomp}
Let $v_0 = w_0+u_0$ with $ w_0\in E^2_\si$ and $u_0\in L^3_{{\mathrm{uloc}},\si}$. Let $(v,p)$ be any local energy solution of \eqref{NS} with initial data $v_0$ in $\mathbb{R}^3 \times (0,T)$, $0<T<\infty$.
Then, for each $x_0\in \mathbb{R}^3$, we can find $q_{x_0}\in L^\frac 32(0,T)$ such that
\[
p(x,t) =\widecheck{p}_{x_0}(x,t) +q_{x_0}(t) \quad \text{in }L^\frac 32((0,T)\times B(x_0, \tfrac32))
\]
where
\EQ{\label{cp}
\widecheck{p}_{x_0}
=&-\frac 13 (|w|^2 + 2w\cdot V)
+ \pv \int_{B(x_0,2)} K_{ij}(\cdot-y) (w_iw_j + V_iw_j+ w_iV_j)(y) dy\\
&+ \int_{B(x_0,2)^c} (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) (w_iw_j + V_iw_j+ w_iV_j)(y) dy \\
&+ \int K_i(\cdot-y)[ (V\cdot \nabla)V_i]\rho_2 (y) dy \\
&+ \int (K_{ij}(\cdot -y)-K_{ij}(x_0-y)) V_iV_j(1-\rho_2)(y)dy \\
&+ \int (K_{i}(\cdot -y)-K_{i}(x_0-y)) V_iV_j(\partial_j\rho_2)(y)dy.
}
Here, $w=v-V$, $V(t)=e^{t\De}u_0$, $K_i = \partial_i K$, $K_{ij}=\partial_{ij}K$, $K(x)= \frac 1{4\pi|x|}$, and $\rho_2 = \Phi(\frac{\cdot-x_0}{2})$.
\end{lemma}
\begin{proof} Consider $(x,t)\in B(x_0, \frac 32)\times (0,T)$.
Let $F_{ij} = w_iw_j + V_iw_j+ w_iV_j$ and $G_{ij} = V_iV_j$.
Substituting $v=V+w$ in \eqref{hp.def}, we get
\begin{align}
\widehat{p}_{x_0} &= p_{x_0}^F + p_{x_0}^G \nonumber\\
\begin{split} \label{pFx0}
p_{x_0}^F &= -\frac 13 \tr F
+ \pv \int_{B(x_0,2)} K_{ij}(\cdot-y) F_{ij}(y) dy
\\
&\quad + \int_{B(x_0,2)^c} (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) F_{ij}(y) dy
\end{split}
\end{align}
and
\EQN{
p_{x_0}^G &= -\frac 13 \tr G
+ \pv \int_{B(x_0,2)} K_{ij}(\cdot-y) G_{ij}(y) dy
\\
&\quad + \int_{B(x_0,2)^c} (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) G_{ij}[\rho_2+ (1-\rho_2)](y) dy
\\
&= -\frac 13 \tr G
+ \pv \int K_{ij}(\cdot-y) G_{ij}\rho_2(y) dy + p_{x_0,\mathrm{far}}^G + \tilde q_{x_0}(t) ,
}
where
\EQN{
p_{x_0,\mathrm{far}}^G &= \int (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) G_{ij}(1-\rho_2)(y) dy,
\\
\tilde q_{x_0}(t) &= -\int_{B(x_0,2)^c} K_{ij}(x_0-y) G_{ij}\rho_2(y) dy.
}
Integrating by parts the principle value integral, we get
\EQN{
p_{x_0}^G &= \int
K_{i}(\cdot-y)\partialrtial_j[ G_{ij}\rho_2(y) ]dy
+ p_{x_0,\mathrm{far}}^G + \tilde q_{x_0}(t).
}
Note $\partialrtial_j[ G_{ij}\rho_2] = (V\cdot \nablabla V_i)\rho_2 + G_{ij} \partialrtial_j \rho_2$. Denote
\[
\widehat q_{x_0}(t)=\int K_{i}(x_0-y) V_iV_j(\partial_j\rho_2)(y)dy.
\]
We get
\EQ{\label{hp.cp}
\widehat{p}_{x_0}(x,t) &= p_{x_0}^F
+ \int K_i(\cdot-y) (V\cdot \nabla)V_i\rho_2 (y) dy + p^G_{x_0,\text{far}} + \tilde q_{x_0}(t)
\\
&\quad + \int (K_{i}(\cdot -y)-K_{i}(x_0-y)) V_iV_j(\partial_j\rho_2)(y)dy+ \widehat q_{x_0}(t)
\\
&=\widecheck{p}_{x_0}(x,t) + \tilde q_{x_0}(t) + \widehat q_{x_0}(t).
}
Thus we have $p(x,t) =\widecheck{p}_{x_0}(x,t) + q_{x_0}(t)$ with
\[
q_{x_0}(t) = c_{x_0}(t) + \tilde q_{x_0}(t) + \widehat q_{x_0}(t).
\]
Note that using $\norm{G}_{U^{\infty,1}_T}\le \norm{V}_{U^{\infty,2}_T}^2$ and $|x_0-y|>2$ for $y\in \supp(\partial_j\rho_2)$, we have
\EQ{\label{q.est}
\norm{\tilde q_{x_0}(t)}_{L^{\infty}(0,T)}
+\norm{\widehat q_{x_0}(t)}_{L^{\infty}(0,T)}
& \lesssim \norm{\int_{B(x_0,3)\setminus B(x_0,2)} |G_{ij}|(y) dy}_{L^{\infty}(0,T)}\\
& \lesssim \norm{G}_{L^{\infty}(0,T;L^1(B(x_0,3)))} \lesssim \norm{V}_{U^{\infty,2}_T}^2.
}
Since $\tilde q_{x_0}(t) + \widehat q_{x_0}(t)$ is in $L^{3/2}(0,T)$, so is $q_{x_0}(t)$.
\end{proof}
Although $\nablabla V$ has spatial decay, it is not uniform in $t$. Thus, to show the spatial decay of $w$, we will first show \eqref{eq1.5}, i.e., the smallness of $w$ in $L^2_{\mathrm{uloc}}$ at far distance for a short time in Lemma \ref{th0730b}. For that we need Lemmas \ref{adm}, \ref{th0803} and \ref{th:LEI.w.0}.
\begin{lemma}\label{adm}
For $u_0 \in L^3(\mathbb{R}^3)$, if $\frac 2s + \frac 3q = 1$ and $3\le q< 9$, then
\[
\norm{ e^{t \De}u_0}_{L^s(0,\infty;L^q(\mathbb{R}^3))} \leq C_q \norm{u_0}_{L^3(\mathbb{R}^3)}.
\]
\end{lemma}
This is proved in Giga \cite[196--197]{Gi86}. The case $q=9$ is also true according to
\cite[Acknowledgment]{Gi86}, but
there is no detailed proof.
\begin{lemma}\label{th0803}
Suppose $u_0 \in L^2_{\mathrm{uloc}}$ and $u_0 \in L^3(B(x_0,3))$. Then, $V =e^{t\De}u_0$ satisfies
\EQ{\label{th0803-1}
\norm{ V}_{L^8(0,T; L^4(B(x_0,\frac 32)))}
\lesssim \norm{ u_0}_{L^3(B(x_0,3))} + T^{\frac 18} \norm{u_0}_{L^2_{\mathrm{uloc}}} .
}
\end{lemma}
\begin{proof}
Let $\phi (x)=\Phi(\frac{x-x_0}2)$. Decompose
\[
u_0= u_0\phi + u_0(1-\phi) =: u_1 + u_2.
\]
By Lemma \ref{adm},
\EQ{\label{est.v1}
\norm{e^{t\De}u_1}_{L^8(0,T; L^4(B(x_0,\frac32)))}
\leq \norm{ e^{t\De} u_1}_{L^8(0,T;L^4(\mathbb{R}^3))}
\lesssim \norm{u_1}_{L^3(\mathbb{R}^3)}
\le \norm{u_0}_{L^3(B(x_0,3))}.
}
On the other hand, we have
\EQN{
\norm{e^{t\De}u_2}_{L^8(0,T; L^4(B(x_0,\frac32)))}
& \lesssim \norm{\nabla e^{t\De}u_2}_{L^8(0,T; L^2(B(x_0,\frac32)))}
+\norm{e^{t\De}u_2}_{L^8(0,T; L^2(B(x_0,\frac32)))}.
}
Obviously,
\[
\norm{e^{t\De}u_2}_{L^8(0,T; L^2(B(x_0,\frac32)))}
\lesssim T^\frac 18 \norm{e^{t\De}u_2}_{L^\infty(0,T; L^2_{\mathrm{uloc}})}
\lesssim T^\frac 18 \norm{u_2}_{L^2_{\mathrm{uloc}}}.
\]
Using $\supp(u_2)\subset B(x_0,2)^c$ and heat kernel estimate, we get
\EQN{
\norm{\nabla e^{t\De}u_2}_{L^8(0,T;L^2(B(x_0,\frac 32)))}
& \lesssim T^\frac 18 \norm{\nabla e^{t\De}u_2}_{L^\infty((0,T)\times B(x_0,\frac32))}
\\
& \lesssim T^{\frac 18}\int_{B(x_0,2)^c}\frac 1{|x_0-y|^4} |u_0(y)|dy\\
& \lesssim T^{\frac 18}\sum_{k=1}^\infty \int_{B(x_0,2^{k+1})\setminus B(x_0,2^{k})}\frac 1{{2^{4k}}} |u_0(y)|dy\\
& \lesssim T^{\frac 18} \norm{u_0}_{L^2_{\mathrm{uloc}}} .
}
Therefore, we obtain
\[
\norm{e^{t\De}u_2}_{L^8(0,T;L^4(B(x_0,\frac 32)))} \lesssim T^\frac 18\norm{u_0}_{L^2_{\mathrm{uloc}}}.
\]
Together with \eqref{est.v1}, we get
\eqref{th0803-1}.
\end{proof}
The perturbation $w=v-V$, $V(t)=e^{t\De}u_0$, satisfies the {\it perturbed Navier-Stokes equations} in the sense of distributions,
\begin{equation}\label{PNS}
\begin{cases}
\partial_t w -\De w + (V+w)\cdot \nabla (V+w) + \nabla p = 0 \\
\mathop{\rm div} w =0 \\
w|_{t=0}=w_0 .
\end{cases}
\end{equation}
It also satisfies the following local energy inequality for test functions supported away from $t=0$.
\begin{lemma}[Local energy inequality for $w$]\label{th:LEI.w.0}
Let $v_0 , u_0 \in L^2_{{\mathrm{uloc}},\si}$.
Let $(v,p)$ be any local energy solution of \eqref{NS} with initial data $v_0$ in $\mathbb{R}^3 \times (0,T)$, $0<T<\infty$.
Then $w(t)=v(t)-e^{t\De}u_0$ satisfies
\EQ{ \label{LEI.w}
\int & |w|^2 \ph(x,t)\, dx
+ 2\int_{0}^t\!\int |\nabla w|^2 \ph\, dxds \\
& \leq
\int_{0}^t\!\int |w|^2 (\partialrtial_s\ph+\De \ph +v\cdot \nabla\ph )\,dxds\\
&\quad + \int_{0}^t\!\int 2 p w \cdot \nabla \ph\, dxds
+ \int_{0}^t\!\int 2V\cdot (v \cdot \nabla )(w\ph)\, dxds,
}
for any non-negative $\ph \in C_c^\infty((0,T)\times \mathbb{R}^3)$
and any $t\in (0,T)$.
\end{lemma}
Note that $\ph$ vanishes near $t=0$. If $\ph$ does not vanish near $t=0$, the last integral in \eqref{LEI.w} may not be defined.
\begin{proof}
Recall that we have the local energy inequality \eqref{LEI} for $(v,p)$. The equivalent form for $(w,p)$ is
exactly
\eqref{LEI.w}.
Indeed, \eqref{LEI} and \eqref{LEI.w} are equivalent because they differ by an equality which is the sum of the weak form of $V$-equation
with $2v \ph$ as the test function and the weak form of the $w$-equation \eqref{PNS} with $2V \ph$ as the test function, after suitable integration by parts.
This equality can be proved because $V$ and $\nablabla V$ are in $L^\infty_{\mathrm{loc}}(0,T; L^\infty(\mathbb{R}^3))$, and $\ph$ has a compact support in space-time.
\end{proof}
For $r>0$, let
\[
\chi_r(x) = 1 - \Phi\bke{\frac xr},
\]
so that $\chi_r(x)=1$ for $|x|\ge 2r$ and $\chi_r(x)=0$ for $|x|\le r$.
\begin{lemma}\label{th0730b}
Let $v_0 = w_0+u_0$ with $ w_0\in E^2_\si$ and $u_0\in L^3_{{\mathrm{uloc}},\si}$.
Let $(v,p)$ be any local energy solution of \eqref{NS} with initial data $v_0$ in $\mathbb{R}^3 \times (0,T)$, $0<T<\infty$.
Then, there exist $T_0=T_0(\norm{v_0}_{L^2_{\mathrm{uloc}}}) \in (0,1)$ and
$C_0=C_0(\norm{w_0}_{L^2_{\mathrm{uloc}}}, \norm{u_0}_{L^3_{\mathrm{uloc}}})>0$ such that
$w(t)=v(t)-e^{t\De} u_0$ satisfies
\EQ{\label{th0730b-2}
\norm{w(t)\chi_R}_{L^2_{\mathrm{uloc}}} \leq C_0 (t^\frac 1{20} + \norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}),
}
for any $R>0$ and any $t \in (0,T_1)$, $T_1 =\min(T_0,T)$.
\end{lemma}
In this lemma, we do not assume the oscillation decay.
\begin{proof}
By Lemma \ref{lemma24} and similar to \eqref{uni.est.ep.li}, we can find $T_0 = T_0 (\norm{v_0}_{L^2_{\mathrm{uloc}}}^2)\in (0,1)$ such that, for $T_1=\min(T_0,T)$,
\EQN{
\norm{w}_{\mathcal{E}_{T_1}} +\norm{V}_{\mathcal{E}_{T_1}}
\lesssim \norm{w_0}_{L^2_{\mathrm{uloc}}}+\norm{u_0}_{L^2_{\mathrm{uloc}}}.
}
By interpolation, it follows that for any $2\leq s\leq \infty$, and $2\leq q\leq 6$ satisfying $\frac 2s+\frac 3q = \frac 32$, we have
\EQN{\label{est.103}
\norm{w}_{U^{s,q}_{T_1}} + \norm{V}_{U^{s,q}_{T_1}}
\lesssim \norm{w_0}_{L^2_{\mathrm{uloc}}}+\norm{u_0}_{L^2_{\mathrm{uloc}}}.
}
On the other hand, by Lemma \ref{th0803}, for any $t\in(0,1)$,
\EQN{\label{V-strong-est}
\norm{V}_{U^{8,4}_t} \lesssim \norm{u_0}_{L^3_{\mathrm{uloc}}}.
}
Let $A = \norm{w_0}_{L^2_{\mathrm{uloc}}}+\norm{u_0}_{L^3_{\mathrm{uloc}}}$. Then, both inequalities can be combined for $t\le T_1$ as
\EQ{\label{bdd.A}
\norm{w}_{\mathcal{E}_{t}} +\norm{V}_{\mathcal{E}_{t}}
+\norm{w}_{U^{\frac{10}3,\frac{10}3}_{t}} + \norm{V}_{U^{\frac{10}3,\frac{10}3}_{t}}+ \norm{V}_{U^{8,4}_t} \lesssim A.
}
Fix $x_0\in\mathbb{R}^3$ and $R>0$, and let
\EQ{\label{xi.def}
\phi_{x_0}= \Phi(\cdot - x_0), \quad \xi = \phi_{x_0}^2 \chi_R^2.
}
Fix $\Theta\in C^\infty(\mathbb{R})$, $\Theta' \ge 0$, $\Theta(t)=1$ for $t>2$, and $\Theta(t)=0$ for $t<1$.
Define $\th_{\ep}\in C_c^\infty(0,T)$ for sufficiently small $\ep>0$ by
\EQ{\label{th_ep.def}
\th_\ep(s) = \Theta\left(\frac{s}{\ep}\right) - \Theta\left(\frac{s-T+3\ep}{\ep}\right).
}
Thus $\th_{\ep}(s)=1$ in $(2\ep,T-2\ep)$ and $\th_{\ep}(s)=0$ outside of $(\ep,T-\ep)$.
We now consider the local energy inequality \eqref{LEI.w} for $w$ with $\ph(x,s) = \xi(x)\th_\ep(s)$. We may replace $p$ by $\widehat{p}_{x_0}$ in \eqref{LEI.w} as supp\,$\xi \subset B(x_0,\frac32)$ and $\iint c_{x_0}(t) w\cdot \nablabla \xi\, dxdt = 0$.
We now take $\ep \to 0_+$. Since $\norm{v(t)-v_0}_{L^2(B_2(x_0))}\to 0$ and $\norm{V(t)-u_0}_{L^2(B_2(x_0))}\to 0$ as $t\to 0^+$, we get
\EQ{\label{app.w0}
\int_0^{2\ep}\int |w|^2 \xi (\th_\e)' \,dxds \to \int |w_0|^2 \xi dx.
}
The last term in \eqref{LEI.w} converges by Lebesgue dominated convergence theorem using
\EQN{
\int_{0}^t\!\int |V\cdot (v \cdot \nabla )(w\xi)|\, dxds
& \lesssim
\norm{V}_{L^8(0,T;L^4(B(x_0,\frac32)))}\norm{v}_{U^{8/3,4}_T}(\norm{\nablabla w}_{U^{2,2}_T}+\norm{w}_{U^{2,2}_T}),
}
where the right hand side of the inequality is bounded independently of $\ep$.
In the limit $\ep \to 0_+$, for any $t\in (0,T)$, we get
\EQ{ \label{LEI.w.0}
\int & |w|^2(x,t) \xi(x)\, dx
+ 2\int_{0}^t\!\int |\nabla w|^2 \xi\, dxds \\
& \leq \int |w_0|^2 \xi\, dx +\int_{0}^t\!\int |w|^2 (\De \xi +v\cdot \nabla\xi )\,dxds\\
&\quad + \int_{0}^t\!\int 2 \widehat{p}_{x_0} w \cdot \nabla \xi\, dxds
+ \int_{0}^t\!\int 2V\cdot (v \cdot \nabla )(w\xi)\, dxds,
}
for $\xi$ given by \eqref{xi.def}. Now, we consider $t\leq T_1$.
Using \eqref{bdd.A}, we have
\[
\int_{0}^t\!\int |w|^2 \De \xi dxds
\lesssim \norm{w}_{U^{2,2}_t}^2 \lesssim A^2 t,
\]
\EQN{
\int_{0}^t\! \int |w|^2(v\cdot \nabla)\xi dxds
& \lesssim \norm{v}_{U^{3,3}_t}\norm{w}_{U^{3,3}_t}^2 \lesssim A^3 t^{\frac 1{10}}.
}
For the convenience, we suppress the indexes $x_0$ and $R$ in $\phi_{x_0}$, $\widehat{p}_{x_0}$ and $\chi_R$.
By additionally using \eqref{est.hp2},
\EQN{
\int_{0}^t\! \int \widehat{p} w \cdot \nabla \xi dxds
& \lesssim \int_{0}^t\! \int_{B(x_0,\frac 32)} |\widehat{p}||w| dxds
\lesssim \norm{\widehat{p} }_{L^{\frac 32}([0,t]\times B(x_0,\frac 32))}
\norm{w}_{U^{3,3}_t}\\
& \lesssim \norm{v}_{U^{3,3}_t}^2 \norm{w}_{U^{3,3}_t} \lesssim A^3 t^{\frac 1{10}}.
}
To estimate the last term in \eqref{LEI.w.0},
we decompose it as
\EQN{
&\int_{0}^t\!\int V\cdot (v \cdot \nabla )(w\xi)\, dxds
= I_1+I_2+ I_3
\\
&= \int_{0}^t\! \int \xi V\cdot (V \cdot \nabla )w\, dxds
+\int_{0}^t\! \int \xi V\cdot (w \cdot \nabla )w\, dxds
+\int_{0}^t\! \int V\cdot w (v \cdot \nabla )\xi\, dxds.
}
We have
\EQN{
|I_1| \lesssim
\norm{V}_{L^4(0,T;L^4(\supp(\xi)))} ^2
\norm{\nabla w}_{U^{2,2}_t}
\lesssim A^3 t^\frac14 .
}
On the other hand, by Poincar\'{e} inequality, we have
\EQN{
\int_0^t \norm{w\phi\chi}_{L^6}^2 ds
& \lesssim \int_0^t \norm{\nabla (w\phi\chi)}_{L^2}^2 ds
+ \int_0^t \norm{w\phi\chi}_{L^2}^2 ds\\
& \lesssim \int_0^t \norm{|\nabla w|\phi\chi}_{L^2}^2 ds
+ \norm{w}_{U^{2,2}_t}^2,
}
which follows that (using Young's inequality $abc \le \e a^2 + \e b^{8/3} + C(\e) c^8$)
\EQN{
|I_2|
\leq& \int_0^t \norm{|\nabla w|\phi\chi}_{L^2} \norm{w\phi\chi}_{L^4} \norm{V}_{L^4(\supp(\xi))} ds\\
\le& \int_0^t \norm{|\nabla w|\phi\chi}_{L^2}\norm{w\phi\chi}_{L^6}^\frac34\norm{w\phi\chi}_{L^2}^\frac14 \norm{V}_{L^4(\supp(\xi))} ds\\
\leq& \
\e \int_0^t \bke{\norm{|\nabla w|\phi\chi}_{L^2}^2 + \norm{w\phi\chi}_{L^6}^2 }ds\\
&+ C(\e) \int_0^t \norm{ V}_{L^4(\supp(\xi))}^8\norm{w\phi\chi}_{L^2}^2ds\\
\le& \ \frac 1{100}\int_0^t \norm{|\nabla w|\phi\chi}_{L^2}^2 ds + A^2 t
+ C\int_0^t \norm{V}_{L^4(\supp(\xi))}^8\norm{w\phi\chi}_{L^2}^2ds
}
by choosing suitable $\e$.
It is easy to control $I_3$:
\[
|I_3| \lesssim t^{\frac 1{10}}\norm{V}_{U^{\frac {10}3,\frac{10}3}_t}\norm{v}_{U^{\frac {10}3,\frac{10}3}_t}\norm{w}_{U^{\frac {10}3,\frac{10}3}_t} \lesssim {A^3} t^{\frac 1{10}}.
\]
Therefore, we obtain
\EQN{
\abs{\int_{0}^t\!\int V\cdot (v \cdot \nabla )(w\xi)\, dxds}
\leq& \norm{|\nabla w| \phi\chi }_{L^2([0,t]\times \mathbb{R}^3)}^2\\ &+C(1+A^3)\bke{t^\frac 1{10}
+ \int_0^t \norm{V}_{L^4(\supp(\xi))}^8\norm{w\phi\chi}_{L^2}^2ds},
}
for some absolute constant $C$. Finally, we combine all the estimates to get from \eqref{LEI.w.0} that
\EQN{
&\norm{w(t)\phi \chi}_{L^2(\mathbb{R}^3)}^2
+\norm{|\nabla w| \phi\chi }_{L^2([0,t]\times \mathbb{R}^3)}^2
\\
&\qquad \lesssim \norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}^2+
(1+A^3)\bke{t^\frac 1{10}
+ \int_0^t \norm{V}_{L^4(\supp(\xi))}^8\norm{w\phi\chi}_{L^2}^2ds}
}
Note that $\norm{w(t)\phi \chi}_{L^2(\mathbb{R}^3)}^2$ is lower semicontinuous in $t$ as $w\phi$ is weakly $L^2$-continuous in $t$.
By Gr\"{o}nwall's inequality and \eqref{bdd.A}, we have
\EQN{
\norm{w(t)\phi \chi}_{L^2(\mathbb{R}^3)}^2
\leq C_0^2(\norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}^2+ t^\frac1{10}),
}
for some $C_0=C_0(A)>0$. Taking supremum in $x_0$, we get
\[
\norm{w(t)\chi_R}_{L^2_{\mathrm{uloc}}} \leq C_0 (t^{\frac1{20}} +\norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}) .
\]
This finishes the proof of Lemma \ref{th0730b}.
\end{proof}
\begin{lemma}[Strong local energy inequality]\label{th:SLEI}
Let $(v,p)$ be a local energy solution in $\mathbb{R}^3\times (0,T)$ to Navier-Stokes equations \eqref{NS} for the initial data $v_0\in L^2_{\mathrm{uloc}}$ constructed in Theorem \ref{loc.ex}, as the limit of approximation solutions $(v^{(k)},p^{(k)})$
of \eqref{reg.NS}. Then there is a subset $\Si \subset (0,T)$ of zero Lebesgue measure such that, for any $t_0 \in (0,T) \setminus \Si$ and any $t \in (t_0,T)$, we have
\EQ{\label{SLEI-v}
\int & |v|^2 \ph(x,t)\, dx
+ 2\int_{t_0}^t\!\int |\nabla v|^2 \ph\, dxds \\
& \leq \int |v|^2 \ph (x,t_0)\, dx
+\int_{t_0}^t\!\int \bket{ |v|^2 (\partialrtial_s \ph +\De \ph) +(|v|^2+2 p)v\cdot \nabla\ph } dxds,
}
for any $\ph \in C^\infty_c(\mathbb{R}^3 \times [t_0,T))$. If, furthermore, for some $u_0\in L^2_{{\mathrm{uloc}},\si}$, $V(t) = e^{t\De}u_0$ and $w=v-V$, then for any $t_0 \in (0,T) \setminus \Si$ and any $t \in (t_0,T)$, we have
\EQ{\label{pre.decay}
\int & |w|^2 \ph(x,t)\, dx
+ 2\int_{t_0}^t\!\int |\nabla w|^2 \ph\, dxds \\
& \leq \int |w|^2 \ph (x,t_0)\, dx
+\int_{t_0}^t\!\int |w|^2 (\partialrtial_s\ph+\De \ph +v\cdot \nabla\ph )dxds\\
&\quad + \int_{t_0}^t\!\int 2 p w \cdot \nabla \ph\, dxds
- \int_{t_0}^t\!\int 2(v \cdot \nabla )V\cdot w\ph\, dxds,
}
for any $\ph \in C^\infty_c(\mathbb{R}^3 \times [t_0,T))$.
\end{lemma}
This lemma is not for general local energy solutions, but only for those constructed by the approximation \eqref{reg.NS}.
Also note that \eqref{SLEI-v} is true for $t_0=0$ since it becomes \eqref{LEI}, but
\eqref{pre.decay} is unclear for $t_0=0$ since the last integral in \eqref{pre.decay} may not be defined without further assumptions; Compare it with \eqref{LEI.w.0}.
\begin{proof} For any $n \in \mathbb{N}$,
the approximation $v^{(k)}$ satisfy
\[
\lim_{k \to \infty} \norm{v^{(k)} -v}_{L^2(0,T; L^2(B_n))} =0.
\]
Thus there is a set $\Si_n \subset (0,T)$ of zero Lebesgue measure such that
\[
\lim_{k \to \infty} \norm{v^{(k)}(t) -v(t)}_{L^2(B_n)} =0, \quad \forall t \in [0,T)\setminus \Si_n.
\]
Let
\[
\Si = \cup _{n=1}^\infty \Si_n, \quad |\Si|=0.
\]
We get
\EQ{\label{eq4.14}
\lim_{k \to \infty} \norm{v^{(k)}(t) -v(t)}_{L^2(B_n)} = 0, \quad \forall t \in [0,T)\setminus \Si, \ \forall n \in \mathbb{N}.
}
The local energy equality of $(v^{(k)},p^{(k)})$ in $[t_0,T]$ is derived similarly to \eqref{LEI.vep00}
\EQ{\label{SLEI-vk}
2\int_{t_0}^T\! \int |\nabla v^{(k)}|^2\psi dxds
=& \int |v^{(k)}|^2 \psi (x,t_0)\, dx +\int_{t_0}^T\!\int |v^{(k)}|^2(\partial_s \psi +\De \psi) dxds \\
&+\int_{t_0}^T\!\int |v^{(k)}|^2\Phi_{(k)} ({\cal J}_{(k)}(v^{(k)}) \cdot \nabla)\psi dxds\\
&+ \int_{t_0}^T\! \int 2 p^{(k)} v^{(k)} \cdot \nabla \psi \, dxds\\
&-\int_{t_0}^T\!\int |v^{(k)}|^2\psi ({\cal J}_{(k)}(v^{(k)})\cdot \nabla)\Phi_{(k)}dxds,
}
for any $\psi \in C^\infty_c(\mathbb{R}^3 \times [0,T))$. By \eqref{eq4.14},
we have
\[
\lim_{k \to \infty} \int |v^{(k)}|^2 \psi(x,t_0)\, dx = \int |v|^2 \psi (x,t_0)\, dx
\]
for $t_0 \in [0,T)\setminus \Si$. Taking limit infimum $k \to \infty$ in \eqref{SLEI-vk}, we get
\EQN{
&2\int_{t_0}^T\!\int |\nabla v|^2 \psi\, dxds
\\
& \leq \int |v|^2 \psi (x,t_0)\, dx
+\int_{t_0}^T\!\int \bket{ |v|^2 (\partialrtial_s \psi +\De \psi) +(|v|^2+2 p)v\cdot \nabla\psi } dxds.
}
By the same argument for \eqref{pre.lei}, we get \eqref{SLEI-v} from the above.
Finally, inequality \eqref{pre.decay} for $t_0>0$ is equivalent to \eqref{SLEI-v} by the same argument of Lemma \ref{th:LEI.w.0}. We have integrated by parts the last term in \eqref{pre.decay}, which is valid since $\nablabla V \in L^\infty(\mathbb{R}^3 \times (t_0,t))$.
\end{proof}
We now prove the main result of this section.
\begin{proposition}[Decay of $w$ and $\widecheck{p}$]\label{dot.E.3}
Let $v_0 = w_0+u_0$ with $ w_0\in E^2_\si$ and $u_0\in L^3_{{\mathrm{uloc}},\si}$,
and
\EQN{
\lim_{|x_0|\to \infty}\int_{B(x_0,1)}| v_0- (v_0)_{B(x_0,1)}| dx =0.
}
Let $(v,p)$ be a local energy solution in $\mathbb{R}^3\times (0,T)$ to Navier-Stokes equations \eqref{NS} for the initial data $v_0\in L^2_{\mathrm{uloc}}$ constructed in Theorem \ref{loc.ex}, as the limit of approximation solutions $(v^{(k)},p^{(k)})$ of \eqref{reg.NS}.
Let $w= v-V$ for $V(t) =e^{t\De}u_0$.
Then, $w$ and $\widecheck{p}_{x_0}$, defined in Lemma \ref{new.decomp}, decay at spatial infinity: For any $t_1\in (0,T)$,
\EQ{\label{dot.E.3-decay}
\lim_{|x_0|\to \infty}\bke{
\norm{w}_{L^\infty_t L^2_x \cap L^3(Q_{x_0})} + \norm{\nablabla w}_{L^2(Q_{x_0})}
+ \norm{\widecheck{p}_{x_0}}_{L^{\frac 32}(Q_{x_0})}
}=0,
}
where $Q_{x_0} = B(x_0,\frac 32)\times(t_1,T) $.
\end{proposition}
Note that we do not assert uniform decay up to $t_1=0$. We assume the approximation \eqref{reg.NS} only to ensure the conclusion of Lemma \ref{th:SLEI}, the strong local energy inequality.
\begin{proof}
Choose $A=A(\norm{w_0}_{L^2_{\mathrm{uloc}}},\norm{u_0}_{L^2_{\mathrm{uloc}}},T)$ such that
\EQN{
\norm{w}_{\mathcal{E}_T} +\norm{V}_{\mathcal{E}_T}
+
\norm{w}_{U^{s,q}_T} + \norm{V}_{U^{s,q}_T}
& \lesssim A,
}
for any $2\leq s\leq \infty$, and $2\leq q\leq 6$ satisfying $\frac 2s+\frac 3q = \frac 32$.
Fix $x_0\in \mathbb{Z}^3$ and $R \in \mathbb{N}$. Let $\phi_{x_0}=\Phi(\cdot-x_0)$, $\chi_R(x)=1-\Phi\left(\frac {x}R\right)$, and
\EQ{\label{xi.def}
\xi = \phi_{x_0}^2 \chi_R^2.
}
For the convenience, we suppress the indexes $x_0$ and $R$ in $\phi_{x_0}$, $\widecheck{p}_{x_0}$ and $\chi_R$.
Let $\Si$ be the subset of $(0,T)$ defined in Lemma \ref{th:SLEI}. For any $ t_0\in (0,t_1)\setminus \Si$ and $t\in (t_0,T)$, choose $\th(t) \in C^\infty_c(0,T)$ with $\th=1$ on $[t_0,t]$. Let $\ph(x,t) = \xi(x) \th(t)$. By \eqref{pre.decay} of Lemma \ref{th:SLEI}, using $t_0\not \in \Si$, we have
\EQ{\label{eq4.19}
\int |w(x,t)|^2 \xi (x)\, dx
+&2\int_{t_0}^t\!\int |\nabla w|^2 \xi\, dxds \\
\leq &\int |w(x,t_0)|^2 \xi (x) \,dx
+\int_{t_0}^t\!\int |w|^2 (\De \xi +(v\cdot \nabla)\xi )\,dxds\\
&+ 2\int_{t_0}^t\!\int \widecheck{p}_{x_0} w \cdot \nabla \xi \,dxds
-2\int_{t_0}^t\!\int (v \cdot \nabla )V\cdot w\xi \,dxds.
}
Above we have replaced $p$ by $\widecheck{p}_{x_0}$ using $\iint q_{x_0}(t) w \cdot \nabla \xi \,dxds=0$.
By the choice of $\xi$, we can easily see that
\[
\int |w(\cdot,t)|^2 \xi dx
+2\int_{t_0}^t\!\int |\nabla w|^2 \xi dxds
\geq
\norm{w(\cdot,t)\chi}_{L^2(B(x_0,1))}^2
+2\norm{|\nabla w| \chi }_{L^2([t_0,t]\times B(x_0,1) )}^2,
\]
\[
\int |w(\cdot,t_0)|^2 \xi dx \lesssim \norm{w(\cdot,t_0)\chi}_{L^2_{\mathrm{uloc}}}^2,
\]
\[
\int_{t_0}^t\!\int |w|^2 \De \xi dxds
\lesssim \norm{w\chi}_{U^{2,2}(t_0,t)}^2
+ \frac 1R \norm{w}_{U^{2,2}_T}^2,
\]
and
\EQN{
\int_{t_0}^t\!\int |w|^2(v\cdot \nabla)\xi dxds
& \lesssim \norm{v}_{U^{3,3}_T}\norm{w\chi}_{U^{3,3}(t_0,t)}^2 + \frac 1R\norm{v}_{U^{3,3}_T}\norm{w }_{U^{3,3}_T}^2\\
& \lesssim _A \norm{w\chi}_{U^{3,3}(t_0,t)}^2 + \frac 1R.
}
The last term can be also estimated by
\EQN{
\left|\int_{t_0}^t\!\int (v \cdot \nabla )V\cdot w\xi dxds\right|
& \lesssim \norm{|\nabla V|\chi}_{U^{\infty,3}(t_0,T)}
\norm{v}_{U^{2,6}_T}
\norm{w\chi}_{U^{2,2}(t_0,t)}\\
& \lesssim _A \norm{w\chi}_{U^{2,2}(t_0,t)}^2
+ \norm{|\nabla V|\chi}_{U^{\infty,3}(t_0,T)}^2.
}
The only remaining term is the one with pressure. Note
\EQN{
\int_{t_0}^t\!\int &\widecheck{p} w \cdot \nabla \xi dxds
\lesssim \int_{t_0}^t\!\int_{B(x_0,\frac 32)} |\widecheck{p}||w| \chi^2 dxds
+\frac 1R \int_{t_0}^t\!\int_{B(x_0,\frac 32)} |\widecheck{p}||w|\chi dxds \\
& \lesssim \norm{\widecheck{p} \chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\norm{w\chi}_{U^{3,3}(t_0,t)}
+ \frac 1R \norm{\widecheck{p}}_{L^{\frac 32}([0,T]\times B(x_0,\frac 32))}
\norm{w\chi}_{U^{3,3}_T}.
}
For the second term, we can use a bound uniform in $x_0$
\EQN{
\norm{\widecheck{p}_{x_0}}_{L^\frac32([0,t]\times B(x_0,\frac 32))} \le C \norm{v}_{U^{3,3}_t}^2 + C(T) \norm{V}_{U^{\infty,2}_T}^2,
}
which follows from \eqref{est.hp2}, \eqref{hp.cp} and \eqref{q.est}. For the first term, although the other factor $\norm{w\chi}_{U^{3,3}(t_0,t)}$ also has decay, it is larger than the left side of \eqref{eq4.19} by itself.
Hence we need to estimate $\norm{\widecheck{p}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}$ and show its decay.
Let $F_{ij}=w_iw_j + w_i V_j + w_jV_i$ and $G_{ij}=V_iV_j$. The local pressure $\widecheck{p}$ defined in Lemma \ref{new.decomp} can be further decomposed as
\[
\widecheck{p}(x,t)
=p^F + p^{G,1}+ p^{G,2} + p^{G,3}
\]
where $p^F=p^F_{x_0}$ is defined as in \eqref{pFx0},
\[
p^{G,1}
=\int K_i(\cdot-y) [\partial_jG_{ij}] \rho_2 (y) dy,
\]
\EQN{
p^{G,2}
=& \int (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) G_{ij}(\rho_\tau-\rho_2)(y)dy\\
&- \int (K_{i}(\cdot-y)-K_{i}(x_0-y)) G_{ij}\partial_j(\rho_\tau- \rho_2)(y)dy,
}
for $\rho_\tau = \Phi\left(\frac{\cdot -x_0}{\tau}\right)$, $\tau>4$, and
\EQN{
p^{G,3} =& \int (K_{ij}(\cdot-y)-K_{ij}(x_0-y))G_{ij}(1-\rho_\tau)(y)dy \\
&+ \int (K_{i}(\cdot-y)-K_{i}(x_0-y)) G_{ij}\partial_j\rho_\tau(y)dy.
}
Recall $p^F=p^F_{x_0}$
\begin{align*}
p^F &= -\frac 13 \tr F
+ \pv \int_{B(x_0,2)} K_{ij}(\cdot-y) F_{ij}(y) dy
\\
&\quad + \int_{B(x_0,2)^c} (K_{ij}(\cdot-y)-K_{ij}(x_0-y)) F_{ij}(y) dy\\
&= p^{F,1} +p^{F,2} + p^{F,3}.
\end{align*}
We estimate $p^{F,i}\chi$, $i=1,2,3$. Obviously, we have
\EQN{
\norm{p^{F,1}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\lesssim \norm{F\chi}_{U^{\frac 32,\frac 32}(t_0,t)}.
}
Using $L^p$-norm preservation of Riesz transfroms and $\norm{\nabla \chi}_\infty \lesssim \frac 1R$,
\EQN{
\norm{p^{F,2}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\le& \norm{\pv \int_{B(x_0,2)} K_{ij}(\cdot-y) F_{ij}(y)\chi(y) dy }_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}\\
&+ \norm{\pv \int_{B(x_0,2)} K_{ij}(\cdot-y) F_{ij}(y)(\chi(\cdot)-\chi(y)) dy}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}\\
\lesssim & \norm{F\chi}_{U^{\frac 32,\frac 32}(t_0,t)}
+ \frac 1R \norm{ \int_{B(x_0,2)} \frac 1{|\cdot-y|^2} |F_{ij}(y)| dy}_{L^{\frac 32}(t_0,t;L^3 (\mathbb{R}^3)))}\\
\lesssim & \norm{F\chi}_{U^{\frac 32,\frac 32}(t_0,t)}
+ \frac 1R \norm{F}_{U^{\frac 32,\frac 32}(t_0,t)}.
}
The last inequality follows from the Riesz potential estimate. Since
\[
|\chi(x)-\chi(y)|\leq \norm{\nabla\chi}_\infty |x-y| \lesssim \frac 1{\sqrt{R}}
\]
for $x\in B(x_0,\frac 32)$ and $y\in B(x_0,\sqrt R)$, and
\[
|x-y| \geq |x_0-y| - |x-x_0| \geq \frac 14 |x_0-y|
\]
for $x\in B(x_0, \frac 32)$ and $y\in B(x_0,2)^c$, we get
\EQN{
\norm{p^{F,3}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\le & \ \norm{\int_{B(x_0,\sqrt{R})\setminus B(x_0,2)} \frac1{|\cdot-y|^4} F_{ij}\chi(y) dy}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}\\
& + \norm{\int_{B(x_0,\sqrt{R})\setminus B(x_0,2)} \frac1{|\cdot-y|^4} F_{ij}(y) (\chi(\cdot)-\chi(y))dy}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}\\
& + \norm{\int_{B(x_0,\sqrt{R})^c} \frac1{|\cdot-y|^4} F_{ij}(y) dy\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}.
}
Thus
\EQN{
\norm{p^{F,3}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\lesssim & \ \sum_{k=1}^\infty \norm{\int_{B(x_0,2^{k+1})\setminus B(x_0,2^k)} \frac1{|x_0-y|^4} |F_{ij}\chi(y)| dy}_{L^{\frac 32}(t_0,t;L^\infty( B(x_0,\frac 32)))}\\
& + \frac 1{\sqrt R}\sum_{k=1}^{\infty} \norm{\int_{B(x_0,2^{k+1})\setminus B(x_0,2^k)} \frac1{|x_0-y|^4} |F_{ij}(y)| dy}_{L^{\infty}([t_0,t]\times B(x_0,\frac 32))}\\
& + \sum_{\lfloor \log_2 \sqrt R \rfloor}^\infty\norm{\int_{B(x_0,2^{k+1})\setminus B(x_0,2^k)} \frac1{|x_0-y|^4} |F_{ij}(y)| dy}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}\\
\lesssim & \ \norm{ F\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
+ \frac 1{\sqrt{R}} \norm{F}_{U^{\infty,1}(t_0,t)}.
}
Combining the estimates for $p^{F,i}\chi$, $i=1,2,3$, we obtain
\EQN{
\norm{p^F\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
& \lesssim _T \norm{F\chi}_{U^{\frac 32,\frac 32}(t_0,t)} + \frac 1R\norm{F}_{U^{\frac 32,\frac 32}(t_0,t)} + \frac 1{\sqrt{R}}\norm{F}_{U^{\infty,1}(t_0,t)}\\
& \lesssim _{A,T}\norm{w\chi}_{U^{3,3}(t_0,t)} + \frac 1{\sqrt{R}}.
}
Now, we consider $p^{G,i}$'s. Since for $x\in B(x_0,\frac 32)$, $p^{G,1}$ satisfies
\EQN{
|p^{G,1}\chi(x,t)|
\leq&
\int_{|x_0-y|\leq 3} |(\nabla K)(x-y)||V||\nabla V|(y,t)(|\chi(y)|+|\chi(x)-\chi(y)|)dy\\
\lesssim &
\int_{B_3(x_0)} \frac 1{|x-y|^2}||V||\nabla V(y,t)|\chi(y)dy+\frac 1R\int_{B_3(x_0)} \frac 1{|x-y|}|V||\nabla V(y,t)|dy
}
using $|\chi(x)-\chi(y)| \lesssim \norm{\nablabla \chi}_\infty |x-y|$,
the estimate for $p^{G,1}\chi$ can be obtained from Young's convolution inequality;
\EQN{
\norm{p^{G,1}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\lesssim &_T
\norm{\int_{|x_0-y|\leq 3} \frac 1{|\cdot-y|^2}||V||\nabla V|(y,t)\chi(y)dy}_{L^{2}([t_0,t]\times \mathbb{R}^3)}\\
&+\frac 1R\norm{\int_{|x_0-y|\leq 3} \frac 1{|\cdot-y|}|V||\nabla V|(y,t)dy}_{L^{\frac {20}{13}}(t_0,t;L^{\frac{30}7}(\mathbb{R}^3))}\\
\lesssim & \norm{\frac 1{|\cdot|^2}}_{\frac 32, \infty}\norm{|\nabla V|\chi}_{L^\infty_t(t_0,T;L^\frac 32(B(x_0,3)))}\norm{V}_{L^2(0,T; L^6(B(x_0,3)))}\\
&+ \frac 1R \norm{\frac 1{|\cdot|}}_{3,\infty} \norm{V}_{L^{\frac {20}3}(0,T; L^\frac52 (B(x_0,3)) )}\norm{\nabla V}_{U^{2,2}_T}\\
\lesssim &_{A,T} \norm{|\nabla V|\chi}_{U^{\infty,\frac 32}(t_0,T)}
+\frac 1R.
}
By integration by parts, for $x \in B(x_0, \frac 32)$, $p^{G,2}$ can be rewritten as
\[
p^{G,2} =\int (K_{i}(\cdot-y)-K_{i}(x_0-y)) V_i\partial_jV_j(y,t)(\rho_\tau-\rho_2)(y)dy
\]
and then it satisfies
\EQN{
|p^{G,2}\chi(x,t)|
& \lesssim \int_{2<|x_0-y|\leq 2\tau} \frac 1{|x_0-y|^3}|V||\nabla V|(y,t)(|\chi(y)|+|\chi(x)-\chi(y)|) dy\\
& \lesssim \sum_{i=1}^{m_\tau} \int_{B_{i+1}\setminus B_i} \frac 1{|x_0-y|^3}|V||\nabla V|(y,t)\left(|\chi(y)|+\frac {\tau}R\right) dy,
}
where $m_\tau = \lceil \ln (2\tau)/\ln 2 \rceil$ and $B_i = B(x_0,2^i)$.
Taking $L^2(t_0,t)$ on it, we have
\EQN{
\norm{p^{G,2}\chi}_{L^2(t_0,t;L^\infty(B(x_0,\frac 32)))}
& \lesssim \norm{
\sum_{i=1}^{m_\tau} \int_{B_{i+1}\setminus B_i} \frac 1{|x_0-y|^3}|V||\nabla V|(y,t)\left(|\chi(y)|+\frac {\tau}R\right) dy}_{L^2(t_0,t)}\\
& \lesssim \sum_{i=1}^{m_\tau} \frac 1 {2^{3i}} \left(\norm{V|\nabla V|\chi}_{L^2(t_0,t;L^1(B_{i+1}))}
+ \frac {\tau}R\norm{V|\nabla V|}_{L^2(t_0,t;L^1(B_{i+1}))}\right)\\
& \lesssim \sum_{i=1}^{m_\tau}\bke{ (\norm{|V||\nabla V|\chi}_{U^{2,1}(t_0,T)}
+ \frac {\tau}R\norm{|V||\nabla V|}_{U^{2,1}_T} }\\
& \lesssim _T \ln \tau \norm{V}_{U^{\infty,2}_T}\norm{|\nabla V|\chi}_{U^{\infty,2}(t_0,T)}+ \frac {\tau\ln \tau}{ R}\norm{V}_{U^{\infty,2}_T}\norm{\nabla V}_{U^{2,2}_T}.
}
Lastly,
\[
|p^{G,3}(x,t)| \le \int_{|x_0-y|\geq \tau } \frac {|V(y,t)|^2}{|x_0-y|^4} dy + \frac 1{\tau}\int_{\tau \leq |x_0-y| \leq 2\tau} \frac {|V(y,t)|^2}{|x_0-y|^3} dy
\le \frac 1{\tau}\norm{V}_{U^{\infty,2}_T}^2 .
\]
Hence
\[
\norm{p^{G,3}\chi}_{L^\frac 32([t_0,t]\times B(x_0, \frac 32))}
\leq \norm{p^{G,3}}_{L^\frac 32([t_0,t]\times B(x_0, \frac 32))}
\lesssim _{A,T} \frac 1{\tau}
\]
To summarize, we have shown
\[
\sum_{i=1}^3\norm{p^{G,i}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\lesssim _{A,T}\ln \tau \norm{|\nabla V|\chi}_{U^{\infty,2}(t_0,T)} + \frac {\tau \ln \tau}R + \frac 1 \tau,
\]
and therefore
\EQ{\label{est.decay.p}
\norm{\widecheck{p}\chi}_{L^{\frac 32}([t_0,t]\times B(x_0,\frac 32))}
\lesssim _{A,T}\norm{w\chi}_{U^{3,3}(t_0,t)} + \frac 1{\sqrt R} +
\ln \tau \norm{|\nabla V|\chi}_{U^{\infty,2}(t_0,T)} + \frac {\tau \ln \tau}R + \frac 1 \tau.
}
Finally, combining all estimates and then taking supremum on \eqref{eq4.19} over $x_0\in \mathbb{R}^3$, we obtain
\EQ{\label{pre.decay.est00}
\norm{w(\cdot,t)\chi}_{L^2_{\mathrm{uloc}}}^2
+&2\norm{|\nabla w| \chi }_{U^{2,2}(t_0,t)}^2\\
\lesssim _{A,T}& \norm{w(\cdot,t_0)\chi}_{L^2_{\mathrm{uloc}}}^2
+\norm{w\chi}_{U^{2,2}(t_0,t)}^2
+\norm{w\chi}_{U^{3,3}(t_0,t)}^2\\
&+
(\ln \tau)^2 \norm{|\nabla V|\chi}_{U^{\infty,3}(t_0,T)}^2 + \frac {(\tau \ln \tau)^2}{R^2} + \frac 1 {\tau^2}+ \frac 1{R}.
}
Using the estimates
\EQ{\label{U33.est}
\norm{w \chi}_{U^{3,3}(t_0,t)}^2
\lesssim \norm{w \chi}_{U^{6,2}(t_0,t)}
\bke{
\norm{w \chi}_{U^{2,2}(t_0,t)}
+\norm{|\nabla w| \chi}_{U^{2,2}(t_0,t)}
+\frac 1R\norm{w}_{U^{2,2}_T}
},
}
and Lemma \ref{th0730b},
it becomes
\EQ{\label{pre.decay.est}
\norm{w(\cdot,t)\chi}_{L^2_{\mathrm{uloc}}}^2
+&\norm{|\nabla w| \chi }_{U^{2,2}(t_0,t)}^2\\
\lesssim _{A,T,C_0}& \
t_0^{\frac 1{10}}+\norm{w_0\chi}_{L^2_{\mathrm{uloc}}}^2
+\norm{w\chi}_{L^6(t_0,t;L^2_{\mathrm{uloc}})}^2\\
&+
(\ln \tau)^2 \norm{|\nabla V|\chi}_{U^{\infty,3}(t_0,T)}^2 + \frac {(\tau \ln \tau)^2}{R^2} + \frac 1 {\tau^2}+ \frac 1{R},
}
where $C_0$ is defined as in Lemma \ref{th0730b}.
Note that $\norm{w(\cdot,t)\chi}_{L^2_{\mathrm{uloc}}}^2$ is lower semicontinuous in $t$ as $w$ is weakly $L^2(B_n)$-continuous in $t$ for any $n$.
By Gr\"{o}nwall inequality,
we have
\EQ{\label{decay.est}
\norm{w\chi}_{L^6(t_0,T;L^2_{\mathrm{uloc}})}^2
\lesssim _{A,T,C_0}& \ t_0^{\frac 1{10}}+\norm{w_0\chi}_{L^2_{\mathrm{uloc}}}^2
\\
&+(\ln \tau)^2 \norm{|\nabla V|\chi}_{U^{\infty,3}(t_0,T)}^2+ \frac {(\tau \ln \tau)^2}{R^2} + \frac 1 {\tau^2}+ \frac 1{R}.
}
We now prove \eqref{dot.E.3-decay}.
Fix $t_1\in (0,T)$.
For every $n\in \mathbb{N}$ we can choose $t_0=t_0(n)\in (0,t_1)\setminus \Si$ satisfying
\[
t_0^{\frac 1{10}}<\tfrac 1n.
\]
At the same time, we pick $\tau=\tau(n)>4$ satisfying $\tau^{-2} \leq 1/n$.
After $t_0$ and $\tau$ are fixed, we can make all the remaining terms small by choosing $R=R(n,\norm{v_0}_{L^2_{\mathrm{uloc}}}, t_0,\tau)$ sufficiently large:
\[
\norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}^2+(\ln \tau)^2 \norm{|\nabla V|\chi_R}_{U^{\infty,3}(t_0,T)}^2+ \frac {(\tau \ln \tau)^2}{R^2} + \frac 1{R}\leq \frac 1n.
\]
Here, the smallness of the second term follows from $\nablabla V$ decay (Lemma \ref{decay.na.U}), using the oscillation decay of $v_0$.
In conclusion, by \eqref{decay.est}, for each $n\in \mathbb{N}$, we can find $t_0$, $\tau$ and $R\gg 1$ so that
\[
\norm{w\chi_R}_{L^6(t_0,T;L^2_{\mathrm{uloc}})}^2
\lesssim _{A,T,C_0}\frac 1n.
\]
By \eqref{pre.decay.est},
\[
\norm{w\chi_R}_{L^\infty(t_0,T;L^2_{\mathrm{uloc}})}^2
+\norm{|\nabla w|\chi_R}_{U^{2,2}(t_0,T)} \lesssim _{A,T,C_0} \frac 1n.
\]
By \eqref{U33.est},
\[
\norm{w\chi_R}_{U^{3,3}(t_0,T)}^2
\lesssim _{A,T,C_0} \frac 1n.
\]
Restricted to the original time interval $(t_1,T)$, the perturbation $w$ satisfies
\[
\lim_{R\to \infty}\norm{w\chi_R}_{U^{3,3}(t_1,T)} = 0,
\]
\[
\lim_{R\to \infty}\norm{w\chi_R}_{L^\infty(t_1,T;L^2_{\mathrm{uloc}})}^2
+\norm{|\nabla w|\chi_R}_{U^{2,2}(t_1,T)} =0.
\]
Using \eqref{est.decay.p}, we also have
\[
\lim_{R\to \infty}\sup_{x_0\in \mathbb{R}^3} \norm{\widecheck{p}_{x_0}\chi_R}_{L^{\frac32}(B(x_0,\frac32)\times(t_1,T))} = 0.
\]
This completes the proof of Proposition \ref{dot.E.3}.
\end{proof}
\begin{corollary}\label{w.E4}
Under the same assumptions of Proposition \ref{dot.E.3}, the perturbed Navier-Stokes flow $w= v-e^{t\De}u_0$ satisfies $w(t)\in E^p(\mathbb{R}^3)$ for almost all $t\in (0,T]$ for any $3\leq p\leq 6.$
\end{corollary}
\begin{proof}
By Proposition \ref{dot.E.3}, for any fixed $x_0\in \mathbb{R}^3$ and $t_1\in (0,T)$,
the perturbed local energy solution $w$ to Navier-Stokes equations satisfies
\[
\norm{w}_{L^3(B_{3/2}(x_0) \times (t_1,T))}
+\norm{\widecheck{p}_{x_0}}_{L^{3/2}(B_{3/2}(x_0) \times (t_1,T))} \to 0 \quad\text{as }|x_0|\to \infty.
\]
Recall that $V\in C^1([\de,\infty)\times \mathbb{R}^3)$ for any $\de>0$.
Then, by Caffarelli-Kohn-Nirenberg criteria \cite{CKN},
for any $t_2\in (t_1,T]$, we can find $R_0>0$ such that if $|x_0|\geq R_0$,
\[
\norm{w}_{L^\infty([t_2,T] \times B_1(x_0))}
\lesssim \norm{w}_{L^3(B_{3/2}(x_0) \times (t_1,T))}
+\norm{\widecheck{p}_{x_0}}_{L^{3/2}(B_{3/2}(x_0) \times (t_1,T))}^{1/2},
\]
and the constant in the inequality is independent of $x_0$. Moreover, $ \norm{w}_{L^\infty([t_2,T] \times B_1(x_0))} \to 0$ as $|x_0|\to \infty$. Although the system \eqref{PNS} satisfied by $w$ is not the original \eqref{NS}, similar proof works since $V \in C^1$. See \cite[Theorem 2.1]{JiaSverak} for more singular $V\in L^m$, $m>1$, but without the source term $V \cdot \nablabla V$.
On the other hand, $w\in \mathcal{E}_T$ implies that \[
w\in L^ s(0,T;L^p(B_{R_0}))
\]
for any $s\in [2,\infty]$ and $p\in [2,6]$ with $\frac 2s + \frac 3p = \frac 32$,
and therefore $w(t)\in E^p$ for a.e.~$t\in (0,T]$.
\end{proof}
\section{Global existence}\label{global.sec}
In this section, we prove Theorem \ref{global.ex}. We first give the following decay estimates.
\begin{lemma}\label{dot.E.3.2}
Let $(v,p)$ be a local energy solution in $\mathbb{R}^3\times [t_0,T]$, $0<t_0<T<\infty$, to the Naiver-Stokes equations \eqref{NS} for the initial data
\[
v|_{t=t_0}=w_* + e^{t_0\De}u_0
\]
where $w_*\in E^2_\si$ and $u_0 \in L^3_{{\mathrm{uloc}},\si}$ satisfies the oscillation decay \eqref{ini.decay}.
Let $V(t)=e^{t\De}u_0$. Then, the perturbation $w= v-V $ also decays at infinity:
\[
\norm{w}_{L^3([t_0,T]\times B(x_0,1) )}+\norm{\widecheck{p}_{x_0}}_{L^{\frac 32}([t_0,T]\times B(x_0,1) )} \to 0,
\]
and
\[
\norm{w}_{L^\infty(t_0,T;L^2(B(x_0,1)))}
+\norm{\nabla w}_{L^2(t_0,T;L^2(B(x_0,1)))} \to 0 ,\quad\text{as }|x_0|\to \infty.
\]
\end{lemma}
{\it Remark.} This $T$ is arbitrarily large, unlike the existence time given in the local existence theorem, Theorem \ref{loc.ex}. We assume $w_*\in E^2$, and we have $V \in C^1(\mathbb{R}^3 \times [t_0,T])$. We no longer need Lemma \ref{th0730b} nor the strong local energy inequality.
\begin{proof}
The proof is almost the same as that of Proposition \ref{dot.E.3} except for the way to estimate $\norm{w(\cdot, t_0)\chi_R}_{L^2_{\mathrm{uloc}}}$ in \eqref{pre.decay.est00}. Indeed, $\lim_{R \to \infty}\norm{w(\cdot, t_0)\chi_R}_{L^2_{\mathrm{uloc}}} = 0$ by the assumption $w(\cdot, t_0)=w_*\in E^2$.
\end{proof}
Now, we prove the main theorem.
\begin{proof}[Proof of Theorem \ref{global.ex}.]
Let $(v,p)$ be a local energy solution to the Naiver-Stokes equations in $\mathbb{R}^3\times[0,T_0]$, $0<T_0<\infty$, for the initial data $v|_{t=0}=v_0$, constructed in Theorem \ref{loc.ex}. By Corollary \ref{w.E4}, there exists $t_0\in (0,T_0)$, arbitrarily close to $T_0$, with $w(t_0)=v(t_0)-e^{t_0\De}u_0\in E^4$. Then, by Lemma \ref{decomp.Ep}, for any small $\de>0$, we can decompose
\[
w(t_0) = W_0 + h_0,
\]
where $W_0\in C_{c,\si}^\infty(\mathbb{R}^3)$ and $h_0\in E^4(\mathbb{R}^3)$ with $\norm{h_0}_{L^4_{\mathrm{uloc}}}<\de$.
To construct a local energy solution $(\tilde v, \tilde p)$ to \eqref{NS} for $t \ge t_0$ with initial data $\tilde v|_{t=t_0} = v(t_0)$, we decompose $(\tilde v, \tilde p)$ as
\[
\tilde v = V + h+ W, \quad \tilde p = p_h + p_W
\]
where $V(t)=e^{t\De} v_0$, $(h,p_h)$ satisfies
\EQ{\label{eqn.h}
\begin{cases}
\partial_t h -\De h + \nabla p_h = -H \cdot \nablabla H, \quad H = V+h,\\
\mathop{\rm div} h =0,\quad
h|_{t=t_0} = h_0,
\end{cases}
}
so that $H$ solves \eqref{NS} with $H(t_0)=e^{t_0\De}u_0+h_0$,
and $(W,p_W)$ satisfies
\EQ{\label{eqn.td.w}
\begin{cases}
\partial_t W -\De W + \nabla p_W
= -[(H+W)\cdot \nabla]W -(W \cdot \nabla)H, \\
\mathop{\rm div} W = 0,\quad
W|_{t=t_0} = W_0.
\end{cases}
}
Our strategy is
to first find, for each $\ep>0$,
a distributional solution $(h^\ep,p_h^\ep)$ and a Leray-Hopf weak solution $(W^\ep,p_W^\ep)$ to
$\ep$-approximations of \eqref{eqn.h} and \eqref{eqn.td.w} for $t\in I$ for some $S=S(\de, V)>0$ uniform in $\ep$. Then, we prove that they have a limit $(\tilde v, \tilde p)$ which is a desired local energy solution to \eqref{NS} on $I$. By gluing two solutions $v$ and $\tilde v$ at $t=t_0$, we can get the extended local energy solution on the time interval $[0, t_0+S]$. Repeating this process, we get a time-global local energy solution. The detailed proof is given below.
\noindent\texttt{Step 1.} Construction of approximation solutions
Let $I=(t_0,t_0+S)$ for some small $S\in (0,1)$ to be decided.
For $0 < \ep <1$, we first consider the fixed point problem for
\EQ{\label{op.eqn.h}
\Psi (h)&=e^{(t-t_0)\De}h_0
-\int_{t_0}^t e^{(t-s)\De}\mathbb{P} \nabla \cdot(\mathcal{J}_\ep H \otimes H \Phi_\ep)(s) ds,
\quad H = V + h,
}
where $\mathcal{J}_\ep(H) = H\ast \eta_\ep$ is mollification of scale $\e$ and $\Phi_\ep(x) = \Phi(\ep x)$ is a localization factor of scale $\ep^{-1}$. We will solve for a fixed point $h=h^\ep$
in the Banach space
\[
\mathcal{F} =\mathcal{F}_{t_0,S} :=\{h \in U^{\infty,4}(I): (t-t_0)^{\frac 38}h(\cdot,t) \in L^\infty(I\times \mathbb{R}^3) \}
\]
for some small $S >0$
with
\[
\norm{h}_{\mathcal{F}} := \norm{h}_{U^{\infty,4}(I)} + \norm{(t-t_0)^\frac 38h(t)}_{L^\infty(I\times \mathbb{R}^3)}.
\]
Denote $M= \norm{V}_{L^{\infty}(I\times \mathbb{R}^3)} \lesssim (1+t_0^{-4/3})\norm{v_0}_{L^2_{\mathrm{uloc}}}$.
By Lemma \ref{lemma23}, we have
\EQN{
\norm{\Psi h}_{U^{\infty,4}(I)}
& \lesssim \norm{h_0}_{L^4_{\mathrm{uloc}}}
+S^\frac 18\norm{h}_{U^{\infty,4}}^2
+ S^\frac 12M\norm{h}_{U^{\infty,4}}
+ S^\frac 12\norm{V}_{L^\infty(I;L^8_{\mathrm{uloc}})}^2
\\
& \lesssim \norm{h_0}_{L^4_{\mathrm{uloc}}}
+S^\frac 18\norm{h}_{\mathcal{F}}^2
+ S^\frac 12M\norm{h}_{\mathcal{F}}
+ S^\frac 12M^2 ,
}
and for $t \in I$,
\EQN{
\norm{\Psi h(t)}_{L^\infty(\mathbb{R}^3)}
& \lesssim (t-t_0)^{-\frac 38}\norm{h_0}_{L^4_{\mathrm{uloc}}}
+\int _{t_0}^t |t-s|^{-1/2} \bke{ \norm{h(s)}_{L^\infty}^2 + M^2} ds
\\
& \lesssim (t-t_0)^{-\frac 38}\norm{h_0}_{L^4_{\mathrm{uloc}}}+(t-t_0)^{-1/4} \norm{h}_{\mathcal{F}}^2 + (t-t_0)^{1/2} M^2.
}
Therefore, we get
\EQN{
\norm{\Phi h}_{\mathcal{F}}& \lesssim
\norm{h_0}_{L^4_{\mathrm{uloc}}}
+S^\frac 18\norm{h}_{\mathcal{F}}^2
+ S^\frac 12M\norm{h}_{\mathcal{F}}
+ S^\frac 12M^2 .
}
Similarly we can show
\EQN{
\norm{\Phi h_1-\Phi h_2}_{\mathcal{F}}& \lesssim
\bket{ S^\frac 18(\norm{h_1}_{\mathcal{F}}+\norm{h_2}_{\mathcal{F}}) + S^\frac 12M} \norm{h_1-h_2}_{\mathcal{F}} .
}
By the Picard contraction theorem, we can find $S=S(\de,\norm{V}_{L^\infty(I\times\mathbb{R}^3)} )\in (0,1)$ such that
a unique fixed point (mild solution) $h^\ep$ to \eqref{op.eqn.h} exists in $\mathcal{F}_{t_0,S}$ with
\EQ{\label{est1.hep}
\norm{h^\ep}_{\mathcal{F}} \leq C\de, \qquad\forall 0<\ep<1.
}
We also have the uniform bound
\EQ{\label{est2.hep}
\norm{h^\ep}_{\mathcal{E}(I)} \lesssim \norm{\mathcal{J}_\ep H^\ep \otimes H^\ep \Phi_\ep}_{U^{2,2}(I)} \lesssim \norm{h^\ep}_{\mathcal{F}}^2 + \norm{V}_{U^{4,4}(I)}^2 \lesssim \de^2+M^2.
}
Now, we define $H^\ep = V+ h^\ep$ and the pressure $p_h^\ep$ by
\EQN{
p_h^\ep &= -\frac 13 \mathcal{J}_\ep H^\ep \cdot H^\ep \Phi_\ep
+\pv \int_{B_2} K_{ij}(\cdot-y)(\mathcal{J}_\ep H^\ep )_i H^\ep _j\Phi_\ep(y,t) dy\\
&+\pv \int_{B_2^c} (K_{ij}(\cdot-y)-K_{ij}(-y))(\mathcal{J}_\ep H^\ep) _i H^\ep _j\Phi_\ep(y,t) dy.
}
It is well defined thanks to the localization factor $\Phi_\ep$.
For each $R>0$, we have a uniform bound
\EQ{\label{est.phep}
\norm{p_h^\ep}_{L^\frac 32(I\times B_R)}
\le C(R)
}
in a similar way to getting \eqref{uni.bdd.p}. The pair $(h^\ep, p^\ep_h)$ solves, with $H^\ep = V + h^\ep$,
\EQ{\label{eqn.hep}
\begin{cases}
\partial_t h^\ep -\De h^\ep + \nabla p_h^\ep =
-(\mathcal{J}_\ep H^\ep \cdot \nabla)(H^\ep \Phi_{\ep}) ,\\
\mathop{\rm div} h^\ep =0,\quad
h^\ep|_{t=t_0} = h_0\in L^4_{\mathrm{uloc}}
\end{cases}
}
in $\mathbb{R}^3 \times I$ in the distributional sense.
We next consider the equation for $W=W^\ep$,
\EQ{\label{eqn.Wep}
\begin{cases}
\partial_t W -\De W + \nabla p_W
= f_W^\ep
\\
f_W^\ep:=- \mathcal{J}_\ep (H^\ep+ W)\cdot \nablabla W - \mathcal{J}_\ep W \cdot \nablabla H^\ep,
\\
\mathop{\rm div} W = 0,\quad
W|_{t=t_0} = W_0 \in C^\infty_{c,\si}.
\end{cases}
}
Note that \eqref{eqn.Wep} is a mollified and perturbed \eqref{NS}, and has no localization factor $\Phi_\e$.
Using $W^\ep$ itself as a test function, we can get an a priori estimate: for $t\in I$,
\[
\norm{W(t)}_{L^2(\mathbb{R}^3)}^2 + 2\norm{\nabla W}_{L^2([t_0,t]\times \mathbb{R}^3)}^2
\le \norm{W_0}_{L^2(\mathbb{R}^3)}^2 + \iint f_W^\ep \cdot W.
\]
Note that $\iint \mathcal{J}_\ep (H+ W)\cdot \nablabla W \cdot W=0$ and $- \iint (\mathcal{J}_\ep W \cdot \nablabla) h^\ep \cdot W=\iint(\mathcal{J}_\ep W \cdot \nablabla) W \cdot h^\ep $. Also recall that
\[
\norm{h^\ep W}_{L^2(Q)}
\lesssim \norm{h^\ep}_{L^\infty(I;L^3_{\mathrm{uloc}})} (\norm{\nabla W}_{L^2(Q)}
+\norm{ W}_{L^2(Q)})
\]
for $Q=[t_0,t]\times \mathbb{R}^3$. Its proof can be found in \cite[page 162]{KS}. Thus
\EQN{
\iint f_W^\ep \cdot W
&= \iint (\mathcal{J}_\ep W \cdot \nablabla) W \cdot h^\ep - \iint (\mathcal{J}_\ep W \cdot \nablabla) V \cdot W
\\
&\le C\norm{\nabla W}_{L^2(Q)} \de (\norm{\nabla W}_{L^2(Q)}
+\norm{ W}_{L^2(Q)})
+ M_1 \norm{W}_{L^2(Q)}^2.
}
where $M_1= \norm{\nablabla V}_{L^{\infty}(I\times \mathbb{R}^3)}$.
By choosing $\de$ sufficiently small, we conclude
\[
\norm{W(t)}_{L^2(\mathbb{R}^3)}^2 + \norm{\nabla W}_{L^2([t_0,t]\times \mathbb{R}^3)}^2
\le \norm{W_0}_{L^2(\mathbb{R}^3)}^2 + C(1+M_1) \norm{W}_{L^2(Q)}^2.
\]
By Gr\"{o}nwall inequality (using that $\norm{W(t)}_{L^2(\mathbb{R}^3)}^2$ is lower semicontinuous), we obtain
\EQ{\label{est.Wep}
\norm{W^\ep}_{L^\infty(I;L^2(\mathbb{R}^3))}^2 &+ \norm{\nabla W^\ep}_{L^2(I\times \mathbb{R}^3)}^2
\leq C(M_1) \norm{W_0}_{L^2(\mathbb{R}^3)}^2.
}
With this uniform a priori bound, for each $0<\ep<1$, we can use Galerkin method to construct a Leray-Hopf weak solution $W^\ep$ on $I \times \mathbb{R}^3$ to \eqref{eqn.Wep}.
Define $F^\ep_{ij} = \mathcal{J}_\ep( W^\ep+ H^\ep) _iW^\ep_j + (\mathcal{J}_\ep W^\ep)_iH^\ep _j $. We have the uniform bound
\[
\norm{F^\ep_{ij}}_{U^{3/2,3/2}(I)} \le C \norm{|V|+|h^\ep|+|W^\ep|}_{U^{3,3}(I)}^2 \le C(M,M_1,\norm{W_0}_{L^2(\mathbb{R}^3)}).
\]
Define
$p^\ep_W (x,t) = \lim_{n \to \infty} p^{\ep,n}_W(x,t) $, and $p^{\ep,n}_W(x,t) $ is defined for $|x|<2^n$ by
\EQN{
p^{\ep,n}_W(x,t)
=& -\frac 13 \tr F^\ep_{ij}(x,t) +\pv \int_{B_2(0)} K_{ij}(x-y)F^\ep_{ij} (y,t) dy\\
&+\bke{\pv \int_{B_{2^{n+1}}\setminus B_2} + \int_{B_{2^{n+1}}^c}} (K_{ij}(x-y)-K_{ij}(-y))F^\ep_{ij}(y,t) dy.
}
For each $R>0$, we have a uniform bound
\EQ{\label{est.pWep}
\norm{p_W^\ep}_{L^\frac 32(I\times B_R)}
\le C(R,M,M_1,\norm{W_0}_{L^2(\mathbb{R}^3)}).
}
By the usual theory for the nonhomogeneous Stokes system in $\mathbb{R}^3$, the pair $(W^\ep,p^\ep_W)$ solves \eqref{eqn.Wep} in distributional sense.
We now define
\[
v^\ep = H^\ep + W^\ep = V + h^\ep + W^\ep,\quad
p^\ep = p_h^\ep + p_W^\ep.
\]
Summing \eqref{eqn.hep} and \eqref{eqn.Wep},
the pair $(v^\ep,p^\ep)$ solves in distributional sense
\EQ{\label{eqn.vep}
\begin{cases}
\partial_t v^\ep-\De v^\ep + \nabla p^\ep
=- \mathcal{J}_\ep v^\ep \cdot \nablabla v^\ep + E^\ep,
\\
\hspace{26.5mm}
E^\ep = \mathcal{J}_\ep H^\ep \cdot \nablabla (H^\ep(1-\Phi_\ep)),
\\
\mathop{\rm div} v^\ep = 0,\quad
v^\ep|_{t=t_0} = v(t_0).
\end{cases}
}
Thanks to the mollification, $h^\ep$ and $W^\ep$ have higher local integrability by the usual regularity theory. Thus we can test \eqref{eqn.vep} by $2v^\ep\xi$,
$\xi \in C^\infty_c([t_0,t_0+S) \times \mathbb{R}^3)$, and integrate by parts to get the identity
\EQ{\label{LEI.vep}
& 2\int_{I} \!\int |\nabla v^\ep |^2\xi \,dxds = \int |v|^2 \xi (x,t_0)\,dx \\
&\quad + \int_{I} \!\int |v^\ep |^2(\partial_s\xi + \De \xi) + (|v^\ep |^2\mathcal{J}_\ep v^\ep +2p^\ep v^\ep )\cdot \nabla \xi +E^\ep \cdot 2v^\ep \xi\,dxds .
}
Note that $v$ in $\int |v|^2 \xi (x,t_0)\,dx$ is the original solution in $[0,T)$.
\noindent\texttt{Step 2.} A local energy solution on $I=(t_0,t_0+S)$
We now show that $(v^\ep,p^\ep)$ has a weak limit $(\tilde v, \tilde p)$ which is a local energy solution on $I$.
Recall the uniform bounds \eqref{est1.hep}, \eqref{est2.hep}, \eqref{est.phep}, \eqref{est.Wep}, and \eqref{est.pWep} for $h^\ep,p_h^\ep,W^\ep$ and $p_W^\ep$.
As in the proof of Theorem \ref{loc.ex}, from the uniform estimates and the compactness argument, we can find a subsequence $(v^{(k)}, p^{(k)})$, $k \in \mathbb{N}$, from $(v^\ep, p^\ep)$ which converges to some $(\tilde v, \tilde p)$ in the following sense: for each $n\in \mathbb{N}$,
\EQN{
v^{(k)} &\stackrel{\ast}{\rightharpoonup} \tilde v \qquad\qquad \text{in } L^\infty(I;L^2(B_{2^n})), \\
v^{(k)} &\rightharpoonup \tilde v \qquad\qquad\text{in }L^2(I;H^1(B_{2^n})),\\
v^{(k)}, {\cal J}_{(k)}v^{(k)} &\rightarrow \tilde v \qquad\qquad\text{in }L^3(I\times B_{2^{n}}), \\
p^{(k)} &\to \tilde{p} \qquad\qquad \text{in }L^{\frac 32}(I\times B_{2^{n}}),
}
where
$\tilde p(x,t) = \lim_{n \to \infty}\tilde p^n(x,t) $, and $\tilde p^n(x,t) $ is defined for $|x|<2^n$ by
\EQN{
\tilde p^n(x,t)
=&-\frac 13 |\tilde v(x,t)|^2 +\pv \int_{ B_2} K_{ij}(x-y) \tilde v_i\tilde v_j(y,t) \, dy
\\
&+\bke{\pv \int_{B_{2^{n+1}}\setminus B_2} + \int_{B_{2^{n+1}}^c}} (K_{ij}(x-y)-K_{ij}(-y)) \tilde v_i\tilde v_j(y,t) \, dy .
}
Taking the limit of the weak form of \eqref{eqn.vep}, we obtain that
$(\tilde v, \tilde p)$ satisfies the weak form of \eqref{NS} for the initial data $\tilde v|_{t=t_0} = v(t_0)$. Furthermore,
the limit of \eqref{LEI.vep} gives us the local energy inequality: For any
$\xi \in C^\infty_c([t_0,t_0+S) \times \mathbb{R}^3)$, $\xi \ge 0$, we have
\EQ{\label{LEI.tdv}
& 2\int_{I} \!\int |\nabla \tilde v |^2\xi \,dxds \le \int |v|^2 \xi (x,t_0)\,dx \\
&\quad + \int_{I} \!\int |\tilde v |^2(\partial_s\xi + \De \xi) + (|\tilde v |^2 +2\tilde p) \tilde v \cdot \nabla \xi \,dxds .
}
Here we have used that
$
\iint E^{(k)} \cdot v^{(k)} \xi
= \iint \mathcal{J}_{(k)} H^{(k)} \cdot \nablabla (H^{(k)}(1-\Phi_{(k)})) \cdot v^{(k)} \xi =0$
for $k$ sufficiently large. In a way similar to the proof of Theorem \ref{loc.ex}, we get the local pressure decomposition for $\tilde p$, weak local $L^2$-continuity of $\tilde v(t)$, and local $L^2$-convergence to initial data. We also get \eqref{LEI.tdv} with the time interval $I$ replaced by $[t_0,t]$ and an additional term $\int |\tilde v|^2 \xi (x,t)\,dx$ in the left side.
We have shown that $(\tilde v, \tilde p)$ is a local energy solution on $\mathbb{R}^3 \times I$ with initial data $\tilde v|_{t=t_0} = v(t_0)$.
\noindent\texttt{Step 3.} To extend to a time-global local energy solution.
We first prove that the combined solution
\[
u = v1_{[0,t_0]}+ \tilde v 1_{I},\quad q = p1_{[0,t_0]}+ \tilde p 1_{I}
\]
is a local energy solution on the extended time interval $[0,T_1]=[0,t_0+S]$. It is obvious that $u$ and $q$ are bounded in $\mathcal{E}_{T_1}$ and $L^\frac 32_{\mathrm{loc}} ([0,T_1]\times \mathbb{R}^3)$, respectively and $q$ satisfies the decomposition at each point $x_0\in \mathbb{R}^3$. Since we have for any $\zeta \in C_c^\infty([t_0,T_1)\times \mathbb{R}^3;\mathbb{R}^3)$
\[
\int_{t_0}^{T_1} -(\tilde v, \partial_t \zeta) + (\nabla \tilde v, \nabla \zeta) + (\tilde v, (\tilde v\cdot \nabla) \zeta) + (\tilde p, \mathop{\rm div} \zeta) dt = (\tilde v,\zeta)(t_0) = (v,\zeta)(t_0),
\]
and for any $\zeta \in C_c^\infty((0,t_0]\times \mathbb{R}^3;\mathbb{R}^3)$
\[
\int_{0}^{t_0} -(v, \partial_t \zeta) + (\nabla v, \nabla \zeta)+ ( v, ( v\cdot \nabla) \zeta) + ( p, \mathop{\rm div} \zeta) dt = -(v,\zeta)(t_0),
\]
from the weak continuity of $\tilde v$ at $t_0$ from the right and that of $v$ at $t_0$, we can prove that $(u,p)$ satisfies \eqref{NS} in the distribution sense: For any $\zeta\in C_c^\infty((0,T_1)\times \mathbb{R}^3;\mathbb{R}^3)$
\[
\int_{0}^{T_1} -(u, \partial_t \zeta) + (\nabla u, \nabla \zeta)+ ( u, ( u\cdot \nabla) \zeta) + ( q, \mathop{\rm div} \zeta) dt = 0.
\]
Also, since we already have local $L^2$-weak continuity of $u$ on $[0,T_1]\setminus \{t_0\}$, it is enough to check it at $t_0$; for any $\ph\in L^2(\mathbb{R}^3)$ with a compact support,
\[
\lim_{t\to t_0^-} (u, \ph)(t)
=\lim_{t\to t_0^-} (v, \ph)(t)
=(v,\ph)(t_0)
=\lim_{t\to t_0^+} (\tilde v, \ph)(t)
=\lim_{t\to t_0^+} (u, \ph)(t).
\]
Finally, we prove the local energy inequality \eqref{LEI}. Indeed, for any $t\in (0,t_0]$, the inequality follows from the one of $v$. For $t\in (t_0,T_1)$, we add the inequality of $v$ in $[0,t_0]$ to the one of $\tilde v$ in $[t_0,t]$ to get, for any non-negative $\xi \in C_c^\infty((0,T_1)\times \mathbb{R}^3)$,
\EQN{
\int |u|^2\xi &(t) dx
+2\int_0^t\! \int |\nabla u|^2\xi dxds
\\
&= \int |\tilde v|^2\xi(t) dx
+2\int_0^{t_0} \int |\nabla v|^2\xi dxds
+2\int_{t_0}^t\!\int |\nabla \tilde v|^2\xi dxds\\
&\leq
\int_0^{t_0}\int |v|^2 (\partial_s \xi + \De \xi) +(|v|^2 +2p)(v\cdot \nabla)\xi dxds\\
&\qquad +\int_{t_0}^t\!\int |\tilde v|^2 (\partial_s \xi + \De \xi) +(|\tilde v|^2 +2\tilde p)(\tilde v\cdot \nabla)\xi dxds\\
&=\int_0^t\!\int |u|^2 (\partial_s \xi + \De \xi) +(|u|^2 +2q)(u\cdot \nabla)\xi dxds.
}
Therefore, $(u,q)$ is a local energy solution on $[0,T_1]$ and is an extension of $(v,p)$.
Then, by Lemma \ref{dot.E.3.2} and the proof of Corollary \ref{w.E4}, we can find $t_1\in (t_0 + \frac 78 S, t_0 + S)$ such that $ u(t_1)-V(t_1) \in E^4$. Repeating the above argument with new initial time $t_1$, we can get a local energy solution in $[0,t_1+S)$. Iterating this process, we get
a local energy solution global in time. Note that $\norm{V}_{L^\infty([t_1,\infty)\times \mathbb{R}^3)}\leq \norm{V}_{L^\infty([t_0,\infty)\times \mathbb{R}^3)}$ whenever $t_1 >t_0$, so that on each step, we can extend the time interval for the existence by at least $\frac78 S$.
\end{proof}
\section{Perturbations of global solutions with no spatial oscillation decay}
\label{sec6}
As mentioned in the introduction, there are many known non-decaying flows like constant flows, spatially periodic flows (flows on torus) and \emph{two-and-a-half dimensional flows}. The last two do not have oscillation decay in general. We do not have a general existence theory for initial data with no oscillation decay. However, the method of this paper can be used to construct perturbations of global solutions with no spatial oscillation decay. The perturbation of a constant flow is already covered by Theorem \ref{global.ex}. The
perturbation of spatially periodic flows and two-and-a-half dimensional flows are covered by the following theorem, which does not assume spatial decay or spatial oscillation decay of initial data.
\begin{theorem}\label{theorem2}
Let $V(x,t)$ be a global in time local energy solution of \eqref{NS} with
\[
V \in L^\infty(0,\infty; L^q_{\mathrm{uloc}}), \quad V|_{t=0}=V_0 \in L^q_{{\mathrm{uloc}},\si},
\]
for some $q>3$. Then for any $w_0 \in E^2_\si$, there is a global-in-time local energy solution $v$ of \eqref{NS} with initial data
$v_0= V_0 + w_0$.
\end{theorem}
\begin{proof} We may assume $3<q<\infty$. Let $P$ be an associated pressure of $V$. Let $w = v-V$ and $q=p-P$. If $(v,p)$ is a solution of \eqref{NS}, then $(w,q)$ should satisfy the perturbed equation
\begin{equation}
\begin{cases}
\partial_t w -\De w + (V+w)\cdot \nabla w + w \cdot \nablabla V + \nabla q = 0 ,\quad
\mathop{\rm div} w =0 \\
w|_{t=0}=w_0 ,
\end{cases}
\end{equation}
which is \eqref{PNS} without the source term $V \cdot \nablabla V$. As a result, we don't need the spatial decay of $\nablabla V$, the strong local energy inequality \eqref{SLEI-v}, or the spatial decay estimate \eqref{pre.decay.est00} with $\nablabla V$. Hence, the proof is much easier.
Since $v_0 \in L^2_{\mathrm{uloc}}$, a local energy solution $v$ to \eqref{NS} exists on the time interval $[0,T]$ for some $T>0$ by Theorem \ref{loc.ex}.
Using Lemma \ref{th:LEI.w.0}, we have the local energy estimate for $w$
\EQ{ \label{LEI,w2}
\int & |w|^2(x,t) \xi(x)\, dx
+ 2\int_{0}^t\!\int |\nabla w|^2 \xi\, dxds \\
& \leq \int |w_0|^2 \xi(x)\, dx
+\int_{0}^t\!\int |w|^2 (\De \xi +v\cdot \nabla\xi )\,dxds\\
&\quad + \int_{0}^t\!\int 2 q_{x_0} w \cdot \nabla \xi\, dxds
+ \int_{0}^t\!\int 2V\cdot (w \cdot \nabla )(w\xi)\, dxds,
}
for any $\xi\in C^\infty_c(\mathbb{R}^3)$, $\xi \ge 0$. Here ${q}_{x_0}$ is defined by
\EQN{
q_{x_0}(x,t) =& -\frac 13 (|w(x,t)|^2+2w\cdot V) + \pv \int_{B(x_0,2)} K_{ij}(x-y)(w_iw_j+V_iw_j+w_iV_j)(y,t)dy\\
&+\int_{B(x_0,2)^c} (K_{ij}(x-y)-K_{ij}(x_0-y))(w_iw_j+V_iw_j+w_iV_j)(y,t)dy,
}
where $K_{ij}(x) = \partial_{ij} \frac 1{4\pi |x|}$. Let $\phi_{x_0}$ and $\chi_R$ be defined as in \eqref{xi.def}.
We first derive an a priori bound from \eqref{LEI,w2} taking $\xi = \phi_{x_0}^2$ and taking sup over $x_0 \in \mathbb{R}^3$ using $q>3$ (compare \eqref{eq6.4} below for the last term of \eqref{LEI,w2})
\EQ{\label{eq6.3}
\sup_{0<t<T}\norm{w(\cdot, t)}_{L^2_{\mathrm{uloc}}}^2
+ \norm{\nabla w}_{U^{2,2}_T}^2 + \norm{w}_{U^{3,3}_T} \leq A,
}
where $A=A(T,\norm{w_0}_{L^2_{\mathrm{uloc}}}, q, \norm{V}_{L^\infty L^q_{{\mathrm{uloc}}}})$.
Next, by the proof of \cite[Section 2]{KS} with $\xi= \phi_{x_0}^2\chi_R^2$, we can prove a spatial decay estimate (easier than \eqref{pre.decay.est00})
\EQ{\label{decay.w.6}
\sup_{0<t<T}\norm{w(\cdot, t)\chi_R}_{L^2_{\mathrm{uloc}}}^2 + \norm{|\nabla w|\chi_R}_{U^{2,2}_T}^2
\leq C\left(\norm{w_0\chi_R}_{L^2_{\mathrm{uloc}}}^2 + R^{-\frac 23}\right),
}
where $C=C(T, A, q,\norm{V}_{L^\infty L^q_{{\mathrm{uloc}}}})$.
Indeed, all terms in \eqref{LEI,w2} except the last one can be estimated in the same way. For the last term,
\begin{align*}
\int_0^T\int &V (w\cdot \nabla) (w\phi_{x_0}^2\chi_R^2) dxdt\\
& \lesssim
\int_0^T\int |V||w|(|\nabla w|\phi_{x_0}^2\chi_R^2
+ |w|\phi_{x_0}\chi_R^2 + \frac 1R |w|\phi_{x_0}^2) dxdt\\
& \lesssim _{A,T,q} \norm{V}_{L^\infty(0,\infty;L^q_{\mathrm{uloc}})}\left[\norm{w\chi_R}_{U^{2,\left( \frac 12-\frac 1q\right)^{-1}}_T}\norm{|\nabla w|\chi_R}_{U^{2,2}_T} + \norm{w\chi_R}_{U^{3,3}_T}^2 + \frac 1R
\right].
\end{align*}
Then, we use the Gagliardo-Nirenberg interpolation inequality to get
\begin{align*}
\norm{w\chi_R}_{U^{2,\left( \frac 12-\frac 1q\right)^{-1}}_T}
& \lesssim \norm{\nabla (w\chi_R)}_{U^{2,2}_T}^\frac 3q \norm{w\chi_R}_{U^{2,2}_T}^{1-\frac 3q} +\norm{w\chi_R}_{U^{2,2}_T}\\
& \lesssim \norm{|\nabla w|\chi_R}_{U^{2,2}_T}^\frac 3q\norm{w\chi_R}_{U^{2,2}_T}^{1-\frac 3q}+\norm{w\chi_R}_{U^{2,2}_T} + \frac {C_q(A,T)}{R^\frac3q},
\end{align*}
and hence (using $q>3$ to get a small constant)
\EQ{\label{eq6.4}
\int_0^T\int V (w\cdot \nabla) (w\phi_{x_0}^2\chi_R^2) dxdt
\leq \frac 1{99} \norm{|\nabla w|\chi_R}_{U^{2,2}_T}^2 + C_q(A,T) \left(\norm{w\chi_R}_{U^{3,3}_T}^2 + \frac 1{R^\frac3q}\right).
}
This is enough to complete the proof for \eqref{decay.w.6}. Finally, as in Corollary \ref{w.E4}, it implies
\begin{align}\label{w2Ep}
w(t)\in E^p(\mathbb{R}^3), \quad \text{for almost all }t\in (0,T]
\end{align} for any $3\leq p\leq 6$.
Now, we repeat the extension argument in Section \ref{global.sec} with the replacement of the heat equation solution by the time-global solution $V$ given in Theorem \ref{theorem2}. Assume that a local energy solution $(v,p)$ to \eqref{NS} for initial data $v_0\in L^2_{\mathrm{uloc}}(\mathbb{R}^2)$ exists on $[0,T_0]$, $T_0\in (0,\infty)$. Then, by \eqref{w2Ep}, we can find $t_0 \in (0,T_0)$, arbitrarily close to $T_0$, such that $w(t_0) = W_0 +h_0$ where $W_0 \in C_{c,\si}^\infty(\mathbb{R}^3)$ and $h_0 \in E^4(\mathbb{R}^4)$ with $\norm{h_0}_{L^4_{\mathrm{uloc}}}<\de$. The construction of a local energy solution $(\tilde v, \tilde p)$ after time $t_0$ proceeds as follows. We decompose the solution
\[
\tilde v = V + h + W, \quad \tilde p = p_V + p_h + p_W,
\]
where $V$ is the given solution with pressure $p_V$, $(h,p_h)$ solves
\begin{align}\label{eqn.h2}
\begin{cases}
\partial_t h -\De h + \nabla p_h = -(V+h)\cdot \nabla h - (h\cdot \nabla)V\\
\mathop{\rm div} h=0, \quad h|_{t=t_0} = h_0,
\end{cases}
\end{align}
and $(W,p_W)$ satisfies \eqref{eqn.td.w} with the given solution $V$. The only difference with \eqref{eqn.h} is that \eqref{eqn.h2} excludes the term $(V\cdot \nabla)V$. With the interior regularity (see e.g.~\cite[Theorem A1]{LuoTsai})
\[
\norm{V}_{L^\infty(\mathbb{R}^3 \times (t_0,\infty))} \le C(t_0, \norm{V}_{L^\infty(0,\infty; L^q_{\mathrm{uloc}})}),
\]
(we need the strict inequality $q>3$ for this uniform estimate),
the rest of the proof is the same as in Section \ref{global.sec}.
\end{proof}
\section*{Acknowledgments}
The research of both Kwon and Tsai was partially supported by NSERC grant 261356-13.
Hyunju Kwon,
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Current address:
Department of Mathematics, Institute for Advanced Study,
Princeton, NJ 08540, USA;
e-mail: [email protected]
Tai-Peng Tsai, Department of Mathematics, University of British Columbia,
Vancouver, BC V6T 1Z2, Canada;
e-mail: [email protected]
\end{document} |
\begin{document}
\title {Entropy and the Combinatorial Dimension}
\author {S. Mendelson\footnote{Research School of
Information Sciences and Engineering, The Australian National
University, Canberra, ACT 0200, Australia, e-mail:
[email protected]} \and
R. Vershynin\footnote{
Department of Mathematical Sciences,
University of Alberta,
Edmonton, Alberta T6G 2G1, Canada,
e-mail: [email protected]}}
\date{}
\maketitle
\begin{abstract}
We solve Talagrand's entropy problem: the $L_2$-covering numbers
of every uniformly bounded class of functions are exponential in
its shattering dimension. This extends Dudley's theorem on classes
of $\{0,1\}$-valued functions, for which the shattering dimension
is the Vapnik-Chervonenkis dimension.
In convex geometry, the solution means that the entropy of a
convex body $K$ is controlled by the maximal dimension of a cube
of a fixed side contained in the coordinate projections of $K$.
This has a number of consequences, including the optimal Elton's
Theorem and estimates on the uniform central limit theorem in the
real valued case.
\end{abstract}
\section{Introduction}
The fact that the covering numbers of a set are exponential in its
linear algebraic dimension is fundamental and simple. Let $A$ be a
class of functions bounded by $1$, defined on a set $\Omega$. If
$A$ is a finite dimensional class then for every probability
measure on $\mu$ on $\Omega$,
\begin{equation} \label{volumetric}
N(A, t, L_2(\mu)) \le \Big( \frac{3}{t} \Big)^{\dim(A)},
\ \ \ \ 0 < t < 1,
\end{equation}
where $\dim(A)$ is the linear algebraic dimension of $A$ and the
left-hand side of \eqref{volumetric} is the covering number of
$A$, the minimal number of functions needed to approximate any
function in $A$ within an error $t$ in the $L_2(\mu)$-norm. This
inequality follows by a simple volumetric argument (see e.g.
\cite{Pi} Lemma 4.10) and is, in a sense, optimal: the dependence
both on $t$ and on the dimension is sharp (except, perhaps, for
the constant $3$).
The linear algebraic dimension of $A$ is often too large for
\eqref{volumetric} to be useful, as it does not capture the
``size'' of $A$ in different directions but only determines in how
many directions $A$ does not vanish. The aim of this paper is to
replace the linear algebraic dimension by a combinatorial
dimension originated from the classical works of Vapnik and
Chervonenkis \cite{VC 71}, \cite{VC 81}.
We say that a subset $\s$ of $\Omega$ is $t$-shattered by a class
$A$ if there exists a level function $h$ on $\s$ such that, given
any subset $\s'$ of $\s$, one can find a function $f \in A$ with
$f(x) \le h(x)$ if $x \in \s'$ and $f(x) \ge h(x) + t$ if $x
\in \s \setminus \s'$. The {\em shattering dimension} of $A$,
denoted by $\vc(A,t)$ after Vapnik and Chervonenkis, is the
maximal cardinality of a set $t$-shattered by $A$. Clearly, the
shattering dimension does not exceed the linear algebraic
dimension, and is often much smaller. Our main result states that
the linear algebraic dimension in \eqref{volumetric} can be
essentially replaced by the shattering dimension.
\begin{theorem} \label{main}
Let $A$ be a class of functions bounded by $1$,
defined on a set $\Omega$.
Then for every probability measure $\mu$ on $\Omega$,
\begin{equation} \label{combinatorial}
N(A, t, L_2(\mu))
\le \Big( \frac{2}{t} \Big)^{K \cdot \vc(A,\, c t)},
\ \ \ \ 0 < t < 1,
\end{equation}
where $K$ and $c$ are positive absolute constants.
\end{theorem}
There also exists a (simple) reverse inequality complementing
\eqref{combinatorial}: for some measure $\mu$, one has $N(A, t,
L_2(\mu)) \ge 2^{K \cdot \vc(A,\, c t)}$, where $K$ and $c$ are
some absolute constants, see e.g. \cite{T 02}.
The origins of Theorem \ref{main} are rooted in the work of Vapnik
and Chervonenkis, who first understood that entropy estimates are
essential in determining whether a class of functions obeys the
uniform law of large numbers. The subsequent fundamental works of
Koltchinskii \cite{K} and Gin\`{e} and Zinn \cite{GZ} enhanced the
link between entropy estimates and uniform limit theorems (see
also \cite{T 96}).
In 1978, R.~Dudley proved Theorem \ref{main} for classes of
$\{0,1\}$-valued functions (\cite{Du}, see \cite{LT} 14.3). This
yielded that a $\{0,1\}$-class obeys the uniform law of large
numbers (and even the uniform Central Limit Theorem) if and only
if its shattering dimension is finite for $0 < t < 1$. The main
difficulty in proving such limit theorems for general classes has
been the absence of a uniform entropy estimate of the nature of
Theorem \ref{main} (\cite{T 88}, \cite{T 92}, \cite{T 96},
\cite{ABCH}, \cite{BL}, \cite{T 02}). However, proving Dudley's
result for general classes is considerably more difficult due to
the lack of the obvious property of the $\{0,1\}$-valued classes,
namely that if a set $\s$ is $t$-shattered for some $0 < t < 1$
then it is automatically $1$-shattered.
In 1992, M.~Talagrand proved a weaker version of Theorem
\ref{main}: under some mild regularity assumptions, $\log N(A, t,
L_2(\mu)) \le K \cdot \vc(A, \,ct) \log^M (\frac{2}{t})$, where
$K$, $c$ and $M$ are some absolute constants (\cite{T 92}, \cite{T
02}). Theorem \ref{main} is Talagrand's inequality with the best
possible exponent $M = 1$ (and without regularity assumptions).
Talagrand's inequality was motivated not only by limit theorems in
probability, but to a great extent by applications to convex
geometry. A subset $B$ of $\R^n$ can be viewed as a class of real
valued functions on $\{1, \ldots, n\}$. If $B$ is convex and, for
simplicity, symmetric, then its shattering dimension $\vc(B,t)$ is
the maximal cardinality of a subset $\s$ of $\{1, \ldots, n\}$
such that $P_\s (B) \supset [-\frac{t}{2}, \frac{t}{2}]^\s$,
where $P_\s$ denotes the orthogonal projection in $\R^n$ onto
$\R^\s$. In the general, non-symmetric, case we allow translations
of the cube $[-\frac{t}{2}, \frac{t}{2}]^\s$ by a vector in
$\R^\s$.
The following entropy bound for convex bodies is then an immediate
consequence of Theorem \ref{main}. Recall that $N(B,D)$ is the
covering number of $B$ by a set $D$ in $\R^n$, the minimal number
of translates of $D$ needed to cover $B$.
\begin{corollary} \label{main corollary}
There exist positive absolute constants $K$ and $c$ such that
the following holds.
Let $B$ be a convex body contained in $[0,1]^n$,
and $D_n$ be the unit Euclidean ball in $\R^n$.
Then for $0 < t < 1$
$$
N(B, t \sqrt{n} D_n)
\le \Big( \frac{2}{t} \Big)^{K d},
$$
where $d$ is the maximal cardinality of a subset $\s$
of $\{1, \ldots, n\}$ such that
$$
P_\s (B) \supseteq h + [0, ct]^\s
\ \ \ \text{for some vector $h$ in $\R^n$.}
$$
\end{corollary}
As M.~Talagrand notices in \cite{T 02}, Theorem \ref{main} is a
``concentration of pathology'' phenomenon. Assume one knows that a
covering number of the class $A$ is large. All this means is that
$A$ contains many well separated functions, but it tells nothing
about the structure these functions form. The conclusion of
\eqref{combinatorial} is that $A$ must shatter a large set $\s$,
which detects a very accurate pattern: one can find functions in
$A$ oscillating on $\s$ in all possible $2^{|\s|}$ ways around
fixed levels. The ``largeness'' of $A$, {\em a priori} diffused,
is {\em a fortiori} concentrated on the set $\s$.
The same phenomenon is seen in Corollary \ref{main corollary}:
given a convex body $B$ with large entropy, one can find an entire
cube in a coordinate projection of $B$, the cube that certainly
{\em witnesses} the entropy's largeness.
When dualized, Corollary \ref{main corollary} solves the problem
of finding the best asymptotics in Elton's Theorem. Let $x_1,
\ldots, x_n$ be vectors in the unit ball of a Banach space, and
$\e_1, \ldots, \e_n$ be Rademacher random variables (independent
Bernoulli random variables taking values $1$ and $-1$ with
probability $1/2$). By the triangle inequality, the expectation
$\E \| \sum_{i=1}^n \e_i x_i\|$ is at most $n$, and assume that
$\E \| \sum_{i=1}^n \e_i x_i \| \ge \d n$ for some number $\d >
0$.
In 1983, J.~Elton \cite{E} proved an important result that there
exists a subset $\s$ of $\{1, \ldots, n\}$ of size proportional to
$n$ such that the set of vectors $(x_i)_{i \in \s}$ is equivalent
to the $\ell_1$ unit-vector basis. Specifically, there exist
numbers $s, t > 0$, depending only on $\d$, such that
\begin{equation} \label{st}
|\s| \ge s^2 n \ \ \ \text{and} \ \ \ \Big\| \sum_{i \in \s} a_i
x_i \Big\|
\ge t \sum_{i \in \s} |a_i|
\ \ \text{for all real numbers $(a_i)$}.
\end{equation}
Several steps have been made towards finding the best possible $s$
and $t$ in Elton's Theorem. A trivial upper bound is $s, t \le \d$
which follows from the example of identical vectors and by
shrinking the usual $\ell_1$ unit-vector basis. As for the lower
bounds, J.~Elton proved \eqref{st} with $s \sim \d / \log(1/\d)$
and $t \sim \d^3$. A.~Pajor \cite{Pa} removed the logarithmic
factor from $s$. M.~Talagrand \cite{T 92}, using his inequality
discussed above, improved $t$ to $\d / \log^M (1/\d)$. In the
present paper, we use Corollary \ref{main corollary} to solve this
problem by proving the optimal asymptotics: $s, t \sim \d$.
\begin{theorem} \label{intro elton}
Let $x_1, \ldots, x_n$ be vectors in the unit
ball of a Banach space, satisfying
$$
\E \Big\| \sum_{i=1}^n \e_i x_i \Big\| \ge \d n
\ \ \ \text{for some number $\d > 0$}.
$$
Then there exists a subset $\s \subset \{ 1, \ldots, n \}$
of cardinality $|\s| \ge c \d^2 n$ such that
$$
\Big\| \sum_{i \in \s} a_i x_i \Big\|
\ge c \d \sum_{i \in \s} |a_i|
\ \ \text{for all real numbers $(a_i)$},
$$
where $c$ is a positive absolute constant.
\end{theorem}
Furthermore, there is an interplay between the size of $\s$ and
the isomorphism constant -- they can not attain their worst
possible values together. Namely, we prove that $s$ and $t$ in
\eqref{st} satisfy in addition to $s,t \gtrsim \d$ also the lower
bound $s \cdot t \log^{1.6}(2/t) \gtrsim \d$, which, as an easy
example shows, is optimal for all $\d$ within the logarithmic
factor. The power 1.6 can be replaced by any number greater than
1.5. This estimate improves one of the main results of the paper
\cite{T 92} where this phenomenon in Elton's Theorem was
discovered and proved with a constant (unspecified) power of
logarithm.
The paper is organized as follows. In the remaining part of the
introduction we sketch the proof of Theorem \ref{main}; the
complete proof will occupy Section \ref{s:proof}. Section
\ref{s:convexity} is devoted to applications to Elton's Theorem
and to empirical processes.
Here is a sketch of the proof of Theorem \ref{main}. Starting with
a set $A$ which is separated with respect to the $L_2(\mu)$-norm,
it is possible find a coordinate $\omega \in \Omega$ (selected
randomly) on which $A$ is diffused, i.e. the values $\{f(\omega),
\; f \in A\}$ are spread in the interval $[-1,1]$. Then there
exist two nontrivial subsets $A_1$ and $A_2$ of $A$ with their set
of values $\{f(\omega), \; f \in A_1\}$ and $\{f(\omega), \; f \in
A_2\}$ well separated from each other on the line. Continuing this
process of separation for $A_1$ and $A_2$, etc., one can construct
a dyadic tree of subsets of $A$, called a separating tree, with at
least $|A|^{1/2}$ leaves. The ``largeness'' of the class $A$ is
thus captured by its separating tree.
The next step evoked from a beautiful idea in \cite{ABCH}. First,
there is no loss of generality in discretizing the class: one can
assume that $\Omega$ is finite (say $|\Omega|=n$) and that the
functions in $A$ take values in $\frac{t}{6}\Z \cap [-1,1]$. Then,
instead of producing a large set $\s$ shattered by $A$ with a
certain level function $h$, one can count the number of different
pairs $(\s,h)$ for which $\s$ is shattered by $A$ with the level
function $h$. If this number exceeds $\sum_{k=0}^d
\binom{n}{k}(\frac{12}{t})^k$ then there must exist a set $\s$ of
size $|\s|>d$ shattered by $A$ (because there are $\binom{n}{k}$
possible sets $\s$ of cardinality $k$, and for such a set there
are at most $(\frac{12}{t})^k$ possible level functions).
The only thing remaining is to bound below the number of pairs
$(\s,h)$ for which $\s$ is shattered by $A$ with a level function
$h$. One can show that this number is bounded below by the number
of the leaves in the separating tree of $A$, which is $|A|^{1/2}$.
This implies that $|A|^{1/2} \leq\sum_{k=0}^d
\binom{n}{k}(\frac{12}{t})^k \sim (\frac{n}{td})^d$, where
$d=\vc(A,ct)$. The ratio $\frac{n}{d}$ can be eliminated from this
estimate by a probabilistic extraction principle which reduces the
cardinality of $\Omega$.
\noindent ACKNOWLEDGEMENTS
\noindent The first author was supported by an Australian Research
Council Discovery grant. The second author thanks Nicole
Tomczak-Jaegermann for her constant support. He also acknowledges
a support from the Pacific Institute of Mathematical Sciences, and
thanks the Department of Mathematical Sciences of the University
of Alberta for its hospitality. Finally, we would like to thank
the referee for his valuable comments and suggestions.
\section{The Proof of Theorem \ref{main}}
\label{s:proof}
For $t > 0$, a pair of functions $f$ and $g$ on $\Omega$ is {\em
$t$-separated in $L_2(\mu)$} if $\|f - g\|_{L_2(\mu)} > t$. A set
of functions is called $t$-separated if every pair of distinct
points in the set is $t$-separated. Let $N_{\rm sep} (A, t,
L_2(\mu))$ denote the maximal cardinality of a $t$-separated
subset of $A$. It is standard and easily seen that
\begin{equation*}
N (A, t, L_2(\mu))
\le N_{\rm sep} (A, t, L_2(\mu))
\le N (A, \frac{t}{2}, L_2(\mu)).
\end{equation*}
This inequality shows that in the proof of Theorem \ref{main} we
may assume that $A$ is $t$-separated in the $L_2(\mu)$ norm, and
replace its covering number by its cardinality.
We will need two probabilistic results, the first of which is
straightforward.
\begin{lemma} \label{l:xx'}
Let $X$ be a random variable
and $X'$ be an independent copy of $X$.
Then
$$
\E |X - X'|^2 = 2 \E |X - \E X|^2 = 2 \inf_a \E |X - a|^2.
$$
\end{lemma}
The next lemma is a small deviation principle. Denote by $\s(X)^2
= \E |X - \E X|^2$ the variance of the random variable $X$.
\begin{lemma} \label{l:small deviation}
Let $X$ be a random variable with nonzero variance.
Then there exist numbers $a \in \R$ and
$0 < \b \le \frac{1}{2}$, so that letting
\begin{align*}
p_1 &= \P \{ X > a + {\textstyle \frac{1}{6}} \s(X) \}
\ \ \ \text{and} \\
p_2 &= \P \{ X < a - {\textstyle \frac{1}{6}} \s(X) \},
\end{align*}
one has either $p_1 \ge 1-\b$ and $p_2 \ge \frac{\b}{2}$,
or $p_2 \ge 1-\b$ and $p_1 \ge \frac{\b}{2}$.
\end{lemma}
\proof Recall that a median of $X$ is a number $M_X$ such that $\P
\{ X \ge M_X \} \ge 1/2$ and $\P \{ X \le M_X \} \ge 1/2$; without
loss of generality we may assume that $M_X = 0$. Therefore $\P \{
X > 0 \} = 1 - \P \{ X \le 0 \} \le 1/2$ and similarly $\P \{ X <
0 \} \le 1/2$.
By Lemma \ref{l:xx'},
\begin{align} \label{integrals}
\s(X)^2
&\le \E |X|^2
= \int_0^\infty \P \{ |X| > \l \} \; d\l^2 \nonumber \\
&= \int_0^\infty \P \{ X > \l \} \; d\l^2
+ \int_0^\infty \P \{ X < -\l \} \; d\l^2
\end{align}
where $d\lambda^2 = 2\lambda \;d\lambda$.
Assume that the conclusion of the lemma fails, and let $c$ be any
number satisfying $\frac{1}{3} < c < \frac{1}{\sqrt{8}}$. Divide
$\R_+$ into intervals $I_k$ of length $c \s(X)$ by setting
$$
I_k = \Big( c \s(X) k, \; c \s(X) (k + 1) \Big], \ \ \ k = 0, 1,
2, \ldots
$$
and let $\b_0, \b_1, \b_2, \ldots$ be the non-negative numbers
defined by
$$
\P \{ X > 0 \} = \b_0 \le 1/2, \ \ \ \ \ P \{X \in I_k \} =
\b_k - \b_{k+1}, \ \ \ k = 0, 1, 2, \ldots
$$
We claim that
\begin{equation} \label{bk}
\text{for all $k \ge 0$,} \ \ \
\b_{k+1} \le \frac{1}{2} \b_k.
\end{equation}
Indeed, assume that $\b_{k+1} > \frac{1}{2} \b_k$ for some $k$ and
consider the intervals $J_1 = \left(-\infty, c \s(X) k \right]$
and $J_2 = \left(c \s(X) (k+1), \infty \right)$. Then $J_1 =
(-\infty, 0] \cup ( \bigcup_{0 \le l \le k-1} I_l)$, so
$$
\P \{ X \in J_1 \} = (1 - \b_0) + \sum_{0 \le l \le k-1} (\b_l -
\b_{l+1}) = 1- \b_k.
$$
Similarly, $J_2 = \bigcup_{l \ge k+1} I_l$ and thus
$$
\P \{ X \in J_2 \} = \sum_{l \ge k+1} (\b_l - \b_{l+1}) = \b_{k+1}
> \frac{1}{2} \b_k.
$$
Moreover, since the sequence $(\b_k)$ is non-increasing by its
definition, then $\b_k \ge \b_{k+1} > \frac{1}{2} \b_k \ge 0$ and
$\b_k \le \b_0 \le \frac{1}{2}$. Then the conclusion of the lemma
would hold with $a$ being the middle point between the intervals
$J_1$ and $J_2$ and with $\b = \b_k$, which contradicts the
assumption that the conclusion of the lemma fails. This proves
\eqref{bk}.
Now, one can apply \eqref{bk} to estimate the first integral in
\eqref{integrals}. Note that whenever $\l \in I_k$,
$$
\P \{ X > \l \} \le \P \{ X > c \s(X) k \} = \P \Big( \bigcup_{l
\ge k} I_l \Big) = \b_k.
$$
Then
\begin{align}
\int_0^\infty \P \{ X > \l \} \; d\l^2
&\le \sum_{k \ge 0} \int_{I_k} \b_k \cdot 2 \l \;d\l \nonumber\\
&\le \sum_{k \ge 0} \b_k \cdot
2 c \s(X) (k+1) \;{\rm length}(I_k). \label{series}
\end{align}
Applying \eqref{bk} inductively, it is evident that $\b_k \le
(\frac{1}{2})^k \b_0 \le \frac{1}{2^{k+1}}$, and since ${\rm
length }(I_k) = c \s(X)$, \eqref{series} is bounded by
$$
2 c^2 \s(X)^2 \sum_{k \ge 0} \frac{k+1}{2^{k+1}} = 4 c^2 \s(X)^2 <
\frac{1}{2} \s(X)^2.
$$
By an identical argument one can show that the second integral in
\eqref{integrals} is also bounded by $\frac{1}{2} \s(X)^2$.
Therefore
$$
\s(X)^2 < \frac{1}{2} \s(X)^2 + \frac{1}{2} \s(X)^2 = \s(X)^2,
$$
and this contradiction completes the proof.
\endproof
\subsection*{Constructing a separating tree}
Let $A$ be a finite class of functions on a probability space
$(\Omega, \mu)$, which is $t$-separated in $L_2 (\mu)$. Throughout
the proof we will assume that $|A| > 1$. One can think of the
class $A$ itself as a (finite) probability space with the uniform
measure on it, that is, each element $x$ in $A$ is assigned
probability $\frac{1}{|A|}$.
\begin{lemma} \label{separationlemma}
Let $A$ be a $t$-separated subset of $L_2(\mu)$. Then, there
exist a coordinate $i$ in $\Omega$
and numbers $a \in \R$ and $0<\beta \leq 1/2$, so that setting
\begin{align*}
N_1 &= | \{ x \in A : \; x(i) > a + {\textstyle \frac{1}{12}} t \}
|
\ \ \ \text{and} \\
N_2 &= | \{ x \in A : \; x(i) < a - {\textstyle \frac{1}{12}} t \}
|,
\end{align*}
one has either $N_1 \ge (1 - \b) |A|$
and $N_2 \ge \frac{\b}{2} |A|$, or vice versa.
\end{lemma}
\begin{proof}
Let $x, x'$ be random points in $A$ selected independently
according to the uniform (counting) measure on $A$. By Lemma
\ref{l:xx'},
\begin{align} \label{exys}
\E \|x - x'\|_{L_2(\mu)}^2
&= \E \int_\Omega |x(i) - x'(i)|^2 \; d\mu(i)
= \int_\Omega \E |x(i) - x'(i)|^2 \; d\mu(i) \nonumber \\
&= 2 \int_\Omega \E |x(i) - \E x(i)|^2 \; d\mu(i) \\ \nonumber
&= 2 \int_\Omega \s(x(i))^2 \; d\mu(i)
\end{align}
where $\s(x(i))^2$ is the variance of the random variable $x(i)$
with respect to the uniform measure on $A$.
On the other hand, with probability $1 - \frac{1}{|A|}$ we have $x
\ne x'$ and, whenever this event occurs, the separation assumption
on $A$ implies that $\|x - x'\|_{L_2(\mu)} \ge t$. Therefore
$$
\E \|x - x'\|^2_{L_2(\mu)} \ge \Big( 1 - \frac{1}{|A|} \Big) t^2
\ge \frac{t^2}{2}
$$
provided that $|A| > 1$.
Together with \eqref{exys} this proves the existence of a
coordinate $i \in \Omega$, on which
\begin{equation} \label{sxi}
\s(x(i)) \ge \frac{t}{2},
\end{equation}
and the claim follows from Lemma \ref{l:small deviation} applied
to the random variable $x(i)$.
\end{proof}
This lemma should be interpreted as a separation lemma for the set
$A$. It means that one can always find two nontrivial subsets of
$A$ and a coordinate in $\Omega$, on which the two subsets are
separated with a ``gap" proportional to $t$.
Based on Lemma \ref{separationlemma}, one can construct a large
separating tree in $A$. Recall that a {\em tree of subsets} of a
set $A$ is a finite collection $T$ of subsets of $A$ such that,
for every pair $B, D \in T$ either $B$ and $D$ are disjoint or one
of them contains the other. We call $D$ a {\em son} of $B$ if $D$
is a maximal (with respect to inclusion) proper subset of $B$ that
belongs to $T$. An element of $T$ with no sons is called a {\em
leaf}.
\begin{definition}
Let $A$ be a class of functions on $\Omega$ and $t > 0$.
A {\em $t$-separating tree $T$ of $A$} is a tree of subsets of $A$
such that every element $B \in T$ which is not a leaf has
exactly two sons $B_+$ and $B_{-}$ and, for some
coordinate $i \in \Omega$,
$$
f(i) > g(i) + t
\ \ \ \text{for all $f \in B_+$, $g \in B_-$.}
$$
\end{definition}
\begin{proposition} \label{thm:separatingtree}
Let $A$ be a finite class of functions on a probability space
$(\Omega,\mu)$. If $A$ is $t$-separated with respect to the
$L_2(\mu)$ norm, then there exists a $\frac{1}{6} t$-separating
tree of $A$ with at least $|A|^{1/2}$ leaves.
\end{proposition}
\begin{proof}
By Lemma \ref{separationlemma}, any finite class $A$ which is
$t$-separated with respect to the $L_2(\mu)$ norm has two subsets
$A_+$ and $A_{-}$ and a coordinate $i \in \Omega$ for which $f(i)
> g(i) + \frac{1}{6} t$ for every $f \in A_+$ and $g \in A_{-}$.
Moreover, there exists some number $0<\beta \leq 1/2$ such that
\begin{equation*}
|A_+|\geq (1-\beta)|A| \ \ \ \text{and} \ \ \ |A_{-}| \geq
\frac{\beta}{2}, \ \ \ \text{or vice versa}.
\end{equation*}
Thus, $A_+$ and $A_{-}$ are sons of $A$ which are both large and
well separated on the coordinate $i$.
The conclusion of the proposition will now follow by induction on
the cardinality of $A$. The proposition clearly holds for $|A|=2$.
Assume it holds for every $t$-separated class of cardinality
bounded by $N$, and let $A$ be a $t$-separated class of
cardinality $N+1$. Let $A_+$ and $A_{-}$ be the sons of $A$ as
above; since $\b > 0$, we have $|A_+|,|A_{-}| \leq N$. Moreover,
if $A_+$ has a $\frac{1}{6} t$-separating tree with $N_+$ leaves
and $A_{-}$ has a $\frac{1}{6} t$-separating tree with $N_{-}$
leaves then, by joining these trees, $A$ has a $\frac{1}{6}
t$-separating tree with $N_+ + N_{-}$ leaves, the number bounded
below by $|A_+|^{1/2}+|A_-|^{1/2}$ by the induction hypothesis.
Since $\b \le 1/2$,
\begin{align*}
|A_+|^{\frac{1}{2}}+|A_{-}|^{\frac{1}{2}}
&\geq \bigl((1-\beta)|A|\bigr)^{\frac{1}{2}}
+\bigl(\frac{\beta}{2}|A|\bigr)^{\frac{1}{2}}\\
&= \Bigl[(1-\beta)^{\frac{1}{2}}
+\bigl(\frac{\beta}{2}\bigr)^{\frac{1}{2}}\Bigr]
|A|^{\frac{1}{2}}
\geq |A|^{\frac{1}{2}}
\end{align*}
as claimed.
\end{proof}
The exponent $1/2$ has no special meaning in Proposition
\ref{thm:separatingtree}. It can be improved to any number smaller
that $1$ at the cost of reducing the constant $\frac{1}{6}$.
\subsection*{Counting shattered sets}
As explained in the introduction, our aim is to construct a large
set shattered by a given class. We will first try to do this for
classes of integer-valued functions.
Let $A$ be a class of integer-valued functions on a set $\Omega$.
We say that a couple $(\s, h)$ is a {\em center} if $\s$ is a
finite subset of $\Omega$ and $h$ is an integer-valued function on
$\s$. We call the cardinality of $\s$ the dimension of the center.
For convenience, we introduce (the only) $0$-dimensional center
$(\emptyset, \emptyset)$, which is the {\em trivial center}.
\begin{definition} \label{shatteredcenter}
The {\em set $A$ shatters a center $(\s, h)$} if the following
holds:
\begin{itemize}
\item{either $(\s, h)$ is trivial and $A$ is nonempty,}
\item{or, otherwise, for every choice of signs
$\theta \in \{-1,1\}^\s$
there exists a function $f \in A$ such that for $i \in \s$
\begin{equation}
\label{discrete shatter}
\begin{cases}
f(i) > h(i) & \text{when $\theta(i) = 1$},\\
f(i) < h(i) & \text{when $\theta(i) = -1$}.
\end{cases}
\end{equation}
}
\end{itemize}
\end{definition}
It is crucial that both inequalities in \eqref{discrete shatter}
are strict: they ensure that whenever a $d$-dimensional center is
shattered by $A$, one has $\vc(A, 2) \ge d$. In fact, it is
evident that $\vc(A,2)$ is the maximal dimension of a center
shattered by $A$.
\begin{proposition} \label{centers vs leaves}
The number of centers shattered by $A$ is
at least the number of leaves in any $1$-separating tree of $A$.
\end{proposition}
\proof Given a class $B$ of integer-valued functions, denote by
$s(B)$ the number of centers shattered by $B$. It is enough to
prove that if $B_+$ and $B_-$ are the sons of an element $B$ of a
$1$-separating tree in $A$ then
\begin{equation} \label{HHH}
s(B) \ge s(B_+) + s(B_-).
\end{equation}
By the definition of the $1$-separating tree, there is a
coordinate $i_0 \in \Omega$, such that $f(i_0) > g(i_0) + 1$ for
all $f \in B_+$ and $g \in B_-$. Since the functions are
integer-valued, there exists an integer $t$ such that
$$
f(i_0) > t \ \ \text{for $f \in B_+$} \ \ \ \text{and} \ \ \
g(i_0) < t \ \ \text{for $g \in B_-$}.
$$
If a center $(\s, h)$ is shattered either by $B_+$ or by $B_-$, it
is also shattered by $B$. Next, assume that $(\s, h)$ is shattered
by both $B_+$ and $B_-$. Note that in this case $i_0 \not\in \s$.
Indeed, if the converse holds then $\s$ contains $i_0$ and hence
is nonempty. Thus the center $(x,\s)$ is nontrivial and there
exist $f \in B_+$ and $g \in B_-$ such that $t < f(i_0) < h(i_0)$
(by \eqref{discrete shatter} with $\theta(i_0) = -1$) and $t >
g(i_0) > h(i_0)$ (by \eqref{discrete shatter} with $\theta(i_0) =
1$), which is impossible. Consider the center $(\s', h') = (\s
\cup \{i_0\}, h \oplus t)$, where $h \oplus t$ is the extension of
the function $h$ onto the set $\s \cup \{i_0\}$ defined by $(h
\oplus t) (i_0) = t$.
Observe that $(\s', h')$ is shattered by $B$. Indeed, since $B_+$
shatters $(\s, h)$, then for every $\theta \in \{-1,1\}^\s \times
\{1\}^{\{i_0\}}$ there exists a function $f \in B_+$ such that
\eqref{discrete shatter} holds for $i \in \s$. Also, since $f \in
B_+$, then automatically $f(i_0) > t = h'(i_0)$. Similarly, for
every $\theta \in \{-1,1\}^\s \times \{-1\}^{\{i_0\}}$, there
exists a function $f \in B_-$ such that \eqref{discrete shatter}
holds for $i \in \s$ and automatically $f(i_0) < t = h'(i_0)$.
Clearly, $(\s', h')$ is shattered by neither $B_+$ nor by $B_-$,
because $f(i_0) > t = h'(i_0)$ for all $f \in B_+$, so
\eqref{discrete shatter} fails if $\theta(i_0) = -1$; a similar
argument holds for $B_-$.
Summarizing, $(\s, h) \to (\s', h')$ is an injective mapping from
the set of centers shattered by both $B_+$ and $B_-$ into the set
of centers shattered by $B$ but not by $B_+$ or $B_-$, which
proves our claim.
\endproof
Combining Propositions \ref{thm:separatingtree} and \ref{centers
vs leaves}, one bounds from below the number of shattered centers.
\begin{corollary} \label{c: lowerbound}
Let $A$ be a finite class of integer-valued functions on
a probability space $(\Omega,\mu)$.
If $A$ is $6$-separated with respect to the $L_2(\mu)$ norm
then it shatters at least $|A|^{1/2}$ centers.
\end{corollary}
To show that there exists a large dimensional center shattered by
$A$, one must assume that the class $A$ is bounded in some sense,
otherwise one could have infinitely many low dimensional centers
shattered by the class. A natural assumption is the uniform
boundedness of $A$, under which we conclude a preliminary version
of Theorem \ref{main}.
\begin{proposition} \label{p: upperbound}
Let $(\Omega,\mu)$ be a probability space,
where $\Omega$ is a finite set of cardinality $n$.
Assume that $A$ is a class of functions on
$\Omega$ into $\{0, 1, \ldots, p \}$,
which is $6$-separated in $L_2(\mu)$.
Set $d$ to be the maximal dimension of a center
shattered by $A$.
Then
\begin{equation} \label{A discrete}
|A| \le \Big( \frac{p n}{d} \Big)^{C d},
\end{equation}
where $C$ is an absolute constant.
In particular, the same assertion holds for $d=\vc(A,2)$.
\end{proposition}
\proof By Corollary \ref{c: lowerbound}, $A$ shatters at least
$|A|^{1/2}$ centers. On the other hand, the total number of
centers whose dimension is at most $d$ that a class of $\{0, 1,
\ldots, p \}$-valued functions on $\Omega$ can shatter is bounded
by $\sum_{k=0}^d \binom{n}{k} p^k$. Indeed, for every $k$ there
exist at most $\binom{n}{k}$ subsets $\s \subset \Omega$ of
cardinality $k$ and, for each $\s$ with $|\s| = k$ there are at
most $p^k$ level functions $h$ for which the center $(\s, h)$ can
be shattered by such a class. Therefore $|A|^{1/2} \le
\sum_{k=0}^d \binom{n}{k} p^k$ (otherwise there would exist a
center of dimension larger than $d$ shattered by $A$,
contradicting the maximality of $d$). The proof is completed by
approximating the binomial coefficients using Stirling's formula.
\endproof
Actually, the ratio $n/d$ can be eliminated from \eqref{A
discrete} (perhaps at the cost of increasing the separation
parameter $6$). To this end, one needs to reduce the size of
$\Omega$ without changing the assumption that the class is ``well
separated". This is achieved by the following probabilistic
extraction principle.
\begin{lemma} \label{extraction}
There is a positive absolute constant $c$ such that the
following holds.
Let $\Omega$ be a finite set with the uniform probability measure
$\mu$ on it.
Let $A$ be a class of functions bounded by $1$, defined
on $\Omega$.
Assume that for some $0 < t < 1$
$$
\text{$A$ is $t$-separated with respect to the $L_2(\mu)$ norm.}
$$
If $|A| \leq \frac{1}{2} \exp(c t^4 k)$
for some positive number $k$,
there exists a subset $\s \subset \Omega$
of cardinality at most $k$ such that
$$
\text{$A$ is $\frac{t}{2}$-separated with respect to the
$L_2(\mu_\s)$
norm,}
$$
where $\mu_\s$ is the uniform probability measure on $\s$.
\end{lemma}
As the reader guesses, the set $\s$ will be chosen randomly in
$\Omega$. We will estimate probabilities using a version of
Bernstein's inequality (see e.g. \cite{VW}, or \cite{LT} 6.3 for
stronger inequalities).
\begin{lemma}[Bernstein's inequality]
\label{thm:bernstein} Let $X_1, \ldots, X_n$ be independent random
variables with zero mean. Then, for every $u>0$,
\begin{equation*}
\P \Big\{ \big| \sum_{i=1}^n X_i \big| > u \Big\}
\leq 2 \exp \Big(-\frac{u^2}{2(b^2 + a u/3)} \Big),
\end{equation*}
where $a=\sup_i \|X_i\|_\infty$ and $b^2=\sum_{i=1}^n \E |X_i|^2$.
\end{lemma}
\noindent {\bf Proof of Lemma \ref{extraction}. } For the sake of
simplicity we identify $\Omega$ with $\{1,2, \ldots, n\}$. The
difference set $S = \{f - g |\; f \not = g, \ f,g \in A\}$ has
cardinality $|S| \leq |A|^2$. For each $x \in S$ we have $|x(i)|
\leq 2$ for all $i \in \{1,...,n\}$ and $\sum_{i=1}^n |x(i)|^2
\geq t^2 n$. Fix an integer $k$ satisfying the assumptions of the
lemma and let $\delta_1, \ldots, \d_n$ be independent
$\{0,1\}$-valued random variables with $\E \delta_i = \frac{k}{2n}
=: \delta$. Then for every $z \in S$
\begin{align*}
\P \Big\{\sum_{i=1}^n \delta_i |x(i)|^2 \leq \frac{t^2\delta n}{2}
\Big\}
& \leq \P \Big\{ \Big| \sum_{i=1}^n \delta_i |x(i)|^2
- \delta \sum_{i=1}^n |x(i)|^2 \Big |
> \frac{t^2 \delta n}{2} \Big\} \\
&= \P \Big\{ \Big|\sum_{i=1}^n (\delta_i -\delta)|x(i)|^2 \Big|
> \frac{t^2 \d n}{2} \Big\} \\
&\le 2 \exp \Big(-\frac{c t^4 \d n}{1+t^2} \Big)
\le 2 \exp (-c t^4 k),
\end{align*}
where the last line follows from Bernstein's inequality for
$a=\sup_i \|X_i\| \leq 2$ and
$$
b^2 = \sum_{i=1}^n \E |X_i|^2
= \sum_{i=1}^n |x(i)|^4 \; \E(\delta_i - \delta)^2 \leq 16 \d n.
$$
Therefore, by the assumption on $k$
$$
\P \Big\{ \exists x \in S :
\Big( \frac{1}{k} \sum_{i = 1}^n \d_i |x(i)|^2 \Big)^{1/2}
\leq \frac{t}{2} \Big\}
\le |S| \cdot 2 \exp(- c t^4 k) < 1/2.
$$
Moreover, if $\s$ is the random set $\{i \,|\; \delta_i =1\}$ then
by Chebyshev's inequality,
$$
\P \{ |\s| > k \} = \P \Big\{ \sum_{i=1}^n \d_i > k \Big\} \le
1/2,
$$
which implies that
$$
\P \big \{ \exists x \in S : \|x\|_{L_2(\mu_\s)} \le \frac{t}{2}
\big\} < 1.
$$
This translates into the fact that with positive probability the
class $A$ is $\frac{t}{2}$-separated with respect to the
$L_2(\mu_\s)$ norm.
\endproof
\qquad
\noindent {\bf Proof of Theorem \ref{main}. } One may clearly
assume that $|A| > 1$ and that the functions in $A$ are defined on
a finite domain $\Omega$, so that the probability measure $\mu$ on
$\Omega$ is supported on a finite number of atoms. Next, by
splitting these atoms (by replacing an atom $\w$ by, say, two
atoms $\w_1$ and $\w_2$, each carrying measure $\frac{1}{2}
\mu(\w)$ and by defining $f(\w_1) = f(\w_2) = f(\w)$ for $f \in
A$), one can make the measure $\mu$ almost uniform without
changing neither the covering numbers nor the shattering dimension
of $A$. Therefore, assume that the domain $\Omega$ is $\{1, 2,
\ldots, n\}$ for some integer $n$, and that $\mu$ is the uniform
measure on $\Omega$.
Fix $0 < t \leq 1/2$ and let $A$ be a $2 t$-separated in the
$L_2(\mu)$ norm. By Lemma \ref{extraction}, there is a set of
coordinates $s \subset \{1,...,n\}$ of size $|\s| \leq
\frac{C\log|A|}{t^4}$ such that $A$ is $t$-separated in
$L_2(\mu_\s)$, where $\mu_\s$ is the uniform probability measure
on $\s$.
Let $p = \lfloor 7/t \rfloor$, define $\tilde{A} \subset
\{0,1,...,p\}^\s$ by
$$
\tilde{A}=\Bigl\{ \Bigl( \Bigl \lfloor\frac{7f(i)}{t} \Big \rfloor
\Bigr)_{i \in \s} \, | \; f \in A \Bigr\},
$$
and observe that $\tilde{A}$ is $6$-separated in $L_2(\mu_\s)$. By
Proposition \ref{p: upperbound},
$$
|A|=|\tilde{A}| \le \Big( \frac{p |\sigma|}{d} \Big)^{C d}
$$
where $d=\vc(\tilde{A},2)$, implying that
$$
|A| \leq \Big(\frac{C\log |A|}{dt^5}\Big)^{Cd}.
$$
By a straightforward computation,
$$
|A| \leq \Bigl(\frac{1}{t}\Bigr)^{Cd},
$$
and our claim follows from the fact that $\vc(\tilde{A},2) \leq
\vc(A, t/7)$.
\endproof
\qquad
\remark Theorem \ref{main} also holds for the $L_p(\mu)$ covering
numbers for all $0 < p < \infty$, with constants $K$ and $c$
depending only on $p$. The only minor modification of the proof is
in Lemma \ref{l:xx'}, where the equations would be replaced by
appropriate inequalities.
\qquad
\section{Applications: Gaussian Processes and Convexity}
\label{s:convexity}
The first application is a bound on the expectation of the
supremum of a Gaussian processes indexed by a set $A$. Such a
bound is provided by Dudley's integral in terms of the $L_2$
entropy of $A$; the entropy, in turn, can be majorized through
Theorem \ref{main} by the shattering dimension of $A$. The
resulting integral inequality improves the main result of
M.~Talagrand in \cite{T 92}.
If $A$ be a class of functions on the finite set $I$, then a
natural Gaussian process $(X_a)_{a \in A}$ indexed by elements of
$A$ is
$$
X_a = \sum_{i \in I} g_i \, a(i)
$$
where $g_i$ are independent standard Gaussian random variables.
\begin{theorem} \label{thm:talagrand}
Let $A$ be a class of functions bounded by $1$,
defined on a finite set $I$ of cardinality $n$.
Then $E = \E \sup_{a \in A} X_a$
is bounded as
$$
E \le K \sqrt{n}
\int_{cE/n}^{1} \sqrt{\vc(A,t) \cdot \log (2/t)}\; dt,
$$
where $K$ and $c$ are absolute positive constants.
\end{theorem}
The nonzero lower limit in the integral will play an important
role in the application to Elton's Theorem.
The first step in the proof is to view $A$ as a subset of $\R^n$.
Dudley's integral inequality can be stated as
$$
E \le K \int_0^\infty \sqrt{\log N(A, t D_n)} \; dt,
$$
where $D_n$ is the unit Euclidean ball in $\R^n$, see \cite{Pi}
Theorem 5.6. The lower limit in this integral can be improved by a
standard argument. This fact was first noticed by A. Pajor.
\begin{lemma} \label{dudley}
Let $A$ be a subset of $\R^n$.
Then $E = \E \sup_{a \in A} X_a$
is bounded as
$$
E \le K \int_{cE/\sqrt{n}}^\infty \sqrt{\log N(A, t D_n)} \; dt,
$$
where $K$ is an absolute constant.
\end{lemma}
\proof Fix positive absolute constants $c_1, c_2$ whose values
will be specified later. There exists a subset $\NN$ of $A$, which
is a $(\frac{c_1 E}{\sqrt{n}})$-net of $A$ with respect to the
Euclidean norm and has cardinality $|\NN| \le N(A, \frac{c_1 E}{2
\sqrt{n}} D_n)$. Then $A \subset \NN + \frac{c_1 E}{2 \sqrt{n}}
D_n$, and one can write
\begin{equation} \label{E}
E = \E \sup_{a \in A} X_a
\le \E \max_{a \in \NN} X_a
+ \E \sup_{a \in \frac{c_1 E}{\sqrt{n}} D_n} X_a.
\end{equation}
The first summand is estimated by Dudley's integral as
\begin{equation} \label{first NN}
\E \max_{a \in \NN} X_a
\le K \int_0^\infty \sqrt{\log N(\NN, t D_n)} \; dt.
\end{equation}
On the interval $(0, \frac{c_2 E}{\sqrt{n}})$,
\begin{align*}
K \int_0^{ \frac{c_2 E}{\sqrt{n}} } \sqrt{\log N(\NN, t D_n)} \;
dt
&\le K \frac{c_2 E}{\sqrt{n}} \cdot \sqrt{\log|\NN|} \\
&\le K \frac{c_2 E}{\sqrt{n}} \cdot
\sqrt{\log N(A, {\textstyle \frac{c_1 E}{2 \sqrt{n}} } D_n)}.
\end{align*}
The latter can be estimated using Sudakov's inequality
\cite{D,Pi}, which states that $\e \sqrt{\log(N, \e D_n)} \le K
\,\E \sup_{a \in A} X_a$ for all $\e
> 0$. Indeed,
$$
K \frac{c_2 E}{\sqrt{n}} \cdot
\sqrt{\log N(A, {\textstyle \frac{c_1 E}{2 \sqrt{n}} } D_n)}
\leq
K_1 (2c_2 / c_1) \,\E \sup_{a \in A} X_a
= K_1 (2c_2 / c_1) E
\le \frac{1}{4} E,
$$
if we select $c_2$ as $c_2 = c_1 / 8 K_1$. Combining this with
\eqref{first NN} implies that
\begin{equation} \label{first sum}
\E \max_{x \in \NN} X_a
\le \frac{1}{4} E + K \int_{ \frac{c_2 E}{\sqrt{n}} }^\infty
\sqrt{\log N(A, t D_n)} \; dt
\end{equation}
because $\NN$ is a subset of $A$.
To bound the second summand in \eqref{E}, we apply the
Cauchy-Schwarz inequality to obtain that for any $t>0$,
$$
\E \sup_{a \in tD_n} X_a \le t \cdot \E \Big( \sum_{i \in I} g_i^2
\Big)^{1/2} \le t \sqrt{n}.
$$
In particular, if $c_1 < 1/4$ then
$$
\E \sup_{a \in \frac{c_1E}{\sqrt{n}}D_n} X_a \leq c_1E \leq
\frac{1}{4}E.
$$
This, \eqref{E} and \eqref{first sum} imply that
$$
E \le K_2 \int_{ \frac{c_2 E}{\sqrt{n}} }^\infty
\sqrt{\log N(A, t D_n)} \; dt,
$$
where $K_2$ is an absolute constant.
\endproof
\qquad
\noindent{\bf Proof of Theorem \ref{thm:talagrand}. } By Lemma
\ref{dudley},
$$
E \le K \int_{cE/\sqrt{n}}^\infty \sqrt{\log N(A, t D_n)} \; dt.
$$
Since $A \subset [-1,1]^n \subset \sqrt{n} D_n$, the integrand
vanishes for $t \ge \sqrt{n}$. Hence, by Theorem \ref{main}
\begin{align*}
E &\le K \int_{cE/\sqrt{n}}^{\sqrt{n}}
\sqrt{\log N(A, t D_n)} \; dt \\
&= K \sqrt{n} \int_{cE/n}^1
\sqrt{\log N(A, t \sqrt{n} D_n)} \; dt \\
&\le K_1\sqrt{n} \int_{cE/n}^1
\sqrt{\vc(A, c_1 t) \cdot \log (2/t)} \; dt.
\end{align*}
The absolute constant $0 < c_1 < 1/2$ can be made $1$ by a further
change of variable.
\endproof
\qquad
The main consequence of Theorem \ref{thm:talagrand} is Elton's
Theorem with the optimal dependence on $\d$.
\begin{theorem} \label{thm:elton}
There is an absolute constant $c$ for which the following
holds.
Let $x_1, \ldots, x_n$ be vectors in the unit
ball of a Banach space.
Assume that
$$
\E \Big\| \sum_{i = 1}^n g_i x_i \Big\| \ge \d n
\ \ \ \text{for some number $\d > 0$}.
$$
Then there exist numbers $s, t \in (c \d, 1)$,
and a subset $\s \subset \{ 1, \ldots, n \}$
of cardinality $|\s| \ge s^2 n$,
such that
\begin{equation} \label{lower l1}
\Big\| \sum_{i \in \s} a_i x_i \Big\|
\ge t \sum_{i \in \s} |a_i|
\ \ \ \text{for all real numbers $(a_i)$}.
\end{equation}
In addition, the numbers $s$ and $t$ satisfy the inequality
$s \cdot t \log^{1.6} (2 / t) \ge c\d$.
\end{theorem}
Before the proof, recall the interpretation of the shattering
dimension of convex bodies. If a set $B \subset \R^n$ is convex
and symmetric then $\vc(B,t)$ is the maximal cardinality of a
subset $\s$ of $\{1, \ldots, n\}$ such that $P_\s(B) \supset
[-\frac{t}{2}, \frac{t}{2}]^\s$. Indeed, every convex symmetric
set in $\R^n$ can be viewed as a class of functions on
$\{1,...,n\}$. If $\s$ is $t$-shattered with a level function $h$
then for every $\s' \subset \s$ there is some $f_{\s'}$ such that
$f_{\s'}(i) \geq h(i)+t$ if $i \in \s'$ and $f_{\s'} \leq h$ on
$\s \backslash \s'$. By selecting for every such $\s'$ the
function $(f_{\s'}-f_{\s \backslash \s'})/2$ and since the class
is convex and symmetric, it follows that $P_\s(B) \supset
[-\frac{t}{2},\frac{t}{2}]^\s$, as claimed.
Taking the polars, this inclusion can be written as $\frac{t}{2}
(B^\circ \cap \R^\s) \subset B_1^n$, where $B_1^n$ is the unit
ball of $\ell_1^n$. Denoting by $\|\cdot\|_{B^\circ}$ the
Minkowski functional (the norm) induced by the body $B^\circ$, one
can rewrite this inclusion as the inequality
$$
\Big\| \sum_{i \in \s} a_i e_i \Big\|_{B^\circ}
\ge \frac{t}{2} \sum_{i \in \s} |a_i|
\ \ \ \text{for all real numbers $(a_i)$},
$$
where $(e_i)$ is the standard basis of $\R^n$. Therefore, to prove
Theorem \ref{thm:elton}, one needs to bound below the shattering
dimension of the dual ball of a given Banach space.
\qquad
\noindent {\bf Proof of Theorem \ref{thm:elton}.} By a
perturbation argument, one may assume that the vectors $(x_i)_{i
\le n}$ are linearly independent. Hence, using an appropriate
linear transformation one can assume that $X = (\R^n, \|\cdot\|)$
and that $(x_i)_{i \le n}$ are the unit coordinate vectors
$(e_i)_{i \le n}$ in $\R^n$. Let $B=(B_X)^\circ$ and note that the
assumption $\|e_i\|_X \leq 1$ implies that $B \subset [-1,1]^n$.
Set
$$
E = \E \Big\|\sum_{i=1}^n g_i x_i \Big\|_X
= \E \sup_{b \in B} \sum_{i=1}^n g_i \,b(i).
$$
By Theorem \ref{thm:talagrand},
\begin{equation*}
\d n \leq E \leq K\sqrt{n} \int_{c \delta}^{1}
\sqrt{\vc(B,t) \cdot \log(2/t)} \; dt.
\end{equation*}
Consider the function
$$
h(t) = \frac{c_0}{t \log^{1.1} (2 / t)}
$$
where the absolute constant $c_0 > 0$ is chosen so that
$\int_0^{1} h(t) \; dt = 1$. It follows that there exits some
$c\d \le t \le 1$ such that
$$
\sqrt{\vc(B, t) / n \cdot \log(2 / t)} \ge \d h(t).
$$
Hence
$$
\vc(B, t) \ge \frac{c_0 \d^2}{t^2 \log^{3.2} (2 / t)} n.
$$
Therefore, letting $s^2 = \vc(B, t) / n$, it follows that $s \cdot
t \log^{1.6} (2 / t) \ge \sqrt{c_0} \d$ as required, and by the
discussion preceding the proof there exists a subset $\s$ of $\{1,
\ldots, n\}$ of cardinality $|\s| \ge s^2 n$ such that
\eqref{lower l1} holds with $t/2$ instead of $t$. The only thing
remaining is to check that $s \gtrsim \d$. Indeed, $s \ge
\frac{\sqrt{c_0} \d}{t \log^{1.6} (2 / t)} \ge c_1 \d$, because $t
\le 1$.
\endproof
\noindent {\bf Remarks. } 1. As the proof shows, the exponent
$1.6$ can be reduced to any number larger than $3/2$.
2. The relation between $s$ and $t$ in Theorem \ref{thm:elton} is
optimal up to a logarithmic factor for all $0 < \d < 1$. This is
seen from by the following example, shown to us by Mark Rudelson.
For $0 < \d < 1 / \sqrt{n}$, the constant vectors $x_i = \d
\sqrt{n} \cdot e_1$ in $X = \R$ show that $s t$ in Theorem
\ref{thm:elton} can not exceed $\d$. For $1 / \sqrt{n} \le \d \le
1$, we consider the body $D = \conv( B_1^n \cup \frac{1}{\d
\sqrt{n}} D_n )$ and let $X = (\R^n, \|\cdot\|_D)$ and $x_i =
e_i$, $i = 1, \ldots, n$. Clearly, $\E \| \sum g_i x_i \|_X \ge
\E \| \sum \e_i e_i \|_D = \d n$. Let $0 < s, t < 1$ be so that
\eqref{lower l1} holds for some subset $\s \subset \{ 1, \ldots, n
\}$ of cardinality $|\s| \ge s^2 n$. This means that $\|x\|_D
\ge t \|x\|_1$ for all $x \in \R^\s$. Dualizing, $\frac{t}{\d
\sqrt{n}} \|x\|_2 \le t \|x\|_{D^\circ} \le \|x\|_\infty$ for
all $x \in \R^\s$. Testing this inequality for $x = \sum_{i \in
\s} e_i$, it is evident that $\frac{t}{\d \sqrt{n}} \sqrt{|\s|}
\le 1$ and thus $s t \le \d$.
\qquad
We end this article with an application to empirical processes. A
key question is when a class of functions satisfies the central
limit theorem uniformly in some sense. Such classes of functions
are called {\it uniform Donsker classes}. We will not define these
classes formally but rather refer the reader to \cite{D,VW} for an
introduction on the subject. It turns out that the uniform Donsker
property is related to uniform estimates on covering numbers via
the Koltchinskii-Pollard entropy integral.
\begin{theorem} \cite{D}
\label{dud} Let $F$ be a class of functions bounded by $1$. If
\begin{equation*}
\int_0^\infty \sup_{n}\sup_{\mu_n} \sqrt{ \log N
\bigl(F,L_2(\mu_n),\e\bigr) } \;d\eps < \infty,
\end{equation*}
then $F$ is a uniform Donsker class.
\end{theorem}
Having this condition in mind, it is natural to try to seek
entropy estimates which are ``dimension free", that is, do not
depend on the size of the sample. In the $\{0,1\}$-valued case,
such bounds where first obtained by Dudley who proved Theorem
\ref{main} for these classes (see \cite{LT} Theorem 14.13) which
implied through Theorem \ref{dud} that every VC class is a uniform
Donsker class.
Theorem \ref{main} solves the general case: the following
corollary extends Dudley's result on the uniform Donsker property
from $\{0,1\}$ classes to classes of real valued functions.
\begin{corollary}
Let $F$ be a class of functions bounded by $1$
and assume that the integral
$$
\int_0^1 \sqrt{ \vc(F,t)\log \frac{2}{t}} \; dt
$$
converges.
Then $F$ is a uniform Donsker class.
\end{corollary}
In particular this shows that if $\vc(F,t)$ is ``slightly better"
than $1/t^2$, then $F$ is a uniform Donsker class.
This result has an advantage over Theorem \ref{dud} because in
many cases it is easier to compute the shattering dimension of the
class rather than its entropy (see, e.g. \cite{AB}).
{\small
}
\end{document} |
\begin{document}
\begin{abstract}
We modify a classical construction of Bousfield and Kan \cite{BousKanSSeq.pdf} to define the Adams tower of a simplicial nonunital commutative algebra over a field $k$. We relate this construction to Radulescu-Banu's cosimplicial resolution \cite{Radulescu-Banu.pdf}, and prove that all connected simplicial algebras are complete with respect to Andr\'e-Quillen homology. This is a convergence result for the unstable Adams spectral sequence for commutative algebras over $k$.
\end{abstract}
\title{Connected simplicial algebras are\Andr\'e-Quillen complete}
Let $s\mathcal{C}$ denote the simplicial model category \cite{QuillenHomAlg.pdf} of simplicial non-unital commutative algebras over a field $k$, and let $X$ be an object of $s\mathcal{C}$. Radulescu-Banu \cite{Radulescu-Banu.pdf} constructed a cosimplicial resolution $\mathcal{X}^\bullet$ of $X$ by generalised Eilenberg-Mac Lane objects, and defined the \emph{completion of $X$ with respect to Andr\'e-Quillen homology} to be the totalization $X\hat{\ }:=\textup{Tot}(\mathcal{X}^\bullet)$. The purpose of the present work is to prove the following conjecture of Radulescu-Banu:
\begin{completenesstheorem}\label{completenesstheorem}
If $X$ is a connected simplicial $k$-algebra, then $X$ is naturally equivalent to its completion $X\hat{\ }$.
\end{completenesstheorem}
Radulescu-Banu's completion functor $X\mapsto X\hat{\ }$ is the analogue of Bousfield and Kan's $R$-completion functor on simplicial sets \cite{BousKanSSeq.pdf}, a construction that has proven extremely useful in classical homotopy theory. As in the classical case, the homotopy of the completion $X\hat{\ }$ is the target of an unstable Adams spectral sequence. The completeness theorem may be viewed as a convergence result for this spectral sequence, which we will study in detail in forthcoming work.
In fact, Bousfield and Kan have defined the unstable Adams spectral sequence of a simplicial set in two different ways. Their earlier approach \cite{BK_pairings.pdf} was to define the \emph{derivation of a functor with respect to a ring}. This approach constructs the \emph{Adams tower} over the simplicial set in question, and lends itself well to connectivity analyses. Their latter approach, \cite{BousKanSSeq.pdf}, to give a cosimplicial resolution of a simplicial set by simplicial $R$-modules, lends itself more to the analysis of the $E^2$ page, and is directly analogous to Radulescu-Banu's construction.
Since the release of \cite{BK_pairings.pdf} and \cite{BousKanSSeq.pdf}, the relationship between the two approaches has been clarified by the introduction of the theory of cubical diagrams to homotopy theory \cite{GoodwillieCalcII}. Our approach to proving the completeness theorem will be to define the Adams tower of a simplicial algebra using a construction analogous to Bousfield and Kan in \cite{BK_pairings.pdf} (Section \ref{sec:derWRTab}), and to use the theory of cubical diagrams to relate it to Radulescu-Banu's construction (Section \ref{sec:relnWithRB}). In Section \ref{sec:connectivityAnalysis}, we perform the necessary connectivity estimates in the Adams tower in order to prove the completeness theorem.
As Radulescu-Banu observed, there is an additional difficulty in constructing the Bousfield-Kan cosimplicial resolution of a simplicial algebra which is not present in the classical context. Namely, since not all simplicial algebras are cofibrant, the naive cosimplicial resolution will not be homotopically correct. Radulescu-Banu's innovation was to explain that the cofibrant replacement functor $c:s\mathcal{C}\longrightarrow s\mathcal{C}$ constructed by Quillen's small object argument \cite{QuillenHomAlg.pdf} admits a comonad diagonal $\psi:c\longrightarrow cc$, and can thus be mixed into the cosimplicial resolution, making it homotopically correct.
Accordingly, we must mix Quillen's cofibrant replacement functor $c$ into our definition of the Adams tower over a simplicial algebra so that it relates as desired to Radulescu-Banu's resolution. The application of these cofibrant replacement functors adds to the difficulty of proving the connectivity estimates of Section \ref{sec:connectivityAnalysis}. We circumvent this difficulty by shifting to the standard comonadic bar construction.
In Section \ref{introToRBwork} we will introduce Radulescu-Banu's cosimplicial resolution and explain a little terminology. In the appendix we will state and prove a useful result on iterated simplicial bar constructions.
This research will form part of the author's PhD thesis. The author would like to thank his advisor, Haynes Miller, for his support and guidance, and John Harper, for sharing helpful insights into the use of cubical diagrams in connectivity analyses such as that of Section \ref{sec:connectivityAnalysis}.
\section{Radulescu-Banu's completion functor}\label{introToRBwork}
A non-unital commutative $k$-algebra is a $k$-vector space $Y$ equipped with an associative and commutative map $\mu:Y\otimes Y\longrightarrow Y$. We will refer to such objects simply as \emph{algebras}. It can be shown that if $Y\in\mathcal{C}$ is a categorical group object, then it is in fact a zero-square algebra, that is $Y^2=\im(\mu)=0$, and the group map $Y\times Y\longrightarrow Y$ is simply the vector space addition. Thus, the abelianization adjunction for $\mathcal{C}$ can be modeled as $Q:s\mathcal{C}\rightleftarrows s\mathsf{Vect}:K$, in which the left adjoint is the \emph{abelianization} or \emph{indecomposables} functor
\[Q:\mathcal{C}\longrightarrow \mathsf{Vect},\qquad Y\mapsto Y/Y^2,\]
and the right adjoint is the \emph{zero-square} functor
\[K:\mathsf{Vect}\longrightarrow\mathcal{C},\qquad V\mapsto V\textup{ with $V^2$ set to zero}.\]
The Andr\'e-Quillen homology of a simplicial algebra $X$ is defined as the homotopy groups of its left derived abelianization:
\[H_*X:=\pi_*(QcX),\]
where $c:s\mathcal{C}\longrightarrow s\mathcal{C}$ is the cofibrant replacement from Quillen's SOA. For an excellent introduction to these ideas, see \cite[\S4]{MR1089001}. Radulescu-Banu constructed \cite{Radulescu-Banu.pdf} a comonad diagonal $\psi:c\longrightarrow cc$, in order to define the coface maps in a (coaugmented) cosimplicial object:
\[\makebox[0cm][r]{\,$\mathcal{X}^\bullet:\qquad $}\vcenter{
\def\scriptstyle{\scriptstyle}
\xymatrix@C=1.5cm@1{
cX\,
\ar[r]
&
\,cKQcX\,
\ar[r];[]
&
\,c(KQc)^2X\,
\ar@<-1ex>[l];[]
\ar@<+1ex>[l];[]
\ar@<+1ex>[r];[]
\ar@<-1ex>[r];[]
&
\,c(KQc)^3X\,\makebox[0cm][l]{\,$\cdots. $}
\ar[l];[]
\ar@<-2ex>[l];[]
\ar@<+2ex>[l];[]
}}\]
The construction is explained and generalized by Blumberg and Riehl \cite{BlumRiehlResolutions.pdf}. The definition of the coface and codegeneracy maps is similar to that in the monadic resolution of $X$ using the adjunction $Q\dashv K$, however, the coface maps must create an extra copy of $c$, and the codegeneracies must destroy a copy. Instead of simply using the unit and counit of the adjunction respectively, one uses the composites
\[c\overset{\psi}{\longrightarrow}cc\overset{c\eta c}{\longrightarrow}cKQc\textup{\quad and\quad }QcK\overset{Q\varepsilonilon K}{\longrightarrow}QK\longrightarrow \textup{id}.\]
Denote by $X\hat{\ }$ the totalization of $\mathcal{X}^\bullet$, naming $X\hat{\ }$ the \emph{completion of $X$ with respect to Andr\'e-Quillen homology}. There is a natural zig-zag $X\overset{\sim}{\longleftarrow} cX\longrightarrow X\hat{\ }$, and one says that $X$ is \emph{complete with respect to Andr\'e-Quillen homology} when the map $cX\longrightarrow X\hat{\ }$ is an equivalence. We will prove that this occurs whenever $X$ is connected. In order to understand the name of the completion functor, note that levelwise application of Andr\'e-Quillen homology to $\mathcal{X}^\bullet$ yields a cosimplicial vector space $H_*\mathcal{X}^\bullet$ which is weakly equivalent to its coaugmentation $H_*X$ (cf. \cite{BlumRiehlResolutions.pdf}).
\section{The Adams tower}\label{sec:derWRTab}
For any functor $F:s\mathcal{C}\longrightarrow s\mathcal{C}$, we define the $r^\textup{th}$ derivation $\dupdown{r}{b}F$ of $F$ with respect to Andr\'e-Quillen homology. The definition is recursive:
\begin{alignat*}{2}
(\dupdown{0}{b}F)(X)
&:=
F(cX)
\\
(\dupdown{s}{b}F)(X)
&:=
\hofib((\dupdown{s-1}{c}F)(cX)\xrightarrow{(\dupdown{s-1}{c}F)(\eta_{cX})} (\dupdown{s-1}{c}F)(KQcX))
\end{alignat*}
where $\eta$ is the unit of the adjunction $Q\dashv K$, i.e.\ the natural surjection onto indecomposables, and $\hofib$ is any fixed functorial construction of the homotopy fiber. These functors fit into a tower via the following composite natural transformation:
\[\delta:\left((\dupdown{s}{c}F)(X)\longrightarrow (\dupdown{s-1}{c}F)(cX)\overset{(\dupdown{s-1}{c}F)(\varepsilonilon)}{\longrightarrow} (\dupdown{s-1}{c}F)(X)\right).\]
We have thus constructed a tower
\[\xymatrix@R=4mm{
\cdots
\ar[r]
&
(\dupdown{2}{c}F)X
\ar[r]
&
(\dupdown{1}{c}F)X
\ar[r]&
(\dupdown{0}{c}F)X=FcX,
}\]
which is natural in the object $X$ and the functor $F$.
The functors $\dupdown{r}{c}F$ are homotopical as long as $F$ preserves weak equivalences between cofibrant objects. Employing the shorthand
\[\dupdown{s}{c}X:=(\dupdown{s}{c}I)X,\]
we define \emph{the Adams tower of $X$} to be the tower
\[\xymatrix@R=4mm{
\cdots
\ar[r]
&
\dupdown{2}{c}X
\ar[r]
&
\dupdown{1}{c}X
\ar[r]&
\dupdown{0}{c}X=cX.
}\]
For example, $(\dupdown{2}{c}F)(X)$ is constructed by the following diagram in which every composable pair of parallel arrows is \emph{defined} to be a homotopy fiber sequence.
\[\def\scriptstyle{\scriptstyle}
\xymatrix@!0@R=30pt@C=43pt{
(\dupdown{2}{c}F)(X)\ar[d]\\
(\dupdown{1}{c}F)(cX) \ar[rr]\ar[d] & &FcccX \ar[rr]\ar[d] & & FcKQccX \ar[d] & \\
(\dupdown{1}{c}F)(KQcX) \ar[rr] & & FccKQcX \ar[rr] & & FcKQcKQcX
}\]
In general, $(\dupdown{n+1}{c}F)(X)$ is the homotopy total fiber of an $(n+1)$-cubical diagram:
\[(\dupdown{n+1}{c}F)(X):=\textup{hototfib} \bigl(({D}_{n+1}^{\smash{\square}}F)X\bigr).\]
See \cite{GoodwillieCalcII}, \cite{LuisGoodwillie.pdf} or \cite{CubicalHomotopyTheory.pdf} for the general theory of cubical diagrams. Before defining the cubical diagram $({D}_{n+1}^{\smash{\square}}F)X$, we set notation: for $n\geq0$ let $[n]=\{0,\ldots,n\}$, and define $\mathcal{P}[n]=\left\{S\subseteq [n]\right\}$ to be the poset category whose morphisms are the inclusions $S\subseteq S'$. Then an $(n+1)$-cube in $s\mathcal{C}$ is a functor $\mathcal{P}[n]\longrightarrow s\mathcal{C}$, and the $(n+1)$-cubical diagram $({D}_{n+1}^{\smash{\square}}F)X:\mathcal{P}[n]\longrightarrow s\mathcal{C}$ is the functor:
\[S\mapsto Fc(KQ)^{\chi_{n}}c(KQ)^{\chi_{n-1}}c\cdots c(KQ)^{\chi_0}cX\quad \textup{where}\quad \chi_i:=\begin{cases}
1,&\textup{if }i\in S;\\
0,&\textup{if }i\notin S,
\end{cases}
\]
where for $S\subseteq S'$, the map $(({D}_{n+1}^{\smash{\square}}F)X)(S)\longrightarrow (({D}_{n+1}^{\smash{\square}}F)X)(S')$ is given by applying the counit $\eta:1\longrightarrow KQ$ in those locations indexed by $S'\setminus S$.
\section{Relationship between the Adams tower and Radulescu-Banu's resolution}\label{sec:relnWithRB}
Radulescu-Banu defines the completion of $X$ to be the totalization
\[X\hat{\ }:=\textup{Tot}(\mathcal{X}^\bullet)=\holim (\textup{Tot}_n(\mathcal{X}^\bullet)),\]
and the unstable Adams spectral sequence to be the spectral sequence of the tower
\[\cdots \longrightarrow\textup{Tot}_n(\mathcal{X}^\bullet)\longrightarrow \textup{Tot}_{n-1}(\mathcal{X}^\bullet)\longrightarrow\cdots \]
under $cX$. Our goal in this section is to prove
\begin{prop}\label{towerIdentification}
There is a natural zig-zag of weak equivalences of towers between $D_{n+1}X$ and $\hofib(cX\longrightarrow\textup{Tot}_{n}(\mathcal{X}^\bullet))$. That is, up to homotopy, the $\textup{Tot}$ tower induces the Adams tower by taking fibers.
\end{prop}
Before giving the proof, we recall a useful relationship between cosimplicial objects and cubical diagrams, explained by Sinha in \cite[Theorem 6.5]{SinhaSpacesOfKnots.pdf}, and expanded on by Munson-Voli\'c \cite{CubicalHomotopyTheory.pdf}.
We will only present that part of the theory that we need. There is a diagram of inclusions of categories
\[\xymatrix@!0@R=10mm@C=13mm{
\mathcal{P}[-1]\ar[rr]^-{\tau}\ar_-{h_{-1}}[drrr]
&&
\mathcal{P}[0]\ar[rr]^-{\tau}\ar^(.65){\!h_{0}}[dr]
&&
\mathcal{P}[1]\ar[rr]^-{\tau}\ar_(.65){h_{1}\!}[dl]
&&
\mathcal{P}[2]\ar[rr]^-{\tau}\ar^-{h_{2}}[dlll]
&&
\cdots \\
&&&\Delta_+\!\!
}\]
As the coaugmented cosimplicial object $\mathcal{X}^\bullet$ is Reedy fibrant (cf.\ \cite[{X.4.9}]{YellowMonster}), there are natural weak equivalences $\hofib(\mathcal{X}^{-1}\longrightarrow\textup{Tot}_n\mathcal{X}^\bullet) \overset{\smash{\sim}}{\longrightarrow} \textup{hototfib}(h_n^*(\mathcal{X}^\bullet))$ under which the tower map
\[\hofib(\mathcal{X}^{-1}\longrightarrow\textup{Tot}_n\mathcal{X}^\bullet)\longrightarrow \hofib(\mathcal{X}^{-1}\longrightarrow\textup{Tot}_{n-1}\mathcal{X}^\bullet)\]
is identified with the map
\[\textup{hototfib}(h_n^*(\mathcal{X}^\bullet))\longrightarrow \textup{hototfib}(\tau^*h_n^*(\mathcal{X}^\bullet))=\textup{hototfib}(h_{n-1}^*(\mathcal{X}^\bullet)).\]
As $\mathcal{X}^{-1}$ equals $cX$, the tower $\hofib(\mathcal{X}^{-1}\longrightarrow\textup{Tot}_n\mathcal{X}^{\bullet})$ is one of the towers in proposition \ref{towerIdentification}.
\begin{proof}[Proof of Proposition \ref{towerIdentification}]
It will suffice to construct a weak equivalence $h_n^*\mathcal{X}^\bullet\longrightarrow (D_n^{\smash{\square}}I)(X)$ of $(n+1)$-cubes. The $(n+1)$-cubical diagram $h_n^*\mathcal{X}^\bullet$ is defined by
\[(h_n^*\mathcal{X}^\bullet)(S):= c(KQc)^{\chi_{n}}(KQc)^{\chi_{n-1}}\cdots (KQc)^{\chi_0}X\quad \textup{where}\quad \chi_i:=\begin{cases}
1,&\textup{if }i\in S,\\
0,&\textup{if }i\notin S,
\end{cases}
\]
where we describe the map $(h_n^*\mathcal{X}^\bullet)(S)\longrightarrow (h_n^*\mathcal{X}^\bullet)(S\sqcup\{i\})$, for $i\notin S$, as follows. Let $j$ be the smallest element of $S\sqcup\{n+1\}$ exceeding $i$, so that
\[(h_n^*\mathcal{X}^\bullet)(S):= \begin{cases}
c(KQc)^{\chi_{n}}\cdots (KQc)^{\chi_{j+1}}(KQ\underline{c})(KQc)^{\chi_{i-1}}\cdots (KQc)^{\chi_0}X,&\textup{if }j\leq n;\\
\underline{c}(KQc)^{\chi_{i-1}}\cdots (KQc)^{\chi_0}X,&\textup{if }j=n+1.
\end{cases}
\]
In the expression for either case, we have distinguished one of the applications of $c$ with an underline, and the map to $(h_n^*\mathcal{X}^\bullet)(S\sqcup\{i\})$ is induced by the composite $\underline{c}\longrightarrow cc\longrightarrow cKQc$ of the diagonal of the comonad $c$ with the unit of the monad $KQ$.
We now define maps $(h_n^*\mathcal{X}^\bullet)(S)\longrightarrow (({D}_{n+1}^{\smash{\square}}I)X)(S)$ for $S=\{j_0<j_1<\cdots<j_r\}\subset\{0,\ldots,n\}$.
The only difference between the domain and codomain is that in $(({D}_{n+1}^{\smash{\square}}I)X)(S)$, all $n+2$ applications of $c$ are present, whereas in $(h_n^*\mathcal{X}^\bullet)(S)$, only $r+2$ appear. The map is then
\[\psi^{n-j_r}KQ\psi^{j_r-j_{r-1}-1}KQ\psi^{j_{r-1}-j_{r-2}-1}KQ\cdots KQ\psi^{j_{1}-j_0-1}KQ\psi^{j_0}X\]
which is to say that we apply the iterated diagonal the appropriate number of times in each $c$ appearing in the domain. As $\psi$ is coassociative, this definition is unambiguous, and the resulting maps assemble to a weak equivalence of $(n+1)$-cubes.
\end{proof}
\section{Connectivity estimates in the Adams tower}
\label{sec:connectivityAnalysis}
We wish to prove the following connectivity result for the Adams tower:
\begin{prop}\label{convergenceProp}
For integers $t\geq2$ and $q\geq0$, the map $\pi_q(\dupdown{2t+q-1}{c}X)\longrightarrow\pi_q(\dupdown{t}{c}X)$ is zero.
\end{prop}
\noindent We defer the proof until the end of this section. Note that proposition \ref{towerIdentification} and \ref{convergenceProp} together imply the completeness theorem:
\begin{proof}[Proof of completeness theorem]
The fiber sequences $\dupdown{n+1}{c}X\longrightarrow cX\longrightarrow \textup{Tot}_n\mathcal{X}^\bullet$ fit together into a tower of fiber sequences. Taking homotopy limits, one obtains a fiber sequence
\[\holim (\dupdown{n}{c}X)\longrightarrow cX\longrightarrow X\hat{\ }.\]
We need to show that $\holim (\dupdown{n}{c}X)$ has zero homotopy groups.
We may replace the tower $\dupdown{n}{c}X$ with a weakly equivalent tower of fibrations in $s\mathcal{C}$ whose set-theoretic inverse limit is the homotopy limit in question. Applying \cite[Proposition 6.14]{goerss-jardine.pdf} to the new tower, there is a short exact sequence
\[0\longrightarrow \textup{lim}^1\pi_{q+1}(\dupdown{n}{c}X)\longrightarrow \pi_q(\holim (\dupdown{n}{c}X))\longrightarrow \textup{lim}\,\pi_{q}(\dupdown{n}{c}X)\longrightarrow 0.\]
Proposition \ref{convergenceProp} implies that for each $q$, the tower $\{\pi_{q}(\dupdown{n}{c}X)\}$ has zero inverse limit, and satisfies the Mittag-Leffler condition (cf.\ \cite[p.264]{YellowMonster}), so that the $\textup{lim}^1$ groups appearing also vanish.
\end{proof}
With an eye to proving proposition \ref{convergenceProp} we define a somewhat less homotopical version $\mathcal{D}up{s}F$ of the derivations $\dupdown{s}{b}F$ of $F$. Again, the definition is recursive:
\begin{alignat*}{2}
(\mathcal{D}up{0}F)(X)
&:=
F(X)
\\
(\mathcal{D}up{s}F)(X)
&:=
\textup{ker}\,((\mathcal{D}up{s-1}F)(bX)\xrightarrow{(\mathcal{D}up{s-1}F)(\eta_{bX})} (\mathcal{D}up{s-1}F)(KQbX))
\end{alignat*}
There are three differences between this definition and that of $D_sF$: here, there is one fewer cofibrant replacement applied, we use the standard simplicial bar construction $b$ (see Appendix \ref{sec:ItSimpBar}) instead of the SOA functor $c$, and we take \emph{strict} fibers, not \emph{homotopy} fibers.
While these functors are not generally homotopical, we define \emph{the modified Adams tower of $X$} to be the tower
\[\xymatrix@R=4mm{
\cdots
\ar[r]
&
\mathcal{D}up{2}X
\ar[r]
&
\mathcal{D}up{1}X
\ar[r]&
\mathcal{D}up{0}X=X,
}\]
where $\mathcal{D}up{s}X$ is again shorthand for $(\mathcal{D}up{s}I)X$, and the tower maps $\delta$ are defined as before.
\begin{prop}\label{prop:modifiedAdamsTower}
There's a natural zig-zag of weak equivalences of towers between the Adams tower of $X$ and the modified Adams tower of $X$. In particular, the modified tower is homotopical.
\end{prop}
\begin{proof}
Let $\mathsf{CR(s\mathcal{C})}$ be the category of cofibrant replacement functors in $s\mathcal{C}$. That is, an object of $\mathsf{CR(s\mathcal{C})}$ is a pair, $(f,\varepsilonilon)$, such that $f:s\mathcal{C}\longrightarrow s\mathcal{C}$ is a functor whose image consists only of cofibrant objects, and $\varepsilonilon:f\Rightarrow I$ is a natural acyclic fibration. Morphisms in $\mathsf{CR(s\mathcal{C})}$ are natural transformations which commute with the augmentations. For any $(f,\varepsilonilon)\in\mathsf{CR(s\mathcal{C})}$ we obtain an alternative definition of the derivations of a functor $F:s\mathcal{C}\longrightarrow s\mathcal{C}$:
\begin{alignat*}{2}
(D_{\smash{0}}^{\smash{f}}F)(X)
:=
F(fX),\quad
(D_{\smash{s}}^{\smash{f}}F)(X)
:=
\hofib((D_{\smash{s-1}}^{\smash{f}}F)(fX)\longrightarrow (D_{\smash{s-1}}^{\smash{f}}F)(KQfX)).
\end{alignat*}
These functors are natural in $f$, so that a morphism in $\mathsf{CR(s\mathcal{C})}$ induces a weak equivalence of towers. Our proposed zig-zag of towers is:
\[D_{\smash{s}}= D_{\smash{s}}^{\smash{c}}I\overset{}{\longleftarrow}D_{\smash{s}}^{\smash{b\relax c}}\!I\overset{}{\longrightarrow}D_{\smash{s}}^{\smash{b}}I\overset{\gamma_s}{\longleftarrow}\mathcal{D}up{s}b\overset{\mathcal{D}up{s}\varepsilonilon}{\longrightarrow}\mathcal{D}up{s}I=\mathcal{D}up{s}\]
The maps with domain $D_{\smash{s}}^{\smash{b\relax c}}I$ are induced by the maps $\varepsilonilon c:b\relax c\longrightarrow c$ and $b\varepsilonilon:b\relax c\longrightarrow b$ and are evidently natural weak equivalences of towers. The map $\gamma_0:(\mathcal{D}up{0}b)X\longrightarrow (D_{\smash{0}}^{\smash{b}}I)X$ is the identity of $bX$, and the map $\mathcal{D}up{0}\varepsilonilon:(\mathcal{D}up{0}b)X\longrightarrow (D_{\smash{0}}^{\smash{b}}I)X$ is $\varepsilonilon:bX\longrightarrow X$. Thereafter, $\gamma_s$ and $\mathcal{D}up{s}\varepsilonilon$ are defined recursively:
\[\qquad \quad \xymatrix@R=4mm{
\makebox[0cm][r]{$(\mathcal{D}up{s+1}I)X:=\,$}\makebox[5cm][l]{$\,\textup{ker}\,((\mathcal{D}up{s}I)(bX)\longrightarrow (\mathcal{D}up{s}I)(KQbX))$}\\
\makebox[0cm][r]{$(\mathcal{D}up{s+1}b)X:=\,$}\makebox[5cm][l]{$\,\textup{ker}\,((\mathcal{D}up{s}b)(bX)\longrightarrow (\mathcal{D}up{s}b)(KQbX))$}\ar[d]^-{\textup{incl.}}_-{}
\ar[u]_-{\textup{induced by }(\mathcal{D}up{s}\varepsilonilon,\mathcal{D}up{s}\varepsilonilon)}^-{}
\\%r1c1
\makebox[5cm][l]{$\,\hofib((\mathcal{D}up{s}b)(bX)\longrightarrow(\mathcal{D}up{s}b)(KQbX))$}\ar[d]^-{\textup{induced by }(\gamma_{s},\gamma_{s})}_-{}
\\%r2c1
\makebox[0cm][r]{$(D_{\smash{s+1}}^{\smash{b}})X:=\,$}\makebox[5cm][l]{$\,\hofib((D_{\smash{s}}^{\smash{b}}I)(bX)\longrightarrow (D_{\smash{s}}^{\smash{b}}I)(KQbX))$}
}\]
\noindent Lemma \ref{towerWithPowers} shows that the kernels taken are actually kernels of surjective maps, and by induction on $s$, the maps $\gamma_s$ and $\mathcal{D}up{s}\varepsilonilon$ are weak equivalences.
\end{proof}
The connectivity result will rely on the observation that any element in the $s^\textup{th}$ level of the modified tower maps down to an $(s+1)$-fold product in $X$. In order to formalise this, let $P^s:s\mathcal{C}\longrightarrow s\mathcal{C}$ be the ``$s^\textup{th}$ power'' functor, the prolongation of the endofunctor $Y\mapsto Y^s$ of $\mathcal{C}$, where $Y^s=\textup{im}(\textup{mult}:Y^{\otimes s}\longrightarrow Y)$. Then we have:
\begin{lem}\label{towerWithPowers}
The functors $\mathcal{D}up{r}$, $\mathcal{D}up{r}b$ and $\mathcal{D}up{r}P^{s}$ preserve surjective maps and there is a commuting diagram of functors:
\[\xymatrix@R=4mm{
\cdots
\ar[r]
&
\mathcal{D}up{r}
\ar[r]
\ar[d]
&
\cdots \ar[r]
&
\mathcal{D}up{2}
\ar[r]
\ar[d]
&
\mathcal{D}up{1}
\ar[r]
\ar[d]&
\mathcal{D}up{0}
\ar@{=}[d]
\\%r1c6
\cdots
\ar[r]
&
P^{r+1}
\ar[r]
&
\cdots
\ar[r]&
P^3
\ar[r]
&P^2
\ar[r]
&
I
}\]
\end{lem}
\begin{proof}
Suppose that $X$ is a simplicial algebra. Then $\mathcal{D}up{r}X$
is constructed as the subalgebra
\[\mathcal{D}up{r}X:= \bigcap_{i=1}^{r}\textup{ker}\,\!\left(b^{r-i}\eta b^{i}:b^{r}X\longrightarrow b^{r-i}KQb^{i}X\right)\]
of $b^rX$. In dimension $n$, this is the following subset of $(b^rX)_n:=(T^{n+1})^rX_n$:
\[(\mathcal{D}up{r}X)_n:=\bigcap_{i=1}^{r}\textup{ker}\,\!\left(T^{(r-i)(n+1)}\eta T^{i(n+1)}:(T^{n+1})^rX_n\longrightarrow (T^{n+1})^{r-i}KQ(T^{n+1})^iX_n\right)\]
As in the appendix, $T$ is the free tensor algebra comonad on $\mathcal{C}$.
The $r$ conditions on elements of $(\mathcal{D}up{r}X)_n$ ensure that their image in $(\mathcal{D}up{0}X)_n=X_n$ under the iterated tower map is a sum of $(r+1)$-fold products. This completes the construction of the tower of functors.
In order to prove the surjectivity statements, we must describe the iterated free construction $T^{r(n+1)}X_n$. A basis of $TX_n$ may be given by the \emph{monomials} in a basis of $X_n$, and $T^{r(n+1)}X_n$ has basis given by taking monomials iteratively, $r(n+1)$ times. The subset $(\mathcal{D}up{r}X)_n$ has basis those iterated monomials in which the monomials formed in the $((n+1)i)^\textup{th}$ iteration have degree at least two for each $1\leq i\leq r$. This simple description of a basis of $(\mathcal{D}up{r}X)_n$ shows that $\mathcal{D}up{r}$ preserves surjections. Similar analysis applies to $\mathcal{D}up{r}b$ and $\mathcal{D}up{r}P^s$.
\end{proof}
We are now able to state and prove the key connectivity result:
\begin{lem}\label{connectivityOfDerivedPowers}
For $A\in s\calC$ connected, and any $t\geq1$ and $s\geq2$, $(\mathcal{D}up{t}P^{s})(A)$ is $(s-t)$-connected.
\end{lem}
\begin{proof}
We will prove this by induction on $t$. When $t=1$:
\[(\mathcal{D}up{1}P^{s})A:=\textup{ker}\,(P^{s}(bA)\longrightarrow P^{s}(QbA))=P^{s}(bA),\]
so we must show that $P^s(bA)$ is $(s-1)$-connected. For this we use a truncation of Quillen's fundamental spectral sequence, as presented in \cite[Thm 6.2]{MR1089001}: the filtration
\[P^s(bA)\supset P^{s+1}(bA)\supset P^{s+2}(bA)\supset\cdots \]
of $P^s(bA)$ yields a convergent spectral sequence $E^0_{p,q}\implies \pi_q(P^s(ba))$, with:
\[E^0_{p,q}=N_q\bigl(P^{p}(bA)/P^{p+1}(bA)\bigr)\textup{ if $p\geq s$, and }E^0_{p,q}=0\textup{ if $p<s$.}\]
Examination of the structure of $bA$ reveals that $P^{p}(bA)/P^{p+1}(bA)\cong (QbA)^{\otimes p}_{\Sigma_p}$, and moreover, $QbA$ is a connected simplicial vector space, as $\pi_0(QbA)=Q(\pi_0bA)=Q(0)=0$. The $t=1$ case now follows from \cite[Satz 12.1]{DoldPuppeSuspension.pdf}: if $V$ is a connected simplicial vector space then $V^{\otimes p}_{\Sigma_p}$ is $(p-1)$-connected.
Now let $t\geq2$ and suppose by induction that $(\mathcal{D}up{t-1}P^{s})(B)$ is $(s-(t-1))$-connected for any connected $B$ and any $s\geq2$. Then by lemma \ref{towerWithPowers}, there's a short exact sequence:
\[\xymatrix{
0\ar[r]&
(\mathcal{D}up{t}P^{s})(A)\ar[r]&
(\mathcal{D}up{t-1}P^{s})(bA)\ar[r]&
(\mathcal{D}up{t-1}P^{s})(QbA)\ar[r]&
0
}\]
in which the rightmost two objects are each $(s-t+1)$-connected. The associated long exact sequence shows that $(\mathcal{D}up{t}P^{s})(A)$ is $(s-t)$-connected.
\end{proof}
Before we can give the proof of proposition \ref{convergenceProp}, we need the following \emph{twisting lemma}, analogous to that of \cite{BK_pairings.pdf}. Before stating it, we note that $(\mathcal{D}up{s}\mathcal{D}up{t})X$ and $\mathcal{D}up{s+t}X$ are equal by construction.
\begin{twistinglemma}
\label{DsDt=Dt+s}
The maps $\mathcal{D}up{i}\delta:\mathcal{D}up{n}X\longrightarrow \mathcal{D}up{n-1}X$ are homotopic for $0\leq i< n$.
\end{twistinglemma}
\begin{proof}
We may reindex the twisting lemma as follows: the maps
\[\mathcal{D}up{s}\delta,\mathcal{D}up{s-1}\delta:\mathcal{D}up{s+t}X\longrightarrow \mathcal{D}up{s+t-1}X\]
are homotopic whenever $s,t\geq1$. Now $\mathcal{D}up{s+t}X$ is constructed as the subalgebra
\[\mathcal{D}up{s+t}X:= \bigcap_{i=1}^{s+t}\textup{ker}\,\!\left(b^{s+t-i}\eta b^{i}:b^{s+t}X\longrightarrow b^{s+t-i}KQb^{i}X\right)\]
of the iterated bar construction $b^{s+t}X$, and for $0\leq i<s+t$, $\mathcal{D}up{i}\delta$ is the restriction of the map $b^i\varepsilonilon b^{s+t-i-1}:b^{s+t}X\longrightarrow b^{s+t-1}X$.
Proposition \ref{IteratedBarConstructionHomotopy} gives an explicit simplicial homotopy between the maps $b^s\varepsilonilon b^{t-1}$ and $b^{s-1}\varepsilonilon b^{t}$. Moreover, the naturality of the construction of proposition \ref{IteratedBarConstructionHomotopy} implies that this homotopy does indeed restrict to a homotopy of maps $\mathcal{D}up{s+t}X\longrightarrow \mathcal{D}up{s+t-1}X$.
\end{proof}
Now that we have the twisting lemma, Proposition \ref{convergenceProp} follows:
\begin{proof}[Proof of Proposition \ref{convergenceProp}]
By proposition \ref{prop:modifiedAdamsTower}, it is enough to prove that $\pi_q(\mathcal{D}up{2t+q-1}X)\longrightarrow\pi_q(\mathcal{D}up{t}X)$ is zero.
Apply $\mathcal{D}up{t}\textup{--}$ to the diagram of functors constructed in \ref{towerWithPowers} and apply the result to $X$ to obtain a commuting diagram of functors
\[\xymatrix@R=4mm{
\mathcal{D}up{2t+q-1}X
\ar[r]^-{\mathcal{D}up{t}\delta}
\ar[d]
&
\cdots \ar[r]^-{\mathcal{D}up{t}\delta}
&
\mathcal{D}up{t+1}X
\ar[r]^-{\mathcal{D}up{t}\delta}
\ar[d]&
\mathcal{D}up{t}X
\ar@{=}[d]
\\%r1c6
\mathcal{D}up{t}P^{t+q}X
\ar[r]
&
\cdots
\ar[r]&
\mathcal{D}up{t}P^2X
\ar[r]
&
\mathcal{D}up{t}P^1X
}\]
By the twisting lemma, \ref{DsDt=Dt+s}, the composite along the top row is homotopic to the map of interest, and factors through $(\mathcal{D}up{t}P^{t+q})(A)$, which is $q$-connected by lemma \ref{connectivityOfDerivedPowers}.
\end{proof}
\appendix
\section{Iterated simplicial bar constructions}\label{sec:ItSimpBar}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\trip}[3]{{#1}_{\smash{#2}}^{\smash{#3}}}
In this section we will be concerned with iterated simplicial bar constructions. The result here applies in general in the category of algebras over a monad, but nonetheless we only state it for commutative algebras. Establishing notation, for any simplicial object $X$ in $\calC$, we'll write
\[\trip{d}{i,q}{X}:X_q\longrightarrow X_{q-1}\textup{\ and\ }\trip{s}{i,q}{X}:X_q\longrightarrow X_{q+1}\]
for the $i^\textup{th}$ face and degeneracy maps out of $X_q$. Suppose $F,G\in\calC^\calC$ are endofunctors, $\Phi:F\longrightarrow G$ is a natural transformation, and $A,B\in\calC$ are objects. Write $[\Phi]:\calC(A,B)\longrightarrow\calC(FA,GB)$ for the operator sending $m:A\longrightarrow B$ to the diagonal composite in the commuting square
\[\xymatrix@R=4mm{
FA
\ar[r]^-{\Phi_A}
\ar[d]_-{Fm}
\ar[dr]|-{[\Phi]m}
&
GA
\ar[d]^-{Gm}
\\%r1c2
FB
\ar[r]_-{\Phi_B}
&
GB
}\]
Write $T:\calC\longrightarrow \calC$ for the comonad of the adjunction $\textup{free}:\calC\rightleftarrows\mathsf{Vect}:\textup{forget}$. Then there is an (augmented) simplicial endofunctor, $\mathfrak{b}\in s\calC^\calC$, derived from the unit and counit of the adjunction:
\[\vcenter{
\def\scriptstyle{\scriptstyle}
\xymatrix@C=1.65cm@1{
{\ I\,}
&
\,T^1\,
\ar@{..>}[l]|(.65){\mathfrak{d}_{0,0}}
\ar[r]|(.65){\mathfrak{s}_{0,0}}
&
\,T^2\,
\ar@<-1ex>[l]|(.65){\mathfrak{d}_{0,1}}
\ar@<+1ex>[l]|(.65){\mathfrak{d}_{1,1}}
\ar@<+1ex>[r]|(.65){\mathfrak{s}_{0,1}}
\ar@<-1ex>[r]|(.65){\mathfrak{s}_{1,1}}
&
\,T^3\,
\ar[l]|(.65){\mathfrak{d}_{1,2}}
\ar@<-2ex>[l]|(.65){\mathfrak{d}_{0,2}}
\ar@<+2ex>[l]|(.65){\mathfrak{d}_{2,2}}
\ar[r]|(.65){\mathfrak{s}_{1,2}}
\ar@<+2ex>[r]|(.65){\mathfrak{s}_{0,2}}
\ar@<-2ex>[r]|(.65){\mathfrak{s}_{2,2}}
&
\,T^4\,\makebox[0cm][l]{\,$\cdots $}
\ar@<-3ex>[l]|(.65){\mathfrak{d}_{0,3}}
\ar@<-1ex>[l]|(.65){\mathfrak{d}_{1,3}}
\ar@<+1ex>[l]|(.65){\mathfrak{d}_{2,3}}
\ar@<+3ex>[l]|(.65){\mathfrak{d}_{3,3}}
}}\]
The \emph{simplicial bar construction} is the cofibrant replacement functor $(b,\varepsilonilon)$ on $s\calC$ which is the diagonal of the bisimplicial object obtained by application of $\mathfrak{b}$ levelwise. That is, for $X\in s\calC$, $bX$ is the simplicial object with $(bX)_q:=T^{q+1}X_q$, and with
\[\trip{d}{i,q}{bX}:=[\mathfrak{d}_{i,q}]\trip{d}{i,q}{X}.\]
The augmentation $\varepsilonilon:b\longrightarrow I$ is defined on level $q$ by
\[\varepsilonilon_q=\mathfrak{d}_{0,0}\mathfrak{d}_{0,1}\cdots \mathfrak{d}_{0,q}:T^{q+1}\longrightarrow I\]
We can now construct the simplicial homotopy needed for the twisting lemma, \ref{DsDt=Dt+s}.
\begin{prop}\label{IteratedBarConstructionHomotopy}
The natural transformations $\varepsilonilon_b$ and $b\varepsilonilon$ from $b^2:s\mathcal{A}\longrightarrow s\mathcal{A}$ to $b:s\mathcal{A}\longrightarrow s\mathcal{A}$ are naturally simplicially homotopic.
\end{prop}
\begin{proof}
Write $K=b^{2}X$ and $L=bX$ for the source and target of these maps respectively. Noting the formulae
\[[\mathfrak{d}_{iq}]^2= [\mathfrak{d}_{q+i,2q}\relax\mathfrak{d}_{i,2q+1}]\ \textup{and}\ [\mathfrak{s}_{iq}]^2= [\mathfrak{s}_{q+i+2,2q+2}\relax\mathfrak{s}_{i,2q+1}],\]
we can describe the simplicial structure maps in $K$ and $L$ as follows:
\begin{alignat*}{2}
\trip{d}{iq}{L}&=[\mathfrak{d}_{iq}]\trip{d}{iq}{X}\\
\trip{s}{iq}{L}&=[\mathfrak{s}_{iq}]\trip{s}{iq}{X}\\
\trip{d}{iq}{K}&=[\mathfrak{d}_{q+i,2q}\relax\mathfrak{d}_{i,2q+1}]\trip{d}{iq}{X}\\
\trip{s}{iq}{K}&=[\mathfrak{s}_{q+i+2,2q+2}\relax\mathfrak{s}_{i,2q+1}]\trip{s}{iq}{X}
\end{alignat*}
We can now state an explicit simplicial homotopy between the two maps of interest. Using precisely the notation of \cite[\S5]{MaySimpObj.pdf}, we define $\trip{h}{jq}{}:K_q\longrightarrow L_{q+1}$, for $0\leq j\leq q$, by the formula
\[\trip{h}{jq}{}:=[\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}]\trip{s}{jq}{X}.\]
We first check that these maps satisfy the defining identities for the notion of simplicial homotopy, numbered (1)-(5) as in \cite[\S5]{MaySimpObj.pdf}. Each identity can be checked in two parts (a)-(b):
{\renewcommand{\relax}{\relax}
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item We must check that $\trip{d}{i,q+1}{L}\relax \trip{h}{j,q}{}=\trip{h}{j-1,q-1}{}\relax \trip{d}{i,q}{K}$ whenever $0\leq i<j\leq q$, i.e.:
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item[({\makebox[.51em][c]{a}})] $\trip{d}{i,q+1}{X}\relax \trip{s}{j,q}{X}=\trip{s}{j-1,q-1}{X}\relax \trip{d}{i,q}{X}$,\textup{ and}
\item[({\makebox[.51em][c]{b}})]
$\mathfrak{d}_{i,q+1}\relax
\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}=
\mathfrak{d}_{j,q+1}\relax\cdots \relax\mathfrak{d}_{j,2q-1}\relax
\mathfrak{d}_{q+i,2q}\relax\mathfrak{d}_{i,2q+1}$.
\end{enumerate}
\item We must check that $\trip{d}{j+1,q+1}{L}\relax \trip{h}{j,q}{}=\trip{d}{j+1,q+1}{L}\relax \trip{h}{j+1,q}{}$ whenever $0\leq j\leq q-1$, i.e.:
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item[({\makebox[.51em][c]{a}})] $\trip{d}{j+1,q+1}{X}\relax \trip{s}{j,q}{X}=\trip{d}{j+1,q+1}{X}\relax \trip{s}{j+1,q}{X}$,\textup{ and}
\item[({\makebox[.51em][c]{b}})]
$\mathfrak{d}_{j+1,q+1}\relax
\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}=
\mathfrak{d}_{j+1,q+1}\relax
\mathfrak{d}_{j+2,q+2}\relax\cdots \relax\mathfrak{d}_{j+2,2q+1}$.
\end{enumerate}
\item We must check that $\trip{d}{i,q+1}{L}\relax \trip{h}{j,q}{}=\trip{h}{j,q-1}{}\relax \trip{d}{i-1,q}{K}$ whenever $0\leq j<i-1\leq q$, i.e.:
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item[({\makebox[.51em][c]{a}})] $\trip{d}{i,q+1}{X}\relax \trip{s}{j,q}{X}=\trip{s}{j,q-1}{X}\relax \trip{d}{i-1,q}{X}$,\textup{ and}
\item[({\makebox[.51em][c]{b}})]
$\mathfrak{d}_{i,q+1}\relax
\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}=
\mathfrak{d}_{j+1,q+1}\relax\cdots \relax\mathfrak{d}_{j+1,2q-1}\relax
\mathfrak{d}_{q+i-1,2q}\relax\mathfrak{d}_{i-1,2q+1}$.
\end{enumerate}
\item We must check that $\trip{s}{i,q+1}{L}\relax \trip{h}{j,q}{}=\trip{h}{j+1,q+1}{}\relax \trip{s}{i,q}{K}$ whenever $0\leq i\leq j\leq q$, i.e.:
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item[({\makebox[.51em][c]{a}})] $\trip{s}{i,q+1}{X}\relax \trip{s}{j,q}{X}=\trip{s}{j+1,q+1}{X}\relax \trip{s}{i,q}{X}$,\textup{ and}
\item[({\makebox[.51em][c]{b}})]
$\mathfrak{s}_{i,q+1}\relax
\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}=
\mathfrak{d}_{j+2,q+3}\relax\cdots \relax\mathfrak{d}_{j+2,2q+3}\relax
\mathfrak{s}_{q+i+2,2q+2}\relax\mathfrak{s}_{i,2q+1}$.
\end{enumerate}
\item We must check that $\trip{s}{i,q+1}{L}\relax \trip{h}{j,q}{}=\trip{h}{j,q+1}{}\relax \trip{s}{i-1,q}{K}$ whenever $0\leq j<i\leq q+1$, i.e.:
\begin{enumerate}\squishlist
\setlength{\parindent}{.25in}
\item[({\makebox[.51em][c]{a}})] $\trip{s}{i,q+1}{X}\relax \trip{s}{j,q}{X}=\trip{s}{j,q+1}{X}\relax \trip{s}{i-1,q}{X}$,\textup{ and}
\item[({\makebox[.51em][c]{b}})]
$\mathfrak{s}_{i,q+1}\relax
\mathfrak{d}_{j+1,q+2}\relax\cdots \relax\mathfrak{d}_{j+1,2q+1}=
\mathfrak{d}_{j+1,q+3}\relax\cdots \relax\mathfrak{d}_{j+1,2q+3}\relax
\mathfrak{s}_{q+i+1,2q+2}\relax\mathfrak{s}_{i-1,2q+1}$.
\end{enumerate}
\end{enumerate}
\noindent Each of these equations follows from the simplicial identities, proving that the $h_{jq}$ form a homotopy.
Finally, we check that this homotopy is indeed a homotopy between the two maps of interest:
\begin{alignat*}{2}
\trip{d}{0,q+1}{L}\trip{h}{0,q}{}
&=
[\mathfrak{d}_{0,q+1}\mathfrak{d}_{1,q+2}\relax\cdots \relax\mathfrak{d}_{1,2q+1}](\trip{d}{0,q+1}{X}\trip{s}{0q}{X})
\\
&=
[\mathfrak{d}_{0,q+1}\mathfrak{d}_{0,q+2}\relax\cdots \relax\mathfrak{d}_{0,2q+1}]\trip{\textup{id}}{X_q}{}
\end{alignat*}
is the action of $\varepsilonilon_{(bX)}$ in level $q$, and similarly,
\begin{alignat*}{2}
\trip{d}{q+1,q+1}{L}\trip{h}{q,q}{}
&=
[\mathfrak{d}_{q+1,q+1}\mathfrak{d}_{q+1,q+2}\relax\cdots \relax\mathfrak{d}_{q+1,2q+1}](\trip{d}{q+1,q+1}{X}\trip{s}{qq}{X})
\\
&=
[\mathfrak{d}_{q+1,q+1}\mathfrak{d}_{q+1,q+2}\relax\cdots \relax\mathfrak{d}_{q+1,2q+1}]\trip{\textup{id}}{X_q}{}
\end{alignat*}
is the action of $b\varepsilonilon_{X}$ in level $q$.
}
\end{proof}
\printbibliography
\end{document} |
\begin{document}
\hypersetup{bookmarksdepth=-1}
\title{On Structural Parameterizations of Hitting Set: \ Hitting Paths in Graphs Using 2-SAT hanks{Supported by NWO Veni grant ``Frontiers in Parameterized Preprocessing'' and NWO Gravity grant ``Networks''.}
\hypersetup{bookmarksdepth=2}
\begin{abstract}
\textsc{Hitting Set}\xspace is a classic problem in combinatorial optimization. Its input consists of a set system~$\ensuremath{\mathcal{F}}\xspace$ over a finite universe~$U$ and an integer~$t$; the question is whether there is a set of~$t$ elements that intersects every set in~$\ensuremath{\mathcal{F}}\xspace$. The \textsc{Hitting Set}\xspace problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of \textsc{Hitting Set}\xspace under various structural parameterizations of the input. Our starting point is the folklore result that \textsc{Hitting Set}\xspace is polynomial-time solvable if there is a tree~$T$ on vertex set~$U$ such that the sets in~$\ensuremath{\mathcal{F}}\xspace$ induce connected subtrees of~$T$. We consider the case that there is a treelike graph with vertex set~$U$ such that the sets in~$\ensuremath{\mathcal{F}}\xspace$ induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph~$G$ with cyclomatic number~$k$, a collection~$\ensuremath{\mathcal{P}}\xspace$ of simple paths in~$G$, and an integer~$t$, determines in time~$2^{5k} (|G| +|\ensuremath{\mathcal{P}}\xspace|)^{{\mathcal{O}}(1)}$ whether there is a vertex set of size~$t$ that hits all paths in~$\ensuremath{\mathcal{P}}\xspace$. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results.
\end{abstract}
\section{Introduction}
\textsc{Hitting Set}\xspace is a classic problem in combinatorial optimization that asks, given a set system~$\ensuremath{\mathcal{F}}\xspace$ over a finite universe~$U$, and an integer~$t$, whether there is a set of~$t$ elements that intersects every set in~$\ensuremath{\mathcal{F}}\xspace$. It was one of the first problems to be identified as NP-complete~\cite{Karp72}. Parameterized complexity theory is a refined view of computational complexity that aims to attack NP-hard problems by algorithms whose running time is exponential in a problem-specific \emph{parameter value}, but polynomial in terms of the overall input size. The standard parameterization of \textsc{Hitting Set}\xspace by the size of the desired solution is unlikely to admit such a fixed-parameter tractable algorithm, as it is W[2]-complete~\cite{DowneyF13}. The goal of this paper is to consider other parameterizations of \textsc{Hitting Set}\xspace, with the aim of obtaining FPT algorithms. Our starting point is the folklore result that \textsc{Hitting Set}\xspace is polynomial-time solvable when there is a tree~$T$ on vertex set~$U$ such that all sets~$S \in \ensuremath{\mathcal{F}}\xspace$ induce connected subtrees of~$T$. The \textsc{Hitting Set}\xspace problem on such an instance can be solved by a greedy strategy (Section~\ref{section:prelims}). Motivated by this result, we consider whether \textsc{Hitting Set}\xspace can be solved efficiently if there is a graph~$G$ that is close to being a tree, such that all~$S \in \ensuremath{\mathcal{F}}\xspace$ induce connected subgraphs of~$G$. We therefore parameterize the problem by measures of closeness of~$G$ to a tree, which forms an example of parameterizing by distance from triviality~\cite{Niedermeier10}.
\textbf{Our results.}
One way to measure how close a connected graph is to a tree is to consider its \emph{cyclomatic number}~$k := m - (n - 1)$. This is the size of a minimum feedback edge set of the graph, i.e., of a minimum set of edges whose removal breaks all cycles in the graph. As a tree has cyclomatic number zero, it is natural to ask if \textsc{Hitting Set}\xspace can be solved efficiently if the set system~$\ensuremath{\mathcal{F}}\xspace$ can be represented by a graph~$G$ on vertex set~$U$ having small cyclomatic number, such that every set~$S \in \ensuremath{\mathcal{F}}\xspace$ induces a connected subgraph of~$G$. To decouple the difficulty of finding a representation of~$\ensuremath{\mathcal{F}}\xspace$ in this form from the problem of exploiting this representation to solve \textsc{Hitting Set}\xspace, we consider the situation when such a representation is given. In this setting, the problem can be phrased more naturally in graph-theoretical terms: given a graph~$G$ of cyclomatic number~$k$, a collection~$\ensuremath{\mathcal{S}}\xspace$ of connected subgraphs of~$G$, and an integer~$t$, is there a vertex set of size~$t$ that hits all subgraphs in~$\ensuremath{\mathcal{S}}\xspace$?
\begin{table}[t]
\caption{Parameterized complexity overview for hitting subgraphs by the minimum number of vertices, parameterized by measures of structure of the host graph.
}
\centering
{
\begin{tabular}{@{}llllll@{}}
\toprule
parameter & \multicolumn{5}{c}{complexity for type of subgraphs to be hit} \\
\cmidrule{2-6}
& \multicolumn{2}{l}{path} & \phantom{abc} & \multicolumn{2}{l}{3-leaf subtree} \\
\midrule
cyclomatic number & FPT, no~$k^{{\mathcal{O}}(1)}$ kernel & thm.~\ref{theorem:pathsingraph:fpt} & & W[1]-hard & thm.~\ref{theorem:hitclaws:whard} \\
feedback vertex number & para-NP-complete & thm.~\ref{theorem:hitpaths:fvs:npc} & & para-NP-complete & thm.~\ref{theorem:hitpaths:fvs:npc} \\
\bottomrule
\end{tabular}
\label{table:summary}
}
\end{table}
Our first result for the parameterization by cyclomatic number is a hardness proof showing this problem to be W[1]-hard. In fact, we prove W[1]-hardness even when all subgraphs in~$\ensuremath{\mathcal{S}}\xspace$ are trees with at most three leaves. To establish this hardness result we prove that a variation of 3-SAT in multiple valued logic (see Section~\ref{section:prelims}) is W[1]-hard, which may be of independent interest. Concretely, we show the following. Given a set of~$n$ variables~$x_1, \ldots, x_n$ that can take values from~$1$ to~$N$, and a formula that is a conjunction of clauses of size at most three, where each literal is of the form~$x_i \geq c$ or~$x_i \leq c$ for~$c \in [N]$, it is W[1]-hard parameterized by~$n$ to determine whether there is an assignment to the variables satisfying all clauses. This parameterized logic problem reduces to the discussed structural parameterization of \textsc{Hitting Set}\xspace in a natural way.
The hardness result motivates us to place further restrictions on the problem in search of fixed-parameter tractable cases. We consider the situation of hitting a set~$\ensuremath{\mathcal{P}}\xspace$ of \emph{simple paths} in a graph~$G$ of cyclomatic number~$k$. This corresponds to \textsc{Hitting Set}\xspace instances where there is a graph~$G$ on~$U$ such that for all sets~$S$ in~$\ensuremath{\mathcal{F}}\xspace$, there is a \emph{simple path} in~$G$ on vertex set~$S$. We prove that this problem is fixed-parameter tractable and can be solved in time~$2^{5k} (|G| + |\ensuremath{\mathcal{P}}\xspace|)^{{\mathcal{O}}(1)}$, which is the main algorithmic result in this paper. The algorithm is based on a reduction to~$2^{5k}$ instances of the 2-SAT problem in multiple valued logic, which is known to be polynomial-time solvable~\cite{BejarHM01,Manya00}. The reduction exploits the fact that in tree-like parts of the graph, the local structure of minimum hitting sets can be determined by greedily computed optimal hitting sets for subtrees of a tree. After branching in~$2^{5k}$ directions to determine the form of a solution, the interaction between such canonical subsolutions is then encoded in a 2-SAT formula in multiple valued logic, which can be evaluated efficiently.
There are several other parameters that measure the closeness of a graph to a tree, such as the \emph{feedback vertex number} and \emph{treewidth} (cf.~\cite{FellowsJR13}). As these parameters have smaller values than the cyclomatic number, one might hope to extend the FPT result mentioned above to these parameters. However, we show that this is impossible, unless P=NP. In particular, we prove that the problem of hitting simple paths in a graph of feedback vertex number~$2$ is NP-complete, showing the parameterizations by feedback vertex number and treewidth to be para-NP-complete. Table~\ref{table:summary} gives an overview of the results in this paper.
\textbf{Related work.} Several authors~\cite{CoppersmithV85,Fiala01,UhlmannW13} have considered problems parameterized by cyclomatic number; this is also known as parameterizing by feedback edge set.
In parameterized complexity, \textsc{Hitting Set}\xspace is often studied when the sets to be hit have constant size. In this setting, several FPT algorithms and kernelizations bounds are known \cite{Abu-Khzam10,DellM14,Wahlstrom07}. The weighted \textsc{Set Cover}\xspace problem, which is dual to \textsc{Hitting Set}\xspace, has been analyzed for tree-like set systems by Guo and Niedermeier~\cite{GuoN06}. Recently, Lu et al.~\cite{LuLTLX14} considered \textsc{Set Cover}\xspace and \textsc{Hitting Set}\xspace for set systems representable as subtrees of a (restricted type of) tree, distinguishing polynomial-time and NP-complete cases.
\textbf{Organization.} Preliminaries are given in~\ref{section:prelims}. The FPT algorithm for hitting paths is developed in Section~\ref{section:fpt}. Section~\ref{section:hardness} contains the hardness proofs.
\section{Preliminaries} \label{section:prelims}
\textbf{Parameterized complexity.}
A parameterized problem is a set~$Q \subseteq \ensuremath{\mathcal{S}}\xspaceigma^* \times \mathbb{N}$, where~$\ensuremath{\mathcal{S}}\xspaceigma$ is a fixed finite alphabet. The second component of a tuple~$(x,k) \in \ensuremath{\mathcal{S}}\xspaceigma^* \times \mathbb{N}$ is the \emph{parameter}. A parameterized problem is (strongly uniformly) \emph{fixed-parameter tractable} if there is an algorithm that decides every input~$(x,k)$ in time~$f(k)|x|^{{\mathcal{O}}(1)}$. Evidence that a problem is not fixed-parameter tractable is given by proving that it is W[1]-hard. We refer to one of the textbooks~\cite{DowneyF13,FlumG06} for more background.
\textbf{Graphs.}
All graphs we consider are simple, undirected and finite. A graph~$G$ consists of a set of vertices~$V(G)$ and edges~$E(G)$. Notation not defined here is standard. For a set of vertices~$S$ we denote by~$N_G(S)$ the set~$\bigcup _{v \in S} N_G(v) \setminus S$. A path in a graph~$G$ is a sequence of distinct vertices such that successive vertices are connected by an edge. The first and last vertices on the path are its endpoints, the remaining vertices are its interior vertices. Given a graph~$G$ and a vertex subset~$S \subseteq V(G)$, the operation of \emph{identifying} the vertices of~$S$ into a new vertex~$z$ is performed as follows: delete the vertices in~$S$ and their incident edges, and insert a new vertex~$z$ that is adjacent to~$N_G(S)$, i.e., to all remaining vertices of~$G$ that were adjacent to at least one member of~$S$.
\begin{proposition} \label{proposition:degtwo}
Let~$G$ be a connected graph of minimum degree at least two with cyclomatic number~$k$. The number of vertices in~$G$ with degree at least three is bounded by~$2k-2$.
\end{proposition}
\begin{proof}
Denote by~$n_2$ and~$n_{\geq 3}$ the number of vertices in~$G$ with degree two and at least three, respectively. Let~$n$ and~$m$ be the total number of vertices and edges in~$G$, and let~$d(v)$ denote the degree of a vertex~$v$. Since~$k = m - (n-1)$ we have~$m = k + (n_2 + n_{\geq 3} - 1)$. The value of~$m$ can also be obtained as half the degree sum of~$G$:
$$m = \frac{1}{2} \sum_{v \in V(G)} d(v) \geq n_2 + \frac{3n_{\geq 3}}{2}.$$
Hence we find:
$$m = k + (n_2 + n_{\geq 3} - 1) \geq n_2 + \frac{3n_{\geq 3}}{2},$$ from which we obtain~$n_{\geq 3} \leq 2k-2$ by subtracting~$n_2 + n_{\geq 3}$ on both sides and multiplying by two.
\qed
\end{proof}
\begin{proposition} \label{proposition:numcomponents}
Let~$G$ be a connected graph of minimum degree at least two with cyclomatic number~$k$ and let~$S$ be the set of vertices of degree at least three. If~$S \neq \emptyset$ then the number of connected components of~$G - S$ is at most~$k + |S| - 1$.
\end{proposition}
\begin{proof}
As~$S$ contains all vertices of degree at least three, every connected component~$C$ of~$G - S$ is a path. Since~$G$ has minimum degree at least two, every endpoint of such a path has a neighbor in~$S$. Hence for every connected component~$C$ of~$G - S$ there are exactly two edges between~$C$ and~$S$. Consider the multigraph~$H$ on vertex set~$S$ defined as follows. For every component~$C$ of~$G - S$, consider the two edges between~$C$ and~$S$ and let~$x,y$ be their endpoints in~$S$. We add an edge between~$x$ and~$y$ to~$H$; if~$x=y$ this becomes a self-loop, and there is the chance of creating parallel edges. Since~$H$ is a connected topological minor of~$G$ it is easy to see that the cyclomatic number~$k'$ of~$H$ does not exceed that of~$G$. Since~$|E(H)| = (|V(H)| - 1) + k'$, we find that~$|E(H)| = |S| - 1 + k' \leq k + |S| - 1$. Since connected components of~$G - S$ are in 1-to-1 correspondence with edges of~$H$, this completes the proof.
\qed
\end{proof}
\textbf{Hitting set.}
A set system~$\ensuremath{\mathcal{F}}\xspace \subseteq 2^U$ can be viewed as a hypergraph whose vertices are~$U$ and whose hyperedges are formed by the sets in~$\ensuremath{\mathcal{F}}\xspace$. A set system~$\ensuremath{\mathcal{F}}\xspace$ is a \emph{hypertree} if there is a tree~$T$ on vertex set~$U$ such that every set in~$\ensuremath{\mathcal{F}}\xspace$ induces a subtree of~$T$. Testing whether a set system is a hypertree, and constructing a tree representation if this is the case, can be done in polynomial time~\cite{Trick87}.
We frequently use the fact that a minimum hitting set for a hypertree can be found in polynomial time (cf.~\cite[\ensuremath{\mathcal{S}}\xspaceSign 2]{GuoN06} for a view from a dual perspective). When a tree representation is known, a greedy algorithm can be used to find a minimum hitting set. If we root the tree at a leaf and find a vertex~$v$ of maximum depth for which there is a set~$S \in \ensuremath{\mathcal{F}}\xspace$ whose members all belong to the subtree rooted at~$v$, then it is easy to show there is a minimum hitting set containing~$v$. Consequently, we may add~$v$ to the solution under construction, remove all sets hit by~$v$, and remove all elements in the subtree rooted at~$v$ from the universe.
This idea can be extended for the following setting. Suppose we have a graph~$G$ that is isomorphic to a simple cycle and a set~$\ensuremath{\mathcal{P}}\xspace$ of paths in~$G$. To find a minimum vertex set that hits all the paths in~$\ensuremath{\mathcal{P}}\xspace$, we try for each vertex~$v$ of~$G$ whether there is a minimum solution containing it. After removing~$v$ and the paths hit by~$v$, the remaining structure is a hypertree since the cycle breaks open when removing~$v$. The minimum over all choices of~$v$ gives an optimal hitting set. We will use this in our FPT algorithm to deal with a corner case.
\textbf{Multiple valued logic.}
The hitting set problems we are interested in turn out to be related to variations of the \textsc{Satisfiability} problem that have been studied in the field of multiple valued logic. In a multiple valued logic, variables can take on more values than just~$0$ and~$1$: there is a \emph{truth value set} containing the possible values. For our application, the truth value set is totally ordered; it is a range of integers~$[N] = \{1, \ldots, N\}$. A \emph{regular sign} is a constraint of the form~$\geq j$ or~$\leq j$ for~$j \in [N]$. By constraining variables with regular signs, resulting in (generalized) literals of the form~$x_i \geq j$ or~$x_i \leq j$, and combining such literals with the usual logical connectives, one creates totally ordered regular signed formulas. As expected, the satisfiability problem for such formulas is to determine whether every variable can be assigned a value in the range~$[N]$ such that the formula is satisfied. We shall be interested in the case of CNF formulas with clauses having at most two (2-SAT) or at most three (3-SAT) literals.
\defparproblem{\textsc{$n$-Totally Ordered Regular Signed 3-SAT}\xspace}
{A totally ordered regular signed 3-CNF formula with~$n$ variables and truth value set~$[N]$.}
{$n$.}
{Is the formula satisfiable?}
For brevity we sometimes refer to this problem as \textsc{$n$-TORS 3-SAT}\xspace. We also consider \ensuremath{\mathcal{T}}\xspaceORSTwoSat, where clauses have at most two literals, which is polynomial-time solvable~\cite{Manya00}. In particular, \ensuremath{\mathcal{T}}\xspaceORSTwoSat can be reduced to the 2-SAT problem in classical logic~\cite[\ensuremath{\mathcal{S}}\xspaceSign 3]{BejarHM01}, which is well-known to be solvable in linear time~\cite{AspvallPT79}. For completeness, we sketch the reduction in Appendix~\ref{app:twosat}.
\section{Algorithms} \label{section:fpt}
The goal of this section is to develop an FPT algorithm for the following parameterized problem.
\defparproblem{\textsc{Hitting Paths in a Graph}\xspace}
{An undirected simple graph~$G$ with cyclomatic number~$k$, an integer~$t$, and a set~$\ensuremath{\mathcal{P}}\xspace$ of simple paths in~$G$.}
{$k$.}
{Is there a set~$X \subseteq V(G)$ of size at most~$t$ that hits all paths in~$\ensuremath{\mathcal{P}}\xspace$?}
The algorithm consists of two reductions. An instance of \textsc{Hitting Paths in a Graph}\xspace is reduced to a hitting set problem on a more structured graph, called a flower. An instance with such a flower structure can be reduced to a polynomial-time solvable 2-SAT problem in multiple valued logic. This section is structured as follows. We first describe the flower structure and the reduction to 2-SAT in Section~\ref{subsection:hit:flowers}. Afterward we show how to build an FPT algorithm from this ingredient, in Section~\ref{subsection:hit:paths}.
\subsection{Hitting Paths in Flowers} \label{subsection:hit:flowers}
The key notion in this section is that of a \emph{flower graph}, which is a graph~$G$ with a distinguished vertex~$z$ called the \emph{core} such that all connected components of~$G-\{z\}$ are paths~$R_1, \ldots, R_n$ of which no interior vertex is adjacent to~$z$. These paths are called \emph{petals} of the flower. When working with flower graphs we will assume an arbitrary but fixed ordering of the petals as~$R_1, \ldots, R_n$, as well as an orientation of each petal~$R_i$ as consisting of vertices~$r_{i,1}, \ldots, r_{i,|V(R_i)|}$. For ease of discussion we will interpret each petal to be laid out from left to right in order of increasing indices. We will give an FPT branching algorithm that reduces \textsc{Hitting Paths in a Graph}\xspace to solving several instances of the following more restricted problem.
\defproblem{\textsc{Hitting Paths in a Flower with Budgets}\xspace}
{A flower graph~$G$ with core~$z$ and petals~$R_1, \ldots, R_n$, a set of simple paths~$\ensuremath{\mathcal{P}}\xspace = \{P_1, \ldots, P_m\}$ in~$G$, and a budget function~$b \colon [n] \to \mathbb{N}_{\geq 1}$.}
{Is there a set~$X \subseteq V(G) \setminus \{z\}$ that hits all paths in~$\ensuremath{\mathcal{P}}\xspace$ such that~$|X \cap V(R_i)| = b(i)$ for all~$i \in [n]$?}
We show that \textsc{Hitting Paths in a Flower with Budgets}\xspace can be solved in polynomial time. The following notion will be instrumental to analyze the structure of solutions to this problem.
\begin{definition} \label{definition:canonical}
Let~$R_i$ be a petal of an instance~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ of \textsc{Hitting Paths in a Flower with Budgets}\xspace and let~$1 \leq \ell \leq |V(R_i)|$. The \emph{canonical $\ell$-th solution} for petal~$R_i$ is defined by the following process.
\begin{enumerate}
\item If there is a path in~$\ensuremath{\mathcal{P}}\xspace$ that is contained entirely within~$\{r_{i,1}, \ldots, r_{i,\ell-1}\}$, then define the canonical $\ell$-th solution to be NIL.\label{step:earlyout}
\item Otherwise, initialize~$X_{i,\ell}$ as the singleton set containing~$r_{i,\ell}$.
\begin{enumerate}
\item While there is a path in~$\ensuremath{\mathcal{P}}\xspace$ that is contained entirely within~$R_i$ and is not intersected by~$X_{i,\ell}$, consider a path among this set that minimizes the index~$j'$ of its right endpoint and add~$r_{i,j'}$ to~$X_{i,\ell}$.\label{step:hitpath}
\item While~$|X_{i,\ell}| < b(i)$ and~$Y := \{r_{i,\ell}, \ldots, r_{i,|V(R_i)|}\} \setminus X_{i,\ell} \neq \emptyset$, add the highest-indexed vertex from~$Y$ to~$X_{i,\ell}$. (Recall that~$b(i)$ is the budget for petal~$R_i$.)\label{step:fillsize}
\item If~$|X_{i,\ell}| = b(i)$, the canonical $\ell$-th solution is~$X_{i,\ell}$. If~$|X_{i,\ell}| \neq b(i)$, define the canonical $\ell$-th solution to be NIL.\label{step:toolarge}
\end{enumerate}
\end{enumerate}
A set~$X_i \subseteq V(R_i)$ is a canonical solution for petal~$R_i$ if there is an integer~$\ell$ for which~$X_i$ is the canonical $\ell$-th solution for~$R_i$. A canonical solution is well defined if it is not NIL. A solution~$X$ to the instance~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ is \emph{globally canonical} if~$X \cap R_i$ is a well-defined canonical solution for all~$i$.
\end{definition}
Figure~\ref{fig:petals} illustrates these concepts. For a set~$X_i \subseteq V(R_i)$ we will denote by~$\max(X_i)$ the highest index of any vertex in~$X_i$, i.e., the index of the rightmost vertex of~$X_i$. Similarly, we denote by~$\min(X_i)$ the index of the leftmost vertex of~$X_i$. The following observations about the procedure will be useful.
\begin{figure}
\caption{(\ref{fig:petal0}
\label{fig:petal0}
\label{fig:petal1}
\label{fig:petal2}
\label{fig:petals}
\end{figure}
\begin{observation} \label{observation:leftmost}
If~$X_{i,\ell}$ is a well-defined canonical solution, then~$\min(X_{i,\ell}) = \ell$.
\end{observation}
\begin{observation} \label{observation:partition}
Let~$X_{i,\ell}$ result from Definition~\ref{definition:canonical}, and assume that Step~\ref{step:earlyout} does not apply and that Step~\ref{step:fillsize} is never triggered during the procedure. Partition the interval~$\{r_{i,\ell}, \ldots, r_{i,\max(X_{i,\ell})}\}$ into~$|X_{i,\ell}|$ maximal subpaths that each end at a vertex of~$X_{i,\ell}$ and contain no other vertices of~$X_{i,\ell}$. Then, for every such subpath~$R'$ except the singleton subpath~$\{r_{i,\ell}\}$, there is a path in~$\ensuremath{\mathcal{P}}\xspace$ contained entirely within~$R'$.
\end{observation}
The main strategy behind our reduction of \textsc{Hitting Paths in a Flower with Budgets}\xspace to \ensuremath{\mathcal{T}}\xspaceORSTwoSat will be as follows. We will show that, if a solution to the hitting set problem exists, then there is a globally canonical solution. Such a solution can be fully characterized by indicating, for each petal, the index of the canonical solution on the petal (i.e., the leftmost vertex of the petal that is in the solution). Hence finding a solution reduces to finding a choice of canonical solutions on the petals. It turns out that for every path~$P \in \ensuremath{\mathcal{P}}\xspace$, one can create a signed 2-clause on the variables controlling the choices on two petals, such that the path is hit by the selected solution if and only if the indices of the canonical subsolutions satisfy the 2-clause. This allows the hitting set problem to be modeled by \ensuremath{\mathcal{T}}\xspaceORSTwoSat. We now formalize these ideas. Let us first get a feeling for canonical solutions by proving the following lemma.
\begin{lemma} \label{lemma:contiguous}
Let~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ be an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace and let~$R_i$ be a petal. The indices for which~$R_i$ has a well-defined canonical solution form a contiguous set of integers.
\end{lemma}
\begin{proof}
Assume for a contradiction that there are~$\ell_1 < \ell_2 < \ell_3$ such that the canonical solutions for~$\ell_1$ and~$\ell_3$ are well-defined, but that for~$\ell_2$ is not. Let us consider why the canonical solution for~$\ell_2$ is not well-defined.
\begin{enumerate}
\item If Step~\ref{step:earlyout} applies for~$\ell_2$, then the path~$P \in \ensuremath{\mathcal{P}}\xspace$ that is contained entirely within~$\{r_{i,1}, \ldots, r_{i,\ell_2}\}$ also causes Step~\ref{step:earlyout} to apply for~$\ell_3 > \ell_2$, contradicting the assumption that there is a well-defined canonical solution for~$\ell_3$.
\item If~$X_{i,\ell_2}$ is too small in Step~\ref{step:toolarge}, then this implies that there are less than~$b(i)$ vertices in~$\{r_{\ell_2}, \ldots, r_{\ell_{|V(R_i)|}}\}$. But this contradicts the fact that~$X_{i,\ell_3}$ has~$b(i)$ vertices and is a subset of~$\{r_{i,\ell_3}, \ldots, r_{i,|V(R_i)|}\}$ for~$\ell_3 > \ell_2$.
\item If~$X_{i,\ell_2}$ is too large in Step~\ref{step:toolarge}, then its size exceeds~$|X_{i,\ell_1}| = b(i)$. Hence the precondition to Step~\ref{step:fillsize} never applied during the procedure for~$\ell_2$. Consider the partition of the interval~$\{r_{i,\ell_2}, \ldots, r_{i,\max (X_{i,\ell_2})}\}$ into~$|X_{i,\ell_2}|$ subpaths as described in Observation~\ref{observation:partition}. Every subpath in the partition contains exactly one vertex of~$X_{i,\ell_2}$, and all vertices of~$X_{i,\ell_2}$ are in one such subpath. Observe that~$r_{i,\ell_1} \in X_{i,\ell_1} \setminus X_{i,\ell_2}$, such that~$X_{i,\ell_1}$ contains at most~$b(i) - 1$ vertices in the interval~$r_{i,\ell_2}, \ldots, r_{i,|V(R_i)|}$. Since~$|X_{i,\ell_2}| > |X_{i,\ell_1}| = b(i)$, there are at least two subpaths in the partition from which~$X_{i,\ell_1}$ contains no vertex. Hence there is such a subpath, say~$R' := \{r_{i,p}, \ldots, r_{i,q}\}$, that is not the singleton path~$\{r_{i,\ell_2}\}$ and that contains no vertices of~$X_{i,\ell_1}$. Then, by Observation~\ref{observation:partition}, there is a target path~$P$ in~$\ensuremath{\mathcal{P}}\xspace$ that is entirely contained within~$R'$. But~$X_{i,\ell_1}$ contains no vertex of this path, showing that~$X_{i,\ell_1}$ does not intersect~$P$, which contradicts the fact that the while-loop of Step~\ref{step:hitpath} terminated when defining~$X_{i,\ell_1}$.
\end{enumerate}
As the cases are exhaustive, this concludes the proof.
\qed
\end{proof}
As the procedure of Definition~\ref{definition:canonical} can be implemented in polynomial time, the set of indices for which a petal has a canonical solution can be computed in polynomial time. We continue describing the structure of canonical solutions.
\begin{lemma} \label{lemma:latersolutionsreachfurther}
Let~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ be an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace and let~$R_i$ be a petal. If~$\ell_1 < \ell_2$, and the $\ell_1$-th and the~$\ell_2$-th canonical solutions are well-defined as~$X_{i,\ell_1}$ and~$X_{i,\ell_2}$, then~$\max(X_{i,\ell_1}) \leq \max(X_{i,\ell_2})$.
\end{lemma}
\begin{proof}
Assume that~$\max(X_{i,\ell_1}) > \max(X_{i,\ell_2})$. We aim to apply Observation~\ref{observation:partition} to derive a contradiction. Since both canonical solutions are well defined, Step~\ref{step:earlyout} does not apply to~$\ell_2$. As our assumption implies that the rightmost vertex of~$R_i$ is not in~$X_{i,\ell_2}$, it follows that Step~\ref{step:fillsize} never applied during the procedure for~$\ell_2$. Since at least one vertex of~$X_{i,\ell_1}$ lies right of~$\max(X_{i,\ell_2})$, and~$X_{i,\ell_1}$ contains vertex~$r_{i,\ell_1}$ that lies left of~$r_{i,\ell_2}$, the partition of~$\{r_{i,\ell_2}, \ldots, r_{i,\max(X_{i,\ell_2})}\}$ into~$b(i)$ subpaths described by Observation~\ref{observation:partition} contains at least two subpaths from which~$X_{i,\ell_1}$ contains no vertex. Hence there is such a subpath~$R'$ that is not intersected by~$X_{i,\ell_1}$ for which there is a target path~$P \in \ensuremath{\mathcal{P}}\xspace$ contained entirely within~$R'$. This contradicts the fact that~$X_{i,\ell_1}$ hits all paths contained entirely within~$R_i$ by Step~\ref{step:hitpath} of Definition~\ref{definition:canonical}.
\qed
\end{proof}
We now establish that the hitting set problem has a globally canonical solution, if it has a solution at all. The proof exploits the fact that, after selecting the leftmost vertex of a petal to be used in the hitting set, removing it from the graph, and removing the paths hit by this vertex from the graph, the remainder of the petal turns into a pendant path that connects to the rest of the graph at vertex~$z$. The hitting set problem has a greedy solution within this resulting path, which reflects the structure of the canonical solution. Formalizing this line of reasoning is tedious but straight-forward.
\begin{lemma} \label{lemma:existscanonical}
Let~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ be an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace having petals~$R_1, \ldots, R_n$. If the instance has a solution~$X'$, then it has a globally canonical solution~$X$.
\end{lemma}
\begin{proof}
Proof by induction on the number~$k$ of petals for which~$X' \cap R_i$ is not a canonical solution. When~$k=0$ the claim is trivial, so assume~$k > 0$ and let~$R_i$ be a petal such that~$X' \cap V(R_i)$ is not a canonical solution. Since~$X'$ is a solution, by the definition of \textsc{Hitting Paths in a Flower with Budgets}\xspace we have~$|X' \cap V(R_i)| = b(i)$. Let~$\ell$ be the index of the leftmost vertex from~$X'$ on~$R_i$. Since~$b(i) \geq 1$, such a vertex exists. Let~$X'_{i,\ell} := X' \cap V(R_i)$.
\begin{claim}
The $\ell$-th canonical solution for~$R_i$ is well defined.
\end{claim}
\begin{proof}
Consider the set~$X_{i,\ell}$ resulting from the process of Definition~\ref{definition:canonical} and assume for a contradiction that the process defines the canonical $\ell$-th solution to be NIL. There are two cases that yield NIL; we treat them consecutively.
\begin{enumerate}
\item If the canonical solution is NIL because there is a path~$P$ in~$\ensuremath{\mathcal{P}}\xspace$ that is contained entirely within~$\{r_{i,1}, \ldots, r_{i,\ell-1}\}$, then since~$\ell$ is the index of the leftmost vertex from~$X'$ on~$R_i$ we have~$X' \cap \{r_{i,1}, \ldots, r_{i,\ell-1}\} = \emptyset$. Consequently, the set~$X'$ does not intersect path~$P$, contradicting the assumption that~$X'$ is a solution.
\item Consider the case that the canonical solution is NIL because the size of~$X_{i,\ell}$ is not equal to~$b(i)$ in Step~\ref{step:toolarge}.
\begin{enumerate}
\item If~$|X_{i,\ell}| < b(i)$, then by the while-loop of Step~\ref{step:fillsize}, all vertices of~$\{r_{i,\ell}, \ldots, \linebreak[1] r_{i,|V(R_i)|}\}$ are in~$X_{i,\ell}$. Since~$X'_{i,\ell} = X' \cap V(R_i)$ and the leftmost vertex of~$X'_{i,\ell}$ on~$R_i$ is~$\ell$, the set~$X'_{i,\ell}$ cannot contain more vertices than~$X_{i,\ell}$. But then~$|X'_{i,\ell}| < b(i)$, showing that~$X'$ is not a solution.
\item If~$|X_{i,\ell}| > b(i)$, Step~\ref{step:fillsize} never applied during the procedure. Consider the partition of the interval~$\{r_{i,\ell}, \ldots, r_{i,\max(X_{i,\ell})}\}$ into~$|X_{i,\ell}|$ subpaths as described in Observation~\ref{observation:partition}. Since every subpath in the partition contains exactly one vertex of~$X_{i,\ell}$, and all vertices of~$X_{i,\ell}$ are in one such subpath, it follows from~$|X_{i,\ell}| > |X'_{i,\ell}| = b(i)$ that there is such a subpath, say~$R' := \{r_{i,p}, \ldots, r_{i,q}\}$, containing no vertices of~$X'_{i,\ell}$. Since~$X'_{i,\ell}$ and~$X_{i,\ell}$ both contain~$r_{i,\ell}$, we know~$R'$ is not~$\{r_{i,\ell}\}$. Then, by Observation~\ref{observation:partition}, there is a target path in~$\ensuremath{\mathcal{P}}\xspace$ that is entirely contained within~$R'$. But~$X'_{i,\ell}$ contains no vertex of this path, showing that~$X'$ is not a solution.
\end{enumerate}
\end{enumerate}
As we covered all cases that lead to the canonical solution being NIL, this concludes the proof.
\claimqed
\end{proof}
In the remainder, let~$X_{i,\ell}$ be the $\ell$-th canonical solution for~$R_i$, which is well defined by the previous claim.
\begin{claim}
$\max(X'_{i,\ell}) \leq \max(X_{i,\ell})$.
\end{claim}
\begin{proof}
Consider the process of Definition~\ref{definition:canonical}. If the loop of Step~\ref{step:fillsize} was executed at least once, then the rightmost vertex of~$R_i$ is in~$X_{i,\ell}$ and the claim is trivially true. So assume that this is not the case, and assume for a contradiction that~$\max(X'_{i,\ell}) > \max(X_{i,\ell})$. Consider the partition of the interval~$\{r_{i,\ell}, \ldots, r_{i,\max(X_{i,\ell})}\}$ as in Observation~\ref{observation:partition}. Since~$|X_{i,\ell}| = |X'_{i,\ell}| = b(i)$ and at least one vertex of~$X'_{i,\ell}$ does not lie in the interval~$\{r_{i,\ell}, \ldots, r_{i,\max(X_{i,\ell})}\}$ since~$\max(X'_{i,\ell}) > \max(X_{i,\ell})$, it follows that there is a subpath~$R'$ in the partition from which~$X'_{i,\ell}$ contains no vertices, and which therefore cannot be the subpath~$\{r_{i,\ell}\}$. As there is a path~$P \in \ensuremath{\mathcal{P}}\xspace$ that is entirely contained within~$R'$ by Observation~\ref{observation:partition}, the fact that~$X'_{i,\ell}$ and therefore~$X'$ contains no vertices from~$R'$ shows that~$X'$ is not a solution; contradiction.
\claimqed
\end{proof}
Using the previous claim we can finish the proof. Consider the set~$X := (X' \setminus X'_{i,\ell}) \cup X_{i,\ell}$, whose size equals that of~$X'$. We show that~$X$ is a valid solution to the instance. To see that, observe that the budget constraints are trivially satisfied since~$|X'_{i,\ell}| = |X_{i,\ell}|$. To see that all paths in~$\ensuremath{\mathcal{P}}\xspace$ are hit by~$X'$, consider a path~$P \in \ensuremath{\mathcal{P}}\xspace$. If~$P$ is hit by~$X' \setminus X'_{i,\ell}$ then it is also hit by~$X$. If~$P$ is contained entirely within~$R_i$, then by Step~\ref{step:hitpath} of Definition~\ref{definition:canonical} the path~$P$ is hit by~$X_{i,\ell}$ and thus by~$X$. If~$P$ is not contained entirely within~$R_i$ and is not hit by~$X' \setminus X'_{i,\ell}$, then it enters the petal at the leftmost or rightmost vertex of the petal and contains a prefix or suffix of the petal. (Here we use the structure of the flower graph: the interior vertices of petal~$R_i$ are not adjacent to any other vertex in the graph, only to their predecessor and successor on~$R_i$.) Since~$X'_{i,\ell}$ hits~$P$, it follows from the structure of the path in the petal that the leftmost or rightmost vertex of~$X'_{i,\ell}$ on~$R_i$ hits~$P$. But since the leftmost vertex of~$X'_{i,\ell}$ is~$r_{i,\ell}$, which is also in~$X_{i,\ell}$, and the rightmost vertex of~$X'_{i,\ell}$ does not have larger index than the rightmost vertex of~$X_{i,\ell}$ by the previous claim, it follows that~$X_{i,\ell}$ also hits~$P$. Hence all paths in~$\ensuremath{\mathcal{P}}\xspace$ are hit by~$X$, which is therefore a valid solution. Since the number of petals for which~$X$ does not contain a canonical solution is less than for~$X'$, by induction it follows that there is a solution for the instance whose intersection with every petal is a canonical solution.
\qed
\end{proof}
\begin{lemma} \label{lemma:createliteral}
Let~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ be an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace. There is a polynomial-time algorithm that, given a path~$P$ (not necessarily contained in~$\ensuremath{\mathcal{P}}\xspace$) which is a suffix or a prefix of a petal~$R_i$, either correctly determines that no well-defined canonical solution for~$R_i$ hits~$P$, or produces a literal of the form~$x_i \geq c$ or~$x_i \leq c$ for~$c \in \mathbb{N}_{\geq 1}$, such that the following holds.
\begin{enumerate}
\item If~$X$ is a globally canonical solution for the instance that hits~$P$ and contains the $\ell$-th canonical solution for petal~$R_i$, then the literal is satisfied by setting~$x_i = \ell$.
\item If~$x_i = \ell$ satisfies the literal and the $\ell$-th canonical solution~$X_{i,\ell}$ is well-defined, then~$P$ is hit by~$X_{i,\ell}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The definition of the literal depends on whether~$P$ is a suffix or a prefix of a petal. First consider the case that~$P$ is a prefix of petal~$R_i$. Observe that a well-defined canonical solution~$X_{i,\ell}$ for~$R_i$ hits~$P$ if and only if~$\ell \leq \max (P)$, since~$\max(P)$ marks the index of the end of the prefix of~$R_i$ used by~$P$, and~$\ell$ is the index of the leftmost vertex of the $\ell$-th canonical solution on the petal by Observation~\ref{observation:leftmost}. Hence for this case we obtain the literal~$x_i \leq \max (P)$.
Now consider the case that~$P$ is a suffix of petal~$R_i$. The situation is similar: a well-defined canonical solution~$X_{i,\ell}$ hits the suffix~$P$ if and only if~$\max(X_{i,\ell}) \geq \min(P)$, i.e., when the rightmost vertex of the canonical solution lies right of the starting point of the suffix~$P$. Since all canonical solutions for~$R_i$ can be computed in polynomial time, we can efficiently find the indices~$\ell$, if any, for which a canonical solution is well defined satisfying~$\max(X_{i,\ell}) \geq \min(P)$. The indices for which a canonical solution is well defined form a contiguous set by Lemma~\ref{lemma:contiguous}. If~$\max(X_{i,\ell}) \geq \min(P)$ holds for some~$\ell$, then for all~$\ell' \geq \ell$ for which a canonical solution is well defined we have~$\max(X_{i,\ell'}) \geq \min(P)$ by Lemma~\ref{lemma:latersolutionsreachfurther}. Hence we can determine the smallest value~$\ell^*$ for which this holds, and find that the canonical solution on~$R_i$ hits~$P$ if and only if its index is at least~$\ell^*$. Hence we obtain the literal~$x_i \geq \ell^*$. In the case that there is no well-defined canonical solution that hits the suffix, we report this instead.
The two correctness properties follow directly from the if-and-only-if nature of our arguments above.
\qed
\end{proof}
Using the lemmata developed so far, we can present a polynomial-time algorithm for the problem in flower graphs.
\begin{theorem} \label{theorem:pathsinflower:poly}
\textsc{Hitting Paths in a Flower with Budgets}\xspace can be solved in polynomial time.
\end{theorem}
\begin{proof}
We show how to reduce an instance~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ with petals~$(R_1, \ldots, R_n)$ to an equivalent instance of the polynomial-time solvable \ensuremath{\mathcal{T}}\xspaceORSTwoSat problem. The main work will be done by Lemma~\ref{lemma:createliteral} to create the literals of the formula. Let~$N := \max _{i \in [n]} |V(R_i)|$ be the maximum size of a petal. The truth value set for our multiple valued logic formula will be~$[N]$. We create a variable~$x_i$ for every petal~$i$. The clauses in the formula are produced as follows.
\begin{enumerate}
\item For every petal index~$i \in [n]$, we compute the values of~$1 \leq \ell \leq |V(R_i)|$ for which the $\ell$-th canonical solution for petal~$R_i$ is well-defined, using the procedure of Definition~\ref{definition:canonical}. By Lemma~\ref{lemma:contiguous} these values form a contiguous interval, say~$\ell_1, \ldots, \ell_2$. We add the singleton clause~$x_i \geq \ell_1$ to the formula, as well as the singleton clause~$x_i \leq \ell_2$. If there is no well-defined canonical solution for~$R_i$ then, by Lemma~\ref{lemma:existscanonical}, the hitting set instance has no solution. In this case we simply output the answer \textsc{no}\xspace.
\item For every path~$P \in \ensuremath{\mathcal{P}}\xspace$ that is not contained entirely within a single petal (i.e., for every path that contains the core vertex~$z$ of the flower) we do the following. If~$P = \{z\}$ is the singleton path containing only vertex~$z$, then we output \textsc{no}\xspace as a solution is not allowed to contain vertex~$z$; this path can never be hit. Otherwise, let~$P_1, P_2$ be the two connected components of~$P - \{z\}$. (In the exceptional case that~$P - \{z\}$ has only a single component because~$P$ has~$z$ as an endpoint, take~$P_1 = P_2$ to be equal to~$P - \{z\}$.) For~$k \in \{1,2\}$ let~$R_{i_k}$ be the petal containing~$P_k$ and invoke Lemma~\ref{lemma:createliteral} on~$P_k$ with~$R_{i_k}$. If the invocations for both values of~$k$ produce a literal, say~$\phi_1$ and~$\phi_2$, then add the disjunction~$\phi_1 \vee \phi_2$ as a 2-clause to the formula. If one invocation concludes that no well-defined canonical solution hits the path, but the other invocation produces a literal, then add a singleton clause with the latter literal. Finally, if neither~$P_1$ nor~$P_2$ produces a literal, then neither of the subpaths of~$P - \{z\}$ are hit by any well-defined canonical solution, and therefore the path~$P$ is not hit by any canonical solution. (Recall that solutions are forbidden to contain~$z$.) Since, by Lemma~\ref{lemma:existscanonical}, a canonical solution exists if a solution exists at all, it follows that we can safely output \textsc{no}\xspace and halt.
\end{enumerate}
The process above results in a totally ordered regular signed 2-SAT formula~$\ensuremath{\mathcal{P}}\xspacehi$ on~$n$ variables with~${\mathcal{O}}(n + |\ensuremath{\mathcal{P}}\xspace|)$ clauses, which is polynomial in the size of the total input. All numbers involved are in the range~$[N]$ which is bounded by the order of the input graph~$G$. The reduction can therefore be performed in polynomial time, and produces an instance of \ensuremath{\mathcal{T}}\xspaceORSTwoSat of polynomial size, even when encoding the numbers in unary. It remains to prove correctness of the reduction.
\begin{claim}
Formula~$\ensuremath{\mathcal{P}}\xspacehi$ is satisfiable if and only if~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ has a solution.
\end{claim}
\begin{proof}
($\Rightarrow$) Suppose that the formula is satisfiable and consider a satisfying assignment to the variables~$x_1, \ldots, x_n$. Since the assignment satisfies the first type of clauses introduced, for every petal index~$i \in [n]$, if~$x_i = \ell$ then the $\ell$-th canonical solution for~$R_i$ is well-defined. Initialize~$X$ as an empty solution set. For each~$i \in [n]$ add the canonical solution for~$R_i$ whose index is given by~$x_i$ to the set~$X$. Since well-defined canonical solutions for~$i$ have size~$b(i)$ by Definition~\ref{definition:canonical}, this satisfies the budget constraints of the problem since the only vertices of~$R_i$ added to~$X$ are those of the canonical solution employed on that petal. As we trivially do not include~$z$ in the solution~$X$, to verify that~$X$ is a valid solution it remains to check that~$X$ intersects all paths in~$\ensuremath{\mathcal{P}}\xspace$. To this end, consider an arbitrary path~$P \in \ensuremath{\mathcal{P}}\xspace$.
\begin{enumerate}
\item If~$P$ is contained entirely within one petal, say~$R_i$, then observe that any well-defined canonical solution for petal~$R_i$ hits~$P$ by Step~\ref{step:hitpath} of Definition~\ref{definition:canonical}. Since a canonical solution for~$R_i$ is included in~$X$, the path~$P$ is hit.
\item If~$P$ is not contained entirely within one petal, then by the structure of flower graphs we know that~$P$ contains vertex~$z$ and was considered in the second phase of the construction. Consider the clause created on account of~$P$ during the construction above. Since the formula satisfies the clause, at least one literal is satisfied; say the literal for the subpath~$P_k$ of~$P - \{z\}$ residing in petal~$R_{i_k}$. Then Lemma~\ref{lemma:createliteral} guarantees that the canonical solution employed on~$R_{i_k}$ hits~$P_k$, and therefore hits the larger path~$P$ as well.
\end{enumerate}
\textsc{no}\xspaceindent As~$X$ hits all paths in~$\ensuremath{\mathcal{P}}\xspace$, this proves the forward direction.
($\Leftarrow$) For the reverse direction, suppose that~$(G,z,\ensuremath{\mathcal{P}}\xspace,b)$ has a solution. By Lemma~\ref{lemma:existscanonical} there is a globally canonical solution~$X$. For every petal index~$i$ let~$\ell_i$ be such that~$X$ includes the $\ell_i$-th canonical solution on~$R_i$, and assign variable~$x_i$ the value~$\ell_i$. Let us check that this assignment satisfies the formula. Every clause of the first type is satisfied by any setting corresponding to the index of a canonical solution, which is clearly the case. For the clauses of the second type that are produced on account of paths~$P \in \ensuremath{\mathcal{P}}\xspace$, observe that~$X$ intersects such a path in a connected component of~$P - \{z\}$, since~$z \textsc{no}\xspacet \in X$. By Lemma~\ref{lemma:createliteral} this implies that the corresponding literal of the clause is satisfied, implying that the entire clause is satisfied. Hence all types of clauses are satisfied, showing the formula to be satisfiable.
\claimqed
\end{proof}
The claim shows that to solve the hitting set problem, it suffices to check the satisfiability of the polynomial-sized \ensuremath{\mathcal{T}}\xspaceORSTwoSat instance. As the latter can be done in polynomial time, this proves Theorem~\ref{theorem:pathsinflower:poly}.
\qed
\end{proof}
\subsection{Hitting Paths in Graphs} \label{subsection:hit:paths}
In this section we will show that an instance of \textsc{Hitting Paths in a Graph}\xspace can be reduced to~$2^{5k}$ instances of \textsc{Hitting Paths in a Flower with Budgets}\xspace. By the results of the previous section, this leads to an FPT algorithm.
We will frequently use the following observation. It formalizes that if~$v$ is a degree-one vertex in~$G$ and we are looking for a set that hits all paths in~$\ensuremath{\mathcal{P}}\xspace$, then either there is a single-vertex path~$P = \{v\} \in \ensuremath{\mathcal{P}}\xspace$, forcing~$v$ to be in any solution, or there is an optimal solution that does not contain~$v$.
\begin{observation} \label{observation:removeleaf}
Let~$(G,k,t,\ensuremath{\mathcal{P}}\xspace)$ be an instance of \textsc{Hitting Paths in a Graph}\xspace and let~$v \in V(G)$ have degree at most one.
\begin{enumerate}
\item If the singleton path~$P = \{v\}$ is contained in~$\ensuremath{\mathcal{P}}\xspace$, then~$(G,k,t,\ensuremath{\mathcal{P}}\xspace)$ is equivalent to the instance obtained by decreasing~$t$ by one, removing~$v$ from the graph, and removing all paths containing~$v$ from~$\ensuremath{\mathcal{P}}\xspace$.
\item Otherwise,~$(G,k,t,\ensuremath{\mathcal{P}}\xspace)$ is equivalent to the instance obtained by removing~$v$ from the graph and replacing every path~$P \in \ensuremath{\mathcal{P}}\xspace$ by~$P \setminus \{v\}$.
\end{enumerate}
The cyclomatic number is not affected by these operations.
\end{observation}
For an instance~$(G,\ensuremath{\mathcal{P}}\xspace,k,t)$ of \textsc{Hitting Paths in a Graph}\xspace and a vertex subset~$S \subseteq V(G)$, the \emph{cost of the subgraph induced by~$S$}, denoted~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(S)$, is defined as the minimum cardinality of a set that hits all paths~$P \in \ensuremath{\mathcal{P}}\xspace$ for which~$V(P) \subseteq S$. Equivalently,~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(S)$ is the minimum cardinality of a set that hits all paths~$\{ P \in \ensuremath{\mathcal{P}}\xspace \mid V(P) \subseteq S\}$ in the graph~$G[S]$. Observe that if~$S$ induces an acyclic subgraph of~$G$, then this value is computable in polynomial time as discussed in Section~\ref{section:prelims}. To reduce the general \textsc{Hitting Paths in a Graph}\xspace problem to the version with budget constraints discussed in the previous section, the following lemma is useful for determining relevant values for the budgets.
\begin{lemma} \label{lemma:budget:on:paths}
Let~$(G,\ensuremath{\mathcal{P}}\xspace,k,t)$ be an instance of \textsc{Hitting Paths in a Graph}\xspace. Let~$S$ be the vertices of degree unequal to two in~$G$. There is a minimum-size hitting set~$X$ for~$\ensuremath{\mathcal{P}}\xspace$ such that, for every connected component~$C$ of~$G - S$, we have~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C) \leq |X \cap V(C)| \leq \ensuremath{\mathrm{\textsc{opt}}}\xspace(C)+1$.
\end{lemma}
\begin{proof}
The fact that~$X \cap V(C) \geq \ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ for all hitting sets of~$\ensuremath{\mathcal{P}}\xspace$ follows trivially since~$X \cap V(C)$ is a hitting set for the induced subinstance. For the other inequality we exploit the structure of the graph.
Let~$X$ be a minimum-size hitting set for~$\ensuremath{\mathcal{P}}\xspace$, whose size may be less than~$t$. We give a proof by induction on the number of components for which~$X \cap V(C)$ exceeds~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 1$. The statement is trivially true if this number is zero. Otherwise, fix a component~$C$ for which~$X \cap V(C) > \ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 1$. As~$C$ is a connected subgraph containing only vertices of degree two, the neighborhood~$N_G(C)$ has size at most two, and is contained within~$S$. Let~$X_C$ be a minimum-cardinality hitting set for the instance induced by~$C$, of size~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$. Consider the set~$X' := (X \setminus V(C)) \cup N_G(C) \cup X_C$, whose intersection with~$C$ has size~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$. Since~$|N_G(C) \cup X_C| \leq \ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 2$ and~$X \cap V(C) \geq \ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 2$, the set~$X'$ is not bigger than~$X$. We show it to be a hitting set as well. To see that, observe that all paths contained in~$G[C]$ are hit by~$X_C$, as it is a solution to the subproblem induced by~$C$. Any path intersecting~$C$ that is not hit by~$X_C$ was not included in the subinstance induced by~$C$, and hence contains a vertex of~$N_G(C)$. Such paths are therefore hit by~$X'$. All paths that do not intersect~$C$ are hit by~$X \setminus V(C)$, and are therefore hit by its superset~$X'$ as well. It follows that~$X'$ is a hitting set of minimum cardinality. As the number of components~$C$ from which it uses at least~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 2$ vertices is strictly smaller than for~$X$ (the vertices of~$N_G(C)$ belong to~$S$ and therefore do not increase this number, as components are taken of~$G-S$), the proof now follows by induction.
\qed
\end{proof}
Using these ingredients we give an algorithm for \textsc{Hitting Paths in a Graph}\xspace.
\begin{theorem} \label{theorem:pathsingraph:fpt}
\textsc{Hitting Paths in a Graph}\xspace parameterized by cyclomatic number can be solved in time~$2^{5k} \cdot (|G| + |\ensuremath{\mathcal{P}}\xspace|)^{{\mathcal{O}}(1)}$.
\end{theorem}
\begin{proof}
When presented with an input~$(G,\ensuremath{\mathcal{P}}\xspace,k,t)$, the algorithm proceeds as follows. First, as a preprocessing step, the algorithm repeatedly removes vertices of degree at most one from the graph using Observation~\ref{observation:removeleaf}. If the resulting graph is empty, then we can simply decide the problem: the answer is \textsc{yes}\xspace if and only if the value of~$t$ was not decreased below zero by these operations. Otherwise we obtain a graph with minimum degree at least two. While this graph is disconnected, add an arbitrary edge between two distinct connected components. This does not change the answer to the instance (the paths~$\ensuremath{\mathcal{P}}\xspace$ to be hit are unchanged) and leaves the cyclomatic number unchanged. From now on we therefore assume that the instance we work with has minimum degree at least two and consists of a connected graph. For ease of notation, we refer to instance resulting from these steps simply as~$(G,\ensuremath{\mathcal{P}}\xspace,k,t)$. If~$G$ consists of just a simple cycle (i.e.,~$G$ is 2-regular) then we can decide the problem in polynomial time as discussed in Section~\ref{section:prelims}, so we focus on the case that the set~$S$ of vertices of degree at least three is nonempty. By Proposition~\ref{proposition:degtwo}, the size of~$S$ is bounded by~$2k$. The main idea of the algorithm is to use branching make two successive guesses.
\begin{itemize}
\item First, we guess which vertices from~$S$ are used in a solution. Concretely, we try all subsets~$S' \subseteq S$ and test whether there is a solution~$X$ for which~$X \cap S = S'$.
\item For every such set~$S'$, we do the following. By Lemma~\ref{lemma:budget:on:paths}, there is a minimum-size hitting set that intersects every component~$C$ of~$G - S$ (which is a path) in either~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ or~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 1$ vertices. Let~$\ensuremath{\mathcal{C}}\xspace$ denote the set of these components. By Proposition~\ref{proposition:numcomponents}, we have~$|\ensuremath{\mathcal{C}}\xspace| < k + |S| \leq 3k$. We now guess the collection~$\ensuremath{\mathcal{C}}\xspace' \subseteq \ensuremath{\mathcal{C}}\xspace$ of components~$C$ for which the solution uses~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ vertices. Every guess~$\ensuremath{\mathcal{C}}\xspace'$ defines a budget~$b(C)$ for each component~$C \in \ensuremath{\mathcal{C}}\xspace$ as follows: $b(C) = \ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ if~$C \in \ensuremath{\mathcal{C}}\xspace'$, and~$b(C) = \ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 1$ otherwise.
\end{itemize}
Having guessed both~$S'$ and~$\ensuremath{\mathcal{C}}\xspace'$, we create an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace to verify whether there is a hitting set~$X$ for the paths~$\ensuremath{\mathcal{P}}\xspace$ such that~$X \cap S = S'$ and for all components~$C$ of~$G-S$ we have~$|X \cap C| = b(C)$. Observe that these constraints on~$X$ completely determine its size, which must be~$|S'| + \sum _{C \in \ensuremath{\mathcal{C}}\xspace} b(C)$. Hence if the size exceeds~$t$, then these guessed sets will not lead to a hitting set of the desired size, and can therefore be skipped. When we have a guess that leads to a hitting set size of at most~$t$, we aim to produce an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace to check whether there is a solution consistent with the guesses. To this end, initialize~$G'$ as a copy of~$G$, and~$\ensuremath{\mathcal{P}}\xspace'$ as a copy of~$\ensuremath{\mathcal{P}}\xspace$. We modify these structures to create an input on a flower graph. Throughout these modifications there will be a clear correspondence between components of~$G' - S$ and those of~$\ensuremath{\mathcal{C}}\xspace$, so that we may refer to the budgets of components~$C$ of~$G' - S$. For each guess~$S'$ and~$\ensuremath{\mathcal{C}}\xspace'$, we proceed as follows.
\begin{enumerate}
\item Remove all the vertices of~$S'$ from the graph~$G'$ and remove all paths hit by~$S'$ from~$\ensuremath{\mathcal{P}}\xspace'$.\label{flower:step:removeshitpaths}
\item For all paths~$P \in \ensuremath{\mathcal{P}}\xspace'$ for which there is a component~$C$ of~$G' - S$ such that all vertices of~$C$ belong to~$P$ and~$b(C) > 0$, remove~$P$ from the set~$\ensuremath{\mathcal{P}}\xspace'$. All hitting sets that contain~$b(C)$ vertices from~$C$ must hit~$P$, so we can drop the constraint~$P$ because we will introduce a budget constraint on~$C$.\label{flower:step:removebudgethitpaths}
\item For all components~$C$ of~$G' - S$ such that~$b(C) = 0$, do the following. Remove the vertices of~$C$ from the graph~$G'$. For every~$P \in \ensuremath{\mathcal{P}}\xspace'$, replace~$P$ by the subgraph~$P - V(C)$. This may cause the elements of~$\ensuremath{\mathcal{P}}\xspace'$ to become disconnected subgraphs, rather than paths, but this will be resolved in the next step.\label{flower:step:splitpaths}
\item The final step identifies several vertices in the graph into a single core vertex, to obtain a flower structure. Concretely, update the graph~$G'$ by identifying all vertices of~$S \setminus S'$ into a single vertex~$z$. Similarly, update every subgraph~$P \in \ensuremath{\mathcal{P}}\xspace'$ by identifying all vertices of~$(S \setminus S') \cap V(P)$ into a single vertex~$z$.\label{flower:step:merge}
\end{enumerate}
\begin{figure}
\caption{(\ref{fig:toflower1}
\label{fig:toflower1}
\label{fig:toflower4}
\label{fig:toflower3}
\label{fig:toflower5}
\label{fig:mergeintoflower}
\end{figure}
Let~$G^*, \ensuremath{\mathcal{P}}\xspace^*$ denote the resulting graph and system of subgraphs. Refer to Figure~\ref{fig:mergeintoflower} for an illustration of these steps.
\begin{numberedclaim}
$G^*$ is a flower with core~$z$ and all subgraphs in~$\ensuremath{\mathcal{P}}\xspace^*$ are simple paths.
\end{numberedclaim}
\begin{proof}
Let us first verify that~$G^*$ is indeed a flower. As every vertex of~$S$ was either removed or merged into~$z$, we find that~$G^* - \{z\}$ is a subgraph of~$G - S$. Since~$S$ is the set of vertices of degree unequal to two, and~$G$ had no vertices of degree at most one after preprocessing, every connected component of~$G - S$ consists of vertices that have degree two in~$G$ and therefore such components form paths. As~$S$ is not empty, every such component has exactly two neighbors in~$S$, which are adjacent to the first and last vertex of the path. Hence no interior vertex of such paths is adjacent to~$S$. Since~$G^* - \{z\}$ is a subgraph of~$G - S$ it follows that all connected components of~$G^* - \{z\}$ are paths and no interior vertex of such a path is adjacent to~$z$. Hence~$G^*$ is a flower with core~$z$.
We continue by proving the second part of the claim. Consider a subgraph~$P^*$ in the final set~$\ensuremath{\mathcal{P}}\xspace^*$, and let~$P \in \ensuremath{\mathcal{P}}\xspace$ be the simple path in~$G$ from which it originated. As~$P^*$ is present after the last step, it follows that~$P$ did not meet the precondition for removal in Step~\ref{flower:step:removeshitpaths} and therefore~$P \cap S' = \emptyset$, showing that~$P$ is a simple path in~$G - S'$. Similarly, as~$P$ was not removed by Step~\ref{flower:step:removebudgethitpaths} we know that~$P$ does not fully contain any component~$C$ of~$G - S$ with positive budget. Consider what happens when deleting components~$C$ with budget zero in Step~\ref{flower:step:splitpaths}. Since all vertices of~$G - S$ have degree two in~$G$, there are at most two components~$C$ of~$G - S$ from which~$P$ contains at least one, but not all vertices: these are the components containing the endpoints of~$P$. Hence if~$P$ is transformed into a disconnected subgraph~$P'$ by Step~\ref{flower:step:splitpaths}, then (1)~$P'$ contains at least one vertex of~$S \setminus S'$, and (2) there are at most two connected components in~$P' - S$, and both these components have (in subgraph~$P'$) a neighbor in~$S \setminus S'$. This shows that when all vertices of~$S \setminus S'$ are merged into a single vertex~$z$ by Step~\ref{flower:step:merge}, then the disconnected subgraph~$P'$ is transformed into a path~$P^*$ containing the core vertex~$z$ in its interior. Hence all subgraphs~$P^*$ in~$\ensuremath{\mathcal{P}}\xspace^*$ are simple paths.
\claimqed
\end{proof}
The claim shows that we can use the structures~$G^*$ and~$\ensuremath{\mathcal{P}}\xspace^*$ resulting from the process above to formulate an instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace. To that end, we use~$G^*$ as the flower graph,~$z$ as the core, and~$\ensuremath{\mathcal{P}}\xspace^*$ as the set of paths to be hit. We number the connected components of~$G^* - \{z\}$, which are the petals of the flower, as~$R_1, \ldots, R_n$. Each such petal corresponds to a connected component of~$G - S$ for which we assigned a budget when guessing~$\ensuremath{\mathcal{C}}\xspace' \subseteq \ensuremath{\mathcal{C}}\xspace$; we define the budget function~$b^*$ for the instance by letting~$b^*(i)$ be~$b(C_i)$ where~$C_i$ is the component of~$G - S$ corresponding to~$R_i$. This results in a valid instance~$(G^*,z,\ensuremath{\mathcal{P}}\xspace^*,b^*)$ of \textsc{Hitting Paths in a Flower with Budgets}\xspace. For the correctness of the algorithm, the following claim is crucial.
\begin{numberedclaim}
For every guess of~$S' \subseteq S$ and~$\ensuremath{\mathcal{C}}\xspace' \subseteq \ensuremath{\mathcal{C}}\xspace$, the following are equivalent.
\begin{enumerate}
\item There is a hitting set~$X$ for the paths~$\ensuremath{\mathcal{P}}\xspace$ in graph~$G$ such that~$X \cap S = S'$ and all components~$C$ of~$G-S$ satisfy~$|X \cap C| = b(C)$.\label{eqv:graphhitset}
\item The produced instance~$(G^*,z,\ensuremath{\mathcal{P}}\xspace^*,b^*)$ of \textsc{Hitting Paths in a Flower with Budgets}\xspace has a solution.\label{eqv:flowerhitset}
\end{enumerate}
\end{numberedclaim}
\begin{proof}
(\ref{eqv:graphhitset}$\Rightarrow$\ref{eqv:flowerhitset}) Suppose there is a hitting set~$X$ satisfying the stated conditions. We claim that~$X^* := X \setminus S$ is a solution for~$(G^*,z,\ensuremath{\mathcal{P}}\xspace^*,b^*)$. By the preconditions, set~$X^*$ satisfies the budget constraints for the petals. Consider a path~$P^* \in \ensuremath{\mathcal{P}}\xspace^*$ derived from a path~$P \in \ensuremath{\mathcal{P}}\xspace$. Set~$X$ does not hit~$P$ in a component of~$G - S$ with~$b(C) = 0$, as~$X$ contains no vertices of such components. Set~$X$ does not intersect~$P$ in a vertex of~$S'$, as such paths are not present in~$\ensuremath{\mathcal{P}}\xspace^*$ due to Step~\ref{flower:step:removeshitpaths}. It follows that~$X$ intersects~$P$ in a vertex~$v$ of~$V(G) \setminus S$ that lies in a connected component of~$G - S$ with positive budget. Hence the component forms a petal~$R_i$ in~$G^*$, and the intersection of~$P$ with the petal is contained in~$P^*$. Hence~$X^*$ hits~$P^*$ at~$v$. Since all paths in~$\ensuremath{\mathcal{P}}\xspace^*$ are hit, the budget constraints are met, and~$z \textsc{no}\xspacet \in X^*$, it follows that~$X^*$ is a solution to~$(G^*,z,\ensuremath{\mathcal{P}}\xspace^*,b^*)$.
(\ref{eqv:flowerhitset}$\Rightarrow$\ref{eqv:graphhitset}) For the reverse direction, suppose the flower problem has a solution~$X^*$. We claim that~$X := X^* \cup S'$ is a hitting set for the paths~$\ensuremath{\mathcal{P}}\xspace$ in~$G$ with~$|X \cap C| = b(C)$ for all components~$C$ of~$G - S$. The latter condition is easily verified: components~$C$ with~$b(C) = 0$ were discarded in Step~\ref{flower:step:splitpaths}, do not occur in~$G^*$, and therefore~$X^*$ contains no vertices of such components. For components with positive budget, which also exist in~$G^*$, the budget constraints in the definition of \textsc{Hitting Paths in a Flower with Budgets}\xspace ensure that~$|X \cap C| = |X^* \cap C| = b(C)$. Let us verify that~$X$ indeed hits all paths~$\ensuremath{\mathcal{P}}\xspace$ in~$G$ by considering an arbitrary~$P \in \ensuremath{\mathcal{P}}\xspace$. If~$P$ contains a vertex of~$S'$ then~$X$ trivially hits~$P$. Similarly, if~$P$ fully contains a component~$C$ of~$G - S$ with~$b(C) > 0$, then~$X^*$ contains at least one vertex of~$C$ and therefore of~$P$; hence~$X \supseteq X^*$ also hits~$P$. In the remaining case, the construction of~$G^*,\ensuremath{\mathcal{P}}\xspace^*$ ensures that~$\ensuremath{\mathcal{P}}\xspace^*$ contains a path~$P^*$ such that~$P^* - \{z\}$ is a subgraph of~$P - S$. Since~$X^*$ hits~$P^*$ in a vertex other than~$z$, this vertex is included in~$X$ and therefore~$X$ hits~$P$. It follows that~$X$ is indeed a hitting set for all paths in~$\ensuremath{\mathcal{P}}\xspace$, which concludes the proof of the equivalence.
\claimqed
\end{proof}
Using the claim, the final part of the algorithm becomes clear. For every guess~$S' \subseteq S$ and~$\ensuremath{\mathcal{C}}\xspace' \subseteq \ensuremath{\mathcal{C}}\xspace$ that leads to a solution of size at most~$t$, we construct the corresponding instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace and solve it using Theorem~\ref{theorem:pathsinflower:poly}. Since the flower instances are not larger than the input instance, this can be done in time~$(|G| + |\ensuremath{\mathcal{P}}\xspace|)^{{\mathcal{O}}(1)}$ for every guess. As there are~$2^{|S|} \cdot 2^{|\ensuremath{\mathcal{C}}\xspace|} \leq 2^{2k} \cdot 2^{3k}$ options for~$S'$ and~$\ensuremath{\mathcal{C}}\xspace'$ to check, the total running time is bounded by~$2^{5k} (|G| + |\ensuremath{\mathcal{P}}\xspace|)^{{\mathcal{O}}(1)}$. If one of the \textsc{Hitting Paths in a Flower with Budgets}\xspace instances has answer \textsc{yes}\xspace, then we output \textsc{yes}\xspace; otherwise we output \textsc{no}\xspace. In one direction, the correctness of this approach follows from the previous claim together with the facts that flower instances are only produced when the size of the resulting hitting sets is at most~$t$. For the other direction, if~$(G,\ensuremath{\mathcal{P}}\xspace,t,k)$ has a hitting set of size at most~$t$, then by Lemma~\ref{lemma:budget:on:paths} there is a minimum-cardinality hitting set~$X$ (whose size is at most~$t$) whose intersection with every component~$C$ of~$G - S$ is either~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ or~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C) + 1$. In the branch where~$S' = X \cap S$ and~$\ensuremath{\mathcal{C}}\xspace'$ consists of the components where we use~$\ensuremath{\mathrm{\textsc{opt}}}\xspace(C)$ vertices, this leads to a \textsc{yes}\xspace-instance of \textsc{Hitting Paths in a Flower with Budgets}\xspace. This concludes the proof of Theorem~\ref{theorem:pathsingraph:fpt}.
\qed
\end{proof}
We remark that, while the previous theorem shows that \textsc{Hitting Paths in a Graph}\xspace is fixed-parameter tractable parameterized by the cyclomatic number, this problem is unlikely to admit a polynomial kernel. The general \textsc{Hitting Set}\xspace problem parameterized by the number of universe elements~$n$ can be reduced to an instance of \textsc{Hitting Paths in a Graph}\xspace with cyclomatic number~${\mathcal{O}}(n^2)$: if we let~$G$ be a complete~$n$-vertex graph, which has cyclomatic number~${\mathcal{O}}(n^2)$, then we can model any subset of the universe as a simple path in~$G$. Hence there is a polynomial-parameter transformation from \textsc{Hitting Set}\xspace parameterized by the universe size to \textsc{Hitting Paths in a Graph}\xspace parameterized by cyclomatic number. Since \textsc{Hitting Set}\xspace parameterized by universe size has no polynomial kernel unless \containment~\cite[Theorem 5.3]{DomLS14}, the same holds for \textsc{Hitting Paths in a Graph}\xspace parameterized by cyclomatic number.
\section{Hardness proofs} \label{section:hardness}
In this section we develop several hardness proofs. It turns out to be convenient to first prove the W[1]-hardness of 3-SAT in multiple valued logic. A similar result concerning the W[1]-hardness of \emph{not-all-equal} 3-SAT was obtained independently by Bringmann et al.~\cite{BringmannHML15}, who studied the problem under the name \textsc{NAE-Integer-$3$-SAT}.
\begin{theorem} \label{thm:threesat:whard}
The problem \textsc{$n$-Totally Ordered Regular Signed 3-SAT}\xspace is W[1]-hard.
\end{theorem}
\begin{proof}
To establish the theorem we give an FPT-reduction from the W[1]-complete \textsc{$k$-Clique}\xspace problem~\cite[Chapter 21]{DowneyF13}. Let~$(G,k)$ be an instance of \textsc{$k$-Clique}\xspace, asking whether the graph~$G$ has a clique of size~$k$. We use an edge representation strategy to encode this problem into an instance of \textsc{$n$-TORS 3-SAT}\xspace whose parameter, the number of variables~$n$ in the formula, is~${\mathcal{O}}(k^2)$. We may assume that~$|E(G)| \geq \binom{k}{2} \geq k \geq 2$, as the instance is trivial otherwise. We may also assume that~$G$ has no isolated vertices. The formula is constructed as follows.
There are variables~$x_1, \ldots x_k$ corresponding to a choice of~$k$ vertices in the clique. In addition, there are~$\binom{k}{2}$ variables~$x_{i,j}$ for~$1 \leq i < j \leq k$ that correspond to the edges between these vertices. The truth value set for the formula is the range of integers from~$1$ to~$|E(G)|$, so~$N := |E(G)|$. Number the vertices of~$G$ as~$v_1, \ldots, v_{|V(G)|}$, and the edges from~$1$ to~$N$, arbitrarily. For every~$1 \leq i < j \leq k$ and for every possible edge index~$\ell \in [N]$, we add four clauses to the formula. Let~$\{v_p,v_q\}$ be the endpoints of the $\ell$-th edge such that~$p < q$. We add the following clauses:
$$(x_{i,j} \leq \ell-1 \vee x_{i,j} \geq \ell+1 \vee x_i \leq p) \quad (x_{i,j} \leq \ell-1 \vee x_{i,j} \geq \ell+1 \vee x_i \geq p),$$
$$(x_{i,j} \leq \ell-1 \vee x_{i,j} \geq \ell+1 \vee x_j \leq q) \quad (x_{i,j} \leq \ell-1 \vee x_{i,j} \geq \ell+1 \vee x_j \geq q).$$
To obtain a valid formula, we omit the literal~$x_{i,j} \leq \ell - 1$ when~$\ell=1$, as do we omit the literal~$x_{i,j} \geq \ell + 1$ when~$\ell = N$. These clauses are automatically satisfied if~$x_{i,j} \neq \ell$, i.e., if~$x_{i,j}$ does \emph{not} select the~$\ell$-th edge. If~$x_{i,j} = \ell$, however, then the clauses force~$x_i$ to have the value~$p$ and~$x_j$ to have value~$q$.
The conjunction of the produced clauses for all valid values of~$i,j$, and~$\ell$ forms the output formula. The construction can be performed in polynomial time and produces an instance of \textsc{$n$-TORS 3-SAT}\xspace whose parameter~$n$ is~$\binom{k}{2} + k \in {\mathcal{O}}(k^2)$, which is suitably bounded. To complete the proof it suffices to show that~$G$ has a $k$-clique if and only if the formula is satisfiable.
\begin{claim}
If~$G$ has a $k$-clique, then the formula is satisfiable.
\end{claim}
\begin{proof}
Consider a $k$-clique in~$G$ and let the indices of its vertices be~$u_1, u_2, \ldots, u_k$ in order of increasing value. For~$i \in [k]$ assign variable~$x_i$ value~$u_i$, and for~$1 \leq i < j \leq k$ assign variable~$x_{i,j}$ the value of the index of the edge between~$v_{u_i}$ and~$v_{u_j}$. As observed above, the clauses that are created for values~$i,j,\ell$ such that~$x_{i,j}$ does not select the~$\ell$-th edge, are satisfied. It is easy to verify that when~$x_{i,j} = \ell$, the third literal of the created clauses is satisfied. Hence all clauses are satisfied and the formula is satisfiable.
\claimqed
\end{proof}
\begin{claim}
If the formula is satisfiable, then~$G$ has a $k$-clique.
\end{claim}
\begin{proof}
Consider an assignment to the variables that satisfies all clauses. Consider the values taken by the variables~$x_1, \ldots, x_k$. Suppose that some variable~$x_i$ with~$i<k$ has a value exceeding~$|V(G)|$. Then consider the value~$\ell$ of variable~$x_{i,i+1}$, and the clauses produced for this combination. Since~$x_{i,i+1} \leq \ell - 1$ is false, as is~$x_{i,i+1} \geq \ell+1$, we must have~$x_i \leq p$ where~$p$ is the lowest-indexed endpoint of the $\ell$-th edge; but this contradicts the assumption that~$x_i > |V(G)|$. A similar contradiction is reached when~$x_i = k$ by considering variable~$x_{i-1, i}$ instead. Hence the variables~$x_1, \ldots, x_k$ represent indices of vertices in~$G$.
Next, assume for a contradiction that there are indices~$1 \leq i < j \leq k$ such that~$x_i = x_j$, and let~$\ell$ be the value of variable~$x_{i,j}$. As observed above, the clauses added for the combination~$i,j,\ell$ are only satisfied if~$x_i$ and~$x_j$ represent the indices of the endpoints of the~$\ell$-th edge. But since~$G$ is a simple graph without self-loops, these indices are distinct and therefore these clauses cannot all be satisfied if~$x_i$ and~$x_j$ coincide. Hence the variables~$x_1, \ldots, x_k$ take~$k$ distinct values in the range of~$1$ to~$|V(G)|$.
We claim that the~$k$ vertices in~$G$ whose indices correspond to the values of~$x_1, \ldots, x_k$ form a clique. To see that all pairs of these vertices are adjacent in~$G$, consider a pair~$1 \leq i < j \leq k$ and the value~$\ell$ taken by variable~$x_{i,j}$. The clauses produced for~$i,j,\ell$ are only satisfied if~$x_i$ is the lower-indexed endpoint of the $\ell$-th edge and~$x_j$ is the higher-index endpoint of that edge. Given the values of~$x_i$ and~$x_j$, the clauses can therefore only be satisfied if~$\ell$ is the index of the edge between vertices with indices~$x_i$ and~$x_j$. Hence the edge connecting this pair must be present in~$G$. As~$i$ and~$j$ were arbitrary, this shows that all vertex pairs are adjacent. Hence the set of vertices with indices~$x_1, \ldots, x_k$ is a $k$-clique in~$G$.
\claimqed
\end{proof}
This concludes the proof of Theorem~\ref{thm:threesat:whard}.
\qed
\end{proof}
Theorem~\ref{thm:threesat:whard} is used as the starting point for the next hardness proof.
\begin{theorem} \label{theorem:hitclaws:whard}
It is W[1]-hard to determine, given a graph~$G$ with cyclomatic number~$k$, a set~$\ensuremath{\mathcal{S}}\xspace$ of subgraphs of~$G$, each isomorphic to a tree with at most three leaves, and an integer~$t$, whether there is a set of~$t$ vertices in~$G$ that intersects all subgraphs in~$\ensuremath{\mathcal{S}}\xspace$.
\end{theorem}
\begin{proof}
We give an FPT-reduction from \textsc{$n$-TORS 3-SAT}\xspace. Consider an instance of that problem, consisting of a signed 3-CNF formula over variables~$x_1, \ldots, x_n$ whose truth value set is~$[N]$. We assume that there are no clauses that are trivially satisfied (that contain literals~$x_i \leq c_1$ and~$x_i \geq c_2$ for~$c_2 \leq c_1 + 1$), as they can be efficiently recognized and removed without changing the answer.
We construct a hitting set problem on a flower graph~$G$ that has a core~$z$ and~$n$ petals~$R_1, \ldots, R_n$. Each petal is a path on~$N$ vertices whose endpoints are adjacent to~$z$. It is easy to see that this gives a cyclomatic number of at most~$k = n$ for the graph~$G$, as removing the~$n$ edges from~$z$ to the last vertex of each petal gives an acyclic graph. We seek a hitting set of size at most~$t := n$.
Signed literals of the formula have the form~$x_i \leq c$ or~$x_i \geq c$ for~$c \in [N]$. We associate every literal to a prefix or suffix of a petal: a literal~$x_i \leq c$ corresponds to the prefix~$\{r_{i,1}, \ldots, r_{i,c}\}$ of petal~$R_i$, while a literal~$x_i \geq c$ corresponds to the suffix~$\{r_{i,c}, \ldots, r_{i,N}\}$. For every clause~$C$ of the formula, we consider the pre/suffixes associated to its literals. We add the subgraph~$S_C$ that is induced by their vertices, together with~$z$, to the set~$\ensuremath{\mathcal{S}}\xspace$ of subgraphs to be hit. Observe that, since there are no clauses that are trivially satisfied, each such subgraph~$S_C$ induces a tree in~$G$ with at most three leaves. In addition, for every petal~$R_i$ we add the path~$R_i$ as a subgraph to~$\ensuremath{\mathcal{S}}\xspace$. This concludes the description of the hitting set instance.
\begin{numberedclaim} \label{claim:hitset:iff:sat}
There is a hitting set of size at most~$t$ if and only if the formula is satisfiable.
\end{numberedclaim}
\begin{proof}
($\Rightarrow$) Suppose there is a hitting set~$X$ of size~$t = n$. Since every petal~$R_i$ is present as a subgraph in~$\ensuremath{\mathcal{S}}\xspace$ that must be hit, and the petals are pairwise disjoint, it follows that~$X$ contains exactly one vertex of each petal. In particular, the core~$z$ is not in~$X$. Consider the assignment that sets the value of variable~$x_i$ to the index of the vertex in~$X \cap V(R_i)$, which is a number in the range~$[N]$. To see that an arbitrary clause~$C$ is satisfied, consider the subgraph~$S_C$ created on account of the clause, which consists of~$z$ together with at most three pre/suffixes of petals, one for each literal of~$C$. As the pre/suffix that is hit by~$X$ corresponds to a literal that is satisfied by the assignment, clause~$C$ is satisfied. As~$C$ was arbitrary, the formula is satisfiable.
($\Leftarrow$) Suppose that the formula is satisfied by a particular assignment to~$x_1, \ldots, \linebreak[1] x_n$. Let~$X$ contain vertex~$r_{i, x_i}$ for all~$i \in [n]$. Then all petals are hit by~$X$, and all subgraphs~$S_C$ added on account of a clause~$C$ are hit at a pre/suffix corresponding to a literal in the clause that is satisfied.
\claimqed
\end{proof}
The claim shows the correctness of the reduction. It is a valid FPT-reduction since it can be executed in polynomial time and the new parameter~$k$ equals the old parameter~$n$. Since \textsc{$n$-Totally Ordered Regular Signed 3-SAT}\xspace is W[1]-hard by Theorem~\ref{thm:threesat:whard}, this concludes the proof.
\qed
\end{proof}
By slightly modifying the construction, we can also obtain the following result which shows that hitting paths in graphs is para-NP-complete~\cite{FlumG06} parameterized by the feedback vertex number of the graph.
\begin{theorem} \label{theorem:hitpaths:fvs:npc}
It is NP-complete to determine, given a graph~$G$ with a feedback vertex set of size two, a set~$\ensuremath{\mathcal{P}}\xspace$ of simple paths in~$G$, and an integer~$t$, whether there is a set of~$t$ vertices in~$G$ that intersects all paths in~$\ensuremath{\mathcal{P}}\xspace$.
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem~\ref{theorem:hitclaws:whard}, so we only mention the key points. An instance of \textsc{$n$-Totally Ordered Regular Signed 3-SAT}\xspace on variables~$x_1, \ldots, x_n$ with truth value set~$[N]$ is reduced to an instance of the hitting set problem as follows. For every variable~$x_i$ we create a new path~$R_i$ on~$N$ vertices in the graph. Finally we add two universal vertices~$z, z'$ to the graph, adjacent to all vertices on all created paths. The resulting graph~$G$ has a feedback vertex set of size two, being~$\{z,z'\}$. For every~$i \in [n]$ we add~$R_i$ to~$\ensuremath{\mathcal{S}}\xspace$ to ensure that a vertex of~$R_i$ is selected in every hitting set. For every clause of the formula, we consider the (at most) three pre/suffixes of the petals~$R_i$ corresponding to its literals, as in Theorem~\ref{theorem:hitclaws:whard}. Since both~$z$ and~$z'$ are universal vertices, there is a simple path in~$G$ consisting of the first pre/suffix, vertex~$z$, the second pre/suffix, the vertex~$z'$, and ending with the last pre/suffix. For every clause we add such a path to~$\ensuremath{\mathcal{P}}\xspace$, which ensures that the clause must be satisfied when all paths are hit. Finally, we set the budget to~$t := n$ to ensure that valid solutions select one value for each variable. Following the argumentation of Theorem~\ref{theorem:hitclaws:whard} it is easy to see that the reduction is correct. Since the \textsc{$n$-Totally Ordered Regular Signed 3-SAT}\xspace problem is NP-complete, the theorem follows.
\qed
\end{proof}
We close this section on hardness by a discussion of subexponential-time algorithms. The construction in Theorem~\ref{theorem:hitpaths:fvs:npc} can be used to reduce an $n$-variable instance of the classical 3-SAT problem (with binary variables) to the problem of hitting simple paths in a graph of cyclomatic number~${\mathcal{O}}(n)$. This implies that, assuming the exponential-time hypothesis~\cite{ImpagliazzoPZ01}, the dependence on~$k$ in Theorem~\ref{theorem:pathsingraph:fpt} cannot be improved to~$2^{o(k)}$.
\section{Conclusion}
We have analyzed the problem of hitting subgraphs of a restricted form within a larger host graph, parameterized by structural measures of the host graph. There are several research directions related to this work that remain unexplored. For example, we have not touched upon the issue of computing, given a generic hitting set instance consisting of a set system~$\ensuremath{\mathcal{F}}\xspace$ over a universe~$U$, how complex graphs on vertex set~$U$ must be in which every set in~$\ensuremath{\mathcal{F}}\xspace$ induces a connected subgraph. What is the complexity of finding, given~$\ensuremath{\mathcal{F}}\xspace$ and~$U$, a graph of minimum cyclomatic number that embeds~$\ensuremath{\mathcal{F}}\xspace$ in this way? Alternatively, what is the complexity of finding the minimum cyclomatic number of a graph~$G$ such that for every set~$S \in \ensuremath{\mathcal{F}}\xspace$, there is a simple path in~$G$ on vertex set~$S$? Efficient algorithms for this task
could be used to transform generic hitting set instances into inputs of \textsc{Hitting Paths in a Graph}\xspace, on which Theorem~\ref{theorem:pathsingraph:fpt} can be applied.
One can also consider aggregate parameterizations of the hitting set problem using the measure of structure introduced here. We have shown that \textsc{Hitting Paths in a Graph}\xspace is FPT parameterized by the cyclomatic number. It is well known that the general \textsc{Hitting Set}\xspace problem is FPT parameterized by the number of sets, as it can be solved by dynamic programming. Suppose we have a \textsc{Hitting Set}\xspace instance where there are~$k_1$ arbitrary sets, and there is a graph~$G$ of cyclomatic number~$k_2$ such that the remaining sets correspond to paths in~$G$. Is \textsc{Hitting Set}\xspace parameterized by~$k_1+k_2$ FPT, when this structure is given?
The complexity of the problem changes significantly when weights are introduced for the elements in the universe and the task is to find a minimum-weight hitting set. A simple reduction from \textsc{Vertex Cover}\xspace shows that finding a minimum-weight set that hits a prescribed set of three-vertex paths in a star graph is already NP-complete. This suggests some topics for further investigation; we list some examples.
\begin{enumerate}
\item Is the problem of finding a minimum-weight vertex set that hits a prescribed set of \emph{directed paths} in a \emph{directed tree} polynomial-time solvable?
\item What is the parameterized complexity of the problem of hitting weighted paths in a tree plus~$k$ edges, when the largest weight value is bounded by a constant?
\end{enumerate}
\textbf{Acknowledgments}. We are grateful to Mark de Berg and Kevin Buchin for interesting discussions that triggered this research.
\appendix
\section{Reducing signed 2-SAT to classic 2-SAT} \label{app:twosat}
Given a totally ordered regular signed 2-SAT formula, replace each literal of the form~$x_i \geq j$ by~$\neg (x_i \leq j-1)$, noting that if~$j = 1$ the clause is always satisfied and can be removed instead. The resulting clauses consist of literals~$x_i \geq j$ and~$\neg (x_i \geq j)$ for~$j \in [N]$. To construct a classical 2-SAT formula, we interpret every term of the form~$x_i \geq j$ as a new variable, such that we have a variable~$x_i \geq 1$, another variable~$x_i \geq 2$, and so on. The resulting 2-SAT formula over this new set of variables consists of the clauses resulting from our conversion process above, together with clauses~$(x_i \geq j+1 \Rightarrow x_i \geq j)$ for all~$i \in [n]$ and~$j \in [N-1]$. Observe that such clauses may also be represented as~$(\neg (x_i \geq j+1) \vee (x_i \geq j))$, which shows they are valid 2-clauses. Finally, we add singleton clauses~$(x_i \geq 1)$ for all~$i \in [n]$. We invite the reader to verify that the resulting classical formula on variables~$(x_1 \geq 1), \ldots, (x_1 \geq N), \ldots, (x_n \geq 1), \ldots, (x_n \geq N)$ is classically satisfiable if and only if the signed formula is satisfiable over truth value set~$[N]$.
\end{document} |
\begin{document}
\begin{titlepage}
\title{Competitive equilibrium \\and the double auction
\thanks{For helpful comments and useful discussions, I would like to thank Jesper Akesson, Miguel Ballester, Adam Brzezinski, David Van Dijcke, Dan Friedman, Steve Gjerstad, Bernhard Kasberger, Erik Kimbrough, John Ledyard, Luke Milsom, Heinrich Nax, Charlie Plott, David Porter and Jasmine Theilgaard. I would also like to thank audiences at Oxford and Michigan State's Quantitative Economics Workshop. Finally, I am grateful to
Aniket Chakravorty for excellent research assistance and to the George Webb Medley Fund for generous financial support.
}}
\author{Itzhak Rasooly\thanks{Sciences Po, the Paris School of Economics, and the University of Oxford.} }
\date{\today}
\maketitle
\vspace*{-0.4cm}
\begin{center}
\end{center}
\vspace*{-0.8cm}
\begin{abstract}
\noindent In this paper, we revisit the common claim that double auctions necessarily generate competitive equilibria. We begin by observing that competitive equilibrium has some counterintuitive implications: specifically, it predicts that monotone shifts in the value distribution can leave prices unchanged. Using experiments, we then test whether these implications are borne out by the data. We find that in double auctions with stationary value distributions, the resulting prices can be far from competitive equilibria. We also show that the effectiveness of our counterexamples is blunted when traders can leave without replacement as time progresses. Taken together, these findings suggest that the `Marshallian path' is crucial for generating equilibrium prices in double auctions.
\noindent \\
\\
\noindent \textsc{Keywords:} double auction, competitive equilibrium, Marshallian path
\\
\noindent\textsc{JEL Codes:} C92, D01, D02, D90\\
\end{abstract}
\setcounter{page}{0}
\thispagestyle{empty}
\end{titlepage}
\pagebreak
\section{Introduction}\label{introduction}
In his `Introduction to Economic Science', \cite{fisher} wrote that `If you want to make a first-class economist, catch a parrot and teach him to say ``supply and demand'' in response to every question you ask him.' Apparently, this joke was considered dated even in 1910 --- it is attributed to a critic of economics writing from a `long time ago' --- and it thus serves to illustrate the dominance of supply and demand in the century before Fisher's publication. However, supply and demand analysis remained popular in Fisher's time; and indeed one of the purposes of Fisher's book was to expound and refine such analysis.
More than a century later, the idea of supply and demand --- or `competitive equilibrium', as we might now call it --- remains pervasive throughout economic theory. Its extension to multiple markets in the form of general equilibrium theory has provided a basis for celebrated welfare theorems \citep{arrow1951extension, debreu1954valuation}, as well as the foundation for much of modern macroeconomics (starting with \cite{lucas1977understanding}, \cite{kydland1982time}, etc.). In addition, partial equilibrium models have been applied to areas as diverse as discrimination \citep{becker1957economics}, marriage \citep{becker1973theory, becker1974theory} and location choice \citep{glaeser2007economics}. Thus, although the notion of competitive equilibrium may not have quite the dominance that it did in Fisher's time --- supplanted in part by the rise of non-cooperative game theory --- it surely remains one of the foundational concepts of economic theorising.
In part, the pervasiveness of competitive equilibrium may be due to the perception that its predictions have been experimentally vindicated by a series of double auction experiments starting with \cite{smith1962experimental}. As \cite{plott1981theories} puts it in an early review of the literature: `The overwhelming result [from these experiments] is that these markets converge to the competitive equilibrium even with very few traders'. In a more recent study of two thousand classroom experiments, \cite{lin2020evidence} reach a similar conclusion, declaring that competitive equilibrium convergence in double auctions appears to be `as close to a culturally universal, highly reproducible outcome as one is likely to get in social science'. They add that such convergence should be considered `as reproducible as the kinds of experiments that are done in a college chemistry laboratory to demonstrate universal chemistry principles'.\footnote{Indeed, existing experimental results appear to be surprisingly robust to changing the number of bidders \citep{smith1965experimental}, changing the cultural context \citep{kachelmeier1992culture}, and even introducing `extreme earnings inequality' at equilibrium \citep{holt1986market, smith2000boundaries, kimbrough2018testing}. Thus, while it may be possible to at least slow convergence to competitive equilibrium through large changes to the underlying environment --- e.g. by allowing for resale as in \cite{dickhaut2012commodity} or market power as in \cite{kimbrough2018testing} --- the existing literature suggests fast convergence to competitive equilibrium in double auction environments similar to those first considered by \cite{smith1962experimental}. See also \cite{ikica2018competitive} for a large-scale replication of competitive equilibrium convergence in standard double auction environments.}
In this paper, we revisit this conclusion. Our starting point is that competitive equilibrium can make highly counterintuitive, but previously unstudied, predictions. To see the basic idea, consider a market with 99 buyers, each with unit demand and with valuations of £1.01, £2.01, £3.01, ..., up to £99.01. Meanwhile, suppose that there are 99 sellers, each possessing just one unit to sell and with valuations (or `costs') of £0.99, £1.99, £2.99, ... up to £98.99. Under such assumptions, one can check that the (essentially unique\footnote{Depending on how one handles ties, prices of £49.99 and £50.01 can also clear the market.}) market-clearing price is £50: at such a price, 50 buyers want to purchase (those with valuations above £50) and 50 sellers want to sell (those with valuations below £50). Imagine now that we decrease the valuations of all sellers whose initial valuation was below £50 by an arbitrary amount, say to £0. Intuitively, one would expect this to drive down the price, both by inducing sellers to accept lower offers and by allowing them to profitably submit lower offers themselves. Despite this, however, the shift does not change the competitive equilibrium price: a price of £50 still generates demand from 50 buyers (since demand has not changed), and still generates supply from the same 50 sellers (who are now even more eager to sell).
In Section \ref{CE preserving shifts}, we begin by generalising the example just given. That is, we identify a broad class of downward shifts to the distributions of buyer and seller valuations which preserve the set of competitive equilibrium prices. As discussed, it is highly counterintuitive that such downward shifts would in fact leave market prices unaltered. As a result, such shifts provide a challenging test for competitive price theory.
We then conduct experimental double auctions to investigate whether such shifts do in fact depress observed prices. One important feature of our initial set of double auction experiments is that they hold the value distributions fixed through use of a queue of buyers and sellers: every time a buyer or seller exits the market, a new buyer or seller takes their place (see \cite{brewer2002behavioral} for a similar methodology). We use this queueing procedure for two reasons. First, it can be justified on grounds of realism: in actual markets, trade does not end after a couple of periods once all willing buyers and sellers have traded. Instead, the market is continually replenished by a new stock of traders. Second, and much more importantly, our design necessarily holds competitive equilibrium fixed, thereby allowing us to rigorously study if double auctions outcomes converge to \textit{the} set of competitive equilibrium prices. In contrast, standard designs do not possess this feature: every time a pair of traders drop out of the market, the supply and demand schedules shift, something which may (or may not) change the set of competitive equilibrium prices.\footnote{Remarkably, this issue was noticed in the first-ever experimental study of double auctions: see footnote 6 of \cite{smith1962experimental}. Despite conceding that `the supply and demand functions continually alter as the trading process occurs', Smith asserts that it is `the \textit{initial} [supply and demand] schedules prevailing at the opening of each trading period' that are of interest to `the theorist'. However, Smith does not give us any reason for privileging the initial demand and supply schedules over any others; and in the absence of such a reason, such privileging would appear to be entirely arbitrary.}
Our initial set of experiments yields three main findings. First, contrary to the predictions of competitive equilibrium theory, our downward shifts in the valuations do markedly decrease observed prices. Second, and partly as a result of the first finding, prices are almost never at the competitive equilibrium. Remarkably, this is the case even for our symmetric treatment, which might have been expected to yield competitive equilibrium prices. Third, prices show very little sign of converging to competitive equilibrium. Taken together, these findings imply that competitive equilibrium is not a good description of double auctions with stationary value distributions.
We then discuss why decreasing valuations and costs might decrease observed prices. While we do not attempt to identify one model as definitively `correct', we do identify a number of models that can rationalise our finding. To take one example, within a zero intelligence model \citep{gode1993allocative}, decreasing valuations leads to stochastically lower distributions of bids and asks, thereby lowering expected prices. More interestingly, perhaps, this effect is also generated by optimising models, including those based both on myopic pay-off maximisation \citep{gjerstad1998price} and more sophisticated optimal stopping \citep{friedman1991simple}.
Our first set of experiments establishes that, if value distributions are held fixed, double auctions need not produce competitive equilibria; and that double auction outcomes are sensitive to the kind of value shifts described previously. It is natural to wonder, however, whether our shifts are still able to move observed prices in more standard double auction environments in which players are allowed to drop out of the market (without replacement) as trade progresses. To investigate this question, we also run a series of double auctions without queues, including some very long sessions involving over an hour's trading in order to give the auctions the best possible chance of equilibrating.
Our findings from these more conventional experiments are more mixed. On the one hand, there is still some evidence that our shifts depress the observed prices; and some of the sessions we run fail to equilibrate even after many periods of trading. On the other hand, there is now a marked tendency towards equilibrium, and some of our sessions do converge to equilibrium despite the marked asymmetry in the designs. Thus, these sessions reveal both the power of our shifts as well as the equilibrating forces first observed in \cite{smith1962experimental}.
Our central pair of findings --- that one can easily `break' competitive equilibrium in environments with a queue of buyers and sellers but much less easily in an environment without a queue --- suggests that the question of whether the value distribution is held fixed is of critical importance. We provide a theoretical explanation as to why this should be the case. We assume that trade follows a `Marshallian path' \citep{brewer2002behavioral}, which means that (i) at any point in time, trade takes place between the active buyer with the highest valuation and the active seller with the lowest cost, and (ii) trade occurs if and only if it is mutually beneficial (we provide a formal definition in Section \ref{The Marshallian Path}). We prove that in standard double auction formats, a Marshallian path implies that final trades must take place at equilibrium prices; and we discuss why this result might also extend to non-final trades. Importantly, this result does \textit{not} extend to double auctions with fixed value distributions, which can help explain why such auctions lack the standard equilibrating tendencies. Therefore, we identify the Marshallian path dynamic as a key driver of equilibration in standard double auctions, thus helping to solve the `scientific mystery' introduced by Vernon Smith 60 years ago.
The remainder of this article is structured as follows. Section \ref{CE preserving shifts} generalises and formalises the idea of competitive equilibrium preserving shifts. Section \ref{Experimental Design} outlines the design of experiments aimed at testing the impacts of such shifts; Section \ref{Results} contains the results of such experiments; and Section \ref{Understanding the monotonicity} discusses which models can rationalise our findings. Section \ref{The Marshallian Path} uses a combination of theory and further experimentation to study whether our findings change once value distributions are no longer held stationary. Finally, Section \ref{Concluding remarks} concludes with an outline of some new areas of research opened up by this work.
\section{CE preserving shifts}\label{CE preserving shifts}
In this section, we generalise and formalise the example from the introduction in order to obtain a better understanding of its structure. To this end, let us consider a unit mass of buyers, each indexed by $i \in [0, 1]$.\footnote{While we work with a continuum of buyers and sellers for convenience, similar results are available in the discrete case.} Each buyer has a valuation $v_i \in [0, \bar{v}_b]$ (where $\bar{v}_b > 0$ is the maximum buyer valuation) and chooses to buy (exactly) one unit of the good if and only if their valuation is at least the market price $p$. The distribution of buyer valuations is described by the cumulative distribution function $F \colon \mathbb{R} \rightarrow [0, 1]$. For simplicity, we assume that (i) $F$ has full support on the interval $[0, \bar{v}_b]$ (ii) $F$ is continuous.
Let $d(p)$ denote market demand at price $p$. Then
\begin{equation}
d(p) = \int_0^1 \mathbbm{1}(v_i \geq p) di = \mathbb{P}(v_i \geq p) = 1 - F(p)
\end{equation}
where $\mathbbm{1}(v_i \geq p)$ is an indicator function. Since, $d(p) = 1 - F(p)$, $d(0) = 1 - F(0) = 1$ and $d(\bar{v}_b) = 1 - F(\bar{v}_b) = 0$. That is, demand starts at $1$ (at a price of $0$) and eventually falls to $0$ (at a price of $\bar{v}_b$). In addition, observe that $d$ is continuous (since it inherits the continuity of $F$) and strictly decreasing over the interval $[0, \bar{v}_b]$ (since $F$ has full support on this interval). In summary, then, we obtain a continuous and strictly decreasing demand function which starts at 1 before falling to 0 when the price equals the maximum valuation.
We treat sellers entirely symmetrically. That is, we have a unit mass of sellers, indexed by $i\in [0, 1]$; and each seller has a valuation (or `cost') $v_i \in [0, \bar{v}_s]$ (where $v_s > 0$ is the maximum seller valuation). Seller $i$ chooses to sell their one unit of the good if and only if $v_i \leq p$. The distribution of seller valuations is described by the cumulative distribution function $G \colon \mathbb{R} \rightarrow [0, 1]$. As before, we assume that (i) $G$ has full support on the interval $[0, \bar{v}_s]$ (ii) $G$ is continuous.
Let $s(p)$ denote market supply at price $p$. Then
\begin{equation}
s(p) = \int_0^1 \mathbbm{1}(v_i \leq p) di = \mathbb{P}(v_i \leq p) = G(p)
\end{equation}
where $\mathbbm{1}(v_i \leq p)$ is again an indicator function. Since $s(p) = G(p)$, observe that $s(0) = G(0) = 0$ and $s(\bar{v}_s) = G(\bar{v}_s) = 1$. Moreover, $s$ is continuous (since $G$ is continuous) and strictly increasing over the interval $[0, \bar{v}_s]$ (since $G$ has full support on that interval).
We define a \textit{competitive equilibrium price} as a price $p^* \in \mathbb{R^+}$ such that $d(p^*) = s(p^*)$. According to competitive price theory, this is the price which will prevail in a market; and the associated quantity traded will be $d(p^*) = s(p^*)$. As an aside, we notice that, although competitive price theory gives us a clear prediction as to the price which should prevail in a market, it does not provide us with an explanation as to \textit{why} such a price should arise. Whether such an explanation can be provided is itself an interesting question.\footnote{\cite{aumann1964markets} and \cite{cripps2006efficiency} are particularly influential attempts to provide a foundation to competitive equilibrium theory. The result we prove in Section \ref{The Marshallian Path} provides a rather different (and much simpler) foundation, albeit one that is most directly applicable to final period trades.}
As our first observation, let us note that, under our assumptions, there is exactly one competitive equilibrium price $p^*$. To see this, define excess demand by $e(p) = d(p) -s(p)$ and note that excess demand is positive at a price of zero (specifically, $e(0) = d(0) - s(0) = 1 - 0 = 1$). Meanwhile, $e(\bar{v}_b) = d(\bar{v}_b) - s(\bar{v}_b) = 0 - G(\bar{v}_b)$, so excess demand becomes negative at a price of $v_b$. Given that excess demand inherits the continuity of $d$ and $s$, this means that there must be some $p^*$ such that $e(p^*) = 0$, i.e there exists an equilibrium price. Moreover, since $e$ is strictly decreasing, we see that $p^*$ is unique.
We now define the central concept of this section.
\begin{definition}
A \textit{competitive equilibrium preserving demand contraction} is a transformation $T_b \colon [0, \bar{v}_b] \rightarrow \mathbb{R}^+$ such that
\begin{enumerate}
\item $T(V_b) \leq V_b$ for all $V_b \leq p^* - \epsilon^-$
\item $T(V_b) = V_b$ for all $V_b \in (p^* - \epsilon^-, p^* + \epsilon^+)$
\item $T(V_b) \in [p^* + \epsilon^+, V_b]$ for all $V_b \geq p^* + \epsilon^+$
\end{enumerate}
for some $\epsilon^+, \epsilon^- > 0$.
\end{definition}
As stated above, competitive equilibrium preserving demand contractions are downward shifts to the distribution of buyer valuations that satisfy three conditions. First, we require that low valuations (specifically those below $p^* - \epsilon^-$) are weakly decreased. Second, we require that there is some (possibly asymmetric) $\epsilon$-ball around $p^*$ at which valuations remain unchanged. Finally, we require that high valuations (those above $p^* + \epsilon^+$) are reduced, but not reduced so much that they are brought below $p^* + \epsilon^+$. Altogether, this amounts to a stochastic reduction to the distribution of buyer valuations, although one that leaves the ranking (i.e. percentile) of valuations in a neighbourhood of $p^*$ unchanged.
We define a competitive equilibrium preserving decrease in the seller valuations analogously.
\begin{definition}
A \textit{competitive equilibrium preserving supply expansion} is a transformation $T_s \colon [0, \bar{v}_s] \rightarrow \mathbb{R}^+$ such that
\begin{enumerate}
\item $T(V_s) \leq V_s$ for all $V_s \leq p^* - \epsilon^-$
\item $T(V_s) = V_s$ for all $V_s \in (p^* - \epsilon^-, p^* + \epsilon^+)$
\item $T(V_s) \in [p^* + \epsilon^+, V_s]$ for all $V_s \geq p^* + \epsilon^+$
\end{enumerate}
for some $\epsilon^+, \epsilon^- > 0$.
\end{definition}
\begin{figure}
\caption{A CE preserving shift}
\label{shifts}
\label{fig2}
\end{figure}
Figure \ref{shifts} plots a competitive equilibrium preserving demand contraction and supply expansion. As can be seen, we have decreased the buyer and seller valuations when the values were previously low. This corresponds to a downward shift in the demand and supply functions in the leftward portion of the diagram. In addition, we have left the buyer and seller valuations around the equilibrium price unchanged, which means that the demand and supply schedules remain unchanged in a neighbourhood of $p^*$. Finally, we have decreased valuations that were initially very high, corresponding to a downward shift in supply and demand in the rightward portion of Figure \ref{shifts}. As can be seen from the diagram, the unique equilibrium price remains at $p^*$ despite these downward shifts.\footnote{As will be clear from Figure \ref{shifts}, we hold valuations fixed within an $\epsilon$-ball of $p^*$ purely to preserve the uniqueness of the competitive equilibrium price. In fact, only one half of the ball is required for this purpose, although we retain the full ball for simplicity of exposition.} We now verify that this result holds in general.\footnote{All proofs are collected in Appendix \ref{proofs}.}
\begin{proposition}\label{prop1}
Let $p^*$ denote the competitive equilibrium price when buyer and seller valuations are distributed according to $V_b$ and $V_s$ respectively. Then if $T_b$ (respectively, $T_s$) is a competitive equilibrium preserving demand contraction (supply expansion), $p^*$ remains the unique competitive equilibrium price when buyer and seller valuations are distributed according to $V_b' = T_b(V_b)$ and $V_s' = T_s(V_s)$.
\end{proposition}
We have thus seen that there is a wide class of downward shifts to the buyer and seller valuations that leave competitive equilibrium predictions unaffected.\footnote{Unsurprisingly, one can also define an analogous class of \textit{upward} shifts to the buyer and seller valuations that leave equilibria unaffected. More generally, any shift that preserves the ranking of valuations in a neighbourhood of $p^*$ unaffected will preserve the unique equilibrium price $p^*$ (although we focus on everywhere upward or everywhere downward shifts since these generate the most counterintuitive implications).} Intuitively, one might expect such shifts to depress prices, either because they encourage buyers to offer lower prices or accept lower offers from sellers, or otherwise because they encourage sellers to offer lower prices or accept lower prices from buyers. That such shifts should leave the predictions of competitive equilibrium unchanged may thus come as a surprise. Whether these counterintuitive predictions are borne out by the data is a topic that we take up in the next section.
\section{Experimental design}\label{Experimental Design}
In order to examine the effect of competitive equilibrium preserving shifts, we ran a series of double auction experiments in Oxford in early June 2022.\footnote{The experiments received IRB Approval from the University of Oxford (ECONCIA21-22-44) and were pre-registered on the AEA registry: \url{https://www.socialscienceregistry.org/trials/9547}} The basic idea of the experiment was straightforward. Buyers and sellers were first endowed with their own (private) values and costs using the technique of induced valuations (see \cite{smith1976experimental} for discussion and elaboration). They then participated in a series of double auctions. In such auctions, buyers may, at any point in time, make `bids' to purchase at a particular price or accept offers that have been made by sellers. Similarly, at any point in time, sellers may offer to sell at a particular price (`making an ask') or accept a bid made by a buyer.
We opted to conduct a series of oral double auctions, which means that subjects made bids/asks verbally instead of submitting them electronically. While this resulted in somewhat slower data collection than one would have obtained from a computerised experiment, it yielded several important advantages. First, based on some pilot experiments, it seemed that subjects found oral double auctions more engaging, and also found the structure of oral double auctions rather easier to understand. Second, using oral double auctions ensured that our results were maximally comparable to classic studies like \cite{smith1962experimental}, \cite{smith1965experimental} and so forth. For this reason, we kept as close as possible to classical experimental economics protocols (see \cite{plott_document} for a helpful document outlining how such experiments were run and \cite{kimbrough2018testing} for a more recent experiment that adheres closely to such protocols).
In this initial set of experiments, we employed a queue in order to keep supply and demand schedules stable over time. This meant that every time a trade was executed, a new buyer and seller entered the market with the valuation and cost of the just departed buyer and seller. As discussed in the introduction, we did this for two reasons. First, such a design is arguably more realistic: actual markets do not typically dissolve after several trades have occurred (as in standard double auction experiments). Instead, they are continually replenished by a steady flow of new buyers and sellers. Second and much more importantly, our queue ensured that the set of competitive equilibria remained fixed over time, allowing us to rigorously study whether prices approached \textit{the} competitive equilibrium set. Standard experiments do not necessarily have this property.
To implement the queue experimentally, we recruited a group of four buyers and four sellers for every session who began each trading period as `inactive' (so they could not engage in market activity). As trade progressed, buyers and sellers were successively drawn from the queue into the main trading area, and explicitly adopted the value and cost of the buyer and seller who had just departed (by sitting down in their place and inspecting the back of the value/cost card that had been left on the table). This was done in full view of the other experimental participants so as to emphasise that the distribution of the values and costs had remained unchanged.
To study the impact of competitive equilibrium preserving shifts, we used two different treatments, each with five `active' buyers and five `active' sellers (in addition to the eight initially inactive traders in the queue). In the `symmetric' treatment, buyer valuations were £12, £32, £52, £72, £92; and seller valuations were £8, £28, £48, £68, £88. One can check that this yields an interval of competitive equilibrium prices from £48 to £52.\footnote{The competitive equilibrium prices of £48 and £52 are only \textit{weak} competitive equilibria: at such prices, there exists at least one trader who lacks any strict incentive to act in a way that clears the market. However, prices of £49, £50 and £51 are strict equilibria.} Our second treatment (the `low values' treatment) was obtained by decreasing the valuations and costs in the symmetric treatment as aggressively as possible in a way that preserves the set of competitive equilibrium prices. To this end, we changed the buyer valuations of £12 and £32 to £0 and reduced the buyer valuations of £72 and £92 to £52. Meanwhile, we reduced the seller valuations of £8 and £28 to £0 and reduced the seller valuations of £68 and £58 to £52. This yielded the new vector of buyer and seller valuations, namely £0, £0, £52, £52, £52 for the buyers and £0, £0, £48, £52, £52 for the sellers. One can verify directly that these distributions yield the very same set of equilibrium prices, namely £48 -- £52.\footnote{Since the low values treatment yields most of the surplus to sellers at the competitive equilibrium, it is reminiscent of the extreme earnings inequality design that has been studied in the literature (see, for example, \cite{smith2000boundaries}). However, it differs in at least two important respects. First, our design has an equal number of buyers and sellers: in contrast, the classic design generates extreme earnings inequality through an imbalance in the number of buyers and sellers. Second, our design generates an interval of strict competitive equilibrium prices (namely, £49, £50 and £51). In contrast, the standard extreme earnings design leads to the non-existence of strict competitive equilibria (the weak competitive equilibrium is computed by finding the price at which one side of the market is indifferent between trading and not trading).}
To control for subject fixed effects, we conducted both treatments sequentially within every experimental session. To get a handle on order effects, we conducted two sessions (Sessions 1 and 2) and varied the order of the treatments within these sessions. We ran the symmetric treatment first in Session 1; and ran the low values treatment first in Session 2 (see Table \ref{overview} for an overview of all experimental sessions).
At the start of each session, the auction rules were presented in written form (see Appendix \ref{instructions} for the rules with which subjects were presented). The rules were then further emphasised through an extensive oral quiz; and subjects were asked if they had any outstanding questions about the auction rules. Finally, subjects were asked to engage in a mock round of trading (which would not be used to calculate payoffs) for didactic purposes. As a result of these measures, subjects' understanding of the rules seemed to be excellent. As one indication of this, 99.5\% of bids and asks made in our experiments were `individually rational' in the sense that they would have made a (non-negative) profit if accepted. This compares favourably to existing auction datasets: for example, the dataset used by \cite{lin2020evidence} involves individual rationality violations in 90\% of rounds (see \cite{ledyard_unpublished} for discussion).\footnote{In our main analysis, we drop the handful of bids and asks that violate individual rationality from our dataset. However, our results are almost entirely unaffected if we include such data points.}
Once the illustrative trading round had concluded, the real trading began. Within each round of trading, active buyers and sellers were free to make or accept offers (`bids' or `asks') at any time. All offers were repeated by the auctioneer and recorded on a whiteboard.\footnote{To make or accept an offer, a buyer would say (for example) `Buyer 2 bids 30' or `Buyer 2 accepts 60'; and this would be duly repeated by the auctioneer. An analogous comment applies to sellers. All offers and acceptances were recorded by a research assistant and double-checked using an audio recording of the experimental sessions.} Trading used the standard improvement rule, which meant that bids needed to get successively higher and asks needed to get successively lower until a transaction occurred (at which point everything was reset and all bids and asks became permissible). Each round continued until the queue had been exhausted (i.e. until four trades had occurred); and we conducted five rounds of trading for each of the two sets of value distributions. In line with recent recommendations \citep{charness2016experimental, azrieli2018incentives} and double auction experiments \citep{ikica2018competitive}, subjects were only paid for one randomly chosen round within a session.\footnote{On average, subjects received £18.36, with a mean absolute deviation of £12.09. In the experiments reported in Section \ref{The Marshallian Path}, average earnings were £16.98 with a mean absolute deviation of £11.35. In general, sessions took around 1.5 hours, about half an hour of which was devoted to carefully explaining the rules to subjects, with the remaining time devoted to trading.}
\section{Results}\label{Results}
We begin by examining the transactions that occurred in our first two experimental sessions. Panel A of Figure \ref{sessions_1_2} displays the buyer valuation (top line), price (middle line) and cost (bottom line) associated with each of the transactions in Session 1. The band of equilibrium prices (£48--£52) is indicated by the dotted lines, and the end of each of the five rounds is indicated by a break. The left half of Panel A shows the transactions from the first half of the experiment (i.e., the symmetric treatment); whereas the right half of Panel A shows the transactions from the second half of the experiment (i.e. the low values treatment). Analogously, Panel B of Figure \ref{sessions_1_2} displays the valuations, prices and costs associated with the transactions from Session 2, which began with the low values treatment and proceeded to the symmetric treatment.
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_1_2}
\end{figure}
Three results are apparent. First, it is clear that shifting valuations and costs downward lowers observed prices, in violation of competitive equilibrium. Comparing the first halves of the separate sessions --- which is perhaps the cleanest comparison since it is uncomplicated by order effects --- we see that average prices are £56.0 (in the symmetric treatment) as opposed to £32.3 (in the low value treatment) ($p < 0.0001$).\footnote{The $p$-values in this and the next paragraph are generated by unpaired $t$-tests of the hypothesis of equal means.} We also see a similar trend within sessions. In the first session, shifting values and costs downward reduces average bids from £56.0 to £42.9 ($p < 0.0001$). In the second session, shifting values and costs upward increased average bids from £32.3 to £40.0 ($p < 0.01$). Therefore, we obtain strong evidence that competitive equilibrium preserving shifts do in fact shift prices, though the effects are substantially larger when comparing across sessions than when comparing within sessions (as one might expect given the `price stickiness' observed in the data).
While Figure \ref{sessions_1_2} solely displays data for the transactions, we can also see a similar pattern when examining the data on bids and asks. Comparing the first halves of the two sessions, we see that average bids/asks are £47.0/£67.9 in the symmetric treatment, as opposed to £26.6/£58.9 in the low values treatment ($p < 0.0001$, $p = 0.03$).\footnote{In all our analyses of the bid and ask data, we remove the handful of (rather optimistic) asks that exceed £1,000 (e.g. we drop one participant's offer to sell for £1 million). Including these asks only strengthens our conclusions.} In the first session, shifting values downward reduces average bids from £47.0 to £38.0 ($p < 0.01$) and reduces average asks from £67.9 to £48.5 ($p < 0.0001$). In the second session, shifting values up increases average bids from £26.6 to £31.7 ($p = 0.03$) and increases average asks from £59.0 to £153.2 ($p = 0.02$). We therefore conclude that, just as shifting valuations and costs downward reduces observed transaction prices, it also tends to reduce the bids and asks made by traders.
Our second main finding, which partially although not entirely follows from the first, is that prices are almost never at competitive equilibrium. In the first session (again, see Figure \ref{sessions_1_2}), prices start consistently above equilibrium, and then fall persistently below it. Moreover, not merely does competitive equilibrium fail as a literal description of the observed prices, but we can also reject a stochastic version of competitive equilibrium that allows for independent errors in every period ($p < 0.0001$, $p < 0.0001$).\footnote{To test this, we evaluate the null hypothesis that the price data were i.i.d. draws from a normal distribution with a mean of the closest competitive equilibrium price and a variance to be estimated from the data. Observe that both of these choices --- choosing the closest competitive equilibrium price along with allowing the variance to be fit ex post --- substantially stacks the deck in competitive equilibrium's favour, which then makes the clear rejection of competitive equilibrium even more striking.} The prices in our second session are also almost never at equilibrium, but in a different way. Now, prices are persistently below even the lowest competitive equilibrium price. Again, not only can we reject a rather literal interpretation of competitive equilibrium, but we can also reject a stochastic version that allows for independent errors ($p < 0.0001$, $p < 0.01$).
Our third main result is that prices do not seem to be converging to competitive equilibrium over time. In Session 1, there is little indication that prices are trending downwards (in the first half) or upwards (in the second half). Indeed, the average price changes are close to zero (£0.17 and -£0.14 in the first and second half respectively) and neither are statistically different from zero ($p = 0.84$, $p = 0.89$). In Session 2, there is a little more indication of an upward drift in prices, but again this is very weak: average changes are again close to zero (£0.83, £0.24) and again statistically insignificant ($p = 0.61$, $p = 0.94$). Even more strikingly, in neither session does competitive equilibrium seem to be an absorbing state. For example, although the price starts at equilibrium in Session 1 (with a first transaction price of £50), prices quickly drift upwards away from the equilibrium set. Similarly, although prices hit competitive equilibrium briefly in the third round of the second half of Session 2, they again move away from it. Thus, there is little appearance of convergence in about an hour's worth of trading.
In summary, we see that in double auctions with stationary value distributions (a property ensured through our use of a queue), resulting prices can remain far from competitive equilibrium even after long periods of trading, and show little sign of converging to equilibrium. Consistent with this, our competitive equilibrium preserving shifts substantially depress observed prices. In the next section, we turn to the question of what might explain this phenomenon.
\section{Understanding the monotonicity}\label{Understanding the monotonicity}
There are a number of double auction models which can rationalise the monotonicity documented in the previous section (i.e., that lower values/costs generate lower prices). To start with, consider the zero intelligence (ZI) model introduced by \cite{gode1993allocative}: buyers bid uniformly between 0 and their valuation, sellers bid uniformly between their valuation and some maximum, and trade occurs when the market bid and market ask cross (at a price equal to the earlier of the two offers). Under such assumptions, decreasing buyer and seller valuations leads to buyer bid distributions and seller ask distributions that are stochastically lower (in the sense of first-order stochastic dominance). As a result, it will tend to decrease observed prices.
To see how this works quantitatively, we conducted an extensive simulation of ZI trading under both of our experimental treatments. To operationalise ZI trading, one needs to specify a maximum ask; and we set this maximum at £100. We also assumed that at every point in time, one trader (either a buyer or seller) is chosen randomly to make an offer; and we then simulated a sequence of 10 million such offers (leading to about 800,000 market prices).\footnote{If the value distributions are held fixed as in our experiment, then nothing changes under ZI trading at the conclusion of a round. Thus, it is easiest to simply simulate a very large number of offers (and examine the resulting prices when these offers lead to trade) instead of simulating a large number of rounds.} The simulation reveals that, as one would expect, average ZI prices in the symmetric treatment are around £50. Meanwhile, average ZI prices in the low values treatment are around £35. Thus, the ZI model predicts that shifting values and costs down in the way done in our experiments should very substantially depress average prices.
While ZI can rationalise the monotonicity we observe in our data, it is doubtful that postulating random bids and asks can be said to explain the source of the monotonicity in any meaningful way. Fortunately, however, such monotonicity is also generated by optimising models. For instance, consider the model developed in \cite{gjerstad1998price}: buyers and sellers choose bids and asks in a way that maximises this period's expected pay-off. Ignoring integer constraints, the optimal bid/ask satisfies a first order condition, inspection of which reveals that optimal bids/asks are increasing in valuation/costs. Thus, the Gjerstad/Dickhaut model also predicts the monotonicity observed in our data.
Finally, we observe that this monotonicity also arises in more complicated optimising models. For example, consider the model of \cite{friedman1991simple}, in which buyers and sellers optimally choose reservation prices so as to balance the benefit of waiting for better bids/offers against the costs of running out of time. As Friedman remarks (p. 60), optimal reservation prices are monotone in valuations: this means, for example, that buyers with lower valuations are happy to accept lower offers. As a result, Friedman's optimal stopping logic also predicts the monotonicity that we observe.
In this section, our goal is not to select one model which is the `true' explanation for the observed monotonicity; and still less to discuss which model can best explain all aspects of double auction experiments (for efforts in this direction, see \cite{cason1996price} and \cite{ledyard_unpublished}). Rather, our goal is simply to argue that the observed monotonicity is nothing very mysterious: indeed, it is a simple consequence of both non-optimising as well as optimising double auction models.
\section{The Marshallian path}\label{The Marshallian Path}
Although the previous results demonstrate that double auctions need not generate equilibrium prices, one might suspect that this has something to do with the queuing procedure used in order to ensure that the value distribution remains fixed. Indeed, if traders are allowed to drop out without replacement as time progresses, then one may expect that prices approach competitive equilibrium due to a Marshallian path dynamic. While this route to equilibration has been discussed informally (see e.g. \cite{brewer2002behavioral}), we now formalise the dynamic in order to obtain a more rigorous understanding of its properties.
To this end, return to the environment in Section \ref{CE preserving shifts}, recalling that $F$ and $G$ denote the distribution of buyer and seller valuations respectively, and that $p^*$ denotes the unique competitive equilibrium price. Consider now a sequence of trades, indexed by $t \in [0, T]$. Let $v_b(t)$, $v_s(t)$ and $p(t)$ denote the buyer valuation, seller valuation, and price associated with trade $t$; and (with some abuse of notation) denote the corresponding functions by $v_b$, $v_s$ and $p$. We can now formalise the concept of a Marshallian path.
\begin{definition}
A \textit{Marshallian path} is a triple $(v_b, v_s, p)$ such that
\begin{enumerate}
\item For all $t \in [0, T]$, $v_b(t) = F^{-1}(1-t)$ and $v_s(t) = G^{-1}(t)$.
\item For all $i \in [0, 1]$, $i \in [0, T]$ if and only if $F^{-1}(1-i) \geq G^{-1}(i)$.
\item For all $t \in [0, T]$, $v_s(t) \leq p(t) \leq v_b(t)$.
\end{enumerate}
\end{definition}
As stated above, a Marshallian path is a sequence of trades that satisfies three conditions. To understand the first condition, start with the simpler equation $v_s(t) = G^{-1}(t)$ and invert it to get $ t = G(v_s(t))$. This says that the fraction of values that are below $v_s(t)$ is $t$; so at time $t$, the valuation of the seller about to engage in trade is at the $t$-th percentile of the seller value distribution.\footnote{Our usage of the word `percentile’ differs from the normal usage by a factor of 100: for example, we say `0.2-th percentile’ to mean the 20th percentile.} Likewise, the other equation reads $v_b(t) = F^{-1}(1-t)$, which may be inverted to yield $t = 1 - F[v_b(t)]$. This says that the fraction of buyer valuations that are \textit{above} $v_b(t)$ is $t$; so at time $t$, the valuation of the buyer about to engage in trade is at the $(1-t)$-th percentile of the buyer value distribution. Taken together, these assumptions say that trade takes place `in order’: at every point in time, trade occurs between the buyer with the highest valuation and the seller with the lowest valuation out of those `active' traders who remain in the market.
The second condition says that for any possible unit, that unit is traded if and only if the buyer's valuation for that unit exceeds the seller's valuation for that unit (assuming that these units are traded `in order’, in line with condition 1). In other words, trades take place if and only if there is some price at which they would be mutually beneficial. We will discuss the plausibility of this claim when we analyse the experimental results of this section.
The final condition says that, in actual fact, trade is mutually beneficial: the price of every trade lies between the valuation of the relevant buyer and the valuation (i.e. `cost') of the relevant seller. Observe that, although there is just one path of trades satisfying conditions 1 and 2, there are multiple mutually beneficial price paths. Thus, as far as prices are concerned, the Marshallian path is indeterminate; which is why we may, for reasons of pedantry, choose to speak of `a' rather than `the' Marshallian path.\footnote{On the other hand, there is a single path of \textit{trades} which counts as Marshallian, which is presumably why one finds discussion of ‘the’ Marshallian path in prior literature \citep{brewer2002behavioral, plott2013marshall}.}
\begin{proposition}\label{prop2}
If $(v_b, v_s, p)$ is a Marshallian path, then $p(T) = p^*$.
\end{proposition}
Proposition \ref{prop2} says that, if trade follows a Marshallian path, then the final trade must be transacted at the equilibrium price. To see why this is true, consider Figure \ref{fig:prop2}. Recall that the demand curve can be interpreted as the valuation of the marginal buyer (once buyers have been sorted from highest valuation to lowest), and the supply curve can be interpreted as the valuation of the marginal seller (assuming sellers have been sorted in order from lowest to highest valuation). As a result, to assume a Marshallian path is simply to assume that trade takes place in the rightward direction, starting with the buyer with the highest valuation and the seller with the lowest cost (condition 1). At every point in time, the price must lie between the buyer and seller valuation (condition 3), which means that the price path must lie between the demand and supply functions. Furthermore, trade occurs if and only if it could be mutually beneficial (condition 2), which means that the final trade is at the intersection of the demand and supply curves. At such a trade, the only mutually beneficial price is $p(T) =p^*$, so we obtain final period equilibration.\footnote{Incidentally, it is unclear whether this argument for equilibration actually appears anywhere in the work of Marshall. \cite{marshall1890principles} does present a theory of equilibration (see Book V, Chapter 3, Section 6), but this theory involves a quantity adjustment at disequilibrium prices. At the risk of giving Marshall credit for yet another idea that he did not originate (see \cite{humphrey1996marshallian}), we retain the phrase `Marshallian path' in deference to previous literature on this issue \citep{brewer2002behavioral, plott2013marshall}.}
\begin{figure}
\caption{A Marshallian path}
\label{fig:prop2}
\end{figure}
The previous result establishes that if trade follows a Marshallian path, then final period prices must be at competitive equilibrium. However, it does not address why trade should take place in exactly the order required for a Marshallian path (condition 1). While we will not address this issue in detail, there are two reasons why it might seem reasonable to make this assumption. First, prior literature shows that at least as far as experimental double auctions go, condition 1 is indeed a rough approximation of reality (for instance, see \cite{plott2013marshall}). Indeed, as we later show, this assumption matches up very well with our own experimental data. Second, condition 1 should arise for precisely the reasons discussed in Section \ref{Understanding the monotonicity}: it is, after all, simply an assertion that offers are monotone in valuations.
We should perhaps also emphasise that this result does \textit{not} extend to settings with stationary value distributions (like those studied experimentally in Section \ref{Results}). In such settings, Marshallian path logic would suggest that trade should take place between the buyer with the highest value (here, £52) and the seller with the lowest cost (here, £0): and indeed this is precisely what we observe in the low values treatment.\footnote{This phenomenon is also observed, although a little less strikingly, in the symmetric treatments.} However, as far as the logic of Proposition \ref{prop2} goes, there is no reason why they should trade at equilibrium prices if they are continually replaced once they leave the market. Moreover, under anything like an equal division of the surplus, we would expect prices to be considerably below equilibrium (as documented in Section \ref{Results}). Thus, not only does this result provide us with a reason to expect final period equilibration, but it provides us with a reason that is conspicuously absent in the case of stationary value distributions.
Motivated by our result, we now report the results of a series of additional experimental sessions held in order to determine whether the Marshallian path is indeed crucial for equilibrium convergence. These experiments followed exactly the same procedure as those outlined in Section \ref{Experimental Design}, with the exception that we removed the queue of buyers and sellers (so traders could drop out without replacement as time progressed). As in classical double auction experiments, each round now concluded once there were no longer any active traders who wanted to make or accept an offer.\footnote{Similarly to \cite{kimbrough2018testing}, the auctioneer asked: `Sellers, would any of you like to make any offers or accept the current market bid? Buyers, would any of you like to make any bids or accept the current market ask? Going once… going twice… the round has concluded'. See Appendix \ref{instructions} for the full instructions for the experiment.}
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_3_4}
\end{figure}
As before, we ran two sessions (Sessions 3 and 4), varying the order in which the sessions were presented. Session 3 began with the symmetric treatment, whereas Session 4 began with the low values treatment. Panels A and B of Figure \ref{sessions_3_4} display the main results for Session 3 and Session 4 respectively. As usual, the buyer valuation, price, and seller cost associated with each transaction are denoted by the top, middle, and bottom lines respectively. The conclusion of a round is indicated by a break; and the set of equilibrium prices is indicated by the dotted lines.
Two features of the panels are especially noteworthy. First, we still obtain evidence that our shifts depress prices, although the evidence is now more mixed. Comparing across the first halves of each session, which is again the cleanest comparison since it is uncomplicated by order effects, we see that average bids are £46.4 in the symmetric treatment as opposed to £35.3 in the low values treatment ($p < 0.001)$. Thus, shifting the values downward appears to substantially depress prices, at least in the short run. Looking within periods, we see that shifting the values and costs upward increased the average bids in Session 4, from £35.4 to £45.8 ($p < 0.001$). On the other hand, no such effect is observed within Session 3: average prices are actually a little higher in the low value treatment, although the difference is not statistically significant ($p = 0.36$).
Our second finding, which again is related to the first, is that the double auctions now demonstrate strong equilibrating tendencies. In Session 3, prices reach equilibrium by the last trade of the second round of the symmetric treatment and remain within the band of equilibrium prices for the rest of the session (even after valuations are shifted downward). In Session 4, prices failed to equilibrate despite half an hour's trading in the low values treatment. However, they do show a clear upward drift, and we do obtain an equilibrium price for one transaction. Moreover, switching to the symmetric treatment leads prices to more or less equilibrate by the end of the second round.
Given the previous findings, it is natural to conjecture that the low values treatment might equilibrate (even without the help of the symmetric treatment) if it were simply given enough time to do so. To check this conjecture, we ran two further experimental sessions (Sessions 5 and 6) which solely studied the low value treatment.\footnote{These experimental sessions were not contained in our original pre-registration.} To give our auctions the best chance of equilibration, we conducted nine rounds within each session (which corresponded to around one hour of trading time per session). To prevent subjects with unfavourable valuations from becoming bored by their lack of profitable trading opportunities, we re-shuffled the values and costs twice (once at the beginning of the fourth round, and once at the beginning of the seventh). We did not explicitly inform subjects that the distribution of values and costs had remained the same, though we also gave them no indication that it had changed.
\begin{figure}
\caption{Valuations, prices, and costs}
\label{sessions_5_6}
\end{figure}
Panels A and B of Figure \ref{sessions_5_6} display the results of our final two sessions. Several features of the data are evident. First, trade very clearly follows the order assumed by the Marshallian path. We find that trade follows the exact Marshallian order in 17 out of the 18 rounds (the corresponding figure is 47 out of 48 rounds if one includes the low valuation treatments from Sessions 3 and 4). These results are consistent with previous evidence on the order of trade within double auctions (see, e.g., \cite{cason1996price}, \cite{plott2013marshall}, \cite{lin2020evidence} and \cite{sherstyuk2021randomized}).\footnote{While other studies observe a similar phenomenon in terms of trading order, their results on trading order are rather less pronounced. Presumably, this is due to the fact that the two highest valuations are the same, and the two lowest costs are also the same, making it much easier to obtain a Marshallian trading order.}
Second, we observe striking evidence of third period equilibration. As predicted by our theory, \textit{all} third trades occur at a competitive equilibrium price. Moreover, whilst not all rounds lead to a third trade, a good portion of them do. As a result, we obtain final period equilibration in 12 out of 18 of the rounds (this corresponding figure is 16 out of 38 rounds if one includes the low values treatments from Sessions 3 and 4). Consistent with our theory, prices are much less likely to be at equilibrium for non-third period trades. For example, in Session 5, we \textit{never} observe equilibrium prices for first and second period trades.
Third, we also obtain evidence that equilibration is possible even for non-final periods. This is certainly not evident in Session 5, where (as noted) we obtain equilibrium prices only for third period trades. On the other hand, in Session 6, we obtain equilibrium prices for all trades in the fifth, sixth, eighth and ninth rounds. We discuss why this may be below.
Altogether, our results are highly consistent with the theoretical result presented above. In essentially all periods, the first trades happen between buyers with the highest valuations (here £52) and sellers with the lowest cost (here £0): this validates condition 1 of the Marshallian path. Once these two trades have been executed, the only remaining mutually beneficial traders are a buyer with a valuation of £52 and a seller with a cost of £48. Thus, if all mutually beneficial trades are going to occur, a deal must be struck in the equilibrium price range of £48 -- £52. This is exactly what we see in our data (again, see Figure \ref{sessions_5_6}): it is as if a force, namely a large jump in the valuation of the marginal seller, is `squeezing' the realised price into the equilibrium set.
While our results are broadly consistent with our theory, they do leave at least two unanswered questions. The first question is why not all mutually beneficial trades are made within certain rounds. From a mechanical point of view, one can see that the obstacle is essentially on the buyers' side. In all but one of the third period trades, seller offers fall to the equilibrium range, and it is the market bid which potentially fails to rise high enough for a deal to be struck (see Figure \ref{bids_asks} for the evolution of the market bids and asks in such situations). The question is then why third period buyers with a valuation of £52 refuse to trade at equilibrium prices, thereby `leaving money on the table'.
While we do not attempt to definitively answer this question, at least two explanations are suggested by comments made by buyers in post-experimental discussion. First, buyers may have had a strategic motive: by refusing to pay equilibrium prices, they may have been attempting to signal a reluctance to pay such prices in future rounds, thereby encouraging lower future offers from sellers.\footnote{Ironically, this would be a reason why repetition of rounds might actually hinder equilibration: such an effect would not be present in a single round double auction.} Second, buyers may have refused to accept sellers' offers of equilibrium prices on the mistaken belief that such offers would deliver almost all the surplus to the relevant seller. In rounds that generated only two trades, prices tended to be in the £35 -- £40 range, with an average of about £37. As a result, buyers might have reasonably formed the expectation that sellers' values are rather low (about £22 under the assumption of equal surplus sharing) and therefore become quite frustrated when the remaining seller started to demand competitive equilibrium prices. In such a situation, a buyer might be willing to forfeit a few pounds in profit in return for the pleasure of punishing what they take to be greediness on a seller's part (see \cite{rabin1993incorporating} and \cite{fehr1999theory} for formal models in this vein).
A final question is why prices can potentially become `stuck' at equilibrium, even for non-third period trades. For concreteness, let us consider Session 6 (see panel B of Figure \ref{sessions_5_6}) since this phenomenon does not arise in Session 5. By Marshallian path logic, one would expect all third period trades to be at equilibrium, and indeed this is precisely what we observe. However, there is no special reason, at least so far as the logic of Proposition \ref{prop2} goes, to expect equilibrium prices for earlier trades; and under anything like equal surplus sharing, one would expect the realised prices to be substantially lower. Strikingly, however, we see that once prices have been pushed to equilibrium repeatedly by the force of the Marshallian path, they may persist at equilibrium (or may not, as in Session 5). The question is then what might explain this.
While we do not attempt to answer this question formally, it is not too hard to see how the Marshallian path logic might extend to trades that do not occur in the final period. Suppose that the first two trades occur at reasonably low prices, say £35, and the third period price jumps (by the logic of Proposition \ref{prop2}) to a competitive equilibrium price. Over time, traders should notice this pattern --- and indeed many traders reported noticing this in post-experimental discussions. Given that they expect the price to rise in the final period, buyers should be especially keen to trade earlier than their competitors. This in turn should induce them to submit higher bids and to accept higher asks for early period offers. Similarly, given that the price is expected to jump in the third period, sellers should be keen to trade late, which should induce them to only offer very high asks or accept very high bids in the early periods. Taken together, these forces explain how the final period equilibration delivered by the Marshallian path might eventually drag earlier round prices up to equilibrium levels.\footnote{Importantly, a similar argument can apply even in rounds which lack a third period trade. Such third periods typically involve a substantial increase in buyer bids and seller asks relative to previous periods. As a result, even those periods that do not lead to a trade can facilitate equilibration.}
\section{Concluding remarks}\label{Concluding remarks}
In this paper, we revisit the link between competitive equilibrium and the double auction. We begin by showing that competitive equilibrium generates a set of highly counterintuitive predictions. Specifically, it predicts that prices can remain unchanged following a certain class of downward shifts to buyer and seller valuations. We find that in double auctions with stationary value distributions, these predictions are strongly falsified; and more generally that competitive equilibrium is a poor description of outcomes in such auctions. On the other hand, we also find that the effectiveness of our counterexamples is blunted in double auctions in which traders may drop out (without replacement) as time progresses. Taken together, this pair of findings imply that whether value distributions are held stationary is a crucial determinant of whether prices converge to competitive equilibrium; which in turn suggests that the Marshallian path is a key driver of equilibration in double auctions.
Despite the long history of double auction research, we believe that our findings open up several new avenues for investigation. First, further experimental work on the potential importance of the stationarity of the value distribution is clearly needed. Indeed, we are aware of only one other paper on this issue \citep{brewer2002behavioral}; and that paper reaches the rather contrasting conclusion that double auctions can converge even with stationary value distributions.\footnote{In part, the apparent difference between this paper and ours may be somewhat illusory. Examining Figures 7 -- 9 in \cite{brewer2002behavioral}, one sees that prices are persistently above equilibrium, and then (following the shifts) persistently below equilibrium. While we would not classify this as `convergence to equilibrium', it does count as convergence under the relatively undemanding definition used by \cite{brewer2002behavioral} (see p. 191 for details).} Given the paucity of studies on this issue and the apparently mixed nature of the evidence, it is vital that further experiments are conducted to verify whether the issue of stationarity is indeed as critical as we claim.
On the theoretical front, it may also be useful to explicitly develop models of double auction behaviour in stationary environments. Classical double auction models, e.g. \cite{easley1986theories}, \cite{friedman1991simple} and \cite{gjerstad1998price}, explicitly address the situation in which traders drop out without replacement as time progresses, and it would be interesting to investigate the extent to which their insights carry over to the stationary setting. Indeed, the stationary setting would appear to be rather more tractable from a modelling perspective, since it may be viewed (at least very roughly) as a repeated version of the much simpler $k$-double auction. As a result, the insights of \cite{chatterjee1983bargaining}, \cite{jackson2005existence}, \cite{reny2006toward}, \cite{fudenberg2007existence} and others may prove highly relevant.
Finally, it might be valuable to further study what precisely hinders equilibration in the `low values treatment' investigated in this paper. We have informally sketched two alternatives: that buyers reject equilibrium offers on the (mistaken) belief that they allocate almost all the surplus to sellers, or otherwise that buyers reject these offers strategically as a means of generating higher offers in subsequent rounds. While we have not attempted to learn which of these is the key driver, this would seem straightforward to check experimentally. For example, the second channel is possible only when rounds are repeated, so can be easily ‘turned off’ by conducting double auctions with a single incentivised round.
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\begin{appendices}
\section{Proofs} \label{proofs}
\begin{proof}[Proof of Proposition \ref{prop1}]
We begin by arguing that $p^*$ remains \textit{an} equilibrium price when valuations are distributed according to $T_b(V_b) = V_b'$ and $T_s(V_s) = V_s'$. To this end, define $\mathcal{B} = (p^* - \epsilon^-, p^* + \epsilon^+)$ and fix some $p \in \mathcal{B}$. By the law of total probability,
\begin{equation}\label{totalprob}
\begin{split}
P(V'_b \leq p) &= P(V'_b \leq p|V_b \leq p^* - \epsilon^-)P(V_b \leq p^* - \epsilon^-) + P(V'_b \leq p|V_b \in \mathcal{B} )P(V_b \in \mathcal{B} ) \\ &+ P(V'_b \leq p |V_b \geq p^* + \epsilon^+)P(V_b \geq p^* + \epsilon^+)
\end{split}
\end{equation}
Since $T_b(V_b) = V_b'$, where $T_b$ is a CE preserving demand contraction,
\begin{itemize}
\item If $V_b \leq p^* - \epsilon^-$, then $V'_b \leq V_b \leq p^* - \epsilon^- \leq p$, so $P(V'_b \leq p|V_b \leq p^* - \epsilon^-) = 1$
\item If $V_b \in \mathcal{B}$, then $V_b' = V_b$ and so $P(V'_b \leq p|V_b \in \mathcal{B} ) = P(V_b \leq p|V_b \in \mathcal{B} )$
\item If $V_b \geq p^* + \epsilon^+$, then $V_b' \geq p^* + \epsilon^+ > p$, so $P(V'_b \leq p |V_b \geq p^* + \epsilon^+) = 0$
\end{itemize}
Inserting these equalities into (\ref{totalprob}), we obtain
\begin{equation}
\begin{split}
P(V'_b \leq p) &= P(V_b \leq p^* - \epsilon^-) + P(V_b \leq p|V_b \in \mathcal{B} )P(V_b \in \mathcal{B} ) \\
&= P(V_b \leq p^* - \epsilon^-) + P(V_b \leq p \wedge V_b \in \mathcal{B} ) \\
&= P(V_b \leq p^* - \epsilon^-) +P(p^* - \epsilon^- < V_b \leq p) \\
&= P(V_b \leq p)
\end{split}
\end{equation}
Let $F'$ and $G'$ denote the distributions of $V_b'$ and $V_s'$; let $d'(p)$ and $s'(p)$ denote the demand and supply functions generated by these distributions; and define $e'(p) \equiv d'(p) - s'(p)$.
The argument just given reveals that $F'(p) = F(p)$ for any $p \in \mathcal{B}$. Therefore, $d'(p) = 1 - F'(p) = 1 - F(p) = d(p)$ (for $p \in \mathcal{B}$). In particular, $d'(p^*) = d(p^*)$. Likewise, if we replace $F$ with $G$ in the previous argument, we see that $G'(p) = G(p)$ for any $p \in \mathcal{B}$. Hence, $s'(p) = G'(p) = G(p) = s(p)$ at such prices; and in particular, $s'(p^*) = s(p^*)$. Since $d(p^*) = s(p^*)$ by definition, this means that $d'(p^*) = s'(p^*)$, i.e. $p^*$ remains a CE.
To see that the CE remains unique, fix some $p < p^*$ and consider the following cases:
Case 1: $p \in \mathcal{B}$. Then $e'(p) = e(p)$ (shown above), and $e(p) > e(p^*) = 0$ (since $p < p^*$ and $e$ is strictly decreasing). This means that $e'(p) > 0$, i.e. no such prices clear the market.
Case 2: $p \notin \mathcal{B}$. Since $p < p^*$, this means that $p \leq p^* - \epsilon^-$. Now define $\bar{p} \equiv p^* - 0.5\epsilon^-$. Plainly, $\bar{p} \in \mathcal{B}$ and $p < p^*$, so $e'(\bar{p}) = e(\bar{p}) > 0$. Moreover, since $e'(p) = 1 - F'(p) - G'(p)$ where $F'$ and $G'$ are CDFs, $e'$ is weakly decreasing over $\mathbb{R}^+$. Thus, if $p \leq p^* - \epsilon^- < \bar{p}$, then $e'(p) \geq e'(\bar{p}) = e(\bar{p}) > 0$, so no such prices can clear the market either.
Since these cases exhaust the possibilities, we see that no prices $p < p^*$ can be CE. By an analogous argument, one can also rule out prices $p > p^*$. We conclude that $p^*$ must remain the unique competitive equilibrium price.\end{proof}
\begin{proof}[Proof of Proposition \ref{prop2}]
First, we show that $v_b(T) = v_s(T)$. To see this, recall condition 2 of the Marshallian path definition: for every $i \in [0, 1]$, $i \in [0, T]$ if and only if $F^{-1}(1-i) \geq G^{-1}(i)$. Define $\phi(i) = F^{-1}(1-i) - G^{-1}(i)$, so our inequality is $\phi(i) \geq 0$. Observe that $\phi(0) = F^{-1}(1) - G^{-1}(0) = \bar{v}_b$ and $\phi(1) = F^{-1}(0) - G^{-1}(1) = -\bar{v}_s$. Also, $\phi$ is continuous and strictly decreasing in $i$. Therefore, letting $i^*$ denote the unique root of $\phi$, the set of $i \in [0, 1]$ such that $\phi(i) \geq 0$ is the set $[0, i^*]$. Hence, $T = i^*$ and so $\phi(T) = 0$, or $F^{-1}(1-T) = G^{-1}(T)$. Using condition 1, this finally yields $v_b(T) = v_s(T)$.
By condition 3, we see that $v_b(T) \geq p(T) \geq v_s(T)$. However, $v_b(T) = v_s(T)$. Therefore, $p(T) = v_b(T) = v_s(T)$.
Finally, invert condition 1 to get $F[v_B(T)] = 1-T$ and $G[p(T)] = T$. Using $p(T) = v_b(T) = v_s(T)$, we infer that $F[p(T)] = 1-T$ and $G[p(T)] = T$, and so
\begin{equation}G[p(T)] = 1 - F[p(T)] \end{equation}
But this is precisely the equation whose unique solution is $p^*!$ Hence, $p(T) = p^*$.\end{proof}
\section{Additional tables and figures} \label{tables_figures}
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\begin{table}[H]
\centering
\caption{Overview of the experimental sessions}\label{overview}
\begin{threeparttable}[h]
\begin{tabular}{cclcc}
\hline
Session & Queue? & \hspace{3em}Treatments & Rounds & Participants \\ \hline
1 & Yes & Symmetric then low values & 10 & 18 \\
2 & Yes & Low values then symmetric & 10 & 18 \\
3 & No & Symmetric then low values & 10 & 10 \\
4 & No & Low values then symmetric & 10 & 10 \\
5 & No & Low values only & 9 & 10 \\
6 & No & Low values only & 9 & 10 \\ \hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \hspace{-0.2em}\textit{Notes}: This table describes the differences between our experimental treatments. The second column specifies whether a treatment used a queue of buyers and sellers, and the third column specifies which treatments were run. The fourth and fifth columns specify the number of rounds and participants in the session.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}
\caption{Market bids and asks}
\label{bids_asks}
\end{figure}
\section{Experimental instructions} \label{instructions}
\subsection{Instruction for Buyers (Sessions 1 and 2)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your buyer number. B1 means `Buyer 1', B2 means ‘Buyer 2’, and so forth. Please wear your ID card visibly at all times.
\end{itemize}
\textit{Roles}
\begin{itemize}
\item There are two types of buyers in this experiment: active buyers and pending buyers. If your buyer number is between 1 and 5 inclusive, then you are an active buyer and will be sitting in the main trading area. On the other hand, if your buyer number is 6 or higher, then you are a pending buyer and will be sitting in the queue.
\item Active buyers are free to trade from the very start of a trading period. However, pending buyers may only trade after active buyers have made a trade and dropped out of the market (see elaboration below).
\end{itemize}
\textit{Valuations}
\begin{itemize}
\item If you are an active buyer, you will have a card marked ‘Valuation’ in front of you. Examine the number on the back of this card, being careful not to let any other participants see this number. This number is your ‘valuation’ for the fictitious commodity to be traded. Memorise your valuation, turn your ID card back face-down (so the valuation is hidden), and do not reveal your valuation to anybody else!
\item If you are a pending buyer, then you have not yet been assigned a valuation. However, you will acquire a valuation as soon as an active buyer has dropped out and you have taken that buyer’s place (and valuation).
\item If you manage to make a trade, you will receive your valuation minus the price that you agreed to pay (this assumes that the trade ‘counts’ — see discussion below). For example, if your valuation is £55 and you agree to a price of £20, your net earnings are £35.
\item Notice that your valuation represents the most that you should be conceivably willing to pay to make a trade. For example, if your valuation is £40, then you should never pay more than £40 to buy a unit: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, active buyers may offer to buy at a particular price – this is called making a ‘bid’. To make a bid, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you want to bid. For example, if you are buyer 3 and you want to bid £40, say ‘buyer 3 bids 40’.
\item In the sequence of market activity leading up to a trade, each bid must be higher than the current bid. For example, if one buyer bids £20, then all subsequent buyers need to bid more than £20.
\item All bids need to be whole numbers. For example, while you can bid £30, you cannot bid £30.14.
\item Just as active buyers may make bids (at any point in time), active sellers can make ‘asks’ (at any point in time). For example, if a seller ‘asks’ for £70, that means that she is willing to sell for £70. Each new ask must be lower than the current ask, so asks must ‘improve’ over time.
\item At any stage, active buyers may accept an ask that has been made by a seller. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current ask. For example, if you are buyer 2 and want to accept an ask of £70, say ‘buyer 2 accepts 70’.
\item If multiple buyers want to make a bid or accept an ask, priority will be given to the buyer who has raised their hand first.
\item If you have made a trade, you become inactive and cannot make any further trades in that round. At this point, you should move to the inactive area and allow your place to be taken by the buyer at the front of the queue (or go to the back of the queue if you were previously a pending buyer). That new buyer will acquire your valuation (so should examine the card on the desk to see what that valuation is).
\item While each bid needs to be higher than the previous bid, everything is reset following a trade. In other words, once a trade has been made, active buyers are free to submit any bid that they choose – even if that bid is lower than previously submitted bids.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Buyer 3 wants to bid £40 so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 3 bids 40’.
\item Buyer 1 wants to outbid her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 1 bids 43’.
\item Seller 4 wants to offer to sell for £95, so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 4 asks 95’.
\item Buyer 2 wants to accept the ask, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 2 accepts 95’. A trade has occurred (at a price of £95).
\item Since buyer 2 has just traded with seller 4, they both become inactive and should move to the inactive area. Two traders from the queue take their place and acquire their valuation and cost respectively. Participants then continue to bargain over prices. Since a trade has just occurred, subsequent bids are no longer constrained to be above £43 (the previous leading bid).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until the auctioneer chooses to end the round.
\item Within each round, you are only allowed to purchase (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new valuation for the fictitious commodity. So even if your current valuation is rather low, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There are two types of buyers: active buyers (IDs 1-5) and pending buyers (IDs 6 or higher). Once active buyers have made a trade, the pending buyer at the front of the queue takes their place and their valuation.
\item Active buyers may make bids or accept asks at any point in time. Similarly, active sellers may make asks or accept bids at any point in time.
\item If you want to make a bid or accept an ask, you need to first raise your hand.
\item Bids must be whole numbers; and new bids must be greater than previous bids (until a trade is made).
\item If you manage to make a trade, you earn your valuation minus the price you agreed to pay (assuming that this trade is selected to ‘count’ for your earnings).
\item You can purchase up to one unit within every round.
\end{itemize}
\subsection{Instructions for Sellers (Sessions 1 and 2)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your seller number. S1 means ‘Seller 1’, S2 means ‘Seller 2’, and so forth. Please wear your ID card visibly at all times.
\end{itemize}
\textit{Roles}
\begin{itemize}
\item There are two types of sellers in this experiment: active sellers and pending sellers. If your seller number is between 1 and 5 inclusive, then you are an active seller and will be sitting in the main trading area. On the other hand, if your seller number is 6 or higher, then you are a pending seller and will be sitting in the queue.
\item Active sellers are free to trade from the very start of a trading period. However, pending sellers may only trade after active sellers have made a trade and dropped out of the market (see elaboration below).
\end{itemize}
\textit{Costs}
\begin{itemize}
\item If you are an active seller, you will have a card marked ‘Cost’ in front of you. Examine the number on the back of this card, being careful not to let any other participants see this number. This number is how much it would cost you to produce and sell a unit of the fictitious commodity to be traded. Memorise your cost, turn your ID card back face-down (so the cost is hidden), and do not reveal your cost to anybody else!
\item If you are a pending seller, then you have not yet been assigned a cost. However, you will acquire a cost as soon as an active seller has dropped out and you have taken that seller’s place (and cost).
\item If you manage to make a trade, you will receive the price paid by the buyer minus your cost (this assumes that the trade ‘counts’ — see discussion below). For example, if you sell for a price of £55 and your cost is £20, your net earnings are £35.
\item Notice that your cost represents the least that you should be conceivably willing to sell for. For example, if your cost is £40, then you should never sell for less than £40: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, active sellers may offer to sell at a particular price – this is called making an ‘ask’. To make an ask, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you are asking for. For example, if you are seller 3 and you want to ask for £60, say ‘seller 3 asks 60’.
\item In the sequence of market activity leading up to a trade, each ask must be lower than the current ask. For example, if one seller asks £80, then all subsequent sellers need to ask for less than £80.
\item All asks need to be whole numbers. For example, while you can ask £70, you cannot ask £70.14.
\item Just as active sellers may make asks (at any point in time), active buyers can make bids (at any point in time). For example, if a buyer bids £70, that means that she is willing to pay £70. Each new bid must be higher than the current bid, so bids must ‘improve’ over time.
\item At any stage, active sellers may accept a bid that has been made by a buyer. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current bid. For example, if you are seller 2 and want to accept a bid of £70, say ‘seller 2 accepts 70’.
\item If multiple sellers want to make an ask or accept a bid, priority will be given to the seller who has raised their hand first.
\item If you have made a trade, you become inactive and cannot make any further trades in that round. At this point, you should move to the inactive area and allow your place to be taken by the seller at the front of the queue (or go to the back of the queue if you were previously a pending seller). That new seller will acquire your cost (so should examine the card on the desk to see what that cost is).
\item While each ask needs to be lower than the previous ask, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any ask that you choose – even if that ask is higher than previously submitted asks.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Seller 3 wants to ask for £60 so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 3 asks 60’.
\item Seller 1 wants to undercut her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 1 asks 57’.
\item Buyer 4 wants to bid £5, so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 4 bids 5’.
\item Seller 2 wants to accept the bid, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 2 accepts 5’. A trade has occurred (at a price of £5).
\item Since seller 2 has just traded with buyer 4, they both become inactive and should move to the inactive area. Two traders from the queue take their place and acquire their valuation and cost respectively. Participants then continue to bargain over prices. Since a trade has just occurred, subsequent asks are no longer constrained to be below £57 (the previous lowest).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until the auctioneer chooses to end the round.
\item Within each round, you are only allowed to sell (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new cost for the fictitious commodity. So even if your current cost is rather high, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There are two types of sellers: active sellers (IDs 1-5) and pending sellers (IDs 6 or higher). Once active sellers have made a trade, the pending seller at the front of the queue takes their place and their cost.
\item Active sellers may make asks or accept bids at any point in time. Similarly, active buyers may make bids or accept asks at any point in time.
\item If you want to make an ask or to accept a bid, you need to first raise your hand.
\item Asks must be whole numbers; and new asks must be lower than previous asks (until a trade is made).
\item If you manage to make a trade, you earn the price paid by the buyer minus your cost (assuming that this trade is selected to ‘count’ for your earnings).
\item You can sell up to one unit within every round.
\end{itemize}
\subsection{Instructions for Buyers (Sessions 3 -- 6)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your buyer number. B1 means ‘Buyer 1’, B2 means ‘Buyer 2’, and so forth. Please keep your ID card visible at all times.
\end{itemize}
\textit{Valuations}
\begin{itemize}
\item Please examine the number of the back of your ID card, being careful not to let any other participants see this number. This number is your ‘valuation’ for the fictitious commodity to be traded. Memorise your valuation, turn your ID card back face-down (so the valuation is hidden), and do not reveal your valuation to anybody else!
\item If you manage to make a trade, you will receive your valuation minus the price that you agreed to pay (this assumes that the trade ‘counts’ — see discussion below). For example, if your valuation is £55 and you agree to a price of £20, your net earnings are £35.
\item Notice that your valuation represents the most that you should be conceivably willing to pay to make a trade. For example, if your valuation is £40, then you should never pay more than £40 to buy a unit: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, you may offer to buy at a particular price – this is called making a ‘bid’. To make a bid, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you want to bid. For example, if you are buyer 3 and you want to bid £40, say ‘buyer 3 bids 40’.
\item In the sequence of market activity leading up to a trade, each bid must be higher than the current bid. For example, if one buyer bids £20, then all subsequent buyers need to bid more than £20.
\item All bids need to be whole numbers. For example, while you can bid £30, you cannot bid £30.14.
\item Just as buyers may make bids (at any point in time), sellers can make ‘asks’ (at any point in time). For example, if a seller ‘asks’ for £70, that means that she is willing to sell for £70. Each new ask must be lower than the current ask, so asks must ‘improve’ over time.
\item At any stage, you may accept an ask that has been made by a seller. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current ask. For example, if you are buyer 2 and want to accept an ask of £70, say ‘buyer 2 accepts 70’.
\item If multiple buyers want to make a bid or accept an ask, priority will be given to the buyer who has raised their hand first.
\item Once you have made a trade, you become inactive and cannot make any further trades in that round. Likewise, each seller is only able to make at most one trade within a round – so it is as if they possess just one unit of the fictitious commodity.
\item While each bid needs to be higher than the previous bid, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any bid that you choose – even if that bid is lower than previously submitted bids.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Buyer 3 wants to bid £40 so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 3 bids 40’.
\item Buyer 1 wants to outbid her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 1 bids 43’.
\item Seller 4 wants to offer to sell for £95, so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 4 asks 95’.
\item Buyer 2 wants to accept the ask, so raises his hand. Once he is pointed at by the auctioneer, he says ‘buyer 2 accepts 95’. A trade has occurred (at a price of £95).
\item Since buyer 2 has just traded with seller 4, they both become inactive and remain silent for the rest of the round. Meanwhile, other participants continue to bargain over prices. Since a trade has just occurred, subsequent bids are no longer constrained to be above £43 (the previous leading bid).
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until there is a pause in market activity for roughly 20 seconds. At that point, the auctioneer will ask buyers if they would like to make any new bids, or would like to accept the current market ask. The auctioneer will then ask sellers if they would like to make any new asks, or would like to accept the current market bid. If all traders remain silent, then the auctioneer will close the market and conclude the round.
\item Within each round, you are only allowed to purchase (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new valuation for the fictitious commodity. So even if your current valuation is rather low, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There is a group of buyers, and a group of sellers. Buyers may make bids or accept asks at any point in time. Similarly, sellers may make asks or accept bids at any point in time.
\item If you want to make a bid or accept an ask, you need to first raise your hand.
\item Bids must be whole numbers; and new bids must be greater than previous bids (until a trade is made).
\item If you manage to make a trade, you earn your valuation minus the price you agreed to pay (assuming that this trade is selected to ‘count’ for your earnings).
\item You can purchase up to one unit within every round.
\end{itemize}
\subsection{Instructions for Sellers (Sessions 3 -- 6)}
Welcome to the experiment! Please read the following instructions as carefully as possible.
\textit{Preliminaries}
\begin{itemize}
\item Please do not talk to your fellow participants at any stage. Talking may result in a loss of experimental earnings.
\item You will have just received an ID card specifying your seller number. S1 means ‘Seller 1’, S2 means ‘Seller 2’, and so forth. Please keep your ID card visible at all times.
\end{itemize}
\textit{Costs}
\begin{itemize}
\item Please examine the number on the back of your ID card, being careful not to let any other participants see this number. This number is how much it would cost you to produce and sell one unit of the fictitious commodity to be traded. Memorise your cost, turn your ID card back face-down (so the cost is hidden), and do not reveal your cost to anybody else!
\item If you manage to make a trade, you will receive the price paid by the buyer minus your cost (this assumes that the trade ‘counts’ — see discussion below). For example, if you sell for a price of £55 and your cost is £20, your net earnings are £35.
\item Notice that your cost represents the least that you should be conceivably willing to sell for. For example, if your cost is £40, then you should never sell for less than £40: doing so would just lose you money!
\end{itemize}
\textit{Trading}
\begin{itemize}
\item At any point, you may offer to sell at a particular price – this is called making an ‘ask’. To make an ask, raise your hand. Once the auctioneer has pointed at you, state your identity along with how much you are asking for. For example, if you are seller 3 and you want to ask for £60, say ‘seller 3 asks 60’.
\item In the sequence of market activity leading up to a trade, each ask must be lower than the current ask. For example, if one seller asks £80, then subsequent sellers need to ask for less than £80.
\item All asks need to be whole numbers. For example, while you can ask for £70, you cannot ask for £70.14.
\item Just as sellers may make asks (at any point in time), buyers can make bids (at any point in time). For example, if a buyer bids £70, that means that she is willing to pay £70. Each new bid must be higher than the current bid, so bids must ‘improve’ over time.
\item At any stage, you may accept a bid that has been made by a buyer. To do this, raise your hand and wait until the auctioneer points at you. Once this has occurred, state your identity and that you want to accept the current bid. For example, if you are seller 2 and want to accept a bid of £70, say ‘seller 2 accepts 70’.
\item If multiple sellers want to make an ask or accept a bid, priority will be given to the seller who has raised their hand first.
\item Once you have made a trade, you become inactive and cannot make any further trades in that round --- so it is as if you possess just one unit of the fictitious commodity. Likewise, each buyer is only able to make at most one trade within a round.
\item While each ask needs to be lower than the previous ask, everything is reset following a trade. In other words, once a trade has been made, you are free to submit any ask that you choose – even if that ask is higher than previously submitted asks.
\end{itemize}
\textit{Example}
\begin{itemize}
\item Seller 3 wants to ask for £60 so raises her hand. Once she is pointed at by the auctioneer, she says ‘seller 3 asks 60’.
\item Seller 1 wants to undercut her, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 1 asks 57’.
\item Buyer 4 wants to bid £5, so raises her hand. Once she is pointed at by the auctioneer, she says ‘buyer 4 bids 5’.
\item Seller 2 wants to accept the bid, so raises his hand. Once he is pointed at by the auctioneer, he says ‘seller 2 accepts 5’. A trade has occurred (at a price of £5).
\item Since seller 2 has just traded with buyer 4, they both become inactive and remain silent for the rest of the round. Meanwhile, other participants continue to bargain over prices. Since a trade has just occurred, subsequent asks are no longer constrained to be below £57 (the previous lowest ask).
\item
\end{itemize}
\textit{Rounds}
\begin{itemize}
\item Trade will continue in this fashion until there is a pause in market activity for roughly 20 seconds. At that point, the auctioneer will ask buyers if they would like to make any new bids, or would like to accept the current market ask. The auctioneer will then ask sellers if they would like to make any new asks, or would like to accept the current market bid. If all traders remain silent, then the auctioneer will close the market and conclude the round.
\item Within each round, you are only allowed to sell (at most) one unit of the commodity. However, each round starts afresh: so even if you have made a trade in a particular round, you are free to make trades in subsequent rounds.
\item After several rounds of trading have concluded, you will be given a new cost for the fictitious commodity. So even if your current cost is rather high, you might be luckier later on!
\item At the end of the experiment, one round will be randomly selected to ‘count’ for calculating your earnings. Since any of the rounds could turn out to be the one that counts, you should do your best to maximise your net earnings within each round.
\end{itemize}
\textit{Summary}
\begin{itemize}
\item There is a group of sellers, and a group of buyers. Sellers may make asks or accept bids at any point in time. Similarly, buyers may make bids or accept asks at any point in time.
\item If you want to make an ask or to accept a bid, you need to first raise your hand.
\item Asks must be whole numbers; and new asks must be lower than previous asks (until a trade is made).
\item If you manage to make a trade, you earn the price paid by the buyer minus your cost (assuming that this trade is selected to ‘count’ for your earnings).
\item You can sell up to one unit within every round.
\end{itemize}
\end{appendices}
\end{document} |
\begin{equation}gin{document}
\title{Noether's Theorem and the Willmore Functional}
\author{Yann Bernard\footnote{Departement Mathematik, ETH-Zentrum, 8093 Z\"urich, Switzerland.}}
\date{ }
\maketitle
{\bf Abstract :} {\it Noether's theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivi\`ere. Several examples are considered in details.}
\reset
\section{Introduction}
Prior to establishing herself as a leading German mathematician of the early 20$^\text{th}$ century through her seminal work in abstract algebra, Emmy Noether had already made a significant contribution to variational calculus and its applications to physics. Proved in 1915 and published in 1918 \cite{Noe}, what was to become known as {\it Noether's theorem}, is a fundamental tool in modern theoretical physics and in the calculus of variations \cite{GM, Kos, Run}. Generalizing the idea of constants of motion found in classical mechanics, Noether's theorem provides a deep connection between symmetries and conservation laws. It is a recipe to construct a divergence-free vector field from a solution of a variational problem whose corresponding action (i.e. energy) is invariant under a continuous symmetry. For example, in 1-dimensional problems where the independent variable represents time, these vector fields are quantities which are conserved in time, such as the total energy, the linear momentum, or the angular momentum. We now precisely state one version of Noether's theorem\footnote{further generalizations may be found {\it inter alia} in \cite{Kos, Run}.}.
Let $\Omega$ be an open subset of $\mathcal{D}\subset\mathbb{R}^s$, and let $\mathcal{M}\subset\mathbb{R}^m$. Suppose that
$$
L\,:\,\Big\{(x,q,p)\,\big|\,(x,q)\in\mathcal{D}\times\mathcal{M}\;,\;p\in T_q\mathcal{M}\otimes T^*_x\mathcal{D} \Big\}\;\longmapsto\;\mathbb{R}
$$
is a continuously differentiable function. Choosing a $C^1$ density measure $d\mu(x)$ on $\Omega$, we can define the {\it action functional}
$$
\mathcal{L}(u)\;:=\;\int_\Omega L(x,u(x),du(x))\,d\mu(x)
$$
on the set of maps $u\in C^1(\Omega,\mathcal{M})$. A tangent vector field $X$ on $\mathcal{M}$ is called an {\it infinitesimal symmetry} for $\mathcal{L}$ if it satisfies
$$
\dfrac{\partial L}{\partial q^i}(x,q,p)X^i(q)+\dfrac{\partial L}{\partial p^i_\alpha}(x,q,p)\dfrac{\partial X^i}{\partial q^j}(q)p^j_\alpha\;=\;0\:.
$$
\begin{equation}gin{Th}
Let $X$ be a Lipschitz tangent vector field on $\mathcal{M}$ which is an infinitesimal symmetry for the action $\mathcal{L}$. If $u:\Omega\rightarrow\mathcal{M}$ is a critical point of $\mathcal{L}$, then
\begin{equation}\lambdabel{noether}
\sum_{\alpha=1}^{s}\,\dfrac{\partial}{\partial x^\alpha}\bigg(\rho(x)X^j(u)\dfrac{\partial L}{\partial p^j_\alpha}(x,u,du) \bigg)\;=\;0\:,
\end{equation}
where $\{x^\alpha\}_{\alpha=1,\ldots,s}$ are coordinates on $\Omega$ such that $d\mu(x)=\rho(x)dx^1\cdot\cdot\cdot dx^s$.
\end{Th}
Equation (\ref{noether}) is the conservation law associated with the symmetry represented by $X$. The quantity
$$
\rho(x)X^j(u)\dfrac{\partial L}{\partial p^j_\alpha}(x,u,du)
$$
is often called {\it Noether current}, especially in the physics literature. \\
Whether in the form given above or in analogous forms, Noether's theorem has long been recognized as a fundamental tool in variational calculus. In the context of harmonic map theory, Noether's theorem was first used by \cite{Raw} in the mid 1980s. A few years later, several authors \cite{YMC, KRS, Sha} have independently used it to replace the harmonic map equation into spheres, where derivatives of the solution appear in a quadratic way, by an equation in divergence form, where derivatives of the solution appear in a linear way. This gives a particularly helpful analytical edge when studying harmonic maps with only very weak regularity hypotheses. Fr\'ed\'eric H\'elein made significant contributions to the analysis of harmonic maps using conservation laws via Noether's theorem \cite{Hel}. In the same vein, Tristan Rivi\`ere used conservation laws to study conformally invariant variational problems \cite{Riv1}. \\
We will in this paper also make use of Noether's theorem\footnote{not directly in the form (\ref{noether}), but the spirit behind our derivations is the same.}, this time in the context of fourth-order geometric problems in connection with the {\it Willmore functional}. We now briefly recall the main historical landmarks that led to the discovery -- and rediscovery, indeed -- of the Willmore functional. \\
Imagine that you had at your disposal the bow of a violin and a horizontal thin metallic plate covered with grains of sand. What would you observe if you were to rub the bow against the edge of the plate? In 1680, the English philosopher and scientist Robert Hooke was the first to try to answer this question (then posed in slightly different experimental terms). Some 120 years later, the German physicist and musician Ernst Chladni repeated the experiment in a systematic way \cite{Chl}. Rubbing the bow with varying frequency, he observed that the grains of sand arrange themselves in remarkable patterns -- nowadays known as {\it Chladni figures}. Those who witnessed Chladni's experiment were fascinated by the patterns, as was in 1809 the French emperor Napol\'eon I. Eager to understand the physical phenomenon at the origin of the Chladni figures, the emperor mandated Pierre-Simon de Laplace of the Acad\'emie des Sciences to organize a competition whose goal would be to provide a mathematical explanation for the figures. The winner would receive one kilogram of solid gold. Joseph-Louis Lagrange discouraged many potential candidates as he declared that the solution of the problem would require the creation of a new branch of mathematics. Only two contenders remained in the race: the very academic Sim\'eon-Denis Poisson and one autodidactic outsider: Sophie Germain. It is unfortunately impossible to give here a detailed account of the interesting events that took place in the following years (see \cite{Dah}). In 1816, Sophie Germain won the prize -- which she never claimed. Although Germain did not answer Napol\'eon's original question, and although she did not isolate the main phenomenon responsible for the Chladni figures, namely resonance, her work proved fundamental, for, as predicted by Lagrange, she laid down the foundations of a whole new branch of applied mathematics: the theory of elasticity of membranes. For the sake of brevity, one could synthesize Germain's main idea by isolating one single decisive postulate which can be inferred from her work \cite{Ger}. Having found her inspiration in the works of Daniel Bernoulli \cite{DBer} and Leonhard Euler \cite{Eul} on the elastica (flexible beams), Sophie Germain postulates that the density of elastic energy stored in a thin plate is proportional to the square of the mean curvature\footnote{Incidentally, the notion of mean curvature was first defined and used in this context; it is a creation which we owe to Germain.} $H$. In other words, the elastic energy of a bent thin plate $\Sigma$ can be expressed in the form
$$
\int_{\Sigma}H^2(p)d\sigma(p)\:,
$$
where $d\sigma$ denotes the area-element. In the literature, this energy is usually referred to as {\it Willmore energy}. It bears the name of the English mathematician Thomas Willmore who rediscovered it in the 1960s \cite{Wil1}. Prior to Willmore and after Germain, the German school of geometers led by Wilhelm Blaschke considered and studied the Willmore energy in the context of conformal geometry. Blaschke observed that minimal surfaces minimize the Willmore energy and moreover that the Willmore energy is invariant under conformal transformations of $\mathbb{R}^3\cup\{\infty\}$. In his nomenclature, critical points of the Willmore energy were called {\it conformal minimal surfaces} \cite{Bla}. Gerhard Thomsen, a graduate student of Blaschke, derived the Euler-Lagrange equation corresponding to the Willmore energy \cite{Tho} (this was further generalized to higher codimension in the 1970s by Joel Weiner \cite{Wei}). It is a fourth-order nonlinear partial differential equation for the immersion. Namely, let $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ be a smooth immersion of an oriented surface $\Sigma$. The pull-back metric $g:=\vec{\Phi}^*g_{\mathbb{R}^3}$ is represented in local coordinates with components $g_{ij}$. We let $\nabla_j$ denote the corresponding covariant derivative. The second fundamental form is the normal valued 2-tensor with components $\vec{h}_{ij}:=\nabla_i\nabla_j\vec{\Phi}$. Its half-trace is the mean curvature vector $\vec{H}:=\dfrac{1}{2}\vec{h}^{j}_{j}$. The {\it Willmore equation} reads
\begin{equation}\lambdabel{will0}
\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\;=\;\vec{0}\:,
\end{equation}
where $\Delta_\perp$ is the negative covariant Laplacian for the connection $\nabla$ in the normal bundle derived from the ambient scalar product in $\mathbb{R}^m$. Note, in passing, that it is not at all clear how one could define a weak solution of (\ref{will0}) using only the requirement that $\vec{H}$ be square-integrable (i.e. that the Willmore energy be finite). \\
The Willmore energy appears in various areas of science: general relativity, as the main contributor to the Hawking mass \cite{Haw} ; in cell biology (see below) ; in nonlinear elasticity theory \cite{FJM} ; in optical design and lens crafting \cite{KR} ; in string theory, in the guise of a string action \`a la Polyakov \cite{Pol}. As mentioned earlier, the Willmore energy also plays a distinguished role in conformal geometry, where it has given rise to many interesting problems and where it has stimulated too many elaborate works to be cited here. We content ourselves with mentioning the remarkable tours de force of Fernando Marques and Andr\'e Neves \cite{MN} to solve the celebrated Willmore conjecture stating that, up to M\"obius transformations, the Clifford torus\footnote{obtained by rotating a circle of radius 1 around an axis located at a distance $\sqrt{2}$ of its center.} minimizes the Willmore energy amongst immersed tori in $\mathbb{R}^3$. \\
Aiming at solving the Willmore conjecture, Leon Simon initiated the ``modern" variational study of the Willmore functional \cite{Sim} when proving the existence of an embedded torus into $\mathbb{R}^{m\ge3}$ minimizing the $L^2$ norm of the second fundamental form. As this norm does not provide any control of the $C^1$ norm of the surface, speaking of ``immersion" is impossible. Simon thus had to weaken the geometric notion of immersion, and did so by using varifolds and their local approximation by biharmonic graphs. In the following years, this successful ``ambient approach" was used by various authors \cite{BK, KS1, KS2, KS3} to solve important questions about Willmore surfaces. \\
Several authors \cite{CDDRR, Dal, KS2, Pal, Rus} have observed that the Willmore equation in codimension 1 is cognate with a certain divergence form. We will prove below (Theorem \ref{Th1}) a pointwise equality to that effect in any codimension. The versions found in the aforementioned works are weaker in the sense that they only identify an integral identity. \\
In 2006, Tristan Rivi\`ere \cite{Riv2} showed that the fourth-order Willmore equation (\ref{will0}) can be written in divergence form and eventually recast as a system two of second-order equations enjoying a particular structure useful to the analysis of the critical points of the Willmore energy. This observation proved to be decisive in the resolution of several questions pertaining to Willmore surfaces \cite{YBer, BR1, BR2, BR3, KMR, MR, Riv2, Riv3, Riv4}. It also led to the so-called ``parametric approach" of the problem. In contrast with the ambient approach where surfaces are viewed as subsets of $\mathbb{R}^m$, the parametric approach favors viewing surfaces as images of (weak) immersions, and the properties of these immersions become the analytical point of focus. \\
We briefly review the results in \cite{Riv2} (in codimension 1), adapting slightly the original notation and statements to match the orientation of our paper. With the same notation as above, the first conservation law in \cite{Riv2} states that a smooth immersion $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^3$ is a critical point of the Willmore functional if and only if
\begin{equation}\lambdabel{ri1}
\nabla_j\big(\nabla^j\vec{H}-2(\vec{n}\cdot\nabla^j\vec{H})\vec{n}+|\vec{H}|^2\nabla^j\vec{\Phi}\big)\;=\;\vec{0}\:,
\end{equation}
where $\vec{n}$ is the outward unit normal. \\
Locally about every point, (\ref{ri1}) may be integrated to yield a function $\vec{L}\in\mathbb{R}^3$ satisfying
$$
|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\;=\;\nabla^j\vec{H}-2(\vec{n}\cdot\nabla^j\vec{H})\vec{n}+|\vec{H}|^2\nabla^j\vec{\Phi}\:,
$$
where $|g|$ is the volume element of the pull-back metric $g$, and $\varepsilonilon^{kj}$ is the Levi-Civita symbol. The following equations hold:
\begin{equation}\lambdabel{ri2}
\left\{\begin{equation}gin{array}{rcl}
\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\times\nabla_k\vec{\Phi}-\vec{H}\times\nabla^j\vec{\Phi}\big)&=&\vec{0}\\[1ex]
\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\big)&=&0\:.
\end{array}\right.
\end{equation}
These two additional conservation laws give rise (locally about every point) to two potentials $\vec{R}\in\mathbb{R}^3$ and $S\in\mathbb{R}$ satisfying
\begin{equation}s
\left\{\begin{equation}gin{array}{rcl}
\nabla_k\vec{R}&=&\vec{L}\times\nabla_k\vec{\Phi}-|g|^{1/2}\varepsilonilon_{kj}\vec{H}\times\nabla^j\vec{\Phi}\\[1ex]
\nabla_k S&=&\vec{L}\cdot\nabla_k\vec{\Phi}\:.
\end{array}\right.
\end{equation}s
A computation shows that these potentials are related to each other via the {system
\begin{equation}\lambdabel{ri3}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]\:.
\end{array}\right.
\end{equation}
This system is linear in $S$ and $\vec{R}$. It enjoys the particularity of being written in flat divergence form, with the right-hand side comprising Jacobian-type terms. The Willmore energy is, up to a topological constant, the $W^{1,2}$-norm of $\vec{n}$. For an immersion $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$, one can show that $S$ and $\vec{R}$ belong to $W^{1,2}$. Standard Wente estimates may thus be performed on (\ref{ri3}) to thwart criticality and regularity statements ensue \cite{BR1, Riv2}. Furthermore, one verifies that (\ref{ri3}) is stable under weak limiting process, which has many nontrivial consequences \cite{BR1,BR3}. \\
In 2013, the author found that the divergence form and system derived by Rivi\`ere can be obtained by applying Noether's principle to the Willmore energy\footnote{The results were first presented at Oberwolfach in July 2013 during the mini-workshop {\it The Willmore functional and the Willmore conjecture}.}. The translation, rotation, and dilation invariances of the Willmore energy yield via Noether's principle the conservations laws (\ref{ri1}) and (\ref{ri2}).
\begin{equation}gin{Th}\lambdabel{Th1}
Let $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^m$ be a smooth immersion of an oriented surface $\Sigma$. Introduce the quantities
$$
\left\{\begin{equation}gin{array}{lcl}
\vec{\mathcal{W}}&:=&\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\\[1ex]
\vec{T}^j&:=&\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\:,
\end{array}\right.
$$
where $\pi_{\vec{n}}$ denotes projection onto the normal space. \\
Via Noether's theorem, the invariance of the Willmore energy by translations, rotations, and dilations in $\mathbb{R}^m$ imply respectively the following three conservation laws:
\begin{equation}\lambdabel{laws}
\left\{\begin{equation}gin{array}{rcl}
\nabla_j\vec{T}^j&=&-\,\vec{\mathcal{W}}\\[1ex]
\nabla_j\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)&=&-\,\vec{\mathcal{W}}\wedge\vec{\Phi}\\[1ex]
\nabla_j\big(\vec{T}^j\cdot\vec{\Phi}\big)&=&-\,\vec{\mathcal{W}}\cdot\vec{\Phi}\:.
\end{array}\right.
\end{equation}
In particular, the immersion $\vec{\Phi}$ is Willmore if and only if the following conservation holds:
\begin{equation}s
\nabla_j\big(\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi} \big)\;=\;\vec{0}\:.
\end{equation}s
\end{Th}
For the purpose of local analysis, one may apply Hodge decompositions in order to integrate the three conservation laws (\ref{laws}). Doing so yields ``potential" functions related to each other in a very peculiar way, which we state below. The somewhat unusual notation -- the price to pay to work in higher codimension -- is clarified in Section \ref{nota}.
\begin{equation}gin{Th}\lambdabel{Th2}
Let $\vec{\Phi}:D^2\rightarrow\mathbb{R}^m$ be a smooth\footnote{in practice, this strong hypothesis is reduced to $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$ without modifying the result.} immersion of the flat unit disk $D^2\subset\mathbb{R}^2$. We denote by $\vec{n}$ the Gauss-map, by $g:=\vec{\Phi}^*g_{\mathbb{R}^m}$ the pull-back metric, and by $\Delta_g$ the associated negative Laplace-Beltrami operator. Suppose that $\vec{\Phi}$ satisfies the fourth-order equation
$$
\Delta_\perp\vec{H}+\big(\vec{h}^{i}_{j}\cdot\vec{H}\big)\vec{h}^{j}_{i}-2|\vec{H}|^2\vec{H}\;=\;\vec{\mathcal{W}}\:,
$$
for some given $\vec{\mathcal{W}}$. Let $\vec{V}$, $\vec{X}$, and $Y$ solve the problems
$$
\Delta_g\vec{V}\;=\;-\,\vec{\mathcal{W}}\qquad,\qquad \Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\qquad,\qquad\Delta_gY\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:.
$$
Then $\vec{\Phi}$ is a solution of the second-order equation
\begin{equation}\lambdabel{eqphi}
|g|^{1/2}\Delta_g\vec{\Phi}\;=\;-\,\varepsilonilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:,
\end{equation}
where $S$ and $\vec{R}$ satisfy the system
\begin{equation}\lambdabel{thesys000}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\:.
\end{array}\right.
\end{equation}
\end{Th}
In the special case when $\vec{\Phi}$ is Willmore, we have $\mathcal{\vec{W}}\equiv\vec{0}$, and we may choose $\vec{V}$, $\vec{X}$, and $Y$ to identically vanish. Then (\ref{thesys000}) becomes the conservative Willmore system found originally in \cite{Riv2}. \\
Although perhaps at first glance a little cryptic, Theorem \ref{Th2} turns out to be particularly useful for local analytical purposes. If the given $\mathcal{\vec{W}}$ is sufficiently regular, the non-Jacobian terms involving $Y$ and $\vec{X}$ on the right-hand side of (\ref{thesys000}) form a subcritical perturbation of the Jacobian terms involving $S$ and $\vec{R}$ (see \cite{BWW1} for details). From an analytic standpoint, one is left with studying a linear system of Jacobian-type. Wente estimates provide fine regularity information on the potential functions $S$ and $\vec{R}$, which may, in turn, be bootstrapped into (\ref{eqphi}) to yield regularity information on the immersion $\vec{\Phi}$ itself. \\
In \cite{DDW}, the authors study Willmore surfaces of revolution. They use the invariances of the Willmore functional to recast the Willmore ODE in a form that is a special case of (\ref{ri1}). Applying Noether's principle to the Willmore energy had already been independently done and used in the physics community \cite{CG, Mue}. As far as the author understands, these references are largely unknown in the analysis and geometry community. One goal of this paper is to bridge the gap, as well as to present results which do not appear in print. The author hopes it will increase in the analysis/geometry community the visibility of results, which, he believes, are useful to the study of fourth-order geometric problems associated with the Willmore energy. \\
For the sake of brevity, the present work focuses only on computational derivations and on examples. A second work \cite{BWW1} written jointly with Glen Wheeler and Valentina-Mira Wheeler will shortly be available. It builds upon the reformulations given in the present paper to derive various local analytical results.
\paragraph{Acknowledgments.} The author is grateful to Daniel Lengeler for pointing out to him \cite{CG} and \cite{Mue}. The author would also like to thank Hans-Christoph Grunau, Tristan Rivi\`ere, and Glen Wheeler for insightful discussions. The excellent working conditions of the welcoming facilities of the Forschungsinstitut f\"ur Mathematik at the ETH in Z\"urich are duly acknowledged.
\section{Main Result}
After establishing some notation in Section \ref{nota}, the contents of Theorem \ref{Th1} and of Theorem \ref{Th2} will be proved simultaneously in Section \ref{proof}.
\subsection{Notation}\lambdabel{nota}
In the sequel, $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ denotes a smooth immersion of an oriented surface $\Sigma$ into Euclidean space. The induced metric is $g:=\vec{\Phi}^*g_{\mathbb{R}^m}$ with components $g_{ij}$ and with volume element $|g|$. The components of the second fundamental form are denoted $\vec{h}_{ij}$. The mean curvature is $\vec{H}:=\dfrac{1}{2}\,g^{ij}\vec{h}_{ij}$. At every point on $\Sigma$, there is an oriented basis $\{\vec{n}_\alpha\}_{\alpha=1,\ldots,m-2}$ of the normal space. We denote by $\pi_{\vec{n}}$ the projection on the space spanned by the vectors $\{\vec{n}_\alpha\}$, and by $\pi_T$ the projection on the tangent space (i.e. $\pi_T+\pi_{\vec{n}}=\text{id}$). The Gauss map $\vec{n}$ is the $(m-2)$-vector defined via
$$
\star\,\vec{n}\;:=\;\dfrac{1}{2}\,|g|^{-1/2}\varepsilonilon^{ab}\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\:,
$$
where $\star$ is the usual Hodge-star operator, and $\varepsilonilon^{ab}$ is the Levi-Civita symbol\footnote{Recall that the Levi-Civita is {\it not} a tensor. It satisfies $\varepsilonilon_{ab}=\varepsilonilon^{ab}$.} with components $\varepsilonilon^{11}=0=\varepsilonilon^{22}$ and $\varepsilonilon^{12}=1=-\varepsilonilon^{21}$. Einstein's summation convention applies throughout. We reserve the symbol $\nabla$ for the covariant derivative associated with the metric $g$. Local flat derivatives will be indicated by the symbol $\partial$. \\
\noindent
As we work in any codimension, it is helpful to distinguish scalar quantities from vector quantities. For this reason, we append an arrow to the elements of $\Lambdambda^p(\mathbb{R}^m)$, for all $p>0$. The scalar product in $\mathbb{R}^m$ is denoted by a dot. We also use dot to denote the natural extension of the scalar product in $\mathbb{R}^m$ to multivectors (see \cite{Fed}).\\
Two operations between multivectors are useful. The interior multiplication $\res$ maps the pair comprising a $q$-vector $\gamma$ and a $p$-vector $\begin{equation}ta$ to the $(q-p)$-vector $\gamma\res\begin{equation}ta$. It is defined via
\begin{equation}s
\lambdangle \gamma\res\begin{equation}ta\,,\alphapha\rangle\;=\;\lambdangle \gamma\,,\begin{equation}ta\wedge\alphapha\rangle\:\qquad\text{for each $(q-p)$-vector $\alpha$.}
\end{equation}s
Let $\alpha$ be a $k$-vector. The first-order contraction operation $\bullet$ is defined inductively through
\begin{equation}s
\alpha\bullet\begin{equation}ta\;=\;\alpha\res\begin{equation}ta\:\:\qquad\text{when $\begin{equation}ta$ is a 1-vector}\:,
\end{equation}s
and
\begin{equation}s
\alpha\bullet(\begin{equation}ta\wedge\gamma)\;=\;(\alpha\bullet\begin{equation}ta)\wedge\gamma\,+\,(-1)^{pq}\,(\alpha\bullet\gamma)\wedge\begin{equation}ta\:,
\end{equation}s
when $\begin{equation}ta$ and $\gamma$ are respectively a $p$-vector and a $q$-vector.
\subsection{Variational Derivations}\lambdabel{proof}
Consider a variation of the form:
\begin{equation}s
\vec{\Phi}_t\;:=\;\vec{\Phi}\,+\,t\big(A^j\nabla_j\vec{\Phi}+\vec{B}\big)\:,
\end{equation}s
for some $A^j$ and some normal vector $\vec{B}$.
We have
\begin{equation}s
\nabla_i\nabla_j\vec{\Phi}\;=\;\vec{h}_{ij}\:.
\end{equation}s
Denoting for notational convenience by $\delta$ the variation at $t=0$, we find:
\begin{equation}s
\delta\nabla_j\vec{\Phi}\;\equiv\;\nabla_j\delta\vec{\Phi}\;=\;(\nabla_jA^s)\nabla_s\vec{\Phi}+A^s\vec{h}_{js}+\nabla_j\vec{B}\:.
\end{equation}s
Accordingly, we find
\begin{equation}gin{eqnarray*}
\pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}&=&2(\nabla_jA^s)\vec{h}^j_{s}+A^s\pi_{\vec{n}}\nabla^j\vec{h}_{js}+\pi_{\vec{n}}\nabla^j\pi_{\vec{n}}\nabla_j\vec{B}+\pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\\
&=&2(\nabla_jA^s)\vec{h}^j_{s}+2A^s\nabla_s\vec{H}+\Delta_\perp\vec{B}+\pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\:,
\end{eqnarray*}
where we have used the definition of the normal Laplacian $\Delta_\perp$ and the contracted Codazzi-Mainardi equation
\begin{equation}s
\nabla^j\vec{h}_{js}\;=\;2\nabla_s\vec{H}\:.
\end{equation}s
Since $\vec{B}$ is a normal vector, one easily verifies that
\begin{equation}s
\pi_T\nabla_j\vec{B}\;=\;-\,(\vec{B}\cdot\vec{h}_j^s)\,\nabla_s\vec{\Phi}\:,
\end{equation}s
so that
\begin{equation}s
\pi_{\vec{n}}\nabla^j\pi_T\nabla_j\vec{B}\;=\;-\,(\vec{B}\cdot\vec{h}_j^s)\,\vec{h}_s^j\:.
\end{equation}s
Hence, the following identity holds
\begin{equation}\lambdabel{eq1}
\pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}\;=\;2(\nabla_jA^s)\vec{h}_{js}+2A^s\nabla_s\vec{H}+\Delta_\perp\vec{B}-(\vec{B}\cdot\vec{h}_j^s)\,\vec{h}_s^j\:.
\end{equation}
Note that
\begin{equation}\lambdabel{eq2}
\delta g^{ij}\;=\;-2\nabla^jA^i+2\vec{B}\cdot\vec{h}^{ij}\:,
\end{equation}
which, along with (\ref{eq1}) and (\ref{eq2}), then gives
\begin{equation}gin{eqnarray*}
\delta|\vec{H}|^2&=&\vec{H}\cdot\delta\nabla^j\nabla_j\vec{\Phi}\;\;=\;\;\vec{H}\cdot\big[(\delta g^{ij})\partial_i\nabla_j\vec{\Phi}+\nabla^j\delta\nabla_j\vec{\Phi}\big]\\[1ex]
&=&\vec{H}\cdot\big[(\delta g^{ij})\vec{h}_{ij}+\pi_{\vec{n}}\nabla^j\delta\nabla_j\vec{\Phi}\big]\\[1ex]
&=&\vec{H}\cdot\big[\Delta_\perp\vec{B}+(\vec{B}\cdot\vec{h}^i_j)\vec{h}^j_i+2A^j\nabla_j\vec{H} \big]\:.
\end{eqnarray*}
Finally, since
\begin{equation}s
\delta|g|^{1/2}\;=\;|g|^{1/2}\big[\nabla_jA^j-2\vec{B}\cdot\vec{H} \big]\:,
\end{equation}s
we obtain
\begin{equation}gin{eqnarray*}
\delta\big(|\vec{H}|^2|g|^{1/2}\big)&=&|g|^{1/2}\Big[\vec{H}\cdot\Delta_\perp\vec{B}+(\vec{B}\cdot\vec{h}^i_j)\vec{h}^j_i-2(\vec{B}\cdot\vec{H})|H|^2+\nabla_j\big(|\vec{H}|^2A^j\big) \Big]\\[1ex]
&=&|g|^{1/2}\Big[\vec{B}\cdot\vec{\mathcal{W}}+\nabla_j\big(\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+|\vec{H}|^2A^j\big) \Big]\:,
\end{eqnarray*}
where
\begin{equation}s
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}^i_j)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\:.
\end{equation}s
Therefore,
\begin{equation}\lambdabel{diffen}
\delta\int_{\Sigma_0}|\vec{H}|^2\;=\;\int_{\Sigma_0}\Big[\vec{B}\cdot\vec{\mathcal{W}}+\nabla_j\big(\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+|\vec{H}|^2A^j\big) \Big]\:.
\end{equation}
This identity holds for every piece of surface $\Sigma_0\subset\Sigma$. We will now consider specific deformations which are known to preserve the Willmore energy (namely translations, rotations, and dilations \cite{BYC3}), and thus for which the right-hand side of (\ref{diffen}) vanishes.
\paragraph{Translations.}
We consider a deformation of the form
\begin{equation}s
\vec{\Phi}_t\;=\;\vec{\Phi}+t\vec{a}\qquad\text{for some fixed $\vec{a}\in\mathbb{R}^m$}\:.
\end{equation}s
Hence
\begin{equation}s
\vec{B}\;=\;\pi_{\vec{n}}\vec{a}\qquad\text{and}\qquad A^j\;=\;\vec{a}\cdot\nabla^j\vec{\Phi}\:.
\end{equation}s
This gives
\begin{equation}gin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&=&\vec{a}\cdot\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})\vec{n}_\alpha+H^\alpha\nabla^j\vec{n}_\alpha+|\vec{H}|^2\nabla^j\vec{\Phi}\Big]\\[1ex]
&=&\vec{a}\cdot\Big[\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\Big]\:,
\end{eqnarray*}
so that (\ref{diffen}) yields
\begin{equation}s
\vec{a}\cdot\int_{\Sigma_0}\vec{\mathcal{W}}+\nabla_j\Big[\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi} \Big]\;=\;0\:.
\end{equation}s
As this holds for all $\vec{a}$ and all $\Sigma_0$, letting
\begin{equation}\lambdabel{defT}
\vec{T}^j\;:=\;\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}
\end{equation}
gives
\begin{equation}\lambdabel{trans}
\nabla_j\vec{T}^j\;=\;-\,\vec{\mathcal{W}}\:.
\end{equation}
This is equivalent to the conservation law derived in \cite{Riv2} in the case when $\vec{\mathcal{W}}=\vec{0}$ and when the induced metric is conformal with respect to the identity. At the equilibrium, i.e. when $\vec{\mathcal{W}}$ identically vanishes, $\vec{T}^j$ plays in the problem the role of stress-energy tensor. \\
For future convenience, we formally introduce the Hodge decomposition\footnote{Naturally, this is only permitted when working locally, or on a domain whose boundary is contractible to a point.}
\begin{equation}\lambdabel{defLL}
\vec{T}^j\;=\;\nabla^j\vec{V}+|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\:,
\end{equation}
for some $\vec{L}$ and some $\vec{V}$ satisfying
\begin{equation}\lambdabel{defV}
-\,\Delta_g\vec{V}\;=\;\vec{\mathcal{W}}\:.
\end{equation}
\paragraph{Rotations.}
We consider a deformation of the form
\begin{equation}s
\vec{\Phi}_t\;=\;\vec{\Phi}+t\star(\vec{b}\wedge\vec{\Phi})\qquad\text{for some fixed $\vec{b}\in\Lambdambda^{m-2}(\mathbb{R}^m)$}\:.
\end{equation}s
In this case, we have
\begin{equation}s
B^\alpha\;=\;-\,\vec{b}\cdot\star(\vec{n}_\alpha\wedge\vec{\Phi})\qquad\text{and}\qquad A^j\;=\;-\,\vec{b}\,\cdot\star\big(\nabla^j\vec{\Phi}\wedge\vec{\Phi}\big)\:.
\end{equation}s
Hence
\begin{equation}gin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&&\hspace{-1cm}=\;\;-\,\vec{b}\cdot\star\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})(\vec{n}_\alpha\wedge\vec{\Phi})+H^\alpha\nabla^j(\vec{n}_\alpha\wedge\vec{\Phi})+|\vec{H}|^2\nabla^j\vec{\Phi}\wedge\vec{\Phi}\Big]\\
&&\hspace{-1cm}=\;\;-\,\vec{b}\cdot\star\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\:,
\end{eqnarray*}
where we have used the tensor $\vec{T}^j$ defined in (\ref{defT}).
Putting this last expression in (\ref{diffen}) and proceeding as in the previous paragraph yields the pointwise equalities
\begin{equation}gin{eqnarray}\lambdabel{ach1}
\vec{\mathcal{W}}\wedge\vec{\Phi}&=&-\,\nabla_j\big(\vec{T}^j\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&\stackrel{\text{(\ref{defLL})}}{\equiv}&-\,\nabla_j\big(\nabla^j\vec{V}\wedge\vec{\Phi}+|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\wedge\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&=&-\,\nabla_j\big(\nabla^j\vec{V}\wedge\vec{\Phi}-|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}+\vec{H}\wedge\nabla^j\vec{\Phi}\big)\nonumber\\
&=&-\Delta_g\vec{V}\wedge\vec{\Phi}-\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}+\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\big)\:.
\end{eqnarray}
Owing to (\ref{defV}), we thus find
\begin{equation}s
\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\big)\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\:.
\end{equation}s
It will be convenient to define two 2-vectors $\vec{X}$ and $\vec{R}$ satisfying the Hodge decomposition
\begin{equation}\lambdabel{defR}
|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\wedge\nabla_k\vec{\Phi}-\vec{H}\wedge\nabla^j\vec{\Phi}\;=\;\nabla^j\vec{X}+|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{R}\:,
\end{equation}
with thus
\begin{equation}\lambdabel{defX}
\Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\:.
\end{equation}
\paragraph{Dilations.}
We consider a deformation of the form
\begin{equation}s
\vec{\Phi}_t\;=\;\vec{\Phi}+t\lambda\vec{\Phi}\qquad\text{for some fixed $\lambda\in\mathbb{R}$}\:,
\end{equation}s
from which we obtain
\begin{equation}s
B^\alpha\;=\;\lambda\,\vec{n}_\alpha\cdot\vec{\Phi}\qquad\text{and}\qquad A^j\;=\;\lambda\,\nabla^j\vec{\Phi}\cdot\vec{\Phi}\:.
\end{equation}s
Hence
\begin{equation}gin{eqnarray*}
&&\vec{H}\cdot\nabla^j\vec{B}-\vec{B}\cdot\nabla^j\vec{H}+ |\vec{H}|^2A^j\\[1ex]
&&\hspace{-1cm}=\;\;\lambda\Big[(\vec{H}\cdot\nabla^j\vec{n}_\alpha-\vec{n}_\alpha\cdot\nabla^j\vec{H})(\vec{n}_\alpha\cdot\vec{\Phi})+H^\alpha\nabla^j(\vec{n}_\alpha\cdot\vec{\Phi})+|\vec{H}|^2\nabla^j\vec{\Phi}\cdot\vec{\Phi} \Big]\\
&&\hspace{-1cm}=\;\;\lambda\,\vec{T}^j\cdot\vec{\Phi}\:,
\end{eqnarray*}
where we have used that $\vec{H}\cdot\nabla^j\vec{\Phi}=0$, and where $\vec{T}^j$ is as in (\ref{defT}). \\
Putting this last expression in (\ref{diffen}) and proceeding as before gives the pointwise equalities
\begin{equation}gin{eqnarray}\lambdabel{ach2}
\vec{\mathcal{W}}\cdot\vec{\Phi}&=&-\,\nabla_j\big(\vec{T}^j\cdot\vec{\Phi}\big)\\
&\equiv&-\,\nabla_j\big(\nabla^j\vec{V}\cdot\vec{\Phi}+|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\cdot\vec{\Phi} \big)\nonumber\\
&=&-\,\Delta_g\vec{V}\cdot\vec{\Phi}-\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}+\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\cdot\nabla_j\vec{\Phi}\big)\nonumber\:.
\end{eqnarray}
Hence, from (\ref{defV}), we find
\begin{equation}s
\nabla_j\big(|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\big)\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:.
\end{equation}s
We again use a Hodge decomposition to write
\begin{equation}\lambdabel{defS}
|g|^{-1/2}\varepsilonilon^{kj}\vec{L}\cdot\nabla_k\vec{\Phi}\;=\;\nabla^jY+|g|^{-1/2}\varepsilonilon^{kj}\nabla_k S\:,
\end{equation}
where
\begin{equation}\lambdabel{defY}
\Delta_g Y\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\:.
\end{equation}
Our next task consists in relating to each other the ``potentials" $\vec{R}$ and $S$ defined above. Although this is the fruit of a rather elementary computation, the result it yields has far-reaching consequences and which, as far as the author knows, has no direct empirical justification.
Recall (\ref{defR}) and (\ref{defS}), namely:
\begin{equation}\lambdabel{defRS}
\left\{\begin{equation}gin{array}{lcl}
\nabla_k\vec{R}&=&\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big)\\[1ex]
\nabla_k S&=&\vec{L}\cdot\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\nabla^jY\:.
\end{array}\right.
\end{equation}
Define the Gauss map
\begin{equation}s
\star\,\vec{n}\;:=\;\dfrac{1}{2}\,|g|^{-1/2}\varepsilonilon^{ab}\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\:.
\end{equation}s
We have
\begin{equation}gin{eqnarray*}
(\star\,\vec{n})\cdot\nabla_k\vec{R}&=&\dfrac{1}{2}\,|g|^{-1/2}\varepsilonilon^{ab}\big(\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\big)\cdot\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big) \Big]\\[1ex]
&=&|g|^{-1/2}\varepsilonilon^{ab}g_{bk}\vec{L}\cdot\nabla_a\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\\[1ex]
&=&|g|^{-1/2}\varepsilonilon^{ab}g_{bk}\big(\nabla_aS+|g|^{1/2}\varepsilonilon_{aj}\nabla^jY \big)\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\varepsilonilon_{bk}\nabla^bS+\nabla_kY\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\cdot\nabla^j\vec{X}\:,
\end{eqnarray*}
where we have used that $\vec{H}$ is a normal vector, along with the elementary identities
\begin{equation}s
|g|^{-1/2}\varepsilonilon^{ab}g_{bk}\;=\;|g|^{1/2}\varepsilonilon_{bk}g^{ab}\qquad\text{and}\qquad\varepsilonilon^{ab}\varepsilonilon_{aj}\;=\;\delta^{b}_{j}\:.
\end{equation}s
The latter implies
\begin{equation}\lambdabel{nablaS}
\nabla^jS\;=\;|g|^{-1/2}\varepsilonilon^{jk}\big((\star\,\vec{n})\cdot\nabla_k\vec{R}-\nabla_kY\big)\,+\,(\star\,\vec{n})\cdot\nabla^j\vec{X}\:.
\end{equation}
Analogously, we find\footnote{$
(\omega_1\wedge\omega_2)\bullet(\omega_3\wedge\omega_4)\;=\;(\omega_2\cdot\omega_4)\omega_1\wedge\omega_3-(\omega_2\cdot\omega_3)\omega_1\wedge\omega_4-(\omega_1\cdot\omega_4)\omega_2\wedge\omega_3+(\omega_1\cdot\omega_3)\omega_2\wedge\omega_4\:.
$}
\begin{equation}gin{eqnarray*}
(\star\,\vec{n})\bullet\nabla_k\vec{R}&=&\dfrac{1}{2}\,|g|^{-1/2}\varepsilonilon^{ab}\big(\nabla_a\vec{\Phi}\wedge\nabla_b\vec{\Phi}\big)\bullet\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big) \Big]\\[1ex]
&=&|g|^{-1/2}\varepsilonilon^{ab}g_{bk}\nabla_a\vec{\Phi}\wedge\vec{L}\,+\,|g|^{-1/2}\varepsilonilon^{ab}(\vec{L}\cdot\nabla_a\vec{\Phi})(\nabla_b\vec{\Phi}\wedge\nabla_k\vec{\Phi})\\
&&-\:\varepsilonilon^{ab}\varepsilonilon_{kb}\nabla_a\vec{\Phi}\wedge\vec{H}\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\varepsilonilon_{kb}\vec{L}\wedge\nabla^b\vec{\Phi}\,-\,(\star\,\vec{n})(\vec{L}\cdot\nabla_k\vec{\Phi})\,-\,\nabla_k\vec{\Phi}\wedge\vec{H}\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\\[1ex]
&=&|g|^{1/2}\varepsilonilon_{kj}\nabla^j\vec{R}+\nabla_k\vec{X}-(\star\,\vec{n})\big(\nabla_kS+|g|^{1/2}\varepsilonilon_{kj}\nabla^jY \big)\,-\,|g|^{1/2}\varepsilonilon_{kj}(\star\,\vec{n})\bullet\nabla^j\vec{X}\:.
\end{eqnarray*}
It hence follows that there holds
\begin{equation}\lambdabel{nablaR}
\nabla^j\vec{R}\;=\;|g|^{-1/2}\varepsilonilon^{kj}\big((\star\,\vec{n})\nabla_kS+(\star\,\vec{n})\bullet\nabla_k\vec{R}-\nabla_k\vec{X}\big)\,+\,(\star\,\vec{n})\nabla^jY\,+\,(\star\,\vec{n})\bullet\nabla^j\vec{X}\:.
\end{equation}
Applying divergence to each of (\ref{nablaS}) and (\ref{nablaR}) gives rise to the {\it conservative Willmore system}:
\begin{equation}\lambdabel{conswillsys}
\left\{\begin{equation}gin{array}{lcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\:,
\end{array}\right.
\end{equation}
where $\partial_j$ denotes the derivative in flat local coordinates. \\
\noindent
This system is to be supplemented with (\ref{defX}) and (\ref{defY}), which, in turn, are solely determined by the value of the Willmore operator $\vec{\mathcal{W}}$ via equation (\ref{defV}). There is furthermore another useful equation to add to the conservative Willmore system, namely one relating the potentials $S$ and $\vec{R}$ back to the immersion $\vec{\Phi}$. We now derive this identity. Using (\ref{defRS}), it easily follows that
\begin{equation}gin{eqnarray*}
&&\hspace{-2cm}\varepsilonilon^{km}\big(\nabla_k\vec{R}+|g|^{1/2}\varepsilonilon_{kj}\nabla^j\vec{X}\big)\bullet\nabla_m\vec{\Phi}\\[1ex]
&=&\varepsilonilon^{km}\big(\vec{L}\wedge\nabla_k\vec{\Phi}-|g|^{1/2}\varepsilonilon_{kj}\vec{H}\wedge\nabla^j\vec{\Phi}\big)\bullet\nabla_m\vec{\Phi}\\[1ex]
&=&\varepsilonilon^{km}\Big[\big(\vec{L}\cdot\nabla_m\vec{\Phi}\big)\nabla_k\vec{\Phi}\,-\,g_{mk}\vec{L}\,-\,|g|^{1/2}\varepsilonilon_{km}\vec{H}\Big]\\[1ex]
&=&\varepsilonilon^{km}\big(\nabla_mS+|g|^{1/2}\varepsilonilon_{mj}\nabla^jY\big)\nabla_k\vec{\Phi}\,-\,2\,|g|^{1/2}\vec{H}\:.
\end{eqnarray*}
Since $\Delta_g\vec{\Phi}=2\vec{H}$, we thus find
\begin{equation}\lambdabel{law4}
\partial_j\Big[\varepsilonilon^{jk}\big(S\partial_k\vec{\Phi}+\vec{R}\bullet\partial_k\vec{\Phi} \big)+|g|^{1/2}\nabla^j\vec{\Phi}\Big]\;=\;|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:.
\end{equation}
At the equilibrium (i.e. when $\vec{\mathcal{W}}=\vec{0}$), the right-hand side of the latter identically vanishes and we recover a conservation law. With the help of a somewhat tedious computation, one verifies that this conservation law follows from the invariance of the Willmore energy by inversion and Noether's theorem. To do so, one may for example consider an infinitesimal variation of the type $\delta\vec{\Phi}\;=\;|\vec{\Phi}|^2\vec{a}-2(\vec{\Phi}\cdot\vec{a})\vec{\Phi}$, for some fixed constant vector $\vec{a}\in\mathbb{R}^m$.\\
We summarize our results in the following pair of systems.
\begin{equation}\lambdabel{side}
\left\{\begin{equation}gin{array}{rcl}\vec{\mathcal{W}}&=&\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\\[1ex]
\Delta_g\vec{V}&=&-\,\vec{\mathcal{W}}\\[1ex]
\Delta_g\vec{X}&=&\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\\[1ex]
\Delta_g{Y}&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}
\end{array}\right.
\end{equation}
and
\begin{equation}\lambdabel{thesys}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big((\star\,\vec{n})\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^jY+(\star\,\vec{n})\bullet\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\varepsilonilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\bullet\nabla_j\vec{\Phi} \big)\:.
\end{array}\right.
\end{equation}
\noindent
In the next section, we will examine more precisely the structure of this system through several examples.
\begin{equation}gin{Rm}
Owing to the identities
\begin{equation}s
\vec{u}\bullet\vec{v}\;=\;(\star\vec{u})\times\vec{v}\quad\text{and}\quad \vec{u}\bullet\vec{w}\;=\;\star\big[(\star\vec{u})\times(\star\vec{w})\big]\:\quad\text{for}\:\:\:\vec{u}\in\Lambdambda^2(\mathbb{R}^3), \vec{v}\in\Lambdambda^1(\mathbb{R}^3), \vec{w}\in\Lambdambda^2(\mathbb{R}^3)\:,
\end{equation}s
we can in $\mathbb{R}^3$ recast the above systems as
\begin{equation}\lambdabel{side1}
\left\{\begin{equation}gin{array}{rcl}\vec{\mathcal{W}}&=&\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\\[1ex]
\Delta_g\vec{V}&=&-\,\vec{\mathcal{W}}\\[1ex]
\Delta_g\vec{X}&=&\nabla^j\vec{V}\times\nabla_j\vec{\Phi}\\[1ex]
\Delta_g{Y}&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}
\end{array}\right.
\end{equation}
and
\begin{equation}\lambdabel{thesys1}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}
\,+\,|g|^{1/2}\nabla_j\big(\vec{n}\cdot\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big(\vec{n}\,\nabla^jY+\vec{n}\times\nabla^j\vec{X}\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\varepsilonilon^{jk}\Big[\partial_jS\,\partial_k\vec{\Phi}+\partial_j\vec{R}\times\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^jY\nabla_j\vec{\Phi}+\nabla^j\vec{X}\times\nabla_j\vec{\Phi} \big)\:.
\end{array}\right.
\end{equation}
In this setting, $\vec{X}$ and $\vec{R}$ are no longer 2-vectors, but rather simply vectors of $\mathbb{R}^3$.
\end{Rm}
\section{Examples}
\subsection{Willmore Immersions}
A smooth immersion $\vec{\Phi}:\Sigma\rightarrow\mathbb{R}^{m\ge3}$ of an oriented surface $\Sigma$ with induced metric $g=\vec{\Phi}^*g_{\mathbb{R}^m}$ and corresponding mean curvature vector $\vec{H}$, is said to be Willmore if it is a critical point of the Willmore energy $\int_\Sigma|\vec{H}|^2d\text{vol}_g$. They are known \cite{Wil2,Wei} to satisfy the Euler-Lagrange equation
$$
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\vec{0}\:.
$$
In the notation of the previous section, this corresponds to the case $\vec{\mathcal{W}}=\vec{0}$. According to (\ref{side}), we have the freedom to set $\vec{V}$, $\vec{X}$, and $Y$ to be identically zero. The Willmore equation then yields the second-order system in divergence form
\begin{equation}\lambdabel{thesyswillmore}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}
\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\varepsilonilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]\:.
\end{array}\right.
\end{equation}
This system was originally derived by Rivi\`ere in \cite{Riv2}. For notational reasons, the detailed computations were carried out only in local conformal coordinates, that is when $g_{ij}=\text{e}^{2\lambda}\delta_{ij}$, for some conformal parameter $\lambda$. The analytical advantages of the Willmore system (\ref{thesyswillmore}) have been exploited in numerous works \cite{BR1, BR2, BR3, Riv2, Riv3, Riv4}. The flat divergence form of the operator $|g|^{1/2}\Delta_g$ and the Jacobian-type structure of the right-hand side enable using fine Wente-type estimates in order to produce non-trivial local information about Willmore immersions (see aforementioned works).
\begin{equation}gin{Rm}
Any Willmore immersion will satisfy the system (\ref{thesyswillmore}). The converse is however not true. Indeed, in order to derive (\ref{thesyswillmore}), we first obtained the existence of some ``potential" $\vec{L}$ satisfying the first-order equation
\begin{equation}\lambdabel{ceteq}
|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\;=\;\nabla^j\vec{H}-2\pi_{\vec{n}}\nabla^j\vec{H}+|\vec{H}|^2\nabla^j\vec{\Phi}\:.
\end{equation}
In doing so, we have gone from the Willmore equation, which is second-order for $\vec{H}$, to the above equation, which is only first-order in $\vec{H}$, thereby introducing in the problem an extraneous degree of freedom. As we shall see in the next section, (\ref{ceteq}) is in fact equivalent to the conformally-constrained Willmore equation, which, as one might suspect, is the Willmore equation supplemented with an additional degree of freedom appearing in the guise of a Lagrange multiplier.
\end{Rm}
\subsection{Conformally-Constrained Willmore Immersions}
Varying the Willmore energy $\int_\Sigma|\vec{H}|^2d\text{vol}_g$ in a fixed conformal class (i.e. with infinitesimal, smooth, compactly supported, conformal variations) gives rise to a more general class of surfaces called {\it conformally-constrained Willmore surfaces} whose corresponding Euler-Lagrange equation \cite{BPP, KL, Sch} is expressed as follows.
Let $\vec{h}_0$ denote the trace-free part of the second fundamental form, namely
\begin{equation}s
\vec{h}_0\;:=\;\vec{h}\,-\,\vec{H} g\:.
\end{equation}s
A conformally-constrained Willmore immersion $\vec{\Phi}$ satisfies
\begin{equation}\lambdabel{cwe}
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\big(\vec{h}_0\big)_{ij}q^{ij}\:,
\end{equation}
where $q$ is a transverse\footnote{i.e. $q$ is divergence-free: $\nabla^j q_{ji}=0\:\:\forall i$.} traceless symmetric 2-form. This tensor $q$ plays the role of Lagrange multiplier in the constrained variational problem. \\
In \cite{KS3}, it is shown that under a suitable ``small enough energy" assumption, a minimizer of the Willmore energy in a fixed conformal class exists and is smooth. The existence of a minimizer without any restriction on the energy is also obtained in \cite{Riv3} where it is shown that the minimizer is either smooth (with the possible exclusion of finitely many branch points if the energy is large enough to grant their formation) or else isothermic\footnote{the reader will find in \cite {Riv5} an interesting discussion on isothermic immersions.}. One learns in \cite{Sch} that non-degenerate critical points of the Willmore energy constrained to a fixed conformal class are solutions of the conformally constrained Willmore equation.
Continuing along the lines of \cite{Riv3}, further developments are given in \cite{Riv6}, where the author shows that if either the genus of the surface satisfies $g\le2$, or else if the Teichm\"uller class of the immersion is not hyperelliptic\footnote{A class in the Teichm\"uller space is said to be {\it hyperelliptic} if the tensor products of holomorphic 1-forms do not generate the vector space of holomorphic quadratic forms.}, then any critical point $\vec{\Phi}$ of the Willmore energy for $C^1$ perturbations included in a submanifold of the Teichm\"uller space is in fact analytic and (\ref{cwe}) is satisfied for some transverse traceless symmetric 2-tensor $q$. \\
The notion of conformally constrained Willmore surfaces clearly generalizes that of Willmore surfaces, obtained via all smooth compactly supported infinitesimal variations (setting $q\equiv0$ in (\ref{cwe})). In \cite{KL}, it is shown that CMC Clifford tori are conformally constrained Willmore surfaces. In \cite{BR1}, the conformally constrained Willmore equation (\ref{cwe}) arises as that satisfied by the limit of a Palais-Smale sequence for the Willmore functional. \\
Minimal surfaces are examples of Willmore surfaces, while parallel mean curvature surfaces\footnote{parallel mean curvature surfaces satisfy $\pi_{\vec{n}} d\vec{H}\equiv\vec{0}$. They generalize to higher codimension the notion of constant mean curvature surfaces defined in $\mathbb{R}^3$. See [YBer].} are examples of conformally-constrained Willmore surfaces\footnote{{\it a non}-minimal parallel mean curvature surface is however {\it not} Willmore (unless of course it is the conformal transform of a Willmore surface ; e.g. the round sphere).}. Not only is the Willmore energy invariant under reparametrization of the domain, but more remarkably, it is invariant under conformal transformations of $\mathbb{R}^m\cup\{\infty\}$.
Hence, the image of a [conformally-constrained] Willmore immersion through a conformal transformation is again a [conformally-constrained] Willmore immersion. It comes thus as no surprise that the class of Willmore immersions [resp. conformally-constrained Willmore immersions] is considerably larger than that of immersions whose mean curvature vanishes [resp. is parallel], which is {\it not} preserved through conformal diffeomorphisms. \\
Comparing (\ref{cwe}) to the first equation in (\ref{side}), we see that $\vec{\mathcal{W}}=-\,\big(\vec{h}_0\big)_{ij}q^{ij}$.
Because $q$ is traceless and transverse, we have
$$
\big(\vec{h}_0\big)_{ij}q^{ij}\;\equiv\;\vec{h}_{ij}q^{ij}-\vec{H}g_{ij}q^{ij}\;=\;\vec{h}_{ij}q^{ij}\;=\;\nabla_j(q^{ij}\nabla_i\vec{\Phi})\:.
$$
Accordingly, choosing $\nabla^j\vec{V}=-\,q^{ij}\nabla_i\vec{\Phi}$ will indeed solve the second equation (\ref{side}). Observe next that
$$
\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;=\;-\,q^{ij}g_{ij}\;=\;0\:,
$$
since $q$ is traceless. Putting this into the fourth equation of (\ref{side}) shows that we may choose $Y\equiv0$. Furthermore, as $q$ is symmetric, it holds
$$
\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\;=\;-\,q^{ij}\nabla_i\vec{\Phi}\wedge\nabla_j\vec{\Phi}\;=\;\vec{0}\:,
$$
so that the third equation in (\ref{side}) enables us to choose $\vec{X}\equiv\vec{0}$. \\
Altogether, we see that a conformally-constrained Willmore immersion, just like a ``plain" Willmore immersion (i.e. with $q\equiv0$) satisfies the system (\ref{thesyswillmore}). In fact, it was shown in \cite{BR1} that to any smooth solution $\vec{\Phi}$ of (\ref{thesyswillmore}), there corresponds a transverse, traceless, symmetric 2-form $q$ satisfying (\ref{cwe}).
\subsection{Bilayer Models}
Erythrocytes (also called red blood cells) are the body's principal mean of transporting vital oxygen to the organs and tissues. The cytoplasm (i.e. the ``inside") of an erythrocyte is rich in hemoglobin, which chemically tends to bind to oxygen molecules and retain them. To maximize the possible intake of oxygen by each cell, erythrocytes -- unlike all the other types of cells of the human body -- have no nuclei\footnote{this is true for all mammals, not just for humans. However, the red blood cells of birds, fish, and reptiles do contain a nucleus.}. The membrane of an erythrocyte is a bilayer made of amphiphilic molecules. Each molecule is made of a ``head" (rather large) with a proclivity for water, and of a ``tail" (rather thin) with a tendency to avoid water molecules. When such amphiphilic molecules congregate, they naturally arrange themselves in a bilayer, whereby the tails are isolated from water by being sandwiched between two rows of heads. The membrane can then close over itself to form a vesicle. Despite the great biochemical complexity of erythrocytes, some phenomena may be described and explained with the sole help of physical mechanisms\footnote{The most celebrated such phenomenon was first observed by the Polish pathologist Tadeusz Browicz in the late 19$^{\text{th}}$ century \cite{Bro}. Using a microscope, he noted that the luminous intensity reflected by a red blood cell varies erratically along its surface, thereby giving the impression of flickering. In the 1930s, the Dutch physicist Frits Zernicke invented the phase-constrast microscope which revealed that the erratic flickering found by Browicz is in fact the result of very minute and very rapid movements of the cell's membrane. These movements were finally explained in 1975 by French physicists Fran\c{c}oise Brochard and Jean-Fran\c{c}ois Lennon: they are the result of a spontaneous thermic agitation of the membrane, which occurs independently of any particular biological activity \cite{BL}.}. For example, to understand the various shapes an erythrocyte might assume, it is sensible to model the red blood cell by a drop of hemoglobin (``vesicle") whose membrane is made of a lipid bilayer \cite{Lip}. Such ``simple" objects, called {\it liposomes}, do exist in Nature, and they can be engineered artificially\footnote{The adjective ``simple" is to be understood with care: stable vesicles with non-trivial topology can be engineered and observed. See \cite{MB} and \cite{Sei}. }. The membrane of a liposome may be seen as a viscous fluid separated from water by two layers of molecules. Unlike a solid, it can undergo shear stress. Experimental results have however shown that vesicles are very resistant and the stress required to deform a liposome to its breaking-point is so great that in practice, vesicles evolving freely in water are never submitted to such destructive forces. One may thus assume that the membrane of a liposome is incompressible: its area remains constant. The volume it encloses also stays constant, for the inside and the outside of the vesicle are assumed to be isolated. As no shearing and no stretching are possible, one may wonder which forces dictate the shape of a liposome. To understand the morphology of liposomes, Canham \cite{Can}, Evans \cite{Eva}, and Helfrich \cite{Hef} made the postulate that, just as any other elastic material does, the membrane of a liposome tends to minimize its elastic energy. As we recalled in the introduction, the elastic energy of a surface is directly proportional to the Willmore energy. Accordingly, to understand the shape of a liposome, one would seek minimizers of the Willmore energy subject to constraints on the area and on the enclosed volume. A third constraint could be taken into account. As the area of the inner layer of the membrane is slightly smaller than the area of its outer layer, it takes fewer molecules to cover the former as it does to cover the latter. This difference, called {\it asymmetric area difference}, is fixed once and for all when the liposome forms. Indeed, no molecule can move from one layer to the other, for that would require its hydrophilic head to be, for some time, facing away from water. From a theoretical point of view, the relevant energy to study is thus:
\begin{equation}\lambdabel{canham}
E\;:=\;\int_{\Sigma}H^2d\text{vol}_g+\alpha A(\Sigma)+\begin{equation}ta V(\Sigma)+\gamma M(\Sigma)\:,
\end{equation}
where $H$ denotes the mean curvature scalar\footnote{in this section, we will content ourselves with working in codimension 1.}, $A(\Sigma)$, $V(\Sigma)$, $M(\Sigma)$ denote respectively the area, volume, and asymmetric area difference of the membrane $\Sigma$, and where $\alpha$, $\begin{equation}ta$, and $\gamma$ are three Lagrange multipliers.
Depending on the authors' background, the energy (\ref{canham}) bears the names Canham, Helfrich, Canham-Helfrich, bilayer coupling model, spontaneous curvature model, among others.\\
This energy -- in the above form or analogous ones -- appears prominently in the applied sciences literature. It would be impossible to list here a comprehensive account of relevant references. The interested reader will find many more details in \cite{Sei} and \cite{Voi} and the references therein. In the more ``analytical" literature, the energy (\ref{canham}) is seldom found (except, of course, in the case when all three Lagrange multipliers vanish). We will, in time, recall precise instances in which it has been studied. But prior to doing so, it is interesting to pause for a moment and better understand a term which might be confusing to the mathematician reader, namely the asymmetric area difference $M(\Sigma)$. In geometric terms, it is simply the total curvature of $\Sigma$:
\begin{equation}\lambdabel{defM}
M(\Sigma)\;:=\;\int_{\Sigma}H\,d\text{vol}_g\:.
\end{equation}
This follows from the fact that the infinitesimal variation of the area is the mean curvature, and thus the area difference between two nearby surfaces is the first moment of curvature. Hence, we find the equivalent expression
\begin{equation}\lambdabel{canam}
E\;=\;\int_{\Sigma}\bigg(H+\dfrac{\gamma}{2}\bigg)^{\!2}d\text{vol}_g+\bigg(\alpha-\dfrac{\gamma^2}{4}\bigg) A(\Sigma)+\begin{equation}ta V(\Sigma)\:.
\end{equation}
This form of the bilayer energy is used, inter alia, in \cite{BWW2, Whe1}, where the constant $-\gamma/2$ is called {\it spontaneous curvature}. \\
From a purely mathematical point of view, one may study the energy (\ref{canham}) not just for embedded surfaces, but more generally for immersions. An appropriate definition for the volume $V$ must be assigned to such an immersion $\vec{\Phi}$. As is shown in \cite{MW}, letting as usual $\vec{n}$ denote the outward unit-normal vector to the surface, one defines
\begin{equation}s
V(\Sigma)\;:=\;\int_\Sigma\vec{\Phi}^*(d\mathcal{H}^3)\;=\;\int_{\Sigma}\vec{\Phi}\cdot\vec{n}\,d\text{vol}_g\:,
\end{equation}s
where $d\mathcal{H}^3$ is the Hausdorff measure in $\mathbb{R}^3$. Introducing the latter and (\ref{defM}) into (\ref{canham}) yields
\begin{equation}\lambdabel{veneer}
E\;=\;\int_{\Sigma}\big(H^2+\gamma H+\begin{equation}ta\,\vec{\Phi}\cdot\vec{n}+\alpha\big)\,d\text{vol}_g\:.
\end{equation}
We next vary the energy $E$ along a normal variation of the form $\delta\vec{\Phi}=\vec{B}\equiv B\vec{n}$. Using the computations from the previous section, it is not difficult to see that
\begin{equation}\lambdabel{varr1}
\delta\int_\Sigma d\text{vol}_g\;=\;-\,2\int_{\Sigma}\vec{B}\cdot\vec{H} \,d\text{vol}_g\qquad\text{and}\qquad \delta\int_\Sigma H\,d\text{vol}_g\;=\;\int_{\Sigma}
(\vec{B}\cdot\vec{n})\bigg(\dfrac{1}{2}h^i_jh^j_i-2H^2\bigg)\,d\text{vol}_g\:.
\end{equation}
With a bit more effort, in \cite{MW}, it is shown that
\begin{equation}\lambdabel{varr2}
\delta\int_\Sigma \vec{\Phi}\cdot\vec{n}\, d\text{vol}_g\;=\;-\int_{\Sigma}\vec{B}\cdot\vec{n}\,d\text{vol}_g\:.
\end{equation}
Putting (\ref{diffen}), (\ref{varr1}), and (\ref{varr2}) into (\ref{veneer}) yields the corresponding Euler-Lagrange equation
\begin{equation}\lambdabel{ELhelf}
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;2\,\alpha\vec{H}+\begin{equation}ta\,\vec{n}-\gamma\bigg(\dfrac{1}{2}h^i_jh^j_i-2H^2 \bigg)\,\vec{n}\:.
\end{equation}
We now seek a solution to the second equation in (\ref{side1}), namely a vector $\vec{V}$ satisfying $\Delta_g\vec{V}=-\vec{\mathcal{W}}$. To do so, it suffices to observe that
\begin{equation}s
2\vec{H}\;=\;\Delta_g\vec{\Phi}\qquad\text{and}\qquad \big(h^i_jh^j_i-4H^2\big)\vec{n}\;=\;\nabla_j\Big[(h^{ij}-2Hg^{ij})\nabla_i\vec{\Phi}\Big]\:,
\end{equation}s
where we have used the Codazzi-Mainardi identity.
Furthermore, it holds
\begin{equation}s
2\,\vec{n}\;=\;|g|^{-1/2}\varepsilonilon^{ij}\nabla_i\vec{\Phi}\times\nabla_j\vec{\Phi}\;=\;\nabla_j\big[-|g|^{-1/2}\varepsilonilon^{ij}\vec{\Phi}\times\nabla_i\vec{\Phi} \big]\:.
\end{equation}s
Accordingly, we may choose
\begin{equation}s
\nabla^j\vec{V}\;=\;-\,\alpha\,\nabla^j\vec{\Phi}\,+\,\dfrac{\begin{equation}ta}{2}\,|g|^{-1/2}\varepsilonilon^{ij}\,\vec{\Phi}\times\nabla_i\vec{\Phi}\,+\,\dfrac{\gamma}{2}\big(h^{ij}-2Hg^{ij}\big)\nabla_i\vec{\Phi}\:.
\end{equation}s
Introduced into the third equation of (\ref{side1}), the latter yields
\begin{equation}gin{eqnarray}
\Delta_g\vec{X}&=&\nabla^j\vec{V}\times\nabla_j\vec{\Phi}\;\;=\;\;\dfrac{\begin{equation}ta}{2}\,|g|^{-1/2}\varepsilonilon^{ij}\big(\vec{\Phi}\times\nabla_i\vec{\Phi}\big)\times\nabla_j\vec{\Phi}\nonumber\\[0ex]
&=&\dfrac{\begin{equation}ta}{2}\,|g|^{-1/2}\varepsilonilon^{ij}\big[(\vec{\Phi}\cdot\nabla_j\vec{\Phi})\nabla_i\vec{\Phi}-g_{ij}\vec{\Phi}\big]\;\;=\;\;\dfrac{\begin{equation}ta}{4}\,|g|^{-1/2}\varepsilonilon^{ij}\,\nabla_j|\vec{\Phi}|^2\,\nabla_i\vec{\Phi}\:,\nonumber
\end{eqnarray}
so that we may choose
\begin{equation}s\lambdabel{XHel}
\nabla^j\vec{X}\;=\;\dfrac{\begin{equation}ta}{4}\,|g|^{-1/2}\varepsilonilon^{ij}|\vec{\Phi}|^2\,\nabla_i\vec{\Phi}\:.
\end{equation}s
Then we find
\begin{equation}\lambdabel{Xprop}
\vec{n}\times\nabla^j\vec{X}\;=\;\dfrac{\begin{equation}ta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi}\qquad\text{and}\qquad\nabla^j\vec{X}\times\nabla^j\vec{\Phi}\;=\;\dfrac{\begin{equation}ta}{2}\,|\vec{\Phi}|^2\,\vec{n}\:.
\end{equation}
Analogously, the fourth equation of (\ref{side1}) gives
\begin{equation}gin{eqnarray}\lambdabel{YHel}
\Delta_gY&=&\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;\;=\;\;-\,2\alpha\,-\,\gamma H\,+\,\dfrac{\begin{equation}ta}{2}\,|g|^{-1/2}\varepsilonilon^{ij}\big(\vec{\Phi}\times\nabla_i\vec{\Phi}\big)\cdot\nabla_j\vec{\Phi}\nonumber\\[0ex]
&=&-\,2\alpha\,-\,\gamma H\,+\,\begin{equation}ta\,\vec{\Phi}\cdot\vec{n}\:.
\end{eqnarray}
With (\ref{Xprop}) and (\ref{YHel}), the system (\ref{thesys1}) becomes
\begin{equation}\lambdabel{thesyshel}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j\vec{n}\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j\vec{n}\,\partial_kS+\partial_j\vec{n}\times\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\bigg(\vec{n}\,\nabla^jY+\dfrac{\begin{equation}ta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi} \bigg)\\[1.5ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\varepsilonilon^{jk}\Big[\partial_jS\,\partial_k\vec{\Phi}+\partial_j\vec{R}\times\partial_k\vec{\Phi}\Big]+|g|^{1/2}\bigg(\nabla_j\vec{\Phi}\nabla^jY+\dfrac{\begin{equation}ta}{2}\,|\vec{\Phi}|^2\,\vec{n} \bigg)\:.
\end{array}\right.
\end{equation}
Under suitable regularity hypotheses (e.g. that the immersion be locally Lipschitz and lie in the Sobolev space $W^{2,2}$), one can show that the non-Jacobian term in the second equation, namely
$$
|g|^{1/2}\nabla_j\bigg(\vec{n}\,\nabla^jY+\dfrac{\begin{equation}ta}{4}\,|\vec{\Phi}|^2\nabla^j\vec{\Phi} \bigg)\:,
$$
is a subcritical perturbation of the Jacobian term. Analyzing (\ref{thesyshel}) becomes then very similar to analyzing the Willmore system (\ref{thesyswillmore}). Details may be found in \cite{BWW1}. \\
In some cases, our so-far local considerations can yield global information. If the vesicle we consider has the topology of a sphere, every loop on it is contractible to a point. The Hodge decompositions which we have performed in Section II to deduce the existence of $\vec{V}$, and subsequently that of $\vec{X}$ and $Y$ hold globally. Integrating (\ref{YHel}) over the whole surface $\Sigma$ then gives the {\it balancing condition}:
$$
2\alpha A(\Sigma)+\gamma M(\Sigma)\;=\;\begin{equation}ta V(\Sigma)\:.
$$
This condition is well-known in the physics literature \cite{CG, Sei}.
\begin{equation}gin{Rm}
Another instance in which minimizing the energy (\ref{veneer}) arises is the isoperimetric problem \cite{KMR, Scy}, which consists in minimizing the Willmore energy under the constraint that the dimensionless isoperimetric ratio $\sigma:=A^3/V^2$ be a given constant in $(0,1]$. As both the Willmore energy and the constraint are invariant under dilation, one might fix the volume $V=1$, forcing the area to satisfy $A=\sigma^{1/3}$. This problem is thus equivalent to minimizing the energy (\ref{veneer}) with $\gamma=0$ (no constraint imposed on the total curvature, but the volume and area are prescribed separately). One is again led to the system (\ref{thesyshel}) and local analytical information may be inferred.
\end{Rm}
\subsection{Chen's Problem}
An isometric immersion $\vec{\Phi}:N^{n}\rightarrow\mathbb{R}^{m>n}$ of an $n$-dimensional Riemannian manifold $N^n$ into Euclidean space is called {\it biharmonic} if the corresponding mean-curvature vector $\vec{H}$ satisfies
\begin{equation}\lambdabel{chen}
\Delta_g\vec{H}\;=\;\vec{0}\:.
\end{equation}
The study of biharmonic submanifolds was initiated by B.-Y. Chen \cite{BYC1} in the mid 1980s as he was seeking a classification of the finite-type submanifolds in Euclidean spaces. Independently, G.Y. Jiang \cite{Jia} also studied (\ref{chen}) in the context of the variational analysis of the biharmonic energy in the sense of Eells and Lemaire. Chen conjectures that a biharmonic immersion is necessarily minimal\footnote{The conjecture as originally stated is rather analytically vague: no particular hypotheses on the regularity of the immersion are {\it a priori} imposed. Many authors consider only smooth immersions.}.
Smooth solutions of (\ref{chen}) are known to be minimal for $n=1$ \cite{Dim1}, for $(n,m)=(2,3)$ \cite{Dim2}, and for $(n,m)=(3,4)$ \cite{HV}. A growth condition allows a simple PDE argument to work in great generality \cite{Whe2}. Chen's conjecture has also been solved under a variety of hypotheses (see the recent survey paper \cite{BYC2}). The statement remains nevertheless open in general, and in particular for immersed surfaces in $\mathbb{R}^m$. In this section, we show how our reformulation of the Willmore equation may be used to recast the fourth-order equation (\ref{chen}) in a second-order system with interesting analytical features. \\
Let us begin by inspecting the tangential part of (\ref{chen}), namely,
\begin{equation}gin{eqnarray}\lambdabel{tgtchen}
\vec{0}&=&\pi_T\Delta_g\vec{H}\;=\;\big(\nabla_k\vec{\Phi}\cdot\nabla_j\nabla^j\vec{H})\nabla^k\vec{\Phi}\nonumber\\[1ex]
&=&\Big[\nabla_j\big(\nabla_k\vec{\Phi}\cdot\nabla^j\vec{H} \big)-\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\;\;=\;\;-\,\Big[\vec{H}\cdot\nabla_j\vec{h}^{j}_{k}+2\,\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\nonumber\\[1ex]
&=&-\,\Big[\nabla_k|\vec{H}|^2+2\,\vec{h}_{jk}\cdot\nabla^j\vec{H} \Big]\nabla^k\vec{\Phi}\:,
\end{eqnarray}
where we have used that $\vec{H}$ is a normal vector, as well as the Codazzi-Mainardi identity. With the help of this equation, one obtains a decisive identity, which we now derive, and which too makes use of the Codazzi-Mainardi equation.
\begin{equation}gin{eqnarray}\lambdabel{deci}
\nabla_j\Big[|\vec{H}|^2\nabla^j\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\nabla_k\vec{\Phi}\Big]&=&-\,\nabla_j|\vec{H}|^2\nabla^j\vec{\Phi}+2|\vec{H}|^2\vec{H}-2(\vec{h}^{jk}\cdot\nabla_j\vec{H})\nabla_k\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}\nonumber\\[1ex]
&\stackrel{\text{(\ref{tgtchen})}}{=}&2|\vec{H}|^2\vec{H}-2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}\:.
\end{eqnarray}
Note that, in general, there holds
\begin{equation}gin{eqnarray}\lambdabel{bidule}
\pi_T\nabla^j\vec{H}&=&\big[\nabla_k\vec{\Phi}\cdot\nabla^j\vec{H}\big]\nabla^k\vec{\Phi}\;\;=\;\;-\,\big[\vec{H}\cdot\vec{h}^{j}_{k}\big]\nabla^k\vec{\Phi}\:.
\end{eqnarray}
An immersion whose mean curvature satisfies (\ref{chen}) has thus the property that
\begin{equation}gin{eqnarray*}
\Delta_\perp\vec{H}&:=&\pi_{\vec{n}}\nabla_j\pi_{\vec{n}}\nabla^j\vec{H}\;\;=\;\;\pi_{\vec{n}}\Delta_g\vec{H}\,-\,\pi_{\vec{n}}\nabla_j\pi_T\nabla^j\vec{H}\;\;=\;\;\pi_{\vec{n}}\nabla_j\Big[\big(\vec{H}\cdot\vec{h}^{j}_{k}\big)\nabla^k\vec{\Phi} \Big]\nonumber\\[0ex]
&=&\big(\vec{H}\cdot\vec{h}^{j}_{k}\big)\vec{h}^{k}_{j}\:.
\end{eqnarray*}
Putting the latter into the first equation of (\ref{side}) yields
\begin{equation}\lambdabel{ELchen}
\vec{\mathcal{W}}\;:=\;\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;2(\vec{H}\cdot\vec{h}^{jk})\vec{h}_{jk}-2|\vec{H}|^2\vec{H}\:.
\end{equation}
We now seek a solution to the second equation in (\ref{side1}), namely a vector $\vec{V}$ satisfying $\Delta_g\vec{V}=-\vec{\mathcal{W}}$. To do so, it suffices to compare (\ref{ELchen}) and (\ref{deci}) to see that we may choose
\begin{equation}s
\nabla^j\vec{V}\;=\;|\vec{H}|^2\nabla^j\vec{\Phi}-2(\vec{H}\cdot\vec{h}^{jk})\nabla_k\vec{\Phi}\:.
\end{equation}s
Introduced into the third equation of (\ref{side1}), the latter yields immediately
\begin{equation}s
\Delta_g\vec{X}\;=\;\nabla^j\vec{V}\wedge\nabla_j\vec{\Phi}\;=\;\vec{0}\:,
\end{equation}s
thereby prompting us to choosing $\vec{X}\equiv\vec{0}$. The fourth equation of (\ref{side1}) gives
\begin{equation}s
\Delta_gY\;=\;\nabla^j\vec{V}\cdot\nabla_j\vec{\Phi}\;=\;-\,2|\vec{H}|^2\:.
\end{equation}s
On the other hand, owing to (\ref{chen}) and (\ref{bidule}), one finds
\begin{equation}s
\Delta_g(\vec{\Phi}\cdot\vec{H})\;=\;-\,2|\vec{H}|^2\:,
\end{equation}s
so that we may choose $Y=\vec{\Phi}\cdot\vec{H}$. Introducing these newly found facts for $\vec{X}$ and $Y$ into the system (\ref{thesys1}) finally gives
\begin{equation}\lambdabel{thesyschen}
\left\{\begin{equation}gin{array}{rcl}
|g|^{1/2}\Delta_g S&=&\varepsilonilon^{jk}\partial_j(\star\,\vec{n})\cdot\partial_k\vec{R}\\[1ex]
|g|^{1/2}\Delta_g\vec{R}&=&\varepsilonilon^{jk}\Big[\partial_j(\star\,\vec{n})\partial_kS+\partial_j(\star\,\vec{n})\bullet\partial_k\vec{R}\Big]+|g|^{1/2}\nabla_j\big((\star\,\vec{n})\nabla^j(\vec{\Phi}\cdot\vec{H})\big)\\[1ex]
|g|^{1/2}\Delta_g\vec{\Phi}&=&-\,\varepsilonilon^{jk}\Big[\partial_jS\partial_k\vec{\Phi}+\partial_j\vec{R}\bullet\partial_k\vec{\Phi}\Big]+|g|^{1/2}\big(\nabla^j(\vec{\Phi}\cdot\vec{H})\nabla_j\vec{\Phi} \big)\:.
\end{array}\right.
\end{equation}
As previously noted, the reformulation of (\ref{trans}) in the form (\ref{thesys1}) is mostly useful to reduce the nonlinearities present in (\ref{trans}). Moreover, a local analysis of (\ref{thesys1}) is possible when the non-Jacobian terms on the right-hand side are subcritical perturbations of the Jacobian terms. This is not {\it a priori} the case for (\ref{thesyschen}), and the equation (\ref{chen}) is linear to begin with. Nevertheless, the system (\ref{thesyschen}) has enough suppleness\footnote{owing mostly to the fact that the function $Y:=\vec{\Phi}\cdot\vec{H}$ satisfies $\Delta_gY\le 0$ and $\Delta_g^2Y\le0$.} and enough structural features to deduce interesting analytical facts about solutions of (\ref{chen}), under mild regularity requirements. This is discussed in detail in a work to appear \cite{BWW1}\footnote{see also \cite{Whe3} which contains interesting estimates for equation (\ref{chen}).}.
\subsection{Point-Singularities}
As was shown in \cite{YBer, BR2, Riv2}, the Jacobian-type system (\ref{side}) is particularly suited to the local analysis of point-singularities. The goal of this section is not to present a detailed account of the local analysis of point-singularities -- this is one of the topics of \cite{BWW1} -- but rather to give the reader a few pertinent key arguments on how this could be done. \\
Let $\vec{\Phi}:D^2\setminus\{0\}\rightarrow\mathbb{R}^m$ be a smooth immersion of the unit-disk, continuous at the origin (the origin will be the point-singularity in question). In order to make sense of the Willmore energy of the immersion $\vec{\Phi}$, we suppose that $\int_{D^2}|\vec{H}|^2d\text{vol}_g<\infty$. Our immersion is assumed to satisfy the problem
\begin{equation}\lambdabel{ptprob}
\Delta_\perp\vec{H}+(\vec{H}\cdot\vec{h}_j^i)\vec{h}^j_i-2|\vec{H}|^2\vec{H}\;=\;\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\setminus\{0\}\:,
\end{equation}
where the vector $\vec{\mathcal{W}}$ is given. It may depend only on geometric quantities (as is the case in the Willmore problem or in Chen's problem), but it may also involve ``exterior" quantities (as is the case in the conformally-constrained Willmore problem). To simplify the presentation, we will not in this paper discuss the integrability assumptions that must be imposed on $\vec{\mathcal{W}}$ to carry out the procedure that will be outlined. The interested reader is invited to consult \cite{BWW1} for more details on this topic. \\
\noindent
As we have shown in (\ref{defT}), equation (\ref{ptprob}) may be rephrased as
\begin{equation}s
\partial_j\big(|g|^{1/2}\vec{T}^j\big)\;=\;-\,\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\setminus\{0\}\:,
\end{equation}s
for some suitable tensor $\vec{T}^j$ defined solely in geometric terms. Consider next the problem
\begin{equation}s
\Delta_g\vec{V}\;=\;-\,\vec{\mathcal{W}}\qquad\text{on}\:\:D^2\:.
\end{equation}s
As long as $\vec{\mathcal{W}}$ is not too wildly behaved, this equation will have at least one solution. Let next $\mathcal{L}_g$ satisfy
\begin{equation}s
\partial_j\big(|g|^{1/2}\nabla^j\mathcal{L}_g\big)\;=\;\delta_0\qquad\text{on}\:\:D^2\:.
\end{equation}s
If the immersion is correctly chosen (e.g. $\vec{\Phi}\in W^{2,2}\cap W^{1,\infty}$), the solution $\mathcal{L}_g$ exists and has suitable analytical properties (see \cite{BWW1} for details). \\
We have
\begin{equation}\lambdabel{poinc1}
\partial_j\Big[|g|^{1/2}\big(\vec{T}^j-\nabla^j\vec{V}-\vec{\begin{equation}ta}\,\nabla^j\mathcal{L}_g\big)\Big]\;=\;\vec{0}\qquad\text{on}\:\:D^2\setminus\{0\}\:,
\end{equation}
for any constant $\vec{\begin{equation}ta}\in\mathbb{R}^m$, and in particular for the unique $\vec{\begin{equation}ta}$ fulfilling the circulation condition that
\begin{equation}\lambdabel{poinc2}
\int_{\partial D^2}\vec{\nu}\cdot\big(\vec{T}^j-\nabla^j\vec{V}-\vec{\begin{equation}ta}\,\nabla^j\mathcal{L}_g\big)\;=\;0\:,
\end{equation}
where $\vec{\nu}\in\mathbb{R}^2$ denotes the outer unit normal vector to the boundary of the unit-disk. This vector $\vec{\begin{equation}ta}$ will be called {\it residue}.\\
Bringing together (\ref{poinc1}) and (\ref{poinc2}) and calling upon the Poincar\'e lemma, one infers the existence of an element $\vec{L}$ satisfying
\begin{equation}
\vec{T}^j-\nabla^j\vec{V}-\vec{\begin{equation}ta}\,\nabla^j\mathcal{L}_g\;=\;|g|^{-1/2}\varepsilonilon^{kj}\nabla_k\vec{L}\:,
\end{equation}
with the same notation as before. We are now in the position of repeating {\it mutatis mutandis} the computations derived in the previous section, taking into account the presence of the residue. We define $\vec{X}$ and $Y$ via:
\begin{equation}\lambdabel{ptside}
\left\{\begin{equation}gin{array}{rcl}
\Delta_g\vec{X}&=&\nabla^j\big(\vec{V}+\vec{\begin{equation}ta}\mathcal{L}_g\big)\wedge\nabla_j\vec{\Phi}\\[1ex]
\Delta_g{Y}&=&\nabla^j\big(\vec{V}+\vec{\begin{equation}ta}\mathcal{L}_g\big)\cdot\nabla_j\vec{\Phi}
\end{array}\right.\qquad\text{on}\:\:D^2\:.
\end{equation}
One verifies that the following equations hold
\begin{equation}s
\left\{\begin{equation}gin{array}{rcl}
\nabla^k\Big[\vec{L}\wedge\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\big(\vec{H}\wedge\nabla^j\vec{\Phi}+\nabla^j\vec{X}\big)\Big]&=&\vec{0}\\[2ex]
\nabla^k\Big[\vec{L}\cdot\nabla_k\vec{\Phi}\,-\,|g|^{1/2}\varepsilonilon_{kj}\nabla^jY\Big]&=&0
\end{array}\right.\qquad\text{on}\:\:D^2\setminus\{0\}\:.
\end{equation}s
Imposing suitable hypotheses on the integrability of $\vec{\mathcal{W}}$ yields that the bracketed quantities in the latter are square-integrable. With the help of a classical result of Laurent Schwartz \cite{Scw}, the equations may be extended without modification to the whole unit-disk. As before, this grants the existence of two potential functions $S$ and $\vec{R}$ which satisfy (\ref{defRS}) and the system (\ref{thesys}) on $D^2$. The Jacobian-type/divergence-type structure of the system sets the stage for a local analysis argument, which eventually yields a local expansion of the immersion $\vec{\Phi}$ around the point-singularity. This expansion involves the residue $\vec{\begin{equation}ta}$. The procedure was carried out in details for Willmore immersions in \cite{BR2}\footnote{An equivalent notion of residue was also identified in \cite{KS2}.}, and for conformally constrained Willmore immersions in \cite{YBer}. Further considerations can be found in \cite{BWW1}.
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\end{thebibliography}
\end{document} |
\begin{document}
\title[Evaluation of Dedekind sums]{Evaluation of Dedekind sums,
Eisenstein cocycles, and special values of $L$-functions}
\newif \ifdraft
\def \makeauthor{
\ifdraft
\draftauthor{Paul E. Gunnells and Robert Sczech}
\else
\author{Paul E. Gunnells}
\address{Department of Mathematics\\
Columbia University\\
New York, NY 10027}
\email{[email protected]}
\author{Robert Sczech}
\address{Department of Mathematics\\
Rutgers University--Newark\\
Newark, NJ 07102}
\email{[email protected]}
\fi
}
\thanks{The first author was partially supported by a Columbia
University Faculty Research grant and the NSF}
\draftfalse
\makeauthor
\ifdraft
\date{\today}
\else
\date{September 23, 1999}
\fi
\subjclass{11F20, 11F75, 11R42, 11R80, 11Y16}
\keywords{Dedekind sums, reciprocity law, Eisenstein cocycle,
continued fraction algorithms, special values of partial zeta
functions}
\begin{abstract}
We define certain higher-dimensional Dedekind sums that generalize the
classical Dedekind-Rademacher sums, and show how to compute them
effectively using a generalization of the continued-fraction
algorithm.
We present two applications. First, we show how to express special
values of partial zeta functions associated to
totally real number fields in terms of these sums via the Eisenstein
cocycle introduced by the second author. Hence we obtain a
polynomial-time algorithm for computing these special values. Second,
we show how to use our techniques to compute certain special values of
the Witten zeta-function, and compute some explicit examples.
\end{abstract}
\maketitle
\section{Introduction}\label{introduction}
\subsection{}\label{intro}
Let $\sigma $ be a square matrix with integral columns $\sigma
_{j}\in {\mathbb Z} ^{n}$ ($j=1,\dots ,n$), and let $L\subset {\mathbb Z} ^{n}$ be a
lattice of rank $\geq 1$. Let $v\in {\mathbb R}^{n}$, and let $e\in {\mathbb Z} ^{n}$
with $e_{j}\geq 1$. Associated to the data $(L,\sigma ,e,v)$ is the
\emph{Dedekind sum}
\begin{equation}\label{Dede.sum}
S = S (L,\sigma ,e,v) := \sideset{}{'}\sum _{x\in L} {{e}}e (\langle
x,v\rangle )\frac{\det \sigma}{\langle x,\sigma _{1}\rangle
^{e_{1}}\cdots \langle x,\sigma _{n}\rangle ^{e_{n}}}.
\end{equation}
Here $\langle x,y\rangle := \sum x_{i}y_{i}$ is the usual scalar
product on ${\mathbb R} ^{n}$, ${{e}}e (t)$ is the character $\exp (2\pi it)$,
and the prime next to the summation means to omit terms for which the
denominator vanishes. This series converges absolutely if all
$e_{j}>1$, but only conditionally if $e_{j}=1$ for some $j$. In
the latter case, we define the value of $S$ by the $Q$-limit
\begin{equation}
\sideset{}{'}\sum _{x\in L} a (x){\mathscr B}igr|_{Q} := \lim _{t\rightarrow \infty
}{\mathscr B}igl(\sideset{}{'}\sum_{\substack{x\in L\\|Q (x)|<t}} a (x) {\mathscr B}igr),
\end{equation}
where $Q$ is any finite product of real-valued linear forms on ${\mathbb R}
^{n}$ that do not vanish on ${\mathbb Q} ^{n}\smallsetminus \{0 \}$. As we
explain in \S\ref{qlimit}, this limit depends on $Q$ in a rather simple way.
Nevertheless, to keep notation to a minimum, we assume for now that
the series \eqref{Dede.sum} converges absolutely.
\subsection{}\label{nature}
The arithmetic nature of the values of $S$ is well known. Up to
a power of $2\pi i$, they are always rational numbers if $v\in {\mathbb Q}
^{n}$. However, the explicit calculation of these values is not easy,
especially if $|\det \sigma |$ is large. In this paper, we
exhibit a polynomial-time algorithm for calculating $S$ efficiently.
Before stating our main result, we briefly review a few special cases.
In the case $n=1$, $\sigma =1$, $L={\mathbb Z}$, we have
\begin{equation}
S (L,\sigma ,e,v) = -\frac{(2\pi i)^{e}}{e!}{\mathscr B} _{e} (v),
\end{equation}
where ${\mathscr B} _{e} (v)$ is the periodic function (with period lattice ${\mathbb Z}
$) that coincides with the classical Bernoulli polynomial $B_{e} (v)$
on the interval $0<v<1$.
More generally, if $L = {\mathbb Z} ^n$, then we
show in Proposition \ref{how.to.sum} that
\begin{equation}\label{class.dede}
S (L,\sigma ,e,v) = \kappa \sum _{r\in L/\sigma L} {\mathscr B} _{e_{1}} (u_{1})\cdots
{\mathscr B} _{e_{n}} (u_{n}),
\end{equation}
where
\begin{equation}
u = \sigma ^{-1} (r+v)\quad \hbox{and}\quad \kappa = \Sign (\det \sigma) \prod
_{j=1}^{n} \frac{(-2\pi i)^{e_{j}}}{e_{j}!}.
\end{equation}
The finite sum on the right of \eqref{class.dede} is a classical
Dedekind sum. Although theoretically satisfying, from a computational
point of view this sum is of interest only if $|\det \sigma |$ (the
number of summands) is relatively small.
\subsection{}
In general, if ${\mathbb R}ank L <n$, no simple analogue of \eqref{class.dede}
seems to exist except for the special case where ${\mathbb R}ank L$ equals the
number of linear forms $\langle x,\sigma _{j}\rangle $ in
\eqref{Dede.sum} that are not proportional to each other when
restricted to the subspace $L\otimes {\mathbb R} $. Dedekind sums of this
special type are called \emph{diagonal}. It is clear that the finite
sum formula \eqref{class.dede} remains valid (modulo obvious modifications)
for all diagonal Dedekind sums.
Let us further say that a diagonal Dedekind sum is \emph{unimodular}
if the corresponding finite sum has exactly one term. As a measure of
the deviation from unimodularity, we introduce an integer-valued
function $\|\phantom{S}\|$, called the \emph{index}, on the set of all
Dedekind sums (Definition~\ref{index}). For example, if $L = {\mathbb Z} ^{n}$, then
$\|S\| = |\det \sigma |$. In particular, $S$ is said to be unimodular
if and only if $\|S\| = 1$.
\subsection{}
We are now ready to describe our main result.
\begin{theorem}\label{mnthm}
Every Dedekind sum $S (L,\sigma ,e,v)$ can be expressed as a finite
rational linear combination of unimodular diagonal sums. If
$n$, ${\mathbb R}ank L$, and $e$ are fixed, then this expression can be
computed in time polynomial in $\log\Index{S}$. Moreover, the number of
terms in this expression is bounded by a polynomial in $\log\Index{S}$.
\end{theorem}
The special case $n=2$ and $e= (1,1)$ has been known for a very long
time. This is the case of the classical Dedekind-Rademacher sums,
where the simple form of their reciprocity law combined with the
Euclidean algorithm immediately yields a polynomial-time algorithm for
computing their values (cf. Example~\ref{quadratic}). Our proof of
Theorem~\ref{mnthm} is similar. First we prove a general reciprocity
law for the sums $S$. Applying and specializing this law repeatedly
yields a representation of $S$ as a linear combination of diagonal
sums. Combining it further with the algorithm of
Ash-Rudolph~\cite{ash.rudolph} finally yields a representation by
unimodular diagonal sums.
\subsection{}
Among the many applications, we review in this paper the problem of
calculating the Eisenstein cocycle~\cite{eisenstein} on $GL_{n} ({\mathbb Q} )$
that, when combined with Theorem~\ref{mnthm}, yields a polynomial-time
algorithm for calculating special values of partial zeta functions of
totally real number fields (\S\S\ref{ec}--\ref{vpzf}). These special
values are of great interest in view of the many conjectures they are
subject to (Leopoldt conjecture, Brumer-Stark conjecture, etc.). We
also review the connection between Witten's zeta function and
Dedekind sums, and use our techniques to give some explicit formulas
for special values of these functions (\S\ref{witten}).
\section{The modular symbol algorithm and Dedekind reciprocity}
\subsection{}
Let $\sigma _{1},\dots ,\sigma _{n}\in {\mathbb Z} ^{n}$
be nonzero primitive points, and let $D=|\det (\sigma _{1},\dots
,\sigma _{n})|$. Let $w\in {\mathbb Z} ^{n}$ be another nonzero primitive
point, and define
\[
D_{i} (w) := |\det (\sigma _{1},\dots ,\widehat \sigma _{i},\dots ,\sigma
_{n}, w)|,\quad \hbox{$i=1,\dots ,n$.}
\]
The following basic result will play a key role:
\begin{proposition}\label{msa}
\cite{ash.rudolph, barvinok} If $D>1$, then there exists $w\in {\mathbb Z}
^{n}\smallsetminus \{0 \}$ such that
\begin{equation}\label{estimate}
0\leq D_{i} (w) < D^{(n-1)/n},\quad \hbox{$i=1,\dots ,n$,}
\end{equation}
and at least one $D_{i} (w)\not =0$. Moreover, for fixed $n$ the
point $w$ can be constructed in polynomial time.
\end{proposition}
\begin{proof}
(Sketch) Here we only show that $w$ exists. Let $P$ be the open
parallelotope
\[
P:={\mathscr B}igl\{\sum \lambda _{i}\sigma _{i}{\mathscr B}igm | |\lambda_{i} |< D^{-1/n}
{\mathscr B}igr\}.
\]
Then $P$ is an $n$-dimensional centrally symmetric convex body with
volume $2^{n}$. By Minkowski's theorem (cf.~\cite[IV.2.6]{fr.tay}),
$P\cap {\mathbb Z} ^{n}$ contains a nonzero point. This is the desired point $w$.
\end{proof}
\begin{remark}
Ash-Rudolph~\cite{ash.rudolph} show that $w$ satisfies $0\leq D_{i}
(w)<D$. They also show how to construct $w$ using the Euclidean
algorithm. The stronger estimate \eqref{estimate} and the statement
about polynomial time are due to Barvinok~\cite[Lemma 5.2]{barvinok}.
In practice, to efficiently construct $w$ such that $D_{i} (w)$ is
small, one may use $LLL$-reduction~\cite[\S3]{experimental}.
\end{remark}
\subsection{}
We now state the Dedekind reciprocity law. For any nonzero point
$v\in {\mathbb R} ^{n}$, let $v^{\perp }$ be the hyperplane $\{x \mid \langle
v,x\rangle =0 \}$. Let $Q$ be a finite product of real-valued linear
forms on ${\mathbb R} ^{n}$ that do not vanish on ${\mathbb Q} ^{n}\smallsetminus \{0
\}$ and recall that the notation
\[
\sideset{}{'}\sum _{x\in L} a (x){\mathscr B}igr|_{Q}
\]
means the sum is to be evaluated using the $Q$-limit (\S\ref{intro},
\S\ref{qlimit}).
\begin{proposition}\label{recip}
Let $\sigma _{0},\dots ,\sigma _{n}\in {\mathbb Z} ^{n}$ be nonzero. For
$j=0,\dots ,n$, let $\sigma ^{j}$ be the matrix with columns $\sigma
_{0},\dots ,\widehat \sigma _{j},\dots ,\sigma _{n}$. Fix $L\subseteq {\mathbb Z}
^{n}$, and assume $e = {\mathbf{1}} := (1,\dots ,1)$. Then for any $v\in {\mathbb R}
^{n}$, we have the following identity among Dedekind sums:
\begin{equation}\label{recip.law}
\sum _{j=0}^{n} (-1)^{j} S (L,\sigma ^{j},{\mathbf{1}} ){\mathscr B}igr|_{Q} =
\sum _{j=0}^{n} (-1)^{j} S (L\cap \sigma _{j}^{\perp },\sigma
^{j},{\mathbf{1}} ){\mathscr B}igr|_{Q}.
\end{equation}
\end{proposition}
\begin{proof}
Let $D_{j}=\det \sigma ^{j}$.
We have an identity for rational functions of $x$
\begin{equation}\label{id1}
\sum _{j=0}^{n}\frac{(-1)^{j}D_{j}}{\prod _{k\not =j}\langle x,\sigma
_{k}\rangle } = 0,
\end{equation}
valid for any $x\in {\mathbb R} ^{n}$ satisfying $\langle x,\sigma _{j}\rangle \not =0$ for
$j=0,\dots ,n$. To see this, consider the $(n+1)\times (n+1)$ matrix
\begin{equation}\label{mat}
\left(\begin{array}{ccc}
\langle x,\sigma _{0}\rangle &\dots &\langle x,\sigma _{n}\rangle \\
\sigma _{0}&\dots &\sigma _{n}
\end{array} \right).
\end{equation}
This matrix is singular, since the first row is a linear combination
of the others. Expanding by minors along the top row, and dividing by
$\prod \langle x,\sigma _{k}\rangle $ yields \eqref{id1}.
To pass from \eqref{id1} to \eqref{recip.law}, we need to incorporate
the exponential character and sum over $L$ using $Q$. There is no
obstruction to doing this, although we must omit terms where
\emph{any} linear form vanishes. We obtain the expression
\begin{equation}\label{eqqq}
\sum _{j=0}^{n} (-1)^{j}\sum {{e}}e (\langle x,v\rangle
)\frac{D_{j}}{\prod _{k\not =j}\langle x,\sigma _{k}\rangle }\biggr|_{Q} = 0,
\end{equation}
where the inner sum is taken over all $x\in L$ with $\langle x,\sigma
_{k}\rangle \not =0$ for $k=0,\dots ,n$.
In \eqref{eqqq}, the $j$th inner sum corresponds with the Dedekind sum $S
(L,\sigma ^{j},{\mathbf{1}} )$ except for the terms with $\langle x,\sigma
_{j}\rangle=0$ and $\langle x,\sigma _{k}\rangle \not =0$ for $k\not
=j$. In other words, to make the $j$th sum into a Dedekind sum, we
must add
\begin{equation}\label{correct}
\sideset{}{'}\sum_{x\in L\cap \sigma _{j}^{\perp }}{{e}}e(\langle
x,v\rangle )\frac{D_{j}}{\prod _{j\not =k}\langle x,\sigma
_{k}\rangle}\biggr|_{Q}.
\end{equation}
Simultaneously adding and subtracting \eqref{correct} to \eqref{id1}
yields \eqref{recip.law}.
\end{proof}
\subsection{}\label{qlimit}
We now recall the $Q$-limit formula from \cite[Theorem
2]{eisenstein}. Let
\begin{equation*}
Q (y) = \prod _{i=1}^{m} Q_{i} (y)
\end{equation*}
be a product of $m\geq 1$ linear forms
\[
Q_{i} (y) = \sum _{j=1}^{n}Q_{ij}y_{j},
\]
with rationally independent real coefficients $Q_{ij}$. We think of
$Q$ as an $m\times n$ matrix with rows $Q_{i}$.
Given a vector $e= (e_{1},\dots ,e_{n})$ of positive integers and a
vector $v\in {\mathbb R} ^{n}$, let
\[
J = \{j\mid \hbox{$e_{j}=1$ and $v_{j}\in {\mathbb Z}$} \}.
\]
If $\#J\equiv 0 \bmod 2$, define
\begin{equation}
{\mathscr B}B_{e} (v,Q) = \frac{1}{m}\sum _{i=1}^{m}\left ( \prod _{j\in J}\frac{\Sign Q_{ij}}{2}\right)\prod _{j\not \in J}{\mathscr B} _{e_{j}} (v_{j}),
\end{equation}
otherwise let ${\mathscr B}B _{e} (v,Q) = 0$. In particular, if $J=\varnothing $, then
\[
{\mathscr B}B_{e} (v,Q) = \prod _{j=1}^{n}{\mathscr B} _{e_{j}} (v_{j}).
\]
Now the $Q$-limit formula can be stated as follows. Let $\Id_{n}$ be
the $n\times n$ identity matrix. Then
\begin{equation}\label{qfmla1}
S ({\mathbb Z} ^{n},\Id_{n},e,v){\mathscr B}igr|_{Q} = \kappa \,{\mathscr B}B_{e} (v, Q),
\end{equation}
where
\begin{equation}\label{qfmla2}
\kappa = \prod _{j=1}^{n}\frac{(-2\pi i)^{e_{j}}}{e_{j}!}.
\end{equation}
\section{Diagonality and unimodularity}
\subsection{}
We begin with some simplifying assumptions to ease the exposition. We
define the \emph{rank} of $S=S (L,\sigma ,e,v)$ to be the rank of the
lattice $L$.
Suppose that the rank of $S$ is $\ell $, and for any $k$ let $Z
^{k}\subseteq {\mathbb Z} ^{n}$ be the sublattice spanned by the first $k$
standard basis vectors. We claim that $S$ can be
computed using $Z ^{\ell }$ instead of $L$. Indeed, writing $L =
gZ^{\ell } $ with a matrix $g\in GL_{n} ({\mathbb Q} ) $, and letting $\sigma '
= g^{t}\sigma $, $v'=g^{t}v$, and $Q' = Qg$, we have
\[
S (L,\sigma ,e,v){\mathscr B}igr|_{Q} = (\det g)^{-1} S (Z ^{\ell },\sigma'
,e,v'){\mathscr B}igr|_{Q'}.
\]
The entries of $\sigma '$ need not be integral, but after multiplying by
an appropriate rational factor we can assume this is true. In fact,
by further multiplication of $\sigma '$ by rational numbers and
permuting columns, we can write
\[
S (L,\sigma ,e,v){\mathscr B}igr|_{Q} = qS (Z ^{\ell },\sigma'
,e,v'){\mathscr B}igr|_{Q'},\quad q\in {\mathbb Q} ^{\times },
\]
where the pair $(Z ^{\ell },\sigma ')$ satisfies the following:
\begin{enumerate}
\item [(i)] For each column $\sigma_{j} '$, the vector $\sigma _{j}'\cap
Z^{\ell }$ is primitive and integral.
\item [(ii)] If two columns of $\sigma '$ induce proportional linear forms on
$Z ^{\ell }$, then these two linear forms coincide on $Z ^{\ell }$,
and are adjacent columns of $\sigma '$.
\end{enumerate}
\begin{definition}\label{normalized}
We say that a rank $\ell $ Dedekind sum is \emph{normalized} if the
conditions above are met.
\end{definition}
\subsection{}
Let $S (Z ^{\ell},\sigma ,e,v)$ be a normalized Dedekind sum, and let
$N = \sum e_{j}$. We claim that without loss of generality, we need
only consider sums for which $e={\mathbf{1}} $. Indeed, let $R ^\ell
\rightarrow R ^{n}\rightarrow {\mathbb R} ^{N}$ be the spans of the first $\ell
$ (respectively $n$) basis vectors in ${\mathbb R}^{N}$, and identify ${\mathbb R}^{n}$
with $R^{n}$. Let $Q'$ be a product of linear forms on ${\mathbb R} ^{N}$ such
that $Q'$ restricted to $R^{n}$ equals $Q$ restricted to ${\mathbb R} ^{n}$.
We claim that we can construct an $N\times N$ matrix $\sigma '$ and
$v'\in {\mathbb R} ^{N}$ such that
\begin{equation}\label{sum2}
S (Z ^{\ell },\sigma ,e,v){\mathscr B}igr|_{Q} = S (Z ^{\ell },\sigma ',{\mathbf{1}}
,v'){\mathscr B}igr|_{Q'}.
\end{equation}
To see this, let $\pi \colon {\mathbb R} ^{N} \rightarrow {\mathbb R} ^{\ell }$ be the
projection on the first $\ell $ components. Given $v\in R ^{n}$, call any $\tilde{v}\in {\mathbb R} ^{N}$
such that $\pi (\tilde{v}) = \pi (v)$ a \emph{lift} of $v$.
Now to construct $S (Z ^{\ell },\sigma ',{\mathbf{1}} ,v')$, let $v'$ be any
lift of $v$. If the restriction of the linear form $\sigma _{j}$ to
$Z^{\ell }$ appears on the left of \eqref{sum2} with multiplicity
$e_{j}$, set $j$ columns of $\sigma '$ to be $j$ different lifts of
$\sigma _{j}$. If we choose these lifts so that $\det \sigma ' = \det
\sigma $, then we obtain \eqref{sum2}.
\begin{definition}\label{prop.embed}
We say that a Dedekind sum with $e={\mathbf{1}} $ is \emph{properly
embedded}.
\end{definition}
\subsection{}
Let $S (Z ^{\ell },\sigma ,{\mathbf{1}} ,v)$ be a properly embedded,
normalized Dedekind sum. Let $\sets{n}$ be the set $\{1,\dots ,n \}$.
\begin{definition}\label{index}
The \emph{index} of $S$, denoted $\Index{S}$, is defined to be
\[
\Max _{I\subset \sets{n}} \,\bigl|\det \bigl(\pi (\sigma
_{i_{1}}), \dots , \pi (\sigma _{i_{\ell }})\bigr)\bigr|,
\]
where the maximum is taken over all subsets $I = \{i_{1},\dots
,i_{\ell }\}$ of cardinality $\ell $. A Dedekind sum is called
\emph{unimodular} if $\Index{S} = 1$.
\end{definition}
\subsection{}\label{part.sec}
Now define a partition
\begin{equation}\label{part}
\sets{n} = \bigsqcup_{k=1}^{s} I_{k}, \quad
\hbox{$\ell \leq s\leq n$}
\end{equation}
as follows. Put
\[
i,j\in I_{k} \quad \hbox{if and only if}\quad \pi
(\sigma _{i}) = \pi (\sigma _{j}).
\]
In other words, two elements of $\sets{n}$ are in the same set of the
partition if the corresponding columns of $\sigma $ induce
the same linear form on $Z ^{\ell }$.
Let $p_{k}=\#I_{k}$. The vector $p(S) = (p_{1},\dots ,p_{s})$ is
called the \emph{type} of $S$. To emphasize the type, we
relabel the columns of $\sigma $ as
\begin{equation}\label{labelling}
(\sigma _{1}^{1},\dots ,\sigma ^{p_{1}}_{1},\sigma _{2}^{1},\dots
,\sigma _{2}^{p_{2}}, \dots , \sigma _{s}^{1},\dots ,\sigma _{s}^{p_{s}}).
\end{equation}
For any $k=1,\dots ,s$ and any $i=1,\dots, p_{k}$, we denote the point
$\pi (\sigma _{k}^{j})\in {\mathbb Z} ^{\ell }$ by $\sigma _{I_{k}}$.
\begin{definition}\label{diag}
A Dedekind sum is called \emph{diagonal} if $p (S)$
has length $\ell$.
\end{definition}
We omit the proof of the following simple lemma:
\begin{lemma}\label{diag.example}
Any normalized, properly embedded rank 1 Dedekind sum is both
diagonal and unimodular.
\end{lemma}
\subsection{}
The virtue of diagonality and unimodularity is the following:
\begin{proposition}\label{how.to.sum}
Keep the notation of the preceding section.
Let $S (Z ^{\ell },\sigma ,{\mathbf{1}} ,v)$ be properly embedded,
normalized, and diagonal. Let $\rho$ be the $\ell \times \ell $
matrix $(\sigma _{I_{1}},\dots ,\sigma _{I_{\ell }})$, and let
$(p_{1},\dots ,p_{\ell })$ be the type of $S$. Then $S|_{Q}$ is
well-defined. Moreover,
\begin{equation}\label{diagsum}
S (Z^{\ell },\sigma ,{\mathbf{1}} ,v){\mathscr B}igr|_{Q} = \frac{\kappa \det{\sigma }}{|\det{\rho }|}\sum _{r\in {\mathbb Z}
^{\ell}/\rho {\mathbb Z}^{\ell }} {\mathscr B}B_{p} (u, Q'),
\end{equation}
where
\begin{align}\label{diagsum2}
u &= \rho^{-1}(r+\pi (v)),\quad Q'= Q''\circ \rho^{-t}\circ \pi, \quad \hbox{and}\\
\kappa &= \prod_{j=1}^{\ell } \frac{(-2\pi i)^{p_{j}}}{p_{j}!}.
\end{align}
Here $Q'' $ is the restriction of $Q$ to ${\mathbb R}^{\ell }$ into $R^{\ell }$.
If $S$ is unimodular, the sum \eqref{diagsum} has only
one term.
\end{proposition}
\begin{proof}
The second statement follows easily from the first, so we focus on the
first. By definition, we have
\begin{equation}\label{eqa}
S (Z^{\ell },\sigma ,{\mathbf{1}} ,v){\mathscr B}igr|_{Q} = \sideset{}{'}\sum _{x\in
Z^{\ell }}{{e}}e (\langle x, v\rangle )\frac{\det \sigma }{\prod _{1\leq
k\leq \ell }\langle x, \sigma^{1} _{k}\rangle ^{p_{k}}}\biggr|_{Q}.
\end{equation}
Letting $Q' = Q''\circ \rho^{-t}\circ \pi$ and $v' = \rho^{-1}\pi (v)$, the right of
\eqref{eqa} becomes
\begin{equation}\label{eqb}
\det (\sigma )\sideset{}{'}\sum_{y\in \rho^{t}{\mathbb Z} ^{\ell }}{{e}}e (\langle
y,v'\rangle )\frac{\det \rho }{\prod _{1\leq k\leq \ell }
y_{k}^{p_{k}}}\biggr|_{Q'}.
\end{equation}
Inserting the character relations
\begin{equation}\label{eqc}
\sum _{r\in {\mathbb Z} ^{\ell }/\rho{\mathbb Z} ^{\ell }} {{e}}e (\langle y,\rho
^{-1}r\rangle ) =
\begin{cases}
0,&y\in {\mathbb Z} ^{\ell} \smallsetminus \rho^{t}{\mathbb Z} ^{\ell },\\
\#({\mathbb Z} ^{\ell }/\rho{\mathbb Z} ^{\ell }),&y\in \rho^{t}{\mathbb Z} ^{\ell },
\end{cases}
\end{equation}
we obtain
\begin{align}\label{eqd}
S (Z^{\ell },\sigma ,{\mathbf{1}} , v) &=\frac{\det\sigma }{|\det \rho|}\sum _{r\in
{\mathbb Z} ^{\ell }/\rho{\mathbb Z} ^{\ell }}\sideset{}{'}\sum _{y\in {\mathbb Z} ^{\ell }} \frac{{{e}}e (\langle y,\rho^{-1} (r+v)\rangle )}{\prod y_{k}^{p_{k}}}\biggr|_{Q'}\\
&=\frac{\det\sigma }{|\det \rho|}\sum _{r\in{\mathbb Z} ^{\ell }/\rho{\mathbb Z} ^{\ell
}} S ({\mathbb Z} ^{\ell },\Id_{n},p,u)\biggr|_{Q'}\label{eqd:2}
\end{align}
where $u=\rho ^{-1} (r+\pi (v))$.
The proposition follows now from the $Q$-limit formula \eqref{qfmla1}.
\end{proof}
\section{Algorithms}\label{algorithms}
\subsection{}
In this section we prove that any Dedekind sum is a ${\mathbb Q} $-linear
combination of diagonal, unimodular sums. We begin with a lemma. For
simplicity, we abbreviate the Dedekind sum to $S (Z ^{\ell },\sigma
)$.
\begin{lemma}\label{index.no.change}
Let $S (Z ^{\ell },\sigma )$ be a normalized, properly embedded Dedekind sum,
and let $\sigma _{i}$ be a column of $\sigma $. Then
\[
\bigl\|S (Z ^\ell\cap \sigma_{i}^{\perp } ,\sigma )\bigr\| \leq
\bigl\|S (Z ^{\ell },\sigma )\bigr\|.
\]
\end{lemma}
\begin{proof}
Without loss of generality, assume that $\sigma _{i} = \sigma _{1}$.
We can represent $\sigma $ as
\begin{equation}\label{mat1}
\sigma =\left( \begin{array}{ccc|ccc|c|cc}
\sigma _{I_{1}}&\cdots&\sigma _{I_{1}}&\sigma _{I_{2}}&\cdots&
\sigma
_{I_{2}}&\cdots &\cdots &\sigma _{I_{s}}\\
*&\cdots&*&*&\cdots&*&\cdots&\cdots&*
\end{array}\right),
\end{equation}
where there are $p_{k}$ columns of the form $(\sigma _{I_{k}},*)^{t}$.
The stars in the last $n-\ell $ rows represent numbers that are
irrelevant, since they don't affect the value of the sum.
Let $\gamma \in GL_{n} ({\mathbb Q} )$ be a matrix that carries $Z ^{\ell }\cap
\sigma _{1}^{\perp }$ onto $Z ^{\ell -1}$. For $k=1,\dots ,\ell $ let
$\sigma' _{I_{k}}=\gamma \sigma _k^{j}$, where $\sigma _{k}^{j}$ is
any lift of $\sigma _{I_{k}}$. Then $\bar\gamma \sigma $ has the form
\begin{equation}\label{mat2}
\bar\gamma\sigma = \left( \begin{array}{ccc|ccc|c|cc}
0&\cdots&0&\sigma' _{I_{2}}&\cdots&
\sigma'_{I_{2}}&\cdots &\cdots &\sigma' _{I_{s}}\\
\varepsilon &\cdots&\varepsilon &0&\cdots&0&\cdots&\cdots&0\\
*&\cdots&*&*&\cdots&*&\cdots&\cdots&*
\end{array}\right).
\end{equation}
Here the row blocks have sizes $\ell -1$, $1$, and $n-\ell $, and
$\varepsilon = \pm 1$.
Now let $d$ be the determinant of any $(\ell -1)\times (\ell -1)$ minor
from the top $\ell -1$ rows of \eqref{mat2}. This
determinant will be the same up to sign as the determinant of an $\ell
\times \ell $ minor containing $\sigma _{1}$ from the top $\ell $ rows
of \eqref{mat1}. Hence $|d|\leq \Index{S}$,
and the proof is complete.
\end{proof}
Now we come to our first main result.
\begin{theorem}\label{th1}
Let $S = S (Z ^{\ell },\sigma )$ be a normalized, properly embedded
Dedekind sum. Then we may write
\begin{equation}\label{theq}
S = \sum_{\varrho \in R} q_{\varrho }S (Z^{\ell},\varrho )+\sum_{\tau
\in T} q_{\tau }S (Z^{\ell-1 },\tau),
\end{equation}
where
\begin{enumerate}
\item the sets $R$ and $T$ are finite,
\item $q_{\varrho }, q_{\tau }\in {\mathbb Q} $,
\item each $S (Z^{\ell },\varrho)$ is diagonal, and
\item each of the Dedekind sums on the right of \eqref{theq} has index
$\leq \Index{S}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Write $\sigma = (\sigma _{1}^{1},\dots ,\sigma ^{p_{1}}_{1},\sigma
_{2}^{1},\dots ,\sigma _{2}^{p_{2}}, \dots , \sigma _{s}^{1},\dots
,\sigma _{s}^{p_{s}})$ as in \eqref{labelling}. Permuting columns if
necessary, we may assume that $\sigma _{I_{1}},\dots ,\sigma _{I_{\ell
}}$ are linearly independent in $Z ^\ell $. We will show that we can
write an expression like \eqref{theq} so that the rank $\ell $ sums on the
right have type
\[
(p_{1},\dots ,p_{i}-1,\dots ,p_{\ell },\dots ,p_{s}+1),
\]
for some $1\leq i\leq \ell $. Iterating this construction proves that
we can have the rank $\ell $ sums on the right of \eqref{theq}
diagonal.
We proceed as follows. Since $\sigma _{I_{1}},\dots ,\sigma _{I_{\ell
}}$ are linearly independent, we can find unique rational numbers
$\alpha _{j}$ such that
\begin{equation}\label{alphas}
\sigma _{I_{s}} = \sum _{j=1}^{\ell }\alpha _{j}\sigma _{I_{j}}.
\end{equation}
Now we want to apply the relation in Proposition~\ref{recip}. For
each $\sigma _{I_{j}}$, we choose $p_{j}$ rational lifts
$\tilde{\sigma} ^{1}_{j}$,\dots ,$\tilde{\sigma} _{j}^{p_{j}}$, not
necessarily equal to the columns of $\sigma $, and use them to form a
matrix $\tilde{\sigma }$. Clearly $S (Z ^{\ell },\tilde{\sigma}) = S
(Z ^{\ell },\sigma )$.
Now define
\[
\tilde{w} = \sum _{j=1}^{\ell }\alpha _{j}\tilde{\sigma} ^{1}_{j}.
\]
Clearly $\pi (\tilde{w}) = \sigma _{I_{s}}$. Write
$\tilde{\sigma} _{i}^{j} (\tilde{w})$ for the matrix made from $\sigma $ by
replacing the column $\sigma _{i}^{j}$ with $\tilde{w}$.
Using the columns of $\tilde{\sigma} $ and $\tilde{w} $ in
\eqref{recip.law}, we find
\begin{multline}\label{star}
S (Z^{\ell },\tilde{\sigma} ) = S (Z ^{\ell }\cap \tilde{w}^{\perp },\tilde{\sigma} ) + \sum (\varepsilon _{i}^{j}) S
\bigl (Z ^{\ell }, \tilde{\sigma} _{i}^{j} (\tilde{w})\bigr)\\
-\sum (\varepsilon _{i}^{j}) S
\bigl(Z ^{\ell }\cap (\tilde{\sigma} _{i}^{j})^{\perp }, \tilde{\sigma} _{i}^{j} (\tilde{w})\bigr),
\end{multline}
where $\varepsilon _{i}^{j} \in \{\pm 1 \}$ is determined by the
reciprocity law, and the sums are over pairs $(i,j)$ satisfying $1\leq
i\leq s$ and $1\leq j\leq p_{i}$.
We claim that the sums in \eqref{star} actually have only $\ell $
terms. This follows since the points
\[
\tilde{\sigma} _{1}^{1},\dots ,\tilde{\sigma} _{\ell }^{1}, \tilde{w}
\]
are dependent. Hence any sum such that $\tilde{\sigma} _{i}^{j }
(\tilde{w})$ contains these points vanishes. Moreover, the sum $S (Z
^{\ell }\cap \tilde{w}^{\perp },\tilde{\sigma} )$ is zero, since by
construction a column of $\tilde{\sigma }$ induces a linear form
vanishing on $Z ^{\ell }\cap \tilde{w}^{\perp }$. Hence \eqref{star}
becomes
\begin{equation}\label{star2}
S (Z^{\ell },\tilde{\sigma} ) = \sum_{i=1}^{\ell } S
\bigl (Z ^{\ell }, \tilde{\sigma} _{i}^{1} (\tilde{w})\bigr)
-\sum_{i=1}^{\ell } S \bigl(Z ^{\ell }\cap (\tilde{\sigma}
_{i}^{1})^{\perp }, \tilde{\sigma} _{i}^{1} (\tilde{w})\bigr).
\end{equation}
Now consider the types of the rank $\ell $ Dedekind
sums on the right of \eqref{star2}. If the type of $S$ was
\[
(p_{1},\dots ,p_{i},\dots ,p_{\ell },\dots ,p_{s}),
\]
then the type of $S ({\mathbb Z} ^{\ell },\tilde{\sigma} _{i}^{1} (\tilde{w}))$ is
\[
(p_{1},\dots ,p_{i}-1,\dots ,p_{\ell },\dots ,p_{s}+1).
\]
Hence by induction we can write $S$ as a finite ${\mathbb Q} $-linear
combination of diagonal rank $\ell$ Dedekind sums plus sums of lower
rank, which proves 1--3 of the statement.
To complete the proof, we must show that the indices of the Dedekind
sums on the right of \eqref{theq} are no larger than $\Index{S}$.
Indeed, Lemma~\ref{index.no.change} implies that the indices on the
right of \eqref{star} are no larger than $\Index{S}$, and so the claim
follows.
\end{proof}
\begin{remark}
There are some simplifications in \eqref{star2} that are worth
mentioning if one wishes to implement the diagonalization algorithm.
First, if we construct $\tilde{\sigma }$ so that $\det \tilde{\sigma
}=1$, then $\det\tilde{\sigma }_{i}^{1} = \alpha _{j}$. This means
that these determinants can be computed when one computes $\sigma
_{I_{s}}$. Second, not all the terms on the right of \eqref{star2}
necessarily appear. In particular, the rank $\ell -1$ sum $S \bigl(Z
^{\ell }\cap (\tilde{\sigma} _{i}^{1})^{\perp }, \tilde{\sigma}
_{i}^{1} (\tilde{w})\bigr)$ appears on the right of \eqref{star2} only
if $p_{i} = 1$.
\end{remark}
\begin{theorem}\label{msa.dede}
With the notation as in Theorem~\ref{th1}, we can write
\begin{equation}\label{lts2}
S = \sum_{\varrho \in R} q_{\varrho }S (Z^{\ell},\varrho )+
\sum_{\tau \in T} q_{\tau }S (Z^{\ell-1 },\tau),
\end{equation}
where the sums of rank $\ell $ on the right are diagonal and
unimodular, and the rank $\ell -1$ sums have index $\leq \Index{S}$.
\end{theorem}
\begin{proof}
By Theorem~\ref{th1}, we may take $S$ to be diagonal. Suppose that
\[
D = |\det (\sigma _{I_{1}},\dots ,\sigma _{I_{\ell }})| > 1.
\]
Then by
Proposition~\ref{msa}, there exists $w\in Z ^{\ell }$ such that
\[
D_{j} =|\det (\sigma _{I_{1}},\dots ,\widehat \sigma _{I_{j}},\dots
,\sigma _{I_{\ell }}, w)|
\]
satisfies $0\leq D_{j}< D^{(\ell -1)/\ell }$. As in the proof of
Theorem~\ref{th1}, write $w = \sum_{j=1}^{\ell } \alpha _{j}\sigma
_{I_{j}}$, and set $\tilde {w} = \sum_{j=1}^{\ell } \alpha _{j} \sigma
^{1}_{j}$.
Now apply Proposition \ref{recip}, using $\tilde{w}$ and the columns
of $\sigma $, to write $S$ as a finite ${\mathbb Q} $-linear combination of new
Dedekind sums. These sums won't be diagonal, but we can apply the
proof of Theorem \ref{th1} with $w$ playing the role of $\sigma
_{I_{s}}$. The resulting sums will include lifts of $w$ in their
columns and will be diagonal. Thus the resulting rank $\ell $ sums
will have index $< \Index{S}$, and the sums of lower rank will satisfy
the conditions in the statement of Theorem~\ref{th1}. By induction on
the index, this completes the proof.
\end{proof}
\begin{corollary}\label{cor}
Any Dedekind sum can be written as a finite ${\mathbb Q} $-linear combination
of diagonal, unimodular sums.
\end{corollary}
\begin{proof}
First normalize and embed properly. By Lemma \ref{diag.example},
any rank 1 Dedekind sum is automatically unimodular and diagonal. The
result follows by applying Theorems \ref{th1} and~\ref{msa.dede} and
descending induction on the rank.
\end{proof}
\section{Complexity}\label{complexity}
\subsection{}
In this section we discuss the computational complexity of Corollary
\ref{cor}. In particular, we show that if $n$ and $\ell $ are fixed,
and $S = S (Z ^{\ell },\sigma )$ is normalized and properly embedded,
then we can form a finite ${\mathbb Q} $-linear combination of diagonal,
unimodular Dedekind sums
\[
S = \sum _{\substack{\varrho \in R\\
k\leq \ell}} q_{\varrho } S (Z ^{k},\varrho ),
\]
where $\#R$ is bounded by a polynomial in
$\log\Index{S}$. As a corollary we obtain that this expression can be
computed in polynomial time.
To do this, we must make a more detailed
analysis of proofs in \S\ref{algorithms}. We begin by analyzing diagonality.
\begin{lemma}\label{diag.const}
Let $S=S (Z ^{\ell },\sigma )$ be a normalized, properly embedded
Dedekind sum, and write
\begin{equation}\label{st3}
S = \sum_{\varrho \in R} q_{\varrho }S (Z^{\ell},\varrho )+\sum_{t\in T} q_{\tau }S (Z^{\ell-1 },\tau)
\end{equation}
as in Theorem~\ref{th1}, so that the rank $\ell $ sums in \eqref{st3}
are diagonal. If $\ell >1$, there exist constants $M_{n,\ell }$ and
$N_{n,\ell }$ such that
\[
\#R \leq M_{n,\ell } \quad \hbox{and}\quad \#T \leq N_{n,\ell }.
\]
\end{lemma}
\begin{proof}
Write $p (S) = (p_{1},\dots ,p_{s})$, where $\ell \leq s\leq n$. By
the proof of Theorem \ref{th1}, we know how to pass from a sum of type
\begin{equation}\label{typea}
(p_{1},\dots ,p_{i},\dots ,p_{\ell },\dots ,p_{s})
\end{equation}
to a linear combination of sums of types
\begin{equation}\label{typeb}
(p_{1},\dots ,p_{i}-1,\dots ,p_{\ell },\dots ,p_{s}+1),\quad
\hbox{$i=1,\dots ,\ell$.}
\end{equation}
By iterating this, we pass from $S$ to a linear combination of sums
with types
\begin{equation}\label{typec}
(p_{1}',\dots ,p'_{i-1},0,p'_{i+1},\dots ,p'_{\ell },\dots ,p'_{s}),
\quad \hbox{$i=1,\dots ,\ell$.}
\end{equation}
We will bound the number of rank $\ell $ (respectively rank $\ell -1$)
sums produced in passing from \eqref{typea} to \eqref{typec} by a
constant $M_{n,\ell }^{(s)}$ (resp. $N_{n,\ell }^{(s)}$). We can
then take
\[
M_{n,\ell } = \prod _{s=\ell +1}^{n}M_{n,\ell }^{(s)},
\]
and similarly for $N_{n,\ell }$.
To describe what happens in going from \eqref{typea} to
\eqref{typec}, we use a geometric construction. Let $B=B (p_{1},\dots
,p_{\ell })$ be the set
\[
B = \bigl\{ (x_{1},\dots ,x_{\ell })\in {\mathbb Z} ^{\ell }\bigm| 0\leq
x_{i}\leq p_{i}, i=1,\dots ,\ell\bigr\}.
\]
The points in $B$ correspond to types of intermediate sums in the
passage from \eqref{typea} to \eqref{typec}. In particular, passing
from \eqref{typea} to \eqref{typeb} can be encoded by moving from
$(x_{1},\dots ,x_{\ell})$ to $(x_{1},\dots ,x_{i}-1,\dots ,x_{\ell })$
in $B$. Moreover, the sums of the form \eqref{typec} correspond to
the subset of points $B_{0}\subset B$ with exactly one coordinate $0$.
(See Figure~\ref{b23.fig}.)
\begin{figure}
\caption{\label{b23.fig}
\label{b23.fig}
\end{figure}
Now the constant $M_{n,\ell }^{(s)}$ will be given by $\Max \#B_{0}$,
as $B (p_{1},\dots ,p_{\ell })$ ranges over all possibilities for
fixed $n$, $\ell $, and $s$. If we allow the $p_{i}$ to become
continuous parameters, then a simple computation shows that the
maximum occurs when $p_{1}=\cdots=p_{\ell } = (n-s+\ell )/\ell $.
With these conditions we have
\begin{equation}\label{mnls}
M_{n,\ell }^{(s)} \leq \ell \left(\frac{n-s+\ell }{\ell }\right)^{\ell -1}.
\end{equation}
The constant $N_{n,\ell }^{(s)}$ can be computed similarly. There are
$(\ell +1)$ sums of rank $\ell -1$ produced for each point in
\[
B_{+} := \bigl\{
(x_{1},\dots ,x_{\ell })\in B \bigm| x_{i} \not = 0\bigr\}.
\]
One finds
again that the maximum occurs when $p_{1}=\cdots=p_{\ell } =
(n-s+\ell )/\ell $, and is
\[
N_{n,\ell }^{(s)} \leq (\ell+1) \left(\frac{n-s+\ell }{\ell }\right)^{\ell}.
\]
\end{proof}
\begin{proposition}\label{polytime1}
If $n$ and $\ell $ are fixed, then \eqref{st3} can be constructed in
constant time, independent of $\Index{S}$.
\end{proposition}
\begin{proof}
This follows easily from the proof of Lemma \ref{diag.const}. Forming
the expression \eqref{st3} is purely combinatorial, and makes no
reference to $\Index{S}$. In particular, the number of steps needed
can be bounded for fixed $n$ and $\ell $.
\end{proof}
\subsection{}
Now we investigate the size of the output in Theorem \ref{msa.dede}.
The main step of the proof of Theorem \ref{msa.dede} shows how given
$S$, one may write
\begin{equation}\label{lts}
S = \sum_{\varrho \in R} q_{\varrho }S (Z^{\ell},\varrho )+ \sum_{\tau
\in T} q_{\tau }S (Z^{\ell-1 },\tau),
\end{equation}
where the sums of rank $\ell $ on the right satisfy $\Index{S
(Z^{\ell},\varrho)} < \Index{S}^{(n-1)/n}$. Denote by $C_{\ell }$
(respectively $C_{\ell -1}$) the number of rank $\ell $ (resp. rank
$\ell -1$) sums on the right of \eqref{lts}.
\begin{lemma}\label{computingC}
We have $C_{\ell } \leq M^{(\ell +1)}_{n+1,\ell }$ and $C_{\ell -1}
\leq N_{n+1,\ell }^{(\ell +1)}. $
\end{lemma}
\begin{proof}
The proof is very similar to the that of Lemma \ref{diag.const}, with
the following twist: we begin with a diagonal sum, increase the number
of distinct linear forms by one, and then make the sums diagonal
again. We can keep track of the number of sums produced using the set
$B$ as in the preceding proof, although we must replace $n$ with $n+1$
to accommodate the extra initial step. We leave the details to the
reader.
\end{proof}
\subsection{}
Now consider the expression \eqref{lts}. To complete the proof of
Theorem \ref{msa.dede}, we repeat the process that produced
\eqref{lts} until all rank $\ell $ sums are diagonal and unimodular,
and we obtain the expression \eqref{lts2}. Using Lemma
\ref{computingC} and the estimate \eqref{estimate}, we can bound the
number of rank $\ell $ sums produced.
\begin{proposition}\label{poly1}
(cf. \cite[Theorem 5.4]{barvinok}) Write $S$ as a sum of diagonal,
unimodular rank $\ell $ sums and lower rank sums as in \eqref{lts2}.
Then the number of rank $\ell $ sums on the right of \eqref{lts2} is
bounded by
\begin{equation}\label{bound1}
C' (\log\Index{S})^{A\log C_{\ell }},
\end{equation}
where $A = \log (n/ (n-1))^{-1}$, and $C'$ is a constant independent of
$S$.
\end{proposition}
\begin{proof}
In \eqref{lts}, we have
\[
0\leq \Index{S (Z ^{\ell },\varrho )} < \Index{S}^{(n-1)/n}.
\]
Thus after $t$ iterations we'll have
\[
0\leq \Index{S (Z ^{\ell },\varrho )} < \Index{S}^{( (n-1)/n)^{t}}.
\]
Since the index of a Dedekind sum is always an integer, the condition
for termination is that for some $\varepsilon >0$, we have
\begin{equation}\label{test}
\Index{S}^{( (n-1)/n)^{t}} \leq 2-\varepsilon, \quad \hbox{or} \quad
t\geq \frac{\log\log \Index{S}-\log\log (2-\varepsilon )}{\log n - \log (n-1)}.
\end{equation}
On the other hand, by Lemma \ref{computingC}, we know that $t$
iterations will produce no more than $C_{\ell }^{t}$ sums of rank
$\ell $. So fix $\varepsilon >0$, set $A = \log n - \log (n-1)$, and
let
\[
C'_{\ell } = \exp\left(\!\left (\frac{-\log\log (2-\varepsilon
)}{A}+1\right)\log C_{\ell }\right).
\]
Then
\[
C_{\ell }^{t}\leq C'_{\ell } \bigl(\log \Index{S}\bigr)^{A\log C_{\ell }}.
\]
Now if we define $C' = \Max _{\ell \leq n} C_{\ell }'$, we obtain
\eqref{bound1}.
\end{proof}
\subsection{}
We are now ready to discuss the complexity of our
algorithms.
\begin{theorem}\label{th3}
Let $S=S (Z ^{\ell },\sigma )$ be a normalized, properly embedded
Dedekind sum. Using Corollary~\ref{cor}, write $S$ as a ${\mathbb Q} $-linear
combination of diagonal unimodular sums. Then there exists a
polynomial $P_{n,\ell }$ such that the number of terms in the
expression is bounded by $P_{n,\ell } (\log \Index{S})$. Moreover, we
have
\[
\deg P_{n,\ell } \leq A \log\left (\frac{\ell !\,n^{\ell (\ell
-1)/2}}{2^{1}3^{2}\cdots\ell^{(\ell -1)}}\right),
\]
where $A = \log (n/ (n-1))^{-1}$.
\end{theorem}
\begin{proof}
Fix $n$. We proceed by induction on $\ell $.
First, if $\ell =1$, then by Lemma \ref{diag.example} the sum $S (Z^{1}
,\sigma )$ is already diagonal and unimodular. Hence we may take
$P_{n,1} \equiv 1$.
Next, assume that the statement is true for sums of rank $\ell -1$, and
let $P_{n,\ell -1}$ be the corresponding polynomial. First we claim
that without loss of generality, we need only consider the case that
$S$ is diagonal. Indeed, apply
Theorem~\ref{th1} and write
\begin{equation}\label{st4}
S = \sum_{\varrho \in R} q_{\varrho }S (Z^{\ell},\varrho )+\sum_{\tau
\in T} q_{\tau }S (Z^{\ell-1 },\tau),
\end{equation}
where the rank $\ell $ Dedekind sums are diagonal, and all the
Dedekind sums have index $\leq \Index{S}$.
By Lemma \ref{diag.const}, the sets $R$ and $T$ have a bounded number
of elements independent of $S$. Hence we
may bound the output for diagonal $S$, and then multiply this by a
constant to obtain our final answer.
Now we apply Theorem~\ref{msa.dede}, and we must count the number of
rank Dedekind sums produced. By Proposition \ref{poly1}, we know
that the total number of rank $\ell $ sums will be bounded by
\begin{equation}\label{est1}
C' (\log\Index{S})^{A\log C_{\ell }}.
\end{equation}
Furthermore, each sum of lower rank produced in the proof of Theorem
\ref{msa.dede} can be written as a sum of $\leq P_{n,\ell -1}
(\Index{S})$ Dedekind sums by induction. To find the total output, we
must count these lower rank sums.
We can do this as follows. Let $Q = Q_{n,\ell }:= C_{\ell
-1}P_{n,\ell -1}$, where $C_{\ell -1}$ is the constant in Lemma
\ref{computingC}. Represent the process of reducing the diagonal sum
$S$ to unimodularity by the following diagram:
\begin{equation}\label{diagram2}
\xymatrix{{\hbox{rank $\ell $:}}&1\ar[dr]\ar[r]&C_{\ell }\ar[dr]\ar[r]&C_{\ell }^{2}\ar[dr]\ar[r]&\cdots\ar[dr]\ar[r]&C_{\ell }^{t}\\
{\hbox{rank $\ell-1 $:}}&&Q&QC_{\ell }&\cdots&QC_{\ell }^{t-1}}
\end{equation}
The top row represents the bound on the number of rank $\ell $ sums at
each step of the algorithm, and the bottom
row is the number of rank $\ell -1$ sums.
According to \eqref{diagram2}, we have produced
\[
Q\sum _{i=0}^{t-1}C_{\ell }^{i} = Q\,\frac{C_{\ell }^{t}-1}{C_{\ell
}-1} \leq Q\,\frac{C'(\log\Index{S})^{A\log C_{\ell }}-1}{C_{\ell
}-1}
\]
sums of ranks $<\ell $. Adding this estimate to \eqref{est1}, we
find that the total number of Dedekind
sums produced will be a polynomial of
degree
\[
\deg Q + A\log C_{\ell } = \deg P_{n,\ell-1 } + A\log C_{\ell }.
\]
To complete the proof, we compute the degree of $P_{n,\ell }$ by
induction. Indeed, using the estimate
\[
C_{\ell } \leq M_{n+1,\ell }^{(\ell +1)} = \ell\left(\frac{n}{\ell }\right)^{\ell -1},
\]
and that $\deg P_{n,1} = 0$,
an easy computation shows that
\[
\deg P_{n,\ell } \leq A \log\left (\frac{\ell !\,n^{\ell (\ell
-1)/2}}{2^{1}3^{2}\cdots\ell^{(\ell -1)}}\right),
\]
as required.
\end{proof}
\begin{example}\label{table}
Here is a table of the bound of $\deg P_{n,\ell }$ for small values of $n$
and $\ell $.
\vskip2em
\begin{center}
\begin{tabular}{l||ccccccccc}
$n,\ell $&2&3&4&5&6&7&8\\
\hline\hline
2 & 1 &&&&&&\\
3 & 2 & 5 &&&&&&\\
4 & 4 & 10 & 15 &&&&\\
5 & 7 & 16 & 25 & 33 &&&\\
6 & 9 & 23 & 37 & 50 & 60 &&\\
7 & 12 & 30 & 50 & 69 & 86 & 99&\\
8 & 15 & 38 & 64 & 90 & 114 & 135 & 150
\end{tabular}
\end{center}
\vskip2em
\end{example}
\begin{corollary}\label{polytime2}
Keeping the same notation as in Theorem \ref{th3}, for fixed $n$ and
$\ell $ we may express $S$ as a finite ${\mathbb Q} $-linear combination of
diagonal unimodular sums in time polynomial in $\log\Index{S}$.
\end{corollary}
\begin{proof}
First, the vector $w$ constructed in Proposition \ref{msa} can be found in
polynomial time in the size of the coefficients of $S$. In fact,
investigation of \cite[Lemma 5.2]{barvinok} shows that the rational
numbers $\alpha _{j}$ in \eqref{alphas} in the proof of Theorem
\ref{th1} can also be constructed in polynomial time. This implies that
$w$ and the $\alpha _{j}$ can be found in time polynomial in
$\log\Index{S}$. The proof of Theorem \ref{th3} then shows that the
final expression can be computed in polynomial time.
\end{proof}
\section{The Eisenstein cocycle}\label{ec}
\subsection{}
In this section, we briefly review the construction of the Eisenstein
cocycle introduced in \cite{eisenstein}. In particular, we show that
this is a finite object that can be calculated effectively using
Corollary \ref{cor}. Roughly speaking, the Eisenstein cocycle
represents a generalization of the classical Bernoulli polynomial
within the arithmetic of the unimodular group ${\mathscr G}amma =GL_{n}({\mathbb Z}) $.
\subsection{}
Let ${\mathscr A} = (A_{1},\dots ,A_{n})$ be an $n$-tuple of matrices $A_{i}\in
GL_{n} ({\mathbb R})$. For an $n$-tuple $d = (d_{1},\dots ,d_{n})$ of integers
$1\leq d_{i}\leq n$, let ${\mathscr A} (d)\subseteq {\mathbb R} ^{n}$ be the subspace
generated by all columns $A_{ij}$ such that $j<d_{i}$. (Here $A_{ij}$
denotes the $j$th column of $A_{i}$.) Writing ${\mathscr A} (d)^{\perp}$ for
the orthogonal complement of ${\mathscr A} (d)$ in ${\mathbb R}^{n}$, we let
\begin{equation}\label{one}
X (d) = {\mathscr A} (d)^{\perp } \smallsetminus \bigcup_{i=1}^{n} \sigma
_{i}^{\perp }, \quad \hbox{where $\sigma _{i} = A_{id_{i}}$.}
\end{equation}
The $n$-tuple ${\mathscr A} $ determines then the stratification
\begin{equation}
{\mathbb R} ^{n} \smallsetminus \{0 \} = \bigsqcup_{d\in D} X (d),
\end{equation}
indexed by the finite set
\[
D = D ({\mathscr A} ) = \{d \mid X (d) \not = \varnothing \}.
\]
Associated to this decomposition is the rational function $\psi ({\mathscr A} )$
on ${\mathbb R} ^{n}\smallsetminus \{0 \}$ defined by
\[
\psi ({\mathscr A} ) (x) = \frac{\det (\sigma _{1},\dots ,\sigma _{n})}{\langle
x,\sigma _{1}\rangle \cdots \langle x,\sigma _{n}\rangle },\quad
\hbox{if $x\in X (d)$.}
\]
\subsection{}
More generally, if $P (X_{1},\dots ,X_{n})$ is any homogeneous
polynomial, we form the differential operator $P (-\partial
_{x_{1}},\dots ,-\partial _{x_{n}})$ in the partial derivatives
$\partial _{x_{i}} := \partial /\partial x_{i}$, and define
\[
\psi ({\mathscr A} ) (P,x) = P (-\partial
_{x_{1}},\dots ,-\partial _{x_{n}}) \psi ({\mathscr A} ) (x).
\]
The last expression can be written more explicitly as
\begin{equation}
\psi ({\mathscr A} ) (P,x) = \det (\sigma )\sum _{r}P_{r} (\sigma )\prod
_{j=1}^{n} \frac{1}{\langle x,\sigma _{j}\rangle ^{1+r_{j}}},
\end{equation}
where $r$ runs over all decompositions of $\deg (P) =
r_{1}+\cdots+r_{n}$ into nonnegative parts $r_{j}\geq 0$, and $P_{r}
(\sigma )$ is the homogeneous polynomial in the $\sigma _{ij}$ defined
by the expansion
\[
P (X\sigma ^{t}) = \sum _{r}P_{r} (\sigma ) \prod
_{j=1}^{n}\frac{X_{j}^{r_{j}}}{r_{j}!}.
\]
In the excluded case $x = 0$, it is convenient to set $\psi ({\mathscr A} )
(P,0) = 0$.
The definition of the Eisenstein cocycle $\Psi $ is now easy to state:
\begin{equation}\label{four}
\Psi ({\mathscr A} ) (P,Q,v) := (2\pi i)^{-n-\deg P}\sum _{x\in {\mathbb Z} ^{n}} {{e}}e (\langle x,v\rangle )
\psi ({\mathscr A} ) (P,x){\mathscr B}igr|_{Q}.
\end{equation}
The series on right converges provided all components $A_{i}$ of ${\mathscr A} $
are in $GL_{n} ({\mathbb Q} )$. However, since the convergence is only
conditional, we are forced to introduce the additional parameter $Q$
specifying the limiting process.
\subsection{}
Let $M$ be the set of all complex-valued functions $f (P,Q,v)$ with
$P$, $Q$, $v$ as above ($v\in {\mathbb R} ^{n}$). Then $M$ is a left ${\mathscr G}amma
$-module under the action
\[
Af (P,Q,v) = \det ( A) f (A^{t}P, A^{-1}Q, A^{-1}v), \quad \hbox{$A\in
{\mathscr G}amma$,}
\]
where the implied ${\mathscr G}amma $-action on homogeneous polynomials is given
by $(AP) (X) = P (XA)$. With respect to this action, the map
$\Psi \colon {\mathscr G}amma ^{n}\rightarrow M$ has the property
\begin{align}
\Psi &(A{\mathscr A} ) = A\Psi ({\mathscr A} ), \quad\hbox{$A\in {\mathscr G}amma $, ${\mathscr A} \in {\mathscr G}amma ^{n}$,}\\
&\sum _{i=0}^{n} \Psi (A_{0},\dots ,\widehat A_{i},\dots ,A_{n})=0, \quad
\hbox{$A_{i} \in {\mathscr G}amma $.}
\end{align}
In other words, $\Psi $ is a homogeneous cocycle on ${\mathscr G}amma $. It is
known that $\Psi $ represents a nontrivial cohomology class in
$H^{n-1} ({\mathscr G}amma ; M)$ \cite[Theorem 4]{eisenstein}.
Combining \eqref{one}--\eqref{four}, we see that $\Psi $ is a finite
linear combination of Dedekind sums,
\[
\Psi ({\mathscr A} ) (P,Q,v) = (2\pi i)^{-n-\deg P}\sum _{d\in D}\sum _{r} P_{r}
(\sigma )S (L,\sigma ,e,v){\mathscr B}igr|_{Q}.
\]
Here $\sigma $ is the matrix with columns $A_{id_{i}}$ for $i=1,\dots ,n$,
$L$ is the lattice ${\mathscr A} (d)^\perp \cap {\mathbb Z} ^{n}$, and $e_{j} = 1+r_{j}$.
The case $P=1$ is of special interest:
\[
\Psi ({\mathscr A} ) (1,Q,v) = (2\pi i)^{-n}\sum _{d\in D} S (L,\sigma ,{\mathbf{1}}
,v){\mathscr B}igr|_{Q}.
\]
This case yields the classical Dedekind-Rademacher sums if $n=2$, and,
more importantly, it corresponds to special values of partial zeta
functions at $s=0$.
\section{Values of partial zeta functions}\label{vpzf}
\subsection{}
Let $F$ be a totally real number field of degree $n$ over ${\mathbb Q} $, and
let ${\mathfrak f}$, ${\mathfrak b}$ be two relatively prime ideals in the ring of
integers ${\mathscr O} _{F}$. The partial zeta function to the ray class ${\mathfrak b}
\bmod {\mathfrak f}$ is defined by
\[
\zeta ({\mathfrak b}, {\mathfrak f}, s) := \sum _{{\mathfrak a} \equiv {\mathfrak b} \bmod {\mathfrak f} } N ({\mathfrak a}
)^{-s}, \quad \hbox{${\mathbb R}e (s)>1$,}
\]
where ${\mathfrak a} $ runs over all all integral ideals in ${\mathscr O} _{F}$ such that
the fractional ideal ${\mathfrak a} {\mathfrak b} ^{-1}$ is a principal ideal generated
by a totally positive number in the coset $1+{\mathfrak f} {\mathfrak b} ^{-1}$.
According to Klingen-Siegel, the special values
$\zeta ({\mathfrak b} ,{\mathfrak f} ,1-s)$, where $s=1,2,3,\dots, $ are well-defined
rational numbers. In this section, we give a formula for calculating
these numbers in terms of the Eisenstein cocycle $\Psi $.
\subsection{}
The formula depends on the choice of a ${\mathbb Z} $-basis $W$ for the
fractional ideal ${\mathfrak f} {\mathfrak b} ^{-1} = \sum {\mathbb Z} W_{j}$, together with the
dual basis $W^{*}$ determined by $\Trace (W_{i}^{*}W_{j}) = \delta
_{ij}$. Here we identify $\alpha \in F$ with the row vector $(\alpha
^{(1)}, \dots , \alpha ^{(n)})\in {\mathbb R} ^{n}$, where the $\alpha ^{(j)}$
are the $n$ different embeddings of $\alpha $ into the field of real
numbers. Then $W$ can be identified with a matrix in $GL_{n} ({\mathbb R} )$
whose $j$th row is the basis vector $W_{j}$. Let
\begin{align*}
P (X)&=N ({\mathfrak b} )\prod _{i}\sum _{j}X_{j}W_{j}^{(i)},\\
Q (X)&= \prod _{i}\sum _{j}X_{j}(W_{j}^{*})^{(i)},
\end{align*}
and let $v\in {\mathbb Q} ^{n}$ be defined by $v_{j} =
\Trace (W^{*}_{j})$.
The formula also depends on the choice of generators $\varepsilon
_{1},\dots ,\varepsilon _{\nu }$, where $\nu = n-1$, for the group
$U\subset {\mathscr O} _{F}^{\times }$ of totally positive units. Using the regular
representation $\rho \colon U\rightarrow {\mathscr G}amma $, defined via $\rho
(\varepsilon ) = W\delta (\varepsilon )W^{-1}$, where $\delta
(\varepsilon )$ is the matrix $\Diag (\varepsilon ^{(1)},\dots
,\varepsilon ^{(n)})$, we identify the units $\varepsilon _{j}$ with
elements $A_{j} = \rho (\varepsilon_{j} )^{t} \in {\mathscr G}amma $. (Note
that $\rho $ is the \emph{row} regular representation.)
\subsection{}
Using the bar notation
\[
[A_{1}|\cdots | A_\nu ] := (1,A_{1},A_{1}A_{2},\dots ,A_{1}\cdots A_{\nu })\in {\mathscr G}amma ^{n},
\]
we have the following proposition expressing the zeta values in terms
of the Eisenstein cocycle:
\begin{proposition}\label{zeta-values}
Let $U_{{\mathfrak f} }$ be the subgroup $U\cap (1+{\mathfrak f} )$, and let $\pi $ run
through all permutations of \hbox{$\{1,\dots ,\nu \}$}. Then for
$s=1,2,3,\dots$,
\[
\zeta ({\mathfrak b} ,{\mathfrak f} ,1-s) = \eta \sum _{\varepsilon \in U/U_{{\mathfrak f} }}
\sum _{\pi } \Sign (\pi) \Psi ([A_{\pi (1) }|\cdots | A_{\pi (\nu )}])
(P^{s-1},Q,\rho (\varepsilon )^{t}v).
\]
Here the sign $\eta = \pm
1$ is determined by
\[
\eta = (-1)^{\nu }\Sign (\det W) \Sign (R),
\]
where $R = \det (\log \varepsilon _{i}^{(j)})$, $1\leq i,j\leq \nu $.
\end{proposition}
\begin{proof}
This follows from \cite[Corollary, p. 595]{eisenstein} by writing the
fundamental cycle of $U_{{\mathfrak f} }$ in terms of the $A_{j}$.
\end{proof}
\begin{example}\label{quadratic}
We work out the above formula in the case of a real quadratic field
$F$. Let $\varepsilon >1$ be the fundamental unit of $U$, the group of
totally positive units in $F$, and let ${\mathbb Z} w_{1} + {\mathbb Z} w_{2} = {\mathfrak f}
{\mathfrak b} ^{-1}$ be a ${\mathbb Z} $-basis of ${\mathfrak f} {\mathfrak b} ^{-1}$. Such a basis
determines a matrix $A=\left(\begin{smallmatrix}
a&b\\
c&d
\end{smallmatrix} \right)\in SL_{2} ({\mathbb Z} )$ and a vector $v\in {\mathbb Q} ^{2}$ via
\[
\left(\begin{array}{c}
\varepsilon w_{1}\\
\varepsilon w_{2}
\end{array} \right) =
\left(\begin{array}{cc}
a&c\\
b&d
\end{array} \right)
\left(\begin{array}{c}
w_{1}\\
w_{2}
\end{array} \right), \quad v_{1}w_{1} + v_{2}w_{2}= 1.
\]
In addition, we get the normforms
\[
P (X) = N (x_{1}w_{1} + x_{2}w_{2}), \quad Q (X) = N (x_{1}w_{2}-x_{2}w_{1}).
\]
Let $p$ be the smallest positive integer such that $(A^{p}-1)v\in {\mathbb Z}
^{2}$. Then
\begin{equation}\label{eqq1}
\zeta_{F} ({\mathfrak b} ,{\mathfrak f} ,1-s) = \eta \sum _{k\bmod p} \Psi (1,A) (P^{s-1},Q,A^{k}v),
\end{equation}
where $\eta = \Sign (w_{2}w_{1}^{(1)}- w_{1}w_{2}^{(1)})$, and, if $s=1$,
\begin{align}\label{}
\Psi \left(\left(\begin{array}{cc}
1&0\\
0&1
\end{array} \right), \left(\begin{array}{cc}
a&b\\
c&d
\end{array} \right) \right) (1,Q,v) &= \frac{a}{2c}{\mathscr B} _{2} (v_{2}) +
\frac{d}{2c}{\mathscr B} _{2} (cv_{1}- av_{2}) \\
&- \sum _{j\bmod c} {\mathscr B} _{1}
(\frac{j+v_{2}}{|c|}){\mathscr B} _{1} (a\frac{j+v_{2}}{c}-v_{1})\label{classicalsum},
\end{align}
with an additional correction term $-\Sign (c)/4$ on the right if $v\in {\mathbb Z} ^{2}$.
The finite sum \eqref{classicalsum} is the classical Dedekind-Rademacher sum
$(2\pi i)^{-2}S \left({\mathbb Z} ^{2}, \left(\begin{smallmatrix}
1&a\\
0&c
\end{smallmatrix} \right), {\mathbf{1}} ,v\right)\bigr|_{Q}$. Note that the
number of terms in that sum equals $|c| = | (\varepsilon
-\varepsilon ')/(w-w')|$, where $w = w_{2}/w_{1}$, and the prime is
Galois conjugation. Depending on
$\varepsilon $, this number can be very large. To get a more
efficient formula for calculating $\Psi $, we apply the euclidean
algorithm to the first column of $A$ an obtain a product decomposition
\[
A = B_{1}\cdots B_{t}, \quad t\geq 1, \quad B_{j} = \left(\begin{array}{cc}
b_{j}&-1\\
1&0
\end{array} \right).
\]
Then
\begin{equation}\label{eqq2}
\Psi (1,A) = \sum _{\ell =0}^{t-1} (B_{1}\cdots B_{\ell })\Psi
(1,B_{\ell +1}).
\end{equation}
Here $t$ is roughly $\log|c|$. Thus the number of terms is
effectively reduced from $|c|$ to $\log|c|$, since
\begin{multline}\label{eqq3}
\Psi \left(\left(\begin{array}{cc}
1&0\\
0&1
\end{array} \right), \left(\begin{array}{cc}
b&-1\\
1&0
\end{array} \right) \right) (P^{s-1}, Q, v) \\
=
\sum _{r}\left[bP_{r}\left(\begin{array}{cc}
0&b\\
1&1
\end{array} \right)\frac{{\mathscr B} _{2s} (v_{2})}{(2s)!} + P_{r}\left(\begin{array}{cc}
1&b\\
0&1
\end{array} \right)\frac{{\mathscr B} _{1+r_{1}} (v_{1}-bv_{2})}{(1+r_{1})!}\frac{{\mathscr B} _{1+r_{2}} (v_{2})}{(1+r_{2})!}\right],
\end{multline}
where the rational numbers $P_{r}\bigl (\begin{smallmatrix}\alpha &\beta \\
\gamma &\delta \end{smallmatrix}\bigr)$ are the coefficients of the
polynomial
\[
P (\alpha X_{1}+\beta X_{2}, \gamma X_{1}+\delta X_{2})^{s-1} = \sum_{r}P_{r}\left(\begin{array}{cc}
\alpha &\beta \\
\gamma &\delta
\end{array} \right)\frac{X_{1}^{r_{1}}}{r_{1}!}\frac{X_{2}^{r_{2}}}{r_{2}!}.
\]
In the exceptional case $s=1$, ${\mathfrak f} = (1)$, the correction term
\[
-\frac{1}{8}\{\Sign (w+b) + \Sign (w'+b) \}
\]
must be added to the right side of \eqref{eqq3}.
As a numerical example, we choose $F={\mathbb Q} (\sqrt{5})$, $\varepsilon =
(3+\sqrt{5})/2$, ${\mathfrak f} = {\mathfrak b} = (1)$, $w_{1}=-\varepsilon $, $w_{2}=1$. Then
$\eta = +1$, $A = \left(\begin{smallmatrix}3&-1\\
1&0\end{smallmatrix} \right)$, $P_{11}\left(\begin{smallmatrix}1&3\\
0&1\end{smallmatrix} \right) = 3$, while $P_{20}\left(\begin{smallmatrix}0&3\\
1&1\end{smallmatrix} \right) = 2$, $P_{11}\left(\begin{smallmatrix}0&3\\
1&1\end{smallmatrix} \right) = -7$, $P_{02}\left(\begin{smallmatrix}0&3\\
1&1\end{smallmatrix} \right) = 2$. Hence, according to \eqref{eqq1}
and \eqref{eqq3}, we get for the value of the Dedekind zeta function
of $F$ at $s=-1$,
\[
\zeta _{F} (-1) = \zeta ((1),(1),-1) = 3 (2-7+2) (-\frac{1}{720})
+ 3 (\frac{1}{12})^{2} = \frac{1}{30}.
\]
\end{example}
\begin{example}\label{cubic}
As a second example, we consider the cubic field ${\mathbb Q} (\theta )$ of
discriminant $148$ given by $\theta ^{3} - \theta ^{2} -3\theta +1=0$.
According to \cite{khan}, the group of totally positive units $U$ is
generated by $\varepsilon _{1} = -3\theta ^{2}+2\theta +10$ and
$\varepsilon _{2} = 5\theta ^{2}+6\theta -2$.
Let ${\mathfrak f} = (2)$ and ${\mathfrak b} = (1)$.
Then ${\mathfrak f} {\mathfrak b} ^{-1} = {\mathbb Z} w_{1}+ {\mathbb Z} w_{2} + {\mathbb Z} w_{3}$, where $w_{1} =
2$, $w_{2} = 2 \theta $, and $w_{3} = 2\theta ^{2}$.
With respect to this basis, we find
\[
A_{1} = \left(\begin{array}{ccc}
{10}&{3}&{1}\\
{2}&{1}&{0}\\
{-3}&{-1}&{0}
\end{array} \right),
A_{2} = \left(\begin{array}{ccc}
{-2}&{-5}&{-11}\\
{6}&{13}&{28}\\
{5}&{11}&{24}
\end{array} \right),
A_{1}A_{2} = \left(\begin{array}{ccc}
{3}&{0}&{-2}\\
{2}&{3}&{6}\\
{0}&{2}&{5}
\end{array} \right).
\]
Then $v = (1/2,0,0)^{t}$ and $\eta = 1$. Let $V\subset {\mathbb Q} ^{3}$ be a
complete set of representatives for the orbit of $v+{\mathbb Z}^3$ under the
action of $U$ (via $A_{1}$ and $A_{2}$) on ${\mathbb Q}^3/{\mathbb Z}^3$. Note that $V$
is a finite set. Thus
\begin{equation}\label{eqq4}
\zeta ({\mathfrak b} , {\mathfrak f} , 0) = \sum_{v\in V} \bigl (\Psi(1, A_{1},
A_{1}A_{2}) - \Psi (1, A_{2}, A_{1}A_{2})\bigr) (1, Q, v).
\end{equation}
Each term on the right of \eqref{eqq4} breaks up into $10$ Dedekind
sums: one of rank $3$, three of rank $2$, and six of rank $1$. Note
that the rank $3$ and rank $1$ sums are diagonal, whereas the rank $2$
sums are not. After making all sums diagonal, we find that each $\Psi
$ in \eqref{eqq4} has $30$ terms. Applying the summation formula for
diagonal sums (Proposition \ref{how.to.sum}), we see that to evaluate
$\Psi $ on any element of $V$, we must sum $76$ terms.
Since $\varepsilon _{2}^{2} \equiv \varepsilon _{1}\varepsilon
_{2}\equiv 1 \bmod {\mathfrak f} $, we can take $V = \{v, A_{2}v\}$.
This yields $152$ terms altogether, all of which sum to
$0$, and thus $\zeta ((1), (2), 0) = 0$. This agrees with
\cite{khan}, and also with the observation that since $-1$ preserves
the congruence class of $1+{\mathfrak f} $, and the norm of $-1$ is $-1$, the
special value at $s=0$ must vanish.
Now let ${\mathfrak f} = (3)$. Then we may take $A_{1}, A_{2}$ as above, and
$v = (1/3,0,0)^{t}$ and $\eta =1$. Since $\varepsilon _{2}^{13}
\equiv \varepsilon _{1}\varepsilon _{2}^{5}\equiv 1 \bmod {\mathfrak f} $, we
must sum $13\cdot 76= 988$ terms, and we find $\zeta ((1), (3), 0) =
2/3$, again in agreement with \cite{khan}.
Note that to compute $\zeta ((1), (N), 0)$ for various $N\in {\mathbb Z} $, we
must only compute $A_{1}$, $A_{2}$, and thus $\Psi
(1,A_{1},A_{1}A_{2}) - \Psi (1,A_{2},A_{1}A_{2})$ once. After this it
is routine to compute special values at $s=0$, and the complexity in
\eqref{eqq4} comes from $\#V$, which can be large, even for small
values of $N$. A table of $\zeta ((1), (N), 0)$ for several rational
integers $N$ is given below. The values for $N=2,3,5,7$ are also in
\cite{khan}.
\vskip2em
\begin{center}
\begin{tabular}{|c|r||c|r||c|r||c|r||c|r|}
\hline
$N$&$N\cdot \zeta$&$N$&$N\cdot \zeta$&$N$&$N\cdot \zeta$&$N$&$N\cdot \zeta$&$N$&$N\cdot \zeta$\\
\hline\hline
&&$11$&$-18$&$21$&$-78$&$31$&$74$&$41$&$382$\\
$2$&$0$&$12$&$5$&$22$&$68$&$32$&$15$&$42$&$-228$\\
$3$&$2$&$13$&$-22$&$23$&$12$&$33$&$62$&$43$&$-366$\\
$4$&$1$&$14$&$-20$&$24$&$23$&$34$&$50$&$44$&$1$\\
$5$&$-4$&$15$&$42$&$25$&$106$&$35$&$-54$&$45$&$254$\\
$6$&$4$&$16$&$7$&$26$&$-24$&$36$&$-43$&$46$&$6$\\
$7$&$2$&$17$&$100$&$27$&$-190$&$37$&$20$&$47$&$-570$\\
$8$&$3$&$18$&$-32$&$28$&$25$&$38$&$22$&$48$&$-13$\\
$9$&$-10$&$19$&$82$&$29$&$242$&$39$&$156$&$49$&$-222$\\
$10$&$-2$&$20$&$4$&$30$&$6$&$40$&$2$&$50$&$178$\\
\hline
\end{tabular}
\end{center}
\begin{center}
$\zeta ((1),(N),0)$ for the cubic field of discriminant 148.
\end{center}
\vskip2em
\end{example}
\section{Witten's zeta function}\label{witten}
\subsection{}
To give another illustration, we recall the definition of Witten's zeta
function \cite{witten}, and show how our algorithms can be used to
compute special values of this function at even integers. For
unexplained notions from representation theory, the reader may consult
\cite{fulton.harris}.
Let ${\mathfrak g}$ be a simple complex lie algebra, and let $R$ be the
associated root system. Let $R^{+}$ (respectively $R^{-}$) be a
subset of positive roots (resp. negative roots), and let $\Delta
\subset R^{+}$ be the set of simple roots.
The roots $R$ generate a lattice $\Lambda _{R}$ in an $\ell
$-dimensional real vector space $E$ endowed with an inner product
$(\phantom{a},\phantom{a})$. Let $\Lambda _{W}$ be the weight
lattice, which is the dual of $\Lambda _{R}$ with respect to this
inner product. We denote by ${\mathscr O}mega \subset E$ the set of fundamental
weights, which is the basis of $\Lambda _{W}$ dual to $\Delta $.
Let $W$ be the Weyl group of $R$. This is a finite group that acts on
$E$ via a reflection representation, and preserves the inner product
and the lattices $\Lambda _{R}$ and $\Lambda _{W}$. There is a
decomposition of $E$ into a finite union of rational polyhedral cones,
and $W$ acts by permuting these cones. Let $C^{+}$ be the closed
top-dimensional cone generated by ${\mathscr O}mega $.
Let $\Pi$ denote the set of isomorphism classes of complex irreducible
representations of ${\mathfrak g} $. It is known that elements of $\Pi $ are
in bijection with the set $\Lambda _{W}\cap C^{+}$, the dominant
weights. Given $\lambda $ from this latter set, we denote the
corresponding representation by $\pi _{\lambda }$. Then the
definition of the zeta function associated to ${\mathfrak g} $ is
\begin{equation}\label{wzf}
\zeta _{{\mathfrak g} } (s) := \sum _{\lambda \in \Lambda _{W}\cap C^{+}} (\dim
\pi _{\lambda })^{-s}.
\end{equation}
\subsection{}
Let $m>1$ be an integer. The special value $\zeta _{{\mathfrak g} } (2m)$
can be computed using a Dedekind sum as follows.
Let $\rho$ be one-half the sum of the positive roots. An application
of the Weyl character formula \cite[Corollary 24.6]{fulton.harris}
shows that for any dominant weight $\lambda $, we have
\begin{equation}\label{wdf}
\dim \pi _{\lambda } = \prod _{\alpha \in R^{+}} \frac{(\rho + \lambda
, \alpha )}{(\rho ,\alpha )}.
\end{equation}
It is known that any dominant weight $\lambda$ can be written as a
nonnegative integral linear combination of the fundamental weights.
Using this in \eqref{wdf}, a computation shows that \eqref{wzf}
becomes
\begin{equation}\label{wzf2}
\zeta _{{\mathfrak g} } (2m) = M^{2m}\sum _{x\in ({\mathbb Z} ^{>0})^{\ell
}}\frac{1}{\prod _{i=1}^{r} \langle a_{i} , x\rangle ^{2m}}.
\end{equation}
Here $M$ is the integer $\prod _{\alpha \in R^{+}} (\rho ,\alpha )$,
$r$ is the number of positive roots, and the $a_{i} \in ({\mathbb Z}
^{>0})^{\ell}$ are the coefficients of the positive roots in terms of
$\Delta $. The pairing $\langle \phantom{a},\phantom{a}\rangle$ is
the usual scalar product on ${\mathbb R} ^{\ell }$.
\subsection{}
We obtain a Dedekind sum by extending the sum \eqref{wzf2} to the
whole lattice. Let $Z ^{\ell }\subset {\mathbb R} ^{r}$ be the span of the
first $\ell$ basis vectors, and let $\sigma = \sigma ({\mathfrak g} ) $ be an
$r\times r$ integral matrix such that $\langle \sigma _{i},x\rangle =
\langle a_{i}, x\rangle $ for $x\in Z^{\ell }$, and such that $\det
\sigma = 1$. Let $e = (2k, \dots ,2k)\in {\mathbb R} ^{r}$.
\begin{proposition}
(cf. \cite[p. 507]{zagier.zeta})
$\zeta _{{\mathfrak g} } (2k) = \frac{M^{2k}}{\#W}S (Z^{\ell },\sigma ({\mathfrak g} ),e,0)$.
\end{proposition}
Hence these special values can be computed in polynomial time using
our techniques. We conclude with two examples:
${\mathfrak s \mathfrak l}_{3}$ and ${\mathfrak s \mathfrak l}_{4}$. We recommend verification of these formulas
to the interested reader for a pleasant combinatorial exercise.
\begin{proposition}\label{sln}
Let $\zeta (s)$ be the Riemann zeta function. Then
\begin{align}
&\frac{6}{2^{2m}}\zeta _{{\mathfrak s \mathfrak l}_{3}} (2m) = 8 \sum _{\substack{0\leq i\leq 2m\\i\equiv 0\bmod{2}}}{\binom{4m-i-1}{2m-1}} \zeta (i)\zeta (6m-i)\label{sl3form}.\\
&\frac{24}{12^{2m}}\zeta _{{\mathfrak s \mathfrak l}_{4}} (2m) = 16 \sum _{0\leq i\leq 2m}{\binom{4m-i-1}{2m-1}} (A+B+C+D),\quad \hbox{where}\\
A=\sum& _{\substack{0\leq j\leq 2m\\0\leq t\leq 4m+i-j\\j,t\equiv 0 \bmod{2}}}\binom{2m+i-j-1}{i-1}\binom{6m+i-j-t-1}{2m-1}\zeta (j)\zeta (t)\zeta (12m-j-t),\\
B=\sum& _{\substack{0\leq j\leq 2m\\0\leq u\leq 2m\\j,u\equiv 0 \bmod{2}}}\binom{2m+i-j-1}{i-1}\binom{6m+i-j-u-1}{4m+i-j-1}\zeta (j)\zeta (u)\zeta (12m-j-u),\\
C=\sum& _{\substack{0\leq k\leq i\\0\leq v\leq 4m+i-k\\k,v\equiv 0 \bmod{2}}}\binom{2m+i-k-1}{i-k}\binom{6m+i-k-v-1}{2m-1}\zeta (k)\zeta (v)\zeta (12m-k-v),\\
D=\sum& _{\substack{0\leq k\leq i\\0\leq w\leq 2m\\k,w\equiv 0 \bmod{2}}}\binom{2m+i-k-1}{i-k}\binom{6m+i-k-w-1}{4m+i-k-1}\zeta (k)\zeta (w)\zeta (12m-k-w).
\end{align}
\end{proposition}
\begin{remark}
The formula \eqref{sl3form} was independently discovered by Zagier,
S. Garoufalidis, and L. Weinstein \cite[p. 506]{zagier.zeta}.
\end{remark}
\begin{example}
Here are some special values of $\zeta _{{\mathfrak s \mathfrak l}_{3}}$ and $\zeta _{{\mathfrak s \mathfrak l}_{4}}$.
\begin{center}
\begin{tabular}[ht]{c||l}
$2m$&$(6m+1)! \cdot 6\cdot \zeta_{{\mathfrak s \mathfrak l} _{3}} (2m)/ ( 2^{2m}\cdot (2\pi) ^{6m})$\\
\hline
\hline
2&$1/(2\cdot 3)$\\
4&$19/ (2\cdot 3\cdot 5)$\\
6&$1031/ (3\cdot 7)$\\
8&$(11\cdot 43\cdot 751)/ (2\cdot 7)$\\
10&$(5\cdot 13 \cdot 27739097)/(3\cdot 11)$\\
12&$(17\cdot 29835840687589)/ (3\cdot 5\cdot 7\cdot 13)$\\
14&$(2\cdot 17\cdot 19\cdot 89\cdot 127\cdot 6353243297)/ 7$\\
16&$(19\cdot 23\cdot 31\cdot 221137132669842886663)/ (2\cdot 5^{2}\cdot 13\cdot 17)$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}[ht]{c||l}
$2m$&$(12m+1)!\cdot (6m+1)\cdot (4m+1)\cdot 24\cdot \zeta _{{\mathfrak s \mathfrak l} _{4}} (2m)/ (12^{2m}\cdot (2\pi) ^{12m})$\\
\hline
\hline
$2$&$23/ 2$\\
$4$&$(3\cdot 7\cdot 14081)/ 2$\\
$6$&$(757409\cdot 23283173)/ (5\cdot 7)$\\
$8$&$(3\cdot 11\cdot 1021\cdot 5529809\cdot 754075957)/ 2$\\
$10$&$(13\cdot 116763209\cdot 1872391681\cdot 3187203549787)/ (5\cdot 11)$\\
$12$&$(17\cdot 1798397149\cdot 5509496891\cdot 6127205846988571484743)/ (3\cdot7\cdot 13)$
\end{tabular}
\end{center}
\end{example}
\begin{comment}
\subsection{Related work}
\begin{enumerate}
\item Khan, student of David Hayes, developed some techniques that
he used to compute special values of Hecke $L$-functions for some
totally real cubic fields~\cite{khan}.
\item Barvinok described an algorithm to count lattice points in a
rational polytope that is polynomial-time when the dimension is fixed
\cite{barvinok}. Barvinok and Pommersheim used this to show that the
higher-dimensional Dedekind sums of Zagier were computable in
polynomial time~\cite{barv.pommersheim}. Their techniques are
different from ours. (Also, our Dedekind sums are more general than
those considered by Zagier.)
\item Szenes showed how to compute (diagonal ?) sums as an iterated
residue, and computed these sums explicitly for certain special cases
(those associated to Witten's $\zeta $-function for $SU (n)$)
\cite{szenes}. (Although he didn't deal with the conditionally
convergent case, and our results cover more sums.)
\item Concerning more about the Witten $\zeta $-function. Witten was
able, using very deep mathematics, to pin down the rationality
properties of his zeta functions at even positive integers. This was
already shown by the second author in~\cite{eisenstein}. Our
results give another proof of this, and provide an efficient algorithm
to calculate these special values. Our results say nothing about
the values at odd positive integers, though.
\item Finally, one of our key transformations appears in Zagier's
paper on multiple $\zeta $-values~\cite{zagier.zeta}.
\end{enumerate}
Despite the sobering growth of $\#V$, the Eisenstein cocycle still
compares quite favorably with Shintani's formula \cite{shintani}. For
example, if $N = 97$ then $\#V = 4704$, and evaluting $\zeta ((1),
(97), 0)$ with $\Psi $ entails summing $357504$ rational numbers. On
the other hand, naively applying Shintani's formula to compute $\zeta
((1), (97), 0)$ involves Dedekind sums with approximately $10^{11170}$
terms.
$32$&$15/32$&$33$&$62/33$&$34$&$25/17$&$35$&$-54/35$&$36$&$-43/36$\\
$37$&$20/37$&$38$&$11/19$&$39$&$4$&$40$&$1/20$&$41$&$382/41$\\
$42$&$-38/7$&$43$&$-366/43$&$44$&$1/44$&$45$&$254/45$&$46$&$3/23$\\
$47$&$-570/47$&$48$&$-13/48$&$49$&$-222/49$&$50$&$89/25$&$51$&$-142/17$\\
$52$&$23/13$&$53$&$586/53$&$54$&$-250/27$&$55$&$918/55$&$56$&$235/56$\\
$57$&$24/19$&$58$&$23/29$&$59$&$658/59$&$60$&$-11/5$&$61$&$-1044/61$\\
$62$&$-14/31$&$63$&$-502/63$&$64$&$31/64$&$65$&$-74/65$&$66$&$-166/33$\\
$67$&$-131/67$&$68$&$36/17$&$69$&$-214/23$&$70$&$4/35$&$71$&$1286/71$\\
$72$&$47/72$&$73$&$418/73$&$74$&$5/37$&$75$&$-136/25$&$76$&$-116/19$\\
$77$&$70/11$&$78$&$-4$&$79$&$-258/79$&$80$&$221/80$&$81$&$134/81$\\
$82$&$-212/41$&$83$&$-698/83$&$84$&$95/28$&$85$&$-698/85$&$86$&$-27/43$\\
$87$&$776/29$&$88$&$199/88$&$89$&$-460/89$&$90$&$131/45$&$91$&$4/13$\\
$92$&$41/23$&$93$&$674/31$&$94$&$210/47$&$95$&$164/19$&$96$&$203/96$\\
$97$&$-1682/97$&$98$&$18/49$&$99$&$-2182/99$&$100$&$-59/25$&\\
\end{comment}
\end{document} |
\begin{document}
\title[Loose Hamiltonian cycles forced by $(k-2)$-degree]{Loose Hamiltonian cycles forced by large $(k-2)$-degree \\ -- sharp version --}
\author[J.~de~O.~Bastos]{Josefran de Oliveira Bastos}
\author[G.~O.~Mota]{Guilherme Oliveira Mota}
\address{Instituto de Matem\'atica e Estat\'{\i}stica, Universidade de
S\~ao Paulo, S\~ao Paulo, Brazil}
\email{\{josefran|mota\}@ime.usp.br}
\author[M.~Schacht]{Mathias Schacht}
\author[J.~Schnitzer]{Jakob Schnitzer}
\author[F.~Schulenburg]{Fabian Schulenburg}
\address{Fachbereich Mathematik, Universit\"at Hamburg, Hamburg, Germany}
\email{[email protected]}
\email{\{jakob.schnitzer|fabian.schulenburg\}@uni-hamburg.de}
\thanks{The first author was supported by CAPES\@.
The second author was supported by FAPESP (Proc. 2013/11431-2 and 2013/20733-2) and CNPq (Proc. 477203/2012-4 and {456792/2014-7}).
The cooperation was supported by a joint CAPES/DAAD PROBRAL (Proc. 430/15).}
\begin{abstract}
We prove for all $k\geq 4$ and $1\ifmmode\ell\else\polishlcross\fieq\ifmmode\ell\else\polishlcross\fi<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph~$\cch$ on~$n$ vertices to contain a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle if $k-\ifmmode\ell\else\polishlcross\fi$ divides~$n$ and~$n$ is sufficiently large.
This extends a result of Han and Zhao for $3$-uniform hypegraphs.
\end{abstract}
\keywords{hypergraphs, Hamiltonian cycles, degree conditions}
\subjclass[2010]{05C65 (primary), 05C45 (secondary)}
\maketitle
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Introduction}
Given $k\geq 2$, a $k$-uniform hypergraph $\cch$ is a pair $(V,E)$ with vertex set $V$ and edge set $E\subseteq V^{(k)}$, where $V^{(k)}$ denotes the set of all $k$-element subsets of $V$.
Given a $k$-uniform hypergraph $\cch=(V,E)$ and a subset $S \in V^{(s)}$, we denote by $d(S)$ the number of edges in $E$ containing~$S$ and we denote by $N(S)$ the $(k-s)$-element sets $T\in V^{(k-s)}$ such that $T\dcup S\in E$, so $d(S)=|N(S)|$.
The \emph{minimum $s$-degree} of $\cch$ is denoted by $\delta_s(\cch)$ and it is defined as the minimum of $d(S)$ over all sets $S\in V^{(s)}$.
We denote by the \textit{size} of a hypergraph the number of its edges.
We say that a $k$-uniform hypergraph $\ccc$ is an \emph{$\ifmmode\ell\else\polishlcross\fi$-cycle} if there exists a cyclic ordering of its vertices such that every edge of $\ccc$ is composed of $k$ consecutive vertices, two (vertex-wise) consecutive edges share exactly $\ifmmode\ell\else\polishlcross\fi$ vertices, and every vertex is contained in an edge.
Moreover, if the ordering is not cyclic, then $\ccc$ is an \emph{$\ifmmode\ell\else\polishlcross\fi$-path} and we say that the first and last~$\ifmmode\ell\else\polishlcross\fi$ vertices are the ends of the path.
The problem of finding minimum degree conditions that ensure the existence of Hamiltonian cycles, i.e.\ cycles that contain all vertices of a given hypergraph, has been extensively studied over the last years (see, e.g., the surveys~\cites{RRsurv,Zhao-survey}).
Katona and Kierstead~\cite{KaKi99} started the study of this problem, posing a conjecture that was confirmed by R\"odl, Ruci\'nski, and Szemer\'edi~\cites{RoRuSz06,RoRuSz08}, who proved the following result:
For every $k\geq 3$, if $\cch$ is a $k$-uniform $n$-vertex hypergraph with $\delta_{k-1}(\cch)\geq {(1/2+o(1))}n$, then $\cch$ contains a Hamiltonian $(k-1)$-cycle.
K\"uhn and Osthus proved that $3$-uniform hypergraphs~$\cch$ with $\delta_2(\cch)\geq {(1/4 +o(1))}n$ contain a Hamiltonian $1$-cycle~\cite{KuOs06}, and H\`an and Schacht~\cite{HaSc10} (see also~\cite{KeKuMyOs11}) generalized this result to arbitrary $k$ and $\ifmmode\ell\else\polishlcross\fi$-cycles with $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi <k/2$.
In~\cite{KuMyOs10}, K\"uhn, Mycroft, and Osthus generalized this result to $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k$, settling the problem of the existence of Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycles in $k$-uniform hypergraphs with large minimum $(k-1)$-degree.
In Theorem~\ref{theorem:asymp} below (see~\cites{BuHaSc13,BaMoScScSc16+}) we have minimum $(k-2)$-degree conditions that ensure the existence of Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycles for $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$.
\begin{theorem}\ifmmode\ell\else\polishlcross\fiabel{theorem:asymp}
For all integers $k\geq 3$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$ and every $\gamma>0$ there exists an $n_0$ such that every $k$-uniform hypergraph $\cch=(V,E)$ on $|V|=n\geq n_0$ vertices with $n\in(k-\ifmmode\ell\else\polishlcross\fi)\bbn$ and
\begin{equation*}
\delta_{k-2}(\cch)\geq\ifmmode\ell\else\polishlcross\fieft(\frac{4(k-\ifmmode\ell\else\polishlcross\fi)-1}{4{(k-\ifmmode\ell\else\polishlcross\fi)}^2}+\gamma\right)\binom{n}{2}
\end{equation*}
contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
\qed
\end{theorem}
The minimum degree condition in Theorem~\ref{theorem:asymp} is asymptotically optimal as the following well-known example confirms.
The construction of the example varies slightly depending on whether~$n$ is an odd or an even multiple of~$k-\ifmmode\ell\else\polishlcross\fi$.
We first consider the case that $n = (2m + 1)(k-\ifmmode\ell\else\polishlcross\fi)$ for some integer~$m$.
Let $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)=(V,E)$ be a $k$-uniform hypergraph on $n$ vertices such that an edge belongs to $E$ if and only if it contains at least one vertex from $A \subset V$, where $|A|=\ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\fifloor \frac{n}{2(k-\ifmmode\ell\else\polishlcross\fi)} \right\rfloor$.
It is easy to see that $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)$ contains no Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle, as it would have to contain $\frac{n}{k-\ifmmode\ell\else\polishlcross\fi}$ edges and each vertex in~$A$ is contained in at most two of them.
Indeed any maximal $\ifmmode\ell\else\polishlcross\fi$-cycle includes all but $k-\ifmmode\ell\else\polishlcross\fi$ vertices and adding any additional edge to the hypergraph would imply a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
Let us now consider the case that $n= 2m(k-\ifmmode\ell\else\polishlcross\fi)$ for some integer~$m$.
Similarly, let $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)=(V,E)$ be a $k$-uniform hypergraph on $n$ vertices that contains all edges incident to $A \subset V$, where $|A|=\frac{n}{2(k-\ifmmode\ell\else\polishlcross\fi)}-1$.
Additionally, fix some $\ell+1$ vertices of $B = V\smallsetminus A$ and let $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)$ contain all edges on $B$ that contain all of these vertices, i.e., an $(\ifmmode\ell\else\polishlcross\fi+1)$-star.
Again, of the $\frac{n}{k-\ifmmode\ell\else\polishlcross\fi}$ edges that a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle would have to contain, at most $\frac{n}{k-\ifmmode\ell\else\polishlcross\fi} - 2$ can be incident to $A$.
So two edges would have to be completely contained in $B$ and be disjoint or intersect in exactly $\ifmmode\ell\else\polishlcross\fi$ vertices, which is impossible since the induced subhypergraph on $B$ only contains an $(\ifmmode\ell\else\polishlcross\fi+1)$-star.
Note that for the minimum $(k-2)$-degree the $(\ifmmode\ell\else\polishlcross\fi+1)$-star on $B$ is only relevant if $\ifmmode\ell\else\polishlcross\fi=1$, in which case this star increases the minimum $(k-2)$-degree by one.
In~\cite{HaZh15b}, Han and Zhao proved the exact version of Theorem~\ref{theorem:asymp} when $k=3$, i.e., they obtained a sharp bound for $\delta_{1}(\cch)$.
We extend this result to $k$-uniform hypergraphs.
\begin{theorem}[Main Result]\ifmmode\ell\else\polishlcross\fiabel{theorem:main}
For all integers $k\geq 4$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$ there exists $n_0$ such that every $k$-uniform hypergraph $\cch=(V,E)$ on $|V|=n\geq n_0$ vertices with $n\in(k-\ifmmode\ell\else\polishlcross\fi)\bbn$ and
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:sharp_minimum_degree}
\delta_{k-2}(\cch)
>
\delta_{k-2}(\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n))
\end{equation}
contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
In particular, if
\begin{equation*}
\delta_{k-2}(\cch)
\geq
\frac{4(k-\ifmmode\ell\else\polishlcross\fi)-1}{4{(k-\ifmmode\ell\else\polishlcross\fi)}^2} \binom{n}{2},
\end{equation*}
then $\cch$ contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
\end{theorem}
The following notion of extremality is motivated by the hypergraph $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)$.
A $k$-uniform hypergraph $\cch=(V,E)$ is called \emph{$(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal} if there exists a partition $V=A\dcup B$ such that $|A|=\ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\ficeil \frac{n}{2(k-\ell)} - 1 \right\rceil$, $|B|=\ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\fifloor \frac{2(k-\ifmmode\ell\else\polishlcross\fi)-1}{2(k-\ifmmode\ell\else\polishlcross\fi)}n + 1 \right\rfloor$ and $e(B)=|E\cap B^{(k)}|\ifmmode\ell\else\polishlcross\fieq \xi \binom{n}{k}$.
We say that $A\dcup B$ is an \emph{$(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal partition} of $V$.
Theorem~\ref{theorem:main} follows easily from the next two results, the so-called \emph{extremal case} (see Theorem~\ref{theorem:extremal} below) and the \emph{non-extremal case} (see Theorem~\ref{theorem:non-extremal}).
\begin{theorem}[Non-extremal Case]\ifmmode\ell\else\polishlcross\fiabel{theorem:non-extremal}
For any $0<\xi<1$ and all integers $k\geq 4$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$, there exists $\gamma>0$ such that the following holds for sufficiently large $n$.
Suppose $\cch$ is a $k$-uniform hypergraph on $n$ vertices with $n\in(k-\ifmmode\ell\else\polishlcross\fi)\bbn$ such that $\cch$ is not $(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal and
\begin{equation*}
\delta_{k-2}(\cch)\geq\ifmmode\ell\else\polishlcross\fieft(\frac{4(k-\ifmmode\ell\else\polishlcross\fi)-1}{4{(k-\ifmmode\ell\else\polishlcross\fi)}^2}-\gamma\right)\binom{n}{2}.
\end{equation*}
Then $\cch$ contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
\qed
\end{theorem}
The non-extremal case was the main result of~\cite{BaMoScScSc16+}.
\begin{theorem}[Extremal Case]\ifmmode\ell\else\polishlcross\fiabel{theorem:extremal}
For any integers $k\geq 3$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$, there exists $\xi>0$ such that the following holds for sufficiently large $n$.
Suppose $\cch$ is a $k$-uniform hypergraph on $n$ vertices with $n\in(k-\ifmmode\ell\else\polishlcross\fi)\bbn$ such that $\cch$ is $(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal and
\begin{equation*}
\delta_{k-2}(\cch)
>
\delta_{k-2}(\ccx_{k,\ifmmode\ell\else\polishlcross\fi}).
\end{equation*}
Then $\cch$ contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle.
\end{theorem}
In Section~\ref{sec:overview} we give an overview of the proof of Theorem~\ref{theorem:extremal} and state Lemma~\ref{lem:mainlemma}, the main result required for the proof.
In Section~\ref{sec:mainproof} we first prove some auxiliary lemmas and then we prove Lemma~\ref{lem:mainlemma}.
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Overview}\ifmmode\ell\else\polishlcross\fiabel{sec:overview}
Let $\cch=(V,E)$ be a $k$-uniform hypergraph and let $X,Y\subset V$ be disjoint subsets.
Given a vertex set $L\subset V$ we denote by $d(L,X^{(i)} Y^{(j)})$ the number of edges of the form $L \cup I \cup J$, where $I \in X^{(i)}$, $J \in Y^{(j)}$, and $|L| + i + j = k$.
We allow for $Y^{(j)}$ to be omitted when $j$ is zero and write $d(v,X^{(i)} Y^{(j)})$ for $d(\{ v \},X^{(i)} Y^{(j)})$.
The proof of Theorem~\ref{theorem:extremal} follows ideas from~\cite{HaZh15}, where a corresponding result with a $(k-1)$-degree condition is proved.
Let $\cch=(V,E)$ be an extremal hypergraph satisfying~\eqref{eq:sharp_minimum_degree}.
We first construct an $\ifmmode\ell\else\polishlcross\fi$-path~$\ccq$ in $\cch$ (see~Lemma~\ref{lem:mainlemma} below) with ends $L_0$ and $L_1$ such that there is a partition $A_{\ast}\dcup B_{\ast}$ of $(V\smallsetminus \ccq) \cup L_0 \cup L_1$ composed only of ``typical'' vertices (see~\ref{it:2-mainlemma} and~\ref{it:3-mainlemma} below).
The set $A_{\ast}\cup B_{\ast}$ is suitable for an application of Lemma~\ref{lem:3.10} below, which ensures the existence of an $\ifmmode\ell\else\polishlcross\fi$-path $\ccq'$ on $A_{\ast}\cup B_{\ast}$ with $L_0$ and $L_1$ as ends.
Note that the existence of a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-cycle in $\cch$ is guaranteed by $\ccq$ and $\ccq'$.
So, in order to prove Theorem~\ref{theorem:extremal}, we only need to prove the following lemma.
\begin{lemma}[Main Lemma]\ifmmode\ell\else\polishlcross\fiabel{lem:mainlemma}
For any $\varrho > 0$ and all integers $k\geq 3$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi <k/2$, there exists a positive~$\xi$ such that the following holds for sufficiently large~$n\in (k-\ifmmode\ell\else\polishlcross\fi)\bbn$.
Suppose that~$\cch=(V,E)$ is an $(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal $k$-uniform hypergraph on~$n$ vertices and
\[
\delta_{k-2}(\cch)
>
\delta_{k-2}(\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)).
\]
Then there exists a non-empty $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$ in $\cch$ with ends $L_0$ and $L_1$ and a partition $A_{\ast}\dcup B_{\ast}=(V\smallsetminus \ccq) \cup L_0 \cup L_1$ where $L_0, L_1\subset B_{\ast}$ such that the following hold:
\begin{enumerate}[label=\rmlabel]
\item\ifmmode\ell\else\polishlcross\fiabel{it:1-mainlemma} $|B_{\ast}|=(2k-2\ifmmode\ell\else\polishlcross\fi-1)|A_{\ast}|+\ifmmode\ell\else\polishlcross\fi$,
\item\ifmmode\ell\else\polishlcross\fiabel{it:2-mainlemma} $d(v,B_{\ast}^{(k-1)})\geq (1-\varrho) \binom{|B_{\ast}|}{k-1}$ for any vertex $v\in A_{\ast}$,
\item\ifmmode\ell\else\polishlcross\fiabel{it:3-mainlemma} $d(v,A_{\ast}^{(1)}B_{\ast}^{(k-2)})\geq (1-\varrho) |A_{\ast}| \binom{|B_{\ast}|}{k-2}$ for any vertex $v\in B_{\ast}$,
\item\ifmmode\ell\else\polishlcross\fiabel{it:4-mainlemma} $d(L_0,A_{\ast}^{(1)} B_{\ast}^{(k - \ifmmode\ell\else\polishlcross\fi - 1)}), d(L_1,A_{\ast}^{(1)}B_{\ast}^{(k - \ifmmode\ell\else\polishlcross\fi - 1)}) \geq (1-\varrho)|A_{\ast}|\binom{|B_{\ast}|}{k - \ifmmode\ell\else\polishlcross\fi - 1}$.
\end{enumerate}
\end{lemma}
The next result, which we will use to conclude the proof of Theorem~\ref{theorem:extremal}, was obtained by Han and Zhao (see~\cite{HaZh15}*{Lemma~3.10}).
\begin{lemma}\ifmmode\ell\else\polishlcross\fiabel{lem:3.10}
For any integers $k\geq 3$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi < k/2$ there exists $\varrho > 0$ such that the following holds.
If $\cch$ is a sufficiently large $k$-uniform hypergraph with a partition $V(\cch)=A_{\ast}\dcup B_{\ast}$ and there exist two disjoint $\ifmmode\ell\else\polishlcross\fi$-sets $L_0,L_1\subset B_{\ast}$ such that~\ref{it:1-mainlemma}--\ref{it:4-mainlemma} hold, then $\cch$ contains a Hamiltonian $\ifmmode\ell\else\polishlcross\fi$-path $\ccq'$ with $L_0$ and $L_1$ as ends.
\qed
\end{lemma}
\@ifstar{\origsection*}{\@startsection{section}{1}\z@{.7\ifmmode\ell\else\polishlcross\fiinespacing\@plus\ifmmode\ell\else\polishlcross\fiinespacing}{.5\ifmmode\ell\else\polishlcross\fiinespacing}{\normalfont\scshape\centering\S}}{Proof of the Main Lemma}\ifmmode\ell\else\polishlcross\fiabel{sec:mainproof}
We will start this section by describing the setup for the proof, which will be fixed for the rest of the paper.
Then we will prove some auxiliary lemmas and finally prove Lemma~\ref{lem:mainlemma}.
Let $\varrho > 0$ and integers $k\geq 3$ and $1\ifmmode\ell\else\polishlcross\fieq \ifmmode\ell\else\polishlcross\fi<k/2$ be given.
Fix constants
\[
\frac{1}{k}, \frac{1}{\ifmmode\ell\else\polishlcross\fi}, \varrho \gg \delta \gg \varepsilon \gg \varepsilonprime \gg \vartheta \gg \xi.
\]
Let $n\in (k-\ifmmode\ell\else\polishlcross\fi)\bbn$ be sufficiently large and let $\cch$ be an $(\ifmmode\ell\else\polishlcross\fi,\xi)$-extremal $k$-uniform hypergraph on $n$ vertices that satisfies the $(k-2)$-degree condition
\begin{equation*}
\delta_{k-2}(\cch)
>
\delta_{k-2}(\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)).
\end{equation*}
Let $A \dcup B=V(\cch)$ be a minimal extremal partition of~$V(\cch)$, i.e.\ a partition satisfying
\begin{equation}
\ifmmode\ell\else\polishlcross\fiabel{eq:partition-sizes}
a = |A| = \ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\ficeil \frac{n}{2(k-\ifmmode\ell\else\polishlcross\fi)} \right\rceil - 1, \quad b = |B| = n - a, \qand e(B) \ifmmode\ell\else\polishlcross\fieq \xi \binom{n}{k},
\end{equation}
which minimises $e(B)$.
Recall that the extremal example $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)$ implies
\begin{equation}
\ifmmode\ell\else\polishlcross\fiabel{eq:minimumDegree}
\delta_{k-2}(\cch)
>
\binom{a}{2} + a(b - k + 2).
\end{equation}
Since $e(B)\ifmmode\ell\else\polishlcross\fieq \xi\binom{n}{k}$, we expect most vertices $v\in B$ to have low degree $d(v,B^{(k-1)})$ into~$B$.
Also, most $v\in A$ must have high degree $d(v,B^{(k-1)})$ into $B$ such that the degree condition for $(k-2)$-sets in~$B$ can be satisfied.
Thus, we define the sets $A_{\eps}$ and $B_{\eps}$ to consist of vertices of high respectively low degree into~$B$ by
\begin{align*}
A_{\eps} &= \ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\fibrace v\in V\colon d(v,B^{(k-1)})\geq (1-\varepsilon)\binom{|B|}{k-1}\right\rbrace,\\
B_{\eps} &=\ifmmode\ell\else\polishlcross\fieft\ifmmode\ell\else\polishlcross\fibrace v\in V\colon d(v,B^{(k-1)})\ifmmode\ell\else\polishlcross\fieq \varepsilon\binom{|B|}{k-1}\right\rbrace,
\end{align*}
and set $V_{\eps}=V\smallsetminus (A_{\eps} \cup B_{\eps} )$.
We will write $a_{\eps} = |A_{\eps}|$, $b_{\eps} = |B_{\eps}|$, and $v_{\eps} = |V_{\eps}|$.
It follows from these definitions that
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:Aeps-Beps-inclusion}
\text{if } A\cap B_{\eps} \neq \varnothing,
\quad \text{then} \quad
B \subset B_{\eps} ,
\quad \text{while otherwise} \quad
A \subset A_{\eps}.
\end{equation}
For the first inclusion, consider a vertex $v \in A \cap B_{\eps}$ and a vertex $w \in B \smallsetminus B_{\eps}$.
Exchanging~$v$ and~$w$ would create a minimal partition with fewer edges in $e(B)$, a contradiction to the minimality of the extremal partition.
The other inclusion is similarly implied by the minimality.
Actually, as we shall show below, the sets $A_{\eps}$ and $B_{\eps}$ are not too different from $A$ and $B$ respectively:
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:Aeps-Beps-sizes}
|A\smallsetminus A_{\eps} |, |B\smallsetminus B_{\eps} |, |A_{\eps} \smallsetminus A|, |B_{\eps} \smallsetminus B|\ifmmode\ell\else\polishlcross\fieq \vartheta b \qand |V_{\eps}|\ifmmode\ell\else\polishlcross\fieq 2\vartheta b.
\end{equation}
Note that by the minimum $(k-2)$-degree
\[
\binom{a}{2}\binom{b}{k-2}+a\binom{b}{k-1}(k-1)
<
\binom{b}{k-2}\delta_{k-2}(\cch)
\ifmmode\ell\else\polishlcross\fieq
\sum_{S\in B^{(k-2)}} d(S).
\]
Every vertex $v \in |A\smallsetminus A_{\eps} |$ satisfies $d(v, B^{(k-1)}) < (1 - \varepsilon)\binom{b}{k-1}$, so we have
\begin{align*}
\sum_{S\in B^{(k-2)}} d(S)
\ifmmode\ell\else\polishlcross\fieq&
\binom{a}{2}\binom{b}{k-2}+a\binom{b}{k-1}(k-1) \\
&+ e(B)\binom{k}{2} - |A \smallsetminus A_{\eps}| \varepsilon \binom{b}{k-1}(k-1).
\end{align*}
Consequently $|A\smallsetminus A_{\eps} |\ifmmode\ell\else\polishlcross\fieq \vartheta b$, as $e(B) < \xi \binom{n}{k}$ and $\xi \ifmmode\ell\else\polishlcross\fil \vartheta, \varepsilon$.
Moreover, $|B\smallsetminus B_{\eps} | \ifmmode\ell\else\polishlcross\fieq \vartheta b$ holds as a high number of vertices in $B \smallsetminus B_{\eps} $ would contradict $e(B) < \xi \binom{b}{k}$.
The other three inequalities~\eqref{eq:Aeps-Beps-sizes} follow from the already shown ones, for example for $|A_{\eps} \smallsetminus A| < \vartheta b$ observe that
\[
A_{\eps} \smallsetminus A = A_{\eps} \cap B \subset B \smallsetminus B_{\eps}.
\]
Although the vertices in $B_{\eps}$ were defined by their low degree into $B$, they also have low degree into the set $B_{\eps}$ itself; for any $v \in B_{\eps}$ we get
\begin{align*}
d(v, B_{\eps}^{(k - 1)})
&\ifmmode\ell\else\polishlcross\fieq
d(v, B^{(k - 1)}) + |B_{\eps}\smallsetminus B| \binom{|B_{\eps}| - 1}{k - 2}\\
&\ifmmode\ell\else\polishlcross\fieq
\varepsilon \binom{b}{k -1 } + \vartheta b |B_{\eps}|^{k -1}\\
&<
2 \varepsilon \binom{|B_{\eps}|}{k - 1}.
\end{align*}
Since we are interested in $\ifmmode\ell\else\polishlcross\fi$-paths, the degree of $\ifmmode\ell\else\polishlcross\fi$-tuples in $B_{\eps}$ will be of interest, which motivates the following definition.
An $\ifmmode\ell\else\polishlcross\fi$-set $L\subset B_{\eps}$ is called $\varepsilon$-\emph{typical} if
\[
d(L,B^{(k - \ifmmode\ell\else\polishlcross\fi)})\ifmmode\ell\else\polishlcross\fieq \varepsilon\binom{|B|}{k-\ifmmode\ell\else\polishlcross\fi}.
\]
If $L$ is not $\varepsilon$-typical, then it is called $\varepsilon$-\emph{atypical}.
Indeed, most $\ifmmode\ell\else\polishlcross\fi$-sets in $B_{\eps}$ are $\varepsilon$-typical; denote by $x$ the number of $\varepsilon$-atypical sets in~$B_{\eps}$.
We have
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:num-typical-sets}
\frac{x\cdot\varepsilon \binom{|B|}{k-\ifmmode\ell\else\polishlcross\fi}}{\binom{k}{\ifmmode\ell\else\polishlcross\fi}} \ifmmode\ell\else\polishlcross\fieq e(B \cup B_{\eps}) \ifmmode\ell\else\polishlcross\fieq \xi \binom{n}{k} + \vartheta {|B|}^k,
\quad \text{implying} \quad
x\ifmmode\ell\else\polishlcross\fieq \varepsilonprime \binom{|B_{\eps}|}{\ifmmode\ell\else\polishlcross\fi}.
\end{equation}
\begin{lemma}\ifmmode\ell\else\polishlcross\fiabel{lem:typical-degree}
The following holds for any $B_{\eps}^{(m)}$-set $M$ if $m \ifmmode\ell\else\polishlcross\fieq k-2$.
\[
d(M,A_{\eps}^{(1)} B_{\eps}^{(k - m - 1)}) + \frac{k-m}{2} d(M, B_{\eps}^{(k - m)})
\geq
\ifmmode\ell\else\polishlcross\fieft(1-\delta\right)|A_{\eps}|\binom{|B_{\eps}| - m}{k-m-1}.
\]
In particular, the following holds for any $\varepsilon$-typical $B^{(\ifmmode\ell\else\polishlcross\fi)}$-set $L$.
\[
d(L,A_{\eps}^{(1)} B_{\eps}^{(k - \ifmmode\ell\else\polishlcross\fi - 1)})\geq (1-2\delta)|A_{\eps}|\binom{|B_{\eps}|-\ifmmode\ell\else\polishlcross\fi}{k-\ifmmode\ell\else\polishlcross\fi-1}.
\]
\end{lemma}
In the proof of the main lemma we will connect two $\varepsilon$-typical sets only using vertices that are unused so far.
Even more, we want to connect two $\varepsilon$-typical sets using exactly one vertex from $A$.
The following corollary of Lemma~\ref{lem:typical-degree} allows us to do this.
\begin{corollary}\ifmmode\ell\else\polishlcross\fiabel{corr:connect-extend-typical}
Let $L$ and $L'$ be two disjoint $\varepsilon$-typical sets in $B_{\eps}$ and $U\subset V$ with~$|U| \ifmmode\ell\else\polishlcross\fieq \varepsilon n$.
Then the following holds.
\begin{enumerate}[label=\alabel]
\item\ifmmode\ell\else\polishlcross\fiabel{it:connect-typical} There exists an $\ifmmode\ell\else\polishlcross\fi$-path disjoint from $U$ of size two with ends $L$ and $L'$ that contains exactly one vertex from $ A_{\eps}$.
\item\ifmmode\ell\else\polishlcross\fiabel{it:extend-typical} There exist $a \in A_{\eps} \smallsetminus U$ and a set $(k - \ifmmode\ell\else\polishlcross\fi -1)$-set $C \subset B_{\eps} \smallsetminus U$ such that $L \cup a \cup C$ is an edge in $\cch$ and every $\ifmmode\ell\else\polishlcross\fi$-subset of $C$ is $\varepsilon$-typical.
\end{enumerate}
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{corr:connect-extend-typical}]
For~\ref{it:connect-typical}, the second part of Lemma~\ref{lem:typical-degree} for $L$ and $L'$ implies that they both extend to an edge with at least $(1 - 2\delta)|A_{\eps}|\binom{|B_{\eps}| - \ifmmode\ell\else\polishlcross\fi}{k - \ifmmode\ell\else\polishlcross\fi - 1}$ sets in $A_{\eps}^{(1)}B_{\eps}^{(k - \ifmmode\ell\else\polishlcross\fi - 1)}$.
Only few of those intersect $U$ and by an averaging argument we obtain two sets $C, C' \in A_{\eps}^{(1)}B_{\eps}^{(k - \ifmmode\ell\else\polishlcross\fi - 1)}$ such that $|C \cap C'| = \ifmmode\ell\else\polishlcross\fi$ and $L \cup C$ as well as $L' \cup C'$ are edges in $\cch$, which yields the required $\ifmmode\ell\else\polishlcross\fi$-path.
In view of~\eqref{eq:num-typical-sets},~\ref{it:extend-typical} is a trivial consequence of the second part of Lemma~\ref{lem:typical-degree}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:typical-degree}]
Let $m \ifmmode\ell\else\polishlcross\fieq k-2$ and let $M \in B_{\eps}^{(m)}$ be an $m$-set.
We will make use of the following sum over all $(k-2)$-sets $D \subset B_{\eps}$ that contain $M$.
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:sumdeg1}
\begin{split}
\sum_{\substack{M \subset D \subset B_{\eps}\\|D| = k - 2}} d(D)
=
\sum_{\substack{M \subset D \subset B_{\eps} \\ |D| = k - 2}} \Big(& d(D, A_{\eps}^{(1)} B_{\eps}^{(1)}) + d(D, {(A_{\eps} \cup V_{\eps})}^{(2)}) \\
& \qquad + d(D,B_{\eps}^{(2)}) + d(D, V_{\eps}^{(1)} B_{\eps}^{(1)})\Big)
\end{split}
\end{equation}
Note that we can relate the sums $\sum d(D, A_{\eps}^{(1)}B_{\eps}^{(1)})$ and $\sum d(D,B_{\eps}^{(2)})$ in~\eqref{eq:sumdeg1} to the terms in question as follows.
\begin{equation}\ifmmode\ell\else\polishlcross\fiabel{eq:sumdeg2}
\begin{split}
d(M, A_{\eps}^{(1)} B_{\eps}^{(k - m - 1)})
&=
\frac{1}{k - m - 1}\sum_{\substack{M \subset D \subset B_{\eps}\\|D| = k - 2}} d(D, A_{\eps}^{(1)} B_{\eps}^{(1)}),
\\
d(M, B_{\eps}^{(k - m)})
&=
\frac{1}{\binom{k-m}{2}}\sum_{\substack{M \subset D \subset B_{\eps}\\|D| = k - 2}} d(D, B_{\eps}^{(2)}).
\end{split}
\end{equation}
We will bound some of the terms on the right-hand side of~\eqref{eq:sumdeg1}.
It directly follows from~\eqref{eq:Aeps-Beps-sizes} that $d(D, {(A_{\eps}\cup V_{\eps})}^{(2)})\ifmmode\ell\else\polishlcross\fieq \binom{a+3\vartheta b}{2}$; moreover, $d(D, V_{\eps}^{(1)} B_{\eps}^{(1)}) \ifmmode\ell\else\polishlcross\fieq 2 \vartheta bb_{\eps}$.
Using the minimum $(k-2)$-degree condition~\eqref{eq:minimumDegree} we obtain
\begin{equation*}
\sum_{\substack{M \subset D \subset B_{\eps}\\|D| = k - 2}} d(D)
>
\binom{b_{\eps} - m}{k - m - 2}\ifmmode\ell\else\polishlcross\fieft(\binom{a}{2} + a(b - k + 2)\right).
\end{equation*}
Combining these estimates with~\eqref{eq:sumdeg1}~and~\eqref{eq:sumdeg2} yields
\begin{align*}
& d(M, A_{\eps}^{(1)} B_{\eps}^{(k - m - 1)})
+ \frac{k-m}{2} d(M, B_{\eps}^{(k - m)})
\\
& \quad \geq
\frac{1}{k - m - 1}\binom{b_{\eps} - m}{k - m - 2}
\ifmmode\ell\else\polishlcross\fieft(\binom{a}{2} + a(b-k+2) - \binom{a+3\vartheta b}{2} - 2\vartheta bb_{\eps} \right)
\\
& \quad \geq
\ifmmode\ell\else\polishlcross\fieft(1-\delta\right)a_{\eps}\binom{b_{\eps} - m}{k - m - 1}.
\end{align*}
For the second part of the lemma, note that the definition of $\varepsilon$-typicality and $\varepsilon \ifmmode\ell\else\polishlcross\fil \delta$ imply that $\frac{k-\ifmmode\ell\else\polishlcross\fi}{2} d(L, B_{\eps}^{(k - \ifmmode\ell\else\polishlcross\fi)})$ is smaller than $\delta a_{\eps} \binom{b_{\eps} - \ifmmode\ell\else\polishlcross\fi}{k - \ifmmode\ell\else\polishlcross\fi - 1}$ for any $\varepsilon$-typical $\ifmmode\ell\else\polishlcross\fi$-set $L$, which concludes the proof.
\end{proof}
For Lemma~\ref{lem:mainlemma}, we want to construct an $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$, such that $V_{\eps} \subset V(\ccq)$ and the remaining sets $A_{\eps}\smallsetminus \ccq$ and $B_{\eps}\smallsetminus \ccq$ have the right relative proportion of vertices, i.e., their sizes are in a ratio of one to $(2k - 2\ifmmode\ell\else\polishlcross\fi - 1)$.
If $|A \cap B_{\eps}| > 0$, then $B \subset B_{\eps}$ (see~\eqref{eq:Aeps-Beps-inclusion}) and so $\ccq$ should cover $V_{\eps}$ and contain the right number of vertices from $B_{\eps}$.
For this, we have to find suitable edges inside $B_{\eps}$, which the following lemma ensures.
\begin{lemma}\ifmmode\ell\else\polishlcross\fiabel{lem:2q-path}
Suppose that $q = |A \cap B_{\eps}| > 0$.
Then there exist $2q + 2$ disjoint paths of size three, each of which contains exactly one vertex from $A_{\eps}$ and has two $\varepsilon$-typical sets as its ends.
\end{lemma}
\begin{proof}
We say that an $(\ifmmode\ell\else\polishlcross\fi - 1)$-set $M \subset B_{\eps}$ is $\emph{good}$ if it is a subset of at least $(1 - \sqrt{\varepsilonprime })b_{\eps}$ $\varepsilon$-typical sets, otherwise we say that the set is \emph{bad}.
We will first show that there are $2q+2$ edges in $B_{\eps}$, each containing one $\varepsilon$-typical and one \emph{good} $(\ifmmode\ell\else\polishlcross\fi - 1)$-set.
Then we will connect pairs of these edges to $\ifmmode\ell\else\polishlcross\fi$-paths of size three.
Suppose that $q = |A \cap B_{\eps}| > 0$.
So $B \subset B_{\eps}$ by~\eqref{eq:Aeps-Beps-inclusion} and consequently $|B_{\eps}| = |B| + q$ and $q \ifmmode\ell\else\polishlcross\fieq \vartheta |B|$.
It is not hard to see from~\eqref{eq:num-typical-sets} that at most a $\sqrt{\varepsilonprime}$ fraction of the $(\ifmmode\ell\else\polishlcross\fi - 1)$-sets in~$B_{\eps}^{(l-1)}$ are bad.
Hence, at least
\[
\ifmmode\ell\else\polishlcross\fieft( 1 - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi}\varepsilonprime - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi - 1}\sqrt{\varepsilonprime} \right) \binom{b}{k-2}
\]
$(k -2)$-sets in $B_{\eps}$ contain no $\varepsilon$-atypical or bad subset.
Let $\ccb \subset B_{\eps}^{(k)}$ be the set of edges inside $B_{\eps}$ that contain such a $(k-2)$-set.
For all $M \in B_{\eps}^{(k - 2)}$, by the minimum degree condition, we have $d(M, B_{\eps}^{(2)}) \geq q(b-k+2) + \binom{q}{2}$ and, with the above, we have
\begin{align*}
|\ccb| & \geq \ifmmode\ell\else\polishlcross\fieft(1 - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi}\varepsilonprime - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi - 1}\sqrt{\varepsilonprime }\right)\binom{b}{k-2}\frac{q(b-k+2)}{\binom{k}{2}} \nonumber \\
& = \ifmmode\ell\else\polishlcross\fieft(1 - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi}\varepsilonprime - \binom{k - 2}{\ifmmode\ell\else\polishlcross\fi - 1}\sqrt{\varepsilonprime }\right)\binom{b}{k-1}\frac{2q}{k} \geq \frac{q}{k} \binom{b}{k-1}.
\end{align*}
On the other hand, for any $v \in B_{\eps}$ we have $d(v, B_{\eps}^{(k - 1)}) < 2 \varepsilon \binom{b_{\eps}}{k - 1}$ which implies that any edge in $\ccb$ intersects at most $2k \varepsilon \binom{b_{\eps}}{k - 1}$ other edges in $\ccb$.
So, in view of $\varepsilon \ifmmode\ell\else\polishlcross\fil \frac{1}{k}$ we may pick a set $\ccb'$ of $2q+2$ disjoint edges in $\ccb$.
We will connect each of the edges in $\ccb'$ to an $\varepsilon$-typical set.
Assume we have picked the first $i-1$ desired $\ifmmode\ell\else\polishlcross\fi$-paths, say $\ccp_1, \dots, \ccp_{i-1}$, and denote by $U$ the set of vertices contained in one of the paths or one of the edges in $\ccb'$.
For the rest of this proof, when we pick vertices and edges, they shall always be disjoint from $U$ and everything chosen before.
Let $e$ be an edge in $\ccb'$ we have not considered yet and pick an arbitrary $\varepsilon$-typical set $L' \subset B_{\eps} \smallsetminus U$.
We will first handle the cases that $2\ifmmode\ell\else\polishlcross\fi + 1 < k$ or that $\ifmmode\ell\else\polishlcross\fi=1$, $k=3$.
In the first case, a $(k-2)$-set that contains no $\varepsilon$-atypical set already contains two disjoint $\varepsilon$-typical sets.
In the second case, an $\ifmmode\ell\else\polishlcross\fi$-set $\{v\}$ is $\varepsilon$-typical for any vertex $v$ in $B_{\eps}$ by the definition of $\varepsilon$-typicality.
Hence in both cases $e$ contains two disjoint $\varepsilon$-typical sets, say $L_0$ and $L_1$.
We can use Corollary~\ref{corr:connect-extend-typical}\,\ref{it:connect-typical}, as $|U| \ifmmode\ell\else\polishlcross\fieq 6kq$, to connect $L_1$ to $L'$ and obtain an $\ifmmode\ell\else\polishlcross\fi$-path $\ccp_i$ of size three that contains one vertex in $A_{\eps}$ and has $\varepsilon$-typical ends $L_0$ and $L'$.
So now assume that $2\ifmmode\ell\else\polishlcross\fi + 1 = k$ and $k > 3$, in particular $k - 2 = 2\ifmmode\ell\else\polishlcross\fi - 1$ and we may split the $(k-2)$-set considered in the definition of $\ccb$ into an $\varepsilon$-typical $\ifmmode\ell\else\polishlcross\fi$-set $L$ and a good $(\ifmmode\ell\else\polishlcross\fi-1)$-set $G$.
Moreover, let $w \in e \smallsetminus (L \cup G)$ be one of the remaining two vertices and set~$N = G \cup {w}$.
First assume that $d(N, A_{\eps}^{(1)} B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)}) \geq \frac{\delta}{3} a_{\eps}\binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi}$.
As $\vartheta \ifmmode\ell\else\polishlcross\fil \delta$, at most $\frac{\delta}{3} a_{\eps} \binom{b}{\ifmmode\ell\else\polishlcross\fi}$ sets in $A_{\eps}^{(1)} B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)}$ intersect $U$.
So it follows from Lemma~\ref{lem:typical-degree} that there exist $A_{\eps}^{(1)} B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)}$-sets $C$, $C'$ such that $N \cup C$ and $L' \cup C'$ are edges, $|C \cap C'| = \ifmmode\ell\else\polishlcross\fi$ and~$|C \cap C' \cap A_{\eps}|=1$.
Now assume that $d(N, A_{\eps}^{(1)} B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)}) < \frac{\delta}{3} a_{\eps}\binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi}$.
As the good set $G$ forms an $\varepsilon$-typical set with most vertices in $B_{\eps}$, there exists $v \in B_{\eps}\smallsetminus U$ such that
\begin{equation*}
d(N \cup \{v\}, A_{\eps}^{(1)} B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi - 1)}) < \delta a_{\eps}\binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi - 1}
\end{equation*}
and $G \cup \{ v \}$ is an $\varepsilon$-typical set.
Lemma~\ref{lem:typical-degree} implies that
\begin{align*}
d(N \cup \{v\}, B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)})
&\geq
\frac{2}{\ifmmode\ell\else\polishlcross\fi} \ifmmode\ell\else\polishlcross\fieft((1-\delta) a_{\eps} \binom{b_{\eps}-(\ell+1)}{\ifmmode\ell\else\polishlcross\fi - 1} - \deltaa_{\eps}\binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi - 1}\right)\\
&\geq
\frac{2}{\ifmmode\ell\else\polishlcross\fi} \ifmmode\ell\else\polishlcross\fieft(\frac{1}{2} - 2\delta\right) a_{\eps} \binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi - 1}\\
&\geq
\delta \binom{b_{\eps}}{\ifmmode\ell\else\polishlcross\fi}.
\end{align*}
So there exists an $\varepsilon$-typical $\ifmmode\ell\else\polishlcross\fi$-set $L^* \subset (B_{\eps} \smallsetminus U)$ such that $N \cup L^* \cup \{ v \}$ is an edge in $\cch$.
Use Lemma~\ref{corr:connect-extend-typical}\,\ref{it:connect-typical} to connect $L^*$ to $L'$ and obtain an $\ifmmode\ell\else\polishlcross\fi$-path $\ccp_i$ of size three that contains one vertex in $A_{\eps}$ and has $\varepsilon$-typical ends $G \cup \{ v \}$ and $L'$.
\end{proof}
If the hypergraph we consider is very close to the extremal example then Lemma~\ref{lem:2q-path} does not apply and we will need the following lemma.
\begin{lemma}\ifmmode\ell\else\polishlcross\fiabel{lem:one-or-two-edges}
Suppose that $B = B_{\eps}$.
If $n$ is an odd multiple of $k-\ell$ then there exists a single edge on $B_{\eps}$ containing two $\varepsilon$-typical $\ell$-sets.
If $n$ is an even multiple of $k-\ell$ then there either exist two disjoint edges on $B_{\eps}$ each containing two $\varepsilon$-typical $\ell$-sets or an $\ell$-path of size two with $\varepsilon$-typical ends.
\end{lemma}
\begin{proof}
For the proof of this lemma all vertices and edges we consider will always be completely contained in $B_{\eps}$.
First assume that there exists an $\varepsilon$-atypical $\ell$-set $L$.
Recall that this means that $d(L,B^{(k - \ifmmode\ell\else\polishlcross\fi)}) > \varepsilon\binom{|B|}{k-\ifmmode\ell\else\polishlcross\fi}$ so in view of~\eqref{eq:num-typical-sets} and $\varepsilonprime \ifmmode\ell\else\polishlcross\fil \varepsilon$ we can find two disjoint $(k-\ifmmode\ell\else\polishlcross\fi)$-sets extending it to an edge, each containing an $\varepsilon$-typical set, which would prove the lemma.
So we may assume that all $\ell$-sets in $B_{\eps}^{(\ifmmode\ell\else\polishlcross\fi)}$ are $\varepsilon$-typical.
We infer from the minimum degree condition that $B_{\eps}$ contains a single edge, which proves the lemma in the case that $n$ is an odd multiple of $k-\ifmmode\ell\else\polishlcross\fi$ and for the rest of the proof we assume that $n$ is an even multiple of $k-\ifmmode\ell\else\polishlcross\fi$.
Assume for a moment that $\ifmmode\ell\else\polishlcross\fi = 1$.
Recall that in this case any $(k-2)$-set in $B$ in the extremal hypegraph $\ccx_{k,\ifmmode\ell\else\polishlcross\fi}(n)$ is contained in one edge.
Consequently, the minimum degree condition implies that any $(k-2)$-set in $B_{\eps}$ extends to at least two edges on $B_{\eps}$.
Fix some edge $e$ in $B_{\eps}$; any other edge on $B_{\eps}$ has to intersect $e$ in at least two vertices or the lemma would hold.
Consider any pair of disjoint $(k-2)$-sets $K$ and $M$ in $B_{\eps} \smallsetminus e$ to see that of the four edges they extend to, there is a pair which is either disjoint or intersect in one vertex, proving the lemma for the case $\ifmmode\ell\else\polishlcross\fi=1$.
Now assume that $\ifmmode\ell\else\polishlcross\fi > 1$.
In this case the minimum degree condition implies that any $(k-2)$-set in $B_{\eps}$ extends to at least one edge on $B_{\eps}$.
Again, fix some edge $e$ in $B_{\eps}$; any other edge on $B_{\eps}$ has to intersect $e$ in at least one vertex or the lemma would hold.
Applying the minimum degree condition to all $(k-2)$-sets disjoint from $e$ implies that one vertex $v \in e$ is contained in at least $\frac{1}{2k^2} \binom{|B_{\eps}|}{k-2}$ edges.
We now consider the $(k-1)$-uniform link hypergraph of $v$ on $B_{\eps}$.
Since any two edges intersecting in $\ifmmode\ell\else\polishlcross\fi-1$ vertices would finish the proof of the lemma, we may assume that there are no such pair of edges.
However, a result of Frankl and Füredi~\cite[Theorem 2.2]{FranklFuredi} guarantees that this $(k-1)$-uniform hypergraph without an intersection of size $\ifmmode\ell\else\polishlcross\fi-1$ contains at most $\binom{|B_{\eps}|}{k-\ifmmode\ell\else\polishlcross\fi-1}$ edges, a contradiction.
\end{proof}
The following lemma will allow us to handle the vertices in $V_{\eps}$.
\begin{lemma}\ifmmode\ell\else\polishlcross\fiabel{lemma:pathV0}
Let $U \subset B_{\eps}$ with $|U| \ifmmode\ell\else\polishlcross\fieq 4k\vartheta$.
There exists a family $\ccp_1, \ifmmode\ell\else\polishlcross\fidots, \ccp_{v_{\eps}}$ of disjoint $\ifmmode\ell\else\polishlcross\fi$-paths of size two, each of which is disjoint from $U$ such that for all $i \in [v_{\eps}]$
\[
|V(\ccp_i) \cap V_{\eps}| = 1 \qand |V(\ccp_i) \cap B_{\eps}| = 2k - \ifmmode\ell\else\polishlcross\fi - 1,
\]
and both ends of $\ccp_i$ are $\varepsilon$-typical sets.
\end{lemma}
\begin{proof}
Let $V_{\eps} = \{ x_1, \dots, x_{v_{\eps}} \}$.
We will iteratively pick the paths.
Assume we have already chosen $\ifmmode\ell\else\polishlcross\fi$-paths $\ccp_1, \dots, \ccp_{i-1}$ containing the vertices $v_1, \dots, v_{i-1}$ and satisfying the lemma.
Let $U'$ be the set of all vertices in $U$ or in one of those $\ifmmode\ell\else\polishlcross\fi$-paths.
From $v_i \notin B_{\eps}$ we get
\[
d(v_i, B_{\eps}^{(k-1)}) \geq d(v_i, B) - |B \smallsetminus B_{\eps}| \cdot \binom{|B|}{k-2} \geq \frac{\varepsilon}{2} \binom{b_{\eps}}{k-1}.
\]
From~\eqref{eq:num-typical-sets} we get that at most $k^\ifmmode\ell\else\polishlcross\fi \varepsilonprime \binom{b_{\eps}}{k-1}$ sets in $B_{\eps}^{(k-1)}$ contain at least one $\varepsilon$-atypical $\ifmmode\ell\else\polishlcross\fi$-set.
Also, less than~$\frac{\varepsilon}{8} \binom{b_{\eps}}{k-1}$ sets in $B_{\eps}^{(k-1)}$ contain one of the vertices of $U'$.
In total, at least $\frac{\varepsilon}{4} \binom{b_{\eps}}{k-1}$ of the $B_{\eps}^{(k-1)}$-sets form an edge with $v_i$.
So we may pick two edges $e$ and $f$ in $V_{\eps}^{(1)} B_{\eps}^{(k-1)}$ that contain the vertex $v_i$ and intersect in $\ifmmode\ell\else\polishlcross\fi$ vertices.
In particular, these edges form an $\ifmmode\ell\else\polishlcross\fi$-path of size two as required by the lemma.
\end{proof}
We can now proceed with the proof of Lemma~\ref{lem:mainlemma}.
Recall that we want to prove the existence of an $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$ in $\cch$ with ends $L_0$ and $L_1$ and a partition
\[
A_{\ast}\dcup B_{\ast}=(V\smallsetminus \ccq)\dcup L_0\dcup L_1
\]
satisfying properties~\ref{it:1-mainlemma}--\ref{it:4-mainlemma} of Lemma~\ref{lem:mainlemma}.
Set $q = |A \cap B_{\eps}|$.
We will split the construction of the $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$ into two cases, depending on whether $q=0$ or not.
First, suppose that $q > 0$.
In the following, we denote by $U$ the set of vertices of all edges and $\ifmmode\ell\else\polishlcross\fi$-paths chosen so far.
Note that we will always have $|U| \ifmmode\ell\else\polishlcross\fieq 20 k \vartheta n$ and hence we will be in position to apply Corollary~\ref{corr:connect-extend-typical}.
We use Lemma~\ref{lem:2q-path} to obtain paths $\ccq_1, \ifmmode\ell\else\polishlcross\fidots, \ccq_{2q+2}$ and then we apply Lemma~\ref{lemma:pathV0} to obtain $\ifmmode\ell\else\polishlcross\fi$-paths $\ccp_1, \ifmmode\ell\else\polishlcross\fidots, \ccp_{v_{\eps}}$.
Every path $\ccq_i$, for $i \in [2q+2]$, contains $3k - 2\ifmmode\ell\else\polishlcross\fi - 1$ vertices from $B_{\eps}$ and one from $A_{\eps}$, while every $\ccp_j$, for $j \in [v_{\eps}]$, contains $2k - \ifmmode\ell\else\polishlcross\fi - 1$ from $B_{\eps}$ and one from $V_{\eps}$.
As the ends of all these paths are $\varepsilon$-typical, we apply Corollary~\ref{corr:connect-extend-typical}\,\ref{it:connect-typical} repeatedly to connect them to one $\ifmmode\ell\else\polishlcross\fi$-path $\ccp$.
In each of the $v_{\eps} + 2q + 1$ steps of connecting two $\ifmmode\ell\else\polishlcross\fi$-paths, we used one vertex from $A_{\eps}$ and $2k - 3\ifmmode\ell\else\polishlcross\fi - 1$ vertices from $B_{\eps}$.
Overall, we have that
\[
|V(\ccp) \cap A_{\eps}| = v_{\eps} + 4q + 3,
\]
as well as
\[
|V(\ccp) \cap B_{\eps}| = (4k - 4\ifmmode\ell\else\polishlcross\fi - 2)v_{\eps} + (5k - 5\ifmmode\ell\else\polishlcross\fi - 2)(2q+2) - (2k - 3\ifmmode\ell\else\polishlcross\fi - 1).
\]
Furthermore $|V(\ccp)| \ifmmode\ell\else\polishlcross\fieq 10 k \vartheta b$.
Using the identities $a_{\eps} + b_{\eps} + v_{\eps} = n$ and $a_{\eps} + q + v_{\eps} = a$, we will now establish property~\ref{it:1-mainlemma} of Lemma~\ref{lem:mainlemma}.
Set $s(\ccp) = (2k - 2\ifmmode\ell\else\polishlcross\fi - 1)|A_{\eps}\smallsetminus V(\ccp)| - |B_{\eps}\smallsetminus V(\ccp)| - 2\ifmmode\ell\else\polishlcross\fi$, so
\begin{align*}
s(\ccp) & = (2k - 2\ifmmode\ell\else\polishlcross\fi - 1)|A_{\eps} \smallsetminus V(\ccp)| - |B_{\eps} \smallsetminus V(\ccp)| - 2\ifmmode\ell\else\polishlcross\fi\\
& = (2k - 2\ifmmode\ell\else\polishlcross\fi - 1)(a_{\eps} - (v_{\eps} + 4q + 3)) - b_{\eps} \\
& \phantom{{} = {}} {} + (4k - 4\ifmmode\ell\else\polishlcross\fi -2)v_{\eps} + (5k - 5\ifmmode\ell\else\polishlcross\fi - 2)(2q+2) - (2k - 3\ifmmode\ell\else\polishlcross\fi - 1) - 2\ifmmode\ell\else\polishlcross\fi \\
& = (2k - 2\ifmmode\ell\else\polishlcross\fi - 1)a_{\eps} - b_{\eps} + (2k - 2\ifmmode\ell\else\polishlcross\fi - 1)v_{\eps} + 2(k - \ifmmode\ell\else\polishlcross\fi)q + 2k - 3\ifmmode\ell\else\polishlcross\fi \\
& = 2(k - \ifmmode\ell\else\polishlcross\fi)(a_{\eps} + v_{\eps} + q + 1) - n - \ifmmode\ell\else\polishlcross\fi \\
& = 2(k - \ifmmode\ell\else\polishlcross\fi)(a + 1) - n - \ifmmode\ell\else\polishlcross\fi.
\end{align*}
If $n/(k - \ifmmode\ell\else\polishlcross\fi)$ is even, $s(\ccp) = -\ifmmode\ell\else\polishlcross\fi$ (see~\eqref{eq:partition-sizes}) and we set $\ccq = \ccp$.
Otherwise $s(\ccp) = k - 2\ifmmode\ell\else\polishlcross\fi$ and we use Corollary~\ref{corr:connect-extend-typical}\,\ref{it:extend-typical} to append one edge to $\ccp$ to obtain $\ccq$.
It is easy to see that one application of Corollary~\ref{corr:connect-extend-typical}\,\ref{it:extend-typical} decreases $s(\ccp)$ by $k - \ifmmode\ell\else\polishlcross\fi$.
Setting $A_{\ast} = A_{\eps}\smallsetminus V(\ccq)$ and $B_{\ast} = (B_{\eps} \smallsetminus V(\ccq)) \cup L_0 \cup L_1$ we get from $s(\ccq) = -\ifmmode\ell\else\polishlcross\fi$ that $A_{\ast}$ and $B_{\ast}$ satisfy~\ref{it:1-mainlemma}.
Now, suppose that $q = 0$.
Apply Lemma~\ref{lemma:pathV0} to obtain $\ifmmode\ell\else\polishlcross\fi$-paths $\ccp_1, \ifmmode\ell\else\polishlcross\fidots, \ccp_{v_{\eps}}$.
If $B=B_{\eps}$, apply Lemma~\ref{lem:one-or-two-edges} to obtain one or two more $\ifmmode\ell\else\polishlcross\fi$-paths contained in $B_{\eps}$.
We apply Corollary~\ref{corr:connect-extend-typical}\,\ref{it:connect-typical} repeatedly to connect them to one $\ifmmode\ell\else\polishlcross\fi$-path $\ccp$.
Since $q = 0$, we have that $B_{\eps} \subset B$ and $a_{\eps} + v_{\eps} = |V\smallsetminus B_{\eps}| = a + |B\smallsetminus B_{\eps}|$.
We can assume without loss of generality that $V_{\eps} \neq \varnothing$, otherwise just take $V_{\eps} = \{ v \}$ for an arbitrary $v \in V (\cch)$.
If $B=B_{\eps}$ let $x$ be $2(k-\ifmmode\ell\else\polishlcross\fi)$ or $k-\ifmmode\ell\else\polishlcross\fi$ depending on whether $n$ is an odd or even multiple of $k-\ifmmode\ell\else\polishlcross\fi$; otherwise let $x=0$.
With similar calculations as before and the same definition of $s(\ccp)$ we get that
\[
s(\ccp) = 2(k-\ifmmode\ell\else\polishlcross\fi)a + x + 2(k-\ifmmode\ell\else\polishlcross\fi)|B \smallsetminus B_{\eps}| - n - \ifmmode\ell\else\polishlcross\fi \equiv -\ifmmode\ell\else\polishlcross\fi \mod(k - \ifmmode\ell\else\polishlcross\fi).
\]
Extend the $\ifmmode\ell\else\polishlcross\fi$-path $\ccp$ to an $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$ by adding $\frac{s(\ccp) + \ifmmode\ell\else\polishlcross\fi}{k - l}$ edges using Corollary~\ref{corr:connect-extend-typical}\,\ref{it:extend-typical}.
Thus $s(\ccq) = -\ifmmode\ell\else\polishlcross\fi$, and we get~\ref{it:1-mainlemma} as in the previous case.
In both cases, we will now use the properties of the constructed $\ifmmode\ell\else\polishlcross\fi$-path $\ccq$ to show~\ref{it:2-mainlemma}-\ref{it:4-mainlemma}.
We will use that $v(\ccq) \ifmmode\ell\else\polishlcross\fieq 20 k \vartheta b$, which follows from the construction.
Since $A_{\ast} \subset A_{\eps}$, for all $v \in A_{\ast}$ we have $d(v, B^{(k-1)}) \geq (1 - \varepsilon)B^{(k-1)}$.
Thus
\[
d(v, B_{\ast}^{(k-1)}) \geq d(v, B^{(k-1)}) - |B_{\ast}\smallsetminus B|\binom{|B_{\ast}| - 1}{k-2} \geq (1 - 2\varepsilon)\binom{|B_{\ast}|}{k-1},
\]
which shows~\ref{it:2-mainlemma}.
For~\ref{it:3-mainlemma}, Lemma~\ref{lem:typical-degree} yields for all vertices $v \in B_{\ast} \subset B_{\eps}$ that
\[
d(v,A_{\eps}^{(1)} B_{\eps}^{(k-2)}) + \frac{k-1}{2} d(v, B_{\eps}^{(k-1)})
\geq
\ifmmode\ell\else\polishlcross\fieft(1-\delta\right)|A_{\eps}|\binom{|B_{\eps}| - 1}{k-2}.
\]
The second term on the left can be bounded from above by $2k\varepsilon\binom{b_{\eps}}{k-1}$.
So, as $\delta, \varepsilon \ifmmode\ell\else\polishlcross\fil \varrho$ and $a_{\eps} - |A_{\ast}| \ifmmode\ell\else\polishlcross\fil \varrho |A_{\ast}| $ as well as $b_{\eps} - |B_{\ast}| \ifmmode\ell\else\polishlcross\fil \varrho |B_{\ast}|$, we can conclude~\ref{it:3-mainlemma}.
By Lemma~\ref{lem:typical-degree}, we know that
\[
d(L_{0}, A_{\eps}^{(1)}B_{\eps}^{(k - 1)}),d(L_{1}, A_{\eps}^{(1)}B_{\eps}^{(k - 1)}) \geq (1 - \delta) a_{\eps}\binom{b_{\eps} - \ifmmode\ell\else\polishlcross\fi}{k - \ifmmode\ell\else\polishlcross\fi - 1}.
\]
As $\delta \ifmmode\ell\else\polishlcross\fil \varrho$ and $a_{\eps} - |A_{\ast}| \ifmmode\ell\else\polishlcross\fil \varrho |A_{\ast}| $ as well as $b_{\eps} - |B_{\ast}| \ifmmode\ell\else\polishlcross\fil \varrho |B_{\ast}|$, we can conclude~\ref{it:4-mainlemma}.
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\begin{document}
\mathtt{i}tle[Coidempotent subcoalgebras and short exact sequences of $2$-representations]{Coidempotent subcoalgebras and short exact sequences of finitary $2$-representations}
\author{Aaron Chan}
\mathrm{op}eratorname{add}ress{Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, JAPAN}
\email{[email protected]}
\urladdr{}
\author{Vanessa Miemietz}
\mathrm{op}eratorname{add}ress{
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK}
\email{[email protected]}
\urladdr{https://www.uea.ac.uk/~byr09xgu/}
\begin{abstract}
In this article, we study short exact sequences of finitary $2$-representations of a weakly fiat $2$-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra $1$-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian $2$-representations.
\end{abstract}
\maketitle
\section*{Introduction}
The subject of $2$-representation theory originated from \cite{CR,KhLa,Ro} and is the higher categorical
analogue of the classical representation theory of algebras. The articles \cite{MM1}--\cite{MM6} develop the $2$-categorical analogue of finite-dimensional algebras and their finite-dimensional modules, by defining and studying finitary $2$-categories and their finitary $2$-representations. One of the fundamental questions in representation theory is to find the simple representations of a given algebra. The question of how to define the $2$-categorical analogue of these was answered in \cite{MM5} where the notion of simple transitive $2$-representations was defined and a Jordan-H\"{o}lder theory for finitary $2$-categories is provided. Since then, there has been considerable effort to classify simple transitive $2$-representations for certain classes of finitary $2$-categories.
Most of the $2$-categories appearing in the categorification of Lie theoretic objects are examples of the so-called weakly fiat $2$-categories. An important defining property of weakly fiat $2$-categories is that, roughly speaking, all $1$-morphisms have adjoints (often called duals for monoidal categories, which, after strictification, can be viewed as $2$-categories with a single object).
The article \cite{MMMT}
shows that every finitary $2$-representation over a weakly fiat $2$-category can be realised as the category of injective right comodules over a coalgebra $1$-morphism.
This gives a new approach to studying finitary $2$-representations.
It is shown in \cite{MMMZ} that a coalgebra $1$-morphism is cosimple if and only if the corresponding $2$-representation is simple transitive.
In other words, classifying simple transitive $2$-representations is equivalent to classifying cosimple coalgebra $1$-morphisms (up to Morita--Takeuchi equivalences).
This article takes a slightly different direction. After all, another important aspect of the theory of modules over algebras is homological algebra, i.e. how to build all representations from simple ones. The $2$-analogue for homological theory associated to finitary $2$-categories has so far only been studied in \cite{CM}, where an analogue of Ext-groups are introduced and studied. In this article, instead, we look back at the definition of short exact sequence of (finitary) $2$-representations used in \cite{CM} (originally from \cite{SVV}), and relate them to comodules categories over coalgebra $1$-morphisms.
The questions we ask are the following.
\begin{itemize}
\item How do we realise a finitary sub-$2$-representation in the language of comodule theory over coalgebra $1$-morphisms?
\item When can we fit the quotient morphism of $2$-representations induced by a subcoalgebra into a short exact sequence of $2$-representation?
\item What is the relation between the coalgebra $1$-morphisms generating the three finitary $2$-representations appearing in a short exact sequence of $2$-representations?
\end{itemize}
It turns out that the answer is closely related to coidempotent subcoalgebras (see Definition \ref{def:coidem}) and recollements of abelian categories. More precisely, our main theorem (Theorem \ref{thm-main}) states that
\begin{itemize}
\item given a coidempotent subcoalgebra $\mathrm{D}$ of a coalgebra $1$-morphism $\mathrm{C}$, we can construct a coalgebra $1$-morphism $\mathrm{A}$ from a certain injective $\mathrm{C}$-comodule $\mathrm{I}$ such that there is a short exact sequence
$$0 \longrightarrow \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{A}) \xrightarrow{-\square_\mathrm{A}\mathrm{I}} \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})\xrightarrow{-\square_\mathrm{C}\mathrm{D}}\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D}) \longrightarrow 0$$
of $2$-representations, where $-\square_\mathrm{Y}\mathrm{X}$ denotes the cotensor product functor;
\item given a short exact sequence of $2$-representations
$$0\longrightarrow\mathbf{N}\longrightarrow\bfM\longrightarrow\bfK\longrightarrow 0$$
and choosing a coalgebra $1$-morphism $\mathrm{C}$ with $\bfM\cong \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$, there exists a subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, unique up to isomorphism and necessarily coidempotent, such that $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ is equivalent to the quotient $2$-representation $\bfK$.
\end{itemize}
Moreover, passing to the abelianised $2$-representations, in the above situation, we have a recollement of abelian categories
\begin{equation*}
\xymatrix{ \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\ar^{-\square_\mathrm{D}\mathrm{D}_\mathrm{C}}[rr]&& \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})\ar^{[\mathrm{I},-]}[rr]\ar@<1ex>@/^/^{-\square_\mathrm{C}\mathrm{D}}[ll]\ar@<-2ex>@/_/_{[\mathrm{D},-]}[ll]&& \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A}),\ar@<1ex>@/^/^{-\square_\mathrm{A}\mathrm{I}}[ll]\ar@<-2ex>@/_/_{[[\mathrm{I},\mathrm{C}],-]}[ll]
}\end{equation*}
where $[\mathrm{X},-]$ denotes the internal hom functor.
The paper is organised as follows. In Section \ref{oldstuff}, we provide a summary of the setup and results from previous articles on the subject which we need for our purposes. In Section \ref{prelims}, we discuss some preliminary results about recollements and functors between comodules categories. We also provide a correspondence between subcoalgebras of a given coalgebra and subcategories of its comodule categories that are closed under subobjects, quotients and closed under the action by the $2$-category, generalising results in \cite{NT}. In Section \ref{coidext}, we define coidempotent subcoalgebras, show that they correspond to Serre subcategories of the category of comodules, and discuss their relationship with recollements. This then leads to the statement and the proof of the main theorem in the final subsection. Finally, we provide some examples in Section \ref{sec:eg}.
{\bf Acknowledgements.} AC is supported by a JSPS International Research Fellowship. Part of this research was carried out during a visit of AC to the University of East Anglia, whose hospitality is gratefully acknowledged.
\section{Recollections.}\label{oldstuff}
Let $\mathcal{B}bbk$ be an algebraically closed field.
\subsection{$2$-categories and $2$-representations}\label{s2.1}
We start by recalling some terminology on finitary categories and $2$-categories. We refer to reader to \cite{Le,McL} for more detail on general $2$-categories and to \cite{MM1,MM2,MM3,MM4,MM5,MM6} for more detail on
$2$-representations of finitary $2$-categories.
A $\mathcal{B}bbk$-linear category is called {\bf finitary} if it is idempotent complete, has only finitely many isomorphism classes of indecomposable objects and all morphism spaces are finite dimensional. The collection of finitary $\mathcal{B}bbk$-linear categories, together with additive $\mathcal{B}bbk$-linear functors and all natural transformations between such functors, forms a $2$-category denoted by $\mathfrak{A}_{\mathcal{B}bbk}^f$.
In \cite{MM1}, a {\bf finitary} $2$-category $\sc\mbox{C}\hspace{1.0pt}$ was defined to be a $2$-category such that
\begin{itemize}
\item $\sc\mbox{C}\hspace{1.0pt}$ has finitely many objects;
\item each morphism category $\sc\mbox{C}\hspace{1.0pt}(\mathtt{i},\mathtt{j})$ is in $\mathfrak{A}_{\mathcal{B}bbk}^f$;
\item horizontal composition is biadditive and bilinear;
\item for each $\mathtt{i}\in\sc\mbox{C}\hspace{1.0pt}$, the identity $1$-morphism $\mathbbm{1}_\mathtt{i}$ is indecomposable.
\end{itemize}
We denote by $\circ_{0}$ and $\circ_{1}$ the horizontal and vertical compositions in $\sc\mbox{C}\hspace{1.0pt}$, respectively.
A finitary $2$-category $\sc\mbox{C}\hspace{1.0pt}$ is called {\bf weakly fiat} if it has a weak
anti-equivalence $(-)^*$ reversing the direction of both $1$- and $2$-morphisms, such that, for a
$1$-morphism $\mathrm{F}$, the pair $(\mathrm{F},\mathrm{F}^*)$ is an adjoint pair, see \cite[Subsection~2.4]{MM1}. It is called {\bf fiat} if $(-)^*$ is weakly involutive. We denote the weak inverse of $(-)^*$ by ${}^*(-)$, obtaining another adjoint pair $({}^*\mathrm{F},\mathrm{F})$.
A {\bf finitary} $2$-representation of $\sc\mbox{C}\hspace{1.0pt}$ is a $2$-functor from $\sc\mbox{C}\hspace{1.0pt}$ to $\mathfrak{A}_{\mathcal{B}bbk}^f$. An important example of a finitary $2$-representation is, for each $\mathtt{i}\in\sc\mbox{C}\hspace{1.0pt}$, the {\bf principal}
$2$-representation $\mathbf{P}_{\mathtt{i}}:=\sc\mbox{C}\hspace{1.0pt}(\mathtt{i},{}_-)$.
We can (injectively) abelianise both the $2$-category $\sc\mbox{C}\hspace{1.0pt}$ and, for a $2$-representation $\bfM$, the category $\mathcal{M}:=\prod_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}}\bfM(\mathtt{i})$ and use the notation $\underline{(-)}$ for the injective abelianisation ($2$)-functor. For the $2$-category, this needs to be done in a rather technical way, see \cite[Section 3.2]{MMMT} to preserve strictness of horizontal composition. Note that, provided that $\sc\mbox{C}\hspace{1.0pt}$ is weakly fiat, composition in $\underline{\cC}$ is left exact in both variables. Indeed, left and right multiplication by $1$-morphisms in $\sc\mbox{C}\hspace{1.0pt}$ is exact thanks to the existence of adjoints, and all $1$-morphisms of $\underline{\cC}$ can be regarded as kernels of $2$-morphisms in $\sc\mbox{C}\hspace{1.0pt}$, whence application of the snake lemma yields the claim, cf. \cite[Subsection 3.1]{MMMZ}.
For $\mathcal{M}$, it is equivalently possible to use the classical diagrammatic abelianisation, see \cite{Fr}, or \cite[Section 3.1]{MMMT} for a presentation adapted to our notation. This induces an {\bf abelian} $2$-representation $\underline{\bfM}$ on $\underline{\mathcal{M}}$.
Both finitary and abelian $2$-representations of $\sc\mbox{C}\hspace{1.0pt}$ form $2$-categories,
denoted $\sc\mbox{C}\hspace{1.0pt}$-afmod and $\sc\mbox{C}\hspace{1.0pt}$-mod, respectively, in which $1$-morphisms are strong
$2$-natural transformations, which we also simply call morphisms of $2$-represenations, and $2$-morphisms are modifications, see \cite[Section 2]{MM3} for details.
In slight abuse of notation, we will, for any $2$-representation $\mathbf{M}$, write $\mathrm{F}\,X$ rather than $\bfM(\mathrm{F})(X)$.
A $2$-representation $\mathbf{M}\in\sc\mbox{C}\hspace{1.0pt}$-afmod is said to be {\bf transitive}, cf. \cite[Subsection 3.1]{MM5},
if, for any indecomposable objects $X,Y\in\mathcal{M}$, there exists a $1$-morphism $\mathrm{F}$ in $\sc\mbox{C}\hspace{1.0pt}$ such that
$Y$ is isomorphic to a direct summand of $\mathrm{F}\,X$. We say that a transitive $2$-representation $\bfM$
is {\bf simple transitive}, cf. \cite[Subsection 3.5]{MM5}, if
$\mathcal{M}$ has no proper $\sc\mbox{C}\hspace{1.0pt}$-invariant ideals.
In \cite[Section 4]{MM5}, it was proved that every $\bfM\in\sc\mbox{C}\hspace{1.0pt}$-afmod has a {\bf weak Jordan-H{\"o}lder series} with transitive subquotients, and the list of their respective simple transitive quotients is unique up to permutation and equivalence.
\subsection{Coalgebra $1$-morphisms and their comodule categories.}
A {\bf coalgebra $1$-morphism} in $\underline{\cC}$ is a coalgebra object in $\coprod_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}}\underline{\cC}(\mathtt{i},\mathtt{i})$, i.e. a direct sum $\mathrm{C}$ of $1$-morphisms in $ \coprod_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}}\underline{\cC}(\mathtt{i},\mathtt{i})$equipped with $2$-morphisms $\mu_\mathrm{C}\colon \mathrm{C}\to\mathrm{C}\mathrm{C}$ and $\varepsilon_\mathrm{C}\colon\mathrm{C}\to\mathbbm{1}=\bigoplus_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}}\mathbbm{1}_\mathtt{i}$, called comultiplication and counit respectively, satisfying coassociativity $(\mu_\mathrm{C}\circ_0\mathrm{id}_\mathrm{C})\circ_1\mu_\mathrm{C} = (\mathrm{id}_\mathrm{C}\circ_0\mu_\mathrm{C})\circ_1\mu_\mathrm{C}$ and counitality $(\mathrm{id}_\mathrm{C}\circ_0\varepsilon_\mathrm{C})\circ_1\mu_\mathrm{C} = \mathrm{id}_\mathrm{C} = (\varepsilon_\mathrm{C}\circ_0\mathrm{id}_\mathrm{C})\circ_1\mu_\mathrm{C}$.
A {\bf right} (resp. {\bf left}) {\bf comodule} over $\mathrm{C}$ is a $1$-morphism $\mathrm{M}$ in $\underline{\cC}$ together with a coaction $\rho_\mathrm{M}\colon \mathrm{M}\to\mathrm{M}\mathrm{C}$ (resp. $\lambda_\mathrm{C}\colon \mathrm{M}\to\mathrm{C}\mathrm{M}$) such that $(\mathrm{id}_\mathrm{M}\circ_0\mu_\mathrm{C})\circ_1\rho_\mathrm{M} = (\rho_\mathrm{M}\circ_0\mathrm{id}_\mathrm{C})\circ_1\rho_\mathrm{M}$ and $(\mathrm{id}_\mathrm{M}\circ_0\varepsilon_\mathrm{C})\circ_1\rho_\mathrm{M} = \mathrm{id}_\mathrm{M}$ (resp. $(\mu_\mathrm{C}\circ_0\mathrm{id}_\mathrm{M})\circ_1\rho_\mathrm{M} = (\mathrm{id}_\mathrm{C}\circ_0\lambda_\mathrm{M})\circ_1\rho_\mathrm{M}$ and $(\varepsilon_\mathrm{C}\circ_0\mathrm{id}_\mathrm{M})\circ_1\rho = \mathrm{id}_\mathrm{M}$). Note that the last condition implies that all coaction maps are monomorphisms in $\underline{\cC}$.
The {\bf cotensor product over $\mathrm{C}$} of a right $\mathrm{C}$-comodule $\mathrm{M}$ with a left $\mathrm{C}$-comodule $\mathrm{N}$ is the kernel of the map $$\mathrm{M}\mathrm{N} \xrightarrow{\rho_\mathrm{M}\circ_0\mathrm{id}_\mathrm{N}-\mathrm{id}_\mathrm{M}\circ_0\lambda_\mathrm{N}}\mathrm{M}\mathrm{C}\mathrm{N}.$$
\subsection{Internal homs and $2$-representations.}\label{coalgrecall} Let $\sc\mbox{C}\hspace{1.0pt}$ be a weakly fiat $2$-category.
This subsection is essentially a summary of \cite[Sections 4]{MMMT}. Note that results there were stated for a fiat $2$-category, but none of the proofs use involutivity of $(-)^*$, hence all proofs go through verbatim for the weakly fiat case.
Let $\mathbf{M}$ be a finitary $2$-representation of $\sc\mbox{C}\hspace{1.0pt}$ and $N\in {\mathcal{M}}$. Recall the internal hom functor $[N,-]\colon \underline{\mathcal{M}} \to \underline{\cC}$, which is defined as the left adjoint to the evaluation of the action on $N$, i.e.
$$\mathrm{Hom}_{\underline{\mathcal{M}}}(-, \mathrm{F} N) \cong \mathrm{Hom}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}([N,-], \mathrm{F}) $$ for all $\mathrm{F} \in \underline{\cC}$. The internal hom $[N,N]$ has the structure of a coalgebra $1$-morphism and for any $M\in \underline{\mathcal{M}}$, $[N,M]$ has the structure of a right $[N,N]$-comodule in $\underline{\sc\mbox{C}\hspace{1.0pt}}$. The category consisting of such right $[N,N]$-comodule in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ carries the structure of an abelian $2$-representation of $\sc\mbox{C}\hspace{1.0pt}$, denoted by $\mathrm{op}eratorname{comod}_{\underline{\ccC}}[N,N]$, and the finitary $2$-representation on the subcategory of injective right $[N,N]$-comodules is denoted by $\mathrm{op}eratorname{inj}_{\underline{\ccC}}[N,N]$. The latter is equivalent to the additive closure in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}[N,N]$ of $\{\mathrm{F}[N,N] \mid \mathrm{F}\in \sc\mbox{C}\hspace{1.0pt}\}$.
Note that, as shown in \cite[Proof of Lemma 6]{MMMT}, $\mathrm{F}[X,Y]\cong [X,\mathrm{F} Y]$ for all $X, Y\in \mathcal{M}$ and all $\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}$; the same holds also for $X, Y\in \underline{\mathcal{M}}$ by the same proof.
Note also that $\mathrm{op}eratorname{comod}_{\underline{\ccC}}[N,N]$ is equivalent to $\underline{\mathrm{op}eratorname{inj}_{\underline{\ccC}}[N,N]}$.
In \cite[Section 4]{MMMT}, it was shown that when $\bfM$ is transitive, then the realisation morphism $[N,-]$ defines an equivalence of abelian $2$-representations between $\underline{\bfM}$ and $\mathrm{op}eratorname{comod}_{\underline{\ccC}}[N,N]$, and also restricts to an equivalence of finitary $2$-representations between $\bfM$ and $\mathrm{op}eratorname{inj}_{\underline{\ccC}}[N,N]$.
In fact, the same proof works for any $\mathbf{G}_\bfM(N)$, i.e.
for arbitrary $N\in \mathcal{M}$, the realisation morphism induces an equivalence of finitary 2-representations between $\mathbf{G}_\bfM(N)$ and $\mathrm{op}eratorname{inj}_{\underline{\ccC}}[N,N]$. In particular, one can always realise a finitary 2-representation as $\mathrm{op}eratorname{inj}_{\underline{\ccC}}[N,N]$ by taking $N$ as the direct sum of all indecomposable objects (up to isomorphisms).
As such, from now on, we do not distinguish between comodules (resp. injective comodules) over a coalgebra 1-morphism and objects of an abstract abelian (respectively, finitary) $2$-representation.
\subsection{Extensions of $2$-representations.}
A sequence
\begin{equation}\label{eq1}
\xymatrix{
0\ar[r]& \mathcal{A}\ar[r]^{\mathrm{F}}& \mathcal{B}\ar[r]^{\mathrm{G}}& \mathcal{C}\ar[r]& 0
}
\end{equation}
in $\mathfrak{A}_{\mathcal{B}bbk}^f$ will be called {\bf short exact} (cf. \cite[Subsection~2.2.1]{SVV})
provided that
\begin{itemize}
\item $\mathrm{F}$ is full and faithful;
\item $\mathrm{G}$ is full and dense;
\item the kernel of $\mathrm{G}$ coincides with the ideal of
$\mathcal{B}$ generated by $\mathrm{F}(\mathcal{A})$.
\end{itemize}
A sequence of morphisms $\Phi,\Psi$ of additive $2$-representations
\begin{equation*}
\xymatrix{
0\ar[r]& \mathbf{N}\ar[r]^{\Phi}& \bfM\ar[r]^{\Psi}& \bfK \ar[r]& 0
}
\end{equation*}
will be called an {\bf extension} of $2$-representations, provided that the underlying sequence
\begin{equation*}
\xymatrix{
0\ar[r]& \mathcal{N}\ar[r]^{\Phi}& \mathcal{M}\ar[r]^{\Psi}& {\mathcal{K}} \ar[r]& 0
}
\end{equation*}
is short exact in $\mathfrak{A}_{\mathcal{B}bbk}^f$, where in the second sequence $\Phi$ and $\Psi$ refer to the underlying additive functors.
\section{Preliminary results}\label{prelims}
In this section, we collect some preliminary results leading towards our main theorem.
\subsection{Recollements of abelian categories.}
Recall that a diagram
\begin{equation*}
\xymatrix{ \mathcal{A}\ar^{i}[rr]&& \mathcal{B}\ar^{e}[rr]\ar@<1ex>@/^/^{p}[ll]\ar@<-2ex>@/_/_{q}[ll]&& \mathcal{C}\ar@<1ex>@/^/^{r}[ll]\ar@<-2ex>@/_/_{l}[ll]
}\end{equation*}
of abelian categories is a recollement provided that
\begin{itemize}
\item $(q,i,p)$ and $(l,e,r)$ are adjoint triples;
\item the functors $l,r$ and $i$ are fully faithful;
\item the image of $i$ is a Serre subcategory, which is the kernel of $e$.
\end{itemize}
\begin{lemma}\label{recolllem}
Let
\begin{equation*}
\xymatrix{ \mathcal{A}\ar^{i}[rr]&& \mathcal{B}\ar^{e}[rr]\ar@<1ex>@/^/^{p}[ll]\ar@<-2ex>@/_/_{q}[ll]&& \mathcal{C}\ar@<1ex>@/^/^{r}[ll]\ar@<-2ex>@/_/_{l}[ll]
}\end{equation*}
be a recollement of abelian categories with enough injectives, where $(q,i,p)$ and $(l,e,r)$ are adjoint triples. Then the sequence given by $r$ and $p$ restricts to a short exact sequence of additive categories
\begin{equation}\label{injrecoll}
0\to \mathrm{op}eratorname{Inj}\mathcal{C}\overset{r}{\to}\mathrm{op}eratorname{Inj}\mathcal{B}\overset{p}{\to}\mathrm{op}eratorname{Inj}\mathcal{A} \to 0.
\end{equation}
between the full subcategories of injective objects.
\end{lemma}
\proof
The sequence restricts since both $r$ and $p$ are right adjoints to exact functors and hence preserve injectives.
By the definition of recollement, $r$ is fully faithful. Since $pi$ is naturally isomorphic to the identity functor on $\mathcal{A}$ (see e.g.\cite[Proposition 2.7(ii)]{PV}), $p$ is necessarily full and dense.
It remains to show that the kernel of $p$ coincides with the ideal $\mathcal{I}$ in $\mathrm{op}eratorname{Inj}\mathcal{B}$ generated by (the full subcategory given by) the essential image of $r$ restricted to $\mathrm{op}eratorname{Inj}\mathcal{C}$.
It is well-known that $pr=0$ (see e.g.\cite[Proposition 2.7(ii)]{PV}), so it immediately follows that, considering the restricted sequence \eqref{injrecoll}, $\mathcal{I}$ is contained in the kernel of $p$. For simplicity, we say that an object is in $\mathcal{I}$ if its identity morphism is in $\mathcal{I}$.
Assume that $Q_1,Q_2\in \mathrm{op}eratorname{Inj}\mathcal{B}$ are both not annihilated by $p$, and hence are not objects in $\mathcal{I}$. We claim that if $f\colon Q_1 \to Q_2$ is annihilated by $p$, it factors over some $I\in \mathcal{I}$.
Indeed, as $ip(M)$ is the maximal subobject of $M$ with composition factors belonging to $i(\mathcal{A})$ for any $M\in\mathcal{B}$, $ip$ is a subfunctor of $\mathcal{I}d_{\mathcal{B}}$. Thus, we have a commutative diagram of solid arrows
$$\xymatrix{
ip(Q_1)\ar@{^{(}->}[d]\ar^{ip(f)=0}[rr]&&ip(Q_2)\ar@{^{(}->}[d]\\
Q_1\ar^{f}[rr]\ar@{->>}[d]&&Q_2\\
Q_1/ip(Q_1)\ar@{-->}^{\bar f}[urr]&&
}$$
meaning that $f$ factors over $Q_1/ip(Q_1)$ as indicated by the dashed arrow $\bar f$.
Considering the exact sequence
$$0\to ip(Q_1)\to Q_1\to re(Q_1)$$
(cf. \cite[Proposition 4.2]{FP}, \cite[Proposition 2.6(ii)]{Ps})
and letting $I'$ be the injective hull of $e(Q_1)\in \mathcal{C}$, we obtain a monomorphism $Q_1/ip(Q_1)\hookrightarrow r(I')$, and hence
the injective hull $I$ of $Q_1/ip(Q_1)$, which is a direct summand of $r(I')$, is in $\mathcal{I}$.
By injectivity of $Q_2$, $\bar f$ now factors over $I$, so $f$ factors over $I\in\mathcal{I}$, as claimed.
\endproof
\subsection{Functors between comodule categories.}
From now on, $\sc\mbox{C}\hspace{1.0pt}$ will denote a weakly fiat $2$-category.
\begin{lemma}\label{adjlemma}
Let $\mathrm{C}, \mathrm{C}'$ be coalgebra $1$-morphisms in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and $Y$ a $\mathrm{C}, \mathrm{C}'$-bicomodule.
\begin{enumerate}[$($i$)$]
\item\label{adjlemma1} For any $M\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C}')$, the internal hom $[Y,M]$ is a right $\mathrm{C}$-comodule in $\underline{\cC}$.
\item\label{adjlemma2} $[Y,-]\colon \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C}') \to \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ is left adjoint to $-\square_\mathrm{C} Y$.
\end{enumerate}
\end{lemma}
\proof
Both statements are proved in exactly the same way as in the classical case of coalgebras over a field, see \cite[12.6, 12.7]{BW}.
\endproof
\begin{lemma}\label{12.8} Let $\mathrm{C}, \mathrm{C}'$ be coalgebra $1$-morphisms in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and $\mathrm{Y}$ a $\mathrm{C}, \mathrm{C}'$-bicomodule. The following statements are equivalent:
\begin{enumerate}[$($a$)$]
\item $\mathrm{Y}\in \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C}')$.
\item $[\mathrm{Y},-]$ is exact.
\end{enumerate}
If either condition is satisfied, we have $[\mathrm{Y},-]\cong -\square_{\mathrm{C}'} [\mathrm{Y},\mathrm{C}']$ as functors from $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C}')$ to $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$
\end{lemma}
\proof
The same proof as in \cite[12.8, 23.7]{BW} shows that $[\mathrm{Y},-]:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C}')\to \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ is exact if and only if $\mathrm{I}\square_{\mathrm{C}}\mathrm{Y}$ is injective for all injective $\mathrm{C}$-comodules $\mathrm{I}$.
In our setting, since every injective $\mathrm{I}$ is direct summand in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ of $\mathrm{F}\mathrm{C}$ for some $1$-morphism $\mathrm{F}$, this is equivalent to $\mathrm{C}\square_{\mathrm{C}}\mathrm{Y}\in \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C}')$, but $\mathrm{C}\square_\mathrm{C} \mathrm{Y}\cong \mathrm{Y}$.
The last statement is proved in the same way as in loc. cit.
\endproof
Since $[\mathrm{C},\mathrm{C}]\cong \mathrm{C}$ by the definition of realisation morphism, an immediate consequence of Lemma \ref{12.8} is the following result.
\begin{corollary}\label{idcohom}
For a coalgebra $1$-morphism $\mathrm{C}$ in $\underline{\sc\mbox{C}\hspace{1.0pt}}$, we have an isomorphism between $[\mathrm{C},-]$ and the identity functor on $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
\end{corollary}
\begin{lemma}\label{Iinfromleft}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in $\underline{\sc\mbox{C}\hspace{1.0pt}}$, $\mathrm{I}\in \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$. Then
$$\mathrm{I}\square_\mathrm{C}[\mathrm{M},\mathrm{C}] \cong [\mathrm{M},\mathrm{I}]$$
for all $\mathrm{M}\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
\end{lemma}
\proof
Any $\mathrm{I}\in \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$ is a direct summand in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ of $\mathrm{F}\mathrm{C}$ for some $1$-morphism $\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}$. Since all functors are additive, the claim follows from
$$\mathrm{F}\mathrm{C}\square_\mathrm{C}[\mathrm{M},\mathrm{C}] \cong \mathrm{F}[\mathrm{M},\mathrm{C}]\cong [\mathrm{M},\mathrm{F}\mathrm{C}].$$
\endproof
\subsection{The comodule category of a subcoalgebra and related functors}
Let $\mathrm{C}=(\mathrm{C},\mu_\mathrm{C},\epsilon_\mathrm{C})$ be a coalgebra $1$-morphism.
By a {\bf subcoalgebra} $\mathrm{D}$ of $\mathrm{C}$, we mean a coalgebra $1$-morphism $\mathrm{D}=(\mathrm{D},\mu_\mathrm{D},\epsilon_\mathrm{D})$ together with a monomorphism $\iota:\mathrm{D}\hookrightarrow\mathrm{C}$ in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ satisfying $\mu_\mathrm{D}\circ_1\iota=(\iota\circ_0\iota)\circ_1\mu_\mathrm{D}$ and $\epsilon_\mathrm{D}=\epsilon_\mathrm{C}\circ_1\iota$.
Note that for any right $\mathrm{D}$-comodule $\mathrm{N}$ with coaction map $\rho_\mathrm{N}^\mathrm{D}:\mathrm{N}\to\mathrm{N}\mathrm{D}$, one naturally obtains a right $\mathrm{C}$-comodule by post-composing $\rho_\mathrm{N}^\mathrm{D}$ with $\mathrm{id}_\mathrm{N}\circ_0\iota$.
This construction give rise to a functor $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ (see the lemma below and \cite[Section 3.4]{MMMZ}).
In particular, a right $\mathrm{C}$-comodule is in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ if its coaction map $\rho_\mathrm{M}^\mathrm{C}:\mathrm{M}\to \mathrm{M}\mathrm{C}$ factors through $\mathrm{id}_\mathrm{M}\circ_0\iota$.
This fact will be used throughout the rest of the article.
\begin{lemma}\label{lemclosed}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and $\mathrm{D} \overset{\iota}{\hookrightarrow} \mathrm{C}$ be a subcoalgebra with cokernel $\mathrm{C} \overset{\pi}{\twoheadrightarrow} \mathrm{J}$. The natural morphism of $2$-representations $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D}) \to\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ given by $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ is fully faithful, exact, and the subcategory it defines is closed under quotients and subobjects.
\end{lemma}
\proof
The fact that $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ is faithful is obvious from the definition.
By injectivity of ${}_\mathrm{D}\mathrm{D}$, it follows from Lemma \ref{12.8} that $[\mathrm{D},-]:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\to \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ is exact and $[\mathrm{D},-]\cong -\square_\mathrm{D} [\mathrm{D},\mathrm{D}]_\mathrm{C} \cong -\square_\mathrm{D} \mathrm{D}_\mathrm{C}$, hence $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ is exact.
To see that it is full consider a morphism $f\colon \mathrm{M}\to \mathrm{N}$ between two objects isomorphic to $\mathrm{M}'\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ and $\mathrm{N}'\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ respectively, i.e. both coaction $\rho_\mathrm{M}$ and $\rho_\mathrm{N}$ factor over $\rho_\mathrm{M}^\mathrm{D}\colon \mathrm{M}\to \mathrm{M}\mathrm{D}$ and $\rho_\mathrm{N}^\mathrm{D}\colon \mathrm{N}\to \mathrm{N}\mathrm{D}$ respectively. Consider the diagram
$$\xymatrix@C=40pt{
\mathrm{M}\ar^{f}[rrr]\ar^{\rho_\mathrm{M}^\mathrm{D}}[rd]\ar_{\rho_\mathrm{M}}[dd]&&&\mathrm{N}\ar[dd]^{\rho_\mathrm{N}}\ar_{\rho_\mathrm{N}^\mathrm{D}}[ld]\\
&\mathrm{M}\mathrm{D}\ar^{f\circ_0\mathrm{id}_\mathrm{D}}[r]\ar^{\mathrm{id}_\mathrm{M}\circ_0\iota}[dl]&\mathrm{N}\mathrm{D}\ar^{\mathrm{id}_\mathrm{N}\circ_0\iota}[dr]&\\
\mathrm{M}\mathrm{C}\ar_{f\circ_0\mathrm{id}_\mathrm{C}}[rrr]&&&\mathrm{N}\mathrm{C}.
}$$
where the triangles, the outer square and the lower trapezium commute. Then
\begin{equation*}
\begin{split}
(\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\rho_\mathrm{N}^\mathrm{D}\circ_1 f&= \rho_\mathrm{N}\circ_1 f = (f\circ_0\mathrm{id}_\mathrm{C})\circ_1\rho_\mathrm{M}\\
&=(f\circ_0\mathrm{id}_\mathrm{C})\circ_1(\mathrm{id}_\mathrm{M}\circ_0\iota)\circ_1\rho_\mathrm{M}^\mathrm{D}\\
&= (\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\rho_\mathrm{M}^\mathrm{D}.
\end{split}
\end{equation*}
Since $(\mathrm{id}_\mathrm{N}\circ_0\iota)$ is mono, $\rho_\mathrm{N}^\mathrm{D}\circ_1 f=(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\rho_\mathrm{M}^\mathrm{D}$, so $f$ is induced from a morphism in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ and $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ is full.
Let $\mathrm{M}$ be isomorphic to an object of the form $\mathrm{M}'\square_\mathrm{D} \mathrm{D}_\mathrm{C}$, i.e. the coaction $\rho_\mathrm{M}\colon \mathrm{M}\to \mathrm{M}\mathrm{C}$ factors over the inclusion $\mathrm{id}_\mathrm{M}\circ_0\iota\colon \mathrm{M}\mathrm{D}\hookrightarrow \mathrm{M}\mathrm{C}$.
To show closure under quotients,
let $f\colon \mathrm{M}\twoheadrightarrow \mathrm{N}$ be an epimorphism in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. Consider the solid part of the diagram
$$\xymatrix@C=40pt{
\mathrm{M}\ar@{->>}^{f}[rrr]\ar_{\rho_\mathrm{M}^\mathrm{D}}[rd]\ar_{\rho_\mathrm{M}}[dd]&&&\mathrm{N}\ar^{\rho_\mathrm{N}}[dd]\ar@{-->}[dl]_{\sigma}\\
&\mathrm{M}\mathrm{D}\ar^{f\circ_0\mathrm{id}_\mathrm{D}}[r]\ar^{\mathrm{id}_\mathrm{M}\circ_0\iota}[dl]&\mathrm{N}\mathrm{D}\ar^{\mathrm{id}_\mathrm{N}\circ_0\iota}[dr]&\\
\mathrm{M}\mathrm{C}\ar^{f\circ_0\mathrm{id}_\mathrm{C}}[rrr]\ar_{\mathrm{id}_\mathrm{M}\circ_0\pi}[d]&&&\mathrm{N}\mathrm{C}\ar_{\mathrm{id}_\mathrm{N}\circ_0\pi}[d]\\
\mathrm{M}\mathrm{J}\ar^{f\circ_0\mathrm{id}_\mathrm{J}}[rrr]&&&\mathrm{N}\mathrm{J}.
}$$
Since $(\mathrm{id}_\mathrm{M}\circ_0\pi)\circ_1\rho_\mathrm{M} = (\mathrm{id}_\mathrm{M}\circ_0\pi)\circ_1(\mathrm{id}_\mathrm{M}\circ_0\iota)\circ_1\rho_\mathrm{M}^\mathrm{D} = (\mathrm{id}_\mathrm{M}\circ_0(\pi\circ_1\iota))\circ_1\rho_\mathrm{M}^\mathrm{D}=0$, we have
$ (\mathrm{id}_\mathrm{N}\circ_0\pi)\circ_1\rho_\mathrm{N}\circ_1 f =(f\circ_0\mathrm{id}_\mathrm{J})\circ_1(\mathrm{id}_\mathrm{M}\circ_0\pi)\circ_1\rho_\mathrm{M} = 0$ and, since $f$ is epi, $ (\mathrm{id}_\mathrm{N}\circ_0\pi)\circ_1\rho_\mathrm{N}=0$. Hence $\rho_\mathrm{N}$ factors over the kernel of $\mathrm{id}_\mathrm{N}\circ_0\pi$, which, by left exactness of horizontal composition with $\mathrm{id}_\mathrm{N}$ is $\mathrm{id}_\mathrm{N}\circ_0\iota$. This yields the dashed arrow $\sigma$. Now we have
\begin{equation*}
\begin{split}
(\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\sigma\circ_1 f &= \rho_\mathrm{N}\circ_1 f =(f\circ_0\mathrm{id}_\mathrm{C})\circ_1\rho_\mathrm{M}\\
&=(f\circ_0\mathrm{id}_\mathrm{C})\circ_1(\mathrm{id}_\mathrm{M}\circ_0\iota)\circ_1\rho_\mathrm{M}^\mathrm{D} \\
&= (\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\rho_\mathrm{M}^\mathrm{D}.
\end{split}
\end{equation*}
As $(\mathrm{id}_\mathrm{N}\circ_0\iota)$ is mono, it follows that $\sigma\circ_1 f =(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\rho_\mathrm{M}^\mathrm{D}$, so the coaction on $\mathrm{N}$ indeed factors over $\mathrm{N}\mathrm{D}$ as claimed.
To show closure under subobjects, let $f\colon \mathrm{N}\hookrightarrow \mathrm{M}$ be a monomorphism.
Consider the solid part of the diagram
$$\xymatrix@C=40pt{
\mathrm{N}\ar@{^{(}->}^{f}[rrr]\ar@{-->}^{\sigma}[rd]\ar_{\rho_\mathrm{N}}[dd]&&&\mathrm{M}\ar[dd]^{\rho_\mathrm{M}}\ar_{\rho_\mathrm{M}^\mathrm{D}}[ld]\\
&\mathrm{N}\mathrm{D}\ar^{f\circ_0\mathrm{id}_\mathrm{D}}[r]\ar^{\mathrm{id}_\mathrm{N}\circ_0\iota}[dl]&\mathrm{M}\mathrm{D}\ar^{\mathrm{id}_\mathrm{M}\circ_0\iota}[dr]&\\
\mathrm{N}\mathrm{C}\ar^{f\circ_0\mathrm{id}_\mathrm{C}}[rrr]\ar_{\mathrm{id}_\mathrm{N}\circ_0\pi}[d]&&&\mathrm{M}\mathrm{C}\ar_{\mathrm{id}_\mathrm{M}\circ_0\pi}[d]\\
\mathrm{N}\mathrm{J}\ar^{f\circ_0\mathrm{id}_\mathrm{J}}[rrr]&&&\mathrm{M}\mathrm{J}.
}$$
As before, $(\mathrm{id}_\mathrm{M}\circ_0\pi)\circ_1\rho_\mathrm{M} =0$, so $(\mathrm{id}_\mathrm{M}\circ_0\pi)\circ_1\rho_\mathrm{M} \circ_1 f=(f\circ_0\mathrm{id}_\mathrm{J})\circ_1(\mathrm{id}_\mathrm{N}\circ_0\pi)\circ_1\rho_\mathrm{N} = 0$ and since $f\circ_0\mathrm{id}_\mathrm{J}$ is a monomorphism (using left exactness of horizontal composition with $\mathrm{id}_\mathrm{J}$), furthermore, $(\mathrm{id}_\mathrm{N}\circ_0\pi)\circ_1\rho_\mathrm{N} = 0$. Hence, as above, $\rho_\mathrm{N} $ factors over $\mathrm{N}\mathrm{D}$, giving the dashed arrow $\sigma$.
Similarly to before,
\begin{equation*}
\begin{split}
(\mathrm{id}_\mathrm{M}\circ_0\iota)\circ_1(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\sigma&=
(f\circ_0\mathrm{id}_\mathrm{C})\circ_1(\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\sigma\\
&=(f\circ_0\mathrm{id}_\mathrm{C})\circ_1\rho_\mathrm{N}=\rho_\mathrm{M}\circ_1 f\\
&=(\mathrm{id}_\mathrm{M}\circ_0\iota)\circ_1\rho_\mathrm{M}^\mathrm{D}\circ_1 f\\
\end{split}
\end{equation*}
and thanks to monicity of $\mathrm{id}_\mathrm{M}\circ_0\iota$, we conclude $(f\circ_0\mathrm{id}_\mathrm{D})\circ_1\sigma = \rho_\mathrm{M}^\mathrm{D}\circ_1 f$.
It is immediate that in both cases that $\sigma$ defines a right coaction on $\mathrm{N}$.
Indeed, in general, if a right $\mathrm{C}$-coaction $\rho_\mathrm{N}\colon \mathrm{N}\to \mathrm{N}\mathrm{C}$ factors over the inclusion $\mathrm{id}_\mathrm{N}\circ_0\iota\colon \mathrm{N}\mathrm{D}\to\mathrm{N}\mathrm{C}$ via a map $\sigma$, we have
\begin{equation*}\begin{split}
(\mathrm{id}_\mathrm{N}\circ_0\iota\circ_0\iota)\circ_1(\sigma\circ_0\mathrm{id}_\mathrm{D})\circ_1\sigma
&= [((\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1 \sigma)\circ_0 \mathrm{id}_\mathrm{C}]\circ_1 (\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\sigma\\
&= (\rho_\mathrm{N}\circ_0\mathrm{id}_\mathrm{C}) \circ_1\rho_\mathrm{N}\\
&= (\mathrm{id}_\mathrm{N}\circ_0\mu_\mathrm{C}) \circ_1\rho_\mathrm{N}\\
&=(\mathrm{id}_\mathrm{N}\circ_0\mu_\mathrm{C}) \circ_1 (\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\sigma\\
&=(\mathrm{id}_\mathrm{N}\circ_0\iota\circ_0\iota)\circ_1(\mathrm{id}_\mathrm{N}\circ_0\mu_\mathrm{D})\circ_1\sigma
\end{split}\end{equation*}
where the first equality uses the interchange law twice, the second and fourth equalities are the definition of $\sigma$ , the third equality comes from $\rho_N$ being a coaction, and the last equality from $\iota$ being a coalgebra map. Cancelling the monomorphism $\mathrm{id}_\mathrm{N}\circ_0\iota\circ_0\iota$ implies the first comodule axiom. For the second, we compute
$$(\mathrm{id}_\mathrm{N}\circ_0\varepsilon_\mathrm{D})\circ_1\sigma = (\mathrm{id}_\mathrm{N}\circ_0\varepsilon_\mathrm{C})\circ_1(\mathrm{id}_\mathrm{N}\circ_0\iota)\circ_1\sigma (\mathrm{id}_\mathrm{N}\circ_0\varepsilon_\mathrm{C})\circ_1\rho_\mathrm{N}=\mathrm{id}_\mathrm{N}.$$
\endproof
\begin{lemma}\label{MMMZCor7Lem8}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in in $\underline{\sc\mbox{C}\hspace{1.0pt}}$, and $\mathrm{D} \overset{\iota}{\hookrightarrow} \mathrm{C}$ a subcoalgebra.
\begin{enumerate}[$($i$)$]
\item\label{MMMZ7} $-\square_\mathrm{D}\mathrm{D}\square_\mathrm{C}\mathrm{D}$ is naturally isomorphic to the identity morphism on $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$.
\item\label{MMMZ8} There is a monic natural transformation from $-\square_{\mathrm{C}}\mathrm{D}\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}$ to the identity morphism on $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
\end{enumerate}
\end{lemma}
\proof
Denote by $\Psi$ the morphism $(-\square_\mathrm{D} \mathrm{D}_\mathrm{C}):\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\to\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ and by $\Phi$ the morphism $-\square_\mathrm{C}\mathrm{D}:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})\to\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$.
Note that $\Psi\cong [\mathrm{D},-]$ as argued in the proof of Lemma \ref{lemclosed}, and $(\Psi\cong [\mathrm{D},-],\Phi=-\square_{\mathrm{C}}\mathrm{D})$ is an adjoint pair by Lemma \ref{adjlemma}.
Now (i) and (ii) are exactly the same as \cite[Corollary 7]{MMMZ} and \cite[Lemma 8]{MMMZ} respectively.
\endproof
\begin{lemma}\label{realise-subcat}
Suppose $\mathcal{S}$ is a full subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ that is $\sc\mbox{C}\hspace{1.0pt}$-stable, subobject-closed, and quotient-closed.
Let $i$ be the (fully faithful exact) embedding of $\mathcal{S}$ into $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ and $p$ be its right adjoint.
Then $\mathrm{D}:=[ip(\mathrm{C}),ip(\mathrm{C})]$ is a subcoalgebra of $\mathrm{C}$ so that $-\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\to\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ induces an equivalence between $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ and $\mathcal{S}$.
\end{lemma}
\begin{proof}
Consider the right $\mathrm{C}$-comodule $\mathrm{B}$ given by the sum of all images of right $\mathrm{C}$-comodule morphisms of the form $f:\mathrm{M}\to \mathrm{C}$ with $\mathrm{M}\in \mathcal{S}$.
Since $\mathcal{S}$ is quotient-closed, we have $\mathrm{B}\in \mathcal{S}$.
In particular, $\mathrm{B}$ coincides with $ip(\mathrm{C})$ (which is the sum of all subobjects of $\mathrm{C}$ in $\mathcal{S}$), and the counit of the adjoint pair $(i,p)$ therefore defines a monomorphism $\iota':\mathrm{B}\to \mathrm{C}$.
For any $\mathrm{F}\in \sc\mbox{C}\hspace{1.0pt}$, there is an exact sequence
\[
0\to \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{F}\mathrm{B},\mathrm{B})\xrightarrow{\mathrm{Hom}(\mathrm{F}\mathrm{B},\iota')=\iota'\circ-} \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{F}\mathrm{B},\mathrm{C})
\]
in $\underline{\cC}$.
Since $\mathcal{S}$ is $\sc\mbox{C}\hspace{1.0pt}$-stable, we have $\mathrm{F}\mathrm{B}\in\mathcal{S}$.
By the construction of $\mathrm{B}$, every morphism from an object of $\mathcal{S}$ to $\mathrm{C}$ factors through $\iota'$, so the morphism in the above exact sequence is surjective and hence an isomorphism.
Using the adjoint pairs $([\mathrm{B},-],-\cdot\mathrm{B})$ and $(\mathrm{F},\mathrm{F}^*)$, and the fact that $\mathrm{F}[X,Y]\cong [X,\mathrm{F} Y]$ for all $X,Y\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ and all $1$-morphism $\mathrm{F}$, we obtain the following commutative diagram
\[
\xymatrix@C=60pt{
\mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{F}\mathrm{B},\mathrm{B}) \ar[r]^{\iota'\circ-}\ar[d]^{\sim} & \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{F}\mathrm{B},\mathrm{C})\ar[d]^{\sim}\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{B},\mathrm{F}\mathrm{B}],\mathbbm{1}) \ar[r]^{-\circ [\iota', \mathrm{F}\mathrm{B}]}\ar[d]^{\sim} & \mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{F}\mathrm{B}],\mathbbm{1}) \ar[d]^{\sim}\\
\mathrm{Hom}_{\underline{\ccC}}(\mathrm{F}[\mathrm{B},\mathrm{B}],\mathbbm{1}) \ar[r]^{-\circ \mathrm{F}[\iota', \mathrm{B}]}\ar[d]^{\sim} & \mathrm{Hom}_{\underline{\ccC}}(\mathrm{F}[\mathrm{C},\mathrm{B}],\mathbbm{1})\ar[d]^{\sim}\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{B},\mathrm{B}],\mathrm{F}^*) \ar[r]^{-\circ[\iota',\mathrm{B}]} & \mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],\mathrm{F}^*).
}
\]
Hence, the bottom row is an isomorphism which holds for any $1$-morphism $\mathrm{F}$.
Thus, $[\iota',\mathrm{B}]\colon[\mathrm{C},\mathrm{B}]\to [\mathrm{B},\mathrm{B}]$ is an isomorphism whose inverse we denote by $\alpha$.
Using that $[\mathrm{C},-]\cong \mathcal{I}d_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}$ by Corollary \ref{idcohom} yields commutative diagram
$$\xymatrix{
[\mathrm{B},\mathrm{B}]\ar^{\alpha }_{\sim}[r]& [\mathrm{C},\mathrm{B}]\ar^{[\mathrm{C},\iota']}[r] \ar_{\sim}[d]&[\mathrm{C},\mathrm{C}]\ar^{\sim}[d]\\
& \mathrm{B}\ar@{^(->}^{\iota'}[r] & \mathrm{C},
}$$ with vertical isomorphisms. In particular, $[\mathrm{C},\iota']$ is mono.
So setting $\mathrm{D}:=[\mathrm{B},\mathrm{B}]$, we obtain a monomorphism $\iota:\mathrm{D} \to \mathrm{C}$ in $\underline{\sc\mbox{C}\hspace{1.0pt}}$.
Showing that $\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}$ is a subcoalgebra is equivalent to showing that $[\mathrm{B},\mathrm{B}] \overset{{\mathtt{h}}eta}{\hookrightarrow}[\mathrm{C},\mathrm{C}]$ is a subcoalgebra, where ${\mathtt{h}}eta:=[\mathrm{C},\iota']\circ\alpha$.
For simplicity, let us denote by $\mu_\mathrm{C},\epsilon_\mathrm{C}$ the comultiplication and counit of $[\mathrm{C},\mathrm{C}]$ throughout the rest of the proof.
We first verify the compatibility of the counit maps of $\mathrm{D}$ and $\mathrm{C}$, i.e. $\epsilon_\mathrm{D}=\epsilon_\mathrm{C}\circ{\mathtt{h}}eta$.
Using the definition ${\mathtt{h}}eta=[\mathrm{C},\iota']\circ \alpha$ and that $\alpha$ is the inverse of $([\iota',\mathrm{B}])^{-1}$, this is equivalent to showing that $\epsilon_\mathrm{C}\circ_1[\mathrm{C},\iota'] = \epsilon_\mathrm{D} \circ_1 [\iota',\mathrm{B}]$.
Recall that, for any $\mathrm{X}\in\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$, the counit of $[\mathrm{X},\mathrm{X}]$ is the map in $\underline{\cC}$ corresponding to $\mathrm{id}_\mathrm{X}$ under the adjunction isomorphism $\mathrm{Hom}_{\underline{\ccC}}([\mathrm{X},\mathrm{X}],\mathbbm{1})\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{X},\mathrm{X})$.
We consider the commutative diagrams
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{C}],\mathbbm{1})\ar@{-}^{\sim}[r]\ar^{-\circ_1[\mathrm{C},\iota']}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{C},\mathrm{C})\ar^{-\circ\iota'}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],\mathbbm{1}) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},\mathrm{C})
}$$
and
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{B},\mathrm{B}],\mathbbm{1})\ar@{-}^{\sim}[r]\ar^{-\circ_1[\iota', \mathrm{B}]}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},\mathrm{B})\ar^{\iota'\circ-}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],\mathbbm{1}) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},\mathrm{C})
}$$
where the second is obtained from
combining the natural transformation $[\iota',-]:[\mathrm{C},-]\to[\mathrm{B},-]$ with the adjoint pairs $([\mathrm{B},-],-\cdot\mathrm{B})$ and $([\mathrm{C},-],-\cdot\mathrm{C})$.
Since $\mathrm{id}_{\mathrm{C}}\circ_1\iota' = \iota' \circ_1 \mathrm{id}_{\mathrm{B}}$, and these two maps correspond to $\epsilon_\mathrm{C} \circ_1[\mathrm{C},\iota']$ and $\epsilon_\mathrm{D} \circ_1 [\iota',\mathrm{B}]$ respectively on the left columns of the diagrams, the latter two maps are equal, as claimed.
To show compatibility of the comultiplications, let us start by recalling some essential facts.
For any $\mathrm{X},\mathrm{Y}\in\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$, the coevaluation map $\mathrm{coev}_{\mathrm{X},\mathrm{Y}}:\mathrm{Y}\to[\mathrm{X},\mathrm{Y}]\mathrm{X}$ is the map corresponding to $\mathrm{id}_{[\mathrm{X},\mathrm{Y}]}$ under the adjunction $\mathrm{Hom}_{\underline{\ccC}}([\mathrm{X},\mathrm{Y}],[\mathrm{X},\mathrm{Y}])\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{Y},[\mathrm{X},\mathrm{Y}]\mathrm{X})$.
The comultiplication of the coalgebra $[\mathrm{X},\mathrm{X}]$ is given by the map in $\underline{\cC}$ corresponding to $(\mathrm{id}_{[\mathrm{X},\mathrm{X}]}\circ_0\mathrm{coev}_{\mathrm{X},\mathrm{X}})\circ_1 \mathrm{coev}_{\mathrm{X},\mathrm{X}}$.
Observe that the following diagram is commutative.
\[
\xymatrix@C=100pt@R=35pt{
\mathrm{B} \ar[r]^{\mathrm{coev}_{\mathrm{B},\mathrm{B}}} \ar[d]_{\mathrm{id}}
& [\mathrm{B},\mathrm{B}]\mathrm{B} \ar[r]^{\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\mathrm{coev}_{\mathrm{B},\mathrm{B}}} \ar[d]_{\mathrm{id}}
& [\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]\mathrm{B} \ar[d]_{\mathrm{id}_\mathrm{D}\circ_0\alpha\circ_0\iota'}\\
\mathrm{B} \ar[r]^{\mathrm{coev}_{\mathrm{B},\mathrm{B}}} \ar[d]_{\mathrm{id}} & [\mathrm{B},\mathrm{B}]\mathrm{B} \ar[r]^{\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\mathrm{coev}_{\mathrm{B},\mathrm{B}}} \ar[d]_{\alpha\circ_0\iota'}& [\mathrm{B},\mathrm{B}][\mathrm{C},\mathrm{B}]\mathrm{C} \ar[d]_{\alpha\circ_0[\mathrm{C},\iota']\circ_0\mathrm{id}_\mathrm{C}}\\
\mathrm{B} \ar[r]^{\mathrm{coev}_{\mathrm{C},\mathrm{B}}} \ar[d]_{\iota'}& [\mathrm{C},\mathrm{B}]\mathrm{C} \ar[r]^{\mathrm{id}_{[\mathrm{C},\mathrm{B}]}\circ_0\mathrm{coev}_{\mathrm{C},\mathrm{C}}} \ar[d]_{[\mathrm{C},\iota']\circ_0\mathrm{id}_\mathrm{C}}& [\mathrm{C},\mathrm{B}][\mathrm{C},\mathrm{C}]\mathrm{C} \ar[d]_{[\mathrm{C},\iota']\circ_0\mathrm{id}_{[\mathrm{C},\mathrm{C}]\mathrm{C}}}\\
\mathrm{C} \ar[r]^{\mathrm{coev}_{\mathrm{C},\mathrm{C}}} & [\mathrm{C},\mathrm{C}]\mathrm{C} \ar[r]^{\mathrm{id}_{[\mathrm{C},\mathrm{C}]}\circ_0\mathrm{coev}_{\mathrm{C},\mathrm{C}}} & [\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]\mathrm{C}
}
\]
Indeed, commutativity of the top left square is trivial; that of the bottom right square is easy, since both maps are just $[\mathrm{C},\iota']\circ_0 \mathrm{coev}_{\mathrm{C},\mathrm{C}}$.
It is also easy to see that commutativity of the top (resp. middle) right square follows immediately from that of the middle (resp. bottom) left square as the former are obtained from the latter by horizontally composing with identity maps.
To see that the middle left square commutes (i.e. $\mathrm{coev}_{\mathrm{C},\mathrm{B}}=(\alpha\circ_0\iota')\circ_1\mathrm{coev}_{\mathrm{B},\mathrm{B}}$), we use the commutative diagrams
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{B}])\ar@{-}^{\sim}[r]\ar_{[\iota',\mathrm{B}]\circ_1-}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{B}]\mathrm{C})\ar^{([\iota',\mathrm{B}]\circ_0\mathrm{id}_\mathrm{C})\circ-}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{B},\mathrm{B}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{B},\mathrm{B}]\mathrm{C})
}$$
and
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{B},\mathrm{B}],[\mathrm{B},\mathrm{B}])\ar@{-}^{\sim}[r]\ar_{-\circ_1[\iota',\mathrm{B}]}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{B},\mathrm{B}]\mathrm{B})\ar^{(\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ-}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{B},\mathrm{B}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{B},\mathrm{B}]\mathrm{C}),
}
$$
as well as $[\iota',\mathrm{B}]\circ_1\mathrm{id}_{[\mathrm{C},\mathrm{B}]} = \mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_1 [\iota',\mathrm{B}]$. Together, these yield
\[
([\iota',\mathrm{B}]\circ_0\mathrm{id}_\mathrm{C})\circ_1\mathrm{coev}_{\mathrm{C},\mathrm{B}} = (\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ_1\mathrm{coev}_{\mathrm{B},\mathrm{B}},
\]
hence
$\mathrm{coev}_{\mathrm{C},\mathrm{B}} = (\alpha\circ_0\mathrm{id}_\mathrm{C})\circ_1 (\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ_1\mathrm{coev}_{\mathrm{B},\mathrm{B}} = (\alpha\circ_0 \iota')\circ_1 \mathrm{coev}_{\mathrm{B},\mathrm{B}}$.
Commutativity of the bottom left square (i.e. $\mathrm{coev}_{\mathrm{C},\mathrm{C}}\circ_1\iota' = ([\mathrm{C},\iota']\circ_0\mathrm{id}_\mathrm{C})\circ_1 \mathrm{coev}_{\mathrm{C},\mathrm{B}}$) follows similarly from the commutative diagrams
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{C}],[\mathrm{C},\mathrm{C}])\ar@{-}^{\sim}[r]\ar^{-\circ_1[\mathrm{C},\iota']}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{C},[\mathrm{C},\mathrm{C}]\mathrm{C})\ar^{-\circ\iota'}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{C}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{C}]\mathrm{C})
}$$
and
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{B}])\ar@{-}^{\sim}[r]\ar^{[\mathrm{C},\iota']\circ_1-}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{B}]\mathrm{C})\ar^{([\mathrm{C},\iota']\circ_0\mathrm{id}_\mathrm{C})\circ-}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{C}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{C}]\mathrm{C}),
}$$
together with $[\mathrm{C},\iota']\circ_1\mathrm{id}_{[\mathrm{C},\mathrm{B}]} = \mathrm{id}_{[\mathrm{C},\mathrm{C}]}\circ_1[\mathrm{C},\iota']$.
Now that we know all six squares commute, composing the maps on the outer boundary of the big square yields
\begin{equation}\label{eq-bigsq}
\mu_\mathrm{C}^\vee \circ_1\iota' = ({\mathtt{h}}eta\circ_0{\mathtt{h}}eta\circ_0\iota') \circ_1\mu_\mathrm{D}^\vee,
\end{equation}
where $\mu_\mathrm{C}^\vee := (\mathrm{id}_{[\mathrm{C},\mathrm{C}]}\circ_0\mathrm{coev}_{\mathrm{C},\mathrm{C}}) \circ_1\mathrm{coev}_{\mathrm{C},\mathrm{C}}$ and $\mu_\mathrm{D}^\vee := (\mathrm{id}_{[\mathrm{B},\mathrm{B}]}\circ_0\mathrm{coev}_{\mathrm{B},\mathrm{B}}) \circ_1\mathrm{coev}_{\mathrm{B},\mathrm{B}}$ are the maps that correspond to $\mu_\mathrm{C}$ and $\mu_\mathrm{D}$ respectively under adjunction.
Using the commutative diagram
$$\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{C}],[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}])\ar@{-}^{\sim}[r]\ar_{-\circ_1[\mathrm{C},\iota']}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{C},[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]\mathrm{C})\ar^{-\circ\iota'}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]\mathrm{C}),
}$$
we can see that the left-hand map $\mu_\mathrm{C}^\vee \circ_1\iota'$ of \eqref{eq-bigsq} corresponds to $\mu_\mathrm{C}\circ_1[\mathrm{C},\iota']$ under the adjunction isomorphism of the bottom row.
We claim that the right-hand map $({\mathtt{h}}eta\circ_0{\mathtt{h}}eta\circ_0\iota')\mu_\mathrm{D}^\vee$ of \eqref{eq-bigsq} corresponds to $({\mathtt{h}}eta\circ_0{\mathtt{h}}eta)\circ_1\mu_\mathrm{D}\circ_1[\iota',\mathrm{B}]$ under the same adjunction isomorphism.
Indeed, using the commutative diagram
$$
\xymatrix{
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{B},\mathrm{B}],[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}])\ar@{-}^{\sim}[r]\ar_{-\circ_1[\iota',\mathrm{B}]}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]\mathrm{B})\ar^{(\mathrm{id}_{[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ_1-}[d]\\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]) \ar@{-}^{\sim}[r]\ar_{({\mathtt{h}}eta\circ_0{\mathtt{h}}eta)\circ_1-}[d]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]\mathrm{C}) \ar^{({\mathtt{h}}eta\circ_0{\mathtt{h}}eta\circ_0\mathrm{id}_\mathrm{C})\circ_1-}[d] \\
\mathrm{Hom}_{\underline{\ccC}}([\mathrm{C},\mathrm{B}],[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]) \ar@{-}^{\sim}[r]& \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})} (\mathrm{B},[\mathrm{C},\mathrm{C}][\mathrm{C},\mathrm{C}]\mathrm{C}),
}
$$
the correspondence between $\mu_\mathrm{D}$ and $\mu_\mathrm{D}^\vee$ on the top row induces the correspondence between
$\mu_D\circ_1[\iota',\mathrm{B}]$ and $(\mathrm{id}_{[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ_1\mu_\mathrm{D}^\vee$ on the second row, which in turn induces a correspondence between $({\mathtt{h}}eta\circ_0{\mathtt{h}}eta)\circ_1\mu_D\circ_1[\iota',\mathrm{B}]$ and $({\mathtt{h}}eta\circ_0{\mathtt{h}}eta\circ_0\mathrm{id}_\mathrm{C})\circ_1 (\mathrm{id}_{[\mathrm{B},\mathrm{B}][\mathrm{B},\mathrm{B}]}\circ_0\iota')\circ_1\mu_\mathrm{D}^\vee = ({\mathtt{h}}eta \circ_0{\mathtt{h}}eta\circ_0\iota')\circ_1\mu_\mathrm{D}^\vee$ on the bottom row.
Thus, \eqref{eq-bigsq} is equivalent to saying that $\mu_\mathrm{C}\circ_1[\mathrm{C},\iota'] =({\mathtt{h}}eta\circ_0{\mathtt{h}}eta)\circ_1\mu_D\circ_1[\iota',\mathrm{B}]$.
Since ${\mathtt{h}}eta = [\mathrm{C},\iota']\circ_1([\iota',\mathrm{B}])^{-1}$, we obtain that $\mu_\mathrm{C}\circ_1{\mathtt{h}}eta =({\mathtt{h}}eta\circ_0{\mathtt{h}}eta)\circ_1\mu_D$.
This completes the proof of the compatibility between comultiplications of $\mathrm{D}$ and $[\mathrm{C},\mathrm{C}]\cong \mathrm{C}$ under ${\mathtt{h}}eta$.
It remains to show the equivalence $-\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\to\mathcal{S}$.
For a $\mathrm{D}$-comodule $\mathrm{M}$, we have an exact sequence $0\to \mathrm{M}\to \mathrm{F}\mathrm{D}$ in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ for some $\mathrm{F}\in \sc\mbox{C}\hspace{1.0pt}$.
Recall that $ip(\mathrm{C})=\mathrm{B}\cong [\mathrm{C},\mathrm{B}] \cong [\mathrm{B},\mathrm{B}]$, so we have isomorphisms of right $\mathrm{C}$-comodules $ip(\mathrm{C})\cong \mathrm{D}_\mathrm{C} \cong \mathrm{C}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}$. In particular, we have $\mathrm{D}_\mathrm{C}\in \mathcal{S}$.
Since $\mathcal{S}$ is $\sc\mbox{C}\hspace{1.0pt}$-stable,
we have $(\mathrm{F}\mathrm{D})\square_\mathrm{D}\mathrm{D}_\mathrm{C}\cong \mathrm{F}\mathrm{D}_\mathrm{C}\in \mathcal{S}$, so it follows from the assumption of $\mathcal{S}$ being closed under subobjects that $\mathrm{M}\square_\mathrm{D}\mathrm{D}_\mathrm{C}\in \mathcal{S}$.
Hence, $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ induces a well-defined functor from $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ to $\mathcal{S}$.
Recall from Lemma \ref{lemclosed} that $-\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}$ is fully faithful. It remains to show that it is dense. Indeed, if $\mathrm{M}\in \mathcal{S}$, then we have an exact sequence $0\to \mathrm{M}\to \mathrm{F}\mathrm{C}$ in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ for some $\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}$, which induces an exact sequence $0\to ip(\mathrm{M})\to ip(\mathrm{F}\mathrm{C})$. By assumption, we have $ip(\mathrm{M})=\mathrm{M}$.
Since by assumption $i(\mathrm{F}\mathrm{M})\cong \mathrm{F} i(\mathrm{M})$ for all $\mathrm{M}\in\mathcal{S}$, we also have $p(\mathrm{F}\mathrm{N})\cong \mathrm{F} p(\mathrm{N})$ for all $\mathrm{N}\in\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$, as can be seen from the chain of isomorphisms
\begin{equation*}
\begin{split}
\mathrm{Hom}_{\mathcal{S}}(\mathrm{M}, p(\mathrm{F}\mathrm{N}))&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(i(\mathrm{M}), \mathrm{F}\mathrm{N})\\
&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}({}^*\mathrm{F} i(\mathrm{M}), \mathrm{N})\\
&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}( i({}^*\mathrm{F}\mathrm{M}), \mathrm{N})\\
& \cong \mathrm{Hom}_{\mathcal{S}}( {}^*\mathrm{F}\mathrm{M}, p(\mathrm{N}))\\
& \cong \mathrm{Hom}_{\mathcal{S}}( \mathrm{M}, \mathrm{F} p(\mathrm{N})),\\
\end{split}
\end{equation*}
which holds for any $\mathrm{M}\in\mathcal{S}$.
We thus have $ip(\mathrm{F}\mathrm{C})\cong \mathrm{F} (ip(\mathrm{C}))\cong \mathrm{F}\mathrm{D}$, which is in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$. Thus, as $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ is closed under subobjects by Lemma \ref{lemclosed} and $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is exact, $\mathrm{M}$ is also in the essential image of $-\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}$.
\end{proof}
This leads us to the following proposition, which generalises \cite[Theorem 4.2(iii)]{NT}.
\begin{proposition}\label{cat-coalg-bij}
The construction in Lemma \ref{realise-subcat} induces a bijection between the set of $\sc\mbox{C}\hspace{1.0pt}$-stable subobject-closed quotient-closed full subcategories of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ up to equivalence and the set of subcoalgebras of $\mathrm{C}$ up to isomorphism.
\end{proposition}
\begin{proof}
Let $\Omega$ be the set of $\sc\mbox{C}\hspace{1.0pt}$-stable subobject-closed quotient-closed full subcategories of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ up to equivalence, and $\Phi$ be the set of subcoalgebras of $\mathrm{C}$ up isomorphism.
By Lemma \ref{realise-subcat}, assigning $\mathcal{S} \mapsto [ip(\mathrm{C}),ip(\mathrm{C})]$, where $i$ is the inclusion of $\mathcal{S}$ into $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ and $p$ is the right adjoint of $i$, defines a map $f:\Omega\to\Phi$.
On the other hand, for a subcoalgebra $\mathrm{D}$, it follows from Lemma \ref{lemclosed} that $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ is equivalent to a subobject-closed quotient-closed full subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. Note that this subcategory is also $\sc\mbox{C}\hspace{1.0pt}$-stable as $\mathrm{D}$ is a coalgebra $1$-morphism in $\underline{\cC}$.
Clearly, isomorphic subcoalgebras define the same full subcategory up to equivalence.
Hence, we have a map $g:\Phi\to \Omega$.
Starting with $\mathcal{S}\in \Omega$, we have $gf(\mathcal{S})=\mathrm{op}eratorname{comod}_{\underline{\ccC}}(f(\mathcal{S}))$, which is equivalent to $\mathcal{S}$ by Lemma \ref{realise-subcat}; this means that $gf=\mathrm{id}_\Omega$.
For $\mathrm{D}\in \Phi$, Lemma \ref{MMMZCor7Lem8} says that the inclusion of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ into $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ and its right adjoint are given by $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ and $-\square_\mathrm{C}\mathrm{D}$ respectively.
Since $\mathrm{C}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}\cong \mathrm{D}_\mathrm{C}$, the subcoalgebra $fg(\mathrm{D})$ is given $[\mathrm{D}_\mathrm{C},\mathrm{D}_\mathrm{C}]$.
By the same argument as in the first two paragraphs in the proof of Lemma \ref{realise-subcat}, we have $[\mathrm{D}_\mathrm{C},\mathrm{D}_\mathrm{C}]\cong [\mathrm{C},\mathrm{D}] \cong \mathrm{D}$.
Therefore, we have $fg(\mathrm{D})\cong \mathrm{D}$, i.e. $fg=\mathrm{id}_\Phi$ as required.
\end{proof}
\section{Coidempotent subalgebras and extensions}\label{coidext}
\subsection{Coidempotent subcoalgebras.}
Following \cite{NT}, we define the following notion, which, in the classical setting, is dual to idempotent quotient algebras $A/AeA$.
\begin{definition}\label{def:coidem}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and $\mathrm{D}$ a subcoalgebra of $\mathrm{C}$. We say that $\mathrm{D}$ an {\bf coidempotent subcoalgebra} of $\mathrm{C}$ if $\mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})=\mathrm{D}$ or, equivalently, for $\mathrm{J}=\mathrm{C}/\mathrm{D}$, the map $\mu_\mathrm{J}:=(\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{J}:\mathrm{J}\to \mathrm{J}\mathrm{J}$ is a monomorphism in $\underline{\cC}$, where $\pi_\mathrm{J}:\mathrm{C}\to\mathrm{J}$ is the natural projection and $\rho_\mathrm{J}$ is the right $\mathrm{C}$-coaction map of $\mathrm{J}$.
\end{definition}
\begin{lemma}\label{seqzero}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and $\mathrm{D}$ a subcoalgebra. Set $\mathrm{J}=\mathrm{C}/\mathrm{D}$ and let $\mathrm{I}$ be the injective hull of $\mathrm{J}$ in $\mathrm{op}eratorname{comod}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})$.
\begin{enumerate}[$($i$)$]
\item\label{seqzero1} $\mathrm{D}$ is coidempotent if and only if $\mathrm{J}\square_{\mathrm{C}}\mathrm{D}=0$.
\item\label{seqzero2} For $\mathrm{Q}\in \mathrm{op}eratorname{inj}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})$, if $\mathrm{Q}\square_{\mathrm{C}}\mathrm{D}=0$, then $\mathrm{Q}\in \mathrm{op}eratorname{add}_{\mathrm{op}eratorname{inj}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})}\{\mathrm{F}\mathrm{I}\mid \mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$. Moreover, the converse holds when $\mathrm{D}$ is coidempotent.
\end{enumerate}
\end{lemma}
\proof
\eqref{seqzero1}
Applying $\mathrm{J}\square_\mathrm{C} -$ to the exact sequence $0\to\mathrm{D}\to\mathrm{C}\overset{\pi}{\to}\mathrm{J}\to 0$ of $\mathrm{C}$-$\mathrm{C}$-bicomodules yields and exact sequence
$$0 \longrightarrow \mathrm{J}\square_\mathrm{C}\mathrm{D}\longrightarrow\mathrm{J}\square_\mathrm{C}\mathrm{C}\overset{\mathrm{id}_\mathrm{J}\square\pi}{\longrightarrow}\mathrm{J}\square_\mathrm{C}\mathrm{J}.$$
Now consider the diagram
$$\xymatrix{
\mathrm{J}\square_\mathrm{C}\mathrm{C} \ar@{^{(}->}_{\alpha}[d] \ar^{\mathrm{id}_\mathrm{J}\square\pi}[r]& \mathrm{J}\square_\mathrm{C}\mathrm{J}\ar@{^{(}->}^{\alpha'}[d]\\
\mathrm{J}\mathrm{C} \ar^{\mathrm{id}_\mathrm{J}\circ_0\pi}[r] \ar_{\mathrm{id}_\mathrm{J}\circ_0\mu_\mathrm{C}-\rho_\mathrm{J}\circ_0\mathrm{id}_\mathrm{C}}[d] & \mathrm{J}\mathrm{J} \ar^{\mathrm{id}_\mathrm{J}\lambda_\mathrm{J}-\rho_\mathrm{J}\circ_0\mathrm{id}_\mathrm{J}}[d] \\
\mathrm{J}\mathrm{C}\mathrm{C} \ar^{\mathrm{id}_{\mathrm{J}\mathrm{C}}\circ_0\pi}[r]&\mathrm{J}\mathrm{C}\mathrm{J},
}$$
where $\lambda_\mathrm{J}$ is the left $\mathrm{C}$-coaction map of $\mathrm{J}$.
Using the interchange law and the induced (left) $\mathrm{C}$-comodule structure of $\mathrm{J}$, the lower square commutes, which yields the commutativity of the upper square.
Since there is an isomorphism $\beta:\mathrm{J} \xrightarrow{\sim}\mathrm{J}\square_\mathrm{C}\mathrm{C}$, we have $\alpha\circ_1\beta=\rho_\mathrm{J}$.
The induced map $\mu_\mathrm{J}\colon\mathrm{J}\to\mathrm{J}\mathrm{J}$ is precisely $(\mathrm{id}_\mathrm{J}\circ_0\pi)\circ_1\rho_\mathrm{J}$.
Hence, we have two exact sequences
$$
\xymatrix{
0\ar[r]& \mathrm{J}\square_\mathrm{C}\mathrm{D} \ar[r]\ar@{.>}[d] & \mathrm{J}\square_\mathrm{C}\mathrm{C} \ar_{\beta^{-1}}^{\sim}[d] \ar^{\mathrm{id}_\mathrm{J}\square\pi}[r]& \mathrm{J}\square_\mathrm{C}\mathrm{J}\ar@{^{(}->}^{\alpha'}[d] \\
0\ar[r]& \ker\mu_\mathrm{J} \ar[r] & \mathrm{J} \ar^{\mu_\mathrm{J}}[r] & \mathrm{J}\mathrm{J}
}
$$
so that the right-hand square commutes.
This implies that $\mathrm{J}\square_\mathrm{C}\mathrm{D}\cong \ker\mu_\mathrm{J}$. The claim follows.
\eqref{seqzero2}
Realise $\mathrm{Q}\in \mathrm{op}eratorname{inj}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})$ as a direct summand (inside $\mathrm{op}eratorname{inj}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})$) of $\mathrm{G} \mathrm{C}$, for some $1$-morphism $\mathrm{G}\in \sc\mbox{C}\hspace{1.0pt}$, with complement $\mathrm{Q}'$. Let $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}\colon \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\hookrightarrow \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ be the morphism from Lemma \ref{lemclosed} given by extending the coaction from $\mathrm{D}$ to $\mathrm{C}$.
Consider the exact sequence
$$0\to \mathrm{G}\mathrm{D}\to \mathrm{Q}\mathrm{op}lus \mathrm{Q}'\to\mathrm{G} \mathrm{J}$$
in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
We claim that if the induced morphism $\alpha\colon\mathrm{G} \mathrm{D}\to \mathrm{Q}$ is nonzero, then $\mathrm{Q}\square_\mathrm{C}\mathrm{D}\neq 0$.
Indeed, as $\mathrm{G}\mathrm{D}$ is in the essential image of $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$, the nonzero image $\mathrm{Z}$ of $\alpha$, as a quotient of $\mathrm{G}\mathrm{D}$, is also in the essential image of $-\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ by Lemma \ref{lemclosed}, and isomorphic to $\mathrm{Z}'\square_\mathrm{D} \mathrm{D}_\mathrm{C}$ for some $\mathrm{Z}'\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$.
On the one hand, applying $-\square_{\mathrm{C}}\mathrm{D}$ to the monomorphism $\mathrm{Z}\hookrightarrow \mathrm{Q}$ yields a monomorphism $\mathrm{Z}\square_\mathrm{C}\mathrm{D}\hookrightarrow \mathrm{Q}\square_\mathrm{C}\mathrm{D}$.
On the other hand, it follows from Lemma \ref{MMMZCor7Lem8}\eqref{MMMZ7} that $Z\square_{\mathrm{C}}\mathrm{D} \cong Z'\square_{\mathrm{D}}\mathrm{D}\square_{\mathrm{C}}\mathrm{D} \cong Z'$ is nonzero.
Thus we obtain that $\mathrm{Q}\square_{\mathrm{C}}\mathrm{D}$ is also nonzero, as claimed.
Therefore, if $\mathrm{Q}\square_\mathrm{C}\mathrm{D}=0$, then $\mathrm{Q}$ is not in the coimage of the first map of the exact sequence above. This implies that $\mathrm{Q}$ is isomorphic to a subobject of $\mathrm{G}\mathrm{J}$, which in turn is a subobject of $\mathrm{G} \mathrm{I}$. Injectivity of $\mathrm{Q}$ implies that it is in fact isomorphic to a direct summand of $\mathrm{G} \mathrm{I}$.
Let us now assume $\mathrm{D}$ is coidempotent and show the converse.
Let $\mathrm{F}_0$ be the injective hull of $\mathrm{J}$ in $\underline{\cC}$ and $\vartheta\colon \mathrm{J}\hookrightarrow \mathrm{F}_0$ the canonical embedding.
Since the induced comultiplication on $\mathrm{J}$ is, by assumption, a monomorphism in $\underline{\cC}$ and composition in $\underline{\cC}$ is left exact, we have monomorphisms $\mathrm{J}\hookrightarrow \mathrm{J}\mathrm{J} \hookrightarrow \mathrm{F}_0\mathrm{J}$ in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
We obtain a commutative diagram
$$\xymatrix{
\mathrm{J} \ar@{^{(}->}_{\mu_\mathrm{J}}[rr] \ar@{^{(}->}[d]^{\rho_\mathrm{J}}&&\mathrm{J} \mathrm{J} \ar@{^{(}->}^{\vartheta\circ_0 \mathrm{id}_\mathrm{J}}[d]\\
\mathrm{J} \mathrm{C} \ar^{\mathrm{id}_\mathrm{J}\circ_0\pi}[urr]\ar@{^{(}->}_{\vartheta\circ_0 \mathrm{id}_\mathrm{C}}[d]&&\mathrm{F}_0\mathrm{J} \\
\mathrm{F}_0\mathrm{C}\ar@{->>}^{\mathrm{id}_{\mathrm{F}_0}\circ_0 \pi}[urr]&&
}$$
in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
By injectivity of $\mathrm{F}_0\mathrm{C}$, the resulting maps $\tau=(\vartheta\circ_0 \mathrm{id}_\mathrm{J})\circ_1\mu_\mathrm{J}$ and $\sigma=(\vartheta\circ_0 \mathrm{id}_\mathrm{C})\circ_1\rho_\mathrm{J}$ in the diagram
$$\xymatrix{
\mathrm{J}\ar@{^{(}->}^{\tau}[rr]\ar@{^{(}->}_{\sigma}[d]&&\mathrm{F}_0\mathrm{J} \ar@{-->}@/_/_{\kappa}[dll]\\
\mathrm{F}_0\mathrm{C}\ar@{->>}@/_/_{\mathrm{id}_{\mathrm{F}_0}\circ_0\pi}[urr]&&
}$$
give rise to the dotted map $\kappa\colon\mathrm{F}_0\mathrm{J} \to \mathrm{F}_0\mathrm{C}$, such that the diagram commutes both ways around. The equality $\kappa\circ_1(\mathrm{id}\circ_0\pi)\circ_1\sigma=\kappa\tau=\sigma$ implies that $\kappa\circ_1(\mathrm{id}\circ_0\pi)$ is the identity on $\mathrm{I}$ as a direct summand of $\mathrm{F}_0\mathrm{C}$ and hence $\mathrm{I}$ is a direct summand of $\mathrm{F}_0\mathrm{J}$.
By part \eqref{seqzero1}, we have $\mathrm{F}_0\mathrm{J}\square_{\mathrm{C}}\mathrm{D}=0$.
In particular, its direct summand $\mathrm{I}\square_{\mathrm{C}}\mathrm{D}$ is also zero, and hence any $Q\in\mathrm{op}eratorname{add}_{\mathrm{op}eratorname{inj}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})}\{\mathrm{F}\mathrm{I}\mid \mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$ satisfies $\mathrm{Q}\square_{\mathrm{C}}\mathrm{D}=0$.
\endproof
\begin{lemma}\label{kill-simple}
Suppose $\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}$ is a coidempotent subcoalgebra.
Let $\mathrm{I}$ be the injective hull of the cokernel of $\iota$ and $\mathrm{M}$ be a simple $\mathrm{C}$-comodule with injective hull $\mathrm{Q}$.
Then the following are equivalent.
\begin{enumerate}[(i)]
\item\label{kill-simple1} $\mathrm{M}\square_\mathrm{C}\mathrm{D}=0$.
\item\label{kill-simple2} $\mathrm{Q}\square_\mathrm{C}\mathrm{D}=0$.
\item\label{kill-simple3} $\mathrm{Q}\in \mathrm{op}eratorname{add}\{\mathrm{F}\mathrm{I}\mid \mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{kill-simple2}${\mathcal{L}}eftrightarrow$\eqref{kill-simple3}: This is Lemma \ref{seqzero} \eqref{seqzero2}.
\eqref{kill-simple2}$\Rightarrow$\eqref{kill-simple1}: Clear by left exactness of $-\square_\mathrm{C}\mathrm{D}$.
\eqref{kill-simple1}$\Rightarrow$\eqref{kill-simple2}:
By Lemma \ref{MMMZCor7Lem8} \eqref{MMMZ8}, $\mathrm{Q}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is a subcomodule of $\mathrm{Q}$, which has simple socle $\mathrm{M}$ in the case when it is non-zero.
Since the smallest non-trivial subcomodule $\mathrm{M}$ of $\mathrm{Q}$ is annihilated by $-\square_\mathrm{C}\mathrm{D}$, it follows that $\mathrm{Q}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}=0$.
But $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is fully faithful, so $\mathrm{Q}\square_\mathrm{C}\mathrm{D}=0$.
\end{proof}
\subsection{Coidempotent subcoalgebras and Serre subcategories}
In this subsection, we provide a correspondence between coidempotent subcoalgebras of a coalgebra $1$-morphism $\mathrm{C}$ and Serre subcategories of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. Throughout this subsection, we let $\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}$ be a subcoalgebra, let $\mathrm{J},\pi_\mathrm{J}$ be defined by the short exact sequence
$$0\to\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}\overset{\pi_\mathrm{J}}{\twoheadrightarrow}\mathrm{J}\to 0$$
and $\mu_\mathrm{J}=(\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{J}$ the induced multiplication on $\mathrm{J}$.
\begin{lemma}\label{coidem-then-Serre}
If $\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}$ is a coidempotent subcolagebra, then the fully faithful exact embedding $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ sends $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ to a Serre subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
\end{lemma}
\proof
By Lemma \ref{lemclosed}, it remains to show closure under extensions.
For any $M\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$, we denote by $\sigma_M$ the composition $(\mathrm{id}_M\circ_0\pi_\mathrm{J})\circ_1\rho_M$, where $\rho_M$ is the coaction map.
Then $M$ being in the essential image of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ is equivalent to $\sigma_M=0$.
Let $0\to X\xrightarrow{f} Y\xrightarrow{g}Z \to 0$ be a short exact sequence of right $\mathrm{C}$-comodule such that $X,Z$ is in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$.
Our aim is to show that $\sigma_Y$ is zero.
Since horizontal composition is left exact, we have commutative diagram
$$
\xymatrix@C=45pt{
0\ar[r] & X\ar[r]^{f}\ar[d]_{\rho_X} & Y\ar[d]_{\rho_Y} \ar[r]^{g} & Z \ar[d]^{\rho_Z} \ar[r] & 0\\
0\ar[r] & X\mathrm{C} \ar[r]_{f\circ_0\mathrm{id}_\mathrm{C}} & Y\mathrm{C} \ar[r]_{g\circ_0\mathrm{id}_\mathrm{C}} & Z\mathrm{C}
}
$$
in $\underline{\cC}$ with exact rows.
This induces a commutative diagram where all $\mathrm{C}$'s and $\rho$'s above are replaced by $\mathrm{J}$ and $\sigma$ respectively.
Hence we have $(g\circ_0\mathrm{id}_\mathrm{J})\circ_1\sigma_Y = \sigma_Z\circ_1g = 0$, which means that the image of $\sigma_Y$ is in the kernel of $g\circ_0\mathrm{id}_\mathrm{J}$.
Exactness of the top row of the diagram implies that there is $\phi:Y\to X\mathrm{J}$ so that $(f\circ_0\mathrm{id}_\mathrm{J})\circ_1\phi = \sigma_Y$.
Thus, we have
\begin{equation*}\begin{split}
(\sigma_Y\circ_0\mathrm{id}_\mathrm{J})\circ_1 \sigma_Y& = (\sigma_Y\circ_0\mathrm{id}_\mathrm{J})\circ_1(f\circ_0\mathrm{id}_\mathrm{J})\circ_1\phi \\
&= ((\sigma_Y\circ_1 f)\circ_0\mathrm{id}_\mathrm{J})\circ_1\phi\\
&= (((f\circ_0\mathrm{id}_\mathrm{J})\circ_1\sigma_X)\circ_0 \mathrm{id}_\mathrm{J}) \circ_1\phi \\
&=0.
\end{split}\end{equation*}
On the other hand, $(Y\xrightarrow{\rho_Y}Y\mathrm{C}\xrightarrow{\rho_Y\circ_0\mathrm{id}_\mathrm{C}}Y\mathrm{C}\mathrm{C} )=(Y\xrightarrow{\rho_Y}Y\mathrm{C}\xrightarrow{\mathrm{id}_Y\circ_0\mu_\mathrm{C}}Y\mathrm{C}\mathrm{C} )$ and this induces $(\sigma_Y\circ_0\mathrm{id}_\mathrm{J})\circ_1 \sigma_Y = (\mathrm{id}_Y\circ_0\mu_\mathrm{J})\circ_1\sigma_Y$. Combining this with the argument in the previous paragraph, we see that $(\mathrm{id}_Y\circ_0\mu_\mathrm{J})\circ_1\sigma_Y=0$. Since $\mathrm{D}$ is coidempotent (i.e. $\mu_\mathrm{J}$ is mono) and horizontal left composition with $\mathrm{id}_Y$ preserves monicity, we obtain that $\mathrm{id}_Y\circ_0\mu_\mathrm{J}$ is mono, which implies $\sigma_Y=0$ as required.
\endproof
\begin{lemma}\label{I/I^2 is A/I-module}
Let $\mathrm{K}$ be the kernel of $\mu_\mathrm{J}$.
Then there is a short exact sequence
$$
0 \to \mathrm{D} \to \mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C}) \to \mathrm{K} \to 0
$$
of right $\mathrm{C}$-comodules, and $\mathrm{K}$ is also a $\mathrm{D}$-comodule.
\end{lemma}
\proof
The right $\mathrm{C}$-comodule map $\pi_\mathrm{J}:\mathrm{C}\twoheadrightarrow\mathrm{J}$ induces a commutative diagram
$$
\xymatrix@C=50pt{
\mathrm{C} \ar[r]^{\mu_\mathrm{C}}\ar[d]_{\pi_\mathrm{J}} & \mathrm{C}\mathrm{C} \ar[d]^{\pi_\mathrm{J}\circ_0\mathrm{id}_\mathrm{C}} \ar[r]^{\mathrm{id}_\mathrm{C}\circ_0\pi_\mathrm{J}} & \mathrm{C}\mathrm{J} \ar[d]^{\pi_\mathrm{J}\circ_0\mathrm{id}_\mathrm{J}}\\
\mathrm{J}\ar[r]_{\rho_\mathrm{J}} &\mathrm{J}\mathrm{C} \ar[r]_{\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J}}& \mathrm{J}\mathrm{J} ,
}
$$
in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. Since $\mu_\mathrm{J}:=(\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{J}$, $\mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})=\ker((\pi_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\mu_\mathrm{C})$ coincides with $\ker(\mu_\mathrm{J}\circ_1\pi_\mathrm{J})$.
We have a commutative diagram
$$
\xymatrix{
& 0\ar[r]\ar[d] & \mathrm{C}\ar[r]^{\mathrm{id}_\mathrm{C}}\ar[d]^{\pi_\mathrm{J}} & \mathrm{C}\ar[r]\ar[d]^{(\pi_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\mu_\mathrm{C}} & 0\\
0 \ar[r] & \mathrm{K} \ar[r]_{\iota_\mathrm{K}} & \mathrm{J} \ar[r]_{\mu_\mathrm{J}} & \mathrm{J}\mathrm{J} &
}
$$
where both rows are exact.
Now the snake lemma provides the required short exact sequence $0\to \mathrm{D}\to \mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})\to \mathrm{K}\to 0$ of right $\mathrm{C}$-comodules.
Consider the following commutative diagram
$$
\xymatrix@C=50pt{
\mathrm{K} \ar[r]^{\iota_\mathrm{K}}\ar[d]_{\rho_\mathrm{K}} & \mathrm{J} \ar[d]^{\rho_\mathrm{J}} \\
\mathrm{K}\mathrm{C} \ar[r]^{\iota_\mathrm{K}\circ_0\mathrm{id}_\mathrm{C}}\ar[d]_{\mathrm{id}_\mathrm{K}\circ_0\pi_\mathrm{J}}& \mathrm{J}\mathrm{C} \ar[d]^{\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J}} \\
\mathrm{K}\mathrm{J} \ar[r]_{\iota_\mathrm{K}\circ_0\mathrm{id}_\mathrm{J}} & \mathrm{J}\mathrm{J}
}
$$
in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
This yields $$(\iota_\mathrm{K}\circ_0\mathrm{id}_\mathrm{J})\circ_1(\mathrm{id}_\mathrm{K}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{K} = (\mathrm{id}_\mathrm{J}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{J}\circ_1\iota_\mathrm{K} = \mu_\mathrm{J}\circ_1\iota_\mathrm{K} = 0.$$
In particular, since $\iota_\mathrm{K}\circ_0\mathrm{id}_\mathrm{J}$ is mono (as, again, horizontal composition inherits monicity of $\iota_\mathrm{K}$), we deduce that $(\mathrm{id}_\mathrm{K}\circ_0\pi_\mathrm{J})\circ_1\rho_\mathrm{K}=0$, as required to show that $\mathrm{K}$ is indeed a right $\mathrm{D}$-comodule.
\endproof
\begin{remark}
All maps in the above proof are in fact morphisms of $\mathrm{C}$-$\mathrm{C}$-bicomodules, so the exact sequence in the statement of the lemma can be interpreted as an exact sequence of $\mathrm{C}$-$\mathrm{C}$-bicomodules. A similar proof shows that $\mathrm{K}$ is also a $\mathrm{D}$-$\mathrm{D}$ bicomodule.
\end{remark}
\begin{proposition}\label{NT-corresp}
Suppose $\mathrm{D}\overset{\iota}{\hookrightarrow}\mathrm{C}$ is a subcoalgebra.
Then the fully faithful embedding $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ sends $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ to a Serre subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ if, and only if, $\mathrm{D}$ is coidempotent.
\end{proposition}
\begin{proof}
If $\mathrm{D}$ is coidempotent, we have already shown in Lemma \ref{coidem-then-Serre} that $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ embeds as a Serre subcategory. It remains to show the converse.
Recall from Lemma \ref{I/I^2 is A/I-module} that we have a short exact sequence of right $\mathrm{C}$-comodules
$$
0\to \mathrm{D} \to \mathrm{D}_2 \to \mathrm{K} \to 0,
$$
with $\mathrm{D}_2=\mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})$. Furthermore, $\mathrm{D}$ and $\mathrm{K}$ are both right $\mathrm{D}$-comodules, meaning their $\mathrm{C}$-coaction map factors through $\mathrm{id}\circ_0\iota$, that is, $\mathrm{D}, \mathrm{K}$ are in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$.
Since a Serre subcategory is extension-closed, we obtain that $\mathrm{D}_2$ is in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$.
Note that $\mathrm{C}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}=\mathrm{D}_\mathrm{C} $ is the maximal subobject of $\mathrm{C}$ that belongs to the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$.
However, $\mathrm{D}_2$ is a subobject of $\mathrm{C}$ (the cokernel being the image of $\mu_\mathrm{J}$), so we deduce that $\mathrm{D}_2\cong \mathrm{D}$, i.e. $\mathrm{D}$ is coidempotent.
\end{proof}
\subsection{Coidempotent subcoalgebras and recollements}
We have now shown $\sc\mbox{C}\hspace{1.0pt}$-stable Serre subcategories of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ can be associated to a coidempotent subcoalgebra of $\mathrm{D}$.
It is natural to ask what the quotient $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})/\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ is, or how the results in the previous subsection fit into the framework of recollements.
\begin{lemma}\label{ker-comod}
Let $\mathrm{I}$ be an injective $\mathrm{C}$-comodule.
The following hold.
\begin{enumerate}[$($i$)$]
\item\label{ker-comod0} Let $\mathrm{M}$ be a simple $\mathrm{C}$-comodule with injective hull $\mathrm{Q}$, then $[\mathrm{I},\mathrm{M}]=0$ if and only if $\mathrm{Q}$ is not in $\mathrm{op}eratorname{add}\{\mathrm{F}\mathrm{I}\mid\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$.
\item\label{ker-comod1} The full subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ given by the $\mathrm{C}$-comodules $\mathrm{M}$ with $[\mathrm{I},\mathrm{M}]=0$ is equivalent to $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ for some coidempotent subcoalgebra $\mathrm{D}$ of $\mathrm{C}$.
\item\label{ker-comod2} Let $\mathrm{D}$ be the subcoalgebra of $\mathrm{C}$ given in \eqref{ker-comod1}, and $\mathrm{M}$ a simple $\mathrm{D}$-comodule.
Then $\mathrm{M}\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}$ is a simple $\mathrm{C}$-comodule whose injective hull is not in $\mathrm{op}eratorname{add}\{\mathrm{F}\mathrm{I}\mid\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}\subset \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$.\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{ker-comod0}: By the defining property of internal homs, $[\mathrm{I},\mathrm{M}]=0$ is equivalent to $\mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{M},\mathrm{F}\mathrm{I})\cong\mathrm{Hom}_{\underline{\ccC}}([\mathrm{I},\mathrm{M}],\mathrm{F})=0$ for all $\mathrm{F}\in \underline{\cC}$.
This is the same as saying that $\mathrm{M}$ is not in the socle of any of the object in $\mathrm{op}eratorname{add}\{\mathrm{F}\mathrm{I}\mid \mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$.
\eqref{ker-comod1}: Let $\mathrm{A}$ be the coalgebra $1$-morphism given by $[\mathrm{I},\mathrm{I}]$. Then $[\mathrm{I},-]:\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})\to\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})$ is exact by Lemma \ref{12.8}. The full subcategory in the claim is then the kernel of an exact functor, hence a Serre subcategory. This subcategory is clearly $\sc\mbox{C}\hspace{1.0pt}$-stable as $[\mathrm{I},-]$ is a morphism of 2-representations.
Now it follows from Lemma \ref{realise-subcat} that this category is equivalent to $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ for some subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, and $\mathrm{D}$ being coidempotent follows from Lemma \ref{NT-corresp}.
\eqref{ker-comod2}: Since $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ embeds (via $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$) as a Serre subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$, $\mathrm{M}\square_{\mathrm{D}}\mathrm{D}_\mathrm{C}$ is a simple $\mathrm{C}$-comodule. By the defining property of this Serre subcategory, $[\mathrm{I}, \mathrm{M}\square_\mathrm{D}\mathrm{D}_\mathrm{C}]=0$, and the claim follows from \eqref{ker-comod0}.
\end{proof}
For a subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, Lemma \ref{adjlemma} tells us that there is an adjoint triple
$$([{}_\mathrm{D}\mathrm{D}_\mathrm{C},-],-\square_\mathrm{D}\mathrm{D}_\mathrm{C} \cong [{}_\mathrm{C}\mathrm{D}_\mathrm{D},-],-\square_\mathrm{C}\mathrm{D})$$ between the comodule categories of these two coalgebras.
It follows from Lemma \ref{lemclosed} that $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is fully faithful.
On the other hand, if we pick an injective $\mathrm{C}$-comodule $\mathrm{I}$ and let $\mathrm{A}$ to be the coalgebra $1$-morphism given by $[\mathrm{I},\mathrm{I}]$, then we obtain another adjoint triple
$$([[\mathrm{I},\mathrm{C}],-],-\square_\mathrm{C} [\mathrm{I},\mathrm{C}]\cong [\mathrm{I},-],-\square_\mathrm{A}\mathrm{I})$$
between the comodule categories of $\mathrm{C}$ and $\mathrm{A}$.
Note that the middle isomorphisms follow from Lemma \ref{12.8}.
Moreover, $-\square_\mathrm{A}\mathrm{I}$ is fully faithful; one can see this by showing $[\mathrm{I},-]\circ (-\square_{\mathrm{A}}\mathrm{I})$ is naturally isomorphic to the identity functor on $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})$.
Indeed, as $[\mathrm{I},-]\cong -\square_{\mathrm{C}}[\mathrm{I},\mathrm{C}]$, the functor is naturally isomorphic to $-\square_{\mathrm{A}}\mathrm{I}\square_{\mathrm{C}}[\mathrm{I},\mathrm{C}] \cong -\square_{\mathrm{A}}[\mathrm{I},\mathrm{I}]=-\square_{\mathrm{A}}\mathrm{A} \cong \mathrm{Id}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})}$, where the first isomorphism uses Lemma \ref{Iinfromleft}.
Similarly, we show that $[\mathrm{I},[[\mathrm{I},\mathrm{C}],-]]\cong \mathrm{Id}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})}$ to demonstrate that $[[\mathrm{I},\mathrm{C}],-]$ is fully faithful. To this end, we compute, for all $\mathrm{M},\mathrm{M}'\in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})$, that
\begin{equation*}\begin{split}
\mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})}([\mathrm{I},[[\mathrm{I},\mathrm{C}],\mathrm{M}]], \mathrm{M}')&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})}([[\mathrm{I},\mathrm{C}],\mathrm{M}]], \mathrm{M}'\square_\mathrm{A} \mathrm{I})\\
&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})}(\mathrm{M}, \mathrm{M}'\square_\mathrm{A} \mathrm{I}\square_\mathrm{C}[\mathrm{I},\mathrm{C}])\\
&\cong \mathrm{Hom}_{\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})}(\mathrm{M}, \mathrm{M}'),\\
\end{split}\end{equation*}
where the last isomorphism uses the same argument as in the previous paragraph.
Suppose $\mathrm{C}$ and either one of $\mathrm{D}$ or $\mathrm{I}$ is given. We would like to understand when the two adjoint triples above defines a recollement
\begin{equation}\label{recollement}
\xymatrix{ \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})\ar^{-\square_\mathrm{D}\mathrm{D}_\mathrm{C}}[rr]&& \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})\ar^{[\mathrm{I},-]}[rr]\ar@<1ex>@/^/^{-\square_\mathrm{C}\mathrm{D}}[ll]\ar@<-2ex>@/_/_{[\mathrm{D},-]}[ll]&& \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A}).\ar@<1ex>@/^/^{-\square_\mathrm{A}\mathrm{I}}[ll]\ar@<-2ex>@/_/_{[[\mathrm{I},\mathrm{C}],-]}[ll]
}\end{equation}
of comodule categories. In other words, we ask under what conditions on $\mathrm{D}$ and $\mathrm{I}$, $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ embeds via $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ as a Serre subcategory and coincides with the kernel (category) of the exact functor $[\mathrm{I},-]$.
\begin{proposition}\label{recollcondition}
Suppose $\mathrm{D}\overset{\iota}{\hookrightarrow} \mathrm{C}$ is a coidempotent subcoalgebra, and $\mathrm{I}$ the injective hull in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ of the cokernel of $\iota$.
Then we have a recollement of the form \eqref{recollement}.
\end{proposition}
\proof
We already know from Proposition \ref{NT-corresp} that the essential image of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ under $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is a Serre subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. It remains to show that this coincides with the full subcategory consisting of $\mathrm{M}\in\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ such that $[\mathrm{I},\mathrm{M}]=0$.
It suffices to check that $[\mathrm{I},\mathrm{M}]=0 \in \mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{A})$ if and only if $\mathrm{M}\cong \mathrm{M}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}$. Furthermore, it suffices to check that these conditions are equivalent for simple objects $\mathrm{M}$.
Let $\mathrm{Q}$ be the injective hull of $\mathrm{M}$.
It follows from Lemma \ref{ker-comod}\eqref{ker-comod0} that $[\mathrm{I},\mathrm{M}]=0$ is equivalent to $\mathrm{Q}$ not being in $\mathrm{op}eratorname{add}\{\mathrm{F}\mathrm{I}\mid\mathrm{F}\in\sc\mbox{C}\hspace{1.0pt}\}$.
By Lemma \ref{kill-simple} this is then equivalent to $\mathrm{M}\square_\mathrm{C}\mathrm{D} \neq 0$.
As $\mathrm{M}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is a subcomodule of $\mathrm{M}$, the assumption of $\mathrm{M}$ being simple means that $\mathrm{M}\square_\mathrm{C}\mathrm{D}\neq 0$ is equivalent to $\mathrm{M}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}\cong \mathrm{M}$.
\endproof
\begin{proposition}\label{fix}
Let $\mathrm{C}$ be a coalgebra 1-morphism, and $\mathrm{I}$ an injective $\mathrm{C}$-comodule. There exists a subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, unique up to isomorphism, which is maximal with respect to $\mathrm{I}\square_\mathrm{C}\mathrm{D}=0$, and such that $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ is equivalent to the quotient $2$-representation $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})/\mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$. Furthermore, $\mathrm{D}$ is coidempotent.
\end{proposition}
\proof
Consider the exact sequence of $2$-representations
$$0 \to \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I}) \to \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C}) \overset{\pi}{\to} \bfK\to 0.$$
The construction in \cite[Section 3.2]{MMMZ} produces, for any full and dense morphism of $2$-representations, an embedding of a subcoalgebra, and this embedding is strict if and only if the full and dense morphism is not an equivalence.
Explicitly, in our situation, this construction defines the coalgebra $1$-morphism $\mathrm{D}$ via
$$\mathrm{Hom}_{\bfK}(\mathrm{C}, \mathrm{F}\mathrm{C})\cong \mathrm{Hom}_{\underline{\ccC}}(\mathrm{D}, \mathrm{F})$$
for all $1$-morphisms $\mathrm{F}$ in $\sc\mbox{C}\hspace{1.0pt}$ and produces an embedding $\iota:\mathrm{D}\hookrightarrow\mathrm{C}$.
Furthermore, $\bfK$ is equivalent to $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ and the full and dense morphism of $2$-representations $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})\twoheadrightarrow\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ corresponding to $\pi$ is given by $-\square_\mathrm{C}\mathrm{D}$ by \cite[Proposition 11]{MMMZ}.
We assert that $\mathrm{D}$ is maximal among subcoalgebras $\mathrm{B}$ of $\mathrm{C}$ with $\mathrm{I}\square_\mathrm{C}\mathrm{B}=0$. Firstly, note that we indeed have $\mathrm{I}\square_\mathrm{C}\mathrm{D}=0$ by exactness of
\begin{equation}\label{prop18ses}0\to \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})\to \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})\xrightarrow{-\square_\mathrm{C}\mathrm{D}}\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D}) \to 0.\end{equation}
Secondly, if $\mathrm{B}$ is another subcoalgebra of $\mathrm{C}$, strictly containing $\mathrm{D}$, then we obtain a full and dense morphism of $2$-representations $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{B})\twoheadrightarrow
\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ that is not an equivalence.
Hence the kernel of $-\square_\mathrm{C}\mathrm{B}$ would be strictly contained in the kernel of $-\square_\mathrm{C}\mathrm{D}$. By exactness of \eqref{prop18ses}, the latter ideal (of $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$) is the same as the ideal generated by $\mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$, so there must be some $\mathrm{Q}\in \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$ so that $\mathrm{Q}\square_\mathrm{C}\mathrm{B}\neq 0$.
But $\mathrm{Q}\square_\mathrm{C}\mathrm{B}$ is a direct summand of $\mathrm{F}\mathrm{I}\square_\mathrm{C}\mathrm{B}$, so we deduce that $\mathrm{I}\square_\mathrm{C}\mathrm{B}\neq 0$.
It remains to show that $\mathrm{D}$ is coidempotent.
We first claim that the essential image in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ of the fully faithful functor $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is closed under extensions.
Indeed, assume $\mathrm{M}_1,\mathrm{M}_2$ are in the full subcategory given by the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$. In particular, for $i=1,2$, we have $\mathrm{M}_i\cong \mathrm{M}_i\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}$, which implies that no composition factor of $\mathrm{M}_i$ is annihilated by $-\square_\mathrm{C}\mathrm{D}$. Consider an extension $0\to\mathrm{M}_1\to \mathrm{M}\to\mathrm{M}_2 \to 0$ in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$. Then also no composition factor of $\mathrm{M}$ is annihilated by $-\square_\mathrm{C}\mathrm{D}$, which implies that there is no map from $\mathrm{M}$ to any injective $\mathrm{Q}\in \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$.
We obtain a commutative diagram with exact rows
$$\xymatrix{
0 \ar[r]&\mathrm{M}_1\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C} \ar^{\sim}[d]\ar[r]&\mathrm{M}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}\ar@{^{(}->}[d]\ar[r]&\mathrm{M}_2\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C}\ar^{\sim}[d]&\\
0\ar[r]&\mathrm{M}_1\ar[r]&\mathrm{M}\ar[r]&\mathrm{M}_2\ar[r]&0
}$$
in $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$.
Let $q \colon \mathrm{M} \to \mathrm{N}$ be the cokernel map of the middle vertical map in the diagram and let
$$\xymatrix{
\mathrm{M} \ar@{^{(}->}^j[r]\ar@{->>}^{q}[d] & \mathrm{Q}_1 \ar^{g}[r]\ar^{q_1}[d] &\mathrm{Q}_2\ar^{q_2}[d] \\
\mathrm{N}\ar@{^{(}->}^{j'}[r]& \mathrm{Q}_1' \ar^{g'}[r] &\mathrm{Q}_2'
}$$
in be a lift of $q$ to an injective presentation. Since $q$ is annihilated by $-\square_\mathrm{C}\mathrm{D}$, the map $q_1 \square_\mathrm{C}\mathrm{D}$ factors over $g\square_\mathrm{C}\mathrm{D}$.
By fullness of $-\square_\mathrm{C}\mathrm{D}$, this implies that there already is a map $h\colon \mathrm{Q}_2\to \mathrm{Q}_1'$ such that setting $q_1'=q_1-hg$, we have $q_1' \square_\mathrm{C}\mathrm{D}=0$. Note that replacing $q_1$ by $q_1'$ and $q_2$ by $q_2-g'h$ defines another lift of $q$ to a map between injective presentations, so without loss of generality, we may assume that we already had $q_1 \square_\mathrm{C}\mathrm{D}=0$ by choosing $q_1$ appropriately. By exactness of \eqref{prop18ses}, this implies that
$q_1$ factors over an object $\mathrm{F}\mathrm{I}$ in $ \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$ and the first two columns of the diagram give rise to a commutative diagram
$$\xymatrix{
\mathrm{M} \ar@{^{(}->}^{j}[r]\ar@{->>}^{q}[dd] & \mathrm{Q}_1 \ar^{q_1}[dd] \ar[rd]& \\
&& \mathrm{F}\mathrm{I}\ar[ld]\\
\mathrm{N}\ar@{^{(}->}^{j'}[r]& \mathrm{Q}_1'.
}$$
Now the fact that there is no nonzero map from $\mathrm{M}$ to any object in $ \mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$ implies that $j'q=q_1j=0$ and by monicity of $j'$, we conclude that $q=0$. Therefore, the embedding $\mathrm{M}\square_\mathrm{C}\mathrm{D}\square_\mathrm{D}\mathrm{D}_\mathrm{C} \hookrightarrow \mathrm{M}$ is an isomorphism, meaning that $\mathrm{M}$ is in the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$.
This finishes the proof for the claim that the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ is extension-closed.
By Lemma \ref{lemclosed}, we already know that $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ embeds $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{D})$ as a full subcategory of $\mathrm{op}eratorname{comod}_{\underline{\ccC}}(\mathrm{C})$ that is closed under subobjects and quotients.
So the essential image of $-\square_\mathrm{D}\mathrm{D}_\mathrm{C}$ being also closed under extensions implies that it is a Serre subcategory.
Now the statement that $\mathrm{D}$ is coidempotent follows from Proposition \ref{NT-corresp}, whereas the uniqueness of $\mathrm{D}$ follows from Proposition \ref{cat-coalg-bij}.\endproof
\subsection{The main result.}
\begin{theorem}\label{thm-main}
\begin{enumerate}[$($i$)$]
\item\label{thm1.1}
Let $\mathrm{C}$ be a coalgebra $1$-morphism in $\underline{\sc\mbox{C}\hspace{1.0pt}}$ and suppose $\mathrm{D}$ is a coidempotent subcoalgebra of $\mathrm{C}$. Set $\mathrm{J}:=\mathrm{C}/\mathrm{D}$. Let $\mathrm{I}$ be the injective hull of $\mathrm{J}$ inside $\mathrm{op}eratorname{comod}_{\underline{\scc\mbox{C}\hspace{1.0pt}}}(\mathrm{C})$ and set $\mathrm{A}=[\mathrm{I},\mathrm{I}]$.Then we have a short exact sequence of $2$-representations
$$0 \longrightarrow \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{A}) \xrightarrow{-\square_\mathrm{A}\mathrm{I}} \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})\xrightarrow{-\square_\mathrm{C}\mathrm{D}}\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D}) \longrightarrow 0.$$
\item\label{thm1.2} Suppose $$0\longrightarrow\mathbf{N}\longrightarrow\bfM\longrightarrow\bfK\longrightarrow 0$$ is a short exact sequence of $2$-representations.
Then, choosing a coalgebra $1$-morphism $\mathrm{C}$ with $\bfM\cong \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$, there exists a subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, unique up to isomorphism, which is maximal with respect to $X\square_\mathrm{C}\mathrm{D}=0$ for all $X\in \coprod_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}} \mathbf{N}(\mathtt{i})$, and such that $\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{D})$ is equivalent to the quotient $2$-representation $\bfK$. Furthermore, $\mathrm{D}$ is coidempotent.
\end{enumerate}
\end{theorem}
\proof
\eqref{thm1.1} The assumption is precisely that of Proposition \ref{recollcondition}, so we obtain a recollement of the form \eqref{recollement}. Now the claim follows from Lemma \ref{recolllem}.
\eqref{thm1.2}
Choose $\mathrm{I}$ such that, under the equivalence $\bfM\cong \mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})$, the $2$-subrepresentation $\mathbf{N}$ corresponds to $\mathbf{G}_{\mathrm{op}eratorname{inj}_{\underline{\ccC}}(\mathrm{C})}(\mathrm{I})$. Then, for a subcoalgebra $\mathrm{D}$ of $\mathrm{C}$, we have $X\square_\mathrm{C}\mathrm{D}=0$ for all $X\in \coprod_{\mathtt{i}\in\scc\mbox{C}\hspace{1.0pt}} \mathbf{N}(\mathtt{i})$ if and only if $\mathrm{I}\square_\mathrm{C}\mathrm{D}=0$.
The claim now follows from Proposition \ref{fix}.
\endproof
\section{Examples}\label{sec:eg}
\subsection{Projective functors over dual numbers}
Let $R=\mathcal{B}bbk[x]/(x^2)$ be the ring of dual numbers.
Consider the $2$-category $\sc\mbox{C}\hspace{1.0pt}_R$ of projective functors on $R\text{-}\mathrm{mod}$, see e.g. \cite[Example 2]{MM1}.
More precisely, this is the $2$-category with one object $\mathtt{i}$, which we identify with a small category $\mathcal{R}$ equivalent to $R\text{-}\mathrm{mod}$, and $\sc\mbox{C}\hspace{1.0pt}_R(\mathtt{i},\mathtt{i})$ is the full subcategory of all endofunctors of $\mathcal{R}$ given by functors isomorphic to tensoring over $R$ with an $R$-$R$-bimodule in $\mathrm{op}eratorname{add}(R\mathrm{op}lus R\otimes_{\mathcal{B}bbk}R)$.
The $2$-category $\sc\mbox{C}\hspace{1.0pt}_R$ has two indecomposable $1$-morphisms $\mathbbm{1}$ and $\mathrm{F}$ corresponding to the identity functor and to tensoring with $R\otimes R$, respectively.
Since the principal $2$-representation $\mathbf{P}=\sc\mbox{C}\hspace{1.0pt}_R(\mathtt{i},-)$ is generated by $\mathbbm{1}$, we have a coalgebra $1$-morphism $\mathrm{C}_\mathbf{P}$ corresponding to $\mathbf{P}$ given by $[\mathbbm{1},\mathbbm{1}]$.
Using the fact that the underlying category of $\underline{\mathbf{P}}$ is precisely $\underline{\cC}_R$, one can see that $[\mathbbm{1},\mathbbm{1}]\cong\mathbbm{1}$ as an object in $\underline{\cC}_R(\mathtt{i},\mathtt{i})$ and comultiplication and counit are both the identity map. Note that this argument applies for any principal $2$-representation of a finitary $2$-category.
There are two simple transitive $2$-representations, denoted by $\mathbf{C}_{{\mathcal{L}}}, \mathbf{C}_{\mathbbm{1}}$, up to equivalence (see \cite{MM5} for details). Here
$\mathbf{C}_{\mathbbm{1}}$ is the trivial $2$-representation, whose underlying category is equivalent to $\mathcal{B}bbk\text{-}\mathrm{mod}$, where $\mathrm{F}$ acts by annihilating everything. On the other hand, $\mathbf{C}_{{\mathcal{L}}}$ is the natural $2$-representation, whose underlying category is equivalent to $R\text{-}\mathrm{proj}$, where $\mathrm{F}$ acts as $R\otimes_{\mathcal{B}bbk}R\otimes_R-$.
It follows from \cite[Theorem 22]{MMMT} that the as an object of $\underline{\cC}_R(\mathtt{i},\mathtt{i})$, the coalgebra $1$-morphism $\mathrm{C}_{\mathcal{L}}:=[R,R]$ is isomorphic to $\mathrm{F}$.
For $\mathbf{C}_{\mathbbm{1}}$, the corresponding coalgebra $1$-morphism $\mathrm{C}_\mathbbm{1}:=[\mathcal{B}bbk,\mathcal{B}bbk]$ can be calculated via the defining adjunction isomorphisms $\mathrm{Hom}_{\underline{\ccC}_R}(\mathrm{C}_\mathbbm{1},\mathrm{G})\cong \mathrm{Hom}_{\underline{\mathbf{C}}_\mathbbm{1}(\mathtt{i})}(\mathcal{B}bbk,\mathrm{G}\mathcal{B}bbk)$ for all indecomposable $1$-morphisms $\mathrm{G}$, which yield that it is is isomorphic to the simple socle $L_\mathbbm{1}$ of $\mathbbm{1}$ in $\underline{\cC}_R(\mathtt{i},\mathtt{i})$.
In fact, $\mathbf{C}_{\mathbbm{1}}$ is a quotient $2$-representation of $\mathbf{P}$, so $\mathrm{C}_\mathbbm{1}$ is a subcoalgebra of $\mathrm{C}_\mathbf{P}$, which implies that the counit and comultiplication maps
defining $\mathrm{C}_\mathbbm{1}$ are both the identity map on $L_\mathbbm{1}$.
There is a short exact sequence of finitary $2$-representations
\[
0 \to \mathbf{C}_{{\mathcal{L}}}\boxtimes \mathcal{A} \to \mathbf{P} \to \mathbf{C}_{\mathbbm{1}} \to 0,
\]
where $\mathbf{C}_{{\mathcal{L}}}\boxtimes \mathcal{A}$ is the inflation of $\mathbf{C}_{{\mathcal{L}}}$ by $\mathcal{A}:=R\text{-}\mathrm{proj}$ (see \cite{MM6} for details about inflations). Computing coalgebra $1$-morphism corresponding to $\mathbf{C}_{{\mathcal{L}}}\boxtimes \mathcal{A}$ via the defining adjunction isomorphism $\mathrm{Hom}_{\underline{\ccC}}([R\otimes R,R\otimes R],\mathrm{F})\cong \mathrm{Hom}_{\underline{\mathbf{C}_{{\mathcal{L}}}\boxtimes \mathcal{A}}}(R\otimes R,\mathrm{F}(R\otimes R))$ shows that its underlying object in $\underline{\cC}_R(\mathtt{i},\mathtt{i})$ is isomorphic to $\mathrm{F}\mathrm{op}lus \mathrm{F}$.
Thus, the coalgebra $1$-morphisms $\mathrm{D}, \mathrm{C}$ in Theorem \ref{thm-main} corresponding to the above short exact sequence are $\mathrm{C}_\mathbbm{1}$, $\mathrm{C}_\mathbf{P}$.
Since the quotient of $\mathbbm{1}$ by $L_\mathbbm{1}$ has simple socle $L_\mathrm{F}$, the coalgebra $1$-morphism is $\mathrm{A}=[\mathrm{F},\mathrm{F}]$ and it has underlying object in $\underline{\cC}_R(\mathtt{i},\mathtt{i})$ given by $\mathrm{F}\mathrm{op}lus \mathrm{F}$.
Let us look at another finitary $2$-representation $\bfM$, whose underlying category is equivalent to ${\mathcal{L}}ambda\text{-}\mathrm{proj}$, where ${\mathcal{L}}ambda:=\mathrm{End}_R(R\mathrm{op}lus \mathcal{B}bbk)$ (the action of $\sc\mbox{C}\hspace{1.0pt}_R$ is naturally induced by that on the natural $2$-representation $\mathbf{C}_{\mathcal{L}}$).
There is a short exact sequence of
\[
0 \to \mathbf{C}_{{\mathcal{L}}} \to \bfM \to \mathbf{C}_{\mathbbm{1}} \to 0.
\]
We already explained that the coalgebra $1$-morphisms $\mathrm{A},\mathrm{D}$ corresponding to the first and last term, respectively, are $\mathrm{F}$ and $L_\mathbbm{1}$.
It is possible to calculate the object in $\underline{\cC}_R(\mathtt{i},\mathtt{i})$ underlying the coalgebra $1$-morphism $\mathrm{C}_\bfM$ corresponding to $\bfM$ as follows.
First note that there is a quotient morphism $\mathbf{P}\to \bfM$, so we can take $\mathrm{C}_\bfM$ is to be the subcoalgebra of $\mathrm{C}_\mathbf{P}$ given by $[M,M]$ with $M$ being the underlying object of $\mathrm{C}_\bfM$ in $\underline{\cC}=\mathbf{P}(\mathtt{i})$.
Note that the internal Hom used here is taken in $\mathbf{P}$ (instead of $\bfM$).
The underlying object of $\mathrm{C}_\mathbf{P}$ is the injective object $\mathbbm{1}$ of $\underline{\cC}_R(\mathtt{i}, \mathtt{i})$, and $\mathbbm{1}$ turns out to be uniserial with four composition factor, $L_\mathrm{F}, L_\mathbbm{1}, L_\mathrm{F}, L_\mathbbm{1}$ from top to socle, where $L_\mathrm{F}$ is the simple socle of $\mathrm{F}$.
By the above short exact sequence, $\mathrm{C}_\mathbbm{1}$ is coidempotent subcoalgebra of $\mathrm{C}_\bfM$ and $\mathrm{C}_\mathbbm{1}\ncong \mathrm{C}_\bfM$, so the underlying object is not $L_\mathbbm{1}$.
Since $\mathrm{C}_\mathbf{P} \ncong \mathrm{C}_\bfM$, the underlying object $M$ of $\mathrm{C}_\bfM$ can only be either the length 2 or the length 3 subobject of $\mathbbm{1}$.
These two objects can be distinguished by the dimension of the Hom-space of maps to $\mathbbm{1}$ in $\underline{\cC}$ - they are of dimension 1 and 2 respectively.
By construction, $M\cong [M,M]=\mathrm{C}_\bfM$ as objects in $\mathbf{P}(\mathtt{i})$ (c.f. Proof of Lemma \ref{realise-subcat}) and $\mathrm{Hom}_{\underline{\ccC}}(\mathrm{C}_\bfM,\mathbbm{1})\cong \mathrm{Hom}_{\underline{\mathbf{P}}}(M, M)$.
Note that $\mathrm{Hom}_{\underline{\mathbf{P}}}(M,M)\cong \mathrm{Hom}_{\underline{\bfM}}(M,M)$ as $\underline{\bfM}(\mathtt{i})\to\underline{\mathbf{P}}(\mathtt{i})$ is a fully faithful embedding.
We claim that $ \mathrm{Hom}_{\underline{\bfM}}(M,M)\cong \mathcal{B}bbk$; in which case, we can conclude that $M$ is the subobject of $\mathbbm{1}$ of length 2.
Indeed, under the equivalence between the underlying category of $\bfM$ and ${\mathcal{L}}ambda\text{-}\mathrm{proj}$, $M$ corresponds to the indecomposable projective ${\mathcal{L}}ambda$-module $P$ such that $\mathrm{F}(P)\notin \mathrm{op}eratorname{add}(P)$. By the construction of 2-representation structure on ${\mathcal{L}}ambda\text{-}\mathrm{proj}$, $\mathrm{F}(P')\in\mathrm{op}eratorname{add} (P')$ for an indecomposable projective ${\mathcal{L}}ambda$-module $P'$ if and only if $P'\cong\mathrm{Hom}_R(R\mathrm{op}lus \mathcal{B}bbk, R)$. So we have $P\cong \mathrm{Hom}_R(R\mathrm{op}lus \mathcal{B}bbk, \mathcal{B}bbk)$, which is a uniserial module with a 1-dimensional endomorphism ring; this means that $\mathrm{Hom}_{\underline{\bfM}}(M,M)=\mathcal{B}bbk$, as claimed.
Note that the comultiplication and counit maps defining $\mathrm{C}_\bfM$ are both identity, since $\mathrm{C}_\bfM$ is a subcoalgebra of $\mathrm{C}_\mathbf{P}$.
Let us summarise, for clarity, the object underlying each coalgebra $1$-morphism corresponding to the $2$-representations mentioned, in the table below.
\[
\begin{array}{ccccc}
\mathbf{P} & \mathbf{C}_\mathbbm{1} & \mathbf{C}_{\mathcal{L}} & \mathbf{C}_{\mathcal{L}}\boxtimes \mathcal{A} & \bfM \\
\hline
\mathbbm{1} & L_\mathbbm{1} & \mathrm{F} & \mathrm{F}\mathrm{op}lus \mathrm{F} & \begin{array}{c} L_\mathrm{F} \\ L_\mathbbm{1} \end{array}
\end{array}
\]
\subsection{Triangular coalgebras}
In this section, we provide a more general class of examples. Let $\sc\mbox{C}\hspace{1.0pt}$ be a weakly fiat $2$-category, $\mathrm{A}$ and $\mathrm{D}$ coalgebra $1$-morphisms in $\sc\mbox{C}\hspace{1.0pt}$ with comultiplications and counits given by $\mu_\mathrm{A},\epsilon_\mathrm{A}$ and $\mu_\mathrm{D},\epsilon_\mathrm{D}$, respectively, and ${}_\mathrm{A}\mathrm{M}_\mathrm{D}$ a $\mathrm{A}$-$\mathrm{D}$-bicomodule with left and right coactions $\lambda$ respectively $\rho$. We define a coalgebra structure $\mu_\mathrm{C},\epsilon_\mathrm{C}$ on $\mathrm{C}:=\mathrm{A}\mathrm{op}lus\mathrm{D}\mathrm{op}lus\mathrm{M}$ by specifying that
\begin{itemize}
\item the restriction of $\mu_\mathrm{C}$ to $\mathrm{X}$ is $\mu_\mathrm{X}$ for $\mathrm{X}\in\{\mathrm{C},\mathrm{D}\}$,
\item the restriction of $\epsilon_\mathrm{C}$ to $\mathrm{X}$ is $\epsilon_\mathrm{X}$ for $\mathrm{X}\in\{\mathrm{C},\mathrm{D}\}$,
\item the restriction of $\mu_\mathrm{C}$ to $\mathrm{M}$ is $\left(\begin{array}{c}\lambda\\ \mu\end{array}\right)\colon \mathrm{M}\to \mathrm{A}\mathrm{M}\mathrm{op}lus \mathrm{M}\mathrm{D}$, and
\item the restriction of $\epsilon_\mathrm{C}$ to $\mathrm{M}$ is zero.
\end{itemize}
It is straightforward to check that this indeed defines a coalgebra structure on $\mathrm{C}$.
We claim that $\mathrm{D}$ is a coidempotent subcoalgebra. It is a subcoalgebra by definition, so we need to check that $\mu_\mathrm{C}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C}) = \mathrm{D}$. Now we have $$\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C}\cong \mathrm{A}\mathrm{D}\mathrm{op}lus \mathrm{D}\mathrm{D}\mathrm{op}lus \mathrm{M}\mathrm{D} + \mathrm{D}\mathrm{A} \mathrm{op}lus\mathrm{D}\mathrm{M}\mathrm{op}lus \mathrm{D}\mathrm{D}.$$
As none of $\mathrm{A}\mathrm{D}, \mathrm{D} \mathrm{A}, \mathrm{D}\mathrm{M}$ is in the range of $\mu_\mathrm{C}$, we have $\mu_\mathrm{C}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})=\mu_\mathrm{C}^{-1}(\mathrm{D}\mathrm{D}\mathrm{op}lus \mathrm{M}\mathrm{D})$.
Since $\mu_\mathrm{C}$ sends $\mathrm{M}$ to $\mathrm{A}\mathrm{M}\mathrm{op}lus \mathrm{M}\mathrm{D}$, but only $\mathrm{M}\mathrm{D}$ is a direct summand of $\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C}$, we get that
$$\mu_{\mathrm{C}}^{-1}(\mathrm{D}\mathrm{D}\mathrm{op}lus \mathrm{M}\mathrm{D})=\mu_\mathrm{D}^{-1}(\mathrm{D}\mathrm{D})\mathrm{op}lus \left(\lambda^{-1}(0)\cap \rho^{-1}(\mathrm{M}\mathrm{D})\right).$$
It follows from the construction that $\mu_\mathrm{D}^{-1}(\mathrm{D}\mathrm{D})=\mathrm{D}$, whereas $\lambda^{-1}(0)=0$ due to $\lambda$ being mono, so we obtain $\mu_{\mathrm{C}}^{-1}(\mathrm{C}\mathrm{D}+\mathrm{D}\mathrm{C})=\mathrm{D}$, i.e. $\mathrm{D}$ is coidempotent as claimed.
\end{document} |
\begin{document}
\date{ }
\title{Fast Low-Rank Matrix Estimation without the Condition Number}
\begin{abstract}
In this paper, we study the general problem of optimizing a convex function $F(L)$ over the set of $p \times p$ matrices, subject to rank constraints on $L$. However, existing first-order methods for solving such problems either are too slow to converge, or require multiple invocations of singular value decompositions. On the other hand, factorization-based non-convex algorithms, while being much faster, require stringent assumptions on the \emph{condition number} of the optimum. In this paper, we provide a novel algorithmic framework that achieves the best of both worlds: asymptotically as fast as factorization methods, while requiring no dependency on the condition number.
We instantiate our general framework for three important matrix estimation problems that impact several practical applications; (i) a \emph{nonlinear} variant of affine rank minimization, (ii) logistic PCA, and (iii) precision matrix estimation in probabilistic graphical model learning. We then derive explicit bounds on the sample complexity as well as the running time of our approach, and show that it achieves the best possible bounds for both cases. We also provide an extensive range of experimental results, and demonstrate that our algorithm provides a very attractive tradeoff between estimation accuracy and running time.
\end{abstract}
\section{Introduction}
\lambdabel{sec:intro}
In this paper, we consider the following optimization problem:
\begin{align}
\underset{L}{\text{min}}
&~~F(L) \lambdabel{opt_prob} \\
\text{s.t.}
&~~\text{rank}(L)\leq r^*, \Big{\|}onumber
\end{align}
where $F(L):{\mathbb{R}}^{p\times p}\rightarrow{\mathbb{R}}$ is a convex smooth function defined over matrices $L\in{\mathbb{R}}^{p\times p}$ with rank $r^*\ll p$.\footnote{For convenience, all our matrix variables will be of size $p \times p$, but our results extend seamlessly to rectangular matrices.}
This problem has recently received significant attention in machine learning, statistics, and signal processing~\cite{chen2015fast,udell2016generalized}. Several applications of this problem abound in the literature, including affine rank minimization~\cite{recht2010guaranteed,tu2016low,jain2010guaranteed}, matrix completion~\cite{candes2009exact}, and collaborative filtering~\cite{jain2013low}.
In most of the above applications, $F(L)$ is typically assumed to be a smooth, quadratic function (such as the squared error). For instance, in machine learning, the squared loss between the pair of observed and predicted outputs would be a natural choice, and indeed most of the works in the matrix estimation literature focus on optimizing such functions.
But there are many cases in which other loss functions are used. For example, in neural network learning, the loss function is usually chosen according to the negative cross-entropy between the distributions of the fitted model and the training samples~\cite{Goodfellow-et-al-2016}.
As another example, in graphical model learning, the goal is usually to estimate the covariance/precision matrix. In this case, the negative log-likelihood function is an appropriate choice. As a third example, in the problem of one-bit matrix completion~\cite{davenport20141} or logistic PCA~\cite{park2016finding}, $F(L)$ is modeled, again, as the log-likelihood of the observations.
From the computational perspective, the traditional approach is to adopt first-order optimization for solving \eqref{opt_prob}. Several different approaches (with theoretical guarantees) have been proposed in recent years. The first group of these methods are related to the convex methods in which the rank constraint is relaxed by the nuclear norm proxy~\cite{fazel2002matrix}, resulting the overall convex problem which can be solved by off-the-shelf solvers. While these methods achieve the best sample complexity, i.e., the minimum required number of samples for achieving the small estimation error, they are computationally expensive and the overall running time can be slow if $p$ is very large.
To alleviate this issue, several non-convex methods have been proposed based on using non-convex regularizers. Non-convex iterative methods~\cite{jain2010guaranteed,jain2014iterative,quanming2017large} typically require less computational complexity per iteration. On the other hand, factorized gradient methods~\cite{chen2015fast,bhojanapalli2016dropping,tu2016low,wang2017unified} are computationally very appealing since they reduce the number of variables from $p^2$ to $pr$ by writing $L$ as $L=UV^T$ where $U,V\in{\mathbb{R}}^{p\times r}$ and $r\ll p$, and removing the rank constraint from problem~\eqref{opt_prob}.
However, the overwhelming majority of existing methods suffer from one or several of the following problems: their convergence rate is slow (typically, sublinear); the computational cost per iteration is high, involving invocations of singular value decomposition; or they have stringent assumptions on the {spectral properties} (such as the condition number) of the solution to~\eqref{opt_prob}.
Our goal in this paper is to propose an algorithm to alleviate the above problems simultaneously. Specifically, we seek an algorithm that exhibits:
(i) \emph{linearly} fast convergence,
(ii) \emph{computationally efficient} per iteration,
(iii) works for a \emph{broad} class of loss functions, and
(iv) \emph{robust} to effects such as matrix condition number.
\subsection{Our contributions}
In this paper, we propose a general \emph{non-convex} algorithmic framework, that we call \emph{MAPLE}, for solving problems of the form~\eqref{opt_prob} for objective functions that satisfy the commonly-studied \emph{Restricted Strongly Convex/Smooth} ({RSC/RSS}) conditions. Our algorithmic approach enjoys the following benefits:
{\textit{\textbf{Linear convergence.}} We provide rigorous analysis to show that our proposed algorithm enjoy a linear convergence rate (no matter how it is initialized).
{\textit{\textbf{Fast per-iteration running time}}}. We provide rigorous analysis to show that our algorithm exhibits fast per-iteration running time. Our method (per-iteration) leverages recent advances in randomized low-rank approximation methods, and their running time is close to optimal for constant $r$\footnote{Our approach is akin to the previous work of~\cite{becker2013randomized}, but strictly improves upon this approach in terms of sample complexity.}.
{\textit{\textbf{No limitations on strong convexity/smoothness constants.}} In a departure from the majority of the matrix optimization literature, our algorithm succeeds under no particular assumptions on the \emph{extent} to which the objective function $F$ is strongly smooth/convex.
(These are captured by properties known as \emph{restricted strong convexity} and \emph{smoothness}, which we elaborate below.)
{\textit{\textbf{No dependence on matrix condition numbers.}} In contrast with several other results in the literature, our proposed algorithm does not depend on stringent assumptions on the condition number (i.e., the ratio of maximum to minimum nonzero singular values) of the solution to~\eqref{opt_prob}.
{\textit{\textbf{Instantiation in applications.}} We instantiate our MAPLE framework to three important and practical applications; nonlinear affine rank minimization, logistic PCA, and precision matrix estimation in probabilistic graphical model learning.
\subsection{Techniques}
\lambdabel{ourtechnique}
Our approach is an adaptation of the algorithm proposed in~\cite{jain2014iterative}. That is an iterative approach that alternates between taking a gradient descent step and thresholding the largest singular values of the optimization variable. The key idea of that work is that each gradient update is projected onto the space of matrices with rank $r$ that is \emph{larger} than $r^*$, the rank parameter in Problem~\eqref{opt_prob}. This trick can greatly alleviate situations where the objective function exhibits poor restricted strong convexity/smoothness properties; more generally,
the overall algorithm can be applied to ill-posed problems. However, their algorithm requires performing a full exact singular value decomposition (SVD) after each gradient descent step. This results in poor overall running time, as the per-iteration cost is \emph{cubic} (${\mathcal{O}}(p^3)$) in the matrix dimension\footnote{Here, one may argue that the running time of the approach in~\cite{jain2014iterative} and the other IHT-type algorithms take ${\mathcal{O}}(p^2r)$ time using truncated SVD (via power iteration or similar). Unfortunately, this is not technically true and seems to be a common misconception in several low-rank matrix recovery papers. Finding a truncated SVD of a matrix only takes ${\mathcal{O}}(p^2r)$ time if the input matrix is exactly rank $r$; more generally, the running time of power method-like iterative approaches scales as ${\mathcal{O}}(p^2r/gap)$ where the denominator denotes the ratio of the $r^{th}$ and $(r+1)^{th}$ singular values which can be very small, and consequently inflates the running time to ${\mathcal{O}}(p^3)$ time.}.
Our method resolves this issue by replacing the exact SVD with a gap-independent \emph{approximate} low-rank projection, while still retaining the idea of projecting onto a larger space. To establish soundness of our approach, we establish a property about (approximate) singular value projections, extending recent new results from non-convex optimization ~\cite{shen2016tight,li2016Nonconvex}. In particular, we prove a new structural result for an $\epsilon$-approximate projection onto the space of rank-$r$ matrices. We prove that such an approximate projection is \emph{nearly non-expansive}, and therefore enjoys similar convergence guarantees as convex projected gradient descent.
To be more precise, we know that for any matrix $A$ and rank-$r'$ matrix $B$, the best rank-$r'$ approximation of $A$, denoted by $H_{r'}(A)$ satisfies the following:
$$ \|H_{r'}(A) - B\|_F\leq2\|A - B\|_F,$$
This bound is very loose (following a simple application of the triangle inequality) and the upper bound is, in fact, never achieved~\cite{shen2016tight}. We prove that the approximation factor 2 can be sharpened to close to $1$ if we use a rank parameter that is sufficiently larger than $r'$. In particular, if $\widetildea$ is an $\epsilon$-approximate singular value projection operator, we prove that:
\begin{align*}
\|\widetildea(A) - B\|_F^2\leq\left(1+\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r'}}{\sqrt{r-r'}}\right)\|A - B\|_F^2,
\end{align*}
where $r>r'$, $\text{rank}(B) = r'$ and $\widetildea$ implements an $\epsilon$-approximate projection onto the set of matrices with rank-$r$. Therefore, by increasing $r$, we (nearly) recover the non-expansivity property of projection, and this helps prove strong results about our proposed projected gradient descent scheme.
Integrating the above result into (projected) gradient descent gives linear convergence of the proposed algorithm for a very broad class of objective functions. Since we use approximate low-rank projections, the running time of the projection step is (almost) linear in the size of the matrix if $r^*$ is sub-linear in $n$.
\subsection{Stylized applications}
We also instantiate our MAPLE framework to three applications of practical interest.
First, we consider a problem that we call \emph{nonlinear affine rank minimization} (NLARM). Formally, we consider an observation model akin to the Generalized Linear Model (GLM)~\cite{kakade2011}:
$$
y = g(\mathcal{A} (L^*)) + e,
$$
where $g$ denotes a nonlinear \emph{link} function, $\mathcal{A}$ denotes a linear observation operator, which we formally define later, and $e\in{\mathbb{R}}^m$ denotes an additive noise vector. The goal is to reconstruct $L^*$ from $y$, given that $L^*$ is of rank at most $r^*$. For this application, we derive the sample complexity of our algorithm, calculate the running time, and analyze statistical error rates. More specifically, we define an specific objective function tailored to $g$ and verify that it is strongly convex/smooth; moreover, we show that $\widetilde{O}(pr^*)$ samples is enough to estimate $L^*$ up to the noise level, and this matches those of the best available methods.
In addition, the running time required to estimate $L^*$ scales as ${\mathcal{O}}tilde(p^2 r^*)$, which is nearly linear with the size of $L^*$ and independent of all other spectral properties of $L^*$ (such as its condition number). This marks a strict improvement over all other comparable existing methods.
Second, we discuss the problem of \emph{logistic PCA}~\cite{park2016finding} in which we observe a binary matrix $Y$ with entries belonging to ${\{}0, 1{\}}$ such that the likelihood of each $Y_{ij}$ is given by $P(Y_{ij} = 1|L_{ij}) = \sigmagma(L^*_{ij})$ where $\sigmagma(x) = \frac{1}{1+\exp(-x)}$ is a sigmoidal nonlinearity. The goal is to estimate an underlying low-rank matrix $L^*$ by trying to find the solution of following optimization problem:
$$F(L) = -\sum_{i,j}\big{(}Y_{ij}\log\sigmagma(L_{ij}) + (1-Y_{ij})\log(1-\sigmagma(L_{ij}))\big{)}.$$ Again, we show how to use our framework to solve this problem with nearly linear running time.
Third, we instantiate our framework in the context of \emph{precision matrix estimation} in probabilistic graphical models. Specifically, the goal is to estimate a low-rank precision matrix $L^*$ based on observed samples $X_i\in{\mathbb{R}}^p$ for $i=1,\ldots,n$. In this setup, the objective function $F(L)$ is given by the negative log likelihood of the observed samples.
We show that with $n = {\mathcal{O}}(pr)$ independent samples, the proposed algorithm returns an estimate up to constant error, and once again, our algorithm exhibits nearly linear running time, independent of how poorly the underlying precision matrix is conditioned. Moreover, we show that the our algorithm provide the best empirical performance (in terms of estimation error) among available competing methods.
\section{Prior Work}
\lambdabel{sec:prior}
Optimization problems with rank constraints arise in several different applications; examples include robust PCA~\cite{candes2011rpca,Venkat2009sparse,netrapalli2014non,yi2016fast}, precision matrix estimation using graphical models~\cite{hsieh2014quic,Venkat2010latent}, phase retrieval~\cite{candes2015phase,candes2013phaselift,netrapalli2013phase}, finding the square root of a PSD matrix~\cite{jain2015computing}, dimensionality reduction techniques~\cite{johnson2014logistic,schein2003generalized}, video denoising~\cite{ji2010robust}, subspace clustering~\cite{liu2013robust}, face recognition~\cite{yang2017nuclear} and many others. We only provide a subset of relevant references here; please refer to the recent survey~\cite{davenport2016overview} and references therein for a more comprehensive discussion.
In general, most optimization approaches to solve \eqref{opt_prob} can be categorized in four groups. In the first group of approaches, the non-convex rank constraint is relaxed into a \emph{nuclear norm} penalty, which results in a convex problem and can be solved by off-the-shelf solvers such as SDP solvers~\cite{cvx}, singular value thresholding and its accelerated versions~\cite{recht2010guaranteed,cai2010singular,GoldsteinStuderBaraniuk:2014}, and active subspace selection methods~\cite{hsieh2014nuclear}. While convex methods are well-known, their usage in the high dimensional regime is prohibitive (incurring cubic, or worse, running time).
The second group of approaches includes non-convex methods, replacing the rank constraint with a more tractable \emph{non-convex} regularizer instead of the nuclear norm.
These include regularization with the smoothly clipped absolute deviation (SCAD)~\cite{fan2001variable}, and iteratively re-weighted nuclear norm (IRNN) minimization~\cite{lu2016nonconvex}.
While these approaches can reduce the computational cost per iteration, from $p^3$ to $p^2r$,
they exhibit sub-linear convergence, and are quite slow in high dimensional regimes; see~\cite{quanming2017large} for details.
The third group of approaches try to solve the non-convex optimization problem~\eqref{opt_prob} based on the factorization approach of~\cite{burer2003nonlinear}. In these algorithms, the rank-$r$ matrix $L$ is factorized as $L = UV^T$, where $U,V\in{\mathbb{R}}^{p\times r}$. Using this idea removes the difficulties caused by the non-convex rank constraint; however, the objective function is not convex anymore. Nevertheless, under certain conditions, such methods succeed and have recently gained in popularity in the machine learning literature, and several papers have developed provable linear-convergence guarantees for both squared and non-squared loss functions
~\cite{tu2016low,bhojanapalli2016dropping,park2016non,chen2015fast,zheng2015convergent,jain2013low}.
Such methods are currently among the fastest available in terms of running time. However, a major drawback is that they may require a careful spectral initialization that usually involves one or multiple full singular value decompositions (SVDs).
To our knowledge, only three recent works in the matrix recovery literature require no full SVDs for their initialization: \cite{bhojanapalli2016global,ge2016matrix,ge2017no}. However,~\cite{bhojanapalli2016global} only discusses about the linear matrix sensing problem, and it only applies to the squared loss which requires that sensing matrix satisfies RIP condition, while our stylized application is for general non-squared loss functions with no assumption for the upper bound of $\frac{M}{m}$ (the ratio of RSS to RSC constant). Also, \cite{ge2016matrix,ge2017no} makes stringent assumptions on the coherence and other spectral properties of ground-truth matrix. For example, the running time of saddle-avoiding local search algorithm used in~\cite{ge2017no} shows polynomial dependency on the condition number (i.e., the ratio of the largest to the smallest non-zero singular values). Furthermore, the instantiation to the linear matrix sensing problem shows strict upper bound on the RIP constant. As a result, the convergence rate depends heavily on the condition number as well as other spectral properties of the optimum. Hence, if the problem is somehow poorly conditioned, their sample complexity and running time can blow up by a significant amount.
The fourth class of methods also includes non-convex methods. Unlike the factorized methods, they do not factorize the optimization variable, $L$, but instead use low-rank projections within classical gradient descent. This approach, also called singular value projection (SVP) or iterative hard thresholding, was introduced by~\cite{jain2010guaranteed} for matrix recovery from linear measurements, and was later modified for general M-estimation problems with well-behaved objective functions~\cite{jain2014iterative}. These methods require multiple invocations of exact singular value decompositions (SVDs). While their computational complexity can be cubic in $p$ (see footnote in section~\ref{ourtechnique}), and consequently very slow in very large-scale problems, these methods do not depend on the condition number of the optimum, and in this sense are more robust than factorized methods.
A similar algorithm to SVP as proposed by~\cite{becker2013randomized} for the squared loss case, which replaces the exact SVD with an approximate one. However, their theoretical guarantees is very restrictive which overshadows any advantage of using an approximate SVD algorithm instead of an exact SVD. That is, in the regime of optimal sample complexity, i.e., $n={\mathcal{O}}(pr)$, their approximate projection should be applied onto a matrix with rank as the order of $p$ in order to have convergence. Furthermore, while the idea of projecting onto the larger set is theoretically backed up in~\cite{jain2014iterative}, and also in this paper, its usage within the factorized approach has been shown to obtain practical improvements; however, currently there is no theory for this~\cite{bhojanapalli2016dropping}.
In addition to the above algorithms, \emph{stochastic} gradient methods for low-rank matrix recovery have also been investigated~\cite{wang2017universal,li2016Nonconvex}. The goal of these methods are to reduce the cost of calculating the full gradient in each iteration which typically requires ${\mathcal{O}}(np^2)$ operations. For instance, \cite{wang2017universal} has combined the factorized method with SVRG~\cite{johnson2013accelerating}, while the authors in~\cite{li2016Nonconvex} have used the SVP algorithm along with SVRG or SAGA~\cite{defazio2014saga} algorithms. However, these algorithms suffer from either heavy computational cost due to the initialization and projection step, or assume stringent conditions on the RSC/RSS conditions. Similar to the factorized method proposed in~\cite{tu2016low}, the method in~\cite{wang2017universal} requires multiple SVDs for the initialization step, and its total running time depends the condition number of the ground truth matrix. In addition, to establish the linear convergence, one needs no limitations on the RSC/RSS conditions. On the other hand, the method in~\cite{li2016Nonconvex} is robust to ill-condition problem and it uses the idea of projection on the set of matrices with larger rank than the true one. However, each iteration of it needs SVD and it may overshadow the benefit of it in alleviating the computation of the gradient.
Finally, we mention a non-iterative algorithm for recovery of low-rank matrices from a set of nonlinear measurements proposed by~\cite{plan2017high}. While this approach does not need to know the nonlinearity of the link function, its recovery performance is limited, and we can only recover the solution of the optimization problem up to a scalar ambiguity.
All the aforementioned algorithms suffer from one (or more) of the following issues: expensive computational complexity, slow convergence rate, and troublesome dependency on the condition number of the optimum. In this paper, we resolve these problems by a renewed analysis of approximate low-rank projection algorithms, and integrate this analysis to obtain a new algorithmic framework for optimizing general convex loss functions with rank constraints.
\section{Algorithm and Analysis}
\lambdabel{sec:model}
In this section, we propose our algorithm and provide the theoretical results to support it. Before that we introduce some notations and definitions.
\subsection{Preliminaries}
\lambdabel{prelim}
We denote the minimum and maximum eigenvalues of matrix $\bar{S}$ by $S_p$ and $S_1$, respectively. We use $\|A\|_2$ and $\|A\|_F$ for spectral norm and Frobenius norm of a matrix $A$, respectively.
We show the maximum and minimum eigenvalues of a matrix $A\in{\mathbb{R}}^{p\times p}$ as $\lambdambda_1(A), \lambdambda_p(A)$, respectively. In addition, for any subspace $W \subset \mathbb{R}^{p \times p}$, we denote ${\mathcal{P}}_{W}$ as the orthogonal projection operator onto it. Finally, the phrase ``with high probability" indicates
an event whose failure rate is exponentially small. Our analysis will rely on the following definition~\cite{negahban2009unified,jain2014iterative}:
\begin{definition} \lambdabel{defRSCRSS}
A function $f$ satisfies the Restricted Strong Convexity (RSC) and Restricted Strong Smoothness (RSS) conditions if for all $L_1,L_2\in\mathbb{R}^{p\times p}$ such that $\text{rank}(L_1)\leq r, \text{rank}(L_2)\leq r$, we have:
\begin{align}
\lambdabel{rscrss}
\frac{m_{2r}}{2}&\|L_2-L_1\|^2_F\leq f(L_2) - f(L_1)- \lambdangle\Big{\|}abla f(L_1) , L_2-L_1\rangle\leq\frac{M_{2r}}{2}\|L_2-L_1\|^2_F,
\end{align}
where $m_{2r}$ and $M_{2r}$ are called the RSC and RSS constants, respectively.
\end{definition}
Let $\mathbb{U}_r$ as the set of all rank-$r$ matrix subspaces, i.e., subspaces of $\mathbb{R}^{p \times p}$ that are spanned by any $r$ atoms of the form $uv^T$ where $u, v \in \mathbb{R}^p$ are unit-norm vectors. We will exclusively focus on low-rank approximation algorithms that satisfy the following two properties:
\begin{definition}[Approximate tail projection]
\lambdabel{taildef}
Let $\epsilon>0$. Then, $\widetildea:{\mathbb{R}}^{p\times p}\rightarrow\mathbb{U}_{r}$ is an approximate tail projection algorithm if for all $L\in \mathbb{R}^{p\times p}$, $\widetildea$ returns a subspace $Z=\widetildea(L)$ that satisfies:
$$\|L -{\mathcal{P}}_Z L\|_F\leq (1+\epsilon)\|L-L_r\|_F,$$
where ${\mathcal{P}}_Z L = ZZ^TL$, and $L_r$ is the optimal rank-$r$ approximation of $L$ in the Frobenius norm.
\end{definition}
\begin{definition}[Per-vector approximation guarantee]
Let $L\in\mathbb{R}^{p\times p}$. Suppose there is an algorithm that satisfies approximate tail projection such that it returns a subspace $Z$ with basis vectors $z_1,z_2,\ldots,z_r$ and approximate ratio $\epsilon$. Then, this algorithm additionally satisfies the per-vector approximation guarantee if
$$
|u_i^TLL^Tu_i - z_iLL^Tz_i|\leq\epsilon\sigmagma_{r+1}^2,
$$
where $u_i$'s are the eigenvectors of $L$.
\end{definition}
In this paper, we focus on the randomized Block Krylov SVD (BKSVD) method for implementation of $\widetildea$. This algorithm has been proposed by~\cite{musco2015randomized} which satisfies both of these properties with probability at least $99/100$. However, one can alternately use a recent algorithm called LazySVD~\cite{allen2016lazysvd} with very similar properties. For constant approximation ratio $\epsilon$, the asymptotic running time of these algorithms is given by $\widetilde{{\mathcal{O}}}(p^2 r)$, \emph{independent} of any spectral properties of the input matrix; however, BKSVD ensures a slightly stronger \emph{per-vector} approximation guarantee.
\begin{algorithm}[!t]
\caption{MAPLE}
\lambdabel{alg:appsvp}
\begin{algorithmic}
\STATE \textbf{Inputs:} rank $r$, step size $\eta$, approximate tail projection $\widetildea$
\STATE \textbf{Outputs:} Estimates $\widehat{L}$
\STATE\textbf{Initialization:} $L^0\leftarrow 0$, $t \leftarrow 0$
\WHILE{$t\leq T$}
\STATE ${L}^{t+1} = \widetildea\left(L^{t} - \eta\Big{\|}abla F(L^t)\right)$
\STATE$t\leftarrow t+1$
{\mathbb{E}}NDWHILE
\STATE\textbf{Return:} $\widehat{L} = L^{T}$
\end{algorithmic}
\end{algorithm}
As we discussed above, our goal is to solve the optimization problem~\eqref{opt_prob}. The traditional approach is to perform projected gradient descent:
$$
{L}^{t+1} = P_r\left(L^{t} - \eta\Big{\|}abla F(L^t)\right),
$$
where $P_r$ denotes an exact projection onto the space of rank-$r$ matrices, and can be accomplished via SVD. However, for large $p$, this incurs cubic running time and can be very challenging. To alleviate this issue, one can instead attempt to replace the full SVD in each iteration with a tail-{approximate} low-rank projection; it is known that such projections can computed in $O(p^2 \log p)$ time~\cite{clarksonwoodruff}.
This is precisely our proposed algorithm, which we call {\emph{Matrix Approximation for Low-rank Estimation} (MAPLE)}, is described in pseudocode form as Algorithm~\ref{alg:appsvp}. This algorithm is structurally very similar to~\cite{jain2014iterative,becker2013randomized}. However, the mechanism of~\cite{jain2014iterative} requires \emph{exact} low-rank projections, and~\cite{becker2013randomized} is specific to the least-squares loss function and with weak guarantees.
Here, we show that for low-rank matrix estimation, an \emph{coarse, approximate} low-rank projection (the $\widetildea$ operator in Algorithm~\ref{alg:appsvp}) is sufficient for estimating the solution of~\eqref{opt_prob}. A key point is that our algorithm uses approximate low-rank projections with parameter $r$ such that $r\geq r^*$. As we show in Theorem~\ref{AppSVD}, the combination of using approximate projection, together with choosing a large enough rank parameter $r$, enables efficient solution of problems of the form \eqref{opt_prob} for \emph{any (given) restricted convexity/smoothness constants} $M,m$.
Specifically, this ability removes any upper bound assumptions on the ration $\frac{M}{m}$, which have appeared in several recent related works, such as~\cite{bhojanapalli2016dropping}. While the output matrix of {MAPLE} may have larger rank than $r^*$, one can easily post-process it with an final hard thresholding step in order to enforce the result to have exactly rank $r^*$.
In Algorithm~\ref{alg:appsvp}, the choice of approximate low-rank projections is flexible, as long as the approximate tail and per-vector approximation guarantee are satisfied. We note that tail-approximate low-rank projection algorithms are widespread in the literature~\cite{clarksonwoodruff_old,drineas_mahoney,tygert}; however, per-vector approximation guarantee algorithms are less common. As will become clear in the proof of Theorem~\ref{AppSVD}, the per-vector guarantee is crucial in our analysis.
In our implementation of MAPLE, we invoke the BKSVD method for low-rank approximation mentioned above\footnote{We note that since the BKSVD algorithm is randomized while the definitions of approximate tail projection and per-vector approximation guarantee are deterministic. Fortunately, the running time of BKSVD depends only logarithmically on the failure probability, and therefore an additional union bound argument is required to precisely prove algorithmic correctness of our method.}. Assuming BKSVD as the approximate low-rank projection of choice,
we now prove a key structural result about the non-expansiveness of $\widetildea$. This result, to the best of our knowledge, is novel and generalizes a recent result reported in~\cite{shen2016tight,li2016Nonconvex}. (We defer the full proof of all theoretical results to the appendix.)
\begin{lemma}~\lambdabel{ApprHTh}
For $r>(1+\frac{1}{1-\epsilon})r^*$ and for any matrices $L, L^* \in{\mathbb{R}}^{p\times p}$ with $\text{rank}(L^*)=r^*$, we have
\begin{align*}
\|\widetildea(L)-L^*\|_F^2\leq\left(1+\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r^*}}{\sqrt{r-r^*}}\right)\|L-L^*\|_F^2,
\end{align*}
where $\widetildea:{\mathbb{R}}^{p\times p}\rightarrow\mathbb{U}_{r}$ denotes the approximate tail projection defined in Definition~\ref{taildef} and $\epsilon>0$ is the corresponding approximation ratio.
\end{lemma}
\begin{proof}[proof sketch]
The proof follows the approach of~\cite{li2016Nonconvex} where it is first given for sparse hard thresholding, and then is generalized to the low-rank case using Von Neumann's trace inequality. First, define $\theta = [\sigmagma_1^2(L),\sigmagma_2^2(L)\ldots,\sigmagma_r^2(L)]^T$. Also let $\theta^* = [\sigmagma_1^2(L^*),\sigmagma_2^2(L^*)\ldots,\sigmagma_{r^*}^2(L^*)]^T$, and $\theta' = \widetildea(\theta)$. Also, let $supp(\theta^*) = \mathcal{I^*}$, $supp(\theta) = \mathcal{I}$, $supp(\theta') = \mathcal{I'}$, and $\theta'' =\theta - \theta'$ with support $I''$.
Now define new sets $\mathcal{I^*}\cap\mathcal{I'} = \mathcal{I}^{*1}$ and $\mathcal{I^*}\cap\mathcal{I''} = \mathcal{I}^{*2}$ with restricted vectors to these sets as $\theta_{\mathcal{I}^{*1}} = \theta^{*1}$, $\theta_{\mathcal{I}^{*2}} = \theta^{*2}$, $\theta'_{\mathcal{I}^{*1}} = \theta^{1*}$, $\theta''_{\mathcal{I}^{*2}} = \theta^{2*}$ such that $|\mathcal{I}^{*2}| = r^{**}$, and $\theta_{\max} = \|\theta^{2*}\|_{\infty}$.
The proof continues by upper bounding the ratio of $\frac{\|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2}{\|\theta- \theta^*\|_2^2}$ in terms of $r,r^*, r^{**}$ and by using the inequality $\widehat{\theta}_{\min}\geq(1-\epsilon)\theta_{\max}$ where $\widehat{\theta}$ denotes the vector of approximate eigenvalues returned back by $\widetildea$. This inequality is resulted by invoking the per-vector guarantee property of $\widetildea$. We can now obtain the desired upper bound to get the final claim.
\end{proof}
We now leverage the above lemma to provide our main theoretical result supporting the algorithmic efficiency of MAPLE.
\begin{theorem}[Linear convergence of MAPLE]
\lambdabel{AppSVD}
Assume that the objective function $F(L)$ satisfies the RSC/RSS conditions with parameters $M_{2r+r^*}$ and $m_{2r+r^*}$. Define $\Big{\|}u = \sqrt{1+\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r^*}}{\sqrt{r-r^*}}}$. Let $J_t$ denote the subspace formed by the span of the column spaces of the matrices $L^t, L^{t+1}$, and $L^*$, the solution of~\eqref{opt_prob}. In addition, assume that $r >\frac{C_1}{1-\epsilon}\left(\frac{M_{2r+r^*}}{m_{2r+r^*}}\right)^4r^*$ for some $C_1>2$. Choose step size as $\eta$ as $\frac{1-\sqrt{\alpha'}}{M_{2r+r^*}}\leq\eta\leq\frac{1+\sqrt{\alpha'}}{m_{2r+r^*}}$ where $\alpha' = \frac{\sqrt{\alpha-1}}{\sqrt{1-\epsilon}\sqrt{\alpha-1}+2}$ {for some $\alpha = \widetildeheta(r/r^*) > 1$}. Then, MAPLE outputs a sequence of estimates $L^t$ such that:
\begin{align}
\lambdabel{linconvApp}
\|L^{t+1} - L^{*}\|_F\leq \rho\|L^{t} - L^{*}\|_F + \Big{\|}u\eta\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^*)\|_F,
\end{align}
where $\rho = \Big{\|}u\sqrt{1+M_{2r+r^*}^2\eta^2 - 2m_{2r+r^*}\eta} < 1$.
\end{theorem}
We have to mention that $L_*$ can be any rank $r$ matrix which of course includes the solution of~\eqref{opt_prob}. Also, Theorem~\ref{AppSVD} guarantees the linear convergence of the MAPLE algorithm up to a given radius of convergence determined by the gradient of $F$ at $L^*$. We note that the contraction factor $\rho$ is not affected by extent to which the objective function $F(L)$ is strongly smooth/convex. In other words, no matter how large the ratio $\frac{M}{m}$ is, its effect is balanced by $\Big{\|}u$ through choosing large enough $r$. Also, the quality of the estimates in Theorem~\ref{AppSVD} is upper-bounded by the gradient term $\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F$ in~\eqref{linconvApp}, within each iteration.
Below, we instantiate the general optimization problem~\eqref{opt_prob} in the context of three estimation problems (NLARM, logistic PCA, and PME). In NLARM and PME, $L^*$ denotes the ground truth which we are looking for to estimate; as a result, the gradient term in~\eqref{linconvApp} represents the statistical aspect of MAPLE. For these problems, we give an upper bound on this term. Also, we show that the loss function $F(L)$ satisfies the RSC/RSS conditions in all three instantiations, and consequently, derive the sample complexity and the running time of MAPLE.
\subsection{Discussion: Main Results and Novelty}
First we note that the randomized SVD approach being used in \cite{becker2013randomized} and MAPLE are algorithmically the same. However, the algorithm in \cite{becker2013randomized} has been analyzed only for squared loss, and its theoretical guarantees are somewhat weak. Specifically, in the regime of parameters required to obtain optimal sample complexity (i.e., $n={\mathcal{O}}(pr)$), the quality of their approximate projection should be the order of ${\mathcal{O}}(1/p)$ in order to have provable convergence. This inflates the running time to cubic, and overshadows the usage of any approximate SVD methods. On the other hand, MAPLE can handle general loss functions that satisfy the RSC/RSS conditions. Specifically, the analysis in MAPLE exploits the novel structural result for approximate rank-$r$ projection onto the space of rank-$r$ matrices ($r\gg r^*$) (Lemma~\ref{ApprHTh}), which shows that each projection step in MAPLE is \emph{nearly non-expansive}. This is a crucial new theoretical result in our paper, and is a geometric property of any partial SVD routine which satisfies a per-vector approximation guarantee (and this can be of independent interest in other low-rank estimation problems as well).
Second, for approximate tail projection, MAPLE uses a \emph{gap-independent} SVD method which guarantees that the running time for calculating the approximation of right singular vectors takes ${\mathcal{O}}tilde(p^2r)$ operations in each iteration. This step is crucial as even projection onto a subspace with rank-$1$ can take ${\mathcal{O}}(p^3)$ time due to the existence of a vary small gap between $r^{th}$ and $(r+1)^{th}$ singular values~\cite{musco2015randomized}. Here, one might ask that the classical methods are better than the gap independent result in~\cite{musco2015randomized} if the approximate ratio, $\epsilon$ is less than the $gap$. However, this is not the case in our setup, since we do not need to be very accurate in computing the approximation of right singular values (achieving very small $\epsilon$). Indeed $\epsilon$ is given by $iter = \widetildeheta(\frac{\log p}{\sqrt{\epsilon}})$ where $iter$ denotes the number of iterations required in BKSVD. In all our experiments, we have chosen $iter=2$ which implies very large $\epsilon$ close to $1$ is sufficient for tail projection. On the other hand, the spectral gap can be a very small number, i.e., $10^{-6}$ for many matrices encountered in practice.
Finally, we highlight the ability of MAPLE for handling the objective functions $F(L)$ with arbitrary large smoothness-to-convexity ratio $\frac{M}{m}$. For functions even with very large condition number $\frac{M}{m}$, MAPLE has the ability to choose a projected rank $r\gg r^*$ to guarantee the convergence. This is the role of $\Big{\|}u$ in the expression of the contraction factor, $\rho$ in~\eqref{linconvApp}; no matter how large $\frac{M}{m}$ is, its effect is balanced by $\Big{\|}u$. To see this, fix $m$, and let $M$ be a given arbitrary large value, then by choosing $r>M^4r^*$, and step size as stated in the theorem, we can guarantee that $\rho<1$; hence, establishing linear convergence.
We note that a good choice of step size (which is constant) does depend on problem parameters, as is the case for many other first order algorithms. In practice, this has to be appropriately tuned. However, assuming this choice is made, the convergence rate is not affected.
\section{Applications}
We now instantiate the MAPLE framework in three low-rank matrix estimation problems of interest.
\subsection{Nonlinear Affine Rank Minimization}
\lambdabel{NLARMsec}
\begin{table}
\caption{Summary of our contributions, and comparison with existing methods for NLARM.
$\kappa$ denotes the condition number of $L^*$, and $\vartheta$ denotes the final optimization error. Also, SC and RT denote sample complexity and running time, respectively. Here we have presented (for each algorithm) the best available running time result.}
\lambdabel{NLARMtable}
\begin{center}
\begin{small}
\renewcommand{1.5}{1.5}
\begin{tabular}{lccr}
\hline
Algorithm & SC & RT & Bounded $\frac{M}{m}$\\
\hline
{Convex~\cite{recht2010guaranteed}} &${\mathcal{O}}tilde(pr^*)$ & ${\mathcal{O}}(\frac{p^3}{\sqrt{\vartheta}})$ & Yes \\
Non-convex Reg~\cite{quanming2017large} & ${\mathcal{O}}tilde(pr^*)$ & ${\mathcal{O}}(\frac{p^2r^*}{\vartheta})$ & Yes \\
Factorized~\cite{bhojanapalli2016dropping} & ${\mathcal{O}}tilde(pr^*)$& ${\mathcal{O}}(p^2(r^*+\log p)\kappa^2\log(\frac{1}{\vartheta}) + p^3)$ & Yes \\
SVP~\cite{jain2014iterative} & ${\mathcal{O}}tilde(pr^*)$ & ${\mathcal{O}}(p^3\log(\frac{1}{\vartheta})) $& No \\
\textbf{MAPLE} &$\mathbf{{{\mathcal{O}}tilde(pr^*)}}$ & $\mathbf{{\mathcal{O}}(p^2r^*\log p\log(\frac{1}{\vartheta}))}$ & {No} \\
\hline
\end{tabular}
\end{small}
\end{center}
\end{table}
Consider the nonlinear observation model $y = g(\mathcal{A}(L^*)) + e$, where $\mathcal{A}$ is a linear operator, $\mathcal{A}:{\mathbb{R}}^{p\times p}\rightarrow{\mathbb{R}}^n$ parametrized by $n$ full rank matrices, $A_i\in{\mathbb{R}}^{p\times p}$ such that $(\mathcal{A}(L^*))_i = \lambdangle A_i,L^*\rangle$ for $i=1,\ldots,n$. Also, $e$ denotes an additive subgaussian noise vector with i.i.d., zero-mean entries that is also assumed to be independent of $\mathcal{A}$ (see appendix for more details). If $g(x)=x$, we have the well-known matrix sensing problem for which a large number of algorithms have been proposed. The goal is to estimate the ground truth matrix $L^*\in{\mathbb{R}}^{p\times p}$ for more general nonlinear link functions.
We assume that link function $g(x)$ is a differentiable monotonic function, satisfying $0<\mu_1\leq g'(x)\leq\mu_2$ for all $x\in\mathcal{D}(g)$ (domain of $g$). This assumption is standard in statistical learning~\cite{kakade2011} and in nonlinear sparse recovery~\cite{negahban2009unified,yang2015sparse,soltani2016fastIEEETSP17}. Also, as we will discuss below, this assumption will be helpful for verifying the RSC/RSS condition for the loss function that we define as follows. We estimate $L^*$ by solving the optimization problem:
\begin{equation} \lambdabel{opt_probNLARM}
\begin{aligned}
& \underset{L}{\text{min}}
\quad F(L) = \frac{1}{n}\sum_{i=1}^{n}{\mathcal{O}}mega(\lambdangle A_i,L\rangle) - y_i\lambdangle A_i,L\rangle\\
& \text{s.t.}\ \ \quad\text{rank}(L)\leq r^*,
\end{aligned}
\end{equation}
where ${\mathcal{O}}mega : \mathbb{R} \rightarrow \mathbb{R}$ is chosen such that ${\mathcal{O}}mega'(x) = g(x)$.\footnote{The objective functioon $F(L)$ in~\eqref{opt_probNLARM} is standard; see~\cite{soltani2016fastIEEETSP17} for an in-depth discussion.}
Due assumption on the derivative of $g$, we see that $F(L)$ is a convex function (actually strongly convex), and can be considered as a special case of general problem in~\eqref{opt_prob}.
We assume that the design matrices $A_i$'s are constructed as follows. Consider a partial Fourier or partial Hadamard matrix $X'\in{\mathbb{R}}^{n\times p^2}$ which is multiplied from the right by a diagonal matrix, $D$, whose diagonal entries are uniformly distributed over $\{-1,+1\}^{p^2}$. Call the resulting matrix $X = X'D$ where each row is denoted by $X_i^T\in{\mathbb{R}}^{p^2}$. If we reshape each of these rows as a matrix, we obtain ``measurement'' (or ``design'') matrices $A_i\in{\mathbb{R}}^{p\times p}$ for $i=1,\ldots,n$. This particular choice of design matrices $A_i$'s is because they support fast matrix-vector multiplication which takes ${\mathcal{O}}(p^2\log(p))$. (The origins of constructing design matrices of this form come from the compressive sensing literature~\cite{modelcsICALP}).
The following theorem gives the upper bound on the term, $\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F$, that appears in Theorem~\ref{AppSVD}. This can be viewed as a ``statistical error'' term, and is zero in the absence of noise.
\begin{theorem}\lambdabel{staterrorNLARM}
Consider the observation model $y = g(\mathcal{A} (L^*)) + e$ as described above. Let the number of samples scale as $n= {\mathcal{O}}(pr~\textrm{polylog}~(p))$, then with high probability, for any given subspace $J$ of $\mathbb{R}^{p \times p}$,
we have for $t=1,\ldots,T$:
\begin{align}
\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F\leq\frac{1+\delta_{2r+r^*}}{\sqrt{n}}\|e\|_2,
\end{align}
where $0<\delta_{2r+r^*}<1$ denotes the RIP constant of $\mathcal{A}$.
\end{theorem}
\begin{corollary}\lambdabel{induclin}
Consider all the assumptions and definitions stated in Theorem~\ref{AppSVD}. If we initialize MAPLE with $L^0 = 0$, then after $T_{iter} = \mathcal{O}\left(\log\left(\frac{\|L^*\|_F}{\vartheta}\right)\right)$ iterations, we obtain:
\begin{align}
\|L^{T+1} - L^{*}\|_F\leq \vartheta + \frac{1}{\sqrt{n}}\frac{\Big{\|}u\eta(1+\delta_{2r+r^*})}{1-\rho}\|e\|_2,
\end{align}
for some $\vartheta>0$.
\end{corollary}
We now provide conditions under which the RSS/RSC assumptions in Theorem~\ref{AppSVD} are satisfied.
\begin{theorem}[RSC/RSS conditions for MAPLE]
Let the number of samples scale as $n= {\mathcal{O}}(pr~\mathrm{polylog}~(p))$. Assume that $\frac{\mu_2^4(1+\omega)^4}{\mu_1^4(1-\omega)^4}\leq C_2(1-\epsilon)\frac{r}{r^*}$ for some $C_2, \omega>0$ and $\epsilon>0$ denotes the approximation ratio in Algorithm~\ref{alg:appsvp}. Then with high probability, the loss function $F(L)$ in~\eqref{opt_probNLARM} satisfies the RSC/RSS conditions with constants $m_{2r+r^*}\geq \mu_1(1-\omega)$ and $M_{2r+r^*}\leq \mu_2(1+\omega)$
in each iteration.
\lambdabel{RSCRSSappNLARM}
\end{theorem}
\textbf{Sample complexity.} By Corollary~\ref{induclin} and Theorem~\ref{RSCRSSappNLARM}, the sample complexity of MAPLE algorithm is given by $n= {\mathcal{O}}(pr~\mathrm{polylog}~(p))$ in order to achieve a specified estimation error. This sample complexity is nearly as good as the optimal rate, ${\mathcal{O}}(pr)$. We note that the leading constant hidden within the ${\mathcal{O}}$-notation depends on $\rho, \eta$, the RIP constant of the linear operator $\mathcal{A}$, and the magnitude of the additive noise. (Since we assume that this noise term is subgaussian, it is easy to show that $\|e\|_2$ scales as ${\mathcal{O}}(\sqrt{n})$ in expectation and with high probability).
\textbf{Time complexity.} Each iteration of MAPLE needs to compute the gradient, plus an approximate tail projection to produce a rank-$r$ matrix. Computing the gradient involves one application of the linear operator $\mathcal{A}$ for calculating $\mathcal{A}(L)$, and one application of the adjoint operator, i.e., $\mathcal{A}^*(y-g(\mathcal{A}(L))$. Let $T_{mult}$ and $T'_{mult}$ denote the required time for these operations, respectively. On the other hand, approximate tail projection takes ${\mathcal{O}}\left(\frac{p^2r\log p}{\sqrt{\varepsilon}}\right)$ operations for achieving the approximate ratio $\epsilon$ according to~\cite{musco2015randomized}. Thanks to the linear convergence of MAPLE, the total number of iterations for achieving $\vartheta$ accuracy is given by $T_{iter} = \mathcal{O}\left(\log\left(\frac{\|L^*\|_F}{\vartheta}\right)\right)$. Let $\pi = \frac{M}{m}$; thus, the overall running time scales as $T = {{\mathcal{O}}}\left(\left(T_{mult} +T'_{mult}+ \frac{p^2r^*\pi^4\log p}{\sqrt{\epsilon}}\right)\left(\log\frac{\|L^*\|_F}{\vartheta}\right)\right)$ by the choice of $r$ according to Theorem~\ref{AppSVD}. If we assume that the design matrices $A_i$'s are implemented via a Fast Fourier Transform, computing $T_{mult} =T'_{mult}$ takes ${{\mathcal{O}}}(p^2\log p)$ operations. As a result, $T = {\mathcal{O}}\left(\left(p^2\log p +\frac{p^2r^*\pi^4\log p}{\sqrt{\epsilon}}\right)\left(\log\frac{\|L^*\|_F}{\vartheta}\right) \right)$.
In Table~\ref{NLARMtable}, for $g(x) = x$ and the linear operator $\mathcal{A}$ defined above, we summarize the sample complexity as well as (asymptotic) running time of several algorithms. {In this table, we assume a constant ratio of $M/m$ for all the algorithms.} We find that all previous methods, while providing excellent sample complexity benefits, suffer from either cubic dependence on $p$, or inverse dependence on the estimation error $\vartheta$, or quadratic dependence on the condition number $\kappa$ of the ground truth matrix. In contrast, MAPLE enjoys (unconditional) ${\mathcal{O}}tilde(p^2 r^*)$ dependence, which is (nearly) linear in the size of the matrix for small enough $r^*$.
\subsection{Logistic PCA}
\lambdabel{LPCA}
Principle component analysis (PCA) is a widely used statistical tool in various applications such as dimensionality reduction, denoting, and visualization, to name a few. While the regular PCA sometimes called linear PCA can be applied for any data type, its usage for binary or categorical observed data is not satisfactory, due to the fact that it tries to minimize a least square objective function.
As a result, applying it to the binary case makes the result less interpretable~\cite{jolliffe1986principal}.
To alleviate this issue, one can assume that each row of the observed binary matrix (a sample data) follows the multivariate Bernoulli distribution such that its maximum variations can be captured by a low-dimensional subspace, and then use the logistic loss to find this low-dimensional representation of the observed data. This problem has been also studied in the context of collaborative filtering on binary data~\cite{johnson2014logistic}, one-bit matrix completion~\cite{davenport20141}, and network sign prediction~\cite{chiang2014prediction}.
Mathematically, consider an observed binary matrix $Y\in{\mathbb{R}}^{p\times p}$ with entries belong to set ${\{}0, 1{\}}$ such that the mean of each $Y_{ij}$ is given by $p_{ij} = P(Y_{ij} = 1|L_{ij}) = \sigmagma(L^*_{ij})$ where $\sigmagma(z) = \frac{1}{1+\exp(-z)}$. The goal is to estimate a low-rank matrix $L^*$ such that $L^*_{ij} = \log(\frac{p_{ij}}{1-p_{ij}}) = \text{logit}(p_{ij})$ by minimizing the following regularized logistic loss:
\begin{equation}\lambdabel{logisticPCA}
\begin{aligned}
&\underset{L}{\text{min}}
\quad F(L) = -\sum_{i,j}\big{(}Y_{ij}\log\sigmagma(L_{ij}) + (1-Y_{ij})\log(1-\sigmagma(L_{ij}))\big{)} + \lambdambda\|L\|_F^2\\
& \text{s.t.}\ \ \quad\text{rank}(L)\leq r^*,
\end{aligned}
\end{equation}
where $\lambdambda>0$ is a tuning parameter.\footnote{Here, $L_*$ denotes the solution of the optimization problem~\eqref{logisticPCA}.} We note that the objective function in~\eqref{logisticPCA} without the regularizer term is only strongly smooth. By adding the Frobenius norm of the optimization variable, we ensure that it is also globally strongly convex (Hence, RSC/RSC conditions are trivially satisfied). Here, we focus on finding the solution of~\eqref{logisticPCA}, $L^*$. Hence, we do not have explicitly the notion of ground truth as previous application.
For solving the optimization problem~\eqref{logisticPCA}, several algorithms have been proposed in recent years. Unfortunately, algorithms such as convex nuclear norm minimization are either too slow, or do not have theoretical guarantees~\cite{chiang2014prediction,johnson2014logistic,davenport20141}. Very recently, a non-convex factorized algorithm proposed by~\cite{park2016finding} has been supported by rigorous convergence analysis. We will compare the performance of this algorithm with the MAPLE in the experimental section. In particular, we show that the running time of MAPLE for solving the above problem is given by ${\mathcal{O}}tilde(rp^2)$ as the dominating term is related to the projection step and gradient calculation takes ${\mathcal{O}}(p^2)$ time.
\subsection{Precision Matrix Estimation (PME)}
\lambdabel{PME}
{Gaussian graphical models} are a popular tool for modeling the interaction of a collection of Gaussian random variables. In Gaussian graphical models, nodes represent random variables and edges model conditional (in)dependence among the variables
~\cite{wainwright2008graphical}. Over the last decade, significant efforts have been directed towards algorithms for learning \emph{sparse} graphical models.
Mathematically, let $\Sigma^*$ denote the positive definite covariance matrix of $p$ Gaussian random variables, and let $\widetildeheta^* = (\Sigma^*)^{-1}$ be the corresponding precision matrix. Then, $\widetildeheta^*_{ij} = 0$ implies that the $i^{\textrm{th}}$ and $j^{\textrm{th}}$ variables are conditionally independent given all other variables and the edge $(i,j)$ does not exist in the underlying graph. The basic modeling assumption is that $\widetildeheta^*$ is sparse, i.e., such graphs possess only a few edges. Such models have been fruitfully used in several applications including astrophysics~\cite{padmanabhan}, scene recognition~\cite{souly}, and genomic analysis~\cite{yin2013}. Numerous algorithms for sparse graphical model learning -- both statistically as well as computationally efficient -- have been proposed in the machine learning literature~\cite{friedman2008sparse,mazumder2012graphical,banerjee2008model,hsieh2011sparse}.
Unfortunately, sparsity is a simplistic first-order model and is not amenable to modeling more complex interactions. For instance, in certain scenarios, only some of the random variables are directly observed, and there could be relevant \textit{latent} interactions to which we do not directly have access.
The existence of latent variables poses a significant challenge in graphical model learning since they can confound an otherwise sparse graphical model with a dense one. This scenario is illustrated in Figure~\ref{lvgfig}. Here, nodes with solid circles denote the observed variables, and solid black edges are the ``true" edges. One can see that the ``true" graph is rather sparse. However, if there is even a single unobserved (hidden) variable denoted by the node with the broken red circle, then it will induce dense, apparent interactions between nodes that are otherwise disconnected; these are denoted by the dotted black lines. A flexible and elegant method to learn latent variables in graphical models was proposed by~\cite{chandrasekaran2012latent}. At its core, the method imposes a {superposition} structure in the observed precision matrix as the sum of \emph{sparse} and \emph{low-rank} matrices, i.e.,
$ \widetildeheta^* = S^{*} + L^{*} $.
Here, $\widetildeheta^*, S^*, L^*$ are $p \times p$ matrices where $p$ is the number of variables. The matrix $S^*$ specifies the conditional observed precision matrix given the latent variables, while $L^*$ encodes the effect of marginalization over the latent variables. The rank of $L^*$, $r^*$, is equal to the number of latent variables and we assume that $r^*$ is much smaller than $p$. The goal is to estimate precision matrix $\widetildeheta^*$. Here, we merely focus on the learning the low-rank part, and assume that the sparse part is a known prior.\footnote{For, instance, if the data obeys the spiked covariance model~\cite{johnstone2001distribution}, the covariance matrix is expressed as the sum of a low-rank matrix and a diagonal matrix. Consequently, by the Woodbury matrix identity, the precision matrix is the sum of a diagonal matrix and a low-rank matrix; $\widetildeheta^* = \bar{S} + L^*$. In addition, problem in~\eqref{opt_probPME} is similar to the latent variable in Gaussian graphical model proposed by~\cite{Venkat2010latent}.}
We cast the estimation of matrix $\widetildeheta^*$ into our framework. Suppose that we observe samples $x_1,x_2,\ldots,x_n \overset{i.i.d}{\thicksim}\mathcal{N}(0,\Sigma)$ where each $x_i\in\mathbb{R}^{p}$. Let $C= \frac{1}{n}\sum_{i=1}^{n}x_ix_i^{T}$ denote the sample covariance matrix, and $\widetildeheta^* = (\Sigma^*)^{-1}$ denote the true precision matrix. Following the formulation of~\cite{han2016fast}, we solve the following minimization of NLL problem:
\begin{equation} \lambdabel{opt_probPME}
\begin{aligned}
& \underset{L}{\text{min}}
& & F(L) = -\log \ \det (\bar{S}+L) + \lambdangle \bar{S}+L,C\rangle\\
& \text{s.t.}
& & \text{rank}(L)\leq r^*, \ L\succeq 0 .
\end{aligned}
\end{equation}
where $\widetildeheta^* = \bar{S} + L^*$ such that $\bar{S}$ is a known positive diagonal matrix (in general, a positive definite matrix) imposed in the structure of precision matrix to make the above optimization problem well-defined. We will exclusively function in the high-dimensional regime where $n \ll p^2$. As an instantiation of the general problem~\eqref{opt_prob}, our goal is to learn the low-rank matrix $L^{*}$ with rank $r^*\ll p$, from samples $x_i$'s. We provide a summary of the theoretical properties of our methods, and contrasts them with other existing methods for PME existing methods in Table~\ref{ComptablePME} (We assume a constant ratio of $M/m$ for all the algorithms).
\begin{table*}
\caption{Summary of our contributions, and comparison with existing methods. Here, $\gamma = \sqrt{\frac{\sigmagma_r}{\sigmagma_{r+1}}-1}$ represents the spectral gap parameter. }
\lambdabel{ComptablePME}
\begin{center}
\begin{small}
\renewcommand{1.5}{1.5}
\begin{tabular}{lccr}
\hline
Algorithm & Running Time & Spectral dependency \\%& Output rank\\
\hline
SDP \cite{chandrasekaran2012latent} &$\text{poly}(p)$ & Yes \\%& $\gg r$ \\
ADMM\cite{ma2013alternating} & $\text{poly}(p)$ & Yes \\%& $\gg r$ \\
QUICDIRTY\cite{yang2013dirty} & ${\mathcal{O}}tilde(p^3)$ & Yes \\%& $\gg r$ \\
SVP\cite{jain2014iterative} & ${\mathcal{O}}tilde(p^3)$ & No \\%& $\gg r$ \\
Factorized\cite{bhojanapalli2016dropping} & ${\mathcal{O}}tilde(p^2r^* / \gamma)$ & Yes \\%& $r$ \\
\textbf{MAPLE}&$\mathbf{{\mathcal{O}}tilde(p^2 r^*)}$ & $\mathbf{No}$ \\%& $\mathbf{r}$ \\
\hline
\end{tabular}
\end{small}
\end{center}
\end{table*}
As an illustration of our results, we first analyze the \textbf{exact projected-gradient approach}, which is a slight variant of the approach of~\cite{li2016Nonconvex}, since its analysis for establishing RSC/RSS is somewhat different from ours. In this setup, the algorithm starts with a zero initialization and proceeds in each iteration as $L^{t+1} = \mathcal{P}_r^+\left(L^{t} - \eta'\Big{\|}abla F(L^t)\right)$ where ${\mathcal{P}}_r^+(\cdot)$ for some $r>r^*$ denotes projection onto the space of rank-$r$ matrices which is implemented through performing an exact eigenvalue decomposition (EVD) of the input and selecting the nonnegative eigenvalues and corresponding eigenvectors~\cite{henrion2012projection}.\footnote{Note that we may not impose a PSD projection within every iteration. If an application requires a PSD matrix as the output (i.e., if proper learning is desired), then we can simply post-process the final estimate $\widehat{L}$ by retaining the nonnegative eigenvalues (and corresponding eigenvectors) through an exact EVD.}
The following theorem shows an upper bound on the estimation error of the low-rank matrix at
each iteration through exact projected-gradient approach.
\begin{figure}
\caption{Illustration of effects of latent variable in graphical model learning. Solid edges represent ``true" conditional dependence, while dotted edges represent apparent dependence due to the presence of the latent variable $h$.
\lambdabel{lvgfig}
\end{figure}
\begin{theorem}[Linear convergence with exact projected-gradient approach]
\lambdabel{ExactSVD}
Assume that the objective function $F(L)$ satisfies the RSC/RSS conditions with corresponding constants as $M_{2r+r^*}$ and $m_{2r+r^*}$. Define $\Big{\|}u' = \sqrt{1+\frac{2\sqrt{r^*}}{\sqrt{r-r^*}}}$. Let $J_t$ denotes the subspace formed by the span of the column spaces of the matrices $L^t, L^{t+1}$, and $L^*$.
In addition, assume that $r >C_1'\left(\frac{M_{3r}}{m_{3r}}\right)^4r^*$ for some $C'_1>0$. Choose step size $\eta'$ as $\frac{1-\sqrt{\beta'}}{M_{2r+r^*}}\leq\eta'\leq\frac{1+\sqrt{\beta'}}{m_{2r+r^*}}$ where $\beta' = \frac{\sqrt{\beta-1}}{\sqrt{\beta-1}+2}$ for some $\beta >1$. Then, exact projected-gradient outputs a sequence of estimates $L^t$ such that:
\begin{align}
\lambdabel{linconvEx}
\|L^{t+1} - L^{*}\|_F\leq \rho'\|L^t-L^*\|_F + \Big{\|}u'\eta'\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^*)\|_F,
\end{align}
where $\rho' = \Big{\|}u'\sqrt{1+M_{3r}^2\eta'^2 - 2m_{3r}\eta'}$.
\end{theorem}
The quality of the estimates in Theorems~\ref{ExactSVD} is upper-bounded by the gradient term $\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F$ in~\eqref{linconvEx} within each iteration. The following theorem establishes this bound:
\begin{theorem}
\lambdabel{BoundGrad}
Under the assumptions of Theorem~\ref{ExactSVD}, for any fixed $t$ we have:
\begin{align}
\lambdabel{BGRExc}
\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F\leq c_2\sqrt{\frac{rp}{n}},
\end{align}
with probability at least $1 - 2\exp(-p)$ where $c_2>0$ is an absolute constant.
\end{theorem}
Next, we verify the RSS/RSC conditions of the objective function defined in~\eqref{opt_probPME}, justifying the assumptions made in Theorem~\ref{ExactSVD} (please see appendix for full expression of the sample complexity in terms of the leading constants).
\begin{theorem}[RSC/RSS conditions for exact projected-gradient approach]
\lambdabel{RSCRSSex}
Let the number of samples scaled as $n={\mathcal{O}}\left(pr\right)$. Also, assume that
$$S_p\leq S_1\leq C_2''(\frac{r}{r^*})^{\frac{1}{8}}S_p - \left(1+\sqrt{r^*}\right)\|L^*\|_2- \delta.$$
Then, the loss function $F(L)$ in~\eqref{opt_probPME} satisfies RSC/RSS conditions with constants $m_{2r+r^*}\geq \frac{1}{(S_1 +\left(1+\sqrt{r}\right)\|L^*\|_2+ \delta)^2}$ and $M_{2r+r^*}\leq \frac{1}{S_p^2}$ that satisfy the assumptions of Theorem~\ref{ExactSVD} in each iteration.
\end{theorem}
The above theorem states that convergence of our method is guaranteed when the eigenvalues of $\bar{S}$ are roughly of the same magnitude, and large when compared to the spectral norm of $L^*$. We believe that this is merely a sufficient condition arising from our proof technique, and our numerical evidence shows that the algorithm succeeds for more general $\bar{S}$ and $L^*$.
\textbf{Time complexity.} Each iteration of exact projected-gradient approach needs a full EVD (similar to IHT-type algorithms), which requires cubic running time (computing the gradient needs only needs ${\mathcal{O}}(pr+r^3)$ operations). Since the total number of iterations is logarithmic, the overall running time scales as $\widetilde{{\mathcal{O}}}(p^3)$.
The above running time is cubic, and can be problematic for very large $p$. Here, we show that MAPLE (without imposing the PSD constraint) can successfully reduce the cubic time complexity to \emph{nearly quadratic} in $p$. All we need to do is to provide conditions under which the assumption of RSC/RSS in Theorem~\ref{AppSVD} are satisfied. We achieve this via the following theorem.
\begin{theorem}[RSC/RSS conditions for MAPLE]
\lambdabel{RSCRSSapp}
Let $n={\mathcal{O}}\left(pr\right)$. Also, assume the followings for some $C_4,C_3''>0$:
\begin{align}
\lambdabel{TrueLapp}
&\|L^*\|_2\leq\frac{1}{1+\sqrt{r^*}}\left(\frac{S_p}{1+C_4\left((1-\epsilon)(\frac{r^*}{r})\right)^{\frac{1}{8}}} - \frac{S_1(C_4((1-\epsilon)(\frac{r^*}{r}))^{\frac{1}{8}})}{1+C_4\left((1-\epsilon)(\frac{r^*}{r})\right)^{\frac{1}{8}}} - \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}\right),\\
&\hspace{18mm}S_p\leq S_1\leq \frac{C_3''}{(1-\epsilon)^{\frac{1}{8}}}(\frac{r}{r^*})^{\frac{1}{8}}(S_p-a^{\prime}) - \left(1+\sqrt{r^*}\right)\|L^*\|_2- \delta^{\prime},
\end{align}
where $0<a^{\prime}\leq\left(1+\sqrt{r^*}\right)\|L^*\|_2 + \delta^{\prime}$ for some $\delta^{\prime}>0$.
Then, the loss function $F(L)$ in~\eqref{opt_probPME} satisfies RSC/RSS conditions with constants $m_{2r+r^*}\geq \frac{1}{(S_1 +\left(1+\sqrt{r}\right)\|L^*\|_2+ \delta')^2}$ and $M_{2r+r^*}\leq \frac{1}{(S_p-a')^2}$ that satisfy the assumptions of Theorem~\ref{AppSVD} in each iteration.
\end{theorem}
Theorem~\ref{RSCRSSapp} specifies a family of true precision matrices $\widetildeheta^* = \bar{S} + L^*$ that can be provably estimated using our approach with an optimal number of samples. Note that since we do not perform PSD projection within MAPLE, it is possible that some of the eigenvalues of $L^t$ are negative. Next, we show that with high probability, the absolute value of the minimum eigenvalue of $L^t$ is small.
\begin{theorem}
\lambdabel{mineigval}
Under the assumptions in Theorem~\ref{RSCRSSapp} on $L^*$, using MAPLE to generate a rank-$r$ matrix $L^t$ for all $t=1,\ldots,T$ guarentees with high probability the minimum eigenvalue of $L^t$ satisfies: $
\lambda_p(L^t)\geq -a^{\prime}$ where $0<a^{\prime}\leq\left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}$.
\end{theorem}
\textbf{Time complexity.} Each iteration of MAPLE needs a tail approximate projection on the set of rank $r$ matrices. According to~\cite{musco2015randomized}, these operations takes $k' = {\mathcal{O}}\left(\frac{p^2r\log p}{\sqrt{\epsilon}}\right)$ for approximate ratio $\epsilon$ (computing gradient needs only needs ${\mathcal{O}}(pr+r^3)$). Since the total number of iterations is once again logarithmic, the overall running time scales as $\widetilde{O}(p^2 r)$.
\textbf{Sample complexity.} Using the upper bounds in~\eqref{BGRExc} and Theorems~\ref{RSCRSSex} and ~\ref{RSCRSSapp}, the sample complexity of MAPLE scales as $n = {\mathcal{O}}(pr)$ to achieve a given level of estimation error. From a statistical perspective, this matches, up to constant factors, the number of degrees of freedom of a $p \times p$ matrix with rank $r$.
\section{Experimental results}
\lambdabel{sec:expe}
We provide a range of numerical experiments supporting our proposed algorithm and comparing with existing approaches. For NLARM and logistic PCA frameworks, we compare our algorithms with factorized gradient descent~\cite{bhojanapalli2016dropping} as well as projected gradient descent (i.e., the SVP algorithm of~\cite{jain2014iterative}). In our results below, FGD denotes factorized gradient descent algorithm, and SVD refers to SVP-type algorithms where exact SVDs are used for the projection step.\footnote{We have also used the more well-known (but gap-dependent) Lanczos approximation method for the projection step, and have obtained the same performance as full SVD.} For the PME application, our comparisons is with the regularized maximum likelihood approach of~\cite{chandrasekaran2012latent}, which we compare with CVX~\cite{cvx}, and a modification of the ADMM-type method proposed by~\cite{ma2013alternating} (SVD denotes the exact projected-gradient approach). We manually tuned step-sizes and regularization parameters in the different algorithms to achieve the best possible performance.
\subsection{Nonlinear Affine Rank Minimization}
\begin{figure*}
\caption{ Comparisons of algorithms with $g(x) = 2x+sin(x)$. (a) Average of the relative error in estimating $L^*$. Parameters: $p =1000$, $r^* = r = 50$, and $n= 4pr$. \textbf{Top:}
\end{figure*}
\begin{figure*}
\caption{Comparison of algorithms for real 2D image with $g(x) = \frac{1-e^{-x}
\end{figure*}
\begin{table}[t]
\caption{Numerical results for the real data experiment illustrating in Figure~\ref{Real2D}. $T$ denotes the number of iterations.}
\lambdabel{TableRealData}
\begin{center}
\begin{small}
\renewcommand{1.5}{1.5}
\begin{tabular}{lccr}
\hline
Algorithm & Relative Error & Running Time & Projected Rank\\
\hline
FGD ($T=300$) & $0.0879$ & $4.9816$ & $30 $ \\
FGD ($T=1000$) & $0.0602$ & $15.9472$ & $30 $ \\
SVD ($T=300$)& $4.4682e-04$ & $19.4700$ & $30$ \\
MAPLE ($T=300$) & $9.7925e-05$ & $4.2375$ & $30$\\
MAPLE ($T=300$) & $9.9541e-05$ & $5.5571$ & $40$\\
MAPLE ($T=300$) & $1.3286e-04$ & $7.1306$ & $50$\\
\hline
\end{tabular}
\end{small}
\end{center}
\end{table}
We report results for all algorithms in Figure~\ref{NLARM}. The link function is set to $g(x) = 2x + \sigman(x)$; this function satisfies the derivative conditions discussed above. We construct the ground truth low-rank matrix $L^*$ with rank $r^*$ by generating a random matrix $U\in{\mathbb{R}}^{p\times r^*}$ with entries drawn from the standard normal distribution. We ortho-normalize the columns of $U$, and set $L^* = UDU^T$ where $D\in{\mathbb{R}}^{r^*\times r^*}$ is a diagonal matrix with $D_{11} = \kappa(L^*)$, and $D_{jj} = 1$ for $j\Big{\|}eq1$. After this, we apply a linear operator $\mathcal{A}$ on $L^*$, i.e., $\mathcal{A}(L^*)_i = \lambdangle A_i,L^*\rangle$ where the choice of $A_i$ has been discussed above. Finally, we obtain the measurements $y=g(\mathcal{A}(L^*))$. When reporting noise robustness, we add a Gaussian noise vector $e\in{\mathbb{R}}^{m}$ to $g(\mathcal{A}(L^*))$.
In Figure~\ref{NLARM}(a), the running time of the four algorithms are compared. For this experiment, we have chosen $p=1000$, and the rank of the underlying matrix $L^*$ to be $50$. We also set the projected rank as $r = 50$. The number of measurements is set to $n= 4pr$. We consider a well-conditioned matrix $L^*$ with $\kappa(L^*)= 1.1$ for top plot and $\kappa = 20$ for the bottom one. Then, we measure the relative error in estimating of $L^*$ in Frobenius norm in log scale versus the CPU time takes for $200$ iterations for all of the algorithms. We run the algorithms for $15$ Monte Carlo trials. As we can see, when $\kappa$ is small, FGD has comparable running time with MAPLE (top plot); on the other hand, when we have ill-posed $L^*$, FGD takes much longer to achieve the same relative error.
Next, we show the performance of the algorithms when the projected rank is changed. The parameters are as $p=300$, $\kappa(L^*)= 1.1$, $r^*=10$, and $n= 4pr$. We set the number of Monte Carlo trials to $50$. In the top plot in Panel (b), we have plotted the relative error as before versus the various $r$ values by averaging over the trials. As we can see, projecting onto the larger space is an effective and practical strategy to achieve small relative error when we do not know the true rank.
Furthermore, the bottom plot of Panel (b) shows the the average running time for either achieving relative error less than $10^{-4}$, or $100$ iterations versus the projected rank. These results suggest that both FGD and MAPLE have the comparable running when we increase the projected rank, while the other SVP algorithms have much longer running time.
Next, we consider the effect of increasing condition number of the underlying low-rank matrix $L^*$ on the performance of the different algorithms. To do this, we set $p=300$, and $r^* = r=10$. The number of measurements is set to $cpr$ where $c = 5,8,11$ for FGD and $5$ for others. Then we run all the algorithms $50$ times with different condition numbers ranging from $\kappa=1$ (well-posed) to i.e., $\kappa=1024$ (highly ill-posed). We define the probability of success as the number of times that the relative error is less than $0.001$. As illustrated in the top plot of panel (c), all SVP-type algorithms are always able to estimate $L^*$ even for large condition number, i.e., $\kappa = 1024$, whereas FGD fails. In our opinion, this feature is a key benefit of MAPLE over the current fastest existing methods for low-rank estimation (based on factorization approaches).
Finally, we consider the noisy scenario in which the observation $y$ is corrupted by different Gaussian noise level. The parameters are set as $p=300$, $r=10,25,40$ for MAPLE and $10$ for the others, $r^*=10$, $n=7pr$, and $\kappa = 2$. The bottom plot in Panel (c) shows the averaged over $50$ trials of the relative error in $L^*$ versus the various standard deviations. From this plot, we see that MAPLE with $r=40$ is most robust, indicating that projection onto the larger subspace is beneficial when noise is present.
We also run MAPLE on a real 2D $512\times 512$ image, assumed to be an approximately low-rank matrix. The choice of $\mathcal{A}$ is as before, but for the link function, we choose the sigmoid $g(x) = \frac{1-e^{-x}}{1+e^{-x}}$. Figure~\ref{Real2D} visualizes the reconstructed image by different algorithms. In Figure~\ref{Real2D}, (a) is the true image and (b) is the same image truncated to its $r^* = 30$ largest singular values. The result of FGD is shown in (c) and (d) where for (d) we let algorithm run for many more iterations. Reconstruction by SVD is shown in (e). Finally, (f), (g), and (h) illustrate the reconstructed image by using MAPLE with various rank parameters. The numerical reconstruction error is given in Table~\ref{TableRealData}. MAPLE is the fastest method among all methods, even when performing rank-$r$ projection with $r$ larger than $r^*$.
\subsection{Logistic PCA}
\begin{figure*}
\caption{ Comparisons of the algorithms for the average of the logarithm of the logistic loss. (a) Parameters: $p =1000$, $r^* = r = 5$. \textbf{Top:}
\end{figure*}
In this section, we provide some representative experimental results for our second application, logistic PCA. We report results for all algorithms in Figure~\ref{LogisticOCAResults}. We construct the ground truth low-rank matrix $L^*$ with rank $r^*$ similar to NLARM case.
In panel (a), the running time of all algorithms are compared. For this experiment, we have chosen $p=1000$, and the rank of the underlying matrix $L^*$ to be $5$. We also set the projected rank as $r = 5$. We consider a well-conditioned matrix $L^*$ with $\kappa(L^*)= 1.1531$ for top plot and $\kappa = 21.4180$ for the bottom one. Then we measure the evolution of the logistic loss defined in~\eqref{logisticPCA} without any regularizer versus the CPU time takes for $50$ iterations for all of the algorithms. We run the algorithms for $20$ Monte Carlo trials, and illustrate the average result. As we can see, when $\kappa$ is small, FGD has comparable running time with MAPLE (top plot); on the other hand, when we have ill-posed $L^*$, FGD takes longer to achieve the same performance.
In panel (b), top plot, we consider the effect of increasing the dimension of the projected space. In this experiment, we set $p=200$, consider the well-posed case where $\kappa(L^*) = 1.4064$, and use $20$ Monte Carlo trials. As we can see all the algorithm show the same trend which verifies that projecting onto the larger space is an effective and practical strategy to achieve small relative error when we do not know the true rank (This is expected according to the theory of MAPLE, while it is not theoretically justified by factorized method).
Finally, the bottom plot in panel (b) shows the effect of increasing condition number of $L^*$. In this experiment, $p=200$, $r^*=r = 5$, and the number of trials equals to $20$. We first let all algorithms run for $50$ iterations, and also consider FGD for more number of iterations, $T=200$ and $T=400$. As it is illustrated, both MAPLE and SVD algorithms are more robust to the large condition number than FGD with $50$ number of iterations. But if we let FGD run longer, it shows the same performance as SVPs which again verifies the dependency of the running time of factorized method to the condition number.
\subsection{Precision Matrix Estimation (PME)}
We start first with synthetic datasets.
We use a diagonal matrix with positive values for the (known) sparse part, $\bar{S}$. For a given number of observed variables $p$, we set $r= 5\%$ as the number of latent variables. We then follow the method proposed in~\cite{ma2013alternating} for generating the sparse and low-rank components $\bar{S}$ and $L^*$. For simplicity, we impose the sparse component to be PSD by forcing it to be positive diagonal matrix. All reported results on synthetic data are the average of 5 independent Monte-Carlo trials.
Our observations comprise $n$ samples, $x_1,x_2,\ldots,x_n \overset{i.i.d}{\thicksim}\mathcal{N}(0,(\bar{S}+L^*)^{-1})$. In our experiments, we used a full SVD as projection step for exact projected-gradient procedure, FGD, ADMM method and nuclear norm minimization. We used CVX to solve nuclear norm minimization; alternatively, one can use other convex approaches.
Panels (a) and (b) in Figure~\ref{allfigsPME} illustrate the comparison of algorithms for PME in terms of the relative error of the estimated $L$ in Frobenius norm versus the ``oversampling" ratio $n/p$. In this experiment, we fixed $p=100$ in (a) and $p=1000$ in (b) and vary $n$. In addition, for both of these results, condition number is given by $\kappa(L^*) = 2.4349$ and $\kappa(L^*) = 2.9666$, respectively. We observe that MAPLE, FGD, and exact projected gradient descent are able to estimate the low-rank matrix even for the regime where $n$ is very small, whereas both ADMM and CVX does not produce very meaningful results.
\begin{figure}
\caption{Comparison of algorithms both in synthetic and real data. (a) relative error of $L$ in Frobenius norm with $p =100$, and $r=r^* = 5$. (b) relative error of $L$ in Frobenius norm with $p =1000$, and $r=r^* = 50$. (c) NLL versus time in Rosetta data set with $p=1000$.}
\end{figure}
We also report the results of several more experiments on synthetic data. In the first experiment, we set $p=100$, $n=400p$, and $r=r^*=5$. Table~\ref{p100n400ptable} lists several metrics that we use for algorithm comparison.
From Table~\ref{p100n400ptable}, we see that MAPLE, FGD, and exact procedure produce better estimates of $L$ compared to ADMM and convex method. As anticipated, the total running time of convex approach is much larger than other algorithms. Finally, the estimated objective function for first three algorithms is very close to the optimal (true) objective function compared to ADMM and CVX.
We increase the dimension to $p=1000$ and reported the same metrics in Table~\ref{p1000n400ptable} similar to Table~\ref{p100n400ptable}. We did not report convex results as it takes long time to be completed. Again, we get the same conclusions as Table~\ref{p100n400ptable}. Important point here is that in this specific application, FGD has better running time compared to MAPLE for both well-condition and ill-condition problem. Here, we did not report the running time for the ill-posed case; however, we observed that FGD is not affected by condition number of ground-truth. We conjecture that FGD delivers a solution for problem~\eqref{opt_probPME} such that its convergence is independent of the condition number of ground-truth similar to~\cite{bhojanapalli2016global} where authors showed that for linear matrix sensing problem, there is no dependency on the condition number if they use FGD method. Proving of this conjecture can be interesting future direction. Also, Tables~\ref{p100n50ptable} and~\ref{p1000n50ptable} show the same experiment discussed in Tables~\ref{p100n400ptable} and~\ref{p1000n400ptable}, but for small number of samples, $n=50p$.
Here, we just evaluate our methods through the \emph{Rosetta} gene expression data set~\cite{hughes2000functional}. This data set includes 301 samples with 6316 variables. We run the ADMM algorithm by~\cite{ma2013alternating} with $p=1000$ variables which have highest variances, and obtained an estimate of the positive definite component $\bar{S}$. Then we used $\bar{S}$ as the input for MAPLE and exact projection procedure. The target rank for all three algorithms is set to be the same as that returned by ADMM. In Figure~\ref{allfigsPME} plot (c), we illustrate the NLL for these algorithms versus wall-clock time (in seconds) over 50 iterations. We observe that all the algorithms demonstrate linear convergence, as predicted in the theory. Among the these algorithms, MAPLE obtains the quickest rate of decrease of the objective function.}
\begin{table*}
\caption{Comparison of different algorithms for $p=100$ and $n=400p$. NLL stands for negative log-likelihood.}
\lambdabel{p100n400ptable}
\vskip 0.1in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\hline
Alg & Estimated NLL & True NLL & Relative error & Total time \\
\hline
FGD & $-9.486278e+01 $ & $-9.485018e+01$ & $2.914218e-01$ & $2.150596e-02$ \\
SVD & $9.485558e+01$ & $-9.485018e+01$ & $3.371867e-01$ & $6.552529e-01$ \\
MAPLE & $-9.485558e+01$ & $-9.485018e+01$ & $3.371742e-01$ & $3.092728e-01$ \\
ADMM & $-9.708976e+01$ & $-9.485018e+01$ & $5.192783e-01$ & $1.475124e+00 $ \\
Convex & $-9.491779e+01$ & $-9.485018e+01$ & $5.192783e-01$ & $7.482316e+02$ \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\end{table*}
\begin{table*}[ht]
\caption{Comparison of different algorithms for $p=1000$ and $n=400p$.}
\lambdabel{p1000n400ptable}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\hline
Alg & Estimated NLL & True NLL & Relative error & Total time \\
\hline
FGD & $-2.638684e+03$ & $-2.638559e+03$ & $3.144617e-01$ & $1.301985e+01$ \\
SVD & $-2.638674e+03 $ &$-2.638559e+03$ & $3.019913e-01$ & $1.584453e+02$ \\
MAPLE & $-2.638675e+03$ & $-2.638559e+03$ & $3.020130e-01$ & $2.565310e+01$ \\
ADMM & $-2.638920e+03$ & $-2.638559e+03$ & $3.921407e-01$ & $3.375073e+02$ \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\end{table*}
\begin{table*}[!t]
\caption{Comparison of different algorithms for $p=100$ and $n=50p$. NLL stands for negative log-likelihood.}
\lambdabel{p100n50ptable}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\hline
Alg & Estimated NLL & True NLL & Relative error & Total time \\
\hline
FGD & $-9.483037e+01$ & $-9.470944e+01$ & $1.034812e+00$ & $2.294928e-02$ \\
SVD & $-9.477855e+01$ & $-9.470944e+01$ & $8.586494e-01$ & $1.026811e+00$ \\
MAPLE & $-9.478611e+01$ & $-9.470944e+01$ & $8.606593e-01$ & $4.854349e-01$ \\
ADMM & $-9.356307e+01$ & $-9.470944e+01$ & $1.823421e+00$ & $3.001534e+00$ \\
Convex & $-9.528296e+01$ & $-9.470944e+01$ & $1.864212e+00$ & $7.046295e+02$ \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\end{table*}
\begin{table*}[ht]
\caption{Comparison of different algorithms for $p=1000$ and $n=50p$.
}
\lambdabel{p1000n50ptable}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccccr}
\hline
Alg & Estimated NLL & True NLL & Relative error & Total time \\
\hline
FGD & $-2.639646e+03$ & $-2.638491e+03$ & $1.155856e+00$ & $1.335701e+01$ \\
SVD & $-2.638804e+03$ & $-2.638491e+03$ & $8.610451e-01$ & $1.567543e+02$ \\
MAPLE & $-2.638878e+03$ & $-2.638491e+03$ & $8.722342e-01$ & $2.606750e+01$ \\
ADMM & $-2.643757e+03$ & $-2.638491e+03$ & $1.517834e+00$ & $4.019458e+02$ \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\end{table*}
\Big{\|}ewpage
\small
\Big{\|}ewpage
\lambdabel{append}
\subsection{Proofs}
We provide full proofs of all theorems discussed in this paper.
Below, $\mathcal{M}(\mathbb{U}_r)$ denotes the set of vectors associated with $\mathbb{U}_r$, the set of all rank-r matrix subspaces. We show the maximum and minimum eigenvalues of a matrix $A\in{\mathbb{R}}^{p\times p}$ as $\lambdambda_{\min}(A), \lambdambda_{\max}(A)$, respectively. Furthermore $\sigmagma_i(A)$ denotes the $i^{th}$ largest singular value of matrix $A$.
We need the following equivalent definitions of restricted strongly convex and restricted strong smoothness conditions.
\begin{definition} \lambdabel{defRSCRSS_app}
A function $f$ satisfies the Restricted Strong Convexity (RSC) and Restricted Strong Smoothness (RSS) conditions if one of the following equivalent definitions is satisfied for all $L_1,L_2, L\in\mathbb{R}^{p\times p}$ such that $\text{rank}(L_1)\leq r, \text{rank}(L_2)\leq r,rank(L)\leq r$:
\begin{align}
&\frac{m_r}{2}\|L_2-L_1\|^2_F \leq f(L_2) - f(L_1) - \lambdangle\Big{\|}abla f(L_1) , L_2-L_1\rangle\leq\frac{M_{r}}{2}\|L_2-L_1\|^2_F, \lambdabel{rscrss_app1}\\
&\hspace{5mm}m_{r}\|L_2-L_1\|^2_F \leq\lambdangle{\mathcal{P}}_U\left(\Big{\|}abla f(L_2) - \Big{\|}abla f(L_1)\right), L_2-L_1\rangle \leq M_{r}\|L_2-L_1\|^2_F, \lambdabel{rscrss_app2}\\
&\hspace{35mm}m_{r} \leq\|{\mathcal{P}}_U\Big{\|}abla^2 f(L)\|_2 \leq M_{r},\lambdabel{rscrss_app3} \\
&\hspace{7mm}m_{r}\|L_2-L_1\|_F \leq\|{\mathcal{P}}_U\left(\Big{\|}abla f(L_2) - \Big{\|}abla f(L_1)\right)\|_F \leq M_{r}\|L_2-L_1\|_F,\lambdabel{rscrss_app4}
\end{align}
where $U$ is the span of the union of column spaces of the matrices $L_1$ and $L_2$. Here, $m_r$ and $M_r$ are the RSC and RSS constants, respectively.
\end{definition}
\subsection{Proof of theorems in section~\ref{prelim} }
Before proving the main theorems, we restate the following hard-thresholding result from lemma $3.18$ in~\cite{li2016Nonconvex}:
\begin{lemma}\lambdabel{HT}
For $r>r^*$ and for any matrix $L\in{\mathbb{R}}^{p\times p}$, we have
\begin{align}
\|H_r(L)-L^*\|_F^2\leq\left(1+\frac{2\sqrt{r^*}}{\sqrt{r-r^*}}\right)\|L-L^*\|_F^2,
\end{align}
where $\text{rank}(L^*)$ = $r^*$, and $H_r(.):{\mathbb{R}}^{p\times p}\rightarrow\mathbb{U}_{r}$ denotes the singular value thresholding operator, which keeps the largest $r$ singular values and sets the others to zero.
\end{lemma}
For proving theorem~\ref{AppSVD}, we cannot use directly lemma~\ref{HT} since $\widetildea$ operator returns an approximation of the top $r$ singular vectors, and using exact projection in the proof of lemma~\ref{HT} is necessary~\cite{li2016Nonconvex}. However, we can modify the proof of lemma~\ref{HT} to make it applicable through the approximate projection approach. Hence, we can prove Lemma~\ref{ApprHTh}:
\begin{proof}[Proof of Lemma~\eqref{ApprHTh}]
The proof is similar to the procedure described in~\cite{li2016Nonconvex} with some modification based on the per-vector guarantee property of approximate projection. In this work, the proof in is given first for sparse hard thresholding, and then is generalized to the low-rank case using Von Neumann's trace inequality, i.e., for two matrices $A,B\in{\mathbb{R}}^{p\times p}$ and corresponding singular values $\sigmagma_i(A)$ and $\sigmagma_i(B)$, respectively, we have:
\begin{align}\lambdabel{VoNeu}
\lambdangle A,B\rangle = \Sigma_{k=1}^{\min\{\text{rank}(A),\text{rank}(B)\}}\sigmagma_k(A)\sigmagma_k(B).
\end{align}
First define $\theta = [\sigmagma_1^2(L),\sigmagma_2^2(L)\ldots,\sigmagma_r^2(L)]^T$. Let $\theta^* = [\sigmagma_1^2(L^*),\sigmagma_2^2(L^*)\ldots,\sigmagma_r^2(L^*)]^T$, and $\theta' = \widetildea(\theta)$. Also, let $supp(\theta^*) = \mathcal{I^*}$, $supp(\theta) = \mathcal{I}$, $supp(\theta') = \mathcal{I'}$, and $\theta'' =\theta - \theta'$ with support $I''$. It follows that $$\|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2\leq 2\lambdangle\theta'',\theta^*\rangle - \|\theta''\|_2^2.$$ Now define new sets $\mathcal{I^*}\cap\mathcal{I'} = \mathcal{I}^{*1}$ and $\mathcal{I^*}\cap\mathcal{I''} = \mathcal{I}^{*2}$ with restricted vectors to these sets as $\theta_{\mathcal{I}^{*1}} = \theta^{*1}$, $\theta_{\mathcal{I}^{*2}} = \theta^{*2}$, $\theta'_{\mathcal{I}^{*1}} = \theta^{1*}$, and $\theta''_{\mathcal{I}^{*2}} = \theta^{2*}$ such that $|\mathcal{I}^{*2}| = r^{**}$. Hence, $\|\theta^{2*}\|_2 = \beta\theta_{\max}$ where $\beta\in[\sqrt{r^{**}}]$ and $\theta_{\max} = \|\theta^{2*}\|_{\infty}$. By these definitions, we have:
$$ \|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2\leq2\|\theta^{2*}\|_2\|\theta^{*2}\|_2 - \|\theta^{2*}\|_2^2.$$
The proof continues to discuss in three cases as:
\begin{enumerate}
\item if $\|\theta^{2*}\|_2\leq \theta_{\max}$, then $\beta=1$.
\item if $\theta_{\max}\leq\|\theta^{2*}\|_2< \sqrt{r^{**}}\theta_{\max}$, then $\beta = \frac{\|\theta^{2*}\|_2}{\theta_{\max}}$.
\item if $\|\theta^{2*}\|_2\geq\sqrt{r^{**}}\theta_{\max}$, then $\beta = \sqrt{r^{**}}$.
\end{enumerate}
In each case, the ratio of $\frac{\|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2}{\|\theta- \theta^*\|_2^2}$ is upper bounded in terms of $r,r^*, r^{**}$ and by using the inequality $|\theta_{\min}|\geq|\theta_{\max}|$ where $|\theta_{\min}|$ is defined as the smallest entry of $\theta^{1*}$. This inequality holds due to the exact hard thresholding. However, it does not necessary hold when approximate projection is used. To resolve this problem, we note that in our framework the approximate tail projection is implemented via any randomized SVD method which supports the so-called per-vector guarantee. Recall from our discussion in section~\ref{prelim}, the per vector guarantee condition means:
$$|u_i^TLL^Tu_i - z_iLL^Tz_i|\leq\epsilon\sigmagma_{r+1}^2\leq\epsilon\sigmagma_{i}^2, \ \ \ \ \ i\in[r].$$ In our implementation, we use randomized block Krylov method (BK-SVD) which supports this condition. In our notations, this condition implies, $|\theta_{\min} - \widehat{\theta}_{\min}|\leq\epsilon\theta_{\min}$ where $\widehat{\theta} = [\widehat{\sigmagma}_1^2(L),\widehat{\sigmagma}_2^2(L)\ldots,\widehat{\sigmagma}_r^2(L)]^T$. By combing with $|\theta_{\min}|\geq|\theta_{\max}|$, we thus have $\widehat{\theta}_{\min}\geq(1-\epsilon)\theta_{\max}$. Now by this modification, we can continue the proof with the procedure described in~\cite{li2016Nonconvex}. Let $I:=\frac{\|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2}{\|\theta- \theta^*\|_2^2}$. For each case, we have::
\begin{itemize}
\item case 1: $I\leq \frac{\theta_{\max}^2}{(r-r^*+r^{**} )(1-\epsilon)\theta^2_{\min}-\theta^2_{\max}}\leq \frac{1}{(r-r^*+r^{**} )(1-\epsilon)-1}$.
\item case 2: $I\leq \frac{r^{**}\theta^2_{\max}}{(r-r^*+r^{**})(1-\epsilon)\theta^2_{\min}}\leq\frac{r^{**}}{(r-r^*+r^{**})(1-\epsilon)}$.
\item case 3: $I\leq\frac{2\gamma\sqrt{r^{**}}\theta^2_{\max}-r^{**}\theta^2_{\max}}{(r-r^*+r^{**})(1-\epsilon)\theta^2_{\min}+r^{**}\theta^2_{\max}+\gamma^2\theta^2_{\max}-2\gamma\sqrt{r^{**}}\theta^2_{\max}}\\
~~~~~~~~~~~~~\overset{e_1}\leq\frac{2\sqrt{r^{**}}}{2\sqrt{(r-r^*)(1-\epsilon) +r^{**}(\frac{5}{4}-\epsilon)}-\sqrt{r^{**}}}$ \ \ , for some $\gamma\geq\sqrt{r^{**}}$.
\end{itemize}
In all the above cases, we have used the fact that $\widehat{\theta}_{\min}\geq(1-\epsilon)\theta_{\max}$. In addition, $e_1$ in case 3 holds due to maximizing the R.H.S. with respect to $\gamma$. After taking derivative w.r.t. $\gamma$, setting to zero, and solving the resulted quadratic equation, we obtain that:
$$ \gamma = \max\Big{\{} \sqrt{r^{**}},\frac{\sqrt{r^{**}}}{2} +\sqrt{(r-r^*)(1-\epsilon) +r^{**}(\frac{5}{4}-\epsilon)} \Big{\}} $$
Now if we plug in the value of $\gamma$ in the R.H.S of case 3, we obtain the claimed bound. Putting the three above bounds all together, we have:
\begin{align*}
\frac{\|\theta'-\theta^*\|_2^2 - \|\theta- \theta^*\|_2^2}{\|\theta- \theta^*\|_2^2}&\leq\max\Big{\{}\frac{1}{(r-r^*+r^{**} )(1-\epsilon)-1},\frac{r^{**}}{(r-r^*+r^{**})(1-\epsilon)}, \\
&\hspace{3cm}\frac{2\sqrt{r^{**}}}{2\sqrt{(r-r^*)(1-\epsilon) +r^{**}(\frac{5}{4}-\epsilon)}-\sqrt{r^{**}}}\Big{\}}\\
&\overset{e_1}\leq\frac{2\sqrt{r^{**}}}{2\sqrt{(r-r^*)(1-\epsilon) +r^{**}(\frac{5}{4}-\epsilon)}-\sqrt{r^{**}}}\\
&\overset{e_2}\leq\frac{2\sqrt{r^{*}}}{2\sqrt{(r-r^*)(1-\epsilon) }-\sqrt{r^{*}}}\\
&\overset{e_3}\leq\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r^*}}{\sqrt{r-r^*}},
\end{align*}
where $e_1$ follows by choosing $r$ sufficiently large and the fact that $\epsilon$ can be chosen arbitrary small (this inceases the running time of the approximate projection by $\log(\frac{1}{\epsilon})$ factor), $e_2$ holds due to $r^{**}\leq r^*$, and finally $e_3$ holds by the assumption on $r$ in the lemma. This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{AppSVD}]
Let $V^t, V^{t+1}$, and $V^{*}$ denote the bases for the column space of $L^t, L^{t+1}$, and $L^*$, respectively. Assume $\Big{\|}u = \sqrt{1+\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r^*}}{\sqrt{r-r^*}}}$. Also, by the definition of the tail projection, we have $L^{t}\in\mathcal{M}(\mathbb{U}_{r})$, and by definition of set $J$ in the theorem, $V^t\cup V^{t+1}\cup V^*\subseteq J_t := J$ such that $rank(J_t)\leq 2r+r^*\leq3r$. Define $b = L^t -\eta{\mathcal{P}}_J\Big{\|}abla F(L^t)$. We have:
\begin{align}
\lambdabel{linApp}
\|L^{t+1} - L^*\|_F&\overset{e_1}{\leq}\Big{\|}u\|b-L^*\|_F \Big{\|}onumber \\
& \leq\Big{\|}u\|L^t - L^*-\eta{\mathcal{P}}_J\Big{\|}abla F(L^t)\|_F \Big{\|}onumber \\
&\overset{e_2}{\leq}\Big{\|}u\|L^t - L^*-\eta{\mathcal{P}}_J\left(\Big{\|}abla F(L^t)- \Big{\|}abla F(L^*) \right)\|_F + \Big{\|}u\eta\|{\mathcal{P}}_J\Big{\|}abla F(L^*)\|_F \Big{\|}onumber \\
&\overset{e_3}{\leq}\Big{\|}u\sqrt{1+M_{2r+r^*}^2\eta^2 - 2m_{2r+r^*}\eta}\|L^t-L^*\|_F + \Big{\|}u\eta\|{\mathcal{P}}_J\Big{\|}abla F(L^*)\|_F
\end{align}
where $e_1$ holds due to applying lemma~\ref{HT}. Moreover, $e_2$ holds by applying triangle inequality and $e_3$ is obtained by combining the lower bound in~\eqref{rscrss_app2} and upper bound in~\eqref{rscrss_app4}, i.e.,
$$\|L^t - L^* - \eta^{\mathbb{P}ime}\left(\Big{\|}abla_J F(L^t) -\Big{\|}abla_J F(L^*)\right)\|_2^2\leq(1+{\eta^{\mathbb{P}ime}}^2M_{2r+r^*}^2-2\eta^{\mathbb{P}ime} m_{2r+r^*})\|L^t-L^*\|_2^2.$$
In order that~\eqref{linApp} implies convergence, we require that
$$\rho = \left(\sqrt{1+\frac{2}{\sqrt{1-\epsilon}}\frac{\sqrt{r^*}}{\sqrt{r-r^*}}}\right)\sqrt{1+M_{2r+r^*}^2\eta^2 - 2m_{2r+r^*}\eta}<1$$.
By solving this quadratic inequality with respect to $\eta$, we obtain:
\begin{align*}
\left(\frac{M_{2r+r^*}}{m_{2r+r^*}}\right)^2\leq1+\frac{\sqrt{r-r^*}\sqrt{1-\epsilon}}{2\sqrt{r^*}},
\end{align*}
As a result, we obtain the the condition $r\geq \frac{C_1}{1-\epsilon}\left(\frac{M_{2r+r^*}}{m_{2r+r^*}}\right)^4r^*$ for some $C_1>0$. Furthermore, since $r = \alpha r^*$ for some $\alpha> 1$, we conclude the condition on step size $\eta$ as $\frac{1-\sqrt{\alpha'}}{M_{2r+r^*}}\leq\eta\leq\frac{1+\sqrt{\alpha'}}{m_{2r+r^*}}$ where $\alpha' = \frac{\sqrt{\alpha-1}}{\sqrt{1-\epsilon}\sqrt{\alpha-1}+2}$. This completes the proof of Theorem~\ref{AppSVD}.
\end{proof}
\subsection{Proof of theorems in section~\ref{NLARMsec} }
We first prove the statistical error rate, staing in Theorem~\ref{staterrorNLARM}.
\begin{proof}[proof of Theorem~\ref{staterrorNLARM}]
Let $b_i = vec(A_i)\in{\mathbb{R}}^{p^2}$ denotes the $i^{th}$ row of matrix $X\in{\mathbb{R}}^{m\times p^2}$, defining in the section~\ref{NLARMsec} for $i = 1,\ldots,n$. Since $X$ is constructed by uniform randomly chosen $m$ rows of a $p^2\times p^2$ DFT matrix multiplied by a diagonal matrix whose diagonal entries are uniformly distributed over $\{-1,+1\}^{P^2}$, $\frac{1}{\sqrt{n}}X$ satisfies the rank-$r$ RIP condition with probability at least $1-\exp(-cn\varpi^2)$ ($c>0$ is a constant) provided that $m= {\mathcal{O}}(\frac{1}{\varpi^2}pr\text{\text{poly}log(p)})$~\cite{candes2011tight}. On the other hand, if a matrix $B$ satisfies the rank-r RIP condition, then~\cite{lee2010admira}
\begin{align}\lambdabel{ripprop}
\Big{\|}{\mathcal{P}}_U\frac{1}{\sqrt{n}}B^*a\Big{\|}_2\leq (1+\delta_r)\|a\|_2, \ \ for \ all \ a\in{\mathbb{R}}^n,
\end{align}
where $U$ denotes the set of rank-$r$ matrices, and $\delta_{r}$ is the RIP constant. As a result, for all $t=1,\ldots,T$ we have:
$$ \Big{\|}\frac{1}{n}{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\Big{\|}_F = \Big{\|}\frac{1}{n}\mathcal{A}^*e\Big{\|}_F = \frac{1}{\sqrt{n}}\Big{\|}{\mathcal{P}}_{J_t}\frac{1}{\sqrt{n}}X^*e\Big{\|}_2\leq \frac{1+\delta_{2r+r^*}}{\sqrt{n}}\|e\|_2,$$
where the last inequality holds due to~\eqref{ripprop} ($\frac{1}{\sqrt{n}}X$ has RIP constant $\delta_r$, and from our definition, $rank(J_t)\leq 2r+r^*$), and the fact that $e\in{\mathbb{R}}^n$.
\end{proof}
\begin{proof}[proof of corollary~\ref{induclin}]
Consider upper bound in~\eqref{linApp}. By using induction, zero initialization, and Theorem~\ref{staterrorNLARM}, we obtain $\vartheta$ accuracy after $T_{iter} = \mathcal{O}\left(\log\left(\frac{\|L^*\|_F}{\vartheta}\right)\right)$ iterations. In other words, after $T_{iter}$ iterations, we obtain:
$$ \|L^{T+1} - L^{*}\|_F\leq \vartheta + \frac{1}{\sqrt{n}}\frac{\Big{\|}u\eta(1+\delta_{2r+r^*})}{1-\rho}\|e\|_2.$$
\end{proof}
The above results shows the linear convergence of APRM if there is no additive noise. We now prove that the objective function defined in problem~\eqref{opt_probNLARM} satisfies the RSC/RSS conditions in each iteration.
\begin{proof}[proof of Theorem~\ref{RSCRSSappNLARM}]
Let $L = L^t$ for all $t=1,\ldots,T$. We follow the approach in~\cite{soltani2016fastIEEETSP17}. hence, we use the the Hessian based definition of RSC/RSC, stating in equation~\eqref{rscrss_app3} in definition~\ref{defRSCRSS_app}. We note that the Hessian of $F(L)$ is given by:
\[
\Big{\|}abla^2F(L) = \frac{1}{n}\sum_{i=1}^{n}A_ig'(\lambdangle A_i,L\rangle)A_i^T,
\]
According to our assumption on the link function, we know $0<\mu_1\leq g'(x)\leq\mu_2$ for all $x\in\mathcal{D}(g)$. As a result $\lambdambda_{\min}(\Big{\|}abla^2F(L))\geq0$ due to the positive semidefinite of $A_iA_i^T$ for all $i=1,\dots,n$. Now let ${\cal L}ambda_{\max} =\max_U \lambdambda_{\max}({\mathcal{P}}_U\Big{\|}abla^2F(L))$ and ${\cal L}ambda_{\min} =\min_U \lambdambda_{\min}({\mathcal{P}}_U\Big{\|}abla^2F(L))$. Moreover, let $W$ be any set of rank-$2r$ matrices such that $U\subseteq W$. We have:
\begin{align}\lambdabel{RIPNLARM}
\mu_1\min_W \lambdambda_{\min}\left({\mathcal{P}}_W\left(\frac{1}{n}\sum_{i=1}^{n}A_iA_i^T\right)\right)\leq{\cal L}ambda_{\min}
\leq{\cal L}ambda_{\max}\leq\mu_2\max_W \lambdambda_{\max}\left({\mathcal{P}}_W\left(\frac{1}{n}\sum_{i=1}^{n}A_iA_i^T\right)\right),
\end{align}
Now, we need to bound the upper bound and the lower bound in the above inequality. To do this, we are using the assumption on the design matrices $A_i$'s, stating in the theorem. According to this, we can write, ${\mathcal{P}}_W\left(\frac{1}{n}\sum_{i=1}^{n}A_iA_i^T\right) = {\mathcal{P}}_W\left(\frac{1}{n}X^TX\right)$. We follow the approach of~\cite{HegdeFastUnionNips2016}. Now fix any set $W$ as defined above. Recall that $X =X'D$, where $X'$ is a partial Fourier or partial Hadamard matrix. Thus, by~\cite{haviv2017restricted}, $X'$ satisfies RIP condition with constant $4\upsilon$ over the set of of $s$-sparse vectors with high probability when $m={\mathcal{O}}\left(\frac{1}{\upsilon^2}s\log^2(\frac{s}{\upsilon})\log(p)\right)$. Also from~\cite{krahmer2011new}, $X$ is a $(1\pm{\mathbf x}i)-$Johnson-Lindenstrauss embedding (with $4\upsilon<{\mathbf x}i$) for set $W$ with probability at least $1-\varsigma$ provided that $s >{\mathcal{O}}(\frac{V}{\varsigma})$, where $V$ is the number of vectors in $W$. In other words, the Euclidean distance between any two vectors (matrix) $\beta_1,\beta_2\in W\in{\mathbb{R}}^{p\times p}$ is preserved up to a $\pm{\mathbf x}i$ by application of $X$. As a result, with high probability
\[
1-{\mathbf x}i\leq\lambdambda_{\min}\left({\mathcal{P}}_W\left(\frac{1}{n}\sum_{i=1}^{n}A_iA_i^T\right)\right)\leq\lambdambda_{\max}\left({\mathcal{P}}_W\left(\frac{1}{n}\sum_{i=1}^{n}A_iA_i^T\right)\right)\leq 1+{\mathbf x}i
\]
Now it remains to argue the final bound in~\eqref{RIPNLARM}. By~\cite{candes2011tight}, we know that the set of $p\times p$ rank-$r$ matrices can be discretized by a $\zeta$-cover $S_r$ such that $|S_r| = (\frac{9}{\zeta})^{(2p+1)r}$. In addition, They show that if a matrix $X$ satisfies JL embedding by constant ${\mathbf x}i$, then $X$ satisfies the rank-$r$ RIP with constant $\omega={\mathcal{O}}({\mathbf x}i)$. As a result, by taking union bound (taking maximum over all set $W$ in~\eqref{RIPNLARM}), we establish RSC/RSS constants such that $M_{2r+r^*}\leq \mu_2(1+\omega)$ and $m_{2r+r^*}\geq \mu_1(1-\omega)$ provided that $s={\mathcal{O}}(pr)$ and $V = |S_r|$ which implies $m= {\mathcal{O}}(pr\text{\text{poly}log(p)})$. Now, In order to satisfy the assumptions in Theorem~\ref{AppSVD}, we need to have $\frac{M_{2r+r^*}^2}{m_{2r+r^*}^4}\leq C_2(1-\epsilon)\frac{r}{r^*}$ for some $C_2>0$ and $\epsilon$ defined in lemma~\ref{ApprHTh}. Thus, we have $\frac{\mu_2^4(1+\omega)^4}{\mu_1^4(1-\omega)^4}\leq C_2(1-\epsilon)\frac{r}{r^*}$ which justifies the assumption in Theorem~\ref{RSCRSSappNLARM}.
\end{proof}
\subsection{Proof of theorems in section~\ref{PME} }
\begin{proof}[Proof of Theorem~\ref{ExactSVD}]
Let $V^t, V^{t+1}$, and $V^{*}$ denote the bases for the column space of $L^t, L^{t+1}$, and $L^*$, respectively. Assume $\Big{\|}u' = \sqrt{1+\frac{2\sqrt{r^*}}{\sqrt{r-r^*}}}$. By definition of set $J$ in the theorem, $V^t\cup V^{t+1}\cup V^*\subseteq J_t := J$ and $\text{rank}(J_t)\leq 2r+r^*$. Define $b = L^t -\eta'{\mathcal{P}}_J\Big{\|}abla F(L^t)$. We have:
\begin{align}
\lambdabel{linExact}
\|L^{t+1} - L^*\|_F&\overset{e_1}{\leq}\Big{\|}u'\|b-L^*\|_F \Big{\|}onumber \\
& \leq\Big{\|}u\|L^t - L^*-\eta'{\mathcal{P}}_J\Big{\|}abla F(L^t)\|_F \Big{\|}onumber \\
&\overset{e_2}{\leq}\Big{\|}u'\|L^t - L^*-\eta'{\mathcal{P}}_J\left(\Big{\|}abla F(L^t)- \Big{\|}abla F(L^*) \right)\|_F + \Big{\|}u'\eta'\|{\mathcal{P}}_J\Big{\|}abla F(L^*)\|_F \Big{\|}onumber \\
&\overset{e_3}{\leq}\Big{\|}u'\sqrt{1+M_{2r+r^*}^2\eta'^2 - 2m_{2r+r^*}\eta'}\|L^t-L^*\|_F + \Big{\|}u'\eta'\|{\mathcal{P}}_J\Big{\|}abla F(L^*)\|_F,
\end{align}
where $e_1$ holds due to applying lemma~\ref{HT}. Moreover, $e_2$ holds by applying triangle inequality and $e_3$ is obtained by combining the lower bound in~\eqref{rscrss_app2} and upper bound in~\eqref{rscrss_app4}, i.e.,
$$\|L^t - L^* - \eta^{\mathbb{P}ime}\left(\Big{\|}abla_J F(L^t) -\Big{\|}abla_J F(L^*)\right)\|_2^2\leq(1+{\eta^{\mathbb{P}ime}}^2M_{2r+r^*}^2-2\eta^{\mathbb{P}ime} m_{2r+r^*})\|L^t-L^*\|_2^2.$$
In order that~\eqref{linExact} implies convergence, we require that
$$\rho' = \sqrt{1+2\frac{\sqrt{r^*}}{\sqrt{r-r^*}}}\sqrt{1+M_{2r+r^*}^2\eta'^2 - 2m_{2r+r^*}\eta'}<1$$.
By solving this quadratic inequality with respect to $\eta$, we obtain:
\begin{align*}
\left(\frac{M_{2r+r^*}}{m_{2r+r^*}}\right)^2\leq1+\frac{\sqrt{r-r^*}}{2\sqrt{r^*}},
\end{align*}
As a result, we obtain the the condition $r\geq C_1'\left(\frac{M_{2r+r^*}}{m_{2r+r^*}}\right)^4r^*$ for some $C_1'>0$. Furthermore, since $r = \alpha r^*$ for some $\beta> 1$, we conclude the condition on step size $\eta'$ as $\frac{1-\sqrt{\beta'}}{M_{2r+r^*}}\leq\eta'\leq\frac{1+\sqrt{\beta'}}{m_{2r+r^*}}$ where $\beta' = \frac{\sqrt{\beta-1}}{\sqrt{\beta-1}+2}$ for some $\beta>1$.
If we initialize at $L^0 = 0$, then we obtain $\vartheta$ accuracy after $T = \mathcal{O}\left(\log\left(\frac{\|L^*\|_F}{\vartheta}\right)\right)$ iterations.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{BoundGrad}]
The proof of this theorem is a direct application of the Lemma 5.4 in \cite{Venkat2009sparse} and we restate it for completeness:
\begin{lemma}
\lambdabel{boundSmCo}
Let $C$ denote the sample covariance matrix, then with probability at least $1 - 2\exp(-p)$ we have $\|C - (S^* + L^*)^{-1}\|_2\leq c_1\sqrt{\frac{p}{n}}$ where $c_1>0$ is a constant.
\end{lemma}
By noting that $\Big{\|}abla F(L^{*}) = C - (S^* + L^*)^{-1}$ and $\text{rank}(J_t)\leq 2r+r^*\leq3r$, we can bound the term on the right hand side in Theorem~\ref{ExactSVD} as:
$$\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F\leq \sqrt{3r}\|\Big{\|}abla F(L^{*})\|_2\leq c_2\sqrt{\frac{rp}{n}} .$$
\end{proof}
The key observation is that the objective function in~\eqref{opt_probPME} is globally strongly convex, and when restricted to any compact psd cone, it also satisfies the smoothness condition. As a result, it satisfies RSC/RSS conditions. Our strategy to prove Theorems~\ref{RSCRSSex} and~\ref{RSCRSSapp} is to establish upper and lower bounds on the spectrum of the sequence of estimates $L^t$ independent of $t$. We use the following lemma.
\begin{lemma}\cite{yuan2013gradient,boyd2004convex}
\lambdabel{hessian}
The Hessian of the objective function $F(L)$ is given by $\Big{\|}abla^2F(L) = \widetildeheta^{-1}\otimes\widetildeheta^{-1}$ where $\otimes$ denotes the Kronecker product and $\widetildeheta = \bar{S} + L$. In addition if $\alpha I\mathbb{P}eceq\widetildeheta\mathbb{P}eceq\beta I$ for some $\alpha$ and $\beta$, then $\frac{1}{\beta^2} I\mathbb{P}eceq\Big{\|}abla^2F(L)\mathbb{P}eceq\frac{1}{\alpha^2} I$.
\end{lemma}
\begin{lemma}[Weyl type inequality]
\lambdabel{weyl}
For any two matrices $A,B\in{\mathbb{R}}^{p\times p}$, we have:
$$\max_{1\leq i\leq p} |\sigmagma_i(A+B) - \sigmagma_i(A)|\leq\|B\|_2.$$
\end{lemma}
If we establish an universal upper bound and lower bound on $\lambdambda_1(\widetildeheta^t)$ and $\lambdambda_p(\widetildeheta^t)$ for all $t=1\dots T$, then we can bound the RSC constant as $m_{2r+r^*}\geq\frac{1}{\lambdambda_1(\widetildeheta^t)^2}$ and the RSS-constant as $M_{2r+r^*}\leq\frac{1}{\lambdambda_p(\widetildeheta^t)^2}$ using Lemma~\ref{hessian} and the definition of RSS/RSC.
\begin{proof}[Proof of Theorem~\ref{RSCRSSex}]
Recall that by Theorem~\ref{ExactSVD}, we have:
$$\|L^{t} - L^{*}\|_F\leq \rho'\|L^{t-1} - L^{*}\|_F + \Big{\|}u'\eta'\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^{*})\|_F,$$
By Theorem~\ref{BoundGrad}, the second term on the right hand side can be bounded by ${\mathcal{O}}(\sqrt{\frac{rp}{n}})$ with high probability. Therefore, recursively applying this inequality to $L^t$ (and initializing with zero), we obtain:
\begin{align}
\lambdabel{firstexactupp}
\|L^{t} - L^{*}\|_F\leq(\rho')^t\|L^*\|_F+ \frac{c_2\Big{\|}u'\eta'}{1-\rho'} \sqrt{\frac{rp}{n}}.
\end{align}
Since $\rho'<1$, then $ (\rho')^t<1$. On the other hand $\|L^*\|_F\leq\sqrt{r^*}\|L^*\|_2$. Hence, $\rho^t\|L^*\|_F\leq\sqrt{r^*}\|L^*\|_2$. Also, by the Weyl inequality, we have:
\begin{align}
\|L^t\|_2 - \|L^*\|_2\leq\|L^t-L^*\|_2\leq\|L^t-L^*\|_F.
\end{align}
Combining~\eqref{firstexactupp} and~\eqref{secondexactupp} and using the fact that $\lambda_1(L^{t})\leq\sigma_1(L^{t})$,
\begin{align}
\lambda_1(L^{t})&\leq\|L^*\|_2+\|L^t-L^*\|_F\Big{\|}onumber \\
&\leq\|L^*\|_2+\sqrt{r^*}\|L^*\|_2+ \frac{c_2\Big{\|}u'\eta'}{1-\rho'} \sqrt{\frac{rp}{n}}.\Big{\|}onumber
\end{align}
Hence for all $t$,
\begin{align}
\lambdabel{secondexactupp}
\lambda_1(\widetildeheta^t) = S_1+\lambda_1(L^t)\leq S_1 +\left(1+\sqrt{r^*}\right)\|L^*\|_2+\frac{c_2\Big{\|}u'\eta'}{1-\rho'} \sqrt{\frac{rp}{n}}.
\end{align}
For the lower bound, we trivially have for all $t$:
\begin{align}
\lambda_p(\widetildeheta^t) = \lambda_p(\bar{S} + L^t)\geq S_p.
\end{align}
If we select $n={\mathcal{O}}\left(\frac{1}{\delta^2}\left(\frac{\Big{\|}u'\eta'}{1-\rho'}\right)^2rp\right)$ for some small constant $\delta>0$, then~\eqref{secondexactupp} becomes:
\begin{align*}
\lambda_1(\widetildeheta^t)\leq S_1 +\left(1+\sqrt{r}\right)\|L^*\|_2+ \delta.
\end{align*}
As mentioned above, we set $m_{2r+r^*}\geq\frac{1}{\lambdambda_1^2(\widetildeheta^t)}$ and $M_{2r+r^*}\leq\frac{1}{\lambdambda_p^2(\widetildeheta^t)}$ which implies $\frac{M_{2r+r^*}}{m_{2r+r^*}}\leq\frac{\lambda_1^2(\widetildeheta^t)}{\lambda_p^2(\widetildeheta^t)}$. In order to satisfy the assumption on the RSC/RSS in theorem~\ref{ExactSVD}, i.e., $\frac{M_{2r+r^*}^4}{m_{2r+r^*}^4}\leq C_2'\frac{r}{r^*}$ for some $C_2'>0$, we need to establish a regime such that $\frac{\lambda_1^8(\widetildeheta^t)}{\lambda_p^8(\widetildeheta^t)}\leq C_2'\frac{r}{r^*}$.
As a result, to satisfy this condition, we need to have the following condition, verifying the assumption in the theorem.
\begin{align}
S_p\leq S_1\leq C_3'(\frac{r}{r^*})^{\frac{1}{8}}S_p - \left(1+\sqrt{r^*}\right)\|L^*\|_2- \delta.
\end{align}
for some constant $C_3'>0$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{RSCRSSapp}]
The proof is similar to the proof of theorem~\ref{RSCRSSex}. Recall that by theorem~\ref{AppSVD}, we have
$$\|L^{t+1} - L^{*}\|_F\leq \rho\|L^{t} - L^{*}\|_F + \Big{\|}u\eta\|{\mathcal{P}}_{J_t}\Big{\|}abla F(L^*)\|_F,$$
As before, the second term on the right hand side is bounded by ${\mathcal{O}}(\sqrt{\frac{rp}{n}})$ with high probability by Theorem~\ref{BoundGrad}. As above, recursively applying this inequality to $L^t$ and using zero initialization, we obtain:
\begin{align*}
\|L^{t} - L^{*}\|_F\leq\rho^t\|L^*\|_F + \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}.
\end{align*}
Since $\rho<1$, then $\rho^t<1$. Now similar to the exact algorithm, $\|L^*\|_F\leq\sqrt{r^*}\|L^*\|
|_2$ and $\rho_1^t\|L^*\|_F\leq\sqrt{r^*}\|L^*\|_2$. , Hence with high probability,
\begin{align}
\lambdabel{secontineqApp}
\lambda_1(L^{t})&\leq\|L^*\|_2+\|L^t-L^*\|_F\Big{\|}onumber \\
&\overset{}{\leq}\left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}},
\end{align}
Hence, for all $t$:
\begin{align}
\lambdabel{firstineqApp}
\lambda_1(\widetildeheta^t) = S_1+\lambda_1(L^t)\leq S_1 + \left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}},
\end{align}
Also, we trivially have:
\begin{align}
\lambda_p(\widetildeheta^t)=\lambda_p(\bar{S} + L^t)\geq S_p - a^{\mathbb{P}ime}, \ \forall t.
\end{align}
By selecting $n={\mathcal{O}}\left(\frac{1}{\delta^{\prime2}}\left(\frac{\Big{\|}u\eta}{1-\rho}\right)^2rp\right)$ for some small constant $\delta^{\prime}>0$, \eqref{firstineqApp} can be written as follows:
\begin{align*}
\lambda_1(\widetildeheta^t)\leq S_1 + \left(1+\sqrt{r^*}\right)\|L^*\|_2+\delta^{\prime},
\end{align*}
In order to satisfy the assumptions in Theorem~\ref{AppSVD}, i.e., $\frac{M_{2r+r^*}^4}{m_{2r+r^*}^4}\leq \frac{C_2''}{1-\epsilon}\frac{r}{r^*}$, we need to guarantee that $\frac{\lambda_1^8(\widetildeheta^t)}{\lambda_p^8(\widetildeheta^t)}\leq \frac{C_2''}{1-\epsilon}\frac{r}{r^*}$. As a result, to satisfy this inequality, we need to have the following condition on $S_1$ and $S_p$:
\begin{align}
S_p\leq S_1&\leq \frac{C_3''}{(1-\epsilon)^{\frac{1}{8}}}(\frac{r}{r^*})^{\frac{1}{8}}(S_p-a^{\prime}) - \left(1+\sqrt{r^*}\right)\|L^*\|_2- \delta^{\prime}.
\end{align}
for some $C_3''>0$. Also, we can choose RSC/RSS constant as previous case.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{mineigval}]
Recall from~\eqref{secontineqApp} that with very high probability,
$$\|L^t\|_2{\leq}\left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}.$$
Also, we always have: $\lambda_p(L^t)\geq-\|L^t\|_2$. As a result:
\begin{align}
\lambda_p(L^t)\geq-\left(1+\sqrt{r^*}\right)\|L^*\|_2- \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}.
\end{align}
Now if the inequality $\left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}< S_p$ is satisfied, then we can select $0<a^{\prime}\leq\left(1+\sqrt{r^*}\right)\|L^*\|_2+ \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}$. The former inequality is satisfied
by the assumption of Theorem~\ref{RSCRSSapp} on $\|L^*\|_2$, i.e.,
\begin{align*}
\|L^*\|_2\leq\frac{1}{1+\sqrt{r^*}}\left(\frac{S_p}{1+C_4\left((1-\epsilon)(\frac{r^*}{r})\right)^{\frac{1}{8}}} - \frac{S_1(C_4((1-\epsilon)(\frac{r^*}{r}))^{\frac{1}{8}})}{1+C_4\left((1-\epsilon)(\frac{r^*}{r})\right)^{\frac{1}{8}}} - \frac{c_2\Big{\|}u\eta}{1-\rho}\sqrt{\frac{rp}{n}}\right).
\end{align*}
\end{proof}
\end{document} |
\begin{document}
\title[Starlike functions]{Radius of Starlikeness for Classes of Analytic Functions}
\alphauthor[K. Khatter]{Kanika Khatter}
\alphaddress{Department of Mathematics, SGTB Khalsa College, University of Delhi,
Delhi--110 007, India}
\email{[email protected]}
\alphauthor[S. K. Lee]{See Keong Lee}
\alphaddress{School of Mathematical Sciences,
Universiti Sains Malaysia,
11800 USM, Penang, Malaysia}
\email{[email protected]}
\alphauthor[V. Ravichandran]{V. Ravichandran}
\alphaddress{Department of Mathematics, NIT Tiruchirappalli,
Tamil Nadu--620015, India}
\email{[email protected]}
\begin{abstract} We consider normalized analytic function $f$ on the open unit disk for which either $\RE f(z)/g(z)>0$, $|f(z) /g(z) - 1|<1$ or $\RE (1-z^2) f(z) /z>0$ for some analytic function $g$ with $\RE (1-z^2) g(z) /z>0$. We have obtained the radii for these functions to belong to various subclasses of starlike functions. The subclasses considered include the classes of starlike functions of order $\alphalpha$, lemniscate starlike functions and parabolic starlike functions.
\end{abstract}
\keywords{starlike functions, exponential function, lemniscate of Bernoulli, radius problems, coefficient estimate}
\subjclass[2010]{30C45, 30C80}
\maketitle
\section{Introduction}
For any two classes $\mathcal{G}$ and $\mathcal{H}$ of analytic functions defined on the unit disk $\mathbb{D}$, the $\mathcal{H}$-radius for the class $\mathcal{G}$, denoted by $\mathcal{R}_{\mathcal{H}}(\mathcal{G})$, is the maximal radius $\rho \leq 1$ such that the $f\in \mathcal{G}$ implies that the function $f_r$, defined by $f_r(z)=f(rz)/r$, belongs to class $\mathcal{H}$ for all $0< r \leq \rho$. Among the radius problems for various subclasses of analytic functions, one direction of study focuses on obtaining the radius for classes consisting of functions characterised by ratio of the function $f$ and another function $g$, where $g$ is a function belonging to some special subclass of $\mathcal{A}$ of all analytic functions on $\mathbb{D}$ normalized by $f(0)=0=f'(0)-1$. MacGregor \cite{Mac,Mac1} obtained the radius of starlikeness for the class of functions $f \in \mathcal{A}$ satisfying either $\RE(f(z)/g(z))>0$ or $|f(z)/g(z)-1|<1$ for some $g \in \mathcal{K}$. Ali \emph{et al.}\cite{Ali} estimated several radii for classes of functions satisfying either (i)~$\RE(f(z)/g(z))>0$, where $\RE(g(z)/z)>0$ or $\RE(g(z)/z)>1/2$; (ii) $|f(z)/g(z)-1|<1$, where $\RE(g(z)/z)>0$ or $g$ is convex; (iii) $|f'(z)/g'(z)-1|<1$, where $\RE g'(z)>0$.
The work is further investigated in \cite{asha}. These classes are related to the Caratheodory class $\mathcal{P}$ consisting of all analytic functions $p$ with $p(0)=1$ and $\RE p(z)>0$ for all $z \in \mathbb{D}$.
Motivated by the aforesaid studies, we consider the following three classes $\mathcal{K}_1$, $\mathcal{K}_2$, and $\mathcal{K}_3$:
\begin{align*}
\mathcal{K}_1&:= \left\{ f \in \mathcal{A} : \frac{f(z)}{g(z)}\in \mathcal{P}, ~~\text{for some}~~ g \in \mathcal{A} ,\ \RE \frac{1-z^2}{z}g(z) >0 \right\},\\
\mathcal{K}_2&:= \left\{ f \in \mathcal{A} : \left| \frac{f(z)}{g(z)} - 1 \right|<1, ~~ \text{for some}~~ g\in \mathcal{A} , \ \RE \frac{1-z^2}{z}g(z) >0 \right\},
\intertext{and}
\mathcal{K}_3&:= \left\{ f \in \mathcal{A} : \RE \frac{1-z^2}{z}f(z) >0 \right\},
\end{align*}
and estimate the radius for the functions in the classes to belong to various subclasses of starlike functions which we discuss below.
Let $f, F$ be analytic on $\mathbb{D}:= \{z \in \mathbb{C}: |z| < 1\}$; the function $f$ is subordinate to $F$, written $f\prec F$, provided $f=F\circ w$ for some analytic self-mapping $w$ of the unit disk $ \mathbb{D}$ that fixes the origin. Subordination is very useful in the study of subclasses of univalent functions. For instance, the concept of Hadamard product and subordination was used in \cite{uni} to introduce the class of all functions $f$ satisfying $z{(k_\alpha * f)'}/{(k_\alpha * f)}\prec h$ where $k_\alpha(z)= z/(1-z)^{\alpha}$, $\alpha \in \mathbb{R}$, $f \in \mathcal{A}$ and $h$ is a convex function.
Later in 1989, Shanmugam \cite{shan} studied the class $\mathcal{S}_g^*(h)$ of all functions $f\in{\mathcal{A}}$ satisfying $z(f*g)'/(f*g) \prec h$ where $h$ is a convex function and $g$ is a fixed function in $\mathcal{A}$. By replacing $g$ with the functions $z/(1-z)$ and $z/(1-z)^2$, we get the subclasses $\mathcal{S}^*(h)$ and $\mathcal{K}(h)$ of Ma-Minda starlike and convex functions, respectively.
In 1992, Ma and Minda \cite{mam} studied the distortion, growth, covering and coefficient estimates for these functions with the weaker assumption of starlikeness on $h$. These classes unifies several subclasses of starlike and convex functions. When $h$ is the mapping of $\mathbb{D}$ onto the right half-plane, $\mathcal{S}^*(h)$ and $\mathcal{K}(h)$ reduce to the class $\mathcal{S}^*$ of starlike and $\mathcal{K}$ of convex functions, respectively. For $h(z)=(1+Az)/(1+Bz)$, with $-1 \leq B <A \leq 1$, the classes become $\mathcal{S}^*[A,B]$ of Janowski starlike functions and $\mathcal{K}[A,B]$ of Janowski convex functions. For $A= 1- 2 \alphalpha$ and $B = -1$ where $0 \leq \alpha < 1$, these subclasses become $\mathcal{S}^*(\alpha)$ of the starlike functions of order $\alpha$ and $\mathcal{K}(\alphalpha)$ of convex functions of order $\alpha$, respectively introduced by Robertson \cite{rob}. For $h (z)= \sqrt{1+z}$, the class $\mathcal{S}^*(h)$ becomes the class $\mathcal{S}^*_{L} $ of the lemniscate starlike functions introduced and studied by Sok\'{o}l and Stankiewicz \cite{sok,sok2}; analytically, $f \in \mathcal{S}^*_{L}$ if $|(z f'(z)/f(z))^2-1|<1$.
Mendiratta \emph{et al.}\cite{sumit,sumit2} studied the classes $\mathcal{S}^*_e = \mathcal{S}^*(e^z)$ and $\mathcal{S}^*_{RL} = \mathcal{S}^*(h_{RL})$, where
\[ h_{RL}:= \sqrt{2} - (\sqrt{2} -1) \sqrt{\frac{1-z}{1+2(\sqrt{2}-1)z}}.\] Indeed, a function $f$ belongs to $\mathcal{S}^*_e$ or to $ \mathcal{S}^*_{RL}$ if $z f'(z)/f(z)$ respectively belongs to $\{ w \in \mathbb{C}: |\log w|<1\}$ or $\{ (w - \sqrt{2})^2 -1 < 1 \}$. Sharma \emph{et al.}\cite{sharma} defined and studied the class of functions defined by $\mathcal{S}^*_c = \mathcal{S}^* (h_c(z))$, where $h_c(z) = 1 + (4/3) z + (2/3) z^2$; a function $f\in \mathcal{S}^*_c$ if $z f'(z)/f(z)\in \{ x + i y : (9 x^2 + 9 y^2 -18 x + 5)^2 -16 (9x^2 + 9 y^2 -6 x +1) = 0 \}$. Cho \emph{et al.}\cite{cho} defined and studied the class $\mathcal{S}^*_{\sin} = \mathcal{S}^*(1 + \sin z)$. Raina and Sokol \cite{raina} defined the class $\mathcal{S}^*_{\leftmoon} = \mathcal{S}^*(h_{\leftmoon})$, where $h_{\leftmoon} = z + \sqrt{1+ z^2}$ and $\mathcal{S}^*_{\leftmoon}$ consists of functions for which $z f'(z)/f(z)$ lies in the the leftmoon region defined by $\Omega_{\leftmoon}= h_{\leftmoon}(\mathbb{D}) :=\{ w\in \mathbb{C}: |w^2 - 1| < 2 |w| \}$. Another particular case is the class $\mathcal{S}^*_R = \mathcal{S}^*(h_R)$ studied in \cite{kumar} where $h_R = 1+ (zk + z^2)/(k^2 - kz)$, and $k = \sqrt{2} + 1$. The subclass $\mathcal{S}_{P} $ of parabolic starlike functions (see the survey \cite{ronning} or \cite{ali,mam1, ganga}) consists of all normalized analytic functions $f$ with $zf'(z)/f(z)$ lying in the parabolic region $(\IM(w))^2< 2 \RE(w)-1$.
\section{Main Results}
The first theorem gives the various radii of starlikeness for the class $\mathcal{K}_1$ which consists of functions $f \in \mathcal{A}$ satisfying $\RE(f(z)/g(z)) > 0$ for some $g \in \mathcal{A}$ and $\RE ((1-z^2)g(z)/z) >0$.
Note that the functions $f_1,\ g_1: \mathbb{D}\rightarrow \mathbb{C}$ defined by
\begin{equation} \label{f1}
f_1(z) = \frac{z(1+ i z)^2}{(1-z^2)(1-i z)^2} \quad \quad ~~~ \quad \text{and}~~~ \quad \quad g_1(z)= \frac{z (1+ i z)}{(1-z^2) (1- i z)}
\end{equation}
satisfy
\begin{equation*}
\RE \frac{f_1(z)}{g_1(z)} = \RE \frac{1-z^2}{z}g_1(z) = \RE \frac{1+ i z}{1- i z} >0.
\end{equation*}
This means the function $f_1 \in \mathcal{K}_1$ and so $\mathcal{K}_1 \neq \phi$. Further we will see that this function $f_1$ serves as an extremal function for many radii problems studied here.
\begin{theorem}\label{th1}
For the class $\mathcal{K}_1$, the following results hold:
\begin{enumerate}
\item The $\mathcal{S}^*(\alpha)$-radius is the smallest positive real root of the equation $r^4(1+\alpha) - 4 r^3-2 r^2 -4 r +(1-\alpha)=0$, \quad $0 \leq \alpha <1$.
\item The $\mathcal{S}^*_L$-radius is ${R}_{\mathcal{S}^*_L}= (\sqrt{5}-2)/(\sqrt{2}+1) \alphapprox 0.0977826$.
\item The $\mathcal{S}_{P}$-radius is the smallest positive real root of the equation $ 3 r^4 - 8r^3 -4 r^2 -8 r +1 = 0 $ i.e. ${R}_{\mathcal{S}_{P}} \alphapprox 0.116675$.
\item The $\mathcal{S}^*_{e}$-radius is the smallest positive real root of the equation $(2r^2 +4 r +4 r^3 -1-r^4)e = r^4 - 1 $ i.e. ${R}_{\mathcal{S}^*_{e}} \alphapprox 0.144684$.
\item The $\mathcal{S}^*_{c}$-radius is the smallest positive real root of the equation $ 4 r^4 - 12 r^3 - 6 r^2 - 12 r + 2 = 0 $ i.e. ${R}_{\mathcal{S}^*_{c}} \alphapprox 0.15182$.
\item The $\mathcal{S}^*_{\leftmoon}$-radius is the smallest positive real root of the equation $ 4 r^3 + 2 r^2 + 4 r + \sqrt{2}(1- r^4) = 2 $ i.e. ${R}_{\mathcal{S}^*_{\leftmoon}} \alphapprox 0.134993$.
\item The $\mathcal{S}^*_{\sin}$-radius is ${R}_{\mathcal{S}^*_{\sin}}= (-2+\sqrt{4+ \sin{1}(2+ \sin{1})})/(2+ \sin{1}) \alphapprox 0.185835$.
\item The $\mathcal{S}^*_{RL}$-radius is ${R}_{\mathcal{S}^*_{RL}} \alphapprox 0.0687813$.
\item The $\mathcal{S}^*_{R}$-radius is the smallest positive real root of the equation $ 4 r^3 + 2 r^2 + 4 r - r^4 -1 = 2 (1- \sqrt{2})(1- r^4) $ i.e. ${R}_{\mathcal{S}^*_{R}} \alphapprox 0.0419413$.
\end{enumerate}
All the radii obtained are sharp.
\end{theorem}
We would use the following lemmas in order to prove our results:
\begin{lemma}[{\cite[Lemma 2.2, p.\ 4]{jain}}]\label{T1}
For $0<a<\sqrt{2}$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
(\sqrt{1-a^2}- (1-a^2))^{1/2}, & \hbox{$0< a \leq 2 \sqrt{2}/3$;} \\
\sqrt{2}-a, & \hbox{$ 2 \sqrt{2}/3 \leq a < \sqrt{2}$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \{w: |w^2-1|<1\}$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 1, p.\ 321]{shan}}]\label{T2}
For $a> 1/2$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
a-1/2, & \hbox{$1/2< a \leq 3/2$;} \\
\sqrt{2a-2}, & \hbox{$ a \geq 3/2$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \{w: \RE w > |w-1|\} = \Omega_{1/2}$. Here, $\Omega_p$ is a parabolic region which is symmetric with respect to the real axis and vertex at $(p,0)$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.2, p.\ 368]{sumit}}]\label{T3}
For $ e^{-1} < a < e$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
a-e^{-1}, & \hbox{$e^{-1}< a \leq (e+e^{-1})/2$;} \\
e - a , & \hbox{$ (e+e^{-1})/2 \leq a < e$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \{w: |\log w|<1\} = \Omega_e$, which is the image of the unit disk $\mathbb{D}$ under the exponential function.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.5, p.\ 926]{sharma}}]\label{T4}
For $ 1/3 < a < 3$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
(3a-1)/3, & \hbox{$1/3< a \leq 5/3$;} \\
3- a , & \hbox{$ 5/3 \leq a \leq 3$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \Omega_c$.
Here $\Omega_c$ is the region bounded by the cardioid $\{x+ i y: (9x^2+9y^2-18x+5)^2 - 16 (9 x^2+ 9 y^2 -6x + 1) = 0 \}$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 3.3, p.\ 7]{cho}}]\label{T5}
For $ 1 - \sin 1 < a < 1+ \sin 1 $, let $r_a= \sin 1 - |a-1|$. Then $\{w: |w-a|< r_a \} \subseteq \Omega_{sin}$. Here $\Omega_{sin}$ is the image of the unit disk $\mathbb{D}$ under the function $1+ \sin z$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.1, p.\ 3]{gandhi}}]\label{T7}
For $ \sqrt{2}-1 < a < \sqrt{2}+1$, let $r_a= 1 - |\sqrt{2}-a|$. Then $\{w: |w-a|< r_a \} \subseteq \Omega_{\leftmoon}= \{w: |w^2 - 1|< 2 |w|\}$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.2, p.\ 202]{kumar}}]\label{T8}
For $ 2 (\sqrt{2}-1) < a < 2$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
a-2(\sqrt{2}-1), & \hbox{$2 (\sqrt{2}-1) < a \leq \sqrt{2}$;} \\
2- a , & \hbox{$\sqrt{2} \leq a < 2$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \Omega_R$, where $\Omega_R$ is the image of the unit disk $\mathbb{D}$ under the function $1+ ((z k + z^2)/(k^2 -k z))$, ~~ $k= \sqrt{2}+1$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 3.2, p.\ 10]{sumit2}}]\label{T9}
For $ 0 < a < \sqrt{2}$, let $r_a$ be given by
\begin{align*}
r_a= \left\{
\begin{array}{ll}
a, & \hbox{$0 < a \leq \sqrt{2}/3$;} \\
\big((1- (\sqrt{2}-a)^2)^{1/2}-(1-(\sqrt{2}-a)^2)\big)^{1/2}, & \hbox{$\sqrt{2}/3 \leq a < \sqrt{2}$.}
\end{array}
\right.
\end{align*}
Then $\{w: |w-a|< r_a \} \subseteq \{ w: \RE w >0, |(w- \sqrt{2})^2-1|< 1 \}= \Omega_{RL}$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2, p.\ 240]{shah}}]\label{T10}
If $ p(z) = 1 + b_n z^n + b_{n+1} z^{n+1} + \cdots $ is analytic and satisfies $ \RE p(z) > \alphalpha$, $0 \leq \alphalpha < 1$, for $|z|< 1$, then
\begin{equation*}
\left| \frac{zp'(z)}{p(z)}\right| \leq \frac{2 n z^n(1-\alphalpha)}{(1- |z|^n)(1+ (1-2 \alphalpha)|z|^n)}.
\end{equation*}
\end{lemma}
With all these tools, we are ready to give the proof of our first result.
\begin{proof}[Proof of Theorem~\ref{th1}]
Let $f \in \mathcal{K}_1$ and the function $g: \mathbb{D} \rightarrow \mathbb{C}$ be chosen such that
\begin{equation}\label{1}
\RE {\frac{f(z)}{g(z)}}>0 \quad \text{and} \quad \RE{\Big( \frac{1-z^2}{z} g(z) \Big)}>0 \quad (z \in \mathbb{D}).
\end{equation}
Let us define $p_1, p_2: \mathbb{D}\rightarrow \mathbb{C}$ as
\begin{equation}\label{2}
p_1(z)= \frac{1-z^2}{z} g(z) \quad \text{and} \quad p_2(z)= \frac{f(z)}{g(z)}
\end{equation}
Therefore, by equation \eqref{1},
$p_1$ and $p_2$ are in $\mathcal{P}$. Equation \eqref{2} yields
\begin{equation*}
f(z) = \frac{z}{(1-z^2)} p_1(z) p_2(z).
\end{equation*}
Take logarithm at both sides and differentiate with respect to $z$ would give
\begin{equation}\label{3}
\frac{z f'(z)}{f(z)}= \frac{1+z^2}{1-z^2}+ \frac{z p_1'(z)}{p_1(z)}+ \frac{z p_2'(z)}{p_2(z)}.
\end{equation}
It can be easily proved that the bilinear transform $w= (1+z^2)/(1-z^2)$ maps the disk $|z|\leq r$ onto the disk
\begin{equation}\label{4}
\left| \frac{1+z^2}{1-z^2}- \frac{1+r^4}{1-r^4} \right| \leq \frac{2r^2}{1-r^4}.
\end{equation}
Now, by Lemma \ref{T10}, for $p \in \mathcal{P}(\alpha) := \{ p \in \mathcal{P} : \RE p(z) > \alpha, z \in \mathbb{D}\}$, we have
\begin{equation}\label{5}
\left| \frac{z p'(z)}{p(z)}\right| \leq \frac{2 (1- \alpha) r}{(1-r)\big(1+ (1-2\alpha)r\big)} \quad (|z|\leq r).
\end{equation}
By using equations \eqref{3}, \eqref{4} and \eqref{5}, we can conclude that a function $f \in \mathcal{K}_1$ maps the disk $|z|\leq r$ onto the disk
\begin{equation}\label{6}
\left| \frac{z f'(z)}{f(z)} - \frac{1+r^4}{1-r^4} \right| \leq \frac{2r (2 r^2 + r + 2)}{1-r^4}.
\end{equation}
In order to solve radius problems for $f \in \mathcal{K}_1$, we are interested in computing the value of $r$ for which the disk in \eqref{6} is contained in the corresponding regions.
The classes we are considering here are all subclasses of starlike functions and therefore, we first determine the radius of starlikeness for $f \in \mathcal{K}_1$. From \eqref{6}, we have
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{r^4 - 4 r^3 - 2 r^2 -4r + 1}{1-r^4} \geq 0.
\end{equation*}
Solving the above inequality for $r$, we get that the function $f \in \mathcal{K}_1$ is starlike in $|z|\leq 0.216845$. Hence, all the radii that we are going to estimate here, will be less than 0.216845. For the function $f_1$ defined in \eqref{f1}, we have
\begin{align*}
\frac{zf'_1(z)}{f_1(z)} & = \frac{1+ 4 i z +2 z^2 -4 i z^3 +z^4}{1- z^4}\\
& = \frac{1+ 4 i z(1- z^2) + 2 z^2 + z^4 }{1 - z^4}
\end{align*}
At $z := r i = (0.216845) i $, we have $zf'_1(z)/f_1(z)\alphapprox 0$, thereby proving that the radius of starlikeness obtained for the class $\mathcal{K}_1$ is sharp.
\begin{enumerate}
\item In order to compute $R_ {\mathcal{S}^*(\alpha)}$, we estimate the value of $r \in (0,1)$ satisfying
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{r^4 - 4 r^3 - 2 r^2 -4r + 1}{1-r^4} \geq \alpha.
\end{equation*}
Therefore, the number $r = R_ {\mathcal{S}^*(\alpha)}$, is the smallest positive real root of the equation $r^4(1+\alpha) - 4 r^3-2 r^2 -4 r +(1-\alpha)=0$ in $(0,1)$. For the function $f_1 \in \mathcal{K}_1$ given by \eqref{f1}, we have
\begin{align} \label{f11}
\frac{z f_1'(z)}{f_1(z)} & = \frac{1+ 4 i z +2 z^2 -4 i z^3 +z^4}{1- z^4}
\end{align}
At $z:= r i = \mathcal{R}_{\mathcal{S}^*(\alphalpha)}$, \eqref{f11} reduces to
\begin{align*}
\frac{z f_1'(z)}{f_1(z)} & = \frac{1- 4 r - 2 r^2 -4 r^3 +r^4}{1- r^4}
= \alphalpha,
\end{align*}
thereby proving that the radius is sharp.
\item We use lemma \ref{T1} to compute the lemniscate starlike radius for the function $f \in \mathcal{K}_1$.
Let $a = (1+r^4)/(1-r^4)$. Then for $0 \leq r < 1$, we have $a \geq 1$. So for $a < \sqrt{2}$, we get $r < \sqrt[4]{(\sqrt{2}-1)/(\sqrt{2}+2)} \alphapprox 0.59018$. On the other hand, consider
\[
\frac{2r(2r^2 + r + 2)}{1-r^4} \leq \sqrt{2} - a = \sqrt{2} - \frac{1+r^4}{1 - r^4}.
\]
From this, let $r^*$ be the smallest positive real roof of the equation $(1+\sqrt{2})r^4 + 4r^3 + 2r^2 + 4r + (1 - \sqrt{2}) = 0$. Then the radius of lemniscate starlikeness for $f\in \mathcal{K}_1$ is
\[
R_{\mathcal{S}^*_L} = \min\left\{ \left( \frac{\sqrt{2}-1}{\sqrt{2}+2}\right)^{1/4}, r^*\right\} = r^* = \frac{\sqrt{5}-2}{\sqrt{2}+1}.
\]
The radius obtained is sharp. Consider the functions $f, g: \mathbb{D}\rightarrow \mathbb{C}$ defined by
\begin{equation} \label{f_1}
f(z) = \frac {z(1-z)}{(1+z)^3} \quad \quad ~~~ \quad \text{and}~~~ \quad \quad g(z)= \frac{z}{(1+z)^2}.
\end{equation}
Then clearly $f \in \mathcal{K}_1$ as
\begin{equation*}
\RE \frac{f(z)}{g(z)} = \RE \frac{1-z^2}{z}g(z) = \RE \frac{1+ z}{1- z} >0.
\end{equation*}
Now, for $z := - r^* = - R_{\mathcal{S}^*_L}$, we have $(z^2 -4 z +1)/(1-z^2) = \sqrt{2}$ and thus
\begin{equation*}
\left| \left(\frac{zf'(z)}{f(z)}\right)^2 -1 \right| = \left| \left(\frac{r^2 -4r+1}{1-r^2}\right)^2 -1 \right|=1,
\end{equation*}
thereby proving that the radius obtained is sharp by the function $f$ in \eqref{f_1}.
\item We use Lemma \ref{T2} to compute the parabolic starlike radius for $f \in \mathcal{K}_1$. Again, let $a = (1+r^4)/(1-r^4)$, which is larger than or equal to $1$ for $0 \leq r < 1$. Note that
\[
a = \frac{1 + r^4}{1 - r^4} = \frac{3}{2} \quad \Leftrightarrow \quad r= \left(\frac{1}{5} \right)^{1/4} \alphapprox 0.66874.
\]
Since the radius we are looking for would be less than $0.216845$, we only consider the case $1/2 < a \leq 3/2$ in Lemma \ref{T2}. So when considering
\[
\frac{2r(2r^2 + r + 2)}{1-r^4} \leq \frac{1 + r^4}{1-r^4} - \frac{1}{2},
\]
let $r^*$ be the smallest positive real root of the equation $3 r^4-8 r^3 -4 r^2 -8 r +1= 0$. Then the radius of parabolic starlikeness for $f\in \mathcal{K}_1$ is
\[
R_{\mathcal{S}_P} = \min\left\{ \left(\frac{1}{5}\right)^{1/4} , r^*\right\} = r^* \alphapprox 0.116675.
\]
\indent We see that the sharpness follows for the function $f_1 \in \mathcal{K}_1$ defined in \eqref{f1}. At $z = ir$, we have
\begin{equation*}
F(r) = \frac{z f_1'(z)}{f_1(z)}\bigg|_{z = ir} = \frac{1- 4r - 2 r^2 -4 r^3 +r^4}{1- r^4}.
\end{equation*}
Then,
\begin{align*}
\left| F(r)- 1 \right| &= \left| \frac{2r (r^3-2r^2-r-2)}{1-r^4}\right|.
\end{align*}
For $z := ir^* = iR_{\mathcal{S}_{P}}$, we have
\begin{align*}
\RE \frac{z f'_1(z)}{f_1(z)} &= \frac{1+ r^4 - 4 r^3 -2 r^2- 4r}{1- r^4} (\alphapprox 0.5)\\
&= \frac{2 r (2+r+2r^2-r^3)}{1 -r^4}= \left| \frac{zf_1'(z)}{f_1(z)}- 1 \right|.
\end{align*}
Thus the radius obtained is sharp for the function $f_1$.
\item By using Lemma \ref{T3} and the argument similar to the above, we get that the exponential starlike radius $R_{\mathcal{S}^*_e}$ for the class $\mathcal{K}_1$ is the smallest positive real root of the equation $(4 r^3 +2 r^2 + 4 r -1 -r^4)e = r^4 -1$.
The radius is sharp for the function $f_1$ defined in \eqref{f1}.
For $z := ir = i{R}_{\mathcal{S}^*_e}$, we have
\begin{equation*}
\left|\log \frac{zf'_1(z)}{f_1(z)} \right| = \left| \log \frac{1 + r^4 - 4r^3 - 2 r^2 - 4r}{1 - r^4}\right|=1.
\end{equation*}
\item
By using Lemma \ref{T4}, and similar argument as before,
the $\mathcal{S}^*_c$-radius for the class $\mathcal{K}_1$ is the smallest positive real root of the equation $2 r^4 - 6 r^3 - 3 r^2 - 6 r +1 = 0$.
The radius is sharp for the function $f_1$ defined in \eqref{f1}.
Indeed, for the function $f_1$ defined in \eqref{f1}, we have
at $z := ir = i \mathcal{R}_{\mathcal{S}^*_c}$,
\begin{equation*}
\frac{zf'_1(z)}{f_1(z)} = \frac{1 + r^4 - 4r^3 - 2 r^2 - 4r}{1 - r^4} =\frac{1}{3}=h_c(-1)\in \partial h_c(\mathbb{D}),
\end{equation*}
where $h_c(z) = 1 + (4/3)z + (2/3)z^2$.
This shows that the result is sharp.
\item To determine the $\mathcal{S}^*_{\leftmoon}$-radius, ${R}_{\mathcal{S}^*_{\leftmoon}}$, we will use Lemma \ref{T7}. After some computations following the idea above, it can be shown that ${R}_{\mathcal{S}^*_{\leftmoon}}$ is the smallest positive real root of the equation $4 r^3 + 2 r^2 + 4r = 2 -\sqrt{2}(1-r^4)$.
The radius is sharp for the function $f_1$ defined in \eqref{f1}, since at $z := ir = i{R}_{\mathcal{S}^*_{\leftmoon}}$, we have
\begin{align*}
\left| \left( \frac{zf_1'(z)}{f_1(z)}\right)^2-1 \right| &= \left| \left( \frac{1+ r^4 - 4 r^3 - 2 r^2 - 4 r }{1-r^4}\right)^2 -1 \right| (\alphapprox 0.134993 )\\&= 2\left| \frac{1+ r^4 - 4 r^3 - 2 r^2 - 4 r }{1-r^4} \right| = 2 \left|\frac{zf_1'(z)}{f_1(z)} \right|.
\end{align*}
\item In order to find the $\mathcal{S}^*_{\sin}$-radius for function $f \in \mathcal{K}_1$, we make use of Lemma \ref{T5}. Similarly as above, with $a = {(1+r^4)}/{(1-r^4)} >1$, it can be shown by arguing similarly as above that the $\mathcal{S}^*_{\sin}$-radius is the smallest positive real root of the equation $(2 + \sin{1})r^4 + 4r^3 + 2r^2 + 4r - \sin{1} = 0$.
The radius is sharp for the function $f_1$ defined in \eqref{f1}.
\item In order to compute the $\mathcal{S}^*_{RL}$- radius for the class $\mathcal{K}_1$, we use Lemma \ref{T9}. As $\sqrt{2}/3 \leq a = {(1+ r^4)}/{(1- r^4)} < \sqrt{2}$, a computation using Lemma \ref{T9} shows that the $\mathcal{S}^*_{RL}$- radius is the smallest positive real root of the equation
\[
4r^2(2r^2 + r + 2)^2 = (1-r^4)\sqrt{\left(\sqrt{2}-1\right) + \left(\sqrt{2}-2\right)r^4} - 2\left( \sqrt{2}-1 + (\sqrt{2}-2)r^4\right).
\]
The radius obtained is sharp for the function $ f \in \mathcal{K}_1$ given by \eqref{f_1}.
At $ z := - r = - {R}_{\mathcal{S}^*_{RL}}$, we have $(z^2 -4 z +1)/(1-z^2) = \sqrt{2}$ and therefore,
\begin{equation*}
\left| \left( \frac{zf'(z)}{f(z)} - \sqrt{2} \right)^2 -1 \right| = \left| \left( \frac{1- 4z + z^2}{1- z^2} - \sqrt{2} \right)^2 -1 \right| = 1.
\end{equation*}
Hence the result.
\item Since $2 (\sqrt{2}-1) < a = {(1+ r^4)}/{(1- r^4)}\leq \sqrt{2}$, by using Lemma \ref{T8}, it can be shown that the $\mathcal{S}_R$- radius is obtained by solving the equation
\begin{equation*}
\left( 2\sqrt{2} - 1\right)r^4 - 4r^3 - 2r^2 - 4r + \left( 3 - 2\sqrt{2}\right)=0.
\end{equation*}
The radius is sharp for the function $f_1$ defined in \eqref{f1}.
Indeed, for the function $f_1$ defined in \eqref{f1}, we have
at $z := ir = i {R}_{\mathcal{S}^*_R}$ that
\begin{equation*}
\frac{zf'_1(z)}{f_1(z)} = \frac{1 + r^4 - 4r^3 - 2 r^2 - 4r}{1 - r^4} = 2\sqrt{2} -2 =h_R(-1)\in \partial h_R(\mathbb{D}).
\end{equation*}
Here, $h_R = 1+ (zk + z^2)/(k^2 - kz)$, and $k = \sqrt{2} + 1$.
\qedhere
\end{enumerate}
\end{proof}
Our next result gives various radii of starlikeness for the $\mathcal{K}_2$, which consists of functions $f \in \mathcal{A}$ satisfying $|(f(z)/g(z))-1|<1$ for some $g \in \mathcal{A}$ and $\RE ((1-z^2)g(z)/z) >0$.
Consider the functions $f_2$, $g_2: \mathbb{D}\rightarrow \mathbb{C}$ defined by
\begin{equation} \label{f2}
f_2(z) = \frac{z(1+ i z)^2}{(1-z^2)(1-i z)} \quad \quad ~~~ \quad \text{and}~~~ \quad \quad g_2(z)= \frac{z (1+ i z)}{(1-z^2) (1- i z)}.
\end{equation}
Clearly,
\begin{equation*}
\left| \frac{f_2(z)}{g_2(z)} - 1 \right| = |i z| = |z|< 1~~~~\quad \text{and}~~~~\quad \RE \frac{1-z^2}{z}g_2(z) = \RE \frac{1+ i z}{1- i z} >0.
\end{equation*}
Therefore, the function $f_2$ is in $\mathcal{K}_2$ and this shows $\mathcal{K}_2 \neq \phi$. Note that this function $f_2$ would serve as an extremal function for several radii-problems that we study here.
\begin{theorem}
For $f \in \mathcal{K}_2$, the following results hold:
\begin{enumerate}
\item The sharp $\mathcal{S}^*(\alpha)$ radius is the smallest positive real root of the equation $ \alpha r^4 -3r (r^2 + r + 1) + (1- \alpha)=0$, \quad $0 \leq \alpha <1$.
\item The $\mathcal{S}^*_L$ radius is ${R}_{\mathcal{S^*_L}}= (\sqrt{2}-1)/(\sqrt{2}+2) \alphapprox 0.12132$.
\item The sharp $\mathcal{S}_{P}$ radius is the smallest positive real root of the equation $ 6 r^3 + 6 r^2 + 6 r -1 - r^4 = 0$ i.e., ${R}_{\mathcal{S}_{P}} \alphapprox 0.1432698$.
\item The sharp $\mathcal{S}^*_{e}$ radius is the smallest positive real root of the equation $(3 r^3 + 3 r^2 + 3 r -1)e + 1 - r^4 = 0$ i.e., ${R}_{\mathcal{S}^*_{e}} \alphapprox 0.174887$.
\item The sharp $\mathcal{S}^*_{c}$ radius is the smallest positive real root of the equation $ 9 r^3 + 9 r^2 + 9 r - 2 - r^4 = 0 $ i.e., ${R}_{\mathcal{S}^*_{c}} \alphapprox 0.182815$.
\item The sharp $\mathcal{S}^*_{\leftmoon}$ radius is the smallest positive real root of the equation $ r^4 (1- \sqrt{2}) + 3 r^3 + 3 r^2 + 3 r = 2 - \sqrt{2}$ i.e., ${R}_{\mathcal{S}^*_{\leftmoon}} \alphapprox 0.164039$.
\item The sharp $\mathcal{S}^*_{\sin}$ radius is ${R}_{\mathcal{S}^*_{\sin}}= \sin{1}/(3+ \sin{1}) \alphapprox 0.219049$.
\item The sharp $\mathcal{S}^*_{R}$ radius is the smallest positive real root of the equation $ 2 r^4 + 3 r^3 + 3 r^2 + 3 r - 3 + 2 \sqrt{2}(1- r^4) =0 $ i.e., ${R}_{\mathcal{S}^*_{R}} \alphapprox 0.0541073$.
\item The $\mathcal{S}^*_{RL}$ radius is ${R}_{\mathcal{S}^*_{RL}} \alphapprox 0.0870259$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $f \in \mathcal{K}_2$ and the function $g: \mathbb{D} \rightarrow \mathbb{C}$ be chosen such that
\begin{equation}\label{2.1}
\left| {\frac{f(z)}{g(z)}}-1\right|< 1 \quad \text{and} \quad \RE{\Big( \frac{1-z^2}{z} g(z) \Big)}>0 \quad (z \in \mathbb{D}).
\end{equation}
Note that $| {f(z)}/{g(z)}-1|< 1 $ holds if and only if $\RE (g(z)/f(z)) > 1/2$
Let define $p_1, p_2: \mathbb{D}\rightarrow \mathbb{C}$ as
\begin{equation}\label{2.2}
p_1(z)= \frac{1-z^2}{z} g(z) \quad \text{and} \quad p_2(z)= \frac{g(z)}{f(z)}.
\end{equation}
Then, by equations \eqref{2.1} and \eqref{2.2}, $p_1 \in \mathcal{P}$ and $p_2 \in \mathcal{P}(1/2)$. Equation \eqref{2.2} also yields
\begin{equation*}
f(z) = \frac{z}{1-z^2} \frac{p_1(z)}{ p_2(z)}.
\end{equation*}
Taking logarithm on both sides and differentiating with respect to $z$ gives
\begin{equation}\label{2.3}
\frac{z f'(z)}{f(z)}= \frac{1+z^2}{1-z^2}+ \frac{z p_1'(z)}{p_1(z)}- \frac{z p_2'(z)}{p_2(z)}.
\end{equation}
By using equations \eqref{4},\eqref{5} and \eqref{2.3}, it can proven that the function $f$ maps the disk $|z|\leq r$ onto the disk
\begin{equation}\label{2.6}
\left| \frac{z f'(z)}{f(z)} - \frac{1+r^4}{1-r^4} \right| \leq \frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4}.
\end{equation}
From \eqref{2.6}, we can get
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{1 - 3 r ( r^2 + r + 1)}{1-r^4} \geq 0.
\end{equation*}
Upon solving for $r$, we can conclude that the function $f \in \mathcal{K}_2$ is starlike in $|z|\leq 0.253077\cdots$.
The classes we are considering here are all subclasses of starlike functions, hence, all the radii we estimate here, will be less than $0.253077\cdots$. For the function $f_2$ defined in \eqref{f2}, we have
\begin{align*}
\frac{zf'_2(z)}{f_2(z)} & = \frac{1 + 3 i z + 3 z^2 - 3 i z^3}{1- z^4}\\
& = \frac{1+ 3 i z (1- z^2) + 3 z^2 }{1 - z^4}
\end{align*}
At $z := ir = i(0.253077)$, we have $zf_2'(z)/f_2(z)\alphapprox 0$, thereby proving that the radius of starlikeness obtained for the class $\mathcal{K}_2$ is sharp.
\begin{enumerate}
\item In order to compute $R_ {\mathcal{S}^*(\alpha)}$, we estimate the value of $r \in [0,1]$ satisfying
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{1 - 3 r ( r^2 + r + 1)}{(1-r^4)} \geq \alpha.
\end{equation*}
Therefore, the number $R_ {\mathcal{S}^*(\alpha)}$, is the root of the equation $\alpha r^4 - 3 r ( r^2 + r + 1) +(1-\alpha)=0$ in $[0,1]$. For the function $f_2 \in \mathcal{K}_2$ given by \eqref{f2}, we have
\begin{align} \label{f22}
\frac{z f'_2(z)}{f_2(z)} & = \frac{1 + 3 i z + 3 z^2 - 3 i z^3}{1- z^4}
\end{align}
At $z:= i r = {R}_{\mathcal{S}^*(\alphalpha)}$, \eqref{f22} reduces to
\begin{align*}
\frac{z f'_2(z)}{f_2(z)} & = \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4}
= \alphalpha,
\end{align*}
thereby proving that the radius is sharp.
\item We would use Lemma \ref{T1} to compute the lemniscate starlike radius for $f \in \mathcal{K}_2$. So, let $a = (1 + r^4)/(1-r^4)$. Then $1 \leq a < \infty$ for $r\in [0,1)$, and $a < \sqrt{2}$ when $r < \left(\left(\sqrt{2} - 1\right)/\left(\sqrt{2} + 1\right)\right)^{1/4}$. From \ref{2.6}, we know that $f\in \mathcal{K}_2$ maps the disk $|z| \leq r$ onto the disk
\begin{equation}
\left| \frac{z f'(z)}{f(z)} - \frac{1+r^4}{1-r^4} \right| \leq \frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4}.
\end{equation}
So, consider
\[
\frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4} \leq \sqrt{2} - \frac{1+r^4}{1-r^4},
\]
and let $r^*$ be the smallest positive real root of the equation
\[
\left( \sqrt{2} + 2\right) r^4 - 4r^3 - 2r^2 - 4r + \left( 3 - 2\sqrt{2}\right) = 0.
\]
Then by Lemma \ref{T1}, the lemniscate starlike radius $R_{\mathcal{S}^*_L}$ for $f \in \mathcal{K}_2$ is given by
\[
R_{\mathcal{S}^*_L} = \min\left\{ \left(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right)^{1/4}, r^*\right\} = r^* = \frac{\sqrt{2} -1}{\sqrt{2} + 2} = 0.12132\cdots
\]
This radius may not be sharp.
\item We use Lemma \ref{T2} to compute the parabolic starlike radius for $f \in \mathcal{K}_2$. For $a = (1 + r^4)/(1-r^4)$, we have $a \leq 3/2$ if $r \leq (1/5)^{1/4} \alphapprox 0.668740305$. by Lemma \ref{T2}, consider
\[
\frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4} \leq \frac{1+r^4}{1-r^4} - \frac{1}{2},
\]
and let $r^*$ be the smallest positive real root of the equation
\[
r^4 - 6r^3 - 6r^2 - 6r + 1 = 0.
\]
Then the $\mathcal{S}_{P}$-radius is
\[
R_{\mathcal{S}_P} = \min \left\{ \left( \frac{1}{5}\right)^{1/4}, r^*\right\} = r^* \alphapprox 0.1432698.
\]
We see that sharpness follows for the function $f_2 \in \mathcal{K}_2$ defined in \eqref{f2}. As shown previously, at $z := i r $, we have
\begin{equation*}
\frac{z f_2'(z)}{f_2(z)} = \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4}.
\end{equation*}
Thus,
\begin{align*}
\left| \frac{zf_2'(z)}{f_2(z)}- 1 \right| &= \left| \frac{r (3 + 3 r + 3 r^2 - r^3)}{1 - r^4}\right|
\end{align*}
For $z := i r = i {R}_{\mathcal{S}_P}$, we have
\begin{align*}
\RE \frac{z f_2'(z)}{f_2(z)} &= \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4} \quad(\alphapprox 0.5)\\
&=\frac{r (3 + 3 r + 3 r^2 - r^3)}{1 - r^4}= \left| \frac{zf_2'(z)}{f_2(z)}- 1 \right|.
\end{align*}
Thus the radius obtained is sharp for the function $f_2$.
\item For the $\mathcal{S}_e$-radius of $f\in \mathcal{K}_2$, we will use Lemma \ref{T3} since if $a = (1 + r^4)/(1-r^4)$, $0\leq r <1$, we have $a < e$ for $r < [(e-1)/(e+1)]^{1/4} \alphapprox 0.82449$. Also, since $a \leq \frac{1}{2}(e + e^{-1})$ for $r < [(e-1)/(e+1)]^{2} \alphapprox 0.213552$, consider
\[
\frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4} \leq \frac{1+r^4}{1-r^4} - \frac{1}{e},
\]
and let $r^*$ be the smallest positive real root of the equation
\[
r^4 - e(3r^3 + 3r^2 + 3r) + e - 1 = 0.
\]
Then the $\mathcal{S}_{e}$-radius is
\[
R_{\mathcal{S}_e} = \min \left\{ \left( \frac{e-1}{e+1}\right)^{2}, r^*\right\} = r^* \alphapprox 0.174887.
\]
\begin{equation*}
\left|\log \frac{zf'_2(z)}{f_2(z)} \right| = \left| \log \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4}\right|=1,
\end{equation*}
hence proving that the exponential starlike radius obtained for the class $\mathcal{K}_2$ is sharp.
\item By using Lemma \ref{T4}, it can be proven similarly as above that the $\mathcal{S}^*_c$-radius $R_{\mathcal{S}^*_c}$ for the class $\mathcal{K}_2$ is the smallest positive real root of the equation $9 r^3 + 9 r^2 + 9 r - 2 - r^4 = 0$, which is $R_{\mathcal{S}^*_c} \alphapprox 0.182815$.
The radius obtained is sharp for the function $f_2$ defined in \eqref{f2} as for $z := ir = i {R}_{\mathcal{S}^*_c}$,
\begin{equation*}
\frac{zf'_2(z)}{f_2(z)} = \frac{1 -3 r - 3 r^2 -3 r^3}{1 - r^4} =\frac{1}{3}=h_c(-1)\in \partial h_c(\mathbb{D}).
\end{equation*}
\item The $\mathcal{S}^*_{\leftmoon}$-radius ${R}_{\mathcal{S}^*_{\leftmoon}}$ for the class $\mathcal{K}_2$ is the smallest positive real root of the equation $r^4 (1- \sqrt{2}) + 3 r^3 + 3 r^2 + 3 r = 2 - \sqrt{2}$. This can be obtained by considering the inequality
\[
\frac{3 r^3 +3 r^2 + 3 r - 1}{1- r^4} \leq 1- \sqrt{2}
\]
and then using Lemma \ref{T7}.
The radius is sharp for the function $f_2$ defined in \eqref{f2}, since at $z := i r = i{R}_{\mathcal{S}^*_{\leftmoon}}$, we have
\begin{align*}
\left| \left( \frac{zf_2'(z)}{f_2(z)}\right)^2 - 1 \right| &= \left| \left( \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4}\right)^2 -1 \right| \quad (\alphapprox 0.82842)\\
&= 2\left| \frac{1 - 3 r - 3 r^2 - 3 r^3}{1- r^4} \right| = 2 \left|\frac{zf_2'(z)}{f_2(z)} \right|.
\end{align*}
\item In order to find the $\mathcal{S}^*_{\sin}$-radius for the function $f \in \mathcal{K}_2$, we make use of Lemma \ref{T5}. It is easy to see that $ 1 - \sin 1 < a = {(1+r^4)}/{(1-r^4)} < 1+ \sin 1 $ for $r < [(\sin{1})/(2 + \sin{1})]^{1/4}$. Since $a>1$, consider
\begin{equation*}\label{11}
\frac{r (r^3 + 3 r^2 + 3 r + 3)}{1-r^4} \leq \sin 1 - \left(\frac{1+r^4}{1-r^4} - 1\right).
\end{equation*}
Then the $\mathcal{S}^*_{\sin}$-radius, $R_{\mathcal{S}^*_{\sin}} (\alphapprox 0.219049)$, is the smallest positive real root of the equation
\[
(3 + \sin{1})r^4 + 3r(r^2 + r +1) = \sin{1}.
\]
The radius obtained is sharp for the function $f_2$ defined in \eqref{f2}.
\item We use Lemma \ref{T8} in order to compute the $\mathcal{S}^*_R$- radius for the class $\mathcal{K}_2$. Since $2 (\sqrt{2}-1) < a = {(1+ r^4)}/{(1- r^4)}\leq \sqrt{2}$ for $r < [(\sqrt{2} - 1)/(\sqrt{2} + 1)]^{1/4} \alphapprox 0.64359$, by Lemma \ref{T8}, we consider
\[
\frac{r(r^3 + 3 r^2 +3 r + 3)}{1-r^4} \leq \frac{1 + r^4}{1 - r^4} - 2(\sqrt{2} - 1).
\]
Then the $\mathcal{S}^*_R$- radius for $\mathcal{K}_2$ can be computed to be $R_{\mathcal{S}^*_{R}} \alphapprox 0.0870259$. The radius obtained is sharp for the function $f_2$ defined in \eqref{f2}.
Indeed, for the function $f_2$ defined in \eqref{f2}, we have
at $z := ir = i {R}_{\mathcal{S}^*_R}$,
\begin{equation*}
\frac{zf'_2(z)}{f_2(z)} = \frac{1 - 3 r - 3 r^2 -3 r^3}{1 - r^4} =2 \sqrt{2} - 2=h_R(-1)\in \partial h_R(\mathbb{D}).
\end{equation*}
This shows that the result is sharp.
\item Finally, for the $\mathcal{S}^*_{RL}$- radius, $R_{\mathcal{S}^*_{RL}}$, for the class $\mathcal{K}_2$, by Lemma \ref{T9}, the value of $R_{\mathcal{S}^*_{RL}} \alphapprox 0.0541073$ is obtained from solving the equation
\begin{align*}
(1- r^4) \big\{ (1- r^4)^2 - \big( (\sqrt{2} - 1)& - (\sqrt{2} + 1) r^4 \big)^2 \big\}^{1/2} = (r^4 + 3r^3 + 3r^2 + 3r)^2 \\& \quad +(1- r^4)^2 - \big((\sqrt{2} - 1)- (\sqrt{2} + 1) r^4 \big)^2. \qedhere
\end{align*}
\end{enumerate}
\end{proof}
The last theorem aims at computing the various radii of starlikeness for the function $f \in \mathcal{K}_3$ that satisfies $\RE ((1-z^2)f(z)/z) >0$.
Consider the function $f_3: \mathbb{D}\rightarrow \mathbb{C}$ defined by
\begin{equation} \label{f3}
f_3(z) = \frac{z(1+ i z)}{(1-z^2)(1- i z)}
\end{equation}
Clearly,
\begin{equation*}
\RE \frac{(1-z^2)}{z}f_3(z) = \RE \frac{1+ i z}{1- i z} >0.
\end{equation*}
Therefore, the function $f_3 \in \mathcal{K}_3$ and $\mathcal{K}_3 \neq \phi$. This function $f_3$ would serve as an extremal function for various radius problems in the following theorem.
\begin{theorem}
For $f \in \mathcal{K}_3$, the following results hold:
\begin{enumerate}
\item The sharp $\mathcal{S}^*(\alpha)$ radius is the smallest positive real root of the equation $ (1+\alpha) r^4 - 2 r (r^2 + r + 1) + (1- \alpha)=0$, \quad $0 \leq \alpha <1$.
\item The sharp $\mathcal{S}^*_L$ radius is ${R}_{\mathcal{S_L}}= (\sqrt{2}-1)/(\sqrt{2}+1) \alphapprox 0.171573$.
\item The sharp $\mathcal{S}_{P}$ radius is the smallest positive real root of the equation $ 4 r^3 + 4 r^2 + 4 r -1 - 3r^4 = 0$ i.e. ${R}_{\mathcal{S}_{P}} \alphapprox 0.2021347$.
\item The sharp $\mathcal{S}^*_{e}$ radius is the smallest positive real root of the equation $(2 r^3 + 2 r^2 + 2 r -1 - r^4)e + 1 - r^4 = 0$ i.e. ${R}_{\mathcal{S}^*_{e}} \alphapprox 0.244259$.
\item The sharp $\mathcal{S}^*_{c}$ radius is the smallest positive real root of the equation $ 3 r^3 + 3 r^2 + 3 r - 1 - 2 r^4 = 0 $ i.e. ${R}_{\mathcal{S}^*_{c}} \alphapprox 0.254726$.
\item The sharp $\mathcal{S}^*_{\leftmoon}$ radius is the smallest positive real root of the equation $ 2 r^3 + 2 r^2 + 2 r - \sqrt{2} r^4 = 2 - \sqrt{2}$ i.e. ${R}_{\mathcal{S}^*_{\leftmoon}} \alphapprox 0.229877$.
\item The sharp $\mathcal{S}^*_{\sin}$ radius is ${R}_{\mathcal{S}^*_{\sin}}= \sin{1}/(2 + \sin{1}) \alphapprox 0.296139$.
\item The sharp $\mathcal{S}^*_{R}$ radius is the smallest positive real root of the equation $ r^4 + 2 r^3 + 2 r^2 + 2 r -3 + 2 \sqrt{2} (1 - r^4)=0 $ i.e. ${R}_{\mathcal{S}^*_{R}} \alphapprox 0.0790749$.
\item The $\mathcal{S}^*_{RL}$ radius is ${R}_{\mathcal{S}^*_{RL}} \alphapprox 0.125145$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let the function $f \in \mathcal{K}_3$. Then
\begin{equation}\label{3.1}
\RE{\Big( \frac{1-z^2}{z} f(z) \Big)}>0 \quad (z \in \mathbb{D}).
\end{equation}
Define $p: \mathbb{D}\rightarrow \mathbb{C}$ as
\begin{equation}\label{3.2}
p(z)= \frac{1-z^2}{z} f(z)
\end{equation}
Therefore, by equation \eqref{3.1}, we have $p \in \mathcal{P}$ and
\begin{equation*}
f(z) = \frac{z}{(1-z^2)} p(z).
\end{equation*}
From this, take logarithm on both sides and then differentiate with respect to $z$:
\begin{equation}\label{3.3}
\frac{z f'(z)}{f(z)}= \frac{1+z^2}{1-z^2}+ \frac{z p'(z)}{p(z)}.
\end{equation}
By using equations \eqref{4},\eqref{5} and \eqref{3.3}, we can prove that the function $f$ maps the disk $|z|\leq r$ onto the disk
\begin{equation}\label{3.6}
\left| \frac{z f'(z)}{f(z)} - \frac{1+r^4}{1-r^4} \right| \leq \frac{2 r (r^2 + r + 1)}{1-r^4}.
\end{equation}
In order to solve radius problems, we are interested in computing the value of $r$ for which the disk in \eqref{3.6} is contained in the corresponding regions.
Again the classes we are considering here are all subclasses of starlike functions and therefore, are defined by the quantity $z f'(z)/f(z)$ lying in some region in the right half plane. In particular, for $f$ to be in $\mathcal{S^*}$, we need
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{1 + r^4 - 2 r ( r^2 + r + 1)}{1-r^4} \geq 0.
\end{equation*}
Thus the function $f \in \mathcal{K}_3$ is starlike in $|z|\leq 0.346014$. With this, now all the radii we estimate here shall be less than $0.346014$.
For the function $f_3$ defined in \eqref{f3}, we have
\begin{align}\label{f33}
\frac{zf'_3(z)}{f_3(z)} & = \frac{1 + 2 i z + 2 z^2 - 2 i z^3 + z^4}{1- z^4}\\ \notag
& = \frac{1+ 2 i z (1- z^2) + 2 z^2 + z^4 }{1 - z^4}.
\end{align}
At $z := ir = i(0.346014)$, we have $zf'_3(z)/f_3(z)\alphapprox 0$, thereby proving that the radius of starlikeness obtained for the class $\mathcal{K}_3$ is sharp.
\begin{enumerate}
\item To determine the radius $R_ {\mathcal{S}^*(\alpha)}$ of starlikeness of order $\alphalpha$, we estimate the value of $r \in [0,1]$ satisfying
\begin{equation*}
\RE \frac{z f'(z)}{f(z)} \geq \frac{1 + r^4 - 2 r ( r^2 + r + 1)}{1-r^4} \geq \alpha.
\end{equation*}
Hence, $R_ {\mathcal{S}^*(\alpha)}$ is the root of the equation $(1 + \alpha) r^4 - 2 r ( r^2 + r + 1) +(1-\alpha)=0 $ in $[0,1]$. From \eqref{f33}, if $z:= i r = i{R}_{\mathcal{S}^*(\alphalpha)}$, then \eqref{f33} reduces to
\begin{align*}
\frac{z f_3'(z)}{f_3(z)} & = \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4}
= \alphalpha,
\end{align*}
which shows that $f_3$ is the extremal function.
\item We can use Lemma \ref{T1} to compute the lemniscate starlike radius for $f \in \mathcal{K}_3$. For $a = (1 + r^4)/(1-r^4)$, we have $a < \sqrt{2}$ when $r < \left(\left(\sqrt{2} - 1\right)/\left(\sqrt{2} + 1\right)\right)^{1/4}$. By \eqref{2.6} and Lemma \ref{T1}, consider
\[
\frac{2r (r^2 + r + 1)}{1-r^4} \leq \sqrt{2} - \frac{1+r^4}{1-r^4},
\]
and let $r^*$ be the smallest positive real root of the equation
\[
\left( \sqrt{2} + 1\right) r^4 + 2r^3 + 2r^2 + 2r + \left( 1 - \sqrt{2}\right) = 0.
\]
Hence, the lemniscate starlike radius $R_{\mathcal{S}^*_L}$ for $f \in \mathcal{K}_3$ is given by
\[
R_{\mathcal{S}^*_L} = \min\left\{ \left(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right)^{1/4}, r^*\right\} = r^* = \frac{1 - \sqrt{2}}{1 + \sqrt{2}} = 0.1715728753\ldots.
\]
For the sharpness, consider the function $\hat{f_3}: \mathbb{D}\rightarrow \mathbb{C}$ defined by
\begin{equation} \label{f_31}
\hat{ f_3}(z) = \frac {z}{(1+z)^2}.
\end{equation}
Clearly,
\begin{equation*}
\RE \frac{1-z^2}{z}\hat{f_3}(z) = \RE \frac{1 - z}{1+ z} >0.
\end{equation*}
So $\hat{f_3} \in \mathcal{K}_3$. Also
\begin{equation*}
\left| \left(\frac{z\hat{f_3}'(z)}{\hat{f_3}(z)}\right)^2 -1 \right| = \left| \left(\frac{1 - z}{1 + z}\right)^2 -1 \right|.
\end{equation*}
Now, for $z := r = {R_{\mathcal{S}^*_L}}$, we have $(1 - z)/(1 + z) = \sqrt{2}$ and
\begin{equation*}
\left| \left(\frac{z\hat{f_3}'(z)}{\hat{f_3}(z)}\right)^2 -1 \right| = \left| \left(\sqrt{2}\right)^2 -1 \right|=1,
\end{equation*}
Therefore, the radius obtained is sharp for the function $\hat{f_3}$.
\item For the parabolic starlike radius for $f \in \mathcal{K}_3$, we would use Lemma \ref{T2}. For $r \leq (1/5)^{1/4}$, we have $a= {(1+r^4)}/{(1-r^4)}\leq 3/2$, and if we consider
\begin{equation}\label{39}
\frac{ 2 r ( r^2 + r + 1)}{1-r^4} \leq \frac{1+r^4}{1-r^4} -\frac{1}{2},
\end{equation}
then the $\mathcal{S}_P$-radius is given by
\[
R_{\mathcal{S}_P} = \min\left\{ \left(\frac{1}{5}\right)^{1/4}, r^*\right\} = r^* \alphapprox 0.2021347,
\]
where $r^*$ is the smallest positive real root of the equation $3r^4 - 4 r^3 - 4 r^2 - 4 r + 1 = 0$.
The sharpness of the result follows for the function $f_3$ defined in \eqref{f3}. As shown previously, at $z := i r$, we have
\begin{equation*}
\frac{z f_3'(z)}{f_3(z)} = \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4}.
\end{equation*}
Then for $z := i r = {R}_{\mathcal{S}_P}$, we have
\begin{align*}
\RE \frac{z f'_3(z)}{f_3(z)} &= \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4} \quad (= 0.5)\\
&= \frac{2 r (1 + r + r^2 - r^3)}{1 - r^4}= \left| \frac{zf_3'(z)}{f_3(z)}- 1 \right|,
\end{align*}
thus illustrates the radius obtained is sharp for the function $f_3$.
\item By using Lemma \ref{T3} and considering
\begin{equation*}
\frac{2 r ( r^2 + r + 1)}{(1-r^4)} \leq \frac{1+r^4}{1-r^4} -\frac{1}{e},
\end{equation*}
it can be proven similarly as above that the exponential starlike radius $R_{\mathcal{S}^*_e}$ for the class $\mathcal{K}_3$ is the smallest positive real root of the equation $(2 r^3 + 2 r^2 + 2 r -1 - r^4)e + 1 - r^4 = 0$. Again, for the function $f_3$ given in \eqref{f3}, at $z := i r = i{R}_{\mathcal{S}^*_e} \alphapprox i(0.244259)$, we have
\begin{equation*}
\left|\log \frac{zf'_3(z)}{f_3(z)} \right| = \left| \log \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4}\right|=1,
\end{equation*}
thereby proving that the result obtained is sharp.
\item For the $\mathcal{S}^*_c$-radius for the class $\mathcal{K}_3$, we use Lemma \ref{T4} by considering
\begin{equation*}
\frac{2 r ( r^2 + r + 1)}{1-r^4} \leq \frac{1+r^4}{1-r^4} -\frac{1}{3}.
\end{equation*}
Then it can be proven similarly as above that $R_{\mathcal{S}^*_c}$ is the smallest positive real root of the equation $$3 r^3 + 3 r^2 + 3 r - 1 - 2 r^4 = 0.$$
The radius obtained is sharp for the function $f_3$ defined in \eqref{f3}.
Indeed, for the function $f_3$, we have
at $z := i r = i {R}_{\mathcal{S}^*_c} \alphapprox i(0.254716)$ that
\begin{equation*}
\frac{zf'_3(z)}{f_3(z)} = \frac{1 - 2 r - 2r^2 -2 r^3 +r^4}{1 - r^4} =\frac{1}{3}=h_c(-1)\in \partial h_c(\mathbb{D}).
\end{equation*}
\item By proving similarly as above, the $\mathcal{S}^*_{\leftmoon}$-radius for the class $\mathcal{K}_3$ is the smallest positive real root of the equation
\[ 2 r^3 + 2 r^2 + 2 r - \sqrt{2} r^4 = 2 - \sqrt{2}.\]
In this case, we would use Lemma \ref{T7} and consider
\begin{equation*}
\frac{2 r ( r^2 + r + 1)}{1- r^4} \leq 1- \sqrt{2} + \frac{1 + r^4}{1 - r^4}.
\end{equation*}
The radius is sharp for the function $f_3$ defined in \eqref{f3}, since at $z := ir = i{R}_{\mathcal{S}^*_{\leftmoon}}$, we have
\begin{align*}
\left| \left( \frac{zf_3'(z)}{f_3(z)}\right)^2-1 \right| &= \left| \left( \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4}\right)^2 -1 \right| \\&= 2\left| \frac{1 - 2 r - 2 r^2 - 2 r^3 + r^4}{1- r^4} \right| = 2 \left|\frac{zf_3'(z)}{f_3(z)} \right|.
\end{align*}
\item In order to find the $\mathcal{S}^*_{\sin}$-radius for the function $f \in \mathcal{K}_3$, we make use of Lemma \ref{T5}, where we would consider
\begin{equation*}\label{11}
\frac{2 r ( r^2 + r + 1)}{(1-r^4)} \leq \sin 1 - \frac{2 r^4}{1-r^4}.
\end{equation*}
The $\mathcal{S}^*_{\sin}$-radius, $R_{\mathcal{S}^*_{\sin}}$, is smallest positive real root of the equation
\[
2r(r^3 + r^2 + r + 1) = (\sin{1})(1 - r^4).
\]
The radius obtained is sharp for the function $f_3$ defined in \eqref{f3}.
\item We use Lemma \ref{T8} to compute the $\mathcal{S}^*_R$- radius for the class $\mathcal{K}_3$. By considering
\begin{equation*}
\frac{2 r ( r^2 + r + 1)}{1-r^4} \leq \frac{1 + r^4}{1 - r^4} - 2(\sqrt{2} - 1),
\end{equation*}
we would obtain $R_{\mathcal{S}^*_{R}}$ to be given by the smallest positive real root of the equation
\[
(2\sqrt{2} - 1)r^4 - 2r(r^2 + r + 1) + (3 - 2\sqrt{2}) = 0.
\]
The radius obtained is sharp for the function $f_3$ defined in \eqref{f3}.
Indeed, for the function $f_3$, we have at $z := i r = i {R}_{\mathcal{S}^*_c} \alphapprox 0.0790749$,
\begin{equation*}
\frac{zf'_3(z)}{f_3(z)} = \frac{1 - 2 r - 2r^2 -2 r^3 + r^4}{1 - r^4} = 2 \sqrt{2}-2=h_R(-1)\in \partial h_R(\mathbb{D}).
\end{equation*}
\item Lastly, the $\mathcal{S}^*_{RL}$- radius for the class $\mathcal{K}_3$ is obtained by using Lemma \ref{T9} and from the equation
\begin{align*}
(1- r^4) \big\{ (1- r^4)^2 - \big( (\sqrt{2} - 1) - (\sqrt{2} + 1) r^4 \big)^2 \big\}^{1/2} &= (2r^3+2r^2+2r)^2 +(1- r^4)^2 \\& \quad - \big((\sqrt{2} - 1)- (\sqrt{2} + 1) r^4 \big)^2. \qedhere
\end{align*}
\end{enumerate}
\end{proof}
\subsection*{\textbf{Acknowledgment.}} The second author gratefully acknowledges support from USM research university grants 1001.PMATHS.8011101.
\end{document} |
\begin{document}
\title{Extending a valuation centered in a local domain to the formal
completion.}
\section{Introduction}
\label{In}
All the rings in this paper will be commutative with 1.
Let $(R,m,k)$ be a local noetherian domain with field of fractions
$K$ and $R_\nu$ a valuation ring, dominating $R$ (not necessarily
birationally). Let $\nu|_K:K^*\twoheadrightarrow\mbox{l{\Phi}space{-.47em}G}amma$ be the
restriction of $\nu$ to $K$; by definition, $\nu|_K$ is centered at
$R$. Let ${\Phi}at R$ denote the $m$-adic completion of $R$. In the
applications of valuation theory to commutative algebra and the
study of singularities, one is often induced to replace $R$ by its
$m$-adic completion ${\Phi}at R$ and $\nu$ by a suitable extension
${\Phi}at\nu_-$ to $\frac{{\Phi}at R}P$ for a suitably chosen prime ideal
$P$, such that $P\cap R=(0)$ (one specific application we have in mind has to do with the approaches to proving the Local Uniformization Theorem in arbitrary characteristic such as \cite{Spi2} and \cite{Te}). The first reason is that the ring
${\Phi}at R$ is not in general an integral domain, so that we can only
hope to extend $\nu$ to a \textit{pseudo-valuation} on ${\Phi}at R$,
which means precisely a valuation ${\Phi}at\nu_-$ on a quotient
$\frac{{\Phi}at R}P$ as above. The prime ideal $P$ is called the
\textit{support} of the pseudo-valuation. It is well known and not
hard to prove that such extensions ${\Phi}at\nu_-$ exist for some
minimal prime ideals $P$ of ${\Phi}at R$. Although, as we shall see, the
datum of a valuation $\nu$ determines a unique minimal prime of
${\Phi}at R$ when $R$ is excellent, in general there are many possible
primes $P$ as above and for a fixed $P$ many possible extensions
${\Phi}at\nu_-$. This is the second reason to study extensions
${\Phi}at\nu_-$.
The purpose of this paper is to give, assuming
that $R$ is excellent, a systematic description of all such
extensions ${\Phi}at\nu_-$ and to identify certain classes of extensions
which are of particular interest for applications. In fact, the only
assumption about $R$ we ever use in this paper is a weaker and more
natural condition than excellence, called the G condition, but we
chose to talk about excellent rings since this terminology seems to
be more familiar to most people. For the reader's convenience, the
definitions of excellent and G-rings are recalled in the
Appendix. Under this assumption, we study
\textbf{extensions to (an
integral quotient of) the completion ${\Phi}at R$ of a valuation $\nu$
and give descriptions of the
valuations with which such extensions are composed. In particular we give criteria
for the uniqueness of the extension if certain simple data on these
composed valuations are fixed}.
\noindent
We conjecture (see statement 5.19 in \cite{Te} and Conjecture \ref{teissier} below for a stronger and
more precise statement) that\par\noindent
\textbf{given an excellent local ring $R$ and a valuation $\nu$ of
$R$ which is positive on its maximal ideal $m$, there exists a prime ideal
$H$ of the $m$-adic completion ${\Phi}at R$ such that $H\bigcap R=(0)$
and an extension of $\nu$ to $\frac{{\Phi}at R}{H}$ which has the same
value group as $\nu$.}
When studying extensions of $\nu$ to the completion of $R$, one is led to
the study of its extensions to the henselization $\tilde R$ of $R$ as a
natural first step. This, in turn, leads to the study of extensions of
$\nu$ to finitely generated local strictly \'etale extensions $R^e$ of
$R$. We therefore start out by letting $\sigma:R\rightarrow R^\dag$
denote one of the three operations of completion, (strict)
henselization, or a finitely generated local strictly \'etale
extension:
\begin{eqnarray}
R^\dag&=&{\Phi}at R\quad\mbox{ or}\label{eq:hatR}\\
R^\dag&=&\tilde R\quad\mbox{ or}\label{eq:tildeR}\\
R^\dag&=&R^e.\label{eq:Re}
\end{eqnarray}
The ring $R^\dag$ is local; let $m^\dag$ denote its maximal ideal.
The homomorphisms
$$
R\rightarrow\tilde R\quad\text{ and }\quad R\rightarrow R^e
$$
are regular for any ring $R$; by definition, if $R$ is an excellent ring
then the completion homomorphism is regular (in fact, regularity of
the completion homomorphism is precisely the defining property of
G-rings; see the Appendix for the definition of regular
homomorphism).
Let $r$ denote the (real) rank of $\nu$. Let $(0)=\Delta_r\subsetneqq
\Delta_{r-1}\subsetneqq\dots\subsetneqq \Delta_0=\mbox{l{\Phi}space{-.47em}G}amma$ be the isolated
subgroups of $\mbox{l{\Phi}space{-.47em}G}amma$ and $P_0=(0)\subsetneqq
P_1\subseteq\dots\subseteq P_r=m$ the prime valuation ideals of $R$,
which need not, in general, be distinct. In this paper, we will assume
that $R$ is excellent. Under this assumption, we will canonically
associate to $\nu$ a chain $H_1\subset H_3\subset\dots\subset
H_{2r+1}=mR^\dag$ of ideals of $R^\dag$, numbered by odd integers from
1 to $2r+1$, such that $H_{2\ell+1}\cap R=P_\ell$ for $0\le \ell\le
r$. We will show that all the ideals $H_{2\ell+1}$ are prime. We will
define $H_{2\ell}$ to be the unique minimal prime ideal of $P_\ell
R^\dag$, contained in $H_{2\ell+1}$ (that such a minimal prime is
unique follows from the regularity of the homomorphism $\sigma$).
We will thus obtain, in the cases (\ref{eq:hatR})--(\ref{eq:Re}), a
chain of $2r+1$ prime ideals
$$
H_0\subset H_1\subset\dots\subset H_{2r}=H_{2r+1}=mR^\dag,
$$
satisfying $H_{2\ell}\cap R=H_{2\ell+1}\cap R=P_\ell$ and such that
$H_{2\ell}$ is a minimal prime of $P_\ell R^\dag$ for $0\le \ell\le
r$. Moreover, if $R^\dag=\tilde R$ or $R^\dag=R^e$, then
$H_{2\ell}=H_{2\ell+1}$. We call $H_i$ the $i${\bf-th implicit prime
ideal} of $R^\dag$, associated to $R$ and $\nu$. The ideals $H_i$
behave well under local blowing ups along $\nu$ (that is, birational
local homomorphisms $R\to R'$ such that $\nu$ is centered in $R'$),
and more generally under \textit{$\nu$-extensions} of $R$ defined
below in subsection \ref{trees}. This means that given any local
blowing up along $\nu$ or $\nu$-extension $R\rightarrow R'$, the
$i$-th implicit prime ideal $H'_i$ of ${R'}^\dag$ has the property
that $H'_i\cap R^\dag=H_i$. This intersection has a meaning in view of Lemma
\ref{factor} below.
For a prime ideal $P$ in a ring $R$, $\kappa(P)$ will denote the residue
field $\frac{R_P}{PR_P}$.
Let $(0)\subsetneqq \mathbf{m}_1\subsetneqq\dots\subsetneqq
\mathbf{m}_{r-1}\subsetneqq\mathbf{m}_r=\mathbf{m}_\nu$ be the prime
ideals of the valuation ring $R_\nu$. By definitions, our valuation
$\nu$ is a composition of $r$ rank one valuations
$\nu=\nu_1\circ\nu_2\dots\circ\nu_r$, where $\nu_\ell$ is a valuation
of the field $\kappa(\mathbf{m}_{\ell-1})$, centered at
$\frac{(R_\nu)_{\mathbf{m}_\ell}}{\mathbf{m}_{\ell -1}}$ (see
\cite{ZS}, Chapter VI, \S10, p. 43 for the definition of composition of
valuations; more information and a simple example of composition is
given below in subsection \ref{trees}, where we interpret each
$\mathbf{m}_\ell$ as the limit of a tree of ideals).
If $R^\dag=\tilde R$, we will prove that there is a unique extension
$\tilde\nu_-$ of $\nu$ to $\frac{\tilde R}{H_0}$. If $R^\dag={\Phi}at
R$, the situation is more complicated. First, we need to discuss the
behaviour of our constructions under $\nu$-extensions.
\subsection{Local blowings up and trees.}\label{trees}
We consider \textit{extensions} $R\rightarrow R'$ of local rings, that
is, injective morphisms such that $R'$ is an $R$-algebra essentially
of finite type and $m'\cap R=m$. In this paper we consider only
extensions with respect to $\nu$; that is, both $R$ and $R'$ are
contained in a fixed valuation ring $R_\nu$. Such extensions form a
direct system $\{R'\}$. We will consider many direct systems of rings
and of ideals indexed by $\{R'\}$; direct limits will always be taken
with respect to the direct system $\{R'\}$. Unless otherwise
specified, we will assume that
\begin{equation}
\lim\limits_{\overset\longrightarrow{R'}}R'=R_\nu.\label{eq:ZariskiRiemann}
\end{equation}
Note that by the fundamental properties of valuation rings
(\cite{ZS}, \S VI), assuming the equality (\ref{eq:ZariskiRiemann})
is equivalent to assuming that
$\lim\limits_{\overset\longrightarrow{R'}}K'=K_\nu$, where $K'$ stands
for the field of fractions of $R'$ and $K_\nu$ for that of $R_\nu$, and
that $\lim\limits_{\overset\longrightarrow{R'}}R'$ is a valuation
ring.
\begin{definition} A \textbf{tree} of $R'$-algebras is a direct system
$\{S'\}$ of rings, indexed by the directed set
$\{R'\}$, where $S'$ is an $R'$-algebra. Note that the maps are not
necessarily injective. A morphism $\{S'\}\to\{T'\}$ of
trees is the datum of a map of $R'$-algebras $S'\to T'$ for each $R'$
commuting with the tree morphisms for each map $R'\to R''$.
\end{definition}
\begin{lemma}\label{factor} Let $R\rightarrow R'$ be an extension of
local rings. We have:\par
\noindent 1) The ideal
$N:=m^\dag\otimes_R1+1\otimes_Rm'$ is maximal in the $R$-algebra
$R^\dag\otimes_RR'$.\par
\noindent 2) The natural map of completions (resp. henselizations)
$R^\dag\to{R'}^\dag$ is injective.
\end{lemma}
\begin{proof} 1) follows from that fact that $R^\dag/m^\dag
=R/m$. The proof of 2) relies on a construction which we shall use
often: the map $R^\dag\to{R'}^\dag$ can be factored as
\begin{equation}
R^\dag\to\left(R^\dag\otimes_RR'\right)_N\to{R'}^\dag,\label{eq:iota}
\end{equation}
where the first map sends $x$ to $x\otimes 1$ and the second is determined
by $x\otimes x'\mapsto {\Phi}at b(x).c(x')$ where ${\Phi}at b$ is the natural
map $R^\dag\to {R'}^\dag$ and $c$ is the canonical map
$R'\to{R'}^\dag$. The first map is injective because $R^\dag$ is a
flat $R$-algebra and it is obtained by tensoring the injection $R\to
R'$ by the $R$-algebra $R^\dag$; furthermore, elements of $R^\dag$
whose image in $R^\dag\otimes_RR'$ lie outside of $N$ are precisely
units of $R^\dag$, hence they are not zero divisors in
$R^\dag\otimes_RR'$ and $R^\dag$ injects in every localization of
$R^\dag\otimes_RR'$.\par
\noindent Since $m'\cap R=m$, we see that the inverse image by the
natural map of $R'$-algebras
$$
\iota\colon R'\to(R^\dag\otimes_RR')_N,
$$
defined by $x'\mapsto1\otimes_Rx'$, of the maximal ideal
$M=(m^\dag\otimes_R1+1\otimes_Rm')(R^\dag\otimes_RR')_N$ of
$(R^\dag\otimes_RR')_N$ is the ideal $m'$ and that $\iota$ induces a
natural isomorphism
$\frac{R'}{{m'}^i}\overset\sim\rightarrow\frac{(R^\dag\otimes_RR')_N}{M^i}$
for each $i$. From this it follows by the universal properties of
completion and henselization that the second map in the sequence
(\ref{eq:iota}) is the completion (resp. the henselization inside the
completion) of $R^\dag\otimes_RR'$ with respect to the ideal $M$. It
is therefore also injective.
\end{proof}
\begin{definition} Let $\{S'\}$ be a tree of $R'$-algebras. For each $S'$,
let $I'$ be an ideal of $S'$. We say that $\{I'\}$ is a tree of
ideals if for any arrow $b_{S'S''}\colon S'\rightarrow S''$ in our
direct system, we have $b^{-1}_{S'S''}I''=I'$. We have the obvious
notion of inclusion of trees of ideals. In particular, we may speak
about chains of trees of ideals.
\end{definition}
\begin{examples}
The maximal ideals of the local rings of our system $\{R'\}$ form a
tree of ideals.
For any non-negative element $\beta\in\mbox{l{\Phi}space{-.47em}G}amma$, the valuation ideals
$\mathcal{P}'_\beta\subset R'$ of value $\beta$ form a tree of ideals of
$\{R'\}$. Similarly, the $i$-th prime valuation ideals $P'_i\subset
R'$ form a tree. If $rk\ \nu=r$, the prime valuation ideals $P'_i$
give rise to a chain
\begin{equation}
P'_0=(0)\subsetneqq P'_1\subseteq\dots\subseteq P'_r=m'\label{eq:treechain'}
\end{equation}
of trees of prime ideals of $\{R'\}$.
\end{examples}
We discuss this last example in a little more detail and
generality in order to emphasize our point of view, crucial
throughout this paper: the data of a composite valuation is
equivalent to the data of its components. Namely, suppose we are
given a chain of trees of ideals as in (\ref{eq:treechain'}),
where we relax our assumptions of the $P'_i$ as follows. We no
longer assume that the chain (\ref{eq:treechain'}) is maximal, nor
that $P'_i\subsetneqq P'_{i+1}$, even for $R'$ sufficiently large;
in particular, for the purposes of this example we momentarily drop
the assumption that $rk\ \nu=r$. We will still assume, however, that
$P'_0=(0)$ and that $P'_r=m'$.
Taking the limit in (\ref{eq:treechain'}), we obtain a chain
\begin{equation}
(0)=\mathbf{m}_0\subsetneqq\mathbf{m}_1\subseteqq\dots\subseteqq
\mathbf{m}_r=\mathbf{m}_\nu\label{eq:treechainlim}
\end{equation}
of prime ideals of the valuation ring $R_\nu$.
Similarly, for each $1\leq\ell\leq r$ one has the equality
$$
\lim\limits_{\overset\longrightarrow{R'}}{\frac{R'}{P'_\ell}}=\frac{R_\nu}{\bf
m_\ell}.
$$
Then \textbf{specifying the valuation $\nu$ is equivalent to
specifying valuations $\nu_0,\nu_1$, \dots, $\nu_r$, where $\nu_0$ is the
trivial valuation of $K$ and, for $1\le \ell\le r$, $\nu_\ell$ is a
valuation of the residue field
$k_{\nu_{\ell-1}}=\kappa(\mathbf{m}_{\ell-1})$, centered at the local
ring
$\lim\limits_{\longrightarrow}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}=
\frac{(R_\nu)_{\mathbf{m}_\ell}}{\mathbf{m}_{\ell-1}}$ and taking its
values in the totally ordered group
$\frac{\Delta_{\ell-1}}{\Delta_\ell}$.}
The relationship between $\nu$ and the $\nu_\ell$ is that $\nu$ is the
composition
\begin{equation}
\nu=\nu_1\circ\nu_2\circ\dots\circ\nu_r.\label{eq:composition1}
\end{equation}
For example, the datum of the valuation $\nu$, or of its valuation ring $R_\nu$,
is equivalent to the datum of the valuation ring
$\frac{R_\nu}{\mathbf{m}_{r-1}}\subset
\frac{(R_\nu)_{\mathbf{m}_{r-1}}}{\mathbf{m}_{r-1}(R_\nu)_{\mathbf{m}_{r-1}}}=\kappa
(\mathbf{m}_{r-1})$ of the valuation $\nu_r$ of the field
$\kappa(\mathbf{m}_{r-1})$ and the valuation ring
$(R_\nu)_{\mathbf{m}_{r-1}}$. If we assume, in addition, that for $R$
sufficiently large the chain (\ref{eq:treechain'}) (equivalently,
(\ref{eq:treechainlim})) is a maximal chain of distinct prime ideals
then $rk\ \nu=r$ and $rk\ \nu_\ell=1$ for each $\ell$.\par
\begin{remark}\label{composite}
Another way to describe the same property of valuations is that, given
a prime ideal $H$ of the local integral domain $R$ one builds all
valuations centered in $R$ having $H$ as one of the $P_\ell$ by
choosing a valuation $\nu_1$ of $R$ centered at $H$, so that
$\mathbf{m}_{\nu_1}\cap R=H$ and choosing a valuation subring
$\overline R_{\overline\nu}$ of the field
$\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$ centered at $R/H$. Then
$\nu=\nu_1\circ \overline\nu$.\par
\noindent Note that choosing a valuation of $R/H$ determines a
valuation of its field of fractions $\kappa(H)$, which is in general
much smaller than $\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$. Given a
valuation of $R$ with center $H$, in order to determine a valuation of
$R$ with center $m$ inducing on $R/H$ a given valuation $\mu$ we must
choose an extension $\overline\nu$ of $\mu$ to
$\frac{R_{\nu_1}}{\mathbf{m}_{\nu_1}}$, and there are in general many
possibilities.\par
This will be used in the sequel. In particular, it will be applied to
the case where a valuation $\nu$ of $R$ extends uniquely to a
valuation ${\Phi}at\nu_-$ of $\frac{{\Phi}at R}{H}$ for some prime $H$ of ${\Phi}at
R$. Assuming that ${\Phi}at R$ is an integral domain, this determines a
unique valuation of ${\Phi}at R$ only if the height ${\Phi}e\ H$ of $H$ in
${\Phi}at R$ is at most one. In all other cases the dimension of ${\Phi}at
R_H$ is at least $2$ and we have infinitely many valuations with which
to compose ${\Phi}at\nu_-$. This is the source of the height conditions we
shall see in \S\ref{extensions}.
\end{remark}
\begin{example} Let $k_0$ be a field and $K=k_0((u,v))$ the
field of fractions of the complete local ring $R=k_0[[u,v]]$. Let
$\mbox{l{\Phi}space{-.47em}G}amma=\mathbf Z^2$ with lexicographical ordering. The isolated
subgroups of $\mbox{l{\Phi}space{-.47em}G}amma$ are $(0)\subsetneqq(0)\oplus\mathbf
Z\subsetneqq\mathbf Z^2$. Consider the valuation
$\nu:K^*\rightarrow\mathbf Z^2$, centered at $R$, given by
\begin{eqnarray}
\nu(v)&=&(0,1)\\
\nu(u)&=&(1,0)\\
\nu(c)&=&0\quad\text{ for any }c\in k_0^*.
\end{eqnarray}
This information determines $\nu$ completely; namely, for any power series
$$
f=\sum\limits_{\alpha,\beta}c_{\alpha\beta}u^\alpha v^\beta \in k_0[[u,v]],
$$
we have
$$
\nu(f)=\min\{(\alpha,\beta)\ |\ c_{\alpha\beta}\ne 0\}.
$$
We have $rk\ \nu=rat.rk\ \nu=2$. Let $\Delta=(0)\oplus\mathbf Z$. Let
$\mbox{l{\Phi}space{-.47em}G}amma_+$ denote the semigroup of all the non-negative elements of
$\mbox{l{\Phi}space{-.47em}G}amma$. Let $k_0[[\mbox{l{\Phi}space{-.47em}G}amma_+]]$ denote the $R$-algebra of power series
$\sum c_{\alpha,\beta}u^\alpha v^\beta$ where $c_{\alpha,\beta}\in
k_0$ and the exponents $(\alpha ,\beta)$ form a well ordered subset of
$\mbox{l{\Phi}space{-.47em}G}amma_+$. By classical results (see \cite{Kap1}, \cite{Kap2}), it is
a valuation ring with maximal ideal generated by all the monomials
$u^\alpha v^\beta$, where $(\alpha,\beta)>(0,0)$ (in other words,
either $\alpha>0,\beta\in\mathbf Z$ or $\alpha=0,\beta>0$). Then
$$
R_\nu=k_0[[\mbox{l{\Phi}space{-.47em}G}amma_+]]\bigcap k_0((u,v))
$$
is a valuation ring of $K$, and contains $k[[u,v]]$; it is the valuation ring of the valuation
$\nu$. The prime ideal $\mathbf{m}_1$ is the ideal of $R_\nu$
generated by all the $uv^\beta$, $\beta\in\mathbf Z$. The valuation
$\nu_1$ is the discrete rank 1 valuation of $K$ with valuation ring
$$
(R_\nu)_{\mathbf{m}_1}=k_0[[u,v]]_{(u)}
$$
and $\nu_2$ is the discrete rank 1 valuation of $k_0((v))$ with
valuation ring $\frac{R_\nu}{\mathbf{m}_1}\cong k_0[[v]]$.
\end{example}
\begin{example} To give a more interesting example, let $k_0$ be
a field of characteristic zero and
$$
K=k_0(x,y,z)
$$
a purely transcendental extension of $k_0$ of degree 3. Let $w$ be an
independent variable and put $k=\bigcup\limits_{j=1}^\infty
k_0\left(w^{\frac 1j}\right)$. Let $\mbox{l{\Phi}space{-.47em}G}amma=\mathbf Z\oplus\mathbf{Q}$ with the
lexicographical ordering and $\Delta=(0)\oplus\mathbf{Q}$ the non-trivial isolated
subgroup of $\mbox{l{\Phi}space{-.47em}G}amma$. Let $u,v$ be new variables and let
$\mu_1:k((u,v))\rightarrow\mathbf{Z}^2_{lex}$ be the valuation of the
previous example. Let $\mu_2$ denote the $x$-adic valuation of $k$ and
put $\mu=\mu_1\circ\mu_2$.
Consider the map $\iota:k_0[x,y,z]\rightarrow
k[[u,v]]$ which sends $x$ to $w$, $y$ to $v$ and $z$ to
$u-\sum\limits_{j=1}^\infty w^{\frac 1j}v^j$. Let $\nu_1=\left.\mu_1\right|_K$ and
$\nu=\mu|_K$.
The valuation $\nu:K^*\rightarrow\mbox{l{\Phi}space{-.47em}G}amma$ is centered at the local ring
$R=k_0[x,y,z]_{(x,y,z)}$; we have
\begin{eqnarray}
\nu(x)&=&(0,1)\\
\nu(y)&=&(1,0),\\
\nu(z)&=&(1,1).
\end{eqnarray}
Write as a composition of two rank 1 valuations:
$\nu=\nu_1\circ\nu_2$. We have natural inclusions $R_{\nu_1}\subset
R_{\mu_1}$ and $k_{\nu_1}\subset k_{\mu_1}=k$. We claim that
$k_{\nu_1}$ is not finitely generated over $k_0$. Indeed, if this were not the
case then there would exist a prime number $p$ such that
$w^{\frac1p}\mbox{$\in$ {\Phi}space{-.8em}/} k_{\nu_1}$. Let
$k'=k_0\left(x^{\frac1{(p-1)!}}\right)$. Let $L= k'(y,z)$. Consier
the tower of field extensions $K\subset L\subset k[[u,v]]$ and let $\nu'$ denote
the restriction of $\mu$ to $L$. Let $\mbox{l{\Phi}space{-.47em}G}amma'$ be the value group of
$\nu'$ and $k_{\nu'}$ the residue field of its valuation ring. Now,
$L$ contains the element $z_p:=z-\sum\limits_{j=1}^{p-1}x^{\frac1j}y^j$ as
well as $\frac{z_p}{y^p}$. We have
\begin{equation}
\nu'(z_p)=\left(p,\frac1p\right),
\end{equation}
$\nu'\left(\frac{z_p}{y^p}\right)=0$ and the natural image of
$\frac{z_p}{y^p}$ in $k_{\mu_1}=k$ is $w^{\frac1p}$. Now,
$p\not|\ [L:K]$, $\left.[\mbox{l{\Phi}space{-.47em}G}amma':\mbox{l{\Phi}space{-.47em}G}amma]\ \right|\
[L:K]$ and $\left.[k_{\nu'}:k_\nu]\ \right|\
[L:K]$. This implies that $z_p\in L$ and $w^{\frac1p}\in k_{\nu_1}$,
which gives the desired contradicion.
It is not hard to show that for each $j$, there exists a local blowing
up $R\rightarrow R'$ of $R$ such that, in the notation of
(\ref{eq:treechain'}), we have
$\kappa(P'_1)=k_0\left(w^{\frac1{j!}}\right)$ and that
$\kappa(\mathbf{m}_1)=\lim\limits_{j\to\infty}\kappa(P'_1)=k$. The
first one is the blowing up of the ideal $(y,z)R$, localized at the
point $y=0, z/y=x$. Then one blows up the ideal $(z/y-x, y)$, and so
on.
Another way to see the valuation $\nu=\nu_1\circ\nu_2$ is to note that
$\nu_1$ is the restriction to $K$ of the $v$-adic valuation under the
inclusion of fields deduced from the inclusion of rings
$$
k_0[[x,y,z]]_{(y,z)}{\Phi}ookrightarrow k\left[\left[v^{{\mathbf Z}_+}\right]\right]
$$
which sends $x$ to $w$, $y$ to $v$ and $z$ to $\sum\limits_{j=1}^\infty
w^{\frac1j}v^j$. Recall that the ring on the right is made of power
series with non negative rational exponents whose set of exponents is
well ordered. We have $k_{\nu_1}=k$.
\end{example}
\begin{remark} The point of the last example is to show that, given a
composed valuation as in (\ref{eq:composition1}), $\nu_\ell$ is a
valuation of the field $k_{\nu_{\ell-1}}$, which may
\textbf{properly} contain $\kappa(P'_{\ell-1})$ for \textbf{every}
$R'\in\mathcal{T}$. This fact will be a source of complication
later on and we prefer to draw attention to it from the beginning.
\end{remark}
Coming back to the implicit prime ideals, we will see that the
implicit prime ideals $H'_i$ form a tree of ideals of $R^\dag$.
We will show that if $\nu$ extends to a valuation of ${\Phi}at\nu_-$ centered at
$\frac{{\Phi}at R}P$ with $P\cap R=(0)$ then the prime $P$ must
contain the minimal prime $H_0$ of ${\Phi}at R$. We will then show
that specifying an extension ${\Phi}at\nu_-$ of $\nu$ as above is
equivalent to specifying a chain of prime valuation ideals
\begin{equation}
\tilde H'_0\subset\tilde H'_1\subset\dots\subset\tilde H'_{2r}=m'{\Phi}at
R'\label{eq:chaintree}
\end{equation}
of ${\Phi}at R'$ such that $H'_\ell\subset\tilde H'_\ell$ for all
$\ell\in\{0,\dots,2r\}$, and valuations
${\Phi}at\nu_1,{\Phi}at\nu_2,\dots,{\Phi}at\nu_{2r}$, where ${\Phi}at\nu_i$ is a
valuation of the field $k_{{\Phi}at\nu_{i-1}}$ (the residue field of the
valuation ring $R_{{\Phi}at\nu_{i-1}}$), arbitrary when $i$ is odd and
satisfying certain conditions, coming from the valuation $\nu_{\frac
i2}$, when $i$ is even.
The prime ideals $H_i$ are defined as follows.\par
\noindent Recall that given a valued ring $(R,\nu)$, that is a subring $R\subseteq R_\nu$ of the valuation ring
$R_\nu$ of a valuation with value group $\mbox{l{\Phi}space{-.47em}G}amma$, one defines for each
$\beta\in \mbox{l{\Phi}space{-.47em}G}amma$ the valuation ideals of $R$ associated to $\beta$:
$$
\begin{array}{lr} {\cal P}_\beta (R)=&\{x\in R/\nu(x){\Gamma}eq\beta\}\cr
{\cal P}^+_\beta (R)=&\{x\in R/\nu(x)>\beta\}\end{array}
$$
and the associated graded ring
$$
{\Phi}box{\rm gr}_\nu R=\bigoplus_{\beta\in \mbox{l{\Phi}space{-.47em}G}amma}\frac{{\cal P}_\beta (R)}
{{\cal P}^+_\beta (R)}=\bigoplus_{\beta\in
\mbox{l{\Phi}space{-.47em}G}amma_+}\frac{{\cal P}_\beta (R)}{{\cal P}^+_\beta (R)}.
$$
The second equality comes from the fact that if
$\beta\in\mbox{l{\Phi}space{-.47em}G}amma_-\setminus\{0\}$, we have ${\cal P}^+_\beta (R)={\cal
P}_\beta (R)=R$. If $R\to R'$ is an extension of local rings such
that $R\subset R'\subset R_\nu$ and $m_\nu\cap R'=m'$, we will write
${\cal P}'_\beta$ for ${\cal P}_\beta(R')$.
Fix a valuation ring $R_\nu$ dominating $R$, and a tree ${\cal
T}=\{R'\}$ of n\oe therian local $R$-subalgebras of $R_\nu$, having
the following properties: for each ring $R'\in\cal{T}$, all the
birational $\nu$-extensions of $R'$ belong to $\cal{T}$. Moreover, we
assume that the field of fractions of $R_\nu$ equals
$\lim\limits_{\overset\longrightarrow{R'}}K'$, where $K'$ is the field
of fractions of $R'$. The tree $\cal{T}$ will stay constant throughout
this paper. In the special case when $R$ happens to have the same
field of fractions as $R_\nu$, we may take $\cal{T}$ to be the tree of
all the birational $\nu$-extensions of $R$.
\begin{notation} For a ring $R'\in\cal T$, we shall denote by
${\cal T}(R')$ the subtree of $\cal T$ consisting of all the
$\nu$-extensions $R''$ of $R'$.
\end{notation}
We now define
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}
\left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal P}'_\beta
{R'}^\dag\right)\bigcap R^\dag\right),\ 0\le\ell\le
r-1\label{eq:defin}
\end{equation}
(in the beginning of \S\ref{basics} we provide some motivation for this
definition and give several elementary examples of $H'_i$ and $\tilde H'_i$).
The questions answered in this paper originally arose from our work on the
Local Uniformization Theorem, where passage to completion is required in
both the approaches of \cite{Spi2} and \cite{Te}. In \cite{Te}, one really
needs to pass to completion for valuations of arbitrary rank. One of the
main intended applications of the theory of implicit prime ideals is
the following conjecture. Let
\begin{equation}
\mbox{l{\Phi}space{-.47em}G}amma{\Phi}ookrightarrow{\Phi}at\mbox{l{\Phi}space{-.47em}G}amma\label{eq:extGamma}
\end{equation}
be an extension of ordered groups of the same rank. Let
\begin{equation}
(0)=\Delta_r\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_0=\mbox{l{\Phi}space{-.47em}G}amma
\label{eq:isolated}
\end{equation}
be the isolated subgroups of $\mbox{l{\Phi}space{-.47em}G}amma$ and
$$
(0)={\Phi}at\Delta_r\subsetneqq{\Phi}at\Delta_{r-1}\subsetneqq\dots\subsetneqq
{\Phi}at\Delta_0={\Phi}at\mbox{l{\Phi}space{-.47em}G}amma
$$
the isolated subgroups of ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma$, so that the inclusion
(\ref{eq:extGamma}) induces inclusions
\begin{eqnarray}
\Delta_\ell&{\Phi}ookrightarrow&{\Phi}at\Delta_\ell\quad\text{ and}\\
\frac{\Delta_\ell}{\Delta_{\ell+1}}&{\Phi}ookrightarrow&
\frac{{\Phi}at\Delta_\ell}{{\Phi}at\Delta_{\ell+1}}.
\end{eqnarray}
Let $G{\Phi}ookrightarrow{\Phi}at G$ be an extension of graded algebras
without zero divisors, such that $G$ is graded by $\mbox{l{\Phi}space{-.47em}G}amma_+$ and ${\Phi}at
G$ by ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma_+$. The graded algebra $G$ is endowed with a natural
valuation with value group $\mbox{l{\Phi}space{-.47em}G}amma$ and similarly for ${\Phi}at G$ and
${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma$. These natural valuations will both be denoted by $ord$.
\begin{definition} We say that the extension $G{\Phi}ookrightarrow{\Phi}at G$ is
\textbf{scalewise birational} if for any $x\in{\Phi}at G$ and
$\ell\in\{1,\dots,r\}$ such that $ord\ x\in{\Phi}at\Delta_\ell$ there
exists $y\in G$ such that $ord\ y\in\Delta_\ell$ and $xy\in G$.
\end{definition}
Of course, scalewise birational implies birational and also that
${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma=\mbox{l{\Phi}space{-.47em}G}amma$.\par
\noindent While the main result of this paper is the primality
of the implicit ideals associated to a valuation, and the subsequent
description of the extensions of the valuation to the completion, the
main conjecture stated here is the following:\par
\begin{conjecture}\label{teissier} Assume that $\dim\ R'=\dim\ R$ for all
$R'\in\mathcal{T}$. Then there exists a tree of prime ideals $H'$
of ${\Phi}at R'$ with $H'\cap R'=(0)$ and a valuation ${\Phi}at\nu_-$,
centered at $\lim\limits_\to\frac{{\Phi}at R'}{H'}$ and having the
following property:\par\noindent For any $R'\in\cal{T}$ the graded
algebra $\mbox{gr}_{{\Phi}at\nu_-}\frac{{\Phi}at R'}{H'}$ is a scalewise birational
extension of $\mbox{gr}_\nu R'$.
\end{conjecture}
The example given in remark 5.20, 4) of \cite{Te} shows that the
morphism of associated graded rings is not an isomorphism in general.
The approach to the Local Uniformization Theorem taken in \cite{Spi2} is to
reduce the problem to the case of rank 1 valuations. The theory of implicit
prime ideals is much simpler for valuations of rank 1 and takes only a
few pages in Section \ref{archimedian}.
The paper is organized as follows. In \S\ref{basics} we define the
odd-numbered implicit ideals $H_{2\ell+1}$ and prove that $H_{2\ell+1}\cap
R=P_\ell$. We observe that by their very definition, the ideals
$H_{2\ell+1}$ behave well under $\nu$-extensions; they form a
tree. Proving that $H_{2\ell+1}$ is indeed prime is postponed until
later sections; it will be proved gradually in
\S\ref{technical}--\S\ref{prime}. In the beginning of \S\ref{basics}
we will explain in more detail the respective roles played by the
odd-numbered and the even-numbered implicit ideals, give several
examples (among other things, to motivate the need for taking the
limit with respect to $R'$ in (\ref{eq:defin})) and say one or two
words about the techniques used to prove our results.
In \S\ref{technical} we prove the primality of the implicit prime
ideals assuming a certain technical condition, called
\textbf{stability}, about the tree $\cal T$ and the operation ${\
}^\dag$. It follows from the noetherianity of $R^\dag$ that there
exists a specific $R'$ for which the limit in (\ref{eq:defin}) is
attained. One of the main points of \S\ref{technical} is to prove
properties of stable rings which guarantee that this limit is attained
whenever $R'$ is stable. We then use the excellence of $R$ to define
the even-numbered implicit prime ideals: for $i=2\ell$ the ideal
$H_{2\ell}$ is defined to be the unique minimal prime of $P_\ell
R^\dag$, contained in $H_{2\ell+1}$ (in the case $R^\dag={\Phi}at R$ it is
the excellence of $R$ which implies the uniqueness of such a minimal
prime). We have
$$
H_{2\ell}\cap R=P_\ell
$$
for $\ell\in\{0,\dots,r\}$. The results of \S\ref{technical} apply
equally well to completions, henselizations and other local \'etale
extensions; to complete the proof of the primality of the implicit
ideals in various contexts such as henselization or completion, it
remains to show the existence of stable $\nu$-extensions in the
corresponding context.
In \S\ref{Rdag} we describe the set of extensions $\nu^\dag_-$ of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{P'{R'}^\dag}$,
where $P'$ is a tree of prime ideals of ${R'}^\dag$ such that $P'\cap
R'=(0)$. We show (Theorem \ref{classification}) that specifying such a
valuation $\nu^\dag_-$ is equivalent to specifying the following data:
(1) a chain (\ref{eq:chaintree}) of trees of prime ideals $\tilde
H'_i$ of ${R'}^\dag$ (where $\tilde H'_0=P'$), such that
$H'_i\subset\tilde H'_i$ for each $i$ and each $R'\in\mathcal{T}$,
satisfying one additional condition (we will refer to the chain
(\ref{eq:chaintree}) as the chain of trees of ideals,
\textbf{determined by} the extension $\nu^\dag_-$)
(2) a valuation $\nu^\dag_i$ of the residue field $k_{\nu^\dag_{i-1}}$
of $\nu^\dag_{i-1}$, whose restriction to the field
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ is
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$.
If $i=2\ell$ is even, the valuation $\nu^\dag_i$ must be of rank 1 and
its restriction to $\kappa(\mathbf{m}_{\ell-1})$ must coincide with $\nu_\ell$.
Notice the recursive nature of this description of $\nu^\dag_-$: in
order to describe $\nu^\dag_i$ we must know $\nu^\dag_{i-1}$ in order
to talk about its residue field $k_{\nu^\dag_{i-1}}$.
In \S\ref{extensions} we address the question of uniqueness of
$\nu^\dag_-$. We describe several classes of extensions $\nu^\dag_-$ which
are particularly useful for the applications: \textbf{minimal} and
\textbf{evenly minimal} extensions, and also those $\nu^\dag_-$ for
which, denoting by ${\Phi}e\ I$ the height of an ideal, we have
\begin{equation}
{\Phi}e\ \tilde H'_{2\ell+1}- {\Phi}e\ \tilde H'_{2_\ell} \le 1\quad\text{ for
}0\le\ell\le r;\label{eq:odd=even4}
\end{equation}
in fact, the special case of (\ref{eq:odd=even4}) which is of most
interest for the applications is
\begin{equation}
\tilde H'_{2\ell}=\tilde H'_{2\ell+1}\quad\text{ for }1\le\ell\le
r.\label{eq:odd=even}
\end{equation}
We prove some necessary and some sufficient conditions under which an
extension $\nu^\dag_-$ whose corresponding ideals $\tilde H'_i$ satisfy
(\ref{eq:odd=even}) is uniquely determined by the ideals $\tilde
H'_i$. We also give sufficient conditions for the graded algebra
$gr_\nu R'$ to be scalewise birational to $gr_{{\Phi}at\nu_-}{\Phi}at R'$ for
each $R'\in\cal{T}$. These sufficient conditions are used in
\S\ref{locuni1} to prove some partial results towards Conjecture \ref{teissier}.
In \S\ref{henselization} we show the existence of $\nu$-extensions in
$\cal T$, stable for henselization, thus reducing the proof of the
primality of $H_{2\ell+1}$ to the results of \S\ref{technical}. We
study the extension of $\nu$ to $\tilde R$ modulo its first prime
ideal and prove that such an extension is unique.
In \S\ref{prime} we use the results of \S\ref{henselization} to prove
the existence of $\nu$-extensions in $\cal T$, stable for
completion. Combined with the results of \S\ref{technical} this proves
that the ideals $H_{2\ell+1}$ are prime.
In \S\ref{locuni1} we describe a possible
approach and prove some partial results towards constructing a
chain of trees (\ref{eq:chaintree}) of prime ideals of ${\Phi}at R'$
satisfying (\ref{eq:odd=even}) and a corresponding valuation ${\Phi}at\nu_-$
which satisfies the conclusion of Conjecture \ref{teissier}. We also
prove a necessary and a sufficient condition for the uniqueness of ${\Phi}at\nu_-$, assuming Conjecture \ref{teissier}.
We would like to acknowledge the paper \cite{HeSa} by Bill Heinzer and Judith
Sally which inspired one of the authors to continue thinking about this subject, as well as the work of S.D. Cutkosky, S. El Hitti and L. Ghezzi: \cite{CG} (which contains results closely related to those of \S\ref{archimedian}) and \cite{CE}.
\section{Extending a valuation of rank one centered in a local domain
to its formal completion.}
\label{archimedian}
Let $(R,M,k)$ be a local noetherian domain, $K$ its field of
fractions, and $\nu:K\rightarrow{\Gamma}_+\cup\{\infty\}$ a rank one valuation,
centered at $R$ (that is, non-negative on $R$ and positive
on $M$).\par Let ${\Phi}at R$ denote the formal completion of $R$. It is
convenient to extend $\nu$ to a valuation centered at $\frac{{\Phi}at R}H$, where
$H$ is a prime ideal of ${\Phi}at R$ such that $H\cap R=(0)$. In this
section, we will assume that $\nu$ is of rank one, so that the value
group ${\Gamma}$ is archimedian. We will explicitly describe a prime ideal
$H$ of ${\Phi}at R$, canonically associated to $\nu$, such that $H\cap
R=(0)$ and such that $\nu$ has a unique extension ${\Phi}at\nu_-$ to $\frac{{\Phi}at R}H$.
Let ${\Phi}=\nu(R\setminus (0))$, let
$\mathcal{P}_\beta$ denote the $\nu$-ideal of $R$ of value $\beta$ and $\mathcal{P}_{\beta}^+$ the
greatest $\nu$-ideal, properly contained in $\mathcal{P}_\beta$. We now
define the main object of study of this section. Let
\begin{equation}
H:=\bigcap\limits_{\beta\in{\Phi}}(\mathcal{P}_\beta{\Phi}at R).\label{tag51}
\end{equation}
\begin{remark}\label{Remark51} Since $R$ is noetherian, we have $\nu
(M)>0$ and since the ordered group $\mbox{l{\Phi}space{-.47em}G}amma$ is archimedian, for
every $\beta\in{\Phi}$ there exists $n\in\mathbf N$ such that
$M^n\subset \mathcal{P}_\beta$. In other words, the $M$-adic topology on $R$
is finer than (or equal to) the $\nu$-adic topology. Therefore an
element $x\in{\Phi}at R$ lies in $\mathcal{P}_\beta{\Phi}at R\iff$ there exists a
Cauchy sequence $\{x_n\}\subset R$ in the $M$-adic topology,
converging to $x$, such that $\nu(x_n){\Gamma}e\beta$ for all $n\iff$ for
{\it every} Cauchy sequence $\{x_n\}\subset R$, converging to $x$,
$\nu(x_n){\Gamma}e\beta$ for all $n{\Gamma}g0$. By the same token, $x\in H\iff$
there exists a Cauchy sequence $\{x_n\}\subset R$, converging to
$x$, such that $\lim\limits_{n\to\infty}\nu(x_n)=\infty\iff$ for
{\it every} Cauchy sequence $\{x_n\}\subset R$, converging to $x$,
$\lim\limits_{n\to\infty}\nu(x_n)=\infty$.
\end{remark}
\begin{example} Let $R=k[u,v]_{(u,v)}$. Then ${\Phi}at
R=k[[u,v]]$. Consider an element $w=u-\sum\limits_{i=1}^\infty
c_iv^i\in{\Phi}at R$, where $c_i\in k^*$ for all $i\in\mathbf N$, such
that $w$ is transcendental over $k(u,v)$. Consider the injective map
$\iota:k[u,v]_{(u,v)}\rightarrow k[[t]]$ which sends $v$ to $t$ and
$u$ to $\sum\limits_{i=1}^\infty c_it^i$. Let $\nu$ be the valuation
induced from the $t$-adic valuation of $k[[t]]$ via $\iota$. The value
group of $\nu$ is $\mathbf Z$ and ${\Phi}=\mathbf N_0$. For each
$\beta\in\mathbf N$,
$\mathcal{P}_\beta=\left(v^\beta,u-\sum\limits_{i=1}^{\beta-1}c_iv^i\right)$. Thus
$H=(w)$.
We come back to the general theory. Since the formal completion
homomorphism $R\rightarrow{\Phi}at R$ is faithfully flat,
\begin{equation}
\mathcal{P}_\beta{\Phi}at R\cap R=\mathcal{P}_\beta\quad\text{for all }\beta\in{\Phi}.\label{tag52}
\end{equation}
Taking the intersection over all $\beta\in{\Phi}$, we obtain
\begin{equation}
H\cap R=\left(\bigcap\limits_{\beta\in{\Phi}}\left(\mathcal{P}_\beta{\Phi}at R\right)\right)\cap
R=\bigcap\limits_{\beta\in{\Phi}}\mathcal{P}_\beta=(0),\label{tag53}
\end{equation}
In other words, we have a natural inclusion $R{\Phi}ookrightarrow\frac{{\Phi}at R}H$.
\end{example}
\begin{theorem}\label{th53}
\begin{enumerate}
\item $H$ is a prime ideal of ${\Phi}at R$.
\item $\nu$ extends uniquely to a valuation ${\Phi}at\nu_-$, centered at
$\frac{{\Phi}at R}H$.
\end{enumerate}
\end{theorem}
\begin{proof} Let $\bar x\in\frac{{\Phi}at R}H\setminus\{0\}$. Pick a
representative $x$ of $\bar x$ in ${\Phi}at R$, so that $\bar x= x\ {\operatorname{mod}}\
H$. Since $x\mbox{$\in$ {\Phi}space{-.8em}/} H$, we have $ x\mbox{$\in$ {\Phi}space{-.8em}/} \mathcal{P}_\alpha{\Phi}at R$ for some
$\alpha\in{\Phi}$.
\begin{lemma}\label{lemma36} {\rm (See \cite{ZS}, Appendix 5, lemma 3)} Let $\nu$ be a valuation of rank one centered in a
local noetherian domain $(R,M,k)$. Let
$$
{\Phi}=\nu(R\setminus (0))\subset{\Gamma}.
$$
Then ${\Phi}$ contains no infinite bounded sequences.
\end{lemma}
\begin{proof} An infinite ascending sequence $\alpha_1<\alpha_2<\dots$ in
${\Phi}$, bounded above by an element $\beta\in{\Phi}$, would give rise to an
infinite descending chain of ideals in $\frac R{\mathcal{P}_\beta}$. Thus it is
sufficient to prove that $\frac R{\mathcal{P}_\beta}$ has finite length.
Let $\delta:=\nu(M)\equiv\min({\Phi}\setminus\{0\})$. Since ${\Phi}$ is
archimedian, there exists $n\in\mathbf N$ such that $\beta\le n\delta$.
Then $M^n\subset \mathcal{P}_\beta$, so that there is a surjective map $\frac
R{M^n}\twoheadrightarrow\frac R{\mathcal{P}_\beta}$. Thus $\frac R{\mathcal{P}_\beta}$ has
finite length, as desired.
\end{proof}
By Lemma \ref{lemma36}, the set $\{\beta\in{\Phi}\ |\ \beta<\alpha\}$
is finite. Hence there exists a unique $\beta\in{\Phi}$ such that
\begin{equation}
x\in \mathcal{P}_\beta{\Phi}at R\setminus \mathcal{P}_{\beta}^+{\Phi}at R.\label{tag54}
\end{equation}
Note that $\beta$ depends only on $\bar x$, but not on the choice of the
representative $ x$. Define the function ${\Phi}at\nu_-:\frac{{\Phi}at
R}H\setminus\{0\}\rightarrow{\Phi}$ by
\begin{equation}
{\Phi}at\nu_-(\bar x)=\beta.\label{tag55}
\end{equation}
By (\ref{tag52}), if $x\in R\setminus\{0\}$ then
\begin{equation}
{\Phi}at\nu_-(x)=\nu(x).
\end{equation}
It is obvious that
\begin{equation}
{\Phi}at\nu_-(x+y){\Gamma}e\min\{{\Phi}at\nu_-(x),{\Phi}at\nu_-(y)\}\label{tag57}
\end{equation}
\begin{equation}
{\Phi}at\nu_-(xy){\Gamma}e{\Phi}at\nu_-(x)+{\Phi}at\nu_-(y)\label{tag58}
\end{equation}
for all $x,y\in\frac{{\Phi}at R}H$. The point of the next lemma is to
show that $\frac{{\Phi}at R}H$ is a domain and that ${\Phi}at\nu_-$ is, in fact, a
valuation (i.e. that the inequality (\ref{tag58}) is, in fact, an equality).
\begin{lemma}\label{lemma54} For any non-zero $\bar x,\bar
y\in\frac{{\Phi}at R}H$, we have $\bar x\bar y\ne0$ and ${\Phi}at\nu_-(\bar
x\bar y)={\Phi}at\nu_-(\bar x)+{\Phi}at\nu_-(\bar y)$.
\end{lemma}
\begin{proof} Let $\alpha={\Phi}at\nu_-(\bar x)$, $\beta={\Phi}at\nu_-(\bar
y)$. Let $ x$ and $ y$ be representatives in ${\Phi}at R$ of $\bar x$ and
$\bar y$, respectively. We have $M\mathcal{P}_\alpha\subset \mathcal{P}_{\alpha}^+$, so that
\begin{equation}
\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+}\cong\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^++M\mathcal{P}_\alpha}\cong
\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+}\otimes_Rk\cong\frac{\mathcal{P}_\alpha}{\mathcal{P}_{\alpha}^+}
\otimes_R\frac{{\Phi}at R}{M{\Phi}at R}\cong\frac{\mathcal{P}_\alpha{\Phi}at
R}{(\mathcal{P}_{\alpha}^++M\mathcal{P}_\alpha){\Phi}at R}\cong\frac{\mathcal{P}_\alpha{\Phi}at
R}{\mathcal{P}_{\alpha}^+{\Phi}at R},\label{tag59}
\end{equation}
and similarly for $\beta$. By (\ref{tag59}) there exist $z\in \mathcal{P}_\alpha$, $w\in
\mathcal{P}_\beta$, such that $z\equiv x\ {\operatorname{mod}}\ \mathcal{P}_{\alpha}^+{\Phi}at R$ and
$w\equiv y\ {\operatorname{mod}}\ \mathcal{P}_{\beta}^+{\Phi}at R$. Then
\begin{equation}
xy\equiv zw\ {\operatorname{mod}}\ \mathcal{P}_{\alpha+\beta}^+{\Phi}at R.\label{tag510}
\end{equation}
Since $\nu$ is a valuation, $\nu(zw)=\alpha+\beta$, so that $zw\in
\mathcal{P}_{\alpha+\beta}\setminus \mathcal{P}_{\alpha+\beta}^+$. By (\ref{tag52}) and
(\ref{tag510}), this proves that $xy\in \mathcal{P}_{\alpha+\beta}{\Phi}at R\setminus
\mathcal{P}_{\alpha+\beta}^+{\Phi}at R$. Thus $xy\mbox{$\in$ {\Phi}space{-.8em}/} H$ (hence $\bar x\bar
y\ne0$ in $\frac{{\Phi}at R}H$) and ${\Phi}at\nu_-(\bar x\bar y)=\alpha+\beta$,
as desired.
\end{proof}
By Lemma \ref{lemma54}, $H$ is a prime ideal of ${\Phi}at R$. By
(\ref{tag57}) and Lemma \ref{lemma54}, ${\Phi}at\nu_-$ is a valuation,
centered at $\frac{{\Phi}at R}H$. To complete the proof of Theorem
\ref{th53}, it remains to prove the uniqueness of ${\Phi}at\nu_-$. Let $x$,
$\bar x$, the element $\alpha\in{\Phi}$ and
\begin{equation}
z\in \mathcal{P}_\alpha\setminus \mathcal{P}_{\alpha}^+\label{tag511}
\end{equation}
be as in the proof of Lemma \ref{lemma54}. Then there exist
\begin{equation}
\begin{array}{rl}
u_1,\ldots , u_n&\in \mathcal{P}_{\alpha}^+\text{ and}\\
v_1,\ldots ,v_n&\in{\Phi}at R
\end{array}\label{tag512}
\end{equation}
such that $ x=z+\sum\limits_{i=1}^nu_i v_i$. Letting $\bar v_i:=v_i\
{\operatorname{mod}}\ H$, we obtain $\bar x=\bar z+\sum\limits_{i=1}^n\bar u_i\bar
v_i$ in $\frac{{\Phi}at R}H$. Therefore, by
(\ref{tag511})--(\ref{tag512}), for any extension of $\nu$ to a
valuation ${\Phi}at\nu '_-$, centered at $\frac{{\Phi}at R}H$, we have
\begin{equation}
{\Phi}at\nu '_-(\bar x)=\alpha={\Phi}at\nu_-(\bar x),\label{tag513}
\end{equation}
as desired. This completes the proof of Theorem \ref{th53}.
\end{proof}
\begin{definition}\label{deft55} The ideal $H$ is called the {\bf implicit prime
ideal} of ${\Phi}at R$, associated to $\nu$. When dealing with more than one
ring at a time, we will sometimes write $H(R,\nu)$ for $H$.
\end{definition}
More generally, let $\nu$ be a valuation centered at $R$, not
necessarily of rank one. In any case, we may write $\nu$ as a
composition $\nu=\mu_2\circ\mu_1$, where $\mu_2$ is centered at a
non-maximal prime ideal $P$ of $R$ and $\mu_1\left|_{\frac RP}\right.$
is of rank one. The valuation $\mu_1\left|_{\frac RP}\right.$ is
centered at $\frac RP$. We define the {\bf implicit prime ideal} of
$R$ with respect to $\nu$, denoted $H(R,\nu)$, to be the inverse image
in ${\Phi}at R$ of the implicit prime ideal of $\frac{{\Phi}at R}P$ with
respect to $\mu_1\left|_{\frac RP}\right.$. For the rest of this
section, we will continue to assume that $\nu$ is of rank one.
\begin{remark}\label{Remark56} By (\ref{tag59}), we have the following natural
isomorphisms of graded algebras:
$$
\begin{array}{rl}
\mbox{gr}_\nu R&\cong\mbox{gr}_{{\Phi}at\nu_-}\frac{{\Phi}at R}H\\
G_\nu&\cong G_{{\Phi}at\nu_-}.
\end{array}
$$
\end{remark}
\par
We will now study the behaviour of $H$ under local blowings up of $R$
with respect to $\nu$ and, more generally, under local homomorphisms. Let
$\pi:(R,M)\rightarrow(R',M')$ be a local homomorphism of local noetherian
domains. Assume that $\nu$ extends to a rank one valuation
$\nu':R'\setminus\{0\}\rightarrow{\Gamma}'$, where ${\Gamma}'\supset{\Gamma}$. The
homomorphism $\pi$ induces a local homomorphism ${\Phi}at\pi:{\Phi}at R\rightarrow{\Phi}at
R'$ of formal completions. Let ${\Phi}'=\nu'(R'\setminus\{0\})$. For
$\beta\in{\Phi}'$, let $\mathcal{P}'_\beta$ denote the $\nu'$-ideal of $R_{\nu'}$
of value $\beta$, as above. Let $H'=H(R',\nu')$.
\begin{lemma}\label{lemma58} Let $\beta\in{\Phi}$. Then
\begin{equation}
\left(\mathcal{P}'_\beta{\Phi}at R'\right)\cap{\Phi}at R=\mathcal{P}_\beta{\Phi}at R.\label{tag516}
\end{equation}
\end{lemma}
\begin{proof} Since by assumption $\nu'$ extends $\nu$ we have
$\mathcal{P}'_\beta\cap R=\mathcal{P}_\beta$ and the inclusion
\begin{equation}
\left(\mathcal{P}'_\beta{\Phi}at R'\right)\cap{\Phi}at R\supseteq \mathcal{P}_\beta{\Phi}at R.\label{tag517}
\end{equation}
We will now prove the opposite inclusion. Take an element
$x\in\left(\mathcal{P}'_\beta{\Phi}at R'\right)\cap{\Phi}at R$. Let $\{x_n\}\subset R$
be a Cauchy sequence in the $M$-adic topology, converging to $x$. Then
$\{\pi(x_n)\}$ converge to ${\Phi}at\pi(x)$ in the $M'$-adic topology of
${\Phi}at R'$. Applying remark \ref{Remark51} to $R'$, we obtain
\begin{equation}
\nu(x_n)\equiv\nu'(\pi(x_n)){\Gamma}e\beta\quad\text{for }n{\Gamma}g0.\label{tag518}
\end{equation}
By (\ref{tag518}) and Remark \ref{Remark56}, applied to $R$, we have
$x\in \mathcal{P}_\beta{\Phi}at R$. This proves the opposite inclusion in
(\ref{tag517}), as desired.
\end{proof}
\begin{corollary}\label{Corollary59} We have
$$
H'\cap{\Phi}at R=H.
$$
\end{corollary}
\begin{proof} Since $\nu'$ is of rank one, ${\Phi}$ is cofinal in ${\Phi}'$. Now
the Corollary follows by taking the intersection over all $\beta\in{\Phi}$
in (\ref{tag516}).
\end{proof}
Let $J$ be a non-zero ideal of $R$ and let $R\rightarrow R'$ be the
local blowing up along $J$ with respect to $\nu$. Take an element
$f\in J$, such that $\nu(f)=\nu(J)$. By the {\bf strict transform} of
$J$ in ${\Phi}at R'$ we will mean the ideal
$$
J^{\text{str}}:=\bigcup\limits_{i=1}^\infty\left(\left(J{\Phi}at
R'\right):f^i\right)\equiv\left(J{\Phi}at R'_f\right)\cap{\Phi}at R'.
$$
If $g$ is another element of $J$ such that $\nu(g)=\nu(J)$ then
$\nu\left(\frac fg\right)=0$, so that $\frac fg$ is a unit in
$R'$. Thus the definition of strict transform is independent of the
choice of $f$.
\begin{corollary}\label{Corollary510} $H^{\text{str}}\subset H'$.
\end{corollary}
\begin{proof} Since $H{\Phi}at R'\subset H'$, we have
$H^{\text{str}}=\left(H{\Phi}at R'_f\right)\cap{\Phi}at R'\subset\left(H'{\Phi}at
R'_f\right)\cap{\Phi}at R'=H'$, where the last equality holds because
$H'$ is a prime ideal of ${\Phi}at R'$, not containing $f$.
\end{proof}
Using Zariski's Main Theorem, it can be proved that $H^{\text{str}}$
is prime. Since this fact is not used in the sequel, we omit the proof.
\begin{corollary}\label{Corollary511} Let the notation and assumptions be as in
corollary \ref{Corollary510}. Then
\begin{equation}
{\Phi}e\ H'{\Gamma}e{\Phi}e\ H.\label{tag519}
\end{equation}
In particular,
\begin{equation}
\dim\frac{{\Phi}at R'}{H'}\le\dim\frac{{\Phi}at R}H.\label{tag520}
\end{equation}
\end{corollary}
\begin{proof} Let $\bar R:=\left({\Phi}at
R\otimes_RR'\right)_{M'{\Phi}at R'\cap({\Phi}at R\otimes_RR')}$. Let $\phi$
denote the natural local homomorphism
$$
\bar R\rightarrow{\Phi}at R'.
$$
Let $\bar H:=H'\cap\bar R$. Now, take $f\in J$ such that
$\nu(f)=\nu(J)$. Then $f\mbox{$\in$ {\Phi}space{-.8em}/} H'$ and, in particular, $f\mbox{$\in$ {\Phi}space{-.8em}/}\bar
H$. Since $R'_f\cong R_f$, we have ${\Phi}at R_f=\bar R_f$. In view of
Corollary \ref{Corollary59}, we obtain $H{\Phi}at R_f\cong\bar H\bar R_f$,
so
\begin{equation}
{\Phi}e \ H={\Phi}e\ \bar H.\label{tag521}
\end{equation}
Now, $\bar R$ is a local noetherian ring, whose formal completion is
${\Phi}at R'$. Hence $\phi$ is faithfully flat and therefore satisfies the going
down theorem. Thus we have ${\Phi}e\ H'{\Gamma}e{\Phi}e\ \bar H$. Combined with
(\ref{tag521}), this proves (\ref{tag519}). As for the last statement
of the Corollary, it follows from the well known fact that dimension
does not increase under blowing up (\cite{Spi1}, Lemma 2.2): we have
$\dim\ R'\le\dim\ R$, hence
$$
\dim\ {\Phi}at R'=\dim R'\le\dim\ R=\dim\ {\Phi}at R,
$$
and (\ref{tag520}) follows from (\ref{tag519}) and from the fact that
complete local rings are catenarian.
\end{proof}
It may well happen that the containment of corollary \ref{Corollary510} and the inequality
in (\ref{tag519}) are strict. The possibility of strict containement in corollary \ref{Corollary510}
is related to the existence of subanalytic functions, which are not
analytic. We illustrate this statement by an example in which
$H^{\text{str}}\subsetneqq H'$ and ${\Phi}e\ H<{\Phi}e\ H'$.
\begin{example} Let $k$ be a field and let
$$
\begin{array}{rl}
R&=k[x,y,z]_{(x,y,z)},\\
R'&=k[x',y',z']_{(x',y',z')},
\end{array}
$$
where $x'=x$, $y'=\frac yx$ and $z'=z$. We have $K=k(x,y,z)$, ${\Phi}at
R=k[[x,y,z]]$, ${\Phi}at R'=k[[x',y',z']]$. Let $t_1,t_2$ be auxiliary variables
and let $\sum\limits_{i=1}^\infty c_it_1^i$ (with $c_i\in k$) be an element of
$k[[t_1]]$, transcendental over $k(t_1)$. Let $\theta$ denote the valuation,
centered at $k[[t_1,t_2]]$, defined by $\theta(t_1)=1$, $\theta(t_2)=\sqrt2$
(the value group of $\theta$ is the additive subgroup of $\mathbf R$, generated
by 1 and $\sqrt2$). Let $\iota:R'{\Phi}ookrightarrow k[[t_1,t_2]]$ denote the
injective map defined by $\iota(x')=t_2$, $\iota(y')=t_1$,
$\iota(z')=\sum\limits_{i=1}^\infty c_it_1^i$. Let $\nu$ denote the
restriction of $\theta$ to $K$, where we view $K$ as a subfield of
$k((t_1,t_2))$ via $\iota$. Let ${\Phi}=\nu(R\setminus\{0\})$;
${\Phi}'=\nu(R'\setminus\{0\})$. For $\beta\in{\Phi}'$, $P'_\beta$ is generated by all
the monomials of the form ${x'}^\alpha{y'}^{\Gamma}amma$ such that
$\sqrt2\alpha+{\Gamma}amma{\Gamma}e\beta$, together with
$z'-\sum\limits_{j=1}^ic_j{y'}^j$, where $i$ is the greatest non-negative
integer such that $i<\beta$.
Let $w':=z'-\sum\limits_{i=1}^\infty c_i{y'}^i$. Then $H'=(w')$, but
$H=H'\cap{\Phi}at R=(0)$, so that $H^{\text{str}}=(0)\subsetneqq H'$ and
${\Phi}e\ H=0<1={\Phi}e\ H'$.
\end{example}
Recall the following basic result of the theory of G-rings:
\begin{proposition}\label{Corollary57} Assume that $R$ is a reduced G-ring. Then ${\Phi}at
R_H$ is a regular local ring.
\end{proposition}
\begin{proof} Let $K=R_{\mathcal{P}_\infty}=\kappa(\mathcal{P}_\infty)$ (here we are using that $R$ is reduced and that $\mathcal{P}_\infty$ is a minimal
prime of $R$). By definition of G-ring, the map
$R\rightarrow{\Phi}at R$ is a regular homomorphism. Then by (\ref{tag53}) ${\Phi}at
R_H$ is geometrically regular over $K$, hence regular.
\end{proof}
\begin{remark} Having extended in a unique manner the valuation $\nu$ to
a valuation ${\Phi}at\nu_-$ of $\frac{{\Phi}at R}{H}$, we see that if $R$ is
a G-ring, by Proposition \ref{Corollary57} there is a unique minimal
prime ${\Phi}at\mathcal{P}_\infty$ of ${\Phi}at R$ contained in $H$, corresponding to
the ideal $(0)$ in ${\Phi}at R_H$. Since $H\cap R=(0)$, we have the
equality ${\Phi}at\mathcal{P}_\infty\cap R=(0)$. Choosing a valuation $\mu$ of
the fraction field of $\frac{{\Phi}at R_H}{{\Phi}at\mathcal{P}_\infty{\Phi}at R_H}$
centered at $\frac{{\Phi}at R_H}{{\Phi}at\mathcal{P}_\infty{\Phi}at R_H}$ whose value
group $\mathcal{P}si$ is a free abelian group produces a composed valuation
${\Phi}at\nu_-\circ\mu$ on $\frac{{\Phi}at R}{{\Phi}at\mathcal{P}_\infty}$ with value group
$\mathcal{P}si\bigoplus\mbox{l{\Phi}space{-.47em}G}amma$ ordered lexicographically, as
follows:\par\noindent Given $x\in \frac{{\Phi}at R}{{\Phi}at\mathcal{P}_\infty}$, let
$\psi=\mu(x)$ and blow up in $R$ the ideal $\mathcal{P}_\psi$ along our
original valuation, obtaining a local ring $R'$. According to what
we have seen so far in this section, in its completion ${\Phi}at R'$ we
can write $x=ye$ with $\mu(e)=\psi$ and $y\in {\Phi}at R'\setminus
H'$. The valuation $\nu$ on $R'$ extends uniquely to a valuation of
$\frac{{\Phi}at R'}{H'}$, which we may still denote by ${\Phi}at\nu_-$
because it induces ${\Phi}at\nu_-$ on $\frac{{\Phi}at R}{H}$. Let us
consider the image $\overline y$ of $y$ in $\frac{{\Phi}at
R'}{H'}$. Setting $({\Phi}at\nu_-\circ\mu)(x)=\psi\bigoplus
{\Phi}at\nu_-(\overline y)\in \mathcal{P}si\bigoplus\mbox{l{\Phi}space{-.47em}G}amma$ determines a
valuation of $\frac{{\Phi}at R}{{\Phi}at\mathcal{P}_\infty}$ as
required.\par
If we drop the assumption that $\mathcal{P}si$ is a free abelian group, the
above construction still works, but the value group ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma$ of
${\Phi}at\nu_-\circ\mu$ need not be isomorphic to the direct sum
$\mathcal{P}si\bigoplus\mbox{l{\Phi}space{-.47em}G}amma$. Rather, we have an exact sequence
$0\rightarrow\mbox{l{\Phi}space{-.47em}G}amma\rightarrow\bar\mbox{l{\Phi}space{-.47em}G}amma\rightarrow\mathcal{P}si\rightarrow0$,
which need not, in general, be split; see {\rm\cite{V1}, Proposition 4.3}.
\noindent In the sequel we shall reduce to the case where ${\Phi}at R$ is an
integral domain, so that ${\Phi}at\mathcal{P}_\infty=(0)$ and we will have
constructed a valuation of ${\Phi}at R$.
\end{remark}
\section{Definition and first properties of implicit ideals.}
\label{basics}
Let the notation be as above. Before plunging into technical details,
we would like to give a brief and informal overview of our
constructions and the motivation for them. Above we recalled the well
known fact that if $rk\ \nu=r$ then for every $\nu$-extension
$R\rightarrow R'$ the valuation $\nu$ canonically determines a flag
(\ref{eq:treechain'}) of $r$ subschemes of $\mbox{Spec}\ R'$. This paper
shows the existence of subschemes of $\mbox{Spec}\ {\Phi}at R$, determined by
$\nu$, which are equally canonical and which become explicit only
after completion. To see what they are, first of all note that the
ideal $P'_l{\Phi}at R'$, for $R'\in\cal{T}$ and $0\le\ell\le r-1$, need
not in general be prime (although it is prime whenever $R'$ is
henselian). Another way of saying the same thing is that the ring
$\frac{R'}{P'_\ell}$ need not be analytically irreducible in
general. However, we will see in \S\ref{prime}
(resp. \S\ref{henselization}) that the valuation $\nu$ picks out in a
canonical way one of the minimal primes of $P'_l{\Phi}at R'$
(resp. $P'_l\tilde R'$). We call this minimal prime $H'_{2\ell}$ for
reasons which will become apparent later. By the flatness of
completion (resp. henselization), we have $H'_{2\ell}\cap
R'=P'_\ell$. We will show that the ideals $H'_{2\ell}$ form a tree.
Let
\begin{equation}
(0)=\Delta_r\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_0=
\mbox{l{\Phi}space{-.47em}G}amma\label{eq:isolated1}
\end{equation}
be the isolated subgroups of $\mbox{l{\Phi}space{-.47em}G}amma$. There are other ideals of ${\Phi}at
R$, apart from the $H_{2\ell}$, canonically associated to $\nu$, whose
intersection with $R$ equals $P_\ell$, for example, the ideal
$\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta{\Phi}at R$. The same is true
of the even larger ideal
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}
\left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal
P}'_\beta{\Phi}at R'\right)\bigcap{\Phi}at R\right),\label{eq:defin2}
\end{equation}
(that $H_{2\ell+1}\cap R=P_\ell$ is easy to see and will be shown
later in this section, in Proposition \ref{contracts}). While the
examples below show that the ideal
$\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta{\Phi}at R$ need not, in
general, be prime, the ideal $H_{2\ell+1}$ always is (this is the main
theorem of this paper; it will be proved in \S\ref{prime}). The ideal
$H_{2\ell+1}$ contains $H_{2\ell}$ but is not, in general equal to
it. To summarize, we will show that the valuation $\nu$ picks out in a
canonical way a generic point $H_{2\ell}$ of the formal fiber over
$P_\ell$ and also another point $H_{2\ell+1}$ in the formal fiber,
which is a specialization of $H_{2\ell}$.
The main technique used to prove these results is to to analyze the
set of zero divisors of $\frac{{R'}^\dag}{P'_\ell{R'}^\dag}$ (where
$R^\dag$ stands for either ${\Phi}at R$, $\tilde R$, or a finite type
\'etale extension $R^e$ of $R$), as follows. We show that the
reducibility of $\frac{{R'}^\dag}{P'_\ell{R'}^\dag}$ is related to the
existence of non-trivial algebraic extensions of $\kappa(P_\ell)$
inside $\kappa(P_\ell)\otimes_RR^\dag$. More precisely, in the next
section we define $R$ to be \textbf{stable} if $\frac{R^\dag}{P_{\ell
+1}R^\dag}$ is a domain and there does not exist a non-trivial
algebraic extension of $\kappa(P_{\ell+1})$ which embeds both into
$\kappa(P_{\ell +1})\otimes_RR^\dag$ and into $\kappa(P'_{\ell+1})$
for some $R'\in\mathcal{T}$. We show that if $R$ is stable then
$\frac{{R'}^\dag}{P'_{\ell+1}{R'}^\dag}$ is a domain for all
$R'\in\cal T$. For $\overline\beta\in\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_{\ell+1}}$,
let
\begin{equation}
\mathcal{P}_{\overline\beta}=\left\{x\in R\ \left|\
\nu(x)\mod\Delta_{\ell+1}{\Gamma}e\overline\beta\right.\right\}\label{eq:pbetamodl}
\end{equation}
If $\mathcal{P}hi$ denotes the semigroup $\nu(R\setminus\{0\})\subset\mbox{l{\Phi}space{-.47em}G}amma$,
which is well ordered since $R$ is noetherian (see [ZS], Appendix 4,
Proposition 2), and
$$
\beta(\ell)=\min\{{\Gamma}amma\in\mathcal{P}hi\ |\ \beta-{\Gamma}amma\in\Delta_{\ell+1}\}
$$
then $\mathcal{P}_{\overline\beta}=\mathcal{P}_{\beta(l)}$.\par\noindent
We have the inclusions
$$
P_\ell\subset \mathcal{P}_{\overline\beta}\subset P_{\ell+1},
$$
and $\mathcal{P}_{\overline\beta}$ is the inverse image in $R$ by the canonical map
$R\to \frac{R}{P_\ell}$ of a valuation ideal $\overline
\mathcal{P}_{\overline\beta}\subset \frac{R}{P_\ell}$ for the rank one
valuation $\frac{R}{P_\ell}\setminus\{0\}\to
\frac{\Delta_\ell}{\Delta_{\ell+1}}$ induced by $\nu_{\ell+1}$.\par
We will deduce from the above that if $R$ is stable then for each
$\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$ and each
$\nu$-extension $R\rightarrow R'$ we have
$\mathcal{P}'_{\overline\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\overline\beta} R^\dag$,
which gives us a very good control of the limit in the definition of
$H_{2\ell+1}$ and of the $\nu$-extensions $R'$ for which the limit is attained.
We then show, separately in the cases when $R^\dag=\tilde R$
(\S\ref{henselization}) and $R^\dag={\Phi}at R$ (\S\ref{prime}), that
there always exists a stable $\nu$-extension $R'\in\cal{T}$.
We are now ready to go into details, after giving several examples of
implicit ideals and the phenomena discussed above.
Let $0\le\ell\le r$. We define our main object of study, the
$(2\ell+1)$-st implicit prime ideal $H_{2\ell+1}\subset R^\dag$, by
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}
\left(\left(\lim\limits_{\overset\longrightarrow{R'}}{\cal P}'_\beta
{R'}^\dag\right)\bigcap R^\dag\right),\label{eq:defin1}
\end{equation}
where $R'$ ranges over $\mathcal{T}$. As usual, we think of
(\ref{eq:defin1}) as a tree equation: if we replace $R$ by any other
$R''\in{\cal T}$ in (\ref{eq:defin1}), it defines the corresponding
ideal $H''_{2\ell +1}\subset{\Phi}at R''^\dag$. Note that for $\ell=r$
(\ref{eq:defin1}) reduces to
$$
H_{2r+1}=mR^\dag.
$$
We start by giving several examples of the ideals $H'_i$ (and also of
$\tilde H'_i$, which will appear a little later in the paper).
\begin{example}{\label{Example31}} Let $R=k[x,y,z]_{(x,y,z)}$. Let $\nu$ be the
valuation with value group $\mbox{l{\Phi}space{-.47em}G}amma=\mathbf Z^2_{lex}$, defined as
follows. Take a transcendental power series $\sum\limits_{j=1}^\infty
c_ju^j$ in a variable $u$ over $k$. Consider the homomorphism
$R{\Phi}ookrightarrow k[[u,v]]$ which sends $x$ to $v$, $y$ to $u$ and $z$
to $\sum\limits_{j=1}^\infty c_ju^j$. Consider the valuation $\nu$,
centered at $k[[u,v]]$, defined by $\nu(v)=(0,1)$ and $\nu(u)=(1,0)$;
its restriction to $R$ will also be denoted by $\nu$, by abuse of
notation. Let $R_\nu$ denote the valuation ring of $\nu$ in $k(x,y,z)$
and let $\mathcal{T}$ be the tree consisting of all the local rings
$R'$ essentially of finite type over $R$, birationally dominated by
$R_\nu$. Let ${}^\dag={\Phi}at{\ }$ denote the operation of formal
completion. Given $\beta=(a,b)\in\mathbf Z^2_{lex}$, we have
${\mathcal{P}}_\beta=x^b\left(y^a,z-c_1y-\dots-c_{a-1}y^{a-1}\right)$. The
first isolated subgroup $\Delta_1=(0)\oplus\mathbf Z$. Then
$\bigcap\limits_{\beta\in(0)\oplus\mathbf Z}\left({\cal P}_\beta{\Phi}at
R\right)=(y,z)$ and
$\bigcap\limits_{\beta\in\mbox{l{\Phi}space{-.47em}G}amma=\Delta_0}\left({\cal P}_\beta{\Phi}at
R\right)=\left(z-\sum\limits_{j=1}^\infty c_jy^j\right)$. It is not
hard to show that for any $R'\in\mathcal{T}$ we have
$H'_1=\left(z-\sum\limits_{j=1}^\infty c_jy^j\right){\Phi}at R'$ and
that $H_3=(y,z){\Phi}at R$. It will follow from the general theory
developed in \S\ref{extensions} that $\nu$ admits a unique extension
${\Phi}at\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}{\Phi}at R'$.
This extension has value group ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma=\mathbf Z^3_{lex}$ and is
defined by ${\Phi}at\nu(x)=(0,0,1)$, ${\Phi}at\nu(y)=(0,1,0)$ and
${\Phi}at\nu\left(z-\sum\limits_{j=1}^\infty c_jy^j\right)=(1,0,0)$. For
each $R'\in\mathcal{T}$ the ideal $H'_1$ is the prime valuation
ideal corresponding to the isolated subgroup $(0)\oplus\mathbf
Z^2_{lex}$ of ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma$ (that is, the ideal whose elements have
values outside of $(0)\oplus\mathbf Z^2_{lex}$) while $H'_3$ is the
prime valuation ideal corresponding to the isolated subgroup
$(0)\oplus(0)\oplus\mathbf Z$.
\end{example}
\begin{example}{\label{Example32}} Let $R=k[x,y,z]_{(x,y,z)}$, $\mbox{l{\Phi}space{-.47em}G}amma=\mathbf
Z^2_{lex}$, the power series $\sum\limits_{j=1}^\infty c_ju^j$ and the
operation ${}^\dag={\Phi}at{\ }$ be as in the previous example. This time,
let $\nu$ be defined as follows. Consider the homomorphism
$R{\Phi}ookrightarrow k[[u,v]]$ which sends $x$ to $u$, $y$ to
$\sum\limits_{j=1}^\infty c_ju^j$ and $z$ to $v$. Consider the
valuation $\nu$, centered at $k[[u,v]]$, defined by $\nu(v)=(1,0)$ and
$\nu(u)=(0,1)$; its restriction to $R$ will be also denoted by
$\nu$. Let $R_\nu$ denote the valuation ring of $\nu$ in $k(x,y,z)$
and let $\mathcal{T}$ be the tree consisting of all the local rings
$R'$ essentially of finite type over $R$, birationally dominated by
$R_\nu$. Given $\beta=(a,b)\in\mathbf Z^2_{lex}$, we have
$\mathcal{P}_\beta=z^a\left(x^b,y-c_1x-\dots-c_{b-1}x^{b-1}\right)$. The first
isolated subgroup $\Delta_1=(0)\oplus\mathbf Z$. Then
$\bigcap\limits_{\beta\in(0)\oplus\mathbf Z}\left({\mathcal{P}}_\beta{\Phi}at
R\right)=\left(y-\sum\limits_{j=1}^\infty c_jx^j,z\right)$ and
$\bigcap\limits_{\beta\in\mbox{l{\Phi}space{-.47em}G}amma=\Delta_0}\left({\cal P}_\beta{\Phi}at
R\right)=(0)$. It is not hard to show that for any $R'\in\cal{T}$ we
have $H'_1=(0)$ and that $H_3=\left(y-\sum\limits_{j=1}^\infty
c_jx^j,z\right){\Phi}at R'$. In this case, the extension ${\Phi}at\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}{\Phi}at R'$ is not unique.
Indeed, one possible extension ${\Phi}at\nu^{(1)}$ has value group
${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma=\mathbf Z^3_{lex}$ and is defined by
${\Phi}at\nu^{(1)}(x)=(0,0,1)$,
${\Phi}at\nu^{(1)}\left(y-\sum\limits_{j=1}^\infty
c_jx^j\right)=(0,1,0)$ and ${\Phi}at\nu^{(1)}(z)=(1,0,0)$. In this case,
for any $R'\in\cal{T}$ the ideal $H'_3$ is the prime valuation ideal
corresponding to the isolated subgroup $(0)\oplus(0)\oplus\mathbf Z$
of ${\Phi}at\mbox{l{\Phi}space{-.47em}G}amma$.
Another extension ${\Phi}at\nu^{(2)}$ of $\nu$ is defined by
${\Phi}at\nu^{(2)}(x)=(0,0,1)$,
${\Phi}at\nu^{(2)}\left(y-\sum\limits_{j=1}^\infty
c_jx^j\right)=(1,0,0)$ and ${\Phi}at\nu^{(2)}(z)=(0,1,0)$. In this case,
the tree of ideals corresponding to the isolated subgroup
$(0)\oplus(0)\oplus\mathbf Z$ is $H'_3$ (exactly the same as for
${\Phi}at\nu^{(1)}$) while that corresponding to $(0)\oplus\mathbf
Z^2_{lex}$ is $\tilde H'_1=\left(y-\sum\limits_{j=1}^\infty
c_jx^j\right)$. The tree $\tilde H'_1$ of prime
${\Phi}at\nu^{(2)}$-ideals determines the extension ${\Phi}at\nu^{(2)}$
completely.
\end{example}
The following two examples illustrate the need for taking the limit
over the tree $\mathcal{T}$.
\begin{example}{\label{Example33}} Let us consider the local domain
$S=\frac{k[x,y]_{(x,y)}}{(y^2-x^2-x^3)}$. There are two distinct valuations
centered in $(x,y)$. Let $a_i\in k,\ i{\Gamma}eq 2$ be such that
$$
\left(y+x+\sum_{i{\Gamma}eq 2}a_ix^i\right)\left(y-x-\sum_{i{\Gamma}eq
2}a_ix^i\right)=y^2-x^2-x^3.
$$
We shall denote by $\nu_+$ the rank one discrete valuation defined by
$$
\nu_+(x)=\nu_+(y)=1,
$$
$$
\nu_+(y+x)=2,
$$
$$
\nu_+\left(y+x+\sum_{i= 2}^{b-1}a_ix^i\right)=b.
$$
Now let $R=\frac{k[x,y,z]_{(x,y,z)}}{(y^2-x^2-x^3)}$. Let $\mbox{l{\Phi}space{-.47em}G}amma =\mathbf Z^2$
with the lexicographical ordering. Let $\nu$ be the composite valuation
of the $(z)$-adic one with $\nu_+$, centered in $\frac R{(z)}$. The
point of this example is to show that
$$
H^*_{2\ell+1}=\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{{\Phi}at R}
$$
does not work as the definition of the $(2\ell+1)$-st implicit prime
ideal because the resulting ideal $H^*_{2\ell+1}$ is not
prime. Indeed, as $\mathcal{P}_{(a,0)}=(z^a)$, we have
$$
H_1^*=\bigcap_{(a,b)\in\mathbf{Z}^2}\mathcal{P}_{(a,b)}{{\Phi}at R}=(0).
$$
Let $f=y+x+\sum\limits_{i{\Gamma}eq 2}a_ix^i,g=y-x-\sum\limits_{i{\Gamma}eq
2}a_ix^i\in{\Phi}at R$. Clearly $f,g\mbox{$\in$ {\Phi}space{-.8em}/} H^*_1=(0)$, but $f\cdot
g=0$, so the ideal $H^*_1$ is not prime.
One might be tempted (as we were) to correct this problem by
localizing at $H^*_{2\ell+3}$. Indeed, if we take the new definition
of $H^*_{2\ell+1}$ to be, recursively in the descending order of
$\ell$,
\begin{equation}
H^*_{2\ell +1}=\left(\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{{\Phi}at R}_{H^*_{2\ell
+3}}\right)\cap{{\Phi}at R},\label{eq:localization}
\end{equation}
then in the present example the resulting ideals $H^*_3=(z,f)$ and
$H^*_1=(f)$ are prime. However, the next example shows that the
definition (\ref{eq:localization}) also does not, in general, give
rise to prime ideals.
\end{example}
\begin{example}{\label{Example34}} Let
$R=\frac{k[x,y,z]_{(x,y,z)}}{(z^2-y^2(1+x))}$. Let $\mbox{l{\Phi}space{-.47em}G}amma=\mathbf
Z^2$ with the lexicographical ordering. Let $t$ be an independent
variable and let $\nu$ be the valuation, centered in $R$, induced by
the $t$-adic valuation of $k\left[\left[t^\mbox{l{\Phi}space{-.47em}G}amma\right]\right]$ under
the injective homomorphism $\iota:R{\Phi}ookrightarrow
k\left[\left[t^\mbox{l{\Phi}space{-.47em}G}amma\right]\right]$, defined by $\iota(x)=t^{(0,1)}$,
$\iota(y)=t^{(1,0)}$ and $\iota(z)=t^{(1,0)}\sqrt{1+t^{(0,1)}}$. The
prime $\nu$-ideals of $R$ are $(0)\subsetneqq P_1\subsetneqq m$, with
$P_1=(y,z)$. We have $\bigcap\limits_{\beta\in\Delta_1}\mathcal{P}_\beta{\Phi}at
R=(y,z){\Phi}at R=P_1{\Phi}at R$ and
$\bigcap\limits_{\beta\in\mbox{l{\Phi}space{-.47em}G}amma}\mathcal{P}_\beta{\Phi}at
R_{(y,z)}=\bigcap\limits_{\beta\in\mbox{l{\Phi}space{-.47em}G}amma}\mathcal{P}_\beta{\Phi}at R=(0)$. Note
that the ideal $(0)$ is not prime in ${\Phi}at R$. Now, let
$R'=R\left[\frac zy\right]_{m'}$, where $m'=\left(x,y,\frac
zy-1\right)$ is the center of $\nu$ in $R\left[\frac zy\right]$. We
have $z-y\sqrt{1+x}\in{\Phi}at R\setminus\mathcal{P}_{(2,0)}{\Phi}at R$. On the other
hand, $z-y\sqrt{1+x}=y\left(\frac zy-\sqrt{1+x}\right)=0$ in ${\Phi}at
R'$; in particular,
$z-y\sqrt{1+x}\in\bigcap\limits_{\beta\in\mbox{l{\Phi}space{-.47em}G}amma}\mathcal{P}'_\beta{\Phi}at
R'$. Thus this example also shows that the ideals $\mathcal{P}_\beta{\Phi}at R$,
$\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta{\Phi}at R$ and
$\bigcap\limits_{\beta\in\Delta_\ell}\mathcal{P}_\beta{\Phi}at R_{H_{2\ell+3}}$ do
not behave well under blowing up.
\end{example}
Note that both Examples \ref{Example33} and \ref{Example34} occur not only for the completion ${\Phi}at
R$ but also for the henselization $\tilde R$.
We come back to the general theory of implicit ideals.
\begin{proposition}\label{contracts} We have $H_{2\ell+1}\cap R=P_\ell$.
\end{proposition}
\begin{proof} Recall that $P_\ell=\left\{x\in R\ \left|\ \
\nu(x)\mbox{$\in$ {\Phi}space{-.8em}/} \Delta_\ell\right.\right\}$. If $x\in P_\ell$ then,
since $\Delta_\ell$ is an isolated subgroup, we have $x\in {\cal
P}_\beta$ for all $\beta\in \Delta_\ell$. The same inclusion holds
for the same reason in all extensions $R'\subset R_\nu$ of $R$, and
this implies the inclusion $P_\ell\subseteq H_{2\ell+1}\cap R$. Now
let $x$ be in $H_{2\ell+1}\cap R$ and assume $x\mbox{$\in$ {\Phi}space{-.8em}/} P_\ell$. Then
there is a $\beta\in \Delta_\ell$ such that $x\mbox{$\in$ {\Phi}space{-.8em}/}{\cal
P}_\beta$. By faithful flatness of $R^\dag$ over $R$ we have ${\cal
P}_\beta R^\dag\cap R={\cal P}_\beta$. This implies that
$x\mbox{$\in$ {\Phi}space{-.8em}/}{\cal P}_\beta R^\dag$, and the same argument holds in all the
extensions $R'\in\mathcal{T}$, so $x$ cannot be in $H_{2\ell+1}\cap
R$. This contradiction shows the desired equality.
\end{proof}
\begin{proposition}\label{behavewell} The ideals $H'_{2\ell+1}$ behave well
under $\nu$-extensions $R\rightarrow R'$ in $\mathcal{T}$. In other
words, let $R\rightarrow R'$ be a $\nu$-extension in $\mathcal{T}$
and let $H'_{2\ell+1}$ denote the $(2\ell+1)$-st implicit prime
ideal of ${\Phi}at R'$. Then $H_{2\ell+1}=H'_{2\ell+1}\cap R^\dag$.
\end{proposition}
\begin{proof} Immediate from the definitions.
\end{proof}
To study the ideals $H_{2\ell+1}$, we need to understand more explicitly the
nature of the limit appearing in (\ref{eq:defin1}). To study the relationship
between the ideals $P_\beta R^{\dag}$ and $P'_\beta {R'}^\dag\bigcap
R^\dag$, it is useful to factor the natural map $R^\dag\rightarrow{R'}^\dag$
as
$R^\dag\rightarrow(R^\dag\otimes_RR')_{M'}\overset\phi\rightarrow{R'}^\dag$
as we did in the proof of Lemma \ref{factor}. In general, the ring
$R^\dag\otimes_RR'$ is not local (see the above examples), but it has
one distinguished maximal ideal $M'$, namely, the ideal generated by
$mR^\dag\otimes1$ and $1\otimes m'$, where $m'$ denotes the maximal
ideal of $R'$. The map $\phi$ factors through the local ring
$\left(R^\dag\otimes_RR'\right)_{M'}$ and the resulting map
$\left(R^\dag\otimes_RR'\right)_{M'}\rightarrow{R'}^\dag$ is either
the formal completion or the henselization; in either case, it is
faithfully flat. Thus
$P'_\beta{R'}^\dag\cap\left(R^\dag\otimes_RR'\right)_{M'}=
P'_\beta\left(R^\dag\otimes_RR'\right)_{M'}$. This shows that we may
replace ${R'}^\dag$ by $\left(R^\dag\otimes_RR'\right)_{M'}$ in
(\ref{eq:defin1}) without affecting the result, that is,
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}\left(
\left(\lim\limits_{\overset\longrightarrow{R'}}
{\cal P}'_\beta\left(R^\dag\otimes_RR'\right)_{M'}\right)\bigcap
R^\dag\right).\label{eq:defin3}
\end{equation}
From now on, we will use (\ref{eq:defin3}) as our working definition of the
implicit prime ideals. One advantage of the expression (\ref{eq:defin3}) is
that it makes sense in a situation more general than the completion and the
henselization. Namely, to study the case of the henselization $\tilde R$, we
will need to consider local \'etale extensions $R^e$ of $R$, which are contained
in $\tilde R$ (particularly, those which are essentially of finite type). The
definition (\ref{eq:defin3}) of the implicit prime ideals makes sense also in
that case.
\section{Stable rings and primality of their implicit ideals.}
\label{technical}
Let the notation be as in the preceding sections. As usual, $R^\dag$
will denote one of ${\Phi}at R$, $\tilde R$ or $R^e$ (a local \'etale
$\nu$-extension essentially of finite type). Take an
$R'\in\mathcal{T}$ and
$\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$. We have the
obvious inclusion of ideals
\begin{equation}
\mathcal{P}_{\overline\beta} R^\dag\subset\mathcal{P}_{\overline\beta}{R'}^\dag\cap
R^\dag\label{eq:stab1}
\end{equation}
(where $\mathcal{P}_{\overline\beta}$ is defined in (\ref{eq:pbetamodl})). A
useful subtree of $\mathcal{T}$ is formed by the $\ell$-stable rings,
which we now define. An important property of stable rings, proved
below, is that the inclusion (\ref{eq:stab1}) is an equality whenever
$R'$ is stable.
\begin{definition}\label{stable} A ring $R'\in\mathcal{T}(R)$ is said to be
$\ell$-\textbf{stable} if the following two conditions hold:
(1) the ring
\begin{equation}
\kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}
\label{eq:extension1}
\end{equation}
is an integral domain and
(2) there do not exist an $R''\in\mathcal{T}(R')$ and a non-trivial
algebraic extension $L$ of $\kappa(P'_\ell)$ which embeds both into
$\kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}$
and $\kappa(P''_\ell)$.
We say that $R$ is \textbf{stable} if it is $\ell$-stable for each
$\ell\in\{0,\dots,r\}$.
\end{definition}
\begin{remark}\label{interchanging} (1) Rings of the form
(\ref{eq:extension1}) will be a basic object of study in this
paper. Another way of looking at the same ring, which we will often
use, comes from interchanging the order of tensor product and
localization. Namely, let $T'$ denote the image of the
multiplicative system $\left(R'\otimes_RR^\dag\right)\setminus M'$
under the natural map
$R'\otimes_RR^\dag\rightarrow\kappa\left(P'_\ell\right)\otimes_RR^\dag$. Then
the ring (\ref{eq:extension1}) equals the localization
$(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^\dag\right)$.
(2) In the special case $R'=R$ in Definition \ref{stable}, we have
$$
\kappa\left(P'_\ell\right)\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}=
\kappa\left(P_\ell\right)\otimes_RR^\dag.
$$
If, moreover, $\frac R{P_\ell}$ is analytically irreducible then the
hypothesis that $\kappa\left(P_\ell\right)\otimes_RR^\dag$ is a domain
holds automatically; in fact, this hypothesis is equivalent to
analytic irreducibility of $\frac R{P_\ell}$ if $R^\dag={\Phi}at R$ or
$R^\dag=\tilde R$.
(3) Consider the special case when $R'$ is Henselian and ${\ }^\dag={\Phi}at{\ }$. Excellent Henselian rings are algebraically closed inside their formal completions, so both (1) and (2) of Definition \ref{stable} hold automatically for this $R'$. Thus excellent Henselian local rings are always stable.
\end{remark}
In this section we study $\ell$-stable rings. We prove that if $R$ is
$\ell$-stable then so is any $R'\in\mathcal{T}(R)$ (justifying the
name ``stable''). The main result of this section, Theorem
\ref{primality1}, says that if $R$ is stable then the implicit ideal
$H'_{2\ell+1}$ is prime for each $\ell\in\{0,\dots,r\}$ and each
$R'\in\mathcal{T}(R)$.
\begin{remark}\label{hypotheses} In the next two sections we will show
that there exist stable rings $R'\in\cal T$ for both $R^\dag={\Phi}at R$
and $R^\dag=R^e$. However, the proof of this is different depending
on whether we are dealing with completion or with an \'etale
extension, and will be carried out separately in two separate
sections: one devoted to henselization, the other to completion.
\end{remark}
\begin{proposition}\label{largeR1} Fix an integer $\ell$, $0\le\ell\le
r$. Assume that $R'$ is $\ell$-stable and let
$R''\in\mathcal{T}(R')$. Then $R''$ is $\ell$-stable.
\end{proposition}
\begin{proof} We have to show that (1) and (2) of
Definition \ref{stable} for $R'$ imply (1) and (2) of
Definition \ref{stable} for $R''$. The ring
\begin{equation}
\kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^\dag\right)_{M''}
\label{eq:extension}
\end{equation}
is a localization of
$\kappa\left(P''_\ell\right)\otimes_R\left(\kappa\left(P'_\ell\right)
\otimes_R\left(R'\otimes_RR^\dag\right)_{M'}\right)$. Hence (1) and
(2) of Definition \ref{stable}, applied to $R'$, imply that
$\kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^\dag\right)_{M''}$
is an integral domain, so (1) of Definition \ref{stable} holds for
$R''$. Replacing $R'$ by $R''$ clearly does not affect the hypotheses
about the non-existence of the extension $L$, so (2) of Definition
\ref{stable} also holds for $R''$.
\end{proof}
Next, we prove a technical result on which much of the rest of the
paper is based. For $\overline\beta\in\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_{\ell+1}}$,
let
\begin{equation}
\mathcal{P}_{\overline\beta+}=\left\{x\in R\ \left|\
\nu(x)\mod\Delta_{\ell+1}>\overline\beta\right.\right\}.\label{eq:pbetamodl+}
\end{equation}
As usual, $\mathcal{P}'_{\overline\beta+}$ will stand for the analogous notion,
but with $R$ replaced by $R'$, etc.
\begin{proposition}\label{largeR2} Assume that $R$ itself is $(\ell+1)$-stable and let
$R'\in\mathcal{T}(R)$.
\begin{enumerate}
\item For any $\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$
\begin{equation}
\mathcal{P}'_{\overline\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab}
\end{equation}
\item For any $\overline\beta\in\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_{\ell+1}}$ the natural map
\begin{equation}\label{eq:gammaversion}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow
\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{\mathcal{P}'_{\overline\beta+}{R'}^\dag}
\end{equation}
is injective.
\end{enumerate}
\end{proposition}
\begin{proof} As explained at the end of the
previous section, since ${R'}^\dag$ is faithfully flat over the ring
$\left(R^\dag\otimes_RR'\right)_{M'}$, we may replace ${R'}^\dag$ by
$\left(R^\dag\otimes_RR'\right)_{M'}$ in both 1 and 2 of the Proposition.
\noi\textbf{Proof of 1 of the Proposition:} It is sufficient to prove that
\begin{equation}
\mathcal{P}'_{\overline\beta}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap
R^\dag=\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab2}
\end{equation}
One inclusion in (\ref{eq:stab2}) is trivial; we must show that
\begin{equation}
\mathcal{P}'_{\overline\beta}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap
R^\dag\subset\mathcal{P}_{\overline\beta} R^\dag.\label{eq:stab3}
\end{equation}
\begin{lemma}\label{injectivity} Let $T'$ denote the image of the
multiplicative set $\left(R'\otimes_RR^\dag\right)\setminus M'$ under
the natural map of $R$-algebras
$R'\otimes_RR^\dag\rightarrow\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}
R'_{P'_{\ell+1}}}\otimes_RR^\dag$. Then the map of $R$-algebras
\begin{equation}
\bar\pi:\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag
\rightarrow(T')^{-1}\left(\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}
R'_{P'_{\ell+1}}}\otimes_RR^\dag\right)
\label{eq:inclusion3}
\end{equation}
induced by $\pi:R\rightarrow R'$ is injective.
\end{lemma}
\begin{proof}\textit{(of Lemma \ref{injectivity})} We start with the field extension
$$
\kappa(P_{\ell+1}){\Phi}ookrightarrow\kappa(P'_{\ell+1})
$$
induced by $\pi$. Since $R^\dag$ is flat over $R$, the induced map
$\pi_1:\kappa(P_{\ell+1})\otimes_RR^\dag\rightarrow\kappa(P'_{\ell+1})\otimes_R
R^\dag$ is also injective. By (1) of Definition \ref{stable},
$\kappa(P'_{\ell+1})\otimes_RR^\dag$ is a domain. In particular,
\begin{equation}
\kappa\left(P'_{\ell+1}\right)\otimes_RR^\dag=
\left(\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_R
R^\dag\right)_{red}.
\label{eq:reduced1}
\end{equation}
The local ring
$\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ is
artinian because it can be seen as the quotient of
$\frac{R'_{P'_{\ell+1}}}{P'_\ell R'_{P'_{\ell+1}}}$ by a valuation
ideal corresponding to a rank one valuation. Since the ring is
noetherian the valuation of the maximal ideal is positive, and since
the group is archimedian, a power of the maximal ideal is contained in
the valuation ideal.\par
Therefore, its only associated prime is its nilradical, the ideal
$\frac{P'_{\ell+1}
R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$; in
particular, the $(0)$ ideal in this ring has no embedded
components. Since $R^\dag$ is flat over $R$,
$\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$
is flat over
$\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ by
base change. Hence the $(0)$ ideal of
$\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$
has no embedded components. In particular, since the multiplicative
system $T'$ is disjoint from the nilradical of
$\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag$,
the set $T'$ contains no zero divisors, so localization by $T'$ is
injective.\par
By the definition of $\mathcal{P}_{\overline\beta}$, the map
$\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\rightarrow
\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}$ is
injective, hence so is
$$
\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag\rightarrow
\frac{R'_{P'_{\ell+1}}}{\mathcal{P}'_{\overline\beta}R'_{P'_{\ell+1}}}\otimes_RR^\dag
$$
by the flatness of $R^\dag$ over $R$. Combining this with the injectivity of the
localization by $T'$, we obtain that $\bar\pi$ is injective, as
desired. Lemma \ref{injectivity} is proved.
\end{proof}
Again by the definition of $\mathcal{P}_{\overline\beta}$, the localization map
$\frac
R{\mathcal{P}_{\overline\beta}}\rightarrow\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}
R_{P_{\ell+1}}}$ is injective, hence so is the map
\begin{equation}
\frac {R}{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow
\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline\beta}R_{P_{\ell+1}}}\otimes_RR^\dag
\label{eq:injective}
\end{equation}
by the flatness of $R^\dag$ over $R$. Combining this with Lemma
\ref{injectivity}, we see that the composition
\begin{equation}
\frac R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow
(T')^{-1}\left(\frac{R'_{P'_\ell}}{\mathcal{P}'_{\overline\beta}
R'_{P'_\ell}}\otimes_RR^\dag\right)
\label{eq:injective1}
\end{equation}
of (\ref{eq:injective}) with $\bar\pi$ is also injective. Now,
(\ref{eq:injective1}) factors through
$\left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'}$
(here we are guilty of a slight abuse of notation: we denote the
natural image of $M'$ in
$\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag$ also by $M'$). Hence
the map
\begin{equation}
\frac
R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\rightarrow
\left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'}
\label{eq:injective2}
\end{equation}
is injective. Since $\frac
R{\mathcal{P}_{\overline\beta}}\otimes_RR^\dag\cong\frac{R^\dag}{\mathcal{P}_{\overline\beta}
R^\dag}$ and
$\left(\frac{R'}{\mathcal{P}'_{\overline\beta}}\otimes_RR^\dag\right)_{M'}\cong
\frac{\left(R'\otimes_RR^\dag\right)_{M'}}{\mathcal{P}'_{\overline\beta}
\left(R'\otimes_RR^\dag\right)_{M'}}$, the injectivity of
(\ref{eq:injective2}) is the same as (\ref{eq:stab3}). This completes
the proof of 1 of the Proposition.
\noi\textbf{Proof of 2 of the Proposition:} We start with the
injective homomorphism
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_R\kappa(P_{\ell+1})
\rightarrow\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R
\kappa(P'_{\ell+1})\label{eq:vspaces}
\end{equation}
of $\kappa(P_{\ell+1})$-vector spaces. Since $R^\dag$ is flat over
$R$, tensoring (\ref{eq:vspaces}) produces an injective homomorphism
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R
\kappa(P_{\ell+1})\rightarrow\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}
\otimes_R\kappa(P'_{\ell+1})\otimes_RR^\dag\label{eq:Rdagmodules}
\end{equation}
of $R^\dag$-modules. Now, the $\kappa(P_{\ell+1})$-vector space
$\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R\kappa(P'_{\ell+1})$
is, in particular, a torsion-free $\kappa(P_{\ell+1})$-module. Since
$\kappa(P_{\ell+1})\otimes_RR^\dag$ is a domain by definition of
$(\ell+1)$-stable and by the flatness of
$R^\dag\otimes_R\kappa(P_{\ell+1})$ over $\kappa(P_{\ell+1})$, the
$R^\dag\otimes_R\kappa(P_{\ell+1})$-module
$\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R\kappa(P'_{\ell+1})
\otimes_RR^\dag$ is also torsion-free; in particular, its localization
map by any multiplicative system is injective. Let $S'$ denote the image of the
multiplicative set $\left(R'\otimes_RR^\dag\right)\setminus M'$ under
the natural map of $R$-algebras
$R'\otimes_RR^\dag\rightarrow\kappa(P'_{\ell+1})\otimes_RR^\dag$. By
the above, the composition
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R
\kappa(P_{\ell+1})\rightarrow(S')^{-1}
\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R
\kappa(P'_{\ell+1})\otimes_RR^\dag\right)\label{eq:Rdagmodules1}
\end{equation}
of (\ref{eq:Rdagmodules}) with the localization by $S'$ is injective.
By the definition of $\mathcal{P}_{\overline\beta}$, the localization map
$\frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\rightarrow
\frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_R\kappa(P_{\ell+1})$
is injective, hence so is the map
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}=
\frac{\mathcal{P}_{\overline\beta}}{\mathcal{P}_{\overline\beta+}}\otimes_RR^\dag\rightarrow
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\otimes_R
\kappa(P_{\ell+1})\label{eq:injectivegamma}
\end{equation}
by the flatness of $R^\dag$ over $R$. Combining this with the
injectivity of (\ref{eq:Rdagmodules1}), we see that the composition
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow
(S')^{-1}\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_R
\kappa(P'_{\ell+1})\otimes_RR^\dag\right)
\label{eq:injective1gamma}
\end{equation}
of (\ref{eq:injectivegamma}) with (\ref{eq:Rdagmodules1}) is also
injective. Now, (\ref{eq:injective1gamma}) factors through
$\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}
\left(R'\otimes_RR^\dag\right)_{M'}$. Hence the map
\begin{equation}
\frac{\mathcal{P}_{\overline\beta}R^\dag}{\mathcal{P}_{\overline\beta+}R^\dag}\rightarrow
\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}
\left(R'\otimes_RR^\dag\right)_{M'}\label{eq:injective2gamma}
\end{equation}
is injective. Since
$\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}{R'}^\dag\cong
\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{\mathcal{P}'_{\overline\beta+}{R'}^\dag}$ and
by faithful flatness of ${R'}^\dag$ over
$\left(R'\otimes_RR^\dag\right)_{M'}$, the injectivity of
(\ref{eq:injective2gamma}) implies the injectivity of the map
(\ref{eq:gammaversion}) required in 2 of the Proposition.
This completes the proof of the Proposition.
\end{proof}
\begin{corollary}\label{stableimplicit} Take an integer
$\ell\in\{0,\dots,r-1\}$ and assume that $R$ is
$(\ell+1)$-stable. Then
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}{\cal P}_\beta
R^\dag.\label{eq:defin5}
\end{equation}
\end{corollary}
\begin{proof} By Lemma 4 of Appendix 4 of \cite{ZS}, the ideals
$\mathcal{P}_{\overline\beta}$ are cofinal among the ideals $\mathcal{P}_\beta$ for
$\beta\in \Delta_\ell$.
\end{proof}
\begin{corollary}\label{stablecontracts} Assume that $R$ is stable. Take an
element $\beta\in\mbox{l{\Phi}space{-.47em}G}amma$. Then $\mathcal{P}'_\beta{R'}^\dag\cap
R^\dag=\mathcal{P}_\beta$.
\end{corollary}
\begin{proof} It is sufficient to prove that for each
$\ell\in\{0,\dots,r-1\}$ and
$\bar\beta\in\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_{\ell+1}}$, we have
\begin{equation}
\mathcal{P}'_{\bar\beta}{R'}^\dag\cap R^\dag=\mathcal{P}_{\bar\beta};\label{eq:scalewisebeta}
\end{equation}
the Corollary is just the special case of (\ref{eq:scalewisebeta})
when $\ell=r-1$. We prove (\ref{eq:scalewisebeta}) by
contradiction. Assume the contrary and take the smallest $\ell$ for
which (\ref{eq:scalewisebeta}) fails to be true. Let
$\mathcal{P}hi'=\nu(R'\setminus\{0\})$. We will denote by
$\frac\mathcal{P}hi{\Delta_{\ell+1}}$ the image of $\mathcal{P}hi$ under the composition
of natural maps
$\mathcal{P}hi{\Phi}ookrightarrow\mbox{l{\Phi}space{-.47em}G}amma\rightarrow\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_{\ell+1}}$ and
similarly for $\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}$. Clearly, if
(\ref{eq:scalewisebeta}) fails for a certain $\bar\beta$, it also
fails for some $\bar\beta\in\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}$; take the
smallest $\bar\beta\in\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}$ with this
property. If we had
$\bar\beta=\min\left\{\left.\tilde\beta\in\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}\
\right|\ \tilde\beta-\bar\beta\in\Delta_\ell\right\}$, then
(\ref{eq:scalewisebeta}) would also fail with $\bar\beta$ replaced by
$\bar\beta\mod\Delta_\ell\in\frac\mbox{l{\Phi}space{-.47em}G}amma{\Delta_\ell}$, contradicting
the minimality of $\ell$. Thus
\begin{equation}\label{eq:notminimum}
\bar\beta>\min\left\{\left.\tilde\beta\in\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}\
\right|\ \tilde\beta-\bar\beta\in\Delta_\ell\right\}.
\end{equation}
Let $\bar\beta-$ denote the immediate predecessor of $\bar\beta$ in
$\frac{\mathcal{P}hi'}{\Delta_{\ell+1}}$. By (\ref{eq:notminimum}), we have
$\bar\beta-\bar\beta-\in\Delta_\ell$. By the choice of $\bar\beta$, we
have $\mathcal{P}_{\bar\beta-}=\mathcal{P}'_{\bar\beta-}{R'}^\dag\cap R^\dag$ but
$\mathcal{P}_{\bar\beta}\subsetneqq\mathcal{P}'_{\bar\beta}{R'}^\dag\cap R^\dag$. This
contradicts Proposition \ref{largeR2}, applied to $\bar\beta-$. The
Corollary is proved.
\end{proof}
Now we are ready to state and prove the main Theorem of this section.
\begin{theorem}\label{primality1} (1) Fix an integer
$\ell\in\{1,\dots,r+1\}$. Assume that there exists
$R'\in\mathcal{T}(R)$ which is $(\ell+1)$-stable. Then the ideal
$H_{2\ell+1}$ is prime.
(2) Let $i=2\ell+2$. There exists an extension $\nu^\dag_{i0}$ of
$\nu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, with
value group
\begin{equation}
\Delta_{i-1,0}=\frac{\Delta_\ell}{\Delta_{\ell+1}},\label{eq:groupequal2}
\end{equation}
whose valuation ideals are described as follows. For an element
$\overline\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$, the
$\nu^\dag_{i0}$-ideal of $\frac{R^\dag}{H_{i-1}}$ of value
$\overline\beta$, denoted by $\mathcal{P}^\dag_{\overline\beta\ell}$, is given
by the formula
\begin{equation}
\mathcal{P}^\dag_{\overline\beta,\ell+1}=\left(\lim\limits_{\overset\longrightarrow{R'}}
\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag}{H'_{i-1}}\right)\cap\frac{R^\dag}{H_{i-1}}.
\label{eq:valideal1}
\end{equation}
\end{theorem}
\begin{remark} Once the even-numbered implicit prime ideals $H'_{2\ell}$
are defined below, we will show that $\nu^\dag_{i0}$ is the unique
extension of $\nu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$,
centered in the local ring
$\lim\limits_{\overset\longrightarrow{R'}}
\frac{R'^\dag_{H'_{2\ell+2}}}{H'_{2\ell+1}R'^\dag_{H'_{2\ell+2}}}$.
\end{remark}
\begin{proof}\textit{(of Theorem \ref{primality1})} Let $R'$ be a stable
ring in $\mathcal{T}(R)$. Once Theorem \ref{primality1} is proved for
$R'$, the same results for $R$ will follow easily by intersecting all
the ideals of ${R'}^\dag$ in sight with $R^\dag$. Therefore from now
on we will replace $R$ by $R'$, that is, we will assume that $R$
itself is stable.
Let $\mathcal{P}hi_\ell$ denote the image of the semigroup
$\nu(R\setminus\{0\})$ in $\frac{\mbox{l{\Phi}space{-.47em}G}amma}{\Delta_{\ell+1}}$. As we saw
above, $\mathcal{P}hi_\ell$ is well ordered. For an element
$\overline\beta\in\mathcal{P}hi_\ell$, let $\overline\beta+$ denote the
immediate successor of $\overline\beta$ in $\mathcal{P}hi_\ell$.
Take any element $x\in R^\dag\setminus H_{i-1}$. By Corollary
\ref{stableimplicit}, there exists (a unique)
$\overline\beta\in\mathcal{P}hi_\ell\cap\frac{\Delta_\ell}{\Delta_{\ell+1}}$
such that
\begin{equation}
x\in{\cal P}_{\overline\beta} R^\dag\setminus{\cal
P}_{\overline\beta+}R^\dag\label{eq:xinbeta}
\end{equation}
(where, of course, we allow $\overline\beta=0$). Let $\bar x$ denote the image of $x$ in $\frac{R^\dag}{H_{i-1}}$.
We define
$$
\nu^\dag_{i0}(\bar x)=\overline\beta.
$$
Next, take another element $y\in R^\dag\setminus H_{2\ell+1}$ and let
${\Gamma}amma\in\mathcal{P}hi_\ell\cap\frac{\Delta_\ell}{\Delta_{\ell+1}}$ be such that
\begin{equation}
y\in{\cal P}_{\overline{\Gamma}amma}R^\dag\setminus{\cal P}_{\overline{\Gamma}amma+}R^\dag.\label{eq:yingamma}
\end{equation}
Let $(a_1,...,a_n)$ be a set of generators of ${\cal P}_{\overline\beta}$ and
$(b_1,...,b_s)$ a set of generators of ${\cal P}_{\overline{\Gamma}amma}$, with
$\nu_{\ell+1}(a_1)=\overline\beta$ and $\nu_{\ell+1}(b_1)=\overline{\Gamma}amma$. Let $R'$ be a local
blowing up along $\nu$ such that $R'$ contains all the fractions
$\frac{a_i}{a_1}$ and $\frac{b_j}{b_1}$. By Proposition \ref{largeR1} and Definition \ref{stable} (1), the ideal
$P'_{\ell+1}{R'}^\dag$ is prime. By construction, we have $a_1\ |\ x$ and $b_1\ |\ y$ in ${R'}^\dag$.
Write $x=za_1$ and $y=wb_1$ in ${R'}^\dag$. The equality
(\ref{eq:stab}), combined with (\ref{eq:xinbeta}) and (\ref{eq:yingamma}), implies that $z,w\mbox{$\in$ {\Phi}space{-.8em}/} P'_{\ell+1}{R'}^\dag$, hence
\begin{equation}
zw\mbox{$\in$ {\Phi}space{-.8em}/} P'_{\ell+1}{R'}^\dag\label{eq:zwnotin}
\end{equation}
by the primality of $P'_{\ell+1}{R'}^\dag$. We obtain
\begin{equation}
xy=a_1b_1zw.\label{eq:xyzw}
\end{equation}
Since $\nu$ is a valuation on $R'$, we have $\left({\cal P}'_{\overline\beta+\overline{\Gamma}amma+}:(a_1b_1)R'\right)\subset
P'_{\ell +1}$. By faithful flatness of ${R'}^\dag$ over $R'$ we obtain
\begin{equation}
\left({\cal P}'_{\overline\beta+\overline{\Gamma}amma+}{R'}^\dag:(a_1b_1){R'}^\dag\right)\subset P'_{\ell+1}{R'}^\dag.
\end{equation}
Combining this with (\ref{eq:zwnotin}) and (\ref{eq:xyzw}), we obtain
\begin{equation}
xy\mbox{$\in$ {\Phi}space{-.8em}/}{\cal P}_{\overline\beta+\overline{\Gamma}amma+}R^\dag,\label{eq:xynotin}
\end{equation}
in particular, $xy\mbox{$\in$ {\Phi}space{-.8em}/} H_{2\ell+1}$. We started with two arbitrary elements $x,y\in R^\dag\setminus H_{2\ell+1}$ and
showed that $xy\mbox{$\in$ {\Phi}space{-.8em}/} H_{2\ell+1}$. This proves (1) of the Theorem.
Furthermore, (\ref{eq:xynotin}) shows that $\nu^\dag_{i0}(\bar x\bar y)=\overline\beta+\overline{\Gamma}amma$, so $\nu^\dag_{i0}$ induces a
valuation of $\kappa(H_{i-1})$ and hence also of $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$.
Equality (\ref{eq:groupequal2}) holds by definition and (\ref{eq:valideal1}) by the assumed stability of $R$.
\end{proof}
Next, we define the even-numbered implicit prime ideals $H'_{2\ell}$. The only information we need to use to define the
prime ideals $H'_{2\ell}\subset H'_{2\ell+1}$ and to prove that $H'_{2\ell-1}\subset H'_{2\ell}$ are the facts that
$H_{2\ell+1}$ is a prime lying over $P_\ell$ and that the ring homomorphism $R'\rightarrow{R'}^\dag$ is regular.
\begin{proposition}\label{H2l} There exists a unique minimal prime ideal $H_{2\ell}$ of
$P_\ell R^\dag$, contained in $H_{2\ell+1}$.
\end{proposition}
\begin{proof} Since $H_{2\ell+1}\cap R=P_\ell$, $H_{2\ell+1}$ belongs to
the fiber of the map $Spec\ R^\dag\rightarrow Spec\ R$ over $P_\ell$. Since
$R$ was assumed to be excellent, $S:={R^\dag}\otimes_R\kappa(P_\ell)$
is a regular ring (note that the excellence assumption is needed only in the case $R^\dag={\Phi}at R$; the ring homomorphism
$R\rightarrow R^\dag$ is automatically regular if $R^\dag=\tilde R$ or $R^\dag=R^e$). Hence its localization $\bar
S:=S_{H_{2\ell+1}S}\cong\frac{R^\dag_{H_{2\ell+1}}}{P_\ell R^\dag_{H_{2\ell+1}}}$ is
a regular {\em local} ring. In particular, $\bar S$ is an integral
domain, so $(0)$ is its unique minimal prime ideal. The set of minimal
prime ideals of $\bar S$ is in one-to-one correspondence with the set of
minimal primes of $P_\ell$, contained in $H_{2\ell+1}$, which shows that such
a minimal prime $H_{2\ell}$ is unique, as desired.
\end{proof}
We have $P_\ell\subset H_{2\ell}\cap R\subset H_{2\ell+1}\cap R=P_\ell$, so $H_{2\ell}\cap R\subset P_\ell$.
\begin{proposition} We have $H_{2\ell-1}\subset H_{2\ell}$.
\end{proposition}
\begin{proof} Take an element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and a stable ring $R'\in\cal T$.
Then $\mathcal{P}'_\beta\subset P'_\ell$, so
\begin{equation}
H'_{2\ell-1}\subset\mathcal{P}'_\beta{R'}^\dag\subset P'_\ell{R'}^\dag\subset H'_{2\ell}.\label{eq:inclusion}
\end{equation}
Intersecting (\ref{eq:inclusion}) back with $R^\dag$ we get the result.
\end{proof}
In \S\ref{henselization} we will see that if $R^\dag=\tilde R$ or $R^\dag=R^e$ then $H_{2\ell}=H_{2\ell+1}$ for all
$\ell$.
Let the notation be the same as in Theorem \ref{primality1}.
\begin{proposition}\label{nu0unique} The valuation $\nu_{i0}^\dag$ is the unique extension of $\nu_\ell$ to a valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered in the local ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}$.
\end{proposition}
\begin{proof} As usual, without loss of generality we may assume that $R$ is stable. Take an element
$x\in R^\dag\setminus H_{2\ell-1}$. Let $\beta=\nu^\dag_{i0}(\bar x)$ and let $R'$ be the blowing up of the ideal
$\mathcal{P}_\beta=(a_1,\dots,a_n)$, as in the proof of Theorem \ref{primality1}. Write
\begin{equation}
x=za_1\label{eq:xza}
\end{equation}
in $R'$. We have $z\in{R'}^\dag\setminus P'_\ell{R'}^\dag$, hence
\begin{equation}
\bar z\in\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\setminus\frac{P'_\ell{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}=
\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\setminus\frac{H'_{2\ell}{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}.
\label{eq:znotinPl}
\end{equation}
If $\nu^*$ is any other extension of $\nu_\ell$ to $\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered in
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}$, then $\nu^*(\bar a_1)=\beta$, $\nu^*(z)=0$
by (\ref{eq:znotinPl}), so $\nu^*(\bar x)=\beta=\nu_{i0}^\dag(\bar x)$. This completes the proof of the uniqueness of
$\nu_{i0}^\dag$.
\end{proof}
\begin{remark}\label{sameresfield} If $R'$ is stable, we have a natural isomorphism of graded algebras
$$
\mbox{gr}_{\nu^\dag_{i0}}\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}{R'}^\dag_{H'_{2\ell}}}\cong
\mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{R'}\kappa(H'_{2\ell}).
$$
In particular, the residue field of $\nu^\dag_{i0}$ is
$k_{\nu^\dag_{i0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})$.
\end{remark}
\section{A classification of extensions of $\nu$ to ${\Phi}at R$.}
\label{Rdag}
The purpose of this section is to give a systematic description of
all the possible extensions $\nu^\dag_-$ of $\nu$ to a quotient of
$R^\dag$ by a minimal prime as compositions of $2r$ valuations,
\begin{equation}
\nu^\dag_-=\nu^\dag_1\circ\dots\circ\nu^\dag_{2r},\label{eq:composition}
\end{equation}
satisfying certain conditions. One is naturally led to consider the
more general problem of extending $\nu$ not only to rings of the form
$\frac{R^\dag}P$ but also to the ring
$\lim\limits_\to\frac{{R'}^\dag}{P'}$, where $P'$ is a tree of prime
ideals of ${R'}^\dag$, such that $P'\cap R'=(0)$. We deal in a uniform
way with all the three cases $R^\dag={\Phi}at R$, $R^\dag=\tilde R$ and
$R^\dag=R^e$, in order to be able to apply the results proved here to
all three later in the paper. However, the reader should think of the
case $R^\dag={\Phi}at R$ as the main case of interest and the cases
$R^\dag=\tilde R$ and $R^\dag=R^e$ as auxiliary and slightly
degenerate, since, as we shall see, in these cases the equality
$H_{2\ell}=H_{2\ell+1}$ is satisfied for all $\ell$ and the extension
$\nu^\dag_-$ will later be shown to be unique.
We will associate to each extension $\nu^\dag_-$ of $\nu$ to $R^\dag$ a chain
\begin{equation}
\tilde H'_0\subset\tilde H'_1\subset\dots\subset\tilde
H'_{2r}=m'{R'}^\dag\label{eq:chaintree''}
\end{equation}
of prime $\nu^\dag_-$-ideals, corresponding to the decomposition
(\ref{eq:composition}) and prove some basic properties of this chain of ideals.
Now for the details. We wish to classify all the pairs $\left(\left\{\tilde
H'_0\right\},\nu^\dag _+\right)$, where $\left\{\tilde H'_0\right\}$
is a tree of prime ideals of ${R'}^\dag$, such that $\tilde H'_0\cap
R'=(0)$, and $\nu^\dag_+$ is an extension of $\nu$ to the ring
$\lim\limits_\to\frac{{R'}^\dag}{\tilde H'_0}$.
Pick and fix one such pair $\left(\left\{\tilde
H'_0\right\},\nu^\dag_+\right)$. We associate to it the following
collection of data, which, as we will see, will in turn determine the
pair $\left(\left\{\tilde H'_0\right\},\nu^\dag_+\right)$.
First, we associate to $\left(\left\{\tilde
H'_0\right\},\nu^\dag_-\right)$ a chain (\ref{eq:chaintree''}) of
$2r$ trees of prime $\nu^\dag_-$-ideals. Let $\mbox{l{\Phi}space{-.47em}G}amma^\dag$ denote the
value group of $\nu^\dag_-$. Defining (\ref{eq:chaintree''}) is
equivalent to defining a chain
\begin{equation}
\mbox{l{\Phi}space{-.47em}G}amma^\dag=\Delta^\dag_0\supset\Delta^\dag_1\supset\dots\supset\Delta^\dag_{2r}=
\Delta^\dag_{2r+1}=(0)\label{eq:groups}
\end{equation}
of $2r$ isolated subroups of $\mbox{l{\Phi}space{-.47em}G}amma^\dag$ (the chain
(\ref{eq:groups}) will not, in general, be maximal, and
$\Delta^\dag_{2\ell +1}$ need not be distinct from $\Delta^\dag_{2\ell}$).
We define the $\Delta^\dag_i$ as follows. For $0\le\ell\le r$, let
$\Delta^\dag_{2\ell}$ and $\Delta^\dag_{2\ell +1}$ denote, respectively,
the greatest and the smallest isolated subgroups of $\mbox{l{\Phi}space{-.47em}G}amma^\dag$ such that
\begin{equation}
\Delta^\dag_{2\ell}\cap\mbox{l{\Phi}space{-.47em}G}amma=\Delta^\dag_{2\ell
+1}\cap\mbox{l{\Phi}space{-.47em}G}amma=\Delta_\ell.\label{eq:Delta}
\end{equation}
\begin{lemma}\label{rank1} We have
\begin{equation}
rk\ \frac{\Delta^\dag_{2\ell-1}}{\Delta^\dag_{2\ell}}=1\label{eq:rank1}
\end{equation}
for $1\le \ell\le r$.
\end{lemma}
\begin{proof} Since by construction
$\Delta^\dag_{2\ell}\neq\Delta^\dag_{2\ell-1}$, equality
(\ref{eq:rank1}) is equivalent to saying that there is no isolated
subgroup $\Delta^\dag$ of $\mbox{l{\Phi}space{-.47em}G}amma^\dag$ which is properly contained in
$\Delta^\dag_{2\ell-1}$ and properly contains
$\Delta^\dag_{2\ell}$. Suppose such an isolated subgroup $\Delta^\dag$
existed. Then
\begin{equation}
\Delta_\ell=\Delta^\dag_{2\ell}\cap\mbox{l{\Phi}space{-.47em}G}amma\subsetneqq\Delta^\dag\cap\mbox{l{\Phi}space{-.47em}G}amma
\subsetneqq\Delta^\dag_{2\ell-1}\cap\mbox{l{\Phi}space{-.47em}G}amma=\Delta_{\ell-1},
\label{eq:noninclusion}
\end{equation}
where the first inclusion is strict by the maximality of
$\Delta^\dag_{2\ell}$ and the second by the minimality of
$\Delta^\dag_{2\ell -1}$. Thus $\Delta^\dag\cap\mbox{l{\Phi}space{-.47em}G}amma$ is an isolated subgroup of
$\mbox{l{\Phi}space{-.47em}G}amma$, properly containing $\Delta_\ell$ and properly contained in
$\Delta_{l-1}$, which is impossible since $rk\
\frac{\Delta_{\ell-1}}{\Delta_\ell}=1$. This is a contradiction, hence
$rk\ \frac{\Delta^\dag_{2\ell -1}}{\Delta^\dag_{2\ell}}=1$, as desired.
\end{proof}
\begin{definition} Let $0\le i\le 2r$. The $i$-th prime ideal
\textbf{determined} by $\nu^\dag_-$ is the prime $\nu^\dag_-$-ideal
$\tilde H'_i$ of ${R'}^\dag$, corresponding to the isolated subgroup
$\Delta^\dag_i$ (that is, the ideal $\tilde H'_i$ consisting of all
the elements of ${R'}^\dag$ whose values lie outside of
$\Delta^\dag_i$). The chain of trees (\ref{eq:chaintree''}) of prime
ideals of ${R'}^\dag$ formed by the $\tilde H'_i$ is referred to as
the chain of trees \textbf{determined} by $\nu^\dag_-$.
\end{definition}
The equality (\ref{eq:Delta}) says that
\begin{equation}
\tilde H'_{2\ell}\cap R'=\tilde H'_{2\ell+1}\cap R'=P'_\ell\label{eq:tildeHcapR}
\end{equation}
By definitions, for $1\le i\le2r$, $\nu^\dag_i$ is a valuation of the field
$k_{\nu^\dag_{i-1}}$. In the sequel, we will find it useful to talk
about the restriction of $\nu^\dag_i$ to a smaller field, namely, the
field of fractions of the ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$; we will denote
this restriction by $\nu^\dag_{i0}$. The field of fractions of
$\frac{{R'}^\dag_{\tilde H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde
H'_i}}$ is $\kappa(\tilde H'_{i-1})$, hence that of
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$ is
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$,
which is a subfield of $k_{\nu^\dag_{i-1}}$. The value group of
$\nu^\dag_{i0}$ will be denoted by $\Delta_{i-1,0}$; we have
$\Delta_{i-1,0}\subset\frac{\Delta^\dag_{i-1}}{\Delta^\dag_i}$. If
$i=2\ell$ is even then
$\frac{R'_{P'_l}}{P'_{l-1}R'_{P'_l}}<\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, so
$\lim\limits_{\overset\longrightarrow{R'}}\frac{R'_{P'_l}}{P'_{l-1}R'_{P'_l}}<
\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. In this case $rk\
\nu^\dag_i=1$ and $\nu^\dag_i$ an extension of the rank 1 valuation
$\nu_\ell$ from $\kappa(P_{\ell-1})$ to $k_{\nu^\dag_{i-1}}$; we have
\begin{equation}
\frac{\Delta_{\ell-1}}{\Delta_\ell}\subset\Delta_{i-1,0}\subset
\frac{\Delta^\dag_{i-1}}{\Delta^\dag_i}.\label{eq:deltalindelati-1}
\end{equation}
\begin{proposition}\label{necessary} Let $i=2\ell$. As usual, for an element
$\overline\beta\in\left(\frac{\Delta_\ell}{\Delta_{\ell+1}}\right)_+$,
let $\mathcal{P}_{\overline\beta}$ (resp. $\mathcal{P}'_{\overline\beta}$) denote
the preimage in $R$ (resp. in $R'$) of the $\nu_{\ell+1}$-ideal of
$\frac R{P_\ell}$ (resp. $\frac{R'}{P'_\ell}$) of value greater
than or equal to $\overline\beta$. Then
\begin{equation}
\bigcap\limits_{\overline\beta\in\left(\frac{\Delta_\ell}{\Delta_{\ell+1}}\right)_+}
\lim\limits_{\overset\longrightarrow{R'}}
\left(\mathcal{P}'_{\overline\beta}{R'}^\dag+\tilde
H'_{i+1}\right){R'}^\dag_{\tilde H'_{i+2}}\cap{R}^\dag\subset\tilde
H_{i+1}.
\label{eq:restriction}
\end{equation}
\end{proposition}
The inclusion (\ref{eq:restriction}) should be understood as a
condition on the tree of ideals. In other words, it is equally valid
if we replace $R'$ by any other ring $R''\in\cal T$.
\begin{proof}\textit{(of Proposition \ref{necessary})} Since
$rk\frac{\Delta^\dag_{i+1}}{\Delta^\dag_{i+2}}=1$ by Lemma
\ref{rank1}, $\frac{\Delta_\ell}{\Delta_{\ell+1}}$ is cofinal in
$\frac{\Delta^\dag_{i+1}}{\Delta^\dag_{i+2}}$. Then for any
$x\in\bigcap\limits_{\overline\beta\in\left(\frac{\Delta_\ell}
{\Delta_{\ell+1}}\right)_+}\lim\limits_{\overset
\longrightarrow{R'}}\left(\mathcal{P}'_{\overline\beta}{R'}^\dag+\tilde
H'_{i+1}\right){R'}^\dag_{\tilde H'_{i+2}}\cap{R}^\dag$ we have
$\nu^\dag_-(x)\mbox{$\in$ {\Phi}space{-.8em}/}\Delta^\dag_i$, hence $x\in\tilde H_{i+1}$, as
desired.
\end{proof}
From now to the end of \S\ref{extensions}, we will assume that $\cal
T$ contains a stable ring $R'$, so that we can apply the results of
the previous section, in particular, the primality of the ideals
$H'_i$.
\begin{proposition}\label{Hintilde} We have
\begin{equation}
H'_i\subset\tilde H'_i\qquad\text{ for all
}i\in\{0,\dots,2r\}.\label{eq:Hintilde}
\end{equation}
\end{proposition}
\begin{proof} For $\beta\in\mbox{l{\Phi}space{-.47em}G}amma^\dag$ and $R'\in\cal T$, let
${\mathcal{P}'_\beta}^\dag$ denote the $\nu^\dag_-$-ideal of ${R'}^\dag$ of value
$\beta$. Fix an integer $\ell\in\{0,\dots,r\}$. For each $R'\in\cal
T$, each $\beta\in\Delta_\ell$ and $x\in\mathcal{P}'_\beta$ we have
$\nu^\dag_-(x)=\nu(x){\Gamma}e\beta$, hence
\begin{equation}
\mathcal{P}'_\beta{R'}^\dag\subset{\mathcal{P}'_\beta}^\dag.\label{eq:PinPdag}
\end{equation}
Taking the inductive limit over all $R'\in\cal T$ and the intersection
over all $\beta\in\Delta_\ell$ in (\ref{eq:PinPdag}), and using the
cofinality of $\Delta_\ell$ in $\Delta^\dag_{2\ell+1}$ and the fact
that $\bigcap\limits_{\beta\in\Delta^\dag_{2\ell}}
\left(\lim\limits_{\overset\longrightarrow{R'}}{\mathcal{P}'_\beta}^\dag\right)=
\lim\limits_{\overset\longrightarrow{R'}}\tilde H'_{2\ell+1}$, we
obtain the inclusion (\ref{eq:Hintilde}) for $i=2\ell+1$. To prove
(\ref{eq:Hintilde}) for $i=2\ell$, note that $\tilde H'_{2\ell}\cap
R'=\tilde H'_{2\ell+1}\cap R'=P_\ell$. By the same argument as in
Proposition \ref{H2l}, excellence of $R'$ implies that there is a
unique minimal prime $H^*_{2\ell}$ of $P'_\ell{R'}^\dag$, contained in
$\tilde H'_{2\ell+1}$ and a unique minimal prime $H^{**}_{2\ell}$ of
$P'_\ell{R'}^\dag$, contained in $\tilde H'_{2\ell}$. Now, Proposition
\ref{H2l} and the facts that $H'_{2\ell+1}\subset\tilde H'_{2\ell+1}$
and $\tilde H'_{2\ell}\subset\tilde H'_{2\ell+1}$ imply that
$H'_{2\ell}=H^*_{2\ell}=H^{**}_{2\ell}$, hence
$H'_{2\ell}=H^{**}_{2\ell}\subset\tilde H'_{2\ell}$, as desired.
\end{proof}
\begin{definition} A chain of trees (\ref{eq:chaintree''}) of prime ideals
of ${R'}^\dag$ is said to be \textbf{admissible} if
$H'_i\subset\tilde H'_i$ and (\ref{eq:tildeHcapR}) and
(\ref{eq:restriction}) hold.
\end{definition}
Equalities (\ref{eq:tildeHcapR}), Proposition \ref{necessary} and
Proposition \ref{Hintilde} say that a chain of trees
(\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$, determined by
$\nu^\dag_-$, is admissible.
Summarizing all of the above results, and keeping in mind the fact
that specifying a composition of $2r$ valuation is equivalent to
specifying all of its $2r$ components, we arrive at one of the main
theorems of this paper:
\begin{theorem}\label{classification} Specifying the valuation $\nu^\dag_-$
is equivalent to specifying the following data. The data will be
described recursively in $i$, that is, the description of
$\nu^\dag_i$ assumes that $\nu^\dag_{i-1}$ is already defined:
(1) An admissible chain of trees (\ref{eq:chaintree''}) of prime
ideals of ${R'}^\dag$.
(2) For each $i$, $1\le i\le 2r$, a valuation $\nu^\dag_i$ of
$k_{\nu^\dag_{i-1}}$ (where $\nu^\dag_0$ is taken to be the
trivial valuation by convention), whose restriction to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ is
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$.
The data $\left\{\nu^\dag_i\right\}_{1\le i\le 2r}$ is subject to the
following additional condition: if $i=2\ell$ is even then $rk\
\nu^\dag_i=1$ and $\nu^\dag_i$ is an extension of $\nu_\ell$ to
$k_{\nu^\dag_{i-1}}$ (which is naturally an extension of $k_{\nu_{\ell-1}}$).
\end{theorem}
In particular, note that such extensions $\nu^\dag_-$ always exist, and
usually there are plenty of them. The question of uniqueness of
$\nu^\dag_-$ and the related question of uniqueness of $\nu^\dag_i$,
especially in the case when $i$ is even, will be addressed in the next section.
\section{Uniqueness properties of $\nu^\dag_-$.}
\label{extensions}
In this section we address the question of uniqueness of the extension
$\nu^\dag_-$. One result in this direction, which will be very useful
here, was already proved in \S\ref{technical}: Proposition
\ref{nu0unique}. We give some necessary and some sufficient conditions
both for the uniqueness of $\nu^\dag_-$ once the chain
(\ref{eq:chaintree''}) of prime ideals determined by $\nu^\dag_-$ has
been fixed, and also for the unconditional uniqueness of
$\nu^\dag_-$. In \S\ref{henselization} we will use one of these
uniqueness criteria to prove uniqueness of $\nu^\dag_-$ in the cases
$R^\dag=\tilde R$ and $R^\dag=R^e$. At the end of this section we
generalize and give a new point of view of an old result of W. Heinzer
and J. Sally (Proposition \ref{HeSal}), which provides a sufficient
condition for the uniqueness of $\nu^\dag_-$; see also \cite{Te}, Remarks 5.22.
For a ring $R'\in\cal T$ let $K'$ denote the field of fractions of
$R'$. For some results in this section we will need to impose an
additional condition on the tree $\cal T$: we will assume that there
exists $R_0\in\cal T$ such that for all $R'\in{\cal T}(R_0)$ the field
$K'$ is algebraic over $K_0$. This assumption is needed in order to be
able to control the height of all the ideals in sight. Without loss of
generality, we may take $R_0=R$.
\begin{proposition}\label{htstable} Assume that for all $R'\in\cal T$ the
field $K'$ is algebraic over $K$. Consider a ring homomorphism
$R'\rightarrow R''$ in $\cal T$. Take an $\ell\in\{0,\dots,r\}$. We
have
\begin{equation}
{\Phi}e\ H''_{2\ell}\le {\Phi}e\ H'_{2\ell}.\label{eq:odddecreases}
\end{equation}
If equality holds in (\ref{eq:odddecreases}) then
\begin{equation}
{\Phi}e\ H''_{2\ell+1}{\Gamma}e{\Phi}e\ H'_{2\ell+1}.\label{eq:odddecreases1}
\end{equation}
\end{proposition}
\begin{proof} We start by recalling a well known Lemma (for a
proof see \cite{ZS}, Appendix 1, Propositions 2 and 3, p. 326):
\begin{lemma}\label{idealheight} Let $R{\Phi}ookrightarrow R'$ be an
extension of integral domains, essentially of finite type. Let $K$
and $K'$ be the respective fields of fractions of $R$ and
$R'$. Consider prime ideals $P\subset R$ and $P'\subset R'$ such
that $P=P'\cap R$. Then
\begin{equation}
{\Phi}e\ P'+tr.deg.(\kappa(P')/\kappa(P))\le\ {\Phi}e\
P+tr.deg.(K'/K).\label{eq:heightdrops}
\end{equation}
Moreover, equality holds in (\ref{eq:heightdrops}) whenever $R$ is
universally catenarian.
\end{lemma}
Apply the Lemma to the rings $R'$ and $R''$ and the prime ideals
$P'_\ell\subset R'$ and $P''_\ell\subset R''$. In the case at hand we
have $tr.deg.(K''/K')=0$ by assumption. Hence
\begin{equation}
{\Phi}e\ P''_\ell\le\ {\Phi}e\ P'_\ell.\label{eq:htP}
\end{equation}
Since $H'_{2\ell}$ is a minimal prime of $P'_\ell{R'}^\dag$ and
${R'}^\dag$ is faithfully flat over $R'$, we have $ht\ P'_\ell=ht\
H'_{2\ell}$. Similarly, ${\Phi}e\ P''_\ell={\Phi}e\ H''_{2\ell}$, and
(\ref{eq:odddecreases}) follows. Furthermore, equality in
(\ref{eq:odddecreases}) is equivalent to equality in (\ref{eq:htP}).
To prove (\ref{eq:odddecreases1}), let $\bar
R=(R''\otimes_{R'}{R'}^\dag)_{M''}$, where $M''=(m''\otimes1+1\otimes
m'{R'}^\dag)$ and let $\bar m$ denote the maximal ideal of $\bar
R$. We have the natural maps ${R'}^\dag\overset\iota\rightarrow\bar
R\overset\sigma\rightarrow{R''}^\dag$. The homomorphism $\sigma$ is
nothing but the formal completion of the local ring $\bar R$; in
particular, it is faithfully flat. Let
\begin{equation}
\bar H=H''_{2\ell+1}\cap\bar R,\label{eq:barH}
\end{equation}
$\bar H_0=H''_0\cap\bar R$. Since $H''_0$ is a minimal prime of
${R''}^\dag$ and $\sigma$ is faithfully flat, $\bar H_0$ is a minimal
prime of $\bar R$.
Assume that equality holds in (\ref{eq:odddecreases}) (and hence also
in (\ref{eq:htP})). Since equality holds in (\ref{eq:htP}), by Lemma
\ref{idealheight} (applied to the ring extension $R'\rightarrow R''$)
the field $\kappa(P'')$ is algebraic
over $\kappa(P')$.
Apply Lemma \ref{idealheight} to the ring extension
$\frac{{R'}^\dag}{H'_0}{\Phi}ookrightarrow\frac{\bar R}{\bar H_0}$ and the
prime ideals $\frac{H'_{2\ell+1}}{H'_0}$ and $\frac{\bar H}{\bar
H_0}$. Since $K''$ is algebraic over $K'$, $\kappa(\bar H_0)$ is
algebraic over $\kappa(H'_0)$. Since $\kappa(P'')$ is algebraic over
$\kappa(P')$, $\kappa(\bar H)$ is algebraic over
$\kappa(H'_{2\ell+1})$. Finally, ${\Phi}at R'$ is universally catenarian
because it is a complete local ring. Now in the case ${\ }^\dag={\Phi}at{\
}$ Lemma \ref{idealheight} says that ${\Phi}e\
\frac{H'_{2\ell+1}}{H'_0}={\Phi}e\ \frac{\bar H}{\bar H_0}$. Since both
${\Phi}at R'$ and $\bar R$ are catenarian, this implies that
\begin{equation}
{\Phi}e\ H'_{2\ell+1}={\Phi}e\ \bar H.\label{eq:htequal}
\end{equation}
In the case where ${\ }^\dag$ stands for henselization or a finite
\'etale extension, (\ref{eq:htequal}) is an immediate consequence of
(\ref{eq:barH}). Thus (\ref{eq:htequal}) is true in all the
cases. Since $\sigma$ is faithfully flat and in view of
(\ref{eq:barH}), ${\Phi}e\bar H\le {\Phi}e\ H''_{2\ell+1}$. Combined with
(\ref{eq:htequal}), this completes the proof.
\end{proof}
\begin{corollary}\label{htstable1} For each $i$, $0\le i\le 2r$, the
quantity ${\Phi}e\ H'_i$ stabilizes for $R'$ sufficiently far out in $\cal T$.
\end{corollary}
The next Proposition is an immediate consequence of Theorem
\ref{classification}.
\begin{proposition}\label{uniqueness2} Suppose given an admissible chain of
trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$. For
each $\ell\in\{0,\dots,r-1\}$, consider the set of all $R'\in\cal T$
such that
\begin{equation}
{\Phi}e\ \tilde H'_{2\ell+1}- {\Phi}e\ \tilde H'_{2_\ell} \le 1\qquad\text{
for all even }i\label{eq:odd=even3}
\end{equation}
and, in case of equality, the 1-dimensional local ring
$\frac{{R'}^\dag_{\tilde H'_{2\ell+1}}}{\tilde
H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$ is unibranch (that is,
analytically irreducible). Assume that for each $\ell$ the set of such
$R'$ is cofinal in $\cal T$. Let $\nu^\dag_{2\ell+1,0}$ denote the
unique valuation centered at
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_{2\ell+1}}}{\tilde H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$.
Assume that for each even $i=2\ell$, $\nu_\ell$ admits a unique
extension $\nu^\dag_{i0}$ to a valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$,
centered in
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. Then specifying
the valuation $\nu^\dag_-$ is equivalent to specifying for each odd $i$,
$2<i<2r$, an extension $\nu_i^\dag$ of the valuation
$\nu^\dag_{i0}$ of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$ to
its field extension $k_{\nu^\dag_{i-1}}$ (in particular, such
extensions $\nu^\dag_-$ always exist). If for each odd $i$, $2<i<2r$,
the field extension
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde
H'_{i-1})\rightarrow k_{\nu^\dag_{i-1}}$ is algebraic and the
extension $\nu_i^\dag$ of $\nu^\dag_{i0}$ to $k_{\nu^\dag_{i-1}}$ is
unique then there is a unique extension $\nu^\dag_-$ of $\nu$ such that
the $\tilde H'_i$ are the prime ideals, determined by $\nu^\dag_-$.
Conversely, assume that $K'$ is algebraic over $K$ and that there
exists a unique extension $\nu^\dag_-$ of $\nu$ such that the $\tilde
H'_i$ are the prime $\nu^\dag_-$-ideals, determined by $\nu^\dag_-$. Then
for each $\ell\in\{0,\dots,r-1\}$ and for all $R'$ sufficiently far
out in $\cal T$ the inequality (\ref{eq:odd=even3}) holds. For each even $i=2\ell$,
$\nu_\ell$ admits a unique extension $\nu^\dag_{i0}$ to a valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa\left(\tilde
H'_{i-1}\right)$, centered in
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$; we have $rk\
\nu^\dag_{i0}=1$. For each odd $i$, the ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$ is a valuation
ring of a (not necessarily discrete) rank 1 valuation. For each odd $i$,
$1\le i<2r$, the field extension
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde
H'_{i-1})\rightarrow k_{\nu^\dag_{i-1}}$ is algebraic and the
extension $\nu_i^\dag$ of $\nu^\dag_{i0}$ to $k_{\nu^\dag_{i-1}}$ is
unique.
\end{proposition}
\begin{remark} We do not know of a simple criterion to decide when, given
an algebraic field extension $K{\Phi}ookrightarrow L$ and a valuation
$\nu$ of $K$, is the extension of $\nu$ to $L$ unique. See
\cite{HOS}, \cite{V2} for more information about this question and an
algorithm for arriving at the answer using MacLane's key
polynomials.
\end{remark}
Next we describe three classes of extensions of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}{R'}^\dag$, which are of
particular interest for applications, and which we call
\textbf{minimal}, \textbf{evenly minimal} and \textbf{tight} extensions.
\begin{definition} Let $\nu^\dag_-$ be an extension of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}{R'}^\dag$ and let the
notation be as above. We say that $\nu^\dag_-$ is \textbf{evenly
minimal} if whenever $i=2\ell$ is even, the following two conditions hold:
(1)
\begin{equation}
\Delta_{i-1,0}=\frac{\Delta_{\ell-1}}{\Delta_\ell}.\label{eq:groupequal}
\end{equation}
(2) For an element $\overline\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$,
the $\nu^\dag_{i0}$-ideal of $\frac{R^\dag_{\tilde H_i}}{\tilde
H_{i-1}R^\dag_{\tilde H_i}}$ of value $\overline\beta$, denoted by
$\mathcal{P}^\dag_{\overline\beta ,\ell}$, is given by the formula
\begin{equation}
\mathcal{P}^\dag_{\overline\beta,\ell}=\left(\lim\limits_{\overset\longrightarrow{R'}}
\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\right)\cap\frac{R^\dag_{\tilde
H_i}}{\tilde H_{i-1}R^\dag_{\tilde H_i}}.\label{eq:valideal}
\end{equation}
We say that $\nu^\dag_-$ is \textbf{minimal} if $\tilde H'_i=H'_i$
for each $R'$ and each $i\in\{0,\dots,2r+1\}$. We say that
$\nu^\dag_-$ is \textbf{tight} if it is evenly minimal and
\begin{equation}
\tilde H'_i=\tilde H'_{i+1}\qquad\text{ for all even }i.\label{eq:odd=even2}
\end{equation}
\end{definition}
\begin{remark} (1) The valuation $\nu^\dag_{i0}$ is uniquely determined by
conditions (\ref{eq:groupequal}) and (\ref{eq:valideal}). Recall
also that if $i=2\ell$ is even and we have:
\begin{eqnarray}
\tilde H'_i&=&H'_i\qquad\text{ and}\label{eq:tilde=H1}\\
\tilde H'_{i-1}&=&H'_{i-1}\label{eq:tilde=H2}
\end{eqnarray}
then $\nu^\dag_{i0}$ is uniquely determined by $\nu_\ell$ by
Proposition \ref{nu0unique}. In particular, if $\nu^\dag_-$ is minimal
(that is, if (\ref{eq:tilde=H1})--(\ref{eq:tilde=H2}) hold for all
$i$) then $\nu^\dag_-$ is evenly minimal.
(2) The definition of evenly minimal extensions can be rephrased as follows in terms of the associated graded algebras of $\nu_\ell$ and $\nu^\dag_{i0}$. First, consider a homomorphism in $\mathcal T$ and the following diagram, composed of natural homomorphisms:
\begin{equation}\label{eq:CDevminimal}
\xymatrix{{\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}{R'}^\dag_{\tilde H'_i}}\ar[r]^-{\lambda'}&{\frac{\mathcal{P}'_{\overline\beta}{R'}^\dag_{\tilde H'_i}}{\mathcal{P}'_{\overline\beta+}{R'}^\dag_{\tilde H'_i}}} \ar[r]^-{\phi'}& {\frac{{\mathcal{P}'_{\overline\beta}}^\dag}{{\mathcal{P}'}_{\overline\beta+}^\dag}}\\
{\ }&{\ }&{\frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}}\ar[u]_-{\psi'}}
\end{equation}
It follows from Nakayama's Lemma that equality (\ref{eq:valideal}) is equivalent to saying that
\begin{equation}\label{eq:equalgraded}
\frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}=\lim\limits_{\overset\longrightarrow{R'}}{\psi'}^{-1}\left((\phi'\circ\lambda')\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}\kappa\left(\tilde H'_i\right)\right)\right).
\end{equation}
Taking the direct sum in (\ref{eq:equalgraded}) over all $\overline\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and passing to the limit on the both sides, we see that the extension $\nu^\dag_-$ is evenly minimal if and only if we have the following equality of graded algebras:
$$
\mbox{gr}_{\nu_\ell}\left(\lim\limits_{\overset\longrightarrow{R'}}\frac{R'}{P'_\ell}\right)\otimes_{R'}\kappa\left(\tilde H'_i\right)=\mbox{gr}_{\nu^\dag_{i0}}\left(\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\right).
$$
\end{remark}
\begin{examples} The extension ${\Phi}at\nu$ of Example \ref{Example31} (page \pageref{Example31}) is
minimal, but not tight. The valuation $\nu$ admits a unique tight
extension ${\Phi}at\nu_2\circ{\Phi}at\nu_3$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_1}$; the
valuation ${\Phi}at\nu$ is the composition of the discrete rank 1
valuation ${\Phi}at\nu_1$, centered in
$\lim\limits_{\overset\longrightarrow{R'}}{\Phi}at R'_{H'_1}$ with
${\Phi}at\nu_2\circ{\Phi}at\nu_3$.
The extension ${\Phi}at\nu^{(1)}$ of Example \ref{Example32} (page \pageref{Example32}) is minimal. The
extension ${\Phi}at\nu^{(2)}$ is evenly minimal but not minimal.
Neither ${\Phi}at\nu^{(1)}$ nor ${\Phi}at\nu^{(2)}$ is tight. The valuation
$\nu$ admits a unique tight extension ${\Phi}at\nu_2\circ{\Phi}at\nu_3$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{R'}{\tilde H'_1}$,
where $\tilde H'_1=\left(y-\sum\limits_{j=1}^\infty c_jx^j\right)$;
the valuation ${\Phi}at\nu^{(2)}$ is the composition of the discrete
rank 1 valuation ${\Phi}at\nu_1$, centered in
$\lim\limits_{\overset\longrightarrow{R'}}{\Phi}at R'_{\tilde H'_1}$
with ${\Phi}at\nu_2\circ{\Phi}at\nu_3$.
\end{examples}
\begin{remark} As of this moment, we do not know of any examples of
extensions ${\Phi}at\nu_-$ which are not evenly minimal. Thus, formally,
the question of whether every extension ${\Phi}at\nu_-$ is evenly minimal
is open, though we strongly suspect that counterexamples do
exist.
\end{remark}
\begin{proposition}\label{resmin} Let $i=2\ell$ be even and let
$\nu^\dag_{i0}$ be the extension of $\nu_\ell$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_{i-1})$,
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, defined by
(\ref{eq:valideal}).
Then
\begin{equation}
k_{\nu^\dag_{i0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde
H'_i).\label{eq:resfield}
\end{equation}
\end{proposition}
\begin{proof} Take two elements
$x,y\in\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag_{\tilde
H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde H'_i}}$, such that
$\nu_{i0}^\dag(x)=\nu_{i0}^\dag(y)$. We must show that the image of
$\frac xy$ in $k_{\nu^\dag_{i0}}$ belongs to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde
H'_i)$. Without loss of generality, we may assume that
$x,y\in\frac{R^\dag_{\tilde H_i}}{\tilde H_{i-1}{R}^\dag_{\tilde
H_i}}$. Let $\beta=\nu_{i0}^\dag(x)=\nu_{i0}^\dag(y)$. Choose
$R'\in\cal T$ sufficiently far out in the direct system so that
$x,y\in\frac{\mathcal{P}'_\beta{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}$. Let $R'\rightarrow R''$ be the
blowing up of the ideal $\mathcal{P}'_\beta R'$. Then in
$\frac{\mathcal{P}''_\beta{R''}^\dag_{\tilde H''_i}}{\tilde
H''_{i-1}{R''}^\dag_{\tilde H''_i}}$ we can write
\begin{eqnarray}
x&=&az\qquad\text{ and }\label{eq:xaz}\\
y&=&aw,
\end{eqnarray}
where $\nu_{i0}^\dag(a)=\beta$ and
$\nu_{i0}^\dag(z)=\nu_{i0}^\dag(w)=0$. Let $\bar z$ be the image of
$z$ in $\kappa(\tilde H''_i)$ and similarly for $\bar w$. Then the
image of $\frac xy$ in $k_{\nu^\dag_{i0}}$ equals $\frac{\bar z}{\bar
w}\in\kappa(\tilde H''_i)$, and the result is proved.
\end{proof}
\begin{remark}\label{minimalexist} Theorem \ref{classification} and the existence of the
extension $\nu^\dag_{2\ell,0}$ of $\nu_\ell$ in the case when
$\tilde H'_{2\ell}=H'_{2\ell}$ and $\tilde
H'_{2\ell-1}=H'_{2\ell-1}$ guaranteed by Theorem \ref{primality1}
(2) allow us to give a fairly explicit description of the totality
of minimal extensions as compositions of $2r$ valuations and, in
particular, to show that they always exist. Indeed, minimal
extensions $\nu^\dag_-$ can be constructed at will, recursively in
$i$, as follows. Assume that the valuations
$\nu^\dag_1,\dots,\nu^\dag_{i-1}$ are already constructed. If $i$ is
odd, let $\nu^\dag_i$ be an arbitrary valuation of the residue field
$k_{\nu^\dag_{i-1}}$ of the valuation ring $R_{\nu^\dag_{i-1}}$. If
$i=2\ell$ is even, let $\nu^\dag_{i0}$ be the extension of $\nu_\ell$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$,
centered at the local ring
$\lim\limits_{\overset\longrightarrow{R'}}
\frac{{R'}^\dag_{H'_i}}{H'_{i-1}{R'}^\dag_{H'_i}}$, whose existence
and uniqueness are guaranteed by Theorem \ref{primality1} (2) and
Proposition \ref{nu0unique}, respectively. Let $\nu^\dag_i$ be an
arbitrary extension of $\nu^\dag_{i0}$ to the field
$k_{\nu^\dag_{i-1}}$. It is clear that all the minimal extensions
$\nu^\dag_-$ of $\nu$ are obtained in this way.
In the next section we will use this remark to show that if
$R^\dag=\tilde R$ or $R^\dag=R^e$ then $\nu$ admits a unique extension
to $\frac{R^\dag}{H_0}$, which is necessarily minimal.
\end{remark}
We end this section by giving some sufficient conditions for the
uniqueness of $\nu^\dag_-$.
\begin{proposition}\label{uniqueness1} Suppose given an admissible chain of
trees (\ref{eq:chaintree''}) of prime ideals of ${R'}^\dag$. For
each $\ell\in\{0,\dots,r-1\}$, consider the set of all $R'\in\cal T$ such that
\begin{equation}
ht\ \tilde H'_{2\ell+1}- ht\ \tilde H'_{2\ell} \le 1\qquad\text{ for
all even }i\label{eq:odd=even1}
\end{equation}
and, in case of equality, the 1-dimensional local ring $\frac{{R'}^\dag_{\tilde
H'_{2\ell+1}}}{\tilde H'_{2\ell}{R'}^\dag_{\tilde H'_{2\ell+1}}}$
is unibranch (that is, analytically irreducible). Assume that for each
$\ell$ the set of such $R'$ is cofinal in $\cal T$.
Let $\nu^\dag_-$ be an extension of $\nu$ such that the $\tilde H'_i$
are prime $\nu^\dag_-$-ideals. Assume that $\nu^\dag_-$ is evenly
minimal. Then there is at most one such extension $\nu^\dag_-$ and
exactly one such $\nu^\dag_-$ if
\begin{equation}
\tilde H'_i=H'_i\quad\text{ for all }i.\label{eq:tildeH=H}
\end{equation}
(in the latter case $\nu^\dag_-$ is minimal by definition).
\end{proposition}
\begin{proof} By Theorem
\ref{primality1} (2) and Proposition \ref{nu0unique}, if
(\ref{eq:tildeH=H}) holds then $\nu^\dag_-$ is minimal and for each even
$i$ the extension $\nu^\dag_{i0}$ exists and is unique. Therefore we
may assume that in all the cases $\nu^\dag_-$ is evenly minimal and that
$\nu^\dag_{i0}$ exists whenever (\ref{eq:tildeH=H}) holds.
The valuation $\nu^\dag_-$, if it exists, is a composition of $2r$ valuations:
$\nu^\dag_-=\nu^\dag_1\circ\nu^\dag_2\circ\dots\circ\nu^\dag_{2r}$,
subject to the conditions of Theorem \ref{classification}. We prove
the uniqueness of $\nu^\dag_-$ by induction on $r$. Assume the result is
true for $r-1$. This means that there is at most one evenly minimal
extension $\nu^\dag_3\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$ of
$\nu_2\circ\nu_3\circ\dots\circ\nu_r$ to
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_2)$, and
exactly one in the case when (\ref{eq:tildeH=H}) holds. To complete
the proof of uniqueness of $\nu^\dag_-$, it is sufficient to show that
both $\nu_1^\dag$ and $\nu_2^\dag$ are unique and that the residue
field of $\nu_2^\dag$ equals
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_2)$.
We start with the uniqueness of $\nu_1^\dag$. If (\ref{eq:odd=even2})
holds then $\nu_1^\dag$ is the trivial valuation. Suppose, on the
other hand, that equality holds in (\ref{eq:odd=even1}). Then the
restriction of $\nu_1^\dag$ to each $R'\in\cal T$ such that the local
ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'_{\tilde
H'_1}}{\tilde H'_0{\Phi}at R'_{\tilde H'_1}}$ is one-dimensional and
unibranch is the unique divisorial valuation centered in that ring (in
particular, its residue field is $\kappa(\tilde H'_1)$). By the assumed
cofinality of such $R'$, the valuation $\nu^\dag_1$ is unique and its
residue field equals
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde H'_1)$. Thus,
regardless of whether or not the inequality in (\ref{eq:odd=even1}) is
strict, $\nu^\dag_1$ is unique and we have the equality of residue
fields
\begin{equation}
k_{\nu_1^\dag}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(\tilde
H'_1)\label{eq:eqres}
\end{equation}
This equality implies that $\nu_2^\dag=\nu^\dag_{20}$. Now, the valuation
$\nu^\dag_2=\nu^\dag_{20}$ is uniquely determined by the conditions
(\ref{eq:groupequal}) and (\ref{eq:valideal}), and its residue field is
\begin{equation}
k_{\nu^\dag_2}=k_{\nu^\dag_{20}}=\lim\limits_{\overset\longrightarrow{R'}}
\kappa(\tilde H'_2).\label{eq:resfield1}
\end{equation}
by Proposition \ref{resmin}. Furthermore, by Theorem \ref{primality1}
exactly one such $\nu_2^\dag$ exists whenever (\ref{eq:tildeH=H})
holds. This proves that there is at most one possibility for
$\nu^\dag_-$: the composition of $\nu_1^\dag\circ\nu^\dag_2$ with
$\nu^\dag_3\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$, and exactly
one if (\ref{eq:tildeH=H}) holds.
\end{proof}
\begin{proposition}\label{tight=scalewise} The extension $\nu^\dag_-$ is tight if and only if for each $R'$ in our direct
system the natural graded algebra extension $\mbox{gr}_\nu
R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational.
\end{proposition}
\begin{remark}\label{rephrasing} Proposition \ref{tight=scalewise} allows us
to rephrase Conjecture \ref{teissier} as follows: the valuation $\nu$
admits at least one tight extension $\nu^\dag_-$.
\end{remark}
\begin{proof}\textit{(of Proposition \ref{tight=scalewise})} ``If'' Assume that for each $R'$ in our direct
system the natural graded algebra extension $\mbox{gr}_\nu
R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational. Then
\begin{equation}\label{eq:gamma=dag}
\mbox{l{\Phi}space{-.47em}G}amma^\dag=\mbox{l{\Phi}space{-.47em}G}amma.
\end{equation}
Together with (\ref{eq:Delta}) this implies that for each
$l\in\{1,\dots,r+1\}$ we have
$\Delta^\dag_{2\ell-2}=\Delta^\dag_{2\ell-1}=\Delta_{\ell-1}$ under
the identification (\ref{eq:gamma=dag}). Then
$\frac{\Delta^\dag_{2\ell-1}}{\Delta^\dag_{2\ell}}=(0)$, so for all
odd $i$ the valuation $\nu^\dag_i$ is trivial. This proves the
equality (\ref{eq:odd=even2}) in the definition of tight. It remains
to show that $\nu^\dag_-$ is evenly minimal. We will prove the even
minimality in the form of equality (\ref{eq:equalgraded}) for each
$\bar\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$. The right hand
side of (\ref{eq:equalgraded}) is trivially contained in the left
hand side; we must prove the opposite inclusion. To do that, take a
non-zero element
$x\in\frac{\mathcal{P}_{\overline\beta}^\dag}{\mathcal{P}_{\overline\beta+}^\dag}$. By
scalewise birationaliy, there exist non-zero elements $\bar y,\bar
z\in\mbox{gr}_\nu R$, with $ord\ \bar y,ord\ \bar z\in\Delta_\ell$, such
that $x\bar y=\bar z$. Let $y$ be a representative of $\bar y$ in
$R$, and similarly for $z$. Let $R\rightarrow R'$ be the local
blowing up with respect to $\nu$ along the ideal $(y,z)$. Then, in
the notation of (\ref{eq:equalgraded}), we have
\begin{equation}\label{eq:equalgraded1}
x={\psi'}^{-1}\left(\left(\phi'\circ\lambda'\right)\left(\frac{\bar z}{\bar y}\otimes_{R'}1\right)\right)\in{\psi'}^{-1}\left((\phi'\circ\lambda')\left(\frac{\mathcal{P}'_{\overline\beta}}{\mathcal{P}'_{\overline\beta+}}\otimes_{R'}\kappa\left(\tilde H'_i\right)\right)\right).
\end{equation}
This proves (\ref{eq:equalgraded}). ``If'' is proved.
``Only if''. Assume that $\nu^\dag_-$ is tight (that is, it is evenly minimal and
(\ref{eq:odd=even2}) holds) and take $R'\in\cal T$. Then the valuation
$\nu_{2\ell+1}$ is trivial for all $\ell$, so
$\nu^\dag_-=\nu^\dag_2\circ\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}$. We
must show that the graded algebra extension $\mbox{gr}_\nu
R'\rightarrow\mbox{gr}_{\nu^\dag_-}{R'}^\dag$ is scalewise birational. Again,
we use induction on $r$. Take an element $x\in{R'}^\dag$. If
$\nu^\dag_-(x)\in\Delta_1$ then
$\mbox{in}_{\nu^\dag_-}x\in\mbox{gr}_{\nu^\dag_4\circ\dots\circ\nu^\dag_{2r}}
\frac{{R'}^\dag}{\tilde H'_2}$, hence by the induction assumption
there exists $y\in R'$ with $\nu^\dag_-(x)\in\Delta_1$ and
$\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu\frac{R'}{P_1}$. In this case, there is
nothing more to prove. Thus we may assume that
$\nu^\dag_-(x)\mbox{$\in$ {\Phi}space{-.8em}/}\Delta_1$. It remains to show that there exists
$y\in R'$ such that $\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu R'$. Since the natural map sending each element of the ring to its image in the graded algebra behaves well with respect to multiplication and division, local
blowings up induce birational transformations of graded algebras, and it
is enough to find a local blowing up $R''\in{\cal T}(R')$ and $y\in
R''$ such that $\mbox{in}_{\nu^\dag_-}(xy)\in\mbox{gr}_\nu R''$.
Now, Proposition \ref{resmin} shows that there exists a local blowing
up $R'\rightarrow R''$ such that $x=az$ (\ref{eq:xaz}), with $z\in
R''$ and $\nu^\dag_2(a)=\nu^\dag_{2,0}(a)=0$. The last equality means
that $\nu^\dag_-(a)\in\Delta_1$, and the result follows from the
induction assumption, applied to $a$.
\end{proof}
The argument above also shows the following. Let
${\mathcal{P}hi'}^\dag=\nu^\dag_-\left({R'}^\dag\setminus\{0\}\right)$, take an
element $\beta\in{\mathcal{P}hi'}^\dag$ and let ${\cal P'}^\dag_\beta$ denote
the $\nu^\dag_-$-ideal of ${R'}^\dag$ of value $\beta$.
\begin{corollary}\label{blup1} Take an element $x\in{\cal P'}^\dag_\beta$. There
exists a local blowing up $R'\rightarrow R''$ such that
$\beta\in\nu(R'')\setminus\{0\}$ and $x\in{\cal P}''_\beta{R''}^\dag$.
\end{corollary}
The next Proposition gives a sufficient condition for the uniqueness
of $\nu^\dag_-$ (this result is due to Heinzer and Sally \cite{HeSa}).
\begin{proposition}\label{HeSal} Assume that $K'$ is algebraic over $K$ for
all $R'\in\cal T$ and that the following conditions hold:
(1) $ht\ H'_1\le1$
(2) $ht\ H'_1+rat.rk\ \nu=\dim\ R'$, where $R'$ is taken to be
sufficiently far out in the direct system.
Let $\nu^\dag_-$ be an extension of $\nu$ to a ring of the form
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$. Then either
\begin{eqnarray}
\tilde H'_0&=&H'_0\qquad\text{ or }\label{eq:tildeH=H0}\\
\tilde H'_0&=&H'_1.\label{eq:tildeH=H1}
\end{eqnarray}
The valuation $\nu$ admits a unique extension to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{H'_0}$ and
a unique extension to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{H'_1}$. The
first extension is minimal and the second is tight.
\end{proposition}
\begin{proof} For $1\le\ell\le r$, let $r_\ell$ denote the rational
rank of $\nu_\ell$. Let $\nu^\dag_-$ be an extension of $\nu$ to a ring
of the form
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$, where $\tilde H'_0$ is a tree of prime ideals of ${R'}^\dag$
such that $\tilde H'_0\cap R'=(0)$.
By Corollary \ref{htstable1} $ht\ H'_i$ stabilizes for $1\le i\le 2r$
and $R'$ sufficiently far out in the direct system. From now on, we
will assume that $R'$ is chosen sufficiently far so that the stable
value of $ht\ H'_i$ is attained. Now, let $i=2\ell$. The valuation
$\nu^\dag_{i0}$ is centered in the local noetherian ring
$\frac{{R'}^\dag_{\tilde H'_i}}{\tilde H'_{i-1}{R'}^\dag_{\tilde
H'_i}}$, hence by Abhyankar's inequality
\begin{equation}
rat.rk\ \nu^\dag_{i0}\le\dim\frac{{R'}^\dag_{\tilde H'_i}}{\tilde
H'_{i-1}{R'}^\dag_{\tilde H'_i}}\le ht\ \tilde H'_i-ht\ \tilde
H'_{i-1}.\label{eq:abhyankar}
\end{equation}
Since this inequality is true for all even $i$, summing over all $i$ we obtain:
\begin{equation}
\begin{array}{rcl}
\dim R'&=&\dim\ {R'}^\dag=\sum\limits_{i=1}^{2r}(ht\ \tilde H'_i-ht\
\tilde H'_{i-1}){\Gamma}e ht\ \tilde H'_1+\sum\limits_{\ell=1}^r(ht\ \tilde
H'_{2\ell}-ht\ \tilde H'_{2\ell-1}){\Gamma}e\\
&{\Gamma}e&ht\ H'_1+\sum\limits_{\ell=1}^rrat.rk\ \nu^\dag_{2\ell,0}{\Gamma}e ht\
H'_1+\sum\limits_{\ell=1}^rr_\ell=ht\ H'_1+rat.rk\ \nu=\dim
R'.\label{eq:inequalities}
\end{array}
\end{equation}
Hence all the inequalities in (\ref{eq:abhyankar}) and
(\ref{eq:inequalities}) are equalities. In particular, we
have
$$
ht\ \tilde H'_1=ht\ H'_1;
$$
combined with Proposition \ref{Hintilde} this shows that
\begin{equation}
\tilde H'_1=H'_1.
\end{equation}
Together with the hypothesis (1) of the Proposition, this already
proves that at least one of (\ref{eq:tildeH=H0})--(\ref{eq:tildeH=H1})
holds. Furthermore, equalities in (\ref{eq:abhyankar}) and
(\ref{eq:inequalities}) prove that
$$
ht\ \tilde H'_i=ht\ \tilde H'_{i-1}
$$
for all odd $i>1$, so that
\begin{equation}
\tilde H'_i=\tilde H'_{i-1}\qquad\text{whenever $i>1$ is
odd}\label{eq:oddiseven}
\end{equation}
and that
\begin{equation}
r_i=ht\ \tilde H'_i-ht\ \tilde H'_{i-1}\label{heightfixed}
\end{equation}
whenever $i$ is even.
Now, consider the special case when $\tilde H'_i=H'_i$ for $i{\Gamma}e1$ and
$\tilde H'_0$ is as in
(\ref{eq:tildeH=H0})--(\ref{eq:tildeH=H1}). According to Proposition
\ref{nu0unique} for each even $i=2\ell$ there exists a unique
extension $\nu^\dag_{i0}$ of $\nu_l$ to a valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{i-1})$, centered
in the local ring
$\lim\limits_{\overset\longrightarrow{R'}}
\frac{{R'}^\dag_{H'_{2\ell}}}{H'_{2\ell-1}}$. Moreover, we have
\begin{equation}
k_{\nu^\dag_{2\ell,0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})
\label{eq:knudag2l}
\end{equation}
by Remark \ref{sameresfield}. By Theorem \ref{classification}, there
exists an extension $\nu^\dag_-$ of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{R'}^\dag}{\tilde
H'_0}$ such that the $\{\tilde H'_i\}$ as above is the chain of
trees of prime ideals, determined by $\nu^\dag_-$. In particular,
(\ref{eq:oddiseven}) and (\ref{heightfixed}) hold with $\tilde H'_i$
replaced by $H'_i$.
Now (\ref{heightfixed}) and Proposition \ref{Hintilde} imply that for
\textit{any} extension $\nu^\dag_-$ we have $\tilde H'_i=H'_i$ for
$i>0$, so that the special case above is, in fact, the only case
possible. Furthermore, by (\ref{eq:oddiseven}) we have
$H'_{2\ell+1}=H'_{2\ell}$ for all $\ell\in\{1,\dots,r\}$. This implies
that for all such $\ell$ the valuation $\nu^\dag_{2\ell+1,0}$ is the
trivial valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})$; in particular,
\begin{equation}
k_{\nu^\dag_{2\ell+1,0}}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_{2\ell})
\label{eq:knudag2l+1}
\end{equation}
for all $\ell\in\{1,\dots,r-1\}$.
If $\tilde H'_0=H'_1=\tilde H'_1$ then the only possibility for
$\nu^\dag_{10}=\nu^\dag_1$ is the trivial valuation of
$\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_1)$; we have
\begin{equation}
k_{\nu^\dag_1}=k_{\nu^\dag_{10}}=\lim\limits_{\overset\longrightarrow{R'}}
\kappa(H'_1).\label{eq:knudag10}
\end{equation}
If $\tilde H'_0=H'_0$ then by the hypothesis (1) of the Proposition
and the excellence of $R$ the ring $\frac{{R'}^\dag_{H'_1}}{H'_0{R'}^\dag_{H'_1}}$ is
a regular one-dimensional local ring (in particular, unibranch), hence
the valuation $\nu^\dag_1=\nu^\dag_{10}$ centered at
$\lim\limits_{\overset\longrightarrow{R'}}
\frac{{R'}^\dag_{H'_1}}{H'_0{R'}^\dag_{H'_1}}$ is unique and
(\ref{eq:knudag10}) holds also in this case.
By induction on $i$, it follows from (\ref{eq:knudag2l}),
(\ref{eq:knudag2l+1}), the uniqueness of $\nu^\dag_{2\ell,0}$ and the
triviality of $\nu^\dag_{2\ell+1,0}$ for $\ell{\Gamma}e1$ that $\nu_i^\dag$
is uniquely determined for all $i$ and
$k_{\nu^\dag_i}=\lim\limits_{\overset\longrightarrow{R'}}\kappa(H'_i)$. This
proves that in both cases (\ref{eq:tildeH=H0}) and
(\ref{eq:tildeH=H1}) the valuation
$\nu^\dag_-=\nu^\dag_1\circ\dots\circ\nu^\dag_{2r}$ is unique. The last
statement of the Proposition is immediate from definitions.
\end{proof}
A related necessary condition for the uniqueness of $\nu^\dag_-$ will be
proved in \S\ref{locuni1}.
\section{Extending a valuation centered in an excellent local domain
to its henselization.}
\label{henselization}
Let $\tilde R$ denote the henselization of $R$, as above. The
completion homomorphism $R\rightarrow{\Phi}at R$ factors through the
henselization: $R\rightarrow\tilde R\rightarrow{\Phi}at R$. In this
section, we will show that $H_1$ is a minimal prime of $\tilde R$,
that $\nu$ extends uniquely to a valuation $\tilde\nu_-$ of rank $r$
centered at $\frac{\tilde R}{H_1}$, and that $H_1$ is the unique
prime ideal $P$ of $\tilde R$ such that $\nu$ extends to a valuation
of $\frac{\tilde R}P$. Furthermore, we will prove that $H_{2\ell+1}$
is a minimal prime of $P_\ell\tilde R$ for all $\ell$ and that these
are precisely the prime $\tilde\nu$-ideals of $\tilde R$.
Studying the implicit prime ideals of $\tilde R$ and the extension of $\nu$ to
$\tilde R$ is a logical intermediate step before attacking the formal
completion, for the following reason. As we will show in the next section, if
$R$ is already henselian in (\ref{eq:defin1}) then $\mathcal{P}'_\beta{\Phi}at
R'_{H'_{2\ell+1}}\cap{\Phi}at R=\mathcal{P}_\beta{\Phi}at R$ for all $\beta$ and $R'$
and thus we have
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left({\cal
P}_\beta{\Phi}at R\right)$.
We state the main result of this section. In the case when $R^e$ is an
\'etale extension of $R$, contained in $\tilde R$, we use
(\ref{eq:defin3}) with $R^\dag=R^e$ as our definition of the implicit
prime ideals.
\begin{theorem}\label{hensel0} Let $R^e$ be a local \'etale extension of
$R$, contained in $\tilde R$. Then:
(1) The ideal $H_{2\ell+1}$ is prime for $0\le l\le r$; it is a minimal prime of
$P_\ell R^e$. In particular, $H_1$ is a minimal prime of $R^e$. We
have $H_{2\ell}=H_{2\ell+1}$ for $0\le l\le r$.
(2) The ideal $H_1$ is the unique prime $P$ of $R^e$ such that there
exists an extension $\nu^e_-$ of $\nu$ to $\frac{R^e}P $; the
extension $\nu^e_-$ is unique. The graded algebra
$\mbox{gr}_{\nu^e_-}\frac{R^e}{H_1}$ is scalewise birational to $\mbox{gr}_\nu
R$; in particular, $rk\ \nu^e_-=r$.
(3) The ideals $H_{2\ell+1}$ are precisely the prime $\nu^e_-$-ideals of
$R^e$.
\end{theorem}
\begin{proof} By assumption, the ring $R^e$ is a direct limit of
local, strict \'etale extensions of $R$ which are essentially of finite
type. All the assertions (1)--(3) behave well under taking direct limits, so it
is sufficient to prove the Theorem in the case when $R^e$ is essentially of
finite type over $R$. From now on, we will restrict attention to this case.
The next step is to describe explicitly those local blowings up
$R\rightarrow R'$ for which $R'$ is $\ell$-stable. Their interest to us is that, according to Proposition \ref{largeR2},
if $R'$ is $\ell$-stable then for all $R''\in{\cal T}(R')$ and all $\beta\in\frac{\Delta_\ell}{\Delta_{\ell+1}}$, we have the
equality
\begin{equation}
{\cal P}''_\beta(R''\otimes_RR^e)\cap R^e={\cal P}_\beta R^e;\label{eq:contracts}
\end{equation}
in particular, the limit in (\ref{eq:defin3}) is attained, that is, we have the equality
\begin{equation}
H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_\ell}\left(\left({\cal P}'_\beta\left(R^e\otimes_RR'\right)_{M'}\right)
\bigcap R^e\right).\label{eq:defin4}
\end{equation}
\begin{lemma}\label{lift} Let $\frac{R}{P_\ell}\to T$ be a finitely generated extension of $\frac {R}{P_\ell}$, contained in
$\frac{R_\nu}{\bf m_\ell}$. Let
$$
{\bf q}=\frac{\bf m_\nu}{\bf m_\ell}\cap T.
$$
There exists a $\nu$-extension $R\to R'$ of $R$ such that $\frac{R'}{P'_\ell}=T_{\bf q}$.
\end{lemma}
\begin{proof} Write $T=\frac R{P_\ell}\left[\overline a_1,\ldots,\overline
a_k\right]$, with $\overline a_i\in\frac{R_\nu}{\bf m_\ell}$, that
is, $\nu_{\ell+1}\left(\overline a_i\right){\Gamma}eq 0,\ 1\leq i\leq
k$. We can lift the $\overline a_i$ to elements $a_i$ in $R_\nu$
such that $\nu\left(a_i\right){\Gamma}eq0$. Let us consider the ring
$R''=R\left[a_1,\ldots,a_k\right]\subset R_\nu$ and its
localization $R'=R''_{{\bf m}_\nu\cap R''}$. The ideal $P'_\ell$
is the kernel of the natural map $R'\rightarrow\frac{R_\nu}{\bf
m_\ell}$. Thus both $\frac{R'}{P'_\ell}$ and $T_{\bf q}$ are equal
to the $\frac R{P_l}$-subalgebra of $\frac{R_\nu}{\bf m_\ell}$,
obtained by adjoining $\overline a_1,\ldots,\overline a_k$ to
$\frac R{P_l}$ inside $\frac{R_\nu}{\bf m_\ell}$ and then
localizing at the preimage of the ideal $\frac{\bf m_\nu}{\bf
m_\ell}$. This proves the Lemma.
\end{proof}
Let us now go back to our \'etale extension $R\to R^e$.
\begin{lemma}\label{anirred1} Fix an integer $l\in\{0,\dots,r\}$. There exists a
local blowing up $R\rightarrow R'$ along $\nu$ having the following
property: let $P'_\ell$ denote the $\ell$-th prime $\nu$-ideal of $R'$. Then
the ring $\frac{R'}{P'_\ell}$ is analytically irreducible; in particular, $\frac{R'}{P'_\ell}\otimes_R R^e$ is an integral
domain.
\end{lemma}
\begin{remark} We are not claiming that there exists $R'\in\cal T$ such that $\frac{R'}{P'_\ell}$ is analytically
irreducible for all $\ell$ (and we do not know how to prove such a
claim), only that for each $\ell$ there exists an $R'$, which may
depend on $\ell$, such that $\frac{R'}{P'_\ell}$ is analytically
irreducible. On the other hand, below we will prove that there exists an $\ell$-stable $R'\in\cal T$. According to
Definition \ref{stable} (2) and Proposition \ref{largeR1}, such a stable $R'$ has the property that
$\kappa\left(P''_\ell\right)\otimes_R\left(R''\otimes_RR^e\right)_{M''}$ is a domain for all $R''\in{\cal T}(R')$. For a given $R''$, this property is weaker than the analytic irreducibility of $R''/P''_\ell$. The latter is equivalent to saying that
$\kappa(P''_\ell)\otimes_R(R''\otimes_RR^\sharp)_{M''}$ is a domain for every local \'etale extension $R^\sharp$ of $R''$.
\end{remark}
\begin{proof}\textit{(of Lemma \ref{anirred1})} Since $R$ is an excellent local ring, every homomorphic image of $R$ is Nagata
\cite{Mat} (Theorems 72 (31.H), 76 (33.D) and 78 (33.H)). Let $\pi:\frac R{P_\ell}\rightarrow S$ be the normalization of
$\frac {R}{P_\ell}$. Then $S$ is a finitely generated $\frac {R}{P_\ell}$-algebra contained in $\frac{R_\nu}{\bf m_\ell}$,
to which we can apply Lemma \ref{lift}. We obtain a $\nu$-extension $R\to R'$ such that the ring
$\frac{R'}{P'_\ell}\cong\frac{R'}{P_\ell R'}$ is a localization of $S$ at a prime ideal, hence it is an excellent normal
local ring. In particular, it is analytically irreducible (\cite{Nag}, Theorem (43.20), p. 187 and Corollary (44.3), p. 189), as desired.
\end{proof}
Next, we fix $\ell\in\{0,\dots,r\}$ and study the ring $(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$, in particular, the structure of
the set of its zero divisors, as $R'$ runs over ${\cal T}(R)$ (here $T'$ is as in Remark \ref{interchanging}). Since $R^e$
is separable algebraic, essentially of finite type over $R$, the ring $(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$ is finite over
$\kappa(P'_\ell)$; this ring is reduced, but it may contain zero
divisors. In fact, it is a direct product of fields which are finite separable extensions of $\kappa(P'_\ell)$ because $R^e$
is separable and essentially of finite type over $R$.\par
Consider a chain $R\rightarrow R'\rightarrow R''$ of $\nu$-extensions in $\cal T$. Let
\begin{eqnarray}
\kappa(P_\ell)\otimes_RR^e&=&\prod\limits_{j=1}^nK_j\\
(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)&=&\prod\limits_{j=1}^{n'}K'_j\\
(T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)&=&\prod\limits_{j=1}^{n''}K''_j
\end{eqnarray}
be the corresponding decompositions as products of finite field extensions of $\kappa(P_\ell)$ (resp. $\kappa(P'_\ell)$,
resp. $\kappa(P''_\ell)$). We want to compare $(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)$ with
$(T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)$.
\begin{remark} The ring $\kappa\left(P'_\ell\right)\otimes_RR^e$ is itself a direct product of finite extensions of
$\kappa\left(P'_\ell\right)$; say $\kappa\left(P'_\ell\right)=\prod\limits_{j\in S'}K'_j$ for a certain set $S'$. In this
situation, localization is the same thing as the natural projection to the product of the $K'_j$ over a certain subset
$\{1,\dots,n'\}$ of $S'$. Thus the passage from $(T')^{-1}\left(\kappa\left(P'_\ell\right)\otimes_RR^e\right)$ to
$(T'')^{-1}\left(\kappa\left(P''_\ell\right)\otimes_RR^e\right)$ can be viewed as follows: first, tensor each $K'_j$ with
$\kappa\left(P''_\ell\right)$ over $\kappa\left(P'_\ell\right)$; then, in the resulting direct product of fields, remove
a certain number of factors.
\end{remark}
Let $\bar K'_1,\dots,\bar K'_{\bar n'}$ be the distinct isomorphism classes of finite extensions of
$\kappa\left(P'_\ell\right)$ appearing among $K'_1,\dots,K'_{n'}$, arranged in such a way that
$\left[\bar K'_j:\kappa\left(P'_\ell\right)\right]$ is non-increasing with $j$, and similarly for
$\bar K''_1,\dots,\bar K''_{\bar n''}$.
\begin{lemma}\label{decrease} We have the inequality
\begin{equation}
\left(\left[\bar K''_1:\kappa\left(P''_\ell\right)\right],\dots,\left[\bar K''_{\bar n''}:\kappa\left(P''_\ell\right)\right],
n''\right)\le
\left(\left[\bar K'_1:\kappa\left(P'_\ell\right)\right],\dots,\left[\bar K'_{\bar n'}:\kappa\left(P'_\ell\right)\right],
n'\right)\label{eq:lex}
\end{equation}
in the lexicographical ordering. Furthermore, either $R'$ is $\ell$-stable or there exists $R''\in\cal T$ such that strict
inequality holds in (\ref{eq:lex}).
\end{lemma}
\begin{proof} Fix a $q\in\{1,\dots,\bar n'\}$ and consider the tensor product $\bar
K'_q\otimes_R\kappa\left(P''_\ell\right)$. Since $\bar K'_q$ is separable over $\kappa\left(P'_\ell\right)$, the ring
$\bar K'_q\otimes_R\kappa\left(P''_\ell\right)=\prod\limits_{j\in S''_q}K''_j$ is a product of fields. Moreover, two cases
are possible:
(a) there exists a non-trivial extension $L$ of $\kappa\left(P'_\ell\right)$ which embeds both into
$\kappa\left(P''_\ell\right)$ and $\bar K'_q$. In this case
\begin{equation}
\left[K''_j:\kappa\left(P''_\ell\right)\right]<\left[\bar K'_q:\kappa\left(P'_\ell\right)\right]\quad\text{ for all }j\in
S''_q.\label{eq:strict}
\end{equation}
(b) there is no field extension $L$ as in (a). In this case $\bar K'_q\otimes_R\kappa\left(P''_\ell\right)$ is a field,
so
\begin{equation}
\#S''_q=1\label{eq:card1}
\end{equation}
and
\begin{equation}
\left[K''_j:\kappa\left(P''_\ell\right)\right]=\left[\bar K'_q:\kappa\left(P'_\ell\right)\right]\quad\text{ for }j\in
S''_q.\label{eq:equal}
\end{equation}
Now, if there exists $q\in\{1,\dots,\bar n'\}$ for which (a) holds, take the smallest such $q$. Then
(\ref{eq:strict})--(\ref{eq:equal}) imply that strict inequality holds in (\ref{eq:lex}). On the other hand, if (b) holds
for all $q\in\{1,\dots,\bar n'\}$ then (\ref{eq:card1}) and (\ref{eq:equal}) imply that
\begin{equation}
\left(\left[\bar K''_1:\kappa\left(P''_\ell\right)\right],\dots,\left[\bar K''_{\bar n''}:\kappa\left(P''_\ell\right)\right]
\right)=
\left(\left[\bar K'_1:\kappa\left(P'_\ell\right)\right],\dots,\left[\bar K'_{\bar n'}:\kappa\left(P'_\ell\right)\right]
\right)\label{eq:lex1}
\end{equation}
and $n''\le n'$, so again (\ref{eq:lex}) holds.
Finally, assume that $R'$ is not $\ell$-stable. If there exists
$R''\in\cal T$ and $q\in\{1,\dots,\bar n'\}$ for which (a) holds, then by the above we have strict inequality in
(\ref{eq:lex}) and there is nothing more to prove. Assume there are no such $R''$ and $q$. Then
$(T')^{-1}(\kappa(P'_\ell)\otimes_RR^e)$ is not a domain, so $n'>1$.
Take $R''\in{\cal T}(R')$ such that $\left(\frac{R''}{P''_l}\otimes_RR^e\right)_{M''}$ is an integral domain; such an $R''$
exists by Lemma \ref{anirred1}. Then $n''=1<n'$, as desired.
\end{proof}
\begin{corollary}\label{anirred2} There exists a stable $R'\in\cal T$. The limit in
(\ref{eq:defin3}) is attained for this $R'$.
\end{corollary}
\begin{proof} In view of Proposition \ref{largeR1}, it is sufficient to prove that there exists $R'\in\cal T$
which which is $\ell$-stable for all $\ell\in\{0,1,\dots,r\}$. First, we fix
$\ell\in\{0,1,\dots,r\}$. Lemma \ref{decrease} implies that there exists $R'\in\mathcal{T}(R)$ which is $\ell$-stable.
By Proposition \ref{largeR1}, repeating the procedure above for each $\ell$ we can successively enlarge $R'$ in such a
way that it becomes stable.
The last statement follows from Proposition \ref{largeR2}.
\end{proof}
We are now in the position to prove Theorem \ref{hensel0}.
By Theorem \ref{primality1} (1), $H_{2\ell-1}$ is prime. By Proposition \ref{contracts}, $H_{2\ell+1}$
maps to $P_\ell$ under the map $\pi^e:\mbox{Spec}\ R^e\rightarrow\mbox{Spec}\ R$.
Since this map is \'etale, its fibers are zero-dimensional, which
shows that $H_{2\ell+1}$ is a minimal prime of $P_\ell$. This proves (1) of Theorem \ref{hensel0}.
By Proposition \ref{Hintilde}, for $0\le i\le2r$, $\tilde H_i$ is a prime ideal of $R^e$, containing
$H_i$. Since the fibers of $\pi^e$ are zero-dimensional, we must have $\tilde H_i=H_i$, so $\tilde H_{2\ell}=\tilde
H_{2\ell+1}=H_{2\ell}=H_{2\ell+1}$ for $0\le\ell\le r$. In particular, $\tilde H_0=H_1$. This shows that the unique prime
$\tilde H_0$ of $R^e$ such that there exists an extension $\nu^e_-$ of $\nu$ to $\frac{R^e}{\tilde H_0}$ is $\tilde H_0=H_1$.
Now (2) of the Theorem is given by Proposition \ref{uniqueness1}.
(3) of Theorem \ref{hensel0} is now immediate. This completes the proof of Theorem \ref{hensel0}.
\end{proof}
We note the following corollary of the proof of (2) of Theorem
\ref{hensel0} and Corollary \ref{blup1}. Let $\mathcal{P}hi^e=\nu^e_-(R^e\setminus\{0\})$, take an element $\beta\in\mathcal{P}hi^e$ and let
${\cal P}^e_\beta$ denote the $\nu^e_-$-ideal of $R^e$ of value $\beta$.
\begin{corollary}\label{blup} Take an element $x\in {\cal P}^e_\beta$. There
exists a local blowing up $R\rightarrow R'$ such that
$\beta\in\nu(R')\setminus\{0\}$ and $x\in {\cal P}'_\beta{R'}^e$.
\end{corollary}
\section{The Main Theorem: the primality of implicit ideals.}
\label{prime}
In this section we study the ideals $H_j$ for ${\Phi}at R$ instead of $\tilde R$. The main result of this section is
\begin{theorem}\label{primality} The ideal $H_{2\ell-1}$ is prime.
\end{theorem}
\begin{proof} For the purposes of this proof, let $H_{2\ell-1}$
denote the implicit ideals of ${\Phi}at R$ and $\tilde H_{2\ell-1}$ the
implicit prime ideals of the henselization $\tilde R$ of $R$.
Let $S$ be a local domain. By \cite{Nag} (Theorem (43.20), p. 187) there exists bijective maps between the set of minimal
prime ideals of the henselization $\tilde S$ and the maximal ideals of the normalization $S^n$. If, in addition, $S$ is excellent, the two above sets also admit a natural bijection to the set of minimal primes of ${\Phi}at S$ \cite{Nag} (Corollary (44.3), p. 189). If $S$ is a henselian local domain, its only minimal prime is the (0) ideal, hence by the above the same is true of ${\Phi}at S$. Thus ${\Phi}at S$ is also a domain.
This shows that any excellent henselian local domain is analytically
irreducible, hence $\tilde H_{2\ell-1}{\Phi}at R$ is prime for all
$\ell\in\{1,\dots,r+1\}$. Let $\tilde\nu_-$ denote the unique
extension of $\nu$ to $\frac{\tilde R}{\tilde H_1}$, constructed in
the previous section. Let $H^*_{2\ell-1}\subset\frac{\tilde
R}{\tilde H_1}$ denote the implicit ideals associated to the
henselian ring $\frac{\tilde R}{\tilde H_1}$ and the valuation
$\tilde\nu_-$.
\noi\textit{Claim.} We have $H^*_{2\ell-1}=\frac{H_{2\ell-1}}{\tilde H_1}$.
\noi\textit{Proof of the claim:} For $\beta\in\mbox{l{\Phi}space{-.47em}G}amma$, let $\tilde
P_\beta$ denote the $\tilde\nu_-$-ideal of $\frac{\tilde R}{\tilde
H_1}$ of value $\beta$. For all $\beta$, we have
$\frac{P_\beta}{\tilde H_1}\subset\tilde P_\beta$, and the same
inclusion holds for all the local blowings up of $R$, hence
$\frac{H_{2\ell-1}}{\tilde H_1}\subset H^*_{2\ell-1}$. To prove the
opposite inclusion, we may replace $\tilde R$ by a finitely
generated strict \'etale extension $R^e$ of $R$. Now let
$\mathcal{P}hi^e=\nu^e_-\left(R^e\setminus\{0\}\right)$ and take an element
$\beta\in\mathcal{P}hi^e\cap\Delta_{\ell -1}$. By Corollary \ref{blup}, there
exists a local blowing up $R\rightarrow R'$ such that $x\in
P'_\beta{R'}^e$. Letting $\beta$ vary over
$\mathcal{P}hi^e\cap\Delta_{\ell-1}$, we obtain that if $x\in H^*_{2\ell-1}$
then $x\in\frac{H_{2\ell-1}}{\tilde H_1}$, as desired. This
completes the proof of the claim.
The Claim shows that replacing $R$ by $\frac{\tilde R}{\tilde H_1}$ in Theorem
\ref{primality} does not change the problem. In other words, we
may assume that $R$ is a henselian domain and, in particular, that
${\Phi}at R$ is also a domain. Similarly, the ring $\frac
R{P_\ell}\otimes_R{\Phi}at R\cong\frac{{\Phi}at R}{P_\ell}$ is a domain, hence
so is its localization $\kappa(P_\ell)\otimes_R{\Phi}at R$.
Since $R$ is a henselian excellent ring, it is algebraically closed
in ${\Phi}at R$ (\cite{Nag}, Corollary (44.3), p. 189 and Corollary
\ref{notnormal} of the Appendix); of course, the same holds for
$\frac R{P_\ell}$ for all $\ell$. Then $\kappa(P_\ell)$ is
algebraically closed in $\kappa(P_\ell)\otimes_R{\Phi}at R$. This shows
that the ring $R$ is stable. Now the Theorem follows from Theorem
\ref{primality1}. This completes the proof of Theorem
\ref{primality}.
\end{proof}
\section{Towards a proof of Conjecture \ref{teissier}, assuming local
uniformization in lower dimension}
\label{locuni1}
Let the notation be as in the previous sections. In this section,
we assume that the Local Uniformization Theorem holds and propose an approach to proving Conjecture \ref{teissier}. We prove a Corollary of Conjecture \ref{teissier} which gives a sufficient condition for ${\Phi}at\nu_-$ to be unique, which also turns out to be necessary under the additional assumption that ${\Phi}at\nu_-$ is minimal. We will assume that all the $R'\in\mathcal T$ are birational to each other, so that all the fraction fields
$K'=K$ and the homomorphisms $R'\rightarrow R''$ are local
blowings up with respect to $\nu$. Finally, we assume that $R$
contains a field $k_0$ and a local $k_0$-subalgebra $S$
essentailly of finite type, over which $R$ is strictly \'etale. In
particular, all the rings in sight are equicharacteristic.
First, we state the Local Uniformization Theorem in the precise form
in which we are going to use it. \begin{definition}\label{lut} We say that
\textbf{the embedded Local Uniformization theorem holds in }
$\mathcal T$ if the following conditions are satisfied.
Take an integer $\ell\in\{1,\dots,r-1\}$. Let
$\mu_{\ell+1}:=\nu_{\ell+1}\circ\nu_{\ell+2}\circ\dots\circ\nu_r$. Consider
a tree $\{H'\}$ of prime ideals of $\frac{{\Phi}at R'}{P'_\ell}$ such that
$H'\cap\frac{R'}{P'_\ell}=(0)$ and a tight extension
${\Phi}at\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'}$.
(1) There exists a local blowing up $\pi:R\rightarrow R'$ in $\mathcal
T$, which induces an isomorphism at the center of $\nu_\ell$, such
that $\frac{R'}{P'_\ell}$ is a regular local ring.
(2) Assume that $\frac{R'}{P'_\ell}$ is a regular local ring. Then
there exists in $\mathcal T$ a sequence $\pi:R\rightarrow R'$ of local
blowings up along non-singular centers not containing the center of
$\nu_l$ such that $\frac{{\Phi}at R'}{H'}$ is a regular local ring.
\end{definition}
It is well known (\cite{A}, \cite{L}, \cite{Z}) that the embedded
Local Uniformization theorem holds if $R$ is an excellent local domain
such that either $char\ k=0$ or $\dim\ R\le3$ (to be precise, (1) of
Definition \ref{lut} is well known and (2) is an easy consequence of
known results). While the Local Uniformization theorem in full
generality is still an open problem, it is widely believed to hold for
arbitrary quasi-excellent local domains. Proving this is an active
field of current research in algebraic geometry. Proving local
uniformization for rings of arbitrary characteristic is one of the
intended applications of Conjecture \ref{teissier}. Note that in Definition \ref{lut} we require only
local uniformization of rings of dimension strictly less than $\dim\
R$; the idea is to use induction on $\dim\ R$ to prove local
uniformization of rings of dimension $\dim\ R$.
We begin by stating a strengthening of Conjecture \ref{teissier} (using Remark
\ref{rephrasing}):
\begin{conjecture}\label{teissier1} The valuation $\nu$ admits at least one
tight extension ${\Phi}at\nu_-$. This tight extension ${\Phi}at\nu_-$ can be
chosen to have the following additional property: for rings $R'$
sufficiently far in the tree $\mathcal T$ we have the equality of
semigroups ${\Phi}at\nu_-\left(\frac{{\Phi}at R'}{\tilde
H'_0}\setminus\{0\}\right)=\nu(R'\setminus\{0\})$ and for
$\beta\in\nu(R'\setminus\{0\})$ the ${\Phi}at\nu_-$-ideal of value $\beta$
is $\frac{\mathcal P_\beta{\Phi}at R'}{\tilde H'_0}$. In particular, we
have the equality of graded algebras $\mbox{gr}_\nu
R'=\mbox{gr}_{{\Phi}at\nu_-}\frac{{\Phi}at R'}{\tilde H'_0}$.
\end{conjecture}
Below, we give an explicit construction of a valuation ${\Phi}at\nu_-$ whose existence is asserted in the Conjecture by describing the trees of ideals $\tilde H'_i$, $0\le i\le 2r$ and, for each $i$, a valuations ${\Phi}at\nu_i$ of the residue field $k_{\nu_{i-1}}$, such that ${\Phi}at\nu_-={\Phi}at\nu_1\circ\dots{\Phi}at\nu_{2r}$. More precisely, for $\ell\in\{0,\dots,r-1\}$, we will construct, recursively in the descending order of $\ell$, a tree $J'_{2\ell+1}$ of prime ideals of $\frac{{\Phi}at R'}{H'_{2\ell}}$, $R'\in\mathcal{T}$, such that
$J'_{2\ell+1}\cap\frac{R'}{P_\ell}=(0)$, and an extension ${\Phi}at\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'\in\mathcal{T}}}\frac{{\Phi}at R'}{J'_{2\ell+1}{\Phi}at R'}$; the valuation
${\Phi}at\mu_2$ will be our candidate for the desired tight extension ${\Phi}at\nu_-$ of $\mu_1=\nu$. Unfortunately, two steps in this construction still remain
conjectural, namely, proving that ${\Phi}at\mu_{2\ell+2}$ is, indeed, a valuation, and that it is tight (this is essentially the content of Conjectures \ref{strongcontainment} and \ref{containment} below). Once these conjectures are proved, our recursive construction will be complete and Conjecture \ref{teissier1} will follow by setting ${\Phi}at\nu_-={\Phi}at\mu_2$.
Let us now describe the construction in detail. According to Corollary \ref{htstable1}, we may
assume that $ht\ H'_i$ is constant for each $i$ after replacing $R$ by
some other ring sufficiently far in $\mathcal T$. From now on, we will
make this assumption without always stating it explicitly.
By (1) of Definition \ref{lut}, applied successively to the trees of ideals
$$
P'_\ell\subset R',\quad\ell\in\{1,\dots,r-1\},
$$
there exists $R''\in\cal T$ such that $\frac{R''}{P''_\ell}$ is regular for all $\ell\in\{1,\dots,r-1\}$. Without loss of generality,
we may also assume that $R''$ is stable.
For $\ell\in\{1,\dots,r-1\}$ and $R''\in\mathcal{T}$, let
$\mathcal{T}_\ell(R'')$ denote the subtree of $\mathcal T$,
consisting of all the local blowings up of $R''$ along ideals not
contained in $P''_\ell$ (such local blowings up induce an
isomorphism at the point $P''_\ell\in\mbox{Spec}\ R''$). Below, we will sometimes
work with trees of rings and ideals indexed by
$\mathcal{T}_\ell(R'')$ for suitable $\ell$ and $R''$ (instead of
trees indexed by all of $\cal T$); the precise tree with which we
are working will be specified in each case.
For $\ell=r-1$, we define $J'_{2r-1}:=H'_{2r-1}$ and ${\Phi}at\mu_{2r}:={\Phi}at\nu_{2r,0}$; according to Proposition
\ref{nu0unique}, ${\Phi}at\nu_{2r,0}={\Phi}at\nu_{2r}$ is the unique extension of $\nu_r$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_{2r-1}{\Phi}at R'}$.
Next, assume that $\ell\in\{1,\dots,r-1\}$, that the tree $J'_{2\ell+1}$ of prime ideals of $\frac{{\Phi}at
R'}{H'_{2\ell}{\Phi}at R'}$ and a tight extension ${\Phi}at\mu_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{J'_{2\ell+1}}$ are already constructed for
$R'\in\mathcal{T}$ and that $J'_{2\ell+1}\cap\frac{R'}{P_\ell}=(0)$. It remains to construct the
ideals $J'_{2\ell-1}\subset\frac{{\Phi}at R'}{H'_{2\ell-2}{\Phi}at R'}$ and a tight extension
${\Phi}at\mu_{2\ell}$ of $\mu_\ell$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{J'_{2\ell-1}}$ for
$R'\in\mathcal{T}$.
We will assume, inductively, that for all $R'\in\mathcal{T}$ the quantity $ht\ J'_{2\ell+1}$ is constant and the following conditions hold:
\begin{enumerate}
\item We have the equality of semigroups ${\Phi}at\mu_{2\ell+2}\left(\frac{{\Phi}at R'}{J'_{2\ell+1}}\setminus\{0\}\right)\cong
\mu_{\ell+1}\left(\frac{R'}{P'_\ell}\setminus\{0\}\right)$.
\item For all $\beta\in\mu_{\ell+1}\left(\frac{R'}{P'_\ell}\setminus\{0\}\right)$ the ${\Phi}at\mu_{2\ell+2}$-ideal of $\frac{{\Phi}at R'}{J'_{2\ell+1}}$ of value $\beta$ is the extension to $\frac{{\Phi}at R'}{J'_{2\ell+1}}$ of the $\mu_{\ell+1}$-ideal of $\frac{R'}{P'_\ell}$ of value $\beta$.
\item In particular, we have a canonical isomorphism $gr_{{\Phi}at\mu_{2\ell+2}}\frac{{\Phi}at R'}{J'_{2\ell+1}}\cong
gr_{\mu_{\ell+1}}\frac{R'}{P'_\ell}$ of graded algebras.
\end{enumerate}
By (2) of Definition \ref{lut} applied to the prime ideals $J'_{2\ell+1}\subset\frac{{\Phi}at R'}{H'_{2\ell}{\Phi}at R'}$, there exists
$R'\in\cal T$ such that both $\frac{R'}{P'_\ell}$ and $\frac{{\Phi}at R'}{J'_{2\ell+1}}$ are regular. The fact that
$\frac{R'}{P'_\ell}$ is regular implies that so is $\frac{{\Phi}at R'}{P'_\ell{\Phi}at R'}$. In particular, $\frac{{\Phi}at
R'}{P'_\ell{\Phi}at R'}$ is a domain, so $H'_{2\ell}=P'_\ell{\Phi}at R'$. Take a regular system of parameters
$$
\bar u'=(\bar u'_1,\dots,\bar u'_{n_\ell})
$$
of $\frac{R'}{P'_\ell}$. Let $k'$ denote the common residue field of $R'$,
$\frac{R'}{P'_{\ell-1}}$ and $\frac{R'}{P'_\ell}$. Fix an isomorphism $\frac{R'}{P'_\ell}\cong k'[[\bar u']]$. Renumbering the variables, if necessary, we may assume that there exists $s_\ell\in\{1,\dots,n_\ell\}$ such that $\bar u'_1,\dots,\bar u'_{s_\ell}$ are $k'$-linearly independent modulo
$({m'}^2+J'_{2\ell+1})\frac{{\Phi}at R'}{H'_{2\ell}}$. Since $\frac{{\Phi}at R'}{J'_{2\ell+1}}$ is regular, the ideal $J'_{2\ell+1}$ is
generated by a set of the form $\bar v'=(\bar v'_{s_\ell+1},\dots,\bar v'_{n_\ell})$, where
$$
\bar v'_j=\bar u'_j-\bar\phi_j(\bar u'_1,\dots,\bar u'_{s_\ell}),\ \bar\phi_j(\bar u'_1,\dots,\bar u'_{s_\ell})\in k'[[\bar
u'_1,\dots,\bar u'_{s_\ell}]].
$$
Let $\bar w'=(\bar w'_1,\dots,\bar w'_{s_\ell})=(\bar
u'_1,\dots,\bar u'_{s_\ell})$. Let $z'$ be a minimal set of
generators of $\frac{P'_\ell}{P'_{\ell-1}}$. Let $k'_0$ be a
quasi-coefficient field of $R'$ (that is, a subfield of $R'$ over
which $k'$ is formally \'etale; such a quasi-coefficient field
exists by \cite{Mat}, moreover, since $R'$ is algebraic over a
finite type algebra over a field by hypotheses, $k'$ is finite
over $k'_0$). By the hypotheses on $R$ and since
$\frac{R'}{P'_\ell}$ is a regular local ring and $\bar u'$ is a minimal
set of generators of its maximal ideal $\frac{m'}{P'_\ell}$, there
exists an ideal $I\subset k'_0[z']$ such that
$\frac{R'}{P'_{\ell-1}}$ is an \'etale extension of
$\frac{k'_0[z',\bar u']_{(z',\bar u')}}I$. By assumptions, we have $ht\
P'_{\ell-1}<ht\ P'_\ell$, so $0<ht\ P'_\ell-ht\
P'_{\ell-1}=ht(z')-ht\ I$, in other words,
\begin{equation}
ht\ I<ht(z').\label{eq:Inotmaximal}
\end{equation}
Next, we prove two general lemmas about ring extensions.
\begin{notation} Let $k_0$ be a field and $(S,m,k)$ a local noetherian
$k_0$-algebra. For a field extension
\begin{equation}
k_0{\Phi}ookrightarrow L\label{eq:k0inktilde}
\end{equation}
such that $k\otimes_{k_0}L$ is a domain, let $S(L)$ denote the
localization of the ring $S\otimes_{k_0}L$ at the prime ideal
$m(S\otimes_{k_0}L)$.
\end{notation}
\begin{lemma}\label{noetherian} Let $k_0$, $(S,m,k)$ and $L$ be as above. The ring $S(L)$ is noetherian.
\end{lemma}
\begin{proof} If the field extension (\ref{eq:k0inktilde})
is finitely generated, the Lemma is obvious. In the general case,
write $L=\lim\limits_{\overrightarrow{i}}L_i$ as a direct limit of
its finitely generated subextensions. For each $L_i$, let $k_i$
denote the residue field of $S(L_i)$; $k_i$ is nothing but the
field of fractions of $k\otimes_{k_0}L_i$. Write ${\Phi}at
S=\frac{k[[x]]}H$, where $x$ is a set of generators of $m$ and $H$
a certain ideal of $k[[x]]$. Then
$\widehat{S(L_i)}\cong\frac{k_i[[x]]}{Hk_i[[x]]}$. Given two
finitely generated extensions $L_i\subset L_j$ of $k_0$, contained
in $L$, we have a commutative diagram
$$
\begin{matrix}
S(L_j)&\overset{\pi_j}\rightarrow&\widehat{S(L_j)}&\cong&\frac{k_j[[x]]}{Hk_j[[x]]}&\\
\psi_{ij}\uparrow&\ &\uparrow&\ &\uparrow\phi_{ij}\\
S(L_i)&\overset{\pi_i}\rightarrow&\widehat{S(L_i)}&\cong&\frac{k_i[[x]]}{Hk_i[[x]]}
\end{matrix}
$$
where $\phi_{ij}$ is the map induced by the natural inclusion
$k_i{\Phi}ookrightarrow k_j$ and the identity map of $x$ to itself.
Let $k_\infty=\lim\limits_{\overrightarrow{i}}k_i$. Then, for each
$i$, we have the obvious faithfully flat map
$\rho_i:\widehat{S(L_i)}\rightarrow\frac{k_\infty[[x]]}{Hk_\infty[[x]]}$,
defined by the natural inclusion $k_i{\Phi}ookrightarrow k_\infty$ and
the identity map of $x$ to itself; the maps $\rho_i$ commute with
the $\phi_{ij}$. Thus, we have constructed a faithfully flat map
$\rho_i\circ\pi_i$ from each element of the direct system $S(L_i)$ to the fixed noetherian ring
$\frac{k_\infty[[x]]}{Hk_\infty[[x]]}$; moreover, the maps
$\rho_i\circ\pi_i$ are compatible with the homomorphisms
$\psi_{ij}$ of the direct system. This implies that the ring
$S(L)=\lim\limits_{\overrightarrow{i}}S(L_i)$ is noetherian.
\end{proof}
\begin{lemma}\label{IS(t)} Let $(S,m,k)$ be a local noetherian ring. Let $t$ be an arbitrary collection of independent variables.
Consider the rings $S[t]$ and $S(t):=S[t]_{mS[t]}$. Let $I$ be an
ideal of $S$. Then
\begin{equation}
IS(t)\cap S[t]=IS[t].\label{eq:IcapSt=I}
\end{equation}
\end{lemma}
\begin{proof} First, assume the collection $t$ consists of a
single variable. Consider elements $f,g\in S[t]$ such that
\begin{equation}
f\mbox{$\in$ {\Phi}space{-.8em}/} mS[t]\label{eq:fnotinmSt}
\end{equation}
and
\begin{equation}
fg\in IS[t].\label{eq:fginISt}
\end{equation}
Proving the equation (\ref{eq:IcapSt=I}) amounts to proving that
\begin{equation}
g\in IS[t].\label{eq:ginISt}
\end{equation}
We prove (\ref{eq:ginISt}) by contradiction. Assume that $g\mbox{$\in$ {\Phi}space{-.8em}/} IS[t]$. Then there exists $n\in\mathbb N$ such that
$g\mbox{$\in$ {\Phi}space{-.8em}/}(I+m^n)S[t]$. Take the smallest such $n$, so that
\begin{equation}
g\in\left(I+m^{n-1}\right)S[t]\setminus(I+m^n)S[t].\label{eq:setminus}
\end{equation}
Write $f=\sum\limits_{j=0}^qa_jt^j$ and
$g=\sum\limits_{j=0}^lb_jt^j$. Let
\begin{eqnarray}
l_0:&=&\max\{j\in\{0,\dots,l\}\ |\ b_j\mbox{$\in$ {\Phi}space{-.8em}/} I+m^n\}\quad\text{ and}\\
q_0:&=&\max\{j\in\{0,\dots,q\}\ |\ a_j\mbox{$\in$ {\Phi}space{-.8em}/} m\}.
\end{eqnarray}
Let $c_{l_0+q_0}$ denote the $(l_0+q_0)$-th coefficient of $fg$. We have
$$
c_{l_0+q_0}=\sum\limits_{i+j=l_0+q_0}a_ib_j=a_{q_0}b_{l_0}+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\
i>q_0\end{array}}a_ib_j+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ j>l_0\end{array}}a_ib_j.
$$
By definition of $l_0$ and $q_0$ and (\ref{eq:setminus}) we have:
\begin{eqnarray}
a_{q_0}b_{l_0}&\mbox{$\in$ {\Phi}space{-.8em}/}&I+m^n\quad\text{and}\\
\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\ i>q_0\end{array}}a_ib_j+\sum\limits_{\begin{array}{c}i+j=l_0+q_0\\
j>l_0\end{array}}a_ib_j&\in&I+m^n.
\end{eqnarray}
Hence $c_{l_0+q_0}\mbox{$\in$ {\Phi}space{-.8em}/} I+m^n$, which contradicts
(\ref{eq:fginISt}). This completes the proof of Lemma \ref{IS(t)}
in the case when $t$ is a single variable. The case of a general
$t$ now follows by transfinite induction on the collection $t$.
\end{proof}
\begin{lemma}\label{contractsto0} There exist sets of representatives
$$
u'=(u'_1,\dots,u'_{n_\ell})
$$
of $\bar u'$ and $\phi_j$ of $\bar\phi_j$, $s_\ell<j\le n_\ell$, in $\frac{{\Phi}at R'}{H'_{2\ell-2}{\Phi}at R'}$, having the
following properties. Let
\begin{eqnarray}
w'=(w'_1,\dots,w'_{s_\ell})&=&(u'_1,\dots,u'_{s_\ell}),\\
v'=(v'_{s_\ell+1},\dots,v'_{n_\ell})&=&
(u'_{s_\ell+1}-\phi_{s_\ell+1},\dots,u'_{n_\ell}-\phi_{n_\ell})\label{eq:defv}.
\end{eqnarray}
Let $J'_{2\ell-1}=\frac{H'_{2\ell-1}}{H'_{2\ell-2}}+(v')\subset\frac{{\Phi}at R'}{H'_{2\ell-2}}$. Then
\begin{equation}
w'\subset\frac{R'}{P'_{\ell-1}}\label{eq:winR}
\end{equation}
and
\begin{equation}
J'_{2\ell-1}\cap\frac{R'}{P'_{\ell-1}}=(0).\label{eq:contractsto0}
\end{equation}
\end{lemma}
\begin{proof}\textit{(of Lemma \ref{contractsto0})} There is no problem choosing $w'$ to satisfy (\ref{eq:winR}).
As for (\ref{eq:contractsto0}), we first prove the Lemma under the
assumption that $k$ is countable. We choose the representatives $u'$
arbitrarily and let $\bar\phi_j(u')\in k[[u']]$ denote the formal
power series obtained by substituting $u'$ for $\bar u'$ in
$\bar\phi_j$. Any representative $\phi_j$ of $\bar\phi_j$,
$s_\ell<j\le n_\ell$ has the form $\phi_j=\bar\phi_j(u')+h_j$ with
$h_j\in(z')\frac{{\Phi}at R'}{H'_{2\ell-2}}$. We define the $h_j$
required in the Lemma recursively in $j$. Take
$j\in\{s_\ell+1,\dots,n_\ell\}$. Assume that
$h_{s_{\ell+1}},\dots,h_{j-1}$ are already defined and that
\begin{equation}
(v'_{s_\ell+1},\dots,v'_{j-1})\cap\frac{R'}{P'_{\ell-1}}=(0),\label{eq:j-1inter0}
\end{equation}
where we view $\frac{R'}{P'_{\ell-1}}$ as a subring of $\frac{{\Phi}at
R'}{H'_{2\ell-1}}$. Since the ring $\frac{R'}{P'_{\ell-1}}$ is
countable, there are countably many ideals in $\frac{{\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, not contained in
$\frac{(z'){\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$,
which are minimal primes of ideals of the form $(f)\frac{{\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, where $f$ is a
non-zero element of $\frac{m'}{P'_{\ell-1}}$. Let us denote these
ideals by $\{I_q\}_{q\in\mathbb N}$; we have
\begin{equation}
ht\ I_q=1\quad\text{ for all }q\in\mathbb N.\label{eq:ht=1}
\end{equation}
We note that
\begin{equation}
\frac{(z'){\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}\not\subset I_q\quad\text{ for all }q\in\mathbb
N.\label{eq:znotin}
\end{equation}
Indeed, by (\ref{eq:Inotmaximal}) and (\ref{eq:j-1inter0}) we have $ht\ \frac{(z'){\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}{\Gamma}e1$. In view of
(\ref{eq:ht=1}), containment in (\ref{eq:znotin}) would imply
equality, which contradicts the definition of $I_q$.
Since $H'_{2\ell-1}\subsetneqq H'_{2\ell}$ and $J'_{2\ell+1}\subsetneqq\frac{m'{\Phi}at R'}{H'_{2\ell}}$, we have
$$
\dim\frac{{\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}{\Gamma}e(ht\ H'_{2\ell}-ht\ H'_{2\ell-1})+ht\
\frac{m'{\Phi}at R'}{H'_{2\ell}}-(j-s_\ell-1){\Gamma}e
$$
\begin{equation}
(ht\ H'_{2\ell}-ht\ H'_{2\ell-1})+ht\ \frac{m'{\Phi}at R'}{H'_{2\ell}}-ht\ J'_{2\ell+1}+1{\Gamma}e3.
\end{equation}
Let $\tilde u_j$ denote the image of $u'_j-\bar\phi_j(u')$ in $\frac{{\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Next, we construct an element $\tilde h_j\in\frac{(z'){\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$ such that
\begin{equation}
\tilde u_j-\tilde h_j\mbox{$\in$ {\Phi}space{-.8em}/}\bigcup\limits_{q=1}^\infty I_q.\label{eq:notinIq}
\end{equation}
The element $\tilde h_j$ will be given as the sum of an infinite
series $\sum\limits_{t=0}^\infty h_{jt}^t$ in $(z')\frac{{\Phi}at
R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, convergent in
the $m'$-adic topology, which we will now construct recursively in
$t$. Put $h_{j0}=0$. Assume that $t>0$, that $h_{j0},\dots,h_{j,t-1}$ are already defined and that for
$q\in\{1,\dots,t-1\}$ we have $u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^qh_{jl}\mbox{$\in$ {\Phi}space{-.8em}/}\bigcup\limits_{l=1}^qI_l$
and $h_{jq}\in(z')\bigcap\left(\bigcap\limits_{l=1}^{q-1}I_l\right)$.
If $u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^{t-1}h_{jl}\mbox{$\in$ {\Phi}space{-.8em}/} I_t$, put $h_{jt}=0$. If
$u'_j-\bar\phi_j(u')-\sum\limits_{l=0}^{t-1}h_{jl}\in I_t$, let $h_{jt}$ be any element of
$(z')\bigcap\left(\bigcap\limits_{l=1}^{t-1}I_l\right)\setminus
I_t$ (such an element exists because $I_t$ is prime, in view of
(\ref{eq:znotin})). This completes the definition of $\tilde h_j$.
Let $h_j$ be an arbitrary representative of $\tilde h_j$ in
$\frac{{\Phi}at R'}{H'_{2\ell-2}}$.
We claim that
\begin{equation}
\left(H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_j)\right)\cap\frac{R'}{P'_{\ell-1}}=(0).\label{eq:jinter0}
\end{equation}
Indeed, suppose the above intersection contained a non-zero element $f$. Then any minimal prime $\tilde I$ of the
ideal $(v'_j)\frac{{\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$ is also a minimal prime of
$(f)\frac{{\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Since
$v_j\mbox{$\in$ {\Phi}space{-.8em}/}\frac{(z'){\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$, we have $\tilde
I\not\subset\frac{(z'){\Phi}at R'}{H'_{2\ell-1}+(v'_{s_\ell+1},\dots,v'_{j-1})}$. Hence $\tilde I=I_q$ for some
$q\in\mathbb N$. Then $v_j\in I_q$, which contradicts (\ref{eq:notinIq}).
Carrying out the above construction for all
$j\in\{s_\ell+1,\dots,n_\ell\}$ produces the elements $\phi_j$
required in the Lemma. This completes the proof of Lemma
\ref{contractsto0} in the case when $k$ is countable.
Next, assume that $k$ is uncountable. Let $u'$ be chosen as above.
By assumption, $\frac{R'}{P'_{\ell-1}}$ contains a
$k_0$-subalgebra $S$ essentially of finite type, over which $\frac{R'}{P'_{\ell-1}}$ is strictly \'etale. Take a
countable subfield $L_1\subset k_0$ such that the algebra $S$ is defined already over $L_1$ (this
means that $S$ has the form
\begin{equation}
S=(S'_1\otimes_{L_1}k_0)_{m'_1(S'_1\otimes_{L_1}k_0)},\label{eq:R'/P'}
\end{equation}
where $(S'_1,m'_1,k'_1)$ is a local $L_1$-algebra essentially of
finite type). Next, let $L_1\subset L_2\subset...$ be an
increasing chain of finitely generated field extensions of $L_1$,
contained in $k_0$, having the following property. Let
$(S'_q,m'_q,k'_q)$ denote the localization of
$S'_1\otimes_{k'_1}L_q$ at the maximal ideal
$m'_1(S'_1\otimes_{k'_1}L_q)$. We require that
$$
k'_\infty:=\bigcup\limits_{q=1}^\infty k'_q
$$
contain all the coefficients of all the formal power series
$\bar\phi_{s_\ell+1},\dots,\bar\phi_{n_\ell}$ and such that the
ideal $\frac{H'_{2\ell-1}}{H'_{2\ell-2}}$ is generated by elements
of $\frac{k'_\infty[[z']]}{I_\infty}[[u']]$, where $I_\infty$ is
the kernel of the natural homomorphism
$k'_\infty[[z']]\rightarrow\frac{{\Phi}at R'}{H'_{2\ell-2}}$. Let
$H'_{2\ell-1,\infty}=\frac{H'_{2\ell-1}}{H'_{2\ell-2}}\cap
\frac{k'_\infty[[z']]}{I_\infty}[[u']]$. We have constructed an increasing chain
$S'_1\subset S'_2\subset...$ of local $L_1$-algebras essentially of finite type such that $k'_q$ is the
residue field of $S'_q$. Then
$S'_\infty:=\bigcup\limits_{q=1}^\infty S'_q$ is a local
noetherian ring whose completion is
$\frac{k'_\infty[[z',u']]}{\left(P'_{\ell-1}\cap
S'_\infty\right)k'_\infty[[z',u']]}$. Let $m'_\infty$ denote the
maximal ideal of $S'_\infty$. The above argument in the countable
case shows that there exist representatives
$\phi_{s_\ell+1},\dots,\phi_{n_\ell}$ of
$\bar\phi_{s_\ell+1},\dots,\bar\phi_{n_\ell}$ in $\frac{{\Phi}at
S'_\infty}{H'_{2\ell-2}\cap{\Phi}at S'_\infty}$ such that, defining
$v'=(v'_{s_\ell+1},\dots,v'_{n_\ell})$ as in (\ref{eq:defv}), we
have
\begin{equation}
\left((v')+H'_{2\ell-1,\infty}\right)\cap S'_\infty=(0).\label{eq:S'infty0}
\end{equation}
Let $L_\infty=\bigcup\limits_{q=1}^\infty L_q$ and let $t$ denote
a transcendence base of $k_0$ over $L_\infty$. Let the notation be
as in Lemma \ref{noetherian} with $k_0$ replaced by $L_\infty$.
For example, $S'_\infty(L_\infty(t))$ will denote the localization
of the ring $S'_\infty\otimes_{L_\infty}L_\infty(t)$ at the prime
ideal ideal $m'_\infty(S'_\infty\otimes_{L_\infty}L_\infty(t))$.
By (\ref{eq:S'infty0}),
\begin{equation}
\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at S'_\infty[t]\cap
S'_\infty[t]=(0).\label{eq:S'infty0t}
\end{equation}
Now Lemma \ref{IS(t)} and the fact that $S'_\infty[t]$ is a domain
imply that
\begin{equation}
\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at
S'_\infty(L_\infty(t))\cap
S'_\infty(L_\infty(t))=(0).\label{eq:capbarS=0}
\end{equation}
Next, let $\tilde L$ be a finite extension of $L_\infty(t)$,
contained in $k_0$; then $S'_\infty(\tilde L)$ is finite over
$S'_\infty(L_\infty(t))$. Since ${\Phi}at S'_\infty(\tilde L)$ is
faithfully flat over ${\Phi}at S'_\infty(L_\infty(t))$ and in view of
(\ref{eq:capbarS=0}), we have
$$
\left(\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at S'_\infty(\tilde
L)\cap S'_\infty(\tilde L)\right)\cap S'_\infty(L_\infty(t))=(0).
$$
Hence $ht\ \left((v')+H'_{2\ell-1,\infty}\right){\Phi}at
S'_\infty(\tilde L)\cap S'_\infty(\tilde L)=0$. Since
$S'_\infty(\tilde L)$ is a domain, this implies that
\begin{equation}
\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at S'_\infty(\tilde L)\cap
S'_\infty(\tilde L)=(0).\label{eq:tildek=0}
\end{equation}
Since $k_0$ is algebraic over $L_\infty(t)$, it is the limit of
the direct system of all the finite extensions of $L_\infty(t)$
contained in it. We pass to the limit in (\ref{eq:tildek=0}). By
(\ref{eq:R'/P'}), we have $S=S'_\infty(k_0)$; we also note that
${\Phi}at S=\frac{{\Phi}at R'}{H'_{2\ell-2}}$.
Since the natural maps ${\Phi}at S'_\infty(\tilde L)\rightarrow{\Phi}at
S'_\infty(k_0)$ are all faithfully flat, we obtain
\begin{equation}
\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at S'_\infty(k_0)\cap
S=(0).\label{eq:capS=0}
\end{equation}
Since ${\Phi}at S=\frac{{\Phi}at R'}{H'_{2\ell-2}}$ is also the formal
completion of ${\Phi}at S'_\infty(k_0)$, it is faithfully flat over
${\Phi}at S'_\infty(k_0)$. Hence
\begin{equation}
J'_{2\ell-1}\cap{\Phi}at
S'_\infty(k_0)=\left((v')+H'_{2\ell-1,\infty}\right)\frac{{\Phi}at
R'}{H'_{2\ell-2}}\cap{\Phi}at
S'_\infty(k_0)=\left((v')+H'_{2\ell-1,\infty}\right){\Phi}at
S'_\infty(k_0).\label{eq:capbarS}
\end{equation}
Combining this with (\ref{eq:capS=0}), we obtain
\begin{equation}
J'_{2\ell-1}\cap S=(0).\label{eq:capS(t)=0}
\end{equation}
Thus the ideal $J'_{2\ell-1}\cap\frac{R'}{P'_{\ell-1}}$ contracts
to $(0)$ in $S$. Since $\frac{R'}{P'_{\ell-1}}$ is \'etale over
$S$, this implies the desired equality (\ref{eq:contractsto0}).
This completes the proof of Lemma \ref{contractsto0}.
\end{proof}
Since $\frac{{\Phi}at R'}{H'_{2\ell}{\Phi}at R'}$ is a complete regular
local ring and $(w',v')$ is a set of representatives of a minimal
set of generators of its maximal ideal $\frac{m'{\Phi}at
R'}{H'_{2\ell}}$, there exists a complete local domain
$R'_\ell$ (not necessarily regular) such that $\frac{{\Phi}at
R'}{H'_{2\ell-1}}\cong R'_\ell[[w',v']]$. Consider the ring
homomorphism
\begin{equation}
R'_\ell[[w',v']]\rightarrow R'_\ell[[w']],\label{eq:homomorphisms}
\end{equation}
obtained by taking the quotient modulo $(v')$. By
(\ref{eq:homomorphisms}), the quotient of $\frac{{\Phi}at
R'}{H'_{2\ell-2}}$ by $J'_{2\ell-1}$ is the integral domain
$R'_\ell[[w']]$, hence $J'_{2\ell-1}$ is prime.
Consider a local blowing up $R'\rightarrow R''$ in $\mathcal T$. Because of the stability assumption on $R$, the ring $\frac{{\Phi}at
R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})$ is finite over $\frac{{\Phi}at R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$; hence the ring $\lim\limits_{\overset\longrightarrow{R''\in\mathcal T}}\left(\frac{{\Phi}at R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})\right)$ is integral over $\frac{{\Phi}at R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$. In particular, there exists a prime ideal in
$$
\lim\limits_{\overset\longrightarrow{R''\in\mathcal T}}\left(\frac{{\Phi}at R''}{H''_{2\ell-2}}\otimes_R\kappa(P''_{l-1})\right),
$$
lying over $J'_{2\ell-1}\frac{{\Phi}at R'}{H'_{2\ell-2}}\otimes_R\kappa(P'_{l-1})$. Pick and fix one such prime ideal. Intersecting this ideal with $\frac{{\Phi}at R''}{H''_{2\ell-2}}$ for each $R''\in\mathcal T$, we obtain a tree $J''_{2\ell-1}$ of prime ideals of $\frac{{\Phi}at R''}{H''_{2\ell-2}}$, $R''\in\mathcal T$.
Our next task is to define the restriction of the valuation
${\Phi}at\mu_{2\ell}$ to the ring $\frac{{\Phi}at R'}{J'_{2\ell-1}}$. By
the induction assumption, ${\Phi}at\mu_{2\ell+2}$ is already defined
on $\lim\limits_{\overset\longrightarrow{R'\in\mathcal{T}}}\frac{{\Phi}at R'}{J'_{2\ell+1}{\Phi}at R'}$.
For all \textit{stable} $R''\in\mathcal{T}$ we have the isomorphism $gr_{{\Phi}at\mu_{2\ell+2}}\frac{{\Phi}at R''}{J''_{2\ell+1}}\cong
gr_{\mu_{\ell+1}}\frac{R''}{P''_\ell}$ of graded algebras (in particular, $gr_{{\Phi}at\mu_{2\ell+2}}\frac{{\Phi}at
R''}{J''_{2\ell+1}}$ is scalewise birational to $gr_{\mu_{\ell+1}}\frac{R''}{P''_\ell}$ for any $R''\in\mathcal{T}$ and
${\Phi}at\mu_{2\ell+2}$ has the same value group $\Delta_\ell$ as $\mu_{\ell+1}$).
Define the prime ideals $\tilde H''_{2\ell-2}=\tilde H''_{2\ell-1}$
to be equal to the preimage of $J''_{2\ell-1}$ in ${\Phi}at R''$
and $\tilde H''_{2\ell}=\tilde H''_{2\ell+1}$ the preimage of $J''_{2\ell+1}$ in ${\Phi}at R''$. By definition of tight extensions,
the valuation ${\Phi}at\nu_{2\ell+1}$ must be trivial. It remains to describe the valuation ${\Phi}at\mu_{2\ell}$ on $\frac{{\Phi}at R''}{J''_{2\ell-1}}$, $R''\in\mathcal T$. We will first define ${\Phi}at\nu_{2\ell}$ and then put ${\Phi}at\mu_{2\ell}={\Phi}at\nu_{2\ell}\circ{\Phi}at\mu_{2\ell+2}$.
By definition of tight extensions, the value group of
${\Phi}at\mu_{2\ell}$ must be equal to $\Delta_{\ell-1}$ and that of ${\Phi}at\nu_{2\ell}$ to $\frac{\Delta_{\ell-1}}{\Delta_\ell}$. For a positive element $\bar\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$, define the candidate for
${\Phi}at\nu_{2\ell}$-ideal of $\frac{{\Phi}at R''_{\tilde H''_{2\ell}}}{\tilde H''_{2\ell-1}{\Phi}at R''_{\tilde H''_{2\ell}}}$ of value $\bar\beta$, denoted
by ${\Phi}at{\mathcal P}''_{\beta\ell}$, by the formula
\begin{equation}
{\Phi}at{\mathcal P}''_{\bar\beta\ell}=\frac{\mathcal{P}''_{\bar\beta}{\Phi}at R''_{\tilde H''_{2\ell}}}{\tilde H''_{2\ell-1}{\Phi}at R''_{\tilde H''_{2\ell}}}.\label{eq:validealmain}
\end{equation}
\begin{conjecture}\label{strongcontainment} The elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen in such a way that the following condition holds. For each positive element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_\ell}$ and each tree morphism $R'\rightarrow R''$ in $\mathcal T$, we have
$$
{\Phi}at{\mathcal P}''_{\beta\ell}\cap{\Phi}at R'_{\tilde H'_{2\ell}}={\Phi}at{\mathcal P}'_{\beta\ell}.
$$
\end{conjecture}
\begin{conjecture}\label{containment} The elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen in such a way that
\begin{equation}
\bigcap\limits_{\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+}
\left(\mathcal{P}'_{\bar\beta}+\tilde H'_{2\ell-1}\right){\Phi}at R'_{\tilde
H'_{2\ell}}\subset\tilde H'_{2\ell-1}. \label{eq:restrictionmain}
\end{equation}
\end{conjecture}
For the rest of this section assume that Conjectures \ref{strongcontainment} and \ref{containment} are true.
For all $\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+$,
we have the natural isomorphism
$$
\lambda_{\bar\beta}:\frac{\mathcal{P}'_{\bar\beta}}{\mathcal{P}'_{\bar\beta+}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde
H'_{2\ell})\longrightarrow\frac{{\Phi}at{\mathcal P}'_{\bar\beta}}{{\Phi}at{\mathcal P}'_{\bar\beta+}}
$$
of $\kappa(\tilde H'_{2\ell})$-vector spaces. The following fact is an easy consequences of Conjecture \ref{strongcontainment}:
\begin{corollary}\label{integraldomain}\textbf{(conditional on Conjecture \ref{strongcontainment})} If the elements $\phi_j$ of Lemma \ref{contractsto0} can be chosen as in Conjecture \ref{strongcontainment} then the graded algebra
$$
\mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde
H'_{2\ell})\cong\bigoplus\limits_{\bar\beta\in\left(\frac{\Delta_{\ell-1}}{\Delta_\ell}\right)_+}\frac{{\Phi}at{\mathcal P}'_{\bar\beta}}{{\Phi}at{\mathcal P}'_{\bar\beta+}}
$$
is an integral domain.
\end{corollary}
For a non-zero element $x\in\frac{{\Phi}at R'_{\tilde H'_{2\ell}}}{\tilde H'_{2\ell-1}}$, let $\operatorname{Val}_\ell(x)=\left\{\left.\beta\in\nu_\ell\left(\frac{R'}{P'_{\ell-1}}\setminus\{0\}\right)\ \right|\ x\in{\Phi}at{\mathcal P}'_{\beta\ell}\right\}$.
We define ${\Phi}at\nu_{2\ell}$ by the formula
\begin{equation}\label{eq:mu2l}
{\Phi}at\nu_{2\ell}(x)=\max\ \operatorname{Val}_\ell(x).
\end{equation}
Since $\nu_\ell$ is a rank 1 valuation, centered in a local noetherian domain $\frac{R'}{P'_{\ell-1}}$, the semigroup $\nu_\ell\left(\frac{R'}{P'_{\ell-1}}\setminus\{0\}\right)$ has order type $\mathbb N$, so by (\ref{eq:restrictionmain}) the set $Val_\ell(x)$ contains a maximal element. This proves that the valuation ${\Phi}at\nu_{2\ell}$ is well defined by the formula (\ref{eq:mu2l}),
and that we have a natural isomorphism of graded algebras
$$
\mbox{gr}_{\nu_\ell}\frac{R'_{P'_\ell}}{P'_{\ell-1}R'_{P'_\ell}}\otimes_{\kappa(P'_{\ell-1})}\kappa(\tilde
H'_{2\ell})\cong\mbox{gr}_{{\Phi}at\nu_{2\ell}}\frac{{\Phi}at R_{\tilde
H'_{2\ell}}}{\tilde H'_{2\ell-1}}.
$$
Since the above construction is valid for all $R\in\mathcal T$, ${\Phi}at\nu_{2\ell}$ extends naturally to a valuation centered in the
ring $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at
R''}{\tilde H''_{2\ell-1}{\Phi}at R''}$ (by abuse of notation, this extension will also be denoted by ${\Phi}at\nu_{2\ell}$).
The extension ${\Phi}at\mu_{2\ell}$ of $\mu_\ell$ to
$\lim\limits_{\overset\longrightarrow{R''\in\mathcal{T}(R')}}\frac{{\Phi}at
R''}{\tilde H''_{2\ell-1}{\Phi}at R'}$ is defined by
${\Phi}at\mu_{2\ell}={\Phi}at\nu_{2\ell}\circ{\Phi}at\mu_{2\ell+2}$.
This completes the proof of Conjecture \ref{teissier1} (assuming Conjectures \ref{strongcontainment} and \ref{containment}) by descending induction on $\ell$.{\Phi}fill$\Box$
The next Corollary of Conjecture \ref{teissier1} gives necessary conditions for ${\Phi}at\nu_-$ to be uniquely determined by
$\nu$; it also shows that the same conditions are sufficient for ${\Phi}at\nu_-$ to be the unique minimal extension of $\nu$, that is, to satisfy
\begin{equation}
\tilde H'_i=H'_i,\quad0\le i\le 2r.\label{eq:tilde=nothing}
\end{equation}
Suppose given a tree $\left\{\tilde H'_0\right\}$ of minimal prime ideals of ${\Phi}at R'$ (in
particular, $R'\cap\tilde H'_0=(0)$). If the valuation $\nu$ admits an extension to a valuation ${\Phi}at\nu_-$ of
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{\tilde H'_0}$, then $\tilde H'_0$ is the 0-th prime ideal of ${\Phi}at R'$, determined by ${\Phi}at\nu_-$. Since $\tilde H'_0$ is assumed to be a \textit{minimal} prime, we have $\tilde H'_0=H'_0$ by Proposition \ref{Hintilde}.
\begin{remark} Let the notation be as in Conjecture \ref{teissier1}. Denote the tree of prime ideals $\{\tilde H'_0\}$ by $\{H'\}$ for short. Consider a homomorphism
\begin{equation}
R'\rightarrow R''\label{eq:R'toR''}
\end{equation}
in $\mathcal T$. Assume that the local rings $R'$ and $\frac{{\Phi}at R'}{H'}$ are regular, and let $V=(V_1,\dots,V_s)$ be a minimal set of generators of $H'$. Then $V$ can be extended to a regular system of parameters for ${\Phi}at R'$. We have an isomorphism ${\Phi}at R'\cong\frac{{\Phi}at R'}{H'}[[V]]$. The morphism (\ref{eq:R'toR''}) induces an isomorphism ${\Phi}at R'_{H'}\cong {\Phi}at R''_{H''}$, so that $V$ induces a regular system of parameters of ${\Phi}at R''_{H''}$. In particular, the $H''$-adic valuation of ${\Phi}at R''_{H''}$ coincides with the $H'$-adic valuation of ${\Phi}at R'_{H'}$. On the other hand, we do not know, assuming that $R''$ and $\frac{{\Phi}at R''}{H''}$ are regular and $ht\ H''=ht\ H'$, whether $V$ induces a minimal set of generators of $H''$; we suspect that the answer is ``no''.
\end{remark}
\begin{corollary}\label{uncond1}\textbf{(conditional on Conjecture \ref{teissier1})} If the valuation $\nu$ admits a unique extension to a valuation ${\Phi}at\nu_-$ of $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_0}$, then the following conditions hold:
(1) $ht\ H'_1\le 1$
(2) $H'_i=H'_{i-1}$ for all odd $i>1$.
Moreover, this unique extension ${\Phi}at\nu_-$ is minimal.
Conversely, assume that (1)--(2) hold. Then there exists a unique minimal extension ${\Phi}at\nu_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_0}$.
\end{corollary}
\begin{proof} The fact that conditions (1), (2) and equations (\ref{eq:tilde=nothing}) determine ${\Phi}at\nu_-$ uniquely is nothing but Proposition \ref{uniqueness1}. Conversely, assume that there exists a unique extension ${\Phi}at\nu_-$ of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_0}$. By Remark \ref{minimalexist}, there exist minimal extensions of $\nu$ to $\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_0}$, hence ${\Phi}at\nu_-$ must be minimal.
Next, by Conjecture \ref{teissier1}, there exists a tree of prime
ideals $\tilde H'$ with $H'\cap R'=(0)$ and a tight extension
${\Phi}at\mu_-$ of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'}$. The
ideals $H'$ are both the the 0-th and the 1-st ideals determined by
${\Phi}at\mu_-$; in particular, we have
\begin{equation}
H'_0\subset H'_1\subset H'\label{eq:inH'}
\end{equation}
by Proposition \ref{Hintilde}. Now, take any valuation $\theta$,
centered in the regular local ring $\frac{R'_{H'}}{H'_0}$, such that
the residue field $k_\theta=\kappa(H')$. Then the composition
${\Phi}at\mu_-\circ\theta$ is an extension of $\nu$ to
$\lim\limits_{\overset\longrightarrow{R'}}\frac{{\Phi}at R'}{H'_0}$,
hence
\begin{equation}
{\Phi}at\mu_-\circ\theta={\Phi}at\nu_-\label{eq:mucirctheta}
\end{equation}
by uniqueness. For $i{\Gamma}e1$, the $i$-th prime ideal, determined by
${\Phi}at\mu_-\circ\theta={\Phi}at\nu_-$ coincides with that determined by
${\Phi}at\mu_-$. Since $\nu$ is minimal and ${\Phi}at\mu_-$ is tight, we
obtain condition (2) of the Corollary. Finally, if we had $ht\
H'>1$, there would be infinitely many choices for $\theta$,
contradicting \ref{eq:mucirctheta} and the uniqueness of
${\Phi}at\nu_-$. Thus $ht\ H'\le1$. Combined with \ref{eq:inH'}, this
proves (1) of the Corollary. This completes the proof of Corollary
(\ref{uncond1}), assuming Conjecture \ref{teissier1}.
\end{proof}
\appendix{Regular morphisms and G-rings.}
In this Appendix we recall the definitions of regular homomorphism, G-rings and excellent and quasi-excellent rings. We also summarize some of their basic properties used in the rest of the paper.
\begin{definition}\label{regmor} (\cite{Mat}, Chapter 13, (33.A), p. 249) Let $\sigma:A\rightarrow B$
be a homomorphism of noetherian rings. We say that $\sigma$ is {\bf regular} if
it is flat, and for every prime ideal $P\subset A$, the ring $B\otimes_A\kappa(P)$ is
geometrically regular over $\kappa(P)$ (this means that for any finite field
extension $\kappa(P)\rightarrow k'$, the ring $B\otimes_Ak'$ is regular).
\end{definition}
\begin{remark} If $\kappa(P)$ is perfect, the ring $B\otimes_A\kappa(P)$ is geometrically regular over $\kappa(P)$ if and
only if it is regular.
\end{remark}
\begin{remark} It is known that a morphism of finite type is regular in the above sense if and only if it is smooth (that is, formally smooth in the sense of Grothendieck with respect to the discrete topology), though we do not use this fact in the present paper.
\end{remark}
Regular morphisms come up in a natural way when one wishes to pass to the formal completion of a local ring:
\begin{definition}\label{Gring} (\cite{Mat}, (33.A) and (34.A)) Let $R$ be a noetherian ring. For a maximal ideal $m$ of $R$, let ${\Phi}at R_m$ denote the $m$-adic completion of $R$. We say that $R$ is a {\bf G-ring} if for every maximal ideal
$m$ of $R$, the natural map $R\rightarrow{\Phi}at R_m$ is a regular homomorphism.
\end{definition}
The property of being a G-ring is preserved by localization and passing to rings essentially of finite type over $R$.
\begin{definition}\label{quasiexcellent} (\cite{Mat}, Definition 2.5, (34.A), p. 259) Let $R$ be a noetherian
ring. We say that $R$ is {\bf quasi-excellent} if the following two
conditions hold:
(1) $R$ is J-2, that is, for any scheme $X$, which is reduced and of
finite type over $\mbox{Spec}\ R$, $Reg(X)$ is open in the Zariski topology.
(2) For every maximal ideal $m\subset R$, $R_m$ is a G-ring.
\end{definition}
It is known \cite{Mat} that a \textit{local} G-ring is automatically J-2, hence automatically quasi-excellent. Thus for local rings ``G-ring'' and ``quasi-excellent'' are one and the same thing.
A ring is said to be \textbf{excellent} if it is quasi-excellent and
universally catenary, but we do not need the catenary condition in this paper.
Both excellence and quasi-excellence are preserved by localization and passing to rings of finite type over $R$
(\cite{Mat}, Chapter 13, (33.G), Theorem 77, p. 254). In
particular, any ring essentially of finite type over a field, $\mathbf Z$, $\mathbf Z_{(p)}$, $\mathbf Z_p$,
the Witt vectors or any other excellent Dedekind domain is
excellent. See \cite{Nag} (Appendix A.1, p. 203) for some examples of
non-excellent rings.
Rings which arise from natural constructions in algebra and geometry are
excellent. Complete and complex-analytic local rings are excellent (see
\cite{Mat}, Theorem 30.D) for a proof that any complete local ring is excellent).
Finally, we remark that the category of quasi-excellent rings is a natural one for doing algebraic geometry, since it is the
largest reasonable class of rings for which resolution of singularities can hold. Namely, let $R$ be a noetherian ring.
Grothendieck (\cite{EGA}, IV.7.9) proves that if all of the irreducible closed subschemes of $\mbox{Spec}\ R$ and all of their
finite purely inseparable covers admit resolution of singularities, then $R$ must be quasi-excellent.
Grothendieck's result means that the largest {\it class} of noetherian
rings, closed under homomorphic images and finite purely inseparable extensions, for which resolution of singularities
could possibly exist, is {\it quasi-excellent} rings.
We now summarize the specific uses we make of quasi-excellence in the present paper. We begin by recalling three results from \cite{Mat} and \cite{Nag}. As a point of terminology, we note that Nagata's ``pseudo-geometric'' rings are now commonly known as Nagata rings. Quasi-excellent rings are Nagata (\cite{Mat}, (33.H), Theorem 78).
\begin{theorem}\label{annormal}(\cite{Mat}, (34.C), Theorem 79) Let $R$ be an excellent normal local ring. Then $R$ is analytically normal (this means that its formal completion ${\Phi}at R$ is normal).
\end{theorem}
\begin{theorem}(\cite{Nag}, (43.20), p. 187) Let $R$ be a local integral domain, $\tilde R$ its Henselization and $R'$ its normalization. There is a natural one-to-one correspondence between the minimal primes of $\tilde R$ and the maximal ideals of $R'$.
\end{theorem}
\begin{proposition}\label{anirred}(\cite{Nag}, Corollary (44.3), p. 189) Let $R$ be a quasi-excellent analytically normal local ring. Then its Henselization $\tilde R$ is analytically irreducible and is algebraically closed in its formal completion.
\end{proposition}
From the above results we deduce
\begin{corollary}\label{notnormal} Let $(R,\mathbf m)$ be a Henselian excellent local domain. Then $R$ is analytically irreducible and is algebraically closed in ${\Phi}at R$.
\end{corollary}
\begin{proof} If, in addition, we assume $R$ to be normal, the result follows from Theorems \ref{annormal} and \ref{anirred}. In the general case, let $R'$ denote the normalization of $R$. Then $R'$ is a Henselian normal quasi-excellent local ring, so it satisfies the conclusions of the Corollary. Consider the commutative diagram
\begin{equation}\label{eq:CD}
\xymatrix{R\ar[d]_-\psi \ar[r]^-\phi & {{\Phi}at R}\ar[d]_-{{\Phi}at\psi}\\
R'\ar[r]^-{\phi'}&{{\Phi}at R'}}
\end{equation}
where ${\Phi}at R'$ stands for the formal completion of $R'$.
Since $R$ is Nagata, $R'$ is a finite $R$-module. Thus $\phi'$ coincides with the $\mathbf m$-adic completion of $R'$, viewed as an $R$-module. Hence ${\Phi}at R'\cong R'\otimes_R{\Phi}at R$. Since $\psi$ is injective and ${\Phi}at R$ is flat over $r$, the map ${\Phi}at\phi$ is also injective. Since $R'$ is analytically irreducible, ${\Phi}at R'$ is a domain, and therefore so is its subring ${\Phi}at R$. This proves that $R$ is analytically irreducible.
To prove that $R$ is algebraically closed in ${\Phi}at R$, take an element $x\in{\Phi}at R$, algebraic over $R$. Since all the maps in \ref{eq:CD} are injective, let us view all the rings involved as subrings of ${\Phi}at R'$. Since $R'$ is algebraically closed in ${\Phi}at R'$, we have $x\in R'$, in particular, we may write $x=\frac ab$ with $a,b\in R$. Now, since $(a){\Phi}at R\subset(b){\Phi}at R$ and ${\Phi}at R$ is faithfully flat over $R$, we have $(a)\subset(b)$ in $R$, so $x=\frac ab\in R$. This proves that $R$ is algebraically closed in ${\Phi}at R$. The Corollary is proved.
\end{proof}
Next we summarize, in a more specific manner, the way in which these results are applied in the present paper. The main applications are as follows.
(1) Let $R$ be an excellent local domain, $P$ a prime ideal of $R$ and $H_i\subset H_{i+1}$ two prime ideals of ${\Phi}at R$ such that
\begin{equation}
H_i\cap R=H_{i+1}\cap R=P.\label{eq:HicontractstoP}
\end{equation}
Then $\frac RP$ is also excellent. Definitions \ref{regmor}, \ref{Gring} and \ref{quasiexcellent} imply that the ring ${\Phi}at R\otimes_R\kappa(P)$ is geometrically regular over $P$, in particular, regular. Moreover, (\ref{eq:HicontractstoP}) implies that the ideal $\frac{H_{i+1}}{P{\Phi}at R}$ is a prime ideal of $\frac{{\Phi}at R}{P{\Phi}at R}$, disjoint from the natural image of $R\setminus P$ in $\frac{{\Phi}at R}{P{\Phi}at R}$. Thus the local ring $\frac{{\Phi}at R_{H_{i+1}}}{P{\Phi}at R_{H_{i+1}}}$ is a localization of ${\Phi}at R\otimes_R\kappa(P)$ at the prime ideal $H_{i+1}({\Phi}at R\otimes_R\kappa(P))$ and so is a local ring, geometrically regular over $\kappa(P)$, in particular, a regular local ring and, in particular, a domain.
(2) Assume, in addition, that $H_i$ is a minimal prime of $P{\Phi}at R$. Since $\frac{{\Phi}at R_{H_{i+1}}}{P{\Phi}at R_{H_{i+1}}}$ is a domain, $H_i$ is the only minimal prime of $P{\Phi}at R$, contained in $H_{i+1}$. We have $P{\Phi}at R_{H_{i+1}}=H_i{\Phi}at R_{H_{i+1}}$.
\end{document} |
\begin{document}
\title{Combinatorial properties of the G-degree}
\renewcommand{\scshape\small}{\scshape\small}
\renewcommand{\itshape\small}{\itshape\small}
\renewcommand{ and }{ and }
\author[1] {Maria Rita Casali}
\author[2] {Luigi Grasselli}
\affil[1] {Department of Physics, Mathematics and Computer Science, University of Modena and Reggio Emilia, \ \ \ \ \ \ \ \ \ Via Campi 213 B - 41125 Modena (Italy), [email protected]}
\affil[2] {Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Via Amendola 2, Pad. Morselli - 42122 Reggio Emilia (Italy), [email protected]}
\maketitle
\abstract {A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor models context.
In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension $d\ge 4$, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of $(d-1)!$.
As a consequence, in even dimension, the terms of the $1/N$ expansion corresponding to odd powers of $1/N$ are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context.
\noindent
In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of ``associated" cyclic permutations,
several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.}
\varepsilonndabstract
\par \noindent
{\bf Keywords}: edge-colored graph; PL-manifold; singular manifold; colored tensor model;
regular genus; Gurau-degree.
\par \noindent
\par \noindent
{\bf 2000 Mathematics Subject Classification}:
57Q15 - 57N13 - 57M15 - 83E99.
\section{Introduction}
\label{intro}
It is well-known that regular edge-colored graphs may encode PL-pseudomanifolds, giving rise to a combinatorial representation theory ({\it crystallization theory}) for singular PL-manifolds of arbitrary dimension (see Section \ref{sec: Preliminaries}).
In the last decade, the strong interaction between the topology of edge-colored graphs and random tensor models has been deeply investigated, bringing insights in both research fields.
The colored tensor models theory arises as a possible approach to the study of Quantum Gravity: in some sense, its aim is to generalize to higher dimension the matrix models theory which, in dimension two, has shown to be quite useful at providing a framework for Quantum Gravity. The key generalization is the recovery of the so called {\it $1/N$ expansion} in the tensor models context. In matrix models, the $1/N$ expansion is driven by the genera of the surfaces represented by Feynman graphs; in the higher dimensional setting of tensor models the $1/N$ expansion is driven by the {\it G-degree} of these graphs (see Definition \ref{G-degree}), that equals the genus of the represented surface in dimension two.
If $(\mathbb{C}^N)^{\otimes d }$ denotes the $d$-tensor product of the $N$-dimensional complex space $\mathbb{C}^N,$ a {\it $(d+1)$-dimensional colored tensor model} is a formal partition function
\begin{equation*}
\mathcal{Z}[N,\{\alpha_{B}\}]:=\int_{\mbox{f}}\frac{dTd\overline{T}}{(2\pi)^{N^d}}\varepsilonxp(-N^{d-1}\overline{T}\cdot T + \sum_{B}\alpha_BB(T,\overline{T})),
\varepsilonnd{equation*}
where $T$ belongs to $(\mathbb{C}^N)^{\otimes d },$ $\overline{T}$ to its dual and $B(T,\overline{T})$ are {\it trace invariants} obtained by contracting the indices of the components of $T$ and $\overline{T}.$
In this framework, colored graphs naturally arise as Feynman graphs encoding tensor trace invariants.
As shown in \cite{BGR}, the {\it free energy} $\frac{1}{N^d}\log \mathcal{Z}[N,\{t_{B}\}]$ is the formal series
\begin{equation} \label{1/N expansion}
\frac{1}{N^d}\log \mathcal{Z}[N,\{t_{B}\}] = \sum_{\omega_G\ge 0}N^{-\frac{2}{(d-1)!}\omega_G}F_{\omega_G}[\{t_B\}]\in \mathbb{C}[[N^{-1}, \{t_{B}\}]],
\varepsilonnd{equation}
\noindent where the coefficients $F_{\omega_G}[\{t_B\}]$ are generating functions of connected bipartite $(d+1)$-colored graphs with fixed G-degree $\omega_G$.
The {\it $1/N$ expansion} of formula \varepsilonqref{1/N expansion} describes the r\^ole of colored graphs (and of their G-degree $\omega_G$) within colored tensor models theory and explains the importance of trying to understand which are the manifolds and pseudomanifolds represented by $(d+1)$-colored graphs with a given G-degree.
A more detailed description of these relationships between Quantum Gravity via tensor models and topology of colored graphs may be found in \cite{BGR}, \cite{Gurau-Ryan}, \cite{Gurau-book}, \cite{Casali-Cristofori-Dartois-Grasselli}.
A parallel tensor models theory, involving {\it real} tensor variables $T\in (\mathbb{R}^N)^{\otimes d}$, has been developed, taking into account also non-bipartite colored graphs (see \cite{Witten}): this is why both bipartite and non-bipartite colored graphs will be considered within the paper.
Section \ref{sec: Preliminaries} contains a quick review of crystallization theory, including the idea of {\it regular embedding} of edge-colored graphs into surfaces, which is crucial for the definitions of {\it G-degree} and {\it regular genus} of graphs (Definition \ref{G-degree}).
In Section \ref{sec: Factorizations}, combinatorial properties concerning Hamiltonian decompositions of the complete graph allow to prove the main results of the paper.
\begin{theorem} \label{Th.1}
For each bipartite $(d+1)$-colored graph $(\Gamma,\gamma)$, with $d$ even, $d \ge 4$, \ $$\omega_G (\Gamma) \varepsilonquiv 0 \mod (d-1)!$$
\varepsilonnd{theorem}
\begin{theorem} \label{Th.2}
For each (bipartite or non-bipartite) $(d+1)$-colored graph $(\Gamma,\gamma)$ representing a singular $d$-manifold, with $d$ even, $d\ge 4,$ \ $$\omega_G (\Gamma) \varepsilonquiv 0 \mod (d-1)!$$
\varepsilonnd{theorem}
Note that the above results turn out to have specific importance in the tensor models framework. In fact Theorem \ref{Th.1} implies that, in the $d$-dimensional complex context, with $d$ even and $d\ge 4$, the only non-null terms in the $1/N$ expansion of formula \varepsilonqref{1/N expansion} are the ones corresponding to even (integer) powers of $1/N.$
On the other hand, Theorem \ref{Th.2} ensures that in the real tensor models framework, where also non-bipartite graphs are involved, the $1/N$ expansion contains colored graphs representing (orientable or non-orientable) singular manifolds - and, in particular, closed manifolds - only in the terms corresponding to even (integer) powers of $1/N.$
Both Theorems extend to arbitrary even dimension a result proved in \cite[Corollary 23]{Casali-Cristofori-Dartois-Grasselli} for graphs representing singular $4$-manifolds.
Section \ref{sec: n=4} is devoted to the 4-dimensional case: in this particular situation, the general results of Section \ref{sec: Factorizations} allow to obtain interesting properties relating
the G-degree with the topology of the associated PL 4-manifolds. In fact, the G-degree of a $5$-colored graph is shown to depend only on the regular genera with respect to an arbitrary pair of ``associated" cyclic permutations (Proposition \ref{iper-reduced G-degree(d=4)}). This fact yields relations between these two genera and the Euler characteristic of the associated PL 4-manifold (Proposition \ref{inequalities Euler characteristic} and Proposition \ref{sum/difference}); moreover, two interesting classes of crystallizations arise in a natural way, whose intersection consists in the known class of semi-simple crystallizations, introduced in \cite{[Basak-Casali 2016]} (see Remark \ref{rem:intersection-classes}).
\section{Edge-colored graphs and G-degree}
\label{sec: Preliminaries}
A {\it singular $d$-manifold} is a compact connected $d$-dimensional polyhedron admitting a simplicial triangulation where the links of vertices are closed connected $(d-1)$-manifolds, while the link of any $h$-simplex, for each $h > 0$, is a PL $(d-h-1)$-sphere. A vertex whose link is not a PL $(d-1)$-sphere is called {\it singular}.
\begin{remark}\label{correspondence-sing-boundary}
{\rm The class of singular $d$-manifolds includes the class of closed $d$-manifolds: in fact, a closed $d$-manifold is a singular $d$-manifold without singular vertices.
Moreover, if $N$ is a singular $d$-manifold, then a compact PL $d$-manifold $\check N$ is obtained by deleting small open neighbourhoods of its singular vertices.
Obviously, $N=\check N$ if and only if $N$ is a closed manifold; otherwise, $\check N$ has a non-empty boundary without spherical components.
Conversely, given a compact PL $d$-manifold $M$, a singular $d$-manifold $\widehat M$ can be obtained by capping off each component of $\partial M$ by a cone over it.
Note that, in virtue of the above correspondence,
a bijection is defined between singular $d$-manifolds and compact PL $d$-manifolds with no spherical boundary components.
}
\varepsilonnd{remark}
\begin{definition} \label{$d+1$-colored graph}
A $(d+1)$-colored graph ($d \ge 2$) is a pair $(\Gamma,\gamma)$, where $\Gamma=(V(\Gamma), E(\Gamma))$ is a regular $d+1$ valent multigraph (i.e. multiple edges are allowed, while loops are forbidden) and $\gamma: E(\Gamma) \rightarrow \Delta_d=\{0,\ldots, d\}$ is a map injective on adjacent edges, called {\it coloration}.
\varepsilonnd{definition}
For every $\mathcal B\subseteq\Delta_d$ let $\Gamma_{\mathcal B}$ be the subgraph obtained from $(\Gamma, \gamma)$ by deleting all the edges colored by $\Delta_d - \mathcal B$. The connected components of $\Gamma_{\mathcal B}$ are called {\it ${\mathcal B}$-residues} or, if $\#\mathcal B = h$, {\it $h$-residues} of $\Gamma$; the symbol $g_{\mathcal B}$ denotes their number. In the following, if $\mathcal B =\{c_1,\ldots,c_h\}$, its complementary set $\Delta_d - \mathcal B$ will be denoted by $\hat c_1\ldots\hat c_h.$
\noindent Given a $(d+1)$-colored graph $(\Gamma, \gamma)$, a $d$-dimensional pseudocomplex $K(\Gamma)$ can be associated by the following rules:
\begin{itemize}
\item for each vertex of $\Gamma$, let us consider a $d$-simplex and label its vertices by the elements of $\Delta_d$;
\item for each pair of $c$-adjacent vertices of $\Gamma$ ($c\in\Delta_d$), let us glue the corresponding $d$-simplices along their $(d-1)$-dimensional faces opposite to the $c$-labeled vertices, so that equally labeled vertices are identified.
\varepsilonnd{itemize}
$|K(\Gamma)|$ turns out to be a {\it $d$-pseudomanifold}\footnote{In fact, $|K(\Gamma)|$ is a {\it quasi-manifold}: see \cite{Gagliardi 1979}.}, which is orientable if and only if $\Gamma$ is bipartite, and $(\Gamma, \gamma)$ is said to {\it represent} it.
Note that, by construction, $K(\Gamma)$ is endowed with a vertex-labeling by $\Delta_d$ that is injective on any simplex.
Moreover, a bijective correspondence exists between the $h$-residues of $\Gamma$ colored by any $\mathcal B\subseteq\Delta_d$ and the $(d-h)$-simplices of $K(\Gamma)$ whose vertices are labeled by $\Delta_d - \mathcal B$.
In particular, for any color $c\in\Delta_d$, each connected component of $\Gamma_{\hat c}$ is a $d$-colored graph representing a pseudocomplex that is PL-homeomorphic to the link of a $c$-labeled vertex of $K(\Gamma)$ in its first barycentric subdivision.
As a consequence, $|K(\Gamma)|$ is a singular $d$-manifold (resp. a closed $d$-manifold) iff for each color
$c\in\Delta_d$, all $\hat c$-residues of $\Gamma$ represent closed $(d-1)$-manifolds (resp. the $(d-1)$-sphere).
In virtue of the bijection described in Remark \ref{correspondence-sing-boundary}, a $(d+1)$-colored graph $(\Gamma,\gamma)$ is said to {\it represent}
a compact PL $d$-manifold $M$ with no spherical boundary components if and only if it represents the associated singular manifold $\widehat M$.
\begin{definition} \label{crystallization}
A {\it crystallization} of a closed PL $d$-manifold $M^d$ is a $(d+1)$-colored graph representing $M^d$, such that each $d$-residue is connected (i.e. $g_{\hat i}=1$ \ $\forall i \in \Delta_d$).
\varepsilonnd{definition}
The following theorem extends to singular manifolds a well-known result - due to Pezzana (\cite{[Pezzana]}) - founding the combinatorial representation theory for closed PL-manifolds of arbitrary dimension via colored graphs (the so called {\it crystallization theory}). See also \cite{Cristofori-Fomynikh-Mulazzani-Tarkaev} and \cite{Cristofori-Mulazzani} for the 3-dimensional case.
\begin{theorem} {\varepsilonm (\cite[Theorem 1]{Casali-Cristofori-Grasselli})} \label{Theorem_gem}
Any singular $d$-manifold - or, equivalently, any compact $d$-manifold with no spherical boundary components - admits a $(d+1)$-colored graph representing it.
In particular, each closed PL $d$-manifold admits a crystallization.
\varepsilonnd{theorem}
It is well known the existence of a particular set of embeddings of a bipartite (resp. non-bipartite) $(d+1)$-colored graph into orientable (resp. non-orientable) surfaces.
\begin{theorem}{\varepsilonm (\cite{Gagliardi 1981})}\label{reg_emb}
Let $(\Gamma,\gamma)$ be a bipartite (resp. non-bipartite) $(d+1)$-colored graph of order $2p$. Then for each cyclic permutation $\varepsilon = (\varepsilon_0,\ldots,\varepsilon_d)$ of $\Delta_d$, up to inverse, there exists a cellular embedding, called \varepsilonmph{regular}, of $(\Gamma,\gamma)$ into an orientable (resp. non-orientable) closed surface $F_{\varepsilon}(\Gamma)$
whose regions are bounded by the images of the $\{\varepsilon_j,\varepsilon_{j+1}\}$-colored cycles, for each $j \in \mathbb Z_{d+1}$.
Moreover, the genus (resp. half the genus) $\rho_{\varepsilon} (\Gamma)$ of $F_{\varepsilon}(\Gamma)$ satisfies
\begin{equation*}
\chi (F_\varepsilon(\Gamma)) = 2 - 2\rho_\varepsilon(\Gamma)= \sum_{j\in \mathbb{Z}_{d+1}} g_{\varepsilon_j\varepsilon_{j+1}} + (1-d)p.
\varepsilonnd{equation*}
No regular embeddings of $(\Gamma,\gamma)$ exist into non-orientable (resp. orientable) surfaces.
\varepsilonnd{theorem}
The \varepsilonmph{Gurau degree} (often called {\it degree} in the tensor models literature) and the {\it regular genus} of a colored graph are defined in terms of the embeddings of Theorem \ref{reg_emb}.
\begin{definition} \label{G-degree}
Let $(\Gamma,\gamma)$ be a $(d+1)$-colored graph.
If $\{\varepsilon^{(1)}, \varepsilon^{(2)}, \dots , \varepsilon^{(\frac {d!} 2)}\}$ is the set of all cyclic permutations of $\Delta_d$ (up to inverse), $ \rho_{\varepsilon^{(i)}}$ ($i=1, \dots , \frac {d!} 2$) is called the \varepsilonmph{regular genus of $\Gamma$ with respect to the permutation $\varepsilon^{(i)}$}. Then, the \varepsilonmph{Gurau degree} (or \varepsilonmph{G-degree} for short) of $\Gamma$, denoted by $\omega_{G}(\Gamma)$, is defined as
\begin{equation*}
\omega_{G}(\Gamma) \ = \ \sum_{i=1}^{\frac {d!} 2} \rho_{\varepsilon^{(i)}}(\Gamma)
\varepsilonnd{equation*}
and the \varepsilonmph{regular genus} of $\Gamma$, denoted by $\rho(\Gamma)$, is defined as
\begin{equation*}
\rho(\Gamma) \ = \ \min\, \Big\{\rho_{\varepsilon^{(i)}}(\Gamma)\ /\ i=1,\ldots,\frac {d!} 2\Big\}.
\varepsilonnd{equation*}
\varepsilonnd{definition}
Note that, in dimension $2$, any bipartite (resp. non-bipartite) $3$-colored graph $(\Gamma,\gamma)$ represents an orientable (resp. non-orientable) surface $|K(\Gamma)|$ and $\rho(\Gamma)= \omega_G(\Gamma)$ is exactly the genus (resp. half the genus) of $|K(\Gamma)|.$
On the other hand, for $d\geq 3$, the G-degree of any $(d+1)$-colored graph (resp. the regular genus of any $(d+1)$-colored graph representing a closed PL $d$-manifold) is proved to be a non-negative {\it integer}, both in the bipartite and non-bipartite case: see \cite[Proposition 7]{Casali-Cristofori-Dartois-Grasselli} (resp. \cite[Proposition A]{Chiavacci-Pareschi}).
\section{Proof of the general results}
\label{sec: Factorizations}
Within combinatorics, the problem of the existence of $m$-cycle decompositions of the complete graph $K_n$, or of the complete multigraph $\lambda K_n$ (i.e. the multigraph with $n$ vertices and with $\lambda$ edges joining each pair of distinct vertices) is long standing: a survey result, for general $m$, $n$ and $\lambda$, is given in \cite[Theorem 1.1]{[Bryant&al]}.
Moreover, the following results hold, concerning Hamiltonian cycles (i.e. $m=n$) in $K_n$, both in the case $n$ odd and in the case $n$ even.
\begin{proposition} \ {\rm \cite[Theorem 1.3]{[Bryant]}} \label{Bryant}
For all odd $n \ge 3$ there exists a partition of all Hamiltonian cycles
of $K_n$ into $(n-2)!$ Hamiltonian cycle decompositions of $K_n$.
\varepsilonnd{proposition}
\begin{proposition} \ {\rm \cite[Theorem 2.2]{Zhao-Kang}} \label{Zhao-Kang}
For all even $n \ge 4$ there exists a partition of all Hamiltonian cycles of $K_n$ into $\frac{(n-2)!}{2}$ classes, so that each edge of $K_n$ appears in exactly two cycles belonging to the same class.
\varepsilonnd{proposition}
Figure 1 describes - as an example of Proposition \ref{Bryant} - the six Hamiltonian cycle decompositions of $K_5$, each of them containing a pair of disjoint Hamiltonian cycles (given by the dashed and continuous edges respectively).
Note that, by labelling the vertices of $K_n$ with the elements of $\Delta_{n-1}$, each Hamiltonian cycle in $K_n$ defines a cyclic permutation of $\Delta_{n-1}$ (together with its inverse), and viceversa.
\begin{figure} \label{figure1}
\centerline{\scalebox{0.4}{\includegraphics{Casali-Grasselli_Figure1.pdf}}}
\caption{The six Hamiltonian cycle decompositions of $K_5$}
\varepsilonnd{figure}
On the other hand, the following statement regarding the G-degree has been recently proved.
\begin{proposition} \ {\rm \cite[Proposition 7]{Casali-Cristofori-Dartois-Grasselli}} \label{ridotto G-degree}
If $(\Gamma, \gamma)$ is a $(d+1)$-colored graph of order $2p$ ($d\geq 3$), then
\begin{equation} \label{formula_G-degree}
\omega_G(\Gamma) \ = \ \frac{(d-1)!}{2} \cdot \Big(d + p \cdot (d-1) \cdot \frac d 2 - \sum_{r,s \in \Delta_d} g_{rs}\Big).\varepsilonnd{equation}
As a consequence, the G-degree of any $(d+1)$-colored graph is a non-negative integer multiple of $\frac {(d-1)!} 2$.
\varepsilonnd{proposition}
The result of Proposition \ref{ridotto G-degree}, which was originally stated in the bipartite case (see \cite{BGR}), suggested the definition, for $d\geq 3$, of the (integer) {\it reduced G-degree}
$$\omega^{\prime}_G(\Gamma) \ = \ \frac {2} {(d-1)!} \cdot \omega_G(\Gamma),$$
which is used by many authors within tensor models theory (see for example \cite{Gurau-book}).\footnote{In fact, the exponents of $N^{-1}$ in the $1/N$ expansion of formula \varepsilonqref{1/N expansion} are all (non-negative) integers $\omega_G^{\prime}$.}
Actually, we are able to prove that, if $d \ge 4$ is even, under
rather weak hypotheses,
the G-degree is multiple of (d-1)! (or, equivalently, the reduced G-degree is even).
\begin{proposition} \label{ridottissimo G-degree}
If $d \ge 4$ is even, and $(\Gamma, \gamma)$ is a $(d+1)$-colored graph such that
each $3$-residue is bipartite and each $d$-residue is either bipartite or non-bipartite with integer regular genus with respect to any permutation,
then
$$\omega_G(\Gamma) \varepsilonquiv 0 \mod (d-1)!$$
\varepsilonnd{proposition}
\noindent \varepsilonmph{Proof. }
Let $(\Gamma,\gamma)$ be a $(d+1)$-colored graph, with $d \ge 4,$ $d$ even.
Since $n = d+1$ is odd, Proposition \ref{Bryant} implies that all $\frac{d!}{2} $ cyclic permutations (up to inverse) of $\Delta_d$ can be partitioned in $(d-1)!$ classes, each containing $d/2$ cyclic permutations,
$\bar \varepsilon^{(1)}, \bar \varepsilon^{(2)}, \dots , \bar \varepsilon^{(\frac {d} 2)}$ say,
so that
$$ \sum_{i=1}^{d/2} \Big[ \sum_{j\in \mathbb{Z}_{d+1}} g_{\bar \varepsilon^{(i)}_j,\bar \varepsilon^{(i)}_{j+1}} \Big] = \sum_{r,s\in \Delta_{d}} g_{r,s}.$$
Hence, by Theorem \ref{reg_emb},
$$2 \cdot \frac d 2 - 2 \cdot \sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}} = \sum_{r,s \in \Delta_d} g_{rs} + p \cdot (1-d) \cdot \frac d 2,$$
and
\begin{equation} \label{eq.sum}
2 \cdot \sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}} = d + p \cdot (d-1) \cdot \frac d 2 - \sum_{r,s \in \Delta_d} g_{rs}.
\varepsilonnd{equation}
This proves that the quantity $\sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}}$ is constant, for each class of the above partition; then,
$$\omega_G (\Gamma)= (d-1)! \cdot \sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}}$$
immediately follows.\footnote{In this way, one can reobtain - for $d \ge 4$ even - relation \varepsilonqref{formula_G-degree}.}
Now, the hypotheses on the $3$-residues and $d$-residues
of $\Gamma$ directly implies, in virtue of Theorem \ref{reg_emb} and of \cite[Lemma 2]{Casali-Cristofori-Grasselli} (originally proved in \cite[Lemma 4.2]{Chiavacci-Pareschi}),
that $\sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}}$ is an integer, and hence $\omega_G (\Gamma) \varepsilonquiv 0 \mod (d-1)!$ \ \ (or equivalently, $\omega_G^{\prime} (\Gamma) = 2 \cdot \frac{\omega_G (\Gamma)}{(d-1)!}$ is even).
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
Then, the results stated in Section \ref{intro} trivially follow.
\noindent {\it Proof of Theorems 1 and 2.}
\par \noindent
Both in the case of $(\Gamma, \gamma)$ bipartite and in the case of $(\Gamma, \gamma)$ representing a singular $d$-manifold,
the residues of $\Gamma$ obviously satisfy the hypotheses of Proposition \ref{ridottissimo G-degree}.
Hence, if $d\ge 4$ is even, $\omega_G(\Gamma) \varepsilonquiv 0 \mod (d-1)!$ holds. \
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
Another particular situation is covered by Proposition \ref{ridottissimo G-degree}, as the following corollary explains.
\begin{corollary}
Let $(\Gamma,\gamma)$ be a $(d+1)$-colored graph, with $d \ge 4,$ $d$ even.
If $(\Gamma,\gamma)$ is a non-bipartite $(d+1)$-colored graph such that each $d$-residue is bipartite,
then $$\omega_G (\Gamma) \varepsilonquiv 0 \mod (d-1)!$$
\vskip-0.3truecm \ \ \\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\varepsilonnd{corollary}
Note that there exist $(d+1)$-colored graphs, with $d$ even, $d \ge 4$ and odd reduced G-degree: of course, in virtue of Theorems 1 and 2, the represented $d$-pseudomanifold must be non-orientable and it can't be a singular $d$-manifold. As an example, for each $d \ge 4$, the $(d+1)$-colored graph $(\Gamma,\gamma)$ of Figure 2 represents the $(d-2)$-th suspension $\Sigma$ of the real projective plane $\mathbb{RP}^2$ and $\omega_G^{\prime}(\Gamma)=d-1.$
It is easy to check that $\Gamma$ is non-bipartite and $\Sigma$ is not a singular manifold, since all $d$-residues of $\Gamma$, with the exception of $\Gamma_{\hat 0}$, do not represent a closed $(d-1)$-manifold.
Moreover, its (unique) $\{0,1,c \}$-residue, for any $c \in \{2, \dots, d\}$, is non-bipartite and represents the non-orientable genus one surface $\mathbb{RP}^2$.
\begin{figure} \label{figure2}
\centerline{\scalebox{0.4}{\includegraphics{Casali-Grasselli_Figure2.pdf}}}
\caption{A $(d+1)$-colored graph representing the $(d-2)$-th suspension of
$\mathbb{RP}^2$}
\varepsilonnd{figure}
\begin{remark} \label{rem:odd} {\rm
In the case $d \ge 3$ odd, Proposition \ref{Zhao-Kang} implies that all $\frac{d!}{2} $ cyclic permutations (up to inverse) of $\Delta_d$ can be partitioned in $\frac{(d-1)!}{2}$ classes, each containing $d$ cyclic permutations,
$\bar \varepsilon^{(1)}, \bar \varepsilon^{(2)}, \dots , \bar \varepsilon^{(d)}$ say,
so that
$$ \sum_{i=1}^{d} \Big[ \sum_{j\in \mathbb{Z}_{d+1}} g_{\bar \varepsilon^{(i)}_j,\bar \varepsilon^{(i)}_{j+1}} \Big] = 2 \sum_{r,s\in \Delta_{d}} g_{r,s}.$$
Hence, a reasoning similar to the one used to prove Proposition \ref{ridottissimo G-degree} yields an alternative proof - for $d \ge 3$ odd - of relation \varepsilonqref{formula_G-degree}:
since $$2 \cdot d - 2 \cdot \sum_{i=1}^d \rho_{\bar \varepsilon^{(i)}} = 2 \sum_{r,s \in \Delta_d} g_{rs} + p \cdot (1-d) \cdot d$$
and
\begin{equation} \label{eq.sum(odd)} \sum_{i=1}^d \rho_{\bar \varepsilon^{(i)}} = d + p \cdot (d-1) \cdot \frac d 2 - \sum_{r,s \in \Delta_d} g_{rs}
\varepsilonnd{equation}
hold, then
$$\omega_G (\Gamma)= \frac{(d-1)!}{2} \cdot \sum_{i=1}^d \rho_{\bar \varepsilon^{(i)}} = \frac{(d-1)!}{2} \cdot \Big(d + p \cdot (d-1) \cdot \frac d 2 - \sum_{r,s \in \Delta_d} g_{rs} \Big)$$
directly follows.}
\varepsilonnd{remark}
\begin{remark} \label{rem:constant-general} {\rm
It is worthwhile to stress that, for $d$ even (resp. odd), formula \varepsilonqref{eq.sum} of Proposition \ref{ridottissimo G-degree} (resp. formula \varepsilonqref{eq.sum(odd)} of Remark \ref{rem:odd}) proves that the sum $\sum_{i=1}^{d/2} \rho_{\bar \varepsilon^{(i)}}$ (resp. $ \sum_{i=1}^d \rho_{\bar \varepsilon^{(i)}}$) of all regular genera with respect to the $d/2$ (resp. $d$) permutations belonging to the same class
is half the reduced G-degree (resp. is the reduced G-degree)
$$ \omega^{\prime}_G(\Gamma) \ = \ d + p \cdot (d-1) \cdot \frac d 2 - \sum_{r,s \in \Delta_d} g_{rs}, $$
i.e. it is a constant which does not depend on the chosen partition class.
Hence, the regular genus $\rho(\Gamma)$ of the graph $\Gamma$ is realized by the (not necessarily unique) permutation $\varepsilon$ which maximizes the difference
$$ \rho_{\hat \varepsilon}(\Gamma) - \rho_{\varepsilon}(\Gamma),$$
where $ \rho_{\hat \varepsilon}(\Gamma)$ denotes the sum of the genera with respect to all other permutations of the same partition class.
}
\varepsilonnd{remark}
\section{The $4$-dimensional case} \label{sec: n=4}
In the $4$-dimensional setting, the above combinatorial properties allow to prove further results about the G-degree.
In fact, it is easy to check that, for each cyclic permutation $\varepsilon = (\varepsilon_0, \varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4)$ of $\Delta_4$, we have
\begin{equation} \label{2-factorization n=4}
\{ (\varepsilon_j,\varepsilon_{j+1}) \ / \ j\in \mathbb{Z}_{5}\} \ \cup \ \{ (\varepsilon_j, \varepsilon_{j+2}) \ / j\in \mathbb{Z}_{5}\} \ = \ \{ (i,j) \ / \ i,j\in \Delta_{5}, i \ne j\}.
\varepsilonnd{equation}
Let us denote by $\varepsilon^{\prime}=(\varepsilon^{\prime}_0, \varepsilon^{\prime}_1, \varepsilon^{\prime}_2, \varepsilon^{\prime}_3, \varepsilon^{\prime}_4)$ the permutation $(\varepsilon_0, \varepsilon_2, \varepsilon_4, \varepsilon_1, \varepsilon_3)$ of $\Delta_4$, which may be said to be {\it associated} to $\varepsilon$.
Note that, when $d=4$, the only partition of all $12$ cyclic permutations of $\Delta_4$ (up to inverse) is given by the six classes containing a given permutation $\varepsilon$ and its associated $\varepsilon^{\prime}$: see Figure 1, where each class of the Hamiltonian cycle decomposition of $K_5$ ensured by Proposition \ref{Bryant} is shown to correspond to a pair $(\varepsilon_i, \varepsilon_i^{\prime})$ of associated permutations, for $i \in \mathbb N_6.$ For example, the first class corresponds to the identical permutation $\varepsilon_1 = (0,1,2,3,4)$ and its associated permutation $\varepsilon_1^{\prime} = (0,2,4,1,3).$
Then, the following result holds.
\begin{proposition} \label{iper-reduced G-degree(d=4)}
For each $5$-colored graph $(\Gamma, \gamma),$ and for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$,
$$ \omega_{G}(\Gamma) \ = \ 6 \Big(\rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma)\Big).$$
\varepsilonnd{proposition}
\noindent \varepsilonmph{Proof. }
Equality \varepsilonqref{2-factorization n=4} directly yields
$$ \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1}} + \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon^{\prime}_j,\varepsilon^{\prime}_{j+1}}= \sum_{i,j\in \Delta_5} g_{i,j}.$$
As a consequence, the sum of all regular genera of $\Gamma$ with respect to the $12$ cyclic permutations (up to inverse) of $\Delta_4$ is six times the sum between the regular genera of $\Gamma$ with respect to any pair $\varepsilon$, $\varepsilon^{\prime}$ of associated permutations:
$$ \omega_{G}(\Gamma) \ = \ 6 \Big(\rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma)\Big).$$
\vskip-0.6truecm \ \ \\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\begin{remark} \label{sum_constant} {\rm
By Proposition \ref{iper-reduced G-degree(d=4)}, for any 5-colored graph the sum between the regular genera of $\Gamma$ with respect to any pair $\varepsilon$, $\varepsilon^{\prime}$ of associated cyclic permutations is
constant (see equality \varepsilonqref{eq.sum} and Remark \ref{rem:constant-general}, for $d=4$):
$$ \rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma) \ = \ \frac 1 2 \omega'_G(\Gamma) = 2+3p - \frac 1 2 \sum_{i,j\in \Delta_{5}} g_{i,j}.$$
Hence,
the regular genus $\rho(\Gamma)$ of the graph $\Gamma$ is realized by the (not necessarily unique) permutation $\varepsilon$ so that $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma)$ is maximal.
}
\varepsilonnd{remark}
Moreover:
\begin{proposition} \label{differences}
\begin{itemize}
\item[(a)]
If $(\Gamma, \gamma)$ is a $5$-colored graph, then for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$,
$$ 2 \Big(\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \Big) \ = \ \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1}} - \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2}}$$
\item[(b)]
If $(\Gamma, \gamma)$ is a $5$-colored graph representing a singular $4$-manifold $M^4$, then for each pair $(\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$,
$$ \rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \ = \ \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1},\varepsilon_{j+2}} - \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2}, \varepsilon_{j+4}}$$
\varepsilonnd{itemize}
\varepsilonnd{proposition}
\noindent \varepsilonmph{Proof. }
Statement (a) is an easy consequence of Theorem \ref{reg_emb}:
$$2-2 \rho_{\varepsilon}(\Gamma) = \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1}} -3p$$
\noindent and
$$ 2-2 \rho_{\varepsilon^{\prime}}(\Gamma) = \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon^{\prime}_j,\varepsilon^{\prime}_{j+1}} -3p= \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2}} -3p.$$
On the other hand, relation $2 g_{r,s,t} = g_{r,s} + g_{r,t} + g_{s,t} - p$
is known to be true for each order $2p$ $5$-colored graph representing a singular $4$-manifold (see \cite[Lemma 21]{Casali-Cristofori-Dartois-Grasselli}. As a consequence we have:
$$ 2 \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1},\varepsilon_{j+2}}= \sum_{r,s\in \mathbb{Z}_{5}} g_{r,s} + \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1}} - 5p$$ and
$$ 2 \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2},\varepsilon_{j+4}}= \sum_{r,s\in \mathbb{Z}_{5}} g_{r,s} + \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2}} - 5p.$$
By making the difference,
$$2 \Big(\sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1},\varepsilon_{j+2}} - \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2},\varepsilon_{j+4}}\Big) = \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+1}} - \sum_{j\in \mathbb{Z}_{5}} g_{\varepsilon_j,\varepsilon_{j+2}}$$ is obtained; so, statement (b) follows, via statement (a).
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
Proposition \ref{differences} enables to obtain
the following improvement of \cite[Proposition 29(a)]{Casali-Cristofori-Dartois-Grasselli}.
\begin{corollary}
Let $(\Gamma,\gamma)$ be a $5$-colored graph. \ Then:
$$ \omega_G(\Gamma) \ = \ 12 \cdot \rho (\Gamma) \ \ \ \Longleftrightarrow$$
$$ \sum_{j\in\mathbb{Z}_{5}} g_{\varepsilon_j, \varepsilon_{j+1}} = \sum_{j\in\mathbb{Z}_{5}} g_{\varepsilon_j, \varepsilon_{j+2}} \ \text{for each cyclic permutation} \ \varepsilon \ \text{of} \ \Delta_4.$$
\varepsilonnd{corollary}
\noindent \varepsilonmph{Proof. }
From \cite[Proposition 29 (a)]{Casali-Cristofori-Dartois-Grasselli}, it is known that
$$ \omega_G(\Gamma) \ = \ 12 \cdot \rho (\Gamma) \ \ \ \Longrightarrow \ \ \ \ \sum_{j \in \mathbb{Z}_{5}} g_{\bar \varepsilon_j, \bar \varepsilon_{j+1}} = \sum_{j\in\mathbb{Z}_{5}} g_{\bar \varepsilon_j, \bar \varepsilon_{j+2}},$$
$\bar \varepsilon$ being the cyclic permutation of $\Delta_4$ such that $\rho(\Gamma) = \rho_{\bar \varepsilon}(\Gamma).$
So, $\rho_{\bar \varepsilon^{\prime}}(\Gamma) - \rho_{\bar \varepsilon}(\Gamma) = 0$ directly follows via Proposition \ref{differences} (a).
Now, since $\rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma)$ is constant for each pair $(\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$ (see Remark \ref{sum_constant}), then:
$$|\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma)| \le \rho_{\bar \varepsilon^{\prime}}(\Gamma) - \rho_{\bar \varepsilon}(\Gamma).$$
Hence, $\rho_{\bar \varepsilon^{\prime}}(\Gamma) - \rho_{\bar \varepsilon}(\Gamma) = 0$ implies $\sum_{j\in\mathbb{Z}_{5}} g_{\varepsilon_j, \varepsilon_{j+1}} = \sum_{j\in\mathbb{Z}_{5}} g_{\varepsilon_j, \varepsilon_{j+2}}$ (i.e. $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) = 0$, in virtue of Proposition \ref{differences} (a)) for each cyclic permutation $\varepsilon$ of $\Delta_4$.
\par \noindent
The reversed implication is straightforward, via Proposition \ref{differences}(a).
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\begin{proposition} \label{inequalities Euler characteristic}
If $(\Gamma, \gamma)$ is an order $2p$ 5-colored graph representing
a singular 4-manifold $M^4$, then, for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$:
$$ \chi(M^4) \ = \ \Big(\rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma)\Big) -p + \sum_{i\in \Delta_{4}} g_{\hat i} -2.$$
\varepsilonnd{proposition}
\noindent \varepsilonmph{Proof. }
It is sufficient to apply Proposition \ref{iper-reduced G-degree(d=4)} to the third equality of \cite[Proposition 22]{Casali-Cristofori-Dartois-Grasselli}.
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
Let us now recall two particular types of crystallizations introduced and studied in \cite{[Basak-Casali 2016]} and \cite{[Basak]}\footnote{Both semi-simple and weak semi-simple crystallizations generalize the notion of {\it simple crystallizations} for simply-connected PL $4$-manifolds: see \cite{[Basak-Spreer 2016]} and \cite{[Casali-Cristofori-Gagliardi JKTR 2015]}.}: they are proved to be ``minimal" with respect to regular genus, among all graphs representing the same PL $4$-manifold.
\begin{definition} {\rm A crystallization of a PL $4$-manifold $M^4$ with \, $rk(\pi_1(M^4))= m \ge 0$ is called a {\varepsilonm semi-simple crystallization}
if \ $g_{j,k,l} = 1 + m \ \ \forall \ j,k,l \in \Delta_4.$
A crystallization of a PL $4$-manifold $M^4$ with \, $rk(\pi_1(M^4))= m $ is called a {\varepsilonm weak semi-simple crystallization}
if \ $g_{i, i+1, i +2} = 1 + m \ \ \forall \ i \in \Delta_4$ (where the additions in subscripts are intended in $\mathbb{Z}_{5}$).
}
\varepsilonnd{definition}
According to \cite{[Basak-Casali 2016]}, for each order $2p$ crystallization $(\Gamma, \gamma)$ of a closed PL 4-manifold $M^4$, with \, $rk(\pi_1(M^4))= m0$, let us set
$$g_{j,k,l}= 1+m + t_{j,k,l}, \ \ \text{with} \ t_{j,k,l} \ \ \forall j,k,l \in \Delta_4.$$
Semi-simple (resp. weak semi-simple) crystallizations turn out to be characterized by $t_{j,k,l} = 0$ \ $\forall j,k,l \in \Delta_4$ (resp. $t_{i, i+1, i +2} = 0$ \ $\forall i \in \mathbb{Z}_{5}$).
In \cite{[Basak-Casali 2016]}, the relation
\begin{equation} \label{p and q}
p= 3 \chi(M^4)+ 5(2m -1) + \sum_{j,k,l \in \Delta_4} t_{j,k,l}
\varepsilonnd{equation}
is proved to hold; hence, $p = \bar p + q$ follows, where $q= \sum_{j,k,l \in \Delta_4} t_{j,k,l} \ge 0$ and $\bar p= 3 \chi(M^4)+ 5(2m -1)$ is the minimum possible half order of a crystallization of $M^4$, which is attained if and only if $M^4$ admits semi-simple crystallizations.
With the above notations, the following results can be obtained.
\begin{proposition} \label{sum/difference}
Let $(\Gamma,\gamma)$ be an order $2p$ crystallization of a closed PL 4-manifold $M^4$, with $rk(\pi_1(M^4))=m.$ Then, for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$, with $\rho_{\varepsilon}(\Gamma) \le \rho_{\varepsilon^{\prime}}(\Gamma)$:
\begin{equation}
\label{eq.difference}
\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) = q - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ \le \ q.
\varepsilonnd{equation}
Moreover, The following statements are equivalent:
\begin{itemize}
\item[(a)] a cyclic permutation $\varepsilon$ of $\Delta_4$ exists, such that $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q;$
\item[(b)] $\Gamma$ is a weak semi-simple crystallization;
\item[(c)] $ \rho(\Gamma)= 2\chi(M^4) + 5 m -4$.\footnote{Note that, in this case, the permutation $\bar \varepsilon$ such that $\rho(\Gamma)= \rho_{\bar \varepsilon}(\Gamma)$ coincides with the permutation $\varepsilon$ of point (a); moreover, $\rho(\Gamma)= \mathcal G(M^4)$ holds, in virtue of the inequality $\mathcal G(M^4) \ge 2\chi(M^4) + 5 m -4$, proved in \cite{[Basak-Casali 2016]}.}
\varepsilonnd{itemize}
\varepsilonnd{proposition}
\noindent \varepsilonmph{Proof. }
In virtue of Proposition \ref{differences}(b), an easy computation proves relation \varepsilonqref{eq.difference}:
$$ \begin{aligned} \rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \ & = \ \sum_{i\in \mathbb{Z}_{5}} (m+1 + t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}} ) - \sum_{i\in \mathbb{Z}_{5}} (m+1 + t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}) \ = \\
\ & = \ \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}} - \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ = \\
\ & = \ \sum_{j,k,l \in \Delta_4} t_{j,k,l} - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ = \\
\ & = \ q - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ \le \ q. \varepsilonnd{aligned}$$
Now, if $\Gamma$ is a weak semi-simple crystallization, by definition itself a cyclic permutation $\varepsilon$ of $\Delta_4$ exists\footnote{$\varepsilon$ turns out to be the permutation of $\Delta_4$ associated to $\varepsilon^{\prime}=(0,1,2,3,4)$.}, so that, for each $i \in \mathbb Z_5,$ \ $g_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}= m+1,$ i.e. $t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}=0.$ So, $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q$ easily follows from relation \varepsilonqref{eq.difference}.
On the other hand, if a cyclic permutation $\varepsilon$ of $\Delta_4$ exists, so that $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q,$ relation \varepsilonqref{eq.difference} yields $t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}=0,$ i.e. $g_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}= m+1,$ which is exactly - up to color permutation - the definition of weak semi-simple crystallization.
Hence, (a) and (b) are proved to be equivalent.
Then, by comparing Proposition \ref{inequalities Euler characteristic} (with the assumption $g_{\hat i}=1$ \ $\forall i \in \Delta_4$) and relation \varepsilonqref{p and q}, we obtain:
\begin{equation} \label{sum_q}
\rho_{\varepsilon^{\prime}}(\Gamma) + \rho_{\varepsilon}(\Gamma) = 2 ( 2 \chi(M^4) + 5m -4) + q.
\varepsilonnd{equation}
By making use of relation \varepsilonqref{eq.difference},
\begin{equation} \label{rho-corrected}
\rho_{\varepsilon}(\Gamma) = 2 \chi(M^4) + 5m -4 + \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}
\varepsilonnd{equation}
easily follows, as well as
\begin{equation} \label{rho-epsilon'} \rho_{\varepsilon^{\prime}}(\Gamma) = 2 \chi(M^4) + 5m -4 + \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}}.
\varepsilonnd{equation}
Relation \varepsilonqref{rho-corrected} directly yields the co-implication between statements (b) and (c).
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
We conclude the paper with a list of remarks, which arise from the previous results.
\begin{remark} {\rm With the above notations, formulae \varepsilonqref{rho-corrected} and \varepsilonqref{rho-epsilon'} immediately give
$$ 2 + \frac {\rho_{\varepsilon}(\Gamma)} 2 - \frac {5 m} 2 - \frac q 4 \ \le \ \chi(M^4) \ \le \ 2 + \frac {\rho_{\varepsilon^{\prime}}(\Gamma)} 2 - \frac {5 m} 2 - \frac q 4$$
for each crystallization of a closed PL 4-manifold $M^4$.
Another double inequality concerning the Euler characteristic of a closed PL 4-manifold $M^4$ and the regular genera of any order $2p$ crystallization of $M^4$ with respect to a pair of associated permutations may be easily obtained from Proposition \ref{inequalities Euler characteristic},
by making use of the assumptions $\sum_{i\in \Delta_{4}} g_{\hat i}=5$ and $\rho_{\varepsilon}(\Gamma) \le \rho_{\varepsilon^{\prime}}(\Gamma):$
$$ 2\rho_{\varepsilon}(\Gamma) -p +3 \ \le \ \chi(M^4) \ \le \ 2 \rho_{\varepsilon^{\prime}}(\Gamma) -p +3.$$
Note that such double inequalities assume a specific relevance in case of ``low" difference between $\rho_{\varepsilon^{\prime}}(\Gamma)$ and $\rho_{\varepsilon}(\Gamma)$ (in particular if $\rho_{\varepsilon^{\prime}}(\Gamma)= \rho_{\varepsilon}(\Gamma)$ occurs, possibly with $\rho(\Gamma) < \rho_{\varepsilon}(\Gamma)$).
On the other hand, Proposition \ref{inequalities Euler characteristic} and formula \varepsilonqref{eq.difference} (resp. formula \varepsilonqref{sum_q}) give
$$\chi(M^4) \ = \ 2 \rho_{\varepsilon}(\Gamma) -p + 3 + (q -2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}})$$
$$ \text{\Big(resp. \ \ } \chi(M^4) \ = \ 2 + \frac {\rho_{\varepsilon}(\Gamma)} 2 - \frac {5 m} 2 - \frac 1 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ \text{\Big)}.$$
Hence, the following double inequalities arise, too, both involving the regular genus with respect to only one cyclic permutation: $$ 2\rho_{\varepsilon}(\Gamma) -p +3 \ \le \ \chi(M^4) \ \le \ 2 \rho_{\varepsilon}(\Gamma) - p + q +3; $$
$$ 2 + \frac {\rho_{\varepsilon}(\Gamma)} 2 - \frac {5 m} 2 - \frac q 4 \ \le \ \chi(M^4) \ \le \ 2 + \frac {\rho_{\varepsilon}(\Gamma)} 2 - \frac {5 m} 2 .$$
}
\varepsilonnd{remark}
\begin{remark} {\rm Note that relation \varepsilonqref{sum_q} exactly corresponds, via Proposition \ref{iper-reduced G-degree(d=4)}, to \cite[Proposition 27]{Casali-Cristofori-Dartois-Grasselli}. Moreover, relation \varepsilonqref{rho-corrected}\footnote{Actually, relation \varepsilonqref{rho-corrected} corrects a trivial error in the proof (and statement) of \cite[Lemma 7]{[Basak]}, not affecting the implications in order to prove the main result of that paper.}
directly implies the inequality $\rho_{\varepsilon}(\Gamma) \ge 2 \chi(M^4) + 5m -4$ (which is one of the upper bounds obtained in \cite{[Basak-Casali 2016]}) and ensures that, for each crystallization $(\Gamma,\gamma)$ of a PL 4-manifold, the regular genus $\rho(\Gamma)$ is realized by the (not necessarily unique) permutation $\varepsilon$ so that $\sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}$ (or, equivalently, $\sum_{i\in \mathbb{Z}_{5}} g_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}$) is minimal.
Finally, the equivalence between items (b) and (c) in the above Proposition \ref{sum/difference} gives a direct proof of \cite[Theorem 2]{[Basak]}.
}
\varepsilonnd{remark}
\begin{remark} \label{rem:intersection-classes} {\rm Semi-simple crystallizations turn out to be the intersection (characterized by $q=0$) between the two classes of weak semi-simple crystallizations and of crystallizations satisfying $\omega_G(\Gamma) \ = \ 12 \cdot \rho (\Gamma)$ (and hence $\rho_{\varepsilon}(\Gamma) = 2 \chi(M^4) + 5m -4 + \frac 1 2 q,$ for each cyclic permutation $\varepsilon$ of $\Delta_4$).
Moreover, it is easy to check that $$ q\le 2 \ \ \ \ \Longrightarrow \ \ \ \ \Gamma \ \text{ is a weak semi-simple crystallization}.$$
In fact, if $q = \sum_{j,k,l \in \Delta_4} t_{j,k,l} \le 2, $ at most two triads $(j,k,l)$ of distinct elements in $\Delta_4$ exist, so that $g_{j,k,l}= 1+m + t_{j,k,l} > 1+m.$ This ensures the existence of a cyclic permutation $\varepsilon$ of $\Delta_4$ so that, for each $i \in \mathbb Z_5,$ \ $g_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}}= m+1,$ which is exactly the requirement for a weak semi-simple crystallization.
}
\varepsilonnd{remark}
\begin{remark} {\rm The formula obtained in \cite[Lemma 4.2]{Gurau-Ryan} for bipartite $(d+1)$-colored graphs and extended to the general case in \cite[Lemma 13]{Casali-Cristofori-Dartois-Grasselli} gives, if $d=4$,
$$\omega_G(\Gamma)= 3 \Big(p + 4 - \sum_{i\in \Delta_{4}} g_{\hat i}\Big) + \sum_{i\in \Delta_{4}} \omega_G(\Gamma_{\hat i}),$$
where, for each $i\in \Delta_{4}$, $\omega_G(\Gamma_{\hat i})$ denotes the sum of the G-degrees of the connected components of $\Gamma_{\hat i}.$
Hence, $\sum_{i\in \Delta_{4}} \omega_G(\Gamma_{\hat i})$ is always a multiple of $3$ (recall Proposition \ref{iper-reduced G-degree(d=4)} and Theorem \ref{Th.1}).
Moreover, if $(\Gamma,\gamma)$ represents a singular $4$-manifold $M^4$, Proposition \ref{iper-reduced G-degree(d=4)} and Proposition \ref{inequalities Euler characteristic} imply:
$$ \sum_{i\in \Delta_{4}} \omega_G(\Gamma_{\hat i}) \ = \ 3 \Big(2 \chi(M^4) + p - \sum_{i\in \Delta_{4}} g_{\hat i}\Big).$$
Note that, if $(\Gamma,\gamma)$ is a crystallization of a closed PL $4$-manifold $M^4$ with $rk(\pi_1(M^4))=m$, relation \varepsilonqref{p and q} gives:
$$\sum_{i\in \Delta_{4}} \omega_G(\Gamma_{\hat i}) \ =
\ 3 \Big[5 \Big(\chi(M^4) + 2m -2 \Big) + q \Big].$$
In particular, if $(\Gamma,\gamma)$ is semi-simple (i.e. $q=0$), $ \sum_{i\in \Delta_{4}} \omega_G(\Gamma_{\hat i}) \ = \ 15 \Big(\chi(M^4) + 2m -2\Big)$ follows, as \cite[Proposition 8]{[Basak-Casali 2016]} trivially implies.
}
\varepsilonnd{remark}
\section*{Acknowledgments}
This work was supported by the {\it ``National Group for Algebraic and Geometric Structures, and their Applications''} (GNSAGA - INDAM) and by University of Modena and Reggio Emilia, projects {\it ``Colored graphs representing pseudomanifolds: an interaction with random geometry and physics"} and {\it ``Applicazioni della Teoria dei Grafi nelle Scienze, nell'Industria e nella Societ\'a"}.
\par \noindent
The authors would like to thank Gloria Rinaldi for her helpful ideas and suggestions about relationship between cyclic permutation properties and Hamiltonian cycle decompositions of complete graphs.
\noindent They also thank the referees for their very helpful suggestions.
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\varepsilonnd{document}
\begin{prop} \label{max difference}
Let $(\Gamma,\gamma)$ be an order $2p$ crystallization of a closed PL 4-manifold $M^4$, with $rk(\pi_1(M^4)=m;$ then, for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$, with $\rho_{\varepsilon}(\Gamma) \le \rho_{\varepsilon^{\prime}}(\Gamma)$:
$$\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \le q,$$
where $q = p- 3 \chi(M^4)- 5(2m-1) \ge 0,$
and
$$ 2\rho_{\varepsilon}(\Gamma) -p +3 \ \le \ \chi(M^4) \ \le \ 2 \rho_{\varepsilon}(\Gamma) - p + q +3.$$
Moreover,
$$ \text{a
permutation} \ \varepsilon \ \text{exists, such that} \ \rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q \ \ \ \ \Longleftrightarrow \ \ \ \ \Gamma \ \text{ is a weak semi-simple crystallization}.$$
\varepsilonnd{prop}
\noindent \varepsilonmph{Proof. }
Let us assume, according to \cite{[Basak-Casali 2016]} or \cite{[Basak]},
$g_{j,k,l}= 1+m + t_{j,k,l},$ with $t_{j,k,l} \ge 0$ for each $j,k,l \in \Delta_4$; it is not difficult to check that $ \sum_{j,k,l \in \Delta_4} t_{j,k,l}=q$ (see the quoted papers).
In virtue of Proposition \ref{differences}(b), we have:
\begin{equation}
\begin{aligned} \rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \ & = \ \sum_{i\in \mathbb{Z}_{5}} (m+1 + t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}} ) - \sum_{i\in \mathbb{Z}_{5}} (m+1 + t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}) \ = \\
\ & = \ [5(m+1) + \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}}] - [5(m+1) + \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}] \ = \\
\ & = \ \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+1},\varepsilon_{i+2}} - \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ = \\
\ & = \ \sum_{j,k,l \in \Delta_4} t_{jkl} - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ = \\
\ & = \ q - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}} \ \le \ q.
\varepsilonnd{aligned}
\varepsilonnd{equation}
On the other hand, $\rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma) = \chi(M^4) +p - 3$ and
$\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) \le q$ yield $ \chi(M^4) \ \le \ 2 \rho_{\varepsilon}(\Gamma) - p + q +3.$ Hence, the double inequality directly follows, by making use of the first one obtained in Proposition \ref{inequalities Euler characteristic}.
Moreover, if $\Gamma$ is a weak semi-simple crystallization, by definition itself a cyclic permutation $\varepsilon$ of $\Delta_4$ Exists\footnote{By definition, $\varepsilon$ turns out to be the permutation of $\Delta_4$ associated to the identical one.}, so that, for each $i \in \mathbb Z_5,$ \ $g_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}= m+1,$ i.e. $t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}=0.$ So, $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q$ easily follows from the above computation.
On the other hand, if a cyclic permutation $\varepsilon$ of $\Delta_4$ exists, so that $\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q,$ the same computation yields $t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}=0,$ i.e. $g_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}= m+1,$ which is exactly - up to color permutation - the definition of weak semi-simple crystallization. \\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\begin{prop}
Let $(\Gamma,\gamma)$ be an order $2p$ crystallization of a closed PL 4-manifold $M^4;$ then, for each pair ($\varepsilon, \varepsilon^{\prime})$ of associated cyclic permutations of $\Delta_4$, with $\rho_{\varepsilon}(\Gamma) \le \rho_{\varepsilon^{\prime}}(\Gamma)$,
$$ 2 + \frac {\rho_{\varepsilon}(\Gamma)} 2 - \frac {5 m} 2 - \frac q 4 \ \le \ \chi(M^4) \ \le \ 2 + \frac {\rho_{\varepsilon^{\prime}}(\Gamma)} 2 - \frac {5 m} 2 - \frac q 4,$$
where $q = p
- 3 \chi(M^4)+ 5(2m-1) \ge 0.$
\varepsilonnd{prop}
\noindent \varepsilonmph{Proof. }
This double inequality is a consequence of the one in Proposition \ref{inequalities Euler characteristic}, by making use of the relation $p = 3 \chi(M^4)+ 5(2m-1) + q$, with $q\ge 0$.
\\ \rightline{$\Box$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\begin{rem} {\rm It is easy to prove also that
$$ \text{a cyclic permutation} \ \varepsilon \ \text{exists, so that} \ \rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) =q \ \ \ \ \Longleftrightarrow \ \ \ \ \mathcal G(M^4) = \rho_{\varepsilon}(\Gamma) = 2\chi(M^4) + 5 m -4,$$
where $ \mathcal G(M^4)$ denotes the so called {\it regular genus of the PL 4-manifold} $M^4$, i.e.
$$\mathcal G(M^4)=\min \{\rho_{\varepsilon}(\Gamma)\ | \ (\Gamma,\gamma)\mbox{ represents} \ M^4 \ \mbox{and $\varepsilon$ is a cyclic permutation of $\Delta_4$}\}.$$
In fact, the equality \varepsilonqref{eq.difference} gives
$\rho_{\varepsilon^{\prime}}(\Gamma) - \rho_{\varepsilon}(\Gamma) = q - 2 \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}}.$
On the other hand, \cite[Proposition 27]{Casali-Cristofori-Dartois-Grasselli} implies
$ \rho_{\varepsilon}(\Gamma) + \rho_{\varepsilon^{\prime}}(\Gamma) = 2 ( 2 \chi(M^4) + 5m -4) + q$.
\par \noindent
Hence,
$$ \rho_{\varepsilon}(\Gamma) = 2 \chi(M^4) + 5m -4 + \sum_{i\in \mathbb{Z}_{5}} t_{\varepsilon_i,\varepsilon_{i+2},\varepsilon_{i+4}},$$
and the statement immediately follows.
As a consequence, by means of the second part of Proposition \ref{max difference}, a direct proof of \cite[Theorem 2]{[Basak]} is obtained.
}
\varepsilonnd{rem}
\varepsilonnd{document} |
\begin{document}
\draft
\title{Statistical Properties of Quantum Graph Spectra}
\author{Yu. Dabaghian}
\address{Department of Physiology,
Keck Center for Integrative Neuroscience,\\
University of California,
San Francisco, California 94143-0444, USA \\
e-mail [email protected]}
\date{\today}
\maketitle
\begin{abstract}
A general analytical approach to the statistical description of quantum
graph spectra based on the exact periodic orbit expansions of quantum levels
is discussed. The exact and approximate expressions obtained in \cite{Anima}
for the probability distribution functions using the spectral hierarchy method
are analyzed. In addition, the mechanism of appearance of the universal
statistical properties of spectral fluctuations of quantum-chaotic systems
is considered in terms of the semiclassical theory of periodic orbits.
\end{abstract}
\pacs{03.65.Sq, 05.45.+b}
\section{Introduction}
A quantum graph system consists of a quantum particle moving along the bonds
of an arbitrary finite graph $G$ \cite{QGT}. In the classical limit, this
system generates a simple stochastic dynamics, which is specified by the
translational motion along the bonds of the graph and stochastic scattering
at its vertices with preset scattering probabilities. This dynamics has many
common features with the dynamics of usual chaotic systems \cite{Gaspard}.
For example, periodic trajectories in such a system are isolated and their
number increases exponentially with the period. At the same time, the statistical
behavior of various spectral characteristics of sufficiently complex quantum
graphs, e.g. the probability distribution of spacings $s_{n}=k_{n}-k_{n-1}$
between the nearest levels of the momentum was numerically shown \cite{QGT}
to follow the predictions of the Random Matrix Theory (RMT) \cite{BGS,Zaslavsky},
as it is usually the case for classically nonintegrable systems. It also turns
out that a great number of problems of classical and quantum dynamics on the
graph allow exact solutions, which makes these systems convenient models in
the context of the analytical theory of ``quantum chaos''. In particular, for
these systems there exist the exact periodic orbit expansions of the quantum
density of states (Gutzwiller formula) \cite{QGT} along with a similar expansion
for the spectral staircase:
\begin{equation}
N(k)\equiv \sum_{j=1}^{\infty }\Theta \left( k-k_{j}\right)=\bar{N}(k)+
\frac{1}{\pi}\mathop{\rm Im}\sum_{p}A_{p}e^{iL_{p}^{(0)}k},
\label{gutzw}
\end{equation}
Here $\bar{N}(k)$ is the average number of levels in the range $\left[0,k\right]$,
$L_{p}^{(0)}$ is the optical length of the periodic trajectory with the index $p$,
and $A_{p}$ is a certain weight factor explicitly defined in terms of the scattering
coefficients at the graph vertices.
It should be emphasized that the existence of the explicit expansions of the global
characteristics such as (\ref{gutzw}) is not equivalent to the ultimate solution of
the spectral problem, which should provide local information about the individual
levels in the form of an explicit dependence $k_{n}=k(n)$. An approach for determining
the quantities $k_{n}$ explicitly was proposed in \cite{Anima}, which is based on
using a finite system of $r+2$ auxiliary ``separators'' $\hat{k}_{n}^{(0)}$,
$\hat{k}_{n+1}^{(1)}$, ..., $\hat{k}_{n}^{(r+1) }$, the first of which is the physical
spectral sequence $k_{n}=\hat{k}_{n}^{(0)}$, and the last one is a globally defined
explicit function of $n$:
\begin{equation}
\hat{k}_{n}^{(r+1) }=\frac{\pi }{L_{0}}\left( n+\frac{1}{2}\right).
\label{mid}
\end{equation}
The key property of these sequences is that they must satisfy the ``bootstrapping''
conditions
\begin{equation}
\hat{k}_{n}^{(j)}<\hat{k}_{n}^{(j-1)}<\hat{k}_{n+1}^{(j)},
\label{bootstrap}
\end{equation}
which guarantee that between every pair of the neighboring points $\hat{k}_{n}^{(j)}$
and $\hat{k}_{n+1}^{(j)}$ (see Fig. 1) there exists a single point $\hat{k}_{n}^{(j-1)}$.
In \cite{Anima} it was also pointed out that due to certain analytical properties of the
spectral determinant $\Delta (k)=1+\sum_{i}a_{i}e^{ikL_{(i)}}$, where $L_{(i)}$, are
different linear combinations for the bond lengths $l_{1},l_{2}$, ..., $l_{N_{B}}$, the
set $\hat{k}_{n}^{(j)}$ can be provided by the sequence of zeros of the $j$-th derivative
of the function $\Delta (k)$ \cite{Anima,LevinBY}. In this case, the quantity $r$ characterizing
the degree of spectral irregularity is defined as the minimal number for which the condition
$\sum_{i}\left\vert a_{i}\left(L_{(i)}/L_{0}\right) ^{r}\right\vert <1$ is satisfied \cite{Anima}.
\begin{figure}
\caption{Bootstrapping of spectral staircases for separating
sequences $\hat{k}
\label{Fig.1}
\end{figure}
In the simplest case of regular graphs when $r = 0$ \cite{Opus,Prima}, only one auxiliary
sequence (\ref{mid}) is required and various spectral characteristics can be calculated using
the formula
\begin{equation}
f\left( k_{n}\right) =\int_{\hat{k}_{n-1}^{(1)}}^{\hat{k}
_{n}^{(1)}}f(k)\rho (k)dk.
\label{leveldelta}
\end{equation}
As pointed out in \cite{Opus,Anima}, this case corresponds to the situation in which the
straight line with the slope $L_{0}/\pi $, representing the Weyl average $\bar{N}(k)$,
``pierces'' the physical spectral staircase $N(k)$, i.e., $\bar{N}(k)$ intersects every
stair step of $N(k)$ at the points $\hat{k}_{n}^{(1)}$.
\section{Statistical properties of the spectra of regular graphs}
Using the Gutzwiller formula in Eq. (\ref{leveldelta}), one can derive the explicit
expansions for various spectral characteristics $f_{n}^{(0)}$, for example, for fluctuations
$\delta _{n}^{(0)}=\frac{L_{0}}{\pi }\left(k_{n}-\bar{k}_{n}\right) $, of the eigenvalues
$k_{n}$ around the Weyl average or for the distances between levels $s_{n,m}=k_{n+m}-k_{n}$.
Such expansions have the form \cite{Opus,Prima}
\begin{equation}
f_{n}^{(0)}=\bar{f}^{(0)}-\sum_{p}C_{p}^{(0)}\,\cos \left(\omega _{p}^{(0)}n+\varphi _{p}^{(0)}\right) ,
\label{fn}
\end{equation}
where the frequencies $\omega _{p}^{(0)}$ are defined via the periodic orbit lengths as,
$\omega _{p}^{(0)}=\pi L_{p}^{(0)}/L_{0}$. The first term of expansion (\ref{fn}) determines
the average value of the quantity $f_{n}^{(0)}$, whereas the following sum describes
fluctuations around the average. Each frequency $\omega_{p}^{(0)}$ is an integer combination
$\omega _{p}^{(0)}=m_{p,1}^{(0)}\Omega_{1}
+m_{p,2}^{(0)}\Omega_{2}+...+m_{p,N_{B}}^{(0)}\Omega_{N_{B}}$, of the quantities $\Omega_{i}$,
which are expressed in terms of the lengths of the graph bonds as $\Omega _{i}=l_{i}/L_{0}$,
and the coefficients $m_{p,i}^{(0)} $ indicate how many times the orbit passes along the bond
$l_{i}$.
The sum $\left\vert m_{p}^{(0)}\right\vert =m_{p,1}^{(0)} +m_{p,2}^{(0)}+...+m_{p,N_{B}-1}^{(0)}$
specifies the total number of scattering events that the particle moving along the trajectory $p$
undergoes at the vertices. If Eq. (\ref{fn}) includes only the orbits for which $|m_{p}^{(0)}|<m$,
we arrive at the $m$-th approximation to the exact value $f_{n}^{(0)}$ \cite{QGT,Opus}.
Since the numbers $\Omega _{i}$ satisfy the condition $\Omega _{1}+\Omega_{2}+...+\Omega _{N_{B}}=1$,
only $N_{B}-1$ of these numbers are independent. Expressing one of them, e.g., $\Omega _{N_{B}}$, in
terms of the others, let us consider the (generic) case when the numbers
$\tilde{\Omega} _{i}=\Omega _{i}-\Omega _{N_{B}}$ are irrational and algebraically independent. Let
us call the orbit $p$ algebraically simple (with the notation $p'$) if the integer coefficients
$\tilde{m}_{p,i}^{(0)}=m_{p,i}^{(0)}-m_{p,N_{B}}^{(0)}$ have no common divisors. Such orbits in general
differ from the {\em dynamically} simple orbits that correspond to single traversals along closed
sequences of bonds during the particle’s motion along the graph \cite{QGT,Gaspard,Anima,Opus,Prima}.
The expansion (\ref{fn}) enables one to pass immediately to the statistical description of the sequence
$f_{n}^{(0)}$. Indeed, it is well known that the sequence of the remainders
$x_{n}=\left[\alpha n\right]_{\mathop{\rm mod}1}$ for any irrational number $\alpha $ and $n=1,2,...$,
is uniformly distributed in the interval $\left[ 0,1\right] $ \cite{Karatsuba}. Since the arguments of
the trigonometric functions appearing in series (\ref{fn}) are defined modulo $2\pi $, parsing through
the values $f_{n}^{(0)}$ yields a sequence which is statistically equivalent to the series
\begin{equation}
f_{x}^{(0)}=\bar{f}^{(0)}-\sum_{p}\tilde{C}_{p}^{(0)}\,\sin
\left( \tilde{m}_{p}^{(0)}x+\varphi _{p}^{(0)}\right),
\label{fxind}
\end{equation}
Here, $\tilde{C}_{p}^{(0)}$ and $\tilde{m}_{p}^{(0)}$ correspond to the coefficients of Eq. (\ref{fn})
in which the condition $\sum_{i}\Omega_i=1$ is taken into account, and $x$ is a set of $N_{B}-1$
independent, uniformly distributed random variables. The distribution of the quantities $\delta f_{x}^{(0)}$
in this case is obtained from the expression $P_{f}^{(0)} =\langle\delta \left(f^{(0)}-f_{x}^{(0)}\right)\rangle$:
\begin{equation}
P_{f}^{(0)} =\int dke^{ik\left( f^{(0)}-\bar{f}^{(0)}\right)}
\int_{0}^{2\pi}\prod_{p}\Lambda_{p}(x)\frac{dx}{2\pi }
\label{pf}
\end{equation}
where every factor $\Lambda _{p}( \vec{x}) =e^{ik\tilde{C}_{p}^{(0)}\,\cos
\left( \tilde{m}_{p}^{(0)}x+\varphi _{p}^{(0)}\right) }$
determines the contribution to the integral from the corresponding periodic orbit $p$. Thus, Eq. (\ref{pf})
gives the exact expression for the distribution $P_{f}^{(0)}$ in terms of the periodic orbit theory.
It is important to point out that the properties of the asymptotic distributions of trigonometric sums
of form (\ref{fxind}) are one of the traditional areas of research of mathematical statistics (see, e.g.,
\cite{Proxorov,Revesz} and references therein). In particular, it is known that separate terms (or groups
of terms) of lacunary trigonometric series of form (\ref{fxind}) can be considered as weakly dependent
random variables, for which one can be establish a generalization of the central limit theorem, and
consequently their sum is asymptotically Gauss distributed according to
\begin{equation}
P_{f}^{(0)} =\frac{1}{\sigma \sqrt{
2\pi }}e^{-\frac{\left( \delta f^{(0)}\right) ^{2}}{2\sigma ^{2}}},
\label{gauss}
\end{equation}
with the variance
\begin{equation}
\sigma ^{2}=\frac{1}{2}\sum_{p}\tilde{C}_{p}^{(0)2}=\left\langle
\left( \delta f_{x}^{(0)}\right) ^{2}\right\rangle.
\label{variance}
\end{equation}
The conclusion about the Gaussian form of the distribution of the fluctuations also appears to be
applicable to this kind of spectral characteristics expansions of most of the regular quantum graphs (and
other scaling systems), which are described by series of form (\ref{fxind}) with constant coefficients.
A hypothesis about the Gaussian nature of the distribution the spectral staircase fluctuations
$\delta N(k)=N(k)-\bar{N}(k)$, confirmed by extensive numerical investigations, was previously
proposed in \cite{ABS} as the universal ``central limit theorem for spectral fluctuations''
applicable to general quantum chaotic systems. Owing to the existence of additional explicit
expansions (\ref{fn}) this hypothesis, corroborated by the relation with the theory of weakly
dependent random variables (trigonometric sums) can actually be extended to a much wider set of
spectral characteristics.
\section{Approximate description of the distribution functions}
Since the contributions of individual orbits to the series $\delta f_{x}^{(0)}$ behave as
weakly dependent random variables, some physical simplifications are possible in Eq. (\ref{pf}).
Expanding the exponentials $\Lambda _{p}( \vec{x})$, one can note that because expansion (\ref{fn})
is made in orthogonal harmonics, most integrals of the cross terms appearing from the product of
the expansions $\Lambda _{p}(\vec{x})$ in Eq. (\ref{pf}). Contributions come only from the ``resonant''
terms for which one of the algebraic sums of the frequencies vanishes. The amplitude of these
contributions decreases rapidly in the orders of the corresponding degrees of $C^{(0)}_{p}$,
that are proportional to the product of the corresponding number of scattering coefficients at
graph vertices \cite{Opus,Prima}.
This argumentation can be used to simplify the integral for $P_{f}^{(0)}$. For example, in a
simple approximation the contributions from resonances between different algebraically simple
orbits can be disregarded. This is equivalent to untangling of the factors $\Lambda _{p'}(x)$
corresponding to different algebraically simple orbits, i.e., to the introducing an independent
set of variables $x_{p'}$ for every algebraically simple orbit. In this case, the distribution
probability is represented in the form
\begin{equation}
P_{f}^{(0)} =\int dke^{ik\left( f^{(0)}-
\bar{f}^{(0)}\right) }\prod_{p'}Q_{p'}\left(k\tilde{C}^{(0)}_{p'}\right) ,
\label{prime}
\end{equation}
where every factor
\begin{equation}
Q_{p'}=\int_{0}^{2\pi}e^{ik\sum_{\nu }\tilde{C}_{p'\nu}^{(0)}\,
\cos \left( \nu \tilde{m}_{p'}x_{p'}+\varphi_{p}\right) }dx_{p'},
\label{primep}
\end{equation}
corresponds to the algebraically simple orbit $p'$ and the sum with respect to $\nu $
in Eq. (\ref{primep}) is calculated over orbits whose indices are multiples of $\tilde{m}_{p’}$.
For a more crude description of the probability distribution profile, one can disregard the resonances
between any distinct orbits, which is equivalent to the introduction of an independent phase $x_{p}$
for every orbit. Under this assumption, the integral in Eq. (\ref{pf}) is separated into independent
integrals and, as a result, we arrive at the simple expression
\begin{equation}
P_{f}^{(0)} =\int dke^{ik\left( f^{(0)}-\bar{f}^{(0)}\right)}
\prod_{p}J_{0}\left( k\tilde{C}_{p}^{(0)}\right) ,
\label{distreg}
\end{equation}
where $J_{0}(x)$ is the zeroth Bessel function.
Distributions of form (\ref{distreg}) appear in communication theory, for example, when analyzing the
intensity of interfering telecommunication channels, the theory of wave propagation in random media,
and other fields where stochastic signal models are used \cite{LevinBR,Paetzold}.
It is also worth noting that, in the approximation of independent random contributions, the conditions
of the Lindeberg--Feller theorem and central limit theorem are satisfied, which establish the normal
distribution law for the sum of independent random variables. For spectral expansions (\ref{fxind})
these conditions on the variances $\sigma_{p}^{2}=\left(\tilde{C}_{p}^{(0)}\right) ^{2}/2$ of individual
contributions are satisfied due to the exponential increase in the number of periodic orbits and the
uniform exponential decrease of the magnitude of the coefficients $\tilde{C}_{p}^{(0)}$. As a result,
in the approximation of independent random contributions, distribution (\ref{distreg}) has the same
Gaussian form (\ref{gauss}), with the variance $\sigma ^{2}=\sum_{p}\tilde{C}_{p}^{(0)2}/2<\infty$ as
that predicted in \cite{Proxorov,Revesz} and \cite{ABS} for the case of weakly dependent variables.
Such description is applicable to the statistical properties of various spectral characteristics of
the regular graphs beginning with their harmonic expansions \cite{Anima,Opus,Prima}. For example one
can consider the fluctuations $\delta _{n}^{(0)}=\frac{L_{0}}{\pi}\left( k_{n}-\bar{k}_{n}\right) $,
of levels around the average value, which have form (\ref{fn}) with $\bar{\delta}^{(0)}=0$,
$\varphi _{p}^{(0)}=-\frac{\pi }{2}$, and the coefficients
\begin{equation}
C_{p}^{(0)}=-\frac{2}{\pi }\frac{A_{p}^{(0)}}{\omega _{p}}\sin \left( \frac{
\omega _{p}}{2}\right) ,
\label{cp}
\end{equation}
or the difference $s_{m,n}^{(0)}=k_{n+m}-k_{n}$ with $\bar{s}_{m,n}^{(0)}=$ $\frac{\pi }{
L_{0}}m$, $\varphi _{p}^{(0)}=\frac{\omega _{p}m}{2}$ and the coefficients
\begin{equation}
D_{p,m}^{(0)}=\frac{4}{L_{0}}\frac{A_{p}^{(0)}}{\omega _{p}}\sin \left(
\frac{\omega _{p}}{2}\right) \sin \left( \frac{\omega _{p}m}{2}\right) .
\label{dp}
\end{equation}
Knowing the distributions of these quantities, one can describe more complex objects such as the
correlation function of fluctuations $\left\langle \delta_{n}^{(0)}\delta_{n+m}^{(0)}\right\rangle$,
autocorrelation function $R_{2}(x)$, and the form factor $K_{2}(\tau)$, given by the expression
\begin{equation}
K_{2}=\frac{\pi }{L_{0}}\sum_{m}\left\langle e^{-is_{mn}\tau}\right\rangle=
\frac{\pi }{L_{0}}\sum_{m}e^{-i\frac{\pi m}{L_{0}}\tau }F_{s_{m}}^{(0)}(k),
\label{k2}
\end{equation}
where $F_{s_{m}}^{(0)}(k)$ is the characteristic function of distributions of form (\ref{pf}),
(\ref{prime}) or (\ref{distreg}), which are obtained from expansion (\ref{fn}) for $s_{m,n}$ with
coefficients (\ref{dp}), and thus,
\begin{equation}
R_{2}(x)=\frac{\pi }{L_{0}}\sum_{m=1}^{\infty }P_{s_{m}}^{(0)}(x).
\label{r2}
\end{equation}
It is important that all above distributions are closed expressions consistently describing the
spectral characteristics in terms of periodic orbit theory.
\section{Spectral hierarchy}
As mentioned above, in general quantum graphs are not regular and so for them the spectral expansions
of form (\ref{fn}) cannot be obtained directly. A generalization to the irregular case can be obtained
by using the relationship between the two neighboring separator systems $\hat{k}_{n}^{(j)}$ and
$\hat{k}_{n}^{(j-1)}$ and by applying Eq. (\ref{leveldelta}) to $f(k)=k$ at the $(j-1)$th level of the
hierarchy:
\begin{equation}
\hat{k}_{n}^{(j-1)}=\int_{\hat{k}_{n-1}^{(j)}}^{\hat{k}_{n}^{(j)}}kdN^{(j-1)}.
\label{kjint}
\end{equation}
Here, $N^{(j)}(k)$ corresponds to the spectral staircase of the sequence $\hat{k}_{n}^{(j)}$.
Bootstrapping of the sequences $\hat{k}_{n}^{(j-1)}$ by $\hat{k}_{n}^{(j)}$ (or $N^{(j-1)}(k)$
by $N^{(j)}(k)$, see Fig. 1) means that $N^{(j-1)}\left( \hat{k}_{n}^{(j)}\right) =n$. Substituting
expansion (\ref{gutzw}) for $N^{(j-1)}\left( \hat{k}_{n}^{(j)}\right)$ into Eq. (\ref{kjint}),
and using $\hat{k}_{n}^{(j)}$ in the form
\begin{equation}
\hat{k}_{n}^{(j)}=\frac{\pi}{L_{0}}\left( n+\delta _{n}^{(j)}\right) ,
\label{knjdecomposition}
\end{equation}
we obtain the oscillating part of $\hat{k}_{n}^{(j-1)}$ in the form
\begin{equation}
\delta _{n}^{(j-1)}= f_{\delta }^{(j-1)}
-\sum_{p}C_{p}^{(j-1)}\sin \left(\omega _{p}^{(j-1)}n+\varphi _{p}^{(j-1)}\right) ,
\label{jfluct}
\end{equation}
Here, the zeroth term
\begin{equation}
f_{\delta }^{(j-1)} =\frac{1}{2}
\left(\delta _{n}^{(j)}-\delta _{n-1}^{(j)}\right) -\frac{1}{2}\left(
(\delta _{n}^{(j)})^{2}-(\delta _{n-1}^{(j)})^{2}\right) ,
\end{equation}
the amplitudes,
\begin{equation}
C_{p}^{(j-1)}=\frac{2}{L_{0}}\frac{A_{p}^{(j-1)}}{\omega^{(j-1)}_{p}}
\sin \frac{\omega^{(j-1)}_{p}}{2}\allowbreak \left(\delta _{n}^{(j)}-\delta
_{n-1}^{(j)}+1\right) ,
\end{equation}
and phases $\varphi_{p}^{(j-1)}=\omega^{(j-1)}_{p}\left(\delta _{n}^{(j)}+\delta _{n-1}^{(j)}-1\right)/2$
for every level $j$ are functions of the fluctuations $\delta _{n}^{(j)}$ and $\delta _{n-1}^{(j)}$ at the
preceding hierarchy level.
Similar expansions are easily obtained for other spectral characteristics, for example, for
$s_{n,m}^{(j-1)}=\hat k_{n+m}^{(j-1)}-\hat k_{n}^{(j-1)}$:
\begin{equation}
s_{n,m}^{(j-1)}=f^{(j-1)}_{s}+\frac{2}{L_{0}}\sum_{p}D_{p,m}^{(j-1)}\cos \omega^{(j-1)}_{p}
\left( n-\frac{m}{2}\varphi^{(j-1)}_{p}\right),
\label{snj}
\end{equation}
with the zeroth term
\begin{eqnarray}
f^{(j-1)}_{s}=s_{n,m}^{(j)}+\left(s_{n,m}^{(j)}-s_{n,m-1}^{(j)}\right) \times
\cr
\left(\pi m/L_{0}-( s_{n,m}^{(j)}+s_{n-1,m}^{(j)})/2 \right)
-\xi_{n}^{(j)}\left( s_{n,m}^{(j)}-s_{n-1,m}^{(j)}\right),
\end{eqnarray}
where $\xi _{n}^{(j)}\,=(\delta _{n}^{(j)}+\delta _{n-1}^{(j)})/2$ and the expansion coefficients
$\tilde D_{p,m}^{(j-1)}$ are obtained from the corresponding expansion for $s_{n,m}^{(j)}$. The
equations relating the neighboring sequences can also be considered as describing the transition
of a single separating sequence $f_{n}^{(j)}$ from one hierarchy level to another.
\section{Statistical description of spectral hierarchy}
As in the case of the regular graphs, the description of the stochastic properties of sequences
such as $\delta _{n}^{(j)}$ or $s_{n,m}^{(j)}$ is based on the observation that parsing through
the indices $n$ in the arguments of harmonic functions (\ref{jfluct}) and (\ref{snj}) leads to
the appearance of random variables $x$. The idea of finding the distribution functions for various
spectral characteristics is based on using the structural relations between the separating sequences
obtained above in order to relate the probability distributions $P_{f}^{(j)}$ at different hierarchy
levels. Beginning with the distribution $P_{f}^{(r)}$ at the regular level, one can determine the
distribution $P_{f}^{(r-1)}$ at the next level and so on, ending with the last, physical level.
\begin{figure}
\caption{Distribution of variances at the odd levels of the
spectral hierarchy for the four-vertex quantum graph with
$r=7$. The solid lines are the Gaussian approximations of the
numerically calculated histograms.}
\label{Fig.2}
\end{figure}
As an example, let us consider the behavior of the sequences $\delta _{n}^{(j)}$. For simplicity,
we treat the fluctuations $\delta _{n}^{(j)}$ and $\delta _{n-1}^{(j)}$ as independent random
variables $\delta _{1}$ and $\delta_{2}$ distributed according to $P_{\delta }^{(j)}$.
Correspondingly, one can write for the density $P_{\delta}^{(j-1)}(\delta)$
\begin{equation}
P_{\delta }^{(j-1)}=\int \delta \left(\delta
-\delta^{(j-1)}_{x}\right)
P_{\delta_{1}}^{(j)} P_{\delta_{2}}^{(j)}d\delta _{1}d\delta _{2}dx.
\end{equation}
Using Eq. (\ref{jfluct}) and representing the delta functional in exponential form, we obtain
\begin{equation}
P_{\delta }^{(j-1)}(\delta) =\int dke^{ik\delta }\left\langle
\prod_{p}\Lambda _{p}^{(j-1)}\left( x,\delta _{1},\delta
_{2}\right) dx\right\rangle _{\Omega^{(j-1)}},
\label{pdeltaj}
\end{equation}
Here, the factors $\Lambda _{p}^{(j)}\left( x,\delta _{1},\delta_{2}\right)$ correspond to the
terms of expansion (\ref{jfluct}), which are now explicit functions of fluctuations at preceding
hierarchy levels, and $\left\langle \ast \right\rangle _{\Omega ^{(j)}}$ denotes averaging over
these fluctuations with the weight
\begin{equation}
\Omega^{(j-1)}\left(\delta _{1},\delta _{2},k\right) =e^{-ikf_{\delta}^{(j-1)}
\left( \delta _{1},\delta _{2}\right) }P_{\delta}^{(j)}\left( \delta _{1}\right)
P_{\delta }^{(j)}\left( \delta _{2}\right).
\end{equation}
The expression (\ref{pdeltaj}) generalizes regular expansions (\ref{pf}),
(\ref{prime}) and (\ref{distreg}) for the single-level hierarchy to the general expressions for
$j>0$, averaged over the disorder at the preceding levels.
\begin{figure}
\caption{Development of the probability distributions for the
distances between the nearest neighbors $s_{n}
\label{Fig.3}
\end{figure}
We note that the argumentation concerning the Gaussian distribution form in Section 2
\cite{Proxorov,Revesz} can be directly applied to the distribution of $\delta_{x}^{(r)}$
at the regular level. However, as shown in Fig. 2, the distribution of $\delta^{(j)}_{x}$
at higher levels $j>0$ is also Gaussian-like. For other spectral characteristics, for
example, $s_{n}^{(j)}$ (see Fig. 3), the sequence of transitions of form
(\ref{pdeltaj}) can lead to asymmetric (non-Gaussian) distributions.
\section{Discussion}
The method proposed in \cite{Anima} for solving the spectral problem is based on establishing
the structural relationships between the sequence of physical levels $k_{n}$ and the regular
sequence $\hat{k}_{n}^{(r+1) }$ specified as an explicit function $\hat{k}_{n}^{(r+1)}=\hat{k}^{(r+1)}(n)$.
For quantum graphs, the regular sequence is given by (\ref{mid}) and relation to $k_{n}$ is
established through the system of auxiliary sequences $\hat{k}_{n}^{(j)}$, bootstrapping $k_{n}$
with $\hat{k}^{(r+1) }(n)$. The spectral hierarchy thus obtained consists of the system of
sequences $\hat{k}_{n}^{(j)}$ and transition equations (\ref{kjint}) from
$\hat{k}_{n}^{(j)}$ to $\hat{k}_{n}^{(j-1)}$.
This approach allows not only the description of the evolution of base sequences $\hat{k}_{n}^{(j)}$
from low to high hierarchy levels, but also the complete probability description of spectral
characteristics in the framework of periodic orbit theory including those that are not directly
described by the Gutzwiller formula. In this case, it is possible to follow the development of
the scales of spectral fluctuations, distributing disorder over the intermediate hierarchy levels,
gradually passing from less to more disordered sequences. While the base sequence is maximally
ordered, the amplitude of fluctuations in each next sequence $\hat{k}_{n}^{(j)}$ increases as
the index $j$ decreases, i.e. with the approach to the physical spectrum \cite{Anima}. The
minimum number of auxiliary sequences $\hat{k}_{n}^{(j)}$ necessary for bootstrapping $\hat{k}_{n}^{(r+1)}$
with $\hat{k}_{n}^{(0)}$ defines to the complexity of the spectral problem with respect to the
given bootstrapping method.
The above relation between the properties of the series of expansions (\ref{jfluct}) and the
properties of weakly dependent random variables \cite{Proxorov,Revesz} reveals the physical
origins of the universality of the distributions of different spectral characteristics following
from the limiting properties of the sums of such quantities.
The existence of a sufficient number of transitions between hierarchy levels of irregular systems
and, correspondingly, of averaging processes over random phases and disordered sequences
$\hat{k}_{n}^{(j)}$ in Eq. (\ref{pdeltaj}) leads not only to the Gaussian shape of the distribution
of probabilities $P_{f}^{(0)}$ [as, e.g., for $\delta N(k)$ and, correspondingly for $\delta _{n}^{(0)}$,
see \cite{ABS} and Fig. 2), but also to the appearance of more complex (e.g., Wignerian, see \cite{BGS}
and Fig. 3) distributions.
It is also important that determining the fluctuation probabilities in form (\ref{pdeltaj})
makes it possible not only to follow the appearance of general, universal statistical relations,
but also to describe in detail the specific features of distributions $P_{f}^{(j)}$, which present
the individual properties of each particular system.
Work supported in part by the Sloan--Swartz Foundation.
\end{document} |
\begin{document}
\mathfrak{m} aketitle
\begin{abstract}
Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuovo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.
\end{abstract}
\section{Introduction}
Let $k$ be a field and $R=k[x_1,\ldots ,x_n]$ be a polynomial ring in $n$ variables and $I$ be a homogeneous ideal of $R$.
Then for $n\geq 1$, the $n$-th symbolic power of $I$ is defined as $I^{(n)}=\displaystyle{\bigcap_{p\in \operatorname{Ass} I}(I^nR_p\cap R)}$. Symbolic powers of an ideal is geometrically an important object of study as by a classical result of Zariski and Nagata $n$-th symbolic power of a given ideal consists of the elements that vanish up to order $n$ on the corresponding variety. In general finding the generators of symbolic power is a difficult job. It is easy to see that $I^n\subseteq I^{(n)}$ for all $n\geq 1$. The opposite containment, however, does not hold in general. Much effort has been invested in to determine for which values of $r$ the containment $I^{(r)}\subseteq I^m$ holds. To answer this question C. Bocci and B. Harbourne in \cite{bocci2010} defined an asymptotic quantity known as resurgence which is defined as $\rho{(I)} =\sup \tiny\{ \frac{s}{t} ~~|~~I^{(s)}\mathfrak{m} athfrak{n} subseteq I^t \tiny\}$ and showed that it exists for radical ideals. Since computing the exact value of resurgence is difficult, another asymptotic invariant $ \widehat{\alpha}{(I)} = \displaystyle{\displaystyle{\lim_{\substack{\longrightarrow\\k}}} m_{s\rightarrow\infty}{} \frac{\alpha{(I^{(s)})}}{s}}~,$ known as Waldschmidt constant was introduced, where $\alpha(I)$ denotes the least generating degree of $I$. In order to measure the difference between $I^{(m)}$ and $I^m$, Gattleo et al in \cite{ggsv} have introduced an invariant known as the symbolic defect which is defined as $ \operatorname{sdefect}(I,m) = \mathfrak{m} u\left(\frac{I^{(m)}}{I^m}\right),$ where $\mathfrak{m} u(I)$ denotes the minimal number of generators of $I$. It counts the minimal number of generators which must be added to $I^m$ to make $I^{(m)}$.
In this paper we investigate these invariants for edge ideal of graphs. Let $G$ be a simple graph with $n$ vertices $x_1,\ldots ,x_n$ and $I=I(G)$ be the edge ideal generated by $\{x_ix_j|~x_ix_j \mathfrak{m} box{ is an edge of } G\}$. In \cite{svv} Simis, Vasconcelos and Villarreal have proved that $G$ is a bipartite graph if and only if $I^s=I^{(s)},$ for every $s\in\mathfrak{m} athbb{N}$. So it is interesting to study the symbolic powers of edge ideal of non-bipartite graphs. The class of non-bipartite graph which was studied first is odd cycle by Janssen et al in \cite{janssen2017comparing}. They have described the generators of the symbolic powers of the edge ideal of an odd cycle by using the concept of minimal vertex cover and calculated the invariants associated to the symbolic powers of the edge ideal of the same graph. In \cite{gu2018symbolic} Gu et al have extended these results for the unicyclic graph by explicitly computing the generators of the symbolic powers. Another important invariant in commutative algebra is Castelnuovo-Mumford regularity. Regularity of the edge ideal and their powers has been extensively studied in literature by many researchers while the regularity of the symbolic powers of edge ideals has not been much explored. It has been conjectured by N. C. Minh that for a finite simple graph $G$
$$\operatorname{reg} I(G)^{(s)}=\operatorname{reg} I(G)^s$$
for $s\in \mathfrak{m} athbb N$. The conjecture is true for bipartite graph. In \cite{gu2018symbolic} Gu et al have proved the conjecture for odd cycles. Recently in \cite{jayanthan} Jayanthan and Kumar have proved the conjecture for certain class of unicyclic graph and in \cite{fakhari} Seyed Fakhari has solved the conjecture for unicyclic graph.\\
The work of this paper is mainly motivated by the papers \cite{janssen2017comparing} and \cite{gu2018symbolic}. In this paper we study the structure of the symbolic powers of edge ideal of clique sum of two different odd length cycles joined at a single vertex and prove Minh's conjecture for the class of complete graphs. In section 2, we recall all the definitions and results that will be required for the rest of the paper. Motivated by the concept of vertex weight described in \cite{janssen2017comparing} by Janssen, in section 3, we find the generators of the symbolic powers of edge ideal of clique sum of two different odd length cycles joined at a single vertex by explicitly choosing a minimal vertex cover. Using the description of the generating set we compute the Waldschmidt constant. In section 4, we describe the generators of the symbolic powers of edge ideal of complete graph and compute the Waldschmidt constant, the resurgence and establish the symbolic defect partially. We close the paper by showing that for a complete graph $G$ and for all $s\geq 1$,
$$\operatorname{reg} I(G)^{(s)}=\operatorname{reg} I(G)^s.$$
\vskip 0.3cm
\section{Preliminaries}
In this section, we collect the notations and terminologies used in this paper. Throughout the paper, $G$ denotes a finite simple graph over the vertex set $V(G)$ and the edge set $E(G).$
\\
\begin{defn}
Let $G$ be a graph.
\begin{enumerate}
\item A collection of the vertices $W \subseteq V (G)$ is called a vertex cover if for any edge $e\in E(G), W\cap e \mathfrak{m} athfrak{n} eq \mathcal{P}hi.$ A vertex cover is called minimal if no proper subset of it is also a vertex cover.
\item The vertex cover number of $G$, denoted by $\tau(G)$, is the smallest size of a minimal vertex cover in G.
\item The graph $G$ is called decomposable if there is a proper partition of its vertices $V(G) = \bigcupdot_{i=1}^{r}{V_i}$ such that $\tau(G) =\sum_{i=1}^{r}{\tau(G[V_i ])}$.
In this case $(G[V_1 ],\dots,G[V_r ])$ is called a decomposition of G. If G is not decomposable then G is said to be indecomposable.
\end{enumerate}
\end{defn}
\begin{defn}
Let $V' \subseteq V(G)=\tiny\{x_1,x_2,\ldots,x_n\tiny\}$ be a set of vertices. For a monomial $ x^{\underline{a}} \in k[x_1,\ldots,x_n]$ with exponent vector $\underline{a} = (a_1,a_2,\ldots ,a_n)$ define the vertex weight $W_{V'}{(x^{\underline{a}})}$ to be $$ W_{V'}{(x^{\underline{a}})} := \sum_{x_i\in V'}{a_i}. $$
\end{defn}
Now we recall some results which describe the symbolic powers of the edge ideal in terms of minimal vertex covers of the graph.
\begin{lem}\cite[Corollary 3.35]{vantuylbook}
Let $G$ be a graph on vertices $\{x_1,\ldots,x_n\}, I=I(G)\subseteq k[x_1,\ldots,x_n]$ be the edge ideal of $G$ and $V_1,\ldots,V_r$ be the minimal vertex covers of $G.$ Let $P_j$ be the monomial prime ideal generated by the variables in $V_j.$ Then $$I=P_1 \cap\operatorname{codim}ots\cap P_r$$ and $$I^{(m)}=P_1^{m} \cap\operatorname{codim}ots\cap P_r^{m}.$$
\end{lem}
In the next lemma, the elements in the symbolic power of edge ideals have been described in terms of minimal vertex cover of the graph.
\begin{lem}\cite[Lemma 2.6]{bocci2016}\label{psymbolic}
Let $I \subseteq R$ be a square free monomial ideal with minimal primary decomposition $I=P_1\cap\operatorname{codim}ots \cap P_r ~~with ~~P_j = (x_{j_1},\dots ,x_{j_{s_j}})$ for $j= 1,\ldots ,r$. Then $ {x_1^{a_1}}\operatorname{codim}ots {x_n^{a_n}} \in I^{(m)} \mathfrak{m} box{ if and only if } a_{j_1}+\dots +a_{j_{s_j}}\geq m$ for $j=1,\dots,r$.
\end{lem}
Using Lemma \ref{psymbolic} and the concept of vertex weight Janssen et al in \cite{janssen2017comparing} described the elements of symbolic powers of edge ideals as follows
$$ I^{(t)} = (\{ x^{\underline{a}} ~|\mathfrak{m} box{ for all minimal vertex covers }V^{\mathcal{P}rime}, W_{V^{\mathcal{P}rime}}(x^{\underline{a}})\geq t \}).$$
Further they have divided the elements of the symbolic powers of edge ideals into two sets written as $I^{(t)}=(L(t))+(D(t)),$ where
$$L(t) = \{ x^{\underline{a}}~ | \deg(x^{\underline{a}})\geq 2t\mathfrak{m} box{ and for all minimal vertex covers } V^{\mathcal{P}rime}, W_{V^{\mathcal{P}rime}}(x^{\underline{a}})\geq t \}$$ and
$$D(t) = \{ x^{\underline{a}}~ | \deg(x^{\underline{a}})< 2t\mathfrak{m} box{ and for all minimal vertex covers } V^{\mathcal{P}rime}, W_{V^{\mathcal{P}rime}}(x^{\underline{a}})\geq t \}.$$
Thus for any graph, if we are able to identify the elements in $L(t)$ and $D(t)$ then we will be able to describe $I^{(t)}$.
\begin{defn}
Let $G$ be a graph with $n$-vertices and let $v=(v_1 ,\dots,v_n )\in \mathfrak{m} athbb{N}^{n} .$ For a vertex $x \in V (G),$ let $N_G (x)=\{y \in V (G)~|~ \{x,y\}\in E(G)\}$ be its neighborhood.
\begin{enumerate}
\item The duplication of a vertex $x\in V(G)$ in $G$ is the graph obtained from $G$ by adding a new vertex $x^{\mathcal{P}rime}$ and all edges $\{x^{\mathcal{P}rime} ,y\}$ for $y \in N_G(x).$
\item The parallelization of G with respect to $v$, denoted by $G^v$ , is the graph obtained from $G$ by deleting the vertex $x_i$ if $v_i = 0$, and duplicating $v_i-1$ times the vertex $x_i$ if $v_i \mathfrak{m} athfrak{n} eq 0.$
\end{enumerate}
\end{defn}
A commonly-used method in commutative algebra when investigating (symbolic) powers
of an ideal is to consider its (symbolic) Rees algebra.
\begin{defn} Let $R$ be a ring and $I$ be an ideal of $R.$ The Rees algebra, denoted by $\mathfrak{m} athcal{R}(I),$ and the symbolic Rees algebra, denoted by $\mathfrak{m} athcal{R}_s(I),$ of $I$ are defined as $$\mathfrak{m} athcal{R}(I):= \displaystyle {\bigoplus_{n\geq 0}{I^nt^n} }\subseteq R[t] \mathfrak{m} box{ and } \mathfrak{m} athcal{R}_s (I) :=\displaystyle {\bigoplus_{n\geq 0}{I^{(n)}t^n}} \subseteq R[t].$$
\end{defn}
If $I=I(G)$ is the edge ideal of a graph then the generators of $\mathfrak{m} athcal{R}_s(I)$ can be described by indecomposable graphs arising from $G.$ The following characterization for $\mathfrak{m} athcal{R}_s(I)$ was given in \cite{bernal}.
\begin{thm}\label{1implosive}
Let $G$ be a graph over the vertex set $V(G)=\{x_1,\dots,x_n\}.$ Let $I=I(G)$ be its edge ideal. Then $$\mathfrak{m} athcal{R}_s(I)=k[x^vt^b|\mathfrak{m} box{ $G^v$ is an indecomposable graph and } b=\tau(G^v) ],$$ where $v=(v_1,\dots,v_n)$ and $ x^vt^b=x_1^{v_1}\operatorname{codim}ots x_n^{v_n}t^b.$
\end{thm}
\begin{defn}
Let $G$ be a simple graph of $n$ vertices with edge ideal $I=I(G).$ Then the graph $G$ is called an implosive graph if the symbolic Rees algebra $\mathfrak{m} athcal{R}_s(I)$ is generated by monomials of the form $x^vt^b,$ where $v\in\{0,1\}^n.$
\end{defn}
Next two theorems gives a class of implosive graphs.
\begin{thm}\cite[Theorem 2.3]{flores}\label{pcycle}
If $G$ is a cycle, then $G$ is implosive.
\end{thm}
\begin{defn}
Let $G_1$ and $G_2$ be graphs. Suppose that $G_1\cap G_2=K_r$ is the complete graph of order $r,$ where $G_1\mathfrak{m} athfrak{n} eq K_r$ and $G_2\mathfrak{m} athfrak{n} eq K_r.$ Then, $G_1\cup G_2$ is called the clique-sum of $G_1$ and $G_2.$
\end{defn}
\begin{thm}\cite[Theorem 2.5]{flores}\label{pimplosive}
The clique-sum of implosive graphs is again implosive.
\end{thm}
\section{Symbolic powers of clique sum of two odd cycles joined at a single vertex}
The work of this section is mainly motived by \cite[Example 3.5]{gu2018symbolic}. Let $G$ be the clique sum of two different odd length cycles joined at single vertex and $I=I(G)$ be the edge ideal of $G$. In this section we will describe the generators for $I^{(t)}$ by explicitly identifying $(L(t))$ and $(D(t)).$ In \cite[Theorem 4.4]{janssen2017comparing} Janssen et al have proved that if $G$ is an odd cycle and $I$ is the edge ideal of $G$ then $(L(t))=I^t$ for $t\geq 1$. In the following example we show that for the clique sum of two different odd length cycles joined at single vertex $(L(t))\mathfrak{m} athfrak{n} ot=I^t$.
\begin{example}
Let $G$ be the clique-sum of two cycles $C_1=(x_1,x_2,x_3,x_4,x_5)$ and $C_2=(x_1,y_2,y_3,y_4,y_5)$ joined at a single vertex $x_1.$ Consider the monomial $m=x_4y_2y_3y_4y_5x_1$. Clearly $m\mathfrak{m} athfrak{n} otin I^3.$ Also for any vertex cover $V$, $W_V(m)\geq 3$, therefore $m\in(L(3)). $ So $I^3\mathfrak{m} athfrak{n} eq (L(3)).$
\end{example}
In this case we will try to understand the generators of $(L(t))$. For this we recall the definition of the optimal form introduced in \cite{janssen2017comparing}.
\begin{defn}
Let $ m \in k[x_1,\ldots,x_{n}]$ be a monomial and $G$ be a finite simple connected graph on the set of vertices $\{x_1,\dots,x_n\}$. Let $\{e_1,e_2,\dots,e_r\}$ denote the set of edges in the graph. We may write $ m=x_1^{a_1}x_2^{a_2} \operatorname{codim}ots x_{n}^{a_{n}}e_1^{b_1}e_2^{b_2}\operatorname{codim}ots e_{r}^{b_{r}} $,
where $b(m) := \sum{b_j}$ is as large as possible, when $m$ is written in this way, we will call this an optimal form of $m$ or we will say that $m$ is expressed in optimal form, or simply $m$ is in optimal form. In addition, each $ x_i^{a_i} $ with $a_i >0 $ in this form will be called an ancillary factor of the optimal form, or just ancillary for short and $x_i$'s are called ancillary vertices.
\end{defn}
Note that optimal form of a monomial need not be unique but $b(m)$ is unique.
\begin{lem}\label{2optimal}
Let $G$ be a finite simple connected graph on the set of vertices $\{x_1,x_2,\dots,x_n\}$. $I=I(G)$ be the edge ideal. Let $m=x_i e_i^{b_i}\operatorname{codim}ots e_{j-1}^{b_{j-1}}\in I(G)$ be a monomial, where $b_k\geq 0$ for $i\leq k \leq j-1$ and $e_i=x_ix_{i+1}.$ Then $mx_j$ will not be an optimal form if and only if the number of vertices within $x_i$ and $x_j$ is even and $b_{i+2h+1}\geq 1$, for $0\leq h\leq \frac{j-i-2}{2}$ with $h\in \mathfrak{m} athbb{Z}.$
\end{lem}
\begin{proof}
Since $m$ contains only one ancillary, so $m$ is in optimal form. Here $b(m)=b_i+b_{i+1}+\dots+b_{j-1}.$ Assume that $mx_j$ is not in optimal form, then $b(mx_j)>b(m)$ so there will be no ancillary in $mx_j,$ which implies $x_i$ will no longer be an ancillary, so $x_i$ has to pair up with $x_{i+1}$ to form an edge. The vertex $x_{i+1}$ can come from two edges $e_i$ or $e_{i+1}.$ If $x_{i+1}$ comes from $e_i$, then $x_i$ will be again an ancillary, so that vertex should come from the edge $e_{i+1}$, which implies that the edge $e_{i+1}$ has to be present in $m$. As $x_i$ form an edge with $x_{i+1}$, then $x_{i+2}$ has to be pair up with some vertex. By similar argument $e_{i+3}$ has to be present in $m$. By repeating this process we get $b_{i+2h+1} \geq 1$ for $0\leq h\leq \frac{j-i-2}{2}.$
As there are no ancillary in $mx_j,$ $x_j$ has to be pair up with $x_{j-1}.$ Then there is only one option to get the edge $x_{j-1}x_j,$ is that $x_{j-1}$ has to come from an edge $e_{i+2h+1}=x_{i+2h+1}x_{i+2h+2}$ for some $h$, which implies $x_{i+2h+2}=x_{j-1}.$ Thus the number of vertices in $[x_i,x_j]$ is $i+2h+3-i+1=2h+4$ which is even and $b_{i+2h+1}\geq 1$ for $0\leq h\leq \frac{j-i-2}{2}$.
Conversely if $mx_j=x_i e_i^{b_i}\operatorname{codim}ots e_{j-1}^{b_{j-1}}x_j$ with $b_{i+2h+1}\geq 1$ for $0\leq h\leq \frac{j-i-2}{2},$ then by \cite[Lemma 3.4]{janssen2017comparing} $mx_j$ is not in optimal form.
\end{proof}
\begin{defn}
Let $G$ be the clique sum of two cycles joined at a single vertex. Let $x_i,x_j \in V(G)$ and the vertices between $x_i$ and $x_j$ is even and let $e_i$ denote the edge $x_ix_{i+1}.$ Then the edges $e_{i+1},e_{i+3},\dots,e_{j-2}$ are called alternating edges.
\end{defn}
In the next lemma we state the key idea of \cite[Theorem 4.4]{janssen2017comparing}, as we will be using this fact in describing the generators of $(L(t)).$
\begin{lem}\label{cycle}
Let $G$ be an odd cycle and $I=I(G)$ be the edge ideal of $G$. Let $m$ be any monomial then there exists a minimal vertex cover $V$ such that $W_V(m)=b(m)$.
\end{lem}
\begin{lem}\label{bm}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n\leq m$ and $I=I(G)$ be the edge ideal of $G$. Let $\underline{m}$ be a monomial such that $\underline{m} \mathfrak{m} athfrak{n} otin (c_1,c_2),$ where $c_1=x_1\operatorname{codim}ots x_{2n+1}$ and $c_2=x_1y_2\operatorname{codim}ots y_{2m+1}$. Then there exists a minimal vertex cover $V$ such that $W_V(\underline m)=b(\underline m)$.
\end{lem}
\begin{center}
\includegraphics[scale=0.55]{image3}
\end{center}
\begin{proof}
Let the monomial $\underline{m}$ is of the form
$\underline{m}=x_{l_1}^{a_{l_1}}x_{l_2}^{a_{l_2}}\operatorname{codim}ots x_{l_k}^{a_{l_k}}y_{r_1}^{a_{r_1}}y_{r_2}^{a_{r_2}}\operatorname{codim}ots y_{r_p}^{a_{r_p}}e_1^{b_1}\operatorname{codim}ots e_{2n+1}^{b_{2n+1}}f_1^{d_1}\operatorname{codim}ots\\ f_{2m+1}^{d_{2m+1}},$ where $e_i=x_ix_{i+1}$ for $1\leq i\leq 2n$ and $e_{2n+1}=x_{2n+1}x_1$ and $f_i=y_iy_{i+1}$ for $2\leq i\leq 2m$ and $f_1=x_1y_2$, $f_{2m+1}=y_{2m+1}x_1.$
If $x_1$ is an ancillary vertex of $\underline{m}$ and $\underline{m}\mathfrak{m} athfrak{n} otin(c_1,c_2)$ then by Lemma \ref{cycle} we can choose minimal vertex cover $V_1$ and $V_2$ from the cycles $C_1$ and $C_2$ such that $V=V_1 \cup V_2$ will not contain $x_1$ and $W_V(\underline{m})=b(\underline{m}).$\\ In the rest of the proof we assume that $x_1$ is not an ancillary of $\underline{m}$. Let $[x_i,x_j]$ denote the set of vertices between $x_i$ and $x_j$ including $x_i$ and $x_j$ along clock-wise path and $|[x_i,x_j]|$ denote the number of vertices in that set. Depending on neighbourhood ancillary vertices of $x_1$ we describe the process to choose the minimal vertex cover $V$ such that $W_V(\underline{m})=b(\underline{m}).$ We have divided the proof into four cases depending on the neighbourhood ancillaries of $x_1$.
\textbf{Case 1:} Suppose there are at least two ancillaries from each cycle. Let $x_{i_1}$ and $x_{j_1}$ are two consecutive ancillaries from the cycle $C_1$ and $y_{i_2}$, $y_{j_2}$ are two consecutive ancillaries from the cycle $C_2$ such that $x_1\in [x_{j_1},x_{i_1}]$ and $x_1\in [y_{j_2},y_{i_2}]$, where $i_1<j_1$ and $i_2<j_2$ .
Let $H_q$ be the induced subgraph of $G$ on $V_{H_q}=\{x_{j_1},x_{j_1+1},\ldots,x_1,\ldots,x_{i_1},y_{j_2},\ldots,y_{2m+1},y_2,\ldots,y_{i_2} \}.$
And let $\underline{m}_q=x_{j_1}^{a_{j_1}}e_{j_1}^{b_{j_1}}\operatorname{codim}ots e_1^{b_1}\operatorname{codim}ots e_{i_1-1}^{b_{i_1-1}}x_{i_1}^{a_{i_1}}y_{j_2}^{c_{j_2}}f_{j_2}^{d_{j_2}}\operatorname{codim}ots f_{2m+1}^{d_{2m+1}}f_1^{d_1}\operatorname{codim}ots f_{i_2-1}^{d_{i_2-1}}y_{i_2}^{c_{i_2}}.$
Let $\underline{m}_{1_{C_1}}= x_{j_1}^{a_{j_1}}e_{j_1}^{b_{j_1}}\operatorname{codim}ots e_{2n+1}^{b_{2n+1}}$, $\underline{m}_{2_{C_1}}= e_1^{b_1}\operatorname{codim}ots e_{i_1-1}^{b_{i_1-1}}x_{i_1}^{a_{i_1}}$,
$\underline{m}_{1_{C_2}}= f_1^{d_1}\operatorname{codim}ots f_{i_2-1}^{d_{i_2-1}}y_{i_2}^{c_{i_2}}$ and
$\underline{m}_{2_{C_2}}= y_{j_2}^{c_{j_2}}f_{j_2}^{d_{j_2}}\operatorname{codim}ots f_{2m+1}^{d_{2m+1}}.$
If $\underline{m}_{1_{C_1}}x_1$, $\underline{m}_{2_{C_1}}x_1$, $\underline{m}_{1_{C_2}}x_1$ and $\underline{m}_{2_{C_2}}x_1$ are in optimal form then $mx_1$ is in optimal form, which means $x_1$ is an ancillary vertex of $mx_1.$ So we can find a minimal vertex cover $V$ such that $W_V(\underline{m}x_1)=b(\underline{m}x_1),$ which implies $W_V(\underline{m})=b(\underline{m}).$ Let us assume that $\underline{m}_{1_{C_1}}x_1$ is not in optimal form.
Then by Lemma \ref{2optimal}, $|[x_{j_1},x_1]|$ is even and $b_{j_1+2h+1}\geq 1$ for $0\leq h\leq \frac{2n-j_1}{2}.$
Now choose the minimal vertex cover for the part $\underline{m}_{1_{C_1}}$ as $C_{1_{V_1}}=\{x_{j_1+1},x_{j_1+3},\ldots,x_1 \}.$
\mathfrak{m} athfrak{n} ewline (a) If $\underline{m}_{1_{C_2}}x_1$ is in optimal form then one of the following condition will hold.
\begin{enumerate}
\item[(i)] $|[x_1,y_{i_2}]|$ is odd. Then $|[x_{j_1},y_{i_2}]|$ is even and one of the alternating edge between $x_{j_1}$ and $y_{i_2}$ will be missing (as $\underline{m}_{1_{C_1}} \underline{m}_{1_{C_2}}$ is in optimal form). Since all the alternating edges in $\underline{m}_{1_{C_1}}$ are present so the alternating edge will be missing from the part $\underline{m}_{1_{C_2}}.$
Therefore $d_{2h+1}=0$ for some $h$, where $0\leq h\leq \frac{i_2-2}{2}.$ Then choose the minimal vertex cover for the part $\underline{m}_{1_{C_2}}$ as $C_{2_{V_1}}= \{ x_1,y_3,\ldots,y_{2h+1},y_{2h+2},y_{2h+4},\ldots,\\y_{i_2-1}\}.$
\item[(ii)] $|[x_1,y_{i_2}]|$ is even and one of the alternating edges will be missing. Then choose the minimal vertex cover as $C_{2_{V_1}}= \{ x_1,y_3,\ldots,y_{i_2-1}\}$ (as $i_2-1$ is odd).
\end{enumerate}
(b) If $\underline{m}_{1_{C_2}}x_1$ is not optimal form then $|[x_1,y_{i_2}]|$ is even, so choose the minimal vertex cover as $ C_{2_{V_1}}=\{x_1,y_3,\ldots,y_{i_2-1} \}.$
Similarly we do the same thing for $\underline{m}_{2_{C_2}}x_1.$ If $\underline{m}_{2_{C_2}}x_1$ is in optimal form and if $|[y_{j_2},x_1]|$ is odd then we can deduce from the previous discussion that $d_{j_2+2h+1}=0$ for some $h$, where $0\leq h \leq \frac{2m-j_2}{2}.$ Then take the minimal vertex cover as $C_{2_{V_2}}=\{y_{j_2+1},y_{j_2+3},\ldots,y_{j_2+2h+1},\\y_{j_2+2h+2},y_{j_2+2h+4}, \ldots,x_1\}.$
In other cases $|[y_{j_2},x_1]|$ will be even then choose the minimal vertex cover as
$C_{2_{V_2}}= \{y_{j_2+1},y_{j_2+3},\ldots,x_1\}$.
Again we will find the minimal vertex cover for the part $\underline{m}_{2_{C_1}}.$ If $|[x_1,x_{i_1}]|$ is odd in clock-wise direction then $b_{2h+1}=0$ for some $h$, where $0\leq 2h\leq i_1-2.$ Then take the minimal vertex cover as $C_{1_{V_2}}=\{ x_1,x_3,\ldots,x_{2h+1},x_{2h+2},x_{2h+4},\ldots,x_{i_1-1}\}$ (as $i_1-1$ is even).
In other cases $|[x_1,x_{i_1}]|$ is even, then take the minimal vertex cover as $C_{1_{V_2}}=\{x_1,x_3,\ldots,x_{i_1-1}\}$ (as $i_1-1$ is odd).
Our aim is here to find a minimal vertex cover $V$ such that $W_V(\underline{m}_q)=b(\underline{m}_q).$ Depending on whether the four monomial part $\underline{m}_{1_{C_1}}x_1,\underline{m}_{2_{C_1}}x_1,\underline{m}_{1_{C_2}}x_1,\underline{m}_{2_{C_2}}x_1$ are in optimal form or not we can choose the minimal vertex cover. Here we will show for one case and the other cases will be similar.
Assume that $\underline{m}_{1_{C_1}}x_1,$ $\underline{m}_{2_{C_2}}x_1$, $\underline{m}_{2_{C_1}}x_1$ are not in optimal form and $\underline{m}_{1_{C_2}}x_1$ is in optimal form. Then depending on $|[x_1,y_{i_2}]|$ is even or odd there are two possibilities of minimal vertex cover. If $|[x_1,y_{i_2}]|$ is odd then choose the minimal vertex cover for $\underline{m}_q$ as $V=\{x_{j_1+1},x_{j_1+3},\ldots,x_1,y_3,\ldots,y_{2h+1},y_{2h+2},y_{2h+4},y_{i_2-1},y_{j_2+1}
,y_{j_2+3},\ldots,\\y_{2m}, x_1,x_3,\ldots,x_{i_1-1}\},$ where $d_{2h+1}=0.$
Now $b(\underline{m}_q)=b_{j_1}+\operatorname{codim}ots+b_{2n+1}+b_1+\operatorname{codim}ots+b_{i_1-1}+d_{j_2}+\operatorname{codim}ots+d_{2m+1}+d_1+\operatorname{codim}ots+d_{i_2-1}$ and $W_V(\underline{m}_q)=b_{j_1}+\operatorname{codim}ots+b_{2n+1}+b_1+d_1+d_{2m+1}+d_2+\operatorname{codim}ots+d_{2h}+2d_{2h+1}+d_{2h+2}+\ldots+d_{i_2-1}
+d_{j_2}+\operatorname{codim}ots+d_{2m}+b_2+b_3+\operatorname{codim}ots+b_{i_1-1}.$ As $d_{2h+1}=0$, $W_V(\underline{m}_q)=b(\underline{m}_q).$
If $|[x_1,y_{i_2}]|$ is even then choose the minimal vertex cover for $\underline{m}_q$ as $V=\{x_{j_1+1},x_{j_1+3},\ldots,x_1,\\y_3,\ldots,y_{i_2-1},y_{j_2+1},y_{j_2+3},\ldots,
y_{2m},x_1,x_3,\ldots,x_{i_1-1}\} $ and note that $W_V(\underline{m}_q)=b(\underline{m}_q).$ Similarly we can write the minimal vertex cover for the remaining cases such that $W_V(\underline{m}_q)=b(\underline{m}_q).$ Now for the vertices of $G$ that are not in $V_{H_q}$, we can choose a minimal vertex cover $V^{\mathcal{P}rime}$ by Lemma \ref{cycle} such that $W_{V^{\mathcal{P}rime}\cup V}(\underline{m})=b(\underline{m}).$
\\
\textbf{Case 2:} Suppose that each cycle contains single ancillary. Let the ancillaries are $x_{j_1}$ and $y_{j_2}$ in $C_1$ and $C_2$ respectively and $\underline{m}_{1_{C_1}}= x_{j_1}^{a_{j_1}}e_{j_1}^{b_{j_1}}\operatorname{codim}ots e_{2n+1}^{b_{2n+1}}$, $\underline{m}_{2_{C_1}}= e_1^{b_1}\operatorname{codim}ots e_{i_1-1}^{b_{i_1-1}}x_{j_1}^{a_{i_1}}$,
$\underline{m}_{1_{C_2}}= f_1^{d_1}\operatorname{codim}ots f_{i_2-1}^{d_{i_2-1}}y_{j_2}^{c_{i_2}}$ and
$\underline{m}_{2_{C_2}}= y_{j_2}^{c_{j_2}}f_{j_2}^{d_{j_2}}\operatorname{codim}ots f_{2m+1}^{d_{2m+1}}.$
Let us assume that
$\underline{m}_{1_{C_1}}x_1$ is not in optimal form. Then $|[x_{j_1},x_1]|$ is even and all the alternating edges will be present. Then choose the minimal vertex cover for $\underline{m}_{1_{C_1}}$as $C_{1_{V_1}}=\{ x_{j_1+1},x_{j_1+3},\ldots,x_1\}.$ As $\underline{m}\mathfrak{m} athfrak{n} otin (c_1,c_2)$, therefore at least one vertex will be missing from each of the cycle in $\underline{m}$. The missing vertex in $C_1$ will be from the part $\underline{m}_{2_{C_1}}$ and will be of the form $x_{2k}$ or $x_{2k+1}.$ If $x_{2k}$ is missing then take the minimal vertex cover for $\underline{m}_{2_{C_1}}$ as (here $b_{2k-1}=0$ and $b_{2k}=0$) $C_{1_{V_2}}= \{ x_1,x_3,\ldots,x_{2k-1},x_{2k},x_{2k+2},\ldots,x_{j_1-1} \}$ (as $j_1-1$ is even).
If $x_{2k+1}$ is missing then take the minimal vertex cover as $C_{1_{V_2}}=\{
x_1,x_3,\ldots,x_{2k+1},x_{2k+2},x_{2k+4},\ldots,x_{j_1-1} \}.$ Now take
$V_1=C_{1_{V_1}}\cup C_{1_{V_2}}.$ Then $V_1$ will be either $\{x_{j_1+1},x_{j_1+3},\ldots,x_1,x_3,\ldots,x_{2k-1},x_{2k},x_{2k+2},\\ \ldots,x_{j_1-1}\}$
or $\{x_{j_1+1},x_{j_1+3},\ldots,x_1,x_3, \ldots,x_{2k+1},x_{2k+2},x_{2k+4},\ldots,x_{j_1-1}\}.$ Now if $\underline{m}_{1_{C_2}}x_1$ is not in optimal form then similarly we can construct a minimal vertex cover $V_2$ containing $x_1$ for $C_2$.
Next assume that $\underline{m}_{1_{C_2}}x_1$ is in optimal form. Then there are two possibilities:\\
\romannumeral 1) $|[x_1,y_{j_2}]|$ is odd. In this case we can choose the minimal vertex cover as $ C_{2_{V_1}}=\{x_1,y_3,\ldots,y_{2h+1},y_{2h+2},y_{2h+4},\ldots,y_{j_2-1}\}$ by the method described in the Case 1. Since $|[y_{j_2},x_1]|$ is even so we can choose the minimal vertex cover for $\underline{m}_{2_{C_2}}$ as $C_{2_{V_2}}=\{y_{j_2+1},y_{j_2+3},\\ \dots ,y_{2m},x_1\}$. \\
\romannumeral 2) $|[x_1,y_{j_2}]|$ is even and one of the alternating edge is missing. In this case we can choose the minimal vertex cover as $ C_{2_{V_1}}=\{x_1,y_3,\ldots,y_{j_2-1}\}.$ Since $|[y_{j_2},x_1]|$ is odd so we can choose the minimal vertex cover for $\underline{m}_{2_{C_2}}$ as $C_{2_{V_2}}=\{y_{j_2+1},y_{j_2+3},\dots ,y_{j_2+2h+1},y_{j_2+2h+2},\\y_{j_2+2h+4},\dots ,x_1\}$ as described in Case 1.
In both the cases consider $V_2=C_{2_{V_1}}\cup C_{2_{V_2}}$ and $V=V_1\cup V_2.$
Then by similar type of calculation observe that $W_V(\underline{m})=b(\underline{m})$.
\\
\textbf{Case 3:} Suppose one cycle contains single ancillary and another cycle contains more than one ancillary, then by Case 1 and Case 2 we can choose the minimal vertex cover $V$ such that $W_V(\underline{m})=b(\underline{m}).$\\
\textbf{Case 4:} Suppose that all the ancillaries are from one cycle and assume that this cycle is $C_1$. It may contain one ancillary or more than one ancillary. Let us do here for the case more than one ancillary, other cases will be similar. Let us assume that $x_{j_1}$ and $x_{i_1}$ are two consecutive ancillaries from the cycle $C_1$ such that $x_1\in[x_{j_1},x_{i_1}],$ where $i_1<j_1$ and $V_{H_q}=\{x_{j_1},x_{j_1+1},\dots,x_1,\dots,x_{i_1},y_2,y_3,\dots,y_{2m+1}\}.$
By Case 1 choose the minimal vertex cover for the monomial parts $\underline{m}_{1_{C_1}}$ and $\underline{m}_{2_{C_1}}.$ Let $V_1$ be the minimal vertex cover for these two parts. Now we will choose the minimal vertex cover for the cycle $C_2.$ Since $\underline{m}\mathfrak{m} athfrak{n} otin (c_1,c_2)$ so at least one vertex will be missing from the cycle $C_2.$
(a) If the vertex $x_1$ is missing from the monomial then we can keep $x_1$ in the minimal vertex cover and take $V_2=\{x_1,y_2,y_4,\dots,y_{2n}\}$ and $V=V_1\cup V_2.$\\
(b) If $x_1$ is not missing from the monomial then we need to find $V_2.$ Here some $y_r$ will be missing from the monomial. Now $V_1$ may or may not contain $x_1.$ First we consider the case where $V_1$ contains $x_1.$
If the missing vertex is of the form $y_{2k}$ then take the minimal vertex cover as $V_2=\{x_1,y_3,\ldots,y_{2k-1},y_{2k},y_{2k+2},\ldots,y_{2m}\},$ or if missing vertex is of the form $y_{2k+1}$ then take the minimal vertex cover as $V_2=\{x_1,y_3,\ldots,y_{2k+1},y_{2k+2},y_{2k+4},\ldots,\\y_{2m}\}.$ Now if $x_1$ is not missing as well as $x_1$ is not in $V_1$ then take $V_2$ as $\{y_2,y_4,\dots,y_{2k},y_{2k+1},\\\dots,y_{2m}\}.$ Then consider $V=V_1\cup V_2$ and by similar type of calculation observe that $W_V(\underline{m}_q)=b(\underline{m}_q).$ Now for the vertices of $G$ that are not in $V_{H_q}$, we can choose a minimal vertex cover $V^{\mathcal{P}rime}$ by Lemma \ref{cycle} such that $W_{V^{\mathcal{P}rime}\cup V}(\underline{m})=b(\underline{m}).$
\end{proof}
\begin{proposition}\label{lt}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n\leq m$ and $I=I(G)$ be the edge ideal of $G$. Let $c_1=x_1\operatorname{codim}ots x_{2n+1}$ and $c_2=x_1y_2\operatorname{codim}ots y_{2m+1}.$ Then $(L(t))\subseteq I^{t} + (c_1,c_2).$
\end{proposition}
\begin{proof} We prove the proposition by contrapositive. Let $\underline{m}$ be a monomial such that $\underline{m}$ $\mathfrak{m} athfrak{n} otin I^{t}$ and $\underline{m} \mathfrak{m} athfrak{n} otin (c_1,c_2).$ By Lemma \ref{bm}, there exist a minimal vertex cover $V$ such that $W_V({\underline{m}})=b({\underline{m}})$. As $\underline{m}\mathfrak{m} athfrak{n} otin I^{t}$, therefore $b(\underline{m})< t.$ As $b(\underline{m})<t$, so $\underline{m}\mathfrak{m} athfrak{n} otin L(t)$. Since $L(t)$ is a generating set of $(L(t)),$ this is sufficient to claim that $\underline{m}\mathfrak{m} athfrak{n} otin (L(t))$ because neither $\underline{m}$, nor any of its divisors whose vertex weights can only be less than that of $\underline{m}$, will be in the generating set.
\end{proof}
In order to describe $I^{(t)}$ we need the description of $(D(t)).$ Next lemma describes the elements of $(D(t)).$
\begin{lem}\label{dt}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n\leq m$. Let $I=I(G)$ be the edge ideal of $G$. Then for $t\geq 2$ {\small $$(D(t))= \langle\{I^i(c_1)^s(c_2)^b \mathfrak{m} id {i+(n+1)s+(m+1)b \geq t \mathfrak{m} box{ and } 2i+(2n+1)s+(2m+1)b \leq 2t-1 } \}\rangle $$}
where $c_1=x_1x_2\operatorname{codim}ots x_{2n+1}$ and $c_2=x_1y_2\operatorname{codim}ots y_{2m+1}$.
\end{lem}
\begin{proof}
Consider the set\\ $T=\{I^i(c_1)^s(c_2)^b \mathfrak{m} id {i+(n+1)s+(m+1)b \geq t \mathfrak{m} box{ and } 2i+(2n+1)s+(2m+1)b \leq 2t-1 } \}.$
From the definition of $D(t)$ it is clear that $T \subseteq D(t).$
Let $x^{\underline{a}}$ be a monomial in $D(t)$. Write $x^{\underline{a}}=(c_1)^s(c_2)^b x^{\underline{d}}$ such that $s$ and $b$ are as large as possible. Now $x^{\underline{d}}$ can be written in the form $E(x^{\underline{d}})A(x^{\underline{d}}),$ where $E(x^{\underline{d}})= e_{i_1}^{b_{i_1}}\operatorname{codim}ots e_{i_k}^{b_{i_k}}f_{j_1}^{b_{j_1}}\operatorname{codim}ots f_{j_k}^{b_{j_k}}$ with $b_{i_1}+\dots+b_{i_k}+b_{j_1}+\dots+b_{j_k}=i,$ and $A(x^{\underline{d}})$ is a product of ancillaries of $x^{\underline{d}}$.
Then from Lemma \ref{bm} we can find a minimal vertex cover $V$ such that $W_V(x^{\underline{d}})=W_V(E(x^{\underline{d}}))=i,$ which implies that $W_V(x^{\underline{a}})=(n+1)s+(m+1)b+i=W_V(E(x^{\underline{d}})(c_1)^s(c_2)^b)\geq t.$ As for any other minimal vertex cover $V^{\mathcal{P}rime}$ we have $W_{V^{\mathcal{P}rime}}(E(x^{\underline{d}})(c_1)^s(c_2)^b)\geq i+(n+1)s+(m+1)b\geq t$, therefore $E(x^{\underline{d}})(c_1)^s(c_2)^b \in D(t).$ Hence $\langle T\rangle=\langle (D(t))\rangle.$
\end{proof}
\begin{cor}
$(D(n+1))=(c_1).\label{3c1}$
\end{cor}
\begin{proof}
From Lemma \ref{dt} it follows that we need to find all the possible values of $(i,s,b)$ such that $i+(n+1)s+(m+1)b \geq n+1$ and $2i+(2n+1)s+(2m+1)b \leq 2n+1.$ Then it is clear that $(0,1,0)$ is the only possible value of $(i,s,b)$. Therefore $D(n+1)$ will contain only $c_1.$
\end{proof}
\begin{cor}\label{t3}
If $C_1$ and $C_2$ are of same length say $2n+1$ then $(D(n+1))=(c_1,c_2).$
\end{cor}
\begin{proof}
In Lemma \ref{dt} put $m=n$, then we need to find all the possible values of $(i,s,b)$ such that $i+(n+1)(s+b) \geq n+1$ and $2i+(2n+1)(s+b) \leq 2n+1.$ Observe that $(0,1,0),(0,0,1)$ are the only possible values of $(i,s,b)$ . Thus $D(n+1)=\{c_1,c_2\}.$
\end{proof}
All the generators of $D(t)$ described in Lemma \ref{dt} need not be minimal. In the next lemma we identify the minimal generators of $D(t)$ for $n+1\leq t\leq m+1$.
\begin{cor}\label{3dmin}
The minimal set of generators of the ideal $(D(t))$ for $n+1\leq t \leq m+1$ is given by the following set $$D_{\mathfrak{m} in}(t)=\{I^i{(c_1)}^s{(c_2)}^b \mathfrak{m} id i+(n+1)s+(m+1)b=t \mathfrak{m} box{ and } i<t\}.$$
\end{cor}
\begin{proof}
If $b\geq 1$ then by Corollary \ref{dt} there is only one possible generator namely $c_2$ corresponding to $(i,s,b)=(0,0,1)$, which is in fact a minimal generator for $(D(t))$ and this is only for $t=m+1.$ To find the remaining minimal generators let us assume that $b=0.$ Then it is clear that $I^ic_1^s$ with $i+(n+1)s=t$ are minimal generators.
Let us now assume that $I^ic_1^s$ with $i+(n+1)s>t$ is a minimal generator. But since $(i-1)+(n+1)s\geq t$ therefore $I^{i-1}c_1^s\in (D(t))$ which will contradict that $I^ic_1^s$ is a minimal generator of $(D(t))$. Hence the proof.
\end{proof}
\begin{thm}
Let $G$ be the clique-sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n\leq m.$ Let $I=I(G)$ be the edge ideal. Then for any $ t\in \mathfrak{m} athbb{N}$, $ \alpha{(I^{(t)})}$
=$ 2t- \Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor.\mathfrak{q} quad$\\
Particularly, the Waldschmidt constant of $I$ is given by \begin{align*}\widehat{\alpha}{(I)}=\frac{2n+1}{n+1}.\end{align*}
\end{thm}
\begin{proof}Since the degree of the elements of $D(t)$ is strictly less than the degree of the elements of $L(t),$ so if $D(t)$ is nonempty then minimal generating degree term will come from $D(t).$ Note that $\alpha(I^{(t)})=\mathfrak{m} in\{2i+(2n+1)s+(2m+1)b \mathfrak{m} id i+(n+1)s+(m+1)b=t\}.$ Let $d=2i+(2n+1)s+(2m+1)b=2t-(s+b).$ Then $d$ will be minimum when $s+b$ will be maximum. Write $t=k(n+1)+r,$ where $0\leq r\leq n.$ In general all the possible choices of $(i,s,b)$ are $(r_1,k_1,k_2),$ where $t=k_1(n+1)+k_2(m+1)+r_1.$ We will prove that $k\geq k_1+k_2.$ Write $k_2(m+1)=k_3(n+1)+r_2,$ where $r_2\leq n.$ Therefore $k_3\geq k_2.$ So $t=(k_1+k_3)(n+1)+(r_1+r_2).$ Again write $r_1+r_2=k_4(n+1)+r_4,$ where $r_4\leq n.$ Then $t=(k_1+k_2+k_3)(n+1)+r_4,$ with $r_4\leq n$, which implies that $k=k_1+k_3+k_4$ and $r=r_4.$ Hence $k\geq k_1+k_3 \geq k_1+k_2.$ Thus $\alpha(I^{(t)})=2t-k=2t- \Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor.$ Therefore $\widehat{\alpha}{(I)} = 2- \frac{1}{n+1}=\frac{2n+1}{n+1}$.
\end{proof}
\begin{lem}\label{ld}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n<m$. Let $I=I(G)$ be the edge ideal of $G$ and $c_1=x_1x_2\operatorname{codim}ots x_{2n+1},$ $c_2=x_1y_2\operatorname{codim}ots y_{2m+1}.$ Let $x^{\underline{a}}\in I^{(t)}$ be a minimal generator of $I^{(t)}$ such that $x^{\underline{a}}\in (c_1)$ and $x^{\underline{a}}\mathfrak{m} athfrak{n} otin I^{t}$ then $x^{\underline{a}}\in (D(t))$ for $n+1\leq t\leq m+1.$
\end{lem}
\begin{proof}Write $x^{\underline{a}}=c_1^sx^{\underline{d}}$ such that $s$ is as large as possible. Now $x^{\underline{d}}$ can be written in the form $E(x^{\underline{d}})A(x^{\underline{d}}),$ where $E(x^{\underline{d}})= e_{i_1}^{b_{i_1}}\operatorname{codim}ots e_{i_k}^{b_{i_k}}f_{j_1}^{b_{j_1}}\operatorname{codim}ots f_{j_k}^{b_{j_k}}$ with $b_{i_1}+\dots+b_{i_k}+b_{j_1}+\dots+b_{j_k}=i$ and $A(x^{\underline{d}})$ is a product of ancillaries of $x^{\underline{d}}$. By Proposition \ref{lt}, we can choose a minimal vertex cover $V^{\mathcal{P}rime}$ excluding the vertices that appear in ancillaries such that $W_{V^{\mathcal{P}rime}}(x^{\underline{d}})=W_{V^{\mathcal{P}rime}}(E(x^{\underline{d}}))=i, \text { which implies that } W_{V^{\mathcal{P}rime}}(x^{\underline{a}})=W_{V^{\mathcal{P}rime}}(c_1^sE(x^{\underline{d}}))=(n+1)s+i\geq t.$
As for any minimal vertex cover $V$, $W_V(c_1^sE(x^{\underline{d}}))\geq W_{V^{\mathcal{P}rime}}(c_1^sE(x^{\underline{d}})),$ so $c_1^sE(x^{\underline{d}})\in I^{(t)}.$ Since we are interested in minimal generators so we can take the monomial as $x^{\underline{a}}=c_1^sE(x^{\underline{d}}).$ Then $\deg(x^{\underline{a}})=(2n+1)s+2i.$ Now from Corollary \ref{3dmin} it follows that $(2n+1)s+2i=2[(n+1)s+i]-s=2t-s<2t.$
\end{proof}
\begin{thm}\label{3differentlength}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n<m$. Let $I=I(G)$ be the edge ideal of $G$. Let $c_1=x_1\operatorname{codim}ots x_{2n+1}$ and $c_2=x_1y_2\operatorname{codim}ots y_{2m+1}$. Then
\begin{enumerate}
\item For $1\leq t\leq n,$ we have $I^{(t)}=I^t.$
\item $I^{(n+1)}= I^{n+1}+(c_1).$
\item For $n+2\leq t\leq m+1$ we have $I^{(t)}=I^{t}+(D(t)).$
\end{enumerate}
\end{thm}
\begin{proof}
\begin{enumerate}
\item The proof follows from \cite[Corollary 4.5]{lam2015associated}.
\item Since $c_2\in I^m$ and $n+1\leq m$ thus $c_2\in I^{n+1}$. Then by Proposition \ref{lt}, $L(n+1)\subseteq I^{n+1}+c_1.$ Thus by Corollary \ref{3c1}, $I^{(n+1)}\subseteq I^{n+1}+c_1.$ Hence $I^{(n+1)}=I^{n+1}+c_1.$
\item From Lemma \ref{ld} and Proposition \ref{lt}, it follows that $(L(t))\subseteq I^{t}+(D(t)).$ So $I^{(t)}\subseteq I^{t}+(D(t)).$ Hence $I^{(t)}=I^{t}+(D(t)).$
\end{enumerate}
\end{proof}
\begin{cor}\label{3samelength}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2n+1})$ of same length joined at a vertex $x_1$. Let $I=I(G)$ be the edge ideal of $G$ and $c_1=x_1\operatorname{codim}ots x_{2n+1}$ and $c_2=x_1y_2\operatorname{codim}ots y_{2n+1}$. Then
\begin{enumerate}
\item $I^{(s)}=I^s$ for $1\leq s\leq n$.
\item $I^{(n+1)}= I^{n+1} + (x_1\operatorname{codim}ots x_{2n+1}) + (x_1y_2\operatorname{codim}ots y_{2n+1}).$
\end{enumerate}
\end{cor}
\begin{proof}
\begin{enumerate}
\item The proof follows from \cite[Corollary 4.5]{lam2015associated}.
\item The proof follows from Proposition \ref{lt} and Corollary \ref{t3}.
\end{enumerate}
\end{proof}
In \cite[Lemma 3.1]{jayanthan} Jayanthan and Kumar have described the structure of the symbolic Rees algebra for the clique sum of same length odd cycles and computed the invariants. In the next theorem we describe the structure of the symbolic Rees algebra for the clique sum of two different length odd cycles joined at a single vertex using the description of $L(t)$ and $D(t)$.
\begin{thm}\label{t2}
\begin{enumerate}
\item
Let $G$ be clique sum of two same length odd cycles $C_1=(x_1,\ldots,\\x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2n+1})$ joined at a single vertex $x_1$. Let $I=I(G)$ be the edge ideal. Let $s\in \mathfrak{m} athbb{N}$ and write $s=k(n+1)+r$ for some $k \in\mathfrak{m} athbb{Z} $ and $0\leq r\leq n.$ Then $$I^{(s)}=\displaystyle {\sum_{p+q=t=0}^{k}{I^{s-t(n+1)}(x_1\operatorname{codim}ots x_{2n+1})^p(x_1y_2\operatorname{codim}ots y_{2n+1})^q}}.$$
\item
\label{t1}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,\\y_{2m+1})$ joined at a vertex $x_1$ with $n<m$. Let $I=I(G)$ be the edge ideal of $G$ and $c_1=x_1\operatorname{codim}ots x_{2n+1}.$ Let $s=k_1(n+1)+r_1$, where $0\leq r_1\leq n$ and $s=k_2(m+1)+r_2$, where $0\leq r_2\leq m.$ Then\\
$$I^{(s)}=\displaystyle{\sum_{\substack{t_1,t_2\\0\leq t_1\leq k_1~and \\0\leq t_2\leq k_2 ~and \\ s-t_1(n+1)-t_2(m+1)\geq 0}}{I^{s-t_1(n+1)-t_2(m+1)}(c_1)^{t_1}(D_{\mathfrak{m} in}(m+1))^{t_2}}}.$$
\end{enumerate}
\end{thm}
\begin{proof} By Theorems \ref{pcycle} and \ref{pimplosive}, $G$ is an implosive graph in either case. Thus symbolic Rees algebra $\mathfrak{m} athcal{R}_s(I)$ of $I$ is generated by the monomials of the form $x^vt^b,$ where $v\in \{0,1\}^{\mathfrak{m} id V(G)\mathfrak{m} id}$ and $G^v$ is induced indecomposable subgraph of $G.$ It can be seen from \cite[Corollary 2a]{Frank} that induced indecomposable subgraphs of $G$ is either an edge or an odd cycle.
\begin{enumerate}
\item
Thus symbolic Rees algebra is minimally generated only in degrees 1 and $n+1$, so we have
\begin{align*} I^{(s)} &= \displaystyle{\sum_{r+k(n+1)=s}{I^r{(I^{(n+1)})}^k}}\\ &=\displaystyle {\sum_{p+q=t=0}^{k}{I^{s-t(n+1)}
(x_1\operatorname{codim}ots x_{2n+1})^p(x_1y_2\operatorname{codim}ots y_{2n+1})}^q}\text{ (By Corollary \ref{3samelength}).}
\end{align*}
\item
Here symbolic Rees algebra is generated in degrees 1, $n+1$ and $m+1.$\\
\begin{align*}
I^{(s)} &= \displaystyle{\sum_{p+q(n+1)+r(m+1)=s}{I^p{(I^{(n+1)})}^q}{(I^{(m+1)})}^r} \\
&= \displaystyle{\sum_{\substack{t_1,t_2\\0\leq t_1\leq k_1~and \\0\leq t_2\leq k_2 ~and \\ s-t_1(n+1)-t_2(m+1)\geq 0}}{I^{s-t_1(n+1)-t_2(m+1)}(c_1)^{t_1}(D_{\mathfrak{m} in}(m+1))^{t_2}}} \text{ (By Theorem \ref{3differentlength}).}
\end{align*}
\end{enumerate}
\end{proof}
\begin{proposition}\label{Dm}
Let $G$ be the clique sum of two odd cycles $C_1=(x_1,\ldots,x_{2n+1})$ and $C_2=(x_1,y_2,\ldots,y_{2m+1})$ joined at a vertex $x_1$ with $n<m$. Let $I=I(G)$ be the edge ideal of $G$ and $k= \Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor .$ Then
\begin{equation*}
\operatorname{sdefect}(I,t)\leq\begin{cases}
\displaystyle{\sum_{i=1}^{k}{{t-i(n+1)+2n+2m+1}\choose{2n+2m+1}}}, & \text{for $n+1\leq t\leq m $}.\\
\displaystyle{\sum_{i=1}^{k}{{t-i(n+1)+2n+2m+1}\choose{2n+2m+1}}}+1, & \text{ for $t=m+1$}.
\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}
From Theorem \ref{3differentlength}, it follows that $I^{(t)}=I^t+(D_{\mathfrak{m} in}(t))$ for $n+1\leq t\leq m+1.$ Therefore to compute symbolic defect we need to count the minimal number of generators of $(D_{\mathfrak{m} in}(t)).$
To find the cardinality of $D_{\mathfrak{m} in}(t),$ we need to find the number of solution of the equation $i+(n+1)s+(m+1)b=t$ for $i<t.$ For $b\geq 1,$ there is only one solution namely $(i,s,b)=(0,0,1)$ and it is for $t=m+1$. So to find the other solutions we assume $b=0$ and count the number of solutions of the equation $i+(n+1)s=t$ with $i<t$. As $0\leq i< t,$ therefore $1\leq s\leq \Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor.$ Thus the cardinality of the set $D_{\mathfrak{m} in}(t)$ is $\Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor $ for $1\leq t\leq m$ and cardinality of $D_{\mathfrak{m} in}(m+1)$ is $\Bigl\lfloor\dfrac{t}{n+1}\Bigr\rfloor +1.$ The elements of $D_{\mathfrak{m} in}(t)$ are of the form $I^i(c_1)^s(c_2)^b.$ If $b=0$ then the elements $D_{\mathfrak{m} in}(t)$ are of the form $I^i(c_1)^s$ and for different $i,s$ there are elements be computed repeatedly. Therefore\\ $\operatorname{sdefect}(I,t)\leq \displaystyle{\sum_{i=1}^{k}{\mathfrak{m} u(I^{t-i(n+1)})}}=\displaystyle{\sum_{i=1}^{k}{{t-i(n+1)+2n+2m+1}\choose{2n+2m+1}}}
\text{ for $n+1\leq t\leq m $ }$ and \\$\operatorname{sdefect}(I,t)\leq \displaystyle{\sum_{i=1}^{k}{\mathfrak{m} u(I^{t-i(n+1)})}}+1 =\displaystyle{\sum_{i=1}^{k}{{t-i(n+1)+2n+2m+1}\choose{2n+2m+1}}}+1$ for $t=m+1.$
\end{proof}
\iffalse
\begin{thm}
Let $G$ be clique sum of two odd cycles of same length $(x_1,\ldots,x_{2n+1})$ and $(x_1,y_2,\ldots,y_{2n+1})$ with a common vertex $x_1$. Let $I=I(G)$ be the edge ideal. Let $s\in \mathfrak{m} athbb{N}$ and write $s=k(n+1)+r$ for some $k \in\mathfrak{m} athbb{Z} $ and $0\leq r\leq n.$ Then $$\operatorname{sdefect}(I,s)=\displaystyle{\sum_{t=1}^{k}\mathfrak{m} u({I^{s-t(n+1)}})}(t+1)=\sum_{t=1}^{k} {s-t(n+1)+4n+1\choose 4n+1}(t+1).$$
\end{thm}
\begin{proof}
Let $I=(g_1,\dots,g_{4n+2}),$ where
$ g_i= x_ix_{i+1} ~\text{for}\ i=1,\dots,2n, $
$ g_{2n+1}= x_{2n+1}x_1,$
$g_{2n+2}= x_{1}y_2 ,$
$ g_i= y_iy_{i+1}~ \text{for}\ i=2n+2,\dots,4n+1,$
and $g_{4n+2} =y_{2m+1}x_1.$
Then $M=(M_{ij})=(\frac{\mathcal{P}artial g_i}{\mathcal{P}artial x_j})$ is an $(4n+2)\times(4n+1)$ with maximal rank. The conclusion now follows from Theorem \ref{t2} and \cite[Proposition5.3]{drabkin2018asymptotic}.
\end{proof}
\fi
\section{Symbolic powers of edge ideals of complete graph}
Throughout this section $G$ is a complete graph with $n$ vertices and $I=I(G)$ be the edge ideal of $G$. In this section we describe the generators of the symbolic powers of $I$ and calculate Waldschmidt constant, the resurgence and find the symbolic defect partially. We prove Minh's conjecture by showing that the regularity of symbolic powers and ordinary powers of $I$ are equal. We know that $I^{(t)}=(L(t))+(D(t))$. In \cite[Theorem 6.4]{janssen2017comparing} it is known that for complete graph $(L(t))=I^t$. Thus it is enough to understand the generators of $(D(t))$. The following lemma describes the generators of $(D(t))$.
\begin{lem} Let $G$ be a complete graph with $n$ vertices and $I=I(G)$ is the edge ideal. Then for $t\geq 2$ we have
\begin{equation*}
\begin{split}
D(t) = \{x_1^{a_1}x_2^{a_2}\operatorname{codim}ots x_n^{a_n}& \mathfrak{m} id a_{i_1}+a_{i_2}+\operatorname{codim}ots+a_{i_{n-1}}\geq t \mathfrak{m} box{ for } \{i_1,i_2,\ldots,i_{n-1}\}\subseteq\{1,2,\ldots,n\} \\
& \mathfrak{m} box{ and } a_1+\operatorname{codim}ots +a_n\leq 2t-1\}
\end{split}
\end{equation*}
\end{lem}
\begin{proof}
Since $G$ is a complete graph in $n$ vertices, any set of $(n-1)$ vertices forms a minimal vertex cover for $G.$ Let $x^{\underline{a}}={x_1^{a_1}x_2^{a_2}\operatorname{codim}ots x_n^{a_n}\in D(t) }$ then $a_1+\dots +a_n\leq 2t-1$ and since $x^{\underline{a}}\in I^{(t)}$ for any minimal vertex cover $V,$ $W_V(x^{\underline{a}})\geq t$ which implies that $a_{i_1}+a_{i_2}+\dots+a_{i_{n-1}}\geq t \mathfrak{m} box{ for } \{i_1,i_2,\dots,i_{n-1}\}\subseteq\{1,2,\dots,n\}$. Hence the result follows.
\end{proof}
Next theorem describes the generators of $\mathfrak{m} athcal{R}_s(I)$. Since complete graph is perfect graph thus by \cite{flores}, $G$ is implosive graph.
Hence by Theorem \ref{1implosive}, we need to find all the induced indecomposable subgraph of $G$.
\begin{thm}\label{4complete}
For $1\leq s\leq n-1$ we get $I^{(s)}= I^s+(D(s)).$ And for any $s\geq n$ we have
$$I^{(s)}=\sum_{\substack{(r_1,\dots,r_{n-1})\\ s=r_1+2r_2+\operatorname{codim}ots+ (n-1)r_{n-1}}}{I^{r_1}{{I^{(2)}}^{r_2}}\operatorname{codim}ots{{I^{(n-1)}}^{r_{n-1}}}}$$
\end{thm}
\begin{proof}
Here all induced subgraphs are induced indecomposable subgraphs. Therefore symbolic Rees algebra is minimally generated in degrees $1,2,\ldots,n-1$. Thus it is enough to find $D(s)$ for $2\leq s\leq n-1$. For $1\leq s\leq n-1$ we get $I^{(s)}= I^s+(D(s)),$ and if $s\geq n$ then $I^{(s)}$ is generated by $I,I^{(2)},\dots,I^{(n-1)}.$ Hence the result follows.
\end{proof}
Next we compute the Waldschmidt constant, resurgence and symbolic defect for $G$.
\begin{thm}\label{4completeinvariant}
Let $G$ be the complete graph on the vertices $\{x_1,\ldots,x_{n}\}$ and $I=I(G)$ be its edge ideal. Then
\begin{enumerate}
\item For any $ s\in \mathfrak{m} athbb{N}$, $ \alpha{(I^{(s)})}$
= $ s+ \Bigl\lceil\dfrac{s}{n-1}\Bigr\rceil.\mathfrak{q} quad$\\
Particularly, the Waldschmidt constant of $I$ is given by \begin{align*}\widehat{\alpha}{(I)} =\frac{n}{n-1}\end{align*}
\item $\alpha{(I^{(s)})} < \alpha{(I^t)}~~ if~~ and~~ only ~~if~~ I^{(s)}\mathfrak{m} athfrak{n} subseteq I^t$
\item The resurgence of $I$ is given by $\rho(I) = \frac{2n-2}{n}$
\end{enumerate}
\end{thm}
\begin{proof}
\begin{enumerate}
\item Since $G$ is a complete graph, any minimal vertex cover of $G$ will be of the form $x_{i_1,}x_{i_2},\ldots ,x_{i_{n-1}} $ where $ \{i_1,i_2,\dots,i_{n-1}\}\subseteq\{1,2,\dots,n\}$. Note that for $1\leq s\leq n-1,$ no monomial of degree $\leq s$ is in $I^{(s)},$ where as $x_1x_2\dots x_{s+1}\in I^{(s)}.$ Therefore $\alpha(I^{(s)})=s+1$ for $1\leq s\leq n-1.$ From Theorem \ref{4complete}, it follows that for $s\geq n,$ $$\alpha(I^{(s)})=\mathfrak{m} in\{2r_1+3r_2+\dots+nr_{n-1}\mathfrak{m} id s=r_1+2r_2+\dots+(n-1)r_{n-1}\}.$$ Now $2r_1+3r_2+\dots+nr_{n-1}=s+r_1+r_2+\dots+r_{n-1}.$ Then it is equivalent to find the minimum of ${r=r_1+r_2\dots+r_{n-1}}$ with the condition $s=r_1+2r_2+\dots+(n-1)r_{n-1}.$ Write $s=k(n-1)+p$ for some $k\in \mathfrak{m} athbb{Z}$ and $0\leq p\leq n-2.$ Then observe that minimum value of $r$ will occur for maximum value of $r_{n-1}$ and the maximum value of $r_{n-1}$ is $k.$ Therefore the minimal generating degree term will come from $I^{(n-1)^k}I^{(p)}.$ Hence
\begin{align*}
\alpha(I^{(s)}) & = \left\{\begin{array}{ll}
kn+(p+1)& \text{ if $p \mathfrak{m} athfrak{n} eq 0$} \\
kn & \text{ if $p=0$ }\\
\end{array} \right.\\
&= \left\{\begin{array}{ll}
s+1+ \Bigl\lfloor\dfrac{s}{n-1}\Bigr\rfloor & \text{ if $p \mathfrak{m} athfrak{n} eq 0$} \\
& \\
s+ \Bigl\lfloor\dfrac{s}{n-1}\Bigr\rfloor & \text{ if $p=0$ .}\\
\end{array} \right.
\end{align*}
Therefore $ \alpha{(I^{(s)})}$
=$ s+ \Bigl\lceil\dfrac{s}{n-1}\Bigr\rceil.$
Moreover $\frac{s}{n-1}\leq\Bigl\lceil\dfrac{s}{n-1}\Bigr\rceil\leq \frac{s}{n-1}+1.$ Thus \begin{align*}\widehat{\alpha}{(I)}=\displaystyle{\lim_{\substack{\longrightarrow\\k}}} m_{s\rightarrow\infty}{} \frac{\alpha{(I^{(s)})}}{s} = 1+ \frac{1}{n-1}=\frac{n}{n-1}.\end{align*}
\item Follows from \cite[Lemma 5.5]{janssen2017comparing}.
\item Let $T =\tiny\{ \frac{s}{t} ~~|~~I^{(s)}\mathfrak{m} athfrak{n} subseteq I^t \tiny\}. $
For any $\frac{s}{t}\in T$ we have $\alpha(I^{(s)})<\alpha(I^t).$ By part (1) it follows that $ s+ \Bigl\lceil\dfrac{s}{n-1}\Bigr\rceil< 2t.$ This implies that $\frac{s}{t}<\frac{2(n-1)}{n}.$ So $\rho(I)\leq \frac{2(n-1)}{n}.$ By \cite[Theorem 1.2]{guardo2013}, we have ${\alpha(I)}/{\operatorname{ht}at{\alpha}(I)}\leq \rho(I).$ Thus by part (1), this gives that $\frac{2(n-1)}{n}\leq \rho(I)$. Hence $\rho(I)=\frac{2(n-1)}{n}$.
\end{enumerate}
\end{proof}
\begin{thm}
Let $G$ be a complete graph with $n$ vertices and $I=I(G)$ be the edge ideal of $G$. Then for $2\leq s\leq n-1$ we have $$\operatorname{sdefect}(I,s)={n \choose s+1}.$$
\end{thm}
\begin{proof}
Since $G$ is a complete graph we have $I^{(s)}=I^s+(D(s))$. Therefore in order to compute $\operatorname{sdefect}(I,s),$ we count the number of minimal generators of the ideal $(D(s)).$ Then the following set give the minimal generators of the ideal $(D(s)),$
\begin{equation*}
\begin{split}
D_{\mathfrak{m} in}(s) = \{x_1^{a_1}x_2^{a_2}\operatorname{codim}ots x_n^{a_n}& \mathfrak{m} id a_{i_1}+a_{i_2}+\dots+a_{i_{n-1}}\geq s \mathfrak{m} box{ for } \{i_1,i_2,\dots,i_{n-1}\}\subseteq\{1,2,\dots,n\} \\
& \mathfrak{m} box{ and } a_1+\dots +a_n= \alpha(I^{(s)})\}.
\end{split}
\end{equation*}
Any other elements in $D(s)$ with condition $a_1+\dots+a_n>\alpha(I^{(s)})$ will be generated by the elements of $D_{\mathfrak{m} in}(s).$
From Theorem \ref{4completeinvariant}, it follows that for $1\leq s\leq n-1,$ $\alpha(I^{(s)})=s+1.$ Therefore $a_i\in\{0,1\}.$ Then it is clear that the monomial $x_1^{a_1}x_2^{a_2}\operatorname{codim}ots x_n^{a_n}$ will be an element of $D_{\mathfrak{m} in}(s)$ if and only if $(s+1)$ co-ordinates in that tuple will be $1$ and rest will be $0$. Hence number of solutions will be $n\choose {s+1}$.
\end{proof}
Now we show that Minh's conjecture is true for complete graphs. In order to prove this we need the following lemma.
\begin{lem}\label{artinian}
Let $G$ be a complete graph with $n$ vertices and $I=I(G)$ is the corresponding edge ideal in the polynomial ring $R=k[x_1,\ldots,x_n]$. Let $\mathfrak{m} =(x_1,\ldots,x_n)$ is the maximal homogeneous ideal of $R$, then $\mathfrak{m} ^{t-1}I^{(t)}\subseteq I^t.$
\end{lem}
\begin{proof}
Let $x^{\underline{a}}=x_1^{a_1}x_2^{a_2}\operatorname{codim}ots x_n^{a_n}\in I^{(t)}$ and $x_1^{b_1}x_2^{b_2}\operatorname{codim}ots x_n^{b_n}\in \mathfrak{m} ^{t-1}$, which implies $a_{i_1}+a_{i_2}+\dots+a_{i_{n-1}}\geq t \mathfrak{m} box{ for } \{i_1,i_2,\dots,i_{n-1}\}\subseteq\{1,2,\dots,n\}$ and $b_1+\dots+b_n\geq t-1.$ Then any monomial in $\mathfrak{m} ^{t-1}I^{(t)}$ will be of the form $x_1^{a_1+b_1}x_2^{a_2+b_2}\operatorname{codim}ots x_n^{a_n+b_n}=x^{\underline a+\underline b}.$ Since $G$ is a complete graph, if any monomial is written in the optimal form then there will be at most one ancillary. Let $x_1$ is the ancillary with degree at least $2$. Let $e_{ij}$ denote the edge between $x_i$ and $x_j.$ Then in the optimal form only the edges of the form $e_{1i}$ will be present. Thus $x^{\underline a+\underline b}= x_1^ce_{12}^{a_2+b_2}e_{13}^{a_3+b_3}\operatorname{codim}ots e_{1n}^{a_n+b_n}$ where $c\geq 2$. Now $b(x_1^ce_{12}^{a_2+b_2}e_{13}^{a_3+b_3}\operatorname{codim}ots e_{1n}^{a_n+b_n})= a_2+\dots+ a_n+b_2+\dots+b_n \geq t$, which implies $x^{\underline a+\underline b}\in I^t$. Hence $\mathfrak{m} ^{t-1}I^{(t)}\subseteq I^t.$
Now consider the case when $x_1$ is an ancillary of degree at most $1. $
We would like to prove that $\deg(x^{\underline{a}})\geq t+1.$
Let us assume that $\deg{(x^{\underline{a}})}< t+1.$ So $\deg(x^{\underline{a}})\leq t,$ but we also have that $a_1+\dots+a_{n-1}\geq t$ which implies that $a_n=0.$ Similarly we can show that $a_1=a_2=\operatorname{codim}ots=a_n=0$ which is a contradiction. Therefore $\deg(x^{\underline{a}})\geq t+1.$ Thus the degree of $x^{\underline a+\underline b}$ is at least $(t-1)+(t+1)=2t.$ Now in $x^{\underline a+\underline b}$ if the ancillary is of degree $0$ then nothing to prove. Let us assume that the ancillary is of degree $1$. Then $x^{\underline a+\underline b}=x_1\mathcal{P}rod_{i,j}e_{ij}^{b_{ij}}.$ which implies that $\deg(\mathcal{P}rod_{i,j}e_{ij}^{b_{ij}})\geq 2t-1$. Hence $\deg(\mathcal{P}rod_{i,j}e_{ij}^{b_{ij}})\geq 2t.$ Thus $x^{\underline a+\underline b}\in I^t.$
\end{proof}
\begin{thm}
Let $G $ be a complete graph with $n$ vertices and $I=I(G)$ be the edge ideal of $G$. Then for any $s\geq 1,$ we have $$\operatorname{reg} I^{(s)}=\operatorname{reg} I^s.$$
\end{thm}
\begin{proof}
By Lemma \ref{artinian} we have $\mathfrak{m} ^{s-1}I^{(s)}\subseteq I^s,$ so $I^{(s)}/I^s$ is an Artinian module. Therefore $\dim {I^{(s)}}/{I^s}=0$ and hence $H_{\mathfrak{m} }^i(I^{(s)}/I^s)=0$ for $i>0.$ Consider the following short exact sequence
$$0\rightarrow I^{(s)}/I^s \rightarrow R/I^s \rightarrow R/I^{(s)}\rightarrow 0. $$
Applying local cohomology functor we get $H_{\mathfrak{m} }^i(R/I^{(s)})\cong H_{\mathfrak{m} }^i(R/I^{s})$ for $i\geq 1$ and the following short exact sequence
\begin{equation} \label{exact}
0\rightarrow H_{\mathfrak{m} }^0(I^{(s)}/I^s) \rightarrow H_{\mathfrak{m} }^0(R/I^s) \rightarrow H_{\mathfrak{m} }^0(R/I^{(s)})\rightarrow 0.
\end{equation}
Now from $(\ref{exact})$ it follows that $a_0(R/I^{(s)})\leq a_0(R/I^s).$
So we can conclude that $\operatorname{reg} R/I^{(s)}=\mathfrak{m} ax\{a_i(R/I^{(s)})+i~|~i\geq 0\}\leq\mathfrak{m} ax\{a_i(R/I^{(s)})+i~|~i\geq 0\}=\operatorname{reg}(R/I^s). $ Therefore $\operatorname{reg} I^{(s)}\leq \operatorname{reg} I^s.$
It follows from \cite[Theorem 4.6]{gu2018symbolic} that $\operatorname{reg} I^{(s)}\geq 2s.$ As complete graph is co-chordal graph, so by \cite{herzog2004} $\operatorname{reg} I^{s}= 2s.$ Thus we have $\operatorname{reg} I^{(s)}=\operatorname{reg} I^s.$
\end{proof}
\end{document} |
\begin{document}
\title{Zero-sum $K_m$ over $\Z$ and the story of $K_4$}
\begin{center}
\begin{multicols}{2}
Yair Caro\\[1ex]
{\small Dept. of Mathematics and Physics\\
University of Haifa-Oranim\\
Tivon 36006, Israel\\
[email protected]}
\columnbreak
Adriana Hansberg\\[1ex]
{\small Instituto de Matem\'aticas\\
UNAM Juriquilla\\
Quer\'etaro, Mexico\\
[email protected]}\\[2ex]
\end{multicols}
Amanda Montejano\\[1ex]
{\small UMDI, Facultad de Ciencias\\
UNAM Juriquilla\\
Quer\'etaro, Mexico\\
[email protected]}\\[4ex]
\end{center}
\begin{abstract}
We prove the following results solving a problem raised in Caro-Yuster \cite{CY3}.
For a positive integer $m\geq 2$, $m\neq 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-1,1\}$ (and hence a weighting function $f: E(K_n)\to \{-1,0,1\}$), such that $\sum_{e\in E(K_n)}f(e)=0$ but, for every copy $H$ of $K_m$ in $K_n$, $\sum_{e\in E(H)}f(e)\neq 0$. On the other hand, for every integer $n\geq 5$ and every weighting function $f:E(K_n)\to \{-1,1\}$ such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-h(n)$, where $h(n)=2(n+1)$ if $n \equiv 0$ (mod $4$) and $h(n)=2n$ if $n \not\equiv 0$ (mod $4$), there is always a copy $H$ of $K_4$ in $K_n$ for which $\sum_{e\in E(H)}f(e)=0$, and the value of $h(n)$ is sharp.
\end{abstract}
\section{Introduction}
Our main source of motivation is a recent paper of Caro and Yuster \cite{CY3}, extending classical zero-sum Ramsey theory to weighting functions $f:E(K_n)\to \{-r,-r+1, \cdots ,0, \cdots , r-1,r\}$ seeking zero-sum copies of a given graph $H$ subject to the obviously necessary condition that $|\sum_{e\in E(K_n)}f(e)|$ is bounded away from $ r\binom{n}{2}$, or even in the extreme case where $|\sum_{e\in E(K_n)}f(e)|=0$.
In zero-sum Ramsey theory, one studies functions $f:E(K_n)\to X$, where $X$ is usually the cyclic group $\mbox{${\mathbb Z}$}_k$ or (less often) an arbitrary finite abelian group. The goal is to show that, under some necessary divisibility conditions imposed on the number of the edges $e(H)$ of a graph $H$ and for sufficiently large $n$, there is always a zero-sum copy of $H$, where by a zero-sum copy of $H$ we mean a copy of $H$ in $K_n$ for which $\sum_{e\in E(H)}f(e)=0$ (where $0$ is the neutral element of $X$). For several results concerning zero-sum Ramsey theory for graphs see \cite{AC,BD1,BD2,C1,C2,CY1,FK,SS},
for zero-sum Ramsey problems concerning matrices and linear algebraic techniques see \cite{BCRY,CY2,WW,W}.
The following notation was introduced in \cite{CY3} and the following zero-sum problems over $\mbox{${\mathbb Z}$}$
were considered. For positive integers $r$ and $q$, an \emph{$(r,q)$-weighting} of the edges of the complete graph $K_n$ is a function $f:E(K_n)\to \{-r, \cdots ,r\}$ such that $|\sum_{e\in E(K_n)}f(e)|<q$. The general problem considered in \cite{CY3} is to find nontrivial conditions on the $(r,q)$-weightings that guarantee the existence of certain bounded-weight subgraphs and even zero-weighted subgraphs (also called \emph{zero-sum} subgraphs). So, given a subgraph $H$ of $K_n$, and a weighting $f:E(K_n)\to \{-r,\cdots ,r\}$, the \emph{weight} of $H$ is defined as $w(H)=\sum_{e\in E(H)}f(e)$, and we say that $H$ is a \emph{zero-sum graph} if $w(H)=0$. Finally, we say that a weighting function $f:E(K_n) \to \{-1,1\}$ is \emph{zero-sum-$H$ free} if it contains no zero-sum copy of $H$.
Among the many results proved in \cite{CY3}, the following theorem and open problem are the main motivation of this paper.
\begin{theorem}[Caro and Yuster, \cite{CY3}]\label{thm:CY}
For a real $\epsilon >0$ the following holds. For $n$ sufficiently large, any weighting $f:E(K_n)\to \{-1,0,1\}$ with $|\sum_{e\in E(K_n)}f(e)|\leq (1-\epsilon)n^2/6$ contains a zero-sum copy of $K_4$.
On the other hand, for any positive integer $m$ which is not of the form $m = 4d^2$, there are infinitely many integers $n$ for which there is a weighting $f:E(K_n)\to \{-1,0,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ without a zero-sum copy of $K_m$.
\end{theorem}
The authors posed the following complementary problem:
\begin{problem}[Caro and Yuster, \cite{CY3}]\label{probl}
For an integer $m = 4d^2$, is it true that, for $n$ sufficiently large, any weighting $f:E(K_n)\to \{-1,0,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ contains a zero-sum copy of $K_m$?
\end{problem}
The main result in this paper is a negative answer to the above problem for any $m \geq 2$ except for $m=4$ already for weightings of the form $f:E(K_n)\to \{-1,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$. On the other hand, concerning the study of the existence of zero-sum copies of $K_4$, we prove a result analogous to Theorem \ref{thm:CY} where the range $\{-1,1\}$ instead of $\{-1,0,1\}$ is considered. Finally, we show that Theorem \ref{thm:CY} can neither be extended to wider ranges. To be more precise, we gather our results in the following theorem.
\begin{theorem}\label{thm:main} \hspace{1cm}
\begin{enumerate}
\item For any positive integer $m \geq2$, $m\neq 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-1,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_m$ free.
\item Let $n$ be an integer such that $n\geq 5$. Define $g(n)=2(n+1)$ if $n \equiv 0$ (mod $4$) and $g(n)=2n$ if $n \not\equiv 0$ (mod $4$). Then, for any weighting $f:E(K_n)\to \{-1,1\}$ such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-g(n)$, there is a zero-sum copy of $K_4$.
\item There are infinitely many values of $n$ such that the following holds:
There is a weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_4$ free.
\end{enumerate}
\end{theorem}
Theorem \ref{thm:main} together with the above Theorem \ref{thm:CY} do not only solve Problem \ref{probl}, they also supply a good understanding of the situation concerning $K_4$ as the value of $g(n)$ is sharp and the upper bound $(1-\epsilon)n^2/6$ in Theorem \ref{thm:CY} is nearly sharp, as already observed in \cite{CY3}.
We will use the following notation. Given a weighting $f:E(K_n)\to \{-r, \cdots, r\}$ and $i\in \ \{-r, \cdots, r\}$, denote by $E(i)$ the set of the $i$-weighted edges, that is, $E(i)=f^{-1}(i)$ and define $e(i)=|E(i)|$. Given a vertex $x\in V(K_n)$ we use $deg_{i}(x)$ to denote the number of $i$-weighted edges incident to $x$, that is, $deg_{i}(x)=|\{u:f(xu)=i\}|$.
In Section \ref{sec:K4} we will prove instances 2 and 3 of Theorem \ref{thm:main}, corresponding to the study of the existence of zero-sum copies of $K_4$. In order to prove instance 2, we will use an equivalent formulation consequence of the following remark.
\begin{remark}\label{rem:eq}
A weighting $f:E(K_n)\to \{-1,1\}$ satisfies $\left|\sum_{e\in E(K_n)}f(e)\right|\leq \binom{n}{2}-g(n)$ if and only if $\min\{e(-1),e(1)\}\geq \frac{1}{2}g(n)$.
\end{remark}
The remark follows from the fact that $e(1)+e(-1)=\binom{n}{2}$, which implies $|\sum_{e\in E(K_n)}f(e)|=|e(1)-e(-1)|=\max\{e(-1),e(1)\}-\min\{e(-1),e(1)\}=\binom{n}{2}-2\min\{e(-1),e(1)\}$.
In Section \ref{sec:K4} we will also prove that instance 2 of Theorem \ref{thm:main} is best posible by exhibiting, for each $n\geq5$, a weighting function $f:E(K_n)\to \{-1,1\}$ with $\min\{e(-1),e(1)\}= \frac{1}{2}g(n)-1$ and no zero-sum copies of $K_4$. Moreover, we will characterize the extremal functions.
Finally, relying heavily on Pell equations and some classical biquadratic Diophantine equations,
in Section \ref{sec:Kk} we will prove instance 1 of Theorem \ref{thm:main}, corresponding to the study of the existence of zero-sum copies of $K_m$ in $0$-weighted weightings, where $m \neq 4$.
\section{The case of $K_4$}\label{sec:K4}
We will use standard graph theoretical notation to denote particular graphs. Having said this, $K_{1,3}$ will stand for the star with three leaves, $K_3 \cup K_1$ for the disjoint union of a triangle and a vertex, $P_k$ for a path with $k$ edges, and $C_k$ for a cycle with $k$ edges.
A weighting function $f:E(K_n) \to \{-1,1\}$ is \emph{zero-sum-$K_4$ free} if and only if the graph induced by $E(-1)$ (or equivalently $E(1)$) is $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free (in the induced sense). The following lemma, which characterizes the $K_3$-free subclass of the family of $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graphs, will be useful in proving the forthcoming results.
We define
\[h(n)= \left\{ \begin{array}{rl}
n+1, & \mbox{ if } n\equiv 0 \mbox{ (mod $4$), and} \\
n, & \mbox{ otherwise. } \\
\end{array}\right.\]
\begin{lemma}\label{lem:triangle-free}
Let $G$ be a $\{K_{1,3}, K_3, P_3\}$-free graph on $n$ vertices. Then each component of $G$ is isomorphic to one of $C_4$, $K_1$, $K_2$ or $P_2$. Moreover, $e(G) \le h(n)-1$, and equality holds if and only if $G \cong J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$.
\end{lemma}
\begin{proof}
Let $J$ be a connected component of $G$. If $J$ has at most $3$ vertices, then it is easy to see that $J \in \{K_1, K_2, P_2\}$. So assume that $J$ has at least $4$ vertices. Then, since $J$ is $\{K_3, P_3\}$-free, we infer that $J$ has no vertex of degree larger than $2$ and so we can deduce that $J \cong C_4$. Further, we note that $e(J) = |J|$ if $J \cong C_4$, and $e(J) = |J|-1$ otherwise. This implies that, among all $\{K_{1,3}, K_3, P_3\}$-free graphs on $n$ vertices, $G$ has maximum number of edges if and only if $G \cong J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$. Since, clearly, $e(J \cup \bigcup_{i=1}^{q} C_4) = h(n) -1$, the proof is complete.
\end{proof}
\begin{lemma}\label{lem:one_is_K3-free}
Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ be a zero-sum-$K_4$ free coloring. Let $G_{-1}$ and $G_1$ be the graphs induced by $E(-1)$ and $E(1)$, respectively. Then at least one of $G_{-1}$ or $G_1$ is triangle-free.
\end{lemma}
\begin{proof}
Suppose for contradiction that both $G_{-1}$ and $G_1$ have a triangle. Let $abc$ be a triangle in $G_{-1}$ and $uvw$ a triangle in $G_1$. Suppose first that $abc$ and $uvw$ have a vertex in common, say $a=u$. Consider the graph $J$ induced by the $(-1)$-edges among vertices in $\{a,b,c,v,w\}$. If a vertex $x \in \{b,c\}$ is neighbor of both $v$ and $w$, then $\{x,v,w,a\}$ would induce a $K_{1,3}$ in $G_{-1}$, which is not possible. If no vertex $x \in \{b,c\}$ is adjacent to some $y \in \{v,w\}$, then $\{a,b,c,y\}$ would induce a $K_3\cup K_1$ in $G_{-1}$, which again is not possible. Hence, $\{b,c,v,w\}$ induces two independent edges. But then $\{b,c,v,w\}$ induces a $P_3$ in $G_{-1}$, a contradiction. Hence, we can assume that any pair of triangles such that one has only $(-1)$-edges and the other only $1$-edges are vertex disjoint. This implies that from any vertex in $\{u,v,w\}$ there is at most one $(-1)$-edge to vertices from $\{a,b,c\}$. Analogously, from any vertex in $\{a,b,c\}$ there is at most one $1$-edge to vertices from $\{u,v,w\}$. But this implies that there are at most $6$ edges between $\{a,b,c\}$ and $\{u,v,w\}$, which is false.
Since in all cases we obtain a contradiction, we conclude that at least one of $G_{-1}$ or $G_1$ is triangle-free.
\end{proof}
By Remark \ref{rem:eq}, the next result is equivalent to instance 2 of Theorem \ref{thm:main}.
\begin{theorem}\label{thm:k4}
Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ such that $\min\{e(-1),e(1)\}\geq h(n)$. Then there is a zero-sum $K_4$.
\end{theorem}
\begin{proof}
Let $f:E(K_n)\to \{-1,1\}$ be such that $\min\{e(-1),e(1)\}\geq h(n)$ and suppose for contradiction that it has no zero-sum $K_4$. Let $G_{-1}$ and $G_1$ be the graphs induced by $E(-1)$ and $E(1)$, respectively. Then both $G_{-1}$ and $G_1$ are $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graphs. By Lemma \ref{lem:one_is_K3-free}, $G_{-1}$ or $G_1$ is $K_3$-free. So we may assume, without loss of generality, that $G_{-1}$ is triangle-free. It follows by Lemma \ref{lem:triangle-free} that $e(-1) = |E(G_{-1})| \le h(n)-1$, which is a contradiction to the hypothesis.
\end{proof}
The following theorem shows that Theorem \ref{thm:k4} is best possible and characterizes the extremal zero-sum-$K_4$ free weightings. We will use Mantel's Theorem, that any graph on $n$ vertices and at least $\frac{n^2}{4}+1$ edges contains a copy of $K_3$.
\begin{theorem}\label{thm:k4_sharp}
Let $n\geq 5$ and $f:E(K_n)\to \{-1,1\}$ such that $e(1) = h(n)-1$. Then $f$ is zero-sum-$K_4$ free if and only if the graph induced by $E(1)$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$.
\end{theorem}
\begin{proof}
If the graph induced by $E(1)$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$, it is easy to check that $f$ is zero-sum-$K_4$ free. Conversely, let $f$ be zero-sum-$K_4$ free. Then the graphs $G_{-1}$ and $G_1$ induced by $E(-1)$ and $E(1)$, respectively, are both $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free. If $n = 5$, it is easy to check that the only $\{K_{1,3}, K_3 \cup K_1, P_3\}$-free graph with $h(5) - 1 = 4$ edges is isomorphic to $C_4 \cup K_1$, and so we are done. Hence, we may assume that $n \ge 6$. Observe that
\[e(-1) = \frac{n(n-1)}{2} - h(n)+1 \ge \frac{n(n-1)}{2} - n = \frac{n(n-3)}{2},\]
whose right-hand side is at least $\frac{n^2}{4}$ for $n \ge 6$. Hence, by Mantel's Theorem, $G_{-1}$ has a triangle, and, by Lemma \ref{lem:one_is_K3-free}, this implies that $G_1$ is triangle-free. It follows that $G_1$ is a $\{K_{1,3}, K_3, P_3\}$-free graph on $n$ vertices and with $h(n) - 1$ edges. Thus, with Lemma \ref{lem:triangle-free}, we obtain that $G_1$ is isomorphic to $J \cup \bigcup_{i=1}^{q} C_4$, where $J \in \{\emptyset, K_1, K_2, P_2\}$ and $q = \lfloor \frac{n}{4} \rfloor$, and we are done.
\end{proof}
The following theorem is instance 2 from Theorem \ref{thm:main}. It shows that, whenever we take a wider range for the weighting function $f$, we cannot hope for a result as in Theorem \ref{thm:k4} anymore.
\begin{theorem}\label{thm:larger_range}
There are infinitely many values of $n$ such that the following holds:
There is a weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ with $\sum_{e\in E(K_n)}f(e)=0$ which is zero-sum-$K_4$ free.
\end{theorem}
\begin{proof}
Let $X \cup Y$ be a partition of the vertex set of $K_n$ and consider the weighting function $f:E(K_n)\to \{-2,-1,0,1,2\}$ such that
\[
f(uv) = \left\{\begin{array}{ll}
-2, & \mbox{if } u,v \in X\\
1, & \mbox{if } u,v \in Y\\
0, & \mbox{otherwise}.
\end{array}
\right.
\]
Clearly, $f$ is zero-sum-$K_4$ free. On the other hand, $\sum_{e\in E(K_n)}f(e)=0$ if and only if
\[
-2\frac{|X|(|X|-1)}{2} + \frac{|Y|(|Y|-1)}{2} = 0,
\]
which is equivalent to $(2|Y|-1)^2-2(2|X|-1)^2 = -1$. Hence, solving the latter equation is equivalent to solve the following Pell's equation
\begin{equation}\label{eq:pell-pythago}
y^2-2x^2 = -1,
\end{equation}
for (odd) integers $x = 2|X|-1$ and $y= 2|Y|-1$. It is well-known that the Diophantine equation $y^2-2x^2=\pm 1$ has infinitely many solutions given by
\[x_{k}=\frac{a^k-b^k}{a-b}=\frac{a^k-b^k}{2\sqrt{2}}, \hspace{2ex} y_k=\frac{a^k+b^k}{2},\]
where $a=1+\sqrt{2}$, $b=1-\sqrt{2}$ and $k \in \mathbb{N}$. Moreover, since $y_k^2 - 2x_k^2 = (-1)^k$, the solutions for equation (\ref{eq:pell-pythago}) are the pairs $(x_k,y_k)$ where $k$ is odd. Observe also that, for odd $k$, $x_k$ and $y_k$ are odd, too. Hence, each odd $k$ gives us a solution $(\frac{x_k+1}{2}, \frac{y_k+1}{2})$ for $(|X|,|Y|)$ and thus for $n = \frac{x_k+y_k}{2}+1$ and we are done.
\end{proof}
For the sake of comprehension, let us compute small values of $n=\frac{x_k+y_k}{2}+1$ and exhibit how the partition $(|X|,|Y|)=(\frac{x_k+1}{2}, \frac{y_k+1}{2})$ gives a zero-sum weighting function $f$ as described in the theorem. Recall that we only want to consider solutions for odd values of $k$.
So we have $(x_1,x_3,x_5,\dots)=(1,5,29,\dots)$ and $(y_1,y_3,y_5,\dots)=(1,7,41,\dots)$, and the corresponding sequence of $n$'s is $(2,7,36,\dots )$. The case of $n=2$ is not interesting for vaquity reasons. For $n=7$, the partition is $(|X|,|Y|)=(3,4)$, thus there will be $ \binom{3}{2}$ edges weighted with $-2$, $ \binom{4}{2}$ edges weighted with $1$ and the rest of edges weighted with $0$, adding up to zero. For $n=36$, the partition is $(|X|,|Y|)=(15,21)$, and the sum of weighted edges is $-2\binom{15}{2}+1\binom{21}{2}=(-2) \cdot 105+1 \cdot 210=0$.
\section{The case of $K_m$, $m\neq 4$}\label{sec:Kk}
A \emph{balanced} $\{-1,1\}$-weighting function $f:E(K_n)\to \{-1,1\}$ is a function for which $e(-1)=e(1)$.
In Section \ref{sec:K4}, we prove that, for $n\geq 5$, any function $f:E(K_n)\to \{-1,1\}$ with sufficiently many edges assigned to each type contains a zero-sum $K_4$. In this section, we prove that this is not true for $K_m$ with $m \in \mathbb{N} \setminus \{1, 4\}$. In other words, we exhibit, for infinitely many values of $n$, the existence of a balanced weighting function $f:E(K_n)\to \{-1,1\}$ without a zero-sum copies of $K_m$, where $m \neq 1,4$. In order to define those functions, consider first the following Pell equation:
\begin{equation}\label{eq:pell}
8x^2-8x+1=y^2.
\end{equation}
It is well known that such a Diophantine equation has infinitely many solutions given by the recursion
\[(x_1,y_1)=(1,1),\]
\[(x_2,y_2)=(3,7),\]
\[y_k=6y_{k-1}-y_{k-2}, \hspace{.5cm} x_{k}=\frac{y_k+x_{k-1}+1}{3}.\]
\begin{lemma}\label{lem:bal1}
Let $n$ be a positive integer and consider the complete graph $K_n$ and a partition $V(K_n) = A \cup B$ of its vertex set. Then the function $f:E(K_n)\to \{-1,1\}$ defined as
\[f(e)= \left\{ \begin{array}{rl}
-1, & \mbox{ if } e\subset A \\
1, & \mbox{ otherwise, } \\
\end{array}\right.\]
is balanced if and only if $n = \frac{1+y_k}{2}$ and $|A| = x_k$ for some $k \in\mathbb{N}$.
\end{lemma}
\begin{proof}
Suppose first that $f$ is balanced and let $|A| = x$. Then
$$e(-1)=\frac{x(x-1)}{2} = \frac{1}{2} \binom{n}{2},$$ which yields
\[
n^2-n-(2x^2-2x)=0,
\]
and therefore $n = \frac{1+\sqrt{8x^2-8x+1}}{2}$. But this is only an integer if $8x^2-8x+1 = y^2$ for some integer $y$, and we obtain equation (\ref{eq:pell}). Hence, $|A| = x = x_k$ and $n = \frac{1+y_k}{2}$ for some $k \in \mathbb{N}$.\\
Conversely, suppose that $n = \frac{1+y_k}{2}$ and $|A| = x_k$ for some $k \in\mathbb{N}$. Then
\[
n = \frac{1+y_k}{2} = \frac{1+\sqrt{y_k^2}}{2} = \frac{1+\sqrt{8x_k^2-8x_k+1}}{2}.
\]
Thus $n$ is the positive root of
\begin{equation}\label{eq:n_k}
n^2-n-(2x_k^2-2x_k)=0,
\end{equation}
which is equivalent to
\[
\frac{x_k(x_k-1)}{2}=\frac{1}{2}\binom{n}{2}.
\]
Since the left hand side of this equation is precisely $e(-1)$ and the right hand side is half the number of the edges of $K_n$, it follows that $f$ is balanced.
\end{proof}
\begin{lemma}\label{lem:bal2}
Let $n$ be a positive integer and consider the complete graph $K_n$ and a partition $V(K_n) = A \cup B$ of its vertex set. Then the function \[f(e)= \left\{ \begin{array}{rl}
-1, & \mbox{ if } e\subset A \mbox{ or } e\subset B \\
1, & \mbox{ otherwise, } \\
\end{array}\right.\]
is balanced if and only if $n = k^2$ and $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ for some $k \in\mathbb{N}$.
\end{lemma}
\begin{proof}
Suppose first that $f$ is balanced and let $|A| = w$. Then
\[
e(1) = w(n-w)=\frac{1}{2} \binom{n}{2},
\]
which is equivalent to
\[
w^2 - nw + \frac{1}{4}n(n-1)=0.
\]
Hence,
\begin{equation}\label{eq:n(w)}
w = \frac{n \pm \sqrt{n}}{2},
\end{equation}
which is an integer if and only if $n = k^2$ for some $k \in \mathbb{N}$. So we obtain $n = k^2$ and $w \in \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$. Since $|B| = n - |A| = k^2 - w$, it follows easily that $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ and we are done.\\
Conversely, suppose that $n = k^2$ and $\{|A|, |B| \} = \{\frac{1}{2}k(k+1), \frac{1}{2}k(k-1)\}$ for some $k \in\mathbb{N}$. Without loss of generality, assume that $|A| = \frac{1}{2}k(k+1)$. Then
\[
e(1) = |A| (n - |A|) = \frac{1}{2}k(k+1) \left( k^2 - \frac{1}{2}k(k+1)\right) =
\frac{1}{4} k^2(k^2-1) = \frac{1}{2} \binom{n}{2},\]
implying that $f$ is balanced.
\end{proof}
We define the set $S_1$ as the set of all integers $n_k = \frac{1+y_k}{2}$, $k \in \mathbb{N}$, where $(x_k,y_k)$ is the k-th solution of (\ref{eq:pell}), that is,
$$S_1 = \left\{\frac{1+y_k}{2} \; | \; k \in \mathbb{N} \right\}.$$
Further, let $S_2$ be the set of all integer squares, that is,
$$S_2 = \left\{k^2 \; | \; k \in \mathbb{N} \right\}.$$
Lemmas \ref{lem:bal1} and \ref{lem:bal2} yield the following corollary.
\begin{corollary}\label{cor:Km_Si}
For any integer $m \in \mathbb{N} \setminus (S_1 \cap S_2)$, there are infinitely many positive integers $n$ such that there exists a balanced function $f:E(K_n)\to \{-1,1\}$ without zero-sum $K_m$.
\end{corollary}
\begin{proof}
Let $m \in \mathbb{N} \setminus (S_1 \cap S_2)$. By Lemmas \ref{lem:bal1} and \ref{lem:bal2}, there is a balanced function $f:E(K_n)\to \{-1,1\}$ for each $n \in S_1 \cup S_2$. Suppose that there is a zero-sum $K_m$ in such a weighting $f$ for a given $n \in S_1 \cup S_2$. Then, the function $f$ restricted to the edges of $K_m$ is a balanced function on $E(K_m)$, which is not possible by Lemmas \ref{lem:bal1} and \ref{lem:bal2} since $m \in \mathbb{N} \setminus S_1 \cap S_2$. Since $S_1 \cup S_2$ has infinitely many elements, it follows that there are infinitely many positive integers $n$ such that there exists a balanced function $f:E(K_n)\to \{-1,1\}$ without zero-sum $K_m$.
\end{proof}
Now we can state the main result of this section, which is equivalent to instance 3 of Theorem \ref{thm:main}.
\begin{theorem}
For any integer $m \in \mathbb{N} \setminus \{1, 4\}$, there are infinitely many positive integers $n$ such that there exists a balanced weighting $f:E(K_n)\to \{-1,1\}$ which is zero-sum-$K_m$ free.
\end{theorem}
\begin{proof}
By Corollary \ref{cor:Km_Si}, for any $m \in \mathbb{N} \setminus (S_1\cap S_2)$, there are infinitely many positive integers $n$, such that there exists a balanced weighting function $f:E(K_n)\to \{-1,1\}$ without a zero-sum $K_m$. We will show that $S_1 \cap S_2=\{1,4\}$. Let $q$ be an integer such that $q^2\in S_1$ (and thus $q^2\in S_1\cap S_2$). Then $q^2$ must be the positive root of equation (\ref{eq:n_k}) for some $x_k$. Thus we need to know for which positive integers $q$ and $x$ the following is possible:
\begin{equation}\label{eq:qx}
q^4-q^2-(2x^2-2x)=0.
\end{equation}
Note that equation (\ref{eq:qx}) can be written as:
\begin{equation}\label{eq:QX}
Q^2-2X^2=-1.
\end{equation}
where $Q=2q^2-1$ and $X=2x-1$. Again (as in the proof of Theorem \ref{thm:larger_range}), we have to deal with the Diophantine equation $Q^2-2X^2=\pm 1$, which has infinitely many solutions given by
\[Q_k=\frac{a^k+b^k}{2}, \hspace{.5cm} X_{k}=\frac{a^k-b^k}{a-b}=\frac{a^k-b^k}{2\sqrt{2}},\]
where $a=1+\sqrt{2}$ and $b=1-\sqrt{2}$. Since we need to solve equation (\ref{eq:QX}) (that is, with $-1$ on the right side), we know that $k$ must be odd. Therefore, according to the definition of $Q$, we need to determine all odd $k$'s such that $$Q_k+1=2q^2,$$ or equivalently, $$2Q_k+2=4q^2,$$ and so,
\begin{equation}\label{eq:ab}
a^k+b^k+a+b=(2q)^2.
\end{equation}
We consider two cases:\\
\noindent
\emph{Case 1.} If $k\equiv 1$ (mod $4$), we will prove that the left side of equation (\ref{eq:ab}) is $4Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}$. Note that $ab=-1$ and, since in this case $\frac{k-1}{2}$ is even, we have $(ab)^{\frac{k-1}{2}}=(-1)^{\frac{k-1}{2}}=1$. Hence,
\begin{align*}
a^k+b^k+a+b&=a^k+b^k+(ab)^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}+a^{\frac{k-1}{2}}b^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}+a^{\frac{k+1}{2}}b^{\frac{k-1}{2}}+a^{\frac{k-1}{2}}b^{\frac{k+1}{2}}\\
&=(a^{\frac{k-1}{2}}+b^{\frac{k-1}{2}})(a^{\frac{k+1}{2}}+b^{\frac{k+1}{2}})\\
&=4Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}.
\end{align*}
Thus, by (\ref{eq:ab}), we conclude that $Q_{\frac{k-1}{2}}Q_{\frac{k+1}{2}}$ is a perfect square. We know that, for all $i$, $Q_i$ and $Q_{i+1}$ are coprimes. Thus, it follows that both $Q_{\frac{k-1}{2}}$ and $Q_{\frac{k+1}{2}}$ are perfect squares. Coming back to equation (\ref{eq:QX}), the following must be satisfied
\begin{equation}\label{eq:YX}
Y^4-2X^2=-1
\end{equation}
where $Q_{\frac{k+1}{2}}=Y^2$. But, the only possible solution for the Diophantine equation (\ref{eq:YX}) is $(Y,X)=(\pm 1,1)$. Hence, $Q_{\frac{k+1}{2}}= 1$, which means that $k=1$, and so $Q = Q_1 = 1$. Since $Q=2q^2-1$ and $q>0$, we conclude that $q=1$.\\
\noindent
\emph{Case 2.} If $k\equiv 3$ (mod $4$), then we will prove that the left side of equation (\ref{eq:ab}) is $8X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}$. Recall that $ab=-1$ and, since in this case $\frac{k+1}{2}$ is even, we have $(ab)^{\frac{k+1}{2}}=(-1)^{\frac{k+1}{2}}=1$. Hence,
\begin{align*}
a^k+b^k+a+b&=a^k+b^k+(ab)^{\frac{k+1}{2}}(a+b)\\
&=a^k+b^k-(ab)^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}-a^{\frac{k-1}{2}}b^{\frac{k-1}{2}}(a+b)\\
&=a^{\frac{k-1}{2}}a^{\frac{k+1}{2}}+b^{\frac{k-1}{2}}b^{\frac{k+1}{2}}-a^{\frac{k+1}{2}}b^{\frac{k-1}{2}}-a^{\frac{k-1}{2}}b^{\frac{k+1}{2}}\\
&=(a^{\frac{k-1}{2}}-b^{\frac{k-1}{2}})(a^{\frac{k+1}{2}}-b^{\frac{k+1}{2}})\\
&=8X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}.
\end{align*}
Thus, by (\ref{eq:ab}), we conclude that $2X_{\frac{k-1}{2}}X_{\frac{k+1}{2}}$ is a perfect square. We know that $X_{\frac{k-1}{2}}$and $X_{\frac{k+1}{2}}$ have different parity. Observe that, for $k\equiv 3$ (mod $4$), $X_{\frac{k-1}{2}}$ is odd and $X_{\frac{k+1}{2}}$ is even. Since for all $i$, $X_i$ and $X_{i+1}$ are coprimes, also $X_{\frac{k-1}{2}}$ and $2X_{\frac{k+1}{2}}$ are coprimes, from which it follows that both $X_{\frac{k-1}{2}}$ and $2X_{\frac{k+1}{2}}$ are perfect squares. Particularly, coming back to equation (\ref{eq:QX}), we obtain
\begin{equation}\label{eq:QW}
Q^2-2W^4=-1
\end{equation}
where $X_{\frac{k-1}{2}}=W^2$. Note that equation (\ref{eq:QW}) is the well known Ljunggren Equation $1+Q^2=2W^4$. Such a Diophantine equation has solutions only for $W=1$ and $W=13$, which correspond respectively to $X_1$ and $X_7$ (because $X_1=1=1^2$ and $X_7=169=13^2$). Therefore, we have two possibilities, either $k=3$ (that is $X_{\frac{3-1}{2}}=X_1$), or $k=15$ (that is $X_{\frac{15-1}{2}}=X_7$). The second case is disclaimed since $X_{\frac{15+1}{2}}=X_8=408=2 \cdot (204)$ and $204$ is not a perfect square. The first case, corresponding to $k=3$, leads to $X_{\frac{3-1}{2}}=X_1=1$ and $X_{\frac{3+1}{2}}=X_2=2$. The solution $(Q_1,X_1)=(1,1)$ gives $q=1$ as we saw in Case 1. The solution $(Q_2,X_2)=(3,2)$ gives $q=2$ (since $Q=2q^2-1$ an $q>0$).
From both cases we conclude that, if $q^2\in S_1\cap S_2$ then either $q=1$ or $q=2$. Hence, $S_1\cap S_2=\{1,4\}$ which concludes the proof.
\end{proof}
\section{Conclusions}
While the situation about zero-sum copies of $K_m$ over $\mbox{${\mathbb Z}$}$-weightings is fairly clear now,
a lot of interesting results can be proved when the graphs in question are not complete graphs.
Several examples are given in \cite{CY3} (for example, certain complete bipartite graphs and many more), and in a forthcoming paper \cite{CHM} which is under preparation.
\end{document} |
\begin{document}
\title{On tempered representations}
\author[David Kazhdan and Alexander Yom Din]{David Kazhdan and Alexander Yom Din}
\begin{abstract}
Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial $V$ being tempered, i.e. to $G$ being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness.
We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered $V$'s, as well as for all tempered $V$'s in the cases of $G := SL_2 ({\mathbb R})$ and of $G = PGL_2 (\Omega)$ for a non-Archimedean local field $\Omega$ of characteristic $0$ and residual characteristic not $2$. We also establish a weaker form of the conjecture, involving only $K$-finite vectors.
In the non-Archimedean case, we give a formula expressing the character of a tempered $V$ as an appropriately-weighted conjugation-average of a matrix coefficient of $V$, generalizing a formula of Harish-Chandra from the case when $V$ is square-integrable.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}\leftarrowbel{sec intro}
\subsection{}
Throughout the paper, we work with a unimodular second countable locally compact group $G$, and fix a Haar measure $dg$ on it. In the introduction, in \S\ref{ssec intro 1} - \S\ref{ssec intro 2.5} $G$ is assumed semisimple over a local field, while in \S\ref{ssec intro 3} - \S\ref{ssec intro 4} there is no such assumption. After the introduction, in \S\ref{sec Kfin} - \S\ref{sec proof of SL2R} $G$ is assumed semisimple over a local field, while in \S\ref{sec bi tempered} - \S\ref{sec ctemp is temp} there is no such assumption. Unitary representations of $G$ are pairs $(V , \pi)$, but for lightness of notation we denote them by $V$, keeping $\pi$ implicit.
\subsection{}\leftarrowbel{ssec intro 1}
Assume that $G$ is a semisimple group over a local field\footnote{So, for example, $G$ can be taken $SL_n ({\mathbb R})$ or $SL_n ({\mathbb Q}_p)$.}. The characterization of temperedness of irreducible unitary $G$-representations in terms of the rate of decrease of $K$-finite matrix coefficients is well-studied (see for example \cite{Wa,CoHaHo,Be}). Briefly, fixing a maximal compact subgroup $K \subset G$, an irreducible unitary $G$-representation $V$ is tempered if and only if for every two $K$-finite vectors $v_1 , v_2 \in V$ there exists $C>0$ such that $$ |\leftarrowngle gv_1 , v_2 \rightarrowngle | \leq C \cdot \Xi_G (g)$$ for all $g \in G$, where $\Xi_G : G \to {\mathbb R}_{\ge 0}$ is Harish-Chandra's $\Xi$-function (see \S\ref{ssec V1 and Xi} for a reminder on the definition of $\Xi_G$). When considering matrix coefficients of more general vectors, differentiating between tempered and non-tempered irreducible unitary $G$-representations becomes more problematic, as the following example shows.
\begin{example}[see Claim \ref{clm counterexample}]\leftarrowbel{rem counterexample}
Let $G := PGL_2 (\Omega)$, $\Omega$ a local field. Denote by $A \subset G$ the subgroup of diagonal matrices. Given a unitary $G$-representation $V$ let us denote $$ {\mathcal M}_V (A) := \left\{ a \mapsto \leftarrowngle a v_1 , v_2 \rightarrowngle \right\}_{v_1 , v_2 \in V} \subset C(A),$$ i.e. the set of matrix coefficients of $V$ restricted to $A$. Let us also denote $$ \widehat{L^1} (A) := \left\{ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d \chi \right\}_{\phi \in L^1 (\hat{A})} \subset C(A),$$ i.e. the set of Fourier transforms of $L^1$-functions on $\hat{A}$. Then for any non-trivial irreducible unitary $G$-representation $V$ we have $$ {\mathcal M}_V (A) = \widehat{L^1} (A).$$
\end{example}
The remedy proposed in this paper is that, instead of analysing the pointwise growth of matrix coefficients, we analyse their ``growth in average", i.e. the behaviour of integrals of norm-squared matrix coefficients over big balls.
\subsection{}
We fix a norm\footnote{In the non-Archimedean case, norms on finite-dimensional vector spaces are discussed, for example, in \cite[Chapter II, \S 1]{We}.} $|| - ||$ on the vector space ${\mathfrak g} := {\rm Lie} (G)$ and consider also the induced operator norm $|| - ||$ on $\textnormal{End} ({\mathfrak g})$. We define the ``radius" function $\mathbf{r} : G \to {\mathbb R}_{\ge 0}$ by $$ \mathbf{r} (g) := \log \left( \max \{ || \textnormal{Ad} (g) ||, || \textnormal{Ad} (g^{-1}) || \} \right)$$ where $\textnormal{Ad} : G \to \textnormal{Aut} ({\mathfrak g})$ is the adjoint representation. We denote then by $G_{<r} \subset G$ the subset of elements $g$ for which $\mathbf{r} ( g) < r$.
\begin{conjecture}[``asymptotic Schur orthogonality relations"]\leftarrowbel{main conj red exact}
Let $V$ be a tempered irreducible unitary $G$-representation. There exist $\mathbf{d}(V) \in {\mathbb Z}_{\ge 0}$ and $\mathbf{f} (V) \in {\mathbb R}_{>0}$ such that for all $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle } \cdot dg}{ r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \cdot \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}.$$
\end{conjecture}
\begin{remark}[see Claim \ref{clm doesnt depend on norm}]\leftarrowbel{rem doesnt depend on norm}
The validity of Conjecture \ref{main conj red exact}, as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$ (and of other similar results/conjectures below - see the formulation of Claim \ref{clm doesnt depend on norm}), do not depend on the choice of the norm $|| - ||$ on ${\mathfrak g}$ (used to construct the subsets $G_{<r}$).
\end{remark}
\begin{remark}[see Remark \ref{rem ctemp red is temp}]
An irreducible unitary $G$-representation $V$ for which the condition of Conjecture \ref{main conj red exact} is verified is tempered.
\end{remark}
\begin{remark}
In the notation of Conjecture \ref{main conj red exact}, $\mathbf{d} (V) = 0$ if and only if $V$ is square-integrable. In that case, $\mathbf{f} (V)$ is the well-known formal degree of $V$.
\end{remark}
\begin{remark}[following from Proposition \ref{prop c-tempered orth rel cross}]
Let $V$ and $W$ be two tempered irreducible unitary $G$-representations for which Conjecture \ref{main conj red exact} holds, and which are non-isomorphic. Then for all $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ one has $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle g w_1 , w_2 \rightarrowngle} \cdot dg}{r^{(\mathbf{d} (V) + \mathbf{d} (W))/2}} = 0.$$
\end{remark}
\subsection{}\leftarrowbel{ssec intro 1.25}
We show the following statement, weaker than Conjecture \ref{main conj red exact}:
\begin{theorem}[see \S\ref{sec Kfin}]\leftarrowbel{thm main Kfin}
Let $V$ be a tempered irreducible unitary $G$-representation and $K \subset G$ a maximal compact subgroup. There exists $\mathbf{d}(V) \in {\mathbb Z}_{\ge 0}$ such that:
\begin{enumerate}
\item if $G$ is non-Archimedean, there exists $\mathbf{f} (V) \in {\mathbb R}_{>0}$ such that for all $K$-finite\footnote{When $G$ is non-Archimedean $K$-finite is the same as smooth (in particular does not depend on $K$).} $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle } \cdot dg}{ r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \cdot \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}.$$
\item If $G$ is Archimedean, for any given non-zero $K$-finite vectors $v_1 , v_2 \in V$ there exists $C(v_1 , v_2) >0$ such that $$ \lim_{r \to +\infty} \frac{\int_{G_{< r}} | \leftarrowngle g v_1 , v_2 \rightarrowngle |^2 \cdot dg}{ r^{\mathbf{d} (V)}} = C (v_1 , v_2).$$
\end{enumerate}
\end{theorem}
\begin{remark}
We expect that it should not be very difficult to establish the statement of item $(1)$ of Theorem \ref{thm main Kfin} also in the Archimedean case, instead of the weaker statement of item $(2)$.
\end{remark}
Concentrating on the non-Archimedean case for simplicity, Theorem \ref{thm main Kfin} has as a corollary the following proposition, a generalization (from the square-integrable case to the tempered case) of a formula of Harish-Chandra (see \cite[Theorem 9]{Ha2}), expressing the character as a conjugation-average of a matrix coefficient.
\begin{definition}
Assume that $G$ is non-Archimedean. We denote by $C^{\infty} (G)$ the space of (complex-valued) smooth functions on $G$ and by $D_c^{\infty} (G)$ the space of smooth distributions on $G$ with compact support. We denote by $C^{-\infty} (G)$ the dual to $D_c^{\infty} (G)$, i.e. the space of generalized functions on $G$ (thus we have an embedding $C^{\infty} (G) \subset C^{-\infty} (G)$). Given an admissible unitary $G$-representation $V$, we denote by $\Theta_V \in C^{-\infty} (G)$ the character of $V$.
\end{definition}
\begin{proposition}[see \S\ref{ssec proof of prop formula ch}]\leftarrowbel{prop formula ch}
Let $V$ be a tempered irreducible unitary $G$-representation. Let $v_1 , v_2 \in V$ be smooth vectors. Denote by $m_{v_1 , v_2} \in C^{\infty}(G) \subset C^{-\infty} (G)$ the matrix coefficient $m_{v_1 , v_2} (g) := \leftarrowngle g v_1 , v_2 \rightarrowngle$. Denoting $({}^g m) (x) := m(g^{-1} x g)$, the limit $$\lim_{r \to +\infty} \frac{\int_{G_{<r}} {}^g m_{v_1 , v_2} \cdot dg}{r^{\mathbf{d} (V)}}$$ exists in $C^{-\infty}(G)$, in the sense of weak convergence of generalized functions (i.e. convergence when paired against every element in $D_c^{\infty} (G)$), and is equal to $$ \frac{\leftarrowngle v_1 , v_2 \rightarrowngle}{\mathbf{f} (V)} \cdot \Theta_V.$$
\end{proposition}
\subsection{}\leftarrowbel{ssec intro 1.5}
We are able to verify Conjecture \ref{main conj red exact} in some cases.
\begin{theorem}[see Theorem \ref{thm V1 c temp}]\leftarrowbel{thm intro slowest}
Conjecture \ref{main conj red exact} is true for the principal series irreducible unitary representation of ``slowest decrease", i.e. the unitary parabolic induction of the trivial character via a minimal parabolic subgroup.
\end{theorem}
Here is the main result of the paper:
\begin{theorem}[see \S\ref{sec proof of SL2R}]\leftarrowbel{thm intro SL2R}
Conjecture \ref{main conj red exact} is true for all tempered irreducible unitary representations of $G := SL_2 ({\mathbb R})$ and of $G := PGL_2 (\Omega)$, where $\Omega$ is a non-Archimedean field of characteristic $0$ and of residual characteristic not equal to $2$.
\end{theorem}
\subsection{}\leftarrowbel{ssec intro 2}
The proposition that follows shows that a seemingly weaker property implies that of Conjecture \ref{main conj red exact}.
\begin{definition}
Given a unitary $G$-representation $V$ and vectors $v_1 , v_2 \in V$ we define $$ \Ms{v_1}{v_2}{r} := \int_{ G_{<r}} |\leftarrowngle g v_1 , v_2 \rightarrowngle |^2 \cdot dg.$$
\end{definition}
\begin{proposition}[see \S\ref{ssec clms red 1}]\leftarrowbel{prop from folner to exact}
Let $V$ be an irreducible unitary $G$-representation. Let $v_0 \in V$ be a unit vector such that the following holds:
\begin{enumerate}
\item For any vectors $v_1 , v_2 \in V$ we have $$ \underset{r \to +\infty}{\limsup} \frac{ \Ms{v_1}{v_2}{r} }{ \Ms{v_0}{v_0}{r} } < +\infty.$$
\item For any vectors $v_1 , v_2 \in V$ and $r^{\prime} > 0$ we have $$ \lim_{r \to +\infty} \frac{ \Ms{v_1}{v_2}{r + r^{\prime}} - \Ms{v_1}{v_2}{r - r^{\prime}} }{ \Ms{v_0}{v_0}{r} } = 0.$$
\end{enumerate}
Then Conjecture \ref{main conj red exact} holds for $V$.
\end{proposition}
\begin{question}
Does item $(1)$ of Proposition \ref{prop from folner to exact} hold for arbitrary irreducible unitary $G$-representations?
\end{question}
\begin{remark}[see Proposition \ref{clm Prop tempered is c-tempered reductive holds}]\leftarrowbel{rem ctemp red is temp}
An irreducible unitary $G$-representation for which there exists a unit vector $v_0 \in V$ such that conditions $(1)$ and $(2)$ of Proposition \ref{prop from folner to exact} are satisfied is tempered.
\end{remark}
\subsection{}\leftarrowbel{ssec intro 2.5}
After finishing writing the current paper, we have found previous works \cite{Mi} and \cite{An}. Work \cite{Mi} intends at giving an asymptotic Schur orthogonality relation for tempered irreducible unitary representations, but we could not understand its validity; on the first page the author defines a seminorm $||-||_p^2$ on $C^{\infty} (G)$ by a limit, but this limit clearly does not always exist. Work \cite{An} (which deals with the more general setup of a symmetric space) provides an asymptotic Schur orthogonality relation for $K$-finite vectors in a tempered irreducible unitary $G$-representation, in the case when $G$ is real and under a regularity assumption on the central character. This work also seems to provide an interpretation of what we have denoted as $\mathbf{f} (V)$ in terms of the Plancherel density (but it would be probably good to work this out in more detail).
\subsection{}\leftarrowbel{ssec intro 3}
Let now $G$ be an arbitrary unimodular second countable locally compact group. We formulate a property of irreducible unitary $G$-representations which we call c-temperedness (see Definition \ref{def c-temp}). The property of c-temperedness is, roughly speaking, an abstract version of properties (1) and (2) of Proposition \ref{prop from folner to exact}. Here $G_{< r} \subset G$ are replaced by a sequence $\{ F_n \}_{n \ge 0}$ of subsets of $G$, which we call a F{\o}lner sequence, whose existence is part of the definition (so that we speak of a representation c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$), while the condition replacing property (2) of Proposition \ref{prop from folner to exact} generalizes, in some sense, the F{\o}lner condition for a group to be amenable (i.e. for the trivial representation to be tempered).
We show in Corollary \ref{cor c-temp are temp} that any c-tempered irreducible unitary $G$-representation is tempered and pose the question:
\begin{question}\leftarrowbel{main question}
For which groups $G$ every tempered irreducible unitary $G$-representation is c-tempered with some F{\o}lner sequence?
\end{question}
As before, c-tempered irreducible unitary $G$-representations enjoy a variant of asymptotic Schur orthogonality relations (see Proposition \ref{prop c-tempered orth rel}): \begin{equation}\leftarrowbel{eq orth rel} \lim_{n \to +\infty} \frac{\int_{F_n} \leftarrowngle g v_1 , v_3 \rightarrowngle \overline{\leftarrowngle gv_2 , v_4 \rightarrowngle} \cdot dg }{\int_{F_n} |\leftarrowngle g v_0 , v_0 \rightarrowngle |^2 \cdot dg} = \leftarrowngle v_1 , v_2 \rightarrowngle \overline{\leftarrowngle v_3 , v_4 \rightarrowngle}\end{equation} for all $v_1 , v_2 , v_3 , v_4 \in V$ and all unit vectors $v_0 \in V$. Also, we have a variant for a pair of non-isomorphic representations (see Proposition \ref{prop c-tempered orth rel cross}).
\begin{definition}
Let us say that two irreducible unitary $G$-representations are twins if their closures in $\hat{G}$ (w.r.t. the Fell topology) coincide.
\end{definition}
\begin{question}
Let $V_1$ and $V_2$ be irreducible unitary $G$-representations and assume that $V_1$ and $V_2$ are twins. Suppose that $V_1$ is c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$.
\begin{enumerate}
\item Is it true that $V_2$ is also c-tempered with F{\o}lner sequence $\{ F_n \}_{n \ge 0}$?
\item If so, is it true that for unit vectors $v_1 \in V_1$ and $v_2 \in V_2$ we have $$ \lim_{n \to +\infty} \frac{\int_{F_n} |\leftarrowngle g v_1 , v_1 \rightarrowngle |^2 \cdot dg}{\int_{F_n} |\leftarrowngle g v_2 , v_2 \rightarrowngle|^2 \cdot dg} = 1?$$
\end{enumerate}
\end{question}
\subsection{}\leftarrowbel{ssec intro 4}
For many groups there exist tempered representations with the slowest rate of decrease of matrix coefficients. For such representations it is often much easier to prove analogs of c-temperedness or of orthogonality relation (\ref{eq orth rel}) than for other representations - as exemplified by Theorem \ref{thm intro slowest} above. See \cite{BoGa} for hyperbolic groups.
\subsection{} Alexander Yom Din's research was supported by the ISRAEL SCIENCE FOUNDATION (grant No 1071/20). David Kazhdan's research was partially supported by ERC grant No 669655. We would like to thank Pavel Etingof for great help with the proof of Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$, which was present in a prior draft of the paper, before we encountered the work \cite{BrCoNiTa}. We would like to thank Vincent Lafforgue for a very useful discussion. We thank Erez Lapid for useful correspondence.
\subsection{}\leftarrowbel{ssec notation} Throughout the paper, $G$ is a unimodular second countable locally compact group. We fix a Haar measure $dg$ on $G$, as well as Haar measures on the other unimodular groups we encounter ($dk$ on the group $K$, etc.). We denote by $\textnormal{vol}_G (-)$ the volume with respect to $dg$.
All unitary $G$-representations are on separable Hilbert spaces.
Given a unitary $G$-representation $V$, vectors $v_1 , v_2 \in V$ and a measurable subset $F \subset G$, we denote $$ \Mss{v_1}{v_2}{F} := \int_{F} |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg.$$ So in the case of a semisimple group over a local field as above, we have set $$ \Ms{v_1}{v_2}{r} := \Mss{v_1}{v_2}{G_{<r}}.$$
We write $L^2 (G) := L^2 (G , dg)$, considered as a unitary $G$-representation via the right regular action.
Given Hilbert spaces $V$ and $W$, we denote by ${\mathcal B} (V; W)$ the space of bounded linear operators from $V$ to $W$, and write ${\mathcal B} (V) := {\mathcal B} (V; V)$.
We write $F_1 \smallsetminus F_2$ for set differences and $F_1 \triangle F_2 := (F_1\smallsetminus F_2) \cup (F_2 \smallsetminus F_1)$ for symmetric set differences.
\section{Notion of c-temperedness}\leftarrowbel{sec bi tempered}
In this section, let $G$ be a unimodular second countable locally compact group. We introduce the notion of a c-tempered (with a given F{\o}lner sequence) irreducible unitary $G$-representation.
\subsection{}
The following definition aims at a generalization of the hypotheses of Proposition \ref{prop from folner to exact}, so as to make them suitable for a general group.
\begin{definition}\leftarrowbel{def c-temp}
Let $V$ be an irreducible unitary $G$-representation. Let $F_0 , F_1 , \ldots \subset G$ be a sequence of measurable pre-compact subsets all containing a neighbourhood of $1$. We say that $V$ is \textbf{c-tempered\footnote{``c" stands for ``matrix coefficients".} with F{\o}lner sequence $F_0 , F_1 , \ldots$} if there exists a unit vector $v_0 \in V$ such that the following two conditions are satisfied:
\begin{enumerate}
\item For all $v_1 , v_2 \in V$ we have\footnote{The notation $\Mss{-}{-}{-}$ is introduced in \S\ref{ssec notation}.} $$ \limsup_{n \to +\infty} \frac{ \Mss{v_1}{v_2}{F_n} }{ \Mss{v_0}{v_0}{F_n} } < +\infty.$$
\item For all $v_1 , v_2 \in V$ and all compact subsets $K \subset G$ we have $$ \lim_{n \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} } = 0.$$
\end{enumerate}
\end{definition}
\begin{example}
The trivial unitary $G$-representation is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ if for any compact $K \subset G$ we have \begin{equation}\leftarrowbel{eq folner} \lim_{n \to +\infty} \sup_{g_1 , g_2 \in K} \frac{\textnormal{vol}_G (F_n \triangle g_2^{-1} F_n g_1)}{\textnormal{vol}_G (F_n)} = 0.\end{equation} By \textbf{F{\o}lner's condition}, the existence of such a sequence is equivalent\footnote{When stating F{\o}lner's condition for the amenability of $G$ it is more usual to consider $g_2^{-1} F_n$ rather than $g_2^{-1} F_n g_1$ in (\ref{eq folner}), i.e. to shift only on one side. However, using, for example, \cite[Theorem 4.1]{Gr} applied to the action of $G \times G$ on $G$, we see that the above stronger ``two-sided" condition also characterizes amenability.} to the trivial irreducible unitary $G$-representation being tempered, i.e. to $G$ being \textbf{amenable}.
\end{example}
\subsection{} Irreducible unitary $G$-representations which are c-tempered satisfy ``asymptotic Schur orthogonality relations":
\begin{proposition}\leftarrowbel{prop c-tempered orth rel}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for all $v_1 , v_2 , v_3 , v_4 \in V$ we have \begin{equation}\leftarrowbel{eq approx schur bi} \lim_{n \to +\infty} \frac{\int_{F_n} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle g v_3 , v_4 \rightarrowngle} \cdot dg}{ \Mss{v_0}{v_0}{F_n} } = \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}.\end{equation}
\end{proposition}
\begin{proof}
First, notice that in order to show that the limit in (\ref{eq approx schur bi}) holds, it is enough to show that for every sub-sequence there exists a further sub-sequence of it on which the limit holds. Replacing our sequence by the sub-sequence, it is therefore enough to show simply that there exists a sub-sequence on which the limit holds - which is what we will do.
Define bilinear maps\footnote{Recall that $L^2 (G)$ denotes $L^2 (G , dg)$, viewed as a unitary $G$-representation via the right regular action.} $$ S_0 , S_1 , \ldots : V \times \overline{V} \to L^2 (G)$$ by $$ S_n (v_1 , v_2) (g) := \begin{cases} \frac{1}{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \cdot \leftarrowngle g v_1 , v_2 \rightarrowngle, & g \in F_n \\ 0, & g \notin F_n \end{cases}.$$ Clearly those are bounded.
\begin{itemize}
\item The bilinear maps $S_n$ are jointly bounded, i.e. there exists $C>0$ such that $||S_n||^2 \leq C$ for all $n$.
\end{itemize}
Indeed, by condition (1) of Definition \ref{def c-temp}, for any fixed $v_1 , v_2 \in V$ there exists $C>0$ such that $||S_n (v_1 , v_2)||^2 \leq C$ for all $n$. By the Banach-Steinhaus theorem, there exists $C>0$ such that $||S_n ||^2 \leq C$ for all $n$.
Next, define quadlinear forms $$ \Phi_1 , \Phi_2 , \ldots : V \times \overline{V} \times \overline{V} \times V \to {\mathbb C}$$ by $$ \Phi_n (v_1 , v_2 , v_3 , v_4) := \leftarrowngle S_n (v_1 , v_2) , S_n (v_3 , v_4) \rightarrowngle$$
\begin{itemize}
\item The quadlinear forms $\Phi_n$ are jointly bounded, in fact $|| \Phi_n || \leq C$ for all $n$.
\end{itemize}
This follows immediately from the above finding $|| S_n ||^2 \leq C$ for all $n$.
\begin{itemize}
\item For all $g_1 , g_2 \in G$ and $v_1 , v_2 , v_3 , v_4 \in V$ we have \begin{equation}\leftarrowbel{eq approx GG-inv} \lim_{n \to +\infty } \left( \Phi_n (g_1 v_1 , g_2 v_2 , g_1 v_3, g_2 v_4 ) - \Phi_n (v_1 , v_2 , v_3 , v_4) \right)= 0.\end{equation}
\end{itemize}
Indeed, $$ | \Phi_n (g_1 v_1 , g_2 v_2 , g_1 v_3 , v_2 v_4) - \Phi_n (v_1 , v_2 , v_3 , v_4 ) | = $$ $$ = \frac{ | \int_{F_n} \leftarrowngle g g_1 v_1 , g_2 v_2 \rightarrowngle \overline{\leftarrowngle g g_1 v_3 , g_2 v_4 \rightarrowngle} \cdot dg - \int_{F_n} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle g v_3 , v_4 \rightarrowngle} \cdot dg |}{ \Mss{v_0}{v_0}{F_n} } \leq $$ $$ \leq \frac{ \int_{F_n \triangle g_2^{-1} F_n g_1 } |\leftarrowngle g v_1 , v_2 \rightarrowngle | \cdot | \leftarrowngle g v_3 , v_4 \rightarrowngle | \cdot dg}{ \Mss{v_0}{v_0}{F_n} } \leq $$ $$ \leq \sqrt{\frac{ \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} }} \cdot \sqrt{\frac{ \Mss{v_3}{v_4}{F_n \triangle g_2^{-1} F_n g_1} }{ \Mss{v_0}{v_0}{F_n} }}$$ and the last expression tends to $0$ as $n \to +\infty$ by condition (2) of Definition \ref{def c-temp}.
\begin{itemize}
\item There exists a sub-sequence $0 \leq m_0 < m_1 < \ldots$ such that $$ \lim_{n \to +\infty} \Phi_{m_n} (v_1 , v_2 , v_3 , v_4) = \leftarrowngle v_1 , v_3 \rightarrowngle \overline{ \leftarrowngle v_2 , v_4 \rightarrowngle } $$ for all $v_1 , v_2 , v_3 , v_4 \in V$.
\end{itemize}
By the sequential Banach-Alaouglu theorem (which is applicable since $V$ is separable), we can find a sub-sequence $0 \leq m_0 < m_1 < \ldots$ and a bounded quadlinear form $$ \Phi : V \times \overline{V} \times \overline{V} \times V \to {\mathbb C}$$ such that $\lim_{n \to +\infty}^{\textnormal{weak-}*} \Phi_{m_n} = \Phi$. Passing to the limit in equation (\ref{eq approx GG-inv}) we obtain that for all $g_1 , g_2 \in G$ and all $v_1 , v_2 , v_3 , v_4 \in V$ we have $$ \Phi (g_1 v_1 , g_2 v_2 , g_1 v_3 , g_2 v_4) = \Phi (v_1 , v_2 , v_3 , v_4).$$ Fixing $v_2 , v_4$, we obtain a bounded bilinear form $\Phi (- , v_2 , - , v_4) : V \times \overline{V} \to {\mathbb C}$ which is $G$-invariant, and hence by Schur's lemma is a multiple of the form $\leftarrowngle - , - \rightarrowngle$, i.e. we have a uniquely defined $c_{v_2 , v_4} \in {\mathbb C}$ such that $$ \Phi (v_1 , v_2 , v_3 , v_4) = c_{v_2 , v_4} \cdot \leftarrowngle v_1 , v_3 \rightarrowngle$$ for all $v_1 , v_3 \in V$. Similarly, fixing $v_1 , v_3$ we see that we have a uniquely defined $d_{v_1 , v_3} \in {\mathbb C}$ such that $$ \Phi (v_1 , v_2 , v_3 , v_4) = d_{v_1 , v_3} \cdot \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}$$ for all $v_2, v_4 \in {\mathbb C}$. Since $\Phi (v_0 , v_0, v_0, v_0) = 1$, plugging in $(v_1 , v_2 , v_3 ,v_4) := (v_0 , v_0 , v_0, v_0)$ in the first equality we find $c_{v_0 , v_0} = 1$. Then plugging in $(v_1 , v_2 , v_3 , v_4) := (v_1 , v_0 , v_3 , v_0)$ in both equalities and comparing, we find $d_{v_1 , v_3} = \leftarrowngle v_1 , v_3 \rightarrowngle$. Hence we obtain $$ \Phi (v_1 , v_2 , v_3 , v_4) = \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle} $$ for all $v_1 , v_2 , v_3 , v_4 \in V$.
Now, writing explicitly $\Phi_{m_n} (v_1 , v_2 , v_3 , v_4)$, we see that the limit in (\ref{eq approx schur bi}) is valid on our sub-sequence, so we are done, as we explained in the beginning of the proof.
\end{proof}
\subsection{} If one unit vector $v_0$ satisfies conditions (1) and (2) of Definition \ref{def c-temp} then all unit vectors do:
\begin{proposition}\leftarrowbel{prop c-tempered all are temp}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for any unit vector $v'_0 \in V$ the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied.
\end{proposition}
\begin{proof}
Let $v'_0 \in V$ be a unit vector. From (\ref{eq approx schur bi}) we get $$ \lim_{n \to +\infty} \frac{ \Mss{v'_0}{v'_0}{F_n} }{ \Mss{v_0}{v_0}{F_n} } = 1.$$ This makes the claim clear.
\end{proof}
\subsection{}
We also have the following version of ``asymptotic Schur orthogonality relations" for a pair of non-isomorphic irreducible representations:
\begin{proposition}\leftarrowbel{prop c-tempered orth rel cross}
Let $V$ and $W$ be irreducible unitary $G$-representations. Assume that $V$ and $W$ are c-tempered with the same F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ and $w_0 \in W$ be unit vectors for which the conditions (1) and (2) of Definition \ref{def c-temp} are satisfied. Then for all $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ we have \begin{equation}\leftarrowbel{eq approx schur bi cross} \lim_{n \to +\infty} \frac{\int_{F_n} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{\leftarrowngle g w_1 , w_2 \rightarrowngle } \cdot dg}{ \sqrt{\Mss{v_0}{v_0}{F_n}} \sqrt{\Mss{w_0}{w_0}{F_n}} } = 0.\end{equation}
\end{proposition}
\begin{proof}
We proceed similarly to the proof of Proposition \ref{prop c-tempered orth rel}. Namely, again it is enough to find a sub-sequence on which the limit holds. We define quadlinear forms $$ \Phi_1 , \Phi_2 , \ldots : V \times \overline{V} \times \overline{W} \times W \to {\mathbb C}$$ by $$ \Phi_n (v_1 , v_2 , w_1 , w_2) := \frac{\int_{F_n} \leftarrowngle g v_1 , v_2 \rightarrowngle \overline{ \leftarrowngle g w_1 , w_2 \rightarrowngle} \cdot dg}{\sqrt{\Mss{v_0}{v_0}{F_n}} \sqrt{\Mss{w_0}{w_0}{F_n}} }.$$ We see that these are jointly bounded, and that for all $g_1 , g_2 \in G$ and $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$ we have $$ \lim_{n \to +\infty} \left( \Phi_n (g_1 v_1 , g_2 v_2 , g_1 w_1 , g_2 w_2) - \Phi_n (v_1 , v_2 , w_1 , w_2) \right) = 0.$$ We then find a bounded quadlinear form $$ \Phi : V \times \overline{V} \times \overline{W} \times W \to {\mathbb C}$$ and a sub-sequence $0 \leq m_0 < m_1 < \ldots$ such that $\lim_{n \to +\infty}^{\textnormal{weak-}*} \Phi_{m_n} = \Phi$. We get, for all $g_1 , g_2 \in G$ and $v_1 , v_2 \in V$ and $w_1 , w_2 \in W$: $$ \Phi (g_1 v_1 , g_2 v_2 , g_1 w_1 , g_2 w_2) = \Phi (v_1 , v_2 , w_1 , w_2).$$ By Schur's lemma we obtain $\Phi = 0$, giving us the desired.
\end{proof}
\subsection{}
It is easy to answer Question \ref{main question} in the case of square-integrable representations:
\begin{proposition}\leftarrowbel{prop sq int}
Let $V$ be a square-integrable irreducible unitary $G$-representation. Then $V$ is c-tempered with F{\o}lner sequence any increasing sequence $F_0 , F_1 , \ldots $ of open pre-compact subsets in $G$, such that $1 \in F_0$ and $\cup_{n \ge 0} F_n = G$.
\end{proposition}
\begin{proof}
Recall, that matrix coefficients of a square-integrable irreducible representation are square integrable. Let $v_0 \in V$ be a unit vector. Let $F_0 , F_1 , \ldots $ be any increasing sequence of open pre-compact subsets in $G$ whose union is $G$ and with $1 \in F_0$. Let $v_1,v_2 \in V$. Condition (1) of Definition \ref{def c-temp} holds because we have $$ \Mss{v_1}{v_2}{F_n} \leq \Mss{v_1}{v_2}{G} \leq \left( \frac{ \Mss{v_1}{v_2}{G} }{ \Mss{v_0}{v_0}{F_1} } \right) \cdot \Mss{v_0}{v_0}{F_n} .$$ As for condition (2) of Definition \ref{def c-temp}, let $\epsilon > 0$ and let $K \subset G$ be compact. There exists $n_0 \ge 0$ such that $$ \Mss{v_1}{v_2}{G \smallsetminus F_{n_0}} \leq \epsilon \cdot \Mss{v_0}{v_0}{F_1} .$$ There exists $n_1 \ge n_0$ such that $K F_{n_0} K^{-1} \subset F_{n_1}$. Let $n \ge n_1$ and let $g_1 , g_2 \in K$. Notice that $ ( F_n \triangle g_2^{-1} F_n g_1 ) \cap F_{n_0} = \emptyset$. Thus we have $$ \Mss{v_1}{v_2}{F_n \triangle g_2^{-1} F_n g_1} \leq \Mss{v_1}{v_2}{G \smallsetminus F_{n_0}} \leq \epsilon \cdot \Mss{v_0}{v_0}{F_1} \leq $$ $$ \leq \epsilon \cdot \Mss{v_0}{v_0}{F_n}.$$
\end{proof}
\section{c-Tempered irreps are tempered}\leftarrowbel{sec ctemp is temp}
In this section, let $G$ be a unimodular second countable locally compact group. We introduce some intermediate concepts, with the goal of showing that c-tempered irreducible unitary $G$-representations are tempered (Corollary \ref{cor c-temp are temp}).
\subsection{}
Let us recall some standard definitions and statements regarding weak containment.
\begin{definition}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item $V$ is \textbf{weakly contained} in $W$ if for every $v \in V$, compact $K \subset G$ and $\epsilon > 0$ there exist $w_1 , \ldots , w_r \in W$ such that $$ | \leftarrowngle gv , v \rightarrowngle - \sum_{1 \leq i \leq r} \leftarrowngle g w_i , w_i \rightarrowngle | \leq \epsilon$$ for all $g \in K$.
\item $V$ is \textbf{Zimmer-weakly contained}\footnote{or ``weakly contained in the sense of Zimmer", following \cite[Remark F.1.2.(ix)]{BeHaVa}.} in $W$ if for every $v_1 , \ldots , v_r \in V$, compact $K \subset G$ and $\epsilon > 0$ there exist $w_1 , \ldots , w_r \in W$ such that $$ |\leftarrowngle gv_i , v_j \rightarrowngle - \leftarrowngle g w_i , w_j \rightarrowngle | \leq \epsilon $$ for all $1 \leq i,j \leq r$ and $g \in K$.
\end{enumerate}
\end{definition}
To facilitate the formulation of the next lemma, let us also give the following intermediate definition:
\begin{definition}
Let $V$ and $W$ be unitary $G$-representations. Let us say that $V$ is \textbf{strongly-weakly contained} in $W$ if for every $v \in V$, compact $K \subset G$ and $\epsilon > 0$ there exists $w \in W$ such that $$ | \leftarrowngle g v , v \rightarrowngle - \leftarrowngle gw , w \rightarrowngle | \leq \epsilon$$ for all $g \in K$.
\end{definition}
\begin{lemma}\leftarrowbel{lem prop of weak cont}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item If $V$ is Zimmer-weakly contained in $W$ then $V$ is strongly-weakly contained in $W$, and if $V$ is strongly-weakly contained in $W$ then $V$ is weakly contained in $W$.
\item If $V$ is weakly contained in $W$ then $V$ is strongly-weakly contained in\footnote{Here, $W^{\textnormal{op}lus \infty}$ stands for the Hilbert direct sum of countably many copies of $W$.} $W^{\textnormal{op}lus \infty}$.
\item If $V$ is weakly contained in $W^{\textnormal{op}lus \infty}$ then $V$ is weakly contained in $W$.
\item If $V$ is irreducible and $V$ is weakly contained in $W$ then $V$ is strongly-weakly contained in $W$.
\item If $V$ is cyclic (in particular, if $V$ is irreducible) and $V$ is strongly-weakly contained in $W$ then $V$ is Zimmer-weakly contained in $W$.
\item If $V$ is strongly-weakly contained in $W$ then $V$ is Zimmer-weakly contained in $W^{\textnormal{op}lus \infty}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Statements $(1)$, $(2)$ and $(3)$ are straight-forward. For statement $(4)$ see, for example, \cite[Proposition F.1.4]{BeHaVa}. For statement $(5)$ see \cite[proof of $(iii)\implies(iv)$ of Proposition 2.2]{Ke}. For statement $(6)$, again see \cite[proof of $(iii)\implies(iv)$ of Proposition 2.2]{Ke} (one writes $V$ as a Hilbert direct sum of countably many cyclic unitary $G$-representations, and uses item $(5)$).
\end{proof}
\begin{corollary}\leftarrowbel{cor temp Zimmer temp}
Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item $V$ is weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W^{\textnormal{op}lus \infty}$.
\item If $V$ is irreducible, $V$ is weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W$.
\end{enumerate}
\end{corollary}
The following definition of temperedness is classical:
\begin{definition}
A unitary $G$-representation $V$ is said to be \textbf{tempered} if $V$ is weakly contained in\footnote{Recall that $L^2 (G)$ denotes $L^2 (G , dg)$, viewed as a unitary $G$-representation via the right regular action.} $L^2 (G)$.
\end{definition}
\begin{remark}\leftarrowbel{rem irr zimmer}
Notice that an irreducible unitary $G$-representation is tempered if and only if it is Zimmer-weakly contained in $L^2 (G)$, by part $(2)$ of Corollary \ref{cor temp Zimmer temp}.
\end{remark}
\subsection{}
The next definitions are related to the idea that one representation is weakly contained in another if there ``almost" exists a $G$-intertwining isometric embedding from the one to the other.
\begin{definition}\leftarrowbel{def asymp emb}
Let $V$ and $W$ be unitary $G$-representations. A sequence $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$ is an \textbf{asymptotic embedding} if the following conditions are satisfied:
\begin{enumerate}
\item The operators $\{ S_n \}_{n \ge 0}$ are jointly bounded, i.e. there exists $C>0$ such that $|| S_n ||^2 \leq C$ for all $n \ge 0$.
\item Given $v_1 , v_2 \in V$ and a compact $K \subset G$ we have $$ \lim_{n \to +\infty} \ \sup_{g \in K} | \leftarrowngle (S_n g - g S_n) v_1 , S_n v_2 \rightarrowngle | = 0.$$
\item Given $v_1 , v_2 \in V$, we have $$ \lim_{n \to +\infty} \ \leftarrowngle S_n v_1 , S_n v_2 \rightarrowngle = \leftarrowngle v_1 , v_2 \rightarrowngle.$$
\end{enumerate}
\end{definition}
\begin{definition} Let $V$ and $W$ be unitary $G$-representations.
\begin{enumerate}
\item We say that $V$ is \textbf{o-weakly contained}\footnote{``o" stands for ``operator".} in $W$ if there exists an asymptotic embedding $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$.
\item We say that $V$ is \textbf{o-tempered} if it is o-weakly contained in $L^2 (G)$.
\end{enumerate}
\end{definition}
\begin{lemma}\leftarrowbel{lem asymp emb uniform}
In the context of Definition \ref{def asymp emb}, if conditions $(1)$ and $(2)$ of Definition \ref{def asymp emb} are satisfied then given compacts $L_1 , L_2 \subset V$ and a compact $K \subset G$ we have $$ \lim_{n \to +\infty} \sup_{v_1 \in L_1 , v_2 \in L_2 , g \in K} | \leftarrowngle (S_n g - g S_n) v_1 , S_n v_2 \rightarrowngle | = 0,$$ and if conditions $(1)$ and $(3)$ of Definition \ref{def asymp emb} are satisfied then given compacts $L_1, L_2 \subset V$ we have $$ \lim_{n \to +\infty} \sup_{v_1 \in L_1 , v_2 \in L_2} |\leftarrowngle S_n v_1 , S_n v_2 \rightarrowngle - \leftarrowngle v_1 , v_2 \rightarrowngle| = 0.$$
\end{lemma}
\begin{proof}
This follows from the well-known fact from functional analysis that pointwise convergence coincides with compact convergence on equi-continuous subsets, see \cite[Proposition 32.5]{Tr}.
\end{proof}
\begin{lemma}\leftarrowbel{lem asymp emb irrep}
In the context of Definition \ref{def asymp emb}, assume that $V$ is irreducible. If conditions $(1)$ and $(2)$ of Definition \ref{def asymp emb} are satisfied then there exists a sub-sequence $0 \leq m_0 < m_1 < \ldots$ and $c \in {\mathbb R}_{\ge 0}$ such that for all $v_1 , v_2 \in V$ we have \begin{equation}\leftarrowbel{eq limit c} \lim_{n \to +\infty} \ \leftarrowngle S_{m_n} v_1 , S_{m_n} v_2 \rightarrowngle = c \cdot \leftarrowngle v_1 , v_2 \rightarrowngle.\end{equation} In particular, if there exists $v \in V$ such that $\liminf_{n \to +\infty} || S_n v ||^2 > 0$ then there exists $d \in {\mathbb R}_{> 0}$ (in fact, $d^{-2} = \lim_{n \to +\infty} || S_{m_n} v ||^2 / || v ||^2$) such that $\{ d S_{m_n} \}_{n \ge 0}$ satisfies condition $(3)$ of Definition \ref{def asymp emb}, i.e. is an asymptotic embedding.
\end{lemma}
\begin{proof}
By the sequential Banach–Alaoglu theorem (applicable as $V$ is separable, and $\{ S_n^* S_n \}_{n \ge 0}$ are jointly bounded by condition $(1)$), there exists a sub-sequence $1 \leq m_0 < m_1 < \ldots$ such that $\{ S_{m_n}^* S_{m_n} \}_{n \ge 0}$ converges in the weak operator topology to some $S \in {\mathcal B} (V)$.
Let us first check that $S$ is $G$-invariant. For $g \in G$ and $v_1 , v_2 \in V$ we have $$ |\leftarrowngle S_n^* S_n g v_1 , v_2 \rightarrowngle - \leftarrowngle S_n^* S_n v_1 , g^{-1} v_2 \rightarrowngle | = | \leftarrowngle S_n g v_1 , S_n v_2 \rightarrowngle - \leftarrowngle S_n v_1 , S_n g^{-1} v_2 \rightarrowngle | \leq $$ $$ \leq |\leftarrowngle (S_n g - g S_n) v_1 , S_n v _2 \rightarrowngle | + |\leftarrowngle S_n v_1 , (g^{-1} S_n - S_n g^{-1}) v_2 \rightarrowngle |$$ and both summands in the last expression converge to $0$ as $n \to +\infty$ by condition $(2)$. Therefore $$ | \leftarrowngle Sg v_1 , v_2 \rightarrowngle - \leftarrowngle g S v_1 , v_2 \rightarrowngle | = \lim_{n \to +\infty} |\leftarrowngle S_{m_n}^* S_{m_n} g v_1 , v_2 \rightarrowngle - \leftarrowngle S_{m_n}^* S_{m_n} v_1 , g^{-1} v_2 \rightarrowngle | = 0 $$ i.e. $\leftarrowngle S g v_1 , v_2 \rightarrowngle = \leftarrowngle g S v_1 , v_2 \rightarrowngle$. Thus, since $v_1$ and $v_2$ were arbitrary, $S g = g S$. This holds for all $g \in G$, i.e. $S$ is $G$-invariant.
By Schur's lemma, we deduce $S = c \cdot \textnormal{Id}_V$ for some $c \in {\mathbb C}$. This translates precisely to (\ref{eq limit c}). The last claim is then straight-forward.
\end{proof}
\begin{remark}\leftarrowbel{rem alt cond}
Using Lemma \ref{lem asymp emb uniform}, it is straight-forward that, assuming condition $(1)$ of Definition \ref{def asymp emb}, conditions $(2)$ and $(3)$ in Definition \ref{def asymp emb} are equivalent to the one condition that for $v_1 , v_2 \in V$ and a compact $K \subset G$ one has $$ \lim_{n \to +\infty} \sup_{g \in K} | \leftarrowngle g S_n v_1 , S_n v_2 \rightarrowngle - \leftarrowngle g v_1 , v_2 \rightarrowngle | = 0.$$ Indeed, let us write \begin{equation}\leftarrowbel{eq alt cond} \leftarrowngle g S_n v_1 , S_n v_2 \rightarrowngle - \leftarrowngle gv_1 , v_2 \rightarrowngle = \leftarrowngle (g S_n - S_n g) v_1 , S_n v_2 \rightarrowngle + ( \leftarrowngle S_n g v_1 , S_n v_2 \rightarrowngle - \leftarrowngle gv_1 , v_2 \rightarrowngle).\end{equation} The current condition gives condition $(3)$ by plugging in $g = 1$, and then (\ref{eq alt cond}) gives condition $(2)$, using the uniformity provided by Lemma \ref{lem asymp emb uniform}. Conversely, (\ref{eq alt cond}) shows immediately (again taking into consideration Lemma \ref{lem asymp emb uniform}) that conditions $(2)$ and $(3)$ imply the current condition.
\end{remark}
\subsection{}
The concept of o-weak containment in fact coincides with that of Zimmer-weak containment:
\begin{proposition}\leftarrowbel{prop o cont is weakly cont}
Let $V$ and $W$ be unitary $G$-representations. Then $V$ is o-weakly contained in $W$ if and only if $V$ is Zimmer-weakly contained in $W$.
\end{proposition}
\begin{proof}
Let $\{ S_n \}_{n \ge 0} \subset {\mathcal B} (V ; W)$ be an asymptotic embedding. Given $v_1 , \ldots , v_r \in V$, by Remark \ref{rem alt cond}, given any compact $K \subset G$ we have $$ \lim_{n \to +\infty} \sup_{g \in K} |\leftarrowngle g S_n v_i , S_n v_j \rightarrowngle - \leftarrowngle g v_i , v_j \rightarrowngle | = 0$$ for all $1 \leq i,j \leq r$, and thus $$ \lim_{n \to +\infty} \sup_{g \in K} \sup_{1 \leq i,j \leq r} |\leftarrowngle g S_n v_i , S_n v_j \rightarrowngle - \leftarrowngle g v_i , v_j \rightarrowngle | = 0.$$ Thus by definition $V$ is Zimmer-weakly contained in $W$.
Conversely, suppose that $V$ is Zimmer-weakly contained in $W$. Let $\{ e_n \}_{n \ge 0}$ be an orthonormal basis for $V$. Let $\{ K_n \}_{n \ge 0}$ be an increasing sequence of compact subsets in $G$, with $1 \in K_0$ and with the property that for any compact subset $K \subset G$ there exists $n \ge 0$ such that $K \subset K_n$. As $V$ is Zimmer-weakly contained in $W$, given $n \ge 0$, let us find $w^n_0 , \ldots , w^n_n \in W$ such that $$ \sup_{g \in K_n} |\leftarrowngle g e_i , e_j \rightarrowngle - \leftarrowngle g w^n_i , w^n_j \rightarrowngle | \leq \frac{1}{n+1}$$ for all $0 \leq i,j \leq n$. Define $S_n : V \to W$ by $$ S_n \left( \sum_{i \ge 0} c_i \cdot e_i \right) := \sum_{0 \leq i \leq n} c_i \cdot w^n_i.$$ We want to check that $\{ S_n \}_{n \ge 0}$ is an asymptotic embedding. As for condition $(1)$, notice that $$ \left\Vert S_n \left( \sum_{i \ge 0} c_i e_i \right) \right\Vert^2 = \left\Vert \sum_{0 \leq i \leq n} c_i w^n_i \right\Vert^2 = \left| \sum_{0 \leq i,j \leq n} c_i \overline{c_j} \cdot \leftarrowngle w^n_i , w^n_j \rightarrowngle \right| \leq $$ $$ \leq \left| \sum_{0 \leq i,j \leq n} c_i \overline{c_j} \cdot \leftarrowngle e_i , e_j \rightarrowngle \right| + \left| \sum_{0 \leq i,j \leq n } c_i \overline{c_j} \cdot \left( \leftarrowngle w_i^n , w_j^n \rightarrowngle - \leftarrowngle e_i , e_j \rightarrowngle \right)\right| \leq $$ $$ \leq \sum_{0 \leq i \leq n} |c_i|^2 + \frac{1}{n+1} \left( \sum_{0 \leq i \leq n} |c_i| \right)^2 \leq2 \sum_{0 \leq i \leq n} |c_i|^2 \leq 2 \cdot \left\Vert \sum_{i \ge 0} c_i e_i \right\Vert^2,$$ showing that $|| S_n ||^2 \leq 2$ for all $n \ge 0$. It is left to show the condition as in Remark \ref{rem alt cond}. Let us thus fix a compact $K \subset G$. Notice that it is straight-forward to see that it is enough to check the condition for vectors in a subset of $V$, the closure of whose linear span is equal to $V$. So it is enough to check that $$ \lim_{n \to +\infty} \sup_{g \in K} |\leftarrowngle g S_n e_i , S_n e_j \rightarrowngle - \leftarrowngle g e_i , e_j \rightarrowngle | = 0$$ for any given $i,j \ge 0$. Taking $n$ big enough so that $K \subset K_n$ and $n \ge \max \{ i,j \}$, we have $$ \sup_{g \in K} |\leftarrowngle g S_n e_i , S_n e_j \rightarrowngle - \leftarrowngle g e_i , e_j \rightarrowngle | = \sup_{g \in K} |\leftarrowngle g w^n_i , w^n_j \rightarrowngle - \leftarrowngle g e_i , e_j \rightarrowngle | \leq \frac{1}{n+1},$$ giving the desired.
\end{proof}
\begin{corollary}\leftarrowbel{cor o temp is temp}
An irreducible unitary $G$-representation is o-tempered if and only if it is tempered.
\end{corollary}
\begin{proof}
This is a special case of Proposition \ref{prop o cont is weakly cont}, taking into account Remark \ref{rem irr zimmer}.
\end{proof}
\subsection{}
Here we give a weaker version of c-temperedness, which is technically convenient to relate to other concepts of this section.
\begin{definition}\leftarrowbel{def right-c-temp}
Let $V$ be an irreducible unitary $G$-representation. Let $F_0 , F_1 , \ldots \subset G$ be a sequence of measurable pre-compact subsets all containing a neighbourhood of $1$. We say that $V$ is \textbf{right-c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$} if there exists a unit vector $v_0 \in V$ such that the following two conditions are satisfied:
\begin{enumerate}
\item For all $v \in V$ we have $$ \limsup_{n \to +\infty} \frac{ \Mss{v}{v_0}{F_n} }{ \Mss{v_0}{v_0}{F_n} } < +\infty.$$
\item For all $v \in V$ and all compact subsets $K \subset G$ we have $$ \lim_{n \to +\infty} \frac{\sup_{g \in K} \Mss{v}{v_0}{F_n \triangle F_n g} }{ \Mss{v_0}{v_0}{F_n} } = 0.$$
\end{enumerate}
\end{definition}
\subsection{}
Finally, we can show that c-tempered irreducible unitary $G$-representations are tempered.
\begin{proposition}\leftarrowbel{prop right-c is op}
Let $V$ be an irreducible unitary $G$-representation. Assume that $V$ is right-c-tempered (with some F{\o}lner sequence). Then $V$ is o-tempered. More precisely, suppose that $V$ is right-c-tempered with F{\o}lner sequence $F_0 , F_1 , \ldots$ and let $v_0 \in V$ be a unit vector for which the conditions (1) and (2) of Definition \ref{def right-c-temp} are satisfied. Then the sequence of operators $$ S_0 , S_1 , \ldots : V \to L^2 (G)$$ given by $$ S_n (v) (x) := \begin{cases} \frac{1}{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \cdot \leftarrowngle x v , v_0 \rightarrowngle , & x \in F_n \\ 0, & x \notin F_n \end{cases}$$ admits a sub-sequence which is an asymptotic embedding.
\end{proposition}
\begin{corollary}\leftarrowbel{cor c-temp are temp}
Every c-tempered irreducible unitary $G$-representation (with some F{\o}lner sequence) is tempered.
\end{corollary}
\begin{proof}
It is clear that c-temperedness implies right-c-temperedness, Proposition \ref{prop right-c is op} says that right-c-temperedness implies o-temperedness, and Corollary \ref{cor o temp is temp} says that o-temperedness is equivalent to temperedness.
\end{proof}
\begin{proof}[Proof (of Proposition \ref{prop right-c is op}).]
Clearly each $S_n$ is bounded. By condition (2) of Definition \ref{def right-c-temp}, for any fixed $v \in V$ there exists $C>0$ such that $||S_n (v)||^2 \leq C$ for all $n$. By the Banach-Steinhaus theorem, this implies that the operators $S_0 , S_1 , \ldots$ are jointly bounded, thus condition $(1)$ of Definition \ref{def asymp emb} is verified.
To verify condition $(2)$ of Definition \ref{def asymp emb}, fix $v \in V$ and a compact $K \subset G$. Given $g \in K$ and a function $f \in L^2 (G)$ of $L^2$-norm one, we have $$ | \leftarrowngle S_n (g v) - g S_n (v) , f \rightarrowngle | = \frac{ \left| \int_{F_n} \leftarrowngle x g v , v_0 \rightarrowngle \overline{f(x)} \cdot dx - \int_{F_n g^{-1}} \leftarrowngle xg v , v_0 \rightarrowngle \overline{f(x)} \cdot dx \right| }{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \leq $$ $$ \leq \frac{ \int_{F_n \triangle F_n g^{-1}} \left| \leftarrowngle xg v , v_0 \rightarrowngle \overline{f(x)} \right| \cdot dx }{\sqrt{ \Mss{v_0}{v_0}{F_n} }} \leq \sqrt{\frac{\int_{F_n \triangle F_n g^{-1}} |\leftarrowngle xg v , v_0 \rightarrowngle |^2 \cdot dx}{ \Mss{v_0}{v_0}{F_n} }} \cdot \sqrt{\int_{G} |f(x)|^2 \cdot dx} = $$ $$ = \sqrt{\frac{ \Mss{v}{v_0}{F_n g \triangle F_n} }{ \Mss{v_0}{v_0}{F_n} }}.$$ Since $f$ was arbitrary, this implies $$ || S_n (gv) - g S_n (v) || \leq \sqrt{\frac{ \Mss{v}{v_0}{F_n g \triangle F_n} }{ \Mss{v_0}{v_0}{F_n} }}$$ for $g \in K$. By condition (2) of Definition \ref{def right-c-temp}, this tends to $0$ as $n \to +\infty$, uniformly in $g \in K$, and hence the desired.
Now, using Lemma \ref{lem asymp emb irrep} we see that some sub-sequence will satisfy condition $(3)$ of Definition \ref{def asymp emb}, once we notice that $|| S_n v_0 ||^2 = 1$ for all $n$ by construction.
\end{proof}
\section{The case of $K$-finite vectors}\leftarrowbel{sec Kfin}
In this section $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. The purpose of this section is to prove Theorem \ref{thm main Kfin}.
\subsection{}
Let us first show that, when $G$ is non-Archimedean, it is enough to establish condition (2) of Theorem \ref{thm main Kfin}, and condition (1) will then follow. So we assume condition (2) and use the notation $C(v_1 , v_2)$ therein.
Let us denote by $\underline{V} \subset V$ the subspace of $K$-finite (i.e. smooth) vectors. By the polarization identity, it is clear that for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$ the limit $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} \leftarrowngle gv_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle} \cdot dg}{r^{\mathbf{d} (V)}}$$ exists, let us denote it by $D(v_1 , v_2 , v_3 , v_4)$, and $D$ is a quadlinear form $$D : \underline{V} \times \overline{\underline{V}} \times \overline{\underline{V}} \times \underline{V} \to {\mathbb C}.$$
Next, we claim that for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$ and all $g_1 , g_2 \in G$ we have $$ D(g_1 v_1, g_2 v_2 , g_1 v_3 , g_2 v_4) = D(v_1 , v_2 , v_3 , v_4).$$ Indeed, again by the polarization identity, it is enough to show that for all $v_1 , v_2 \in \underline{V}$ and all $g_1 , g_2 \in G$ we have \begin{equation}\leftarrowbel{eq Kfin nonArch C is G inv} C(g_1 v_1 , g_2 v_2) = C(v_1 , v_2).\end{equation} There exists $r_0 \ge 0$ such that $$ G_{<r-r_0} \subset g_2^{-1} G_{<r} g_1 \subset G_{<r+r_0}.$$ We have: $$ \int_{G_{<r}} |\leftarrowngle g g_1 v_1 , g_2 v_2 \rightarrowngle |^2 \cdot dg = \int_{g_2^{-1} G_{<r} g_1} |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg$$ and therefore $$ \int_{G_{<r-r_0}} |\leftarrowngle g v_1 , v_2 \rightarrowngle |^2 \cdot dg \leq \int_{G_{<r}} |\leftarrowngle g g_1 v_1 , g_2 v_2 \rightarrowngle |^2 \cdot dg \leq \int_{G_{<r+r_0}} |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg.$$ Dividing by $r^{\mathbf{d} (V)}$ and taking the limit $r \to +\infty$ we obtain (\ref{eq Kfin nonArch C is G inv}).
Now, by Schur's lemma (completely analogously to the reasoning with $\Phi$ in the proof of Proposition \ref{prop c-tempered orth rel}), we obtain that for some $C >0$ we have $$ D(v_1 , v_2 , v_3 , v_4) = C \cdot \leftarrowngle v_1 , v_ 3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}$$ for all $v_1 , v_2 , v_3 , v_4 \in \underline{V}$.
\subsection{}
Thus, we aim at establishing condition (2) of Theorem \ref{thm main Kfin} in either the non-Archimedean or the Archimedean cases. Since a complex group can be considered as a real group and the formulation of the desired theorem will not change, we assume that we are either in the real case or in the non-Archimedean case.
Also, notice that to show Theorem \ref{thm main Kfin} for all maximal compact subgroups it is enough to show it for one maximal compact subgroup (in the non-Archimedean case because the resulting notion of $K$-finite vectors does not depend on the choice of $K$ and in the real case since all maximal compact subgroups are conjugate).
\subsection{}
Let us fix some notation. We choose a maximal split torus $A \subset G$ and a minimal parabolic $P \subset G$ containing $A$. We denote $$ {\mathfrak a} := {\textnormal{Hom}}_{{\mathbb Z}} (X^* (A) , {\mathbb R}).$$ We let $L \subset {\mathfrak a}$ to be ${\mathfrak a}$ itself in the real case and the lattice in ${\mathfrak a}$ corresponding to $X_* (A)$ in the non-Archimedean case. We let $\exp : L \to A$ be the exponential map constructed in the usual way:
\begin{itemize}
\item If $G$ is real, we let $\exp$ to be the composition $L = {\mathfrak a} \cong {\rm Lie} (A) \to A$ where the last map is the exponential map from the Lie algebra to the Lie group, while the isomorphism is the identification resulting from the map $X^* (A) \to {\rm Lie} (A)^*$ given by taking the differential at $1 \in A$.
\item If $G$ is non-Archimedean, we let $\exp$ be the composition $L \cong X_* (A) \to A$ where the last map is given by sending $\chi$ to $\chi (\varpi^{-1})$, where $\varpi$ is a uniformizer.
\end{itemize}
We denote by $$ \Delta \subset \widetilde{\Delta} \subset X^* (A) \subset {\mathfrak a}^*$$ the set of simple roots $\Delta$ and the set of positive roots $\widetilde{\Delta}$ (resulting from the choice of $P$). We identify ${\mathfrak a}$ with ${\mathbb R}^{\Delta}$ in the clear way. We set $$ {\mathfrak a}^+ := \{ x \in {\mathfrak a} \ | \ \alpha (x) \ge 0 \ \forall \alpha \in \Delta \}$$ and $L^+ := L \cap {\mathfrak a}^+$.
Let us in the standard way choose a maximal comapct subgroup $K \subset G$ ``in good relative position" with $A$. In the real case this means ${\rm Lie} (A)$ sitting in the $(-1)$-eigenspace of a Cartan involution whose $1$-eigenspace is ${\rm Lie} (K)$ and in the non-Archimedean case it is as in \cite[V.5.1., Th{\' e}or{\` e}me]{Re}. In the non-Archimedean case let us also, to simplify notation, assume that $G = K \exp (L^+) K$ (in general there is a finite subset $S \subset Z_G (A)$ such that $G = \coprod_{s \in S} K \exp (L^+) s K$ and one proceeds with the obvious modifications).
Let us denote $\rho := \frac{1}{2}\sum_{\alpha \in \widetilde{\Delta}} \mu_{\alpha} \cdot \alpha \in {\mathfrak a}^*$ where $\mu_{\alpha} \in {\mathbb Z}_{\ge 1}$ is the multiplicity of the root $\alpha$.
Fixing Haar measures, especially denoting by $dx$ a Haar measure on $L$, we have a uniquely defined continuous $\omega : L^+ \to {\mathbb R}_{\ge 0}$ such that the following integration formula holds: $$ \int_{G} f(g) \cdot dg = \int_{K \times K} \left( \int_{L^+} \omega (x) f(k_1 \exp (x) k_2) \cdot dx \right) \cdot dk_1 dk_2.$$ Regarding the behaviour of $\omega (x)$, we can use \cite[around Lemma 1.1]{Ar} as a reference. In the real case there exists $C>0$ such that
\begin{equation}\leftarrowbel{eq omega is like rho Arch}
\frac{\omega (x)}{e^{2\rho (x)}} = C \cdot \prod_{\alpha} \left( 1 - e^{-2\alpha (x)} \right)
\end{equation} where $\alpha$ runs over $\widetilde{\Delta}$ according to multiplicities $\mu_{\alpha}$. In the non-Archimedean case, for every $\Theta \subset \Delta$ there exists $C_{\Theta} > 0$ such that
\begin{equation}\leftarrowbel{eq omega is like rho nonArch}
\frac{\omega (x)}{e^{2\rho (x)}} = C_{\Theta}
\end{equation} for all $x \in L^+$ satisfying $\alpha (x) = 0$ for all $\alpha \in \Theta$ and $\alpha (x) \neq 0$ for all $\alpha \in \Delta \smallsetminus \Theta$.
Since, by Claim \ref{clm doesnt depend on norm}, we are free in our choice of the norm $||-||$ on ${\mathfrak g}$, let us choose $||-||$ to be a supremum norm in coordinates gotten from an $A$-eigenbasis. Then\footnote{Recall the notation $\mathbf{r}$ from \S\ref{sec intro}.} $$ \mathbf{r} (\exp (x)) = \log q \cdot \max_{\alpha \in \widetilde{\Delta} } |\alpha (x)|$$ where $q$ is the residual cardinality in the non-Archimedean case and $q := e$ in the real case. Let us denote $$ {\mathfrak a}_{<r} := \{ x \in {\mathfrak a} \ | \ | \alpha (x)| < \tfrac{r}{\log q} \ \forall \alpha \in \widetilde{\Delta} \}$$ and ${\mathfrak a}^+_{<r} := {\mathfrak a}_{<r} \cap {\mathfrak a}^+$ and similarly $L_{<r} := L \cap {\mathfrak a}_{<r}$, $L^+_{<r} := L^+ \cap L_{<r}$. Then $ L_{<r} = \exp^{-1} (G_{<r})$. Hence there exists $r_0 \ge 0$ such that
\begin{equation}\leftarrowbel{eq Gr is like Lr}
K \exp (L^+_{<r-r_0}) K \subset G_{<r} \subset K \exp(L^+_{<r+r_0}) K.
\end{equation}
\subsection{}
Let now $V$ be a tempered irreducible unitary $G$-representation. Let us denote by $\underline{V} \subset V$ the subspace of $K$-finite vectors. Given $v_1 , v_2 \in V$, we will denote by $f_{v_1 , v_2}$ the continuous function on $L^+$ given by $$f_{v_1 , v_2} (x) := e^{\rho (x)} \leftarrowngle \exp (x) v_1 , v_2 \rightarrowngle.$$ We have $$ \Ms{v_1}{v_2}{r} = \int_{G_{<r}} |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg = $$ $$= \int_{K \times K} \left( \int_{L^+ \cap \exp^{-1} (k_2 G_{<r} k_1^{-1})} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2.$$ In view of (\ref{eq Gr is like Lr}), in order to prove Theorem \ref{thm main Kfin} it is enough to show:
\begin{claim}\leftarrowbel{clm Kfin ess}
There exists $\mathbf{d} (V) \in {\mathbb Z}_{\ge 0}$ such that for every non-zero $v_1 , v_2 \in \underline{V}$ there exists $C(v_1 , v_2)>0$ such that $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}}| f_{k_1 v_1 , k_2 v_2} (x) |^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = C(v_1 , v_2).$$
\end{claim}
\subsection{} We have the following:
\begin{claim}\leftarrowbel{clm growth exists}\
\begin{enumerate}
\item Given $v_1 , v_2 \in \underline{V}$, either $f_{v_1 , v_2} = 0$ in which case we set $\mathbf{d} (v_1 , v_2) := -\infty$, or there exist $\mathbf{d} (v_1 , v_2) \in {\mathbb Z}_{\ge 0}$ and $C(v_1 , v_2)>0$ such that $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx = C(v_1 , v_2).$$
\item In the real case, we have $\mathbf{d} (v_1 , X v_2) \leq \mathbf{d} (v_1 , v_2)$ for all $v_1 , v_2 \in \underline{V}$ and $X \in {\mathfrak g}$.
\item Denoting $\mathbf{d}(V) := \sup_{v_1 , v_2 \in \underline{V}} \mathbf{d} (v_1 , v_2)$, we have neither $\mathbf{d} (V) = -\infty$ nor $\mathbf{d} (V) =+\infty$ (i.e. $\mathbf{d} (V) \in {\mathbb Z}_{\ge 0}$).
\end{enumerate}
\end{claim}
Let us establish Claim \ref{clm Kfin ess} given Claim \ref{clm growth exists}:
\begin{proof}[Proof (of Claim \ref{clm Kfin ess} given Claim \ref{clm growth exists}).]
Let us first handle the non-Archimedean case. Let us notice that we can replace $v_1$ and $v_2$ by $g_1 v_1$ and $g_2 v_2$ for any $g_1,g_2 \in G$. Indeed, for some $r_0 \ge 0$ we have $$g_2^{-1} G_{<r-r_0} g_1 \subset G_{<r} \subset g_2^{-1} G_{<r+r_0} g_1$$ and thus $$ \int_{G_{<r-r_0}} | \leftarrowngle g g_1 v_1 , g_2 v_2 \rightarrowngle |^2 \cdot dg \leq \int_{G_{<r}} |\leftarrowngle g v_1 , v_2 \rightarrowngle |^2 \cdot dg \leq \int_{G_{<r+r_0}} | \leftarrowngle g g_1 v_1 , g_2 v_2 \rightarrowngle |^2 \cdot dg,$$ from which the claim clearly follows. Since $G \cdot v_1$ spans $\underline{V}$ and $G \cdot v_2$ spans $\underline{V}$, we deduce that by replacing $v_1$ and $v_2$ we can assume that $\mathbf{d} (v_1 , v_2) = \mathbf{d} (V)$. Now, since the integral $$ \int_{K \times K} \left( \int_{L^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1, k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2$$ over $K \times K$ is simply a finite linear combination the claim is clear.
Let us now handle the real case. First, we would like to see that for some $k_1 , k_2 \in K$ we have $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$. To that end, let us denote by ${\mathfrak n}$ and ${\mathfrak n}^-$ the Lie algebras of $N$ and $N^-$ (the unipotent radicals of $P$ and of $P^-$, the opposite to $P$ with respect to $A$) and identify ${\mathfrak a}$ with the Lie algebra of $A$ as before. Since ${\mathcal U} ({\mathfrak n}^-) {\mathcal U} ({\mathfrak a}) K v_1$ spans $\underline{V}$ and ${\mathcal U} ({\mathfrak n}) {\mathcal U} ({\mathfrak a}) K v_2$ spans $\underline{V}$, we can find $k_1 , k_2 \in K$ and some elements $v'_1 \in {\mathcal U} ({\mathfrak n}^-) {\mathcal U} ({\mathfrak a}) k_1 v_1$ and $v'_2\in {\mathcal U} ({\mathfrak n}) {\mathcal U} ({\mathfrak a}) k_2 v_2$ such that $\mathbf{d} (v'_1, v'_2) = \mathbf{d} (V)$. By Claim \ref{clm growth exists}(2) this forces $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$.
Next, given two continuous functions $f_1 , f_2$ on ${\mathfrak a}^+$ and $d \in {\mathbb Z}_{\ge 0}$ let us denote $$ \leftarrowngle f_1 , f_2 \rightarrowngle_d := \lim_{r \to +\infty} \frac{ \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} f_1(x) \overline{f_2 (x)} \cdot dx}{r^d}$$ if the limit exists, and $|| f ||^2_d := \leftarrowngle f , f \rightarrowngle_d$.
We claim that the function $(k_1 , k_2) \mapsto || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V)}$ on $K \times K$ is continuous and that $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = \int_{K \times K} || f_{k_1 v_1 , k_2 v_2 } ||^2_{\mathbf{d} (V)} \cdot dk_1 dk_2.$$ Then the right hand side is non-zero since we have seen that $\mathbf{d} (k_1 v_1 , k_2 v_2) = \mathbf{d} (V)$ for some $k_1 , k_2 \in V$, and we are done.
Let $(v_1^i)$ be a basis for the ${\mathbb C}$-span of $\{ k v_1 \}_{k \in K}$ and let $(v_2^j)$ be a basis for the ${\mathbb C}$-span of $\{ k v_2 \}_{k \in K}$. Let us write $k v_1 = \sum_i c_i(k) v_1^i$ and $k v_2 = \sum_j d_j (k) v_2^j$, so that $c_i$ and $d_j$ are continuous ${\mathbb C}$-valued functions of $K$. Then $$ \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx = $$ $$ = \sum_{i_1 , i_2 , j_1 , j_2} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \int_{{\mathfrak a}^+_{<r} } \frac{\omega (x)}{e^{2 \rho (x)}} \cdot f_{v_1^{i_1} , v_2^{j_1}} (x) \cdot \overline{f_{v_1^{i_2} , v_2^{j_2}} (x)} \cdot dx.$$ Therefore $$ || f_{k_1 v_1 , k_2 v_2}||^2_{\mathbf{d} (V)} = \sum_{i_1 , i_2 , j_1 , j_2} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \leftarrowngle f_{v_1^{i_1} , v_2^{j_1}} , f_{v_1^{i_2} , v_2^{j_2}} \rightarrowngle_{\mathbf{d} (V)} $$ so $(k_1 , k_2) \mapsto || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V)}$ is indeed continuous. Also, it is now clear that we have $$ \lim_{r \to +\infty} \frac{\int_{K \times K} \left( \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{k_1 v_1 , k_2 v_2} (x)|^2 \cdot dx \right) \cdot dk_1 dk_2}{r^{\mathbf{d} (V)}} = $$ $$ = \sum_{i_1 , i_2 , j_1 , j_2} \leftarrowngle f_{v_1^{i_1} , v_2^{j_1}} , f_{v_1^{i_2} , v_2^{j_2}} \rightarrowngle_{\mathbf{d} (V)} \int_{K \times K} c_{i_1} (k_1) \overline{c_{i_2} (k_1)} \overline{d_{j_1} (k_2)} d_{j_2} (k_2) \cdot dk_1 dk_2 = $$ $$ = \int_{K \times K} || f_{k_1 v_1 , k_2 v_2} ||^2_{\mathbf{d} (V) } \cdot dk_1 dk_2$$
\end{proof}
\subsection{}
Let us now explain Claim \ref{clm growth exists} in the case when $G$ is non-Archimedean. Let $v_1 , v_2 \in \underline{V}$. Let us choose a positive integer $k$ large enough so that $k \cdot {\mathbb Z}_{\ge 0}^{\Delta} \subset L$. By enlarging $k$ even more if necessary, by \cite[Theorem 4.3.3.]{Ca} for every $\Theta \subset \Delta$ and every $y \in ( {\mathbb R}_{\ge 0}^{\Theta} \times {\mathbb R}_{>k}^{\Delta \smallsetminus \Theta}) \cap L^+ $ the function $$ f_{v_1 , v_2 , \Theta , y} : k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta} \to {\mathbb C}$$ given (identifying ${\mathbb R}^{\Delta \smallsetminus \Theta}$ with a subspace of ${\mathbb R}^{\Delta}$ in the clear way) by $x \mapsto f_{v_1 , v_2}(y+x)$, can be written as $$ \sum_{1 \leq i \leq p} c_i \cdot e^{\leftarrowmbda_i (x_{\Delta \smallsetminus \Theta})} q_i (x_{\Delta \smallsetminus \Theta})$$ where $c_i \in {\mathbb C} \smallsetminus \{ 0\}$, $\leftarrowmbda_i$ is a complex-valued functional on ${\mathbb R}^{\Delta \smallsetminus \Theta}$, and $q_i$ is a monomial on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. Here $x_{\Delta \smallsetminus \Theta}$ is the image of $x$ under the natural projection ${\mathbb R}^{\Delta} \to {\mathbb R}^{\Delta \smallsetminus \Theta}$. We can assume that the couples in the collection $\{ (\leftarrowmbda_i , q_i) \}_{1 \leq i \leq p}$ are pairwise different. Since $V$ is tempered, by ``Casselman's criterion" we in addition have that for every $1 \leq i \leq p$, ${\rm Re} (\leftarrowmbda_i)$ is non-negative on ${\mathbb R}_{\ge 0}^{\Delta \smallsetminus \Theta}$.
By Claim \ref{clm appb lat}, either $p = 0$, equivalently $f_{v_1 , v_2 , \Theta , y} = 0$ (in which case we set $d_{v_1 , v_2 , \Theta , y} := -\infty$), or there exists $d_{v_1 , v_2 , \Theta , y} \in {\mathbb Z}_{\ge 0}$ such that the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d_{v_1 , v_2 , \Theta , y}}} \sum_{x \in (k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta}) \cap L^+_{<r}} |f_{v_1 , v_2}(y+x)|^2$$ exists and is strictly positive.
Now, given $y \in L^+$ let us denote $\Theta_y := \{ \alpha \in \Delta \ | \ y_{\alpha} \leq k \}$ where by $y_{\alpha}$ we denote the coordinate of $y \in {\mathbb R}^{\Delta}$ at the $\alpha$-place. Let $Y \subset L^+$ be the subset of $y \in L^+$ for which $y_{\alpha} \leq 2k$ for all $\alpha \in \Delta$. Then $Y$ is a finite set, and we have \begin{equation}\leftarrowbel{eq break L into subsets} L^+ = \coprod_{y \in Y} \left( y + k \cdot {\mathbb Z}_{\ge 0}^{\Delta \smallsetminus \Theta_y} \right).\end{equation} Notice also that $\omega (x) / e^{2 \rho (x)}$ is a positive constant on each one of the subset of which we take union in (\ref{eq break L into subsets}). We set $\mathbf{d} (v_1 , v_2) := \max_{y \in Y} d_{v_1 , v_2 , \Theta_y , y}$. We see that either $f_{v_1 , v_2} = 0$ (then $\mathbf{d} (v_1 , v_2) = -\infty$) or the limit $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \sum_{x \in L^+_{<r} } \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2$$ exists and is strictly positive.
That $\mathbf{d} (V)$ is finite follows from $\mathbf{d} (v_1 , v_2)$ being controlled by finitely many Jacquet modules, with the finite central actions on them.
\subsection{}
Let us now explain Claim \ref{clm growth exists} in the case when $G$ is real. Using \cite{CaMi} we know that, fixing $k >0$, given $\Theta \subset \Delta$ the restriction of $\frac{\omega (x)^{1/2}}{e^{\rho (x)}} f_{v_1 , v_2} (x)$ to ${\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} \times [0,k]^{\Theta}$ can be written as $$ \sum_{1 \leq i \leq p} e^{\leftarrowmbda_i (x_{\Delta \smallsetminus \Theta})} q_i (x_{\Delta \smallsetminus \Theta}) \phi_i (x) $$ where the notation is as follows. First, $\leftarrowmbda_i$ is a complex-valued functional on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. Next, $q_i$ is a monomial on ${\mathbb R}^{\Delta \smallsetminus \Theta}$. The couples $(\leftarrowmbda_i , q_i)$, for $1 \leq i \leq p$, are pairwise distinct. The function $\phi_i$ is expressible as a composition $$ [0,k]^{\Theta} \times {\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} \xrightarrow{{\rm id} \times {\rm ei}} [0,k]^{\Theta} \times ({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta} \xrightarrow{\phi_i^{\circ}} {\mathbb C}$$ where ${\rm ei}$ is the coordinate-wise application of $x \mapsto e^{-x}$ and $\phi_{i}^{\circ}$ is a continuous function such that, for every $b \in [0,k]^{\Theta}$, the restriction of $\phi_i^{\circ}$ via $({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta} \xrightarrow{z \mapsto (b,z)} [0,k]^{\Theta} \times ({\mathbb C}_{|-|<1})^{\Delta \smallsetminus \Theta}$ is holomorphic. Lastly, the function $b \mapsto \phi_{i}^{\circ} (b , \{0\}^{\Delta \smallsetminus \Theta} ) $ on $[0,k]^{\Theta}$ is not identically zero. Since $V$ is tempered, by ``Casselman's criterion" we in addition have that for every $1 \leq i \leq p$, ${\rm Re} (\leftarrowmbda_i)$ is non-negative on ${\mathbb R}_{\ge 0}^{\Delta \smallsetminus \Theta}$.
If $p = 0$, we set $d_{v_1 , v_2 , \Theta} := -\infty$. Otherwise, Claim \ref{clm appb} provides a number $d_{v_1 , v_2 , \Theta} \in {\mathbb Z}_{\ge 0}$, described concretely in terms of $\{ (\leftarrowmbda_i , q_{i} , \phi_{i} ) \}_{1 \leq i \leq p}$, such that the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d_{v_1 , v_2 , \Theta}}} \int_{ ( [0,k]^{\Theta} \times {\mathbb R}_{\ge k}^{\Delta \smallsetminus \Theta} ) \cap {\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx $$ exists and is strictly positive. We set $\mathbf{d} (v_1 , v_2) := \max_{\Theta \subset \Delta} d_{v_1 , v_2 , \Theta}$. Then either $f_{v_1 , v_2} = 0$ or $\mathbf{d} (v_1 , v_2) > 0$ and the limit $$ \lim_{r \to +\infty} \frac{1}{r^{\mathbf{d} (v_1 , v_2)}} \int_{{\mathfrak a}^+_{<r}} \frac{\omega (x)}{e^{2 \rho (x)}} |f_{v_1 , v_2} (x)|^2 \cdot dx$$ exists and is strictly positive. That $\mathbf{d} (V)$ is finite follows from $\mathbf{d} (v_1 , v_2)$ being controlled by finitely many data, as in \cite{CaMi}. Part (2) of Claim \ref{clm growth exists} follows easily from the concrete description of $d_{v_1 , v_2 , \Theta}$ in Claim \ref{clm appb}.
\section{Proofs for Remark \ref{rem counterexample}, Remark \ref{rem doesnt depend on norm} , Proposition \ref{prop formula ch}, Proposition \ref{prop from folner to exact} and Remark \ref{rem ctemp red is temp}.}\leftarrowbel{sec clms red}
In this section, $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. We explain Remark \ref{rem counterexample} (in Claim \ref{clm counterexample}), explain Remark \ref{rem doesnt depend on norm} (in Claim \ref{clm doesnt depend on norm}), prove Proposition \ref{prop formula ch} (in \S\ref{ssec proof of prop formula ch}), prove Proposition \ref{prop from folner to exact} (in \S\ref{ssec clms red 1}) and explain Remark \ref{rem ctemp red is temp} (in Claim \ref{clm Prop tempered is c-tempered reductive holds}).
\subsection{}\leftarrowbel{ssec clms red 1}
\begin{lemma}\leftarrowbel{lem red temp is ctemp}
Let $V$ be an irreducible unitary $G$-representation and suppose that there exists a unit vector $v_0 \in V$ satisfying properties (1) and (2) of Proposition \ref{prop from folner to exact}. Let $0 < r_0 < r_1 < \ldots$ be a sequence such that $\lim_{n \to +\infty} r_n = +\infty$. Then $V$ is c-tempered with F{\o}lner sequence $G_{< r_0} , G_{<r_1}, \ldots$.
\end{lemma}
\begin{proof}
Property (1) of Definition \ref{def c-temp} is immediate from property (1) of Proposition \ref{prop from folner to exact}. Let us check property (2) of Definition \ref{def c-temp}. Thus, let $v_1 , v_2 \in V$ and let $K \subset G$ be a compact subset. Fix $r^{\prime} \ge 0$ big enough so that $K \subset G_{< r^{\prime}}$ and $K^{-1} \subset G_{< r^{\prime}}$. We then have, for all $r>0$ and all $g_1 , g_2 \in K$: $$G_{< r} \triangle g_2^{-1} G_{< r} g_1 \subset G_{< r + 2 r^{\prime}} \smallsetminus G_{< r - 2 r^{\prime}}.$$ Therefore, using property (2) of Proposition \ref{prop from folner to exact}, $$ \limsup_{r \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{G_{< r} \triangle g_2^{-1} G_{< r} g_1}}{ \Mss{v_0}{v_0}{G_{<r}} } \leq \limsup_{r \to +\infty} \frac{ \Mss{v_1}{v_2}{G_{< r + 2 r^{\prime}} \smallsetminus G_{\leq r - 2 r^{\prime} }} }{ \Mss{v_0}{v_0}{G_{<r}}} = 0$$ and therefore also $$ \lim_{n \to +\infty} \frac{\sup_{g_1 , g_2 \in K} \Mss{v_1}{v_2}{G_{< r_n} \triangle g_2^{-1} G_{< r_n} g_1} }{ \Mss{v_0}{v_0}{G_{< r_n}}} = 0.$$
\end{proof}
\begin{proof}[Proof (of Proposition \ref{prop from folner to exact}).]
Let us fix a $K$-finite unit vector $v'_0 \in V$, for some maximal compact subgroup $K \subset G$. Let $0 < r_0 < r_1 < \ldots$ be a sequence such that $\lim_{n \to +\infty} r_n = +\infty$. By Lemma \ref{lem red temp is ctemp} $V$ is c-tempered with F{\o}lner sequence $G_{<r_0} , G_{<r_1} , \ldots$ and hence by Proposition \ref{prop c-tempered orth rel} we obtain $$ \lim_{n \to +\infty} \frac{\int_{g \in G_{<r_n}} \leftarrowngle gv_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle} \cdot dg}{\Ms{v'_0}{v'_0}{r_n}} = \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle}$$ for all $v_1 , v_2 , v_3 , v_4 \in V$. Since this holds for any such sequence $\{ r_n \}_{n \ge 0}$, we obtain \begin{equation}\leftarrowbel{eq formula again} \lim_{r \to +\infty} \frac{\int_{g \in G_{<r}} \leftarrowngle gv_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle} \cdot dg}{\Ms{v'_0}{v'_0}{r}} = \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle} \end{equation} for all $v_1 , v_2 , v_3 , v_4 \in V$. By Theorem \ref{thm main Kfin} we have $$ \lim_{r \to +\infty} \frac{\Ms{v'_0}{v'_0}{r}}{r^{\mathbf{d}(V)}} = C$$ for some $C>0$. This enables to rewrite (\ref{eq formula again}) as $$ \lim_{r \to +\infty} \frac{\int_{g \in G_{<r}} \leftarrowngle gv_1 , v_2 \rightarrowngle \overline{\leftarrowngle gv_3 , v_4 \rightarrowngle} \cdot dg}{r^{\mathbf{d} (V)}} = C \cdot \leftarrowngle v_1 , v_3 \rightarrowngle \overline{\leftarrowngle v_2 , v_4 \rightarrowngle} $$ for all $v_1 , v_2 , v_3 , v_4 \in V$, as desired.
\end{proof}
\subsection{}
\begin{claim}\leftarrowbel{clm doesnt depend on norm}
The validity of Conjecture \ref{main conj red exact}, as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$, of Theorem \ref{thm main Kfin} as well as the resulting invariants $\mathbf{d} (V)$ and $\mathbf{f} (V)$ (the latter in the non-Archimedean case), and of Proposition \ref{prop from folner to exact}, do not depend on the choice of the norm $|| - ||$ on ${\mathfrak g}$.
\end{claim}
\begin{proof}
Let $|| - ||^{\prime}$ be another norm on ${\mathfrak g}$, let $\mathbf{r}^{\prime} : G \to {\mathbb R}_{\ge 0}$ be the resulting function, and let $G_{<r}^{\prime} \subset G$ be the resulting subsets. There exists $r_0 \ge 0$ such that $$ e^{-r_0} \cdot || X || \leq ||X ||^{\prime} \leq e^{r_0} \cdot || X||, \quad \forall X \in {\mathfrak g}$$ and therefore $$ e^{-2r_0} \cdot ||{\rm Ad} (g)|| \leq || {\rm Ad} (g) ||' \leq e^{2 r_0} \cdot || {\rm Ad} (g) ||, \quad \forall g \in G.$$ Then $$ G^{\prime}_{< r} \subset G_{<r + 2r_0}, \quad \forall r\ge 0$$ and $$ G_{< r} \subset G^{\prime}_{<r + 2r_0}, \quad \forall r\ge 0.$$ These ``sandwich" relations readily imply the independence claims.
\end{proof}
\subsection{}
\begin{claim}\leftarrowbel{clm Prop tempered is c-tempered reductive holds}
An irreducible unitary $G$-representation for which there exists a unit vector $v_0 \in V$ such that conditions (1) and (2) of Proposition \ref{prop from folner to exact} are satisfied is tempered.
\end{claim}
\begin{proof}
Clear from Lemma \ref{lem red temp is ctemp} coupled with Corollary \ref{cor c-temp are temp}.
\end{proof}
\subsection{}
\begin{claim}\leftarrowbel{clm counterexample}
Let $G := PGL_2 (\Omega)$, $\Omega$ a local field. Let $A \subset G$ be the subgroup of diagonal matrices. Then, for every non-trivial irreducible unitary $G$-representation $V$, the set of matrix coefficients of $V$ restricted to $A$ is equal to the set of function on $A$ of the form $$ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d\chi$$ as $\phi$ runs over $ L^1 (\hat{A})$.
\end{claim}
\begin{proof}
Denote by $B \subset G$ the subgroup of upper-triangular matrices and by $N \subset B$ its unipotent radical.
Let us recall that, by Mackey theory, there is a unique (up to isomorphism) infinite-dimensional irreducible unitary $B$-representation $W$, and the rest of irreducible unitary $B$-representations are killed by $N$. The restriction ${\rm Res}^B_A W$ is isomorphic to the right regular unitary $A$-representation $L^2 (A)$.
Let now $V$ be a non-trivial irreducible unitary $G$-representation. Recall that by the Howe-Moore theorem (or by a step in one of its usual proofs) $V$ does not contain non-zero $N$-invariant vectors. By decomposing the restriction ${\rm Res}^G_B V$ into a direct integral of irreducible unitary $B$-representations, and using the fact that $V$ admits no non-zero $N$-invariant vectors, we see that ${\rm Res}^G_B V$ is a multiple of $W$. Hence, we deduce that ${\rm Res}^G_A A$ is a multiple of the right regular unitary $A$-representation $L^2 (A)$.
Now, the matrix coefficients of a multiple of the right regular unitary $A$-representation $L^2 (A)$ are easily seen to be the functions on $A$ of the form $$ a \mapsto \int_{\hat{A}} \chi (a) \cdot \phi (\chi) \cdot d\chi$$ where $\phi \in L^1 (\hat{A})$.
\end{proof}
\subsection{}\leftarrowbel{ssec proof of prop formula ch}
\begin{proof}[Proof (of Proposition \ref{prop formula ch}).]
Fix $d \in D_c^{\infty} (G)$. Let $K \subset G$ be an open compact subgroup such that $d$ is invariant under $K$ both on left and on right. Let us denote by $e_1 , \ldots , e_n$ an orthonormal basis of $V^K$, and let us denote by $\pi_K : V \to V^K$ the orthonormal projection. Let us denote by $[-,-] : C^{-\infty} (G) \times D_c^{\infty} (G) \to {\mathbb C}$ the canonical pairing. We have
$$
[{}^g m_{v_1 , v_2} , d] = [m_{g v_1 , g v_2} , d] = \leftarrowngle d g v_1 , g v_2 \rightarrowngle = \leftarrowngle d \pi_K (g v_1) , \pi_K (g v_2) \rightarrowngle = $$ $$ = \sum_{1 \leq i,j \leq n} \leftarrowngle g v_1 , e_i \rightarrowngle \overline{\leftarrowngle g v_2 , e_j \rightarrowngle} \leftarrowngle d e_i , e_j \rightarrowngle.
$$
Hence $$ \frac{\int_{G_{<r}} [{}^g m_{v_1 , v_2} , d] \cdot dg}{r^{\mathbf{d} (V)}} = \sum_{1 \leq i,j \leq n} \leftarrowngle d e_i , e_j \rightarrowngle \cdot \frac{\int_{G_{<r}} \leftarrowngle g v_1 , e_i \rightarrowngle \overline{\leftarrowngle g v_2 , e_j \rightarrowngle} \cdot dg}{r^{\mathbf{d} (V)}}$$ and therefore $$ \lim_{r \to +\infty} \frac{\int_{G_{<r}} [{}^g m_{v_1 , v_2} , d] \cdot dg}{r^{\mathbf{d} (V)}} = \frac{1}{\mathbf{f} (V)} \sum_{1 \leq i,j \leq n} \leftarrowngle d e_i , e_j \rightarrowngle \cdot \leftarrowngle v_1 , v_2 \rightarrowngle \overline{\leftarrowngle e_i , e_j \rightarrowngle} = $$ $$ = \frac{1}{\mathbf{f} (V)} \sum_{1 \leq i \leq n} \leftarrowngle d e_i , e_i \rightarrowngle \cdot \leftarrowngle v_1 , v_2 \rightarrowngle = \frac{\leftarrowngle v_1 , v_2 \rightarrowngle}{\mathbf{f} (V)} \Theta_V (d).$$
\end{proof}
\section{The case of the principal series representation $V_1$ of slowest decrease}\leftarrowbel{sec proof of V1}
In this section $G$ is a semisimple group over a local field. We continue with notations from \S\ref{sec intro}. Our goal is to prove Theorem \ref{thm intro slowest} (restated as Theorem \ref{thm V1 c temp} below).
\subsection{}\leftarrowbel{ssec V1 and Xi}
We fix a minimal parabolic $P \subset G$ and a maximal compact subgroup $K \subset G$ such that $G = PK$. We consider the principal series unitary $G$-representation $V_1$ consisting of functions $f : G \to {\mathbb C}$ satisfying $$ f(pg) = \Delta_P (p)^{1/2} \cdot f(g) \quad \forall p \in P , g \in G$$ where $\Delta_P : P \to {\mathbb R}^{\times}_{>0}$ is the modulus function of $P$. The $G$-invariant inner product on $V_1$ can be taken to be $$ \leftarrowngle f_1 , f_2 \rightarrowngle = \int_K f_1 (k) \cdot \overline{f_2 (k)} \cdot dk$$ (where we normalize the Haar measure on $K$ to have total mass $1$). Recall that $V_1$ is irreducible. We denote by $f_0 \in V_1$ the spherical vector, determined by $f_0 (k) = 1$ for all $k \in K$. We also write $$ \Xi_G (g) := \leftarrowngle g f_0 , f_0 \rightarrowngle.$$
\begin{lemma}\leftarrowbel{lem growth Xi}
Given $r' \ge 0$ we have $$ \lim_{r \to +\infty} \frac{\int_{G_{<r+r'} \smallsetminus G_{<r-r'}} \Xi_G (g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} = 0$$ and $$ \lim_{r \to +\infty} \frac{\int_{G_{<r+r'}} \Xi_G (g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} = 1.$$
\end{lemma}
\begin{proof}
The second equality follows from the first, and the first is immediately implied by Theorem \ref{thm main Kfin}.
\end{proof}
\subsection{} The main result of this section is:
\begin{theorem}\leftarrowbel{thm V1 c temp}
Let $V$ be an irreducible tempered unitary $G$-representation. Suppose that there exist a unit vector $v_0 \in V$ such that \begin{equation}\leftarrowbel{eq good vector}
\underset{r \to +\infty}{\limsup} \frac{ \int_{G_{<r}} \Xi_G (g)^2 \cdot dg }{ \Ms{v_0}{v_0}{r} } < +\infty.
\end{equation} Then Conjecture \ref{main conj red exact} holds for $V$. In particular, Conjecture \ref{main conj red exact} holds for $V_1$.
\end{theorem}
\subsection{}
We will prove Theorem \ref{thm V1 c temp} using the following result:
\begin{claim}\leftarrowbel{clm est using Xi}
Let $V$ be a tempered unitary $G$-representation. Then for all unit vectors $v_1,v_2 \in V$ and all measurable $K$-biinvariant subsets $S \subset G$ we have $$ \int_S |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg \leq \int_S \Xi_G (g)^2 \cdot dg.$$
\end{claim}
\begin{proof}[Proof (of Theorem \ref{thm V1 c temp} given Claim \ref{clm est using Xi}).]
To show that Conjecture \ref{main conj red exact} holds for $V$ we will use Proposition \ref{prop from folner to exact}, applied to our $V$ and our $v_0$.
There exists $r_0 \ge 0$ such that $K G_{<r} K \subset G_{< r + r_0}$ for all $r \ge 0$.
Let us verify condition (1) of Proposition \ref{prop from folner to exact}. For unit vectors $v_1 , v_2 \in V$ we have $$ \frac{\Ms{v_1}{v_2}{r}}{\Ms{v_0}{v_0}{r}} \leq \frac{\int_{G_{<r+r_0}} \Xi_G (g)^2 \cdot dg}{\Ms{v_0}{v_0}{r}}$$ and therefore condition (1) of Proposition \ref{prop from folner to exact} follows from (\ref{eq good vector}) and Lemma \ref{lem growth Xi}. Let us now verify condition (2) of Proposition \ref{prop from folner to exact}. For unit vectors $v_1 , v_2 \in V$ and $r' \ge 0$ we have $$ \frac{ \Ms{v_1}{v_2}{r+r'} - \Ms{v_1}{v_2}{r-r'}}{\Ms{v_0}{v_0}{r}} \leq \frac{\int_{G_{<r+r'+r_0} \smallsetminus G_{<r-(r'+r_0)}} \Xi _G(g)^2 \cdot dg}{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg} \cdot \frac{\int_{G_{<r}} \Xi_G (g)^2 \cdot dg}{\Ms{v_0}{v_0}{r}}$$ and therefore condition (2) of Proposition \ref{prop from folner to exact} follows from (\ref{eq good vector}) and Lemma \ref{lem growth Xi}.
\end{proof}
\subsection{}
We will prove Claim \ref{clm est using Xi} using the following result:
\begin{claim}\leftarrowbel{clm norm of conv}
Let $\phi \in L^2 (G)$ be zero outside of a measurable $K$-biinvariant subset $S \subset G$ of finite volume. Denote by $T_{\phi} : L^2 (G) \to L^2 (G)$ the operator of convolution $\psi \mapsto \phi \star \psi$. Then\footnote{Here $|| \phi ||$ stands for the $L^2$-norm of $\phi$.} $$ || T_{\phi} ||^2 \leq \left( \int_{S} \Xi_G (g)^2 \cdot dg \right) \cdot || \phi ||^2.$$
\end{claim}
\begin{proof}[Proof (of Claim \ref{clm est using Xi} given Claim \ref{clm norm of conv}).]
We can clearly assume that $S$ has finite volume. Let us denote $$ \phi (g) := {\rm ch}_{S} (g) \cdot \overline{\leftarrowngle gv_1 , v_2 \rightarrowngle},$$ where ${\rm ch}_S$ stands for the characteristic function of $S$. Let us denote by $S_{\phi} : V \to V$ the operator $$ v \mapsto \int_G \phi (g) \cdot gv \cdot dg.$$ Since $V$ is tempered, we have $ ||S_{\phi}|| \leq || T_{\phi}||$. Therefore $$ \int_{S} |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg = \int_G \phi (g) \cdot \leftarrowngle gv_1 , v_2 \rightarrowngle \cdot dg = \leftarrowngle S_{\phi} v_1 , v_2 \rightarrowngle \leq || S_{\phi} || \leq || T_{\phi} || \leq $$ $$ \leq \left( \sqrt{\int_S \Xi_G (g)^2 \cdot dg} \right) \cdot || \phi || = \left( \sqrt{\int_S \Xi_G (g)^2 \cdot dg} \right) \cdot \left( \sqrt{\int_S |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg} \right)$$ thus $$\int_S |\leftarrowngle gv_1 , v_2 \rightarrowngle |^2 \cdot dg \leq \int_S \Xi_G (g)^2 \cdot dg$$ as desired.
\end{proof}
\subsection{}
Finally, let us prove Claim \ref{clm norm of conv}, following \cite{ChPiSa}.
\begin{proof}[Proof (of Claim \ref{clm norm of conv}).]
By\footnote{In the lemma we refer to it is assumed that $\phi$ is continuous but the arguments there apply to our $\phi$ without any modification.} \cite[Lemma 3.5]{ChPiSa} we can assume that $\phi$ is $K$-biinvariant and non-negative. By \cite[Proposition 4.3]{ChPiSa} we have $$ || T_{\phi} || = \int_G \Xi_G (g) \cdot \phi (g) \cdot dg.$$ Applying the Cauchy-Schwartz inequality, we obtain $$ ||T_{\phi} ||^2 \leq \left( \int_S \Xi_G (g)^2 \cdot dg\right) \cdot|| \phi||^2,$$ as desired.
\end{proof}
\section{The proof of Theorem \ref{thm intro SL2R}}\leftarrowbel{sec proof of SL2R}
In this section we let $G$ be either $SL_2 ({\mathbb R})$ or $PGL_2 (\Omega)$, where $\Omega$ is a non-Archimedean local field of characteristic $0$ and residual characteristic not equal to $2$. We prove Theorem \ref{thm intro SL2R}.
\subsection{}
If $G = PGL_2 (\Omega)$, we denote by $\varpi$ a uniformizer in $\Omega$, by ${\mathcal O}$ the ring of integers in $\Omega$, by $p$ the residual characteristic of $\Omega$ and $q:= |{\mathcal O} / \varpi {\mathcal O} |$.
\subsection{}\leftarrowbel{ssec SL2 notation}
We denote by $A \subset G$ the subgroup of diagonal matrices and by $U \subset G$ the subgroup of unipotent upper-triangular matrices. If $G = SL_2 ({\mathbb R})$ we define the isomorphism $$ \mathbf{a} : {\mathbb R}^{\times} \to A, \quad t \mapsto \mtrx{t}{0}{0}{t^{-1}}$$ and if $G = PGL_2 (\Omega)$ we define the isomorphism $$ \mathbf{a} : \Omega^{\times} \to A, \quad t \mapsto \mtrx{t}{0}{0}{1}.$$ We denote $A^+ := \{ a \in A \ | \ |\mathbf{a}^{-1} (a) | \ge 1 \}$.
If $G = SL_2 ({\mathbb R})$ then we can (and will) take $||- ||$ on ${\mathfrak g}$ to be such that $$\mathbf{r} \left( k_1 \mtrx{t}{0}{0}{t^{-1}} k_2 \right) = \log \max \{ |t|^2 , |t|^{-2} \}$$ where $t\in {\mathbb R}^{\times}$ and $k_1 , k_2 \in SO (2)$. If $G = PGL_2 (\Omega)$ then we can (and will) take $|| - ||$ on ${\mathfrak g}$ to be such that $$\mathbf{r} \left( k_1 \mtrx{t}{0}{0}{s} k_2 \right) = \log \max \{ |t/s| , |s/t| \}$$ where $t,s \in \Omega^{\times}$ and $k_1 , k_2 \in PGL_2 ({\mathcal O})$. Let us denote $A^+_{<r} := A^+ \cap G_{<r}$.
If $G = SL_2 ({\mathbb R})$ we set $K := SO (2) \subset G$. If $G = PGL_2 (\Omega)$ we choose a non-square $\zeta \in {\mathcal O}^{\times}$ and set $K \subset G$ to be the subgroup of elements of the form $\mtrx{a}{\zeta b}{b}{a}$, $(a,b) \in \Omega^2 \smallsetminus \{ (0,0) \}$ (so $K$ is a closed compact subgroup in $G$, but not open, and in particular not maximal).
We set $\omega : A^+ \to {\mathbb R}_{\ge 0}$ to be given by $\omega (\mathbf{a} (t)) := |t^2 - t^{-2} |$ if $G = SL_2 ({\mathbb R})$ and $\omega (\mathbf{a} (t)) := |t - t^{-1} |$ if $G = PGL_2 (\Omega)$. Then, taking the Haar measure on $K$ to have total mass $1$ and appropriately normalizing the Haar measure on $A$, for all non-negative-valued measurable functions $f$ on $G$ we have $$ \int_G f(g) \cdot dg = \int_{A^+} \omega (a) \left( \int_{K \times K} f(k_2 a k_1) \cdot dk_1 dk_2 \right) da.$$
Given a unitary $G$-representation $V$, vectors $v_1 , v_2 \in V$ and $a \in A^+$, we write $$ \ms{v_1}{v_2}{a} := \int_{K \times K} | \leftarrowngle k_2 a k_1 v_1 , v_2 \rightarrowngle |^2 \cdot dk_1 dk_2.$$ We have \begin{equation}\leftarrowbel{eq M in term of Mcirc} \Ms{v_1}{v_2}{r} := \int_{A^+_{<r}} \omega (a) \cdot \ms{v_1}{v_2}{a} \cdot da \end{equation} (where $M_{v_1 , v_2} (r)$ was already defined in \S\ref{sec intro}).
Given a unitary character\footnote{${\rm U} (1)$ denotes the subgroup of ${\mathbb C}^{\times}$ consisting of complex numbers with absolute value $1$.} $\chi: A \to {\rm U}(1) $ we consider the principal series unitary $G$-representation $V_{\chi}$, consisting of functions $f : G \to {\mathbb C}$ satisfying $$ f(u a g) = \chi (a) \cdot \Delta (a)^{1/2} \cdot f(g) \quad \forall a \in A, u \in U , g \in G,$$ where $\Delta (a) = |\mathbf{a}^{-1} (a)|^2$ if $G = SL_2 ({\mathbb R})$ and $\Delta (a) = |\mathbf{a}^{-1} (a)|$ if $G = PGL_2 (\Omega)$. Here $G$ acts by $(g^{\prime}f)(g) := f(gg^{\prime})$. The $G$-invariant inner product on $V_{\chi}$ can be expressed as $$ \leftarrowngle f_1 , f_2 \rightarrowngle = \int_{K} f_1 (k) \cdot \overline{f_2 (k)} \cdot dk.$$ For $\theta \in \hat{K}$, let $h^{\chi}_{\theta} \in V_{\chi}$ denote the unique vector determined by $h^{\chi}_{\theta} (k) = \theta (k)$ for $k \in K$, if it exists, and write $\textnormal{types} (V_{\chi}) \subset \hat{K}$ for the subset of $\theta$'s for which it exists. Thus $(h^{\chi}_{\theta})_{\theta \in \textnormal{types} (V_{\chi})}$ is a Hilbert basis for $V_{\chi}$.
\subsection{}
Let us now give several preparatory remarks.
First, we do not establish Conjecture \ref{main conj red exact} directly but, rather, establish conditions (1) and (2) of Proposition \ref{prop from folner to exact} (which suffices by that proposition).
Second, for a square-integrable irreducible unitary $G$-representation $V$, establishing conditions (1) and (2) of Proposition \ref{prop from folner to exact} with any unit vector $v_0 \in V$ is straight-forward (see the proof of Proposition \ref{prop sq int} for a spelling-out). As is well-known, a tempered irreducible unitary $G$-representation which is not square-integrable is a direct summand in some $V_{\chi}$. Therefore, we establish conditions (1) and (2) of Proposition \ref{prop from folner to exact} for irreducible direct summands in $V_{\chi}$.
Third, if $\chi = 1$ when $G = SL_2 ({\mathbb R})$ or if $\chi^2 = 1$ when $G = PGL_2 (\Omega)$, $V_{\chi}$ satisfies Conjecture \ref{main conj red exact} by Theorem \ref{thm V1 c temp}. So we assume throughout:
\begin{equation}\leftarrowbel{eq condition} \chi \neq 1 \textnormal{ if } G=SL_2 ({\mathbb R}), \quad \quad \quad \chi^2 \neq 1 \textnormal{ if } G = PGL_2 (\Omega). \end{equation}
\subsection{}
We reduce Conjecture \ref{main conj red exact} for an irreducible summand in $V_{\chi}$ to the following two claims.
\begin{claim}\leftarrowbel{clm SL2 2}
Fix $\chi$ satisfying (\ref{eq condition}). Let $V$ be an irreducible direct summand in $V_{\chi}$. There exist $f \in V$, $r_0 \ge 0$ and $D>0$ such that for all $r \ge r_0$ we have \begin{equation}\leftarrowbel{eq from below} \Ms{f}{f}{r} \geq D \cdot r \end{equation}
\end{claim}
\begin{claim}\leftarrowbel{clm SL2 2b}
Fix $\chi$ satisfying (\ref{eq condition}). There exist $r_0 > 0$ and $C>0$ (depending on $\chi$) such that for all $a \in A^+ \smallsetminus A^+_{<r_0}$ we have \begin{equation}\leftarrowbel{eq conj SL2 2} \ms{f_1}{f_2}{a} \leq C \cdot \omega (a)^{-1} \cdot ||f_1||^2 \cdot ||f_2||^2 \quad \quad \forall f_1, f_2 \in V_{\chi}.\end{equation}
\end{claim}
\begin{proof}[Proof (of Conjecture \ref{main conj red exact} for summands in $V_{\chi}$ given Claim \ref{clm SL2 2} and Claim \ref{clm SL2 2b} for $\chi$).]
Let $V$ be an irreducible direct summand in $V_{\chi}$. Let $f$, $r_0$, $D$ and $C$ be as in Claim \ref{clm SL2 2} and as in Claim \ref{clm SL2 2b} (taking $r_0$ to be the maximum of the values from the two statements).
In order to verify Conjecture \ref{main conj red exact} for $V$, we will verify the conditions (1) and (2) of Proposition \ref{prop from folner to exact}, where for $v_0$ we take our $f$.
Using (\ref{eq conj SL2 2}) we obtain the existence of $E,E'>0$ such that for all $r_0 \leq r_1 < r_2$ we have $$ \Ms{f_1}{f_2}{r_2} - \Ms{f_1}{f_2}{r_1} \leq E \cdot \textnormal{vol}_A ( A^+_{<r_2} \smallsetminus A^+_{<r_1} ) \cdot || f_1||^2 \cdot || f_2 ||^2 \leq $$ $$ \leq E' \cdot (1 + (r_2 - r_1)) \cdot || f_1||^2 \cdot || f_2 ||^2.$$ From this and (\ref{eq from below}) the conditions (1) and (2) of Proposition \ref{prop from folner to exact} are immediate.
\end{proof}
\subsection{}\leftarrowbel{sec proof of clm SL2 2}
Let us prove Claim \ref{clm SL2 2}.
\begin{proof}[Proof (of Claim \ref{clm SL2 2}).]
Let $V$ be an irreducible direct summand of $V_{\chi}$.
Let us first treat the case $G = PGL_2 (\Omega)$. We use the (normalized) Jacquet $A$-module $J(-)$ with respect to $G \hookleftarrow AU \twoheadrightarrow A$. We denote by $\underline{V} \subset V$ the subspace of smooth vectors. $J(\underline{V})$ is isomorphic to ${\mathbb C}_{\chi} \textnormal{op}lus {\mathbb C}_{\chi^{-1}}$. We consider $v \in \underline{V}$ whose projection under the canonical $\underline{V} \twoheadrightarrow J(\underline{V})$ is non-zero and is an $A$-eigenvector with eigencharacter $\chi$. By Casselman's canonical pairing theory there exists a non-zero $\alpha \in J(\underline{V})^*$ which is $A$-eigenvector with eigencharacter $\chi^{-1}$ such that $\leftarrowngle a v , v \rightarrowngle = |a |^{-1/2} \alpha (a v)$ whenever $a \in A^+ \smallsetminus A^+_{<r_0}$, for large enough $r_0 \ge 0$. Since we have $\alpha (a v) = \chi (a) \cdot \alpha (v)$ and $\alpha (v) \neq 0$, we deduce that for some $C>0$ we have $|\leftarrowngle a v , v \rightarrowngle|^2 = C \cdot |a|^{-1} $ for $a \in A^+ \smallsetminus A^+_{<r_0}$. Let $K_v \subset K$ be an open compact subgroup, small enough so that $K_v v = v$. We have, again for $a \in A^+ \smallsetminus A^+_{< r_0}$: $$ \ms{v}{v}{a} = \int_{K \times K} |\leftarrowngle k_2 a k_1 v , v \rightarrowngle |^2 \cdot dk_1 dk_2 \ge $$ $$ \ge \int_{K_v \times K_v} |\leftarrowngle k_2 a k_1 v , v \rightarrowngle |^2 \cdot dk_1 dk_2 = C^{\prime} \cdot |\leftarrowngle a v , v \rightarrowngle |^2$$ for some $C^{\prime}>0$ and so $\ms{v}{v}{a} \ge C^{\prime \prime} \cdot |a|^{-1}$ for some $C^{\prime \prime}>0$. From this we obtain the desired.
Let us now treat the case $G = SL_2 ({\mathbb R})$. Fix any $\theta \in {\rm types} (V_{\chi})$. The leading asymptotic of $K$-finite vectors are well-known, and can be computed from explicit expressions in terms of the hypergeometric function (see \cite[\S 6.5]{KlVi}). In the case $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} \neq 1$, denoting by $0 \neq s \in {\mathbb R}$ the number for which $\chi (\mathbf{a} (t)) = t^{is}$ for all $t \in {\mathbb R}^{\times}_{>0}$, we have $$ \leftarrowngle \mathbf{a} (e^x) h^{\chi}_{\theta} , h^{\chi}_{\theta} \rightarrowngle \sim e^{-x} \cdot (E_1 \cdot e^{-isx} + E_2 \cdot e^{isx} + o(1)) \quad\quad (x \to +\infty)$$ for some non-zero $E_1$ and $E_2$ and so $$ |\leftarrowngle \mathbf{a} (e^x) h^{\chi}_{\theta} , h^{\chi}_{\theta} \rightarrowngle|^2 \sim e^{-2x} \left( D + E_3 \cdot e^{-2isx} + E_4 \cdot e^{2isx} + o(1) \right) \quad\quad (x \to +\infty)$$ for some $D>0$, $E_3$ and $E_4$. From this we obtain the desired. In the case $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} = 1$, and so $\chi (\mathbf{a} (-1)) = - 1$, we have $$ \leftarrowngle \mathbf{a} (e^x) h_{\theta}^{\chi} , h_{\theta}^{\chi} \rightarrowngle \sim E \cdot e^{-x} \quad\quad (x \to +\infty) $$ for some non-zero $E$ and so $$| \leftarrowngle \mathbf{a} (e^x) h_{\theta}^{\chi} , h_{\theta}^{\chi} \rightarrowngle |^2 \sim D \cdot e^{-2x} \quad\quad (x \to +\infty) $$ for some $D>0$. From this we obtain the desired.
\end{proof}
\subsection{}
We further reduce Claim \ref{clm SL2 2b}.
\begin{claim}\leftarrowbel{clm SL2 3}
Fix $\chi$ satisfying (\ref{eq condition}). There exist $r_0 > 0$ and $C>0$ (depending on $\chi$) such that for all $\theta,\eta \in \textnormal{types} (V_{\chi})$ and all $a \in A^+ \smallsetminus A^+_{<r_0}$ we have \begin{equation}\leftarrowbel{eq thm SL2R 3} |\leftarrowngle a h^{\chi}_{\theta} , h^{\chi}_{\eta} \rightarrowngle|^2 \leq C \cdot \omega(a)^{-1} .\end{equation}
\end{claim}
\begin{proof}[Proof (of Claim \ref{clm SL2 2b} for $\chi$ given Claim \ref{clm SL2 3} for $\chi$).]
Let $f_1 , f_2 \in V_{\chi}$ and write $$ f_1 = \sum_{\theta \in \textnormal{types} (V_{\chi})} c_{\theta} \cdot h^{\chi}_{\theta}, \quad f_2 = \sum_{\theta \in \textnormal{types} (V_{\chi})} d_{\theta} \cdot h^{\chi}_{\theta}$$ with $c_{\theta} , d_{\theta} \in {\mathbb C}$. Using Fourier expansion of the function $(k_1 , k_2) \mapsto \leftarrowngle a k_1 f_1 , k_2 f_2 \rightarrowngle$ on $K \times K$ we have, for $a \in A^+ \smallsetminus A^+_{<r_0}$: $$ \ms{f_1}{f_2}{a} = \int_{K\times K} \left| \leftarrowngle a k_1 f_1 , k_2 f_2 \rightarrowngle \right|^2 \cdot dk_1 dk_2 = $$ $$ = \sum_{\theta,\eta \in \textnormal{types} (V_{\chi})} |c_{\theta}|^2 \cdot |d_{\eta}|^2 \cdot \left| \leftarrowngle a h^{\chi}_{\theta} , h^{\chi}_{\eta} \rightarrowngle \right|^2 \leq C \cdot \omega (a)^{-1} \cdot ||f_1||^2 \cdot ||f_2||^2.$$
\end{proof}
\subsection{}
Let us now establish Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$:
\begin{proof}[Proof (of Claim \ref{clm SL2 3} in the case $G = SL_2 ({\mathbb R})$).]
In the case $\chi |_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} \neq 1$, this is the contents of \cite[Theorem 2.1]{BrCoNiTa} (which contains a stronger claim, incorporating $\chi$ into the inequality). Let us therefore assume $\chi|_{\mathbf{a} ({\mathbb R}^{\times}_{>0})} = 1$ and so $\chi (\mathbf{a} (-1)) = -1$. For $n \in {\mathbb Z}$, let us denote by $\theta_n$ the character of $K$ given by $\mtrx{c}{-s}{s}{c} \mapsto (c+is)^n$. We want to see that $$ \cosh (x) \left| \leftarrowngle \mathbf{a} (e^x) h_{\theta_n}^{\chi} , h_{\theta_m}^{\chi} \rightarrowngle \right|$$ is bounded as we vary $x \in [0,+\infty)$ and $m,n \in 1 + 2 {\mathbb Z}$. We have $V_{\chi} = V_{\chi}^- \textnormal{op}lus V_{\chi}^+$, where $V_{\chi}^-$ and $V_{\chi}^+$ are irreducible unitary $G$-representations, ${\rm types} (V_{\chi}^-) = \{ \theta_n \ : \ n \in -1 - 2 {\mathbb Z}_{\ge 0}\}$ and ${\rm types} (V_{\chi}^+) = \{ \theta_n \ : \ n \in 1 + 2 {\mathbb Z}_{\ge 0}\}$. Since the matrix coefficient in question vanishes when $m \in -1 - 2 {\mathbb Z}_{\ge 0}$ and $n \in 1 + 2 {\mathbb Z}_{\ge 0}$ or vice versa, and since the matrix coefficient in question does not change when we replace $(n ,m)$ by $(-n , -m)$, we can assume $m,n \in 1 + 2 {\mathbb Z}_{ \ge 0}$. Furthermore, by conjugating by $\mtrx{0}{1}{-1}{0}$ it is straight-forward to see that we can assume that $m \ge n$. We denote $k := \tfrac{m-n}{2}$. Let us use the well-known concrete expression of matrix coefficients in terms of the hypergeometric function (see \cite[\S 6.5]{KlVi}): $$ \cosh (x) \cdot \leftarrowngle \mathbf{a} (e^x) h_{\theta_n}^{\chi} , h_{\theta_m}^{\chi} \rightarrowngle = \tanh (x)^k \cdot \frac{(1+\frac{n-1}{2})_k}{k!} \cdot {}_2 F_1 \left( \frac{n-1}{2}+k+1 , \ -\frac{n-1}{2} , \ k+1 , \ \tanh (x)^2 \right).$$ We want to show that this expression is bounded as we vary $x \in [0,+\infty)$, $n \in 1+2 {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$. Performing a change of variables $\frac{1-t}{2} := \tanh (x)^2$, denoting $r := \tfrac{n-1}{2}$ and interpreting in terms of Jacobi polynomials, we can rewrite this last expression as: $$Q^k_r (t) := (\tfrac{1-t}{2})^{k/2} \cdot P^{(k,0)}_{r} (t).$$ We want to see that $Q^k_r (t)$ is bounded as we vary $t \in [-1,1]$, $r \in {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$. But, it is known that under a suitable interpretation of the variable $t$, $Q^k_r (t)$ is equal to a matrix coefficient of unit vectors in an irreducible unitary representation of ${\rm SU} (2)$, see, for example, \cite[\S 6.3]{KlVi}. Therefore, we have $ |Q^k_r (t)| \leq 1$ for all $t \in [-1,1]$, $r \in {\mathbb Z}_{\ge 0}$ and $k \in {\mathbb Z}_{\ge 0}$, as desired. See also \cite[Equation (20)]{HaSc} for a direct proof of this last inequality.
\end{proof}
\subsection{}
Finally, we want to establish Claim \ref{clm SL2 3} in the case $G = PGL_2 (\Omega)$. We will use the following proposition:
\begin{proposition}\leftarrowbel{prop padic estimate}
There exists $C>0$ such that the following holds. Let $\psi_1$ and $\psi_2$ be unitary characters of $\varpi {\mathcal O}$. Let $\alpha_1 , \alpha_2 \in {\mathcal O}^{\times} x + x ^2\Omega [[x ]] \subset \Omega [[x]]$ be power series. Let $\chi$ be a non-trivial unitary character of $\Omega^{\times}$. Denote by $c(\chi)$ the number $0$ if $\chi|_{{\mathcal O}^{\times}} = 1$ and otherwise the smallest number $c \in {\mathbb Z}_{\ge 1}$ for which $\chi|_{1+\varpi^c {\mathcal O}} = 1$. Also, denote by $d(\chi)$ the number $1/|1-\chi (\varpi)|$ if $c(\chi) = 0$ and the number $0$ if $c(\chi) \neq 0$. Let $0 < m_1 \leq m_2 \leq n$ be integers. Then $$ \left| \int_{\varpi^{m_1} {\mathcal O} \smallsetminus \varpi^{m_2} {\mathcal O}} \psi_1 (\alpha_1 (x)) \psi_2 (\alpha_2 (\varpi^n x^{-1})) \chi (x) \cdot \frac{dx}{|x|} \right| \leq C(c(\chi)+d(\chi) + 1).$$
\end{proposition}
\begin{proof}[Proof (of Claim \ref{clm SL2 3} in the case $G = PGL_2 (\Omega)$ given Proposition \ref{prop padic estimate}).]
Let us calculate more concretely the inner product appearing in Claim \ref{clm SL2 3}. We can normalize the inner product on $V_{\chi}$ so that for $f_1, f_2 \in V_{\chi}$ we have $$ \leftarrowngle f_1 , f_2 \rightarrowngle = \int_{\Omega} f_1 \mtrx{1}{0}{x}{1} \overline{f_2 \mtrx{1}{0}{x}{1}} \cdot dx.$$ We then calculate $$ \left\leftarrowngle \mtrx{t}{0}{0}{1} h^{\chi}_{\theta} , h^{\chi}_{\mu} \right\rightarrowngle = |t|^{-1/2} \chi (t) \int_{\Omega} h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} \overline{h^{\chi}_{\mu} \mtrx{1}{0}{t^{-1} x}{1}} \cdot dx $$ and thus we want to see that $$ I_{\theta , \mu} (t) := \int_{\Omega} h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} \overline{h^{\chi}_{\mu} \mtrx{1}{0}{t^{-1} x}{1}} \cdot dx $$ is bounded independently of $\theta , \mu \in \hat{K}$ and $t \in F$ satisfying $|t| \ge 1$. In general, for $\theta \in \hat{K}$ and $x \in \Omega$, we have $$ h^{\chi}_{\theta} \mtrx{1}{0}{x}{1} = \frac{\chi^{-1} (1 - \zeta x^2)}{|1 - \zeta x^2|^{1/2}} \theta \mtrx{1}{\zeta x}{x}{1}.$$ And so we obtain \begin{equation}\leftarrowbel{eq for I} I_{\theta , \mu} (t) = \int_{\Omega} \frac{\chi^{-1} (1 - \zeta x^2)}{|1 - \zeta x^2|^{1/2}} \frac{\chi (1 - \zeta t^{-2} x^2)}{|1 - \zeta t^{-2} x^2|^{1/2}} \theta \mtrx{1}{\zeta x}{x}{1} \mu^{-1} \mtrx{1}{\zeta t^{-1} x}{t^{-1} x}{1} \cdot dx. \end{equation} If for $D \subset F$ we denote by $I^D_{\theta , \mu} (t)$ the same expression as that for $I_{\theta , \mu} (t)$ in (\ref{eq for I}) but where integration is performed over $D$, we see $$ | I^{\varpi {\mathcal O}}_{\theta , \mu} (t) | \leq \int_{\varpi {\mathcal O}} dx = 1/q$$ and thus this is bounded. Furthermore, $$ |I^{\Omega \smallsetminus t {\mathcal O}}_{\theta , \mu} (t)| \leq \int_{\Omega \smallsetminus t {\mathcal O}} \frac{dx}{|x| \cdot |t^{-1} x|} = 1/q $$ and thus this is bounded. Finally, for any specific $- \log_q |t| \leq m \leq 0$, we have $$ |I^{\varpi^{m} {\mathcal O}^{\times}}_{\theta , \mu} (t)| \leq \int_{\varpi^{m} {\mathcal O}^{\times}} \frac{dx}{|x|} =1-1/q.$$ Therefore, denoting by $k \in {\mathbb Z}_{\ge 1}$ a number such that $\chi|_{1+\varpi^{2k} {\mathcal O}} = 1$, it is enough to bound $I^{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}}_{\theta , \mu} (t)$. We have $$I^{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}}_{\theta , \mu} (t) = $$ $$ = \chi^{-1} (- \zeta) \theta \mtrx{0}{\zeta}{1}{0} \int_{\varpi^k t {\mathcal O} \smallsetminus \varpi^{-k} {\mathcal O}} \chi^{-2} (x) \theta \mtrx{1}{x^{-1}}{\zeta^{-1} x^{-1} }{1} \mu^{-1} \mtrx{1}{\zeta t^{-1} x}{t^{-1} x}{1} \cdot \frac{dx}{|x|}.$$ Let us denote by $K^{\prime} \subset K$ the subgroup consisting of $\mtrx{x}{\zeta y}{y}{x}$ for which $|y| \leq |x| \cdot |p|$. We have an isomorphism of topological groups $$ e : p {\mathcal O} \to K^{\prime}$$ given by $$ e (y) := \exp \mtrx{0}{\zeta y}{y}{0}.$$ Let us denote by $\alpha : p {\mathcal O} \to p {\mathcal O}$ the map given by $\alpha (y) := e^{-1} \mtrx{1}{\zeta y}{y}{1}$. We have a power series expansion $\alpha (y) = y + \zeta y^3 / 3 + \zeta^2 y^5 / 5 + \ldots$. Let us now denote by $\widetilde{\theta}$ the unitary character of $p {\mathcal O}$ satisfying $\theta|_{K'} \circ e = \widetilde{\theta}$ and by $\widetilde{\mu}$ the unitary character of $p {\mathcal O}$ satisfying $\mu|_{K'} \circ e = \widetilde{\mu}$. Returning to our integral, we can take $k$ big enough so that $\varpi^k \in p {\mathcal O}$. Substituting $x^{-1}$ in place of $x$ in the integral we have, we see that we need to show that $$ \int_{\varpi^{k+1} {\mathcal O} \smallsetminus \varpi^{-k+1} t^{-1} {\mathcal O}} \widetilde{\theta} (\alpha (\zeta^{-1} x)) \widetilde{\mu}^{-1} (\alpha (t^{-1} x^{-1})) \cdot \chi^2 (x) \cdot \frac{dx}{|x|}$$ is bounded independently of $\widetilde{\theta} , \widetilde{\mu} \in \widehat{p {\mathcal O}}$ and $t \in F$ satisfying $|t| \ge 1$. This is implied by Proposition \ref{prop padic estimate}.
\end{proof}
\begin{remark}
We see from the proof that we have a more precise version of Claim \ref{clm SL2 3}: There exists $C>0$ such that for $\chi$ satisfying (\ref{eq condition}), $\theta , \mu \in {\rm types} (V_{\chi})$ and $t \in \Omega^{\times}$ satisfying $|t| \ge 1$, we have $$ \left| \left\leftarrowngle \mtrx{t}{0}{0}{1} h^{\chi}_{\theta} , h^{\chi}_{\mu} \right\rightarrowngle \right| \leq C \cdot |t|^{-1/2} \cdot (c(\chi) +d(\chi)+1).$$ Here $c(\chi)$ and $d(\chi)$ are as in the formulation of Proposition \ref{prop padic estimate} (when we identify $\chi$ with a character of $\Omega^{\times}$ via the isomorphism ${\boldsymbol{a}} : \Omega^{\times} \to A$).
\end{remark}
Let us prove Proposition \ref{prop padic estimate}.
\begin{proof}[Proof (of Proposition \ref{prop padic estimate}).]
We can assume that the $1$-th coefficients of $\alpha_1$ and $\alpha_2$ are equal to $1$. Let us fix a unitary character $\psi$ of $\Omega$ satisfying $\psi|_{{\mathcal O}} = 1$ and $\psi|_{\varpi^{-1} {\mathcal O}} \neq 1$. For $i \in \{ 1 , 2 \}$, let $a_i \in \Omega$ be such that $\psi (a_i x) = \psi_i (x)$ for all $x \in \varpi {\mathcal O}$ ($a_i$ are defined up to addition of elements in $\varpi^{-1} {\mathcal O}$, so in particular we can assume that $a_i \neq 0$). Given $0 < m < n$ we define $$ J^m := \int_{\varpi^m {\mathcal O}^{\times}} \psi \left( a_1 \alpha_1 (x) + a_2 \alpha_2 (\varpi^n x^{-1}) \right) \cdot \chi (x) \cdot \frac{dx}{|x|} = $$ $$ = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times} } \psi \left( a_1 \alpha_1 (\varpi^m x) + a_2 \alpha_2 (\varpi^{n-m} x^{-1}) \right) \cdot \chi (x) \cdot dx,$$ so that the integral in question equals $\sum_{m_1 \leq m < m_2} J^m$. Let us abbreviate $b := (a_2 \varpi^{n-m}) / (a_1 \varpi^m)$. As we vary $m$, let us divide into cases.
\begin{enumerate}
\item Assume that $ |\varpi^m | < q^{-c (\chi)}$ and that $|b| < q^{-c(\chi)}$. Set $$\beta (x) := \varpi^{-m} \alpha_1 (\varpi^m x) + b \cdot \varpi^{m-n} \alpha_2 (\varpi^{n-m} x^{-1}).$$ Then $\beta$ gives a well-defined invertible analytic map ${\mathcal O}^{\times} \to {\mathcal O}^{\times}$, whose derivative is everywhere a unit. Moreover, if ${c(\chi)} > 0$, we have $\beta^{-1} (x_0 (1 + \varpi^{c(\chi)} {\mathcal O})) = x_0 (1 + \varpi^{c(\chi)} {\mathcal O})$ for any $x_0 \in {\mathcal O}^{\times}$. Hence $$ J^m = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times}} \psi (a_1 \varpi^m \beta (x)) \cdot \chi (x) \cdot dx = $$ $$ = \chi (\varpi)^{m} \int_{ {\mathcal O}^{\times} } \psi (a_1 \varpi^m x) \cdot \chi (x) \cdot dx.$$ Therefore:
\begin{enumerate}
\item Suppose that $|a_1 \varpi^m| \leq 1$. Then $J^m = \chi (\varpi)^{m} (1-1/q)$ if $\chi$ is unramified and $J^m = 0$ if $\chi$ is ramified.
\item Suppose that $|a_1 \varpi^m \varpi^{c(\chi)} | > q^{-1}$. Then $J^m = 0$ if $\chi$ is unramified. If $\chi$ is ramified, we write $$ J^m = \chi (\varpi)^{m} \sum_{x_0 \in {\mathcal O}^{\times} / (1+\varpi^{c(\chi)} {\mathcal O})} \psi (a_1 \varpi^m x_0) \chi (x_0) \int_{\varpi^{c(\chi)} {\mathcal O}} \psi (a_1 \varpi^m x) \cdot dx$$ and each integral here is equal to $0$, so that also in that case we obtain $J^m = 0$.
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)+1$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case.
\end{enumerate}
\item Assume that $|\varpi^{n-m}| < q^{-c(\chi)}$ and $|b^{-1}| < q^{-c ( \chi)}$. This case is dealt with analogously to the previous one; one denotes $$ \beta (x) := \omega^{m-n} \alpha_2 (\varpi^{n-m} x^{-1}) + b^{-1} \varpi^{-m} \alpha_1 (\varpi^m x)$$ and gets $$ J^m = \chi (\varpi)^m \int_{{\mathcal O}^{\times}} \psi (a_2 \varpi^{n-m} x) \cdot \chi (x) \cdot dx.$$ And thus:
\begin{enumerate}
\item Suppose that $|a_2 \varpi^{n-m}| \leq 1$. Then $J^m = \chi (\varpi)^{m} (1-1/q)$ if $\chi$ is unramified and $J^m = 0$ if $\chi$ is ramified.
\item Suppose that $|a_2 \varpi^{n-m} \varpi^{c(\chi)} | > q^{-1}$. Then $J^m = 0$ if $\chi$ is unramified. If $\chi$ is ramified, we write $$ J^m = \chi (\varpi)^{m} \sum_{x_0 \in {\mathcal O}^{\times} / (1+\varpi^{c(\chi)} {\mathcal O})} \psi (a_2 \varpi^{n-m} x_0) \chi (x_0) \int_{\varpi^{c(\chi)} {\mathcal O}} \psi (a_2 \varpi^{n-m} x) \cdot dx$$ and each integral here is equal to $0$, so that also in that case we obtain $J^m = 0$.
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)+1$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case.
\end{enumerate}
\item The case when neither of these two cases is satisfied corresponds to only finitely many values of $m$, whose number is linearly bounded in terms of $c(\chi)$, and so we can be content with the crude estimate $|J^m| \leq 1$ in this case.
\end{enumerate}
As our integral in question is equal to $\sum_{m_1 \leq m < m_2} J^m$, it is straight-forward that the findings above give the boundedness as desired.
\end{proof}
\appendix
\section{Auxiliary claims regarding polynomial growth of exponential integrals and sums}
\subsection{Some notation}
We denote $[n] := \{1 , 2, \ldots , n \}$. We denote $${\mathbb C}_{\leq 0} := \{ z \in {\mathbb C} \ | \ {\rm Re} (z) \leq 0\}, \quad D := \{ z \in {\mathbb C} \ | \ |z| \leq 1\}.$$ Given $x = (x_1 , \ldots , x_n) \in {\mathbb R}_{\ge 0}^n$ and $m = (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}^n$, we write $x^m := x_1^{m_1} \ldots x_n^{m_n}$. Given $\leftarrowmbda \in {\mathbb C}_{\leq 0}^n$ we denote $$J_{\leftarrowmbda} := \{ 1 \leq j \leq n \ | \ {\rm Re} (\leftarrowmbda_j) = 0\}.$$ Given $(\leftarrowmbda , m) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$, we denote $d(\leftarrowmbda , m) := \sum_{j \in J_{\leftarrowmbda}} (1+m_j)$. Given $J \subset [n]$ and some set $X$, let us denote by ${\rm res}_{J} : X^n \to X^{J}$ the natural restriction and by ${\rm ext}^{J} : X^{J} \to X^{n}$ the natural extension by zero.
We fix a finite set ${\mathcal I} \subset {\mathbb R}_{\ge 0}^n$ with the property that given $j \in [n]$ there exists $v \in {\mathcal I}$ such that $\leftarrowngle v , e_j \rightarrowngle \neq 0$, where $e_j$ the $j$-th standard basis vector. We denote $$ P_{< r} := \{ x \in {\mathbb R}_{\ge 0}^n \ | \ \leftarrowngle v , x \rightarrowngle < r \ \forall v \in {\mathcal I} \}.$$ Given $J \subset [n]$, we denote by $P_J \subset {\mathbb R}_{\ge 0}^J$ the convex pre-compact subset $\{ y \in {\mathbb R}_{\ge 0}^J \ | \ {\rm ext}^{J} (y) \in P_{<1}\}$.
In \S\ref{ssec growth integral} we will also use the following notations. We consider a compact space $B$ equipped with a nowhere vanishing Radon measure $db$. Let us say that a function $\phi : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ is \textbf{nice} if it is expressible as $$ B \times {\mathbb R}_{\ge 0}^n \xrightarrow{{\rm id}_B \times {\rm ei}} B \times D^n \xrightarrow{\phi^{\circ}} {\mathbb C}$$ where ${\rm ei}(x_1 , \ldots , x_n) := (e^{-x_1} , \ldots , e^{-x_n})$ and $\phi^{\circ}$ is continuous and holomorphic in the second variable (in the sense that when we fix the variable in $B$ it is the restriction of a holomorphic function on a neighbourhood of $D^n$). Given $J \subset [n]$ we denote by ${\rm res}_{J} \phi : B \times {\mathbb R}_{\ge 0}^J \to {\mathbb C}$ the function given by ${\rm res}_{J} \phi (b , y) := \phi^{\circ} (b , {\rm ext}^{J} ({\rm ei} (y)))$. We also write $\phi (b , +\infty)$ for $\phi^{\circ} (b , 0)$ etc.
\subsection{Growth - the case of summation over a lattice}
\begin{lemma}\leftarrowbel{lem appb lat 1}
Let $\leftarrowmbda := (\leftarrowmbda_1 , \ldots , \leftarrowmbda_n) \in {\mathbb C}_{\leq 0}^n$ and $m := (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}$. Let $K \subset {\mathbb R}_{\ge 0}^n$ be a compact subset. Assume that ${\rm Re} (\leftarrowmbda) = 0$ and $\leftarrowmbda \notin (2\pi i) {\mathbb Z}^n$. We have $$ \sup_{Q \subset K} \left| \frac{1}{r^n} \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r \leftarrowngle \leftarrowmbda , x \rightarrowngle} \right| = O( r^{-1} )$$ as $r \to +\infty$, where $Q$ denote convex subsets.
\end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $\leftarrowmbda_1 \notin 2\pi i {\mathbb Z}$. Let us write $x = (x_1 , x')$ where $x' = (x_2 , \ldots , x_n)$ and analogously write $m'$ et cetera. Given a convex subset $Q \subset K$ and $x' \in {\mathbb R}_{\ge 0}^{n-1}$ let us denote by $Q^{x'} \subset {\mathbb R}_{\ge 0}$ the subset consisting of $x_1$ for which $(x_1 , x') \in Q$ (it is an interval). Let us enlarge $K$ for convenience, writing it in the form $K = K_1 \times K'$ where $K_1 \subset {\mathbb R}_{\ge 0}$ is a closed interval and $K' \subset {\mathbb R}_{\ge 0}^{n-1}$ is the product of closed intervals.
We have $$ \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r\leftarrowngle \leftarrowmbda , x \rightarrowngle} = \sum_{ x' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{n-1} \cap K' } (x')^{m'} e^{r \leftarrowngle \leftarrowmbda' , x' \rightarrowngle }\left( \sum_{ x_1 \in \frac{1}{r} {\mathbb Z}_{\ge 0} \cap Q^{x'}} x_1^{m_1} e^{r \leftarrowmbda_1 x_1} \right).$$ We have $Q^{x'} \subset K'$ and it is elementary to see that $$ \sup_{R \subset K'} \left| \sum_{x_1 \in \frac{1}{r} {\mathbb Z}_{\ge 0} \cap R} x_1^{m_1} e^{r \leftarrowmbda_1 x_1} \right| = O(1)$$ as $r \to +\infty$, where $R$ denote intervals. Therefore we obtain, for some $C>0$ (not depending on $Q$) and all $r \ge 1$: $$ \left| \frac{1}{r^n} \sum_{x \in \frac{1}{r} {\mathbb Z}_{\ge 0}^n \cap Q} x^m e^{r\leftarrowngle \leftarrowmbda , x \rightarrowngle} \right| \leq C \left( \frac{1}{r^{n-1}} \sum_{ x' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{n-1} \cap K' } (x')^{m'} \right) r^{-1}.$$ Since the expression in brackets is clearly bounded independently of $r$, we are done.
\end{proof}
\begin{lemma}\leftarrowbel{lem appb lat 2} Let $(\leftarrowmbda , m ) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$. Then the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d(\leftarrowmbda , m)}} \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} x^m e^{\leftarrowngle \leftarrowmbda , x \rightarrowngle} dx$$ exists, equal to $0$ if ${\rm res}_{J_{\leftarrowmbda}} (\leftarrowmbda) \notin 2\pi i \cdot {\mathbb Z}^{J_{\leftarrowmbda}}$ and otherwise equal to $$\left( \int_{P_{J_{\leftarrowmbda}}} y^{{\rm res}_{J_{\leftarrowmbda}} (m)} dy \right) \left( \sum_{z \in {\mathbb Z}_{\ge 0}^{J_{\leftarrowmbda}^c}} z^{{\rm res}_{J_{\leftarrowmbda}^c} (m)} e^{\leftarrowngle {\rm res}_{J_{\leftarrowmbda}^c} (\leftarrowmbda) , z \rightarrowngle } \right)$$ (the sum converging absolutely).
\end{lemma}
\begin{proof}
Let us abbreviate $J := J_{\leftarrowmbda}$. Let us denote $\leftarrowmbda' := {\rm res}_J (\leftarrowmbda)$ and $\leftarrowmbda'' := {\rm res}_{J^c} (\leftarrowmbda)$, and similarly for $m$. Given $x'' \in {\mathbb Z}_{\ge 0}^{J_c}$ let us denote by $P_{(<r)}^{x''} \subset {\mathbb R}_{\ge 0}^{J}$ the subset consisting of $y'$ for which ${\rm ext}^{J} (r y') + {\rm ext}^{J^c} (x'') \in P_{<r}$.
We have $$ \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} x^m e^{\leftarrowngle \leftarrowmbda , x \rightarrowngle} = r^{d(\leftarrowmbda , m) - |J|} \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap P^{x''}_{(<r)}} (y')^{m'} e^{r \leftarrowngle \leftarrowmbda' , y' \rightarrowngle} := \triangle.$$
Let us assume first that $\leftarrowmbda' \notin 2 \pi i \cdot {\mathbb Z}^{J}$. Then by Lemma \ref{lem appb lat 1} there exists $C>0$ such that for all convex subsets $Q \subset P_J$ and all $r \ge 1$ we have $$ \left| \frac{1}{r^{|J|}} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap Q} (y')^{m'} e^{r \leftarrowngle \leftarrowmbda' , y' \rightarrowngle} \right| \leq C \cdot r^{-1}.$$ Therefore $$ | \triangle | \leq C r^{d(\leftarrowmbda , m) - 1} \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle {\rm Re} (\leftarrowmbda'') , x'' \rightarrowngle}, $$ giving the desired.
Now we assume $\leftarrowmbda' \in 2 \pi i \cdot {\mathbb Z}^J$. It is not hard to see that $$ \lim_{r \to +\infty} \frac{1}{r^{|J|}} \sum_{y' \in \frac{1}{r} {\mathbb Z}_{\ge 0}^{J} \cap P^{x''}_{(<r)}} (y')^{m'} = \int_{P_J} (y')^{m'} dy'.$$ Hence we have (by dominated convergence) $$ \lim_{r \to +\infty} \frac{1}{r^{d(\leftarrowmbda,m)}} \triangle = \sum_{x'' \in {\mathbb Z}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle} \int_{P_J} (y')^{m'} dy' .$$
\end{proof}
\begin{claim}\leftarrowbel{clm appb lat}
Let $p \ge 1$, let $\{ ( \leftarrowmbda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}_{\leq 0}^{n} \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples and let $\{ c^{(\ell)} \}_{\ell \in [p]} \subset {\mathbb C} \smallsetminus \{ 0 \}$ be a collection of non-zero scalars. Denote $d := \max_{\ell \in [p]} d(2{\rm Re} (\leftarrowmbda^{(\ell)}) , 2 m^{(\ell)})$.
The limit $$ \lim_{r \to +\infty} \frac{1}{r^d} \sum_{x \in {\mathbb Z}_{\ge 0}^n \cap P_{<r}} \left| \sum_{\ell \in [p]} c^{(\ell)} x^{m^{(\ell)}} e^{\leftarrowngle \leftarrowmbda^{(\ell)} , x \rightarrowngle } \right|^2 $$ exists and is strictly positive.
\end{claim}
\begin{proof}
Let us break the integrand into a sum following $$\left| \sum_{\ell \in [p]} A_{\ell} \right|^2 = \sum_{\ell_1 , \ell_2 \in [p]} A_{\ell_1} \overline{A_{\ell_2}}.$$ Using Lemma \ref{lem appb lat 2} we see the that resulting limit breaks down as a sum, over $(\ell_1 ,\ell_2) \in [p]^2$, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the $(\ell_1 , \ell_2)$ place is zero unless $d(\leftarrowmbda^{(\ell_1)} , m^{(\ell_1)} ) = d$, $d(\leftarrowmbda^{(\ell_2)} , m^{(\ell_2)} ) = d$, $J_{\leftarrowmbda^{(\ell_1)}} = J_{\leftarrowmbda^{(\ell_2)}}$ and ${\rm res}_{J^{(\ell_1)}} (\leftarrowmbda^{(\ell_2)}) - {\rm res}_{J^{(\ell_1)}} (\leftarrowmbda^{(\ell_1)}) \in 2\pi i \cdot {\mathbb Z}^J$. We thus can reduce to the case when, for a given $J \subset [n]$, we have $J_{\leftarrowmbda^{(\ell)}} = J$ for all $\ell \in [p]$, we have $d(\leftarrowmbda^{(\ell)} , m^{(\ell)}) = d$ for all $\ell \in [p]$, and we have ${\rm res}_{J} (\leftarrowmbda^{(\ell_2)}) - {\rm res}_{J} (\leftarrowmbda^{(\ell_1)}) \in 2\pi i \cdot {\mathbb Z}^J$ for all $\ell_1 , \ell_2 \in [p]$. We then obtain, using Lemma \ref{lem appb lat 2}, that our overall limit equals $$ \sum_{z \in {\mathbb Z}_{\ge 0}^{J^c}} \int_{P_J} \left| \sum_{\ell \in [p]} c^{(\ell)} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z \rightarrowngle } \right|^2 dy.$$ It is therefore enough to check that $$ \sum_{\ell \in [p]} c^{(\ell)} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z \rightarrowngle }, $$ a function in $(z,y ) \in {\mathbb Z}_{\ge 0}^{J^c} \times P_J$, is not identically zero. By the local linear independence of powers of $y$, we can further assume that ${\rm res}_J (m^{(\ell)})$ is independent of $\ell \in [p]$, and want to check that $$\sum_{\ell \in [p]} c^{(\ell)} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z \rightarrowngle }, $$ a function in $z \in {\mathbb Z}_{\ge 0}^{J^c}$, is not identically zero. Notice that, by our assumptions, the elements in the collection $ \{ ( {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , {\rm res}_{J^c} (m^{(\ell)}) ) \}_{\ell \in [p]}$ are pairwise different. Thus the non-vanishing of our sum is clear (by linear algebra of generalized eigenvectors of shift operators on ${\mathbb Z}^{J^c}$).
\end{proof}
\subsection{Growth - the case of an integral}\leftarrowbel{ssec growth integral}
\begin{lemma}\leftarrowbel{lem appb 1}
Let $\leftarrowmbda := (\leftarrowmbda_1 , \ldots , \leftarrowmbda_n) \in {\mathbb C}_{\leq 0}^n$ and $m := (m_1 , \ldots , m_n) \in {\mathbb Z}_{\ge 0}$. Let $K \subset {\mathbb R}_{\ge 0}^n$ be a compact subset. Assume that ${\rm Re} (\leftarrowmbda) = 0$ and $\leftarrowmbda \neq 0$. We have $$ \sup_{Q \subset K} \left| \int_Q x^m e^{r \leftarrowngle \leftarrowmbda , x \rightarrowngle} dx \right| = O( r^{-1} )$$ as $r \to +\infty$, where $Q$ denote convex subsets.
\end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $\leftarrowmbda_1 \neq 0$. Let us write $x = (x_1 , x')$ where $x' = (x_2 , \ldots , x_n)$ and analogously write $m'$ etcetera. Given a convex subset $Q \subset K$ and $x' \in {\mathbb R}_{\ge 0}^{n-1}$ let us denote by $Q^{x'} \subset {\mathbb R}_{\ge 0}$ the subset consisting of $x_1$ for which $(x_1 , x') \in Q$ (it is an interval). Let us enlarge $K$ for convenience, writing it in the form $K = K_1 \times K'$ where $K_1 \subset {\mathbb R}_{\ge 0}$ is a closed interval and $K' \subset {\mathbb R}_{\ge 0}^{n-1}$ is the product of closed intervals.
Using Fubini's theorem $$ \int_Q x^m e^{r\leftarrowngle \leftarrowmbda , x \rightarrowngle} dx = \int_{ K' } (x')^{m'} e^{r \leftarrowngle \leftarrowmbda' , x' \rightarrowngle }\left( \int_{Q^{x'}} x_1^{m_1} e^{r \leftarrowmbda_1 x_1} dx_1 \right) dx'.$$ We have $Q^{x'} \subset K'$ and it is elementary to see that $$ \sup_{R \subset K'} \left| \int_{R} x_1^{m_1} e^{r \leftarrowmbda_1 x_1} dx_1 \right| = O(r^{-1})$$ as $r \to +\infty$, where $R$ denote intervals. Therefore we obtain, for some $C>0$ and all $r \ge 1$: $$ \left| \int_Q x^m e^{r\leftarrowngle \leftarrowmbda , x \rightarrowngle} dx \right| \leq C \left( \int_{K'} (x')^{m'} dx' \right) r^{-1},$$ as desired.
\end{proof}
\begin{lemma}\leftarrowbel{lem appb 2} Let $(\leftarrowmbda , m ) \in {\mathbb C}_{\leq 0}^n \times {\mathbb Z}_{\ge 0}^n$ and let $\phi : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ be a nice function. Then the limit $$ \lim_{r \to +\infty} \frac{1}{r^{d(\leftarrowmbda , m)}} \int_B \int_{P_{<r}} x^m e^{\leftarrowngle \leftarrowmbda , x \rightarrowngle} \phi (b,x)dxdb$$ exists, equal to $0$ if ${\rm res}_{J_{\leftarrowmbda}} \leftarrowmbda \neq 0$ and otherwise equal to $$\left( \int_{P_{J_{\leftarrowmbda}}} y^{{\rm res}_{J_{\leftarrowmbda}} (m)} dy \right) \left( \int_B \int_{{\mathbb R}_{\ge 0}^{J_{\leftarrowmbda}^c}} z^{{\rm res}_{J_{\leftarrowmbda}^c} (m)} e^{\leftarrowngle {\rm res}_{J_{\leftarrowmbda}^c} (\leftarrowmbda) , z\rightarrowngle} {\rm res}_{J_{\leftarrowmbda}^c} \phi (b , z) dz db \right)$$ (the double integral converging absolutely).
\end{lemma}
\begin{proof}
Let us re-order the variables, assuming that $J := J_{\leftarrowmbda} = [k]$. Let us write $x = (x',x'')$ where $x'$ consists of the first $k$ components and $x''$ consists of the last $k$ components. Let us write analogously $m' , \leftarrowmbda'$ etc.
First, let us notice that if $k \neq 0$, we can write $$ \phi (b , x ) = e^{-x_1} \phi_0 (b , x) + \phi_1 (b , x)$$ where $\phi_0 , \phi_1 : B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$ are nice functions and $\phi_1$ does not depend on $x_1$. Dealing with $ e^{-x_1} \phi_0 (b,x)$ instead of $\phi (b,x)$ makes us consider $\leftarrowmbda$ with smaller set $J_{\leftarrowmbda}$ and thus $(\leftarrowmbda , m)$ with a smaller $d(\leftarrowmbda , m)$ and from this, reasoning inductively, we see that we can assume that $\phi$ only depends on $(b,x'')$. Let us write $\phi'' := {\rm res}_{J^c} \phi$.
Let us perform a change of variables $y' := \frac{1}{r} x'$. Let $P_{(<r)} \subset {\mathbb R}_{\ge 0}^n$ denote the transform of $P_{<r}$ under this changes of variables (i.e. $(x',x'') \in P_{<r}$ if and only if $(y',x'') \in P_{(<r)}$). We obtain $$ \int_B \int_{P_{<r}} x^m e^{\leftarrowngle \leftarrowmbda , x \rightarrowngle} \phi (b , x) dx db = $$ $$ = r^{d} \int_B \int_{P_{(<r)}} (y')^{m'} e^{r \leftarrowngle \leftarrowmbda' , y' \rightarrowngle } (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle} \phi'' (b , x'') dy' dx'' db =: \triangle.$$
Given $x'' \in {\mathbb R}_{\ge 0}^{J^c}$, let us denote by $P_{(<r)}^{x''} \subset {\mathbb R}_{\ge 0}^{J}$ the set consisting of $y'$ for which $(y',x'') \in P_{(<r)}$. Notice that $P_{(<r_1)}^{x''} \subset P_{(<r_2)}^{x''}$ for $r_1 < r_2$ and $\cup_{r} P_{(<r)}^{x''} = P_{J}$. Using Fubini's theorem
$$ \triangle = r^{d} \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle } \phi'' (b , x'') \left( \int_{ P_{(<r)}^{x''}} (y')^{m'} e^{r \leftarrowngle \leftarrowmbda' , y' \rightarrowngle } dy' \right) dx'' db.$$
If $\leftarrowmbda' \neq 0$, by Lemma \ref{lem appb 1} there exists $C>0$ such that for all convex subsets $Q \subset P_J$ and all $r \ge 1$ we have $$ \left| \int_Q (y')^{m'} e^{r \leftarrowngle \leftarrowmbda' , y' \rightarrowngle } dy' \right| \leq C \cdot r^{-1}.$$ We have therefore $$ \left| \triangle \right| \leq C \cdot r^{d - 1} \cdot \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle {\rm Re} (\leftarrowmbda'' ) , x'' \rightarrowngle } | \phi'' (b , x'')| dx'' db $$ and thus indeed the desired limit is equal to $0$.
Now we assume $\leftarrowmbda' = 0$. Using Lebesgue's dominated convergence theorem we have $$ \lim_{r \to +\infty} \frac{1}{r^d} \triangle = \lim_{r \to +\infty} \int_B \int_{{\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle } \phi'' (b , x'') \left( \int_{ P_{(<r)}^{x''}} (y')^{m'} dy' \right) dx'' db = $$ $$ = \int_B \int_{ {\mathbb R}_{\ge 0}^{J^c}} (x'')^{m''} e^{\leftarrowngle \leftarrowmbda'' , x'' \rightarrowngle } \phi'' (b , x'') \left( \int_{P_J} (y')^{m'} dy' \right) dx'' db$$ as desired.
\end{proof}
\begin{claim}\leftarrowbel{clm appb}
Let $\{ ( \leftarrowmbda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}_{\leq 0}^{n} \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples. Let $\{ \phi^{(\ell)} \}_{\ell \in [p]}$ be a collection of nice functions $B \times {\mathbb R}_{\ge 0}^n \to {\mathbb C}$, such that for every $\ell \in [p]$ the function $b \mapsto \phi^{(\ell)} (b , +\infty)$ on $B$ is not identically zero. Denote $d := \max_{\ell \in [p]} d(2{\rm Re} (\leftarrowmbda^{(\ell)}) , 2 m^{(\ell)})$.
The limit $$ \lim_{r \to +\infty} \frac{1}{r^d} \int_B \int_{P_{<r}} \left| \sum_{\ell \in [p]} x^{m^{(\ell)}} e^{\leftarrowngle \leftarrowmbda^{(\ell)} , x \rightarrowngle } \phi^{(\ell)} (b , x)\right|^2 dx db$$ exists and is strictly positive.
\end{claim}
\begin{proof}
Let us break the integrand into a sum following $$\left| \sum_{\ell \in [p]} A_{\ell} \right|^2 = \sum_{\ell_1 , \ell_2 \in [p]} A_{\ell_1} \overline{A_{\ell_2}}.$$ Using Lemma \ref{lem appb 2} we see the that resulting limit breaks down as a sum, over $(\ell_1 ,\ell_2) \in [p]^2$, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the $(\ell_1 , \ell_2)$ place is zero unless $d(\leftarrowmbda^{(\ell_1)} , m^{(\ell_1)} ) = d$, $d(\leftarrowmbda^{(\ell_2)} , m^{(\ell_2)} ) = d$, $J_{\leftarrowmbda^{(\ell_1)}} = J_{\leftarrowmbda^{(\ell_2)}}$ and ${\rm res}_{J^{(\ell_1)}} (\leftarrowmbda^{(\ell_1)}) = {\rm res}_{J^{(\ell_1)}} (\leftarrowmbda^{(\ell_2)})$. We thus can reduce to the case when, for a given $J \subset [n]$, we have $J_{\leftarrowmbda^{(\ell)}} = J$ for all $\ell \in [p]$, we have $d(\leftarrowmbda^{(\ell)} , m^{(\ell)}) = d$ for all $\ell \in [p]$, and we have ${\rm res}_{J} (\leftarrowmbda^{(\ell_1)}) = {\rm res}_{J} (\leftarrowmbda^{(\ell_2)})$ for all $\ell_1 , \ell_2 \in [p]$. We then obtain, using Lemma \ref{lem appb 2}, that our overall limit equals $$ \int_B \int_{{\mathbb R}_{\ge 0}^{J^c}} \int_{P_J} \left| \sum_{\ell \in [p]} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z\rightarrowngle} {\rm res}_{J^c} \phi^{(\ell)} (b , z) \right|^2 dy dz db.$$ It is therefore enough to check that $$ \sum_{\ell \in [p]} y^{{\rm res}_J (m^{(\ell)})} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z\rightarrowngle} {\rm res}_{J^c} \phi^{(\ell)} (b , z),$$ a function in $(b,z,y ) \in B \times {\mathbb R}_{\ge 0}^{J^c} \times P_J$, is not identically zero. By the local linear independence of powers of $y$, we can further assume that ${\rm res}_J (m^{(\ell)})$ is independent of $\ell \in [p]$, and want to check that $$ \sum_{\ell \in [p]} z^{{\rm res}_{J^c} (m^{(\ell)})} e^{\leftarrowngle {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , z\rightarrowngle} {\rm res}_{J^c} \phi^{(\ell)} (b , z),$$ a function in $(b,z) \in B \times {\mathbb R}_{\ge 0}^{J^c}$, is not identically zero. Notice that, by our assumptions, the elements in the collection $ \{ ( {\rm res}_{J^c} (\leftarrowmbda^{(\ell)}) , {\rm res}_{J^c} (m^{(\ell)}) ) \}_{\ell \in [p]}$ are pairwise different and for every $\ell \in [p]$, the function $b \mapsto \phi^{(\ell)} (b , {\rm ext}^{J^c} (+\infty))$ on $B$ is not identically zero. Considering the partial order on ${\mathbb C}^{J^c}$ given by $\mu_1 \leq \mu_2$ if $\mu_2 - \mu_1 \in {\mathbb Z}_{\ge 0}^{J^c}$, we can pick $\ell \in [p]$ for which ${\rm res}_{J^c} (\leftarrowmbda^{(\ell)})$ is maximal among the $\{ {\rm res}_{J^c} (\leftarrowmbda^{(\ell')}) \}_{\ell' \in [p]}$. We can then pick $b \in B$ such that $\phi^{(\ell)} (b , {\rm ext}^{J^c} (+\infty)) \neq 0$. We then boil down to Lemma \ref{lem appb 4} that follows.
\end{proof}
In the end of the proof of Claim \ref{clm appb} we have used the following:
\begin{lemma}\leftarrowbel{lem appb 4}
Let $\{ (\leftarrowmbda^{(\ell)} , m^{(\ell)}) \}_{\ell \in [p]} \subset {\mathbb C}^n \times {\mathbb Z}_{\ge 0}^n$ be a collection of pairwise different couples. Let $\{ \phi^{(\ell)}\}_{\ell \in [p]}$ be a collection of nice functions ${\mathbb R}_{\ge 0}^n \to {\mathbb C}$ (so here $B = \{ 1 \}$). Suppose that $\phi^{(\ell)} (+\infty) \neq 0$ for some $\ell \in [p]$ for which $\leftarrowmbda^{(\ell)}$ is maximal among the $\{ \leftarrowmbda^{(\ell')} \}_{\ell' \in [p]}$ with respect to the partial order $\leftarrowmbda_1 \leq \leftarrowmbda_2$ if $\leftarrowmbda_2 - \leftarrowmbda_1 \in {\mathbb Z}_{\ge 0}^n$. Then the function $$ x \mapsto \sum_{\ell \in [p]} x^{m^{(\ell)}} e^{\leftarrowngle \leftarrowmbda^{(\ell)} , x \rightarrowngle} \phi^{(\ell)} (x)$$ on ${\mathbb R}_{\ge 0}^n$ is not identically zero.
\end{lemma}
\begin{proof}
We omit the proof - one develops the $\phi^{(\ell)}$ into power series in $e^{-x_1} , \ldots , e^{-x_n}$ and uses separation by generalized eigenvalues of the partial differentiation operators $\partial_{x_1} , \ldots , \partial_{x_n}$.
\end{proof}
\end{document} |
\begin{document}
\blacksquaref
\title[Calabi-Yau objects in triangulated categories]{Calabi-Yau objects in triangulated categories}
\author[C. Cibils, P. Zhang] {Claude Cibils$^a$ and Pu Zhang$^{b, *}$}
\thanks{The second named author is supported by the CNRS of France, the NSF of
China and of Shanghai City (Grant No. 10301033 and ZR0614049).}
\thanks{$^*$ The corresponding author}
\keywords{Serre functor, Calabi-Yau object, Auslander-Reiten
triangle, stable module category, self-injective Nakayama algebra}
\maketitle
\begin{center}
$^A$D\'epartement de Math\'ematiques, \ \ Universit\'e de Montpellier 2\\
F-34095, Montpellier Cedex 5, France\ \ \
Claude.Cibils$
\blacksquareymbol{64}$math.univ-montp2.fr\\
$^B$Department of Mathematics, \ \ Shanghai Jiao Tong University\\
Shanghai 200240, P. R. China\ \ \ \ pzhang$
\blacksquareymbol{64}$sjtu.edu.cn
\end{center}
\begin{abstract} We introduce the Calabi-Yau (CY) objects in a
Hom-finite Krull-Schmidt triangulated $k$-category, and notice
that the structure of the minimal, consequently all the CY
objects, can be described. The relation between indecomposable CY
objects and Auslander-Reiten triangles is provided. Finally we
classify all the CY modules of self-injective Nakayama algebras,
determining this way the self-injective Nakayama algebras
admitting indecomposable CY modules. In particular, this result
recovers the algebras whose stable categories are Calabi-Yau,
which have been obtained in [BS].
\end{abstract}
\vskip10pt
\blacksquareection {\bf Introduction}
Calabi-Yau (CY) categories have been introduced by Kontsevich
[Ko]. They provide a new insight and a wide framework for topics
as in mathematical physics ([Co]), non-commutative geometry
([B], [Gin1], [Gin2]), and representation theory of Artin algebras
([BS], [ES], [IR], [Ke], [KR1], [KR2]).
Triangulated categories with Serre dualities ([BK], [RV]) and CY
categories have important global naturality. On the other hand,
even in non CY categories, inspired by [Ko], one can introduce CY
objects. It turns out that they arise naturally in non CY
categories and enjoy ``local naturality'' and interesting
properties (Prop. 4.4, Theorems 3.2, 4.2, 5.5 and 6.1).
\vskip10pt
The first aim of this paper is to study the properties of such
objects in a Hom-finite Krull-Schmidt triangulated $k$-category
with Serre functor $F$. We give the relation between
indecomposable CY objects and the Auslander-Reiten triangles ($\S
3$), and describe all the $d$-th CY objects via the minimal ones,
which are exactly the direct sum of all the objects in finite
$\langle [-d]\circ F\rangle$-orbits of
$\operatorname{Ind}(\mathcal A)$ ($\S 4$). We classify all the
$d$-th CY modules of self-injective Nakayama algebras for any
integer $d$ ($\S 5$). Finally, we determine all the self-injective
Nakayama algebras which admit indecomposable CY modules. In
particular, this recovers the algebras whose stable categories are
Calabi-Yau ($\S 6$), included in the work of Bialkowski and
Skowro\'nski [BS]. Note that the CY modules are invariant under
stable equivalences between self-injective algebras, with a very
few exceptions (Prop.3.1). Consequently our results on
self-injective Nakayama algebras extend to the one on the
wreath-like algebras ([GR]), which contains the Brauer tree
algebras ([J]).
This also raises an immediate question. Let $\mathcal A$ be a
Hom-finite Krull-Schmidt triangulated $k$-category with a Serre
functor. If all objects are $d$-th CY with the same $d$, whether
$\mathcal A$ is a Calabi-Yau category?
\vskip10pt
\blacksquareection {\bf Backgrounds and Preliminaries}
\blacksquareubsection{} Let $k$ be a field and $\mathcal{A}$ a Hom-finite
$k$-category. Recall from Bondal and Kapranov [BK]
that a $k$-linear functor $F \colon
\mathcal{A} \to \mathcal{A}$ \ is {\em a right Serre functor} if
there exist $k$-isomorphisms $$\eta_{A, B}: \ \
\operatorname{Hom}_{\mathcal{A}}(A, B) \longrightarrow D
\operatorname{Hom}_{\mathcal{A}}(B, FA), \ \ \forall \ \ A, \ B\in
\mathcal{A},$$ which are natural both in $A$ and $B$, where $D =
\operatorname{Hom}_{k}(-, k)$. Such an $F$ is unique up to a natural isomorphism, and fully-faithful;
if it is an equivalence, then a quasi-inverse $F^{-1}$ is a left Serre
functor; in this case we call $F$ a Serre functor. Note that
$\mathcal{A}$ has a Serre functor if and only if it has both right
and left Serre functor. See Reiten and Van den Bergh [RV].
\vskip10pt
For triangulated categories we refer to [Har], [V], and [N]. Let
$\mathcal{A}$ be a Hom-finite triangulated $k$-category. Following
Happel [Hap1], {\em an Auslander-Reiten triangle} $X
\blacksquaretackrel{f}
{\longrightarrow} Y
\blacksquaretackrel{g} {\longrightarrow} Z
\blacksquaretackrel{h}
{\longrightarrow} X[1]$ of $\mathcal{A}$ is a distinguished
triangle satisfying:
(AR1) \ $X$ and $Z$ are indecomposable;
(AR2) \ $h\ne 0$;
(AR3) \ If $t: Z'\longrightarrow Z$ is not a retraction, then
there exists $t': Z'\longrightarrow Y$ such that $t = gt'$.
\vskip10pt
Note that (AR3) is equivalent to
(AR4) \ If $Z'$ is indecomposable and $t: Z'\longrightarrow Z$ is
a non-isomorphism, then $ht = 0$.
Under (AR1) and (AR2), (AR3) is equivalent to
(AR3') \ If $s: X\longrightarrow X'$ is not a section, then there exists
$s': Y\longrightarrow X'$ such that $s = s'f$.
Also, (AR3') is equivalent to
(AR4') \ If $X'$ is indecomposable and $s: X\longrightarrow X'$ is
a non-isomorphism, then $s\circ h[-1] = 0$.
\vskip10pt
In an Auslander-Reiten triangle $X {\longrightarrow} Y
{\longrightarrow} Z {\longrightarrow} X[1]$, the object $X$ is
uniquely determined by $Z$. Write $X = \tau_\mathcal A Z$. In
general $\tau_\mathcal A$ is {\em not} a functor. By definition
$\mathcal A$ has right Auslander-Reiten triangles if
there exists an
Auslander-Reiten triangle $X {\longrightarrow} Y {\longrightarrow}
Z {\longrightarrow} X[1]$ for any indecomposable $Z$; and $\mathcal A$ has
Auslander-Reiten triangles if $\mathcal{A}$ has right and
left Auslander-Reiten triangles. We refer to [Hap1], [XZ] and [A] for the
Auslander-Reiten quiver of a triangulated category.
\vskip10pt
A Hom-finite $k$-category is {\em Krull-Schmidt} if the
endomorphism algebra of any indecomposable is local. In this case
any object is uniquely decomposed into a direct sum of
indecomposables, up to isomorphisms and up to the order of
indecomposable direct summands (Ringel [R], p.52).
Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category. Theorem I.2.4 in [RV] says that $\mathcal{A}$ has a
right Serre functor $F$ if and if $\mathcal{A}$ has right
Auslander-Reiten triangles. In this case, $F$ coincides with
$[1]\circ \tau_\mathcal A$ on objects, up to isomorphisms.
\vskip10pt
\blacksquareubsection{}Let $\mathcal{A}$ be a Hom-finite
triangulated $k$-category with Serre functor $F$. Denote by $[1]$
the shift functor of $\mathcal A$. Following Kontsevich [Ko],
$\mathcal{A}$ is {\em a Calabi-Yau category} if there is a natural
isomorphism $F \cong [d]$ of functors for some $d\in\Bbb Z$.
Denote by $o([1])$ the order of $[1]$. If $o([1]) = \infty$ then
the integer $d$ above is unique, and is called {\em the CY dimension} of
$\mathcal{A}$; if $o([1])$ is finite then we call the minimal non-negative integer $d$
such that $F\cong [d]$ {\em the CY dimension} of
$\mathcal{A}$. Denote by $\operatorname{CYdim}(\mathcal A)$ the CY
dimension of $\mathcal A$.
\vskip10pt
For example, if $A$ is a symmetric algebra and $\mathcal P$ is the
category of projective modules, then the homotopy category
$K^b(\mathcal P)$ is of CY dimension $0$. Moreover, if $\mathcal
A$ is of CY dimension $d$, then $\operatorname {Ext}_\mathcal
A^i(X, Y)\cong D\circ \operatorname {Ext}_\mathcal A^{d-i}(Y, X),
\ X, Y\in\mathcal A, \ i\in\Bbb Z$, where $\operatorname
{Ext}_\mathcal A^i(X, Y): =\operatorname {Hom}_\mathcal A(X,
Y[i])$. Thus, if $A$ is a CY algebra ([B], [Gin2]), i.e. the
bounded derived category $D^b(A\mbox{-mod})$ is Calabi-Yau of CY
dimension $d$, then $\operatorname {gl.dim}(A\mbox{-mod})=d$ (see
[B]).
\vskip10pt
\blacksquareubsection{} Let $\mathcal{A}$ and $\mathcal{B}$ be triangulated categories.
{\em A triangle functor} from $\mathcal{A}$ to $\mathcal{B}$ is a pair $(F, \eta^F)$, where
$F\colon \mathcal{A} \to \mathcal{B}$ is an additive functor and
$\eta^F: \ F\circ [1] \longrightarrow [1]\circ F$ is a natural
isomorphism, such that if $X
\blacksquaretackrel{f} {\longrightarrow} Y
\blacksquaretackrel{g} {\longrightarrow} Z
\blacksquaretackrel{h} {\longrightarrow}
X[1]$ is a distinguished triangle of $\mathcal{A}$ then $FX
\blacksquaretackrel{Ff} {\longrightarrow} FY
\blacksquaretackrel{Fg} {\longrightarrow}
FZ
\blacksquaretackrel{\eta^F_X \circ Fh} {\longrightarrow} (FX)[1]$ is a
distinguished triangle of $\mathcal{B}$. Triangle functors $(F, \
\eta^F)$ and $(G, \ \eta^G)$ are {\em natural isomorphic} if there
is a natural isomorphism $\xi: \ F\longrightarrow G$ such that the
following diagram commutes for any $A\in\mathcal{A}$
\[\xymatrix{
F(A[1]) \ar[rr]^{\eta^F_A} \ar[d]_{\xi_{A[1]}} && F(A)[1]
\ar[d]^-{\xi_A[1]}\\
G(A[1]) \ar[rr]^{\eta^G_A} && G(A)[1].}\]
As Keller pointed out, the pair $([n], \ (-1)^n{\rm Id}_{[n+1]}):
\ \mathcal{A}\longrightarrow\mathcal{A}$ is a triangle functor for
$n\in\Bbb Z$. However, $([n], \ {\rm Id}_{[n+1]})$ may be {\em
not}.
We need the following important result. A nice proof given by Van
den Bergh is in the Appendix of [B].
\begin {lem} \ \ (Bondal-Kapranov {\em [BK]}; Van den
Bergh {\em [B]}) \ \ Let $F$ be a Serre functor of a Hom-finite
triangulated $k$-category $\mathcal{A}$. Then there exists a
natural isomorphism $\eta^F: \ F\circ [1] \longrightarrow [1]\circ
F$ such that $(F, \ \eta^F): \
\mathcal{A}\longrightarrow\mathcal{A}$ is a triangle functor.
\end{lem}
>From [Ke, 8.1] and [B, A.5.1] one has the following
\begin{prop} (Keller; Van den Bergh)\ Let $\mathcal{A}$ be a
Hom-finite triangulated $k$-category with Serre functor $F$. Then
$\mathcal{A}$ is a Calabi-Yau category if and only if there exists
a natural isomorphism $\eta^F: F\circ [1] \longrightarrow [1]\circ
F,$ such that $(F, \ \eta^F)$ is a triangle functor and $(F,
\eta^F) \longrightarrow ([d], (-1)^d\operatorname {Id}_{[d+1]})$\
is a natural isomorphism of triangle functors, for some integer
$d$. \end{prop}
\noindent {\bf Proof.} For convenience we justify the ``only if''
part. By assumption we have a natural isomorphism $\xi: F \cong
[d]$. Define $\eta^F_A \colon F(A[1]) \longrightarrow (FA)[1]$ for
$A\in\mathcal{A}$ by \ $\eta^F_A:=(-1)^d(\xi_A)^{-1}[1]\circ
\xi_{A[1]}.$\ Then $\xi_A[1] \circ \eta_A^F = (-1)^d \xi_{A[1]}$.
The naturality of $\eta^F: \ F\circ [1] \longrightarrow [1]\circ
F$ follows from the one of $\xi$. It remains to show that $(F,
\eta^F)\colon \mathcal{A}\to \mathcal{A}$ is a triangle functor.
Let $X
\blacksquaretackrel{f} {\longrightarrow} Y
\blacksquaretackrel{g}
{\longrightarrow} Z
\blacksquaretackrel{h} {\longrightarrow} X[1]$ be a
distinguished triangle. Since \ $X[d]
\blacksquaretackrel{f[d]}
{\longrightarrow}Y[d]
\blacksquaretackrel{g[d]} {\longrightarrow} Z[d]
\blacksquaretackrel{(-1)^d h[d]} {\longrightarrow}X[d+1]$\ \ is a
distinguished triangle, it suffices to prove that the following
diagram is commutative
\[\xymatrix{ F(X)\ar[r]^-{F(f)} \ar[d]_{\xi_X}^-{\wr} & F(Y)
\ar[r]^-{F(g)}\ar[d]_{\xi_Y}^-{\wr}& F(Z)\ar[rr]^-{\eta_X^F \circ
F(h)}\ar[d]_{\xi_Z}^-{\wr}&&
(F(X))[1]\ar[d]\ar[d]_{\xi_X[1]}^-{\wr}\\
X[d]\ar[r]^-{f[d]} & Y[d]\ar[r]^-{g[d]}& Z[d]\ar[rr]^-{(-1)^d
h[d]}&& X[d+1].}\] By the naturality of $\xi$ the first and the second square are
commutative. We also have
$$\xi_X[1]\circ \eta_X^F \circ F(h) = (-1)^d \xi_{X[1]} \circ F(h) = (-1)^d h[d]\circ \xi_Z.
\ \ \ \blacksquare $$
\vskip10pt
\blacksquareubsection{} Let $A$ be a self-injective $k$-algebra,
$A$-mod the category of finite-dimensional left $A$-modules,
and $A\underline {\mbox{-mod}}$ the stable category of
$A$-mod modulo projective modules. Then the Nakayama functor
$\mathcal N: = D( A)\otimes_A-$, Heller's syzygy functor $\Omega$,
and the Auslander-Reiten translate $\tau
\cong \Omega^2\circ \mathcal N\cong\mathcal N\circ \Omega^2$
([ARS], p.126), and are endo-equivalences of
$A\underline {\mbox{-mod}}$ ([ARS], Chap. IV). Note that
$A\underline {\mbox{-mod}}$ is a Hom-finite Krull-Schmidt
triangulated $k$-category with $[1] = \Omega^{-1}$ ([Hap1], p.16).
By the bi-naturality of the Auslander-Reiten isomorphisms ([AR])
$$ \underline {\operatorname {Hom}}(X, Y) \cong D\circ\operatorname
{Ext}_A^1(Y, {\tau} X) \cong D\circ\underline {\operatorname
{Hom}}(Y, [1]\circ {\tau} X),$$ where $\underline {\operatorname
{Hom}}(X, Y): = \operatorname {Hom} _{A\underline
{\mbox{-mod}}}(X, Y)$, one gets the Serre functor
$F:=[1]\circ {\tau} \cong \Omega\circ \mathcal N$ of $A\underline {\mbox{-mod}}$.
It follows that $A\underline {\mbox{-mod}}$ is
Calabi-Yau if and only if $\mathcal N \cong
\Omega^{-(d+1)}$ for some $d$ ([Ke, 8.3]). In this case denote by
$\operatorname{CYdim}(A)$ the CY dimension of $A\underline
{\mbox{-mod}}$. Note that $\Omega, \ F, \ \mathcal N,
\ \tau$ are pairwise commutative
as functors of $A\underline {\mbox{-mod}}$. This follows from Lemma 2.1.
\vskip10pt
\blacksquareubsection{} Let $A$ be a finite-dimensional
$k$-algebra. Recall that $A$ is a Nakayama algebra if any
indecomposable is uniserial, i.e. it has a unique composition
series ([ARS], p.197). In this case $A$ is representation-finite.
If $k$ is algebraically closed then any connected self-injective
Nakayama algebra is Morita equivalent to $\Lambda(n, t),$ $n\ge 1, \
t\ge 2$ ([GR], p.243), which is defined below.
Let $\Bbb Z_n$ be the cyclic quiver with vertices indexed by the
cyclic group $\Bbb Z/n\Bbb Z$ of order $n$, and with arrows $a_i:
\ i \longrightarrow i+1, \ \forall \ i\in \Bbb Z/n\Bbb Z$. Let
$k\Bbb Z_n$ be the path algebra of the quiver $\Bbb Z_n$, $J$ the
ideal generated by all arrows, and $\Lambda = \Lambda(n, t): =k\Bbb Z_n/J^t$
with $t\ge 2$. Denote by $\gamma^l_i$ the path starting at vertex
$i$ and of length $l$, and $e_i: = \gamma^0_i$. We write the
conjunction of paths from right to left. Then $\{\gamma^l_i \ | \
0\le i\le n-1, \ 0\le l\le t-1\}$ is a basis of $\Lambda$; while
$\{P(i): = \Lambda e_i\ | \ 0\le i\le n-1\}$ is the set of pairwise
non-isomorphic indecomposable projective modules, and $\{I(i): =
D(e_{i}\Lambda)\ | \ 0\le i\le n-1\}$ is the set of pairwise
non-isomorphic indecomposable injective modules, with $P(i) \cong
I(i+t-1)$. Note that $\Lambda$ is a Frobenius algebra, and $\Lambda$ is
symmetric if and only if $n\mid (t-1)$. Write $S(i): =
P(i)/\operatorname{rad}P(i)$, and $S^l_i: =
\Lambda\gamma_{i+l-t}^{t-l}$. Then $S_i^l$ is the indecomposable with
top $S(i)$ and the Loewy length $l$, and $\{ S^l_i \ | \ 0\le i\le
n-1, \ 1\le l\le t \}$ is the set of pairwise non-isomorphic
indecomposable modules, with $S^{t}_i = P(i)$ and
$\operatorname{soc}(S^l_i) = S(i+l-1)$. For the Auslander-Reiten
quiver of $\Lambda$ see [GR], Section 2, and [ARS], p.197. In
particular, the stable Auslander-Reiten quiver of $\Lambda$ is $\Bbb Z
A_{t-1}/\langle\tau^n\rangle.$
\vskip10pt
\blacksquareection{\bf Indecomposable Calabi-Yau objects}
The purpose of this section is to introduce the Calabi-Yau objects
and to give the relation between indecomposable
Calabi-Yau objects and Auslander-Reiten triangles.
\vskip10pt
\blacksquareubsection{} Let $\mathcal{A}$ be a Hom-finite
triangulated $k$-category. A non-zero object $X$ is called \emph
{a Calabi-Yau object} if there exists a natural isomorphism
\begin{align}\operatorname{Hom}_{\mathcal A}(X, -)\cong D\circ \operatorname{Hom}_{\mathcal A}(-,
X[d])\end {align} for some integer $d$.
By Yoneda Lemma, such a $d$ is unique up to a multiple of the
relative order $o([1]_X)$ of $[1]$ respect to $X$. Recall that
$o([1]_X)$ is the minimal positive integer such that
$X[o([1]_X)]\cong X$, otherwise $o([1]_X) = \infty$. If
$o([1]_X)=\infty$ then $d$ in $(3.1)$ is unique and is called {\em
the CY dimension} of $X$. If $o([1]_X)$ is finite then the minimal
non-negative integer $d$ in $(3.1)$ is called {\em the CY
dimension} of $X$. We denote $\operatorname{CYdim}(X)$ the CY
dimension. Thus, if $o([1]) < \infty$ then $o([1]_X)\mid o([1])$
and $0\le \operatorname{CYdim}(X) < o([1]_X).$
Let $A$ be a finite-dimensional self-injective algebra. An
$A$-module $M$ without projective direct summands is called {\em a
Calabi-Yau module} of CY dimension $d$, if it is a Calabi-Yau
object of $A\underline{\mbox{-mod}}$ with $\operatorname{CYdim}(M) = d$.
\vskip10pt
Note that $\operatorname{CYdim}(X)$ is usually not easy to
determine. In case $(3.1)$ holds for some $d$, we say that $X$ is a {\em $d$-th CY object}.
Of course, if $o([1]_X) <
\infty$ then $o([1]_X)\mid (d - \operatorname{CYdim}(X))\ge 0.$
\vskip10pt
If $\mathcal A$ has right Serre functor $F$, then by Yoneda Lemma
a non-zero object $X$ is a $d$-th CY object if and only if $F(X)\cong X[d]$, or equivalently,
$F(X)[-d]\cong X$. Thus, a non-zero $A$-module $M$
without projective direct summands is a $d$-th CY module
if and only if $\mathcal N(M)\cong
\Omega^{-(d+1)}(M)$ in $A\underline{\mbox{-mod}}$ (in fact, this
isomorphism can be taken in $A$-mod).
\vskip10pt
\blacksquareubsection{} We have the following basic property.
\vskip10pt
\begin {prop} \ $(i)$ \ The Calabi-Yau property for a category or an object,
is invariant under triangle-equivalences.
\vskip10pt
$(ii)$ \ The Calabi-Yau property for a module is {\em ``usually"}
invariant under stable equivalences between self-injective
algebras. Precisely, let $A$ and $B$ be self-injective
algebras, $G: A\underline {\mbox{-mod}}\longrightarrow B\underline
{\mbox{-mod}}$ a stable equivalence, and $X$ a CY $A$-module of
dimension $d$. If $A\ncong \Lambda(n, 2),$ or if $A$ and $B$ are
symmetric algebras, then $G(X)$ is a CY $B$-module of dimension
$d$.
\end{prop}
\noindent{\bf Proof.} \ $(i)$ \ Let $\mathcal A$ be a Calabi-Yau
category with $F_\mathcal A\cong [d]$, where $F_\mathcal A$ is the
Serre functor. Clearly $F_\mathcal B: = G \circ
F_\mathcal A\circ G^{-1}$ is a Serre functor of $\mathcal B$ (if
$\mathcal B$ has already one, then it is natural
isomorphic to $F_\mathcal B$). By the natural
isomorphism $(\xi^G)^d: G\circ [d] \longrightarrow [d]\circ G$,
which is the composition $G\circ [d] \longrightarrow [1]\circ
G\circ [d-1] \longrightarrow \cdots \longrightarrow [d]\circ G$
(A.2 in [B]), we see that $\mathcal B$ is a Calabi-Yau
category with $F_\mathcal B\cong [d].$ If $X$ is a calabi-Yau object
with a natural isomorphism $\eta$ as in
$(3.1)$, then we have natural isomorphism \
$\operatorname{Hom}_\mathcal B(-, (\xi^G)^d_X)\circ G\circ
\eta\circ G^{-1}: \ \operatorname{Hom}_{\mathcal B}(G(X), -)\cong
D\circ \operatorname{Hom}_{\mathcal B}(-, (GX)[d]),$ \ which
implies that $G(X)$ is a Calabi-Yau object of $\mathcal B$.
$(ii)$ \ Recall that an equivalence $G: A\underline
{\mbox{-mod}}\longrightarrow B\underline {\mbox{-mod}}$ of
categories is called a stable equivalence. Note that in general
$G$ is not induced by an exact functor (cf. [ARS], p.339), hence
$G$ may be not a triangle-equivalence (cf. [Hap1], Lemma 2.7,
p.22. Note that the converse of Lemma 2.7 is also true). One may
assume that $A$ is connected. If $A\ncong \Lambda(n, 2),$ or if $A$ and
$B$ are symmetric algebras, then by Corollary 1.7 and Prop. 1.12
in [ARS], p.344, we know that $G$ commutes with $\tau$ and
$\Omega$ on modules, hence we have isomorphism
\begin{align*}\Omega_B^{-1}\circ \tau_B (G(X)) \cong
G(\Omega_A^{-1}\circ \tau_A) (X)) \cong G(\Omega_A^{-d}(X))\cong
\Omega_B^{-d}(G(X)),\end{align*} which implies that $G(X)$ is a
Calabi-Yau $B$-module of CY dimension $d$.
$\blacksquare$
\vskip10pt
It seems that the Calabi-Yau property for the stable category
is also invariant under stable equivalence $G$
between self-injective algebras. However, this need natural isomorphiams between
$G$ and $\tau$, and $G$ and $\Omega$, which are not clear to us.
\vskip10pt
\blacksquareubsection{} The main result of this section
is as follows.
\vskip10pt
\begin{thm} Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category, and $X$ an indecomposable object of $\mathcal A$.
Then $X$ is a $d$-th CY object if and
only if there exists an Auslander-Reiten triangle of the form
\begin{align}X[d-1]
\blacksquaretackrel f \longrightarrow Y
\blacksquaretackrel g \longrightarrow X
\blacksquaretackrel h\longrightarrow
X[d].\end{align}
Moreover, $Y$ is also a $d$-th CY object.
\end{thm}
\vskip10pt
\blacksquareubsection{}The proof of the first part of Theorem 3.2
follows an argument of Reiten and Van den Bergh in [RV]. For the convenience
we include a complete proof.
\vskip10pt
\begin{lem} \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category, and $X$ a non-zero object of $\mathcal A$. Then $X$
is a $d$-th CY object if and only if for
any indecomposable $Z$ there exists a non-degenerate bilinear form
\begin{align}(-,-)_{Z}: \
\operatorname{Hom}_{\mathcal A}(X, Z)\times
\operatorname{Hom}_{\mathcal A}(Z, X[d]) \longrightarrow
k\end{align} such that for any \ $u\in
\operatorname{Hom}_{\mathcal A}(X, Z), \ \ v\in
\operatorname{Hom}_{\mathcal A}(Z, W)$, \ and \ $w\in
\operatorname{Hom}_{\mathcal A}(W, X[d]),$ there holds
\begin{align}(u, wv)_Z=(vu, w)_W.\end{align}
\end{lem}
\noindent{\bf Proof.} \ If $X$ is a $d$-th CY object, then we
have $\eta_Z: \ \operatorname{Hom}_{\mathcal A}(X, Z)\cong D\circ
\operatorname{Hom}_{\mathcal A}(Z, X[d]), \ \forall \ Z\in\mathcal
A$, which are natural in $Z$. Each isomorphism $\eta_Z$ induces a
non-degenerate bilinear form $(-,-)_{Z}$ in $(3.3)$ by $(u, z)_Z:
= \eta_Z(u)(z),$ and $(3.4)$ follows from the naturality of
$\eta_Z$ in $Z$. Conversely, if we have $(3.3)$ and $(3.4)$ for
any indecomposable $Z$, then we have isomorphism $\eta_Z: \
\operatorname{Hom}_{\mathcal A}(X, Z)\cong D\circ
\operatorname{Hom}_{\mathcal A}(Z, X[d])$ given by $\eta_Z(u)(z):
= (u, z)_Z, \ \forall \ z\in \operatorname{Hom}_{\mathcal A}(Z,
X[d])$. By $(3.4)$ $\eta_Z$ are natural in $Z$. Since $\mathcal
A$ is Krull-Schmidt, it follows that we have isomorphisms $\eta_Z$
for any $Z$ which are natural in $Z$. This means that $X$ is a
$d$-th CY object.
$\blacksquare$
\vskip10pt
The following Lemma in [RV] will be used.
\vskip10pt
\begin{lem} \ \ ({\em [RV]}, Sublemma I.2.3)\ Let $\mathcal{A}$ be
a Hom-finite Krull-Schmidt triangulated $k$-category,
$\tau_{\mathcal A}(X) \longrightarrow Y
\longrightarrow X
\blacksquaretackrel h\longrightarrow
\tau_\mathcal A(X)[1]$ an Auslander-Reiten triangle of $\mathcal
A$, and $Z$ an indecomposable in $\mathcal A$. Then
$(i)$ \ For any non-zero $z\in \operatorname{Hom}_{\mathcal A}(Z,
\tau_{\mathcal A}(X)[1])$ there exists $u\in
\operatorname{Hom}_{\mathcal A}(X, Z)$ such that $zu = h$.
$(ii)$ For any non-zero $u\in \operatorname{Hom}_{\mathcal A}(X,
Z)$ there exists $z\in \operatorname{Hom}_{\mathcal A}(Z,
\tau_{\mathcal A}(X)[1])$ such that $zu = h$.
\end{lem}
\vskip10pt
In a Hom-finite Krull-Schmidt triangulated $k$-category without
Serre functor (e.g., by [Hap2] and [RV] if
$\operatorname{gl.dim}(A) = \infty$ then $D^b(A\mbox{-mod})$ has
no Serre functor), one may use {\em the generalized Serre functor}
introduced by Chen [Ch].
\vskip10pt
\begin{lem} (Chen [Ch]) Let $\mathcal A$ be a Hom-finite Krull-Schmidt triangulated
$k$-category. Consider the full subcategories of $\mathcal A$
given by $$\mathcal{\mathcal A}_r:=\{X \in \mathcal{A}\; |\;
D\circ\operatorname{Hom}_{\mathcal A}(X, -) \mbox{ is
representable} \}$$ and $$\mathcal{\mathcal A}_l:=\{X \in
\mathcal{A}\;|\; D\circ \operatorname{Hom}_\mathcal A(-, X) \mbox{
is representable}\}.$$ Then both $\mathcal{A}_r$ and
$\mathcal{A}_l$ are thick triangulated subcategories of $\mathcal
A$. Moreover, one has
$(i)$ \ There is a unique $k$-functor $S: \mathcal{A}_r
\longrightarrow \mathcal{A}_l$ which is an equivalence, such that
there are natural isomorphisms
\begin{align}\operatorname{Hom}_\mathcal A(X, -)
\blacksquareimeq
D\circ\operatorname{Hom}_\mathcal A(-, S(X)), \ \ \forall \ X\in
\mathcal {A}_r,\end{align} which are natural in $X$. $S$ is called
the generalized Serre functor, with range $\mathcal A_r$ and
domain $\mathcal A_l$.
$(ii)$ \ There exists a natural isomorphism $\eta^S: S\circ
[1]\longrightarrow [1]\circ S$ such that the pair $(S, \eta^S):
\mathcal{A}_r \longrightarrow \mathcal{A}_l$ is an
triangle-equivalence.
\end{lem}
In this terminology, a non-zero object $X$ is a $d$-th CY object
if and only if $X\in\mathcal A_r$ and $S(X)\cong X[d]$, by $(3.5)$
and Yoneda Lemma.
\vskip10pt
\blacksquareubsection{} {\bf Proof of Theorem 3.2.}\quad Let
$X$ be an indecomposable $d$-th CY object. By Lemma 3.3 we have a
non-degenerate bilinear from $(-,-)_X: \
\operatorname{Hom}_{\mathcal A}(X, X)\times
\operatorname{Hom}_{\mathcal A}(X, X[d]) \longrightarrow k.$ It
follows that there exists $0\ne h\in \operatorname{Hom}_{\mathcal
A}(X, X[d])$ such that $(\operatorname{radHom}_{\mathcal A}(X, X),
h)_X=0.$ Embedding $h$ into a distinguished triangle as in
$(3.2)$. We claim that it is an Auslander-Reiten triangle. For
this it remains to prove (AR4) in 2.1. Let $X'$ be indecomposable
and $t: X'\longrightarrow X$ a non-isomorphism. Then by $(3.4)$
for any $u\in \operatorname{Hom}_{\mathcal A}(X, X')$ we have $(u,
ht)_{X'} = (tu, h)_X = 0.$ Since $(-,-)_{X'}$ is non-degenerate,
it follows that $ht=0$.
Conversely, let $(3.2)$ be an Auslander-Reiten triangle. In order
to prove that $X$ is a $d$-th CY object, by Lemma 3.3 it suffices
to prove that for any indecomposable $Z$ there exists a
non-degenerate bilinear form $(-,-)_{Z}$ as in $(3.3)$ satisfying
$(3.4)$. For this, choose an arbitrary linear function
$\operatorname {tr}\in D\circ \operatorname{Hom}_{\mathcal A}(X,
X[d])$ such that $\operatorname {tr}(h)\ne 0$, and define $(u,
z)_Z = \operatorname{tr}(zu).$ Then $(3.4)$ is automatically
satisfied. It remains to prove that $(-, -)_Z$ is non-degenerate.
In fact, for any $0\ne z\in \operatorname{Hom}_{\mathcal A}(Z,
X[d])$, by Lemma 3.4 there exists $u\in
\operatorname{Hom}_{\mathcal A}(X, Z)$ such that $zu = h$. So $(u,
z)_Z = \operatorname{tr}(zu) = \operatorname{tr}(h)\ne 0$.
Similarly, for any $0\ne u\in \operatorname{Hom}_{\mathcal A}(X,
Z)$ we have $z\in \operatorname{Hom}_{\mathcal A}(Z, X[d])$ such
that $(u, z)_Z \ne 0$. This proves the non-degenerateness of $(-,
-)_Z$.
\vskip10pt
Now we prove that $Y$ in $(3.2)$ is also a $d$-th CY object. We make use of the
generalized Serre functor in [Ch]
(For the reader prefer Serre functor, one can assume the existence, and use Lemma 2.1).
Since $X, \ X[d-1]\in\mathcal A_r$, it follows from Lemma 3.5 that
$Y\in\mathcal A_r$. Applying the
generalized Serre functor $(S, \eta^S)$ to $(3.2)$ we get the
distinguished triangle (by Lemma 3.5$(ii)$)
$$S(X[d-1])
\blacksquaretackrel {S(f)}\longrightarrow S(Y)
\blacksquaretackrel {S(g)}\longrightarrow
S(X)
\blacksquaretackrel {\eta^S_{X[d-1]}\circ S(h)}\longrightarrow
S(X[d-1])[1].\eqno(*)$$ Also, we have the Auslander-Reiten
triangle
$$X[2d-1]
\blacksquaretackrel {f[d]}
{\longrightarrow}Y[d]
\blacksquaretackrel {g[d]}
{\longrightarrow}X[d]
\blacksquaretackrel {(-1)^d h[d]}{\longrightarrow}
X[2d].$$ Since $X$ is a $d$-th CY object it
follows that we have an isomorphism $w: S(X)\longrightarrow X[d]$.
Note that $S(h)\ne 0$ means that $S(g)$ is not a retraction
([Hap1], p.7). Thus, by (AR3) there exists $v: S(Y)\longrightarrow
Y[d]$ such that $w\circ S(g) = g[d]\circ v$. By the definition of a triangulated category
we get $u:
S(X[d-1])\longrightarrow X[2d-1]$ such that the following diagram
is commutative
\[\xymatrix{ S(X[d-1])\ar[r]^-{S(f)}
\ar[d]_{u} & S(Y) \ar[r]^-{S(g)}\ar[d]_{v}&
S(X)\ar[rr]^-{\eta_{X[d-1]}^S \circ S(h)}\ar[d]_{w}&&
S(X[d-1])[1]\ar[d]\ar[d]_{u[1]}\\
X[2d-1]\ar[r]^-{f[d]} & Y[d]\ar[r]^-{g[d]}& X[d]\ar[rr]^-{(-1)^d
h[d]}&& X[2d].}\eqno(**)\] We claim that $u: S(X[d-1])\longrightarrow
X[d]$ is an isomorphism, hence by the property of a triangulated
category we know that $v: S(Y)\longrightarrow Y[d]$ is also an
isomorphism, i.e. $Y$ is a $d$-th CY object.
Otherwise $u: S(X[d-1])\longrightarrow X[2d-1]$ is not an
isomorphism. Note that $S$ is {\em only} defined on $\mathcal A_r$, it follows that
we do not know if $(*)$ is an Auslander-Reiten triangle.
Since $X[d-1]$ is a $d$-th CY object, it
follows that we have an isomorphism $\alpha:
X[2d-1]\longrightarrow S(X[d-1])$, hence $\alpha\circ u\in
\operatorname{Hom}_\mathcal A(S(X[d-1]), S(X[d-1]))$. So we have
$u'\in \operatorname{Hom}_\mathcal A(X[d-1], X[d-1])$ such that $
\alpha\circ u = S(u')$, and $u'$ is also a non-isomorphism. Since
$X[d-1]
\blacksquaretackrel f \longrightarrow Y
\blacksquaretackrel g \longrightarrow
X
\blacksquaretackrel h\longrightarrow X[d]$ is an Auslander-Reiten triangle
it follows from (AR4') that $u'\circ h[-1] = 0$, or equivalently,
$S(u')\circ S(h[-1]) = 0$ (note that $h[-1]\in \mathcal A_r$).
Thus we have
$$u[1] \circ S(h[-1])[1]\circ \eta^S_{X[-1]} = 0$$
where $\eta^S_{X[-1]}: S(X)\longrightarrow S(X[-1])[1]$ is an
isomorphism. By the naturality of $\eta^S$ we have the
commutative diagram
\[\xymatrix{
S(X) \ar[rr]^{\eta_{X[-1]}^S} \ar[d]_{S(h)}&&
S(X[-1])[1] \ar[d]^-{S(h[-1])[1]}\\
S(X[d]) \ar[rr]^{\eta_{X[d-1]}^S} && S(X[d-1])[1].}\] It follows
that we have $u[1]\circ \eta^S_{X[d-1]}\circ S(h) = 0,$ and hence
by the commutative diagram $(**)$ we get a contradiction \ $(-1)^dh[d]\circ w = u[1]\circ
\eta^S_{X[d-1]}\circ S(h) = 0.$ This completes the proof.
$\blacksquare$
\vskip10pt
\blacksquareubsection{} {\bf Remark 3.6.} Let $\mathcal A$ be a Hom-finite Krull-Schmidt triangulated
$k$-category. If every indecomposable $X$ in $\mathcal A$ is a
$d_X$-th CY object, then $\mathcal A$ has a Serre functor $F$ with
$F(X)\cong X[d_X]$.
In fact, by Theorem 3.2 $\mathcal A$ has right and left Auslander-Reiten triangles,
and then by Theorem I.2.4 in [RV] $\mathcal A$ has Serre functor
$F$. By Prop.I.2.3 in [RV] and $(3.2)$ we have $F(X)\cong
\tau_\mathcal A(X)[1] = X[d_X-1][1] = X[d_X]$ for any indecomposable
$X$.
\vskip10pt
However, even if all indecomposables are $d$-th CY objects with
the same $d$, we do not know whether $\mathcal A$ is a Calabi-Yau
category, although $F$ and $[d]$ coincide on objects. The examples
we know have a positive answer to this question.
$\blacksquare$
\vskip10pt
\blacksquareection{\bf Minimal Calabi-Yau objects}
The purpose of this section is to describe all the Calabi-Yau objects of a
Hom-finite Krull-Schmidt triangulated $k$-category with a Serre functor.
\blacksquareubsection{} Let $\mathcal{A}$ be a
Hom-finite Krull-Schmidt triangulated $k$-category. A $d$-th CY
object $X$ is said to be {\em minimal} if
any {\em proper} direct summand of $X$ is {\em not} a $d$-th CY
object.
\vskip10pt
\begin{lem} \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category with right Serre functor $F$. Then a non-zero object
$X$ is a minimal $d$-th CY object if and
only if the following are satisfied:
\vskip10pt
1. The indecomposable direct summands of $X$ can be ordered as $X
= X_1\oplus \cdots \oplus X_r$ such that
\begin{align} F(X_1) \cong X_2[d],
\ F(X_2) \cong X_3[d], \ \cdots, \ F(X_{r-1}) \cong X_r[d], \
F(X_r)\cong X_1[d].\end{align}
We call the cyclic order arising from this property {\em a canonical order} of $X$
(with respect to $F$ and $[d]$).
2. \ $X$ is multiplicity-free, i.e. its indecomposable direct
summands are pairwise non-isomorphic.
\end{lem}
\noindent {\bf Proof.}\quad In the following we often use that a
non-zero object $X$ is a $d$-th CY object if
and only if $F(X)\cong X[d]$.
Let $X= X_1\oplus \cdots \oplus X_r$ be a minimal $d$-th CY
object, with each $X_i$ indecomposable and
$r\ge 2$. Then $F(X_1) \oplus\cdots\oplus F(X_r)\cong
X_1[d]\oplus\cdots\oplus X_r[d]$. Since $\mathcal{A}$ is
Krull-Schmidt, it follows that there exists a permutation $
\blacksquareigma$
of $1, \cdots, r$, such that $F(X_i)\cong X_{
\blacksquareigma(i)}[d]$ for
each $i$. Write $
\blacksquareigma$ as a product of disjoint cyclic
permutations. Since $X$ is minimal, it follows that $
\blacksquareigma$ has
to be a cyclic permutation of length $r$. By reordering the
indecomposable direct summands of $X$, one may assume that $
\blacksquareigma
= (12\cdots r)$. Thus, $X$ satisfies the condition 1.
Now, we consider a canonical order $X= X_1\oplus \cdots \oplus
X_r$. If $X_i\cong X_j$ for some $1\le i < j\le r$, then
$F(X_{j-1})\cong X_j[d]\cong X_i[d]$, it follows that $X_i\oplus
\cdots\oplus X_{j-1}$ is already a $d$-th CY object,
which contradicts the minimality of $X$. This
proves that $X$ is multiplicity-free.
\vskip10pt Conversely, assume that a multiplicity-free object $X=
X_1\oplus \cdots \oplus X_r$ is in a canonical order. By $(4.1)$
we have $F(X)\cong X[d]$. So $X$ is a $d$-th CY object.
It remains to show the minimality.
If not, then there exists a proper direct summand $X_{i_1}\oplus
\cdots\oplus X_{i_t}$ of $X$ which is a minimal $d$-th CY
object, so $1\le t< r$. By what we have proved above
we may assume that this is a canonical order. Then
$$F(X_{i_1})\cong X_{i_2}[d],\ \cdots, \ F(X_{i_{t-1}})\cong
X_{i_t}[d], \ F(X_{i_t})\cong X_{i_1}[d].$$ While $X= X_1\oplus
\cdots \oplus X_r$ is also in a canonical order, it follows that
(note that we work on indices modulo $r$, e.g. if $i_1 =
r$ then $i_1+1$ is understood to be $1$)
$$F(X_{i_1})\cong X_{i_1+1}[d], \ \cdots, \ F(X_{i_{t-1}})\cong
X_{i_{t-1}+1}[d], \ F(X_{i_t})\cong X_{i_t+1}[d].$$ Since $X$ is
multiplicity-free, it follows that (considering indices modulo
$r$)
$$i_2 = i_1+1, \ \cdots, \ i_t = i_{t-1}+1, \ i_1 = i_t + 1,$$
hence $i_1 = i_1+t$, which means $r\mid t$. This is impossible
since $1\le t< r.$
$\blacksquare$
\vskip10pt
\blacksquareubsection{} Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category with Serre functor $F$. For each $d\in\Bbb Z$,
consider the triangle-equivalence $G:=[-d]\circ F\cong
F\circ [-d]: \mathcal A\longrightarrow \mathcal A$. For each
indecomposable $M\in\mathcal A$, denote by $o(G_M)$ the relative
order of $G$ respect to $M$, that is, $r:=o(G_M)$ is the
minimal positive integer such that $G^r(M)\cong M$, otherwise $o(G_M) =
\infty$. Denote by
$\operatorname{Aut}(\mathcal A)$ the group of the
triangle-equivalences of $\mathcal A$, and by $\langle G\rangle$
the cyclic subgroup of $\operatorname{Aut}(\mathcal A)$ generated
by $G$. Then $\langle G\rangle$ acts naturally on
$\operatorname{Ind}(\mathcal A)$, the set of the isoclasses of
indecomposables of $\mathcal A$. Denote by $\mathcal O_M$ the
$G$-orbit of an indecomposable $M$. Then $|\mathcal O_M| = o(G_M)$.
If $|\mathcal O_M| < \infty$ then the set $\mathcal O_M$ is a finite
$G$-orbit.
\vskip10pt
Denote by $\operatorname{Fin}\mathcal{O}(\mathcal A, d)$ the set
of all the finite $G$-orbits of $\operatorname{Ind}(\mathcal A)$,
and by $\operatorname{MinCY}(\mathcal A, d)$ the set of isoclasses
of minimal $d$-th CY objects. We
have the following
\vskip10pt
\begin {thm} \ \ Let $\mathcal{A}$ be a Hom-finite Krull-Schmidt triangulated
$k$-category with Serre functor $F$. Then
\vskip10pt
$(i)$ \ Every $d$-th CY object is a direct sum of finitely many
minimal $d$-th CY objects.
\vskip10pt
$(ii)$ \ With the notations above, for each $d\in\Bbb Z$ the map
\begin{align}\mathcal O_M \ \mapsto \
\bigoplus\limits_{X\in \mathcal O_M} X = M\oplus G(M)\oplus\cdots
\oplus G^{o(G_M)-1}(M)\end{align} \noindent gives a one-to-one
correspondence between the sets
$\operatorname{Fin}\mathcal{O}(\mathcal A, d)$ and
$\operatorname{MinCY}(\mathcal A, d)$, where $G: = [-d]\circ F$.
Thus, a minimal $d$-th CY object is
exactly the direct sum of all the objects in a finite $G$-orbit
of $\operatorname{Ind}(\mathcal A)$.
\vskip10pt
$(iii)$ Non-isomorphic minimal $d$-th CY objects are disjoint, i.e. they have no isomorphic
indecomposable direct summands. \end {thm}
\noindent {\bf Proof.}\quad Let $X$ be a $d$-th CY object.
If $X$ is not minimal, then $X = Y\oplus Z$
with $Y\ne 0\ne Z$ such that $F(Y)\oplus F(Z) \cong Y[d]\oplus
Z[d]$ and $F(Y)\cong Y[d]$. Since $\mathcal{A}$ is Krull-Schmidt,
it follows that $F(Z) \cong Z[d]$, i.e. $Z$ is also a $d$-th CY
object. Then $(i)$ follows by induction.
Thanks to Lemma 2.1, $(4.1)$ becomes $X_i = G^{i-1}(X_1), \ 1\le
i\le r.$ Then $(ii)$ is a reformulation of Lemma 4.1. Moreover
$(iii)$ follows from $(ii)$.
$\blacksquare$
\vskip10pt
\begin {cor} \ \ Let $A$ be a finite-dimensional self-injective
algebra. Then $X$ is a minimal $d$-th CY module
if and only if $X$ is of the form $X \cong
\bigoplus \limits_{0\le i\le r-1}G^i(M)$, where $M$ is an
indecomposable non-projective $A$-module with $r: = o(G_M)<\infty$,
and $G: = \Omega^{d+1}\circ \mathcal N$. \end {cor}
\noindent {\bf Proof.}\quad Note that in this case $[-d]\circ F =
\Omega^{d+1}\circ \mathcal N$.
$\blacksquare$
\vskip10pt
\blacksquareubsection{} As an example, we describe all the
Calabi-Yau objects in $D^b(kQ\mbox{-mod})$, the bounded derived
category of $kQ\mbox{-mod}$, where $Q$ is a finite quiver without
oriented cycles.
Note that indecomposable objects of $D^b(kQ\mbox{-mod})$ are
exactly stalk complexes of indecomposable $kQ$-modules. The category
$D^b(kQ\mbox{-mod})$ has Serre functor $F =[1]\circ \tau_D$, where
$\tau_D$ is the Auslander-Reiten translation of
$D^b(kQ\mbox{-mod})$. Recall that $\tau_D$ is given by
$$\tau_D(M) =
\begin{cases}\tau(M), & \ \mbox{if} \ M \ \mbox{is an
indecomposable non-projective}; \\ I[-1], & \ \mbox{if} \ P \
\mbox{is an indecomposable projective},
\end{cases}$$ where $I$ is the indecomposable injective with \ $\operatorname{soc}(I) =
P/\operatorname{rad} P$, and $\tau$ is the Auslander-Reiten
translation of $kQ\mbox{-mod}$ ([Hap1], p.51). Note that
$D^b(kQ\mbox{-mod})$ is {\em not} a Calabi-Yau category except
that $Q$ is the trivial quiver with one vertex and no arrows.
However, the cluster category $\mathcal C_{kQ}$
introduced in [BMRRT], which is the orbit category of
$D^b(kQ\mbox{-mod})$ respect to the functor $\tau_D^{-1}\circ
[1]$, is a Calabi-Yau category of CY dimension $2$ ([BMRRT]).
Let $M$ be a minimal Calabi-Yau object of $D^b(kQ\mbox{-mod})$ of
CY dimension $d$ (in this case $o([1]_M) = \infty$, hence $d$
is unique). By shifts we may assume that $M$ is a $kQ$-module. By
$F(M)=\tau_D(M[1]) = \tau_D(M)[1] = M[d]$ we see $d = 1$ or $0$.
Note that $kQ$ admits an indecomposable projective-injective
module if and only if $Q$ is of type $A_n$ with the linear
orientation. However, in this case the unique indecomposable
projective-injective module $P = I$ does not satisfy the relation
$\operatorname{soc}(I) = P/\operatorname{rad} P$. It follows that
$d\ne 0$. Thus $d=1$ and $\tau(M) = M$. Consequently, $Q$ is an
affine quiver and $M$ is a $\tau$-periodic (regular) module of
period $1$. All such modules are well-known, by the
classification of representations of affine quivers (see
Dlab-Ringel [DR]). Thus we have
\vskip10pt
\begin{prop} \ \ $D^b(kQ\mbox{-mod})$ admits a Calabi-Yau object $M$ if and only if
$Q$ is an affine quiver. In this case, $M$ is minimal if and
only if $M$ is an indecomposable in a homogeneous tube of
the Auslander-Reiten quiver of $kQ$-mod, or $M$ is the direct sum of all the indecomposables of
same quasi-length in a non-homogeneous tube of the Auslander-Reiten quiver of $kQ$-mod, up to
shifts. Moreover, all such $M$'s have CY dimension $1$.
\end{prop}
\vskip10pt
\blacksquareection{\bf Calabi-Yau modules of self-injective Nakayama algebras}
This purpose of this section is to classify all the $d$-th CY
modules of self-injective Nakayama algebras $\Lambda(n, t), \ n\ge 1, \
t\ge 2$, where $d$ is any given integer. By Theorem 4.2$(i)$ it
suffices to consider the minimal $d$-th CY $\Lambda$-modules. By
Corollary 4.3 this reduces to computing the relative order $o(G_M)$
of $G: = \Omega^{d+1}\circ \mathcal N$ respect to any
indecomposable $\Lambda(n, t)$-module $M=S_i^l, \ 1\le l\le t-1$, for
any integer $d$.
\vskip10pt
\blacksquareubsection{} Recall that $\Lambda(n, t)$ is the quotient of the path algebra of the
cyclic quiver with $n$ vertices by the truncated ideal $J^t$,
where $J$ is the two-sided ideal generated by the arrows. From now
on we write $\Lambda$ instead of $\Lambda(n, t)$.
We keep the notations introduced in 2.5. Note that the
indecomposable module $S^l_i$ \ ($i\in\Bbb Z/n\Bbb Z,$ \ $1\le
l\le t-1$) \ has a natural $k$-basis, consisting of all the paths
of quiver $\Bbb Z_n$ starting at the vertex $i+l-t$ and of lengths
at least $t-l$:
$$\gamma^{t-l}_{i+l-t}, \ \gamma^{t-l+1}_{i+l-t}, \ \cdots, \ \gamma^{t-1}_{i+l-t}.$$
For $1\le l\le t-1$ denote by $
\blacksquareigma_{i}^{l}:
S_{i}^{l}\longrightarrow S_{i-1}^{l+1}$ the inclusion by embedding
the basis above; and for $2\le l\le t$ denote by $p_{i}^{l}:
S_{i}^{l}\longrightarrow S_{i}^{l-1}$ the $\Lambda$-epimorphism given
by the right multiplication by arrow $a_{i+l-t-1}$. These
$
\blacksquareigma_i^l$'s and $p_i^l$'s are all the irreducible maps of
$\Lambda$-mod, up to scalars.
\vskip10pt
We need the explicit actions of functors $\mathcal N$ and
$\Omega^{-(d+1)}$ of $\Lambda\underline {\mbox{-mod}}$. By the exact
sequences $0 \longrightarrow S_{i}^l \longrightarrow
I(i+l-1)\longrightarrow S_{i+l-t}^{t-l}\longrightarrow 0$ (with
the canonical maps), via the basis above one has the actions of
functor $\Omega^{-1}$ for $i\in\Bbb Z/n\Bbb Z, \ \ 1\le l \le
t-1$:
\begin{align*}\Omega^{-1}(S^l_i) =
S_{i+l-t}^{t-l}, \ \ \ \ \Omega^{-1}(
\blacksquareigma^l_i) = p_{i+l-t}^{t-l},
\ \ \ \ \Omega^{-1}(p^l_i) =
\blacksquareigma_{i+l-t}^{t-l}.\end{align*} By
induction one has in $\Lambda\underline {\mbox{-mod}}$ for any integer
$m$ (even negative):
\begin{align} \Omega^{-(2m-1)}(S_i^l) = S_{i+l-mt}^{t-l}; \ \ \ \
\Omega^{-2m}(S_i^l) = S_{i-mt}^{l};\end{align}
and $$\Omega^{-(2m-1)}(
\blacksquareigma_i^l) = p_{i+l-mt}^{t-l}, \ \ \ \
\Omega^{-(2m-1)}(p_i^l) =
\blacksquareigma_{i+l-mt}^{t-l};\eqno(*)$$ and
$$\Omega^{-2m}(
\blacksquareigma_i^l) =
\blacksquareigma_{i-mt}^{l}, \ \ \ \
\Omega^{-2m}(p_i^l) = p_{i-mt}^{l}.\eqno(**)$$ In particular, we
have
$$o([1]) = \begin{cases} n, & \ t = 2;\\ 2m, & \ t\ge 3,\end{cases}\eqno(***)$$
where $m$ is the minimal positive integer such that $n\mid mt.$
\vskip10pt
\blacksquareubsection{} Again using the natural basis of $S_i^l$ one has the following commutative
diagrams in $\Lambda\underline {\mbox{-mod}}$:
\[\xymatrix{\mathcal N(S_{i}^l)\ar[rr]^{\mathcal N(
\blacksquareigma_{i}^l)} \ar[d]_{\theta_i^l}^-{\wr}
&& \mathcal N(S_{i-1}^{l+1})\ar[d]_{\theta_{i-1}^{l+1}}^-{\wr}\\
S_{i+1-t}^l \ar[rr]^{
\blacksquareigma_{i+1-t}^l} && S_{i-t}^{l+1};} \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \xymatrix{ \mathcal
N(S_{i}^l)\ar[rr]^{\mathcal N(p_{i}^l)} \ar[d]_{\theta_i^l}^-{\wr}
&&
\mathcal N(S_{i}^{l-1})\ar[d]_{\theta_i^{l-1}}^-{\wr}\\
S_{i+1-t}^l \ar[rr]^{p_{i+1-t}^l} && S_{i+1-t}^{l-1}.}\]
\vskip10pt We justify the commutative diagrams above. Note that for
any finite quiver $Q$ the bimodule structure of $D(kQ)$ is given
by (using the dual basis)
$$p^*a = \begin{cases} b^*, \ & \mbox{if} \ ab = p, \\ 0, \ &
\mbox{otherwise;}\end{cases} \ \ \ \
\ \ \ \ \ \ \ \mbox{and} \ \ \ \ \ \ \ \ \ \ \ ap^* = \begin{cases} b^*, \ & \mbox{if} \ ba = p, \\
0, \ & \mbox{otherwise.}\end{cases}$$ for any paths $p$ and $a$.
Note that $N(S_{i}^l) = D(\Lambda)\otimes_\Lambda S_{i}^l$ is spanned by
$(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}$, where
$j\in\Bbb Z/n\Bbb Z, \ 1\le l'\le t-1, \ 1\le u\le l.$ By
$(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}=
(\gamma_j^{l'})^*\gamma_{i+l-t}^{t-u}\otimes_\Lambda e_{i+l-t}$ we see
that if $(\gamma_j^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}\ne 0$
then $j = i+l-l'-u$; and in this case we have
$$(\gamma_{i+l-l'-u}^{l'})^*\otimes_\Lambda \gamma_{i+l-t}^{t-u}= (\gamma_{i+l-l'-u}^{l'})^*
\gamma_{i+l-t}^{t-u}\otimes_\Lambda e_{i+l-t}
=(\gamma_{i+l-(l'+u)}^{l'+u-t})^*\otimes_\Lambda e_{i+l-t}.$$ This makes
sense only if $l'+u\ge t$. So we have a basis of $N(S_{i}^l)$:
$$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}, \ \ 0\le v\le l-1.$$
Using the natural basis of $S_{i+1-t}^l$ given in 5.1 we have a
$\Lambda$-isomorphism $\theta_i^l: \ \mathcal N(S_{i}^l)\longrightarrow
S_{i+1-t}^l$ for any $i\in\Bbb Z/n\Bbb Z$ and $1\le l\le t-1$:
$$\theta_i^l: \ (\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}\mapsto
\gamma_{i+1+l-2t}^{t-(v+1)}, \ \ 0\le v\le l-1.$$ (One
checks that this is indeed a left $\Lambda$-map.) Note that
$N(
\blacksquareigma_{i}^l): \ \mathcal N(S_{i}^l)\longrightarrow \mathcal
N(S_{i-1}^{l+1})$ is a natural embedding given by
$$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda
e_{i+l-t}\mapsto (\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}, \ \
0\le v\le l-1;$$ and that $N(p_{i}^l): \ \mathcal
N(S_{i}^l)\longrightarrow \mathcal N(S_{i}^{l-1})$ is a
$\Lambda$-epimorphism given by
$$(\gamma_{i+l-t-v}^v)^*\otimes_\Lambda e_{i+l-t}\mapsto
(\gamma_{i+l-t-v}^{v-1})^*\otimes_\Lambda e_{i+l-1-t}, \ \ 0\le v\le
l-1$$ where $\gamma_{i+l-1-t}^{-1}$ is understood to be $0$. Then
one easily checks the following
$$
\blacksquareigma_{i+1-t}^l\circ \theta_i^l = \theta_{i-1}^{l+1}\circ \mathcal N(
\blacksquareigma_i^l); \ \ \ \
p_{i+1-t}^l\circ \theta_i^l = \theta_{i}^{l-1}\circ \mathcal
N(p_i^l).$$ This justifies the commutative diagrams.
Since all these $\theta_i^l$ depend only on $i$ and $l$, which
means that they do not depend on whatever the maps $
\blacksquareigma_i^l$ or
$p_i^l$ are (this is important for the bi-naturality of a
Calabi-Yau category), it follows, without loss of the generality,
that we can specialize these maps to identities. Thus we have
\begin{align} \mathcal N(S^l_i) = S_{i+1-t}^{l}, \ \ \ \mathcal
N(
\blacksquareigma^l_i) =
\blacksquareigma_{i+1-t}^{l}, \ \ \ \mathcal N(p^l_i) =
p_{i+1-t}^{l}.\end{align}
\vskip10pt
\blacksquareubsection{} By $(***)$ we have $o([1]) <
\infty$, it follows, without loss of generality, that we can
assume $d\ge 0$. For convenience, set $d(t): = 1 +
\frac{(d-1)t}{2}\in \frac {1}{2}\Bbb Z$, whatever $d$ is even or
odd; denote by $N=N(d, n, t)$ the minimal positive integer such
that
\begin{align}
\begin{cases} n\mid N d(t), \ &\mbox{if} \ (d-1)t \ \mbox{is
even}; \\ n\mid N(2d(t)), \ &\mbox{if} \ (d-1)t \ \mbox{is odd}.
\end{cases}\end{align}
(When $2d(t)$ is odd, we will write it together in the following.)
\vskip10pt By $(5.1)$ we have
\ $\Omega^{(2m-1)}(S_i^l) = S_{i+l+(m-1)t}^{t-l}$ \ and \ $
\Omega^{2m}(S_i^l) = S_{i+mt}^{l}$, hence by
$(5.2)$ we have (remember $G: = \Omega^{d+1}\circ \mathcal N$ and $d\ge 0$)
\begin{align*} G(S_i^l) = S_{i+1+(m-1)t}^{l}, \ \mbox{if} \ d =
2m-1\end{align*} and
\begin{align*} G(S_i^l) = S_{i+l+1+(m-1)t}^{t-l}, \ \mbox{if} \ d = 2m.\end{align*}
Thus, by induction we have
\begin{align} G^{m'}(S_i^l) = S_{i+m'(1+(m-1)t)}^{l} = S_{i+m'd(t)}^{l}, \ \mbox{if} \ d =
2m-1;\end{align} and
\begin{align} G^{2m'}(S_i^l) = S_{i+m'(2+(2m-1)t)}^{l} = S_{i+m'(2d(t))}^{l}, \ \mbox{if} \ d =
2m,
\end{align}
and
\begin{align} G^{2m'+1}(S_i^l) = S_{i+(m'+1)(2d(t))+l-1-mt}^{t-l}, \ \mbox{if} \ d =
2m,
\end{align}
for $m'\ge 0$.
\blacksquareubsection{} If $d= 2m-1\ge1$, then by $(5.4)$ we see that
$o(G_{S_i^l}) = N$, where $N$ is as given in $(5.3)$, i.e. $N$ is the
minimal positive integer such that $n\mid N(1+(m-1)t)$. It follows
from Corollary 4.3 and $(5.4)$ that we have
\vskip10pt
\begin{lem} Let $d = 2m-1\ge 1$.
Then $M$ is a minimal $d$-th CY $\Lambda$-module
if and only if $M$ is isomorphic to one of the following
\begin{align} S_i^l\oplus S_{i+d(t)}^l\oplus S_{i+2d(t)}^l\oplus \cdots \oplus
S_{i + (N-1)d(t)}^l, \ \ 1\le l\le t-1, \ \ i\in\Bbb Z/n\Bbb
Z.\end{align}
\vskip10pt In particular, all the minimal $d$-th CY modules
have the same number $N = N(d, n, t)$ of
indecomposable direct summands.
\end{lem}
\vskip10pt
\blacksquareubsection{} If $d =
2m\ge 0$ and $t$ is odd, then by $(5.6)$ we see
$G^{2m'+1}(S_i^l)\ne S_i^l$ for any $i, l$ (since $t-l\ne l$).
Note that in this case $(d-1)t$ is odd. It follows from $(5.5)$ that
$o(G_{S_i^l}) = 2N$, where $N$ is as given in $(5.3)$, i.e. $N$
is the minimal positive integer such that $n\mid N(2d(t))$. It
follows from Corollary 4.3, $(5.5)$ and $(5.6)$ that we have
\vskip10pt
\begin{lem} Let $t\ge 3$ be an odd integer and $d = 2m\ge 0$. Then $M$ is a
minimal $d$-th CY $\Lambda$-module if and only
if $M$ is isomorphic to one of the following
\begin{align} S_i^l\oplus S_{i+l' + 2d(t)}^{t-l} \oplus
S_{i+2d(t)}^l\oplus S_{i+l'+ 4d(t)}^{t-l} \oplus \cdots \oplus
S_{i + 2d(t)(N-1)}^l\oplus S_{i + l'+ 2d(t)N}^{t-l}, \end{align}
where $l': = l-1-mt,$ \ $1\le l\le t-1$ \ and \ $i\in\Bbb Z/n\Bbb
Z$.
In particular, any minimal $d$-th CY modules has $2N = 2N(d, n, t)$ indecomposable direct
summands.\end{lem}
\vskip10pt
\blacksquareubsection{} Let $d = 2m\ge 0$ and $t=2s$. Then $d(t) = 1+ (2m-1)s\in\Bbb Z$.
First, we consider $o(G_{S^s_i})$.
In this case $(5.5)$ and $(5.6)$ can be written in a unified way:
\begin{align} G^{m'}(S_i^s) = S_{i+m'd(t)}^{s}, \ m'\ge 0, \ \mbox{if} \ d =
2m, \ t=2s.\end{align} So we have $o(G_{S_i^s}) = N$, where $N$ is
as given in $(5.3)$, i.e. $N$ is the minimal positive integer such
that $n\mid N(1+(2m-1)s)$.
\vskip10pt
Now, we consider $o(G_{S^l_i})$ with $l\ne s, \ 1\le l\le t-1$.
In this case $(5.5)$ and $(5.6)$ are written respectively as:
\begin{align} G^{2m'}(S_i^l) = S_{i+2m'd(t)}^{l}, \ \mbox{if} \ d =
2m, \ t=2s, \ l\ne s,\end{align}
and
\begin{align} G^{2m'+1}(S_i^l) = S_{i+(2m'+1)d(t)+l-s}^{t-l}, \ \mbox{if} \ d =
2m, \ t=2s, \ l\ne s\end{align} for $m'\ge 0.$ Since $l\ne t-l$
for $l\ne s$, it follows that $G^{2m'+1}(S_i^l) \ne S_i^l$. So by
$(5.10)$ we see $o(G_{S_i^l}) = 2N',$ where $N'$ is the minimal
positive integer such that $n\mid 2N'd(t)$. In order to determine
$N'$, we divided into two cases.
{\em Case} 1. \ If $N = N(d, n, t) = N(2m, n, 2s)$ is even, then
$o(G_{S_i^l}) = N$ for any $1\le l\le t-1$. It follows from
Corollary 4.3, $(5.9)$, $(5.10)$, and $(5.11)$ that we have the
following (note that in this case $(5.9)$ is exactly $(5.10)$
together with $(5.11)$, by taking $l=s$)
\vskip10pt
\begin{lem} Let $t = 2s$ and $d = 2m\ge 0$.
Assume that $N = N(d, n, t)$ is even. Then $M$ is a minimal
$d$-th CY $\Lambda$-module if and only if
$M$ is isomorphic to one of the following
\begin{align} S_i^l\oplus S_{i+d(t)+l-s}^{t-l} \oplus
S_{i+2d(t)}^l\oplus S_{i+3d(t)+l-s}^{t-l}\oplus \cdots \oplus S_{i
+ (N-2)d(t)}^l\oplus S_{i + (N-1)d(t)+l-s}^{t-l}\end{align} where
$1\le l\le t-1, \ i\in\Bbb Z/n\Bbb Z$.
\vskip10pt In particular, all the minimal $d$-th CY modules
have the same number $N$ of indecomposable
direct summands.
\end{lem}
\vskip10pt
It remains to deal with
{\em Case} 2. Let $N = N(d, n, t) = N(2m, n, 2s)$ be odd. Since by
definition $N$ is the minimal positive integer such that $n\mid
Nd(t)$, it follows that $N < o(G_{S_i^l}) = 2N' \le 2N.$ It is
easy to see $N'=N$: otherwise $1\le 2N' - N\le N-1$ and $n\mid
(2N' - N)d(t)$, which contradicts the minimality of $N$. It
follows from Corollary 4.3, $(5.9)$, $(5.10)$, and $(5.11)$ that
we have
\vskip10pt
\begin{lem} Let $t = 2s$ and $d = 2m\ge 0$. Assume that $N=N(d, n, t)$ is odd.
Then $M$ is a minimal $d$-th CY $\Lambda$-module if and only if $M$ is isomorphic to one of the following
\begin{align} S_i^s\oplus S_{i+d(t)}^{s} \oplus
S_{i+2d(t)}^s\oplus \cdots \oplus S_{i +
(N-1)d(t)}^{s}\end{align} where $i\in\Bbb Z_n$, and
\begin{align}\begin{matrix} S_i^l \oplus & S_{i+d(t)+l-s}^{t-l}
\oplus & S_{i+2d(t)}^l \oplus
& S_{i+3d(t)+l-s}^{t-l} \oplus \cdots \oplus & S_{i + (N-1)d(t)}^{l}\\
\oplus S_{i+l-s}^{t-l} \oplus & S_{i+d(t)}^{l} \oplus &
S_{i+2d(t)+l-s}^{t-l} \oplus & S_{i+3d(t)}^{l} \oplus \cdots
\oplus & S_{i + (N-1)d(t)+l-s}^{t-l}\end{matrix}\end{align}
where $l\ne s$, \ $1\le l\le t-1$ and $i\in\Bbb Z_n$.
\vskip10pt In particular, all the minimal $d$-th CY modules have
either $N$, or $2N$ indecomposable direct summands. \end{lem}
\vskip10pt
\blacksquareubsection{} By Lemmas 5.1-5.4 all the
minimal $d$-th CY modules of self-injective Nakayama algebras have been classified,
where $d$ is any given integer.
The main result of
this section is as follows.
\vskip10pt
\begin{thm} For any $n\ge 1, \ t\ge 2, \ d\ge 0$, let $N=N(d, n, t)$ be as in $(5.3)$.
Then $M$ is a minimal $d$-th CY \ $\Lambda$-module if and only if $M$
is isomorphic to one of the following
\vskip10pt
$(i)$ \ \ The modules in $(5.7)$, when $d=2m-1$;
$(ii)$ \ \ The modules in $(5.8)$, when $d=2m$ and $t$ is odd;
$(iii)$ \ \ The modules in $(5.12)$, when $d=2m$, $t=2s$, and
$N(d, n, t)$ is even;
$(iv)$ \ \ The modules in $(5.13)-(5.14)$, when $d=2m$, $t=2s$,
and $N(d, n, t)$ is odd.
\vskip10pt
In particular, any minimal $d$-th CY $\Lambda$-module has either $N$ or
$2N$ indecomposable direct summands; and
\begin{align}\operatorname{min} \{d\ge 0 \ |\ N = c(M), \mbox{or}
\ 2N = c(M) \} \ \le \operatorname{CYdim}(M) < o([1]_M) \le 2n
\end{align} for any minimal Calabi-Yau $\Lambda$-modules $M$, where $c(M)$ is
the number of indecomposable direct summands of $M$.
\end{thm}
\noindent {\bf Proof.}\ \ By Lemmas 5.1-5.4 the CY dimension $d$
of any minimal Calabi-Yau module $M$ satisfies $N(d, n, t) = c(M)
\ \mbox{or} \ \ 2N(d, n, t) = c(M).$
$\blacksquare$
\vskip10pt
\begin{rem} The modules in $(5.7)-(5.8)$ and
$(5.12)-(5.14)$ have overlaps. This is because $d$ is not
uniquely determined by a minimal $d$-th CY module. A general
formula of the CY dimensions of the minimal Calabi-Yau modules
seems to be difficult to obtain.
Note that the inequality on the left hand side in $(5.15)$ can not
be an equality in general. For example, take $n = 2, \ t=4, \ m =
2, \ d=2m-1=3$. Then $d(t) = 5$, \ $N = N(3, 2, 4) = 2$, and
$S_i^l\oplus S_{i+1}^l, \ 1\le l\le 3$, are all the minimal $3$-th
CY modules of $CY$ dimension $0$ if $l = 2$, and $1$ if $l=1, \
3$. However, the left hand side in $(5.15)$ is $0$ since $N(0, 2,
4) = 2$. \end{rem}
\vskip10pt
\blacksquareection{\bf Self-injective Nakayama algebras with indecomposable Calabi-Yau modules}
In this section we determine all the self-injective Nakayama
algebras $\Lambda = \Lambda(n, t), \ n\ge 1, \ t\ge 2$, which admit
indecomposable Calabi-Yau modules.
\vskip10pt
Note that Erdmann and Skowro\'nski have proved in $\S 2$ of [ES]
that self-injective algebras $A$ such that $A\underline
{\mbox{-mod}}$ is Calabi-Yau of CY dimension $0$ (resp. $1$) are
the algebras Morita equivalent to $\Lambda(n, 2)$ for some $n\ge 1$
(resp. $\Lambda(1, t)$ for some $t\ge 3$). So we assume that $t\ge 3$.
\vskip10pt
\begin{thm} \ \ Let $t\ge 3$. Then $\Lambda$ has
an indecomposable Calabi-Yau module if and only if $n$ and $t$
satisfy one of the following conditions
$(i)$ \ $g.c.d. \ (n, \ t) = 1$. This is exactly the case where
$\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category. In this
case we have $\operatorname{CYdim}(\Lambda) = 2m-1$, where $m$ is the
minimal positive integer such that $n\mid (m-1)t + 1$.
\vskip10pt
$(ii)$ \ $g.c.d. \ (n,\ t)\ne 1$, \ $t = 2s$, \ and \
$g.c.d. \ (n,\ s) = 1$. This is exactly the case where $\Lambda\underline
{\mbox{-mod}}$ is not a Calabi-Yau category but admits
indecomposable Calabi-Yau modules.
In this case, we have $g.c.d.\ (n,\ t) = 2$ and
$(a)$\ \ $S_i^s, \ i\in \Bbb Z/n\Bbb Z,$ \ are all the
indecomposable Calabi-Yau modules;
$(b)$ \ \ $S_i^l\oplus S_{i+l-s}^{t-l}, \ 1\le l\le s-1, \
i\in\Bbb Z/n\Bbb Z,$ \ are all the decomposable minimal $2m$-th CY
modules, where $m$ is the minimal non-negative integer such that
$n\mid (2m-1)s + 1$;
$(c)$ \ All of these modules in $(a)$ and $(b)$ have the same CY
dimension $2m$.
\end{thm}
\vskip10pt
{\bf Remark.} Bialkowski and Skowro\'nski [BS] have classified
representation-finite self-injective algebras whose stable
categories are Calabi-Yau. This includes the assertion $(i)$ of
Theorem 6.1.
\vskip10pt
\noindent {\bf Proof.}\ \ If $\Lambda$ has an indecomposable $d$-th CY
module $S_i^l$, then $\mathcal N(S^l_i)
\cong \Omega^{-(d+1)}(S^l_i)$. By
$(5.2)$ and $(5.1)$ we have
\begin{align}1-t\equiv -mt \ (\mbox{mod} \ n), \ \mbox{if} \ d+1 =2m,\end{align}
or
\begin{align} t= t-l, \ 1-t\equiv l-mt \ (\mbox{mod} \ n), \ \mbox{if} \ d+1 =2m-1.\end{align}
In the first case we get $g.c.d.\ (n, \ t) = 1.$ In the second case we get
$t=2s, \ l=s$ and $g.c.d.\ (n, \ s) = 1.$ Excluding the overlap
situations we conclude that either $g.c.d.\ (n, \ t) = 1;$ \ or \
$g.c.d.\ (n, \ t) \ne 1, \ t= 2s, \ g.c.d.\ (n, \ s) =
1$.
\vskip10pt
Assume that $g.c.d. \ (n, \ t) = 1$. Then there exists an integer
$m$ such that $n\mid (m-1)t+1$. We chose a positive (otherwise,
add $n(-m+2)t$), and minimal $m$. Set $d: = 2m-1$. Then the same
computation shows that every indecomposable is a $d$-th CY module.
We claim that $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau
category. For this, it remains to show $\mathcal N(f) =
\Omega^{-(d+1)}(f) =\Omega^{-2m}(f)$ for any morphism $f$ between
indecomposables of $\Lambda\underline {\mbox{-mod}}$. Since $n\mid
(m-1)t +1$, it follows from $(**)$ in $\S 5$ that
\begin{align}\mathcal N(
\blacksquareigma_i^l) =
\Omega^{-2m}(
\blacksquareigma^l_i), \ \ \mathcal N(p_i^l) =
\Omega^{-2m}(p^l_i).\end{align} Since $\Lambda$ is
representation-finite, it follows that $f$ is a $k$-combination of
compositions of irreducible maps $
\blacksquareigma^l_i$'s and $p^l_i$'s,
hence $\mathcal N(f) = \Omega^{-2m}(f)$ by $(6.3)$. We stress here
that, this argument relies on the fact that all the isomorphisms
$\theta_i^l$ in 5.2 depend only on $i$ and $l$, which means that
they do not depend on whatever the maps $
\blacksquareigma_i^l$ or $p_i^l$
are. Otherwise we can not take them as identities, and then we can
not get the naturality for the Calabi-Yau category of this case.
This proves the claim, hence $\Lambda\underline {\mbox{-mod}}$ is a
Calabi-Yau category of CY dimension $D$, with $0\le D \le d =
2m-1$. We claim $D=d$. In fact, since every indecomposable is a
$D$-th CY module, and since we have $l$ such that $t\ne t-l$, it
follows from $(6.1)$ and $(6.2)$ that $D = 2m'-1$ with $n\mid
(m'-1)t+1$, hence $D = d$, by the minimality of $m$.
The argument above also proves
that if $\Lambda\underline {\mbox{-mod}}$ is a Calabi-Yau category then
$g.c.d. \ (n, \ t) = 1$.
\vskip10pt
Assume that $g.c.d. \ (n,\ t)\ne 1$, \ $t = 2s$, \ and
$g.c.d. \ (n, \ s) = 1$. Then there exists an integer $m'$ such
that $n\mid (m'-1)s+1$. We choose a positive $m'$. Since $g.c.d.\
(n, \ t) \ne 1$, it follows that $m'$ is even, say $m' =
2m$ with $m\ge 1$. Let $m$ be the the minimal positive integer
such that $n\mid (2m-1)s+1$, and set $d: = 2m$. Then
the same computation shows that \ $S_i^s, \ i\in \Bbb Z/n\Bbb
Z,$ \ are all the indecomposable Calabi-Yau modules.
This proves $(a)$.
By applying Lemma 5.4 to $d$ given above (note that the
corresponding $N = N(n, d, t) = 1$ in this case), we know that $S_i^l\oplus
S_{i+l-s}^{t-l}, \ 1\le l\le t-1, \ l\ne s, \ i\in\Bbb Z/n\Bbb
Z,$ \ are all the decomposable minimal $2m$-th CY modules. By
symmetry one can consider $1\le l\le s-1:$ if $l>s$ then one can replace
$l$ by $t-l$, since $i = (i+l-s) + (t-l)-s.$ This proves $(b)$.
It remains to prove $(c)$. Let $\operatorname{CYdim}(S^s_i) = d'$.
Since $g.c.d. \ (n,\ t)\ne 1$, it follows
that $d'$ has to be an even integer $2m'\ge 0$ with $n\mid
(2m'-1)s +1$. it follows that $d' = d$, by the minimality of $m$.
Let $\operatorname{CYdim}(S_i^l\oplus S_{i+l-s}^{t-l}) = d'$. Then
we have $\mathcal N(S^l_i) = \Omega^{-(d'+1)}(S_{i+l-s}^{t-l})$.
Since $l\ne s$ it follows from $(5.2)$ and $(5.1)$ that $d'$ has
to be $2m'\ge 0$ with $n\mid (2m'-1)s+1$. Again
by the minimality of $m$ we have $d' = d$. This completes the
proof.
$\blacksquare$
\vskip10pt
\begin{rem} $(i)$ $\operatorname{CYdim}(X)$ usually differs from
$\operatorname{CYdim}(\mathcal A)$ in a Calabi-Yau category
$\mathcal A$.
For example, if $n=3, \ t=4$, then $o([1]) = 6$,
$\operatorname{CYdim}(\Lambda) = 5$, while $\operatorname{CYdim}(S_i^2)
= 2$ and $o([1]_{S_i^2}) = 3, \ \forall \ i\in\Bbb Z/3\Bbb Z$.
However, for $t\ge 3$ and \ $g.c.d.\ (n,\ t) = 1$, if $X$ is
indecomposable and $\operatorname{CYdim}(X)$ is odd then
$\operatorname{CYdim}(X) = \operatorname{CYdim}(\Lambda)$. In fact,
since $o([1]) < \infty$ it follows that $\operatorname{CYdim}(X) =
2m'-1\ge 1$, and $n\mid 1+(m'-1)t$. By Theorem 6.1$(i)$
$\operatorname{CYdim}(\Lambda) = 2m-1$, where $m$ is the minimal
positive integer such that $n\mid 1+(m-1)t$. It follows that
$m'\ge m$, hence $\operatorname{CYdim}(X) \ge
\operatorname{CYdim}(\Lambda)$. On the other hand we have
$\operatorname{CYdim}(X) \le \operatorname{CYdim}(\Lambda)$ by
definition.
$(ii)$ \ Consider the algebra $A(t): = kA_\infty ^\infty/J^t$ where
$A_\infty^\infty$ is the infinite quiver
$$ \cdots \longrightarrow \bullet
\longrightarrow \bullet \longrightarrow \bullet \longrightarrow
\bullet \longrightarrow \bullet\longrightarrow \cdots.$$ Then
$A(t)\underline {\mbox{-mod}}$ has a Serre functor, and there is a
natural covering functor $A(t)\underline {\mbox{-mod}}$
$\longrightarrow \Lambda(n, t)\underline {\mbox{-mod}}$ ([Gab], 2.8).
But one can prove that in any case $A(t)\underline {\mbox{-mod}}$
is not a Calabi-Yau category.
\end{rem}
\vskip10pt
{\bf Acknowledgements.} This work is done during a visit of the
second named author at Universit\'e de Montpellier 2, supported by
the CNRS of France. He thanks the first named author and his group
for the warm hospitality, the D\'epartement de Math\'ematiques of
Universit\'e de Montpellier 2 for the working facilities, and the
CNRS for the support. We thank Bernhard Keller for helpful
conversations.
\vskip30pt
\end{document} |
\begin{document}
\title{The G-convex Functions Based on the Nonlinear Expectations Defined by G-BSDEs$^*$}
\footnote[0]{${}^{*}$The Project-sponsored by NSFC (11301068), NSFC
(11171062), NSFC(11371362) and the Fundamental Research Funds for the Central Universities No. 2232014D3-08.}
\author[K. He]{Kun He }
\date{}
\keywords{}
\muaketitle
\begin{center}
{\footnotesize {\it hekun\symbol{64}dhu.edu.cn\\
Department of Mathematics\\
Donghua University\\
2999 North Renmin Rd., Songjiang\\
Shanghai 201620, P.R. China
}}
\end{center}
\begin{abstract}
In this paper, generalizing the definition of $G$-convex functions defined by Peng \cite{Peng2010} during the construction of $G$-expectations and related properties, we define a group of $G$-convex functions based on the Backward Stochastic Differential Equations driven by $G$-Brownian motions.
\end{abstract}
{\bf Key words:} G-expectations, G-BSDEs, G-convex functions, Nonlinear expectations\\
{\bf AMS 2000 subject classifications: } 60H10, 60H30
\section{Introduction}
As we all know, Jensen's inequality is an important result in the theory of linear expectations.
For nonlinear expectations, Jiang \cite{CKJ1,CKJ2,J} talked about a type of $g-$expectation and found out that if this group of
$g-$expectations satisfy the related Jensen's inequality, then the corresponding generator $g$ should satisfies
the propositions of positive homogenous and subadditivity. After this in 2010, Jia and Peng \cite{JP}
defined a new group of functions as $g-$convexity and gave a necessary and sufficient condition for a $\muathbb{C}^2$ function being
a $g-$convex function.
\par
After the construction of the G-expectations by Peng's work from 2005 to 2010 \cite{Peng2005,Peng2007a,Peng2008a,Peng2008b,Peng2010},
another series of work \cite{STZ,Song1,Song2} aims at solving an opening problem, a G-martingale $M$ which can be decomposed into a sum of a symmetric G-martingale $\bar{M}$ and a decreasing G-martingale $K$, and this problem was solved by \cite{PSZ} in 2012. Then Hu, Ji, Peng and Song defined a new type of
Backward Stochastic Differential Equation driven by G-Brownian motion (G-BSDE) \cite{HJPS1}, proved a related comparison theorem and defined the group of nonlinear expectations by the solutions of G-BSDEs \cite{HJPS2}. Based on their definition of this group of nonlinear expectations and the related comparison theorem of G-BSDEs, He and Hu \cite{HH} proved a representation
theorem for this group of nonlinear expectations and proved some related equivalent conditions between the generator and
related nonlinear expectations. In this paper, we will talk about the $G-$convex function, defined in Peng \cite{Peng2010}, under the framework of the nonlinear expectations defined by the G-BSDEs \cite{HJPS1}. In section \ref{Sec:Preliminary}, we recall some fundamental definitions
and results about G-expectations and G-BSDEs. In section \ref{Sec:Main}, we will prove our main result, giving the equivalent condition of G-convex function under the framework of G-BSDEs.
\section{Preliminary}\label{Sec:Preliminary}
Let us recall some notations for the related spaces of random variables, definitions and results in the construction of G-Brownian motions and G-expectations. The readers may refer to \cite{Peng2007a,Peng2008a,Peng2008b,Peng2010,HJPS1}.
Throughout the paper, for $x\in\muathbb{R}^d$, we denote $|x|=\sqrt{x\cdot x}$ and $\langle x,x\rangle=x\cdot x$.
\begin{definition}\label{def2.1}
Let $\Omega$ be a given set and let $\muathcal{H}$ be a vector
lattice of real valued functions defined on $\Omega$, namely $c\in \muathcal{H}$
for each constant $c$ and $|X|\in \muathcal{H}$ if $X\in \muathcal{H}$.
$\muathcal{H}$ is considered as the space of random variables. A sublinear
expectation $\muathbb{\hat{E}}$ on $\muathcal{H}$ is a functional $\muathbb{\hat
{E}}:\muathcal{H}\rightarrow \muathbb{R}$ satisfying the following properties:
for all $X,Y\in \muathcal{H}$, we have
\item[(a)] Monotonicity: If $X\geq Y$ then $\muathbb{\hat{E}}[X]\geq
\muathbb{\hat{E}}[Y]$;
\item[(b)] Constant preservation: $\muathbb{\hat{E}}[c]=c$;
\item[(c)] Sub-additivity: $\muathbb{\hat{E}}[X+Y]\leq \muathbb{\hat{E}
}[X]+\muathbb{\hat{E}}[Y]$;
\item[(d)] Positive homogeneity: $\muathbb{\hat{E}}[\lambda X]=\lambda
\muathbb{\hat{E}}[X]$ for each $\lambda \geq 0$.
$(\Omega,\muathcal{H},\muathbb{\hat{E}})$ is called a sublinear expectation space.
\end{definition}
\begin{definition}
\label{def2.2} Let $X_{1}$ and $X_{2}$ be two $n$-dimensional random vectors
defined respectively in sublinear expectation spaces $(\Omega_{1}
,\muathcal{H}_{1},\muathbb{\hat{E}}_{1})$ and $(\Omega_{2},\muathcal{H}
_{2},\muathbb{\hat{E}}_{2})$. They are identically distributed, denoted
by $X_{1}\overset{d}{=}X_{2}$, if $\muathbb{\hat{E}}_{1}[\varphi(X_{1}
)]=\muathbb{\hat{E}}_{2}[\varphi(X_{2})]$, for all$\ \varphi \in C_{b.Lip}
(\muathbb{R}^{n})$, where $C_{b.Lip}(\muathbb{R}^{n})$ denotes the space of
bounded and Lipschitz functions on $\muathbb{R}^{n}$.
\end{definition}
\begin{definition}
\label{def2.3} In a sublinear expectation space $(\Omega,\muathcal{H}
,\muathbb{\hat{E}})$, a random vector $Y=(Y_{1},\cdot \cdot \cdot,Y_{n})$,
$Y_{i}\in \muathcal{H}$, is said to be independent of another random vector
$X=(X_{1},\cdot \cdot \cdot,X_{m})$, $X_{i}\in \muathcal{H}$ under $\muathbb{\hat
{E}}[\cdot]$, denoted by $Y\bot X$, if for every test function $\varphi \in
C_{b.Lip}(\muathbb{R}^{m}\times \muathbb{R}^{n})$ we have $\muathbb{\hat{E}
}[\varphi(X,Y)]=\muathbb{\hat{E}}[\muathbb{\hat{E}}[\varphi(x,Y)]_{x=X}]$.
\end{definition}
\begin{definition}
\label{def2.4} ($G$-normal distribution) A $d$-dimensional random vector
$X=(X_{1},\cdot \cdot \cdot,X_{d})$ in a sublinear expectation space
$(\Omega,\muathcal{H},\muathbb{\hat{E}})$ is called $G$-normally distributed if
for each $a,b\geq0$ we have
\[
aX+b\bar{X}\overset{d}{=}\sqrt{a^{2}+b^{2}}X,
\]
where $\bar{X}$ is an independent copy of $X$, i.e., $\bar{X}\overset{d}{=}X$
and $\bar{X}\bot X$. Here the letter $G$ denotes the function
\[
G(A):=\frac{1}{2}\muathbb{\hat{E}}[\langle AX,X\rangle]:\muathbb{S}
_{d}\rightarrow \muathbb{R},
\]
where $\muathbb{S}_{d}$ denotes the collection of $d\times d$ symmetric matrices.
\end{definition}
Peng \cite{Peng2008b} showed that $X=(X_{1},\cdot \cdot \cdot,X_{d})$ is $G$-normally
distributed if and only if for each $\varphi \in C_{b.Lip}(\muathbb{R}^{d})$,
$u(t,x):=\muathbb{\hat{E}}[\varphi(x+\sqrt{t}X)]$, $(t,x)\in \lbrack
0,\infty)\times \muathbb{R}^{d}$, is the solution of the following $G$-heat
equation:
\[
\partial_{t}u-G(D_{x}^{2}u)=0,\ u(0,x)=\varphi(x).
\]
The function $G(\cdot):\muathbb{S}_{d}\rightarrow \muathbb{R}$ is a monotonic,
sublinear mapping on $\muathbb{S}_{d}$ and $G(A)=\frac{1}{2}\muathbb{\hat{E}
}[\langle AX,X\rangle]\leq \frac{1}{2}|A|\muathbb{\hat{E}}[|X|^{2}]$ implies
that there exists a bounded, convex and closed subset $\Gamma \subset
\muathbb{S}_{d}^{+}$ such that
\[
G(A)=\frac{1}{2}\sup_{\gamma \in \Gamma}\muathrm{tr}[\gamma A],
\]
where $\muathbb{S}_{d}^{+}$ denotes the collection of non-negative elements in
$\muathbb{S}_{d}$.
In this paper, we only consider non-degenerate $G$-normal distribution, i.e.,
there exists some $\underline{\sigma}^{2}>0$ such that $G(A)-G(B)\geq
\underline{\sigma}^{2}\muathrm{tr}[A-B]$ for any $A\geq B$.
\begin{definition}
\label{def2.5} i) Let $\Omega=C_{0}^{d}(\muathbb{R}^{+})$ denote the space of
$\muathbb{R}^{d}$-valued continuous functions on $[0,\infty)$ with $\omega
_{0}=0$ and let $B_{t}(\omega)=\omega_{t}$ be the canonical process. Set
\[
L_{ip}(\Omega):=\{ \varphi(B_{t_{1}},...,B_{t_{n}}):n\geq1,t_{1},...,t_{n}
\in \lbrack0,\infty),\varphi \in C_{b.Lip}(\muathbb{R}^{d\times n})\}.
\]
Let $G:\muathbb{S}_{d}\rightarrow \muathbb{R}$ be a given monotonic and sublinear
function. $G$-expectation is a sublinear expectation defined by
\[
\muathbb{\hat{E}}[X]=\muathbb{\tilde{E}}[\varphi(\sqrt{t_{1}-t_{0}}\xi_{1}
,\cdot \cdot \cdot,\sqrt{t_{m}-t_{m-1}}\xi_{m})],
\]
for all $X=\varphi(B_{t_{1}}-B_{t_{0}},B_{t_{2}}-B_{t_{1}},\cdot \cdot
\cdot,B_{t_{m}}-B_{t_{m-1}})$, where $\xi_{1},\cdot \cdot \cdot,\xi_{n}$ are
identically distributed $d$-dimensional $G$-normally distributed random
vectors in a sublinear expectation space $(\tilde{\Omega},\tilde{\muathcal{H}
},\muathbb{\tilde{E}})$ such that $\xi_{i+1}$ is independent of $(\xi_{1}
,\cdot \cdot \cdot,\xi_{i})$ for every $i=1,\cdot \cdot \cdot,m-1$. The
corresponding canonical process $B_{t}=(B_{t}^{i})_{i=1}^{d}$ is called a
$G$-Brownian motion.
ii) For each fixed $t\in \lbrack0,\infty)$, the conditional $G$-expectation
$\muathbb{\hat{E}}_{t}$ for $\xi=\varphi(B_{t_{1}}-B_{t_{0}},B_{t_{2}}
-B_{t_{1}},\cdot \cdot \cdot,B_{t_{m}}-B_{t_{m-1}})\in L_{ip}(\Omega)$, without
loss of generality we suppose $t_{i}=t$, is defined by
\[
\muathbb{\hat{E}}_{t}[\varphi(B_{t_{1}}-B_{t_{0}},B_{t_{2}}-B_{t_{1}}
,\cdot \cdot \cdot,B_{t_{m}}-B_{t_{m-1}})]
\]
\[
=\psi(B_{t_{1}}-B_{t_{0}},B_{t_{2}}-B_{t_{1}},\cdot \cdot \cdot,B_{t_{i}
}-B_{t_{i-1}}),
\]
where
\[
\psi(x_{1},\cdot \cdot \cdot,x_{i})=\muathbb{\hat{E}}[\varphi(x_{1},\cdot
\cdot \cdot,x_{i},B_{t_{i+1}}-B_{t_{i}},\cdot \cdot \cdot,B_{t_{m}}-B_{t_{m-1}
})].
\]
\end{definition}
For each fixed $T>0$, we set
\[
L_{ip}(\Omega_{T}):=\{ \varphi(B_{t_{1}},...,B_{t_{n}}):n\geq1,t_{1}
,...,t_{n}\in \lbrack0,T],\varphi \in C_{b.Lip}(\muathbb{R}^{d\times n})\}.
\]
For each $p\geq1$, we denote by $L_{G}^{p}(\Omega)$ (resp. $L_{G}^{p}
(\Omega_{T})$) the completion of $L_{ip}(\Omega)$ (resp. $L_{ip}(\Omega_{T})$)
under the norm $\Vert \xi \Vert_{p,G}=(\muathbb{\hat{E}}[|\xi|^{p}])^{1/p}$. It
is easy to check that $L_{G}^{q}(\Omega)\subset L_{G}^{p}(\Omega)$ for $1\leq
p\leq q$ and $\muathbb{\hat{E}}_{t}[\cdot]$ can be extended continuously to
$L_{G}^{1}(\Omega)$.
For each fixed $\muathbf{a}\in \muathbb{R}^{d}$, $B_{t}^{\muathbf{a}}
=\langle \muathbf{a},B_{t}\rangle$ is a $1$-dimensional $G_{\muathbf{a}}
$-Brownian motion, where $G_{\muathbf{a}}(\alpha)=\frac{1}{2}(\sigma
_{\muathbf{aa}^{T}}^{2}\alpha^{+}-\sigma_{-\muathbf{aa}^{T}}^{2}\alpha^{-})$,
$\sigma_{\muathbf{aa}^{T}}^{2}=2G(\muathbf{aa}^{T})$, $\sigma_{-\muathbf{aa}^{T}
}^{2}=-2G(-\muathbf{aa}^{T})$. Let $\pi_{t}^{N}=\{t_{0}^{N},\cdots,t_{N}^{N}
\}$, $N=1,2,\cdots$, be a sequence of partitions of $[0,t]$ such that $\muu
(\pi_{t}^{N})=\muax \{|t_{i+1}^{N}-t_{i}^{N}|:i=0,\cdots,N-1\} \rightarrow0$,
the quadratic variation process of $B^{\muathbf{a}}$ is defined by
\[
\langle B^{\muathbf{a}}\rangle_{t}=\lim_{\muu(\pi_{t}^{N})\rightarrow0}
\sum_{j=0}^{N-1}(B_{t_{j+1}^{N}}^{\muathbf{a}}-B_{t_{j}^{N}}^{\muathbf{a}}
)^{2}.
\]
For each fixed $\muathbf{a}$, $\muathbf{\bar{a}}\in \muathbb{R}^{d}$, the mutual
variation process of $B^{\muathbf{a}}$ and $B^{\muathbf{\bar{a}}}$ is defined by
\[
\langle B^{\muathbf{a}},B^{\muathbf{\bar{a}}}\rangle_{t}=\frac{1}{4}[\langle
B^{\muathbf{a}+\muathbf{\bar{a}}}\rangle_{t}-\langle B^{\muathbf{a}
-\muathbf{\bar{a}}}\rangle_{t}].
\]
\begin{definition}
\label{def2.6} For fixed $T>0$, let $M_{G}^{0}(0,T)$ be the collection of
processes in the following form: for a given partition $\{t_{0},\cdot
\cdot \cdot,t_{N}\}=\pi_{T}$ of $[0,T]$,
\[
\eta_{t}(\omega)=\sum_{j=0}^{N-1}\xi_{j}I_{[t_{j},t_{j+1})}(t),
\]
where $\xi_{j}\in L_{ip}(\Omega_{t_{j}})$, $j=0,1,2,\cdot \cdot \cdot,N-1$. For
$p\geq1$, we denote by $H_{G}^{p}(0,T)$, $M_{G}^{p}(0,T)$ the completion of
$M_{G}^{0}(0,T)$ under the norms $\Vert \eta \Vert_{H_{G}^{p}}=\{ \muathbb{\hat
{E}}[(\int_{0}^{T}|\eta_{s}|^{2}ds)^{p/2}]\}^{1/p}$, $\Vert \eta \Vert
_{M_{G}^{p}}=\{ \muathbb{\hat{E}}[\int_{0}^{T}|\eta_{s}|^{p}ds]\}^{1/p}$ respectively.
\end{definition}
For each $\eta \in M_{G}^{1}(0,T)$, we can define the integrals $\int_{0}
^{T}\eta_{t}dt$ and $\int_{0}^{T}\eta_{t}d\langle B^{\muathbf{a}}
,B^{\muathbf{\bar{a}}}\rangle_{t}$ for each $\muathbf{a}$, $\muathbf{\bar{a}}
\in \muathbb{R}^{d}$. For each $\eta \in H_{G}^{p}(0,T;\muathbb{R}^{d})$ with
$p\geq1$, we can define It\^{o}'s integral $\int_{0}^{T}\eta_{t}dB_{t}$. In the following $\langle B\rangle$ denotes the quadratic variation of $B$ (refer to \cite{Peng2010,HJPS1,PSZ}).
Let $S_{G}^{0}(0,T)=\{h(t,B_{t_{1}\omegaedge t},\cdot \cdot \cdot,B_{t_{n}\omegaedge
t}):t_{1},\ldots,t_{n}\in \lbrack0,T],h\in C_{b,Lip}(\muathbb{R}^{n+1})\}$. For
$p\geq1$ and $\eta \in S_{G}^{0}(0,T)$, set $\Vert \eta \Vert_{S_{G}^{p}}=\{
\muathbb{\hat{E}}[\sup_{t\in \lbrack0,T]}|\eta_{t}|^{p}]\}^{\frac{1}{p}}$.
Denote by $S_{G}^{p}(0,T)$ the completion of $S_{G}^{0}(0,T)$ under the norm
$\Vert \cdot \Vert_{S_{G}^{p}}$.
We consider the following type of $G$-BSDEs (in this paper we always use
Einstein convention):
\begin{equation}\label{eq:GBSDE1}
Y_{t} =\xi+\int_{t}^{T}g(s,Y_{s},Z_{s})ds+\int_{t}^{T}f(s,Y_{s}
,Z_{s})d\langle B\rangle_{s}
-\int_{t}^{T}Z_{s}dB_{s}-(K_{T}-K_{t}),
\end{equation}
where
\[
g(t,\omega,y,z),f(t,\omega,y,z):[0,T]\times \Omega_{T}\times
\muathbb{R}\times \muathbb{R}\rightarrow \muathbb{R}
\]
satisfy the following properties:
\begin{itemize}
\item[(H1)] There exists some $\beta>1$ such that for any $y,z$,
$g(\cdot,\cdot,y,z),f(\cdot,\cdot,y,z)\in M_{G}^{\beta}(0,T)$.
\item[(H2)] There exists some $L>0$ such that
\[
|g(t,y,z)-g(t,y^{\prime},z^{\prime})|+|f(t,y,z)-f(t,y^{\prime},z^{\prime})|\leq L(|y-y^{\prime}|+|z-z^{\prime}|).
\]
\end{itemize}
For simplicity, we denote by $\muathfrak{S}_{G}^{\alpha}(0,T)$ the collection
of processes $(Y,Z,K)$ such that $Y\in S_{G}^{\alpha}(0,T)$, $Z\in
H_{G}^{\alpha}(0,T;\muathbb{R})$, $K$ is a decreasing $G$-martingale with
$K_{0}=0$ and $K_{T}\in L_{G}^{\alpha}(\Omega_{T})$.
\begin{definition}
\label{def3.1} Let $\xi \in L_{G}^{\beta}(\Omega_{T})$, $g$ and $f$ satisfy (H1) and
(H2) for some $\beta>1$. A triplet of processes $(Y,Z,K)$ is called a solution
of equation (\ref{eq:GBSDE1}) if for some $1<\alpha \leq \beta$ the following
properties hold:
\begin{itemize}
\item[(a)] $(Y,Z,K)\in \muathfrak{S}_{G}^{\alpha}(0,T)$;
\item[(b)] $Y_{t}=\xi+\int_{t}^{T}g(s,Y_{s},Z_{s})ds+\int_{t}^{T}
f(s,Y_{s},Z_{s})d\langle B\rangle_{s}-\int_{t}^{T}Z_{s}
dB_{s}-(K_{T}-K_{t})$.
\end{itemize}
\end{definition}
\begin{lemma}
\label{the1.1} (\cite{HJPS1}) Assume that $\xi \in L_{G}^{\beta}(\Omega_{T})$
and $g$, $f$ satisfy (H1) and (H2) for some $\beta>1$. Then equation
(\ref{eq:GBSDE1}) has a unique solution $(Y,Z,K)$. Moreover, for any $1<\alpha<\beta$
we have $Y\in S_{G}^{\alpha}(0,T)$, $Z\in H_{G}^{\alpha}(0,T;\muathbb{R})$
and $K_{T}\in L_{G}^{\alpha}(\Omega_{T})$.
\end{lemma}
In this paper, we also need the following assumptions for $G$-BSDE
(\ref{eq:GBSDE1}).
\begin{itemize}
\item[(H3)] For each fixed $(\omega,y,z)\in \Omega_{T}\times \muathbb{R}
\times \muathbb{R}$, $t\rightarrow g(t,\omega,y,z)$ and $t\rightarrow
f(t,\omega,y,z)$ are continuous.
\item[(H4)] For each fixed $(t,y,z)\in \lbrack0,T)\times \muathbb{R}
\times \muathbb{R}$, $g(t,y,z)$, $f(t,y,z)\in L_{G}^{\beta}(\Omega
_{t})$ and
\[
\lim_{\varepsilon \rightarrow0+}\frac{1}{\varepsilon}\muathbb{\hat{E}}[\int
_{t}^{t+\varepsilon}(|g(u,y,z)-g(t,y,z)|^{\beta}+
|f(u,y,z)-f(t,y,z)|^{\beta})du]=0.
\]
\item[(H5)] $K_t=\int_0^t\eta_sd\langle B\rangle_s-2\int_0^tG(\eta_s)ds$,
where $\eta\in M_G^p(0,T)$, $p\geq 1$.
\item[(H6)] For each $(t,\omega,y)\in \lbrack0,T]\times \Omega_{T}
\times \muathbb{R}$, $g(t,\omega,y,0)=f(t,\omega,y,0)=0$.
\end{itemize}
Assume that $\xi \in L_{G}^{\beta}(\Omega_{T})$, $g$ and $f$ satisfy (H1) and
(H2) for some $\beta>1$. Let $(Y^{T,\xi},Z^{T,\xi},K^{T,\xi})$ be the
solution of $G$-BSDE (\ref{eq:GBSDE1}) corresponding to $\xi$, $g$ and $f$
on $[0,T]$. It is easy to check that $Y^{T,\xi}=Y^{T^{\prime},\xi}$ on $[0,T]$
for $T^{\prime}>T$. Following (\cite{HJPS2}), we define
a nonlinear expectation as
\[
\muathbb{\muathcal{E}}_{s,t}[\xi]=Y_{s}^{t,\xi}\text{\quad for \quad }0\leq s\leq t\leq T.
\]
\begin{remark}
In \cite{HJPS2,HH} they both define the nonlinear expectation under the assumption(H1), (H2) and (H6), their consistent nonlinear
expectation was defined by $\muathcal{{E}}_{t}[\xi]=Y_{t}^{T,\xi}\text{ for }t\in \lbrack0,T]$. As described by \cite{HJPS2}, under assumption (H6), the nonlinear expectation satisfies, for $T_1<T_2$, $\muathcal{E}_{t,T_1}[\xi]=\muathcal{E}_{t,T_2}[\xi]$. Then the $\muathcal{E}_t[\xi]=\muathcal{E}_{t,T}[\xi]=Y_t^{T,\xi}$ notation is used.
\end{remark}
The classical g-expectations possess many properties that are useful
in finance and economics and became an important risk measure tool in financial mathematics under the complete market case. The nonlinear expectations derived by the G-BSDEs is a useful
generalization of $g$-expectations defined on an incomplete market case.
\section{Main result}\label{Sec:Main}
\begin{definition}\label{def:1}
For $\xi\in \muathbb{L}_G^{\infty}(\Omega_s), ~ 0\leq t\leq s\leq T$, we define the function $h\in C^2(\muathbb{R})$ be $G$-convex, if $\muathcal{E}_{t,s}[h(\xi)]\geq h[\muathcal{E}_{t,s}(\xi)]$.
\end{definition}
\begin{lemma}\label{le:1}(see \cite{HJPS1}).
Let $\xi\in L_T^{\beta}(\Omega_T)$ and $g, f$ satisfy (H1) and (H2) for some $\beta>1$. Assume that $(Y,Z,K)$ satisfies $(Y,Z)\in S_G^{\alpha}(0,T)\times H_G^{\alpha}(0,T;\muathbb{R}^d)$ and K is a decreasing G-martingale with $K_0=0$ and
$K_T\in L_G^{\alpha}(\Omega_T)$ for some $1<\alpha<\beta$ is a solution of \eqref{eq:GBSDE1}. Then, there exists a constant $C_{\alpha}>0$ depending on $\alpha, T, G$ and $L$ such that
\begin{equation}\label{eq:Y-esti}
|Y_t|^{\alpha}\leq C_{\alpha}\hat{{\muathbb E}}_t\left[|\xi|^{\alpha}+\left(\int_t^T|h_s^0|ds\right)^{\alpha}\right],
\end{equation}
\begin{equation}\label{eq:Z-esti}
\hat{{\muathbb E}}\left[\left(\int_0^T|Z_s|^2ds\right)^{\alpha/2}\right]\leq C_{\alpha}\left\{\hat{{\muathbb E}}\left[\sup_{t\in[0,T]}|Y_t|^{\alpha}\right]+
\left(\hat{{\muathbb E}}\left[\sup_{t\in[0,T]}|Y_t|^{\alpha}\right]\right)^{1/2}
\left(\hat{{\muathbb E}}\left[\left(\int_0^Th_s^0ds\right)^{\alpha}\right]\right)^{1/2}\right\},
\end{equation}
where $h_s^0=|g(s,0,0)|+|f(s,0,0)|$.
\end{lemma}
\begin{lemma}\label{le:2}(see \cite{HJPS1,Song1})
Let $\alpha\geq 1$ and $\delta>0$ be fixed. Then, there exists a constant C depending on $\alpha$ and $\delta$ such that
\begin{equation}\label{eq:xi-esti}
\hat{{\muathbb E}}\left[\sup_{t\in[0,T]}\hat{E}_t[|\xi|^{\alpha}]\right]\leq C\left\{\left(\hat{{\muathbb E}}\left[|\xi|^{\alpha+\delta}\right]\right)^{\alpha/(\alpha+\delta)}+
\hat{{\muathbb E}}\left[|\xi|^{\alpha+\delta}\right]\right\},
\end{equation}
$\forall \xi\in L_G^{\alpha+\delta}(\Omega_T)$.
\end{lemma}
Following Theorem 12 in \cite{HH}, we have the following representation.
\begin{lemma}\label{le:3}
Suppose (H1)-(H4) satisfied. Take a polynomial growth function $\Phi\in C^2_b(\muathbb{R})$, $s\in[t,t+\epsilon]$.
$$Y_s=\Phi(B_{t+\epsilon}-B_t)+\int_s^{t+\epsilon}g(r,Y_r,Z_r)dr+\int_s^{t+\epsilon}f(r,Y_r,Z_r)d\langle B\rangle_r-
\int_s^{t+\epsilon}Z_rdB_r-(K_{t+\epsilon}-K_s).$$
Then
\begin{equation}\label{eq:g-represent}
L^2_G-\lim_{\epsilon\rightarrow 0+}\frac{\muathcal{E}_{t,t+\epsilon}[\Phi(B_{t+\epsilon}-B_t)]-\Phi(0)}{\epsilon}
=g(t,\Phi(0),\Phi'(0))+2G(f(t,\Phi(0),\Phi'(0))+\frac12\Phi''(0)).
\end{equation}
\end{lemma}
{\bf Proof: } Let $\tilde{Y}_s=Y_s-\Phi(B_s-B_t)$, then $\tilde{Y}_t=Y_t-\Phi(0)$ and $\tilde{Y}_{t+\epsilon}=
Y_{t+\epsilon}-\Phi(B_{t+\epsilon}-B_t)=0$ hold.
By using Ito's formula,
\begin{equation*}\begin{split}
-d\tilde{Y}_s=&-dY_s+d\Phi(B_s-B_t)\\
=&g(s,Y_s,Z_s)ds+f(s,Y_s,Z_s)d\langle B\rangle_s-Z_sdB_s-dK_s\\
&+\Phi'(B_s-B_t)dB_s+\frac 12\Phi''(B_s-B_t)d\langle B\rangle_s
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\tilde{Y}_s=&0+\int_s^{t+\epsilon}g(r,Y_r,Z_r)dr+\int_s^{t+\epsilon}(f(r,Y_r,Z_r)+\frac12\Phi''(B_r-B_t))d\langle B\rangle_r\\
&-\int_s^{t+\epsilon}(Z_r-\Phi'(B_r-B_t))dB_r-(K_{t+\epsilon}-K_s).
\end{split}\end{equation*}
Let $\tilde{Z}_s=Z_s-\Phi'(B_s-B_t)$ and $\tilde{K}_s=K_s$. Then $(\tilde{Y}_s,\tilde{Z}_s,\tilde{K}_s)$
satisfies the G-BSDE:
\begin{equation*}\begin{split}
\tilde{Y}_s=&0+\int_s^{t+\epsilon}g(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r+\Phi'(B_r-B_t))dr\\
&+\int_s^{t+\epsilon}(f(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r+\Phi'(B_r-B_t))+\frac12\Phi''(B_r-B_t))d\langle B\rangle_r\\
&-\int_s^{t+\epsilon}\tilde{Z_r}dB_r-(\tilde{K}_{t+\epsilon}-K_s).
\end{split}\end{equation*}
From Lemma \ref{le:1},
\begin{equation*}\begin{split}
|\tilde{Y}^{\epsilon}_s|^{\alpha}&\leq C_{\alpha}\hat{{\muathbb E}}_s\left[\left(\int_s^{t+\epsilon}(
|g(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))|\right.\right.\\
&\left.\left.+|f(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))|+\frac12|\phi''(B_r-B_t)|
)dr\right)^{\alpha}\right],
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\hat{{\muathbb E}}\left[\left(\int_t^{t+\epsilon}|\tilde{Z}_r^{\epsilon}|^2dr\right)^{\alpha/2}\right]&\leq
C_{\alpha}\left\{\hat{{\muathbb E}}\left[\left(\int_t^{t+\epsilon}(|g(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))|+
\frac12|\Phi''(B_r-B_t)|\right.\right.\right.\\
&\left.\left.\left.+|f(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))|)dr\right)^{\alpha}\right]
+\hat{{\muathbb E}}\left[\sup_{s\in[t,t+\epsilon]}|\tilde{Y}^{\epsilon}_s|^{\alpha}\right]\right\}
\end{split}\end{equation*}
hold for some constant $C_{\alpha}>0$, only depending on $\alpha, T, G$ and $L$.
\begin{multline*}
\int_t^{t+\epsilon}\left(|g(r,0,0)|^{\beta}+|f(r,0,0)|^{\beta}\right)dr\leq 2^{\beta-1}\left\{
\epsilon\left(|g(t,0,0)|^{\beta}+|f(t,0,0)|^{\beta}\right)\right.\\
\left.
+\int_t^{t+\epsilon}\left(
|g(r,0,0)-g(t,0,0)|^{\beta}+|f(r,0,0)-f(t,0,0)|^{\beta}\right)dr\right\}.
\end{multline*}
Together with Lemma \ref{le:2} and assumption (H4), we get
\begin{equation}\label{eq:YZ-esti}
\hat{{\muathbb E}}\left[\sup_{s\in[t,t+\epsilon]}|\tilde{Y}_s^{\epsilon}|^{\alpha}+
\left(\int_t^{t+\epsilon}|\tilde{Z}^{\epsilon}_r|^2\right)^{\alpha/2}\right]\leq C_3\epsilon^{\alpha},
\end{equation}
where $C_3$ depends on $x, y, p,\alpha, \beta, T, G$ and $L$.
Now we prove \eqref{eq:g-represent}. Dividing $\epsilon$, take conditional G-expectations and take limits on both sides of the equation in $L_G^2$ norm, then
$\forall \Phi\in C_b^2(\muathbb{R})$,
\begin{equation*}
\begin{split}
\lim_{\epsilon\rightarrow 0+}\frac{\tilde{Y}_t}{\epsilon}=&\lim_{\epsilon\rightarrow 0+}\frac1{\epsilon}
\hat{E}_t[\tilde{Y}_t+(\tilde{K}_{t+\epsilon}-\tilde{K}_t)]\\
=&\lim_{\epsilon\rightarrow 0+}\frac1{\epsilon}\hat{{\muathbb E}}_t\left[\int_t^{t+\epsilon}g(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r
+\Phi'(B_r-B_t))dr\right.
\\
&\left.+\int_t^{t+\epsilon}\left(f(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r+\Phi'(B_r-B_t))
+\frac12\Phi''(B_r-B_t)\right)d\langle B\rangle_r\right]\\
=&\lim_{\epsilon\rightarrow 0+}\frac1{\epsilon}\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}g(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))dr\right.\\
&\left.+\int_t^{t+\epsilon}\left(f(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))+\frac12 \Phi''(B_r-B_t)\right)d\langle B\rangle_r
\right]+L_{\epsilon}
\end{split}\end{equation*}
where
\begin{equation*}\begin{split}
L_{\epsilon}=&\frac1{\epsilon}\left\{\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}
g(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r+\Phi'(B_r-B_t))dr\right.\right.\\ +&\left.\int_t^{t+\epsilon}\left(f(r,\tilde{Y}_r+\Phi(B_r-B_t),\tilde{Z}_r+\Phi'(B_r-B_t))+
\frac12\Phi''(B_r-B_t)\right)d\langle B\rangle_r\right]\\
-&\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}g(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))dr\right.\\
+&\left.\left.\int_t^{t+\epsilon}\left(f(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))+\frac12\Phi''(B_r-B_t)\right)d\langle B\rangle_r\right]\right\}
\end{split}\end{equation*}
It can be verified that $|L_{\epsilon}|\leq (C_4/\epsilon)\hat{{\muathbb E}}[\int_t^{t+\epsilon}(|\tilde{Y}_r|+|\tilde{Z}_r|)dr]$,
where $C_4$ depends on $G, L$ and $T$. By \eqref{eq:YZ-esti}, we have
\begin{equation*}\begin{split}
\hat{{\muathbb E}}[|L_{\epsilon}|^{\alpha}]\leq&
\frac{C_4^{\alpha}}{\epsilon^{\alpha}}\hat{{\muathbb E}}\left[\left(\int_t^{t+\epsilon}(|\tilde{Y}_r|+
|\tilde{Z}_r|)dr\right)^{\alpha}\right]\\
\leq & \frac{2^{\alpha-1}C_4^{\alpha}}{\epsilon^{\alpha}}\hat{{\muathbb E}}\left[\left(\int_t^{t+\epsilon}|\tilde{Y}_r|dr\right)^{\alpha}+
\left(\int_t^{t+\epsilon}|\tilde{Z}_r|dr\right)^{\alpha}\right]\\
\leq &
2^{\alpha-1}C_4^{\alpha}\left\{\hat{{\muathbb E}}\left[\sup_{s\in[t,t+\epsilon]}|\tilde{Y}_s|^{\alpha}\right]+
\epsilon^{-\alpha/2}\hat{{\muathbb E}}\left[\left(\int_t^{t+\epsilon}|\tilde{Z}_r|^2dr\right)^{\alpha/2}\right]\right\}\\
\leq &
2^{\alpha-1}C_4^{\alpha}C_3(\epsilon^{\alpha}+\epsilon^{\alpha/2}),
\end{split}\end{equation*}
then $L_G^{\alpha}-\lim_{\epsilon\rightarrow 0+}L_{\epsilon}=0$.
We set
\begin{equation*}\begin{split}
M_{\epsilon}=&\frac1{\epsilon}\left\{\hat{{\muathbb E}}_t\left[\int_t^{t+\epsilon}g(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))dr\right.\right.\\
+&\left.\int_t^{t+\epsilon}f(r,\Phi(B_r-B_t),\Phi'(B_r-B_t))+\frac12\Phi''(B_r-B_t)d\langle B\rangle_r\right]\\
-&\left.\hat{{\muathbb E}}_t\left[\int_t^{t+\epsilon}g(r,\Phi(0),\Phi'(0))dr+\int_t^{t+\epsilon}\left(f(r,\Phi(0),\Phi'(0))
+\frac12\Phi''(0)\right)d\langle B\rangle_r
\right]\right\}
\end{split}\end{equation*}
By the Lipschitz condition of function $g$ and $f$, and the polynomial growth of $\Phi\in C_b^2(\muathbb{R})$,
we have $L_{G}^{\alpha}-\lim_{\epsilon\rightarrow 0+}M_{\epsilon}=0$.
Further we set
\begin{equation*}\begin{split}
N_{\epsilon}=&\frac1{\epsilon}\left\{\hat{{\muathbb E}}_t\left[\int_t^{t+\epsilon}g(r,\Phi(0),\Phi'(0))dr
+\int_t^{t+\epsilon}\left(f(r,\Phi(0),\Phi'(0))+\frac12\Phi''(0)\right)d\langle B\rangle_r\right]\right.\\
&\left.-\hat{{\muathbb E}}_t\left[\int_t^{t+\epsilon}g(r,\Phi(0),\Phi'(0))dr+\int_t^{t+\epsilon}\left(
f(r,\Phi(0),\Phi'(0))+\frac12\Phi''(0)\right)
d\langle B\rangle_r\right]\right\}
\end{split}\end{equation*}
You can check that
\begin{equation*}\begin{split}
|N_{\epsilon}|\leq& (C_7/\epsilon)\hat{{\muathbb E}}_t[\int_t^{t+\epsilon}(|g(r,\Phi(0),\Phi'(0))-
g(t,\Phi(0),\Phi'(0))|+\\
&|f(r,\Phi(0),\Phi'(0))-f(t,\Phi(0),\Phi'(0))|)^{\alpha}dr],
\end{split}\end{equation*}
where $C_7$ depends on $G$.
Then,
\begin{equation*}\begin{split}
\hat{{\muathbb E}}[|N_{\epsilon}|^{\alpha}]\leq & C_7^{\alpha} \frac1{\epsilon}\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}(
|g(r,\Phi(0),\Phi'(0))-g(t,\Phi(0),\Phi'(0))|\right.\\
&\left.+|f(r,\Phi(0),\Phi'(0))-f(t,\Phi(0),\Phi'(0))|
)^{\alpha}dr\right]\\
\leq & C_7^{\alpha}\left(\frac1{\epsilon}\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}(|g(r,\Phi(0),\Phi'(0))-g(t,\Phi(0),\Phi'(0))|
\right.\right.\\
&\left.\left.+|f(r,\Phi(0),\Phi'(0))-f(t,\Phi(0),\Phi'(0))|)^{\beta}dr\right]\right)^{\alpha/\beta}.
\end{split}\end{equation*}
Take limits from both sides of the above inequality and use assumption (H4), then we have
\begin{equation*}
L_G^{\alpha}-\lim_{\epsilon\rightarrow 0+}N_{\epsilon}=0.
\end{equation*}
At the same time,
\begin{equation*}\begin{split}
\hat{{\muathbb E}}\left[\int_t^{t+\epsilon}g(r,\Phi(0),\Phi'(0))dr+
\int_t^{t+\epsilon}(f(t,\Phi(0),\Phi'(0))+\frac12\Phi''(0))d\langle B\rangle_r\right]\\
=g(t,\Phi(0),\Phi'(0))\epsilon+\hat{{\muathbb E}}_t\left[f(t,\Phi(0),\Phi'(0))\left(\langle B\rangle_{t+\epsilon}-\langle B\rangle_t\right)\right]\\
=\left[g(t,\Phi(0),\Phi'(0))+2G\left((f(t,\Phi(0),\Phi'(0))+\frac12\Phi''(0))\right)\right]\epsilon.
\end{split}\end{equation*}
Then we have
\begin{equation*}\begin{split}
L_G^{\alpha}-\lim_{\epsilon\rightarrow 0+}\frac{\tilde{Y}_t}{\epsilon}=& \lim_{\epsilon\rightarrow 0+}\frac1{\epsilon}\{Y_t-\Phi(0)\}\\
=& g(t,\Phi(0),\Phi'(0))+2G\left(
(f(r,\Phi(0),\Phi'(0)+\frac12\Phi''(0))\right).
\end{split}\end{equation*}
The proof is finished.
\begin{theorem}\label{thm:main}
Suppose (H1)-(H4) satisfied. Take a function $h\in C^2$, $\phi\in C_b^2(\muathbb{R})$ is polynomial growth function and $h(\phi)\in C_b^2(\muathbb{R})$.
Then $h$ is a G-convex function that is equivalent with
\begin{multline}\label{eq:G-convex-equivalent}
g(t,h(y),h'(y)z)+2G(f(t,h(y),h'(y)z)+\frac12h''(y)z^2+\frac12h'(y)A)\geq\\
h'(y)g(t,y,z)+2h'(y)G(f(t,y,z)+\frac12A),\quad\textrm{ for all } y,z\in\muathbb{R} \textrm{ and } A\in\muathbb{R}.
\end{multline}
\end{theorem}
{\bf Proof: Necessary condition: }
Take a function $h\in C^2$ and $\phi\in C^2_b(\muathbb{R})$, with $H(\phi)\in C^2_b(\muathbb{R})$. By Lemma \ref{le:3} we have
\begin{equation}\label{eq:e-h-phi}\begin{split}
L^2_G-& \lim_{\epsilon\rightarrow 0+}\frac{\muathcal{E}_{t,t+\epsilon}[h(\phi(B_{t+\epsilon}-B_t))]-h(\phi(0))}{\epsilon}=
g(t,h(\phi(0)),h'(\phi(0))\phi'(0))\\
&+2G\left(f(t,h(\phi(0)),h'(\phi(0))\phi'(0))+\frac12h''(\phi(0))(\phi'(0))^2+\frac12h'(\phi(0)
)\phi''(0)\right)
\end{split}\end{equation}
and
\begin{equation}\label{eq:phi-lim-rep}\begin{split}
&L_G^2-\lim_{\epsilon\rightarrow 0+}\frac{\CE_{t,t+\epsilon}[\phi(B_{t+\epsilon}-B_t)]-\phi(0)}{\epsilon}\\
&=g(t,\phi(0),\phi'(0))+2G(f(t,\phi(0),\phi'(0))+\frac12\phi''(0)).
\end{split}\end{equation}
Based on \eqref{eq:phi-lim-rep}, we have
\begin{equation}\label{eq:h-e-phi}\begin{split}
&L_G^2-\lim_{\epsilon\rightarrow 0+}\frac{h(\CE_{t,t+\epsilon}[\phi(B_{t+\epsilon}-B_t)])-h(\phi(0))}{\epsilon}\\
&=h'(\phi(0))\left(g(t,\phi(0),\phi'(0))+2G(f(t,\phi(0),\phi'(0))+\frac12\phi''(0))\right).
\end{split}\end{equation}
If $h$ is a G-convex function, from Definition \ref{def:1}, $h$ satisfies $\muathcal{E}_{t,t+\epsilon}[h(\phi(B_{t+\epsilon}-B_t))]\geq h\{\muathcal{E}_{t,t+\epsilon}[\phi(B_{t+\epsilon}-B_t)]\}$.
From \eqref{eq:e-h-phi} and \eqref{eq:h-e-phi} we have
\begin{equation*}\begin{split}
g(t,h(\phi(0)),h'(\phi(0))\phi'(0))&+2G\left(f(t,h(\phi(0)),h'(\phi(0))\phi'(0))+
\frac12h''(\phi(0))(\phi'(0))^2+\frac12h'(\phi(0)
)\phi''(0)\right)\\
&\geq
h'(\phi(0))\left(g(t,\phi(0),\phi'(0))+2G(f(t,\phi(0),\phi'(0))+\frac12\phi''(0))\right)
\end{split}\end{equation*}
Where $(\phi(0),\phi'(0),\phi''(0))$ are arbitrary values in $\muathbb{R}^3$. Then we get \eqref{eq:G-convex-equivalent}.
\par
{\bf Sufficient condition:}
Following a series of work of Soner, Touzi and Zhang \cite{STZ}, Song \cite{Song1,Song2}, the work of Peng, Song and Zhang \cite{PSZ}
proved a representation theorem of G-martingales in a complete subspace of $L_G^{\alpha}(\Omega_T)$ $(\alpha\geq 1)$. They proved
the decomposition of G-martingale of $\hat{{\muathbb E}}_t[\xi]$ can be uniquely represented $K_t=\int_0^t\eta_sd\langle B\rangle_s-\int_0^t2G(\eta_s)ds$. And then use the similar Picard approximation approach used in \cite{HJPS1} we can get the corresponding theorem as Theorem \ref{thm:main} for a normal decreasing martingale with $K_0=0$ and $K_T\in \muathbb{L}^{\alpha}(\Omega_T)$.
Take $\xi\in L^{\infty}_G(\phi(B_t))$,
we need to prove
$$\muathcal{E}_{s,t}[h(\xi)]\geq h[\muathcal{E}_{s,t}(\xi)].$$
Take
$$Y_u=\xi+\int_u^tg(r,Y_r,Z_r)dr+\int_u^tf(r,Y_r,Z_r)d\langle B\rangle_r-\int_u^tZ_rdB_r-(K_t-K_u).$$
Applying Ito's formula,
$$-dh(Y_r)=h'(Y_r)\left[g(r,Y_r,Z_r)dr+f(r,Y_r,Z_r)d\langle B\rangle_r-Z_rdB_r-dK_r\right]-\frac12h''(Y_r)|Z_r|^2d\langle B\rangle_r,$$
then
\begin{equation*}
\begin{split}
h(Y_u)=&h(\xi)+\int_u^th'(Y_r)g(r,Y_r,Z_r)dr
+\int_u^t\left[h'(Y_r)f(r,Y_r,Z_r)-\frac12h''(Y_r)|Z_r|^2\right]d\langle B\rangle_r\\
&-\int_u^th'(Y_r)Z_rdB_r-\int_u^th'(Y_r)dK_r\\
=&h(\xi)+\int_u^tg(r,h(Y_r),h'(Y_r)Z_r)dr+\int_u^tf(r,h(Y_r),h'(Y_r)Z_r)d\langle B\rangle_r-\int_u^th'(Y_r)Z_rdB_r\\
&+\int_u^t\left(h'(Y_r)g(r,Y_r,Z_r)-g(r,h(Y_r),h'(Y_r)Z_r)\right)dr\\
&+\int_u^t\left(h'(Y_r)f(r,Y_r,Z_r)-\frac12h''(Y_r)|Z_r|^2-f(r,h(Y_r),h'(Y_r)Z_r)\right)d\langle B\rangle_r
-\int_u^th'(Y_r)dK_r.
\end{split}
\end{equation*}
Since the decreasing process $K_r$, G-martingale, satisfies (H5), we have
\begin{equation*}\begin{split}
=&h(\xi)+\int_u^tg(r,h(Y_r),h'(Y_r)Z_r)dr+\int_u^tf(r,h(Y_r),h'(Y_r)Z_r)d\langle B\rangle_r-\int_u^th'(Y_r)Z_rdB_r\\
&+\int_u^t\left[h'(Y_r)g(r,Y_r,Z_r)-g(r,h(Y_r),h'(Y_r)Z_r)+2h'(Y_r)G(\eta_r)\right]dr\\
&-\int_u^t\left[-h'(Y_r)f(r,Y_r,Z_r)+\frac12h''(Y_r)|Z_r|^2+f(r,h(Y_r),h'(Y_r)Z_r)+h'(Y_r)\eta_r\right]d\langle B\rangle_r
\end{split}\end{equation*}
\begin{equation*}\begin{split}
=&h(\xi)+\int_u^tg(r,h(Y_r),h'(Y_r)Z_r)dr+\int_u^tf(r,h(Y_r),h'(Y_r)Z_r)d\langle B\rangle_r-\int_u^th'(Y_r)Z_rdB_r\\
&+\int_u^t\left[h'(Y_r)g(r,Y_r,Z_r)-g(r,h(Y_r),h'(Y_r)Z_r)+2h'(Y_r)G(\eta_r)\right.\\
&\left.-2G\left(f(r,h(Y_r),h'(Y_r)Z_r)+
\frac12h''(Y_r)|Z_r|^2+h'(Y_r)(\eta_r-f(r,Y_r,Z_r))\right)\right]dr-(\tilde{K}_t-\tilde{K}_u),
\end{split}\end{equation*}
where
\begin{equation*}\begin{split}
\tilde{K}_t=&-\left\{\int_0^t\left[f(r,h(Y_r),h'(Y_r)Z_r)+\frac12h''(Y_r)|Z_r|^2+h'(Y_r)\left(\eta_r-f(r,Y_r,Z_r)\right)\right]d\langle B\rangle_r\right.\\
&\left.-2\int_0^tG\left[f(r,h(Y_r),h'(Y_r)Z_r)+\frac12h''(Y_r)|Z_r|^2+h'(Y_r)\left(\eta_r-f(r,Y_r,Z_r)\right) \right] \right\}
\end{split}\end{equation*}
is a decreasing G-martingale.
Denote $\tilde{Y}_u=h(Y_u)$ and $\tilde{Z}_u=h'(Y_u)Z_u$, then
\begin{equation}\label{eq:G-BSDE-G-conv-1}\begin{split}
\tilde{Y}_u=&h(\xi)+\int_u^tg(r,\tilde{Y}_r,\tilde{Z}_r)dr+
\int_u^tf(r,\tilde{Y}_r,\tilde{Z}_r)d\langle B\rangle_r-\int_u^t\tilde{Z}_rdB_r\\
&+\int_u^t\left[h'(Y_r)g(r,Y_r,Z_r)-g(r,\tilde{Y}_r,\tilde{Z}_r)+2h'(Y_r)G(\eta_r)\right.\\
&\left.-2G\left(f(r,\tilde{Y}_r,\tilde{Z}_r)+
\frac12h''(Y_r)|Z_r|^2+h'(Y_r)(\eta_r-f(r,Y_r,Z_r))\right)\right]dr-(\tilde{K}_t-\tilde{K}_u),
\end{split}\end{equation}
we know from the inequality \eqref{eq:G-convex-equivalent} that the fourth integral is less or equal to 0.
\par
On the other hand, $\CE_{s,t}[h(\xi)]$ is the solution of the following G-BSDE,
\begin{equation}\label{eq:G-BSDE-G-conv-2}
\bar{Y}_u=h(\xi)+\int_u^tg(r,\bar{Y}_r,\bar{Z}_r)dr+\int_u^tf(r,\bar{Y}_r,\bar{Z}_r)d\langle B\rangle_r-\int_u^t\bar{Z}_rdB_r
-(\bar{K}_t-\bar{K}_u)
\end{equation}
Applying the comparison theorem of G-BSDEs \cite{HJPS2}, we have
$$\tilde{Y}_u\leq\bar{Y}_u.$$
Since $\tilde{Y}_u=h(Y_u)=h(\CE_{u,t}[\xi])$ and $\bar{Y}_u=\CE_{u,t}[h(\xi)]$, then
$$\CE_{u,t}[h(\xi)]\geq h(\CE_{u,t}[\xi]).$$
\begin{remark}
In this paper, we covered how G-Brownian motion is 1-dimensional case. In fact, the n-dimensional case is also satisfied. The proof does not have any great difference.
\end{remark}
\end{document} |
\begin{document}
\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lmm}[thm]{Lemma}
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\theoremstyle{remark}\newtheorem{Rem}{Remark}
\title{Counting Phylogenetic Networks with Few Reticulation Vertices: A Second Approach}
\author{Michael Fuchs\\
Department of Mathematical Sciences\\
National Chengchi University\\
Taipei 116\\
Taiwan \and
En-Yu Huang\\
Department of Mathematical Sciences\\
National Chengchi University\\
Taipei 116\\
Taiwan \and
Guan-Ru Yu\\
Department of Mathematics\\
National Kaohsiung Normal University\\
Kaohsiung 824\\
Taiwan}
\maketitle
\begin{abstract}
Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices has the first-order asymptotics
\[
{\rm TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad (\ell\rightarrow\infty).
\]
Moreover, they also computed $c_k$ for $k=1,2,$ and $3$. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for $c_k$, namely, $c_k=2^{k-1}\sqrt{2}/k!$ for all $k\geq 0$.
\end{abstract}
\section{Introduction and Results}\label{intro}
Over the last decades, phylogenetic networks have become more and more popular as models in evolutionary biology. As a result, apart from biological and algorithmic studies, recently also combinatorial and probabilistic studies have been undertaken for many of the fundamental classes of phylogenetic networks; e.g., see \cite{BiLaSt,BoGaMa,DiSeWe,FuGiMa1,FuGiMa2,FuYuZh1,FuYuZh2,St}.
This short note is concerned with tree-child networks which have been introduced by Cardona et al. in \cite{CaRoVa}. We recall how they are defined. First, a (bifurcating and bicombining) rooted {\it phylogenetic network} on $\ell$ leaves is defined as a directed acyclic graph (DAG) with no double edges which has the following vertices:
\begin{itemize}
\item[(i)] a unique {\it root} with indegree $0$ and outdegree $1$;
\item[(ii)] {\it leaves} with indegree $1$ and outdegree $0$ which are bijectively labeled by $\{1,\ldots,\ell\}$;
\item[(iii)] {\it tree vertices} with indegree $1$ and outdegree $2$;
\item[(iv)] {\it reticulation vertices} with indegree $2$ and outdegree $1$.
\end{itemize}
Such a rooted phylogenetic network is called a {\it tree-child network} if every non-leaf vertex has at least one child which is not a reticulation vertex, i.e., (a) a reticulation vertex is followed by a tree vertex or a leaf; (b) not both of the children of a tree vertex are reticulation vertices; and (c) the root is followed by a tree vertex or a leaf; see Figure~\ref{tc-ex}-(a) for an example.
Denote by ${\rm TC}_{\ell,k}$ the number of tree-child networks with $\ell$ leaves and $k$ reticulation vertices. Note that if $k=0$, then this number counts phylogenetic trees (i.e., rooted, binary trees with $\ell$ labeled leaves); see \cite{OEIS}.
In \cite{FuGiMa1}, the authors obtained the following first-order asymptotics of ${\rm TC}_{\ell,k}$: for any fixed $k$, there exists a constant $c_k>0$ such that, as $\ell\rightarrow\infty$,
\begin{equation}\label{asymp-exp}
{\rm TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1};
\end{equation}
see also \cite{FuGiMa2} for some corrections of the proof of \cite{FuGiMa1}. Moreover, the authors also computed $c_k$ for small values of $k$ and obtained that
\[
c_1=\sqrt{2},\qquad c_2=\sqrt{2},\qquad c_3=\frac{2\sqrt{2}}{3}.
\]
(For $k=0$, one easily finds $c_0=\sqrt{2}/2$; see for instance the introduction of \cite{DiSeWe}.) The method of \cite{FuGiMa1} yields $c_k$ also for larger values of $k$, however, the computation becomes more and more cumbersome; see the remark at the end of Section 3 in \cite{FuGiMa1}, where it was claimed that there is probably no closed-form expression for $c_k$. This claim turns out to be wrong; see our main theorem below.
The purpose of this note is to give a second proof of (\ref{asymp-exp}) which is based on the recent algorithmic approach to the counting of tree-child networks due to Cardona and Zhang \cite{CaZh}. Moreover, this approach is capable of giving a simple, closed-form expression for $c_k$ for all $k\geq 0$.
\begin{thm}\label{main-result}
For the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices, for fixed $k$ and $\ell\rightarrow\infty$,
\[
{\rm TC}_{\ell,k}\sim \frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1}.
\]
\end{thm}
A second class of phylogenetic networks which was treated in \cite{FuGiMa1} was the class of normal networks. Here, a tree-child network is called a {\it normal network} if the two parents of each reticulation vertex are incomparable with respect to the ancestor-descendant relation, e.g., the tree-child network in Figure~\ref{tc-ex}-(a) is not normal.
It was shown in \cite{FuGiMa1} that, for fixed $k$ and $\ell\rightarrow\infty$, we have ${\rm TC}_{\ell,k}\sim N_{\ell,k}$ where $N_{\ell,k}$ denotes the number of normal networks with $\ell$ leaves and $k$ reticulation vertices. Thus, we have the following corollary.
\begin{cor}
For the number $N_{\ell,k}$ of normal networks with $\ell$ leaves and $k$ reticulation vertices, for fixed $k$ and $\ell\rightarrow\infty$,
\[
N_{\ell,k}\sim \frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1}.
\]
\end{cor}
\begin{Rem}
Instead of considering phylogenetic networks with just the leaves labeled, some authors also considered phylogenetic networks with all non-root vertices bijectively labeled; see \cite{DiSeWe,FuGiMa1,FuGiMa2}.
Denote by $\widehat{{\rm TC}}_{n,k}$ (resp. $\widehat{N}_{n,k}$) the number of such tree-child (resp. normal) networks with $n$ non-root vertices and $k$ reticulation vertices. This number is closely related to ${\rm TC}_{\ell,k}$ (resp. ${\rm N}_{\ell,k}$); see Section 5 in \cite{FuGiMa1}. From this relationship and the above results, we obtain that for fixed $k$ and $n\rightarrow\infty$,
\begin{equation}\label{cor-node}
\widehat{{\rm TC}}_{n,k}\sim\widehat{N}_{n,k}\sim\frac{\sqrt{2}}{4^k k!}\left(1-(-1)^n\right)\left(\frac{\sqrt{2}}{e}\right)^n n^{n+2k-1}.
\end{equation}
\end{Rem}
We end the introduction by giving a short outline of the structure of this note. In the next section, we will recall the approach from \cite{CaZh} which was used to (a) compute values of $\mathrm{TC}_{\ell,k}$ for small $\ell$ and $k$ and (b) obtain formulas for all $\ell$ and $k=1$ and $k=2$. In fact, this approach is also useful for obtaining asymptotic results as will be shown in Section~\ref{aa} which contains the proof of our main result. Then, in a last section, we will give a brief comparison of the asymptotic approach introduced in this paper with the one from the previous publications \cite{FuGiMa1,FuGiMa2}.
\section{The Method of Cardona and Zhang}\label{meth-CaZh}
\begin{figure}
\caption{(a) A tree-child network (which is not normal) with $6$ leaves and $4$ reticulation vertices (in gray). The encircled trees are obtained by deleting incoming edges of reticulation vertices; suppressing vertices of indegree and outdegree $1$ gives the tree components. (b) The corresponding component graph.}
\label{tc-ex}
\end{figure}
One of the goals of \cite{CaZh} was to obtain a formula for $\mathrm{TC}_{\ell,k}$ which can be utilized to (algorithmically) compute these numbers for small values of $\ell$ and $k$. For this purpose, the authors in \cite{CaZh} proposed the notion of a \textit{component graph} whose definition we will recall next.
First, for a given tree-child network $N$ with $\ell$ leaves and $k$ reticulation vertices, deleting all incoming edges of reticulation vertices and suppressing resulting vertices with indegree and outdegree $1$ gives a forest consisting of $k+1$ trees each of which contains at least one of the $\ell$ leaves. (The latter follows from the tree-child property.) The components of this forest are called \textit{tree components}. The component graph of the tree-child network $N$ is then defined as follows: its vertex set is the set of the tree components of $N$ and edges between vertices mean that the tree components are connected via the deleted edges; see Figure~\ref{tc-ex}-(b) for an example.
Note that this definition implies that the set of component graphs is the set of rooted DAGs where double edges are allowed and each non-root vertex has indegree exactly equal to two. It is an easy exercise to show that each such DAG has at least one double edge starting from the root (two in the example from Figure~\ref{tc-ex}-(b)). Moreover, these DAGs can also be counted; see Theorem~15 in \cite{CaZh} for a formula for the number of component graphs with $m$ vertices.
All tree-child networks with $\ell$ leaves and $k$ reticulation vertices can now be obtained as follows: first list all the component graphs with $k+1$ vertices (see Figure~8 and Figure~9 in \cite{CaZh} for the lists for $k=0,1,2,3,4$); then, for each fixed component graph with $k+1$ vertices, replace the vertices by phylogenetic trees $T_1,\ldots, T_{k+1}$ and add back the deleted edges (by choosing suitable edges, adding a vertex inside them and connecting this new vertex to the root of a tree component); finally, partition the set $\{1,\ldots,\ell\}$ into $k+1$ blocks with block sizes equal to the number of leaves of $T_1,\ldots, T_{k+1}$, respectively, and re-label the leaves of the phylogenetic trees with the labels from their corresponding blocks in an order-consistent way. This gives a formula for the number of tree-child networks with $\ell$ leaves and $k$ reticulation vertices; see Theorem 16 of \cite{CaZh}.
A different way (which was also used in \cite{CaZh}) to perform the above procedure is to work with \textit{one-component tree-child networks} which are tree-child networks with each reticulation vertex immediately followed by a leaf. (The name comes from the fact that one-component tree-child networks have only one non-trivial tree-component.)
The following formula for the number of one-component tree-child networks was proved in \cite{CaZh}.
\begin{lmm}[Theorem 13 in \cite{CaZh}]\label{number-o}
Let $\mathrm{O}_{\ell,k}$ denote the number of one-component tree-child networks with $\ell$ leaves and $k$ reticulation vertices where the labels of the leaves below the reticulation vertices are from the set $\{1,\ldots,k\}$. Then,
\[
\mathrm{O}_{\ell,k}=\frac{(2\ell-2)!}{2^{\ell-1}(\ell-k-1)!},\qquad (k\leq\ell-1).
\]
\end{lmm}
\begin{Rem}\label{rem}
\begin{itemize}
\item[(i)] For $k=0$, this gives the well-known formula $(2\ell-3)!!=(2\ell-3)\cdots 5\cdot 3\cdot 1$ for the number of phylogenetic trees on $\ell$ leaves.
\item[(ii)] Note that for any one-component tree-child network, the number of reticulation vertices is strictly smaller than the number of leaves. (This more generally holds for any tree-child network.)
\end{itemize}
\end{Rem}
Using these networks, we can now give a second construction procedure for building all tree-child networks with $\ell$ leaves and $k$ reticulation vertices from component graphs with $k+1$ vertices.
First, we will explain a reduction procedure for a component graph $C$ with $k+1$ vertices. We assume that $C$ has $1\leq t\leq k$ double edges starting from the root. First, remove all the single edges starting from the root and merge vertices with only one incoming (simple) edge with their parents. The resulting structure consists of a root with $t$ double edges which is on top of a DAG $\tilde{C}$ which has $t$ roots and again every non-root vertex has indegree exactly equal to two; see Step (i) of Figure~\ref{red-comp} for examples. In the next step, remove edges of $\tilde{C}$ until it decomposes into $t$ connected DAGs each of which is rooted at one of the $t$ roots of $\tilde{C}$. Note that this is always possible: for instance for each non-root vertex in $\tilde{C}$ whose two incoming edges are not a double edge, pick one of the edges and remove it. Next, again merge vertices with only one incoming edge with their parents; see Step (ii) of Figure~\ref{red-comp} for examples. We call the resulting DAG after these two steps a {\it reduced component graph} of $C$: it consists of a root with $t$ outgoing double edges and the children of this root are the roots of DAGs $D_1,\ldots,D_t$ which are again component graphs with $\beta_1,\ldots,\beta_t$ non-root vertices. Set $m:=\beta_1+\cdots+\beta_t$ and note that (i) $t+m\leq k$ and (ii) exactly $k-t-m$ edges have been removed in the above two steps.
\begin{figure}
\caption{Three component graphs and their reductions constructed via the two steps from Section~\ref{meth-CaZh}
\label{red-comp}
\end{figure}
It is important to note that more than one possible choice of a reduced component graph might be possible for a component graph. Also, different component graphs might yield the same reduced component graph; see Figure~\ref{red-comp}.
Now, in order to construct all tree-child networks with $\ell$ leaves and $k$ reticulation vertices, we start again from the component graphs which are first reduced. Then, for a reduced component graph, the root with its outgoing double edges is replaced by a one-component tree-child network $O$ with $t$ reticulation vertices which are followed by leaves which are labeled by $\{1,\ldots,t\}$ (notation is as above). Here, the incoming edges of the reticulation nodes correspond to the outgoing double edges and the leaves below them correspond to the roots of $D_1,\ldots,D_t$. Then, these leaves are replaced by tree-child networks $N_1,\ldots,N_t$ whose component graphs are given by $D_1,\ldots,D_t$. Next, the $k-t-m$ removed edges from the reduction steps above are re-attached (by choosing two suitable edges, adding a vertex inside them and connecting the added vertices). Finally, the set $\{1,\ldots,\ell\}$ is partitioned into $t+1$ blocks the first of which has size equal to the number of leaves of $O$ which are not below reticulation vertices and the remaining have sizes equal to the number of leaves of $N_1,\ldots,N_t$. Then, the leaves of the above constructed network are re-labeled with their corresponding blocks in an order-consistent way.
In order to give an example for the construction just described, consider the component graph in Figure~\ref{tc-ex}-(b) whose reduced component graph is displayed in Figure~\ref{red-comp}-(a). Note that $t=2$ and $D_1,D_2$ consist both of a single root (thus, $m=0$ and consequently $2$ edges have been removed). Now, we need to pick a one-component tree-child network with $2$ reticulation vertices and replace the two leaves below the reticulation vertices both by phylogenetic trees. Finally, we add an edge between the one-component tree-child network and one of the phylogenetic trees and an edge between the two phylogenetic trees. Then, re-labeling gives, e.g., the tree-child network in Figure~\ref{tc-ex}-(a).
This approach was used in \cite{CaZh} to find formulas for ${\rm TC}_{\ell,k}$ for $k=1,2$ and all $\ell$; see the proofs of the results in Section~5 in \cite{CaZh}. We will use this approach to prove our main result. The strategy is as follows: we will identify one (special) component graph which gives the main contribution; see Proposition~\ref{main-asymp} below. For all the other component graphs, we will give an upper bound on the number of corresponding tree-child networks arising from these component graphs from the above procedure (by allowing that networks might be counted multiple times) and this upper bound will turn out to be asymptotically negligible; see Proposition~\ref{neg-asymp} below.
Before going on, we recall a simple lemma which has been stated before and whose simple proof we leave to the interested reader. This lemma will be used to count the number of ways of adding back edges in the construction above. Recall that an edge of a phylogenetic network is called \textit{reticulation edge} if its tail is a reticulation vertex and \textit{tree edge} otherwise.
\begin{lmm}\label{tree-edges}
A phylogenetic network with $\ell$ leaves and $k$ reticulation vertices has $2k$ reticulation edges and $2\ell+k-1$ tree edges; consequently, the total number of edges is $2\ell+3k-1$.
\end{lmm}
\section{Asymptotic Analysis}\label{aa}
In this section, we will prove our main result (Theorem~\ref{main-result}) for the number $\mathrm{TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices. For this purpose, we will use the second construction procedure from the previous section. The main observation is that the main term of the asymptotics will arise from the tree-child networks that are constructed from the \textit{star component graph of size $k$} which is the component graph consisting of a root which has $k$ children; see Figure~\ref{comp-graphs}. This case will be treated in the first paragraph below. Then, in the second paragraph, we will finish our asymptotic analysis by showing that the contribution of all other component graphs is asymptotically negligible.
\paragraph{Star Component Graph.} Denote by $\mathrm{S}_{\ell,k}$ the number of tree-child networks with $\ell$ leaves and $k$ reticulation vertices that arise from the star component graph of size $k$. We have the following formula.
\begin{figure}
\caption{The star component graph of size $4$.}
\label{comp-graphs}
\end{figure}
\begin{lmm}
For $k\geq 1$, we have
\begin{equation}\label{S-l-k}
\mathrm{S}_{\ell,k}=\frac{\ell!}{2^{\ell-1}(k-1)!}\sum_{j=1}^{\ell-k}\frac{(2j+2k-2)!}{ j!(j-1)!}\cdot\frac{(2\ell-2j-k-1)!}{(\ell-j-k)!(\ell-j)!}.
\end{equation}
\end{lmm}
\begin{proof}
We use the second construction from the previous section. First, note that the reduced component graph of the star component graph of size $k$ is the graph itself. Thus, $t=k$ and $\beta_1=\cdots=\beta_k=0$. Consequently, we first need to pick a one-component tree-child network with $k$ reticulation vertices; then, we replace all the leaves below the reticulation vertices by phylogenetic trees; and finally, we re-label (in an order consistent way) all leaves of the resulting network such that only labels from the set $\{1,\ldots,\ell\}$ are used.
Based on this, we can give the following formula:
\begin{equation}\label{star-formula}
\mathrm{S}_{\ell,k}=\sum_{j=1}^{\ell-k}\binom{\ell}{j}\frac{(2j+2k-2)!}{2^{j+k-1}(j-1)!}\cdot\frac{1}{k!}(\ell-j)![z^{\ell-j}]T(z)^{k},
\end{equation}
where $T(z)$ is the exponential generating function of the number of phylogenetic trees (see Remark~\ref{rem}, (ii)) and $[z^n]f(z)$ denotes the $n$-th coefficient of the Maclaurin series of $f(z)$.
The factors inside the above sum are explained as follows: the factor $(2j+2k-2)!/(2^{j+k-1}(j-1)!)$ counts the number of one-component tree-child networks with $k$ reticulation vertices and $k+j$ leaves of which the ones with labels $\{1,\ldots,k\}$ are below the $k$ reticulation vertices; see Lemma~\ref{number-o}. The latter leaves are replaced by a set of $k$ phylogenetic trees whose number is given by $((\ell-j)![z^{\ell-j}]T(z)^{k})/k!$ since $T(z)^{k}$ enumerates sequences of $k$ phylogenetic trees and the factor $1/k!$ removes the order. Finally, the binomial coefficient takes care of all the possibilities of re-labeling the leaves.
Now, since we know from Lemma~\ref{number-o} (see also the remark after the lemma) that
\[
[z^{\ell}]T(z)=\frac{(2\ell-2)!}{2^{\ell-1}(\ell-1)!}=(2\ell-3)!!,
\]
a straightforward computation gives
\[
T(z)=\sum_{\ell\geq 1}(2\ell-3)!!\cdot\frac{z^{\ell}}{\ell!}=1-\sqrt{1-2z}
\]
which satisfies the equation $T(z)=z+T(z)^2/2$. (This equation could also be derived directly from the definition of phylogenetic trees and symbolic combinatorics; see Section 3.9 in \cite{FlSe}.)
Thus, from the Lagrange inversion formula (see, e.g., p126 in \cite{FlSe}),
\begin{align*}
[z^{\ell-j}]T(z)^k&=\frac{k}{(\ell-j)}[\omega^{\ell-j-k}](1-\omega/2)^{-(\ell-j)}\\
&=\frac{k}{(\ell-j)}2^{j+k-\ell}\binom{-(\ell-j)}{\ell-j-k}(-1)^{\ell-j-k}\\
&=\frac{k}{(\ell-j)}2^{j+k-\ell}\binom{2\ell-2j-k-1}{\ell-j-k}.
\end{align*}
Plugging this into (\ref{star-formula}) and re-arrangement gives the claimed result.
\end{proof}
We apply now the Laplace method (see, e.g., Section 4.7 in \cite{FlSe}) to the sum from the last lemma in order to obtain the asymptotics of $\mathrm{S}_{\ell,k}$.
\begin{lmm}
For $k\geq 1$, we have, as $\ell\rightarrow\infty$,
\[
\mathrm{S}_{\ell,k}\sim \frac{\sqrt{2}d_k}{2(k-1)!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},
\]
where
\[
d_k:=\sum_{j\geq 0}\frac{(2j+k-1)!}{j!(j+k)!}4^{-j}.
\]
\end{lmm}
\begin{proof}
It is sufficient to consider
\begin{align}
\Sigma_{\ell,k}&:=\sum_{j=1}^{\ell-k}\frac{(2j+2k-2)!}{ j!(j-1)!}\cdot\frac{(2\ell-2j-k-1)!}{(\ell-j-k)!(\ell-j)!}\nonumber\\
&=\sum_{j=0}^{\ell-k-1}\frac{(2\ell-2j-2)!}{(\ell-j-k)!(\ell-j-k-1)!}\cdot\frac{(2j+k-1)!}{j!(j+k)!}\label{laplace-sum}
\end{align}
since the asymptotics of the factor in front of the sum of (\ref{S-l-k}) is easily derived from Stirling's formula; see (\ref{asymp-fac}) below.
The main contribution to the sum (\ref{laplace-sum}) comes from the terms with small values of $j$. Thus, we expand the first term inside (\ref{laplace-sum}) which gives, as $\ell\rightarrow\infty$,
\[
\frac{(2\ell-2j-2)!}{(\ell-j-k)!(\ell-j-k-1)!}=\frac{1}{\sqrt{\pi}}4^{\ell-j-1}\ell^{2k-3/2}\left(1+{\mathcal O}\left(\frac{1+j^2}{\ell}\right)\right)
\]
uniformly in $j$ with $j=o(\sqrt{\ell})$. In addition note that
\[
\frac{(2\ell-2j-2)!}{(\ell-j-k)!(\ell-j-k-1)!}
\]
is a decreasing sequence in $j$ for all $k\geq 1$.
Thus, by a standard application of the Laplace method, as $\ell\rightarrow\infty$,
\[
\Sigma_{\ell,k}\sim\frac{d_k}{\sqrt{\pi}}4^{\ell-1}\ell^{2k-3/2}
\]
which multiplied with
\begin{equation}\label{asymp-fac}
\frac{\ell!}{2^{\ell-1}(k-1)!}\sim\frac{2\sqrt{2\pi}}{(k-1)!}\left(\frac{1}{2e}\right)^{\ell}\ell^{\ell+1/2},\qquad (\ell\rightarrow\infty).
\end{equation}
gives the claimed result.
\end{proof}
The final step is to simplify the constant $d_k$ from the last lemma.
\begin{lmm}
For $k\geq 1$, we have
\[
\sum_{j\geq 0}\frac{(2j+k-1)!}{j!(j+k)!}4^{-j}=\frac{2^k}{k}.
\]
\end{lmm}
\begin{proof}
Set
\[
A(z):=\sum_{j\geq 0}\frac{(2j+k-1)!}{(j+k)!j!}z^{j}.
\]
Our goal is to find $A(1/4)$. To this end, note that
\begin{align*}
[z^{j}]A(z)&=\frac{(2j+k-1)!}{(j+k)!j!}\\
&=\frac{1}{(j+k)}\binom{2j+k-1}{j}\\
&=\frac{1}{(j+k)}[\omega^{j}](1-\omega)^{-(j+k)}\\
&=\frac{1}{k}\cdot\frac{k}{(j+k)}[\omega^{j+k-k}](1-\omega)^{-(j+k)}\\
&=\frac{1}{k}[z^{j+k}]C(z)^{k}=\frac{1}{k}[z^j]\left(\frac{C(z)}{z}\right)^k,
\end{align*}
where $C(z)$ is a function which satisfies the equation $C(z)-C(z)^2=z$ and the second last equality follows by an application of the Lagrange inversion formula (in reversed direction). Thus,
\[
C(z)=\frac{1-\sqrt{1-4z}}{2}.
\]
and consequently
\[
A(z)=\frac{1}{k}\cdot\left(\frac{1-\sqrt{1-4z}}{2z}\right)^k
\]
which yields $A(1/4)=2^k/k$ as claimed.
\end{proof}
Combining the last two lemmas, we have the following asymptotic result for $S_{\ell,k}$.
\begin{pro}\label{main-asymp}
For the number $\mathrm{S}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices arising from the star component graph of size $k$, as $\ell\rightarrow\infty$,
\[
\mathrm{S}_{\ell,k}\sim\frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1}.
\]
\end{pro}
\paragraph{Remaining Component Graphs.} Denote by $\mathrm{R}_{\ell,k}$ the number of tree-child networks arising from all non-star component graphs; see Figure~\ref{red-comp} for some examples. The proof of Theorem~\ref{main-result} will be finished by showing that $\mathrm{R}_{\ell,k}$ contributes asymptotically less than $\mathrm{S}_{\ell,k}$.
\begin{pro}\label{neg-asymp}
For the number $\mathrm{R}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices arising from all component graphs except the star component graph of size $k$, we have
\[
\mathrm{R}_{\ell,k}={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-3/2}\right).
\]
\end{pro}
For the proof, we will use induction on $k$. However, we first need two (simple) lemmas. The first gives an upper bound for the number $\mathrm{O}_{\ell,k}$ from Lemma~\ref{number-o}.
\begin{lmm}\label{o-est}
We have
\[
\mathrm{O}_{\ell,k}={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+k-1}\right)
\]
\end{lmm}
\begin{proof}
This is a consequence of Stirling's formula.
\end{proof}
The second lemma gives a bound for certain sums.
\begin{lmm}\label{sum-est}
Let $\alpha_0,\ldots,\alpha_t$ be real numbers none of which equal to $-1$. Set
\[
s:=\#\{0\leq j\leq t\ :\ \alpha_j>-1\}
\]
Then,
\[
\sum_{\ell_0+\cdots+\ell_t=\ell}\ell_0^{\alpha_0}\cdots\ell_t^{\alpha_t}={\mathcal O}\left(\ell^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right),
\]
where the sum runs over all positive integers $\ell_0,\ldots,\ell_t$.
\end{lmm}
\begin{proof}
We first assume that $0<s<t+1$. Then, we can write the sum as
\[
\sum_{\ell_0+\cdots+\ell_t=\ell}\ell_0^{\alpha_0}\cdots\ell_t^{\alpha_t}=\sum_{i=1}^{\ell-1}\left(\sum_{\sum_{\alpha_j<-1}\ell_j=\ell-i}\left(\prod_{\alpha_j<-1}\ell_j^{\alpha_j}\right)\right)
\left(\sum_{\sum_{\alpha_j>-1}\ell_j=i}\left(\prod_{\alpha_j>-1}\ell_j^{\alpha_j}\right)\right).
\]
We will start by estimating the two terms inside this sum.
For the first term, we have
\begin{equation}\label{est-1}
\sum_{\sum_{\alpha_j<-1}\ell_j=\ell-i}\left(\prod_{\alpha_j<-1}\ell_j^{\alpha_j}\right)={\mathcal O}\left((\ell-i)^{\alpha}\right),
\end{equation}
where $\alpha=\max\{\alpha_j\ :\ \alpha_j<-1\}$. This follows because at least one of the $\ell_j$'s with $\sum_{\alpha_j<-1}\ell_j=\ell-i$ is at least $(\ell-i)/(t+1-s)$ (giving the claimed upper bound) and the series $\sum_{\ell=1}^{\infty}\ell^{\beta}$ converges for all $\beta<-1$ (giving a constant upper bound for the remaining $\ell_j$'s).
For the second term, by approximating by an integral,
\begin{align}
\sum_{\sum_{\alpha_j>-1}\ell_j=i}\left(\prod_{\alpha_j>-1}\ell_j^{\alpha_j}\right)&={\mathcal O}\left(i^{s-1+\sum_{\alpha_j>-1}\alpha_j}\int_{\sum_{\alpha_j>-1}x_j=1}\left(\prod_{\alpha_j>-1}x_j^{\alpha_j}\right){\rm d}{\bf x}\right)\nonumber\\
&={\mathcal O}\left(i^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right)\label{est-2}
\end{align}
where the integral is ${\mathcal O}(1)$ since it converges.
Finally, by combining the estimates of the two terms:
\begin{align*}
\sum_{\ell_0+\cdots+\ell_t=\ell}\ell_0^{\alpha_0}\cdots\ell_t^{\alpha_t}
&={\mathcal O}\left(\sum_{i=1}^{\ell-1}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}(\ell-i)^{\alpha}\right)\\
&={\mathcal O}\left(\sum_{1\leq i\leq\ell/2}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}(\ell-i)^{\alpha}+\sum_{\ell/2\leq i\leq\ell-1}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}(\ell-i)^{\alpha}\right).
\end{align*}
For the first sum, we have
\begin{align*}
\sum_{1\leq i\leq\ell/2}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}(\ell-i)^{\alpha}&={\mathcal O}\left(\ell^{\alpha}\sum_{1\leq i\leq\ell/2}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right)\\
&={\mathcal O}\left(\ell^{\alpha+s+\sum_{\alpha_j>-1}\alpha_j}\int_{0}^{1/2}x^{s-1+\sum_{\alpha_j>-1}\alpha_j}{\rm d}x\right)\\
&={\mathcal O}\left(\ell^{\alpha+s+\sum_{\alpha_j>-1}\alpha_j}\right).
\end{align*}
For the second sum, we have
\[
\sum_{\ell/2\leq i\leq\ell-1}i^{s-1+\sum_{\alpha_j>-1}\alpha_j}(\ell-i)^{\alpha}={\mathcal O}\left(\ell^{s-1+\sum_{\alpha_j>-1}\alpha_j}\sum_{j\geq 1}j^{\alpha}\right)={\mathcal O}\left(\ell^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right).
\]
Thus,
\[
\sum_{\ell_0+\cdots+\ell_t=\ell}\ell_0^{\alpha_0}\cdots\ell_t^{\alpha_t}={\mathcal O}\left(\ell^{\alpha+s+\sum_{\alpha_j>-1}\alpha_j}\right)+{\mathcal O}\left(\ell^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right)={\mathcal O}\left(\ell^{s-1+\sum_{\alpha_j>-1}\alpha_j}\right)
\]
which is the claimed result for $0<s<t$.
For the missing cases $s=0$ and $s=t+1$, the result is already implied by (\ref{est-1}) and (\ref{est-2}), respectively. This concludes the proof.
\end{proof}
We are now ready to prove Proposition~\ref{neg-asymp}.
\vspace*{0.35cm}\noindent\textit{Proof of Proposition~\ref{neg-asymp}.} We use induction on $k$. The claim is trivial for $k=1$ since $\mathrm{R}_{\ell,1}=0$.
Next, assume that the claim holds for all $k'<k$. Note that because of the induction hypothesis and Proposition~\ref{main-asymp}, we have
\begin{equation}\label{ind-hyp}
\mathrm{TC}_{\ell,k'}={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k'-1}\right).
\end{equation}
We will now prove the claim for $k$.
Fix a non-star component graph with $k+1$ vertices and consider its reduced component graph. Using the same notation as in Section~\ref{meth-CaZh}, let $t$ be the number of double edges starting from the root of the reduced component graph and $m:=\beta_1+\cdots+\beta_t$ where $\beta_j$ is the number of non-root vertices of the DAG rooted at the $j$-th child of the root in the reduced component graph (note that there are no edges between the DAGs rooted at the children because they have been removed when constructing the reduced component graph). A crucial fact used below is that $t<k$ since $t=k$ only holds for the star component graph of size $k$ (and this component graph is not considered).
Now, using the second construction from the previous section, we obtain (up to a constant) the following upper bound for the number of tree-child networks arising from the fixed component graph
\[
U_{\ell,k}:=\sum_{\ell_0+\cdots+\ell_t=\ell}\binom{\ell}{\ell_0,\ldots,\ell_t}\mathrm{O}_{\ell_0+t,t}\mathrm{TC}_{\ell_1,\beta_1}\cdots\mathrm{TC}_{\ell_t,\beta_t}\ell_0^{\delta_0}\ell_1^{\delta_1}\cdots\ell_t^{\delta_t}
\]
since we pick a one-component tree-child network $O$ with $t$ reticulation vertices labeled $1$ to $t$ whose leaves are replaced by tree-child networks with $\beta_1,\ldots,\beta_t$ reticulation vertices. Moreover, the binomial coefficient inside the sum takes care of the re-labeling and $\ell_0^{\delta_0}\cdots\ell_t^{\delta_t}$ upper bounds (up to a constant) the number of ways of re-attaching the edges which where removed from the component graph (the ends of each edge can be assigned to vertices from that graph; $\delta_0$ and $\delta_j$ then count the number of ends contained in the networks counted by $\mathrm{O}_{\ell_0+t,t}$ and $\mathrm{TC}_{\ell_j,\beta_j}$, respectively); see Lemma~\ref{tree-edges}. Here, $\delta_0,\ldots,\delta_t$ are non-negative integers such that
\begin{equation}\label{rel}
\delta_0+\ldots+\delta_t=2(k-t-m)
\end{equation}
since exactly $k-t-m$ edges are re-attached. (Note that this is indeed just an upper bound since some networks might be constructed multiple times when re-attaching edges. Moreover, since the leaves of $O$ are replaced by any tree-child networks with $\beta_1,\ldots,\beta_t$ reticulation vertices, this formula also counts networks whose component graph might not be the component graph we started out with.)
In order to estimate the above sum, by Lemma~\ref{o-est}, (\ref{ind-hyp}) and Stirling's formula,
\begin{align*}
U_{\ell,k}&=\ell!\sum_{\ell_0+\cdots+\ell_t=\ell}\frac{\ell_0^{\delta_0}\mathrm{O}_{\ell_0+t,t}}{\ell_0!}\cdot\frac{\ell_1^{\delta_1}\mathrm{TC}_{\ell_1,\beta_1}}{\ell_1!}\cdots\frac{\ell_t^{\delta_t}\mathrm{TC}_{\ell_t,\beta_t}}{\ell_t!}\\
&={\mathcal O}\left(\left(\frac{1}{e}\right)^{\ell}\ell^{\ell+1/2}\sum_{\ell_0+\cdots+\ell_t=\ell}2^{\ell_0}\ell_0^{2t+\delta_0-3/2}\cdot2^{\ell_1}\ell_1^{2\beta_1+\delta_1-3/2}\cdots2^{\ell_t}\ell_t^{2\beta_t+\delta_t-3/2}\right)\\
&={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+1/2}\sum_{\ell_0+\cdots+\ell_t=\ell}\ell_0^{2t+\delta_0-3/2}\cdot\ell_1^{2\beta_1+\delta_1-3/2}\cdots\ell_t^{2\beta_t+\delta_t-3/2}\right).
\end{align*}
Note that $2t+\delta_0-3/2>-1$ (since $t\geq 1$) and $2\beta_j+\delta_j-3/2>-1$ if and only if $\beta_j>0$ or $\delta_j>0$ and at least one these $t$ numbers is $>-1$ since $t<k$; see (\ref{rel}). Set
\[
s':=\#\{1\leq j\leq t\ :\ \beta_j>0\ \text{or}\ \delta_j>0\}\geq 1.
\]
Then, by Lemma~\ref{sum-est}, where the $s$ in that lemma is $s'+1$, we have
\[
U_{\ell,k}={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+1/2}\ell^{s'+2t+\delta_0-3/2+\sum_{j=1}^{t}(2\beta_j+\delta_j)-3t/2+3(t-s')/2}\right)\\
={\mathcal O}\left(\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1-s'/2}\right)
\]
which gives the required bound for $U_{\ell,k}$ since $s'\geq 1$.
Finally, summing over all non-star component graphs with $k+1$ vertices (whose number just depends on $k$) gives the claimed result. (Recall that we do not care about double-counting since we only need a bound which is smaller than the first-order asymptotics from Proposition~\ref{main-asymp}.){\quad\rule{1mm}{3mm}\,}
\section{Conclusion}
The main purpose of this short note was to provide a second approach to the counting of tree-child networks with few reticulation vertices. This second approach is based on a recent approach of Cardona and Zhang \cite{CaZh} which was devised for the purpose of algorithmically solving the counting problem for tree-child networks. We showed that this approach can be used to solve the asymptotic counting problem, too. Moreover, this approach is capable of yielding the multiplicative constant of the first-order asymptotics, something which was left open by the previous approach from \cite{FuGiMa1,FuGiMa2}.
We are going to end the paper by giving a brief comparison between the previous approach and the approach from this note. In the previous approach, tree-child networks where built from so-called \textit{sparsened skeletons}, whereas in the current note, we used the component graphs from \cite{CaZh}. As pointed out in \cite{FuGiMa1} the sparsened skeletons which give the main contribution to the asymptotics are all rooted, non-plane trees with $k$ leaves whose number does not permit an easy formula. Because of this, the authors from \cite{FuGiMa1} expected that the multiplicative constant in the first-order asymptotics has no simple form; see the end of Section 3 in \cite{FuGiMa1}. However, this turned out to be wrong as shown in this note. In fact, using component graphs simplifies the situation because the main contribution to the asymptotics just comes from a single component graph, namely the star component graph; see our analysis in Section~\ref{aa}.
Both approaches are also capable of giving exact formulas for small values of the number $k$ of reticulation vertices; for the approach from \cite{FuGiMa1} this was shown in \cite{FuGiMa2} and for the approach from \cite{CaZh}, such results were contained in that publication.
Overall our note shows that the approach from \cite{CaZh} seems to be more flexible than the approach from \cite{FuGiMa1} for the counting of tree-child networks with fixed $k$. However, the approach from \cite{FuGiMa1} has still one advantage over the one from \cite{CaZh}, namely, it also applies, \textit{mutatis mutandis}, to the counting of normal networks which is not the case for the approach from \cite{CaZh} (because it is not clear how to avoid the creation of edges that violate the normal condition). In fact, the approach from \cite{FuGiMa1} was used to solve a question concerning the counting of normal networks from \cite{CaZh}; see \cite{FuGiMa2}.
\section*{Acknowledgments}
We thank Apoorva Khare who after seeing (\ref{cor-node}) for $k=1,2,3$ during a talk of the first author at the \textit{6th India-Taiwan Conference on Combinatorics} in Varanasi suggested that the multiplicative constant might have the claimed form. We also thank the anonymous referees for their many insightful suggestions.
The first and second author acknowledge financial support by the Ministry of Science and Technology, Taiwan under the grants MOST-107-2115-M-009-010-MY2 and MOST-109-2115-M-004-003-MY2; the third author was partially supported by the grant MOST-110-2115-M-017-003-MY3.
\end{document} |
\begin{document}
\title{The One-Phase Bifurcation For The $p$-Laplacian}
\author{Alaa Akram Haj Ali \& Peiyong Wang\footnote{Peiyong Wang is partially supported by a Simons Collaboration Grant.}\\
\footnotesize Department of Mathematics\\
\footnotesize Wayne State University\\
\footnotesize Detroit, MI 48202\\
\normalsize}
\date{}
\maketitle
\begin{abstract}
A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the $p$-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.
\end{abstract}
\textbf{AMS Classifications:} 35J92, 35J25, 35J62, 35K92, 35K20, 35K59
\textbf{Keywords:} bifurcation, phase transition, $p$-Laplacian, Mountain Pass Theorem, Palais-Smale condition, critical point, critical boundary data, convergence of evolution.
\section{Introduction}\label{introduction}
In this paper, one considers the phase transition problem of minimizing the $p$-functional
\begin{equation}\label{p-functional}
J_{p,\varepsilon}(u) = \int_{\Omega}\frac{1}{p}|\nabla u(x)|^p + Q(x)\Gamma_{\varepsilon}(u(x))\,dx\ \ \ (1<p<\infty)
\end{equation}
which is a singular perturbation of the one-phase problem of minimizing the functional associated with the $p$-Laplacian
\begin{equation}\label{p-functional_original}
J_p(u) = \int_{\Omega}\frac{1}{p}|\nabla u(x)|^p + Q(x)\chi_{\{u(x)>0\}}\,dx,
\end{equation}
where $\Gamma_{\varepsilon}(s) = \Gamma(\frac{s}{
\varepsilon})$ for $\varepsilon > 0$ and for a $C^{\infty}$ function $\Gamma$ defined by
$$\Gamma(s) = \left\{\begin{array}{ll}
0 &\ \text{\ if\ }s\leq 0\\
1 &\ \text{\ if\ }s\geq 1,
\end{array}\right.$$
and $0\leq\Gamma(s)\leq 1$ for $0<s<1$, and $Q\in W^{2,2}(\Omega)$ is a positive continuous function on $\Omega$ such that $\inf_{\Omega}Q(x) > 0$. Let $\beta_{\varepsilon}(s) = \Gamma'_{\varepsilon}(s) = \frac{1}{
\varepsilon}\beta(\frac{s}{\varepsilon})$ with $\beta = \Gamma'$. The domain $\Omega$ is always assumed to be smooth in this paper for convenience. As in the following we will fix the value of $\varepsilon$ unless we specifically examine the influence of the value of $\varepsilon$ on the critical boundary data and will not use the notation $J_p$ for a different purpose, we are going to abuse the notation by using $J_p$ for the functional $J_{p,\varepsilon}$ from now on.
The Euler-Lagrange equation of (\ref{p-functional}) is
\begin{equation}\label{eulereq}
-\bigtriangleup_p u + Q(x)\beta_{\varepsilon}(u) = 0\ \ x\in\Omega
\end{equation}
One imposes the boundary condition
\begin{equation}\label{bdrycondition}
u(x) = \sigma(x),\ \ x\in\partial\Omega
\end{equation}
on $u$, for $\sigma\in C(\partial\Omega)$ with $\min_{\partial\Omega}\sigma > 0$, to form a boundary value problem.
In this paper, we take on the task of establishing in the general case when $p\ne 2$ the results proved in \cite{CW} for the Laplacian when $p=2$. The main difficulty in this generalization lies in the lack of sufficient regularity and the singular-degenerate nature of the $p$-Laplacian when $p\ne 2$. A well-known fact about $p$-harmonic functions is the optimal regularity generally possessed by them is $C^{1,\alpha}$ (e.\,g.\,\cite{E} and \cite{Le}). Thus we need to employ more techniques associated with the $p$-Laplacian, and in a case or two we have to make our conclusion slightly weaker. Nevertheless, we follow the overall scheme of approach used in \cite{CW}. In the second section, we prove the bifurcation phenomenon through the Mountain Pass Theorem. In the third section, we establish a parabolic comparison principle. In the last section, we show the convergence of an evolution to a stable steady state in accordance with respective initial data.
\section{A Third Solution}\label{thirdsolution}
We first prove if the boundary data is small enough, then the minimizer is nontrivial. More precisely, let $u_0$ be the trivial solution of (\ref{eulereq}) and (\ref{bdrycondition}), being $p$-harmonic in the weak sense, and $u_2$ be a minimizer of the $p$-functional (\ref{p-functional}), and
set $$\sigma_M = \max_{\partial\Omega}\sigma(x)\ \ \text{ and\ }\ \ \sigma_m = \min_{\partial\Omega}\sigma(x).$$ If $\sigma_M$ is small enough, then $u_0\neq u_2$.
In fact, we pick $u\in W^{1,p}(\Omega)$ so that
\begin{equation}
\left\{\begin{array}{ll}
u = 0 &\ \ \text{ in $\Omega_{\delta}$}\\
u = \sigma &\ \ \text{ on $\partial\Omega$,\ \ \ \ and }\\
-\bigtriangleup_p u = 0 &\ \ \text{ in $\Omega\backslash\bar{\Omega}_{\delta}$,}\end{array}\right.
\end{equation}
where $\Omega_{\delta} = \{x\in\Omega\colon dist(x,\partial\Omega) > \delta\}$ and $\delta > 0$ is a small constant independent of $\varepsilon$ and $\sigma$ so that $\int_{\Omega_{\delta}}Q(x)\,dx$ has a positive lower bound which is also independent of $\varepsilon$ and $\sigma$. Using an approximating domain if necessary, we may assume $\Omega_{\delta}$ possesses a smooth boundary. Clearly,
\begin{equation*}
J_p(u_0) = \int_{\Omega}\frac{1}{p}|\nabla u_0|^p + Q(x)\,dx
\geq \int_{\Omega}Q(x)\,dx.
\end{equation*}
It is well-known that
\begin{equation*}
\int_{\Omega\backslash\Omega_{\delta}}|\nabla u|^p \leq C\sigma^{\,p}_M\delta^{1-p}\ \ \text{ for $C = C(n,p,\Omega)$},
\end{equation*}
so that
\begin{alignat*}{1}
&J_p(u) = \int_{\Omega\backslash \Omega_{\delta}}\frac{1}{p}|\nabla u|^p +
\int_{\Omega\backslash\Omega_{\delta}}Q(x)\,dx\\
&\leq C\sigma^{\,p}_M\delta^{1-p} +
\int_{\Omega\backslash\Omega_{\delta}}Q(x)\,dx.
\end{alignat*}
So, for all small $\varepsilon > 0$,
\begin{equation*}
J_p(u) - J_p(u_0) \leq C\sigma^{\,p}_M\delta^{1-p} - \int_{\Omega_{\delta}} Q(x)\,dx < 0
\end{equation*}
if $\sigma_M\leq \sigma_0$ for some small enough $\sigma_0 = \sigma_0(\delta, \Omega, Q)$.
Let $\mathfrak{B}$ denote the Banach space $W^{1,p}_0(\Omega)$ we will work with. For every $v\in\mathfrak{B}$, we write $u = v + u_0$ and adopt the norm $\|v\|_{\mathfrak{B}} = \left(\int_{\Omega}|\nabla v|^p\right)^{\frac{1}{p}} = \left(\int_{\Omega}|\nabla u - \nabla u_0|^p\right)^{\frac{1}{p}}$. We define the functional
\begin{equation}
I[v] = J_p(u) - J_p(u_0) = \int_{\Omega}\frac{1}{p}|\nabla u|^p - \int_{\{u < \varepsilon\}}Q(x)\left(1 - \Gamma_{\varepsilon}(u)\right) - \int_{\Omega}\frac{1}{p} |\nabla u_0|^p
\end{equation}
Set $v_2 = u_2 - u_0$. Clearly, $I[0] = 0$ and $I[v_2] \leq 0$ on account of the definition of $u_2$ as a minimizer of $J_p$. If $I[v_2] < 0$ which is the case if $\sigma_M$ is small, we will apply the Mountain Pass Lemma to prove there exists a critical point of the functional $I$ which is a weak solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}).
The Fr\'{e}chet derivative of $I$ at $v\in \mathfrak{B}$ is given by
\begin{equation}
I'[v]\varphi = \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(u)\varphi\ \ \ \ \varphi\in\mathfrak{B}
\end{equation}
which is obviously in the dual space $\mathfrak{B}^*$ of $\mathfrak{B}$ in light of the H\"{o}lder's inequality. Equivalently
\begin{equation}
I'[v] = -\bigtriangleup_p (v+u_0) + Q(x)\beta_{\varepsilon}(v + u_0)\in\mathfrak{B}^*.
\end{equation}
We see that $I'$ is Lipschitz continuous on any bounded subset of $\mathfrak{B}$ with Lipschitz constant depending on $\varepsilon$, $p$, and $\sup Q$. In fact, for any $v$, $w$, and $\varphi\in \mathfrak{B}$,
\begin{alignat*}{1}
&\ \ \left|I'[v]\varphi - I'[w]\varphi\right| = |\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(v+u_0)\\
&- |\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot\nabla\varphi - Q(x)\beta_{\varepsilon}(w+u_0)| \\
&\leq \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot\nabla\varphi -
|\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot\nabla\varphi\right| \\
&+ \left|\int_{\Omega}Q(x)\beta_{\varepsilon}(v+u_0) - Q(x)\beta_{\varepsilon}(w+u_0)\right|
\end{alignat*}
Furthermore,
\begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}Q(x)\beta_{\varepsilon}(v+u_0) - Q(x)\beta_{\varepsilon}(w + u_0)\right|\\
&= \left|\int_{\Omega}Q(x)\int^1_0\beta'_{\varepsilon}((1-t)w + tv + u_0)\,dt\,(v(x) - w(x))\,dx \right|\\
&\leq \sup|\beta'_{\varepsilon}|\int_{\Omega}\left|Q(x)\left(v(x) - w(x)\right)\right|\,dx \\
&\leq \frac{C}{\varepsilon^2}\left(\int_{\Omega}Q^{p'}(x)\right)^{\frac{1}{p'}}\left(\int_{\Omega}|v(x) - w(x)|^p\,dx\right)^{\frac{1}{p}}
\end{alignat*}
and
\begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v + \nabla u_0)\cdot \nabla \varphi - |\nabla w + \nabla u_0|^{p-2}(\nabla w + \nabla u_0)\cdot \nabla \varphi\right|\\
&\leq \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v - \nabla w)\cdot \nabla \varphi\right| \\
&\ \ \ \ + \left|\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-2} - |\nabla w + \nabla u_0|^{p-2}\right)(\nabla w + \nabla u_0)\cdot \nabla \varphi\right|.
\end{alignat*}
In addition,
\begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}|\nabla v + \nabla u_0|^{p-2}(\nabla v - \nabla w)\cdot \nabla \varphi\right|\\
&\leq \left(\int_{\Omega}|\nabla v + \nabla u_0|^p\right)^{\frac{p-2}{p}}\left(\int_{\Omega}|\nabla\varphi|^p\right)^{\frac{1}{p}}\left(\int_{\Omega}|\nabla v - \nabla w|^p\right)^{\frac{1}{p}},
\end{alignat*}
and
\begin{alignat*}{1}
&\ \ \ \ \left|\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-2} - |\nabla w + \nabla u_0|^{p-2}\right)\left(\nabla w + \nabla u_0\right)\cdot\nabla\varphi\right|\\
&\leq C(p)\int_{\Omega}\left(|\nabla v + \nabla u_0|^{p-3} + |\nabla w + \nabla u_0|^{p-3}\right)|\nabla v - \nabla w||\nabla w + \nabla u_0||\nabla\varphi|\\
&\leq C(p)\left(\|\nabla v\|_{L^p} + \|\nabla w\|_{L^p} + \|\nabla u_0\|_{L^p}\right)^{p-2}\|\nabla v - \nabla w\|_{L^p(\Omega)}\|\nabla\varphi\|_{L^p(\Omega)}.
\end{alignat*}
Therefore $I'$ is Lipschitz continuous on bounded subsets of $\mathfrak{B}$.
We note that $f\in\mathfrak{B}^*$ if and only if there exist $f^0$, $f^1$, $f^2$, ..., $f^n\in L^{p'}(\Omega)$, where $\frac{1}{p} + \frac{1}{p'} = 1$, such that
\begin{alignat}{1}
&<f,u>\ = \int_{\Omega}f^0u + \sum^n_{i=1}f^iu_{x_i} \ \ \text{ holds for all $u\in\mathfrak{B}$; and}\label{repre}\\
&\|f\|_{\mathfrak{B}^*} = \inf\left\{\left(\int_{\Omega}\sum^n_{i=0}|f^i|^{p'}\,dx\right)^{\frac{1}{p'}}\colon \text{(\ref{repre}) holds.}\right\}
\end{alignat}
Next we justify the Palais-Smale condition on the functional $I$. Suppose $\{v_k\}\subset\mathfrak{B}$ is a Palais-Smale sequence in the sense that
\begin{equation*}
\left|I[v_k]\right|\leq M\ \ \ \ \text{and\ \ }\ \ I'[v_k]\rightarrow 0\ \ \ \ \text{in $\mathfrak{B}^*$}
\end{equation*}
for some $M > 0$. Let $u_k = v_k + u_0\in W^{1,p}(\Omega)$, $k = 1, 2, 3, ...$.
That $Q(x)\beta_{\varepsilon}(v + u_0)\in W^{1,p}_0(\Omega)$ implies that the mapping $v\mapsto Q(x)\beta_{\varepsilon}(v + u_0)$ from $W^{1,p}_0(\Omega)$ to $\mathfrak{B}^*$ is compact due to the fact $W^{1,p}_0(\Omega)\subset\subset L^p(\Omega)\subset \mathfrak{B}^*$ following from the Rellich-Kondrachov Compactness Theorem. Then there exists $f\in L^p(\Omega)\subset\mathfrak{B}^*$ such that for a subsequence, still denoted by $\{v_k\}$, of $\{v_k\}$, it holds that
\begin{equation*}
Q(x)\beta_{\varepsilon}(u_k)\rightarrow -f\ \ \text{ in $L^p(\Omega)$.}
\end{equation*}
Recall that
\begin{equation*}
\left|I'[v_k]\varphi\right| = \sup_{\|\varphi\|_{\mathfrak{B}}\leq 1}\left|\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\cdot\nabla\varphi + Q(x)\beta_{\varepsilon}(u_k) \varphi\right|\rightarrow 0.
\end{equation*}
As a consequence,
\begin{equation}\label{test1}
\sup_{\|\varphi\|_{\mathfrak{B}}\leq M}\left|\int_{\Omega}|\nabla u_k|^{p-2}\nabla u_k\cdot\nabla \varphi - f\varphi\right| \rightarrow 0\ \ \ \ \text{for any $M\geq 0$.}
\end{equation}
Obviously, that $\{I[v_k]\}$ is bounded implies that a subsequence of $\{v_k\}$, still denoted by $\{v_k\}$ by abusing the notation without confusion, converges weakly in $\mathfrak{B} = W^{1, p}_0(\Omega)$. In particular,
\begin{equation*}
\int_{\Omega}fv_k - fv_m\rightarrow 0\ \ \ \ \text{as $k$, $m\rightarrow\infty$.}
\end{equation*}
Then by setting $\varphi = v_k - v_m = u_k - u_m$ in (\ref{test1}), one gets
\begin{equation}\label{conv}
\left|\int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{p-2}\nabla u_m\right)\cdot \nabla (u_k - u_m)\right| \rightarrow 0\ \ \ \ \text{as $k$, $m\rightarrow\infty$,}
\end{equation}
since
\begin{equation*}
\|u_k - u_m\|^p_{\mathfrak{B}} = \|v_k - v_m\|^p_{\mathfrak{B}} \leq 2pM + 2J_p[u_0].
\end{equation*}
In particular, if $p = 2$, $\{v_k\}$ is a Cauchy sequence in $W^{1,2}_0(\Omega)$ and hence converges.
We will apply the following elementary inequalities associated with the $p$-Laplacian, \cite{L}, to the general case $p\neq 2$:
\begin{alignat}{1}
&<|b|^{p-2}b - |a|^{p-2}a,\,b - a> \geq (p-1)|b-a|^2(1 + |a|^2 + |b|^2)^{\frac{p-2}{2}},\ \ 1\leq p\leq 2;\label{ele1}\\
&\text{and}\ \ \ \ <|b|^{p-2}b - |a|^{p-2}a,\,b - a> \geq 2^{2-p}|b-a|^p,\ \ p\geq 2.\label{ele2}
\end{alignat}
We assume first $1 < p < 2$. Let $K = 2pM + 2J_p[u_0]$. Then the first elementary inequality (\ref{ele1}) implies
\begin{alignat*}{1}
&\ \ \ \ \ (p-1)\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1+|\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\\ &\leq \int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{p-2}\nabla u_m\right)\cdot \nabla (u_k - u_m) \rightarrow 0
\end{alignat*}
Meanwhile H\"{o}lder's inequality implies
\begin{alignat*}{1}
&\ \ \ \ \ \int_{\Omega}|\nabla v_k - \nabla v_m|^p = \int_{\Omega}|\nabla u_k - \nabla u_m|^p \\
&\leq \left(\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\right)^{\frac{p}{2}}
\left(\int_{\Omega}\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p}{2}}\right)^{\frac{2-p}{2}} \\
&\leq C(p)\left(|\Omega| + K\right)^{\frac{2-p}{2}} \left(\int_{\Omega}|\nabla u_k - \nabla u_m|^2\left(1 + |\nabla u_k|^2 + |\nabla u_m|^2\right)^{\frac{p-2}{2}}\right)^{\frac{p}{2}}
\end{alignat*}
Therefore, $\{v_k\}$ is a Cauchy sequence in $\mathfrak{B}$ and hence converges.
Suppose $p > 2$. The second elementary inequality (\ref{ele2}) implies
\begin{alignat*}{1}
&\ \ \ \ \ \int_{\Omega}|\nabla v_k - \nabla v_m|^p = \int_{\Omega}|\nabla u_k - \nabla u_m|^p \\
&\leq 2^{p-2}\int_{\Omega}\left(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_m|^{ p-2}\nabla u_m\right)\cdot \left(\nabla u_k - \nabla u_m\right),
\end{alignat*}
which in turn implies $\{v_k\}$ is a Cauchy sequence in $\mathfrak{B}$ and hence converges, on account of (\ref{conv}). The Palais-Smale condition is verified for $1 < p < \infty$ for the functional $I$ on the Banach space $W^{1,p}_0(\Omega)$.
Before we continue the main proof, let us state an elementary result closely related to the $p$-Laplacian, which follows readily from the Fundamental Theorem of Calculus.
\begin{lemma}\label{p-inequalities}
For any $a$ and $b\in\mathbb{R}^n$, it holds
\begin{equation}\label{ele3}
|b|^p \geq |a|^p + p<|a|^{p-2}a, b-a> +\, C(p)|b - a|^p\ \ \ \ (p\geq 2)
\end{equation}
where $C(p) > 0$.
If $1 < p < 2$, then
\begin{equation}\label{ele4}
|b|^p \geq |a|^p + p<|a|^{p-2}a, b-a> +\, C(p)|b-a|^2\int^1_0\int^t_0\left|(1-s)a+sb\right|^{p-2}\,dsdt,
\end{equation}
where $C(p) = p(p-1)$.
\end{lemma}
We are now in a position to show there is a closed mountain ridge around the origin of the Banach space $\mathfrak{B}$ that separates $v_2$ from the origin with the energy $I$ as the elevation function, which is the content of the following lemma.
\begin{lemma}
For all small $\varepsilon > 0$ such that $C\varepsilon \leq \frac{1}{2}\sigma_m$ for a large universal constant $C$, there exist positive constants $\delta$ and $a$ independent of $\varepsilon$, such that, for every $v$ in $\mathfrak{B}$ with $\|v\|_{\mathfrak{B}} = \delta$, the inequality $I[v] \geq a$ holds.
\end{lemma}
\begin{pf}
It suffices to prove $I[v] \geq a > 0$ for every $v\in C^{\infty}_0(\Omega)$ with $\|v\|_{\mathfrak{B}} = \delta$ for $\delta$ small enough, as $I$ is continuous on $\mathfrak{B}$, and $C^{\infty}_0(\Omega)$ is dense in $\mathfrak{B}$.
Let $\Lambda = \{x\in\Omega\colon u(x)\leq\varepsilon\}$, where $u = v + u_0$. We claim that $\Lambda = \emptyset$ if $\delta$ is small enough. If not, one may pick $z\in\Lambda$. Let $\mathcal{AC}([a,b], S)$ be the set of absolutely continuous functions $\gamma\colon [a,b]\rightarrow S$, where $S\subseteq\mathbb{R}^n$. For each $\gamma\in\mathcal{AC}([a,b], S)$, we define its length to be $L(\gamma) = \int^b_a|\gamma'(t)|\,dt$. For $x_0\in\partial\Omega$, we define the distance from $x_0$ to $z$ to be
\begin{equation*}
d(x_0,z) = \inf\{L(\gamma): \gamma\in\mathcal{AC}([0,1],\bar{\Omega}),
\ \text{s.t.\ }\gamma(0) = x_0, \ \text{and\ }\gamma(1)=z\}
\end{equation*}
As shown in \cite{CW}, there is a minimizing path $\gamma_{x_0}$ for the distance $d(x_0, z)$.
Suppose the domain $\Omega$ is convex or star-like about $z$. For any $x_0\in\partial\Omega$, let $\gamma = \gamma_{x_0}$ be a minimizing path of $d(x_0, z)$.
Then it is clear that $\gamma$ is a straight line segment and $\gamma(t)\neq z$
for $t\in [0,1)$. Furthermore, for any two distinct points $x_1$ and $x_2\in\partial\Omega$, the
corresponding minimizing paths do not intersect in $\Omega\backslash\{z\}$. For this reason,
we can carry out the following computation. Clearly $v(x_0) = 0$
and $v(\gamma(1)) = \varepsilon - u_0(\gamma(1))\leq \varepsilon - \sigma_m < 0$. So the
Fundamental Theorem of Calculus
\begin{equation*}
v(\gamma(1)) - v(\gamma(0)) = \int^1_0\nabla v(\gamma(t))\cdot\gamma'(t)dt
\end{equation*}
implies
\begin{equation}\label{ineq-ftc}
\sigma_m - \varepsilon \leq \int^1_0|\nabla v(\gamma(t))||\gamma'(t)|dt.
\end{equation}
For each $x_0\in\partial\Omega$, let $e(x_0)$ be the unit vector in the direction of $x_0 - z$ and $\nu(x_0)$ the outer normal to $\partial\Omega$ at $x_0$. Then $\nu(x_0)\cdot e(x_0) > 0$ everywhere on $\partial\Omega$. Hence the above inequality (\ref{ineq-ftc}) implies
\begin{alignat*}{1}
&\ \ \ \ (\sigma_m - \varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0)\\
&\leq \int_{\partial\Omega}\int^1_0|\nabla v(\gamma(t))| |\gamma'(t)|\,dt\,\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \\
&\leq \int_{\partial\Omega}\left(\int^1_0|\gamma'(t)|\,dt\right)^{\frac{1}{p'}}\left(\int^1_0|\nabla v(
\gamma(t))|^p|\gamma'(t)|\,dt\right)^{\frac{1}{p}}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0),\\
&\hspace{3.5in}\text{ where $\frac{1}{p} + \frac{1}{p'} = 1$,} \\
&= \int_{\partial\Omega} L(\gamma_{x_0})^{\frac{1}{p'}}\left(\int^1_0|\nabla v(\gamma(t))|^p|\gamma'(t)|
\,dt\right)^{\frac{1}{p}}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \\
&\leq \left(\int_{\partial\Omega}L(\gamma_{x_0})\nu(x_0)\cdot e(x_0)\,dH^{n-1}\right)^{\frac{1}{p'}}\left(\int_{\partial\Omega}
\int^1_0|\nabla v(\gamma(t))|^p|\gamma'(t)|\nu \cdot e\,dt\,dH^{n-1}\right)^{\frac{1}{p}} \\
&= C|\Omega|^{\frac{1}{p'}}\left(\int_{\Omega}|\nabla v|^p\,dx\right)^{\frac{1}{p}}\\
&\leq C|\{u > \varepsilon\}|^{\frac{1}{p'}}\delta \leq C|\{u>0\}|^{\frac{1}{p'}}\delta,
\end{alignat*}
where the second and third inequalities are due to the application of the H\"{o}lder's
inequality, and the constant $C$ depends on $n$ and $p$. The second equality follows from the two representation formulas
\begin{equation*}
\left|\Omega\right| = C(n)\int_{\partial\Omega}L(\gamma_{x_0})\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0)
\end{equation*}
and
\begin{equation*}
\int_{\Omega}\left|\nabla v(x)\right|^p\,dx = C(n)\int_{\partial\Omega}\int^1_0\left|\nabla v(\gamma_{x_0}(t))\right|^p\,\left|\gamma'_{x_0}(t)\right|\nu(x_0) \cdot e(x_0)\,dt\,dH^{n-1}(x_0).
\end{equation*}
If we take $\delta$ sufficiently small and independent of $\varepsilon$ in the preceding inequality
\begin{equation*}
(\sigma_m - \varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \leq C|\{u>0\}|^{\frac{1}{p'}}\delta,
\end{equation*}
the measure $|\{u > 0\}|$ of the positive domain would be greater than that of $\Omega$, which is
impossible, provided that
\begin{equation}\label{direction-normal-ineq}
\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \geq C,
\end{equation}
for a constant $C$ which depends on $n$, $p$ and $|\Omega|$, but not on $z$ or $v$.
Hence $\Lambda$ must be empty. So we need to justify the inequality (\ref{direction-normal-ineq}). To fulfil that condition, for $e = e(x_0)$, we set $l(e,z) = l(e) = L(\gamma_{x_0})$. Then \begin{equation*}
\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) = \int_{e\in \partial B}\left(l(e)\right)^{n-1}\,d\sigma(e),
\end{equation*}
where $B$ is the unit ball about $z$ and $d\sigma(e)$ is the surface area element on the unit sphere $\partial B$ which is invariant under rotation and reflection. Clearly,
\begin{equation*}
\left(\int_{\partial B}\left(l(e)\right)^{n-1}\,d\sigma(e)\right)^{\frac{2}{n-1}}\geq C(n)\int_{\partial B}l^2(e)\,d\sigma(e)
\end{equation*}
Consequently, in order to prove (\ref{direction-normal-ineq}), one needs only to prove
\begin{equation}\label{equiv-integ}
\int_{\partial B}l^2(e)\,d\sigma(e) \geq C(n, p, |\Omega|).
\end{equation}
Next, we show the integral on the left-hand-side of (\ref{equiv-integ}) is minimal if $\Omega$ is a ball while its measure is kept unchanged. In fact, this is almost obvious if one notices the following fact. Let $\pi$ be any hyperplane passing through $z$, and $x_1$ and $x_2$ be the points on $\partial\Omega$ which lie on a line perpendicular to $\pi$. Let $x^*_1$ and $x^*_2$ be the points on the boundary $\partial\Omega_{\pi}$, where $\Omega_{\pi}$ is the symmetrized image of $\Omega$ about the hyperplane $\pi$, which lie on the line $\overline{x_1x_2}$. Let $2a = |\overline{x_1x_2}| = |\overline{x^*_1x^*_2}|$ and $d$ be the distance from $z$ to the line $\overline{x_1x_2}$. Then for some $t$ in $-a \leq t \leq a$, it holds that
\begin{equation*}
L^2(\gamma_{x_1}) + L^2(\gamma_{x_2}) = \left(d^2+(a-t)^2\right) + \left(d^2+(a+t)^2\right) \geq 2(d^2 + a^2) = 2\left(L^*(\gamma_{x^*_1})\right)^2.
\end{equation*}
As a consequence, if $\Omega^*$ is the symmetrized ball with measure equal to that of $\Omega$, then
\begin{equation*}
\int_{\partial B}l^2(e)\,d\sigma(e) \geq \int_{\partial B}\left(l^*(e)\right)^2\,d\sigma(e) = C(n,|\Omega|),
\end{equation*}
where $l^*$ is the length from $z$ to a point on the boundary $\partial\Omega^*$ which is constant.
This finishes the proof of the fact that $\Lambda = \emptyset$.
In case the domain $\Omega$ is not convex, the minimizing paths of $d(x_1, z)$ and
$d(x_2, z)$ for distinct $x_1$, $x_2\in\partial\Omega$ may partially coincide. We form the set $\mathcal{DA}(\partial\Omega)$ of the
points $x_0$ on $\partial\Omega$ so that a minimizing path $\gamma$ of $d(x_0, z)$
satisfies $\gamma(t)\in\Omega\backslash\{z\}$ for $t\in (0,1)$. We call a point in $\mathcal{DA}(\partial\Omega)$ a \textbf{directly accessible} boundary point. Let $\Omega_1$ be the union of these minimizing paths for the directly accessible boundary points. It is not difficult to see that $|\Omega_1| > 0$ and hence $H^{n-1}(\mathcal{DA}(\partial\Omega)) > 0$. Then we may apply the above computation to the star-like domain $\Omega_1$ with minimal modification. We have
\begin{equation}
(\sigma_m - C\varepsilon)\int_{\partial\Omega}\nu(x_0)\cdot e(x_0)\,dH^{n-1}(x_0) \leq C|\Omega_1|^{\frac{1}{p'}}\delta \leq C|\Omega|^{\frac{1}{p'}}\delta.
\end{equation}
For small enough $\delta$, this raises a contradiction $|\Omega| > |\Omega|$.
So $\Lambda = \emptyset$.
Finally we prove that $\|v\|_{\mathfrak{B}} = \delta$ implies
\begin{equation}
I[v]= \int_{\Omega}\frac{1}{p}|\nabla v + \nabla u_0|^p - \frac{1}{p}|\nabla u_0|^p \geq a\ \ \text{for a certain $a > 0$.}
\end{equation}
If $p\geq 2$, then the elementary inequality (\ref{ele3}) implies that
\begin{alignat*}{1}
I[v] &= \int_{\Omega}\frac{1}{p}\left|\nabla v+ \nabla u_0\right|^p - \frac{1}{p}\left|\nabla u_0\right|^p \\
&\geq \int_{\Omega}<\left|\nabla u_0\right|^{p-2}\nabla u_0, \nabla v> + C(p)\left|\nabla v\right|^p \\
&= C(p)\int_{\Omega}\left|\nabla v\right|^p = C(p)\delta^p > 0,
\end{alignat*}
while if $1 < p < 2$, then the elementary inequality (\ref{ele4}) implies
\begin{alignat*}{1}
I[v] &\geq p(p-1)\int_{\Omega}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left|\nabla u_0 + s\nabla v\right|^{2-p}}\,dsdtdx \\
&\geq p(p-1)\int_{\Omega}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx.
\end{alignat*}
If $\int_{\Omega}|\nabla u_0|^p = 0$, then $I[v] = \frac{1}{p}\delta^p > 0$. So in the following, we assume $\int_{\Omega}|\nabla u_0|^p > 0$.
Let $S = S_{\lambda} = \{x\in\Omega\colon |\nabla v| > \lambda\delta\}$, where the constant $\lambda = \lambda(p,|\Omega|)$ is to be taken. Then
\begin{alignat*}{1}
\delta^p &= \int_{\Omega}|\nabla v|^p = \int_{\{|\nabla v|\leq \lambda\delta\}}|\nabla v|^p + \int_S|\nabla v|^p \\
&\leq (\lambda\delta)^p|\Omega| + \int_S|\nabla v|^p
\end{alignat*}
and hence
\begin{equation*}
\int_S|\nabla v|^p \geq \delta^p\left(1 - \lambda^p|\Omega|\right) \geq \frac{1}{2}\delta^p,\ \ \text{if $\lambda$ satisfies\ }\frac{1}{4} < \lambda^p|\Omega| \leq \frac{1}{2}.
\end{equation*}
Meanwhile, for $1 < p < 2$, it holds that
\begin{alignat*}{1}
I[v] &\geq C(p)\int_{S}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx \\
&=C(p)\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(|\nabla u_0| + s|\nabla v|\right)^{2-p}}\,dsdtdx \right. \\
&\ \ \ \ + \left.\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx\right).
\end{alignat*}
The first integral on the right satisfies
\begin{alignat*}{1}
&\ \ \ \ \int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(|\nabla u_0| + s|\nabla v|\right)^{2-p}}\,dsdtdx \\
&\geq \int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^p\int^1_0\int^t_0\frac{1}{\left(1 + s\right)^{2-p}}\,dsdtdx \\
&= C(p)\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}\left|\nabla v\right|^p\,dx,
\end{alignat*}
while the second integral on the right satisfies
\begin{alignat*}{1}
&\ \ \ \ \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^2\int^1_0\int^t_0\frac{1}{\left(\left|\nabla u_0\right| + s\left|\nabla v\right|\right)^{2-p}}\,dsdtdx \\
&\geq \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{\left|\nabla v\right|^2}{|\nabla u_0|^{2-p}}\int^1_0\int^t_0\frac{ds\,dt}{(1+s)^{2-p}}\,dx \\
&= C(p) \int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{\left|\nabla v\right|^2}{|\nabla u_0|^{2-p}}\,dx.
\end{alignat*}
The H\"{o}lder's inequality applied with exponents $\frac{2}{p}$ and $\frac{2}{2-p}$ implies that
\begin{equation*}
\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p \leq \left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{|\nabla v|^2}{|\nabla u_0|^{2-p}}\right)^{\frac{p}{2}}\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}|\nabla u_0|^p\right)^{\frac{2-p}{2}},
\end{equation*}
or equivalently
\begin{alignat*}{1}
\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\frac{|\nabla v|^2}{|\nabla u_0|^{2-p}} &\geq \frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}|\nabla u_0|^p\right)^{\frac{2-p}{p}}} \\
&\geq \frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}}.
\end{alignat*}
Consequently,
\begin{alignat*}{1}
I[v] &\geq C(p)\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p + C(p)\frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}} \\
&\geq C(p)\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}} + C(p)\frac{\left(\int_{S\cap\{|\nabla u_0| > |\nabla v|\}}\left|\nabla v\right|^p\right)^{\frac{2}{p}}}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}},\ \ \text{as $\delta$ is small} \\
&\geq C(p)A(u_0)\left(\left(\int_{S\cap\{|\nabla u_0|\leq |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}} + \left(\int_{S\cap\{|\nabla u_0|> |\nabla v|\}}|\nabla v|^p\right)^{\frac{2}{p}}\right) \\
&\geq C(p)A(u_0)\left(\int_{S}|\nabla v|^p\right)^{\frac{2}{p}} = C(p)A(u_0)\delta^2,
\end{alignat*}
where the last inequality is a consequence of the elementary inequality
\begin{equation*}
a^{\frac{2}{p}} + b^{\frac{2}{p}} \geq C(p)\left(a+b\right)^{\frac{2}{p}}\ \ \text{for\ }\ a, b\geq 0,
\end{equation*}
and the constant
\begin{equation*}
A(u_0) = \min\left\{1,\frac{1}{\left(\int_{\Omega}|\nabla u_0|^p\right)^{\frac{2-p}{p}}}\right\}.
\end{equation*}
So we have proved $I[v] \geq a > 0$ for some $a> 0$ whenever $v\in C^{\infty}_0(\Omega)$ satisfies $\|v\|_{\mathfrak{B}} = \delta$, for any $p\in (1, \infty)$.
\end{pf}
Let
\begin{equation*}
\mathcal{G} = \{\gamma\in C([0,1],H): \gamma(0) = 0\ \text{and\ }\gamma(1) = v_2\}
\end{equation*}
and
\begin{equation*}
c = \inf_{\gamma\in\mathcal{G}}\max_{0\leq t\leq 1}I[\gamma(t)].
\end{equation*}
The verified Palais-Smale condition and the preceding lemma allow us to apply the Mountain Pass Theorem as stated, for example, in \cite{J} to conclude that there is a $v_1\in \mathfrak{B}$ such that $I[v_1] = c$,
and $I'[v_1] = 0$ in $\mathfrak{B}^*$.
That is
\begin{equation*}
\int_{\Omega}\left|\nabla u_1\right|^{p-2}\nabla u_1\cdot\nabla \varphi + Q(x)\beta_{\varepsilon}(u_1)\varphi dx = 0
\end{equation*}
for any $\varphi\in \mathfrak{B} = W^{1,p}_0(\Omega)$, where $u_1 = v_1 + u_0$. So $u_1$ is a weak solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}). In essence, the Mountain Pass Theorem is a way to produce a saddle point solution. Therefore, in general, $u_1$ tends to be an unstable solution in contrast to the stable solutions $u_0$ and $u_2$.
In this section, we have proved the following theorem.
\begin{theorem}
If $\varepsilon << \sigma_m$ and $J_p(u_2) < J_p(u_0)$, then there exists a third weak solution $u_1$ of the problem (\ref{eulereq}) and (\ref{bdrycondition}). Moreover, $J_p(u_1) \geq J_p(u_0) + a$, where $a$ is independent of $\varepsilon$.
\end{theorem}
\section{A Comparison Principle for Evolution}\label{comparison}
In this section, we prove a comparison theorem for the following evolution problem.
\begin{equation}\label{evolution2}
\left\{\begin{array}{ll}
w_t - \bigtriangleup_p w + \alpha(x,w) = 0 &\ \text{in\ }\Omega\times(0, T)\\
w(x,t) = \sigma(x) &\ \text{on\ }\partial \Omega\times (0, T)\\
w(x,0) = v_0(x) &\ \text{for\ }x\in\bar{\Omega},\end{array}\right.
\end{equation}
where $T>0$ may be finite or infinite, and $\alpha$ is a continuous function satisfying $0 \leq \alpha(x,w) \leq Kw$ and
\begin{equation*}
\left|\alpha(x,r_2) - \alpha(x,r_1)\right| \leq K\left|r_2 - r_1\right|
\end{equation*}
for all $x\in\Omega$, $r_1$ and $r_2\in\mathbb{R}$, and some $K \geq 0$. Let us introduce the notation $H_pw = w_t - \bigtriangleup_p w + \alpha(x,w)$. We recall a weak sub-solution $w\in L^2(0,T; W^{1,p}(\Omega))$ satisfies
\begin{equation*}
\left.\int_Vw\varphi\ \right|_{t_1}^{t_2} + \int^{t_2}_{t_1}\int_V-w\varphi_t + |\nabla w|^{p-2}\nabla w\cdot\nabla\varphi + \alpha(x,w)\varphi \leq 0
\end{equation*}
for any region $V\subset\subset\Omega$ and any test function $\varphi\in L^2(0,T; W^{1,p}(\Omega))$ such that $\varphi_t\in L^2(\Omega\times\mathbb{R}_T)$ and $\varphi\geq 0$ in $\Omega\times\mathbb{R}_T$, where $L^2_0(0,T; W^{1,p}(\Omega))$ is the subset of $L^2(0,T; W^{1,p}(\Omega))$ that contains functions which is equal zero on the boundary of $\Omega\times\mathbb{R}_T$, where $\mathbb{R}_T = [0, T]$. For convenience, we let $\mathfrak{T}_+$ denote this set of test functions in the following.
In particular, it holds that
\begin{equation*}
\int^T_0\int_{\Omega}-w\varphi_t + <|\nabla w|^{p-2}\nabla w, \nabla\varphi> + \alpha(x,w)\varphi \leq 0
\end{equation*}
for any test function $\varphi\in L^2_0(0,T; W^{1,p}(\Omega))$ such that $\varphi_t\in L^2(\Omega\times\mathbb{R}_T)$ and $\varphi\geq 0$ in $\Omega\times\mathbb{R}_T$.
The comparison principle for weak sub- and super-solutions is stated as follows.
\begin{theorem}\label{paraboliccomparison}
Suppose $w_1$ and $w_2$ are weak sub- and super-solutions of the evolutionary problem
(\ref{evolution2}) respectively with $w_1\leq w_2$ on the parabolic boundary $(\bar{\Omega}\times\{0\})
\cup(\partial \Omega\times (0, +\infty))$. Then $w_1\leq w_2$ in $\mathcal{D}$.
\end{theorem}
Uniqueness of a weak solution of (\ref{evolution2}) follows from the comparison principle, Theorem \ref{paraboliccomparison}, immediately.
\begin{lemma}
For $T > 0$ small enough, if $H_pw_1 \leq 0 \leq H_pw_2$ in the weak sense in
$\Omega\times \mathbb{R}_T$ and $w_1 < w_2$ on $\partial_p(\Omega \times
\mathbb{R}_T)$, then $w_1\leq w_2$ in $\Omega\times\mathbb{R}_T$.
\end{lemma}
\begin{pf}
For any given small number $\delta > 0$, we define a new function
$\tilde{w}_1$ by $$\tilde{w}_1(x,t) = w_1(x,t) - \frac{\delta}{T-t},$$ where
$x\in\bar{\Omega}$ and $0\leq t < T$. In order to prove $w_1\leq
w_2$ in $\Omega\times\mathbb{R}_T$, it suffices to prove $\tilde{w}_1\leq w_2$ in
$\Omega\times\mathbb{R}_T$ for all small $\delta > 0$. Clearly, $\tilde{w}_1 <
w_2$ on $\partial_p(\Omega\times \mathbb{R}_T)$, and $\lim_{t\rightarrow
T}\tilde{w}_1(x,t) = -\infty$ uniformly on $\Omega$. Moreover, the following holds for any $\varphi\in\mathfrak{T}_+$:
\begin{alignat*}{1}
&\ \ \ \ \ \ \int^T_0\int_{\Omega}-\tilde{w}_1\varphi_t + <|\nabla\tilde{w}_1|^{p-2}\nabla\tilde{w}_1, \nabla\varphi> + \alpha(x,\tilde{w}_1)\varphi \\
&= \int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1, \nabla\varphi> + \frac{\delta}{T-t}\varphi_t + \left(\alpha(x,\tilde{w}_1) - \alpha(x, w_1)\right)\varphi \\
&\leq \int^T_0\int_{\Omega}\frac{\delta}{T-t}\varphi_t + K\frac{\delta}{T-t}\varphi, \ \ \text{as $w_1$ is a weak sub-solution}\\
&= \int^T_0\int_{\Omega}\left(-\frac{\delta}{(T-t)^2} + K\frac{\delta}{T-t}\right)\varphi \\
&\leq \int^T_0\int_{\Omega} -\frac{\delta}{2(T-t)^2}\varphi,\ \ \text{for $T\leq\frac{1}{2K}$ so that $2K\leq \frac{1}{T-t}$} \\
&< 0,
\end{alignat*}
i.\,e.\,
\begin{equation*}
H_p\tilde{w}_1 \leq -\frac{\delta}{2(T-t)^2} \leq -\frac{\delta}{2T^2} < 0\ \ \text{in the weak sense.}
\end{equation*}
That is, if we abuse the notation a little by denoting $\tilde{w}_1$ by $w_1$ in the following for convenience, it holds for any $\varphi\in\mathfrak{T}_+$,
\begin{equation*}
\int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1, \nabla\varphi> + \alpha(x,w_1)\varphi \leq \int^T_0\int_{\Omega}-\frac{\delta}{2T^2}\varphi < 0.
\end{equation*}
Meanwhile, for any $\varphi\in \mathfrak{T}_+$, $w_2$ satisfies
\begin{equation*}
\int^T_0\int_{\Omega}-w_2\varphi_t + <|\nabla w_2|^{p-2}\nabla w_2, \nabla\varphi> + \alpha(x,w_2)\varphi \geq 0.
\end{equation*}
Define, for $j = 1, 2$, $v_j(x,t) = e^{-\lambda t}w_j(x, t)$, where the constant $\lambda > 2K$. Then $w_j(x,t) = e^{\lambda t}v_j(x, t)$, and it is clear
that $w_1\leq w_2$ in $\Omega\times\mathbb{R}_T$ is equivalent to $v_1\leq v_2$ in $\Omega\times\mathbb{R}_T$. In addition, for any $\varphi\in \mathfrak{T}_+$, the following inequalities hold:
\begin{alignat*}{1}
&\int^T_0\int_{\Omega}-e^{\lambda t}v_1\varphi_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1, \nabla\varphi> + \alpha(x,e^{\lambda t}v_1)\varphi \leq -\int^T_0\int_{\Omega}\frac{\delta}{2T^2}\varphi \\
&\text{and\ }\ \int^T_0\int_{\Omega}-e^{\lambda t}v_2\varphi_t + e^{\lambda(p-1)t}<|\nabla v_2|^{p-2}\nabla v_2, \nabla\varphi> + \alpha(x,e^{\lambda t}v_2)\varphi \geq 0.
\end{alignat*}
Consequently, it holds for any $\varphi\in \mathfrak{T}_+$
\begin{alignat*}{1}
&\int^T_0\int_{\Omega}-e^{\lambda t}(v_1 - v_2)\varphi_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla\varphi> \\
&\ \ \ \ \ \ \ \ + \left( \alpha(x, e^{\lambda t}v_1) - \alpha(x, e^{\lambda t}v_2)\right)\varphi \leq -\int^T_0\int_{\Omega}\frac{\delta}{2T^2}\varphi.
\end{alignat*}
We take $\varphi = \left(v_1 - v_2\right)^+ = \max\{v_1 - v_2, 0\}$ as the test function, since it vanishes on the boundary of $\Omega\times\mathbb{R}_T$. Then
\begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}-e^{\lambda t}(v_1 - v_2)(v_1 - v_2)_t + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla v_1 - \nabla v_2> \\
&\ \ \ \ \ \ \ \ + \left(\alpha(x,e^{\lambda t}v_1) - \alpha(x,e^{\lambda t}v_2)\right)(v_1 - v_2) \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2).
\end{alignat*}
Since
\begin{equation*}
\{v_1>v_2\}\subset\Omega\times(0,T)\ \ \text{due to the facts $v_1\leq v_2$ on $\partial_p(\Omega\times\mathbb{R}_T)$ and $v_1\rightarrow -\infty$ as $t\uparrow T$,}
\end{equation*}
the divergence theorem implies
\begin{equation*}
\int^T_0\int_{\{v_1 > v_2\}}-e^{\lambda t}(v_1-v_2)(v_1-v_2)_t = \int^T_0\int_{\{v_1>v_2\}}\lambda e^{\lambda t}\frac{1}{2}(v_1-v_2)^2.
\end{equation*}
On the other hand,
\begin{equation*}
\left(\alpha(x,e^{\lambda t}v_1) - \alpha(x,e^{\lambda t}v_2)\right)(v_1 - v_2)\geq -Ke^{\lambda t}(v_1-v_2)^2\ \ \text{on $\{v_1 > v_2\}$.}
\end{equation*}
As a consequence, it holds that
\begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + e^{\lambda(p-1)t}<|\nabla v_1|^{p-2}\nabla v_1 - |\nabla v_2|^{p-2}\nabla v_2, \nabla v_1 - \nabla v_2> \\
& \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2).
\end{alignat*}
We call into play two elementary inequalities (\cite{L}) associated with the $p$-Laplacian:
\begin{equation*}
<|b|^{p-2}b - |a|^{p-2}a, b-a> \geq (p-1)|b-a|^2\left(1 + |b|^2 + |a|^2\right)^{\frac{p-2}{2}}\ \ (1\leq p\leq 2),
\end{equation*}
and
\begin{equation*}
<|b|^{p-2}b - |a|^{p-2}a, b-a> \geq 2^{2-p}|b-a|^p\ \ (p\geq 2)\ \ \text{for any $a$, $b\in\mathbb{R}^n$.}
\end{equation*}
By applying them with $b = \nabla v_1$ and $a = \nabla v_2$ in the preceding inequalities, we obtain
\begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + (p-1)e^{\lambda(p-1)t}\left|\nabla v_1 - \nabla v_2\right|^2\left( 1 + |\nabla v_1|^2 + |\nabla v_2|^2\right)^{\frac{p-2}{2}} \\
& \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2)\ \ \ \ \text{for $1 < p <2$}
\end{alignat*}
and
\begin{alignat*}{1}
&\int^T_0\int_{\{v_1>v_2\}}\left(\frac{\lambda}{2} - K\right)e^{\lambda t}(v_1 - v_2)^2 + 2^{2-p}e^{\lambda(p-1)t}\left|\nabla v_1 - \nabla v_2\right|^p \\
& \leq -\frac{\delta}{2T^2}\int^T_0\int_{\{v_1>v_2\}}(v_1-v_2)\ \ \ \ \text{for $p\geq 2$.}
\end{alignat*}
One can easily see in either case the respective inequality is true only if the measure of the set $\{v_1>v_2\}$ is zero. The proof is complete.
\end{pf}
In the next lemma, we show the strict inequality on the boundary data can be relaxed to a non-strict one.
\begin{lemma}
For $T > 0$ sufficiently small, if $H_pw_1 \leq 0 \leq H_pw_2$ in the weak sense in
$\Omega\times\mathbb{R}_T$ and $w_1\leq w_2$ on $\partial_p(\Omega \times
\mathbb{R}_T)$, then $w_1\leq w_2$ on $\overline{\Omega\times\mathbb{R}_T}$.
\end{lemma}
\begin{pf}
For any $\delta > 0$, take $\tilde{\delta} > 0$ such that $\tilde{\delta} \leq\frac{\delta}{4K}$ and define
\begin{equation*}
\tilde{w}_1(x,t) = w_1(x,t) - \delta t - \tilde{\delta}\ \ (x,t)\in\bar{\Omega}\times\mathbb{R}^n.
\end{equation*}
Then $\tilde{w}_1 < w_1 \leq w_2$ on $\partial_p(\Omega\times\mathbb{R}^n)$, and for any $\varphi\in\mathfrak{T}_+$, the following holds:
\begin{alignat*}{1}
&\ \ \ \ \int^T_0\int_{\Omega}-\tilde{w}_1\varphi_t + <|\nabla\tilde{w}_1|^{p-2}\nabla\tilde{w}_1,\nabla\varphi> + \alpha(x,\tilde{w})\varphi \\
&= \int^T_0\int_{\Omega}-w_1\varphi_t + <|\nabla w_1|^{p-2}\nabla w_1,\nabla\varphi> + \alpha(x,w_1)\varphi\\
&\ \ \ \ \ \ \ \ \ \ - \delta \varphi + \left(\alpha(x,w_1 - \delta t - \tilde{\delta}) - \alpha(x,w_1)\right)\varphi \\
&\leq \int^T_0\int_{\Omega} -\delta\varphi + K\left(\delta t + \tilde{\delta}\right)\varphi \leq \int^T_0\int_{\Omega} -\delta\varphi + K\left(\delta T + \tilde{\delta}\right)\varphi \\
&\leq \int^T_0\int_{\Omega}\left(-\delta + \frac{\delta}{2} + \frac{\delta}{4}\right)\varphi\ \ \text{for $T$ small} \\
&= -\frac{\delta}{4}\int^T_0\int_{\Omega}\varphi.
\end{alignat*}
The preceding lemma implies $\tilde{w}_1\leq w_2$ in $\overline{\Omega\times\mathbb{R}_T}$ for small $T$ and for any small $\delta > 0$, and whence the conclusion of this lemma.
\end{pf}
Now the parabolic comparison theorem (\ref{paraboliccomparison}) follows from
the preceding lemma quite easily as shown by the following argument: Let $T_0 > 0$ be
any small value of $T$ in the preceding lemma so that the conclusion of the preceding
lemma holds. Then $w_1\leq w_2$ on $\overline{\Omega\times (0, T_0)}$. In particular,
$w_1 \leq w_2$ on $\partial_p(\Omega\times (T_0,2T_0))$. The preceding lemma may be
applied again to conclude that $w_1 \leq w_2$ on $\overline{\Omega\times (T_0, 2T_0)}$.
And so on. This recursion allows us to conclude that $w_1 \leq w_2$ on $\overline{\Omega\times\mathbb{R}_T}$.
\section{Convergence of Evolution}
Define $\mathfrak{S}$ to be the set of weak solutions of the stationary problem (\ref{eulereq}) and (\ref{bdrycondition}). The $p$-harmonic function $u_0$ is the maximum element in $\mathfrak{S}$, while $u_2$ denotes the least solution which may be constructed as the infimum of super-solutions. We also use the term \textit{non-minimal solution} with the same definition in \cite{CW}. That is, $u$ a non-minimal solution of the problem (\ref{eulereq}) and (\ref{bdrycondition}) if it is a viscosity solution but not a local minimizer in the sense that for any $\delta > 0$, there exists $v$ in the admissible set of the functional $J_p$ with $v = \sigma$ on $\partial\Omega$ such that
$\|v - u\|_{L^{\infty}} < \delta$, and
$J_p(v) < J_p(u)$.
In this section, we consider the evolutionary problem
\begin{equation}\label{evolution}
\left\{\begin{array}{ll}
w_t - \bigtriangleup_p w + Q(x)\beta_{\varepsilon}(w) = 0 &\ \text{in\ }\Omega\times(0,+\infty)\\
w(x,t) = \sigma(x) &\ \text{on\ }\partial \Omega\times (0, +\infty)\\
w(x,0) = v_0(x) &\ \text{for\ }x\in\bar{\Omega},\end{array}\right.
\end{equation}
and will apply the parabolic comparison principle (\ref{paraboliccomparison}) proved in Section \ref{comparison} to prove the following convergence of evolution theorem. One just notes that the parabolic problem (\ref{evolution2}) includes the above problem (\ref{evolution}) as a special case so that the comparison principle (\ref{paraboliccomparison}) applies in this case.
\begin{theorem}\label{convergence} If the initial data $v_0$ falls into any of the
categories specified below, the corresponding conclusion of convergence holds.
\begin{enumerate}
\item If $v_0 \leq u_2$ on $\bar{\Omega}$, then $\lim_{t\rightarrow +\infty}w(x,t) = u_2(x)$ locally uniformly
for $x\in\bar{\Omega}$;
\item Define
\begin{equation*}
\bar{u}_2(x) = \inf_{u\in\mathfrak{S}, u \geq u_2, u \neq u_2}u(x),\ x\in\bar{\Omega}.
\end{equation*}
If $\bar{u}_2 > u_2$, then for $v_0$ such that $u_2 < v_0 < \bar{u}_2$, $\lim_{t\rightarrow +\infty}w(x,t) = u_2(x)$ locally uniformly for $x\in\bar{\Omega}$;
\item Define $\bar{u}_0(x) = \sup_{u\in\mathfrak{S}, u \leq u_0, u\neq u_0}u(x)$, $x\in\bar{\Omega}$. If $\bar{u}_0
< u_0$, then for $v_0$ such that $\bar{u}_0 < v_0 < u_0$, $\lim_{t\rightarrow +\infty}w(x,t) =
u_0(x)$ locally uniformly for $x\in\bar{\Omega}$;
\item If $v_0 \geq u_0$ in $\bar{\Omega}$, then $\lim_{t\rightarrow +\infty}w(x,t) = u_0(x)$ locally uniformly
for $x\in\bar{\Omega}$;
\item Suppose $u_1$ is a non-minimal solution of (\ref{eulereq}) and (\ref{bdrycondition}). For any small $\delta > 0$, there exists $v_0$ such that $\|v_0 - u_1\|_{L^{\infty}(\Omega)} < \delta$
and the solution $w$ of the problem (\ref{evolution}) does not satisfy $$\lim_{t\rightarrow
\infty} w(x,t) = u_1(x)\ \text{\ in\ } \Omega.$$
\end{enumerate}
\end{theorem}
\begin{pf}
We first take care of case 4. We may take new initial data a smooth function $\tilde{v}_0$ so that $D^2\tilde{v}_0 < -KI$ and $|\nabla \tilde{v}_0| \geq \delta > 0$ on $\bar{\Omega}$. According to the parabolic comparison principle (\ref{paraboliccomparison}), it suffices to prove the solution $\tilde{w}$ generated by the initial data $\tilde{v}_0$ converges locally uniformly to $u_0$ if we also take $\tilde{v}_0$ large than $v_0$, which can easily be done. So we use $v_0$ and $w$ for the new functions $\tilde{v}_0$ and $\tilde{w}$ without any confusion.
For any $V\subset\subset\Omega$ and any nonnegative function $\varphi$ which is independent of the time variable $t$ and supported in $V$, it holds that
\begin{equation*}
\begin{split}
\int_V|\nabla v_0|^{p-2}\nabla v_0\cdot\nabla\varphi &= \int_V - div\left(|\nabla v_0|^{p-2}\nabla v_0\right)\varphi \\
&\geq \int_V M\varphi\ \ \ \ \text{for some $M = M(n,p,K,\delta) > 0$.}
\end{split}
\end{equation*}
The H\"{o}lder continuity of $\nabla w$ up to $t = 0$ as stated in \cite{DiB}, then implies
\begin{equation*}
\int_V|\nabla w|^{p-2}\nabla w\cdot\nabla\varphi \geq \frac{M}{2}\int_V\varphi
\end{equation*}
for any small $t$ in $(0,t_0)$, and any nonnegative function $\varphi$ which is independent of $t$, supported in $V$ and subject to the condition
\begin{equation}\label{conda}
\frac{\int_V|\nabla \varphi|}{\int_V\varphi}\leq A
\end{equation}
for a fixed constant $A > 0$ and some $t_0 > 0$ dependent on $A$. Then the sub-solution condition on $w$
\begin{equation*}
\left.\int_Vw\varphi\right|_{t=t_2} - \left.\int_Vw\varphi\right|_{t=t_1} + \int^{t_2}_{t_1}\int_V|\nabla w|^{p-2}\nabla w\cdot\nabla\varphi \leq 0
\end{equation*}
implies that
\begin{equation*}
\left.\int_Vw\varphi\right|_{t=t_2} - \left.\int_Vw\varphi\right|_{t=t_1} \leq -\frac{M}{2}(t_2-t_1)\int_V\varphi
\end{equation*}
for any small $t_2 > t_1$ in $(0, t_0)$, and any nonnegative function $\varphi$ which is independent of $t$, supported in $V$ and subject to (\ref{conda}).
In particular, $\left.\int_Vw\varphi\right|^{t_2}_{t_1} \leq 0$ for any nonnegative function $\varphi$ independent of $t$, supported in $V$ and subject to (\ref{conda}). So $$w(x, t_2) \leq w(x, t_1)$$ for any $x\in\Omega$ and $0\leq t_1\leq t_2$. Then the parabolic comparison principle readily implies $w$ is decreasing in $t$ for $t$ in $[0, \infty)$. Therefore $w(x,t)\rightarrow u^{\infty}(x)$ locally uniformly as $t\rightarrow\infty$ and hence $u^{\infty}$ is a solution of (\ref{eulereq}) and (\ref{bdrycondition}). Furthermore, the parabolic comparison principle also implies $w(x,t)\geq u_0(x)$ at any time $t > 0$. Consequently, $u^{\infty} = u_0$ as $u_0$ is the greatest solution of (\ref{eulereq}) and (\ref{bdrycondition}).
Next, we briefly explain the proof for case 1. We may take a new smooth initial data $\tilde{v}_0$ such that $\tilde{v}_0$ is very large negative, $D^2\tilde{v}_0 \geq KI$ and $|\nabla \tilde{v}_0| \geq\delta$ on $\bar{\Omega}$ for large constant $K>0$ and constant $\delta > 0$. It suffices to prove the solution $\tilde{w}$ generated by the initial data $\tilde{v}_0$ converges to $u_2$ locally uniformly on $\bar{\Omega}$ as $t\rightarrow\infty$. Following a computation exactly parallel to that in case 4, we can prove $w$ is increasing in $t$ in $[0, \infty)$. So $w$ converges locally uniformly to a solution $u^{\infty}$ of (\ref{eulereq}) and (\ref{bdrycondition}). As $u^{\infty}\leq u_2$ and $u_2$ is the least solution of (\ref{eulereq}) and (\ref{bdrycondition}), we conclude $u^{\infty} = u_2$.
In case 2, we may replace $v_0$ by a strict super-solution of $\bigtriangleup_p v - Q\beta_{\varepsilon}(v) = 0$ in $\bar{\Omega}$ between $u_2$ and $\bar{u}_2$, by employing the fact that $u_2$ is the infimum of super-solutions of (\ref{eulereq}) and (\ref{bdrycondition}). Using $v_0$ as the initial data, we obtain a solution $w(x,t)$ of (\ref{evolution}). Then one argues as in case 4 that for any $V\subset\subset\Omega$, there exist constants $A > 0$ and $t_0 > 0$ such that for $t_1 < t_2$ with $t_1$, $t_2\in [0, t_0)$, $\int_Vw\varphi\,|^{t_2}_{t_1} \leq 0$ for any nonnegative function $\varphi$ independent of $t$, supported in $V$ and subject to the condition $\frac{\int_V|\nabla\varphi|}{\int_V\varphi} \leq A$. As a consequence, $w(x,t_1) \geq w(x,t_2)\ \ (x\in\Omega)$. Then the parabolic comparison principle implies $w$ is decreasing in $t$ over $[0, +\infty)$. Therefore $w(x,t)$ converges locally uniformly to some function $u^{\infty}$ as $t\rightarrow\infty$ which solves (\ref{eulereq}) and (\ref{bdrycondition}). Clearly $u_2(x) \leq w(x,t) \leq \bar{u}_2(x)$ from which $u_2(x) \leq u^{\infty}(x) \leq \bar{u}_2(x)$ follows. As $w$ is decreasing in $t$ and $v_0\neq \bar{u}_2$, $u^{\infty} \neq \bar{u}_2$. Hence $u^{\infty} = u_2$.
The proof of case 3 is parallel to that of case 2 with the switch of sub- and super-solutions. Hence we skip it.
In case 5, we pick $v_0$ with $\|v_0 - u_1\|_{L^{\infty}} < \delta$ and $J_p(v_0) < J_p(u_1)$. Let $w$ be the solution of (\ref{evolution}) with $v_0$ as the initial data. Clearly, we may change the value of $v_0$ slightly if necessary so that it is not a solution of the equation
$$-\nabla\cdot\left(\left(\varepsilon + |\nabla u|^2\right)^{p/2-1}\nabla u\right) + Q(x)\beta(u) = 0$$ for any small $\varepsilon > 0$.
Let $w^{\varepsilon}$ be the smooth solution of the uniformly parabolic boundary-value problem
\begin{equation*}
\left\{\begin{array}{ll}
w_t - \nabla\cdot\left(\left(\varepsilon + |\nabla w|^2\right)^{p/2-1}\nabla w\right) + Q\beta(w) = 0 &\ \ \text{in $\Omega\times (0, +\infty)$}\\
w(x,t) = \sigma(x) &\ \ \text{on $\partial\Omega\times (0, +\infty)$}\\
w(x,0) = v_0(x) &\ \ \text{on $\bar{\Omega}$.}
\end{array}\right.
\end{equation*}
$w^{\varepsilon}$ converges to $w$ in $W^{1,p}(\Omega)$ for every $t\in [0, \infty)$ as $\varepsilon\rightarrow 0$.
We define the functional
$$J_{\varepsilon, p}(u) = \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla u|^2\right)^{p/2} + Q(x)\Gamma(u)\,dx.$$
It is easy to see that
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 - \nabla\cdot\left(\left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2-1}\nabla w^{\varepsilon}\right)w^{\varepsilon}_t + Q\beta(w^{\varepsilon})w^{\varepsilon}_t = 0.$$
As $w^{\varepsilon}_t = 0$ on $\partial\Omega\times (0,\infty)$, one gets
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2-1}\nabla w^{\varepsilon}\cdot \nabla w^{\varepsilon}_t + Q(x)\Gamma(w^{\varepsilon})_t = 0,$$
which implies
$$\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \frac{1}{p}\left(\left(\varepsilon + |\nabla w^{\varepsilon}|^2\right)^{p/2}\right)_t + Q(x)\Gamma(w^{\varepsilon})_t = 0.$$
Consequently, it holds
\begin{equation*}
\begin{split}
&\ \ \ \ \int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla w^{\varepsilon}(x,t)|^2\right)^{p/2} + Q\Gamma(w^{\varepsilon}(x,t)) \\
&= \frac{1}{p}\int_{\Omega}\left(\varepsilon + |\nabla w^{\varepsilon}(x,0)|^2\right)^{p/2} + Q\Gamma(w^{\varepsilon}(x,0))
\end{split}
\end{equation*}
i.\,e.\,
\begin{equation*}
\int^t_0\int_{\Omega}\left(w^{\varepsilon}_t\right)^2 + J_{\varepsilon, p}(w^{\varepsilon}(\cdot,t)) = J_{\varepsilon, p}(w^{\varepsilon}(\cdot,0)).
\end{equation*}
Therefore
$$J_{\varepsilon, p}(w^{\varepsilon}(\cdot, t) \leq J_{\varepsilon, p}(v_0),$$
which in turn implies
$$J_p(w(\cdot, t) \leq J_p(v_0) < J_p(u_1).$$
In conclusion, $w$ does not converge to $u_1$ as $t\rightarrow\infty$.
\end{pf}
\end{document} |
\begin{document}
\date{
\vskip 20pt}
\begin{abstract} Let $G$ be a locally compact
topological group, $G_0$ the connected component of
its identity element, and $\mathrm{comp}(G)$ the union of all compact
subgroups. A topological group will be called
inductively monothetic if any subgroup generated (as
a topological group) by finitely many elements is generated
(as a topological group) by a single element.
The space ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ of all closed subgroups of $G$ carries
a compact Hausdorff topology called the Chabauty topology.
Let ${\mathcal{F}_1}(G)$, respectively, ${\mathcal{R}_1}(G)$, denote the subspace of
all discrete subgroups isomorphic to $\mathbb{Z}$, respectively,
all subgroups isomorphic to $\mathbb{R}$.
It is shown that a necessary and sufficient
condition for $G\in\overline{{\mathcal{F}_1}(G)}$ to hold is that $G$ is
abelian, and either that
$G\cong \mathbb{R}\times \mathrm{comp}(G)$ and $G/G_0$ is inductively
monothetic, or else that $G$ is discrete and isomorphic to a subgroup
of $\mathbb{Q}$. It is further shown that a necessary and sufficient
condition for $G\in\overline{{\mathcal{R}_1}(G)}$ to hold is that
$G\cong\mathbb{R}\times C$ for a compact connected abelian group $C$.\\
\textit{MSC 2010:} 22B05, 54E45.
\end{abstract}
\maketitle
\vskip-15pt
\centerlineterline{Authors' addresses:}
\hbox{\vtop{\hsize=.45\hsize
Hatem Hamrouni\\
Faculty of Sciences at Sfax\\
Department of Mathematics\\
Sfax University\\
B.P. 1171. 3000 Sfax, Tunisia\\
{\tt [email protected]}}
\vtop{\hsize=.45\hsize
Karl H. Hofmann\\
Fachbereich Mathematik\\
Technische Universit\"at Darmstadt\\
Schlossgartenstrasse 7\\
Darmstadt 64289, Germany\\
{\tt [email protected]}}}
\centerlineterline{Running title:}
\centerlineterline{Locally Compact Groups Approximable by $\mathbb{Z}$ and $\mathbb{R}$}
\section{Preface}
\label{s:preface}
The simplest group arising directly from the
activity of counting is the group $\mathbb{Z}$ of integers.
On the other hand, one of the more sophisticated
concepts of group theory is that of a locally compact
topological group; it evolved widely and deeply
since David Hilbert in 1900 posed
the question whether a locally euclidean
topological group might be parametrized differentiably
so that the group operations become differentiable.
Bringing $\mathbb{Z}$ and locally compact groups together
in a topologically systematic fashion is made possible
by a compact Hausdorff space
${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ attached to a locally compact group $G$
in a very natural fashion, namely, as the set of closed subgroups
endowed with a suitably defined topology. Since $G$ itself
is a prominent element of ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$, we pose and answer
completely the question under which circumstances
$G$ can be approximated in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ by those subgroups
of $G$ which are isomorphic to $\mathbb{Z}$. A topological parameter
attached to a topological space $X$ is the smallest cardinal
of a basis for the collection of all open subsets;
this parameter is called the weight $w(X)$ of $X$.
If the weight of $X$ is not bigger than the first infinite
cardinal, one says that $X$ satisfies the Second Axiom of
Countability.
Our findings about the approximability of $G$ by
subgroups isomorphic to $\mathbb{Z}$ will show a fact that one
would not expect at first glance: Such an approximabilty does
not impose any bound whatsoever on the weight $w(G)$ of $G$.
If one ascends from the group $\mathbb{Z}$ of counting numbers to
the group $\mathbb{R}$ of real numbers which permits us to measure lengths
and distances, then the completely analogous question
suggests itself, asking indeed which locally compact groups $G$ can be
approximated in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ by subgroups isomorphic to $\mathbb{R}$.
Again we answer that question completely and find, that the
answer to the second question yields a simpler structure than
the answer to the first question.
The answers are described in the Abstract which precedes our text.
It is relatively immediate that
our discussion will usher us into the domain of abelian locally compact
groups. But the final proofs do lead us more deeply
into the structure of these groups than one might anticipate
and are, therefore, also longer than expected.
With these remarks, we now turn to the details.
\section{Preliminaries}
\label{s:prelim}
\subsection{Basic concepts and definitions} \label{ss:basics}
Let $G$ be a locally compact group and $\mathrm{comp}(G)$ the union
of its compact subgroups.
By Weil's Lemma (\cite[Proposition 7.43]{hofmorr}),
an element $g\in G$ is either contained in $\mathrm{comp}(G)$
or else the group $\langle g\rangle$ is isomorphic
as a topological group to $\mathbb{Z}$. Subgroups of this kind we shall
call \textit{integral}.
A subgroup $E$ of $G$ is called \textit{real} if it is isomorphic
to $\mathbb{R}$ as a topological group.
We denote by $\c{G}$ the space of closed subgroups of $G$
equipped with the \textit{Chabauty topology};
this is a compact space. In this space,
each closed subgroup~$H$ of~$G$ has a neighborhood base consisting of sets
\begin{equation}
\label{eq:Chabauty_base}
\mathcal{U}(H; K, W){\buildrel\mathrm{def}\over=}\left\{L{\in}{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G) \mid L{\cap}K{\subseteq}WH
\hbox{ and }H{\cap}K{\subseteq}WL\right\},
\end{equation}
where~$K$ ranges through the set $\mathcal K$ of all compact subsets
of~$G$ and~$W$ through the set $\mathcal U(e)$ of all neighborhoods
of the identity.
In particular $G\in{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ and the singleton subgroup $E=\{1\}$ have bases for their
respective neighborhoods of the form
\begin{eqnarray}
\label{eq:G_base}
\mathcal{U}(G; K, W) &=& \left\{L{\in}{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\mid K{\subseteq}WL\right\},\label{eq:G_base}\\
\mathcal{U}(E; K, W) &=&\left\{L{\in}{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\mid L\cap K\subseteq W\right\},\label{eq:E_base}
\end{eqnarray}
$K\in{\mathcal K}$ and $W\in{\mathcal U}(e)$.
\begin{rem} \label{countability} Returning for a moment
to Equation (\ref{eq:Chabauty_base})
we assume that $G$ is $\sigma$-compact and satisfies the First Axiom
of Countability. Then $G$ contains a sequence $(K_m)_{m\in\mathbb{N}}$
of compact subsets whose interiors for an scending sequence of open sets
covering $G$, and there is a sequence $(W_n)_{n\in\mathbb{N}}$ of open
identity neighborhoods forming a basis of the filter of identity neighborhoods.
Then each element $H\in{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ has a countable basis
$\big(\mathcal{U}(H; K_m, W_n)\big)_{m,n\in \mathbb{N}}$ for its neighborhood filter
according to Equation (\ref{eq:Chabauty_base}).
Thus ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ satisfies the First Axiom of Countability.
\end{rem}
We denote by ${\mathcal{F}_1}(G)$ the subspace of ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ containing all
$\langle g\rangle$ with $g\in G\setminus\mathrm{comp}(G)$, that is,
all subgroups isomorphic to the discrete group $\mathbb{Z}$ of all integers,
the {\it free group of rank} $1$.
\begin{example}[The additive group ${\mathbb R}$] \label{e:R}
The mapping $\phi_{{\mathbb R}}\colon [0,\infty]\to{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}({\mathbb R})$ defined by
$$\phi_\mathbb{R}(r)=\begin{cases}
\frac 1 r{\cdot}\mathbb{Z} &\mbox{if } 0<r<\infty,\cr
\{0\} &\mbox{if } r=0, \cr
\mathbb{R} &\mbox{if } r=\infty
\end{cases}$$
is a homeomorphism (see Proposition 1.7 of \cite{Hattel1}).
Here $\mathrm{comp}(G)=\{0\}$, and ${\mathcal{F}_1}(\mathbb{R})=\{\langle r\rangle| 0<r\}$,
and if $(r_n)_{n\in \mathbb{N}}$ is a sequence
of real numbers converging to $0$, then
$(\langle r_n\rangle)_{n\in \mathbb{N}}$ converges to ${\mathbb R}$.
Therefore, $\mathbb{R}\in\overline{{\mathcal{F}_1}(\mathbb{R})}$
\end{example}
\begin{defn} \label{integral-subgroups}
A locally compact group $G$ is said to be
\textit{integrally approximable}
if $G\in\overline{{\mathcal{F}_1}(G)}$.
\end{defn}
Here is an equivalent way of expressing that $G$ is
integrally approximable:
{\it There is a net
$(S_j)_{j\in J}$ of subgroups isomorphic to $\mathbb{Z}$ in $G$
such that $G=\lim_{j\in J} S_j$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$.}
\begin{rem}\label{r:immediate}
If $G$ is integrally approximable, $\overline{{\mathcal{F}_1}(G)}\ne\emptyset$,
and so $G\ne\mathrm{comp}(G)$. In particular, $G$ is not singleton.
\end{rem}
Our objective is to describe precisely which locally compact groups
are integrally approximable. The outcome is anticipated in the
abstract. It may be instructive for our intuition
to consider some examples right now.
\subsection{Some examples}\label{ss:examples}
We noticed in Example \ref{e:R} above that $\mathbb{R}$ is integrally approximable.
A bit more generally, we record
\begin{example} \label{p:vector-groups}
For $n\in\mathbb{N}$ and $G=\mathbb{R}^n$, the following statements are equivalent:
\begin{itemize}
\item[(1)] $G$ is integrally approximable.
\item[(2)] $n=1$.
\end{itemize}
\end{example}
\begin{proof} By Example \ref{e:R}, (2) implies (1).
For proving the reverse implication, we define the elements
$e_k=(\delta_{km})_{m=1,\dots,n}$, $k=1,\dots, n$ for the Kronecker
deltas
$$\delta_{km}=\begin{cases}
1 &\mbox{if }\; k=m \\
0 &\mbox{otherwise.}\\
\end{cases}$$
Now we assume (1) and $n{\fam\euffam g}e2$ and propose to derive a contradiction.
We let
$W\in{\mathcal U}((0,\dots,0))$ be the open ball of
radius $\frac 1 2$ with respect to the euclidean metric on
$\mathbb{R}^n$, and let $K$ be the closed ball of radius 2 around $(0,\dots,0)$.
By (1) there is an
integral subgroup $S$ in the neighborhood
$\mathcal U(G;K,W)$ of $G$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$.
In view of Equation (\ref{eq:G_base}) this means
$K\subset W+S$. Since $n{\fam\euffam g}e 2$ we know that
$e_1$ and $e_2$ are contained in $K$ and therefore in $W+S$, that is
there are elements $w_1, w_2\in W$ such that $s_1=e_1-w_1$
and $s_2=e_2-w_2$ are contained $S$. Now the euclidean
distance of $s_m$ from $e_m$ for $m=1,2$ is $<{\frac 1 2}$
in the euclidean plane $E_2\cong\mathbb{R}^2$ spanned by $e_1$ and $e_2$.
Therefore, the two elements $s_1$ and $s_2$ are linearly
independent. On the
other hand, being elements of the subgroup $S\cong \mathbb{Z}$,
they must be linearly dependent.
This contradiction proves that (1) implies (2).
\end{proof}
\begin{example} \label{e:rational} The group $\mathbb{Q}$ (with the discrete topology) is integrally approximable.
\end{example}
\begin{proof} For each natural number $n$ set $H_n=\frac{1}{n!}\mathbb{Z}$.
Since $(H_n)$ is an increasing sequence and
\begin{equation*}
\bigcup_{n\in \mathbb{N}} H_n = \mathbb{Q},
\end{equation*}
we also have $\lim_{n\in \mathbb{N}} H_n = \mathbb{Q}$ as we may conclude
directly using (\ref{eq:G_base}) or by invoking
Proposition 2.10 of \cite{Hamr-Sad-JOLT}. This proves the claim.
\end{proof}
\begin{example} \label{e:R-times-four} The group $G=\mathbb{R}\times\mathbb{Z}(p)^n$
for any prime $p$ and any natural number $n{\fam\euffam g}e2$
is not approximable by integral subgroups. The smallest example
in this class is $\mathbb{R}\times\mathbb{Z}(2)^2$.
\end{example}
\begin{proof} We note that $\mathbb{Z}(p)^n$ is a vector space $V$ of
dimension $n$ over the field $\mathbb{F}=\mathrm{GF}(p)$.
By way of contradiction suppose that
$G$ is integrally approximable. Let $W=[-1,1]\times\{0\}$
and let $K=\{0\}\times B$ for a basis $B$ of $V$.
Since $G$ is integrally approximable, there is a $Z\in {\mathcal{F}_1}(G)$
such that $Z\in \mathcal U(G; K, W)$, that is,
$K\subseteq W+Z$. There is an $r\in\mathbb{R}$ and a $v\in V$
such that $Z=\mathbb{Z}{\cdot}(r,v)$. Thus
\begin{multline*}
\{0\}\times B\subseteq ([-1,1]\times\{0\})+\mathbb{Z}{\cdot}(r,v)\\
=\bigcup_{n\in\mathbb{Z}} \{nr+[-1,1]\}\times \{n{\cdot}v\}
\subseteq \mathbb{R}\times \mathbb{F}{\cdot}v,
\end{multline*}
and thus $B\in \mathbb{F}{\cdot}v$.
This implies $\dim_\mathbb{F} V\le 1$ in contradiction to
the assumption $n{\fam\euffam g}e2$.
\end{proof}
For any prime $p$ we recall the basic groups
$\mathbb{Z}(p^n)$, $n=1,\dots,\infty$, and $\mathbb{Z}_p\subset\mathbb{Q}_p$,
where $\mathbb{Z}(p^\infty)$ is the divisible Pr\"ufer group and
$\mathbb{Z}_p$ its character group, the group of $p$-adic integers
(see. e.g.\ \cite{hofmorr}, Example 1.38(i), p.~27) and where
$\mathbb{Q}_p$ denotes the group of $p$-adic rationals
(see loc.~cit. Exercise E1.16, p.~27). Then Example \ref{e:R-times-four}
raises at once the following question:
\begin{ques} \label{q:question} Are the locally compact
groups $G=\mathbb{R}\times C$ for $C=\mathbb{Z}(p^n)$, $C=\mathbb{Z}(p^\infty)$, $C=\mathbb{Z}_p$, or
$C=\mathbb{Q}_p$ integrally approximable?
\end{ques}
In the light of the negative Example \ref{e:R-times-four}, this
may not appear so simple a matter to answer. Our results will show that
they all are integrally approximable.
While the group $\mathbb{Z}$ is integrally approximable trivially from
the definition, we have, in the context of the group $\mathbb{Z}$,
the following lemma, whose proof we
can handle as an exercise directly from the definitions and which serves
as a further example of the particular role played by $\mathbb{Z}$ in
the context of integrally approximable groups.
\begin{lem} \label{l:G=Z} Let $A$ be a locally compact
abelian group and assume that $A\times \mathbb{Z}$
is integrably approximable. Then $|A|=1$.
\end{lem}
\begin{proof} By way of contradiction assume that there is an
$a\ne 0$ in $A$. Then there is a zero-neighborhood $W=-W$ in $A$
such that $a\notin W+W$. Set $G=A\times\mathbb{Z}$ and
$K=\{(a,1),(0,1)\}\subseteq G$.
Since $G$ is integrally approximable there is
a $Z\cong\mathbb{Z}$ in ${\mathcal{F}_1}(G)$ such that
$$Z\in{\mathcal U}(G; K,W\times\{0\}).$$
Since $Z$ is an integral subgroup, there are
elements $b\in A$ and $0<n\in\mathbb{Z}$ such that
$Z=\mathbb{Z}{\cdot}(b,n)$. So $(a,1)$ and $(0,1)$ are contained in
$(W\times\{0\})+\mathbb{Z}{\cdot}(b,n)$. Therefore
there are elements
$w_a, w_0\in W$
and integers $m_a, m_0\in\mathbb{Z}$
such that
\begin{enumerate}
\item[$(\mathrm{i})$] $w_0+m_0{\cdot}b=0$,
\item[$(\mathrm{ii})$] $w_a+m_a{\cdot}b=a$, and
\item[$(\mathrm{iii})$] $m_0n=1$.
\item[$(\mathrm{iv})$] $m_an=1$.
\end{enumerate}
Equations (iii) and (iv), holding in $\mathbb{Z}$, imply
$m_0=m_a=n=1$.
Thus equation (i) implies $b\in -W=W$. Then from (ii) it follows that
$a\in W+W$. This is a contradiction, which proves our claim for the
example.
\end{proof}
\section{The class of approximable groups}
\label{s:class}
As a first step towards the main results it will be helpful
to observe the available closure properties of the class of
integrally approximable groups.
\subsection{Preservation properties} \label{ss:preserv}
\begin{prop}\label{p:class(appro-integral)}
The class of integrally approximable locally compact groups
is closed under the following operations:
\begin{enumerate}
\item[(OS)] Passing to open nonsingleton subgroups,
\item[(QG)] passing to quotients modulo compact subgroups,
\item[(QO)] passing to torsion-free quotients modulo open subgroups,
\item[(DU)] forming directed unions of closed subgroups,
\item[(PL)] forming strict projective limits of quotients
$G/N$ modulo compact subgroup $N$.
\end{enumerate}
\end{prop}
\begin{proof} (OS) Let $G$ be a locally compact group
$G$ that is approximable by integral subgroups and let $U$ be
an open nonsingleton subgroup. Let $K$ be a compact subspace of
$U$ and $W$ an identity neighborhood contained in $U$; we
may take $K\ne\{1\}$ and $W$ small enough so that $K\not\subseteq W$.
We must
find a subgroup $Z\cong\mathbb{Z}$ inside $U$ such that
$Z\in \mathcal U(U;K,W)$, that is, $K\subseteq WZ$.
However, $G$ is integrally approximable, and so there is a
subgroup $E\cong\mathbb{Z}$ of $G$ such that $E\in \mathcal U(G;K,W)$,
that is, $K\subseteq WE$. Then $K=K\cap U\subseteq WE\cap U$.
By the modular law, $W(E\cap U)=WE\cap U$ and so
$K\subseteq W(E\cap U)$. The condition $K\not\subseteq W$
rules out the possibility that $E\cap U=\{1\}$. Since
$E\cong\mathbb{Z}$ we know that $E\cap U\cong\mathbb{Z}$ and so we may take
$Z=E\cap U$ and thus obtain $K\subseteq WZ$ which is what
we have to prove.
(QG) Let $N$ be a compact subgroup of a locally compact
group $G$ approximable by integral subgroups and
let $\pi\colon G\to H$, $H=G/N$, be the quotient morphism. Then
the continuity of the
map $A\mapsto \pi(A):\c{G}\to \c{H}$ (see Corollary 2.4 of \cite{Hamr-Kad-JOLT}) implies that $H$
is approximable by integral subgroups.
(QO) Let $U$ be an open subgroup of a locally compact
group $G$ approximable by integral subgroups
so that $H\buildrel\mathrm{def}\over= G/U$
is torsion-free, and
let $\pi\colon G\to H$ be the quotient morphism.
Since $U$ is open, $H$ is discrete, and the singleton
set containing the identity $\widetilde e =U$ in $H=G/U$ is an
identity neighborhood. So for a given compact, hence finite,
subset $\widetilde K$ of $H$ we have to find a $\widetilde Z\in {\mathcal{F}_1}(H)$
with $\widetilde Z \subseteq \mathcal U(H;\widetilde K,\{\widetilde e\})$,
that is $\widetilde K\subseteq \widetilde Z$ according to
Equation (\ref{eq:G_base}). We may and will assume that there
is at least one $\widetilde k\in \widetilde K$ such that $\widetilde k\ne\widetilde e$.
Now by the local compactness of $G$ we find a compact
set $K$ of $G$ such that $\pi(K)= \widetilde K$.
Since $U$ is an identity neighborhood in $G$
and since $G$ is integrally approximable, there is a
subgroup $Z\in{\mathcal{F}_1}(H)$ contained in
$\mathcal{U}(G; K, U)$, that is $K\subseteq UZ$. Applying $\pi$,
we get $\widetilde K\subseteq \pi(Z)$. In particular,
$\widetilde e\ne\widetilde k\in \pi(Z)$. Since $H$ is torsion-free
and $Z\cong\mathbb{Z}$ we conclude that $\pi(Z)\cong\mathbb{Z}$, and so
we can set $\widetilde Z=\pi(Z)$, getting $\widetilde K\subseteq \widetilde Z$,
which we had to show.
(DU) Let $(G_i)_{i\in I}$ be a directed family of closed
subgroups of a locally compact group $G$ such that
$G=\overline{\bigcup_{i\in I} G_i}$.
Then from Proposition 2.10 of \cite{Hamr-Sad-JOLT} we know
\begin{equation} \label{eq:no-one}
G=\lim_i G_i\mbox{ in $\c{G}$}.
\end{equation}
Now assume
\begin{equation} \label{eq:no-two}
(\forall i\in I)\ ~G_i \mbox{ is approximable by integral
subgroups.}
\end{equation}
Let $\mathcal U$ be an open
neighborhood of $G$ in $\c{G}$. Then by (\ref{eq:no-one}) there is some $j\in I$ with
$G_j\in {\mathcal U}$, and so ${\mathcal U}$ is also an open neighborhood
of $G_j$ in $\c{G}$. Then by (\ref{eq:no-two}) there is a closed subgroup $Z\cong\mathbb{Z}$
in $G$ with $Z\in{\mathcal U}$. This proves that $G$ is approximable
by integral subgroups.
(PL) Let the locally compact group $G$ be a strict projective
limit $G=\lim_{N\in{\mathcal N}}G/N$ of integrally approximable
quotient groups modulo compact
normal subgroups $N$. Then $G$ has arbitrarily small open identity
neighborhoods $W$ for which there is an $N\in{\mathcal N}$ such that
$W=NW$. Let $K\in\mathcal K$ be a compact subspace of $G$. If $W$ is
given, we note that $NK$ is still compact, and so we assume that
$NK=K$ as well. We aim to show that there is a subgroup $Z\cong\mathbb{Z}$
of $G$ such that $K\subseteq WZ$ which will show that
$Z\in{\mathcal U}(G;K,W)$ as in Equation (\ref{eq:G_base}), and this will
complete the proof.
Now we assume that for all $M\in\mathcal N$ the group $G/M$
is approximable by integral subgroups. Then, in particular,
$G/N$ is approximable by integral subgroups. Therefore
we find a subgroup $Z_N\subseteq G/N$ such that $Z_N\cong\mathbb{Z}$ and
$Z_N\in {\mathcal U}(G/N; K/N, W/N)$ according to
(\ref{eq:G_base}). This means that
\begin{equation} \label{eq:in N}
K/N\subseteq (W/N){\cdot}Z_N.
\end{equation}
Let $z\in G$ be such that $Z_N={\fam\euffam g}en{zN}$. Set $Z={\fam\euffam g}en z$, then
$Z\cong\mathbb{Z}$ and $Z_N=ZN/N$. Then (\ref{eq:in N}) is equivalent
to $K/N\subseteq (W/N)(ZN/N)=WZ/N$ which in turn is equivalent
to $K\subseteq WZ$ and this is what we had to show.
\end{proof}
\begin{rem}
In the proof of (QO) it is noteworthy that we did not invoke
an argument claiming the continuity of the function
$A\mapsto AU/U:{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\to {\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(H)$. Indeed
let $\pi\colon\mathbb{R}\times \mathbb{Z}\to \mathbb{Z}$ be the projection. The sequence
$(\langle(n, 1)\rangle)_{n\in\mathbb{N}}$
converges to the trivial subgroup $\{(0,0)\}$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$, but
its image is the constant sequence with value $\langle 1\rangle=\mathbb{Z}$
and therefore converges to $\mathbb{Z}$. This shows that the induced
map ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\pi)\colon{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)\to{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(H)$ need not be continuous in general.
\end{rem}
\section{The case of discrete groups}
\label{s:discr}
In order to demonstrate the workings of some of the operations
discussed in Proposition \ref{p:class(appro-integral)} we show
\begin{lem} \label{l:discreteness}
Assume that an integrally approximable locally compact
group $G$ has a compact identity component $G_0$. Then $G$ is discrete.
\end{lem}
\begin{proof} By way of contradiction suppose that $G$ is not
discrete.
Since $G_0$ is compact, there is a compact open subgroup $U$
(see \cite{Montgomery--Zippin}, Lemma 2.3.1 on p.~54). Since $G$ is not
discrete, $U\ne\{1\}$. Since
$G$ is integrally approximable, so is $U$ by
Proposition \ref{p:class(appro-integral)} (OS).
Then $U\ne\mathrm{comp}(U)$ by Remark \ref{r:immediate}, but $U$ being
compact we have $U=\mathrm{comp}(U)$ and this is a contradiction which
proves the lemma.
\end{proof}
\subsection{Monothetic and inductively monothetic groups}
\label{ss:mon}
Before we proceed we need to recall some facts around
monothetic groups. A topological group $G$ is {\it monothetic}
if there is an element $g\in G$ such that $G={\fam\euffam g}en g$.
\begin{defn}[Inductively monothetic group]\label{d:ind-mon} A topological group $G$ is called
{\em inductively monothetic} if (and only if) every finite
subset $F\subseteq G$ there is an element $g\in G$
such that ${\fam\euffam g}en F ={\fam\euffam g}en g$.
\end{defn}
The circle group $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ contains a unique element
$t=2^{-1}+\mathbb{Z}\in\mathbb{T}$ such that $2{\cdot}t=0$. The group $\mathbb{T}^2$
is monothetic but not inductively monothetic,
since the subgroup $\langle\{(t,0), (0,t)\}\rangle$ is finitely
generated but not monothetic. The discrete additive group $\mathbb{Q}$
is inductively monothetic but is not monothetic.
In \cite{HHR}, Theorem 4.12 characterizes inductively monothetic
locally compact groups. Before we cite this result we recall
that Braconnier (see \cite{Braconnier}) called a locally
compact group $G$ a {\it local product}
$\prod_{j\in J}^{\mathrm{loc}}(G_j,C_j)$ for a family $(G_j)_{j\in J}$
of locally compact groups $G_j$ if each of them has a compact open
subgroup $C_j$ such that an element $g=(g_j)_{j\in J}\in\prod_{j\in J}G_j$
is in $G$ iff there is a finite subset $F_g\subseteq J$ such that
$g_j\in C_j$ whenever $j\notin F_g$.
\begin{prop}[Classification of inductively monothetic groups]\label{p:ind-mon} A locally compact group $G$
is inductively monothetic if one of the following conditions is
satisfied:
\begin{itemize}
\item[(1)] $G$ is a one-dimensional compact connected group,
\item[(2)] $G$ is discrete and is isomorphic to
a subgroup of $\mathbb{Q}$.
\item[(3)] $G$ is isomorphic to a local product
$\prod_{p\ \mathrm{prime}}^{\mathrm{loc}}(G_p,C_p)$
where each of its characteristic $p$-primary components $G_p$
is either $\cong \mathbb{Z}(p^n)$, $n=0,1,\dots,\infty$, or $\mathbb{Z}_p$,
or $\mathbb{Q}_p$.
\end{itemize}
\end{prop}
It follows, in particular, that a \emph{totally disconnected compact
monothetic group is inductively monothetic} so that the concept
of an inductively monothetic locally compact group is more general
than that of a locally compact monothetic group \emph{in the
totally disconnected domain}.
We remark that a locally compact group is called {\it periodic} if
it is totally disconnected and has no subgroups isomorphic to $\mathbb{Z}$.
Thus the class (3) of Proposition \ref{p:ind-mon} covers precisely the
periodic inductively monothetic groups.
Our present stage of information allows us to clarify on an elementary level
the discrete side of our project:
\begin{thm} \label{th:discrete-case} Let $G$ be a locally compact group
such that $G_0$ is a compact group.
Then the following assertions are equivalent:
\begin{enumerate}
\item[$(1)$] $G$ is integrally approximable.
\item[$(2)$] $G$ is discrete and isomorphic to a nonsingleton
subgroup of $\mathbb{Q}$.
\end{enumerate}
\end{thm}
\begin{proof} In Example \ref{e:rational}
we saw that $\mathbb{Q}$ is integrally approximable. Then from
Proposition \ref{p:class(appro-integral)} (OS) if follows
that every nonsingleton subgroup of $\mathbb{Q}$ is integrally
approximable. Thus (2) \hbox{$\Rightarrow$} (1).
We have to show $(1)\hbox{$\Rightarrow$} (2)$:\quad
Thus we assume (1). In particular, $G$ is nonsingleton.
Since the subgroup $G_0$ is compact, Lemma \ref{l:discreteness}
applies and shows that $G$ is discrete.
Then by Equation \ref{eq:G_base} in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ the element
$G$ has a basis of neighborhoods
$\mathcal U(G; F, \{0\})=\{H\in{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G): F\subseteq H\}$ as
$H$ ranges through the finite subsets of $G$.
Since $G$ is integrally approximable, there exists
a $Z\in {\mathcal{F}_1}(G)$ such that $Z\in\mathcal U(G;F,\{0\})$,
that is, $F\subseteq Z$. Then $\langle F\rangle$ is infinite
cyclic as a subgroup of a
group $\cong\mathbb{Z}$. Therefore $G$ is discrete, torsion-free,
and inductively monothetic.
Then Proposition \ref{p:ind-mon} shows that $G$ is isomorphic to
a subgroup of $\mathbb{Q}$.
\end{proof}
\section{Necessary conditions}
\label{s:necessary}
For the remainder of the effort to classify integrally approximable
groups we may therefore concentrate on nondiscrete groups, and
indeed on locally compact groups $G$ whose identity component
$G_0$ is noncompact.
\subsection{Background on abelian
locally compact groups}
\label{ss:commutativity}
We first point out why we have to focus on commutative
locally compact groups.
Indeed in \cite[Proposition 3.4]{Harp1} the
following fact was established:
\begin{prop}\label{space-abelian-closed}
Let $G$ be a locally compact group.
Then the space $\cab{G}$ of closed
abelian subgroups of $G$ is closed in $\c{G}$.
\end{prop}
We have ${\mathcal{F}_1}(G)\subseteq\cab{G}$ and thus
$\overline{{\mathcal{F}_1}(G)}\subseteq \cab{G}$.
Accordingly, in view of Definition \ref{integral-subgroups}
we therefore have
\begin{cor} \label{focus-abelian}
An integrally approximable locally compact group is abelian.
\end{cor}
Thus we now focus on locally compact abelian groups and their
duality theory. As a consequence we shall henceforth write
the groups we discuss in additive notation.
An example is the following result of Cornulier's
(see \cite{Cornulier-LCA-Chabauty}, Theorem 1.1):
\begin{prop}[Pontryagin-Chabauty Duality] \label{Pon-Cha-duality}
Let $G$ be an abelian locally compact group. Then the
annihilator map
\begin{equation*}
\ H\mapsto H^\bot : \c{G}\to \cbig{\widehat G}
\end{equation*}
is a homeomorphism.
\end{prop}
This will allow us to apply the so-called {\it annihilator mechanism}
as discussed e.g. in \cite{hofmorr}, 7.12 ff., pp.314 ff..
What will be relevant in our present context is a main
structure theorem for abelian groups (see \cite{hofmorr},
Theorem 7.57, pp. 345 ff.).
\begin{prop} \label{p:vector-splitting} Every
locally compact abelian group $G$ is algebraic\-al\-ly and topologically of
the form $G=E\oplus H$ for a subgroup $E\cong \mathbb{R}^n$ and a locally compact
abelian subgroup $H$ which has the following properties
\begin{itemize}
\item[\rm(a)] $H$ contains a compact subgroup which is open in $H$.
\item[\rm(b)] $H$ contains $\mathrm{comp}(G)$.
\item[\rm(c)] $H_0=(\mathrm{comp}(G))_0=\mathrm{comp}(G_0)$ is the unique maximal
compact connected subgroup of $G$.
\item[\rm(d)] The subgroup $G_1\buildrel\mathrm{def}\over= G_0+\mathrm{comp}(G)$ is an open,
hence closed, fully characteristic
subgroup which is isomorphic to $\mathbb{R}^n\times \mathrm{comp}(G)$.
\item[\rm(e)] $G/G_1$ is a discrete torsion-free group and
$G_1$ is the smallest open subgroup with this property.
\end{itemize}
\end{prop}
\begin{gather}
\begin{aligned}
\xymatrix{
&&&G \ar@{-}[dr] \ar@{-}[dl]\\
&&G_1 \ar@{-}[dr] \ar@{-}[dl]&&H\ar@{-}[dl]\\
&G_0 \ar@{-}[dr] \ar@{-}[dl] &&C\ar@{-}[dl]\\
E\ar@{-}[dr]&&C_0 \ar@{-}[dl]\\
&0
}\end{aligned}
\label{Fig1}
\end{gather}
\begin{center}
{$C=\mathrm{comp}(G),\quad G_1=G_0+C=E\oplus C.$}
\end{center}
\subsection{Necessity}
\label{ss:nec}
These results allow us to narrow our scope onto integrally
approximable groups $G$ further and to derive necessary
conditions for $G$ to be integrally approximable.
\begin{prop} \label{p:vector-splitting-int-appr} Every nondiscrete
integrally approximable locally compact abelian group $G$
is algebraically and topologically of
the form $G=E\oplus\mathrm{comp}(G)$ for a subgroup $E\cong \mathbb{R}$ and
the locally compact abelian subgroup $\mathrm{comp}(G)$.
\end{prop}
\begin{proof} (i) Using the notation of
Proposition \ref{p:vector-splitting} (d) and (e) above we first claim
$G=G_1$. By (d) the subgroup $G_1=G_0+\mathrm{comp}(G)$ is open, and by
(e) the factor group $G/G_1$ is torsion free. Suppose our claim
is false. Then we find an element $g\in G\setminus G_1$. Then
$Z\buildrel\mathrm{def}\over= \langle g\rangle$ is cyclic and $(Z+G_1)/G_1
\cong Z/(Z\cap G_1)$ is torsion-free by (e), and so $X\cap G_1=\{0\}$.
Then it follows from (d), since $G_1$ is open, that $Z$ is discrete.
Hence $U=G_1 +Z$ is a nonsingleton open subgroup $\cong G_1\times \mathbb{Z}$
by (e). Then by Lemma \ref{l:G=Z} we have $G_1=\{0\}$ which forces
$G$ to be discrete, contrary to our hypothesis.
(ii) Now by Proposition \ref{p:vector-splitting} again,
we may identify $G$ with $\mathbb{R}^n\times H$ where $H_0$ is
compact and $H=\mathrm{comp}(H)$. Now $H$ contains a compact
open subgroup $U$ (cf.\ the proof of Lemma \ref{l:discreteness})
and $\mathbb{R}^n\times U$ is an open subgroup of $G$ which is nonsingleton
since $G$ is nondiscrete. Hence \break $\mathbb{R}^n\times U$ is integrally
approximable by Proposition \ref{p:class(appro-integral)}.
Then the projection $\mathbb{R}^n\times U\to\mathbb{R}^n$ is covered by part
(QG) of Proposition \ref{p:class(appro-integral)} and thus
we know that $\mathbb{R}^n$ is integrally approximable.
Then Example \ref{p:vector-groups} shows that $n=1$.
\end{proof}
\begin{rem} \label{r:iso} If the relation $G=G_0+\mathrm{comp}(G)$
holds for a locally compact abelian group $G$, then
$G/G_0\cong \mathrm{comp}(G)/\mathrm{comp}(G)_0$. Moreover,
$\mathrm{comp}(G)$ is the union of compact subgroups being open
in $\mathrm{comp}(G)$.
\end{rem}
From here on we concentrate on groups of the form
$\mathbb{R}\times H$ where $H$ is an abelian locally compact
group with $H=\mathrm{comp}(H)$.
\begin{defn} \label{d:periodic} We call a locally compact group
$H$ {\it periodic} if it is totally disconnected and satisfies
$H=\mathrm{comp}(H)$.
\end{defn}
Periodic abelian groups have known structure due to Braconnier
(see \cite{Braconnier}; cf.\ also \cite{HHR}).
Indeed, a periodic group $G$ is (isomorphic to) a local product
$$\prod_{p\ \mathrm{prime}}^{\mathrm{loc}}(G_p,C_p)$$
for the $p$-primary components (or $p$-Sylow subgroups)
$G_p$.
\begin{lem} \label{l:monothetic}
Let $G$ be an integrally approximable locally compact
group such that $\mathrm{comp}(G)$ is periodic and compact.
Then $\mathrm{comp}(G)$ is monothetic.
\end{lem}
\begin{proof} Write $H=\mathrm{comp}(G)$. Then we may identify
$H$ with the product $\prod_p H_p$ of its compact $p$-primary
components (see also \cite{hofmorr}, Proposition 8.8(ii)).
A compact group $H$ is monothetic iff there is a morphism
$\mathbb{Z}\to H$ with dense image iff (dually) there is an injective morphism
$\widehat H\to \mathbb{T}$. As $H$ is totally disconnected, $\widehat H$ is
a torsion group (see \cite{hofmorr}, Corollary 8.5, p.~377), and so the group
$\widehat H$ is embeddable into $\mathbb{T}$ iff it is embeddable into
the torsion group $\mathbb{Q}/\mathbb{Z}=\bigoplus_p\mathbb{Z}(p^\infty)$ of $\mathbb{T}$ (see
\cite{hofmorr}, Corollary A1.43(ii) on p.~694) iff each $\widehat H_p$
is embeddable into $\mathbb{Z}(p^\infty)$. Hence $H$ is monothetic iff
each $H_p$ has $p$-rank $\le 1$. By way of contradiction suppose
that this is not the case and that there is a prime $p$ such that
the $p$-rank of $H_p$ is ${\fam\euffam g}e2$. So the compact
group $H/p{\cdot}H$ has exponent $p$ and is isomorphic to a power
$\mathbb{Z}(p)^I$ with $\card I{\fam\euffam g}e2$. Therefore we have a projection of
$H/p{\cdot}H\cong\mathbb{Z}(p)^I$ onto $\mathbb{Z}(p)^2$. This provides a surjective
morphism $H\to H/p{\cdot}H\cong \mathbb{Z}(p)^I\to \mathbb{Z}(p)^2$.
Thus Proposition \ref{p:vector-splitting-int-appr}
tells us that $G=\mathbb{R}\times H$ has a quotient group $\mathbb{R}\times\mathbb{Z}(p)^2$ modulo
a compact kernel. By Proposition \ref{p:class(appro-integral)} (QG)
this quotient group is integrally approximable
which we know to be impossible by
Example \ref{e:R-times-four}. This contradiction proves
the lemma.
\end{proof}
\begin{lem} \label{l:ind-mon} Let $G$ be a totally disconnected
locally compact abelian group satisfying $G=\mathrm{comp}(G)$.
Assume that every compact open subgroup of $G$ is monothetic.
Then $G$ is inductively monothetic.
\end{lem}
\begin{proof}
Let $F$ be a finite subset of $G$; we must show that
${\fam\euffam g}en F$ is monothetic.
Let $\mathcal S$ be the $\subseteq$-directed set of
all compact open (and therefore monothetic) subgroups of $G$.
Since $G=\bigcup{\mathcal S}$ (see Remark \ref{r:iso})
for each $x\in F$ there is a $C_x\in{\mathcal S}$ such that
$x\in C_x$. Since ${\mathcal S}$ is directed and $F$ is finite
there is an $K\in{\mathcal S}$ such that
$\bigcup_{x\in F}C_x\subseteq K$. Then $F\subseteq K$ and so
${\fam\euffam g}en F\subseteq K$. Since $G$ is totally disconnected, the same
is true for $K$. By hypothesis, $K$ is monothetic, and so
the comment following Proposition \ref{p:ind-mon}
shows that $K$ is inductively
monothetic, whence ${\fam\euffam g}en F$ is monothetic.
\end{proof}
\begin{cor} \label{c:ind-mon-2} Let $G$ be an integrally
approximable group such that $\mathrm{comp}(G)$ is periodic.
Then $G/G_0\cong \mathrm{comp}(G)$ is inductively monothetic.
\end{cor}
\begin{proof} Let $U$ be a compact-open subgroup of
$\mathrm{comp}(G)$. Then $G_0+U\cong \mathbb{R}\times U$ is an open subgroup of
$G=G_0+\mathrm{comp}(G)=\mathbb{R}\times\mathrm{comp}(G)$. Hence it is integrally
approximable by Proposition \ref{p:class(appro-integral)} (OG).
So $U$ is monothetic by Lemma \ref{l:monothetic}. Now
Lemma \ref{l:ind-mon} shows that $\mathrm{comp}(G)$ is inductively
monothetic.
\end{proof}
Now we have a necessary condition on a locally compact group to
be integrally approximable:
\begin{thm} \label{th:necessary} Let $G$ be a nondiscrete integrally
approximable locally compact group. Then
\begin{itemize}
\item[\rm(a)] $G\cong \mathbb{R}\times \mathrm{comp}(G)$ and
\item[\rm(b)] $G/G_0\cong \mathrm{comp}(G)/\mathrm{comp}(G)_0$ is inductively monothetic.
\end{itemize}
\end{thm}
\begin{gather}
\begin{aligned}
\xymatrix{
&&G \ar@{-}[dr] \ar@{-}[dl]&&\\
&G_0 \ar@{-}[dr] \ar@{-}[dl] &&C\ar@{-}[dl]\\
\mathbb{R}\ar@{-}[dr]&&C_0 \ar@{-}[dl]\\
&0
}\end{aligned}
\label{Fig2}
\end{gather}
\begin{center}
{$C=\mathrm{comp}(G),\quad G=G_0+C=\mathbb{R}\oplus C.$}
\end{center}
\begin{rem} \label{r:detail} The Classification
of locally compact inductively monothetic groups (Proposition \ref{p:ind-mon}) yields that the group
$G/G_0$ is of type (3).
Indeed, it is a local product of inductively monothetic $p$-Sylow
subgroups of type $\mathbb{Z}(p^n)$, $\mathbb{Z}(p^\infty)$, $\mathbb{Z}_p$, or $\mathbb{Q}_p$.
\end{rem}
\begin{proof} As is described
in Proposition \ref{p:vector-splitting-int-appr} $G$
has a compact characteristic subgroup
$N\buildrel\mathrm{def}\over=\mathrm{comp}(G_0)=\mathrm{comp}(G)_0$, the unique largest compact connected
subgroup. So by Proposition \ref{p:class(appro-integral)} (QG),
the quotient $G/N$ is also integrally approximable, and
$\mathrm{comp}(G/N)$ is totally disconnected and therefore is periodic.
As Corollary \ref{c:ind-mon-2} applies to $G/N$, its
factor $\mathrm{comp}(G/N)$ is inductively monothetic. However,
$G/N\cong \mathbb{R}\times \mathrm{comp}(G/N)=\mathbb{R}\times\mathrm{comp}(G)/N$ and
$G_0\cong \mathbb{R}\times N$ we see that $G/G_0\cong \mathrm{comp}(G/N)$.
Thus $G/G_0$ is inductively monothetic.
\end{proof}
It is noteworthy that there is no limitation on the size of
the compact connected abelian group $\mathrm{comp}(G)_0=\mathrm{comp}(G_0)$.
The locally compact abelian group $\mathrm{comp}(G)$ is an extension of
the compact group $\mathrm{comp}(G)_0$ by an inductively monothetic
group.
\section{Sufficient conditions}
\label{s:suff}
In this section we shall prove the following complement to
Theorem \ref{th:necessary} and thereby complete the proof of the main
theorem formulated in the abstract.
\begin{thm} \label{th:sufficient} Let $G=\mathbb{R}\times H$ for a
locally compact abelian group $H$ satisfying the following conditions:
\begin{itemize}
\item[\rm(a)] $H=\mathrm{comp}(H)$ and
\item[\rm(b)] $H/H_0\cong G/G_0$ is inductively monothetic.
\end{itemize}
Then $G$ is integrally approximable.
\end{thm}
\subsection{Various reductions}
\label{ss:reduct}
We shall achieve the proof by reducing the problem step by step.
Firstly, every inductively monothetic group is the directed
union of monothetic subgroups by Proposition \ref{p:ind-mon}.
and so if $H$ satisfies (b) it is of the form
$H=\bigcup_{i\in I}H_i$ with a directed family of subgroups
$H_i\supseteq H_0$ such that $H_i/H_0$ is compact monothetic.
Then, by Proposition \ref{p:class(appro-integral)} (DU),
$\mathbb{R}\times H$ is integrally approximable if all $\mathbb{R}\times H_i$
are integrally approximable for $i\in I$.
Thus from here on,
in place of condition (b),
we shall assume that $H$ satisfies
\begin{itemize}
\item[(c)] $H/H_0$ is monothetic.
\end{itemize}
After condition (c), $H$ is compact.
Every locally compact abelian group is a strict
projective limit of Lie groups.
This applies to $H$. Clearly condition (a) holds for all
quotient groups. If $N$ is a compact normal subgroup of $H$
Then $(H/N)_0=H_0N/N$ and thus $(H/N)/(H/N)_0\cong H/H_0N$,
whence $(H/N)/(H/N)_0$ is a quotient group of $H/H_0$
and is therefore inductively monothetic.
Thus if we can show that all for all abelian Lie groups $H$
satisfying (a) and (c), the groups $\mathbb{R}\times H$
are integrally approximable, then
the Proposition \ref{p:class(appro-integral)}(PL)
will show that $\mathbb{R}\times H$ is integrally approximable.
Therefore we need to prove Theorem \ref{th:sufficient} for
a compact Lie group $H$ satisfying (a) and (c).
\begin{lem} \label{l:lie-case} Let $H$ be a compact abelian Lie group
satisfying $\mathrm{(a)}$ and $\mathrm{(c)}$.
Then there is are nonnegative integers $m$ and $n$
$$ H \cong \mathbb{T}^m\times \mathbb{Z}(n).$$
In particular, $H$ is monothetic.
\end{lem}
\begin{proof} $H$ is a compact abelian Lie group such that $H/H_0$
is cyclic. Then $H$ has the form asserted (see e.g.\ \cite{hofmorr},
Proposition 2,42 on p.~48 or Corollary 7.58 (iii) on p. 356).
We have seen in the proof of Lemma \ref{l:monothetic} that a compact group
is monothetic if and only if its character group can be injected
into the discrete circle group $\mathbb{T}_d=\mathbb{R}_d\oplus\mathbb{Q}/\mathbb{Z}$. Since
$\widehat H \cong \mathbb{Z}^m\oplus \mathbb{Z}(n)$, this condition is satisfied.
\end{proof}
Thus for a proof of Theorem \ref{th:sufficient} it will suffice
to prove
\begin{lem} \label{l:R-plus-mon} If $H$ is monothetic, then
$\mathbb{R}\times H$ is integrally approximable.
\end{lem}
We shall use the Bohr compactification
of the group $\mathbb{Z}$ of integers. This group is also called the {\it
universal monothetic group}.
Here we shall identify $\widehat{{\mathrm b}\Z}$ with $\mathbb{T}_d$ (for $\mathbb{T}=\mathbb{R}/\mathbb{Z}$)
and consider the elements $\chi$ of ${\mathrm b}\Z$ as characters of $\mathbb{T}_d$.
There is one distinguished character, namely the identity morphism
$\mathrm{id}_\mathbb{T}\colon\mathbb{T}_d\to\mathbb{T}$, and we map $\mathbb{Z}$ naturally and bijectively
onto a dense subgroup of
${\mathrm b}\Z$ via the map
\begin{equation}\label{defn:rho}
\rho\colon\mathbb{Z}\to{\mathrm b}\Z,\quad
m\mapsto m{\cdot}\mathrm{id}.
\end{equation}
The following Lemma shows that for a proof of Lemma \ref{l:R-plus-mon}
it suffices to prove it for $H={\mathrm b}\Z$.
\begin{lem} \label{l:reduction} The group $G=\mathbb{R}\times H$ is integrally approximable for any
monothetic compact subgroup $H$ if and only if it is so for $H={\mathrm b}\Z$.
\end{lem}
\begin{proof} Obviously the latter condition is a necessary one for the
former, so we have to show that it is sufficient. Let $H$ be a
monothetic group. Then there is a morphism
$f\colon \mathbb{Z}\to H$ with dense image.
We have the canonical dense morphism
$\rho\colon \mathbb{Z}\to{\mathrm b}\Z$. By
the universal property of the Bohr compactification there is a unique
morphism $\pi\colon {\mathrm b}\Z\to H$ such that $f=\pi\circ\rho$.
Since $f$ has a dense image, this holds for $\pi$, whence by the compactness
of ${\mathrm b}\Z$, the morphism is a surjective morphism between compact groups and therefore
is a quotient morphism with a compact kernel.
Therefore there is a quotient morphism with compact
kernel
$\mathbb{R}\times {\mathrm b}\Z\to \mathbb{R}\times H$. Hence
by Proposition \ref{p:class(appro-integral)} (QG), $\mathbb{R}\times H$ is
integrally approximable if $\mathbb{R}\times{\mathrm b}\Z$ is integrally
approximable.
\end{proof}
Now the following lemma will complete the proof of
Theorem \ref{th:sufficient} and thereby conclude the section with
a proof of the main result of the article.
\begin{lem}[First Key Lemma] \label{l:R-plus-b}
The group $\mathbb{R}\times {\mathrm b}\Z$
is approximable by a sequence of integral subgroups.
\end{lem}
\subsection{Proving the First Key Lemma}
\label{ss:keylem}
The proof of the First Key Lemma requires some technical preparations
in which we use the duality of locally compact abelian groups.
In the process we need to consider the charachter group of
$\mathbb{R}\times\mathrm {\mathrm b}\Z$. Here is a reminder of the determination of
the character group of a product:
\begin{lem}\label{dual-group-cartesian-product}
Let $A$ and $B$ be locally compact abelian groups.
Then there is an isomorphism
$\phi\colon\widehat A\times\widehat B\to(A\times B)\widehat{\phantom w}$
such that
\centerlineterline{$\phi(\chi_A,\chi_B)(a,b)=\chi_A(a)-\chi_B(b).$}
\end{lem}
We apply this with $A=\mathbb{R}$ and $B={\mathrm b}\Z$.
For the simplicity of notation we shall denote
the coset $r+\mathbb{Z}\in\mathbb{T}$ of $r\in\mathbb{R}$ by $\overline r$.
We consider $\mathbb{R}$ also as the character group of $\mathbb{R}$ by
letting $r(s)=\overline{rs}\in\mathbb{T}$.
We also identify $\widehat\mathbb{T}$ with $\mathbb{Z}$ by considering
$k\in \mathbb{Z}$ as the character defined by $k(\overline r)=\overline{kr}$.
Recall that we consider ${\mathrm b}\Z$ as the character group of $\mathbb{T}_d$.
In the spirit of Lemma \ref{dual-group-cartesian-product}, we
identify $\mathbb{R}\times \mathbb{T}_d$ with the character group $\widehat G$ of
$G=\mathbb{R}\times \widehat{\mathbb{T}_d}=\mathbb{R}\times{\mathrm b}\Z$
by letting $(r,\overline s)$ denote the character of $G$ defined by
$(r,\overline s)(x,\chi)=\overline{rx}-\chi(\overline s)\in\mathbb{R}/\mathbb{Z}$.
We recall that the identity function $\mathrm{id}_\mathbb{T}\colon\mathbb{T}_d\to\mathbb{T}$ is
a particular character of $\mathbb{T}_d$ and thus is an element of ${\mathrm b}\Z$;
indeed $\mathrm{id}_\mathbb{T}$ is the distinguished generator of ${\mathrm b}\Z$.
Now we define
\begin{equation}
\label{eq:Zn}
Z_n=\mathbb{Z}{\cdot}(\frac 1 n,\mathrm{id}_\mathbb{T})\in {\mathcal{F}_1}(G), \quad G=\mathbb{R}\times{\mathrm b}\Z.
\end{equation}
Accordingly, $(r,\overline s)\in \mathbb{R}\times\mathbb{T}_d$ belongs to the annihilator
$Z_n^\perp=(\frac 1 n,\mathrm{id}_\mathbb{T})^\perp$
iff $\overline{\frac r n}-\overline s=0$, that is, iff
$\overline{\frac r n}=\overline s$. This means $\frac r n +\mathbb{Z}=s+\mathbb{Z}$,
and so
\begin{equation}
\label{eq:annih}
(r,\overline s)\in\mathbb{R}\times \mathbb{T}_d\mbox{\quad is in $Z_n^\perp$ iff\quad }
\frac r n-s\in \mathbb{Z}.
\end{equation}
In an effort to show that $\lim_n Z_n^\perp=\{0\}$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\widehat G)$
we consider the following
\begin{lem}[Convergence to the trivial subgroup]\label{convergence-trivial-subgp}
Let $\Gamma$ be a
locally compact group with identity $e$, and let $(H_n)_{n\in \mathbb{N}}$ be
a sequence of closed subgroups. Then the following statements are equivalent:
\begin{enumerate}
\item[$(a)$] For each subnet $(H_{n_j})_{j\in J}$ of $(H_n)_{n\in \mathbb{N}}$
and each convergent
net $(h_{n_j})_{j\in J}$ with $h_{n_j}\in H_{n_j}$ and limit $h$ we have $h=e$.
\item[$(b)$] $\lim_{n\in \mathbb{N}} H_n=\{e\}$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$.
\end{enumerate}
\end{lem}
\begin{proof}
$(a)\mathbb{R}ightarrow (b)$:\quad We argue by contradiction.
Suppose that there is a compact subset $K$ of $G$ and
an open neighborhood $U$ of the identity such that
for each $\alpha\in \mathbb{N}$ there
is an $n_\alpha\in \mathbb{N}$ with $\alpha \leq n_\alpha$ such that
$H_{n_\alpha}\not\in \mathcal{U}(\{e\}; K, U)$. That is,
$H_{n_\alpha}\cap K \not\subseteq U$ by equation (\ref{eq:E_base}), and so
there is an $h_{n_\alpha}\in H_{n_\alpha}$ such that
$h_{n_\alpha}\in K\setminus U$. As $K\setminus U$
is a compact subset of $G$, the net $(h_{n_\alpha})$
admits a subnet converging to a point $a$ of $K\setminus U$. Since
$e\not\in K\setminus U$, $a\not= e$, which is a contradiction.
$(b)\mathbb{R}ightarrow (a)$: Let $(h_{n_j})_{j\in J}$ be a net
converging to $h$ and assume $h_{n_j}\in H_{n_j}$ for every $j\in J$.
Suppose $h\ne e$. Now let $K$ be a compact neighborhood of $h$ not containing $e$.
We may assume that $h_{n_j}\in K$ for all $j\in J$.
As $\lim_{n\in \mathbb{N}} H_n=\{e\}$, there exists $N\in \mathbb{N}$ such that for each
$n{\fam\euffam g}e N$ we have $H_n\in\mathcal U(\{e\};K,\Gamma\setminus K)$, that is,
$H_n\cap K\subseteq\Gamma\setminus K$ by equation (\ref{eq:E_base}), and this is a contradiction.
\end{proof}
In order to appreciate this lemma, consider the condition
\qquad $(a')$ {\em For each convergent
net $(h_{i})_{i\in I}$ with $h_{i}\in H_{i}$ and limit $h$ we have
$h=e$.}
The following example will show that the implication
$(a')\mathbb{R}ightarrow (b)$ fails.
\begin{example}
In ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\mathbb{R})$, let
$$H_n=\left\{
\begin{array}{ll}
n\mathbb{Z}, & \hbox{if $n$ is even;} \\
&\\
\frac{1}{n}\mathbb{Z}, & \hbox{if $n$ is odd.}
\end{array}
\right.$$
Let $(h_n)$ be a sequence in $\mathbb{R}$ converging
to $h$ and such that, for each $n$,\ $h_n\in H_n$.
For any $n\in \mathbb{N}$, there is $k_n\in \mathbb{Z}$ such that
$h_{2n}= 2n k_n$. As the subsequence $(h_{2n})$ converges
to $h$, $h=0$. So $(a')$ is satisfied.
However,
as the subsequence $(H_{2n})$ converges to $\{0\}$
and the subsequence $(H_{2n+1})$ converges to $\mathbb{R}$, the
sequence $(H_n)$ is divergent and so $(b)$ fails.
\end{example}
\begin{lem} \label{l:null-convergence}
$\lim_{n\in\mathbb{N}} Z_n^\perp =\{0\}$.
\end{lem}
\begin{proof}
We shall apply Lemma \ref{convergence-trivial-subgp} and assume
that we have a net $(n_j)_{j\in J}$ cofinal in $\mathbb{N}$ such that
$(r_j,\overline s_j)_{j\in J}$ is a convergent net with
\centerlineterline{$(r_j,\overline{s_j})\in Z_{n_j}^\perp$ for all $j\in J$
and with $(r,\overline s)=\lim_{j\in J} (r_j,\overline{s_j})$.}
From Equation (\ref{eq:annih}) we know that
$\frac{r_j}{n_j}-s_j\in \mathbb{Z}$. Since $r_j\to r$, $s_j\to s$,
and $n_j\to\infty$, we conclude $s\in \mathbb{Z}$ and thus $\overline s=0$.
Further $s\in \mathbb{Z}$ and $r_j/n_j\to 0$ imply the existence
of a $j_0$ such that
$j_0\le j$ implies $s_j=s$. Then $r_j/n_j$ is an integer,
and so for large enough $j$
we have $r_j=0$ which implies $r=0$.
Thus $\lim_{j\in J}(r_j,\overline{s_j})=0$
and by Lemma \ref{convergence-trivial-subgp} this shows that
$\lim_{n\in\mathbb{N}}Z_n^\perp =\{0\}$ which we had to show.
\end{proof}
Now we are ready for proof of the First Key Lemma:
There is a sequence $(Z_n)_{n\in \mathbb{N}}$ in ${\mathcal{F}_1}(G)$,
$Z_n=\mathbb{Z}{\cdot}(\frac 1 n,\mathrm{id}_\mathbb{T})$
for which $(Z_n^\perp)_{n\in \mathbb{N}}$ converges to $\{0\}$
in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\widehat G)$ by Lemma \ref{l:null-convergence}.
Now we apply Pontryagin-Chabauty Duality in the form
of Proposition \ref{Pon-Cha-duality} and conclude
\begin{equation}
\label{eq:sequential-appro}
\lim_{n\in\mathbb{N}}Z_n =G\mbox{ in }{\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)
\mbox{ for }G=\mathbb{R}\times{\mathrm b}\Z.
\end{equation}
This concludes the proof of the First Key Lemma \ref{l:R-plus-b}
and thus also finishes the proof of Theorem \ref{th:sufficient}.
We can summarize the main result as follows:
\begin{thmA} A locally compact group $G$ is integrally
approximable if and only if it is either discrete, in which case it is isomorphic
to a nonsingleton subgroup of $\mathbb{Q}$, or else
it is abelian and of the form $G\cong \mathbb{R}\times \mathrm{comp}(G)$
where $G/G_0\cong \mathrm{comp}(G)/\mathrm{comp}(G)_0$ is periodic
inductively monothetic.
\end{thmA}
We recall that the groups $\mathrm{comp}(G)_0$ range through all compact connected abelian groups
and that the periodic inductively monothetic groups were classified in
Proposition \ref{p:ind-mon} (3).
We mention in passing that Theorem A yields a characterisation of the the group
$\mathbb{R}$ in the class of locally compact groups. For this purpose let us call
a topological group {\it compact-free} if it does not contain a nonsingleton
compact subgroup.
\begin{cor} \label{c:compactfree} For a locally compact group $G$ the
following conditions are equivalent:
\begin{enumerate}
\item $G\cong\mathbb{R}$ or else is isomorphic to a nonsingleton subgroup of the
discrete group $\mathbb{Q}$.
\item $G$ is compact-free and integrally approximable.
\end{enumerate}
\end{cor}
A similar characterization using the property of being
compact-free was suggested by Chu in \cite{chu}.
\section{Complement: Groups approximable by real subgroups}
We classified locally compact groups $G$ for which every neighborhood of
$G\in {\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ contains a subgroup $H$ of $G$ isomorphic to $\mathbb{Z}$. We called
such groups {\it integrally approximable}. Now we
shall do the same for groups $G$ with the same property except that
$\mathbb{Z}$ is replaced by $\mathbb{R}$. We need a name for these groups that are
approximated by subgroups of (real) {\it numbers}. For this
purpose let us denote by ${\mathcal{R}_1}(G)$ the subspace of ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$ containing all
subgroups isomorphic to the group $\mathbb{R}$ of all real numbers, the
{\it real vector group of dimension} $1$.
\begin{defn} \label{numeral-subgroups}
A locally compact group $G$ is said to be
\textit{numerally approximable}
if $G\in\overline{{\mathcal{R}_1}(G)}$.
\end{defn}
Trivially, $\mathbb{R}$ is numerally approximable.
The theory and classification of numerally approximable groups is
in most ways simpler than that of integrally approximable groups.
The basic aspects are completely analogous to the former and
that allows us now to proceed more expeditiously.
\begin{example} \label{p:vector-groups-again}
For $n\in\mathbb{N}$ and $G=\mathbb{R}^n$, the following statements are equivalent:
\begin{itemize}
\item[(1)] $G$ is numerally approximable.
\item[(2)] $n=1$.
\end{itemize}
\end{example}
\begin{proof} Trivially, (2) implies (1). Conversely,
if (1) is satisfied, an inspection of the proof of (1)\hbox{$\Rightarrow$}(2)
for Example \ref{p:vector-groups} shows that it applies to
the next to the last sentence, where $\mathbb{Z}$ needs to be replaced
by $\mathbb{R}$ to applie literally.
\end{proof}
\begin{lem}\label{encadrement}
Let $G$ be a locally compact group and
$(A_i)_{i\in I}$, $(B_i)_{i\in I}$ two nets
converging to $A$ and $B$ respectively.
If $A_i\subseteq B_i$ holds eventually, then $A\subseteq B$.
\end{lem}
\begin{proof} By way of contradiction, suppose that $A$ is not a
subgroup of $B$. Let $x\in A\setminus B$ and $U$ a
relatively compact open neighborhood of $e$ such that
$\overline U x\cap B =\emptyset$. As $B_i \to B$, there
exists $i_0\in I$ such that for any $i{\fam\euffam g}e i_0$ we
have $\overline{U} x\cap
B_i=\emptyset$ and so $Ux \cap A_i =\emptyset$,
which is a contradiction because
the set if all closed subgroups meeting the open set $Ux$ is
an open neighborhood of $A$.
\end{proof}
In particular, this lemma implies that the limit of
a net of connected closed subgroups of a locally compact group $G$
is contained in the identity component $G_0$ of $G$, whence
a group $G$ which is approximable by real subgroups is necessarily
connected. Using also Proposition \ref{space-abelian-closed},
we conclude:
\begin{lem}\label{l:realappro:connected}
Every numerally approximable locally compact group
is abelian and connected.
\end{lem}
In view of Proposition \ref{p:vector-splitting} we may rephrase this
as follows:
\begin{thm} \label{th:necessary-numeral} Let $G$ be a numerally
approximable locally compact group. Then $\mathrm{comp}(G)$ is
compact and connected and
$$G\cong \mathbb{R}\times \mathrm{comp}(G).$$
\end{thm}
\vskip-20pt
\begin{gather}
\begin{aligned}
\xymatrix{
&G \ar@{-}[dr] \ar@{-}[dl]&&\\
\mathbb{R}\ar@{-}[dr] && C\ar@{-}[dl] \\
&0
}\end{aligned}
\label{Fig3}
\end{gather}
\begin{center}
{$C=\mathrm{comp}(G),\quad G=\mathbb{R}\oplus C.$}
\end{center}
The second part of the structure theorem for numerally approximable
groups, saying that every group $\mathbb{R}\times C$ with any compact connected abelian group $C$
is numerally approximable is proved in a reduction procedure similar to the one
we used for integrally approximable groups.
\begin{thm} \label{th:sufficient-numeral} Let $G=\mathbb{R}\times C$ for
a compact connected abelian group $C$. Then $G$ is numerally approximable.
\end{thm}
\begin{proof}
Every compact connected group is a directed union of compact connected
monothetic groups (see e.g. \cite{hofmann-tran-amer}, Theorem I or \cite{hofmorr},
Theorem 9.36(ix), pp.~479f.). Thus $G$ is the directed union of
subgroups $\mathbb{R}\times M$ where $M$ is compact connected monothetic.
The closure lemma Proposition \ref{p:class(appro-integral)} (DU)
is easily seen to apply to numerally approximable groups in place of
integrally approximable groups. Therefore
it is no loss of generality to assume that $C$ is compact connected
monothetic. Then it is a quotient of ${\mathrm b}\R$, the Bohr compactification
of $\mathbb{R}$. (Indeed $\widehat C$ is a discrete torsion free group of
rank $\le 2^{\aleph_0}$ and thus is a subgroup of $\mathbb{R}_d$ (the discrete
reals), and thus $C$ is a quotient of $\widehat{\mathbb{R}_d}\cong {\mathrm b}\R$.)
Since the closure lemma Proposition \ref{p:class(appro-integral)} (QG)
again applies to numerally approximable groups in place of integrally
approximable groups, the proof will be
complete if it is shown that $\mathbb{R}\times{\mathrm b}\R$ is numerally approximable.
This will be done in the Second Key Lemma that follows.
\end{proof}
The proof of the Theorem is therefore reduced to showing that
one special group in numerally approximated:
\begin{lem}[Second Key Lemma] \label{l:key-lemma-2}
The group $G=\mathbb{R}\times{\mathrm b}\R$ is numerally
approximable.
\end{lem}
Before we prove this Second Key Lemma, we review the duality aspects of
the present situation.
From Lemma \ref{dual-group-cartesian-product} we recall
that the dual $\widehat G$ of $G$ may be identified with $\mathbb{R}\times\mathbb{R}_d$
where, as before we identify $\widehat \mathbb{R}$ and $\mathbb{R}$.
Let $f\colon\mathbb{R}\to{\mathrm b}\R$ the canonical one-parameter
subgroup of ${\mathrm b}\R$, namely the dual of $\mathrm{id}_\mathbb{R}\colon\mathbb{R}_d\to\mathbb{R}$.
For each natural number $n\in\mathbb{N}$, the morphism
$n{\cdot}f$ defined by $(n{\cdot}f)(r)=n{\cdot}f(r)$ (in the additively
written) abelian group ${\mathrm b}\R$) is the dual of the
morphism $n{\cdot}\mathrm{id}_\mathbb{R}\colon \mathbb{R}_d\to\mathbb{R}$ which is
just multiplication by $n$.
We let the subgroup $R_n\le \mathbb{R}\times{\mathrm b}\R$
be the graph of $n{\cdot}f$, that is,
$$R_n= \set{(r,n{\cdot}f(r))}{r\in\mathbb{R}}.$$
Clearly, $R_n$ is isomorphic to $\mathbb{R}$, since
the projection of a graph of a morphism onto its domain
is always an isomorphism.
We claim that $G=\lim_n R_n$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(G)$; this claim
will finish the proof of the Second Key Lemma.
We shall prove the claim
by showing that $\lim_n R_n^\perp=\{(0,0)\}$ in
${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\widehat G)$.
This will prove the claim by Pontryagin-Chabauty-Duality
in the form of Proposition \ref{Pon-Cha-duality}.
We need information on the annihilator of a graph:
\begin{lem}\label{GraphAdjoint} Let $A$ and $B$ be locally compact groups and
$f\colon A\to B$ a morphism. Define $\Gamma=\set{(a,f(a))}{a\in A}
\subseteq A\times B$ to denote the graph of $f$. We identify
$(A\times B)\widehat{\phantom w}$
with $\widehat A\times \widehat B$ (via $\rho$ as in Lemma \ref{dual-group-cartesian-product}).
Then the graph $\set{(\widehat f(b),b)}{b\in B}\subseteq A\times B$ of the
adjoint morphism $\widehat f\colon \widehat B\to \widehat A$
is the annihilator $\Gamma^\perp$ of $\Gamma$.
\end{lem}
\begin{proof} An element $(\chi_A,\chi_B)\in (A\times B)^\times$ is in
$\Gamma^\perp$ if and only if, for any $a\in A$,
$$0= (\chi_A,\chi_B)(a,f(a))
=\chi_A(a)-(\chi_B\circ f)(a)=(\chi_A-\widehat f(\chi_B))(a),$$
that is $\chi_A=\widehat f(\chi_B)$.
\end{proof}
Applying this lemma to the graph $R_n$ we see that
\begin{equation} \label{annihilation}
R_n^\perp=\{(nr,r):r\in\mathbb{R}\}=
\{(r,\frac r n):r\in\mathbb{R}\}
\subseteq \mathbb{R}\times\mathbb{R}_d.\end{equation}
Note that $R_n^\perp={\fam\euffam g}raph(r\mapsto \frac r n)$.
\begin{lem} \label{l:null-convergence-2} We have $\lim_n R_n^\perp=\{(0,0)\}$ in ${\mathcal{S\hskip-.5pt U\hskip-.9pt B}}(\widehat G)$.
\end{lem}
\begin{proof} As in the proof of Lemma \ref{l:null-convergence} we invoke
Lemma \ref{convergence-trivial-subgp} and consider
a net $(n_j)_{j\in J}$ cofinal in $\mathbb{N}$ such that
$(r_j,s_j)_{j\in J}$ is a convergent net in $\widehat G=\mathbb{R}\times\mathbb{R}_d$
such that according to Equation (\ref{annihilation}) we have
\begin{itemize}
\item[(a)] $(r_j,s_j)\in R_{n_j}^\perp=\{(r,\frac r n_j): r\in\mathbb{R}\}$
for all $j\in J$, and
\item[(b)] $(r,s)=\lim_{j\in J} (r_j,s_j)$ in $\widehat G=\mathbb{R}\times\mathbb{R}_d$.
\end{itemize}
We must show that this implies $r=s=0$; then
Lemma \ref{convergence-trivial-subgp} completes the proof of
the lemma.
Now (a) implies $s_j=r_j/n_j$ for all $j\in J$, and (b) yields,
firstly, that $r=\lim_j r_j$ in $\mathbb{R}$ and, secondly, that in view
of the discreteness of $\mathbb{R}_d$ the net of the $r_j/n_j=s_j$ in $\mathbb{R}_d$ are
eventually constant, say $=t$ for $j>j_0$ for some $j_0$.
For these $j$ we now have $r_j=tn_j$, and so $r=\lim_j tn_j$.
Since the $n_j$ increase beyond all bounds, this implies $r=t=0$,
and so $r_j=0$ for $j>j_0$. Accordingly, $s_j=0$ for $j>j_0$,
and thus $s=\lim_j s_j=0$ as well. This completes the proof.
\end{proof}
By our earlier remarks, this shows $\lim_n R_n=G$ and thus completes
the proof of the Second Key Lemma \ref{l:key-lemma-2} and thereby also
completes the proof of Theorem \ref{th:sufficient-numeral}.
We can summarize the material on numerally approximable groups as follows:
\begin{thmB} A locally compact group $G$ is
numerally approximable if and only if it is of the form $G\cong\mathbb{R}\times C$
with a compact connected abelian group $C$.
\end{thmB}
By Pontryagin Duality, the groups $C$ range through a class equivalent to
the class of all torsion free abelian (discrete) groups.
A comparison of Main Theorems A and B allow us to draw the following
conclusion:
\begin{cor} \label{c:final} If a locally compact group is
numerally approximable, then it is integrally approximable, while the reverse
is not generally true.
\end{cor}
\begin{rem} Locally compact groups of the form $\mathbb{R}\times C$ for a
compact connected group $C$ have been called {\it two-ended}
by Freudenthal (\cite{freud}).
\end{rem}
\section{An alternate proof of Corollary \ref{c:final}}
Corollary 6.10 was proved above in a roundabout fashion. We therefore present
a different direct way to arrive at the same conclusion.
\subsection{Iterated Limit Theorem}
The Iterated Limit Theorem for nets deals with the following
data:
Let $I$ be a directed set and $(J_i)_{i\in I}$ a family of
directed sets indexed by $I$. Assume that for each $i\in I$
we are given a converging net $(x_{ij})_{j\in J_i}$ in a topological
space $X$.
The limits $r_i=\lim_{j\in J_i}x_{ij}$, $i\in I$ form a net $(r_i)_{i\in I}$
which may or may not converge. If it does, we have what is sometimes called
an iterated limit
$$ r=\lim_{i\in I}\lim_{j\in J_i} x_{ij}.$$
There is no harm in our assuming that the sets $J_i$ are pairwise disjoint.
(If necessary we can replace each $J_i$ by $\{i\}\times J_i$!). Now
$D\buildrel\mathrm{def}\over=\dot\bigcup_{k\in I}J_k$ is a disjoint union of the fibers $J_i$
of the fibration $\pi\colon D\to I$, $\pi(d)=i$ iff $d\in J_i$.
A function $\sigma\colon I\to D$ is a {\it section} if
$\pi\circ \sigma=\mathrm{id}_I$, the identity function of $I$. The
set of sections is $\prod_{i\in I} J_i$.
The set $D$ is partially ordered lexicographically:
$$d\le d'\hbox{ if either }\pi(d)< \pi(i'),
\hbox{ or }\pi(i)=\pi(i')\hbox{ and }d\le d'
\hbox{ in }J_{\pi(d)}.\leqno(1)$$
These data taken together result in
a net $(x_d)_{d\in D}$ which converges fiberwise,
and the limits
along the fibers form a convergent net.
In applications such as ours it is desirable to construct from the given
data a subnet
$(y_p)_{p\in P}$ of the net $(x_d)_{d\in D}$ in $X$
for some directed set $P$ such that
$$ r=\lim_{p\in P} y_p.$$
\begin{lem} Let $P=I\times\prod_{k\in I}J_k$. For
$p=(i,\sigma)\in P$
we define $d_p =\sigma(i)\in D$.
Then
$p\mapsto d_p:P\to D$ is a cofinal function between directed sets,
that is, for each $d\in D$ there is a $p_0\in P$ such that $p_0\le p$ implies
$d\le d_p$.
\end{lem}
\begin{proof} Let $d\in D$, that is, $d\in J_{\pi(d)}$.
We have to find a
$p_0=(i,\sigma)\in P$ such that $p_0\le p=(k,\tau)$ in $P$
implies $\sigma(i)\le \tau(k)$ in $D$.
Now for $k\ne i$ use the Axiom of Choice to select an arbitrary $j_k\in J_k$.
Define $\sigma\in\prod_{i'\in I}J_{i'}$ by
$$\sigma(k)=\begin{cases}\sigma(\pi(d)), & \mbox{if } k=\pi(i)\\
j_k. & \mbox{otherwise.}
\end{cases}
$$
Set $p_0=(\pi(d),\sigma)$. Now if $p_0\le p=(k,\tau)$ with
$\pi(d)\le k$ and $\sigma\le \tau$
we claim that $d=\sigma(\pi(d)))\le d_p=\tau(k))$. Indeed,
if $\pi(d)<k$ then $d\le \tau(k)=d_p$ by the order on $D$, and if
$i=k$ then $d=\sigma(\pi(d)) \le \tau(\pi(d))$ since $\sigma\le \tau$.
This completes the proof.
\end{proof}
The subnet is constructed so as to be convergent if the iterated
limit exists and to have the same limit.
This is the so-called
{\it Iterated Limit Theorem} in whose formulation we use the
notation introduced above.
\begin{thm}\label{IteratedLimitTheorem}
Assume that the net $(x_d)_{d\in D}$ converges fiberwise
and that the fiberwise limits converge as well.
Then it has a subnet $(y_p)_{p\in P}$,
$y_p=x_{d_p}$, $d_{i,\sigma}=\sigma(i)$,
such that
$$\lim_{p\in P}y_p=\lim_{i\in I}\lim_{d\in J_i}x_d.$$
\end{thm}
\begin{proof}
For a proof see \cite{kelley}, p.~69.
\end{proof}
In this remarkable fact about nets and their convergence it
is noteworthy that the index set $P$ is vastly larger
than the already large index set $D$.
A frequent special case is that the index sets $J_i$ all agree with
one and the same index set $J$ in which case we have
$D=I\times J$, $J_i=\{i\}\times J$, and $P=I\times J^I$.
Now we present an alternate proof of Corollary \ref{c:final}.
\begin{proof} Let $G$ be a
numerally aproximable locally compact group and let $(R_j)_{j\in J}$ be a net such that
$G=\lim_{j\in J} R_j$ and $R_j\cong \mathbb{R}$. For each $j\in J$ we have $R_j=\lim_{n\in \mathbb{N}} Z_{(j,n)}$ for a
sequence $Z_{(j, n)}\cong \mathbb{Z}$. Therefore
\begin{equation*}
G=\lim_{j\in J}\lim_{n\in\mathbb{N}} Z_{(j,n)}.
\end{equation*}
Then there exists a subnet $(Z_p)_{p\in P}$
of the net $(Z_{(j, n)})_{(j,n)\in J\times \mathbb{N}}$
such that
\begin{equation*}
G=\lim_{p\in P} Z_{p}.
\end{equation*}
by the
Theorem of the Iterated Limit.
\end{proof}
A review of the iterated limit theorem together with an application of it in
the context of our present topic may be of independent interest.
\end{document} |
\begin{document}
\title{Observation of the tradeoff between internal quantum nonseparability and external classical correlations }
\author{Jie Zhu}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China, and CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China}
\author{Yue Dai}
\affiliation{School of Physical Science and Technology, Ningbo University, Ningbo, 315211, China}
\affiliation{School of Physical Science and Technology, Soochow University, Suzhou, 215006, China}
\author{S. Camalet}
\email{[email protected]}
\affiliation{Sorbonne Universit\'{e}, CNRS, Laboratoire de Physique Th\'{e}orique de la Mati\`{e}re Condens\'{e}e,
LPTMC, F-75005 Paris, France}
\author{Cheng-Jie Zhang}
\email{[email protected]}
\affiliation{School of Physical Science and Technology, Ningbo University, Ningbo, 315211, China}
\author{Bi-Heng Liu}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China, and CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China}
\author{Chuan-Feng Li}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China, and CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China}
\author{Guang-Can Guo}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China, and CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China}
\author{Yong-Sheng Zhang}
\email{[email protected]}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China, and CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China}
\begin{abstract}
\noindent The monogamy relations of entanglement are highly significant. However, they involve only amounts of entanglement shared by different subsystems. Results on monogamy relations between entanglement and other kinds of correlations, and particularly classical correlations, are very scarce. Here we experimentally observe a tradeoff relation between internal quantum nonseparability and external total correlations in a photonic system and found that even purely classical external correlations have a detrimental effect on internal nonseparability. The nonseparability we consider, measured
by the concurrence, is between different degrees of freedom within the same photon, and the external classical correlations, measured by the standard quantum mutual information, are generated between the photons of
a photon pair using the time-bin method.
Our observations show that to preserve the internal entanglement in a system, it is necessary to maintain low external correlations, including classical ones, between the system and
its environment.
\end{abstract}
\date{\today}
\maketitle
\noindent{\bf INTRODUCTION}
\noindent Since the Einstein-Podolsky-Rosen (EPR) paradox \cite{epr} was first proposed in 1935, quantum entanglement \cite{HHHH} has drawn considerable attention. Subsequently, the Bell inequality \cite{bellinequality} provided a further demonstration that quantum entanglement differs from any classical correlation. As part of the ongoing progress in the research on quantum entanglement, the properties of multipartite quantum systems must be characterized. Most explorations of multipartite systems to date have focused on quantum entanglement. For example, the important traditional entanglement monogamy relation for three qubits \cite{CKWmonogamy,gilad2018},
say $A$, $B$ and $C$, states that the entanglement shared by qubits $A$ and $B$ and that shared by qubits $A$ and $C$ limit each other. This monogamy relation has been generalized to the multipartite systems case \cite{CKWmonogamy2}. Recently, Camalet derived a new type of monogamy relation that is a tradeoff between the internal entanglement within one system and the external entanglement of that system with another system \cite{C3,C4}, which has been observed experimentally \cite{ZHDBCZLGZ}.
\begin{figure}
\caption{\textbf{Theoretical sketch.}
\label{theory}
\end{figure}
On the other hand, relations between quantum entanglement and other kinds of correlations, and particularly classical correlations, have
attracted far less attention.
It is well known that, while the distribution of quantum correlations, such as entanglement, is constrained, classical correlations can be shared freely. The question of whether any limitation exists between quantum correlations
(e.g. entanglement or nonseparability) and classical correlations then naturally arises. This issue has been addressed for the three-system scenario mentioned above \cite{KW} and, more recently,
for a composite system correlated to another system \cite{C}. In this last case,
it has been demonstrated that internal entanglement has
a tradeoff relation not only
with external entanglement but also with other forms of
external correlations, total correlations for instance, as
illustrated in Fig. \ref{theory}.
In particular, even purely classical external correlations limit internal entanglement and vice versa.
In this study, we provide an experimental demonstration of this monogamy relation in a photonic system. Instead of internal entanglement, we consider the analogue for two
degrees of freedom of a single system, a photon in our experiments, known as quantum nonseparability
\cite{ZHDBCZLGZ,schumacher1991,hasan2020}.
The time-bin method \cite{timebinfirst,francesco2018} is used to prepare purely classical correlations between two photons.
This is a novel using of this technique for that purpose.
The considered tradeoff relation
may play an important role in the evolution of open systems \cite{ines2017,simon2020}, where the internal entanglement (or nonseparability) of the open system
and the external correlations with the system environment,
influence each other continuously.
Therefore, our results show that it is necessary to maintain low external correlations, including classical correlations, to allow more of the internal entanglement or nonseparability of the system to be preserved.
\noindent{\bf THEORETICAL BACKGROUND}
\noindent Consider any finite-dimensional system $A$, which
consists of two subsystems (or has two degrees of freedom),
and any other system $B$, see Fig.\ref{theory}.
The entanglement (or the quantum nonseparability) between the two subsystems (or degrees of freedom) of $A$ and the total correlations between $A$ and $B$ limit each other \cite{C}. This relation can be described quantitatively using an inequality that involves an entanglement monotone $E$ and a correlation monotone $C$. Monotone $C$ is required to vanish for product states and to be non-increasing under local operations, i.e., operations that do not affect either $A$ or $B$ \cite{C,C2}. Monotone $E$ must vanish for separable states and be non-increasing under local operations
and classical communication \cite{V,HHHH}.
The inequality described above can be written more specifically as
\begin{equation}
E(\rho_A) \le \xi [C(\rho)] , \label{mineq}
\end{equation}
where $\rho$ is the global state of $A$ and $B$, $\rho_A$ is the state of $A$, and $\xi$ is a non-increasing function. For any number of correlations $c$, there are states $\rho$ such that $C(\rho)=c$ and the two sides of the inequality are infinitely close to each other.
We note that $C$ in Eq. \eqref{mineq} can be an entanglement monotone because such a measure is also a correlation monotone. This inequality thus generalizes the relation between the internal and external entanglements studied in Refs. \cite{C3,ZHDBCZLGZ}.
Equation \eqref{mineq} has been obtained by assuming that $E$ is convex and that $C$ satisfies the following requirement.
When $A$ and $B$ are in a pure state $|\psi\rangle$,
$C(|\psi\rangle \langle \psi|)$ is only dependent on the nonzero
eigenvalues $\lambda_1 (\rho_A), \ldots$ of $\rho_A$, i.e.,
$C(|\psi\rangle \langle \psi|)=f(\lambda_1 (\rho_A), \ldots)$.
Equation \eqref{mineq} can be derived when the function $f$ is continuous.
Because inequality \eqref{mineq} holds when $C$ is a measure of the total correlations, it implies that even purely classical correlations between $A$ and $B$ have a detrimental effect on the internal entanglement of $A$. This can be seen clearly in the case where systems $A$ and $B$ are in a classical-classical state
\begin{equation}
\rho=\sum_{i,j} p_{ij}
|i\rangle \langle i| \otimes|j\rangle \langle j| ,
\label{ccs}
\end{equation}
where $\{|i\rangle\}$ (\{$|j\rangle$\}) denote
orthonormal states of $A$ ($B$) and $ p_{ij}$ are probabilities that sum to unity. For such a state $\rho$, not only there is no entanglement between $A$ and $B$, but also the quantum discord measures vanish \cite{MBCPV}.
The classical-classical states \eqref{ccs} obey Eq. \eqref{mineq} with $\xi$ being replaced with a non-increasing function $\zeta$, which is lower than $\xi$, see Fig. \ref{theoreticalexper} \cite{C}.
The total correlations are usually quantified using the mutual information
\begin{equation}
I(\rho)=S(\rho_A)+S(\rho_B)-S(\rho), \label{mi}
\end{equation}
where $S$ is the von Neumann entropy, which is readily computable for any global state $\rho$. For general states,
the mutual information is not larger than $2\ln d$, where $d$ is the Hilbert space dimension of $A$. For the classical-classical states \eqref{ccs}, the mutual information cannot exceed $\ln d$. This measure is a correlation monotone \cite{MPSVW}. We use it in the following to evaluate the correlations between $A$ and $B$. In this case, one has $\zeta(\ln d)=0$ for any $E$. In other words, for a classical-classical state $\rho$, the internal entanglement
$E(\rho_A)$ must necessarily vanish when $I(\rho)$ reaches the corresponding
maximum value of $\ln d$. Note that here inequality \eqref{mineq} with $C=I$ and $\zeta$ in place of $\xi$ is actually valid for all separable states $\rho$.
For a system $A$ that consists of two two-level subsystems (or two two-level degrees of freedom), the concurrence is a familiar entanglement monotone that can be evaluated for any state $\rho_A$ \cite{W}. Because it is a convex roof measure, it is convex \cite{V}. Its maximum value is $1$. When $E$ is the concurrence and $C$ is the mutual information \eqref{mi},
the function $\zeta$ can be obtained using the results of
Ref. \cite{VAD}; see the Supplementary Materials \cite{SM}.
This function vanishes in the interval $[\ln (2\sqrt{3}),2\ln 2]$.
In the interval $[0,\ln (2\sqrt{3})]$,
its inverse function is given by
\begin{eqnarray}
&&\zeta^{-1}(e)=\max\Big\{ \mu(1+e)+\mu(1-e)+(1-e)\ln (3)/2,\nonumber\\
&&\ \ \mu(1+e-\kappa(e))+\mu(1-e-\kappa(e))
-\kappa(e)\ln \kappa(e) \Big\} ,
\label{IC}
\end{eqnarray}
where $\mu(e)=-e \ln(e/2)/2$ and $\kappa(e)=(\sqrt{4-3e^2}-1)/3$.
The function $\zeta$ is shown as a blue line in Fig.\ref{theoreticalexper}.
\begin{figure}
\caption{\textbf{Relationship between the internal quantum nonseparability and the external classical correlations.}
\label{theoreticalexper}
\end{figure}
We now consider a system $A$ that consists of
two two-level subsystems (or two two-level degrees of freedom)
and a four-level system $B$ in a classical-classical state of the form
\begin{multline}
\rho^{(p)}=p^2 |00\rangle \langle 00| \otimes
|\alpha \rangle \langle \alpha|
+(1-p)^2 |+ \rangle \langle + | \otimes |\beta \rangle \langle \beta|\\
+p(1-p) \Big(| 11 \rangle \langle 11 | \otimes
|\gamma \rangle \langle \gamma|
+ |- \rangle \langle - | \otimes |\delta \rangle \langle \delta| \Big) ,
\label{expst}
\end{multline}
where $p \in [0,1]$, $|\alpha \rangle$, $|\beta \rangle$,
$|\gamma \rangle$, and $|\delta \rangle$ are orthonormal states, and
$| 00 \rangle=| 0 \rangle \otimes | 0 \rangle$,
$| 11 \rangle=| 1 \rangle \otimes | 1 \rangle$, and
$|\pm\rangle
=(|0 \rangle \otimes | 1\rangle\pm|1 \rangle \otimes |0\rangle)/\sqrt{2}$
with orthonormal states $|0\rangle$
and $|1 \rangle$ of the subsystems
of $A$. Below, we will see how states \eqref{expst} can be prepared experimentally. The corresponding mutual information \eqref{mi} is given by
\begin{equation}
I\big(\rho^{(p)}\big)=-2p\ln p - 2(1-p)\ln (1-p) ,
\label{Iexp}
\end{equation}
which reaches all values in the interval $[0,2\ln 2]$ when $p$ varies from $0$ to $1/2$. The concurrence of the reduced density operator $\rho_A^{(p)}$ is given by
\begin{equation}
E\big(\rho_A^{(p)}\big)
= \max \Big\{ 0, (1-2p)(1-p)-2\sqrt{p^3(1-p)} \Big\} ,
\label{Eexp}
\end{equation}
which decreases from $1$ to $0$ as $p$ increases and vanishes when $p > 0.302$; see the Supplementary Materials \cite{SM}.
The curve described by $(I\big(\rho^{(p)}\big),E\big(\rho_A^{(p)}\big))$, as $p$
varies from $0$ to $1/2$, is shown as a red line in Fig. \ref{theoreticalexper}. The extreme cases where $p=0$ and $p=1$ can be
readily understood from expression \eqref{expst}.
For the global state
$\rho^{(0)}
=|+ \rangle \langle + | \otimes |\beta \rangle \langle \beta|$,
$A$ and $B$ are uncorrelated and the two subsystems of $A$ are maximally entangled, which means that $I=0$ and $E=1$. For
$\rho^{(1)}
=| 00 \rangle \langle 00| \otimes |\alpha \rangle \langle \alpha|$,
$A$ and $B$ are uncorrelated and the two subsystems of $A$ are also uncorrelated, which means that $I=E=0$. For $p=1/2$, $\rho_A^{(1/2)}$ is the maximally mixed state of $A$ and thus the two subsystems of $A$ are uncorrelated and $E=0$. For any $p$, the mutual information \eqref{Iexp} and the concurrence \eqref{Eexp} satisfy inequality \eqref{mineq} with the function $\zeta$ given by Eq. \eqref{IC}, i.e., in Fig. \ref{theoreticalexper}, the red line is entirely in the light blue shaded region.
In the Supplementary Materials, a two-parameter family of
states that generalizes Eq. \eqref{expst} is considered. The blue
dots in Fig. \ref{theoreticalexper} correspond to such states.
\begin{figure*}
\caption{\textbf{Experimental setup.}
\label{setup}
\end{figure*}
\noindent{\bf EXPERIMENT}
\noindent To test the relation between
internal quantum nonseparability and external correlations experimentally, we first prepare some states as per the form of Eq. \eqref{expst} and measure them to evaluate the corresponding concurrence and mutual information. Polarization and path degrees of freedom are used for the preparation of these states. As Fig. \ref{setup} shows, pairs of entangled photons are generated, and the two photons of a pair travel along two different paths via single-mode fibers. The upper path belongs to Alice, which receives photon $A$, and the lower path belongs to Bob, which receives photon $B$. In the following, we will describe these two parts of the setup in detail.
In the entangled-photon-pair source, the entangled state $|\psi\rangle = \cos\theta|HH\rangle
+\sin\theta|VV\rangle$, where $|H\rangle$
and $|V\rangle$ denote, respectively, horizontal and vertical polarization states of a photon, is produced via a type I phase-matching spontaneous parametric down-conversion (SPDC) process in a joint $\beta$-barium-borate (BBO) crystal \cite{P.G.Kwiat1999}. The angle $\theta$ is
adjustable.
The laser used here is a semiconductor laser with a wavelength of 404 nm and power of approximately 100 mW. Then, a suitably long quartz plate causes this state to be completely decohered into
the mixed state $\rho_d = \cos^2\theta|HH\rangle \langle HH| + \sin^2\theta|VV\rangle\langle VV|$.
The two photons are subsequently distributed to the
Alice's and Bob's parts of the setup,
where operators act on them to produce the target states as shown in Eq. (\ref{expst}). In both Alice's and Bob's parts, there are
an up-path and a down-path.
In the following, we use two conventions for both
photons A and B: (\textrm{i}): both $|H\rangle$ and the path state $|\text{up-path}\rangle$ represent $|0\rangle$; and (\textrm{ii}): both $|V\rangle$ and the path state $|\text{down-path}\rangle$ represent $|1\rangle$.
In Alice's part, two operations are performed on photon A with different probabilities. The first operation is $U_1: (|0\rangle, |1\rangle) \rightarrow (|11\rangle, |+\rangle)$, which has probability $1-p$. The second operation is
$U_2: (|0\rangle, |1\rangle) \rightarrow (|00\rangle, |-\rangle)$, which has probability
$p$. We fix the two half-wave plates (HWPs) after the first beam displacer (BD) at angles of $45^\circ$ and $22.5^\circ$, respectively. Note that the beam displacers used in our experiments always shift the photons with the horizontal polarization upward and keep the photons with the vertical polarization on the original path. Therefore, after the first two BDs, operation $U_1$ has been fulfilled. The beam splitter (BS) then gives two paths of different lengths $\ell_1$ and $\ell_2$.
On the shorter path, an operation $U$ such that $U_2=UU_1$ is performed, while the longer path involves direct reflection to the measurement device. The ratio between these two paths is adjusted using a movable attenuator. The two HWPs after the first BS are fixed at angles of $-45^\circ$ and $45^\circ$, respectively.
In Bob's part, two operations are also enacted on the photons with probabilities $1-p$ and $p$ and the lengths of
the two paths are also $\ell_1$ and $\ell_2$. The operation
$V_1: (|0\rangle, |1\rangle) \rightarrow (|00\rangle, |10\rangle)$ is performed on the longer path and the operation
$V_2: (|0\rangle, |1\rangle) \rightarrow (|01\rangle, |11\rangle)$ is performed on the shorter path.
The photon B states $|01\rangle$,
$|10\rangle$, $|00\rangle$, and $|11\rangle$ correspond, respectively, to the states $|\alpha \rangle$, $|\beta \rangle$,
$|\gamma \rangle$, and $|\delta \rangle$ in Eq. \eqref{expst}. Unlike $U_1$ and $U_2$, the operations $V_1$ and $V_2$ can be completed using only BSs. The ratio between the two operations
can also be tuned via a movable attenuator.
Here we use the time-bin method,
that is, the coincidence detection only records the photon pairs who arrive the coincidence counter within the time window, to prepare the classical-classical states. As described above, when both photons of a pair travel the longer (shorter) paths, the operation $U_1~(U_2)$ is performed
on photon $A$, and the operation $V_1~(V_2)$ is performed on
photon $B$. Therefore, the state of the photon pair becomes
$\rho = (1-p)(U_1\otimes V_1)\rho_d (U^\dag_1\otimes V^\dag_1)
+ p(U_2\otimes V_2)\rho_d (U^\dag_2\otimes V^\dag_2)$,
that is the same as Eq. (\ref{expst}). It should be noted that the time-bin method has been employed for coherently synthesizing quantum entangled states extensively in many previous works. However, here we can use it for incoherently producing classical correlations with suitable postselection.
After state preparation, determination of the method required to measure the prepared states is also a crucial problem.
They are four-qubit states encoded in polarization and path degrees of freedom \cite{ZHDBCZLGZ,chaozhang2019, yunfenghuang2004, ojimenez2012}. As shown in Fig.({\ref{setup}}), the experimental setup contains four standard polarization tomography setups (SPTSs) that each consist of one half-wave plate (HWP), one quarter-wave plate (QWP), and a polarizing beam splitter (PBS). The first SPTS, on each side, is used to measure the polarization states and
collapses these states to $|H\rangle$ so that the polarization information is erased. Then, the final BDs, on both sides,
convert the path information into polarization information to help the subsequent SPTSs to measure the path states.
Using the four SPTSs, full quantum state tomography can be performed and the complete $16\times16$ density matrix $\rho$ can be reconstructed \cite{DanielF.V.James2001}. The reduced density matrices $\rho_A$ and $\rho_B$ are then derived from the experimentally determined
state $\rho$, and the mutual information \eqref{mi} and
the concurrence of the state $\rho_A$ are calculated.
By adjusting the angle $\theta$ of the HWP before the BBO crystal, and the transmissivities of Alice's and Bob's
attenuators so that $p=\text{cos}^2(\theta)$,
we prepared 12 states. We used the values
$0$, $\pi/16$, $\pi/8$, $3\pi/16$, $\pi/4$, $9\pi/32$, $11\pi/32$, $3\pi/8$, $13\pi/32$, $7\pi/16$, $15\pi/32$, and $\pi/2$ for $\theta$. The average fidelity of these states, as described using Eq. (\ref{expst}), is beyond $95\%$, see the
Supplementary Materials \cite{fidelitysquare,SM}.
Fig. \ref{Ic} shows the theoretical and experimental values of the internal concurrence and external mutual information as functions of $p$.
The experimental results for $\theta$ from $\pi/4$ to
$\pi/2$, are represented as red dots with error bars in Fig. \ref{theoreticalexper}. They are all in the allowed region
determined by inequality \eqref{mineq} with the function $\zeta$ given by Eq. \eqref{IC}, which experimentally demonstrates the
tradeoff relation between internal nonseparability and external
purely classical correlations.
The errors in the experiment mainly resulted from the quality of the interferometer, for which the visibility was approximately 50:1, and fluctuations in the photon count.
\begin{figure}
\caption{\textbf{Experimental data.}
\label{Ic}
\end{figure}
\noindent {\bf DISCUSSION}
\noindent The entanglement and classical correlation tradeoff relation observed in the present experiment may be useful in
the exploration of a number of fields, ranging from quantum communication \cite{nicolas2007} and quantum computation \cite{david1995} to open systems and many-body physics \cite{luigi2008}.
This relation does not apply solely to
the considered optical system;
it can also be observed in several other physical systems, including cold atoms and trapped ions. This tradeoff relation is thus a fundamental result for the development of quantum information science, particularly for quantum communication network.
For open systems \cite{opensystem}, external correlations
between the system and the environment is a major concern because
they may damage the entanglement inside the system. According to our results, external total correlations and internal quantum nonseparability limit each other.
Even purely classical external correlations can have a detrimental effect on the internal quantum nonseparability. Therefore, to preserve the internal entanglement or nonseparability of the system as much as possible, the correlations, including the classical ones, between the system and the environment, must be reduced as low as possible.
In conclusion, we have presented the tradeoff relation between
internal quantum nonseparability and external classical correlations in a photonic system experimentally. It is remarkable that the realization involves use of the time-bin method to produce purely classical correlations between
the photons of a photon pair. Furthermore, polarization
and path degrees of freedom of one of the photons of a pair
have been entangled to realize the internal nonseparability experimentally. The classical-classical states are of major significance in this work and we have proposed a convenient and efficient method to prepare these states.
\noindent {\bf DATA AVAILABILITY}
\noindent The data that support the findings of this study are available from the corresponding author upon reasonable
request.
\noindent{\bf ACKNOWLEDGEMENTS}
\noindent We thank Qiongyi He for helpful discussions. This work is funded by the National Natural Science Foundation of China (Grants Nos.~ 11674306, 92065113 and 11734015), National Key R$\&$D Program (No. 2016YFA0301300 and No. 2016A0301700), Anhui Initiative in Quantum Information Technologies, and the K.C. Wong Magna Fund in Ningbo University.
\noindent{\bf AUTHOR CONTRIBUTIONS}
\noindent S. Camalet, C.-J. Zhang and Y.-S. Zhang start the project and design the experiment. J. Zhu perform the experiment and complete the data analysis. S. Camalet and C.-J. Zhang provide the theoretical calculations. Y.-S. Zhang, C.-F. Li and G.-C. Guo supervise the project. J. Zhu, Y.-S. Zhang, C.-J. Zhang and S. Camalet write the manuscript and all authors participate in discussions.
\noindent{\bf ADDITIONAL INFORMATION}
\noindent The authors declare that they have no competing interests.
\noindent {\bf REFERENCES}
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\section{Supplementary Material}
\section{Derivation of equation (4)}
When $A$ consists of two two-level systems, $E$ is the concurrence
and $C$ is the mutual information, the function $\zeta$ mentioned
in the main text is given by
$$ \zeta(c)=\max_{\boldsymbol \lambda \in \Lambda
|h(\boldsymbol \lambda)=c} \max \Big\{ 0,
\lambda_1-\lambda_3 - 2\sqrt{\lambda_2 \lambda_4} \Big\} , $$
where $\Lambda$ refers to the set of tuples of four probabilities
summing to unity, arranged in decreasing order, and $h$ is the Shannon
entropy, i.e., $h(\boldsymbol \lambda)=-\sum_i \lambda_i \ln \lambda_i$ \cite{C,VAD}.
To determine $\zeta$, we first consider $\chi$ defined by
$$\chi(e)=\max_{\boldsymbol \lambda \in \Lambda
|k(\boldsymbol \lambda)=e} h(\boldsymbol \lambda) , $$ where
$k(\boldsymbol \lambda)
=\lambda_1-\lambda_3 - 2\sqrt{\lambda_2 \lambda_4}$. To find
the maximum value of $h(\boldsymbol \lambda)$ under the constraints
$k(\boldsymbol \lambda)=e$, $\sum_i \lambda_i=1$, and
$0 \le \lambda_4 \le \lambda_3 \le \lambda_2 \le \lambda_1 \le 1$,
we introduce $x$ and $y$ such that $0 \le x \le y \le 1$ and let
$\lambda_4=x^2$ and $\lambda_2=y^2$. The conditions
$k(\boldsymbol \lambda)=e$
and $\sum_i \lambda_i=1$ can be rewritten as $\lambda_{2 \mp 1}
=[1\pm e -(x \mp y)^2]/2$, and the above inequality requirements
on the probabilities $\lambda_i$ determine the $e$-dependent domain
${\cal D}_e$ of $(x,y)$ which exists for $e \in [-1/2,1]$. More precisely,
${\cal D}_e$ is given by $x \ge 0$, $y \le -x+(1-e-2x^2)^{1/2}$,
$y \le x/3+(3+3e-2x^2)^{1/2}/3$, and $y \ge -x/3+(3-3e-2x^2)^{1/2}/3$.
The first and second order derivatives of $h$ with respect to $x$ and $y$
can be written as
\begin{eqnarray}
\partial_z h &=& -4z\ln z +(z-\bar z) \ln \lambda_1 + (z+\bar z) \ln \lambda_3 ,
\nonumber \\
\partial^2_z h &=& -4 \ln z -(1+e)/\lambda_1-(1-e)/\lambda_3
+\ln (\lambda_1 \lambda_3),
\nonumber \\
\partial^2_{z \bar z} h &=& (1+e)/\lambda_1-(1-e)/\lambda_3
+\ln (\lambda_3/\lambda_1),
\nonumber
\end{eqnarray}
where $z=x$ or $y$, $\bar x=y$, and $\bar y=x$. The function $h$ has a
critical point in the interior of ${\cal D}_e$ for some values of $e$, but it is
not a maximum, as shown by the fact that the Hessian determinant,
$\partial^2_x h \partial^2_y h-(\partial^2_{xy} h)^2$, is negative at this point.
For $e>0$, the boundary of ${\cal D}_e$ contains a line segment on the $y$-axis.
On this line segment, the maximum value of $h$ is at $y=\kappa(e)^{1/2}$ and is
equal to $\mu(1+e-\kappa(e))+\mu(1-e-\kappa(e))-\kappa(e)\ln \kappa(e)$.
The functions $\kappa$ and $\mu$ are defined in the main text.
On the boundary of ${\cal D}_e$ and for $x>0$, the maximum value of $h$
is at $x=y=\sqrt{(1-e)/6}$ and is equal to
$\mu(1+e)+\mu(1-e)+(1-e)\ln (3)/2$, or is reached in the limit $x \rightarrow 0$.
Consequently, $\chi$ is given by the right side of eq. (4) in the main text.
Since $\chi$ is a continuous and strictly decreasing function on
$[-1/2,1]$, $\chi(-1/2)=2\ln 2$ and $\chi(1)=0$, it has an inverse
function $\chi^{-1}$ with domain $X=[0,2\ln 2]$. Consider any
$c \in X$ and define $e_c=\chi^{-1}(c)$. As seen above, there is
$\boldsymbol \lambda \in \Lambda$ such that
$k(\boldsymbol \lambda)=e_c=\chi^{-1}(c)$ and
$h(\boldsymbol \lambda)=\chi(e_c)=c$. Let now $\boldsymbol \lambda$
be any tuple of $\Lambda$ such that $h(\boldsymbol \lambda)=c$ and
let $e=k(\boldsymbol \lambda)$. Assuming $e>e_c$ implies
$\chi(e)<\chi(e_c)=c$, and so, by definition of $\chi(e)$,
$h(\boldsymbol \lambda)<c$. As this last inequality cannot hold,
one has necessarily $k(\boldsymbol \lambda) \le \chi^{-1}(c)$, and hence
$$\max_{\boldsymbol \lambda \in \Lambda |h(\boldsymbol \lambda)=c}
k(\boldsymbol \lambda)=\chi^{-1}(c). $$
When $\chi^{-1}(c) \le 0$, i.e., for $c \in [\ln (2\sqrt{3}),2\ln 2]$,
$k(\boldsymbol \lambda)$ is non positive for any
$\boldsymbol \lambda \in \Lambda$ such that $h(\boldsymbol \lambda)=c$,
and so $\zeta(c)=0$. When $\chi^{-1}(c) > 0$, $\zeta(c)$ is equal to
$\chi^{-1}(c)$, as given by eq. (4) in the main text.
\section{Derivation of equations (6) and (7)}
Consider a system $A$ consisting of two two-level systems and
a four-level system $B$ in a classical-classical state of the form
$$\rho=p(1-q) | 00 \rangle \langle 00|
\otimes |\alpha \rangle \langle \alpha|
+(1-p)q |+ \rangle \langle + | \otimes |\beta \rangle \langle \beta|
+pq |11\rangle \langle 11| \otimes |\gamma \rangle \langle \gamma|
+ (1-p)(1-q)|- \rangle \langle - | \otimes |\delta \rangle \langle \delta| , $$
where $p,q \in [0,1]$, $|\alpha \rangle$, $|\beta \rangle$,
$|\gamma \rangle$ and $|\delta \rangle$ are orthonormal states,
$| 00 \rangle=| 0 \rangle \otimes | 0 \rangle$,
$| 11 \rangle=| 1 \rangle \otimes | 1 \rangle$ and
$|\pm\rangle
=(|0 \rangle \otimes | 1\rangle\pm|1 \rangle \otimes |0\rangle)/\sqrt{2}$
with $|0\rangle$ and $|1 \rangle$ denoting orthonormal states of
the subsystems of $A$. Since $S(\rho)=S(\rho_A)=S(\rho_B)$,
the corresponding mutual information between $A$ and $B$ is
$$I(\rho)=-p\ln p -(1-p)\ln (1-p) -q\ln q-(1-q)\ln (1-q) , $$
which simplifies to eq. (6) in the main text for $q=1-p$.
Note that it is invariant under the changes $p \leftrightarrow 1-p$
and $q \leftrightarrow 1-q$.
The concurrence $E$ between the two subsystems of $A$ is determined
by the eigenvalues $\mu_i$ of
$\rho_A \sigma \otimes \sigma \rho_A^* \sigma \otimes \sigma$
where $\sigma=-|1\rangle\langle 0| +|0\rangle\langle 1|$ and
$\rho_A^*=\rho_A$ is the complex conjugate of $\rho_A$ written in
the standard basis $\{ | 00 \rangle,| 01 \rangle,| 10 \rangle,| 11 \rangle \}$.
It reads as
$E(\rho_A)=\max \{ 0, 2\max_i \sqrt{\mu_i}-\sum_i \sqrt{\mu_i} \}$.
The eigenvalues $\mu_i$ are $q^2(1-p)^2$, $(1-q)^2(1-p)^2$, and twice
$p^2q(1-q)$. They are given by the same expressions with $q$ replaced by
$\tilde q=\min \{ q, 1-q \}=(1-|2q-1|)/2$. Since the last one is doubly
degenerate, $2\max_i \sqrt{\mu_i}-\sum_i \sqrt{\mu_i}$ can be positive
only when $\max_i \sqrt{\mu_i}=(1- \tilde q)(1-p)$, and so
$$E(\rho_A)=\max \{ 0, (1-2\tilde q)(1-p)-2p\sqrt{\tilde q(1-\tilde q)} \} , $$
which simplifies to eq. (7) in the main text for $q=1-p$. Clearly, $E(\rho_A)$ is
also invariant under the change $q \leftrightarrow 1-q$.
\section{Fidelities of prepared states}
\setlength{\tabcolsep}{1.5mm}
\begin{table}[!h]
\renewcommand\arraystretch{1.5}
\caption{Fidelities of prepared states}
\centering
\scalebox{1}{
\begin{tabular*}{15cm}{ccccccc}
\hline
\hline
$\theta$ &0 & $\frac{1}{16}\pi$ & $\frac{1}{8}\pi$ & $\frac{3}{16}\pi$ & $\frac{1}{4}\pi$ & $\frac{9}{32}\pi$ \\
\hline
fidelities &$98.81\pm0.11\%$ & $95.66\pm0.35\%$ & $95.25\pm0.24\%$ & $95.30\pm0.31\%$ & $95.27\pm0.33\%$ & $95.80\pm0.22\%$\\
\hline
\hline
$\theta$ &$\frac{11}{32}\pi$ & $\frac{3}{8}\pi$ & $\frac{13}{32}\pi$ &
$\frac{7}{16}\pi$ & $\frac{15}{32}\pi$ & $\frac{1}{2}\pi$ \\
\hline
fidelities &$95.01\pm0.48\%$ & $95.55\pm0.42\%$ & $94.70\pm0.28\%$ &
$94.68\pm0.37\%$ & $95.64\pm0.26\%$ & $97.20\pm0.15\%$ \\
\hline
\end{tabular*} }
\end{table}
\end{document} |
\begin{document}
\title{Hong-Ou-Mandel interferometry on a biphoton beat note}
\author{Yuanyuan Chen}
\email{[email protected]}
\affiliation{
Institute for Quantum Optics and Quantum Information - Vienna (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.}
\affiliation{Vienna Center for Quantum Science \& Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria}
\affiliation{State Key Laboratory for Novel Software Technology, Nanjing University, Xianlin Avenue 163, Nanjing 210046, China.}
\author{Matthias Fink}
\affiliation{
Institute for Quantum Optics and Quantum Information - Vienna (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.}
\affiliation{Vienna Center for Quantum Science \& Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria}
\author{Fabian Steinlechner}
\email{[email protected]}
\affiliation{Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Strasse 7, 07745 Jena, Germany.}
\affiliation{Friedrich Schiller University Jena, Abbe Center of Photonics, Albert-Einstein-Str. 6, 07745 Jena, Germany.}
\author{Juan P. Torres}
\affiliation{
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain}
\affiliation{Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain}
\author{Rupert Ursin}
\email{[email protected]}
\affiliation{
Institute for Quantum Optics and Quantum Information - Vienna (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.}
\affiliation{Vienna Center for Quantum Science \& Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria}
\begin{abstract}
Hong-Ou-Mandel interference, the fact that identical photons that arrive simultaneously on different input ports of a beam splitter bunch into a common output port, can be used to measure optical delays between different paths. It is generally assumed that great precision in the measurement requires that photons contain many frequencies, i.e., a large bandwidth. Here we challenge this ``well-known'' assumption and show that the use of two well-separated frequencies embedded in a quantum entangled state (discrete color entanglement) suffices to achieve great precision. We determine optimum working points using a Fisher Information analysis and demonstrate the experimental feasibility of this approach by detecting thermally-induced delays in an optical fiber. These results may significantly facilitate the use of quantum interference for quantum sensing, by avoiding some stringent conditions such as the requirement for large bandwidth signals.
\end{abstract}
\maketitle
\section{Introduction}
The exploitation of quantum interference promises to enhance sensing technologies beyond the possibilities of classical physics. Hong-Ou-Mandel (HOM) interference is a prototypical example of such a quantum phenomenon, that lacks any counterpart in classical optics. When two identical photons in a global pure state impinge on a beam splitter from separate input modes, they both leave the beam splitter through the same output port, as a consequence of their bosonic nature \cite{hong1987measurement}. On the other hand, if the input photons are not identical, or they are independent but not in a pure state, the ``bunching'' probability is directly related to the photons' level of indistinguishability, or its degree of purity \cite{mosley2008heralded}. This effect enables a wide range of quantum information processing tasks, ranging from the characterization of ideally identical single-photon emitters \cite{aharonovich2016solid}, the implementation of photonic Bell state measurements for entanglement swapping and quantum teleportation \cite{pan2012multiphoton}, or in tailoring high-dimensional entangled states of light \cite{zhang2016engineering,ndagano2019entanglement}.
HOM interferometry also holds great promise for sensing schemes that require precise knowledge of optical delays. When the relative arrival time of two photons is varied, the coincidence rate exhibits a characteristic dip with a width that is related to the photons' coherence time. Notably, and unlike other interferometric approaches based on first-order interference, HOM interference is not affected by variations in the optical phase. As a consequence, a HOM interferometer maintains its ability to measure time delays, even when fluctuations of path length difference are on the order of the wavelength. This feature has resulted in proposals for HOM-based time delay sensors with an ultrahigh timing resolution \cite{lyons2018attosecond} and novel protocols such as Quantum Optical Coherence Tomography (QOCT) that benefit from other quantum features, such as the cancellation of some deleterious dispersion effects \cite{nasr2003demonstration}.
In the context of such applications, the broad consensus has been that the width of the dip, i.e. the coherence time, imposes the ultimate limit on the precision. As a consequence, ultra-broad-band photon sources have long been hailed as a vital prerequisite for ultra-precise HOM interferometry.
Here we embark on an alternative route towards ultra-precise HOM interferometry using superpositions of two well-separated and entangled discrete frequency modes and coincidence detection on the bi-photon beat note. The manifestation of this fourth-order spatial beating effect is an oscillation within a typically Gaussian envelope that is determined by the coherence time of the two photons, as a direct result of relative phase shift between distinct colors \cite{ou1988observation,rarity1990two}. We explore the sensitivity limits as a function of the difference frequency of color-entangled states, as imposed by the Quantum Cram$\acute{e}$r-Rao (QCR) bound, and find that the precision with which the delays can be measured is mainly determined not by the coherence time of photons, but by the separation of the center frequencies of the state. Aside from promising improved precision, the approach allows to increase the dynamic range of a HOM-based sensor, provided the required frequency non-degenerate states can be generated in a tunable manner.
We show how suitable frequency entangled states are readily obtained with comparatively little technological effort by employing a variation of the source scheme recently developed in Ref. \cite{chen2018polarization}. Building on the measurement and estimation strategy by analyzing the Fisher information to determine the optimum working points for frequency-degenerate HOM interference, recently proposed in Ref. \cite{lyons2018attosecond}, we experimentally demonstrate an optimized HOM sensor that we use to detect delays introduced by temperature drifts in an optical fiber.
The results obtained in this proof of concept experiment show that quantum interference of unconventional frequency states on a beam splitter provides a simple way of enhancing the timing resolution in HOM-based sensors and may also indicate a new direction towards fully harnessing HOM interference in quantum sensing and quantum information processing.
\section{Results}
\subsection{HOM interfereometry with frequency entangled states}
Let us search for the ultimate limits to the precision of a HOM-based sensor. We consider the generic task of estimating an unknown parameter $\tau$ of a physical system. We prepare a probe state $\ket{\Psi_0}$ that is transformed as $\ket{\Psi_0}\rightarrow\ket{\Psi(\tau)}$ upon interaction with the physical system. The transformed state is then subjected to a particular measurement strategy to obtain an estimator of $\tau$. Irrespective of the specifics of the final measurement step, we may already state a fundamental limit for the precision of estimation $\delta \tau$ \cite{helstrom1969quantum,fujiwara1995quantum}:
\begin{equation}
\begin{split}
\delta\tau\geq \frac{1}{2\sqrt{NQ}}=\delta\tau_{QCR},
\end{split}
\end{equation}
where
\begin{equation} \label{eq:q}
\begin{split}
Q=\langle\frac{\partial\Psi(\tau)}{\partial\tau}|\frac{\partial\Psi(\tau)}{\partial\tau}\rangle-|\langle\Psi(\tau)|\frac{\partial\Psi(\tau)}{\partial\tau}\rangle|^2
\end{split}
\end{equation}
and $N$ is the number of independent trials of the experiment. The generality of this statement, known as the Quantum Cram$\acute{e}$r-Rao bound, is remarkable: no matter what ingenious measurement procedure the experimenter may contrive, she will never achieve a precision better than $\delta \tau_{QCR}$. Since the QCR bound is attached to a particular quantum state, it is clear that the appropriate choice of the probe state is of the utmost importance \cite{pirandola2018advances}.
Let us now consider an experimental configuration where paired photons (signal and idler), with central frequencies $\omega_1^0$ and $\omega_2^0$, originate from a parametric down-conversion process (SPDC) pumped by a CW pump with frequency $\omega_p=\omega_1^0+\omega_2^0$. Each photon of the pair is injected into one of the two arms of a HOM interferometer. The time delay of interest is one that may occur due to an imbalance between the two arms of the interferometer. Even though the common case in HOM interferometry is to consider signal and idler photons with the same central frequency, in the following we allow for a more general configuration where the state of interest is a discrete or continuous frequency entangled state \cite{ramelow2009discrete}:
\begin{equation}
\begin{split}
\ket{\Psi(\tau)}=&\frac{1}{\sqrt{2}}\int d\Omega f(\Omega) \times \\
&[e^{i(\Delta+2\Omega) \tau}a_1^\dag(\omega_1^0+\Omega)a_2^\dag(\omega_2^0-\Omega) - \\
&a_1^\dag(\omega_2^0+\Omega)a_2^\dag(\omega_1^0-\Omega)]\ket{vac},
\end{split}
\end{equation}
where $\Delta=\omega_1^0-\omega_2^0$ is the difference frequency of two well-separated center frequency bins, $\ket{vac}$ is the vacuum state, and $f(\Omega)$ is an the Gaussian spectral amplitude function with $\int d\Omega|f(\Omega)|^2=1$. For this state, the Quantum Cram$\acute{e}$r-Rao limit on the estimation of time delays writes
\begin{equation}\label{eq:quantum cramer rao bound}
\begin{split}
\delta\tau_{QCR}= \frac{1}{N^{1/2}}\frac{1}{(\Delta^2+4\sigma^2)^{1/2}},
\end{split}
\end{equation}
where $\sigma=\sqrt{\langle\Omega^2\rangle-\langle\Omega\rangle^2}$ is the RMS (root mean square) bandwidth of SPDC photons. The dependence of the QCR bound on frequency detuning $\Delta$ gives us a first indication to the potential use of non-degenerate frequency entanglement as an alternative to large bandwidth for enhanced resolution HOM interferometry.
Up until now, we have only considered limitations that are inherent to the particular choice of the quantum state. We must confirm that we can experimentally realize this potential benefit using an appropriate measurement strategy, i.e. one that allows us to saturate equation \eqref{eq:quantum cramer rao bound}. As we shall see in the following, this can be accomplished via coincidence detection in the output ports of a balanced beam splitter. The beam splitter transforms the bi-photon state (see Methods for details) to
\begin{equation}
\ket{\Psi(\tau)}\rightarrow\ket{\Psi_A(\tau)}+\ket{\Phi(\tau)},
\end{equation}
where $\ket{\Psi_A(\tau)}$ and $\ket{\Phi(\tau)}$ correspond to the events that two photons emerge in opposite and identical outports, respectively. The normalized coincidence detection probability $P_c(\tau)=|\braket{\Psi(\tau)}{\Psi_A(\tau)}|^2$ reads
\begin{equation}\label{eq:interference probability}
\begin{split}
P_c(\tau)=\frac{1}{2}[1+ cos(\Delta\tau+\phi)exp(-2\sigma^2\tau^2)],
\end{split}
\end{equation}
where $\phi$ is a relative phase factor.
In the case of a real HOM interferometer, that is subject to photon loss $\gamma$ and imperfect experimental visibility $\alpha$, there are three possible measurement outcomes; either both photons are detected, one photon is detected, or no photon detected. The corresponding probability distributions read
\begin{equation}\label{eq:loss probability}
\begin{split}
P_2(\tau)&=\frac{1}{2}(1-\gamma)^2[1+\alpha cos(\Delta\tau)e^{-2\sigma^2\tau^2}]\\
P_1(\tau)&=\frac{1}{2}(1-\gamma)^2[\frac{1+3\gamma}{1-\gamma}-\alpha cos(\Delta\tau)e^{-2\sigma^2\tau^2}]\\
P_0(\tau)&=\gamma^2,
\end{split}
\end{equation}
where subscripts 0, 1 and 2 denote the number of detectors that click, corresponding to total loss, bunching and coincidence, respectively. For a more detailed discussion refer to \cite{lyons2018attosecond}. The outcome probabilities in this measurement can now be used to construct an \emph{estimator} for the value of $\tau$.
An estimator $\tilde{\tau}$ is a function of the experimental data that allows us to infer the value of the unknown time delay using a particular statistical model for the probability distribution of measurement outcomes. It is thus itself a random variable, that can be constructed from the probability distributions $P_i(\tau)$ as a function of time delay. The average of an \emph{unbiased estimator} corresponds to the real time delay. For any such estimator, classical estimation theory states standard deviation is lower bounded by
\begin{equation}\label{eq:cramer rao bound}
\begin{split}
\delta\tau_{CR}= \frac{1}{(N F_\tau)^{1/2}} \geq \delta\tau_{QCR},
\end{split}
\end{equation}
where the Fisher information $F_\tau$ reads
\begin{equation}
\label{eq: Fisher information}
F_\tau=\frac{(\partial_\tau P_2(\tau))^2}{P_2(\tau)}+\frac{(\partial_\tau P_1(\tau))^2}{P_1(\tau)}+\frac{(\partial_\tau P_0(\tau))^2}{P_0(\tau)}.
\end{equation}
This limit is known as the Cram$\acute{e}$r Rao bound. It is tied to a particular quantum state and a specific measurement strategy. Evaluating the Fisher information for this set of probabilities, we find that its upper bound is achieved in ideal case ($\gamma=0$, $\alpha=1$) at position of $\tau\rightarrow0$ as
\begin{equation}
\label{eq: maximal Fisher information}
\begin{split}
\lim_{\tau\rightarrow0}F_\tau=\Delta^2+4\sigma^2.
\end{split}
\end{equation}
In the case of zero loss and perfect visibility we recover the Quantum Cram$\acute{e}$r Rao Bound, thus confirming that the measurement strategy is indeed optimal.
While equation \eqref{eq:cramer rao bound} provides an ultimate bound on the achievable precision of estimation that can be achieved, the approach does not yet tell us how to construct a suitable estimator for $\tau$. To this end a widely used analytical technique is maximum likelihood estimation (MLE). The likelihood function $\mathcal{L}(\tau)$ is defined from measurement outcomes, whose logarithm can be maximized by using optimization algorithm such as Gradient Descent to predict the parameter $\tau$ that we want to infer. In our framework, the likelihood function is a multinomial distribution as $\mathcal{L}(N_0,N_1,N_2|\tau)\propto P_0(\tau)^{N_0}P_1(\tau)^{N_1}P_2(\tau)^{N_2}$, where $N_0$, $N_1$ and $N_2$ denote the numbers of events that no, only one and two detector(s) click(s), respectively. Note that $P_0(\tau)$, being independent of $\tau$, results in a constant scale factor that is of no relevance to the final calculation of the Fisher information and parameter estimation. The likelihood is extremized as \cite{lyons2018attosecond}:
\begin{equation}
\begin{split}
0&=:(\partial_\tau log\mathcal{L})_{\tilde{\tau}_{MLE}}\\
&=\frac{N_0P_0(\tau)^\prime}{P_0(\tau)}+\frac{N_1P_1(\tau)^\prime}{P_1(\tau)}+\frac{N_2P_2(\tau)^\prime}{P_2(\tau)},
\end{split}
\end{equation}
and solving this equation enables us to predict an optimal estimator as $\tilde{\tau}_{MLE}$.
\subsection{Experiment}
\textbf{Experimental realization of bi-photon beat note.}
\begin{figure}
\caption{Experimental setup for temperature sensor through beating of frequency entanglement. LD: laser diode; HWP: half wave plate; WP: wave plate; ppKTP: type-II periodically poled potassium titanyl phosphate crystal; TEC: temperature controller; LP: long pass filter; PBS: polarizing beam splitter; Oven: computer-controlled heating device; PC: polarization controller; BS: beam splitter; Detectors: single photon detectors.}
\label{figure_1}
\end{figure}
We generate photon pairs via spontaneous parametric down-conversion pumped with a continuous-wave pump laser. The experimental configuration implemented to generate the desired frequency entangled state of distant frequency modes (i.e. signal and idler frequencies that are separated by more than the spectral bandwidth $\omega_{s}-\omega_{i}\gg \Delta \omega$) is a modified crossed-crystal configuration \cite{kwiat1999ultrabright,steinlechner2012high} shown in the inset of Fig.\ \ref{figure_1}. In this configuration, two nonlinear crystals for type-II SPDC are placed in sequence, whereby the optical axis of the second crystal is rotated by 90$^\circ$ with respect to the first. Balanced pumping of the two crystals ensures equal probability amplitudes for SPDC emission $\ket{V,\omega_p}\rightarrow \ket{V,\omega_s}\ket{H,\omega_i}$ in the first-, or $\ket{H,\omega_p}\rightarrow \ket{H,\omega_s}\ket{V,\omega_i}$ in the second crystal, where $H/V$ denote horizontal and vertical polarizations. The photons are guided to a PBS, which maps the orthogonally polarized photon pairs into two distinct spatial modes ($1,2$) in the desired frequency entangled state
\begin{equation}\label{eq:frequency state}
\begin{split}
\ket{\psi}_\omega^- \rightarrow
(\frac{\ket{\omega_s}_{1}\ket{\omega_i}_{2} - \ket{\omega_i}_{1}\ket{\omega_s}_{2}}{\sqrt{2}})\otimes\ket{H}_1\ket{V}_2.
\end{split}
\end{equation}
The frequency entangled photons are routed to the input ports of a beam splitter. After operation of HOM interference, we only focus on the situation that two detectors indiscriminately register coincidence, i.e., exiting via different ports, as a direct consequence of anti-bunching effect of photons entangled in the form of anti-symmetric state.
\begin{figure}
\caption{ Two photon HOM interference of frequency entanglement with different frequency detunings of (a) $\unit[3.65]{THz}
\label{figure_2}
\end{figure}
As the central wavelengths of down-converted photons are related to the phase-matching temperature of nonlinear crystals, our source has the ability to produce color tunable frequency entangled photon pairs. We analyze the HOM signal for various frequency detunings to demonstrate this flexibility (see Fig.\ \ref{figure_2}). By fitting these interference fringes to normalized coincidence probability as equation \eqref{eq:interference probability}, we are able to estimate single photon frequency bandwidth to be $\unit[0.253]{THz}$, which corresponds to a bandwidth in wavelength of $\unit[0.55]{nm}$ and a coherence time of $\unit[3.5]{ps}$. Frequency detunings are much larger than single photon bandwidth such that two frequency bins could be separated completely. The visibilities of these experimentally measured frequency entangled photon pairs can reach $0.85\pm0.05$. The maximal frequency detuning we have measured is $\unit[17.08]{THz}$ at temperature of $\unit[100]{^\circ C}$, which is about 68 times the single photon frequency bandwidth.
\textbf{Fisher information analysis.}
\begin{figure*}
\caption{Experimental description of Fisher information. Hong-Ou-Mandel interference of frequency entanglement with detunings of (a) \unit[1.75]{THz}
\label{figure_3}
\end{figure*}
Figure \ref{figure_3} demonstrates the explicit procedure of parameter estimation and their corresponding Fisher information in experiment, from which we see that frequency detuning can facilitate the achievement of higher resolution and precision. The oscillation of Fisher information within two-photon coherence time is a key signature of discrete frequency entanglement \cite{ramelow2009discrete}. Here the maximal Fisher information we have obtained is $\unit[245]{ps^{-2}}$ for frequency detuning of $\unit[5.34]{THz}$, which inversely reveals the highest precision of $\unit[639]{as}$, i.e., relative path delay of $\unit[192]{nm}$, for experimental trials of $O(10^4)$. It is noticed that the quadratic dependence of Fisher information as a function of frequency detuning could be used to further enhance the Fisher information with respect to the frequency degenerate case, where values of $\sim \unit[8]{ps^{-2}}$ have already been reported \cite{lyons2018attosecond}.
\textbf{Experimental application as a temperature sensor.}
\begin{figure*}
\caption{Experimental demonstration of thermal characteristics of jacket optical fiber. (a) Two-fold coincidence probability and (b) corresponding shifted phase as a function of heating temperature of sensing fiber versus different frequency detunings. The shaded regions bounded by two smoothed curves represent the standard deviation of experimental results estimated by statistical methods assuming a Poisson distribution.}
\label{Fig4.sub.1}
\label{Fig4.sub.2}
\label{figure_4}
\end{figure*}
In order to demonstrate the viability principle of employing our HOM sensor, we performed a proof of concept experiment in which we estimate the time delay due to linear expansion of a jacket optical fiber.
In order to verify the conclusion that quantum metrology based on frequency entanglement with larger frequency detuning has higher precision, we experimentally measure two-fold coincidence probabilities and predict the thermal coefficient by heating the sensing fiber to vary relative phase shifts (see Fig.\ \ref{figure_4}).
In principle, the relative phase shift varies almost linearly with fiber length and is described as $\beta=N_gkL$, where $L$ is sensing fiber length, $N_g$ is the material group index and $k$ is the light wave number \cite{lagakos1981temperature}. Since the input frequency entangled state of HOM sensor is highly sensitive to transmission time, the relative phase shift, introduced by the length extension of fiber, can be expressed as a function of heating temperature, and resulting in the thermal coefficient as
\begin{equation}\label{eq:coefficient}
\frac{d\beta}{dT}\approx(\frac{2\pi}{\lambda_s}-\frac{2\pi}{\lambda_i})\frac{dN}{dT}L_o+(\frac{2\pi}{\lambda_s}N_o^{\lambda_s}-\frac{2\pi}{\lambda_i}N_o^{\lambda_i})\frac{dL}{dT},
\end{equation}
where $\lambda_{s/i}$ is center wavelength of signal or idler photons, $N_o^{\lambda_{s/i}}$ and $L_o$ are the corresponding parameters at room temperature, $T$ is heated temperature, $\frac{dN}{dT}$ and $\frac{dL}{dT}$ are thermal coefficients of material group index and fiber length, respectively.
We notice that the thermal coefficient of shifted phase is related to frequency detuning, which agrees well with the experimental measurement results (see Fig.\ \ref{Fig4.sub.2}), and results in the coincidence probability varies as cosine function (see Fig.\ \ref{Fig4.sub.1}). The measured thermal coefficients is $\unit[0.13]{rad/deg}$, $\unit[0.2]{rad/deg}$, $\unit[0.3]{rad/deg}$ and $\unit[0.48]{rad/deg}$ for frequency detunings of $\unit[3.7]{THz}$, $\unit[7.4]{THz}$, $\unit[11.2]{THz}$ and $\unit[17.1]{THz}$, respectively. The refractive index of pure silica is wavelength dependent, and its first derivative with respect to temperature is about $\unit[1\times10^{-5}]{/deg}$ \cite{bruckner1970properties}. Then we are able to estimate the thermal coefficient of linear expansion of jacket optical fiber to be $\frac{dL}{dT} \sim \unit[4.8\times10^{-7}]{m/deg}$, which agrees well with the results reported in Ref. \cite{tateda1980thermal,priest1997thermal}. Accordingly the maximal frequency detuning that we observed in this proof-of-principle experiment enables us to achieve temperature resolution of $\unit[0.12]{deg}$.
\section{Discussion}
We have demonstrated a new approach to HOM interferometry based on discrete frequency entanglement of well separated frequency modes and detection of a beat note coincidence signal.
Previous HOM-interferometric sensing schemes required perfect frequency degenerate and ultra-broad-band SPDC emission. Any wavelength distinguishability decreases visibility of the HOM dip and correspondingly diminishes the resolution. Providing suitable quantum sources for this case is a significant challenge, as it either requires the engineering of aperiodic poling structures or the use of very short nonlinear crystals, at the cost of efficiency. In contrast, the approach outlined here requires only a sufficiently large non-degeneracy, whereby the spectral bandwidth can be small. We have experimentally demonstrated how to generate suitable discrete frequency-entangled states, in a manner that can be readily extended to larger wavelength separations. For example, $\lambda_s$= $\unit[1,500]{nm}$ and $\lambda_2$ = $\unit[800]{nm}$ ($\unit[1,000]{THz}$ angular difference frequency) a timing sensitivity of 9 as could already be obtained for only $N = 10^4$ detection events. Backed by the results of our proof-of-concept experiment, this shows that the approach can provide higher resolution and highly sensitive measurement, and makes it an ideal candidate for more quantum enhanced metrology applications.
Although this work only reports the advantages of our approach in estimating delays, similar great enhancement can also be achieved for a variety of applications like state discrimination or hypothesis testing.
In conclusion, we believe that fully harnessing HOM interference and frequency entanglement will provide additional tools, e.g. for frequency shaping of photons and interference phenomena in general, ultimately broadening the path towards practical quantum applications.
\\
\section{Methods}
\textbf{Entangled Photon source.}
In our experimental realization of flexible frequency entanglement source \cite{chen2018polarization}, two mutually orthogonally oriented 10-mm-long ppKTP crystals are manufactured to provide collinear phase matching with pump (p), signal (s) and idler (i) photons at center wavelengths of $\lambda_p \approx \unit[405]{nm}$ and $\lambda_{s,i} \approx \unit[810]{nm}$. They are pumped with a $\unit[405]{nm}$ continuous wave grating-stabilized laser diode. To achieve the desired diagonally and anti-diagonally polarization states for simplifying alignment, we design an oven with V-groove such that two crossed crystals are oriented at $\unit[45]{^\circ}$. Since the pump beam is horizontally polarized, it is equally likely to generate a photon pair in the first or second crystal, resulting in a state of equation \eqref{eq:frequency state}. The relative phase factor is compensated by tilting a half wave plate. Long pass filter is set to block pump beam. Then PBS routes a pair of photons into two distinct spatial modes according to orthogonal polarizations.
\textbf{HOM interferometer.}
In spatial mode 1, a translation stage introduces a relative path delay to accomplish the task of scanning HOM interference fringes. Polarization controllers are required to compensate polarization difference of biphotons such that only frequency entanglement can make contributions to the interference effect. Finally the anti-bunched photons are detected by silicon avalanche photon diodes, and two-fold events are identified using a fast electronic AND gate when two photons arrive at the detectors within a coincidence window of $\sim \unit[3]{ns}$.
\textbf{Coincidence signal with frequency-entangled states.}
An optimal measurement procedure may allow us to saturate the limit set by equation \eqref{eq:cramer rao bound}. In the ideal, lossless with perfect visibility, case, such a measurement can be accomplished by interference on a balanced beam splitter. The beam splitter transformation on the input modes can be expressed by
\begin{equation}
\begin{split}
\hat{a}_1^\dag(\omega_1)=\frac{1}{\sqrt{2}}[\hat{a}_3^\dag(\omega_1)+\hat{a}_4^\dag(\omega_1)]\\
\hat{a}_2^\dag(\omega_2)=\frac{1}{\sqrt{2}}[\hat{a}_3^\dag(\omega_2)-\hat{a}_4^\dag(\omega_2)],
\end{split}
\end{equation}
where $\omega_1$ and $\omega_2$ denote the signal or idler frequency mode that are incident from opposite ports, and subscripts 1/2 (3/4) represent two input (output) ports of that beam splitter. Accordingly the state is transformed as
\begin{equation}
\ket{\Psi(\tau)}\rightarrow\ket{\Psi_A(\tau)}+\ket{\Phi(\tau)},
\end{equation}
where these state contributions can be expressed as
\begin{equation}
\begin{split}
\ket{\Psi_A(\tau)}=&\frac{1}{2}\int d\Omega f(\Omega)(1+e^{-i\Delta\tau})\\
&[\hat{a}_3^\dag(\omega_1^0+\Omega)\hat{a}_4^\dag(\omega_2^0-\Omega)\\
&-\hat{a}_3^\dag(\omega_2^0-\Omega)\hat{a}_4^\dag(\omega_1^0+\Omega)]\ket{vac}\\
\ket{\Phi(\tau)}=&\frac{1}{2}\int d\Omega f(\Omega)(1-e^{-i\Delta\tau})\\
&[\hat{a}_3^\dag(\omega_1^0+\Omega)\hat{a}_3^\dag(\omega_2^0-\Omega)\\
&-\hat{a}_4^\dag(\omega_1^0+\Omega)\hat{a}_4^\dag(\omega_2^0-\Omega)]\ket{vac}.
\end{split}
\end{equation}
Due to HOM interference on the beam splitter coincidence detection in distinct spatial modes projects onto the state component $\ket{\Psi_A(\tau)}$.
\textbf{Fisher information.}
In a specific experiment (measurement strategy), with outcomes $x_i$, and corresponding probability distributions $P_i(\tau )$, any unbiased estimator will fulfill equation \eqref{eq:cramer rao bound}, where the Fisher information $F_\tau$ quantifies the information that a particular measurement can reveal about the unknown parameter of interest. Note that optimizing over all probability distributions results we recover the QCR bound. The outcomes of this measurement are sufficient to obtain an estimator for the value of $\tau$.
By substituting equation \eqref{eq: Fisher information} with the corresponding probabilities from equation \eqref{eq:loss probability}, we could calculate the Fisher information as
\begin{equation}
\begin{split}
F_\tau=\frac{(1-\gamma^2)[\alpha\Delta sin(\Delta\tau)+4\alpha\sigma^2\tau cos(\Delta\tau)]^2e^{-4\sigma^2\tau^2}}{4P_1(\tau)P_2(\tau)/(1-\gamma)^4}.
\end{split}
\end{equation}
We note that the Fisher information is undefined at position of $\tau = 0$ in ideal case since the denominator will be zero.
\iffalse
\textbf{Sensitivity.}
In a particular measurement, we measure an observable $\hat{S}$ and the mean value of the operator $\langle\hat{S}\rangle$ that depends on $\tau$ to predict the value of target parameter. The precision of estimation $\delta\tau$ can be calculated with the help of the calculus of error propagation as
\begin{equation}\label{eq:sensitivity}
\begin{split}
\delta\tau=\sqrt{\langle(\Delta\hat{S})^2\rangle}/|\frac{\partial\langle\hat{S}\rangle}{\partial\tau}|,
\end{split}
\end{equation}
where
\begin{equation}
\langle(\Delta\hat{S})^2\rangle=\langle\hat{S}^2\rangle-\langle\hat{S}\rangle^2,
\end{equation}
the numerator is the variance of the operator and the denominator is the sensitivity of mean value of the operator to target parameter $\tau$. By combining equation \eqref{eq:loss probability} and equation \eqref{eq:sensitivity}, we can calculate the general form of sensitivity in a particular measurement as
\begin{equation}\label{eq:calculated sensitivity}
\begin{split}
\delta\tau=&\{[1+\alpha cos(\Delta\tau)e^{-2\sigma^2\tau^2}]\\
&\times[2-(1-\gamma)^2(1+\alpha cos(\Delta\tau)e^{-2\sigma^2\tau^2})]\}^{1/2}\\
&/(1-\gamma)[\alpha\Delta sin(\Delta\tau)+4\alpha cos(\Delta\tau)\sigma^2\tau]e^{-2\sigma^2\tau^2}.
\end{split}
\end{equation}
In ideal case ($\gamma=0$, $\alpha=1$), we can obtain the lower bound given if we go to $\tau\rightarrow0$ as
\begin{equation}
\label{eq: maximal sensitivity}
\begin{split}
\lim_{\tau\rightarrow0}\delta\tau=\frac{1}{(\Delta^2+4\sigma^2)^{1/2}}.
\end{split}
\end{equation}
CR bound represents the ultimate limit of sensitivity that can be achieved, which means any experiment that we could perform cannot provide a better result than this.
\fi
\textbf{Maximum-likelihood estimator.} Since no prior knowledge is provided, we can apply maximum likelihood estimation approach to predict the target parameter. We extremized the likelihood function as
\begin{equation}
\begin{split}
0&=:(\partial_\tau log\mathcal{L})_{\tilde{\tau}_{MLE}}=\frac{N_1P_1(\tau)^\prime}{P_1(\tau)}+\frac{N_2P_2(\tau)^\prime}{P_2(\tau)}.
\end{split}
\end{equation}
Based on the calculation in equation \eqref{eq:loss probability}, we know $P_1(\tau)^\prime=-P_2(\tau)^\prime$ such that
\begin{equation}
N_1P_2(\tau)|_{\tilde{\tau}_{MLE}}=N_2P_1(\tau)|_{\tilde{\tau}_{MLE}}.
\end{equation}
For the sake of simplicity, $\tilde{\tau}$ in term of $e^{-2\sigma^2\tau^2}$ can be considered as a constant value, i.e., coarse sensing position $\tau_s$ where Fisher information is highest. Thus we get an optimal estimator to variable relative time delay as
\begin{equation}
\begin{split}
\tilde{\tau}_{MLE}=\arccos(\frac{\frac{1+3\gamma}{1-\gamma}N_2-N_1}{\alpha(N_1+N_2)e^{-2\sigma^2\tau_s^2}})/\Delta,
\end{split}
\end{equation}
and the values of parameters $\tau_s$, $\gamma$, $\alpha$, $\sigma$ and $\Delta$ need to be separately estimated before the measurements begin.
\textbf{Shifted phase of temperature sensor.}
The introduced phase shift in frequency entangled state can be expressed as a function of heating temperature as
\begin{equation}\label{eq:linear phase}
\begin{split}
\beta=&\frac{2\pi}{\lambda_s}(\frac{dN}{dT}L_o+\frac{dL}{dT}N_o^{\lambda_s}+\frac{dN}{dT}\frac{dL}{dT}T)T\\
&-\frac{2\pi}{\lambda_i}(\frac{dN}{dT}L_o+\frac{dL}{dT}N_o^{\lambda_i}+\frac{dN}{dT}\frac{dL}{dT} T)T.
\end{split}
\end{equation}
Since $\frac{dN}{dT}\frac{dL}{dT}$ is in the order of much smaller magnitude, we ignore the term of $\frac{dN}{dT}\frac{dL}{dT}T$ in equation \eqref{eq:linear phase}.
\textbf{See supplementary materials for more information.}
\section*{Data availability}
Data available on request from the authors.
\section*{Contributions}
F.S. developed the initial idea for this work. Y.C. conducted the experiment under supervision from F.S. and R.U. Theoretical analysis was carried out by J.T. and Y.C.. Y.C. and F.S. wrote the first draft and all authors contributed to the final version of the manuscript.
\section*{Competing interests}
The authors declare that there are no competing interests.
\end{document} |
\begin{document}
\title{Existence of weak solutions to
stochastic heat equations driven
by truncated $lpha$-stable white noises with
non-Lipschitz coefficients}
\begin{abstract}
We consider a class of stochastic heat
equations driven by truncated $\alpha$-stable
white noises for $\alpha\in(1,2)$ with noise coefficients that are
continuous but not necessarily Lipschitz and satisfy globally linear growth conditions.
We prove the existence of weak solution, taking values in two different forms under different conditions, to such an equation using a weak
convergence argument on solutions to the approximating stochastic heat equations.
More precisely,
for $\alpha\in(1,2)$ there exists a measure-valued weak solution.
However,
for $\alpha\in(1,5/3)$ there exists
a function-valued weak solution,
and in this case
we further show that for $p\in(\alpha,5/3)$
the uniform $p$-th moment in $L^p$-norm of the weak solution is finite,
and that the weak solution is uniformly stochastic
continuous in $L^p$ sense.
\noindent\textbf{Keywords:}
Non-Lipschitz noise coefficients;
Stochastic heat equations;
Truncated $\alpha$-stable white noises;
Uniform $p$-th moment;
Uniform stochastic continuity.
\noindent{{\bf MSC Classification (2020):} Primary: 60H15;
Secondary: 60F05, 60G17}
\end{abstract}
\section{Introduction}
\label{sec1}
In this paper we study the existence of weak solution to the
following non-linear stochastic heat equation
\begin{equation}
\label{eq:originalequation1}
\left\{\begin{array}{lcl}
\dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2}
\dfrac{\partial^2u(t,x)}{\partial x^2}+
\varphi(u(t-,x))\dot{L}_{\alpha}(t,x),
&& (t,x)\in (0,\infty)
\times(0,L),\\[0.3cm]
u(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm]
u(t,0)=u(t,L)=0,&& t\in[0,\infty),
\end{array}\right.
\end{equation}
where $L$ is an arbitrary positive constants,
$\dot{L}_{\alpha}$
denotes a truncated $\alpha$-stable white
noise on $[0,\infty)\times[0,L]$ with $\alpha\in(1,2)$,
the noise coefficient
$\varphi:\mathbb{R}\rightarrow \mathbb{R}$
satisfies the hypotheses given below,
and the initial function
$u_0$ is random and measurable.
Before studying the equation
of particular form (\ref{eq:originalequation1}), we first consider
a general stochastic heat equation
\begin{equation}
\label{GeneralSPDE}
\dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2}\dfrac{\partial^2u(t,x)}{\partial
x^2}+G(u(t,x))+H(u(t,x))\dot{F}(t,x),
\quad t\geq0, x\in\mathbb{R},
\end{equation}
in which $G:\mathbb{R}\rightarrow \mathbb{R}$
is Lipschitz continuous, $H:\mathbb{R}\rightarrow
\mathbb{R}$
is continuous and $\dot{F}$ is a
space-time white noise.
When $\dot{F}$ is a Gaussian
white noise, there is a growing literature on stochastic partial
differential equations (SPDEs for short) related to (\ref{GeneralSPDE})
such as the
stochastic Burgers equations (see, e.g., Bertini and Cancrini \cite{Bertini:1994}, Da Prato et al. \cite{Daprato:1994}), SPDEs with reflection
(see, e.g., Zhang \cite{Zhang:2016}),
Parabolic Anderson Model (see, e.g., {G\"{a}rtner and Molchanov \cite{Gartner:1990}), etc.
In particular, such a SPDE arises from super-processes (see, e.g., Konno and Shiga \cite{Konno:1988}, Dawson \cite{Dawson:1993} and Perkins \cite{P1991} and references therein). For $G\equiv 0$ and
$H(u)=\sqrt{u}$, the solution to (\ref{GeneralSPDE}) is the density field of a one-dimensional super-Brownian motion.
For $H(u)=\sqrt{u(1-u)}$ (stepping-stone model in population genetics),
Bo and Wang \cite{Bo:2011}
considered a stochastic interacting model
consisting of equations (\ref{GeneralSPDE})
and proved the existence of weak solution
to the system by using a weak convergence argument.
In the case that
$\dot{F}$
is a Gaussian colored noise that is white in time and colored in space, for continuous function
$H$ satisfying the linear growth condition,
Sturm \cite{Sturm:2003} proved the existence of a pair $(u,F)$ satisfying
(\ref{GeneralSPDE}), the so-called weak
solution, by first establishing the existence and uniqueness of lattice systems of SDEs driven by correlated Brownian motions with non-Lipschitz diffusion coefficients that describe branching particle systems in random environment in which the motion process has a discrete Laplacian generator and the branching mechanism is affected by a colored Gaussian random field,
and then applying an approximation procedure.
Xiong and Yang \cite{Xiong:2023}
proved the existence of weak solution
$(u,F)$ to (\ref{GeneralSPDE}) in a finite spatial domain with different boundary conditions
by considering the weak
limit of a sequence of approximating SPDEs of (\ref{GeneralSPDE}). They further
proved the existence and uniqueness of the strong solution under additional H\"{o}lder continuity assumption on $H$.
If $\dot{F}$ is a L\'{e}vy space-time white noise
with Lipschitz continuous coefficient $H$,
Albeverio et al. \cite{Albeverio:1998}
first proved the existence and uniqueness of the solution when
$\dot{F}$ is a Poisson white noise.
Applebaum and Wu \cite{Applebaum:2000} extended
the results to a general L\'{e}vy space-time
white noise. For a stochastic fractional Burgers type non-linear equation that is similar to equation (\ref{GeneralSPDE}) and driven by L\'{e}vy space-time white noise on multidimensional space variables, we refer to Wu and Xie \cite{Wu:2012} and references therein.
In particular, when $\dot{F}$ is
an $\alpha$-stable white noise for $\alpha\in (0,1)\cup(1,2)$,
Balan \cite{Balan:2014} studied SPDE
(\ref{GeneralSPDE}) with $G\equiv 0$ and Lipschitz coefficient $H$ on a bounded domain in $\mathbb{R}^d$ with
zero initial condition and Dirichlet
boundary, and proved the existence
of random field solution $u$ for the given noise $\dot{F}$
(the so-called strong solution). The approach in \cite{Balan:2014} is
to first solve the equation with truncated noise
(by removing the big jumps, the jumps size exceeds
a fixed value $K$, from $\dot{F}$),
yielding a solution $u_K$,
and then show that for $N\geq K$
the solutions $u_N=u_K$
on the event $t\leq\tau_K$, where
$\{\tau_K\}_{K\geq1}$ is a sequence of stopping
times which tends to infinity
as $K$ tends to infinity.
Such a localization method
which is also applied in
Peszat and Zabczyk
\cite{Peszat:2006} to show the existence of weak Hilbert-space valued solution.
For $\alpha\in(1,2)$, Wang et al. \cite{Wang:2023} studied the existence and pathwise uniqueness of strong function-valued solution of (\ref{GeneralSPDE}) with Lipschitz coefficient $H$ using a localization method, and showed a comparison principle of solutions to such equation with different initial functions and drift coefficients.
Yang and Zhou \cite{Yang:2017} found sufficient conditions on
pathwise uniqueness of solutions to a class of SPDEs (\ref{GeneralSPDE}) driven by $\alpha$-stable white noise without negative jumps and with non-decreasing H\"{o}lder continuous noise coefficient $H$. But the existence of weak solution to (\ref{GeneralSPDE}) with general non-decreasing H\"{o}lder continuous noise coefficient is left open.
For stochastic heat equations driven by general heavy-tailed
noises with Lipschitz noise coefficients, we refer to Chong \cite{Chong:2017} and references therein.
When $G=0$, $H(u)=u^{\beta}$ with $0<\beta<1$
(non-Lipschitz continuous) in (\ref{GeneralSPDE}) and
$\dot{F}$ is an $\alpha$-stable ($\alpha\in(1,2)$) white noise on
$[0,\infty)\times\mathbb{R}$ without negative
jumps, it is shown in
Mytnik \cite{Mytnik:2002}
that for $0<\alpha\beta<3$
there exists a weak solution $(u,F)$ satisfying
(\ref{GeneralSPDE}) by constructing a sequence of approximating processes that is tight with its limit solving the associated martingale problem,
and that in the case of $\alpha\beta=1$
the weak uniqueness
of solution to (\ref{GeneralSPDE}) holds.
The pathwise uniqueness is shown in \cite{Yang:2017}
for $\alpha\beta=1$ and $1<\alpha<\sqrt{5}-1$.
For $\alpha$-stable colored noise $\dot{F}$ without negative jumps and with H\"{o}lder continuous coefficient $H$, Xiong and Yang \cite{Xiong:2019} proved the existence of week solution $(u,F)$ to (\ref{GeneralSPDE}) by showing the weak convergence of solutions to SDE systems on rescaled lattice with discrete Laplacian and driven by common stable random measure, which is similar to \cite{Sturm:2003}.
In both \cite{Sturm:2003} and \cite{Xiong:2019} the dependence of colored noise helps with establishing the existence of weak solution.
Inspired by work in the above mentioned literature,
we are interested in the stochastic heat equation
(\ref{eq:originalequation1}) in which
the noise coefficient $\varphi$
satisfies the following more general hypothesis:
\begin{hypothesis}
\label{Hypo}
$\varphi:\mathbb{R}\rightarrow \mathbb{R}$
is continuous and globally linear growth,
and there exists a sequence of Lipschitz
continuous functions
$\varphi^n:\mathbb{R}\rightarrow \mathbb{R}$
such that
\begin{itemize}
\item[(i)] $\varphi^n$ uniformly converges to
$\varphi$ as $n\rightarrow\infty$;
\item[(ii)] for each $n\geq1$, there exists a
constant $C_n$ such that
$$\vert\varphi^n(x)-\varphi^n(y)\vert
\leq C_n\vert x-y\vert,\,\,\forall x,y\in\mathbb{R};$$
\end{itemize}
\end{hypothesis}
The main contribution of this paper
is to prove the
existence and regularity of weak solutions to equation
(\ref{eq:originalequation1}) under Hypothesis \ref{Hypo}. To this end, we consider two types of weak solutions that are
measure-valued and function-valued, respectively.
In addition,
we also study the uniform $p$-moment and
uniform stochastic continuity
of the weak solution to
equation (\ref{eq:originalequation1}).
In the case that
$\varphi$ is Lipschitz continuous, the existence
of the solution can be usually obtained by standard
Picard iteration
(see, e.g., Dalang et al.
\cite{Dalang:2009}, Walsh \cite{Walsh:1986})
or Banach fixed point principle (see, e.g.,
Truman and Wu \cite{Truman:2003},
Bo and Wang \cite{Bo:2006}).
We thus mainly consider the case
that $\varphi$ is non-Lipschitz continuous.
Since the classical approaches of
Picard iteration and Banach fixed point principle
fail for SPDE (\ref{eq:originalequation1}) with non-Lipschitz
$\varphi$,
to prove the existence of a weak solution
$(u,L_{\alpha})$ to
(\ref{eq:originalequation1}), we first construct
an approximating SPDE sequence with
Lipschitz continuous noise coefficients
$\varphi^n$, and prove the existence and
uniqueness of strong solutions
to the approximating SPDEs.
We then proceed to show that
the sequence of solution is tight
in appropriate spaces. Finally, we prove that
there exists a weak solution of
(\ref{eq:originalequation1}) by using a weak
convergence procedure.
The rest of this paper is organized as follows.
In the next section, we introduce some notation
and the main theorems on the existence, uniform $p$-moment
and uniform stochastic continuity of weak solution to
(\ref{eq:originalequation1}).
Section \ref{sec3} is devoted to the proof of the existence of
measure-valued weak solution to (\ref{eq:originalequation1}).
In Section \ref{sec4}, for $\alpha\in(1,5/3)$
we prove that there exists a weak solution to (\ref{eq:originalequation1}) as an
$L^p$-valued process
with $p\in(\alpha,5/3)$,
and that the weak solution
has the finite uniform $p$-th moment
and the uniform stochastic continuity in the $L^p$ norm with $p\in(\alpha,5/3)$.
\section{Notation and main results}
\label{sec2}
\subsection{Notation}
\label{sec2.1}
Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq0}, \mathbb{P})$ be a complete probability
space with filtration
$(\mathcal{F}_t)_{t\geq0}$ satisfying the usual
conditions, and let
$
N(dt,dx,dz): [0,\infty)\times[0,L]\times
\mathbb{R}\setminus\{0\}\rightarrow \mathbb{N}
\cup\{0\}\cup\{\infty\}
$
be a Poisson random measure on
$(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq0},
\mathbb{P})$ with intensity measure
$dtdx\nu_{\alpha}(dz)$, where $dtdx$ denotes the Lebesgue measure on $[0,\infty)\times[0,L]$ and the
jump size measure $\nu_{\alpha}(dz)$ for $ \alpha\in(1,2) $ is given by
\begin{align}
\label{eq:smalljumpsizemeasure}
\nu_{\alpha}(dz):=(c_{+}z^{-\alpha-1}1_{(0,K]}(z)+c_{-}(-z)^{-\alpha-1}1_{[-K,0)}(z)
)dz,
\end{align}
where $c_{+}+c_{-}=1$ and
$K>0$ is an arbitrary constant.
Define
\begin{align*}
\tilde{N}(dt,dx,dz):=
N(dt,dx,dz)-dtdx\nu_{\alpha}(dz).
\end{align*}
Then $\tilde{N}(dt,dx,dz)$
is the compensated Poisson random measure
(martingale measure)
on $[0,\infty)\times[0,L]\times
\mathbb{R}\setminus\{0\}$.
As in Balan \cite[Section 5]{Balan:2014},
define a martingale measure
\begin{align}
\label{def:stablenoise}
L_{\alpha}(dt,dx):=
\int_{\mathbb{R}\setminus\{0\}}z\tilde{N}(dt,dx,dz)
\end{align}
for $(t, x)\in [0,\infty)\times[0,L]$.
Then the corresponding distribution-valued derivative
$\{\dot{L}_{\alpha}(t,x):t\in[0,\infty),x\in[0,L]\}$
is a truncated $\alpha$-stable
white noise.
Write $\mathcal{G}^{\alpha}$ for the class of almost
surely $\alpha$-integrable random functions defined by
\begin{align*}
\mathcal{G}^{\alpha}:=\left\{f\in\mathbb{B}:
\int_0^t\int_0^L\vert f(s,x)\vert ^{\alpha}dxds<\infty,
\mathbb{P}\text{-a.s.}\,\,\text{for all} \,\,t\in[0,\infty)\right\},
\end{align*}
where $\mathbb{B}$ is the space of progressively
measurable functions on
$[0,\infty)\times[0,L]\times\Omega$. Then it
holds by Mytnik \cite[Section 5]{Mytnik:2002} that the
stochastic integral with respect to
$\{L_{\alpha}(dx,ds)\}$
is well defined for all
$f\in \mathcal{G}^{\alpha}$.
Throughout this paper, $C$ denotes the arbitrary
positive constant whose value might vary
from line to line.
If $C$ depends on some parameters such as $p,T$,
we denote it by $C_{p,T}$.
Let $G_t(x,y)$ be the fundamental
solution of heat equation
$\frac{\partial u}{\partial t}
=\frac{1}{2}\frac{\partial^2 u}{\partial x^2}$
on the domain $[0,\infty)\times[0,L]\times[0,L]$
with Dirichlet boundary conditions
(the subscript $t$ is not a derivative but a variable).
Its explicit formula
(see, e.g., Feller \cite[Page 341]{Feller:1971})
is given by
\begin{equation*}
G_t(x,y)=\dfrac{1}{\sqrt{2\pi t}}\sum_{k=-\infty}^{+\infty}\left\{
\exp\left(-\dfrac{(y-x+2kL)^2}{2t}\right)
-\exp\left(-\dfrac{(y+x+2kL)^2}{2t}
\right)\right\}
\end{equation*}
for $t\in(0,\infty),x,y\in[0,L]$; and $\lim_{t\downarrow0}G_t(x,y)=\delta_y(x)$,
where
$\delta$ is the Dirac delta distribution.
Moreover, it holds by Xiong and Yang
\cite[Lemmas 2.1-2.3]{Xiong:2023}
that for $s,t\in[0,\infty)$ and $x,y,z\in[0,L]$
\begin{equation}
\label{eq:Greenetimation0}
G_t(x,y)=G_t(y,x),\,\,
\int_0^L|G_t(x,y)|dy+\int_0^L|G_t(x,y)|dx\leq C,
\end{equation}
\begin{equation}
\label{eq:Greenetimation1}
\int_0^LG_s(x,y)G_t(y,z)dy=G_{t+s}(x,z),
\end{equation}
\begin{equation}
\label{eq:Greenetimation2}
\int_0^L\vert G_t(x,y)\vert ^pdy\leq Ct^{-\frac{p-1}{2}},\,\, p\geq1.
\end{equation}
Given a topological space $V$,
let $D([0,\infty),V)$
be the space of c\`{a}dl\`{a}g paths from
$[0,\infty)$
to $V$ equipped with the Skorokhod topology.
For given $p\geq1$
we denote by $v_t\equiv\{v(t,\cdot),t\in[0,\infty)\}$
the $L^p([0,L])$-valued
process equipped with norm
\begin{equation*}
\vert\vert v_t\vert\vert_{p}=\left(\int_0^L\vert v(t,x)\vert^pdx\right)^{\frac{1}{p}}.
\end{equation*}
For any $p\geq1$ and $T>0$
let $L_{loc}^p([0,\infty)\times[0,L])$ be the
space of measurable functions $f$ on
$[0,\infty)\times[0,L])$ such that
\begin{equation*}
\vert \vert f\vert \vert _{p,T}=\left(\int_0^T\int_0^L
\vert f(t,x)\vert ^pdxdt\right)^{\frac{1}{p}}<\infty,\,\,
\forall\,\, 0<T<\infty.
\end{equation*}
Let $B([0,L])$ be the space of all Borel functions on $[0,L]$, and
let $\mathbb{M}([0,L])$ be the space of finite
Borel measures on $[0,L]$ equipped with
the weak convergence topology.
For any $f\in B([0,L])$ and
$\mu\in\mathbb{M}([0,L])$ define
$
\langle f,\mu\rangle:=\int_0^L f(x)\mu(dx)
$
whenever it exists.
With a slight abuse of notation, for any $f,g\in B([0,L])$ we also denote by
$
\langle f,g\rangle=\int_0^L f(x)g(x)dx.
$
\subsection{Main results}
\label{sec2.2}
By a solution to
equation (\ref{eq:originalequation1})
we mean a process $u_t\equiv\{u(t,\cdot),t\in[0,\infty)\}$ satisfying the
following weak (variational) form equation:
\begin{align}
\label{eq:variationform}
\langle u_t,\psi\rangle
&=\langle u_0,\psi\rangle+\dfrac{1}{2}\int_0^t
\langle u_s,\psi{''}\rangle ds
+\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\varphi(u(s-,x))\psi(x)z\tilde{N}(ds,dx,dz)
\end{align}
for all $t\in[0,\infty)$ and for any $\psi\in C^{2}([0,L])$
with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ or equivalently
satisfying the
following mild form equation:
\begin{align}
\label{eq:mildform}
u(t,x)=
\int_0^LG_t(x,y)u_0(y)dy
+\int_0^{t+}\int_0^L
\int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y)
\varphi(u(s-,y))z\tilde{N}(ds,dy,dz)
\end{align}
for all $t\in [0, \infty)$ and
for a.e. $x\in [0,L]$, where the last terms in above equations follow from
(\ref{def:stablenoise}).
For the equivalence between the weak form (\ref{eq:variationform}) and mild form (\ref{eq:mildform}),
we refer to Walsh \cite{Walsh:1986} and references therein.
We first give the definition
(see also in Mytnik \cite{Mytnik:2002})
of a weak solution
to stochastic heat equation (\ref{eq:originalequation1}).
\begin{definition}
Stochastic heat equation (\ref{eq:originalequation1})
has a weak solution with initial function $u_0$
if there exists a pair $(u,L_{\alpha})$
defined on some filtered probability space such that
${L}_{\alpha}$ is a truncated $\alpha$-stable martingale measure on
$[0,\infty)\times[0,L]$ and
$(u,L_{\alpha})$ satisfies either
equation (\ref{eq:variationform})
or equation (\ref{eq:mildform}).
\end{definition}
We now state the main theorems in this paper. The first theorem is on the existence of weak solution in $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ with $p\in(\alpha,2]$
that was first considered in
Mytnik \cite{Mytnik:2002}.
\begin{theorem}
\label{th:mainresult}
If the initial function $u_0$
satisfies $\mathbb{E}[\vert\vert u_0\vert\vert_p^p]<\infty$
for some $p\in(\alpha,2]$, then under
{\rm Hypothesis \ref{Hypo}}
there exists a weak solution
$(\hat{u}, {\hat{L}}_{\alpha})$ to equation (\ref{eq:originalequation1})
defined on a filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0},
\hat{\mathbb{P}})$ such that
\begin{itemize}
\item[\rm (i)]
$\hat{u}\in
D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$;
\item[\rm (ii)] ${\hat{L}}_{\alpha}$ is a truncated
$\alpha$-stable martingale measure
with the same distribution as ${L}_{\alpha}$.
\end{itemize}
Moreover, for any $T>0$ we have
\begin{equation}
\label{eq:momentresult}
\hat{\mathbb{E}}\left[\vert\vert\hat{u}\vert\vert_{p,T}^p\right]=
\hat{\mathbb{E}}\left[\int_0^T\vert\vert\hat{u}_t\vert\vert_p^pdt\right]<\infty.
\end{equation}
\end{theorem}
The proof of Theorem \ref{th:mainresult} is deferred to Section \ref{sec3}.
Under additional assumption on $\alpha$,
we can show that there exists a weak solution in
$D([0,\infty),L^p([0,L]))$,
$p\in(\alpha,5/3)$
with better regularity.
\begin{theorem}
\label{th:mainresult2}
Suppose that $\alpha\in (1,5/3)$.
If the initial function $u_0$
satisfies $\mathbb{E}[||u_0||_p^p]<\infty$
for some $p\in(\alpha,5/3)$,
then under {\rm Hypothesis \ref{Hypo}}
there exists a weak solution
$(\hat{u}, {\hat{L}}_{\alpha})$ to equation (\ref{eq:originalequation1})
defined on a filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0},
\hat{\mathbb{P}})$ such that
\begin{itemize}
\item[\rm (i)]
$\hat{u}\in D([0,\infty),L^p([0,L]))$;
\item[\rm (ii)] ${\hat{L}}_{\alpha}$ is a
truncated
$\alpha$-stable martingale measure
with the same distribution as ${L}_{\alpha}$.
\end{itemize}
Furthermore, for any $T>0$ we have the following uniform $p$-moment and uniform stochastic continuity, that is,
\begin{equation}
\label{eq:momentresult2}
\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert\vert\hat{u}_t\vert\vert_p^p\right]<\infty,
\end{equation}
and that
for each $0\leq h\leq\delta$
\begin{equation}
\label{eq:timeregular}
\lim_{\delta\rightarrow0}\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\vert\vert
\hat{u}_{t+h}-\hat{u}_t\vert\vert_p^p\right]=0.
\end{equation}
\end{theorem}
The proof of Theorem \ref{th:mainresult2} is deferred to Section \ref{sec4}.
We present a specific
stochastic heat equation to illustrate our
results in the following example.
\begin{example}
Given $0<\beta<1$,
consider the equation (\ref{eq:originalequation1})
with $\varphi(u)=\vert u\vert^{\beta}$ for $u\in\mathbb{R}$,
that is,
\begin{equation*}
\label{eq:example}
\left\{\begin{array}{lcl}
\dfrac{\partial u(t,x)}{\partial t}=\dfrac{1}{2}
\dfrac{\partial^2u(t,x)}{\partial x^2}+
\vert u(t-,x)\vert^{\beta}\dot{L}_{\alpha}(t,x),
&& (t,x)\in (0,\infty)
\times(0,L),\\[0.3cm]
u(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm]
u(t,0)=u(t,L)=0,&& t\in[0,\infty).
\end{array}\right.
\end{equation*}
It is clear that $\vert u \vert^{\beta}$ is a non-Lipschitz
continuous function with globally linear growth.
We can construct
a sequence of
Lipschitz continuous
functions $(\varphi^n)_{n\geq1}$
of the form
\begin{equation*}
\varphi^n(u)=(\vert u \vert\vee\varepsilon_n)^{\beta},
u\in\mathbb{R},
\end{equation*}
where $\varepsilon_n\downarrow0$ as
$n\uparrow\infty$,
such that $\varphi^n$ satisfies {\rm Hypothesis \ref{Hypo}}. One can then apply {\rm Theorems \ref{th:mainresult} and \ref{th:mainresult2}} to establish the existence of weak solutions to the above stochastic heat equation.
\end{example}
Finally, we provide some discussions on our main results in the following remarks.
\begin{remark}
Note that the globally linear growth of
$\varphi$
in {\rm Hypothesis \ref{Hypo}} guarantees the global existence
of weak solutions. One can remove this condition
if one only needs the existence of a weak solution up to the
explosion time.
On the other hand, the uniqueness of the solution to equation (\ref{eq:originalequation1}) is still an open problem because
$\varphi$ is non-Lipschitz continuous.
\end{remark}
\begin{remark}
The weak solutions of equation (\ref{eq:originalequation1}) in {\rm Theorems \ref{th:mainresult}} and {\rm\ref{th:mainresult2}} are proved by showing the tightness of the approximating solution sequence
$(u^n)_{n\geq1}$ of equation (\ref{eq:approximatingsolution}); see
{\rm Propositions \ref{th:tightnessresult}} and {\rm\ref{prop:tightnessresult2}}
in {\rm Sections \ref{sec3}} and {\rm \ref{sec4}}, respectively.
To show that the equation (\ref{eq:originalequation1}) has a function-valued
weak solution, it is necessary to restrict
$\alpha\in(1,5/3)$ due to a technical reason
that Doob's maximal inequality can not be directly applied to show the uniform $p$-moment estimate of
$(u^n)_{n\geq1}$ that is key to the proof
of the tightness for $(u^n)_{n\geq1}$. To this end,
we apply the factorization method in {\rm Lemma
\ref{lem:uniformbound}} for transforming the stochastic integral such that the uniform $p$-moment of $(u^n)_{n\geq1}$ can be obtained.
In order to remove this restriction and consider the case of $\alpha\in(1,2)$, we apply
another tightness criteria, i.e.,
{\rm Lemma \ref{lem:tightcriterion0}}, to show the
tightness of $(u^n)_{n\geq1}$. However,
the weak solution of equation (\ref{eq:originalequation1}) is a
measure-valued process in this situation.
We also note that
the existence of function-valued
weak solution of equation (\ref{eq:originalequation1}) in the case of
$\alpha\in[5/3,2)$ is still an unsolved problem.
\end{remark}
\begin{remark}
If we remove the restriction of the bounded jumps for the $\alpha$-stable white noise
$\dot{L}_{\alpha}$ in
equation (\ref{eq:originalequation1}),
the jump size measure
$\nu_{\alpha}(dz)$ in (\ref{eq:smalljumpsizemeasure})
becomes
\begin{align*}
\nu_{\alpha}(dz)=(c_{+}z^{-\alpha-1}1_{(0,\infty)}(z)+c_{-}(-z)^{-\alpha-1}1_{(-\infty,0)}(z)
)dz
\end{align*}
for $\alpha\in(1,2)$ and $c_{+}+c_{-}=1$.
As in Wang et al. \cite[Lemma 3.1]{Wang:2023}
we can construct a sequence of truncated
$\alpha$-stable white noise $\dot{L}^K_{\alpha}$
with the jumps size measure given by
(\ref{eq:smalljumpsizemeasure}) and a sequence
of stopping times $(\tau_K)_{K\geq1}$ such that
\begin{equation}
\label{eq:stoppingtimes1}
\lim_{K\rightarrow+\infty}
\tau_K=\infty,\,\,\mathbb{P}\text{-a.s.}.
\end{equation}
Similar to equation (\ref{eq:originalequation1}),
for given $K\geq1$,
we can
consider the following non-linear stochastic heat equation
\begin{equation}
\label{eq:truncatedSHE}
\left\{\begin{array}{lcl}
\dfrac{\partial u_K(t,x)}{\partial t}=\dfrac{1}{2}
\dfrac{\partial^2u_K(t,x)}{\partial x^2}+
\varphi(u_K(t-,x))\dot{L}^K_{\alpha}(t,x),
&& (t,x)\in (0,\infty)
\times(0,L),\\[0.3cm]
u_K(0,x)=u_0(x),&&x\in[0,L],\\[0.3cm]
u_K(t,0)=u_K(t,L)=0,&& t\in[0,\infty).
\end{array}\right.
\end{equation}
If $\varphi$ is Lipschitz continuous,
similar to the proof of
{\rm Proposition \ref{th:Approximainresult}} in
Wang et al. \cite[Proposition 3.2]{Wang:2023}
one can show that there exists a unique strong solution
$u_K=\{u_K(t,\cdot),t\in[0,\infty)\}$ to equation
(\ref{eq:truncatedSHE}) by using the
Banach fixed point principle. On the other hand,
by Wang et al. \cite[Lemma 3.4]{Wang:2023},
it holds for each $K\leq N$ that
\begin{align*}
u_K=u_N
\,\,\mathbb{P}\text{-} \,a.s.
\,\,\text{on}\,\{t<\tau_K\}.
\end{align*}
By setting
\begin{align*}
u=u_K,\,0\leq t<\tau_K,
\end{align*}
and by the fact (\ref{eq:stoppingtimes1}),
we obtain the strong (weak) solution
$u$ to equation
(\ref{eq:originalequation1}) with noise of
unbounded
jumps via letting $K\uparrow+\infty$.
If $\varphi$ is non-Lipschitz continuous,
for any $K\geq1$,
{\rm Theorem \ref{th:mainresult}} or
{\rm Theorem \ref{th:mainresult2}}
shows that there exists a weak solution
$(\hat{u}_K, {\hat{L}}^K_{\alpha})$ to equation (\ref{eq:truncatedSHE})
defined on a filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0},
\hat{\mathbb{P}})_K$.
However, we can not
show that for each $K\leq N$
\begin{align*}
(\hat{u}_K, {\hat{L}}^K_{\alpha})=(\hat{u}_N, {\hat{L}}^N_{\alpha})
\,\,\mathbb{P}\text{-} \,a.s.
\,\,\text{on}\,\,\{t<\tau_K\}
\end{align*}
due to the non-Lipschitz continuity of
$\varphi$. Therefore, we do not know
whether there exists a common probability
space $(\hat{\Omega}, \hat{\mathcal{F}}, \{\hat{\mathcal{F}}_t\}_{t\geq0},
\hat{\mathbb{P}})$ on which all of the
weak solutions $((\hat{u}_K, {\hat{L}}^K_{\alpha}))_{K\geq1}$ are defined.
Hence, the localization
method in Wang et al. \cite{Wang:2023}
becomes invalid,
and the existence of the weak solution
to equation (\ref{eq:originalequation1}) with untruncated $\alpha$-stable noise remains an unsolved problem.
\end{remark}
\section{Proof of Theorem \ref{th:mainresult}}\label{sec3}
The proof of Theorem \ref{th:mainresult}
proceeds in the following three steps.
We first
construct a sequence of the approximating SPDEs with globally Lipschitz continuous noise coefficients
$(\varphi^n)_{n\geq1}$
satisfying Hypothesis \ref{Hypo},
and show that for each fixed $n\geq1$ there exists
a unique strong
solution $u^n$ in $D([0,\infty),L^p([0,L])$ with $p\in(\alpha,2]$
of the approximating SPDE; see Proposition \ref{th:Approximainresult}.
We then prove that the
approximating solution sequence $(u^n)_{n\geq1}$ is tight in both $D([0,\infty),\mathbb{M}([0,L]))$ and $L_{loc}^p([0,\infty)\times[0,L])$
for all $p\in(\alpha,2]$;
see Proposition \ref{th:tightnessresult}.
Finally, we proceed to show that there exists a weak solution $(\hat{u},\hat{L}_{\alpha})$
to equation (\ref{eq:originalequation1})
defined on another probability space
$(\hat{\Omega}, \hat{\mathcal{F}},
\{\hat{\mathcal{F}}_t\}_{t\geq0},
\hat{\mathbb{P}})$
by applying a weak convergence argument.
For each fixed $n\geq1$,
we construct the approximate SPDE
of the form
\begin{equation}
\label{eq:approximatingsolution}
\left\{\begin{array}{lcl}
\dfrac{\partial u^n(t,x)}{\partial t}=\dfrac{1}{2}\dfrac{\partial^2u^n(t,x)}
{\partial x^2}+\varphi^n(u^n(t-,x))\dot{L}_{\alpha}(t,x),&& (t,x)\in (0,\infty)
\times (0,L),\\[0.3cm]
u^n(0,x)=u_0(x),&&x\in [0,L],
\\[0.3cm]
u^n(t,0)=u^n(t,L)=0,&&t\in[0,\infty),
\end{array}\right.
\end{equation}
where the coefficient $\varphi^n$
satisfies Hypothesis \ref{Hypo}.
Given $n\geq1$, by a solution to
equation (\ref{eq:approximatingsolution})
we mean a process $u^n_t\equiv\{u^n(t,\cdot), t\in[0,\infty)\}$
satisfying the following
weak form equation:
\begin{align}
\label{eq:approxivariationform}
\langle u^n_t,\psi\rangle
&=\langle u_0,\psi\rangle+\dfrac{1}{2}\int_0^t
\langle u^n_s,\psi{''}\rangle ds
+\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\varphi^n(u^n(s-,x))\psi(x)z\tilde{N}(ds,dx,dz)
\end{align}
for all $t\in[0,\infty)$ and for any $\psi\in C^{2}([0,L])$
with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$ or equivalently
satisfying the
following mild form equation:
\begin{align}
\label{mildformapproxi0}
u^n(t,x)&=\int_0^LG_t(x,y)u_0(y)dy
+\int_0^{t+}\int_0^L
\int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y)
\varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz)
\end{align}
for all $t\in [0, \infty)$ and for a.e. $x\in [0, L]$.
We now present the definition
(see also in Wang et al. \cite{Wang:2023})
of a strong solution
to stochastic heat equation (\ref{eq:approximatingsolution}).
\begin{definition}
Given $p\geq 1$, the stochastic heat equation (\ref{eq:approximatingsolution})
has a strong solution in $D([0,\infty),L^p([0,L]))$
with initial function $u_0$
if for a given truncated $\alpha$-stable
martingale measure $L_{\alpha}$ there exists
a process
$u^n_t\equiv\{u^n(t,\cdot),t\in[0,\infty)\}$ in $D([0,\infty),L^p([0,L]))$ such that
either
equation (\ref{eq:approxivariationform})
or equation (\ref{mildformapproxi0}) holds.
\end{definition}
Note that for each $n\geq1$ the noise coefficient
$\varphi^n$ is not only Lipschitz continuous but also of globally linear growth. Indeed,
for a given $\epsilon>0$ and $n_0\in\mathbb{N}$
large enough, Hypothesis \ref{Hypo} (i) and the
globally linear growth of $\varphi$ imply that
\begin{equation}
\label{eq:glo-lin-growth}
|\varphi^n(x)|\leq|\varphi^n(x)-\varphi(x)|
+|\varphi(x)|\leq\epsilon+C(1+|x|),\,\,
\forall n\geq n_0,\, \forall x\in\mathbb{R}.
\end{equation}
Therefore, we can use the classical Banach fixed point
principle to show the existence and pathwise
uniqueness of the strong solution to equation (\ref{eq:approximatingsolution}). Since the proof is standard, we just state the main result in the following
proposition. For more details of the proof, we refer to Wang et al. \cite[Proposition 3.2]{Wang:2023} and references
therein.
Also note that
the same method was applied in
Truman and Wu \cite{Truman:2003}
and in Bo and Wang \cite{Bo:2006}
where the stochastic Burgers equation
and the stochastic {C}ahn-{H}illiard equation
driven by L\'{e}vy space-time white noise were studied,
respectively.
\begin{proposition}
\label{th:Approximainresult}
Given any $n\geq1$, if the initial
function $u_0$
satisfies $\mathbb{E}[||u_0||_p^p]<\infty$
for some $p\in(\alpha,2]$,
then under {\rm Hypothesis \ref{Hypo}}
there exists a pathwise unique strong
solution
$u^n_t\equiv\{u^n(t,\cdot),t\in[0,\infty)\}$
to equation (\ref{eq:approximatingsolution}) such that for any $T>0$
\begin{equation}
\label{eq:Approximomentresult}
\sup_{n\geq1}\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert u^n_t\vert \vert _p^p\right]<\infty.
\end{equation}
\end{proposition}
\begin{remark}
By {\rm Hypothesis \ref{Hypo} (ii)}, (\ref{eq:glo-lin-growth}) and
estimate (\ref{eq:Approximomentresult}),
the stochastic integral on the right-hand side of
(\ref{mildformapproxi0}) is well defined.
\end{remark}
We are going to prove that the approximating
solution sequence
$(u^n)_{n\geq1}$ is tight in both
$D([0,\infty),\mathbb{M}([0,L]))$ and
$L_{loc}^p([0,\infty)\times[0,L])$
for all $p\in(\alpha,2]$
by using the following tightness criteria;
see, e.g., Xiong and Yang \cite[Lemma 2.2]{Xiong:2019}.
Note that this tightness criteria can be obtained by
Ethier and Kurtz \cite[Theorems 3.9.1, 3.9.4 and 3.2.2]{Ethier:1986}.
\begin{lemma}
\label{lem:tightcriterion0}
Given a complete and separable metric space $E$,
let $(X^n=\{X^n(t),t\in[0,\infty)\})_{n\geq1}$ be a sequence of stochastic processes
with sample paths in $D([0,\infty),E)$, and let $C_a$ be a subalgebra and dense subset of
$C_b(E)$ (the bounded continuous functions space on $E$).
Then the sequence $(X^n)_{n\geq1}$ is tight in $D([0,\infty),E)$
if both of the following conditions hold:
\begin{itemize}
\item[\rm (i)] For every $\varepsilon>0$ and $T>0$
there exists a compact set
$\Gamma_{\varepsilon,T}\subset E$ such that
\begin{equation}
\label{eq:tightcriterion1}
\inf_{n\geq1}\mathbb{P}[X^n(t)\in\Gamma_{\varepsilon,T}\,\, \text{for all}\,\, t\in[0,T]
]\geq1-\varepsilon.
\end{equation}
\item[\rm (ii)]
For each $f\in C_a$, there exists a process $g_n\equiv\{g_n(t),t\in[0,\infty)\}$
such that
\begin{equation*}
f(X^n(t))-\int_0^tg_n(s)ds
\end{equation*}
is an $(\mathcal{F}_t)$-martingale and
\begin{align}
\label{eq:tight-moment}
\sup_{0\leq t\leq T}\mathbb{E}\left[\vert f(X^n(t))\vert +\vert g_n(t)\vert \right]<\infty
\end{align}
and
\begin{align}
\label{eq:tight-moment2}
\sup_{n\geq1}\mathbb{E}\left[\left(\int_0^T\vert g_n(t)\vert ^qdt\right)^{\frac{1}{q}}\right]<\infty
\end{align}
for each $n\geq1, T>0$ and $q>1$.
\end{itemize}
\end{lemma}
Before showing the tightness of solution sequence
$(u^n)_{n\geq1}$, we first find a uniform moment estimate in the following lemma.
\begin{lemma}
\label{le:uniformbounded}
For each $n\geq1$ let $u^n$ be the strong solution
to equation (\ref{eq:approximatingsolution})
given by
{\rm Proposition \ref{th:Approximainresult}}.
Then for given $T>0$ and
$\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$
and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, we have for $p\in(\alpha,2]$
that
\begin{align}
\label{eq:p-uniform}
\sup_{n\geq1}\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]<\infty.
\end{align}
\end{lemma}
\begin{proof}
By (\ref{eq:approxivariationform}), it holds that for each $n\geq1$
\begin{align*}
\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]\leq C_p(A_1+A_2+A_3),
\end{align*}
where
\begin{align*}
A_1&=\mathbb{E}\left[\left\vert \int_0^Lu_0(x)\psi(x)dx\right\vert ^p\right],
\\
A_2&=\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_0^{t}\int_0^Lu^n(s,x)\psi^{''}(x)dxds
\right\vert ^p\right],
\\
A_3&=\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi^n(u^n(s-,x))\psi(x)z\tilde{N}(ds,dx,dz)
\right\vert ^p\right].
\end{align*}
For $p\in(\alpha,2]$ we separately estimate $A_1$, $A_2$ and $A_3$ as follows.
For $A_1$, it holds by H\"{o}lder's inequality that
\begin{align*}
A_1&\leq C_p\left(\int_0^L\vert \psi(x)\vert ^{\frac{p}{p-1}}dx\right)^{\frac{p(p-1)}{p}}
\mathbb{E}\left[\int_0^L\vert u_0(x)\vert ^pdx\right]
\leq C_p\mathbb{E}[\vert \vert u_0\vert \vert _p^p]
\leq C_p
\end{align*}
due to $\psi\in C^{2}([0,L])$ and $\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty$.
For $A_2$, it holds by $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$
and H\"{o}lder's inequality that
\begin{align*}
A_2&\leq C_p\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_0^{t}\int_0^Lu^n(s,x)\psi(x)dxds
\right\vert ^p\right]
\leq C_{p,T}\int_0^{T}
\mathbb{E}\left[\sup_{0\leq r\leq s}
\left\vert \int_0^Lu^n(r,x)\psi(x)dx
\right\vert ^p\right]ds.
\end{align*}
For $A_3$, the Doob maximal inequality and the Burkholder-Davis-Gundy inequality
imply that
\begin{align*}
A_3&\leq C_p \mathbb{E}\left[\left|\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s-,x))
\psi(x)z\vert^2N(ds,dx,dz)\right|^{\frac{p}{2}}\right]
\\
&\leq C_p \mathbb{E}\left[\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s-,x))
\psi(x)z\vert ^pN(ds,dx,dz)\right]
\\
&=C_p \mathbb{E}\left[\int_0^{T}\int_0^L\int_{\mathbb{R}\setminus\{0\}}\vert \varphi^n(u^n(s,x))
\psi(x)z\vert ^pdsdx\nu_{\alpha}(dz)\right],
\end{align*}
where the second inequality follows from the fact that
\begin{align}
\label{eq:element-inequ}
\left\vert \sum_{i=1}^ka_i^2\right\vert ^{\frac{q}{2}}\leq
\sum_{i=1}^{k}\vert a_i\vert ^q
\end{align}
for $a_i\in\mathbb{R}, k\geq1$,
and $q\in(0,2]$.
By (\ref{eq:smalljumpsizemeasure}), it holds that for $p>\alpha$
\begin{align}
\label{eq:jumpestimate}
\int_{\mathbb{R}\setminus\{0\}}\vert z\vert ^p\nu_{\alpha}
(dz)=c_{+}\int_0^Kz^{p-\alpha-1}dz+c_{-}\int_{-K}^0(-z)^{p-\alpha-1}dz
=\frac{K^{p-\alpha}}{p-\alpha},
\end{align}
then there exists a constant $C_{p,K,\alpha}$ such that
\begin{align*}
A_3\leq C_{p,K,\alpha}
\mathbb{E}\left[\int_0^{T}\int_0^L\vert \varphi^n(u^n(s,x))\psi(x)\vert ^pdsdx\right].
\end{align*}
By (\ref{eq:glo-lin-growth}),
$\psi\in C^{2}([0,L])$
and (\ref{eq:Approximomentresult}) in
Proposition \ref{th:Approximainresult},
it is easy to see that
\begin{align*}
A_3&\leq C_{p,K,\alpha,T}\left(1+\sup_{0\leq s\leq T}
\mathbb{E}[\vert \vert u^n_s\vert \vert _p^p]\right)\leq
C_{p,K,\alpha,T}.
\end{align*}
Combining the estimates $A_1,A_2$ and $A_3$, we have
\begin{align*}
\mathbb{E}&\left[\sup_{0\leq t\leq T}
\left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]
\leq
C_{p,K,\alpha,T}+C_{p,T}\int_0^{T}\mathbb{E}\left[\sup_{0\leq r\leq s}
\left\vert \int_0^Lu^n(r,x)\psi(x)dx\right\vert ^p\right]ds
\end{align*}
Therefore, it holds by Gronwall's lemma that for $p\in(\alpha,2]$
\begin{align*}
\sup_{n\geq1}\mathbb{E}\left[\sup_{0\leq t\leq T}
\left\vert \int_{0}^Lu^n(t,x)\psi(x)dx\right\vert ^p\right]<\infty,
\end{align*}
which completes the proof.
$\Box$
\end{proof}
Note that for any function
$v\in L^q[0,L]$ with $q\geq1$,
we can identify $L^q([0,L])$ as a subset
of $\mathbb{M}([0,L])$ by using the following correspondence
$$v(x)\mapsto v(x)dx.$$
Then for each $n\geq1$ we can identify the
$D([0,\infty),L^p([0,L]))$-valued random variable $u^n$
as a $D([0,\infty),\mathbb{M}([0,L]))$-valued random variable (still denoted by $u^n$).
We now show the tightness of
$(u^n)_{n\geq1}$ in the following proposition.
\begin{proposition}
\label{th:tightnessresult}
The solution sequence
$(u^n)_{n\geq1}$ to
equation (\ref{eq:approximatingsolution}) given by
{\rm Proposition \ref{th:Approximainresult}}
is tight in both
$D([0,\infty),\mathbb{M}([0,L]))$ and
$L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$.
Let $u$ be an arbitrary limit point of $u^n$.
Then
\begin{align}
\label{eq:tightresult}
u\in D([0,\infty),\mathbb{M}([0,L]))
\cap L_{loc}^p([0,\infty)\times [0,L])
\end{align}
for $p\in(\alpha,2]$.
\end{proposition}
\begin{proof}
For each $n\geq1, t\geq0$ and
$\psi\in C^{2}([0,L])$ with
$\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$
and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, let us define
\begin{align*}
\langle u^n,\psi\rangle:=\langle u_t^n,\psi\rangle=\int_0^Lu^n(t,x)\psi(x)dx.
\end{align*}
We first prove that the sequence $(\langle u^n,\psi\rangle)_{n\geq1}$
is tight in $D([0,\infty),\mathbb{R})$ by using Lemma \ref{lem:tightcriterion0}.
It is easy to see that the condition (i) in Lemma \ref{lem:tightcriterion0}
can be verified by Lemma \ref{le:uniformbounded} .
In the following we mainly
verify the condition (ii) in
Lemma \ref{lem:tightcriterion0}.
For each $f\in C_b^2(\mathbb{R})$ ($f,f^{'},f^{''}$ are bounded and uniformly continuous)
with compact supports, it holds
by (\ref{eq:approxivariationform}) and It\^{o}'s formula that
\begin{align}
\label{eq:ito}
f(\langle u_t^n,\psi\rangle)&=f(\langle u_0^n,\psi\rangle)
+\int_0^tf^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle) ds
\nonumber\\
&\quad+\int_0^t\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z)
dsdx\nu_{\alpha}(dz)+\text{mart.},
\end{align}
where
$
\mathcal{D}(u,v)=f(u+v)-f(u)-vf^{'}(u)
$
for $u,v\in\mathbb{R}$.
Since $f,f^{'},f^{''}$ are bounded
and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$, then
\begin{align}
\label{eq:estimate1}
\vert f^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle\vert
\leq C\left\vert \int_0^Lu^n({s},x)\psi(x)dx\right\vert .
\end{align}
By Taylor's formula, one can show that
$\vert \mathcal{D}(u,v)\vert \leq C(\vert v\vert\wedge \vert v\vert ^2)$,
which also implies that $\vert \mathcal{D}(u,v)\vert \leq C(\vert v\vert \wedge \vert v\vert ^p)$
for $p\in(\alpha,2]$. Thus we have for $p\in(\alpha,2]$,
\begin{align}
\label{eq:estimate2}
&\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\vert \mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z)\vert
dx\nu_{\alpha}(dz)
\nonumber\\
&\leq C\left(\int_{\mathbb{R}\setminus\{0\}}\vert z\vert \wedge \vert z\vert ^p\nu_{\alpha}(dz)\right)
\int_0^L(\vert \varphi^n(u^n(s,x))\psi(x)\vert +\vert \varphi^n(u^n(s,x))\psi(x)\vert ^p)dx
\nonumber\\
&\leq C_{p,K,\alpha}\int_0^L(\vert \varphi^n(u^n(s,x))\psi(x)\vert +\vert \varphi^n(u^n(s,x))\psi(x)\vert ^p)dx,
\end{align}
where by (\ref{eq:smalljumpsizemeasure}),
\begin{align*}
\int_{\mathbb{R}\setminus\{0\}}\vert z\vert \wedge \vert z\vert ^p\nu_{\alpha}(dz)
&=c_{+}\int_0^1z^{p-\alpha-1}dz
+c_{-}\int_{-1}^0(-z)^{p-\alpha-1}dz
+c_{+}\int_1^Kz^{-\alpha}dz
\\
&\quad
+c_{-}\int_{-K}^{-1}(-z)^{-\alpha}dz
\\
&=\frac{1}{p-\alpha}+\dfrac{1-K^{1-\alpha}}{\alpha-1}\leq C_{p,K,\alpha}.
\end{align*}
For given $n\geq1$ let us define
\begin{align*}
g_n(s)&:=f^{'}(\langle u_s^n,\psi\rangle)\langle u_s^n,\psi^{''}\rangle)
+\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\mathcal{D}(\langle u_s^n,\psi\rangle,\varphi^n(u^n(s,x))\psi(x)z)
dx\nu_{\alpha}(dz).
\end{align*}
By (\ref{eq:ito}), it is easy to see that
\begin{align*}
f(\langle u_t^n,\psi\rangle)-\int_0^{t}g_n(s)ds
\end{align*}
is an $(\mathcal{F}_t)$-martingale.
Now we verify the moment estimates
(\ref{eq:tight-moment})
and (\ref{eq:tight-moment2}) of the condition (ii) in
Lemma \ref{lem:tightcriterion0}.
For each $t\in[0,T]$, it holds by the boundedness of $f$, estimates
(\ref{eq:estimate1})-(\ref{eq:estimate2}) and (\ref{eq:glo-lin-growth}) that
\begin{align*}
\mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right]
&\leq C\left(1+\mathbb{E}\left[\left\vert \int_0^Lu^n(t,x)\psi(x)dx\right\vert \right]\right)
\\
&\quad
+C_{p,K,\alpha}\mathbb{E}\left[\int_0^L(\vert \psi(x)\vert +\vert u^n(t,x)\psi(x)\vert )dx\right]
\\
&\quad+C_{p,K,\alpha}\mathbb{E}\left[\int_0^L(\vert \psi(x)\vert ^p+
\vert u^n(t,x)\psi(x)\vert ^p)dx\right].
\end{align*}
Since $\psi\in C^2([0,L])$ implies that $\psi$ is bounded,
then it holds by H\"{o}lder's inequality and
(\ref{eq:p-uniform}) that for $p\in(\alpha,2]$
\begin{align*}
\mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right]
&\leq C_{p,K,\alpha}\left(1+\left(\sup_{0\leq t\leq T}
\mathbb{E}[\vert \vert u^n_t\vert \vert_p^p]\right)^{\frac{1}{p}}
+\sup_{0\leq t\leq T}
\mathbb{E}[\vert \vert u^n_t\vert \vert_p^p]\right),
\end{align*}
and so by (\ref{eq:Approximomentresult}),
\begin{align*}
\sup_{0\leq t\leq T}\mathbb{E}\left[\vert f(\langle u_t^n,\psi\rangle)\vert +\vert g_n(t)\vert \right]<\infty,
\end{align*}
which verifies the estimate (\ref{eq:tight-moment}).
To verify (\ref{eq:tight-moment2}), it suffices to show that for each $n\geq1$
\begin{align*}
\mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]<\infty
\end{align*}
for some $q>1$.
By the estimates (\ref{eq:estimate1})-(\ref{eq:estimate2}) and (\ref{eq:glo-lin-growth}), we have
\begin{align*}
\mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]
&\leq C_{q}\mathbb{E}\left[\int_0^T\left\vert \int_0^Lu^n(t,x)\psi(x)dx
\right\vert ^qdt\right]
\\
&\quad+C_{p,q,K,\alpha}\mathbb{E}\left[\int_0^T\left\vert \int_0^L(\vert \psi(x)\vert +
\vert u^n(t,x)\psi(x)\vert )dx
\right\vert ^qdt\right]
\\
&\quad+C_{p,q,K,\alpha}\mathbb{E}\left[\int_0^T\left\vert \int_0^L(\vert \psi(x)\vert ^p+
\vert u^n(t,x)\psi(x)\vert ^p)dx
\right\vert ^qdt\right].
\end{align*}
Taking $1<q<2/p$, the H\"{o}lder inequality
and boundedness of
$\psi$ imply that
\begin{align*}
\mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]
&\leq C_{p,q,K,\alpha,T}\left(1+\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert u^n_t\vert\vert_q^q]
+\sup_{0\leq t\leq T}\mathbb{E}
\left[\vert \vert u^n_t\vert \vert_{pq}^{pq}\right]\right),
\end{align*}
and so by (\ref{eq:Approximomentresult}),
\begin{align*}
\sup_{n\geq1}\mathbb{E}\left[\int_0^T\vert g_n(t)\vert ^qdt\right]<\infty,
\end{align*}
which verifies the estimate (\ref{eq:tight-moment2}).
Therefore, for each $\psi\in C^{2}([0,L])$ with $\psi(0)=\psi(L)=\psi^{'}(0)=\psi^{'}(L)=0$
and $\vert \psi^{''}(x)\vert \leq C\psi(x),x\in[0,L]$,
the sequence $(\langle u^n,\psi\rangle)_{n\geq1}$
is tight in $D([0,\infty),\mathbb{R})$, and so
it holds by Mitoma's theorem
(see, e.g., Walsh \cite[pp.361--365]{Walsh:1986}) that
$(u^n)_{n\geq1}$
is tight in $D([0,\infty),\mathbb{M}([0,L]))$.
On the other hand, by (\ref{eq:Approximomentresult})
we have for each $T>0$
\begin{align*}
\sup_{n\geq1}\mathbb{E}
\left[\int_0^T\int_0^L\vert u^n(t,x)\vert ^pdxdt\right]
\leq C_T\sup_{n\geq1}\sup_{0\leq t\leq T}
\mathbb{E}[\vert \vert u^n_t\vert \vert _p^p]<\infty
\end{align*}
for $p\in(\alpha,2]$.
The Markov's inequality implies that
for each $\varepsilon>0,T>0$
there exists a constant $C_{\varepsilon,T}$ such that
\begin{align*}
\sup_{n\geq1}\mathbb{P}\left[\int_0^T\int_0^L\vert u^n(t,x)\vert
^pdxdt>C_{\epsilon,T}\right]<\varepsilon
\end{align*}
for $p\in(\alpha,2]$.
Therefore, the sequence $(u^n)_{n\geq1}$ is also tight
in $L^p_{loc}([0,\infty)\times[0,L])$ for $p\in(\alpha,2]$, and the conclusion
(\ref{eq:tightresult}) holds.
$\Box$
\end{proof}
\begin{Tproof}\textbf{~of Theorem
\ref{th:mainresult}.}
We are going to prove
Theorem \ref{th:mainresult} by applying weak
convergence arguments. For each
$n\geq1$, let $u^n$ be the
strong solution
of equation (\ref{eq:approximatingsolution}) given by Proposition \ref{th:Approximainresult}.
It can also be regarded as an element in
$D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$
with $p\in(\alpha,2]$.
By Proposition \ref{th:tightnessresult}, there exists
a $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$-valued random variable $u$ such that
$u^n$ converges to $u$ in distribution in
$D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$.
On the other hand, the Skorokhod Representation Theorem
(see, e.g., Either and Kurtz \cite[Theorem 3.1.8]{Ethier:1986})
yields that
there exists another filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t\geq0},\hat{\mathbb{P}})$
and on it a further subsequence
$(\hat{u}^n)_{n\geq1}$ and
$\hat{u}$
which have the same distribution as
$(u^n)_{n\geq1}$ and $u$,
so that
$\hat{u}^n$ almost surely converges to
$\hat{u}$ in $D([0,\infty),\mathbb{M}([0,L]))\cap L_{loc}^p([0,\infty)\times [0,L])$ for $p\in(\alpha,2]$.
For each $t\geq0,n\geq1$ and any test function
$\psi\in C^{2}([0,L])$ with
$\psi(0)=\psi(L)=0$ and
$\psi^{'}(0)=
\psi^{'}(L)=0$,
let us define
\begin{align*}
\hat{M}^n_{t}(\psi)&:=\int_0^L\hat{u}^n(t,x)
\psi(x)dx-\int_0^L\hat{u}_0(x)
\psi(x)dx
-\frac{1}{2}\int_0^{t}
\int_0^L\hat{u}^n(s,x)\psi^{''}(x)dxds.
\end{align*}
Since
$\hat{u}^n$ almost surely converges to
$\hat{u}$ in the Skorokhod topology
as $n\rightarrow\infty$, then
\begin{align}
\label{eq:M^n_t}
\hat{M}^n_{t}(\psi)&
\overset{\mathbf{\hat{P}}\text{-a.s.}}{\longrightarrow}
\int_0^L\hat{u}(t,x)\psi(x)dx-
\int_0^L\hat{u}_0(x)\psi(x)dx
-\frac{1}{2}\int_0^{t}
\int_0^L\hat{u}(s,x)
\psi^{''}(x)
dxds
\end{align}
in the Skorokhod topology as $n\rightarrow\infty$.
By (\ref{eq:approxivariationform}) and the fact that
$\hat{u}^n$ has the same distribution as
$u^n$ for each $n\geq1$, we have
\begin{align*}
\hat{M}^n_{t}(\psi)
\overset{D}=&
\int_0^Lu^n(t,x)
\psi(x)dx-\int_0^Lu_0(x)\psi(x)dx
-\frac{1}{2}\int_0^{t}\int_0^L
u^n(s,x)\psi^{''}(x)dxds
\nonumber\\
=&\int_0^{t+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}\psi(x)
\varphi^n(u^n(s-,x))z\tilde{N}(ds,dx,dz),
\end{align*}
where $\overset{D}=$ denotes the identity
in distribution.
The Burkholder-Davis-Gundy inequality,
(\ref{eq:element-inequ})-(\ref{eq:jumpestimate})
and (\ref{eq:glo-lin-growth}) imply that
for $p\in(\alpha,2]$
\begin{align*}
\hat{\mathbb{E}}[\vert \hat{M}^n_{t}(\psi)\vert ^p]
&=\mathbb{E}
\left[\left\vert \int_0^{t+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}\psi(x)
\varphi^n(u^n(s-,x))z\tilde{N}(ds,dx,dz)
\right\vert ^p\right]
\\
&\leq C_p\mathbb{E}
\left[\int_0^t\int_0^L
\int_{\mathbb{R}\setminus\{0\}}
\vert \psi(x)\vert ^p(1+\vert u^n(s,x)\vert )^p
\vert z\vert ^pdsdx\nu_{\alpha}(dz)
\right]
\\
&\leq C_{p,K,\alpha,T}
\left(\int_0^L\vert \psi(x)\vert ^pdx+\bigg\vert \sup_{x\in[0,L]}\psi(x)\bigg\vert ^p
\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert u^n_t\vert \vert _p^p\right]\right).
\end{align*}
Then by $\psi\in C^{2}([0,L])$ and (\ref{eq:Approximomentresult}),
we have for each $T>0$
\begin{align*}
\sup_{n\geq1}\sup_{0\leq t\leq T}
\hat{\mathbb{E}}[\vert \hat{M}^n_{t}(\psi)\vert ^p]<\infty.
\end{align*}
Therefore, it holds by (\ref{eq:M^n_t}) that
there exists an
$(\hat{\mathcal{F}}_t)$-martingale
$\hat{M}_{t}(\psi)$ such that $\hat{M}^n_{t}(\psi)$
converges weakly to
$\hat{M}_{t}(\psi)$
as $n\rightarrow\infty$, and for each $t\geq0$
\begin{align}
\label{martingle1}
\hat{M}_{t}(\psi)&=
\int_0^L\hat{u}(t,x)\psi(x)dx-
\int_0^L
\hat{u}_0(x)\psi(x)dx-\frac{1}{2}\int_0^{t}
\int_0^L\hat{u}(s,x)
\psi^{''}(x)dxds.
\end{align}
By Hypothesis \ref{Hypo} (i),
the quadratic variation of
$\{\hat{M}^n_{t}(\psi),t\in[0,\infty)\}$ satisfies that
\begin{align*}
\langle \hat{M}^n(\psi), \hat{M}^n(\psi)
\rangle_t&=\int_0^{t}\int_0^L
\int_{\mathbb{R}\setminus\{0\}}\varphi^n({u}^n(s,x))^2
\psi(x)^2z^2dsdx\nu_{\alpha}(dz)
\\
&\overset{D}=\int_0^{t}\int_0^L
\int_{\mathbb{R}\setminus\{0\}}\varphi^n(\hat{u}^n(s,x))^2
\psi(x)^2z^2dsdx\nu_{\alpha}(dz)
\\
&\overset{\mathbb{P}-a.s.}
\rightarrow\int_0^{t}\int_0^L
\int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s,x))^2
\psi(x)^2z^2dsdx\nu_{\alpha}(dz),\,\,t\in[0,T],
\end{align*}
as $n\rightarrow\infty$.
We denote by
$\{\langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t,t\in[0,\infty)\}$
the quadratic variation process
\begin{align*}
\langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t=\int_0^{t}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s,x))^2
\psi(x)^2z^2dsdx\nu_{\alpha}(dz),\,\,t\geq0.
\end{align*}
Similar to
Konno and Shiga \cite[Lemma 2.4]{Konno:1988},
$\langle \hat{M}(\psi),\hat{M}(\psi)\rangle_t$
corresponds to an orthonormal
martingale measure $\hat{M}(dt,dx,dz)$
defined on the filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}},
(\hat{\mathcal{F}_t})_{t\geq0},
\hat{\mathbb{P}})$
in the sense of Walsh
\cite[Chapter 2]{Walsh:1986} whose
quadratic measure is given by
\begin{align*}
\varphi(\hat{u}(t,x))^2z^2dtdx\nu_{\alpha}(dz).
\end{align*}
Let $\{\dot{\bar{L}}_{\alpha}(t,x):t\in[0,\infty),x\in[0,L]\}$ be another
truncated $\alpha$-stable white noise,
defined possibly on
$(\hat{\Omega}, \hat{\mathcal{F}},
(\hat{\mathcal{F}_t})_{t\geq0},
\hat{\mathbb{P}})$,
independent of
$\hat{M}(dt,dx,dz)$ and define
\begin{align*}
\hat{L}_{\alpha}(t,\psi)&:=
\int_0^{t+}\int_{0}^L\int_{\mathbb{R}\setminus\{0\}}
\dfrac{1}{\varphi(\hat{u}(s-,x))}
1_{\{\varphi(\hat{u}(s-,x))\neq0\}}\psi(x)z\hat{M}(ds,dx,dz)
\\
&\quad+\int_0^{t+}\int_{0}^L
\psi(x)
1_{\{\varphi(\hat{u}(s-,x))=0\}}
\bar{L}_{\alpha}(ds,dx).
\end{align*}
Then
$\{\hat{L}_{\alpha}(t,\psi): \,t\in[0,\infty),\, \psi\in C^2([0, L]), \psi(0)=\psi(L)=0, \psi^{'}(0)=\psi^{'}(L)=0\}$
determines a truncated
$\alpha$-stable white noise
$\dot{\hat{L}}_{\alpha}(t,x)$ on
$(\hat{\Omega}, \hat{\mathcal{F}},
(\hat{\mathcal{F}_t})_{t\geq0},
\hat{\mathbb{P}})$ with the same distribution as
$\dot{L}_{\alpha}(t,x)$
such that
\begin{align*}
\hat{M}_t(\psi)&=\int_0^{t+}
\int_0^L\varphi(\hat{u}(s-,x))
\psi(x)
\hat{L}_{\alpha}(ds,dx)
=\int_0^{t+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}\varphi(\hat{u}(s-,x))
\psi(x)z\widetilde{\hat{N}}(ds,dx,dz),
\end{align*}
where $\widetilde{\hat{N}}(dt,dx,dz)$ denotes
the compensated Poisson random measure
associated to the truncated $\alpha$-stable martingale
measure $\hat{L}_{\alpha}(t,x)$.
Hence, it holds by (\ref{martingle1}) that
$(\hat{u},\hat{L}_{\alpha})$
is a weak solution to
(\ref{eq:originalequation1}) defined on
$(\hat{\Omega}, \hat{\mathcal{F}},
(\hat{\mathcal{F}_t})_{t\geq0},
\hat{\mathbb{P}})$.
On the other hand,
since $\hat{u}^n$ has the same distribution as
$u^n$ for each $n\geq1$, then the moment estimates (\ref{eq:Approximomentresult}) in Proposition \ref{th:Approximainresult} can be replaced by
\begin{equation*}
\sup_{n\geq1}\sup_{0\leq t\leq T}
\hat{\mathbb{E}}\left[\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty
\end{equation*}
for $p\in(\alpha,2]$.
For moment estimate
(\ref{eq:momentresult}), the Fatou's Lemma
implies that for $p\in(\alpha,2]$
\begin{align*}
\hat{\mathbb{E}}\left[\vert \vert \hat{u}\vert \vert _{p,T}^p\right]&=
\hat{\mathbb{E}}\left[\int_0^T\vert \vert
\hat{u}_t\vert \vert _p^pdt\right]
\leq\liminf_{n\rightarrow\infty}C_T
\sup_{0\leq t\leq T}\hat{\mathbb{E}}\left[\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty,
\end{align*}
which completes the proof.
$\Box$
\end{Tproof}
\section{Proof of Theorem \ref{th:mainresult2}}\label{sec4}
The proof of
Theorem \ref{th:mainresult2} is
similar to that of Theorem \ref{th:mainresult}.
The main difference between them is that in
the current proof
we need to prove the solution sequence
$(u^n)_{n\geq1}$
to equation (\ref{eq:approximatingsolution}), obtained from
Proposition \ref{th:Approximainresult}, is tight
in $D([0,\infty),L^p([0,L]))$ for $p\in(\alpha,5/3)$.
To this end,
we need the following tightness criteria;
see, e.g., Ethier and Kurtz \cite[Theorem 3.8.6 and Remark (a)]
{Ethier:1986}.
Note that the same criteria was
also applied in Sturm \cite {Sturm:2003} with
Gaussian colored noise setting.
\begin{lemma}
\label{lem:tightcriterion}
Given a complete and separable metric
space $(E,\rho)$,
let $(X^n)$ be a sequence of stochastic processes
with sample paths in $D([0,\infty),E)$.
The sequence is tight in $D([0,\infty),E)$ if the
following conditions hold:
\begin{itemize}
\item[\rm (i)] For every $\varepsilon>0$ and
rational $t\in[0,T]$, there exists a compact set $
\Gamma_{\varepsilon,T}\subset E$ such that
\begin{equation}
\label{eq:tightcriterion1}
\inf_{n}\mathbb{P}[X^n(t)\in\Gamma_{\varepsilon,T}]
\geq1-\varepsilon.
\end{equation}
\item[\rm (ii)]
There exists $p>0$ such that
\begin{equation}
\label{eq:tightcriterion2}
\lim_{\delta\rightarrow0}\sup_n\mathbb{E}
\left[\sup_{0\leq t\leq T}\sup_{0\leq u\leq
\delta}(\rho(X^n_{t+u},X^n_t)\wedge1)^p\right]=0.
\end{equation}
\end{itemize}
\end{lemma}
To verify condition (i) of Lemma
\ref{lem:tightcriterion}, we need the following
characterization of the relatively compact set
in $L^p({[0,L]}),p\geq1$; see, e.g., Sturm \cite[Lemma 4.3]{Sturm:2003}.
\begin{lemma}
\label{lem:compactcriterion}
A subset $\Gamma\subset L^p({[0,L]})$ for $p\geq1$
is relatively compact if and only if
the following conditions hold:
\begin{itemize}
\item[\rm (a)] $\sup_{f\in\Gamma}
\int_0^L\vert f(x)\vert ^pdx<\infty$,
\item[\rm (b)] $\lim_{y\rightarrow0}\int_0^L\vert
f(x+y)-f(x)\vert ^pdx=0$ uniformly for all
$f\in\Gamma$,
\item[\rm (c)] $\lim_{\gamma\rightarrow\infty}
\int_{(L-\frac{L}{\gamma},L]}\vert f(x)\vert ^pdx=0$
for all $f\in\Gamma$.
\end{itemize}
\end{lemma}
The proof of the tightness of $(u^n)_{n\geq1}$
is accomplished
by verifying conditions (i) and (ii) in
Lemma \ref{lem:tightcriterion}.
To this end, we need some estimates on
$(u^n)_{n\geq1}$, that is, the uniform bound
estimate in Lemma
\ref{lem:uniformbound},
the temporal difference estimate in Lemma
\ref{lem:temporalestimation}
and the spatial difference estimate in Lemma
\ref{lem:spatialestimation}, respectively.
\begin{lemma}
\label{lem:uniformbound}
Suppose that $\alpha\in(1,5/3)$ and
for each $n\geq1$ $u^n$ is the solution
to equation (\ref{eq:approximatingsolution})
given by
{\rm Proposition \ref{th:Approximainresult}}.
Then for given $T>0$
there exists a constant
$C_{p,K,\alpha,T}$ such that
\begin{equation}
\label{eq:unformlybounded}
\sup_n\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T},\,\,\,
\text{for}\,\,\,p\in(\alpha,5/3).
\end{equation}
\end{lemma}
\begin{proof}
For each $n\geq 1$, by (\ref{mildformapproxi0})
it is easy to see that
$$\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq C_p(A_1+A_2),$$
where
\begin{align*}
A_1&=\mathbb{E}\left[\sup_{0\leq t\leq T}\Bigg\vert \Bigg\vert \int_0^LG_{t}(\cdot,y)u_0(y)dy\Bigg\vert \Bigg\vert _p^p\right],\\
A_2&=\mathbb{E}\left[\sup_{0\leq t\leq T}\Bigg\vert \Bigg\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
G_{t-s}(\cdot,y)\varphi^n(u^n(s-,y))z
\tilde{N}(ds,dy,dz)\Bigg\vert \Bigg\vert _p^p\right].
\end{align*}
We separately estimate $A_1$ and $A_2$ as follows.
For $A_1$, it holds by Young's
convolution inequality
and (\ref{eq:Greenetimation0}) that
\begin{align*}
\label{eq:A_1}
A_1
&\leq C\mathbb{E}\left[\int_0^L
\sup_{0\leq t\leq T}\left(\int_0^L|G_{t}
(x,y)|dx\right)
\vert u_0(y)\vert ^pdy\right]\leq C_T\mathbb{E}[\vert\vert u_0\vert \vert _p^p].
\end{align*}
By Proposition \ref{th:Approximainresult},
we have $\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty$
for $p\in(\alpha,2]$, and so there exists a
constant $C_{p,T}$ such that $A_1\leq C_{p,T}$.
For $A_2$, we use the factorization method;
see, e.g., Da Prato et al. \cite{Prato:1987}, which is based on the fact that for $0<\beta<1$ and $\,0\leq s\leq t$,
\begin{equation*}
\int_s^t(t-r)^{\beta-1}(r-s)^{-\beta}dr=\dfrac{\pi}{\sin(\beta\pi)}.
\end{equation*}
For any function $v: [0,\infty)\times {[0,L]}
\rightarrow \mathbb{R}$ define
\begin{align*}
&\mathcal{J}^{\beta}v(t,x)
:=\dfrac{\sin(\beta\pi)}{\pi}\int_0^t\int_0^L(t-s)^{\beta-1}G_{t-s}(x,y)v(s,y)dyds,\\
&\mathcal{J}^n_{\beta}v(t,x):=\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
(t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(v(s-,y))z\tilde{N}(ds,dy,dz).
\end{align*}
By the stochastic Fubini Theorem and (\ref{eq:Greenetimation1}), we have
\begin{align*}
\mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n(t,x)
&=\dfrac{\sin(\beta\pi)}{\pi}\int_0^t\int_0^L
(t-s)^{\beta-1}G_{t-s}(x,y)\Bigg(\int_0^{s+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}
(s-r)^{-\beta}\\
&\quad\quad\times G_{s-r}(y,m)
\varphi^n(u^n(r-,m))z\tilde{N}(dr,dm,dz)\Bigg)dyds
\\
&=\dfrac{\sin(\beta\pi)}{\pi}\int_0^{t+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\Bigg[\int_r^{t}(t-s)^{\beta-1}(s-r)^{-\beta}\\
&\quad\quad\times\Bigg(\int_0^LG_{t-s}(x,y)
G_{s-r}(y,m)dy\Bigg)ds\Bigg]
\varphi^n(u^n(r-,m))z\tilde{N}(dr,dm,dz)
\\
&=\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}G_{t-s}(x,y)\varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz).
\end{align*}
Thus, $$A_2=\mathbb{E}\left[\sup_{0\leq t\leq T}\vert \vert \mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right].$$
Until the end of the proof we fix a
$0<\beta<1$ satisfying
\begin{align}
\label{eq:factorization}
1-\frac{1}{p}<\beta<\frac{3}{2p}-\frac{1}{2},
\end{align}
which requires that
$$\frac{3}{2p}-\frac{1}{2}-(1-\frac{1}{p})>0.$$
Therefore, we need the assumption $p<5/3$ for this lemma.
Back to our main proof, to estimate $A_2$
we first estimate
$\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]$.
For $p\in(\alpha,5/3)$, the Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}) and
(\ref{eq:glo-lin-growth}) imply that
\begin{align*}
\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]
=&\int_0^L\mathbb{E}\left[\left\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
(t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(u^n(s-,y))z\tilde{N}(ds,dy,dz)\right\vert ^p\right]dx\nonumber\\
\leq&C_p\int_0^L\int_0^{t}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\mathbb{E}\left[\vert (t-s)^{-\beta}G_{t-s}(x,y)\varphi^n(u^n(s,y))z\vert ^p\right]\nu_{\alpha}(dz)dydsdx\nonumber\\
\leq&C_{p,K,\alpha}\int_0^L\int_0^{t}\int_0^L
\mathbb{E}\left[1+\vert u^n(s,y)\vert ^p\right]\vert t-s\vert ^{-\beta p}\vert G_{t-s}(x,y)\vert ^pdydsdx.
\end{align*}
Combine (\ref{eq:Greenetimation2}), we have
\begin{align*}
\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\leq&C_{p,K,\alpha}\left(L+\mathbb{E}\left[\sup_{0\leq s\leq t}\vert \vert u^n_s\vert \vert _p^p\right]\right)
\int_0^Ts^{-(\frac{p-1}{2}+\beta p)}ds.
\end{align*}
For $p<5/3$, by (\ref{eq:factorization}) we have
$$\int_0^Ts^{-(\frac{p-1}{2}+\beta p)}ds<\infty.$$
Therefore,
there exists a constant $C_{p,K,\alpha,T}$ such that
\begin{equation}
\label{eq:momentestimation}
\mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]\leq
C_{p,K,\alpha,T}\left(1+\mathbb{E}\left[\sup_{0\leq s\leq t}\vert \vert u^n_s\vert \vert _p^p\right]\right).
\end{equation}
We now estimate $A_2=\mathbb{E}[\sup_{0\leq t\leq T}\vert \vert \mathcal{J}^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]$. The Minkowski inequality implies that
\begin{align}
\label{eq:A_2_1}
A_2
=&\mathbb{E}\left[\sup_{0\leq t\leq T}\dfrac{\sin(\pi\beta)}{\pi}\Bigg\vert \Bigg\vert \int_0^{t}\int_0^L(t-s)^{\beta-1}
G_{t-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds \Bigg\vert \Bigg\vert _p^p\right]\nonumber\\
\leq&\dfrac{\sin(\pi\beta)}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^{t}(t-s)^{\beta-1}
\Bigg\vert \Bigg\vert \int_0^LG_{t-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _pds\right)^p\right].
\end{align}
By the H\"{o}lder inequality and
(\ref{eq:Greenetimation0}), we have
\begin{align}
\label{eq:A_2_2}
&\Bigg\vert \Bigg\vert \int_0^LG_{t-s}(\cdot-y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _p\nonumber\\
&\quad=\left(\int_0^L\left\vert \int_0^L\vert G_{t-s}(x,y)\vert ^{\frac{p-1}{p}}\vert G_{t-s}(x,y)\vert ^{\frac{1}{p}}\mathcal{J}^n_{\beta}u^n(s,y)
dy\right\vert ^pdx\right)^{\frac{1}{p}}\nonumber\\
&\quad\leq\left(\int_0^L\left\vert \left(\int_0^L\vert G_{t-s}(x,y)\vert dy\right)^{\frac{p-1}{p}}
\left(\int_0^LG_{t-s}(x,y)\vert \mathcal{J}^n_{\beta}u^n(s,y)\vert ^pdy\right)^{\frac{1}{p}}\right\vert ^pdx\right)^{\frac{1}{p}}\nonumber\\
&\quad\leq\left(\sup_{x\in[0,L]}\int_0^L|G_{t-s}(x,y)|dy\right)^{\frac{p-1}{p}}
\left(\int_0^L\int_0^L|G_{t-s}(x,y)|\vert \mathcal{J}^n_{\beta}u^n(s,y)\vert ^pdxdy\right)^{\frac{1}{p}}\nonumber\\
&\quad\leq C_T\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p.
\end{align}
Therefore, it follows from (\ref{eq:A_2_1}), (\ref{eq:A_2_2}), and the H\"{o}lder inequality that
\begin{align*}
\label{eq:A_2}
A_2\leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^t(t-s)^{\beta-1}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _pds\right)^p\right]\nonumber\\
\leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\mathbb{E}\left[\sup_{0\leq t\leq T}\left(\int_0^{t}1^{\frac{p}{p-1}}ds\right)^{p-1}
\left(\int_0^{t}(t-s)^{(\beta-1)p}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^pds\right)\right]
\nonumber\\
\leq&\dfrac{\sin(\pi\beta)C_{p,T}}{\pi}\int_0^{T}
(T-s)^{(\beta-1)p}
\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p]ds.
\end{align*}
By (\ref{eq:momentestimation}),
it also holds that
\begin{align}
A_2
\leq&\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi}
\int_0^{T}(T-s)^{(\beta-1)p}\left(1+\mathbb{E}
\left[\sup_{0\leq r\leq s}\vert \vert u^n_r\vert \vert
_p^p\right]\right)ds\nonumber\\
\leq&\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi}
\left(1+\int_0^{T}(T-s)^{(\beta-1)p}\mathbb{E}
\left[\sup_{0\leq r\leq s}\vert \vert u^n_r
\vert \vert _p^p\right]ds\right).
\end{align}
Combining (\ref{eq:A_2}) and the estimate for $A_1$,
we have for each $T>0$,
\begin{align*}
&\mathbb{E}\left[\sup_{0\leq t \leq T}\vert \vert u^n_t\vert \vert _p^p\right]\leq
C_{p,T}
+\dfrac{\sin(\pi\beta)C_{p,K,\alpha,T}}{\pi}
\int_0^{T}(T-s)^{(\beta-1)p}\mathbb{E}
\left[\sup_{0\leq r\leq s}\vert \vert u^n_r\vert \vert
_p^p\right]ds.
\end{align*}
Since $\beta>1-1/p$,
applying a generalized Gronwall's Lemma
(see, e.g., Lin \cite[Theorem 1.2]{Lin:2013}), we have
\begin{align*}
\sup_{n}\mathbb{E}\left[\sup_{0\leq t \leq T}\vert \vert
u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T},
\,\,\,\text{for}\,\,\,p\in(\alpha,5/3),
\end{align*}
which completes the proof.
$\Box$
\end{proof}
\begin{lemma}
\label{lem:temporalestimation}
Suppose that $\alpha\in(1,5/3)$ and
for each $n\geq1$ $u^n$ is the solution
to equation (\ref{eq:approximatingsolution})
given by {\rm Proposition \ref{th:Approximainresult}}.
Then for given $T>0$, $0\leq h\leq\delta$
and $p\in(\alpha,5/3)$
\begin{equation}
\label{eq:temporalestimation}
\lim_{\delta\rightarrow0}\sup_n\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\vert \vert
u^n_{t+h}-u^n_t\vert \vert _p^p\right]=0.
\end{equation}
\end{lemma}
\begin{proof}
For each $n\geq 1$, by the factorization
method in the proof of Lemma
\ref{lem:uniformbound}, we have
$$\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq
h\leq \delta}\vert \vert u^n_{t+h}-u^n_t\vert \vert
_p^p\right]\leq
C_p(B_1+B_2),$$
where
\begin{align*}
B_1&=\mathbb{E}\left[\sup_{0\leq t\leq T}
\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert
\int_0^L(G_{t+h}(\cdot-y)-G_{t}(\cdot-
y))u_0(y)dy\Bigg\vert \Bigg\vert _p^p\right],\\
B_2&=\mathbb{E}\left[\sup_{0\leq t\leq T}
\sup_{0\leq h\leq \delta}\vert \vert \mathcal{J}^{\beta}
\mathcal{J}^n_{\beta}u^n_{t+h}-\mathcal{J}
^{\beta}\mathcal{J}^n_{\beta}u^n_t\vert \vert
_p^p\right].
\end{align*}
For $B_1$, Young's convolution inequality
and (\ref{eq:Greenetimation0}) imply that
\begin{align*}
B_1
&\leq \mathbb{E}\left[\int_0^L\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left(\int_0^L(|G_{t+h}(x,y)|+|G_{t}(x,y)|)dx\right)\vert u_0(y)\vert ^pdy\right]
\leq C_T\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty.
\end{align*}
Therefore, it holds by Lebesgue's
dominated convergence theorem that
$B_1$ converges to 0 as $\delta\rightarrow0$.
For $B_2$, it is easy to see that
$$B_2\leq \frac{\sin(\beta\pi)C_p}{\pi}
(B_{2,1}+B_{2,2}+B_{2,3}),$$
where
\begin{align*}
B_{2,1}&=\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert
\int_0^{t}\int_0^L(t-s)^{\beta-1}
(G_{t+h-s}(\cdot,y)-G_{t-s}(\cdot,y))
\mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\Bigg],\\
B_{2,2}&=\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert
\int_0^{t}\int_0^L((t+h-s)^{\beta-1}-(t-s)^{\beta-1})G_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\Bigg],\\
B_{2,3}&=\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\Bigg\vert \Bigg\vert
\int_{t}^{t+h}\int_0^L(t+h-s)^{\beta-1}G_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dyds\Bigg\vert \Bigg\vert _p^p\right].
\end{align*}
By the assumption $p\in(\alpha,5/3)$ of this lemma we can choose a $0<\beta<1$ satisfying $1-1/p<\beta<3/2p-1/2$.
By Lemma \ref{lem:uniformbound} and (\ref{eq:momentestimation}), there exists a constant $C_{p,K,\alpha,T}$ such that
\begin{equation}
\label{ineq:0}
\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}.
\end{equation}
To estimate $B_{2,1}$,
we set $G^h_t(x,y)=G_{t+h}(x,y)-G_{t}(x,y)$.
Similar to the estimates for
(\ref{eq:A_2_1}) and (\ref{eq:A_2_2}) in the proof
of Lemma \ref{lem:uniformbound}, we have
\begin{align*}
B_{2,1}\leq&\mathbb{E}\Bigg[\sup_{0\leq t\leq T}
\sup_{0\leq h\leq
\delta}\Bigg(\int_0^{t}(t-s)^{\beta-1}
\Bigg(\sup_{x\in [0,L]}
\int_0^LG^h_{t-s}(x,y)dy\Bigg)^{\frac{p-1}{p}}
\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert
_pds\Bigg)^p\Bigg].
\end{align*}
It also follows from the H\"{o}lder inequality and (\ref{ineq:0}) that
\begin{align*}
B_{2,1}\leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}\left[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p\right]
\sup_{0\leq h\leq \delta}
\left(\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG_{t-s}^h(x,y)dy\right)^{p-1}ds\right)\nonumber\\
\leq&C_{p,K,\alpha,T}\sup_{0\leq h\leq \delta}
\left(\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG^h_{t-s}(x,y)dy\right)^{p-1}ds\right).
\end{align*}
Moreover, since $\beta>1-1/p$, it holds by (\ref{eq:Greenetimation0}) that
\begin{align*}
&\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in [0,L]}\int_0^LG_{t-s}^h(x,y)dy\right)^{p-1}ds\\
&\quad\leq\int_0^{T}s^{(\beta-1)p}\left(\sup_{x\in
[0,L]}\left(\int_0^L|G_{t+h-s}(x,y)|dy+\int_0^L|G_{t-s}(x,y)|dy\right)\right)^{p-1}ds\\
&\quad\leq
C_{p,T}\int_0^{T}s^{(\beta-1)p}ds<\infty.
\end{align*}
Thus, Lebesgue's Dominated Convergence Theorem implies that
$B_{2,1}$ converges to 0 as $\delta\rightarrow0$.
For $B_{2,2}$, the Minkowski inequality and Young's convolution inequality imply that
\begin{align*}
B_{2,2}\leq&\mathbb{E}\Bigg[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}
\Bigg(\int_0^{t}((t+h-s)^{\beta-1}-(t-s)^{\beta-1})
\Bigg\vert \Bigg\vert \int_0^LG_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Bigg\vert \Bigg\vert _p ds\Bigg)^p\Bigg]\nonumber\\
\leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left(\int_0^{t}((t+h-s)^{\beta-1}-(t-s)^{\beta-1})
\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p ds\right)^p\right].\nonumber
\end{align*}
By the H\"{o}lder inequality and (\ref{ineq:0}) we have for
$\beta>1-1/p$,
\begin{align*}
B_{2,2}\leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}
\int_0^{t}\vert (t+h-s)^{\beta-1}-(t-s)^{\beta-1}\vert ^{p}
\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p ds\right]\nonumber\\
\leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\int_0^{T}\vert (s+\delta)^{\beta-1}-s^{(\beta-1)}\vert ^pds\nonumber\\
\leq&C_{p,K,\alpha,T}\int_0^{T}\vert (s+\delta)^{\beta-1}-s^{(\beta-1)}\vert ^pds<\infty.
\end{align*}
Therefore, by Lebesgue's dominated convergence theorem,
we know that $B_{2,2}$ converges to 0 as $\delta\rightarrow0$.
For $B_{2,3}$, similar to $B_{2,2}$, we get
\begin{align}
\label{ineq:B_{2_3}}
B_{2,3}\leq&\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}
\left(\int_{t}^{t+h}(t+h-s)^{\beta-1}\Big\vert \Big\vert \int_0^LG_{t+h-s}(\cdot,y)\mathcal{J}^n_{\beta}u^n(s,y)dy\Big\vert \Big\vert _p
ds\right)^p\right]\nonumber\\
\leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}\left(
\int_{t}^{t+h}(t+h-s)^{\beta-1}\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p ds\right)^p\right]\nonumber\\
\leq&C_{p,T}\mathbb{E}\left[\sup_{0\leq t\leq T}\sup_{0\leq h\leq \delta}
\int_{t}^{t+h}\vert (t+h-s)^{\beta-1}\vert ^p\vert \vert \mathcal{J}^n_{\beta}u^n_s\vert \vert _p^p ds\right]\nonumber\\
\leq&C_{p,T}\sup_{0\leq t\leq T}\mathbb{E}[\vert \vert \mathcal{J}^n_{\beta}u^n_t\vert \vert _p^p]\sup_{0\leq h\leq \delta}
\int_{t}^{t+h}\vert (t+h-s)^{\beta-1}\vert
\leq C_{p,K,\alpha,T}\int_0^{\delta}s^{(\beta-1)p}ds.
\end{align}
Since $\beta>1-1/p$, we can conclude that the right-hand side
of (\ref{ineq:B_{2_3}}) converges to 0 as $\delta\rightarrow0$. Therefore, by
the estimates of $B_{2,1}, B_{2,2}, B_{2,3}$ and $B_1$, the
desired result (\ref{eq:temporalestimation}) holds,
which completes the proof.
$\Box$
\end{proof}
\begin{lemma}
\label{lem:spatialestimation}
For each $n\geq1$ let $u^n$ be the solution
to equation (\ref{eq:approximatingsolution})
given by {\rm Proposition \ref{th:Approximainresult}}.
Then for given $t\in[0,\infty)$,
$0\leq\vert x_1\vert \leq\delta$
and $p\in(\alpha,2]$
\begin{equation}
\label{eq:spatialestimation}
\lim_{\delta\rightarrow 0}\sup_n\mathbb{E}\left[
\sup_{\vert x_1\vert \leq \delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t,
\cdot)\vert \vert _p^p\right]=0.
\end{equation}
\end{lemma}
\begin{proof}
Since the shift operator is continuous in $L^p([0,L])$, then for each $n\geq1$
and $\delta>0$
there exists a pathwise
$x_1^{n,\delta}(t)\in\mathbb{R}$ such that
$\vert x_1^{n,\delta}(t)\vert \leq\delta$ and
$$\sup_{\vert x_1\vert \leq \delta}\vert \vert
u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p=\vert \vert u^n(t,\cdot+x_1^{n,\delta}(t))-u^n(t,\cdot)\vert \vert _p^p.$$
As before, it is easy to see that
$$\mathbb{E}[\vert \vert
u^n(t,\cdot+x_1^{n,\delta}(t))-u^n(t,\cdot)
\vert \vert _p^p]\leq C_p(C_1+C_2),$$
where
\begin{align*}
C_1&=\mathbb{E}\left[
\bigg\vert \bigg\vert \int_0^L
(G_{t}(\cdot+x_1^{n,\delta}(t),y)
-G_{t}(\cdot,y))u_0(y)dy\bigg\vert \bigg\vert _p^p
\right],
\\
C_2&=\mathbb{E}\bigg[\bigg\vert \bigg\vert
\int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
(G_{t-s}(\cdot+{x_1^{n,\delta}(t)},y)
-G_{t-s}(\cdot,y))
\varphi^n(u^n(s-,y))
z\tilde{N}(dz,dy,ds)\bigg\vert \bigg\vert _p^p\bigg].
\end{align*}
For $C_1$, Young's convolution inequality
and (\ref{eq:Greenetimation0}) imply that
\begin{align*}
C_1&\leq\mathbb{E}\left[\int_0^L
\left(\int_0^L(|G_{t}(x+x_1^{n,
\delta}(t),y)|+|G_{t}(x,y))|dx\right)
\vert u_0(y)\vert ^pdy\right]
\leq C_T\mathbb{E}[\vert \vert u_0\vert \vert _p^p]<\infty.
\end{align*}
Thus, the Lebesgue dominated convergence theorem implies that $C_1$ converges to 0 as $\delta\rightarrow0$.
For $C_2$, it follows from the Burkholder-Davis-Gundy inequality, (\ref{eq:element-inequ})-(\ref{eq:jumpestimate}), (\ref{eq:glo-lin-growth})
and (\ref{eq:Greenetimation2}) that for $p\in(\alpha,2]$
\begin{align*}
C_2
=&\int_0^L\mathbb{E}\bigg[\bigg\vert \int_0^{t+}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
(G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\varphi^n(u^n(s-,y))z
\tilde{N}(ds,dy,dz)\bigg\vert ^p\bigg]dx\nonumber\\
\leq&C_p\int_0^L\int_0^{t}\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\mathbb{E}[\vert (G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\varphi^n(u^n(s,y))z\vert ^p]\nu_{\alpha}(dz)dydsdx\nonumber\\
\leq&C_{p,K,\alpha}\int_0^L\int_0^{t}\int_0^L\mathbb{E}[(1+\vert u^n(s,y)\vert )^p]
\vert (G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y))\vert ^pdydsdx\nonumber\\
\leq&C_{p,K,\alpha}\left(\int_0^L\int_0^{t}\vert G_{t-s}(x+{x_1^{n,\delta}(t)},y)-G_{t-s}(x,y)\vert ^pdsdx\right)
\left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right)
\\
\leq&
C_{p,K,\alpha}\left(\int_0^L\int_0^{t}(\vert G_{t-s}(x+{x_1^{n,\delta}(t)},y)\vert ^p+\vert G_{t-s}(x,y)\vert ^p)dsdx\right)
\left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right)
\\
\leq&
C_{p,K,\alpha}\left(\int_0^t(t-s)^{-\frac{p-1}{2}}ds\right)
\left(L+\sup_{0\leq s\leq t}\mathbb{E}\left[\vert \vert u^n_s\vert \vert _p^p\right]\right)
\end{align*}
Therefore, it holds
by (\ref{eq:Approximomentresult}) and
Lebesgue's dominated convergence theorem
that $C_2$ converges to 0 as $\delta\rightarrow0$.
Hence, by the
estimates of $C_1$ and $C_2$, we obtain
\begin{align*}
\lim_{\delta\rightarrow 0}\sup_n\mathbb{E}\bigg[\sup_{\vert x_1\vert \leq
\delta}\vert \vert u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p\bigg]&=0,
\end{align*}
which completes the proof.
$\Box$
\end{proof}
\begin{proposition}
\label{prop:tightnessresult2}
Suppose that $\alpha\in(1,5/3)$.
The sequence of solutions
$(u^n)_{n\geq1}$ to equation
(\ref{eq:approximatingsolution})
given by {\rm Proposition \ref{th:Approximainresult}}
is tight in $D([0,\infty),L^p([0,L]))$
for $p\in(\alpha,5/3)$.
\end{proposition}
\begin{proof}
From (\ref{eq:Approximomentresult}) and
Markov's inequality, for each $\varepsilon>0$, $p\in(\alpha,2]$ and $T>0$
there exists a $N\in\mathbb{N}$ such that
\begin{align*}
\sup\limits_n\mathbb{P}\left[\vert \vert u^n_t\vert \vert
_p^p>N\right]\leq\dfrac{\varepsilon}{3},\quad t\in [0,T].
\end{align*}
Let $\Gamma^1_{\varepsilon,T}$ be a closed set
defined by
\begin{align}
\label{eq:Gamma1}
\Gamma^1_{\varepsilon,T}:=\{v_t\in L^p([0,L]):
\vert \vert v_t\vert \vert _p^p\leq N,t\in[0,T]\}.
\end{align}
By Lemma \ref{lem:spatialestimation}
and Markov's inequality, it holds that for each
$\varepsilon>0$, $p\in(\alpha,2]$ and $T>0$
\begin{equation*}
\lim_{\delta\rightarrow 0}\sup_n\mathbb{P}\left[
\sup_{\vert x_1\vert \leq \delta}\vert \vert
u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert
_p^p>\varepsilon\right]=0,\quad t\in[0,T].
\end{equation*}
Then for $k\in\mathbb{N}$
we can choose a sequence
$(\delta_k)_{k\geq1}$ with
$\delta_k\rightarrow0$ as $k\rightarrow\infty$
such that
\begin{align*}
\sup\limits_n\mathbb{P}\left[\sup_{\vert x_1\vert \leq
\delta_k}\vert \vert
u^n(t,\cdot+x_1)-u^n(t,\cdot)\vert \vert _p^p>\frac{1}{k}
\right]\leq\dfrac{\varepsilon}{3}2^{-k},\quad t\in[0,T].
\end{align*}
Let $\Gamma^2_{\varepsilon,T}$ be a closed set
defined by
\begin{align}
\label{eq:Gamma2}
\Gamma^2_{\varepsilon,T}:=\bigcap_{k=1}^{\infty}
\left\{v_t\in L^p([0,L]):
\sup_{\vert x_1\vert \leq \delta_k}\vert \vert
v(t,\cdot+x_1)-v(t,\cdot)\vert \vert _p^p\leq\frac{1}{k},t\in[0,T]
\right\}.
\end{align}
We next prove that for each $\varepsilon>0$ and $p\in(\alpha,2]$ ,
\begin{equation}
\label{eq:compactset1}
\lim_{\gamma\rightarrow \infty}\sup_n\mathbb{P}\left[\int_{(L-\frac{L}{\gamma},L]}\vert u^n(t,x)\vert ^pdx>\varepsilon\right]=0.
\end{equation}
It is easy to see that
\begin{align*}
\mathbb{E}\bigg[\int_{(L-\frac{L}{\gamma},L]}\vert u^n(t,x)\vert ^pdx\bigg]
&=\mathbb{E}\bigg[\int_0^L
\vert u^n(t,x)\vert ^p1_{(L-\frac{L}{\gamma},L]}(x)dx\bigg]
\leq C_p(D_1+D_2),
\end{align*}
where
\begin{align*}
D_1&=\int_0^L\mathbb{E}\left[\left\vert \int_0^LG_{t}
(x,y)u_0(y)dy\right\vert ^p\right]
1_{(L-\frac{L}{\gamma},L]}(x)dx,
\\
D_2&=\int_0^L\mathbb{E}\Bigg[\Bigg\vert \int_0^{t+}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}
G_{t-s}(x,y)
\varphi^n(u^n(s-,y))z
\tilde{N}(ds,dy,dz)
\Bigg\vert ^p\Bigg]1_{(L-\frac{L}{\gamma},L]}(x)dx.
\end{align*}
It is easy to prove that $D_1$ converges to 0 as
$\gamma\rightarrow\infty$ by using Young's convolution
inequality and Lebesgue's dominated convergence
theorem. For $D_2$, it holds by the
Burkholder-Davis-Gundy inequality,
(\ref{eq:element-inequ})-(\ref{eq:jumpestimate})
and
(\ref{eq:glo-lin-growth}) that
for $p\in(\alpha,2]$
\begin{align*}
D_2
\leq &C_p\int_0^L\int_0^{t}
\int_0^L\int_{\mathbb{R}\setminus\{0\}}
\mathbb{E}[\vert G_{t-s}(x,y)\varphi^n(u^n(s,y))z\vert ^p]
1_{(L-\frac{L}{\gamma},L]}(x)\nu_{\alpha}(dz)dydsdx
\nonumber\\
\leq&C_{p,K,\alpha}\int_0^L\int_0^{t}
\int_0^L\mathbb{E}(1+\vert u^n(s,y)\vert )^p\vert G_{t-s}(x,y)\vert
^p1_{(L-\frac{L}{\gamma},L]}(x)dydsdx
\nonumber\\
\leq&C_{p,K,\alpha,T}\left(L+\sup_{0\leq t\leq T}\mathbb{E}
\left[\vert \vert u^n_t\vert \vert
_p^p\right]\right)
\int_0^L\int_0^{t}(t-s)^{-\frac{p-1}{2}}
1_{(L-\frac{L}{\gamma},L]}(x)dsdx.
\end{align*}
Since $p\leq2$,
it holds that
\begin{align*}
D_2\leq C_{p,K,\alpha,T}
\left(\int_0^L1_{(L-\frac{L}{\gamma},L]}(x)dx\right)\left(L+\sup_{0\leq t\leq T}\mathbb{E}
\left[\vert \vert u^n_t\vert \vert _p^p\right]\right).
\end{align*}
By (\ref{eq:Approximomentresult}), $D_2$
converges to 0 as
$\gamma\rightarrow\infty$.
Therefore,
(\ref{eq:compactset1}) is obtained from the
estimates of $D_1$ and $D_2$ and Markov's
inequality.
For any $k\in\mathbb{N}$ and $T>0$ we can choose a sequence
$(\gamma_k)_{k\geq1}$ with $\gamma_k\rightarrow\infty$ as
$k\rightarrow\infty$
such that
\begin{align*}
\sup\limits_n\mathbb{P}\left
[\int_{(L-\frac{L}{\gamma_k},L]}\vert u^n(t,x)\vert ^pdx>\frac{1}{k}\right]
\leq\dfrac{\varepsilon}{3}2^{-k},\quad t\in[0,T].
\end{align*}
Let $\Gamma^3_{\varepsilon,T}$ be a closed set
defined by
\begin{align}
\label{eq:Gamma3}
\Gamma^3_{\varepsilon,T}:=\bigcap_{k=1}^{\infty}
\left\{v_t\in L^p([0,L]): \int_{(L-\frac{L}{\gamma_k},L]}\vert v(t,x)\vert ^pdx\leq\frac{1}{k}, t\in[0,T]
\right\}.
\end{align}
Combining (\ref{eq:Gamma1}), (\ref{eq:Gamma2})
and (\ref{eq:Gamma3}) to define
\begin{align*}
\Gamma_{\varepsilon,T}:=\Gamma^1_{\varepsilon,T}
\cap\Gamma^2_{\varepsilon,T}
\cap\Gamma^3_{\varepsilon,T},
\end{align*}
then $\Gamma_{\varepsilon,T}$ is a closed set
in $L^p([0,L]),p\in(\alpha,2]$. For any function $f\in\Gamma_{\varepsilon,T}$
the definition of $\Gamma_{\varepsilon,T}$
implies that the conditions (a)-(c) in
Lemma \ref{lem:compactcriterion}
hold, and so $\Gamma_{\varepsilon,T}$ is a
relatively compact set in
$L^p([0,L]),p\in(\alpha,2]$.
Combing the closeness and relatively
compactness, we know that
$\Gamma_{\varepsilon,T}$ is a
compact set in $L^p([0,L]),p\in(\alpha,2]$.
Moreover, the definition of
$\Gamma_{\varepsilon,T}$
implies that
\begin{align*}
\inf_n\mathbb{P}
[u^n_t\in\Gamma_{\varepsilon,T}]&\geq1-
\frac{\varepsilon}{3}\left(1+2\sum_{k=1}^{\infty}
2^{-k}\right)=1-\varepsilon,
\end{align*}
which verifies condition (i) of Lemma
\ref{lem:tightcriterion}. Condition (ii) of
Lemma \ref{lem:tightcriterion} is verified by
Lemma \ref{lem:temporalestimation} with $p\in(\alpha,5/3)$.
Therefore, $(u^n)_{n\geq1}$ is tight in
$D([0,\infty),L^p([0,L]))$ for $p\in(\alpha,5/3)$,
which completes the proof.
$\Box$
\end{proof}
\begin{Tproof}\textbf{~of Theorem
\ref{th:mainresult2}.}
According to Proposition \ref{prop:tightnessresult2},
there exists a $D([0,\infty),L^p([0,T]))$-valued random variable
$u$ such that $u^n$ converges to $u$ in distribution in
the Skorohod topology.
The Skorohod Representation Theorem
yields that
there exists another filtered probability space
$(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t\geq0},\hat{\mathbb{P}})$
and on it a further subsequence
$(\hat{u}^n)_{n\geq1}$ and
$\hat{u}$
which have the same distribution as
$(u^n)_{n\geq1}$ and $u$,
so that
$\hat{u}^n$ almost surely converges to
$\hat{u}$ in the Skorohod topology.
The rest of the proofs,
including the construction
of a truncated $\alpha$-stable measure $\hat{L}_{\alpha}$ such that
$(\hat{u},\hat{L}_{\alpha})$ is a weak solution to equation
(\ref{eq:originalequation1}),
is same as the proof of Theorem \ref{th:mainresult} and
we omit them.
Since $\hat{u}^n$ has the same distribution as
$u^n$ for each $n\geq1$,
the moment estimate (\ref{eq:unformlybounded}) in Lemma \ref{lem:uniformbound} can be
written as
$$
\sup_{n\geq1}\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}^n_t\vert \vert _p^p\right]\leq C_{p,K,\alpha,T}.
$$
Hence, by Fatou's Lemma,
$$
\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}_t\vert \vert _p^p\right]\leq\liminf_{n\rightarrow\infty}
\hat{\mathbb{E}}\left[\sup_{0\leq t\leq T}\vert \vert \hat{u}^n_t\vert \vert _p^p\right]<\infty.
$$
This yields the uniform $p$-moment estimate
(\ref{eq:momentresult2}). Similarly, we can obtain the
uniform stochastic
continuity (\ref{eq:timeregular}) by
Lemma \ref{lem:temporalestimation}.
$\Box$
\end{Tproof}
\noindent \textbf{Acknowledgements}
This work is supported by the National Natural Science Foundation of China (NSFC) (Nos. 11631004, 71532001), Natural Sciences and Engineering Research Council of Canada (RGPIN-2021-04100).
\end{document} |
\begin{equation}gin{document}
\title[Asymptotic behavior on the hyperbolic plane]{Asymptotic behavior of the steady Navier-Stokes equation on the hyperbolic plane.}
\author[Chan]{Chi Hin Chan}
\address{Department of Applied Mathematics, National Chiao Tung University,1001 Ta Hsueh Road, Hsinchu, Taiwan 30010, ROC}
\email{[email protected]}
\author[Chen]{Che-Kai Chen}
\address{Department of Applied Mathematics, National Chiao Tung University,1001 Ta Hsueh Road, Hsinchu, Taiwan 30010, ROC}
\email{ckchen7537\@gmail.com}
\author[Czubak]{Magdalena Czubak}
\address{Department of Mathematics\\
University of Colorado Boulder\\ Campus Box 395, Boulder, CO, 80309, USA}
\email{[email protected]}
\begin{equation}gin{abstract}
We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to $0$ at infinity. We also address the decay rate for the vorticity and the behavior of the pressure.
\end{abstract}
\subjclass[2010]{58J05, 76D05, 76D03;}
\keywords{Exterior domain, Stationary Navier-Stokes, asymptotics, hyperbolic plane}
\maketitle
\section{Introduction}
Exterior domain is one of the fundamental domains studied in fluid mechanics.
The problem to be described has a satisfactory answer in three dimensions in the Euclidean setting, but there are questions that remain open in two dimensions, and they have been open since the work of Leray \cite{LerayExt}. In this article, we show these questions can be answered if we pose them on the hyperbolic plane. We begin by describing the problem and providing historical background.
Let $K$ be a compact set, an obstacle, in $\mathbb R^2$. Consider a fluid surrounding $K$, where the behavior of the fluid is governed by the stationary Navier-Stokes equation. Then the exterior domain problem in the $\mathbb R^2$ setting consists of finding a
smooth solution $u : \mathbb{R}^{2} - K \rightarrow \mathbb{R}^{2}$, and the pressure $p: \mathbb R^2-K \rightarrow \mathbb R$ satisfying
\begin{equation}gin{equation}\label{ext}
\begin{equation}gin{split}
-\triangle u + u\cdot \nabla u + \nabla p &= 0 ,\\
\dv u & = 0,\\
u|_{\partial K} &= 0,
\end{split}
\end{equation}
and $\int_{\mathbb{R}^{2} - K } |\nabla u|^{2} < \infty, \ \mbox{and such that}\quad u(x) \rightarrow \overline{u}_{\infty}\ \mbox{ as}\quad |x| \rightarrow \infty,$
where $\overline{u}_{\infty}\in \mathbb R^2$ is a given constant vector. $\overline{u}_{\infty}$ represents the behavior of the flow at the far range.
The history of the problem and the settlement of the analogous problem in three dimensions begins with the work of Leray \cite{LerayExt}. The method of Leray leads to a solution in three dimensions, but meets with a hurdle in 2D.
The idea of Leray was to obtain a solution $(u_R, p_R$) in $\{|x| \leq R\}\cap \mathbb{R}^{2} -K$ satisfying $u_{R}|_{\{|x| = R\}} = \overline{u}_{\infty}$ and the finite Dirichlet property in $\{|x| \leq R\}\cap \mathbb{R}^{2} -K$. Then while the limiting solution, denoted by $u_{L}$, was shown to satisfy the finite Dirichlet norm in $\mathbb{R}^{2} - K $, the behavior of $u_{L}$ at infinity was not known. This was also an issue in 3D, but Finn \cite{Finn}, Ladyzhenskaya \cite{Lbook}, and Babenko \cite{Babenko} were able to bring the 3D problem to a positive conclusion. The reason for this is that in 3D, the homogeneous $\dot H^1$ norm controls the $L^6$ norm of the difference $u_L-\overline u_\infty$ as well as $\int_{\mathbb R^3-K}\frac{\abs{u_L-\overline u_\infty}^2}{r^2}$. In 2D, the following holds
\[
\int_{\mathbb R^2-K}\frac{\abs{u_L-\overline u_\infty}^2}{r^2(\log r)^2}\leq C(1 + \int_{\mathbb R^2-K}\abs{\nabla u_L}^2).
\]
Unlike the 3D estimates, this estimate does not preclude $u_L$ from being trivial. Essentially, the failure of the energy method to produce good estimates in 2D is the source of the difficulty in completing the 2D problem.
Important progress was made by Gilbarg and Weinberger \cite{GilbargWeinberger1974, GilbargWeinberger1978}, who in particular showed that a typical solution,
$u : \mathbb{R}^{2}-K \rightarrow \mathbb{R}^{2}$ to \eqref{ext} with $\int_{\mathbb{R}^{2}-K} |\nabla u|^{2} < \infty$ (so not necessarily obtained by Leray's method)
satisfies the following
\begin{equation}gin{equation}\label{log}
\lim_{r\rightarrow \infty } \frac{|u(r,\theta )|^{2}}{\log r} = 0.
\end{equation}
However, from \eqref{log} is not clear if $|u| \in L^{\infty }(\mathbb{R}^{2}-K )$.
\eqref{log} is based on the finite Dirichlet norm of $u$ and a standard energy estimate. On the other hand, they showed that the Leray solution $u_L$ has to be in $L^\infty$, and if $\overline u_\infty$ is trivial, then so is $u_L$ at infinity.
Subsequent breakthrough came from Amick \cite{Amick}, who indicated that one cannot improve \eqref{log} without taking into account the structure of the equation \eqref{ext}. Amick was able
to prove that the properties found by Gilbarg and Weinberger for Leray solutions hold for all solutions. Moreover, he showed that the solution converges to
\emph{some} nonzero vector ${u}_{\infty}$ in the far range for symmetric flows, and in certain sectors of the plane if the flow is not symmetric.
However, whether ${u}_{\infty}$ coincides with the prescribed $\overline{u}_{\infty}$, and if the pointwise convergence can be proved in general are questions that are still open.
In this paper, we answer these questions on the hyperbolic plane.\footnote{The 3D problem on the hyperbolic space will be considered in a forthcoming work by the second author.} More precisely, let $a, R_0>0$, and consider $$\Omega (R_0) = \mathbb{H}^2(-a^2) -\overline{ B_O(R_0)},$$ where $B_O(R_0)$ is a geodesic ball in a hyperbolic plane $\mathbb H^{2}(-a^{2})$ with constant sectional curvature $-a^2$, and $O$ is a fixed base point in $\mathbb H^{2}(-a^{2})$. We study the following stationary Navier-Stokes equation on $\Omega (R_0)$,
\begin{equation}gin{equation}\label{StatNSforvelocityDecay}
\begin{equation}gin{split}
2 \Def^* \Def v + \nabla_v v + \dd P & = 0 , \\
\dd^* v & = 0,
\end{split}
\end{equation}
where $P$ is a smooth function on $\Omega (R_0)$, and $2\Def^\ast\Def v=-2\dv \Def v,$ and $\Def$ is the deformation tensor, which can be written in coordinates as
\[
(\Def v)_{ij}=\frac 12(\nabla_i v_j+\nabla_j v_i).
\]
Moroever, a computation using Ricci identity shows for divergence free $v$ that on the hyperbolic plane
\[
2\Def^\ast\Def v=-\Delta v -2\mathbb Ric v=-\Delta v + 2a^2 v,
\]
where $-\Delta$ is the Hodge Laplacian. We use this operator as we believe this is the correct form of the equations on a Riemannian manifold as indicated in \cite{EbinMarsden}. For an extended discussion about the possible forms of the equations, see \cite{CCD16}.
We assume that just like on $\mathbb R^2$, $v$ satisfies the finite Dirichlet property
\begin{equation}gin{equation}\label{FDIforVelDecay}
\int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} < \infty .
\end{equation}
Without prescribing any conditions on the boundary of the obstacle, we show that $v$ must vanish at infinity.
\begin{equation}gin{thm}\label{VelocityDecayThm}
Let $R_0 > 0$, and suppose $v$ is a smooth $1-$form that solves \eqref{StatNSforvelocityDecay} on $\Omega (R_0)$, and
satisfies the finite Dirichlet norm property \eqref{FDIforVelDecay}.
Then, it follows that we have the following decay property of $v$ in the far range.
\begin{equation}gin{equation}\label{DecayofVel}
\lim_{\rho (x) \rightarrow \infty } \big \|v\big \|_{L^{\infty} (B_{x} (r(a)) )} = 0 ,
\end{equation}
where $\rho (x)$ is the geodesic distance of $x$ from the center $O$ of the obstacle $B_O(R_0)$ in $\mathbb{H}^2(-a^2)$ and
\begin{equation}gin{equation*}
r(a) = \frac{1}{a} \log \Big( \frac{1+3e^a}{3+e^a } \Big ).
\end{equation*}
\end{thm}
(The reason for the form of $r(a)$ is explained in Section \ref{prelim}.) Then together with smoothness of $v$, we immediately get
\begin{equation}gin{cor} Let $R_1>R_0$.
Suppose $v$ is a smooth $1-$form that solves \eqref{StatNSforvelocityDecay} on $\Omega (R_0)$, and
satisfies \eqref{FDIforVelDecay}. Then $v\in L^\infty(\Omega(R_1)).$
\end{cor}
We also address the decay of the vorticity at infinity.
\begin{equation}gin{thm}\label{FinalThmforExpdecay}
Let $R_1>R_0$, and let $v$ be a smooth $1-$form that solves \eqref{StatNSforvelocityDecay} on $\Omega (R_0)$, which satisfies \eqref{FDIforVelDecay}. Let $\omega = * \dd v$ be the associated vorticity of $v$. We consider the positive constant
\begin{equation}gin{equation}\label{definedelta}
\delta\equiv\delta (a, \|v\|_{L^\infty(\Omega(R_1))}) = \frac{1}{4} \Big \{ \Big ( ( \|v\|_{L^\infty(\Omega(R_1))} - a )^2 + 8a^2 \Big )^{\frac{1}{2}} - (\|v\|_{L^\infty(\Omega(R_1))} - a) \Big \}.
\end{equation}
Then, the following apriori estimate holds for any $x \in \Omega (R_1)$.
\begin{equation}gin{equation}\label{Finalexpdecay}
-A e^{-\delta \rho (x)} \leq \omega (x) \leq A e^{-\delta \rho (x)} ,
\end{equation}
where
\begin{equation}gin{equation}\label{definitionofA}
A = \exp ( \delta R_1 ) \big \| \omega \big \|_{L^{\infty} (\partial B_O(R_1))}.
\end{equation}
\end{thm}
Gilbarg and Weinberger use the vorticity equation and first establish decay rates for the vorticity, and then move on to showing $L^\infty$ bounds for $v$. What we found is that in the hyperbolic setting, the $L^\infty$ bounds are easier to obtain due to better estimates than in the Euclidean 2D setting. The key idea is the use of a Poincar\'e type inequality on an exterior domain to obtain a uniform control on the $L^2$ norm of the solution. Such inequality on the whole hyperbolic space was established by the first and third author in \cite{CC15}. To show it here, we follow the approach from \cite{CC15} combined with test functions used by Gilbarg and Weinberger \cite{GilbargWeinberger1978}.
Initial attempts to adapt the proof for the vorticity decay as in \cite{GilbargWeinberger1978} to the hyperbolic plane were not successful, so we ended up using a geometric approach inspired by the work of Anderson and Schoen \cite{AndersonSchoen}. There, Perron's method with barrier function $e^{-\delta \rho (x)}$ is applied to the Laplacian on a negatively curved manifold. We apply that idea to an elliptic equation for the vorticity that can be obtained by taking $* \dd$ on both sides of the first line of \eqref{StatNSforvelocityDecay}. The equation is
\begin{equation}gin{equation}\label{vorticityeq}
-\Delta \omega + 2a^2 \omega + g(v , \nabla \omega) = 0,
\end{equation}
where $g$ is the metric on the hyperbolic plane.
So we consider the elliptic operator
\begin{equation}gin{equation}\label{eoperator}
L (f) = \Delta f - 2a^2 f - g(v ,\nabla f),
\end{equation}
and construct subsolutions and supersolutions $\pm A e^{-\delta \rho (x)}$.
Finally we show that the property of the pressure obtained by Gilbarg and Weinberger \cite{GilbargWeinberger1978} cannot be expected in general.
\begin{equation}gin{thm}\label{thm_p}
Let $R_0>0$. There exist $(v, P)$ that satisfy \eqref{StatNSforvelocityDecay} on $\Omega (R_0)$, are both smooth, and such that $v$ has finite Dirichlet property \eqref{FDIforVelDecay}, but there exist no constant $L$ such that
\[
\lim_{\rho (x) \rightarrow \infty } |P(x)-L|=0.
\]
\end{thm}
\subsection{Organization of the paper}
In Section \ref{prelim} we set up the Poincar\'e model for the hyperbolic plane, and introduce the function spaces that will be used throughout the paper. Section \ref{section_Linfty} is devoted to showing the solution to the Stokes equation can be estimated locally in $L^\infty$. The strategy here is to rely on the well-developed theory of a priori estimates in the Euclidean setting. Therefore, we start with the intrinsic Stokes equation on the hyperbolic plane, and then we write it in terms of the Euclidean derivatives on the Poincar\'e disk (see equation \eqref{Equation3.14NEW}). In Section \ref{section_v_decay} we derive the Poincar\' e type estimate on the exterior domain, and then apply it together with the result of Section \ref{section_Linfty} to prove Theorem \ref{VelocityDecayThm}, the decay of the velocity at infinity. The decay rate for the vorticity is obtained in Section \ref{section_vorticity}, and Section \ref{section_p} discusses the pressure. In the appendix \ref{appendixa} we include what should be a standard material for the $L^\infty$ bound for the solution of the Stokes equation.
\subsection{Acknowledgments}
The first and third author would like to thank Vladim{\'i}r \v{S}ver{\'a}k for introducing us to the problem of the exterior domain. C. H. Chan is partially supported by a
grant from the National Science Council of Taiwan (NSC 101-2115-M-009-016-MY2).
M. Czubak is partially supported by a grant from the Simons Foundation \# 246255, and would like to also thank MSRI, where part of this work was carried out.
\section{Preliminaries}\label{prelim}
\subsection{Hyperboloid model}\label{Hyperboloid}
The hyperboloid model for the hyperbolic space $\Bbb{H}^{2}(-a^{2})$ is given by
\begin{equation}gin{equation}
\mathbb{H}^2(-a^2) = \big\{ (x_0, x_1 , x_2 ) : x_0^2 - x_1^2 - x_2^2 = \frac{1}{a^2} , x_0 > 0 \big\}\subset \mathbb R^3.
\end{equation}
For each $x = (x_0, x_1, x_2) \in \mathbb{R}^3$, the tangent space $T_x\mathbb{R}^3$ can be equipped with the following symmetric quadratic form
\begin{equation}gin{equation}\label{lorentz}
\ip{v, w} = -v_0w_0 + v_1w_1 + v_2w_2 , \quad v,w \in T_x\mathbb{R}^3.
\end{equation}
Then the Riemannian metric $g(\cdot ,\cdot )$ on $\mathbb{H}^2(-a^2)$ is induced through the restriction of $\ip{\cdot ,\cdot}$ onto the tangent bundle of the submanifold $\mathbb H^{2}(-a^{2})$.
In other words, for each point $x \in \mathbb{H}^2(-a^2)$, $g(\cdot , \cdot )_x$ is given by the following relation
\begin{equation}gin{equation}
g(\cdot , \cdot )_{x} = \ip{\cdot , \cdot }\big |_{x} .
\end{equation}
From now on, we write a point $x = (x_0, x_1, x_2)$ as $x = (x_0 , x')$, with $x' = (x_1 , x_2)$.
In general, the geodesic ball at $x$ with radius $R$ in $\Bbb H^{2}(-a^{2})$ will be denoted by $$B_{x}(R)=\{y\in \mathbb H^{2}(-a^{2}):\rho(x,y)< R\},$$
where $\rho(x,y)$ is the geodesic distance between $x$ and $y$ in $\Bbb{H}^{2}(-a^{2}).$ For any $x\in \Bbb R^{2}$ and $R>0$, the Euclidean open ball centered at $x$ with radius $R$ will be denoted by $$D_{x}(R)=\{y\in \Bbb R^{2}:|x-y|<R\}.$$
Next, we consider the unit disc $D_{0}(1)$ in $\Bbb R^{2}$ and the smooth mapping $Y: \mathbb H^{2}(-a^{2})\rightarrow D_{0}(1)$ defined by $$Y(x)=\frac{x'}{x_{0}+\frac{1}{a}},\quad x=(x_{0},x')\in \Bbb H^{2}(-a^{2}).$$
The map $Y$ maps $\Bbb H^{2}(-a^{2})$ bijectively onto $D_{0}(1)$ with a smooth inverse, so $Y$ can be chosen as a coordinate system on the manifold $\Bbb H^{2}(-a^{2}).$
The inverse map $Y^{-1}: D_{0}(1)\rightarrow \Bbb H^{2}(-a^{2})$ is given by $$Y^{-1}(y)=\bigg(\frac{2}{a(1-|y|^{2})}-\frac{1}{a},\frac{2y_{1}}{a(1-|y|^{2})},\frac{2y_{2}}{a(1-|y|^{2})}\bigg),\quad y=(y_1,y_2)\in D_{0}(1). $$
Using $Y$ we can identify $\mathbb H^{2}(-a^{2})$ with $D_0(1)$ equipped with the metric $\frac{4}{a^2(1-\abs{y}^2)^2}dy^i\otimes dy^i$. So this is the Poincar\'e disk model. Now, let $\tilde y \in D_0(1)$ with $\abs{\tilde y}=r$, then by parametrizing the straight line connecting $0$ and $\tilde y$, we see that the geodesic distance between $0$ and $\tilde y$ is (see for example \cite{redbook})
\begin{equation}\label{gdist}
\rho(0,\tilde y)=\frac 1a\int^r_0 \frac{2}{1-t^2} d t =\frac 1a \log(\frac{1+r}{1-r}).
\end{equation}
So if we would like to talk about a geodesic ball $B_O(R)\subset \mathbb H^{2}(-a^{2})$, and relate it to a Euclidean ball in the unit disk, then we need to find $r$ such that
\[
\frac 1a \log(\frac{1+r}{1-r})=R.
\]
A computation shows that $$r=\tanh(\frac{a}{2}R),$$ so $Y$ maps a geodesic ball of radius $R$ onto the Euclidean ball of radius $\tanh(\frac{a}{2}R)$, i.e., $Y\big(B_{O}(R)\big)=D_{0}\big(\tanh(\frac{a}{2}R)\big).$ The way this is employed is that we will start with a ball of radius $1$ on the hyperbolic plane, so that means doing estimates on the Euclidean ball of radius $\tanh(\frac{a}{2})$. Then at some point we go from the estimates on the ball of radius $\tanh(\frac{a}{2})$ to $\frac 12$ of $\tanh(\frac{a}{2})$ (e.g. when applying
\eqref{BootstrapFinalNEW}), so when we go back to the hyperbolic plane, this maps to a ball of radius
\[
\frac 1a \log \Big(\frac{1+\frac 12\tanh(\frac{a}{2})}{1-\frac 12\tanh(\frac{a}{2})}\Big)= \frac{1}{a} \log \Big ( \frac{1+3 e^a}{3+e^a} \Big ).
\]
This explains the reason for the choice of $r(a)$ in Theorem \ref{VelocityDecayThm}.
We now introduce several function spaces, which will be used in this article.
\subsection{Function spaces}
Let $M$ be a Riemannian manifold with a Riemannian mteric $g_{_{M}}$, and let $\nabla^{M}$ be the Levi-Civita connection on $M$. Consider a domain $\Omega$ in $M$. We define the following function spaces:
\begin{equation}gin{itemize}
\item $\bigwedge^{k}(\Omega)$ is the space of all smooth $k$-forms in $\Omega$.
\item $\bigwedge^{k}_{c}(\Omega)$ is the space of all smooth $k$-forms with compact support in $\Omega$.
\item$\bigwedge^{k}_{\sigma}(\Omega)$ is the space of all smooth, $\dd^{*}$-closed, $k$-forms on $\Omega.$
\item $L^{k,p}(\Omega)$ is the space of all weakly differentiable $1$-forms $v$ with $(\nabla^M)^{k}v\in L^{p}(\Omega)$. $L^{k,p}(\Omega)$ is equipped with the semi-norm $\|v\|_{_{L^{k, p}(\Omega)}}=\|(\nabla^M)^{k}v\|_{_{L^{p}(\Omega)}},$ and $L^{k,p}_{0}(\Omega)$ is the closure of $\bigwedge^1_c(\Omega)$ in $L^{k,p}(\Omega).$
\item $W^{k,p}(\Omega)$ is the Sobolev space which consists of all weakly differentiable $1$-forms $v$ with $(\nabla^M)^{\alpha}v\in L^{p}(\Omega)$ for all $0 \leq \alpha\leq k$. $W^{k,p}(\Omega)$ is equipped with the norm $\|v\|_{_{W^{k, p}(\Omega)}}=\sum_{\alpha=0}^{k}\|(\nabla^M)^{\alpha}v\|_{_{L^{p}(\Omega)}},$ and $W^{k,p}_{0}(\Omega)$ is the closure of $\bigwedge^1_c(\Omega)$ in $W^{k,p}(\Omega).$
\end{itemize}
For the case of $p=2$, we write $W^{k,2}(\Omega)=H^{k}(\Omega)$, $W^{k,2}_{0}(\Omega)=H^{k}_{0}(\Omega).$
In order to simplify our notation, the Levi-Civita connection $\nabla^{\Bbb H^{2}(-a^{2})}$ on the hyperbolic space $\mathbb{H}^2(-a^2)$ will be denoted by $\nabla$. We use $C_0$ to denote an absolute constant in each inequality estimate which could change from line to line.
\section{Local $L^\infty$ bound on $v$}\label{section_Linfty}
The purpose of this section is to show we can obtain a bound on $L^\infty$ norm of $v$ on a small enough ball in the hyperbolic plane, where $v$ is a solution to the Stokes equation. First we consider a general $u$, not necessarily a solution to the Stokes equation, and prove a bound on the Dirichlet norm of the pull-back of $u$ to the Poincar\'e disk. The bound is in terms of the intrinsic $L^2$ and Dirichlet norms.
\begin{equation}gin{lemma}\label{Lemma3.1NEW}
The following estimate holds for any $1-$form $u \in H^1(B_{O}(1))$, where $u^{\sharp}$ is the pull back of $u$ via the map $Y^{-1}$.
\begin{equation}gin{equation}\label{Equation3.1NEW}
\big \|\nabla^{\mathbb{R}^2} u^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))}^2 \leq 32\Big \{ \frac{1}{a^2} \cosh^4\Big ( \frac{a}{2}\Big ) \big \|\nabla u \big \|_{L^2(B_O(1))}^2 +\sinh^2a \big \| u \big \|_{L^2(B_O(1))}^2 \Big \} .
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Now, for any $1-$form $u$ on $(B_{O}(1))$, the pull back of $u$ is given by $$ u^{\sharp}:=(Y^{-1})^{*}u=(u_{1}\circ Y^{-1})dy^{1}+(u_{2}\circ Y^{-1})dy^{2}.$$ Write $u_{\alpha}^{\sharp}=u_{\alpha}\circ Y^{-1}$ for $\alpha=1, 2,$ and let $\nabla$ be the induced Levi-Civita connection acting on smooth $1$-forms on $\Bbb{H}^{2}(-a^{2})$. Then (see \cite[Appendix]{CC13})
\begin{equation}gin{equation}\label{identity3.2NEW}
\begin{equation}gin{split}
\nabla u= &\bigg\{\frac{\partial u_{1}}{\partial Y^{1}}-\frac{2Y^{1}u_{1}}{1-|Y|^{2}}+\frac{2Y^{2}u_{2}}{1-|Y|^{2}}\bigg\}dY^{1}\otimes dY^{1}\\
&\quad+\bigg\{\frac{\partial u_{2}}{\partial Y^{1}}-\frac{2Y^{2}u_{1}}{1-|Y|^{2}}-\frac{2Y^{1}u_{2}}{1-|Y|^{2}}\bigg\}dY^{1}\otimes dY^{2}\\
&\quad+\bigg\{\frac{\partial u_{1}}{\partial Y^{2}}-\frac{2Y^{2}u_{1}}{1-|Y|^{2}}-\frac{2Y^{1}u_{2}}{1-|Y|^{2}}\bigg\}dY^{2}\otimes dY^{1}\\
&\quad+\bigg\{\frac{\partial u_{2}}{\partial Y^{2}}+\frac{2Y^{1}u_{1}}{1-|Y|^{2}}-\frac{2Y^{2}u_{2}}{1-|Y|^{2}}\bigg\}dY^{2}\otimes dY^{2}.\\
\end{split}
\end{equation}
We consider the orthonormal
frame $\{e_1^*, e_2^* \}$ of $T^*(\mathbb{H}^2(-a^2))$ given by
\begin{equation}gin{equation}\label{Dualframe}
e_j^* = \frac{2}{a(1-|Y|^2)} \dd Y^j, \quad j=1, 2.
\end{equation}
Hence $\{e_i^* \otimes e_j^* : 1\leq i,j\leq 2\}$ constitutes an orthonormal frame on $T^*(\mathbb{H}^2(-a^2))\otimes T^*(\mathbb{H}^2(-a^2))$, and it follows that
\begin{equation}\label{normdY}
\abs{dY^j\otimes dY^k}=\frac{a^2(1-\abs{Y}^2)^2}{4}\delta^{jk}.
\end{equation}
To obtain \eqref{Equation3.1NEW}, we have to estimate the absolute value of the partial derivatives of $u_{\begin{equation}ta}^{\sharp}$ with respect to $y^{\alpha}$ for all $\alpha$ and $\begin{equation}ta$ equal $1$ or $2$. We just estimate $|\partial_{y_{1}}u_{1}^{\sharp}|$ to illustrate the idea, then the estimates for all other terms follow basically in the same manner.
First, we observe that by \eqref{normdY}
$$|\nabla u|_{a}\geq \frac{a^2(1-|Y|^{2})^2}{4}\Big |\frac{\partial u_{1}}{\partial Y^{1}}-\frac{2Y^{1}u_{1}}{1-|Y|^{2}}+\frac{2Y^{2}u_{2}}{1-|Y|^{2}}\Big|.$$
Thus, by the triangle inequality $$\Big|\frac{\partial u_{1}}{\partial Y^{1}}\Big|\leq \Big |\frac{\partial u_{1}}{\partial Y^{1}}-\frac{2Y^{1}u_{1}}{1-|Y|^{2}}+\frac{2Y^{2}u_{2}}{1-|Y|^{2}}\Big|+\frac{4|Y||u|}{1-|Y|^{2}}\leq
\frac{4}{a^2(1-|Y|^{2})^2}|\nabla u|_{a}+\frac{4|Y||u|}{1-|Y|^{2}}.$$
These imply the following pointwise estimate on $D_{0}(\tanh(\frac{a}{2}))$,
\begin{equation}gin{equation}\label{PointwiseestimateNEW}
\Big|\frac{\partial u^{\sharp}_{1}}{\partial y^{1}}\Big|\leq \frac{4}{a^2(1-|y|^{2})^2}|\nabla u|_{a}\circ Y^{-1}+\frac{4|y||u^{\sharp}|}{1-|y|^{2}}.
\end{equation}
Next, using $(a+b)^2\leq 2a^2+2b^2$, $\cosh^2\theta-\sinh^2\theta=1$ and the definition of the integration on manifolds
\begin{equation}gin{align}
\int_{D_{0}(\tanh (\frac{a}{2}))}\Big|\frac{\partial u^{\sharp}_{1}}{\partial y^{1}}\Big|^{2}dy^{1}\wedge dy^{2}&\leq 2\int_{D_{0}(\tanh (\frac{a}{2}))} \Big(\frac{4}{a^2(1-|y|^{2})^2}\Big)^2|\nabla u |^2_{a} \circ Y^{-1}dy^{1}\wedge dy^{2}\nonumber\\
&\qquad+2\int_{D_{0}(\tanh (\frac{a}{2}))} \frac{16|y|^2|u^\sharp|^2}{(1-|y|^{2})^2}dy^{1}\wedge dy^{2}\nonumber\\
&\leq \frac{8}{a^2}\cosh^4(\frac a2)\int_{D_{0}(\tanh (\frac{a}{2}))} \frac{4}{a^2(1-|y|^{2})^2}|\nabla u |^2_{a} \circ Y^{-1}dy^{1}\wedge dy^{2}\nonumber\\
&\qquad+32\int_{D_{0}(\tanh (\frac{a}{2}))} \frac{|y|^2|u^\sharp|^2}{(1-|y|^{2})^2}dy^{1}\wedge dy^{2}\nonumber\\
&\leq\quad \frac{8}{a^2}\cosh^4(\frac a2)\int_{B_O(1)} |\nabla u |^2_{a} \frac{4}{a^2(1-|Y|^{2})^2} dY^{1}\wedge dY^{2}\nonumber\\
&\qquad +32\tanh^{2}\Big(\frac{a}{2}\Big)\cosh^{4}\Big(\frac{a}{2}\Big)\int_{D_{0}(\tanh (\frac{a}{2}))} |u^\sharp|^2 dy^{1}\wedge dy^{2}\nonumber\\
&=\quad \frac{8}{a^2}\cosh^4(\frac a2)\int_{B_O(1)} |\nabla u |^2_{a} \mathbb Vol_{\mathbb H^{2}(-a^{2})}\nonumber\\
&\qquad + 8\sinh^2(a)\int_{D_{0}(\tanh (\frac{a}{2}))}|u^{\sharp}|^{2}dy^{1}\wedge dy^{2} \nonumber
\end{align}
The above estimate still works if $\frac{\partial u^{\sharp}_{1}}{\partial y^{1}}$ is replaced by $\frac{\partial u^{\sharp}_{i}}{\partial y^{j}}$ for any $1 \leq i,j \leq 2$.
Hence \eqref{Equation3.1NEW} follows.
\end{proof}
We are now ready to consider the Stokes equation.
\begin{equation}gin{lemma}\label{Supnormelliptic}
Consider a smooth $1$-form $v \in \Lambda^1_{\sigma}(B_O(1))$ and a smooth function $P \in C^{\infty} (B_O(1))$ which satisfy the following Stokes equation on $B_O(1)$
\begin{equation}gin{equation}\label{LinearStokeswithForcingNEW}
\begin{equation}gin{split}
2\Def^* \Def v + \dd P & = F , \\
\dd^* v & = 0 ,
\end{split}
\end{equation}
where $F \in \Lambda^1(B_O(1)) \cap L^{\frac{4}{3}}(B_O(1))$. Let
\begin{equation}gin{equation}\label{quitesimpler}
r(a) = \frac{1}{a} \log \Big ( \frac{1+3 e^a}{3+e^a} \Big ) ,
\end{equation}
where $r(a)$ is such that $Y(B_O(r(a)))=D_0(\frac{1}{2} \tanh \big(\frac{a}{2}\big))$ (using \eqref{gdist}). Then, it follows that $v$ satisfies the following a priori estimate.
\begin{equation}gin{equation}\label{Supnormestimate}
\big \| v \big \|_{L^{\infty} (B_O(r(a)))} \leq C_0 \Big \{ A_1(a) \big \| F \big \|_{L^{\frac{4}{3}} (B_O(1)) } + A_2(a) \big \|v \big \|_{L^2(B_O(1))} + A_3(a) \big \| \nabla v \big \|_{L^2(B_O(1))} \Big \} ,
\end{equation}
where $C_0 > 0$ is an absolute constant which is independent of $a$, and where the constants $A_1(a)$, $A_2(a)$, $A_3(a)$ can be given explicitly as follows.
\begin{equation}gin{equation}\label{aboutA}
\begin{equation}gin{split}
A_1(a) & = a^{-\frac{1}{2}} \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a, \\
A_2(a) & = a\bigg \{ \tanh \Big(\frac{a}{2}\Big) \Big ( \cosh^4 \Big(\frac{a}{2}\Big) \cosh^2 a + \sinh^2 a \Big ) + \coth \Big(\frac{a}{2}\Big) + \sinh a \bigg \} ,\\
A_3(a) & =\cosh^2\Big(\frac{a}{2}\Big) \bigg \{ \tanh \Big(\frac{a}{2}\Big) \sinh a \cdot\Big ( \cosh^2\Big(\frac{a}{2}\Big) \cosh a + 1 \Big ) + 1 \bigg \}.
\end{split}
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Consider a $1$-form $v \in \Lambda^1(B_O(1))$ and a smooth function $P \in C^{\infty}(B_O(1)),$ which satisfy equation \eqref{LinearStokeswithForcingNEW}. Under the coordinate system $Y : \mathbb{H}^2(-a^2) \rightarrow D_0(1)$, we express $v$ as $v = v_1 \dd Y^1 + v_2 \dd Y^2$. We also express $F$ as $F = F_1 \dd Y^1 + F_2 \dd Y^2$.
For each $j =1,2$, we define the function $v_j^{\sharp}$ by $v_j^{\sharp} = v_j \circ Y^{-1}$, and the function $F_j^{\sharp}$ by $F_j^{\sharp} = F_j \circ Y^{-1}$. We also write $P^{\sharp} = P \circ Y^{-1}$. Then, saying that the pair $(v, P)$ satisfies \eqref{LinearStokeswithForcingNEW} on the geodesic ball $B_O(1)$ is equivalent to saying that the $\mathbb{R}^2$-valued function $v^{\sharp} = (v_1^{\sharp} , v_2^{\sharp} )$ and the function $P^{\sharp}$ satisfy the following system of equations on the Euclidean disc $D_0\big(\tanh \big(\frac{a}{2}\big)\big)$ (see \cite{CC13}).
\begin{equation}gin{equation}\label{Equation3.14NEW}
\begin{equation}gin{split}
\frac{a^2 (1-|y|^2)^2}{4}\Big(-\Delta^{\Bbb R^{2}}v^{\sharp}_{1}+\frac{4y^2(\partial_{2}v^{\sharp}_{1}-\partial_{1}v^{\sharp}_{2})}{1-|y|^2} \Big) + 2a^2 v_1^{\sharp} +\partial_1 P^\sharp &= F_1^{\sharp} ,\\
\frac{a^2 (1-|y|^2)^2}{4}\Big(-\Delta^{\Bbb R^{2}}v^{\sharp}_{2}+\frac{4y^1(\partial_{2}v^{\sharp}_{1}-\partial_{1}v^{\sharp}_{2})}{1-|y|^2} \Big ) + 2a^2 v_2^{\sharp}+\partial_2 P^\sharp&=F_2^{\sharp},\\
\text{div}\ v^{\sharp}&=0 .
\end{split}
\end{equation}
By a direct computation, we get
\begin{equation}gin{equation}
\begin{equation}gin{split}
\int_{B_O(1)} \big | F \big |_a^{\frac{4}{3}} \mathbb Vol_{\mathbb{H}^2(-a^2)} & = \int_{D_0(\tanh (\frac{a}{2}))} \big ( (F_1^{\sharp})^2 + (F_2^{\sharp})^2 \big )^{\frac{2}{3}} \bigg ( \frac{4}{a^2 (1-|y|^2)^2} \bigg )^{\frac{1}{3}} \dd y^1 \wedge \dd y^2 \\
& \geq \frac{4^{\frac{1}{3}}}{a^{\frac{2}{3}}} \int_{D_0(\tanh (\frac{a}{2}))} \big | F^{\sharp}\big |^{\frac{4}{3}} \mathbb Vol_{\mathbb{R}^2},
\end{split}
\end{equation}
which immediately gives
\begin{equation}gin{equation}\label{FL-12NEW}
\big \| F^{\sharp} \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \leq \Big(\frac{a}{2}\Big)^{\frac{1}{2}}
\big \| F\big \|_{L^{\frac{4}{3}}(B_O(1))} .
\end{equation}
Next, for convenience we rephrase \eqref{Equation3.14NEW} as
\begin{equation}gin{equation}\label{Equation1NEW}
\frac{a^2 (1-|y|^2)^2}{4}\big(-\Delta^{\Bbb R^{2}}v^{\sharp} \big ) +a^2 (1-|y|^2) (\partial_2v_1^{\sharp} - \partial_1 v_2^{\sharp} ) y^{\perp} + 2a^2 v^{\sharp} + \nabla P^\sharp = F^{\sharp},
\end{equation}
where $y^{\perp} = (y^2 , y^1)$.
Next, we have to estimate $\big \|\nabla P \big \|_{L^{-1,2}(D_0(\tanh (\frac{a}{2})))}$. To that end, we first estimate
\begin{equation}gin{equation*}
\bigg \| \frac{a^2 (1-| \cdot |^2)^2}{4} \big ( -\Delta^{\Bbb R^{2}}v^{\sharp} \big ) \bigg \|_{L^{-1,2}(D_0(\tanh (\frac{a}{2})))}.
\end{equation*}
Let $\varphi \in C^\infty_{c} (D_0(\tanh (\frac{a}{2})))$, then
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& \bigg | \bigg < \frac{a^2 (1-| \cdot |^2)^2}{4} \big ( -\Delta^{\Bbb R^{2}}v^{\sharp} \big ) , \varphi \bigg > _{L^{-1,2}(D_0(\tanh (\frac{a}{2}))) \otimes L^{1,2}_0(D_0(\tanh (\frac{a}{2})))} \bigg | \\
= & \bigg | \int_{D_0(\tanh (\frac{a}{2}))} \frac{a^2 (1-| y |^2)^2}{4} \big ( -\Delta^{\Bbb R^{2}}v^{\sharp} \big ) \cdot \varphi \bigg | \\
= & \bigg | \int_{D_0(\tanh (\frac{a}{2}))} \nabla^{\Bbb R^{2}} v^{\sharp} : \nabla^{\Bbb R^{2}} \bigg \{ \frac{a^2 (1-|y|^2)^2}{4} \cdot \varphi \bigg \} \bigg | \\
\leq & \bigg | \int_{D_0(\tanh (\frac{a}{2}))} \nabla^{\Bbb R^{2}} v^{\sharp} : \nabla^{\Bbb R^{2}} \bigg ( \frac{a^2 (1-|y|^2)^2}{4}\bigg ) \cdot \varphi \bigg | + \bigg | \int_{D_0(\tanh (\frac{a}{2}))}\frac{a^2 (1-|y|^2)^2}{4} \nabla^{\Bbb R^{2}} v^{\sharp} : \nabla^{\Bbb R^{2}} \varphi \bigg | \\
\leq & \bigg | \int_{D_0(\tanh (\frac{a}{2}))} a^2 \big (1-| y |^2 \big ) \nabla^{\Bbb R^{2}} v^{\sharp} : y \varphi \bigg | + \frac{a^2}{4}
\big \| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))}\cdot \big \| \nabla^{\Bbb R^{2}} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
\leq & a^2 \tanh \Big (\frac{a}{2}\Big ) \int_{D_0(\tanh (\frac{a}{2}))} \big | \nabla^{\Bbb R^{2}} v^{\sharp}\big | \big | \varphi \big | + \frac{a^2}{4}
\big \| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big \| \nabla^{\Bbb R^{2}} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
\leq & a^2 \tanh \Big (\frac{a}{2}\Big ) \big \| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big \| \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} + \frac{a^2}{4}
\big \| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big \| \nabla^{\Bbb R^{2}} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
\leq & C_0 a^{2} \Big ( 1+ \tanh^2 \Big (\frac{a}{2}\Big )\Big ) \big\| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big\| \nabla^{\Bbb R^{2}} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} .
\end{split}
\end{equation*}
In the last line of the above estimate, we employed the standard Poincar\'e inequality
\begin{equation}gin{equation}\label{PoincareinequalityNEW}
\big\| \varphi \big\|_{L^2(D_0(r))} \leq C_0 r \big\| \nabla^{\mathbb{R}^2} \varphi \big\|_{L^2(D_0(r))}.
\end{equation}
To summarize, we have
\begin{equation}gin{equation}\label{Estimate1NEW}
\bigg \| \frac{a^2 (1-| \cdot |^2)^2}{4} \big ( -\Delta^{\Bbb R^{2}}v^{\sharp} \big ) \bigg \|_{L^{-1,2}(D_0(\tanh (\frac{a}{2})))} \leq C_0A_0(a) \big\| \nabla^{\Bbb R^{2}} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} ,
\end{equation}
where the absolute constant $C_0$ is independent of $a$, and
\[
A_0(a)= a^{2} \Big ( 1 + \tanh^2 \Big(\frac{a}{2}\Big) \Big ) .
\]
Also, it follows from \eqref{FL-12NEW} that
the following estimate holds for any $\varphi \in C^\infty_{c} \big(D_0 \big(\tanh \big(\frac{a}{2}\big)\big)\big) .$
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\bigg | \int_{D_0(\tanh (\frac{a}{2}))} F^{\sharp} \cdot \varphi \bigg | & \leq \big \| F^{\sharp} \big \|_{L^{\frac{4}{3}} (D_0(\tanh (\frac{a}{2})))} \big \| \varphi \big \|_{L^4 (D_0 (\tanh (\frac{a}{2} )))} \\
& \leq \big \| F^{\sharp} \big \|_{L^{\frac{4}{3}} (D_0(\tanh (\frac{a}{2})))} C_0 \big \| \varphi \big \|_{L^2 (D_0 (\tanh (\frac{a}{2} )))}^{\frac{1}{2}} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2 (D_0 (\tanh (\frac{a}{2} )))}^{\frac{1}{2}} \\
& \leq C_0 \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \| F^{\sharp} \big \|_{L^{\frac{4}{3}} (D_0(\tanh (\frac{a}{2})))} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2 (D_0 (\tanh (\frac{a}{2} )))} \\
& \leq C_0 \Big (\frac{a}{2} \Big )^{\frac{1}{2}} \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \|F\|_{L^{\frac{4}{3}} (B_O(1))} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2 (D_0 (\tanh (\frac{a}{2} )))} ,
\end{split}
\end{equation*}
which gives
\begin{equation}gin{equation}\label{L-12estimate2NEW}
\big \| F^{\sharp} \big \|_{L^{-1,2} (D_0 (\tanh (\frac{a}{2})))} \leq C_0\Big (\frac{a}{2} \Big )^{\frac{1}{2}} \Big ( \tanh\Big (\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \|F \big \|_{L^{\frac{4}{3}} (B_O(1))} .
\end{equation}
Next, estimate
\begin{equation}gin{align}
& \bigg | \int_{D_0(\tanh (\frac{a}{2}))} a^2 (1-|y|^2) (\partial_2 v_1^{\sharp} - \partial_1 v_2^{\sharp} ) y^{\perp} \cdot \varphi \bigg | \nonumber\\
&\quad\leq 2a^2 \int_{D_0 (\tanh (\frac{a}{2}))} \big | \nabla^{\mathbb{R}^2} v^{\sharp} \big | |y| |\varphi| \nonumber\\
&\quad\leq 2a^2 \tanh \Big(\frac{a}{2}\Big) \int_{D_0(\tanh (\frac{a}{2}))} \big | \nabla^{\mathbb{R}^2} v^{\sharp} \big | |\varphi| \nonumber\\
&\quad\leq 2a^2 \tanh \Big(\frac{a}{2}\Big) \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2 (D_0 (\tanh (\frac{a}{2})))} \big \| \varphi \big \|_{L^2(D_0 (\tanh (\frac{a}{2})))}\nonumber \\
&\quad\leq C_0 a^2 \tanh^2 \Big(\frac{a}{2}\Big) \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2 (D_0 (\tanh (\frac{a}{2})))} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2(D_0 (\tanh (\frac{a}{2})))} \label{L-12ThreeNEW},
\end{align}
where in the last line we again used \eqref{PoincareinequalityNEW}.
Using the easy fact that $ \big \| v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} = \big \| v \big \|_{L^2(B_O(1))}$, we also have
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\bigg | 2a^2 \int_{D_0(\tanh (\frac{a}{2}))} v^{\sharp} \varphi \bigg | & \leq
2a^2 \big \| v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big \| \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
& \leq C_0 a^2 \tanh \Big(\frac{a}{2}\Big) \big \| v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
& = C_0 a^2 \tanh \Big(\frac{a}{2}\Big) \big \| v \big \|_{L^2(B_O(1))} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2(D_0(\tanh (\frac{a}{2})))}.
\end{split}
\end{equation*}
So
\begin{equation}gin{equation}\label{TrivialL-12NEW}
\big \| 2a^2 v^{\sharp} \big \|_{L^{-1,2} (D_0(\tanh (\frac{a}{2})))} \leq C_0 a^2 \tanh \Big(\frac{a}{2}\Big) \big \| v \big \|_{L^2(B_O(1))}.
\end{equation}
By combining estimates \eqref{Estimate1NEW}, \eqref{L-12estimate2NEW}, \eqref{L-12ThreeNEW}, and \eqref{TrivialL-12NEW}, we deduce
\begin{equation}gin{equation}\label{L-12pressure}
\begin{equation}gin{split}
\big \| \nabla^{\mathbb{R}^2} P^\sharp \big \|_{L^{-1,2} (D_0(\tanh (\frac{a}{2})))}
\leq & C_0 a^2 \Big ( 1 + \tanh^2 \Big(\frac{a}{2}\Big) \Big ) \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2 (D_0(\tanh (\frac{a}{2})))} \\
& + C_0 a^2 \tanh\Big (\frac{a}{2}\Big) \big \| v \big \|_{L^2(B_O(1))}\\
& + C_0 a^{\frac{1}{2}} \Big ( \tanh\Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} .\\
\end{split}
\end{equation}
Now, by combining \eqref{Equation3.1NEW} and \eqref{L-12pressure}, we obtain
\begin{equation}gin{equation}\label{EstimatenablapressureNEW}
\begin{equation}gin{split}
\big \| \nabla^{\mathbb{R}^2} P^{\sharp} \big \|_{L^{-1,2} (D_0(\tanh (\frac{a}{2})))}
\leq & C_0 a^2 \bigg \{ \Big ( 1 + \tanh^2 \Big(\frac{a}{2}\Big) \Big) \sinh a + \tanh \Big(\frac{a}{2}\Big) \bigg \} \big \| v \big \|_{L^2(B_O(1))} \\
& + C_0 a \Big( \cosh^2\Big ( \frac{a}{2} \Big) + \sinh^2\Big (\frac{a}{2}\Big) \Big ) \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& + C_0 a^{\frac{1}{2}} \Big ( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \|F \|_{L^{\frac{4}{3}} (B_O(1))} .
\end{split}
\end{equation}
At this point, we employ the following fact from the regularity theory for Navier-Stokes equation \cite{Seregin}.
\begin{equation}gin{itemize}
\item For any $R > 0$, and any $P \in L^1_{loc} ( D_0(R) )$ which satisfies $\nabla^{\mathbb{R}^2} P \in L^{-1,2} (D_0(R))$, it follows that there exists some $c \in \mathbb{R}$ such that
\begin{equation}gin{equation}
\big \| P - c\big \|_{L^2 (D_0(R))} \leq C_0 \big \| \nabla^{\mathbb{R}^2} P \big \|_{L^{-1,2} (D_0(R))} ,
\end{equation}
where the absolute constant $C_0 > 0$ is \emph{independent} of $R$. (We note the similarity with \eqref{EASYPressureNEW}. We just want to stress the independence of $C_0$ from $R$).
\end{itemize}
So, it follows from \eqref{EstimatenablapressureNEW} that, we can find some $c \in \mathbb{R}$ such that $P^{\sharp}-c$ satisfies
\begin{equation}gin{equation}\label{EstimatePminusLNEW}
\begin{equation}gin{split}
\big \| P^{\sharp} - c \big \|_{L^2 (D_0(\tanh (\frac{a}{2})))}
\leq & C_0 a^2 \bigg \{\Big( 1 + \tanh^2 \Big(\frac{a}{2}\Big)\Big) \sinh a + \tanh\Big (\frac{a}{2}\Big) \bigg \} \big \| v \big \|_{L^2(B_O(1))} \\
& + C_0 a \Big ( \cosh^2\Big( \frac{a}{2} \Big ) + \sinh^2 \Big(\frac{a}{2}\Big) \Big ) \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& + C_0 a^{\frac{1}{2}} \Big( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \|F \|_{L^{\frac{4}{3}} (B_O(1))}.
\end{split}
\end{equation}
Next, rearranging \eqref{Equation1NEW} we get
\begin{equation}gin{equation*}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} v^{\sharp} + \nabla^{\mathbb{R}^2} \bigg \{\frac{4}{a^2 (1-|y|^2)^2} \cdot (P^{\sharp}-c) \bigg\}& = \Psi , \\
\dv v^{\sharp} & = 0,
\end{split}
\end{equation*}
where
\[
\Psi = \frac{4}{a^2 (1-|y|^2)^2} F^{\sharp} + \frac{16}{a^2} \frac{1}{(1-|y|^2)^3} (P^{\sharp} -c ) y - \frac{8}{(1-|y|^2)^2} v^{\sharp} - \frac{4}{(1-|y|^2)} (\partial_2v_1^{\sharp} - \partial_1 v_2^{\sharp}) y^{\perp}.
\]
By applying Lemma \ref{LinfityrescaledNEW} directly to $v^{\sharp}$, it follows that $v^{\sharp}$ satisfies the following estimate
\begin{equation}gin{equation}\label{straightforwardNEW}
\begin{equation}gin{split}\displaystyle
\big \| v^{\sharp} \big \|_{L^{\infty}(D_0(\frac{1}{2} \tanh (\frac{a}{2})))} \leq &
C_0 \bigg \{ \Big( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))}\\
&\quad+ \displaystyle\coth\Big(\frac{a}{2}\Big) \big \| v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))}
+ \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2}) ))} \bigg \}.
\end{split}
\end{equation}
Next we use \eqref{Equation3.1NEW} in \eqref{straightforwardNEW} to get
\begin{equation}gin{equation}\label{supnormvsharpNEW}
\begin{equation}gin{split}
&\big \| v^{\sharp} \big \|_{L^{\infty}(D_0(\frac{1}{2} \tanh (\frac{a}{2})))} \leq
C_0 \bigg \{ \Big ( \tanh \Big(\frac{a}{2} \Big) \ \Big)^{\frac{1}{2}} \big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \\
& \qquad+ \Big( \coth \Big(\frac{a}{2}\Big) + \sinh a \Big) \big \|v \big \|_{L^2(B_O(1))} + \frac{1}{a} \cosh^2 \Big(\frac{a}{2} \Big)
\big \| \nabla v \big \|_{L^2(B_O(1))} \bigg \}.
\end{split}
\end{equation}
Now, observe that the following estimate holds for any $y \in D_0(\tanh (\frac{a}{2}))$.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
|\Psi (y)| & \leq \frac{4}{a^2}\cosh^4\Big(\frac{a}{2}\Big) \big | F^{\sharp} \big | + \frac{16}{a^2}\big | P^{\sharp}-c \big |\tanh \Big(\frac{a}{2}\Big) \cosh^6 \Big(\frac{a}{2}\Big)+ 8\cosh^4 \Big(\frac{a}{2}\Big) \big | v^{\sharp}\big |\\
&\qquad + 8 \tanh\Big (\frac{a}{2}\Big)\cosh^2\Big(\frac{a}{2}\Big) \big | \nabla^{\mathbb{R}^2}v^{\sharp}\big | \\
& = \frac{4}{a^2}\cosh^4\Big(\frac{a}{2}\Big) \big | F^{\sharp} \big | + \frac{8}{a^2}\sinh a \cosh^4 \Big(\frac{a}{2}\Big) \big | P^{\sharp}-c \big |
+ 8 \cosh^4 \Big(\frac{a}{2}\Big) \big | v^{\sharp} \big | + 4 \sinh a \big | \nabla^{\mathbb{R}^2} v^{\sharp} \big |.
\end{split}
\end{equation*}
Then applying \eqref{FL-12NEW} and the Holder's inequality $\|f\|_{L^{\frac{4}{3}}(D_0(r))} \leq |D_0(r)|^{\frac{1}{4}}\|f\|_{L^2(D_0(r))}$, it follows directly from the above estimate that
\begin{equation}gin{equation*}\label{psicomplicatedone}
\begin{equation}gin{split}
& \big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \\
&\ \leq 8\cosh^4\Big(\frac{a}{2}\Big) \bigg \{ \frac{1}{a^2} \big \|F^{\sharp} \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))}
+ \frac{\sinh a}{a^2} \big \| P^{\sharp} - c \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \\
&\quad + \big \|v^{\sharp} \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \bigg \} + 4 \sinh a \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^{\frac{4}{3}} (D_0(\tanh (\frac{a}{2})))} \\
&\ \leq \frac{C_0}{a^{\frac{3}{2}}} \cosh^4 \Big(\frac{a}{2}\Big) \big \| F \big \|_{L^{\frac{4}{3}}(B_O(1))}
+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}}
\bigg \{ \frac{\sinh a }{a^2} \big \|P^{\sharp} - c \big \|_{L^2(D_0 (\tanh (\frac{a}{2})) )} \\
& \quad+ \big \|v^{\sharp} \big \|_{L^2(D_0 (\tanh (\frac{a}{2})))} \bigg \} + C_0 \sinh a \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}}
\big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))}
\end{split}
\end{equation*}
We now use \eqref{EstimatePminusLNEW} in the above estimate to obtain
\begin{equation}gin{equation}\label{DelicatepsiNEW}
\begin{equation}gin{split}
& \big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \\
&\quad\leq \frac{C_0}{a^{\frac{3}{2}}} \cosh^4 \Big(\frac{a}{2}\Big) \big \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \qquad\quad+ C_0 \cosh^4\Big(\frac{a}{2}\Big) \Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \sinh a \bigg \{ \Big(1+ \tanh^2\Big(\frac{a}{2}\Big)\Big) \sinh a + \tanh \Big(\frac{a}{2}\Big) \bigg \}
\big \|v \big \|_{L^2(B_O(1))} \\
& \qquad\quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \frac{\sinh a}{a} \cosh a \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& \qquad\quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \tanh \Big(\frac{a}{2}\Big) \frac{\sinh a}{a^{\frac{3}{2}}} \big \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \qquad \quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big( \tanh\Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \| v \big \|_{L^2(B_O(1))} \\
& \qquad\quad+ C_0 \sinh a \Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} \\
&\quad\leq \frac{C_0}{a^{\frac{3}{2}}} \cosh^4\Big (\frac{a}{2}\Big) \Big \{ 1+ \tanh \Big(\frac{a}{2}\Big) \sinh a \Big \} \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \qquad\quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big)\Big ( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \bigg \{ \Big( 1 + \tanh^2\Big(\frac{a}{2}\Big) \Big ) \sinh^2 a + \tanh \Big(\frac{a}{2}\Big) \sinh a + 1 \bigg \} \big \| v \big \|_{L^2(B_O(1))} \\
& \qquad\quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big)\Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \frac{\sinh (2a)}{a} \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& \qquad\quad+ C_0 \sinh a \Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} .
\end{split}
\end{equation}
To simplify the above computations, we observe that the following relations hold.
\begin{equation}gin{equation}\label{hyptrgidentity}
\begin{equation}gin{split}
1+ \tanh \Big(\frac{a}{2}\Big) \sinh a & = \cosh a ,\\
\Big( 1 + \tanh^2\Big(\frac{a}{2}\Big) \Big) \sinh^2 a + \tanh \Big(\frac{a}{2}\Big) \sinh a + 1 & = \cosh a \Big( 1 + 4 \sinh^2\Big(\frac{a}{2}\Big) \Big) \\
& \leq 2 \cosh a \Big( 1 + 2 \sinh^2 \Big(\frac{a}{2}\Big)\Big ) \\
& = 2 \cosh^2 a .
\end{split}
\end{equation}
So, by using the relations in \eqref{hyptrgidentity}, we can now greatly simplify estimate \eqref{DelicatepsiNEW} as follows.
\begin{equation}gin{equation}\label{psisimpleONENEW}
\begin{equation}gin{split}
\big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))} \leq & \frac{C_0}{a^{\frac{3}{2}}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
&\quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big ( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \cosh^2 a \big \| v \big \|_{L^2(B_O(1))} \\
&\quad + C_0 \cosh^4 \Big(\frac{a}{2}\Big)\Big ( \tanh \Big(\frac{a}{2}\Big)\Big )^{\frac{1}{2}} \frac{\sinh (2a)}{a} \big \| \nabla v \big \|_{L^2(B_O(1))} \\
&\quad + C_0 \sinh a \Big ( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \big \| \nabla^{\mathbb{R}^2} v^{\sharp} \big \|_{L^2(D_0(\tanh (\frac{a}{2})))} .
\end{split}
\end{equation}
Now, by applying estimate \eqref{Equation3.1NEW} in \eqref{psisimpleONENEW} we get
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))}
\leq & \frac{C_0}{a^{\frac{3}{2}}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \quad+ C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big( \tanh \Big(\frac{a}{2}\Big)\Big )^{\frac{1}{2}} \cosh^2 a \big \| v \big \|_{L^2(B_O(1))} \\
&\quad + C_0 \cosh^4 \Big(\frac{a}{2}\Big) \Big( \tanh\Big (\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \frac{\sinh (2a)}{a} \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& \quad+ C_0 \sinh a \Big ( \tanh\Big (\frac{a}{2}\Big)\Big)^{\frac{1}{2}} \frac{1}{a} \cosh^2 \Big(\frac{a}{2}\Big) \big \| \nabla v \big \|_{L^2(B_O(1))} \\
& \quad+ C_0 \sinh a \Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \sinh a \big \| v \big \|_{L^2(B_O(1))} ,
\end{split}
\end{equation*}
which is equivalent to the following estimate
\begin{equation}gin{equation}\label{PsiFinalEstimate}
\begin{equation}gin{split}
&\big \| \Psi \big \|_{L^{\frac{4}{3}}(D_0(\tanh (\frac{a}{2})))}
\leq \frac{C_0}{a^{\frac{3}{2}}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
&\qquad + C_0 \Big( \tanh \Big(\frac{a}{2}\Big) \Big )^{\frac{1}{2}} \Big \{ \cosh^4 \Big(\frac{a}{2}\Big) \cosh^2 a + \sinh^2 a \Big \} \big \| v \big \|_{L^2(B_O(1))} \\
&\qquad + \frac{C_0}{a} \Big( \tanh \Big(\frac{a}{2}\Big)\Big )^{\frac{1}{2}} \sinh a \cosh^2\Big (\frac{a}{2}\Big) \Big \{ \cosh^2 \Big(\frac{a}{2}\Big) \cosh a + 1 \Big \} \big \| \nabla v \big \|_{L^2(B_O(1))} .
\end{split}
\end{equation}
Next, by combining \eqref{supnormvsharpNEW} with \eqref{PsiFinalEstimate}, we deduce
\begin{equation}gin{equation}\label{QuiteTedious}
\begin{equation}gin{split}
& \big \| v^{\sharp} \big \|_{L^{\infty} (D_0 (\frac{1}{2} \tanh (\frac{a}{2})))} \\
&\quad\leq \frac{C_0}{a^{\frac{3}{2}}} \Big( \tanh \Big(\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \quad\qquad+ C_0 \tanh \Big(\frac{a}{2}\Big) \Big \{ \cosh^4 \Big(\frac{a}{2}\Big) \cosh^2 a + \sinh^2 a \Big \} \big \| v \big \|_{L^2(B_O(1))} \\
& \quad\qquad+ \frac{C_0}{a} \tanh \Big(\frac{a}{2}\Big) \sinh a \cosh^2 \Big(\frac{a}{2}\Big) \bigg \{ \cosh^2\Big (\frac{a}{2}\Big) \cosh a + 1 \bigg \} \big \| \nabla v \big \|_{L^2(B_O(1))} \\
&\quad\qquad + \Big (\coth\Big(\frac{a}{2}\Big) + \sinh a \Big ) \big \| v \big \|_{L^2 (B_O(1))} \\
& \quad\qquad+ \frac{1}{a} \cosh^2\Big(\frac{a}{2}\Big) \big \| \nabla v \big \|_{L^2(B_O(1))} \\
&\quad= \frac{C_0}{a^{\frac{3}{2}}}\Big( \tanh \Big(\frac{a}{2}\Big)\Big)^{\frac{1}{2}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
& \qquad+ C_0 \bigg \{ \tanh\Big (\frac{a}{2}\Big) \Big( \cosh^4 \Big(\frac{a}{2}\Big) \cosh^2 a + \sinh^2 a \Big) + \coth\Big(\frac{a}{2}\Big) + \sinh a \bigg \} \big \| v \big \|_{L^2(B_O(1))} \\
& \qquad+ \frac{C_0}{a} \cosh^2\Big(\frac{a}{2}\Big) \bigg \{ \tanh \Big(\frac{a}{2}\Big) \sinh a \Big ( \cosh^2\Big(\frac{a}{2}\Big) \cosh a + 1 \Big ) + 1 \bigg \} \big \| \nabla v \big \|_{L^2(B_O(1))}.
\end{split}
\end{equation}
Now, we recall the definition for the positive number $r(a) > 0$ given in \eqref{quitesimpler}
\[
r(a) = \frac{1}{a} \log \Big ( \frac{1 + 3 e^a }{3+e^a } \Big ) .
\]
Note that we have the following relation, which holds on $D_0(1)$.
\begin{equation}gin{equation}\label{Quitestraightforward}
| v |_a \circ Y^{-1} = \frac{a (1-|y|^2)}{2}| v^{\sharp} | \leq \frac{a}{2} | v^{\sharp} | .
\end{equation}
So, \eqref{QuiteTedious} and \eqref{Quitestraightforward} together give the following estimate:
\begin{equation}gin{equation*}
\begin{equation}gin{split}
&\big \| v \big \|_{L^{\infty} (B_O(r(a)))} \\
&\quad \leq\frac{a}{2} \big \| v^{\sharp} \big \|_{L^{\infty} (D_0 (\frac{1}{2} \tanh (\frac{a}{2})))} \\
&\quad\leq \frac{C_0}{a^{\frac{1}{2}}} \Big ( \tanh\Big (\frac{a}{2}\Big) \Big)^{\frac{1}{2}} \cosh^4\Big(\frac{a}{2}\Big) \cosh a \| F \big \|_{L^{\frac{4}{3}}(B_O(1))} \\
&\qquad + C_0 a \bigg \{ \tanh \Big(\frac{a}{2}\Big) \Big ( \cosh^4 \Big(\frac{a}{2}\Big) \cosh^2 a + \sinh^2 a \Big ) + \coth\Big(\frac{a}{2}\Big) + \sinh a \bigg \} \big \| v \big \|_{L^2(B_O(1))} \\
&\qquad + C_0 \cosh^2\Big(\frac{a}{2}\Big) \bigg \{ \tanh \Big(\frac{a}{2}\Big) \sinh a \bigg ( \cosh^2\Big(\frac{a}{2}\Big) \cosh a + 1 \bigg ) + 1 \bigg \} \big \| \nabla v \big \|_{L^2(B_O(1))} ,
\end{split}
\end{equation*}
which is exactly estimate \eqref{Supnormestimate} as required in the conclusion of Lemma \ref{Supnormelliptic}.
\end{proof}
\section{Pointwise decay of the velocity profile}\label{section_v_decay}
Starting here, we will consider, for each $R > 0$, the exterior domain $$\Omega (R) = \mathbb{H}^2(-a^2) - \overline{B_O(R)}.$$ We first establish the following lemma. The proof is similar to the proof of an estimate on the full hyperbolic space as it was established in \cite{CC15}, but to show it on the exterior domain, we need to use the cut-off functions from Gilbarg and Weinberger \cite{GilbargWeinberger1978}.
\begin{equation}gin{lemma}\label{GlobalH1giveL2}
Let $R_0 > 0$ to be given. Consider now a divergence-free $1$-form $v \in \Lambda^1_{\sigma} (\Omega (R_0) ),$ which satisfies the following finite Dirichlet integral property.
\begin{equation}gin{equation}\label{FDIproperty}
\int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} < \infty .
\end{equation}
Then, it follows that $v$ satisfies the following a priori estimate for each $R_1 > R_0$.
\begin{equation}gin{equation}\label{GlobalH1inducesL2}
\int_{\Omega (R_1)} \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \leq \frac{2}{a^2} \bigg \{ 2 + \frac{18}{a^2} \bigg ( \frac{4}{(R_1-R_0)} \bigg )^2 \bigg \}\int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
Let $v \in \Lambda^1 (\Omega (R_0) ) \cap C^0 (\overline{\Omega (R_0) })$ satisfy \eqref{FDIproperty}. We now consider a cut off function $\varphi_1 \in C^{\infty} ([0,\infty ))$ which satisfies the following.
\begin{equation}gin{equation}\label{CutoffpropertyONE}
\begin{equation}gin{split}\displaystyle
& \chi_{_{[R_1 , \infty )}} \leq \varphi_1 \leq \chi_{_{[\frac{R_0 + R_1}{2} , \infty )}}, \\
& \big |\varphi_1' \big | \leq \frac{4}{(R_1-R_0)} \chi_{_{[\frac{R_0 + R_1}{2} , R_1 ]}} .
\end{split}
\end{equation}
Next, we also need another cut off function $\varphi_2 \in C_c^{\infty} ([0,2 ))$ such that
\begin{equation}gin{equation}\label{CutoffpropertyTWO}
\begin{equation}gin{split}
& \chi_{_{[0,1]}} \leq \varphi_2 \leq \chi_{_{[0,2]}} ,\\
& \big | \varphi_2' \big | \leq 2 \cdot \chi_{_{[1,2]}} .
\end{split}
\end{equation}
Now, let us select a fixed $R_1 > R_0$, and let $R \geq \max \{ R_1 , 1 \}$, with respect to which we consider the cut-off function
$\eta_{_{R}} \in C_c^{\infty} (\Omega (R_0))$ defined as follows.
\begin{equation}gin{equation*}
\eta_{_{R}} (x) = \varphi_1 (\rho (x)) \cdot \varphi_2 \Big(\frac{\rho (x)}{R}\Big) ,
\end{equation*}
where $\rho (x)$ is the geodesic distance in $\mathbb{H}^2(-a^2)$ from $O$ to $x$. Then by the definition of $\eta_{_{R}}$
\begin{equation}gin{equation}\label{GradEtaR}
\nabla \eta_{_{R}} (x) = \varphi_1'(\rho (x)) \nabla \rho (x) \varphi_2\Big (\frac{\rho (x)}{R}\Big) + \varphi_1(\rho (x))\varphi_2' \Big(\frac{\rho (x)}{R}\Big) \frac{\nabla \rho (x)}{R} .
\end{equation}
Since $\abs{\nabla \rho(x)}_a=1$, \eqref{CutoffpropertyONE}, \eqref{CutoffpropertyTWO} and \eqref{GradEtaR} give
\begin{equation}gin{equation}\label{estimateGradEtaR}
\big | \nabla \eta_{_{R}} (x) \big |_a \leq \frac{4}{(R_1 -R_0)} + \frac{2}{R},
\end{equation}
for all $x \in \Omega (R_0)$.
We now recall the Bochner-Weitzenb\"ock formula
\[
\nabla^\ast\nabla=\dd^\ast \dd+\dd\dd^\ast-\mathbb Ric,
\]
so for the divergence-free $1$-form $v$ on $\Omega(R_0)$ we get
\begin{equation}gin{equation*}
\nabla^* \nabla v = \dd^*\dd v + a^2 v ,
\end{equation*}
from which we yield
\begin{equation}gin{equation}\label{VERYTRIVIAL}
\int_{\Omega (R_0)} g ( \nabla^* \nabla v , \eta_{_{R}}^2 v ) \mathbb Vol_{\mathbb{H}^2(-a^2)} = \int_{\Omega (R_0)} g (\dd^* \dd v , \eta_{_{R}}^2 v ) \mathbb Vol_{\mathbb{H}^2(-a^2)}
+ a^2 \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{equation}
Since $\eta_{_{R}}^2 v \in \Lambda_c^1 (\Omega (R_0))$, we can do integration by parts as follows.
\begin{equation}gin{equation}\label{Integrationbypart1}
\begin{equation}gin{split}
& \int_{\Omega (R_0)} g ( \nabla^* \nabla v , \eta_{_{R}}^2 v ) \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
= & \int_{\Omega (R_0)} g (\nabla v , \nabla (\eta_{_{R}}^2 v ) ) \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
= & \int_{\Omega (R_0)} g ( \nabla v , 2 \eta_{_{R}} \dd \eta_{_{R}} \otimes v ) \mathbb Vol_{\mathbb{H}^2(-a^2)} + \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation}
In the same way, we have
\begin{equation}gin{equation}\label{Integrationbypart2}
\begin{equation}gin{split}
\int_{\Omega (R_0)} g ( \dd^* \dd v , \eta_{_{R}}^2 v ) \mathbb Vol_{\mathbb{H}^2(-a^2)}
& = \int_{\Omega (R_0)} g ( \dd v , 2 \eta_{_{R}} \dd \eta_{_{R}} \wedge v ) \mathbb Vol_{\mathbb{H}^2(-a^2)} + \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | \dd v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{split}
\end{equation}
Recall that we have the following standard estimates
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big | \dd v \big |_a & \leq \big | \nabla v \big |_a , \\
\big | \dd \eta_{_{R}} \wedge v \big |_a & \leq 2 \big | \nabla \eta_{_{R}} \big |_a \cdot \big | v \big |_a .
\end{split}
\end{equation*}
So, \eqref{VERYTRIVIAL}, \eqref{Integrationbypart1}, and \eqref{Integrationbypart2} together give the following estimate.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& a^2 \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 2 \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + 6 \int_{\Omega (R_0)} \eta_{_{R}} \big |\nabla \eta_{_{R}} \big |_a
\big | v \big |_a \big |\nabla v \big |_a \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + 3 \varepsilon \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + \frac{3}{\varepsilon} \int_{\Omega (R_0)} \big |\nabla \eta_{_{R}} \big |_a^2 \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation*}
By taking $\varepsilon = \frac{a^2}{6}$ in the above estimate, it follows through applying \eqref{estimateGradEtaR} that the following estimate holds
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& \frac{a^2}{2} \int_{\Omega (R_0)} \eta_{_{R}}^2 \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + \frac{18}{a^2} \bigg ( \frac{4}{(R_1-R_0)} + \frac{2}{R} \bigg )^2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad= \bigg \{ 2 + \frac{18}{a^2} \bigg ( \frac{4}{(R_1-R_0)} + \frac{2}{R} \bigg )^2 \bigg \} \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation*}
By taking $R$ to $\infty$ we get the estimate \eqref{GlobalH1inducesL2} as needed.
\end{proof}
We are now ready to establish Theorem \ref{VelocityDecayThm}.
\subsection{Proof of Theorem \ref{VelocityDecayThm}}
\begin{equation}gin{proof}
Let $v$ be as stated in the hypotheses. Since $v$ satisfies \eqref{FDIforVelDecay}, we can apply Lemma \ref{GlobalH1giveL2} to get
\begin{equation}gin{equation*}
\int_{\Omega (R_0 + 1)} \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \leq \frac{2}{a^2} \Big ( 2 + \frac{18\cdot 16}{a^2} \Big ) \int_{\Omega (R_0)}
\big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{equation*}
Now, we take a smooth function $\varphi \in C^{\infty} ([0,\infty ))$ which satisfies the following properties.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& \chi_{_{[R_0 + 2 ,\infty )}} \leq \varphi \leq \chi_{_{[R_0 +1 , \infty )}} ,\\
& \big | \varphi ' \big | \leq 2 \chi_{_{[R_0 + 1 ,R_0 + 2 ]}} .
\end{split}
\end{equation*}
Next, we consider the radially symmetric cut-off function $\eta \in C^{\infty} (\mathbb{H}^2(-a^2))$
\begin{equation}gin{equation*}
\eta (x) = \varphi (\rho (x)).
\end{equation*}
Let $w = \eta v$. Notice that the support of $\eta$ lies in $\Omega (R_0 + 1)$. This tells us that $w$ can be regarded as a globally defined smooth $1$-form on the whole space-form $\mathbb{H}^2(-a^2)$ and that the support of $w$ also lies in $\Omega (R_0 + 1)$. That is, we have $w \in \Lambda^1 (\mathbb{H}^2(-a^2))$. Then, $w$ clearly satisfies the following properties.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\int_{\mathbb{H}^2(-a^2)} \big | w \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} & \leq \int_{\Omega (R_0 + 1)} \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} , \\
\int_{\mathbb{H}^2(-a^2)} \big | \nabla w \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} & \leq 2 \int_{\Omega (R_0 + 1)} \big | \dd \eta \big |_a^2 \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + 2 \int_{\Omega (R_0 + 1)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
& \leq 8\int_{\Omega (R_0 + 1)} \big | v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + 2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation*}
So $w \in H^1(\mathbb{H}^2(-a^2))$. The hyperbolic Ladyzhenskaya inequality gives (see for example \cite{CC13})
\begin{equation}gin{equation*}
\big \| w \big \|_{L^4(\mathbb{H}^2(-a^2))} \leq C_a \Big \{ \big \| w \big \|_{L^2 (\mathbb{H}^2(-a^2))} + \big \| \nabla w \big \|_{L^2 (\mathbb{H}^2(-a^2))} \Big \} .
\end{equation*}
Now, notice that $w(x) = v (x)$ holds for all $x \in \Omega (R_0 + 2)$. So, we have the following straightforward estimate.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\int_{\Omega (R_0 +2)} \big | \nabla_v v \big |_a^{\frac{4}{3}} \mathbb Vol_{\mathbb{H}^2(-a^2)} & \leq \int_{\mathbb{H}^2(-a^2)} \big | \nabla_w w \big |^{\frac{4}{3}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
& \leq \int_{\mathbb{H}^2(-a^2)} \big | w \big |_a^{\frac{4}{3}} \big | \nabla w \big |_a^{\frac{4}{3}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
& \leq \Big ( \int_{\mathbb{H}^2(-a^2)} \big | w \big |_a^4 \mathbb Vol_{\mathbb{H}^2(-a^2)} \Big )^{\frac{1}{3}} \Big ( \int_{\mathbb{H}^2(-a^2)} \big |\nabla w \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \Big )^{\frac{2}{3}} .
\end{split}
\end{equation*}
Now, let us take an arbitrary point $x_0 \in \Omega (R_0 + 8)$. Then, applying Lemma \ref{Supnormelliptic} over the geodesic ball $B_{x_0} (1)$, we immediately obtain \begin{equation}gin{equation}\label{SecondbeforeLast}
\big \|v\big \|_{L^{\infty} (B_{x_0} (r(a)) )} \leq C_0 \Big \{ A_1(a) \big \| \nabla_v v \big \|_{L^{\frac{4}{3}} (B_{x_0}(1))}
+ A_2(a) \|v \|_{L^2(B_{x_0}(1))} + A_3 (a) \big \| \nabla v \big \|_{L^2(B_{x_0}(1))} \Big \}.
\end{equation}
Now, since we have $\nabla_v v \in L^{\frac{4}{3}}(\Omega (R_0 +2))$, $v \in L^2(\Omega (R_0 + 1 )) $ and $\nabla v \in L^2(\Omega (R_0))$, it follows that we have the following decay properties.
\begin{equation}gin{equation}\label{EsayDECAY}
\begin{equation}gin{split}
\lim_{\rho(x_0) \rightarrow \infty } \big \| \nabla_v v \big \|_{L^{\frac{4}{3}}(B_{x_0}(1))} & = 0 , \\
\lim_{\rho(x_0) \rightarrow \infty } \big \| v \big \|_{L^2(B_{x_0}(1))} & = 0 ,\\
\lim_{\rho(x_0) \rightarrow \infty } \big \| \nabla v \big \|_{L^2(B_{x_0}(1))} & = 0.
\end{split}
\end{equation}
So, by combining \eqref{SecondbeforeLast} with \eqref{EsayDECAY}, we have
\begin{equation}gin{equation}
\lim_{\rho (x_0) \rightarrow \infty } \big \|v\big \|_{L^{\infty} (B_{x_0} (r(a)) )} = 0.
\end{equation}
This completes the proof of Theorem \ref{VelocityDecayThm}.
\end{proof}
\section{About the vorticity.}\label{section_vorticity}
In this section, we show vorticity is in $H^1$, which is used to establish the rate of decay in the far range, Theorem \ref{FinalThmforExpdecay}. As in the last section, we will use the notation $\Omega (R) = \mathbb{H}^2(-a^2) - \overline{B_O(R)}$, for any $R > 0$. We start with the $H^1$ property.
\subsection{ $H^1$-property of the vorticity.} The statement of the following theorem and the proof is based on Gilbarg and Weinberger \cite[Lemma 2.3]{GilbargWeinberger1978}.
\begin{equation}gin{thm}\label{L2propertyvorticity}
Let $R_0 > 0$. Consider a smooth $1$-form $v \in \Lambda^1_{\sigma} (\Omega (R_0)),$ which satisfies the following stationary Navier-Stokes equation on $\Omega (R_0)$, with $P$ to be some smooth function on $\Omega (R_0)$.
\begin{equation}gin{equation}\label{NSforvorticity}
\begin{equation}gin{split}
2 \Def^* \Def v + \nabla_v v + \dd P & = 0 ,\\
\dd^* v & = 0.
\end{split}
\end{equation}
Suppose that $v$ also satisfies
\begin{equation}gin{equation}\label{FDNpropertyVort}
\int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} < \infty.
\end{equation}
Consider $\omega \in C^{\infty} (\Omega (R_0))$ to be the function defined as follows.
\begin{equation}gin{equation}\label{DefofVorticity}
\omega = * \dd v .
\end{equation}
Then, for any $R_1 > R_0$, the following a priori estimate holds.
\begin{equation}gin{equation}\label{Vorticity2.23}
\begin{equation}gin{split}
& \int_{\Omega (R_1)} \big | \nabla \omega \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 2a^2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + C(a,R_0 , R_1 ) \int_{A\big(\frac{R_0 + R_1}{2} ,R_1\big )} \big | \omega \big |_a^2 \big ( 1 + \big | v \big |_a \big ) \mathbb Vol_{\mathbb{H}^2(-a^2)},
\end{split}
\end{equation}
where $A\big(\frac{R_0 + R_1}{2} , R_1\big) = \big\{x \in \mathbb{H}^2(-a^2) : \frac{R_0 + R_1}{2} < \rho (x) < R_1 \big\}$ .
\end{thm}
\begin{equation}gin{proof}
Here, we closely follow the main ideas of a lemma by Gilbarg and Weinberger \cite[Lemma 2.3]{GilbargWeinberger1978}. So, we take an arbitrary positive number $L > 0$, which plays the role of the level of truncation. With respect to $L$, we consider the associated function $h_{_{L}} \in C^1 (\mathbb{R})$ which is defined as follows.
\begin{equation}gin{equation}\label{DefofhL}
h_{_{L}}(\lambda ) = \lambda^2 \chi_{_{\{|\lambda| \leq L \}} } + \big \{ 2 L ( |\lambda | - L ) + L^2 \big \} \chi_{_{ \{ |\lambda | > L \}} } .
\end{equation}
Hence, it follows that
\begin{equation}gin{equation*}
h_{_{L}}'(\lambda ) = -2L \chi_{_{\{ \lambda < -L \}} } + 2 \lambda \chi_{_{ \{ |\lambda | \leq L \} }} + 2L \chi_{_{\{ \lambda > L \}}} ,
\end{equation*}
and that
\begin{equation}gin{equation*}
h_{_{L}}''(\lambda ) = 2\chi_{_{\{ |\lambda | < L \}}} .
\end{equation*}
Now, take any $R_1 > R_0$.
As in the proof of Lemma \ref{GlobalH1giveL2}, we consider the very same cut off functions $\varphi_1 \in C^{\infty} ([0,\infty )) $ , $\varphi_2 \in C^{\infty} ( [0,2 ))$ which are characterized by conditions \eqref{CutoffpropertyONE} and \eqref{CutoffpropertyTWO} respectively. We also use the same $\eta_R$, for each $R \geq \max \{ R_1 , 1 \}$, which was given by
\begin{equation}gin{equation}\label{DefETA_R}
\eta_{_{R}} (x) = \varphi_1 (\rho (x) ) \varphi_2 \Big ( \frac{\rho (x)}{R} \Big ).
\end{equation}
Now, notice that we have
\begin{equation}gin{equation*}
\eta_{_{R}} \chi_{_{\{ R \leq \rho \leq 2 R \}}} = \varphi_2 \Big (\frac{\rho}{R}\Big ) \chi_{_{ \{ R \leq \rho \leq 2 R \}} }.
\end{equation*}
Hence, it follows
\begin{equation}gin{equation}\label{Vorticity2.7}
\nabla \eta_{_{R}} \chi_{_{\{ R \leq \rho \leq 2R \}}} = \varphi_2' \Big ( \frac{\rho}{R} \Big ) \frac{\nabla \rho }{R} \chi_{_{\{ R \leq \rho \leq 2R \}}} ,
\end{equation}
which directly gives
\begin{equation}gin{equation}\label{Vorticity2.8}
\big |\nabla \eta_{_{R}} \big | \chi_{_{\{ R \leq \rho \leq 2R \}}} \leq \frac{2}{R} \chi_{_{\{ R \leq \rho \leq 2R \}}} .
\end{equation}
Identity \eqref{Vorticity2.7} leads to
\begin{equation}gin{equation*}
\Delta \eta_{_{R}} \chi_{_{\{ R \leq \rho \leq 2R \}}} = \bigg ( \varphi_2'' \Big(\frac{\rho}{R} \Big) \frac{1}{R^2} + \varphi_2' \Big(\frac{\rho}{R} \Big) \frac{\Delta \rho}{R} \bigg ) \chi_{_{\{ R \leq \rho \leq 2R \}}} .
\end{equation*}
Since $\Delta \rho = a \coth (a \rho )$ holds on $\mathbb{H}^2(-a^2)-{O}$, it follows from the above identity that
\begin{equation}gin{equation}\label{Vorticity2.9}
\begin{equation}gin{split}
\big | \Delta \eta_{_{R}} \big | \chi_{_{\{ R \leq \rho \leq 2R \}}} & \leq \bigg ( \frac{1}{R^2} \big \|\varphi_2'' \big \|_{L^{\infty}([0,2))} +
\frac{2a\coth (a\rho )}{R} \bigg ) \chi_{_{\{ R \leq \rho \leq 2R \}}} \\
& \leq \bigg ( \frac{1}{R^2} \big \|\varphi_2'' \big \|_{L^{\infty}([0,2))} +
\frac{2a\coth (a R )}{R} \bigg ) \chi_{_{\{ R \leq \rho \leq 2R \}}} ,
\end{split}
\end{equation}
where the second inequality follows from the fact that $\coth (t)$ is monotone decreasing in $t \in (0,\infty )$.\\
Recall $\omega=\ast \dd v$, which by definition is equivalent to
\begin{equation}gin{equation*}
\dd v = \omega \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{equation*}
So using the estimate
\begin{equation}gin{equation*}
\big | \dd v \big |_a \leq \big | \nabla v \big |_a,
\end{equation*}
and the finite Dirichlet-norm property \eqref{FDNpropertyVort} we get
\begin{equation}gin{equation}\label{Vorticity2.10}
\begin{equation}gin{split}
\int_{\Omega (R_0)} \big | \omega \big |^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} & = \int_{\Omega (R_0)} \big | \dd v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \leq \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{split}
\end{equation}
Now, as in the paper by Gilbarg and Weinberger, we carry out the following computation
\begin{equation}gin{equation}\label{Vorticity2.11}
\begin{equation}gin{split}
{2}\eta_{_{R}} \chi_{_{\{ |\omega | < L \}}} \big | \nabla \omega \big |_a^2
=& \eta_{_{R}} h_{_{L}}''(\omega ) g( \nabla \omega , \nabla \omega) \\
= & \eta_{_{R}} g( \nabla (h_{_{L}}'(\omega )) ,\nabla \omega) \\
= & \dv \big \{ \eta_{_{R}} h_{_{L}}'(\omega ) \nabla \omega \big \} - g(\nabla \eta_{_{R}} , h_{_{L}}'(\omega ) \nabla \omega) - \eta_{_{R}} h_{_{L}}'(\omega ) \Delta \omega \\
= & \dv \big \{ \eta_{_{R}} h_{_{L}}'(\omega ) \nabla \omega \big \} - g(\nabla \eta_{_{R}} , \nabla \big ( h_{_{L}}(\omega ) \big )) - \eta_{_{R}} h_{_{L}}'(\omega ) \Delta \omega .
\end{split}
\end{equation}
Next, taking $* \dd$ on both sides of the first line of \eqref{NSforvorticity}, we obtain the following equation satisfied by the vorticity function $\omega$ on $\Omega (R_0)$.
\begin{equation}gin{equation}\label{Vorticity2.12}
-\Delta \omega + 2a^2 \omega + g(v , \nabla \omega) = 0 .
\end{equation}
By using \eqref{Vorticity2.12}, it follows from identity \eqref{Vorticity2.11} that
\begin{equation}gin{equation}\label{Vorticity2.13}
\begin{equation}gin{split}
&{2}\int_{\Omega (R_0)} \big | \nabla \omega \big |_a^2 \eta_{_{R}} \chi_{_{\{ |\omega | < L \}} } \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
= & \int_{\Omega (R_0)} -g(\nabla \eta_{_{R}} , \nabla \big ( h_{_{L}}(\omega )\big ) )- \eta_{_{R}} h_{_{L}}'(\omega ) \big ( 2a^2 \omega + g(v, \nabla \omega \big )) \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
= & \int_{\Omega (R_0)} -g(\nabla \eta_{_{R}} , \nabla \big ( h_{_{L}}(\omega )\big )) - \eta_{_{R}} g( v , \nabla \big ( h_{_{L}}(\omega )\big ) )- 2a^2 h_{_{L}}'(\omega ) \eta_{_{R}} \omega \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
= & \int_{\Omega (R_0)} \big ( \Delta \eta_{_{R}} +g( v, \nabla \eta_{_{R}} \big ) h_{_{L}}(\omega ) )\mathbb Vol_{\mathbb{H}^2(-a^2)} - 2a^2 \int_{\Omega (R_0)} h_{_{L}}'(\omega ) \eta_{_{R}} \omega \mathbb Vol_{\mathbb{H}^2(-a^2)} ,
\end{split}
\end{equation}
where the last equality follows from integration by parts and the divergence-free property $\dd^* v = 0$ of $v$. \\
Since we know the following straightforward estimate
\begin{equation}gin{equation*}
\big | h_{_{L}} ' (\omega ) \big | \leq 2\min \{ |\omega | , L \} ,
\end{equation*}
it follows, by taking \eqref{Vorticity2.10} into our account, that we have
\begin{equation}gin{equation}\label{Vorticity2.14}
\begin{equation}gin{split}
\bigg | \int_{\Omega (R_0)} h_{_{L}}'(\omega ) \eta_{_{R}} \omega \mathbb Vol_{\mathbb{H}^2(-a^2)} \bigg | & \leq 2 \int_{\Omega (R_0)} \big | \omega \big |^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \leq 2 \int_{\Omega(R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation}
Recall
\begin{equation}gin{equation*}
A(r_1 , r_2) = \big \{ x \in \mathbb{H}^2(-a^2) : r_1 \leq \rho(x) \leq r_2 \big \},
\end{equation*}
and observe that we have
\begin{equation}gin{equation}\label{Vorticity2.15}
\begin{equation}gin{split}
& \bigg | \int_{\Omega (R_0)} \big ( \Delta \eta_{_{R}} + g(v, \nabla \eta_{_{R}} )\big ) h_{_{L}}(\omega ) \mathbb Vol_{\mathbb{H}^2(-a^2)} \bigg | \\
&\quad \leq \int_{A\big( \frac{R_0 + R_1}{2} , R_1 \big)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big |\Delta \big (\varphi_1(\rho)\big ) \big | + \big | v \big |_a \big | \nabla \big ( \varphi_1 (\rho ) \big ) \big |_a \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\qquad\quad + \int_{A(R, 2R)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big | \Delta \eta_{_{R}} \big | + \big | g(v, \nabla \eta_{_{R}} )\big | \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation}
By using the obvious relation $0 \leq h_{_{L}}(\lambda ) \leq \lambda^2$, the first integral which appears on the right-hand side of \eqref{Vorticity2.15} will be controlled as follows.
\begin{equation}gin{equation}\label{simplesimplesimple1000}
\begin{equation}gin{split}
& \int_{A\big( \frac{R_0 + R_1}{2} , R_1 \big)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big |\Delta \big ( \varphi_1 (\rho ) \big ) \big | + \big | v \big |_a \big | \nabla \big ( \varphi_1(\rho ) \big ) \big |_a \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq C(a, R_0 , R_1) \int_{A\big( \frac{R_0 + R_1}{2} , R_1 \big)} \big | \omega \big |^2 \big ( 1 + \big | v \big |_a \big ) \mathbb Vol_{\mathbb{H}^2(-a^2)} < \infty ,
\end{split}
\end{equation}
where the absolute constant $C(a, R_0, R_1)$ is just
\begin{equation}gin{equation*}
C(a, R_0 , R_1) = \big \| \Delta \big ( \varphi_1 (\rho ) \big ) \big \|_{L^{\infty} (\mathbb{H}^2(-a^2))} + \big \| \nabla \big ( \varphi_1 (\rho ) \big ) \big \|_{L^{\infty} (\mathbb{H}^2(-a^2))} .
\end{equation*}
Next, we have to prove that the second integral which appears on the right-hand side of \eqref{Vorticity2.15} tends to $0$ as $R$ goes to infinity. We now achieve this as follows. First, notice that we have the following straightforward estimate, which holds for any $\lambda \in \mathbb{R}$.
\begin{equation}gin{equation}\label{estimateofh_{_{L}}}
\big | h_{_{L}}( \lambda ) \big | \leq \min \big \{ \lambda ^2 , 2 L |\lambda | \big \} .
\end{equation}
So, by combining \eqref{Vorticity2.8}, \eqref{Vorticity2.9} with \eqref{estimateofh_{_{L}}}, it follows that
\begin{equation}gin{equation}\label{Vorticity2.17}
\begin{equation}gin{split}
& \int_{A(R, 2R)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big | \Delta \eta_{_{R}} \big | + \big | g(v ,\nabla \eta_{_{R}} )\big | \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq \bigg ( \frac{1}{R^2} \big \| \varphi_2'' \big \|_{L^{\infty} ([0,2))} + \frac{2a \coth (aR)}{R} \bigg ) \int_{A(R, 2R)} \big | \omega \big |^2
\mathbb Vol_{\mathbb{H}^2(-a^2)} \\
& \qquad\qquad\qquad\qquad+ \frac{4}{R} \int_{A(R, 2R)} L |\omega | \big | v \big |_a \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq \bigg ( \frac{1}{R^2} \big \| \varphi_2'' \big \|_{L^{\infty} ([0,2))} + \frac{2a \coth (aR)}{R} \bigg ) \int_{A(R, 2R)} \big | \omega \big |^2
\mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\qquad\qquad\qquad\qquad + \frac{4L}{R} \big \| \omega \big \|_{L^2(A(R, 2R))} \big \| v \big \|_{L^2(A(R, 2R))} .
\end{split}
\end{equation}
Independently, we also observe that since we have $\omega \in L^2(\Omega (R_0))$ and $ v \in L^2(\Omega (R_0))$, it must hold that
\begin{equation}gin{equation*}
\lim_{R \rightarrow \infty } \bigg \{ \int_{A(R, 2R)} |\omega |^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + \int_{A(R, 2R)} |v|^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \bigg \} = 0 .
\end{equation*}
So, it follows from \eqref{Vorticity2.17} that we have
\begin{equation}gin{equation}\label{Vorticity2.18}
\lim_{R\rightarrow \infty} R \int_{A(R, 2R)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big | \Delta \eta_{_{R}} \big | + \big | g(v ,\nabla \eta_{_{R}} )\big | \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} = 0.
\end{equation}
Actually, \eqref{Vorticity2.18} immediately implies the following weaker conclusion.
\begin{equation}gin{equation}\label{Vorticity2.19}
\lim_{R\rightarrow \infty} \int_{A(R, 2R)} \big | h_{_{L}}(\omega ) \big | \Big \{ \big | \Delta \eta_{_{R}} \big | + \big | g(v , \nabla \eta_{_{R}} )\big | \Big \} \mathbb Vol_{\mathbb{H}^2(-a^2)} = 0.
\end{equation}
\eqref{Vorticity2.19} allows us to pass to the limit on both sides of \eqref{Vorticity2.15}, and then using \eqref{simplesimplesimple1000} we have
\begin{equation}gin{equation}\label{Vorticity2.20}
\begin{equation}gin{split}
& \limsup_{R\rightarrow \infty} \bigg | \int_{\Omega (R_0)} \big ( \Delta \eta_{_{R}} + g(v , \nabla \eta_{_{R}} ) \big ) h_{_{L}}(\omega ) \mathbb Vol_{\mathbb{H}^2(-a^2)} \bigg | \\
&\quad\leq C(a, R_0 , R_1) \int_{A \big(\frac{R_0 + R_1}{2} , R_1 \big )} |\omega |^2 \big ( 1 + \big | v \big |_a \big ) \mathbb Vol_{\mathbb{H}^2(-a^2)}.
\end{split}
\end{equation}
Now, from the definition of $\varphi_1$ and $\eta_R$ we get
\begin{equation}gin{align*}
&2\int_{\Omega (R_1)} \big | \nabla \omega \big |_a^2 \chi_{_{\{ |\omega | < L \}}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq {2}\int_{\Omega (R_0)} \big | \nabla \omega \big |_a^2 \varphi_1 (\rho ) \chi_{_{\{ |\omega | < L \}}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad= 2 \lim_{R \rightarrow \infty } \int_{\Omega (R_0)} \big | \nabla \omega \big |_a^2 \eta_{_{R}} \chi_{_{\{ |\omega | < L \}}} \mathbb Vol_{\mathbb{H}^2(-a^2)},
\end{align*}
so by means of \eqref{Vorticity2.14} and \eqref{Vorticity2.20}, we now pass to the limit on both sides of \eqref{Vorticity2.13} to deduce
\begin{equation}gin{equation}\label{Vorticity2.21}
\begin{equation}gin{split}
& 2\int_{\Omega (R_1)} \big | \nabla \omega \big |_a^2 \chi_{_{\{ |\omega | < L \}}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq2 \lim_{R \rightarrow \infty } \int_{\Omega (R_0)} \big | \nabla \omega \big |_a^2 \eta_{_{R}} \chi_{_{\{ |\omega | < L \}}} \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 4a^2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + \limsup_{R\rightarrow \infty} \bigg | \int_{\Omega (R_0)} \big ( \Delta \eta_{_{R}} + v \nabla \eta_{_{R}} \big ) h_{_{L}}(\omega ) \mathbb Vol_{\mathbb{H}^2(-a^2)} \bigg | \\
&\quad\leq 4a^2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + C(a, R_0 , R_1) \int_{A \big(\frac{R_0 + R_1}{2} , R_1 \big)} |\omega |^2 \big ( 1 + \big | v \big |_a \big ) \mathbb Vol_{\mathbb{H}^2(-a^2)} .
\end{split}
\end{equation}
Finally, we take $L \rightarrow \infty$ on both sides of \eqref{Vorticity2.21} to obtain
\begin{equation}gin{equation*}
\begin{equation}gin{split}
& \int_{\Omega (R_1)} \big | \nabla \omega \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} \\
&\quad\leq 2a^2 \int_{\Omega (R_0)} \big | \nabla v \big |_a^2 \mathbb Vol_{\mathbb{H}^2(-a^2)} + C(a, R_0 , R_1) \int_{A \big(\frac{R_0 + R_1}{2} , R_1 \big )} |\omega |^2 \big ( 1 + \big | v \big |_a \big ) \mathbb Vol_{\mathbb{H}^2(-a^2)} ,
\end{split}
\end{equation*}
which is exactly estimate \eqref{Vorticity2.23} as required in the statement of Theorem \ref{L2propertyvorticity}.
\end{proof}
\subsection{About the pointwise decay of the vorticity in the far range.}
As before, take a fixed $R_0 > 0$, and consider a smooth $1$-form $v \in \Lambda_{\sigma}^1 (\Omega (R_0)),$ which is a solution to \eqref{NSforvorticity} on the exterior domain $\Omega (R_0) = \mathbb{H}^2(-a^2)-\overline{B_O(R_0)}$. Let $\omega = * \dd v$ be the associated vorticity function of $v$. Then, as before it follows that $\omega$ satisfies \begin{equation}gin{equation}\label{VorticityeqNEW}
- \Delta \omega + 2a^2 \omega + g(v ,\nabla \omega) = 0.
\end{equation}
Also, estimate \eqref{Vorticity2.23} of Theorem \ref{L2propertyvorticity} informs us that $\omega$ has the following property.
\begin{equation}gin{equation*}
\nabla \omega \in L^2 (\Omega (R_0 + 1 )).
\end{equation*}
Next, we take any $x_0 \in \Omega (R_0 +2)$, so that we have $\rho (x_0) > R_0 +2$, and note
\begin{equation}gin{equation*}
B_{x_0}(1) \subset \Omega (R_0 + 1 ) .
\end{equation*}
Now, since $\omega \in L^2(\Omega (R_0))$, and $\nabla \omega \in L^2(\Omega (R_0 + 1 ))$, we can at once deduce that
\begin{equation}gin{equation}\label{limitingW12}
\lim_{\rho (x_0)\rightarrow \infty} \big \| \omega \big \|_{W^{1,2}(B_{x_0}(1))} = 0 .
\end{equation}
Since $\mathbb{H}^2(-a^2)$ is homogeneous in that its spatial structure around one reference point is \emph{identical} to its spatial structure around any other reference point, up to some isometric transformation on $\mathbb{H}^2(-a^2)$, we can simply regard the base point $x_0$ as the vertex $(\frac{1}{a}, 0 , 0)$ of the hyperboloid model of $\mathbb{H}^2(-a^2)$. Under this identification of $x_0$ with the point $(\frac{1}{a} , 0 , 0)$ in the hyperboloid model, the coordinate system
$Y : \mathbb{H}^2(-a^2) \rightarrow D_0(1)$ as defined in subsection \ref{Hyperboloid} now maps $x_0$ to the center $0$ of the unit disc $D_0(1)$. It is equally obvious that $Y$ maps the geodesic ball $B_{x_0}(1)$ onto $D_0(\tanh (\frac{a}{2}))$.
Now, under the following local coordinate system
\begin{equation}gin{equation*}
Y : B_{x_0}(1) \rightarrow D_0(\tanh (\frac{a}{2})),
\end{equation*}
the vorticity equation \eqref{VorticityeqNEW} as restricted on $B_{x_0} (1)$ now will have the following local representation on the Euclidean disc $D_0(\tanh (\frac{a}{2}))$.
\begin{equation}gin{equation}\label{localversionofVorteq}
\Delta^{\mathbb{R}^2} \omega^{\sharp} + \frac{8}{\big ( 1 - |y|^2\big )^2} \omega^{\sharp} + \sum_{1\leq i \leq 2} v_i^{\sharp} \partial_i \omega^{\sharp} = 0 ,
\end{equation}
where $\omega^{\sharp} = \omega \circ Y^{-1}$ and $v_j^{\sharp} = v_j \circ Y^{-1}$ (Recall that $v = v_1 \dd Y^1 + v_2 \dd Y^2$).
Also, a computation shows
\begin{equation}gin{equation}\label{boringONE}
\frac{1}{C_a} \big \|\omega \big \|_{H^1 (B_{x_0}(1))} \leq \big \| \omega^{\sharp} \big \|_{H^1(D_0(\tanh (\frac{a}{2})))} \leq C_a \big \| \omega \big \|_{H^1(B_{x_0}(1))}
\end{equation}
for some absolute constant $C_a > 1$, which depends only on $a$.
Now, we can apply standard local elliptic regularity \cite[Section 8.9, Thm 8.24]{GilbargTrudinger} directly to $\omega^{\sharp}$, as a solution to equation \eqref{localversionofVorteq}, to deduce that there exists a constant $\widetilde{C}(a, \|v\|_{\infty}) >0$ such that $\omega^{\sharp}$ satisfies the following apriori estimate.
\begin{equation}gin{equation}\label{boringTWO}
\big \| \omega^{\sharp} \big \|_{L^{\infty}(D_0(\frac{1}{2} \tanh(\frac{a}{2})))} \leq \widetilde{C}(a, \|v\|_{\infty}) \big \| \omega^{\sharp} \big \|_{L^2(D_0(\tanh(\frac{a}{2})))}.
\end{equation}
Recall that
\begin{equation}gin{equation}
r(a) = \frac{1}{a} \log \Big ( \frac{1+ 3e^a}{3+ e^a} \Big ) ,
\end{equation}
which satisfies the property that the coordinate chart $Y$ maps the geodesic disc $B_{x_0} (r(a))$ diffeomorphically onto $D_0(\frac{1}{2}\tanh (\frac{a}{2}))$.
So, through combining \eqref{boringONE} with \eqref{boringTWO}, we yield the following estimate.
\begin{equation}gin{equation}
\begin{equation}gin{split}
\big \| \omega \big \|_{L^{\infty} (B_{x_0}(r(a))) } & = \big \| \omega^{\sharp} \big \|_{L^{\infty} (D_0(\frac{1}{2}\tanh (\frac{a}{2})))} \\
& \leq \widetilde{C}(a, \|v\|_{\infty}) \big \| \omega^{\sharp} \big \|_{L^2(D_0(\tanh(\frac{a}{2})))}.\\
& \leq \widetilde{C}(a, \|v\|_{\infty}) C_a \big \| \omega \big \|_{H^1 (B_{x_0}(1))}.
\end{split}
\end{equation}
So, the limiting property \eqref{limitingW12} together with the above estimate gives
\begin{equation}gin{equation}\label{supdecay}
\lim_{\rho (x_0) \rightarrow \infty } \big \| \omega \big \|_{L^{\infty} (B_{x_0}(r(a))) } = 0,
\end{equation}
which confirms the fact that $\omega (x) \rightarrow 0$, as $\rho (x) \rightarrow \infty$.\\
Armed with \eqref{supdecay} we are finally ready to deduce the exponential decay rate for $\omega (x)$, as $\rho (x) \rightarrow \infty$.
\subsection{Proof of Theorem \ref{FinalThmforExpdecay}}
\begin{equation}gin{proof}
To begin, using the identity $\Delta \rho = a \coth (a \rho)$, and $g(\nabla \rho, \nabla \rho)=1$, we compute
\begin{equation}gin{equation}
\begin{equation}gin{split}
\Delta \big ( e^{-\delta \rho } \big ) & =\dv (\nabla e^{-\delta \rho } )\\
&=\dv (-\delta e^{-\delta \rho} \nabla \rho )\\
&=-\delta g(\nabla e^{-\delta\rho},\nabla \rho)-\delta e^{-\delta \rho}\Delta \rho\\
&= \delta^2 e^{-\delta \rho} - \delta a e^{-\delta \rho} \coth (a \rho ) \\
& \leq e^{-\delta \rho} \big \{ \delta^2 - \delta a \big \}.
\end{split}
\end{equation}
Next, smoothness of $v$, and Theorem \ref{VelocityDecayThm} imply
\begin{equation}gin{equation*}
\big \| v \big \|_{L^{\infty}(\Omega (R_1) )} < \infty.
\end{equation*}
In what follows, we use the abbreviation $\|v\|_{\infty}$ for $\big \| v \|_{L^{\infty}(\Omega(R_1) )}$. Observe the following straightforward estimate holds pointwise on $\Omega (R_1)$.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big |g( v, \nabla e^{-\delta \rho } )\big | & = \big | \delta e^{-\delta \rho}g( v, \nabla \rho) \big | \\
& \leq \delta \big \| v \big \|_{\infty} e^{-\delta \rho} .
\end{split}
\end{equation*}
Hence
\begin{equation}gin{equation}\label{quiteboringest1}
\begin{equation}gin{split}
L \big ( e^{-\delta \rho } \big ) \leq e^{-\delta \rho} \cdot \Big \{ \delta^2 + \delta \big ( \|v\|_{\infty} - a \big ) -2 a^2 \Big \} ,
\end{split}
\end{equation}
holds pointwise on $\Omega (R_1)$, where $L$ is the elliptic operator specified in \eqref{eoperator}.
Note that the two distinct roots of the quadratic equation $t^2 + \big( \|v\|_{\infty} - a \big )t -2a^2 = 0$ are given by
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\tau_1 & = -\frac{1}{2}\Big \{ \sqrt{( \|v\|_{\infty} - a )^2 + 8a^2 } + (\|v\|_{\infty} - a) \Big \} ,\\
\tau_2 & = \frac{1}{2} \Big \{ \sqrt{ ( \|v\|_{\infty} - a )^2 + 8a^2 } - (\|v\|_{\infty} - a) \Big \} .
\end{split}
\end{equation*}
It is obvious that $\tau_1 < 0 < \tau_2$, and that the relation $t^2 + \big( \|v\|_{\infty} - a \big )t -2a^2 < 0$ holds for any $t \in (\tau_1 , \tau_2)$. So, by just taking $t$ to be the constant $\delta (a, \|v\|_{\infty})$ as specified in \eqref{definedelta}, we yield the following estimate.
\begin{equation}gin{equation}\label{quitetrivialest1}
\big ( \delta (a, \|v\|_{\infty}) \big )^2 + \big( \|v\|_{\infty} - a \big )\cdot \delta (a, \|v\|_{\infty}) - 2a^2< 0 .
\end{equation}
So, \eqref{quiteboringest1} and \eqref{quitetrivialest1} give
\begin{equation}gin{equation}\label{beingsupsolution}
L \big ( e^{-\delta \rho } \big ) < 0 .
\end{equation}
Consider now the positive constant $A$ which is specified in \eqref{definitionofA}. Then the functions $Ae^{-\delta \rho}$ and $-Ae^{-\delta \rho}$ are supersolution and subsolution of $L$, respectively.
Next, by definition of $A$, the desired estimate \eqref{Finalexpdecay} holds so far for any
$x \in \partial B_O(R_1)$, so
\begin{equation}gin{equation}\label{boundaryproperty}
\begin{equation}gin{split}
-A e^{-\delta (R_1)} \leq \omega \big |_{\partial B_O(R_1)} \leq -A e^{-\delta (R_1)} .
\end{split}
\end{equation}
We note that we would like \eqref{Finalexpdecay} to hold for all $x\in \Omega(R_1)$. To see that this is in fact the case, we recall we have
\[
\lim_{|x| \rightarrow \infty } \omega (x) = 0
\]
so this allow us to use the comparison principle for the operator $L$ to deduce that estimate \eqref{Finalexpdecay} does hold for any $x \in \Omega (R_1)$. This completes the proof of Theorem \ref{FinalThmforExpdecay}.
\end{proof}
\section{Pressure: proof of Theorem \ref{thm_p}}\label{section_p}
Due to the work of Anderson \cite{Anderson} and Sullivan \cite{Sullivan}, we know there exists a smooth and bounded harmonic function $F$ that comes from a continuous boundary data $\phi$ at infinity (see also \cite{AndersonSchoen}). If the boundary data is chosen to be non-constant, then $F$ is nontrivial. Now let $v=\dd F$, and $P=-2a^2F-\frac 12\abs{\dd F}_a^2$, then \eqref{StatNSforvelocityDecay} is satisfied since (more details for these computations can be found in \cite{CC10})
\begin{equation}gin{align*}
2\Def^*\Def +\nabla_v v&=-\Delta v-2 \mathbb Ric(v)+ \nabla_v v\\
&=-\Delta (\dd F)-2\mathbb Ric(\dd F)+\frac 12\dd \abs{\dd F}_a^2\\
&=2a^2\dd F+\frac 12\dd \abs{\dd F}_a^2,
\end{align*}
and as shown in \cite{CC10}, $\abs{\dd F}_a \rightarrow 0$ at infinity, so $P\rightarrow-2a^2F=-2a^2 \phi\neq $ constant as needed.
\appendix
\section{Standard sup norm estimates}\label{appendixa}
The following is a derivation of what should be a standard $L^\infty$ estimate for the solution of the Stokes equation, and we only include it here for completeness. It is based on \cite{Seregin}, and we write it in the form that we apply it in the paper.
For each $r > 0$, we consider the Eucldiean disc $D_0(r) = \{ x \in \mathbb{R}^2 : |x| < r\}$. Consider a vector valued function $u \in C^{\infty} (D_0(1))$, and a function $P \in C^{\infty}(D_0(1))$ which satisfies the Stokes equation
\begin{equation}gin{equation}\label{EASYLINEARSTOKESNEW}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} u + \nabla^{\mathbb{R}^2} P & = F ,\\
\dv u & = 0 ,
\end{split}
\end{equation}
where the external force $F \in C^{\infty}(D_0(1)) \cap L^{\frac{4}{3}} (D_0(1))$. Our goal here is to derive an a priori estimate for
\begin{equation}gin{equation*}
\big \| \big ( \nabla^{\mathbb{R}^2} \big )^2 u \big \|_{L^{\frac{4}{3}} (D_0(\frac{1}{2}))}
\end{equation*}
in terms of $\|F\|_{L^{\frac{4}{3}} (D_0(1))}$, $\|u\|_{L^2(D_0(1))}$ and $\|\nabla^{\mathbb{R}^2} u\|_{L^2(D_0(1))}$.
To this end, we first carry out the following estimate, which holds for any test vector field $\varphi \in C^{\infty}_{c} ( D_0(1))$.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\bigg | \int_{D_0(1)} F \cdot \varphi \bigg | & \leq \big \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))} \big \| \varphi \big \|_{L^4 (D_0(1))} \\
& \leq \big \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))} C_0 \big \| \varphi \big \|_{L^2 (D_0(1))}^{\frac{1}{2}} \big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2 (D_0(1))}^{\frac{1}{2}} \\
& \leq C_0\big \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))}\big \| \nabla^{\mathbb{R}^2} \varphi \big \|_{L^2 (D_0(1))} ,
\end{split}
\end{equation*}
which gives
\begin{equation}gin{equation}\label{EASYFNEW}
\big \| F \big \|_{L^{-1,2} (D_0(1))} \leq C_0 \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))}.
\end{equation}
It is plain to see that we have
\begin{equation}gin{equation}\label{TRIVIALNEW}
\big \| -\Delta^{\mathbb{R}^2} u \big \|_{L^{-1,2}(D_0(1))} \leq \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))} .
\end{equation}
So \eqref{EASYLINEARSTOKESNEW}, \eqref{EASYFNEW} and \eqref{TRIVIALNEW} give
\begin{equation}gin{equation}\label{EASYnablaPressureNEW}
\big \| \nabla^{\mathbb{R}^2} P \big \|_{L^{-1,2} (D_0 (1))} \leq C_{0} \Big(\big \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))}\Big). \end{equation}
The fact that $\nabla^{\mathbb{R}^2} P \in L^{-1,2} (D_0(1))$ implies that there exists some $c \in \mathbb{R}$ for which the following a priori estimate holds \cite{Seregin}.
\begin{equation}gin{equation}\label{EASYPressureNEW}
\big \| P - c \big \|_{L^2 (D_0(1))} \leq C_0\big \| \nabla^{\mathbb{R}^2} P \big \|_{L^{-1,2} (D_0 (1))} .
\end{equation}
Hence \eqref{EASYnablaPressureNEW} and \eqref{EASYPressureNEW} together give
\begin{equation}gin{equation}\label{pressureLTWONEW}
\big \| P - c \big \|_{L^2 (D_0(1))} \leq C_{0} \Big(\big \|F\big \|_{L^{\frac{4}{3}} (D_0 (1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))} \Big).
\end{equation}
We can now rephrase the Stokes equation \eqref{EASYLINEARSTOKESNEW} as follows, with $P$ replaced by $P-c$.
\begin{equation}gin{equation}\label{EASYLINEARSTOKESNEWV2}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} u + \nabla^{\mathbb{R}^2} \big ( P-c \big ) & = F ,\\
\dv u & = 0 .
\end{split}
\end{equation}
Next, to localize $u$ to the ball $D_0(\frac{1}{2})$, we take a radially symmetric bump function $\eta \in C^{\infty} (D_0(1))$ which satisfies
$\chi_{_{D_0(\frac{1}{2})}} \leq \eta \leq \chi_{_{D_0(1)}}$. Then, it follows that $w = \eta u\in C^{\infty}(D_0(1))$ satisfies the following system of equations.
\begin{equation}gin{equation}\label{PDEforCoerciveEstNEW}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} w + \nabla^{\mathbb{R}^2} \big\{(P-c) \eta \big \} & = \eta F -2 \nabla^{\mathbb{R}^2} u \cdot \nabla^{\mathbb{R}^2}\eta - u \Delta^{\mathbb{R}^2} \eta + (P-c)\nabla^{\mathbb{R}^2} \eta ,\\
\dv w & = u\cdot \nabla^{\mathbb{R}^2}\eta .
\end{split}
\end{equation}
By applying the Cattabriga-Solonnikov estimate \cite{Seregin} to system \eqref{PDEforCoerciveEstNEW}, we deduce that $w$ satisfies
\begin{equation}gin{align}
& \big \| \big ( \nabla^{\mathbb{R}^2}\big )^2 w \big \|_{L^{\frac{4}{3}}(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} \big\{(P-c) \eta \big \}\big \|_{L^{\frac{4}{3}} (D_0(1))} \nonumber\\
\leq & C_0 \big \| \eta F -2 \nabla^{\mathbb{R}^2} u \cdot \nabla^{\mathbb{R}^2}\eta - u \Delta^{\mathbb{R}^2} \eta + (P-c)\nabla^{\mathbb{R}^2} \eta \big \|_{L^{\frac{4}{3}} (D_0(1)) } + C_0 \big \| \nabla^{\mathbb{R}^2} \big ( u \cdot \nabla^{\mathbb{R}^2}\eta\big ) \big \|_{L^{\frac{4}{3}} (D_0(1))} \nonumber\\
\leq &C_0 \Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| \nabla^{\mathbb{R}^2} \eta \big \|_{L^{\infty} (D_0(1))} \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| \big ( \nabla^{\mathbb{R}^2} \big )^2 \eta \big \|_{L^{\infty} (D_0(1))} \big \| u \big \|_{L^{\frac{4}{3}} (D_0(1))} \nonumber\\
& \qquad+ \big \| \nabla^{\mathbb{R}^2} \eta \big \|_{L^{\infty} (D_0(1))} \big \| P- c \big \|_{L^{\frac{4}{3}}(D_0(1))} \Big \} \nonumber\\
\leq & C_0 \Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))}
+ \big \| P - c \big \|_{L^2(D_0(1))} \Big \}, \label{CoerciveL4over3NEWEASY}
\end{align}
where in the last line we use the Holder's estimate $\|f\|_{L^{\frac{4}{3}}(D_0(1))} \leq \big |D_0(1) \big |^{\frac{1}{4}} \|f\|_{L^2(D_0(1))}$. Now, \eqref{pressureLTWONEW} and \eqref{CoerciveL4over3NEWEASY} together with the fact that $\eta\equiv 1$ on $D_0(\frac 12)$ we have
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big \| \big ( \nabla^{\mathbb{R}^2}\big )^2 u \big \|_{L^{\frac{4}{3}}(D_0(\frac{1}{2}))}
& \leq \big \| \big ( \nabla^{\mathbb{R}^2}\big )^2 w \big \|_{L^{\frac{4}{3}}(D_0(1))} \\
& \leq C_0 \Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))} \Big \} .
\end{split}
\end{equation*}
By the standard Sobolev embedding, the above estimate gives
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big \| \nabla^{\mathbb{R}^2} u \big \|_{L^4(D_0(\frac{1}{2}))} & \leq C_0 \Big \{ \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^{\frac{4}{3}}(D_0(\frac{1}{2}))} + \big \| \big ( \nabla^{\mathbb{R}^2} \big )^2 u \big \|_{L^{\frac{4}{3}}(D_0(\frac{1}{2}))}\Big \} \\
& \leq C_0 \Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))} \Big \}.
\end{split}
\end{equation*}
Of course, the standard Sobolev embedding also gives
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big \| u \big \|_{L^4(D_0(\frac{1}{2}))} & \leq C_0 \Big \{ \big \| u \big \|_{L^{\frac{4}{3}}(D_0(\frac{1}{2}))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^{\frac{4}{3}}(D_0(\frac{1}{2}))}\Big \} \\
& \leq C_0 \Big \{ \big \| u \big \|_{L^{2}(D_0(\frac{1}{2}))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^{2}(D_0(\frac{1}{2}))} \Big \}. \end{split}
\end{equation*}
Since we have the Morrey's type embedding $W^{1,4} (D_0(\frac{1}{2})) \subset C^{0,\frac{1}{2}} (D_0(\frac{1}{2}))$, it follows from the above two estimates that
\begin{equation}gin{equation}\label{yetanothersimplebutusefulestimate}
\begin{equation}gin{split}
\big \| u \big \|_{C^{0,\frac{1}{2}} (D_0(\frac{1}{2}))} & \leq C_0 \Big \{ \big \| u \big \|_{L^4(D_0(\frac{1}{2}))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^4(D_0(\frac{1}{2}))} \Big \} \\
& \leq C_0 \Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))}\Big \} .
\end{split}
\end{equation}
In the above argument, we have already established the following useful lemma.
\begin{equation}gin{lemma}\label{LinfityregularityNEW}
Consider a vector field $u \in C^{\infty} (D_0(1)) \cap W^{1,2} (D_0(1))$, and a function $P \in C^{\infty} (D_0(1))$ which together satisfy the following Stokes equation, with the external force $F \in C^{\infty}(D_0(1)) \cap L^{\frac{4}{3}} (D_0(1))$.
\begin{equation}gin{equation}\label{EASYLINEARSTOKESNEWTWO}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} u + \nabla^{\mathbb{R}^2} P & = F ,\\
\dv u & = 0 .
\end{split}
\end{equation}
Then, it follows that $u$ satisfies the following a priori estimate, with $C_0 > 0$ to be some absolute constant which depends only on the dimension of $\mathbb{R}^2$.
\begin{equation}gin{equation}\label{BootstrapNEW}
\big \| u \big \|_{L^{\infty} (D_0(\frac{1}{2}))} \leq C_0\Big \{ \big \|F \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(1))} \Big \} .
\end{equation}
\end{lemma}
\begin{equation}gin{remark}
Indeed, in the estimate \eqref{yetanothersimplebutusefulestimate}, $ \big \| u \big \|_{C^{0,\frac{1}{2}} (D_0(\frac{1}{2}))}$ can be replaced by $\big \| u \big \|_{L^{\infty} (D_0(\frac{1}{2}))}$. We decide to drop the Holder's semi-norm, since this will help us get a cleaner estimate in the process of rescaling a solution to \eqref{EASYLINEARSTOKESNEWTWO}.
\end{remark}
Now, fix $R > 0$. Suppose that we have a vector field $u \in C^{\infty} (D_0(R)) \cap H^1 (D_0(R))$ and a function $P \in C^{\infty} (D_0(R))$, which satisfy the linear Stokes equation \eqref{EASYLINEARSTOKESNEW} on $D_0(R)$, with an external force $F \in C^{\infty} (D_0(R))\cap L^{\frac{4}{3}} (D_0(R))$. Now, we consider the rescaled functions
$u_{_{R}} : D_0(1) \rightarrow \mathbb{R}^2$ and $P_{_{R}} : D_0(1) \rightarrow \mathbb{R}$ defined by
\begin{equation}gin{equation*}
\begin{equation}gin{split}
u_{_{R}}(y) & = R^{-2} u(R y) ,\\
P_{_{R}}(y) & = R^{-1} P (Ry) .
\end{split}
\end{equation*}
Then, the pair $(u_{_{R}} , P_{_{R}} )$ is a solution to the following linear Stokes equation on $D_0(1)$.
\begin{equation}gin{equation*}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} u_{_{R}} + \nabla^{\mathbb{R}^2} P_{_{R}} & = F_{_{R}} ,\\
\dv u_{_{R}} & = 0 ,
\end{split}
\end{equation*}
where $F_{_{R}} : D_0(1) \rightarrow \mathbb{R}^2$ is given by $F_{_{R}}(y) = F (R\cdot y )$, for all $y \in D_0(1)$. By applying estimate \eqref{BootstrapNEW} in Lemma \ref{LinfityregularityNEW} directly to the pair $(u_{_{R}} , P_{_{R}})$, we yield the following estimate
\begin{equation}gin{equation}\label{BootstrapRescaled}
\big \| u_{_{R}} \big \|_{L^{\infty} (D_0(\frac{1}{2}))} \leq C_0 \Big \{ \big \|F_{_{R}} \big \|_{L^{\frac{4}{3}} (D_0(1))} + \big \| u_{_{R}} \big \|_{L^2(D_0(1))} + \big \| \nabla^{\mathbb{R}^2} u_{_{R}} \big \|_{L^2(D_0(1))} \Big \} .
\end{equation}
Observe that we have
\begin{equation}gin{equation*}
\begin{equation}gin{split}
\big \|F_{_{R}} \big \|_{L^{\frac{4}{3}} (D_0(1))} & = R^{-\frac{3}{2}} \big \|F \big \|_{L^{\frac{4}{3}}(D_0(R))}, \\
\big \| u_{_{R}} \big \|_{L^2(D_0(1))} & = R^{-3} \big \| u \big \|_{L^2(D_0(R))}, \\
\big \| \nabla^{\mathbb{R}^2} u_{_{R}}\big \|_{L^2(D_0(1))} & = R^{-2} \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(R))}, \\
\big \| u_{_{R}} \big \|_{L^{\infty} (D_0(\frac{1}{2}))} & = R^{-2} \big \| u \big \|_{L^{\infty} (D_0(\frac{R}{2}))} .
\end{split}
\end{equation*}
In light of the above scaling properties, we can rephrase \eqref{BootstrapRescaled} as follows.
\begin{equation}gin{equation*}
\big \| u \big \|_{L^{\infty} (D_0(\frac{R}{2}))} \leq C_0 \Big \{ R^{\frac{1}{2}} \big \|F \big \|_{L^{\frac{4}{3}} (D_0(R))} + R^{-1}\big \| u \big \|_{L^2(D_0(R))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(R))} \Big\} .
\end{equation*}
The above argument clearly gives the following rescaled version of Lemma \ref{LinfityregularityNEW}.
\begin{equation}gin{lemma}\label{LinfityrescaledNEW}
Consider a vector field $u \in C^{\infty} (D_0(R)) \cap W^{1,2} (D_0(R))$, and a function $P \in C^{\infty} (D_0(R))$ which together satisfy the following Stokes equation, with the external force $F \in C^{\infty}(D_0(R)) \cap L^{\frac{4}{3}} (D_0(R))$.
\begin{equation}gin{equation}\label{EASYLINEARSTOKESNEWThree}
\begin{equation}gin{split}
-\Delta^{\mathbb{R}^2} u + \nabla^{\mathbb{R}^2} P & = F ,\\
\dv u & = 0.
\end{split}
\end{equation}
Then, it follows that $u$ satisfies the following a priori estimate, with $C_0 > 0$ to be some absolute constant which depends only on the dimension of $\mathbb{R}^2$.
\begin{equation}gin{equation}\label{BootstrapFinalNEW}
\big \| u \big \|_{L^{\infty} (D_0(\frac{R}{2}))} \leq C_0 \Big \{ R^{\frac{1}{2}}\big \|F \big \|_{L^{\frac{4}{3}} (D_0(R))} + R^{-1} \big \| u \big \|_{L^2(D_0(R))} + \big \| \nabla^{\mathbb{R}^2} u \big \|_{L^2(D_0(R))} \Big \} .
\end{equation}
\end{lemma}
\end{document} |
\begin{document}
\title{Unavoidable Subtournaments in Tournaments with Large Chromatic Number}
\begin{abstract}
For a set $\mathcal{H}$ of tournaments,
we say $\mathcal{H}$ is \emph{heroic}
if every tournament,
not containing any member of $\mathcal{H}$ as a subtournament,
has bounded chromatic number.
In~\cite{hero}, Berger et al. explicitly characterized all heroic sets containing one tournament.
Motivated by this result, we study heroic sets containing two tournaments.
We give a necessary condition for a set containing two tournaments to be heroic.
We also construct infinitely many minimal heroic sets of size two.
\end{abstract}
\section{Introduction}\label{SEC:intro}
All graphs and digraphs in this paper are simple.
For a graph $G$, the \emph{chromatic number} of $G$, denoted by $\chi(G)$, is the minimum number of colors needed to color vertices of $G$ in such a way that there are no adjacent vertices with the same color.
Since the chromatic number of $G$ is lower bounded by its clique number $\omega(G)$ (the maximum number of pairwise adjacent vertices of $G$), there has been great interest in a class of graphs whose chromatic number is bounded by some function of its clique number.
If $\mathcal{C}$ is a class of graphs closed under induced subgraphs, and there exists a function $f$ such that $\chi(G)\le f(\omega(G))$ for every $G\in \mathcal{C}$, then we say $\mathcal{C}$ is \emph{$\chi$-bounded by a $\chi$-bounding function $f$}.
A well-known example of a $\chi$-bounded class is the class of \emph{perfect} graphs.
(A perfect graph is a graph with the property that, $\chi(H)=\omega(H)$ for every its induced subgraph $H$.)
Clearly, the identity function is a $\chi$-bounding function for the class of perfect graphs.
There are many results and conjectures about $\chi$-bounded classes which are obtained by forbidding certain families of graphs.
A well-known example is the strong perfect graph theorem~\cite{perfect} which states that the set of graphs $G$, such that neither $G$ nor its complement contains an induced odd cycle of length at least five, is $\chi$-bounded by the identity function.
Recently, three conjectures of Gy\'arf\'as~\cite{gyarfas3}, regarding $\chi$-bounded classes of graphs forbidding some infinite sets of cycles, were proved in a series of papers by Chudnovsky, Scott, Seymour and Spirkl.
For a graph $G$ and a set $\mathcal{G}$ of graphs, we say $G$ is \emph{$\mathcal{G}$-free} if $G$ contains no members of $\mathcal{G}$ as induced subgraphs.
The three conjectures, which are now theorems, state as follows:
If $\mathcal{G}$ is either one of the following classes, then the set of all $\mathcal{G}$-free graphs is $\chi$-bounded.
\begin{itemize}
\item the set of all odd holes of length at least five (Scott and Seymour~\cite{hole1});
\item the set of all holes of length at least $\ell$ for some $\ell$ (Chudnovsky, Scott and Seymour~\cite{hole3});
\item the set of all odd holes of length at least $\ell$ for some $\ell$ (Chudnovsky, Scott, Seymour and Spirkl~\cite{hole8}).
\end{itemize}
Another conjecture due to Gy\'arf\'as~\cite{gyarfas}, independently proposed by Sumner~\cite{sumner}, deals with $\chi$-bounded classes obtained by forbidding finite family $\mathcal{F}$ of graphs. By the random construction of Erd\H{o}s~\cite{erdos}, we know that for each $c$ and $g$, there exists a graph $G$ with $\chi(G)\ge c$ and minimum cycle length at least $g$. This implies that for the class of $\mathcal{F}$-free graphs to be $\chi$-bounded, it is necessary that $\mathcal{F}$ contains a forest. The Gy\'arf\'as-Sumner conjecture asserts that the necessary condition is also sufficient.
\begin{GS}
Let $K$ be a complete graph and $F$ a forest.
Then, there exists $c$ such that every $\{K,F\}$-free graph has chromatic number at most $c$.
\end{GS}
This conjecture is known to be true for several classes of forests~\cite{gyarfas5, gyarfas2, gyarfas7, gyarfas6, hole12}, but is mostly wide open.
In this paper, we are interested in a similar question to the Gy\'arf\'as-Sumner conjecture for tournaments.
A \emph{tournament} is a digraph of which underlying graph is a complete graph. For a tournament $T$ and a set $S\preceqseteq V(T)$, we denote by $T|S$ the subtournament of $T$ induced on $S$.
We say $S\preceqseteq V(T)$ is \emph{transitive} if $T|S$ has no directed cycle.
For a tournament $T$ and its vertices $u$ and $v$, if $uv \in E(T)$, then we say \emph{$u$ is adjacent to $v$} or \emph{$v$ is adjacent from $u$}.
For two disjoint subsets $X$ and $Y$ of $V(T)$, if every vertex in $X$ is adjacent to every vertex in $Y$, then we say \emph{$X$ is complete to $Y$}, and write $X \Rightarrow Y$.
If $T$ is obtained from the disjoint union of tournaments $T_1$ and $T_2$ by adding all edges from $V(T_1)$ to $V(T_2)$, we write $T=T_1\Rightarrow T_2$.
For tournaments $T_1$ and $T_2$, if $T_2$ is isomorphic to a subtournament of $T_1$, then we say \emph{$T_1$ contains $T_2$}, and if $T_1$ does not contain $T_2$, we say $T_1$ is \emph{$T_2$-free}.
If $\mathcal{H}$ is a set of tournaments and a tournament $T$ is $H$-free for every $H\in \mathcal{H}$, then we say $T$ is \emph{$\mathcal{H}$-free}.
For a positive integer $k$ and a tournament $T$,
a \emph{$k$-coloring of $T$} is a map $\phi:V(G)\to C$ with $|C|=k$ such that $\phi^{-1}(c)$ is transitive for $c \in C$.
The \emph{chromatic number} of a tournament $T$, denoted by $\chi(T)$, is the minimum $k$ such that $T$ admits a $k$-coloring. This tournament invariant was first introduced by Neumann Lara~\cite{tournamentcoloring}.
In this paper, we study the tournament version of the Gy\'arf\'as-Sumner conjecture, that is, we investigate
a class $\mathcal{H}$ of tournaments where every $\mathcal{H}$-free tournament has bounded chromatic number.
Such a set is called \emph{heroic}. (A heroic set for graphs can be defined similarly. We direct the interested reader to~\cite{graphhero}.)
\begin{DEF}
A set $\mathcal{H}$ of tournaments is \emph{heroic} if there exists $c$ such that every $\mathcal{H}$-free tournament has chromatic number at most $c$.
\end{DEF}
For example, if $\mathcal{H}$ contains a cyclic triangle, then $\mathcal{H}$ is heroic since every tournament with chromatic number at least three contains a cyclic triangle.
\preceqsection{Tournaments with large chromatic number}\label{sec:construction}
Several graph classes with large chromatic number
are known~\cite{My1955,Kv1989,Lo1968,NeRo1989,AlKo2016}.
A complete graph is a trivial example, and
a Mycielski graph is a non-trivial example, which have clique number two but arbitrarily large chromatic number.
In contrast to graphs, few such classes of tournaments have been developed. One is introduced in~\cite{hero}, as follows:
\\
\\
{\bf Construction of $D_n$.}
If $T$ is a tournament and $(X,Y,Z)$ is a partition of $V(T)$ such that $X \Rightarrow Y$, $Y \Rightarrow Z$ and $Z \Rightarrow X$,
we call $(X,Y,Z)$ a \emph{trisection} of $T$,
and if $T|X$, $T|Y$ and $T|Z$ are isomorphic to tournaments $A$, $B$ and $C$, respectively, we write $T=\Delta(A,B,C)$. We denote by $I$ a one-vertex tournament.
We construct tournaments $D_n$ as follows: $D_1=I$, and for $n\ge 2$, $D_n =\Deltaelta(I, D_{n-1},D_{n-1})$. See Figure~\ref{Dn_fig}.
In section~\ref{SEC:class_unbounded}, we prove that the chromatic number of $D_n$ is equal to $n$.
\begin{figure}
\caption{$D_n$ for $n=1,2,3$}
\label{Dn_fig}
\end{figure}
In this section,
we introduce another class of tournaments, which are denoted by $A_n$,
with large chromatic number.
For a tournament $T$ and an integer $n\ge 2$,
if $(X_1,X_2,\ldots,X_{2n-1})$ is a partition of $V(T)$ such that
for $1\le i < j \le 2n-1$,
\begin{itemize}
\item $V_j$ is complete to $V_i$ if both $i$ and $j$ are odd, and
\item $V_i$ is complete to $V_j$ if either $i$ or $j$ is even,
\end{itemize}
then we call $(X_1,X_2,\ldots,X_{2n-1})$ a \emph{$\Delta$-partition} of $T$, and we write $T=\Deltaelta(T_1,T_2,\ldots,T_{2n-1})$ where $T_i=T|V_i$ for $1\le i \le 2n-1$. Note that every trisection of a tournament is a $\Deltaelta$-partition of it.
\\
\\
{\bf Construction of $A_n$.}
$A_1$ is a one-vertex tournament, and for $n \ge 2$, $A_n=\Delta(I^{(1)},A_{n-1}^{(1)},I^{(2)},A_{n-1}^{(2)},\ldots,A_{n-1}^{(n-1)},I^{(n)})$ where each $I^{(i)}$ is isomorphic to $I$ (a one-vertex tournament) and $A_{n-1}^{(i)}$ is isomorphic to $A_{n-1}$. See Figure~\ref{An_fig}.
\begin{figure}
\caption{$A_n$ for $n=1,2,3$.
}
\label{An_fig}
\end{figure}
In section~\ref{SEC:class_unbounded}, we prove that $\chi(A_n)=n$.
\preceqsection{Tournaments contained in heroic sets}\label{SEC:intro-contained}
Let $\mathcal{T}$ be a set of tournaments.
If, for every tournament $T\in \mathcal{T}$, every subtournament of $T$ is contained in $\mathcal{T}$, then $\mathcal{T}$ is said to be \emph{hereditary}.
If for every $n$, there exists a tournament in $\mathcal{T}$ with chromatic number larger than $n$,
we say \emph{$\mathcal{T}$ has unbounded chromatic number}.
It is easy to see that if $\mathcal{T}$ is a hereditary class of tournaments and has unbounded chromatic number, then every heroic set contains a tournament in $\mathcal{T}$.
\begin{PROP}\label{PRO:hereditary}
Let $\mathcal{T}$ be a hereditary class of tournaments. If the chromatic number of $\mathcal{T}$ is unbounded, then every heroic set meets $\mathcal{T}$.
\end{PROP}
For a set $\mathcal{T}$ of tournaments, the \emph{closure of $\mathcal{T}$} is the minimal hereditary class of tournaments containing $\mathcal{T}$. We define two classes of tournaments as follows:
\begin{itemize}
\item $\mathcal{D}$ is the closure of $\{D_n\mid n\ge 1\}$.
\item $\mathcal{A}$ is the closure of $\{A_n\mid n\ge1\}$.
\end{itemize}
Since $\mathcal{D}$ and $\mathcal{A}$ have unbounded chromatic number, Proposition~\ref{PRO:hereditary} implies the following.
\begin{THM}\label{THM_meet1}
Every heroic set intersects with $\mathcal{D}$ and $\mathcal{A}$.
\end{THM}
It is easy to see that two sets $\mathcal{D}$ and $\mathcal{A}$ are minimal in the
sense that there is no proper hereditary subset of $\mathcal{D}$ or $\mathcal{A}$ with unbounded chromatic number.
\preceqsection{Forest tournaments}
In the previous section, we constructed two (minimal) classes of tournaments intersecting with all heroic sets. In this section, we introduce another class of tournaments, which are called forest tournaments, intersecting with all finite heroic sets.
If $S$ is a finite set and $\sigma$ is an ordering of $S$, then for $a,b \in S$, we write $a<_{\sigma} b$ if $a$ comes before $b$ in $\sigma$.
For example, if $\sigma=s_1,s_2,\ldots,s_n$, then $s_i <_{\sigma} s_j$ for every $1\le i<j\le n$.
If $S'$ is a subset of $S$, then $\sigma|S'$ is the sub-ordering of $S$ on $S'$.
We denote by $\sigma\setminus S'$ the ordering $\sigma|(S-S')$.
For a tournament $T$ and an ordering $\sigma=v_1,v_2,\ldots,v_n$ of $V(T)$, an edge $v_iv_j$ of $T$ is called a {\em backward edge (under $\sigma$)} if $i>j$. The \emph{backedge graph $B_{\sigma}(T)$ of $T$ with respect to $\sigma$} is the ordered (undirected) graph with vertex set $V(T)$ and vertex ordering $\sigma$ such that $uv \in E(B_{\sigma}(T))$ if and only if either $uv$ or $vu$ is a backward edge of $T$ under $\sigma$. See Figure~\ref{fig:backedge}.
\begin{figure}
\caption{$T$ and $B_{\sigma}
\label{fig:backedge}
\end{figure}
The definition of a forest tournament is as follows:
\begin{DEF}
For a tournament $T$, a {\em forest ordering of $V(T)$} is an ordering $\sigma=v_1,v_2,\ldots,v_n$ of $V(T)$ such that
\begin{itemize}
\item there exists $i$ such that no two edges of $B_{\sigma}(T)$ between $\{v_1,v_2,\ldots,v_i\}$ and $\{v_{i+1},\ldots,v_n\}$ are in the same component of $B_{\sigma}(T)$,
and sub-orderings $v_1,v_2,\ldots,v_i$ and $v_{i+1},\ldots,v_n$ of $\sigma$ are forest orderings of $T|\{v_1,v_2,\ldots,v_i\}$ and $T|\{v_{i+1},\ldots,v_n\}$, respectively.
\end{itemize}
If such an ordering exists, we say $T$ is a {\em forest tournament} and the partition $(\{v_1,v_2,\ldots,v_i\}, \{v_{i+1},\ldots,v_n\})$ of $V(T)$ is a {\em forest cut of $T$ (under $\sigma$)}.
\end{DEF}
For example, in Figure~\ref{forestexample}, $T$ is a forest tournament with forest cut $(\{v_1,v_2,v_3\}, \{v_4,v_5,v_6,v_7\})$
since $T|\{v_1,v_2,v_3\}$ and $T|\{v_4,v_5,v_6,v_7\}$ are forest tournaments with forest cut $(\{v_1\},\{v_2,v_3\})$ and $(\{v_4,v_5\}, \{v_6,v_7\})$, respectively.
In section~\ref{SEC:forest}, we will show the following theorem.
\begin{THM}\label{THM_meet2}
Every finite heroic set contains a forest tournament.
\end{THM}
\begin{figure}
\caption{A tournament $T$ with ordering $\sigma$ and its backward edges.}
\label{forestexample}
\end{figure}
We will also show some properties of forest tournaments. For example, we will show that the backedge graph $B_{\sigma}(T)$ of a forest tournament $T$ with forest ordering $\sigma$ is a forest, which is the reason that we call this tournament a forest tournament.
We also prove that every forest tournament has chromatic number at most two, which shows the existence of (infinite) heroic sets not containing any forest tournaments. (e.g. the set of all tournaments with chromatic number three is heroic, but it does not contain any forest tournaments.)
\preceqsection{Small Heroic sets}
A tournament $H$ is called a \emph{hero} if every $H$-free tournament has bounded chromatic number, that is, $H$ is a hero if and only if $
\{H\}$ is heroic. In~\cite{hero}, Berger et al. explicitly characterized every \emph{hero} as follows:
\begin{THM}[Berger et al.~\cite{hero}]\label{hero}
Let $H$ be a tournament.
\begin{itemize}
\item[(1)] $H$ is a hero if and only if every strong component of $H$ is a hero.
\item[(2)] If $H$ is strongly connected, then $H$ is a hero if and only if $H=\Delta(I,H_1,H_2)$ where $H_1$ and $H_2$ are heroes and one of them is transitive.
\end{itemize}
\end{THM}
This result motivated us to the study of small heroic sets, in particular, heroic sets containing two tournaments.
For a set $\mathcal{H}$ consisting of two tournaments to be heroic,
it must contain some tournament $D$ in $\mathcal{D}$ by Theorem~\ref{THM_meet1}.
If $D$ is a hero, then no matter what the other tournament in $\mathcal{H}$ is, $\mathcal{H}$ is heroic. Thus, the only interesting case is when $D$ is a non-hero.
Every non-hero in $\mathcal{D}$ is characterized as follows:
\begin{restatable}{LEM}{reD}\label{lem:nonhero-d3}
For a tournament $D\in \mathcal{D}$, $D$ is a non-hero if and only if $D$ contains $D_3$.
\end{restatable}
We will prove this lemma in section~\ref{SEC:nonhero}.
Let $\mathcal{D}'=\{D\in \mathcal{D} \mid \text{$D$ contains $D_3$}\}$, that is, the set of all non-heroes in $\mathcal{D}$.
For a tournament $F$, we say a tournament $H$ is an \emph{$F$-hero} if there exists $c$ such that every $\{F,H\}$-free tournament $T$ has chromatic number at most $c$.
For example, if $F$ is a hero, then every tournament is an $F$-hero.
By answering the following question, we can characterize all heroic sets consisting of two tournaments.
\begin{QUE}
Let $D\in \mathcal{D}'$. Which tournaments are $D$-heroes?
\end{QUE}
In this paper, we give a necessary condition for a tournament $H$ to be a $D$-hero for $D\in \mathcal{D}'$. Let $L_k$ be a transitive tournament with $k$ vertices.
\begin{THM}\label{THM:main2}
Let $D$ be a tournament in $\mathcal{D}'$. If a tournament $H$ is a $D$-hero, then $H$ is isomorphic to one of the following. (See Figure~\ref{FIG:maintheorem}.)
\begin{itemize}
\item[1)] $I$;
\item[2)] $H_1\Rightarrow H_2$ for some $D$-heroes $H_1$ and $H_2$;
\item[3)] $\Deltaelta(I,L_k,H')$ or $\Deltaelta(I,H',L_k)$ for some integer $k$ and some $D$-hero $H'$;
\item[4)] $\Delta(L_{k_1},I,L_{k_2},L_{k_3}, I)$ or $\Delta(I,L_{k_1},L_{k_2},I, L_{k_3})$ for some integers $k_1,k_2,k_3$;
\item[5)] $\Delta(I,H',L_{k_1},L_{k_2},I)$ or $\Delta(I,L_{k_1},L_{k_2},H',I)$ for some integers $k_1$, $k_2$ and some $D$-hero $H'$;
\item[6)] $\Delta(I,L_{k_1}, L_{k_2},I,L_{k_3},L_{k_4},I)$ for some integers $k_1,k_2,k_3,k_4$.
\end{itemize}
\end{THM}
We prove Theorem~\ref{THM:main2} in section~\ref{SEC:mainproof}.
\begin{figure}
\caption{Tournaments in Theorem~\ref{THM:main2}
\label{FIG:maintheorem}
\end{figure}
\preceqsection{$D$-heroes for $D\in \mathcal{D}'$}
The first result of Theorem~\ref{hero} also holds for $D$-heroes.
\begin{THM}\label{THM:growing1}
Let $D$ be a tournament in $\mathcal{D}'$.
Then, a tournament $H$ is a $D$-hero if and only if every strong component of $H$ is a $D$-hero.
\end{THM}
Theorem~\ref{THM:growing1} is straightforward by the following lemma proved in~\cite{hero}.
\begin{LEM}[Berger et al. \cite{hero}]\label{lemma1}
Let $\mathcal{H}_1,\mathcal{H}_2$ be sets of tournaments such that every member of $\mathcal{H}_1\cup \mathcal{H}_2$ has at most $c\,(\ge 3)$ vertices. Let $\mathcal{H}=\{H_1\Rightarrow H_2 \mid H_1\in \mathcal{H}_1, H_2\in \mathcal{H}_2\}$.
For every $\mathcal{H}$-free tournament $T$, if every $\mathcal{H}_1$-free subtournament of $T$ and $\mathcal{H}_2$-free subtournament of $T$ has chromatic number at most $c$, then the chromatic number of $T$ is at most $(2c)^{4c^2}$.
\end{LEM}
We remark that Lemma~\ref{lemma1} also implies (1) of Theorem~\ref{hero}.
In contrast to (1) of Theorem~\ref{hero}, the second result does not hold for $D$-heroes in general.
(Theorem~\ref{THM:U3} will give an example of a $D$-hero which is strongly connected but does not admit a trisection.)
However, it turns out that
(2) of Theorem~\ref{hero} holds for $D$-heroes admitting a trisection.
\begin{THM}\label{THM:growing2}
Let $D$ be a tournament in $\mathcal{D}'$.
Let $H$ be a tournament admitting a trisection.
Then, $H$ is a $D$-hero if and only if $H$ is either $\Delta(I,H',L_k)$ or $\Delta(I,L_k,H')$ where $k$ is a positive integer and $H'$ is a $D$-hero.
\end{THM}
Alghough the proof of Theorem~\ref{THM:growing2} is the same as that of (2) of Theorem~\ref{hero} in~\cite{hero}, we give the proof in section~\ref{SEC:growing} for reader's convenience.
The smallest tournament in the list of Theorem~\ref{THM:main2}, which cannot be obtained from Theorem~\ref{THM:growing1} or Theorem~\ref{THM:growing2}, is $\Delta(I,I, I,I,I)$.
We simply denote this tournament by $U_3$.
In the following theorem, we show that $U_3$ is a $D$-hero for every $D\in \mathcal{D}'$.
\begin{THM}\label{THM:U3}
Let $D\in \mathcal{D}'$. Then, $U_3$ is a $D$-hero.
\end{THM}
The proof will be given in section~\ref{SEC:proofU3}.
Generalizing the definition of $U_3$,
let $U_n=\Delta(I^{(1)},I^{(2)}, \ldots,I^{(2n-1)})$, that is, the tournament with $V(U_n)=\{v_1,v_2,\ldots,v_{2n-1}\}$ such that for $1\le i<j \le 2n-1$, $v_j$ is adjacent to $v_i$ if and only if both $i$ and $j$ are odd. See Figure~\ref{Un_fig}.
If $n\ge 5$, then $U_n$ is not contained in the list in Theorem~\ref{THM:main2}, and if $n\le 2$, then $U_n$ is either a one-vertex tournament or a cyclic triangle, which is a trivial $D$-hero. And by Theorem~\ref{THM:U3}, we know that $U_3$ is a $D$-hero for every $D\in \mathcal{D}'$. The only remaining case is that $n=4$.
So, we finish this section with the following question.
\begin{figure}
\caption{$U_n$ for $n=1,2,3,4$.
}
\label{Un_fig}
\end{figure}
\begin{QUE}\label{question}
For which tournaments $T \in \mathcal{D'}$, is $U_4$ a $T$-hero? In particular, is $U_4$ a $D_3$-hero?
\end{QUE}
\section{Classes of tournaments with unbounded chromatic number}\label{SEC:class_unbounded}
In this section, we prove that $\chi(D_n)=\chi(A_n)=n$, which directly implies Theorem~\ref{THM_meet1}.
\begin{PROP}\label{LEM:Dn}
For every positive integer $n$, $\chi(D_n)=n$.
\end{PROP}
\begin{proof}
We proceed by induction on $n$.
If $n=1$, then $|V(D_1)|=1$, so $\chi(D_1)=1$.
Let $n\ge 2$, and suppose $\chi(D_k)=k$ for all $k <n$.
Let $(X_1,X_2,X_3)$ be a trisection of $D_n$ such that $|X_1|=1$ and $X_2$ and $X_3$ induce $D_{n-1}$. Let $X_1=\{x_1\}$ and $\phi_i:X_i \to [n-1]$ be an $(n-1)$-coloring of $D_n|X_i$ for $i=2,3$. Such colorings exist by the induction hypothesis. Let $\phi:V(D_n)\to [n]$ be a map such that $\phi(x_1)=n$ and for $v\in V_i$, $\phi(v)=\phi_i(v)$ for $i=1,2$.
Then, clearly, $\phi$ is an $n$-coloring of $D_n$, so $\chi(D_n) \le n$.
To show $\chi(D_n)\ge n$, suppose there exists an $(n-1)$-coloring $\psi: V(D_n) \to [n-1]$ of $D_n$. Since $\chi(D_{n-1})=n-1$, it follows that $|\psi(X_2)|=|\psi(X_3)|=n-1$. We may assume that $\psi(x_1)=n-1$. Let $x_2\in X_2$ and $x_3 \in X_3$ be vertices with $\psi(x_2)=\psi(x_3)=n-1$. Such $x_2$ and $x_3$ exist as $|\psi(X_2)|=|\psi(X_3)|=n-1$.
Then, $\{x_1,x_2,x_3\}$ induces a monochromatic cyclic triangle in $D_n$ which yields a contradiction. Therefore $D_{n}$ is not $(n-1)$-colorable, implying that $\chi(D_n)= n$. This completes the proof.
\end{proof}
\begin{PROP}\label{LEM:An}
For every positive integer $n$, $\chi(A_n)=n$.
\end{PROP}
\begin{proof}
We proceed by induction on $n$.
If $n=1$, then $|V(A_1)|=1$, so $\chi(A_1)=1$.
Let $n\ge 2$, and assume the chromatic number of $A_k$ is equal to $k$ for all $k<n$.
Let $(\{v_1\},X_1,\{v_2\},X_2,\ldots,X_{n-1},\{v_n\})$ be a $\Delta$-partition of $A_n$ where $A_n|X_j$ is isomorphic to $A_{n-1}$ for $1\le j \le n-1$.
Let $\phi_j: X_j \to [n-1]$ be an $(n-1)$ coloring of $A_n|X_{j}$ for $j=1,2,\ldots,n-1$.
Then, the map $\phi:V(A_n) \to [n]$ defined as, $\phi(v_i)=n$ for $i=1,2,\ldots,n$ and $\phi(v)=\phi_j(v)$ if $v\in X_j$ for $j=1,2,\ldots,n-1$, is an $n$-coloring of $A_n$.
To prove that $A_n$ is not $(n-1)$-colorable, let us assume that there exists an $(n-1)$-coloring $\psi:V(A_n) \to [n-1]$ of $A_n$.
Since $\psi$ is an $(n-1)$-coloring, there exist two vertices $v_p,v_q$ with $\psi(v_p)=\psi(v_q)$ and $p<q$ by the pigeonhole principle.
We may assume that $\psi(v_p)=\psi(v_q)=n-1$.
Since $A_{n}|X_p$ is not $(n-2)$-colorable, it follows that $\psi(X_p)=[k-1]$, and there exists $y \in X_p$ such that $\psi(y)=n-1$.
Then, $\{y,v_p,v_q\}$ induces a monochromatic cyclic triangle, a contradiction.
This completes the proof.
\end{proof}
In the remaining of this section, we investigate properties of tournaments in $\mathcal{A}$.
\begin{PROP}\label{PROP:upartition}
If a tournament $T\in \mathcal{A}$ is strongly connected, then there exists a $\Delta$-partition of $T$.
\end{PROP}
\begin{proof}
Take the minimal $n$ such that $A_n$ contains $T$.
We consider a $\Delta$-partition $(\{v_1\},X_1,\{v_2\},X_2,\ldots,X_{n-1},\{v_n\})$ of $A_n$ where $A_n|X_j$ is isomorphic to $A_{n-1}$ for $1\le j \le n-1$.
Let $B=V(T)\cap \{v_1,v_2,\ldots,v_n\}$.
If $B$ is empty,
then let $m$ be the minimum such that $V(T)\cap X_m \neq \emptyset$.
Since $V(T) \not \preceqseteq X_m$ by the minimality of $n$, it follows that $V(T) \setminus X_m$ is not empty.
So, $V(T)\cap X_m$ is complete to $V(T)\setminus X_m$ in $T$, which yields a contradiction that $T$ is strongly connected. Therefore, $B\neq \emptyset$.
Let $B=\{v_{i_1},v_{i_2},\ldots,v_{i_k}\}$ with $i_1<i_2<\cdots <i_k$.
Observe that $V(T)\setminus B \preceqseteq \bigcup_{j=i_1}^{i_k-1}X_{j}$ since $T$ is strongly connected.
For $1\le j \le k-1$,
let $Y_j=V(T)\cap \left( \bigcup_{s=i_j}^{i_{j+1}-1}X_s\right)$.
Then, $v_{i_j}$ is complete to $Y_{j'}$ for $j\le j'\le k-1$
and complete from $Y_{j''}$ for $1\le j'' \le j-1$.
So, $(\{v_{i_1}\},Y_1 ,\{v_{i_2}\},Y_2,\ldots,Y_{k-1},\{v_{i_k}\})$ is a $\Delta$-partition of $T$. This completes the proof.
\end{proof}
For a tournament $T$, a subset $S \preceqseteq V(T)$ and a vertex $v$ outside of $S$, we say \emph{$v$ is mixed on $S$}, if $v$ has both an out-neighbor and an in-neighbor in $S$. A subset $S$ of $V(T)$ with $1<|S|<|V(T)|$ is called \emph{homogeneous} if every vertex outside of $S$ is not mixed on $S$.
\begin{PROP}\label{Uhomo}
Let $T$ be a strong tournament in $\mathcal{A}$. If $S$ is a maximal homogeneous set of $T$ and $(\{v_1\},X_1,\{v_2\},X_2,\ldots,X_{n-1},\{v_n\})$ is a $\Delta$-partition of $T$, then $S=X_k$ for some $1\le k \le n-1$.
\end{PROP}
\begin{proof}
Let $B=\{v_1,v_2,\ldots,v_{n}\}$. Clearly, $S\not\preceqseteq B$.
Choose the smallest $m$ such that $X_m \cap S \neq \emptyset$, and let $x \in X_m \cap S$.
We claim $S\cap B= \emptyset$.
Suppose $S \cap B\neq \emptyset$, and let $y \in S\cap B$. Since $v_1$ and $v_n$ are mixed on $\{x,y\}$, they belong to $S$.
By the definition of a homogeneous set, there exists $z\in V(T)\setminus S$, but $z$ is mixed on $\{v_1,v_n\}$, a contradiction.
Therefore $S \cap B=\emptyset$.
If $S \not \preceqseteq X_m$, then $T|S$ is not strongly connected since $S\cap X_m$ is complete to $S\setminus X_m$. So, it follows that $S\preceqseteq X_m$.
Lastly, since $X_m$ is homogeneous and $S$ is maximal, $S=X_m$.
This completes the proof.
\end{proof}
A tournament is \emph{prime} if it does not have homogeneous sets. Observe that if a tournament $T$ has at least three vertices and is prime, then $T$ is strongly connected.
Recall that $U_n=\Deltaelta(I^{(1)},I^{(2)},\ldots,I^{(2n+1)})$ where $I^{(i)}$ is a one-vertex tournament. It is easy to see that $U_n$ is prime.
\begin{PROP}\label{Uprime}
Let $T \in \mathcal{A}$ be a tournament with at least three vertices. Then, $T$ is prime if and only if $T$ is isomorphic to $U_{n}$ for some integer $n\ge2$.
\end{PROP}
\begin{proof}
The `if' part is clear.
For the `only if' part,
if $T$ is prime, then $T$ is strongly connected, and by Proposition~\ref{PROP:upartition}, there exists a $\Delta$-partition $(\{v_1\},X_1,\{v_2\},\ldots,X_{n-1},\{v_{n}\})$ of $T$.
If $|X_i| \ge2$ for some $1\le i\le n$, then $X_i$ is homogeneous, so, $|X_i| = 1$ for every $i$. Therefore, $T$ is isomorphic to $U_n$.
\end{proof}
\section{Proof of Theorem~\ref{THM_meet2}}\label{SEC:forest}
Let $\mathcal{F}$ be the set of all forest tournaments.
First, we show that $\mathcal{F}$ is hereditary.
\begin{PROP}\label{closed}
Let $T$ be a forest tournament with at least two vertices and forest ordering $\sigma$. Then, for every $v\in V(T)$, $T\setminus v$ is a forest tournament and $\sigma\setminus v$ is its forest ordering.
\end{PROP}
\begin{proof}
We use induction on the number of vertices of $T$. Let $|V(T)|=n$ and let $\sigma'=\sigma\setminus v$ and $T'=T\setminus v$.
If $n=2$, we are done since $B_{\sigma'}(T')$ has no edge.
Let $n>2$ and assume that Proposition~\ref{closed} is true for every forest tournament with less than $n$ vertices.
Let $(V_1,V_2)$ be a forest cut of $T$ under $\sigma$, so $T|V_i$ is a forest tournament with forest ordering $\sigma|V_i$ for $i=1,2$.
Without loss of generality, let $v \in V_1$. Let $\sigma_1=\sigma|V_1$.
If $V_1=\{v\}$, then $T'$ is $T|V_2$ which is a forest tournament with forest ordering $\sigma'=\sigma|V_2$.
Thus we may assume that $|V_1|>1$.
Then, $T|V_1$ is a forest tournament with forest ordering $\sigma_1$, and by the induction hypothesis, $T|(V_1\setminus v)$ is a forest tournament with forest ordering $\sigma_1\setminus v$.
Therefore, $(V_1\setminus v,V_2)$ is a forest cut of $T'$ under $\sigma'$, and so $T'$ is a forest tournament with forest ordering $\sigma'$. This completes the proof.
\end{proof}
Next, we prove that for a forest tournament $T$ and its forest ordering $\sigma$, $B_{\sigma}(T)$ does not contain a cycle as an induced subgraph.
For an ordered graph $G$ with at least two vertices and vertex ordering $\sigma=v_1,\ldots,v_n$, the {\em thickness of $G$ (under $\sigma$)} is the minimum number of edges between $\{v_1,v_2,\ldots,v_i\}$ and $\{v_{i+1},\ldots,v_n\}$ over all $i$'s.
\begin{PROP}\label{onecomponent}
Let $T$ be a forest tournament with forest ordering $\sigma$. If $B_{\sigma}(T)$ is connected, then the thickness of $B_{\sigma}(T)$ is one.
\end{PROP}
\begin{proof}
Since $B_{\sigma}(T)$ is connected, its thickness is at least one.
Let $(V_1,V_2)$ be a forest cut of $T$ under $\sigma$. Since $B_{\sigma}(T)$ has one component, there is exactly one edge between $V_1$ and $V_2$ in $B_{\sigma}(T)$. So, the thickness of $B_{\sigma}(T)$ is one.
\end{proof}
\begin{COR}\label{nocycle}
Let $T$ be a forest tournament and $\sigma$ its forest ordering. Then, $B_{\sigma}(T)$ does not contain a cycle as an induced subgraph.
\end{COR}
\begin{proof}
Suppose there exists $V' \preceqseteq V(T)$ such that $B_{\sigma}(T)|V'$ is a cycle. Let $T'=T|V'$ and $\sigma' =\sigma|V'$.
Then, $T'$ is a forest tournament and $\sigma'$ is its forest ordering by Proposition~\ref{closed}.
Since a cycle is connected, Proposition~\ref{onecomponent} implies that the thickness of $B_{\sigma'}(T')$ is one. However, for every partition $(V_1,V_2)$ of $V(B_{\sigma'}(T'))$, there exist at least two edges between $V_1,V_2$ since $B_{\sigma'}(T')$ is a cycle, a contradiction. This completes the proof.
\end{proof}
A forest (undirected graph) has chromatic number at most two. It holds for a forest tournament as well.
\begin{PROP}\label{twocolor}
Every forest tournament has chromatic number at most two.
\end{PROP}
\begin{proof}
Let $T$ be a forest tournament with forest ordering $\sigma$.
By Corollary~\ref{nocycle}, $B_{\sigma}(T)$ is a forest,
in particular, it is 2-colorable (as a graph coloring).
So, $V(T)$ can be partitioned into two sets $(X,Y)$ such that no pair of adjacent vertices in the same set.
Then, there is no backward edge in $T|X$ (resp. $T|Y$) under $\sigma|X$ (resp. $\sigma|Y$), which implies that $X$ and $Y$ are transitive sets in $T$. So, $\chi(T)\le 2$.
\end{proof}
The remaining of this section is devoted to proving Theorem~\ref{THM_meet2}.
We start with some definitions.
For a tournament $T$ and an injective map $\phi :V(T)\to \mathbb{Z}^+$, let $\sigma_{\phi}$ be the ordering $v_1,v_2,\ldots,v_n$ of $V(T)$ such that $\phi(v_i) < \phi(v_j)$ for every $i<j$. For a backward edge $e$ of $T$ under $\sigma_{\phi}$, if its end vertices are $x$ and $y$, we define $\phi(e)=|\phi(x)-\phi(y)|$. For integers $r,s \ge 1$ and distinct $e,f \in E(B_{\sigma_{\phi}}(T))$ we say $e$ and $f$ are {\em $(r,s)$-comparable (under $\phi$)} if
\begin{itemize}
\item there exists a path in $B_{\sigma_{\phi}}(T)$ with at most $s$ edges containing $e$ and $f$, and
\item $ \frac{1}{r} \le \frac{\phi(e)}{\phi(f)} \le r$.
\end{itemize}
For positive integers $r$ and $s$, we denote by $\mathcal{C}_{(r,s)}$ the class of tournaments $T$ such that there exists an injective map $\phi$ from $V(T)$ to $\mathbb{Z}$ such that no two edges of $B_{\sigma_{\phi}} (T)$ are $(r,s)$-comparable.
It is easy to see that $\mathcal{C}_{(r,s)}$ is hereditary.
The following is proved in~\cite{hero}.
\begin{LEM}[Berger et al.~\cite{hero}]\label{crs}
For integers $r,s \ge 1$, the chromatic number of $\mathcal{C}_{(r,s)}$ is unbounded. So, every heroic set meets $\mathcal{C}_{(r,s)}$.
\end{LEM}
Lemma~\ref{crs} provides infinitely many hereditary classes $\mathcal{C}(r,s)$ of tournaments meeting every heroic set.
This implies that if $\mathcal{H}$ is a finite heroic set, then $\mathcal{H}$ contains a tournament $H$ belonging to $\mathcal{C}(r,s)$ for infinitely many pairs $(r,s)$.
Observe that for $e,f \in E(T)$,
if $e$ and $f$ are not $(r,s)$-comparable under $\phi$,
then they are not $(r',s')$-comparable under $\phi$ for every positive integers $r' (\le r)$ and $s' (\le s)$.
So, it follows that $\mathcal{C}_{(r,s)} \preceqseteq \mathcal{C}_{(r',s')}$, and
it directly leads to the following lemma.
Let $\mathcal{C}=\bigcap_{r,s \in \mathbb{Z}^+} \mathcal{C}_{(r,s)}$.
\begin{LEM}\label{comparable}
If $\mathcal{H}$ is a finite heroic set, then it contains $H$ such that $H \in \mathcal{C}_{(r,s)}$ for every positive integers $r$ and $s$. That is,
$\mathcal{H}\cap \mathcal{C} \neq \emptyset$.
\end{LEM}
For a tournament $T$ and a positive integer $r$, we say an injective map $\phi:V(T)\to \mathbb{Z}^+$ is \emph{$r$-incomparable}, if
for every pair $(e,f)$ of edges of $B_{\sigma_{\phi}}(T)$ in the same component, $\frac{\phi(e)}{\phi(f)}$ is either greater than $r$ or less than $\frac{1}{r}$.
We note that for $r\ge r'$, if $\phi$ is $r$-incomparable, then it is $r'$-incomparable.
We say a vertex ordering $\sigma$ of $T$ is \emph{incomparable} if for every positive integer $r$, there exists an $r$-incomparable injective map $\phi:V(T)\to \mathbb{Z}^+$ such that $\sigma=\sigma_{\phi}$.
\begin{LEM}\label{stableordering}
Let $T$ be a tournament.
Then, $T$ belongs to $\mathcal{C}$ if and only if there exists an incomparable vertex ordering of $T$.
\end{LEM}
\begin{proof}
The `if' part is clear by the definitions of $\mathcal{C}$ and an incomparable vertex ordering.
For the `only if' part, let $|V(T)|=n$.
For each integer $r\ge 1$, let $\phi_r$ be an injective map from $V(T)$ to $\mathbb{Z}^+$ with the property that
no two edges of $B_{\sigma_{\phi_r}} (T)$ are $(r,n-1)$-comparable. Such $\phi_r$ exists by the definition of $\mathcal{C}$.
Since for every pair $(e,f)$ of edges in the same component of $B_{\sigma_{\phi_r}}(T)$, there exists a path $P$ with at most $n-1$ edges, with $e,f \in E(P)$,
it follows that $\frac{\phi_r(e)}{\phi_r(f)}$ is either greater than $r$ or less than $\frac{1}{r}$.
So, $\phi_r$ is $r$-incomparable.
Since there are finitely many orderings of $V(T)$, there exists an ordering $\sigma$ of $V(T)$ which is equal to $\sigma_{\phi_r}$ for infinitely many positive integers $r$.
We claim $\sigma$ is incomparable.
For every integer $r'\ge 1$, there exists $r\ge r'$ such that $\sigma=\sigma_{\phi_r}$.
Since $\phi_r$ is $r'$-incomparable, $\sigma$ is incomparable.
\end{proof}
We are ready to prove Theorem~\ref{THM_meet2}.
\begin{proof}[Proof of Theorem~\ref{THM_meet2}]
We will show that for a tournament $T$ and a vertex ordering $\sigma=v_1,v_2,\ldots,v_n$ of $T$, $\sigma$ is incomparable if and only if $\sigma$ is a forest ordering. Then, by Lemma~\ref{comparable}, Theorem~\ref{THM_meet2} is straightforward.
We use induction on $n$. It is clear when $n=1$. Let $n>1$ and assume that the statement holds for every tournament with less than $n$ vertices.
Let $\sigma$ be incomparable. Let $\phi: V(T)\to \mathbb{Z}^+$ be a map such that
\begin{itemize}
\item[(1)] $\sigma=\sigma_{\phi}$, and
\item[(2)] for every $e,f \in B_{\sigma_{\phi}}(T)$ in the same component, $\frac{\phi(e)}{\phi(f)}$ is either greater than $n-1$ or less than $\frac{1}{n-1}$.
\end{itemize}
Take $v_i$ such that $|\phi(v_i)-\phi(v_{i+1})|$ is maximized. Let $V_1=\{v_1,\ldots,v_i\}$ and $V_2=\{v_{i+1},\ldots,v_n\}$.
We claim that $(V_1,V_2)$ is a forest cut of $T$ under $\sigma$.
Suppose there exist two edges $e$ and $f$ between $V_1$ and $V_2$ contained in the same component of $B_{\sigma}(T)$.
Without loss of generality, we may assume that $\frac{\phi(e)}{\phi(f)}>n-1$, so $\phi(e)>(n-1)\phi(f)$.
Then, it follows by (2) that
$$
\phi(v_n)-\phi(v_1) \ge \phi(e) > (n-1) \phi(f)
\ge (n-1) \left(\phi(v_{i+1})-\phi(v_i)\right).
$$
Since $\phi(v_{i+1})-\phi(v_i) \ge \phi(v_{j+1})-\phi(v_j)$ for every $j=1,2,\ldots,n-1$, it follows that $(n-1) \left(\phi(v_{i+1})-\phi(v_i)\right)
\ge \phi(v_n)-\phi(v_1)$, which yields a contradiction.
Therefore, no two edges between $V_1$ and $V_2$ are in the same component of $B_{\sigma}(T)$.
Moreover, for $i=1,2$, $\sigma|V_i$ is an incomparable vertex ordering of $T|V_i$, so $\sigma|V_i$ is a forest ordering of $T|V_i$ by the induction hypothesis. Hence, $\sigma$ is a forest ordering of $T$ with forest cut $(V_1,V_2)$.
Conversely, suppose $\sigma$ is a forest ordering of $T$ with a forest cut $(V_1,V_2)$ where $V_1=\{v_1,v_2,\ldots,v_i\}$ and $V_2=\{v_{i+1},\ldots,v_n\}$.
Let $T_1=T|V_1$ and $T_2=T|V_2$. By the induction hypothesis, $\sigma|V_i$ is an incomparable vertex ordering of $T|V_i$ for $i=1,2$.
It is enough to show that for each $r \ge 1$, there exists an injective map $\phi:V(T)\to \mathbb{Z}^+$ satisfying the conditions (1) and (2) above.
For $i=1,2$, let $\phi_i$ be an $r$-incomparable map from $V_i$ to $\mathbb{Z}^+$ with $\sigma_{\phi_i}=\sigma|V_i$.
Let $\phi_1(v_i)-\phi_1(v_1)=a$ and $\phi_2(v_n)-\phi_2(v_{i+1})=b$. We define an injective map $\phi:V(T)\to \mathbb{Z}^+$ as follows:
$\phi(v_j)=\phi_1(v_j)$ for $1\le j \le i$ and $\phi(v_j)=\phi_1(v_i)+ab(r+1)^2+a(r+1)\phi_2(v_j)$ for $i+1\le j \le n$.
Then, obviously, $\sigma=\sigma_{\phi}$.
We claim that $\phi$ is $r$-incomparable.
Let $e$ and $f$ be edges of $B_{\sigma}(T)$ in the same component.
If either $e,f \in E(B_{\sigma_{\phi}|V_1}(T_1))$ or $e,f \in E(B_{\sigma_{\phi}|V_2}(T_2))$, we are done.
If one is an edge of $B_{\sigma_{\phi}|V_1}(T_1)$ and the other is an edge of $B_{\sigma_{\phi}|V_2}(T_2)$, say $e \in E(B_{\sigma_{\phi}|V_1}(T_1))$ and $f\in E(B_{\sigma_{\phi}|V_2}(T_2))$, then $\frac{\phi(f)}{\phi(e)} \ge r+1>r$ since $\phi(e)\le \phi(v_i)-\phi(v_1)=a$ and $\phi(f) \ge a(r+1)$.
So, we may assume that either $e$ or $f$ is an edge between $V_1$ and $V_2$, say $f$. Then, $e$ is contained in $B_{\sigma_{\phi}|V_1}(T_1)$ or $B_{\sigma_{\phi}|V_2}(T_2)$ since $e$ and $f$ are contained in the same component of $B_{\sigma_{\phi}}(T)$.
Note that $\phi(f) \ge \phi(v_{i+1})-\phi(v_i) = ab(r+1)^2+a(r+1)\phi_2(v_{i+1})>ab(r+1)^2$, and $\phi(e)\le \max\{\phi_1(v_i)-\phi_1(v_1),a(r+1)\left(\phi_2(v_n)-\phi_2(v_{i+1})\right)\} \le ab(r+1)$.
So, $\frac{\phi(f)}{\phi(e)} \ge r+1>r$. Therefore, $\phi$ is $r$-incomparable.
This completes the proof.
\end{proof}
\section{Minimal non-heroes}\label{SEC:nonhero}
Since every subtournament of a hero is a hero, it is interesting to characterize minimal non-heroes. In~\cite{hero}, the authors showed that there are only five minimal non-heroes.
Let $N$ be the tournament with five vertices $\{v_1,v_2,v_3,v_4,v_5\}$ such that $v_i$ is adjacent to $v_j$ for $2\le i<j\le 5$ and $v_1$ is complete to $\{v_2,v_4\}$ and complete from $\{v_3,v_5\}$.
Let $S_n$ be the tournament with $(2n-1)$ vertices $v_1,v_2,\ldots,v_{2n-1}$ such that $v_i$ is adjacent to $v_j$ if and only if $j-i \in \{1,2,\ldots,n-1\} \mod (2n-1)$.
Let $\Delta_2=\Delta(L_2,L_2,L_2)$.
\begin{THM}[Berger et al. \cite{hero}]
A tournament $H$ is a hero if and only if $H$ does not contain $D_3$, $U_3$, $N$, $S_3$ or $\Delta_2$.
\end{THM}
\begin{figure}
\caption{Minimal non-heroes}
\end{figure}
In this section, we prove the following lemmas,
which will be used to prove Theorem~\ref{THM:main2}.
\reD*
\begin{LEM}\label{LEM:A}
$\mathcal{A}$ contains $U_3$ and $\Delta_2$, but does not contain $D_3,N$ or $S_3$.
\end{LEM}
\begin{LEM}\label{LEM:F}
$\mathcal{F}$ contains $U_3$ and $N$, but does not contain $D_3,S_3$ or $\Delta_2$.
\end{LEM}
We remark that Lemma~\ref{lem:nonhero-d3} is equivalent to that $D_3$ is the only minimal non-hero contained in $\mathcal{D}$.
Before proving the lemmas, we note that $U_3, N$ and $S_3$ are prime, that is, they do not contain homogeneous sets. We also note that every minimal non-hero is strongly connected.
\begin{proof}[Proof of Lemma~\ref{lem:nonhero-d3}]
Clearly, $D_3 \in \mathcal{D}$ by the definition of $\mathcal{D}$.
Suppose either $U_3$, $N$, $S_3$ or $\Delta_2$ is contained in $\mathcal{D}$, say $X$.
Let $k$ be the minimum integer such that $D_k$ contains $X$.
Since $X$ is contained in $D_k$ but not in $D_{k-1}$, and $X$ is strongly connected,
it follows that $X$ has a trisection $(A,B,C)$ with $|A|=1$, say $A=\{a\}$.
So, $X\setminus a$ is not strongly connected.
Since there is no such vertex $a$ in $\Delta_2$, $X$ is either $U_3$, $N$ or $S_3$, which implies that $X$ is prime.
So, $B$ and $C$ also contain only one vertex, which yields a contradiction since $|V(X)|=5$.
This completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{LEM:A}]
Clearly, $U_3,\Delta_2 \in \mathcal{A}$ since $A_3$ contains $U_3$ and $A_5$ contains $\Delta_2$.
It is also trivial that $N, S_3 \not \in \mathcal{A}$ by Proposition~\ref{Uprime} since $N$ and $S_3$ are prime but not isomorphic to $U_3$.
To show that $D_3 \not\in \mathcal{A}$,
suppose $D_3 \in \mathcal{A}$. Since $D_3$ is strongly connected, it has a $\Delta$-partition $(\{v_1\},X_1,\{v_2\},X_2,\ldots,X_{n-1},\{v_n\})$ by Proposition~\ref{PROP:upartition}.
Let $(\{x\},Y,Z)$ be a trisection of $D_3$ where $D_3|Y$ and $D_3|Z$ are cyclic triangles.
Since $Y$ and $Z$ are maximal homogeneous sets of $D_3$, it follows that $Y=X_i$ and $Z=X_j$ for some $i<j$ by Proposition~\ref{Uhomo}.
Thus, we obtain that $n\ge 3$, which is a contradiction
since $7=|V(D_3)|\ge n+ |X_i|+|X_j| \ge 9$.
Therefore, $D_3$ does not belong to $\mathcal{A}$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{LEM:F}]
Observe that $U_3$ has a forest ordering $v_1,v_2,v_3,v_4,v_5$ with backward edges $\{v_1v_4,v_2v_5 \}$,
and $N$ has a forest ordering $u_1,u_2,u_3,u_4,u_5$ with backward edges $\{u_1u_3,u_1u_5\}$.
Hence, $U_3, N\in \mathcal{F}$.
Since $D_3$ has chromatic number three, $D_3$ is not a forest tournament by Proposition~\ref{twocolor}.
To prove $S_3 \not \in \mathcal{F}$, suppose $S_3$ is a forest tournament, and $\sigma$ is its forest ordering.
Let $V(S_3)= \{v_1,v_2,\ldots,v_5\}$ such that $v_i$ is adjacnet to $v_j$ if and only if $j-i \equiv \{1,2\} \mod 5$.
Since $S_3$ is vertex-transitive,
we may assume that $v_1$ is the first vertex in $\sigma$. The in-neighbors of $v_1$ are $v_4$ and $v_5$, so $v_4<_{\sigma}v_5$ since otherwise, $\{v_1,v_4,v_5\}$ induces a triangle in $B_{\sigma}(S_3)$.
If $v_2<_{\sigma}v_5$, then $B_{\sigma}(S_3)|\{v_1,v_2,v_4,v_5\}$ has thickness two, but it has only one component which yields a contradiction by Proposition~\ref{onecomponent}. Thus, $v_5<_{\sigma}v_2$, and we have $\sigma|\{v_1,v_2,v_4,v_5\}=v_1,v_4,v_5,v_2$.
Then, no matter where $v_3$ is, $B_{\sigma}(S_3)$ is connected and has thickness two,
again a contradiction by Proposition~\ref{onecomponent}.
Hence, $S_3 \not \in \mathcal{F}$.
Suppose $\Delta_2$ is a forest tournament. Let $(\{v_1,v_2\},\{v_3,v_4\},\{v_5,v_6\})$ be a trisection of $\Delta_2$ where $v_i$ is adjacent to $v_{i+1}$ for $i=1,3,5$,
and $\sigma$ be a forest ordering of $\Delta_2$.
We may assume that either $v_1$ or $v_2$ is the first vertex in $\sigma$.
Let $v_1$ be the first vertex in $\sigma$. Then, $v_5 <_{\sigma} v_6$, since otherwise $\{v_1,v_5,v_6\}$ induces a triangle in $B_{\sigma}(\Delta_2)$.
If $v_6<_{\sigma} v_3$, then $\{v_1,v_3,v_5,v_6\}$ induces a cycle in $B_{\sigma}(\Delta_2)$. So, $v_3 <_{\sigma} v_6$, and, by the same reason, $v_4 <_{\sigma} v_6$.
If $v_2 <_{\sigma} v_5$, then $B_{\sigma}(\Delta_2)|\{v_1,v_2,v_5,v_6\}$ is a cycle of length four.
If $v_5 <_{\sigma} v_2<_{\sigma} v_6$, then the induced subgraph $B_{\sigma}(\Delta_2)|\{v_1,v_2,v_3, v_5,v_6\}$ is connected and has thickness two.
So, Corollary~\ref{nocycle} and Proposition~\ref{onecomponent} imply
$v_6 <_{\sigma}v_2$, and so $v_3 <_{\sigma} v_4$ since otherwise, $\{v_2,v_3,v_4\}$ induces a triangle in $B_{\sigma}(\Delta_2)$.
If $v_5<_{\sigma} v_4 <_{\sigma}v_6$, then $B_{\sigma}(\Delta_2)$ is connected and has thickness two, so Proposition~\ref{onecomponent} implies that $v_3<_{\sigma} v_4 <_{\sigma} v_5$.
Now we have $\sigma=v_1,v_3,v_4,v_5,v_6,v_2$ and the backward edges $v_5v_1,v_6v_1,v_2v_3,v_2v_4$.
However, in this case, there is no forest cut of $\Delta_2$ under $\sigma$. It is a contradiction.
Hence, $v_2$ is the first vertex in $\sigma$. Again, $v_5 <_{\sigma} v_6$. If $v_1 <_{\sigma} v_6$, then $\{v_1,v_2,v_6\}$ induces a triangle in $B_{\sigma}(\Delta_2)$. Hence, $v_6 <_{\sigma}v_1$. Then, no matter where $v_3$ is, $B_{\sigma}(\Delta_2)|\{v_1,v_2,v_3,v_5,v_6\}$ is connected and has thickness two, a contradiction by Proposition~\ref{onecomponent}. Therefore $\Delta_2 \not \in \mathcal{F}$.
\end{proof}
\section{Characterization of tournaments in $\mathcal{A}\cap \mathcal{F}$}\label{SEC:mainproof}
In this section, we characterize all tournaments in $\mathcal{A}\cap \mathcal{F}$.
We simply write $\mathcal{AF}$ for $\mathcal{A}\cap \mathcal{F}$.
\begin{THM}\label{THM:main2-1}
Let $H$ be a tournament. Then, $H \in \mathcal{AF}$ if and only if it is isomorphic to one of the following.
\begin{itemize}
\item[1)] $I$;
\item[2)] $H_1\Rightarrow H_2$ for $H_1,H_2 \in \mathcal{AF}$;
\item[3)] $\Deltaelta(I,L_k,H')$ or $\Deltaelta(I,H',L_k)$ for an integer $k$ and $H'\in \mathcal{AF}$;
\item[4)] $\Delta(L_{k_1},I,L_{k_2},L_{k_3}, I)$ or $\Delta(I,L_{k_1},L_{k_2},I, L_{k_3})$ for integers $k_1,k_2,k_3$;
\item[5)] $\Delta(I,H',L_{k_1},L_{k_2}, I)$ or $\Delta(I,L_{k_1},L_{k_2},H', I)$ for integers $k_1$, $k_2$ and $H' \in \mathcal{AF}$;
\item[6)] $\Delta(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)$ for integers $k_1,k_2,k_3,k_4$.
\end{itemize}
\end{THM}
Theorem~\ref{THM:main2-1} implies Theorem~\ref{THM:main2} as follows.
\begin{proof}[Proof of Theorem~\ref{THM:main2}, assuming Theorem~\ref{THM:main2-1}]
Let $D\in \mathcal{D}'$ and $H$ be a $D$-hero.
Lemma~\ref{LEM:A} together with Lemma~\ref{LEM:F} imply $D_3 \notin \mathcal{A} \cup \mathcal{F}$. So, since $D$ contains $D_3$, $D \notin A \cup F$.
So, by Theorem~\ref{THM_meet1} and Theorem~\ref{THM_meet2}, $H$ must belong to $\mathcal{AF}$.
Therefore, $H$ is a tournament in the list of Theorem~\ref{THM:main2-1}.
This completes the proof.
\end{proof}
In order to prove Theorem~\ref{THM:main2-1}, we need the following lemmas. Observe that $\mathcal{AF}$ is hereditary since both $\mathcal{A}$ and $\mathcal{F}$ are hereditary.
We denote by $C$ a cyclic triangle.
\begin{LEM}\label{AF3}
Let $H=\Deltaelta(G_1,G_2,G_3)$ for some tournaments $G_1,G_2,G_3$. If $H\in \mathcal{AF}$, then, either $G_1$, $G_2$ or $G_3$ is a one-vertex tournament, and one of the others is transitive.
\end{LEM}
\begin{proof}
If $|V(G_i)|\ge2$ for every $i=1,2,3$, then $H$ contains $\Delta_2$, which is a contradiction since $\Delta_2 \not \in \mathcal{F}$ by Lemma~\ref{LEM:F}. So, either $|V(G_1)|,|V(G_2)|$ or $|V(G_3)|$ is equal to one.
We may assume $|V(G_1)|=1$.
If both of $G_2$ and $G_3$ contain a cyclic triangle,
then $T$ contains $D_3$, a contradiction since $D_3 \not \in \mathcal{AF}$. So, either $G_2$ or $G_3$ is a transitive tournament.
This completes the proof.
\end{proof}
\begin{LEM}\label{AF5}
Let $H=\Delta(G_1,G_2,G_3,G_4,G_5)$ for tournaments $G_i$. If $H\in \mathcal{AF}$ then $(G_1,G_2,G_3,G_4,G_5)$ is either
\begin{itemize}
\item $(L_{k_1},I,L_{k_2},L_{k_3}, I)$, $(I,L_{k_1},L_{k_2},I, L_{k_3})$,
\item $(I,H',L_{k_1},L_{k_2},I)$ or $(I,L_{k_1},L_{k_2},H',I)$
\end{itemize}
for some integers $k_1,k_2,k_3$ and some $H'\in \mathcal{AF}$.
\end{LEM}
\begin{proof}
First, we claim that $G_1,G_3$ and $G_5$ are transitive tournaments.
\\
\noindent (1) {\em $G_1,G_3,G_5$ does not contain a cyclic triangle.}
\\
\\
Let $H_1=\Delta(C,I,I,I,I)$, $H_2=\Delta(I,I,C,I,I)$ and $H_3=\Delta(I,I,I,I,C)$.
Since $\mathcal{AF}$ is hereditary,
it is enough to prove that neither $H_1$, $H_2$ nor $H_3$ is contained in $\mathcal{A}$.
Suppose $H_k \in \mathcal{A}$ for some $k\in \{1,2,3\}$.
Note that $V(H_k)$ contains a maximal homogeneous set $S$ with three vertices inducing a cyclic triangle, and there are two vertices complete to $S$ and there are two vertices complete from $S$.
Since $H_k$ is strongly connected, there exists a $\Delta$-partition of $V(H_k)$, $(\{v_1\},X_1,\{v_2\},X_2,\ldots,X_{n-1},\{v_{n}\})$, by Proposition~\ref{PROP:upartition}.
By Proposition~\ref{Uhomo}, there exists $r$ such that $S=X_r$. Observe that $|X_i|= 1$ for every $i(\neq r)$ since $S$ is the only maximal homogeneous set of $H_k$. So, we obtain the following inequality:
$$
7=|V(H_k)|=n+\sum_{i=1}^{n-1} |X_i| \ge n+3+(n-2)=2n+1.
$$
So, $n\le 3$. If $n=2$ then $r=1$, and $|V(H_k)|=n+|X_1|=5<7$, a contradiction.
Hence, $n=3$.
If $X_1=S$ (resp. $X_2=S$), then there exists only one vertex complete to $S$ (resp. complete from $S$). This yields a contradiction since in $H_k$, there are two vertices complete to $S$ and two vertices complete from $S$.
Therefore, $H_k \not \in \mathcal{A}$ for $k=1,2,3$. This proves (1).
\\
\noindent (2) {\em $\Delta(I,C,I,I,L_2), \Delta(L_2,I,I,C,I) \not \in \mathcal{F}$. }
\\
\\
The complement of a forest tournament is a forest tournament.
So, it is enough to prove that $\Delta(I,C,I,I,L_2) \not \in \mathcal{F}$ since $\Delta(L_2,I,I,C,I)$ is the complement of $\Delta(I,C,I,I,L_2)$,
Let $K=\Delta(I,C,I,I,L_2)$ and $V(K)=\{v_1,v_2,\ldots,v_8\}$ such that for $1\le i<j\le 8$, $v_i$ is adjacent from $v_j$ if and only if $(i,j)=(1,5)$, $(1,7)$, $(1,8)$, $(2,4)$, $(5,7)$, $(5,8)$ and $(7,8)$. Figure~\ref{pic1} describes all backward edges of $K$ under $\sigma=v_1,v_2,\ldots,v_8$.
\begin{figure}
\caption{All backward edges of $K$ under $\sigma$}
\label{pic1}
\end{figure}
Suppose $K$ is a forest tournament with forest ordering $\sigma$.
Since $\{v_2,v_3,v_4\}$ induces a cyclic triangle, there exists a backward edge $v_b v_a$ with $a,b \in \{2,3,4\}$ under $\sigma$.
Since $v_1$ is complete to $\{v_2,v_3,v_4\}$, if $v_b <_{\sigma} v_1$, then $\{v_1,v_a,v_b\}$ induces a cyclic triangle in $B_{\sigma}(K)$. So, $v_1 <_{\sigma} v_b$.
Similarly, for $i=5,6,7,8$, since $v_i$ is complete from $\{v_2,v_3,v_4\}$, $v_a <_{\sigma} v_i$.
Suppose $v_i <_{\sigma} v_1$ for some $i=5,6,7,8$. Then, $v_a <_{\sigma} v_i <_{\sigma} v_1 <_{\sigma}v_b$, and $K|\{v_a,v_i,v_1,v_b\}$ has thickness two under $\sigma| \{v_a,v_i,v_1,v_b\}$, which yields a contradiction by Proposition~\ref{onecomponent}. Hence, $v_1 <_{\sigma} v_i$ for $i=5,6,7,8$.
Since $B_{\sigma}(K)$ is a forest and $v_1 <_{\sigma} v_i$ for $i=5,7,8$,
it follows that $v_8 <_{\sigma} v_7 <_{\sigma} v_5$.
Let us look at $v_6$.
If $v_7 <_{\sigma} v_6$, then $\{v_1,v_8,v_7,v_6\}$ induces a cycle of length four in $B_{\sigma}(K)$.
So, $v_1 <_{\sigma} v_6 <_{\sigma}v_7$. However, in this case, $K|\{v_1,v_6,v_8,v_7,v_5\}$ has thickness two under $\sigma|\{v_1,v_6,v_8,v_7,v_5\}$, which is a contradiction.
Therefore, $K$ is not a forest tournament. This prove (2).
\\
By (1), $G_1$, $G_3$ and $G_5$ are transitive tournaments.
\noindent {\bf Case 1: $|V(G_1)|\ge 2$.}
Then $G_2$ and $G_5$ are one-vertex tournaments, since otherwise, $V(G_1) \cup V(G_2) \cup V(G_5)$ induces a subtournament of $H$ containing $\Delta_2$ which is not a forest tournament by Lemma~\ref{LEM:F}.
By (2), $G_4$ does not contain a cyclic triangle. This implies that $(G_1,G_2,G_3,G_4,G_5)=(L_{k_1}, I, L_{k_2}, L_{k_3},I)$ for some positive integers $k_1,k_2$ and $k_3$.
\noindent {\bf Case 2: $|V(G_5)|\ge 2$.}
Similar to Case 1, we have $(G_1,G_2,G_3,G_4,G_5)=(I,L_{k_1}, L_{k_2},I, L_{k_3})$ for some positive integers $k_1,k_2$ and $k_3$.
\noindent {\bf Case 3: $|V(G_1)|=|V(G_1)|=1$.}
If both $G_2$ and $G_4$ contain a cyclic triangle, then $H$ contains $A_3$, which is not a forest tournament since $\chi(A_3)=3$.
Thus, either $G_2$ or $G_4$ is a transitive tournament.
Finally, since $\mathcal{AF}$ is hereditary, it follows that $G_2, G_4 \in \mathcal{AF}$, so
$(G_1,G_2,G_3,G_4,G_5)= (I,H',L_{k_1},L_{k_2},I)$ or $(I,L_{k_1},L_{k_2},H',I)$ for positive integers $k_1,k_2$ and $H'\in \mathcal{AF}$. This completes the proof.
\end{proof}
\begin{LEM}\label{AF7}
Let $H=\Delta(G_1,G_2,G_3,G_4,G_5,G_6,G_7)$ for tournaments $G_i$s. If $H\in \mathcal{AF}$ then $G_i$ is transitive for $i=1,\ldots,7$ and $|V(G_j)|=1$ for $j=1,4,7$.
\end{LEM}
\begin{proof}
For some $j=1,4,7$, if $|V(G_j)|\ge 2$, then $H$ contains $\Delta_2$ which is not a forest tournament by Lemma~\ref{LEM:F}. So, $G_j$ is a one-vertex tournament for $j=1,4,7$.
If either $G_2, G_3, G_5$ or $G_6$ contains a cyclic triangle, then $T$ contains either $\Delta(I,I,C,I,I)$, $\Delta(I,C,I,I,L_2)$ or $\Delta(L_2,I,I,C,I)$, which does not belong to $\mathcal{AF}$ by Lemma~\ref{AF5}. So, $G_i$ is a transitive tournament for $i=2,3,5,6$.
This completes the proof.
\end{proof}
If $T_1$ and $T_2$ are tournaments with at least two vertices,
then for a vertex $v \in V (T_1)$, we say \emph{a tournament $T$ is obtained from $T_1$ by substituting $T_2$
for $v$} if $V(T) = V(T_1) \cup V (T_2) \setminus \{v\}$ and $xy \in E(T)$ if and only if one of
the following holds.
\begin{itemize}
\item $xy \in E(T_1 \setminus v)$ or $xy \in E(T_2)$,
\item $x \in V (T_1)$, $y\in V (T_2)$ and $xv \in V (T_1)$,
\item $x \in V (T_2)$, $y \in V (T_1)$, and $vy \in V (H_1)$.
\end{itemize}
We remark that every non-prime tournament can be obtained from a prime tournament by substitutions.
Now we are ready to prove Theorem~\ref{THM:main2-1}.
\begin{proof}[Proof of Theorem~\ref{THM:main2-1}]
First, we prove the `only if' part.
Let $H \in \mathcal{AF}$.
\\
\\
(1) \emph{If $H$ is prime, then $H$ is isomorphic to $I,L_2, U_2(=C), U_3$ or $U_4$.}
\\
\\
If $|V(H)|\le 2$, then $H$ is isomorphic to either $I$ or $L_2$, so we are done.
Assume that $|V(H)|\ge 3$.
Since $H$ is prime and belongs to $\mathcal{A}$,
Proposition~\ref{Uprime} implies that $H$ is isomorphic to $U_n$ for some $n\ge 2$.
If $n\ge 5$, then $U_n$ contains $\Delta_2$, so $U_n \not\in \mathcal{F}$ by Lemma~\ref{LEM:F}.
Therefore, $H$ is isomorphic to either $U_2 (=C), U_3$ or $U_4$.
\\
Since $I \in \mathcal{AF}$, we can obtain $L_2$ from 2), $U_2$ from 3), $U_3$ from 4) and $U_4$ from 6).
Let us consider the case that $H$ is not prime, that is,
$H$ can be obtained from some prime tournament $G_0$ with $|V(G_0)|>1$ by substituting $G_1,G_2,\ldots,G_n$ for vertices $v_1,v_2,\ldots,v_n$ of $G_0$.
For each $i$, since $G_i$ is a subtournament of $H$, it also belongs to $\mathcal{AF}$. In particular, $G_0 \in \mathcal{AF}$.
So, by (1), $G_0$ is isomorphic to either $L_2, U_2, U_3$ or $U_4$.
Then, Lemma~\ref{AF3}, Lemma~\ref{AF5} and Lemma~\ref{AF7} imply that $H$ is isomorphic to one of the tournaments in the list of Theorem~\ref{THM:main2-1}.
Now we prove the `if' part, that is, every tournament $H$ in the list of Theorem~\ref{THM:main2-1} belongs to $\mathcal{AF}$.
Since the complement of a tournament in $\mathcal{AF}$ belongs to $\mathcal{AF}$, we need to consider the following six cases.
\noindent{\bf Case 1.} It is trivial when $H=I$.
\\
\\
\noindent{\bf Case 2.} $H=H_1\Rightarrow H_2$ for some $H_1,H_2 \in \mathcal{AF}$.
\\
\\
To show $H\in \mathcal{A}$,
choose $k_i$ such that $A_{k_i}$ contains $H_i$ for $i=1,2$. Let $K=\max\{k_1,k_2\}$. Then, $A_{K+1}$ contains $H$ since $A_{K+1}$ contains two copies $A_K^{(1)}$ and $A_K^{(2)}$ of $A_K$ where $V(A_K^{(1)})$ is complete to $V(A_K^{(2)})$.
To show that $H\in \mathcal{F}$, let $\sigma_i$ be a forest ordering of $H_i$ for $i=1,2$. Let $\sigma=\sigma_1,\sigma_2$, that is, for $u,v \in V(H)$, $v<_{\sigma} u$ if either
\begin{itemize}
\item $v\in V(H_1)$ and $u\in V(H_2)$,
\item $v,u \in V(H_1)$ and $v<_{\sigma_1} u$, or
\item $v,u \in V(H_2)$ and $v<_{\sigma_2} u$.
\end{itemize}
Then, $\sigma$ is a forest ordering of $H$, and so $H$ is a forest tournament. Therefore $H\in \mathcal{AF}$.
\\
\\
\noindent{\bf Case 3.} $H=\Delta(I,L_k,H')$
for an integer $k$ and $H' \in \mathcal{AF}$.
\\
\\
To show $H \in \mathcal{A}$, let $M$ be a positive integer such that $A_{M-1}$ contains $H'$ and $M>k$. Then, clearly, $A_{M}$ contains $ \Delta(I,L_k,H')$.
To show $H\in \mathcal{F}$, let $v_1,v_2,\ldots, v_n$ be a forest ordering of $H'$, and $V(L_k)=\{u_i\mid 1\le i\le k\}$ with $u_i \to u_j$ for $1\le i <j \le k$.
Then, the ordering $u_1,u_2,\ldots,u_k,v_1,v_2,\ldots,v_n,w$ is a forest ordering of $\Deltaelta(I,L_k,H)$, so, $H$ is a forest tournament. Therefore, $H\in \mathcal{AF}$.
\\
\\
\noindent{\bf Case 4.} $H=\Delta(L_{k_1},I,L_{k_2},L_{k_3}, I)$
for integers $k_1,k_2,k_3$;
\\
\\
To prove $H \in \mathcal{A}$, let $K=k_1+k_2+k_3$. Then, $U_K$ contains $\Delta(L_{k_1},I,L_{k_2},L_{k_3}, I)$
and so, $\Delta(L_{k_1},I,L_{k_2},L_{k_3},I)$ belongs to $\mathcal{A}$.
Let $\sigma=a,b,y_1,y_2,\ldots,y_{k_2},x_1,x_2,\ldots,x_{k_1},z_1,z_2,\ldots,z_{k_3}$ be a vertex ordering of $\Delta(L_{k_1},I,L_{k_2},L_{k_3},I)$ with backward edges $\{x_{i}a \mid 1\le i \le k_1\} \cup \{z_jb\mid 1\le j \le k_3\}$. Then, $\sigma$ is a forest ordering of $H$. So $H$ is a forest tournament.
\\
\\
\noindent{\bf Case 5.} $H=\Delta(I,H',L_{k_1},L_{k_2}, I)$
for integers $k_1$, $k_2$ and $H' \in \mathcal{AF}$.
\\
\\
Let $M>k_1+k_2$ be a positive integer such that $A_{M-1}$ contains $H'$. Then, $A_M$ contains $H$, so $H \in \mathcal{A}$.
To prove $H \in \mathcal{F}$, let $\sigma'=v_1,v_2,\ldots,v_n$ be a forest ordering of $H'$. Let $\sigma=a,v_1,v_2,\ldots,v_n, b,x_1,x_2,\ldots,x_{k_1},y_1,\ldots,y_{k_2}$ be a vertex ordering of $H$ with backward edge set the union of the set of backward edges of $H'$ under $\sigma'$, $\{ab\}$, $\{x_ia\mid 1\le i \le k_1\}$ and $\{y_ib \mid 1\le i \le k_2\}$. Then, $\sigma$ is a forest ordering. So, $H \in \mathcal{F}$.
\\
\\
\noindent{\bf Case 6.} $H=\Delta(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)$ for integers $k_1,k_2,k_3,k_4$.
\\
\\
Let
$$
\sigma=x_1,\ldots,x_{k_1},c,z_1,\ldots,z_{k_3},b,y_1,\ldots,y_{k_2},a,w_1,\ldots,w_{k_4}
$$
be a vertex ordering of $\Delta(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)$ where the set of backward edges is $\{(a,x_i) \mid i\in [k_1]\} \cup \{(w_i,c)\mid i\in [k_4]\} \cup \{(b,c)\} \cup \{(b,z_i)\mid i\in [k_3]\} \cup \{(y_i,b)\mid i\in [k_2]\} \cup \{(a,b)\}$.
Then, $\sigma$ is a forest ordering, so $\Delta(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)$ is a forest tournament.
To prove $\Delta(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)\in \mathcal{A}$, let $K=k_1+k_2+k_3+k_4+1$. Then, $U_4(I,L_{k_1}, L_{k_2}, I,L_{k_3},L_{k_4},I)$ is contained in $A_K$.
This completes the proof.
\end{proof}
\section{Constructions of heroes}\label{SEC:growing}
In this section, we prove Theorem~\ref{THM:growing2} and Theorem~\ref{THM:U3}.
\preceqsection{Proof of Theorem~\ref{THM:growing2}}
We start with the following observations.
\begin{OBS}\label{OBS:hero_property}
Let $D\in \mathcal{D}'$, and $\mathcal{H}$ be the set of all $D$-heroes.
\begin{itemize}
\item[(1)] $\mathcal{H}$ is hereditary since for a tournament $T$ and its subtournament $T'$, every $T'$-free tournament is $T$-free.
\item[(2)] $\mathcal{H}$ is closed under taking complement since every tournament has the same chromatic number with its complement.
\end{itemize}
\end{OBS}
We need a lemma from~\cite{hero} in order to
prove Theorem~\ref{THM:growing2}.
We first give the following definitions.
For tournaments $G, H$ and an integer $a$, an {\em $(a,G,H)$-jewel} is a tournament $T$ with $|V(T)|=a$ such that every partition $(A,B)$ of $V(T)$, either $T|A$ contains $G$ or $T|B$ contains $H$.
We say a tournament $T$ \emph{contains an $(a,G,H)$-jewel chain of length $n$} if there exist vertex disjoint subtournaments $J_1,J_2,\ldots,J_n$ of $T$, which are $(a,G,H)$-jewels, such that $V(J_i)$ is complete to $V(J_j)$ for $1\le i<j\le n$.
\begin{LEM}[Berger et al. \cite{hero}]\label{lemma2}
Let $H$, $K$ be tournaments and $a\ge 1$ an integer. If either $H$ or $K$ is transitive, then there is a map $f_{H,K}:\mathbb{Z}^+\times \mathbb{Z}^+ \to \mathbb{Z}^+$ satisfying the following property.
For every $\Deltaelta(I,H,K)$-free tournament $G$, if
\begin{itemize}
\item $c_1$ is an integer such that every $H$-free subtournament of $G$ and $K$-free subtournament of $G$ has chromatic number at most $c_1$, and
\item $c_2$ is an integer such that every subtournament of $G$ containing no $(a,H,K)$-jewel-chain of length four has chromatic number at most $c_2$,
\end{itemize}
then $G$ has chromatic number at most $f_{H,K}(c_1,c_2)$.
\end{LEM}
We also need the following result of Stearns~\cite{transitive}.
\begin{THM}[Stearns~\cite{transitive}]\label{THM:Ramsey}
For every integer $k\ge 1$, every tournament with at least $2^{k-1}$ vertices contains $L_k$.
\end{THM}
\begin{proof}[Proof of Theorem~\ref{THM:growing2}]
Let $D \in \mathcal{D}'$ and $H$ a tournament admitting a trisection.
Suppose $H$ is a $D$-hero. Then, $H$ belongs to $\mathcal{AF}$, and by Lemma~\ref{AF3}, $H$ is isomorphic to $\Delta(I,H', L_k)$ or $\Delta(I,L_k,H')$ for some positive integer $k$ and $H' \in \mathcal{AF}$. Since $H'$ is a subtournament of $H$, it is a $D$-hero by Observation~\ref{OBS:hero_property} (1).
This proves the `only if' part.
For the `if' part, it is enough to show that $H=\Delta(I,H',L_k)$ is a $D$-hero by Observation~\ref{OBS:hero_property} (2). Let $c$ be an integer such that every $H'$-free tournament has chromatic number at most $c$.
We show that there exists $d$ such that every $H$-free tournament $T$ has chromatic number at most $d$.
Let $a=2^k|V(H')|$.
\\
\\
(1) For every tournament $T'$, if $T'$ contains no $(a,H',L_k)$-jewels, then $\chi(T')$ is less than $a+c$.
\\
\\
If $T'$ is $H'$-free, then $\chi(T') \le c$. So, we may assume that $T'$ contains $H'$.
Let $H_1,H_2,\ldots,H_m$ be vertex disjoint subtournaments of $T'$ isomorphic to $H'$ with $m$ maximum.
Let $J=\bigcup_{i=1}^{\min\{m,2^k\}} V(H_i)$.
If $m\ge 2^k$, then $T'|J$ is an $(H',L_k)$-jewel. (For every partition $(X,Y)$ of $J$, if $T'|X$ is $H'$-free, then $Y$ meets $V(H_i)$ for every $i=1,2,\ldots,2^k$, which implies $|Y| \ge 2^k$. So, $T'|Y$ contains $L_k$ by Theorem~\ref{THM:Ramsey}.)
Thus, $m<2^k$.
Note that $T'\setminus J$ is $H'$-free by the maximality of $m$, so it has chromatic number at most $c$.
Therefore, the chromatic number of $T'$ is at most
$$
\chi(T'|J)+\chi(T'\setminus J) \le |J|+c < a+c.
$$
This proves (1).
\\
By (1), we may assume that $T$ contains $(a,H',L_k)$-jewels.
Let $\mathcal{J}$ be the set of all $(a,H',L_k)$-jewels, $\mathcal{J}_1=\{J'\Rightarrow J''|J',J'' \in \mathcal{J}\}$
and $\mathcal{J}_2=\{J_1' \Rightarrow J_1''|J_1',J_1'' \in \mathcal{J}_1\}$.
Since every $\mathcal{J}$-free subtournament of $T$ has chromatic number at most $a+c$ by (1),
every $\mathcal{J}_1$-free subtournament of $T$ has chromatic number at most some constant $c_1$ by Lemma~\ref{lemma1} with $\mathcal{H}_1=\mathcal{H}_2=\mathcal{J}$.
By applying Lemma~\ref{lemma1} again,
there exists $c_2$ such that every $\mathcal{J}_2$-free subtournament of $T$
has chromatic number at most $c_2$.
Since every tournament not containing $(a,H',L_k)$-jewel-chain of length four is $\mathcal{J}_2$-free, it has chromatic number at most $c_2$.
Hence, by Lemma~\ref{lemma2}, there exists $d$ such that every $H$-free tournament has chromatic number at most $d$. This completes the proof.
\end{proof}
\preceqsection{Proof of Theorem~\ref{THM:U3}}\label{SEC:proofU3}
To prove Theorem~\ref{THM:U3}, we need the following result of Liu~\cite{gaku}.
(Recall that $S_n$ is a tournament defined at the beginning of Section~\ref{SEC:nonhero}.)
\begin{THM}\label{gaku}(Liu~\cite{gaku})
Let $T$ be a prime tournament. Then, $T$ is $U_3$-free if and only if $T$ is isomorphic to $S_n$ for some $n\ge1$ or $V(T)$ can be partitioned into sets $X_1,X_2,X_3$ such that $X_1\cup X_2$, $X_2\cup X_3$ and $X_3\cup X_1$ are transitive.
\end{THM}
\begin{proof}[Proof of Theorem~\ref{THM:U3}]
We prove that for a tournament $T$, if $T$ is $\{D_n,U_3\}$-free for some $n\ge 2$, then $T$ is $3^{n-2}$-colorable.
This implies Theorem~\ref{THM:U3} since for every $D\in \mathcal{D}'$, there exists $D_n$ containing $D$, and every $\{D,U_3\}$-free tournament is $\{D_n,U_3\}$-free.
We use induction on $|V(T)|$.
The base case is that either $n=2$ or $T$ is prime.
If $n=2$, then $\chi(T) =1$ since $T$ is $D_2$-free.
If $T$ is prime and $n>2$, then Theorem~\ref{gaku} implies that $G$ is two colorable, so we are done.
Suppose $T$ is not prime, and assume the statement is true for every graph with less than $|V(T)|$ vertices.
Let $T$ be $\{D_n, U_3\}$-free for some $n>2$.
Since $T$ is not prime, $T$ is obtained from some prime tournament $G_0$ by substitutions, and Theorem~\ref{gaku} implies that either
\begin{itemize}
\item $G_0$ is isomorphic to $S_m$ for some $m\ge 2$, or
\item $V(G_0)$ can be partitioned into sets $X_1,X_2,X
_3$ such that $X_1\cup X_2$, $X_2\cup X_3$ and $X_3\cup X_1$ are transitive.
\end{itemize}
For the first case, let $V(G_0)=\{v_1,v_2,\ldots,v_{2m-1}\}$ with $v_iv_j \in E(G_0)$ for every $i,j$ with $j-i \in \{1,2,\ldots,m-1\} \mod{2m-1}$, and let $T$ be obtained from $G_0$ by substituting $G_i$ for $v_i$ for $i=1,2,\ldots,2m-1$.
For every edge $v_iv_j$ of $G_0$, there exists a vertex $v_k$ such that $v_j \to v_k$ and $v_k\to v_i$.
Hence, if both $G_i$ and $G_j$ contain $D_{n-1}$, then for every $v\in V(G_k)$, $V(G_i) \cup V(G_j) \cup \{v\}$ induces a subtournament of $T$ containing $D_n$, which yields a contradiction.
Therefore, all tournaments $G_1,G_2,\ldots,G_{2m-1}$ but one are $D_{n-1}$-free.
Without loss of generality, let $G_1,G_2,\ldots,G_{2m-2}$ be $D_{n-1}$-free.
By the induction hypothesis,
there exist a $3^{n-3}$-coloring $\phi_i:V(G_i) \to \{1,2,3\}^{n-3}$ of $G_i$ for $i=1,2,\ldots,2m-2$
and a $3^{n-2}$-coloring $\phi_{2m-1}:V(G_{2m-1}) \to \{1,2,3\}^{n-2}$ of $G_{2m-1}$.
We define a map $\phi: V(T)\to \{1,2,3\}^{n-2}$ as follows:
\begin{itemize}
\item for $v\in V(G_{2m-1})$, $\phi(v)=\phi_{2m-1}(v)$;
\item for $i=1,2,\ldots,m-1$ and $v\in V(G_i)$, $\phi(v)=\{1\} \times \phi_i(v)$, and
\item for $i=m,m+1,\ldots,2m-2$ and $v\in V(G_i)$,
$\phi(v)=\{2\} \times \phi_i(v)$.
\end{itemize}
We claim that $\phi$ is a $3^{n-2}$-coloring of $T$.
Suppose there exist three vertices $u\in V(G_i)$, $v\in V(G_j)$ and $w\in V(G_k)$ inducing a monochromatic cyclic triangle in $T$. Clearly, $i$, $j$ and $k$ are all distinct. Indeed, $\{i,j,k\}$ intersects with only one of $\{1,2,\ldots,m-1\}$ and $\{m,m+1,\ldots, 2m-2\}$ by the definition of $\phi$.
However, in either case, $\{u,v,w\}$ is transitive, a contradiction.
Therefore, $\phi$ is a $3^{n-2}$-coloring of $T$.
For the second case, let $\{G_v \mid v \in V(G_0)\}$ be tournaments such that
$T$ is obtained from $G_0$ by substituting $G_v$ for $v$ for every $v\in G_0$.
Clearly, $G_v$ is $\{D_n, U_3\}$-free for $v \in V(G_0)$.
For each $v\in V(G_0)$,
let $\phi_v: V(G_v) \to \{1,2,3\}^{n-3}$ be a $3^{n-3}$-coloring of $G_v$
if $G_v$ is $D_{n-1}$-free,
and $\phi_v:V(G_v) \to \{1,2,3\}^{n-2}$ be a $3^{n-2}$-coloring of $G_v$
if $G_v$ contains $D_{n-1}$.
We define a map $\phi:V(G)\to \{1,2,3\}^{n-2}$ as follows.
\begin{itemize}
\item for $v\in V(G_0)$ and $u\in G_v$, if $G_v$ contains $D_{n-1}$, then let $\phi(u)=\phi_v(u)$;
\item for $v\in v(G_0)$ and $u\in G_v$, if $G_v$ is $D_{n-1}$-free and $v \in X_i$,
then let $\phi(u)=\{i\} \times \phi_v(u)$.
\end{itemize}
We claim that $\phi$ is a $3^{n-2}$-coloring of $T$.
Suppose there exist three vertices $u_1\in V(G_{v_1})$, $u_2\in V(G_{v_2})$ and $u_3\in V(G_{v_3})$ inducing a monochromatic cyclic triangle in $T$. Clearly, $v_1$, $v_2$ and $v_3$ are all distinct vertices of $G_0$.
If two of $G_{v_1}$, $G_{v_2}$ and $G_{v_3}$ contain $D_{n-1}$, then $V(G_{v_1})\cup V(G_{v_2}) \cup V(G_{v_3})$ induces a subtournament of $T$ containing $D_n$, which yields a contradiction. So, without loss of generality, let $G_{v_1}$ and $G_{v_2}$ be $D_{n-1}$-free.
Then, for $u_1$ and $u_2$ to have the same color in $\phi$, $v_1$ and $v_2$ belong to the same $X_{\ell}$ for some $\ell =1,2,3$.
Then, by the condition that $X_1 \cup X_2$, $X_2 \cup X_3$ and $X_3\cup X_1$ are transitive in $G_0$, $\{v_1,v_2,v_3\}$ is transitive in $G_0$ and so $\{u_1,u_2,u_3\}$ is transitive in $T$, a contradiction. Therefore, $\phi$ is a $3^{n-2}$-coloring of $G$. This completes the proof.
\end{proof}
\cleardoublepage
\ifdefined\phantomsection
\phantomsection
\else
\fi
\addcontentsline{toc}{chapter}{Bibliography}
\end{document} |
\begin{document}
\title[On the Weak Maximizing Properties]{On the Weak Maximizing Properties}
\author[L. Garc\'ia-Lirola]{Luis C. Garc\'ia-Lirola}
\address{Departamento de Matem\'aticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain}
\email{[email protected]}
\author[C. Petitjean]{Colin Petitjean}
\address{LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vall\'ee, France}
\email{[email protected]}
\overline{\delta}ate{November, 2020}
\begin{abstract}
Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the weak$^*$ topology). We
provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair $(X,Y)$ enjoys a certain maximizing property. This approach not only allows us to (re)obtain as a direct consequence that the pair $(\ell_p,\ell_q)$ has the WMP but also provides many more natural examples of pairs having a given maximizing property.
\end{abstract}
\subjclass{Primary 46B20; Secondary 46B04, 54E50}
\keywords{Norm attainment, Banach Space, Weak Maximizing Property, Asymptotic Uniform Smoothness, Asymptotic Uniform Convexity}
\maketitle
\section{Introduction}
Let $X,Y$ be two real Banach spaces and let $T\colon X \to Y$ be a bounded linear operator (we will write $T \in \mathcal{L}(X,Y)$). A maximizing sequence for $T$ is a sequence $(x_n)_n \subset X$ with $\|x_n\|=1$ for every $n \in \mathbb{N}$ and such that $\lim\limits_{n \to \infty} \|Tx_n\|_Y = \|T\|$. Next, we say as usual that $T\colon X \to Y$ attains its norm whenever there exists a vector $x \in X$ of norm 1 such that $\|Tx\|_Y = \|T\|$.
Now following \cite{Aron_WMP19}, a pair of Banach spaces $(X,Y)$ is said to have the weak maximizing property (WMP) if for any bounded linear operator $T\colon X \to Y$, the existence of a non-weakly null maximizing sequence for $T$ implies that $T$ attains its norm.
For instance, it is proved in \cite[Theorem~1]{Pellegrino_09} that the pair $(\ell_p,\ell_q)$ has the WMP whenever $1<p<\infty$ and $1 \leq q < \infty$. That result was extended in \cite[Proposition 2.2]{Aron_WMP19} to the pair $(\ell_p(\Gamma_1),\ell_q(\Gamma_2))$ where $\Gamma_1,\Gamma_2$ are arbitrary index sets.
Needless to say that the theory of norm attaining operators finds many applications in both pure and applied mathematics. A motivation for the study of the WMP is the following application which was noted in \cite{Aron_WMP19} (and extends a former result due to J.~Kover in the Hilbert case \cite{Kover2005}): if $(X,Y)$ has the weak maximizing property, $T \in \mathcal{L}(X,Y)$ and $K\colon X \to Y$ is a compact operator such that $\|T\| < \|T+K\|$, then $T+K$ is norm attaining.
As a consequence, the authors could deduce that a pair $(X,Y)$ has the weak maximizing property for some $Y \neq \set{0}$ if and only if $X$ is reflexive.
In this note, we provide a new (possibly simpler) approach based on a comparison between the modulus of asymptotic uniform convexity of $X$ and the modulus of asymptotic uniform smoothness of $Y$, and conversely; see Section~\ref{section_AUSAUC} for precise definitions. Moreover, we consider the following related and quite natural properties.
\begin{definition}
Let $X$ and $Y$ be Banach spaces. Let $\tau_X$ and $\tau_Y$ be any topology on $X$ and $Y$ respectively.
\begin{itemize}[leftmargin=*]
\item We say that a pair $(X,Y)$ has the $\tau_X$-to-$\tau_Y$ maximizing property ($\tau_X$-to-$\tau_Y$MP) if for any $T \in \mathcal{L}(X,Y)$ which is $\tau_X$-to-$\tau_Y$ continuous, the existence of a non $\tau_X$-null maximizing sequence for $T$ implies that $T$ attains its norm.
\end{itemize}
In this paper, $\tau_X$ and $\tau_Y$ will mainly be the usual weak or weak$^*$ topologies. Since any $T \in \mathcal{L}(X,Y)$ is weak-to-weak continuous, it is clear that the WMP corresponds to the weak-to-weakMP. However, there are operators $T \in \mathcal{L}(X^*,Y^*)$ which are not weak$^*$-to-weak$^*$ continuous, so we will also consider the next property:
\begin{itemize}[leftmargin=*]
\item We say that a pair $(X^*,Y)$ has the weak$^*$ maximizing property (W$^*$MP) if for any $T \in \mathcal{L}(X^*,Y)$, the existence of a non-weak$^*$ null maximizing sequence for $T$ implies that $T$ attains its norm.
\end{itemize}
Notice that if the pair $(X^*,Y)$ has the WMP then it also has the W$^*$MP, and consequently the weak$^*$-to-weakMP. Also, if $X$ is reflexive, then a pair $(X,Y)$ has the WMP if and only if $(X,Y)$ has the W$^*$MP if and only if $(X,Y)$ has the weak$^*$-to-weakMP. However, we will prove that these three properties do not coincide in general.
\end{definition}
\begin{remark}
The definition of the above properties can be stated in terms of nets instead of sequences. Indeed, given an operator $T\colon X\to Y$, the following properties are equivalent:
\begin{itemize}
\item[i)] There exists a non $\tau_X$-null maximizing sequence for $T$.
\item[ii)] There exists a non $\tau_X$-null maximizing net for $T$.
\item[iii)] There exists a net which is maximizing for $T$ and does not admit $0$ as a $\tau_X$-cluster point.
\end{itemize}
Clearly, i)$\mathbb{R}ightarrow$ii).
To see that iii)$\mathbb{R}ightarrow$ i), let $(x_\alpha)_\alpha\subset S_{X}$ be a net such that $0$ is not a $\tau_X$-cluster point and $\lim_\alpha\norm{Tx_\alpha}=\norm{T}$. Pick inductively $\alpha_n$ such that $\norm{Tx_{\alpha_n}}\geq 1-\frac{1}{n}$ and $\alpha_n\geq \alpha_m$ if $n\geq m$. Then the sequence $(x_{\alpha_n})_n$ is not $\tau_X$-null, and it is maximizing for $T$.
Finally, assume that ii) holds and let's prove iii). Let $(x_\alpha)_\alpha$ be a non $\tau_X$-null net maximizing for $T$. If $0$ is the only $\tau_X$-cluster point of $(x_\alpha)_\alpha$, then all the subnets of $(x_\alpha)_\alpha$ converge to $0$, which is a contradiction (see e.g. \cite{AliprantisBorder}).
Thus, $0$ is not the only $\tau_X$-cluster point of $(x_\alpha)_\alpha$, that is, there is a subnet $(y_\beta)_\beta$ convergent to $y\neq 0$, and $(y_\beta)_\beta$ is also maximizing for \nolinebreak$T$.
\end{remark}
\begin{remark}
Replacing the WMP by the weak$^*$-to-weak$^*$MP, one can follow the lines of \cite[Proposition 2.4]{Aron_WMP19} to obtain the following result:\\
Suppose that $(X^*,Y^*)$ has the weak$^*$-to-weak$^*$MP. Let $T,K \colon X^* \to Y^*$ be weak$^*$-to-weak$^*$ continuous linear operators such that $K$ is compact.
If $\norm{T} < \norm{T+K}$ then $T+K$ is norm attaining.
\end{remark}
We now describe the main findings of this paper. Throughout the paper, $X$ and $Y$ will denote real Banach spaces while $X^*$ denotes, as usual, the topological dual of $X$.
After recalling the definition of the modulus $\overline{\delta}_{X^*}^*(t)$ of weak$^*$ asymptotic uniform convexity and of the modulus $\overline{\rho}_Y(t)$ of asymptotic uniform smoothness in Section~\ref{section_AUSAUC}, we prove our first theorem in Section~\ref{section-mainresults}.
\begin{T1} \label{T1}
Let $X,Y$ be Banach spaces. Assume that for every $t>0$, $\overline{\overline{\delta}elta}_{X^*}^*(t)\geq \overline{\rho}_Y(t)$ and, for all $t\geq 1$,
$\overline{\overline{\delta}elta}_{X^*}^*(t)>t-1$. Then,
\begin{enumerate}
\item the pair $(X^*,Y)$ has the weak$^*$-to-weakMP.
\item If moreover $Y\equiv Z^*$ is a dual space, then the pair $(X^*,Z^*)$ has the weak$^*$-to-weak$^*$MP.
\item If $\overline{\delta}_{X^*}^*(t)=t$, then $(X^*,Z)$ has the W$^*$MP for any Banach space $Z$.
\end{enumerate}
In particular, if $X$ is reflexive then the pair $(X^*,Y)$ has the WMP.
\end{T1}
Since for any infinite sets $\Gamma_1$ and $\Gamma_2$ and for $1<p<q<\infty$, it is well known that $\overline{\delta}_{\ell_p(\Gamma_1)}^*(t) = (1+t^p)^{1/p} - 1 > (1+t^q)^{1/q} - 1 = \overline{\rho}_{\ell_q(\Gamma_2)}$, our last theorem provides a new proof of the fact that the pair $(\ell_p(\Gamma_1), \ell_q(\Gamma_2))$ has the WMP. Note that the case $p=q$ also follows from the last theorem.
In both papers \cite{Aron_WMP19,Pellegrino_09}, the proof for the case $1\leq q<p<\infty$
follows from Pitt's theorem asserting that any bounded operator $T\colon \ell_p \to \ell_q$ is compact and thus attains its norm. In fact, Pitt's theorem can also be generalised using the modulus $\overline{\rho}_X(t)$ of asymptotic uniform smoothness of $X$ and the modulus $\overline{\delta}_Y(t)$ of asymptotic uniform convexity of $Y$. Namely, if there exists $t>0$ such that $\overline{\rho}_X(t) < \overline{\delta}_Y(t)$, then every bounded linear operator from $X$ to $Y$ is compact (see \cite[Proposition~2.3]{JLPS02}, where this is stated only for $0<t<1$, but the
proof works for any $t >0$). As an easy consequence we deduce the next result.
\begin{T2}
Let $X,Y$ be Banach spaces and $\eqnorma$ be an equivalent norm on $X^*$. If there exists $t>0$ such that $\overline{\rho}_{\eqnorma}(t) < \overline{\delta}_Y(t)$, then
\begin{enumerate}
\item the pair $(X^*,Y)$ has the weak$^*$-to-weakMP.
\item If moreover $Y \equiv Z^*$ is a dual space, then $(X^*,Z^*)$ has the weak$^*$-to-weak$^*$MP.
\end{enumerate}
In particular, if $X$ is reflexive then $(X^*,Y)$ has the WMP.
\end{T2}
In Section~\ref{section-renorming}, we take advantage of the renorming theory to enlarge the range of applications of Theorem~\ref{TheoremWeakStarMP}. Notably, we deduce that if $X$ belongs to a special class of Banach spaces, one can find an equivalent norm $\eqnorma$ on $X^*$ as well as some $q \geq 1$ such that the pair $\big((X^*,\eqnorma),\ell_q\big)$ has the weak$^*$-to-weak$^*$MP. In particular, if $X$ is moreover reflexive then $\big((X^*,\eqnorma),\ell_q\big)$ has the WMP. Furthermore, there is a strong relationship between asymptotic moduli and lower/upper estimates for spaces having finite dimensional decompositions (shortened FDDs). This allows us to prove the next theorem.
\begin{T3}
Let $X, Y$ be Banach spaces with shrinking FDDs. Assume that the norm of $X^*$ is $p$-AUC* and the norm of $Y$ is $q$-AUS for some $1<p \leq q<\infty$. Then there exist equivalent norms $\eqnorma_X$ on $X$ and $\eqnorma_Y$ on $Y$ such that the pair $\big((X^*,\eqnorma_{X}^*), (Y,\eqnorma_Y)\big)$ has the weak*-to-weakMP.
\end{T3}
Finally, in Section~\ref{section-applications}, we deal with some classical Banach spaces.
For instance, it is readily seen that Schur spaces are the best range spaces for the WMP: the pair $(X,Y)$ has the WMP for any reflexive space $X$ and any Schur space $Y$ (Proposition~\ref{prop-Schur-range}). Also, thanks to Theorem~\ref{TheoremWeakStarMP}~(3) and $\overline{\delta}_{\ell_1}
^*(t) = t$, the space $\ell_1=c_0^*$ is also a very good domain for the W$^*$MP since
$(\ell_1,Y)$ has the W$^*$MP for every Banach space $Y$ (see Corollary~\ref{Cor-l1-domain}). The latter result does not hold for every Schur space as it is shown for instance by Example~\ref{Example1-Schur}. To finish, we discuss the case of Dunford--Pettis spaces, James sequence spaces, Orlicz spaces and we also add some comments about the pair $(L_p([0,1]),L_q([0,1]))$.
\section{Preliminaries: Asymptotic uniform smoothness and convexity}
\label{section_AUSAUC}
Consider a real Banach space $(X,\norm{\cdot})$ and let $S_X$ be its unit sphere.
The \emph{modulus of asymptotic uniform convexity} of $X$ is given by
\[ \forall t >0, \quad \overline{\delta}_X(t) = \inf_{x\in S_X} \sup_{\overline{\delta}im(X/Y)<\infty}\inf_{y\in S_Y} \norm{x+ty}-1\,. \]
The space $(X,\norm{\cdot})$ is said to be \emph{asymptotically uniformly convex} (AUC for short) if $\overline{\delta}_X(t)>0$ for each $t>0$. When $X$ is a dual space and $Y$ runs through all finite-codimensional weak$^*$-closed subspaces of $X$, the corresponding modulus is denoted by $\overline{\delta}_X^*(t)$.
Then the space $X$ is said to be \emph{weak$^*$ asymptotically uniformly convex} (AUC*) if $\overline{\delta}_X^*(t)>0$ for each $t>0$.
The \emph{modulus of asymptotic uniform smoothness} of $X$ is given by
\[ \forall t >0, \quad \overline{\rho}_X(t) = \sup_{x\in S_X} \inf_{\overline{\delta}im(X/Y)<\infty}\sup_{y\in S_Y} \norm{x+ty}-1\,. \]
The space $(X,\norm{\cdot})$ is said to be \emph{asymptotically uniformly smooth} (AUS for short) if $\lim_{t\to 0} t^{-1}\overline{\rho}_X(t) = 0$.
We refer the reader to~\cite{JLPS02} and the references therein for a detailed study of these properties.
Let $p,q\in [1,\infty)$. We say that $X$ is \emph{weak$^*$ $p$-asymptotic uniformly convex} (abbreviated by $p$-AUC$^*$) if there exists $C>0$ so that $\overline{\overline{\delta}elta}_X^*(t)\geq Ct^p$ for all $t\in [0,1]$. Similarly, we say that $X$ is $q$-asymptotic uniformly smooth (abbreviated by $q$-AUS) if there exists $C >0$ so that $\overline{\rho}_X(t) \leq C t^q$ for all $t \in [0,1]$.
Let us highlight that the following is proved in \cite[Corollary 2.4]{DKLR}.
\begin{proposition}\label{duality} Let $X$ be a Banach space.
\begin{enumerate}[(i)]
\item Then $\norm{\cdot}_X$ is AUS if and and only if $\norm{\cdot}_{X^*}$ is AUC$^*$.
\item If $p\in (1,\infty]$ and $q\in [1,\infty)$ are conjugate exponents, then $\norm{\cdot}_X$ is $p$-AUS if and only if $\norm{\cdot}_{X^*}$ is $q$-AUC$^*$.
\end{enumerate}
\end{proposition}
It is worth mentioning that if $X\equiv Z^*$ and the dual norm $\norm{\cdot}_{Z^*}$ is AUS, then $X$ must be reflexive (see for instance Proposition 2.6 in \cite{CauseyLancien}). The following proposition is elementary.
\begin{proposition}\label{as-sequences} Let $X$ be a Banach space. For any weakly null net $(x_\alpha)_\alpha$ in $X$
and any $x \in X\setminus\{0\}$ we have:
\[
\limsup_{\alpha} \norm{x+x_\alpha} \leq \norm{x}\left(1+\overline{\rho}_X\left(\frac{\limsup_\alpha \norm{x_\alpha}}{\norm{x}}\right)\right).
\]
For any weak$^*$-null net $(x^*_\alpha)_\alpha \subset X^*$ and for any $x^* \in X^*\setminus \set{0}$ we have
\[
\liminf_{\alpha} \norm{x^*+x^*_\alpha} \geq \norm{x^*}\left(1+\overline{\overline{\delta}elta}_X^*\left(\frac{\liminf_\alpha \norm{x_\alpha^*}}{\norm{x^*}}\right)\right).
\]
\end{proposition}
Assume now that $\varphi\colon [0,\infty) \to [0,\infty)$ is a 1-Lipschitz convex function with $\lim_{t\to \infty}\varphi(t)/t=1$ and $\varphi(t)\geq t-1$ for all $t\geq 0$.
Consider for $(s,t) \in \mathbb{R}^2$,
\[
N_2^\varphi(s,t)=
\begin{cases}\abs{s}+\abs{s}\varphi(\abs{t}/\abs{s}) & \text{ if }s\neq 0,\\
\abs{t}& \text{ if }s=0.
\end{cases}
\]
The following is stated
in \cite{KaltonTAMS2013} (see Lemma~4.3 and its preparation), we include a proof for completeness.
\begin{lemma}\label{Orlicz-Kalton} The function $N_2^\varphi$ is an absolute (or lattice) norm on $\mathbb{R}^2$, meaning that $N_2^\varphi(s_1,s_2)\le N_2^\varphi(t_1,t_2)$, whenever $|s_i|\le |t_i|$ for all $i\le 2$.
\end{lemma}
\begin{proof} First, note that if $0<u<v$ then
\[ \frac{\varphi(v)-\varphi(u)}{v-u} \leq 1\leq \frac{1+\varphi(v)}{v}\]
since $\varphi$ is $1$-Lipschitz and $\varphi(v)\geq v-1$. It follows that $\frac{1}{v}(1+\varphi(v))\leq \frac{1}{u}(1+\varphi(u))$. Thus,
\[ N_2^\varphi(s,1)=s(1+\varphi(1/s))\leq s'(1+\varphi(1/s')) = N_2^\varphi(s',1) \quad \text{for all } 0<s<s'.\]
From that and the positive homogeneity of $N_2^\varphi$, its clear that
\begin{equation}\label{eq:increasing}
N_2^\varphi(s,t)\leq N_2^\varphi(s',t') \quad \text{for all } |s|\leq |s'| \text{ and } |t|\leq |t'|.
\end{equation}
Now, assume that $0<s,s', t,t'$. The convexity of $\varphi$ yields
\begin{align*}
N_2^\varphi(s+s',t+t')&=(s+s')\left(1+\varphi\left(\frac{t+t'}{s+s'}\right)\right) \\
&= s+s'+ (s+s')\varphi\left(\frac{s}{s+s'}\frac{t}{s}+\frac{s'}{s+s'}\frac{t'}{s'}\right) \\
&\leq s+s'+s\varphi(t/s)+s'\varphi(t'/s') = N_2^\varphi(s,t)+N_2^\varphi(s',t').
\end{align*}
From that and \eqref{eq:increasing}, we get
\begin{align*} N_2^\varphi(s+s',t+t')&=N_2^\varphi(|s+s'|,|t+t'|)\leq N_2^\varphi(|s|+|s'|, |t|+|t'|)\\
&\leq N_2^\varphi(|s|, |t|)+N_2^\varphi(|s'|,|t'|)
=N_2^\varphi(s,t)+N_2^\varphi(s',t')
\end{align*}
whenever $s,s'\neq 0$. The case where $s=0$ or $s'=0$ follows from the continuity of $N_2^\varphi$. This shows that $N_2^\varphi$ defines a norm on $\mathbb R^2$, and clearly it is absolute.
\end{proof}
\begin{remark}
In the proof of \cite[Proposition~4.1]{KaltonCompact}, it is assumed that $\varphi$ is such that $t \in (0,\infty) \mapsto t^{-1}(\varphi(t)+1)$ is monotone decreasing, which is obtained in the previous proof under the stronger hypothesis that $\varphi(t)\geq t-1$. For our purposes, there is no loss of generality. Indeed, when $X$ is a Banach space, it is easy to see that $\overline{\rho}_X$ is a 1-Lipschitz convex function such that $\lim_{t\to \infty}\overline{\rho}_X(t)/t=1$. Moreover, $\overline{\rho}_X(t)\geq \overline{\rho}_{c_0}(t)=\max\{0,t-1\}$ so that one can consider the norm $N_2^{\overline{\rho}_X}$.
\end{remark}
We will use in the sequel the following reformulation of Proposition~\ref{as-sequences} in terms of the norm $N_2^{\overline{\rho}_X}$.
\begin{lemma}\label{l:asymptotic-Nnorm}
Let $X$ be a Banach space. If $(x_\alpha)_\alpha \subset X$ is weakly-null and $x\in X$, then
\[ \limsup_{\alpha} \| x+x_\alpha \| \leq N_2^{\overline{\rho}_X}(\norm{x},\limsup_\alpha \norm{x_\alpha}).\]
\end{lemma}
\begin{proof}
If $x=0$ there is nothing to do, so we may assume that $x\neq 0$.
By application of Proposition~\ref{as-sequences} we see that
\[
\begin{split}
\limsup_{\alpha}\norm{x+x_\alpha}&\leq \norm{x}\left(1+\overline{\rho}_X\left(\frac{\limsup_\alpha\norm{x_\alpha}}{\norm{x}}\right)\right) \\
&=N_2^{\overline{\rho}_X}(\norm{x},\limsup_\alpha \norm{x_\alpha}).
\end{split}
\]
\end{proof}
\begin{example}\label{ex:ausauc} Let us recall the asymptotic moduli for some classical spaces, the three first examples are mostly taken from \cite{Milman71}.
\begin{enumerate}[(i)]
\item Let $1\leq p<\infty$. If $X=(\sum_{n=1}^\infty E_n)_p$, where $\overline{\delta}im(E_n)<\infty$, then $\overline{\delta}_{X}(t)=\overline{\rho}_{X}(t)=(1+t^p)^{1/p}-1$ for all $t>0$.
\item For any infinite set $\Gamma$ and $1\leq p<\infty$, $\overline{\delta}_{\ell_p(\Gamma)}(t)=\overline{\rho}_{\ell_p(\Gamma)}(t)=(1+t^p)^{1/p}-1$ for all $t>0$.
\item For any infinite set $\Gamma$, $\overline{\delta}_{c_0(\Gamma)}(t)=\overline{\rho}_{c_0(\Gamma)}(t)=\max\{0,t-1\}$ for all $t>0$.
\item Let $J$ be the James space. Then $\overline{\delta}_{J}(t)=(1+t^2)^{1/2}-1$ for all $t>0$, and there is an equivalent norm $|\cdot|$ on $J$ so that $\overline{\rho}_{|\cdot|}(t)\leq (1+t^2)^{1/2}-1$ (see Proposition~\ref{Jsmooth}).
\item If $X$ has a Schauder basis $(e_n)_{n=1}^\infty$ which satisfies a lower (resp. upper) $p$-estimate with constant one, then $\overline{\delta}_{X}(t)\geq (1+t^p)^{1/p}-1$ (resp. $\overline{\rho}_{X}(t)\leq (1+t^p)^{1/p}-1$). For instance, the Lorentz sequence space $d(w,p)$ satisfy an upper $p$-estimate with constant 1 \cite[p.~177]{LT77}. Moreover, $d(w,p)$ contains almost isometric copies of $\ell_p$ so $\overline{\rho}_{d(w,p)}(t)=(1+t^p)^{1/p}-1$ for all $t>0$.
\item Given a Banach space $X$ with a $1$-unconditional Schauder basis $(e_n)_{n=1}^\infty$ and $1\leq p<\infty$, its $p$-convexification $X^p$ is defined as
\[X^p=\{(x_n)_{n=1}^\infty : \sum_{n=1}^\infty |x_n|^p e_n\in X\} \]
with the norm $\norm{(x_n)_{n=1}^\infty}_p=\norm{\sum_{n=1}^\infty |x_n|^p e_n}^{1/p}$. It follows from the triangle inequality that $X^p$ satisfies an upper $p$-estimate with constant $1$ which readily implies that $\overline{\rho}_{X^p}(t)\leq (1+t^p)^{1/p}-1$.
\item The predual $JT_*$ of the James tree space $JT$ as well as its dual $JT^*$ are asymptotically uniformly convex \cite{Girardi}. Moreover, there exists a positive constant $c$ so that $\overline{\delta}_{JT_*}(t) \geq ct^3$.
\item The Tsirelson space $\mathrm{T}$ can be renormed for every $p>1$ in such a way that $\overline{\delta}_\mathrm{T}(t) \geq c_p t^p$ \cite[Remarks 7.2]{KOS99}. By duality, we have that $\mathrm{T}^*$ can be renormed for every $q>1$ in such a way that $\overline{\rho}_{\mathrm{T}^*}(t) \leq c_q' t^q$. On the other hand, $\mathrm{T}$ does not have an equivalent AUS norm since otherwise it would follow from \cite{KOS99} (see also \cite{DKLR}) that the Szlenk index of $\mathrm{T}$ is $\omega$, which is not the case thanks to \cite{Odell2007} (see also \cite{Raja2013})).
\end{enumerate}
\end{example}
\section{Proof of the main results} \label{section-mainresults}
Let us begin with the proof of our main theorem.
\begin{theorem} \label{TheoremWeakStarMP}
Let $X,Y$ be Banach spaces. Assume that for every $t>0$, $\overline{\overline{\delta}elta}_{X^*}^*(t)\geq \overline{\rho}_Y(t)$ and, for all $t\geq 1$,
$\overline{\overline{\delta}elta}_{X^*}^*(t)>t-1$. Then,
\begin{enumerate}
\item the pair $(X^*,Y)$ has the weak$^*$-to-weakMP.
\item If moreover $Y\equiv Z^*$ is a dual space, then the pair $(X^*,Z^*)$ has the weak$^*$-to-weak$^*$MP.
\item If $\overline{\delta}_{X^*}^*(t)=t$ then $(X^*,Z)$ has the W$^*$MP for any Banach space $Z$.
\end{enumerate}
In particular, if $X$ is reflexive then the pair $(X^*,Y)$ has the WMP.
\end{theorem}
\begin{proof}
We will only prove the first assertion since the proof of the second and third one are very similar (replacing the weak topology in $Y$ by the weak$^*$ topology in $Y$).
Let $T\colon X^* \to Y$ be a bounded operator which is weak$^*$-to-weak continuous. Without loss of generality, we may assume that $T$ has norm 1. Let $(x_n)$ be a normalized maximizing sequence in $X^*$ which is not weak$^*$-null. Let $(x_\alpha)$ be a subnet which is weak$^*$ convergent to $x \neq 0$. By extracting a subnet again, we may assume that $\lim_\alpha \norm{x_\alpha-x}$ and $\lim_\alpha \norm{Tx-Tx_\alpha}$ exists.
Since $T$ is weak$^*$-to-weak continuous, we have that $Tx_\alpha \overset{w}{\longrightarrow} Tx$. Using Lemma~\ref{l:asymptotic-Nnorm}, we thus have the following estimates:
\begin{eqnarray*}
1 &=& \norm T = \lim\limits_{\alpha} \norm{Tx_\alpha} = \lim\limits_{\alpha} \norm{Tx+Tx_\alpha - Tx} \\
&\leq& N_2^{\overline{\rho}_Y}(\norm{Tx},\lim_\alpha \norm{Tx_\alpha -Tx}).
\end{eqnarray*}
Now notice that (we use Lemma~\ref{Orlicz-Kalton})
\begin{eqnarray*}
N_2^{\overline{\rho}_Y}(\norm{Tx},\lim_\alpha \norm{Tx_\alpha -Tx}) &\leq& N_2^{\overline{\rho}_Y}(\norm{Tx},\lim_\alpha \norm{x_\alpha -x}) \\
&\leq& N_2^{\overline{\rho}_Y}(\norm{x},\lim_\alpha \norm{x_\alpha -x}).
\end{eqnarray*}
Since $x \neq 0$, we deduce from the definition of $N_2^{\overline{\rho}_Y}$ the following estimates:
\begin{eqnarray*}
N_2^{\overline{\rho}_Y}(\norm{x},\lim_\alpha \norm{x_\alpha -x}) &\leq& \|x\| +\|x\|\overline{\rho}_Y\left( \frac{\lim_\alpha \norm{x_\alpha -x}}{\|x\|}
\right) \\
&\leq& \|x\| +\|x\|\overline{\overline{\delta}elta}_{X^*}^*\left( \frac{\lim_\alpha \norm{x_\alpha -x}}{\|x\|} \right) \\
&\leq& \lim_\alpha \norm{x +x_\alpha -x} \\
&=& 1.
\end{eqnarray*}
This implies that all the previous inequalities are in fact equalities, in particular
\begin{eqnarray*} N_2^{\overline{\rho}_Y}(\norm{Tx},\lim_\alpha \norm{x_\alpha -x}) = N_2^{\overline{\rho}_Y}(\norm{x},\lim_\alpha \norm{x_\alpha -x})=1 = N_2^{\overline{\rho}_Y}(0,1).
\end{eqnarray*}
This means that the points $(\norm{Tx},\lim_\alpha \norm{x_\alpha -x})$, $(\norm{x},\lim_\alpha \norm{x_\alpha -x})$ and $(0,1)$ are aligned in $\mathbb R^2$. If $\norm{Tx}=\norm{x}$ or $\lim_\alpha \norm{x-x_\alpha}=0$, then $T$ attains its norm at $x$ and we are done. Otherwise, it follows that $\lim_\alpha\norm{x_\alpha-x}=1$ and
\[ 1= N_2^{\overline{\rho}_Y}(\norm{x},1) = \norm{x}+\norm{x}\overline{\rho}_Y(1/\norm{x}) = \norm{x}+\norm{x}\overline{\delta}_{X^*}^*(1/\norm{x}). \]
That is, $\overline{\delta}_{X^*}(\frac{1}{\norm{x}})=\overline{\rho}_Y(\frac{1}{\norm{x}})=\frac{1}{\norm{x}}-1$, which contradicts our assumptions.
\end{proof}
\begin{remark}\label{rem:failing} Take $X=\mathbb R\oplus_{\infty} \ell_2$ and $Y=c_0$. Then $\overline{\delta}_X(t)=\max\{0,t-1\}=\overline{\rho}_Y(t)$ for all $t>0$. However, the operator $T\colon X\to Y$ given by $T((0,e_n))=\frac{n}{n+1}e_n$ and $T((1,0))=0$ does not attain the norm and admits the non-weakly null maximizing sequence $(x_n)_{n=1}^\infty$ given by $x_n=(1,e_n)$. Thus, the pair $(X, Y)$ fails the WMP.
\end{remark}
\begin{remark} Since $\overline{\rho}_Y$ is 1-Lipschitz and $\overline{\rho}_Y(t) \geq \max\{0,t-1\}$, the condition $\overline{\rho}_Y(1)=0$ is equivalent to $\overline{\rho}_Y(t)=\max\{0,t-1\}$ for all $t>0$. These spaces are called metric weak$^*$ Kadec-Klee spaces. In the separable case, they are precisely those spaces which are $(1+\varepsilon)$-isomorphic to a subspace of $c_0$, for every $\varepsilon>0$ \cite{GKL00}.
\end{remark}
The next result follows the idea of \cite[Theorem 2]{Pellegrino_09}, where the same is obtained in the case $X^*=\ell_p$, $Y=\ell_q$, $p\neq q$.
\begin{corollary} \label{cor:strongconv}
Let $X,Y$ be Banach spaces. Assume that $X$ is separable and
$\overline{\overline{\delta}elta}_{X^*}^*(t) > \overline{\rho}_Y(t)$ for all $t>0$. Let $T\colon X^*\to Y$ be a weak$^*$-to-weak continuous operator. Then any non-weak$^*$ null maximizing sequence for $T$ has a convergent subsequence.
\end{corollary}
\begin{proof}
Let $(x_n)_{n=1}^\infty \subset S_{X^*}$ be a non-weak$^*$ null maximizing sequence for $T$. By extracting a subsequence, we may assume that $(x_n)$ is weak$^*$ convergent to $x\neq 0$ and the limit $\lim_n \norm{x_n-x}$ exists. The proof of Theorem~\ref{TheoremWeakStarMP} shows that $T$ attains its norm at $x$, and moreover
\[ \overline{\rho}_Y\left(\frac{\lim_n \norm{x_n-x}}{\norm{x}}\right)=\overline{\delta}^*_{X^*}\left(\frac{\lim_n\norm{x_n-x}}{\norm{x}}\right). \]
Thus $\lim_n\norm{x_n-x}=0$.
\end{proof}
Note that, as it is shown in \cite{Pellegrino_09}, there are pairs $(X,Y)$ with the WMP such that the conclusion of Corollary~\ref{cor:strongconv} fails. Indeed consider $X=Y=\ell_2$. Then $(\frac{e_1+e_n}{\sqrt{2}})_{n\geq1}$ is a non-weakly null maximizing sequence for the identity operator $I\colon \ell_2\to\ell_2$ which has no norm convergent subsequence.
We will now turn to the case when the modulus of asymptotic uniform smoothness of $X$ is bounded above by the modulus of asymptotic uniform convexity of $Y$. To this aim, we will need the following two lemmata.
\begin{lemma}[Proposition 2.3 in \cite{JLPS02}] \label{CompactOperator}
Let $X,Y$ be Banach spaces. If there exists $t>0$ such that $\overline{\rho}_X(t) < \overline{\delta}_Y(t)$
then every linear operator from $X$ to $Y$ is compact.
\end{lemma}
The next one is classical and easy to prove.
\begin{lemma} \label{CompactAttainment}
Let $X, Y$ be Banach spaces and let $\tau_Y$ be any Hausdorff topology on $Y$ which is coarser than the norm topology. If $T\colon X^*\to Y$ is a compact operator which is weak$^*$-to-$\tau_Y$ continuous, then $T$ attains its norm.
\end{lemma}
\begin{proof}
Let $(x_\alpha^*)$ be a net in $S_{X^*}$ such that $\lim_\alpha\norm{Tx^*_\alpha}=\norm{T}$, we may assume that $(x^*_\alpha)$ is weak$^*$-convergent to $x^*\in B_{X^*}$. Then $(Tx^*_\alpha)$ is $\tau_Y$-convergent to $Tx^*$. Since the norm topology and $\tau_Y$ agree on the norm-compact set $\overline{T(B_{X^*})}$, we have that $\norm{T}=\lim_\alpha\norm{Tx^*_\alpha}=\norm{Tx^*}$.
\end{proof}
We are now ready to prove the desired theorem.
\begin{theorem} \label{TheoremCompactWMP}
Let $X,Y$ be Banach spaces and $\eqnorma$ be an equivalent norm on $X^*$. If there exists $t>0$ such that $\overline{\rho}_{\eqnorma}(t) < \overline{\delta}_Y(t)$, then
\begin{enumerate}
\item the pair $(X^*,Y)$ has the weak$^*$-to-weakMP.
\item If moreover $Y \equiv Z^*$ is a dual space, then $(X^*,Z^*)$ has the weak$^*$-to-weak$^*$MP.
\end{enumerate}
In particular, if $X$ is reflexive then $(X^*,Y)$ has the WMP.
\end{theorem}
\begin{proof}
We only prove the first assertion, the proof of the second one is similar and the last statement is quite obvious. Thanks to Lemma~\ref{CompactOperator}, we know that every bounded linear operator from $(X^*,\eqnorma)$ to $Y$ is compact.
Since $\eqnorma$ is an equivalent norm on $X^*$, this readily implies that every bounded linear operator from $X^*$ to $Y$ is compact.
According to Lemma~\ref{CompactAttainment}, we deduce that every weak$^*$-to-weak continuous operator from $X^*$ to $Y$ attains its norm.
\end{proof}
We highlight some consequences of the previous theorems for particular spaces, more examples will be given in Section~\ref{section-applications}. In all the cases the proof follows from Theorem~\ref{TheoremWeakStarMP} or Theorem~\ref{TheoremCompactWMP} and Example~\ref{ex:ausauc}.
\begin{corollary} \label{CorWMP}
Any of the following conditions ensures that the pair $(X,Y)$ has the WMP:
\begin{enumerate}[a)]
\item $X$ is a reflexive AUC space with $\overline{\delta}_X(t)>t-1$ (e.g. $X=\ell_p, \ell_p(\Gamma)$) and $Y=c_0$.
\item $X=\ell_p(\Gamma)$ and $Y=\ell_q(\Gamma')$, $1<p<\infty, 1\leq q<\infty$.
\item $X=(\sum_{n=1}^\infty E_n)_p$ where $\overline{\delta}im(E_n)<\infty$ and $Y=(\sum_{n=1}^\infty F_n)_p$ where $\overline{\delta}im(F_n)<\infty$, $1<p<\infty, 1\leq q<\infty$.
\item $X=\ell_p$ and either $Y=d(w,q)$ (as in Example~\ref{ex:ausauc}~$\mathrm{(v)}$) or $Y= Z^q$ (as in Example~\ref{ex:ausauc}~$\mathrm{(vi)}$), with $1<p\leq q<\infty$.
\end{enumerate}
\end{corollary}
\section{Renorming results} \label{section-renorming}
First, we wish to underline that for a special class of Banach spaces, namely those $X$ with $\operatorname{Sz}(X) = \omega$, one can find an equivalent norm $\eqnorma$ on $X^*$ and a reflexive Banach space $Y$ such that the pair $\big((X^*,\eqnorma),Y\big)$ has the weak$^*$-to-weak$^*$MP. By $\operatorname{Sz}(X)$ we mean the Szlenk index of $X$ and $\omega$ stands for the first countable ordinal. We refer the reader to \cite{LancienSurvey} for the definition of the Szlenk index as well as its basic properties and main applications to the geometry of Banach spaces.
\begin{corollary}
If $X$ is separable with $\operatorname{Sz}(X) = \omega$, then there exists an equivalent norm $\eqnorma$ on $X^*$ and $q \geq 1$ such that the pair $\big((X^*,\eqnorma),\ell_q\big)$ has the weak$^*$-to-weak$^*$MP.
In particular if $X\equiv Z^*$ is a separable reflexive space with $\operatorname{Sz}(Z)= \omega$, then there exists an equivalent norm $\eqnorma$ on $X$ and there exists $q \geq 1$ such that the pair $\big((X,\eqnorma),\ell_q\big)$ has the WMP.
\end{corollary}
\begin{proof}
Let us start by recalling the fundamental renorming result for spaces with Szlenk index equal to $\omega$. The result is due to H.~Knaust, E.~Odell and Th.~Schlumprecht \cite[Corollary 5.3]{KOS99}: if $X$ is a separable Banach space such that $\operatorname{Sz}(X) =\omega$, then there exists $p\in (1,\infty)$ such that $X^*$ admits an equivalent $p$-AUC* norm $\eqnorma$, moreover $\overline{\delta}^*_{\eqnorma}(t) \geq (1+t^p)^{\frac{1}{p}}-1$.
Let us consider $q>p$. As we already mentioned in Example~\ref{ex:ausauc}, $\overline{\rho}_{\ell_q}(t) = (1+t^q)^{\frac{1}{q}}-1$. Thus for every $t>0$ we have that $\overline{\delta}^*_{\eqnorma}(t) > \overline{\rho}_{\ell_q}(t)$. The result now readily follows from Theorem~\ref{TheoremWeakStarMP}.
\end{proof}
Unlike Theorem~\ref{TheoremCompactWMP}, the condition on the moduli in Theorem~\ref{TheoremWeakStarMP} is stated for every $t>0$. This can be problematic in some concrete situations, even if $X$ is $p$-AUC and $Y$ is $q$-AUS with $p<q$. Nevertheless, it is sometimes possible to define equivalent norms on $X$ and $Y$ for which the required inequality is satisfied for every $t>0$. Our next goal is to outline a setting where such renormings exist.
Recall that a sequence $\{E_n\}$ of finite-dimensional subspaces of a Banach space $X$ is called a \emph{finite-dimensional decomposition} (FDD in short) if every $x\in X$ has a unique representation of the form $x=\sum_{n=1}^\infty x_n$ with $x_n\in E_n$. An FDD determines a sequence $\{P_n\}$ of commuting projections, given by $P_n(\sum_{i=1}^\infty x_i)=\sum_{i=1}^n x_i$, in particular $E_n=(P_n-P_{n-1})X$. The sequence $\{(P_n^*-P_{n-1}^*)X^*\}$ is an FDD of its closed linear span in $X^*$; it is often denoted $\{E_n^*\}$ and called the dual FDD of $\{E_n\}$. In the case when the latter closed linear span is the whole $X^*$ (equivalently, if $\lim_n\norm{P_n^*x^*-x^*}=0$ for every $x^*\in X^*$), it is said that $\{E_n\}$ is a \emph{shrinking FDD}. Next an FDD $\{E_n\}$ is called \textit{boundedly complete} if for every $(x_n)_n \in \{E_n\}$ with $\sup_{n \in \mathbb{N}} \| \sum_{i=1}^n x_i\| < \infty $, the series $\sum_{i=1}^{\infty} x_i$ converges in $X$. Any space with a boundedly complete FDD is naturally a dual space and moreover the concepts of shrinking and boundedly complete are dual properties, in the sense that an FDD is shrinking if and only if its dual FDD is boundedly complete. We refer the reader to \cite{Biorthogonal} for more details about FDDs (we also recommend \cite[Section~3.2]{albiackalton} where Schauder bases are considered but the proofs easily extend to FDDs).
\begin{theorem}\label{thm:renorming} Let $X, Y$ be Banach spaces with shrinking FDDs. Assume that the norm of $X^*$ is $p$-AUC* and the norm of $Y$ is $q$-AUS for some $1<p\leq q<\infty$. Then there exist equivalent norms $\eqnorma_X$ on $X$ and $\eqnorma_Y$ on $Y$ such that the pair $((X^*,\eqnorma_{X}^*), (Y,\eqnorma_Y))$ has the weak*-to-weakMP.
\end{theorem}
The proof of Theorem \ref{thm:renorming} follows from the relations between asymptotic uniform convexity and lower estimates, and asymptotic uniform smoothness and upper estimates (see \cite[Proposition 2.10]{JLPS02}). Given $p>1$, an FDD $\{E_n\}$ is said to satisfy an upper (lower, respectively) $p$-estimate provided there is a constant $C > 0$ so that for any sequence of consecutively supported vectors $x_1<\cdots<x_m$ in $X$,
$\|\sum_{k=1}^m x_k\|\leq C(\sum_{k=1}^m \|x_k\|^p)^{1/p}$ ($\|\sum_{k=1}^m x_k\|\geq C^{-1}(\sum_{k=1}^m \|x_k\|^p)^{1/p}$, respectively).
Recall that a \emph{blocking} of an FDD $\{E_n\}$ is a sequence of the form $\{E_{n_i}+\ldots+E_{n_{i+1}-1}\}$ with $1=n_1<n_2<\ldots$. Note that each blocking of $\{E_n\}$ corresponds to a subsequence of $\{P_n\}$. Thus, each blocking of a shrinking FDD is also shrinking. Moreover, each blocking of $\{(P_n^*-P_{n-1}^*)X^*\}$ is the dual of a blocking of $\{E_n\}$.
\begin{lemma}\label{lemma:deltaestimate} Let $X$ be a Banach space with a shrinking FDD $\{E_n\}$.
Assume that $\overline{\overline{\delta}elta}^*_{X^*}$ has power type $p$. Then there is a blocking $\{G_n\}$ of $\{E_n\}$ such that the dual FDD $\{G_n^*\}$ satisfies a lower $p$-estimate.
\end{lemma}
\begin{proof}
Take $C>0$ and $t_0>0$ such that $\overline{\overline{\delta}elta}_{X^*}^*(t)\geq Ct^p$ for all $0<t\leq t_0$. First, note that we may assume $t_0\geq 4$. Indeed, if $t_0\leq t\leq 4$ then $\overline{\overline{\delta}elta}^*_{X^*}(t)\geq \overline{\overline{\delta}elta}^*_{X^*}(t_0)\geq C t_0^p \geq C' t^p$ where $C'=\frac{Ct_0^p}{4^p}$.
Now, let $0<M<\min\{C, 4^{-p}(3^p-1)\}$. We claim that for each $n\in \mathbb N$ there is $m>n$ such that
\[ \norm{x^*+y^*}^p \geq \norm{x^*}^p+M\norm{y^*}^p\]
whenever $x^*\in \operatorname{span}\{E_i^*: i=1,\ldots, n\}$ and $y^*\in \overline{\operatorname{span}}\{E_i^*: i\geq m\}$. If $\norm{y^*}> 4\norm{x^*}$, then
\[
\begin{split}
\norm{x^*+y^*}^p &\geq \left|\norm{y^*}-\norm{x^*}\right|^p = \norm{y^*}^p \left|1-\frac{\norm{x^*}}{\norm{y^*}}\right|^p\geq \frac{3^p}{4^p}\norm{y^*}^p \\
&\geq \norm{x^*}^p+ \frac{3^p-1}{4^p}\norm{y^*}^p
\end{split}
\]
so we may assume $\norm{y^*}\leq 4\norm{x^*}$. Moreover, by homogeneity, we may assume $x^*\in S_{X^*}$.
Assume that the claim does not hold. Then there exists $n$ such that for each $m>n $ there is $x_m^*\in \operatorname{span}\{E_i: i=1,\ldots, n\}$ and $y_m^* \in \overline{\operatorname{span}}\{E_i: i\geq m\}$ such that $\norm{x_m^*}=1$, $\norm{y_m^*}\leq 4$, and $\norm{x_m^*+y_m^*}^p <\norm{x_m^*}^p+M\norm{y_m^*}^p$. By extracting a subsequence we may assume $x_m^*\to x^*\in S_{X^*}$. Since $\{E_n\}$ is an FDD for $X$, the sequence $\{y_m^*\}_{m\geq n}$ is weak$^*$-null. Moreover, $\limsup\norm{y_m^*}\leq 4$. Thus,
\[ \lim_{m\to\infty}\norm{x^*+y_m^*} \geq 1+\overline{\overline{\delta}elta}^*_{X^*}(\limsup_m\norm{y_m^*})\geq 1+C\limsup_m\norm{y_m^*}^p.\]
On the other hand,
\[
\begin{split}
\lim_{m\to\infty}\norm{x^*+y_m^*} &= \lim_{m\to\infty}\norm{x_m^*+y_m^*}\leq (1+M\limsup_m\norm{y_m^*}^p)^{1/p}\\
&\leq 1+M\limsup_m\norm{y_m^*}^p.
\end{split}
\]
From that, we got $C\leq M$, a contradiction.
It follows from the claim that there is an increasing sequence $1=n_0<n_1<n_2<...$ such if $x^*\in \operatorname{span}\{E_i^*: 1\leq i\leq n_i\}$ and $y^*\in \overline{\operatorname{span}}\{E_i^* : i\geq n_{i+1}\}$ then $\norm{x^*+y^*}^p\geq \norm{x^*}^p+M\norm{y^*}^p$. It follows easily that the blocking $\{F_n^*\}$ of $\{E_n^*\}$ given by $F_i^* = \operatorname{span}\bigcup_{k=n_i}^{n_{i+1}-1} E_k^*$ satisfies a skipped lower $p$-estimate. Moreover, $\{F_n^*\}$ is boundedly complete since it is a blocking of a boundedly complete FDD. Finally, a result due to Johnson \cite{Johnson77} (see also \cite[Lemma 2.51]{Biorthogonal}) provides a further blocking $\{G_n^*\}$ of $\{F_n^*\}$ satisfying a lower $p$-estimate.
\end{proof}
It is well known that an FDD satisfies a lower estimate if and only if the dual FDD satisfies an upper estimate \cite[Fact~2.19]{Biorthogonal}. In fact, sometimes we can keep the constants, as the following lemma shows.
\begin{lemma}\label{lemma:dualestimate} Let $\{E_n\}$ be a shrinking FDD. Assume that the dual FDD $\{E_n^*\}$ satisfies a lower $p$-estimate with constant $1$. Then $\{E_n\}$ satisfies an upper $p'$-estimate with constant $1$, where $1/p+1/p'=1$.
\end{lemma}
\begin{proof}
Let $x_1,\ldots , x_n\in X$ be consecutively supported vectors such that $\| \sum_{i=1}^n x_i \| = 1$. Using Hahn--Banach theorem, we may consider $x^*\in S_{X^*}$ such that $\langle x^*, \sum_{i=1}^n x_i\rangle= 1$. Since $\{E_n\}$ is shrinking, $\{E_n^*\}$ is an FDD in $X^*$ so we may write $x^* = \sum_{i=1}^{\infty} x_i^*$
with $\supp(x_i^*)=\supp(x_i)$ whenever $i \leq n$. For $i >n$, we let $x_i = 0$ in what follows:
\begin{align*}
1 &= \langle x^*, \sum_{i=1}^n x_i\rangle = \sum_{i=1}^n x_i^*(x_i) = \sum_{i=1}^{\infty} x_i^*(x_i) \leq \sum_{i=1}^\infty \norm{x_i^*}\norm{x_i} \\
&\leq \left(\sum_{i=1}^{\infty}\norm{x_i^*}^p\right)^{\frac{1}{p}}\left(\sum_{i=1}^{\infty} \norm{x_i}^{p'}\right)^{\frac{1}{p'}}
\leq \Big\|\sum_{i=1}^{\infty} x_i^* \Big\| \left(\sum_{i=1}^n \norm{x_i}^{p'}\right)^{\frac{1}{p'}} \\
&\leq \left(\sum_{i=1}^n \norm{x_i}^{p'}\right)^{\frac{1}{p'}}.
\end{align*}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:renorming}]
By Lemma~\ref{lemma:deltaestimate} there is a blocking $(E_n)_n$ of the FDD in $X$ whose dual FDD satisfies a lower $p$-estimate. Now, consider the norm on $X^*$ given by
\[\norm{x^*}_{(p)}=\sup\left\{\left(\sum_{i=1}^k \norm{(P_{n_{i+1}}^*-P_{n_i}^*)(x^*)}^p\right)^{1/p} : 1\leq n_1<n_2<\ldots<n_k\right\}\]
where $\{P_n\}$ are the associated projections to $\{E_n\}$. This is the norm considered by Prus in \cite[Remark~3.2 and Lemma~3.3]{Prus87}, where it is proved that it is an equivalent norm on $X^*$ and that the FDD $\{E_n^*\}$ satisfies a lower $p$-estimate with constant $1$. Moreover, the function $\norm{\cdot}_{(p)}$ is weak*-lower semicontinuous since it is the supremum of weak*-lower semicontinuous functions, so $\norm{\cdot}_{(p)}$ is a dual equivalent norm on $X^*$. It follows that $\overline{\delta}^*_{(X^*, \norm{\cdot}_{(p)})}(t)\geq (1+t^p)^{1/p}-1$ for all $t>0$.
Now, we focus on the space $Y$. Since the norm is $q$-AUS, the dual norm is $q'$-AUC*, where $1/q+1/q'=1$. Lemma~4.3 provides a blocking $\{F_n\}$ of the FDD of $Y$ such that the dual FDD satisfies a lower $q'$-estimate in $Y^*$. The argument above provides a dual equivalent norm $\norm{\cdot}_{(q')}$ on $Y^*$ such that $\{F_n^*\}$ satisfies a lower $q'$-estimate with constant one. Then, $\{F_n\}$ satisfies an upper $q$-estimate in $Y$ endowed with the predual norm $\norm{\cdot}_{(q')_*}$ with the same constant (one) by Lemma~\ref{lemma:dualestimate}. Thus, $\overline{\rho}_{(Y,\norm{\cdot}_{(q')_*})}(t)\leq (1+t^q)^{1/q}-1$ for all $t>0$. It follows that, given $t\geq 0$, either $\overline{\delta}^*_{(X^*, \norm{\cdot}_{(p)})}(t)>\overline{\rho}_{(Y,\norm{\cdot}_{(q')_*})}(t)$, or $p=q$ and $\overline{\delta}^*_{(X^*, \norm{\cdot}_{(p)})}(t)=\overline{\rho}_{(Y,\norm{\cdot}_{(q')_*})}(t)=(1+t^p)^{1/p}-1$, which is greater than $t-1$ for $t\geq 1$. Finally, the conclusion follows from Theorem~\ref{TheoremWeakStarMP}.
\end{proof}
Note that we may avoid one of the renormings in Theorem~\ref{thm:renorming} in case that $\overline{\delta}_{X^*}^*(t)=(1+t^p)^{1/p}-1$ or $\overline{\rho}_{Y}(t)=(1+t^q)^{1/q}-1$. As a consequence, we get for instance that for every $1<p<\infty$ there is an equivalent norm $\eqnorma$ on the Tsirelson space $\mathrm{T}$ so that the pair $((\mathrm{T}, \eqnorma), \ell_p)$ has the WMP.
\section{Application to some classical Banach spaces}
\label{section-applications}
\subsection{The case of \texorpdfstring{$\ell_1$}{l1} and Schur spaces}
Recall that a Banach space $Y$ is said to have the Schur property if every weakly convergent sequence is also norm-convergent. Our first goal is to show that Schur spaces are the best range spaces for the WMP. In the sequel, we will need to following well-known fact, its easy proof is left to the reader.
\begin{lemma} \label{lemmaNA}
Let $T :X \to Y$ be a bounded operator. If $T$ attains it norm then $T^* : Y^* \to X^*$ attains its norm. Conversely, if $T^*$ attains its norm and if $X$ is reflexive, then $T$ also attains its norm.
\end{lemma}
\begin{proposition} \label{prop-Schur-range}
If $X$ is reflexive and $Y$ is Schur, then every operator $T\colon X\to Y$ is compact. Therefore $(X,Y)$ has the WMP and $(Y^*, X^*)$ has the weak$^*$-to-weak$^*$MP.
\end{proposition}
\begin{proof}
The first part of the statement can be obtained as a particular case of each of the following facts:
\begin{itemize}
\item Every weakly precompact operator taking values in a Schur space is compact (obvious). Recall that an operator $T\colon X \to Y$ is called weakly precompact if $T(B_X)$ is weakly precompact, i.e., every sequence in $T(B_X)$ admits a weakly Cauchy subsequence.
\item Every completely continuous operator defined on a Banach space not containing $\ell_1$ is compact (apply Rosenthal’s $\ell_1$-theorem). Recall that an operator $T\colon X \to Y$ is called completely continuous if it is weak-to-norm sequentially continuous.
\end{itemize}
The second part of the statement follows from the fact that any compact operator on a reflexive space attains its norm and Lemma~\ref{lemmaNA}.
\end{proof}
The last proposition allows us to deduce for instance that the pair $(\ell_2, \ell_1)$ has the WMP while $(\ell_\infty, \ell_2)$ has the weak$^*$-to-weak$^*$MP.
Next, it is well known that $\ell_1 \equiv c_0^*$ is weak$^*$ asymptotically uniformly convex with modulus $\overline{\delta}_{\ell_1}^*(t)=t$, and the same is true for $Z^*$ instead of $\ell_1$ whenever $Z$ is a subspace of $c_0$. Consequently,
the next result is a direct consequence of Theorem~\ref{TheoremWeakStarMP}. Let us mention that if a dual Banach space $X \equiv Z^*$ has modulus $\overline{\delta}_{X}^*(t)=t$,
then $Z$ is asymptotically uniformly flat (that is there exists $t_0>0$ such that for every $t<t_0$, $\overline{\rho}_Z(t) = 0$).
This is actually equivalent to the fact that $Z$ is isomorphic to a subspace of $c_0$ (see \cite[Theorem~2.9]{JLPS02}).
\begin{corollary} \label{Cor-l1-domain}
If $Z$ is a closed subspace of $c_0$,
then the pair $(Z^{*},Y)$ has the W$^*$MP for every Banach space $Y$.
In particular $(\ell_1,Y)$ has the W$^*$MP for every Banach space $Y$ \emph{(with the weak$^*$ topology given by $c_0$)}.
\end{corollary}
However, the previous result is not true in general if one considers another weak$^*$ topology on $\ell_1$, or if one replaces $\ell_1$ by any dual space with the Schur property. This is shown by the next examples, we thank Gilles Lancien and Antonin Proch\'azka for presenting the first one to us. Notice that this proves in particular that the W$^*$MP is not an isomorphic property, in the sense that there are isomorphic spaces $X_1$, $X_2$ so that only one of the pairs $(X_i^*, Y)$ has the W*MP.
\begin{example}\label{ex:failsw*w*MP}
Consider $T\colon \mathcal{C}[0,\omega]^* \to c_0^*$ such that $T(e_n) = (1-\frac{1}{n})e_n$ and $T(e_{\omega}) = 0$. Of course $c_0^* \equiv \ell_1$ while $ \mathcal{C}[0,\omega]^* \equiv \ell_1([0,\omega]) \equiv \ell_1$. It is readily seen that $\|T\| = 1$ and that $T$ does not attain its norm. Now the sequence $(e_n)_n \in \mathcal{C}[0,\omega]^*$ is a maximizing sequence for $T$, but this sequence is not weak$^*$ null since it converges to $e_\omega$. Thus the pair $(\mathcal{C}[0,\omega]^* , c_0^*)$ fails the W$^*$MP. In fact, $T$ is the dual operator of $S\colon c_0 \to \mathcal{C}[0,\omega]$ given by $Sx(n)=(1-\frac{1}{n})x_n$ and $Sx(\omega) = 0$ for every $x=(x_n)_n \in c_0$. Therefore the pair $(\mathcal{C}[0,\omega]^* , c_0^*)$ even fails the weak$^*$-to-weak$^*$MP.
\end{example}
A very similar operator $T\colon \mathcal{C}[0,\omega]^* \to \mathbb{R}$, defined by $T(e_n) = (1-\frac{1}{n})$ and $T(e_{\omega}) = 0$, actually yields that the pair $(\mathcal{C}[0,\omega]^* , \mathbb{R})$ fails the W$^*$MP. We choose to present another example of a different nature.
\begin{example} \label{Example1-Schur}
There exists a dual space $Z=X^*$ with the Schur property such that the pair $(X^*, \mathbb{R})$ fails the W$^*$MP. In fact $Z$ is going to be the Lipschitz-free space $\mathcal{F}(M)$ where $M = \{0 \}\cup \{x_n \; : \; n \in \mathbb{N} \} \subset c_0$ is the pointed metric space given by $x_1 = 2e_1$, and $x_n = e_1+(1+\frac{1}{n})e_n$ for $n \geq 2$, where $(e_n)$ is the canonical basis of $c_0$. Then it is known \cite{Kalton04} that $\mathcal{F}(M)$ has the Schur property as $M$ is uniformly discrete and bounded. Indeed, the sequence $(\overline{\delta}elta(x_n))$ is a Schauder basis for $\mathcal{F}(M)$ which is equivalent to the standard $\ell_1$-basis. Therefore $\mathcal{F}(M)$ is actually isomorphic to $\ell_1$ (see \cite[Proposition 4.4]{Kalton04} for more details). It is also known \cite[Example~5.6]{GPPR_2018} that $\mathcal{F}(M)\equiv X^*$ where
\[ X = \{f \in \mathrm{Lip}_0(M) \; : \; \lim\limits_{n \to \infty} f(x_n) = \frac{1}{2} f(x_1)\}\]
is isomorphic to $c_0$. It is readily seen that the sequence $((1+\frac{1}{n})^{-1}\overline{\delta}elta(x_n))_n$ weak$^*$ converges to $\frac{1}{2}\overline{\delta}elta(x_1) \neq 0$. Now let $f \colon M \to \mathbb{R}$ be the Lipschitz map defined by $f(0)=f(x_1)=0$ and $f(x_n)=1$ for every $n \geq 2$. It is obvious that $\|f\|_L = \lim\limits_{n \to \infty} \frac{f(x_n) -f(0)}{d(x_n,0)} = 1$. Now using the linearization property of free spaces, we may consider the bounded operator $\overline{f} \colon \mathcal{F}(M) \to \mathbb{R}$ such that $\overline{f}(\overline{\delta}elta(x_n)) = f(x_n)$. Observe that $((1+\frac{1}{n})^{-1}\overline{\delta}elta(x_n))_n$ is a maximizing sequence which is not weak$^*$-null. To finish, we just need to show that $\overline{f}$ does not attain its norm. Assume that $\overline{f}(\mu)=1$ for $\mu=\sum_{n=1}^\infty a_n \overline{\delta}elta(x_n)\in S_{\mathcal F(M)}$. First note that
\[ 1=\overline{f}(\mu) = \sum_{n=2}^\infty a_n.\]
Let $A=\{n\geq 2: a_n>0\}$ and consider the one-Lipschitz function $g\colon M\to\mathbb R$ given by $g(x_n)=1$ if $n\in A$ and $g(x_n)=0$ otherwise. By evaluating the linear extension of $g$ to $\mathcal F(M)$ at $\mu$ we get $\sum_{n\in A} a_n\leq 1$. It follows that $a_n\geq 0$ for all $n\geq 2$.
Finally, consider the one-Lipschitz function $j\colon M\to\mathbb R$ given by $j(0)=j(x_1)=0$ and $j(x_n)=1+1/n$. Then,
\begin{align*}
1&=\norm{\mu} \geq \langle\overline{j}, \mu\rangle
= \sum_{n=2}^\infty a_n\left(1+\frac{1}{n}\right) = 1+ \sum_{n=2}^\infty \frac{a_n}{n}.
\end{align*}
Thus, $a_n=0$ for all $n\geq 2$. Then $\overline{f}(\mu)=0$, a contradiction.
\end{example}
Notice that the Banach space $\mathcal F(M)$ in the previous example is not AUC*: indeed $\frac{1}{2}\overline{\delta}elta(x_1)$ is an extreme point of $B_{\mathcal F(M)}$ \cite[Theorem~3.2]{AP20} but not a preserved extreme point \cite[Theorem~4.1]{AG19}, in particular it does not belong to slices of small diameter.
\subsection{Dunford--Pettis spaces}
We recall that a Banach space $Y$ has the Dunford--Pettis property if and only if for every sequence $(y_n)_{n}$ in $Y$ converging weakly to $0$ and every sequence $(y_n^*)_n$ in $Y^*$ converging weakly to 0, the sequence of scalars $(y_n^*(y_n))_n$ converges to 0.
It is well-known and easy to see that the weak convergence of $(y_n)_n$ (or of $(y_n^*)_n$) can be changed by weakly Cauchy convergence in this definition (see \cite{DiestelDP} e.g.).
Classical spaces with the Dunford--Pettis property include Schur spaces, $c_0$, $\mathcal{C}(K)$ spaces (where $K$ is a compact Hausdorff space) and $L_1(\mu)$ (where $\mu$ is a $\sigma$-finite measure), see e.g.~\cite{DiestelDP}. However reflexive (infinite dimensional) spaces never have it.
One goal of the section is to take advantage of the Dunford--Pettis property to analyse the possible relationships between $(X,Y)$ having the WMP and $(Y^*,X^*)$ having the weak$^*$-to-weak$^*$MP. Let us begin with the next observation.
\begin{lemma} \label{LemmaMaxSeqDUAL}
If $X$ is reflexive, $Y$ has the Dunford--Pettis property and $T\colon X \to Y$ is a nonzero operator, then every maximizing sequence for $T^*\colon Y^* \to X^*$ is non-weakly null.\\
Moreover, if $T^*$ admits a weakly Cauchy maximizing sequence then both $T$ and $T^*$ attain their norm.
\end{lemma}
\begin{proof}
Let us prove the first statement. Let $(y_n^*)_n \subset S_{Y^*}$ be a maximizing sequence for $T^*$. We define $(x_n)_n \subset S_{X}$ so that $T^*y_n^*(x_n) > \|T^*y_n^*\| - \frac{1}{n}$. It is easy to see that
\[\|Tx_n\| \geq |y_n^*(Tx_n)| = T^*y_n^*(x_n) > \|T^*y_n^*\| - \frac{1}{n},\]
and taking the limit $n \to \infty$ yields $\lim\limits_{n \to \infty} \|Tx_n\| \geq \|T^*\|$. Since $\|T^*\| = \|T\|$ we deduce that $(x_n)_n$ is a maximizing sequence for $T$.
By weak-compactness, we may extract a subsequence of $(x_n)_n$ (still denoted the same way) which weakly converges to some $x$. Now as $T$ is weakly continuous, we have that $(T x_n)_n$ weakly converges to $Tx$. We claim that $(y_n^*)_n$ is not weakly null. Indeed, aiming for a contradiction, assume that $(y_n^*)_n$ is weakly null. As $Y$ has the Dunford--Pettis property, we have that $\lim\limits_{n \to \infty} y_n^*(Tx_n) =\lim\limits_{n \to \infty} y_n^*(Tx_n-Tx) = 0$, contraducting our definition of $(y_n^*)_n$ and $(x_n)_n$.
Finally, if $(y_n^*)_n$ is a weakly Cauchy maximizing sequence for $T^*$, we may build an associated maximizing sequence $(x_n)_n$ for $T$ as described above. Up to extracting a subsequence, we may assume that $(x_n)_n$ is weakly convergent to some $x \in X$. Of course, $\|x\| \leq 1$ by lower semi-continuity of the norm. Next, since $Y$ has the Dunford--Pettis property and $T$ is weakly continuous, $\lim\limits_n \langle y_n^* , Tx_n -Tx \rangle = 0$. By construction we know that $\lim\limits_n \langle y_n^* , Tx_n \rangle = \|T\|$, which in turn implies that $\lim\limits_n \langle y_n^* , Tx \rangle = \|T\|$, and so we obtain that $\|Tx\| \geq \|T\|$. Consequently $\|x\|=1$ and $T$ attains its norm at $x$. The fact that $T^*$ also attains its norm follows from Lemma~\ref{lemmaNA}.
\end{proof}
\begin{remark}
By Rosenthal's $\ell_1$ theorem,
if $T^*$ does not admit a weakly Cauchy maximizing sequence (as it is assumed in the second part of Lemma~\ref{LemmaMaxSeqDUAL}), then every maximizing sequence for $T^*$ has a subsequence equivalent to the $\ell_1$ canonical basis. Let us point out that $Y^*$ contains a copy of $\ell_1$ whenever $Y$ is infinite-dimensional and has the Dunford--Pettis property (see \cite[Corollary~7]{Castillo} e.g.).
\end{remark}
\begin{remark}
Assume this time that $X$ has the Dunford--Pettis property while $Y$ is reflexive. Similarly as above, one can show that every bounded operator $T\colon X \to Y$ admits a non weakly-null maximizing sequence. However, since $X$ is non reflexive, there is no chance that the pair $(X,Y)$ enjoys the WMP.
\end{remark}
We recall that a Banach space $Y$ has the weak$^*$-Dunford--Pettis property if for all weakly null sequences $(y_n)_n$ in $Y$ and all weak$^*$ null sequences $(y_n^*)_n$ in $Y^*$, the sequence $(y_n^*(y_n))_n$ converges to 0. Let us provide some results from \cite{Jaramillo2000} concerning this property. It is clear that the weak$^*$-Dunford--Pettis property implies the classical Dunford--Pettis property. For instance $\ell_1$ (and any Schur space), $\ell_\infty$ and $\ell_1 \oplus \ell_\infty$ have this property. Moreover, if $Y$ has the weak$^*$-Dunford--Pettis property and if the unit ball of $Y^*$ is weak$^*$-sequentially compact (for instance when $Y$ is separable), then $Y$ is a Schur space.
A typical example of a space having the Dunford--Pettis property but lacking the weak$^*$-Dunford--Pettis property is $c_0$. The following result is a modified version of Lemma~\ref{LemmaMaxSeqDUAL}.
\begin{lemma} \label{LemmaMaxSeqDUALstar}
If $X$ is reflexive, $Y$ has the weak$^*$-Dunford--Pettis property and $T\colon X \to Y$ is a nonzero operator, then every maximizing sequence for $T^* \colon Y^* \to X^*$ is non-weak$^*$ null.\\
If moreover the unit ball of $Y^*$ is weak$^*$-sequentially compact (in particular Y is a Schur space) then every maximizing sequence for a nonzero operator $T\colon X \to Y$ is non-weakly null.
\end{lemma}
\begin{proof}
The proof of the first statement follows the same lines as in Lemma~\ref{LemmaMaxSeqDUAL}, so we shall omit the details. Let us prove the second statement. Suppose that $Y^*$ is weak$^*$-sequentially compact and let $(x_n)_n$ be a maximizing sequence for a non-zero operator $T:X \to Y$. Since $X$ is reflexive, up to an extraction, we may assume that $(x_n)_n$ weakly converges to some $x \in X$. Let $(y_n^*)_n$ be a sequence in $S_{Y^*}$ such that
\[ T^*y_n^*(x_n) = y_n^*(Tx_n) > \|Tx_n\| - \frac{1}{n},\]
then $(y_n^*)_n$ is a maximizing sequence for $T^*$.
By assumption, up to an extraction, we may assume that $(y_n^*)_n$ weak$^*$ converges to some $y^* \in Y$. On the one hand $\lim\limits_{n} \langle y_n^*, Tx_n \rangle = \|T\|$ by construction. On the other hand, $\lim\limits_{n} \langle y_n^*, Tx_n \rangle = \langle y^* , Tx \rangle$ since $Y$ has the weak$^*$-Dunford--Pettis property. This clearly yields that $(x_n)_n$ is a non-weakly null maximizing sequence for $T$.
\end{proof}
\begin{proposition} \label{prop-DP-Duality}
Let $X$ be a reflexive space and let $Y$ be a Banach space with the weak$^*$-Dunford--Pettis property.
Then the following assertions are equivalent:
\begin{enumerate}
\item[(i)] $(Y^*,X^*)$ has the weak$^*$-to-weak$^*$MP,
\item[(ii)] Every dual operator $T^* \colon Y^* \to X^*$ attains its norm,
\item[(iii)] Every operator $T \colon X \to Y$ attains its norm.
\end{enumerate}
In particular, any of the above imply that $(X,Y)$ has the WMP.
\end{proposition}
\begin{proof}
According to Lemma~\ref{LemmaMaxSeqDUALstar}, for every bounded operator $T \colon X \to Y$, there exists a non-weak$^*$ null maximizing sequence for $T^* \colon Y^* \to X^*$. So if we assume $(i)$, by definition we clearly obtain $(ii)$. Also, it is trivial that $(ii) \implies (i)$. Now $(ii) \iff (iii)$ since $X$ is reflexive, see Lemma~\ref{lemmaNA}. Finally, the assertion $(iii)$ readily yields that $(X,Y)$ has the WMP.
\end{proof}
\begin{remark}
If moreover the unit ball of $Y^*$ is assumed to be weak$^*$-sequentially compact then any of the previous statements is equivalent to:
\begin{enumerate}
\item[(iv)] $(X,Y)$ has the WMP.
\end{enumerate}
Indeed, in this case $Y$ is a Schur space so that Proposition~\ref{prop-Schur-range} ensures that $(X,Y)$ has the WMP while $(Y^*,X^*)$ has the weak$^*$-to-weak$^*$MP. Of course this could also be proved directly $(iv) \implies (i)$: If $(X,Y)$ has the WMP, we consider $(y_n^*)_n$ a non-weak$^*$ null maximizing sequence for $T^* \colon Y^* \to X^*$. By assumption, $Y^*$ is weak$^*$-sequentially compact so we may apply the second part of Lemma~$\ref{LemmaMaxSeqDUALstar}$ which provides a non-weakly null maximizing sequence for $T$. Therefore $T$ attains its norm, and so does $T^*$ thanks to Lemma~\ref{lemmaNA}.
\end{remark}
Note that the hypothesis on the weak$^*$-Dunford--Pettis property is actually needed in Propostion~\ref{prop-DP-Duality}. Indeed, the pair $(\ell_1, \mathbb R\oplus_1 \ell_2)$ has the W*MP (and so the weak*-to-weak*MP) by Theorem~\ref{TheoremWeakStarMP}. However, $(\mathbb R\oplus_\infty \ell_2, c_0)$ fails the WMP as we have seen in Remark~\ref{rem:failing}.
On the other hand, there are spaces $X, Y$ so that $(Y^*, X^*)$ has the WMP (so the weak$^*$-to-weak$^*$MP) but $(X, Y)$ and fails the WMP: just take $Y=\mathbb R$ and $X$ any non-reflexive space.
The reverse situation is also of particular interest:
\begin{question}
If $(X,Y)$ has the WMP, does it follow that $(Y^*,X^*)$ has the weak$^*$-to-weak$^*$MP?
\end{question}
It is known that a dual Banach space $X^*$ has the Schur property if and only if $X$ has the Dunford--Pettis property and does not contain any isomorphic copy of $\ell_1$. So, if $Y$ is reflexive, $X$ has the Dunford--Pettis property and does not contain any isomorphic copy of $\ell_1$, then $(Y^*,X^*)$ has the WMP. Notice that it is not possible to remove the assumption on the non-containment of $\ell_1$ for $X$. Indeed, $\ell_1$ has the Dunford--Pettis property and $\ell_2$ is of course reflexive but the pair $(\ell_2 , \ell_\infty)$ fails the WMP, as we shall explain in the next example. In fact our example tells us a bit more.
Indeed, we prove that $(\ell_2,\ell_\infty)$ even fails the weak$^*$-to-weak$^*$MP, while $(\ell_2,c_0)$ has the WMP and $(\ell_1,\ell_2)$ has the W$^*$MP (so the weak$^*$-to-weak$^*$MP).
It was suggested to us by R. Aron to consider the following variation of the WMP.
\begin{definition}
We say that a pair $(X,Y)$ has the bidual WMP if for every operator $T \colon X \to Y$, the existence of a non weakly-null maximizing sequence for $T$ implies that $T^{**} \colon X^{**} \to Y^{**}$ attains its norm.
\end{definition}
It is obvious from the definitions that if $(X^{**},Y^{**})$ has the weak$^*$-to-weak$^*$MP then $(X,Y)$ has the bidual WMP. However, the converse does not hold in general as witnessed by the following example.
\begin{example}
First of all, $(\ell_2, c_0)$ has the WMP thanks to Corollary~\ref{CorWMP}~a), thus it also has the bidual WMP. Next $(\ell_1,\ell_2)$ has the W$^*$MP thanks to Corollary~\ref{Cor-l1-domain}.
It only remains to prove that $(\ell_2,\ell_\infty)$ fails the weak$^*$-to-weak$^*$MP (and consequently also fails the WMP and the W$^*$MP).
Let $T \colon \ell_2 \to \ell_{\infty}$ defined for every $x=(x_n)_n \in \ell_2$ by
\[Tx = \left(x_1 , \; x_1 + \Big(1-\frac{1}{2}\Big)x_2 , \; x_1 + \Big(1-\frac{1}{3}\Big)x_3 , \; \ldots\; , \; x_1 + \Big(1-\frac{1}{n}\Big)x_n , \; \ldots\right).\]
In other words, $Te_1 = \mathbbm{1}$ (the constant equal to 1 sequence) and $Te_n = (1-\frac{1}{n})e_n$. Our claim follows from the items below:
\begin{itemize}
\item $\|T\| = \sqrt{2}$. Indeed,
\begin{align*} \norm{Tx}_\infty&=\sup\big\{|x_1+(1-1/n)x_n| : n\geq 1 \big\}\\
&\leq
\sup\big\{(|x_1|^2+|x_n|^2)^{1/2}(1+(1-1/n)^2)^{1/2}: n\geq 2\big\} \\
&\leq \sqrt{2}\norm{x}_2.
\end{align*}
Moreover,
$$\Big\|T\Big(\frac{1}{\sqrt{2}}(e_1+e_n)\Big) \Big\|\to \sqrt{2},$$ so $\norm{T}=\sqrt{2}$ the sequence $\big(\frac{1}{\sqrt{2}}(e_1+e_n)\big)_n$ is a normalized maximizing sequence which is not weakly$^*$ null (it weakly$^*$ converges to $\frac{1}{\sqrt{2}}e_1$).
\item $T$ is weak*-to-weak*-continuous: one can easily check that $T=S^*$ where $S\colon \ell_1\to\ell_2$ is given by $Sx=(\sum_{n=1}^\infty x_n)e_1+\sum_{n=1}^\infty (1-1/n)x_n e_n$.
\item $T$ does not attain its norm: if $\norm{Tx}_\infty=\sqrt{2}$ for some $x$ with $\norm{x}_2=1$, then the above estimation implies that $\limsup_n (|x_1|^2+|x_n|^2)^{1/2} = 1$. Thus $|x_1|=1$, which implies that $x=\overline{\rho}m e_1$, a contradiction.
\end{itemize}
\end{example}
\subsection{The case of the James sequence spaces}
Let $p\in (1,\infty)$. We now recall the definition and some basic properties of the James space $\mathcal{J}_p$. We refer the reader to \cite[Section 3.4]{albiackalton} and references therein for more details on the classical case $p=2$. The James space $\mathcal{J}_p$ is the real Banach space of all sequences $x=(x(n))_{n\in \mathbb{N}}$ of real numbers with finite $p$-variation and satisfying $\lim_{n \to \infty} x(n) =0$. The space $\mathcal{J}_p$ is endowed with the following norm
\[\|x\|_{\mathcal{J}_p} = \sup \Big \{ \big (\sum_{i=1}^{k-1} |x(p_{i+1})-x(p_i)|^p \big )^{1/p} \; \colon \; 1 \leq p_1 < p_2 < \ldots < p_{k} \Big \}. \]
This is the historical example, constructed for $p=2$ by R.~C.~James, of a quasi-reflexive Banach space which is isomorphic to its bidual. In fact $\mathcal{J}_p^{**}$ can be seen as the space of all sequences $x=(x(n))_{n\in \mathbb{N}}$ of real numbers with finite $p$-variation, which is $\mathcal{J}_p \oplus \mathbb{R} \mathbbm{1}$, where $\mathbbm{1}$ denotes the constant sequence equal to $1$.
The standard unit vector basis $(e_n)_{n=1}^{\infty}$ ($e_n(i)=1$ if $i = n$ and $e_n(i)=0$ otherwise) is a
monotone shrinking basis for $\mathcal{J}_p$. Hence, the sequence $(e_n^*)_{n=1}^{\infty}$ of the associated coordinate
functionals is a basis of its dual $\mathcal{J}_p^*$. Then the weak topology $\sigma(\mathcal{J}_p,\mathcal{J}_p^*)$ is easy to
describe: a sequence $(x_n)_{n=1}^{\infty}$ in $\mathcal{J}_p$ converges to $0$ in the $\sigma(\mathcal{J}_p,\mathcal{J}_p^*)$ topology
if and only if it is bounded and $\lim_{n \to \infty} x_n(i)=0$ for every $i \in \mathbb{N}$.
For $x\in \mathcal{J}_p$, we define $\supp x = \{i \in \mathbb{N} \, : \, x(i) \neq 0 \}$.
The detailed proof of the following proposition can be found in \cite[Corollary~2.4]{Netillard}.
This a consequence of the fact that the basis of $\mathcal J_p$ satisfies an upper $p$-estimate.
\begin{proposition}\label{Jsmooth} There exists an equivalent norm $|\cdot|$ on $\mathcal{J}_p$ such that it has the following property: for any $x,y \in \mathcal{J}_p$ such that $\max \supp x < \min \supp y$, we have that
\[|x+y|^p\leq |x|^p+|y|^p.\]
In particular, the modulus of asymptotic uniform smoothness of $\widetilde{\mathcal{J}_p}:=(\mathcal{J}_p,|\cdot|)$ is $\overline{\rho}_{\widetilde{\mathcal{J}_p}}(t) \leq (1+t^p)^{\frac{1}{p}} - 1$ for all $t\geq 0$.
\end{proposition}
There is also a natural weak$^*$ topology on $\mathcal{J}_p$. Indeed, the summing basis $(s_n)_{n=1}^{\infty}$ ($s_n(i)=1$ if $i \leq n$ and $s_n(i)=0$ otherwise) is a monotone and boundedly complete basis for $\mathcal{J}_p$. Thus, $\mathcal{J}_p$ is naturally isometric to a dual Banach space: $\mathcal{J}_p = X^*$ with $X$ being the closed linear span of the biorthogonal functionals $(e_n^* - e_{n+1}^*)_{n=1}^{\infty}$ in $\mathcal{J}_p^*$ associated with $(s_n)_{n=1}^{\infty}$. Note that $X=\{x^*\in \mathcal{J}_p^*,\ \sum_{n=1}^\infty x^*(n)=0\}$. Thus, a sequence $(x_n)_{n=1}^{\infty}$ in $\mathcal{J}_p$ converges to $0$ in the $\sigma(\mathcal{J}_p,X)$ topology if and only if it is bounded and $\lim_{n \to \infty} \big(x_n(i) - x_n(j)\big) = 0$ for every $i \neq j \in \mathbb{N}$.
The next lemma is classical, we include its proof for completeness.
\begin{lemma}\label{Jconvex}
Let $(x_n)_{n=1}^\infty$ be a weak$^*$-null sequence in $\mathcal{J}_p$. Then, for every $x \in \mathcal J_p$, we have
\[ \limsup_{n\to\infty} \norm{x +x _n}^p \geq \|x\|^p + \limsup_{n\to\infty} \|x_n\|^p. \]
Consequently, the modulus of weak$^*$ asymptotic uniform convexity of $\mathcal{J}_p$ is given by
\[\overline{\delta}^*_{\mathcal{J}_p}(t) \geq (1+t^p)^{\frac{1}{p}} - 1.\]
\end{lemma}
\begin{proof}
Let $x=(x(n))_n \in \mathcal{J}_p$, $(x_n)_{n=1}^\infty$ be a weak$^*$-null sequence and $\varepsilon >0$ be fixed. We may and do assume that $\|x\|=1$ and $x$ is finitely supported. Let $(p_i)_{i=1}^{N} \subset \mathbb{N}$ be an increasing family such that \[\sum_{i=1}^{N-1} |x(p_{i+1})-x(p_i)|^p =1\]
and take $N_1 > p_N$ such that $x(n)=0$ for $n\geq N_1$.
Now, since $(x_n)_{n=1}^\infty$ is a weak$^*$-null sequence, it is bounded by some $M>0$ and there exists $N_2 \in \mathbb{N}$ large enough so that for every $n \geq N_2$ and for any increasing finite sequence $(r_i)_{i=1}^{l}$ with $r_l<N_1$:
\[\sum_{i=1}^{l-1} |x_n(r_{i+1})-x_n(r_i)|^p \leq \frac{\varepsilon}{3}
\quad \text{ and } \quad |x_n(r_{l})-x_n(N_1)|^p \leq \varepsilon' , \]
where $\varepsilon'$ is chosen so that $||x|-|y||^p \geq |x|^p - \frac{\varepsilon}{3}$ for every $x\in [0,M]$ and $y \in [0,\varepsilon']$ (the map $f(x,y)=\abs{x-y}^p$ is Lipschitz on $[0, M]\times [0,M]$, so $\varepsilon'=\min\{\frac{\varepsilon}{3\Lip(f)},M\}$ does the work). Note also that in such a case one also has
\begin{align*}
|x_n(r) - x_n(N_1)|^p &\geq \abs{|x_n(r) - x_n(r_l)| - |x_n(r_l) - x_n(N_1)|}^p \\
&\geq |x_n(r) - x_n(r_l)|^p - \frac{\varepsilon}{3},
\end{align*}
for any $r\geq N_1$.
Now, let $n \geq N_2$. It follows from the above estimates that we may find a fixed increasing sequence $(q_i)_{i=1}^{k}$ with $q_1\geq N_1$ and
\[\sum_{i=1}^{k-1} |x_n(q_{i+1})-x_n(q_i)|^p \geq \|x_n\|^p- \varepsilon.\]
Note that $p_N<N_1\leq q_1$ by construction so that
\begin{multline*}
(\star) \quad \|x+x_n\|^p \geq \sum_{i=1}^{N-1}|x(p_{i+1}) -x(p_i) + x_n(p_{i+1}) -x_n(p_i)|^p \\
+ \sum_{i=1}^{k-1}|x(q_{i+1}) -x(q_i) + x_n(q_{i+1}) -x_n(q_i)|^p .
\end{multline*}
But
\begin{multline*}
\Big( \sum_{i=1}^{N-1}|x(p_{i+1}) -x(p_i) + x_n(p_{i+1}) -x_n(p_i)|^p \Big)^{\frac 1p} \\
\geq \Big( \sum_{i=1}^{N-1} |x(p_{i+1}) -x(p_i)|)^p \Big)^{\frac 1p} - \varepsilon
= 1 - \varepsilon,
\end{multline*}
which implies that
$\sum_{i=1}^{N-1}|x(p_{i+1}) -x(p_i) + x_n(p_{i+1}) -x_n(p_i)|^p \geq 1 - p\varepsilon.$ \\
Also, since $x(n)=0$ for $n>N_1$,
\[\sum_{i=1}^{N-1}|x(q_{i+1}) -x(q_i) + x_n(q_{i+1}) -x_n(q_i)|^p \geq \|x_n\|^p - \varepsilon.\]
Finally, we obtain from $(\star)$ the following last estimate
\[ \|x+x_n\|^p \geq 1 + \|x_n\|^p-(1+p)\varepsilon.\]
\end{proof}
As an application of Theorem~\ref{TheoremWeakStarMP} and the two previous lemmata, we obtain the following corollary.
\begin{corollary} \label{CorJamesWMP}
If $1<p\leq q< \infty$ then $(\mathcal J_p, \widetilde{\mathcal J_q})$ has the weak$^*$-to-weak$^*$MP.
\end{corollary}
We now turn to the reverse situation:
\begin{corollary} \label{CorJamesWMP2}
If $1< q< p <\infty$ then every bounded operator from $\mathcal J_p$ to $\mathcal{J}_q$ is compact.
In particular, the pair $(\mathcal J_p,\mathcal J_q)$ has the weak$^*$-to-weak$^*$MP.
\end{corollary}
\begin{proof}
Since $\overline{\rho}_{\widetilde{\mathcal{J}_p}}(t) \leq (1 +t^p)^{1/p} - 1$, $\overline{\delta}_{\mathcal{J}_q}^*(t) \geq (1 +t^q)^{1/q} - 1$ and $\overline{\delta}_{\mathcal{J}_q}(t) \geq \overline{\delta}_{\mathcal{J}_q}^*(t)$, we obtain from Lemma~\ref{CompactOperator} that every bounded operator from $\widetilde{\mathcal{J}_p}$ to $\mathcal{J}_q$ is compact. Since $\widetilde{\mathcal{J}_p}$ and $\mathcal{J}_p$ are isomorphic, this yields the same conclusion for every bounded operator from $\mathcal{J}_p$ to $\mathcal{J}_q$. To conclude, Theorem~\ref{TheoremCompactWMP}~(2) provides the fact that the pair $(\mathcal J_p,\mathcal J_q)$ has the weak$^*$-to-weak$^*$MP.
\end{proof}
\begin{remark}
Quite surprisingly, the pair $(\mathcal J_2, \mathbb{R})$ fails the W$^*$MP (but has the weak$^*$-to-normMP thanks to Corollary~\ref{CorJamesWMP}). Indeed, consider $T \colon \mathcal{J}_2 \to \mathbb{R}$ be the linear operator given by
\[ \forall x=\big(x(j)\big)_{j=1}^{\infty} \in \mathcal{J}_2, \quad Tx := \frac{-x(1)}{2} + \sum_{j=2}^{\infty} \frac{x(j)}{j^2}.\]
That is, $T=(\frac{-1}{2},\frac{1}{2^2}, \frac{1}{3^2}, \ldots ) \in \mathcal{J}_2^*$.
Note that $T$ is not weak$^*$ continuous since $T(s_n) \not\to 0$ while the summing basis $(s_n)$ is weak$^*$-null. Since the standard unit vector basis $(e_n)_{n=1}^{\infty}$ is monotone and shrinking, it follows that for every $x^{**} \in \mathcal{J}_2^{**}$, $\|x^{**}\|_{\mathcal{J}_2^{**}} = \sup_n \|(x^{**}(e_1^*),x^{**}(e_2^*), \ldots,x^{**}(e_n^*), 0 ,\ldots )\|_{\mathcal{J}_2}$ (see e.g. Proposition 4.14 in \cite{FHHMZ11}).
We claim that any norming functional $x^{**}$ for $T$ is of the form
$$x^{**} = \frac{1}{\sqrt{(t_2-t_1)^2+t_2^2}}(t_1,t_2,t_2,\ldots).$$
Indeed, fix any $x^{**}:=(c_n)_n \in S_{\mathcal{J}_2^{**}}$ such that $\langle x^{**} , T \rangle = \|T\|$, and let $s : = \sup_{n \geq 2} |c_n| $. Notice that
$$(\star) \quad 1 = \|x^{**}\| = \sup_n \|(c_1 , \ldots , c_n , 0 , \ldots)\|_{\mathcal{J}_2} \geq \sqrt{(s-c_1)^2 + s^2}.$$
Then some easy computations show that
$$\|T\|=\langle x^{**} , T \rangle
= \frac{-c_1}{2} + \sum_{j=2}^{\infty} \frac{c_j}{j^2}
\leq \frac{-c_1}{2} + \sum_{j=2}^{\infty} \frac{s}{j^2}.
$$
Now the right-hand side of this inequality is equal to $\langle z , T\rangle$, where $z := (c_1,s,s,\ldots)$ belongs to $B_{\mathcal{J}_2^{**}}$ thanks to $(\star)$. Therefore $\langle z , T\rangle \geq \|T\|$ and so $z$ is a norming functional of the desired form. Moreover, if $x^{**} \neq z$ then the last inequality above would be strict and so we would get $\|T\| < \langle z , T\rangle \leq \|T\|$ as a contradiction.
Consequently, in order to maximize $T(x^{**})$ it suffices to maximize the function
\begin{eqnarray*}
g(t_1,t_2) = \frac{\frac{-t_1}{2} + t_2(\frac{\overline{\rho}i^2}{6}-1)}{\sqrt{(t_2-t_1)^2+t_2^2}} = \frac{\frac{-1}{2} \frac{t_1}{t_2} + (\frac{\overline{\rho}i^2}{6}-1)}{\sqrt{(1-\frac{t_1}{t_2})^2+(\frac{t_1}{t_2})^2}}.
\end{eqnarray*}
We set $t = \frac{t_1}{t_2}$ and we are now looking for the maximum of the following map:
\[ f(t) = \frac{\frac{-1}{2}t + (\frac{\overline{\rho}i^2}{6}-1)}{\sqrt{(1-t)^2+t^2}}.\]
A basic study of the map $f$ shows that it attains its maximum at $t_{max} = \frac{\overline{\rho}i^2-9}{2\overline{\rho}i^2-15}$ (and $f(t_{max}) \simeq 0.66$). To conclude, fix $t_1,t_2 \in \mathbb{R}$ such that $\frac{t_1}{t_2} = t_{max}$ and $\sqrt{(t_2-t_1)^2+t_2^2}=1$. Now for every $n \in \mathbb{N}$ let $x_n \in \mathcal{J}_2$ be such that $x_n(1)=t_1$, $x_n(2)= \ldots = x_n(n) = t_2$ and $x_n(j) = 0$ whenever $j>n$ (that is, $x_n = (t_1, t_2 ,\ldots ,t_2 , 0 ,\ldots)$). Then $(x_n)_{n=1}^\infty$ is a normalized maximizing sequence for $T$ which is non-weak$^*$ null. However, $T$ does not attain its norm on $\mathcal{J}_2$. Indeed, we proved that any norming functional for $T$ must be of the form $x=(t_1,t_2,t_2, \ldots)$. Such a functional $x$ belongs to $\mathcal{J}_2$ if and only if $t_2 = 0$ (as every element in $\mathcal{J}_2$ is a sequence that must converge to 0). So one has $|\langle x , T \rangle| = |\frac{-t_1}{2}| = \frac{|t_1|}{2} = \frac{\|x\|}{2} \leq \frac{1}{2} < f(t_{\max} )$ and so a norming functional for $T$ cannot belong to $J_2$.
\end{remark}
Having in mind the previous corollaries, it is quite natural to wonder the following:
\begin{question}
Does the pair $(\mathcal J_p, \mathcal J_q)$ has the weak$^*$-to-weak$^*$MP for $1<p\leq q< \infty$?
\end{question}
\subsection{Orlicz spaces}
Given an Orlicz function $\varphi\colon [0,+\infty)\to[0,+\infty)$ (that is, $\varphi$ is a continuous convex unbounded function with $\varphi(0)=0$), the Orlicz sequence space $\ell_\varphi$ is the space of all real sequences $x=(x_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varphi(|x_n|/\lambda)<\infty$. It is a Banach space when equipped with the Luxemburg norm:
\[ \norm{x}_{\varphi}=\inf\{\lambda>0 : \sum_{n=1}^\infty \varphi(|x_n|/\lambda)\leq 1\}.\]
The closed linear span of $\{e_n : n\in \mathbb N\}$ in $\ell_{\varphi}$ is denoted $h_{\varphi}$. The space $h_\varphi$ coincides with $\ell_\varphi$ precisely if $\varphi$ satisfies the $\Delta_2$ condition at zero, i.e. $\limsup_{t\to 0}\varphi(2t)/\varphi(t)<\infty$. The space $h_\varphi$ (or $\ell_\varphi$) is reflexive if and only if both $\varphi$ and $\varphi^*$
satisfy the $\Delta_2$ condition at zero, where $\varphi^*(t)=\sup\{st-\varphi(s)\}$ is the convex conjugate of $\varphi$. The Boyd indices of an Orlicz function $\varphi$ are defined as follows:
\begin{align*} \alpha_\varphi &= \sup\{p>0 : \sup_{0<u,t\leq 1} \frac{\varphi(tu)}{\varphi(u)t^p}<\infty\}, \\ \beta_\varphi &= \inf\{p>0 : \inf_{0<u,t\leq 1} \frac{\varphi(tu)}{\varphi(u)t^p}>0\}.
\end{align*}
It is known that $\beta_\varphi<\infty$ precisely if $\varphi$ satisfies the $\Delta_2$ condition at $0$. The asymptotic moduli of the space $h_\varphi$ is linked to the Boyd indices: $h_\varphi$ is AUS (resp. AUC) if and only if $\alpha_\varphi>1$ (resp. $\beta_\varphi<\infty$). Moreover $\alpha_\varphi$ is the supremum of the numbers $\alpha$ such that $\overline{\rho}_{h_\varphi}$ has power type $\alpha$ \cite{GJT07}. In addition, $\beta_\varphi$ is the infimum of the numbers $\beta$ such that $\overline{\delta}_{h_\varphi}$ has power type $\beta$ \cite{BM10}.
In order to apply Theorem~\ref{TheoremWeakStarMP} to the case of Orlicz spaces we need an estimation of $\overline{\rho}_{h_\varphi}(t)$ and $\overline{\delta}_{h_\varphi}(t)$ for all $t>0$. To this end, consider the following indices:
\begin{align*} p_\varphi &= \sup\{p>0 : u^{-p} \varphi(u) \text{ is non-decreasing for all } 0<u\leq\varphi^{-1}(1)\} \\
q_\varphi &= \inf\{p>0 : u^{-p} \varphi(u) \text{ is non-increasing for all } 0<u\leq\varphi^{-1}(1)\}.
\end{align*}
Clearly $1\leq p_{\varphi}\leq q_{\varphi}\leq \infty$. Moreover, $\varphi$ (resp. $\varphi^*$) satisfies $\Delta_2$ condition at $0$ if and only if $q_\varphi<\infty$ (resp. $p_\varphi>1$); see \cite{Maligranda85} or \cite{Delpech09}. Delpech \cite{Delpech09} showed that if $q_\varphi<\infty$ then
\[ (1+t^{q_\varphi})^{1/q_\varphi}-1\leq \overline{\delta}_{h_\varphi}(t)\]
for all $t\in (0,1]$, but actually the proof works for all $t>0$. Analogously, one can show that
\[ \overline{\rho}_{h_\varphi}(t)\leq (1+t^{p_\varphi})^{1/p_\varphi}-1\]
for all $t>0$.
\begin{remark} In general $[\alpha_\varphi, \beta_\varphi]\subset [p_\varphi, q_\varphi]$, but the inclusion may be strict \cite{Maligranda85}. Thus the previous inequalities are not tight. Also, recall that, given any $p\in [\alpha_\varphi, \beta_\varphi]$, the space $h_\varphi$ contains almost isometric copies of $\ell_p$. Thus $\overline{\delta}_{\ell_\varphi}(t)\leq (1+t^{\alpha_\varphi})^{1/\alpha_\varphi}-1$ and $\overline{\rho}_{\ell_\varphi}(t)\geq (1+t^{\beta_\varphi})^{1/\beta_\varphi}-1$ for all $t>0$.
\end{remark}
\begin{corollary} Let $\varphi, \overline{\rho}si$ be Orlicz functions. Assume that $h_\varphi, h_\overline{\rho}si$ are reflexive and $q_{\varphi}\leq p_{\overline{\rho}si}$. Then the pair $(h_\varphi, h_\overline{\rho}si)$ has the WMP.
\end{corollary}
\begin{remark}
Given Orlicz functions $\varphi, \overline{\rho}si$ such that that $h_\varphi, h_\overline{\rho}si$ are reflexive, if $\alpha_\varphi>\beta_\overline{\rho}si$ then every operator from $h_\varphi$ to $h_\overline{\rho}si=\ell_\overline{\rho}si$ is compact \cite{AO97}, and so $(h_\varphi, h_\overline{\rho}si)$ has the WMP.
\end{remark}
\subsection{Remarks about the pair \texorpdfstring{$(L_p,L_q)$}{(Lp,Lq)}}
In what follows, $L_p$ stands for $L_p([0,1])$ with $1<p<\infty$. It is known (see \cite{Milman71} at page 117 for instance) that if $X = L_p$ then there exist constants $C_1(p),C_2(p)$ such that for $1<p<2$ we have
\begin{eqnarray*}
C_1(p) t^2 \leq &\overline{\delta}_X(t)& \leq (p-1) t^2 \\
\frac 1p t^p \leq &\overline{\rho}_X(t)& \leq \frac 2p t^p
\end{eqnarray*}
and for $2<p<\infty$ we have that
\begin{eqnarray*}
C_1(p) t^p \leq &\overline{\delta}_X(t)& \leq \frac 1p t^p \\
(p-1) t^2 \leq &\overline{\rho}_X(t)& \leq C_2(p) t^2.
\end{eqnarray*}
Consequently, for $1<p<q<\infty$, we cannot apply Theorem~\ref{TheoremWeakStarMP} to prove that the pair $(L_p,L_q)$ has the WMP. Nevertheless, thanks to Theorem \ref{thm:renorming} and the above estimations we can still say something for a particular renorming of $L_p$ and $L_q$.
\begin{corollary}
Assume that $1<p\leq2$ and $2\leq q<\infty$. Then there are equivalent norms $\eqnorma_p$ on $L_p$ and $\eqnorma_q$ on $L_q$ such that the pair $((L_p, \eqnorma_p), (L_q, \eqnorma_q))$ has the WMP.
\end{corollary}
Replacing the weak topology by the topology $\tau_m$ of convergence in measure, we can apply similar techniques to prove the following related results.
\begin{proposition} \label{propConvMeasure}
Let $1<p<q<\infty$ and let $T \colon L_p \to L_q$ be a bounded operator which is $\tau_m$-to-$\tau_m$ continuous. If there exists a maximizing sequence $(x_n)_n \subset L_p$ for $T$ which $\tau_m$-converges to some $x \neq 0$, then $T$ attains its norm at $x$.
\end{proposition}
\begin{proof}
The proof follows the same lines as in Theorem~\ref{TheoremWeakStarMP}. In fact, $L_p$ has the following property (see \cite{KW} e.g.): if $(x_n)_n \subset L_p$ converges to $0$ in measure, then for every $x \in L_p$
\[(\star) \qquad \limsup_n \|x+x_n \| = \big(\|x\|^p+\limsup_n \|x_n\|^p \big)^{1/p}.\]
Let $T \colon L_p \to L_q$ be a bounded operator which is $\tau_m$-to-$\tau_m$ continuous and let $(x_n)_n \subset L_p$ be a maximizing sequence for $T$ which $\tau_m$-converges to some $x \neq 0$. Without loss of generality, we may assume that $T$ has norm 1.
Since $T$ is $\tau_m$-to-$\tau_m$ continuous, we have that $Tx_n \overset{\tau_m}{\underset{n}{\longrightarrow}} Tx$. Using $(\star)$, we thus have the following estimates:
\begin{eqnarray*}
1 &=& \norm T = \lim\limits_{n} \norm{Tx_n} = \lim\limits_{n} \norm{Tx+Tx_n - Tx} \\
&=& \big(\|Tx\|^q +\lim\limits_{n} \|Tx_n - Tx\|^q \big)^{\frac{1}{q}} \\
&\leq& \big(\|Tx\|^q +\lim\limits_{n} \|x_n - x\|^q \big)^{\frac{1}{q}} \\
&\leq& \big(\|Tx\|^p +\lim\limits_{n} \|x_n - x\|^p \big)^{\frac{1}{p}} \\
&=& \big( \|Tx\|^p + 1-\|x\|^p \big)^{\frac{1}{p}}
\end{eqnarray*}
Therefore, we deduce that $\|Tx\| \geq \|x\|$, which finishes the proof.
\end{proof}
The lack of $\tau_m$-compactness of the unit ball of $L_p$ forces us to consider pairs $(X,L_q)$ where $X$ is a subspace of $L_p$ whose unit ball $B_X$ is $\tau_m$-compact. Let us point out that, given $X\subset L_p$, $1<p<\infty$, the following statements are equivalent:
\begin{enumerate}[i)]
\item $B_X$ is $\tau_m$-compact;
\item $X$ embeds almost isometrically into $\ell_p$;
\item $B_X$ is compact for the topology inherited by the $L_1$-norm.
\end{enumerate}
Indeed, it is clear that iii)$\mathbb{R}ightarrow$ i). i)$\mathbb{R}ightarrow$ ii) follows from the following well-known facts for a finite measure space $(\Omega,\Sigma,\mu)$:
\begin{itemize}
\item If $H \subset L_p(\mu)$ is bounded, then it is uniformly integrable as a subset of $L_1(\mu)$ (this is a well-known, easy application of Holder’s inequality).
\item If $H \subset L_1(\mu)$ is uniformly integrable and relatively compact in measure, then it is relatively norm compact in $L_1(\mu)$. Indeed, this is an immediate application of Vitali’s convergence theorem.
\end{itemize} Finally, it is proved in \cite{KW} that ii) and iii) are equivalent.
\begin{corollary}
Let $1<p<q<\infty$. If $X$ is a subspace of $L_p$ such that $B_X$ is $\tau_m$-compact, then the pair $(X,L_q)$ has the $\tau_m$-to-$\tau_m$MP.
\end{corollary}
In the case of almost everywhere convergence, a related result is obtained from the celebrated Br\'ezis-Lieb lemma, which says that if $(f_n)\subset L_p$ is bounded and converges a.e. to $f$, then $\lim_n\norm{f_n}^p_p-\norm{f-f_n}_p^p=\norm{f}_p^p$. As a consequence, if $1\leq p\leq q<\infty$ and $T\in \mathcal L(L_p,L_q)$ admits a maximizing sequence $(f_n)$ with $f_n\to f$ a.e., $f\neq 0$, and $Tf_n\to Tf$ a.e., then $T$ attains its norm at $f$ (see \cite{BL}).
Let us briefly discuss the case when $1<q<p<\infty$. For the pair $(\ell_p,\ell_q)$, this was handled by Pitt’s Theorem (or more generally by Theorem~\ref{TheoremCompactWMP}) saying that every operator from $\ell_p$ to $\ell_q$ is compact. This approach fails for $(L_p,L_q)$. Indeed,
H.~P.~Rosenthal characterized in \cite[Theorem~A.2]{Rosenthal69} when every operator from $L_p(\mu)$ to $L_q(\nu)$ is compact. That is never the case for $L_p[0,1]$ and $L_q[0,1]$. However, if $q<p$, $1\leq q<2$, then every operator from $L_p[0,1]$ to $\ell_q$ is compact, and if $q<p$ and $2<p$ then every operator from $\ell_p$ to $L_q[0,1]$ is compact.
\section*{Acknowledgments}
Part of this work was carried out during a visit of the two named authors in Murcia (Spain) in January 2020.
They are deeply grateful to Mat\'ias Raja and the ``Universidad de Murcia" for the hospitality and excellent working conditions there.
The authors would like to thank Richard M. Aron, Jos\'e Rodr\'iguez and Abraham Rueda Zoca for very useful conversations, and to the referee for the careful reading of the paper.
The first author is supported in part by the grants MTM2017-83262-C2-2-P and Fundaci\'on S\'eneca Regi\'on de Murcia 20906/PI/18 also supported by a postdoctoral grant from Fundaci\'on S\'eneca.
\end{document} |
\begin{document}
\def\forallyes{\mathcal{A}_{\text{yes}}}
\def\forallno{\mathcal{A}_{\text{no}}}
\defp{p}
\defq{q}
\def\foralllg{\textsc{Alg}}
\def\mathcal{V}_{\text{yes}}{\mathcal{V}_{\text{yes}}}
\def\mathcal{V}_{\text{no}}{\mathcal{V}_{\text{no}}}
\def\mathcal{B}_{\text{yes}}{\mathcal{B}_{\text{yes}}}
\def\mathcal{B}_{\text{no}}{\mathcal{B}_{\text{no}}}
\title{Sample-based high-dimensional convexity testing}
\begin{abstract}
In the problem of \varepsilonilonmph{high-dimensional convexity testing}, there is an unknown set $S \subseteq \R^n$ which is promised to be either convex or $\varepsilonilonps$-far from every convex body with respect to the standard multivariate normal distribution $\normal^n$. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set $S$ as possible.
In this work we consider \varepsilonilonmph{sample-based} testing algorithms, in which the testing algorithm only has access to labeled samples $(\bx,S(\bx))$ where each $\bx$ is independently drawn from $\normal^n$. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant $\varepsilonilonps$, our results show that the sample complexity of one-sided convexity testing is $2^{\tilde{\Theta}(n)}$ samples, while for two-sided convexity testing it is $2^{\tilde{\Theta}(\sqrt{n})}$.
\varepsilonilonnd{abstract}
\thispagestyle{empty}
\setcounter{page}{1}
\section{Introduction}
Over the past few decades the field of property testing has developed into a fertile area with many different branches of active research. Several distinct lines of work have studied the testability of various kinds of \varepsilonilonmph{high-dimensional} objects, including probability distributions (see e.g. \cite{BKR:04long,RubinfeldServedio:05,AAK+07,RX10,ACS10,BFRV11,AcharyaDK15}), Boolean functions (see e.g. \cite{BLR93,PRS02,Blaisstoc09,MORS:10,KMS15} and many other works), and various types of codes and algebraic objects (see e.g. \cite{AKKLRtit,GoldreichSudan06,KaufmanSudan08,BKSSZ10} and many other works). These efforts have collectively yielded significant insight into the abilities and limitations of efficient testing algorithms for such high-dimensional objects. A distinct line of work has focused on testing (mostly low-dimensional) \varepsilonilonmph{geometric properties}. Here too a considerable body of work has led to a good understanding of the testability of various low-dimensional geometric properties, see e.g. \cite{CSZ:00,CS:01,Raskhodnikova:03,BMR16icalp,BMR16socg,BMR16fsttcs}.
This paper is about a topic which lies at the intersection of the two general strands (high-dimensional property testing and geometric property testing) mentioned above: we study the problem of \varepsilonilonmph{high-dimensional convexity testing.} Convexity is a fundamental property which is intensively studied in high-dimensional geometry (see e.g. \cite{convex-geometry,Ball:intro-convex,Szarek06} and many other references) and has been studied in the property testing of images (the two-dimensional case) \cite{Raskhodnikova:03,BMR16icalp,BMR16socg,BMR16fsttcs}, but as we discuss in Section~\ref{sec:relatedwork} below, very little is known about high-dimensional convexity testing.
We consider $\R^n$ endowed with the standard normal distribution $\normal^n$ as our underlying space, so the distance $\dist(S,C)$ between two subsets $S,C \subseteq \R^n$ is $\Pr_{\bx \leftarrow \normal^n}[\bx \in S \bigtriangleup C]$, where $S\bigtriangleup C$ denotes their symmetric difference. The standard normal distribution is arguably one of the most natural, and certainly one of the most studied, distributions on $\R^n$. Several previous works have studied property testing over $\R^n$ with respect to the standard normal distribution, such as the work on testing halfspaces of \cite{MORS:10,BBBY12} and the work on testing surface area of \cite{KNOW:14,Neeman:14}.
\subsection{Our results}
In this paper we focus on \varepsilonilonmph{sample-based} testing algorithms for convexity. Such an algorithm has access to independent draws $(\bx,S(\bx)) \in \R^n \times \{0,1\},$ where $\bx$ is drawn from $\normal^n$ and $S \subseteq \R^n$ is the unknown set being tested for convexity (so in particular the algorithm cannot select points to be queried) with $S(\bx)=1$ if $\bx\in S$.
We say such an algorithm is an \varepsilonilonmph{$\varepsilonilonps$-tester for convexity} if it accepts $S$ with probability at least $2/3$
when $S$ is convex and rejects with probability at least $2/3$ when it is $\varepsilonilonps$-far from convex,
i.e., $\dist(S,C)\ge \varepsilonilonps$ for all convex sets $C\subseteq \R^n$.
The model of sample-based testing was originally introduced
by Goldreich, Goldwasser, and Ron almost two decades ago \cite{GGR98}, where it was referred to as ``passive testing;'' it has received significant attention over the years \cite{KR00,GGLRS,BBBY12,GoldreichRon16}, with an uptick in research activity in this model over just the past year or so \cite{AHW16, BlaisYoshida16,BMR16icalp,BMR16socg,BMR16fsttcs}.
We consider sample-based testers for convexity that are allowed both one-sided (i.e., the
algorithm always accepts $S$ when it is convex) and two-sided error. In each case, for constant $\varepsilonilonps>0$ we give nearly matching upper and lower bounds on sample complexity. Our
results are as follows:
\begin{theorem} [One-sided lower bound] \label{thm:1slb}
\red{Any one-sided sample-based algorithm that is an $\varepsilonilonps$-tester for convexity over $\normal^n$
for some $\varepsilonilonps<1/2$
must use $2^{\Omega(n)}$ samples.}
\varepsilonilonnd{theorem}
\begin{theorem} [One-sided upper bound] \label{thm:1sub}
For any $\varepsilonilonps > 0$, there is a one-sided sample-based $\varepsilonilonps$-tester for convexity over
$\normal^n$ which uses $(n/\varepsilonilonps)^{O(n)}$ samples.
\varepsilonilonnd{theorem}
\begin{theorem} [Two-sided lower bound] \label{thm:2slb}
\red{There exists a positive constant $\varepsilonilonps_0$ such that any
two-sided sample-based algorithm that is an $\varepsilonilonps$-tester for convexity over $\normal^n$
for some $\varepsilonilonps\le \varepsilonilonps_0$ must use $2^{\Omega(\sqrt{n})}$ samples.}
\varepsilonilonnd{theorem}
\begin{theorem} [Two-sided upper bound] \label{thm:2sub}
For any $\varepsilonilonps > 0$, there is a two-sided sample-based $\varepsilonilonps$-tester for convexity over
$\normal^n$ which uses $n^{O(\sqrt{n}/\varepsilonilonps^2)}$ samples.
\varepsilonilonnd{theorem}
We will prove Theorems~\ref{thm:1slb}, \ref{thm:1sub}, \ref{thm:2slb} and \ref{thm:2sub} in Sections
\ref{sec:1slb}, \ref{sec:1sub}, \ref{sec:2slb} and~\ref{sec:2sub} respectively. These results are summarized above in Table~\ref{table:F2}.
\begin{table}[t!]
\renewcommand{1.6}{1.6}
\centering
\begin{tabular}{|m{5.5em}|m{13em}|m{5.7em}|}
\hline
Model & Sample complexity bound & Reference \\ \hline \hline
One-sided & $2^{\Omega(n)}$ samples (for $\varepsilonilonps < 1/2$) & Theorem~\ref{thm:1slb} \\ \hline
& $2^{O(n\log(n/\varepsilonilonps))}$ samples & Theorem~\ref{thm:1sub} \\ \hline \hline
Two-sided & $2^{\Omega(\sqrt{n})}$ samples (for $\varepsilonilonps < \varepsilonilonps_0$) & Theorem~\ref{thm:2slb} \\ \hline
& $2^{O({\sqrt{n}} \log(n)/\varepsilonilonps^2)}$ samples & Theorem~\ref{thm:2sub} \\ \hline
\varepsilonilonnd{tabular}
\caption{Sample complexity bounds for sample-based convexity testing. In line four,
${ \varepsilonilonps_0}>0$ is some absolute constant.}
\label{table:F2}
\varepsilonilonnd{table}
\subsection{Related work} \label{sec:relatedwork}
\noindent {\bf Convexity testing.}
As mentioned above, \cite{Raskhodnikova:03,BMR16fsttcs,BMR16socg,BMR16icalp} studied the testing
of $2$-dimensional convexity under the uniform distribution, either
within a compact body such as $[0,1]^2$ \cite{BMR16fsttcs,BMR16socg} or over
a discrete grid $[n]^2$ \cite{Raskhodnikova:03,BMR16icalp}.
The model of \cite{BMR16fsttcs,BMR16socg} is more closely related to ours:
\cite{BMR16socg} showed that $\Theta(\varepsilonilonps^{-4/3})$
samples are necessary and sufficient for one-sided sample-based testers, while
\cite{BMR16fsttcs} gave a one-sided general tester (which can make adaptive queries to
the unknown set) for $2$-dimensional convexity
with only $O(1/\varepsilonilonps)$ queries.
The only prior work that we are aware of that deals with testing high-dimensional convexity is that of \cite{Vempala}. However, the model
considered in \cite{Vempala} is different from ours in the following important aspects.
First, the goal of an algorithm in their model is to determine whether an unknown $S\subseteq \R^n$ is not convex
or is $\varepsilonilonps$-close to convex in the following sense: the (Euclidean) volume of $S\bigtriangleup C$, for some convex $C$, is at most
an $\varepsilonilonps$-fraction of the volume of $S$. Second, in their model an algorithm both can make membership
queries (to determine whether a given point $x$ belongs to $S$), and can receive
samples which are guaranteed to be drawn independently and uniformly at random from $S$.
The main result of \cite{Vempala} is an algorithm which uses $(cn/\varepsilonilonps)^n$ many random samples \varepsilonilonmph{drawn from $S$},
for some constant $c$,
and $\text{poly}(n)/\varepsilonilonps$ membership queries.
\paragraph{Sample-based testing.} A wide range of papers have studied sample-based testing from several different perspectives, including the recent works \cite{BMR16icalp,BMR16socg,BMR16fsttcs} which study sample-based testing of convexity over two-dimensional domains. In earlier work on sample-based testing, \cite{BBBY12} showed that the class of linear threshold functions can be tested to constant accuracy under $\normal^n$ with $\tilde{\Theta}(n^{1/2})$ samples drawn from $\normal^n$. (Note that a linear threshold function is a convex set of a very simple sort, as every convex set can be expressed as an intersection of (potentially infinitely many) linear threshold functions.) The work \cite{BBBY12} in fact gave a characterization of the sample complexity of (two-sided) sample-based testing, in terms of a combinatorial/probabilistic quantity called the ``passive testing dimension.'' This is a distribution-dependent quantity whose definition involves both the class being tested and the distribution from which samples are obtained; it is not \varepsilonilonmph{a priori} clear what the value of this quantity is for the class of convex subsets of $\R^n$ and the standard normal distribution $\normal^n$. Our upper and lower bounds (Theorems~\ref{thm:2sub} and~\ref{thm:2slb}) may be interpreted as giving bounds on the passive testing dimension of the class of convex sets in $\R^n$ with respect to the $\normal^n$ distribution.
\subsection{Our techniques}
\paragraph{One-sided lower bound.} Our one-sided lower bound has a simple proof using only elementary geometric and probabilistic arguments. It follows from the fact (see Lemma~\ref{lem:rand-all-on-hull}) that if $q=2^{\Theta(n)}$ many points are drawn independently from $\normal^n$, then with probability $1-o(1)$ no one of the points lies in the convex hull of the $q-1$ others. This can easily be shown to imply that more than $q$ samples are required (since given only $q$ samples, with probability $1-o(1)$ there is a convex set consistent with any labeling and thus a one-sided algorithm cannot reject).
\blue{
\paragraph{Two-sided lower bound.} At a high-level, the proof of our two-sided lower bound
uses the~following standard approach. We first define two distributions $\mathcal{D}yes$ and $\mathcal{D}no$
over sets in $\R^n$ such that
(i) $\mathcal{D}yes$ is a distribution over convex sets only, and (ii)
$\mathcal{D}no$ is a distribution such that
$\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ is
$\varepsilonilonps_0$-far from convex with probability at least $1-o(1)$ for some positive constant $\varepsilonilonps_0$.
We then show that every sample-based, $q$-query algorithm $A$ with $q=2^{0.01n}$
must have
\begin{equation}\label{hehe200}
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes;\hspace{0.05cm}\bx} \big[\text{$A $ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big]-
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no ;\hspace{0.05cm}\bx} \big[\text{$A $ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big]\le o(1),
\varepsilonilonnd{equation}
where $\bx$ denotes a sequence of $q$ points drawn from $\normal^n$ independently and
$(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$ denotes the $q$ labeled samples from $\mathcal{E}_{\textsf{no}}^*S$.
Theorem \ref{thm:2slb} follows directly from (\ref{hehe200}).
To draw a set $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$, we sample a sequence of $N=2^{\sqrt{n}}$ points
$\mathbf{y}_1,\ldots,\mathbf{y}_N$ from the sphere $S^{n-1}(r)$ of radius $r$ for some $r=\Theta(n^{1/4})$.
Each $\mathbf{y}_i$ defines a halfspace $\bh_i=\{x:x\cdot \mathbf{y}_i\le r^2\}$.
$\mathcal{E}_{\textsf{no}}^*S$ is then the intersection of all $\bh_i$'s. \red{(This is essentially a construction used by Nazarov \cite{Nazarov:03} to exhibit a convex set that has large Gaussian surface area, and used by \cite{KOS:07} to lower bound the sample complexity of learning convex sets under the Gaussian distribution.)} The most challenging part of the two-sided lower bound proof is to show that,
with $q$ points $\bx_1,\ldots,\bx_q\leftarrow \normal^n$,
the $q$ bits $\mathcal{E}_{\textsf{no}}^*S(\bx_1),\ldots,\mathcal{E}_{\textsf{no}}^*S(\bx_q)$ with $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$ are ``almost'' independent.
More formally, the $q$ bits $\mathcal{E}_{\textsf{no}}^*S(\bx_1),\ldots,\mathcal{E}_{\textsf{no}}^*S(\bx_q)$ with $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$
have $o(1)$-total variation distance from
$q$ independent bits with the $i$th bit drawn from the marginal distribution
of $\mathcal{E}_{\textsf{no}}^*S(\bx_i)$ as $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$.
On the other hand, it is relatively easy to define a distribution~$\mathcal{D}no$ that satisfies~(ii)
and at the same time, $\mathcal{E}_{\textsf{no}}^*S(\bx_1),\ldots,\mathcal{E}_{\textsf{no}}^*S(\bx_q)$ when $\mathcal{E}_{\textsf{no}}^*S\leftarrow\mathcal{D}no$ has $o(1)$-total variation
distance from the same product distribution.
(\ref{hehe200}) follows by combining the two parts.
}
\paragraph{Structural result.} Our algorithms rely on a new structural result which we establish for convex sets in $\R^n$. Roughly speaking, this result gives an upper bound on the Gaussian volume of the ``thickened surface'' of any bounded convex subset of $\R^n$; it is inspired by, and builds on, the classic result of Ball \cite{Ball:93} that upperbounds the Gaussian surface area of any convex subset of $\R^n$.
\paragraph{One-sided upper bound.} Our one-sided testing algorithm employs a ``gridding-based'' approach to decompose the relevant portion of $\R^n$ (namely, those points which are not too far from the origin) into a collection of disjoint cubes. It draws samples and identifies a subset of these cubes as a proxy for the ``thickened surface'' of the target set; by the structural result sketched above, if the Gaussian volume of this thickened surface is too high, then the one-sided algorithm can safely reject (as the target set cannot be convex). Otherwise the algorithm does random sampling to probe for points which are inside the convex hull of positive examples it has received but are labeled negative (there should be no such points if the target set is indeed convex, so if such a point is identified, the one-sided algorithm can safely reject). If no such points are identified, then the algorithm accepts.
\paragraph{Two-sided upper bound.} Finally, the main tool we use to obtain our two-sided testing algorithm is a \varepsilonilonmph{learning} algorithm for convex sets with respect to the normal distribution over $\R^n.$ The main result of \cite{KOS:07} is an (improper) algorithm which learns the class of all convex subsets of $\R^n$ to accuracy $\varepsilonilonps$ using $n^{O(\sqrt{n}/\varepsilonilonps^2)}$ independent samples from $\normal^n$. Using the structural result mentioned above, we show that this can be converted into a \varepsilonilonmph{proper} algorithm for learning convex sets under $\normal^n$, with essentially no increase in the sample complexity. Given this proper learning algorithm, a two-sided algorithm for testing convexity follows from the well-known result of \cite{GGR98} which shows that proper learning for a class of functions implies (two-sided) testability.
\section{Preliminaries and Notation} \label{sec:prelims}
\paragraph{Notation.}
We use boldfaced letters such as $\bx, \boldf,\bA$, etc. to denote random variables (which may be real-valued, vector-valued, function-valued, set-valued, etc; the intended type will be clear from the context).
We write ``$\bx \leftarrow \calD$'' to indicate that the random variable $\bx$ is distributed according to probability distribution $\calD.$ Given $a,b,c\in \R$ we use $a=b\pm c$ to indicate that $b-c\le a\le b+c$.
\paragraph{Geometry.}
For $r >0$, we write $S^{n-1}(r)$ to denote the origin-centered sphere of radius $r$ in $\R^n$
and $\mathrm{Ball}(r)$ to denote the origin-centered ball of radius $r$ in $\R^n$, i.e.,
$$
S^{n-1}(r) = \big\{x \in \R^n: \|x\|=r\big\}\quad\text{and}\quad
\mathrm{Ball}(r) = \big\{x \in \R^n : \|x\| \leq r\big\},
$$
where $\|x\|$ denotes the $\varepsilonilonll_2$-norm $\|\cdot \|_2$ of $x\in \R^n$.
We also write $S^{n-1}$ for the unit sphere $S^{n-1}(1)$.
Recall that a set $C \subseteq \R^n$ is convex if $x,y \in C$ implies $\alpha\hspace{0.03cm}x + (1-\alpha) y \in C$ for all $\alpha\in [0,1].$
We write $\calC_\mathrm{convex}$ to denote the class of all convex sets in $\R^n.$
Recall that convex sets are Lebesgue measurable.
Given a set $C \subseteq \R^n$ we write $\Conv(C)$ to denote the convex hull of $C$.
For sets $A,B \subseteq \R^n$, we write $A + B$ to denote the Minkowski sum $\{a + b: a \in A\ \text{and}\ b \in B\}.$ For a set $A \subseteq \R^n$ and $r > 0$ we write $rA$ to denote the set $\{ra : a \in A\}$.
Given a point $a$ and a set $B\subseteq \R^n$, we use $a+B$ and $B-a$ to denote
$\{a\}+B$ and $B+\{-a\}$ for convenience. For a convex set $C$,
we write $\partial C$ to denote its \varepsilonilonmph{boundary}, i.e. the set of points $x \in \R^n$ such that for all $\partial ta > 0$,
the set $x + \mathrm{Ball}(\partial ta)$ contains at least one point in $C$ and at least one point outside $C$.
\paragraph{Probability.}
We use $\normal^n$ to denote the standard $n$-dimensional Gaussian distribution with zero mean and identity covariance matrix. We also recall that the probability density function for the one-dimensional Gaussian distribution is $$\varphi(x) = {\frac 1 {\sqrt{2 \pi}}}\cdot \varepsilonilonxp(-x^2/2).$$
Sometimes we denote $\normal^n$ by ${\mathcal{N}(0,1)^n}s$ for convenience.
The squared norm $\|\bx\|^2$ of $\bx \leftarrow \normal^n$ is distributed according to the chi-squared distribution $\chi_n^2$
with $n$ degrees of freedom. The following tail bound for $\chi_n^2$ (see \cite{Johnstone01}) will be useful:
\begin{lemma} [Tail bound for the chi-squared distribution] \label{lem:johnstone}\label{johnstone}
Let $\bX \leftarrow \chi_n^2$.
Then we have
$$\Pr\big[|\bX-n| \geq tn\big] \leq e^{-(3/16)nt^2},\quad\text{for all $t \in [0, 1/2)$.}$$
\varepsilonilonnd{lemma}
All target sets $S\subseteq\R^n$ to be tested for convexity are assumed to be
Lebesgue measurable and we write $\Vol(S)$ to denote $\Pr_{\bx \leftarrow {\mathcal{N}(0,1)^n}s}[\bx \in S]$, the \varepsilonilonmph{Gaussian volume} of $S\subseteq \R^n$. Given two Lebesgue measurable subsets $S,C \subseteq \R^n$, we view $\Vol(S \bigtriangleup C)$ as the \varepsilonilonmph{distance} between $S$ and $C$, where $S\bigtriangleup C$ is the symmetric
difference of $S$ and $C$.
Given $S\subseteq \R^n$, we abuse the notation and use $S$ to denote the
indicator function of the set, so we may write ``$S(x)=1$'' or ``$x \in S$'' to mean the same thing.
We say that a subset $\calC$ of $\calC_\mathrm{convex}$ is a \varepsilonilonmph{$\tau$-cover of $\calC_\mathrm{convex}$} if for every $C \in \calC_\mathrm{convex}$, there exists a set $C' \in \calC$ such that $\Vol(C \bigtriangleup C') \leq \tau.$
\red{Given a convex set $C$ and a real number $h>0$, we let
$C_h$ denote the set of points in $\R^n$ whose distance from $C$ do not exceed $h$.
We recall the following theorem of Ball \cite{Ball:93} (also see \cite{Nazarov:03}).
\begin{theorem}[\cite{Ball:93}]\label{balllemma}
For any convex set $C\subseteq \R^n$ and $h>0$, we have
$$
\frac{\Vol(C_h\setminus C)}{h}\le 4\hspace{0.03cm}n^{1/4}.
$$
\varepsilonilonnd{theorem}
}
\paragraph{Sample-based property testing.}
Given a point $x\in \R^n$, we refer to $(x,S(x))\in \R^n\times \{0,1\}$ as a
\varepsilonilonmph{labeled sample} from a set $S\subseteq \R^n$.
A \varepsilonilonmph{sample-based testing algorithm for convexity} is a randomized algorithm which is given
as input an accuracy parameter $\varepsilonilonps>0$ and
access to an oracle that, each time it is invoked, generates a labeled sample $(\bx,S(\bx))$ from the unknown (Lebesgue measurable) \varepsilonilonmph{target set}
$S \subseteq \R^n$ with $\bx$ drawn independently each time from ${\mathcal{N}(0,1)^n}$.
When run with any Lebesgue measurable $S \subseteq \R^n,$ such an algorithm must output ``accept'' with probability at least 2/3 (over the draws it gets from the oracle and its own internal randomness) if $S \in \calC_\mathrm{convex}$ and must output ``reject'' with probability at least $2/3$ if $S$ is $\varepsilonilonps$-\varepsilonilonmph{far} from being convex, meaning that for every $C \in \calC_\mathrm{convex}$ it is the case that $\Vol(S \bigtriangleup C) \geq \varepsilonilonps$. (We also refer to an algorithm as an $\varepsilonilonps$-tester for convexity
if it works for a specific accuracy parameter $\varepsilonilonps$.) Such a testing algorithm is said to be \varepsilonilonmph{one-sided} if whenever it is run on a convex set $S$ it always outputs ``accept;'' equivalently, such an algorithm can only output ``reject'' if the labeled samples it receives are not consistent with any convex set. A testing algorithm which is not one-sided is said to be \varepsilonilonmph{two-sided}.
Throughout the rest of the paper we reserve the symbol $S$ to denote the unknown target set (a measurable subset of $\R^n$) that is being tested for convexity. If $S(x) = 1$ then we say that $x$ is a \varepsilonilonmph{positive point}, and if $S(x)=0$ we say $x$ is a \varepsilonilonmph{negative point}.
Given a finite set $T$ of labeled samples $(x,b)$ with $x\in \R^n$ and $b \in \{0,1\}$,
we say $x$ is a \varepsilonilonmph{positive} point in $T$ if $(x,1)\in T$ and is a \varepsilonilonmph{negative} point
in $T$ if $(x,0)\in T$.
We use
$T^+$ to denote the set of positive points $\{x: (x,1) \in T\}$, and $T^-$ to denote the
set of negative points $\{x: (x,0) \in T\}.$
\section{A useful structural result: Bounding the volume of the\\ thickened boundary of bounded convex bodies} \label{sec:structural}
For a \varepsilonilonmph{bounded} convex set $C$ in $\R^n$ (i.e., $\sup_{c\in C}\|c\|\le K$ for some real $K$) we may view $\partial C + \mathrm{Ball}(\alpha)$ as the ``$\alpha$-\varepsilonilonmph{thickened boundary}'' of $C$. In this section, we use Theorem~\ref{balllemma} of \cite{Ball:93} to give an upper bound on the volume of the $\alpha$-thickened boundary of such a set:
\begin{theorem}\label{thm:surfacevolume}
If $C\subset \R^n$ is convex and $\sup_{c\in C} \|c\| \leq K$ for some $K >1$, then we have
$$\Vol\big(\partial C + \mathrm{Ball}(\alpha)\big) \leq 20\hspace{0.03cm} n^{\red{5/8}}\hspace{0.03cm}K \sqrt{\alpha},\quad\text{\red{for any $0 < \alpha < n^{-3/4}$.}}$$
\varepsilonilonnd{theorem}
Having such a bound will be useful to us in two different contexts. First, it plays an important role in the proof of correctness of our one-sided algorithm for testing convexity (see Section~\ref{sec:1sub}). Second, as an easy consequence of the theorem, we get an algorithm which, for any $\tau>0$, constructs a $\tau$-cover of $\calC_\mathrm{convex}$ (this is Corollary~\ref{cor:epscover}, which we defer to later as its proof employs a ``gridding'' argument which we introduce in Section~\ref{sec:1sub}). This cover construction algorithm plays an important role in our two-sided algorithm for testing convexity (see Section~\ref{sec:2sub}).
\subsection{Proof of Theorem~\ref{thm:surfacevolume}}
Let $C\subset \R^n$ be a bounded convex set that satisfies $\sup_{c\in C} \|c\| \leq K$ for some $K >1$.
The proof has two cases and uses Lemmas
\ref{lem:noball}, \ref{lem:small}, and \ref{lem:large} to be proved later.
\noindent {\bf Case I: } $C$ contains no ball of radius $\rho := \sqrt{\alpha}/n^{3/8}$. In this case we have
\begin{align*}
\Vol(\partial C + \mathrm{Ball}(\alpha)) \leq \Vol(C + \mathrm{Ball}(\alpha))
& \leq 2(n\rho + \alpha) \tag*{(Lemma~\ref{lem:noball})}\\
& \leq 3\hspace{0.03cm}n^{5/8}\hspace{0.03cm}\sqrt{\alpha} \tag*{\red{(using $\alpha < n^{-3/4}$)}}\\
& < 20 \hspace{0.03cm}n^{5/8}\hspace{0.03cm}K\hspace{0.03cm}\sqrt{\alpha} \tag*{(using $K > 1$)}
\varepsilonilonnd{align*}
\noindent {\bf Case II:} $C$ contains some ball of radius $\rho$. We let $z^*$ be the center of
such a ball and let $D = \partial C +$ $\mathrm{Ball}(\alpha).$
To upperbound $\Vol(D)$, we define a set that contains $D$ and then upperbound its volume.
To this end, we first shift $C$ to get $C'=C-z^*$ (so that the ball of radius $\rho$ is now centered
at the origin).
By triangle inequality we have $\sup_{c\in C'} \|c\|\le 2K$.
Let $\beta = n^{3/8}\sqrt{\alpha} = \alpha/\rho$,
and observe that since $\alpha<n^{-3/4}$ we have $\beta < 1.$
Let $D'=D-z^*=\partial C'+\mathrm{Ball}(\alpha)$. By Lemma~\ref{lem:small}, we have
$$C_0' := (1-\beta)C'=(1-\beta)(C-z^*)$$
contains no point of $D'$,
and then \red{by Lemma~\ref{lem:large} the set
$C_1' := (C_0')_h$ with $h={4}\beta K + \alpha$ contains~all of $D'$.}\footnote{Recall that $(C_0')_h$
is the set of all points that have distance at most $h$ to $C_0'$. Also note that
the coefficient of $\beta K$ in our choice of $h$ is $4$ instead of $2$ since we have $\sup_{c\in C'}\|c\|\le 2K$ instead of $K$.}
As a result, $D'\subseteq C_1'\setminus C_0'$ and it suffices to upperbound
$\Vol(z^*+C_1'\setminus C_0')$, which is at most $4hn^{1/4}$ by Theorem \ref{balllemma}
(since $C_0'$ is convex).
Combining everything together, we have
\begin{align*}
\Vol(D) \leq \Vol(z^*+ C_1' \setminus C_0') \le
(4 \beta K + \alpha)\hspace{0.03cm} (4n^{1/4} ) \leq
20\hspace{0.03cm}n^{5/8}\hspace{0.03cm}K\hspace{0.03cm}\sqrt{\alpha}.\varepsilonilonnd{align*}
(again using $K>1$ and $\alpha < n^{-3/4}$ for the last inequality). \qed
It remains to prove Lemmas~\ref{lem:noball}, \ref{lem:small}, and \ref{lem:large}. We prove these lemmas in Appendix~\ref{ap:lemmas}.
\section{One-sided upper bound: Proof of Theorem~\ref{thm:1sub}} \label{sec:1sub}
Recall Theorem~\ref{thm:1sub}:
\begin{reptheorem}{thm:1sub}
For any $\varepsilonilonps > 0$, there is a one-sided sample-based $\varepsilonilonps$-tester for convexity over
$\normal^n$ which uses $(n/\varepsilonilonps)^{O(n)}$ samples.
\varepsilonilonnd{reptheorem}
In Section~\ref{sec:setup} we show that it suffices to test convex bodies contained in a large ball $B$ centered at the origin (rather than all of $\R^n$) and give some useful preliminaries. \red{Section~\ref{sec:bound-vol} then builds~on Theorem~\ref{thm:surfacevolume} (the upper bound on the volume of the ``thickened boundary'' of any \blue{bounded} convex body) to give an upper bound, in the case that $S$ is convex \blue{and contained in $B$}, on the total volume of certain ``boundary cubes'' (defined in Section~\ref{sec:setup}).} In Section~\ref{sec:alg} we present the one-sided testing algorithm and establish its correctness, thus proving Theorem~\ref{thm:1sub}.
\subsection{Setup} \label{sec:setup}
Let $n'$ be the following parameter (that depends on both $n$ and $\varepsilonilonps$):
$$
n':= \left(n + 4\sqrt{n\ln(4/\varepsilonilonps)}\right)^{1/2}.
$$
Let $\calC'_\mathrm{convex}$ denote the set of convex bodies in $\R^n$ that are contained in $\mathrm{Ball}(n')$, equivalently,
\[
\calC'_\mathrm{convex} = \big\{ C \cap \mathrm{Ball}(n'): C \in \calC_\mathrm{convex}\big\}.
\]
We prove the following claim that helps us focus on testing of $\calC'_\mathrm{convex}$ instead $\calC_\mathrm{convex}$.
\begin{claim} \label{claim:reduction}
Suppose that there is a one-sided sample-based $\varepsilonilonps$-testing algorithm $A'$ which, given any Lebesgue measurable target set $S$ contained in $\mathrm{Ball}(n')$, uses $(n/\varepsilonilonps)^{O(n)}$ samples drawn from $\normal^n$ to test whether $S \in \calC'_\mathrm{convex}$ versus $S$ is $\varepsilonilonps$-far from $\calC'_\mathrm{convex}$. Then this implies Theorem~\ref{thm:1sub}.
\varepsilonilonnd{claim}
\begin{proof}
Given $A'$ for $\calC'_\mathrm{convex}$, we consider an algorithm $A$ which works as follows to test whether~an arbitrary Lebesgue measurable subset $S$ of $\R^n$ is convex or $\varepsilonilonps$-far from $\calC_\mathrm{convex}$: algorithm $A$ runs $A'$ with parameter $\varepsilonilonps/2$, but with the following modification: each time $A'$ receives from the oracle a labeled sample $(x,b)$ with $x \notin \mathrm{Ball}(n')$, it replaces the label $b$ with $0$ and gives the modified labeled sample to $A'$. When the run of $A'$ is complete $A$ returns the output of $A'$.
If $S \subseteq \R^n$ is the target set, then it is clear that the above modification results in running $A'$ on $S \cap \mathrm{Ball}(n')$. If $S$ is convex, then $S \cap \mathrm{Ball}(n')$ is also convex. As $A'$ commits only one-sided error, it will always output ``accept,'' and hence so will $A$. On the other hand, suppose that $S$ is $\varepsilonilonps$-far from $\calC_\mathrm{convex}$. We claim that
$\Vol(\mathrm{Ball}(n')) \geq 1 - \varepsilonilonps/4$ (this will be shown below); given this claim, it must be the case that $S \cap \mathrm{Ball}(n')$ is at least $(3\varepsilonilonps/4)$-far from $\calC_\mathrm{convex}$ and at least $(3\varepsilonilonps/4)$-far from $\calC'_\mathrm{convex}$ as well. Consequently $A'$ will output ``reject'' with probability at least $2/3$, and hence so will $A$.
To bound
$\Vol(\mathrm{Ball}(n'))$, observe that it is the probability that an $\bx \leftarrow\normal^n$ has
$$\|\bx\|^2\leq n + 4\sqrt{n\ln(4/\varepsilonilonps)}.$$
It follows from Lemma~\ref{lem:johnstone} that the
probability is at least $1-\varepsilonilonps/4$ as claimed.
\varepsilonilonnd{proof}
Given Claim~\ref{claim:reduction}, it suffices to prove the following slight variant of Theorem~\ref{thm:1sub}:
\begin{theorem} \label{thm:1submodif}
There is a one-sided sample-based $\varepsilonilonps$-testing algorithm $A'$ which, given any Lebesgue measurable target set $S$ contained in $\mathrm{Ball}(n')$, uses $(n/\varepsilonilonps)^{O(n)}$ samples from $\normal^n$ to test whether $S \in \calC'_\mathrm{convex}$ versus $S$ is $\varepsilonilonps$-far from $\calC'_\mathrm{convex}$.
\varepsilonilonnd{theorem}
In the rest of this section we prove Theorem~\ref{thm:1submodif}. We start with some terminology and concepts that we use in the description and analysis of our algorithm. \red{Some of the notions that we introduce below, such as the notions of ``boundary'' cubes and ``internal'' cubes, are inspired by related notions that arise in earlier works such as \cite{Kern,Raskhodnikova:03}.}
Fix $\varepsilonilonll := \varepsilonilonps^3/n^4$ in the rest of the section, and let $\mathrm{Cube}_0$ denote the following set
\[
\mathrm{Cube}_0 := [-\varepsilonilonll/2,\varepsilonilonll/2)^{n} \subset \R^n
\]
of side length $\varepsilonilonll$ that is centered at the origin. We say that a \varepsilonilonmph{cube} is a subset of $\R^n$ of the form $\mathrm{Cube}_0 + \varepsilonilonll \cdot (i_1,\dots,i_n)$, where each $i_j \in \Z$, which contains at least one point of $\mathrm{Ball}(2n').$
We use $\mathrm{Cube}Set$ to denote the set of all such cubes.
It is easy to see that
\[
\mathrm{Ball}(n') \subset \text{union of all cubes in $\mathrm{Cube}Set$} \subset \mathrm{Ball}(2n' + \varepsilonilonll \sqrt{n}) \subset \mathrm{Ball}(3n').
\]
Fix an $S \subseteq \mathrm{Ball}(n')$ as the target set being tested for membership in $\calC'_\mathrm{convex}.$ Additionally fix a finite set $T=\{(x^1,S(x^1)),\dots,(x^M,S(x^M))\}$ of labeled samples according to $S$, for some~positive integer $M$. (The set $T$ will correspond to the set of labeled samples that the testing algorithm receives.) We classify cubes in the $\mathrm{Cube}Set$ based on $T$ in the following way:
\begin{figure}
\centering
\includegraphics[width=7cm]{bb-fig.pdf}
\caption{A 2D example of the different types of cubes induced by a set of labeled samples.~The target set $S$ is a disk, and the solid and hollow dots are positive and negative samples, respectively. The hollow, hatched, and shaded boxes are external, boundary, and internal cubes, respectively. }
\label{fig-boxes}
\varepsilonilonnd{figure}
\begin{flushleft}
\begin{itemize}
\item A cube \(\mathrm{Cube}\) is said to be an \varepsilonilonmph{external cube} if $\mathrm{Cube} \cap T^+ = \varepsilonilonmptyset$ (i.e., no positive point of $T$ lies in $\mathrm{Cube}$). We let $EC$ denote the union of all the external cubes.
\item Any cube which is not an external cube (equivalently, any cube that contains at least one positive point of $T$) is said to be a \varepsilonilonmph{positive cube}.
\item We say that two cubes $\mathrm{Cube},\mathrm{Cube}'$ are \varepsilonilonmph{adjacent} if for any $\kappa>0$ there exist $x \in \mathrm{Cube}$ and $y \in \mathrm{Cube}'$ that have Euclidean distance at most $\kappa$ (in other words, two cubes are adjacent if their closure ``touch anywhere, even only at a vertex;'' note that each cube is adjacent to itself). If a cube is both (i) a positive cube and (ii) is adjacent to a cube (including itself) that contains at least one negative point of $T$, then we call it a \varepsilonilonmph{boundary cube}. We use $BC$ to denote the union of all boundary cubes.
\item We say that a positive cube which is not a boundary cube is an \varepsilonilonmph{internal cube}. (Equivalently, a cube is internal if and only if \red{it contains at least one positive point} and all the points in $T$ that are contained in any of its adjacent cubes, including itself, are positive.) We use $IC$ to denote the union of all internal cubes.
\varepsilonilonnd{itemize}
\varepsilonilonnd{flushleft}
We note that since each cube is either external, internal, or boundary, the set $\mathrm{Ball}(n')$ is contained in the (disjoint) union of $EC,BC$ and $IC.$ Figure~\ref{fig-boxes} illustrates the different types of cubes.
We will use the following useful property of internal cubes:
\begin{lemma}\label{lem:i} Suppose a finite set of labeled samples $T$ is such that every cube in $\mathrm{Cube}Set$ contains at least one point of $T$. Then every internal cube is contained in $\Conv(T^+).$
\varepsilonilonnd{lemma}
The lemma is a direct consequence of the following claim by setting $H=T^+$:
\begin{claim} \label{claim:internal}
Let $H \subseteq \R^n$ be any set that contains at least one point in each cube that is adjacent to $\mathrm{Cube}_0$. Then $\mathrm{Cube}_0$ is contained in $\Conv(H)$.
\varepsilonilonnd{claim}
\begin{proof}
We prove the claim by induction on the dimension $n$. When $n=1$ the claim is trivial since $\mathrm{Cube}_0$ is simply the interval $[-\varepsilonilonll/2,\varepsilonilonll/2)$ and by assumption, there is at least one point of $H$ in $[-3\varepsilonilonll/2,-\varepsilonilonll/2)$ and at least one point of $H$ in $[\varepsilonilonll/2,3\varepsilonilonll/2).$
For $n > 1$, let $P = \{p \in H \mid p_n \geq {\varepsilonilonll}/{2}\}$ and $P' = \{p' \in H \mid p_n' \leq {-\varepsilonilonll}/{2}\}$ be two subsets of $H$. Intuitively, the convex hulls of $P$ and $P'$ ``cover'' $\mathrm{Cube}_0$ on both sides (by induction), so the convex hull of their union will contain the whole $\mathrm{Cube}_0$. More formally, let $x$ be any point in $\mathrm{Cube}_0$. By projecting $P,P'$ and $x$ onto the first $n-1$ dimensions and using the inductive hypothesis\footnote{Observe that after projecting out the last coordinate, the assumed property of $H$ (that it has at least one sample point in each adjacent cube) will still hold in $n-1$ dimensions.}, we can find points $y \in \Conv(P)$ and $y' \in \Conv(P')$ such that $y_i = y_i' = x_i$ for all $i\in [n-1]$. Since we~have $p_n \geq 1/{2}$ and $p_n' \leq -1/{2}$ for all $p \in P$ and $p' \in P'$, respectively, it follows directly that $y_n \geq {1}/{2}$ and $y_n' \leq -{1}/{2}$. As $x\in \mathrm{Cube}_0$, $x$ is on the line segment between $y$ and $y'$ and thus is in the convex hull of $H$. Hence all of $\mathrm{Cube}_0$ is contained in $\Conv(H)$.
\varepsilonilonnd{proof}
\subsection{Bounding the total volume of boundary cubes} \label{sec:bound-vol}
Before presenting our algorithm we record the following useful corollary of Theorem~\ref{thm:surfacevolume}, which allows the one-sided tester to reject bodies as non-convex if it detects too much volume in boundary cubes.
\red{(Note that we do not assume below that $T$ satisfies the condition of Lemma \ref{lem:i}, i.e.,
that $T$ has at least one point in each cube in $\mathrm{Cube}Set$, though this will be
the case when we use it later.)}
\begin{corollary}\label{cor:smallconvbdry}
Let $S$ be a convex set in $\calC'_\mathrm{convex}$ and
$T$ be any finite set of labeled samples according to $S$,
which defines sets $EC,IC$ and $BC$ as discussed earlier. Then we have
$$\Vol(BC) \leq 20\hspace{0.03cm}n^{5/8}\hspace{0.03cm}n'\hspace{0.03cm}\sqrt{2\varepsilonilonll\sqrt{n}} =\red{o(\varepsilonilonps)}.$$
\varepsilonilonnd{corollary}
\begin{proof}
Let $\mathrm{Cube}$ be a boundary cube. Then by definition, there is a positive point of $T$ (call~it~$t$) in $\mathrm{Cube}$, and there is a $\mathrm{Cube}'$ adjacent to $\mathrm{Cube}$ that contains a negative point of $T$ (call it $t'$).~It follows that there must be a boundary point of $\partial S$ (call it $t^*$) in the segment between $t$ and $t'$, and we have $\mathrm{Cube} \in t^* + \mathrm{Ball}(2 \varepsilonilonll \sqrt{n}).$ It follows that $BC \subseteq \partial S + \mathrm{Ball}(2 \varepsilonilonll \sqrt{n})$, and hence
$$\Vol(BC) \leq \Vol\big( \partial S + \mathrm{Ball}(2 \varepsilonilonll \sqrt{n})\big)\le 20 \hspace{0.03cm}n^{5/8}\hspace{0.03cm} n' \sqrt{2\varepsilonilonll\sqrt{n}} =\red{o(\varepsilonilonps)}$$ by Theorem~\ref{thm:surfacevolume} (and using $\varepsilonilonll\sqrt{n}\ll n^{-3/4}$ by our choice of $\varepsilonilonll=\varepsilonilonps^3/n^4$).
\varepsilonilonnd{proof}
\subsection{The one-sided testing algorithm} \label{sec:alg}
Now we describe and analyze
the one-sided testing algorithm $A'$ mentioned in Theorem~\ref{thm:1submodif}.
Algorithm $A'$ works by performing $O(1/\varepsilonilonps)$ independent runs of the algorithm $A^*$,
which we describe in Figure \ref{fig:main}. If any of the runs of $A^*$ output ``reject'' then algorithm $A'$ outputs ``reject,'' and otherwise it outputs ``accept.''
\begin{figure}[t!]
\begin{framed}
\noindent {\bf Algorithm $A^*$:} Given access to independent draws $(\bx,S(\bx))$ where $\bx \leftarrow \normal^n$ and the\\ target set $S$ is a Lebesgue measurable set that \red{is contained in $\mathrm{Ball}(n')$}.
\begin{flushleft}\begin{enumerate}
\item Draw a set $\bT$ of $s:= (n/\varepsilonilonps)^{O(n)}$ labeled samples $(\bx,S(\bx))$, where each $\bx \leftarrow \normal^n$.
\item If any cube does not contain a point of $\bT$, then halt and output ``accept.''
\item If $\Vol(BC)\ge\varepsilonilonps/4$ (the volume of the union of boundary cubes),
halt and output ``reject.''
\item Define $\bI \subseteq \R^n$ to be $\Conv(\bT^+)$, the convex hull of all positive points in $\bT$.
\item Draw a single fresh labeled sample $(\mathbf{y},S(\mathbf{y}))$, where $\mathbf{y}\leftarrow \normal^n$. If $\mathbf{y} \in \bI$ but $S(\mathbf{y})=0$ then halt and output ``reject.'' Otherwise, halt and output ``accept.''
\varepsilonilonnd{enumerate}\varepsilonilonnd{flushleft}
\varepsilonilonnd{framed}
\caption{Description of the algorithm $A^*$}\label{fig:main}\varepsilonilonnd{figure}
In words, Algorithm~$A^*$ works as follows: first, in Step~1 it draws enough samples so that (with very high probability) it will receive at least one sample in each cube (if the low-probability event that this does not occur takes place, then the algorithm outputs ``accept'' since it can only reject if it is impossible for $S$ to be convex). If the region ``close to the boundary'' of $S$ (as measured by $\Vol(BC)$ in Step~3) is too large, then the set cannot be convex (by Corollary \ref{cor:smallconvbdry}) and the algorithm rejects. Finally, the algorithm checks a freshly drawn point; if this point is in the convex hull of the positive samples but is labeled negative, then the set cannot be convex and the algorithm rejects. Otherwise, the algorithm accepts.
To establish correctness and prove Theorem~\ref{thm:1submodif} we must show that (i) algorithm $A^*$ never rejects if the target set $S$ is a Lebesgue measurable set that belongs to $\calC'_\mathrm{convex}$, and (ii) if $S$ is $\varepsilonilonps$-far from $\calC'_\mathrm{convex}$ then algorithm $A^*$ rejects with probability at least $\Omega(\varepsilonilonps).$
Part (i) is trivial as $A^*$ only rejects if either
(a) $\Vol(BC) \ge \varepsilonilonps/4$ or (b) step~5 identifies a negative point in the convex hull of the positive points in $\bT$.
For both cases we conclude (using Corollary \ref{cor:smallconvbdry} for (a))
that $S\notin \calC'_\mathrm{convex}$.
For (ii) suppose that $S$ is $\varepsilonilonps$-far from $\calC'_\mathrm{convex}$. Let
$E$ be the following event (over the draw of $\bT$):
\begin{flushleft}\begin{quote}
Event $E$: Every cube in $\mathrm{Cube}Set$ contains at least one point of $\bT$ (so the\\ algorithm does not
accept in Step 2) and moreover, every $\mathrm{Cube}$ with
$$\frac{\Vol(\mathrm{Cube} \cap S)}{\Vol(\mathrm{Cube})} \geq \varepsilonilon/4$$
contains at least one positive point in $\bT$ and thus, is not external.
\varepsilonilonnd{quote}\varepsilonilonnd{flushleft}
It is easy to show that the probability mass of each cube in $\mathrm{Cube}Set$ is
at least $( \varepsilonilonps/n)^{O(n)}$ (since its volume is $(\varepsilonilonps/n)^{O(n)}$ and the density
function of the Gaussian is at least $(1/\varepsilonilonps)^{O(n)}$ using our~choice of $n'$),
it follows from a union bound over $\mathrm{Cube}Set$ that,
for a suitable choice of $s=(n/\varepsilonilonps)^{O(n)}$ (with a
large enough coefficient in the exponent),
$E$ occurs with probability $1-o(1)$.
Assuming that $E$ occurs, we show below that either $\Vol(BC)\ge \varepsilonilonps/4$ or $A^*$ rejects in Step 5
with probability $\Omega(\varepsilonilonps)$.
For this purpose, we assume below that both $E$ occurs and $\Vol(BC)<\varepsilonilonps/4$.
Note that the set $I$ is convex and is contained in $\mathrm{Ball}(n')$.
Thus it belongs to $\calC'_\mathrm{convex}$ and consequently $\Vol(I \bigtriangleup S) \geq \varepsilonilonps$
(since $S$ is assumed to be $\varepsilonilonps$-far from $\calC'_\mathrm{convex}$), which implies that
$$
\Vol(S\setminus I)+\Vol(I\setminus S)\ge \varepsilonilonps.
$$
It suffices to show that $\Vol(S \setminus I) \leq \varepsilonilon/2$, since $\Vol(I \setminus S)$
is exactly the probability that algorithm $A^*$ rejects in Step 5.
To see that $\Vol(S \setminus I) \leq \varepsilonilon/2$, observe that by Lemma~\ref{lem:i}, $\Vol(S \setminus I)$ is at most $\Vol(S \cap BC) + \Vol(S \cap EC)$.
On the one hand, $\Vol(S\cap BC)\le \Vol(BC)<\varepsilonilonps/4$ by assumption.
On the other hand, given the event $E$, every external cube has at most
$(\varepsilonilonps/4)$-fraction of its volume in $S$ and thus, $\Vol(S\cap EC)\le \varepsilonilonps/4$
(as the total volume of $EC$ is at most $1$).
Hence $\Vol(S\setminus I)\le \varepsilonilonps/2$.
This concludes the proof of Theorem~\ref{thm:1submodif}.
\section{Two-sided lower bound} \label{sec:2slb}
We recall Theorem \ref{thm:2slb}:
\begin{reptheorem}{thm:2slb}
\red{There exists a positive constant $\varepsilonilonps_0$ such that any
two-sided sample-based algorithm that is an $\varepsilonilonps$-tester for convexity over $\normal^n$
for some $\varepsilonilonps\le \varepsilonilonps_0$ must use $2^{\Omega(\sqrt{n})}$ samples.}
\varepsilonilonnd{reptheorem}
\newcommand{\mathcal{D}}{\mathcal{D}}
\newcommand{\mathrm{cap}}{\mathrm{cap}}
\newcommand{\mathrm{fsa}}{\mathrm{fsa}}
Let $q=2^{0.01\sqrt{n}}$ and let $\varepsilonilonps_0$ be a~positive constant to be specified later.
To prove Theorem \ref{thm:2slb}, we show that no sample-based, $q$-query (randomized) algorithm $A$ can achieve the following goal:
\begin{flushleft}\begin{quote}
Let $S\subset \R^n$ be a target set that is Lebesgue measurable.
Let $\bx_1,\ldots,\bx_q$ be a sequence of $q$ samples drawn from ${\mathcal{N}(0,1)^n}$.
Upon receiving $((\bx_i,S(\bx_i)):i\in [q])$,
$A$ accepts with probability at least $2/3$ when $S$ is convex
and rejects with probability at least $2/3$ when $S$ is $\varepsilonilonps_0$-far from convex.
\varepsilonilonnd{quote}\varepsilonilonnd{flushleft}
Recall that a pair $(x,b)$ with $x\in \R^n$ and $b\in \{0,1\}$ is a {labeled sample}.
Thus, a sample-based algorithm $A$ is simply a randomized map from a sequence of $q$ labeled samples to $\{\text{``accept'',``reject''}\}$.
\subsection{Proof Plan}
Assume for contradiction that there is a
$q$-query (randomized) algorithm $A$ that accomplishes the task above.
In Section \ref{sec:dist} we define two probability distributions $\mathcal{D}yes$ and $\mathcal{D}no$ such that
(1) $\mathcal{D}yes$ is a distribution over convex sets in $\R^n$ ($\mathcal{D}yes$
is a distribution over certain convex polytopes that are the intersection of many randomly drawn halfspaces), and (2)
$\mathcal{D}no$ is a probability distribution over sets in $\R^n$ that are Lebesgue measurable
($\mathcal{D}no$ is actually supported over a finite number of measurable sets in $\R^n$) such that
$\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ is $\varepsilonilonps_0$-far from convex with probability at least $1-o(1)$.
Given a sequence $x=(x_1,\ldots,x_q)$ of points, we abuse the notation and write
$$S(x)=(S(x_1),\ldots,S(x_q))$$
and use $(x,S(x))$ to denote the sequence of $q$ labeled samples
$(x_1,S(x_1)),\ldots,(x_q,S(x_q))$.
It then follows from our assumption on $A$ that
\begin{align*}
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes;\hspace{0.05cm}\bx \leftarrow ({\mathcal{N}(0,1)^n}s)^q} \big[\text{$A$ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big] &\ge 2/3\quad \text{and} \\[0.5ex]
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no ;\hspace{0.05cm}\bx \leftarrow ({\mathcal{N}(0,1)^n}s)^q} \big[\text{$A$ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big] &\le 1/3+o(1).
\varepsilonilonnd{align*}
where we use $\bx\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ to denote a sequence of $q$ points sampled independently from ${\mathcal{N}(0,1)^n}s$
and we usually skip the $\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ part in the subscript when it is clear from the context.
Since $A$ is~a mixture of deterministic algorithms,
there exists a deterministic sample-based, $q$-query algorithm $A'$ (equivalently,
a deterministic map from sequences of $q$ labeled samples to $\{\textsf{``Yes''}, \textsf{``No''}\}$) with
\begin{equation}\label{hehe2}
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes;\hspace{0.05cm}\bx} \big[\text{$A'$ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big]-
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no ;\hspace{0.05cm}\bx} \big[\text{$A'$ accepts $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$}\big]\ge 1/3-o(1).
\varepsilonilonnd{equation}
Let $\calE_{\text{yes}}es$ (or $\calE_{\text{no}}o$) be the distribution of $(\bx,\mathcal{E}_{\textsf{no}}^*S(\bx))$, where $\bx\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ and
$\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$ (or $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$, respectively). Both of them are distributions over sequences of $q$
labeled samples.
Then the LHS of (\ref{hehe2}), for any deterministic sample-based, $q$-query algorithm $A'$, is
at most the total variation distance between $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o$.
We prove the following key lemma, which leads to a contradiction.
\begin{lemma}\label{maintechlemma}
The total variation distance between $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o$ is $o(1)$.
\varepsilonilonnd{lemma}
To prove Lemma \ref{maintechlemma}, it is convenient for us to introduce a third distribution $\mathcal{E}_{\textsf{no}}^*$ over
sequences of $q$ labeled samples,
where $(\bx,\mathbf{b})\leftarrow \mathcal{E}_{\textsf{no}}^*$ is drawn by first sampling a sequence of $q$ points $\bx=(\bx_1,\ldots,\bx_q)$
from ${\mathcal{N}(0,1)^n}s$ independently and then for each $\bx_i$, its label $\mathbf{b}_i$ is set to be $1$ independently with
a probability that depends only on $\|\bx_i\|$
(see Section \ref{sec:dist}).
Lemma \ref{maintechlemma} follows from the following two lemmas by the triangle inequality.
\begin{lemma}\label{techlemma1}
The total variation distance between $\calE_{\text{no}}o$ and $\calE_{\text{no}}o^*$ is $o(1)$.
\varepsilonilonnd{lemma}
\begin{lemma}\label{techlemma2}
The total variation distance between $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o^*$ is $o(1)$.
\varepsilonilonnd{lemma}
The rest of the section is organized as follows.
We define the distributions $\mathcal{D}yes,\mathcal{D}no$ (which are used to define $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o$) as well as $\mathcal{E}_{\textsf{no}}^*$ in Section \ref{sec:dist}
and prove the necessary properties about $\mathcal{D}yes$ and $\mathcal{D}no$ as well as Lemma \ref{techlemma1}.
We prove Lemma \ref{techlemma2} in Sections \ref{finalsec} and \ref{hehefinal}.
\subsection{The Distributions}\label{sec:dist}
Let $r=\Theta(n^{1/4})$ be a parameter to be fixed later, and let $N=2^{\sqrt{n}}$.
We start with the definition of $\mathcal{D}yes$.
A random set $\mathcal{E}_{\textsf{no}}^*S\subset \R^n$ is drawn from $\mathcal{D}yes$ using the following procedure:
\begin{flushleft}\begin{enumerate}
\item We sample a sequence of $N$ points $\mathbf{y}_1,\ldots,\mathbf{y}_N$ from $\red{S^{n-1}(r)}$ independently and uniformly at random.
Each point $\mathbf{y}_i$ defines a halfspace $$\bh_i=\big\{x\in \R^n: x\cdot \mathbf{y}_i\le r^2\big\}.$$
\item The set $\mathcal{E}_{\textsf{no}}^*S$ is then the intersection of $\bh_i$, $i\in [N]$ (this is always nonempty as
indeed $\mathrm{Ball}(r)$ is contained in $\mathcal{E}_{\textsf{no}}^*S$).
\varepsilonilonnd{enumerate}\varepsilonilonnd{flushleft}
It is clear from the definition that $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$ is always a convex set.
Next we define $\red{\mathcal{E}_{\textsf{no}}^*}$ (instead of $\red{\mathcal{D}no}$), a distribution over \red{sequences of $q$} labeled samples $(\bx,\mathbf{b})$.
To this end, we use $\mathcal{D}yes$ to define a function \(\rho: \R_{\ge 0} \to [0, 1]\) as follows:
$$
\rho(t) = \Ppr_{\mathcal{E}_{\textsf{no}}^*S \leftarrow \mathcal{D}_\text{yes}}\Big[(t, 0, \ldots,0) \in \mathcal{E}_{\textsf{no}}^*S\Big].
$$
Due to the symmetry of $\mathcal{D}yes$ and ${\mathcal{N}(0,1)^n}s$, the value $\rho(t)$ is indeed the probability that a point $x\in \R^n$ at distance $t$
from the origin lies in $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$.
To draw a sequence of $q$ labeled samples $(\bx,\mathbf{b})\leftarrow \mathcal{E}_{\textsf{no}}^*$, we first independently draw $q$ random points $\bx_1,\dots,\bx_q \leftarrow {\mathcal{N}(0,1)^n}s$ and then independently set each $\mathbf{b}_i=1$ with probability $\rho(\|\bx_i\|)$ and $\mathbf{b}_i=0$ with probability $1-\rho(\|\bx_i\|)$.
Given $\mathcal{D}yes$ and $\red{\mathcal{E}_{\textsf{no}}^*}$,
Lemma \ref{techlemma2} shows that information-theoretically no sample-based algorithm
can distinguish a sequence of $q$ labeled samples $(\bx,\mathbf{b})$ with $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$,
$\bx\leftarrow ({\mathcal{N}(0,1)^n}s)^q$, and $\mathbf{b}=\mathcal{E}_{\textsf{no}}^*S(\bx)$
from a sequence of $q$ labeled samples drawn from $\red{\mathcal{E}_{\textsf{no}}^*}$.
While the marginal distribution of each labeled sample is the same for the two cases,
the former is generated in a correlated fashion using the underlying random convex $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$ while the latter
is generated independently.
Finally we define the distribution $\red{\mathcal{D}no}$, prove Lemma \ref{techlemma1}, and
show that a set drawn from $\red{\mathcal{D}no}$ is far from convex with high probability.
To define $\red{\mathcal{D}no}$, we let $M\ge 2^{\sqrt{n}}$ be a large enough integer to be specified later.
With $M$ fixed, we use
$$
0=t_0<t_1<\cdots<t_{M-1}<t_M=\red{2\sqrt{n}}
$$
to denote a sequence of numbers such that the origin-centered ball $\mathrm{Ball}({\red{2\sqrt{n}}})$ is partitioned into
$M$ \varepsilonilonmph{shells} $\mathrm{Ball}(t_i)\setminus \mathrm{Ball}(t_{i-1})$, $i\in [M]$,
and all the $M$ shells have the same probability mass under ${\mathcal{N}(0,1)^n}s$.
By spherical coordinates, it means that
the following integral takes the same value for all $i$:
\begin{equation}\label{tttt}
\int_{t_{i-1}}^{t_i} \phi(x,0,\ldots,0) x^{n-1} dx,
\varepsilonilonnd{equation}
where $\phi$ denotes the density function of ${\mathcal{N}(0,1)^n}s$.
We show below that when $M$ is large enough, we have
\begin{equation}\label{ofofof}
|\rho(x)-\rho(t_i)|\le 2^{-\sqrt{n}},
\varepsilonilonnd{equation}
for any $i\in [M]$ and any $x\in [t_{i-1},t_i]$.
We will fix such an $M$ and use it to define $\red{\mathcal{D}no}$. (Our results are not affected by the size of $M$ as a function of $n$; we only need it to be finite, given $n$.)
To show that (\ref{ofofof}) holds when $M$ is large enough, we need the continuity of the function $\rho$, which follows
directly from the explicit expression for $\rho$ given later in
(\ref{eq:rhodef}).
\begin{lemma}\label{continuity}
The function $\rho:\R_{\ge 0}\rightarrow [0,1]$ is continuous.
\varepsilonilonnd{lemma}
Since $\rho$ is continuous, it is continuous over $[0,\red{2\sqrt{n}}]$.
Since $[0,\red{2\sqrt{n}}]$ is compact,
$\rho$ is also uniformly continuous over $[0,\red{2\sqrt{n}}]$.
Also note that $\max_{i\in [M]} (t_{i}-t_{i-1})$ goes to $0$ as $M$ goes to $+\infty$.
It follows that (\ref{ofofof}) holds when $M$ is large enough.
With $M\ge 2^{\sqrt{n}}$ fixed,
a random set $\mathcal{E}_{\textsf{no}}^*S\leftarrow \red{\mathcal{D}no}$ is drawn as follows.
We start with $\mathcal{E}_{\textsf{no}}^*S=\varepsilonilonmptyset$ and for each $i\in [M]$,
we add the $i$th shell $\mathrm{Ball}(t_i)\setminus \mathrm{Ball}(t_{i-1})$ to $\mathcal{E}_{\textsf{no}}^*S$ independently
with probability $\rho(t_i)$.
Thus an outcome of $\mathcal{E}_{\textsf{no}}^*S$ is a union of some of the shells and
$\red{\mathcal{D}no}$ is supported over $2^M$ different sets.
Recall the definition of $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o$ using $\mathcal{D}yes$ and $\mathcal{D}no$. We now prove Lemma \ref{techlemma1}.
\begin{proof}[Proof of Lemma \ref{techlemma1}]
Let $x=(x_1,\ldots,x_q)$ be a sequence of $q$ points in $\R^n$.
We say $x$ is \varepsilonilonmph{bad} if either (1) at least one point lies outside of $\mathrm{Ball}(2\sqrt{n})$ or (2)
there are two points that lie in the same shell of $\mathcal{D}no$; we say $x$ is \varepsilonilonmph{good} otherwise.
We first claim that $\bx\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ is bad with probability $o(1)$.
To see this, we have from Lemma \ref{johnstone} that event (1) occurs with probability $o(1)$, and
from $M\ge 2^{\sqrt{n}}$ and $q=2^{0.01\sqrt{n}}$ that event (2) occurs with probability $o(1)$.
The claim follows from a union bound.
Given that $\bx\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ is good with probability $1-o(1)$, it suffices to show that
for any good $q$-tuple $x$, the total variation distance between
(1) $\mathcal{E}_{\textsf{no}}^*S(x)$ with $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ and (2)
$\mathbf{b} = (\mathbf{b}_1,\dots,\mathbf{b}_q)$ with each bit $\mathbf{b}_i$ being $1$ with probability $\rho(\|x_i\|)$ independently, is $o(1)$.
Let $\varepsilonilonll_i \in [M]$ be the index of the shell that $x_i$ lies in.
Since $x$ is good (and thus, all points lie in different shells),
$\mathcal{E}_{\textsf{no}}^*S(x)$ has the $i$th bit being $1$ independently with probability $\rho(t_{\varepsilonilonll_i})$;
for the other distribution, the probability is $\rho(\|x_i\|)$.
Using the subadditivity of total variation distance (i.e., the fact that
the $d_{\text{TV}}$ between two sequences of independent random variables
is upper bounded by the sum of the $d_{\text{TV}}$ between each pair) as well as (\ref{ofofof}), we have
$
\smash{d_{\text{TV}} (\mathcal{E}_{\textsf{no}}^*S(x),\mathbf{b} )\le q\cdot 2^{-\sqrt{n}}=o(1).}
$
This finishes the proof.
\varepsilonilonnd{proof}
The next lemma shows that $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ is $\varepsilonilonps_0$-far from convex with probability $1-o(1)$,
for some positive constant $\varepsilonilonps_0$.
In the proof of the lemma we fix both the constant $\varepsilonilonps_0$ and our choice of $r=\Theta(n^{1/4})$.
(We remind the reader that $\rho$ and $\mathcal{D}no$ both depend on the value of $r$.)
\begin{lemma}
There exist a real value $r=\Theta(n^{1/4})$ with $e^{r^2/2}\ge N/n$ and a positive constant $\varepsilonilonps_0$ such
that a set $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ is $\varepsilonilonps_0$-far from convex with probability at least $1-o(1)$.
\varepsilonilonnd{lemma}
\begin{proof}
We need the following claim but delay its proof to the end of the subsection:
\begin{claim}\label{claim1}
There exist an $r=\Theta(n^{1/4})$ with $e^{r^2/2}\ge N/n$ and a constant $c\in (0,1/2)$ such that
$$c < \rho(x) < 1-c,\quad\text{for all $x\in \left[\sqrt{n} - \red{10}, \sqrt{n} + \red{10}\right]$}. $$
\varepsilonilonnd{claim}
Let $K\subset [M]$ denote the set of all integers $k$ such that
$[t_{k-1},t_k]\subseteq [\sqrt{n}-10,\sqrt{n}+10]$ (note that $K$ is a set of consecutive integers).
Observe that (1) the total probability mass of all shells $k\in K$ is at least
$\Omega(1)$ (by Lemma \ref{johnstone}), and (2) the size $|K|$ is at least $\Omega(M)$ (which follows
from (1) and the fact that all shells have the same probability mass).
\def\boldsymbol{T}{\boldsymbol{T}}
Consider the following $1$-dimensional scenario.
We have $|K|$ intervals $[t_{k-1},t_k]$ and~draw a set $\boldsymbol{T}$ by including each interval independently
with probability $\rho(t_k)$.
We prove the following~claim:
\begin{claim}\label{oror}
The random set $\boldsymbol{T}$ satisfies the following property with probability at least $1-o(1)$:
For any interval $I \subseteq \R_{\geq 0}$, either $I$ contains $\Omega(M)$ intervals $[t_{k-1},t_k]$ that are not included in $\boldsymbol{T}$,
or $\overline{I}$ contains $\Omega(M)$ intervals $[t_{k-1},t_k]$ included in $\boldsymbol{T}$.
\varepsilonilonnd{claim}
\begin{proof}
First note that it suffices to consider intervals $I\subseteq \cup_{k\in K} [t_{k-1},t_k]$ and
moreover, we may further assume that both endpoints of $I$ come from endpoints of $[t_{k-1},t_k]$, $k\in K$.
(In other words, for a given outcome $T$ of $\boldsymbol{T}$, if there exists an interval $I$ that violates the condition, i.e.,
both $I$ and $\overline{I}$ contain fewer than $\Omega(M)$ intervals,
then there is such an interval $I$ with both ends from end points of $[t_{k-1},t_k]$).
This assumption allows us to focus on $|K|^2\le M^2$ many possibilities for $I$ (as we will see below, our argument applies a union bound over these $K^2$ possibilities).
Given a candidate such interval $I$, we consider two cases.
If $I$ contains $\Omega(M)$ intervals $[t_{k-1},t_k]$, $k\in K$, then
it follows from Claim \ref{claim1} and a Chernoff bound that $I$
contains at least $\Omega(M)$ intervals not included in $\boldsymbol{T}$ with probability $1-2^{-\Omega(M)}$.
On the other hand, if $\overline{I}$ contains $\Omega(M)$ intervals,
then the same argument shows that $\overline{I}$ contains $\Omega(M)$ interals
included in $\boldsymbol{T}$ with probability $1-2^{-\Omega(M)}$.
The claim follows from a union bound over all the $|K|^2$ possibilities for $I$.
\varepsilonilonnd{proof}
We return to the $n$-dimensional setting and
consider the intersection of $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}no$ with a ray starting from the origin.
Note that the intersection of the ray and any convex set is an interval on the ray.
As a result, Claim \ref{oror} shows that with probability at least $1-o(1)$ (over the draw of $\mathcal{E}_{\textsf{no}}^*S\leftarrow\mathcal{D}no$),
the intersection of any convex set with any ray either
contains $\Omega(M)$ intervals $[t_{k-1},t_k]$ such that shell $k\in K$ is not included in $\mathcal{E}_{\textsf{no}}^*S$,
or misses $\Omega(M)$ intervals $[t_{k-1},t_k]$ such that shell $k\in K$ is included in $\mathcal{E}_{\textsf{no}}^*S$.
Since by (1) above shells $k\in K$ together have $\Omega(1)$ probability mass under ${\mathcal{N}(0,1)^n}s$
and each shell contains the same probability mass, we have that with probability $1-o(1)$,
$\mathcal{E}_{\textsf{no}}^*S$ is $\varepsilonilonps_0$-far from any convex set for some constant $\varepsilonilonps_0>0$.
(A more formal argument can be given by performing integration using spherical coordinates
and applying (\ref{tttt}).)
\varepsilonilonnd{proof}
\begin{proof}[Proof of Claim \ref{claim1}]
We start with the choice of $r$. Let
$$
\alpha=\sqrt{n}-\red{10}\quad\text{and}\quad \beta=\sqrt{n}+\red{10}.
$$
Let $\mathrm{cap}(t)$ denote the fractional surface area of the spherical cap $S^{n-1}\cap \{x:x_1\ge t\}$, i.e.,
$$
\mathrm{cap}(t)=\Pr_{\bx\leftarrow S^{n-1}} \big[\bx_1\ge t\big].
$$
So $\mathrm{cap}$ is a continuous, strictly decreasing function over $[0,1]$.
Since $\mathrm{cap}(0)=1/2$ and $\mathrm{cap}(1)=0$, there is a unique $r\in (0,\red{\alpha})$ such that
$\red{\mathrm{cap}}(r/\alpha)=1/N=2^{-\sqrt{n}}$.
Below we show that $r=\Theta(n^{1/4})$ and fix it in the rest of the proof.
First recall the following explicit expression (see e.g. \cite{KOS:07}):
$$
\mathrm{cap}(t)=a_n\int_{t}^1\left(\sqrt{1-z^2}\right)^{n-3} dz,
$$
where $a_n = \Theta(n^{1/2})$ is a parameter that only depends on $n$.
Also recall the following inequalities from \cite{KOS:07} about $\mathrm{cap}(t)$:
\begin{equation}\label{KOSinequality}
\mathrm{cap}(t)\le e^{-nt^2/2},\quad \text{~for all $t \in [0,1]$}; \quad \quad
\mathrm{cap}(t)\ge \Omega\left( t\cdot e^{-nt^2/2}\right),\quad \text{for $t=O(1/n^{1/4})$}.
\varepsilonilonnd{equation}
By our choice of $\alpha$ and the monotonicity of the cap function,
this implies that $r=\Theta(n^{1/4})$ and
$$
1/N=\mathrm{cap}(r/\alpha)\ge \Omega(1/n^{1/4})\cdot e^{-n(r/\alpha)^2/2}\ge \Omega(1/n^{1/4})
\cdot e^{-(r^2/2)(1+O(1/\sqrt{n}))}=\Omega(1/n^{1/4})\cdot e^{-r^2/2}
$$
(using $r = \Theta(n^{1/4})$ for the last inequality), and thus, we have $e^{r^2/2}\ge N/n$.
Next, using the function $\mathrm{cap}$ we have the following expression for $\rho$:
\begin{equation} \label{eq:rhodef}
\rho(x) = \left(1 - \mathrm{cap}\left(\frac{r}{x}\right)\right)^N.
\varepsilonilonnd{equation}
As a side note, $\rho$ is continuous and thus, Lemma \ref{continuity} follows.
Since $\mathrm{cap}$ is strictly decreasing, we have that $\rho$ is strictly decreasing as well.
To finish the proof it suffices to show that there is a constant $c\in (0,1/2)$ such that
$\rho(\alpha)<1-c$ and $\rho(\beta)\ge c$.
The first part is easy since
$$\rho(\alpha)=\left(1-1/N\right)^N\approx e^{-1} $$by our choice of $r$.
In the rest of the proof we show that
\begin{equation}\label{lefteq}
\mathrm{cap}\left(\frac{r}{\beta}\right) \leq a \cdot\mathrm{cap}\left(\frac{r}{\alpha}\right)=\frac{a}{N},
\varepsilonilonnd{equation}
for some positive constant $a$.
It follows immediately that
$$\rho(\beta)=\left(1-\mathrm{cap}\left(\frac{r}{\beta}\right)\right)^N\ge
\left(1-\frac{a}{N}\right)^N\ge \left(e^{-2a/ N }\right)^N=e^{-2a},
$$
using $1-x\ge e^{-2x}$ for $0\le x\ll 1$,
and this finishes the proof of the claim.
\begin{figure}[t]
\centering
\includegraphics[width=15cm]{rho-standalone.pdf}
\caption{A plot of the integrand $(\sqrt{1-z^2})^{(n-3)}$. Area $A$ is $\mathrm{cap}(r/\beta)-\mathrm{cap}(r/\alpha)$ and area $B$ is $\mathrm{cap}(r/\alpha)$. The rectangles on the right are an upper bound of $A$ and a lower bound of $B$.}
\label{fig-rho}
\varepsilonilonnd{figure}
Finally we prove (\ref{lefteq}).
Let
$$
w=\frac{r}{\alpha}-\frac{r}{\beta}=\Theta\left(\frac{1}{n^{3/4}}\right)
$$
since $r=\Theta(n^{1/4})$.
Below we show that
\begin{equation}\label{riririr}
\int_{r/\beta}^{r/\alpha} \left(\sqrt{1-z^2}\right)^{n-3} dz
\le a'\cdot \int_{r/\alpha}^{r/\alpha \red{+}w} \left(\sqrt{1-z^2}\right)^{n-3} dz,
\varepsilonilonnd{equation}
for some positive constant $a'$.
It follows that
$$
\mathrm{cap}\left(\frac{r}{\beta}\right)-\mathrm{cap}\left(\frac{r}{\alpha}\right)
\le a'\cdot \mathrm{cap}\left(\frac{r}{\alpha}\right)
$$
and implies (\ref{lefteq}) by setting $a=a'+1$.
For (\ref{riririr}), note that the ratio of the $[r/\beta,r/\alpha]$-integration over
the $[r/\alpha,r/\alpha\red{+}w]$-integration is at most
$$
\left(\frac{\sqrt{1-(r/\beta)^2}}{\sqrt{1-(r/\beta+2w)^2}}\right)^{n-3}
$$
as the length of the two intervals are the same and the function $(\sqrt{1-z^2})^{n-3}$
is strictly decreasing. Figure~\ref{fig-rho} illustrates this calculation.
Let $\tau=r/\beta=\Theta(1/n^{1/4})$.
We can rewrite the~above~as
$$
\left(
\frac{1-\tau^2}{1-(\tau+2w)^2}
\right)^{(n-3)/2}
=\left(1+\frac{4\tau w+4w^2}{1-(\tau+2w)^2}\right)^{(n-3)/2}
=\left(1+O\left(\frac{1}{n}\right)\right)^{(n-3)/2}=O(1).
$$
This finishes the proof of the claim.
\varepsilonilonnd{proof}
\subsection{Distributions $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o^*$ are close}\label{finalsec}
In the rest of the section we show that the total
variation distance between $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o^*$ is $o(1)$ and thus prove Lemma \ref{techlemma2}.
Let $z=(z_1,\ldots,z_q)$ be a sequence of $q$ points in $\R^n$.
We use $\calE_{\text{yes}}es(z)$ to denote the distribution of labeled samples from $\calE_{\text{yes}}es$,
conditioning on the samples being $z$, i.e., $(z,\mathcal{E}_{\textsf{no}}^*S(z))$ with $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$.
We let $\calE_{\text{no}}o^*(z)$ denote the distribution of labeled samples from $\calE_{\text{no}}o^*$,
conditioning on the samples being $z$, i.e., $(z,\mathbf{b})$ where each $\mathbf{b}_i$ is $1$ independently with probability
$\rho(\|z_i\|)$.
Then
\begin{equation}\label{fuif}d_{\text{TV}}(\calE_{\text{yes}}es,\calE_{\text{no}}o^*)
=\E_{\mathbf{z}\leftarrow ({\mathcal{N}(0,1)^n}s)^q} \Big[d_{\text{TV}} (\calE_{\text{yes}}es(\mathbf{z}),\calE_{\text{no}}o^*(\mathbf{z}))\Big].
\varepsilonilonnd{equation}
We split the proof of Lemma \ref{techlemma2} into two steps.
We first introduce the notion of \varepsilonilonmph{typical} sequences $z$ of $q$ points
and show in this subsection that with probability $1-o(1)$, $\mathbf{z}\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ is typical.
In the next subsection we show that $d_{\text{TV}}(\calE_{\text{yes}}es(z),\calE_{\text{no}}o^*(z))$ is $o(1)$ when $z$ is typical.
It follows from (\ref{fuif}) that $d_{\text{TV}}(\calE_{\text{yes}}es,\calE_{\text{no}}o^*)$ is $o(1)$.
We start with the definition of typical sequences.
Given a point $z\in \R^n$, we are interested in the \varepsilonilonmph{fraction} of points $y$
(in terms of the area) in $S^{n-1}(r)$ such that $z\cdot y>r^2$. This is because if any such point $y$ is sampled
in the construction of $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$, then $z\notin \mathcal{E}_{\textsf{no}}^*S$.
This is illustrated in Figure \ref{fig:pic}.
We refer to the set of such points $y$ as the \varepsilonilonmph{\varepsilonilonmph{(}spherical\varepsilonilonmph{)} cap covered by $z$}
and we write $\text{cover}(z)$ to denote it.
(Note that $\text{cover}(z)=\varepsilonilonmptyset$ if $\|z\|\le r$.)
Given a subset $H$ of $S^{n-1}(r)$ (such as $\text{cover}(z)$), we use
$\mathrm{fsa}(H)$ to denote the fractional surface area of $H$ with respect to $S^{n-1}(r)$.
Using Figure \ref{fig:pic} and elementary geometry, we have the following connection between
the fractional surface area of $\text{cover}(z)$ and the cap function (for $S^{n-1}$):
\begin{equation} \label{eq:capcover}
\mathrm{fsa}\big(\text{cover}(z)\big)=\mathrm{cap}\big(r/\|z\|\big).
\varepsilonilonnd{equation}
We are now ready to define typical sequences.
\begin{figure}[t!]
\centering
\includegraphics{fig-similar.pdf}
\caption{The fractional surface area of $\text{cover}(z)$, $\mathrm{fsa}(\text{cover}(z))$, is the fraction of $S^{n-1}(r)$ to the right of the dashed line.
By similarity of triangles $0az$ and $0ba$, scaling down to the unit sphere, we get (\ref{eq:capcover}).}\label{fig:pic}
\varepsilonilonnd{figure}
\begin{definition}
We say a sequence $z=(z_1,\ldots,z_q)$ of $q$ points in $\R^n$ is \varepsilonilonmph{typical} if
\begin{enumerate}
\item For every point $z_i$, we have
\begin{equation}\label{lulala}
\mathrm{fsa}\big(\text{cover}(z_i)\big) \in \left[e^{-0.51\hspace{0.03cm} r^2 }, e^{-0.49\hspace{0.03cm}r^2}\right].
\varepsilonilonnd{equation}
\item For every $i \neq j$, we have
$$\mathrm{fsa}\big(\text{cover}(z_i) \cap \text{cover}(z_j)\big) \le
e^{-0.96\hspace{0.03cm}r^2} .$$
\varepsilonilonnd{enumerate}
\varepsilonilonnd{definition}
The first condition of typicality essentially says that every $z_i$ is not too
close to and not too far away from the origin (so that we have a relatively tight bound on the fractional surface area
of the cap covered by $z_i$).
The second condition says that the caps covered by two points $z_i$ and $z_j$ have very little intersection.
We prove the following lemma:
\begin{lemma} \label{thm:most-z-are-typical}
$\mathbf{z}\leftarrow ({\mathcal{N}(0,1)^n}s)^q$ is typical with probability at least \(1 - o(1)\).
\varepsilonilonnd{lemma}
\begin{proof}
We show that $\mathbf{z}$ satisfies each of the two conditions with probability $1-o(1)$.
The lemma then follows from a union bound.
For the first condition, we let $c^*=0.001$ be a sufficiently small constant.
We have from Lemma \ref{johnstone} and a union bound that every
$\mathbf{z}_i$ satisfies $(1-c^*)\sqrt{n}\le \|\mathbf{z}_i\|\le (1+c^*)\sqrt{n}$ with probability $1-o(1)$.
When this happens, we have (\ref{lulala}) for every $\mathbf{z}_i$ using
(\ref{KOSinequality}) and the upper bound of $\mathrm{cap}(t)\le e^{-nt^2/2}$.
For the second condition, we first note that the argument used in the first part implies that
$$
\E_{\bz_i\leftarrow{\mathcal{N}(0,1)^n}s}\Big[\mathrm{fsa}\big(\text{cover}(\bz_i)\big)\Big]\le e^{-0.49\hspace{0.03cm}r^2}.
$$
Let $x_0$ be a fixed point in $S^{n-1}(r)$.
Viewing the fractional surface area as the following probability
$$
\mathrm{fsa}\big(\text{cover}(z_i)\big) = \Pr_{\bx\leftarrow S^{n-1}(r)} \big[\bx\in \text{cover}(z_i)\big],
$$
we have
\begin{align}
e^{-0.49\hspace{0.03cm}r^2}&\ge \E_{\bz_i\leftarrow {\mathcal{N}(0,1)^n}s}\Big[\mathrm{fsa}\big(\text{cover}(\bz_i)\big)\Big] \label{eq:zzzz}\\
&=\E_{\bz_i}\Big[\Pr_{\bx\leftarrow S^{n-1}(r)} \big[\bx\in \text{cover}(\mathbf{z}_i)\big]\Big] \nonumber \\[0.5ex]
&=\Pr_{\bx,\mathbf{z}_i}\big[\bx\in \text{cover}(\mathbf{z}_i)\big] =\Pr_{\mathbf{z}_i}\big[x_0\in \text{cover}(\mathbf{z}_i)\big], \nonumber
\varepsilonilonnd{align}
where the last equation follows by sampling $\bx$ first and spherical and Gaussian symmetry.
Similarly we can express the fractional surface area of
$\text{cover}(z_i)\cap \text{cover}(z_j)$ as
$$\mathrm{fsa}\big(\text{cover}(z_i)\cap \text{cover}(z_j)\big)=\Pr_{\bx\leftarrow S^{n-1}(r)} \big[\bx\in \text{cover}(z_i)\ \text{and}\
\bx\in \text{cover}(z_j)\big].$$
We consider the expectation over $\mathbf{z}_i$ and $\mathbf{z}_j$ drawn independently from ${\mathcal{N}(0,1)^n}s$:
\begin{align*}
&\E_{\mathbf{z}_i,\mathbf{z}_j}\Big[\mathrm{fsa}\big(\text{cover}(\mathbf{z}_i)\cap \text{cover}(\mathbf{z}_j)\big)\Big]\\
&=\E_{\mathbf{z}_i,\mathbf{z}_j}\Big[\Pr_{\bx\leftarrow S^{n-1}(r)} \big[\bx\in \text{cover}(\mathbf{z}_i)\ \text{and}\
\bx\in \text{cover}(\mathbf{z}_j)\big]\Big]\\
&=\Pr_{\bx,\mathbf{z}_i,\mathbf{z}_j}\big[\bx\in \text{cover}(\mathbf{z}_i)\ \text{and}\
\bx\in \text{cover}(\mathbf{z}_j)\big]=\Pr_{\mathbf{z}_i}\big[x_0\in \text{cover}(\mathbf{z}_i)\big]\cdot \Pr_{\mathbf{z}_j}\big[x_0\in \text{cover}(\mathbf{z}_j)\big],
\varepsilonilonnd{align*}
where the last equation follows by sampling $\bx$ first, independence of $\mathbf{z}_i$ and $\mathbf{z}_j$, and symmetry.
By (\ref{eq:zzzz}), the expectation of $\mathrm{fsa}(\text{cover}(\mathbf{z}_i)\cap \text{cover}(\mathbf{z}_j))$ is
at most $\smash{e^{-0.98\hspace{0.03cm}r^2}}$, and hence by Markov's inequality,
the probability of it being at least $\smash{e^{-0.96\hspace{0.03cm}r^2}}$
is at most $\smash{e^{-0.02\hspace{0.03cm}r^2}}$.
Using $\smash{e^{r^2}\ge (N/n)^2}$ and a union bound, the probability of one of the pairs having
the $\mathrm{fsa}$ at least $\smash{e^{-0.96\hspace{0.03cm}r^2}}$ is at most
$$
q^2 \cdot e^{-0.02r^2}\le 2^{0.02\hspace{0.03cm}\sqrt{n}}\cdot (n/N)^{0.04}=o(1),
$$
since $q=2^{0.01\sqrt{n}}$ and $N=2^{\sqrt{n}}$.
This finishes the proof of the lemma.
\varepsilonilonnd{proof}
We prove the following lemma in Section \ref{hehefinal} to finish the proof of Lemma \ref{techlemma2}.
\begin{lemma} \label{thm:typical-z-are-good}
For every typical sequence $z$ of $q$ points, we have \(d_{\text{TV}}\big(\calE_{\text{yes}}es(z), \calE_{\text{no}}o^*(z)\big) = o(1)\).
\varepsilonilonnd{lemma}
\subsection{Proof of Lemma~\ref{thm:typical-z-are-good}}\label{hehefinal}
Fix a typical $z=(z_1,\ldots,z_q)$.
Our goal is to show that the total variation distance of $\calE_{\text{yes}}es(z)$ and $\calE_{\text{no}}o^*(z)$ is $o(1)$.
To this end, we define a distribution $\calF$ over pairs $(\mathbf{b},\mathbf{d})$ of strings in $\{0,1\}^q$ (as a coupling
of $\calE_{\text{yes}}es(z)$ and $\calE_{\text{no}}o^*(z)$), where the
marginal distribution of $\mathbf{b}$ as $(\mathbf{b},\mathbf{d})\leftarrow \calF$ is the same as $\calE_{\text{yes}}es(z)$ and
the marginal distribution of $\mathbf{d}$ is the same as $\red{\calE_{\text{no}}o^\ast}(z)$.
Our goal follows by establishing
\begin{equation}\label{maineq}
\Prx_{(\mathbf{b},\mathbf{d})\leftarrow \calF} \big[\mathbf{b}\ne \mathbf{d}\big]=o(1).
\varepsilonilonnd{equation}
To define $\calF$, we use $\bM$ to denote the $q\times N$ $\{0,1\}$-valued random matrix
derived from $z$ and $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$ (recall that $\mathcal{E}_{\textsf{no}}^*S$ is the intersection of
$N$ random halfspaces $\bh_j$, $j\in [N]$):
the $(i,j)$th entry $\bM_{i,j}$ of $\bM$ is $1$ if $\bh_j(z_i)=1$ (i.e., $z_i\in \bh_j$)
and is $0$ otherwise.
We use $\bM_{i,*}$ to denote the $i$th row of $\bM$,
$\bM_{*,j}$ to denote the $j$th column of $\bM$,
and $\bM^{(i)}$ to denote the $i\times N$ sub-matrix of $\bM$ that consists of the first $i$ rows of $\bM$.
(We note that $\bM$ is derived from $\mathcal{E}_{\textsf{no}}^*S$ and they are defined over the same probability space.
So we may consider the (conditional) distribution of $\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes$
conditioning on an event involving $\bM$, and we may consider the conditional distribution of $\bM$
conditioning on an event involving $\mathcal{E}_{\textsf{no}}^*S$.)
\def\mathbf{b}{\mathbf{b}}
\def\mathbf{d}{\mathbf{d}}
We now define the distribution $\calF$.
A pair $(\mathbf{b},\mathbf{d})\leftarrow \calF$ is drawn using the following randomized procedure.
The procedure has $q$ rounds and generates the $i$th bits $\mathbf{b}_i$ and $\mathbf{d}_i$ in the $i$th round:
\begin{flushleft}\begin{enumerate}
\item In the first round, we draw a random real number $\mathbf{r}_1$ from $[0,1]$ uniformly at random.
We set $\mathbf{b}_1=1$ if $\mathbf{r}_1\le \Pr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes} [\mathcal{E}_{\textsf{no}}^*S(z_1)=1]$ and set $\mathbf{b}_1=0$ otherwise.
We then set $\mathbf{d}_1=1$ if $\mathbf{r}_1\le \rho(\|z_1\|)$ and set $\mathbf{d}_1=0$ otherwise.
(Note that for the first round, the two thresholds are indeed the same
so we always have $\mathbf{b}_1=\mathbf{d}_1$.)
At the end of the first round, we also draw a row vector $\mathbf{N}_{1,*}$ according to the
distribution of $\bM_{1,*}$ conditioning on $\mathcal{E}_{\textsf{no}}^*S(z_1)=\mathbf{b}_1$.
\item In the $i$th round, for $i$ from $2$ to $q$, we draw a random real number $\mathbf{r}_i$ from $[0,1]$ uniformly at random.
We set $\mathbf{b}_i=1$ if we have $$\mathbf{r}_i\le \Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes}\Big[\mathcal{E}_{\textsf{no}}^*S(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm}\bM^{(i-1)}=\mathbf{N}^{(i-1)}\Big]$$
and set $\mathbf{b}_i=0$ otherwise.
We then set $\mathbf{d}_i=1$ if $\mathbf{r}_i\le \rho(\|z_i\|)$ and set $\mathbf{d}_i=0$ otherwise.
At the end of the $i$th round, we also draw a row vector $\mathbf{N}_{i,*}$ according to the
distribution of $\bM_{i,*}$ conditioning on $\bM^{(i-1)}=\mathbf{N}^{(i-1)}$ and $\mathcal{E}_{\textsf{no}}^*S(z_i)=\mathbf{b}_i$.
\varepsilonilonnd{enumerate}\varepsilonilonnd{flushleft}
It is clear that the marginal distributions of $\mathbf{b}$ and $\mathbf{d}$, as $(\mathbf{b},\mathbf{d})\leftarrow \calF$,
are $\calE_{\text{yes}}es$ and $\calE_{\text{no}}o^*$ respectively.
To prove (\ref{maineq}), we introduce the following notion of \varepsilonilonmph{nice} and \varepsilonilonmph{bad} matrices.
\begin{definition}
Let $M$ be an $i\times N$ $\{0,1\}$-valued matrix for some $i\in [q]$.
We say $M$ is \varepsilonilonmph{nice} if
\begin{enumerate}
\item $M$ has at most $\sqrt{N}$ many $0$-entries; and
\item Each column of $M$ has at most one 0-entry.
\varepsilonilonnd{enumerate}
We say $M$ is \varepsilonilonmph{bad} otherwise.
\varepsilonilonnd{definition}
We prove the following two lemmas and use them to prove (\ref{maineq}).
\begin{lemma}\label{lem1}
$\Pr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes }\big[\bM\ \text{is bad}\big]=o(1/q)$.
\varepsilonilonnd{lemma}
Note that when $\bM$ is nice, we have by definition that $\bM^{(i)}$ is also nice for every $i\in [q]$.
\begin{lemma}\label{lem2}
For any nice $(i-1)\times N$ $\{0,1\}$-valued matrix $M^{(i-1)}$, we have
\begin{equation}\label{huha}
\Prx_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes}\Big[\mathcal{E}_{\textsf{no}}^*S(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm}
\bM^{(i-1)}=M^{(i-1)}\Big]=\rho(\|z_i\|)\pm o(1/q).
\varepsilonilonnd{equation}
\varepsilonilonnd{lemma}
Before proving Lemma \ref{lem1} and \ref{lem2},
we first use them to prove (\ref{maineq}).
Let $\bI_i$ denote the indicator random variable that is $1$ if $(\mathbf{b},\mathbf{d})\leftarrow \calE$ has
$\mathbf{b}_i\ne \mathbf{d}_i$ and is $0$ otherwise, for each $i\in [q]$.
Then (\ref{maineq}) can be bounded from above by
$\sum_{i\in [q]} \Pr[\bI_i=1]$.
To bound each $\Pr[\bI_i=1]$ we split the event into
$$
\sum_{M^{(i-1)}} \Pr\big[\mathbf{N}^{(i-1)}=M^{(i-1)}\big]\cdot
\Pr\big[\bI_i=1\hspace{0.06cm}|\hspace{0.06cm}\mathbf{N}^{(i-1)}=M^{(i-1)}\big],
$$
where the sum is over all $(i-1)\times N$ $\{0,1\}$-valued matrices $M^{(i-1)}$,
and further split the sum into two sums over nice and
bad matrices $M^{(i-1)}$.
As $\mathbf{N}^{(i-1)}$ has the same
distribution as $\bM^{(i-1)}$, it follows from Lemma \ref{lem1} (and the fact that $\bM$ is bad when
$\bM^{(i-1)}$ is bad) that the sum over bad $M^{(i-1)}$ is
at most $o(1/q)$.
On the other hand, it follows from Lemma \ref{lem2} that the sum over nice $ M^{(i-1)}$
is $o(1/q)$.
As a result, we have $\Pr[\bI_i=1]=o(1/q)$ and thus, $\sum_{i\in [q]} \Pr[\bI_i=1]=o(1)$.
We prove Lemmas \ref{lem1} and \ref{lem2} in the rest of the section.
\begin{proof}[Proof of Lemma \ref{lem1}]
We show that the probability of $\bM$ violating each of the two conditions in the definition
of nice matrices is $o(1/q)$. The lemma
then follows by a union bound.
For the first condition, since $z$ is typical the probability of $\bM_{i,j}=0$ is
$$
\mathrm{fsa}\big(\text{cover}(z_i)\big)\le e^{-0.49\hspace{0.03cm}r^2}.
$$
By linearity of expectation, the expected number of $0$-entries in $\bM$ is at most
$$qN\cdot e^{-0.49\hspace{0.03cm}r^2} = o(\sqrt{N}/q),$$
using $e^{r^2/2}\ge N/n$, $N=2^{\sqrt{n}}$ and $q=2^{0.01\sqrt{n}}$.
It follows directly from Markov's inequality that the probability of
$\bM$ having more than $\sqrt{N}$ many $0$-entries is $o(1/q)$.
For the second condition, again since $z$ is typical, the probability of $\bM_{i,j}=\bM_{i',j}=1$ is
$$
\mathrm{fsa}\big(\text{cover}(z_i)\cap \text{cover}(z_i')\big)\le e^{-0.96\hspace{0.03cm}r^2}.
$$
By a union bound, the probability of $\bM_{i,j}=\bM_{i',j}=1$ for some $i,i',j$ is at most
$$q^2N\cdot e^{\red{-0.96r^2}\hspace{0.03cm}}=o(1/q).$$
This finishes the proof of the lemma.
\varepsilonilonnd{proof}
Finally we prove Lemma \ref{lem2}.
Fix a nice $(i-1)\times N$ matrix $M$ (we henceforth omit the superscript $(i-1)$ since the
number of rows of $M$ is fixed to be $i-1$).
Recall that $\mathcal{E}_{\textsf{no}}^*S(z_i)=1$ if and only if $\bh_j(z_i)=1$ for all $j\in [N]$.
As a result, we have
$$
\Ppr_{\mathcal{E}_{\textsf{no}}^*S\leftarrow \mathcal{D}yes}\Big[\mathcal{E}_{\textsf{no}}^*S(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm}
\bM^{(i-1)}=M\Big]
=\prod_{j\in [N]} \hspace{0.05cm}\Ppr_{\bh_j}\Big[\bh_j(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm}
\bM^{(i-1)}_{*,j}=M_{*,j}\Big].
$$
On the other hand, letting $\tau=\mathrm{fsa}(\text{cover}(z_i))=\mathrm{cap}(r/\|z_i\|)$, we have
$\rho(\|z_i\|)=(1-\tau)^N$.
In the next two claims we compare
$$
\Ppr_{\bh_j}\Big[\bh_j(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm}
\bM^{(i-1)}_{*,j}=M_{*,j}\Big]
$$
with $1-\tau$ for each $j\in [N]$ and show that they are very close.
The first claim works on $j\in [N]$ with no $0$-entry in $M_{*,j}$ and
the second claim works on $j\in [N]$ with one $0$-entry in $M_{*,j}$.
(These two possibilities cover all $j\in [N]$ since the matrix $M$ is nice.)
Below we omit $\bM^{(i-1)}_{*,j}$ in writing the conditional probabilities.
\begin{claim}\label{lem-goodj}
For each $j\in [N]$ with no $0$-entry in the $j$th column $M_{*,j}$, we have
$$\Prx_{\bh_j}\Big[\bh_j(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm} M_{*,j}\Big]
= (1-\tau)\left(1\pm \frac{o(1)}{qN}\right).$$
\varepsilonilonnd{claim}
\begin{proof}
Let
$\partial ta$ be the probability of $\bh_j(z_i)=0$ conditioning on $M_{*,j}$ (which is all-$1$).
Then
$$
\partial ta=\frac{\mathrm{fsa}\left(\text{cover}(z_i)-\bigcup_{j<i} \text{cover}(z_j)\right)}{
1-\mathrm{fsa}\left(\bigcup_{j<i} \text{cover}(z_j)\right)}.
$$
Using $e^{-0.51\hspace{0.03cm}r^2}\le \mathrm{fsa}(\text{cover}(z_j))\le e^{-0.49\hspace{0.03cm}r^2}$ and $\mathrm{fsa}(\text{cover}(z_i)\cap \text{cover}(z_j))\le e^{-0.96
\hspace{0.03cm}r^2}$, we have
\begin{align*}
\partial ta &\leq \frac{\tau}{1-q\cdot e^{-0.49\hspace{0.03cm}r^2}}<\tau(1+2q\cdot e^{-0.49\hspace{0.03cm}r^2})
=\tau+2\tau q\cdot e^{-0.49\hspace{0.03cm}r^2}.
\varepsilonilonnd{align*}
Using $\tau\le e^{-0.49\hspace{0.03cm}r^2}$ and $e^{r^2/2}\ge N/n$, we have
$$
1-\partial ta\ge 1-\tau -2\tau q\cdot e^{-0.49\hspace{0.03cm}r^2}
\ge 1-\tau-o\big(1/(qN)\big)\ge (1-\tau)\big(1-o(1/(qN))\big).
$$
On the other hand, we have
$\partial ta \geq \tau -q\cdot e^{-0.96\hspace{0.03cm}r^2}$ and thus,
$$
1-\partial ta\le 1-\tau+q\cdot e^{-0.96\hspace{0.03cm}r^2}\le 1-\tau+o\big(1/(qN)\big)
=(1-\tau)\big(1+o(1/(qN))\big).
$$
This finishes the proof of the claim.
\varepsilonilonnd{proof}
\begin{claim}\label{lem-badj}
For each $j\in [N]$ with one $0$-entry in the $j$th column $M_{*,j}$, we have
$$\Pr_{\bh_j}\Big[\bh_j(z_i)=1\hspace{0.06cm}\big|\hspace{0.06cm} M_{*,j}\Big] \geq 1-O\big(e^{-0.45\hspace{0.03cm}r^2}\big).$$
\varepsilonilonnd{claim}
\begin{proof}
Let $i'$ be the point with $M_{i',j}=1$ and $\partial ta$
be the conditional probability of $\bh_j(z_i)=0$. Then
$$
\partial ta\le \frac{\mathrm{fsa}\big(\text{cover}(z_i)\cap \text{cover}(z_{i'})\big)}{\mathrm{fsa}\left(\text{cover}(z_i')-
\bigcup_{j<i:\hspace{0.03cm}j\ne i'} \text{cover}(z_j)\right)}
\le \frac{e^{-0.96\hspace{0.03cm}r^2}}{e^{-0.51\hspace{0.03cm}r^2}-q\cdot e^{-0.96\hspace{0.03cm}r^2}}
=O\big(e^{-0.45\hspace{0.03cm}r^2}\big),
$$
by our choice of $q$.
This finishes the proof of the claim.
\varepsilonilonnd{proof}
We combine the two claims to prove Lemma \ref{lem2}.
\begin{proof}[Proof of Lemma \ref{lem2}]
Let $h$ be the number of $0$-entries in $M$.
We have $h\le \sqrt{N}$ since $M$ is nice.
By Claims~\ref{lem-goodj}, the conditional probability of $\mathcal{E}_{\textsf{no}}^*S(z_i)=1$ is at most
\begin{align*}
\left((1-\tau)\left(1+o\left(\frac{1}{qN}\right)\right)\right)^{N-h}
&=\rho(\|z_i\|)\cdot \frac{1}{(1-\tau)^h}\cdot \left(1+o\left(\frac{1}{qN}\right)\right)^{N-h} \\
&\le \rho(\|z_i\|) \cdot (1+2\tau)^h\cdot \left(1+o\left(\frac{1}{qN}\right)\right)^{N}\\[0.3ex]
&\le \rho(\|z_i\|)\cdot \varepsilonilonxp\big(2\tau h+ o(1/q)\big)\\[0.6ex]
&=\rho(\|z_i\|)\cdot \varepsilonilonxp\big(o(1/q)\big)=\rho(\|z_i\|)+o(1/q).
\varepsilonilonnd{align*}
Similarly, the conditional probability of $\mathcal{E}_{\textsf{no}}^*S(z_i)=1$ is at least
\begin{align*}
&\left((1-\tau)\left(1-o\left(\frac{1}{qN}\right)\right)\right)^{N-h}\left(1-O\left(e^{-0.45\hspace{0.03cm}r^2}\right)\right)^{h}\\
&\hspace{1cm}\ge \rho(\|z_i\|)\cdot \left(1-o\left(\frac{1}{qN}\right)\right)^{N-h}
\left(1-O\left(e^{-0.45\hspace{0.03cm}r^2}\right)\right)^{h}\\[0.6ex]
&\hspace{1cm}\ge \rho(\|z_i\|)\cdot \big(1-o(1/q)\big)\ge \rho(\|z_i\|)-o(1/q).
\varepsilonilonnd{align*}
This finishes the proof of the lemma.
\varepsilonilonnd{proof}
\section{One-sided lower bound} \label{sec:1slb}
We recall Theorem~\ref{thm:1slb}:
\begin{reptheorem}{thm:1slb}
\red{Any one-sided sample-based algorithm that is an $\varepsilonilonps$-tester for convexity over $\normal^n$
for some $\varepsilonilonps<1/2$
must use $2^{\Omega(n)}$ samples.}
\varepsilonilonnd{reptheorem}
We say a finite set $\{x^1,\dots,x^M\} \subset \R^n$ is \varepsilonilonmph{shattered} by $\calC_\mathrm{convex}$ if for every $(b_1,\dots,b_M) \in \zo^M$ there is a convex set $C \in \calC_\mathrm{convex}$ such that $C(x^i)=b_i$ for all $i \in [M].$
Theorem~\ref{thm:1slb} follows from the following lemma:
\begin{lemma} \label{lem:any-labeling}There is an absolute constant $c>0$ such that for $M=2^{c n}$, it holds that
\[
\Prx_{\bx^i\leftarrow\normal^n}\big[\{\bx^1,\dots,\bx^M\} \text{~is shattered by~}\calC_\mathrm{convex}\hspace{0.03cm}\big] \geq 1 - o(1).
\]
\varepsilonilonnd{lemma}
\begin{proof}[Proof of Theorem~\ref{thm:1slb} using Lemma~\ref{lem:any-labeling}.]
Suppose that $A$ were a one-sided sample-based algorithm for $\varepsilonilonps$-testing $\calC_\mathrm{convex}$ using at most $M$ samples. Fix a set $S$ that is $\varepsilonilonps$-far from $\calC_\mathrm{convex}$ to be the unknown target subset of $\R^n$ that is being tested.\footnote{An example of such a subset $S$ is as follows (we define it as
a function $S:\R^n\rightarrow \{0,1\}$): Given an odd integer $N > (1/2 - \varepsilonilonps)^{-1} - 1$, let $-\infty = \tau_0 < \tau_1 < \cdots < \tau_N < \tau_{N+1}=+\infty$ be values such that
$
\Pr_{\bz \leftarrow \normal}[\bz \leq \tau_i] = i/( {N+1}),
$
and let $S: \R^n \to \{0,1\}$ be the function defined by $S(x_1,\dots,x_n)=\mathbf{1}[i$ is even$]$, where $i \in \{0,\dots,N\}$ is the unique value such that $\tau_i \leq x_1 < \tau_{i+1}.$ Fix any $z = (z_2,\dots,z_n) \in \R^{n-1}$ and we let $S_z: \R \to \{0,1\}$ be the function defined as $S_z(x_1)=S(x_1,z_2,\dots,z_n)$. An easy argument gives that $S_z$ is $\left(1/2 - { 1 /({N+1})}\right)$-far (and hence $\varepsilonilonps$-far) from every convex subset of $\R$, and it follows by averaging (using the fact that the restriction of any convex subset of $\R^n$ to a line is a convex subset of $\R$) that $S$ is $\varepsilonilonps$-far from $\calC_\mathrm{convex}.$}
Since $S$ is $\varepsilonilonps$-far from convex, it must be the case that
\begin{equation}\label{haui}
\Prx_{\bx^i\leftarrow \normal^n}\big[A \text{\ rejects when run on $(\bx^1,S(\bx^1)),\dots,
(\bx^M,S(\bx^M))$}\big] \geq 2/3.
\varepsilonilonnd{equation}
But Lemma~\ref{lem:any-labeling} together with the one-sidedness of $A$ imply that
\begin{align*}
&\Prx_{\bx^i\leftarrow\normal^n}\big[\hspace{0.03cm}\text{for any $(b^1,\dots,b^M) \in \zo^M$,
$A$ rejects when run on~$(\bx^1,b^1),\dots,
(\bx^M,b^M)$}\hspace{0.02cm}\big] \leq o(1),
\varepsilonilonnd{align*}
as $A$ can only reject if the labeled samples are not consistent with any convex set,
which implies that $A$ cannot reject when $\{\bx^1,\ldots,\bx^M\}$ is shattered by $\calC_\mathrm{convex}$.
This contradicts with (\ref{haui}).\varepsilonilonnd{proof}
In the next subsection we prove Lemma~\ref{lem:any-labeling} for $c=1/500.$
\subsection{Proof of Lemma~\ref{lem:any-labeling}}
Let $M=2^{cn}$ with $c=1/500$.
We prove the following lemma:
\begin{lemma} \label{lem:rand-all-on-hull}
For $\bx^1,\dots,\bx^M$ drawn independently from $\normal^n$, with probability $1-o(1)$ it is the case that for all $i \in [M],$ no $\bx^i$ lies in $\Conv(\{\bx^j:j \in [M] \setminus i\}).$
\varepsilonilonnd{lemma}
If $\bx^1,\dots,\bx^M$ are such that no $\bx^i$ lies in $\Conv(\{\bx^j:j \in [M] \setminus i\})$, then given any $(b^1,\dots,b^M)$,
by taking $C=\Conv(\{\bx^i: b^i = 1\})$ we see that there is a convex set $C$ such that $C(\bx^i)=b^i$ for all $i \in [M]$.
Thus to establish Lemma~\ref{lem:any-labeling} it suffices to prove Lemma~\ref{lem:rand-all-on-hull}.
To prove Lemma~\ref{lem:rand-all-on-hull}, it suffices to show that for each fixed $j \in [M]$ we have
\begin{equation}
\Prx_{\bx^i\leftarrow \normal^n}\big[\hspace{0.03cm}\bx^j \in \Conv(\{\bx^k:k \in [M]\setminus \{j\}\})\big] \leq M^{-2}
\label{eq:A}
\varepsilonilonnd{equation}
since given this a union bound implies that
\[
\Prx_{\bx^i\leftarrow \normal^n}\big[\hspace{0.03cm}\text{for some $j \in [M]$, }\bx^j \text{~lies in~} \Conv(\{\bx^k :k \in [M]\setminus \{j\}\})\big] \leq M^{-1}=o(1).
\]
By symmetry, to establish (\ref{eq:A}) it suffices to show that
\begin{equation} \label{eq:B}
\Prx_{\bx^i \leftarrow \normal^n}\big[\bx^M \in \Conv(\{\bx^1,\dots,\bx^{M-1}\})\big] \leq M^{-2}.
\varepsilonilonnd{equation}
In turn (\ref{eq:B}) follows from the following inequalities
($v\in \R^n$ is a fixed unit vector in the second)
\begin{equation}\label{twoinequalities}
\Prx_{\bx \leftarrow \normal^n}\big[\|\bx\| \leq \sqrt{n}/10\big] < {\frac 1 2} M^{-2}\quad\text{and}\quad
\Pr_{\bx \leftarrow \normal^n}\big[\bx \cdot v \geq \sqrt{n}/10\big] < {\frac 1 2}M^{-3}.
\varepsilonilonnd{equation}
The first inequality follows directly from Lemma \ref{lem:johnstone} using $c=1/500$.
For the second, by the spherical symmetry of $\normal^n$ we may take $v=(1,0,\dots,0).$
Recall the standard Gaussian tail bound $$\Pr_{\bz \leftarrow \normal}\big[\bz\ge t\big] \leq e^{-t^2/2}$$ for $t\geq 0$. This gives us that
$$\Pr_{\bx \leftarrow \normal^n}\big[\bx \cdot v \geq \sqrt{n}/10\big] \leq e^{-n/200} < {\frac 1 2} M^{-3},$$ again using that $M=2^{cn}$ and $c=1/500.$
Finally,
to see that (\ref{eq:B}) follows from (\ref{twoinequalities}), we observe first that by the first inequality we may assume that $\|\bx^M\| > \sqrt{n}/10$ (at the cost of failure probability at most $ M^{-2}/2$ towards (\ref{eq:B})); fix~any such outcome $x^M$ of $\bx^M.$ By a union bound over $\bx^1,\dots,\bx^{M-1}$ and the second inequality, we have
\[
\Prx_{\bx^i\leftarrow\normal^n}\left[\text{\hspace{0.03cm}any $i \in [M-1]$ has $\bx^i \cdot {\frac {x^M}{\|x^M\|}} \geq \sqrt{n}/10$}\right] < {\frac 1 2} M^{-2}.
\]
But if every $\bx^i$ has $\bx^i \cdot ({x^M}/{\|x^M\|})<\sqrt{n}/10 < \|x^M\|$, then $x^M\notin\Conv(\{\bx^1,\dots,\bx^{M-1}\}).$
\section{Two-sided upper bound} \label{sec:2sub}
Recall Theorem \ref{thm:2sub}:
\begin{reptheorem}{thm:2sub}
For any $\varepsilonilonps > 0$, there is a two-sided sample-based $\varepsilonilonps$-tester for convexity over
$\normal^n$ using $n^{O(\sqrt{n}/\varepsilonilonps^2)}$ samples.
\varepsilonilonnd{reptheorem}
We begin by recalling some definitions from learning theory. Let $\calC$ be a class of subsets of $\R^n$ (such as $\calC_\mathrm{convex}$).
We say an algorithm learns $\calC$ to error $\varepsilonilonps$ with confidence $1-\partial ta$ under $\normal^n$ if,
given a set of labeled samples $(\bx,S(\bx))$ from an unknown set $S\in \calC$ with $\bx$'s drawn
independently from $\normal^n$, the algorithm outputs with probability at least $1-\partial ta$
a hypothesis set $H\subseteq \R^n$~with $\Vol(S\bigtriangleup H)\le \varepsilonilonps$.
We say it is a \varepsilonilonmph{proper} learning algorithm if
it always outputs a hypothesis $H$ that belongs to $\calC$.
Next we recall the main algorithmic result of \cite{KOS:07}:
\begin{theorem} [Theorem~5 of \cite{KOS:07}] \label{thm:KOSlearn}
There is an algorithm $A$ that learns the class $\calC_\mathrm{convex}$ of all convex subsets of $\R^n$ to error $\varepsilonilonps$ with confidence $1-\partial ta$ under $\normal^n$ using
$$n^{O(\sqrt{n}/\varepsilonilonps^2)} \cdot \log(1/\partial ta)$$ samples\footnote{Theorem~5 as stated in \cite{KOS:07} gives a sample complexity upper bound of
$\smash{n^{O(\sqrt{n}/\varepsilonilonps^4)}}$ for \varepsilonilonmph{agnostic} learning, but inspection of the proof gives the theorem as stated here, with an upper bound of $\smash{n^{O(\sqrt{n}/\varepsilonilonps^2)}}$ for non-agnostic learning.}
drawn from $\normal^n$.
\varepsilonilonnd{theorem}
Next we recall the result of Goldreich, Goldwasser and Ron which relates proper learnability of a class $\calC$ to the testability of $\calC$.
\begin{theorem} [Proposition~3.1.1 of \cite{GGR98}, adapted to our context] \label{thm:GGRlearntest}
Let $\calC$ be a class of subsets of $\R^n$ that has a proper learning algorithm $A$ which uses $m_A(n,\varepsilonilonps,\partial ta)$ samples from $\normal^n$ to learn $\calC$ to error $\varepsilonilonps$ with confidence $1-\partial ta$. Then there is a property testing algorithm $A_{\mathrm{test}}$ for $\calC$ under the distribution $\normal^n$ that uses
\[
m_A\big(n,\varepsilonilonps/2,\partial ta/2\big) + O\big(\log(1/\partial ta)/\varepsilonilonps\big)
\]
samples drawn from $\normal^n.$
\varepsilonilonnd{theorem}
By Theorem~\ref{thm:GGRlearntest}, to obtain Theorem~\ref{thm:2sub} it suffices to have a \varepsilonilonmph{proper} learning analogue of~Theorem \ref{thm:KOSlearn}. We establish the required result, as a corollary of Theorem~\ref{thm:KOSlearn}, in the next subsection:
\begin{corollary} \label{cor:KOSlearn}
There is a proper learning algorithm $A'$ for the class $\calC_\mathrm{convex}$ of all convex subsets of $\R^n$ that uses $n^{O(\sqrt{n}/\varepsilonilonps^2)} \cdot \log(1/\partial ta)$ samples from $\normal^n$ to learn to error $\varepsilonilonps$ with confidence $1-\partial ta$.
\varepsilonilonnd{corollary}
We remark that while algorithm~$A$ from Theorem~\ref{thm:KOSlearn} runs in time $n^{O(\sqrt{n}/\varepsilonilonps^2)}$ and uses
$n^{O(\sqrt{n}/\varepsilonilonps^2)}$ samples, the algorithm $A'$ of Corollary~\ref{cor:KOSlearn} presented below has a much larger running time (at least $(n/\varepsilonilonps)^{O(n)}$); however, its sample complexity is essentially no larger than that of algorithm~$A$.
\subsection{Proof of Corollary~\ref{cor:KOSlearn}}
The idea behind the proof of Corollary~\ref{cor:KOSlearn} is simple. Let $S \subseteq \R^n$ be the unknown target convex set that is to be learned. Algorithm $A'$ first runs algorithm $A$ with error parameter \blue{$\varepsilonilonps/5$} and confidence parameter $\partial ta/2$ to obtain, with probability $1-(\partial ta/2)$, a hypothesis $H\subseteq \R^n$ with $\Vol(H \bigtriangleup S) \leq \varepsilonilonps/\blue{5}.$
In the rest of the algorithm we find with high probability a convex set $C^*$ with
$\Vol(H\bigtriangleup C^*)\le \blue{4\varepsilonilonps/5}$ and thus, we have $\Vol(S\bigtriangleup C^*)\le
\blue{\varepsilonilonps/5+4\varepsilonilonps/5=\varepsilonilonps}$.
(Note that this part of the algorithm does not require any labeled samples
$(\bx,S(\bx))$ from the oracle for $S$.)
\def\calC_{\mathrm{cover}}{\calC_{\mathrm{cover}}}
For this purpose let $\calC_{\mathrm{cover}}\subset \calC_\mathrm{convex}$ be a \varepsilonilonmph{finite $\blue{(\varepsilonilonps/5)}$-cover} of $\calC_\mathrm{convex}$. (We show in Corollary \ref{cor:epscover} below that there is an algorithm the finds a finite $(\varepsilonilonps/5)$-cover of $\calC_\mathrm{convex}$.)
Next, the algorithm $A'$ enumerates over all elements $C \in \calC_{\mathrm{cover}}$ and for each such $C$ uses random sampling from $\normal^n$ to estimate $\Vol(H \bigtriangleup C)$ to within an additive error of $\varepsilonilonps/5$, with success probability $1-\partial ta/(2|\calC_{\mathrm{cover}}|)$ for each $C$. (Note that this does not require any labeled samples $(\bx,S(\bx))$ from the oracle for $S$, since $A'$ can generate its own draws from $\normal^n$ and for each such $\bx$ it can compute $H(\bx)$ and $C(\bx)$ on its own.) $A'$ outputs the $C^* \in \calC_{\mathrm{cover}}$ for which the estimate of $\Vol(H \bigtriangleup C^*)$ is smallest.
The fact that this works follows a standard argument. Since $$\Vol(H \bigtriangleup S) \leq \varepsilonilonps/5\quad \text{and}\quad \Vol(S \bigtriangleup C') \leq \varepsilonilonps/5$$ for some set $C' \in \calC_{\mathrm{cover}}$, it holds that $\Vol(H \bigtriangleup C') \leq 2\varepsilonilonps/5$ and hence the estimate of $\Vol(H \bigtriangleup C')$ will be at most $3\varepsilonilonps/5$. Thus the element $C^*$ of $\calC_{\mathrm{cover}}$ that is selected will have its estimated value of $\Vol(H \bigtriangleup C^\ast)$ being at most $3\varepsilonilonps/5,$ which implies that its actual value of $\Vol(H \bigtriangleup C^\ast)$ will be at most $4\varepsilonilonps/5$ (since each estimate is within $\pm \varepsilonilonps/5$ of the true value).
Given the above analysis, to finish the proof of Corollary~\ref{cor:KOSlearn} it suffices to establish the following corollary of structural results proved in Sections \ref{sec:structural} and \ref{sec:setup},
which shows that indeed it is possible for $A'$ to enumerate over the elements of $\calC_{\mathrm{cover}}$ as described above:
\begin{corollary}\label{cor:epscover}
\hspace{-0.03cm}There is an algorithm that, on inputs $\varepsilonilonps$ and $n$, outputs a finite $\varepsilonilonps$-cover of $\calC_\mathrm{convex}$.
\varepsilonilonnd{corollary}
\begin{proof}
We recall the material and parameter settings from Section~\ref{sec:setup}.
Since every convex set in $\R^n$ is $(\varepsilonilon/4)$-close to a set in $\calC'_{\mathrm{convex}}$,
it suffices to describe a finite family $\calC$ of convex sets $C_1,C_2,\dots$ such that every $C \in \calC'_{\mathrm{convex}}$ is
$(3\varepsilonilon/4)$-close to some $C_i$ in $\calC$.
We claim that
\[
\calC = \big\{\hspace{0.02cm}\Conv(\cup_{\mathrm{Cube} \in Q} \mathrm{Cube}) \mid Q \subseteq \mathrm{Cube}Set\hspace{0.03cm}\big\}
\]
is such a family. To see this, fix any convex body $C \in \calC'_{\mathrm{convex}}$.
Let $$Q_C = \big\{\hspace{0.02cm}\mathrm{Cube} \in \mathrm{Cube}Set \mid \mathrm{Cube} \subseteq C\hspace{0.02cm}\big\},$$ the set of cubes that are entirely contained in $C$. Note that $\Conv(Q_C)$ is a subset of $C$. If a $\mathrm{Cube}$ contains at least one point in $C$ and at least one point outside $C$, then every point in~$\mathrm{Cube}$ has distance at most $\varepsilonilonll\sqrt{n}$ from the boundary of $C$ (since any two points in a given $\mathrm{Cube}$ have distance at most $\varepsilonilonll\sqrt{n}$). Thus, the missing volume $C \setminus \Conv(Q_C)$ is completely contained in $\partial C + \mathrm{Ball}(\varepsilonilonll\sqrt{n})$, whose Gaussian volume, by Theorem~\ref{thm:surfacevolume}, is at most $20\hspace{0.03cm}
n^{5/8}\hspace{0.03cm} n'\sqrt{\varepsilonilonll \sqrt{n}} \ll 3\varepsilonilonps/4.$
\varepsilonilonnd{proof}
\begin{flushleft}
\varepsilonilonnd{flushleft}
\appendix
\section{Proof of Lemmas~\ref{lem:noball}, \ref{lem:small}, and \ref{lem:large}} \label{ap:lemmas}
\begin{lemma}\label{lem:noball}
If $C \subset \R^n$ is convex and contains no ball of radius $\rho$, then we have $$\Vol\big(C + \mathrm{Ball}(\alpha)\big) \leq 2(n\rho + \alpha).$$
\varepsilonilonnd{lemma}
\begin{proof}
By the theorem of John \cite{John48} (see also Theorem~3.1 of \cite{Ball:intro-convex}), there is a unique ellipsoid contained in $C$ that has maximal Euclidean volume; let us denote this by $E(C).$
Since $C$ does not contain a ball of radius $\rho$, $E(C)$ must have some axis $u$ which has length less than $\rho$. Let us translate $C$ so that the center of $E(C)$ lies at the origin.
Again by the theorem from~John \red{(see the discussion in \cite{Ball:intro-convex} on pages 13 and 16),} we have that $C \subseteq nE(C)$. Now consider the set $H$ of all points $v \in \R^n$ whose projection onto the $u$ direction has magnitude at most $n\rho + \alpha$. This is a ``thickened hyperplane'' which contains $C + \mathrm{Ball}(\alpha),$ and its Gaussian volume is given by $$\Vol(H)=\int_{-(n \rho + \alpha)}^{(n\rho + \alpha)} \varphi(x)\,dx,$$ where $\varphi(x)$ is the density function of a univariate normal distribution as defined in Section~\ref{sec:prelims}. We know that $\phi$ is bounded from above by $1$ so this integral is at most $2(n\rho + \alpha)$. It is also easy to see that the same volume upper bound must hold upon undoing the translation of $C$ back to its original position, and the lemma is proved.
\varepsilonilonnd{proof}
\begin{lemma}\label{lem:small}
Let $C$ be a bounded convex subset of $\R^n$ that contains $\mathrm{Ball}(\rho)$, the origin-centered ball of radius $\rho$, for some $\blue{\rho > \alpha}$. Then the distance between $(1- ({\alpha}/{\rho}))C$ and $\partial C$ is at least $\alpha$.
\varepsilonilonnd{lemma}
\begin{proof}
This is essentially Lemma~2.2 of \cite{Kern}; for completeness we give the simple proof here.
Let $\beta=\alpha/\rho$. Let $z \in \partial C$ be a point on the boundary of $C$. Since $C$ is convex and contains the origin, there exists a vector $v$ for which $v \cdot z = 1$ but for all $x \in C$ we have $v \cdot x \leq 1$ (intuitively, one can think of $v$ as defining the tangent hyperplane at $z$). Then for any $y \in (1 - \beta) C$ we have $v \cdot y \leq 1 - \beta$, which implies that $v (z - y) \geq \beta.$ Since ${{\rho v} /{\|v\|}} \in \mathrm{Ball}(\rho) \subseteq C$, it must be the case that $v \cdot {({\rho v} /{\|v\|})} = \rho \|v\| \leq 1$, which means that $\|v\| \leq 1/ \rho$ and thus (as $v (z - y) \geq \beta$) $\|z-y\| \geq \alpha.$
\varepsilonilonnd{proof}
\begin{lemma}\label{lem:large}
Let $C \subset \R^n $ be a convex set that satisfies $\sup_{c\in C} \|c\| \leq K$ for some $K >1$. Then for any $0 < \beta < 1$, every point $v \in \partial C + \mathrm{Ball}(\alpha)$ is within distance $\blue{2K\beta + \alpha}$ of a point in $(1 - \beta)C$.
\varepsilonilonnd{lemma}
\begin{proof}
We have that $v = c + y$ for some $c \in \partial C$ and $y$ with $\|y\| \leq \alpha$.
While $v$ may not lie in $C$ (as $C$ might be an open set), we know for any $\varepsilonilonps>0$ there is
a point $c'\in C$ and $\|c'-c\|\le \varepsilonilonps$.
Take such a point $c'$ with $\varepsilonilonps=\beta K$.
Then $(1-\beta)c'\in (1-\beta)C$ and
$$
\|(1-\beta)c'-v\|=\|(1-\beta)c'-c-y\|\le \|c'-c\|+\beta\|c'\|+\|y\|\le \beta K+\beta K+\alpha
=2\beta K+\alpha.
$$
This finishes the proof of the lemma.
\varepsilonilonnd{proof}
\varepsilonilonnd{document} |
\begin{document}
\tolerance2500
\title{\Large{\textbf{On some groupoids of small orders with Bol-Moufang type of identities}}}
\author{\normalsize {Vladimir Chernov, Alexander Moldovyan, Victor Shcherbacov}
}
\maketitle
\begin{abstract}
We count number of groupoids of order 3 with some Bol-Moufang type identities.
\noindent \textbf{2000 Mathematics Subject Classification:} 20N05 20N02
\noindent \textbf{Key words and phrases:} groupoid, Bol-Moufang type identity.
\end{abstract}
\section{Introduction}
A binary groupoid $(G, \cdot)$ is a non-empty set $G$ together with a binary operation \lq\lq $\cdot$\rq\rq. This definition is very general, therefore usually groupoids with some identities are studied. For example, groupoids with identity associativity (semi-groups) are researched.
We continue the study of groupoids with some Bol-Moufang type identities \cite{NOVIKOV_08, VD, 2017_Scerb}.
Here we present results published in \cite{CHErnov, CHErnov_2018}.
{\bf Definition.} \label{Bol_Moufang_TYpe_Id}
Identities that involve three variables, two of which appear once on both sides of the equation and one of which appears twice on both sides are called Bol-Moufang type identities. \index{identity!Bol-Moufang type}
Various properties of Bol-Moufang type identities in quasigroups and loops are studied in \cite{Fenyves_1, Ph_2005, Cote, AKHTAR}.
Groupoid $(Q, \ast)$ is called a quasigroup, if the following conditions are true \cite{VD}:
$(\forall u, v \in Q) (\exists ! \, x, y \in Q) (u * x = v \, \& \, y * u = v)$.
For groupoids the following natural problems are researched: how many groupoids with some identities of small order there exist?
A list of numbers of semigroups of orders up to 8 is given in \cite{Satoh}; a list of numbers of quasigroups up to 11 is given in \cite{HOP, WIKI_44}.
\section{Some results}
Original algorithm is elaborated and corresponding program is written for generating of groupoids of small (2 and 3) orders with some Bol-Moufang identities, which are well known in quasigroup theory.
To verify the correctness of the written program the number of semigroups of order 3 was counted. Obtained result coincided with well known, namely, there exist 113 semigroups of order 3.
The following identities have the property that any of them define a commutative Moufang loop \cite{BRUCK_46, VD, HOP, 2017_Scerb} in the class of loops: left (right) semimedial identity, Cote identity and its dual identity, Manin identity and its dual identity or in the class of quasigroups (identity (\ref{Comm_Muf_quas_Id}) and its dual identity).
\subsection{Groupoids with left semi-medial identity}
Left semi-medial identity in a groupoid $(Q, \ast)$ has the following form: $xx*yz=xy*xz$.
Bruck \cite{BRUCK_46, VD, 2017_Scerb} uses namely this identity to define commutative Moufang loops in the class of loops.
There exist 10 left semi-medial groupoids of order 2. There exist 7 non-isomorphic left semi-medial groupoids of order 2. The first five of them are semigroups \cite{WIKI_44}.
\[
\begin{array}{lcrr}
\begin{array}{l|ll}
\ast&1&2\\
\hline
1&1&1\\
2&1&1\\
\end{array}
&
\begin{array}{l|ll}
\star&1&2\\
\hline
1&1&1\\
2&1&2\\
\end{array}
&
\begin{array}{l|ll}
\circ&1&2\\
\hline
1&1&1\\
2&2&2\\
\end{array}
&
\begin{array}{l|ll}
\cdot&1&2\\
\hline
1&1&2\\
2&1&2\\
\end{array}
\end{array}
\]
\[
\begin{array}{lcr}
\begin{array}{l|ll}
\diamond&1&2\\
\hline
1&1&2\\
2&2&1\\
\end{array}
&
\begin{array}{l|ll}
\odot&1&2\\
\hline
1&2&1\\
2&2&1\\
\end{array}
&
\begin{array}{l|ll}
\bullet&1&2\\
\hline
1&2&2\\
2&1&1\\
\end{array}
\end{array}
\]
There exist 399 left semi-medial groupoids of order 3.
The similar results are true for groupoids with right semi-medial identity $xy*zz=xz*yz$. It is clear that the identities of left and right semi-mediality are dual. In other language they are (12)-parastrophes of each other \cite{VD, 2017_Scerb}.
It is clear that groupoids with dual identities have similar properties, including the number of groupoids of a fixed order.
\subsection{Groupoids with Cote identity}
Identity $x(xy*z) = (z*xx)y$ is discovered in \cite{Cote}. Here we name this identity Cote identity.
There exist 6 groupoids of order 2 with Cote identity. There exist 3 non-isomorphic in pairs groupoids of order 2 with Cote identity.
There exist 99 groupoids of order 3 with Cote identity.
The similar results are true for groupoids with the following identity $(z\ast yx)x = y(xx \ast z)$. The last identity is (12)-parastrophe of Cote identity.
\subsection{Groupoids with Manin identity}
The identity $x(y*xz) = (xx*y)z$ we call Manin identity \cite{MANIN}. The following identity is dual identity to Manin identity: $(zx\ast y)x = z(y\ast xx)$.
There exist 10 groupoids of order 2 with Manin identity. There exist 7 non-isomorphic in pairs groupoids of order 2 with Manin identity.
There exist 167 groupoids of order 3 with Manin identity.
\subsection{Groupoids with identity $(xy\ast x)z = (y\ast xz) x$ \label{Comm_Muf_quas_Id} (identity (\ref{Comm_Muf_quas_Id}))}
Some properties of identity (\ref{Comm_Muf_quas_Id}) are given in \cite{VS_2014_Kiev, 2017_Scerb}.
The following identity is dual identity to identity (\ref{Comm_Muf_quas_Id}): $z(x\ast yx) = x(zx\ast y)$.
There exist 6 groupoids of order 2 with identity (\ref{Comm_Muf_quas_Id}).
There exist 3 non-isomorphic in pairs groupoids of order 2 with (\ref{Comm_Muf_quas_Id}) identity. Any of these groupoids is a semigroup.
There exist 117 groupoids of order 3 with identity (\ref{Comm_Muf_quas_Id}).
\subsection{Number of groupoids of order 3 with some identities}
We count number of groupoids of order 3 with some identities.
We use list of Bol-Moufang type identities given in \cite{Cote}. In Table 1 we present number of groupoids of order 3 with the respective identity.
\begin{table}
\centering
\caption{Number of groupoids of order 3 with some identities.}
\footnotesize{
\[
\begin{array}{|c||c| c| c| c|}
\hline
Name & Abbreviation & Identity & Number \\
\hline\hline
Semigroups & SGR & x(yz) = (xy)z & 113\\
\hline
Extra & EL & x(y(zx)) = ((xy)z)x & 239\\
\hline
Moufang & ML & (xy)(zx) = (x(yz))x & 196\\
\hline
Left Bol & LB & x(y(xz)) = (x(yx))z & 215\\
\hline
Right Bol & RB & y((xz)x) = ((yx)z)x & 215\\
\hline
C-loops & CL & y(x(xz)) = ((yx)x)z & 133\\
\hline
LC-loops & LC & (xx)(yz) = (x(xy))z & 220\\
\hline
RC-loops & RC & y((zx)x) = (yz)(xx) & 220\\
\hline
Middle Nuclear Square & MN & y((xx)z) = (y(xx))z & 350\\
\hline
Right Nuclear Square & RN & y(z(xx)) = (yz)(xx) & 932\\
\hline
Left Nuclear Square & LN & ((xx)y)z = (xx)(yz) & 932\\
\hline
Comm. Moufang & CM & (xy)(xz) = (xx)(zy) & 297\\
\hline
Abelian Group & AG & x(yz) = (yx)z & 91\\
\hline
Comm. C-loop & CC & (y(xy))z = x(y(yz)) & 169\\
\hline
Comm. Alternative & CA & ((xx)y)z = z(x(yx)) & 110\\
\hline
Comm. Nuclear square & CN & ((xx)y)z = (xx)(zy) & 472\\
\hline
Comm. loops & CP & ((yx)x)z = z(x(yx)) & 744\\
\hline
Cheban \, 1 & C1 & x((xy)z) = (yx)(xz) & 219\\
\hline
Cheban \, 2 & C2 & x((xy)z) = (y(zx))x & 153\\
\hline
Lonely \, I & L1 & (x(xy))z = y((zx)x) & 117\\
\hline
Cheban\, I\, Dual & CD & (yx)(xz) = (y(zx))x & 219\\
\hline
Lonely \, II & L2 & (x(xy))z = y((xx)z) & 157\\
\hline
Lonely \, III & L3 & (y(xx))z = y((zx)x) & 157\\
\hline
Mate \, I & M1 & (x(xy))z = ((yz)x)x & 111\\
\hline
Mate \, II & M2 & (y(xx))z = ((yz)x)x & 196\\
\hline
Mate \, III & M3 & x(x(yz)) = y((zx)x) & 111\\
\hline
Mate \, IV & M4 & x(x(yz)) = y((xx)z) & 196\\
\hline
Triad \, I & T1 & (xx)(yz) = y(z(xx)) & 162\\
\hline
Triad \, II & T2 & ((xx)y)z = y(z(xx)) & 180\\
\hline
Triad \, III & T3 & ((xx)y)z = (yz)(xx) & 162\\
\hline
Triad \, IV & T4 & ((xx)y)z = ((yz)x)x & 132\\
\hline
Triad \, V & T5 & x(x(yz)) = y(z(xx)) & 132\\
\hline
Triad \, VI & T6 & (xx)(yz) = (yz)(xx) & 1419\\
\hline
Triad \, VII & T7 & ((xx)y)z = ((yx)x)z & 428\\
\hline
Triad \, VIII & T8 & (xx)(yz) = y((zx)x) & 120\\
\hline
Triad \, IX & T9 & (x(xy))z = y(z(xx)) & 102\\
\hline
Frute & FR & (x(xy))z = (y(zx))x & 129\\
\hline
Crazy Loop & CR & (x(xy))z = (yx)(xz) & 136\\
\hline
Krypton & KL & ((xx)y)z = (x(yz))x & 268\\
\hline
\end{array}
\]}
\end{table}
\textbf{Acknowledgments.} Authors thank Dr. V.D. Derech for his information on semigroups of small orders.
\begin{center}
\begin{parbox}{118mm}{\footnotesize
Vladimir Chernov$^{1}$, Nicolai Moldovyan$^{2}$,
Victor Shcherbacov$^{3}$
\noindent $^{1}$Master/Shevchenko Transnistria State University
\noindent Email: [email protected]
\noindent $^{2}$Professor/St. Petersburg Institute for Informatics and Automation of Russian Academy of Sciences
\noindent Email: [email protected]
\noindent $^{3}$Principal Researcher/Institute of Mathematics and Computer Science of Moldova
\noindent Email: [email protected]
}
\end{parbox}
\end{center}
\end{document} |
\begin{document}
\title{\bf Efficient coordination mechanisms\\for unrelated machine scheduling\thanks{A preliminary version of the results of this paper appeared in {\em Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}
\begin{abstract}
We present three new coordination mechanisms for scheduling $n$ selfish jobs on $m$ unrelated machines.
A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules
by defining a local scheduling policy on each machine. The scheduling policies induce a game among
the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum
given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of
the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the
efficiency of pure Nash equilibria (price of anarchy) of the induced game.
Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily
process the jobs in an uninterrupted way and may introduce some idle time.
Our first coordination mechanism has approximation ratio $\Theta(\log m)$ and always guarantees that the induced game has pure Nash equilibria
to which the system converges in at most $n$ rounds. This result improves a bound of $O(\log^2 m)$ due to Azar, Jain,
and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of
the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are
anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs
that have different load characteristics.
Our second mechanism handles anonymous jobs and has approximation ratio $O\left(\frac{\log m}{\log \log m}\right)$
although the game induced is not a potential game and, hence, the existence of pure Nash
equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower
bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles
anonymous jobs and has a nice ``cost-revealing'' potential function.
We use this potential function in order, not only to prove the existence of equilibria, but also to upper-bound the price of stability of the induced game by $O(\log m)$ and
the price of anarchy by $O(\log^2m)$. Our third coordination
mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential
game and has bounded price of anarchy. In order to obtain the above bounds, our coordination mechanisms use $m$ as a parameter. Slight variations of these mechanisms in which this information is not necessary achieve approximation ratios of $O\left(m^{\epsilon}\right)$, for any constant $\epsilon>0$.
\end{abstract}
\section{Introduction}
We study the classical problem of {\em unrelated machine scheduling}.
In this problem, we have $m$ parallel machines and $n$ independent jobs. Job $i$ induces a (possibly infinite)
positive processing time (or load) $w_{ij}$ when processed by machine $j$. The load of a machine is the total
load of the jobs assigned to it. The quality of an assignment of jobs to machines is measured by the makespan (i.e., the maximum) of the machine loads
or, alternatively, the maximum completion time among all jobs. The optimization problem of computing an assignment
of minimum makespan is a fundamental APX-hard problem, quite well-understood in terms of its offline \cite{LST93}
and online approximability \cite{AAFPW93,ANR95}.
The approach we follow in this paper is both algorithmic and game-theoretic. We assume that each job is owned by a selfish agent.
This gives rise to a {\em selfish scheduling} setting where each agent aims to minimize the completion time of her job
with no regard to the global optimum. Such a selfish behaviour can lead to inefficient schedules from which no agent
has an incentive to unilaterally deviate in order to improve the completion time of her job. From the algorithmic point of view, the designer of such a system can define a {\em coordination mechanism} \cite{CKN04}, i.e., a
{\em scheduling policy} within each machine in order to ``coordinate'' the selfish behaviour of the jobs.
Our main objective is to design coordination mechanisms that guarantee that the assignments reached by the selfish
agents are {\em efficient}.
\noindent {\bf The model.}
A scheduling policy simply defines the way jobs are scheduled within a machine and can be either {\em non-preemptive} or
{\em preemptive}. Non-preemptive scheduling policies process jobs uninterruptedly according to some order. Preemptive
scheduling policies do not necessarily have this feature and can also introduce some idle time (delay). Although this seems
unnecessary at first glance, as we show in this paper, it is a very useful tool in order to guarantee coordination.
A coordination mechanism is a set of scheduling policies running on the machines. In the sequel, we use the terms coordination
mechanisms and scheduling policies interchangeably.
A coordination mechanism defines (or induces) a game with the job owners as players. Each job has all machines as possible {\em strategies}.
We call an {\em assignment} (of jobs to machines) or {\em state} any set of strategies selected by the players, with one strategy
per player. Given
an assignment of jobs to machines, the cost of a player is the completion time of her job on the machine it has
been assigned to; this completion time depends on the scheduling policy on that machine and the characteristics of all jobs
assigned to that machine. Assignments in which no player has an incentive to change her strategy in order to decrease
her cost given the assignments of the other players are called {\em pure Nash equilibria}. The global objective that is used in order to assess the efficiency of
assignments is the {\em maximum completion time} over all jobs. A related quantity is the {\em makespan} (i.e., the maximum
of the machine loads). Notice that when preemptive scheduling policies are used, these two quantities may not be
the same (since idle time contributes to the completion time but not to the load of a machine). However, the optimal makespan is a lower bound on the optimal maximum completion time. The {\em price of anarchy} \cite{P01} is
the maximum over all pure Nash equilibria of the ratio of the maximum completion time among all jobs over the optimal makespan.
The {\em price of stability} \cite{ADKTWR04} is the minimum over all pure Nash equilibria of the ratio of the maximum completion time among
all jobs over the optimal makespan. The {\em approximation ratio} of a coordination mechanism is the maximum of the price
of anarchy of the induced game over all input instances.
Four natural coordination mechanisms are the {\sf Makespan}, {\sf Randomized}, {\sf LongestFirst}, and {\sf ShortestFirst}.
In the {\sf Makespan} policy, each machine processes the jobs assigned to it ``in parallel'' so that the completion time of each
job is the total load of the machine. {\sf Makespan} is obviously a preemptive coordination mechanism.
In the {\sf Randomized} policy, the jobs are scheduled non-preemptively in random order. Here, the cost of each player
is the expected completion time of her job. In the {\sf ShortestFirst} and {\sf LongestFirst} policies, the jobs assigned
to a machine are scheduled in non-decreasing and non-increasing order of their processing times, respectively. In case of
ties, a {\em global ordering} of the jobs according to their distinct IDs is used. This is necessary by any deterministic
non-preemptive coordination mechanism in order to be well-defined. Note that no such information is required
by the {\sf Makespan} and {\sf Randomized} policies; in this case, we say that they handle {\em anonymous jobs}. According to
the terminology of \cite{AJM08}, all these
four coordination mechanisms are {\em strongly local} in the sense that the only information required by each machine
in order to compute a schedule are the processing times of the jobs assigned to it. A {\em local} coordination mechanism
may use all parameters (i.e., the load vector) of the jobs assigned to the same machine.
Designing coordination mechanisms with as small approximation ratio as possible is our main concern. But there are other
issues related to efficiency. The price of anarchy is meaningful only in games where pure Nash equilibria exist. So, the primary goal
of the designer of a coordination mechanism should be that the induced game {\em always has}
pure Nash equilibria. Furthermore, these equilibria should be {\em easy to find}. A very interesting class of games in which
the existence of pure Nash equilibria is guaranteed is that of {\em potential} games. These games have the property that
a {\em potential function} can be defined on the states of the game so that in any two states differing in the strategy of a
single player, the difference of the values of the potential function
and the difference of the cost of the player have the same sign. This property guarantees that the state with minimum potential
is a pure Nash equilibrium. Furthermore, it guarantees that, starting from any state, the system will reach (converge to)
a pure Nash equilibrium after a finite number of {\em selfish moves}. Given a game, its {\em Nash dynamics}
is a directed graph with the states of the game as nodes and edges connecting two states differing in the strategy of a single player if
that player has an incentive to change her strategy according to the direction of the edge. The Nash dynamics of potential
games do not contain any cycle. Another desirable property here is {\em fast} convergence,
i.e., convergence to a pure Nash equilibrium in a polynomial number of selfish moves. A particular type of selfish moves
that have been extensively considered in the literature \cite{AAEMS08,CMS06,FFM08,MV04} is that of {\em best-response} moves.
In a best-response move, a player having an incentive to change her strategy selects the strategy that yields the maximum
decrease in her cost.
Potential games are strongly related to {\em congestion games} introduced by Rosenthal \cite{R73}. Rosenthal presented a potential
function for these games with the following property: in any two states differing in the strategy of a single player,
the difference of the values of the potential function {\em equals} the difference of the cost of the player. Monderer and Shapley
\cite{MS96} have proved that each potential game having this property is isomorphic to a congestion game. We point out that potential
functions are not the only way to guarantee the existence of pure Nash equilibria. Several generalizations of congestion games such as those with
player-specific latency functions \cite{M96} are not potential games but several subclasses of them provably have pure Nash equilibria.
\noindent {\bf Related work.} The study of the price of anarchy of games began with the seminal work of Koutsoupias and Papadimitriou \cite{KP99}
and has played a central role in the recently emerging field of Algorithmic Game Theory \cite{NRTV07}. Several papers provide bounds on the price of anarchy
of different games of interest. Our work follows a different direction where the price of anarchy is the {\em objective
to be minimized} and, in this sense, it is similar in spirit to studies where the main question is how to change the
rules of the game at hand in order to improve the price of anarchy. Typical examples are the introduction of taxes or tolls in congestion games \cite{CKK06,CDR06,FJM04,KK04,S07},
protocol design in network and cost allocation games \cite{CRV08,KLO95}, Stackelberg routing strategies \cite{KS06,KLO97,KM02,R04,S07}, and network design \cite{R06}.
Coordination mechanisms were introduced by Christodoulou, Koutsoupias, and Nanavati in \cite{CKN04}. They study the case where each player
has the same load on each machine and, among other results, they consider the {\sf LongestFirst} and {\sf ShortestFirst} scheduling policies.
We note that the {\sf Makespan} and {\sf Randomized} scheduling policies were used in \cite{KP99} as models of selfish behaviour in scheduling,
and since that paper, the {\sf Makespan} policy has been considered as standard in the study of selfish scheduling games in models simpler than
the one of unrelated machines and is strongly related to
the study of congestion games (see \cite{V07,R07} and the references therein). Immorlica et al. \cite{ILMS05} study these four scheduling policies
under several scheduling settings including
the most general case of unrelated machines. They prove that the {\sf Randomized} and {\sf ShortestFirst} policies have
approximation ratio $O(m)$ while the {\sf LongestFirst} and {\sf Makespan} policies have unbounded approximation ratio.
Some scheduling policies are also related to earlier studies of local-search scheduling heuristics. So, the fact that the
price of anarchy of the induced game may be unbounded follows by the work of Schuurman and Vredeveld \cite{SV01}.
As observed in \cite{ILMS05}, the equilibria of the game induced by {\sf ShortestFirst} correspond to the solutions of the {\sf ShortestFirst}
scheduling heuristic which is known to be $m$-approximate \cite{IK77}.
The {\sf Makespan} policy is known to induce potential games \cite{EKM03}. The {\sf ShortestFirst} policy also induces potential
games as proved in \cite{ILMS05}. In Section \ref{sec:b-cm}, we present examples showing that the scheduling policies {\sf LongestFirst} and {\sf Randomized} do not
induce potential games.\footnote{After the appearance of the conference version of the paper, we became aware of two independent proofs that {\sf Longest-First} may induce games that do not have pure Nash equilibria \cite{DT09,FRS10}.}
Azar et al. \cite{AJM08} study non-preemptive coordination mechanisms for unrelated machine scheduling. They prove that
any local non-preemptive coordination mechanism is at least $\Omega(\log m)$-approximate\footnote{The corresponding proof of \cite{AJM08} contained a error which has been recently fixed by Fleischer and Svitkina \cite{FS10}.} while any strongly local
non-preemptive coordination mechanism is at least $\Omega(m)$-approximate; as a corollary, they solve an old open
problem concerning the approximation ratio of the {\sf ShortestFirst} heuristic.
On the positive side, the authors of \cite{AJM08} present a non-preemptive
local coordination mechanism (henceforth called {\sf AJM-1}) that is $O(\log m)$-approximate although it may induce games without pure Nash equilibria.
The extra information used by this scheduling policy is the {\em inefficiency} of jobs (defined in the next section).
They also present a technique that transforms this coordination mechanism to a preemptive one that induces potential games
with price of anarchy $O(\log^2 m)$. In their mechanism, the players converge to a pure Nash equilibrium in $n$ rounds of best-response
moves. We will refer to this coordination mechanism as {\sf AJM-2}. Both {\sf AJM-1} and {\sf AJM-2} use the IDs of the jobs.
\noindent {\bf Our results.} We present three new coordination mechanisms for unrelated machine scheduling.
Our mechanisms are deterministic, preemptive, and local. The schedules in each machine are computed as functions of the characteristics
of jobs assigned to the machine, namely the load of jobs on the machine and their inefficiency. In all cases, the
functions use an integer parameter $p\geq 1$; the best
choice of this parameter for our coordination mechanisms is $p=O(\log m)$. Our analysis is heavily based on the convexity of simple polynomials and geometric inequalities for Euclidean norms.
\begin{table}
\centerline{\footnotesize
\begin{tabular}{|l||c|c|c|c|l|}
\hline
Coordination & & & & & \\
mechanism & PoA & Pot. & PNE & IDs & Characteristics \\\hline\hline
{\sf ShortestFirst} & $\Theta(m)$ & Yes & Yes & Yes & Strongly local, non-preemptive\\\hline
{\sf LongestFirst} & unbounded & No & No & Yes & Strongly local, non-preemptive\\\hline
{\sf Makespan} & unbounded & Yes & Yes & No & Strongly local, preemptive\\\hline
{\sf Randomized} & $\Theta(m)$ & No & ? & No & Strongly local, non-preemptive\\\hline
{\sf AJM-1} & $\Theta(\log m)$ & No & No & Yes & Local, non-preemptive\\\hline
{\sf AJM-2} & $O(\log^2m)$ & Yes & Yes & Yes & Local, preemptive, uses $m$ \\\hline\hline
{\sf ACOORD} & $\Theta(\log m)$ & Yes & Yes & Yes & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & Yes & Yes & Yes & Local, preemptive \\\hline
{\sf BCOORD} & $O\left(\frac{\log m}{\log\log m}\right)$ & No & ? & No & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & No & ? & No & Local, preemptive \\\hline
{\sf CCOORD} & $O(\log^2m)$ & Yes & Yes & No & Local, preemptive, uses $m$ \\
& $O(m^\epsilon)$ & Yes & Yes & No & Local, preemptive \\\hline
\end{tabular}}
\label{tab:comparison}
\caption{Comparison of our coordination mechanisms to previously known ones with respect to the price of anarchy of the
induced game (PoA), whether they induced potential games or not (Pot.), the existence of pure Nash equilibria (PNE),
and whether they use the job IDs or not.}
\end{table}
Motivated by previous work, we first consider the scenario where jobs have distinct IDs.
Our first coordination mechanism {\sf ACOORD} uses this information and is superior to the known coordination mechanisms
that induce games with pure Nash equilibria. The game induced
is a potential game, has price of anarchy $\Theta(\log m)$, and the players converge to pure Nash equilibria in at most $n$
rounds. Essentially, the equilibria of the game induced by {\sf ACOORD}
can be thought of as the solutions produced by the application of a particular online algorithm, similar to the greedy online
algorithm for minimizing the $\ell_p$ norm
of the machine loads \cite{AAFPW93,C08}. Interestingly, the local objective of the greedy online algorithm for the $\ell_p$ norm
may not translate to a completion time of jobs in feasible schedules; the online algorithm implicit by {\sf ACOORD} uses
a different local objective that meets this constraint. The related results are presented in Section \ref{sec:a-cm}.
Next we address the case where no ID information is associated to the jobs (anonymous jobs). This scenario is relevant
when the job owners do not wish to reveal their identities or in large-scale settings where distributing IDs to jobs is infeasible.
Definitely, an advantage that could be used for coordination is lost in this way but this makes the problem of designing coordination mechanisms
more challenging. In Section \ref{sec:b-cm}, we present our second coordination mechanism {\sf BCOORD} which induces a simple congestion game with player-specific polynomial latency
functions of a particular form. The price of anarchy of this game is only $O\left(\frac{\log m}{\log \log m}\right)$.
This result demonstrates that preemption may be useful in order to beat the $\Omega(\log m)$ lower bound of \cite{AJM08}
for non-preemptive coordination mechanisms. On the negative side, we show that the game induced may not be a potential
game by presenting an example where the Nash dynamics have a cycle.
Our third coordination mechanism {\sf CCOORD} is presented in Section \ref{sec:c-cm}. The scheduling policy on each machine
uses an interesting function on the loads of the jobs assigned to the machine and their inefficiency. The game induced by {\sf CCOORD} is a potential
game; the associated potential function is ``cost-revealing'' in the sense that it can be used to upper-bound the cost of equilibria.
In particular, we show that the price of stability
of the induced game is $O(\log m)$ and the price of anarchy is $O(\log^2 m)$. The coordination
mechanism {\sf CCOORD} is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential
game and has bounded price of anarchy. Table \ref{tab:comparison} compares our coordination mechanisms to the previously known ones.
Observe that the dependence of the parameter $p$ on $m$ requires that our mechanisms use the number of machines as input. By setting $p$ equal to an appropriately large constant, our mechanisms achieve price of anarchy $O(m^\epsilon)$ for any constant $\epsilon>0$. In particular, the coordination mechanisms {\sf ACOORD} and {\sf CCOORD} are the first ones that do not use the number of machines as a parameter, induce games with pure Nash equilibria, and have price of anarchy $o(m)$.
We remark that the current paper contains several improvements compared to its conference version. There, the three coordination mechanisms had the restriction that a job with inefficiency more than $m$ on some machine has infinite completion time when assigned to that machine. Here, we have removed this restriction and have adapted the analysis accordingly. A nice consequence of the new definition is that the coordination mechanisms can now be defined so that they do not use the number of machines as a parameter.
Furthermore, the definition of {\sf ACOORD} has been significantly simplified. Also, the analysis of the price of anarchy of the coordination mechanism {\sf BCOORD} in the conference version used a technical lemma which is implicit in \cite{STZ04}. In the current version, we present a different self-contained proof that is based on convexity properties of polynomials and Minkowski inequality; the new proof has a similar structure with the analysis of the price of anarchy of mechanism {\sf ACOORD}.
We begin with preliminary technical definitions in Section \ref{sec:prelim} and conclude with interesting open questions in Section
\ref{sec:open}.
\section{Preliminaries}\label{sec:prelim}
In this section, we present our notation and give some statements that will be useful later.
We reserve $n$ and $m$ for the number of jobs and machines, respectively, and the indices $i$ and $j$
for jobs and machines, respectively. Unless specified otherwise, the sums $\sum_i$ and $\sum_j$ run over
all jobs and over all machines, respectively. Assignments are denoted by $N$ or $O$. With some abuse in
notation, we use $N_j$ to denote both the set of jobs assigned to machine $j$ and the set of their loads
on machine $j$. We use the notation $L(N_j)$ to denote the load of machine $j$ under the assignment $N$.
More generally, $L(A)$ denotes the sum of the elements for any set of non-negative reals $A$.
For an assignment $N$ which assigns job $i$ to machine $j$, we denote the completion time
of job $i$ under a given scheduling policy by ${\cal P}(i,N_j)$.
Note that, besides defining the completion times, we do not discuss the particular way the jobs are scheduled by
the scheduling policies we present. However, we require that {\em feasible} schedules are computable efficiently. A natural
sufficient and necessary condition is the following: for any job $i\in N_j$, the total load of jobs with
completion time at most ${\cal P}(i,N_j)$ is at most ${\cal P}(i,N_j)$.
Our three coordination mechanisms use the inefficiency of jobs in order to compute schedules. We
denote by $w_{i,\min}$ the minimum load of job $i$ over all machines. Then, its inefficiency $\rho_{ij}$
on machine $j$ is defined as $\rho_{ij}=w_{ij}/w_{i,\min}$.
Our proofs are heavily based on the convexity of simple polynomials such as $z^k$ for $k\geq 1$ and on the relation
of Euclidean norms of the machine loads and the makespan. Recall that the $\ell_k$ norm of the machine loads for an
assignment $N$ is $\left(\sum_j{L(N_j)^k}\right)^{1/k}$. The proof of the next lemma is trivial.
\begin{lemma}\label{lem:lp-norm}
For any assignment $N$,
$\max_j{L(N_j)} \leq \left(\sum_j{L(N_j)^k}\right)^{1/k} \leq m^{1/k} \max_j{L(N_j)}$.
\end{lemma}
In some of the proofs, we also use the Minkowski inequality (or the triangle inequality for the $\ell_p$ norm).
\begin{lemma}[Minkowski inequality]\label{lem:minkowski}
$\left(\sum_{t=1}^s{(a_t+b_t)^k}\right)^{1/k} \leq \left(\sum_{t=1}^s{a_t^k}\right)^{1/k}+\left(\sum_{t=1}^s{b_t^k}\right)^{1/k}$, for any $k\geq 1$ and $a_t,b_t\geq 0$.
\end{lemma}
The following two technical lemmas are used in some of our proofs. We include them here for easy reference.
\begin{lemma}\label{lem:convexity}
Let $r\geq 1$, $t\geq 0$ and $a_i\geq 0$, for $i=1, ..., k$. Then,
\[\sum_{i=1}^k{\left(\left(t+a_i\right)^r-t^r\right)}\leq
\left(t+\sum_{i=1}^k{a_i}\right)^r-t^r\]
\end{lemma}
\begin{proof}
The case when $a_i=0$ for $i=1, ..., k$ is trivial. Assume otherwise
and let $\xi=\sum_{i=1}^{k}{a_i}$ and $\xi_i=a_i/\xi$. Clearly,
$\sum_{i=1}^k{\xi_i}=1$. By the convexity of function $z^r$ in
$[0,\infty)$, we have that
\begin{eqnarray}\nonumber
(t+a_i)^r &= &
\left((1-\xi_i)t+\xi_i\left(t+\sum_{i=1}^k{a_i}\right)\right)^r\\\label{eq:convexity}
&\leq &
(1-\xi_i)t^r+\xi_i\left(t+\sum_{i=1}^k{a_i}\right)^r
\end{eqnarray}
for $i=1, ..., k$. Using (\ref{eq:convexity}), we obtain
\begin{eqnarray*}
\sum_{i=1}^k{\left((t+a_i)^r-t^r\right)} &\leq &
t^r\left(\sum_{i=1}^k{(1-\xi_i)}-k\right)+\left(t+\sum_{i=1}^k{a_i}\right)^r\sum_{i=1}^k{\xi_i}\\
&=& \left(t+\sum_{i=1}^k{a_i}\right)^r-t^r
\end{eqnarray*}
\qed\end{proof}
\begin{lemma}\label{lem:slope}
For any $z_0\geq 0$, $\alpha\geq 0$, and $p\geq 1$, it holds \[(p+1)\alpha z_0^p \leq (z_0+\alpha)^{p+1}-z_0^{p+1}\leq (p+1)\alpha (z_0+\alpha)^p.\]
\end{lemma}
\begin{proof}
The inequality trivially holds if $\alpha=0$. If $\alpha>0$, the inequality follows since, due to the convexity of the function $z^{p+1}$, the slope of the line that crosses points $(z_0,z_0^{p+1})$ and $(z_0+\alpha,(z_0+\alpha)^{p+1})$ is between its derivative at points $z_0$ and $z_0+\alpha$.\qed\end{proof}
We also refer to the multinomial and binomial theorems. \cite{HLP52} provides an extensive overview of the
inequalities we use and their history (see also {\tt wikipedia.org} for a quick survey).
\section{The coordination mechanism {\sf ACOORD}}\label{sec:a-cm}
The coordination mechanism {\sf ACOORD} uses a global ordering of the jobs according to their distinct IDs.
Without loss of generality, we may assume that the index of a job is its ID.
Let $N$ be an assignment and denote by $N^i$ the restriction of $N$ to the jobs with the $i$ smallest IDs.
{\sf ACOORD} schedules job $i$ on machine $j$ so that it completes at time
\[{\cal P}(i,N_j) = \left(\rho_{ij}\right)^{1/p} L(N^i_j).\]
Since $\rho_{ij}\geq 1$, the schedules produced are always feasible.
Consider the sequence of jobs in increasing order of their IDs and assume that each job plays a best-response
move. In this case, job $i$ will select that machine $j$ so that the quantity $\left(\rho_{ij}\right)^{1/p} L(N^i_j)$
is minimized. Since the completion time of job $i$ depends only on jobs with smaller IDs,
no job will have an incentive to change its strategy and the resulting assignment is a pure Nash
equilibrium. The following lemma extends this observation in a straightforward way.
\begin{lemma}\label{lem:a-cm-potential-convergence}
The game induced by the coordination mechanism {\sf ACOORD} is a potential game.
Furthermore, any sequence of $n$ rounds of best-response moves
converges to a pure Nash equilibrium.
\end{lemma}
\begin{proof}
Notice that since a job does not affect the completion time of jobs with smaller IDs, the vector of completion times of the jobs (sorted in increasing order of their IDs) decreases
lexicographically when a job improves its cost by deviating to another strategy and, hence, it is a potential
function for the game induced by the coordination mechanism {\sf ACOORD}.
Now, consider $n$ rounds of best-response moves of the jobs in the induced game such that each job plays at least
once in each round. It is not hard to see that after round $i$, the job $i$ will have selected that machine $j$
so that the quantity $\left(\rho_{ij}\right)^{1/p} L(N^i_j)$ is minimized. Since the completion time of job
$i$ depends only on jobs with smaller IDs, job $i$ has no incentive to move after round $i$ and, hence, no job
will have an incentive to change its strategy after the $n$ rounds. So, the resulting assignment is a pure Nash equilibrium.
\qed
\end{proof}
The sequence of best-response moves mentioned above can be thought of as an online algorithm that processes the jobs in increasing
order of their IDs. The local objective is slightly different that the local objective of the greedy online
algorithm for minimizing the $\ell_{p+1}$ norm of the machine loads \cite{AAG+95,C08}; in that algorithm, job $i$ is assigned
to a machine $j$ so that the quantity $(L(N^{i-1}_j)+w_{ij})^{p+1}-L(N^{i-1}_j)^{p+1}$ is minimized. Here, we remark that we do not see how the
local objective of that algorithm could be simulated by a scheduling policy that always produces feasible schedules.
This constraint is trivially satisfied by the coordination mechanism {\sf ACOORD}. The next lemma bounds the maximum completion time at pure Nash equilibria in terms of the $\ell_{p+1}$ norm of the machine loads and the optimal makespan.
\begin{lemma}\label{lem:completion-acoord}
Let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanisms {\sf ACOORD} and let $O$ be an optimal assignment. Then
\[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.\]
\end{lemma}
\begin{proof}
Let $i^*$ be the job that has the maximum completion time in assignment $N$. Denote by $j_1$ the machine $i^*$ uses in $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf ACOORD} yields
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\
&=& L(N_{j_1}^{i^*})\\
&\leq & L(N_{j_1})\\
&\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}.
\end{eqnarray*}
Otherwise, since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, we have
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\
&\leq & {\cal P}(i^*,N_{j_2}\cup \{w_{i^*j_2}\})\\
&=& L(N_{j_2}^{i^*})+w_{i^*j_2}\\
&\leq & L(N_{j_2})+\min_j{w_{i^*j}}\\
&\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.
\end{eqnarray*}
\qed
\end{proof}
Next we show that the approximation ratio of {\sf ACOORD} is $O(\log m)$ (for well-selected values of the
parameter $p$). The analysis borrows and extends techniques
from the analysis of the greedy online algorithm for the $\ell_p$ norm in \cite{C08}.
\begin{theorem}\label{thm:a-cm-poa}
The price of anarchy of the game induced by the coordination mechanism {\sf ACOORD} with $p=\Theta(\log m)$ is $O(\log m)$. Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf ACOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$.
\end{theorem}
\begin{proof}
Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, by the definition of {\sf ACOORD} we have that
\begin{eqnarray*}\left(\rho_{ij_1}\right)^{1/p} L(N_{j_1}^i) &\leq & \left(\rho_{ij_2}\right)^{1/p} \left(L(N_{j_2}^{i-1})+w_{ij_2}\right).
\end{eqnarray*}
Equivalently, by raising both sides to the power $p$ and multiplying with $w_{i,\min}$, we have that
\begin{eqnarray*}w_{ij_1} L(N_{j_1}^i)^p &\leq & w_{ij_2} \left(L(N_{j_2}^{i-1})+w_{ij_2}\right)^p.\end{eqnarray*}
Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether job $i$ is assigned to machine $j$ in the assignment $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$), respectively,
or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express this last inequality as follows.
\begin{eqnarray*}
\sum_j{x_{ij}w_{ij}L(N_{j}^i)^p} &\leq &\sum_j{y_{ij}w_{ij}\left(L(N_{j}^{i-1})+w_{ij}\right)^{p}}
\end{eqnarray*}
By summing over all jobs and multiplying with $(e-1)(p+1)$, we have
\begin{eqnarray}\nonumber
& & (e-1)(p+1) \sum_i\sum_j{x_{ij}w_{ij}L(N_j^i)^p}\\\nonumber
&\leq & (e-1)(p+1) \sum_i\sum_j{y_{ij}w_{ij}\left(L(N_j^{i-1})+w_{ij}\right)^p}\\\nonumber
&\leq & (e-1)(p+1) \sum_j\sum_i{y_{ij}w_{ij}\left(L(N_j)+w_{ij}\right)^p}\\\nonumber
&= & (e-1)(p+1) \sum_j\sum_i{y_{ij}w_{ij}\left(L(N_j)+y_{ij}w_{ij}\right)^p}\\\nonumber
&\leq & \sum_j\sum_i{\left(\left(L(N_j)+ey_{ij}w_{ij}\right)^{p+1}-\left(L(N_j)+y_{ij}w_{ij}\right)^{p+1}\right)}\\\nonumber
&\leq & \sum_j\sum_i{\left(\left(L(N_j)+ey_{ij}w_{ij}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber
&\leq & \sum_j{\left(\left(L(N_j)+e\sum_i{y_{ij}w_{ij}}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber
&=& \sum_j{\left(L(N_j)+eL(O_j)\right)^{p+1}}-\sum_j{L(N_j)^{p+1}}\\\label{eq:a-minkowski}
&\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+e\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}}.
\end{eqnarray}
The second inequality follows by exchanging the sums and since $L(N_j^{i-1})\leq L(N_j)$, the first equality follows since $y_{ij}\in \{0,1\}$, the third inequality follows by applying Lemma \ref{lem:slope} with $\alpha=(e-1)y_{ij}w_{ij}$ and $z_0=L(N_j)+y_{ij}w_{ij}$, the fourth inequality is obvious, the fifth inequality follows by Lemma \ref{lem:convexity}, the second equality follows since the definition of the variables $y_{ij}$ implies that $L(O_j)=\sum_i{y_{ij}w_{ij}}$, and the last inequality follows by Minkowski inequality (Lemma \ref{lem:minkowski}).
Now, we will relate the $\ell_{p+1}$ norm of the machines loads of assignments $N$ and $O$. We have
\begin{eqnarray*}
(e-1)\sum_j{L(N_j)^{p+1}} &=& (e-1)\sum_j{L(N_j^n)^{p+1}}\\
&=& (e-1)\sum_{i=1}^n{\sum_j{\left(L(N_j^i)^{p+1}-L(N_j^{i-1})^{p+1}\right)}}\\
&=& (e-1)\sum_{i=1}^n{\sum_j{\left(L(N_j^i)^{p+1}-(L(N_j^{i})-x_{ij}w_{ij})^{p+1}\right)}}\\
&\leq & (e-1)(p+1)\sum_i\sum_j{x_{ij}w_{ij}L(N_j^i)^p}\\
&\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+e\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}}.
\end{eqnarray*}
The first two equalities are obvious (observe that $L(N_j^0)=0$), the third one follows by the definition of variables $x_{ij}$, the first inequality follows by applying Lemma \ref{lem:slope} with $\alpha=x_{ij}w_{ij}$ and $z_0=L(N_j^{i})-x_{ij}w_{ij}$, and the last inequality follows by inequality (\ref{eq:a-minkowski}).
So, the above inequality yields
\begin{eqnarray*}
\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} &\leq & \frac{e}{e^{\frac{1}{p+1}}-1} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\
&\leq & e(p+1) \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\
&\leq & e(p+1)m^{\frac{1}{p+1}} \max_j{L(O_j)}.
\end{eqnarray*}
The second inequality follows since $e^z\geq z+1$ for $z\geq 0$ and the third one follows by Lemma \ref{lem:lp-norm}.
Now, using Lemma \ref{lem:completion-acoord} and this last inequality, we obtain that
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &\leq &\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}\\
&\leq & \left(e(p+1)m^{\frac{1}{p+1}}+1\right) \max_j{L(O_j)}.
\end{eqnarray*}
The desired bounds follow by setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively. \qed
\end{proof}
Our logarithmic bound is asymptotically tight; this follows by the connection to online algorithms mentioned above and the lower bound of \cite{ANR95}.
\section{The coordination mechanism {\sf BCOORD}}\label{sec:b-cm}
We now turn our attention to coordination mechanisms that handle anonymous jobs. We define the coordination mechanism {\sf BCOORD} by slightly changing the definition of {\sf ACOORD} so that the completion time of a job does not depend on its ID. So, {\sf BCOORD} schedules job $i$ on machine $j$ so that it finishes at time $${\cal P}(i,N_j)=\left(\rho_{ij}\right)^{1/p}L(N_j).$$ Since $\rho_{ij}\geq 1$, the schedules produced are always feasible.
The next lemma bounds the maximum completion time at pure Nash equilibria (again in terms of the $\ell_{p+1}$ norm of the machine loads and the optimal makespan).
\begin{lemma}\label{lem:completion-bcoord}
Let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanisms {\sf BCOORD} and let $O$ be an optimal assignment. Then
\[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.\]
\end{lemma}
\begin{proof}
The proof is almost identical to the proof of Lemma \ref{lem:completion-acoord}; we include it here for completeness. Let $i^*$ be the job that has the maximum completion time in assignment $N$. Denote by $j_1$ the machine $i^*$ uses in $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf BCOORD} yields
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\
&=& L(N_{j_1})\\
&\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}.
\end{eqnarray*}
Otherwise, since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, we have
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\
&\leq & {\cal P}(i^*,N_{j_2}\cup \{w_{i^*j_2}\})\\
&=& L(N_{j_2})+w_{i^*j_2}\\
&= & L(N_{j_2})+\min_j{w_{i^*j}}\\
&\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}.
\end{eqnarray*}\qed\end{proof}
We are ready to present our upper bounds on the price of anarchy of the induced game.
\begin{theorem}
The price of anarchy of the game induced by the coordination mechanism {\sf BCOORD} with $p=\Theta(\log m)$ is $O\left(\frac{\log m}{\log \log m}\right)$.
Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf BCOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$.
\end{theorem}
\begin{proof}
Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that
is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, we have that
\begin{eqnarray*}(\rho_{ij_1})^{1/p}L(N_{j_1}) &\leq & (\rho_{ij_2})^{1/p}(L(N_{j_2})+w_{ij_2}).
\end{eqnarray*}
Equivalently, by raising both sides to the power $p$ and multiplying both sides with $w_{i,\min}$, we have that
\begin{eqnarray*}w_{ij_1}L(N_{j_1})^p &\leq &w_{ij_2}(L(N_{j_2})+w_{ij_2})^p.\end{eqnarray*}
Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether job $i$ is assigned to machine $j$ in the assignments $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$), respectively,
or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express this last inequality as follows:
\begin{eqnarray*}
\sum_j{x_{ij}w_{ij}L(N_j)^p} &\leq & \sum_j{y_{ij}w_{ij}(L(N_j)+w_{ij})^p}.
\end{eqnarray*}
By summing over all jobs and multiplying with $p$, we have
\begin{eqnarray}\nonumber
& & p\sum_i{\sum_j{x_{ij}w_{ij}L(N_j)^p}}\\\nonumber
&\leq & p\sum_i{\sum_j{y_{ij}w_{ij}(L(N_j)+w_{ij})^p}}\\\nonumber
&=& p\sum_j{\sum_i{y_{ij}w_{ij}(L(N_j)+y_{ij}w_{ij})^p}}\\\nonumber
&\leq & \sum_j\sum_i{\left(\left(L(N_j)+\frac{2p+1}{p+1}y_{ij}w_{ij}\right)^{p+1}-\left(L(N_j)+y_{ij}w_{ij}\right)^{p+1}\right)}\\\nonumber
&\leq & \sum_j\sum_i{\left(\left(L(N_j)+\frac{2p+1}{p+1}y_{ij}w_{ij}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber
&\leq & \sum_j{\left(\left(L(N_j)+\frac{2p+1}{p+1}\sum_i{y_{ij}w_{ij}}\right)^{p+1}-L(N_j)^{p+1}\right)}\\\nonumber
&=& \sum_j{\left(\left(L(N_j)+\frac{2p+1}{p+1}L(O_j)\right)^{p+1}-L(N_j)^{p+1}\right)}\\\label{eq:b-minkowski}
&\leq& \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+\frac{2p+1}{p+1}\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}-\sum_j{L(N_j)^{p+1}}
\end{eqnarray}
The first equality follows by exchanging the sums and since $y_{ij}\in \{0,1\}$, the second inequality follows by applying Lemma \ref{lem:slope} with $\alpha=\frac{p}{p+1}y_{ij}w_{ij}$ and $z_0=L(N_j)+y_{ij}w_{ij}$, the third inequality is obvious, the fourth inequality follows by applying Lemma \ref{lem:convexity}, the second equality follows since the definition of variables $y_{ij}$ implies that $L(O_j)=\sum_i{y_{ij}w_{ij}}$, and the last inequality follows by applying Minkowski inequality (Lemma \ref{lem:minkowski}).
Now, we relate the $\ell_{p+1}$ norm of the machine loads of assignments $N$ and $O$. We have
\begin{eqnarray*}
(p+1)\sum_j{L(N_j)^{p+1}} &=& p\sum_j{L(N_j)^{p+1}}+ \sum_j{L(N_j)^{p+1}}\\
&=& p\sum_i{\sum_j{x_{ij}w_{ij}L(N_j)^p}} + \sum_j{L(N_j)^{p+1}}\\
&\leq & \left(\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}}+\frac{2p+1}{p+1}\left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\right)^{p+1}.
\end{eqnarray*}
The first equality is obvious, the second one follows by the definition of variables $x_{ij}$ and the inequality follows by inequality (\ref{eq:b-minkowski}).
So, the above inequalities yield
\begin{eqnarray*}
\left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} &\leq & \frac{2p+1}{p+1} \frac{1}{(p+1)^{\frac{1}{p+1}}-1} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\
&\leq & \frac{2p+1}{\ln{(p+1)}} \left(\sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}\\
&\leq & \frac{2p+1}{\ln{(p+1)}} m^{\frac{1}{p+1}} \max_j{L(O_j)}.
\end{eqnarray*}
The second inequality follows since $e^z\geq z+1$ for $z\geq 0$ and the third one follows by Lemma \ref{lem:lp-norm}.
Now, using Lemma \ref{lem:completion-bcoord} and our last inequality we have
\begin{eqnarray*}
\max_{j,i\in N_j}{{\cal P}(i,N_j)} &\leq & \left(\sum_j{L(N_j)^{p+1}}\right)^{\frac{1}{p+1}} +\max_j{L(O_j)}\\
&\leq & \left(1+\frac{2p+1}{\ln{(p+1)}}m^{\frac{1}{p+1}}\right)\max_j{L(O_j)}.
\end{eqnarray*}
The desired bounds follows by setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively.\qed
\end{proof}
Note that the game induced by {\sf BCOORD} with $p=1$ is the same with the
game induced by the coordination mechanism {\sf CCOORD} (with $p=1$) that we present in the next section. As such,
it also has a potential function (also similar to the potential function of \cite{FKS05} for linear weighted congestion
games) as we will see in Lemma \ref{lem:psi-potential}. In this way, we obtain a coordination mechanism that induces a potential game, handles anonymous jobs, and has aproximation ratio $O(\sqrt{m})$. Unfortunately,
the next theorem demonstrates that, for higher values of $p$, the Nash dynamics of the game induced
by {\sf BCOORD} may contain a cycle.
\begin{theorem}\label{thm:no-potential}
The game induced by the coordination mechanism {\sf BCOORD} with $p=2$ is not a potential game.
\end{theorem}
Before proving Theorem \ref{thm:no-potential}, we show that the games induced by the coordination mechanisms
{\sf LongestFirst} and {\sf Randomized} may not be potential games either. All the instances presented in the following consist of four machines
and three basic jobs $A$, $B$, and $C$. In each case, we show that the Nash dynamics contain a cycle of moves
of the three basic jobs.
First consider the {\sf LongestFirst} policy and the instance depicted in the following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|}
\hline
& A & B & C \\
$1$ & $14$ & $\infty$ & $5$ \\
$2$ & $\infty$ & $10$ & $\infty$ \\
$3$ & $3$ & $9$ & $10$ \\
$4$ & $7$ & $8$ & $9$ \\
\hline
\end{tabular}}
\
The cycle is defined on the following states:
\begin{eqnarray*}
& & (C,\underline{B},A,) \rightarrow(C,,\underline{A}B,)\rightarrow(C,,\underline{B},A)\rightarrow(C,,,\underline{A}B)\rightarrow (A\underline{C},,,B)\rightarrow\\
& & (\underline{A},,C,B)\rightarrow(,,A\underline{C},B)\rightarrow(,,A,\underline{B}C)\rightarrow (,B,A,\underline{C})\rightarrow(C,B,A,).
\end{eqnarray*}
Notice that the first and last assignment are the same. In each state, the player that moves next is underlined. Job $B$ is at machine $2$
in the first assignment and has completion time $10$. Hence, it has an incentive to move to machine $3$ (second assignment) where its completion
time is $9$. Job $A$ has completion time $12$ in the second assignment since it is scheduled after job $B$ which has higher load on machine $3$.
Moving to machine $4$ (third assignment), it decreases its completion time to $7$. The remaining moves in the cycle can be verified accordingly.
The instance for the {\sf Randomized} policy contains four additional jobs $D$, $E$, $F$, and $G$ which are always scheduled on
machines $1$, $2$, $3$, and $4$, respectively (i.e., they have infinite load on the other machines). It is depicted in the
following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
& A & B & C & D & E & F & G\\
$1$ & $80$ & $\infty$ & $100$ & $2$ & $\infty$ & $\infty$ & $\infty$ \\
$2$ & $\infty$ & $171$ & $\infty$ & $\infty$ & $2$ & $\infty$ & $\infty$ \\
$3$ & $2$ & $154$ & $124$ & $\infty$ & $\infty$ & $32$ & $\infty$ \\
$4$ & $2$ & $76$ & $10$ & $\infty$ & $\infty$ & $\infty$ & 184\\
\hline
\end{tabular}}
\
The cycle is defined by the same moves of the basic jobs as in the case of {\sf LongestFirst}:
\begin{eqnarray*}
& & (CD,\underline{B}E,AF,G) \rightarrow(CD,E,\underline{A}BF,G)\rightarrow(CD,E,\underline{B}F,AG)\rightarrow(CD,E,F,\underline{A}BG)\rightarrow\\
& & (A\underline{C}D,E,F,BG)\rightarrow(\underline{A}D,E,CF,BG)\rightarrow(D,E,A\underline{C}F,BG)\rightarrow(D,E,AF,\underline{B}CG)\rightarrow\\
& & (D,BE,AF,\underline{C}G)\rightarrow(CD,BE,AF,G).
\end{eqnarray*}
Recall that (see \cite{ILMS05,KP99}) the expected completion time of a job $i$ which is scheduled on machine $j$ in an assignment $N$ is $\frac{1}{2}(w_{ij}+L(N_j))$ when the {\sf Randomized} policy is used.
In each state, the player that moves next is underlined. It can be easily verified that each player in this cycle improves her cost by exactly $1$.
For example, job $B$ has expected completion time $\frac{1}{2}(171+171+2)=172$ at machine $2$ in the first assignment and, hence, an incentive to move to machine $3$
in the second assignment where its completion time is $\frac{1}{2}(154+2+154+32)=171$.
\paragraph{Proof of Theorem \ref{thm:no-potential}.}
Besides the three basic jobs, the instance for the {\sf BCOORD} policy with $p=2$ contains two additional jobs $D$ and $E$ which are always scheduled on
machines $3$ and $4$, respectively. The instance is depicted in the following table.
\
\centerline{
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& A & B & C & D & E \\
$1$ & $4.0202$ & $\infty$ & $4.0741$ & $\infty$ & $\infty$ \\
$2$ & $\infty$ & $8.2481$ & $\infty$ & $\infty$ & $\infty$ \\
$3$ & $0.0745$ & $0.6302$ & $0.3078$ & $29.1331$ & $\infty$ \\
$4$ & $2.4447$ & $5.1781$ & $2.4734$ & $\infty$ & $2.7592$ \\
\hline
\end{tabular}}
\
The cycle is defined by the same moves of the basic jobs as in the previous cases:
\[(C,\underline{B},AD,E)\rightarrow (C,,\underline{A}BD,E)\rightarrow(C,,\underline{B}D,AE)\rightarrow(C,,D,\underline{A}BE)\rightarrow(A\underline{C},,D,BE)\rightarrow\]
\[(\underline{A},,CD,BE)\rightarrow(,,A\underline{C}D,BE)\rightarrow(,,AD,\underline{B}CE)\rightarrow(,B,AD,\underline{C}E)\rightarrow(C,B,AD,E).\]
Notice that, instead of considering the
completion time $\left(\rho_{ij}\right)^{1/p}L(N_j)$ of a job $i$ on machine $j$ in an assignment $N$, it is equivalent to consider its cost as $w_{ij}L(N_j)^p$.
In this way, we can verify that in any of the moves in the above cycle, the job that moves improves its cost. For example, job $B$
has cost $8.2481^3=561.127758090641$ on machine $2$ in the first assignment and cost $0.6302(0.0745+0.6302+29.1331)^2=561.063473430968$ on machine $3$ in the second assignment.
\qed
\section{The coordination mechanism {\sf CCOORD}}\label{sec:c-cm}
In this section we present and analyze the coordination mechanism {\sf CCOORD} that handles anonymous jobs
and guarantees that the induced game has pure Nash equilibria, price of anarchy at most $O(\log^2 m)$, and price
of stability $O(\log m)$. In order to define the scheduling policy, we first define an interesting family of functions.
\begin{defn}
For integer $k\geq 0$, the function $\Psi_k$ mapping finite sets of reals to the reals is defined as follows:
$\Psi_k(\emptyset)=0$ for any integer $k\geq 1$, $\Psi_0(A)=1$ for any (possibly empty) set $A$, and for any
non-empty set $A=\{a_1, a_2, ..., a_n\}$ and integer $k\geq 1$,
\[\Psi_k(A)=k! \sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}.\]
\end{defn}
So, $\Psi_k(A)$ is essentially the sum of all possible monomials of total degree $k$ on the elements of $A$. Each
term in the sum has coefficient $k!$. Clearly, $\Psi_1(A)=L(A)$. For $k\geq 2$, compare $\Psi_k(A)$ with $L(A)^k$
which can also be expressed as the sum of the same terms, albeit with different coefficients in $\{1, ..., k!\}$,
given by the multinomial theorem.
The coordination mechanism {\sf CCOORD} schedules job $i$ on machine $j$ in an assignment $N$ so that its completion time
is $${\cal P}(i,N_j) = \left(\rho_{ij}\Psi_p(N_j)\right)^{1/p}.$$
Our proofs extensively use the properties in the next lemma; its proof is given in appendix. The first inequality implies that the schedule defined by {\sf CCOORD}
is always feasible.
\begin{lemma}\label{lem:properties}
For any integer $k\geq 1$, any finite set of non-negative reals $A$, and any non-negative real $b$ the following hold:
\[\begin{array}{l l}
\mbox{a. } L(A)^k \leq \Psi_k(A) \leq k! L(A)^k & \mbox{d. } \Psi_k(A\cup\{b\}) -\Psi_k(A) = k b\Psi_{k-1}(A\cup \{b\})\\
\mbox{b. } \Psi_{k-1}(A)^{k} \leq \Psi_{k}(A)^{k-1} & \mbox{e. } \Psi_k(A) \leq kL(A)\Psi_{k-1}(A)\\
\mbox{c. } \Psi_k(A\cup\{b\}) = \sum_{t=0}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)} &\mbox{f. } \Psi_k(A\cup \{b\}) \leq \left(\Psi_k(A)^{1/k}+\Psi_k(\{b\})^{1/k}\right)^k
\end{array}\]
\end{lemma}
The second property implies that $\Psi_k(A)^{1/k} \leq \Psi_{k'}(A)^{1/k'}$ for any integer $k'\geq k$.
The third property suggests an algorithm for computing $\Psi_k(A)$ in time polynomial in $k$ and $|A|$
using dynamic programming.
A careful examination of the definitions of the coordination mechanisms {\sf BCOORD} and {\sf CCOORD} and property (a) in the above lemma, reveals that {\sf CCOORD} makes the completion time of a job assigned to machine $j$ dependent on the approximation $\Psi_p(N_j)^{1/p}$ of the load $L(N_j)$ of the machine instead of its exact load as {\sf BCOORD} does. This will be the crucial tool in order to guarantee that the induced game is a potential game without significantly increasing the price of anarchy. The next lemma defines a potential function on the states of the induced game that will be very useful later.
\begin{lemma}\label{lem:psi-potential}
The function $\Phi(N)=\sum_j{\Psi_{p+1}(N_j)}$ is a potential function for the game induced by the coordination mechanism {\sf CCOORD}.
Hence, this game always has a pure Nash equilibrium.
\end{lemma}
\begin{proof}
Consider two assignments $N$ and $N'$ differing in the strategy of the player controlling job $i$. Assume that job
$i$ is assigned to machine $j_1$ in $N$ and to machine $j_2\not=j_1$ in $N'$. Observe that $N_{j_1}=N'_{j_1}\cup \{w_{ij_1}\}$ and
$N'_{j_2}=N_{j_2}\cup \{w_{ij_2}\}$. By Lemma \ref{lem:properties}d, we have that $\Psi_{p+1}(N_{j_1})-\Psi_{p+1}(N'_{j_1})=(p+1)w_{ij_1}\Psi_p(N_{j_1})$
and $\Psi_{p+1}(N'_{j_2})-\Psi_{p+1}(N_{j_2})=(p+1)w_{ij_2}\Psi_p(N'_{j_2})$. Using these properties and the definitions of the coordination
mechanism {\sf CCOORD} and function $\Phi$,
we have
\begin{eqnarray*}
\Phi(N)-\Phi(N') &=& \sum_j{\Psi_{p+1}(N_j)-\sum_j{\Psi_{p+1}(N'_j)}}\\
&=& \Psi_{p+1}(N_{j_1})+\Psi_{p+1}(N_{j_2})-\Psi_{p+1}(N'_{j_1})-\Psi_{p+1}(N'_{j_2})\\
&=& (p+1)w_{ij_1}\Psi_{p}(N_{j_1})-(p+1)w_{ij_2}\Psi_{p}(N'_{j_2})\\
&=& (p+1)w_{i,\min} \left({\cal P}(i,N_{j_1})^p-{\cal P}(i,N'_{j_2})^p\right)
\end{eqnarray*}
which means that the difference of the potentials of the two assignments and the difference of
the completion time of player $i$ have the same sign as desired.
\qed
\end{proof}
The next lemma relates the maximum completion time of a pure Nash equilibrium to the optimal makespan provided that their potentials are close.
\begin{lemma}\label{lem:completion-time}
Let $O$ be an optimal assignment and let $N$ be a pure Nash equilibrium of the game induced by the coordination mechanism {\sf CCOORD} such that $\left(\Phi(N)\right)^{\frac{1}{p+1}}\leq \gamma\left(\Phi(O)\right)^{\frac{1}{p+1}}$. Then,
\[\max_{j,i\in N_j}{{\cal P}(i,N_j)} \leq \left(\gamma (p+1)m^{\frac{1}{p+1}}+p\right)\max_j{L(O_j)}.\]
\end{lemma}
\begin{proof}
Let $i^*$ be the job that has the maximum completion time in $N$. Denote by $j_1$ the machine $i^*$ uses in assignments $N$ and let $j_2$ be a machine such that $\rho_{i^*j_2}=1$. If $j_1=j_2$, the definition of the coordination mechanism {\sf CCOORD} and Lemma \ref{lem:properties}b yield
\begin{eqnarray}\nonumber
\max_{j, i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\\nonumber
&= & \Psi_p(N_{j_1})^{1/p}\\\nonumber
&\leq & \Psi_{p+1}(N_{j_1})^{\frac{1}{p+1}}\\\label{eq:c-same-machines}
&\leq & \left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}.
\end{eqnarray}
Otherwise, since player $i$ has no incentive to use machine $j_2$ instead of $j_1$, we have
\begin{eqnarray}\nonumber
\max_{j, i\in N_j}{{\cal P}(i,N_j)} &=& {\cal P}(i^*,N_{j_1})\\\nonumber
&\leq & {\cal P}(i^*, N_{j_2} \cup \{w_{i^*j_2}\}) \\\nonumber
&=& \Psi_p(N_{j_2}\cup\{w_{i^*j_2}\})^{1/p}\\\nonumber
&\leq & \Psi_p(N_{j_2})^{1/p}+\Psi_p(\{w_{i^*j_2}\})^{1/p}\\\nonumber
&\leq & \Psi_{p+1}(N_{j_2})^{\frac{1}{p+1}}+(p!)^{1/p}w_{i^*j_2}\\\nonumber
&= & \Psi_{p+1}(N_{j_2})^{\frac{1}{p+1}}+(p!)^{1/p}\min_j{w_{i^*j}}\\\label{eq:c-diff-machines}
&\leq & \left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}.
\end{eqnarray}
The first two equalities follows by the definition of {\sf CCOORD}, the first inequality follows since player $i^*$ has no incentive to use machine $j_2$ instead of $j_1$, the second inequality follows by Lemma \ref{lem:properties}f, the third inequality follows by Lemma \ref{lem:properties}b and the definition of function $\Psi_p$, the third equality follows by the definition of machine $j_2$ and the last inequality is obvious.
Now, observe that the term in parenthesis in the rightmost side of inequalities (\ref{eq:c-same-machines}) and (\ref{eq:c-diff-machines}) equals the potential $\Phi(N)$. Hence, in any case, we have
\begin{eqnarray*}
\max_{j, i\in N_j}{{\cal P}(i,N_j)} &\leq& (\Phi(N))^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\
&\leq & \gamma (\Phi(O))^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\
&=& \gamma\left(\sum_j{\Psi_{p+1}(O_j)}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\
&\leq &\gamma\left((p+1)! \sum_j{L(O_j)^{p+1}}\right)^{\frac{1}{p+1}}+p\max_j{L(O_j)}\\
&\leq & \left(\gamma (p+1)m^{\frac{1}{p+1}}+p\right)\max_j{L(O_j)}.
\end{eqnarray*}
The second inequality follows by the inequality on the potentials of assignments $N$ and $O$, the equality follows by the definition of the potential function $\Phi$, the third inequality follows by Lemma \ref{lem:properties}a and the last one follows by Lemma \ref{lem:lp-norm}.
\qed
\end{proof}
A first application of Lemma \ref{lem:completion-time} is in bounding the price of stability of the induced game.
\begin{theorem}\label{lem:stability}
The game induced by the coordination mechanism {\sf CCOORD} with $p=\Theta(\log m)$ has price of stability at most $O(\log m)$.
\end{theorem}
\begin{proof}
Consider the optimal assignment $O$ and the pure Nash equilibrium $N$ of minimum potential. We have
$\left(\Phi(N)\right)^{\frac{1}{p+1}}\leq \left(\Phi(O)\right)^{\frac{1}{p+1}}$ and, using Lemma \ref{lem:completion-time},
we obtain that the maximum completion time in $N$ is at most $(p+1)m^{\frac{1}{p+1}}+p$ times the makespan of $O$.
Setting $p=\Theta(\log m)$,
the theorem follows.
\qed
\end{proof}
A second application of Lemma \ref{lem:completion-time} is in bounding the price of anarchy. In order to apply it, we need a relation between the potential of an equilibrium and the potential of an optimal assignment; this is provided by the next lemma.
\begin{lemma}\label{lem:equilibrium}
Let $O$ be an optimal assignment and $N$ be a pure Nash equilibrium of the game induced by the coordination mechanism {\sf CCOORD}. Then,
\[\left(\Phi(N)\right)^{\frac{1}{p+1}} \leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}}.\]
\end{lemma}
\begin{proof}
Consider a pure Nash equilibrium $N$ and an optimal assignment $O$. Since no job has an incentive to change her strategy from $N$, for any job $i$ that
is assigned to machine $j_1$ in $N$ and to machine $j_2$ in $O$, we have that
\begin{eqnarray*}\left(\rho_{ij_1}\Psi_p(N_{j_1})\right)^{1/p} &\leq & \left(\rho_{ij_2}\Psi_p(N_{j_2}\cup \{w_{ij_2}\})\right)^{1/p}.
\end{eqnarray*}
Equivalently, by raising both sides to the power $p$ and multiplying both sides with $w_{i,\min}$, we have that
\begin{eqnarray*}
w_{ij_1}\Psi_p(N_{j_1}) &\leq & w_{ij_2}\Psi(N_{j_2}\cup \{w_{ij_2}\}).
\end{eqnarray*}
Using the binary variables $x_{ij}$ and $y_{ij}$ to denote whether
job $i$ is assigned to machine $j$ in the assignment $N$ ($x_{ij}=1$) and $O$ ($y_{ij}=1$) or not ($x_{ij}=0$ and $y_{ij}=0$, respectively), we can express the last inequality
as follows:
\begin{eqnarray*}
\sum_j{x_{ij}w_{ij}\Psi_p(N_{j})} &\leq & \sum_j{y_{ij}w_{ij}\Psi(N_{j}\cup \{w_{ij}\})}
\end{eqnarray*}
By summing over all jobs, we have
\begin{eqnarray*}
\sum_i{\sum_j{x_{ij}w_{ij}\Psi_p(N_{j})} } &\leq & \sum_i{\sum_j{y_{ij}w_{ij}\Psi(N_{j}\cup \{w_{ij}\})}}
\end{eqnarray*}
By exchanging the double sums and since $\sum_i{x_{ij}w_{ij}}=L(N_j)$, we obtain
\begin{eqnarray}\label{ineq:delta}
\sum_j{L(N_j)\Psi_p(N_j)} &\leq & \sum_j{\sum_{i}{y_{ij}w_{ij}\Psi_p(N_j \cup \{w_{ij}\})}}
\end{eqnarray}
We now work with the potential of assignment $N$. We have
\begin{eqnarray*}
2\Phi(N) &= & \Phi(N)+\sum_j{\Psi_{p+1}(N_j)}\\
&\leq & \Phi(N)+(p+1)\sum_j{L(N_j)\Psi_p(N_j)}\\
&\leq & \Phi(N)+ (p+1)\sum_j{\sum_i{y_{ij}w_{ij}}\Psi_p(N_j\cup \{w_{ij}\})}\\
&=& \Phi(N)+(p+1)\sum_j{\sum_i{y_{ij}w_{ij}}\sum_{t=0}^{p}{\frac{p!}{(p-t)!}\Psi_{p-t}(N_j)w_{ij}^t}}\\
&=& \Phi(N)+\sum_j{\sum_{t=0}^{p}{\frac{(p+1)!}{(p-t)!}\Psi_{p-t}(N_j)\sum_i{y_{ij}w_{ij}^{t+1}}}}\\
&\leq & \Phi(N)+\sum_j{\sum_{t=0}^{p}{\frac{(p+1)!}{(p-t)!(t+1)!}\Psi_{p-t}(N_j)\Psi_{t+1}(O_j)}}\\
&=& \Phi(N)+\sum_j{\sum_{t=1}^{p+1}{\left(\begin{array}{c}p+1\\t\end{array}\right)\Psi_{p+1-t}(N_j)\Psi_{t}(O_j)}}\\
&\leq & \Phi(N)+\sum_j{\sum_{t=1}^{p+1}{\left(\begin{array}{c}p+1\\t\end{array}\right)\Psi_{p+1}(N_j)^{\frac{p+1-t}{p+1}}\Psi_{p+1}(O_j)^{\frac{t}{p+1}}}}\\
&=& \Phi(N)+\sum_j{\left(\left(\Psi_{p+1}(N_j)^{\frac{1}{p+1}}+\Psi_{p+1}(O_j)^{\frac{1}{p+1}}\right)^{p+1}-\Psi_{p+1}(N_j)\right)}\\
&=& \Phi(N)+\sum_j{\left(\Psi_{p+1}(N_j)^{\frac{1}{p+1}}+\Psi_{p+1}(O_j)^{\frac{1}{p+1}}\right)^{p+1}}-\sum_j{\Psi_{p+1}(N_j)}\\
&\leq & \left(\left(\sum_j{\Psi_{p+1}(N_j)}\right)^{\frac{1}{p+1}}+\left(\sum_j{\Psi_{p+1}(O_j)}\right)^{\frac{1}{p+1}}\right)^{p+1}\\
&=& \left(\left(\Phi(N)\right)^{\frac{1}{p+1}}+\left(\Phi(O)\right)^{\frac{1}{p+1}}\right)^{p+1}
\end{eqnarray*}
The first inequality follows by Lemma \ref{lem:properties}e, the second inequality follows by inequality (\ref{ineq:delta}), the second equality follows by Lemma \ref{lem:properties}c, the third equality follows by exchanging the sums,
the third inequality follows since the jobs $i$ assigned to machine $j$ are those for which $y_{ij}=1$ and by the definition of function $\Psi_{t+1}$ which yields that
$\Psi_{t+1}(O_j)\geq (t+1)! \sum_i{y_{ij}w_{ij}^{t+1}}$, the fourth equality follows by updating the limits of the sum over $t$, the fourth inequality follows by Lemma \ref{lem:properties}b,
the fifth equality follows by the binomial theorem, the sixth equality is obvious, the fifth inequality follows by Minkowski inequality (Lemma \ref{lem:minkowski}) and by the definition of the potential $\Phi(N)$, and the last equality follows by the definition of the potentials $\Phi(N)$ and $\Phi(O)$.
By the above inequality, we obtain that
\begin{eqnarray*}
\left(\Phi(N)\right)^{\frac{1}{p+1}} &\leq & \frac{1}{2^{\frac{1}{p+1}}-1} \left(\Phi(O)\right)^{\frac{1}{p+1}}\leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}}
\end{eqnarray*}
where the last inequality follows using the inequality $e^z\geq z+1$.
\qed
\end{proof}
We are now ready to bound the price of anarchy.
\begin{theorem}\label{thm:c-cm-poa}
The price of anarchy of the game induced by the coordination mechanism {\sf CCOORD} with $p=\Theta(\log m)$ is $O\left(\log^2 m\right)$. Also, for every constant $\epsilon\in (0,1/2]$, the price of anarchy of the game induced by the coordination mechanism {\sf CCOORD} with $p=1/\epsilon-1$ is $O\left(m^\epsilon\right)$.
\end{theorem}
\begin{proof}
Consider a pure Nash equilibrium $N$ and let $O$ be the optimal assignment.
Using Lemma \ref{lem:equilibrium}, we have that $\left(\Phi(N)\right)^{\frac{1}{p+1}} \leq \frac{p+1}{\ln{2}} \left(\Phi(O)\right)^{\frac{1}{p+1}}$.
Hence, by Lemma \ref{lem:completion-time}, we obtain that the maximum completion time in $N$ is at most $\frac{(p+1)^2}{\ln 2}m^{\frac{1}{p+1}}+p$ times the makespan of $O$. By setting $p=\Theta(\log m)$ and $p=1/\epsilon-1$, respectively, the theorem follows.
\qed
\end{proof}
\section{Discussion and open problems}\label{sec:open}
Our focus in the current paper has been on pure Nash equilibria. It is also interesting to generalize the bounds on the price of anarchy of the games induced by our coordination mechanisms for mixed Nash equilibria. Recently, Roughgarden \cite{R09} defined general smoothness arguments that can be used to bound the price of anarchy of games having particular properties. Bounds on the price of anarchy over pure Nash equilibria that are proved using smoothness arguments immediately imply that the same bounds on the price of anarchy hold for mixed Nash equilibria as well. We remark that the arguments used in the current paper in order to prove our upper bounds are not smoothness arguments. At least in the case of the coordination mechanism {\sf BCOORD}, smoothness arguments cannot be used to prove a bound on the price of anarchy as small as $O\left(\frac{\log m}{\log\log m}\right)$ since the price of anarchy over mixed Nash equilibria is provably higher in the case. We demonstrate this using the following construction. Czumaj and Voecking \cite{CV02} present a game induced by the {\sf Makespan} policy on related machines which has price of anarchy over mixed Nash equilibria at least $\Omega\left(\frac{\log m}{\log \log\log m}\right)$. The instance used in \cite{CV02} consists of $n$ jobs and $m$ machines. Each machine $j$ has a speed $\alpha_j\geq 1$ with $\alpha_1=1$ and each job $i$ has a weight $w_i$. The processing time of job $i$ on machine $j$ is $w_{ij}=\alpha_j w_i$ (i.e., the inefficiencies of the jobs are the same on the same machine). Now, consider the game induced by the coordination mechanism {\sf BCOORD} for the instance that consists of the same machines and jobs in which the processing time of job $i$ on machine $j$ is defined by $w'_{ij}=\alpha_j^{\frac{p}{p+1}} w_i$, i.e., the inefficiency of any job on machine $j$ is $\alpha_j^{\frac{p}{p+1}}$. Here, $p$ is the parameter used by {\sf BCOORD}. By the definition of {\sf BCOORD}, we can easily see that the game induced is identical with the game induced by {\sf Makespan} on the original instance of \cite{CV02}. Also note that, in our instance, the processing time of the jobs is not increased (and, hence, the optimal makespan is not larger than that in the original instance of \cite{CV02}). Hence, the lower bound of \cite{CV02} implies a lower bound on the price of anarchy over mixed Nash equilibria of the game induced by the coordination mechanism {\sf BCOORD}.
Our work reveals several other interesting questions. First of all, it leaves open the question of whether coordination mechanisms with constant approximation
ratio exist. In particular, is there any coordination mechanism that handles anonymous jobs,
guarantees that the induced game has pure Nash equilibria, and has constant price of anarchy?
Based on the lower bounds of \cite{AJM08,FS10}, such a coordination mechanism (if it exists) must
use preemption. Alternatively, is the case of anonymous jobs provably more difficult than the case
where jobs have IDs? Investigating the limits of non-preemptive mechanisms is still interesting.
Notice that {\sf AJM-1} is the only non-preemptive coordination mechanism
that has approximation ratio $o(m)$ but it does not guarantee that the induced game has
pure Nash equilibria; furthermore, the only known non-preemptive coordination mechanism that induces a potential
game with bounded price of anarchy is {\sf ShortestFirst}. So, is there any non-preemptive (deterministic or randomized)
coordination mechanism that is simultaneously $o(m)$-approximate and induces a potential game?
We also remark that Theorem \ref{thm:no-potential} does not necessarily exclude a game
induced by the coordination mechanism {\sf BCOORD} from having pure Nash equilibria. Observe that
the examples in the proof of Theorem \ref{thm:no-potential} do not consist of best-response moves and,
hence, it is interesting to investigate whether best-response moves converge to pure Nash equilibria
in such games.
Furthermore, we believe that the games induced by the coordination mechanism
{\sf CCOORD} are of independent interest. We have proved that these games belong to the class
{\sf PLS} \cite{JPY88}. Furthermore, the result of Monderer and Shapley \cite{MS96} and the proof
of Lemma \ref{lem:psi-potential} essentially show that each of these games is isomorphic to a congestion
game. However, they have a beautiful definition as games on parallel machines that gives them a particular
structure. What is the complexity of computing pure Nash equilibria in such games?
Even in case that these games are {\sf PLS}-complete (informally, this would mean that computing a pure Nash
equilibrium is as hard as finding any object whose existence is guaranteed by a potential function
argument) like several variations of congestion games that were considered recently \cite{ARV06,FPT04,SV08},
it is still interesting to study the convergence time to efficient assignments.
A series of recent papers \cite{AAEMS08,CMS06,FFM08,MV04} consider adversarial
rounds of best-response moves in potential games so that each player is given at least one chance to
play in each round (this is essentially our assumption in Lemma \ref{lem:a-cm-potential-convergence}
for the coordination mechanism {\sf ACOORD}). Does the game induced by the coordination mechanism
{\sf CCOORD} converges to efficient assignments after a polynomial number of adversarial rounds of best-response moves?
Although it is a potential game, it does not have the particular properties considered in \cite{AAEMS08}
and, hence, proving such a statement probably requires different techniques.
Finally, recall that we have considered the maximum completion time as the measure of the efficiency of schedules. Other measures such as the weighted sum of completion times that is recently studied in \cite{CGM10} are interesting as well. Of course, considering the application of coordination mechanisms to settings different than scheduling is an important research direction.
\small\paragraph{Acknowledgments.} I would like to thank Chien-Chung Huang for helpful comments on an earlier draft of the paper.
\small
\appendix
\normalsize\section{Proof of Lemma \ref{lem:properties}}
The properties clearly hold if $A$ is empty or $k=1$. In the following, we assume that $k\geq 2$ and $A=\{a_1, ..., a_n\}$ for integer $n\geq 1$.
\paragraph{a.} Clearly,
\begin{eqnarray*}
L(A)^k &=& \left(\sum_{t=1}^k{a_t}\right)^k = \sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\zeta(d_1, ..., d_k)\prod_{t=1}^{k}{a_{d_t}}}
\end{eqnarray*}
where $\zeta(d_1, ..., d_k)$ are multinomial coefficients on $k$ and, hence, belong $\{1, ..., k!\}$. The property then follows by the definition of $\Psi_k(A)$.
\paragraph{b.} We can express $\Psi_{k-1}(A)^k$ and $\Psi_k(A)^{k-1}$ as follows:
\begin{eqnarray*}
\Psi_{k-1}(A)^k &=& \left((k-1)!\right)^k \left(\sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}\right)^k \\
&= & \left((k-1)!\right)^k \sum_{1\leq d_1 \leq ... \leq d_{k(k-1)} \leq n}{\zeta_1(d_1, ..., d_{k(k-1)})\prod_{t=1}^{k(k-1)}{a_{d_t}}}.\\
\Psi_k(A)^{k-1} &=& \left(k!\right)^{k-1} \left(\sum_{1\leq d_1 \leq ... \leq d_k \leq n}{\prod_{t=1}^{k}{a_{d_t}}}\right)^{k-1}\\
&= & \left(k!\right)^{k-1} \sum_{1\leq d_1 \leq ... \leq d_{k(k-1)} \leq n}{\zeta_2(d_1, ..., d_{k(k-1)})\prod_{t=1}^{k(k-1)}{a_{d_t}.}}
\end{eqnarray*}
So, both $\Psi_{k-1}(A)^k$ and $\Psi_k(A)^{k-1}$ are sums of all monomials of degree $k(k-1)$ over the elements of $A$ with different coefficients.
The coefficient $\zeta_1(d_1, ..., d_{k(k-1)})$ is the number of different ways to partition the multiset $D=\{d_1, ..., d_{k(k-1)}\}$ of size $k(k-1)$
into $k$ disjoint ordered multisets each of size $k-1$ so that the union of the ordered multisets yields the original multiset. We refer to
these partitions as $(k,k-1)$-partitions. The
coefficient $\zeta_2(d_1, ..., d_{k(k-1)})$ is the number of different ways to partition $D$ into $k-1$ disjoint ordered multisets
each of size $k$ (resp. $(k-1,k)$-partitions). Hence, in order to prove the property, it suffices to show that for any multiset $\{d_1, ..., d_{k(k-1)}\}$,
\begin{eqnarray}\label{ineq:partitions}
\frac{\zeta_1(d_1, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d_{k(k-1)})} \leq \frac{k^{k-1}}{(k-1)!}.
\end{eqnarray}
Assume that some element of $D$ has multiplicity more than one and consider the new multiset $D'=\{d_1, ..., d'_i, ..., d_{k(k-1)}\}$ that
replaces one appearance $d_i$ of this element with a new element $d'_i$ different than all elements in $D$. Then, in order to generate all
$(k,k-1)$-partitions of $D'$, it suffices to consider the $(k,k-1)$-partitions of $D$ and, for each of them, replace $d_i$ with $d'_i$ once for each
of the ordered sets in which $d_i$ appears. Similarly, we can generate all $(k-1,k)$-partitions of $D'$ using the $(k-1,k)$-partitions of $D$.
Since the number of sets in $(k,k-1)$-partitions is larger than the number of sets in $(k-1,k)$-partitions, we will have that
\begin{eqnarray*}
\frac{\zeta_1(d_1, ..., d'_i, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d'_i, ..., d_{k(k-1)})} \geq \frac{\zeta_1(d_1, ..., d_{k(k-1)})}{\zeta_2(d_1, ..., d_{k(k-1)})} .
\end{eqnarray*}
By repeating this argument, we obtain that the ratio at the left-hand side of inequality (\ref{ineq:partitions})
is maximized when all $d_i$'s are distinct. In this case, both $\zeta_1$ and $\zeta_2$ are given by the multinomial coefficients
\[\zeta_1(d_1, ..., d_{k(k-1)}) = \left(\begin{array}{c}k(k-1)\\\underbrace{k-1, ..., k-1}_{\mbox{\tiny $k$ times}}\end{array}\right) = \frac{(k(k-1))!}{((k-1)!)^{k}}\]
and
\[\zeta_1(d_1, ..., d_{k(k-1)}) = \left(\begin{array}{c}k(k-1)\\\underbrace{k, ..., k}_{\tiny \mbox{$k-1$ times}}\end{array}\right) = \frac{(k(k-1))!}{(k!)^{k-1}}\]
and their ratio is exactly the one at the right-hand side of the inequality (\ref{ineq:partitions}).
\paragraph{c.} The property follows easily by the definition of function $\Psi_k$ by observing that all the monomials of degree $k$ over
the elements of $A$ that contain $b^t$ are generated by multiplying $b^t$ with the terms of $\Psi_{k-t}(A)$.
\paragraph{d.} By property (c), we have
\begin{eqnarray*}
\Psi_k(A\cup\{b\}) -\Psi_k(A) &=& \sum_{t=1}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)}\\
&=& kb\sum_{t=1}^{k}{\frac{(k-1)!}{(k-t)!}b^{t-1}\Psi_{k-t}(A)}\\
&=& kb\sum_{t=0}^{k-1}{\frac{(k-1)!}{(k-1-t)!}b^{t}\Psi_{k-1-t}(A)}\\
&=& kb\Psi_{k-1}(A\cup\{b\}).
\end{eqnarray*}
\paragraph{e.} Working with the right-hand side of the inequality and using the definitions of $L$ and $\Psi_{k-1}$, we have
\begin{eqnarray*}
k L(A)\Psi_{k-1}(A) &=& k!\left(\sum_{t=1}^n{a_t}\right)\cdot \sum_{1\leq d_1 \leq ... \leq d_{k-1}\leq n}{\prod_{t=1}^{k-1}{a_{d_t}}}\\
&\geq & k!\sum_{1\leq d_1 \leq ... \leq d_{k}\leq n}{\prod_{t=1}^{k}{a_{d_t}}}\\
&=& \Psi_{k}(A).
\end{eqnarray*}
The equalities follow obviously by the definitions. To see why the inequality holds, observe that the multiplication of the sum of all monomials of degree $1$ with the sum of all monomials of degree $k-1$
will be a sum of all monomials of degree $k$, each having coefficient at least $1$. \qed
\paragraph{f.} The proof follows by the derivation below in which we use property (c), the fact that $\Psi_t(\{b\})=t! b^t$ by the definition of function $\Psi_t$, property (b), and the binomial theorem. We have
\begin{eqnarray*}
\Psi_k(A\cup\{b\}) &=& \sum_{t=0}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)}\\
&=& \sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(A)\Psi_t(\{b\})}\\
&\leq & \sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k}(A)^\frac{k-t}{k}\Psi_k(\{b\})^\frac{t}{k}}\\
&=& \left(\Psi_k(A)^{1/k}+\Psi_k(\{b\})^{1/k}\right)^k.
\end{eqnarray*}
\qed
\end{document} |
{\mathbf e}gin{document}
\title[\tiny Approximation with respect to polynomials with constant coefficients]{The best m-term approximation with respect to polynomials with constant coefficients}
\alphauthor{Pablo M. Bern\'a}
\alphaddress{Pablo M. Bern\'a
\\
Instituto Universitario de Matem\'atica Pura y Aplicada
\\
Universitat Polit\`ecnica de Val\`encia
\\
46022 Valencia, Spain} \email{[email protected]}
\alphauthor{\'Oscar Blasco}
\alphaddress{\'Oscar Blasco
\\
Departamento de An\'alisis Matem\'atico
\\
Universidad de Valencia, Campus de Burjassot
\\
46100 Valencia, Spain} \email{[email protected]}
\thanks{The first author is partially supported by GVA PROMETEOII/2013/013 and 19368/PI/14 (\textit{Fundaci\'on S\'eneca}, Regi\'on de Murcia, Spain). The second author is partially supported by MTM2014-53009-P (MINECO, Spain).}
\subjclass{41A65, 41A46, 46B15.}
\keywords{thresholding greedy algorithm; m-term approximation;
weight-greedy basis. }
{\mathbf e}gin{abstract}
In this paper we show that that greedy bases can be defined as those where the error term using $m$-greedy approximant is uniformly bounded by the best $m$-term approximation with respect to polynomials with constant coefficients in the context of the weak greedy algorithm and weights.
\end{abstract}
\maketitle
\section{Introduction }
Let $(\SX,\Vert \cdot \Vert)$ be an infinite-dimensional real Banach space and let $\mathscr B = (e_n)_{n=1}^\infty$ be a normalized Schauder basis of $\SX$ with biorthogonal functionals $(e_n^*)_{n=1}^\infty$. Throughout the paper, for each finite set $A\subset \SN$ we write $|A|$ for the cardinal of the set $A$, $1_A=\sum_{j\in A} e_j$ and $P_A(x)=\sum_{n\in A}e_n^*(x) e_n$. Given a collection of signs $(\eta_j)_{j\in A}\in\lambdabrace\pm 1\rbrace$ with $|A|<\infty$, we write $1_{\eta A} = \sum_{n\in A}\eta_j e_j\in \SX$ and we use the notation $[1_{\eta A}]$ and $[e_n, n\in A]$ for the one-dimensional subspace and the $|A|-$dimensional subspace generated by generated by $1_{\eta A}$ and by $\lambdabrace e_n, n\in A\rbrace$ respectively. For each $x\in\SX$ and $m\in \SN$, S.V. Konyagin and V.N. Temlyakov defined in \cite{VT} the \textbf{$m$-th greedy approximant} of $x$ by
$$\mathcal{G}_m(x) = \sum_{j=1}^m e_{\rho(j)}^*(x)e_{\rho(j)},$$
where $\rho$ is a greedy ordering, that is $\rho : \SN \lambdaongrightarrow \SN$ is a permutation such that $supp(x) = \lambdabrace n: e_n^*(x)\neq 0\rbrace \subseteq \rho(\SN)$ and $\vert e_{\rho(j)}^*(x)\vert \geq \vert e_{\rho(i)}^*(x)\vert$ for $j\lambdaeq i$. The collection $(\mathcal{G}_m)_{m=1}^\infty$ is called the \textbf{Thresholding Greedy Algorithm} (TGA).
This algorithm is usually a good candidate to obtain the \textbf{best m-term approximation} with regard to $\mathscr B$, defined by
$$ \mathop{\rm sign}ma_m(x,\mathscr B)_\SX =\mathop{\rm sign}ma_m(x) := \inf\lambdabrace d(x,[e_n, n\in A]) : A\subset \SN, \vert A\vert = m\rbrace.$$
The bases satisfying
{\mathbf e}gin{equation} \lambdaabel{old}\Vert x-\mathcal{G}_m(x)\Vert \lambdaeq C\mathop{\rm sign}ma_m(x),\;\; \forall x\in\SX, \forall m\in\SN,\end{equation}
where $C$ is an absolute constant are called \textbf{greedy bases} (see \cite{VT}).
The first characterization of greedy bases was given by S.V. Konyagin and V. N. Temlyakov in \cite{VT} who established that a basis is greedy if and only if it is unconditional and democratic (where a basis is said to be democratic if there exists $C>0$ so that $\|1_A\|\lambdae C \|1_B\|$ for any pair of finite sets $A$ and $B$ with $|A|=|B|$).
Let us also recall two possible extensions of the greedy algorithm and the greedy basis. The first one consists in taking the $m$ terms with near-biggest coefficients and generating the Weak Greedy Algorithm (WGA) introduced by V.N. Temlyakov in \cite{T}.
For each $t\in (0,1]$,
a finite set $\Ga\subset\SN$ is called a $t$-greedy set for $x\in\SX$, for short $\Ga\in\mathscr G(x,t)$, if
\[
\min_{n\in \Ga}|{\mathbf e}n(x)|\,\geq\,t\max_{n\notin\Ga}|{\mathbf e}n(x)|,
\]
and write $\Ga\in\mathscr G(x,t,N)$ if in addition $|\Ga|=N$. A \textbf{$t$-greedy operator of order $N$} is a mapping $G^t:\SX\to\SX$ such that
\[
G^t(x)=\sum_{n\in \Ga_x}{\mathbf e}n(x){\mathbf e}_n, \quad \mbox{for some }\Ga_x\in \mathscr G(x,t,N).
\]
A basis is called \textbf{$t$-greedy} if there exists $C(t)>0$ such that
{\mathbf e}gin{eqnarray}
\Vert x-G^t(x)\Vert \lambdaeq C(t)\mathop{\rm sign}ma_m(x)\; \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m).
\end{eqnarray}
It was shown that a basis is $t$-greedy for some $0<t\lambdae 1$ if and only if it is $t$-greedy for all $0<t\lambdae 1$. From the proof it follows that
greedy basis are also $t$-greedy basis with constant $C(t)= O(1/t)$ as $t\to 0$.
The second one consists in replacing $|A|$ by $w(A)=\sum_{n\in A} w_n$ and it was considered by G. Kerkyacharian, D. Picard and V.N. Temlyakov in \cite{KPT} (see also \cite[Definition 16]{Tem}). Given a weight sequence $\omega = \lambdabrace \omega_n\rbrace_{n=1}^\infty, \omega_n >0$ and a positive real number $\delta>0$, they defined
$$\mathop{\rm sign}ma_\delta ^\omega (x) = \inf \lambdabrace d(x,[e_n, n\in A]) : A\subset \SN, \omega(A)\lambdaeq \delta\rbrace$$
where $\omega(A) := \sum_{n\in A}\omega_n$, with $A\subset\mathbb{N}$.
They called \textbf{weight-greedy bases} ($\omega$- greedy bases) to those bases satisfying
{\mathbf e}gin{eqnarray}\lambdaabel{wt}
\Vert x-\mathcal{G}_m(x)\Vert \lambdaeq C \mathop{\rm sign}ma_{\omega(A_m)}^\omega (x),\; \forall x\in\SX, \forall m\in\SN,
\end{eqnarray}
where $C>0$ is an absolute constant and $A_m = supp(\mathcal{G}_m(x))$. Moreover, they proved in \cite{KPT} that $\mathscr B$ is a $\omega$- greedy basis if and only if it is unconditional and $w$-democratic (where a basis is $w$-democratic whenever there exists $C>0$ so that $\|1_A\|\lambdae C \|1_B\|$ for any pair of finite sets $A$ and $B$ with $w(A)\lambdae w(B)$).
This generalization was motivated by the work of A. Cohen, R.A. DeVore and R. Hochmuth in \cite{CDH} where the basis was indexed by dyadic intervals and $w_\alphalpha (\Lambda)=\sum_{I\in \Lambda}|I|^\alphalpha$. Later in 2013, similar considerations were considered by E. Hern\'andez and D. Vera to prove some inclusions of approximation spaces (see \cite{HV}).
Let us summarize and use the following combined definition.
{\mathbf e}gin{defi} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$, $0<t\lambdae 1$ and weight sequence $\omega = \lambdabrace \omega_n\rbrace_{n=1}^\infty$ with $\omega_n >0$.
We say that $\mathscr B$ is \textbf{$(t,\omega)$-greedy} if there exists $C(t)>0$ such that
{\mathbf e}gin{equation}\lambdaabel{g}
\Vert x-G^t(x)\Vert \lambdaeq C(t)\mathop{\rm sign}ma^w_{m(t)}(x)\; \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m)
\end{equation}
where $A_m(t)=supp (G^t(x))$ and $m(t)=w(A_m(t))$.
\end{defi}
The authors introduced (see \cite{BB}) the best $m$-term approximation with respect to polynomials with constant coefficients as follows:
$$\mathcal{D}^*_m(x) := \inf \lambdabrace d(x,[1_{\eta A}]) : A\subset \SN, (\eta_n)\in \{\pm 1\}, \vert A\vert = m\rbrace.$$
Obviously, $\mathop{\rm sign}ma_m(x)\lambdaeq \mathcal{D}^*_m(x)$ but, while $\mathop{\rm sign}ma_m(x)\to 0$ as $m\to\infty$ it was shown that for orthonormal bases in Hilbert spaces we have $\mathcal{D}^*_m(x)\to \|x\|$ as $m\to\infty$.
The following result establishes a new description of greedy bases using the best $m$-term approximation with respect to polynomials with constant coefficients.
{\mathbf e}gin{theorem} (\cite[Theorem 3.6]{BB})
Let $\SX$ be a Banach space and $\mathscr B$ a Schauder basis of $\SX$.
(i) If there exists $C>0$ such that
{\mathbf e}gin{equation}\lambdaabel{new}\Vert x-\mathcal{G}_m(x)\Vert \lambdaeq C\mathcal{D}^*_m(x),\; \forall x\in \SX,\; \forall m\in \mathbb{N},\end{equation}
then $\mathscr B$ is $C$-suppression unconditional and $C$-symmetric for largest coefficients.
(ii) If $\mathscr B$ is $K_s$-suppression unconditional and $C_s$-symmetric for largest coefficients then
$$\Vert x-\mathcal{G}_m(x)\Vert \lambdaeq (K_s C_s)\mathop{\rm sign}ma_m(x),\; \forall x\in \SX,\; \forall m\in \mathbb{N}.$$
\end{theorem}
The concepts of suppression unconditional and symmetric for largest coefficients bases can be found in \cite{BB,AA2,AW,DKOSS,VT}. We recall here that a basis is \textbf{$K_s$-suppression unconditional} if the projection operator is uniformly bounded, that is to say
$$\Vert P_A(x)\Vert \lambdaeq K_s\Vert x\Vert,\; \forall x\in\SX,\forall A\subset \SN$$
and $\mathscr B$ is \textbf{$C_s$-symmetric for largest coefficients} if
$$\Vert x+t1_{\varepsilon A}\Vert \lambdaeq C_s\Vert x+t1_{\varepsilon' B}\Vert,$$
for any $\vert A\vert = \vert B\vert$, $A\cap B=\emptyset$, $supp(x) \cap (A\cup B) = \emptyset$, $(\varepsilon_j), (\varepsilon'_j) \in \lambdabrace \pm 1\rbrace$ and $t = \max\lambdabrace \vert e_n^*(x)\vert : n\in supp(x)\rbrace$.
In this note we shall give a direct proof of the equivalence between condition (\ref{old}) and (\ref{new}) even in the setting of $(t,w)$-greedy basis.
Let us now introduce our best $m$-term approximation with respect to polynomials with constant coefficients associated to a weight sequence and the basic property to be considered in the paper.
{\mathbf e}gin{defi} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$, $0<t\lambdae 1$ and a weight sequence $\omega = \lambdabrace \omega_n\rbrace_{n=1}^\infty$ with $\omega_n >0$. We denote by $$\mathcal{D}_{\delta}^\omega (x) := \inf \lambdabrace d(x,[1_{\eta A}]) : A\subset \SN, (\eta_n)\in \{\pm 1\}, \omega(A)\lambdaeq \delta\rbrace.$$
The basis $\mathscr B$ is said to $(t,w)$-greedy for polynomials with constant coefficients, denoted to have {\bf $(t,w)$-PCCG property}, if there exists $D(t)>0$ such that
{\mathbf e}gin{equation}\lambdaabel{ng}\Vert x-G^t(x)\Vert \lambdaeq D(t)\mathcal{D}_{m(t)}^\omega (x), \forall x\in\SX, \forall m\in\SN, \forall G^t\in \mathscr G(x,t,m)\end{equation}
where $A_m(t)=supp(G^t(x))$ and $m(t)=\omega(A_m(t))$.
In the case $t=1$ and $w(A)=|A|$ we simply call it the {\bf PCCG property}.
\end{defi}
Of course $\mathop{\rm sign}ma_\delta ^\omega(x) \lambdaeq \mathcal{D}_\delta^\omega(x)$ for all $\delta>0$, hence if the basis is $(t,\omega)$-greedy then (\ref{ng}) holds with the $D(t)=C(t)$. We now formulate our main result which produces a direct proof of the result in \cite{BB} and give the extension to $t$-greedy and weighted greedy versions.
{\mathbf e}gin{theorem} Let $\mathscr B$ be a normalized Schauder basis in $\mathbb{X}$ and let $\omega = \lambdabrace \omega_n\rbrace_{n=1}^\infty$ be a weight sequence with $\omega_n >0$ for all $n\in \mathbb N$. The following are equivalent:
(i) There exist $0<s\lambdae 1$ such that $\mathscr B$ has the $(s,w)$-PCCG property.
(ii) $\mathscr B$ is $(t,\omega)$-greedy for all $0<t\lambdae 1$.
\end{theorem}
{\mathbf e}gin{proof} Only the implication (i) $\Longrightarrow$ (ii) needs a proof.
Let us assume that (\ref{ng}) holds for some $0<s\lambdae 1$. Let $0<t\lambdae 1$, $x \in\SX$, $m\in \mathbb N$ and $G^t\in \mathscr G(x,t,m)$. We write $G^t(x) = P_{A_m(t)}(x)$ with $A_m(t)\in \mathcal G(x,t,m)$. For each $\varepsilon >0$ we choose $z = \sum_{n\in B}e_n^*(x)e_n$ with $\omega(B)\lambdaeq \omega(A_m(t))$ and $\Vert x-z\Vert \lambdaeq \mathop{\rm sign}ma_{\omega(A_m)}^\omega (x) + \varepsilon$.
We write
$$x- P_{A_m(t)}(x)= x- P_{A_m(t)\cup B} (x) +P_{B\setminus A_m(t)}(x).$$
Taking into account that $P_{B\setminus A_m(t)}(x)\in co(\lambdabrace S 1_{\eta (B\setminus A_m(t))} : \vert \eta_j\vert = 1\rbrace)$ for any $S\ge \underset{j\in B\setminus A_m(t)}{\max}\vert e_j^*(x)\vert$, it suffices to show that there exists $R\ge 1$ and $C(t)>0$ such that
{\mathbf e}gin{equation}\lambdaabel{final}
\|x- P_{A_m(t)\cup B} (x) + R\gamma 1_{\eta(B\setminus A_m(t))}\|\lambdae C(t) \|x-z\|
\end{equation}
for any choice of signs $(\eta_j)_{j\in B\setminus A_m(t)}$ where $\gamma = \underset{j\in B\setminus A_m(t)}{\max}\vert e_j^*(x)\vert$.
Let us assume first that $t\geq s$. We shall show that
{\mathbf e}gin{equation} \lambdaabel{two}\Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert \lambdae D(s)\Vert x- P_B(x)\Vert\end{equation}
for any choice of signs $(\eta_j)_{j\in B\setminus A_m(t)}$.
Given $(\eta_j)_{j\in B\setminus A_m(t)}$ we consider
$$y_{\eta}= x- P_B(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))}=\sum_{n\notin B} e_n^*(x)e_n+ \sum_{n\in B\setminus A_m(t)}\frac{t}{s}\gamma\eta_n e_n .$$
Note that $$\min_{n\in A_m(t)\setminus B}|e_n^*(y_\eta)|= \min_{n\in A_m(t)\setminus B}|e_n^*(x)|\ge \min_{n\in A_m(t)}|e_n^*(x)|$$ and $$s\max_{n\in (A_m(t)\setminus B)^c}|e_n^*(y_\eta)|=\max\{ s\max_{n\notin A_m(t)}|e_n^*(x)|, t\gamma\} .$$
Therefore, since $t\ge s$, we conclude that
$$\min_{n\in A_m(t)\setminus B}|e_n^*(y_\eta)|\ge s \max_{n\in (A_m(t)\setminus B)^c}|e_n^*(y_\eta)|.$$
Hence $A_m(t)\setminus B \in \mathcal G(y_\eta, s, N)$ with $N = \vert A_m(t)\setminus B\vert$. We write $G^s(y_\eta) = P_{A_m(t)\setminus B}(x)$ and notice that
$$ y_\eta- G^s(y_\eta)= x- P_{A_m(t)\cup B}(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))}. $$
Since $\omega(B)\lambdaeq \omega(A_m(t))$ we have also that $\omega(B\setminus A_m(t))\lambdaeq \omega(A_m(t)\setminus B)$.
Hence for $N(s)=\omega(A_m(t)\setminus B)$ we conclude
{\mathbf e}gin{eqnarray*}
\Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert &\lambdaeq& D(s)\mathcal{D}_{N(s)}^\omega (y_\eta)\\\nonumber
&\lambdaeq& D(s) \Vert y_\eta-\frac{t}{s}\gamma1_{\eta B\setminus A_m(t)}\Vert\\\nonumber
&=&D(s)\Vert x- P_B(x)\Vert.
\end{eqnarray*}
Now, let $y=x-z+ \mu 1_B$ for $\mu = s \, \underset{j\notin B}{\max}\vert e_j^*(x-z)\vert +\underset{j\in B}{\max}\vert e_j^*(x-z)\vert.$\newline
Then $$\min_{j\in B} |\mu + e_n^*(x-z)|\ge s\max_{j\notin B} |e_n^*(x-z)|,$$ which gives that $B\in \mathcal G(y,s, |B|)$ and we obtain
$G^s(y) = P_B(x-z)+\mu 1_{B}$. Hence
{\mathbf e}gin{equation}\lambdaabel{three}
\Vert x-P_B(x)\Vert = \Vert y-G^s(y)\Vert \lambdae D(s)\Vert y - \mu 1_B\Vert= D(s)\Vert x-z\Vert.
\end{equation}
Therefore, by $\eqref{two}$ and $\eqref{three}$ we obtain
$$\Vert x- P_{(A_m(t)\cup B)}(x)+\frac{t}{s}\gamma 1_{\eta B\setminus A_m(t)}\Vert\lambdae D(s)^2\|x-z\|.$$
Then, for $s\lambdae t$ we obtain that $\mathscr B$ is $(t,w)$-greedy with constant $C(t)\lambdae D(s)^2$.
We now consider the case $s>t$. We use the following estimates:
$$\Vert x- P_{(A_m(t)\cup B)}(x)+\gamma 1_{\eta B\setminus A_m(t)}\Vert\lambdae \Vert x- P_{ B}(x)\Vert+\Vert P_{A_m(t)\setminus B}(x)\Vert+\gamma \Vert1_{\eta B\setminus A_m(t)}\Vert.$$
Arguing as above, using now
$$\tilde y_{\eta}= P_{A_m(t)\setminus B}(x)+ \frac{t}{s}\gamma 1_{\eta(B\setminus A_m(t))},$$
we conclude that $\frac{t}{s}\gamma \Vert1_{\eta B\setminus A_m(t)}\Vert\lambdae D(s)\|P_{A_m(t)\setminus B}(x)\|$.
The argument used to show (\ref{three}) gives $ \|z- P_C z\|\lambdae D(s) \|z\|$ for all $z\in \mathbb{X}$ and finite set $C$. Therefore
$$\Vert P_{A_m(t)\setminus B}(x)\Vert= \Vert P_{A_m(t)}(x-P_{B}x)\Vert\lambdae (1+ D(s))\|x-P_Bx\|.$$
Putting all together we have
$$\Vert x- P_{(A_m(t)\cup B)}(x)+\gamma 1_{\eta B\setminus A_m(t)}\Vert\lambdae (2+\frac{t+s}{t}D(s))\|x-P_Bx\|,$$
and therefore $\mathscr B$ is $(t,w)$-greedy with constant $C(t)\lambdae (2+\frac{t+s}{t}D(s))D(s).$
\end{proof}
{\mathbf e}gin{corollary}
If $t=1$ and $\omega(A) = \vert A\vert$, then $\mathscr B$ has the PCCG property if and only if $\mathscr B$ is greedy.
\end{corollary}
{\mathbf e}gin{corollary}
If $\omega(A) = \vert A\vert$, then $\mathscr B$ has the $t$-PCCG property if and only if $\mathscr B$ is $t$-greedy.
\end{corollary}
\section{A remark on the Haar system}
Throughout this section $|E|$ stands for the Lebesgue measure of a set in $[0,1]$, $\mathcal D$ for the family of dyadic intervals in $[0,1]$ and $card(\Lambda)$ for the number of dyadic elements in $\Lambda$.
We denote by $\mathcal{H}:=\lambdabrace H_I\rbrace$ the Haar basis in $[0,1]$, that is to say
$$H_{[0,1]} (x) = 1\; \text{for}\; x\in [0,1),$$
and for $I\in \mathcal D$ of the form $I = [(j-1)2^{-n}, j2^{-n})$, $j=1,..,2^n$, $n= 0,1,...$ we have
{\mathbf e}gin{displaymath}
H_{I}(x) = \lambdaeft\{ {\mathbf e}gin{array}{ll}
2^{n/2} & \mbox{if $x\in [(j-1)2^{-n}, (j-\frac{1}{2})2^{-n})$,} \\
-2^{n/2} & \mbox{if $x\in [(j-\frac{1}{2})2^{-n}, j2^{-n})$,} \\
0 & \mbox{otherwise.}
\end{array}
\right.
\end{displaymath}
We write $$c_I(f) := \lambdaangle f,H_I\rangle = \int_0^1 f(x)H_I(x)dx \hbox{ and } c_I(f,p):= \Vert c_I(f)H_I\Vert_p, \quad 1\lambdae p<\infty.$$
It is well known that $\mathcal H$ is an orthonormal basis in $L^2([0,1])$ and for $1<p<\infty$ we can use the Littlewood-Paley's Theorem which gives
{\mathbf e}gin{equation}\lambdaabel{lp} c_p \lambdaeft\Vert \lambdaeft( \sum_I \vert c_I(f,p)\frac{H_I}{\Vert H_I\Vert_p}\vert^2\right)^{1/2}\right\Vert_p \lambdaeq \Vert f\Vert_p \lambdaeq C_p\lambdaeft\Vert \lambdaeft( \sum_I \vert c_I(f,p)\frac{H_I}{\Vert H_I\Vert_p}\vert^2\right)^{1/2}\right\Vert_p
\end{equation}
to conclude that $(\frac{H_I}{\Vert H_I\Vert_p})_I$ is an unconditional basis in $L^p([0,1])$. Denoting $f<<_p g$ whenever $c_I(f,p)\lambdae c_I(g,p)$ for all dyadic intervals $I$ we obtain from (\ref{lp}) the existence of a constant $K_p$ such that
{\mathbf e}gin{equation}\lambdaabel{uncon}
\|f\|_p\lambdae K_p \|g\|_p \quad \forall f, g\in L^p([0,1]) \hbox{ with } f<<_p g,
\end{equation}
and also
{\mathbf e}gin{equation}\lambdaabel{h}
\|P_\Lambda g\|\lambdae K_p \|g\|_p \quad \forall g\in L^p \quad \forall \Lambda\subset \mathcal D.
\end{equation}
Regarding the greedyness of the Haar basis it was
V. N. Temlyakov the first one who proved (see \cite{T}) that the every wavelet basis $L_p$-equivalent to the Haar basis is $t$-greedy in $L_p([0,1])$ with $1<p<\infty$ for any $0<t\lambdae 1$.
Let $\omega:[0,1]\to \mathbb R^+$ be a measurable weight and, as usual, we denote $\omega(I)=\int_I \omega(x)dx$ and $m_I(\omega)=\frac{\omega(I)}{|I|}$ for any $I\in \mathcal D$. In the space $L^p(\omega)=L^p([0,1],\omega)$ we denote $\|f\|_{p, \omega}=(\int_0^1 |f(x)|^p\omega(x) dx)^{1/p}$ and $$ c_I(f,p,\omega):= \Vert c_I(f)H_I\Vert_{p,\omega}= |c_I(f)|\frac{\omega(I)^{1/p}}{|I|^{1/2}}. $$ Recall that $\omega$ is said to be a
dyadic $A_p$-weight (denoted $\omega \in A^{d}_p$) if
{\mathbf e}gin{equation}
A^{d}_p(\omega)= \sup_{I\in \mathcal D} m_I(\omega) \mathcal Big( m_I(\omega^{-1/(p-1)})\mathcal Big)^{p-1}<\infty.
\end{equation}
As one may expect, Littlewood-Paley theory holds for weights in the dyadic $A_p$-class.
{\mathbf e}gin{theorem} (see \cite{ABM, I} for the multidimensional case) If $\omega \in A^{d}_p$ then
{\mathbf e}gin{eqnarray}\lambdaabel{uncon} \Vert f\Vert_{p,\omega} \alphapprox\lambdaeft\Vert \lambdaeft( \sum_I \vert c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}\vert^2\right)^{1/2}\right\Vert_{p,\omega}.
\end{eqnarray}
In particular $(\frac{H_I}{\Vert H_I\Vert_{p,\omega}})_I$ is an unconditional basis in $L^p(\omega)$ for $1<p<\infty$.
\end{theorem}
The greedyness of the Haar basis in $L^p(\omega)$ goes back to M. Izuki (see \cite{I, IS}) who showed that this holds for weights in the class $A_p^d$.
We shall use the ideas in these papers to show that the Haar basis satisfies the PCCG property for certain spaces defined using the Littlewood-Paley theory.
{\mathbf e}gin{defi} Let $\omega:[0,1]\to \mathbb R^+$ be a measurable weight and $1\lambdae p<\infty$. For each finite set of dyadic intervals $\Lambda$ we define
$f_\Lambda=\sum_{I\in \Lambda} c_I(f)H_I=\sum_{I\in \Lambda} c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}$ and write $$\|f\|_{X^p(\omega)}= \lambdaeft\Vert\lambdaeft( \sum_{I\in \Lambda} \vert c_I(f,p,\omega)\frac{H_I}{\Vert H_I\Vert_{p,\omega}}\vert^2\right)^{1/2}\right\Vert_{p,\omega}.$$
The closure of $span(f_\Lambda: card(\Lambda)<\infty)$ under this norm will be denoted $X^p(\omega)$.
\end{defi}
From the definition $(\frac{H_I}{\Vert H_I\Vert_{p,\omega}})_I$ is an unconditional basis with constant 1 in $X^p(\omega)$ and due to (\ref{uncon}) $X^p(\omega)=L^p(\omega)$ whenever $\omega\in A_p^d$. Our aim is to analyze conditions on the weight $\omega$ for the basis to be greedy. For such a purpose we do not need the weight to belong to $A_p^d$. In fact analyzing the proof in \cite{I, IS} one notices that only the dyadic reverse doubling condition (see \cite[p. 141]{GCRF}) was used. Recall that a weight $\omega$ is said to satisfies {\bf the dyadic reverse doubling condition} if there exists $\delta<1$ such that
{\mathbf e}gin{equation}\lambdaabel{dc}\omega(I')\lambdae \delta \omega (I), \forall I,I'\in \mathcal D \hbox{ with } I'\subsetneq I.
\end{equation}
Let us introduce certain weaker conditions.
{\mathbf e}gin{defi} Let $\alphalpha>0$ and $\omega$ be a measurable weight. We shall say that $\omega$ satisfies {\bf the dyadic reverse Carleson condition} of order $\alphalpha$ with constant $C>0$ whenever
{\mathbf e}gin{equation}\lambdaabel{cc}\sum_{I\in \mathcal D, J\subseteq I}\omega(I)^{-\alphalpha}\lambdae C \omega(J)^{-\alphalpha}, \forall J\in \mathcal D .
\end{equation}
\end{defi}
{\mathbf e}gin{defi} Let $\alphalpha>0$ and two sequences $(w_I)_{I\in \mathcal D}$ and $(v_I)_{I\in \mathcal D}$ of positive real numbers. We say that the pair $\mathcal Big((w_I)_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\mathcal Big)$ satisfies $\alphalpha-{\bf DRCC }$ with constant $C>0$ whenever
{\mathbf e}gin{equation}\lambdaabel{cc1}\sum_{I\in \mathcal D, J\subseteq I}w_I^{-\alphalpha}\lambdae C v_J^{-\alphalpha}, \forall J\in \mathcal D .
\end{equation}
\end{defi}
{\mathbf e}gin{Remark} \lambdaabel{n} (i) If $\omega\in \cup_{p> 1}A_p^w$ then $\omega$ satisfies the dyadic reverse doubling condition (see \cite[p 141]{GCRF}).
(ii) If $\omega$ satisfies the dyadic reverse doubling condition then $\omega$ satisfies the dyadic reverse Carleson condition of order $\alphalpha$ with constant $\frac{1}{1-\delta^\alphalpha}$ for any $\alphalpha>0$.
Indeed, $$\sum_{J\subseteq I}\omega(I)^{-\alphalpha}\lambdae \omega(J)^{-\alphalpha}+ \omega(J)^{-\alphalpha}\sum_{m=1}^{\infty} \delta^{m\alphalpha}\lambdae \frac{1}{1-\delta^\alphalpha}\omega(J)^{-\alphalpha}.$$
(iii) If $\omega$ satisfies the dyadic reverse Carleson condition of order $\alphalpha$ and $w_I=\omega(I)$ for each $I\in \mathcal D$ then
$\mathcal Big((w_I)_{I\in \mathcal D},(w_I)_{I\in \mathcal D}\mathcal Big)$ satisfies $\alphalpha$-{\bf DRCC }.
\end{Remark}
We need the following lemmas, whose proofs are essentially included in \cite{CDH, I, IS}.
{\mathbf e}gin{lemma} \lambdaabel{1c} Let $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ be a sequence of positive real numbers such that
$\mathcal Big((v_I)_{I\in \mathcal D}, (\omega(I))_{I\in \mathcal D}\mathcal Big)$ satisfies $ 1$-{\bf DRCC } with constant $C$.
Then
{\mathbf e}gin{equation}\lambdaabel{dem}
\lambdaeft(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p}\lambdae C\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}, \forall 1\lambdae p<\infty.
\end{equation}
\end{lemma}
{\mathbf e}gin{proof} We first write
{\mathbf e}gin{equation}\lambdaabel{main} \lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}= \lambdaeft(\int_0^1
(\sum_{I\in \Lambda} \omega(I)^{-2/p}\chi_I)^{p/2}\omega(x) dx\right)^{1/p}. \end{equation}
Let $I(x)$ denote the minimal dyadic interval in $\Lambda$ with regard to the inclusion relation that contains $x$. Now we use that $$\sum_{I\in \mathcal D, I(x)\subseteq I}v_I^{-1}\lambdae C \omega(I(x))^{-1}$$ to conclude that
{\mathbf e}gin{eqnarray*}
(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I})^{1/p}&=& \mathcal Big(\sum_{I\in\Lambda} \int_{I}v_I^{-1}\omega(x)dx\mathcal Big)^{1/p}= \mathcal Big(\int_0^1(\sum_{I\in \Lambda}v_I^{-1}\chi_I(x))\omega(x)dx\mathcal Big)^{1/p}\\
&\lambdae & C\mathcal Big(\int_0^1 \omega(I(x))^{-1}\omega(x)dx\mathcal Big)^{1/p}\lambdae C\mathcal Big(\int_0^1(\sum_{I\in \Lambda}\omega(I)^{-2/p}\chi_I(x))^{p/2}\omega(x)dx\mathcal Big)^{1/p}\\
&=& C\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_p}\|_{X^p(\omega)}.
\end{eqnarray*}
The proof is complete.
\end{proof}
{\mathbf e}gin{lemma} \lambdaabel{2p}Let $1<p<\infty$, $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ of positive real numbers. If
$\mathcal Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\mathcal Big)$ satisfies $2/p$-{\bf DRCC } with constant $C>0$ then
{\mathbf e}gin{equation}
\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\lambdae C \lambdaeft(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p}
\end{equation}
for all finite family $\Lambda$ of dyadic intervals.
\end{lemma}
{\mathbf e}gin{proof}
Let $E = \cup_{I\in\Lambda}I$. As above
$I(x)$ stands for the minimal dyadic interval in $\Lambda$ with regard to the inclusion relation that contains $x$. From (\ref{cc1}) we have that
{\mathbf e}gin{equation}\lambdaabel{1}
\sum_{I\in \Lambda}\omega(I)^{-2/p}\chi_I(x)\lambdae Cv_{I(x)}^{-2/p}, \quad x\in E.\end{equation}
Now denote for each $I\in \Lambda$, $\tilde I=\{x\in E: I(x)=I\}$. Clearly $\tilde I\subseteq I$ and $E=\cup_{I\in \Lambda}\tilde I $. Hence applying (\ref{main}) and (\ref{1}) we obtain {\mathbf e}gin{eqnarray*}
\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_p}\right\|_{X^p(\omega)}&\lambdae& C\lambdaeft(\int_E
v_{I(x)}^{-1}\omega(x) dx\right)^{1/p} = C\lambdaeft(\int_{\cup_{I\in \Lambda} \tilde I}
v_{I(x)}^{-1}\omega(x) dx\right)^{1/p} \\
&\lambdae& C \mathcal Big(\sum_{I\in \Lambda} \int_{\tilde I}v_{I}^{-1}\omega(x)dx\mathcal Big)^{1/p}\lambdae C \mathcal Big(\sum_{I\in \Lambda} v_I^{-1}\int_{I}\omega(x)dx\mathcal Big)^{1/p}\\
&=& C(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I})^{1/p}.
\end{eqnarray*}
The proof is now complete.
\end{proof}
Combining Remark \ref{n} and Lemmas \ref{1c} and \ref{2p} we obtain the following corollary.
{\mathbf e}gin{corollary} Let $1<p<\infty$, $\omega$ be a weight satisfying the dyadic reverse doubling condition
then
{\mathbf e}gin{equation} \lambdaabel{basic}
\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\alphapprox card(\Lambda)^{1/p}
\end{equation}
for all finite family $\Lambda$ of dyadic intervals.
\end{corollary}
{\mathbf e}gin{corollary} Let $1<p<\infty$, $\omega$ be a weight and $(v_I)_{I\in \mathcal D}$ of positive real numbers. If
$\mathcal Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\mathcal Big)$ satisfies $2/p'$-{\bf DRCC } with constant $C>0$ then
{\mathbf e}gin{equation}\lambdaabel{dem0}
\lambdaeft(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p}\lambdae C \mathcal Big(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\mathcal Big)\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}
\end{equation}
for all finite family $\Lambda$ of dyadic intervals.
\end{corollary}
{\mathbf e}gin{proof}
Note that, using Lemma \ref{2p}, we have
{\mathbf e}gin{eqnarray*}
\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}&=& \int_0^1 (\sum_{I\in\Lambda} v_I^{-1}\chi_I(x))\omega(x)dx\\
&\lambdae & \int_0^1 (\sum_{I\in\Lambda} \omega(I)^{-2/p}\chi_I)^{1/2}(\sum_{I\in\Lambda} v_I^{-2}\omega(I)^{2/p}\chi_I(x))^{1/2}\omega(x)dx\\
&\lambdae & \mathcal Big(\int_0^1 (\sum_{I\in\Lambda} \omega(I)^{-2/p}\chi_I)^{p/2}\omega(x)dx\mathcal Big)^{1/p}\mathcal Big(\int_0^1(\sum_{I\in\Lambda} v_I^{-2}\omega(I)^{2/p}\chi_I(x))^{p'/2}\omega(x)dx\mathcal Big)^{1/p'}\\
&\lambdae& \lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)} \lambdaeft\|\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\frac{H_I}{\Vert H_I\Vert_{p',\omega}}\right\|_{X^{p'}(\omega)}\\
&\lambdae& \lambdaeft(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\right)\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p',\omega}}\right\|_{X^{p'}(\omega)}\\
&\lambdae& C\lambdaeft(\max_{I\in \Lambda}\frac{\omega(I)}{v_I}\right)\lambdaeft\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\right\|_{X^p(\omega)}\lambdaeft(\sum_{I\in \Lambda} \frac{\omega(I)}{v_I}\right)^{1/p'}.
\end{eqnarray*}
The result now follows.
\end{proof}
Taking into account that dyadic reverse Carleson condition of order $\alphalpha$ implies dyadic reverse Carleson condition of order ${\mathbf e}ta$ for ${\mathbf e}ta>\alphalpha$ we obtain the following fact.
{\mathbf e}gin{corollary} Let $1<p<\infty$, $\omega$ be a weight satisfying the dyadic reverse Carleson condition of order $\min\{2/p', 2/p\}$
then
{\mathbf e}gin{equation} \lambdaabel{basic}
\|\sum_{I\in \Lambda} \frac{H_I}{\Vert H_I\Vert_{p,\omega}}\|_{X^p(\omega)}\alphapprox card(\Lambda)^{1/p}
\end{equation}
for all finite family $\Lambda$ of dyadic intervals.
\end{corollary}
{\mathbf e}gin{theorem} Let $1<p<\infty$, $0<t\lambdae 1$, $(w_I)_{I\in \mathcal D}$ be a sequence of real numbers such that $$0<m_0=\inf_{I\in \mathcal D}w_I\lambdae \sup_{I\in \mathcal D}w_I=M_0<\infty$$ and let $\omega$ be a weight satisfying the dyadic reverse Carleson condition of order $\min\{1, 2/p\}$ with constant $C>0$.
Then the Haar basis has the $(t, w_I)$-PCCG property in $X^p(\omega)$.
\end{theorem}
{\mathbf e}gin{proof}
Let $f\in X^p(\omega)$ and let $\Lambda^t_m$ be a set of $m$ dyadic intervals where $$\min_{I\in \Lambda^t_m}c_I(f,p, \omega)\ge t \max_{I'\notin \Lambda^t_m}c_{I'}(f,p, \omega) .$$
For each $\alphalpha\in \mathbb R$, $(\varepsilon_n)\in \{\pm 1\}$ and $\Lambda'$ with $\sum_{J\in \Lambda'}w_J\lambdae \sum_{I\in \Lambda_m^t}w_I$ we need to show that $\|f-P_{\Lambda^t_m}(f)\|_{X^p(\omega)}\lambdae C(t) \|f-\alphalpha 1_{\varepsilon \Lambda'}\|_{X^p(w)}$ for some constant $C(t)>0$.
From triangular inequality
$$\Vert f-P_{\Lambda^t_m}(f)\Vert_{X^p(\omega)} \lambdaeq \Vert P_{(\Lambda_m \cup \Lambda')^c}(f-\alphalpha 1_{\varepsilon \Lambda'})\Vert_{X^p(\omega)} + \Vert P_{\Lambda'\setminus \Lambda^t_m}(f)\Vert_{X^p(\omega)}$$
and the fact $\Vert P_{\Lambda}(f-\alphalpha 1_{\varepsilon B})\Vert_{X^p(\omega)}\lambdae \Vert f-\alphalpha 1_{\varepsilon B}\Vert_{X^p(\omega)}$ for any $\Lambda$
we only need to show that there exists $C>0$ such that
$$\Vert P_{\Lambda'\setminus \Lambda^t_m}(f)\Vert_{X^p(\omega)}\lambdae C \|f-\alphalpha 1_{\varepsilon \Lambda'}\|_{X^p(\omega)}.$$
Set $v_I=\frac{\omega(I)}{w_I}$ and observe that $\mathcal Big((\omega(I))_{I\in \mathcal D}, (v_I)_{I\in \mathcal D}\mathcal Big)$ satisfies $2/p$-{DRCC } with constant $M_0C$ and $\mathcal Big( (v_I)_{I\in \mathcal D}, \omega(I)_{I\in \mathcal D}\mathcal Big)$ satisfies $1$-{DRCC } with constant $C/m_0$.
Note that $\sum_{J\in \Lambda'}w_J\lambdae \sum_{I\in \Lambda_m^t}w_I$ implies that
$$ \sum_{J\in \Lambda'\setminus \Lambda_m^t}\frac{\omega(J)}{v_{J}}\lambdae \sum_{I\in \Lambda_m^t\setminus \Lambda'}\frac{\omega(I)}{v_{I}} $$
and then, invoking Lemma \ref{2p} and Lemma \ref{1c}, we get the estimates
{\mathbf e}gin{eqnarray*}
\Vert P_{\Lambda'\setminus \Lambda_m^t}(f)\Vert_{X^p(\omega)} &\lambdaeq& \Vert \underset{I\in \Lambda'\setminus A_m}{\max} c_I(f,p,\omega)1_{\Lambda'\setminus A_m}\Vert_{X^p(\omega)}\\
&\lambdae& C M_0 \underset{I\in \Lambda'\setminus \Lambda_m^t}{\max} c_I(f,p,\omega) (\sum_{J\in \Lambda'\setminus \Lambda_m^t}\frac{\omega(J)}{v_{J}})^{1/p}\\ &\lambdaeq&t^{-1} C M_0\underset{I\in \Lambda_m^t\setminus \Lambda'}{\min}c_I(f,p,\omega)(\sum_{I\in \Lambda_m^t\setminus \Lambda'}\frac{\omega(I)}{v_I})^{1/p} \\
&\lambdaeq& \frac{C^2 M_0}{tm_0}\Vert \underset{I\in \Lambda_m^t\setminus \Lambda'}{\min} c_I(f,p,\omega)1_{\Lambda_m^t\setminus \Lambda'}\Vert_{X^p(\omega)}\\
&\lambdaeq& \frac{C^2 M_0}{tm_0}\Vert P_{\Lambda_m^t\setminus \Lambda'}(f)\Vert_{X^p(\omega)} \\
&=& \frac{C^2 M_0}{tm_0}\Vert P_{\Lambda_m^t\setminus \Lambda'}(f-\alphalpha 1_{\varepsilon B})\Vert_{X^p(\omega)} \\
&\lambdaeq &\frac{C^2 M_0}{tm_0} \|f-\alphalpha 1_{\varepsilon \Lambda'}\|_{X^p(\omega)}.
\end{eqnarray*}
This completes the proof with $C(t)= 1+\frac{C^2 M_0}{tm_0}$.
\end{proof}
{\mathbf e}gin{corollary}
(i) If $\omega\in A_p^d$ then
the Haar basis has the $t$-PCCG property (and hence is $t$-greedy) in $L^p(\omega)$ with $1<p<\infty$.
(ii) The Haar basis has the $(t,w_I)$-PCCG property (and hence is $(t, w_I)$-greedy) in $L^p([0,1])$ for any sequence $(w_I)_{I\in\mathcal{D}}$ with $0<\inf w_I\lambdae \sup w_I <\infty.$
\end{corollary}
\noindent{\it \bf Acknowledgment:} The authors would like to thank to G. Garrig\'os and E. Hern\'andez for useful conversations during the elaboration of this paper.
\vspace*{-1.5cm}
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\end{document} |
\begin{document}
\title{Variational Perturbation Theory for Density Matrices}
\author{M. Bachmann, H. Kleinert, and A. Pelster}
\address{Institut f\"ur Theoretische Physik, Freie Universit\"at Berlin, Arnimallee 14, 14195 Berlin}
\date{\today}
\maketitle
\begin{abstract}
We develop convergent variational perturbation theory
for quantum statistical density matrices.
The theory is applicable
to polynomial as well as nonpolynomial
interactions.
Illustrating the power of the theory,
we calculate the temperature-dependent density of a particle
in a double-well
and of the electron in a hydrogen atom.
\end{abstract}
\pacs{}
\section{Introduction}
Variational perturbation theory~\cite{kl213,PI}
transforms divergent perturbation expansions into convergent ones.
The convergence extends to infinitely strong couplings~\cite{conv},
a property which has recently been
used to derive critical exponents in field theory
without renormalization group methods~\cite{kl257,kl263}.
The theory has first been
developed in quantum mechanics for the path integral representation
of the free energy
of the anharmonic oscillator~\cite{kl220}
and the hydrogen atom~\cite{PI,kl267}.
Local quantities such as quantum statistical density matrices
have been
treated so far only to lowest-order for the anharmonic oscillator
and the hydrogen atom~\cite{kl145,kl153}.
There has also been a related first-order treatment in
classical phase space~\cite{cucc1} for systems with
dissipation~\cite{cucc2}.
The purpose of this paper is to develop a
systematic convergent variational perturbation theory for
the path integral representation
of density matrices
of a point particle moving in a polynomial or nonpolynomial
potential.
As a first application we calculate the particle density in a double-well
and then the electron
density in a hydrogen atom.
\section{General Features}
\label{genfeat}
Variational perturbation theory approximates a
quantum statistical system
by perturbation expansions around harmonic oscillators
with
trial frequencies which are optimized
differently for each order of the expansions.
When dealing with the free energy,
it is essential to give a special treatment to the
fluctuations of the path average $\overline x\equiv (k_B T/\hbar)\int _0^{\hbar/k_BT }d\tau \,x(\tau )$,
since this performs violent fluctuations at high
temperatures $T$. These cannot be treated by any expansion,
unless the potential is close to harmonic.
The effect of these fluctuations may, however, easily be calculated at the end
by a single numerical fluctuation integral.
For this reason, variational perturbation expansions
are performed for each position $x_0$ of the path average
separately, yielding an $N$th order approximation
$W_N(x_0)$ to the local free energy $V_{\rm eff,cl}(x_0)$, called the
{\em effective classical potential\/}\cite{effcl}.
The name indicates that one may obtain the full quantum partition function
$Z$
from this object by a simple integral over $x_0$
just as in classical statistics,
\begin{equation}
\label{zmpfeff}
Z=\int_{-\infty}^{+\infty}\frac{dx_0}{\sqrt{2\pi\hbar^2/Mk_BT}}\,\exp\left\{-V_{\rm eff,cl}(x_0)/k_BT\right\}.
\end{equation}
Having calculated $W_N(x_0)$, we obtain the $N$th-order approximation to the partition function
\begin{equation}
Z_N=\int\limits_{-\infty}^{+\infty}\frac{dx_0}{\sqrt{2\pi\hbar^2/M k_B T}} e^{-
W_N(x_0)/k_BT}.
\label{gf1}
\end{equation}
The separate treatment
of the path average is important to ensure a
fast convergence at larger temperatures. In the high-temperature limit,
$W_N(x_0)$
converges against the initial potential $V(x_0)$ for any order $N$.
Before embarking upon the theory, it is useful to
visualize some characteristic properties of path fluctuations.
Consider the euclidean path integral
over all periodic paths
$x(\tau )$, with $x(0)=x(\hbar/k_BT)$,
for a harmonic oscillator with minimum at
$x_m$, where the action is
\begin{equation}
\label{zmharmaction}
{\cal A}_{\Omega,x_m}[x]=\int_0^{\hbar/k_BT}d\tau\,\left\{\frac{1}{2}M\dot{x}^2(\tau)+\frac{1}{2}M\Omega^2[x(\tau)-x_m]^2 \right\}.
\end{equation}
Its partition function is
\begin{equation}
\label{zmpf}
Z^{\Omega,x_m}=\oint {\cal D}x\,\exp\left\{-{\cal A}_{\Omega,x_m}[x]/\hbar \right\}=\frac{1}{2\sinh\hbar\Omega/2k_BT}
\end{equation}
and the correlation functions
of local quantities $O_1(x)$, $O_2(x)$, \dots are given by the expectation values
\begin{equation}
\label{meanval}
\langle\,
O_1(x(\tau))
O_2(x(\tau))\cdots\,
\,\rangle^{\Omega,x_m}=\frac{1}{Z^{\Omega,x_m}}\oint {\cal D}x\,O_1(x(\tau_1))O_2(x(\tau_2))\,\cdots \exp\left\{-{\cal A}_{\Omega,x_m}[x]/\hbar \right\}.
\end{equation}
The particle distribution of the oscillator is given by
\begin{equation}
\label{zmdfQS}
P_H(x)\equiv\langle\,\delta(x-x(\tau))\,\rangle^{\Omega,x_m}
=\frac{1}{\sqrt{2\pi a^2_{\rm H}}}\exp\left[-\frac{(x-x_m)^2}{2 a_{\rm H}^2} \right],
\end{equation}
which is a Gaussian distribution
of width
\begin{equation}
\label{zmwidQS}
a_{\rm H}^2=\frac{\hbar}{2M\Omega}\coth\frac{\hbar\Omega}{2 k_BT},
\end{equation}
the subscript
indicating that we are dealing with a harmonic oscillator.
At zero temperature,
this is equal to the square of the ground-state wave function
of the harmonic oscillator, whose width is
\begin{equation}
\label{qmflwidth}
a^2_{\rm H\,0}=\frac{\hbar}{2M\Omega}.
\end{equation}
\begin{figure}
\caption{\label{wid}
\label{wid}
\end{figure}
In the limit $\hbar\to 0$, we obtain from (\ref{meanval}), (\ref{zmdfQS}) the classical distribution
\begin{equation}
\label{zmdfcl}
P_{\rm H\,cl}(x)=\frac{1}{\sqrt{2\pi a^2_{\rm H\,cl}}}\exp\left[-\frac{(x-x_m)^2}{2 a^2_{\rm H\,cl}} \right]
\end{equation}
with
\begin{equation}
\label{zmwidcl}
a^2_{\rm H\,cl}= \frac{k_BT}{M\Omega^2}.
\end{equation}
The linear growth of this classical width is the origin of the
famous Dulong-Petit law
for the specific heat of a harmonic system.
The classical fluctuations are governed by the integral over the Boltzmann factor
\begin{equation}
\label{boltzmann}
e^{-M\Omega^2(x-x_m)^2/2k_BT},
\end{equation}
in the classical partition function
\begin{equation}
Z_{\rm H\,cl}=\int \limits_{-\infty}^{+\infty}\frac{dx}{\sqrt{2\pi\hbar^2/M k_B T}} e^{-M\Omega^2(x-x_m)^2/2k_B T}.
\label{gf2}
\end{equation}
From this we obtain the classical distribution (\ref{zmdfcl}) as the expectation value
\begin{equation}
\label{gf3}
P_{\rm H\,cl}(x)\equiv \langle\,\delta(x-\overline x)\,\rangle^{\Omega,x_m}_{\rm cl}=Z_{\rm cl}^{-1}\int\limits_{-\infty}^{+\infty} \frac{d\overline x}{\sqrt{2\pi\hbar^2/M k_B T}}\, \delta (x-\overline x)\,e^{-M\Omega^2(\overline x-x_m)^2/2k_B T}
=
\frac{1}{\sqrt{2\pi a^2_{\rm cl}}}\exp\left[-\frac{(x-x_m)^2}{2 a^2_{\rm cl}} \right].
\end{equation}
Variational perturbation theory avoids the divergence of the harmonic width $a_{\rm H}^2$ at high temperatures (\ref{zmwidcl}) by
the separate
treatment of the fluctuations of the path average $\overline x$, as explained above.
The average is fixed at some value $x_0$
with the help of a delta function $\delta(\overline{x}-x_0)$.
For each $x_0$ we introduce local expectation values
\begin{equation}
\label{zmcffk}
\langle\,O_1(x(\tau_1))O_2(x(\tau_2))\cdots\,\rangle^{\Omega,x_m}_{x_0}=\frac{\langle\,\delta(\overline{x}-x_0)\,O_1(x(\tau_1))\,O_2(x(\tau_2))\cdots \,\rangle^{\Omega,x_m}}{\langle\,\delta(\overline{x}-x_0) \,\rangle^{\Omega,x_m}}.
\end{equation}
The original quantum statistical distribution
of the harmonic oscillator
(\ref{zmdfQS})
collects fluctuations of $\overline x=x_0$ and those around $x_0$, and can therefore written as a convolution
\begin{equation}
\label{zmconv}
P_{\rm H}(x)=\int_{-\infty}^{+\infty}dx_0\,P_{x_0}(x-x_0)\,P_{\rm cl}(x_0),
\end{equation}
over the classical distribution (\ref{zmdfcl}) and the
local one:
\begin{equation}
\label{zmdffk}
P_{x_0}(x)=\langle\,\delta(x-x(\tau))\,\rangle^{\Omega,x_m}_{x_0}=\frac{1}{\sqrt{2\pi a_{x_0}^2}}\,\exp\left[-\frac{(x-x_0)^2}{2a_{x_0}^2} \right].
\end{equation}
Such a convolution of Gaussian distributions (\ref{zmconv}) leads to
another Gaussian distribution with added
widths, so that the width of the local
distribution is
given by the difference
\begin{equation}
\label{zmwidfk}
a^2_{x_0}=a^2_{\rm H}-a^2_{\rm cl}=
\frac{\hbar}{2M\Omega}\left(\coth\frac{\hbar\Omega}{2k_BT}-\frac{2k_BT}{\hbar\Omega} \right),
\end{equation}
which starts out at a finite value for $T=0$, and goes to zero for $T\to\infty$,
\begin{equation}
\label{wqmlim}
\lim_{T\to\infty}a^2_{x_0}=\frac{\hbar\Omega}{12 k_BT}.
\end{equation}
The latter property suppresses all fluctuations around $\overline x$ and guarantees that $\lim_{T\to\infty}
W_N(x_0)=V(x_0)$ for all $N$ (see Fig. \ref{wid}).
With this separation of the path average,
the partition function
\begin{equation}
\label{zmgenpf}
Z=\oint{\cal D}x\,\exp\left\{-{\cal A}[x]/\hbar \right\}
\end{equation}
for the general particle action
\begin{equation}
\label{zmgenact}
{\cal A}[x]=\int_0^{\hbar/k_BT}d\tau\,\left[\frac{1}{2}M\dot{x}^2(\tau)+V(x(\tau)) \right]
\end{equation}
possesses the effective classical representation (\ref{zmpfeff}) with the
effective classical potential
\begin{equation}
\label{zmeffpot}
V_{\rm eff,cl}(x_0) = -k_BT\,\ln\left(\sqrt{\frac{2\pi\hbar^2}{Mk_BT}}\oint{\cal D}x\,\delta(x_0-\overline{x})\,\exp\left\{-{\cal A}[x]/\hbar \right\} \right).
\end{equation}
In variational perturbation theory, this is expanded perturbatively around an $x_0$-dependent harmonic system with trial frequency $\Omega(x_0)$, whose optimization leads to the approximation $
W_N(x_0)$ for $V_{\rm eff,cl}(x_0)$.
\section{Density Matrix of Harmonic Oscillator}
How can this method be extended to density matrices?
Their path integral representation is
\begin{equation}
\label{zmdm}
\rho(x_b,x_a)=\frac{1}{Z}\tilde{\rho}(x_b,x_a)
\end{equation}
where $\tilde{\rho}(x_b,x_a)$ is the path integral
\begin{equation}
\label{zmprop}
\tilde{\rho}(x_b,x_a)=\int\limits_{(x_a,0)\leadsto (x_b,\hbar/k_BT)}{\cal D}x\,\exp\left\{-{\cal A}[x]/\hbar \right\}
\end{equation}
over all paths with the fixed endpoints $x(0)=x_a$ and $x(\hbar/k_BT)=x_b$.
The partition function is found from the trace of $\tilde{\rho}(x_b,x_a)$:
\begin{equation}
\label{pfi}
Z = \int\limits_{-\infty}^{+\infty}dx\,\tilde{\rho}(x,x).
\end{equation}
For a harmonic oscillator centered at $x_m$ (\ref{zmharmaction}), the path integral (\ref{zmprop}) can be easily done with the result~\cite{PI}
\begin{equation}
\label{harmdens}
\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)=\sqrt{\frac{M\Omega}{2\pi\hbar \sinh\hbar\Omega/k_BT}}\exp\left\{-\frac{M\Omega}{2\hbar \sinh{\hbar\Omega/k_BT}}\left[(\tilde{x}_b^2+\tilde{x}_a^2)\cosh{\hbar\Omega/k_BT}-2 \tilde{x}_b \tilde{x}_a\right]\right\}
\end{equation}
where
\begin{equation}
\label{displ}
\tilde{x}(\tau)=x(\tau)-x_m.
\end{equation}
At fixed endpoints $x_b,x_a$, the quantum mechanical correlation functions are
\begin{equation}
\label{zmdmmv}
\mean{O_1(x(\tau_1))\,O_2(x(\tau_2))\cdots}=\frac{1}{
\tilde{\rho}_0^{~\Omega,x_m}(x_b, x_a)}\,\int\limits_{(x_a,0)\leadsto (x_b,\hbar/k_BT)}{\cal D}x\,O_1(x(\tau_1))\,O_2(x(\tau_2))\cdots\,\exp\left\{-{\cal A}_{\Omega,x_m}[x]/\hbar \right\}
\end{equation}
and the distribution function is given by
\begin{equation}
\label{zmdmQS}
p_{\rm H}(x,\tau)\equiv \mean{\delta(x-x(\tau))} = \frac{1}{\sqrt{2\pi b^2_{\rm H}(\tau)}}\exp\left[-\frac{(\tilde{x}-x_{\rm cl})^2}{2 b^2_{\rm H}(\tau)} \right].
\end{equation}
The classical path of a particle in a harmonic potential is
\begin{equation}
\label{zmclpath}
x_{\rm cl}(\tau) = \frac{\tilde{x}_b\sinh \Omega\tau+\tilde{x}_a\sinh\Omega(\hbar/k_BT-\tau) }{{\sinh \hbar\Omega/k_BT}}
\end{equation}
and the time-dependent width $b^2_{\rm H}(\tau)$ is found to be
\begin{equation}
\label{zmdmQSwid}
b^2_{\rm H}(\tau)=\frac{\hbar}{2M\Omega}\left\{\coth \frac{\hbar\Omega}{k_BT} -\frac{\cosh[\Omega(2\tau-\hbar/k_BT)]}{\sinh \hbar\Omega/k_BT}\right\}.
\end{equation}
Since the euclidean time $\tau$ lies in the interval $0\le\tau\le \hbar/k_BT$, the width (\ref{zmdmQSwid}) is bounded by
\begin{equation}
b_{\rm H}^2(\tau)\le \frac{\hbar}{2M\Omega}\tanh\frac{\hbar\Omega}{2k_BT},
\end{equation}
thus remaining finite at all temperatures. The temporal average of (\ref{zmdmQSwid}) is
\begin{equation}
\label{zmdmQSwidav}
b^2_{\rm H}=\frac{k_BT}{\hbar}\int_0^{\hbar/k_BT}d\tau\,b^2_{\rm H}(\tau)=\frac{\hbar}{2M\Omega}\left(\coth\frac{\hbar\Omega}{k_BT}-\frac{k_BT}{\hbar\Omega}\right).
\end{equation}
Just as $a_{x_0}^2$, this goes to zero for $T\to\infty$ with an asymptotic behaviour $\hbar\Omega/6k_BT$, which is twice as big as that of $a_{x_0}^2$ (see Fig.~\ref{wid}).
\section{Variational Perturbation Theory for Density Matrices}
\label{theory}
To obtain a variational approximation for the density matrix,
it is useful to separate the general action (\ref{zmgenact})
into a trial one for which the euclidean propagator is known,
and a remainder containing the original potential.
If we were to proceed in complete analogy with the treatment
of the partition function,
we would expand the euclidean path integral around
a trial harmonic one with fixed end points $x_b,x_a$ and
a fixed path average $x_0$, and with a trial frequency $\Omega(x_b,x_a;x_0)$. The result would be
an
effective classical potential $
W_N(x_b,x_a;x_0)$ to be optimized in $\Omega(x_b,x_a;x_0)$.
After that we would have to perform a final integral in $x_0$ over the Boltzmann factor $\exp[-
W_N(x_b,x_a;x_0)/k_BT]$.
\begin{figure}
\caption{\label{widB}
\label{widB}
\end{figure}
But, because of the finiteness of the fluctuation width $b_{\rm H}^2$ at all temperatures which is similar to that of $a_{x_0}^2$, the special treatment of $\overline x=x_0$ becomes superfluous for paths with fixed endpoints $x_b,x_a$.
While the separation of $x_0$ was necessary to deal with the diverging fluctuation width of the path average $\overline x$, paths with fixed ends have fluctuations of the path average which are governed by the distribution
\begin{equation}
\label{vpt1}
p_{x_0}(x_b,x_a;x_0)\equiv\mean{\delta(x-\overline x)}=\frac{1}{\sqrt{2\pi b^2_{x_0}}}\exp\left\{-\frac{1}{2b_{x_0}^2}\left[\tilde{x}_0-\frac{1}{2}(\tilde{x}_b+\tilde{x}_a)\frac{2k_BT}{\hbar\Omega}\tanh{\frac{\hbar\Omega}{2 k_BT}} \right]^2\right\}
\end{equation}
with the width
\begin{equation}
\label{vpt2}
b_{x_0}^2=\frac{k_BT}{M\Omega^2}\left[1-\frac{2k_BT}{\hbar\Omega}\tanh{\frac{\hbar\Omega}{2k_BT}} \right],
\end{equation}
which goes to zero for both limits $T\to 0$ and $T\to\infty$ (see Fig.~\ref{widB}).
At each euclidean time, $x(\tau)$ fluctuates narrowly around the classical path $x_{\rm cl}(\tau)$ connecting $x_b$ and $x_a$. This is the reason why we may treat the fluctuations of $\overline x=x_0$ by variational perturbation theory, just as the other fluctuations. As a remnant of the extra treatment of $x_0$ we must, however, perform the initial perturbation expansion around the
minimum of the
effective classical potential which will lie at some point $x_m$ determined by the endpoints $x_b,x_a$, and by the minimum of the potential $V(x)$. Thus we shall use the euclidean path integral for the density matrix of the harmonic oscillator centered at $x_m$ as the trial system around which to perform the variational perturbation theory, treating the fluctuations of $x_0$ around $x_m$ on the same footing as the remaining fluctuations. The position $x_m$ of the minimum is a function $x_m=x_m(x_b,x_a)$, and has to be optimized with respect to the trial frequency, which itself is a function $\Omega=\Omega(x_b,x_a)$
to be optimized.
Hence we start by decomposing the action (\ref{zmgenact}) as
\begin{equation}
\label{actsep}
{\cal A}[x]={\cal A}_{\Omega,x_m}[x]+{\cal A}_{\rm int}[x]
\end{equation}
with an interaction
\begin{equation}
\label{acti}
{\cal A}_{\rm int}[x(\tau)]=\int_0^{\hbar \beta}d\tau\,V_{\rm int}(x(\tau)),
\end{equation}
where the interaction potential is the difference between the original one $V(x)$ and the inserted displaced harmonic oscillator:
\begin{equation}
\label{potint}
V_{\rm int}(x(\tau))=V(x(\tau))-\frac{1}{2}M\Omega^2[x(\tau)-x_m]^2.
\end{equation}
For brevity, we have introduced the inverse temperature
in natural units
$\beta\equiv 1/k_BT$ in (\ref{acti}).
Now we evaluate the path integral for the euclidean propagator (\ref{zmprop})
by treating the interaction (\ref{acti}) as a perturbation, leading to a moment expansion
\begin{equation}
\label{momexp}
\tilde{\rho}(x_b,x_a)=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)\left[1-\frac{1}{\hbar}\mean{{\cal A}_{\rm int}[x]}+\frac{1}{2\hbar^2}\mean{{\cal A}_{\rm int}^2[x]}-\ldots \right],
\end{equation}
with expectation values defined in (\ref{zmdmmv}). The zeroth order consists of the harmonic contribution (\ref{harmdens}) and higher orders contain harmonic averages of the interaction (\ref{acti}). The correlation functions in (\ref{momexp}) can be decomposed into connected ones by going over to cumulants, yielding
\begin{equation}
\label{nd}
\tilde{\rho}(x_b,x_a)=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)\exp\left[-\frac{1}{\hbar}\cum{{\cal A}_{\rm int}[x]}+\frac{1}{2\hbar^2}\cum{{\cal A}_{\rm int}^2[x]}-\ldots\right],
\end{equation}
where the first cumulants are defined as usual
\begin{eqnarray}
\label{cums}
\cum{O_1(x(\tau_1))}&=&\mean{O_1(x(\tau_1))},\nonumber\\
\cum{O_1(x(\tau_1))O_2(x(\tau_2))}&=&\mean{O_1(x(\tau_1))O_2(x(\tau_2))}-\mean{O_1(x(\tau_1))}\mean{O_2(x(\tau_2))},\nonumber\\
&\vdots&\quad .
\end{eqnarray}
The series (\ref{nd}) is truncated after the $N$-th term, resulting in the $N$-th order approximant for the quantum statistical density matrix
\begin{equation}
\label{ndx}
\tilde{\rho}_N^{~\Omega,x_m}(x_b,x_a)=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)\exp\left[\sum\limits_{n=1}^N\,\frac{(-1)^n}{n! \hbar^n}\,\cum{{\cal A}_{\rm int}^n[x]}\right],
\end{equation}
which explicitly depends on both variational parameters $\Omega$ and $x_m$.
In analogy
to classical statistics, where the Boltzmann distribution
in configuration space is controlled
by the classical potential $V(x)$ according to
\begin{equation}
\label{dcl}
\tilde{\rho}_{\rm cl}(x)=\sqrt{\frac{M}{2\pi\hbar^2\beta}}\exp\left[-\beta V(x) \right],
\end{equation}
we now
introduce a new type of
{\em effective classical potential\/}
$V_{\rm eff,cl}(x_a,x_b)$ which governs
the unnormalized density matrix
\begin{equation}
\label{deff}
\tilde{\rho}(x_b,x_a)=\sqrt{\frac{M}{2\pi\hbar^2\beta}}\exp\left[-\beta V_{\rm eff,cl}(x_b,x_a) \right].
\end{equation}
Its $N$th order approximation is
obtained from (\ref{harmdens}), (\ref{ndx}), and (\ref{deff})
voa the cumulant expansion
\begin{equation}
\label{veff}
W_N^{\Omega,x_m}(x_b,x_a)=\frac{1}{2\beta}\ln{\frac{\sinh{\hbar\beta\Omega}}{\hbar\beta\Omega}}+\frac{M\Omega}{2\hbar\beta\sinh{\hbar\beta\Omega}}\left\{(\tilde{x}_b^2+\tilde{x}_a^2)\cosh{\hbar\beta\Omega}-2\tilde{x}_b\tilde{x}_a\right\}-\frac{1}{\beta}\sum\limits_{n=1}^N\,\frac{(-1)^n}{n!\hbar^n}\,\cum{{\cal A}_{\rm int}^n[x]},
\end{equation}
which is optimized for each set of endpoints $x_b,x_a$
in the variational parameters $\Omega^2$ and $x_m$,
the result being denoted by
$W_N(x_b,x_a)$.
The optimal values $\Omega^2(x_a,x_b)$ and $x_m(x_a,x_b)$ are determined from the
extremality conditions\begin{equation}
\label{mincond}
\frac{\partial
W_N^{\Omega,x_m}(x_b,x_a)}
{\partial \Omega^2}\stackrel{!}{=}0,\quad\frac{\partial
W_N^{~\Omega,x_m}(x_b,x_a)}{\partial x_m}\stackrel{!}{=}0.
\end{equation}
The solutions are
denoted by ${\Omega^2}^{N},x_m^N $, both being functions
of $x_b,\,x_a$.
If no extrema are found, one has to look for the flattest region of the function (\ref{veff}), where the lowest higher-order derivative disappears. Eventually the normalized density matrix is obtained from
\begin{equation}
\label{normdx}
\rho_N(x_b,x_a)=Z_N^{-1}
\tilde{\rho}_N^{{~\Omega^2}^{N},x_m^N }(x_b,x_a ),
\end{equation}
where
\begin{equation}
\label{vptpf}
Z_N=\int_{-\infty}^{+\infty}dx\,
\tilde{\rho}_N^{~{\Omega^2}^{N},x_m^N }(x_b,x_a ),
\end{equation}
In principle, one could also optimize the entire ratio (\ref{normdx}), but this would be harder to do in practice. Moreover, the optimization of the unnormalized density matrix is the only option, if the normalization diverges due to singularities of the potential.
This will be seen in Sect.~\ref{csect}.
\section{Smearing Formula for Density Matrices}
\label{smform}
In order to calculate the connected correlation functions in the variational perturbation expansion (\ref{ndx}), we must find efficient formulas for evaluating expectation values (\ref{zmdmmv}) of any power of the interaction (\ref{acti})
\begin{equation}
\label{nmean}
\mean{{\cal A}^n_{\rm int}[x]}=\frac{1}{\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)}\int\limits_{\tilde{x}_a,0}^{\tilde{x}_b,\hbar\beta}{\cal D}\tilde{x}\,\prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\,V_{\rm int}(\tilde{x}(\tau_l)+x_m)\right]\,\exp\left\{-\frac{1}{\hbar}{\cal A}_{\Omega,x_m}[\tilde{x}+x_m] \right\}.
\end{equation}
This can be done by an extension of the smearing formalism, developed in Ref.~\cite{kl267}.
We rewrite the interaction potentials as
\begin{equation}
V_{\rm int}(\tilde{x}(\tau_l)+x_m)=\int\limits_{-\infty}^{+\infty}dz_l\,V_{\rm int}(z_l+x_m)\int\limits_{-\infty}^{+\infty}\frac{d\lambda_l}{2\pi}\,\exp\{i\lambda_lz_l\}\,\exp\left[-\int_0^{\hbar\beta}d\tau\,i\lambda_l\delta(\tau-\tau_l)\tilde{x}(\tau) \right]
\end{equation}
and introduce a current
\begin{equation}
\label{jall}
J(\tau)=\sum\limits_{l=1}^n\,i\hbar\lambda_l\delta(\tau-\tau_l),
\end{equation}
so that (\ref{nmean}) becomes
\begin{equation}
\label{nmeanB}
\mean{{\cal A}^n_{\rm int}[x]}=\frac{1}{\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)}\prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\,\int_{-\infty}^{+\infty}dz_l\,V_{\rm int}(z_l+x_{\rm min})\,\int_{-\infty}^{+\infty}\frac{d\lambda_l}{2\pi}\,\exp\{i\lambda_l z_l \}\right]\,
K^{\Omega,x_m}[J].
\end{equation}
The kernel $K^{\Omega,x_m}[J]$ represents the generating functional for all correlation functions of the displaced harmonic oscillator
\begin{equation}
\label{kernel}
K^{\Omega,x_m}[J]=\int\limits_{\tilde{x}_a,0}^{\tilde{x}_b,\hbar\beta}{\cal D}\tilde{x}\,\exp\left\{-\frac{1}{\hbar}\int_0^{\hbar\beta}d\tau\,\left[\frac{m}{2}\dot{\tilde{x}}^2(\tau)+\frac{1}{2}M\Omega^2\tilde{x}^2(\tau)+J(\tau)\,\tilde{x}(\tau) \right]\right\}.
\end{equation}
For zero current $J$, this generating functional reduces the euclidean harmonic propagator (\ref{harmdens}):
\begin{equation}
K^{\Omega,x_m}[J=0]=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a).
\end{equation}
For nonzero $J$, the solution of the functional integral (\ref{kernel}) is given by
\begin{equation}
\label{solkern}
K^{\Omega,x_m}[J]=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)\exp\left[-\frac{1}{\hbar}\int_0^{\hbar\beta}d\tau\,J(\tau)\,x_{\rm cl}(\tau)+\frac{1}{2\hbar^2}\int_0^{\hbar\beta}d\tau\,\int_0^{\hbar\beta}d\tau'\,J(\tau)\,
G^\Omega(\tau,\tau')\,J(\tau') \right],
\end{equation}
where $x_{\rm cl}(\tau)$ denotes the classical path (\ref{zmclpath}) and $G^\Omega(\tau,\tau')$ the harmonic Green function
\begin{equation}
\label{ggreenh}
G^\Omega(\tau,\tau')=\frac{\hbar}{2M\Omega}\frac{\cosh{\Omega(|\tau-\tau'|-\hbar\beta)}-\cosh{\Omega(\tau+\tau'-\hbar\beta)}}{\sinh{\hbar\beta\Omega}}.
\end{equation}
The expression (\ref{solkern}) can be simplified by using the explicit expression (\ref{jall}) for the current $J$. This leads to a generating functional
\begin{equation}
\label{solkernB}
K^{\Omega,x_m}[J]=\tilde{\rho}_0^{~\Omega,x_m}(x_b,x_a)\,\exp\left(-i\mbox{\boldmath$\lambda$}^T{\bf x}_{\rm cl} -\frac{1}{2}\,\mbox{\boldmath$\lambda$}^T\,G\,\mbox{\boldmath$\lambda$} \right),
\end{equation}
where we have introduced the $n$--dimensional vectors $\mbox{\boldmath$\lambda$}=(\lambda_1,\ldots,\lambda_n)^T$, ${\bf x}_{\rm cl}=(x_{\rm cl}(\tau_1),\ldots,x_{\rm cl}(\tau_n))^T$
with the superscript $T$ denoting transposition, and the symmetric $n\times n$-matrix $G$ whose elements are
$G_{kl}=G^\Omega(\tau_k,\tau_l)$. Inserting (\ref{solkernB}) into (\ref{nmeanB}), and performing the integrals with respect to $\lambda_1,\ldots,\lambda_n$, we obtain the $n$-th order smearing formula for the density matrix
\begin{eqnarray}
\label{smear}
\mean{{\cal A}_{\rm int}^n[x]}&=&\prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\int_{-\infty}^{+\infty}dz_l\,V_{\rm int}(z_l+x_m)\right]\nonumber\\
& &\times\frac{1}{\sqrt{(2\pi)^n\,{\rm det}\,G}}\,\exp\left\{-\frac{1}{2}\sum\limits_{k,l=1}^n\,[z_k-x_{\rm cl}(\tau_k)]\,G^{-1}_{kl}\,[z_l-x_{\rm cl}(\tau_l)] \right\}.
\end{eqnarray}
The integrand contains an $n$-dimensional Gaussian distribution describing both thermal and quantum fluctuations around the harmonic classical path $x_{\rm cl}(\tau)$ of Eq. (\ref{zmclpath}) in a trial oscillator centered at $x_m$, whose width is governed by the Green functions (\ref{ggreenh}).
For closed paths with coinciding endpoints ($x_b=x_a$), formula (\ref{smear}) leads to the $n$-th order smearing formula for particle densities
\begin{equation}
\label{normd}
\rho(x_a)=\frac{1}{Z}\tilde{\rho}(x_a,x_a)=\frac{1}{Z}\oint{\cal D}x\,\delta(x(\tau=0)-x_a)\,\exp\{-{\cal A}[x]/\hbar\},
\end{equation}
which can be written as
\begin{equation}
\label{smeard}
\meandA{{\cal A}_{\rm int}^n[x]}=\frac{1}{\rho_0^{\Omega,x_m}(x_a)}\prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\int_{-\infty}^{+\infty}dz_l\,V_{\rm int}(z_l+x_m)\right]\frac{1}{\sqrt{(2\pi)^{n+1}\,{\rm det}\,a^2}}\,\exp\left(-\frac{1}{2}\sum\limits_{k,l=0}^n\,z_k\,a^{-2}_{kl}\,z_l \right)
\end{equation}
with $z_0=\tilde{x}_a$. Here $a$ denotes a symmetric $(n+1)\times(n+1)$-matrix whose elements $a^2_{kl}=a^2(\tau_k,\tau_l)$ are obtained from the harmonic Green function for
periodic paths $
G^{\Omega,{\rm p}}(\tau,\tau')$ as (see Chapters 3 and 5 in \cite{PI})
\begin{equation}
\label{greenh}
a^2(\tau,\tau')\equiv \frac{\hbar}{M}G^{\Omega,{\rm p}}(\tau,\tau')=\frac{\hbar}{2M\Omega}\frac{\cosh{\Omega(|\tau-\tau'|-\hbar\beta/2)}}{\sinh{\hbar\beta\Omega/2}}.
\end{equation}
The diagonal elements $a^2=a(\tau,\tau)$ represent the fluctuation width (\ref{zmwidQS}) which behaves in the classical limit like (\ref{zmwidcl}) and at zero temperature like (\ref{qmflwidth}).
Both smearing formulas (\ref{smear}) and (\ref{smeard}) allow in principle to determine all harmonic expectation values for the variational perturbation theory of density matrices in terms of ordinary Gaussian integrals. Unfortunately, in many applications containing nonpolynomial potentials, it is impossible to solve neither the spatial nor the temporal integrals analytically. This circumstance drastically increases the numerical effort in higher-order calculations.
\section{First-Order Variational Results}
\label{fstlim}
The first-order variational approximation gives usually a reasonable estimate for any desired quantity. Let us investigate the classical and the quantum mechanical limit of this approximation. To facilitate the discussion, we first derive a new representation for the first-order smearing formula (\ref{smeard}) which allows a direct evaluation of the imaginary time integral. The resulting expression will depend only on temperature, whose low- and high-temperature limits can easily be extracted.
\subsection{Alternative Formula for First-Order Smearing}
\label{newrep}
For simplicity, we restrict ourselves to the case of particle densities and allow only symmetric potentials $V(x)$ centered at the origin. If $V(x)$ has only one minimum at the origin, then also $x_m$ will be zero. If $V(x)$ has several symmetric minima, then $x_m$ goes to zero only at sufficiently high temperatures (see Ref.~\cite{PI}).
To first order, the smearing formula (\ref{smeard}) reads
\begin{equation}
\label{smearfirst}
\meand{{\cal A}_{\rm int}[x]}=\frac{1}{\rho_0^\Omega(x_a)}\int\limits_0^{\hbar\beta}d\tau\,\int\limits_{-\infty}^{+\infty}\frac{dz}{2\pi}\,V_{\rm int}(z)\,\frac{1}{\sqrt{a_{00}^2-a_{01}^2}}\exp\left\{-\frac{1}{2}\frac{(z^2+x_a^2)a_{00}-2z x_a a_{01}}{a_{00}^2-a_{01}^2} \right\},
\end{equation}
so that Mehler's summation formula
\begin{equation}
\label{mehler}
\frac{1}{\sqrt{1-b^2}}\exp\left\{-\frac{(x^2+x'^2)(1+b^2)-4xx'b}{2(1-b^2)}\right\}=\exp\left\{-\frac{1}{2}(x^2+x'^2) \right\}\sum\limits_{n=0}^\infty\,\frac{b^n}{2^n n!}\,H_n(x) H_n(x')
\end{equation}
leads to an expansion in terms of Hermite polynomials $H_n(x)$, whose temperature dependence stems from the diagonal elements of the harmonic Green function (\ref{greenh}):
\begin{equation}
\label{fstexp}
\meand{{\cal A}_{\rm int}[x]}=\sum\limits_{n=0}^\infty\,\frac{\hbar\beta}{2^n n!}\,C^{(n)}_\beta\,H_n\left(x_a/\sqrt{2a^2_{00}}\right)\,\int\limits_{-\infty}^{+\infty}\frac{dz}{\sqrt{2\pi a^2_{00}}}\,V_{\rm int}(z)\,e^{-z^2/2 a^2_{00}}\,H_n\left(z/\sqrt{2 a^2_{00}}\right).
\end{equation}
Here the dimensionless functions $C_\beta^{(n)}$ are defined by
\begin{equation}
\label{timeint}
C_\beta^{(n)}=\frac{1}{\hbar\beta}\int\limits_0^{\hbar\beta}d\tau\,\left(\frac{a^2_{01}}{a^2_{00}}\right)^n.
\end{equation}
Inserting (\ref{greenh}) and performing the integral over $\tau$, we obtain
\begin{equation}
\label{cpoly}
C_\beta^{(n)}=\frac{1}{2^n\cosh^n\hbar\beta\Omega/2}
\sum\limits_{k=0}^n\,\left(
n\atop k
\right)\,
\frac{\sinh\hbar\beta\Omega(n/2-k)}{\hbar\beta\Omega(n/2-k)}.
\end{equation}
At high temperatures, these functions of $\beta$ go all to unity,
\begin{equation}
\label{polyhigh}
\lim_{\beta\to 0}C_\beta^{(n)}=1,
\end{equation}
whereas at zero temperature:
\begin{equation}
\label{polylow}
\lim_{\beta\to\infty}
C_\beta^{(n)}=\left\{
\begin{array}{cl}
1, &~~~~~~ n=0,
\\
\displaystyle
\frac{2}{\hbar\beta\Omega n},&~~~~~~ n>0. \\
\end{array}
\right.
\end{equation}
\begin{figure}
\caption{\label{cnbeta}
\label{cnbeta}
\end{figure}
According to (\ref{veff}), the
first-order approximation to the
new effective potential (\ref{veff}) is given by
\begin{equation}
\label{Wfirst}
W_1^\Omega(x_a)=\frac{1}{2\beta}\ln\frac{\sinh{\hbar\beta\Omega}}{\hbar\beta\Omega}+\frac{M\Omega}{\hbar\beta}x_a^2\tanh\frac{\hbar\beta\Omega}{2}+V_{a^2}^\Omega(x_a)
\end{equation}
with the smeared interaction potential
\begin{equation}
\label{smepot}
V_{a^2}^\Omega(x_a)=\frac{1}{\hbar\beta}\,\meand{{\cal A}_{\rm int}[x]}.
\end{equation}
It is instructive to discuss separately the limits $\beta\to 0$ and $\beta\to \infty$ of dominating thermal and quantum fluctuations, respectively.
\subsection{Classical Limit of Effective Classical Potential}
\label{clli}
In the classical limit $\beta\to 0$, the first-order
effective classical potential (\ref{Wfirst}) reduces to
\begin{equation}
\label{Wbzero}
W_1^{\Omega,{\rm cl}}(x_a)=\frac{1}{2}M\Omega^2x_a^2+\lim_{\beta\to 0}\,V_{a^2}^\Omega(x_a).
\end{equation}
The second term is determined by inserting the high-temperature limit of the fluctuation width (\ref{zmwidcl}) and of the polynomials (\ref{polyhigh}) into the expansion (\ref{fstexp}), leading to
\begin{equation}
\label{vazeroA}
\lim_{\beta\to 0}\,V_{a^2}^\Omega(x_a)=\lim_{\beta\to 0}\sum\limits_{n=0}^\infty\,\frac{1}{2^n n!}\,H_n\left(\sqrt{M\Omega^2\beta/2}\,x_a\right)\int\limits_{-\infty}^{+\infty}\frac{dz}{\sqrt{2\pi/M\Omega^2\beta}}\,V_{\rm int}(z)\,e^{-M\Omega^2\beta\,z^2/2}H_n\left(\sqrt{M\Omega^2\beta/2}\,z\right).
\end{equation}
Then we make use of the completeness relation for Hermite polynomials
\begin{equation}
\label{complherm}
\frac{1}{\sqrt{\pi}} e^{-x^2}\sum\limits_{n=0}^\infty\,\frac{1}{2^n n!}\,H_n(x)\,H_n(x')=\delta(x-x'),
\end{equation}
which may be derived from Mehler's summation formula (\ref{mehler}) in the limit $b\to 1^-$, to reduce the smeared interaction potential $V_{a^2}^\Omega(x_a)$ to the pure interaction potential (\ref{potint}):
\begin{equation}
\label{vazeroB}
\lim_{\beta\to 0}\,V_{a^2}^\Omega(x_a)=V_{\rm int}(x_a).
\end{equation}
Recalling (\ref{potint}) we see that the first-order
effective classical potential (\ref{Wbzero}) approaches the classical one:
\begin{equation}
\label{effpotzero}
\lim_{\beta\to 0}\,W_1^{\Omega,{\rm cl}}(x_a)=V(x_a).
\end{equation}
This is a consequence of the vanishing fluctuation width $b_{\rm H}^2$ of the paths around the classical orbits. This property is universal to all higher-order approximations to the
effective classical potential (\ref{veff}). Thus all corrections terms with $n>1$ must disappear in the limit $\beta\to 0$,
\begin{equation}
\label{highzero}
\lim_{\beta\to 0}\,\frac{-1}{\beta}\sum_{n=2}^\infty\,\frac{(-1)^n}{n! \hbar^n}\,\cumd{{\cal A}_{\rm int}^n[x]}=0.
\end{equation}
\subsection{Zero-Temperature Limit}
\label{quli}
At low temperatures, the first-order
effective classical potential (\ref{Wfirst}) becomes
\begin{equation}
\label{quli1}
W^{\Omega,{\rm qm}}_1(x_a)=\frac{\hbar\Omega}{2}+\lim_{\beta\to\infty}V_{a^2}^\Omega(x_a).
\end{equation}
The zero-temperature limit of the smeared potential
in the second term defined in (\ref{smepot})
follows from Eq.~(\ref{fstexp}) by taking into account the limiting procedure for the polynomials $C_\beta^{(n)}$ in (\ref{polylow}) and for the fluctuation width $a^2_{\rm qm}$ (\ref{qmflwidth}). Thus we obtain with $H_0(x)=1$ and the inverse length $\kappa=\sqrt{M\Omega/\hbar}$:
\begin{equation}
\label{quli2}
\lim_{\beta\to\infty} V_{a^2}^\Omega(x_a)=\int\limits_{-\infty}^{+\infty}dz\,\sqrt{\frac{\kappa^2}{\pi}}H_0(\kappa z)^2\exp\{-\kappa^2z^2\}\,V_{\rm int}(z).
\end{equation}
Introducing the harmonic eigenvalues
\begin{equation}
\label{harmen}
E^\Omega_n=\hbar\Omega\left(n+\frac{1}{2}\right),
\end{equation}
and the harmonic eigenfunctions
\begin{equation}
\label{harmwave}
\psi_n^\Omega(x)=\frac{1}{\sqrt{n! 2^n}}\,\left(\frac{\kappa^2}{\pi}\right)^{1/4}\,e^{-\frac{1}{2}\kappa^2x^2}\,H_n(\kappa x),
\end{equation}
we can reexpress the zero-temperature limit of the first-order
effective classical potential (\ref{quli1}) with (\ref{quli2}) by
\begin{equation}
\label{quli3}
W_1^{\Omega,{\rm qm}}(x_a)=E_0^\Omega+\langle\,\psi_0^\Omega\,|\,V_{\rm int}\,|\,\psi_0^\Omega\,\rangle.
\end{equation}
This is recognized as the first-order harmonic Rayleigh-Schr\"odinger perturbative result for the ground state energy.
For the discussion of the quantum mechanical limit of the first-order normalized density,
\begin{equation}
\label{denszeroA}
\rho_1^\Omega(x_a)=\frac{\tilde{\rho}_1^{~\Omega}(x_a)}{Z}=\rho_0^\Omega(x_a)\,\frac{\exp\left\{-\frac{1}{\hbar} \meand{{\cal A}_{\rm int}[x]}\right\}}{\int_{-\infty}^{+\infty} dx_a\,
\rho_0^\Omega(x_a)\exp\left\{-\frac{1}{\hbar}\meand{{\cal A}_{\rm int}[x]}\right\}},
\end{equation}
we proceed as follows.
First we expand (\ref{denszeroA}) up to first order in the interaction, leading to
\begin{equation}
\label{dmfirst1}
\rho_1^\Omega(x_a)=\rho_0^\Omega(x_a)
\left[1-\frac{1}{\hbar}\left(\meand{{\cal A}_{\rm int}[x]}-\int\limits_{-\infty}^{+\infty}d x_a\,
\rho_0^\Omega(x_a)\,\meand{{\cal A}_{\rm int}[x]} \right) \right].
\end{equation}
Inserting (\ref{displ}) and (\ref{fstexp}) into the third term in (\ref{dmfirst1}), and assuming $\Omega$ not to depend explicitly on $x_a$, the $x_a$-integral reduces to the orthonormality relation for Hermite polynomials
\begin{equation}
\label{ortho}
\frac{1}{2^nn!\sqrt{\pi}}\int\limits_{-\infty}^{+\infty}dx_a H_n(x_a)H_0(x_a)e^{-x_a^2}=\delta_{n0},
\end{equation}
so that the third term in (\ref{dmfirst1}) eventually becomes
\begin{equation}
-\int\limits_{-\infty}^{+\infty}d x_a\,\rho_0^\Omega(x_a)\,\meand{{\cal A}_{\rm int}[x]}=-\beta\int\limits_{-\infty}^{+\infty}dz\,\sqrt{\frac{\kappa^2}{\pi}}\,V_{\rm int}(z)\,\exp\{-\kappa^2z^2\}\,H_0(\kappa z).
\end{equation}
But this is just the $n=0$~-term of (\ref{fstexp}) with an opposite sign, thus cancelling the zeroth component of the second term in (\ref{dmfirst1}), which would have been divergent for $\beta\to\infty$.
The resulting expression for the first-order normalized density is
\begin{equation}
\label{dmfirst2}
\rho^\Omega_1(x_a)=\rho_0^\Omega(x_a)\left[1-\sum\limits_{n=1}^\infty\,\frac{\beta}{2^n n!}\,C_\beta^{(n)}\,H_n(\kappa x_a)\int\limits_{-\infty}^{+\infty}dz\,\sqrt{\frac{\kappa^2}{\pi}}\,V_{\rm int}(z)\,\exp(-\kappa^2z^2)\,H_n(\kappa z)\right].
\end{equation}
The zero-temperature limit of $c_\beta^{(n)}$ is from (\ref{polylow}) and (\ref{harmen})
\begin{equation}
\lim_{\beta\to\infty}\beta C_\beta^{(n)}=\frac{2}{E_n^\Omega-E_0^\Omega},
\end{equation}
so that we obtain from (\ref{dmfirst2}) the limit
\begin{equation}
\label{dmfirst3}
\rho_1^\Omega(x_a)=\rho_0^\Omega(x_a)\,\Bigg[1-2\sum\limits_{n=1}^\infty\,\frac{1}{2^n n!}\frac{1}{E_n^\Omega-E_0^\Omega}H_n(\kappa x_a )\int\limits_{-\infty}^{\infty}dz\,\sqrt{\frac{\kappa^2}{\pi}}\,V_{\rm int}(z)\,\exp\{-\kappa^2z^2 \} H_n(\kappa z)\,H_0(\kappa z )\Bigg].
\end{equation}
Taking into account the harmonic eigenfunctions (\ref{harmwave}), we can rewrite (\ref{dmfirst3}) as
\begin{equation}
\label{dmfirst4}
\rho_1^\Omega(x_a)=|\psi_0(x_a)|^2=[\psi_0^\Omega(x_a)]^2-2\psi_0^\Omega(x_a)\sum\limits_{n>0}\psi_n^\Omega(x_a)\frac{\langle\,\psi_n^\Omega\,|\,V_{\rm int}\,|\,\psi_0^\Omega\,\rangle}{E_n^\Omega-E_0^\Omega}
\end{equation}
which is just equivalent to the harmonic first-order Rayleigh-Schr\"odinger result for particle densities.
Summarizing the results of this section, we have shown that our method has properly reproduced the high- and low-temperature limits. Because of relation (\ref{dmfirst4}), the variational approach for particle densities can be used to determine approximately the ground state wave function $\psi_0(x_a)$ for the system of interest.
\section{Smearing Formula in Higher Spatial Dimensions}
\label{highdim}
Most physical systems possess many degrees of freedom. This requires an extension of our method to higher spatial dimensions. In general, we must consider anisotropic harmonic trial systems, in which the previous variational parameter $\Omega^2$ becomes a $D\times D$--matrix $\Omega^2_{\mu\nu}$ with $\mu,\nu=1,2,\ldots,D$.
\subsection{Isotropic Approximation}
\label{highiso}
An isotropic trial ansatz
\begin{equation}
\label{isoOmega}
\Omega^2_{\mu\nu}=\Omega^2\delta_{\mu\nu}
\end{equation}
can give rough initial estimates for the properties of the system.
In this case, the $n$-th order smearing formula (\ref{smeard}) generalizes directly to
\begin{equation}
\label{smeariso}
\meaniso{{\cal A}_{\rm int}^n[{\bf r}]}=\frac{1}{\rho_0^\Omega({\bf r}_a)}\prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\int d^Dz_l\,V_{\rm int}({\bf z}_l)\right]\,\frac{1}{\sqrt{(2\pi)^{n+1}\,{\rm det}\,a^2}^D}\,\exp\left[-\frac{1}{2}\sum\limits_{k,l=0}^n\,{\bf z}_k\,a^{-2}_{kl}\,{\bf z}_l \right]
\end{equation}
with the $D$--dimensional vectors ${\bf z}_l=(z_{1l},z_{2l},\ldots,z_{Dl})^T$. Note, that greek labels $\mu,\nu,\ldots=1,2,\ldots,D$ specify spatial indices and latin labels $k,l,\ldots=0,1,2,\ldots,n$ refer to the different imaginary times. The vector ${\bf z}_0$ denotes ${\bf r}_a$, the matrix $a^2$ is the same as in Section~\ref{smform}. The harmonic density reads
\begin{equation}
\label{isoharmdens}
\rho_0^{\Omega}({\bf r})=\sqrt{\frac{1}{2\pi a^2_{00}}}^D\,\exp\left[-\frac{1}{2\,a^2_{00}}\sum\limits_{\mu=1}^{D}\,x_\mu^2\right].
\end{equation}
\subsection{Anisotropic Approximation}
In the discussion of the anisotropic approximation, we shall consider only radially-symmetric potentials $V({\bf r})=V(|{\bf r}|)$ for simplicity and their major occurence in physics. The trial frequencies decompose naturally into a radial frequency $\Omega_L$ and a transverse one $\Omega_T$ (see Ref.~\cite{PI}):
\begin{equation}
\label{anisoOmega}
\Omega^2_{\mu\nu}=\Omega_L^2\,\frac{{x_a}_\mu {x_a}_\nu}{r_a^2}+\Omega^2_T\left(\delta_{\mu\nu}-\frac{{x_a}_\mu {x_a}_\nu}{r_a^2}\right)
\end{equation}
with $r_a=|{\bf r}_a|$. For practical reasons we rotate the coordinate system by $\bar{{\bf x}}_n=U\,{\bf x}_n$ so that $\overline{\bf r}_a$ points along the first coordinate axis,
\begin{equation}
\label{ra}
(\bar{\bf r}_a)_\mu\equiv \bar{z}_{\mu 0}=\left\{\begin{array}{cc}r_a, & \mu=1,\\ 0, & 2\le \mu\le D, \end{array} \right.
\end{equation}
and $\Omega^2$-matrix is diagonal:
\begin{equation}
\label{rotOmega}
\overline{\Omega^2}=\left(\begin{array}{ccccc}
\Omega^2_L & 0 & 0 & \cdots & 0\\
0 & \Omega_T^2 & 0 & \cdots & 0\\
0 & 0 &\Omega^2_T & \cdots &0\\
\vdots & \vdots &\vdots &\ddots &\vdots\\
0&0&0&\cdots& \Omega^2_T
\end{array}\right)=U\,\Omega^2\,U^{-1}.
\end{equation}
After this rotation, the {\it anisotropic} $n$-th order smearing formula in $D$ dimensions reads
\begin{eqnarray}
\label{smearaniso}
\lefteqn{\meananiso{{\cal A}_{\rm int}^n[{\bf r}]}=\frac{1}{
\rho_0^{\Omega_{L,T}}(\bar{\bf r}_a)} \prod\limits_{l=1}^n\left[\int_0^{\hbar\beta}d\tau_l\,\int d^D\bar{z}_l\,V_{\rm int}(|\bar{\bf z}_l|)\right]\,(2\pi)^{-D(n+1)/2}({\rm det}\,a^2_L)^{-1/2}\,({\rm det}\,a^2_T)^{-(D-1)/2}}\hspace{140pt}\nonumber\\
&\times& \exp\left\{-\frac{1}{2}\sum\limits_{k,l=0}^n\,\bar{z}_{1k}{a_L}_{kl}^{-2}\bar{z}_{1l}\right\}\,\exp\left\{-\frac{1}{2}\sum\limits_{\mu=2}^D\sum_{k,l=1}^n\,\bar{z}_{\mu k}{a_T}_{kl}^{-2}\bar{z}_{\mu l} \right\}.
\end{eqnarray}
The components of the longitudinal and transversal matrices $a^2_L$ and $a^2_T$ are
\begin{equation}
\label{LTmatrix}
{a^2_{L}}_{kl}=a^2_{L}(\tau_k,\tau_l),\quad {a^2_{T}}_{kl}=a^2_{T}(\tau_k,\tau_l)
\end{equation}
where the frequency $\Omega$ in (\ref{greenh}) must be substituted by the new variational parameters $\Omega_L,\Omega_T$, respectively. For the harmonic density in the rotated system we find
\begin{equation}
\label{LTharmdens}
\rho_0^{\Omega_{L,T}}(\bar{\bf r})=\sqrt{\frac{1}{2\pi {a^2_L}_{00}}}\,\sqrt{\frac{1}{2\pi {a^2_T}_{00}}}^{D-1}\,\exp\left[-\frac{1}{2\, {a^2_L}_{00}}\bar{x}_1^2-\frac{1}{2\,{a^2_T}_{00}}\sum\limits_{\mu=2}^{D}\bar{x}_\mu^2\right]
\end{equation}
which is used to normalize (\ref{smearaniso}).
The anisotropic smearing formula (\ref{smearaniso}) will be applied to the Coulomb problem below. The anisotropy becomes significant only at low temperatures, where radial and transversal quantum fluctuations have quite different weights. The
effect of anisotropy disappears completely in the classical limit.
\section{Applications}
\label{applics}
In this section we apply the theory to calculate the electron density of the hydrogen atom.
For simplicity, we
shall employ
natural units with
$\hbar=k_B=M=1$. In order to
develop some feeling how the approximations work,
we first determine the particle density in
a double-well potential to second order.
\subsection{The Double-Well}
\label{dwsect}
In the case of the double-well potential
\begin{equation}
\label{dwell}
V(x)=-\frac{1}{2}\omega^2x^2+\frac{1}{4}g x^4+\frac{1}{4g}
\end{equation}
with coupling constant $g$, we obtain for the expectation of the interaction (\ref{fstexp}) to first order, also setting $\omega^2=1$,
\begin{eqnarray}
\label{smeartwo}
\meandA{{\cal A}_{\rm int}[x]}&=&\frac{1}{2}\beta g_0+\frac{1}{2}g_1C_\beta^{(1)}H_1\left((x_a-x_m)/\sqrt{2 a_{00}^2}\right)+\frac{1}{4}g_2 C_\beta^{(2)}H_2\left((x_a-x_m)/\sqrt{2 a_{00}^2}\right)\nonumber\\
& &+\frac{1}{8}g_3C_\beta^{(3)}H_3\left((x_a-x_m)/\sqrt{2 a_{00}^2}\right)+\frac{1}{16}g_4C_\beta^{(4)}H_4\left((x_a-x_m)/\sqrt{2 a_{00}^2}\right)
\end{eqnarray}
with
\begin{eqnarray}
\label{abbtwo}
g_0&=&-a_{00}^2(\Omega^2+1)+\frac{3}{2}g a_{00}^4+3g a_{00}^2 x_m^2+\frac{1}{2}gx_m^4+\frac{1}{2g}-\frac{1}{2}x_m^2\nonumber\\
g_1&=&-\sqrt{2a_{00}^2}x_m+\frac{3}{4}g(2 a_{00}^2)^{3/2}x_m+g\sqrt{2 a_{00}^2}x_m^3\nonumber\\
g_2&=&-a_{00}^2(\Omega^2+1)+3ga_{00}^4+3ga_{00}^2x_m^2\nonumber\\
g_3&=&g(2a_{00}^2)^{3/2}x_m\nonumber\\
g_4&=&ga_{00}^4\nonumber.
\end{eqnarray}
Inserting (\ref{smeartwo}) in (\ref{smepot}), we obtain the unnormalized double-well density
\begin{equation}
\label{densdwAtwo}
\tilde{\rho}_1^{~\Omega,x_m}(x_a)=\frac{1}{\sqrt{2\pi\beta}}\exp[-\beta W^{\Omega,x_m}_1(x_a)]
\end{equation}
with the first-order
effective classical potential
\begin{equation}
\label{densW1two}
W^{\Omega,x_m}_1(x_a)=\frac{1}{2}\ln\frac{\sinh{\beta\Omega}}{\beta\Omega}+\frac{\Omega}{\beta}(x_a-x_m)^2\tanh\frac{\beta\Omega}{2}+\frac{1}{\beta}\,\meandA{{\cal A}_{\rm int}[x]}.
\end{equation}
After optimizing $W_1^{\Omega,x_m}(x_a)$, the normalized first-order
particle density $\rho_1(x_a)$ is found by dividing $\tilde{\rho}_1(x_a)$
by the first-order partition function
\begin{equation}
\label{partfunc1}
Z_1=\frac{1}{\sqrt{2\pi\beta}}\int\limits_{-\infty}^{+\infty}dx_a\,\exp[-\beta
W_1(x_a)].
\end{equation}
Subjecting $W_1^{~\Omega,x_m} (x_a)$
to the extremality conditions (\ref{mincond}), we
obtain optimal values for $ \Omega^2(x_a) $ and $x_m(x_a)$.
Usually there is a unique minimum,
but sometimes
this does not exist and a turning point or a
vanishing higher derivative must be used for optimization.
Fortunately, the first case is often realized. Fig.~\ref{w3dplot}
shows the dependence of the first-order
effective classical potential $W^{\Omega,x_m}_1(x_a)$ at $\beta=10$ and $g=0.4$ for three fixed values of position $x_a$ as a function of the variational parameters $\Omega^2(x_a)$ and $x_m(x_a)$ in a three-dimensional plot and its corresponding density plot. Thereby in both representations, the darker the region the smaller the value of $W_1^{\Omega,x_m}$. We can distinguish between deep valleys (darkgray), in which the global minimum resides, and hills (lightgray).
After having determined roughly
the area around the expected minimum, one
solves numerically the extremality conditions (\ref{mincond})
with some nearby starting values, to find
the
exact locations of the minimum. The example in Fig.~\ref{w3dplot} gives an
impression of the general features of the
minimization process. First we note that for symmetry
reasons,
\begin{equation}
\label{xmprop}
x_m(x_a)=-x_m(-x_a),
\end{equation}
and
\begin{equation}
\label{omprop}
\Omega^2(x_a)=\Omega^2(-x_a).
\end{equation}
\begin{figure}
\caption{\label{w3dplot}
\label{w3dplot}
\end{figure}
Some first-order approximations
to the effective classical potential
$W_1(x_a)$ are shown in Fig.~\ref{w1}
obtained by optimizing in $\Omega^2(x_a)$ and $x_m(x_a)$.
The sharp maximum ocurring for weak-coupling
is a consequence of the reflection
property (\ref{xmprop})
enforcing a vanishing $x_m(x_a=0)$.
In the strong-coupling regime, on the other hand,
where $x_m(x_a=0)\approx 0$, the sharp top is absent.
This behaviour is illustrated in the right-hand parts
of Figs.~\ref{omandxm01} and \ref{omandxm10} at different temperatures.
\begin{figure}
\caption{\label{w1}
\label{w1}
\end{figure}
\newcommand{8}{8}
\newcommand{7}{7}
\begin{figure}
\caption{\label{omandxm01}
\label{omandxm01}
\end{figure}
\begin{figure}
\caption{\label{omandxm10}
\label{omandxm10}
\end{figure}
The influence of the center parameter $x_m$ diminuishes
for for increasing values of $g$
and decreasing height $1/4g$ of the central barrier.
The same thing is true
at high temperatures and large values of $x_a$,
where
the precise knowledge of the optimal value of $x_m$ is irrelevant.
In these limits,
the particle density can be determined without optimizing in $x_m$,
setting simply $x_m=0$, where
the expectation value Eq.~(\ref{smeartwo}) reduces to
\begin{equation}
\label{smeardw}
\meand{{\cal A}_{\rm int}[x]}=\frac{1}{4}C_\beta^{(2)}H_2\left(x_a/\sqrt{2a_{00}^2}\right)(g_1+3g_2)+\frac{1}{16}\,g_2\, C_\beta^{(4)}H_4\left(x_a/\sqrt{2 a_{00}^2}\right)+\beta\left(\frac{1}{2}g_1+\frac{3}{4}g_2 +g_3\right),
\end{equation}
with
the abbreviations
\begin{equation}
\label{abbdw}
g_1=-a_{00}^2(\Omega^2+1),\quad g_2=g a_{00}^4,\quad g_3=\frac{1}{4g}.
\end{equation}
Inserting (\ref{smeardw}) in (\ref{smepot}) we obtain the unnormalized double-well density
\begin{equation}
\label{densdwA}
\tilde{\rho}_1^{~\Omega}(x_a)=\frac{1}{\sqrt{2\pi\beta}}\exp[-\beta W^\Omega_1(x_a)]
\end{equation}
with the first-order effective classical potential
\begin{equation}
\label{densW1}
W^\Omega_1(x_a)=\frac{1}{2}\ln\frac{\sinh{\beta\Omega}}{\beta\Omega}+\frac{\Omega}{\beta}x_a^2\tanh\frac{\beta\Omega}{2}+\frac{1}{\beta}\,\meand{{\cal A}_{\rm int}[x]}.
\end{equation}
The
optimization at $x_m=0$
gives reasonable results
for moderate temperatures at couplings as low as $g=0.4$,
as shown in Fig.~\ref{dens10} by a
comparison with the exact density obtained from
numerical solutions of Schr\"odinger equation.
An additional optimization in $x_m$ cannot be distinguished on the plot.
An example where the second variational parameter $x_m$
does become important is shown in Fig.~\ref{secvar}, where
e we compare the first-order approximation
with one ($\Omega$) and two variational parameters ($\Omega,x_m$)
with the exact density for different temperatures
at the smaller coupling strength $g=0.1$.
In Fig.~\ref{omandxm01} we see that for $x_a>0$, the optimal $x_m$-values lie close to the right hand minimum of the double-well potential, which we only want to consider here. The minimum is located at $1/\sqrt{g}\approx 3.16$. We observe, that
with two variational parameters the first-order approximation is nearly exact for all temperatures, in contrast to the results with only one variational parameter at low temperatures (see the curve for $\beta=20$).
Also
for a small valuein the case $g=0.4$, the optimization in $\Omega^2$ only gives reasonable results in the temperature region away from high- and low-temperature limits, as shown in Fig.~\ref{dens10} in comparison with the exact density obtained from numerical solution of Schr\"odinger equation. An optimization in two variational parameters gives no better result. A very instructive example, where the second variational parameter $x_m$ becomes important is shown in Fig.~\ref{secvar}. There, we compare the first-order approximation for the double-well density with one ($\Omega$) and two variational parameters ($\Omega,x_m$)
with the exact density for different temperatures at a coupling strength $g=0.1$. This value of $g$ no longer allows to neglect $x_m$ as in the case $g=0.4$. We see from Fig.~\ref{omandxm01} that for $x_a>0$, the optimal $x_m$-values lie close to the right hand minimum of the double-well potential, which we only want to consider here. The minimum is located at $1/\sqrt{g}\approx 3.16$. We observe, that
with two variational parameters the first-order approximation is nearly exact for all temperatures, in contrast to the results with only one variational parameter at low temperatures (see the curve for $\beta=20$).
\begin{figure}
\caption{\label{dens10}
\label{dens10}
\end{figure}
\begin{figure}
\caption{\label{secvar}
\label{secvar}
\end{figure}
In second-order variational perturbation theory, the differences between the optimization procedures using one or two variational parameters become less significant. Thus, we restrict ourselves to the optimization in $\Omega(x_a)$.
The second-order density
\begin{equation}
\label{dwdens2}
\tilde{\rho}_2^{~\Omega}(x_a)=\frac{1}{\sqrt{2\pi\beta}}\,\exp[-\beta\,W^\Omega_2(x_a)]
\end{equation}
with the second-order approximation of the effective classical potential
\begin{equation}
W^\Omega_2(x_a)=\frac{1}{2}\ln\frac{\sinh{\beta\Omega}}{\beta\Omega}+\frac{\Omega}{\beta}x_a^2\tanh\frac{\beta\Omega}{2}+\frac{1}{\beta}\,\meand{{\cal A}_{\rm int}[x]}-\frac{1}{2\beta}\,\cumd{{\cal A}_{\rm int}^2[x]}
\end{equation}
requires evaluating the smearing formula (\ref{smear}) for $n=1$ which is given in (\ref{smeardw}) and $n=2$ to be calculated. Going immediately to the cumulant we have
\begin{eqnarray}
\label{dwcum}
\cumd{{\cal A}_{\rm int}^2[x]}=\int\limits_0^{\hbar\beta}d\tau_1\int\limits_0^{\hbar\beta}d\tau_2\,&& \Bigg\{ \frac{1}{4}(\Omega^2+1)^2\left[I_{22}(\tau_1,\tau_2)-I_2(\tau_1)I_2(\tau_2) \right]-\frac{1}{4}g(\Omega^2+1)\left[I_{24}(\tau_1,\tau_2)-I_2(\tau_1)I_4(\tau_2) \right] \nonumber\\
& &+\frac{1}{16}g^2\left[I_{44}(\tau_1,\tau_2)-I_4(\tau_1)I_4(\tau_2) \right]\Bigg\}
\end{eqnarray}
with
\begin{equation}
\label{gfA}
I_m(\tau_k)=(a_{00}^4-a_{0k}^4)^m\,\frac{\partial^m}{\partial j^m}\,\exp\left[\frac{j^2+2x_a a_{0k}^2 j}{2a_{00}^2(a_{00}^4-a_{0k}^4)} \right]_{j=0},\qquad k=1,2
\end{equation}
and
\begin{eqnarray}
\label{gfB}
I_{mn}(\tau_1,\tau_2)&=&(-{\rm det}\,A)^{m+n}\frac{\partial^m}{\partial j_1^m}\frac{\partial^n}{\partial j_2^n}\,\exp\left[\frac{F(j_1,j_2)}{2a_{00}^2({\rm det}\,A)^2}\right]_{j_1=j_2=0}\\
{\rm det}\,A&=&a_{00}^6+2a_{01}^2a_{02}^2a_{12}^2-a_{00}^2(a_{01}^4+a_{02}^4+a_{12}^4)\nonumber.
\end{eqnarray}
The generating function is
\begin{eqnarray}
F(j_1,j_2)&=&a_{00}^4(j_1^2+j_2^2)-2a_{00}^6(a_{01}^2j_1+a_{02}^2j_2)x_a+2 a_{00}^2(a_{12}^2j_1j_2+(a_{01}^4+a_{02}^4+a_{12}^4)(a_{01}^2j_1+a_{02}^2j_2)x_a)\nonumber\\
& &-(a_{01}^2j_1+a_{02}^2j_2)(a_{01}^2j_1+a_{02}^2j_2+4a_{01}^2a_{02}^2a_{12}^2x_a).
\end{eqnarray}
\begin{figure}
\caption{\label{dwsec}
\label{dwsec}
\end{figure}
All necessary derivatives and the imaginary time integrations in (\ref{dwcum})
have been calculated analytically. After optimizing the unnormalized second-order density (\ref{dwdens2}) in $\Omega$ we obtain the results depicted in Fig.~\ref{dwsec}. Comparing the second-order results
with the exact densities obtained from numerical solutions of the Schr\"odinger equation, we see that the deviations are strongest in the region of intermediate $\beta$, as expected. Quantum mechanical limits are reproduced very well, classical limits exactly.
\subsection{Distribution Function for the Electron in a Hydrogen Atom}
\label{csect}
With the insights gained in the last section we are prepared to apply our method to the more physical problem of an electron in a hydrogen atom,
with the attractive Coulomb interaction
\begin{equation}
\label{coulpot}
V({\bf r})=-\frac{e^2}{r}.
\end{equation}
Apart from the physical significance,
the
theoretical
interest in this problem originates
from
the non-polynomial bature of
the interaction. This makes
the above-developed smearing formula is essential
for finding variational perturbation expansions.
Restricting our attention to the first-order approximation for the unnormalized density, we must calculate the harmonic expectation value of the action
\begin{equation}
\label{coulintact}
{\cal A}_{\rm int}[{\bf r}]=\int\limits_0^{\hbar\beta}d\tau_1\,V_{\rm int}({\bf r}(\tau_1))
\end{equation}
with the interaction potential
\begin{equation}
\label{intpot}
V_{\rm int}({\bf r})=-\left(\frac{e^2}{r}+\frac{1}{2}{\bf r}^T\,\Omega^2\,{\bf r}\right),
\end{equation}
where the matrix $\Omega^2_{\mu\nu}$ has the form (\ref{anisoOmega}). We do not consider three more variational parameters ${\bf r}_m$ here, because this will not be relevant in a strong-coupling case like the Coulomb interaction, as we know from the last section. After optimization in $\Omega$, we compare our results for the radial distribution function
\begin{equation}
\label{raddist}
g({\bf r})=\sqrt{2\pi\beta}^3\,\tilde{\rho}({\bf r})
\end{equation}
with the precise numerical results of Storer \cite{storer}.
For the Coulomb potential,
the optimization procedure can be simplified
by setting the second optimization parameter
$x_m$ equal to zero from the outset.
This is justified by observation
made for
the double-well potential,
that the importance of knowing
$x_m$ diminuishes for decreasing height of the central barrier.
Since the Coulomb potential has no central barrier, we may set $x_m=0$.
\subsubsection{Isotropic First-Order Approximation}
Applying the isotropic smearing formula (\ref{smeariso}) for $N=1$ to the harmonic term in (\ref{coulintact}) we easily find
\begin{equation}
\label{isoharm}
\meaniso{{\bf r}^2(\tau_1)}=3\frac{a^4_{00}-a^4_{01}}{a^2_{00}}+\frac{a_{01}^4}{a_{00}^4}\,r_a^2.
\end{equation}
For the Coulomb potential we obtain the local average
\begin{equation}
\label{isocoul}
\meaniso{\frac{e^2}{r(\tau_1)}}=\frac{e^2}{r_a}\frac{a^2_{00}}{a^2_{01}}\,{\rm erf}\left(\frac{a^2_{01}}{\sqrt{2a^2_{00}(a_{00}^4-a_{01}^4)}}r_a\right).
\end{equation}
The time integration in (\ref{coulintact}) cannot be done in an analytical manner and must be performed numerically. Alternatively we can use the expansion method introduced in Subsection~\ref{newrep} for evaluating the smearing formula in three dimensions which yields
\begin{equation}
\label{altsmearA}
\meaniso{{\cal A}_{\rm int}[{\bf r}]}=[\rho_0^\Omega({\bf r}_a)]^{-1}\frac{e^{-r_a^2/2 a^2_{00}}}{\pi^2 a^2_{00} r_a}\,\sum\limits_{n=0}^\infty\,\frac{H_{2n+1}(r_a/\sqrt{2 a^2_{00}})}{2^{2n+1}(2n+1)!}\,C_\beta^{(2n)} \int\limits_0^\infty dy\,y\, V_{\rm int}(\sqrt{2 a^2_{00}}\,y)e^{-y^2}H_{2n+1}(y).
\end{equation}
This can be rewritten in terms of Laguerre polynomials $L_n^\mu(r)$ as
\begin{equation}
\label{altsmearB}
\meaniso{{\cal A}_{\rm int}[{\bf r}]}=\sqrt{\frac{2 a^2_{00}}{\pi}}\frac{1}{r_a}\sum\limits_{n=0}^{\infty}\,\frac{(-1)^n n!}{(2n+1)!}C_\beta^{(2n)}H_{2n+1}(r_a/\sqrt{2 a^2_{00}})\int\limits_0^\infty dy\,y^{1/2}V_{\rm int}(\sqrt{2 a^2_{00}}\,y^{1/2})e^{-y}L_n^{1/2}(y)L_0^{1/2}(y).
\end{equation}
\begin{figure}\label{dist}
\end{figure}
Using the integral formula \cite[Eq.~2.19.14.15]{prud}
\begin{equation}
\label{lagint}
\int\limits_0^\infty dx\,x^{\alpha-1}e^{-cx}L_m^\gamma(cx)L_n^\lambda(cx)=\frac{(1+\gamma)_m (\lambda-\alpha+1)_n \Gamma(\alpha)}{m! n! c^\alpha}\, _3F_2(-m,\alpha,\alpha-\lambda;\gamma+1,\alpha-\lambda-n;1),
\end{equation}
where the $(\alpha)_n$ are Pochhammer symbols, $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x)$ denotes the confluent hypergeometric function, and $\Gamma(x)$ is the Gamma function, we apply the smearing formula to the interaction potential (\ref{intpot}) and find
\begin{eqnarray}
\label{cintact}
\meaniso{{\cal A}_{\rm int}[{\bf r}]}&=&-\frac{e^2}{\sqrt{\pi}r_a} \sum\limits_{n=0}^\infty\,\frac{(-1)^n(2n-1)!!}{2^n (2n+1)!} C_\beta^{(2n)}H_{2n+1}(r_a/\sqrt{2 a^2_{00}})\nonumber\\
& &-\frac{3}{4}\sqrt{2 a_{00}^6 \Omega^4}\frac{1}{r_a}\left\{C_\beta^{(0)}H_1(r_a/\sqrt{2 a^2_{00}})+\frac{1}{6}C_\beta^{(2)}H_3(r_a/\sqrt{2 a^2_{00}}) \right\}.
\end{eqnarray}
The first term comes from the Coulomb potential, the second from the harmonic potential. The resulting first-order isotropic form of the radial distribution function (\ref{raddist}), which can be written as
\begin{equation}
\label{rd1}
g_1^\Omega({\bf r}_a)=\exp[-\beta W_1^{\Omega}({\bf r}_a)]
\end{equation}
with the isotropic first-order approximation of the effective classical potential
\begin{equation}
\label{rdveff1}
W^\Omega_1({\bf r}_a)=\frac{3}{2\beta}{\rm ln}\,\frac{\sinh\beta\Omega}{\beta\Omega}+\frac{\Omega}{\beta}\,r_a^2\,\tanh\frac{\beta\Omega}{2}+\frac{1}{\beta}\meaniso{{\cal A}_{\rm int}[{\bf r}]},
\end{equation}
is shown in Fig.~\ref{dist} for various temperatures. The results compare well with Storer's curves \cite{storer}. Near the origin, our results are better than those obtained with an earlier approximation derived from lowest-order effective classical potential $
W_1(x_0)$ \cite{kl153}.
\subsubsection{Anisotropic First-Order Approximation}
The above results can be improved by taking care of the anisotropy of the problem. For the harmonic part of the action (\ref{coulintact}),
\begin{equation}
\label{anisoact}
{\cal A}_{\rm int}[{\bf r}]={\cal A}_\Omega[{\bf r}]+{\cal A}_C[{\bf r}]
\end{equation}
the smearing formula (\ref{smearaniso}) yields the expectation value
\begin{equation}
\label{anisoharm}
\meananiso{{\cal A}_\Omega[{\bf r}]}=-\frac{1}{2}\left\{\Omega_L^2 {a_L^2}_{00}\left(C_\beta^{(0)}+\frac{1}{2}C_{\beta,L}^{(2)}H_2(r_a/\sqrt{2 {a^2_L}_{00}})\right)+2\Omega_T^2 {a^2_T}_{00}(C_\beta^{(0)}-C_{\beta,T}^{(2)}) \right\},
\end{equation}
where the $C_{\beta,L(T)}^{(n)}$ are the polynomials (\ref{cpoly}) in which $\Omega$ is replaced by the longitudinal or transverse frequency. For the Coulomb part of action, the smearing formula (\ref{smearaniso}) leads to a double integral
\begin{equation}
\label{anisocoul}
\meananiso{{\cal A}_C[{\bf r}]}=-e^2\int\limits_0^{\hbar\beta}d\tau_1\,\sqrt{\frac{2}{\pi {a^2_L}_{00}(1-a_L^4)}}\int\limits_0^1 d\lambda\,\left\{1+\lambda^2\left[\frac{{a^2_T}_{00}(1-a_T^4)}{{a^2_L}_{00}(1-a_L^4)}-1\right] \right\}^{-1}\,\exp\left\{-\frac{r_a^2 a_L^4 \lambda^2}{2 {a^2_L}_{00}(1-a_L^4)} \right\}
\end{equation}
with the abbreviations
\begin{equation}
\label{shnot}
a^2_{L}=\frac{{a^2_{L}}_{01}}{{a^2_{L}}_{00}},\qquad a^2_{T}=\frac{{a^2_{T}}_{01}}{{a^2_{T}}_{00}}.
\end{equation}
The integrals must be done numerically and the first-order approximation of the radial distribution function can be expressed by
\begin{equation}
\label{rd1aniso}
g_1^{\Omega_{L,T}}({\bf r}_a)=\exp[-\beta W_1^{\Omega_{L,T}}({\bf r}_a)]
\end{equation}
with
\begin{equation}
\label{veffaniso}
W_1^{\Omega_{L,T}}({\bf r}_a)=\frac{1}{\beta}{\rm ln}\,\frac{\sinh\beta\Omega_L}{\beta\Omega_L}+\frac{1}{2\beta}{\rm ln}\,\frac{\sinh\beta\Omega_T}{\beta\Omega_T}+\frac{\Omega_L}{\beta}\,r_a^2\,\tanh\frac{\beta\Omega_L}{2}+\frac{1}{\beta}\meananiso{{\cal A}_{\rm int}[{\bf r}]}.
\end{equation}
This is optimized in $\Omega_L({\bf r}_a),\Omega_T({\bf r}_a)$
with the results shown in Fig.~\ref{dist}. The anisotropic approach improves the isotropic result for temperatures below $10^4$ K.
\section{Summary}
\label{discus}
We have presented variational perturbation theory for density matrices. A generalized smearing formula which accounts for the effects of quantum fluctuations was essential for the treatment of nonpolynomial interactions. We applied the theory to calculate the particle density in a double-well potential, and the electron density in a Coulomb potential, the latter as an example for nonpolynomial application. In both cases, the approximations were
satisfactory.
\acknowledgements
The work of one of us (M.B.) is supported by the Studienstiftung des
deutschen Volkes.
\end{document} |
\begin{document}
\title{$\mathcal L$-invariants and local-global compatibility for the group $\GL_2/F$}
\author{Yiwen Ding}
\address{Department of Mathematics, Imperial College London}
\email{[email protected]}
\maketitle
\begin{abstract}Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $\rho$ be a $2$-dimensional $p$-adic representation of $\Gal(\overline{F}/F)$ which appears in the \'etale cohomology of quaternion Shimura curves (thus $\rho$ is associated to Hilbert eigenforms). When the restriction $\rho_{\wp}:=\rho|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $\rho_{\wp}$ the so-called Fontaine-Mazur $\mathcal L$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $\mathcal L$-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil's results \cite{Br10} in the $\GL_2/\mathbb Q$-case.
\end{abstract}
\tableofcontents
\section*{Introduction}
Let $F$ be a totally real number field, $B$ a quaternion algebra of center $F$ such that there exists only one real place of $F$ where $B$ is split. One can associate to $B$ a system of quaternion Shimura curves $\{M_K\}_{K}$, proper and smooth over $F$, indexed by open compact subgroups $K$ of $(B\otimes_{\mathbb Q} \mathbb A^{\infty})^{\times}$. We fix a prime number $p$, and suppose that there exists only one prime $\wp$ of $F$ above $p$. Suppose $B$ is split at $\wp$, i.e. $(B\otimes_{\mathbb Q} \mathbb Q_p)^{\times} \mathcal Ong \GL_2(F_{\wp})$ (where $F_{\wp}$ denotes the completion of $F$ at $\wp$). Let $E$ be a finite extension of $\mathbb Q_p$ sufficiently large with $\mathcal O_E$ its ring of integers and $\varpi_E$ a uniformizer of $\mathcal O_E$.
Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that $\rho$ appears in the \'etale cohomology of $M_K$ for $K$ sufficiently small (so $\rho$ is associated to Hilbert eigenforms). By the theory of completed cohomology of Emerton (\cite{Em1}), one can associate to $\rho$ a unitary admissible Banach representation $\widehat{\Pi}(\rho)$ of $\GL_2(F_{\wp})$ as follows: put
\begin{equation*}
\widehat{H}^1(K^p,E):=\Big(\varprojlim_{n}\varinjlim_{K_p'} H^1_{\et}\big(M_{K^pK_p'} \times_{F} \overline{F}, \mathcal O_E/\varpi_E^n\big)\Big)\otimes_{\mathcal O_E} E
\end{equation*}
where $K^p$ denotes the component of $K$ outside $p$, and $K_p'$ runs over open compact subgroups of $\GL_2(F_{\wp})$. This is an $E$-Banach space equipped with a continuous action of $\GL_2(F_{\wp}) \times \Gal(\overline{F}/F) \times \mathcal H^{p}$ where $\mathcal H^{p}$ denotes the $E$-algebra of Hecke operators outside $p$. Put
\begin{equation*}
\widehat{\Pi}(\rho):=\Hom_{\Gal(\overline{F}/F)}\big(\rho, \widehat{H}^1(K^{p},E)\big).
\end{equation*}
The representation $\widehat{\Pi}(\rho)$ is supposed to be (a finite direct sum of) the right representation of $\GL_2(F_{\wp})$ corresponding to $\rho_{\wp}:=\rho|_{\Gal(\overline{F_{\wp}}/F_{\wp})}$ in the $p$-adic Langlands program (cf. \cite{Br0}). In nowadays, we know quite little about $\widehat{\Pi}(\rho)$, e.g. we don't know wether it depends only on the local Galois representation $\rho_{\wp}$. By local-global compatibility of the classical local Langlands correspondence for $\GL_2/F$ (for $\ell=p$), one can indeed describe the locally algebraic vectors of $\widehat{\Pi}(\rho)$ in terms of the Weil-Deligne representation $\WD(\rho_{\wp})$ associated to $\rho_{\wp}$ and the Hodge-Tate weights $\HT(\rho_{\wp})$ of $\rho_{\wp}$ via the local Langlands correspondence. However, in general, (unlike the $\ell\neq p$ case), when passing to $\big(\WD(\rho_{\wp}), \HT(\rho_{\wp})\big)$, a lot of information about $\rho_{\wp}$ is lost. Finding the lost information in $\widehat{\Pi}(\rho)$ is thus one of the key problems in $p$-adic Langlands program (this is in fact the starting point of Breuil's initial work on $p$-adic Langlands program, cf. \cite{Br080}).
In this paper, we consider the case where $\rho_{\wp}$ is semi-stable non-crystalline and non-critical (i.e. $\rho_{\wp}$ satisfies the hypothesis \ref{hyp: clin-aq0}). In this case, the missing data, when passing from $\rho_{\wp}$ to $\big(\WD(\rho_{\wp}), \HT(\rho_{\wp})\big)$, can be explicitly described by the so-called \emph{Fontaine-Mazur $\mathcal L$-invariants} $\underline{\mathcal L}_{\Sigma_{\wp}}=(\mathcal L_{\sigma})_{\sigma\in \Sigma_{\wp}}\in E^d$ associated to $\rho_{\wp}$ (e.g. see \S \ref{sec: clin-ene}), where $\Sigma_{\wp}$ denotes the set of $\mathbb Q_p$-embeddings of $F_{\wp}$ in $\overline{\mathbb Q_p}$. Using these $\mathcal L$-invariants, Schraen has associated to $\rho_{\wp}$ a locally $\mathbb Q_p$-analytic representation $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$ of $\GL_2(F_{\wp})$ over $E$ (cf. \cite[\S 4.2]{Sch10}, see also \S \ref{sec: clin-4.2}), which generalizes Breuil's theory \cite{Br04} in $\GL_2(\mathbb Q_p)$-case. Note that one can indeed recover $\rho_{\wp}$ from $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$. The main result of this paper is the
\begin{theorem}[cf. $\text{Thm.\ref{thm: clin-sio}}$]\label{thm: clin0}
Keep the above notation and suppose that $\rho$ is absolutely irreducible modulo $\varpi_E$, there exists a continuous injection of $\GL_2(F_{\wp})$-representations
\begin{equation*}
\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big) \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an},
\end{equation*}
where $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$ denotes the locally $\mathbb Q_p$-analytic vectors of $\widehat{\Pi}(\rho)$.
\end{theorem}
Such a result is called local-global compatibility, since the left side of this injection depends only on the local representation $\rho_{\wp}$ while the right side is globally constructed. Moreover, one can prove the ``uniqueness" (in the sense of Cor.\ref{cor: clin-ape}) of $\Sigma\big(\WD(\rho_{\wp}), \HT(\rho_{\wp}), \underline{\mathcal L}_{\Sigma_{\wp}}\big)$ as subrepresentation of $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$. As a result, we see the local Galois representation $\rho_{\wp}$ is determined by $\widehat{\Pi}(\rho)$. Such a result in the $\mathbb Q_p$-case, proved by Breuil (\cite{Br10}), was the first discovered local-global compatibility in the $p$-adic local Langlands correspondence. In fact, the $\mathcal L$-invariants appearing in the automorphic representation side are often referred to as \emph{Breuil's $\mathcal L$-invariants}. The theorem \ref{thm: clin0} thus shows the equality of Fontaine-Mazur $\mathcal L$-invariants and Breuil's $\mathcal L$-invariants. Our approach is by using some $p$-adic family arguments on both $\GL_2$-side and Galois side, thus different from that of Breuil (by using modular symbols).
In the following (of the introduction), we sketch how we manage to ``find" $\{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}$ in $\widehat{\Pi}(\rho)$. For simplicity, suppose $\rho_{\wp}$ is of Hodge-Tate weights $(-1,0)_{\Sigma_{\wp}}$ (thus $\rho_{\wp}$ is associated to Hilbert eigenforms of weights $(2,\cdots, 2; 0)$ in the notation of \cite{Ca2}). Let $\tau\in \Sigma_{\wp}$, it's enough to find $\mathcal L_{\tau}$ in $\widehat{\Pi}(\rho)_{\tau-\an}$ \big(the maximal locally $\tau$-analytic subrepresentation of $\widehat{\Pi}(\rho)$\big) in the sense of (\ref{equ: clin-utau}) below:
Denote by $Z_1:=\bigg\{\begin{pmatrix}
a & 0 \\ 0 & a
\end{pmatrix}\ \Big|\ a\in 1+2\varpi \mathcal O_{\wp}\bigg\}$ (where $\mathcal O_{\wp}$ denotes the ring of integers of $F_{\wp}$ and $\varpi$ is a uniformizer of $\mathcal O_{\wp}$), consider $\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}$ \big(where ``$(\cdot)^{Z_1}$" signifies the vectors fixed by $Z_1$, and ``$\tau-\an$" signifies the locally $\tau$-analytic subrepresentation\big). By applying Jacquet-Emerton functor, one gets an essentially admissible locally $\tau$-analytic representation of $T(F_{\wp})$: $J_B(\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1})$, which is moreover equipped with an action of $\mathcal H^p$ commuting with that of $T(F_{\wp})$. Following Emerton, one can construct an eigenvariety $\mathcal V_{\tau}$ from $J_B(\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1})$, which is in particular a rigid space finite over $\widehat{T}_{\tau}$, the rigid space parameterizing locally $\tau$-analytic characters of $T(F_{\wp})$ (cf. Thm.\ref{thm: clin-cjw}). A closed point of $\mathcal V_{\tau}$ can be written as $(\chi,\lambda)$ where $\chi$ is a locally $\tau$-analytic character of $T(F_{\wp})$ and $\lambda$ is a system of Hecke eigenvalues (for $\mathcal H^p$).
One can associate to $\rho$ an $E$-point $z_{\rho}=(\chi_{\rho}, \lambda_{\rho})$ of $\mathcal V_{\tau}$, where $\chi_{\rho}=\unr(\alpha/q)\otimes \unr(q\alpha)$ \big($\unr(a)$ denotes the unramified character of $F_{\wp}^{\times}$ sending $\varpi$ to $a$\big), $\lambda_{\rho}$ denotes the system of eigenvalues of $\mathcal H^p$ associated to $\rho$ (via the Eichler-Shimura relations), $\{\alpha,q\alpha\}$ are the eigenvalues of $\varphi^{d_0}$ on $D_{\st}(\rho_{\wp})$ (where $d_0$ is the degree of the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp}$, $q:=p^{d_0}$). Moreover, by multiplicity one result on automorphic representations of $(B\otimes_{\mathbb Q} \mathbb A)^{\times}$, one can prove as in \cite[\S 4.4]{Che11} that $\mathcal V_{\tau}$ is smooth at $z_{\rho}$ (cf. Thm.\ref{thm: clin-elt}, note that by the hypothesis \ref{hyp: clin-aq0}, $z_{\rho}$ is in fact a \emph{non-critical} point).
Let $t:\Spec E[\epsilon]/\epsilon^2 \rightarrow \mathcal V_{\tau}$ be a non-zero element in the tangent space of $\mathcal V_{\tau}$ at $z_{\rho}$, via the composition
\begin{equation*}
t: \Spec E[\epsilon]/\epsilon^2 \longrightarrow \mathcal V_{\tau} \longrightarrow \widehat{T}_{\tau},
\end{equation*}
one gets a character $\widetilde{\chi}_{\rho}=\widetilde{\chi}_{\rho,1}\otimes \widetilde{\chi}_{\rho,2}: T(F_{\wp})^{\times} \rightarrow (E[\epsilon]/\epsilon^2)^{\times}$, which is in fact an extension of $\chi_{\rho}$ by $\chi_{\rho}$. One key point is that, by applying an adjunction formula in family for the Jacquet-Emerton functor (see \cite[Lem.4.5.12]{Em1} for the $\GL_2(\mathbb Q_p)$-case) to the tangent space of $\mathcal V_{\tau}$ at $z_{\rho}$, one gets a non-zero continuous morphism of $\GL_2(F_{\wp})$-representations (where $\overline{B}(F_{\wp})$ denotes the group of lower triangular matrices) (see (\ref{equ: clin-pqgi}))
\begin{equation}\label{equ: clin-pf2}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an} \longrightarrow \widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}[\lambda_{\rho}]
\end{equation}
where $\delta:=\unr(q^{-1}) \otimes \unr(q)$ and we refer to \cite[\S 2]{Sch10} for locally $\tau$-analytic parabolic inductions, and where the right term denotes the generalized $\lambda_{\rho}$-eigenspace of $\widehat{H}^1(K^p,E)_{\tau-\an}^{Z_1}$.
Another key point is that one can describe the character $\widetilde{\chi}_{\rho}$ in term of $\mathcal L_{\tau}$:
\begin{lemma}[$\text{cf. Lem.\ref{thm: clin-aue}}$]\label{lem: clin-htn}
There exists an additive character $\chi$ of $F_{\wp}^{\times}$ in $E$ such that $\widetilde{\chi}_{\rho}$ (as a $2$-dimensional representation of $T(F_{\wp})$ over $E$) is isomorphic to $\chi_{\rho}\otimes_E \psi(\mathcal L_{\tau},\chi)$ where
\begin{equation*}
\psi(\mathcal L_{\tau},\chi)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix}1 & \log_{\tau,-\mathcal L_{\tau}}(ad^{-1})+\chi (ad) \\ 0 & 1\end{pmatrix},
\end{equation*}
and $\log_{\tau,\mathcal L}$ denotes the additive character of $F_{\wp}^{\times}$ such that $\log_{\tau,\mathcal L}|_{\mathcal O_{\wp}^{\times}}=\tau \circ \log$ and $\log_{\tau,\mathcal L}(p)=\mathcal L$.
\end{lemma}
To prove this lemma, one considers the $p$-adic family of Galois representations over $\mathcal V_{\tau}$. In fact, there exist an admissible neighborhood $U$ of $z_{\rho}$ in $\mathcal V_{\tau}$ and a continuous representation $\rho_U: \Gal(\overline{F}/F) \rightarrow \GL_2(\mathcal O_U)$ such that the evaluation of $\rho_U$ at any classical point of $U$ (which thus corresponds to certain Hilbert eigenforms $h$) is just the Galois representation associated to $h$. Via the map $t$, one gets a continuous representation $\widetilde{\rho}: \Gal(\overline{F}/F) \rightarrow \GL_2(E[\epsilon]/\epsilon^2)$ which satisfies $\widetilde{\rho}\equiv \rho \pmod{\epsilon}$. By the theory of global triangulation \cite{KPX}, one can obtain an exact sequence (cf. (\ref{equ: clin-2no})):
\begin{equation*}
0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}\big(\unr(q)\widetilde{\chi}_{\rho,1}\big) \rightarrow D_{\rig}(\widetilde{\rho}_{\wp}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}\Big(\widetilde{\chi}_{\rho,2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\Big)\rightarrow 0,
\end{equation*}
where $\widetilde{\rho}_{\wp}:=\widetilde{\rho}|_{\Gal(\overline{F_{\wp}}/F_{\wp})}$. The lemma then follows by applying the formula in \cite[Thm.1.1]{Zhang} (which generalizes Colmez's formula \cite{Colm10} in $\mathbb Q_p$-case) to $\widetilde{\rho}_{\wp}$.
Return to the map (\ref{equ: clin-pf2}). We know $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an}$ lies in an exact sequence
\begin{equation*}
0 \rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an}\rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\rho}\delta^{-1}\big)^{\tau-\an} \xrightarrow{s} \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an} \rightarrow 0.
\end{equation*}
where $s$ depends on $\mathcal L_{\tau}$ and $\chi$ as in the lemma \ref{lem: clin-htn}. On the other hand, it's known that $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\rho}\delta^{-1}\big)^{\tau-\an}$ admits a unique finite dimensional subrepresentation $V(\alpha):=\unr(\alpha)\circ \dett$. Put $\Sigma(\alpha,\mathcal L_{\tau}):=s^{-1}(V(\alpha))/V(\alpha)$ (cf. \cite[\S 4.2]{Sch10}), which turns out to be \emph{independent} of the character $\chi$ in Lem.\ref{lem: clin-htn} and thus depends \emph{only} on $\mathcal L_{\tau}$.
At last, one can prove that (\ref{equ: clin-pf2}) induces actually a continuous injection of locally $\tau$-analytic representations of $\GL_2(F_{\wp})$
\begin{equation}\label{equ: clin-utau}
\Sigma(\alpha,\mathcal L_{\tau}) \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\tau-\an}.
\end{equation}
It seems this argument might work for some other groups and some other Shimura varieties. For example, in the $\GL_2/\mathbb Q$-case (with Coleman-Mazur eigencurve, reconstructed by Emerton \cite[\S 4]{Em1} using completed cohomology of modular curves), by restricting the map \cite[(4.5.9)]{Em1} to the tangent space at a semi-stable non-crystalline point, one can obtain a map as in (\ref{equ: clin-pf2}). On the other hand, one can prove a similar result as in Lem.\ref{lem: clin-htn} by Kisin's theory in \cite{Ki} and Colmez's formula \cite{Colm10}. Combining them together, one can actually reprove Breuil's result in \cite{Br10} for locally analytic representations and thus obtain directly the equality of Fontaine-Mazur $\mathcal L$-invariant and Breuil's $\mathcal L$-invariant without using Darmon-Orton's $\mathcal L$-invariant (as in Breuil's original proof \cite{Br10}).
We refer to the body of the text for more detailed and more precise statements.
After the results of this paper was firstly announced, Yuancao Zhang informed us that he had proved the existence of $\mathcal L$-invariants in $\widehat{\Pi}(\rho)$ in certain cases by using some arguments as in \cite[\S 5]{BE}, however, the equality between these $\mathcal L$-invariants and Fontaine-Mazur $\mathcal L$-invariants was not proved.
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
\section{Notations and preliminaries}\label{sec: clin-1}Let $F$ be a totally real field of degree $d$ over $\mathbb Q$, denote by $\Sigma_{\infty}$ the set of real embeddings of $F$. For a finite place $\mathfrak l$ of $F$, we denote by $F_{\mathfrak l}$ the completion of $F$ at $\mathfrak l$, $\mathcal O_{\mathfrak l}$ the ring of integers of $F_{\mathfrak l}$ with $\varpi_{\mathfrak l}$ a uniformiser of $\mathcal O_{\mathfrak l}$. Denote by $\mathbb A$ the ring of adeles of $\mathbb Q$ and $\mathbb A_F$ the ring of adeles of $F$. For a set $S$ of places of $\mathbb Q$ (resp. of $F$), we denote by $\mathbb A^S$ (resp. by $\mathbb A_F^S$) the ring of adeles of $\mathbb Q$ (resp. of $F$) outside $S$, $S_F$ the set of places of $F$ above that in $S$, and $\mathbb A_F^S:=\mathbb A_F^{S_F}$.
Let $p$ be a prime number, suppose there exists only one prime $\wp$ of $F$ lying above $p$. Denote by $\Sigma_{\wp}$ the set of $\mathbb Q_p$-embeddings of $F_{\wp}$ in $\overline{\mathbb Q_p}$; let $\varpi$ be a uniformizer of $\mathcal O_{\wp}$, $F_{\wp,0}$ the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp}$, $d_0:=[F_{\wp,0}:\mathbb Q_p]$, $e:=[F_{\wp}:F_{\wp,0}]$, $q:=p^{d_0}$ and $\upsilon_{\wp}$ a $p$-adic valuation on $\overline{\mathbb Q_p}$ normalized by $\upsilon_{\wp}(\varpi)=1$. Let $E$ be a finite extension of $\mathbb Q_p$ big enough such that $E$ contains all the $\mathbb Q_p$-embeddings of $F$ in $\overline{\mathbb Q_p}$, $\mathcal O_E$ the ring of integers of $E$ and $\varpi_E$ a uniformizer of $\mathcal O_E$.
Let $B$ be a quaternion algebra of center $F$ with $S(B)$ the set (of even cardinality) of places of $F$ where $B$ is ramified, suppose $|S(B)\cap \Sigma_{\infty}|=d-1$ and $S(B)\cap \Sigma_{\wp}=\emptyset$, i.e. there exists $\tau_{\infty}\in \Sigma_{\infty}$ such that $B\otimes_{F,\tau_{\infty}} \mathbb R\mathcal Ong M_2(\mathbb R)$, $B\otimes_{F,\sigma} \mathbb R \mathcal Ong \mathbb H$ for any $\sigma\in \Sigma_{\infty}$, $\sigma\neq \tau_{\infty}$, where $\mathbb H$ denotes the Hamilton algebra, and $B\otimes_{\mathbb Q} \mathbb Q_p\mathcal Ong M_2(F_{\wp})$. We associate to $B$ a reductive algebraic group $G$ over $\mathbb Q$ with $G(R):=(B\otimes_{\mathbb Q} R)^{\times}$ for any $\mathbb Q$-algebra $R$. Set $\mathbb S:=\mathbb Res_{\mathbb C/\mathbb R}\mathbb G_m$, and denote by $h$ the morphism
\begin{equation*}
h: \mathbb S(\mathbb R)\mathcal Ong \mathbb C^{\times} \longrightarrow G(\mathbb R)\mathcal Ong \GL_2(\mathbb R)\times (\mathbb H^*)^{d-1}, \ a+bi\mapsto \bigg(\begin{pmatrix}a&b\\ -b&a\end{pmatrix}, 1, \cdots, 1\bigg).
\end{equation*}
The space of $G(\mathbb R)$-conjugacy classes of $h$ has a structure of complex manifold, and is isomorphic to $\mathfrak h^{\pm}:=\mathbb C \setminus \mathbb R$ (i.e. $2$ copies of the Poincar\'e's upper half plane). We get a projective system of Riemann surfaces indexed by open compact subgroups of $G(\mathbb A^{\infty})$:
\begin{equation*}
M_K(\mathbb C):=G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K)\big)
\end{equation*}
where $G(\mathbb Q)$ acts on $\mathfrak h^{\pm}$ via $G(\mathbb Q)\hookrightarrow G(\mathbb R)$ and the transition map is given by
\begin{equation}\label{equ: clin-sab}
G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K_1)\big) \longrightarrow G(\mathbb Q)\setminus \big(\mathfrak h^{\pm} \times (G(\mathbb A^{\infty})/K_2)\big), \ (x,g)\mapsto (x,g),
\end{equation} for $K_1\subseteq K_2$. It's known that $M_K(\mathbb C)$ has a canonical proper smooth model over $F$ (via the embedding $\tau_{\infty}$), denoted by $M_K$, and these $\{M_K\}_{K}$ form a projective system of proper smooth algebraic curves over $F$ (i.e. the transition map (\ref{equ: clin-sab}) admits also an $F$-model).
One has a natural isomorphism $G(\mathbb Q_p) \xlongrightarrow{\sim} \GL_2(F_{\wp})$.
Let $K_{0,\wp}:=\GL_2(\mathcal O_{\wp})$, in the following, we fix an open compact subgroup $K^{p}$ of $G(\mathbb A^{\infty,p})$ small enough such that the open compact subgroup $K^pK_{0,\wp}$ of $G(\mathbb A^{\infty})$ is neat (cf. \cite[Def.4.11]{New}). Denote by $S(K^{p})$ the set of finite places $\mathfrak l$ of $F$ such that $\mathfrak l \nmid p$, that $B$ is split at $\mathfrak l$, i.e. $B\otimes_F F_{\mathfrak l}\xlongrightarrow{\sim} \mathrm{M}_2(F_{\mathfrak l})$, and that $K^{p}\cap \GL_2(F_{\mathfrak l})\mathcal Ong \GL_2(\mathcal O_{\mathfrak l})$. Denote by $\mathcal H^p$ the commutative $\mathcal O_E$-algebra generated by the double coset operators $[\GL_2(\mathcal O_{\mathfrak l}) g_{\mathfrak l} \GL_2(\mathcal O_{\mathfrak l})]$ for all $g_{\mathfrak l}\in \GL_2(F_{\mathfrak l})$ with $\dett(g_{\mathfrak l})\in \mathcal O_{\mathfrak l}$ and for all $\mathfrak l \in S(K^p)$. Set
\begin{eqnarray*}T_{\mathfrak l}&:=&\bigg[\GL_2(\mathcal O_{\mathfrak l})\begin{pmatrix} \varpi_{\mathfrak l} & 0 \\ 0 & 1\end{pmatrix}\GL_2(\mathcal O_{\mathfrak l})\bigg], \\
S_{\mathfrak l} &:=& \bigg[\GL_2(\mathcal O_{\mathfrak l})\begin{pmatrix} \varpi_{\mathfrak l} & 0 \\ 0 & \varpi_{\mathfrak l}\end{pmatrix}\GL_2(\mathcal O_{\mathfrak l})\bigg],
\end{eqnarray*}
then $\mathcal H^p$ is the polynomial algebra over $\mathcal O_E$ generated by $\{T_{\mathfrak l}, S_{\mathfrak l}\}_{\mathfrak l \in S(K^p)}$.
Denote by $Z_0$ the kernel of the norm map $\mathscr N: \mathbb Res_{F/\mathbb Q} \mathbb G_m \rightarrow \mathbb G_m$ which is a subgroup of $Z=\mathbb Res_{F/\mathbb Q} \mathbb G_m$. We set $G^c:=G/Z_0$.
Denote by $\Art_{F_{\wp}}: F_{\wp}^{\times}\xrightarrow{\sim} W_{F_\wp}^{\ab}$ the local Artin map normalized by sending uniformizers to geometric Frobenius elements \big(where $W_{F_{\wp}}\subset \Gal(\overline{\mathbb Q_p}/F_{\wp})$ denotes the Weil group\big).
Let $\sigma\in \Sigma_{\wp}$, denote by $\log_{\sigma}$ the composition $\mathcal O_{\wp}^{\times} \xrightarrow{\log} \mathcal O_{\wp} \xrightarrow{\sigma} E$. For $\mathcal L\in E$, denote by $\log_{\sigma,\mathcal L, \varpi}$ the (additive) character of $F_{\wp}^{\times}$ such that $\log_{\sigma,\mathcal L,\varpi}|_{\mathcal O_{\wp}^{\times}}=\log_{\sigma}$ and $\log_{\sigma,\mathcal L,\varpi}(\varpi)=\mathcal L$. Denote by $\log_{\sigma,\mathcal L}$ the (additive) character of $F_{\wp}^{\times}$ in $E$ satisfying $\log_{\sigma,\mathcal L}|_{\mathcal O_{\wp}^{\times}}=\log_\sigma$ and $\log_{\sigma,\mathcal L}(p)=\mathcal L$. Let $\mathcal L(\varpi):=e^{-1}\big(\mathcal L-\log_{\sigma}\big(\frac{p}{\varpi^{e}}\big)\big)$, thus one has
\begin{equation*}
\log_{\sigma,\mathcal L}=\log_{\sigma, \mathcal L(\varpi), \varpi}.
\end{equation*}
Denote by $\unr(a)$ the unramified character of $F_{\wp}^{\times}$ sending $\varpi$ to $a$.
Let $V$ be an $E$-vector space equipped with an $E$-linear action of $A$ (with $A$ a set of operators), $\chi$ a system of eigenvalues of $A$, denote by $V^{A=\chi}$ the $\chi$-eigenspace, $V[A=\chi]$ the generalized $\chi$-eigenspace, $V^A$ the vector space of $A$-fixed vectors.
Let $S\subseteq \Sigma_\wp$, $k_{\sigma}\in \mathbb Z_{\geq 2}$ for all $\sigma\in S$, denote by $W(\underline{k}_S):=\otimes_{\sigma\in S} \big(\Sym^{k_{\sigma}-2} E^2\big)^{\sigma}$ the algebraic representation of $G(\mathbb Q_p)\mathcal Ong\GL_2(F_{\wp})$ with $\GL_2(F_{\wp})$ acting on $\big(\Sym^{k_{\sigma}-2} E^2\big)^{\sigma}$ via $ \GL_2(F_{\wp}) \xrightarrow{\sigma} \GL_2(E)$ for $\sigma\in S$. Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$, put $W(\underline{k}_{\Sigma_{\wp}},w):=\otimes_{\sigma\in \Sigma_{\wp}} \big(\Sym^{k_{\sigma}-2} E^2\otimes \dett^{\frac{w-k_{\sigma}+2}{2}}\big)^{\sigma}$.
Denote by $B(F_{\wp})$ \big(resp. $\overline{B}(F_{\wp})$\big) the subgroup of $\GL_2(F_{\wp})$ of upper (resp. lower) triangular matrices, $T(F_{\wp})$ the group of diagonal matrices, $N(F_{\wp})$ the group of unipotent elements in $B(F_{\wp})$, $N_0:=N(F_{\wp})\cap \GL_2(\mathcal O_{\wp})$, $Z':=T(F_{\wp})\cap \SL_2(F_{\wp})$, $K_{1,\wp}:=\{g\in \GL_2(\mathcal O_{\wp})\ |\ g\equiv 1 \pmod{2\varpi} \}$, $Z_{1}$ the center of $K_{1,\wp}$, $Z_{1}':=Z'\cap K_{1,\wp}$. Put $\delta:=\unr(q^{-1}) \otimes \unr(q)$ being a character of $T(F_{\wp})$ (which is in fact the modulus character of $B(F_{\wp})$).
\subsection*{Locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$}
Recall some notions on locally $\mathbb Q_p$-analytic representations. Let $V$ be a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ over $E$, i.e. a locally analytic representation of $\GL_2(F_{\wp})$ with $\GL_2(F_{\wp})$ viewed as a $p$-adic $\mathbb Q_p$-analytic group, $V$ is naturally equipped with a $\mathbb Q_p$-linear action of the Lie algebra $\mathfrak g$ of $\GL_2(F_{\wp})$ \big(thus an $E$-linear action of $\mathfrak g_{\Sigma_{\wp}}:=\mathfrak g\otimes_{\mathbb Q_p} E$\big) given by
\begin{equation*}
\mathfrak x\cdot v:=\frac{d}{dt}\exp(t\mathfrak x)(v)|_{t=0}.
\end{equation*} Using the isomorphism
\begin{equation}\label{equ: clin2-grwm}
F_{\wp}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} E, \ a \otimes b \mapsto (\sigma(a)b)_{\sigma: F_{\wp}\rightarrow E},
\end{equation}
one gets a decomposition $\mathfrak g_{\Sigma_{\wp}}\xrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} \mathfrak g_{\sigma}$ with $\mathfrak g_{\sigma}:=\mathfrak g \otimes_{F_{\wp},\sigma}E$. Let $J\subseteq \Sigma_{\wp}$, a vector $v\in V$ is called \emph{locally $J$-analytic} if the action of $\mathfrak g_{\Sigma_{\wp}}$ on $v$ factors through $\mathfrak g_{J}:=\prod_{\sigma\in J} \mathfrak g_{\sigma}$ (we put $\mathfrak g_{\emptyset}:=\{0\}$), in other words, if $v$ is killed by $\mathfrak g_{\Sigma_{\wp}\setminus J}$ (cf. \cite[Def.2.4]{Sch10}); $v$ is called \emph{quasi-$J$-classical} if there exist a finite dimensional representation $U$ of $\mathfrak g_J$ and a $\mathfrak g_J$-invariant map
\begin{equation*}\label{equ: drp-var}
U \lhook\joinrel\longrightarrow V
\end{equation*}
whose image contains $v$, if the $\mathfrak g_J$-representation $U$ can moreover give rise to an algebraic representation of $\GL_2(F_{\wp})$, then we say that $v$ is \emph{$J$-classical}. In particular, $v$ is $\Sigma_{\wp} \setminus J$-classical if $v$ is locally $J$-analytic. Note that $v$ is $\emptyset$-analytic is equivalent to that $v$ is a smooth vector for the action of $\GL_2(F_{\wp})$ \big(i.e. $v$ is fixed by certain open compact subgroup of $\GL_2(F_{\wp})$\big) which implies in particular $v$ is $\Sigma_{\wp}$-classical.
Let $V$ be a Banach representation of $\GL_2(F_{\wp})$ over $E$, denote by $V_{\mathbb Q_p-\an}$ the $E$-vector subspace generated by the locally $\mathbb Q_p$-analytic vectors of $V$, which is stable by $\GL_2(F_{\wp})$ and hence is a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$. If $V$ is moreover admissible, by \cite[Thm.7.1]{ST03}, $V_{\mathbb Q_p-\an}$ is an admissible locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ and dense in $V$. For $J\subseteq \Sigma_{\wp}$, denote by $V_{J-\an}$ the subrepresentation generated by locally $J$-analytic vectors of $V_{\mathbb Q_p-\an}$, put $V_{\infty}:=V_{\emptyset-\an}$.
Let $\chi$ be a continuous (thus locally $\mathbb Q_p$-analytic) character of $F_{\wp}^{\times}$ (or any open compact subgroup of $F_{\wp}^{\times}$) over $E$, then $\chi$ induces a natural $\mathbb Q_p$-linear map (where $F_{\wp}$ is viewed as the Lie algebra of $F_{\wp}^{\times}$)
\begin{equation*}
F_{\wp}\longrightarrow E,\ \mathfrak x \mapsto \frac{d}{dt}\chi\big(\exp(t\mathfrak x)\big)|_{t=0},
\end{equation*}
and hence an $E$-linear map $d_{\chi}: F_{\wp}\otimes_{\mathbb Q_p} E\mathcal Ong \prod_{\sigma\in \Sigma_{\wp}} E \rightarrow E$. So there exist $k_{\chi,\sigma}\in E$, called the $\sigma$-weight of $\chi$, for all $\sigma\in \Sigma_{\wp}$ such that $d_\chi\big((a_{\sigma})_{\sigma\in \Sigma_{\wp}}\big)=\sum_{\sigma\in \Sigma_{\wp}} a_{\sigma}k_{\chi,\sigma}$.
Let $\chi=\chi_1 \otimes \chi_2 $ be a locally $\mathbb Q_p$-analytic character of $T(F_{\wp})$ over $E$. Put
\begin{equation}\label{equ: clin-cchi}C(\chi):=\{\sigma\in \Sigma_{\wp}\ |\ k_{\chi_1,\sigma}-k_{\chi_2,\sigma}\in \mathbb Z_{\geq 0}\}.\end{equation}
Denote by $\mathfrak t$ the Lie algebra of $T(F_{\wp})$, the character $\chi$ induces a character $d\chi$ of $\mathfrak t_{\Sigma_{\wp}}:=\mathfrak t\otimes_{\mathbb Q_p} E$ given by $d\chi: \mathfrak t_{\Sigma_{\wp}} \rightarrow E$, $d\chi \begin{pmatrix}
a_{\sigma} & 0 \\ 0 & d_{\sigma}
\end{pmatrix}=a_{\sigma}k_{\chi_1,\sigma}+d_{\sigma}k_{\chi_2,\sigma}$ for $ \begin{pmatrix}
a_{\sigma} & 0 \\ 0 & d_{\sigma}
\end{pmatrix}\in \mathfrak t_{\sigma}:=\mathfrak t \otimes_{F_\wp,\sigma} E$, $\sigma\in \Sigma_{\wp}$.
\addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
\section{Completed cohomology of quaternion Shimura curves}\label{sec: clin-2}
Recall some facts on completed cohomology of quaternion Shimura curves, following \cite{Em1} and \cite{New}.
\subsection{Generalities}\label{sec: clin-2.1}
Let $W$ be a finite dimensional algebraic representation of $G^c$ over $E$, as in \cite[\S 2.1]{Ca2}, one can associate to $W$ a local system $\mathcal V_W$ of $E$-vector spaces over $M_K$. Let $W_0$ be $\mathcal O_E$-lattice of $W$, denote by $\mathcal S_{W_0}$ the set (ordered by inclusions) of open compact subgroups of $G(\mathbb Q_p)\mathcal Ong \GL_2(F_{\wp})$ which stabilize $W_0$. For any $K_{p}\in S_{W_0}$, one can associate to $W_0$ \big(resp. to $W_0/\varpi_E^s$ for $s\in \mathbb Z_{\geq 1}$\big) a local system $\mathcal V_{W_0}$ \big(resp. $\mathcal V_{W_0/\varpi_E^s}$\big) of $\mathcal O_E$-modules \big(resp. of $\mathcal O_E/\varpi_E^s$-modules\big) over $M_{K^pK_p}$. Following Emerton (\cite{Em1}), we put
\begin{eqnarray*}
H^i_{\et}(K^p,W_0)&:=&\varinjlim_{K_p\in \mathcal S_{W_0}} H^i_{\et} \big( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0}\big) \\
&\mathcal Ong& \varinjlim_{K_p\in \mathcal S_{W_0}} \varprojlim_{s} H^i_{\et} ( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s});\\
\widetilde{H}^i_{\et}(K^p,W_0)&:=& \varprojlim_{s}\varinjlim_{K_p\in \mathcal S_{W_0}}H^i_{\et} ( M_{K_pK^p,\overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s});\\
H^i_{\et}(K^p,W_0)_E &:=& H^i_{\et}(K^p,W_0) \otimes_{\mathcal O_E}E; \\
\widetilde{H}^i_{\et}(K^p,W_0)_E &:=&\widetilde{H}^i_{\et}(K^p,W_0) \otimes_{\mathcal O_E}E.
\end{eqnarray*}
All these groups ($\mathcal O_E$-modules or $E$-vector spaces) are equipped with a natural topology induced from the discrete topology on the finite group $H^i_{\et} \big( M_{K_pK^p, \overline{\mathbb Q}}, \mathcal V_{W_0/\varpi_E^s}\big)$, and equipped with a natural continuous action of $\mathcal H^p\times \Gal(\overline{\mathbb Q}/F)$ and of $K_p\in \mathcal S_{W_0}$. Moreover, for any $\mathfrak l\in S(K^p)$, the action of $\Gal(\overline{F_{\mathfrak l}}/F_{\mathfrak l})$ (induced by that of $\Gal(\overline{\mathbb Q}/F)$) is unramified and satisfies the Eichler-Shimura relation:
\begin{equation}\label{equ: clin-llf-}
\mathbb Frob_{\mathfrak l}^{-2} -T_{\mathfrak l}\mathbb Frob_{\mathfrak l}^{-1}+\ell^{f_{\mathfrak l}}S_{\mathfrak l}=0
\end{equation}
where $\mathbb Frob_{\mathfrak l}$ denotes the arithmetic Frobenius, $\ell$ the prime number lying below $\mathfrak l$, $f_{\mathfrak l}$ the degree of the maximal unramified extension (of $\mathbb Q_{\ell}$) in $F_{\mathfrak l}$ over $\mathbb Q_{\ell}$ (thus $\ell^{f_{\mathfrak l}}= |\mathcal O_{\mathfrak l}/\varpi_{\mathfrak l}|$). Note that $\widetilde{H}^i_{\et}(K^p,W_0)_E$ is an $E$-Banach space with norm defined by the $\mathcal O_E$-lattice $\widetilde{H}^i_{\et}(K^p,W_0)$.
Consider the ordered set (by inclusion) $\{W_0\}$ of $\mathcal O_E$-lattices of $W$, following \cite[Def.2.2.9]{Em1}, we put
\begin{eqnarray*}
H^i_{\et}(K^p,W)&:=& \varinjlim_{W_0} H^i_{\et}(K^p,W_0)_E, \\
\widetilde{H}^i_{\et}(K^p,W)&:=& \varinjlim_{W_0} \widetilde{H}^i_{\et}(K^p,W_0)_E,
\end{eqnarray*}
where all the transition maps are topological isomorphisms (cf. \cite[Lem.2.2.8]{Em1}). These $E$-vector spaces are moreover equipped with a natural continuous action of $\GL_2(F_{\wp})$.
\begin{theorem}[$\text{cf. \cite[Thm.2.2.11 (i), Thm.2.2.17]{Em1}}$]\label{thm: clin-ecs}(1) The $E$-Banach space $\widetilde{H}^i_{\et}(K^p,W)$ is an admissible Banach representation of $\GL_2(F_{\wp})$. If $W$ is the trivial representation, the representation $\widetilde{H}^i_{\et}(K^p,W)$ is unitary.
(2) One has a natural isomorphism of Banach representations of $\GL_2(F_{\wp})$ invariant under the action of $\mathcal H^p \times \Gal(\overline{F}/F)$:
\begin{equation}\label{equ: clin-wpkw}
\widetilde{H}^i_{\et}(K^p,W) \xlongrightarrow{\sim} \widetilde{H}^i_{\et}(K^p,E)\otimes_E W.
\end{equation}
(3) One has a natural $\GL_2(F_{\wp})\times \mathcal H^p \times \Gal(\overline{F}/F)$-invariant map
\begin{equation}\label{equ: clin-ehiw}
H^i_{\et}(K^p,W) \longrightarrow \widetilde{H}^i_{\et}(K^p,W).
\end{equation}
\end{theorem}
\subsection{Localization at a non-Eisenstein maximal ideal}Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that $\rho$ is unramified at all $\mathfrak l\in S(K^p)$. Let $\rho_0$ be a $\Gal(\overline{F}/F)$-invariant lattice of $\rho$, and $\overline{\rho}^{\sss}$ the semisimplification of $\rho_0/\varpi_E$, which is in fact independent of the choice of $\rho_0$. To $\overline{\rho}^{\sss}$, one can associate a maximal ideal of $\mathcal H^p$, denoted by $\mathfrak{m}(\overline{\rho}^{\sss})$, as the kernel of the following morphism
\begin{equation*}
\mathcal H^p \longrightarrow k_E:=\mathcal O_E/\varpi_E, \ T_{\mathfrak l} \mapsto \tr(\mathbb Frob_{\mathfrak l}^{-1}),\ S_{\mathfrak l} \mapsto \dett(\mathbb Frob_{\mathfrak l}^{-1})
\end{equation*}
for all $\mathfrak l\in S(K^p)$.
\begin{notation}
For an $\mathcal H^p$-module $M$, denote by $M_{\overline{\rho}^{\sss}}$ the localisation of $M$ at $\mathfrak{m}(\overline{\rho}^{\sss})$.
\end{notation}
Keep the notation in \S \ref{sec: clin-2.1}. As in \cite[\S 5.2, 5.3]{Em4}, one can show that $\widetilde{H}_{\et}^1(K^p,W)_{\overline{\rho}^{\sss}}$ is a direct summand of $\widetilde{H}_{\et}^1(K^p,W)$.
Suppose in the following that $\rho$ is absolutely irreducible modulo $\varpi_E$ and put $\overline{\rho}:=\overline{\rho}^{\sss}$.
\begin{proposition}[$\text{\cite[Prop.5.2]{New}}$]\label{prop: clin-trn}
The map (\ref{equ: clin-ehiw}) induces an isomorphism
\begin{equation*}
H^1_{\et}(K^p,W)_{\overline{\rho}} \xlongrightarrow{\sim} \widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}, \infty},
\end{equation*}
where $ \widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}, \infty}$ denotes the smooth vectors (for the action of $\GL_2(F_{\wp})$) in $\widetilde{H}^1_{\et}(K^p,W)_{\overline{\rho}}$.
\end{proposition}
\begin{proposition}[$\text{\cite[Cor.5.8]{New}}$]\label{prop: clin-pfo}
Let $H$ be a open compact prop-$p$ subgroup of $K_{0,\wp}$, then there exists $r\in \mathbb Z_{\geq 1}$ such that
\begin{equation*}
\widetilde{H}_{\et}^1(K^p,E)_{\overline{\rho}} \xlongrightarrow{\sim} \mathcal C\Big(H \big/\overline{(Z(\mathbb Q)\cap K^{p}H)_{p}}, E\Big)^{\oplus r}
\end{equation*}
as representations of $H$, where $\mathcal C\Big(H \big/\overline{(Z(\mathbb Q)\cap K^pH)_p}, E\Big)$ denotes the space of continuous functions from $H \big/\overline{(Z(\mathbb Q)\cap K^pH)_p}$ to $E$, on which $H$ acts by the right regular action, $\overline{(Z(\mathbb Q)\cap K^pH)_p}$ the closure of $(Z(\mathbb Q)\cap K^pH)_p$ in $G(\mathbb Q_p)$, and $(Z(\mathbb Q)\cap K^pH)_p$ the image of $Z(\mathbb Q)\cap K^pH$ in $G(\mathbb Q_p)$ via the projection $G(\mathbb A^{\infty}) \twoheadrightarrow G(\mathbb Q_p)$.
\end{proposition}
Let $\psi$ be a continuous character of $Z_1$ over $E$. We see $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\psi}$
is also an admissible Banach representation of $\GL_2(F_{\wp})$ stable under the action $\Gal(\overline{F}/F) \times \mathcal H^p$. Put
\begin{equation}\label{equ: clin-2ls}U_1:=\{g_{\wp}\in K_{1,\wp}\ |\ \dett(g_{\wp})=1\}.\end{equation} Let $H_{\wp}:=Z_1U_1$ which is a open compact subgroup of $K_{1,\wp}$, ($H_{\wp}=K_{1,\wp}$ when $p\neq 2$), we see the center of $H_{\wp}$ is $Z_1$. By Prop.\ref{prop: clin-pfo} applied to $H=H_{\wp}$, one has \Big(note that $\overline{(Z(\mathbb Q)\cap K^pH_\wp)_p}$ is a subgroup of $Z_1$\Big)
\begin{corollary}\label{cor: clin-ost}
Let $\psi$ be a continuous character of $Z_1$ such that $\psi|_{\overline{(Z(\mathbb Q)\cap K^{p}H_{\wp})_{p}}}=1$, then one has an isomorphism of $H_{\wp}$-representations
\begin{equation*}
\widetilde{H}_{\et}^1(K^{p},E)_{\overline{\rho}}^{Z_1=\psi}\xlongrightarrow{\sim} \mathcal C\big(U_1, E\big)^{\oplus r}
\end{equation*}
where $Z_1$ acts on $\mathcal C\big(U_1, E\big)^{\oplus r}$ by the character $\psi$, and $U_1$ by the right regular action.
\end{corollary}
\section{Eigenvarieties}
\subsection{Generalities}\label{sec: clin-3.1}
Consider the admissible locally $\mathbb Q_p$-analytic representation
$ \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}$,
by applying the functor of Jacquet-Emerton (cf. \cite{Em11}), one obtains an essentially admissible locally $\mathbb Q_p$-analytic representation $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)$ of $T(F_{\wp})$
(cf. \cite[\S 6.4]{Em04}). Denote by $\widehat{T}_{\Sigma_{\wp}}$ the rigid space over $E$ parameterizing the locally $\mathbb Q_p$-analytic characters of $T(F_{\wp})$.
By definition (of essentially admissible locally $\mathbb Q_p$-analytic representations, cf. \emph{loc. cit.}), the action of $T(F_{\wp})$ on $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b$ (where "$b$" signifies the strong topology) can extend to a continuous action of $\mathcal O(\widehat{T}_{\Sigma_{\wp}})$ (being a Fr\'echet-Stein algebra) such that $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b$ is a coadmissible $\mathcal O(\widehat{T}_{\Sigma_{\wp}})$-module. Thus there exists a coherent sheaf $\mathcal M_0$ on $\widehat{T}_{\Sigma_{\wp}}$ such that \begin{equation*}\mathcal M_0\big(\widehat{T}_{\Sigma_{\wp}}\big)\xlongrightarrow{\sim} J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{\vee}_b.\end{equation*}
The action of $\mathcal H^p$ on $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)$ induces a natural $\mathcal O_{\widehat{T}_{\Sigma_{\wp}}}$-linear action of $\mathcal H^p$ on $\mathcal M_0$. Following Emerton, one can construct an \emph{eigenvariety} $\mathcal V(K^p)$ from the triple
$\Big\{\mathcal M_0, \widehat{T}_{\Sigma_{\wp}}, \mathcal H^p\Big\}$:
\begin{theorem}[$\text{cf. \cite[\S 2.3]{Em1}}$]\label{thm: clin-cjw}
There exists a rigid analytic space $\mathcal V(K^p)$ over $E$ together with a finite morphism of rigid spaces
\begin{equation*}
i: \mathcal V(K^p) \longrightarrow \widehat{T}_{\Sigma_{\wp}}
\end{equation*}
and a morphism of $E$-algebras with dense image (see Rem.\ref{rem: linv-yden} below)
\begin{equation}\label{equ: clin-oct}
\mathcal H^p\otimes_{\mathcal O_E} \mathcal O\big(\widehat{T}_{\Sigma_{\wp}}\big) \longrightarrow \mathcal O\big(\mathcal V(K^p)\big)
\end{equation}
such that
\begin{enumerate}
\item a closed point $z$ of $\mathcal V(K^p)$ is uniquely determined by its image $\chi$ in $\widehat{T}_{\Sigma_{\wp}}(\overline{E})$ and the induced morphism $\lambda: \mathcal H^p\rightarrow \overline{E}$, called a system of eigenvalues of $\mathcal H^p$, so $z$ would be denoted by $(\chi,\lambda)$;
\item for a finite extension $L$ of $E$, a closed point $(\chi,\lambda)\in \mathcal V(K^p)(L)$ if and only if the corresponding eigenspace
\begin{equation*}
J_B\big(\widetilde{H}^1_{\et}(K^{p},E)_{\mathbb Q_p-\an}\otimes_E L\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda}
\end{equation*}
is non zero;
\item there exists a coherent sheaf over $\mathcal V(K^p)$, denoted by $\mathcal M$, such that $i_* \mathcal M \mathcal Ong \mathcal M_0$ and that for an $L$-point $z=(\chi, \lambda)$, the special fiber $\mathcal M\big|_z$ is naturally dual to the (finite dimensional) $L$-vector space
\begin{equation*}
J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda}.
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{remark}\label{rem: linv-yden}
Indeed, by construction as in \cite[\S 2.3]{Em1}, for any affinoid admissible open $U=\Spm A$ in $\widehat{T}_{\Sigma_{\wp}}$, one has $i^{-1}(U)\mathcal Ong \Spm B$ where $B$ is the affinoid algebra over $A$ generated by the image of $\mathcal H^p \rightarrow \End_A(\mathcal M_0(U))$, from which we see (\ref{equ: clin-oct}) has a dense image.
\end{remark}
Denote by $\mathcal V(K^p)_{\red}$ the reduced closed rigid subspace of $\mathcal V(K^p)$.
\begin{lemma}
The image of $\mathcal H^p$ in $\mathcal O\big(\mathcal V(K^p)_{\red}\big)$ via (\ref{equ: clin-oct}) lies in \begin{equation*}\mathcal O\big(\mathcal V(K^p)_{\red}\big)^{0}\\ :=\Big\{f\in \mathcal O\big(\mathcal V(K^p)_{\red}\big)\ \Big|\ \|f(x)\|\leq 1,\ \forall x\in \mathcal V(K^p)(\overline{E})\Big\}.\end{equation*}
\end{lemma}
\begin{proof}
It's sufficient to prove for any closed point $(\chi,\lambda)$ of $\mathcal V(K^p)$, the morphism $\lambda: \mathcal H^p \rightarrow \overline{E}$ factors through $\overline{\mathcal O_E}$. But this is clear since $\widetilde{H}^1_{\et}(K^p,E)$ has an $\mathcal H^p$-invariant $\mathcal O_E$-lattice (see \S \ref{sec: clin-2.1}).
\end{proof}
Since the rigid space $\widehat{T}_{\Sigma_{\wp}}$ is nested, by \cite[Lem.7.2.11]{BCh}, one has
\begin{proposition}\label{prop: clin-ide}
The rigid space $\mathcal V(K^p)$ is nested, and $\mathcal O\big(\mathcal V(K^p)_{\red}\big)^{0}$ is a compact subset of $\mathcal O\big(\mathcal V(K^p)_{\red}\big)$ (where we refer to \emph{loc. cit.} for the topology).
\end{proposition}
It would be conviennent to fix a central character (in the quaternion Shimura curve case), let $w\in \mathbb Z$, consider the (essentially admissible) locally $\mathbb Q_p$-analytic representation
\begin{equation}\label{equ: linv-tepn}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\Big)\mathcal Ong J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\big)^{Z_1=\mathscr N^{-w}}.
\end{equation}
One can construct an eigenvariety, denoted by $\mathcal V(K^p,w)$, in the same way as in Thm.\ref{thm: clin-cjw} which satisfies all the properties in Thm.\ref{thm: clin-cjw} with $\widetilde{H}^1_{\et}(K^p,E)$ replaced by $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}$. Denote by $\widehat{T}_{\Sigma_{\wp}}(w)$ the closed rigid subspace of $\widehat{T}_{\Sigma_{\wp}}$ such that \begin{equation}\label{equ: clin-tapwe}\widehat{T}_{\Sigma_{\wp}}(w)(\overline{E})=\big\{\chi\in \widehat{T}_{\Sigma_{\wp}}\ \big|\ \chi|_{Z_1}=\mathscr N^{-w}\big\},\end{equation} moreover, if we denote by $\mathcal M_0(w)$ the coherent sheaf over $\widehat{T}_{\Sigma_{\wp}}(w)$ associated to $J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\Big)$, by (\ref {equ: linv-tepn}), one has $\mathcal M_0(w)\mathcal Ong \mathcal M_0 \otimes_{\mathcal O(\widehat{T}_{\Sigma_{\wp}})} \mathcal O\big(\widehat{T}_{\Sigma_{\wp}}(w)\big)$ and thus (by the construction in \cite[\S 2.3]{Em1}) $\mathcal V(K^p,w)\mathcal Ong \mathcal V(K^p) \times_{\widehat{T}_{\Sigma_{\wp}}} \widehat{T}_{\Sigma_{\wp}}(w)$.
\subsection{Classicality and companion points}Let $T(F_{\wp})^+:=\bigg\{\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix}\in T(F_{\wp})\ \bigg|\ \upsilon_{\wp}(a) \geq \upsilon_{\wp}(d)\bigg\}$,
one can equip $V^{N_0}$ with a continuous action of $T(F_{\wp})^+$ by
\begin{equation*}
\pi_t(v):=\big|N_0/tN_0t^{-1}\big|^{-1} \sum_{n\in N_0/tN_0t^{-1}} (nt)(v).
\end{equation*}
One has a natural $T(F_{\wp})^+$-invariant injection
\begin{equation}\label{equ: clin-pvJ0n}
J_B(V) \lhook\joinrel\longrightarrow V^{N_0}
\end{equation}
which induces a bijection (cf. \cite[Prop.3.4.9]{Em11})
\begin{equation}\label{equ clin-wmv0i}
J_B(V)^{T(F_{\wp})=\chi} \xlongrightarrow{\sim} V^{N_0, T(F_{\wp})^+=\chi}
\end{equation}
for any continuous (locally $\mathbb Q_p$-analytic) characters $\chi$ of $T(F_{\wp})$. In fact, by the same argument in \cite[Prop.3.2.12]{Em11}, one can show (\ref{equ: clin-pvJ0n}) induces a bijection between generalized eigenspaces \big(note that the action of $T(F_{\wp})^+$ on $V^{N_0}[T(F_{\wp})^+=\chi]$ extends naturally to an action of $T(F_{\wp})$\big)
\begin{equation}\label{equ: clin-0nvi}
J_B(V)[T(F_{\wp})=\chi] \xlongrightarrow{\sim} V^{N_0}[T(F_{\wp})^+=\chi].
\end{equation}
\begin{definition}\label{def: clin-ltn}
For an $L$-point $z=(\chi,\lambda)$ of $\mathcal V(K^p)$, $S \subseteq \Sigma_{\wp}$, $z$ is called $S$-classical (resp. quasi-$S$-classical) if there exists a non-zero vector
\[v\in \big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, T(F_{\wp})^+=\chi, \mathcal H^p=\lambda}\] such that $v$ is $S$-classical (resp. quasi-$S$-classical).
We call $z$ classical (quasi-classical) if $z$ is $\Sigma_{\wp}$-classical (resp. quasi-$\Sigma_{\wp}$-classical).
\end{definition}
\begin{definition}Let $z=(\chi_1\otimes \chi_2,\lambda)$ be a closed point in $\mathcal V(K^p)$, for $S\subseteq C(\chi)$ (cf. (\ref{equ: clin-cchi})), put
\begin{equation}\label{equ: clin-s2cd}
\chi_S^c=\chi_{1,S}^c\otimes \chi_{2,S}^c:=\chi_1 \prod_{\sigma \in S} \sigma^{k_{\chi_2,\sigma}-k_{\chi_1,\sigma}-1} \otimes \chi_2 \prod_{\sigma\in S} \sigma^{k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+1};
\end{equation}
we say that $z$ admits an $S$-companion point if $z_S^c:=(\chi_S^c, \lambda)$ is also a closed point in $\mathcal V(K^p)$. If so, we say the companion point $z_S^c$ is effective if it is moreover quasi-$C(\chi)\setminus S$-classical \big(note that $C(\chi_S^c)=C(\chi)\setminus S$\big).
\end{definition}As in \cite[Lem.6.2.24]{Ding}, one has
\begin{proposition}\label{prop: clin-vka}
Let $z=(\chi, \lambda)$ be an $L$-point in $\mathcal V(K^p)$, $\sigma\in C(\chi)$, suppose there exists a non quasi-$\sigma$-classical vector (see \S \ref{sec: clin-1} for $d\chi$)
\begin{equation*}
v\in \big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an} \otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda],
\end{equation*}
then $z$ admits a $\sigma$-companion point. Moreover, there exists $S\subseteq C(\chi)$ containing $\sigma$ such that $z$ admits an effective $S$-companion point.
\end{proposition}
\begin{proof}
We sketch the proof. Let $k_\sigma:=k_{\chi_1,\sigma}-k_{\chi,\sigma}+2\in \mathbb Z_{\geq 2}$ (since $\sigma\in C(\chi)$), if $v$ is not quasi-$\sigma$-classical, we deduce that $v_{\sigma}^c=X_{-,\sigma}^{k_{\sigma}-1} \cdot v\neq 0$ \bigg(where $X_{-,\sigma}:=\begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix}\in \mathfrak g_{\sigma}$\bigg), moreover, as in \cite[Lem.6.3.15]{Ding}, one can prove $v_{\sigma}^c$ is a generalised $(\chi_\sigma^c,\lambda)$-eigenvector for $T(F_{\wp})\times \mathcal H^p$. From which we deduce $z$ admits a $\sigma$-companion point. If $v_{\sigma}^c$ is not quasi-$\sigma'$-classical for some $\sigma'\in C(\chi)\setminus \{\sigma\}=C(\chi_{\sigma}^c)$, one can repeat this argument to find companion points of $z_{\sigma}^c$ until one gets $S\subseteq C(\chi)$ and an effective $S$-companion point of $z$.
\end{proof}
\begin{remark}
One can also deduce this proposition from the adjunction formula in \cite[Thm.4.3]{Br13}.
\end{remark}As in \cite[Prop.6.2.27]{Ding}, one has
\begin{theorem}[Classicality]\label{thm: clin-etn}
Let $z=(\chi=\chi_1\otimes \chi_2, \lambda)$ be an $L$-point in $\mathcal V(K^p)$.
For $\sigma\in C(\chi)$, put $k_{\sigma}:=k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+2\in \mathbb Z_{\geq 2}$. Let $S\subseteq C(\chi)$, if
\begin{equation*}
\upsilon_{\wp}(q\chi_1(\varpi)) <\inf_{\sigma\in S}\{k_{\sigma}-1\},
\end{equation*}
then any vector in
\begin{equation*}
\big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda]
\end{equation*}
in quasi-$S$-classical, in particular, the point $z$ is quasi-$S$-classical.
\end{theorem}
\begin{proof}We sketch the proof. For $\sigma \in S$, if there exists a non quasi-$\sigma$-classical vector
\begin{equation*}
v\in \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E L\big)^{N_0, \mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda],
\end{equation*}
by Prop.\ref{prop: clin-vka}, $z$ admits an effective $S'$-companion point $z_{S'}^c$ with $S'\subseteq C(\chi)$ containing $\sigma$. Using \cite[Prop.6.2.23]{Ding}, this point would induce a continuous injection from a locally $\mathbb Q_p$-analytic parabolic induction twisted with certain algebraic representation \big(as in \emph{loc. cit.} by replacing $J$, $S$, $C_{\overline{B}}(\chi)$ by $\Sigma_{\wp}$, $S'$, $C(\chi)$ respectively\big) into $\widetilde{H}^1_{\et}(K^p,E)\otimes_E L$. Since $\widetilde{H}^1_{\et}(K^p,E)$ is unitary, one can apply \cite[Prop.5.1]{Br}, and get (as in \cite[Cor.6.2.24]{Ding}) $\upsilon_{\wp}(q\chi_1(\varpi))\geq \sum_{\sigma\in S'} (k_{\sigma}-1)$, a contradiction.
\end{proof}
\begin{corollary}\label{cor: clin-caw-}
Let $w\in \mathbb Z$, $z=(\chi=\chi_1\otimes \chi_2, \lambda)$ be an $L$-point in $\mathcal V(K^p,w)$ with $C(\chi)=\Sigma_{\wp}$, $k_{\sigma}:=k_{\chi_1,\sigma}-k_{\chi_2,\sigma}+2\in 2 \mathbb Z_{\geq 1}$ such that $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma \in \Sigma_{\wp}$. There exist thus smooth characters $\psi_1$, $\psi_2$ such that (note that $k_{\chi_1,\sigma}+k_{\chi_2,\sigma}=-w$)
\begin{equation*}
\chi_1 \otimes \chi_2=\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}} \psi_1 \otimes \prod_{\sigma \in \Sigma_{\wp}} \sigma^{-\frac{w+k_{\sigma}-2}{2}} \psi_2
\end{equation*}
Let $S\subseteq \Sigma_{\wp}$, if
\begin{equation*}
\upsilon_{\wp}(q \psi_1(\varpi))<\sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{\sigma}+2}{2} + \inf_{\sigma\in S} \{k_{\sigma}-1\},
\end{equation*}
then the point $z$ is $S$-classical.
\end{corollary}
\begin{remark}
We invite the reader to compare this corollary with conjectures of Breuil in \cite{Br00} and results of Tian-Xiao in \cite{TX}.
\end{remark}
\subsection{Localization at a non-Eisenstein maximal ideal}\label{sec: clin-3.3}
Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$, suppose that $\rho$ is absolutely irreducible modulo $\varpi_E$ and there exists an irreducible algebraic representation $W$ of $G^c$ such that $H^1_{\et}(K^p,W)_{\overline{\rho}}$ ($\rho$ is thus called \emph{modular}). It's known that there exist $w\in \mathbb Z$, $k_{\sigma}\in Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$ such that $W\mathcal Ong W(\underline{k}_{\Sigma_{\wp}},w)$. We fix this $w$ in the following. Consider the essentially admissible locally $\mathbb Q_p$-analytic representation $J_B\big(\widetilde{H}^1_{\et}(K^p,E)_{\overline{\rho}}^{Z_1=\mathscr N^{-w}}\big)$, whose strong dual gives rise to a coherent sheaf $\mathcal M_0(K^p,w)_{\overline{\rho}}$ over $\widehat{T}_{\Sigma_{\wp}}$. As in Thm.\ref{thm: clin-cjw}, one can obtain an eigenvariety $\mathcal V(K^p,w)_{\overline{\rho}}$ together with a coherent sheaf $\mathcal M(K^p,w)_{\overline{\rho}}$ over $\mathcal V(K^p,w)_{\overline{\rho}}$, which satisfies the properties in Thm.\ref{thm: clin-cjw} with $\widetilde{H}^1_{\et}(K^p,E)$ replaced by $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$. Since $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$ is a direct summand of $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}$, $\mathcal V(K^p,w)_{\overline{\rho}}$ is a closed rigid subspace of $\mathcal V(K^p,w)$ (cf. \cite[Lem.6.2.6]{Ding}). By Prop.\ref{prop: clin-trn}, one can describe the classical vectors of $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}}$ as follows:
\begin{corollary}\label{prop: clin-ernr}
With the notation in Cor.\ref{cor: clin-caw-}, suppose moreover $z$ in $\mathcal V(K^p,w)_{\overline{\rho}}$, let $v$ be a vector in
\begin{equation}\label{equ: clin-weht} \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an, \overline{\rho}}\otimes_E L\big)^{N_0, Z_1=\mathscr N^{-w},\mathfrak t_{\Sigma_{\wp}}=d\chi}[T(F_{\wp})^+=\chi, \mathcal H^p=\lambda],\end{equation}
if $v$ is classical, then $v$ lies in (see Rem.\ref{rem: clin-ging} below)
\begin{equation}\label{equ: clin-kpwg}
\big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho}}\otimes_E L\big)^{N_0, Z_1=\psi_1\psi_2}[T(F_{\wp})^+=\psi_1 \otimes \psi_2, \mathcal H^p=\lambda] \otimes_E \chi(\underline{k}_{\Sigma_{\wp}},w),
\end{equation}
with $\chi(\underline{k}_{\Sigma_{\wp}},w):=\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}}\otimes \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w+k_{\sigma}-2}{2}}$ \big(being a character of $T(F_{\wp})$\big).
\end{corollary}
\begin{remark}\label{rem: clin-ging}
Note that $T(F_{\wp})$ acts on $\big(W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}\big)^{N_0}$ via $\chi(\underline{k}_{\Sigma_{\wp}},w)$, the embedding of the vector space (\ref{equ: clin-kpwg}) into (\ref{equ: clin-weht}) is obtained by taking $N_0$-invariant vectors of the following $\GL_2(F_{\wp}) \times \mathcal H^p \times \Gal(\overline{F}/F)$-invariant injection (cf. Prop.\ref{prop: clin-trn})
\begin{multline*}
H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho}} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \otimes_E L\\ \xlongrightarrow{\sim} \big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an,\overline{\rho}} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\infty}\otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \otimes_E L \\ \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an, \overline{\rho}} \otimes_E L.
\end{multline*}
\end{remark}
We study in details the structure of $\mathcal V(K^p,w)_{\overline{\rho}}$. Let \begin{equation}\label{equ: clin-t'b}T':=Z_1'\times \begin{pmatrix}
\varpi & 0 \\ 0 & 1
\end{pmatrix}^{\mathbb Z} \times \begin{pmatrix}
\varpi & 0 \\ 0 & \varpi
\end{pmatrix}^{\mathbb Z} \\ \times \bigg\{ \begin{pmatrix}
z_{1} & 0 \\ 0 & z_{2}
\end{pmatrix}\ \bigg|\ \ z_{i}\in \mathcal O_{\wp}^{\times}, \ z_{i}^{q-1}=1\bigg\}.\end{equation}
One has thus a finite morphism of rigid spaces (which is moreover an isomorphism when $p\neq 2$)
\begin{equation}\label{equ: clin-st1}
\widehat{T}_{\Sigma_{\wp}} \xlongrightarrow{(\pr_1,\pr_2)} \widehat{(T')}_{\Sigma_{\wp}} \times \widehat{(Z_1)}_{\Sigma_{\wp}}, \ \chi\mapsto (\chi|_{T'},\chi|_{Z_1}),
\end{equation}
where $\widehat{(T')}_{\Sigma_{\wp}}$ and $\widehat{(Z_1)}_{\Sigma_{\wp}}$ denote the rigid spaces parameterizing locally $\mathbb Q_p$-analytic characters of $T'$ and $Z_1$ respectively.
Note that $\mathcal M_1:=\pr_{1,*}\mathcal M_0(K^p,w)_{\overline{\rho}}$ is in fact a coherent sheaf over $\widehat{(T')}_{\Sigma_{\wp}}$: the support of $\mathcal M_0(K^p,w)_{\overline{\rho}}$ is contained in $\widehat{T}_{\Sigma_{\wp}}(w)$ \big(as a closed subspace of $\widehat{T}_{\Sigma_{\wp}}$\big), which is finite over $\widehat{(T')}_{\Sigma_{\wp}}$.
Put $\Pi:=\begin{pmatrix}\varpi & 0 \\ 0 & 1\end{pmatrix}$, let $R_{\wp}$ be a (finite) set of representatives of $T(F_{\wp})/T'Z_1$ in $T(F_{\wp})$ (note $T(F_{\wp})=T'Z_1$ when $p\neq 2$), let $\mathcal H$ be the $\mathcal O_E$-algebra generated by $\mathcal H^p$ and $\begin{pmatrix}
\varpi & 0 \\ 0 & \varpi
\end{pmatrix}$, $\bigg\{\begin{pmatrix}
z_{1} & 0 \\ 0 & z_{2}
\end{pmatrix}\ \bigg|\ z_{i}\in \mathcal O_{\wp}^{\times},\ z_{i}^{q-1}=1\bigg\}$, and the elements in $R_{\wp}$. Denote by $\mathcal W_{1,\Sigma_{\wp}}$ the rigid space ove $E$ which parameterizes locally $\mathbb Q_p$-analytic characters of $1+2\varpi \mathcal O_{\wp}\mathcal Ong Z_1'$, by the decomposition of groups (\ref{equ: clin-t'b}), one gets a natural projection $\pr: \widehat{(T')}_{\Sigma_{\wp}}\rightarrow \mathcal W_{1,{\Sigma_{\wp}}}\times \mathbb G_m, \ \chi \mapsto (\chi|_{Z_1'}, \chi(\Pi))$. By Cor.\ref{cor: clin-ost} and the argument in the proof of \cite[Prop.4.2.36]{Em11} (e.g. see \cite[(4.2.43)]{Em11}), we see $\mathcal M_1\big(\widehat{(T')}_{\Sigma_{\wp}}\big)$ is a coadmissible $\mathcal O(\mathcal W_{1,\Sigma_{\wp}})\{\{X,X^{-1}\}\}$-module with $X$ acting on $\mathcal M_1\big(\widehat{(T')}_{\Sigma_{\wp}}\big)$ by the operator $\Pi$. Denote by $\mathcal M_2$ the associated coherent sheaf over $\mathcal W_{1,\Sigma_{\wp}} \times \mathbb G_m$, which is equipped with an $\mathcal O_{\mathcal W_{1,\Sigma_{\wp}}\times \mathbb G_m}$-linear action of $\mathcal H$. One can thus construct $\mathcal V(K^p,w)_{\overline{\rho}}$ from the triple $\big\{\mathcal M_2, \widehat{(T')}_{\Sigma_{\wp}}, \mathcal H\big\}$ as in \cite[\S 2.3]{Em1}.
Let $\{\Spm A_i\}_{i \in I}$ be an admissible covering of $\mathcal W_{1,\Sigma_{\wp}}$ by increasing affinoid opens, by Cor.\ref{cor: clin-ost}, \cite[(4.2.43)]{Em11} and the results in \cite[\S 5.A]{Ding}, for any $i\in I$, there exists a Fredholm series $F_i(z)\in 1+zA_i\{\{z\}\}$ (which is hence a global section over $\Spm A_i \times \mathbb G_m$) such that the coherent sheaf $\mathcal M_2|_{\Spm A_i \times \mathbb G_m}$ is supported at $\mathcal Z_i$ where $\mathcal Z_i$ is the closed rigid subspace of $\Spm A_i \times \mathbb G_m$ defined by $F_i(z)$. Moreover, it's known that (cf. \cite[\S 4]{Bu}) $\mathcal Z_i$ admits an admissible covering $\{U_{ij}\mathcal Ong \Spm A_i[z]/P_j(z)\}$ such that
\begin{itemize}
\item $P_j(z)\in 1+zA_i[z]$ is a polynomial of degree $d_j$ with leading coefficient being a unit,
\item there exists $Q_j(z)\in 1+zA_i\{\{z\}\}$ such that $F_i(z)=P_j(z)Q_j(z)$ and that $(P_j(z),Q_j(z))=1$.
\end{itemize}
As in the proof of \cite[Prop.5.A.6]{Ding}, one can show $\mathcal M_2(U_{ij})$ is a finite locally free $A_i$-module of rank $d_j$, equipped with an $A_i$-linear action of $\mathcal H$ such that the characteristic polynomial of $\Pi$ is given by $P_j(z)$, and that $Q_j(\Pi)$ acts on $\mathcal M_2(U_{ij})$ via an invertible operator. Denote by $\mathcal H_{ij}$ the $A_i[z]/P_j(z)$-algebra generated by the image of the natural map
\begin{equation*}
\mathcal H \longrightarrow \End_{A_i[z]/P_j(z)}(\mathcal M_2(U_{ij})),
\end{equation*}which is also the $A_i$-algebra generated by the image of the map $\mathcal H \rightarrow \End_{A_i}(\mathcal M_2(U_{\ij}))$ (since $\Pi\in \mathcal H$). The restriction $\mathcal V(K^p,w)_{\overline{\rho}} \big|_{U_{ij}}$ is thus isomorphic to $\Spm \mathcal H_{ij}$. In particular, we see that the construction of $\mathcal V(K^p,w)_{\overline{\rho}}$ coincides with the construction of eigenvarieties by Coleman-Mazur (formalized by Buzzard in \cite{Bu}). Since $\mathcal W_{1,\Sigma_{\wp}}$ is equidimensional of dimension $d$, by \cite[Prop.6.4.2]{Che}, we have
\begin{proposition}
The rigid analytic space $\mathcal V(K^p,w)_{\overline{\rho}}$ is equidimensional of dimension $d$.
\end{proposition}
\begin{definition}For a character $\chi$ of $T(F_{\wp})$, we say that $\chi$ is spherically algebraic if $\chi$ is the twist of an algebraic character by an unramified character. We call a closed point $z=(\chi,\lambda)$ of $\mathcal V(K^p,w)$ semi-stable classical if $z$ is classical and $\chi$ is spherically algebraic.
\end{definition}
Denote by $C(w)$ the set of semi-stable classical points in $\mathcal V(K^p,w)_{\overline{\rho}}$. By the same argument as in the proof of \cite[Prop.6.2.7, Prop.6.4.6]{Che}, the following proposition follows from Cor.\ref{cor: clin-caw-}.
\begin{proposition}\label{thm: clin-adtf}(1) Let $z=(\chi,\lambda)$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, suppose moreover $\chi$ spherically algebraic, then the set $C(w)$ accumulates over the point $z$, i.e. for any admissible open $U$ containing $z$, there exists an admissible open $V\subseteq U$, $z\in V(\overline{E})$ such that $C(w)\cap V(\overline{E})$ is Zariski-dense in $V$.
(2) The set $C(w)$ is Zariski-dense in $\mathcal V(K^p,w)_{\overline{\rho}}$.
\end{proposition}
\subsection{Families of Galois representations}
\subsubsection{Families of Galois representations on eigenvarieties}
Keep the notation in \S \ref{sec: clin-3.3}. For $\mathfrak l\in S(K^p)$, denote by $\mathfrak a_{\mathfrak l}\in \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$ \Big(resp. $\mathfrak b_{\mathfrak l}\in \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$\Big) the image of $T_{\mathfrak l}\in \mathcal H^p$ \big(resp. $S_{\mathfrak l}\in \mathcal H^p$\big) via the natural morphism $\mathcal H^p \rightarrow \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$. Denote by $\mathcal S$ the complement of $S(K^p)$ in the set of finite places of $F$, which is hence a finite set. Denote by $F^{\mathcal S}$ the maximal algebraic extension of $F$ which is unramified outside $\mathcal S$.
For any $z\in C(w)$, by \cite{Ca2} (and Cor.\ref{prop: clin-ernr}), there exists a $2$-dimensional continuous representation $\rho_z$ of $\Gal(\overline{F}/F)$ over $k(z)$, the residue field at $z$, which is unramified outside $S$ and hence a representation of $\Gal(F^{\mathcal S}/F)$, such that (see also (\ref{equ: clin-llf-}))
\begin{equation*}
\mathbb Frob_{\mathfrak l}^{-2} -\mathfrak a_{\mathfrak l,z}\mathbb Frob_{\mathfrak l}^{-1}+\ell^{f_{\mathfrak l}}\mathfrak b_{\mathfrak l,z}=0
\end{equation*}where $\mathfrak a_{\mathfrak l,z}, \mathfrak b_{\mathfrak l,z}\in k(z)$ denote the respective evaluation of $\mathfrak a_{\mathfrak l}$ and $\mathfrak b_{\mathfrak l}$ at $z$. In particular, one has $\tr(\mathbb Frob_{\mathfrak l}^{-1})=\mathfrak a_{\mathfrak l,z}$. Denote by $\mathcal T_z: \Gal(F^{\mathcal S}/F) \rightarrow k(z)$, $g\mapsto \tr(\rho_z(g))$, which is thus a $2$-dimensional continuous pseudo-character of $\Gal(F^{\mathcal S}/F)$ over $k(z)$. By \cite[Prop.7.1.1]{Che} and Prop.\ref{prop: clin-ide}, one has
\begin{proposition}
There exists a unique $2$-dimensional continuous pseudo-character $\mathcal T: \Gal(F^{\mathcal S}/F) \rightarrow \mathcal O \big(\mathcal V(K^p,w)_{\overline{\rho},\red}\big)$ such that the evaluation of $\mathcal T$ at $z\in C(w)$ equals to $\mathcal T_z$.
\end{proposition}
Let $z$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, denote by $\mathcal T_z:=\mathcal T|_z$, which is a $2$-dimensional continuous pseudo-character of $\Gal(F^{\mathcal S}/F)$ over $k(z)$. By \cite[Thm.1(2)]{Ta}, there exists a unique $2$-dimensional continuous semi-simple representation $\rho_z$ of $\Gal(F^{\mathcal S}/F)$ such that $\tr(\rho_z)=\mathcal T_z$. By Eichler-Shimura relations, one has $\overline{\rho_z}\mathcal Ong \overline{\rho}$, in particular, $\rho_z$ is absolutely irreducible. By \cite[Lem.5.5]{Bergd}, one has
\begin{proposition}\label{prop: clin-kzc}
For any closed point $z$ of $\mathcal V(K^p,w)_{\overline{\rho}}$, there exist an admissible open affinoid $U$ containing $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\red}$ and a continuous representation
\begin{equation*}
\rho_U: \Gal(F^{\mathcal S}/F) \longrightarrow \GL_2(\mathcal O_U)
\end{equation*}
such that $\rho_U|_{z}\mathcal Ong \rho_{z'}$ for any $z'\in U(\overline{E})$.
\end{proposition}
In general, by \cite[Lem.7.8.11]{BCh}, one has
\begin{proposition}\label{prop: clin-wpw}
Let $U$ be an open affinoid of $\mathcal V(K^p,w)_{\overline{\rho},\red}$, there exist a rigid space $\widetilde{U}$ over $U$, and an $\mathcal O_{\widetilde{U}}$-module $\mathcal M$ locally free of rank $2$ equipped with a continuous $\mathcal O_{\widetilde{U}}$-linear action of $\Gal(F^{\mathcal S}/F)$ such that
\begin{enumerate}
\item the morphism $g: \widetilde{U}\rightarrow U$ factors through a rigid space $U'$ such that $\widetilde{U}$ is a blow-up over $U'$ of $U'\setminus U''$ with $U''$ an Zariski-open Zariski-dense subspace of $U'$ and that $U'$ is finite, dominant over $U$;
\item for any $z\in \widetilde{U}(\overline{E})$, the $2$-dimensional representation $\mathcal M|_z$ of $\Gal(F^{\mathcal S}/F)$ is isomorphic to $\rho_{g(z)}$.
\end{enumerate}
\end{proposition}
\begin{remark}\label{rem: clin-dsu}
Let $Z$ be a Zariski-dense subset of closed points in $U$, then $g^{-1}(U)$ is Zariski-dense in $\widetilde{U}$: denote by $g': U'\rightarrow U$ the morphism as in (1), by \cite[Lem.6.2.8]{Che}, we see $(g')^{-1}(Z)$ is Zariski-dense in $U$, so $(g')^{-1}(Z)\cap U''(\overline{E})$ is Zariski-dense in $U''$ and hence Zariski-dense in $\widetilde{U}$.
\end{remark}
\subsubsection{Trianguline representations} Consider the restriction $\rho_{z,\wp}:=\rho_z|_{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$. Denote by $F_{\wp,\infty}:=\cup_{n} F_{\wp}(\zeta_{p^n})$ where $\zeta_{p^n}$ is a root of unity primitive of order $p^n$. Set $\Gamma:=\Gal(F_{\wp,\infty}/F_{\wp})$, $H_{\wp}:=\Gal(\overline{\mathbb Q_p}/F_{\wp,\infty})$. One has a ring $B_{\rig}^{\dagger}$ (cf. \cite[\S 3.4]{Berger}) which is equipped with an action of $\varphi$ and $\Gal(\overline{\mathbb Q_p}/\mathbb Q_p)$ such that $B_{\rig,F_{\wp}}^{\dagger}:=(B_{\rig}^{\dagger})^{H_{F_{\wp}}}$ is naturally isomorphic to the Robba ring with coefficients in $F_{\wp}'$ where $F_{\wp}'$ denotes the maximal unramified extension of $\mathbb Q_p$ in $F_{\wp,\infty}$ (which is finite over $F_{\wp,0}$). For an $n$-dimensional continuous representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$, $D_{\rig}(V):=(B_{\rig}^{\dagger} \otimes_{\mathbb Q_p} V)^{H_{\mathbb F_{\wp}}}$ is an \'etale $(\varphi,\Gamma)$-module of rank $n$ over $\mathcal R_E:= B_{\rig,F_{\wp}}^{\dagger} \otimes_{\mathbb Q_p} E$ (i.e. an \'etale $(\varphi,\Gamma)$-module over $B_{\rig,F_{\wp}}^{\dagger}$ equipped with an action of $E$ which commutes with that of $\varphi$ and $\Gamma$) (cf. \cite[Prop.3.4]{Berger}). Let $\delta: F_{\wp}^{\times} \rightarrow E^{\times}$ be a continuous character, following \cite[\S 1.4]{Na}, one can associate to $\delta$ a $(\varphi,\Gamma)$-module, denoted by $\mathcal R_E(\delta)$, free of rank $1$ over $\mathcal R_E$. The converse is also true, i.e. for any $(\varphi,\Gamma)$-module $D$ free of rank $1$ over $\mathcal R_E$, there exists a continuous character $\delta: F_{\wp}^{\times} \rightarrow E^{\times}$ such that $D\mathcal Ong \mathcal R_E(\delta)$.
\begin{definition}[$\text{cf. \cite[Def.4.1]{Colm2}, \cite[Def.1.15]{Na}}$]
Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$, $\rho$ is called trianguline if there exist continuous characters $\delta_1$, $\delta_2$ of $F_{\wp}^{\times}$ over $E$ such that $D_{\rig}(\rho)$ lies in an exact sequence as follows:
\begin{equation*}
0 \rightarrow \mathcal R_E(\delta_1) \rightarrow D_{\rig}(V) \rightarrow \mathcal R_E(\delta_2) \rightarrow 0.
\end{equation*}
Such an exact sequence is called a triangulation, denoted by $(\rho,\delta_1,\delta_2)$, of $D_{\rig}(\rho)$ (and of $\rho$).
\end{definition}
We refer to \cite{Na} for a classification of $2$-dimensional trianguline representations of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$. Note that if $\rho$ is semi-stable, then $\rho$ is trianguline, if $\rho$ is moreover non-crystalline, then the triangulation of $\rho$ is unique.
\begin{definition}[$\text{cf. \cite[Def.4.3.1]{Liu}}$]
Let $\rho$ be a $2$-dimensional trianguline representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$ with $(\rho, \delta_1,\delta_2)$ a triangulation of $\rho$. For $\sigma\in \Sigma_{\wp}$, we say that $\rho$ is non-$\sigma$-critical if $k_{\delta_1,\sigma}-k_{\delta_2,\sigma}\in \mathbb Z_{\geq 1}$. More generally, for $J\subseteq \Sigma_{\wp}$, we say $\rho$ is non-$J$-critical if $\rho$ is non-$\sigma$-critical for all $\sigma\in J$, we say $\rho$ is non-critical if $\rho$ is non-$\Sigma_{\wp}$-critical.
\end{definition}For a closed point $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2}, \lambda_z)$ of $\mathcal V(K^p,w)_{\overline{\rho}}$, put $k_{z,\sigma}:= k_{\chi_{z,1},\sigma}-k_{\chi_{z,2},\sigma}+2\in \mathbb Z_{\geq 2}$, for $\sigma\in C(\chi_z)$. If $z\in C(w)$, we see $k_{z,\sigma}\equiv w \pmod{2}$ for all $\sigma \in \Sigma_{\wp}$ (note that $C(\chi_z)=\Sigma_{\wp}$ in this case).
The following theorem can be easily deduced from the results in \cite{Sa} (together with Cor.\ref{prop: clin-ernr} and the results of \cite{Na} on triangulations of semi-stable representations, see \cite[Prop.6.2.44]{Ding} for the unitary Shimura curves case).
\begin{theorem}\label{thm: clin-zgn}
Let $z=\big(\chi_{z,1}\otimes \chi_{z,2}, \lambda_z\big)\in C(w)$, then $\rho_{z,\wp}$ is semi-stable (hence trianguline) with a triangulation given by
\begin{equation}\label{equ: clin-dgz}
0 \rightarrow \mathcal R_{k(z)}(\delta_{z,1}) \rightarrow D_{\rig}(\rho_{z,\wp}) \rightarrow \mathcal R_{k(z)}\big(\delta_{z,2}\big)\rightarrow 0
\end{equation}
where
\begin{equation*}
\begin{cases}
\delta_{z,1}=\unr(q)\chi_{z,1} \prod_{\sigma\in \Sigma_{z}} \sigma^{1-k_{z,\sigma}} \\
\delta_{z,2}=\chi_{z,2}\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}\prod_{\sigma\in \Sigma_{z}} \sigma^{k_{z,\sigma}-1}
\end{cases}
\end{equation*}
with $\Sigma_{z}\subseteq \Sigma_{\wp}$ (maybe empty). Put $\psi_{z,1}:=\chi_{z,1}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{\frac{w-k_{z,\sigma}+2}{2}}$ (being an unramified character of $F_{\wp}^{\times}$), for $S\subseteq \Sigma_{\wp}$, if one has
\begin{equation*}
\upsilon_{\wp}\big(q\psi_{z,1}(\varpi)\big) < \sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{z,\sigma}+2}{2}+\inf_{\sigma\in S}\{k_{z,\sigma}-1\},
\end{equation*}then $\Sigma_{z}\cap S=\emptyset$, in particular, in this case the triangulation (\ref{equ: clin-dgz}) is non-$S$-critical.
\end{theorem}
Denote by $C(w)_0$ the subset of $C(w)$ of points $z$ such that
\begin{equation}\label{equ: clin-man}
\upsilon_{\wp}\big(q\psi_{z,1}(\varpi)\big) < \sum_{\sigma\in \Sigma_{\wp}} \frac{w-k_{z,\sigma}+2}{2}+\inf_{\sigma\in \Sigma_{\wp}}\{k_{z,\sigma}-1\}.
\end{equation}
As in \cite[Prop.6.2.7, Prop.6.4.6]{Che}, one can prove $C(w)_0$ is Zariski-dense in $\mathcal V(K^p,w)_{\overline{\rho}}$, and is an accumulation subset (cf. \cite[\S 3.3.1]{BCh}).
By the theory of global triangulation, one has
\begin{theorem}\label{thm: clin-iz2d}
Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2}, \lambda)$ be a closed point of $\mathcal V(K^p,w)_{\overline{\rho}}$, then the representation $\rho_{z,\wp}$ is trianguline with a triangulation given by
\begin{equation*}
0 \rightarrow \mathcal R_{k(z)}(\delta_{z,1}) \rightarrow D_{\rig}(\rho_{z,\wp}) \rightarrow \mathcal R_{k(z)}\big(\delta_{z,2}\big)\rightarrow 0
\end{equation*}
where
\begin{equation*}
\begin{cases}
\delta_{z,1}=\unr(q)\chi_{z,1}\prod_{\sigma\in \Sigma_{z}} \sigma^{1-k_{z,\sigma}} \\
\delta_{z,2}=\chi_{z,2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\prod_{\sigma\in \Sigma_{z}} \sigma^{k_{z,\sigma}-1}
\end{cases}
\end{equation*}
with $\Sigma_{z}$ a subset (maybe empty) of $C(\chi_z)\cap \Sigma_{\wp}$.
\end{theorem}
\begin{proof}
Since $\mathcal V(K^p,w)_{\overline{\rho},\red}$ is nested, and $C(w)_0$ is Zariski-dense, there exists an irreducible affinoid neighborhood of $z$ such that $C(w)_0\cap U(\overline{E})$ is Zariski-dense in $U$ (e.g. see \cite[Lem.7.2.9]{BCh}). Denote by $g: \widetilde{U}\rightarrow U$ the rigid space as in Prop. \ref{prop: clin-wpw}, thus $g^{-1}\big(C(w)_0\cap U(\overline{E})\big)$ is Zariski-dense in $\widetilde{U}$. The theorem then follows from \cite[Thm.6.3.13]{KPX} and \cite[Ex.6.3.14]{KPX} (see also \cite[Thm.4.4.2]{Liu}).
\end{proof}
\begin{corollary}\label{cor: clin-aat}Keep the notation in Thm.\ref{thm: clin-iz2d}, suppose moreover
\begin{equation}\label{equ: clin-iz-}\unr(q^{-1}) \chi_{z,1}^{-1}\chi_{z,2}\neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}} \text{ for all $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$},\end{equation} let $S\subseteq C(\chi_z)$, if $\Sigma_{z}\cap S=\emptyset$, then $z$ does not have $S'$-companion point for any $S'\subseteq S$, $S'\neq \emptyset$. As a result, the point $z$ is quasi-$S$-classical.
\end{corollary}
\begin{proof}
The second part follows from the first part and Prop.\ref{prop: clin-vka}. We prove the first part. Let $S'\subseteq S$, $S'\neq \emptyset$, suppose $z$ admits an $S'$-companion point $z_{S'}^c$, by applying Thm.\ref{thm: clin-iz2d} to the point $z_{S'}^c$, one can get a triangulation $\big(\rho_{z_{S'}^c,\wp}, \delta_{z_{S'}^c ,1}, \delta_{z_{S'}^c 2}\big)$ for $\rho_{z_{S'}^c,\wp}\mathcal Ong \rho_{z,\wp}$
Note that $S'\cap C((\chi_z)_{S'}^c)=\emptyset$, so $S'\cap \Sigma_{z_{S'}^c,\wp}=\emptyset$. By the hypothesis (\ref{equ: clin-iz-}) and \cite[Thm.3.7]{Na}, one can check the triangulations $\big(\rho_{z_{S'}^c,\wp}, \delta_{z_{S'}^c ,1}, \delta_{z_{S'}^c ,2}\big)$ and $\big(\rho_{z,\wp}, \delta_{z,1}, \delta_{z,2}\big)$ are the same. As a result, one sees $S'\subseteq \Sigma_{z}$, a contradiction.
\end{proof}
\begin{corollary}\label{cor: clin-siz}Keep the notation in Thm.\ref{thm: clin-iz2d}, suppose $\chi_z$ is spherically algebraic \big(thus $\chi_{i,z}=\psi_{i,z}\prod_{\sigma\in \Sigma_{\wp}} \sigma^{k_{\chi_i,\sigma}}$ with $\psi_{i,z}$ an unramified character of $F_{\wp}^{\times}$\big) and satisfies
\begin{equation*}
\psi_{1,z}(p)^{-1}\psi_{2,z}(p) q^{-e} \neq 1
\end{equation*}
\big(note this condition is slightly stronger than the hypothesis (\ref{equ: clin-iz-})\big), then there exists an open affinoid neighborhood $U$ of $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\red}$ containing $z$ such that for any closed point $z'=(\chi_{z'},\lambda')\in U(\overline{E})$, $\Sigma_{z'}=\emptyset$ and $z'$ does not have companion point.
\end{corollary}
\begin{proof}
As in the proof of \cite[Prop.6.2.7]{Che}, one can prove $C(w)_0$ accumulates over $z$. Thus one can choose an open affinoid neighborhood $U_0$ of $z$ such that
\begin{enumerate}\item $C(w)_0\cap U_0(\overline{E})$ is Zariski-dense in $U_0$,
\item $\unr(q^{-1}) \chi_{z',1}^{-1} \chi_{z',2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$, $z'\in U_0(\overline{E})$ (see Lem.\ref{lem: clin-cte}(1) below).\end{enumerate}
By \cite[Thm.6.3.9]{KPX}, $Z_{U_0}:=\{z'\in U_0(\overline{E})\ |\ \Sigma_{z'}\neq \emptyset \}$ is a Zariski-closed subset of $U_0$ and $z\notin Z_{U_0}$. So there exists an open affinoid $U$ of $U_0$ containing $z$ such that $Z_{U}$ (defined in the same way as $Z_{U_0}$ by replacing $U_0$ by $U$) is empty. The corollary follows.
\end{proof}
\begin{lemma}\label{lem: clin-cte}
Let $\chi_1\otimes \chi_2$ be a spherically algebraic character of $T(F_{\wp})$ (which can be seen as a closed point of $\widehat{T}_{\Sigma_{\wp}}$), and let $\psi_i:=\chi_i\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_1,\sigma}}$.
(1) Suppose $\psi_{1}(p)^{-1}\psi_{2}(p) q^{-e} \neq 1$, then there exists an admissible neighborhood $U$ of $\chi_1\otimes \chi_2$ in $\widehat{T}_{\Sigma_{\wp}}$ such that $\unr(q^{-1}) (\chi_{1}')^{-1} \chi_{2}' \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ and $\chi_1'\otimes \chi_2'\in U(\overline{E})$.
(2) Suppose $\psi_1(\varpi)^{-1} \psi_2(\varpi)^{-1}q^{-1} \neq 1$, then there exists an admissible neighborhood $U$ of $\chi_1\otimes \chi_2$ in $\widehat{T}_{\Sigma_{\wp}}$ such that $\unr(q^{-1}) (\chi_{1}')^{-1} \chi_{2}' \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq k_{\chi_2,\sigma}-k_{\chi_1,\sigma}}^{d}$ and $\chi_1'\otimes \chi_2'\in U(\overline{E})$.
\end{lemma}
\begin{proof}
Denote by $\widehat{Z}_{\Sigma_{\wp}}$ the rigid space parameterizing locally $\mathbb Q_p$-analytic characters of $F_{\wp}^{\times}$. One has a morphism of rigid spaces:
\begin{equation}\label{equ: clin-pwz}
\widehat{T}_{\Sigma_{\wp}} \longrightarrow \widehat{Z}_{\Sigma_{\wp}}, \ (\chi_1')^{-1}\otimes \chi_2'\mapsto \chi_1'\chi_2'.
\end{equation}
Let $\psi_0:=\unr(q^{-1}) \psi_1^{-1}\psi_2$, we claim that \begin{itemize}\item if $\psi_0(p)\neq 1$ (resp. $\psi_0(\varpi)\neq 1$), then there exists an admissible open $U_0$ of $\widehat{Z}_{\Sigma_{\wp}}$ containing $\psi_0$ (where $\psi_0$ is seen as a closed point of $\widehat{Z}_{\Sigma_{\wp}}$\big) such that $\prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}\notin U(\overline{E})$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ \big(resp. $\prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}} \notin U(\overline{E})$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq 0}^d$\big).
\end{itemize}
Assuming this claim, and let $U$ be the preimage of the admissible open $\big(\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_1,\sigma}+k_{\chi_2,\sigma}}\big) U_0$ \big(of $\widehat{Z}_{\Sigma_{\wp}}$\big) in $\widehat{T}_{\Sigma_{\wp}}$ via (\ref{equ: clin-pwz}). Thus $U$ satisfies the property in the lemma (1) (resp. (2)).
We prove the claim. Consider the projection $\widehat{Z}_{\Sigma_{\wp}}\rightarrow \mathcal W_{\Sigma_{\wp}} \times \mathbb G_m$, $\chi\mapsto \big(\chi|_{\mathcal O_{\wp}^{\times}}, \chi(p)\big)$ \Big(resp. the isomorphism $\widehat{Z}_{\Sigma_{\wp}} \xrightarrow{\sim} \mathcal W_{\Sigma_{\wp}} \times \mathbb G_m$, $\chi \mapsto \big(\chi|_{\mathcal O_{\wp}^{\times}}, \chi(\varpi)\big)$\Big), set $a:=\psi_0(p)$ (resp. $a:=\psi_0(\varpi)$), which is the image of $\psi_0$ in $\mathbb G_m$. We discuss in the following two cases:
If $\upsilon_{\wp}(a)\neq 0$, then choose $n\in \mathbb Z_{\geq 1}$ such that $\upsilon_{\wp}(a)\notin p^n\mathbb Z$; let $U_1$ be an admissible open in $\mathcal W_{\Sigma_{\wp}}$ containing the trivial character such that if $\prod_{\sigma \in \Sigma_{\wp}} \sigma^{n_{\sigma}}\big|_{\mathcal O_{\wp}^{\times}}\in U_1(\overline{E})$ then $p^n|n_{\sigma}$ for all $\sigma$, $U_2$ be an admissible open in $\mathbb G_m$ containing $a$ such that $\upsilon_{\wp}(a')=\upsilon_{\wp}(a)$ for all $a'\in U(\overline{E})$, one easily check the admissible open $U_0:=U_1\times U_2$ satisfies the property in the claim.
If $\upsilon_{\wp}(a)=0$, since $a\neq 1$ by hypothesis, let $U_2$ be an admissible open in $\mathbb G_m$ such that $a\in U_2(E)$, $1\notin U_2(E)$ and for all $a'\in U_2(\overline{E})$, $\upsilon_{\wp}(a')=0$; put $U_0:=\mathcal W_{\Sigma_{\wp}} \times U_2$, we see if $\chi'=\prod_{\sigma\in \Sigma_L} \sigma^{n_{\sigma}}\in U_0(\overline{E})$ for $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^d$ \big(resp. for $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z_{\geq 0}^d$\big), thus $\upsilon_{\wp}(\chi'(p))=0$ (resp. $\upsilon_{\wp}(\chi'(\varpi))=0$), thus $\sum_{\sigma\in \Sigma_L}n_{\sigma}=0$, so $\chi'(p)=1\notin U_2(\overline{E})$ \big(resp. $n_{\sigma}=0$ for all $\sigma\in \Sigma_{\wp}$ hence $\chi'(\varpi)=1\notin U_2(\overline{E})$\big), a contradiction, so $U_0$ satisfies the property in the claim.
\end{proof}
\subsubsection{\'Etaleness of eigenvarieties at non-critical classical points}\label{sec: clin-3.4.3}
Let $z=\big(\chi_z=\chi_{z,1} \otimes \chi_{z,2}, \lambda_z\big)$ be a semi-stable classical point of $\mathcal V(K^p,w)_{\overline{\rho}}$, for $\sigma\in \Sigma_{\wp}$, we say that $z$ is \emph{non-critical} if
\begin{enumerate} \item the triangulation $(\rho_{z,\wp},\delta_{z,1},\delta_{z,2})$ (cf. Thm.\ref{thm: clin-iz2d}) is non-critical (i.e. $\Sigma_{z}=\emptyset$),\\
\item $\unr(q^{-1}) \chi_{z,1}^{-1} \chi_{z,2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$.
\end{enumerate}
Let $\psi_{z,1}$, $\psi_{z,2}$ be unramified characters of $F_{\wp}^{\times}$ such that $\chi_{z,i}=\psi_{z,i}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{k_{\chi_{z,i},\sigma}}$, then the condition (2) is equivalent to $\psi_{z,1}(q\varpi)\neq \psi_{z,2}(\varpi)$. If one considers the Galois representation $\rho_{z,{\wp}}$ (which is semi-stable), this condition means the eigenvalues of $\varphi^{d_0}$ on $D_{\st}(\rho_{z,\wp})$ are different.
Consider the natural morphism
\begin{equation*}
\kappa: \mathcal V(K^p,w)_{\overline{\rho}} \longrightarrow \widehat{T}_{\Sigma_{\wp}} \longrightarrow \mathcal W_{1,\Sigma_{\wp}},
\end{equation*}
where the last map is induced by the inclusion $Z_1' \rightarrow T(F_{\wp})$ (see also (\ref{equ: clin-st1})). This section is devoted to prove the following result.
\begin{theorem}\label{thm: clin-elt}Let $z$ be a non-critical semi-stable classical point of the rigid space $ \mathcal V(K^p,w)_{\overline{\rho}}$, then $\mathcal V(K^p,w)_{\overline{\rho}}$ is \'etale over $\mathcal W_{1,\Sigma_{\wp}}$ at $z$.
\end{theorem}
The theorem follows by the same argument as in the proof of \cite[Thm.4.8]{Che11}.
Let $z=(\chi_z=\chi_{z,1}\otimes \chi_{z,2},\lambda_z):\Spec \overline{E} \rightarrow \mathcal V(K^p,w)_{\overline{\rho}}$ be a non-critical semi-stable classical point of $C(w)$, by the construction of $ \mathcal V(K^p,w)_{\overline{\rho}}$ as in \S \ref{sec: clin-3.3}, one can find a connected affinoid neighborhood $U$ of $\kappa(z)$ in $\mathcal W_{1,\Sigma_{\wp}}$ and a finite locally free $\mathcal O(U)$-module $M$ equipped with an $\mathcal O(U)$-linear action of $\mathcal H$ such that (see also the proof of \cite[Thm.4.8]{Che11})
\begin{enumerate}
\item the affinoid spectrum $V$ of $\Ima\big(\mathcal O(U) \otimes_{\mathcal O_E} \mathcal H \rightarrow \End_{\mathcal O(U)}(M)\big)$ is an affinoid neighborhood of $z$ in $ \mathcal V(K^p,w)_{\overline{\rho}}$ \big(thus one has $\mathcal M(K^p,w)_{\overline{\rho}}(\mathcal O(V))\mathcal Ong M$ as $\mathcal O(V)$-module\big);
\item for each continuous character $\chi\in \mathcal W_{1,\Sigma_{\wp}}(\overline{E})$, there is a $T(F_{\wp})\times \mathcal H^p$-invariant isomorphism
\begin{equation*}
M\otimes_{\mathcal O(U), \chi} \overline{E}\mathcal Ong
\bigoplus_{(\chi_{z'},\lambda_{z'})\in \kappa^{-1}(\chi)} \Big(J_B\big(\widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an}\big) \otimes_E \overline{E}\Big)^{Z_1=\mathscr N^{-w},Z_1'=\chi}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}]^{\vee};
\end{equation*}
\item $\kappa^{-1}(\kappa(z))^{\red}=\{z\}$ and the natural surjection $\mathcal O(V) \rightarrow k(z)$ has a section.
\end{enumerate}
Let $Z_0\subseteq U$ be the set of closed points $\chi$ such that any point in $\kappa^{-1}(\chi)\cap V(\overline{E})$ is classical, thus $Z_0$ is Zariski-dense in $U$ \big(by Thm.\ref{thm: clin-etn}, note that $\upsilon_{\wp}(\chi_{z',1}(\varpi))$ is bounded for $z'\in V(\overline{E})$\big), and $Z:=\kappa^{-1}(Z_0) \cup \{z\}$ is Zariski-dense in $V$. Up to shrinking $Z_0$, one can assume that for any $z'\in Z$,
\begin{enumerate}[label=(\alph*)]
\item $\Sigma_{z'}=\emptyset$ (since $z$ is supposed to be non-critical, for $z'\neq z$, this would follow from Thm.\ref{thm: clin-zgn}),
\item $\unr(q^{-1}) \chi_{z',1}^{-1} \chi_{z',2} \neq \prod_{\sigma\in \Sigma_{\wp}} \sigma^{n_{\sigma}}$ for any $\underline{n}_{\Sigma_{\wp}}\in \mathbb Z^{d}$ \big(e.g. by Lem.\ref{lem: clin-cte} (2), since by shrinking $Z_0$, one can assume $k_{\chi_{z',2},\sigma}-k_{\chi_{z',1},\sigma}\geq k_{\chi_{z,2},\sigma}-k_{\chi_{z,1},\sigma}$ for all $\sigma\in \Sigma_{\wp}$, and $z'\in Z_0$\big).
\end{enumerate}Let $z'=(\chi_{z'},\lambda_{z'})\in Z$, denote by $k(z')$ the residue field at $z'$. One has an isomorphism (cf. (\ref{equ: clin-0nvi}))
\begin{multline*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{Z_1=\mathscr N^{-w}, Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}] \\ \xlongrightarrow{\sim} \Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}].
\end{multline*}
\begin{lemma}\label{lem: clin-adb}
Keep the above notation, any vector in $$\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}]$$ is classical.
\end{lemma}
\begin{proof}
Suppose there exists a non-classical vector $v$, thus there exists $\sigma\in \Sigma_{\wp}$, such that $v$ is non-$\sigma$-classical. By Prop.\ref{prop: clin-vka}, one can prove $z'$ admits a $\sigma$-companion point, which would lead to a contradiction by the same argument as in the proof of Cor.\ref{cor: clin-aat}.
\end{proof}
Keep the above notation (so $z'\in Z$), put $\psi_{z'}:=\chi_{z'}\big(\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_{1,z'},\sigma}} \otimes \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{\chi_{2,z'},\sigma}}\big)$ (being a smooth character of $T(F_{\wp})$), $T_0:=Z_1Z_1'$, by Lem.\ref{lem: clin-adb}, Cor.\ref{prop: clin-ernr}, one has an isomorphism of $k(z')$-vector spaces
\begin{multline*}
\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{N_0,Z_1=\mathscr N^{-w},Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'}, \mathcal H^p=\lambda_{z'}]
\\ \xlongrightarrow{\sim} \Big(H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\Big)^{N_0, T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}],
\end{multline*}
where $k_{\sigma}:=k_{\chi_{1,z'},\sigma}-k_{\chi_{2,z'},\sigma}+2$ for all $\sigma\in \Sigma_{\wp}$. So
\begin{multline*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}\otimes_E k(z')\Big)^{Z_1=\mathscr N^{-w}, Z_1'=\kappa(z')}[T(F_{\wp})=\chi_{z'},\mathcal H^p=\lambda_{z'}] \\\xlongrightarrow{\sim} J_B\Big(H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\Big)^{T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}].
\end{multline*}
Denote by $\delta(z')$ the dimension of the above vector space over $k(z')$. Set
\begin{equation*}
H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big):=\varinjlim_{(K^p)'}H^1_{\et}\big((K^p)',W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E \overline{E}
\end{equation*}
where $(K^p)'$ runs over open compact subgroups of $K^p$, this is a smooth admissible representation of $G(\mathbb A^{\infty})$ equipped with a continuous action of $\Gal(\overline{F}/F)$. One has a decomposition of $G(\mathbb A^{\infty}) \times \Gal(\overline{F}/F)$-representations
\begin{equation*}
H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big) \mathcal Ong \bigoplus_{\pi} \rho(\pi) \otimes \pi
\end{equation*}
where $\pi$ runs over irreducible smooth admissible representations of $G(\mathbb A^{\infty})$. It's known that if $\rho(\pi)\neq 0$, then $\dim_{\overline{E}}\rho(\pi)=2$ (e.g. see \cite[\S 2.2.4]{Ca2}). A necessary condition for $\rho(\pi)$ to be non-zero is that there exists an admissible representation $\pi_{\infty}$ of $G(\mathbb R)$ such that $\pi_{\infty}\otimes \pi$ is an automorphic representation of $G(\mathbb A)$ (we fix an isomorphism $\overline{E}\xrightarrow{\sim} \mathbb C$). Note that one has
\begin{equation*}
H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_{E} \overline{E} \xlongrightarrow{\sim} H^1_{\et}\big(W(\underline{k}_{\Sigma_{\wp}},w)\big)^{K^p}\mathcal Ong \bigoplus_{\pi} \rho(\pi) \otimes \pi^{K^p}.
\end{equation*}
For an irreducible smooth admissible representation $\pi$ of $G(\mathbb A^{\infty})$, $\pi$ admits thus a decomposition $\pi\mathcal Ong \otimes_{\mathfrak l} \pi_{\mathfrak l}$ with $\pi_{\mathfrak l}$ an irreducible smooth admissible representation of $(B\otimes_F F_{\mathfrak l})^{\times}$, where $\mathfrak l$ runs over the finite places of $F$. Recall (e.g. see \cite[Thm.VI.1.1(4)]{HT})
\begin{proposition}
Let $\pi_1$, $\pi_2$ be two automorphic representations of $G(\mathbb A)$, if $\pi_{1,\mathfrak l}\mathcal Ong \pi_{2,\mathfrak l}$ for all but finitely many places $\mathfrak l$ of $F$, then $\pi_1\mathcal Ong \pi_2$.
\end{proposition}
By this proposition (and the above discussions), there exists a unique irreducible smooth admissible representation $\pi_{z'}$ of $G(\mathbb A^{\infty})$ such that the action of $\mathcal H^p$ on $(\pi_{z'})^{K^p}$ is given by $\lambda_{z'}$ and that $\rho(\pi_{z'})\neq 0$ \big(one has in fact $\rho_{z'}\mathcal Ong \rho(\pi_{z'})$\big). Thus
\begin{equation*}
H^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)[\mathcal H^p=\lambda_{z'}]\mathcal Ong \rho_{z'} \otimes\pi_{z',\wp} \otimes \Big(\Big(\bigotimes_{\mathfrak l \neq \wp} \pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}\Big)[\mathcal H^p=\lambda_{z'}]\Big).
\end{equation*}
We have the following facts:
\begin{itemize}
\item $\dim_{\overline{E}} J_B(\pi_{z',\wp})[T(F_{\wp})=\psi_{z'}]=1$ (by classical Jacquet module theory and the condition (b)),
\item $\dim_{\overline{E}} \pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}=1$, for all $\mathfrak l\in S(K^p)$,
\end{itemize}
from which we deduce
\begin{multline*}
J_B\big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\big)^{T(F_{\wp})=\psi_{z'}, \mathcal H^p=\lambda_{z'}}\\ \xlongrightarrow{\sim}J_B\big(H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\otimes_E k(z')\big)^{T_0=\psi_{z'}}[T(F_{\wp})=\psi_{z'},\mathcal H^p=\lambda_{z'}].
\end{multline*}
Denote by $S'$ the complement of $S(K^p) \cup \{\wp\}$ in the set of finite places of $F$ (thus $S'$ is a finite set), we also deduce (compare with \cite[(4.21)]{Che11})
\begin{equation*}
\delta(z')=2\sum_{\mathfrak l \in S'} \dim_{\overline{E}}\big(\pi_{z',\mathfrak l}^{(K^p)_{\mathfrak l}}\big).
\end{equation*}
By the same argument as in the proof of \cite[Thm.4.8]{Che11}, one can prove $\delta(z')\geq \delta(z)$ for all $z'\in Z$, and then deduce that $\mathcal O(V)\mathcal Ong \mathcal O(U)\otimes_E k(z)$. The theorem follows.
\begin{remark}\label{rem: clin-rat}
Keep the above notation, if $z$ is moreover an $E$-point of $ \mathcal V(K^p,w)_{\overline{\rho}}$ (in practice, one can always enlarge $E$ if necessary), thus one has $\mathcal O(V) \mathcal Ong \mathcal O(U)$. So the action of $T(F_{\wp})$ on $M$ is given by the character $T(F_{\wp})\rightarrow \mathcal O(V)^{\times}\mathcal Ong \mathcal O(U)^{\times}$ induced by the natural morphism $V\rightarrow \widehat{T}_{\Sigma_{\wp}}$.
\end{remark}
\section{$\mathcal L$-invariants and local-global compatibility}
\subsection{Fontaine-Mazur $\mathcal L$-invariants}\label{sec: clin-ene}
Recall Fontaine-Mazur $\mathcal L$-invariants for $2$-dimensional semi-stable non-crystalline representations of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$.
Let $k_{1,\sigma}, k_{2,\sigma}\in \mathbb Z$, $k_{1,\sigma}<k_{2,\sigma}$ for all $\sigma\in \Sigma_{\wp}$; let $\rho$ be a $2$-dimensional semi-stable non-crystalline representation of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E$ of Hodge-Tate weights $(-k_{2,\sigma}, -k_{1,\sigma})_{\sigma\in \Sigma_{\wp}}$. By Fontaine's theory (cf. \cite{Fon94}, \cite{FO}), one can associate to $\rho$ a filtered $(\varphi,N)$-module $(D_0, D)$ where $D_0:=D_{\st}(\rho):= (B_{\st}\otimes_{\mathbb Q_p} \rho)^{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is a free $F_{\wp,0}\otimes_{\mathbb Q_p} E$-module of rank $2$ equipped with a bijective \big($F_{\wp,0}$-semi-linear and $E$-linear\big) endomorphism $\varphi$ and a nilpotent $F_{\wp,0}\otimes_{\mathbb Q_p} E$-linear operator $N$ such that $N\varphi=p\varphi N$, and that $D:=D_0\otimes_{F_{\wp,0}}F_{\wp} \mathcal Ong D_{\dR}(\rho):=(B_{\dR}\otimes_{\mathbb Q_p} \rho)^{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is a free $F_{\wp}\otimes_{\mathbb Q_p} E$-module of rank $2$ equipped with a decreasing exhaustive separated filtration by $F_{\wp} \otimes_{\mathbb Q_p} E$-submodules.
Using the isomorphism
\begin{equation*}
F_{\wp,0}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} E, \ a \otimes b \mapsto \big(\sigma_0(a)b\big)_{\sigma_0: F_{\wp,0}\rightarrow E},
\end{equation*}
one can decompose $D_0$ as $D_0\xrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} D_{\sigma_0}$. Each $D_{\sigma_0}$ is an $E$-vector space of rank $2$ equipped with an $E$-linear action of $\varphi^{d_0}$ and $N$, moreover, the operator $\varphi$ (on $D_0$) induces a bijection: $D_{\sigma_0}\xrightarrow{\sim} D_{\sigma_0\circ \varphi^{-1}}$. It's known that $\Ker(N)$ is a free $F_{\wp,0}\otimes_{\mathbb Q_p} E$-module of rank $1$, and thus admits a decomposition $\Ker(N)\xrightarrow{\sim} \prod_{\sigma_0: F_{\wp,0}\rightarrow E} \Ker(N)_{\sigma_0}$. Let $e_{0,\sigma_0}\in D_{\sigma_0}$ such that $Ee_{0,\sigma_0}=\Ker(N)_{\sigma_0}$. In fact, one can choose $e_{0,\sigma_0}$ such that \begin{equation}\label{equ: clin-e0e}\varphi(e_{0,\sigma_0})=e_{0,\sigma_0\circ \varphi^{-1}}.\end{equation} Since $\Ker(N)_{\sigma_0}$ is stable by $\varphi^{d_0}$, there exists $\alpha\in E^{\times}$ such that $\varphi^{d_0}(e_{0,\sigma_0})=\alpha e_{0,\sigma_0}$ \big(by (\ref{equ: clin-e0e}), we see $\alpha$ is independent of $\sigma_0$\big). Since $N\varphi=p\varphi N$, there exists a unique $e_{1,\sigma_0}\in D_{\sigma_0}$ such that $Ne_{1,\sigma_0}=e_{0,\sigma_0}$ and $\varphi^{d_0}(e_{1,\sigma_0})=q\alpha e_{1,\sigma_0}$ (thus $D_{\sigma_0}=E e_{0,\sigma_0} \oplus E e_{1,\sigma_0}$).
Using the isomorphism
\begin{equation*}
F_{\wp}\otimes_{\mathbb Q_p} E \xlongrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} E, \ a \otimes b \mapsto (\sigma(a)b)_{\sigma\in \Sigma_{\wp}},
\end{equation*}
one can decompose $D$ as $D\xrightarrow{\sim} \prod_{\sigma\in \Sigma_{\wp}} D_{\sigma}$. One has $D_{\sigma_0}\otimes_{F_{\wp,0}} F_{\wp} \mathcal Ong \prod_{\substack{\sigma\in \Sigma_{\wp} \\ \sigma|_{F_{\wp,0}}=\sigma_0}} D_{\sigma}$ for any $\sigma_0: F_{\wp,0}\rightarrow E$. For $\sigma\in \Sigma_{\wp}$, $i=0,1$, let $e_{i,\sigma}\in D_{\sigma}$, such that
\begin{equation*}
e_{i,\sigma_0} \otimes 1 = \big(e_{i,\sigma}\big)_{\substack{\sigma\in \Sigma_{\wp} \\ \sigma|_{F_{\wp,0}}=\sigma_0}}.
\end{equation*}
Since $\rho$ is of Hodge-Tate weights $(-k_{2,\sigma}, -k_{1,\sigma})_{\sigma\in \Sigma_{\wp}}$, for all $\sigma\in \Sigma_{\wp}$, there exists $(a_{\sigma},b_{\sigma})\in E\times E \setminus \{(0,0)\}$ such that
\begin{equation*}
\mathbb Fil^{i} D_{\sigma}=\begin{cases}
D_{\sigma} & i\leq k_{1,\sigma} \\
E\big(a_{\sigma}e_{1,\sigma}+ b_{\sigma}e_{0,\sigma}\big) & k_{1,\sigma}<i \leq k_{2,\sigma} \\
0 & i > k_{2,\sigma}
\end{cases}.
\end{equation*}
We suppose $\rho$ satisfies the following hypothesis.
\begin{hypothesis}\label{hyp: clin-aq0}
For all $\sigma\in \Sigma_{\wp}$, $a_{\sigma}\neq 0$.
\end{hypothesis}
\begin{remark}
This hypothesis is automatically satisfied when $F_{\wp}=\mathbb Q_p$ by the weak admissibility of $(D_0,D)$.
\end{remark}
Pose $\mathcal L_{\sigma}:=b_{\sigma}/a_{\sigma}$, for $\sigma\in \Sigma_{\wp}$. One sees easily that $\mathcal L_{\sigma}$ is independent of the choice of $e_{0,\sigma}$. An important fact is that one can recover $\big(D_{\st}(\rho),D_{\dR}(\rho)\big)$ (and hence $\rho$) by the data:
\begin{equation*}
\Big\{(-k_{2,\sigma}, -k_{1,\sigma}\big)_{\sigma\in \Sigma_{\wp}};\ \alpha,q\alpha;\ \{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}\Big\}.
\end{equation*}
Note that by the hypothesis \ref{hyp: clin-aq0}, $\rho$ admits a unique triangulation given by
\begin{equation*}
0 \rightarrow \mathcal R_E\Big(\unr(\alpha)\prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{1,\sigma}}\Big) \rightarrow D_{\rig}(\rho) \rightarrow \mathcal R_E\Big(\unr(q\alpha) \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-k_{2,\sigma}}\Big) \rightarrow 0,
\end{equation*}
in particular, $\rho$ is non-critical.
\subsection{$\mathcal L$-invariants and locally $\mathbb Q_p$-analytic representations}\label{sec: clin-4.2} Keep the above notation, following \cite{Sch10}, one can associate to $\rho$ a locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$. We recall the construction (note that the log maps that we use are slightly different from those in \cite{Sch10}) and introduce some notations. Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$ for all $\sigma\in \Sigma_{\wp}$ such that $k_{\sigma}\equiv w \pmod{2}$, suppose $\rho$ is of Hodge-Tate weights $\big(-\frac{w+k_{\sigma}}{2}, -\frac{w-k_{\sigma}+2}{2}\big)_{\sigma\in \Sigma_{\wp}}$. Put
\begin{equation*}
\chi(\underline{k}_{\Sigma_{\wp}},w; \alpha):= \unr(\alpha) \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-\frac{w-k_{\sigma}+2}{2}} \otimes \unr(\alpha) \prod_{\sigma \in \Sigma_{\wp}}\sigma^{-\frac{w+k_{\sigma}-2}{2}} ,
\end{equation*}
which is a continuous character of $T(F_{\wp})$ over $E$. Consider the parabolic induction $$\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an},$$ we have the following facts
\begin{itemize}
\item the unique finite dimensional subrepresentation of $\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}$ is $V(\underline{k}_{\Sigma_{\wp}},w;\alpha):=\big(\unr(\alpha)\circ \dett\big)\otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}$;
\item the maximal locally algebraic subrepresentation of the quotient
\begin{equation*}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big):=\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big/V(\underline{k}_{\Sigma_{\wp}},w;\alpha)
\end{equation*}
is $\St(\underline{k}_{\Sigma_{\wp}},w;\alpha):=\St \otimes_E V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$, which is also the socle of $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ (where $\St$ denotes the Steinberg representation).
\end{itemize}
Let $\psi(\underline{\mathcal L}_{\Sigma_{\wp}})$ be the following $(d+1)$-dimensional representation of $T(F_{\wp})$ over $E$
\begin{equation*}
\psi(\underline{\mathcal L}_{\Sigma_{\wp}})\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix} =\begin{pmatrix}
1& \log_{\sigma_1, -\mathcal L_{\sigma_1}}(ad^{-1}) & \log_{\sigma_2, -\mathcal L_{\sigma_2}}(ad^{-1}) & \cdots &\log_{\sigma_{d}, -\mathcal L_{\sigma_{d}}}(ad^{-1}) \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 &\cdots & 0\\
\vdots & \vdots &\vdots &\ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{pmatrix}.
\end{equation*}
One gets thus an exact sequence of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$:
\begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E \psi(\underline{\mathcal L}_{\Sigma_{\wp}})\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s}\Big(\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big)^{\oplus d} \longrightarrow 0.
\end{multline*}
Following Schraen \cite[\S 4.2]{Sch10}, put
\begin{equation}\label{equ: clin-akp}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big):=s^{-1}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus d}\big)/V(\underline{k}_{\Sigma_{\wp}},w;\alpha).
\end{equation}
\begin{remark}\label{rem: clin-tagia} (1) By \cite[Prop.4.13]{Sch10}, $\Sigma\big(\underline{k'}_{\Sigma_{\wp}},w'; \alpha';\underline{\mathcal L'}_{\Sigma_{\wp}}\big) \mathcal Ong \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$ if and only if $\underline{k'}_{\Sigma_{\wp}}=\underline{k}_{\Sigma_{\wp}}$, $w'=w$, $\alpha'=\alpha$ and $\underline{\mathcal L'}_{\Sigma_{\wp}}=\underline{\mathcal L}_{\Sigma_{\wp}}$.
(2) For $\sigma\in \Sigma_{\wp}$, denote by $\psi(\mathcal L_{\sigma})$ the following $2$-dimensional representation of $T(F_{\wp})$:
\begin{equation*}
\psi(\mathcal L_{\sigma})\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix} =\begin{pmatrix}
1& \log_{\sigma, -\mathcal L_{\sigma}}(ad^{-1}) \\ 0 & 1
\end{pmatrix}.
\end{equation*}
One has thus an exact sequence
\begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E \psi(\mathcal L_{\sigma})\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s_{\sigma}}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow 0.
\end{multline*}
Put $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_\sigma\big):=s_{\sigma}^{-1}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)/V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$. One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$:
\begin{multline}\label{equ: clin-al1h}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_1}\big)\oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_2}\big)\oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\\ \cdots \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\sigma_{d}}\big) \xlongrightarrow{\sim} \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big).
\end{multline}
\end{remark}
Let $\chi_i$ be a locally $\sigma_i$-analytic (additive) character of $F_{\wp}^{\times}$ in $E$, replacing the term $\log_{\sigma_i, -\mathcal L_{\sigma_i}}(ad^{-1})$ by $\log_{\sigma_i, -\mathcal L_{\sigma_i}}(ad^{-1}) + \chi_i \circ \dett $, one can construct a representation $\Sigma'\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$ exactly the same way as $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)$. By cohomology arguments as in \cite[\S 4.3]{Sch10}, one can actually prove
\begin{lemma}\label{lem: linv-nmfyc}One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$:
\begin{equation}\label{equ: clin-agia}
\Sigma'\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}\big)\xlongrightarrow{\sim}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\underline{\mathcal L}_{\Sigma_{\wp}}
\big).\end{equation}
\end{lemma}
\begin{proof}We use $\Ext^1$ to denote the extensions in the category of admissible locally $\mathbb Q_p$-analytic representations.
By the same argument as in \cite[\S 4.3]{Sch10}, replacing $\overline{G}$, $\overline{T}$ by $\GL_2(F_{\wp})$, $T(F_{\wp})$ respectively, one has (see in particular \cite[Lem.4.8]{Sch10} and the discussion which follows)
\begin{equation*}\Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha), \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an}\Big)\mathcal Ong \Hom_{\mathbb Q_p-\an}(T(F_{\wp}),E),\end{equation*}
which is hence of dimension $2(d+1)$ over $E$.
On the other hand, one can prove
\begin{equation}\label{equ: linv-2pwv1}\Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),V(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)\mathcal Ong \Hom_{\mathbb Q_p-\an}(F_{\wp}^{\times},E).\end{equation}
Indeed, put $V:=V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ for simplicity, then by \cite[Prop.3.5]{Sch10}, one has
\begin{equation*}
\Ext^1_{\GL_2(F_{\wp})}(V,V)\mathcal Ong H^1_{\an}\big(\GL_2(F_{\wp}), V\otimes_E V^{\vee}\big);
\end{equation*}
for any finite dimensional algebraic representation $W$ of $\mathbb Res_{L/\mathbb Q_p} \GL_2$ over $E$, by \cite[Thm.3]{CW}, one has
\begin{equation*}
H^1_{\an}(\GL_2(F_{\wp}), W)\mathcal Ong H^1(\mathfrak g \otimes_{\mathbb Q_p} E, W).
\end{equation*}
Using K\"unneth formula \big(with respect to the decomposition $\mathfrak g \mathcal Ong \mathfrak s \times \mathfrak z$, where $\mathfrak s$ denotes the Lie algebra of $\SL_2(F_{\wp})$ and $\mathfrak z$ the Lie algebra of the center $Z(F_{\wp})$ of $\GL_2(F_{\wp})$\big) and the first Whitehead lemma (cf. \cite[Cor.7.8.10]{Wei}), one can show $H^1(\mathfrak g\otimes_{\mathbb Q_p} E,W)=0$, if $W$ is irreducible non-trivial; and $H^1(\mathfrak g\otimes_{\mathbb Q_p} E,E)\mathcal Ong H^1(\mathfrak z\otimes_{\mathbb Q_p} E, E)\mathcal Ong H^1_{\an}(F_{\wp}^{\times}, E)\mathcal Ong \Hom_{\mathbb Q_p-\an}(F_{\wp}^{\times},E)$. Since the trivial representation has multiplicity one in $V\otimes_E V^{\vee}$, one gets the isomorphism in (\ref{equ: linv-2pwv1}).
From the exact sequence
\begin{equation*}
0 \rightarrow V(\underline{k}_{\Sigma_{\wp}},w;\alpha) \rightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an} \rightarrow \Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha) \rightarrow 0
\end{equation*}
one gets\begin{multline*}
0 \longrightarrow \Ext^{1}_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha) , V(\underline{k}_{\Sigma_{\wp}},w;\alpha) \big) \\ \longrightarrow \Ext^1_{\GL_2(F_{\wp})}\Big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)^{\mathbb Q_p-\an}\Big) \\
\xlongrightarrow{j} \Ext^1_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big).
\end{multline*} So $\dim_E \Ima(j)=d+1$. This, combined with the discussion above \cite[Prop.4.10]{Sch10}, shows that the natural injection (cf. \cite[Prop.3.5]{Sch10}, where $\PGL_2:=\GL_2/Z$)
\begin{equation*}
\Ext_{\PGL_2(F_{\wp})}^1\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big) \lhook\joinrel\longrightarrow \Ext^1_{\GL_2(F_{\wp})}\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)
\end{equation*}
induces a bijection between $\Ext_{\PGL_2(F_{\wp})}^1\big(V(\underline{k}_{\Sigma_{\wp}},w;\alpha),\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)\big)$ and $\Ima(j)$, from which the isomorphism (\ref{equ: clin-agia}) follows.
\end{proof}
\subsection{Local-global compatibility}Let $w\in \mathbb Z$, $k_{\sigma}\in \mathbb Z_{\geq 2}$, $k_{\sigma}\equiv w \pmod{2}$ for all $\sigma\in \Sigma_{\wp}$. Let $\rho$ be a $2$-dimensional continuous representation of $\Gal(\overline{F}/F)$ over $E$ such that
\begin{enumerate} \item $\rho$ is absolutely irreducible modulo $\varpi_E$;
\item $\rho_{\wp}:=\rho|_{\Gal(\overline{\mathbb Q_p}/F_{\wp})}$ is semi-stable non-crystalline of Hodge-Tate weights $\big(-\frac{w+k_{\sigma}}{2}, -\frac{w-k_{\sigma}+2}{2}\big)_{\sigma\in \Sigma_{\wp}}$ satisfying the Hypothesis \ref{hyp: clin-aq0} with $\{\mathcal L_{\sigma}\}_{\sigma\in \Sigma_{\wp}}$ the associated Fontaine-Mazur $\mathcal L$-invariants and $\{\alpha,q\alpha\}$ the eigenvalues of $\varphi^{d_{0}}$ over $D_{\st}(\rho_{\wp})$
\item $\Hom_{\Gal(\overline{F}/F)}\Big(\rho, H^1_{\et}\big(K^{p}, W(\underline{k}_{\Sigma_{\wp}},w)\big)\Big)\neq 0$.
\end{enumerate}
Denote by $\lambda_{\rho}$ the system of eigenvalues of $\mathcal H^p$ associated to $\rho$ (via the Eichler-Shimura relations), put
\begin{equation*}
\widehat{\Pi}(\rho):=\Hom_{\Gal(\overline{F}/F)}\Big(\rho, \widetilde{H}^1_{\et}(K^p, E)^{\mathcal H^p=\lambda_{\rho}}\Big).
\end{equation*}
Note that one has
\begin{equation*}
\widehat{\Pi}(\rho)\xlongrightarrow{\sim} \Hom_{\Gal(\overline{F}/F)}\Big(\rho, \widetilde{H}^1_{\et}(K^p, E)_{\overline{\rho}}^{\mathcal H^p=\lambda_{\rho}}\Big).
\end{equation*}
One can deduce from the isomorphism (cf. Thm.\ref{thm: clin-ecs}(2))
\begin{equation*}
\widetilde{H}^1_{\et}\big(K^p,W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\mathbb Q_p-\an} \xlongrightarrow{\sim} \widetilde{H}^1_{\et}(K^p, E)_{\mathbb Q_p-\an} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)
\end{equation*}
a natural injection (cf. Prop.\ref{prop: clin-trn} and \cite[Prop.4.2.4]{Em04})
\begin{equation}\label{equ: clin-apwgi}
H^1_{\et}\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)_{\overline{\rho},\mathbb Q_p-\an} \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee} \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p, E)_{\overline{\rho},\mathbb Q_p-\an},
\end{equation}
thus $\widehat{\Pi}(\rho)$ is non-zero \big(by the condition (3)\big). Moreover, by Thm.\ref{thm: clin-ecs}(1), $\widehat{\Pi}(\rho)$ is a unitary admissible Banach representation of $\GL_2(F_{\wp})$ over $E$. In fact, $\widehat{\Pi}(\rho)$ is supposed to be the right representation of $\GL_2(F_{\wp})$ corresponding to $\rho_{\wp}$ in the $p$-adic Langlands program (cf. \cite{Br0}). By the local-global compatibility in the classical local Langlands correspondence for $\ell=p$, and Prop.\ref{prop: clin-trn} (see also \cite[Thm.5.3]{New}), one can show that there exists $r\in \mathbb Z_{\geq 1}$, such that (cf. \S \ref{sec: clin-4.2})
\begin{equation}\label{equ: clin-wrs}
\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus r}\xlongrightarrow{\sim} \widehat{\Pi}(\rho)_{\lalg},
\end{equation}where $\widehat{\Pi}(\rho)_{\lalg}$ denotes the locally algebraic vectors of $\widehat{\Pi}(\rho)$.We can now announce the main result of this article.
\begin{theorem}\label{thm: clin-sio}Keep the above notation and hypothesis,
the natural restriction map
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big) \longrightarrow \Hom_{\GL_2(F_{\wp})}\Big( \St(\underline{k}_{\Sigma_{\wp}},w;\alpha), \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big)
\end{equation*}is bijective. In particular, one has a continuous injection of $\GL_2(F_{\wp})$-representations
\begin{equation*}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)^{\oplus r} \lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}.
\end{equation*}
which induces an isomorphism between the locally algebraic subrepresentations.
\end{theorem}Such a result is called local-global compatibility, since the $\Pi(\rho_{\wp})$ are constructed by the local parameters (i.e. parameters of $\rho_{\wp}$) while $\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}$ is a global object. By the isomorphism (\ref{equ: clin-al1h}), the theorem \ref{thm: clin-sio} would follow from the following proposition.
\begin{proposition}\label{prop: clin-igla}
For any $\tau\in \Sigma_{\wp}$, the restriction map
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big) \longrightarrow \Hom_{\GL_2(F_{\wp})}\Big( \St(\underline{k}_{\Sigma_{\wp}},w;\alpha), \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\Big)
\end{equation*}is bijective.
\end{proposition}
Before proving this proposition, we give a corollary on the uniqueness of $\mathcal L$-invariants (suggested by Breuil):
\begin{corollary}\label{cor: clin-ape}Keep the notation in Prop.\ref{prop: clin-igla}, let $\mathcal L'_{\tau}\in E$, if there exists a continuous injection of $\GL_2(F_{\wp})$-representations
\begin{equation*}
i: \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an},
\end{equation*}
then $\mathcal L'_{\tau}=\mathcal L_{\tau}$.
\end{corollary}
\begin{proof}By Prop.\ref{prop: clin-igla}, the restriction on $i$ to $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ gives rise to a continuous injection
\begin{equation*}
j: \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}.
\end{equation*}
Suppose $\mathcal L'_{\tau}\neq \mathcal L_{\tau}$, one can thus deduce from $i$ and $j$ an injection
\begin{equation}\label{equ: clin-apwap}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big) \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)\lhook\joinrel\longrightarrow \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}.
\end{equation}
Put for simplicity \begin{equation*}V:=\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}'\big) \oplus_{\Sigma(\underline{k}_{\Sigma_{\wp}},w;\alpha)}\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\end{equation*}
the key point is the locally algebraic subrepresentation $V_{\lalg}$ contains an extension of $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ by $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$, which would contradict to (\ref{equ: clin-wrs}):
Denote by $\psi(\mathcal L_{\tau}',\mathcal L_{\tau})$ the following $3$-dimensional representation of $T(F_{\wp})$:
\begin{equation*}
\psi(\mathcal L_{\tau}',\mathcal L_{\tau})\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix} 1 & \log_{\tau,-\mathcal L_{\tau}'}(ad^{-1}) & \log_{\tau, -\mathcal L_{\tau}}(ad^{-1}) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},
\end{equation*}
thus one has an exact sequence
\begin{multline*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\otimes_E\psi(\mathcal L_{\tau}',\mathcal L_{\tau}))\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s'}\Big(\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an}\Big)^{\oplus 2} \longrightarrow 0.
\end{multline*}
It's straightforward to see \begin{equation*}V\xlongrightarrow{\sim} (s')^{-1}(V(\underline{k}_{\Sigma_{\wp}},w;\alpha)^{\oplus 2})/V(\underline{k}_{\Sigma_{\wp}},w;\alpha).\end{equation*}
On the other hand, $\psi(\mathcal L_{\tau}',\mathcal L_{\tau})$ admits a smooth subrepresentation
\begin{equation*}
\psi_0(\mathcal L_{\tau}',\mathcal L_{\tau})\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}=\begin{pmatrix}1 & (\mathcal L'_\tau-\mathcal L_{\tau})\upsilon_{\wp}(ad^{-1}) \\ 0 & 1\end{pmatrix},
\end{equation*}
one has thus an exact sequence of smooth representations of $\GL_2(F_{\wp})$
\begin{equation*}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\Big)^{\infty} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\otimes_E \psi_0(\mathcal L_{\tau}',\mathcal L_{\tau})\Big)^{\infty} \xlongrightarrow{s''}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\Big)^{\infty} \longrightarrow 0,
\end{equation*}
where $\chi_{\alpha}:=\unr(\alpha) \otimes \unr(\alpha)$. Note that $\unr(\alpha)\circ \dett$ is the socle of $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi_{\alpha}\big)^{\infty}$, and one can check
$$V':=\big((s'')^{-1}\big(\unr(\alpha)\circ \dett\big)/\unr(\alpha)\circ \dett\big) \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}$$
is a locally algebraic subrepresentation of $V$, which is an extension (non-split) of $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ by $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$. We deduce from (\ref{equ: clin-apwap}) an injection $V'\hookrightarrow \widehat{\Pi}(\rho)_{\lalg}$, a contradiction with (\ref{equ: clin-wrs}).
\end{proof}
\begin{remark}
By Thm.\ref{thm: clin-sio} and Cor.\ref{cor: clin-ape}, we see that the local Galois representation $\rho_{\wp}$ can be determined by $\widehat{\Pi}(\rho)$.
\end{remark}
The following lemma has a straightforward proof that is omitted.
\begin{lemma}\label{lem: clin-nfo}
Let $V$ be an admissible locally $\mathbb Q_p$-analytic representation of $\GL_2(F_{\wp})$ over $E$, there exists a natural bijection
\begin{equation*}
\Hom_{\GL_2(F_{\wp})}\big(V, \widehat{\Pi}(\rho)_{\mathbb Q_p-\an}\big) \xlongrightarrow{\sim}\\ \Hom_{\Gal(\overline{F}/F)}\Big(\rho, \Hom_{\GL_2(F_{\wp})}\big(V, \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\big)\Big).
\end{equation*}
\end{lemma}
The Prop.\ref{prop: clin-igla} thus follows from
\begin{proposition}\label{prop: clin-awdl}With the notation in Prop.\ref{prop: clin-igla}, the restriction map
\begin{multline}\label{equ: clin-gpwp}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big) \\
\longrightarrow \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big)
\end{multline}is bijective.
\end{proposition}
The rest of this paper is devoted to the proof of Prop.\ref{prop: clin-awdl}. Given an injection (whose existence follows from (\ref{equ: clin-wrs})) $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \hookrightarrow \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$, by applying the Jacquet-Emerton functor and Thm.\ref{thm: clin-cjw}, one gets a closed $E$-point (associate to $\rho$) in $\mathcal V(K^p,w)_{\overline{\rho}}$ given by $$z:=\big(\chi:= \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha) \delta,\lambda_{\rho}\big).$$
\begin{lemma}\label{lem: clin-pvle}
The restriction map (\ref{equ: clin-gpwp})
is injective.
\end{lemma}
\begin{proof}
The proof is the same as in \cite[Prop.6.3.9]{Ding}. Let $f$ be in the kernel of (\ref{equ: clin-gpwp}), suppose $f\neq 0$, thus $f$ would induce an injection $$V_{\wp}\lhook\joinrel\longrightarrow\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$$ with $V_{\wp}$ an irreducible constituent of $\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big)$ different from $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ (since $f$ lies in the kernel of (\ref{equ: clin-gpwp})) and from $V(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ (by (\ref{equ: clin-wrs})), from which, by applying the Jacquet-Emerton functor, one would get a companion point of $z$, which contradicts to the fact that $z$ is non-critical (thus does not admit companion points, cf. Cor.\ref{cor: clin-aat}).
\end{proof}
In the following, we prove the surjectivity of (\ref{equ: clin-gpwp}), which is the key of this paper.
By assumption, we know the point $z$ is non-critical, thus one may find an open neighborhood $\mathcal U$ of $z$ in $\mathcal V(K^p,w)_{\overline{\rho}}$ such that (cf. Cor.\ref{cor: clin-siz} and \cite[Lem.6.3.12]{Ding})
\begin{enumerate}
\item $\mathcal U$ is strictly quasi-Stein (\cite[Def.2.1.17(iv)]{Em04});
\item for any $z'\in \mathcal U(\overline{E})$, $z'$ does not have companion points.
\end{enumerate}
Denote by $\mathcal M:=\mathcal M(K^p,w)_{\overline{\rho}}$ for simplicity, the natural restriction (with dense image since $\mathcal U$ is strictly quasi-Stein)
\begin{equation*}
J_B\Big(\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}\Big)^{\vee}_b \mathcal Ong \mathcal M(\mathcal V(K^p,w)_{\overline{\rho}})\longrightarrow \mathcal M(\mathcal U)
\end{equation*}
induces a continuous injection of locally $\mathbb Q_p$-analytic representations of $T$ (invariant under $\mathcal H^p$)
\begin{equation}\label{equ: clin-wbi}
\mathcal M(\mathcal U)_{b}^{\vee} \lhook\joinrel\longrightarrow J_B\Big(\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}\Big)
\end{equation}
where $\mathcal M(\mathcal U)_b^{\vee}$ denotes the strict dual of $\mathcal M(\mathcal U)$. As in \cite[Lem.6.3.13, 6.3.14]{Ding}, one can show \big(see \cite[Lem.4.5.12]{Em1} and the proof of \cite[Thm.4.5.7]{Em1} for $\GL_2(\mathbb Q_p)$-case\big)
\begin{itemize}
\item $\mathcal M(\mathcal U)_b^{\vee}$ is an allowable subrepresentation of $J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)$ (cf. \cite[Def.0.11]{Em2}) (this follows from the fact $\mathcal U$ is strictly quasi-Stein);
\item the map (\ref{equ: clin-wbi}) is balanced (cf. \cite[Def.0.8]{Em2}) (this follows from the fact any closed point in $\mathcal U$ does not have companion points);
\item $\mathcal M(\mathcal U)$ is a torsion free $\mathcal O(\mathcal W_{1,\Sigma_{\wp}})$-module (cf. \S \ref{sec: clin-3.3}).
\end{itemize}Denote by $\overline{\mathfrak t}\subset \mathfrak t$ the Lie algebra of $Z_1'\subset T(F_{\wp})$, by \cite[Cor.5.3.31]{Ding} \Big(note that \cite[Cor.5.3.31]{Ding} still holds with $\mathfrak t$ replaced by $\overline{\mathfrak t}$, and note that $\mathcal M(\mathcal U)^{\vee}_b$ is a divisible $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})$-module since $\mathcal M(\mathcal U)$ is a torsion free $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})$-module with $\text{U}(\overline{\mathfrak t}_{\Sigma_{\wp}})\hookrightarrow \mathcal O(\mathcal W_{1,\Sigma_{\wp}})$, e.g. see \cite[\S 5.1.3, Prop.5.1.12]{Ding}\Big), the map (\ref{equ: clin-wbi}) induces a continuous $\GL_2(F_{\wp})\times\mathcal H^p$-invariant map \big(see \cite[(4.5.9)]{Em1} for $\GL_2(\mathbb Q_p)$-case\big)
\begin{equation}\label{equ: clin-p2w}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]\big)^{\mathbb Q_p-\an} \longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an},
\end{equation}
where $ \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]$ denotes the twist of $\mathcal M(\mathcal U)^{\vee}_b$ by $\delta^{-1}$. We would deduce Prop.\ref{prop: clin-awdl} from this map.
For $\sigma\in \Sigma_{\wp}$, denote by $\widehat{T}_{\sigma}$ (resp. $\mathcal W_{1,\sigma}$) the rigid space over $E$ parameterizing locally $\sigma$-analytic characters of $T(F_{\wp})$ (resp. $1+2\varpi\mathcal O_{\wp}$), which is hence a closed subspace of $\widehat{T}_{\Sigma_{\wp}}$ (resp. $\mathcal W_{1,\Sigma_{\wp}}$) (e.g. see \cite[\S 5.1.4]{Ding}).
The character $\chi$ induces a closed embedding
\begin{equation*}
\chi: \mathcal W_{1,\tau}\lhook\joinrel\longrightarrow \mathcal W_{1,\Sigma_{\wp}}, \ \chi'\mapsto \chi|_{Z_{1}'} \chi'.
\end{equation*}
Recall that we have a natural morphism $\kappa: \mathcal V(K^p,w)_{\overline{\rho}} \rightarrow \mathcal W_{1,\Sigma_{\wp}}$ which is \'etale at $z$ (cf. Thm.\ref{thm: clin-elt}). Put $\mathcal V(K^p,w)_{\overline{\rho},\tau}:=\mathcal V(K^p,w)_{\overline{\rho}}\times_{\mathcal W_{1,\Sigma_{\wp}},\chi} \mathcal W_{1,\tau}$, one has thus a Cartesian diagram
\begin{equation*}
\begin{CD}
\mathcal V(K^p,w)_{\overline{\rho},\tau} @>>>\mathcal W_{1,\tau} \\
@VVV @V \chi VV\\
\mathcal V(K^p,w)_{\overline{\rho}} @>>> \mathcal W_{1,\Sigma_{\wp}}
\end{CD}
\end{equation*}
Denote still by $z$ the preimage of $z$ in $\mathcal V(K^p,w)_{\overline{\rho},\tau}$, $\kappa$ the natural morphism $\mathcal V(K^p,w)_{\overline{\rho},\tau} \rightarrow \mathcal W_{1,\tau}$, thus $\kappa$ is \'etale at $z$. By results in \S \ref{sec: clin-3.4.3}, one can choose an open affinoid $V$ of $\mathcal V(K^p,w)$ containing $z$ such that $V$ is \'etale over $\mathcal W_{1,\tau}$, and that any point in $V$ does not have companion points. Denote by $V_{\tau}$ the preimage of $V$ in $\mathcal V(K^p,w)_{\overline{\rho},\tau}$. We see $V_{\tau}$ is in fact a smooth curve. By Prop.\ref{prop: clin-kzc} and shrinking $V$ (and hence $V_{\tau}$) if necessary, one gets a continuous representation
$$\rho_{V_{\tau}}: \Gal(\overline{F}/F) \rightarrow \GL_2(\mathcal O(V))\rightarrow \GL_2(\mathcal O(V_{\tau})).$$
Denote by $\chi_{V_{\tau}}=\chi_{V_{\tau},1}\otimes \chi_{V_{\tau},2}: T(F_{\wp})\rightarrow \mathcal O(V_{\tau})^{\times}$ the character induced by the natural morphism $V_{\tau} \rightarrow \mathcal V(K^p,w)_{\overline{\rho},\tau} \rightarrow \mathcal V(K^p,w)_{\overline{\rho}}\rightarrow \widehat{T}_{\Sigma_{\wp}}$. By \cite[Thm.6.3.9]{KPX} applied to the smooth affinoid curve $V_{\tau}$, together with the fact that any point in $V_{\tau}$ does not have companion points, one has
\begin{lemma}\label{lem: clin-foia}
There exists an exact sequence of $(\varphi,\Gamma)$-modules over $\mathcal R_{\mathcal O(V_{\tau})}:=B_{\rig,F_{\wp}}^{\dagger} \widehat{\otimes}_{\mathbb Q_p} \mathcal O(V_{\tau})$ \big(see for example \cite[Thm.2.2.17]{KPX} for $D_{\rig}(\rho_{V_{\tau},\wp})$, and \cite[Const.6.2.4]{KPX} for $(\varphi,\Gamma)$-modules of rank $1$ associate to continuous character of $F_{\wp}^{\times}$ with values in $\mathcal O(V_{\tau})^{\times}$\big):
\begin{equation}\label{equ: clin-grqp}
0 \rightarrow \mathcal R_{\mathcal O(V_{\tau})}\big(\unr(q)\chi_{V_{\tau},1}\big) \rightarrow D_{\rig}(\rho_{V_{\tau},\wp}) \rightarrow \mathcal R_{\mathcal O(V_{\tau})}\big(\chi_{V_{\tau},2}\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-1}\big) \rightarrow 0.
\end{equation}
\end{lemma}
Let $t_{\tau}: \Spec E[\epsilon]/\epsilon^2 \rightarrow \mathcal W_{1,\tau}$ be a \emph{non-zero} element in the tangent space of $\mathcal W_{1,\tau}$ at the identity point (corresponding to the trivial character), since $V_{\tau}$ is \'etale over $\mathcal W_{1,\tau}$, $t_{\tau}$ gives rise to a non-zero element, still denoted by $t_{\tau}$, in the tangent space of $\mathcal V(K^p,w)_{\overline{\rho},\tau}$ at the point $z$. The following composition
\begin{equation}\label{equ: clin-amgi}
t_{\tau}: \Spec E[\epsilon]/\epsilon^2 \longrightarrow \mathcal V(K^p,w)_{\overline{\rho},\tau} \longrightarrow \mathcal V(K^p,w)_{\overline{\rho}} \longrightarrow \widehat{T}_{\Sigma_{\wp}}
\end{equation}
gives rise to a character $\widetilde{\chi}_{\tau}: T(F_{\wp}) \rightarrow (E[\epsilon]/\epsilon^2)^{\times}$. We have in fact $\widetilde{\chi}_{\tau}=t_\tau\circ \chi_{V_{\tau}}$ \big($t_\tau: \mathcal O(V_{\tau}) \rightarrow E[\epsilon]/\epsilon^2$\big). We know $\widetilde{\chi}_{\tau}\equiv \chi \pmod{\epsilon}$. Since the image of (\ref{equ: clin-amgi}) lies in $\widehat{T}_{\Sigma_{\wp}}(w)$ (cf. (\ref{equ: clin-tapwe})), we see $\widetilde{\chi}_{\tau}|_{Z_1}=\mathscr N^{-w}$ and thus $(\widetilde{\chi}_{\tau}\chi^{-1})|_{Z_1}=1$.
\begin{lemma}
There exist $\gamma$, $\eta\in E$, $\mu\in E^{\times}$ such that $$\psi_{\tau}:=\widetilde{\chi}_{\tau}\chi^{-1}=\unr(1+\gamma\epsilon) (1-\mu\epsilon \log_{\tau,0,\varpi}) \otimes \unr(1+\eta \epsilon) (1+\mu\epsilon\log_{\tau,0,\varpi}).$$
\end{lemma}
\begin{proof}
The lemma is straightforward. Note that $\mu\neq 0$ since $t_{\tau}$ (as an element in the tangent space) is non-zero.
\end{proof}
By multiplying $\epsilon$ by constants, we assume $\mu=1$ and thus $$\psi_{\tau}=\unr(1+\gamma\epsilon) (1-\epsilon\log_{\tau,0,\varpi})\otimes \unr(1+\eta\epsilon) (1+\epsilon\log_{\tau,0,\varpi}).$$
The following lemma, which describes the character $\widetilde{\chi}_{\tau}$ in terms of the $\mathcal L$-invariants, is one of the key points in the proof of Prop.\ref{prop: clin-awdl}.
\begin{lemma}\label{thm: clin-aue}$(\eta-\gamma)/2=-e^{-1}\big(\mathcal L_{\tau}+\log_{\tau}\big(\frac{p}{\varpi^{e}}\big)\big)=(-\mathcal L_{\tau})(\varpi)$ (cf. \S \ref{sec: clin-1}).
\end{lemma}
\begin{proof}
Denote by $\widetilde{\rho}_{z,\wp}:=t_{\tau}\circ \rho_{V_{\tau},\wp}:\Gal(\overline{\mathbb Q_p}/F_{\wp}) \rightarrow \GL_2(\mathcal O(V_{\tau}))\rightarrow \GL_2(E[\epsilon]/\epsilon)$, from (\ref{equ: clin-grqp}), one gets an exact sequence of $(\varphi,\Gamma)$-modules over $\mathcal R_{E[\epsilon]/\epsilon^2}$:
\begin{equation}\label{equ: clin-2no}
0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} \big(\unr(q)\widetilde{\chi}_{\tau,1} \big) \rightarrow D_{\rig}(\widetilde{\rho}_{z,\wp}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} \Big(\widetilde{\chi}_{\tau,2} \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}\Big) \rightarrow 0.
\end{equation}
For $\sigma\in \Sigma_{\wp}$, denote by $\varepsilon_{\sigma,\varpi}$ the character of $F_{\wp}^{\times}$ with $\varepsilon_{\sigma,\varpi}|_{\mathcal O_{\wp}^{\times}}=\sigma|_{\mathcal O_{\wp}^{\times}}$ and $\varepsilon_{\sigma,\varpi}(\varpi)=1$. Recall $\chi=\unr(q^{-1}\alpha)\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-\frac{w-k_{\sigma}+2}{2}} \otimes \unr(q\alpha) \prod_{\sigma\in \Sigma_{\wp}}\sigma^{-\frac{w+k_{\sigma}-2}{2}}$. We have thus
\begin{multline*}
\delta_1:=\unr(q)\widetilde{\chi}_{\tau,1}=\unr\big(\alpha(1+\gamma\epsilon)\big)(1-\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}}\sigma^{-{\frac{w-k_{\sigma}+2}{2}}}\\ =\unr\Big(\alpha(1+\gamma\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}\Big) (1-\epsilon\log_{\tau,0,\varpi}) \prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{-\frac{w-k_{\sigma}+2}{2}},
\end{multline*}
\begin{equation*}
\delta_2:=\widetilde{\chi}_{\tau,2} \prod_{\sigma\in \Sigma_{\wp}} \sigma^{-1}=\unr\Big(q\alpha(1+\eta\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w+k_{\sigma}-1}{2}}\Big) (1+\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}} \varepsilon_{\sigma,\varpi}^{-\frac{w+k_{\sigma}}{2}}.
\end{equation*}
Let $\chi_0:=(1-\epsilon\log_{\tau,0,\varpi}) \prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{-\frac{w-k_{\sigma}+2}{2}}$, one can view $\chi_0$ as a character of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ over $E[\epsilon]/[\epsilon^2]$ via the local Artin map $\Art_{F_{\wp}}$. Denote by $\widetilde{\rho}:=\widetilde{\rho}_{z,\wp}\otimes_{E[\epsilon]/\epsilon^2}\chi_0^{-1}$, \begin{eqnarray*}\delta_1'&:=&\delta_1\chi_0^{-1}=\unr\Big(\alpha(1+\gamma\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}\Big) \\
\delta_2'&:=& \delta_2\chi_0^{-1}=\unr\Big(q\alpha(1+\eta\epsilon)\prod_{\sigma\in \Sigma_{\wp}}\sigma(\varpi)^{-\frac{w+k_{\sigma}-1}{2}}\Big) (1+2\epsilon\log_{\tau,0,\varpi})\prod_{\sigma\in \Sigma_{\wp}}\varepsilon_{\sigma,\varpi}^{1-k_{\sigma}}
\end{eqnarray*}
Thus one has
\begin{equation}\label{equ: linv-1drgd}
0 \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2} (\delta_1') \rightarrow D_{\rig}(\widetilde{\rho}) \rightarrow \mathcal R_{E[\epsilon]/\epsilon^2}(\delta_2')\rightarrow 0.
\end{equation}
Denote by $\widetilde{\alpha}:=\alpha(1+\gamma \epsilon)\prod_{\sigma\in \Sigma_{\wp}} \sigma(\varpi)^{-\frac{w-k_{\sigma}+2}{2}}$, by (\ref{equ: linv-1drgd}), $(B_{\cris}\otimes_{\mathbb Q_p} \widetilde{\rho})^{\Gal(\overline{\mathbb Q_p}/F_{\wp}), \varphi^{d_0}=\widetilde{\alpha}}$ is free of rank $1$ over $F_{\wp,0}\otimes_{\mathbb Q_p}E[\epsilon]/\epsilon^2$, and that the $E$-representation $\overline{\widetilde{\rho}} \pmod{\epsilon}$ of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ is semi-stable non-crystalline of Hodge-Tate weights $(1-k_{\sigma},0)_{\sigma\in \Sigma_{\wp}}$, and has the same $\mathcal L$-invariants as $\rho_{\wp}$. By applying the formula in \cite[Thm.1.1]{Zhang}, one gets
\begin{equation*}
\frac{\gamma}{d_0}+\big(-\frac{\gamma+\eta}{2d_0}\big)-\frac{1}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big)-\frac{1}{d} \mathcal L_{\tau}=0.
\end{equation*}
In fact, with the notation of \emph{loc. cit.}, one has
\begin{equation*}
\log_{\tau,0,\varpi}=-\frac{1}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big) \psi_1 +1_{\tau}\psi_2
\end{equation*}
where $1_{\tau}\in F_{\wp}\otimes_{\mathbb Q_p} E\mathcal Ong \prod_{\sigma\in \Sigma_{\wp}} E$ such that $(1_{\tau})_{\sigma}=0$ if $\sigma\neq \tau$ and $(1_{\tau})_{\tau}=1$, and where we view $\log_{\tau,0,\varpi}$ as an additive character of $\Gal(\overline{\mathbb Q_p}/F_{\wp})$ via the local Artin map $\Art_{F_{\wp}}$. Thus one can apply the formula in \cite[Thm.1.1]{Zhang} to
\begin{equation*}\{V, \alpha,\delta,\kappa\}=\Big\{\widetilde{\rho}, \widetilde{\alpha},\Big(-\frac{\gamma+\eta}{d_0}-\frac{2}{d} \log_{\tau}\big(\frac{p}{\varpi^{e}}\big) \Big) \epsilon,2_{\tau} \epsilon\Big\}.
\end{equation*}
The lemma follows.
\end{proof}
The following lemma follows directly from Lem.\ref{thm: clin-aue}.
\begin{lemma}
As representations of $T(F_{\wp})$ (of dimension $2$) over $E$, one has
\begin{equation*}
\widetilde{\chi}_{\tau}\delta^{-1} \mathcal Ong \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha) \otimes_E \psi(\mathcal L_{\tau})',
\end{equation*}
where $\psi(\mathcal L_{\tau})'\begin{pmatrix}
a & 0 \\ 0 & d
\end{pmatrix}=\begin{pmatrix}
1 & \log_{\tau,-\mathcal L_{\tau}}(a d^{-1}) +\frac{\gamma+\eta}{2}\upsilon_{\wp}(ad) \\ 0 & 1
\end{pmatrix}$.
\end{lemma}
The parabolic induction $\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an}$ lies thus in an exact sequence as follows
\begin{multline}\label{equ: clin-pw2p}
0 \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow \Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\Big)^{\mathbb Q_p-\an}\\ \xlongrightarrow{s_{\tau}'}\Big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi(\underline{k}_{\Sigma_{\wp}},w;\alpha)\Big)^{\mathbb Q_p-\an} \longrightarrow 0.
\end{multline}
By Lem.\ref{lem: linv-nmfyc}, one has
\begin{lemma}\label{lem: clin-pcssf}
One has an isomorphism of locally $\mathbb Q_p$-analytic representations of $\GL_2(F_{\wp})$:
\begin{equation*}
(s_{\tau}')^{-1}\big(V\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)\big)/V\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \mathcal Ong \Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big).
\end{equation*}
\end{lemma}
Consider the composition $t_{\tau}: \Spec E[\epsilon]/\epsilon^2\xrightarrow{t_{\tau}} \mathcal V(K^p,w)_{\overline{\rho},\tau}\hookrightarrow \mathcal V(K^p,w)_{\overline{\rho}}$, and $(t_{\tau}^* \mathcal M)^{\vee}$ being a finite dimensional $E$-vector space equipped with a natural action of $T(F_{\wp})\times \mathcal H^p$. We claim there exists $n\in \mathbb Z_{\geq 1}$ such that \big(as $T(F_{\wp})$-representations\big)\begin{equation}\label{equ: clin-mcee}(t_{\tau}^*\mathcal M)^{\vee}\mathcal Ong \widetilde{\chi}_{\tau}^{\oplus n}.\end{equation}In fact, as in \S \ref{sec: clin-3.4.3}, there exists open affinoids $V'$ of $\mathcal V(K^p,w)_{\overline{\rho}}$ and $U$ of $\mathcal W_{1,\Sigma_{\wp}}$ such that $V'$ lies over $U$, $\mathcal O(V')\mathcal Ong \mathcal O(U)$, and that $\mathcal M(V')$ is a locally free $\mathcal O(U)$-module. The group $T(F_{\wp})$ acts on $\mathcal M(V')$ via the character $T(F_{\wp}) \rightarrow \mathcal O(V')^{\times} \mathcal Ong \mathcal O(U)^{\times}$ \big(with the first map induced by the natural morphism $V'\rightarrow \widehat{T}_{\Sigma_{\wp}}$\big), the claim follows. We also see that $\mathcal H^p$ acts on $\mathcal M(V')$ via the natural morphism $\mathcal H^p \rightarrow \mathcal O(V') \mathcal Ong \mathcal O(U)$. Thus $\mathcal H^p$ acts on $(t_{\tau}^*\mathcal M)^{\vee}$ via $\mathcal H^p \rightarrow \mathcal O(V') \rightarrow E[\epsilon]/\epsilon^2$, in particular $(t_{\tau}^*\mathcal M)^{\vee}$ is a generalized $\lambda_{\rho}$-eigenspace for $\mathcal H^p$ (one can view $t_{\tau}$ as a thickening of the point $z$).
Since $\mathcal U$ is strictly quasi-Stein, the restriction map $\mathcal M(\mathcal U)\rightarrow t_{\tau}^*\mathcal M$ is surjective, so we have injections
\begin{equation*}
(z^*\mathcal M)^{\vee} \lhook\joinrel\longrightarrow (t_{\tau}^*\mathcal M)^{\vee} \lhook\joinrel\longrightarrow \mathcal M(\mathcal U)^{\vee}_b.
\end{equation*}Firstly note that a non-zero map $f$ in the right term of (\ref{equ: clin-gpwp}) corresponds to a non-zero vector $v\in (z^*\mathcal M)^{\vee}$ in a natural way:
\begin{multline}\label{equ: clin-pjpk}
(z^* \mathcal M)^{\vee} \xlongrightarrow{\sim} J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}
\\ \xlongrightarrow{\sim} J_B\big(H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big) \otimes_E W(\underline{k}_{\Sigma_{\wp}},w)^{\vee}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}
\\ \xlongrightarrow{\sim} J_B\big(H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)\big)^{T(F_{\wp})=\psi, \mathcal H^p=\lambda_{\rho}} \otimes_E \chi(\underline{k}_{\Sigma_{\wp}},w)
\\ \xlongrightarrow{\sim} \Hom_{\GL_2(F_{\wp})}\Big(\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \psi\delta^{-1}\big)^{\infty}, H_{\et}^1\big(K^p, W(\underline{k}_{\Sigma_{\wp}},w)\big)^{\mathcal H^p=\lambda_{\rho}}\Big)
\\ \xlongrightarrow{\sim} \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}\Big),
\end{multline}where the first isomorphism follows from Thm.\ref{thm: clin-cjw}, the second from the fact that any vector in the second term is classical (see also Cor.\ref{prop: clin-ernr}), $\psi:=\chi \chi(\underline{k}_{\Sigma_{\wp}},w)^{-1}$ (cf. Cor.\ref{prop: clin-ernr}), the fourth from the adjunction formula for the classical Jacquet functor, and the last isomorphism follows from (\ref{equ: clin-apwgi}) and (\ref{equ: clin-wrs}) (and \cite[Cor.5.1.6]{Ding}).
By the isomorphism (\ref{equ: clin-mcee}), there exists $\widetilde{v}\in (t_{\tau}^*\mathcal M)^{\vee}$ such that $(E[\epsilon]/\epsilon^2)\cdot \widetilde{v} \mathcal Ong \widetilde{\chi}$ and that $v\in \big(E[\epsilon]/\epsilon^2\big)\cdot \widetilde{v}$. By multiplying $\widetilde{t}$ by scalars in $E$, one can assume $v=\epsilon\widetilde{v}$. The $T(F_{\wp})$-invariant map, by mapping a basis to $\widetilde{v}$,
\begin{equation*}
\widetilde{\chi}\lhook\joinrel\longrightarrow \mathcal M(\mathcal U)^{\vee}_b[\mathcal H^p=\lambda_{\rho}]
\end{equation*}
induces a $\GL_2(F_{\wp})$-invariant map denoted by $\widetilde{v}$
\begin{multline}\label{equ: clin-pqgi}
\widetilde{v}: \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an} \lhook\joinrel\longrightarrow \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \mathcal M(\mathcal U)^{\vee}_b[\delta^{-1}]\big)^{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\\ \xlongrightarrow{\text{(\ref{equ: clin-p2w})}} \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}].
\end{multline}
Similarly, the $T(F_{\wp})$-invariant map $\chi\hookrightarrow (\mathcal M(\mathcal U)^{\vee}_b)^{\mathcal H^p=\lambda_{\rho}}$, by mapping a basis to $v$, induces a $\GL_2(F_{\wp})$-invariant map, denoted by $v$
\begin{equation*}
v: \big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})}\chi\delta^{-1}\big)^{\mathbb Q_p-\an} \longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}, \mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}.
\end{equation*}
It's straightforward to see that the following diagram commutes
\begin{equation}\label{equ: clin-pagip}
\begin{CD}
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \chi\delta^{-1}\big)^{\mathbb Q_p-\an} @> v >> \widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w},\mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an} \\
@VVV @VVV \\
\big(\Ind_{\overline{B}(F_{\wp})}^{\GL_2(F_{\wp})} \widetilde{\chi}_{\tau}\delta^{-1}\big)^{\mathbb Q_p-\an} @> \widetilde{v} >>\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]
\end{CD}
\end{equation}
where the left arrow is induced by $\chi\hookrightarrow \widetilde{\chi}$, and the right arrow is the natural injection.
By the same argument as in the proof of Lem.\ref{lem: clin-pvle}, one can prove the map $v$ factors through an injection
\begin{equation*}
v:\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big) \lhook\joinrel\longrightarrow \widetilde{H}^1_{\et}(K^p,E\big)^{Z_1=\mathscr N^{-w}, \mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}.
\end{equation*}
Moreover, we claim the restriction $f_v:=v|_{\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)}$ is equal to $f$. In fact, by taking Jacquet-Emerton functor, one sees both the maps $f$ and $f_v$ give rise to the same eigenvector $$v\in J_B\big(\widetilde{H}_{\et}^1(K^p,E)_{\mathbb Q_p-\an}^{Z_1=\mathscr N^{-w}}\big)^{T(F_{\wp})=\chi, \mathcal H^p=\lambda_{\rho}}\mathcal Ong (z^*\mathcal M)^{\vee},$$ from which the claim follows.
By the commutative diagram (\ref{equ: clin-pagip}) and Lem.\ref{lem: clin-pcssf}, we see $\widetilde{v}$ induces a continuous $\GL_2(F_{\wp})$-invariant injection
\begin{equation}\label{equ: clin-walge}
\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big) \lhook\joinrel\longrightarrow\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w}}_{\overline{\rho}, \mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}],
\end{equation}
whose restriction to $\St(\underline{k}_{\Sigma_{\wp}},w;\alpha)$ equals to $f$ (by the above discussion and commutative diagram (\ref{equ: clin-pagip})\big). It's sufficient to prove the map (\ref{equ: clin-walge}) factors through $\widetilde{H}^1_{\et}(K^p,E)^{Z_1=\mathscr N^{-w},\mathcal H^p=\lambda_{\rho}}_{\overline{\rho}, \mathbb Q_p-\an}$.
By the same argument as in the proof of Lem.\ref{lem: clin-pvle}, one can prove the following restriction map is injective:
\begin{multline*}
\Hom_{\GL_2(F_{\wp})}\Big(\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big),\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big) \\
\longrightarrow \Hom_{\GL_2(F_{\wp})}\Big(\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big), \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big).
\end{multline*}
For any $X\in \mathcal H^p$, we know the restriction of the map $(X-\lambda_{\rho}(X))\widetilde{v}$ to $\St\big(\underline{k}_{\Sigma_{\wp}},w;\alpha\big)$ is zero (since the image $f$ lies in the $\lambda_{\rho}$-eigenspace), hence $(X-\lambda_{\rho}(X))\widetilde{v}=0$ \Big(here $X\widetilde{v}$ signifies the composition
$\Sigma\big(\underline{k}_{\Sigma_{\wp}},w;\alpha;\mathcal L_{\tau}\big) \xrightarrow{\text{(\ref{equ: clin-walge})}}\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}] \xrightarrow{X}\widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}[\mathcal H^p=\lambda_{\rho}]\Big)$, in other words, $\Ima(\widetilde{v})\in \widetilde{H}^1_{\et}(K^p,E)_{\mathbb Q_p-\an}^{\mathcal H^p=\lambda_{\rho}}$. This concludes the proof of Prop.\ref{prop: clin-awdl}.
\end{document} |
\begin{document}
\begin{abstract}
Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic, concave equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric which accounts for a critical non-local drift. We prove a $C^{\s+\a}$ estimate in the spatial variable and a $C^{1,\a/\s}$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.
\end{abstract}
\title{$C^{\s+\a}
\section{Introduction}
In this work we are interested in studying regularity of solutions of
\begin{align}\label{eqintro}
u_t- \inf_{L \in \cL} Lu=0 \text{ in $B_1\times(-1,0]$},
\end{align}
where,
\begin{align*}
Lu(x) &:= (2-\s)\int \d u(x;y)\frac{K(y)}{|y|^{n+\s}}dy + b\cdot Du(x),\\
\d u(x;y) &:= u(x+y) - u(x) - Du(x)\cdot y\chi_{B_1}(y).
\end{align*}
The kernel $(2-\s)\frac{K(y)}{|y|^{n+\s}}$ is comparable to the fractional laplacian of order $\s \in [1,2)$ and it is non necessarily symmetric. As it was discussed in a previous paper \cite{CD3}, the odd part of the kernel brings drift terms after rescaling the equation. That is the reason why the drift term $b\cdot Du$ is included above. Contrasting to the second order case, where the lower order drift might be absorbed by estimates proven for pure second order equation at sufficiently small scales, in our case the drift remains comparable to the diffusion as the scales go to zero, making it critical.
This type of equations appear naturally when studying stochastic control problems (see \cite{Soner}), ergodic control problem (see \cite{MR}) and economic applications (see \cite{DP}), in which the the random part is given by a purely jump process, which is most of the time non necessarily symetric. The particular concave case can be seen as a one-player stochastic game, which at each step he can choose a strategy to minimize the expected value of some fixed function evaluated at the first exit point of a given domain.
In the local case ($\s=2$) this problem was first studied independently by L. Evans and N. Krylov (see \cite{Evans}, \cite{Krylov} and also the recent proof by L. Caffarelli and L. Silvestre in \cite{C4}). They obtain $C^{2,\a}$ a priori estimates and therefore the existence of classical solutions by the continuity method.
L. Caffarelli and L. Silvestre proved in \cite{C3} that solutions of the elliptic problem $Iu=0$, where $I$ is a concave operator with smooth kernels, are $C^{\s+\a}$. It relies on the theory of viscosity solutions developed in \cite{C1} and \cite{C2}. The regularity obtained is enough to evaluate the operator in the classical sense. Moreover, the estimates obtained are independent of the order of the equation and extends the theory to the classical case.
A recent improvement of the previous work, done by J. Serra \cite{Serra14-2}, allowed to remove the smoothness condition for symmetric kernels in order to prove $C^{\s+\a}$ estimates. It proceeds by a compactness argument that blows-up the solution, reducing the problem to a Lioville type of result.
Regularity for parabolic nonlocal equations has been studied by the authors in \cite{CD}, \cite{CD2} and \cite{CD3} in which H\"older estimates are proven for general equations like \eqref{eqintro} with a non zero right hand side. Recent advances include the work of J. Serra \cite{Serra14} for $C^{1,\a}$ estimates with rough kernels; and the work of T. Jin, and J. Xiong, \cite{Jin14} for higher order, optimal Schauder estimates in the linear case.
We extend the ideas of \cite{C3} to the parabolic nonlocal case to prove the desired $C^{\s+\a}$ interior regularity. The order $\s$ is assumed at least one in order for the drift to be at most comparable with the diffusion. On the other hand, for $\s\in(0,1]$, the $C^{1,\a}$ estimates established in \cite{CD3} already give classical solutions. In contrast with the classical theory, where drift terms might be absorbed by estimates proven for pure second order equations at sufficiently small scales, our operator keeps the drift comparable to the diffusion as the scale go to zero providing new challenges.
We assume the boundary data to be at least differentiable in time. This is way to ensure that the solution is $C^{1,\a}$ in time in the interior. Keep in mind that for general boundary data one cannot expect $C^{1,\a}$ regularity in time, even for the fractional heat equation, an example is discussed in \cite{CD}. Whenever a weaker condition on the boundary data implies $C^{1,\a}$ regularity in time in the interior remains an open question.
Here is our main Theorem.
\begin{theorem}\label{thmintro}
Let $\s \in [1,2)$, $\cL \ss \cL_2^\s(\l,\L,\b)$ (sufficently smooth kernels to be defined) and $u$ satisfies in the viscosity sense,
\begin{align*}
u_t - \inf_{L \in \cL}Lu = 0 \text{ in $B_1\times(-1,0]$}.
\end{align*}
Then there is some $\a \in (0,1)$ and $C>0$, depending only on $n, \l, \L$ and $\b$ such that,
\begin{align*}
\|u\|_{C^{\s+\a}(B_{1/2}\times(-1,0])} \leq C(\|u\|_{L^1((-1,0]\mapsto L^1(\w_\s))}+[u\chi_{B_1^c}]_{C^{0,1}((-1,0]\mapsto L^1(\w_\s))})).
\end{align*}
\end{theorem}
The paper is divided as follows. In Section \ref{VSP} we introduce the family of operators we are considering, the notion of viscosity solution and recall some properties. We also state some previous results that we need for the rest of this work.
We use the concavity of the non-linearity in Section \ref{EQLU} to determine an equation for the average of a given solution, in particular we get an equation for the fractional laplacian. In Section \ref{EstimateforLap} we use the previous equation to obtain a weak $C^\s$ estimate on the laplacian of the solution. Finally in Section \ref{Further} we prove a diminish of oscillation lemma for the fractional laplacian which implies our main theorem.
\section{Preliminaries and Viscosity Solutions}\label{VSP}
The cylinder of radius $r$, height $\t$ and center $(x,t)$ in $\R^n\times\R$ is denoted by $C_{r,\t}(x,t) := B_r(x)\times(t-\t,t]$. Whenever we omit the center we are assuming that they get centered at the origin in space and time.
Given the scaling properties of linear operators with non symmetric kernels discussed in \cite{Chang12} and \cite{CD3}, it is reasonable enlarge the family of linear operators to include (classical) drift terms. In this sense lets introduce the following notation where the time variable has been omitted as it is irrelevant for the computation:
\begin{align*}
L_{K,b}^\s u(x) &:= (2-\s)\int\d u(x;y)\frac{K(y)}{|y|^{n+\s}}dy + b\cdot Du(x),\\
\d u(x;y) &:= u(x+y) - u(x) - Du(x)\cdot y\chi_{B_1}(y).
\end{align*}
Initially we may consider kernels bounded from above and away from zero:
\begin{align*}
0 < \l \leq K \leq \L < \8.
\end{align*}
The drift comes not only from the term $b\cdot D$ but also from the odd part of the kernel after rescaling. We assume that they are controlled in the following way:
\begin{align*}
\sup_{r \in (0,1)} \left|b+(2-\s)\int_{B_1\sm B_r} \frac{yK(y)}{|y|^{n+\s}}dy\right| \leq \b.
\end{align*}
We denote by $\cL_0^\s(\l,\L,\b)$ the family of all linear operators with the two conditions given above and suppress some its parameters in the notation to follow whenever they are clear from the context, usually we write just $\cL_0$. Sufficient regularity/integrability to evaluate $L_{K,b}^\s u(x)$ is $u \in C^{1,1}(x) \cap L^1(\w_\s)$ where $\w_\s(y) = \min(1,|y|^{-(n+\s)})$.
More regular kernels can be considered in order to prove higher regularity estimates. This corresponds to the initial approach taken in \cite{C1}, \cite{C2} and \cite{C3} in order to use integration by parts techniques to control rough oscillations of the boundary data of the solution. This work follows uses the same technique for which we define the family $\cL_1^\s(\l,\L,\b) \ss \cL_0$ such that for each kernel,
\begin{align*}
|DK(y)| \leq \L|y|^{-1}.
\end{align*}
Moreover, let $\cL_2^\s(\l,\L,\b) \ss \cL_1$ such that for each kernel,
\begin{align*}
|D^2K(y)| \leq \L|y|^{-2}.
\end{align*}
Lets remind that the smoothness hypothesis of the previous works have been lifted in \cite{Serra14} and \cite{Serra14-2} for symmetric kernels. Their techniques applies also if drifts or lower order terms are included because of scaling considerations. In the present case however, an odd kernel renews the drift a may keep it comparable to the diffusion even as the scales go to zero, so our result is not clearly contained in such work.
Given $\cL\in\cL_0$, a non linearity $I$ is given by a function $I:\W\times(t_1,t_2]\times\R^\cL\to \R$ such that,
\begin{align*}
Iu(x,t) := I(x,t,(Lu(x,t))_{L\in\cL}).
\end{align*}
$I$ is considered to be elliptic if it is increasing in $\R^\cL$.
The nonlinearity in our main Theorem is constructed from $\cL \ss \cL_2$ such that,
\begin{align*}
Iu = \cM^-_\cL u := \inf_{L \in \cL} Lu.
\end{align*}
It satisfies the following uniform ellipticity relation with the extremal ope-rators,
\begin{align*}
\cM^-_\cL (u-v) &\leq Iu - Iv \leq \cM^+_\cL(u-v).
\end{align*}
where $\cM^+_{\cL}:= \sup_{L\in\cL}L$.
\subsection{Viscosity solutions}
We recall some definitions pertaining to viscosity solutions $u$ for the equation $u_t-Iu=f$. A test function $\varphi$ needs to be sufficiently smooth/integrable about the contact point where the equation is tested. Moreover, qualitative properties as the continuity of $Iu$, require for the tail of $u$ to be at least continuous in time in the following integrable sense.
\begin{definition}
The space $C((t_1,t_2] \mapsto L^1(\w_\s))$ consists of all measurable functions $u:\R^n\times(t_1,t_2] \to \R$ such that for every $t \in (t_1,t_2]$,
\begin{enumerate}
\item $\|u(\cdot,t)^-\|_{L^1(\w_\s)} < \8.$
\item $\lim_{\t\nearrow0}\|u(\cdot,t) - u(\cdot,t-\t)\|_{L^1(\w_\s)} = 0$.
\end{enumerate}
\end{definition}
\begin{definition}[Test functions]\label{def:test_function}
A test function is defined as a pair $(\varphi, C_{r,\t}(x,t))$, such that $\varphi \in C^{1,1}_xC^1_t(C_{r,\t}(x,t)) \cap C((t-\t,t]\mapsto L^1(\w_\s))$.
\end{definition}
Whenever the cylinder in the Definition \ref{def:test_function} becomes irrelevant we will refer to the test function $(\varphi, C_{r,\t}(x,t))$ just by $\varphi$.
\begin{definition}
Given a function $u$ and a test function $\varphi$, we say that $\varphi$ touches $u$ from below at $(x,t)$ if,
\begin{enumerate}
\item $\varphi(x,t)=u(x,t)$,
\item $\varphi(y,s) \leq u(y,s)$ for $(y,s)\in \R^n\times(t-\t,t]$.
\end{enumerate}
\end{definition}
A similar definition for contact from above will be considered too.
\begin{definition}[Viscosity (super) solutions]\label{viscosity}
Given an elliptic operator $I$ and a function $f$, a function $u \in C(\W\times(t_1,t_2]) \cap C((t_1,t_2]\mapsto L^1(\w_\s))$ is said to be a viscosity super solution to $u_t - Iu \geq f$ in $\W\times(t_1,t_2]$, if for every lower continuous test function $(\varphi,C_{r,\t}(x,t))$ touching $u$ from below at $(x,t) \in \W\times(t_1,t_2]$, we have that $\varphi_{t^-}(x,t) - I\varphi(x,t) \geq f(x,t)$.
\end{definition}
Recall that $\varphi_{t^-}$ denotes the left time derivative of $\varphi$ natural for time evolution problems.
The definition of $u$ being a viscosity \textit{sub} solution to $u_t - Iu \leq f$ in $\W\times(t_1,t_2]$ is done similarly to the definition of super solution replacing contact from below by contact from above and reversing the last inequality. A viscosity solution to $u_t - Iu = f$ in $\W\times(t_1,t_2]$ is a function which is a super and a sub solution simultaneously.
\subsection{Previous Results}
Several qualitative results for viscosity solutions of our parabolic equations such as the stability, a comparison principle and the existence of (viscosity) solutions have been stablished in \cite{CD}, \cite{CD2}, \cite{CD3}. We recall at this point some quantitative estimates for the solutions which will be used in this work.
\begin{theorem}[Point Estimate]\label{PE}
Let $\s \in [1,2)$. Suppose $u \geq 0$ satisfies
\begin{align*}
u_t - \cM^-_{\cL_0}u &\geq -f(t) \text{ in $C_{2r,2r^\s}(0,r^\s)$}.
\end{align*}
Then, for every $s \geq 0$,
\begin{align*}
\frac{|\{u > s\} \cap C_{r,r^\s}|}{|C_{r,r^\s}|} \leq C\1\inf_{C_{r,r^\s}(0,r^\s)}u + r^\s\fint_{-r^\s}^{r^\s}f^+(s)ds\2^{\e}s^{-\e},
\end{align*}
for some constants $\e$ and $C$ depending only on $n, \l, \L$ and $\b$.
\end{theorem}
The Oscillation Lemma provided in \cite{CD3} controls the point-wise size of a non negative sub solution in terms of an integral norm.
\begin{lemma}[Oscillation Lemma]\label{lem:oscillation}
Let $\cL\ss \cL_0$, $I:\W\times(t_1,t_2]\times\R^\cL\to\R$ uniformly elliptic and such that $I0=0$. Let $u$ satisfies,
\begin{align*}
u_t - I u &\leq f \text{ in $\W\times(t_1,t_2]$}.
\end{align*}
Then for every $\W'\times(t_1',t_2] \cc \W\times(t_1,t_2]$,
\begin{align*}
\sup_{\W'\times(t_1',t_2]} u^+ \leq C\1\|u^+\|_{L^1((t_1,t_2] \mapsto L^1(\w_\s))} + \|f^+\|_{L^1((t_1,t_2]\mapsto L^\8(\W))}\2,
\end{align*}
for some universal $C>0$, independent of $\s \in [1,2)$, depending on the domains.
\end{lemma}
\begin{theorem}[H\"older regularity]
Let $u$ satisfies
\begin{alignat*}{2}
u_t - \cM^+_{\cL_0}u &\leq f(t) &&\text{ in $C_{1,1}$},\\
u_t - \cM^-_{\cL_0}u &\geq -f(t) &&\text{ in $C_{1,1}$},
\end{alignat*}
Then there is some $\a \in (0,1)$ and $C>0$, depending only on $n$, $\l$, $\L$ and $\b$, such that for every $(y,s), (x,t) \in C_{1/2,1/2}$
\begin{align*}
\frac{|u(y,s) - u(x,t)|}{(|x-y| + |t-s|^{1/\s})^\a} \leq C\1 \|u\|_{L^1((-1,0]\mapsto L^1(\w_\s))} + \|f\|_{L^1(0,1)}\2.
\end{align*}
\end{theorem}
\begin{theorem}[Regularity for translation invariant operators]\label{C1aold}
Let $\cL \ss \cL_1$, $I:\R^\cL\to\R$ be uniformly elliptic, translation invariant and such that $I0=0$. Let $u$ satisfies
\begin{alignat*}{2}
u_t - Iu = f(t) \text{ in $C_1$}.
\end{alignat*}
Then there is some $\a \in (0,1)$ and $C>0$, depending only on $n$, $\l$, $\L$ and $\b$, such that for every $(y,s), (x,t) \in C_{1/2,1/2}$,
\begin{align*}
|Du(x,t)|+\frac{|Du(x,t)-Du(y,s)|}{(|x-y| + |t-s|^{1/\s})^\a} \leq C\1\|u\|_{L^1((-1,0]\mapsto L^1(\w_\s))} + \|f\|_{L^1(0,1)}\2.
\end{align*}
\end{theorem}
The previous Theorem does not give more regularity in time even if $I$ is translation invariant in time and $f \equiv 0$. In \cite{CD} the authors gave an example of a function, not better than Lipschitz in its time variable, solving the fractional heat equation. However, further regularity in time can be retrieved via the Oscillation Lemma if the Dirichlet data has a smoothness condition controlled by,
\begin{align*}
[u]_{C^{0,1}((t_1,t_2]\mapsto L^1(\w_\s))} &:= \sup_{(t-\t,t] \ss (t_1,t_2]} \frac{\|u(t)-u(t-\t)\|_{L^1(\w_\s)}}{\t}.
\end{align*}
\begin{theorem}[Further regularity in time]\label{furthertime}
Let $\cL \ss \cL_0$, $I:\R^\cL\to\R$ be uniformly elliptic, translation invariant such that $I0=0$. Let $u$ satisfies
\begin{align*}
u_t-Iu = 0 \text{ in $C_{1,1}$}.
\end{align*}
Then there is some $\a \in (0,1)$ and $C>0$, depending only on $n$, $\l$, $\L$ and $\b$, such that for every $(x,t), (y,s)\in C_{1/2,1/2}$ we have
\begin{align*}
|u_t(x,t)| + \frac{|u_t(x,t)-u_t(y,s)|}{(|x-y|+|t-s|^{1/\s})^\a}\leq C[u]_{C^{0,1}((-1,0]\mapsto L^1(\w_\s))}.
\end{align*}
\end{theorem}
\section{Equations for $Lu$ by concavity and translation invariance}\label{EQLU}
For this part we fix $\s\in[1,2)$, $\cL \ss \cL_2$, and $u$ such that,
\begin{align*}
&u_t - \cM^-_\cL u = 0 \text{ in $C_{8,3}$},\\
&\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))} \leq 1.
\end{align*}
We can assume that $u$ is a classical solution with smooth boundary and initial data. Otherwise, we approximate $u$ by a sequence of classical solutions with smooth boundary and initial data and recover the estimates of this chapter in the limit by the regularization procedure described in \cite{CD2}. The only thing we need to be careful about is that those a priori estimates are independent of (fractional) derivatives of $u$ which are not accessible by the viscosity solutions.
Many results in this and the following sections can be obtained by controlling $\|u\|_{L^1((-5,0] \mapsto L^1(\w_\s))}$, instead of the $L^\8$ norm; however, when it is coupled with the bound for $[u]_{C^{0,1}((-5,0] \mapsto L^1(\w_\s))}$ it actually implies $L^\8$ bound.
It is convenient for this section to introduce the following notation, given $K(y) \geq 0$ let
\begin{align*}
K^\s(y) := (2-\s)\frac{K(y)}{|y|^{n+\s}}.
\end{align*}
We denote the convolution by,
\begin{align*}
v \ast w(x) &:= \int w(x+y)v(y)dy.
\end{align*}
In particular, given that $K \geq 0$ goes to zero about the origin with at least a quadratic rate, then we can decompose a linear operator as:
\begin{align*}
L_{K,b}^\s u = \1K^\s \ast {} - \|K^\s\|_1 - \1\int_{B_1} yK^\s(y)dy - b\2\cdot D\2u.
\end{align*}
\begin{property}
Let $\a\in\R$, $b \in \R^n$ and $\eta \geq 0 \in L^1(\R^n)$. Then the following holds for any regular function $v$,
\begin{enumerate}
\item \textbf{Homogeneity:} $\cM^\pm_\cL(\a v) = \a\cM^\pm_\cL v$.
\item \textbf{Translation:} $\cM^-_\cL (b\cdot D v) \leq b\cdot D \cM^\pm_\cL v \leq \cM^+_\cL (b\cdot D v)$.
\item \textbf{Concavity:} $\eta\ast\cM^-_\cL v \leq \cM^\pm_\cL(\eta\ast v) \leq \eta\ast\cM^+_\cL v$.
\end{enumerate}
\end{property}
\begin{corollary}\label{cor:convolution}
For $K \geq 0$, $b \in \R^n$ and $\varphi\in C^\8_0(B_2\mapsto[0,1])$ such that $\varphi = 1$ in $B_1$, it holds that,
\begin{align*}
(L_{K,b}^\s u)_t - \cM^+_\cL(L_{K,b}^\s u) \leq ([(1-\varphi)K^\s]\ast u)_t - \cM^-_\cL ([(1-\varphi)K^\s]\ast u) \text{ in $C_{6,3}$}.
\end{align*}
In particular, if $\supp K \ss B_1$, then,
\begin{align*}
(L_{K,b}^\s u)_t - \cM^+_\cL(L_{K,b}^\s u) \leq 0 \text{ in $C_{6,3}$}.
\end{align*}
\end{corollary}
\begin{proof}
Let, for $\e \in (0,1)$, $K_\e := \chi_{B_\e^c}K$. We decompose the operator $L_{K_\e,b}^\s$ as a sum of a local and a non-local operator, the non-local being the one appearing on the right hand side of the conclusion of the lemma,
\begin{align*}
L_{K_\e,b}^\s &= L + NL,\\
&:= (L_{K_\e,b}^\s - K_\e^\s(1-\varphi)\ast) + K_\e^\s(1-\varphi)\ast,\\
&= \1\varphi K_\e^\s\ast{} - \|K_\e^\s\|_1 - \1\int_{B_1}yK_\e^\s(y)dy - b\2 \cdot D \2 + K_\e^\s(1-\varphi)\ast.
\end{align*}
Then,
\begin{align*}
(L_{K_\e,b}^\s u)_t - \cM^+_\cL(L_{K_\e,b}^\s u) &\leq \1(Lu)_t - \cM^+_\cL(Lu)\2 + \1(NLu)_t - \cM^-_\cL(NLu)\2,\\
&\leq L\1u_t - \cM^-_\cL u\2 + \1(NLu)_t - \cM^-_\cL(NLu)\2.
\end{align*}
In $C_{6,3}$ the first term is zero as the local operator $L$ does not take into account the values of $\1u_t - \cM^+_\cL u\2$ outside of $B_8$. The result now follows in the limit as $\e\searrow0$ by stability.
\end{proof}
\begin{property}[Integration by parts]
Let $K \geq 0$, $b \in \R^n$, $(\bar K(y), \bar b) := (K(-y), -b)$ and for $L = L_{K,b}^\s$, $\bar L = L_{\bar K,\bar b}^\s$. Then the following holds for any pair of regular/integrable functions $v$ and $w$,
\begin{align*}
\int vLw = \int w\bar L v.
\end{align*}
In particular,
\begin{align*}
L(v\ast w) = v\ast(Lw) = (\bar L v)\ast w.
\end{align*}
\end{property}
\begin{corollary}\label{cor:convolution2}
For $L_{K,b}^\s \in \cL_2$ it holds that,
\begin{align*}
(L_{K,b}^\s u)_t - \cM^+_\cL(L_{K,b}^\s u) &\leq C \text{ in $C_{6,3}$}.
\end{align*}
for some universal constant $C>0$.
\end{corollary}
\begin{proof}
Corollary \ref{cor:convolution} tells us that it suffices to estimate $([(1-\varphi)K^\s]\ast u)_t - \cM^-_\cL ([(1-\varphi)K^\s]\ast u)$ in $C_{6,3}$,
\begin{align*}
([(1-\varphi)K^\s]\ast u)_t &= [(1-\varphi)K^\s]\ast u_t,\\
&\leq C[u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))},\\
&=C,\\
\cM^-_\cL ([(1-\varphi)K^\s]\ast u) &\geq \inf_{L \in \cL_2}L([(1-\varphi)K^\s]\ast u),\\
&= \inf_{L \in \cL_2}(\bar L [(1-\varphi)K^\s]\ast u),\\
&\geq -C.
\end{align*}
In the last inequality we used that $|DK(y)| \leq \L|y|^{-1}$, $|D^2K(y)| \leq \L|y|^{-2}$ and $\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} \leq 1$.
\end{proof}
From now on we denote, for $r_1 > r_2 > 0$, $\psi_{r_1,r_2}\in C^\8_0(B_{r_1} \to [0,1])$ such that $\psi_{r_1,r_2} = 1$ in $B_{r_2}$.
\begin{corollary}\label{cor:convolution3}
Let $6 \geq r_1 > r_2 >0$, $K \geq 0$, $b\in\R^n$ such that either $L_{K,b}^\s \in \cL_2$ or $|b| \leq \b'$, $\supp K \ss B_1$ and $K(y) \in [0,\L']$, then,
\begin{align*}
(\psi_{r_1,r_2} L_{K,b}^\s u)_t-\cM^+_\cL(\psi_{r_1,r_2} L_{K,b}^\s u) &\leq C \text{ in $C_{r_2,3}$}.
\end{align*}
for some universal constant $C>0$ depending also on $r_1,r_2,\b'$ and $\L'$.
\end{corollary}
\begin{proof}
We use either Corollary \ref{cor:convolution} or \ref{cor:convolution2} to get that $\psi_{r_1,r_2} L_{K,b}^\s u$ satisfies the following inequality in $C_{r_2,3}$,
\begin{align*}
(\psi_{r_1,r_2} L_{K,b}^\s u)_t-\cM^+_\cL(\psi_{r_1,r_2} L_{K,b}^\s u) &\leq C + \sup_{L \in \cL_2} L((1-\psi_{r_1,r_2})L_{K,b}^\s u),\\
&= C + \sup_{L \in \cL_2} K_L \ast((1-\psi_{r_1,r_2})L_{K,b}^\s u)
\end{align*}
Where $K_L$ is the kernel associated to $L \in \cL_2$, notice the cancellations provided by the fact that $(1-\psi_{r_1,r_2})$ and its gradient are zero in $B_{r_2}$. Now we take a closer look at $[K_L \ast((1-\psi_{r_1,r_2})L_{K,b}^\s u)](x,t)$ for $(x,t) \in C_{r_2,3}$,
\begin{align*}
[K_L\ast((1-\psi_{r_1,r_2})L_{K,b}^\s u)](x,t) &= [(K_L(1-\psi_{r_1,r_2}(x+\cdot))) \ast L_{K,b}^\s u](x,t),\\
&= [L_{\bar K, \bar b}^\s (K_L(1-\psi_{r_1,r_2}(x+\cdot))) \ast u](x,t),\\
&\leq C.
\end{align*}
In the last inequality we used that $|DK_L(y)| \leq \L|y|^{-1}$, $|D^2K_L(y)| \leq \L|y|^{-2}$ and $\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} \leq 1$.
\end{proof}
\section{Estimate for $\D^{\s/2}u$}\label{EstimateforLap}
We keep the same assumptions as before in this part: $\s\in[1,2)$, $\cL \ss \cL_2$, and $u$ such that,
\begin{align*}
&u_t - \cM^-_\cL u = 0 \text{ in $C_{8,3}$},\\
&\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))} \leq 1.
\end{align*}
\begin{lemma}\label{lem:bound_laplacian}
For $K(y) \in [0,\L], b \in B_\b$,
\begin{align*}
\|L_{K,b}^\s u\|_{L^\8(C_{1,1})} \leq C,
\end{align*}
for some universal constant $C$.
\end{lemma}
\begin{proof}
We do it in several steps. Here is a summary of the strategy:
\begin{enumerate}
\item For $L \in \cL$, we bound $Lu$ from below by using the equation for $u$ and the control we have for $u_t$ inside the domain.
\item For $L \in \cL$, we integrate by parts to control $\|Lu(t)\|_{L^1(\w_\s)}$ and then apply Lemma \ref{lem:oscillation} to bound $Lu$ from above.
\item For general $K$ and $b$, we use $L^2$ theory to control $\|L_{K,b}^\s u(t)\|_{L^1(\w_\s)}$ and then apply Lemma \ref{lem:oscillation} to bound $L_{K,b}^\s u$ from above.
\item For general $K$ and $b$, we apply the previous step to
\begin{align*}
(K'',b'') = \L(K',b') - \l(K,b),
\end{align*}
with $L_{K',b'}^\s \in \cL$ to bound $L_{K,b}^\s u$ from below.
\end{enumerate}
\textbf{Step 1:} $L \in \cL$, then $Lu \geq -C$ in $C_{8,3}$.
It follows from the equation for $u$ and the regularity in time,
\begin{align*}
Lu \geq \cM^-_\cL u = u_t \geq -[u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))}.
\end{align*}
\textbf{Step 2:} $L \in \cL$, then $Lu \leq C$ in $C_{4,2}$.
We apply the Oscillation Lemma to $\psi_{6,5} Lu$. By Corollary \ref{cor:convolution3}, $\psi_{6,5} Lu$ satisfies,
\begin{align*}
(\psi_{6,5} Lu)_t-\cM^+_\cL(\psi_{6,5} Lu) &\leq C \text{ in $C_{5,3}$}.
\end{align*}
We estimate now $\|(\psi_{6,5} Lu)^+\|_{L^1((-3,0]\mapsto L^1(\w_\s))}$. As $\psi_{6,5} Lu$ is bounded from below and compactly supported all we need is to control the following integral
\begin{align*}
\int_{-3}^0 \int \psi_{6,5} Lu &= \int_{-3}^0 \int \1\bar L\psi_{6,5}\2u \leq C.
\end{align*}
By Lemma \ref{lem:oscillation}, we have that $\psi_{6,5} Lu$ is bounded from above in $C_{4,2}$ where it coincides with $Lu$.
\textbf{Step 3:} Given $K(y) \in [0,\L']$ and $b \in B_{\b'}$ then $L_{K,b}^\s u \leq C$ in $C_{1,1}$.
Given $L \in \cL$ the previous steps tell us that $Lu$ is bounded in $C_{4,2}$, from Fourier analysis techniques we get then that (see Theorem 4.3 in \cite{C4}),
\begin{align*}
&\|L_{K,b}^\s u(t)\|_{L^2(B_2)} \leq C\|Lu(t)\|_{L^2(B_3)} \leq C,\\
\Rightarrow \quad &\|L_{K\chi_{B_1},b}^\s u(t)\|_{L^1(B_2)} \leq C + \|L_{K\chi_{B_1^c},0}^\s u(t)\|_{L^1(B_2)},\\
\Rightarrow \quad &\|\psi_{3,2}L_{K\chi_{B_1},b}^\s u\|_{L^1((-2,0]\mapsto L^1(\w_\s))} \leq C.
\end{align*}
By Corollary \ref{cor:convolution3}, $\psi_{3,2} L_{K\chi_{B_1},b}^\s u$ satisfies,
\begin{align*}
(\psi_{3,2} L_{K\chi_{B_1},b}^\s u)_t-\cM^+_\cL(\psi_{3,2} L_{K\chi_{B_1},b}^\s u) &\leq C \text{ in $C_{2,2}$},\\
\int_{-4}^0 f(t)dt &\leq C.
\end{align*}
By Lemma \ref{lem:oscillation}, $\psi_{3,2} L_{K\chi_{B_1},b}^\s u$ gets bounded from above in $C_{1,1}$. By the hypoteses we also obtain the bound for $\psi_{3,2} L_{K,b}^\s u$ in $C_{1,1}$ where it coincides with $L_{K,b}^\s u$,
\begin{align*}
\psi_{3,2} L_{K,b}^\s u \leq C + \psi_{3,2} L_{K\chi_{B_1^c},b}^\s u \leq C + \|u\|_{L^\8((-1,0] \mapsto L^1(\w_\s))}.
\end{align*}
\textbf{Step 4:} Given $K(y) \in [0,\L]$ and $b \in B_{\b}$ then $L_{K,b}^\s u \geq -C$ in $C_{1,1}$.
Consider $L_{K',b'}^\s \in \cL$ and $L_{K'',b''}^\s := \L L_{K',b'}^\s - \l L_{K,b}^\s$ such that $|b''| \leq (\L+\l)\b$,
and $K''(y) \in [0,\L^2]$. Given the result from the second step, it suffices to show that $L_{K'',b''}^\s u \geq -C$ in $C_{1,1}$. This now is just a consequence of applying the third step to $L_{K'',b''}^\s u$.
\end{proof}
\begin{corollary}
There is a universal constant $C>0$ such that,
\begin{align*}
(2-\s)\int\frac{|\d u(x,t;y)|}{|y|^{n+\s}}dy \leq C \text{ in $C_{1,1}$}.
\end{align*}
\end{corollary}
In particular, by Morrey estimates, we have that $u \in C^\a_x(C_{1,1})$ for every $\a \in [1,\s)$, see \cite{Stein}.
\begin{proof}
Using $K(y) := \L(2-\s)|y|^{-(n+\s)}$ in the previous Lemma we get,
\begin{align*}
(2-\s)\int\frac{\d u(x,t;y)}{|y|^{n+\s}}dy \geq -C \text{ in $C_{1,1}$}.
\end{align*}
Fixing $(x,t) \in C_{1,1}$ and using $K(y) := \L(2-\s)\sign(\d u(x,t;y))|y|^{-(n+\s)}$ in the previous Lemma we get,
\begin{align*}
(2-\s)\int\frac{\d^+u(x,t;y)}{|y|^{n+\s}}dy \leq C.
\end{align*}
Adding them up we conclude the Corollary.
\end{proof}
\section{Further regularity}\label{Further}
Regularity $C^{2,\a}$ can be reduced to H\"older regularity of the laplacian. The same holds with respect to $C^{\s+\a}$ regularity and $(-\D)^{\s/2}$, which suits well for non-local equations. On the other hand, $(-\D)^{\s/2}u$ can be thought as a difference of an average of $u$ with itself which relates with the concavity of $\cM^-_\cL$ in a proper way. We will exploit these two facts in this section to prove our $C^{\s+\a}$ regularity result.
We keep the previous hypothesis for this section, $\cL \ss \cL_2$ and $u$ satisfies,
\begin{align*}
&u_t - \cM^-_\cL u = 0 \text{ in $C_{8,3}$},\\
&\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))} \leq 1.
\end{align*}
In particular we know by now that, for $K(y) \in [0,\L], b \in B_\b$,
\begin{align*}
\|L_{K,b}^\s u\|_{L^\8(C_{1,1})} \leq C.
\end{align*}
Given $A\ss B_1$, let
\begin{align*}
K_A^\s(y) := (2-\s)\frac{\chi_A(y)}{|y|^{n+\s}},
\end{align*}
Fix $\varphi \in C_0^\8(B_1 \to [0,1])$ such that $\varphi = 1$ in $B_{1/2}$ and define
\begin{align*}
w_A(x) :=\varphi(x)\int (\d u(x;y) - \d u(0;y))K_A^\s(y)dy.
\end{align*}
By the properties deduced in the previous sections we have that $w_A$ is glo-bally bounded and satisfies in $C_{1,1}$,
\begin{align*}
(w_A)_t-\cM^+_\cL w_A\leq C.
\end{align*}
Lets consider also the extremal functions,
\begin{align*}
P(x) := \sup_{A\ss B_1} w_A &= (2-\s)\varphi(x)\int_{B_{1/2}}\frac{(\d u(x;y)-\d u(0;y))^+}{|y|^{n+\a}}dy,\\
N(x) := \sup_{A\ss B_1} (-w_A) &= (2-\s)\varphi(x)\int_{B_{1/2}}\frac{(\d u(x;y)-\d u(0;y))^-}{|y|^{n+\a}}dy.
\end{align*}
Our goal is to prove a diminish of oscillation lemma for $P+N$. This implies that $(-\D)^{\s/2}u$ is H\"older continuous and therefore the $C^{\s+\a}$ regula-rity. We start by proving that $P$ and $N$ are comparable modulus a controlled error.
\begin{lemma}\label{lema91}
There exist universal constants $C>0$ and $\a \in (0,1)$ such that for $(x,t)\in C_{1/8,1/2}$ we have,
\begin{align*}
\frac{\l}{\L}N - C|x|^\a\leq P \leq \frac{\L}{\l}N + C|x|^\a.
\end{align*}
\end{lemma}
\begin{proof}
For $x\in B_{1/8}$, let $u_x(y) = u(x+y)$. Since $u$ solves $u_t-\cM^-_\cL u=0$ in $C_{1,1}$, then difference $(u_x-u)$ satisfies in $C_{7/8,1}$,
\begin{align*}
(u_x-u)_t-\cM^+_\cL(u_x-u)&\leq 0,\\
(u_x-u)_t-\cM^-_\cL(u_x-u)&\geq 0.
\end{align*}
To recover $P$ and $N$ from the previous relations we consider for $L=L_{K,0}^\s\in\cL_2$,
\begin{align*}
L(u_x-u)(0) &= \int(\d u(x;y)-\d u(0;y))K^\s(y)dy,\\
\l P(x) - \L N(x) &\leq \int_{B_1}(\d u(x;y)-\d u(0;y))K^\s(y)dy \leq \L P(x) - \l N(x).
\end{align*}
By change of variables,
\begin{align*}
\int_{B^c_1} = &\int u(y)\1K^\s(y-x)\chi_{B^c_1}(y-x) - K^\s(y)\chi_{B^c_1}(y)\2dy\\
&{} + (u(x)-u(0))\int_{B^c_1} K^\s(y)dy.
\end{align*}
By Theorem \ref{C1aold}, the last term is of order $|x|$. The first term can be estimated using the smoothness hypothesis of $K$,
\begin{align*}
&\int |K^\s(y-x)\chi_{B^c_1}(y-x) - K^\s(y)\chi_{B^c_1}(y)|dy,\\
\leq &\int_{B^c_{1/2}}|K^\s(y-x) - K^\s(y)|dz \leq C|x|.
\end{align*}
On the other hand we have the estimate $\|(u_x-u)_t\|_\8\leq C|x|^\a$ from Theorem \ref{furthertime}. Therefore,
\begin{align*}
[(u_x-u)_t-L(u_x-u)](0,t) &\geq -C|x|^\a - \L P(x) + \l N(x).
\end{align*}
Taking the infimum over $L\in\cL_2$ and using the equation for $(u_x-u)$ we get
\begin{align*}
0 &\geq -C|x|^\a - \L P(x) + \l N(x)
\end{align*}
A similar computation with $(u_x-u)_t-\cM^-_\cL(u_x-u)\geq 0$ provides the other inequality.
\end{proof}
The next result is a diminish of oscillation lemma. As we have learned from \cite{C1,C2,C3,CD,CD2} it is important to strengthen the hypothesis of being just bounded and allow some growth at infinity. This allows to iterate the lemma taking into account that the tails grow in a controlled way.
By rescaling we can further assume that for $\e_1>0$ sufficiently small (to be fixed) and for every set $K \ss \R^n$
\begin{alignat}{2}
\label{resc0} |w_K| &\leq 1/2 &&\text{ in $C_{1,1}$},\\
\label{resc1} |w_K| &\leq |x|^{1/2} &&\text{ in $B_1^c\times[-1,0]$},\\
\label{resc3} (w_K)_t-\cM^+_\cL w_K &\leq \e_1 &&\text{ in $C_{1,1}$}.
\end{alignat}
Additionally, by the previous lemma, we assume that in $C_{1/2,1}$,
\begin{align}
\label{resc4} \frac{\l}{\L}N(x,y)-\e_1|x|^\a &\leq P(x,t)\leq \frac{\L}{\l}N(x,t)+\e_1|x|^\a.
\end{align}
\begin{lemma}\label{Ca for P}
Assume \eqref{resc0}, \eqref{resc1}, \eqref{resc3} and \eqref{resc4}. There are constants $\kappa,\theta > 0$, sufficiently small, such that in $C_{\kappa,\kappa^\s}$
\begin{align*}
P \leq \frac{1}{2}-\theta.
\end{align*}
\end{lemma}
\begin{remark}\label{rescaling_of_osc_lemma}
We should ask ourselves how small should $\kappa$ and $\theta$ in order to be able to iterate the lemma. We need the rescaled $\tilde w_K$, given by
\begin{align*}
\tilde w_K(x,t) = \frac{w_K(\kappa x, \kappa^\s t)}{1-\theta},
\end{align*}
to satisfy the same hypothesis \eqref{resc0}, \eqref{resc1}, \eqref{resc3} and \eqref{resc4}. \eqref{resc0} is immediate, \eqref{resc1} holds if $(1-\theta) - \kappa^{1/2} \geq \theta/2 > 0$ which is reasonable as $\kappa,\theta$ can be chosen even smaller. \eqref{resc3} holds if $(1-\theta) > \kappa^\s$ which was already contained in the previous inequality as $\s > 1/2$. \eqref{resc4} holds if $\kappa^{\s-\a} \leq (1-\theta)$ which is possible because $\s > 1 > \a$.
\end{remark}
\begin{proof}
Assume by contradiction that for some $(x_0,t_0) \in C_{\kappa,\kappa^\s}$, $P(x_0,t_0) > (1/2-\theta)$. There is then some set $A$ such that $w_A(x_0,t_0) > (1/2-\theta)$. The function $v_A$, given by the following truncation,
\begin{align*}
v_A := \1\frac{1}{2} - w_A\2^+,
\end{align*}
satisfies an equation coming from \eqref{resc3}. As usual the truncation introduces an error that can be controlled in the interior
\begin{align*}
(v_A)_t - \cM_{\cL}^-v_A \geq -C \text{ in $C_{1/2,1}$}.
\end{align*}
We use the Point Estimate \ref{PE} to control the distribution of $v_A$ in $C_{\kappa,\kappa^\s}(0,-\kappa^\s)$,
\begin{align}\label{G}
\frac{|\{v_A > s\theta\} \cap C_{\kappa,\kappa^\s}(0,-\kappa^\s)|}{|C_{\kappa,\kappa^\s}(0,-\kappa^\s)|} \leq C(\theta + \kappa^\s)^\e(s\theta)^{-\e}.
\end{align}
By choosing $\kappa^\s\leq\theta$ we will make the right hand side $Cs^{-\e}$ sufficiently small, independently of $\theta$, by taking $s$ sufficiently large. This makes
\begin{align*}
G := \{w_A \geq (1/2 - s\theta)\} \cap C_{\kappa,\kappa^\s}(0,-\kappa^\s)
\end{align*}
to cover a fraction of $C_{\kappa,\kappa^\s}(0,-\kappa^\s)$ close to one.
In $G$, $w_A$ and $P$ are close to $1/2$. By \eqref{resc4}, $N$ can be forced also to be strictly positive in $G$, say larger than $\l/(4\L)$ by making $\e_1 + \theta \leq 1/4$. Also in $G$ and for $B = B_1\sm A$, $w_B$ has to be close to $-N$. This is because $w_A+w_B=P-N$, then
\begin{align*}
0 \leq N+w_B=P-w_A \leq s\theta.
\end{align*}
This allows us to make $w_B \leq -\l/(8\L)$ in $G$ by choosing $\theta < \l/(8s\L)$.
Now we use the Oscillation Lemma to obtain the contradiction. Consider, for $\eta \in (0,1)$, $v_B$ given by,
\begin{align*}
v_B(x,t) = \1w_B(\kappa\eta x, (\kappa\eta)^\s t-\kappa^\s)+\frac{\l}{8\L}\2^+.
\end{align*}
It still satisfies in $C_{(\kappa\eta)^{-1},(\kappa\eta)^{-\s}}$,
\begin{align*}
(v_B)_t-\cM^+_\cL(v_B) \leq \e_1(\kappa\eta)^\s \leq \e_1.
\end{align*}
Also, from \eqref{G}, we know that,
\begin{align*}
|\{v_B > 0\} \cap C_{\eta^{-1},\eta^{-\s}}| \leq C\eta^{-(n+\s)}s^{-\e}.
\end{align*}
By the Oscillation Lemma,
\begin{align*}
\frac{\l}{8\L} = v_B(0,0) &\leq C\1 \e_1 + \eta^{-(n+\s)}s^{-\e} + \sup_{t\in[-\eta^{-\s},0]}\int_{B^c_{\eta^{-1}}} \frac{|v_B(y,t)|}{|y|^{n+\s}}dy\2.
\end{align*}
Changing variables,
\begin{align*}
\int_{B^c_{\eta^{-1}}} \frac{|v_B(y,t)|}{|y|^{n+\s}}dy &= (\kappa\eta)^\s\int_{B^c_\kappa} \frac{\1w_B(y, (\kappa\eta)^\s t-\kappa^\s)+\frac{\l}{8\L}\2^+}{|y|^{n+\s}}dy,\\
&\leq C\eta^\s,
\end{align*}
where the last inequality holds by the bounds \eqref{resc0} and \eqref{resc1}. Putting it back in the estimate we obtain,
\begin{align*}
\frac{\l}{8\L} \leq C\1\e_1 + \eta^{-(n+\s)}s^{-\e} + \eta^\s\2.
\end{align*}
This gives us a contradiction by choosing $\e_1,\eta^\s < \l/(100C\L)$ and then $s^\e > (100C\L)/(\l\eta^{n+\s})$.
\end{proof}
We are now able to prove the parabolic nonlocal Evans-Krylov Theorem.
\begin{theorem}[Classical solutions]
Let $\cL \ss \cL_2$, $u$ be a bounded function in $\R^n\times(-1,0]$ solving
\begin{align*}
u_t-\cM^-_\cL u=0 \text{ in viscosity in $C_{1,1}$},
\end{align*}
Then $(-\D)^\s u$ is H\"older continuous with the following estimate
\begin{align*}
\|(-\D)^\s u\|_{C^\a(C_{1/2,1/2})} \leq C(\|u\|_{L^\8((-1,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-1,0] \mapsto L^1(\w_\s))}).
\end{align*}
\end{theorem}
\begin{proof}
The case $\s\leq 1$ is contained in \cite{CD3}. By the regularization procedure of \cite{CD2} we can assume that $(-\D)^\s u$ is continuous, all we need to show is the estimate at the origin. As usual we re-normalize $u$ in order to have $\|u\|_{L^\8((-1,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-1,0] \mapsto L^1(\w_\s))} \leq 1$
By the definitions of $P$ and $N$ we have the following identity in $B_{1/8}\times(-1,0]$
\begin{align*}
&(-\D)^\s u(0)-(-\D)^\s u(x) =\\
&C\1 P(x)+N(x)+(2-\s)\int_{B^c_1}\frac{\d u(x;y)-\d u(0;y)}{|y|^{n+\s}}dy\2.
\end{align*}
The third term can be bounded by $C|x|$ as in the proof of Lemma \ref{lema91}.
Lemma \ref{Ca for P} and the Remark \ref{rescaling_of_osc_lemma} gives a geometric decay for $P$ around the origin which implies a H\"older modulus of continuity for it. By Lemma \ref{lema91} this is equivalent to a similar modulus of continuity for $N$. Then, the first two terms above can be bounded by $C|x|^{\a}$, for some universal $\a$, which concludes the proof.
\end{proof}
\end{document} |
\begin{document}
\title{The pleasures and pains of studying the two-type Richardson model}
\author{Maria Deijfen \thanks{Department of Mathematics, Stockholm University,
106 91 Stockholm, Sweden. E-mail: [email protected]} \and Olle H\"{a}ggstr\"{o}m
\thanks{Department of Mathematical Sciences, Chalmers University of
Technology, 412 96 G\"oteborg, Sweden. E-mail: [email protected]}}
\date{July 2007}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\noindent This paper provides a survey of known results and open
problems for the two-type Richardson model, which is a stochastic
model for competition on $\mathbb{Z}^d$. In its simplest
formulation, the Richardson model describes the evolution of a
single infectious entity on $\mathbb{Z}^d$, but more recently the
dynamics have been extended to comprise two competing growing entities.
For this
version of the model, the main question is whether there is a
positive probability for both entities to simultaneously grow to
occupy infinite parts of the lattice, the conjecture being that
the answer is yes if and only if the entities have the same
intensity. In this paper attention focuses on the two-type model,
but the most important results for the one-type version are also
described.
\noindent \emph{Keywords:} Richardson model, first-passage
percolation, asymptotic shape, competing growth, coexistence.
\noindent AMS 2000 Subject Classification: 60K35, 82B43.
\end{abstract}
\section{Introduction}
Consider an interacting particle system in which, at any time $t$,
each site $x\in\mathbb{Z}^d$ is in either of two states, denoted by
0 and 1. A site in state 0 flips to a 1 at rate proportional to the
number of nearest neighbors in state 1, while a site in state 1
remains a 1 forever. We may think of sites in state 1 as being occupied
by some kind of infectious entity, and the model then describes the
propagation of
an infection where each infected site tries to infect each of its
nearest neighbors on $\mathbb{Z}^d$ at some constant rate $\lambda>0$.
More precisely, if at time $t$ a vertex $x$ is infected and a neighboring
vertex $y$ is uninfected, then, conditional on the dynamics up to time
$t$, the probability that $x$ infects $y$ during a short time
window $(t, t+h)$ is $\lambda h + o(h)$. Here and
in what follows, sites in state 0 and 1 are referred to as
uninfected and infected respectively. This is the intuitive description
of the model; a formal definition is given in Section \ref{sect:one-type}.
The model is a special case of a class of
models introduced by Richardson (1973), and is commonly referred
to as the Richardson model. It has several cousins among processes
from mathematical biology, see e.g.\ Eden (1961), Williams and
Bjerknes (1972) and Bramson and Griffeath (1981). The model is
also a special case of so called first-passage percolation, which
was introduced in Hammersley and Welsh (1965) as a model for
describing the passage of a fluid through a porous medium. In
first-passage percolation, each edge of the $\mathbb{Z}^d$-lattice
is equipped with a random variable representing the time it takes
for the fluid to traverse the edge, and the Richardson model is
obtained by letting these passage times be i.i.d.\ exponential.
Since an infected site stays infected forever, the set of infected
sites in the Richardson model increases to cover all of
$\mathbb{Z}^d$ as $t\to\infty$, and attention focuses on
\emph{how} this set grows. The main result is roughly that the
infection grows linearly in time in each fixed direction and that,
scaled by a factor $1/t$, the set of infected points converges to a
non-random asymptotic shape as $t\to\infty$. To prove that the
growth is linear in a fixed direction involves
Kingman's subadditive ergodic theorem -- in fact, the study of
first-passage percolation was one of the main motivations for the
development of subadditive ergodic theory. That
the linear growth is preserved when all directions are considered
simultaneously is stated in the celebrated shape theorem
(Theorem \ref{thm:shape} in Section \ref{sect:two-type_bounded})
which originates from Richardson (1973).
Now consider the following extension of the Richardson model,
known as the two-type Richardson model and
introduced in H\"{a}ggstr\"{o}m and Pemantle (1998). Instead of
two possible states for the sites there are three states, which we
denote by 0, 1 and 2. The process then evolves in such a way that, for
$i=1,2$, a site in state 0 flips to state $i$ at rate $\lambda_i$
times the number of nearest neighbors in state $i$ and once in
state 1 or 2, a site remains in that state forever. Interpreting
states 1 and 2 as two different types of infection
and state 0 as absence of infection, this gives rise to a model
describing the simultaneous spread of two infections on
$\mathbb{Z}^d$. To rigorously define
the model requires a bit more work; see Section \ref{sect:two-type_bounded}.
In what follows we will always assume that $d\geq 2$;
the model makes sense also for $d=1$ but the questions
considered here become trivial.
A number of similar extensions of (one-type) growth models to
(two-type) competition models appear in the literature; see for instance
Neuhauser (1992), Durrett and Neuhauser (1997), Kordzakhia and Lalley (2005)
and Ferrari et al.\ (2006). These tend to require somewhat different
techniques, and results
tend not to be easily translated from these other models to the two-type
Richardson model (and vice versa). Closer to the latter are (non-Markovian)
competition models based on first-passage percolation models with
non-exponential passage time variables -- Garet and Marchand (2005),
Hoffman (2005:1), Hoffman (2005:2), Garet and Marchand (2006),
Gou\'er\'e (2007),
Pimentel (2007) -- and a certain continuum model -- Deijfen et al.\ (2004),
Deijfen and H\"aggstr\"om (2004), Gou\'er\'e (2007). For ease
of exposition, we shall not consider these variations even in cases
where results generalize.
The behavior of the two-type Richardson model depends on the
initial configuration of the infection and on the ratio between
the intensities $\lambda_1$ and $\lambda_2$ of the infection
types. Assume first, for simplicity, that the model is started at
time 0 from two single sites, the origin being type 1 infected and
the site $(1,0,\ldots,0)$ next to the origin being type 2
infected. Three different scenarios for the development of
the infection are conceivable:
\begin{itemize}
\item[(a)] The type 1 infection at some point completely surrounds
type 2, thereby preventing type 2 from growing any further.
\item[(b)] Type 2 similarly strangles type 1.
\item[(c)] Both infections grow to occupy infinitely many sites.
\end{itemize}
\noindent It is not hard to see that, regardless of the
intensities of the infections, outcomes
(a) and (b) where one of the infection
types at some point encloses the other have positive
probability regardless of $\lambda_1$ and $\lambda_2$. This is because
each of (a) and (b) can be guaranteed through some finite initial
sequence of infections. In contrast, scenario (c) -- referred to as
infinite coexistence -- can never be guaranteed from
any finite sequence of infections, and is therefore
harder to deal with: the main challenge
is to decide whether, for given values of the parameters
$\lambda_1$ and $\lambda_2$, this event (c) has positive probability
or not. Intuitively, infinite coexistence represents some kind of
power balance between the infections, and it seems reasonable to
suspect that such a balance is possible if and only if the
infections are equally powerful, that is, when
$\lambda_1=\lambda_2$. This is Conjecture \ref{samex_conj} in
Section \ref{sect:two-type_bounded}, which goes back to
H\"{a}ggstr\"{o}m and Pemantle (1998), and, although a lot of
progress have been made, it is not yet fully proved. We describe
the state of the art in Sections \ref{sect:symmetric} and
\ref{sect:nonsymmetric}.
As mentioned above, apart from the intensities, the development of
the infections in the two-type model also depends on the initial
state of the model. However, if we are only interested in deciding
whether the event of infinite coexistence has positive probability
or not, it turns out that, as long as the initial configuration is
bounded and one of the sets does not completely surround the
other, the precise configuration does not matter, that is, whether
infinite coexistence is possible or not is determined only by the
relation between the intensities. This is proved in Deijfen and
H\"{a}ggstr\"{o}m (2006:1); see Theorem \ref{th:startomr} in
Section \ref{sect:two-type_bounded} for a precise formulation.
Of course one may also consider unbounded initial configurations.
Starting with both infection types occupying infinitely many sites
means -- apart from in very labored cases -- that they will both
infect infinitely many sites. A more interesting case is when one
of the infection types starts from an infinite set and the other
one from a finite set. We may then ask if outcomes where the
finite type infects infinitely many sites have positive
probability or not. This question is dealt with in Deijfen and
H\"{a}ggstr\"{o}m (2007), and we describe the results in Section
\ref{sect:unbounded}.
The dynamics of the two-type Richardson model is deceptively
simple, and yet gives rise to intriguing
phenomena on a global scale. In this lies a large part of the pleasure
indicated in the title. Furthermore,
proofs tend to involve elegant probabilistic
techniques such as coupling, subadditivity and stochastic
comparisons, adding more pleasure. The pain alluded to (which
by the way is not so severe that it should dissuade readers from
entering this field) comes from the stubborn resistance that some
of the central problems have so far put up against attempts to solve them.
A case in point is the ``only if'' direction of the aforementioned
Conjecture \ref{samex_conj}, saying that infinite coexistence
starting from a bounded initial configuration does not occur
when $\lambda_1 \neq \lambda_2$.
\section{The one-type model} \label{sect:one-type}
As mentioned in the introduction, the one-type Richardson model is
equivalent to first-passage percolation with i.i.d.\ exponential
passage times. To make the construction of the model more precise, first
define $E_{\mathbb{Z}^d}$ as the edge set for the $\mathbb{Z}^d$ lattice
(i.e., each pair of vertices $x,y \in \mathbb{Z}^d$ at Euclidean distance
$1$ from each other have an edge $e \in E_{\mathbb{Z}^d}$ connecting them).
Then attach i.i.d.\ non-negative
random variables $\{\tau(e)\}_{e \in E_{\mathbb{Z}^d}}$
to the edges. We take each $\tau(e)$ to be exponentially distributed with
parameter $\lambda>0$, meaning that
\[
P(\tau(e)>t) = \exp(-\lambda t)
\]
for all $t\geq 0$. For $x,y \in \mathbb{Z}^d$, define
\begin{equation} \label{eq:path_time}
T(x,y) = \inf_\Gamma \sum_{e \in \Gamma} \tau(e)
\end{equation}
where the infimum is over all paths $\Gamma$ from $x$ to $y$.
The Richardson model with a given set
$S_0 \subset \mathbb{Z}^d$ of initially infected sites is now defined
by taking the set $S_t$ of sites infected at time $t$ to be
\begin{equation} \label{eq:infected_at_time_t}
S_t = \{ x \in \mathbb{Z}^d : T(y,x) \leq t\mbox{ for some } y \in S_0\} \, .
\end{equation}
It turns out that the infimum in (\ref{eq:path_time}) is a.s.\ a minimum
and attained by a unique path. That $S_t$ grows in the
way described in the introduction is a consequence of the memoryless property
of the exponential distribution: for any $s,t >0$ we have that
$P(\tau(e)>s+t \, | \tau(e)>s) = \exp(-\lambda t)$.
Note that for any $x,y,z \in\mathbb{Z}^d$ we have
$T(x,y)\leq T(x,z)+T(z,y)$. This subadditivity property
opens up for the use of subadditive ergodic theory in analyzing
the model. To formulate the basic result, let $T(x)$ be the time
when the point $x\in\mathbb{Z}^d$ is infected when starting from a
single infected site at
the origin and write $\mathbf{n}=(n,0,\ldots,0)$.
It then follows from the subadditive ergodic theorem -- see
e.g.\ Kingman (1968) -- that there is a constant $\mu_\lambda$
such that $T(\mathbf{n})/n\to\mu_\lambda$ almost surely and in
$L_1$ as $n\to\infty$. Furthermore, a simple time scaling argument
implies that $\mu_\lambda=\lambda\mu_1$ and hence, writing
$\mu_1=\mu$, we have that
\begin{equation}\label{eq:time_constant}
\lim_{n\to\infty}\frac{T(\mathbf{n})}{n}=\lambda\mu\quad\textrm{a.s. and in }L_1.
\end{equation}
\noindent The constant $\mu$ indicates the inverse asymptotic
speed of the growth along the axes in a unit rate process and is
commonly referred to as the time constant. It turns out that $\mu>0$,
so that indeed the growth is linear in time. Similarly,
an analog of (\ref{eq:time_constant}) holds in any direction,
that is, for any $x\in\mathbb{Z}^d$, there is a constant $\mu(x)>0$
such that $T(nx)/n\to\lambda\mu(x)$. The infection hence grows
linearly in time in each fixed direction and the asymptotic speed
of the growth in a given direction is an almost sure constant.
We now turn to the shape theorem, which asserts roughly that the
linear growth of the infection is preserved also when all
directions are considered simultaneously. More precisely, when
scaled down by a factor $1/t$ the set $S_t$ converges to a
non-random shape $A$. To formalize this, let $\tilde{S}_t\subset
\mathbb{R}^d$ be a continuum version of $S_t$ obtained by
replacing each $x\in S_t$ by a unit cube centered at $x$.
\begin{theorem}[Shape Theorem] \label{thm:shape}
There is a compact convex set $A$ such that, for any $\varepsilon>0$,
almost surely
$$
(1-\varepsilon)\lambda A\subset\frac{\tilde{S}_t}{t} \subset
(1+\varepsilon)\lambda A
$$
for large $t$.
\end{theorem}
\noindent In the above form, the shape theorem was proved in
Kesten (1973) as an improvement
on the original ``in probability" version, which appears
already in Richardson (1973). See also Cox and Durrett (1988) and
Boivin (1991) for generalizations to first-passage percolation
processes with more general passage times. Results concerning
fluctuations around the asymptotic shape can be found, e.g., in Kesten
(1993), Alexander (1993) and Newman and Piza (1995), and, for
certain other passage time distributions, in Benjamini et al.\
(2003).
Working out exactly, or even approximately, what the asymptotic
shape $A$ is has turned out to be difficult.
Obviously the asymptotic shape inherits all symmetries of the
$\mathbb{Z}^d$ lattice -- invarince under reflection and
permutation of coordiante hyperplanes -- and it is known to be
compact and convex,
but, apart from this, not much is known about its qualitative
features. These difficulties with characterizing the shape revolve
around the fact that $\mathbb{Z}^d$ is not rotationally invariant,
which causes the growth to behave differently in different
directions. For instance, simulations on $\mathbb{Z}^2$
indicate that the asymptotic growth is
slightly faster along the axes as compared to the diagonals. There
is however no formal proof of this.
Before proceeding with the two-type model, we mention some work
concerning properties of
the time-minimizing paths in (\ref{eq:path_time}), also
known as geodesics. Starting at time $0$ with a single infection
at the origin ${\bf 0}$, we denote by $\Gamma(x)$ the (unique) path $\Gamma$
for which the infimum $T({\bf 0}, x)$ in (\ref{eq:path_time}) is
attained. Define
${\bf P}si=\cup_{x\in\mathbb{Z}^d} \Gamma(x)$, making ${\bf P}si$ a
graph specifying which paths the infection actually takes. It is
not hard to see that ${\bf P}si$ is a tree spanning all of
$\mathbb{Z}^d$ and hence there must be at least one semi-infinite
self-avoiding path from the origin (called an end) in ${\bf P}si$.
The issue of whether ${\bf P}si$ has more than one end was noted by
H\"aggstr\"om and Pemantle (1998) to
be closely related to the issue of infinite coexistence in the two-type
Richardson model with $\lambda_1=\lambda_2$: such infinite coexistence
happens with positive probability starting from a finite initial
configuration if and only if ${\bf P}si$ has at least two ends with
positive probability.
We say that an infinite path $x_1,x_2,\ldots$ has
asymptotic direction $\hat{x}$ if $x_k/|x_k|\to\hat{x}$ as
$k\to\infty$. In $d=2$, it has been conjectured that every end in
${\bf P}si$ has an asymptotic direction and that, for every
$x\in\mathbb{R}^2$, there is at least one end (but never more than two)
in ${\bf P}si$ with asymptotic direction $\hat{x}$.
In particular, this would mean that ${\bf P}si$ has
uncountably many ends. For results supporting
this conjecture, see Newman (1995) and Newman and Licea (1996).
In the former of these papers, the conjecture is shown to
be true provided an unproven but highly plausible
assumption on the asymptotic shape $A$, saying roughly that the
boundary is sufficiently smooth. See also Lalley (2003) for
related work.
Results not involving unproven assumptions
are comparatively weak: The coexistence result of H\"aggstr\"om and
Pemantle (1998) shows for $d=2$ that ${\bf P}si$ has at least two ends with
positive probability. This was later improved to ${\bf P}si$
having almost surely at least $2d$ ends, by
Hoffman (2005:2) for $d=2$ and by Gou\'er\'e (2007) for higher dimensions.
\section{Introducing two types}
\label{sect:two-type_bounded}
The definition of the two-type Richardson model turns out to be
simplest in the symmetric case $\lambda_1=\lambda_2$, where the
same passage time variables $\{\tau(e)\}_{e \in E_{\mathbb{Z}^d}}$ as
in the one-type model can be used, with $\lambda= \lambda_1=\lambda_2$.
Suppose we start with an initial configuration
$(S^1_0, S^2_0)$ of infected sites, where $S^1_0 \subset \mathbb{Z}^d$
are those initially containing type $1$ infection, and
$S^2_0 \subset \mathbb{Z}^d$
are those initially containing type $2$ infection. We wish to define
the sets $S_t^1$ and $S_t^2$ of type 1 and type $2$ infected sites for
all $t>0$. To this end, set $S_0=S^1_0 \cup
S^2_0$, and take the set $S_t=S_t^1 \cup S_t^2$ of infected sites
at time $t$ to be given by precisely the same formula
(\ref{eq:infected_at_time_t}) as in the one-type model; a vertex $x\in S_t$
is then assigned infection $1$ or $2$ depending on whether
the $y \in S_0$ for which
\[
\inf\{T(y,x):y\in S_0\}
\]
is attained is in $S_0^1$ or $S_0^2$.
As in the one-type model, it is a straightforward exercise involving
the memoryless property of the exponential distribution to verify that
$(S_t^1, S_t^2)_{t \geq 0}$ behaves in terms of infection intensities as
described in the introduction.
This construction demonstrates an intimate link between the one-type
and the symmetric two-type Richardson model: if we watch the two-type model
wearing a pair of of glasses preventing us from distinguishing the two types
of infection, what we see behaves exactly as the one-type model. The link
between infinite coexistence in the two-type model and the number of ends
in the tree of infection ${\bf P}si$ of the one-type model claimed in
the previous section is also a consequence of the construction.
In the asymmetric case $\lambda_1 \neq \lambda_2$, the two-type model
is somewhat
less trivial to define due to the fact that the time it takes for
infection to spread along a path depends on the type of infection.
There are various ways to deal with this, one being to assign, independently
to each $e \in E_{{\mathbb{Z}^d}}$, two independent random variables
$\tau_1(e)$ and $\tau_2(e)$, exponentially distributed with
respective parameters $\lambda_1$ and $\lambda_2$, representing the time
it takes for infections $1$ resp.\ $2$ to traverse $e$. Starting
from an intial configuration $(S^1_0, S^2_0)$, we may picture the infections
as spreading along the edges, taking time $\tau_1(e)$ or $\tau_2(e)$
to cross $e$
depending on the type of infection, with the extra condition that
once a vertex becomes hit by one type of infection it becomes inaccessible
for the other type. This is intuitively clear, but readers with a taste for
detail may require a more rigorous definition, which however we
refrain from here; see H\"aggstr\"om and Pemantle (2000) and
Deijfen and H\"aggstr\"om (2006:1).
We now move on to describing conjectures
and results. Write $G_i$ for the event that type
$i$ infects infinitely many sites on $\mathbb{Z}^d$ and define
$G=G_1\cap G_2$. The question at issue is:
\begin{equation}\label{eq:coex?}
\textrm{Does $G$ have positive probability?}
\end{equation}
\noindent A priori, the answer to this question may depend both on
the initial configuration -- that is, on the choice of the sets
$S_0^1$ and $S_0^2$ -- and on the ratio between the
infection intensities $\lambda_1$ and
$\lambda_2$. However, it turns out that, if we
are not interested in the actual value of the probability of $G$,
but only in whether it is positive or not, then the initial
configuration is basically irrelevant, as long as neither of the
initial sets completely surrounds the other.
This motivates the following definition.
\begin{defn}
Let $\xi_1$ and $\xi_2$ be two
disjoint finite subsets of $\mathbb{Z}^d$. We say that one of the
sets ($\xi_i$) \emph{strangles} the other ($\xi_j$) if there
exists no infinite self-avoiding path in $\mathbb{Z}^d$ that
starts at a vertex in $\xi_j$ and that does not intersect $\xi_i$.
The pair $(\xi_1,\xi_2)$ is said to be \emph{fertile} if neither
of the sets strangles the other.
\end{defn}
Now write $P^{\lambda_1,\lambda_2}_{\xi_1,\xi_2}$ for
the distribution of a two-type process started from $S_0^1=\xi_1$
and $S_0^2=\xi_2$. We then have the following result.
\begin{theorem}\label{th:startomr}
Let $(\xi_1,\xi_2)$ and $(\xi_1',\xi_2')$ be two fertile pairs of
disjoint finite subsets of $\mathbb{Z}^d$, where $d\geq 2$. For all choices of
$(\lambda_1,\lambda_2)$, we have
$$
P^{\lambda_1,\lambda_2}_{\xi_1,\xi_2}(A)>0\Leftrightarrow
P^{\lambda_1,\lambda_2}_{\xi_1',\xi_2'}(A)>0.
$$
\end{theorem}
For connected initial sets $\xi_1$ and $\xi_2$ and $d=2$, this
result is proved in H\"{a}ggstr\"{o}m and Pemantle (1998). The
idea of the proof in that case is that, by controlling the passage
times of only finitely many edges, two processes started from
$(\xi_1,\xi_2)$ and $(\xi'_1,\xi'_2)$ respectively can be made to
evolve to the same total infected set after some finite time, with
the same configuration of the infection types on the boundary.
Coupling the processes from this time on and observing that the
development of the infections depends only on the boundary
configuration yields the result. This argument however breaks down
when the initial sets are not connected (since it is then not sure
that the same boundary configuration can be obtained in the two
processes) and it is unclear whether it applies for $d\geq 3$. Theorem
\ref{th:startomr} is proved in full generality in Deijfen and
H\"{a}ggstr\"{o}m (2006:1), using a more involved coupling
construction.
It follows from Theorem \ref{th:startomr} that the answer to
(\ref{eq:coex?}) depends only on the value of the intensities
$\lambda_1$ and $\lambda_2$. Hence it is sufficient to consider a
process started from $S_0^1=\mathbf{0}$ and $S_0^2=\mathbf{1}$
(recall that $\mathbf{n}=(n,0,\ldots,0)$),
and in this case we drop subscripts and write $P^{\lambda_1,\lambda_2}$ for
$P^{\lambda_1,\lambda_2}_{{\bf 0}, {\bf 1}}$.
Also, by time-scaling,
we may assume that $\lambda_1=1$. The following conjecture, where
we write $\lambda_2=\lambda$, goes back to H\"{a}ggstr\"{o}m and
Pemantle (1998).
\begin{conj}\label{samex_conj}
In any dimension $d\geq 2$, we have that $P^{1,\lambda}(G)>0$
if and only if $\lambda=1$.
\end{conj}
\noindent The conjecture is no doubt true, although
proving it has turned out to be a difficult task. In fact, the
``only if'' direction is not yet fully established. In the
following two sections we describe
the existing results for $\lambda=1$ and $\lambda\neq 1$ respectively.
\section{The case $\lambda=1$} \label{sect:symmetric}
When $\lambda=1$, we are dealing with two equally powerful
infections and Conjecture \ref{samex_conj} predicts a positive
probability for infinite coexistence. This part of the conjecture
has been proved:
\begin{theorem}\label{th:lambda=1}
If $\lambda=1$, we have, for any $d\geq 2$, that $P^{1,\lambda}(G)>0$.
\end{theorem}
\noindent This was first proved in the special case $d=2$ by
H\"{a}ggstr\"{o}m and Pemantle (1998). That proof has a very ad hoc flavor,
and heavily exploits not only the two-dimensionality but also
other specific properties of the square lattice,
including a lower
bound on the time constant $\mu$ in (\ref{eq:time_constant})
that just happens to be good enough. When eventually the result
was generalized to higher dimensions, which was done simultaneously
and independently by Garet and Marchand (2005) and Hoffman
(2005:1), much more appealing proofs were obtained. Yet another distinct proof
of Theorem \ref{th:lambda=1} was given by Deijfen and H\"{a}ggstr\"{o}m
(2007). All four proofs are different, though if you inspect them for
a smallest common denominator you find that they all make critical
use of the fact that the time constant $\mu$ is strictly positive.
We will give the Garet--Marchand proof below. In Hoffman's proof
ergodic theory is applied to the tree of infection ${\bf P}si$ and a so-called
Busemann function which is shown to exhibit contradictory behavior
under the assumption that infinite coexistence has probability zero. The
Deijfen--H\"{a}ggstr\"{o}m proof proceeds via the two-type Richardson model
with certain infinite initial configurations
(cf.\ Section \ref{sect:unbounded}).
\noindent
{\bf Proof of Theorem \ref{th:lambda=1}:} The following argument
is due to Garet and Marchand (2005), though our presentation follows more
closely the proof of an analogous result in a continuum setting in
Deijfen and H\"{a}ggstr\"{o}m (2004) -- a paper that, despite the
publication dates, was preceded by and also heavily influenced by
Garet and Marchand (2005).
Fix a small $\varepsilon>0$.
By Theorem \ref{th:startomr}, we are free to choose any finite starting
configuration we want, and here it turns out convenient to begin with
a single type $1$ infection at the origin ${\bf 0}$, and a single
type $2$ infection at a vertex ${\bf n}=(n,0,\ldots, 0)$, where
$n$ is large enough so that
\begin{description}
\item{(i)}
$E[T({\bf 0}, {\bf n})] \leq (1 + \varepsilon) n \mu$, and
\item{(ii)}
$P(T({\bf 0}, {\bf n}) < (1 - \varepsilon) n \mu) < \varepsilon$;
\end{description}
note that both (i) and (ii) hold for $n$ large enough due to the
asymptotic speed result (\ref{eq:time_constant}). The reader may
easily check,
for later reference, that (i) and (ii) together with the
nonnegativity of $T({\bf 0}, {\bf n})$ imply for any event $B$
with $P(B)=\alpha$ that
\begin{equation} \label{eq:key_estimate_GM}
E[T({\bf 0}, {\bf n})\, | \, \neg B] \,
\leq \, \left(1 + \frac{3\varepsilon}{1-\alpha}\right) n \mu \, .
\end{equation}
Next comes an important telescoping idea: for any positive integer $k$ we have
\begin{eqnarray*}
E[T({\bf 0}, k{\bf n})] & = &
E[T({\bf 0}, {\bf n})]
+ E[T({\bf 0}, 2{\bf n}) - T({\bf 0}, {\bf n})]
+ E[T({\bf 0}, 3{\bf n}) - T({\bf 0}, 2{\bf n})] \\
& & + \ldots +
E[T({\bf 0}, k{\bf n}) - T({\bf 0}, (k-1){\bf n})] \, .
\end{eqnarray*}
Since $\lim_{k \rightarrow \infty}k^{-1}E[T({\bf 0}, k{\bf n})] = n \mu$,
there must exist arbitrarily large $k$ such that
\[
E[T({\bf 0}, (k+1){\bf n}) - T({\bf 0}, k{\bf n})] \geq (1 - \varepsilon)n \mu \, .
\]
By taking ${\bf m}= k{\bf n}$, and
by translation and reflection invariance, we may deduce that
\begin{equation} \label{eq:for_arbitrarily_large_m}
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})] \geq (1 - \varepsilon)n \mu
\end{equation}
for some arbitrarily large $m$. We will pick such an $m$; how large will
soon be specified.
The goal is to show that $P(G)>0$, so we may assume for contradiction
that $P(G)=0$. By symmetry of the initial configuration, we then have that
$P(G_1)=P(G_2)=\frac{1}{2}$. This implies that
\[
\lim_{m \rightarrow \infty} P({\bf -m} \mbox{ gets infected by type 2})=
\lim_{m \rightarrow \infty} P(T({\bf n}, -{\bf m}) < T({\bf 0}, -{\bf m}))
=\frac{1}{2}
\]
so let us pick $m$ in such a way that
\begin{equation} \label{eq:our_chosen_m}
P(T({\bf n}, -{\bf m}) < T({\bf 0}, -{\bf m})) \geq \frac{1}{4}
\end{equation}
while also (\ref{eq:for_arbitrarily_large_m}) holds. Write $B$ for
the event in (\ref{eq:our_chosen_m}). The expectation
$E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})]$ may be
decomposed as
\begin{eqnarray*}
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m})] & = &
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | B]P(B) \\
& & +E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B]P(\neg B) \\
& \leq & E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B]P(\neg B) \\
& \leq & \frac{3}{4}
E[T({\bf n}, -{\bf m}) - T({\bf 0}, -{\bf m}) \, | \neg B] \\
& \leq & \frac{3}{4} E[T({\bf n}, {\bf 0}) | \neg B] \\
& \leq & \frac{3}{4} (1 + 4 \varepsilon) n \mu
\end{eqnarray*}
where the second-to-last inequality is due to the triangle
inequality
$T({\bf n}, -{\bf m}) \leq T({\bf n}, {\bf 0}) + T({\bf 0}, - {\bf m})$,
and the last one uses (\ref{eq:key_estimate_GM}). For small $\varepsilon$,
this contradicts (\ref{eq:for_arbitrarily_large_m}), so the proof is
complete. ${
\Box}$
\section{The case $\lambda\neq 1$} \label{sect:nonsymmetric}
Let us move on to the case when $\lambda\neq 1$, that is, when the
type 2 infection has a different intensity than type 1. It then
seems unlikely that the kind of equilibrium which is necessary for
infinite coexistence to occur would persist in the long run.
However, this part of Conjecture \ref{samex_conj} is not proved.
The best result to date is the following theorem from
H\"{a}ggstr\"{o}m and Pemantle (2000).
\begin{theorem}\label{th:pain} For any $d\geq 2$, we have
$P^{1,\lambda}(G)=0$ for all but at most countably many values of $\lambda$.
\end{theorem}
\noindent We leave it to the reader to decide whether this
is a very strong or a very weak result: it is very strong in the sense
of showing that infinite coexistence has probability $0$ for
(Lebesgue)-almost all $\lambda$, but very weak in the sense that infinite
coexistence is not ruled out for any given $\lambda$.
The result may seem a bit peculiar at first sight and
we will spend some time explaining where it comes from and where
the difficulties arise when one tries to strengthen it. Indeed, as
formulated in Conjecture \ref{samex_conj}, the belief is that the
set $\{\lambda:P^{1,\lambda}(G)>0\}$ in fact consists of the
single point $\lambda=1$, but Theorem \ref{th:pain} only asserts
that the set is countable.
First note that, by time-scaling and symmetry, we have
$P^{1,\lambda}(G)=P^{1,1/\lambda}(G)$ and hence it is enough to
consider $\lambda\leq 1$. An essential ingredient in the proof of
Theorem \ref{th:pain} is a coupling of the two-type processes
$\{P^{1,\lambda}\}_{\lambda\in(0,1]}$ obtained by associating two
independent exponential mean 1 variables $\tau_1(e)$ and
$\tau_2'(e)$ to each edge $e\in\mathbb{Z}^d$ and then letting the
type 2 passage time at parameter value $\lambda$ be given by
$\tau_2(e)=\lambda^{-1}\tau_2'(e)$ and the type 1 time (for any
$\lambda$) by $\tau_1(e)$. Write $Q$ for the probability measure
underlying this coupling and let $G^\lambda$ be the event that
infinite coexistence occurs at parameter value $\lambda$. Theorem
\ref{th:pain} is obtained by showing that
\begin{equation} \label{eq:in_the_coupling}
\begin{array}{l}
\mbox{with $Q$-probability 1 the event $G^\lambda$ occurs} \\
\mbox{for at most one value of $\lambda\in(0,1]$}.
\end{array}
\end{equation}
Hence, $Q(G^\lambda)$ can be positive for at most countably
many $\lambda$, and Theorem
\ref{th:pain} then follows by noting that
$P^{1,\lambda}(G)=Q(G^\lambda)$.
But why is (\ref{eq:in_the_coupling}) true?
Let $G^\lambda_i$ be the event that the type $i$ infection grows
unboundedly at parameter value $\lambda$. Then the coupling
defining $Q$ can be shown to be monotone in the sense that $G^\lambda_1$ is
decreasing in $\lambda$ -- that is, if $G^\lambda_1$ occurs then
$G^{\lambda'}_1$ occurs for all $\lambda'<\lambda$ as well -- and
$G^\lambda_2$ is increasing in $\lambda$. This kind of monotonicity
of the coupling is
crucial for proving (\ref{eq:in_the_coupling}), as is
the following result, which asserts that, on the event that the
type 2 infection survives, the total infected set in a two-type
process with distribution $P^{1,\lambda}$, where $\lambda<1$,
grows to a first approximation
like a one-type process with intensity $\lambda$. More precisely,
the speed of the growth in the two-type process is determined by
the weaker type 2 infection type. We take $\tilde{S}_t^i$ to denote the
union of all unit cubes centered at points in $S_t^i$ and $A$ is
the limiting shape for a one-type process with rate 1.
\begin{theorem}\label{th:svag_bestammer} Consider a two-type process
with distribution $P^{1,\lambda}$ for some $\lambda\leq 1$. On the
event $G_2$ we have, for any $\varepsilon>0$, that almost surely
$$
(1-\varepsilon)\lambda A\subset\frac{\tilde{S}_t^1\cup
\tilde{S}_t^2}{t}\subset (1+\varepsilon)\lambda A
$$
for large $t$.
\end{theorem}
\noindent Theorem \ref{th:pain} follows readily from this result and the
monotonicity properties of the coupling $Q$. Indeed, fix
$\varepsilon>0$ and suppose $G^\lambda$ occurs. Then Theorem \ref{th:svag_bestammer}
guarantees that on level $\lambda$ the type 1 infection is eventually
contained in $(1+ \varepsilon)\lambda tA$, a conclusion that
extends to all $\lambda'>\lambda$, because increasing the type 2 infection
rate does not help type 1. On the other hand, for any
$\lambda'>\lambda$ we get on level $\lambda'$ that
the union of the two infections will -- again by
Theorem \ref{th:svag_bestammer} -- eventually contain
$(1- \varepsilon)\lambda' tA$, so by taking $\varepsilon$ sufficiently small we see that
the type 1 infection is strangled on level $\lambda'$, implying
(\ref{eq:in_the_coupling}), and Theorem \ref{th:pain} follows.
We will not prove Theorem \ref{th:svag_bestammer}, but
mention that the hard work in proving it lies in
establishing a certain key
result (Proposition 2.2 in H\"{a}ggstr\"{o}m and Pemantle (2000))
that asserts that if the strong infection type reaches outside
$(1 + \varepsilon) \lambda tA$ infinitely often, then the weak type is doomed.
The proof of this uses geometrical arguments, the most important
ingredient being a certain spiral construction, emanating from the part
of the
strong of infection reaching beyond $(1 + \varepsilon) \lambda tA$,
and designed to allow the strong type to
completely surround the weak type before the weak type catches up
from inside.
How would one go about to strengthen Theorem \ref{th:pain}
and rule out infinite coexistence for all $\lambda \neq 1$?
One possibility would be to try to derive a
contradiction with Theorem \ref{th:svag_bestammer} from the
assumption that the strong infection type grows unboundedly. For
instance, intuitively it seems likely that the strong type
occupying a positive fraction of the boundary of the infected set
would cause the speed of the growth to exceed the speed prescribed
by the weak infection type. This type of argument is indeed used
in Garet and Marchand (2007) to show, for $d=2$, that on the event
of infinite coexistence the fraction of infected sites occupied
by the strong infection will tend to $0$ as
$t\rightarrow \infty$. This feels like a strong indication that
infinite coexistence does not happen.
Another approach to strengthening Theorem
\ref{th:pain} in order to obtain the only-if direction of Conjecture
\ref{samex_conj} is based on the
observation that, since coexistence represents a power balance
between the infections, it is reasonable to expect that
$P^{1,\lambda}(G)$ decreases as $\lambda$ moves away from 1. We may
formulate that intuition as a conjecture:
\begin{conj}\label{conj:monotonitet}
For the two-type Richardson model on $\mathbb{Z}^d$ with $d\geq
2$, we have, for $\lambda<\lambda'\in(0,1]$, that
$P^{1,\lambda}(G)\leq P^{1,\lambda'}(G)$.
\end{conj}
\noindent A confirmation of this conjecture would, in combination
with Theorem \ref{th:pain}, clearly establish the only-if
direction of Conjecture \ref{samex_conj}: If $P^{1,\lambda}(G)>0$
for some $\lambda<1$, then, according to Conjecture
\ref{conj:monotonitet}, we would have $P^{1,\lambda'}(G)>0$ for
all $\lambda'\in(\lambda,1]$ as well. But the interval
$(\lambda,1]$ is uncountable, yielding a contradiction to Theorem
\ref{th:pain}.
Although Conjecture
\ref{conj:monotonitet} might seem close to obvious, it has turned
out to be very difficult to prove. A natural first
attempt would be to use coupling. Consider for instance the
coupling $Q$ described above. As pointed out, the events
$G^\lambda_1$ and $G^\lambda_2$ that the individual infections
grow unboundedly at parameter value $\lambda$ are then monotone in
$\lambda$, but one of them is increasing and the other is decreasing,
so monotonicity of their intersection $G^\lambda$ does not follow.
Hence more sophisticated arguments are needed.
Observing how our colleagues react during seminars and corridor chat,
we have noted that it is very tempting to go about trying to prove
Conjecture \ref{conj:monotonitet} by abstract and ``easy'' arguments,
here meaning arguments that do not involve any specifics about the geometry
or graph structure of $\mathbb{Z}^d$. To warn against such attempts,
Deijfen and H\"aggstr\"om (2006:2) constructed graphs on which
the two-type Richardson model fails to exhibit the monotonicity behavior
predicted in Conjecture \ref{conj:monotonitet}. Let us briefly
explain the results.
The dynamics of the
two-type Richardson model can of course be defined on graphs other than
the $\mathbb{Z}^d$ lattice.
For a graph $\mathcal{G}$, write Coex$(\mathcal{G})$ for the set
of all $\lambda\geq 1$ such that there exists a finite initial
configuration $(\xi_1, \xi_2)$
for which the two-type Richardson model with infection
intensities $1$ and $\lambda$
started from $(\xi_1, \xi_2)$
yields infinite coexistence with positive
probability. Note that, by time-scaling and interchange of the
infections, coexistence is possible at parameter value $\lambda$
if and only if it is possible at $\lambda^{-1}$, so no
information is lost by restricting to $\lambda\geq 1$. In Deijfen
and H\"aggstr\"om (2006:2) examples of graphs $\mathcal{G}$ are
given that demonstrate that, among others, the following kinds of
coexistence sets Coex$(\mathcal{G})$ are possible:
\begin{itemize}
\item[(i)] Coex$(\mathcal{G})$ may be an interval $(a,b)$ with
$1<a<b$.
\item[(ii)] For any positive integer $k$ the set
Coex$(\mathcal{G})$ may consist of exactly $k$ points.
\item[(iii)] Coex$(\mathcal{G})$ may be countably infinite.
\end{itemize}
\noindent All these phenomena show that the monotonicity suggested
in Conjecture \ref{conj:monotonitet} fails for general graphs.
However, a reasonable guess is that Conjecture
\ref{conj:monotonitet} is true on transitive graphs. Indeed, all
counterexamples provided by Deijfen and H\"aggstr\"om
are highly non-symmetric (one might even say ugly)
with certain parts of
the graph being designed specifically with propagation of type 1
in mind, while other parts are meant for type 2. We omit the details.
\section{Unbounded initial configurations}
\label{sect:unbounded}
Let us now go back to the $\mathbb{Z}^d$ setting and describe some
results from our most recent paper, Deijfen and H\"aggstr\"om (2007),
concerning the two-type model with unbounded initial configurations.
Roughly, the model will be started from
configurations where one of the infections occupies a single site
in an infinite ``sea" of the other type. The dynamics is as before
and also the question at issue is the same: can both infection
types simultaneously infect infinitely many sites? With both types
initially occupying infinitely many sites the answer is (apart
from in particularly silly cases) obviously yes, so we will focus
on configurations where type 1 starts with infinitely many sites
and type 2 with finitely many -- for simplicity only one. The
question then becomes whether type 2 is able to survive.
To describe the configurations in more detail, write
$(x_1,\ldots,x_d)$ for the coordinates of a point
$x\in\mathbb{Z}^d$, and define $\mathcal{H}=\{x:x_1=0\}$ and
$\mathcal{L}=\{x:x_1 \leq 0\textrm{ and }x_i=0\textrm{ for }i =2,\ldots, d\}$.
We will consider the following starting configurations.
\begin{equation}\label{initial_configurations}
\begin{array}{rl}
I(\mathcal{H}): & \mbox{all points in $\mathcal{H}\backslash\{{\bf 0}\}$ are type 1 infected and}\\
& {\bf 0} \mbox{ is type 2 infected, and}\\
I(\mathcal{L}): & \mbox{all points in $\mathcal{L}\backslash\{{\bf 0}\}$ are type 1 infected and}\\
& {\bf 0} \mbox{ is type 2 infected.}
\end{array}
\end{equation}
\noindent Interestingly, it turns out that the set of parameter values for
which type 2 is able to grow indefinitely is slightly different
for these two configuration. First note that, as before, we may
restrict to the case $\lambda_1=1$. Write
$P^{1,\lambda}_{\mathcal{H},\mathbf{0}}$ and
$P^{1,\lambda}_{\mathcal{L},\mathbf{0}}$ for the distribution of
the process started from $I(\mathcal{H})$ and $I(\mathcal{L})$
respectively and with type 2 intensity $\lambda$. The following
result, where $G_2$ denotes the event that type 2 grows
unboundedly, is proved in Deijfen and H\"aggstr\"om (2007).
\begin{theorem}\label{th:unbounded} For the two-type Richardson model in
$d\geq 2$ dimensions, we have
\begin{itemize}
\item[\rm{(a)}] $P^{1,\lambda}_{\mathcal{H},{\bf 0}}(G_2)>0$ if and only
if $\lambda>1$;
\item[\rm{\rm{(b)}}]$P^{1,\lambda}_{\mathcal{L},{\bf 0}}(G_2)>0$ if and
only if $\lambda\geq 1$.
\end{itemize}
\end{theorem}
\noindent In words, a strictly stronger type 2 infection will be able to
survive in both configurations, but, when the infections have the
same intensity, type 2 can survive only in the configuration
$I(\mathcal{L})$.
The proof of the if-direction of Theorem \ref{th:unbounded} (a) is
based on a lemma stating roughly that the speed of a hampered
one-type process, living only inside a tube which is bounded in
all directions except one, is close to the speed of an unhampered
process when the tube is large. For a two-type process started
from $I(\mathcal{H})$, this lemma can be used to show that, if the
strong type 2 infection at the origin is successful in the
beginning of the time course, it will take off along the
$x_1$-axis and grow faster than the surrounding type 1
infection inside a tube around the $x_1$-axis, thereby escaping
eradication. The same scenario -- that the type 2 infection rushes
away along the $x_1$-axis -- can, by different means, be proved to
have positive probability in a process with $\lambda=1$ started
from $I(\mathcal{L})$. Infinite growth for type 2 when
$\lambda<1$ is ruled out by the key proposition from H\"aggstr\"om
and Pemantle (2000) mentioned in Section 3. Proving that type 2
cannot survive in a process with $\lambda=1$ started from
$I(\mathcal{H})$ is the most tricky part. The idea is basically to
divide $\mathbb{Z}^d$ in different levels, the $l$-th level being
all sites with $x_1$-coordinate $l$, and then show that the
expected number of type 2 infected sites at level $l$ is constant
and equal to 1. It then follows from a certain comparison with a
one-type process on each level combined with an application of
Levy's 0-1 law that the number of type 2 infected sites at the
$l$-th level converges almost surely to 0 as $l\to\infty$.
Finally we mention a question formulated by Itai Benjamini as well
as by an anonymous referee of Deijfen and H\"aggstr\"om (2007).
We have seen that, when $\lambda=1$, the type 2 infection at the
origin can grow unboundedly from $I(\mathcal{L})$ but not from
$I(\mathcal{H})$. It is then natural to ask what happens if we
interpolate between these two configurations. More precisely, instead
of letting type 1 occupy only the negative $x_1$-axis (as in
$I(\mathcal{L})$), we let it occupy a cone
of constant slope around the same
axis. The question then is what the critical slope is for this
cone such that there is a positive probability for type 2 to grow
unboundedly. That type 2 cannot survive when the cone occupies
the whole left half-space follows from Theorem \ref{th:unbounded},
as this situation is equivalent to starting the process from
$I(\mathcal{H})$. It seems likely, as suggested by Itai Benjamini,
that this is actually also the critical case, that is,
infinite growth for type 2 most likely have positive probability
for any smaller type 1 cone. This however remains to be proved.
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\end{document} |
\begin{document}
\title{One class genera of lattice chains over number fields}
\author{Markus Kirschmer}
\email{[email protected]}
\author{Gabriele Nebe}
\email{[email protected]}
\address{Lehrstuhl D f\"ur Mathematik, RWTH Aachen University, 52056 Aachen, Germany}
\begin{abstract}
We classify all one-class genera of admissible
lattice chains of length at least $2$ in hermitian spaces over number fields.
\end{abstract}
\maketitle
1.73205080756887729352744634ection{Introduction}
Kantor, Liebler and Tits \cite{KLT}
classified discrete groups $\Gamma $ with a type preserving chamber transitive action
on the affine building $\mathcal{B} ^+$ of a simple adjoint algebraic group
of relative rank $r\geq 2$.
Such groups are very rare and hence this situation is
an interesting phenomenon.
Except for two cases in characteristic 2 (\cite[case (v)]{KLT}) and
the exceptional group $\textnormal{G}_2(\mathbb{Q}_2) $ (\cite[case (iii)]{KLT}, Section \ref{g2})
the groups arise from classical groups $\textnormal{U}_p$ over $\mathbb{Q}_p$ for $p=2,3 $.
Moreover $\Gamma $ is a subgroup of the $S$-arithmetic group
$\Gamma _{max} := \mathbb{A}ut (L \otimes _{\mathbb{Z} } \mathbb{Z} [\frac{1}{p} ]) $ (so $S = \{ p \}$)
for a suitable lattice $L$ in some hermitian space $(V,\Phi )$ and
$\textnormal{U}_p = \textnormal{U} (V_p,\Phi )$ is the completion of the unitary group $\textnormal{U}(V,\Phi )$
(see Remark \ref{completiongroup}).
This paper uses the classification of one- and two-class genera of hermitian lattices
in \cite{Kirschmer} to obtain
these $S$-arithmetic groups $\Gamma _{max} $.
Instead of the thick building $\mathcal{B}^+$
we start with the affine building $\mathcal{B}$ of admissible lattice chains
as defined in \cite{AN}.
The points in the building ${\mathcal B}$ correspond to
homothety classes of certain $\mathbb{Z} _p$-lattices in $V_p $.
The lattices form a simplex in ${\mathcal B} $, if and only if
representatives in these classes can be chosen to form an
admissible chain of lattices in $V_p$.
In particular the maximal simplices of ${\mathcal B} $ (the so called chambers)
correspond to the fine admissible lattice
chains in $V_p$ (for the thick building
$\mathcal{B} ^+ $ one might have to apply the oriflamme construction
as explained in Remark \ref{oriflamme}).
Any fine admissible lattice chain $\mathcal{L} _p$ in $V_p$ arises as
the completion of a lattice chain $\mathcal{L} '$ in $(V,\Phi )$.
After rescaling and applying the reduction operators from Section \ref{normalizedgenera}
we obtain a fine $p$-admissible lattice chain
${\mathcal L} = (L_0,\ldots , L_r )$ in
$(V,\Phi )$ (see Definition \ref{admissible}) such that
$\mathbb{A}ut ({\mathcal L}) 1.73205080756887729352744634upseteq \mathbb{A}ut (\mathcal{L} ') $ and such that
the completion of $\mathcal{L}$ at $p$ is $\mathcal{L}_p$.
The $S$-arithmetic group $\mathbb{A}ut(L_0 \otimes \mathbb{Z}[\frac{1}{p} ] ) = \mathbb{A}ut(L_i \otimes \mathbb{Z}[\frac{1}{p}])
=: \mathbb{A}ut(\mathcal{L} \otimes \mathbb{Z}[\frac{1}{p}] ) $ contains $\mathbb{A}ut (\mathcal{L}' \otimes \mathbb{Z}[\frac{1}{p}] )$.
Therefore we call this group closed.
The closed $\{ p \} $-arithmetic group $\mathbb{A}ut(L_0 \otimes \mathbb{Z}[\frac{1}{p} ] ) $
acts chamber transitively on ${\mathcal B}$, if the lattice $L_0$ represents a genus of class
number one
and $\mathbb{A}ut (L_0 )$ acts transitively on the fine flags
of (isotropic) subspaces in the hermitian space $\overline{L_0}$ (see Theorem \ref{main}).
If we only impose chamber transitivity on
the thick building $\mathcal{B} ^+$, then
we also have to take two-class genera of lattices $L_0$ into account.
To obtain a complete classification of all chamber transitive actions of
closed $S$-arithmetic groups on the thick building $\mathcal{B} ^+$
we would have to apply our strategy to the still unknown classification of
proper special genera of lattices $L_0$ with class number one (see Theorem \ref{classtwo}).
Already taking only the one-class genera of lattices $L_0$ we find
all the groups from \cite{KLT} and one additional case (described in Proposition \ref{hf} (1)).
Hence our computations
confirm and supplement the classification of \cite{KLT}.
A list of the corresponding buildings and groups $\textnormal{U}_p $ is
given in Section \ref{buildtable}.
1.73205080756887729352744634ection{Lattices in hermitian spaces}
Let $K$ be a number field.
Further, let $E/K$ be a field extension of degree at most $2$ or let $E$ be a quaternion skewfield over $K$.
The canonical involution of $E/K$ will be denoted by $1.73205080756887729352744634igma \colon E \to E$. In particular, $K$ is the fixed field of
$1.73205080756887729352744634igma$ and hence the involution $1.73205080756887729352744634igma$ is the identity if and only if $K=E$.
A hermitian space over $E$ is a finitely generated (left) vector space $V$ over $E$ equipped with a non-degenerate
sesquilinear form $\Phi \colon V \times V \to E$ such that
\begin{itemize}
\item $\Phi(x+x', y) = \Phi(x,y) + \Phi(x', y)$ for all $x,x',y \in V$.
\item $\Phi(\alpha x, \beta y) = \alpha \Phi(x,y) 1.73205080756887729352744634igma(\beta)$ for all $x,y \in V$ and $\alpha, \beta \in E$.
\item $\Phi(y,x) = 1.73205080756887729352744634igma(\Phi(x, y))$ for all $x,y \in V$.
\end{itemize}
The unitary group
$\textnormal{U} (V,\Phi ) $
of $\Phi $ is the group of all $E$-linear endomorphisms of $V$ that
preserve the hermitian form $\Phi$.
The {\em special unitary group} is defined as
$$\textnormal{SU}(V,\Phi ):= \{ g\in \textnormal{U}(V,\Phi ) \mid \det(g) = 1 \} $$
if $E$ is commutative and $\textnormal{SU}(V,\Phi ):= \textnormal{U}(V,\Phi) $ if $E$ is a
quaternion algebra.
We denote by $\mathbb{Z}_K$ the ring of integers of the field $K$ and we fix some maximal order $\overline{\textnormal{M}}_vO$ in $E$. Further, let $d$ be the dimension of $V$ over $E$.
\begin{definition}
An {\em $\overline{\textnormal{M}}_vO $-lattice} in $V$ is a finitely generated
$\overline{\textnormal{M}}_vO $-submodule of $V$ that contains an $E$-basis of $V$.
If $L$ is an $\overline{\textnormal{M}}_vO $-lattice in $V$ then its {\em automorphism group}
is
$$\mathbb{A}ut(L):= \{ g\in \textnormal{U}(V,\Phi ) \mid Lg = L \} .$$
\end{definition}
1.73205080756887729352744634ubsection{Completion of lattices and groups}
Let $\mathfrak{P}$ be a maximal two sided ideal of $\overline{\textnormal{M}}_vO$ and let $\mathfrak{p} = \mathfrak{P} \cap K$.
The completion $\textnormal{U}_{\mathfrak{p}} := \textnormal{U}(V\otimes _K K_{\mathfrak{p} } ,\Phi )$ is an algebraic group over the $\mathfrak{p}$-adic completion $K_{\mathfrak{p} }$ of $K$.
Let $L\leq V$ be some $\overline{\textnormal{M}}_vO $-lattice in $V$.
We define the $\mathfrak{p} $-adic completion of $L$ as $L_{\mathfrak{p} }:=L\otimes _{\mathbb{Z}_K} \mathbb{Z}_{K_{\mathfrak{p}}} $ and we let
$$L(\mathfrak{p}):=\{ X\leq V \mid X_{\mathfrak{q}} = L_{\mathfrak{q} } \mbox{ for
all prime ideals } \mathfrak{q} \neq \mathfrak{p} \}, $$
be the set of all $\overline{\textnormal{M}}_vO $-lattices in $V$ whose $\mathfrak{q} $-adic
completion coincides with the one of $L$ for all prime ideals $\mathfrak{q} \neq \mathfrak{p}$.
\begin{remark} \label{localglobal}
By the local global principle, given a lattice $X$ in $V_{\mathfrak{p} }$,
there is a unique lattice $M \in L(\mathfrak{p}) $ with $M_{\mathfrak{p}} = X $.
\end{remark}
To describe the groups $\textnormal{U}_{\mathfrak{p}}$ in the respective cases,
we need some notation:
Let $R$ be one of $E,K,\mathbb{Z}_K ,\overline{\textnormal{M}}_vO $ or a suitable completion.
A hermitian module $\mathbb{H} (R)$ with $R$-basis $(e,f)$ satisfying
$\Phi (e,f) = 1 , \Phi(e,e) = \Phi(f,f) = 0 $ is called a
{\em hyperbolic plane}.
By \cite[Theorem (2.22)]{Kneser} any hermitian space over $E$
is either anisotropic (i.e.
$\Phi (x,x) \neq 0 $ for all $x\neq 0$) or has a hyperbolic plane as an orthogonal
direct summand.
\begin{remark}\label{completiongroup}
In our situation the following cases are possible:
\begin{itemize}
\item $E=K$: Then $(V\otimes _K K_{\mathfrak{p} } ,\Phi ) $ is a quadratic space and hence isometric to
$\mathbb{H} (K_{\mathfrak{p} } ) ^r \perp (V_0,\Phi _0)$ with $(V_0,\Phi_0)$ anisotropic.
The rank of $\textnormal{U}_{\mathfrak{p}} $ is $r$.
The group that acts type preservingly on the
thick Bruhat-Tits building $\mathcal{B} ^+$ defined in Section \ref{Secoriflamme} is
$$\textnormal{U}^+_{\mathfrak{p} }:= \{ g\in \textnormal{U}_{\mathfrak{p}} \mid \det (g) = 1, \theta (g) \in K^2 \} $$
the subgroup of the special orthogonal group with trivial spinor norm $\theta$.
\item $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P})$. Then $E\otimes _K K_{\mathfrak{p}} \cong K_{\mathfrak{p}} \oplus K_{\mathfrak{p} }$
where the involution interchanges the two components and $\textnormal{U}_{\mathfrak{p}} \cong \textnormal{GL} _d (K_{\mathfrak{p}}) $ has
rank $r=d-1$.
As $\mathfrak{P}$ is assumed to be a maximal $2$-sided ideal of $\overline{\textnormal{M}}_vO $, the case that
$E$ is a quaternion algebra is not possible here.
Here we let $$\textnormal{U}^+_{\mathfrak{p} } = \{
g\in \textnormal{U}_{\mathfrak{p}} \mid \det (g) = 1 \}
= \textnormal{SL} _d(K_{\mathfrak{p}}) .$$
\item $[E:K] =4$ and $\mathfrak{P} =\mathfrak{p} \overline{\textnormal{M}}_vO$. Then $E_\mathfrak{p} \cong K_\mathfrak{p}^{2 \times 2}$ and for $x \in E_\mathfrak{p}$, $1.73205080756887729352744634igma(x)$ is simply the adjugate of $x$ as $1.73205080756887729352744634igma(x) x \in K$.
Let $e^2=e \in E_{\mathfrak{p}} $ such that $1.73205080756887729352744634igma(e) = 1-e$. Then
$V_\mathfrak{p} = e V_\mathfrak{p} \bigoplus (1-e) V_\mathfrak{p}$.
The hermitian form $\Phi$ gives rise to a skew-symmetric form
\begin{align*} \Psi \colon eV_\mathfrak{p} \times eV_\mathfrak{p} &\to e E_\mathfrak{p} (1-e) \cong K_\mathfrak{p},\\ (ex, ey) &\mapsto \Phi(ex,ey) = e \Phi(x,y) (1-e) \:. \end{align*}
From $E_\mathfrak{p} = E_\mathfrak{p} e E_\mathfrak{p}$ we conclude that $V_\mathfrak{p} = E_\mathfrak{p} e E_\mathfrak{p} V$.
Hence we can recover the form $\Phi$ from $\Psi$ and thus $\textnormal{U}_\mathfrak{p} \cong \textnormal{U}(eV, \Psi) \cong \textnormal{Sp}_{2d}(K_\mathfrak{p})$ has rank $r=d$.
Here the full group $\textnormal{U}_{\mathfrak{p} }$ acts type preservingly on
$\mathcal{B}^+$ and we put $\textnormal{U}^+_{\mathfrak{p} } := \textnormal{U}_{\mathfrak{p} } $.
\item
In the remaining cases $E\otimes K_{\mathfrak{p} } = E_{\mathfrak{P} }$ is a skewfield,
which is ramified over $K_{\mathfrak{p}} $ if and only if $\mathfrak{P} ^2 = \mathfrak{p} \overline{\textnormal{M}}_vO $.
In all cases $\textnormal{U}_{\mathfrak{p}} $ is isomorphic to a unitary group over $E_{\mathfrak{P}} $.
Hence it admits a decomposition $\mathbb{H}(E_\mathfrak{P})^r \perp (V_0,\Phi _0)$ with $(V_0,\Phi_0)$ anisotropic
where $r$ is the rank of $\textnormal{U}_{\mathfrak{p}} $. If
$E_{\mathfrak{p} } $ is commutative, we define
$$\textnormal{U}^+_{\mathfrak{p} }:= \{ g \in \textnormal{U}_{\mathfrak{p} } \mid \det(g) =1 \}
= \textnormal{SU}_{\mathfrak{p} } $$
and put $\textnormal{U}^+_{\mathfrak{p} } = \textnormal{SU} _{\mathfrak{p}} := \textnormal{U} _{\mathfrak{p} }$ in the non-commutative
case.
\end{itemize}
\end{remark}
1.73205080756887729352744634ubsection{The genus of a lattice}
To shorten notation, we introduce the adelic ring $A= A(K) = \prod_v K_v$ where
$v$ runs over the set of all places of $K$.
We denote the adelic unitary group of the
$A\otimes _K E$-module $V_A = A \otimes_K V$ by $\textnormal{U} (V_A, \Phi )$.
The normal subgroup
$$\textnormal{U} ^+(V_A , \Phi ) := \{ (g_{\mathfrak{p} } ) _{\mathfrak{p}} \in \textnormal{U} (V_A, \Phi ) \mid
g_{\mathfrak{p} } \in \textnormal{U}^+_{\mathfrak{p} } \} \leq \textnormal{U} (V_A, \Phi ) $$
is called the special adelic unitary group.
The adelic unitary group acts on the set of all $\overline{\textnormal{M}}_vO $-lattices in $V$
by letting $Lg = L'$ where $L'$ is the unique lattice in $V$ such that
its $\mathfrak{p} $-adic completion $(L')_\mathfrak{p} = L_{\mathfrak{p}} g_{\mathfrak{p}}$
for all maximal ideals $\mathfrak{p} $ of $\mathbb{Z} _K$.
\begin{definition}
Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$. Then
\[ \textnormal{genus}(L):= \{ Lg \mid g \in \textnormal{U} (V_A, \Phi ) \} \]
is called the \emph{genus} of $L$. \\
Two lattices $L$ and $M$ are said to be \emph{isometric} (respectively {\em properly isometric}), if $L = Mg$ for some $g \in \textnormal{U}(V,\Phi)$
(resp. $g\in \textnormal{SU}(V,\Phi ) $).
\\
Two lattices $L$ and $M$ are said to be in the same
{\em proper special genus},
if there exist $g\in \textnormal{SU}(V,\Phi )$ and $h\in \textnormal{U}^+(V_A,\Phi )$
such that $Lgh = M$.
\end{definition}
Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$.
It is well known that $\textnormal{genus}(L)$ is a finite union of isometry classes, c.f. \cite[Theorem 5.1]{Borel_finite}.
The number of isometry classes in $\textnormal{genus}(L)$ is called the class number
$h(L) $ of (the genus of) $L$.
Similarly the proper special genus is a finite union of proper isometry
classes, the proper class number will be denoted by $h^+(L)$.
1.73205080756887729352744634ubsection{Normalized genera} \label{normalizedgenera}
\begin{definition}
Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$. Then $L^\# = \{x \in V \mid \Phi(x,L) 1.73205080756887729352744634ubseteq \overline{\textnormal{M}}_vO \}$ is called the \emph{dual} lattice of $L$.
If $\mathfrak{p}$ is a maximal ideal of $\mathbb{Z}_K$, then the unique \mbox{$\overline{\textnormal{M}}_vO$-lattice} $X
\in L(\mathfrak{p} )$ such that
$ X_\mathfrak{p} = L_\mathfrak{p}^\# $
is called the \emph{partial dual of $L$ at $\mathfrak{p}$}. It will be denoted by $L^{\#, \mathfrak{p}}$.
\end{definition}
\begin{definition}\label{squarefree}
Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$.
Further, let $\mathfrak{P}$ be a maximal two sided ideal of $\overline{\textnormal{M}}_vO$ and set $\mathfrak{p} = \mathfrak{P} \cap K$.
If $E_\mathfrak{p} \cong K_\mathfrak{p} \oplus K_\mathfrak{p}$ then $L_\mathfrak{p}$ is called \emph{square-free} if $L_\mathfrak{p} = L_\mathfrak{p}^\#$.
In all other cases, $L_\mathfrak{p}$ is called square-free if $\mathfrak{P} L_\mathfrak{p}^\# 1.73205080756887729352744634ubseteq L_\mathfrak{p} 1.73205080756887729352744634ubseteq L_\mathfrak{p}^\#$.
The lattice $L$ is called square-free if $L_\mathfrak{p}$ is square-free for all maximal ideals $\mathfrak{p}$ of $\mathbb{Z}_K$.
\end{definition}
Given a maximal two sided ideal $\mathfrak{P}$ of $\overline{\textnormal{M}}_vO$, we define an operator $\rho_\mathfrak{P}$ on the set of all $\overline{\textnormal{M}}_vO$-lattices as follows:
\[ \rho_\mathfrak{P}(L) =
\begin{cases}
L + (\mathfrak{P}^{-1} L \cap L^\#) & \text{if }\mathfrak{P} \ne 1.73205080756887729352744634igma(\mathfrak{P}),\\
L + (\mathfrak{P}^{-1} L \cap \mathfrak{P} L^\#) & \text{otherwise.}
\end{cases} \]
The operators generalize the maps defined by L. Gerstein in \cite{Gerstein} for quadratic spaces.
They are similar in nature to the \emph{$p$-mappings} introduced by G.~Watson in \cite{pMap}.
The maps satisfy the following properties:
\begin{remark}\label{RhoRemark} Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$. Let $\mathfrak{P}$ be a maximal two sided ideal of~$\overline{\textnormal{M}}_vO$ and set $\mathfrak{p}= \mathfrak{P} \cap \mathbb{Z}_K$.
\begin{enumerate}
\item $\rho_\mathfrak{P}(L) \in L(\mathfrak{p} )$.
\item If $L_\mathfrak{p}$ is integral, then $(\rho_\mathfrak{P}(L))_\mathfrak{p} = L_\mathfrak{p} \iff L_\mathfrak{p}$ is square-free.
\item If $\mathfrak{Q}$ is a maximal two sided ideal of $\overline{\textnormal{M}}_vO$, then $\rho_{\mathfrak{P}} \circ \rho_\mathfrak{Q} = \rho_{\mathfrak{Q}} \circ \rho_\mathfrak{P}$.
\item If $L$ is integral, there exist a sequence
of not necessarily distinct
maximal two sided ideals $\mathfrak{P}_1, \dots, \mathfrak{P}_s$ of $\overline{\textnormal{M}}_vO$ such that
\[ L':= (\rho_{\mathfrak{P}_1} \circ \ldots \circ \rho_{\mathfrak{P}_s})(L) \]
is square-free. Moreover, the genus of $L'$ is uniquely determined by the genus of $L$.
\end{enumerate}
\end{remark}
\begin{prop}
Let $L$ be an $\overline{\textnormal{M}}_vO$-lattice in $V$ and let $\mathfrak{P}$ be a maximal two sided ideal of~$\overline{\textnormal{M}}_vO$.
Then the class number of $\rho_\mathfrak{P}(L)$ is at most the class number of $L$.
\end{prop}
\begin{proof}
The definition of $\rho_\mathfrak{P}(L)$ only involves taking sums and intersections of multiples of $L$ and its dual.
Hence $ \rho_\mathfrak{P}(L)g = \rho_\mathfrak{P} (Lg) $ for all $g \in \textnormal{U}(V, \Phi)$ and similar for $g \in \textnormal{U}(V_A, \Phi)$.
In particular, $\rho_\mathfrak{P}$ maps lattices in the same genus (isometry class) to ones in the same genus (isometry class).
The result follows.
\end{proof}
\begin{definition}
Let $\mathfrak{A}$ be a two sided $\overline{\textnormal{M}}_vO$-ideal.
An $\overline{\textnormal{M}}_vO$-lattice $L$ is called $\mathfrak{A}$-maximal, if $\Phi(x,x) \in \mathfrak{A}$ for all $x \in L$ and no proper overlattice of $L$ has that property.
Similarly, one defines maximal lattices in $V_\mathfrak{p}$ for a maximal ideal $\mathfrak{p}$ of $\mathbb{Z}_K$.
\end{definition}
\begin{definition}\label{normalised}
Let $\mathfrak{P}$ be a maximal two sided ideal of~$\overline{\textnormal{M}}_vO$ and set $\mathfrak{p} = \mathfrak{P} \cap K$.
We say that an $\overline{\textnormal{M}}_vO $-lattice
$L$ is $\mathfrak{p}$-\emph{normalized} if $L$ satisfies the following conditions:
\begin{itemize}
\item $L$ is square-free.
\item If $E=K$ then $L_\mathfrak{p} \cong \mathbb{H}(\mathbb{Z}_{K _\mathfrak{p}})^r \perp M_0$
where $M_0 = \rho_\mathfrak{p}^\infty(M)$ and $M$ denotes a \mbox{$2\mathbb{Z}_{K_\mathfrak{p}}$-maximal} lattice in an anisotropic quadratic space over $K_\mathfrak{p}$.
\item If $E_\mathfrak{p}/K_\mathfrak{p}$ is a quadratic field extension with different $\mathcal{D}(E_\mathfrak{p}/K_\mathfrak{p})$,
then $L_\mathfrak{p} \cong \mathbb{H}(\overline{\textnormal{M}}_vO _\mathfrak{p})^r \perp M_0 $ where $M_0 = \rho_\mathfrak{p}^\infty(M)$ and $M$ denotes a $\mathcal{D}(E_\mathfrak{p}/K_\mathfrak{p})$-maximal lattice in an anisotropic hermitian space over $E_\mathfrak{p}$.
\item If $[E:K] = 4$, then $L_\mathfrak{p} = L_\mathfrak{p}^\#$.
\end{itemize}
Here $\rho_\mathfrak{P}^\infty(M)$ denotes the image of $M$ under repeated application of $\rho_\mathfrak{P}$ until this process becomes stable.
\end{definition}
\begin{remark}
Let $\mathfrak{P}, \mathfrak{p}$ and $L$ be as in Definition \ref{normalised}. Then the isometry class of $L_\mathfrak{p}$ is uniquely determined by $(V_\mathfrak{p},\Phi)$.
\end{remark}
\begin{proof}
There is nothing to show if $[E:K]=4$. Suppose now $E=K$.
The space $K M_0$ is a maximal anisotropic subspace of $(V_\mathfrak{p}, \Phi)$. By Witt's theorem \cite[Theorem 42:17]{OMeara} its isometry type is uniquely determined by $(V_\mathfrak{p}, \Phi)$.
Further, $M_0$ is the unique $2\mathbb{Z}_{K_\mathfrak{p}}$-maximal $\mathbb{Z}_{K_\mathfrak{p}}$-lattice in $K M_0$, see \cite[Theorem 91:1]{OMeara}.
Hence the isometry type of $\rho_\mathfrak{p}^{\infty}(M_0)$ depends only on $(V_\mathfrak{p},\Phi)$.
The case $[E:K]=2$ is proved similarly.
\end{proof}
1.73205080756887729352744634ection{Genera of lattice chains}
\begin{definition}
Let $\mathcal{L} := (L_1,\ldots , L_m )$ and $\mathcal{L}' := (L'_1,\ldots ,L'_m)$
be two $m$-tuples of $\overline{\textnormal{M}}_vO$-lattices in $V$.
Then $\mathcal{L} $ and $\mathcal{L}'$ are
{\em isometric}, if there is some $g\in \textnormal{U} (V,\Phi )$
such that $L_ig = L_i' $ for all $i=1,\ldots,m $.
They are in the same {\em genus}
if there is such an element $g\in \textnormal{U} (V_A,\Phi )$.
Let
$$[\mathcal{L} ] := \{ \mathcal{L}' \mid \mathcal{L}' \mbox{ is isometric to } \mathcal{L} \} $$
and
$$\textnormal{genus} (\mathcal{L}) := \{ \mathcal{L}' \mid \mathcal{L}' \mbox{ and } \mathcal{L} \mbox{ are in the same genus} \} $$
denote the isometry class and the genus of $\mathcal{L} $, respectively.
The {\em automorphism group} of $\mathcal{L} $ is the stabilizer of $\mathcal{L} $
in $\textnormal{U}(V,\Phi )$, i.e.
$$\mathbb{A}ut (\mathcal{L} ) = \bigcap _{i=1}^m \mathbb{A}ut (L_i ) .$$
\end{definition}
It is well known \cite[Theorem 5.1]{Borel_finite} that any genus of a single lattice
contains only finitely many isometry classes.
This is also true for finite tuples of lattices in $V$:
\begin{lemma}
Let $\mathcal{L} = (L_1,\ldots , L_m )$ be an $m$-tuple of $\overline{\textnormal{M}}_vO$-lattices in $V$.
Then $\textnormal{genus} (\mathcal{L}) $ is the disjoint union of finitely
many isometry classes.
The number of isometry classes in $\textnormal{genus}(\mathcal{L} ) $ is
called the class number of $\mathcal{L} $.
\end{lemma}
\begin{proof}
The case $m=1$ is the classical case.
So assume that $m\geq 2$ and let
$\textnormal{genus} (L_1) := [M_1] \uplus \ldots \uplus [M_h] $,
with $M_i =L_1 g_i $ for suitable $g_i \in \textnormal{U}(V_A,\Phi )$.
We decompose
$\textnormal{genus} (\mathcal{L} ) = \mathcal{G} _1 \uplus \ldots \uplus \mathcal{G} _h $
where
$\mathcal{G} _i := \{ (L'_1,\ldots ,L'_m) \in \textnormal{genus} (\mathcal{L} ) \mid
L'_1 \cong M_i \} $.
It is clearly enough to show that each $\mathcal{G} _i$
is the union of finitely many isometry classes.
By construction, any isometry class in $\mathcal{G} _i$ contains a representative
of the form $(M_i,L'_2,\ldots,L'_m )$ for some lattices $L'_j$ in the
genus of $L_j$.
As all the $L_j$ are lattices in the same vector space $V$, there are
$a,b \in \mathbb{Z}_K$ such that
$$ b L_1 1.73205080756887729352744634ubseteq L_j 1.73205080756887729352744634ubseteq \frac{1}{a} L_1 \mbox{ for all } 1\leq j \leq m.$$
As
$(M_i,L'_2,\ldots,L'_m ) = \mathcal{L} g$ for some $g\in \textnormal{U}(V_A,\Phi )$
we also have
$$ b M_i 1.73205080756887729352744634ubseteq L'_j 1.73205080756887729352744634ubseteq \frac{1}{a} M_i \mbox{ for all } 2\leq j \leq m .$$
So there are only finitely many possibilities for such lattices $L'_j$.
Hence
the set of all $m$-tuples $(M_i,L'_2,\ldots,L'_m ) \in \textnormal{genus} (\mathcal{L} )$
is finite and so is the class number.
\end{proof}
\begin{remark}
If $\mathcal{L} ' 1.73205080756887729352744634ubseteq \mathcal{L} $ then
the class number of
$\mathcal{L}' $ is at most the class number of $\mathcal{L}$.
\end{remark}
1.73205080756887729352744634ubsection{Admissible lattice chains}
\begin{definition}\label{admissible}
Let $\mathfrak{P}$ be a maximal 2-sided ideal of $\overline{\textnormal{M}}_vO$ and $\mathfrak{p} := K \cap \mathfrak{P}$.
A lattice chain
$$ \mathcal{L} : = \{ L_01.73205080756887729352744634upset L_{1} 1.73205080756887729352744634upset \ldots 1.73205080756887729352744634upset L_{m-1} 1.73205080756887729352744634upset L_m \} $$
is called {\em admissible} for $\mathfrak{P}$, if
\begin{enumerate}
\item $L_0 1.73205080756887729352744634ubseteq L_0^{\#, \mathfrak{p}} $,
\item $ \mathfrak{P} L_0 1.73205080756887729352744634ubset L_m $,
\item $\mathfrak{P} L_m^{\#, \mathfrak{p}} 1.73205080756887729352744634ubseteq L_m$ if $\mathfrak{P} = 1.73205080756887729352744634igma(\mathfrak{P}) $.
\end{enumerate}
We call a $\mathfrak{P}$-admissible chain {\em fine}, if
$L_0$ is normalized for $\mathfrak{p}$ in the sense of Definition \ref{normalised},
$L_i$ is a maximal sublattice of $L_{i-1}$ for all $i=1,\ldots , m$
and
either
\begin{itemize}
\item[(a)] $\mathfrak{P} = 1.73205080756887729352744634igma (\mathfrak{P} )$ and
$L_m/\mathfrak{P} L_m^{\#, \mathfrak{p}}$ is an anisotropic space over $\overline{\textnormal{M}}_vO/\mathfrak{P} $
\item[(b)] $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P}) $ and
$\mathfrak{P} L_0 $ is a maximal sublattice of $L_m$.
\end{itemize}
\end{definition}
\begin{remark}
In the case that $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P} )$ the
{\em length} $m$ of a fine admissible lattice chain
is just $m=r=\dim _E (V)-1$.
Also if $\mathfrak{P} = 1.73205080756887729352744634igma(\mathfrak{P}) $, then
$m=r$, where $r$ is the rank of the $\mathfrak{p}$-adic group defined in
Remark \ref{completiongroup}.
\end{remark}
Note that any admissible chain $\mathcal{L}$ contains a unique maximal
integral lattice which we will always denote by $L_0$.
\begin{remark}\label{masschain}
Let $\mathcal{L} =(L_0,\ldots , L_r)$ be a fine
admissible lattice chain for $\mathfrak{P} $.
\begin{itemize}
\item[(a)] If $\mathfrak{P} = 1.73205080756887729352744634igma(\mathfrak{P}) $
then $\overline{L_0} :=
L_0/\mathfrak{P} L_0^{\#,\mathfrak{p} }$ is a hermitian space
over $\overline{\textnormal{M}}_vO/\mathfrak{P} $ and the spaces
$V_j:=\mathfrak{P} L_j ^{\#,\mathfrak{p} }/ \mathfrak{P} L_0^{\#,\mathfrak{p} }$
($j=1,\ldots , r$) define
a maximal chain of isotropic subspaces of this hermitian space.
We call the chain $(V_1,\ldots , V_{r-1})$
{\em truncated}.
\item[(b)] If $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P}) $ then
$\overline{L_0}:=L_0/\mathfrak{P} L_0 $ is a vector space
over $\overline{\textnormal{M}}_vO/\mathfrak{P} $ and the spaces $V_j:=L_j/\mathfrak{P} L_0 $
($j=r,\ldots , 1$)
form a maximal chain of subspaces.
Here we call the chain $(V_{r-1},\ldots , V_1) $
{\em truncated}.
\end{itemize}
\end{remark}
For the different hermitian spaces $\overline{L_0}$, the number of
such chains of isotropic subspaces
can be found by recursively applying the formulas in
\cite[Exercises 8.1, 10.4, 11.3]{Taylor}.
\begin{corollary}\label{einklassig}
The fine admissible lattice chain $\mathcal{L} $ represents a one-class genus of lattice chains
if and only if $L_0$ represents a one-class genus of lattices
and $\mathbb{A}ut(L_0)$ is transitive on the
maximal chains of (isotropic) subspaces of $\overline{L_0}$.
\end{corollary}
1.73205080756887729352744634ection{Chamber transitive actions on affine buildings.}
Kantor, Liebler and Tits \cite{KLT}
classified discrete groups acting chamber transitively and type preservingly
on the affine building of a simple adjoint algebraic group
of relative rank $\geq 2$
over a locally compact local field.
Such groups are very rare and hence this situation is
an interesting phenomenon, further studied in
\cite{Ka1}, \cite{Ka2}, \cite{KMW1}, \cite{MW1}, \cite{KMW90},
and \cite{M} (and many more papers by these authors)
where explicit constructions of the groups are given.
One major disadvantage of the existing literature is that the
proof in \cite{KLT} is very sketchy, essentially the authors
limit the possibilities that need to be checked to a finite number.
From the classification of the one-class genera of
admissible fine lattice chains in Section \ref{oneclasschains}, we obtain
a number theoretic construction of the groups
in \cite{KLT} over fields of characteristic 0.
It turns out that we find essentially all these
groups and that our construction allows to find one more
case:
The building of $\textnormal{U}_5(\mathbb{Q}_3(1.73205080756887729352744634qrt{-3}) )$
of type $C-BC_2$, see Proposition \ref{hf} (1),
which, to our best knowledge, has not appeared in the literature before.
1.73205080756887729352744634ubsection{S-arithmetic groups}
We assume that
$(V,\Phi )$ is a totally positive definite hermitian space, i.e.
$K$ is totally real and
$\Phi (x,x) \in K $ is totally positive for all non-zero $x\in V$.
Let $S =\{ \mathfrak{p}_1,\ldots , \mathfrak{p}_m \} $ be a finite set
of prime ideals of $\mathbb{Z}_K$.
For a prime ideal $\mathfrak{p} $ we denote by $\nu _{\mathfrak{p}}$ the
$\mathfrak{p} $-adic valuation of $K$.
Then the ring of $S$-integers in $K$ is
$$ \mathbb{Z}_S :=
\{ a\in K \mid \nu _{\mathfrak{q}} (a) \geq 0 \text{ for all prime ideals }
\mathfrak{q} \notin S \} .$$
Let $L$ be some $\overline{\textnormal{M}}_vO $-lattice in $(V,\Phi )$ and put
$L_S:= L\otimes_{\mathbb{Z}_K} \mathbb{Z}_S$.
Then the group
$$\mathbb{A}ut (L_S ) := \{ g\in \textnormal{U}(V,\Phi ) \mid L_S g = L_S \} $$
is an $S$-arithmetic subgroup of $\textnormal{U}(V,\Phi )$.
\begin{remark}
For any prime ideal $\mathfrak{p}$, the group $\textnormal{U}(V,\Phi )$
(being a subgroup of $\textnormal{U}_{\mathfrak{p} }$)
acts on the Bruhat-Tits building $\mathcal{B}$ of the group $\textnormal{U}_{\mathfrak{p} }$
defined in Remark \ref{completiongroup}.
Assume that the rank of $\textnormal{U}_{\mathfrak{p} }$ is at least 1.
The action of the subgroup $\mathbb{A}ut (L_S) $ is discrete and cocompact
on $\mathcal{B}$, if and only if $\mathfrak{p} \in S$ and $(V_{\mathfrak{q}} ,\Phi )$ is
anisotropic for all $\mathfrak{p}\neq \mathfrak{q} \in S $.
\\
Note that anisotropic spaces only exist up to dimension 4.
So if we assume that $\textnormal{U}_{\mathfrak{p}} ^+ $ is simple modulo scalars
and has rank $\geq 2$, then
$\mathbb{A}ut (L_S) $ is discrete and cocompact
on $\mathcal{B}$, if and only if $S=\{ \mathfrak{p} \}$
\end{remark}
1.73205080756887729352744634ubsection{The action on the building of $\textnormal{U}_{\mathfrak{p}}$}
In the following we fix a prime ideal $\mathfrak{p} $ and assume
that $S=\{ \mathfrak{p} \} $.
A lattice class model for the affine building
$\mathcal{B}$ has been described in
\cite{AN}. Note that \cite{AN} imposes the assumption that the
residue characteristic of $K_{\mathfrak{p} }$ is $p\neq 2$.
This is only necessary to obtain a proof of the building axioms
that is independent from Bruhat-Tits theory.
For $p=2$, the dissertation \cite{FrischDiss} contains
the analogous description of the Bruhat-Tits building
for orthogonal groups.
For all residue characteristics,
the chambers in $\mathcal{B} $ correspond
to certain fine lattice chains in the natural $\textnormal{U}_{\mathfrak{p}} $-module $W_{\mathfrak{p}} $.
Let $L$ be a fixed $\mathfrak{p} $-normalized lattice in $V$
and put $V_{\mathfrak{p}} := V\otimes _K K_{\mathfrak{p}} $.
In the case that $E\otimes _K K_{\mathfrak{p} }$ is a skewfield,
we decompose the completion
$$L_{\mathfrak{p} } = \mathbb{H}(\overline{\textnormal{M}}_vO _\mathfrak{p})^r \perp M_0
= \bigperp _{i=1}^r \langle e_i,f_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{p}}} \perp M_0$$
as in Definition \ref{normalised}.
Then $ V_\mathfrak{p} = V_0\perp
\langle e_1,\ldots , e_r,f_1,\ldots , f_r \rangle _{K_{\mathfrak{p}}} $
where $V_0 = K_{\mathfrak{p} } M_0$ is anisotropic.
Then the standard chamber corresponding to $L$ and the
choice of this hyperbolic basis is represented by
the admissible fine lattice chain
$$\mathcal{L} = (L=L_0,L_1,\ldots , L_{r} ) $$
where $L_j \in L(\mathfrak{p} ) $ is the unique lattice in $V$ such that
$$(L_j) _{\mathfrak{p}} = \bigperp _{i=1}^{j} \langle \pi e_i,f_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{p}}} \perp
\bigperp _{i=j+1}^{r} \langle e_i,f_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{p}}}
\perp M_0.$$
Now assume that $E\otimes _K K_{\mathfrak{p} } \cong K_{\mathfrak{p}}^{2\times 2}$
and $W_{\mathfrak{p} } = eV_{\mathfrak{p} } $ for some primitive idempotent $e$ such that
$1.73205080756887729352744634igma(e) = 1-e$ as in Remark \ref{completiongroup}.
Then $W_{\mathfrak{p} }$
carries a symplectic form $\Psi $ and the
lattice
$L_{\mathfrak{p} }e $ has a symplectic basis $(e_1,f_1,\ldots , e_r,f_r )$, i.e.
$$L_{\mathfrak{p} }e =
\bigperp _{i=1}^r \langle e_i,f_i \rangle _{\mathbb{Z} _{K_\mathfrak{p}}} $$
with $\Psi (e_i,f_i ) = 1$.
The standard chamber corresponding to $L$ and the
choice of this symplectic basis is represented by
the admissible fine lattice chain
$$\mathcal{L} = (L=L_0,L_1,\ldots , L_{r} ) $$
where $L_j \in L(\mathfrak{p} ) $ is the unique lattice in $V$ such that
$$(L_j) _{\mathfrak{p}} = \bigperp _{i=1}^{j} \langle \pi e_i,f_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{p}}} \perp
\bigperp _{i=j+1}^{r} \langle e_i,f_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{p}}}
.$$
In the last and most tricky case
$E\otimes _K K_{\mathfrak{p} } \cong K_{\mathfrak{p}} \oplus K_{\mathfrak{p}}$. Then $W_{\mathfrak{p}} = V_{\mathfrak{p} } e_{\mathfrak{P}} $ for any of
the two maximal ideals $\mathfrak{P} $ of $\overline{\textnormal{M}}_vO $ that contain $\mathfrak{p}$,
$\textnormal{U}_{\mathfrak{p} } 1.73205080756887729352744634upseteq \textnormal{SL}(W_{\mathfrak{p}}) $ and
$M_{\mathfrak{p}} := L _{\mathfrak{p}} e_{\mathfrak{P} } $ is a lattice in $W_{\mathfrak{p} }$.
To define the standard chamber fix some
$\mathbb{Z}_{K_\mathfrak{p}} $-basis $(e_1,\ldots,e_r)$ of $M_{\mathfrak{p}}$.
Then the fine admissible lattice chain
$$\mathcal{L} = (L=L_0,L_1,\ldots , L_{r} ) $$
where $L_j$ is the unique lattice in $V$ such that
\begin{itemize}
\item
$(L_j)_{\mathfrak{Q}} = L_{\mathfrak{Q}}$ for all prime ideals $ \mathfrak{Q} \neq \mathfrak{P} $ of $\overline{\textnormal{M}}_vO $
\item
$(L_j) _{\mathfrak{P}} = \bigoplus _{i=1}^{j} \langle \pi e_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{P}} } \oplus
\bigoplus _{i=j+1}^{r} \langle e_i \rangle _{\overline{\textnormal{M}}_vO _{\mathfrak{P}}} .$
\end{itemize}
\begin{lemma}\label{uniqueunimod}
Assume that $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P}) $, so $E\otimes _K K_{\mathfrak{p} } \cong K_{\mathfrak{p}} \oplus K_{\mathfrak{p}}$ and keep the notation from above.
Let $M$ be some $\overline{\textnormal{M}}_vO $-lattice in $V$.
Then
$
\{ X \in M(\mathfrak{p} )
\mid e _{\mathfrak{P}} X_{\mathfrak{p}} =e_{\mathfrak{P}}M_{\mathfrak{p}} \} $
contains a unique lattice $Y$
with $Y=Y^{\# ,\mathfrak{p} }$.
\end{lemma}
\begin{proof}
As $Y\in M(\mathfrak{p} )$ it is enough to define
$Y_{\mathfrak{p} } = e_{\mathfrak{P}}M_{\mathfrak{p}} \oplus (1-e_{\mathfrak{P}}) X_{\mathfrak{p}} $. This $\overline{\textnormal{M}}_vO _{\mathfrak{p}}$-lattice is unimodular if and only if
\[ (1-e_{\mathfrak{P}}) X_{\mathfrak{p} } = \{ x\in (1-e_{\mathfrak{P}}) V
\mid \Phi ( e_{\mathfrak{P}}M_{\mathfrak{p}}, x) 1.73205080756887729352744634ubseteq \overline{\textnormal{M}}_vO _{\mathfrak{p}} \} .\]
\end{proof}
Thus for $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P} )$ the stabilizer in the $S$-arithmetic
group $\mathbb{A}ut(L_S)$ of a vertex in the
building $\mathcal{B} $ is the automorphism group of a $\mathfrak{p} $-unimodular lattice.
Also if $\mathfrak{P} = 1.73205080756887729352744634igma(\mathfrak{P} )$,
any vertex in the building $\mathcal{B} $ corresponds to
a unique homothety class of lattices $[M_{\mathfrak{p}}]
=\{ a M_{\mathfrak{p} } \mid a\in K_{\mathfrak{p}}^* \} $.
So by Remark \ref{localglobal} there is a unique
lattice $X\in L(\mathfrak{p})$
with $X_{\mathfrak{p}} = M_{\mathfrak{p}} $.
Hence the stabilizers of the vertices in $\mathcal{B} $ are
exactly the automorphism groups of the respective lattices in $V$.
In particular these are finite groups.
\begin{remark}
As $\textnormal{U}_{\mathfrak{p}} $ acts transitively on the chambers
of $\mathcal{B} $, any other chamber (i.e. $r$-dimensional simplex)
in $\mathcal{B}$ corresponds to
some lattice chain in the genus of $\mathcal{L} = (L_0,\ldots , L_r) $.
The $(r-1)$-dimensional simplices are the $\textnormal{U}_{\mathfrak{p} }$-orbits of
the subchains
$\mathcal{L} _j := ( L_i \mid i\neq j ) $ of $\mathcal{L} $ for $j=0,\ldots , r$.
We call these simplices {\em panels} and $j$ the cotype of the
panel $\mathcal{L} _j$.
\end{remark}
\begin{theorem}\label{main}
Let $\mathcal{L} = (L_0,\ldots , L_r)$ be a fine admissible
lattice chain for $\mathfrak{P} $
of class number one.
Put $L:=L_0$ and $S:=\{ \mathfrak{p} \}$.
Then $\mathbb{A}ut (L_S )$ acts chamber
transitively on the (weak) Bruhat-Tits building $\mathcal{B}$ of the
completion $\textnormal{U}_{\mathfrak{p} }$.
\end{theorem}
\begin{proof}
We use the characterization of Lemma \ref{einklassig}.
Let $\mathcal{C} $ be the chamber of $\mathcal{B} $ that corresponds to $\mathcal{L} $
by the construction above and let $\mathcal{D} $ be some other chamber
in $\mathcal{B} $. Then there is some element $g\in \textnormal{U}_{\mathfrak{p} }$ with
$\mathcal{C} g = \mathcal{D} $. As the
genus of $L$ consists only of one class, there
is some $h \in \mathbb{A}ut(L_S) $ such that
$g h \in \textnormal{U}_{\mathfrak{p} }$ stabilizes the vertex $v$ that corresponds to
$L$. So $gh \in \textnormal{Stab} _{\textnormal{U}_{\mathfrak{p}}} (L_{\mathfrak{p} }) $
and $\mathcal{D} h $ is some chamber in $\mathcal{B} $
containing the vertex $v$. Now $\mathbb{A}ut (L) $ acts transitively
on the set of all fine admissible lattice chains for $\mathfrak{P} $ starting in
$L$, so there is some $h'\in \mathbb{A}ut(L) $ such that
$\mathcal{D} h h' = \mathcal{C} $.
Thus the element $h h' \in \mathbb{A}ut(L_S) $ maps $\mathcal{D} $ to $\mathcal{C} $.
\end{proof}
1.73205080756887729352744634ubsection{The oriflamme construction} \label{Secoriflamme}
The buildings $\mathcal{B}$ described above are in general not thick buildings,
i.e. there are panels that are only contained in exactly two chambers.
Such panels are called thin.
To obtain a thick building $\mathcal{B} ^+$ (with a type preserving action by
the group $\textnormal{U}_{\mathfrak{p}}^+$ defined in Remark \ref{completiongroup})
we need to apply a generalization of the
oriflamme construction as described in \cite[Section 8]{AN}.
In particular \cite[Section 8.1]{AN} gives the precise situations
which panels are thin for the case that $p\neq 2$.
Also for $p=2$ only the panels of cotype $0$ and $r$ can be thin.
We refrain from describing the situations for $p=2$ in general,
but refer to the individual examples below.
\begin{remark}\label{oriflamme}
Assume that $\mathfrak{P} = 1.73205080756887729352744634igma(\mathfrak{P} )$.
\begin{enumerate}
\item[(a)]
Assume that there are only two lattices
$L_0$ and $L_0'$ in the genus of $L_0$ such
that
$$L_1 1.73205080756887729352744634ubseteq L_0,L_0' 1.73205080756887729352744634ubseteq L_1^{\#,\mathfrak{p} }.$$
Then $\mathcal{L} $
and $\mathcal{L}' := (L_0', L_1,\ldots , L_{r}) $ are
the only chambers in $\mathcal{B} $ that contain the
panel $\mathcal{L}_0 = (L_1,\ldots , L_{r}) $ and hence
this panel is thin.
Then we replace the vertex represented by
$L_1$ by the one represented by $L_0'$.
\item[(b)]
Assume that there are only two lattices
$L_r$ and $L_r'$ in the genus of $L_r$ such
that
$$\mathfrak{P} L_{r-1}^{\#,\mathfrak{p} } 1.73205080756887729352744634ubseteq L_r,L_r' 1.73205080756887729352744634ubseteq L_{r-1}.$$
Then $\mathcal{L} $
and $\mathcal{L}' := (L_0, L_1,\ldots , L'_{r}) $ are
the only chambers in $\mathcal{B} $ that contain the
panel $\mathcal{L}_r = (L_0,\ldots , L_{r-1}) $ and hence
this panel is thin.
Then we replace the vertex represented by
$L_{r-1}$ by the one represented by $L_r'$.
\item[(c)]
After this construction the standard chamber ${\mathcal{L}}^+ $
in the
thick building $\mathcal{B}^+$ is either represented by
$\mathcal{L} $, $ (L_0,L_0',L_2,\ldots , L_r )$,
$(L_0,L_1,\ldots , L_{r-2},L_r,L_r' )$, or
$(L_0,L_0',L_2,\ldots , L_{r-2},L_r,L_r' )$.
Note that by construction the chain ${\mathcal{L} }$ can
be recovered from ${\mathcal{L}} ^+ $, so
the stabilizer of
$\mathcal{L} $ is equal to the stabilizer of
all lattices in $\mathcal{L}^+$.
Moreover every element in $\textnormal{U}_{\mathfrak{p} }$
mapping the chain ${\mathcal{L} }$ to some other chain ${\mathcal{L} }'$
maps the chamber ${\mathcal{L} }^+$ to the chamber $(\mathcal{L} ')^+$ .
\end{enumerate}
For more details we refer to \cite[Section 8.3]{AN}.
\end{remark}
In particular by part (c) of the previous remark we find the
important corollary.
\begin{corollary}
In the situation of Theorem \ref{main} the
group $\mathbb{A}ut(L_S)$ also acts chamber transitively (not necessarily type preservingly)
on the thick building $\mathcal{B}^+$.
\end{corollary}
\begin{remark}\label{splitoriflamme}
Also in the situation where $\mathfrak{P} \neq 1.73205080756887729352744634igma(\mathfrak{P} ) $,
i.e. $E_{\mathfrak{p} } = K_{\mathfrak{p}} \oplus K_{\mathfrak{p} }$,
the stabilizers of the points in the building are not
the stabilizers of the lattices in the lattice chain.
By Lemma \ref{uniqueunimod}
the lattices $L_i$ ($i=1,\ldots ,r$)
need to be replaced by the uniquely defined
lattices $Y_i\in L_i(\mathfrak{p} )$, such that $(Y_i)_\mathfrak{p}$ is unimodular
(as in Lemma \ref{uniqueunimod}) and $Y_i\cap L_0 = L_i$.
We refer to this construction as a variant of the oriflamme
construction in the examples below.
\end{remark}
\begin{theorem}\label{classtwo}
Let $\mathcal{L}= (L_0 , \dots, L_r)$ be a fine $\mathfrak{P}$-admissible lattice chain for some maximal two sided
ideal $\mathfrak{P} $ of $\overline{\textnormal{M}}_vO$ such that $1.73205080756887729352744634igma(\mathfrak{P}) = \mathfrak{P}$.
Suppose that the oriflamme construction replaces $\mathcal{L}$ by some sequence of lattices $\mathcal{L}^+$ which is one of
$$ \mathcal{L}, (L_0,L_0',L_2,\ldots , L_r ),
(L_0,L_1,\ldots , L_{r-2},L_r,L_r' ) \mbox{ or }
(L_0,L_0',L_2,\ldots , L_{r-2},L_r,L_r' ). $$
Then $L_0$ and $L_0'$ as well as $L_r$ and $L'_r$ are in the same genus but not in the
same proper special genus.
Put $L:=L_0$ and $S:= \{ \mathfrak{p} \} $.
Then $\mathbb{A}ut ^+(L_S):=\mathbb{A}ut(L_S) \cap \textnormal{SU} (V,\Phi )$ acts type preservingly on the thick building $\mathcal{B} ^+$.
This action is chamber transitive if and only if
$h^+(L) = 1$ and $\mathbb{A}ut ^+(L) $
is transitive on the maximal chains (in the first two cases)
respectively truncated maximal chains (in the last two cases)
of isotropic subspaces of $\overline{L}$
defined in Remark \ref{masschain}.
\end{theorem}
\begin{proof}
The proof that the action is chamber transitive in all cases
is completely analogous to the proof of Theorem \ref{main}.
We only need to show that $h^+(L) = 1$.
So let $M$ be some lattice in the same proper special genus as $L $.
By strong approximation for $\textnormal{U}^+(V_A,\Phi )$ (see \cite{KneserApp}),
there is some element $g\in \textnormal{U}_{\mathfrak{p} }^+ $ and $h\in \textnormal{SU}(V,\Phi )$
such that $Mh=Lg$. As $\mathbb{A}ut ^+(L_S) $ is chamber transitive and type preserving,
there is some $f\in \mathbb{A}ut ^+(L_S) $ such that $Lf = Mh $ so $M=Lfh^{-1} $ is
properly isometric to $L$.
\end{proof}
To obtain a classification of all chamber transitive discrete actions on $\mathcal{B}^+$
we hence need a classification of all proper spinor genera with proper class number
one. The thesis \cite{Kirschmer} only lists the genera of class number one and two.
In some cases, $h(L) = h^+(L) $ for every square-free lattice $L$, for example if:
\begin{itemize}
\item[(a)] $E=K$, $\dim(V) \geq 5$ and $K$ has narrow class number one
(\cite[Theorem 102.9]{OMeara}),
\item[(b)] $[E:K] = 2$ and $\dim_E(V) $ is odd (\cite{ShimuraUnitary}),
\item[(c)] or $[E:K] = 4$.
\end{itemize}
1.73205080756887729352744634ection{The one-class genera of fine admissible lattice chains}\label{oneclasschains}
We split this section into three subsections dealing with the
different types of hermitian spaces
($[E:K] = 1,2,4 $).
The fourth subsection comments on the exceptional groups.
Suppose $\mathcal{L} = (L_0,\dots,L_r)$ is a fine $\mathfrak{P}$-admissible lattice chain
of class number one, where $\mathfrak{P}$ is a maximal two sided ideal of $\overline{\textnormal{M}}_vO $.
Then $\mathfrak{p} := \mathfrak{P} \cap \mathbb{Z}_K$ together with $L_0$ determines the
isometry class of $\mathcal{L} := \mathcal{L}(L_0,\mathfrak{p} )$.
Moreover $L_0$ is a $\mathfrak{p} $-normalized lattice in $(V,\Phi )$ of class number one
and by Corollary \ref{einklassig} the
finite group $\mathbb{A}ut(L_0)$ acts transitively on the fine chains of (isotropic) subspaces of $\overline{L_0}$ as in Remark \ref{masschain}.
The one- and two-class genera of lattices in hermitian spaces $(V,\Phi )$
have been classified in \cite{Kirschmer}.
For all such lattices $L_0$ and all prime ideals $\mathfrak{p} $, for which $L_0$ is $\mathfrak{p}$-normalized,
we check by computer if $\mathbb{A}ut (L_0)$ acts transitively on the
fine chains of (isotropic) subspaces of $\overline{L_0}$.
Note that the number of such chains grows with the norm of $\mathfrak{p}$, so the
order of $\mathbb{A}ut(L_0)$ gives us a bound on the possible prime ideals $\mathfrak{p} $.
We also checked weaker conditions (similar to the ones in Theorem \ref{classtwo}) that would imply a chamber transitive action on the thick building $\mathcal{B} ^+$,
i.e. $h(L_0) \leq 2 $ and transitivity only on the truncated
maximal chains. The cases $h(L_0) = 2$ never gave a transitive action
on the chambers of $\mathcal{B} ^+$.
For any non-empty subset $T$ of $\{1,2,\dots,r\}$ we list the automorphism group $G_T$ of the subchain $(L_i)_{i \in T}$.
With our applications on the action on buildings in mind, we also
give the order of $$G_T^+:= G_T \cap \textnormal{U}_{\mathfrak{p}}^+ $$
where $\textnormal{U}_{\mathfrak{p}}^+$ is given in
Remark \ref{completiongroup}.
Note that we will always assume that the rank of the group $\textnormal{U}_{\mathfrak{p} }$
is $r\geq 2$.
1.73205080756887729352744634ubsection{Quadratic forms}
In this section suppose that $E=K$.
We denote by $\mathbb{A}_n, \mathbb{B}_n, \mathbb{D}_n, \mathbb{E}_n$ the root lattices of the same type over $\mathbb{Z}_K$.
If $L$ is a lattice and $a\in K$ we denote by
${}^{(a)} L $ the lattice $L$ with form rescaled by $a$.
Sometimes we identify lattices over number fields using
the trace lattice. For instance $(\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}} $
denotes a hermitian lattice over $\mathbb{Z} [ \frac{1+1.73205080756887729352744634qrt{-3}}{2} ]$ of dimension
4 whose trace lattice over $\mathbb{Z} $ is isometric to $\mathbb{E}_8$.
1.73205080756887729352744634ubsubsection{Quadratic forms in more than four variables}
If $E=K$, $\dim _K(V) \geq 5$ and $(V,\Phi )$ contains
a one-class genus of lattices, then by \cite[Section 7.4]{Kirschmer}
either $K=\mathbb{Q} $ or $K=\mathbb{Q}[1.73205080756887729352744634qrt{5}]$ where one has essentially
one one-class genus of lattices of dimension 5 and 6 each.
The rational lattices have been classified in
\cite{KL} and are available electronically from \cite{homepage}.
\begin{prop}\label{qf}
If $E=K$, $\dim _K(V) \geq 5$ and $(V,\Phi )$ contains
a fine $\mathfrak{p}$-admissible lattice chain $\mathcal{L} (L_0,\mathfrak{p} ) $ of class number one for some prime ideal $\mathfrak{p} $,
then $K=\mathbb{Q} $ and $\mathcal{L} (L_0,\mathfrak{p} )$ is one of the following
nine essentially different chains:
\begin{enumerate}
\item $\mathcal{L}(\mathbb{E}_8, 2) = (\mathbb{E}_8, \mathbb{D}_8, \mathbb{D}_4 \perp \mathbb{D}_4, {}^{(2)}\mathbb{D}_8^\#, {}^{(2)} \mathbb{E}_8)$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{E}_8$}] (0) at (-1,2.5) {$0$};
\node[mynode, label=right:{$\mathbb{E}_8'$}] (00) at (1,2.5) {$0'$};
\node (1) at (0,1.25) {$\bullet$};
\node[mynode, label=right:{$\mathbb{D}_4 \perp \mathbb{D}_4$}] (2) at (0,0) {$2$};
\node (3) at (0,-1.25) {$\bullet$};
\node[mynode, label=left:{${}^{(2)} \mathbb{E}_8$}] (4) at (-1,-2.5) {$4$};
\node[mynode, label=right:{${}^{(2)} \mathbb{E}_8'$}] (44) at (1,-2.5) {$4'$};
\draw (0) -- (1.center) -- (2) -- (3.center) -- (4);
\draw (00) -- (1.center);
\draw (3.center) -- (44);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows
\[
\begin{array}{ccl}
T & G_T & \# G_T^+ \\ \hline
\{i\} & 2.\textnormal{O}^+_8(2).2 & 2^{13} \cdot 3^5 \cdot 5^2 \cdot 7 \\
\{2\} & \mathbb{A}ut(\mathbb{D}_4) \wr C_2 & 2^{13} \cdot 3^4 \\
\{i,j\}& 2^{1+6}_+.S_8 & 2^{13} \cdot 3^2 \cdot 5 \cdot 7 \\
\{2,i\}& N.(S_3 \times S_3 \wr C_2) & 2^{13} \cdot 3^3\\
\{i,j,k\} & 2^{1+6}_+.(C_3^2 . \textnormal{PSL}_2(7)) & 2^{13} \cdot 3 \cdot 7\\
\{2,i,j\} & N.(C_2 \times S_3 \wr C_2) & 2^{13} \cdot 3^2\\
\{0,0',4,4'\} & 2^{1+6}_+.(C_2^3:S_4) & 2^{13} \cdot 3 \\
\{2,i,j,k\} & N.(C_2^3 \times S_3) & 2^{13} \cdot 3 \\
\{0,0',2,4,4'\} & N.C_2^3 & 2^{13}
\end{array}
\]
where $N=O_2(G_{\{2\}}) \cong 2_+^{1+4} \times 2_+^{1+4}$ and $i,j,k \in \{0,0',2,4,4'\}$ with $\#\{i,j,k\} = 3$.
\item $\mathcal{L}(\mathbb{E}_7, 2) = (\mathbb{E}_7, \mathbb{D}_6 \perp \mathbb{A}_1, \mathbb{D}_4 \perp {}^{(2)}\mathbb{B}_3, {}^{(2)}\mathbb{B}_7)$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{E}_7$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$\mathbb{E}_7'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node[mynode, label=right:{$\mathbb{D}_4 \perp {}^{(2)} \mathbb{B}_3$}] (2) at (0,0) {$2$};
\node[mynode, label=right:{${}^{(2)}\mathbb{B}_7$}] (3) at (0,-1.4) {$3$};
\draw (0) -- (1.center) -- (2) -- (3);
\draw (00) -- (1.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows
\[
\begin{array}{ccl}
T & G_T & \# G_T^+ \\ \hline
\{i\} & C_2 \times \textnormal{PSp}_6(2) & 2^9 \cdot 3^4 \cdot 5 \cdot 7 \\
\{2\} & \mathbb{A}ut(\mathbb{D}_4) \times C_2 \wr S_3 & 2^{9} \cdot 3^3 \\
\{3\} & C_2 \wr S_7 & 2^{9} \cdot 3^2 \cdot 5 \cdot 7 \\
\{0,0'\} & C_2^6.S_6 & 2^9 \cdot 3^2 \cdot 5 \\
\{i,2\} & N.S_3^2 & 2^9 \cdot 3^2 \\
\{i,3\} & C_2^7.\textnormal{PSL}_2(7) & 2^9 \cdot 3 \cdot 7 \\
\{2,3\} & N.(C_2 \times S_3^2) & 2^{9} \cdot 3^2 \\
\{0,0',2\}, \{i,2,3\} & N.D_{12} & 2^9 \cdot 3 \\
\{0,0',3\} & C_2^7.S_4 & 2^9 \cdot 3 \\
\{0,0',2,3\} & N.C_2^2 & 2^9
\end{array}
\]
where $N:= O_2(G_{\{2\}}) \cong 2^{1+4}_+ \times Q_8$ and $i \in \{0,0'\}$.
The one-class chain
$$\mathcal{L}(\mathbb{B}_7, 2) = \{ \mathbb{B}_7, {}^{(2)}(\mathbb{D}_4^\# \perp B_3), {}^{(2)}\mathbb{D}_6^\# \perp \mathbb{B}_1, {}^{(2)} \mathbb{E}_7^\#\}$$ yields the same stabilizers.
\item $\mathcal{L}(\mathbb{A}_6, 2) = \{ \mathbb{A}_6, X, {}^{(2)} X^{\#,2}, {}^{(2)} \mathbb{A}_6 \}$. Here $X$ is an indecomposable lattice with $\mathbb{A}ut(X) = (C_2^4 \times C_3).D_{12}$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{A}_6$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$\mathbb{A}_6'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node (2) at (0,0) {$\bullet$};
\node[mynode, label=left:{${}^{(2)} \mathbb{A}_6$}] (3) at (-1,-1) {$3$};
\node[mynode, label=right:{${}^{(2)}\mathbb{A}_6'$}] (33) at (1,-1) {$3'$};
\draw (0) -- (1.center) -- (2.center) -- (3);
\draw (00) -- (1.center);
\draw (33) -- (2.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\#T = 1 & C_2 \times S_7 & 2^3 \cdot 3^2 \cdot 5 \cdot 7 & - \\
\{0,0'\}, \{3,3'\} & C_2 \times S_3 \times S_4 & 2^3 \cdot 3^2 & 43 \\
\{0,3\}, \{0,3'\},\{0',3\}, \{0 ',3'\} & C_2 \times \textnormal{PSL}_2(7) & 2^3 \cdot 3 \cdot 7 & 42\\
\# T = 3 & C_2 \times S_4 & 2^3 \cdot 3 & 12\\
\{0,0', 3,3'\} & C_2 \times D_8 & 2^3 & 3
\end{array}
\]
Here, and in the following tables, the column sgdb gives the label of
$G_T^+$
as defined by the small group database (\cite{SGDB}).
The admissible one-class chain
$$\mathcal{L}({}^{(7)}\mathbb{A}_6^\#,2) = \{ {}^{(7)}\mathbb{A}_6^\#, {}^{(7)} X^{\#, 7}, {}^{(14)} X^\#, {}^{(14)}\mathbb{A}_6^\# \}$$ yields the same groups.
\item $\mathcal{L}(\mathbb{E}_6, 2) = \{ \mathbb{E}_6, Y_0, \mathbb{D}_4 \perp {}^{(2)}\mathbb{A}_2 \}$.
Here $Y_0$ is the even sublattice of $\mathbb{B}_5 \perp {}^{(3)} \mathbb{B}_1$. It is indecomposable and $\mathbb{A}ut(Y_0) = \mathbb{A}ut(\mathbb{B}_5 \perp {}^{(3)} \mathbb{B}_1 ) \cong C_2 \times C_2 \wr S_5$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{E}_6$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$\mathbb{E}_6'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node[mynode, label=right:{$\mathbb{D}_4 \perp {}^{(2)} \mathbb{A}_2$}] (2) at (0,0) {$2$};
\draw (0) -- (1.center) -- (2);
\draw (00) -- (1.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{i\} & C_2 \times \textnormal{U}_4(2).2 & 2^6 \cdot 3^4 \cdot 5& - \\
\{2\} & \mathbb{A}ut(\mathbb{D}_4) \times D_{12} & 2^6 \cdot 3^3 & - \\
\{0,0'\} & C_2 \wr S_5 & 2^6 \cdot 3 \cdot 5 & 11358 \\
\{i,2\} & N.S_3^2 & 2^6 \cdot 3^2 & 8277 \\
\{0,0',2\} & N.D_{12} & 2^6 \cdot 3 & 201
\end{array}
\]
where $N=O_2(G_3) \cong 2_+^{1+4} \times C_2$ and $i \in \{0,0'\}$.
The admissible one-class chains
\begin{align*}
\mathcal{L}(\mathbb{A}_2 \perp \mathbb{D}_4, 2) &= \{ \mathbb{A}_2 \perp \mathbb{D}_4, {}^{(2)} Y^{\#,2}, {}^{(2)} \mathbb{E}_6 \} \\
\mathcal{L}({}^{(3)} (\mathbb{A}_2^\# \perp \mathbb{D}_4) ,2) &= \{ {}^{(3)} (\mathbb{A}_2^\# \perp \mathbb{D}_4), {}^{(6)} Y^\#, {}^{(6)} \mathbb{E}_6 \} \\
\mathcal{L}({}^{(3)} \mathbb{E}_6^\#, 2) &= \{ {}^{(3)} \mathbb{E}_6^\#, {}^{(3)} Y^{\#, 3}, {}^{(3)} (\mathbb{A}_2 \perp \mathbb{D}_4)^{\#,3} \}
\end{align*}
yield the same stabilizers.
\item $\mathcal{L}(\mathbb{D}_6, 2) = \{ \mathbb{D}_6, \mathbb{D}_4 \perp {}^{(2)} \mathbb{B}_2, {}^{(2)}\mathbb{B}_6 \}$.
Here the application of the oriflamme construction is not necessary.
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb\\ \hline
\{0\}, \{2\} & C_2 \wr S_7 & 2^8 \cdot 3^2 \cdot 5 & - \\
\{1\} & \mathbb{A}ut(\mathbb{D}_4) \perp C_2 \wr S_2 & 2^8 \cdot 3^2 & - \\
\{ 0,1 \}, \{1,2\} & C_2^6.(C_2\times S_4) & 2^8 \cdot 3 & 1086007 \\
\{0,2\} & C_2^6.(C_2 \times S_4) & 2^8 \cdot 3 & 1088660 \\
\{0,1,2\} & C_2^6.(C_2 \times D_8) & 2^8 & 6331
\end{array}
\]
\item $\mathcal{L}(\mathbb{E}_6, 3) = \{ \mathbb{E}_6, \mathbb{A}_2^3, {}^{(3)}\mathbb{E}_6 \}$.
Here the application of the oriflamme construction is not necessary.
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{0\}, \{2\} & C_2 \times \textnormal{U}_4(2).2 & 2^6 \cdot 3^4 \cdot 5 & - \\
\{1\} & D_{12} \wr S_3 & 2^5 \cdot 3^4 & - \\
\{0,2\} & 3^{1+2}_+.(C_2 \times \textnormal{GL}_2(3)) & 2^3 \cdot 3^4 & 533 \\
\{0,1\}, \{1,2\} & N.(C_2^2 \times S_4) & 2^3 \cdot 3^4 & 704\\
\{0,1,2\} & N.(C_2^2 \times S_3) & 2 \cdot 3^4 & 10 \\
\end{array}
\]
where $N=O_3(G_{\{1\}}) \cong C_3^3$.
\item $\mathcal{L}(\mathbb{B}_5 \perp {}^{(3)} \mathbb{B}_1, 3) = \{ \mathbb{B}_5 \perp {}^{(3)} \mathbb{B}_1, \mathbb{B}_2 \perp \mathbb{A}_2 \perp {}^{(3)} \mathbb{B}_2; \mathbb{B}_1 \perp {}^{(3)} \mathbb{B}_5 \}$.
Here the application of the oriflamme construction is not necessary.
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{0\}, \{2\} & C_2 \times C_2 \wr S_5 & 2^6 \cdot 3 \cdot 5 & - \\
\{1\} & C_2 \wr S_2 \times D_{12} \times C_2 \wr S_2 & 2^5 \cdot 3 & 144 \\
\{0,2\} & C_2^2\times \textnormal{GL}_2(3) & 2^3 \cdot 3 & 3 \\
\{0,1\}, \{1,2\} & C_2^2 \times D_8 \times S_3 & 2^3 \cdot 3 & 8 \\
\{0,1,2\} & C_2^3 \times S_3 & 2 \cdot 3 & 2
\end{array}
\]
For $0\le i \le 2$ let $Y_i$ be the even sublattice of $\mathcal{L}(\mathbb{B}_5 \perp {}^{(3)} \mathbb{B}_1, 3)_i$, see also part (4).
Then the admissible one-class chains
$$\mathcal{L}(Y_0, 3) = \{ Y_0, Y_1, Y_2 \} \mbox{ and } \mathcal{L}({}^{(2)} Y_0^{\#,2}, 3) = ( {}^{(2)} Y_0^{\#,2}, {}^{(2)} Y_1^{\#,2}, {}^{(2)} Y_2^{\#,2} )$$
yield the same groups.
\item $\mathcal{L}(\mathbb{A}_5, 2) = \{\mathbb{A}_5, {}^{(2)} \mathbb{B}_1 \perp Z, {}^{(2)} (\mathbb{A}_2 \perp \mathbb{B}_3)\}$.
Here $Z$ is the even sublattice of $\mathbb{B}_3 \perp {}^{(3)} \mathbb{B}_1$ and $\mathbb{A}ut(Z) = \mathbb{A}ut( \mathbb{B}_3 \perp {}^{(3)} \mathbb{B}_1 )$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{A}_5$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$\mathbb{A}_5'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node[mynode, label=right:{${}^{(2)} (\mathbb{A}_2 \perp \mathbb{B}_3)$}] (2) at (0,0) {$2$};
\draw (0) -- (1.center) -- (2);
\draw (00) -- (1.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{0\}, \{0'\} & C_2 \times S_6 & 2^3 \cdot 3^2 \cdot 5 & 118 \\
\{2\} & D_{12} \times C_2 \wr S_3 & 2^3 \cdot 3^2 & 43 \\
\# T = 2 & C_2^2 \times S_4 & 2^3 \cdot 3 & 12 \\
\{0,0',2\} & C_2^2 \times D_8 & 2^3 & 3
\end{array}
\]
The admissible one-class chains
\begin{align*}
\mathcal{L}({}^{(3)}\mathbb{A}_5^{\#,3},2) &= \{ {}^{(3)}\mathbb{A}_5^{\#,3}, {}^{(6)}\mathbb{B}_1 \perp {}^{(3)} Z^{\#, 3}, {}^{(6)} (\mathbb{A}_2^\# \perp \mathbb{B}_3) \} \\
\mathcal{L}(\mathbb{A}_2 \perp \mathbb{B}_3 ,2) &= \{ \mathbb{A}_2 \perp \mathbb{B}_3 , \mathbb{B}_1 \perp {}^{(2)} Z^{\#,2}, {}^{(2)} \mathbb{A}_5^{\#, 2} \} \\
\mathcal{L}( {}^{(3)}(\mathbb{A}_2^\# \perp \mathbb{B}_3), 2) &= \{ {}^{(3)}(\mathbb{A}_2^\# \perp \mathbb{B}_3), {}^{(3)}\mathbb{B}_1 \perp {}^{(6)} Z^\#, {}^{(6)} \mathbb{A}_5^\# \}
\end{align*}
yield the same stabilizers.
\item $\mathcal{L}(\mathbb{B}_5, 3) = \{ \mathbb{B}_5, \mathbb{B}_2 \perp \mathbb{A}_2 \perp {}^{(3)} \mathbb{B}_1, \mathbb{B}_1 \perp {}^{(3)} \mathbb{B}_4 \}$.
After applying the oriflamme construction, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$\mathbb{B}_5$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$\mathbb{B}_5'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node[mynode, label=right:{$(\mathbb{B}_1 \perp {}^{(3)}\mathbb{B}_4)$}] (2) at (0,0) {$2$};
\draw (0) -- (1.center) -- (2);
\draw (00) -- (1.center);
\end{tikzpicture}
\end{center}
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{i\} & C_2 \wr S_5 & 2^6 \cdot 3 \cdot 5 & 11358 \\
\{2\} & C_2 \wr S_4 \times C_2 & 2^5 \cdot 3 & 204 \\
\{0,0'\} & (C_2 \times D_8) \times S_3 & 2^3 \cdot 3 & 3\\
\{i,2\} & C_2 \times \textnormal{GL}_2(3) & 2^3 \cdot 3 & 8 \\
\{0,0',2\} & C_2^2 \times S_3 & 2 \cdot 3 & 2 \\
\end{array}
\]
where $i \in \{0,0'\}$. The admissible one-class chain
\[ \mathcal{L}( \mathbb{B}_4 \perp {}^{(3)} \mathbb{B}_1 ,3) = \{ \mathbb{B}_4 \perp {}^{(3)} \mathbb{B}_1, \mathbb{B}_1 \perp \mathbb{A}_2 \perp {}^{(3)} \mathbb{B}_2, {}^{(3)} \mathbb{B}_5 \} \]
yields the same stabilizers.
\end{enumerate}
\end{prop}
1.73205080756887729352744634ubsubsection{Quadratic forms in four variables}
Now assume that $K=E$ and $\dim _K(V) = 4$.
By \cite[Theorem 7.4.1]{Kirschmer} there are up to similarity exactly 481
one-class genera of lattices if $K=\mathbb{Q}$ and additionally
604 such genera over 21 other base fields where the largest degree is
$[K:\mathbb{Q} ] =5$ (\cite[Theorem 7.4.2]{Kirschmer}).
As we are only interested in the case where
the rank of $\textnormal{U}_{\mathfrak{p} }$ is $2$, we only need to consider pairs
$(L,\mathfrak{p} )$ where $L$ is one of these 1095 lattices and $\mathfrak{p} $
a prime ideal such that
$V_{\mathfrak{p}} \cong \mathbb{H} (K_{\mathfrak{p}} ) \perp \mathbb{H} (K_{\mathfrak{p} }) $.
In this case the building ${\mathcal B}$ of $\textnormal{U}_{\mathfrak{p}} $ is of type $A_1 \oplus A_1 $
and not connected even after oriflamme construction.
We will not list the groups acting chamber transitively on ${\mathcal B}^+$,
also because of the numerous cases of one-class lattice chains
in this situation.
To list the lattices we
need some more notation.
We denote by $\mathcal{Q}:= \mathcal{Q}_{\alpha, \infty, \mathfrak{p}_1,\dots, \mathfrak{p}_s}$ a definite quaternion algebra over $K= \mathbb{Q}(\alpha)$ which ramifies exactly at the finite places $\mathfrak{p}_1,\dots,\mathfrak{p}_s$ of $K$.
Given an integral ideal $\mathfrak{a}$ of $\mathbb{Z}_K$ coprime to all $\mathfrak{p}_i$, then
$\mathcal{O}_{\alpha, \infty, \mathfrak{p}_1,\dots, \mathfrak{p}_s; \mathfrak{a}}$ denotes an Eichler order of level $\mathfrak{a}$ in $\mathcal{Q}$.
We omit the subscript $\alpha$ whenever $K=\mathbb{Q}$.
Similarly, the subscript $\mathfrak{a}$ is omitted, if $\mathfrak{a}=\mathbb{Z}_K$, i.e. the order is maximal.
Then $\mathcal{O}_{\alpha, \infty, \mathfrak{p}_1,\dots, \mathfrak{p}_s; \mathfrak{a}}$ with the reduced norm form of $\mathcal{Q}$ yields a quaternary lattice over $\mathbb{Z}_K$.
By \cite[Corollary 4.6]{Nebe} this lattice is unique in its genus, if and only if all Eichler orders of level $\mathfrak{a}$ in $\mathcal{Q}$ are conjugate.
Hence we identify such orders with their quaternary lattices.
\begin{prop}
Let $L$ be a $\mathfrak{p}$-normalized, quaternary lattice over $\mathbb{Z}_K$ such that $\mathcal{L}(L, \mathfrak{p})$ is a
fine $\mathfrak{p}$-admissible lattice chain of length $2$ and
class number one.
Then one of the following holds.
\begin{enumerate}
\item $K=\mathbb{Q}$ and either
\begin{itemize}
\item $\mathfrak{p} = 2$ and $L \cong \mathcal{O}_{\infty,3} \cong \mathbb{A}_2 \perp \mathbb{A}_2$ or $\mathcal{O}_{\infty, 5}$.
\item $\mathfrak{p} \in \{3,5,11\}$ and $L \cong \mathcal{O}_{\infty,2} \cong \mathbb{D}_4$.
\item $\mathfrak{p} = 3$ and $L \cong \mathbb{B}_4$.
\end{itemize}
\item $K = \mathbb{Q}(1.73205080756887729352744634qrt{5})$ and either
\begin{itemize}
\item $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) \in \{ 4,5,9,11,19,29,59 \}$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty}$. This lattice is called $H_4$ in \cite{Scharlau}.
\item $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) \in \{ 5, 11\}$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty; 2 \mathbb{Z}_K} \cong \mathbb{D}_4$.
\item $\mathfrak{p} = 2\mathbb{Z}_K$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty; \mathfrak{a}} \cong\mathcal{L}( \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty} , \mathfrak{a})_2$ with $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{a}) \in \{5,11\} $.
\end{itemize}
\item $K = \mathbb{Q}(1.73205080756887729352744634qrt{2})$ and either
\begin{itemize}
\item $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) \in \{ 2, 7, 23 \}$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{2},\infty}$.
\item $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) = 7$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{2},\infty; 1.73205080756887729352744634qrt{2} \mathbb{Z}_K} \cong \mathcal{L}( \mathcal{O}_{1.73205080756887729352744634qrt{2},\infty} )_2$ or $L$ is isometric to a unimodular lattice of norm $1.73205080756887729352744634qrt{2}\mathbb{Z}_K$ in $(V,\Phi) \cong \langle 1,1,1,1\rangle$.
By \cite[IX:93]{OMeara}, the genus of the latter lattice is uniquely determined and it has class number one by \cite{Kirschmer}.
\item $\mathfrak{p} = 1.73205080756887729352744634qrt{2}\mathbb{Z}_K$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty; \mathfrak{a}} \cong \mathcal{L}( \mathcal{O}_{1.73205080756887729352744634qrt{5},\infty}, \mathfrak{a} )_2$ with $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{a}) = 7$.
\end{itemize}
\item $K = \mathbb{Q}(1.73205080756887729352744634qrt{3})$ and either
\begin{itemize}
\item $\mathfrak{p} = 1.73205080756887729352744634qrt{3} \mathbb{Z}_K$ and $L\cong \mathcal{O}_{1.73205080756887729352744634qrt{3},\infty;\mathfrak{p}_2}$ or $L$ is isometric to a unimodular lattice of norm $\mathfrak{p}_2$ in $(V,\Phi) \cong \langle 1,1,1,1\rangle$. Again, this lattice is unique up to isometry.
\item $\mathfrak{p} = \mathfrak{p}_2$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{3},\infty;1.73205080756887729352744634qrt{3} \mathbb{Z}_K}$.
\end{itemize}
\item $K = \mathbb{Q}(1.73205080756887729352744634qrt{13})$ and $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) = 3$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{13},\infty}$. This lattice is called $D_4^1.73205080756887729352744634im$ in \cite{Scharlau}.
\item $K = \mathbb{Q}(1.73205080756887729352744634qrt{17})$ and $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) = 2$ and $L \cong \mathcal{O}_{1.73205080756887729352744634qrt{17},\infty}$. This lattice is called $(2A_2)^1.73205080756887729352744634im$ in \cite{Scharlau}.
\item $K = \mathbb{Q}(\theta_9)$ is the maximal totally real subfield of the cyclotomic field $\mathbb{Q}(\zeta_9)$ and $\mathfrak{p} = 2\mathbb{Z}_K$ and $L \cong \mathcal{O}_{\theta,\infty, \mathfrak{p}_3}$.
\item $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[X]/(X^3-X^2-3X+1)$ is the unique totally real number field of degree $3$ and discriminant $148$.
Then either $\mathfrak{p}= \mathfrak{p}_5$ and $L \cong \mathcal{O}_{\alpha,\infty;\mathfrak{p}_2}$ or $\mathfrak{p}= \mathfrak{p}_2$ and $L \cong \mathcal{O}_{\alpha,\infty;\mathfrak{p}_5}$.
\item $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[X]/(X^3-X^2-4X+2)$ is the unique totally real number field of degree $3$ and discriminant $316$.
Then $\mathfrak{p}=\mathfrak{p}_2$ and $L \cong \mathcal{O}_{\alpha, \infty; \mathfrak{p}_4}$.
\item $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[X]/(X^4-X^3-3X^2+X+1)$ is the unique totally real number field of degree $4$ and discriminant $725$.
Then $L \cong \mathcal{O}_{\alpha, \infty}$ and $\textnormal{Nr}_{K/\mathbb{Q}}(\mathfrak{p}) \in \{11, 19\}$ or $\mathfrak{p}$ is the ramified prime ideal of norm $29$.
\item $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[X]/(X^4-4X^2-X+1)$ is the unique totally real number field of degree $4$ and discriminant $1957$. Then $\mathfrak{p} = \mathfrak{p}_3$ and $L \cong \mathcal{O}_{\alpha, \infty}$.
\item $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[X]/(X^4-X^3-4X^2+X+2)$ is the unique totally real number field of degree $4$ and discriminant $2777$.
Then $\mathfrak{p} = \mathfrak{p}_2$ and $L \cong \mathcal{O}_{\alpha, \infty}$.
\end{enumerate}
Here $\mathfrak{p}_q$ denotes a prime ideal of $\mathbb{Z}_K$ of norm $q$.
Conversely, in all these cases the chain $\mathcal{L}(L, \mathfrak{p})$ is $\mathfrak{p}$-admissible and has class number one.
\end{prop}
1.73205080756887729352744634ubsection{Hermitian forms}
In this section we treat the case that $[E:K]=2$, so
$E$ is a totally complex extension of degree 2 of the totally
real number field $K$. All hermitian lattices with class number
$\leq 2$ are classified in \cite[Section 8]{Kirschmer} and
listed explicitly for $n\geq 3$ in \cite[pp 129-140]{Kirschmer}.
\begin{prop}\label{hf}
Let $\mathcal{L}(L_0,\mathfrak{p})$ be a fine $\mathfrak{P}$-admissible chain of class number one and of length at least $2$.
Then $K =\mathbb{Q}$, $d:= \dim_E(V) \in \{3,4,5\}$ and one the following holds:
\begin{enumerate}
\item $E=\mathbb{Q}(1.73205080756887729352744634qrt{-3})$, $\mathfrak{p}=3\mathbb{Z}$ and $L_0 \cong \mathbb{B}_5 \otimes_\mathbb{Z} \mathbb{Z}[\tfrac{1+1.73205080756887729352744634qrt{-3}}{2}] \cong (\mathbb{A}_2^5)_{1.73205080756887729352744634qrt{-3}}$:
$$\mathcal{L}(L_0, 3) = \{ L_0, (\mathbb{A}_2^2 \perp {}^{(3)} \mathbb{E}_6^\# )_{1.73205080756887729352744634qrt{-3}} , (\mathbb{A}_2 \perp \mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}} \}.$$
Here the application of the oriflamme construction is not necessary.
The automorphism groups are as follows:
\[
\begin{array}{ccc}
T & G_T & \# G_T^+ \\ \hline
\{0\} & C_6 \wr S_5 & 2^7 \cdot 3^5 \cdot 5 \\
\{1\} & C_6 \wr S_2 \times {}_{1.73205080756887729352744634qrt{-3}} [ \pm 3^{1+2}_+.\textnormal{SL}_2(3)]_3 & 2^6 \cdot 3^5 \\
\{2\} & C_6 \times {}_{1.73205080756887729352744634qrt{-3}}[\textnormal{Sp}_4(3) \times C_3]_4 & 2^7 \cdot 3^5 \cdot 5 \\
\{0,1\} & C_6 \wr S_2 \times {}_{1.73205080756887729352744634qrt{-3}} [ \pm 3^{1+2}_+.C_6 ]_3 & 2^4 \cdot 3^5 \\
\{1,2\} & C_6 \times {}_{1.73205080756887729352744634qrt{-3}} [\pm(3_+^{1+2}.\textnormal{SL}_2(3) \times C_3) ]_4 & 2^4 \cdot 3^5 \\
\{0,1\} & C_6 \times {}_{1.73205080756887729352744634qrt{-3}} [\pm 3^3 : S_4 \times C_3]_4 & 2^4 \cdot 3^5 \\
\{1,2\} & C_6 \times {}_{1.73205080756887729352744634qrt{-3}} [\pm(3_+^{1+2}.\textnormal{SL}_2(3) \times C_3) ]_4 & 2^4 \cdot 3^5 \\
\{0,1,2\}&C_6 \times {}_{1.73205080756887729352744634qrt{-3}} [\pm 3^{1+2}_+.C_6 \times C_3]_4 & 2^2 \cdot 3^5
\end{array}
\]
\item $E=\mathbb{Q}(1.73205080756887729352744634qrt{-7})$, $\mathfrak{p}=2\mathbb{Z}$ and $L_0 \cong (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-7}}$:
$$\mathcal{L}( (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-7}}, 2) = \{ (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-7}} , (\mathbb{D}_8)_{1.73205080756887729352744634qrt{-7}}, (\mathbb{D}_4 \perp \mathbb{D}_4)_{1.73205080756887729352744634qrt{-7}}, ({}^{(2)}\mathbb{D}_8)_{1.73205080756887729352744634qrt{-7}} \}.$$
After applying the variant of the
oriflamme construction described in Remark \ref{splitoriflamme}, the lattice chain becomes
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$(\mathbb{E}_8)_{1.73205080756887729352744634qrt{-7}} \quad \cong \quad $}] (0) at (0,0) {$0$};
\node[mynode] (33) at (2,0) {$3'$};
\node[mynode] (00) at (4,0) {$0'$};
\node[mynode] (x) at (6,0) {$3$};
\node (1) at (0,-1) {$\bullet$};
\node (2) at (0,-2) {$\bullet$};
\node (3) at (0,-3) {$\bullet$};
\draw (0) -- (1.center) -- (2.center) -- (3.center);
\draw (33) -- (1.center);
\draw (00) -- (2.center);
\draw (3.center) -- (x);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows:
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ &sgdb \\ \hline
\#T = 1 & 2.\mathbb{A}lt_7 & 2^4 \cdot 3^2 \cdot 5 \cdot 7 & - \\
\{0,0'\}, \{3,3'\} & \textnormal{SL}_2(3) \times C_3 :2 & 2^4 \cdot 3^2 & 124 \\
\{0,3\}, \{0,3'\},\{0',3\}, \{0 ',3'\} & \textnormal{SL}_2(7) & 2^4 \cdot 3 \cdot 7 & 114 \\
\# T = 3 & 2.S_4 & 2^4 \cdot 3 & 28 \\
\{0,0', 3,3'\} & Q_{16} & 2^4 & 9
\end{array}
\]
\item $E=\mathbb{Q}(1.73205080756887729352744634qrt{-3})$, $\mathfrak{p}=2\mathbb{Z}$ and $L_0 \cong (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}$:
$$\mathcal{L}( (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}, 2) = \{ (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}, (\mathbb{D}_4 \perp \mathbb{D}_4)_{1.73205080756887729352744634qrt{-3}}, ({}^{(2)}\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}\}.$$
Here the application of the oriflamme construction is not necessary.
The automorphism groups are as follows:
\[
\begin{array}{ccl}
T & G_T & \# G_T^+ \\ \hline
\{0\}, \{2\} & \ _{1.73205080756887729352744634qrt{-3}}[\textnormal{Sp}_4(3) \times C_3]_4 & 2^7 \cdot 3^4 \cdot 5 \\
\{1\} & \ _{1.73205080756887729352744634qrt{-3}}[\textnormal{SL}_2(3) \times C_3]_2^2 & 2^7 \cdot 3^3 \\
\{0,2\} & 2^{1+4}_-.\mathbb{A}lt_5 \times C_3 & 2^7 \cdot 3 \cdot 5 \\
\{0,1\}, \{1,2\} & \textnormal{SL}_2(3) \wr C_2 \times C_3 & 2^7 \cdot 3^2 \\
\{0,1,2\} & (Q_8 \wr S_2):C_3 \times C_3 & 2^7 \cdot 3
\end{array}
\]
\item $E = \mathbb{Q}(1.73205080756887729352744634qrt{-1})$, $\mathfrak{p} = 2\mathbb{Z}$ and $L_0 \cong (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-1}}$:
$$\mathcal{L}( (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-1}}, 2) = \{ (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-1}}, (\mathbb{D}_8)_{1.73205080756887729352744634qrt{-1}}, (\mathbb{D}_4\perp \mathbb{D}_4) _{1.73205080756887729352744634qrt{-1}} \}.$$
Here the application of the oriflamme construction is not necessary.
After applying the oriflamme construction, one obtains the following lattices
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$(\mathbb{E}_8)_{1.73205080756887729352744634qrt{-1}}$}] (0) at (-1,2) {$0$};
\node[mynode, label=right:{$(\mathbb{E}_8)_{1.73205080756887729352744634qrt{-1}}'$}] (00) at (1,2) {$0'$};
\node (1) at (0,1) {$\bullet$};
\node[mynode, label=right:{$(\mathbb{D}_4\perp \mathbb{D}_4) _{1.73205080756887729352744634qrt{-1}}$}] (2) at (0,0) {$2$};
\draw (0) -- (1.center) -- (2);
\draw (00) -- (1.center);
\end{tikzpicture}
\end{center}
\[
\begin{array}{ccl}
T & G_T & \# G_T^+ \\ \hline
\{0\}, \{0'\} & {}_i[(2^{1+4}_+ {\textsc Y} C_4).S_6]_4 & 2^9 \cdot 3^2 \cdot 5 \\
\{2\} & \ _{i}[(D_8 {\textsc Y} C_4 ).S_3]_2^2 & 2^9 \cdot 3^2 \\
\{ 0,2 \}, \{0',2\} & & 2^9 \cdot 3 \\
\{0,0'\} & & 2^9 \cdot 3 \\
\{0,2,0'\} & & 2^9
\end{array}
\]
\item $E = \mathbb{Q}(1.73205080756887729352744634qrt{-3})$, $\mathfrak{p} = 3\mathbb{Z}$ and $L_0 = \mathbb{B}_4 \otimes_\mathbb{Z} \mathbb{Z}[\frac{1+1.73205080756887729352744634qrt{3}}{1}] \cong (\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}$:
Here the application of the oriflamme construction is not necessary.
$$\mathcal{L}( L_0, 3) = \{ (\mathbb{A}_2^4)_{1.73205080756887729352744634qrt{-3}} , (\mathbb{A}_2 \perp {}^{(3)}\mathbb{E}_6^\#)_{1.73205080756887729352744634qrt{-3}}, ({}^{(3)}\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}} \}.$$
After applying the oriflamme construction, the chain becomes:
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$({}^{(3)}\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}$}] (2) at (-1,-2) {$2$};
\node[mynode, label=right:{$({}^{(3)}\mathbb{E}_8)_{1.73205080756887729352744634qrt{-3}}'$}] (22) at (1,-2) {$2'$};
\node (1) at (0,-1) {$\bullet$};
\node[mynode, label=right:{$(\mathbb{A}_2^4) _{1.73205080756887729352744634qrt{-3}}$}] (0) at (0,0) {$0$};
\draw (0) -- (1.center) -- (2);
\draw (22) -- (1.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows:
\[
\begin{array}{cclc}
T & G_T & \# G_T^+ & sgdb \\ \hline
\{0\} & C_6 \wr S_4 & 2^6 \cdot 3^4 & - \\
\{2\}, \{2'\} & \ _{1.73205080756887729352744634qrt{-3}}[\textnormal{Sp}_4(3) \times C_3]_4 & 2^7 \cdot 3^4 \cdot 5 & - \\
\{0,2\}, \{0,2'\} & (\pm C_3^4).S_4 & 2^4 \cdot 3^4 & 3085 \\
\{2,2'\}& (C_6 \times 3_+^{1+2}).S_3 & 2^4 \cdot 3^4 & 2895 \\
\{0,2,2'\} & (C_6 \times C_3 \wr C_3).2 & 2^2 \cdot 3^4 & 68
\end{array}
\]
\item
$E=\mathbb{Q}(1.73205080756887729352744634qrt{-7})$, $\mathfrak{p}=2\mathbb{Z}$ and $L_0 = ({}^{(7)} \mathbb{A}_6^\#)_{1.73205080756887729352744634qrt{-7}}$.
After applying the variant of the oriflamme construction described in
Remark \ref{splitoriflamme}, the chain becomes:
\begin{center}
\begin{tikzpicture}[scale=0.75,-,mynode/.style={circle,draw,inner sep=0pt, minimum size=15pt}]
\node[mynode, label=left:{$ ({}^{(7)} \mathbb{A}_6^\#)_{1.73205080756887729352744634qrt{-7}} \quad \cong \quad $}] (0) at (0,0) {$0$};
\node[mynode] (11) at (2,0) {$0'$};
\node[mynode] (00) at (4,0) {$0''$};
\node (1) at (0,-1) {$\bullet$};
\node (2) at (0,-2) {$\bullet$};
\draw (0) -- (1.center) -- (2.center);
\draw (11) -- (1.center);
\draw (00) -- (2.center);
\end{tikzpicture}
\end{center}
The automorphism groups are as follows:
\[
\begin{array}{ccl}
T & G_T & \# G_T^+ \\ \hline
\# T = 1 & \pm C_7 : 3 & 3 \cdot 7 \\
\# T = 2 & C_6 & 3 \\
\{0,0',0''\} & C_2 & 1
\end{array}
\]
\end{enumerate}
\end{prop}
1.73205080756887729352744634ubsection{Quaternionic hermitian forms}
In this section we treat the case that $[E:K]=4$, so
$E$ is a totally definite quaternion algebra over the totally
real number field $K$. All quaternionic hermitian lattices with class number
$\leq 2$ are classified in \cite[Section 9]{Kirschmer} and
listed explicitly for $n\geq 2$ in \cite[pp 147-150]{Kirschmer}.
\begin{prop}\label{qaf}
Suppose $E$ is a definite quaternion algebra and let $\mathcal{L}(L_0,\mathfrak{p})$
be a fine $\mathfrak{P}$-admissible chain of length at least 2 and of class number one.
Then $K = \mathbb{Q}$, $d:= \dim_E(V) = 2$ and one of the following holds:
\begin{enumerate}
\item $E \cong \mathcal{Q}_{\infty, 2}$, the rational quaternion algebra ramified at $2$ and $\infty$, $\mathfrak{p}=3\mathbb{Z}$ and $L_0 \cong (\mathbb{E}_8)_{\infty, 2}$ is
the unique $\overline{\textnormal{M}}_vO$-structure of the $\mathbb{E}_8$-lattice whose automorphism group is called $ _{\infty,2}[ 2^{1+4}_-.\mathbb{A}lt_5]_2$ in \cite{Nebe}.
The oriflamme construction is not necessary and
the automorphism groups are
\[
\begin{array}{cclc}
T & G_T & \# G_T & sgdb \\ \hline
\{0\}, \{2\} & _{\infty,2}[ 2^{1+4}_-.\mathbb{A}lt_5]_2 & 2^7 \cdot 3 \cdot 5 & - \\
\{1\} & Q_8 : \textnormal{SL}_2(3) & 2^6 \cdot 3 & 1022\\
\{0,1\}, \{1,2\} & C_2 \times \textnormal{SL}_2(3) & 2^4 \cdot 3 & 32 \\
\{0,2\} & C_3:SD_{16} & 2^4 \cdot 3 & 16 \\
\# T = 3 & C_2 \times C_6 & 2^2 \cdot 3 & 9
\end{array}
\]
\item $E \cong \mathcal{Q}_{\infty, 3}$ and $\mathfrak{p}=2\mathbb{Z}$ and $L_0 \cong (\mathbb{E}_8)_{\infty,3}$ is the unique $\overline{\textnormal{M}}_vO$-structure of the
$\mathbb{E}_8$-lattice whose automorphism group is called
$ _{\infty,3}[ \textnormal{SL}_2(9) ]_2$ in \cite{Nebe}.
The oriflamme construction is not necessary and
the automorphism groups are
\[
\begin{array}{cclc}
T & G_T & \# G_T & sgdb \\ \hline
\{0\}, \{2\} & _{\infty,3}[ \textnormal{SL}_2(9) ]_2 & 2^4 \cdot 3^2 \cdot 5 & 409 \\
\{1\} & \textnormal{SL}_2(3).S_3 & 2^4 \cdot 3^2 & 124 \\
\#T = 2 & C_2.S_4 & 2^4 \cdot 3 & 28 \\
\{0,1,2\} & Q_{16} & 2^4 & 9
\end{array}
\]
\end{enumerate}
Note that the above quaternion algebras only have one conjugacy class of maximal orders and for any such order $\overline{\textnormal{M}}_vO$, the above $\overline{\textnormal{M}}_vO$-lattice $L_0$ is uniquely determined up to isometry.
\end{prop}
1.73205080756887729352744634ubsection{The exceptional groups} \label{g2}
The exceptional groups have been dealt with in \cite[Chapter 10]{Kirschmer},
where it is shown that only the group $\textnormal{G}_2$ admits one-class genera
defined by a coherent family of parahoric subgroups.
In all cases the number field is the field of rational numbers.
The one-class genera of lattice chains correspond
to the coherent families of parahoric subgroups $(P_q ) _{q \text{ prime}} $
where for one prime $p$ the
parahoric subgroup $P_p$ is the Iwahori subgroup,
a stabilizer of a chamber in the corresponding $p$-adic building.
Hence \cite[Theorem 10.3.1]{Kirschmer}
shows directly that there is
a unique $S$-arithmetic group of type $\textnormal{G}_2$ with a discrete and
chamber transitive action.
It is given by the $\mathbb{Z} $-form ${\bf G}_2$ where each
parahoric subgroup $P_q$ is hyperspecial.
This integral model of $\textnormal{G}_2$ is described
in \cite{Gross} (see also \cite{CNP} for more one-class genera of $\textnormal{G}_2$).
Here ${\bf G}_2(\mathbb{Z} ) \cong \textnormal{G}_2(2) $ and the
$S$-arithmetic group is ${\bf G}_2(\mathbb{Z} [\frac{1}{2}] )$ (so $S=\{ 2 \} $).
The extended Dynkin diagram of $\textnormal{G}_2$ is as follows.
\begin{center}
\begin{tikzpicture}[-,node distance=3cm,thick,mynode/.style={circle,draw},scale = .75]
\node[mynode] (1) {0};
\node[mynode] (2) [right of = 1] {1};
\node[mynode] (3) [right of = 2] {2};
\draw[style={postaction={decorate},decoration={markings,mark=at position 0.65 with {\arrow{angle 60}}},double distance=6pt}]
(2) -- (3);
\draw (1) -- (2) -- (3);
\end{tikzpicture}
\end{center}
The stabilizers $G_T$ of the simplices $T 1.73205080756887729352744634ubseteq \{ 0,1,2 \}$
in the corresponding
building of $\textnormal{G}_2(\mathbb{Q}_2)$ are given in \cite[Section 10.3]{Kirschmer}:
\begin{table}[H]
$$
\begin{array}{cclc}
T & G_T& \# G_T & sgdb\\ \hline
\{0\} & \textnormal{G}_2(2) & 2^6 \cdot 3^3 \cdot 7 &- \\
\{2\} & 2^3.\textnormal{GL}_3(2) & 2^6 \cdot 3 \cdot 7 &814 \\
\{1\} & 2^{1+4}_+.((C_3 \times C_3).2) & 2^6 \cdot 3^2 & 8282 \\
\{1,2\} & 2_+^{1+4}.S_3 & 2^6 \cdot 3 &1494 \\
\{0,2\} & ((C_4 \times C_4).2).S_3 & 2^6 \cdot 3 &956 \\
\{0,1\} & 2_+^{1+4}.S_3 & 2^6 \cdot 3 &988 \\
\{0,1,2\}& \textnormal{Syl}_2(\textnormal{G}_2(2)) &2^6 & 134 \\
\end{array}
$$
\end{table}
One may visualize the chamber transitive action of ${\bf G}_2(\mathbb{Z}[\frac{1}{2}])$ on the Bruhat-Tits building of $\textnormal{G}_2(\mathbb{Q}_2)$ by indicating the
three generators $x,y,z$ of ${\bf G}_2(\mathbb{Z}[\frac{1}{2}])$ of order 3 mapping the
standard chamber to one of the (three times) two neighbors.
\def1.73205080756887729352744634{1.73205080756887729352744634}
\def1.73205080756887729352744634cale{3}
\begin{center}
\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (1.73205080756887729352744634cale,0);
\coordinate (C) at (2*1.73205080756887729352744634cale,0);
\coordinate (D) at (1.73205080756887729352744634cale,1/3*1.73205080756887729352744634*1.73205080756887729352744634cale);
\coordinate (E) at (1.73205080756887729352744634cale,-1/3*1.73205080756887729352744634*1.73205080756887729352744634cale);
\coordinate (F) at (0.5*1.73205080756887729352744634cale,0.5*1.73205080756887729352744634cale*1.73205080756887729352744634);
\coordinate (G) at (1.8*1.73205080756887729352744634cale,0.5*1/3*1.73205080756887729352744634*1.73205080756887729352744634cale);
\coordinate (H) at (0.8*1.73205080756887729352744634cale,-1/3*1.73205080756887729352744634*0.5*1.73205080756887729352744634cale);
\coordinate (I) at (0.6*1.73205080756887729352744634cale,1/3*1.73205080756887729352744634*1.1*1.73205080756887729352744634cale);
\node at (-1/8*1.73205080756887729352744634cale,0) {$1.73205080756887729352744634ubstack{\textnormal{G}_2(2) \\ 2^6\cdot3^3\cdot 7}$};
\node at (1.73205080756887729352744634cale+0.05*1.73205080756887729352744634cale,1/3*1.73205080756887729352744634cale*1.73205080756887729352744634+0.1*1.73205080756887729352744634cale) {$1.73205080756887729352744634ubstack{2^3.\mathrm{L}_3(2) \\ 2^6\cdot3\cdot7}$};
\node at (1.73205080756887729352744634cale+0.15*1.73205080756887729352744634cale,-0.27*1/4*1.73205080756887729352744634cale) {{1.73205080756887729352744634criptsize $2^6\! \cdot 3^2$}};
\node at (0.8*1.73205080756887729352744634cale,0.4*1/3*1.73205080756887729352744634*1.73205080756887729352744634cale) {{1.73205080756887729352744634criptsize $2^6$}};
\node at (1.73205080756887729352744634cale+0.15*1.73205080756887729352744634cale,0.4*1/3*1.73205080756887729352744634*1.73205080756887729352744634cale) {{1.73205080756887729352744634criptsize $2^6\! \cdot 3$}};
\node at (0.7*1.73205080756887729352744634cale,-0.27*1/4*1.73205080756887729352744634cale) {{1.73205080756887729352744634criptsize $ 2^6\! \cdot 3$}};
\node at (0.6*1.73205080756887729352744634cale,0.63*1/3*1.73205080756887729352744634cale*1.73205080756887729352744634+0.45*1/4*1.73205080756887729352744634cale) {{1.73205080756887729352744634criptsize $ 2^6\! \cdot 3$}};
\node at (0.87*1.73205080756887729352744634cale,0.65*1/3*1.73205080756887729352744634cale*1.73205080756887729352744634) {1.73205080756887729352744634criptsize $z$};
\node at (0.3*1.73205080756887729352744634cale,0.06*1.73205080756887729352744634cale) {1.73205080756887729352744634criptsize $x$};
\node at (0.4*1.73205080756887729352744634cale,0.5*1/3*1.73205080756887729352744634cale*1.73205080756887729352744634+0.04*1.73205080756887729352744634cale) {1.73205080756887729352744634criptsize $y$};
\draw [ultra thick] (A) to (B);
\draw (B) to (C);
\draw [ultra thick] (B) to (D);
\draw (D) to (E);
\draw [ultra thick] (A) to (D);
\draw (E) to (A);
\draw (D) to (C);
\draw (F) to (A);
\draw (F) to (D);
\draw [dashed] (B) to (G);
\draw [dashed] (D) to (G);
\draw [dashed] (A) to (H);
\draw [dashed] (B) to (H);
\draw [dashed] (A) to (I);
\draw [dashed] (D) to (I);
\draw [dashed] (1.1*1.73205080756887729352744634cale,0.65*1/3*1.73205080756887729352744634cale*1.73205080756887729352744634) arc[x radius=0.1*1.73205080756887729352744634cale, y radius=0.1*1.73205080756887729352744634cale, start angle=20, end angle=190];
\draw [dashed] (0.4*1.73205080756887729352744634cale,0.08*1.73205080756887729352744634cale) arc[x radius=0.08*1.73205080756887729352744634cale, y radius=0.08*1.73205080756887729352744634cale, start angle=90, end angle=270];
\draw [dashed] (0.5*1.73205080756887729352744634cale,0.5*1/3*1.73205080756887729352744634cale*1.73205080756887729352744634+0.06*1.73205080756887729352744634cale) arc[x radius=0.08*1.73205080756887729352744634cale, y radius=0.08*1.73205080756887729352744634cale, start angle=90, end angle=270];
\end{tikzpicture}
\end{center}
Using a suitable embedding $\textnormal{G}_2 \hookrightarrow \textnormal{O}_7$ we find matrices for the three
generators
1.73205080756887729352744634mall
$$
x:=\left(\begin{array}{@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}}
0 & 1& 1&-1&-1&-1&\phantom{-}0 \\
0 & 0& 0& 0& 0& 0& 1 \\
0& 1&-1& 0& 0&-1& 1 \\
1& 1& 0&-1& 0&-1& 0 \\
0& 0&-1& 0& 0& 0& 1 \\
0&-1& 1& 0& 0& 0& 0 \\
0& 1& 0& 0&-1&-1& 0
\end{array} \right) ,\
y:=\left(\begin{array}{@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}}
1&\phantom{-}1& 0&-1&-1&-1& 0 \\
1& 1&-1&-1& 0&-1& 0 \\
1& 1&-1& 0& 0&-1& 0 \\
1& 0&-1& 0& 0&-1& 0 \\
1& 1& 0&-1& 0&-1&-1 \\
0& 1& 0&-1&-1& 0& 0 \\
1& 0& 0& 0& 0& 0& 0
\end{array} \right) ,\
z:=\frac{1}{2}\left(\begin{array}{@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}}
2& 2&\phantom{-}0&-1& 0&-2&-1 \\
1& 0& 2& 0&-1& 0&-1 \\
2& 2& 2&-3&-2&-2&-1 \\
2& 2& 0&-1&-2&-2&-1 \\
0& 0& 4&-2&-2& 0&-2 \\
2&-2& 0& 1& 2& 0&-1 \\
0& 2& 0&-1& 0&-2& 1
\end{array} \right) . $$
\normalsize
To obtain a presentation in these generators, one only needs to
compute the relations between the pairs of generators that hold in the finite
group generated by the two matrices (in the stabilizer of a vertex).
1.73205080756887729352744634ection{Chamber transitive actions on $p$-adic buildings.}\label{buildtable}
In this section we tabulate the chamber transitive actions
on the $p$-adic buildings obtained from the
one-class genera of lattice chains given in the previous section.
We use the names and the local Dynkin diagrams as given in \cite{Tits}.
The name for $\textnormal{U}_{\mathfrak{p}} $ usually does not give the
precise type of the $p$-adic group.
For instance the lattices $\mathbb{E}_6$ and $\mathbb{D}_6 $ define two non isomorphic
non-split forms of the algebraic
group $\textnormal{O}_6$ over $\mathbb{Q} _2$ which we both denote by $\textnormal{O}_6^-(\mathbb{Q}_2)$ to avoid clumsy
notation. Computing the discriminant of the invariant form (which is $-3$ respectively
$-1$) we see that for $\mathbb{E}_6$ the orthogonal group splits over the
unramified extension, whereas for $\mathbb{D}_6$ the group splits over
the ramified extension $\mathbb{Q}_2(1.73205080756887729352744634qrt{-1})$ of $\mathbb{Q}_2$.
Note that the isomorphism $\textnormal{O}_6^- \cong \textnormal{U}_4$ is given by the action of
$\textnormal{O}_6$ on the even part of the Clifford algebra. So we find the
one-class genera of lattice chains also in a hermitian geometry,
for $\mathcal{L} (\mathbb{E} _6,2 )$ (from \ref{qf} (3)) we get the same stabilizers as for
$\mathcal{L} ((\mathbb{E}_8 ) _{1.73205080756887729352744634qrt{-3}},2 ) $ (from \ref{hf} (2)) in the
projective group. Such coincidences
are indicated by listing the lattices $L_0$ and the
corresponding references (ref) in Table \ref{table1}.
The last column of Table \ref{table1}
refers to a construction of the respective chamber transitive
action in the literature.
For a more detailed description
of the different unitary groups $\textnormal{U}_{p }$ associated
to the various types of local Dynkin diagrams we refer the reader to
\cite[Section 4.4]{Tits}.
\begin{table}\caption{Buildings with chamber transitive discrete actions} \label{table1}
\begin{longtable}{|B{0.3}|B{0.3}|B{2.1}|B{1.2}|B{2.5}|B{1.8}|B{3}|B{1}|}\hline
$p$ & $r$ & $L_0$ & ref & $\textnormal{U}_{p} $ & name & local Dynkin & Lit \\ \hline
2 & 4 & $\mathbb{E}_8 $ & \ref{qf} (1) & $\textnormal{O}_8^+(\mathbb{Q}_2) $ & $\tilde{D}_4$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (1.5,0.5) {$0$};
\node (2) at (1.5,-0.5) {$0'$};
\node (3) at (0,0) {$2$};
\node (4) at (-1.5,0.5) {$4$};
\node (5) at (-1.5,-0.5) {$4'$};
\draw (1) -- (3) -- (2) -- (3);
\draw (4) -- (3) -- (5) -- (3);
\end{tikzpicture} & \cite{KLT} \cite{Ka1}
\\ \hline
2 & 3 & $\mathbb{E}_7 $ & \ref{qf} (2) & $\textnormal{O}_7(\mathbb{Q}_2) $ & $\tilde{B}_3$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (1.5,0.5) {$0$};
\node (2) at (1.5,-0.5) {$0'$};
\node (3) at (0,0) {$2$};
\node (4) at (-1.5,0) {$3$};
\draw (1) -- (3) -- (2) -- (3);
\draw[double] (3) --node{$<$} (4);
\end{tikzpicture} & \cite{KLT} \cite{Ka1}
\\ \hline
2 & 3 & $\mathbb{A}_6 $, \ $(\mathbb{E} _8)_{1.73205080756887729352744634qrt{-7}}$ & \ref{qf} (3) \ref{hf} (2) & $\textnormal{O}_6^+(\mathbb{Q}_2) \cong \textnormal{SL}_4(\mathbb{Q}_2) $ & $\tilde{A}_3$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (0,0) {$0$};
\node (2) at (2,0) {$3$};
\node (3) at (1,-1) {$0'$};
\node (4) at (-1,-1) {$3'$};
\draw (1) -- (2) -- (3) -- (4) -- (1);
\end{tikzpicture} & \cite{KLT} \cite{Ka2} \\ \hline
2 & 2 & $\mathbb{E}_6 $, \ $(\mathbb{E} _8)_{1.73205080756887729352744634qrt{-3}}$ & \ref{qf} (4) \ref{hf} (3) & $\textnormal{O}_6^-(\mathbb{Q}_2) \cong \textnormal{U}_4(\mathbb{Q}_2(1.73205080756887729352744634qrt{-3})) $ & $B-C_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$>$} (3) --node{$<$} (2);
\end{tikzpicture} & \cite{KLT} \cite{MW1}
\\ \hline
2 & 2 & $\mathbb{D}_6 $, \ $(\mathbb{E} _8)_{1.73205080756887729352744634qrt{-1}}$ & \ref{qf} (5) \ref{hf} (4) & $\textnormal{O}_6^-(\mathbb{Q}_2) \cong \textnormal{U}_4(\mathbb{Q}_2(1.73205080756887729352744634qrt{-1})) $ & $C-B_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$<$} (3) --node{$>$} (2);
\end{tikzpicture} & \cite{KLT} \cite{Ka1}
\\ \hline
3 & 2 & $\mathbb{E}_6 $, $\mathbb{B}_5\perp \ ^{(3)} \mathbb{B}_1$, $(\mathbb{E} _8)_{1.73205080756887729352744634qrt{-3}}$ & \ref{qf} (6) \ref{qf} (7) \ref{hf} (5) & $\textnormal{O}_6^-(\mathbb{Q}_3) = \textnormal{O}_6^-(\mathbb{Q}_3) \cong \textnormal{U}_4(\mathbb{Q}_3(1.73205080756887729352744634qrt{-3})) $ & $C-B_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$<$} (3) --node{$>$} (2);
\end{tikzpicture} & \cite{KLT} \cite{KMW90}
\\ \hline
2 & 2 & $\mathbb{A}_5 $, \ \ $(\mathbb{E} _8)_{\infty, 3}$ & \ref{qf} (8) \ref{qaf} (1) & $\textnormal{O}_5(\mathbb{Q}_2) \cong \textnormal{Sp}_4(\mathbb{Q}_2) $ & $\tilde{C}_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$>$} (3) --node{$<$} (2);
\end{tikzpicture} & \cite{KLT} \cite{MW1}
\\ \hline
3 & 2 & $\mathbb{B}_5 $, \ \ $(\mathbb{E} _8)_{\infty, 2}$ & \ref{qf} (9) \ref{qaf} (2) & $\textnormal{O}_5(\mathbb{Q}_3) \cong \textnormal{Sp}_4(\mathbb{Q}_3) $ & $\tilde{C}_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$>$} (3) --node{$<$} (2);
\end{tikzpicture} & \cite{KLT} \cite{KMW90}
\\ \hline
2 & 2 & $ (\ ^{(7)} \mathbb{A}_6^{\#})_{1.73205080756887729352744634qrt{-7}} $
& \ref{hf} (6) & $\textnormal{SL}_3(\mathbb{Q}_2) $ & $\tilde{A}_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (0,0) {$0$};
\node (3) at (-1,-1) {$0''$};
\node (2) at (1,-1) {$0'$};
\draw (1) -- (3) -- (2) -- (1);
\end{tikzpicture}
& \cite{KLT} \cite{KMW1} \cite{Mu}
\\ \hline
3 & 2 & $(\mathbb{A}_2^5)_{1.73205080756887729352744634qrt{-3}}$ & \ref{hf} (1) & $\textnormal{U}_5(\mathbb{Q}_3(1.73205080756887729352744634qrt{-3})) $ & $C-BC_2$ &
\begin{tikzpicture}[scale=0.75]
\node (1) at (-1.5,0) {$0$};
\node (3) at (0,0) {$2$};
\node (2) at (1.5,0) {$0'$};
\draw[double] (1) --node{$<$} (3) --node{$<$} (2);
\end{tikzpicture} & new
\\ \hline
2 & 2 & ${\bf G}_2(\mathbb{Z}[\frac{1}{2}])$ & \ref{g2} & $\textnormal{G}_2(\mathbb{Q}_2) $ & $\tilde{\textnormal{G}}_2$ &
\begin{tikzpicture}[scale=0.75]
\node (0) at (-1.5,0) {$0$};
\node (1) at (0,0) {$1$};
\node (2) at (1.5,0) {$2$};
\draw (0) -- (1);
\draw[double, double distance=3pt] (1) -- node{$>$} (2);
\draw (2) -- (1);
\end{tikzpicture} & \cite{KLT} \cite{Ka1}
\\ \hline
\end{longtable}
\end{table}
\end{document} |
\begin{document}
\begin{frontmatter}
\pretitle{Research Article}
\title{Drifted Brownian motions governed by fractional tempered derivatives}
\author[a]{\inits{M.}\fnms{Mirko}~\snm{D'Ovidio}\thanksref{cor1}\ead
[label=e1]{[email protected]}}
\author[b]{\inits{F.}\fnms{Francesco}~\snm{Iafrate}\ead
[label=e2]{[email protected]}}
\author[b]{\inits{E.}\fnms{Enzo}~\snm{Orsingher}\ead
[label=e1]{[email protected]}}
\thankstext[type=corresp,id=cor1]{Corresponding author.}
\address[a]{SBAI, \institution{Sapienza University of Rome}, \cny{Italy}}
\address[b]{DSS, \institution{Sapienza University of Rome}, \cny{Italy}}
\markboth{M. D'Ovidio et al.}{Drifted Brownian motions governed by
fractional tempered derivatives}
\begin{abstract}
Fractional equations governing the
distribution of reflecting drifted Brownian motions are presented. The equations are
expressed in terms of tempered Riemann--Liouville type derivatives. For
these operators a Marchaud-type form is obtained and a Riesz tempered
fractional derivative is examined, together with its Fourier transform.
\end{abstract}
\begin{keywords}
\kwd{Tempered fractional derivatives}
\kwd{drifted Brownian motion}
\end{keywords}
\begin{keywords}[MSC2010]
\kwd{34A08}
\kwd{60J65}
\end{keywords}
\received{\sday{24} \smonth{4} \syear{2018}}
\revised{\sday{28} \smonth{6} \syear{2018}}
\accepted{\sday{29} \smonth{8} \syear{2018}}
\publishedonline{\sday{19} \smonth{9} \syear{2018}}
\end{frontmatter}
\section{Introduction}
In this paper we consider various forms of tempered fractional
derivatives. For a function $ f $ continuous and compactly supported on the
positive real line, let us consider the Marchaud type operator defined by
\begin{equation}
\label{eq:march-def} \bigl(\mathscr{D}^{\alpha, \eta}f\bigr) (x) =
\int
_{0}^{\infty} \bigl(f(x) - f(x-y) \bigr) \, \varPi(
\mathrm{d} y)
\end{equation}
where
\begin{equation}
\varPi(\mathrm{d} y) = \frac{\alpha}{\varGamma(1-\alpha)} \frac
{e^{-\eta y }}{y^{\alpha+1}} dy ,\quad y>0\xch{,}{.}
\label{LevyMH}
\end{equation}
with $ \eta>0, 0 < \alpha<1 $. The operator \eqref{eq:march-def}
coincides with the classical Marchaud derivative for $ \eta= 0 $.
The Laplace transform of the fractional operator \eqref{eq:march-def} reads
\begin{align}
\int_{0}^{\infty} e^{ - \lambda x} \, \bigl(
\mathscr{D}^{\alpha, \eta} f \bigr) (x) \, \mathrm{d} x &= \Biggl(
\int
_{0}^{\infty} \bigl( 1 - e^{ - \lambda y} \bigr) \varPi(
\mathrm{d} y) \Biggr) \tilde{f} (\lambda)
\nonumber
\\
&= \bigl( (\eta+ \lambda)^{\alpha}- \eta^{\alpha} \bigr) \tilde{f}(
\lambda) . \label{symbolH}
\end{align}
Throughout the work we denote by $\tilde{f}$ the Laplace transform of
$f$. In the Fourier analysis the factor $ (\eta+ i \lambda)^{\alpha
}- \eta^{\alpha}$ is the multiplier of the Fourier transform of
$ f $~\xch{\cite{meersc15}}{(\cite{meersc15})}. Tempered fractional derivatives emerge in the study
of equations driving the tempered subordinators \xch{\cite{beghin,meersc15}}{(\cite{beghin,meersc15})}.
In particular, the operator \eqref{eq:march-def} is the generator of
the subordinator $ H_{t}, t>0 $, with L\'{e}vy measure \eqref{LevyMH}
and density law whose Laplace transform is given by \eqref{symbolH},
that is,
\begin{align}
\mathbb{E} e^{- \lambda H_{t}} &= e^{-t ( (\eta+ \lambda)^{\alpha}-
\eta^{\alpha} ) } = e^{ - t \int_{0}^{\infty} ( 1 - e^{-\lambda y})
\varPi(\mathrm{d} y)}, \quad\lambda
>0.
\nonumber
\end{align}
The process $ H_{t} $ is called relativistic subordinator and
coincides, for $ \eta= 0 $, with a
positively skewed L\'{e}vy process, that is a stable subordinator.
Tempered stable subordinators can be viewed as the limits of Poisson
random sums with tempered power law jumps \xch{\cite{meersc15}}{(\cite{meersc15})}.
The fractional operator $ \mathscr{D}^{\alpha, \eta} f$ defined in
\eqref{eq:march-def} is related
to the tempered upper Weyl derivatives defined by
\begin{equation}
\label{eq:up-weyl-intro} \bigl(\hat{\mathscr{D}}^{\alpha, \eta
}_{+}f\bigr) (x) =
\frac{1}{\varGamma(1-\alpha)} \frac{\mathrm{d}}{\mathrm{d} x} \int
_{-\infty}^{x}
\frac{f(t)}{(x-t)^{\alpha}} e^{- \eta(x-t)} \, \mathrm{d} t.
\end{equation}
By combining \eqref{eq:up-weyl-intro} with the lower Weyl tempered
derivatives we obtain
the Riesz tempered fractional derivatives $ \frac{\partial^{\alpha,
\eta} f}{\partial|x|^{\alpha}} $
from which we obtain the explicit
Fourier transform in \eqref{eq:temp-riesz-fou}.
We consider the Dzherbashyan--Caputo derivative of order $ \frac{1}{2} $,
that is,
\begin{equation}
\bigl( D^{\frac{1}{2}} f \bigr) (t) = \frac{1}{\sqrt{\pi}} \int
_{0}^{t} f'(s) (t-s)^{-\frac{1}{2}} \,
\mathrm{d} s
\end{equation}
with the Laplace transform
\begin{align*}
\int_{0}^{\infty}e^{-\lambda t} \bigl(
D^{\frac{1}{2}} f \bigr) (t)\, \mathrm{d} t = \lambda^{\frac{1}{2}}
\tilde{f}(
\lambda) - \lambda^{\frac{1}{2}-1} f(0), \quad\lambda>0.
\end{align*}
The relationship between the Riemann--Liouville and the
Dzherbashyan--Caputo de\-ri\-vative can be given as follows,
\begin{equation}
\bigl( \mathscr{D} ^{\frac{1}{2}} f \bigr) (t) = \bigl( D^{\frac{1}{2}} f
\bigr) (t) + \frac{t^{\frac{1}{2} -1}}{\varGamma(\frac{1}{2})} f(0),
\end{equation}
from which we observe that
\begin{align}
\int_{0}^{\infty}e^{-\lambda t} \bigl( \mathscr{D}
^{\frac{1}{2}} f \bigr) (t) \, \mathrm{d} t = \lambda^{\frac{1}{2}}
\tilde{f}(
\lambda). \label{Ltransf-RL}
\end{align}
We remark that the problems
\begin{align*}
\left\lbrace
\begin{array}{@{}ll@{}}
\displaystyle\bigl(D^{\frac{1}{2}} u\bigr)(t) = - \frac{\partial u}{\partial y},
\quad t>0, y>0\\[3pt]
\displaystyle u(0,y)=\delta(y)
\end{array}
\right. \quad\text{\textrm{and}} \quad\left\lbrace
\begin{array}{@{}ll@{}}
\displaystyle\bigl(\mathscr{D}^{\frac{1}{2}} u\bigr)(t) = - \frac{\partial
u}{\partial y}, & t>0, y>0\\[3pt]
\displaystyle u(0,y)=\delta(y)\\[3pt]
\displaystyle u(t,0)= \frac{1}{\sqrt{\pi t}}, & t>0
\end{array}
\right.
\end{align*}
have a unique solution given by the density law of an inverse to a
stable subordinator, say $L_{t}$ (see for example \cite[formulas 3.4
and 3.5]{dovidio}). It is well known that $L_{t}$ (with $L_{0}=0$) is
identical in law to a folded Brownian motion $|B_{t}|$ (with
$B_{0}=0$), that is, $u$ is the unique solution to the problem
\begin{align*}
\left\lbrace
\begin{array}{@{}ll@{}}
\displaystyle\frac{\partial u}{\partial t} = \frac{\partial^{2}
u}{\partial y^{2}}, \quad t>0,\, y>0,\\[3pt]
\displaystyle u(0, y) = \delta(y),\\[3pt]
\displaystyle\frac{\partial u}{\partial y}(t, 0) =0.
\end{array}
\right.
\end{align*}
Thus, by considering the theory of time changes, there exist
interesting connections between fractional Cauchy problems and the
domains of the generators of the base processes. In our view,
concerning the drifted Brownian motion, the present paper gives new
results also in this direction.
We denote by
\begin{equation}
\label{eq:temp-rl-def} \mathscr{D} _{t}^{\frac{1}{2}, \eta}f :=
e^{-\eta t} \mathscr{D} _{t}^{\frac{1}{2}} \bigl(
e^{\eta t} f \bigr) - \sqrt{\eta}f
\end{equation}
the tempered Riemann--Liouville type derivative. The equality between
definitions \eqref{eq:temp-rl-def} and \eqref{eq:march-def} can be
verified by comparing the corresponding Laplace transforms. Indeed,
from \eqref{Ltransf-RL},
\begin{align*}
\int_{0}^{\infty}e^{ - \lambda t}\, \mathscr{D}
_{t}^{\frac{1}{2}, \eta}f \, \mathrm{d} t ={}& \int_{0}^{\infty}e^{
-(\lambda+ \eta) t}
\mathscr{D} _{t}^{\frac{1}{2}} ( g ) \, \mathrm{d} t - \sqrt{\eta}
\tilde{f} (\lambda)
\\
={}& \sqrt{\lambda+\eta} \int_{0}^{\infty}e^{-(\lambda+\eta) t}
g(t)\, \mathrm{d} t - \sqrt{\eta}\tilde{f} (\lambda)
\end{align*}
where $g(t) = e^{\eta t} f(t)$.
Let $B $ represent a Brownian motion starting at the origin with
generator $\Delta$. In the paper we show that the transition density $
u = u(x,y,t) $ of the 1-dimensional process
\begin{equation*}
B^{\mu}(t) = B(t) + \mu t + x, \quad\mu>0, \, x \in\mathbb{R},
\end{equation*}
satisfies the fractional equation on $(0, \infty) \times\mathbb{R}^{2}$
\begin{equation}
\label{eq:frac-eq-u}
\left\{
\begin{aligned} &\mathscr{D} _{t}^{\frac{1}{2}, \eta} u +
\sqrt{\eta} \,u= a(x,y) \biggl( \frac{\partial u}{\partial x} + \sqrt
{\eta} u \biggr) = -
a(x,y) \biggl( \frac{\partial u}{\partial y} - \sqrt{\eta} u \biggr),
\\
&u(x,y,0) = \delta(x-y)
\end{aligned}
\right.
\end{equation}
where
\begin{equation*}
a(x,y) = \mathbh{1}_{(-\infty,y] } (x) - \mathbh{1} _{(y, \infty) } (x)
\end{equation*}
and
\begin{align*}
\eta= \frac{\mu^{2}}{4}.
\end{align*}
A different result concerns the reflected process
\begin{equation*}
|B^{\mu}(t) + \mu t| + x = |\hat{B}|^{\mu}(t)
\end{equation*}
whose transition density $v = v(x,y,t) $ satisfies the equation
\begin{equation}
\label{eq:frac-eq-v} \mathscr{D} _{t}^{\frac{1}{2}, \eta} v + \sqrt
{\eta} \, v=
\frac{\partial v}{\partial x} + \sqrt{\eta} \tanh\bigl(\sqrt{\eta}
(y-x) \bigr) v,
\quad t >0,\; y > x >0,
\end{equation}
with initial and boundary conditions
\begin{align*}
v(x,y,0) ={}& \delta( y - x),
\\
v(x,x,t) ={}& \frac{e^{-\eta t}}{\sqrt{\pi t}}, \quad t>0,
\end{align*}
and
\begin{align*}
\eta= \frac{\mu^{2}}{4}.
\end{align*}
The fractional equation governing the iterated Brownian motion $ B^{\mu
_{2}} ( | B^{\mu_{1}} (t) | )$\break ($ B^{\mu_{j}} $ being independent)
has been studied in \cite{iafrate18} and in the special case $ B^{\mu
}( | B (t) | ) $ explicitly derived. For the iterated Bessel process a
similar analysis is performed in~\cite{DOVIDIO2011441}. A~general
presentation of tempered fractional calculus can be found in the paper \cite{meersc15}.
Many processes like Brownian motion, iterated Brownian motion, Cauchy
process have transition functions
satisfying different partial differential equations and also are
solutions of fractional equations
of different forms with various fractional derivatives. We here show
that a similar situation arises when
drifted reflecting Brownian motion is considered but in this case the
corresponding fractional equations
involve tempered Riemann--Liouville type derivatives.
\section{A generalization of the tempered Marchaud derivative}
In this section we study the tempered Weyl derivatives (upper and lower
ones) and construct the Riesz tempered derivative.
We are able to obtain the Fourier transform of the Riesz tempered
derivatives and thus to solve some generalized fractional diffusion
equation.
We start by giving the explicit forms of the tempered Weyl derivative
\begin{align}
& \bigl(\hat{\mathscr{D}}^{\alpha, \eta}_{+}f\bigr) (x)
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \frac{\mathrm{d}}{\mathrm{d} x} \int
_{-\infty}^{x}
\frac{f(t)}{(x-t)^{\alpha}} e^{- \eta(x-t)} \, \mathrm{d} t
\\
&= \frac{1}{\varGamma(1-\alpha)} \frac{\mathrm{d}}{\mathrm{d} x} \int
_{0}^{\infty}
\frac{f(x-t)}{t^{\alpha}} e^{-\eta t} \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \frac{\mathrm{d}}{\mathrm{d} x} \int
_{0}^{\infty}
f(x-t) e^{-\eta t } \int_{t}^{\infty} \alpha
w^{-\alpha-1} \, \mathrm{d} w \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, \mathrm{d} w \int_{0}^{w}
f'(x-t) e^{-\eta t} \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, \mathrm{d} w \int_{x-w}^{x}
f'(t) e^{-\eta(x-t)} \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, e^{-\eta x} \Biggl\{ f(t) e^{\eta t}
|_{x-w}^{x} - \eta\int_{x-w}^{x}
f(t) e^{\eta t} \, \mathrm{d} t \Biggr\} \, \mathrm{d} w
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, e^{-\eta x} \bigl[ f(x) e^{\eta x} -
f(x-w)e^{\eta(x-w)} \bigr] \, \mathrm{d} w
\nonumber
\\
& \quad- \frac{\eta}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \int_{x-w}^{x} f(t)
e^{-\eta(x-t)}\, \mathrm{d} w\, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, \bigl[ f(x) - f(x-w)e^{-\eta w} \bigr] \, \mathrm{d} w
\nonumber
\\
& \quad- \frac{\eta}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \int_{0}^{w} f(x-t)
e^{-\eta t}\, \mathrm{d} w \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \, \bigl[ f(x) + f(x) e^{-\eta w} - f(x) e^{-\eta w} -
f(x-w)e^{-\eta w} \bigr] \, \mathrm{d} w
\nonumber
\\
& \quad- \frac{\eta}{\varGamma(1-\alpha)} \int_{0}^{\infty} f(x-t)
e^{-\eta t} \int_{t}^{\infty}\alpha
w^{-\alpha-1} \, \mathrm{d} w \, \mathrm{d} t
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \bigl(f(x) -
f(x-w) \bigr) \, \alpha\frac{e^{-\eta w}}{w^{\alpha+1}} \,\mathrm{d}
w \, \mathrm{d} t
\nonumber
\\
& \quad+ \frac{f(x)}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \bigl(1-e^{-\eta w} \bigr)\, \mathrm{d} w -
\frac{\eta}{\varGamma(1-\alpha)} \int_{0}^{\infty} f(x-t)
\frac{e^{-\eta t}}{t^{\alpha}} \, \mathrm{d} t
\nonumber
\\
&= \int_{0}^{\infty} \bigl(f(x) - f(x-w) \bigr) \varPi
(\mathrm{d} w) + \eta\int_{0}^{\infty} \bigl(f(x) -
f(x-w) \bigr) \frac{e^{-\eta w }}{w^{\alpha}\varGamma(1-\alpha)} \,
\mathrm{d} w
\nonumber
\end{align}
The derivative $ \hat{\mathscr{D}}_{+} ^{\alpha, \eta}$ can be
expressed in terms of $ \mathscr{D}^{\alpha, \eta} $ as follows:
\begin{equation*}
\hat{\mathscr{D}}^{\alpha, \eta}_{+}f = \mathscr{D}^{\alpha, \eta}
f -
\eta\, \mathscr{D}^{\alpha-1, \eta} f.
\end{equation*}
In the same way we can obtain the upper Weyl derivative in the Marchaud
form as
\begin{align}
& \bigl(\hat{\mathscr{D}}^{\alpha, \eta}_{-}f\bigr) (x)
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \frac{\mathrm{d}}{\mathrm{d} x} \int
_{x}^{\infty}
\frac{f(t)}{(x-t)^{\alpha}} e^{- \eta(x-t)} \, \mathrm{d} t
\\
&= \int_{0}^{\infty} \frac{\alpha w ^{-\alpha-1}}{\varGamma(1-\alpha
)} \bigl\{
e^{-\eta w} f(x+w) - f(x)\bigr\} \,\mathrm{d} w + \eta\int
_{0}^{\infty} \frac{e^{\eta w} f(x+w)}{\varGamma(1-\alpha) w^{\alpha
} } \,\mathrm{d} w
\nonumber
\\
&= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{\infty} \bigl[ f(x+w) -
f(x)\bigr] \frac{\alpha e^{-\eta w}}{w^{\alpha+1} }
\nonumber
\\
& \quad+ \frac{f(x)}{\varGamma(1-\alpha)} \int_{0}^{\infty} \alpha
w^{-\alpha-1} \bigl(e^{-\eta w } -1\bigr) \, \mathrm{d} w + \eta\int
_{0}^{\infty} f(x+t) \frac{e^{-\eta w}}{\varGamma(1-\alpha)
t^{\alpha}} \, \mathrm{d} t
\nonumber
\\
&= \int_{0}^{\infty} \bigl[ f(x+w) - f(x)\bigr] \varPi
(\mathrm{d} w) + \eta\int_{0}^{\infty} \bigl[ f(x+w) -
f(x)\bigr] \frac{e^{-\eta w}}{\varGamma(1-\alpha) w^{\alpha}} \,
\mathrm{d} w .
\nonumber
\end{align}
For $ 0 < \alpha<1 $ the Riesz fractional derivative writes
\begin{align}
\label{eq:riesz-der} \frac{\partial^{\alpha}f}{\partial|x|^{\alpha
}} &= - \frac{1}{2 \cos\frac{\alpha\pi}{2} \, \varGamma(1-\alpha)}
\int
_{-\infty}^{+\infty} \frac{f(t)}{|x-t|^{\alpha}} \, \mathrm{d} t
\\
&= - \frac{1}{2 \cos\frac{\alpha\pi}{2} \, \varGamma(1-\alpha)}
\Biggl[ \frac{\mathrm{d}}{\mathrm{d} x} \int_{-\infty}^{x}
\frac{f(t)}{(x-t)^{\alpha}} \, \mathrm{d} t - \frac{\mathrm{d}}{\mathrm
{d} x} \int_{x }^{\infty}
\frac{f(t)}{(t-x)^{\alpha}} \, \mathrm{d} t \Biggr].
\nonumber
\end{align}
In the same way we define the tempered Riesz derivative as
\begin{align*}
\frac{\partial^{\alpha, \eta} f}{\partial|x|^{\alpha}} = C_{\alpha
, \eta} \Biggl[ \frac{\mathrm{d}}{\mathrm{d} x} \int
_{-\infty}^{x} \frac{f(t)}{(x-t)^{\alpha}} \frac{e^{- \eta(x-t)} }{
\varGamma(1-\alpha)}\,
\mathrm{d} t - \frac{\mathrm{d}}{\mathrm{d} x} \int_{x }^{\infty}
\frac{f(t)}{(t-x)^{\alpha}} \frac{e^{- \eta(t-x)} }{ \varGamma
(1-\alpha)} \, \mathrm{d} t \Biggr]
\end{align*}
where $ C_{\alpha,\eta} $ is a suitable constant which will be
defined below.
In view of the previous calculations we have that
\begin{align}
\frac{\partial^{\alpha, \eta} f}{\partial|x|^{\alpha}} &=
C_{\alpha, \eta} \Biggl[ \int_{0}^{\infty}
\bigl(f(x) - f(x-w) \bigr) \frac{\alpha e^{-\eta w} \mathrm{d}
w}{\varGamma(1-\alpha) w^{\alpha+ 1} }
\nonumber
\\
&\quad + \eta\int_{0}^{\infty} \bigl(f(x) - f(x-w) \bigr)
\frac{e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha) w^{\alpha}}
\\
&\quad - \int_{0}^{\infty} \bigl(f(x+w) - f(x) \bigr)
\frac{\alpha e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha) w^{\alpha+
1} }
\nonumber
\\
&\quad - \eta\int_{0}^{\infty} \bigl(f(x+w) - f(x) \bigr)
\frac{e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha) w^{\alpha}}
\Biggr]
\nonumber
\\
& = C_{\alpha, \eta} \Biggl[ \int_{0}^{\infty} \bigl( 2
f(x) - f(x-w) - f(x+w) \bigr) \frac{\alpha e^{-\eta w}\mathrm{d}
w}{\varGamma(1-\alpha) w^{\alpha+ 1} }
\nonumber
\\
&\quad+ \eta\int_{0}^{\infty} \bigl( 2 f(x) - f(x-w)
- f(x+w) \bigr) \frac{ e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha)
w^{\alpha} } \Biggr].
\nonumber
\end{align}
We now evaluate the Fourier transform of the tempered Riesz derivative
\begin{align}
\label{eq:temp-riesz-fou} \int_{-\infty}^{+\infty} e^{i\gamma x}
\frac{\partial^{\alpha, \eta} f}{\partial|x|^{\alpha}} \, \mathrm
{d} x ={}& C_{\alpha, \eta} \Biggl\{ \hat{F}(\gamma
) \int_{0}^{\infty} \bigl( 1 - e^{i\gamma w} \bigr)
\frac{\alpha e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha) w^{\alpha+
1} }
\\
& + \eta\hat{F}(\gamma) \int_{0}^{\infty} \bigl( 1 -
e^{i\gamma w} \bigr) \frac{ e^{-\eta w}\mathrm{d} w}{\varGamma
(1-\alpha) w^{\alpha} }
\nonumber
\\
& - \hat{F}(\gamma) \int_{0}^{\infty} \bigl(
e^{i\gamma w} -1 \bigr) \frac{\alpha e^{-\eta w}\mathrm{d} w}{\varGamma
(1-\alpha) w^{\alpha+ 1} }
\nonumber \\
& - \eta\hat{F}(\gamma) \int_{0}^{\infty} \bigl(
e^{i\gamma w} -1 \bigr) \frac{ e^{-\eta w}\mathrm{d} w}{\varGamma
(1-\alpha) w^{\alpha} } \Biggr\}
\nonumber
\\
={}& C_{\alpha, \eta} \hat{F} ( \gamma) \Biggl\{ 2 \int
_{0}^{\infty} (1 - \cos\gamma w) \frac{\alpha e^{-\eta w}\mathrm{d}
w}{\varGamma(1-\alpha) w^{\alpha+ 1} }
\nonumber
\\
& + 2 \eta\int_{0}^{\infty} (1 - \cos\gamma w)
\frac{ e^{-\eta w}\mathrm{d} w}{\varGamma(1-\alpha) w^{\alpha} }
\Biggr\}
\nonumber
\\
={}& C_{\alpha, \eta} \hat{F} ( \gamma) \Biggl\{ -2 w^{-\alpha}
(1-\cos
\gamma w) \frac{e^{-\eta w}}{\varGamma(1-\alpha)} |_{0}^{\infty}
\nonumber
\\
& - 2 \eta\int_{0}^{\infty} (1 - \cos\gamma w)
\frac{e^{-\eta w} \mathrm{d} w}{w^{\alpha}\varGamma(1-\alpha)}
\nonumber
\\
& + 2 \gamma\int_{0}^{\infty} \frac{e^{-\eta w} \sin\gamma
w}{w^{\alpha}\varGamma(1-\alpha)} \,
\mathrm{d} w
\nonumber
\\
& + 2 \eta\int_{0}^{\infty} (1 - \cos\gamma w)
\frac{e^{-\eta w}}{w^{\alpha}\varGamma(1-\alpha)} \mathrm{d} w
\Biggr\}
\nonumber
\\
={}& C_{\alpha, \eta} \hat{F} ( \gamma) 2 |\gamma| \int
_{0}^{\infty}
\frac{e^{-\eta w} \sin|\gamma| w}{w^{\alpha}\varGamma(1-\alpha)}
\, \mathrm{d} w
\nonumber
\\
={}& C_{\alpha, \eta} \hat{F} ( \gamma) \frac{ 2| \gamma| }{ (
\eta^{2} + \gamma^{2})^{ \frac{1 - \alpha}{2}}} \sin\biggl((1-
\alpha) \arctan\frac{|\gamma|}{\eta} \biggr).
\nonumber
\end{align}
In the last step we used the following formula (\cite{gradtable}, p.~490, formula 5)
\begin{equation*}
\int_{0}^{\infty} x^{\mu- 1 } e^{ - \beta x}
\sin\delta x \, \mathrm{d} x = \frac{\varGamma(\mu)}{(\beta^{2} +
\delta^{2}) ^{ \frac{\mu}{2}}} \, \sin\biggl( \mu\arctan
\frac{\delta}{\mu} \biggr)
\end{equation*}
with $\mathrm{Re } \, \mu> -1$, $\mathrm{Re } \, \beta\geq\mathrm{Im
} \, \delta$.
\begin{remark}
For $ \eta\to0 $ we have that
\begin{align*}
\lim_{\eta\to0} \sin\biggl((1-\alpha) \arctan\frac{|\gamma|}{\eta}
\biggr) = \cos\biggl( \frac{\pi\alpha}{2} \biggr).
\end{align*}
Therefore
\begin{align*}
\lim_{\eta\to0} \int_{- \infty}^{+\infty}
e^{ i \gamma x} \, \frac{\partial^{\alpha, \eta} f}{\partial
|x|^{\alpha}} \, \mathrm{d} x= 2 C_{\alpha, 0} \, |
\gamma|^{\alpha}\cos\biggl( \frac{\pi\alpha}{2} \biggr)\, \hat
{F}(\gamma)
\end{align*}
and thus the normalizing constant must be $ C_{\alpha, 0} = - ( 2 \cos
\frac{\pi\alpha}{2} ) ^{-1} $.
This means that for $ \eta\to0 $ we obtain from \eqref
{eq:temp-riesz-fou} the Fourier transform of the Riesz fractional
derivative \eqref{eq:riesz-der}. This result shows that symmetric
stable processes are governed by equations
\begin{equation*}
\frac{\partial u}{\partial t } = \frac{\partial^{\alpha}u}{\partial
|x|^{\alpha}}
\end{equation*}
see, for example, \cite{toaldo14}, where the interplay between stable
laws, including subordinators and inverse
subordinators, and fractional equations is considered.
\end{remark}
\begin{remark}
For fractional equations of the form
\begin{equation}
\left\{
\begin{aligned}
&\frac{\partial u}{\partial t } = \frac{\partial^{\alpha, \eta}
u}{\partial|x|^{\alpha}},
& & t> 0, \;
x \in\mathbb{R},
\\
& u(x,0) = \delta(x), & & x \in\mathbb{R},
\end{aligned}
\right.
\end{equation}
the Fourier transform of the solution reads
\begin{align*}
& \int_{-\infty}^{+\infty} e^{i \gamma x} u(x,t) \,
\mathrm{d} x
\\
&= \exp\biggl\{ t \, C_{\alpha, \eta} \frac{2 |\gamma| }{ ( \eta
^{2} + \gamma^{2})^{ \frac{1 - \alpha}{2} } } \sin\biggl( (1-\alpha
) \arctan\frac{|\gamma|}{\eta} \biggr) \biggr\}
\\
&= \exp\biggl\{ t \, C_{\alpha, \eta} \frac{2 |\gamma| }{ ( \eta
^{2} + \gamma^{2})^{1 - \frac{ \alpha}{2} } } \biggl[ |\gamma| \cos
\biggl( \alpha\arctan\frac{|\gamma|}{\eta} \biggr) - \eta\sin
\biggl( \alpha
\arctan\frac{|\gamma|}{\eta} \biggr) \biggr] \biggr\} .
\end{align*}
\end{remark}
\section{Fractional equations governing the drifted Brownian motion}
The law of the drifted Brownian motion started at $x$ satisfies the equations
\begin{equation*}
\frac{\partial u}{\partial t } = \frac{\partial^{2} u }{\partial
y^{2}} - \mu\frac{\partial u}{\partial y}, \quad t>0, y
\in\mathbb{R},
\end{equation*}
and
\begin{equation*}
\frac{\partial u}{\partial t } = \frac{\partial^{2} u }{\partial
x^{2}} + \mu\frac{\partial u}{\partial x}, \quad t>0,\; x
\in\mathbb{R}.
\end{equation*}
We show here that the drifted Brownian motion is related to time
fractional equations with tempered derivatives. Let us consider the process
\begin{equation}
B^{\mu}(t) = B(t) + \mu t + x, \quad\mu\in\mathbb{R},\; x \in
\mathbb{R}.
\end{equation}
The law $ u = u(x,y,t) $ of the process $ B^{\mu}$ is given by
\begin{equation}
\label{key} u ( x, y, t ) = \frac{
e^{ - \frac{(y-x-\mu t)^{2}}{4t} }
}{\sqrt{4 \pi t}} = \frac{ e^{ - \frac{ (y-x)^{2}}{ 4t} } }{\sqrt{4 \pi t}}
e^{ - \mu^{2} \frac{t}{4} + \frac{\mu}{2} (y-x)}, \quad t>0,\; x,y
\in\mathbb{R}.
\end{equation}
\begin{thm}
The law of $B^{\mu}$ solves the Cauchy problem
\begin{equation}
\label{eq:frac-eq-u-thm}
\left\{
\begin{aligned} &\mathscr{D} _{t}^{\frac{1}{2}, \eta} u +
\sqrt{\eta} \, u = a(x,y) \biggl( \frac{\partial u}{\partial x} + \sqrt
{\eta} \, u
\biggr), \quad t>0,\; x,y, \in\mathbb{R},
\\
& \textcolor{white} {\mathscr{D} _{t}^{\frac{1}{2}, \eta} u + \sqrt
{\eta}
\, u} = - a(x,y) \biggl( \frac{\partial u}{\partial y} - \sqrt{\eta}
\, u \biggr), \quad
t>0,\; x,y, \in\mathbb{R},
\\
&u(x,y,0) = \delta(x-y)
\end{aligned}
\right.
\end{equation}
with
\begin{align*}
\eta= \frac{\mu^{2} }{4}.
\end{align*}
\end{thm}
\begin{proof}
We start by computing the Laplace--Fourier transform of the function
\begin{equation*}
g(x,y,t) = \frac{e^{ - \frac{ (y-x)^{2}}{4t}}}{\sqrt{4 \pi t }},
\end{equation*}
that is,
\begin{align*}
\hat{\tilde{ g}} (y,\xi, \lambda) &= \int_{0}^{\infty}
e^{ - \lambda t} \int_{ -\infty}^{+\infty} e^{i \xi x}
g(x,y,t) \, \mathrm{d} x \,\mathrm{d}t
\\
&= \int_{0}^{\infty}e^{- \lambda t} e^{i \xi y \,- \,\xi^{2} t}
\,\mathrm{d} t
\\
&= \frac{e^{i \xi y}}{\lambda+ \xi^{2}} .
\end{align*}
By using the fact that
\begin{align}
\tilde{g}(x,y,\lambda) = \frac{e^{-|y-x|\sqrt{\lambda}}}{2 \sqrt
{\lambda}} =
\left\lbrace
\begin{array}{@{}ll}
\displaystyle\frac{e^{-(y-x)\sqrt{\lambda}}}{2 \sqrt{\lambda}},
\quad y>x,\\[9pt]
\displaystyle\frac{e^{-(x-y)\sqrt{\lambda}}}{2 \sqrt{\lambda}},
\quad y \leq x,
\end{array}
\right.
\label{lapBYUSING}
\end{align}
we now compute the double transform of $ a(x,y) \frac{ \partial g
}{\partial x} $.
\begin{align}
\label{eq:fou-lap-der} &\int_{-\infty}^{\infty} e^{ i \xi x}
\bigl[ \mathbh{1}_{(-\infty,y]}(x)- \mathbh{1} _{ (y, \infty)}(x)
\bigr]
\frac{\partial\tilde{g}}{\partial x} (x,y,\lambda) \, \mathrm{d} x
\\
&= \frac{1}{2} \Biggl( \int_{ -\infty}^{y}
e^{i \xi y} e^{-(y-x)\sqrt{\lambda}} \,\mathrm{d} x + \int_{y}^{\infty
}e^{i \xi y}
e^{-(x-y)\sqrt{\lambda}} \,\mathrm{d} x \Biggr)
\nonumber
\\
&= \frac{e^{i\xi y}}{2} \Biggl( \int_{0}^{\infty}e^{- i \xi x}
e^{-x \sqrt{\lambda}} \,\mathrm{d} x + \int_{0}^{\infty}e^{ i \xi x}
e^{-x \sqrt{\lambda}} \,\mathrm{d} x \Biggr)
\nonumber
\\
&= \frac{e^{i\xi y}}{2} \biggl( \frac{1}{i \xi+ \sqrt{\lambda}} +
\frac{1}{ - i \xi+ \sqrt{\lambda}} \biggr)
\nonumber
\\
&= \sqrt{\lambda}\frac{e^{i \xi y}}{\lambda+ \xi^{2}} = \sqrt
{\lambda}\hat{\tilde{g}} .
\nonumber
\end{align}
This implies, by inverting the Fourier transform, that
\begin{equation}
\label{eq:inv-fou-g} a(x,y) \frac{\partial\tilde{ g}}{\partial x}=
\sqrt{\lambda}\tilde{g}.
\end{equation}
We recall that
\begin{equation}
\int_{0}^{\infty} e^{-\lambda t} \mathscr{D}
^{\frac{1}{2}}_{t} g \, \mathrm{d} t = \sqrt{\lambda}\tilde{g},
\end{equation}
thus by inverting the Laplace transform in \eqref{eq:inv-fou-g} we obtain
\begin{equation}
\label{eq:g-eq} \mathscr{D} ^{\frac{1}{2}}_{t} g = a(x,y)
\frac{\partial g}{\partial x}
\end{equation}
and, by considering the same arguments (see \eqref{lapBYUSING}),
\begin{align*}
\mathscr{D} ^{\frac{1}{2}}_{t} g = -a(x,y) \frac{\partial g}{\partial y}.
\end{align*}
Returning to our initial problem, by using \eqref{eq:g-eq} and \eqref
{eq:temp-rl-def} we have that
\begin{align*}
\mathscr{D} _{t}^{\frac{1}{2}, \frac{\mu^{2} }{4}} u &= e^{- \frac{\mu
^{2} }{4} t } \mathscr{D}
^{\frac{1}{2}}_{t} \bigl( e^{ \frac{\mu^{2}t }{4}} u \bigr) -
\frac{\mu}{2} u
\\
& = e^{+\frac{\mu}{2} (y-x) \,- \,\frac{\mu^{2}}{4} t } \, \mathscr
{D} ^{\frac{1}{2}}_{t} g -
\frac{\mu}{2} u
\\
& = e^{- \frac{\mu^{2}}{4} t} \, e^{\frac{\mu}{2} (y-x) } \, \,
a(x,y) \frac{\partial g}{\partial x} -
\frac{\mu}{2} u
\\
&= a(x,y) \biggl( \frac{\partial u}{\partial x} + \frac{\mu}{2} u
\biggr) -
\frac{\mu}{2} u
\end{align*}
and
\begin{align*}
\mathscr{D} _{t}^{\frac{1}{2}, \frac{\mu^{2} }{4}} u &= e^{- \frac{\mu
^{2} }{4} t } \mathscr{D}
^{\frac{1}{2}}_{t} \bigl( e^{ \frac{\mu^{2}t }{4}} u \bigr) -
\frac{\mu}{2} u
\\
& = e^{+\frac{\mu}{2} (y-x) \,- \,\frac{\mu^{2}}{4} t } \, \mathscr
{D} ^{\frac{1}{2}}_{t} g -
\frac{\mu}{2} u
\\
& = -e^{- \frac{\mu^{2}}{4} t} \, e^{\frac{\mu}{2} (y-x) } \, \,
a(x,y) \frac{\partial g}{\partial y} -
\frac{\mu}{2} u
\\
&= a(x,y) \biggl( -\frac{\partial u}{\partial y} + \frac{\mu}{2} u
\biggr) -
\frac{\mu}{2} u.
\end{align*}
This completes the proof.
\end{proof}
The drifted Brownian motion has therefore a transition function satisfying
a time fractional equation where the fractional derivative is a
tempered Riemann--Liouville derivative with parameter $ \eta$ which is
related to the drift by the relationship $ \sqrt{\eta}= \frac{\mu}{2} $.
\section{Fractional equation governing the folded drifted Brownian motion}
We here consider the process
\begin{equation}
|B(t) + \mu t | + x = |B^{\mu}(t) | + x, \quad x>0.
\end{equation}
This process has distribution
\begin{align*}
P\bigl(| B(t) + \mu t| + x < y\bigr) ={}& P\bigl(x-y-\mu t < B(t) <
y-x-\mu t
\bigr)
\\
={}& \int_{x-y-\mu t}^{y-x-\mu t} \frac{e^{ - \frac{w^{2}}{4t }}}{\sqrt
{4 \pi t }} \,
\mathrm{d} w
\end{align*}
and therefore its transition function is
\begin{align}
\label{eq:v-tran-foo} P\bigl(|B ^{\mu}(t)| + x \in\mathrm{d} y\bigr) /
\mathrm{d} y &= \frac{
e^{ - \frac{(y-x-\mu t)^{2}}{4t} }
}{\sqrt{ 4 \pi t}} + \frac{
e^{ - \frac{(y-x+\mu t)^{2}}{4t} }
}{\sqrt{ 4 \pi t}}
\\
&= \frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt{4 \pi t}} e^{ - \mu^{2}
\frac{t}{4} } \bigl[ e^{ - \frac{\mu}{2} (y-x) } +
e^{ \frac{\mu}{2} (y-x) } \bigr]
\nonumber
\\
&= v(x,y,t)
\nonumber
\end{align}
for $ y>x $ and $t>0$.
We now prove the following theorem.
\begin{thm}
The law $ v $ of $ | B^{\mu}(t) | +x $ satisfies the fractional equation
\begin{equation}
\label{eq:frac-eq-v-thm} \mathscr{D} _{t} ^{ \frac{1}{2} , \eta} v = -
\frac{\partial v }{\partial y} + v \, \sqrt{\eta}\tanh\bigl(\sqrt
{\eta}( y-x)\bigr) -
\frac{\mu}{2} v, \quad y > x > 0,
\end{equation}
with initial and boundary conditions
\begin{align*}
v(x,y,0) ={}& \delta( y - x),
\\
v(x,x,t) ={}& \frac{e^{-\eta t}}{\sqrt{\pi t}}, \quad t>0,
\end{align*}
and
\begin{align*}
\eta= \frac{\mu^{2}}{4}.
\end{align*}
\end{thm}
\begin{proof}
From \eqref{eq:v-tran-foo} and \eqref{eq:temp-rl-def} we have that
\begin{align*}
\mathscr{D} _{t}^{\frac{1}{2} } \bigl( e^{ \frac{\mu^{2} t}{4} } v
\bigr) &= 2
\cosh\biggl( \frac{\mu}{2} (y-x) \biggr) \,\, \mathscr{D}
_{t}^{ \frac{1}{2} } \biggl( \frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt
{4 \pi t}} \biggr)
\end{align*}
Let $E_{\frac{1}{2}}$ be the Mittag-Leffler function of order $1/2$ and
$g$ be the function
\begin{align*}
g(x,y,t) = \frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt{4 \pi t}}, \quad y>x>0.
\end{align*}
Since
\begin{align*}
\int_{0}^{\infty}e^{-\lambda t} \int
_{x}^{\infty}e^{-\xi y} g(x,y,t) \, \mathrm{d} y\,
\mathrm{d} t ={}& e^{-\xi x} \int_{0}^{\infty}e^{-\lambda t}
E_{\frac{1}{2}}\bigl(- \xi t^{\frac{1}{2}}\bigr) \, \mathrm{d} t
\\
={}& e^{-\xi x} \frac{\lambda^{\frac{1}{2}-1}}{\xi+ \lambda^{\frac{1}{2}}}
\end{align*}
we obtain that
\begin{align*}
\mathscr{D}^{\frac{1}{2}}_{t} g(x,y,t) = - \frac{\partial g}{\partial
y} \quad
\text{\textrm{with boundary condition }} g(x,x, t) = \frac{1}{\sqrt
{4\pi t}}.
\end{align*}
Then, for $y>x$,
\begin{align*}
& \mathscr{D} _{t}^{\frac{1}{2} } \bigl( e^{ \frac{\mu^{2} t}{4} } v
\bigr)
\\
& = 2 \cosh\biggl( \frac{\mu}{2} (y-x) \biggr) \,\, \biggl( -
\frac{\partial}{\partial y} \biggl( \frac{e^{ - \frac{(y-x)^{2}}{4t}
}}{\sqrt{4 \pi t}} \biggr) \biggr)
\\
&= -\frac{\partial}{\partial y} \biggl( 2 \cosh\biggl( \frac{\mu}{2}
(y-x) \biggr)
\frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt{4 \pi t}} \biggr) + \frac{\mu
}{2} \cdot2 \sinh\biggl(
\frac{\mu}{2} (y-x) \biggr) \frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt
{4 \pi t}}
\\
&= -\frac{\partial}{\partial y} \bigl( e^{\frac{\mu^{2}}{4}t }
v(x,y,t) \bigr) +\mu\, \sinh
\biggl( \frac{\mu}{2} (y-x) \biggr) \frac{e^{ - \frac{(y-x)^{2}}{4t}
}}{\sqrt{4 \pi t}}
\end{align*}
with boundary condition
\begin{align*}
e^{\frac{\mu^{2}}{4}t} v(x,x,t)= \frac{1}{\sqrt{4\pi t}} + \frac
{1}{\sqrt{4\pi t}} .
\end{align*}
In view of \eqref{eq:temp-rl-def} we obtain that
\begin{align*}
\mathscr{D} ^{\frac{1}{2}, \eta}_{t} v + \frac{\mu}{2} v & =
e^{ - \frac{ \mu^{2}}{4}t } \mathscr{D} _{t}^{\frac{1}{2}} \bigl(
e^{ \frac{\mu^{2}}{4}t } v \bigr)
\\
&= -\frac{\partial v}{\partial y} - \mu e^{ - \frac{\mu^{2}}{4}t }
\sinh\biggl( \frac{\mu}{2}
(y-x) \biggr) \frac{e^{ - \frac{(y-x)^{2}}{4t} }}{\sqrt{4 \pi t}}
\\
&= -\frac{\partial v}{\partial y} - \frac{\mu}{2} \, \tanh\biggl(
\frac{\mu}{2}
(y-x) \biggr) v.
\qedhere
\end{align*}
\end{proof}
This result shows that the structure of the governing equation of the
process $ |B(t) + \mu t | + x $ is substantially different from that of
$ B(t) + \mu t + x $.
The difference between
\eqref{eq:frac-eq-u-thm} and \eqref{eq:frac-eq-v-thm}
consists in the non-constant coefficient $ \tanh\frac{\mu}{2} (y-x) $
which converges to one as the difference $ |y-x| $ tends to infinite.
Thus the two equations emerging in this analysis coincide for
$ |x-y| \to\infty$.
\end{document} |
\begin{document}
\title{Variational Inference in Nonconjugate Models}
\author{\name Chong Wang \email [email protected] \\
\addr Machine Learning Department\\
Carnegie Mellon University\\
Pittsburgh, PA, 15213, USA
\AND
\name David M. Blei \email [email protected] \\
\addr Department of Computer Science\\
Princeton University\\
Princeton, NJ, 08540, USA
}
\editor{}
\maketitle
\begin{abstract}
Mean-field variational methods are widely used for approximate
posterior inference in many probabilistic models. In a typical
application, mean-field methods approximately compute the posterior
with a coordinate-ascent optimization algorithm. When the model is
conditionally conjugate, the coordinate updates are easily derived
and in closed form. However, many models of interest---like the
correlated topic model and Bayesian logistic regression---are
nonconjuate. In these models, mean-field methods cannot be directly
applied and practitioners have had to develop variational algorithms
on a case-by-case basis. In this paper, we develop two generic
methods for nonconjugate models, Laplace variational inference and
delta method variational inference. Our methods have several
advantages: they allow for easily derived variational algorithms
with a wide class of nonconjugate models; they extend and unify some
of the existing algorithms that have been derived for specific
models; and they work well on real-world datasets. We studied our
methods on the correlated topic model, Bayesian logistic regression,
and hierarchical Bayesian logistic regression.
\end{abstract}
\begin{keywords}
Variational inference, Nonconjugate models, Laplace approximations,
The multivariate delta method
\end{keywords}
\section{Introduction}
Mean-field variational inference lets us efficiently approximate
posterior distributions in complex probabilistic
models~\citep{Jordan:1999,Wainwright:2008}. Applications of
variational inference are widespread. As examples, it has been
applied to Bayesian mixtures~\citep{Attias:2000,Adrian:2001},
factorial models~\citep{Ghahramani:1997}, and probabilistic topic
models~\citep{Blei:2003b}.
The basic idea behind mean-field inference is the following. First
define a family of distributions over the hidden variables where each
variable is assumed independent and governed by its own parameter.
Then fit those parameters so that the resulting distribution is close to
the conditional distribution of the hidden variables given the
observations. Closeness is measured with the Kullback-Leibler divergence.
Inference becomes optimization.
In many settings this approach can be used as a ``black box''
technique. In particular, this is possible when we can easily compute
the conditional distribution of each hidden variable given all of the
other variables, both hidden and observed. (This class contains the
models mentioned above.) For such models, which are called
\textit{conditionally conjugate} models, it is easy to derive a
coordinate ascent algorithm that optimizes the parameters of the
variational distribution~\citep{Beal:2003,Bishop:2006}. This is the
principle behind software tools like VIBES~\citep{Bishop:2003} and
Infer.NET~\citep{InferNET10}, which allow practitioners to define
models of their data and immediately approximate the corresponding
posterior with variational inference.
Many models of interest, however, do not enjoy the properties required
to take advantage of this easily derived algorithm. Such
\textit{nonconjugate} models\footnote{ \citet{Carlin:1991} coined the
term ``nonconjugate model'' to describe a model that does not enjoy
full conditional conjugacy.} include Bayesian logistic
regression~\citep{Jaakkola:1997a}, Bayesian generalized linear
models~\citep{Wells:2001}, discrete choice models~\citep{Braun:2010},
Bayesian item response models~\citep{Clinton:2004,Fox:2010}, and
nonconjugate topic models~\citep{Blei:2006d,Blei:2007}. Using
variational inference in these settings requires algorithms tailored
to the specific model at hand. Researchers have developed a variety
of strategies for a variety of models, including
approximations~\citep{Braun:2010,Ahmed:2007}, alternative
bounds~\citep{Jaakkola:1997a,Blei:2006d,Blei:2007,Khan:2010}, and
numerical quadrature~\citep{Honkela:2004}.
In this paper we develop two approaches to mean-field variational
inference for a large class of nonconjugate models. First we develop
\textit{Laplace variational inference}. This approach embeds Laplace
approximations---an approximation technique for continuous
distributions~\citep{Tierney:1989,MacKay:1992}---within a variational
optimization algorithm. We then develop \textit{delta method
variational inference}. This approach optimizes a Taylor
approximation of the variational objective. The details of the
algorithm depend on how the approximation is formed. Formed one way,
it gives an alternative interpretation of Laplace variational
inference. Formed another way, it is equivalent to using a
multivariate delta approximation~\citep{Bickel:2007} of the
variational objective.
Our methods are generic. Given a model, they can be derived nearly as
easily as traditional coordinate-ascent inference. Unlike traditional
inference, however, they place fewer conditions on the model,
conditions that are less restrictive than conditional conjugacy. Our
methods significantly expand the class of models for which mean-field
variational inference can be easily applied.
We studied our algorithms with three nonconjugate models: Bayesian
logistic regression~\citep{Jaakkola:1997a}, hierarchical logistic
regression~\citep{Gelman:2007}, and the correlated topic
model~\citep{Blei:2007}. We found that our methods give better results
than those obtained through special-purpose techniques. Further, we
found that Laplace variational inference usually outperforms delta
method variational inference, both in terms of computation time and
the fidelity of the approximate posterior.
\paragraph{Related work.} We have described the various approaches
that researchers have developed for specific models. There have been
other efforts to examine generic variational inference in nonconjugate
models. \cite{Paisley:2012} proposed a variational inference approach
using stochastic search for nonconjugate models, approximating the
intractable integrals with Monte Carlo methods. \cite{Gershman:2012a}
proposed a nonparametric variational inference algorithm, which can be
applied to nonconjugate models. \citet{Knowles:2011} presented a
message passing algorithm for nonconjugate models, which has been
implemented in Infer.NET~\citep{InferNET10}; their technique applies
to a subset of models described in this paper.\footnote{It may be
generalizable to the full set. However, one must determine how to
compute the required expectations.}
Laplace approximations have been used in approximate inference in more
complex models, though not in the context of mean-field variational
inference. \cite{Smola:2003} used them to approximate the
difficult-to-compute moments in expectation
propagation~\citep{Minka:2001}. \cite{Rue:2009} used them for
inference in latent Gaussian models. Here we want to use them for
variational inference, in a method that can be applied to a wider
range of nonconjugate models.
Finally, we note that the delta method was first used in variational
inference by~\cite{Braun:2010} in the context of the discrete choice
model. Our method generalizes their approach.
\paragraph{Organization of this paper.} In
\mysec{nonconjugate-models} we review mean-field variational inference
and define the class of nonconjugate models to which our algorithms
apply. In \mysec{inference}, we derive Laplace and delta-method
variational inference and present our full algorithm for nonconjugate
inference. In \mysec{example}, we show how to use our generic method
on several example models and in \mysec{experiments} we study its
performance on these models. In \mysec{discussion}, we summarize and
discuss this work.
\section{Variational Inference and a Class of Nonconjugate Models}
\label{sec:nonconjugate-models}
We consider a generic model with observations $x$ and hidden variables
$\theta$ and $z$,
\begin{align}\label{eq:joint}
p(\theta, z, x) = p(x|z) p(z|\theta) p(\theta).
\end{align}
The distinction between the two hidden variables will be made clear
below.
The inference problem is to compute the posterior, the conditional
distribution of $\theta$ and $z$ given $x$,
\begin{equation*}
p(\theta, z|x) = \frac{p(\theta, z, x)}{\int p(\theta, z,
x) dz d\theta}.
\end{equation*}
This is intractable for many models because the denominator is
difficult to compute; we must approximate the distribution. In
variational inference, we approximate the posterior by positing a
simple family of distributions over the latent variables $q(\theta,
z)$ and then finding the member of that family which minimizes the
Kullback-Leibler (KL) divergence to the true
posterior~\citep{Jordan:1999,Wainwright:2008}.\footnote{In this paper,
we focus on mean-field variational inference where we minimize the
KL divergence to the posterior. We note that there are other kinds
of variational inference, with more structured variational
distributions or with alternative objective
functions~\citep{Wainwright:2008,Barber:2012}. In this paper, we use
``variational inference'' to indicate mean-field variational
inference that minimizes the KL divergence.}
In this section we review variational inference and discuss mean-field
variational inference for the class of conditionally conjugate models.
We then define a wider class of nonconjugate models for which
mean-field variational inference is not as easily applied. In the
next section, we derive algorithms for performing mean-field
variational inference in this larger class of models.
\subsection{Mean-field Variational Inference}
Mean-field variational inference is simplest and most widely used variational
inference method. In mean-field variational inference we posit a fully
factorized variational family,
\begin{align}
\label{eq:mean-field-family}
q(\theta, z) = q(\theta) q(z).
\end{align}
In this family of distributions the variables are independent and each
is governed by its own distribution. This family usually does not
contain the posterior, where $\theta$ and $z$ are dependent. However,
it is very flexible---it can capture any set of marginals of the
hidden variables.
Under the standard variational theory, minimizing the KL divergence
between $q(\theta, z)$ and the posterior $p(\theta,z|x)$ is equivalent
to maximizing a lower bound of the log marginal likelihood of the
observed data $x$. We obtain this bound with Jensen's inequality,
\begin{align}
\log p(x) &= \textstyle \log \int p(\theta, z, x) {\rm d} z {\rm d} \theta
\nonumber \\
&\,\vert\,eq \expect{q}{\log p(\theta, z, x)} - \expect{q}{\log q(\theta,
z)}
\nonumber \\
&\triangleq \mathcal{L}(q), \label{eq:lower-bound}
\end{align}
where $\expect{q} {\cdot}$ is the expectation taken with respect to
$q$ and note the second term is the entropy of $q$. We call ${\cal
L}(q)$ the variational objective.
Setting ${\partial \mathcal{L}(q)}/ {\partial q} = 0$ shows that the optimal
solution satisfies the following,
\begin{align}
q^*(\theta) &\propto \textstyle \exp\left\{\expect{q(z)}{\log
p(z|\theta)p(\theta)} \right\} \label{eq:q-theta} \\
q^*(z) & \propto \textstyle \exp\left\{\expect{q(\theta)} {\log
p(x|z)p(z|\theta)} \right\} \label{eq:q-z}.
\end{align}
Here we have combined the optimal conditions from~\cite{Bishop:2006}
with the particular factorization of \myeq{joint}. Note that the
variational objective usually contains many local optima.
These conditions lead to the traditional coordinate ascent algorithm
for variational inference. It iterates between holding $q(z)$ fixed to
update $q(\theta)$ from \myeq{q-theta} and holding $q(\theta)$ fixed
to update $q(z)$ from \myeq{q-z}. This converges to a local optimum
of the variational objective~\citep{Bishop:2006}.
When all the nodes in a model are \textit{conditionally conjugate},
the coordinate updates of \myeq{q-theta} and \myeq{q-z} are available
in closed form. A node is conditionally conjugate when its
conditional distribution given its Markov blanket (i.e., the set of
random variables that it is dependent on in the posterior) is in the
same family as its conditional distribution given its parents (i.e.,
its factor in the joint distribution). For example, in \myeq{joint}
suppose the factor $p(\theta)$ is a Dirichlet and both factors
$p(z\,\vert\,\theta)$ and $p(x\,\vert\, z)$ are multinomials. This means that the
conditional $p(\theta|z)$ is also a Dirichlet and the conditional $p(z
\,\vert\, x, \theta)$ is also a multinomial. This model, which is latent
Dirichlet allocation~\citep{Blei:2003b}, is conditionally
conjugate. Many applications of variational inference have been
developed for this type of model~\citep{Bishop:1999, Attias:2000,
Beal:2003}.
However, if there exists any node in the model that is not
conditionally conjugate then this coordinate ascent algorithm is
not available. That setting arises in many practical models and
does not permit closed-form updates or easy calculation of the
variational objective. We will develop generic variational
inference algorithms for a wide class of nonconjugate models.
First, we define that class.
\subsection{A Class of Nonconjugate Models}
\label{sec:condition}
We present a wide class of nonconjugate models, still assuming the
factorization of~\myeq{joint}.
\begin{enumerate}
\item We assume that $\theta$ is real-valued and the distribution
$p(\theta)$ is twice differentiable with respect to $\theta$. If we
require $\theta>\theta_0$ ($\theta_0$ is a constant), we may define
a distribution over $\log(\theta-\theta_0)$. These assumptions
cover exponential families, such as the Gaussian, Poisson and
gamma, as well as more complex distributions, such as a
student-t.
\item We assume the distribution $p(z\,\vert\,\theta)$ is in the exponential
family~\citep{Brown:1986},
\begin{align}\label{eq:z-dist}
\textstyle p(z\,\vert\,\theta) = h(z) \exp\left\{\eta(\theta)^\top t(z) -
a(\eta(\theta))\right\},
\end{align}
where $h(z)$ is a function of $z$; $t(z)$ is the sufficient
statistic; $\eta(\theta)$ is the natural parameter, which is a
function of the conditioning variables; and $a(\eta(\theta))$ is the
log partition function. We also assume that $\eta(\theta)$ is twice
differentiable; since $\theta$ is real-valued, this is satisfied in
most statistical models. Unlike in conjugate models, these
assumptions do not restrict $p(\theta)$ and $p(z \,\vert\, \theta)$ to be a
conjugate pair; the conditional distribution $p(\theta \,\vert\, z)$ is not
necessarily in the same family as the prior $p(\theta)$.
\item The distribution $p(x\,\vert\, z)$ is in the exponential family,
\begin{align}\label{eq:x-dist}
\textstyle p(x\,\vert\, z) = h(x) \exp\left\{t(z)^\top \langle t(x), 1
\rangle\right\}.
\end{align}
We set up this exponential family so that the natural parameter for
$x$ is all but the last component of $t(z)$ and the last component
is the negative log normalizer $-a(\cdot)$. Thus, the distribution
of $z$ is conjugate to the conditional distribution of $x$; the
conditional $p(z \,\vert\, \theta, x)$ is in the same family as $p(z \,\vert\,
\theta)$~\citep{Bernardo:1994}.
\end{enumerate}
Our terminology follows these assumptions: $\theta$ is the
\textit{nonconjugate variable}, $z$ is the \textit{conjugate
variable}, and $x$ is the \textit{observation}.
This class of models is larger than the class of conditionally
conjugate models. Our expanded class also includes nonconjugate
models like the correlated topic model~\citep{Blei:2007}, dynamic
topic model~\citep{Blei:2006d}, Bayesian logistic
regression~\citep{Jaakkola:1997a,Gelman:2007}, discrete choice
models~\citep{Braun:2010}, Bayesian ideal point
models~\citep{Clinton:2004}, and many others. Further, the methods we
develop below are easily adapted more complicated graphical models,
those that contain conjugate and nonconjugate variables whose
dependencies are encoded in a directed acyclic graph. Appendix A
outlines how to adapt our algorithms to this more general case.
\paragraph{Example: Hierarchical language modeling.} We introduce the
hierarchical language model, a simple example of a nonconjugate model
to help ground our derivation of the general algorithms. Consider the
problem of unigram language modeling. We are given a collection of
documents ${\cal D} = x_{1:D}$ where each document $x_d$ is a vector
of word counts, observations from a discrete vocabulary of length
$V$. We model each document with its own distribution over words and
place a Dirichlet prior on that distribution. This model is used, for
example, in the language modeling approach to information
retrieval~\citep{Croft:2003}.
We want to place a prior on the Dirichlet parameters, a positive
$V$-vector, that govern each document's distribution over terms.
In theory, every exponential family distribution has a conjugate
prior~\citep{Bernardo:1994} and the prior to the Dirichlet is the
multi-gamma distribution~\citep{Kotz:2000}. However, the
multi-gamma is difficult to work with because its log normalizer
is not easy to compute. As an alternative, we place a log normal
distribution on the Dirichlet parameters. This is not the
conjugate prior.
The full generative process is as follows:
\begin{enumerate*}
\item Draw log Dirichlet parameters $\theta \sim \mathcal{N}(0, I)$.
\item For each document $d$, $1\leq d \leq D$:
\begin{enumerate*}
\item Draw multinomial parameter $z_d \,\vert\, \theta \sim {\rm
Dirichlet}(\exp\{\theta\})$.
\item Draw word counts $x_{d} \sim {\rm Multinomial}(N, z_d)$.
\end{enumerate*}
\end{enumerate*}
Given a collection of documents, our goal is to compute the
posterior distribution $p(\theta, z_{1:D} \,\vert\, x_{1:D})$.
Traditional variational or Gibbs sampling methods cannot be
easily used because the normal prior on the parameters $\theta$
is not conjugate to the $\textrm{Dirichlet}(\exp\{\theta\})$
likelihood.
This language model fits into our model class. In the notation of the
joint distribution of \myeq{joint}, $\theta = \theta$, $z = z_{1:D}$,
and $x = x_{1:D}$. The per-document multinomial parameters $z$ and
word counts $x$ are conditionally independent given the Dirichlet
parameters $\theta$,
\begin{align*}
p(z \,\vert\, \theta) &= \textstyle \prod_d p(z_d \,\vert\, \theta) \\
p(x \,\vert\, z) &= \textstyle \prod_d \prod_n p(x_{dn} \,\vert\, z_d).
\end{align*}
In this case, the natural parameter $\eta(\theta) =
\exp\{\theta\}$. This model satisfies the assumptions: the log normal
$p(\exp\{\theta\})$ is not conjugate to the Dirichlet $p(z_d \,\vert\,
\exp\{\theta\})$ but is twice differentiable; the Dirichlet is
conjugate to the multinomial $p(x_d \,\vert\, z_d)$ and the multinomial is in
the exponential family.
Below we will use various components of the exponential family form of
the Dirichlet:
\begin{align}
h(z_d) =\textstyle \prod_i z_{di}^{-1}; \ \ t(z_d) = \log z_d; \ \
a_d(\eta(\theta)) = \sum_i \log\Gamma (\exp\{\theta_i\} -
\log\Gamma \left(\sum_i\exp\{\theta_i\}\right).\label{eq:example-z}
\end{align}
We will return to this model as a simple running example.
\section{Laplace and Delta Method Variational Inference}
\label{sec:inference}
We have defined a class of nonconjugate models. Variational inference
is difficult to derive for these models because $p(\theta)$ is not
conjugate to $p(z\,\vert\,\theta)$. Specifically, the update in
\myeq{q-theta} does not necessarily have the form of an exponential
family we can work with and it is difficult to use
$\expect{q(\theta)}{\log p(z\,\vert\, \theta)}$ in the update of \myeq{q-z}.
We will develop two variational inference algorithms for this class:
\textit{Laplace variational inference} and \textit{delta method
variational inference}. Both use coordinate ascent to optimize the
variational parameters, iterating between updating $q(\theta)$ and
$q(z)$. They differ in how they update the variational distribution
of the nonconjugate variable $q(\theta)$. In Laplace variational
inference, we use Laplace
approximations~\citep{MacKay:1992,Tierney:1989} within the coordinate
ascent updates of \myeq{q-theta} and \myeq{q-z}. In delta method
variational inference, we apply Taylor approximations to approximate
the variational objective in \myeq{lower-bound} and then derive the
corresponding updates. Different ways of taking the Taylor
approximation lead to different algorithms. Formed one way, this
recovers the Laplace approximation. Formed another way, it is
equivalent to using a multivariate delta
approximation~\citep{Bickel:2007} of the variational objective
function.
In both variants, the variational distribution is the mean-field
family in \myeq{mean-field-family}. The variational distribution of
the nonconjugate variable $q(\theta)$ is a Gaussian; the variational
distribution of the conjugate variable $q(z)$ is in the same family as
$p(z\,\vert\,\eta(\theta))$. In Laplace inference, these forms emerge from
the derivation. In delta method inference, they are assumed. The
complete variational family is,
\begin{align*}
q(\theta, z) = q(\theta \,\vert\, \mu, \Sigma) q(z \,\vert\, \phi).
\end{align*}
where $(\mu, \Sigma)$ are the parameters for a Gaussian distribution
and $\phi$ is a natural parameter for $z$. For example, in the
hierarchical language model of \mysec{condition}, $\phi$ is a
collection of $D$ Dirichlet parameters. We will sometimes supress the
parameters, writing $q(\theta)$ for $q(\theta \,\vert\, \mu, \Sigma)$.
Our algorithms are coordinate ascent algorithms, where we iterate
between updating the nonconjugate variational distribution $q(\theta)$
and updating the conjugate variational distribution $q(z)$. In the
subsections below, we derive the update for $q(\theta)$ in each
algorithm. Then, for both algorithms, we derive the update for
$q(z)$. The full procedure is described in \mysec{alg} and
\myfig{lp-algorithm}.
\subsection{Laplace Variational Inference}\label{sec:embed-laplace}
We first review the Laplace approximation. Then we show how to use it
in variational inference.
\paragraph{The Laplace Approximation.} Laplace approximations use a
Gaussian to approximate an intractable density. Consider
approximating an intractable posterior $p(\theta\,\vert\, x)$. (There is no
hidden variable $z$ in this set up.) Assume the joint distribution
$p(x,\theta)=p(x\,\vert\,\theta)p(\theta)$ is easy to compute. Laplace
approximations use a Taylor approximation around the maximum a
posterior (MAP) point to construct a Gaussian proxy for the posterior.
They are used for continuous distributions.
First, notice the posterior is proportional to the exponentiated log
joint
\[
p(\theta\,\vert\, x) = \exp \{ \log p(\theta\,\vert\, x) \} \propto \exp \{\log p(\theta,x) \}.
\]
Let $\hat{\theta}$ be the MAP of $p(\theta\,\vert\, x)$, found by maximizing $\log
p(\theta,x)$. A Taylor expansion around $\hat{\theta}$ gives
\begin{align}
\log p(\theta\,\vert\, x) \approx \log p(\hat{\theta}\,\vert\, x) +\textstyle \frac{1}{2}
(\theta-\hat{\theta})^\top
H(\hat{\theta}) (\theta-\hat{\theta}). \label{eq:laplace}
\end{align}
The term $H(\hat{\theta})$ is the Hessian of $\log p(\theta\,\vert\, x)$ evaluated at
$\hat{\theta}$, $H(\hat{\theta}) \triangleq \triangledown^2 \log
p(\theta\,\vert\, x)
|_{\theta=\hat{\theta}}$.
In the Taylor expansion of \myeq{laplace}, the first-order term
$(\theta-\hat{\theta})^\top \triangledown \log
p(\theta\,\vert\, x)|_{\theta=\hat{\theta}} $ equals zero. The reason is that
$\hat{\theta}$ is the maximum of $\log p(\theta\,\vert\, x)$ and so its
gradient $\triangledown \log p(\theta\,\vert\, x)|_{\theta=\hat{\theta}}$ is
zero. Exponentiating \myeq{laplace} gives the approximate Gaussian posterior
\begin{align*}
\textstyle p(\theta\,\vert\, x) \approx \frac{1}{C}\exp\left \{-\frac{1}{2}
(\theta-\hat{\theta})^\top \left(-H(\hat{\theta})\right)
(\theta-\hat{\theta})\right\},
\end{align*}
where $C$ is a normalizing constant. In other words, $p(\theta\,\vert\, x)$ can be
approximated by
\begin{align} \label{eq:laplace-approximation}
p(\theta\,\vert\, x) \approx \mathcal{N} (\hat{\theta}, -H(\hat{\theta})^{-1}).
\end{align}
This is the Laplace approximation. While powerful, it is difficult to
use in multivariate settings, for example, when there are discrete
hidden variables. Now we describe how we use Laplace approximations as
part of a variational inference algorithm for more complex models.
\paragraph{Laplace updates in variational inference.} We adapt the
idea behind Laplace approximations to update the variational
distribution $q(\theta)$. First, we combine the coordinate update
in~\myeq{q-theta} with the exponential family assumption
in~\myeq{z-dist},
\begin{align}\label{eq:q-theta-2}
\textstyle q(\theta) \propto \exp\left\{ \eta(\theta)^\top
\expect{q(z)}{t(z)} - a(\eta(\theta)) + \log p(\theta) \right \}.
\end{align}
Define the function $f(\theta)$ to contain the terms inside the
exponent of the update,
\begin{align}
f(\theta) \triangleq \eta(\theta)^\top \expect{q(z)}{t(z)} -
a(\eta(\theta)) + \log p(\theta).\label{eq:f-laplace}
\end{align}
The terms of $f(\theta)$ come from the model and involve $q(z)$ or
$\theta$. Recall that $q(z)$ is in the same exponential family as
$p(z \,\vert\, \theta)$, $t(z)$ are its sufficient statistics, and $\phi$ is
the variational parameter. We can compute $\expect{q(z)}{t(z)}$ from
a basic property of the exponential family~\citep{Brown:1986},
\begin{equation*}
\expect{q(z)}{t(z)} = \nabla a(\phi).
\end{equation*}
Seen another way, $f(\theta) = \expect{q(z)}{\log p(\theta, z)}$.
This function will be important in both Laplace and delta method
inference.
The problem with nonconjugate models is that we cannot update
$q(\theta)$ exactly using \myeq{q-theta-2} because $q(\theta) \propto
\exp\{f(\theta)\}$ cannot be normalized in closed form. We
approximate the update by taking a second-order Taylor approximation
of $f(\theta)$ around its maximum, following the same logic as from
the original Laplace approximation in \myeq{laplace}. The Taylor
approximation for $f(\theta)$ around $\hat{\theta}$ is
\begin{align}\label{eq:taylor-f-theta}
f(\theta) \approx \textstyle f(\hat{\theta}) + \triangledown
f(\hat{\theta})(\theta-\hat{\theta}) + \frac{1}{2}(\theta-\hat{\theta})^\top
\triangledown^2 f(\hat{\theta})(\theta-\hat{\theta}),
\end{align}
where $\triangledown^2 f(\hat{\theta})$ is the Hessian matrix
evaluated at $\hat{\theta}$. Now let $\hat{\theta}$ be the value that
maximizes $f(\theta)$. This implies that $\triangledown
f(\hat{\theta}) =0$ and \myeq{taylor-f-theta} simplifies to
\begin{equation*}
\textstyle q(\theta) \propto \exp\{f(\theta)\}
\approx \exp\left\{ f(\hat{\theta}) + \frac{1}{2}(\theta -
\hat{\theta})^\top \triangledown^2 f(\hat{\theta})
(\theta-\hat{\theta}) \right\}.
\end{equation*}
Thus the approximate update for $q(\theta)$ is to set it to
\begin{align}\label{eq:lap-var}
q(\theta) \approx \mathcal{N}\left(\hat{\theta}, -\triangledown^2
f(\hat{\theta})^{-1} \right).
\end{align}
Note we did not assume $q(\theta)$ is Gaussian. Its Gaussian form
stems from the Taylor approximation.
The update in \myeq{lap-var} can be used in a coordinate ascent
algorithm for a nonconjugate model. We iterate between holding $q(z)$
fixed while updating $q(\theta)$ from \myeq{lap-var}, and holding
$q(\theta)$ fixed while updating $q(z)$. (We derive the second update
in \mysec{q-z}.) Each time we update $q(\theta)$ we must use numerical
optimization to obtain $\hat{\theta}$, the optimal value of
$f(\theta)$.
We return to the hierarchical language model of \mysec{condition},
where $\theta$ are the log of the parameters to the Dirichlet
distribution. Implementing the algorithm to update $q(\theta)$
involves forming $f(\theta)$ for the model at hand and deriving an
algorithm to optimize it.
With the model equations in~\myeq{example-z}, we have
\begin{align}
f(\theta) = \textstyle \exp(\theta)^\top \expect{q(z)}{t(z)} -
D\left(\sum_i \log\Gamma (\exp(\theta_i) - \log\Gamma
\left(\sum_i\exp(\theta_i)\right) \right) -(1/2)\theta^\top \theta.
\label{eq:unigram-f}
\end{align}
The expected sufficient statistics of the conjugate variable are
\begin{eqnarray*}
\expect{q(z)}{t(z)} &=& \textstyle \sum_d \expect{q(z_d)}{t(z_d)} \\
&=& \sum_d\Psi\!\left(\phi_d\right) - \Psi\!\left(\textstyle
\sum_i \phi_{di}\right),
\end{eqnarray*}
where $\Psi(\cdot)$ is the digamma function, the first derivative of
$\log \Gamma(\cdot)$. (This function will also arise in the
gradient.) It is straightforward to use numerical methods, such as
conjugate gradient~\citep{Bertsekas:1999}, to optimize
\myeq{unigram-f}. We can then use \myeq{lap-var} to update the
nonconjugate variable.
\subsection{Delta Method Variational Inference}
\label{sec:lower-bound}
In Laplace variational inference, the variational distribution
$q(\theta)$ \myeq{lap-var} is solely a function of $\hat{\theta}$, the
maximum of $f(\theta)$ in \myeq{f-laplace}. A natural question is,
would other values of $\theta$ be suitable as well? To consider such
alternatives, we describe a different technique for variational
inference. We approximate the variational objective $\mathcal{L}$ in
\myeq{lower-bound} and then optimize that approximation.
Again we focus on updating $q(\theta)$ in a coordinate ascent
algorithm and postpone the discussion of updating $q(z)$. We set the
variational distribution $q(\theta)$ to be a Gaussian
$\mathcal{N}(\mu, \Sigma)$, where the parameters are free
variational parameters fit to optimize the variational objective.
(Note that in Laplace inference, this Gaussian family came out of the
derivation.) We isolate the terms of the objective in
\myeq{lower-bound} related to $q(\theta)$, and we substitute the
exponential family form of $p(z\,\vert\, \theta)$ in \myeq{z-dist},
\begin{align*}
\textstyle \mathcal{L}(q(\theta)) =
\expect{q(\theta)}{\eta(\theta)^\top \expect{q(z)}{t(z)} -
a(\eta(\theta)) + \log p(\theta)} + \frac{1}{2} \log |\Sigma|.
\end{align*}
The second term comes from the entropy of the Gaussian,
\begin{equation*}
- \expect{q(\theta)}{\log q(\theta)} = \frac{1}{2} \log |\Sigma|+C,
\end{equation*}
where $C$ is a constant and is excluded from the objective. The first
term is $\expect{q(\theta)}{f(\theta)}$, where $f(\cdot)$ is the same
as defined for Laplace inference in \myeq{f-laplace}. Thus,
\begin{align*}
\mathcal{L}(q(\theta)) = \textstyle \expect{q(\theta)}{f(\theta)} +
\frac{1}{2} \log |\Sigma|.
\end{align*}
We cannot easily compute the expectation in the first term. So we use
a Taylor approximation of $f(\theta)$ around a chosen value
$\hat{\theta}$ (\myeq{taylor-f-theta}) and then take the expectation,
\begin{align}
\textstyle \mathcal{L}(q(\theta)) \approx f(\hat{\theta}) + \triangledown
f(\hat{\theta})^\top(\mu-\hat{\theta}) + & \textstyle
\frac{1}{2}(\mu-\hat{\theta})^\top
\triangledown^2 f(\hat{\theta})(\mu-\hat{\theta}) ]] \nonumber \\
& + \textstyle \frac{1}{2}\left( {\rm Tr}\left\{\triangledown^2 f(\hat{\theta})
\Sigma\right\} + \log |\Sigma|\right), \label{eq:l-approx}
\end{align}
where ${\rm Tr}(\cdot)$ is the Trace operator. In the coordinate
update of $q(\theta)$, this is the function we optimize with respect
to its variational parameters $\{\mu, \Sigma\}$.
To fully specify the algorithm we must choose $\hat{\theta}$, the
point around which to approximate $f(\theta)$. We will discuss three
choices. The first is to set $\hat{\theta}$ to be the maximum of
$f(\theta)$. With this choice, maximizing the approximation in
\myeq{l-approx} gives $\mu=\hat{\theta}$ and $\Sigma = -
\triangledown^2 f(\hat{\theta})^{-1}$. Notice this is the update
derived in \mysec{embed-laplace}. We have given a different
derivation of Laplace variational inference.
The second choice is to set $\hat{\theta}$ as the mean of the
variational distribution from the previous iteration of coordinate
ascent. If the prior $p(\theta)$ is Gaussian, this recovers the
updates derived in~\cite{Ahmed:2007} for the correlated topic
model.\footnote{This is an alternative derivation of their algorithm.
They derived these updates from the perspective of generalized
mean-field theory~\citep{Xing:2003}.} In our study, we found this
algorithm did not work well. It did not always converge, possibly due
to the difficulty of choosing an appropriate initial $\hat{\theta}$.
The third choice is to set $\hat{\theta} = \mu$, i.e., the mean of the
variational distribution $q(\theta)$. With this choice, the variable
around which we center the Taylor approximation becomes part of the
optimization problem. The objective is
\begin{align}\label{eq:lb-II}
\mathcal{L}(q(\theta)) &\textstyle \approx f(\mu) + \frac{1}{2} {\rm
Tr}\left\{\triangledown^2 f(\mu) \Sigma\right\} + \frac{1}{2} \log
|\Sigma|.
\end{align}
This is the multivariate delta method for evaluating
$\expect{q(\theta)}{f(\theta)}$~\citep{Bickel:2007}.
\textit{Delta method variational inference} optimizes this
objective in the coordinate update of $q(\theta)$ .
In more detail, we first optimize $\mu$ with gradient methods and then
optimize $\Sigma$ in closed form $ \Sigma = - \triangledown^2
f(\mu)^{-1}.$ Note this is more expensive than Laplace variational
inference because optimizing \myeq{lb-II} requires the third
derivative $\triangledown^3 f(\theta)$. \citet{Braun:2010} were the
first to use the delta method in a variational inference algorithm,
developing this technique for the discrete choice model. If we assume
the prior $p(\theta)$ is Gaussian then we recover their algorithm.
With the ideas presented here, we can now use this strategy in many
models.
We return briefly to the unigram language model. The delta method
update for $q(\theta)$ optimizes \myeq{lb-II}, using the specific
$f(\cdot)$ found in \myeq{unigram-f}. While Laplace inference
required the digamma function and $\log \Gamma$ function, delta
method inference will further require the trigamma function.
\subsection{Updating the Conjugate Variable}
\label{sec:q-z}
We derived variational updates for $q(\theta)$ using two methods. We
now turn to the update for the variational distribution of the
conjugate variable $q(z)$. We show that both Laplace inference
(\mysec{embed-laplace}) and delta method inference
(\mysec{lower-bound}) lead to the same update. Further, we have
implicitly assumed that $\expect{q(z)}{t(z)}$ in \myeq{f-laplace} is
easy to compute. We will confirm this as well.
We first derive the update for $q(z)$ when using Laplace inference.
We apply the exponential family form in \myeq{z-dist} to the exact
update of~\myeq{q-z},
\begin{align*}
\log q(z) = \log p(x\,\vert\, z) + \log h(z) +
\expect{q(\theta)}{\eta(\theta)}^\top t(z) + C,
\end{align*}
where $C$ is a constant not depending on $z$. Now we use $p(x\,\vert\, z)$ from
\myeq{x-dist} to obtain
\begin{align}\label{eq:q-z-exact2}
\textstyle q(z) \propto h(z) \exp\left
\{\left(\expect{q(\theta)}{\eta(\theta)} + t(x)\right)^\top
t(z) \right \},
\end{align}
which is in the same family as $p(z\,\vert\,\theta)$ in \myeq{z-dist}. This
is the update for $q(z)$.
Recall that $\eta(\theta)$ maps the nonconjugate variable $\theta$ to
the natural parameter of the conjugate variable $z$. The update for
$q(z)$ requires computing $\expect{q(\theta)}{\eta(\theta)}$. For
some models, this expectation is computable. If not, we can take a
Taylor approximation of $\eta(\theta)$ around the variational
parameter $\mu$,
\begin{align*}
\eta_i(\theta) \approx \textstyle \eta_i(\mu) + \triangledown
\eta(\mu)_i^\top(\theta-\mu) + \frac{1}{2}(\theta-\mu)^\top
\triangledown^2
\eta_i(\mu)(\theta-\mu),
\end{align*}
where $\eta(\theta)$ is a vector and $i$ indexes the $i$th component.
This requires $\eta(\theta)$ is twice differentiable, which is
satisfied in most models. Since $q(\theta) = \mathcal{N}\left(\mu,
\Sigma \right)$, this means that
\begin{align}
\label{eq:eta-theta-approx}
\expect{q(\theta)}{\eta_i(\theta)} \approx \textstyle \eta_i(\mu) +
\frac{1}{2}
{\rm Tr}\left\{\triangledown^2 \eta_i(\mu)\Sigma\right\}.
\end{align}
(Note that the linear term $\expect{q(\theta)}{\triangledown
\eta_i(\mu)^T(\theta - \mu)} = 0$.)
Using delta method variational inference to update $q(\theta)$, the
update for $q(z)$ is identical to that in Laplace variational
inference. We isolate the relevant terms in \myeq{lower-bound},
\begin{align}
\textstyle \mathcal{L}(q(z)) = &\textstyle \expect{q(z)}{ \log
p(x\,\vert\, z) + \log h(z) + \expect{q(\theta)}{\eta(\theta)}^\top t(z)}
- \expect{q(z)}{\log q(z)}.
\label{eq:delta-q-z}
\end{align}
Setting the partial gradient $\partial\mathcal{L}(q(z))/\partial q(z)
= 0$ gives the same optimal $q(z)$ of \myeq{q-z}. Computing this
update reduces to the approach for Laplace variational inference in
\myeq{q-z-exact2}.
We return again to the unigram language model with log normal priors
on the Dirichlet parameters. In this model, we can compute
$\expect{q(\theta)}{\eta(\theta)}$ exactly by using properties of the
log normal,
\begin{align*}
\expect{q(\theta)}{\eta(\theta)} = \expect{q(\theta)}{\exp\{\theta\}} =
\exp\{\mu + {\rm diag}(\Sigma)/2\}.
\end{align*}
Recall that $x_d$ are the word counts for document $d$ and note that
it is its own sufficient statistic in a multinomial count model.
Given the calculation of $\expect{q(\theta)}{\eta(\theta)}$ and the
model-specific calculations in ~\myeq{example-z}, the update for
$q(z_d)$ is
\begin{align*}
q(z_d) = \textstyle {\rm Dirichlet}\left(\exp(\mu + {\rm diag}(\Sigma)/2) +
x_{d}\right).
\end{align*}
This completes our derivation in the example model. To implement
nonconjugate inference we need this update for $q(z)$ and the
definition of $f(\cdot)$ in \myeq{unigram-f}.
\subsection{Nonconjugate Variational Inference}
\label{sec:alg}
\begin{figure}
\caption{Nonconjugate variational inference \label{fig:lp-algorithm}
\label{fig:lp-algorithm}
\end{figure}
We now present the full algorithm for nonconjugate variational
inference. In this section, we will be explicit about the
variational parameters. Recall that the variational distribution of
the nonconjugate variable is a Gaussian $q(\theta \,\vert\, \mu, \Sigma)$;
the variational distribution of the conjugate variable is $q(z \,\vert\,
\phi)$, where $\phi$ is a natural parameter in the same family as $p(z
\,\vert\, \eta(\theta))$.
The algorithm is as follows. Begin by initializing the variational
parameters. Iterate between updating $q(\theta)$ and updating $q(z)$
until convergence. Update $q(\theta)$ by either \myeq{lap-var}
(Laplace inference) or optimizing \myeq{lb-II} (Delta method
inference). Update $q(z)$ from \myeq{q-z-exact2}. Assess convergence
by measuring the $L_2$ norm of the mean of the nonconjugate variable,
$\expect{q}{\theta}$.
This algorithm is summarized in \myfig{lp-algorithm}. In either
Laplace or delta method inference, we have reduced deriving
variational updates for complicated nonconjugate models to mechanical
work---calculating derivatives and calling a numerical optimization
library. We note that Laplace inference is simpler to derive because
it only requires second derivatives of the function
in~\myeq{f-laplace}; delta method inference requires third
derivatives. We study the empirical difference between these methods
in \mysec{experiments}.
Our algorithm (in either setting) is based on approximately optimal
coordinate updates for the variational objective, but we cannot
compute that objective. However, we can compute an approximate
objective at each iteration with the same Taylor approximation used in
the coordinate steps, and this can be monitored as a proxy. The
approximate objective is
\begin{align}
\mathcal{L} \approx f(\hat{\theta}) + \triangledown
f(\hat{\theta})^\top(\mu-\hat{\theta}) +\textstyle & \textstyle
\frac{1}{2}(\mu-\hat{\theta})^\top
\triangledown^2 f(\hat{\theta})(\mu-\hat{\theta}) \nonumber \\
&+ \textstyle \frac{1}{2}\left( {\rm Tr}\left\{\triangledown^2
f(\hat{\theta}) \Sigma\right\} + \log |\Sigma|\right) -
\expect{q(z)}{\log q(z)}
\label{eq:monitor}
\end{align}
where $f(\theta)$ is defined in~\myeq{f-laplace} and $\hat{\theta}$ is
defined as for Laplace or delta method inference.\footnote{We note
again that \myeq{monitor} is not the function we are optimizing.
Even the simpler Laplace approximation is not clearly minimizing a
well-defined distance function between the approximate Gaussian and
true posterior~\citep{MacKay:1992}. Thus, while this approach is an
approximate coordinate ascent algorithm, clearly characterizing the
corresponding objective function is an open problem.}
\myfig{convergence} shows this score at each iteration for two runs of
inference in the correlated topic model. (See \mysec{ctm} for details
about the model.) The approximate objective increases as the
algorithm proceeds, and these plots were typical. In practice, as did
\citet{Braun:2010} in their setting, we found that this is a good
score to monitor.
\begin{figure*}
\caption{The approximate variational objective from \myeq{monitor}
\label{fig:convergence}
\end{figure*}
\section{Example Models}\label{sec:example}
We have described a generic algorithm for approximate posterior
inference in nonconjugate models. In this section we derive this
algorithm for several nonconjugate models from the research
literature: the correlated topic model~\citep{Blei:2007}, Bayesian
logistic regression~\citep{Jaakkola:1997a}, and hierarchical Bayesian
logistic regression~\citep{Gelman:2007}. For each model, we identify
the variables---the nonconjugate variable $\theta$, conjugate variable
$z$, and observations $x$---and we calculate $f(\theta)$ from
\myeq{f-laplace}. (The calculations of $f(\theta)$ are in the
appendices.) In the next section, we study how our algorithms
perform when analyzing data under these models.\footnote{Python
implementations of our algorithms are available at
\url{http://www.cs.cmu.edu/~chongw/software/nonconjugate_inference.tar.gz}.}
\subsection{The Correlated Topic Model}\label{sec:ctm}
\begin{figure}
\caption{The graphical representation of the correlated topic model
(CTM). The nonconjugate variable is $\theta$; the conjugate
variable is the collection $z = z_{1:N}
\label{fig:ctm}
\end{figure}
Probabilistic topic models are models of document collections. Each
document is treated as a group of observed words that are drawn from a
mixture model. The mixture components, called ``topics,'' are
distributions over terms that are shared for the whole collection;
each document exhibits them with individualized proportions.
Conditioned on a corpus of documents, the posterior topics place high
probabilities on words that are associated under a single theme; for
example, one topic may contain words like ``bat,'' ``ball,'' and
``pitcher.'' The posterior topic proportions reflect how each
document exhibits those themes; for example, a document may combine
the topics of \textit{sports} and \textit{health}. This posterior
decomposition of a collection can be used for summarization,
visualization, or forming predictions about a document.
See~\cite{Blei:2012} for a review of topic modeling.
The per-document topic proportions are a latent variable. In latent
Dirichlet allocation (LDA)~\citep{Blei:2003b}---which is the simplest
topic model---these are given a Dirichlet prior, which makes the model
conditionally conjugate. Here we will study the correlated topic
model (CTM)~\citep{Blei:2007}. The CTM extends LDA by replacing the
Dirichlet prior on the topic proportions with a logistic normal
prior~\citep{Aitchison:1982}. This is a richer prior that can capture
correlations between occurrences of the components. For example, a
document about \textit{sports} is more likely to also be about
\textit{health}. The CTM is not conditionally conjugate. But it is a
more expressive model: it gives a better fit to texts and provides new
kinds of exploratory structure.
Suppose there are $K$ topic parameters $\beta_{1:K}$, each of which is
a distribution over $V$ terms. Let $\pi(\theta)$ denote the
multinomial logistic function, which maps a real-valued vector to a
point on the simplex with the same dimension, $\pi(\theta) \propto
\exp\{\theta\}$. The CTM assumes a document is drawn as follows:
\begin{enumerate*}
\item Draw log topic proportions $\theta \sim \mathcal{N}(\mu_0,
\Sigma_0)$.
\item For each word $n$:
\begin{enumerate*}
\item Draw topic assignment $z_n \,\vert\, \theta \sim {\rm Mult}(\pi(\theta))$.
\item Draw word $x_n \,\vert\, z_n, \beta \sim {\rm Mult}(\beta_{z_n})$.
\end{enumerate*}
\end{enumerate*}
\myfig{ctm} shows the graphical model. The topic proportions
$\pi(\theta)$ are drawn from a logistic normal distribution; their
correlation structure is captured in its covariance matrix $\Sigma_0$.
The topic assignment variable $z_n$ indicates from which topic the
$n$th word is drawn.
Holding the topics $\beta_{1:K}$ fixed, the main inference problem in
the CTM is to infer the conditional distribution of the document-level
hidden variables $p(\theta, z_{1:N} \,\vert\, x_{1:N}, \beta_{1:K})$. This
calculation is important in two contexts: it is used when forming
predictions about new data; and it is used as a subroutine in the
variational expectation maximization algorithm for fitting the topics
and logistic normal parameters (mean $\mu_0$ and covariance
$\Sigma_0$) with maximum likelihood. The corresponding per-document
inference problem is straightforward to solve in LDA, thanks to
conditional conjugacy. In the CTM, however, it is difficult because
the logistic normal on $\theta$ is not conjugate to the multinomial on
$z$. \citet{Blei:2007} used a Taylor approximation designed
specifically for this model. Here we apply the generic algorithm from
\mysec{inference}.
In terms of the earlier notation, the nonconjugate variable is the
topic proportions $\theta$, the conjugate variable is the collection
of topic assignments $z = z_{1:N}$, and the observation is the
collection of words $x = x_{1:N}$. The variational distribution for
the topic proportions $\theta$ is Gaussian, $q(\theta) =
\mathcal{N}(\mu, \Sigma)$; the variational distribution for the topic
assignments is discrete, $q(z) = \prod_{n} q(z_n \,\vert\, \phi_n)$ where
each $\phi_n$ is a distribution over $K$ elements. In delta method
inference, as in~\cite{Braun:2010}, we restrict the variational
covariance $\Sigma$ to be diagonal to simplify the derivative of
\myeq{lb-II}. Laplace variational inference does not require this
simplification. Appendix B gives the detailed derivations of the
algorithm.
Besides the CTM, this approach can be adapted to a variety of
nonconjugate topic models, including the topic evolution
model~\citep{Xing:2005}, Dirichlet-multinomial regression
~\citep{Mimno:2008}, dynamic topic
models~\citep{Blei:2006d,Wang:2008}, and the discrete infinite
logistic normal distribution~\citep{Paisley:2012b}.
\subsection{Bayesian Logistic Regression}
\label{sec:logistic-regression}
\begin{figure}
\caption{The graphical representation of hierarchical logistic
regression. (When $M=1$, this is standard Bayesian logistic
regression.)The nonconjugate variable is the vector of
coefficients $\theta_m$, the conjugate variable is the collection
of observed classes for each data point, $z_m = z_{m,1:N}
\label{fig:bayesian-lg}
\end{figure}
Bayesian logistic regression is a well-studied model for binary
classification~\citep{Jaakkola:1997a}. It places a Gaussian prior on a
set of coefficients and draws class labels, conditioned on covariates,
from the corresponding logistic. Let $t_n$ is be a $p$-dimensional
observed covariate vector for the $n$th sample and $z_n$ be its class
label (an indicator vector of length two). Let $\theta$ be the
real-valued coefficients in $\mathbb{R}^p$; there is a coefficient for
each feature. Bayesian logistic regression assumes the following
conditional process:
\begin{enumerate*}
\item Draw coefficients $\theta \sim \mathcal{N} (\mu_0, \Sigma_0)$.
\item For each data point $n$ and its covariates $t_n$, draw its class
label from
\begin{equation*}
z_n \,\vert\, \theta, t_n \sim \textrm{Bernoulli}
\left(\sigma(\theta^\top t_n)^{z_{n,1}}
\sigma(-\theta^\top t_n)^{z_{n,2}}\right),
\end{equation*}
where $\sigma(y) \triangleq 1/\left(1+\exp(-y)\right)$ is the logistic
function.
\end{enumerate*}
\myfig{bayesian-lg} shows the graphical model. Given a dataset of
labeled feature vectors, the posterior inference problem is to compute
the conditional distribution of the coefficients $p(\theta \,\vert\, z_{1:N},
t_{1:N})$. The issue is that the Gaussian prior on the coefficients
is not conjugate to the conditional likelihood of the label.
This is a subset of the model class in \mysec{condition}. The
nonconjugate variable $\theta$ is identical and the variable $z$ is
the collection of observed classes of each data point, $z_{1:N}$.
Note there is no additional observed variable $x$ downstream. The
variational distribution need only be defined for the coefficients,
$q(\theta) = \mathcal{N}(\mu, \Sigma)$. Using Laplace variational
inference, our approach recovers the standard Laplace approximation
for Bayesian logistic regression~\citep{Bishop:2006}. This gives a
connection between standard Laplace approximation and
variational inference. Delta method variational inference provides an
alternative. Appendix C gives the detailed derivations.
An important extension of Bayesian logistic regression is hierarchical
Bayesian logistic regression~\citep{Gelman:2007}. It simultaneously
models related logistic regression problems, and estimates the
hyperparameters of the shared prior on the coefficients. With $M$
related problems, we construct the following hierarchical model:
\begin{enumerate*}
\item Draw the global hyperparameters,
\begin{align}
\Sigma_0^{-1} & \sim {\rm Wishart}(\nu,
\Phi_0) \label{eq:multi-lgr-prior} \\
\mu_0 &\sim \mathcal{N}(0, \Phi_1) \label{eq:multi-lgr-prior1}
\end{align}
\item For each problem $m$:
\begin{enumerate*}
\item Draw coefficients $\theta_m \sim \mathcal{N} (\mu_0,
\Sigma_0)$.
\item For each data point $n$ and its covariates $t_{mn}$, draw its
class label,
\begin{equation*}
z_{mn} \,\vert\, \theta_m, t_{mn} \sim
\textrm{Bernoulli}(\sigma(\theta_m^\top t_{mn})^{z_{mn,1}}
\sigma(-\theta_m^\top t_{mn})^{z_{mn,2}}).
\end{equation*}
\end{enumerate*}
\end{enumerate*}
As for the CTM, we use nonconjugate inference as a subroutine in a
variational EM algorithm (where the M step is regularized). We
construct $f(\theta_m)$ in \myeq{f-laplace} separately for each
problem $m$, and fit the hyperparameters $\mu_0$ and $\Sigma_0$ from
their approximate expected sufficient statistics~\citep{Bishop:2006}.
This amounts to MAP estimation with priors as specified above. See
Appendix C for the complete derivation.
Finally, we note that logistic regression is a generalized linear
model with a binary response and canonical link
function~\citep{McCullagh:1989}. It is straightforward to use our
algorithms with other Bayesian generalized linear models (and their
hierarchical forms).
\section{Empirical Study}
\label{sec:experiments}
We studied nonconjugate variational inference with correlated topic
models and Bayesian logistic regression. We found that nonconjugate
inference is more accurate than the existing methods tailored to
specific models. Between the two nonconjugate inference algorithms,
we found that Laplace inference is faster and more accurate than delta
method inference.
\subsection{The Correlated Topic Model}
\label{sec:ctm-study}
We studied Laplace inference and delta method inference in the CTM.
We compared it to the original inference algorithm of
\citet{Blei:2007}.
We analyzed two collections of documents. The \textit{Associated
Press} (AP) collection contains 2,246 documents from the {\it
Associated Press}. We used a vocabulary of 10,473 terms, which
gave a total of 436K observed words. The {\it New York Times}
(NYT) collection contains 9,238 documents from the {\it New York
Times}. We used a vocabulary of 10,760 terms, which gave a total
of 2.3 million observed words. For each corpus we used 80\% of
the documents to fit models and reserved 20\% to test them.
We fitted the models with variational EM. At each iteration, the
algorithm has a set of topics $\beta_{1:K}$ and parameters to the
logistic normal $\{\mu_0, \Sigma_0\}$. In the E-step we perform
approximate posterior inference with each document, estimating its
topic proportions and topic assignments. In the M-step, we
re-estimate the topics and logistic normal parameters. We fit models
with different kinds of E-steps, using both of the nonconjugate
inference methods from \mysec{inference} and the original approach
of~\citet{Blei:2007}. To initialize nonconjugate inference we set the
variational mean parameter $\mu=0$ for log topic proportions $\theta$
and computed the corresponding updates for the topic assignments $z$.
We initialize the topics in variational EM to random draws from a
uniform Dirichlet.
With nonconjugate inference in the E-step, variational EM
approximately optimizes a bound on the marginal probability of the
observed data. We can calculate an approximation of this bound with
\myeq{monitor} summed over all documents. We monitor this quantity as
we run variational EM.
To test our fitted models, we measured predictive performance on
held-out data with predictive distributions derived from the posterior
approximations. We follow the testing framework
of~\citet{Asuncion:2009} and~\citet{Blei:2007}. We fix fitted topics
and logistic normal parameters $M = \{\beta_{1:K}, \mu_0, \Sigma_0\}$.
We split each held-out document in to two halves $(\bm w_{1}, \bm
w_{2})$ and form the approximate posterior log topic proportions
$q_{\bm w_{1}}(\theta)$ using one of the approximate inference
algorithms and the first half of the document $\bm w_1$. We use
this to form an approximate predictive distribution,
\begin{eqnarray*}
p(w \,\vert\, \bm w_{1}, M) &\approx&
\textstyle \int_{\theta} \sum_{z} p(w \,\vert\, z, \beta_{1:K}) q_{\bm w_1}(\theta)
d\theta \\
&\approx& \textstyle \sum_{k=1}^{K} \beta_{kw} \pi_k,
\end{eqnarray*} where $\pi_k \propto
\exp\{\expect{q}{\theta_k}\}$. Finally, we evaluate the log
probability of the second half of the document
using that predictive distribution; this is the {\it heldout
log likelihood}. A better model and inference method will give
higher predictive probabilities of the unseen words. Note that
this testing framework puts the approximate posterior
distributions on the same playing field. The quantities are
comparable regardless of how the approximate posterior is formed.
\myfig{train-test-likelihood} shows the per-word approximate bound and
the per-word heldout likelihood as functions of the number of
topics. \myfig{train-test-likelihood} (a) indicates that the
approximate bounds from nonconjugate inference generally go up as the
number of topics increases. This is a property of a good
approximation because the marginal certainly goes up as the number of
parameters increases. In contrast, Blei and Lafferty's (2007)
objective (which is a true bound on the marginal of the data) behaves
erratically. This is illustrated for the \textit{New York Times}
corpus; on the {\it Associated Press} corpus, it does not come close
to the approximate bound and is not plotted.
\myfig{train-test-likelihood} (b) shows that on heldout data, Blei and
Lafferty's approach, tailored for this model, performed worse than
both of our algorithms. Our conjecture is that while this method
gives a strict lower bound on the marginal, it might be a loose bound
and give poor predictive distributions. Our methods use an
approximation which, while not a bound, might be closer to the
objective and give better predictive distributions. The heldout
likelihood plots also show that when the number of topics increases
the algorithms eventually overfit the data. Finally, note that
Laplace variational inference was always better than both other
algorithms.
Finally, \myfig{train-test-time} shows the approximate bound and the
heldout log likelihood as functions of running time.\footnote{We did
not formally compare the running time of \cite{Blei:2007}'s method
because we used the authors' C implementation, while ours is in
Python. We observed that their method took more than five times
longer than ours.} From \myfig{train-test-time} (a), we see that
even though variational EM is not formally optimizing this approximate
objective (see \myeq{monitor}), the increase at each iteration
suggests that the marginal probability is also increasing. The plot
also shows that Laplace inference converges faster than delta method
inference. \myfig{train-test-time} (b) confirms that Laplace inference
is both faster and gives better predictive performance.
\begin{figure*}
\caption{ Laplace variational inference is ``Lap-Var''; delta
method variational inference is ``Delta-Var''; Blei and
Lafferty's method is ``BL.'' (a) Approximate per-word lower
bound against the number of topics. A good approximation will
go up as the number of topics increases, but not necessarily
indicate a better predictive performance on the heldout data.
(b) Per-word held-out log likelihood against the number of
topics. Higher numbers are better. Both nonconjugate
methods perform better than Blei and Lafferty's method.
Laplace inference performs best. Blei and Lafferty's method
was erratic in both collections. (It is not plotted for the
AP collection.)
}
\label{fig:train-test-likelihood}
\end{figure*}
\begin{figure*}
\caption{In this figure, we set the number of topics as $K=60$.
(Others are similar.) (a) The per-word approximate bound during
model fitting with variational EM. Though it is an
approximation of the variational EM objective, it converges in
practice. (b) The per-word heldout likelihood during the model
fitting with variational EM. Laplace inference performs best in
terms of speed and predictive performance.}
\label{fig:train-test-time}
\end{figure*}
\subsection{Bayesian Logistic Regression}
We studied our algorithms on Bayesian logistic regression in both
standard and hierarchical settings. In the standard setting, we
analyzed two datasets. With the {\it Yeast}
data~\citep{Elisseeff:2001}, we form a predictor of gene functional
classes from features composed of micro-array expression data and
phylogenetic profiles. The dataset has 1,500 genes in the training set
and 917 genes in the test set. For each gene there are 103 covariates
and up to 14 different gene functional classes (14 labels). This
corresponds to 14 independent binary classification problems. With
the {\it Scene} data~\citep{Boutell:2004}, we form a predictor of
scene labels from image features. It contains 1,211 images in the
training set and 1,196 images in the test set. There are 294 images
features and up to 6 scene labels per image. This corresponds to 6
independent binary classification problems.\footnote{The {\it Yeast}
and {\it Scene} data are at
http://mulan.sourceforge.net/datasets.html.}
We used two performance measures. First we measured accuracy, which is
the proportion of test-case examples correctly labeled. Second, we
measured average log predictive likelihood. Given a test-case input
$t$ with label $z$, we compute the log predictive likelihood,
\begin{equation*}
\log p(z \,\vert\, \mu, t) = z_1\log \sigma(\mu^\top t) + z_2 \log
\sigma(-\mu^\top t),
\end{equation*}
where $\mu$ is the mean of variational distribution $q(\theta) =
\mathcal{N}(\mu, \Sigma)$. Higher likelihoods indicate a better fit.
For both accuracy and predictive likelihood, we used cross validation
to estimate the generalization performance of each inference
algorithm. We set the priors $\mu_0 = 0$ and $\Sigma_0 = I$.
We compared Laplace inference (\mysec{embed-laplace}), delta method
inference (\mysec{lower-bound}), and the method
of~\cite{Jaakkola:1997a}. Jaakkola and Jordan's (1996) method
preserves a lower bound on the marginal likelihood with a first-order
Taylor approximation and was developed specifically for Bayesian
logistic regression. (We note that Blei and Lafferty's
bound-preserving method for the CTM was built on this technique.)
Table~\ref{tab:standard-task} gives the results. To compare methods
we compute the difference in score (accuracy or log likelihood) on the
independent binary classification problems, and then perform a
standard t-test (at level $0.05$) to test if the mean of the
differences is larger than 0. Laplace inference and delta method
inference gave slightly better accuracy than Jaakkola and Jordan's
method, and much better log predictive likelihood.\footnote{Previous
literature, e.g., . \citet{Xue:2007} and \citet{Archambeau:2011}
treat {\it Yeast} and {\it Scene} as multi-task problems. In our
study, we found that our standard Bayesian logistic regression
algorithms performed the same as the algorithms developed in these
papers.} The t-test showed that both Laplace and delta method
inference are better than Jaakkola and Jordan's method.
We next examined a dataset of student performance in a collection of
schools. With the {\it School} data, our goal is to use various
features of a student to predict whether he or she will perform above
or below the median on a standardized exam.\footnote{The data is
available at http://multilevel.ioe.ac.uk/intro/datasets.html.} The
data came from the Inner London Education Authority. It contains
examination records from 139 secondary schools for the years 1985,
1986 and 1987. It is a random 50\% sample with 15,362 students. The
students' features contain four student-dependent features and
school-dependent features. The student dependent features are the
year of the exam, gender, VR band (individual prior attainment data),
and ethnic group; the school-dependent features are the percentage of
students eligible for free school meals, percentage of students in VR
band 1, school gender, and school denomination. We coded the binary
indicator of whether each was below the median (``bad'') or above
(``good''). We use the same 10 random splits of the data
as~\cite{Argyriou:2008}.
In this data, we can either treat each school as a separate
classification problem, pool all the schools together as a single
classification problem, or analyze them with hierarchical logistic
regression (\mysec{logistic-regression}). The hierarchical model
allows the predictors for each school to deviate from each other, but
shares statistical strength across them.
Let $p$ be the number of covariates. We set the prior on the
hyperparameters to the coefficients to $\nu = p+100$,
$\Phi_0=0.01I$, and $\Phi_1 = 0.01I$ ( see \myeq{multi-lgr-prior}
and \myeq{multi-lgr-prior1}) to favor sparsity. We initialized
the variational distributions to $q(\theta)=\mathcal{N}(0, I)$.
Table~\ref{tab:multi-task} gives the results. A standard t-test (at
level $0.05$) showed that the hierarchical models are better than the
non-hierarchical models both in terms of accuracy and predictive
likelihood. With predictive likelihood, Laplace variational inference
in the hierarchical model is significantly better than all other
approaches.
\begin{table*}[t]
\begin{center}
\begin{tabular}{lcc|cc}
\hline
& \textrm{Mult}icolumn{2}{c}{\textit{Yeast}} &
\textrm{Mult}icolumn{2}{c}{\textit{Scene}} \\
& Accuracy & Log Likelihood & Accuracy & Log
Likelihood \\
\hline
Jaakkola and Jordan (1996) & 79.7\% & -0.678 & 87.4\% & -0.670 \\
Laplace inference & \textbf{80.1\%} & {\bf -0.449} & \textbf{89.4\%} & {\bf -0.259} \\
Delta method inference & {\bf 80.2\%} & {\bf -0.450} & {\bf 89.5\%} & -0.265 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of the different methods for Bayesian logistic
regression using accuracy and averaged log predictive likelihood.
Higher numbers are better. These results are averaged from five
random starts. (The variance is too small to report.) Bold results
indicate significantly better performance using a standard
t-test. Laplace and delta method inference perform best.}
\label{tab:standard-task}
\end{table*}
\begin{table*}[t]
\begin{center}
\begin{tabular}{ll|cc}
\hline
&& Accuracy & Log Likelihood \\
\hline
\textit{Separate} &&&\\
& Jaakkola and Jordan (1996) & 70.5\% & -0.684 \\
& Laplace inference & 70.8\% & -0.569 \\
& Delta inference & 70.8\%& -0.571 \\
\textit{Pooled} &&& \\
& Jaakkola and Jordan (1996) & 71.2\% & -0.685 \\
& Laplace inference & 71.3\% & -0.557 \\
& Delta inference & 71.3\% & -0.557 \\
\textit{Hierarchical} &&& \\
& Jaakkola and Jordan (1996) & 71.3\% & -0.685 \\
& Laplace inference & {\bf 71.9\%}& {\bf -0.549} \\
& Delta inference & {\bf 71.9\%} & -0.559 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of the different methods on the \textit{School}
data using accuracy and averaged log predictive likelihood. Results
are averaged from 10 random splits. (The variance is too small to
report.) We compared Laplace inference, delta inference and
Jaakkola and Jordan's (1996) method in three settings: separate
logistic regression models for each school, a pooled logistic
regression model for all schools, and the hierarchical logistic
regression model in \mysec{logistic-regression}. Bold indicates
significantly better performance by a standard t-test (at level 0.05). The
hierarchical model performs best. \label{tab:multi-task}}
\end{table*}
\section{Discussion}
\label{sec:discussion}
We developed Laplace and delta method variational inference, two
strategies for variational inference in a large class of nonconjugate
models. These methods approximate the variational objective function
with a Taylor approximation, each in a different way. We studied them
in two nonconjugate models and showed that they work well in practice,
forming approximate posteriors that lead to good predictions. In the
examples we analyzed, our methods worked better than methods tailored
for the specific models at hand. Between the two, Laplace inference
was better and faster than delta method inference. These methods
expand the scope of variational inference.
\section*{Appendix A: Generalization to Complex Models}
We describe how we can generalize our approaches to more complex
models. Suppose we have a directed probabilistic model with
latent variables $\theta = \theta_{1:m}$ and observations $x$.
(We will not differentiate notation between conjugate and
nonconjugate variables.) The log joint likelihood of all latent
and observed variables is
\begin{equation*}
\log p(\theta, x) =
\sum_{i=1}^{m} \log p(\theta_i \,\vert\, \theta_{\pi_i}) + \log p(x \,\vert\, \theta),
\end{equation*}
where $\pi_i$ are the indices of the parents of $\theta_i$, the
variables it depends on.
Our goal is to approximate the posterior distribution $p(\theta \,\vert\,
x)$. Similar to the main paper, we use mean-field variational
inference~\citep{Jordan:1999}. We posit a fully-factorized
variational family
$$q(\theta) = \prod_{i=1}^{m} q(\theta_i)$$
and optimize the each factor $q(\theta_i)$ to find the member closest
in KL-divergence to the posterior.
As in the main paper, we solve this optimization problem with
coordinate ascent, iteratively optimizing each variational factor
while holding the others fixed. Recall that \cite{Bishop:2006} shows
that this leads to the following update
\begin{equation}
q(\theta_i) \propto \exp\left\{\E{-i}{\log p(\theta,x)}\right\},
\label{eq:bishop-update}
\end{equation}
where $\E{-i}{\cdot}$ denotes the expectation with respect to
$\prod_{j,j\neq i}q(\theta_j)$.
Many of the terms of the log joint will be constant with respect to
$\theta_i$ and absorbed into the constant of proportionality. This
allows us to simplify the update in~\myeq{bishop-update} to be
$q(\theta_i) \propto \exp\left\{f(\theta_i)\right\}$ where
\begin{equation}
f(\theta_i) =
\E{-i}{\log p(\theta_i \,\vert\, \theta_{\pi_i})} +
\sum_{\{j: i \in \pi_j\}} \E{-i}{\log p(\theta_j \,\vert\, \theta_{\pi_j})} +
\E{-i}{\log p(x \,\vert\, \theta)}.
\label{eq:general-laplace-f}
\end{equation}
As in the main paper, this update is not tractable in general. We use
Laplace variational inference (\mysec{embed-laplace}) to approximate
it, although delta method variational inference (\mysec{lower-bound})
is also applicable. In Laplace variational inference, we take a Taylor
approximation of $f(\theta_i)$ around its maximum $\hat{\theta}_i$.
This naturally leads to $q(\theta_i)$ as a Gaussian factor,
\begin{equation*}
q^*(\theta_i) \approx {\cal N}(\hat{\theta}_i, -\triangledown^2
f(\hat{\theta}_i)^{-1}).
\end{equation*}
The main paper considers the case where $\theta$ is a single random
variable and updates its variational distribution. In the more general
coordinate ascent setting considered here, we need to compute or
approximate the expected log probabilities (and their derivatives) in
\myeq{general-laplace-f}.
Now suppose each factor is in the exponential family. (This is weaker
than the conjugacy assumption, and describes most graphical models
from the literature.) The log joint likelihood becomes
\begin{equation}
\log p(\theta, x) =
\sum_{i=1}^{m} \left(
\eta(\theta_{\pi_i})^\top t(\theta_i) -
a(\eta(\theta_{\pi_i}))
\right) +
\log p(x \,\vert\, \theta), \label{eq:log-joint-appendix}
\end{equation}
where $\eta(\cdot)$ are natural parameters, $t(\cdot)$ are sufficient
statistics, and $a(\eta(\cdot))$ are log normalizers. (All are
overloaded.) Substituting the exponential family assumptions into
$f(\theta_i)$ gives
\begin{align*}
f(\theta_i) &= \E{-i}{\eta(\theta_{\pi_i})}^\top t(\theta_i) \nonumber \\
& + \textstyle \sum_{\{j: i \in \pi_j\}} \left( \E{-i}{\eta(\theta_{\pi_j})}^\top
\E{-i}{t(\theta_j)} - \E{-i}{a(\eta(\theta_{\pi_j}))} \right) \\
& + \E{-i}{t(\theta)}^\top t(x) - \E{-i}{a(\eta(\theta))}. \nonumber
\end{align*}
Here we can use further Taylor approximations of the natural
parameters $\eta(\cdot)$, sufficient statistics $t(\cdot)$, and log
normalizers $a(\cdot)$ in order to easily take their expectations.
Finally, for some variables we may be able to exactly compute
$f(\theta_i)$ and form the $q^*(\theta_i)$ without further
approximations. (These are conjugate variables for which the complete
conditional $p(\theta_i \,\vert\, \theta_{-i}, x)$ is available in closed
form.) These variables were separated out in the main paper; here we
note that they can be updated exactly in the coordinate ascent
algorithm.
\section*{Appendix B: The Correlated Topic Model}
The correlated topic model is described in \mysec{ctm}. We identify
the quantities from~\myeq{z-dist} and~\myeq{x-dist} that we need to
compute $f(\theta)$ in~\myeq{f-laplace},
\begin{align*}
h(z) &=1, \ \ t(z) = \textstyle \sum_n z_n, \\
\eta(\theta) &= \textstyle
\theta-\log\left\{\sum_k\exp\{\theta_k\}\right\} \\
a(\eta(\theta)) &= 0.
\end{align*}
With this notation,
\begin{align*}
f(\theta) = \textstyle \eta(\theta)^\top\expect{q(z)}{t(z)} -\frac{1}{2}
(\theta-\mu_0)^\top\Sigma_0^{-1}(\theta - \mu_0),
\end{align*}
where $\expect{q(z)}{t(z)}$ is the expected word counts of each topic under the
variational distribution $q(z)$.
Let $\pi = \eta(\theta)$ be the topic proportions. Using ${\partial
\pi_i }/ {\partial \theta_j} = \pi_i(1_{[i=j]} - \pi_j)$, we obtain
the gradient and Hessian of the function $f(\theta)$ in the CTM,
\begin{align*}
\triangledown f(\theta) &= \textstyle \expect{q(z)}{t(z)} - \pi
\sum_{k=1}^K\left[\expect{q(z)}{t(z)} \right]_k -\Sigma_0^{-1}(\theta - \mu_0), \\
\triangledown^2 f(\theta)_{ij} &= \textstyle (-\pi_i 1_{[i=j]} + \pi_i \pi_j)
\sum_{k=1}^K\left[\expect{q(z)}{t(z)}\right]_k-(\Sigma_0^{-1})_{ij}.
\end{align*}
where $1_{[i=j]}=1$ if $i=j$ and $0$ otherwise. Note that
$\triangledown f(\theta)$ is all we need for Laplace inference.
In delta method variational inference, we also need to compute
the gradient of
\begin{align*}
{\rm Trace}\left\{\triangledown^2 f(\theta) \Sigma\right\} &=
\textstyle \left(-\sum_{k=1}^K \pi_k \Sigma_{kk} + \pi^T \Sigma
\pi\right) \sum_{k=1}^K\left[\expect{q(z)}{t(z)}\right]_k - {\rm
Trace}(\Sigma_0^{-1}\Sigma).
\end{align*}
Following~\citep{Braun:2010}, we assume $\Sigma$ is diagonal in the
delta method. (In Laplace inference, we do not need this assumption.)
This gives
\begin{align*}
& \frac{\partial {\rm Trace}\left\{\triangledown^2 f(\theta)
\Sigma\right\}} { \partial \theta_i} = \textstyle
\pi_i(1-2\pi_i)(\sum_k \pi_k\Sigma_{kk} -1).
\end{align*}
These quantities let us implement the algorithm
in~\myfig{lp-algorithm} to infer the per-document posterior of the CTM
hidden variables.
As we discussed \mysec{ctm}, we use this algorithm in variational EM
for finding maximum likelihood estimates of the model parameters. The
E-step runs posterior inference on each document. Since the
variational family is the same, the M-step is as described in
~\citet{Blei:2007}.
\section*{Appendix C: Bayesian Logistic Regression}
Bayesian logistic regression is described in
\mysec{logistic-regression}.
The distribution of the observations $z_{1:N}$ fit into the
exponential family as follows,
\begin{align*}
h(z) &=1, \ \ t(z) = \textstyle [z_1, \dots, z_N], \\ \eta(\theta) &
= [\log\sigma(\theta^\top t_n), \log\sigma(-\theta^\top
t_n)]_{n=1}^N \\ a(\eta(\theta)) &= 0.
\end{align*}
In this set up, $t(z)$ represents the whole set of labels. Since $z$ is
observed, its ``expectation'' is just itself. With this notation,
$f(\theta)$ from ~\myeq{f-laplace} is
\begin{align*}
f(\theta) = \textstyle \eta(\theta)^\top t(z) -\frac{1}{2}
(\theta-\mu_0)^\top\Sigma_0^{-1}(\theta - \mu_0).
\end{align*}
The gradient and Hessian of $f(\theta)$ are
\begin{align*}
\triangledown f(\theta) &= \textstyle \sum_{n=1}^N t_n
\left(z_{n,1}-\sigma(\theta^T t_n)\right) -\Sigma_0^{-1}(\theta -
\mu_0), \\
\triangledown^2 f(\theta) &= \textstyle -\sum_{n=1}^N\sigma(\theta^T
t_n)\sigma(-\theta^T t_n) t_nt_n^T -\Sigma_0^{-1}.
\end{align*}
This is the standard Laplace approximation to Bayesian logistic
regression~\citep{Bishop:2006}.
For delta variational inference, we also need the gradient for ${\rm
Trace}\left\{\triangledown^2 f(\theta) \Sigma\right\}$. It is
\begin{align*}
& \frac{\partial {\rm Trace}\left\{\triangledown^2 f(\theta)
\Sigma\right\}} { \partial \theta_i} =
-\sum_{n=1}^N\sigma(\theta^T t_n)\sigma(-\theta^T t_n)
(1-2\sigma(\theta^T t_n)) t_n t_n^T\Sigma t_n.
\end{align*}
Here we do not need to assume $\Sigma$ is diagonal. This Hessian is
already diagonal.
\paragraph{Hierarchical logistic regression.} Here we describe how we
update the global hyperparameters $(\mu_0, \Sigma_0)$ (Eq.
\ref{eq:multi-lgr-prior} and \ref{eq:multi-lgr-prior1}) in
hierarchical logistic regression. At each iteration, we first compute
the variational distribution of coefficients $\theta_m$ for each
problem $m = 1,..., M$,
\begin{align*}
q(\theta_m) = \mathcal{N}(\mu_m, \Sigma_m).
\end{align*}
We then estimate the global hyperparameters $(\mu_0, \Sigma_0)$ using
the MAP estimate. These come from the following update equations,
\begin{align*}
\mu_0 & =\left( \frac{\Sigma_0\Phi_1^{-1}}{M} +
I_p \right)^{-1}\frac{\sum_{m=1}^M \mu_m}{M}, \\
\Sigma_0 & =\frac{\Phi_0^{-1} + \sum_{m=1}^M
(\mu_m - \mu_0)(\mu_m - \mu_0)^\top}{M + \nu - p -1},
\end{align*}
where $p$ is the dimension of coefficients $\theta_m$.
\paragraph{Acknowledgements.} We thank Jon McAuliffe and the anonymous
reviewers for their valuable comments. Chong Wang was supported by
Google Ph.D. and Siebel Scholar Fellowships. David M. Blei is
supported by NSF IIS-0745520, NSF IIS-1247664, NSF IIS-1009542, ONR
N00014-11-1-0651, and the Alfred P. Sloan foundation.
\vskip 0.2in
\end{document} |
\begin{document}
\title[On some combinatorial properties of generalized cluster algebras]{On some combinatorial properties of \\generalized cluster algebras}
\author{Peigen Cao}
\address{Department
of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang
310027, P.R.China}
\curraddr{}
\email{[email protected]}
\thanks{}
\author{Fang Li}
\address{Department
of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang
310027, P.R.China}
\curraddr{}
\email{[email protected]}
\thanks{}
\subjclass[2010]{13F60 }
\date{}
\dedicatory{}
\keywords{Generalized cluster algebra, exchange graph}
\begin{abstract}
In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of $\mathcal A(\mathcal S)$; (ii) there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster algebra, if their initial exchange matrices satisfying a mild condition. Moreover, this bijection preserves the set of clusters of these two generalized cluster algebras.
As applications of the second result, we prove some properties of the components of the $d$-vectors of a generalized cluster algebra and we give a characterization for the clusters of a generalized cluster algebra.
\end{abstract}
\maketitle
\section{Introduction}
Cluster algebras were introduced by Fomin and Zelevinsky in \cite{FZ}. The motivation was to create a common framework for the phenomena occurring in connection
with total positivity and canonical bases. Since then, numerous connections between cluster algebras and other branches of mathematics have been discovered, for example, Poisson geometry, discrete dynamical systems, higher Teichm\"uller spaces, representation theory of quivers and finite-dimensional algebras.
Cluster algebras are commutative algebras whose generators and relations are constructed in a recursive manner. The generators of a cluster algebra are called {\em cluster variables}, which are grouped into overlapping {\em clusters} of the same size. One remarkable feature of cluster algebras is that they have the Laurent phenomenon, which says that for any given cluster ${\bf x}_{t_0}=\{x_{1;t_0},\cdots,x_{n;t_0}\}$, any cluster variable $x_{i;t}$ can be written as a Laurent polynomial in $x_{1;t_0},\cdots,x_{n;t_0}$.
Generalized cluster algebras were introduced in \cite{CS} by Chekhov and Shapiro,
which are the generalization of the classic cluster algebras introduced by Fomin and Zelevinsky
in \cite{FZ}. In the classic case, a product of cluster variables, one known and one
unknown, is equal to a binomial in other known variables. These binomial exchange relations are replaced by polynomial exchange relations in generalized cluster algebras.
The generalized cluster structures naturally appear in the Teichm\"{u}ller spaces of Riemann surfaces with orbifold points \cite{CS}, WKB analysis \cite{IN}, representations of quantum affine algebras \cite{G}, Drinfeld double of $GL_n$ \cite{GSV1}. The generalized cluster algebras share many common properties with the classic cluster algebras, for example, the Laurent phenomenon, finite type classification \cite{CS}, tropical dualities phenomenon between $C$-matrices and $G$-matrices \cite{NT}, the existence of greedy bases in rank $2$ case \cite{RD}. One can also refer to \cite{NR,CL}.
In this paper we will provide some other similarity between generalized cluster algebras and classic cluster algebras. Now we introduce our main results in this paper.
Firstly, we prove that the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of $\mathcal A(\mathcal S)$ (see Theorem \ref{thmgraph}).
Secondly, we prove that there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster algebra, if their initial exchange matrices satisfying a mild condition. Moreover, this bijection preserves the set of clusters of these two generalized cluster algebras (see Theorem \ref{promain} for details).
As applications of Theorem \ref{promain}, we prove some properties of the components of the $d$-vectors of a generalized cluster algebra (see Theorem \ref{thmdvec}) and we give a characterization for the clusters of a generalized cluster algebra (see Theorem \ref{thmlast}).
Note that Theorem \ref{thmgraph}, Theorem \ref{thmdvec}, Theorem \ref{thmlast} for classic cluster algebras have been given in \cite{CL1}. These results provide new similarity between generalized cluster algebras and classic cluster algebras.
This paper is organized as follows. In Section 2, some basic definitions, notations and known results are introduced. In Section 3, we give the proof of Theorem Theorem \ref{thmgraph} and Theorem \ref{promain}. In Section 4, we give the applications of Theorem \ref{promain}.
\section{Preliminaries}
\subsection{Generalized cluster algebras}
Recall that $(\mathbb P, \oplus, \cdot)$ is a {\bf semifield} if $(\mathbb P, \cdot)$ is an abelian multiplicative group endowed with a binary operation of auxiliary addition $\oplus$ which is commutative, associative and satisfies that the multiplication distributes over the auxiliary addition.
The {\bf tropical semifield} $\mathbb P=Trop(y_1,\cdots,y_m)$ is the free (multiplicative) abelian group generated by $y_1,\cdots,y_m$
with auxiliary addition $\oplus$ defined by
$$\prod\limits_{i}y_i^{a_i}\oplus\prod\limits_{i}y_i^{b_i}=\prod\limits_{i}y_i^{min(a_i,b_i)},$$
The multiplicative group of any semifield $\mathbb P$ is torsion-free \cite{FZ}, hence its group ring $\mathbb Z\mathbb P$ is a domain.
We take an ambient field $\mathcal F$ to be the field of rational functions in $n$ independent variables with coefficients in $\mathbb Z\mathbb P$.
An integer matrix $B_{n\times n}=(b_{ij})$ is called {\bf skew-symmetrizable} if there is a positive integer diagonal matrix $S$ such that $SB$ is skew-symmetric, where $S$ is said to be a {\bf skew-symmetrizer} of $B$.
\begin{definition}[Seed and mutation pair]
(i) A {\bf (labeled) seed} in $\mathcal F$ is a triple $(B,{\bf x},{\bf y})$, where
\begin{itemize}
\item $B=(b_{ij})$ is an $n\times n$ integer skew-symmetrizable matrix, called an {\bf exchange matrix};
\item ${\bf x}=(x_1,\dots, x_n)$ is an $n$-tuple such that $X=\{x_1,\dots, x_n\}$ is a free generating set of $\mathcal F$ over $\mathbb{ZP}$. We call ${\bf x}$ the {\bf cluster} and $x_1,\dots,x_n$ the {\bf cluster variables} of $({\bf x},{\bf y},B)$;
\item ${\bf y}=(y_1,\cdots,y_n)$ is an $n$-tuple of elements in $\mathbb P$, where $y_1,\cdots,y_n$ are called {\bf coefficients}.
\end{itemize}
(ii) An {\bf (labeled) mutation pair} in $\mathcal F$ is pair $(R,{\bf z})$, where
\begin{itemize}
\item $R=diag(r_1,\cdots,r_n)$ is a diagonal integer matrix with $r_i>0$, called a {\bf mutation degree matrix};
\item
${\bf z}=(z_{i,s})_{i=1,\cdots,n;~s=1,\cdots,r_i-1}$ a family of elements in $\mathbb P$ satisfying the reciprocity condition
$$z_{i,s}=z_{i,r_i-s}$$ for $s=1,\cdots,r_i-1$, which are called {\bf frozen coefficients}. In addition, we denote $$z_{i,0}=z_{i,r_i}=1$$ for $i=1,\cdots,n$.
\end{itemize}
\end{definition}
Each mutation pair $(R, {\bf z})$ naturally corresponds to a collection of polynomials ${\bf Z}=(Z_1,\cdots,Z_n)$, where
$$Z_i(u)=z_{i,0}+z_{i;1}u+\cdots+z_{i,r_i-1}u^{r_i-1}+z_{i;r_i}u^{r_i}\in\mathbb {ZP}[u].$$
We call $Z_1,\cdots,Z_n$ the {\bf mutation polynomials} of $(R, {\bf z})$.
\begin{definition}[$(R,{\bf z})$-seed mutation]
Let $(R, {\bf z})$ be a mutation pair, and ${\bf Z}=(Z_1,\cdots,Z_n)$ be the collection of mutation polynomials of $(R, {\bf z})$. Let $(B,{\bf x},{\bf y})$ be a seed, we define the {\bf $(R,{\bf z})$-seed mutation} at $k\in\{1,\cdots,n\}$ by $\mu_k(B,{\bf x},{\bf y})=(B^\prime, {\bf x}^\prime,{\bf y}^\prime)$, where
\begin{eqnarray}
b_{ij}^\prime&=&\begin{cases} -b_{ij},&\text{if }i=k\text{ or } j=k;\\b_{ij}+r_k(b_{ik}[-b_{kj}]_++[b_{ik}]_+b_{kj}),&\text{otherwise}.\end{cases}\nonumber\\
x_i^\prime&=&\begin{cases}x_i,&\text{if } i\neq k;\\ x_k^{-1}\left(\prod\limits_{j=1}^nx_j^{[-b_{jk}]_+}\right)^{r_k}\frac{Z_k(\hat y_k)}{Z_k|_{\mathbb P}(y_k)},&\text{if }i=k,\end{cases} \;\;\;\;\;\;\text{where }\hat y_k=y_k\prod\limits_{i=1}^nx_i^{b_{ik}}.\nonumber\\
y_i^\prime&=&\begin{cases} y_k^{-1}~,&\text{if } i=k;\\ y_i\left(y_k^{[b_{ki}]_+}\right)^{r_k}\left(Z_k|_{\mathbb P}(y_k)\right)^{-b_{ki}}~,&\text{if } i\neq k.\nonumber
\end{cases}
\end{eqnarray}
When the mutation degree matrix $R$ is given, we also denote $B^\prime=\mu_k(B)$, which is called the {\bf $R$-mutation} at $k$.
\end{definition}
It can be seen that $\mu_k(B,{\bf x},{\bf y})=(B^\prime, {\bf x}^\prime,{\bf y}^\prime)$ is also a seed.
\begin{remark}
(i) If $(R,{\bf z})=(I_n,\phi)$, the $(R,{\bf z})$-seed mutations are just the {\bf (classic) seed mutations} by Fomin and Zelevinsky. We will use $\mu_k^\circ$ to denote the (classic) seed mutations.
(ii) On the level of matrix mutations, we can see $$\mu_k(B)R=\mu_k^\circ(BR),$$ where $\mu_k$ is $R$-mutation, and $\mu_k^\circ$ is $I_n$-mutation.
\end{remark}
\begin{proposition}\cite{NR} The $(R,{\bf z})$-seed mutation $\mu_k$ is an involution.
\end{proposition}
Let $\mathbb T_n$ be the $n$-regular tree, and label the edges of $\mathbb T_n$ by $1,\dots,n$ such that the $n$ different edges adjacent to the same vertex of $\mathbb T_n$ receive different labels.
\begin{definition}
(i) An {\bf $(R,{\bf z})$-cluster pattern} ${\mathcal S}$ is an assignment of a seed $(B_t,{\bf x}_t,{\bf y}_t)$ to every vertex $t$ of the $n$-regular tree $\mathbb T_n$ such that $$(B_t^\prime,{\bf x}_{t}^\prime,{\bf y}_t^\prime)=\mu_{k}(B_t,{\bf x}_t,{\bf y}_t)$$ for any edge $t^{~\underline{\quad k \quad}}~ t^{\prime}$, where $\mu_k$ is the $(R,{\bf z})$-seed mutation at $k$.
(ii) Let $\mathcal S$ be an $(R,{\bf z})$-cluster pattern, the {\bf $(R,{\bf z})$-cluster algebra} $\mathcal A(\mathcal S)$ (also known as {\bf generalized cluster algebra}) associated with $\mathcal S$ is the $\mathbb {ZP}$-subalgebra of $\mathcal F$ generated by all the cluster variables of $\mathcal S$.
\end{definition}
\begin{remark}
If $(R,{\bf z})=(I_n,\phi)$, the $(R,{\bf z})$-cluster algebras are just the {\bf (classic) cluster algebras} by Fomin and Zelevinsky.
\end{remark}
For the seed $(B_t,{\bf x}_t,{\bf y}_t)$, we always write
$$B_t=(b_{ij}^t),\;\;{\bf x}_t=(x_{1;t},\cdots,x_{n;t}),\;\;{\bf y}_t=(y_{1;t},\cdots,y_{n;t}).$$
\begin{theorem}\cite{CS}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra and $(B_{t_0},{\bf x}_{t_0},{\bf y}_{t_0})$ be a seed of $\mathcal A(\mathcal S)$, then any cluster variable $x_{i;t}$ can be written as a Laurent polynomial in $\mathbb {ZP}[x_{1;t_0}^{\pm 1},\cdots,x_{n;t_0}^{\pm 1}]$.
\end{theorem}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, ${\bf x}_t$ and ${\bf x}_{t_0}$ be two clusters of $\mathcal A(\mathcal S)$. We know that each $x_{i;t}$ can be viewed as a rational functions in $x_{1;t_0},\cdots,x_{n;t_0}$ with coefficients in $\mathbb {ZP}$, so we can define the corresponding Jacobi matrix as follows.
$$J^t_{t_0}=\begin{pmatrix} \frac{\partial x_{1;t}}{\partial x_{1;t_0}}&\frac{\partial x_{2;t}}{\partial x_{1;t_0}}&\cdots &\frac{\partial x_{n;t}}{\partial x_{1;t_0}}\\ \frac{\partial x_{1;t}}{\partial x_{2;t_0}}&\frac{\partial x_{2;t}}{\partial x_{2;t_0}}&\cdots &\frac{\partial x_{n;t}}{\partial x_{2;t_0}}\\ \vdots &\vdots& &\vdots\\ \frac{\partial x_{1;t}}{\partial x_{n;t_0}}&\frac{\partial x_{2;t}}{\partial x_{n;t_0}}&\cdots &\frac{\partial x_{n;t}}{\partial x_{n;t_0}} \end{pmatrix}.$$
Let $H^t_{t_0}=diag(x_{1;t_0},\cdots, x_{n;t_0})J^t_{t_0}diag(x_{1;t}^{-1}, \cdots, x_{n;t}^{-1})$, which is called the {\bf $H$-matrix} of ${\bf x}_t$ with respect to ${\bf x}_{t_0}$.
\begin{theorem}[Cluster formula \cite{CL}]
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, $(B_t,{\bf x}_t,{\bf y}_t)$ and $(B_{t_0}$,${\bf x}_{t_0},{\bf y}_{t_0})$ be two seeds of $\mathcal A(\mathcal S)$. Then we have
$$H_{t_0}^{t}(B_tR^{-1}S^{-1}) (H_{t_0}^{t})^{\rm T}=B_{t_0}R^{-1}S^{-1}\;\;\;\;\;\;\; \text{and}\;\;\;\;\;\;\; det(H_{t_0}^{t})=\pm 1,$$
where $S$ is a skew-symmetrizer of $RB_{t_0}$.
\end{theorem}
\subsection{$D$-matrices and exchange graph}
By the Laurent phenomenon, each cluster variable can be written as
\begin{eqnarray}
x_{i;t}=\frac{f(x_{1;t_0},\cdots,x_{n,t_0})}{x_{1;t_0}^{d_1}\cdots x_{n;t_0}^{d_n}},\nonumber
\end{eqnarray}
where $f$ is a polynomial in $x_{1;t_0},\cdots,x_{n;t_0}$ with coefficients in $\mathbb {ZP}$ with $x_{j;t_0}\nmid f$ for any $j=1,\cdots,n$.
The vector ${\bf d}_{i;t}^{t_0}=(d_1,\cdots,d_n)^{\rm T}\in\mathbb Z^n$ is called the {\bf $d$-vector} of $x_{i;t}$ with respect to ${\bf x}_{t_0}$. The matrix
$D_t^{t_0}=({\bf d}_{1;t}^{t_0},\cdots,{\bf d}_{n;t}^{t_0})$ is called the {\bf $D$-matrix} of ${\bf x}_t$ with respect to ${\bf x}_{t_0}$.
\begin{proposition}\cite{CL}\label{prodmut}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed at $t_0$. Then the $D$-matrix $D_t^{t_0}=({\bf d}_{1;t}^{t_0},\cdots,{\bf d}_{n;t}^{t_0})$ is uniquely determined by the initial condition $D_{t_0}^{t_0}=-I_n$, together with the following relation:
\begin{equation}\label{mutationD}
{\bf d}_{j;t^\prime}^{t_0}=\begin{cases}{\bf d}_{j;t}^{t_0} & \text{if } j\neq k;\\ -{\bf d}_{k;t}^{t_0}+max\{\sum\limits_{b_{lk}^t>0}{\bf d}_{l;t}^{t_0} b_{lk}^tr_k, \sum\limits_{b_{lk}^t<0} -{\bf d}_{l;t}^{t_0}b_{lk}^tr_k\} &\text{if } j=k.\end{cases}\nonumber
\end{equation}
for any $t,t'\in\mathbb T_n$ with edge $t^{~\underline{\quad k \quad}} ~t^{\prime}$.
\end{proposition}
The following result is a direct corollary of Proposition \ref{prodmut}.
\begin{corollary}\label{cordmat}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed $(B_{t_0},{\bf x}_{t_0},{\bf y}_{t_0})$, and $\mathcal A(\overline{\mathcal S})$ be an $(\overline R,\overline{\bf z})$-cluster algebra with initial seed $(\overline B_{t_0},\overline{\bf x}_{t_0},\overline{\bf y}_{t_0})$. If $B_{t_0}R=\overline B_{t_0}\overline R$, then for any two vertices $w,v\in\mathbb T_n$, we have
$$D_v^w=\overline D_v^w,$$
where $D_v^w$ is the $D$-matrix of $\mathcal A(\mathcal S)$ and $\overline D_v^w$ is the $D$-matrix of $\mathcal A(\overline{\mathcal S})$.
\end{corollary}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed at $t_0$. Let $(B_{t_1},{\bf x}_{t_1},{\bf y}_{t_1})$ and $(B_{t_2},{\bf x}_{t_2},{\bf y}_{t_2})$ be two seeds of $\mathcal A(\mathcal S)$. We say that the two seeds $(B_{t_1},{\bf x}_{t_1},{\bf y}_{t_1})$ and $(B_{t_2},{\bf x}_{t_2},{\bf y}_{t_2})$ are {\bf equivalent} if there exists a permutation $\sigma$ of $\{1,\cdots,n\}$ such that $$x_{i;t_2}=x_{\sigma(i);t_1},\;y_{i;t_2}=y_{\sigma(i);t_1},\;b_{ij}^{t_2}=b_{\sigma(i)\sigma(j)}^{t_1},$$
for any $i,j=1,\cdots,n$.
\begin{definition}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, the {\bf exchange graph} ${\bf EG}(\mathcal A(\mathcal S))$ of $\mathcal A(\mathcal S)$ is a graph satisfying that
\begin{itemize}
\item the set of vertices of ${\bf EG}(\mathcal A(\mathcal S))$ is in bijection with the set of seeds (up to equivalence) of $\mathcal A(\mathcal S)$;
\item two vertices joined by an edge if and only if the corresponding two seeds (up to equivalence) are obtained from each other by once mutation.
\end{itemize}
\end{definition}
\subsection{Generalized cluster algebras with principal coefficients}
Now we give the definition of principal coefficients $(R,{\bf z})$-cluster algebra.
Let us temporarily regard ${\bf y}=(y_1,\cdots,y_n)$, and ${\bf z}=(z_{i,s})_{i=1,\cdots,n;~s=1,\cdots,r_i-1}$
with $z_{i,s}=z_{i,r_i-s}$ as formal variables. Let $\mathbb P_{pr}:=Trop({\bf y},{\bf z})$
be the tropical semifield of ${\bf y}$ and ${\bf z}$, and $\mathcal F_{pr}$ be the field of rational functions in $n$ independent variables with coefficients in $\mathbb Z\mathbb P_{pr}$.
\begin{definition}
An $(R,{\bf z})$-cluster algebra $\mathcal A(\mathcal S)$ in $\mathcal F_{pr}$ is said to be with {\bf principal coefficients} at $t_0$, if ${\bf y}_{t_0}={\bf y}$.
\end{definition}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$. By the Laurent phenomenon, each cluster variable $x_{i;t}$ can be expressed as $$X_{i;t}({\bf x}_{t_0},{\bf y},{\bf z})=\mathbb {ZP}_{pr}[{\bf x}_{t_0}^{\pm1}]=\mathbb Z[{\bf x}_{t_0}^{\pm1},{\bf y}^{\pm1},{\bf z}^{\pm 1}].$$ We call $X_{i;t}$ the {\bf $X$-function} of $x_{i;t}$.
\begin{proposition}\cite{NT}
Each $X$-function $X_{i;t}$ is a Laurent polynomial in $\mathbb Z[{\bf x}_{t_0}^{\pm1},{\bf y},{\bf z}]$.
\end{proposition}
The {\bf $F$-polynomial} $F_{i;t}$ of $x_{i;t}$ is defined by
$F_{i;t}=X_{i;t}|_{x_{1;t_0}=\cdots=x_{n;t_0}=1}\in\mathbb Z[{\bf y},{\bf z}]$.
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$, we introduce a $\mathbb Z^n$-grading on $\mathbb Z[{\bf x}_{t_0}^{\pm1},{\bf y},{\bf z}]$ as follows:
$$deg(x_{i;t_0})={\bf e}_i,~~deg(y_i)=-{\bf b}_i,~~deg(z_{i;s})=0,$$
where ${\bf e}_i$ is the $i$-th column vector of $I_n$, and ${\bf b}_i$ is the $i$-th column vector of $B_{t_0}$.
\begin{proposition}\cite{NT}
Each $X$-function $X_{i;t}$ is homogeneous with respect to the $\mathbb Z^n$-grading on $\mathbb Z[{\bf x}_{t_0}^{\pm1},{\bf y},{\bf z}]$.
\end{proposition}
Keep the above notations. The vector $g(x_{i;t}):=deg(X_{i;t})\in\mathbb Z^n$ is called the {\bf $g$-vector} of $x_{i;t}$ and the matrix
$$G_t=(g(x_{1;t}),\cdots,g(x_{n;t}))$$
is called the {\bf $G$-matrix} of ${\bf x}_t$.
\begin{proposition-definition}\cite{NT}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$. Then each $y_{i;t}$ is a Laurent monomial of ${\bf y}$ with coefficient $1$, namely, $y_{i;t}$ has the form of
$$y_{i;t}=\prod\limits_{j=1}^ny_j^{c_{ji}^t}.$$
The resulting vector ${\bf c}_{i;t}=(c_{1i}^t,\cdots,c_{ni}^t)^{\rm T}$ is called a {\bf $c$-vector} and the matrix $C_t=({\bf c}_{1;t},\cdots,{\bf c}_{n;t})$ is called a {\bf $C$-matrix}.
\end{proposition-definition}
\begin{proposition}\cite{NT,CL}\label{procg}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$, and $S$ be a skew-symmetrizer of $RB_{t_0}$, then
$$SRC_tR^{-1}S^{-1}G_t^{\rm T}=I_n.$$
\end{proposition}
\begin{theorem}\label{thmnt}
\cite[Theorem 3.22 and 3.23]{NT} Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with coefficients semifield $\mathbb P$ and initial seed at $t_0$. Then
\begin{eqnarray}
y_{i;t}&=&\prod\limits_{j=1}^ny_{j;t_0}^{c_{ji}^t}\prod\limits_{j=1}^n\left(F_{j;t}|_{\mathbb P}({\bf y}_{t_0},{\bf z})\right)^{b_{ji}^t},\nonumber\\
x_{i;t}&=&\left(\prod\limits_{j=1}^nx_{j;t_0}^{g_{ji}^t}\right)\frac{F_{i;t}|_{\mathcal F}(\hat {\bf y}_{t_0},{\bf z})}{F_{i;t}|_{\mathbb P}({\bf y}_{t_0},{\bf z})}.\nonumber
\end{eqnarray}
\end{theorem}
\section{Main results}
In this section, we give our main results.
\begin{lemma}\cite[Lemma 4.20]{CL}\label{lemCL}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$, and $D_t^{t_0}=({\bf d}_{1;t}^{t_0},\cdots,{\bf d}_{n;t}^{t_0})$ be the $D$-matrix of ${\bf x}_t$ with respect to ${\bf x}_{t_0}$. If there exists a permutation $\sigma$ of $\{1,\cdots,n\}$ such that
${\bf d}_{j;t}^{t_0}={\bf d}_{\sigma(j);t_0}^{t_0}$ for $j=1,\cdots,n$, then $x_{j;t}=x_{\sigma(j);t_0}$ holds for $j=1,\cdots,n$.
\end{lemma}
\begin{proposition}\label{prokey}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, $(B_{t_0}$,${\bf x}_{t_0},{\bf y}_{t_0})$ and $(B_t,{\bf x}_t,{\bf y}_t)$ be two seeds of $\mathcal A(\mathcal S)$. Let $D_t^{t_0}=({\bf d}_{1;t}^{t_0},\cdots,{\bf d}_{n;t}^{t_0})$ be the $D$-matrix of ${\bf x}_t$ with respect to ${\bf x}_{t_0}$, and $S$ be a skew-symmetrizer of $RB_{t_0}$. If there exists a permutation $\sigma$ of $\{1,\cdots,n\}$ such that ${\bf d}_{j;t}^{t_0}={\bf d}_{\sigma(j);t_0}^{t_0}$ for any $j=1,\cdots,n$. Then for each $k\in\{1,\cdots,n\}$, we have
(i) $r_k=r_{\sigma(k)},\;s_k=s_{\sigma(k)}$ and $z_{k,s}=z_{\sigma(k),s}$, where $s=1,\cdots,r_k-1$. In particular, the mutation polynomials $Z_k$ and $Z_{\sigma(k)}$ are equal;
(ii) $x_{k;t}=x_{\sigma(k);t_0},\; y_{k;t}=y_{\sigma(k);t_0}$ and $b_{ik}^t=b_{\sigma(i)\sigma(k)}^{t_0}$,
where $i=1,\cdots,n$.
\end{proposition}
\begin{proof}
Let $\mathcal A(\mathcal S^{pr})$ be an $(R,{\bf z})$-cluster algebra with principal coefficients at the seed $(B_{t_0}^{pr},{\bf x}_{t_0}^{pr},{\bf y}_{t_0}^{pr})$ satisfying $B_{t_0}^{pr}=B_{t_0}$. Let $(D_t^{t_0})^{pr}$ be the $D$-matrix of ${\bf x}_t^{pr}$ with respect to ${\bf x}_{t_0}^{pr}$.
By Corollary \ref{cordmat}, we know that $(D_t^{t_0})^{pr}=D_t^{t_0}$.
Since ${\bf d}_{j;t}^{t_0}={\bf d}_{\sigma(j);t_0}^{t_0}$ holds for any $j=1,\cdots,n$, we know that $({\bf d}_{j;t}^{t_0})^{pr}=({\bf d}_{\sigma(j);t_0}^{t_0})^{pr}$ holds for $j=1,\cdots,n$. If we can prove the results in (i), (ii) hold for $\mathcal A(\mathcal S^{pr})$, then by Theorem \ref{thmnt}, we can get that they also hold for $\mathcal A(\mathcal S)$. Thanks to this, we can safely assume that $\mathcal A(\mathcal S)$ itself is an $(R,{\bf z})$-cluster algebra with principal coefficients at $t_0$.
By Lemma \ref{lemCL}, we get that $$x_{j;t}=x_{\sigma(j);t_0}$$ holds for $j=1,\cdots,n$. So the $G$-matrix and the $H$-matrix of ${\bf x}_t$ are given by
$$G_t=({\bf e}_{\sigma(1)},\cdots,{\bf e}_{\sigma (n)})=H_{t_0}^t,$$ where ${\bf e}_i$ is the $i$-th column vector of $I_n$.
By Proposition \ref{procg}, we can get the $C$-matrix of $(B_t,{\bf x}_t,{\bf y}_t)$ is given by
\begin{eqnarray}\label{eqncmat}
\hspace{14mm}C_t=R^{-1}S^{-1}(G_t^{\rm T})^{-1}SR=(c_{ij}^t),\;\;\text{where }\;c_{ij}^t=\begin{cases}\frac{r_j}{r_{\sigma(j)}}\cdot \frac{s_j}{s_{\sigma(j)}},&i=\sigma(j);\\0,&i\neq \sigma(j).\end{cases}
\end{eqnarray}
By the cluster formula, we know that $H_{t_0}^t(B_tR^{-1}S^{-1})(H_{t_0}^t)^{\rm T}=B_{t_0}R^{-1}S^{-1}$. By comparing the $(\sigma(i),\sigma(k))$-entry of both sides, we get $b_{ik}^tr_k^{-1}s_k^{-1}=b_{\sigma(i)\sigma(k)}^{t_0}r_{\sigma(k)}^{-1}s_{\sigma(k)}^{-1}$, i.e., we have
\begin{eqnarray}\label{eqn0}
b_{ik}^t=b_{\sigma(i)\sigma(k)}^{t_0}\cdot \frac{r_k}{r_{\sigma(k)}}\cdot \frac{s_k}{s_{\sigma(k)}}.
\end{eqnarray}
We write ${\bf x}_{t}=(x_1,\cdots,x_n)$, then we know that $x_{\sigma(j);t_0}=x_{j;t}=x_j$, where $j=1,\cdots,n$.
Now we fix a $k\in \{1,\cdots,n\}$, let $t^{~\underline{\quad k \quad}}~ t^{\prime}$ and $t_0^{~\underline{\quad \sigma(k) \quad}}~ t_1$ be the subgraph of $\mathbb T_n$.
By the definition of $(R,{\bf z})$-seed mutation, we have the following equalities.
\begin{eqnarray}
\label{eqn1}
x_{k;t^\prime}x_{k;t}&=&\left(\prod\limits_{i=1}^nx_{i;t}^{[-b_{ik}^t]_+}\right)^{r_k}\frac{Z_k(\hat y_{k;t})}{Z_k|_{\mathbb P}(y_{k;t})};\\
\label{eqn2} x_{\sigma(k);t_1}x_{\sigma(k);t_0}&=&\left(\prod\limits_{i=1}^nx_{i;t_0}^{[-b_{i\sigma(k)}^{t_0}]_+}\right)^{r_{\sigma(k)}}\frac{Z_{\sigma(k)}(\hat y_{\sigma(k);t_0})}{Z_{\sigma(k)}|_{\mathbb P}(y_{\sigma(k);t_0})},
\end{eqnarray}
where $Z_k$ and $Z_{\sigma(k)}$ are the corresponding mutation polynomials. Denote by $$U_{k;t}=y_{k;t}\prod\limits_{i=1}^nx_{i;t}^{[b_{ik}^t]_+},\;V_{k;t}=\prod\limits_{i=1}^nx_{i;t}^{[-b_{ik}^t]_+},$$ then we know that
\begin{eqnarray}
P&:=&\prod\limits_{i=1}^n(x_{i;t}^{[-b_{ik}^t]_+})^{r_k}Z_k(\hat y_{k;t})=V_{k;t}^{r_k}+z_{k,1}V_{k;t}^{r_k-1}U_{k;t}+\cdots+z_{k;r_k-1}V_{k;t}U_{k;t}^{r_k-1}+U_{k;t}^{r_k},\nonumber\\
Q&:=&\prod\limits_{i=1}^n(x_{i;t_0}^{[-b_{i\sigma(k)}^{t_0}]_+})^{r_{\sigma(k)}}Z_{\sigma(k)}(\hat y_{\sigma(k);t_0})=V_{\sigma(k);t_0}^{r_{\sigma(k)}}+z_{\sigma(k),1}V_{\sigma(k);t_0}^{r_{\sigma(k)}-1}U_{\sigma(k);t_0}+\cdots+
U_{\sigma(k);t_0}^{r_{\sigma(k)}}.\nonumber
\end{eqnarray}
Note that both $P$ and $Q$ are polynomials in $\mathbb {ZP}[x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n]$.
Now the equalities (\ref{eqn1}) and (\ref{eqn2}) can be expressed as follows:
\begin{eqnarray}
x_{k;t^\prime}x_k&=&\frac{P}{Z_k|_{\mathbb P}(y_{k;t})};\nonumber\\
x_{\sigma(k);t_1}x_k&=&\frac{Q}{Z_{\sigma(k)}|_{\mathbb P}(y_{\sigma(k);t_0})},\nonumber
\end{eqnarray}
We can get that
\begin{eqnarray}
\label{eqn3}
x_{k;t^\prime}&=&\frac{Z_{\sigma(k)}|_{\mathbb P}(y_{\sigma(k);t_0})}{Z_k|_{\mathbb P}(y_{k;t})}\cdot\frac{P}{Q}\cdot x_{\sigma(k);t_1},\\
\label{eqn4}
x_{\sigma(k);t_1}&=&\frac{Z_k|_{\mathbb P}(y_{k;t})}{Z_{\sigma(k)}|_{\mathbb P}(y_{\sigma(k);t_0})}\cdot\frac{Q}{P}\cdot x_{k;t^\prime}.
\end{eqnarray}
The equality (\ref{eqn3}) is the expansion of $x_{k;t^\prime}$ with respect to ${\bf x}_{t_1}$ and the equality (\ref{eqn4}) is the expansion of
$x_{\sigma(k);t_1}$ with respect to ${\bf x}_{t^\prime}$. By the Laurent phenomenon, we can get both $\frac{P}{Q}$ and $\frac{Q}{P}$ are Laurent polynomial in $\mathbb{ZP}[x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n]$. Thus we get $\frac{P}{Q}$ is a Laurent monomial in $\mathbb{ZP}[x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n]$.
Since both $P$ and $Q$ can not be divided by any $x_j$. We can get $\frac{P}{Q}=1$, i.e., we have $$P=Q.$$
In the following proof, we divide two cases. Case (a): the $k$-th column vector of $B_t$ is a zero vector; Case (b): the $k$-th column vector of $B_t$ is a nonzero vector.
Case (a). In this case, the $k$-th column vector of $B_t$ is a zero vector. So the rank $1$ generalized cluster algebra generated by $x_{k;t}=x_k$ would split off from $\mathcal A(\mathcal S)$. In this case, $x_k=x_{\sigma(k)}$ actually implies that $k=\sigma(k)$. So $r_k=r_{\sigma(k)}, s_k=s_{\sigma(k)}$ and $z_{k,s}=z_{\sigma(k),s}$ hold. By the equality (\ref{eqncmat}), we know that the $k$-th column vector of $C_t$ is ${\bf e}_{\sigma(k)}={\bf e}_{k}$. So ${ y}_{k;t}=y_k=y_{k;t_0}=y_{\sigma(k);t_0}$.
Since the seed at $t_0$ is obtained from the seed at $t$ by a sequence of $(R,{\bf z})$-seed mutations, we get the $\sigma(k)=k$-th column vector of $B_{t_0}$ is also a zero vector. In particular, $b_{ik}^t=b_{\sigma(i)\sigma(k)}^{t_0}$ holds for $i=1,\cdots,n$.
Case (b). In this case, the $k$-th column vector of $B_t$ is a nonzero vector. So there exists $i_0$ such that $b_{i_0k}^t\neq 0$. Without loss of generality, we can assume that $b_{i_0k}^t>0$. By the equality (\ref{eqn0}), we also have $b_{\sigma(i_0)\sigma(k)}^{t_0}>0$. We just view $P$ and $Q$ as polynomials in $x_{i_0;t}=x_{i_0}$. We know that $P$ is a sum of $r_k+1$ distinct monomials, while $Q$ is a sum of $r_{\sigma(k)}+1$ distinct monomials. By $P=Q$, we can get $$r_k=r_{\sigma(k)}.$$
The highest exponent of $x_{i_0;t}=x_{i_0}$ in $P$ is $b_{i_0k}^tr_k$, while the highest exponent of $x_{\sigma(i_0);t_0}=x_{i_0;t}=x_{i_0}$ in $Q$
is $b_{\sigma(i_0)\sigma(k)}^{t_0}r_{\sigma(k)}$. By $P=Q$, we get $b_{i_0k}^tr_k=b_{\sigma(i_0)\sigma(k)}^{t_0}r_{\sigma(k)}$, i.e., we have
$$b_{i_0k}^t=b_{\sigma(i_0)\sigma(k)}^{t_0}\cdot\frac{r_{\sigma(k)}}{r_k}=b_{\sigma(i_0)\sigma(k)}^{t_0}>0.$$
By the equality (\ref{eqn0}), we know that $$b_{i_0k}^t=b_{\sigma(i_0)\sigma(k)}^{t_0}\cdot \frac{r_k}{r_{\sigma(k)}}\cdot \frac{s_k}{s_{\sigma(k)}}=b_{\sigma(i_0)\sigma(k)}^{t_0}\cdot \frac{s_k}{s_{\sigma(k)}}.$$
By comparing the above two equalities, we get $\frac{s_{\sigma(k)}}{s_k}=1$, i.e., $s_k=s_{\sigma(k)}$.
By $r_k=r_{\sigma(k)},\;s_k=s_{\sigma(k)}$ and the equality (\ref{eqn0}), we get that $b_{ik}^t=b_{\sigma(i)\sigma(k)}^{t_0}$ holds for $i=1,\cdots,n$.
By the equality (\ref{eqncmat}), we know that the $k$-th column vector is ${\bf e}_{\sigma(k)}={\bf e}_{k}$. So ${ y}_{k;t}=y_k=y_{k;t_0}=y_{\sigma(k);t_0}$.
By $r_k=r_{\sigma(k)},\;y_{k;t}=y_{\sigma(k);t_0},\;b_{ik}^t=b_{\sigma(i)\sigma(k)}^{t_0}$ for any $i$, and by comparing the coefficients before the monomials in $P$ and $Q$, we can get $z_{k,s}=z_{\sigma(k),s}$, where $s=1,\cdots,r_k-1$. This completes the proof.
\end{proof}
\begin{theorem}\cite[Theorem 6.3]{CL1}\label{thmdposi} Let $\mathcal A(\mathcal S)$ be an $(I_n,\phi)$-cluster algebra with initial seed at $t_0$, and ${\bf d}_{i;t}^{t_0}= (d_1,\cdots,d_n)^{\rm T}$ be the $d$-vector of the cluster variable $x_{i;t}$ with respect to the cluster ${\bf x}_{t_0}$ of $\mathcal A(\mathcal S)$. Then for each $k\in\{1,\cdots,n\}$,
(i) $d_k$ depends only on $x_{i;t}$ and $x_{k;t_0}$, not on the clusters containing $x_{k;t_0}$;
(ii) $d_k\geq -1$ for $k=1,\cdots,n$, and in details,
\begin{eqnarray}
d_k=\begin{cases} -1~,& \text{iff}\;\; x_{i,t}=x_{k,t_0};\\ 0~,& \text{iff}\;\;x_{i,t}\not=x_{k,t_0}\;\;\text{and}\;\; x_{i,t}, x_{k,t_0}\in {\bf x}_{t^\prime} \; \text{for some}\; t^\prime;\\ \text{a positive integer}~, & \text{iff}\;\; \text{there exists no cluster } {\bf x}_{t^\prime} \text{ containing both }x_{i,t} \text{ and }x_{k,t_0} .\end{cases}\nonumber
\end{eqnarray}
In particular, if $x_{i;t}\notin{\bf x}_{t_0}$, then ${\bf d}_{i;t}^{t_0}$ is a nonnegative vector.
\end{theorem}
\begin{theorem}\cite{CL1}\label{thmconnect}
Let $\mathcal A(\mathcal S)$ be an $(I_n,\phi)$-cluster algebra (i.e. classic cluster algebra), then the seeds
whose clusters contain particular cluster variables form a connected subgraph of the exchange graph
of $\mathcal A(\mathcal S)$.
\end{theorem}
\begin{remark}
Note that the statements in Theorem \ref{thmdposi} and Theorem \ref{thmconnect} come from \cite[Conjecture 7.4]{FZ3} and \cite[Conjecture 4.14(3)]{FZ2} respectively.
\end{remark}
Let $I$ be a subset of $\{1,\cdots,n\}$. We say that $(k_1,\cdots,k_s)$ is an {\bf $I$-sequence}, if $k_j\in I$ for $j=1,\cdots,s$.
\begin{theorem}\label{thmgraph}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, then the seeds
whose clusters contain particular cluster variables form a connected subgraph of the exchange graph ${\bf EG}(\mathcal A(\mathcal S))$
of $\mathcal A(\mathcal S)$.
\end{theorem}
\begin{proof}
For any $t_0\in \mathbb T_n$ and any subset $J$ of the cluster variables in ${\bf x}_{t_0}$, we consider the seeds of $\mathcal A(\mathcal S)$
whose clusters contain $J$. We need to check that if these seeds form a connected
subgraph of ${\bf EG}(\mathcal A(\mathcal S))$. Without loss of generality, we can assume that $J=\{x_{p+1;t_0},\cdots,x_{n;t_0}\}$.
Let $(B_{t},{\bf x}_{t},{\bf y}_{t})$ be a seed of $\mathcal A(\mathcal S)$ and the cluster ${\bf x}_{t}$ contains the cluster variables $x_{p+1;t_0},\cdots,x_{n;t_0}$. It suffices to find a vertex $u\in\mathbb T_n$ such that $u$ is connected with $t_0$ by a $\{1,\cdots,p\}$-sequence on $\mathbb T_n$ and the seed $(B_u,{\bf x}_u,{\bf y}_u)$ is equivalent to the seed $(B_{t},{\bf x}_{t},{\bf y}_{t})$.
Let $\mathcal A(\mathcal S^\circ)$ be an $(I_n,\phi)$-cluster algebra (i.e. classic cluster algebra) with initial seed $(B_{t_0}^\circ,{\bf x}_{t_0}^\circ,{\bf y}_{t_0}^\circ)$, where $B_{t_0}^\circ=B_{t_0}R$.
For any $v,w\in\mathbb T_n$, let $D_v^{w}=({\bf d}_{1;v}^{w},\cdots,{\bf d}_{n;v}^{w})$ be the $D$-matrix of ${\bf x}_v$ with respect to ${\bf x}_{w}$, and $(D_v^{w})^\circ$ be the $D$-matrix of ${\bf x}_v^\circ$ with respect to ${\bf x}_{w}^\circ$. By Corollary \ref{cordmat}, we know that $$D_v^{w}=(D_v^{w})^\circ.$$
Since the cluster ${\bf x}_{t}$ contains the cluster variables $x_{p+1;t_0},\cdots,x_{n;t_0}$, we know that the $D$-matrix $D_t^{t_0}=(D_t^{t_0})^\circ$ contains the $d$-vectors ${\bf d}_{p+1;t_0}^{t_0},\cdots,{\bf d}_{n;t_0}^{t_0}$. Note that ${\bf d}_{i;t_0}^{t_0}=-{\bf e}_i$ which is a non-positive vector. Then by Theorem \ref{thmdposi}, we know that the cluster ${\bf x}_t^\circ$ contains the cluster variables $x_{p+1;t_0}^\circ,\cdots,x_{n;t_0}^\circ$. Then by Theorem \ref{thmconnect}, there exists a vertex $u$ of $\mathbb T_n$ satisfying that
(a) $u$ is connected with $t_0$ by an $\{1,\cdots,p\}$-sequence on $\mathbb T_n$, i.e., we have
$$t_0^{~\underline{ \quad k_1\quad }}~ t_1^{~\underline{\quad k_2 \quad}} ~t_2^{~\underline{\quad k_3 \quad}} ~\cdots ~t_{s-1} ^{~\underline{~\quad k_{s} \quad}}~ t_s=u,$$
where $k_j\leq p$ for $j=1,\cdots,s$;
(b) the two seeds $(B_u^\circ,{\bf x}_u^\circ,{\bf y}_u^\circ)$ and $(B_{t}^\circ,{\bf x}_{t}^\circ,{\bf y}_{t}^\circ)$ are equivalent via a permutation $\sigma$. In particular, $x_{j;t}^\circ=x_{\sigma(j);u}^\circ$ for $j=1,\cdots,n$.
By (b), we know that $({\bf d}_{j;t}^u)^\circ=({\bf d}_{\sigma(j);u}^u)^\circ$ for $j=1,\cdots,n$. By $D_t^u=(D_t^u)^\circ$, we get that
$${\bf d}_{j;t}^u={\bf d}_{\sigma(j);u}^u, \;\text{where }j=1,\cdots,n.$$
Then by Proposition \ref{prokey}, we get that the seeds $(B_u,{\bf x}_u,{\bf y}_u)$ and $(B_{t},{\bf x}_{t},{\bf y}_{t})$ are equivalent via the permutation $\sigma$.
On the other hand, we know that each seed of $\mathcal A(\mathcal S)$ appearing in the following subgraph
$$t_0^{~\underline{ \quad k_1\quad }}~ t_1^{~\underline{\quad k_2 \quad}} ~t_2^{~\underline{\quad k_3 \quad}} ~\cdots ~t_{s-1} ^{~\underline{~\quad k_{s} \quad}}~ t_s=u$$
contains the cluster variables $x_{p+1,t_0},\cdots,x_{n;t_0}$, by $k_j\leq p$ for $j=1,\cdots,s$. So the seeds of $\mathcal A(\mathcal S)$
whose clusters contain the cluster variables $x_{p+1,t_0},\cdots,x_{n;t_0}$ form a connected subgraph of the exchange graph ${\bf EG}(\mathcal A(\mathcal S))$
of $\mathcal A(\mathcal S)$.
\end{proof}
\begin{theorem}\label{promain}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed $(B_{t_0},{\bf x}_{t_0},{\bf y}_{t_0})$, and $\mathcal A(\overline{\mathcal S})$ be an $(\overline R,\overline{\bf z})$-cluster algebra with initial seed $(\overline B_{t_0},\overline{\bf x}_{t_0},\overline{\bf y}_{t_0})$. Let $\mathcal X(\mathcal S)$ be the set of cluster variables of $\mathcal A(\mathcal S)$, and $\mathcal X(\overline{\mathcal S})$ be the set of cluster variables of $\mathcal A(\overline{\mathcal S})$. If $B_{t_0}R=\overline B_{t_0}\overline R$, then the following statements hold.
(i) For any $t_1,t\in\mathbb T_n$ and any $i_{0}, j_0\in\{1,\cdots,n\}$, $x_{i_0;t_1}=x_{j_0;t}$ if and only if $\overline x_{i_0;t_1}=\overline x_{j_0;t}$, where $x_{i_0;t_1},x_{j_0;t}$ are cluster variables of $\mathcal A(\mathcal S)$ and $\overline x_{i_0;t_1}, \overline x_{j_0;t}$ are the corresponding cluster variables of $\mathcal A(\overline{\mathcal S})$.
(ii) There exists a bijection $\alpha:\mathcal X(\mathcal S)\rightarrow \mathcal X(\overline{\mathcal S})$ given by
$\alpha(x_{i;t})=\overline x_{i;t}$, which induces a bijection from the set of clusters of $\mathcal A(\mathcal S)$ to the set of clusters of $\mathcal A(\overline{\mathcal S})$.
\end{theorem}
\begin{proof} (i) The proof is similar to that of Theorem \ref{thmgraph}.
For any $v,w\in\mathbb T_n$, let $D_v^{w}=({\bf d}_{1;v}^{w},\cdots,{\bf d}_{n;v}^{w})$ be the $D$-matrix of ${\bf x}_v$ with respect to ${\bf x}_{w}$, and
$\overline D_v^{w}$ be the $D$-matrix of $\overline {\bf x}_v$ with respect to $\overline {\bf x}_{w}$. By Corollary \ref{cordmat}, we know that $$D_v^{w}=\overline D_v^{w}.$$
If $x_{i_0;t_1}=x_{j_0;t}$, then by Theorem \ref{thmgraph}, there exists a vertex $u$ of $\mathbb T_n$ satisfying that
(a) $u$ is connected with $t_1$ by an $\{1,\cdots,i_0-1,i_0+1,\cdots,n\}$-sequence on $\mathbb T_n$, i.e., we have
$$t_1^{~\underline{ \quad k_1\quad }}~ t_2^{~\underline{\quad k_2 \quad}} ~t_3^{~\underline{\quad k_3 \quad}} ~\cdots ~t_{s-1} ^{~\underline{~\quad k_{s-1} \quad}}~ t_s=u,$$
where $k_j\neq i_0$ for $j=1,\cdots,s-1$;
(b) the two seeds $(B_u,{\bf x}_u,{\bf y}_u)$ and $(B_{t},{\bf x}_{t},{\bf y}_{t})$ are equivalent via a permutation $\sigma$. In particular, $x_{j;t}=x_{\sigma(j);u}$ for $j=1,\cdots,n$.
Since $k_j\neq i_0$ for $j=1,\cdots,s-1$, we can get $$x_{i_0;t_1}=x_{i_0;u}\;\;\text{and } \;\overline x_{i_0;t_1}=\overline x_{i_0;u}.$$
By the facts $x_{j_0;t}=x_{i_0;t_1}=x_{i_0;u}$ and $x_{j;t}=x_{\sigma(j);u}$, where $j=1,\cdots,n$, we get that $i_0=\sigma(j_0)$.
By (b), we know that ${\bf d}_{j;t}^u={\bf d}_{\sigma(j);u}^u$ for $j=1,\cdots,n$. Since $D_t^u=\overline D_t^u$, we get
$$\overline{\bf d}_{j;t}^u=\overline {\bf d}_{\sigma(j);u}^u, \;\text{where }j=1,\cdots,n.$$
Then by Proposition \ref{prokey}, we know that $\overline x_{j_0;t}=\overline x_{\sigma(j_0);u}=\overline x_{i_0;u}$. By $\overline x_{i_0;t_1}=\overline x_{i_0;u}$, we get $\overline x_{j_0;t_1}=\overline x_{i_0;t}$
Similarly, if $\overline x_{j_0;t_1}=\overline x_{i_0;t}$, we can show that $x_{i_0;t_1}=x_{j_0;t}$.
(ii) follows from (i).
\end{proof}
\section{Applications of Theorem \ref{promain}}
In this section, we give two applications of of Theorem \ref{promain}. To be more precisely, we prove some properties of the components of the $d$-vectors in Theorem \ref{thmdvec} and we give a characterization for the clusters of a generalized cluster algebra in Theorem \ref{thmlast}.
\begin{theorem}\label{thmdvec}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed at $t_0$, and ${\bf d}_{i;t}^{t_0}= (d_1,\cdots,d_n)^{\rm T}$ be the $d$-vector of the cluster variable $x_{i;t}$ with respect to the cluster ${\bf x}_{t_0}$ of $\mathcal A(\mathcal S)$. Then for each $k\in\{1,\cdots,n\}$,
(i) $d_k$ depends only on $x_{i;t}$ and $x_{k;t_0}$, not on the clusters containing $x_{k;t_0}$;
(ii) $d_k\geq -1$ for $k=1,\cdots,n$, and in details,
\begin{eqnarray}
d_k=\begin{cases} -1~,& \text{iff}\;\; x_{i,t}=x_{k,t_0};\\ 0~,& \text{iff}\;\;x_{i,t}\not=x_{k,t_0}\;\;\text{and}\;\; x_{i,t}, x_{k,t_0}\in {\bf x}_{t^\prime} \; \text{for some}\; t^\prime;\\ \text{a positive integer}~, & \text{iff}\;\; \text{there exists no cluster } {\bf x}_{t^\prime} \text{ containing both }x_{i,t} \text{ and }x_{k,t_0} .\end{cases}\nonumber
\end{eqnarray}
In particular, if $x_{i;t}\notin{\bf x}_{t_0}$, then ${\bf d}_{i;t}^{t_0}$ is a nonnegative vector.
\end{theorem}
\begin{proof}
Let $\mathcal A(\mathcal S^\circ)$ be an $(I_n,\phi)$-cluster algebra (i.e. classic cluster algebra) with initial seed $(B_{t_0}^\circ,{\bf x}_{t_0}^\circ,{\bf y}_{t_0}^\circ)$, where $B_{t_0}^\circ=B_{t_0}R$. By Corollary \ref{cordmat}, we know that ${\bf d}_{i;t}=({\bf d}_{i;t}^{t_0})^\circ$ is also the $d$-vector of $x_{i;t}^\circ$ with respect to ${\bf x}_{t_0}$.
(i) By Theorem \ref{thmdposi} (i), $d_k$ depends only on $x_{i;t}^\circ$ and $x_{k;t_0}^\circ$, not on the clusters containing $x_{k;t_0}^\circ$. Then by Theorem \ref{promain} (ii), we can get $d_k$ depends only on $x_{i;t}$ and $x_{k;t_0}$, not on the clusters containing $x_{k;t_0}$.
(ii) By Theorem \ref{thmdposi} (ii), we know that $d_k\geq -1$ for $k=1,\cdots,n$, and in details,
\begin{eqnarray}
d_k=\begin{cases} -1~,& \text{iff}\;\; x_{i,t}^\circ=x_{k,t_0}^\circ;\\ 0~,& \text{iff}\;\;x_{i,t}^\circ\not=x_{k,t_0}^\circ\;\;\text{and}\;\; x_{i,t}^\circ, x_{k,t_0}^\circ\in {\bf x}_{t^\prime}^\circ \; \text{for some}\; t^\prime;\\ \text{a positive integer}~, & \text{iff}\;\; \text{there exists no cluster } {\bf x}_{t^\prime}^\circ \text{ containing both }x_{i,t}^\circ \text{ and }x_{k,t_0}^\circ .\end{cases}\nonumber
\end{eqnarray}
Then by Theorem \ref{promain} (ii), we can get
\begin{eqnarray}
d_k=\begin{cases} -1~,& \text{iff}\;\; x_{i,t}=x_{k,t_0};\\ 0~,& \text{iff}\;\;x_{i,t}\not=x_{k,t_0}\;\;\text{and}\;\; x_{i,t}, x_{k,t_0}\in {\bf x}_{t^\prime} \; \text{for some}\; t^\prime;\\ \text{a positive integer}~, & \text{iff}\;\; \text{there exists no cluster } {\bf x}_{t^\prime} \text{ containing both }x_{i,t} \text{ and }x_{k,t_0} .\end{cases}\nonumber
\end{eqnarray}
\end{proof}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra, and $\mathcal X(\mathcal S)$ be the set of cluster variables of $\mathcal A(\mathcal S)$. Two cluster variables $x$ and $w$ are said to be {\bf compatible}, if there exists a cluster ${\bf x}_t$ of $\mathcal A(\mathcal S)$ containing both $x$ and $w$. A subset $M\subseteq \mathcal X(\mathcal S)$ is called a {\bf compatible set} of $\mathcal A(\mathcal S)$, if $x$ and $w$ are compatible for any $x,w\in M$.
The theorem is a conjecture in \cite[Conjecture 5.5]{FST} by Fomin, Shapiro and Thurston, which has been proved by the authors in \cite{CL1} for classic cluster algebras.
\begin{lemma}\cite[Theorem 7.4]{CL1}\label{lemcompatible}
Let $\mathcal A(\mathcal S)$ be an $(I_n,\phi)$-cluster algebra (i.e. classic cluster algebra), and $\mathcal X(\mathcal S)$ be the set of cluster variables of $\mathcal A(\mathcal S)$. Then
(i) a subset $M\subseteq \mathcal X(\mathcal S)$ is a compatible set of $\mathcal A(\mathcal S)$ if and only if $M$ is a subset of some cluster (as a set) of $\mathcal A(\mathcal S)$;
(ii) a subset $M\subseteq \mathcal X(\mathcal S)$ is a maximal compatible set of $\mathcal A(\mathcal S)$ if and only if $M$ is a cluster (as a set) of $\mathcal A(\mathcal S)$.
\end{lemma}
\begin{theorem}\label{thmlast}
Let $\mathcal A(\mathcal S)$ be an $(R,{\bf z})$-cluster algebra with initial seed at $t_0$, and $\mathcal X(\mathcal S)$ be the set of cluster variables of $\mathcal A(\mathcal S)$. Then
(i) a subset $M\subseteq \mathcal X(\mathcal S)$ is a compatible set of $\mathcal A(\mathcal S)$ if and only if $M$ is a subset of some cluster (as a set) of $\mathcal A(\mathcal S)$;
(ii) a subset $M\subseteq \mathcal X(\mathcal S)$ is a maximal compatible set of $\mathcal A(\mathcal S)$ if and only if $M$ is a cluster (as a set) of $\mathcal A(\mathcal S)$.
\end{theorem}
\begin{proof}
(i) Let $\mathcal A(\mathcal S^\circ)$ be an $(I_n,\phi)$-cluster algebra (i.e. classic cluster algebra) with initial seed $(B_{t_0}^\circ,{\bf x}_{t_0}^\circ,{\bf y}_{t_0}^\circ)$, where $B_{t_0}^\circ=B_{t_0}R$. Let $\mathcal X(\mathcal S^\circ)$ be the set of cluster variables of $\mathcal A(\mathcal S^\circ)$.
Let $\alpha:\mathcal X(\mathcal S)\rightarrow \mathcal X(\mathcal S^\circ)$ be the bijection given in Theorem \ref{promain} (ii).
By Theorem \ref{promain} (ii), we know that a subset $M\subseteq \mathcal X(\mathcal S)$ is a compatible set of $\mathcal A(\mathcal S)$ if and only if
$M^\circ:=\alpha(M)\subseteq \mathcal X(\mathcal S^\circ)$ is a compatible set of $\mathcal A(\mathcal S^\circ)$. By Lemma \ref{lemcompatible}, $M^\circ$ is a compatible set of $\mathcal A(\mathcal S^\circ)$ if and only if $M^\circ$ is a subset of some cluster ${\bf x}_t^\circ$ (as a set) of $\mathcal A(\mathcal S^\circ)$. By Theorem \ref{promain} (ii), $M^\circ$ is a subset of ${\bf x}_t^\circ$ (as a set) of $\mathcal A(\mathcal S^\circ)$ if and only if $M$ is a subset of ${\bf x}_t$ (as a set) of $\mathcal A(\mathcal S)$. Hence, we obtain that a subset $M\subseteq \mathcal X(\mathcal S)$ is a compatible set of $\mathcal A(\mathcal S)$ if and only if $M$ is a subset of some cluster (as a set) of $\mathcal A(\mathcal S)$.
(ii) follows from (i).
\end{proof}
{\bf Acknowledgements:}\; The authors are grateful to M. Shapiro and M. Gekhtman for inspiring discussions during the ``Cluster Algebras 2019" (June 3-June 21, 2019) at RIMS, Kyoto University.
\end{document} |
\begin{document}
\title{Semi-perfect 1-Factorizations of the Hypercube}
\begin{abstract}
A 1-factorization $\mathcal{M} = \{M_1,M_2,\ldots,M_n\}$ of a graph $G$ is called perfect if the union of any pair of 1-factors $M_i, M_j$ with $i \ne j$ is a Hamilton cycle. It is called $k$-semi-perfect if the union of any pair of 1-factors $M_i, M_j$ with $1 \le i \le k$ and $k+1 \le j \le n$ is a Hamilton cycle.
We consider 1-factorizations of the discrete cube $Q_d$. There is no perfect 1-factorization of $Q_d$, but it was previously shown that there is a 1-semi-perfect 1-factorization of $Q_d$ for all $d$.
Our main result is to prove that there is a $k$-semi-perfect 1-factorization of $Q_d$ for all $k$ and all $d$, except for one possible exception when $k=3$ and $d=6$.
This is, in some sense, best possible.
We conclude with some questions concerning other generalisations of perfect 1-factorizations.
\end{abstract}
Mathematics Subject Classification: 05C70.
Keywords: Factorization; Hypercube; Semi-perfect; Hamilton cycles.
\section{Introduction}
A 1-factorization of a graph $H$ is a partition of the edges of $H$ into disjoint perfect matchings $\{M_1,M_2,\ldots,M_n\}$, also known as 1-factors.
Let $\mathcal{M} = \{M_1,M_2,\ldots,M_n\}$ be such a 1-factorization.
We say that $\mathcal{M}$ is a perfect factorization if every pair $M_i \cup M_j$ with $i,j$ distinct forms a Hamilton cycle.
A 1-factorization $\mathcal{M}$ is called \emph{semi-perfect} if $M_1 \cup M_i$ forms a Hamilton cycle for all $i \ne 1$.
Kotzig \cite{Kotzig} conjectured that the complete graph $K_{2n}$ has a perfect 1-factorization for all $n \ge 2$.
This has long been outstanding and has so far only been shown to hold for $n$ prime and $2n - 1$ prime (independently by Anderson and Nakamura \cite{Anderson, Nakamura}), as well as certain other small values of $n$ (see \cite{Wallis} for references).
The existence or non-existence of perfect or semi-perfect 1-factorizations has been studied for various other families of graphs, in particular for the hypercube $Q_d$, for $d \ge 2$.
The hypercube graph $Q_d$ has vertices the subsets of $\{1,2,\ldots,d\}$ and two vertices joined by an edge if they differ in a single element.
We say a vertex of $Q_d$ is even if the set contains an even number of elements, and odd if not. Note that every edge of $Q_d$ goes from an odd vertex to an even vertex and so $Q_d$ is bipartite with one vertex class of odd vertices and one vertex class of even vertices, each of size $2^{d-1}$.
We say an edge is in direction $i$ if its two endpoints differ in element $i$. This allows us to define some natural 1-factors of $Q_d$, called the \emph{directional matchings}: for each direction $i = 1,\ldots,d$ let $D_i$ be all edges in direction $i$. The collection of all directional matchings is a 1-factorization of $Q_d$, and note that the union of any pair $D_i \cup D_j$, with $i,j$ distinct, is a disjoint union of 4-cycles. Thus any perfect or semi-perfect 1-factorization of $Q_d$ must be in some sense far from this.
Craft \cite{Craft} conjectured that for every integer $d \ge 2$ there is a semi-perfect 1-factorization of $Q_d$. This was proved independently by Gochev and Gotchev \cite{Gochev} and by Kr\'{a}lovi\v{c} and Kr\'{a}lovi\v{c} \cite{Kralovic} in the case where $d$ is odd, and settled for $d$ even by Chitra and Muthusamy \cite{Chitra}.
Gochev and Gotchev in fact went further and defined $\mathcal{M}$ to be $k$-semi-perfect if $M_i \cup M_j$ forms a Hamilton cycle for every $1\le i \le k$ and $k+1 \le j \le d$.
They proved that there is a $k$-semi-perfect factorization of $Q_d$ whenever $k$ and $d$ are both even with $k<d$.
This leads us to wonder how close to a perfect factorization we can get. Is there a k-semi-perfect factorization of $Q_d$ for all $k<d$? Is there a perfect factorization of $Q_d$? If not, what is the maximal number of pairs of 1-factors whose union is a Hamilton cycle?
Let us introduce some definitions.
For a 1-factorization $\mathcal{M}= \{M_1,M_2,\ldots,M_d\}$ of $Q_d$, we define a graph $G[\mathcal{M}]$
with vertices labelled
$M_1, \ldots, M_d$ and an edge between $M_i$ and $M_j$ if $M_i \cup M_j$ is a Hamilton cycle on $H$.
Note that the definitions above can be easily restated using $G[\mathcal{M}]$:
$\mathcal{M}$ is perfect if $G[\mathcal{M}]$ is complete,
$\mathcal{M}$ is semi-perfect if $G[\mathcal{M}]$ contains $K_{1,d-1}$ as a subgraph,
and $\mathcal{M}$ is $k$ semi-perfect if $G[\mathcal{M}]$ contains $K_{k,d-k}$ as a subgraph.
With this new notation, we can rephrase our questions and ask what is the maximal number of edges that $G[\mathcal{M}]$ can contain if $\mathcal{M}$ is a 1-factorization of $Q_d$? Which graphs can $G[\mathcal{M}]$ be isomorphic to?
It is in fact not possible for $G[\mathcal{M}]$ to be complete when $d > 2$ (i.e. $\mathcal{M}$ cannot be perfect). More than this, we can show that $G[\mathcal{M}]$ must be bipartite.
\begin{theorem}[\cite{Laufer}] \label{thm:Bipartite}
Let $H$ be a bipartite graph on two vertex classes each of size $n$, where $n$ is even.
Let $\mathcal{M}$ be a partition of $H$ into perfect matchings. Then $G[\mathcal{M}]$ must be bipartite.
\end{theorem}
A version of Theorem \ref{thm:Bipartite} with the weaker conclusion that $G[\mathcal{M}]$ is not complete has been, according to Bryant, Maenhaut and Wanless \cite{Bryant} proved many times, including by Laufer in 1980 \cite{Laufer}. We re-prove it here for a few reasons, the main one being that we extend the argument slightly to show that $G[\mathcal{M}]$ is bipartite.
The proof also introduces ideas that we will be using later (in Theorem \ref{thm:why3hard}).
In addition, it is hard to find the theorem and its proof in the literature -- in particular, when making the conjecture that there is a semi-perfect 1-factorization of $Q_d$, Craft also asked whether a \emph{perfect} 1-factorization of $Q_d$ could be found. Theorem \ref{thm:Bipartite} is not mentioned in any of the papers that proved Craft's semi-perfect conjecture.
\begin{proof}
Let $X$ and $Y$ be the vertex classes of $H$.
A perfect matching $M$ naturally induces a function ${M: X \rightarrow Y}$, where $(x,M(x))$ is an edge of $M$.
For two perfect matchings $M_i$ and $M_j$, let $\pi_{j,i}$ be the permutation $M_j^{-1}M_i$ on $X$.
Note that $\pi_{i,i} = id$, $\pi_{i,j} = \pi_{j,i}^{-1}$ and
$\pi_{k,j}\pi_{j,i} = \pi_{k,i}$.
Note further that if $M_iM_j$ is an edge of $G[\mathcal{M}]$ then $M_i \cup M_j$ is a Hamilton cycle and so $\pi_{j,i}$ is a cycle of length $n$ on $X$.
Suppose for a contradiction that $G[\mathcal{M}]$ contains a odd cycle and let $M_{i_1}$, $M_{i_2}$, $\ldots$, $M_{i_k}$, $M_{i_1}$ be such a cycle.
The permutations $\pi_{i_2,i_1}, \pi_{i_3,i_2}, \ldots, \pi_{i_k,i_{k-1}}, \pi_{i_1,i_k}$ are all cycles of length $n$. Since $n$ is even, all of these are odd permutations. Now,
\begin{align*}
1 = \sign(\pi_{i_1,i_1}) &= \sign(\pi_{i_1,i_k}\pi_{i_k,i_{k-1}} \pi_{i_{k-1},i_{k-2}} \ldots \pi_{i_3,i_2} \pi_{i_2,i_1}) \\
&= \sign(\pi_{i_1,i_k})\sign(\pi_{i_k,i_{k-1}})\ldots \sign(\pi_{i_3,i_2})\sign(\pi_{i_2,i_1})\\
&= (-1)^k = -1
\end{align*}
We have a contradiction, hence $G[\mathcal{M}]$ contains no odd cycles.
\end{proof}
In the light of Theorem \ref{thm:Bipartite}, the only remaining question is whether for any $k,d$ there is a 1-factorization $\mathcal{M}$ of $Q_d$ such that $G[\mathcal{M}]$ is isomorphic to the complete bipartite graph $K_{k,d-k}$. (Equivalently, whether there is a $k$-semi-perfect 1-factorization of $Q_d$ for every $k$ and $d$, in the language of Gochev and Gotchev.) Section \ref{sec:main} of this paper fully resolves this problem, except for whether $G[\mathcal{M}]$ can be isomorphic to $K_{3,3}$.
We also explain, in section \ref{sec:direction}, why the $K_{3,3}$ case cannot be resolved with our methods. In particular, the 1-factorizations we construct in the proof of the main theorem have a direction respecting property. We show that any 1-factorization $\mathcal{M}$ of $Q_{6}$ satisfying this direction respecting property cannot have $G[\mathcal{M}]$ is isomorphic to $K_{3,3}$.
We finish with some open questions.
\section{Main Theorem}
\label{sec:main}
\begin{theorem} For $k,l \in \mathbb{N}$ not both equal to 3, there is a 1-factorization $\mathcal{M}$ of the hypercube $Q_{k+l}$ such that $G[\mathcal{M}]$ is isomorphic to the complete bipartite graph $K_{k,l}$. \label{thm:main}
\end{theorem}
To prove the theorem, we will use the following result due to Stong, which concerns the symmetric directed hypercube $\overleftrightarrow{Q_d}$, obtained from $Q_d$ by replacing each edge with two directed edges, one in each direction.
\begin{theorem}[\cite{Stong}] \label{thm:Stong}
For $d \ne 3$, the symmetric directed hypercube $\overleftrightarrow{Q_d}$ can be partitioned into $d$ directed Hamilton cycles.
\end{theorem}
Stong's result applies to directed cubes, but the following corollary allows us to use it for undirected cubes.
\begin{corollary}\label{cor:2Stong}
For $d \ne 3$, the cube $Q_d$ can be partitioned into 1-factors $A_1, A_2, \ldots, A_d$ and also partitioned into 1-factors $B_1, B_2, \ldots, B_d$ such that $A_i \cup B_i$ is a Hamilton cycle for all $i = 1,2,\ldots,d$.
\end{corollary}
\begin{proof}
Using Theorem \ref{thm:Stong}, partition $\overleftrightarrow{Q_d}$ into directed Hamilton cycles $H_1,H_2,\ldots,H_d$.
Let $E$ be the even vertices of $\overleftrightarrow{Q_d}$ and $O$ the odd vertices, so that $\overleftrightarrow{Q_d}$ is bipartite with respect to the vertex classes $E$ and $O$. For each $H_i$, we define $A_i$ to be the edges of $H_i$ that go from $E$ to $O$, and $B_i$ to be the edges that go from $O$ to $E$.
Since $H_1,H_2,\ldots,H_d$ partition $\overleftrightarrow{Q_d}$ , every edge from $E$ to $O$ is in a unique $A_i$ and every edge from $O$ to $E$ is in a unique $B_j$.
If we now ignore the directions on the edges, every edge of $Q_d$ is in a unique $A_i$ and a unique $B_j$. It is clear that $A_i$ and $B_i$ are perfect matchings and $A_i \cup B_i$ is a Hamilton cycle by construction.
\end{proof}
Note that we have slightly abused notation in the case $d=1$, since $A_1 = B_1 = Q_1$ and so $A_1 \cup B_1$ is a single edge rather than a cycle. This will not matter in the cases $k \ne 3, l=1$, and we will consider the case $k=3,l=1$ separately.
Corollary \ref{cor:2Stong} together with a theorem of Gochev and Gotchev \cite[Theorem~3.1]{Gochev} is enough to show that it is possible to have $G[\mathcal{M}]$ isomorphic to $K_{k,n-k}$ for all $k\ne 3$ and all even $n-k$. We will improve on their arguments to deal with all but one of the remaining cases.
We will split the theorem for three different cases and prove each separately. Before we do so, let us outline the ideas involved.
We can view the hypercube $Q_{k+l}$ as a $k$-dimensional hypercube whose `vertices' are copies of $Q_l$ (i.e. as the Cartesian product of $Q_k$ and $Q_l$).
Let us formalise this idea:
Label the vertices of $Q_k$ as subsets of $\{1,2,\ldots,k\}$ in the usual way.
For each vertex $u$ of $Q_k$,
we define a different copy of $Q_l$ within $Q_{l+k}$:
let $Q_l^u$ be the induced subgraph of $Q_{k+l}$ on all vertices $w$ where $w \cap \{1,2,\ldots,k\} = u$.
Conversely, we can view $Q_{k+l}$ as a $l$-dimensional hypercube whose `vertices' are copies of $Q_k$.
This time, label the vertices of $Q_l$ as subsets of $\{k+1,k+2,\ldots,k+l\}$ in the natural way.
For each vertex $v$ of $Q_l$,
we define a different copy of $Q_k$ within $Q_{l+k}$:
let $Q_k^v$ be the induced subgraph of $Q_{k+l}$ on all vertices $x$ with $x \cap \{k+1,k+2,\ldots k+l\} = v$.
The most straightforward case of the theorem is when neither $k$ nor $l$ is equal to $3$, proved in Proposition \ref{case:neither3}.
To prove this we use a generalisation of Gochev and Gotchev's construction \cite{Gochev}.
The idea of the proof is as follows:
first, we construct $k$ disjoint matchings that use only edges in directions $1,\ldots k$. The matchings used within the $Q_k^v$s are those obtained from applying Corollary \ref{cor:2Stong} to $Q_k$.
Next we construct $l$ disjoint matchings that use only edges in directions $k+1,\ldots, k+l$. Similarly, the matchings used within the $Q_l^u$s are those obtained from applying Corollary \ref{cor:2Stong} to $Q_l$.
We then prove that taking the union of a matching of the first kind and a matching of the second kind gives a Hamilton cycle.
The second case of the theorem is when $k=3$ and $l$ is not equal to $1$ or $3$, proved in Proposition \ref{case:k=3}. We use a similar construction to the first case, the only difference being that while we can use Corollary \ref{cor:2Stong} on $Q_l$, we cannot apply it to $Q_3$. We will instead take directional matchings on the copies of $Q_3$; it turns out this can be made to work here.
Finally, we are left with two cases: $(k,l) = (3,1)$ and $(k,l) =(3,3)$. The first of these is proved in Proposition \ref{case:k=3,l=1} by means of an explicit example. The case $(k,l) =(3,3)$ is left unsolved. The difficulty of these final two cases is discussed in the section \ref{sec:direction}.
The following useful notation is common to the proofs of propositions \ref{case:neither3} and \ref{case:k=3}.
For a perfect matching $M$ and a vertex $v$, we define $M(v)$ to be the other endpoint of the edge containing $v$ in $M$. (Note that this clashes slightly with our notation in Theorem \ref{thm:Bipartite}: by that notation we are here conflating $M$ and $M^{-1}$.)
\begin{prop} When neither $k$ nor $l$ is equal to 3, there is a 1-factorization $\mathcal{M}$ of the hypercube $Q_{k+l}$ such that $G[\mathcal{M}]$ is isomorphic to the complete bipartite graph $K_{k,l}$.
\label{case:neither3}
\end{prop}
\begin{proof}
Using Corollary \ref{cor:2Stong} partition $Q_k$ into matchings $A_1,A_2,\ldots,A_k$ and matchings $B_1,B_2,\ldots,B_k$ such that $A_i \cup B_i$ is a Hamilton cycle for all $i$.
For $i = 1,2,\ldots,k$ define $M_i$ to be the matching on $Q_{k+l}$ defined by taking the following edges:
$$\begin{cases}
A_i & \text{ on } Q_k^{\emptyset} \\
B_i & \text{ on } Q_k^v \text{ for } v\ne \emptyset
\end{cases}
$$
Note that the $M_i$ are all disjoint, and they only use edges in directions $1,2,\ldots,k$.
Also partition $Q_l$ into matchings $X_1,X_2,\ldots,X_l$ and matchings $Y_1,Y_2,\ldots,Y_l$ such that $X_j \cup Y_j$ is a Hamilton cycle for all $j$.
For $j=1,2,\ldots,l$ define $N_j$ to be the matching on $Q_{k+l}$ defined by taking the following edges:
$$\begin{cases}
X_j & \text{ on } Q_l^u \text{ for } u \text{ even} \\
Y_j & \text{ on } Q_l^u \text{ for } u \text{ odd}
\end{cases}
$$
Another way to think of $N_j$ is as containing edges between copies of $Q_k^v$. From an even vertex in $Q_k^v$ we add an edge to the corresponding vertex in $Q_k^{X_j(v)}$, and from an odd vertex in $Q_k^v$ we add an edge to the corresponding vertex in $Q_k^{Y_j(v)}$.
Note that the $N_j$ are all disjoint, and they only use edges in directions $k+1,k+2,\ldots,k+l$. Thus the matchings $\{M_i\}_{i=1}^k \cup \{N_j\}_{j=1}^l$ are all disjoint and form a 1-factorization of $Q_{k+l}$.
\begin{figure}
\caption{$M_i$}
\caption{$N_j$}
\caption{An example when $k=2$ and $l=4$}
\label{fig:general_k,l}
\end{figure}
All that is left is to show that $M_i \cup N_j$ is a Hamilton cycle for all $i,j$.
Consider following the cycle starting at a vertex $u$ that lies in $Q_k^{\emptyset}$ and alternating between edges first in $N_j$ and then in $M_i$.
Every time we travel along an edge in $M_i$ the parity of the vertex in $Q_k^v$ switches, and so we will alternate using edges from $X_j$ and edges from $Y_j$ in $N_j$.
As $X_j \cup Y_j$ is a Hamilton cycle, the first time the cycle returns to $Q_k^{\emptyset}$ we will have travelled through each other $Q_k^v$ exactly once.
Each time we travel through a different $Q_k^v$ we use an edge from $B_i$ within it.
After passing through $2^l - 1$ copies of $Q_k^v$ we will have bounced between $u$ and $B_i(u)$ an odd number of times, so the first vertex we encounter in our return to $Q_k^{\emptyset}$ is $B_i(u)$.
The next vertex would then be $A_i(B_i(u))$.
After passing through $2(2^l)$ distinct vertices (two in each $Q_k^v$) we have moved from $u$ to $A_i(B_i(u))$, i.e. made two steps of the Hamilton cycle $A_i \cup B_i$ within $Q_k^{\emptyset}$.
Thus the first time we will return to $u \cup \emptyset$ is after passing through $2^k2^l$ vertices, which is the total number of vertices in the graph. Hence we have a Hamilton cycle.
\end{proof}
\begin{prop} For $l$ not equal to $1$ or $3$, there is a 1-factorization $\mathcal{M}$ of the hypercube $Q_{3+l}$ such that $G[\mathcal{M}]$ is isomorphic to the complete bipartite graph $K_{3,l}$.
\label{case:k=3}
\end{prop}
\begin{proof}
Using Corollary \ref{cor:2Stong}, partition $Q_l$ into matchings $A_1,A_2,\ldots,A_l$ and $B_1,B_2,\ldots, B_l$ such that $A_i\cup B_i$ is a Hamilton cycle for all $j$.
Let $X_1, X_2$ and $X_3$ be the three directional matchings of $Q_3$ -- that is, $X_j$ contains all edges in direction $j$.
For $i=1,2,\ldots,l$ define $M_i$ to be the matching on $Q_{3+l}$ defined by taking the following edges:
$$\begin{cases}
A_i & \text{ on } Q_l^{\emptyset}, Q_l^{\{1,2\}}, Q_l^{\{1,3\}}, Q_l^{\{2,3\}} \text{ and } Q_l^{\{1,2,3\}} \\
B_i & \text{ on } Q_l^{\{1\}}, Q_l^{\{2\}} \text{ and } Q_l^{\{3\}}
\end{cases}
$$
For $j = 1,2,3$ define $N_j$ to be the matching on $Q_{3+l}$ defined by taking the following edges, where the subscripts for the $X$s are taken modulo $3$:
$$\begin{cases}
X_{j} & \text{ on } Q_3^v \text{ for $v$ odd} \\
X_{j+1} & \text{ on } Q_3^v \text{ for $v$ even and } v\ne \emptyset \\
X_{j+2} & \text{ on } Q_3^{\emptyset}
\end{cases}
$$
\begin{figure}
\caption{$M_i$}
\caption{$N_1$}
\caption{Sketch of cycle ${M_i\cup N_1}
\caption{An example when $l=2$}
\label{subfig:sketch}
\label{fig:l=3}
\end{figure}
Now $\{M_i\}_{i=1}^l \cup \{N_j\}_{j=1}^3$ is a set of $3+l$ disjoint perfect matchings. It remains to show that $M_i \cup N_j$ is a Hamilton cycle for any $i$ and $j$.
Note that $\{M_i\}$ is invariant under the permutation that cycles directions 1,2 and 3. Since $N_2$ and $N_3$ are obtained from $N_1$ by such cyclic permutations, we can without loss of generality assume that $j=1$.
Consider $M_i \cup N_1$ with the edges in $Q_3^{\emptyset}$ removed; that is, the edges $\emptyset \{3\}$, $\{1\}\{1,3\}$, $\{2\}\{2,3\}$ and $\{1,2\}\{1,2,3\}$.
We will show that the resulting graph comprises four paths, from $\emptyset$ to $\{2\}$, from $\{2,3\}$ to $\{1,2,3\}$, from $\{1,2\}$ to $\{1\}$ and from $\{1,3\}$ to $\{3\}$. Thus when we add back the four edges in direction 3, we get a Hamilton cycle. See figure \ref{subfig:sketch} for an example.
View $Q_{3+l}$ as an $l$-dimensional hypercube whose `vertices' are copies of $Q_3$. Starting at a vertex in $Q_3^\emptyset$ and following the path from it, we will not return to $Q_3^\emptyset$ until we have made $2^l$ steps around $A_i \cup B_i$.
A path starting at $\emptyset$ will move in directions according to $A_i$ then $X_1$ then $B_i$ then $X_2$ and then repeat this pattern. It will return to $Q_3^\emptyset$ after $2^l$ moves from $A_i \cup B_i$ and $2^l- 1$ moves from $X_1 \cup X_2$. Since $l \ge 2$, this means we end at the vertex $\{2\}$, and the path contains $2(2^l)$ vertices.
The same argument works to show that there is a path from $\{1,2\}$ to $\{1\}$ containing $2(2^l)$ vertices.
A path starting at $\{2,3\}$ will move in directions according to $A_i$ then $X_1$ then $A_i$, ending at the vertex $\{1,2,3\}$ and containing $4$ vertices.
A path starting at $\{1,3\}$ will move in directions according to $A_i, X_1, B_i, X_2, A_i, X_1, A_i, X_2$, and then repeat this pattern.
It will return to $Q_3^\emptyset$ after $2(2^l)-2$ moves from $A_i \cup B_i$ and $2(2^l)- 3$ moves from $X_1 \cup X_2$. Thus we end at the vertex $\{3\}$, and the path contains $4(2^l)-4$ vertices.
The sum of the lengths of these paths is $8(2^l)$, and so every vertex is contained in one of these paths.
\end{proof}
\begin{prop} There is a 1-factorization $\mathcal{M}$ of the hypercube $Q_{4}$ such that $G[\mathcal{M}]$ is isomorphic to the complete bipartite graph $K_{3,1}$.
\label{case:k=3,l=1}
\end{prop}
\begin{proof}
The four matchings are shown in figure \ref{fig:3-1}. It is easy to check that the top matching forms a Hamilton cycle with any of the three bottom matchings (in fact, by symmetry you need only check one pair).
\begin{figure}
\caption{The matchings for $k=1$ and $l=3$}
\label{fig:3-1}
\end{figure}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Combine the results of Propositions \ref{case:neither3}, \ref{case:k=3} and \ref{case:k=3,l=1}.
\end{proof}
\section{Direction Respecting 1-Factorizations}
\label{sec:direction}
The only case not covered by Theorem \ref{thm:main} is whether $G[\mathcal{M}]$ can be isomorphic to $K_{3,3}$. This case cannot be resolved with our methods alone. To explain why, we will introduce a notion of direction respecting 1-factorizations.
Fix $k$ and $l$ and let $\mathcal{M} = M_1,M_2,\ldots M_{k+l}$ be a 1-factorization of $Q_{k+l}$. We call the 1-factorization $\mathcal{M}$ \emph{direction respecting} if $M_1,M_2,\ldots, M_k$ only use edges in directions $1,\ldots,k$
and $M_{k+1},M_{k+2},\ldots M_{k+l}$ only use edges in directions $k+1,\ldots k+l$.
Note that the matchings constructed in Propositions \ref{case:neither3} and \ref{case:k=3} were direction respecting for the appropriate $k$ and $l$. However, the 1-factorisation given in the proof of proposition \ref{case:k=3,l=1} was not direction respecting. We shall prove that there is no direction respecting 1-factorization $\mathcal{M}$ with $G[\mathcal{M}]$ isomorphic to $K_{3,3}$ or $K_{3,1}$.
\begin{theorem} \label{thm:why3hard}
There is a direction respecting 1-factorization $\mathcal{M}$ of $Q_{k+l}$ with $G[\mathcal{M}] = K_{k,l}$ if and only if $k,l$ are not $3,1$ or $3,3$.
\end{theorem}
\begin{proof}
First note that the proof of Theorem \ref{thm:main} shows that such a 1-factorization exists when $k,l$ are not $1,3$ or $3,3$.
Let $d = 3+l$ where $l$ is 3 or 1.
For $M$ a perfect matching on $Q_d$, think of $M$ as a bijection from the odd vertices of $Q_d$ to the even vertices (as in Theorem \ref{thm:Bipartite}). If $M$ and $N$ are perfect matchings then $M N^{-1}$ is a permutation on the even vertices of $Q_d$.
We define the \emph{sign} of a 1-factorization $\{M_i\}_{i=1}^{d}$ of $Q_{d}$ to be the product of the signs of the permutations $M_i M_j^{-1}$ for all $i < j$. That is,
$$\sign(\mathcal{M}) = \prod_{i<j} \sign\left(M_i M_j^{-1}\right).$$
Let $\mathcal{D}^{(d)} = \{D^{(d)}_i\}_{i=1}^{n}$ be the directional matchings of $Q_d$, where $D^{(d)}_i$ contains all edges in direction $i$. For $i \ne j$ the permutation $D^{(d)}_i \left(D^{(d)}_j\right)^{-1}$ consists of $2^{d-2}$ disjoint 4-cycles. Thus $\sign\left(D^{(d)}_i \left(D^{(d)}_j \right)^{-1}\right) = (-1)^{2^{d-2}} = 1$ for all $i,j$, and so $\sign(\mathcal{D}^{(d)})=1$.
Suppose
$\mathcal{M} = \{M_i\}_{i=1}^3\cup\{N_j\}_{j=1}^l$
is a 1-factorization of $Q_d$ where $M_i \cup N_j$ is a Hamilton cycle for all $i,j$. The permutation $M_i N_j^{-1}$ is a cycle of length $2^{d-1}$ and so has sign $-1$. Note that $\sign\left(M_i M_s^{-1}\right) = \sign\left(M_i N_j^{-1}\right)\left((M_s N_j^{-1})^{-1}\right) = (-1)(-1) = 1$, and similarly $\sign(N_j N_t^{-1}) = 1$.
Thus $\sign(\mathcal{M}) = (-1)^{3l} = -1$ for any $\mathcal{M}$ with $G[\mathcal{M}] = K_{3,l}$.
We will define a switching operation on 1-factorizations that preserves their sign. We will further show that any direction respecting 1-factorization $\mathcal{M}$ can be obtained from $\mathcal{D}^{(d)}$ using a series of switches. Since the sign of $\mathcal{D}^{(d)}$ is $1$, this is enough to show that $G[\mathcal{M}] \ne K_{3,l}$.
Let $\mathcal{M} = \{M_i\}_{i=1}^d$ be a 1-factorization of $Q_d$. Take a 4-cycle $x,y,v,w$ in $Q_{d}$ and suppose that the edges $xy$ and $vw$ are in matching $M_s$ and $vy$ and $xw$ are in matching $M_t$. A switch on $w,v,y,w$ replaces $\mathcal{M}$ by the 1-factorization $\mathcal{M'} = \{M'_i\}_{i=1}^n$ where
$M'_s = M_s \cup \{ vy, xw\} \setminus \{ xy, vw\}$, $M_t' = M_t \cup \{ xy, vw\} \setminus \{vy, xw\}$, and $M'_i = M_i$ for $i \ne s,t$.
Viewing the 1-factors $M_i$ as bijections from the even vertices to the odd vertices, we have composed $M_s$ and $M_t$ with the function swapping $x$ and $v$, where $x$ and $v$ are the even vertices of $x,y,v,w$. Therefore the permutations $M_s M_i^{-1}$ and $M'_s M_i^{-1}$, where $i \ne s,t$, differ from each other by the transposition $(a,c)$ and so have opposite sign. Similarly $M_t M_i^{-1}$ and $M'_t M_i^{-1}$ have opposite sign.
From this second interpretation of the switch it is clear that:
\begin{align*}
\sign(\mathcal{M'}) &= \prod_{i<j} \sign\left(M'_i (M'_j)^{-1}\right))
\\
&= \prod_{\substack{\text{exactly one of} \\ \text{$i,j$ is $s$ or $t$}}} - \sign\left(M_i M_j^{-1}\right)
\prod_{\substack{\text{neither or both of} \\ \text{$i,j$ is $s$ or $t$}}} \sign\left(M_i M_j^{-1}\right)
\\
&= (-1)^{2(d-2)} \sign(\mathcal{M}) = \sign(\mathcal{M})
\end{align*}
All that is left to show is that a 1-factorization satisfying the conditions of the theorem can be obtained from $\mathcal{D}^{(d)}$ by a series of switches. We will use the following claim.
\begin{claim} Let $D^{(3)}_1,D^{(3)}_2,D^{(3)}_3$ be the directional matchings on $Q_3$ and let $A_1,A_2,A_3$ be another 1-factorization of $Q_3$.
Then there are a series of switches that transform $D^{(3)}_1$, $D^{(3)}_2$, $D^{(3)}_3$ into $A_1,A_2,A_3$, respecting the ordering.
\end{claim}
\begin{proof}[Proof of Claim]
It is easy to check that there are only 4 ways to partition $Q_3$ into perfect matchings, up to ordering -- one way uses three directional matchings and the other three ways each use one directional matching. Without loss of generality say that $A_1$ is a directional matching.
Note that we can use switches to re-order $D^{(3)}_1,D^{(3)}_2,D^{(3)}_3$.
To swap $D^{(3)}_i$ and $D^{(3)}_j$ switch on $\emptyset, \{i\}, \{j\}, \{i,j\}$
and on $\{k\}, \{i,k\}, \{j,k\}, \{i,j,k\}$, where $i,j,k$ is $1,2,3$ in some order.
Thus we can assume without loss of generality that $A_1 = D^{(3)}_1$.
If $A_2$ and $A_3$ are also directional matchings then we are done. If not, then we can switch on $\emptyset, \{2\},\{2,3\},\{3\}$ to make them both directional matchings.
\end{proof}
Let $\mathcal{M} = \{M_i\}_{i=1}^3\cup\{N_j\}_{j=1}^l$
be a 1-factorization of $Q_d$ satisfying the conditions of the theorem.
As in Theorem \ref{thm:main}, we can view $Q_{3+l}$ as an $l$ dimensional hypercube whose `vertices' are copies of $Q_3$.
For $v \subset \{3+1,\ldots, 3+l\}$
let $Q_3^v$ be the induced subgraph of $Q_{3+l}$ on vertices of the form $u \cup v$ for all
$u \subset \{1,2,3\}$.
For each $v$ in turn, apply the claim to $Q_3^v$ and $M_1,M_2,M_3$ restricted to $Q_3^v$. In this way we obtain a series of switches that turns $D^{(d)}_1,D^{(d)}_2,D^{(d)}_3$ into $M_1,M_2,M_3$.
If $l= 1$, $N_1 = D^{(n)}_4$ and we are done. If $l = 3$, apply an analagous process to above to find switches that turn $D^{(d)}_4,D^{(d)}_5,D^{(d)}_6$ into $N_1,N_2,N_3$. Note that these switches will be only on edges in directions 4,5,6 and so will not interfere with $M_1,M_2,M_3$ in any way.
\end{proof}
\section{Open Questions}
\label{sec:questions}
The most obvious question is the missing case from Theorem \ref{thm:main}.
\begin{question}
Is it possible to find a 1-factorization $\mathcal{M}$ of $Q_6$ such that $G[\mathcal{M}] = K_{3,3}$?
\end{question}
Theorem \ref{thm:why3hard} and its proof show that any such matching $\mathcal{M}$ cannot be obtained from applying a series of switches to the directional matchings. However, there is an example where $G[\mathcal{M}] = K_{3,1}$, and computer checking suggests that in 4 or 5 dimensions there are many other ways to 1-factorize $Q_d$ and get complete bipartite graphs than the way shown in this paper.
We know from Theorem \ref{thm:Bipartite} that we cannot have a perfect 1-factorization of $Q_d$ for $d>2$. In fact, the maximum possible number of pairs of 1-factors whose union forms a Hamilton cycle is $\left\lfloor \frac{d^2}{4} \right\rfloor$, obtained when $G[\mathcal{M}] = K_{\lfloor d/2 \rfloor,\lceil d/2 \rceil}$. What can be said about the other pairs -- can their union be close to a Hamilton cycle in some way?
\begin{question}
Let $\mathcal{M} = \{M_i\}_{i=1}^d$ be a 1-factorization of $Q_d$.
Is it possible for $M_i \cup M_j$ to contain a cycle of length $(1-o(1))2^d$ for every $i\ne j$?
\end{question}
\begin{question}
Let $\mathcal{M} = \{M_i\}_{i=1}^d$ be a 1-factorization of $Q_d$. Is it possible for $M_i \cup M_j$ to consist of at most 2 cycles for every $i\ne j$?
\end{question}
Computer checking shows that for $n \le 5$ the answer to the latter question is `yes' and in dimensions 4 and 5 there are actually several different 1-factorizations that work.
One could also phrase more general versions of these questions in terms of finding bounds on an appropriate minimax or maximin function. For example,
\begin{question}
For a 1-factorization $\mathcal{M} = \{M_i\}_{i=1}^d$ of $Q_d$,
let $c_{i,j}$ be the length of the longest cycle in $M_i \cup M_j$
and let $f(\mathcal{M}) = min_{i\ne j}(c_{i,j})$.
Can one find bounds on $max_{\mathcal{M}}(f(\mathcal{M}))$ in terms of $d$?
\end{question}
We can prove that $max_{\mathcal{M}}(f(\mathcal{M}))$ is non-decreasing with $d$. We suspect that it grows exponentially in $d$, but we cannot yet prove it is even better than constant.
A different way of thinking of Hamilton cycles is as connected 2-factors. Thus a different generalisation of the problem would be to ask about the connectivity of other $r$-factors. For example,
\begin{question} For each $d$, let $r=r(d)$ be minimal subject to there existing a 1-factorization $\mathcal{M}$ of $Q_d$ where the union of any $r$ distinct 1-factors is connected. What is the value of $r(d)$?
\end{question}
Theorem \ref{thm:Bipartite} shows that $r(d)$ is greater than $2$ for $d>2$.
The 1-factorization given by Theorem \ref{thm:main} in the case $k = \left\lfloor \frac{d}{2} \right\rfloor$ and $l = \left\lceil \frac{d}{2} \right\rceil$ has the property that the union of any $\left( \left\lceil \frac{d}{2} \right\rceil + 1 \right)$ 1-factors is connected, hence $r(d) \le \left\lceil \frac{d}{2} \right\rceil + 1$ for $d \ne 6$. It seems possible that $r$ is constant and it could be even as small as 3.
\nocite{*}
{}
\end{document} |
\begin{document}
\title{Macroscopic and microscopic structures of the family tree for
decomposable critical branching processes}
\author{Vladimir Vatutin\thanks{
Steklov Mathematical institute RAS, Gubkin str. 8, Moscow, 119991,
Russia; e-mail: [email protected]}}
\date{}
\maketitle
\begin{abstract}
A decomposable strongly critical Galton-Watson branching process with $N$
types of particles labelled $1,2,...,N$ is considered in which a type~$i$
parent may produce individuals of types $j\geq i$ only. This model may be
viewed as a stochastic model for the sizes of a geographically structured
population occupying $N$ islands, the location of a particle being
considered as its type. The newborn particles of island $i\leq N-1$ either
stay at the same island or migrate, just after their birth to the islands $
i+1,i+2,...,N$. Particles of island $N$ do not migrate. We investigate the
structure of the family tree for this process, the distributions of the
birth moment and the type of the most recent common ancestor of the
individuals existing in the population at a distant moment $n.$
\end{abstract}
\section{Introduction and main results}
We consider a Galton-Watson branching process with $N$ types of particles
labelled $1,2,...,N$ \ and denote by
\begin{equation*}
\mathbf{Z}(n)=(Z_{1}(n),...,Z_{N}(n)),\quad \mathbf{Z}(0)=(1,0,...,0)
\end{equation*}
the population vector at time $n\in \mathbb{Z}_{+}=\left\{ 0,1,...\right\} $
. Along with $\mathbf{Z}(n)$ we deal with the process \
\begin{equation*}
\mathbf{Z}(m,n)=(Z_{1}(m,n),...,Z_{N}(m,n)),
\end{equation*}
where $Z_{i}(m,n)$ is the number of type $i$ particles existing in $\mathbf{Z
}(\cdot )$ at moment $m<n$ and having nonempty number of descendants at
moment $n$. We agree to write $Z_{i}(n,n)=Z_{i}(n)$.
The process $\mathbf{Z}( \cdot ,n) $ is called a reduced branching process
and can be thought of as the family tree relating the individuals alive at
time $n$. An important characteristic of the reduced process is the birth
moment $\beta _{n}$ of the most recent common ancestor (MRCA) of all
individuals existing in the population at moment $n$ defined as \
\begin{equation*}
\beta _{n}=\max \left\{ m\leq n-1:Z_{1}( m,n) +Z_{2}( m,n)
+...+Z_{N}(m,n)=1\right\} .
\end{equation*}
The structure of the family tree and the asymptotic distribution of the
birth moment of the MRCA for single-type Galton-Watson branching processes
have been studied in \cite{FP},\cite{FZ},\cite{LS} and \cite{Zub}. The case
of multitype indecomposable critical Markov branching processes was
considered in \cite{Yak83}. Family trees for more general models of
branching processes were investigated in \cite{BorVat}, \cite{FVat99},\cite
{Sag85},\cite{Sag87},\cite{Sag95},\cite{Vat2003},\cite{VD06},\cite{VV79}.
However, the reduced processes for decomposable branching processes have not
been analyzed yet. We fill this gap in the present paper and study various
properties of the family tree for a particular case of the decomposable
Galton-Watson branching processes. Namely, we consider the Galton-Watson
branching process with $N$ types of particles labelled $1,2,...,N$ in which
a type $i$ parent may produce individuals of types $j\geq i$ only. This
model may be viewed as a stochastic model for the sizes of a geographically
structured population occupying $N$ islands, the location of a particle
being considered as its type. The reproduction laws of particles depend on
the island on which the particles are located. The newborn particles of
island $i\leq N-1$ either stay at the same island or migrate, just after
their birth to the islands $i+1,i+2,...,N$. Particles of island $N$ do not
migrate.
We investigate the structure of the family tree of this process, the
distributions of the birth moment $\beta _{n}$ and the type $\zeta _{n}$ of
the MRCA. It is shown, in particular, that, as $n\rightarrow \infty $ the
conditional reduced process
\begin{equation*}
\left\{ \mathbf{Z}(n^{t}\log n,n),0\leq t<1|\mathbf{Z}(n)\neq \mathbf{0}
\right\}
\end{equation*}
converges in a certain sense to an $N-$dimensional inhomogeneous branching
process $\left\{ \mathbf{R}(t),0\leq t<1\right\} $ which, for $t\in \lbrack
0,2^{-(N-1)})$ consists of a single particle of type $1$ only and for $t\in
\lbrack 2^{-(N-i+1)},2^{-(N-i)}),i=2,...,N$ consists of type $i$ particles
only. These particles are born at moment $t=2^{-(N-i+1)}$ and die at moment $
t=2^{-(N-i)}$ producing at this moment a random number of descendants having
type $\min (i+1,N)$. This gives a macroscopic view on the structure of the
family tree of the process.
On the other hand, for each $i=1,2,...,N-1$ the conditional process
\begin{equation*}
\left\{ \mathbf{Z}((y+(\log n)^{-1})n^{2^{-(N-i)}},n),0<y<\infty \big|\,
\mathbf{Z}(n)\neq \mathbf{0}\,\right\}
\end{equation*}
converges in a certain sense, as $n\rightarrow \infty $ to a continuous-time
homogeneous Markov branching process $\left\{ \mathbf{U}_{i}(y),0\leq
y<\infty \right\} $ which is initiated at time $y=0$ by a random number of
type $i$ particles. These type $i$ particles have an exponential life-length
distribution. Dying each of them produces either two particles of type $i$
or one particle of type $i+1$ (both options with probability 1/2). Particles
of type $i+1$ in this process are immortal and produce no offspring. This
provides a microscopic view on the structure of the family tree.
To present our results in a more formal way we need some notation. Let $
\mathbf{e}_{i}$ be a vector whose $i$-th component is equal to one while the
remaining are zeros. The first moments of the components of $\mathbf{Z}(n)$
will be denoted as
\begin{equation*}
m_{ij}(n)=\mathbf{E}\left[ Z_{j}(n)|\mathbf{Z}\left( 0\right) =\mathbf{e}_{i}
\right]
\end{equation*}
with $m_{ij}=m_{ij}(1)$ \thinspace being the average number of children of
type $j$ produced by a particle of type $i$.
Since $m_{ij}=0$ if $i>j$, the mean matrix $\mathbf{M}$ of the decomposable
Galton-Watson branching process has the form
\begin{equation}
\mathbf{M=}(m_{ij})_{i,j=1}^{N}=\left(
\begin{array}{ccccc}
m_{11} & m_{12} & ... & ... & m_{1N} \\
0 & m_{22} & ... & ... & m_{2N} \\
0 & 0 & m_{33} & ... & ... \\
... & ... & ... & ... & ... \\
... & ... & ... & ... & ... \\
0 & 0 & ... & 0 & m_{NN}
\end{array}
\right) . \label{matrix1}
\end{equation}
To go further it is convenient to deal with the probability generating
functions for the reproduction laws of particles
\begin{equation}
h_{i}(s_{1},...,s_{N})=\mathbf{E}\left[ s_{i}^{\eta _{ii}}...\,s_{N}^{\eta
_{iN}}\right] ,\ i=1,2,...,N, \label{DefNONimmigr}
\end{equation}
where $\eta _{ij}$ represent the numbers of daughters of type $j$ a mother
of type $i$.
We say that \textbf{Hypothesis A} is valid if the $N-$type decomposable
process is strongly critical, i.e. (see \cite{FN2}),
\begin{equation}
m_{ii}=\mathbf{E}\left[ \eta _{ii}\right] =1,\ i=1,2,...,N, \label{Matpos}
\end{equation}
and, in addition,
\begin{equation}
m_{i,i+1}=\mathbf{E}\left[ \eta _{i,i+1}\right] \in ( 0,\infty ) ,\
i=1,2,...,N-1, \label{Maseq}
\end{equation}
and
\begin{equation}
\mathbf{E}\left[ \eta _{ij}\eta _{ik}\right] <\infty ,\,i=1,...,N;\
k,j=i,i+1,...,N \label{FinCovar}
\end{equation}
with
\begin{equation}
b_{i}=\frac{1}{2}Var\left[ \eta _{ii}\right] \in ( 0,\infty ) ,\ i=1,2,...,N.
\label{FinVar}
\end{equation}
Thus, a particle of the process under consideration is able to produce the
direct descendants of its own type, of the next in the order type, and (not
necessarily, as direct descendants) of all the remaining in the order types,
but not any preceding ones.
To simplify the presentation we fix, from now on $N\geq 2$ and use, when it
is convenient the notation
\begin{equation*}
\gamma _{0}=0,\ \gamma _{i}=\gamma _{i}(N)=2^{-(N-i)},\ i=1,2,...,N.
\end{equation*}
We also suppose (if otherwise is not stated) that $\mathbf{Z}(0)=\mathbf{e}
_{1}$, i.e., assume that the branching process under consideration is
initiated at time zero by a single particle of type $1$.
Let $\xi ^{(i)}(j),i=1,2,...,N;j=1,2,...$ be a tuple of independent
identically distributed random variables with probability generating function
\begin{equation*}
f(s)=\mathbf{E}\left[ s^{\xi ^{(i)}(j)}\right] =1-\sqrt{1-s}.
\end{equation*}
By means of the tuple we give a detailed construction of an $N-$type
decomposable branching process $\mathbf{R}(t)=(R_{1}(t),...,R_{N}(t)),0\leq
t<1,$ where $R_{i}(t)$ is the number of type $i$ individuals in the
population at moment~$t$. It is this process describes the macroscopic
structure of the family tree $\left\{ \mathbf{Z}(m,n),0\leq m\leq n\right\}$
as $n\rightarrow \infty $.
Let $\mathbf{R}(t)=\mathbf{e}_{1}$ for $\gamma _{0}\leq t<\gamma _{1}$
meaning that the branching process $\mathbf{R}(t)$ starts at $t=0$ by a
single individual of type $1$ which survives up to (but not at) moment $
\gamma _{1}$ without reproduction. If $\gamma _{i}\leq t<\gamma
_{i+1},i=1,2,...,N-1$ then
\begin{equation*}
R_{k}(t)=\left\{
\begin{array}{ccc}
\sum_{j=1}^{R_{i}(\gamma _{i}-0)}\xi ^{(i)}(j) & \text{if} & k=i+1 \\
& & \\
0 & \text{if} & k\neq i+1
\end{array}
\right. .
\end{equation*}
Thus, within the interval $\gamma _{i}\leq t<\gamma _{i+1}$ the population
consists of type $i+1$ particles only. These particles were born at moment $
\gamma _{i}-0$ by particles of type $i$ evolving without reproduction within
the interval $\gamma _{i-1}\leq t<\gamma _{i}$. More precisely, the $j-$th
particle of type $i$ produces at its death moment $\gamma _{i}-0$ a random
number $\xi ^{(i)}(j)$ children of type $i+1$ and no particles of other
types.
In what follows we use the symbol $\Longrightarrow $ to denote convergence
in the space $D_{[a,b)}(\mathbb{Z}_{+}^{N})$ of cadlag functions $\mathbf{x}
(t),a\leq t<b$ with values in \ $\mathbb{Z}_{+}^{N}$ endowed with the metric
of Skorokhod topology. Besides, we agree to consider $\mathbf{Z}(x,n)$ as $
\mathbf{Z}([x],n),$ where $[x]$ is the integer part of $x$.
For $0\leq t\leq 1$ put
\begin{equation*}
g_{n}(t)=1_{\{0\leq t<\gamma _{1}\}}+g_{n}1_{\{\gamma _{1}\leq t\leq 1\}}
\end{equation*}
where $g_{n}$ is a positive monotone increasing sequence such that
\begin{equation*}
\lim_{n\rightarrow \infty }g_{n}=\infty \text{ and }\lim_{n\rightarrow
\infty }n^{-\varepsilon }g_{n}=0\text{ for any }\varepsilon >0.
\end{equation*}
\begin{theorem}
\label{T_SkorohConst}Let Hypothesis A be valid. Then, as $n\rightarrow
\infty $
1) the finite-dimensional distributions of the process
\begin{equation*}
\left\{ (\mathbf{Z}(n^{t}g_{n}(t),n),0\leq t<1)|\mathbf{Z}(n)\neq \mathbf{0}
\right\}
\end{equation*}
converge to the finite-dimensional dis\-tri\-bu\-tions of $\left\{ \mathbf{R}
(t),0\leq t<1\right\};$
2) for any $i=0,1,2,...,N-1$
\begin{equation*}
\mathcal{L}\left\{ (\mathbf{Z}(n^{t}g_{n}(t),n),\gamma _{i}\leq t<\gamma
_{i+1})\,|\,\mathbf{Z}(n)\neq \mathbf{0}\right\} \Longrightarrow \mathcal{L}
\left\{ \mathbf{R}(t),\gamma _{i}\leq t<\gamma _{i+1}\right\} .
\end{equation*}
\end{theorem}
\textbf{Remark 1.} Theorem \ref{T_SkorohConst} shows that the passage to
limit under the macroscopic time-scaling $n^{t}g_{n}(t)$ transforms the
reduced process into an inhomogeneous branching process which consists at
any given moment of particles of a single type only. In particular, the
phase transition from type $i$ to type $i+1$ in the prelimiting process
happens, roughly speaking, at moment $n^{\gamma _{i}}$. This gives a
macroscopic view on the family tree of the reduced process. The microscopic
structure of the family tree described by Theorem \ref{T_Skhod1} below
clarifies the nature of the revealed phase transition.
Let $c_{ji},1\leq j\leq i\leq N$ be a tuple of positive numbers in which $
c_{ii}=b_{i}^{-1}$ for $i=1,2,...,N$ and
\begin{equation}
c_{ji}=\sqrt{b_{j}^{-1}m_{j,j+1}c_{j+1,i}}\text{ for }\ j\leq i-1,\quad
C_{i}=c_{1i}. \label{Const1}
\end{equation}
It is not difficult to check that
\begin{equation}
c_{iN}=\left( \frac{1}{b_{N}}\right) ^{1/2^{N-i}}\prod_{j=i}^{N-1}\left(
\frac{m_{j,j+1}}{b_{j}}\right) ^{1/2^{j-i+1}}. \label{Const2}
\end{equation}
We now define a tuple of continuous time Markov processes
\begin{eqnarray*}
\mathbf{U}_{i}(y) &=&(U_{i1}(y),...,U_{iN}(y)),\ 0\leq y<\infty ,\
i=1,2,...,N-1, \\
&& \\
\mathbf{U}_{N}(x) &=&(U_{N1}(x),...,U_{NN}(x)),\ 0\leq x<1.
\end{eqnarray*}
First we describe the structure of the processes $\mathbf{U}_{i}(y),1\leq
i\leq N-1$. In this case $U_{ij}(y)\equiv 0,~0\leq y<\infty ,~j\neq i,i+1,$
while the pair
\begin{equation*}
(U_{ii}(y),U_{i,i+1}(y)),0\leq y<\infty ,
\end{equation*}
constitutes a two-type continuous-time homogeneous Markov branching process
with particles of types $i$ and $i+1$. This two-type process is initiated at
time $y=0$ by a random number $R_{i}$ of type $i$ particles whose
distribution is specified by the probability generating function
\begin{equation}
\mathbf{E}\left[ s_{i}^{R_{i}}\right] =\mathbf{E}\left[ s_{i}^{U_{ii}(0)}
\right] =1-(1-s_{i})^{1/2^{i-1}} \label{Dist_ro}
\end{equation}
(in particular, $U_{11}(0)=1$ with probability 1). The life-length
distribution of type $i$ particles is exponential with parameter $
2b_{i}c_{iN}$. Dying each particle of type $i$ produces either two particles
of its own type or one particle of type $i+1$ (each option with probability
1/2). Particles of type $i+1$ of $\mathbf{U}_{i}(\cdot )$ are immortal and
produce no children.
The structure of the $N-$ dimensional process $\mathbf{U}_{N}(x),0\leq x<1$
is different. If $j<N$ then $U_{Nj}(x)\equiv 0,~0\leq x<1,$ while the
component $U_{NN}(\cdot )$ is a single-type inhomogeneous Markov branching
process initiated at time $x=0$ by a random number $R_{N}$ of type $N$
individuals distributed in accordance with probability generating function
\begin{equation}
\mathbf{E}\left[ s_{N}^{R_{N}}\right] =\mathbf{E}\left[ s_{N}^{U_{NN}(0)}
\right] =1-(1-s_{N})^{1/2^{N-1}}. \label{DefroN}
\end{equation}
The life-length of each of $R_{N}$ type $N$ initial particles is uniformly
distributed on the interval $\left[ 0,1\right] $. Dying such a particle
produces exactly two children of type~$N$ and nothing else. If the death
moment of the parent particle is $x$ then the life length of each of its
offspring has the uniform distribution on the interval $[x,1]$
(independently of the behavior of other particles and the prehistory of the
process). Dying each particle of the process produces exactly two
individuals of type $N$ and so on...\thinspace .
We are now ready to formulate one more important result of the paper,
describing the microscopic structure of the family tree.
Let $l_{n}$ be a monotone decreasing sequence such that
\begin{equation*}
\lim_{n\rightarrow \infty }l_{n}=0\text{ and }\lim_{n\rightarrow \infty
}n^{\varepsilon }l_{n}=\infty \text{ for any }\varepsilon >0.
\end{equation*}
\begin{theorem}
\label{T_Skhod1}Let Hypothesis A be valid. Then, as $n\rightarrow \infty $
1) for each $i=1,2,...,N-1$
\begin{equation*}
\mathcal{L}\left\{ (\mathbf{Z}\left( (y+l_{n})n^{\gamma _{i}},n\right)
,0\leq y<\infty )\big|\,\mathbf{Z}(n)\neq \mathbf{0}\right\} \Longrightarrow
\mathcal{L}_{R_{i}}\left\{ \mathbf{U}_{i}(y),0\leq y<\infty \,\right\} ,
\end{equation*}
where $\mathcal{L}_{R_{i}}$ means that $\mathbf{U}_{i}(\cdot )$ is initiated
at time $y=0$ by a random number $R_{i}$ particles of type $i$ (with $
R_{1}\equiv 1)$;
\begin{equation*}
\mathit{2)}\,\mathcal{L}\left\{(\mathbf{Z}((x+l_{n})n,n),0\leq x<1)\,|\,
\mathbf{Z}(n)\neq \mathbf{0}\right\} \Longrightarrow \mathcal{L}
_{R_{N}}\left\{ \mathbf{U}_{N}(x),0\leq x<1\,\right\},
\end{equation*}
where $\mathcal{L}_{R_{N}}$ means that $\mathbf{U}_{N}(\cdot)$ is initiated
at time $x=0$ by a random number $R_{N}$ particles of type $N.$
\end{theorem}
\textbf{Remark 2.} Theorems \ref{T_SkorohConst} and \ref{T_Skhod1} reveal an
interesting phenomenon in the development of the critical decomposable
branching processes which may be expressed in terms of the "island"
interpretation of the processes as follows: If the population survives up to
a distant moment $n$, then all surviving individuals are located at this
moment on island $N$ and, moreover, at each moment in the past their
ancestors were (asymptotically) located not more than on two specific
islands.
Basing on the conclusions of Theorems \ref{T_SkorohConst} and \ref{T_Skhod1}
we give in the next theorem an answer to the following important question:
what is the asymptotic distribution of the birth moment of the MRCA for the
population survived up to a distant moment $n$?
\begin{theorem}
\label{T_mrcaMany}Let Hypothesis A be valid. Then
1)
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\ll n^{\gamma _{1}}
\big|\,\mathbf{Z}(n)\neq \mathbf{0}\right) =0;
\end{equation*}
2) if $y\in (0,\infty )$ then for $i=1,2,...,N-1$
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\leq yn^{\gamma _{i}}
\big|\,\mathbf{Z}(n)\neq \mathbf{0}\right) =1-\frac{1}{2^{i}}-\frac{1}{2^{i}}
e^{-2b_{i}c_{iN}y};
\end{equation*}
3) for $i=1,2,...,N-1$
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\ll n^{\gamma _{i}}
\big|\,\mathbf{Z}(n)\neq \mathbf{0}\right) =1-\frac{1}{2^{i-1}};
\label{recent_i}
\end{equation}
3a) for $i=1,2,...,N-1$
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{P}\left( n^{\gamma _{i}}\ll \beta _{n}\ll
n^{\gamma _{i+1}}\big|\,\mathbf{Z}(n)\neq \mathbf{0}\right) =0;
\label{NoMRCA}
\end{equation}
4) for any $x\in (0,1)$
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}(\beta _{n}\leq xn|\mathbf{Z}(n)\neq
\mathbf{0})=1-\frac{1}{2^{N-1}}(1-x).
\end{equation*}
\end{theorem}
\textbf{Remark 3.} As we see by (\ref{NoMRCA}), there are time-intervals of
increasing orders within each of which the probability to find the MRCA of
the population survived up to moment $n\rightarrow \infty $ is negligible
compared to the probability for the population to survive up to this moment.
Moreover, these time-intervals are separated from each other by the
time-intervals of increasing orders within each of which the probability to
find the MRCA is strictly positive. Such a phenomena has no analogues for
the indecomposable Galton-Watson processes.
Along with the distribution of the birth moment of the MRCA, the type $\zeta
_{n}$ of the MRCA of the population survived up to moment $n$ is of
interest. The distribution of this random variable is described by the
following theorem.
\begin{theorem}
\label{T_type}Let Hypothesis A be valid. Then, for $i=1,2,...,N$
\begin{equation*}
p_{i}=\lim_{n\rightarrow \infty }\mathbf{P}(\zeta _{n}=i|\mathbf{Z}(n)\neq
\mathbf{0})=\frac{1}{2^{i}}(1-\delta _{iN})+\frac{1}{2^{N-1}}\,\delta _{iN},
\end{equation*}
where $\delta _{ij}$ is the Kroneker symbol.
\end{theorem}
Observe that $p_{N-1}=p_{N}$.
\textbf{Remark 4.} The authors of paper \cite{FN2}, which contains several
results used in the proofs of our Theorems \ref{T_SkorohConst}-\ref{T_type},
considered a more general case of the strongly critical branching processes.
Namely, they prove a number of conditional limit theorems for the case when
by a suitable labelling the types of the multitype Galton-Watson process can
be grouped into $N\geq 2$ partially ordered classes $\mathcal{C}
_{1}\rightarrow \mathcal{C}_{2}\rightarrow ...\rightarrow \mathcal{C}_{N}$
possessing the following properties:
1) particle types belonging to any given class, say $\mathcal{C}_{i},$
constitute an indecomposable critical branching process with $r_{i}\geq 1$
types;
2) each class $\mathcal{C}_{i}$ contains a type whose representatives are
able to produce offspring in the next class in the order with a positive
probability;
3) particles with types from $\mathcal{C}_{i},i\geq 2,$ are unable to
produce offspring belonging to the classes $\mathcal{C}_{1},...,\mathcal{C}
_{i-1}$.
The methods used in the present paper may be applied to investigate, for
instance, the asymptotic distribution of $\beta _{n}$ for such processes.
Since the needed arguments are too cumbersome and contain no new ideas, we
prefer to concentrate on the case when each class $\mathcal{C}_{i}$ includes
a single type only.
The remainder of the paper is organized as follows. Section \ref{Sec2}
contains some preliminary results. In particular, we recall the statements
from \cite{FN} and \cite{FN2} describing the asymptotic behavior of the
survival probability and the distribution of the number of particles in a
strongly critical decomposable branching process. Section \ref{Sec3} gives a
detailed description of the limiting processes. In Sections \ref{Sec4} and
\ref{Sec5} we check convergence of one-dimensional and finite-dimensional
distributions of the prelimiting processes to the limiting ones. Section \ref
{Sec6} contains the proofs of Theorems \ref{T_SkorohConst} and \ref{T_Skhod1}
. Finally, Section \ref{Sec7} is devoted to the proofs of Theorems \ref
{T_mrcaMany} and \ref{T_type}.
\section{Auxiliary results\label{Sec2}}
For any vector $\mathbf{s}=(s_{1},...,s_{p})$ (the dimension will usually be
clear from the context) and an integer valued vector $\mathbf{k}
=(k_{1}.....k_{p})$ define
\begin{equation*}
\mathbf{s}^{\mathbf{k}}=s_{1}^{k_{1}},...,\,s_{p}^{k_{p}}.
\end{equation*}
Further, let $\mathbf{1}=(1,...,1)$ be a vector of units. It will be
sometimes convenient to write $\mathbf{1}^{(i)}$ for the $i-$dimensional
vector with all its components equal to one.
Let
\begin{equation*}
H_{n}^{(i,N)}(\mathbf{s})=\mathbf{E}\left[ \mathbf{s}^{\mathbf{Z}(n)}|
\mathbf{Z}(0)=\mathbf{e}_{i}\right] =\mathbf{E}\left[ s_{i}^{Z_{i}(n)}...
\,s_{N}^{Z_{N}(n)}|\mathbf{Z}(0)=\mathbf{e}_{i}\right]
\end{equation*}
be the probability generating function for $\mathbf{Z}(n)$ given the process
is initiated at time zero by a single particle of type $i\in \left\{
1,2,...,N\right\}.$ Clearly (recall (\ref{DefNONimmigr})), $H_{1}^{(i,N)}(
\mathbf{s})=h_{i}(\mathbf{s}),\ i=1,...,N$. Denote
\begin{equation*}
Q_{n}^{(i,N)}(\mathbf{s})=1-H_{n}^{(i,N)}(\mathbf{s}
),~Q_{n}^{(i,N)}=1-H_{n}^{(i,N)}(\mathbf{0}),
\end{equation*}
put
\begin{equation*}
\mathbf{H}_{n}(\mathbf{s})=(H_{n}^{(1,N)}(\mathbf{s}),...,H_{n}^{(N,N)}(
\mathbf{s})),~\mathbf{Q}_{n}(\mathbf{s})=(Q_{n}^{(1,N)}(\mathbf{s}
),...,Q_{n}^{(N,N)}(\mathbf{s}))
\end{equation*}
and set
\begin{equation*}
b_{jk}(n)=\mathbf{E}\left[ Z_{j}(n)Z_{k}(n)-\delta _{jk}Z_{j}(n)|\mathbf{Z}
(0)=\vec{e}_{j}\right].
\end{equation*}
The starting point of our arguments is the following theorem being a
simplified combination of the respective results from \cite{FN} and \cite
{FN2}:
\begin{theorem}
\label{T_Foster}Let $\mathbf{Z}(n),n=0,1,..$ be a strongly critical
decomposable multitype branching process satisfying (\ref{matrix1}), (\ref
{Matpos}), (\ref{Maseq}), and (\ref{FinCovar}). Then, as $n\rightarrow
\infty $
\begin{eqnarray}
m_{jj}(n) &=&1,\ m_{ij}(n)\sim a_{ij}n^{j-i},\ i<j, \label{MomentSingle3} \\
b_{jk}(n) &\sim &\hat{a}_{jk}n^{k-j+1},\ j\leq k, \label{Momvariance}
\end{eqnarray}
where $a_{ij}$ and $\hat{a}_{jk}$ are positive constants known explicitly
(see \cite{FN2}, Theorem 1).
Besides \textit{(}see \cite{FN}, Theorem 1), as $n\rightarrow \infty $
\begin{equation}
Q_{n}^{(i,N)}=1-H_{n}^{(i,N)}(\mathbf{0})=\mathbf{P}(\mathbf{Z}( n) \neq
\mathbf{0}|\mathbf{Z}( 0) =\mathbf{e}_{i})\sim c_{iN}n^{-1/2^{N-i}},
\label{SurvivSingle}
\end{equation}
where the constants $c_{iN}$ are the same as in (\ref{Const2}).
\end{theorem}
In the sequel we prove the following Yaglom-type limit theorem being a
compliment to Theorem \ref{T_Foster}.
\begin{theorem}
\label{T_Yaglom}Under the conditions of Theorem \ref{T_Foster}, for any $
\lambda >0$
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{E}\left[ \exp \left\{ -\lambda \frac{
Z_{N}(n)}{b_{N}n}\right\} \Big|\mathbf{Z}( n) \neq \mathbf{0};\mathbf{Z}( 0)
=\mathbf{e}_{i}\right] =1-\Big( \frac{\lambda }{1+\lambda }\Big)
^{1/2^{N-i}}. \label{Yag}
\end{equation}
\end{theorem}
Set $d_{ii}=\sqrt{b_{i}^{-1}m_{i,i+1}}$ , $i=1,2,...,N-1$\ and, for $
j=1,2,...,i-1$ let
\begin{equation}
d_{ji}=\sqrt{b_{j}^{-1}m_{j,j+1}d_{j+1,i}},\quad D_{i}=d_{1i}.
\label{Dreccur}
\end{equation}
Observe that (see (\ref{Const1})) for $k=0,1,2,...,i-1$
\begin{equation}
d_{i-k,i}=(b_{i}m_{i,i+1})^{1/2^{k+1}}c_{i-k,i},\quad
D_{i}=(b_{i}m_{i,i+1})^{1/2^{i}}c_{1i}=(b_{i}m_{i,i+1})^{1/2^{i}}C_{i}.
\label{DcConnection}
\end{equation}
Let $\mathbf{Z}(0)=\mathbf{e_{1}}$ and denote by
\begin{equation*}
T_{i}=\min \left\{ n\geq 1:Z_{1}(n)+Z_{2}(n)+...+Z_{i}(n)=0\right\}
\end{equation*}
the extinction moment of the population generated by the particles of the
first $i$ in order types. Let $\eta _{rj}\left( k,l\right) $ be the number
of daughters of type $j$ of the $l-$th mother of type $r$ belonging to the $
k-$th generation and
\begin{equation*}
W_{pij}=\sum_{r=p}^{i}\sum_{k=0}^{T_{i}}\sum_{q=1}^{Z_{r}(k)}\eta
_{rj}\left( k,q\right)
\end{equation*}
be the total amount of daughters of type $j\geq i+1$ produced by all
particles of types $p,p+1,...,i$ ever born in the process if the process is
initiated at time $n=0$ by a single particle of type $p\leq i.$ Finally, put
\begin{equation*}
W_{pi}=\sum_{j=i+1}^{N}W_{pij}=\sum_{j=i+1}^{N}\sum_{r=p}^{i}
\sum_{k=0}^{T_{i}}\sum_{q=1}^{Z_{r}(k)}\eta _{rj}\left( k,q\right) .
\end{equation*}
We know by (\ref{SurvivSingle}) that
\begin{equation}
Q_{n}^{(1,i)}=\mathbf{P}( T_{i}>n) \sim c_{1i}n^{-2^{-(i-1)}}. \label{Asqq}
\end{equation}
The next lemma describes the tail distributions of $W_{1i,i+1}$ and $W_{1i}$.
\begin{lemma}
\label{L_Laplace} Let Hypothesis A be valid. Then, as $\lambda \downarrow 0$
\begin{equation}
1-\mathbf{E}\left[ e^{-\lambda W_{1i,i+1}}\,|\mathbf{Z}(0)=\mathbf{e}_{1}
\right] \sim d_{1i}\lambda ^{1/2^{i}}=D_{i}\lambda ^{1/2^{i}} \label{Tot1}
\end{equation}
and there exists a constant $F_{i}>0$ such that
\begin{equation}
1-\mathbf{E}\left[ e^{-\lambda W_{1i}}|\mathbf{Z}(0)=\mathbf{e}_{1}\right]
\sim F_{i}\lambda ^{1/2^{i}}. \label{Tot2}
\end{equation}
\end{lemma}
\textbf{Proof.} Set
\begin{equation*}
W_{pi,i+1}(n)=\sum_{r=p}^{i}\sum_{k=0}^{n}\sum_{q=1}^{Z_{r}(k)}\eta
_{rj}\left( k,q\right) ,
\end{equation*}
denote
\begin{equation*}
K_{pi,n}(\mathbf{s};t)=\mathbf{E}\left[
s_{p}^{Z_{p}(n)}...s_{i}^{Z_{i}(n)}t^{W_{pi,i+1}(n)}|\mathbf{Z}(0)=\mathbf{e}
_{p}\right] ,\,K_{pi,n}(t)=K_{pi,n}(\mathbf{1}^{(i-p+1)};t)
\end{equation*}
and put
\begin{equation*}
K_{pi}(t)=\mathbf{E}\left[ t^{W_{pi,i+1}}\big|\,\mathbf{Z}(0)=\mathbf{e}_{p}
\right] =\lim_{n\rightarrow \infty }K_{pi,n}(t)
\end{equation*}
(this limit exists since the random variables $W_{pi,i+1}(n),p=1,2,...,i$
are nondecreasing in $n$). Clearly, to prove the lemma it is sufficient to
show that, as~$t\uparrow 1$
\begin{equation*}
1-K_{1i}(t)=1-\mathbf{E}\left[ t^{W_{1i,i+1}}\,|\mathbf{Z}(0)=\mathbf{e}_{1}
\right] {}\mathbf{\sim {}}d_{1i}(1-t)^{1/2^{i}}.
\end{equation*}
Using properties of branching processes it is not difficult to check that
\begin{equation*}
K_{pi,n+1}(\mathbf{s};t)=h_{p}\left( K_{pi,n}(\mathbf{s};t),...,K_{ii,n}(
\mathbf{s};t),t,\mathbf{1}^{(N-i-1)}\right)
\end{equation*}
implying
\begin{equation*}
K_{pi,n+1}(t)=h_{p}\left( K_{pi,n}(t),...,K_{ii,n}(t),t,\mathbf{1}
^{(N-i-1)}\right) .
\end{equation*}
and
\begin{equation*}
K_{pi}(t)=h_{p}\left( K_{pi}(t),...,K_{ii}(t),t,\mathbf{1}^{(N-i-1)}\right) .
\end{equation*}
In particular,
\begin{equation*}
K_{ii}(t)=h_{i}\left( K_{ii}(t),t,\mathbf{1}^{(N-i-1)}\right) .
\end{equation*}
Since $\mathbf{E}\eta _{ii}=1$ and $b_{i}=\frac{1}{2}Var\eta _{ii}\in
(0,\infty )$, it follows that, as $t\uparrow 1$
\begin{eqnarray*}
1-K_{ii}(t) &=&1-h_{i}\left( K_{ii}(t),t,\mathbf{1}^{(N-i-1)}\right) \\
&=&1-K_{ii}(t)-b_{i}(1-K_{ii}(t))^{2}(1+o(1))+m_{i,i+1}(1-t)
\end{eqnarray*}
or
\begin{equation*}
1-K_{ii}(t)\sim \sqrt{b_{i}^{-1}m_{i,i+1}(1-t)}.
\end{equation*}
This, in particular, proves the statement of the lemma for $i=1$.
Now we use induction and assume that
\begin{equation*}
1-K_{qi}(t)\sim d_{qi}(1-t)^{1/2^{i-q+1}},q=p+1,...,i.
\end{equation*}
Then
\begin{eqnarray*}
1-K_{pi}(t) &=&1-h_{p}\left( K_{pi}(t),...,K_{ii}(t),t,\mathbf{1}
^{(N-i-1)}\right) \\
&=&1-K_{pi}(t)-b_{p}(1-K_{pi}(t))^{2}(1+o(1)) \\
&&+(1+o(1))\left( m_{p,p+1}(1-K_{p+1,i}(t))+\sum_{q=p+2}^{i}m_{pq}\left(
1-K_{qi}(t)\right) \right) \\
&&+(1+o(1))m_{p,i+1}(1-t)
\end{eqnarray*}
implying
\begin{eqnarray*}
1-K_{pi}(t) &\sim &\sqrt{b_{p}^{-1}m_{p,p+1}(1-K_{p+1,i}(t))} \\
&\sim &\sqrt{b_{p}^{-1}m_{p,p+1}d_{p+1,i}}\left( 1-t\right)
^{1/2^{i-p+1}}=d_{pi}\left( 1-t\right) ^{1/2^{i-p+1}}
\end{eqnarray*}
and proving (\ref{Tot1}).
To prove (\ref{Tot2}) it is necessary to use similar arguments. We omit the
details.
Lemma \ref{L_Laplace} is proved.\
From now on and till the end of this section we suppose that
\begin{equation}
s_{k}=\exp (-\lambda _{k}n^{-2^{-(N-k)}})=\exp (-\lambda _{k}n^{-\gamma
_{k}}),\lambda _{k}>0,\,k=1,2,...,N \label{Sasymp}
\end{equation}
and, keeping in mind this assumption, study in Lemmas \ref{L_OneOnly}-\ref
{L_MultiSharp} the asymptotic behavior of the difference $1-H_{m}^{(j,N)}(
\mathbf{s})$ when $m,n\rightarrow \infty .$
\begin{lemma}
\label{L_OneOnly}If
\begin{equation}
m\ll n^{2^{-(N-j)}}=n^{\gamma _{j}} \label{Mnegl}
\end{equation}
then for $N>j$
\begin{equation*}
\lim_{n\rightarrow \infty }n^{\gamma _{j}}Q_{m}^{(j,N)}(\mathbf{s})=\lambda
_{j}.
\end{equation*}
\end{lemma}
\textbf{Proof.} Clearly, it is sufficient to prove the statement for $j=1$
only. Let $r$ be a positive integer such that
\begin{equation*}
1-H_{r}^{(1,1)}(0)\leq 1-s_{1}\leq 1-H_{r-1}^{(1,1)}(0).
\end{equation*}
Since $1-s_{1}\sim \lambda _{1}n^{-\gamma _{1}}$ and $1-H_{r}^{(1,1)}(0)\sim
(b_{1}r)^{-1}\,$as $n,r\rightarrow \infty $, it follows that $r\sim
(b_{1}\lambda _{1})^{-1}n^{\gamma _{1}}$. By the branching property of
probability generating functions we have for $m\ll n^{\gamma _{1}}:$
\begin{eqnarray*}
Q_{m}^{(1,N)}(\mathbf{s}) &\geq &1-H_{m}^{(1,1)}(s_{1})\geq
1-H_{m}^{(1,1)}(H_{r}^{(1,1)}(0)) \\
&=&1-H_{m+r}^{(1,1)}(0)\sim b_{1}^{-1}(m+r)^{-1}\sim \lambda _{1}n^{-\gamma
_{1}}.
\end{eqnarray*}
Besides,
\begin{eqnarray*}
Q_{m}^{(1,N)}(\mathbf{s}) &\leq &1-H_{m}^{(1,1)}(s_{1})+\mathbf{E}\left[
\left( 1-s_{2}^{Z_{2}(m)}...\,s_{N}^{Z_{N}(m)}\right) |\mathbf{Z}(0)=\mathbf{
e}_{1}\right] \\
&\leq &1-H_{m+r-1}^{(1,1)}(0)+\sum_{k=2}^{N}(1-s_{k})\mathbf{E}\left[
Z_{k}(m)|\mathbf{Z}(0)=\mathbf{e}_{1}\right] .
\end{eqnarray*}
We know by (\ref{MomentSingle3}) and (\ref{Sasymp}) that, for a positive
constant $C$
\begin{equation*}
\sum_{k=2}^{N}(1-s_{k})\mathbf{E}\left[ Z_{k}(m)|\mathbf{Z}(0)=\mathbf{e}_{1}
\right] \leq C\sum_{k=2}^{N}\lambda _{k}n^{-\gamma _{k}}m^{k-1}
\end{equation*}
which, in view of (\ref{Mnegl}) is negligible with respect to
\begin{equation*}
C\max_{2\leq i\leq N}\lambda _{i}\times \sum_{k=2}^{N}n^{-\gamma
_{k}}(n^{\gamma _{1}})^{k-1}=C\max_{2\leq i\leq N}\lambda _{i}\times
\sum_{k=2}^{N}n^{(k-1)2^{-(N-1)}-2^{-(N-k)}}.
\end{equation*}
Since $k2^{-(N-1)}-2^{-(N-k)}=2^{-(N-1)}(k-2^{k-1})\leq 0$ for $k\geq 2,$ we
have
\begin{equation*}
n^{2^{-(N-1)}}\sum_{k=2}^{N}n^{(k-1)2^{-(N-1)}-2^{-(N-k)}}=
\sum_{k=2}^{N}n^{k2^{-(N-1)}-2^{-(N-k)}}\leq N-1.
\end{equation*}
Consequently, $Q_{m}^{(1,N)}(\mathbf{s})\sim 1-H_{m}^{(1,1)}(s_{1})\sim
\lambda _{1}n^{-\gamma _{1}}$ as $n\rightarrow \infty $.
This proves the lemma.
In order to formulate the next lemma we introduce a tuple of functions $\phi
_{i}=\phi _{i}( \lambda _{1},\lambda _{2}) ,\ i=1,2,...,N-1$ solving in the
domain $\left\{ \lambda _{1}\geq 0,\lambda _{2}\geq 0\right\} $ the
differential equations
\begin{equation*}
\lambda _{1}\frac{\partial \phi _{i}}{\partial \lambda _{1}}+2\lambda _{2}
\frac{\partial \phi _{i}}{\partial \lambda _{2}}=-b_{i}\phi _{i}^{2}+\phi
_{i}+m_{i,i+1}\lambda _{2}
\end{equation*}
with the initial conditions
\begin{equation*}
\phi _{i}(\mathbf{0})=0,\ \frac{\partial \phi _{i}( \mathbf{0}) }{\partial
\lambda _{1}}=1,\ \frac{\partial \phi _{i}( \mathbf{0}) }{\partial \lambda
_{2}}=m_{i,i+1}.
\end{equation*}
One may check that, for any $y>0$
\begin{equation}
\frac{\phi _{i}(\lambda _{1}y,\lambda _{2}y^{2})}{y}=\sqrt{\frac{
m_{i,i+1}\lambda _{2}}{b_{i}}}\frac{b_{i}\lambda _{1}+\sqrt{
b_{i}m_{i,i+1}\lambda _{2}}\tanh (y\sqrt{b_{i}m_{i,i+1}\lambda _{2}})}{
b_{i}\lambda _{1}\tanh (y\sqrt{b_{i}m_{i,i+1}\lambda _{2}})+\sqrt{
b_{i}m_{i,i+1}\lambda _{2}}}. \label{DefSimpl}
\end{equation}
\begin{lemma}
\label{L_twoOnly}Let condition (\ref{Sasymp}) be valid. If $m\sim yn^{\gamma
_{i}},$ $y>0$ then
\begin{equation*}
\lim_{n\rightarrow \infty }n^{\gamma _{i}}Q_{m}^{(i,N)}(\mathbf{s}
)=y^{-1}\phi _{i}(\lambda _{i}y,\lambda _{i+1}y^{2}).
\end{equation*}
\end{lemma}
\textbf{Proof.} As in the previous lemma, it is sufficient to consider the
case $i=1$ only. It follows from Theorem 2 in \cite{FN2} that for $\lambda
_{k}\geq 0,k=1,2,...,N$
\begin{eqnarray*}
&&\lim_{m\rightarrow \infty }m\left( 1-\mathbf{E}\left[ \exp \left\{
-\sum_{k=1}^{N}\lambda _{k}\frac{Z_{k}(m)}{m^{k}}\right\} \right] \right) \\
&&\qquad \qquad =\lim_{m\rightarrow \infty }m(1-H_{m}^{(1,N)}(e^{-\lambda
_{1}/m},e^{-\lambda _{2}/m^{2}},...,e^{-\lambda _{N}/m^{N}})) \\
&&\qquad \qquad =\Phi (\lambda _{1},\lambda _{2},...,\lambda _{N}),
\end{eqnarray*}
where $\Phi =\Phi (\lambda _{1},\lambda _{2},...,\lambda _{N})$ solves the
differential equation
\begin{equation*}
\sum_{k=1}^{N}k\lambda _{k}\frac{\partial \Phi }{\partial \lambda _{k}}
=-b_{1}\Phi ^{2}+\Phi +\sum_{k=2}^{N}f_{k}\lambda _{k}
\end{equation*}
with the initial conditions
\begin{equation*}
\Phi (\mathbf{0})=0,\ \frac{\partial \Phi (\mathbf{0})}{\partial \lambda _{1}
}=1,\ \frac{\partial \Phi (\mathbf{0})}{\partial \lambda _{k}}=\frac{1}{k-1}
f_{k},\ k=2,...,N
\end{equation*}
and
\begin{equation*}
f_{k}=\frac{1}{(k-2)!}\prod_{j=1}^{k-1}m_{j,j+1},\ k=2,...,N.
\end{equation*}
Since $m^{2^{k-1}}=m^{k}$ for $k=1,2$ and $m^{2^{k-1}}\gg m^{k}$ for $k>2,$
we conclude by the continuity of $\Phi $ at point $\mathbf{0}$ that
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }n^{\gamma _{1}}Q_{m}^{(1,N)}(\mathbf{s}
)=y^{-1}\lim_{m\rightarrow \infty }mQ_{m}^{(1,N)}(\mathbf{s}) \\
&&\quad =y^{-1}\lim_{m\rightarrow \infty }m\left( 1-\mathbf{E}\left[ \exp
\left\{ -\sum_{k=1}^{N}\lambda _{k}\frac{Z_{k}(m)}{n^{1/2^{N-k}}}\right\}
\right] \right) \\
&&\quad =y^{-1}\lim_{m\rightarrow \infty }m\left( 1-\mathbf{E}\left[ \exp
\left\{ -\sum_{k=1}^{N}\lambda _{k}y^{2^{k-1}}\frac{Z_{1}(m)}{m^{2^{k-1}}}
\right\} \right] \right) \\
&&\quad =y^{-1}\Phi (\lambda _{1}y,\lambda _{2}y^{2},0,...,0)=y^{-1}\phi
_{1}(\lambda _{1}y,\lambda _{2}y^{2}).
\end{eqnarray*}
Lemma \ref{L_twoOnly} is proved.\
\begin{lemma}
\label{L_inbetween}Let condition (\ref{Sasymp}) be valid. If, for some $
i\leq N-1$
\begin{equation}
n^{\gamma _{i}}\ll m\ll n^{\gamma _{i+1}} \label{msmall}
\end{equation}
then
\begin{equation*}
\lim_{n\rightarrow \infty }n^{\gamma _{1}}Q_{m}^{(1,N)}(\mathbf{s}
)=D_{i}\left( \lambda _{i+1}\right) ^{1/2^{i}}.
\end{equation*}
\end{lemma}
\textbf{Proof.} It follows from (\ref{Asqq}) and (\ref{msmall}) that
\begin{equation*}
\mathbf{P}(T_{i}>m)\sim c_{1i}m^{-2^{-(i-1)}}=o(n^{-\gamma _{1}}).
\end{equation*}
Therefore,
\begin{eqnarray*}
Q_{m}^{(1,N)}(\mathbf{s}) &=&\mathbf{E}\left[
1-s_{1}^{Z_{1}(m)}s_{2}^{Z_{2}(m)}...\,s_{N}^{Z_{N}(m)}\right] \\
&=&\mathbf{E}\left[ \left(
1-s_{i+1}^{Z_{i+1}(m)}...\,s_{N}^{Z_{N}(m)}\right) ;T_{i}\leq m\right]
+o(n^{-\gamma _{1}}) \\
&=&1-H_{m}^{(1,N)}\left( \mathbf{1}^{(i)},s_{i+1},...,s_{N}\right)
+o(n^{-\gamma _{1}}).
\end{eqnarray*}
It is not difficult to check that for our decomposable branching process
\begin{eqnarray*}
&&H_{m}^{(1,N)}\left( \mathbf{1}^{(i)},s_{i+1},...,s_{N}\right) \\
&&\quad =\mathbf{E}\left[ \prod_{k=0}^{m-1}\prod_{r=1}^{i}
\prod_{l=1}^{Z_{r}(k)}\prod_{j=i+1}^{N}\left( H_{m-k}^{(j,N)}(\mathbf{s}
)\right) ^{\eta _{rj}\left( k,l\right) }\right] \\
&&\quad =\mathbf{E}\left[ \prod_{k=0}^{m-1}\prod_{r=1}^{i}
\prod_{l=1}^{Z_{r}(k)}\prod_{j=i+1}^{N}\left( H_{m-k}^{(j,N)}(\mathbf{s}
)\right) ^{\eta _{rj}\left( k,l\right) };T_{i}\leq \sqrt{mn^{\gamma _{i}}}
\right] \\
&&\qquad +O\left( \mathbf{P}\left( T_{i}> \sqrt{mn^{\gamma _{i}}}\right)
\right) .
\end{eqnarray*}
Observing that $\lim_{m\to\infty}H_{m-k}^{(j,N)}(\mathbf{s})\rightarrow 1$
for $j\geq i+1$ and $k\leq T_{i}\leq \sqrt{mn^{\gamma _{i}}}=o(m),$ we get
on the set $T_{i}\leq \sqrt{mn^{\gamma _{i}}}$
\begin{eqnarray*}
&&\prod_{k=0}^{m-1}\prod_{r=1}^{i}\prod_{l=1}^{Z_{r}(k)}\prod_{j=i+1}^{N}
\left( H_{m-k}^{(j,N)}(\mathbf{s})\right) ^{\eta _{rj}\left( k,l\right) } \\
&&\quad =\exp \left\{
-\sum_{r=1}^{i}\sum_{k=0}^{T_{i}}\sum_{l=1}^{Z_{r}(k)}\sum_{j=i+1}^{N}\eta
_{rj}\left( k,l\right) Q_{m-k}^{(j,N)}(\mathbf{s})(1+o(1))\right\} .
\end{eqnarray*}
If $j\geq i+1$ then Lemma \ref{L_OneOnly} and the estimates $m\ll n^{\gamma
_{i+1}}\leq n^{\gamma _{j}}$ yield
\begin{equation*}
Q_{m-k}^{(j,N)}(\mathbf{s})\sim Q_{m}^{(j,N)}(\mathbf{s})\sim \lambda
_{j}n^{-\gamma _{j}}.
\end{equation*}
Hence it follows that on the set $T_{i}\leq \sqrt{mn^{\gamma _{i}}}
=o(m)=o(n^{\gamma _{i+1}})$
\begin{eqnarray*}
&&\sum_{r=1}^{i}\sum_{k=0}^{T_{i}}\sum_{l=1}^{Z_{r}(k)}\sum_{j=i+1}^{N}\eta
_{rj}\left( k,l\right) Q_{m-k}^{(j,N)}(\mathbf{s}) \\
&&\quad =(1+o(1))\sum_{j=i+1}^{N}Q_{m}^{(j,N)}(\mathbf{s})\sum_{r=1}^{i}
\sum_{k=0}^{T_{i}}\sum_{l=1}^{Z_{r}(k)}\eta _{rj}\left( k,l\right) \\
&&\quad =(1+o(1))\sum_{j=i+1}^{N}W_{1ij}Q_{m}^{(j,N)}(\mathbf{s}) \\
&&\quad =(1+o(1))W_{1i,i+1}Q_{m}^{(i+1,N)}(\mathbf{s})+O\left(
Q_{m}^{(i+2,N)}(\mathbf{s})\right) \sum_{j=i+2}^{N}W_{1ij} \\
&&\quad =(1+o(1))W_{1i,i+1}\lambda _{i+1}n^{-\gamma _{i+1}}+O_{n}(n^{-\gamma
_{i+2}}W_{1i}).
\end{eqnarray*}
Using the estimates
\begin{eqnarray*}
0 &\leq &\mathbf{E}\left[ \exp \left\{ -(1+o(1))W_{1i,i+1}\lambda
_{i+1}n^{-\gamma _{i+1}}\right\} \right] \\
&&-\mathbf{E}\left[ \exp \left\{ -(1+o(1))W_{1i,i+1}\lambda _{i+1}n^{-\gamma
_{i+1}}-O(n^{-\gamma _{i+2}}W_{1i})\right\} \right] \\
&\leq &1-\mathbf{E}\left[ \exp \left\{ -O(n^{-\gamma _{i+2}}W_{1i})\right\}
\right] =O\left( \left( n^{-\gamma _{i+2}}\right) ^{1/2^{i}}\right) =O\left(
n^{-\gamma _{2}}\right)
\end{eqnarray*}
where, for the penultimate equality we applied (\ref{Tot2}), we conclude by (
\ref{Tot1}) that
\begin{eqnarray*}
&&1-H_{m}^{(1,N)}\left( \mathbf{1}^{(i)},s_{i+1},...,s_{N}\right) \\
&&\quad =(1+o(1))\mathbf{E}\left[ 1-\exp \left\{ -(1+o(1))W_{1i,i+1}\lambda
_{i+1}n^{-\gamma _{i+1}}\right\} \right] \\
&&\qquad +O\left( \mathbf{P}\left( T_{i}>\sqrt{mn^{\gamma _{i}}}\right)
\right) \\
&&\quad =(1+o(1))D_{i}\left( \lambda _{i+1}n^{-\gamma _{i+1}}\right)
^{1/2^{i}}+o(n^{-\gamma _{1}})\sim D_{i}(\lambda _{i+1})^{1/2^{i}}n^{-\gamma
_{1}}
\end{eqnarray*}
as desired. \
\begin{lemma}
\label{L_MultiSharp}If $m\sim yn^{\gamma _{i}}$ for some $i\in \left\{
2,3,...,N-1\right\} $ then
\begin{equation*}
\lim_{n\rightarrow \infty }n^{\gamma _{1}}Q_{m}^{(1,N)}(\mathbf{s}
)=D_{i-1}(y^{-1}\phi _{i}(\lambda _{i}y,\lambda _{i+1}y^{2}))^{1/2^{i-1}}.
\end{equation*}
\end{lemma}
\textbf{Proof.} If $m\sim yn^{\gamma _{i}}$ and $j\geq i$ then $n^{\gamma
_{j}}\sim (y^{-1}m)^{2^{j-i}}$ and, therefore,
\begin{equation*}
s_{j}=\exp \left\{ -\lambda _{j}n^{-\gamma _{j}}\right\} =\exp \left\{
-(1+o(1))\lambda _{j}y^{2^{j-i}}m^{-2^{j-i}}\right\} .
\end{equation*}
Hence we may apply Lemma \ref{L_twoOnly} to get, as $n\rightarrow \infty $
\begin{equation*}
n^{\gamma _{i}}Q_{m}^{(i,N)}(\mathbf{s})\sim
y^{-1}mQ_{m}^{(i,N)}(s_{i},s_{i+1},...,s_{N})\sim y^{-1}\phi _{i}(\lambda
_{i}y,\lambda _{i+1}y^{2}).
\end{equation*}
Further, as in the previous lemma we have
\begin{equation*}
Q_{m}^{(1,N)}(\mathbf{s})=1-H_{m}^{(1,N)}\left( \mathbf{1}
^{(i-1)},s_{i},...,s_{N}\right) +o(n^{-\gamma _{1}})
\end{equation*}
and on the set $T_{i-1}\leq \sqrt{mn^{\gamma _{i-1}}}\ll m\sim yn^{\gamma
_{i}}$
\begin{eqnarray*}
&&\sum_{r=1}^{i-1}\sum_{k=0}^{T_{i-1}}\sum_{l=1}^{Z_{r}(k)}\sum_{j=i}^{N}
\eta _{rj}\left( k,l\right) Q_{m-k}^{(j,N)}(\mathbf{s}) \\
&&\quad \quad =(1+o(1))\sum_{j=i}^{N}W_{1,i-1,j}Q_{m}^{(j,N)}(\mathbf{s}) \\
&&\quad =(1+o(1))W_{1,i-1,i}Q_{m}^{(i,N)}(\mathbf{s})+O\left(
Q_{m}^{(i+1,N)}(\mathbf{s})\right) \sum_{j=i+1}^{N}W_{1,i-1,j} \\
&&\quad =(1+o(1))W_{1,i-1,i}(y^{-1}\phi _{i}(\lambda _{i}y,\lambda
_{i+1}y^{2}))^{1/2^{i-1}}n^{-\gamma _{i+1}} \\
&&+O_{n}(n^{-\gamma _{i+2}}W_{1,i-1}).
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&&1-H^{(1,N)}_{m}\left( \mathbf{1}^{(i-1)},s_{i},...,s_{N}\right) \\
&&\quad =\mathbf{E}\left[ 1-\exp \left\{ -(1+o(1))W_{1,i-1,i}y^{-1}\phi
_{i}(\lambda _{i}y,\lambda _{i+1}y^{2})\,n^{-\gamma _{i}}\right\} \right] \\
&&\qquad +O\left( \mathbf{P}\left( T_{i-1}\geq \sqrt{mn^{\gamma _{i+1}}}
\right) \right) \\
&&\quad =(1+o(1))D_{i-1}\big(y^{-1}\phi _{i}(\lambda _{i}y,\lambda
_{i+1}y^{2})\,n^{-\gamma _{i}}\big)^{1/2^{i-1}}+o(n^{-\gamma _{1}}) \\
&&\quad \sim D_{i-1}(y^{-1}\phi _{i}(\lambda _{i}y,\lambda
_{i+1}y^{2}))^{1/2^{i-1}}n^{-\gamma _{1}}.
\end{eqnarray*}
The lemma is proved.
\begin{lemma}
\label{L_DC}For all $i=1,2,...,N-1$
\begin{equation}
C_{N}=C_{i}(m_{i,i+1}b_{i}c_{i+1,N})^{1/2^{i}}=D_{i}(c_{i+1,N})^{1/2^{i}}.
\label{CD}
\end{equation}
\end{lemma}
\textbf{Proof.} Using (\ref{Const1}) we have
\begin{equation*}
c_{iN}=\sqrt{b_{i}^{-1}m_{i,i+1}c_{i+1,N}}=b_{i}^{-1}\sqrt{
b_{i}m_{i,i+1}c_{i+1,N}}=c_{ii}\sqrt{b_{i}m_{i,i+1}c_{i+1,N}}
\end{equation*}
leading in view of (\ref{Const2}) and (\ref{DcConnection}) to
\begin{eqnarray*}
C_{N} &=&c_{1N}=\left( \frac{1}{b_{N}}\right)
^{1/2^{N-1}}\prod_{j=1}^{N-1}\left( \frac{m_{j,j+1}}{b_{j}}\right)
^{1/2^{j}}= \\
&=&c_{1i}(b_{i}m_{i,i+1})^{1/2^{i}}\left( \left( \frac{1}{b_{N}}\right)
^{1/2^{N-i}}\prod_{j=i+1}^{N-1}\left( \frac{m_{j,j+1}}{b_{j}}\right)
^{1/2^{j-i}}\right) ^{1/2^{i}} \\
&=&c_{1i}(b_{i}m_{i,i+1}c_{i+1,N})^{1/2^{i}}=D_{i}(c_{i+1,N})^{1/2^{i}}
\end{eqnarray*}
as desired.
\section{Properties of the limiting processes\label{Sec3}}
In this section we give a more detailed description of the properties of the
limiting processes. It follows from the definition of $\mathbf{R}(t)$ that
if
\begin{equation*}
\mathbf{S}_{i}=(s_{i1},s_{i2},...,s_{iN})\in \left[ 0,1\right] ^{N}\text{
and }t_{i}\in \lbrack \gamma _{i-1},\gamma _{i}),i=1,2,...,N,
\end{equation*}
then
\begin{equation*}
\mathbf{E}\left[ \prod_{i=1}^{N}\mathbf{S}_{i}^{\mathbf{R}(t_{i})}\right]
=\Omega _{N}(s_{11},s_{22},...,s_{NN}),
\end{equation*}
where $\Omega _{1}(s)=s$ and
\begin{equation}
\Omega _{i+1}(s_{1},s_{2},...,s_{i+1})=s_{1}\left( 1-\sqrt{1-\Omega
_{i}(s_{2},...,s_{i+1})}\right) ,\,i=1,2,.... \label{DefOmega}
\end{equation}
If now some intervals $[\gamma _{i-1},\gamma _{i})$ contain more than one
point of observation over the process $\mathbf{R}(\cdot )$, say, $\gamma
_{i-1}\leq t_{i1}<t_{i2}<...<t_{ik_{i}}<\gamma _{i},i=1,2,...,N,$ and $_{j}
\mathbf{S}_{i}=\left( _{j}s_{i1},_{j}s_{i2},...,_{j}s_{iN}\right) \in \left[
0,1\right] ^{N}\text{ }$ then, clearly,
\begin{equation*}
\mathbf{E}\left[ \prod_{i=1}^{N}\prod_{j=1}^{k_{i}}(_{j}\mathbf{S}_{i})^{
\mathbf{R}(t_{ij})}\right] =\Omega _{N}\left(
\prod_{j=1}^{k_{1}}\,_{j}s_{11},\prod_{j=1}^{k_{2}}\,_{j}s_{22},...,
\prod_{j=1}^{k_{N}}\,_{j}s_{NN}\right) .
\end{equation*}
To describe the characteristics of the processes $\mathbf{U}_{i}(\cdot
),i=1,...,N-1$, let, for $(s_{i},s_{i+1})\in \lbrack 0,1]^{2}$
\begin{equation}
\varphi _{i}(y;s_{i},s_{i+1})=\sqrt{1-s_{i+1}}\frac{(1-s_{i})+\sqrt{1-s_{i+1}
}\tanh (b_{i}c_{iN}y\sqrt{1-s_{i+1}})}{(1-s_{i})\tanh (b_{i}c_{iN}y\sqrt{
1-s_{i+1}})+\sqrt{1-s_{i+1}}} \label{Deffi}
\end{equation}
with the natural agreement $\varphi _{i}(y;1,1)=0$ and
\begin{equation*}
\varphi _{i}(y;s_{i},1)=\frac{1-s_{i}}{b_{i}c_{iN}y(1-s_{i})+1}.
\end{equation*}
Denote
\begin{eqnarray*}
X_{i}(y;\mathbf{s}) &=&X_{i}(y;s_{i},s_{i+1})=\mathbf{E}\left[ \mathbf{s}^{
\mathbf{U}_{i}(y)}|\mathbf{U}_{i}(0)=\mathbf{e}_{i}\right] \\
&=&\mathbf{E}\left[ s_{i}^{U_{ii}(y)}s_{i+1}^{U_{i,i+1}(y)}|\mathbf{U}
_{i}(0)=\mathbf{e}_{i}\right]
\end{eqnarray*}
and set
\begin{equation*}
\bar{X}_{R_{i}}(y;\mathbf{s})=\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=\mathbf{E}
_{R_{i}}\left[ \mathbf{s}^{\mathbf{U}_{i}(y)}\right] =\mathbf{E}_{R_{i}}
\left[ s_{i}^{U_{ii}(y)}s_{i+1}^{U_{i,i+1}(y)}\right] ,
\end{equation*}
where the symbol $\mathbf{E}_{R_{i}}[\cdot ]$ means that the process starts
by a random number of type $i$ particles distributed as $R_{i}$ in (\ref
{Dist_ro}).
It follows from the description of the branching mechanism for $\mathbf{U}
_{i}(\cdot )$ and the general theory of branching processes (see, for
instance, \cite{AN72}, p. 201) that $X_{i}(y;s_{i},s_{i+1})$ solves the
differential equation
\begin{eqnarray*}
\frac{\partial }{\partial y}X_{i}(y;s_{i},s_{i+1}) &=&2b_{i}c_{iN}\left(
\frac{1}{2}X_{i}^{2}(y;s_{i},s_{i+1})-X_{i}(y;s_{i},s_{i+1})+\frac{1}{2}
s_{i+1}\right) , \\
X_{i}(0;s_{i},s_{i+1}) &=&s_{i}.
\end{eqnarray*}
Direct calculations show that
\begin{equation}
X_{i}(y;s_{i},s_{i+1})=1-\varphi _{i}(y;s_{i},s_{i+1}) \label{ExplX}
\end{equation}
and, as a result
\begin{equation}
\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=1-(\varphi
_{i}(y;s_{i},s_{i+1}))^{1/2^{i-1}}\text{.} \label{ExplXstar}
\end{equation}
One may check by (\ref{Deffi}) and (\ref{ExplXstar}) that
\begin{equation}
\lim_{y\downarrow 0}\bar{X}_{R_{i}}(y;s_{i},s_{i+1})=1-(1-s_{i})^{1/2^{i-1}}
\text{ } \label{LxLeft}
\end{equation}
and
\begin{equation}
\lim_{y\uparrow \infty }\bar{X}
_{R_{i}}(y;s_{i},s_{i+1})=1-(1-s_{i+1})^{1/2^{i}}. \label{liminf}
\end{equation}
For $y_{k}\in \lbrack 0,\infty ),\,(s_{ki},s_{k,i+1})\in \left[ 0,1\right]
^{2},\ k=1,2,...,p;\ i=1,...,N-1$ denote $\mathbf{y}_{l,p}=(y_{l},...,y_{p})$
and $\mathbf{S}
_{l,p}^{(i)}=(s_{li},s_{l,i+1},s_{l+1,i},s_{l+1,i+1},...,s_{pi},s_{p,i+1}).$
Using (\ref{ExplX}) set
\begin{equation*}
X_{i}^{(2)}\left( \mathbf{y}_{1,2};\mathbf{S}_{1,2}^{(i)}\right)
=X_{i}\left(
y_{1};s_{1i}X_{i}(y_{2};s_{2i},s_{2,i+1}),s_{1,i+1}s_{2,i+1}\right)
\end{equation*}
and, by induction
\begin{equation*}
X_{i}^{(p)}\left( \mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\right)
=X_{i}\left( y_{1};s_{1i}X_{i}^{(p-1)}\left( \mathbf{y}_{2,p};\mathbf{S}
_{2,p}^{(i)}\right) ,\prod_{r=1}^{p}s_{r,i+1}\right) .
\end{equation*}
Finally, recalling (\ref{ExplXstar}) put
\begin{equation*}
\bar{X}_{R_{i}}\left( \mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\right)
=1-\left( 1-X_{i}^{(p)}\left( \mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\right)
\right) ^{1/2^{i-1}}.
\end{equation*}
It is not difficult to check that
\begin{equation*}
\bar{X}_{R_{i}}\left( \mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\right) =
\mathbf{E}_{R_{i}}\left[
s_{1i}^{U_{ii}(y_{1})}s_{1,i+1}^{U_{i,i+1}(y_{1})}...s_{pi}^{U_{ii}(y_{p})}s_{p,i+1}^{U_{i,i+1}(y_{p})}
\right] .
\end{equation*}
To complete the description of the limiting processes we are interesting in
introduce the function
\begin{equation*}
\psi (x;s)=\frac{1}{x+(1-x)/(1-s)},\quad s\in \lbrack 0,1],\,x\in \lbrack
0,1],
\end{equation*}
and consider an $N-$dimensional process $\mathbf{U}_{N}(\cdot
)=(U_{N1}(\cdot ),...,U_{NN}(\cdot ))$ in which the first $N-1$ components
are equal to zero while $U_{NN}(\cdot )$ may be obtained by a time-change
from the following single-type continuous time Markov process $\sigma
(t),0\leq t<\infty $. The life-length distribution of particles in $\sigma
(\cdot )$ is exponential with parameter 1. Dying each particle produces
exactly two children. One may check (compare, for instance, with Example 3,
Section 8, Chapter 1 in \cite{Sev74}) that
\begin{equation*}
\mathbf{E}\left[ s^{\sigma (t)}|\sigma (0)=1\right] =1-\psi (1-e^{-t};s).
\end{equation*}
Assuming that $\sigma (0)\overset{d}{=}R_{N}$ (recall (\ref{DefroN})) and
making the change of time $x=1-e^{-t},0\leq t<\infty ,$ we obtain an
inhomogeneous single-type branching process, denoted by $U_{NN}(\cdot )$
such that
\begin{equation*}
\bar{G}_{R_{N}}(x;s)=\mathbf{E}_{R_{N}}\left[ s^{U_{NN}(x)}\right] =1-(\psi
(x;s))^{1/2^{N-1}}
\end{equation*}
and
\begin{equation*}
\mathbf{E}\left[ s^{U_{NN}(x+\Delta )}|U_{NN}(x)=1\right] =1-\psi \left(
\frac{\Delta }{1-x};s\right) ,\ 0<x+\Delta <1.
\end{equation*}
Let, further, for $x_{j}\in \lbrack 0,1)$ and $\mathbf{S}
_{j,p}=(s_{j},...,s_{p}),\,j=1,2,...,p$
\begin{equation*}
G^{(1)}(x_{1};s_{1})=G(x_{1};s_{1})=1-\psi (x_{1};s_{1})
\end{equation*}
and, by induction
\begin{equation*}
G^{(p)}\left( \mathbf{x}_{1,p};\mathbf{S}_{1,p}\right) =G\left(
x_{1};s_{1N}G^{(p-1)}\left( \frac{\mathbf{x}_{2,p}}{1-x_{1}};\mathbf{S}
_{2,p}\right) \right) .
\end{equation*}
One may check that
\begin{eqnarray*}
\bar{G}_{R_{N}}(\mathbf{x}_{1,p};\mathbf{S}_{1,p}) &=&\mathbf{E}_{R_{N}}
\left[ s_{1}^{U_{NN}(x_{1})}s_{2}^{U_{NN}(x_{2})}...\,s_{p}^{U_{NN}(x_{p})}
\right] \\
&=&1-(1-G^{(p)}(\mathbf{x}_{1,p};\mathbf{S}_{1,p}))^{1/2^{N-1}}.
\end{eqnarray*}
\section{Convergence of one-dimensional distributions\label{Sec4}}
As the first step in proving the main results of the paper we establish
convergence of one-dimensional distributions of $\{ \mathbf{Z}(m,n),0\leq
m\leq n\} $ given $\mathbf{Z}(n)\neq \mathbf{0}$. Let
\begin{eqnarray*}
H_{m,n}^{\,(k,N)}(\mathbf{s})=\mathbf{E}\left[ \mathbf{s}^{\mathbf{Z}(m,n)}|
\mathbf{Z}(0)=\mathbf{e}_{k}\right], J_{m,n}^{\,(k,N)}(\mathbf{s})=\mathbf{E}
\left[ \mathbf{s}^{\mathbf{Z}(m,n)}|\mathbf{Z}(n)\neq \mathbf{0},\mathbf{Z}
(0)=\mathbf{e}_{k}\right],
\end{eqnarray*}
\begin{eqnarray*}
\mathbf{H}_{m,n}(\mathbf{s}) =\left( H_{m,n}^{\,(1,N)}(\mathbf{s}
),...,H_{m,n}^{\,(N,N)}(\mathbf{s})\right),\ \mathbf{J}_{m,n}(\mathbf{s}
)=\left( J_{m,n}^{\,(1,N)}(\mathbf{s}),...,J_{m,n}^{\,(N,N)}(\mathbf{s}
)\right).
\end{eqnarray*}
For $\mathbf{x}=(x_{1},...,x_{N})$ and $\mathbf{y}=(y_{1},...,y_{N})$ put $
\mathbf{x}\otimes \mathbf{y=}(x_{1}y_{1},x_{2}y_{2},...,x_{N}y_{N})$ and
denote
\begin{eqnarray}
s_{k}^{\prime }
&=&s_{k}Q_{n-m}^{(k,N)}+(1-Q_{n-m}^{(k,N)})=1-(1-s_{k})Q_{n-m}^{(k,N)},
\notag \\
&& \notag \\
\mathbf{s}^{\prime } &=&(s_{1}^{\prime },...,s_{N}^{\prime })=\mathbf{1}-(
\mathbf{1}-\mathbf{s})\otimes \mathbf{Q}_{n-m}. \label{sprimeVect}
\end{eqnarray}
It is not difficult to understand that
\begin{equation*}
H_{m,n}^{(k,N)}(\mathbf{s})=H_{m}^{(k,N)}(\mathbf{s}^{\prime
})=H_{m}^{(k,N)}(\mathbf{1}-(\mathbf{1}-\mathbf{s})\otimes \mathbf{Q}_{n-m})
\end{equation*}
and that
\begin{equation}
J_{m,n}^{\,(k,N)}(\mathbf{s})=\mathbf{E}\left[ \mathbf{s}^{\mathbf{Z}(m,n)}|
\mathbf{Z}(n)\neq \mathbf{0},\mathbf{Z}(0)=\mathbf{e}_{k}\right] =1-\frac{
Q_{m}^{\,(k,N)}(\mathbf{s}^{\prime })}{Q_{n}^{\,(k,N)}}. \label{BranProp2}
\end{equation}
\begin{theorem}
\label{T_familyMany}Let Hypothesis A be valid.
1) If $m\ll n^{\gamma _{1}}$ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(1,N)}(\mathbf{s})=\lim_{n\rightarrow
\infty }\mathbf{E}\left[ \mathbf{s}^{\mathbf{Z}(m,n)}|\mathbf{Z}(n)\neq
\mathbf{0},\mathbf{Z}(0)=\mathbf{e}_{1}\right] =s_{1}. \label{LimSing}
\end{equation}
2) If $n^{\gamma _{i}}\ll m\ll n^{\gamma _{i+1}}$ for some $i\in \left\{
1,2,...,N-1\right\} $ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(1,N)}(\mathbf{s}
)=1-(1-s_{i+1})^{1/2^{i}}. \label{LimSIngLate}
\end{equation}
3) If $m=(y+l_{n})n^{\gamma _{i}},\,y\in \lbrack 0,\infty )$ for some $i\in
\left\{ 1,2,...,N-1\right\} $ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(1,N)}(\mathbf{s})=\bar{X}
_{R_{i}}(y;s_{i},s_{i+1}). \label{LimTWo}
\end{equation}
4) If $m=(x+l_{n})n$, $x\in \lbrack 0,1)$ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{{}\,(1,N)}(\mathbf{s})=\bar{G}
_{R_{N}}(x;s_{N}). \label{LimOLD}
\end{equation}
\end{theorem}
\textbf{Proof.} We start by observing that if $m\ll n$ then
\begin{eqnarray*}
1-s_{i}^{\prime } &=&(1-s_{i})Q_{n-m}^{(i,N)}\sim (1-s_{i})Q_{n}^{(i,N)} \\
&\sim &1-\exp \left\{ -(1-s_{i})Q_{n}^{(i,N)}\right\} \sim 1-\exp \left\{
-(1-s_{i})c_{iN}n^{-\gamma _{i}}\right\} .
\end{eqnarray*}
This representation allows us to use the previous results with $s_{i}$ and $
\lambda _{i}$ replaced by $s_{i}^{\prime }$ and $(1-s_{i})c_{iN}$,
respectively.
Recalling (\ref{SurvivSingle}) and applying Lemma \ref{L_OneOnly} we get
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{Q_{m}^{(1,N)}(\mathbf{s}^{\prime}) }{
Q_{n}^{(1,N)}}=\lim_{n\rightarrow \infty }n^{2^{-(N-1)}}\frac{
1-H_{m}^{(1,N)}( \mathbf{s}^{\prime }) }{C_{N}}=1-s_{1}.
\end{equation*}
Hence (\ref{LimSing}) follows.
Applying Lemma \ref{L_inbetween} with $n^{\gamma _{i}}\ll m\ll n^{\gamma
_{i+1}}$ and recalling Lemma \ref{L_DC} we conclude
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{Q_{m}^{(1,N)}(\mathbf{s}^{\prime })}{
Q_{n}^{(1,N)}}=\frac{D_{i}}{C_{N}}
((1-s_{i+1})c_{i+1,N})^{1/2^{i}}=(1-s_{i+1})^{1/2^{i}}
\end{equation*}
leading to (\ref{LimSIngLate}).
\textbf{Proof of (\ref{LimTWo}).} If $y=0$ then the needed statement follows
from~(\ref{LimSing}) and~(\ref{LimSIngLate}). If $i\in \left\{
1,2,...,N-1\right\} $ is fixed and $m\sim yn^{\gamma _{i}},y>0,$ then for $
j\geq i$
\begin{eqnarray*}
1-s_{j}^{\prime } &\sim &1-\exp \left\{ -(1-s_{j})c_{jN}n^{-\gamma
_{j}}\right\} \\
&\sim &1-\exp \left\{ -(1-s_{j})c_{jN}y^{2^{j-i}}m^{-2^{j-i}}\right\} .
\end{eqnarray*}
Hence, by (\ref{SurvivSingle}) and Lemmas \ref{L_twoOnly} and \ref
{L_MultiSharp} we get
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{Q_{m}^{(1,N)}(\mathbf{s}^{\prime })}{
Q_{n}^{(1,N)}}=\frac{D_{i-1}}{C_{N}}\left( \frac{\phi
_{i}(c_{iN}(1-s_{i})y,c_{i+1,N}(1-s_{i+1})y^{2})}{y}\right) ^{1/2^{i-1}}
\end{equation*}
where we agree to write $D_{0}=1$. By~(\ref{DefSimpl}) and~(\ref{Const1})
\begin{eqnarray*}
&&\frac{\phi _{i}(c_{iN}(1-s_{i})y,c_{i+1,N}(1-s_{i+1})y^{2})}{y} \\
&&\quad =\sqrt{\frac{m_{i,i+1}c_{i+1,N}(1-s_{i+1})}{b_{i}}}\times \\
&&\qquad \times \frac{b_{i}c_{iN}(1-s_{i})+\sqrt{
b_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}\tanh y\sqrt{
b_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}}{b_{i}c_{iN}(1-s_{i})\tanh y\sqrt{
b_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}+\sqrt{b_{i}m_{i,i+1}c_{i+1,N}(1-s_{i+1})}
} \\
&&\quad =c_{iN}\sqrt{1-s_{i+1}}\times \frac{b_{i}c_{iN}(1-s_{i})+b_{i}c_{iN}
\sqrt{1-s_{i+1}}\tanh (yb_{i}c_{iN}\sqrt{1-s_{i+1}})}{b_{i}c_{iN}(1-s_{i})
\tanh (yb_{i}c_{iN}\sqrt{1-s_{i+1}})+b_{i}c_{iN}\sqrt{1-s_{i+1}}} \\
&&\quad =c_{iN}\sqrt{1-s_{i+1}}\times \frac{1-s_{i}+\sqrt{1-s_{i+1}}\tanh
(yb_{i}c_{iN}\sqrt{1-s_{i+1}})}{(1-s_{i})\tanh (yb_{i}c_{iN}\sqrt{1-s_{i+1}}
)+\sqrt{1-s_{i+1}}}.
\end{eqnarray*}
To complete the proof of (\ref{LimTWo}) it remains to recall (\ref{CD}).
\textbf{Proof of (\ref{LimOLD}).} If $x=0$ then (\ref{LimOLD}) follows from (
\ref{LimSIngLate}). Consider now the case $m\sim xn,0<x<1$. Observe that for
$\mathbf{s}=(s_{1},s_{2},...,s_{N})\in \left[ 0,1\right] ^{N}$
\begin{eqnarray}
H_{m}^{(1,N)}(\mathbf{1}^{(N-1)},s_{N})-H_{m}^{(1,N)}(\mathbf{s}) &=&\mathbf{
E}\left[ \left( 1-s_{1}^{Z_{1}(m)}...\,s_{N-1}^{Z_{N-1}(m)}\right)
s_{N}^{Z_{N}(m)}\right] \notag \\
\quad \leq \mathbf{E}\left[ 1-s_{1}^{Z_{1}(m)}...\,s_{N-1}^{Z_{N-1}(m)}
\right] &\leq &\mathbf{P}(T_{N-1}>m)\leq cm^{-2^{-(N-2)}}. \label{Neglig}
\end{eqnarray}
Thus,
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s})=1-H_{m}^{(1,N)}\left( \mathbf{1}
^{(N-1)},s_{N}\right) +\varepsilon _{m,n}(\mathbf{s})Q_{m}^{(1,N)}
\end{equation*}
where $\varepsilon _{m,n}(\mathbf{s})\rightarrow 0$ as $n\rightarrow \infty
, $ $m\sim xn$ $\ $uniformly in $\mathbf{s}\in \left[ 0,1\right] ^{N}.$
Therefore,
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })=1-H_{m}^{(1,N)}\left( \hat{\mathbf{s}}
,1-(1-s_{N})Q_{n-m}^{(N,N)}\right) +\varepsilon _{m,n}^{\prime }(\mathbf{s}
)Q_{n}^{(1,N)}
\end{equation*}
where $\varepsilon _{m,n}^{\prime }(\mathbf{s})\rightarrow 0$ as $
n\rightarrow \infty ,$ $m\sim xn$ $\ $uniformly in $\hat{\mathbf{s}}=\left(
s_{1}^{\prime },...,s_{N-1}^{\prime }\right) \in \left[ 0,1\right] ^{N-1}$.
We now select an integer $r=r(m,n)\in \mathbb{N}^{\ast }=\left\{
1,2,...,\right\} $ in such a way that
\begin{equation*}
H_{r-1}^{(N,N)}(0)\leq 1-(1-s_{N})Q_{n-m}^{(N,N)}\leq H_{r}^{(N,N)}(0)
\end{equation*}
or
\begin{equation*}
Q_{r}^{(N,N)}=1-H_{r}^{(N,N)}(0)\leq (1-s_{N})Q_{n-m}^{(N,N)}\leq
Q_{r-1}^{(N,N)}=1-H_{r-1}^{(N,N)}(0).
\end{equation*}
This is possible, since by (\ref{SurvivSingle})
\begin{equation}
Q_{n-m}^{(N,N)}\sim \frac{1}{(n-m)b_{N}}\rightarrow 0,\ n-m\rightarrow
\infty . \label{ASRep}
\end{equation}
In particular,
\begin{equation}
r\sim \frac{n-m}{1-s_{N}}. \label{ASSR}
\end{equation}
Under our choice of $r$, for any $\hat{\mathbf{s}}\in \left[ 0,1\right]
^{N-1}$
\begin{equation*}
H_{m}^{(1,N)}\left( \hat{\mathbf{s}},H_{r-1}^{(N,N)}(0)\right) \leq
H_{m}^{(1,N)}\left( \hat{\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\right) \leq
H_{m}^{(1,N)}\left( \hat{\mathbf{s}},H_{r}^{(N,N)}(0)\right) .
\end{equation*}
Letting $\hat{\mathbf{s}}=\left( H_{r}^{(1,N)}(\mathbf{0}
),...,H_{r}^{(N-1,N)}(\mathbf{0})\right) $ we get by the branching property
of generating functions the estimate
\begin{equation*}
H_{m}^{(1,N)}\left( \hat{\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\right) \leq
H_{m}^{(1,N)}(\mathbf{H}_{r}(\mathbf{0}))=H_{m+r}^{(1,N)}(\mathbf{0})
\end{equation*}
implying
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })\geq 1-H_{m+r}^{(1,N)}(\mathbf{0}
)+\varepsilon _{m,n}^{\prime }Q_{n}^{(1,N)}=Q_{m+r}^{(1,N)}+\varepsilon
_{m,n}^{\prime }Q_{n}^{(1,N)},
\end{equation*}
where $\varepsilon _{m,n}^{\prime }\rightarrow 0$ as $n\rightarrow \infty ,$
$m\sim xn,$ while $\hat{\mathbf{s}}=(H_{r-1}^{(1,N)}(\mathbf{0}
),...,H_{r-1}^{(N-1,N)}(\mathbf{0}))$ gives the inequality
\begin{equation*}
H_{m}^{(1,N)}\left( \hat{\mathbf{s}},1-(1-s_{N})Q_{n-m}^{(N,N)}\right) \geq
H_{m}^{(1,N)}(\mathbf{H}_{r}(\mathbf{0}))=H_{m+r-1}^{(1,N)}(\mathbf{0})
\end{equation*}
leading in the range under consideration to
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })\leq Q_{m+r-1}^{(1,N)}+\varepsilon
_{m,n}^{\prime }Q_{n}^{(1,N)}.
\end{equation*}
Hence
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })=Q_{m+r}^{(1,N)}+\varepsilon
_{m,n}^{\prime \prime }Q_{n}^{(1,N)}
\end{equation*}
where $\varepsilon _{m,n}^{\prime \prime }\rightarrow 0$ as $n\rightarrow
\infty ,$ $m\sim xn$. We now conclude by (\ref{SurvivSingle}) that
\begin{equation*}
1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })\sim Q_{m+r}^{(1,N)}\sim
C_{N}(m+r)^{-2^{-(N-1)}}.
\end{equation*}
Hence, on account of (\ref{ASSR}) and $m\sim xn,0<x<1,$ we get (recall (\ref
{Const1}))
\begin{eqnarray*}
\lim_{n\rightarrow \infty }\frac{1-H_{m}^{(1,N)}(\mathbf{s}^{\prime })}{
Q_{n}^{(1,N)}} &=&\lim_{n\rightarrow \infty }\left( \frac{n}{
nx+n(1-x)/(1-s_{N})}\right) ^{2^{-(N-1)}} \\
&=&\left( \frac{1}{x+(1-x)/(1-s_{N})}\right) ^{2^{-(N-1)}}
\end{eqnarray*}
completing the proof of (\ref{LimOLD}).
Theorem \ref{T_familyMany} is proved.\
\textbf{Proof} \textbf{of Theorem \ref{T_Yaglom}}. Since our process is
decomposable and strongly critical, it is sufficient to check (\ref{Yag})
for $i=1$ only. For $\hat{s}_{N}=\exp (-\lambda /(nb_{N}))$ we have
\begin{equation*}
\mathbf{E}\left[ \exp \left\{ -\lambda \frac{Z_{N}(n)}{b_{N}n}\right\} \Big|
\mathbf{Z}(n)\neq \mathbf{0}\right] =1-\frac{1-H_{n}^{(1,N)}(\mathbf{1}
^{(N-1)},\hat{s}_{N})}{Q_{n}^{(1,N)}}.
\end{equation*}
We now select an integer $r=r(\lambda ,n)\in \mathbb{N}^{\ast }=\left\{
1,2,...,\right\} $ in such a way that
\begin{equation*}
H_{r-1}^{(N,N)}(0)\leq \hat{s}_{N}\leq H_{r}^{(N,N)}(0).
\end{equation*}
It follows from (\ref{ASRep})\ that $r\sim n\lambda ^{-1}$. Letting $
s_{i}=H_{r}^{(i,N)}(\mathbf{0}),i=1,2,...,N,$ and setting $\mathbf{s}
=(s_{1},...,s_{N})$ we get by (\ref{Neglig}) after evident estimates that
\begin{equation*}
\left\vert H_{n}^{(1,N)}(\mathbf{1}^{(N-1)},\hat{s}_{N})-H_{n}^{(1,N)}(
\mathbf{s})\right\vert \leq cn^{1/2^{N-2}}.
\end{equation*}
Hence, using (\ref{SurvivSingle}) with $i=1$ we obtain
\begin{eqnarray*}
\frac{1-H_{n}^{(1,N)}(\mathbf{1}^{(N-1)},\hat{s}_{N})}{Q_{n}^{(1,N)}} &\sim &
\frac{1-H_{n}^{(1,N)}(\mathbf{s})}{Q_{n}^{(1,N)}}=\frac{Q_{r+n}^{(1,N)}}{
Q_{n}^{(1,N)}} \\
&\sim &\left( \frac{n}{r+n}\right) ^{1/2^{N-1}}\sim \left( \frac{\lambda }{
1+\lambda }\right) ^{1/2^{N-1}}
\end{eqnarray*}
as desired.\
\section{Convergence of finite-dimensional distributions\label{Sec5}}
In this section we study the limiting behavior of the finite-dimensional
distributions of the reduced process $\left\{ \mathbf{Z}( m,n) ,0\leq m\leq
n\right\} $. Our first theorem deals with the case $m\ll n$.
\begin{theorem}
\label{T_findim} Let Hypothesis A be valid and $\mathbf{S}
_{l}=(s_{l1},...,s_{lN}),l=1,2,...,p$.
1) If, for a fixed $i\in \left\{ 0,1,...,N-1\right\}$
\begin{equation*}
n^{\gamma _{i}}\ll m_{l}\ll n^{\gamma _{i+1}},\quad l=1,...,p
\end{equation*}
then
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{
\mathbf{Z}(m_{l},n)}\,\Big|\,\mathbf{Z}(n)\neq 0\right] =1-\left(
1-\prod_{l=1}^{p}s_{l,i+1}\right) ^{1/2^{i}}. \label{Lim_diskr}
\end{equation}
2) Let $0=Y_{1}<Y_{2}<...<Y_{p}<\infty $ be a tuple of nonnegative numbers
with $y_{1}=0,$ $y_{l}=Y_{l}-Y_{l-1},\,l=2,...,p.$ If, for a fixed $i\in
\left\{ 1,2,...,N-1\right\} $
\begin{equation*}
m_{1}\sim l_{n}n^{\gamma _{i}},\quad m_{l}\sim Y_{l}n^{\gamma
_{i}},\,l=2,...,p
\end{equation*}
then
\begin{equation}
\lim_{n\rightarrow \infty }\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{
\mathbf{Z}(m_{l},n)}\,\Big|\,\mathbf{Z}(n)\neq 0\right] =\bar{X}_{R_{i}}\Big(
\mathbf{y}_{1,p};\mathbf{S}_{1,p}^{(i)}\Big). \label{Lim_Multi}
\end{equation}
\end{theorem}
The second theorem is devoted to the finite-dimensional distributions of the
reduced process when $m$ is of order $n$.
\begin{theorem}
\label{T_endpoint}Let\textbf{\ }Hypothesis A be valid and $
0=X_{1}<X_{2}<...<X_{p}<1$ be a tuple of nonnegative numbers with $x_{1}=0,$
$x_{l}=X_{l}-X_{l-1},\,l=2,...,p$. If
\begin{equation*}
m_{1}\sim l_{n}n,\quad m_{l}\sim X_{l}n,\,l=2,...,p
\end{equation*}
then
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{
\mathbf{Z}(m_{l},n)}\,\Big|\,\mathbf{Z}(n)\neq 0\right] =\bar{G}_{R_{N}}\Big(
\mathbf{x}_{1,p};\mathbf{S}_{1,p;N}\Big),
\end{equation*}
where $\mathbf{S}_{1,p;N}=(s_{1N},s_{2N},...,s_{pN})$.
\end{theorem}
To prove Theorems \ref{T_findim} and \ref{T_endpoint} we need additional
notation.
For $0\leq m_{0}<m_{1}<...<m_{p}\leq n$ set $\mathbf{m}
=(m_{0},m_{1},...,m_{p}),$ put $\Delta _{i}=m_{i}-m_{i-1}$, and denote
\begin{eqnarray*}
\hat{J}_{m_{0},m_{1},...,m_{p},n}^{\,(i,N)}(\mathbf{S}_{1},...,\mathbf{S}
_{p}) &=&\hat{J}_{\mathbf{m},n}^{\,(i,N)}(\mathbf{S}_{1},...,\mathbf{S}_{p})
\\
&=&\mathbf{E}\left[ \prod_{l=1}^{p}\mathbf{S}_{l}^{\mathbf{Z}(m_{l},n)}\,
\Big|\,\mathbf{Z}(m_{0},n)=\mathbf{e}_{i}\right]
\end{eqnarray*}
and
\begin{equation*}
\mathbf{\hat{J}}_{\mathbf{m},n}(\mathbf{S}_{1},...,\mathbf{S}_{p})=\left(
\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},...,\mathbf{S}_{p}),...,\hat{
J}_{\mathbf{m},n}^{\,(N,N)}(\mathbf{S}_{1},...,\mathbf{S}_{p})\right) .
\end{equation*}
The next statement is a simple observation following from Corollary 2 in~
\cite{VD06}.
\begin{lemma}
\label{L_convol}For any $0\leq m_{0}<m_{1}<...<m_{p}\leq n$ we have
\begin{eqnarray*}
&&\hat{J}_{\mathbf{m},n}^{\,\,(1,N)}(\mathbf{S}_{1},...,\mathbf{S}_{p})=\hat{
J}_{m_{0},m_{1},n}^{\,(1,N)}\left( \mathbf{S}_{1}\otimes \mathbf{\hat{J}}
_{m_{1},m_{2},...,m_{p},n}(\mathbf{S}_{2},...,\mathbf{S}_{p})\right) \\
&&\quad=J_{\Delta _{1},n-m_{0}}^{\,(1,N)}\left( \mathbf{S}_{1}\otimes
\mathbf{J}_{\Delta _{2},n-m_{1}}\left( \mathbf{S}_{2}\otimes ...(\mathbf{S}
_{p-1}\otimes \mathbf{J}_{\Delta _{p},n-m_{p-1}}(\mathbf{S}_{p})\right)
....)\right) .
\end{eqnarray*}
In particular, if $\mathbf{m}=(0,m_{1},m_{2})$ then
\begin{equation*}
\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},\mathbf{S}
_{2})=J_{m,n}^{\,(1,N)}(\mathbf{S}_{1}\otimes \mathbf{J}_{\Delta
_{2},n-m_{1}}(\mathbf{S}_{2}))
\end{equation*}
and if $\mathbf{m}=(m_{0},m_{1})$ then for $\mathbf{s}=(s_{1},...,s_{N})$
\begin{equation}
\hat{J}_{m_{0},m_{1},n}^{\,(k,N)}(\mathbf{s})=J_{\Delta
_{1},n-m_{0}}^{\,(k,N)}(\mathbf{s})=1-\frac{1-H_{\Delta _{1}}^{(k,N)}\left(
\mathbf{1}-(\mathbf{1}-\mathbf{s})\otimes \mathbf{Q}_{n-m_{1}}\right) }{
Q_{n-m_{0}}^{(k,N)}}. \label{Derivat}
\end{equation}
\end{lemma}
Using (\ref{Derivat}) we prove the following statement.
\begin{lemma}
\label{L_derivative}If $m_{0}=(Y_{0}+l_{n})n^{\gamma
_{i}}<m_{1}=(Y_{1}+l_{n})n^{\gamma _{i}}$ then for any $j\geq i$ there
exists a constant $\chi \in (0,\infty )$ such that for all $n\geq n_{0}$ \
\begin{equation*}
\mathbf{P}(\mathbf{Z}(m_{1},n)=\mathbf{e}_{j}|\mathbf{Z}(m_{0},n)=\mathbf{e}
_{j})\geq 1-\chi (Y_{1}-Y_{0}).
\end{equation*}
\end{lemma}
\textbf{Proof.} By the decomposability assumption and the condition $
m_{jj}=1 $ implying
\begin{equation*}
m_{jj}(\Delta _{1})=\frac{\partial H_{\Delta _{1}}^{(j,N)}(\mathbf{s})}{
\partial s_{j}}\left\vert _{\mathbf{s}=\mathbf{1}}\right. =1
\end{equation*}
we get
\begin{equation*}
1-\frac{\partial H_{\Delta _{1}}^{(j,N)}(\mathbf{s})}{\partial s_{j}}
\left\vert _{\mathbf{s}=\mathbf{H}_{n-m_{1}}(\mathbf{0})}\right. \leq
\sum_{k=j}^{N}\mathbf{E}Z_{j}(\Delta _{1})(Z_{k}(\Delta _{1})-\delta
_{kj})Q_{n-m_{1}}^{(k,N)}.
\end{equation*}
Recalling (\ref{Momvariance}) and (\ref{SurvivSingle}) and setting $
h=Y_{1}-Y_{0}$ we obtain
\begin{eqnarray*}
\mathbf{E}Z_{j}(\Delta _{1})Z_{k}(\Delta _{1})Q_{n-m_{1}}^{(k,N)} &\leq
&c_{0}(n-m_{1})^{-1/2^{N-k}}(\Delta _{1})^{k-j+1} \\
&\leq &c_{0}(n-m_{1})^{-1/2^{N-k}}(hn^{1/2^{N-i}})^{k-j+1} \\
&\leq &\chi hn^{-1/2^{N-k}}(n^{1/2^{N-i}})^{k-j+1}
\end{eqnarray*}
for some constants $0<c_{0}\leq \chi <\infty $. On account of $k\geq j\geq i$
we have
\begin{equation*}
\frac{k-j+1}{2^{N-i}}-\frac{1}{2^{N-k}}=\frac{1}{2^{N-i}}(k-j+1-2^{k-i})\leq
0.
\end{equation*}
Thus,
\begin{equation*}
1-\frac{\partial H_{\Delta _{1}}^{(j,N)}(\mathbf{s})}{\partial s_{j}}
\left\vert _{\mathbf{s}=\mathbf{H}_{n-m_{1}}(\mathbf{0})}\right. \leq \chi h.
\end{equation*}
Hence, using the previous lemma and monotonicity of $Q_{r}^{(j,N)}$ in $r$
we get
\begin{eqnarray}
\mathbf{P}(\mathbf{Z}(m_{1},n)=\mathbf{e}_{j}|\mathbf{Z}(m_{0},n)=\mathbf{e}
_{j}) &=&\frac{Q_{n-m_{1}}^{(j,N)}}{Q_{n-m_{0}}^{(j,N)}}\frac{\partial
H_{\Delta _{1}}^{(j,N)}(\mathbf{s})}{\partial s_{j}}\left\vert _{\mathbf{s}=
\mathbf{H}_{n-m_{1}}(\mathbf{0})}\right. \label{ExplCoeff} \\
&\geq &\frac{\partial H_{\Delta _{1}}^{(j,N)}(\mathbf{s})}{\partial s_{j}}
\left\vert _{\mathbf{s}=\mathbf{H}_{n-m_{1}}(\mathbf{0})}\right. \geq 1-\chi
h. \notag
\end{eqnarray}
Lemma \ref{L_derivative} is proved. \
\textbf{Proof} \textbf{of Theorem \ref{T_findim}}. Using (\ref{BranProp2})
and Theorem \ref{T_familyMany} we see that
1) if $m\ll n^{\gamma _{k}}$ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(k,N)}( \mathbf{s}) =s_{k};
\label{LeftEnd}
\end{equation}
2) if $m=(y+l_{n})n^{\gamma _{k}}=(y+l_{n})n^{1/2^{( N-k) }},\,y\in \lbrack
0,\infty )$ then
\begin{equation*}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(k,N)}( \mathbf{s}) =X_{k}(
y;s_{k},s_{k+1}) ;
\end{equation*}
3) if $m=(x+l_{n})n,\,x\in \left[ 0,1\right] $ then
\begin{equation}
\lim_{n\rightarrow \infty }J_{m,n}^{\,(N,N)}(\mathbf{s})=G\left(
x;s_{N}\right) . \label{End1}
\end{equation}
\textbf{Proof of (\ref{Lim_diskr})}. Consider first the case $p=2$ and take $
\mathbf{m}=(0,m_{1},m_{2})$. By Lemma \ref{L_convol}
\begin{equation}
\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},\mathbf{S}
_{2})=J_{m_{1},n}^{\,(1,N)}(\mathbf{S}_{1}\otimes \mathbf{J}_{\Delta
_{2},n-m_{1}}(\mathbf{S}_{2})). \label{JointDist}
\end{equation}
It follows from (\ref{LimSIngLate}) that, given $n^{\gamma _{i}}\ll m_{1}\ll
n^{\gamma _{i+1}}$
\begin{equation*}
J_{m_{1},n}^{\,(1,N)}( \mathbf{S}_{1}) \rightarrow 1-( 1-s_{1,i+1})
^{1/2^{i}}
\end{equation*}
as $n\rightarrow \infty.$ Further, in view of $\Delta _{2}=m_{2}-m_{1}\ll
n^{\gamma _{i+1}}$ and~(\ref{LeftEnd}) $J_{\Delta _{2},n-m_{1}}^{\,(i+1,N)}(
\mathbf{S}_{2})\rightarrow s_{2,i+1}$ as $n\rightarrow \infty.$ Hence, using
the continuity of the functions under consideration and (\ref{JointDist}) we
get
\begin{equation*}
\lim_{n\rightarrow \infty }\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},
\mathbf{S}_{2})=1-( 1-s_{1,i+1}s_{2,i+1}) ^{1/2^{i}}.
\end{equation*}
The validity of (\ref{Lim_diskr}) for any $p>3$ may be checked by induction
using Lemma~\ref{L_convol}.
\textbf{Proof of (\ref{Lim_Multi})}. Consider again the case $p=2$ only. It
follows from (\ref{LimTWo}) that, given $m_{l}\sim Y_{l}n^{\gamma
_{i}},\,l=1,2,$ with $Y_{1}=y_{1}$
\begin{equation*}
J_{m_{1},n}^{\,(1,N)}(\mathbf{s})\rightarrow \bar{X}
_{R_{i}}(y_{1};s_{i},s_{i+1})
\end{equation*}
as $n\rightarrow \infty $ and
\begin{equation*}
\lim_{n\rightarrow \infty }J_{\Delta _{2},n-m_{1}}^{\,(i,N)}(\mathbf{S}
_{2})=X_{i}(y_{2};s_{2i},s_{2,i+1}),\quad \lim_{n\rightarrow \infty
}J_{\Delta _{2},n-m_{1}}^{\,(i+1,N)}(\mathbf{S}_{2})=s_{2,i+1}.
\end{equation*}
Hence, using the continuity of the functions involved and (\ref{JointDist})
we get
\begin{equation*}
\lim_{n\rightarrow \infty }\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},
\mathbf{S}_{2})=\bar{X}_{R_{i}}\left(
y_{1};s_{1i}X_{i}(y_{2};s_{2i},s_{2,i+1}),s_{1,i+1}s_{2,i+1}\right)
\end{equation*}
proving (\ref{Lim_Multi}) for $p=2$.
To justify (\ref{Lim_Multi}) for $p>3$ it is necessary to use Lemma \ref
{L_convol} and induction arguments. We omit the respective details. \
\textbf{Proof} \textbf{of Theorem \ref{T_endpoint}}. We consider the case $
p=2$ only and to this aim take $\mathbf{m}
=(0,(x_{1}+l_{n})n,(x_{1}+x_{2}+l_{n})n)$. By (\ref{JointDist}), (\ref
{LimOLD}) and (\ref{End1})
\begin{eqnarray*}
\lim_{n\rightarrow \infty }\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}_{1},
\mathbf{S}_{2}) &=&\lim_{n\rightarrow \infty }J_{(x_{1}+l_{n})n,n}^{\,(1,N)}(
\mathbf{S}_{1}\otimes \mathbf{J}_{x_{2}n,n(1-x_{1}-l_{n})}(\mathbf{S}_{2}))
\\
&=&\bar{G}_{R_{N}}\left( x_{1};s_{1N}G\left( \frac{x_{2}}{1-x_{1}}
;s_{2N}\right) \right) =\bar{G}_{R_{N}}(\mathbf{x}_{1,2};\mathbf{S}_{1,2;N}).
\end{eqnarray*}
The desired statement for $p>2$ follows by induction. \
\textbf{Proof} \textbf{of point 1) of Theorem \ref{T_Skhod1}}. Let $
0=t_{0}<t_{1}<...<t_{p}<1$. If $\gamma _{i-1}\leq t_{1}<t_{p}<\gamma _{i}$
for some $i\in \left\{ 1,2,...,N\right\} $ then the needed convergence of
finite-dimensional distributions follows from (\ref{Lim_diskr}). We now
consider another extreme case, namely, take a tuple $
0=t_{0}<t_{1}<...<t_{N}<1$ such that $\gamma _{i-1}\leq t_{i}<\gamma _{i}$
for all $i=1,2,...,N$. Then for $m_{i}\sim n^{t_{i}}g_{n}(t_{i})$ we have
\begin{equation*}
n^{\gamma _{i-1}}\ll m_{i}\ll n^{\gamma _{i}},\quad \Delta
_{i}=m_{i}-m_{i-1}\sim m_{i},\quad n-m_{i}\sim n.
\end{equation*}
These relations, (\ref{LimSIngLate}), (\ref{LeftEnd}), and the continuity of
the respective probability generating functions imply (recall (\ref{DefOmega}
))
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\hat{J}_{\mathbf{m},n}^{\,(1,N)}(\mathbf{S}
_{1},...,\mathbf{S}_{N}) \\
&&\quad =\lim_{n\rightarrow \infty }J_{m_{1},n}^{\,(1,N)}(\mathbf{S}
_{1}\otimes \mathbf{J}_{m_{2},n}(\mathbf{S}_{2}\otimes ...(\mathbf{S}
_{N-1}\otimes \mathbf{J}_{m_{N},n}(\mathbf{S}_{N}))...)) \\
&&\quad =s_{11}\left( 1-\sqrt{1-\lim_{n\rightarrow \infty
}J_{m_{2},n}^{\,(2,N)}(\mathbf{S}_{2}\otimes ...(\mathbf{S}_{N-1}\otimes
\mathbf{J}_{m_{N},n}(\mathbf{S}_{N}))...)}\,\right) \\
&&\quad =s_{11}\left( 1-\sqrt{1-s_{22}\left( 1-\sqrt{1-\Omega
_{N-2}(s_{33},...,s_{NN})}\,\right) }\,\,\right) \\
&&\quad =...=\Omega _{N}(s_{11},s_{22},...,s_{NN})
\end{eqnarray*}
as required.
The case when several values among $t_{j}$ are contained in a subinterval $
[\gamma _{i-1},\gamma _{i})$ may be considered by combining the previous
arguments. We omit the respective details. \
\section{Tightness\label{Sec6}}
Denote by $\mathbf{z}^{(i,i+1)},1\leq i\leq N-1,$ the $(N-2)$-dimensional
vector obtained from $\mathbf{z}=(z_{1},...,z_{N})\in \mathbb{Z}_{+}^{N}$ by
deleting the coordinates $i$ and $i+1$ and by $\mathbf{z}^{(i)},1\leq i\leq
N-1,$ the $(N-1)$-dimensional vector obtained from $\mathbf{z}$ by deleting
the $i$-th coordinate. Let $\left\Vert \mathbf{x}\right\Vert $ be the sum of
absolute values of all coordinates of the vector $\mathbf{x}$.
Set $\mathcal{C}_{i}=\left\{ \mathbf{z}\in \mathbb{Z}_{+}^{N}:\left\Vert
\mathbf{z}^{(i)}\right\Vert >0\right\} ,\,\mathcal{B}_{i}=\mathbb{Z}
_{+}^{N}\backslash \mathcal{C}_{i}$ and
\begin{equation*}
\mathcal{C}_{i,i+1}=\left\{ \mathbf{z}\in \mathbb{Z}_{+}^{N}:\left\Vert
\mathbf{z}^{(i,i+1)}\right\Vert >0\right\} .
\end{equation*}
Put \b{Z}$_{i}(m)=Z_{1}(m)+...+Z_{i}(m)$ and denote
\begin{equation*}
\text{\b{Z}}_{i}(m,n)=\sum_{k=1}^{i}Z_{k}(m,n),\quad \bar{Z}
_{i}(m,n)=\sum_{k=i}^{N}Z_{k}(m,n).
\end{equation*}
In what follows it will be convenient to write $\mathbf{P}_{n}(\mathcal{B})$
for $\mathbf{P}(\mathcal{B}|\mathbf{Z}(n)\neq \mathbf{0},\mathbf{Z}(0)=
\mathbf{e}_{1})$ for any admissible event~$\mathcal{B}$.
We start checking the desired tightness of the prelimiting processes in
Theorems \ref{T_SkorohConst} and~\ref{T_Skhod1} by proving two important
lemmas.
Let $A_{i}(n) =\left\{ m:n^{\gamma _{i}}g_{n}(\gamma _{i})\leq m<n^{\gamma
_{i+1}-\varepsilon }g_{n}(\gamma _{i+1}-\varepsilon
)\right\},\,\varepsilon>0.$
\begin{lemma}
\label{L_Negli} For any $i=0,1,2,...,N-1$ and $\varepsilon\in (0,\gamma_1)$
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}_{n}( \exists m\in A_{i}( n) :\mathbf{Z}
( m,n) \in \mathcal{C}_{i+1}) =0.
\end{equation*}
\end{lemma}
\textbf{Proof.} If $m\in A_{i}(n)$ then $\bar{Z}_{i+2}(m,n)\leq \bar{Z}
_{i+2}(n^{\gamma _{i+1}-\varepsilon }g_{n}(\gamma _{i+1}-\varepsilon ),n)$
and
\begin{equation*}
\left\{ \text{\b{Z}}_{i}(m,n)>0\right\} \Rightarrow \left\{ \text{\b{Z}}
_{i}(m)>0\right\} \Rightarrow \left\{ \text{\b{Z}}_{i}(n^{\gamma
_{i}}g_{n}(\gamma _{i}))>0\right\} .
\end{equation*}
Thus,
\begin{eqnarray*}
\mathbf{P}_{n}(\exists m\in A_{i}(n):\mathbf{Z}(m,n)\in \mathcal{C}_{i+1})
&\leq &\mathbf{P}_{n}(\bar{Z}_{i+2}(n^{\gamma _{i+1}-\varepsilon
}g_{n}(\gamma _{i+1}-\varepsilon ),n)>0) \\
&&+\mathbf{P}_{n}\left( \text{\b{Z}}_{i}(n^{\gamma _{i}}g_{n}(\gamma
_{i}))>0\right) .
\end{eqnarray*}
Letting $n$ tend to infinity we see that the first summand at the right-hand
side of the inequality vanishes by (\ref{LimSIngLate}), while the second one
is zero for $i=0$ and tends to zero for $1\leq i\leq N-1$ in view of
\begin{eqnarray*}
\mathbf{P}_{n}\left( \text{\b{Z}}_{i-1}(n^{\gamma _{i}}g_{n}(\gamma
_{i}))>0\right) &=&\frac{\mathbf{P}(T_{i}>n^{\gamma _{i}}g_{n}(\gamma _{i}))
}{\mathbf{P}(T_{N}>n)} \\
&\sim &\frac{c_{1i}}{c_{1N}}\frac{n^{1/2^{N-1}}}{(n^{1/2^{N-i}}g_{n}(\gamma
_{i}))^{1/2^{i-1}}}=\frac{c_{1i}}{c_{1N}}\frac{1}{(g_{n}(\gamma
_{i}))^{1/2^{i-1}}}.
\end{eqnarray*}
The lemma is proved.\
\begin{lemma}
\label{L_Skor1}If $N\geq 3$ then for any $i=1,2,...,N-1$
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}_{n}( \exists m\in \lbrack n^{3\gamma
_{i-1}},n^{3\gamma _{i}}]:\mathbf{Z}( m,n) \in \mathcal{C}_{i,i+1}) =0.
\end{equation*}
\end{lemma}
\textbf{Proof.} By the same arguments as in Lemma \ref{L_Negli}, we conclude
\begin{eqnarray*}
&&\mathbf{P}_{n}(\exists m\in \lbrack n^{3\gamma _{i-1}},n^{3\gamma _{i}}]:
\mathbf{Z}(m,n)\in \mathcal{C}_{i,i+1}) \\
&&\qquad \leq \mathbf{P}_{n}(\bar{Z}_{i+2}(n^{3\gamma _{i}},n)>0)+\mathbf{P}
_{n}(\text{\b{Z}}_{i-1}(n^{3\gamma _{i-1}})>0).
\end{eqnarray*}
According to point 3) of Theorem \ref{T_familyMany} the first summand tends
to zero as $n\rightarrow \infty $ while the second is, by definition zero
for $i=1$ and is evaluated as
\begin{equation*}
\frac{\mathbf{P}(T_{i-1}>n^{3\gamma _{i-1}})}{\mathbf{P}(T_{N}>n)}\sim \frac{
c_{1,i-1}}{c_{1N}}\frac{n^{1/2^{N-1}}}{(n^{3/2^{N-i+1}})^{1/2^{i-2}}}\sim
\frac{c_{1,i-1}}{c_{1N}}\frac{1}{n^{1/2^{N-2}}}
\end{equation*}
for $i\geq 2$. \ This completes the proof of the lemma.
\subsection{Macroscopic view\label{Sec51}}
In this section we prove Theorem \ref{T_SkorohConst}\textbf{\ }which
describes the macroscopic structure of the family tree. Convergence of the
finite-dimensional distributions of $\left\{ \mathbf{Z}(n^{t}g_{n}(t),n),0
\leq t<1\right\} $ to the respective finite-dimensional distributions of $
\left\{ \mathbf{R}(t),0\leq t<1\right\} $ has been established in (\ref
{Lim_diskr}). Thus, we concentrate on proving the tightness.
Since $\mathbf{Z}(n^{t}g_{n}(t),n)$ has integer-valued components we need to
check\ for each interval $A_{i}=\left[ \gamma _{i},\gamma _{i+1}-\varepsilon
\right] ,i=0,1,...,N-1,$ that (see \cite{Bil68}, Theorem 15.3)
1) for any positive $\eta $ there exists $L$ such that
\begin{equation}
\mathbf{P}_{n}\left( \sup_{t\in A_{i}}\left\Vert \mathbf{Z}
(n^{t}g_{n}(t),n)\right\Vert >L\right) \leq \eta ,~n\geq 1; \label{Bi0}
\end{equation}
2) for any positive $\eta $ there exist $\delta >0$ and $n_{0}$ such that,
for all $n\geq n_{0}$
\begin{equation}
\mathbf{P}_{n}\left( \max \left( \min_{k=1,2}\left\Vert \mathbf{Z}
(n^{t}g_{n}(t),n)-\mathbf{Z}(n^{t_{k}}g_{n}(t_{k}),n)\right\Vert \right)
\neq 0\right) \leq \eta , \label{Bi01}
\end{equation}
where the $\max $ is taken over all $\gamma _{i}\leq t_{1}\leq t\leq
t_{2}\leq \gamma _{i+1}-\varepsilon $ such that $t_{2}-t_{1}\leq \delta ;$
\begin{equation}
\mathbf{P}_{n}(\exists t,s\in \left[ \gamma _{i},\gamma _{i}+\delta \right] :
\mathbf{Z}(n^{t}g_{n}(t),n)\neq \mathbf{Z}(n^{s}g_{n}(s),n)\,)\leq \eta ,
\label{Bi02}
\end{equation}
and
\begin{equation}
\mathbf{P}_{n}(\exists t,s\in \lbrack \gamma _{i+1}-\delta -\varepsilon
,\gamma _{i+1}-\varepsilon ]:\mathbf{Z}(n^{t}g_{n}(t),n)\neq \mathbf{Z}
(n^{s}g_{n}(s),n)\,\,)\leq \eta . \label{Bi03}
\end{equation}
The fact that the random variable $\left\Vert \mathbf{Z}( n^{t}g_{n}(t),n)
\right\Vert $ is monotone in $t$ for fixed $n$ essentially simplifies the
proof.
Indeed, in this case
\begin{equation*}
\mathbf{P}_{n}\left( \sup_{t\in A_{i}}\left\Vert \mathbf{Z}
(n^{t}g_{n}(t),n)\right\Vert >L\right) \leq \mathbf{P}_{n}(\left\Vert
\mathbf{Z}(n^{1-\varepsilon }g_{n}(1-\varepsilon ),n)\right\Vert >L\,)
\end{equation*}
and (\ref{Bi0}) follows from the one-dimensional convergence established in (
\ref{LimSIngLate}) for $i=N-1$.
To prove (\ref{Bi01})-(\ref{Bi03}) we introduce the events
\begin{eqnarray*}
\mathcal{D}_{i} &=&\left\{ \forall t\in A_{i}:\mathbf{Z}(n^{t}g_{n}(t),n)\in
\mathcal{B}_{i+1}\right\} , \\
\mathcal{F}_{i}(a,b) &=&\left\{ \exists t,s\in \lbrack
a,b]:Z_{i+1}(n^{t}g_{n}(t),n)\neq Z_{i+1}(n^{s}g_{n}(s),n)\right\} ,
\end{eqnarray*}
take a sufficiently small $\delta >0$ and observe that if $\left[ a,b\right]
\subset \left[ \gamma _{i},\gamma _{i+1}-\varepsilon \right] $ then
\begin{eqnarray*}
&&\mathbf{P}_{n}(\exists t,s\in \lbrack a,b]:\mathbf{Z}(n^{t}g_{n}(t),n)\neq
\mathbf{Z}(n^{s}g_{n}(s),n)\,) \\
&&\qquad \leq \mathbf{P}_{n}(\exists t\in A_{i}:\mathbf{Z}
(n^{t}g_{n}(t),n)\in \mathcal{C}_{i+1})+\mathbf{P}_{n}(\mathcal{D}_{i}\cap
\mathcal{F}_{i}(\gamma _{i},\gamma _{i+1}-\varepsilon )).
\end{eqnarray*}
By Lemma \ref{L_Negli} the first term at the right-hand side tends to zero
as $n\rightarrow \infty $.
Further, for $i\geq 1$
\begin{equation*}
\mathbf{P}_{n}( \mathcal{D}_{i}\cap \mathcal{F}_{i}( \gamma _{i},\gamma
_{i+1}-\varepsilon ) ) \leq \mathbf{P}_{n}( Z_{i+1}( n^{\gamma _{i}}g_{n},n)
\neq Z_{i+1}( n^{\gamma _{i+1}-\varepsilon }g_{n},n) ) \rightarrow 0
\end{equation*}
by (\ref{Lim_diskr}). This justifies (\ref{Bi02})-(\ref{Bi03}).
To check the validity of (\ref{Bi01}) it remains to note that
\begin{eqnarray*}
&&\mathbf{P}_{n}\left( \max \left( \min_{k=1,2}\left\Vert \mathbf{Z}
(n^{t}g_{n}(t),n)-\mathbf{Z}(n^{t_{k}}g_{n}(t_{k}),n)\right\Vert \right)
\neq 0\right) \\
&&\qquad \leq \mathbf{P}_{n}(\exists t,s\in \lbrack \gamma _{i},\gamma
_{i+1}-\varepsilon ]:\mathbf{Z}(n^{t}g_{n}(t),n)\neq \mathbf{Z}
(n^{s}g_{n}(s),n)\,)
\end{eqnarray*}
and to use the same arguments as before.
Theorem \ref{T_SkorohConst} is proved.
\subsection{Microscopic view\label{Sec52}}
We follow in this section the ideas of paper \cite{FZ} and to this aim
formulate a particular and slightly modified case of Theorem 6.5.4 in \cite
{GS71} giving a convergence criterion in Skorokhod topology for a class of
Markov processes.
Let $\mathbf{K}_{n}(y),n=1,2,...$ be a sequence of Markov processes with
values in $\mathbb{Z}_{+}^{N}$ whose trajectories belong with probability 1
to the space $D_{[a,b]}(\mathbb{Z}_{+}^{N})$ of cadlag functions on $[a,b]$.
\begin{theorem}
\label{T_skoroh}If the finite-dimensional distributions of $\left\{ \mathbf{K
}_{n}(y),a\leq y\leq b\right\} $ converge, as $n\rightarrow \infty ,$ to the
respective finite-dimensional distributions of a process $\left\{ \mathbf{K}
(y),a\leq y\leq b\right\} $ and there exists a partition $\mathbb{Z}_{+}^{N}=
\mathcal{B}\cup \mathcal{C},\mathcal{B}\cap \mathcal{C}=\varnothing $ such
that
\begin{equation*}
\lim_{h\downarrow 0}\overline{\lim_{n\rightarrow \infty }}\sup_{0\leq
s-y\leq h}\sup_{\mathbf{z}\in \mathcal{B}}\mathbf{P}(\mathbf{K}_{n}(s)\neq
\mathbf{K}_{n}(y)|\mathbf{K}_{n}(y)=\mathbf{z})=0,
\end{equation*}
and
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}(\exists y\in \lbrack a,b]:\mathbf{K}
_{n}(y)\in \mathcal{C})=0
\end{equation*}
then, as $n\rightarrow \infty $
\begin{equation*}
\mathcal{L}\left\{ \mathbf{K}_{n}(y),a\leq y\leq b\right\} \Longrightarrow
\mathcal{L}\left\{ \mathbf{K}(y),a\leq y\leq b\right\} .
\end{equation*}
\end{theorem}
In view of Lemma \ref{L_convol} the law\textbf{\ }$\mathbf{P}_{n}(\left\{
\mathbf{Z}(m,n),0\leq m\leq n\right\} \in (\cdot )|\mathbf{Z}(n)\neq \mathbf{
0})$ specifies, for each fixed $n$ an inhomogeneous Markov branching
process. We denote its transition probabilities by $\mathbf{P}_{n}(m_{1},
\mathbf{z};m_{2},(\cdot ))$.
Proving the tightness of $\mathbf{U}_{i}\left( \cdot \right) ,$ $
i=1,2,...,N, $ we need to construct an appropriate partition of $\mathbb{Z}
_{+}^{N}$ and to use Theorem \ref{T_skoroh} for each $\left[ 0,b\right]
\subset \lbrack 0,\infty )$.
Observe that if $\mathbf{w}=( w_{1},...,w_{N}) \leq \mathbf{z}=(
z_{1},...,z_{N}) $ (where the inequality is understood componentwise) then
\begin{equation*}
\mathbf{P}_{n}( m_{0},\mathbf{w};m_{1},\left\{ \mathbf{w}\right\} ) \geq
\mathbf{P}_{n}( m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}\right\} ) .
\end{equation*}
Let $\ \mathcal{C}(k)=\left\{ \mathbf{z}\in \mathbb{Z}_{+}^{N}:\left\Vert
\mathbf{z}\right\Vert \leq k\right\} ,$
\begin{equation*}
\mathcal{C}_{i}(k)=\left\{ \mathbf{z}\in \mathbb{Z}
_{+}^{N}:z_{1}+...+z_{i-1}>0;\left\Vert \mathbf{z}\right\Vert \leq k\right\}
,~\mathcal{J}_{i}(k)=\mathcal{C}(k)\backslash \mathcal{C}_{i}(k).
\end{equation*}
Fix $i\in \{1,...,N-1\}$ and denote $m_{j}=( Y_{j}+l_{n}) n^{\gamma
_{i}},j=1,2$.
\begin{lemma}
\label{L_ceretain event.}Under Hypothesis A for any fixed $k$ and $
0<b<\infty $
\begin{equation*}
\lim_{h\downarrow 0}\overline{\lim_{n\rightarrow \infty}}\sup_{\substack{
0\leq Y_{1}-Y_{0}\leq h, \\ Y_{1},Y_{0}\in \lbrack 0,b]}}\sup_{\mathbf{z}
\in \mathcal{J}_{i}(k)}\mathbf{P}_{n}(\mathbf{Z}(m_{1};n)\neq \mathbf{z}|
\mathbf{Z}(m_{0};n)=\mathbf{z})=0.
\end{equation*}
\end{lemma}
\textbf{Proof.} By the branching property, the decomposability assumption,
and the positivity of the offspring number of each particle in the reduced
process we have for all $m_{1}\geq m_{0}$ and $\mathbf{z}\in \mathcal{J}
_{i}(k)$
\begin{eqnarray*}
\mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}\right\} )
&=&\prod_{j=i}^{N}(\mathbf{P}_{n}(m_{0},\mathbf{e}_{j};m_{1},\left\{ \mathbf{
e}_{j}\right\} ))^{z_{j}} \\
&\geq &\prod_{j=i}^{N}(\mathbf{P}_{n}(m_{0},\mathbf{e}_{j};m_{1},\left\{
\mathbf{e}_{j}\right\} ))^{k}.
\end{eqnarray*}
Using Lemma \ref{L_derivative} we get for $m_{0}=(Y_{0}+l_{n})n^{\gamma
_{i}} $ and $m_{1}=(Y_{1}+l_{n})n^{\gamma _{i}}$:
\begin{equation}
\inf_{\substack{ 0\leq Y_{1}-Y_{0}\leq h, \\ Y_{0},Y_{1}\in \left[ 0,b
\right] }}\mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}\right\}
)\geq (1-\chi h)^{Nk}. \label{kapBel}
\end{equation}
This implies the claim of the lemma. \
\begin{lemma}
\label{L_ratio}If $m_{j}=(Y_{j}+l_{n})n^{\gamma _{i}},~j=0,1,2,$ and $0\leq
Y_{0}<Y_{1}<Y_{2}$ with $Y_{1}-Y_{0}\leq h$, then for all $n\geq n_{0}$
\begin{eqnarray*}
&&\mathbf{P}_{n}(\mathbf{Z}(m_{1},n)=\mathbf{z}|\mathbf{Z}(m_{0},n)=\mathbf{z
};\left\Vert \mathbf{Z}(m_{2},n)\right\Vert \leq k) \\
&&\qquad \geq \mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}
\right\} )\frac{\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k))}{
\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k))\mathbf{+}\chi Nkh}.
\end{eqnarray*}
\end{lemma}
\textbf{Proof.} We have
\begin{eqnarray*}
&&\mathbf{P}_{n}(\mathbf{Z}(m_{1},n)=\mathbf{z}|\mathbf{Z}(m_{0},n)=\mathbf{z
},\left\Vert \mathbf{Z}(m_{2},n)\right\Vert \leq k) \\
&&\qquad =\mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}\right\} )
\frac{\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k))}{\mathbf{P}
_{n}(m_{0},\mathbf{z};m_{2},\mathcal{C}(k))}.
\end{eqnarray*}
In view of (\ref{kapBel})
\begin{eqnarray*}
\mathbf{P}_{n}(m_{0},\mathbf{z};m_{2},\mathcal{C}(k)) &=&\sum_{\mathbf{z}
^{\prime }}\mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}^{\prime
}\right\} )\mathbf{P}_{n}(m_{1},\mathbf{z}^{\prime };m_{2},\mathcal{C}(k)) \\
&\leq &1-\mathbf{P}_{n}(m_{0},\mathbf{z};m_{1},\left\{ \mathbf{z}\right\} )+
\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k)) \\
&\leq &1-(1-\chi h)^{Nk}+\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}
(k)) \\
&\leq &\chi Nkh+\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k)).
\end{eqnarray*}
Hence the needed statement follows.
\begin{lemma}
\label{L_Kskrokh}Under Hypothesis A for any fixed $k,$ $0<b<\infty ,$ and $
m_{0}=( Y_{0}+l_{n}) n^{\gamma _{i}},\ m_{1}=( Y_{1}+l_{n}) n^{\gamma
_{i}},\ m_{2}=2bn^{\gamma _{i}}$ we have
\begin{equation*}
\lim_{h\downarrow 0}\overline{\lim_{n\rightarrow \infty }}\sup_{\substack{
0\leq Y_{1}-Y_{0}\leq h \\ Y_{1},Y_{0}\in \lbrack 0,b]}}\sup_{\mathbf{z}\in
\mathcal{J}_{i}(k)}\mathbf{P}_{n}( \mathbf{Z}( m_{1};n) \neq \mathbf{z}|
\mathbf{Z}( m_{0};n) =\mathbf{z},\left\Vert \mathbf{Z}( m_{2},n) \right\Vert
\leq k) =0.
\end{equation*}
\end{lemma}
\textbf{Proof.} By (\ref{kapBel}) and Lemma \ref{L_ratio} for $
m_{0}=(Y_{0}+l_{n})n^{\gamma _{i}}$ and $m_{1}=(Y_{1}+l_{n})n^{\gamma _{i}}$
\begin{eqnarray*}
&&\mathbf{P}_{n}(\mathbf{Z}(m_{1},n)=\mathbf{z}|\mathbf{Z}(m_{0},n)=\mathbf{z
};\left\Vert \mathbf{Z}(m_{2},n)\right\Vert \leq k) \\
&&\qquad \geq (1-\chi h)^{Nk}\frac{\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},
\mathcal{C}(k))}{\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k))
\mathbf{+}\chi Nkh}.
\end{eqnarray*}
Using the decomposability hypothesis and Lemma \ref{L_derivative} we obtain
\begin{eqnarray*}
&&\mathbf{P}_{n}(m_{1},\mathbf{z};m_{2},\mathcal{C}(k))\geq \mathbf{P}
_{n}(m_{1},\mathbf{z};m_{2},\left\{ \mathbf{z}\right\} ) \\
&&\qquad =\prod_{j=i}^{N}(\mathbf{P}_{n}(m_{1},\mathbf{e}_{j};m_{2},\left\{
\mathbf{e}_{j}\right\} ))^{z_{j}}\geq \prod_{j=i}^{N}\mathbf{P}
_{n}^{k}(l_{n}n^{\gamma _{i}},\mathbf{e}_{j};2bn^{\gamma _{i}},\left\{
\mathbf{e}_{j}\right\} ).
\end{eqnarray*}
It follows from Theorem~\ref{T_findim} that
\begin{equation*}
\lim_{n\rightarrow \infty }\prod_{j=i}^{N}\mathbf{P}_{n}^{k}(l_{n}n^{\gamma
_{i}},\mathbf{e}_{j};2bn^{\gamma _{i}},\left\{ \mathbf{e}_{j}\right\} )=
\mathbf{P}^{k}(\mathbf{U}_{i}(2b)=\mathbf{e}_{i}|\mathbf{U}_{i}(0)=\mathbf{e}
_{i})=B>0.
\end{equation*}
Hence we get
\begin{eqnarray*}
&&\varliminf_{n\rightarrow \infty }\inf_{\substack{ 0\leq Y_{1}-Y_{0}\leq h
\\ Y_{1},Y_{0}\in \lbrack 0,b]}}\inf_{\mathbf{z}\in \mathcal{J}_{i}(k)}
\mathbf{P}_{n}(\mathbf{Z}(m_{1};n)=\mathbf{z}|\mathbf{Z}(m_{0};n)=\mathbf{z}
,\left\Vert \mathbf{Z}(m_{2},n)\right\Vert \leq k) \\
&&\qquad \qquad \geq (1-\chi h)^{Nk}\frac{B}{B\mathbf{+}\chi Nkh}.
\end{eqnarray*}
Letting $h\downarrow 0$ completes the proof of the lemma. \
\begin{corollary}
\label{C_skk}Under the conditions of Lemma \ref{L_Kskrokh}
\begin{eqnarray*}
&&\mathcal{L}\left\{ \mathbf{Z}((y+l_{n})n^{1/2^{N-i}},n),0\leq y\leq b\,
\Big|\,\left\Vert \mathbf{Z}(m_{2},n)\right\Vert \leq k,\mathbf{Z}(n)\neq
\mathbf{0}\right\} \\
&&\qquad\qquad\qquad\qquad\qquad\quad\,\Longrightarrow \mathcal{L}
_{R_{i}}\left\{ \mathbf{U}_{i}(y),0\leq y\leq b|\,\left\Vert \mathbf{U}
_{i}(2b)\right\Vert \leq k\right\} .
\end{eqnarray*}
\end{corollary}
\textbf{Proof.} Convergence of finite-dimensional distributions follows from
the respective results for the convergence of the processes established in
point 1) of Theorem \ref{T_Skhod1}. Tightness follows from Lemma \ref
{L_Kskrokh} and Theorem \ref{T_skoroh} by taking $\mathcal{B}=\mathcal{J}
_{i}(k)$ and $\mathcal{C=C}_{i}(k)$ and observing that
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\mathbf{Z}(l_{n}n^{\gamma
_{i}},n)\in \mathcal{C}_{i}(k)|\left\Vert \mathbf{Z}(m_{2},n)\right\Vert
\leq k) \\
&&\qquad \leq \lim_{n\rightarrow \infty }\mathbf{P}_{n}\left( \text{\b{Z}}
_{i-1}(l_{n}n^{\gamma _{i}})>0|\left\Vert \mathbf{Z}(m_{2},n)\right\Vert
\leq k\right) =0.
\end{eqnarray*}
\textbf{Proof} \textbf{of Theorem \ref{T_Skhod1}.} Let for $c>b$
\begin{eqnarray*}
\mathbf{P}_{n,i}(b;(\cdot )) &=&\mathbf{P}_{n}(\left\{ \mathbf{Z}
((y+l_{n})n^{\gamma _{i}},n),0\leq y\leq b\right\} \in (\cdot )), \\
\mathbf{P}_{n,i}^{(k)}(b,c;(\cdot )) &=&\mathbf{P}_{n}(\left\{ \mathbf{Z}
((y+l_{n})n^{\gamma _{i}},n),0\leq y\leq b\right\} \in (\cdot )|\left\Vert
\mathbf{Z}(cn^{\gamma _{i}},n)\right\Vert \leq k), \\
\mathbf{\bar{P}}_{n,i}^{(k)}(b,c;(\cdot )) &=&\mathbf{P}_{n}(\left\{ \mathbf{
Z}((y+l_{n})n^{\gamma _{i}},n),0\leq y\leq b\right\} \in (\cdot )|\left\Vert
\mathbf{Z}(cn^{\gamma _{i}},n)\right\Vert >k)
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal{P}_{i}(b;(\cdot )) &=&\mathbf{P}_{R_{i}}(\left\{ \mathbf{U}
_{i}(y),0\leq y\leq b\right\} \in (\cdot )), \\
\mathcal{P}_{i}^{(k)}(b,c;(\cdot )) &=&\mathbf{P}_{R_{i}}(\left\{ \mathbf{U}
_{i}(y),0\leq y\leq b\right\} \in (\cdot )|\left\Vert \mathbf{U}
_{i}(c)\right\Vert \leq k).
\end{eqnarray*}
Then for $0<b<\infty $ and a continuous real function $\psi $ on $D_{[0,b]}(
\mathbb{Z}_{+}^{N})$ such that $\left\vert \psi \right\vert \leq q$ for a
positive $q$ we have
\begin{eqnarray*}
\int \psi (x)\mathbf{P}_{n,i}(b;dx) &=&\mathbf{P}_{n}(\left\Vert \mathbf{Z}
(2bn^{\gamma _{i}},n)\right\Vert >k)\int \psi (x)\mathbf{\bar{P}}
_{n,i}^{(k)}(b,2b;dx) \\
&&+\mathbf{P}_{n}(\left\Vert \mathbf{Z}(2bn^{\gamma _{i}},n)\right\Vert \leq
k)\int \psi (x)\mathbf{P}_{n,i}^{(k)}(b,2b;dx).
\end{eqnarray*}
For the first summand we get
\begin{eqnarray*}
&&\lim \sup_{n\rightarrow \infty }\mathbf{P}_{n}(\left\Vert \mathbf{Z}
(2bn^{\gamma _{i}},n)\right\Vert >k)\int \psi (x)\mathbf{\bar{P}}
_{n,i}^{(k)}(b,2b;dx) \\
&&\quad \leq q\lim \sup_{n\rightarrow \infty }\mathbf{P}_{n}(\left\Vert
\mathbf{Z}(2bn^{\gamma _{i}},n)\right\Vert >k)=q\mathbf{P}
_{R_{i}}(\left\Vert \mathbf{U}_{i}(2b)\right\Vert >k)=o(1)
\end{eqnarray*}
as $k\rightarrow \infty $ by the properties of $\mathbf{U}_{i}(\cdot )$. \
On the other hand, letting first $n\rightarrow \infty $ and than $
k\rightarrow \infty $ we obtain
\begin{eqnarray*}
&&\lim_{k\rightarrow \infty }\lim_{n\rightarrow \infty }\mathbf{P}
_{n}(\left\Vert \mathbf{Z}(2bn^{\gamma _{i}},n)\right\Vert \leq k)\int \psi
(x)\mathbf{P}_{n,i}^{(k)}(b,2b;dx) \\
&&\quad =\lim_{k\rightarrow \infty }\mathbf{P}_{R_{i}}(0<\left\Vert \mathbf{U
}_{i}(2b)\right\Vert \leq k)\int \psi (x)\mathcal{P}_{i}^{(k)}(b,2b;dx) \\
&&\quad =\lim_{k\rightarrow \infty }\int_{\left\{ 0<\left\Vert \mathbf{U}
_{i}(2b)\right\Vert \leq k\right\} }\psi (x)\mathcal{P}_{i}(b,2b;dx)=\int
\psi (x)\mathcal{P}_{i}(b;dx).
\end{eqnarray*}
Thus,
\begin{equation*}
\lim_{n\rightarrow \infty }\int \psi (x)\mathbf{P}_{n}(\mathbf{Z}((\cdot
+l_{n})n^{\gamma _{i}},n)\in dx)=\int \psi (x)\mathcal{P}_{i}(b;dx)
\end{equation*}
for any bounded continuous function on $D_{[0,b]}(\mathbb{Z}_{+}^{N})$
proving point 1) of Theorem~\ref{T_Skhod1}.
The proof of point 2) of Theorem \ref{T_Skhod1} needs only a few changes in
comparison with the proof of the respective theorem in \cite{FZ} and we omit
it. \
\section{Proofs of Theorems \protect\ref{T_mrcaMany} and \protect\ref{T_type}
\label{Sec7}}
\textbf{Proof} \textbf{of Theorem \ref{T_mrcaMany}.} Our arguments are based
on the following simple observation
\begin{equation*}
\left\{ \bar{Z}_{1}(m,n)=1\right\} \Leftrightarrow \left\{ \beta _{n}\geq
m\right\} .
\end{equation*}
\textbf{Proof of 1).} According to (\ref{LimSing}) for $m\ll n^{\gamma _{1}}$
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\bar{Z}_{1}(m,n)=1)=\lim_{n
\rightarrow \infty }\mathbf{P}_{n}(Z_{1}(m,n)=1) \\
&&\quad +\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\bar{Z}_{2}(m,n)=1)=1+0=1.
\end{eqnarray*}
\textbf{Proof of 2).} Observe that by point 2) of Theorem \ref{T_familyMany}
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\beta _{n}\geq yn^{\gamma
_{i}})=\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\bar{Z}_{1}(yn^{\gamma
_{i}},n)=1) \\
&&\quad =\lim_{n\rightarrow \infty }\mathbf{P}_{n}(Z_{i}(yn^{\gamma
_{i}},n)+Z_{i+1}(yn^{\gamma _{i}},n)=1) \\
&&\quad =\lim_{n\rightarrow \infty }\mathbf{P}_{n}(Z_{i}(yn^{\gamma
_{i}},n)=1)+\lim_{n\rightarrow \infty }\mathbf{P}_{n}(Z_{i+1}(yn^{\gamma
_{i}},n)=1).
\end{eqnarray*}
Direct calculations show that
\begin{equation*}
-\frac{\partial \varphi _{i}(y;s_{i},s_{i+1})}{\partial s_{i}}\left\vert
_{s_{i}=s_{i+1}=0}\right. =\frac{1-\tanh (yb_{i}c_{iN})}{1+\tanh
(yb_{i}c_{iN})}=e^{-2yb_{i}c_{iN}}
\end{equation*}
and
\begin{equation*}
-\frac{\partial \varphi _{i}(y;s_{i},s_{i+1})}{\partial s_{i+1}}\left\vert
_{s_{i}=s_{i+1}=0}\right. =\frac{\tanh (yb_{i}c_{iN})}{1+\tanh (yb_{i}c_{iN})
}=\frac{1}{2}-\frac{1}{2}\,e^{-2yb_{i}c_{iN}}.
\end{equation*}
Thus,
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\mathbf{P}_{n}(Z_{i}(yn^{\gamma
_{i}},n)=1;\beta _{n}\geq yn^{\gamma _{i}}) \\
&&\qquad \qquad =-\frac{\partial (\varphi _{i}(y;s_{i},s_{i+1}))^{1/2^{i-1}}
}{\partial s_{i}}\left\vert _{s_{i}=s_{i+1}=0}\right. =\frac{1}{2^{i-1}}
\,e^{-2yb_{i}c_{iN}}
\end{eqnarray*}
and
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }\mathbf{P}_{n}(Z_{i+1}(yn^{\gamma
_{i}},n)=1;\beta _{n}\geq yn^{\gamma _{i}}) \\
&&\qquad \quad =-\frac{\partial (\varphi _{i}(y;s_{i},s_{i+1}))^{1/2^{i-1}}}{
\partial s_{i+1}}\left\vert _{s_{i}=s_{i+1}=0}\right. =\frac{1}{2^{i}}
(1-e^{-2yb_{i}c_{iN}}).
\end{eqnarray*}
Combining the previous estimates yields
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\beta _{n}\leq yn^{\gamma _{i}})=1-
\frac{1}{2^{i}}-\frac{1}{2^{i}}e^{-2yb_{i}c_{iN}}.
\end{equation*}
\textbf{Proof of 3).} This is evident.
\textbf{Proof of 4).} The needed statement follows from the equality
\begin{equation*}
-\frac{\partial }{\partial s_{N}}\left( \frac{1}{x+(1-x)/(1-s_{N})}\right)
^{2^{-(N-1)}}\left\vert _{s_{N}=0}\right. =\frac{1}{2^{N-1}}(1-x).
\end{equation*}
\textbf{Proof} \textbf{of Theorem \ref{T_type}.} Consider the case $N\geq 4$
and $i\in \left\{ 2,3,..,N-2\right\} $ only. For $N=2,3$ or $N\geq 4$ and $
i\in \left\{ 1,N-1\right\} $ some of the random variables (events) below do
not exist (are empty) and the needed arguments become shorter.
Since the total number of particles of all types in the reduced process does
not decrease with time, $\mathbf{P}_{n}(\beta _{n}<m)=\mathbf{P}_{n}\left(
\text{\b{Z}}_{1}(m,n)\geq 2\right) $. We now take
\begin{equation*}
m_{i}=n^{\gamma _{i}(1+\gamma _{i})},\ i=1,2,...,N-1,
\end{equation*}
and denote $\mathcal{H}_{i}=\left\{ m:m_{i-1}\leq m\leq m_{i}\right\} $.
Since $\bar{Z}_{i}(k,n)$ is monotone increasing in $k$ for each fixed $n,$
Theorem \ref{T_familyMany} and~(\ref{SurvivSingle}) imply, as $n\rightarrow
\infty $
\begin{eqnarray*}
&&\mathbf{P}_{n}(\zeta _{n}=i;\beta _{n}\notin \mathcal{H}_{i})\leq \mathbf{P
}_{n}(\exists k<m_{i-1}:Z_{i}(k,n)>0) \\
&&\qquad +\mathbf{P}_{n}(\exists k>m_{i}:Z_{i}(k,n)>0) \\
&&\quad \leq \mathbf{P}_{n}(\bar{Z}_{i}(m_{i-1},n)>0)+\mathbf{P}
_{n}(Z_{i}(m_{i})>0)=o(1).
\end{eqnarray*}
By the same statements we conclude, as $n\rightarrow \infty $
\begin{eqnarray*}
&&\mathbf{P}_{n}(\zeta _{n}\notin \{i,i+1\};\beta _{n}\in \mathcal{H}
_{i})\leq \mathbf{P}_{n}\left( \exists k\in \mathcal{H}_{i}:\text{\b{Z}}
_{i-1}(k,n)+\bar{Z}_{i+2}(k,n)>0\right) \\
&&\quad \leq \mathbf{P}_{n}\left( \exists k\in \mathcal{H}_{i}:\text{\b{Z}}
_{i-1}(k)+\bar{Z}_{i+2}(k,n)>0\right) \\
&&\quad \leq \mathbf{P}_{n}(\exists k\in \mathcal{H}_{i}:\text{\b{Z}}
_{i-1}(k)>0)+\mathbf{P}_{n}\left( \exists k\in \mathcal{H}_{i}:\bar{Z}
_{i+2}(k,n)>0\right) \\
&&\quad \leq \mathbf{P}_{n}(\text{\b{Z}}_{i-1}(m_{i-1})>0)+\mathbf{P}_{n}(
\bar{Z}_{i+2}(m_{i},n)>0)=o(1).
\end{eqnarray*}
Hence, as $n\rightarrow \infty $
\begin{eqnarray}
\mathbf{P}_{n}(\zeta _{n}=i) &=&\mathbf{P}_{n}(\zeta _{n}=i;\beta _{n}\in
\mathcal{H}_{i})+o(1) \notag \\
&=&\mathbf{P}_{n}(\beta _{n}\in \mathcal{H}_{i})-\mathbf{P}_{n}(\zeta
_{n}=i+1;\beta _{n}\in \mathcal{H}_{i})+o(1). \label{Type}
\end{eqnarray}
Introduce the event
\begin{equation*}
\mathcal{G}_{i}(j,n)=\left\{ \text{\b{Z}}_{i}(j;n)+\bar{Z}
_{i+2}(j+1,n)=0;Z_{i+1}(j,n)=1\right\} .
\end{equation*}
Clearly,
\begin{eqnarray*}
&&\mathbf{P}_{n}(\zeta _{n}=i+1;\beta _{n}\in \mathcal{H}_{i})=
\sum_{j=m_{i-1}}^{m_{i}}\mathbf{P}_{n}(\zeta _{n}=i+1;\beta _{n}=j) \\
&&\,=\sum_{j=m_{i-1}}^{m_{i}}\mathbf{P}_{n}(\mathcal{G}_{i}(j,n),\bar{Z}
_{i+1}(j+1,n)\geq 2) \\
&&\,=o(1)+\sum_{j=m_{i-1}}^{m_{i}}\mathbf{P}_{n}(\mathcal{G}_{i}(j,n))
\mathbf{P}_{n}(Z_{i+1}(j+1,n)\geq 2|\mathbf{Z}(j,n)=\mathbf{e}_{i+1}).
\end{eqnarray*}
It is not difficult to check (recall (\ref{DefNONimmigr}), (\ref{Derivat})
and (\ref{ExplCoeff})) that
\begin{eqnarray*}
\mathbf{P}_{n}(Z_{i+1}(j+1,n)=1|\mathbf{Z}(j,n)=\mathbf{e}_{i+1}) &=&\frac{
Q_{n-j-1}^{(i+1,N)}}{Q_{n-j}^{(i+1,N)}}\frac{dh_{i+1}(s,\mathbf{1}^{(N-i-1)})
}{ds}\left\vert _{s=H_{n-j-1}^{(i+1,N)}(\mathbf{0})}\right. \\
&\geq &\frac{dh_{i+1}(s,\mathbf{1}^{(N-i-1)})}{ds}\left\vert
_{s=H_{n-j-1}^{(i+1,N)}(\mathbf{0})}\right. \\
&\geq &1-2b_{i+1}Q_{n-j-1}^{(i+1,N)} \\
&\geq &1-2b_{i+1}Q_{n-m_{i}}^{(i+1,N)}.
\end{eqnarray*}
Hence, using the estimate
\begin{eqnarray*}
\mathbf{P}_{n}(Z_{i+1}(j+1,n)\geq 2|\mathbf{Z}(j,n)=\mathbf{e}_{i+1}) &=&1-
\mathbf{P}_{n}(Z_{i+1}(j+1,n)=1|\mathbf{Z}(j,n)=\mathbf{e}_{i+1}) \\
&\leq &2b_{i}Q_{n-m_{i}}^{(i+1,N)}
\end{eqnarray*}
we conclude
\begin{eqnarray*}
\mathbf{P}_{n}(\zeta _{n}=i+1;\beta _{n}\in \mathcal{H}_{i})
&=&o(1)+O(m_{i}Q_{n-m_{i}}^{(i+1,N)}) \\
&=&o(1)+O(n^{\gamma _{i}(1+\gamma _{i})}n^{-\gamma _{i+1}})=o(1).
\end{eqnarray*}
This, on account of (\ref{recent_i}) and (\ref{Type}) gives
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}_{n}(\zeta _{n}=i)=\lim_{n\rightarrow
\infty }\mathbf{P}_{n}(\beta _{n}\in \mathcal{H}_{i})=\lim_{n\rightarrow
\infty }\mathbf{P}_{n}(n^{\gamma _{i}}\ll \beta _{n}\ll n^{\gamma _{i+1}})=
\frac{1}{2^{i}}
\end{equation*}
as desired.
Finally,
\begin{equation*}
\lim_{n\rightarrow \infty }\mathbf{P}_{n}( \zeta _{n}=N) =1-\sum_{i=1}^{N-1}
\frac{1}{2^{i}}=\frac{1}{2^{N-1}}.
\end{equation*}
Theorem \ref{T_type} is proved.
\textbf{Acknowledgement}. This work was partially supported by the Russian
Foundation for Basic Research, project N14-01-00318. The author would also
like to thank prof. A.M.Zubkov for valuable remarks.
\end{document} |
\begin{document}
\title{A {De~Giorgi} Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances}
\author{Jun~Zheng$^{1}$\thanks{$^{1}$School of Civil Engineering and School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, P. R. of China 611756
{\tt\small [email protected]}}and Guchuan~Zhu$^{2}$,~\IEEEmembership{Senior~Member,~IEEE}\thanks{$^{2}$Department of Electrical Engineering, Polytechnique Montr\'{e}al, P.O. Box 6079, Station Centre-Ville, Montreal, QC, Canada H3T 1J4
{\tt\small [email protected]}}
\thanks{\textcolor{blue}{This paper has been accepted for publication by IEEE TAC, and is available at http://dx.doi.org/10.1109/TAC.2018.2880160}}
}
\markboth{Manuscript Submitted to IEEE Trans. on Automatic Control}
{Zheng \MakeLowercase{\textit{et al.}}: }
\maketitle
\begin{abstract}
This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)}.
\end{abstract}
\begin{IEEEkeywords}
ISS, De~Giorgi iteration, boundary disturbance, in-domain disturbance, Burgers' equation, unstable reaction-diffusion equation.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}\label{Sec: Introduction}
Extending the theory of ISS, which was originally developed for finite-dimensional nonlinear systems \cite{Sontag:1989,Sontag:1990}, to infinite dimensional systems has received a considerable attention in the recent literature. In particular, there are significant progresses on the establishment of ISS estimates with respect to disturbances \cite{Argomedo:2013, Argomedo:2012, Dashkovskiy:2010, Dashkovskiy:2013, jacob2016input, Karafyllis:2014, Karafyllis:2016, Karafyllis:2016a, Karafyllis:2018, Logemann:2013, Mazenc:2011, Mironchenko:2018, Prieur:2012, Tanwani:2017,Zheng:2017} for different types of {PDEs}.
It is noticed that most of the earlier work on this topic dealt with disturbances distributed over the domain. It was demonstrated that the method of Lyapunov functionals is a well-suited tool for dealing with a wide rang of problems of this category. Moreover, it is shown in \cite{Argomedo:2012} that the method of Lyapunov functionals can be readily applied to some systems with boundary disturbances by transforming the later ones to a distributed disturbance. {However, ISS estimates obtained by such a method may include time derivatives of boundary disturbances, which is not strictly in the original form of ISS formulation.
The problems with disturbances acting on the boundaries usually lead to a formulation involving unbounded operators. It is shown in \cite{jacob2016input,Jacob:2017} that for a class of linear PDEs, the exponential stability plus a certain admissibility implies the ISS and iISS (integral input-to-state stability \cite{jacob2016input,Sontag:1998}) w.r.t. boundary disturbances. However, it may be difficult to assess this property for nonlinear PDEs. To resolve this concern while not invoking unbounded operators in the analysis, it is proposed in \cite{Karafyllis:2016,Karafyllis:2016a,karafyllis2017siam} to derive the ISS property directly from the estimates of the solution to the considered PDEs using the method of spectral decomposition and finite-difference. ISS in $L^2$-norm and in weighted $L^\infty$-norm for PDEs with a Sturm-Liouville operator is established by applied this method in \cite{Karafyllis:2016,Karafyllis:2016a,karafyllis2017siam}. However, spectral decomposition and finite-difference schemes may involve heavy computations for nonlinear PDEs or problems on multidimensional spatial domains. It is introduced in \cite{Mironchenko:2017} a monotonicity-based method for studying the ISS of nonlinear parabolic equations with boundary disturbances. It is shown that with the monotonicity the ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to the ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary conditions. As an application of this method, the ISS properties in $L^p$-norm ({$\forall p>2$}) for some linear parabolic systems have been established.
In a recent work \cite{Zheng:2017}, the classical method of Lyapunov functionals is applied to establish ISS properties in $L^2$-norm w.r.t. boundary disturbances for a class of semilinear parabolic PDEs. Some technical inequalities have been developed, which allows dealing directly with items on the boundary points in proceeding on ISS estimates. {}{The result of \cite{Zheng:2017}} shows that the method of Lyapunov functionals is still effective in obtaining ISS properties for some linear and nonlinear PDEs with Neumann or Robin boundary conditions. However, the technique used may not be suitable for problems with Dirichlet boundary conditions.
{}{The present work is dedicated to the establishment of ISS properties for Burgers' equation that is one of the most popular PDEs in mathematical physics \cite{Hopf:1991}. Burgers' equation is considered as a simplified form of the Navier-Stokes equation and can be used to approximate the Saint-Venant equation. Therefore, the study on the control of the Burgers' equation is an important and natural step for flow control and many other fluid dynamics inspired applications. Indeed, there exists a big amount of work on the control of the Burgers' equation, e.g., just to cite a few, \cite{Azmi:2016_SIAM,Burns:1991, Byrnes:1998, Krstic:1999, Krstic:2008_IEEE_TAC, Liu:2000, Liu:2001}.}
{}{The problem dealt with in this work can be seen as a complementary setting compared to that considered in \cite{Zheng:2017} in the sense that the problem is subject to Dirichlet boundary conditions.} The method developed in this note consists first in splitting the original system into two subsystems: one system with boundary disturbances and a zero-initial condition, and another one with no boundary disturbances, but with homogenous boundary conditions and a non-zero initial condition. Note that the in-domain disturbances can be placed in either of these two subsystems. Then, ISS properties in $L^\infty$-norm for the first system will be deduced by the technique of De~Giorgi iteration, and ISS properties in $L^2$-norm (or $L^\infty$-norm) for the second system will be established by the method of Lyapunov functionals. Finally, the ISS properties in $L^2$-norm (or $L^\infty$-norm) for the original system are obtained by combining the ISS properties of the two subsystems. With this method, we established the ISS in $L^2$-norm for Burgers' equation with boundary and in-domain disturbances. Moreover, using the techniques of transformation, splitting and De~Giorgi iteration, we established the ISS in $L^\infty$-norm for a 1-$D$ {}{linear unstable reaction-diffusion equation} with boundary feedback control including actuation errors. Note that although the De~Giorgi iteration is a classic method in regularity analysis of elliptic and parabolic PDEs, it is the first time, to the best of our knowledge, that it is introduced in the investigation of ISS properties for PDEs. Moreover, the technique of De~Giorgi iteration may be applicable for certain {}{nonlinear PDEs and }problems on multidimensional spatial domains.
The rest of the note is {}{organized} as follows. Section~\ref{Sec: Preliminaries} introduces briefly the technique of De~Giorgi iteration and presents some preliminary inequalities needed for the subsequent development. Section~\ref{Sec: Main results} presents the considered problems and the main results. Detailed development on the establishment of ISS properties for Burgers' equation is given Section~\ref{Sec: Burgers Eq}. The application of De~Giorgi iteration in the establishment of ISS in $L^\infty$-norm for a 1-$D$ {}{linear unstable reaction-diffusion equation} is illustrated in Section~\ref{Sec: reaction-diffusion Eq}. Finally, some concluding remarks are provided in Section~\ref{Sec: Conclusion}.
\section{Preliminaries}\label{Sec: Preliminaries}
\subsection{{De~Giorgi} iteration}\label{Sec: De Giorgi iteration}
De~Giorgi iteration is an important tool for regularity analysis of elliptic and parabolic PDEs. In his famous work on linear elliptic equations published in 1957 \cite{DeGiorgi:1957}, De~Giorgi established local boundedness and H\"{o}lder continuity for functions satisfying certain integral inequalities, known as the De~Giorgi class of functions, which completed the solution of Hilbert's 19$^{\text{th}}$ problem. The same problem has been resolved independently by Nash in 1958 \cite{Nash1958}. It was shown later by Moser that the result of De~Giorgi and Nash can be obtained using a different formulation \cite{Moser1961}. In the literature, this method is often called the De~Giorgi-Nash-Moser theory.
{}{Let $\mathbb{R}:=(-\infty,+\infty)$}, $\Omega\subset \mathbb{R}^n(n\geq 1)$ be an open bounded set, and $\gamma $ be a constant. The De~Giorgi class $DG^+(\Omega,\gamma)$ consists of functions $u\in W^{1,2}(\Omega)$ which satisfy, for every ball $B_r(y)\subset \Omega$, every $0<r'<r$, and every $k \in \mathbb{R}$, the following Caccioppoli type inequality:
\begin{align*}
\int_{B_{r'}(y)}|\nabla (u-k)_+|^2\text{d}x\leq \frac{\gamma}{(r-r')^2}\int_{B_{r}(y)}| (u-k)_+|^2\text{d}x,
\end{align*}
where $ (u-k)_+=\max\{u-k,0\}$. The class $DG^-(\Omega,\gamma)$ is defined in a similar way. The main idea of De~Giorgi iteration is to estimate $ |A_k|$, the measure of $\{x\in \Omega;u(x)\geq k\}$, and derive $|A_k|=0$ with some $k$ for functions $u$ in De~Giorgi class by using the iteration formula given below.
\begin{lemma}[{}{\cite[Lemma 4.1.1]{Wu2006}}]\label{iteration}
Suppose that $\varphi$ is a non-negative decreasing function on $[k_0,+\infty)$ satisfying
\begin{align*}
\varphi(h)\leq \bigg(\frac{M}{h-k}\bigg)^\alpha\varphi^\beta(k),\ \ \forall h>k\geq k_0,
\end{align*}
where $M>0,\alpha>0,\beta>1$ are constants. Then the following holds
\begin{align*}
\varphi(k_0+l_0)=0,
\end{align*}
with $l_0=2^{\frac{\beta}{\beta-1}}M(\varphi(k_0))^{\frac{\beta-1}{\alpha}}$.
\end{lemma}
The method of De~Giorgi iteration can be generalized to some linear parabolic PDEs and PDEs with a divergence form (see, e.g., \cite{DiBenedetto:2010,Wu2006}). However, this method in its original formulation cannot be applied directly in the establishment of ISS properties for infinite dimensional systems. The main reason is that the obtained boundedness of a solution depends always on {some data that is increasing in $t$ rather than a class $\mathcal {K}\mathcal {L}$ function associated with $u_0 $ and $t$}, even for linear parabolic PDEs \cite{DiBenedetto:2010,Wu2006}, {which is not under the form of ISS}. To overcome this difficulty, we developed in this work an approach that amounts first to splitting the original problem into two subsystems and then to applying the De~Giorgi iteration together with the technique of Laypunov functionals to obtain the ISS estimates of the solutions expressed in the standard formulation of the ISS theory.
\subsection{Preliminary inequalities}\label{Sec: preliminary results}
{}{Let $\mathbb{R}_+:=(0,+\infty)$ and $\mathbb{R}_{\geq 0} := [0,+\infty)$. For notational simplicity, we always denote $\|\cdot\|_{L^{2}(0,1)}$ by $\|\cdot\|$ in this note.} We present below two inequalities needed for the subsequent development.
\newline
\begin{lemma}\label{Lemma 3}
Suppose that $u\in C^{1}([a,b];\mathbb{R})$, then for any $p\geq 1$, {}{one has}
\begin{align}\label{eq: Sobolev embedding}
\bigg(\int_{a}^b|u|^p\text{d}x\bigg)^{\frac{1}{p}}\leq (b-a)^{\frac{1}{p}}\bigg(\frac{2}{b-a}\|u\|^2+(b-a)\|u_x\|^2\bigg)^{\frac{1}{2}}.
\end{align}
\end{lemma}
\begin{IEEEproof}
We show first that
\begin{align}
u^2(c)\leq \dfrac{2}{b-a}\|u\|^2+(b-a)\|u_x\|^2,\ \forall c\in[a,b].\label{001}
\end{align}
Denote $(u_z(z))^2$ by $u_z^2(z)$. For each $c\in [a,b]$, let $g(x)={\int_{c}^xu^2_z(z)\text{d}z}$. Note that $g'(x)=u^2_x(x)$. By H\"{o}lder's inequality (see \cite[Appendix B.2.e]{Evans:2010}), we have
\begin{align*}
\displaystyle\left(\int_{x}^cu_z(z)\text{d}z\right)^2 \leq \left|(c-x)\int_{x}^cu^2_z(z)\text{d}z\right|
=(x-c)g(x).
\end{align*}
It follows
\begin{align*}
u^2(c)&=\bigg(u(x)+{\int_{x}^cu_z(z)\text{d}z}\bigg)^2\notag\\
&\leq 2u^2(x)+2\bigg({\int_{x}^cu_z(z)\text{d}z}\bigg)^2\notag\\
&\leq 2u^2(x)+2(x-c)g(x).
\end{align*}
Integrating over $[a,b]$ and
noting that
\begin{align*}
&\int_{a}^b(x-c) g(x)\text{d}x\notag\\
=&\bigg[\frac{(x-c)^2}{2}g(x)\bigg]\bigg|_{x=a}^{x=b}-\int_{a}^b\frac{(x-c)^2}{2} u^2_x(x) \text{d}x\notag\\
\leq &\frac{(b-c)^2}{2}\int_{c}^bu^2_x(x)\text{d}x- \frac{(a-c)^2}{2}\int_{c}^au^2_x(x)\text{d}x\notag\\
\leq &\frac{(b-a)^2}{2}\int_{a}^bu^2_x(x)\text{d}x,
\end{align*}
we get
$
u^2(c)(b-a)\leq 2 \|u\|^2 + (b-a)^2\|u_x\|^2,
$
which yields \eqref{001}.
Now by \eqref{001}, we have
\begin{align*}
\bigg(\int_{a}^b|u|^p\text{d}{x}\bigg)^{\frac{1}{p}}&\leq \bigg(\int_{a}^b\max_{x\in[a,b]}|u|^p\text{d}{x}\bigg)^{\frac{1}{p}}\\
&= (b-a)^{\frac{1}{p}}\max_{x\in[a,b]}|u|\\
&\leq
(b-a)^{\frac{1}{p}}\bigg(\frac{2}{b-a}\|u\|^2+(b-a)\|u_x\|^2\bigg)^{\frac{1}{2}}.
\end{align*}
\end{IEEEproof}
\begin{remark}
{}{Note first that \eqref{eq: Sobolev embedding} is a variation of Sobolev embedding inequality, which will be used in the De~Giorgi iteration in the analysis of the Burgers' equation with in-domain and Dirichlet boundary disturbances. Moreover, the inequality~\eqref{001} is an essential technical result for the establishment of the ISS w.r.t. boundary disturbances for PDEs with Robin or Neumann boundary conditions (see, e.g., \cite{Zheng:2017}). Therefore, these two inequalities play an important role in the establishment of the ISS for PDEs w.r.t. boundary and in-domain disturbances with either Robin, or Neumann, or Dirichlet boundary conditions.}
\end{remark}
\section{Problem Formulation and Main Results}\label{Sec: Main results}
\subsection{Problem formulation and well-posedness analysis}\label{Sec: Problem formulations}
In this work, we address ISS properties for Burgers' equation with Dirichlet boundary conditions:
\begin{subequations}\label{++28}
\begin{align}
&u_t-\mu u_{xx}+\nu uu_x=f(x,t)\ \ {\text{in}\ (0,1)\times\mathbb{R}_{+}},\label{++28'}\\
&u(0,t)=0,u(1,t)=d(t),\label{++2}\\
&u(x,0)=u_0(x),\label{++3}
\end{align}
\end{subequations}
where $\mu>0$, $\nu>0$ are constants, $d(t)$ is the disturbance on the boundary, which can represent actuation and sensing errors, and the function $f(x,t)$ is the disturbance distributed over the domain. {}{Throughout this note, we always assume that $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$ and $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$ for some $\theta\in (0,1)$.}
{}{We refer to \cite[Chapter~1, pages 7-9]{Ladyzenskaja:1968} for the definition on H\"{o}lder type function spaces $ \mathcal {H}^{l}([0,1])$, $ \mathcal {H}^{l}([0,T])$, $\mathcal {H}^{l,\frac{l}{2}}([0,1]\times[0,T])$, $C^1([0,1])$, $C^{2,1}([0,1]\times[0,T])$, $\mathcal {H}^{l}(\mathbb{R}_{\geq 0})$, $\mathcal {H}^{l,\frac{l}{2}}([0,1]\times \mathbb{R}_{\geq 0})$ and $C^{2,1}([0,1]\times\mathbb{R}_{\geq 0})$, where $l>0$ is a nonintegral number and $T>0$.} {}{We also refer to \cite[Chapter~1, page~12]{Ladyzenskaja:1968} for the statement of classical solutions of Cauchy problems.}
{}{The result for well-posedness assessment of \eqref{++28} is given below, which is guaranteed by \cite[Theorem 6.1, pages 452-453]{Ladyzenskaja:1968}.
\begin{proposition}\label{Proposition 1}
Assume that $u_0\in \mathcal {H}^{2+\theta}([0,1])$ with {$u_0(0)=0$, $u_0(1)=d(0)$, $\mu u_0{''}(0)+f(0,0)=0$ and $\mu u_0{''}(1)+f(1,0) =d{'}(0)$}. For any $T>0$, there exists a unique classical solution $u\in \mathcal {H}^{2+\theta,1+\frac{\theta}{2}}( [0,1]\times [0,T])\subset C^{2,1}( [0,1]\times[0,T])$ of \eqref{++28}.
\end{proposition}}
\begin{remark}
The proof of Proposition~\ref{Proposition 1} follows from Theorem 6.1 in \cite[pages 452-453]{Ladyzenskaja:1968}, which establishes the existence of a unique solution in the H\"{o}lder space of functions {$ \mathcal {H}^{2+\theta,1+\frac{\theta}{2}}( [0,1]\times [0,T])$} for a more general quasilinear parabolic equations with Dirichlet boundary conditions. It should be noticed that the proof of Theorem~6.1 in \cite[pages 452-453]{Ladyzenskaja:1968} is based on the linearization of the considered system and the application of the Leray-Schauder theorem on fixed points. Since {$\mathcal {H}^{2+\theta,1+\frac{\theta}{2}}( [0,1]\times [0,T])\subset C^{2,1}( [0,1]\times[0,T])$}, we can obtain the existence of the unique classical solution in the time interval $[0,T]$, where $T>0$ can be arbitrarily large.
\end{remark}
\subsection{{Main results on ISS estimates for Burgers' equation}}
Let $\mathcal {K}=\{\gamma : \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}|\ \gamma(0)=0,\gamma$ is continuous, strictly increasing$\}$; $ \mathcal {K}_{\infty}=\{\theta \in \mathcal {K}|\ \lim\limits_{s\rightarrow\infty}\theta(s)=\infty\}$; $ \mathcal {L}=\{\gamma : \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}|\ \gamma$ is continuous, strictly decreasing, $\lim\limits_{s\rightarrow\infty}\gamma(s)=0\}$; $ \mathcal {K}\mathcal {L}=\{\beta : \mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}|\ \beta(\cdot,t)\in \mathcal {K}, \forall t \in \mathbb{R}_{\geq 0}$, and $\beta(s,\cdot)\in \mathcal {L}, \forall s \in {}{\mathbb{R}_{+}}\}$.
{\begin{definition}
System~\eqref{++28} is said to be input-to-state stable (ISS) in $L^q$-norm ($2\leq q\leq +\infty$) {w.r.t. {boundary disturbances} $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$ and {in-domain disturbances} $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$}, if there exist functions $\beta\in \mathcal {K}\mathcal {L}$ and $ \gamma_1, \gamma_2,\in \mathcal {K}$ such that the solution to \eqref{++28} satisfies
\begin{align}\label{Eq: ISS def2}
\begin{split}
\|{}{u(\cdot,t)}\|_{L^{q}(0,1)}\leq & \beta\left( \|{u_0}\|_{L^{q}(0,1)},t\right)+\gamma_1\left(\max_{s\in [0,t]}|d(s)|\right)\\
&+\gamma_2\left(\max_{(x,s)\in [0,1]\times [0,t]}|f(x,s)|\right),\ \forall t\geq 0.
\end{split}
\end{align}
System~\eqref{++28} is said to be ISS {w.r.t. boundary disturbances $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$, and integral input-to-state stable (iISS) w.r.t. in-domain disturbances $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$}, in $L^q$-norm ($2\leq q\leq +\infty$),
if there exist functions $\beta\in \mathcal {K}\mathcal {L},\theta\in \mathcal {K}_{\infty} $, and $\gamma_1 ,\gamma_2 \in \mathcal {K}$ such that the solution to \eqref{++28} satisfies
\begin{align}\label{Eq: iISS def2}
\begin{split}
\|{}{u(\cdot,t)}\|_{L^{q}(0,1)} \leq & \beta\left( \|{u_0}\|_{L^{q}(0,1)},t\right)
+\gamma_1\left(\max_{s\in [0,t]}|d(s)|\right) \\
&+\theta\left(\!\!\int_{0}^t\!\!\gamma_2(\|f(\cdot,s)\|)\text{d}s\right),\ \forall t\geq 0.
\end{split}
\end{align}
Moreover, System~\eqref{++28} is said to be exponential input-to-state stable (EISS), or exponential integral input-to-state stable (EiISS), w.r.t. {boundary disturbances $d(t)$, or in-domain disturbances $f(x,t)$}, if there exist $\beta'\in \mathcal {K}_{\infty}$ and a constat $\lambda > 0$ such that $\beta( \|{u_0}\|_{L^{q}(0,1)},t) \leq \beta'(\|{u_0}\|_{L^{q}(0,1)})e^{-\lambda t}$ in \eqref{Eq: ISS def2} or \eqref{Eq: iISS def2}.
\end{definition}}
In order to apply the technique of splitting and the method of De~Giorgi iteration in the investigation of the ISS properties for the considered problem, {while guaranteeing the well-posedness by Proposition~\ref{Proposition 1} for every system}, we assume that the compatibility condition {$u_0(0)=u_0''(0)=u_0(1)=u_0''(1)=d(0)=d'(0)=f(0,0)=f(1,0)=0$} always holds {}{in Section \ref{Sec: Main results} and \ref{Sec: Burgers Eq}}. Furthermore, unless} stated, we always take a certain function in {}{$C^{2,1}( [0,1]\times\mathbb{R}_{\geq 0})$} as the unique solution of a considered system. Then the ISS properties w.r.t. boundary and in-domain disturbances for System~\eqref{++28} are stated in the following {theorems}.
\begin{theorem} \label{Theorem 11}
{System \eqref{++28} is {}{EISS} in $L^2$-norm w.r.t. {boundary disturbances} $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$ and {in-domain disturbances} $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$ satisfying $\sup\limits_{s\in \mathbb{R}_{\geq 0}} |d(s)| + \frac{4\sqrt{2}}{\mu}\sup\limits_{(x,s)\in[0,1]\times \mathbb{R}_{\geq 0}}|f(x,s)| < \frac{\mu}{\nu}$}, having the following estimate for any $t>0$:
\begin{align*}
\|u(\cdot,t)\|^2\leq & 2\|u_0\|^2 e^{-\mu t} +4\max\limits_{s\in [0,t)} |d(s)|^2 \nonumber \\
&+\frac{128}{\mu^2}\max\limits_{(x,s)\in[0,1]\times [0,t]}|f(x,s)|^2.
\end{align*}
\end{theorem}
\begin{theorem} \label{Theorem 11-2}
{System \eqref{++28} is {}{EISS} in $L^2$-norm w.r.t. {boundary disturbances} $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$ satisfying $\sup\limits_{t\in \mathbb{R}_{\geq 0}} |d(t)|<\frac{\mu}{\nu}$, and {}{EiISS} w.r.t. {in-domain disturbances} $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$,} having the following estimate for any $t>0$:
\begin{align*}
\|u(\cdot,t)\|^2 \leq& 2\|u_0\|^2 e^{-(\mu -\varepsilon)t}+2\max\limits_{s\in [0,t]}|d(s)|^2 \nonumber \\
&+\frac{2}{\varepsilon} \int_{0}^t\|f(\cdot,s)\|^2\text{d}s,\ \forall\varepsilon\in (0,\mu).
\end{align*}
\end{theorem}
\begin{remark}
In general, the boundness of the disturbances is a reasonable assumption for nonlinear PDEs in the establishment of ISS properties \cite{Mironchenko:2016}. However, as shown in Section \ref{Sec: reaction-diffusion Eq}, the boundedness of the disturbances may be not a necessary condition for ISS properties of linear PDEs.
\end{remark}
{}{\begin{remark}
{}{As pointed out in \cite{Karafyllis:2016a}, the} assumptions on the continuity of $f$ and $d$ are required for assessing the existence of a classical solution of the considered system. However, they are only sufficient conditions and can be weakened {}{if solutions in a weak sense are considered. Moreover, for the establishment of ISS estimates, the assumptions on the continuity of $f$ and $d$ can eventually be relaxed.}
\end{remark}}
\section{Proofs of ISS Estimates for Burgers' Equation}\label{Sec: Burgers Eq}
\subsection{Proof of Theorem~\ref{Theorem 11}}
In this section, we establish the ISS estimates for Burgers' equation w.r.t. boundary and in-domain disturbances described in Theorem~\ref{Theorem 11} by using the technique of splitting. Specifically, let $w$ be the unique solution of the following system:
\begin{subequations}\label{++29}
\begin{align}
&w_t-\mu w_{xx}+\nu ww_x=f(x,t)\ \ \text{in}\ (0,1)\times\mathbb{R}_{+},\\
&w(0,t)=0,w(1,t)=d(t),\\
&w(x,0)=0.
\end{align}
\end{subequations}
Then $v=u-w$ is the unique solution of the following system:
\begin{subequations}\label{++31}
\begin{align}
&v_t-\mu v_{xx}+\nu vv_x+\nu {(wv )_x}=0 \ \text{in}\ (0,1)\times\mathbb{R}_{+}, \\
&v(0,t)=v(1,t)=0,\\
&v(x,0)=u_0(x).
\end{align}
\end{subequations}
For System~\eqref{++29}, we have the following estimate.
\begin{lemma} \label{Theorem 12} Suppose that $\mu>0,\nu>0$. For every $t>0$, {}{one has}
\begin{align}\label{++13}
&\max\limits_{(x,s)\in[0,1]\times [0,t]} |w(x,s)| \notag\\
&\;\;\; \leq \max\limits_{s\in [0,t]}|d(s)| + \frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in[0,1]\times [0,t]}|f(x,s)|.
\end{align}
\end{lemma}
For System~\eqref{++31}, we have the following estimate.
\begin{lemma} \label{Theorem 13}{Suppose that $\mu>0,\nu>0$, and ${\sup\limits_{t\in \mathbb{R}_{\geq 0}}} |d(t)|+ \frac{4\sqrt{2}}{\mu} {\sup\limits_{(x,t)\in [0,1]\times\mathbb{R}_{\geq 0}}} |f(x,t)|<\frac{\mu}{\nu}$}. For every $t>0$, {}{one has}
\begin{align*}
\|v(\cdot,t)\|^2\leq {\|u_0\|^2 e^{-\mu t}.}
\end{align*}
\end{lemma}
Then the result of Theorem~\ref{Theorem 11} is a consequence of {Lemma}~\ref{Theorem 12} and {Lemma}~\ref{Theorem 13}.
\begin{IEEEproof}[Proof of Theorem~\ref{Theorem 11}] Note that $u=w+v$, we get by {Lemma}~\ref{Theorem 12} and {Lemma}~\ref{Theorem 13}:
\begin{align*}
\|u(\cdot,t)\|^2
\leq & 2\|w(\cdot,t)\|^2+2\|v(\cdot,t)\|^2\\
\leq & 2\left(\max\limits_{(x,s)\in [0,1]\times[0,t]} |w(x,s)|\right)^2+2\|v(\cdot,t)\|^2\\
\leq & 2\|u_0\|^2 e^{-\mu t}\notag\\
&+2\left(\!\!\max\limits_{s\in [0,t]}|d(s)|+\frac{4\sqrt{2}}{\mu}\!\! \max\limits_{(x,s)\in [0,1]\times[0,t]} \!\!|f(x,s)|\!\!\right)^2.
\end{align*}
\end{IEEEproof}
In the following, we use De~Giorgi iteration and Lyapunov method to prove {Lemma}~\ref{Theorem 12} and {Lemma}~\ref{Theorem 13}, respectively.
\begin{IEEEproof}[Proof of Lemma~\ref{Theorem 12}] In order to apply the technique of De~Giorgi iteration, we shall define some quantities.
For any $t>0$, let $k_0=\max\Big\{\max\limits_{s\in[0,t]}d(s),0\Big\}$. For any $k\geq k_0$, let $ \eta(x,s)=(w(x,s)-k)_+\chi_{[t_1,t_2]}(
s)$, where $\chi_{[t_1,t_2]}(s) $ is the character function on $[t_1,t_2]$ and $0\leq t_1<t_2\leq t$. Let $A_{k}(s)=\{x\in (0,1);w(x,s)>k\}$ and {}{$\varphi_{k}=\sup\limits_{s\in(0,t)}|A_{k}(s)|$}, where $|B|$ denotes the 1-dimensional Lebesgue measure of a set $B\subset(0,1)$. For any $p>2$, let $l_0= \frac{1}{\mu}2^{\frac{5p-8}{2p-4}}\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)| \varphi_{k_0}$. The main idea of De~Giorgi iteration is to show that $|A_{k_0+l_0}(s)|=0 $ for almost every $s\in [0,t]$, which yields $~~~~{\text{ess}}\!\!\!\!\!\!\!\!\sup\limits_{\!\!\!\!\!\!\!\!(x,s)\in[0,1]\times [0,t]} w(x,s)\leq k_0+l_0 $. The lower boundedness of $w(x,s)$ can be obtained in a similar way. Then the desired result is guaranteed by the continuity of $w$ and its lower and upper boundedness.
{}{Although the computation of De~Giorgi iteration can follow a standard process (see, e.g., the case of linear parabolic equations presented in \cite[Theorem 4.2.1, \S4.2.2]{Wu2006}), we provide the details for completeness.} Multiplying \eqref{++29} by $\eta$, and {}{ noting that $(w(0,s)-k)_+ =(w(1,s)-k)_+=0$ for $k\geq k_0$ and $s\in [0,t]$}, we get
\begin{align}\label{+16}
&\int_{0}^t\int_{0}^1(w-k)_t(w-k)_+\chi_{[t_1,t_2]}(s)\text{d}x\text{d}s \notag\\
&+\mu\int_{0}^t\int_{0}^1|((w-k)_+)_x |^2\chi_{[t_1,t_2]}(s)\text{d}x\text{d}s\notag\\
&+\nu\int_{0}^t\int_{0}^1ww_x(w-k)_+\chi_{[t_1,t_2]}(s) \text{d}x\text{d}s \notag\\
=&\int_{0}^t\int_{0}^1f(w-k)_+\chi_{[t_1,t_2]}(s) \text{d}x\text{d}s.
\end{align}
Let $I_k(s)=\int_{0}^1((w(x,s)-k)_+)^2\text{d}x$, which is absolutely continuous on $[0,t]$. Suppose that $I_k(t_0)=\max\limits_{s\in[0,t]}I_k(s)$ with some $t_0\in [0,t]$. Due to $I_k(0)=0$ and $I_k(s)\geq 0 $, we can assume that $t_0>0$ without loss of generality.
For $ \varepsilon>0$ small enough, choosing $t_1=t_0-\varepsilon$ and $t_2=t_0 $, it follows
\begin{align*}
&\frac{1}{2\varepsilon}\int_{t_0-\varepsilon}^{t_0}\frac{d}{dt}\int_{0}^1((w-k)_+)^2\text{d}x\text{d}s \notag\\
&+\frac{\mu}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1|((w-k)_+)_x |^2\text{d}x\text{d}s \notag\\
&+\frac{\nu}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1ww_x(w-k)_+ \text{d}x\text{d}s \notag\\
\leq & \frac{1}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1|f|(w-k)_+\text{d}x\text{d}s.
\end{align*}
Note that
\begin{align*}
\frac{1}{2\varepsilon}\int_{t_0-\varepsilon}^{t_0}\frac{d}{dt}
\int_{0}^1((w-k)_+)^2\text{d}x\text{d}s&=\frac{1}{2\varepsilon}( I_k(t_0)-I_k(t_0-\varepsilon))\notag\\
&\geq 0.
\end{align*}
We have
\begin{align*}
& \frac{\mu}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1|((w-k)_+)_x |^2\text{d}x\text{d}s \notag\\
&+\frac{\nu}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1ww_x(w-k)_+ \text{d}x\text{d}s \notag\\
\leq & \frac{1}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1|f|(w-k)_+\text{d}x\text{d}s.
\end{align*}
Letting $ \varepsilon\rightarrow 0^+$, and noting that
\begin{align}
&\lim_{\varepsilon\rightarrow 0^+}\frac{1}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1ww_x(w-k)_+ \text{d}x\text{d}s \notag\\
= &\int_{0}^1w(x,t_0)w_x(x,t_0)(w(x,t_0)-k)_+ \text{d}x \notag\\
=& \int_{0}^1(w(x,t_0)-k)_+((w(x,t_0)-k)_+)_x(w(x,t_0)-k)_+ \text{d}x \notag\\
&+\int_{0}^1k((w(x,t_0)-k)_+)_x(w(x,t_0)-k)_+ \text{d}x\notag\\
=& \frac{1}{3}((w(x,t_0)-k)_+)^{3}|^{x=1}_{x=0}+\frac{k}{2}((w(x,t_0)-k)_+)^{2}|^{x=1}_{x=0}\notag\\
=&0,\label{+201803}
\end{align}
we get
\begin{align}
&\mu\int_{0}^1|((w(x,t_0)-k)_+)_x |^2\text{d}x \notag\\
\leq & \int_{0}^1|f(x,t_0)|(w(x,t_0)-k)_+\text{d}x.\label{+20181}
\end{align}
We deduce by Lemma~\ref{Lemma 3}, Poincar\'{e}'s inequality \cite[Chap.~2, Remark~2.2]{Krstic:2008}, and \eqref{+20181} that for any $p>2$,
\begin{align*}
&\bigg(\int_{0}^1|(w(x,t_0)-k)_+ |^p\text{d}x\bigg)^{\frac{2}{p}} \notag\\
\leq & 2 \int_{0}^1|((w(x,t_0)-k)_+)_x |^2\text{d}x \notag\\
\leq & \frac{2}{\mu} \int_{0}^1|f(x,t_0)|(w(x,t_0)-k)_+\text{d}x.
\end{align*}
Then we have
\begin{align*}
&\bigg(\int_{A_{k}(t_0)}|(w(x,t_0)-k)_+ |^p\text{d}x\bigg)^{\frac{2}{p}} \notag\\
\leq & \frac{2}{\mu} \int_{A_{k}(t_0)}|f(x,t_0)|(w(x,t_0)-k)_+\text{d}x.
\end{align*}
By H\"{o}lder's inequality (see \cite[Appendix B.2.e]{Evans:2010}), it follows
\begin{align*}
&\bigg(\int_{A_{k}(t_0)}|(w(x,t_0)-k)_+ |^p\text{d}x\bigg)^{\frac{2}{p}} \notag\\
\leq &\frac{2}{\mu} \bigg(\int_{A_{k}(t_0)}|(w(x,t_0)-k)_+|^p\text{d}x\bigg)^{\frac{1}{p}}\bigg(\int_{0}^1|f(x,t_0)|^q\text{d}x\bigg)^{\frac{1}{q}},
\end{align*}
where $\frac{1}{p}+\frac{1}{q}=1$.
Thus
\begin{align}\label{++14}
&\bigg(\int_{A_{k}(t_0)}|(w(x,t_0)-k)_+ |^p\text{d}x\bigg)^{\frac{1}{p}}\notag\\
\leq & \frac{2}{\mu} \bigg(\int_{A_{k}(t_0)}|f(x,t_0)|^q\text{d}x\bigg)^{\frac{1}{q}}\notag\\
\leq & \frac{2}{\mu} |{A_{k}(t_0)}|^{\frac{1}{q}} \max_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\notag\\
\leq & \frac{2}{\mu} \max_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\varphi_{k}^{\frac{1}{q}}.
\end{align}
Now for $I_k(t_0)$, we get by H\"{o}lder's inequality and \eqref{++14}
\begin{align*}
I_k(t_0)&\leq \bigg(\int_{A_{k}(t_0)}|(w(x,t_0)-k)_+ |^p\text{d}x \bigg)^{\frac{2}{p}}|{A_{k}(t_0)}|^{\frac{p-2}{p}}\notag\\
&\leq \bigg(\frac{2}{\mu} \max_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\bigg)^2\varphi_{k}^{3-\frac{4}{p}}.
\end{align*}
Recalling the definition of $I_k(t_0)$, for any $s\in [0,t]$ we conclude that
\begin{align}
{I_k(s)}\leq I_k(t_0)\leq \bigg(\frac{2}{\mu} \max_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\bigg)^2\varphi_{k}^{3-\frac{4}{p}}.\label{++15}
\end{align}
Note that for any $h>k$ and $s\in [0,t]$ the following holds
\begin{align}
{I_k(s)}\geq \int_{A_{h}({}{s})}\!\!\!\!\!\!\!\!((w(x,{}{s})-k)_+)^2 \text{d}x\geq (h-k)^2{|A_h(s)|}.\label{+201802}
\end{align}
Then we infer from \eqref{++15} and \eqref{+201802} that
\begin{align*}
(h-k)^2\varphi_h\leq \bigg(\frac{2}{\mu} \max_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\bigg)^2\varphi_{k}^{3-\frac{4}{p}},
\end{align*}
which is
\begin{align*}
\varphi_h\leq \left(\frac{2}{\mu}\frac{\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|}{h-k}\right)^2\varphi_{k}^{3-\frac{4}{p}}.
\end{align*}
As $p>2$, we have $ 3-\frac{4}{p}>1$. By Lemma~\ref{iteration}, we obtain
\begin{align*}
\varphi_{k_0+l_0}=\sup_{s\in[0,t]}|A_{k_0+l_0}|=0,
\end{align*}
where $l_0=2^{\frac{3p-4}{2p-4}}\frac{2}{\mu} \max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\varphi_{k_0}^{1-\frac{2}{p}}\leq \frac{1}{\mu}2^{\frac{5p-8}{2p-4}}\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|$.
By the definition of $A_k$, for almost {every} $(x,s)\in [0,1]\times [0,t]$, one has
\begin{align*}
w(x,s)
\leq & k_0+\frac{1}{\mu}2^{\frac{5p-8}{2p-4}}\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|\notag\\
= &\max\Big\{\max\limits_{s\in[0,t]}d(s),0\Big\} +\frac{1}{\mu}2^{\frac{5p-8}{2p-4}}\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|.
\end{align*}
By continuity of $w(x,s) $, for every $(x,s)\in [0,1]\times [0,t]$, the following holds
\begin{align*}
w(x,s)
\leq \max\Big\{\max\limits_{s\in[0,t]}d(s),0\Big\}\frac{1}{\mu}2^{\frac{5p-8}{2p-4}}\max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|.
\end{align*}
Letting $p\rightarrow +\infty$, we get for every $(x,s)\in [0,1]\times [0,t]$
\begin{align}
w(x,s)
\leq & \max\Big\{\max\limits_{s\in[0,t]}d(s),0\Big\}\notag\\
&+\frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|.\label{++16}
\end{align}
To conclude on the inequality \eqref{++13}, we need also to prove the lower boundedness of $w(x,t)$. Indeed, setting $\overline{w}=-w$, we get
\begin{align*}
&\overline{w}_t-\mu \overline{w}_{xx}-\nu \overline{w}\overline{w}_x=-f(x,t),\\
&\overline{w}(0,t)=0,\overline{w}(1,t)=-d(t),\\
&\overline{w}(x,0)=0.\notag
\end{align*}
Proceeding as above and noting \eqref{+201803}, the following equality holds in the process of De~Giorgi iteration:
\begin{align*}
\lim_{\varepsilon\rightarrow 0^+}\frac{1}{\varepsilon}\int_{t_0-\varepsilon}^{t_0}\int_{0}^1-\overline{w}\overline{w}_x(\overline{w}-k)_+ \text{d}x\text{d}s =0.
\end{align*}
Then for every $(x,s)\in [0,1]\times [0,t]$ we have
\begin{align}
- w(x,s)=&\overline{w}(x,s) \notag\\
\leq & \max\Big\{\max\limits_{s\in[0,t]}-d(s),0\Big\}
+\frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in [0,1]\times[0,t]}|f(x,s)|.\label{++17''}
\end{align}
Finally, \eqref{++13} follows from \eqref{++16} and \eqref{++17''}.
\end{IEEEproof}
\begin{IEEEproof}[Proof of {Lemma}~\ref{Theorem 13}]
Multiplying \eqref{++31} by $v$ and integrating over $(0,1)$, we get
\begin{align*}
\int_{0}^1\!\!v_tv\text{d}x+\mu\int_{0}^1v^2_{x}\text{d}x+\nu\int_{0}^1v^2v_x\text{d}x+\nu\int_{0}^1(wv)_xv\text{d}x =0.
\end{align*}
Note that $\int_{0}^1v^2v_x\text{d}x=\frac{1}{3}v^3|^{x=1}_{x=0}=0$ and
\begin{align*}
\int_{0}^1(wv)_xv\text{d}x = wv^2 |^{x=1}_{x=0}-\int_{0}^1wvv_x\text{d}x=-\int_{0}^1wvv_x\text{d}x.
\end{align*}
By Young's inequality (see \cite[Appendix B.2.d]{Evans:2010}), H\"{o}lder's inequality (see \cite[Appendix B.2.e]{Evans:2010}), {Lemma}~\ref{Theorem 12}, and {the assumption on $d$}, we deduce that
\begin{align}
&\frac{1}{2}\frac{d}{dt}\|v(\cdot,t)\|^2+\mu\|v_x(\cdot,t)\|^2 \leq \nu\int_{0}^1|wvv_x|\text{d}x\notag\\
\leq &\frac{\nu}{2}\max\limits_{(x,s)\in[0,1]\times [0,t]} |w(x,s)|(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2) \notag\\
\leq &\frac{\nu}{2}\bigg(\max\limits_{s\in [0,t]}|d(s)|+\frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in [0,1]\times[0,t]} |f(x,s)|\bigg)\notag\\
&\times(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2)\notag\\
\leq &\frac{\nu}{2}\times\frac{\mu}{\nu}(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2)
\notag\\
=&\frac{\mu}{2}(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2).\label{+201804}
\end{align}
By Poincar\'{e}'s inequality, we have
\begin{align}
\mu\|v_x(\cdot,t)\|^2&=\frac{\mu}{2}\|v_x(\cdot,t)\|^2
+\frac{\mu}{2}\|v_x(\cdot,t)\|^2\notag\\
&\geq \frac{\mu}{2}\|v_x(\cdot,t)\|^2+\mu\|v(\cdot,t)\|^2.\label{+201805}
\end{align}
Then by \eqref{+201804} and \eqref{+201805}, it follows
\begin{align*}
\frac{\text{d}}{\text{d}t}\|v(\cdot,t)\|^2
\leq-\mu\|v(\cdot,t)\|^2,
\end{align*}
which together with Gronwall's inequality (\cite[Appendix B.2.j]{Evans:2010}) yields
\begin{align*}
\|v(\cdot,t)\|^2
&\leq\|v(\cdot,0)\|^2 e^{-\mu t}=\|u_0\|^2 e^{-\mu t}.
\end{align*}
\end{IEEEproof}
\subsection{Proof of Theorem \ref{Theorem 11-2}}
In order to prove Theorem~\ref{Theorem 11-2}, we consider the following two systems:
\begin{subequations}\label{+++32}
\begin{align}
&w_t-\mu w_{xx}+\nu ww_x=0\ \ \text{in}\ (0,1)\times\mathbb{R}_{+},\\
&w(0,t)=0,w(1,t)=d(t),\\
&w(x,0)=0,
\end{align}
\end{subequations}
and
\begin{subequations}\label{+++33}
\begin{align}
&v_t-\mu v_{xx}+\nu vv_x+\nu (wv + vw)_x=f(x,t)\ \ \text{in}\ (0,1)\times\mathbb{R}_{+},\\
&v(0,t)=v(1,t)=0,\\
&v(x,0)=u_0(x),
\end{align}
\end{subequations}
where $v=u-w$.
For System~\eqref{+++32}, we have the following estimate, which is a special case (i.e. $f(x,t)=0$) of {Lemma}~\ref{Theorem 12}.
\begin{lemma} \label{Theorem 14} Suppose that $\mu>0,\nu>0$. For every $t>0$, {}{one has}
\begin{align}\label{++32''}
\max\limits_{(x,s)\in[0,1]\times [0,t]} |w(x,s)|\leq \max\limits_{s\in [0,t]}|d(s)|.
\end{align}
\end{lemma}
For System~\eqref{+++33}, we have the following estimate.
\begin{lemma} \label{Theorem 15}{Suppose that $\mu>0,\nu>0$, and ${\sup\limits_{t\in \mathbb{R}_{\geq 0}}} |d(t)|<\frac{\mu}{\nu}$}. For every $t>0$, {}{one has}
\begin{align*}
\|v(\cdot,t)\|^2\leq \|u_0\|^2 e^{-(\mu -\varepsilon)t}+\frac{1}{\varepsilon} \int_{0}^t\|f(\cdot,s)\|^2\text{d}s,\ \forall \varepsilon\in (0,\mu).
\end{align*}
\end{lemma}
Note that $u=w+v$. Then the result of Theorem \ref{Theorem 11-2} is a consequence of {Lemma}~\ref{Theorem 14} and {Lemma}~\ref{Theorem 15}, which can be proven as in Theorem~\ref{Theorem 11}.
\begin{IEEEproof}[Proof of {Lemma}~\ref{Theorem 15}] Multiplying \eqref{+++33} by $v$ and integrating over $(0,1)$, we get
\begin{align*}
&\int_{0}^1v_tv\text{d}x+\mu\int_{0}^1v^2_{x}\text{d}x+\nu\int_{0}^1v^2v_x\text{d}x+\nu\int_{0}^1(wv)_xv\text{d}x \\
=&\int_{0}^1f(x,t)v\text{d}x.
\end{align*}
{}{Arguing as in \eqref{+201804},} we get
\begin{align}
&\frac{1}{2}\frac{d}{dt}\|v(\cdot,t)\|^2+\mu\|v_x(\cdot,t)\|^2\notag \\
\leq &\nu\int_{0}^1|wvv_x|\text{d}x+\int_{0}^1f(x,t)v\text{d}x\notag \\
\leq &\frac{\nu}{2}\max\limits_{s\in [0,t]}|d(s)|(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2)\notag\\
&+\frac{1}{2\varepsilon}\|f(\cdot,t)\|^2
+\frac{\varepsilon}{2}\|v(\cdot,t)\|^2\notag \\
\leq & \frac{\nu}{2}\frac{\mu}{\nu}(\|v(\cdot,t)\|^2+\|v_x(\cdot,t)\|^2)
+\frac{1}{2\varepsilon}\|f(\cdot,t)\|^2+\frac{\varepsilon}{2}\|v(\cdot,t)\|^2\notag \\
=&\frac{1}{2}(\varepsilon+\mu)\|v(\cdot,t)\|^2
+\frac{\mu}{2}\|v_x(\cdot,t)\|^2+\frac{1}{2\varepsilon}\|f(\cdot,t)\|^2,\label{+201806}
\end{align}
where we choose $0<\varepsilon <\mu$.
By \eqref{+201805} and \eqref{+201806}, we get
\begin{align*}
\frac{d}{dt}\|v(\cdot,t)\|^2
\leq-(\mu -\varepsilon)\|v(\cdot,t)\|^2+\frac{1}{\varepsilon}\|f(\cdot,t)\|^2.
\end{align*}
By Growall's inequality (see \cite[Appendix B.2.j]{Evans:2010}), we have
\begin{align*}
\|v(\cdot,t)\|^2
&\leq\|v(\cdot,0)\|^2 e^{-(\mu -\varepsilon)t}+\frac{1}{\varepsilon} \int_{0}^t\|f(\cdot,s)\|^2\text{d}s\\
&=\|u_0\|^2 e^{-(\mu -\varepsilon)t}+\frac{1}{\varepsilon} \int_{0}^t\|f(\cdot,s)\|^2\text{d}s.
\end{align*}
\end{IEEEproof}
\section{Application to a 1-$D$ {}{Linear} Unstable Reaction-Diffusion Equation with Boundary Feedback Control}\label{Sec: reaction-diffusion Eq}
In this section, we {}{illustrate the application of the developed method in the study of the ISS property for the following 1-$D$ {}{linear} reaction-diffusion equation with an unstable term}:
\begin{align}\label{++9181}
u_t-\mu u_{xx}+a(x)u=f(x,t) \ \ \ \ \text{in}\ \ (0,1)\times\mathbb{R}_{+},
\end{align}
where $\mu >0$ is a constant, $a\in C^1([0,1])$ and $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$. The system is subject to the boundary and initial conditions
\begin{subequations}\label{++9182}
\begin{align}
&u(0,t)=0,u(1,t)=U(t),\\
&u(x,0)=u_0(x),
\end{align}
\end{subequations}
where $U(t)\in \mathbb{R}$ is the control input. {}{Note that the control input can be placed on either ends of the boundary. Nevertheless, it can be switched to the other end by a spatial variable transformation $x\rightarrow 1-x$. The ISS properties of this system w.r.t. boundary disturbances, i.e., $f(x,t)\equiv 0$, have been addressed in \cite{Karafyllis:2016,Karafyllis:2016a,Mironchenko:2017}.}
The stabilization of \eqref{++9181} in a disturbance-free setting with $\mu =1$ and $f(x,t)\equiv 0$ is presented in \cite{Krstic:2008,Liu:2003,Smyshlyaev:2004}. {}{The exponential stability is achieved by means of a backstepping boundary feedback control of the form}
\begin{align}
U(t)=\int_{0}^1k(1,y)u(y,t)\text{d}y,\ \forall t\geq 0,\label{++9183}
\end{align}
where $k\in C^2([0,1]\times[0,1])$ can be obtained as the Volterra kernel of a Volterra integral transformation
\begin{align}
w(x,t)=u(x,t)-\int_{0}^xk(x,y)u(y,t)\text{d}y,\label{+2018032701}
\end{align}
which transforms \eqref{++9181}, \eqref{++9182}, and \eqref{++9183} to the problem \begin{align*}
w_t-\mu w_{xx}+\nu w=0 \ \ \ \ \text{in}\ \ (0,1)\times\mathbb{R}_{+},
\end{align*}
with $\nu> 0$, subject to the boundary and initial conditions
\begin{align*}
&w(0,t)=w(1,t)=0,\\
&w(x,0)=w_0(x)=u_0(x)-\int_{0}^xk(x,y)u_0(y)\text{d}y.
\end{align*}
When $\mu >0$, $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$, and in the presence of actuation errors represented by the disturbance $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$, the applied control action is of the form \cite{Karafyllis:2016,Karafyllis:2016a,Mironchenko:2017}
\begin{align}
U(t)=d(t)+\int_{0}^1k(1,y)u(y,t)\text{d}y,\ \forall t\geq 0.\label{++9188}
\end{align}
We can use the Volterra integral transformation \eqref{+2018032701} to transform \eqref{++9181}, \eqref{++9182}, and \eqref{++9188} to the following system
\begin{align}\label{++9185}
w_t-\mu w_{xx}+\nu w=f(x,t) \ \ \ \ \text{in}\ \ (0,1)\times\mathbb{R}_{+},
\end{align}
with $\nu>0$, subject to the boundary and initial conditions
\begin{subequations}\label{++9189}
\begin{align}
&w(0,t)=0,w(1,t)=d(t),\\
&w(x,0)=w_0(x)=u_0(x)-\int_{0}^xk(x,y)u_0(y)\text{d}y.
\end{align}
\end{subequations}
Then the solution to \eqref{++9181}, \eqref{++9182}, and \eqref{++9188} can be found by the
inverse Volterra integral transformation
\begin{align}
u(x,t)=w(x,t)+\int_{0}^xl(x,y)w(y,t)\text{d}y,\label{++9187}
\end{align}
where $l\in C^2([0,1]\times[0,1])$ is an appropriate kernel. Indeed, the existence of
the kernels $k\in C^2([0,1]\times[0,1])$ and $l\in C^2([0,1]\times[0,1])$ can be obtained in the same way as in \cite{Liu:2003,Smyshlyaev:2004}.
For the system~\eqref{++9181} with \eqref{++9182} and \eqref{++9188}, we have the following ISS estimate.
\begin{proposition}\label{Theorem 16}
Suppose that $\mu >0$, $a\in C^1([0,1])$, {$d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$, $ f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$}, and $u_0\in \mathcal {H}^{2+\theta}([0,1])$ for some $\theta\in (0,1)$, with the compatibility conditions:
\begin{align*}
&{u_0(0)=d(0)=d'(0)=f(0,0)=f(1,0)=0,}\\
&u_0(1)=\int_{0}^1k(1,y)u_0(y)\text{d}y,\\
&{u_0(1)\frac{dk(x,x)}{dx}\bigg|_{x=1}+u_0'(1)k(1,1)=0.}
\end{align*}
{System} \eqref{++9181} with \eqref{++9182} and \eqref{++9188} is EISS in $L^\infty$-norm {w.r.t. {boundary disturbances} $d\in \mathcal {H}^{1+\frac{\theta}{2}}(\mathbb{R}_{\geq 0})$ and {in-domain disturbances} $f\in \mathcal {H}^{\theta,\frac{\theta}{2}}([0,1]\times \mathbb{R}_{\geq 0})$}, having the following estimate:
\begin{align*}
\max_{x\in[0,1]}|u(x,t)|\leq &C_0 \max_{x\in [0,1]}|u_0|e^{-\nu t}+C_1\bigg(\max\limits_{s\in [0,t]}|d(s)|\notag\\ &+ \frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in[0,1]\times [0,t]}|f(x,s)|\bigg),
\end{align*}
where $\nu>0$ is the same as in \eqref{++9185}, $C_1= \Big(1+\max\limits_{(x,y)\in[0,1]\times[0,1] }|l(x,y)|\Big)$ and $C_0= C_1\Big(1+ \max\limits_{(x,y)\in[0,1]\times[0,1] }|k(x,y)|\Big)$ are positive constants.
\end{proposition}
\begin{IEEEproof}
{Note that by the compatibility conditions, it follows $w_0(0)=w_0(1)=w_0''(0)=w_0''(1)=f(0,0)=f(1,0)=d(0)=d'(0)=0 $. Therefore, we can use the technique of splitting and De~Giorgi iteration to establish the ISS estimate for System~\eqref{++9185} with \eqref{++9189}. Let} $g$ be the unique solution of the following system:
\begin{subequations}\label{subequ.1}
\begin{align}
&g_t-\mu g_{xx}+\nu g=f(x,t),\ \ \ \ (0,1)\times\mathbb{R}_{+},\\
&g(0,t)=0,g(1,t)=d(t),\\
&g(x,0)=0,
\end{align}
\end{subequations}
and let $h=w-g$ be the unique solution of the following system:
\begin{subequations}\label{subequ.2}
\begin{align}
&h_t-\mu h_{xx}+\nu h=0,\ \ \ \ (0,1)\times\mathbb{R}_{+},\\
&h(0,t)=h(1,t)=0,\\
&h(x,0)=h_0(x)=w_0(x).
\end{align}
\end{subequations}
For \eqref{subequ.1} and \eqref{subequ.2}, we claim that for any $t\in\mathbb{R}_{\geq 0}$:
\begin{align}\label{estimate1}
&\max\limits_{(x,s)\in[0,1]\times [0,t]} |g(x,s)| \notag\\
&\;\;\; \leq \max\limits_{s\in [0,t]}|d(s)| + \frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in[0,1]\times [0,t]}|f(x,s)|,
\end{align}
and
\begin{align}\label{estimate2}
\max_{x\in [0,1]}|h(x,t)|\leq \max_{x\in [0,1]}|h_0(x)| e^{- \nu t}.
\end{align}
We prove \eqref{estimate1} by De~Giorgi iteration. Indeed, for any fixed $t>0$, letting $k_0,k$, $ \eta(x,s)$, and $t_0$ be defined as in the proof of Theorem~\ref{Theorem 11} (replace $w$ by $h$) and taking $\eta(x,s)$ as a test function, we get
\begin{align*}
&\int_{0}^t\int_{0}^1(g-k)_t(g-k)_+\chi_{[t_1,t_2]}(s)\text{d}x\text{d}s \notag\\
&+\mu\int_{0}^t\int_{0}^1\chi_{[t_1,t_2]}(s)|((g-k)_+)_x |^2\text{d}x\text{d}s\notag\\
&+\nu\int_{0}^t\int_{0}^1g(g-k)_+\chi_{[t_1,t_2]}(s) \text{d}x\text{d}s \notag\\
=&\int_{0}^t\int_{0}^1f(g-k)_+\chi_{[t_1,t_2]}(s) \text{d}x\text{d}s.
\end{align*}
Noting that $\nu\int_{0}^t\int_{0}^1g(g-k)_+\chi_{[t_1,t_2]}(s) \text{d}x\text{d}s\geq 0$, it can be seen that \eqref{+20181} still holds (replace $w$ by $h$), which leads to \eqref{estimate1}.
For the proof of \eqref{estimate2}, we choose the following Lyapunov functional
\begin{align*}
E(t)=\int_{0}^{1}|h(\cdot,t)|^{2p}\text{d}x,\ \forall p\geq 1,\forall t\in\mathbb{R}_{\geq 0}.
\end{align*}
Applying Poincar\'{e}'s inequality, it follows $\|h^{p}(\cdot,t)\|^2\leq \frac{p^2}{2} \|h^{p-1}h_x(\cdot,t)\|^2$. Then by direct computations, we get
\begin{align*}
\frac{\text{d}}{\text{d}t}E(t)\leq -2p\bigg( \nu +\frac{2\mu (2p-1)}{p^2}\bigg)E(t),
\end{align*}
{}{which together with Gronwall's inequality yields}
\begin{align}\label{estimate2'}
\|h(\cdot,t)\|_{L^{2p}(0,1)}^{2p}\leq \|h_0\|_{L^{2p}(0,1)}^{2p} e^{-2p\big( \nu +\frac{2\mu (2p-1)}{p^2}\big)t},\ \forall p\geq 1.
\end{align}
Taking the $2p$-th root of \eqref{estimate2'} and letting $p\rightarrow +\infty$, it follows
\begin{align}
\|h(\cdot,t)\|_{L^{\infty}(0,1)}\leq \|h_0\|_{L^{\infty}(0,1)} e^{- \nu t},\ \forall t\in\mathbb{R}_{\geq 0} .\label{estimate2''}
\end{align}
Finally, we obtain \eqref{estimate2} by \eqref{estimate2''} and the continuity of $h$ and $h_0$.
As a consequence of \eqref{estimate1} and \eqref{estimate2}, the following estimate holds for any $t\in\mathbb{R}_{\geq 0}$:
\begin{align}\label{estimate3}
\max\limits_{x\in[0,1]} |w(x,t)|\leq &\max\limits_{(x,s)\in[0,1]\times [0,t]} |g(x,s)|+\max_{x\in [0,1]}|h(x,t)| \notag\\
\leq &\max\limits_{s\in [0,t]}|d(s)| + \frac{4\sqrt{2}}{\mu} \max\limits_{(x,s)\in[0,1]\times [0,t]}|f(x,s)|\notag\\
&+\max_{x\in [0,1]}|w_0(x)| e^{- \nu t}.
\end{align}
Note that
\begin{align}
\max_{x\in [0,1]}|w_0(x)|&\leq \max_{x\in [0,1]}\bigg|u_0-\int_{0}^xk(x,y)u_0(y)\text{d}y \bigg|\notag\\
&\leq \max_{x\in [0,1]}|u_0|+\max_{x\in [0,1]}\bigg|\int_{0}^xk(x,y)u_0(y)\text{d}y \bigg|\notag\\
&\leq \bigg(1+ \max\limits_{(x,y)\in[0,1]\times[0,1] }|k(x,y)|\bigg)\max_{x\in [0,1]}|u_0|.\label{estimate4}
\end{align}
{}{Finally, the desired result follows from \eqref{++9187}, \eqref{estimate3}, and \eqref{estimate4}.}
\end{IEEEproof}
\begin{remark}
If we put $f(x,t)$ in \eqref{subequ.2} instead of in \eqref{subequ.1}, and proceed as in the proof of Theorem~\ref{Theorem 11-2}, we can prove that the system \eqref{++9181} with \eqref{++9182} and \eqref{++9188} is EISS w.r.t. {boundary disturbances} and EiISS w.r.t. {in-domain disturbances}.
\end{remark}
\begin{remark}
In the case where $a(x)\equiv a$ is a constant, the ISS in $L^2$-norm and $L^p$-norm ($\forall p>2$) for the system \eqref{++9181} with \eqref{++9182} w.r.t. actuation errors for boundary feedback control \eqref{++9188} is established in \cite{Karafyllis:2016a} by the technique of eigenfunction expansion, and in \cite{Mironchenko:2017} by the monotonicity method, respectively.
\end{remark}
\begin{remark}
The ISS in a weighted $L^\infty$-norm w.r.t. boundary and in-domain disturbances for solutions to PDEs associated with a Sturm-Liouville operator is established in \cite{karafyllis2017siam} by the method of eigenfunction expansion and a finite-difference scheme. We established in this note the ISS in $L^\infty$-norm for a similar setting with considerably simpler computations by De~Giorgi iteration and Lyapunov method. Moreover, the ISS in $L^2$-norm for solutions to certain semilinear parabolic PDEs with Neumann or Robin boundary disturbances is established in \cite{Zheng:2017} by Lyapunov method. These achievements show that the techniques and tools developed in this note and \cite{Zheng:2017} are effective for the application of Lyapunov method to the analysis of the ISS for certain linear and nonlinear PDEs with different type of boundary disturbances.
\end{remark}
\begin{remark}
The method developed in this work can be also applied to linear problems with multidimensional spatial variables, e.g.,
\begin{subequations}
\begin{align*}
&u_t-\mu \Delta u+c(x,t)u=f(x,t),\ \ \text{in }\ \Omega\times \mathbb{R}_{+},\\
&u(x,t)=0\ \ \text{on }\ \Gamma_0,\ u(x,t)=d(t)\ \ \text{on }\ \Gamma_1,\label{}\\
&u(x,0)=u_0(x),\ \ \text{in }\ \Omega,
\end{align*}
\end{subequations}
where $\Omega\subset\mathbb{R}^n (n\geq 1)$ is an open bounded domain with smooth boundary $\partial \Omega=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $ c(x,t)$ is a smooth function in $ \Omega\times \mathbb{R}_{\geq 0}$ with $0< m\leq c(x,t)\leq M$, $\Delta$ is the Laplace operator, and $\mu>0$ is a constant.
Under appropriate assumptions on $\mu,m,M$ and by the technique of splitting and De~Giorgi iteration, it can be shown that the following estimates hold:
\begin{align*}
\|u(\cdot,t)\|_{L^2(\Omega)}\leq & C_0\|u_0\|_{L^2(\Omega)} e^{-\lambda t}
+ C_1\max\limits_{s\in [0,t]}|d(s)| \notag\\
&+ C_2 \max\limits_{(x,s)\in\overline{Q}_t}|f(x,s)|,
\end{align*}
and \begin{align*}
\|u(\cdot,t)\|_{L^2(\Omega)}^2 \leq& C_0\|u_0\|_{L^2(\Omega)}^2 e^{-\lambda t}+ C_1\max\limits_{s\in [0,t]}|d(s)|^2 \nonumber \\
&+ C_2 \int_{0}^t\|f(\cdot,s)\|^2\text{d}s,
\end{align*}
where $\overline{Q}_t=\overline{\Omega}\times [0,t]$, $C_0$, $C_1$, $C_2$, and $\lambda$ are {}{some positive constants independent of $t$}.
\end{remark}
\section{Concluding Remarks}\label{Sec: Conclusion}
This work applied the technique of De~Giogi iteration to the establishment of ISS properties for nonlinear PDEs. The ISS estimates in $L^2$-norm w.r.t. boundary and in-domain disturbances for Burgers' equation with Dirichlet boundary conditions have been obtained. {The considered setting is a complement of the problems dealt with in \cite{Zheng:2017}, where the ISS in $L^2$-norm has been established for some {semilinear PDEs} with Robin (or Neumann) boundary conditions. It is worth pointing out that the method developed in this note can be generalized for some problems on multidimensional spacial domain and for dealing with ISS properties of PDEs while considering weak solutions (see, e.g., \cite[Ch.~4]{Wu2006}). Finally, as the method of De~Giogi iteration is a well-established tool for regularity analysis of PDEs, we can expect that the method developed in this work is applicable in the study of a wider class of nonlinear PDEs, such as {Chaffee-Infante equation, Fisher-Kolmogorov equation, generalized Burgers' equation, Kuramoto-Sivashinsky equation, and linear or nonlinear Schr\"{o}dinger equations}.}
\ifCLASSOPTIONcaptionsoff
\fi
\end{document} |
\begin{document}
\title{Geometric optics for surface waves in nonlinear elasticity}
\author{Jean-Fran\c{c}ois {\sc Coulombel}\thanks{CNRS and Universit\'e de Nantes, Laboratoire de math\'ematiques
Jean Leray (UMR CNRS 6629), 2 rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France. Email:
{\tt [email protected]}. Research of J.-F. C. was supported by ANR project BoND,
ANR-13-BS01-0009-01.}
\& Mark {\sc Williams}\thanks{University of North Carolina, Mathematics Department, CB 3250, Phillips Hall,
Chapel Hill, NC 27599. USA. Email: {\tt [email protected]}. Research of M.W. was partially supported by
NSF grant DMS-1001616.}}
\maketitle
\begin{abstract}
\emph{\quad} This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a
half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system.
Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an \emph{approximate} Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which we refer to as ``the amplitude equation", is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^{\varepsilon}$ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ${\varepsilon}$, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^{\varepsilon}$ on a time interval independent of ${\varepsilon}$. The paper focuses mainly on the case of Rayleigh waves that are \emph{pulses}, which have profiles with continuous Fourier spectrum, but our method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.
\end{abstract}
\tableofcontents
\chapter{General introduction}
\emph{\quad}This work is devoted to the rigorous justification of weakly nonlinear geometric optics expansions for
a class of second order hyperbolic initial boundary value problems. These are evolution problems for
which we ask the two main questions of geometric optics:
\begin{enumerate}
\item Does an exact solution, say $u_{\varepsilon}$, exist for ${\varepsilon} \in (0,1]$ on a fixed time interval $[0,T_0]$
independent of ${\varepsilon}$, where ${\varepsilon}$ represents the small wavelength of the data?
\item Suppose the answer to the first question is yes. If we let $u^{app}_{\varepsilon}$ denote an approximate
solution on $[0,T_0]$ constructed by the methods of nonlinear geometric optics (that is, solving eikonal
equations for phases and suitable transport equations for profiles), how well does $u^{app}_{\varepsilon}$
approximate $u_{\varepsilon}$ for ${\varepsilon}$ small? For example, is it true that
\begin{equation*}
\lim_{{\varepsilon}\to 0} {\varepsilon}^{-\alpha} \, \| u_{\varepsilon}-u^{app}_{\varepsilon} \|_{L^\infty}\to 0 \, ?
\end{equation*}
where $\alpha$ is a scaling parameter such that both $u_{\varepsilon}$ and $u^{app}_{\varepsilon}$ have $O({\varepsilon}^\alpha)$
$L^\infty$ norm (in that case $u_{\varepsilon}-u^{app}_{\varepsilon}$ is meant to be a remainder of higher order in ${\varepsilon}$
than ${\varepsilon}^\alpha$).
\end{enumerate}
Pursuing a long-term program, we deal here with the above two issues in the context of elastodynamics
in a fixed domain, with boundary conditions giving rise to Rayleigh waves. These are
hyperbolic initial boundary value problems for which the so-called uniform Lopatinskii condition\footnote{This is often referred to as the uniform Kreiss-Lopatinkii condition; problems satisfying this condition are said to be ``strongly" or ``uniformly" stable.}
fails in a manner that is manifested by the presence of surface waves.\footnote{We use the phrase ``surface waves" as a general term that includes both surface wavetrains and surface pulses.}
At the linear level such waves were identified long ago, and they display an exponential decay with respect to the normal variable to the boundary of
the space domain. Our goal here is to understand the behavior of small amplitude, highly oscillating
surface waves in the so-called weakly nonlinear regime, where nonlinearity is visible at the leading
order of the solution, meaning that both $u_{\varepsilon}$ and $u^{app}_{\varepsilon}$ are constructed by solving
nonlinear evolution equations. More precisely, the phrase ``weakly nonlinear" indicates that the $L^\infty$ norm of $u^{app}_{\varepsilon}$, say ${\varepsilon}^\alpha$, is taken to be as small as it can be while still satisfying the requirement that the evolution equation for the leading term of $u^{app}_{\varepsilon}$ be nonlinear. This particular choice of amplitude ${\varepsilon}^\alpha$ (for waves oscillating with wavelength ${\varepsilon}$) is what we mean by the ``weakly nonlinear scaling".
Before reviewing earlier works on weakly nonlinear surface waves, let us recall several facts on hyperbolic
initial boundary value problems and geometric optics expansions. Well-posedness of linear, and subsequently
nonlinear, hyperbolic initial boundary value problems depends on whether or not the uniform
Lopatinskii condition is satisfied, and if not, \emph{how} it fails; see, e.g., \cite{K,sakamoto,CP} and \cite[Chapter 4]{BS}. In the most favorable
case where this condition is satisfied, the corresponding initial boundary value problems satisfy {\it maximal}
energy estimates with no loss of derivatives from the data to the solution, including for the trace of the solution
at the boundary. Nonlinear problems may then be solved by fairly standard Picard iteration techniques. For uniform Lopatinskii problems the
regime of weakly nonlinear geometric optics corresponds to the same scaling as for the Cauchy problem
in the whole space. The main features of geometric optics in that case include: propagation at some given
group velocities inside the space domain, and reflection of (outgoing) oscillating wave packets at the boundary.
The reflection coefficients which determine incoming oscillating wave packets in terms of outgoing ones are
{\it finite} as a byproduct of the uniform Lopatinskii condition. A complete justification of the high
frequency WKB expansions for quasilinear problems is provided in the references \cite{W,CGW1,Hernandez}
for wavetrains, and in \cite{CW1} for pulses. The difference between the two problems is that for wavetrains,
the data, exact and approximate solutions depend in a periodic way on the fast variables $\Phi/{\varepsilon}$ (here $\Phi$
represents the phases of the oscillations arising in the solution), while for pulses the fast variables all lie
in ${\mathbb R}$ with `some' decay property at infinity. In problems with several interacting phases the pulse setting is slightly more favorable because pulses
associated with different phases do not interact at leading order, while wavetrains may interact at leading
order when {\it resonances} occur between several phases. A consequence is that, for pulses, one can usually
derive a rate of convergence between exact and approximate solutions \cite{CW1,CW2}, while this is usually out of reach
for interacting wavetrains
\cite{JMR1,JMR}. An offsetting difficulty in the pulse setting is that it is usually much harder to construct correctors, and almost never possible to construct many correctors.
In this work we study a situation where the uniform Lopatinskii condition fails, but in a controlled way.
The degeneracy of this condition may occur in different ways, and we consider here the case where
surface waves of finite energy occur, see \cite[Chapter 7]{BS}. Let us mention right away the work of Marcou \cite{Mar}
for first order nonlinear systems in the surface wavetrain case. In \cite{Mar} Marcou provided a complete justification of weakly nonlinear geometric optics expansions for surface wavetrains arising in first order conservation laws with linear, homogeneous boundary conditions.
In contrast our main focus will be on the second order hyperbolic systems with fully nonlinear, nonhomogeneous boundary conditions arising in elasticity theory. We shall study surface pulses (or more precisely, Rayleigh pulses) for the most part, and these require very different methods. Moreover, as we explain below, it seems that the rigorous justification even of wavetrain solutions in nonlinear elasticity requires methods different from those of \cite{Mar}.
As pointed out by Serre
\cite{SerreJFA}, surface waves arise in a rather systematic way in second order hyperbolic problems
with a variational formulation, and we shall therefore consider this framework, especially in sections {\rm Re }\, f{var} and {\rm Re }\, f{wna}, before returning to elasticity in section {\rm Re }\, f{isoe} and for the rest of the paper.
Since the uniform Lopatinskii condition fails, well-posedness of the initial boundary value problem is an issue
because maximal energy estimates are known to fail. For a class of isotropic, hyperelastic, Neumann type problems in two space dimensions, a class that includes
the Saint-Venant Kirchhoff system \eqref{a0} as a basic representative,
Sabl\'e-Tougeron \cite{S-T} has
proved precise {\it microlocal} energy estimates which take advantage of the fact that the uniform
Lopatinskii condition degenerates in the {\it elliptic} region (Definition {\rm Re }\, f{lopa}) of the frequency domain.\footnote{As we explain in section {\rm Re }\, f{higherD},
there is still a serious obstacle to deriving such estimates in dimensions $d\geq 3$.}
The main feature of the energy estimates proved in \cite{S-T} is that there is no loss of regularity from
the initial data and interior source term to the solution. The only loss with respect to the strongly stable
case previously mentioned is a loss of one derivative from the boundary data to the trace of the solution, but one recovers an optimal control in the
interior of the space domain due to the exponential decay of the surface waves.
This is consistent with
one of the main results in \cite{SerreJFA}, which shows that a related class of linear initial boundary value problems
with \emph{homogeneous} boundary conditions may be solved, at least in cases of time-independent coefficients, by the Hille-Yosida Theorem.
The class treated in \cite{SerreJFA} consists of linear problems that arise as in \eqref{eqint}, \eqref{cl} from a stored energy function $W({\nabla}bla u)$ that is quadratic in ${\nabla}bla u$, where $u$ is the displacement; but the Saint Venant-Kirchhoff energy function is fourth order in ${\nabla}bla u$, and it leads to more complicated linearized problems. Only the quadratic part of $W({\nabla}bla u)$ contributes to the linearization at ${\nabla}bla u=0$, but to solve the nonlinear problem studied here, we like \cite{S-T} must consider the linearizations at nearby nonzero states, where the higher order terms of $W$ do affect the linearized problem. In the literature the phrase ``linearized elasticity" or ``classical linearized elasticity" is often used to refer to the problem arising from a quadratic $W({\nabla}bla u)$, or from one of the form $W(x,{\nabla}bla u)$, where $W$ is quadratic in ${\nabla}bla u$ \cite{HM,Ci,CC,SerreJFA}. These notions of linearized problem are \emph{different} from the one we must work with here; this is clear from a quick inspection of the principal symbol \eqref{u0} of the Saint Venant-Kirchhoff system linearized about about a state ${\nabla}bla u\neq 0$.
By combining precise estimates for the linearized problem with a clever iteration scheme based on considering the original nonlinear system together with two auxiliary systems obtained from it by differentiation, \cite{S-T} was able to prove a short-time existence theorem for the so-called ``traction" boundary problem in nonlinear elasticity \eqref{a0} where Rayleigh waves appear. In spite of the loss of derivatives, an essentially classical iteration scheme did the job.
Our work
builds on this approach, rather than on alternative approaches for nonlinear elasticity, see, e.g., \cite{Kato, SN}. If one applies the result of \cite{S-T} directly to the nonlinear problem with highly oscillatory boundary data considered here, one obtains a time of existence $T_{\varepsilon}$ converging to zero with the wavelength ${\varepsilon}$. The main new challenge addressed in this paper is to obtain estimates that are \emph{uniform} with respect to small ${\varepsilon}ilon$, and which hold on a time interval independent of ${\varepsilon}$.
We recall now some of the earlier works devoted to constructing approximate, oscillatory surface waves. The appropriate
scaling and leading order amplitude equation were identified in pioneering works going back, at least,
to Lardner, Parker and others, see for instance
\cite{Lardner1,Lardner2,ParkerTalbot,Parker,HamiltonIlinskyZabolotskaya}. The analogous theory in the case
of first order hyperbolic systems was initiated by Hunter \cite{H}.
In both first and second order problems, the leading amplitude equation takes
the form of a nonlinear integro-differential evolution equation of nonlocal Burgers type. The solution of this equation determines the trace of the leading part of
the approximate solution $u^{app}_{\varepsilon}$. Once the trace is known, the leading profile of $u^{app}_{\varepsilon}$ is obtained by a simple lifting procedure used for example in \cite{Le}.
Thus, the main step in constructing the approximate solution
is to prove a well-posedness result for
such nonlocal versions of the more standard Burgers equation. This well-posedness problem has
been solved in \cite{Hunter2006,B}, so constructing the leading profile of $u^{app}_{\varepsilon}$ will not be our main
concern here. However, in order to have a functional framework in which we can show that approximate solutions are close to exact solutions, we shall need to extend somewhat the constructions in \cite{Lardner1,Lardner2} and
recent generalizations in \cite{BC,BCproc}.
Unlike most earlier works we consider a WKB ansatz which incorporates
all possible slow and fast variables. One needs at least the tangential slow spatial variables and the slow time variable in order to specify exactly along which space-time curve on the boundary a Rayleigh pulse propagates. In addition we must also include the slow variable normal to the boundary to have profiles that can be used in the rigorous error analysis.
The analysis of section {\rm Re }\, f{wna}, which works in all space dimensions $d\geq 2$, computes the exact group velocity of Rayleigh waves and shows that the propagation curves are characteristics of the Lopatinski determinant. In addition we try to unify previous works and clarify the arguments
that are used to derive the leading order amplitude equation. In particular, we uncover a cancellation property \eqref{annulation}, \eqref{cancel} that was overlooked by \cite{Lardner1,Lardner2}, and which greatly simplifies the derivation of the amplitude equation.
This is achieved in Chapter {\rm Re }\, f{chapter2}
below. Chapter {\rm Re }\, f{chapter2} is the most formal one, in which we keep the discussion in a
rather broad variational context.
Consistently with earlier references from the physics literature, in section {\rm Re }\, f{isoe} we identify the
nonlocal amplitude equation for the class of isotropic hyperelastic models, which includes the Saint Venant-Kirchhoff model.
We analyze the WKB ansatz in the case of pulses, that is, when the fast variables lie in ${\mathbb R}$.
Similar arguments can be applied in the case of wavetrains, and we leave these rather small modifications
to the interested reader.
Hyperbolic boundary problems in which the uniform Lopatinski condition fails typically exhibit some kind of amplification of solutions \cite{CGW2,CW2,CW4} at the boundary.
In the second order problem with first order boundary condition \eqref{a2}-\eqref{a3} considered here, interior forcing of size ${\varepsilon}$ (in $L^\infty$) with zero boundary and zero initial data gives rise to a surface wave solution of size $O({\varepsilon}^2)$; here the response is the same size as for the forward problem in the whole space, or for a boundary problem of the same order that satisfies the uniform Lopatinski condition. This is compatible with the absence of any loss of derivatives from interior forcing to solution in the linearized estimates. On the other hand, boundary forcing of size $O({\varepsilon}^2)$ with zero interior forcing and zero initial data gives rise to a surface wave solution of size $O({\varepsilon}^2)$, while for a boundary problem of the same order that satisfies the uniform Lopatinski condition, the solution (which would not be a surface wave!) is of size $O({\varepsilon}^3)$. This discrepancy is compatible with the loss of one derivative from boundary forcing to solution in the linearized estimates. Thus, there is a sense, perhaps slightly contorted, in which ``amplification" does occur here.
At this stage, it remains unclear whether or not the leading order amplitude, which we know to exist on a time interval that is
independent of the small wavelength ${\varepsilon}$, is a reasonable approximation of the exact solution.
It is also unclear whether or not the exact solution exists on a time interval that is independent of the small
wavelength ${\varepsilon}$. This part of the analysis is achieved for wavetrains in first order systems in \cite{Mar} by
using high order approximate solutions to construct nearby exact solutions.
This type of argument dates back to
\cite{Gues}. In this case the Fourier transform of profiles with respect to the periodic fast variable $\theta$ is a function of $k\in{\mathbb Z}$ rather than $k\in {\mathbb R}$. Marcou \cite{Mar} took advantage of this discrete Fourier spectrum to construct arbitrarily many correctors yielding an arbitrarily high order approximate solution, and then added a small remainder to this to obtain a nearby exact solution on a fixed time interval independent of ${\varepsilon}$. In particular, her approximate solutions involved profiles that decayed at a rate $e^{-\delta z}$ in the fast variable $z=x_2/{\varepsilon}$ normal to the boundary for a \emph{fixed} $\delta$ independent of $k\in \mathbb{Z}\setminus \{0\}$. In the pulse setting the presence of Fourier spectrum arbitrarily close to $k=0$ makes it impossible to construct such ``strongly evanescent" profiles; indeed, now $\delta=\delta(k)\to 0$ as $k\to 0$.
In the case of surface waves given by pulses, as in other problems involving pulses \cite{CW1,CW2}, there is no hope of constructing high order approximate solutions.
Indeed, in the pulse setting it is usually impossible to construct high order correctors even in problems where the uniform Lopatinski condition is satisfied \cite{CW1}.
In the Rayleigh pulse problem studied here, we are actually able to construct only a \emph{single} corrector.
The first-order systems in \cite{Mar} were assumed to be symmetric with maximal dissipative boundary conditions; moreover, the boundary conditions were linear and \emph{homogeneous}, so she was able to use the well-posedness estimates without loss of derivatives which hold for such systems to rigorously justify the high-order expansions. We see no way to use estimates without loss in a similar way in the traction problem of nonlinear elasticity, even if one is trying to justify high order approximate \emph{wavetrain} solutions.
To sum up, for several reasons a quite different approach is needed to both of the open questions described above. Our alternative approach to both questions depends on the study of singular systems. This is an idea that goes back to \cite{JMR} for problems in free space, and to \cite{W} for problems on domains with boundary.
As we explain in more detail in the introduction to Chapter {\rm Re }\, f{chapter3}, if one looks for an \emph{exact} solution $U^{\varepsilon}(t,x)$ to the system of 2D nonlinear elasticity \eqref{a2} in the form
\begin{align}\langle bel{z1}
U^{\varepsilon}(t,x)=u^{\varepsilon}(t,x,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}}, \text{ where }\beta=(\beta_0,\beta_1) \text{ and }x=(x_1,x_2),
\end{align}
then by plugging the ansatz \eqref{z1} into \eqref{a2}, one obtains a similar system for the function $u^{\varepsilon}(t,x,\theta)$, except that the derivatives $\partial_t$, $\partial_{x_1}$ are now replaced by
\begin{align}\langle bel{z2}
\partial_{t,{\varepsilon}}:=\partial_t+\beta_0\frac{\partial_\theta}{{\varepsilon}},\;\;\partial_{x_1,{\varepsilon}}:=\partial_{x_1}+\beta_1\frac{\partial_\theta}{{\varepsilon}}
\end{align}
wherever they occur. We refer to this new problem \eqref{a5} as the \emph{singular system} associated with \eqref{a2}. One gain is immediately apparent: whereas high Sobolev norms of the boundary data ${\varepsilon}^2 G(t,x_1,\frac{\beta\cdot (t,x_1)}{{\varepsilon}})$ in \eqref{a2} clearly blow up as ${\varepsilon}\to 0$, the Sobolev norms of the boundary data ${\varepsilon}^2 G(t,x_1,\theta)$ in \eqref{a5} \emph{go to zero} as ${\varepsilon}\to 0$. A significant price is also apparent. The problem \eqref{a5} is now singular in two different senses; first because of the factors $1/{\varepsilon}$ that appear in the derivatives, and second because derivatives now occur in the linear combinations \eqref{z2}. Even if ${\varepsilon}$ is fixed, the second sense still makes the singular system hard to study. The first sense of singularity implies that when the boundary, $x_2=0$, is noncharacteristic, one cannot hope to control norms of normal ($\partial_{x_2}$) derivatives of solutions, uniformly with respect to ${\varepsilon}$, simply by employing the classical device of first controlling tangential derivatives and then using the equation.
Our decision to build on the approach of \cite{S-T} to the traction problem, rather than that of \cite{Kato} or \cite{SN}, was based on the fact that we saw no way to implement either of the latter approaches in the associated \emph{singular} traction problem.
In Chapter {\rm Re }\, f{chapter3} we prove the existence of exact solutions to the original nonlinear Saint Venant-Kirchhoff system \eqref{a2} on a fixed time interval independent of ${\varepsilon}$ as a consequence of such an existence theorem for the associated singular system \eqref{a5}. In this approach the exact solution is obtained without any reliance on the approximate solution. Employing an idea used by \cite{S-T} in the nonsingular setting, we consider not just \eqref{a5} but the trio of coupled singular systems \eqref{a7}-\eqref{a9}, where \eqref{a7} and \eqref{a8} are obtained from the original system by differentiating it, and \eqref{a9} is essentially the same as \eqref{a5}. In contrast to \cite{S-T}, it turns out that we are not able to use an iteration scheme modeled on the one employed in that paper, or indeed any iteration scheme at all, to prove estimates uniform with respect to ${\varepsilon}$.
The uniform estimates depend on knowing the relation $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{{\varepsilon}}$, which holds (for short times) between the solution $v^{\varepsilon}$ of \eqref{a7}-\eqref{a8} and $u^{\varepsilon}$ of \eqref{a9}, and this relation holds only for exact solutions \emph{not} for iterates.\footnote{Here ${\nabla}bla_{\varepsilon}=(\partial_{x_1,{\varepsilon}},\partial_{x_2})$.}
Instead, we use a continuous induction argument based on local existence and continuation theorems for singular systems with ${\varepsilon}$ fixed (Propositions {\rm Re }\, f{localex} and {\rm Re }\, f{continuation}) and the uniform a priori estimate of Proposition {\rm Re }\, f{c5} for the coupled systems. The proof of the latter proposition, based on \emph{simultaneous} estimation of the trio of modified singular systems \eqref{c1}-\eqref{c3}, is the core of the rigorous analysis of this paper. The continuous induction argument is summarized in more detail in the introduction to Chapter {\rm Re }\, f{chapter3}, and our uniform existence result for exact solutions is stated in Theorem {\rm Re }\, f{uniformexistence}.
The analysis of singular systems in this paper, for example the proof of the basic estimates for the linearized systems corresponding to \eqref{a7}-\eqref{a9} given in Proposition {\rm Re }\, f{basicest}, uses two main tools. The first is the calculus of singular pseudodiffererential operators for pulses constructed in \cite{CGW}, and the second is the collection of estimates proved in section {\rm Re }\, f{nonlinear} of singular norms of nonlinear functions of $u$.
The calculus of \cite {CGW} is a calculus for symbols of finite $(t,x,\theta)$ regularity which allows us to compose and take adjoints of operators that are, roughly speaking, pseudodifferential with respect to the vector fields $\partial_{t,{\varepsilon}}$, $\partial_{x_1,{\varepsilon}}$ \eqref{z2}. The operators are defined in \eqref{singularpseudop}.\footnote{In \cite{CGW} a singular calculus for wavetrains was also constructed that was an improvement over the one first constructed in \cite{W}.} This is a ``first-order" calculus in the sense that only the principal symbols of compositions or adjoints are given; but there are explicit formulas for the error terms which give a clear picture of how big they are as ${\varepsilon}\to 0$. Using this calculus, one can for example construct a Kreiss symmetrizer for a singular problem simply by taking the classical symbol of the Kreiss symmetrizer for the corresponding nonsingular problem and quantizing it in the singular calculus by the process described in \eqref{sect8}. The main results of this calculus are recalled in Appendix {\rm Re }\, f{calculus}. The calculus was originally created with applications to first order systems in mind. Here we deal with second order singular systems, so commutators arise, sometimes involving operators of fractional order, which can not be treated using the results of \cite{CGW}. Thus we have had to extend the calculus of \cite{CGW} in several ways, and these extensions are given in section {\rm Re }\, f{commutator}.
A quick inspection of the estimates of Proposition {\rm Re }\, f{basicest} for the linearized singular problems shows that a variety of ``singular norms" occur there. These are norms
of the form
$$
|\Lambda^r_D u|_{H^m(t,x,\theta)},
$$
where $\Lambda^r_D$ is the singular operator associated to the symbol $\langle ngle \xi'+\beta \frac{k}{{\varepsilon}},\gamma\rangle ngle^r$,
and $r$ and $m$ are (usually) nonnegative constants. Clearly, to apply these estimates to nonlinear problems we need to be able to estimate singular norms
$$
|\Lambda^r_D f(u)|_{H^m}
$$
of nonlinear functions of $u$ in terms of singular norms of $u$. Estimates of this kind, which are new, are proved in section {\rm Re }\, f{nonlinear} for analytic functions $f(u)$; both tame and simpler non-tame estimates are given.\footnote{The tame estimates are needed mainly for the continuation result, Proposition {\rm Re }\, f{continuation}.} In addition, to take advantage of the extra microlocal precision in the estimates provided, for example, by the terms involving the singular pseudodifferential cutoffs $\phi_{j,D}$ in \eqref{b3} and \eqref{b5}, we need to show that in some cases extra singular \emph{microlocal} regularity of $u$ is preserved under nonlinear functions $f(u)$. A result of this type, which can be viewed as a singular version of the classical Rauch's lemma \cite{R}, is given in Proposition {\rm Re }\, f{f5}.
Once we have the leading order approximate solution and the exact solution, the error analysis
leading to the full justification of geometric optics relies on the construction of an appropriate
{\it corrector}. More precisely, we need to add a small corrector to the leading order approximate
solution in order to be able to control its difference with the exact solution. In Chapter {\rm Re }\, f{chapter2} equations are derived \eqref{bkwordre2} that one ``would like" the corrector to satisfy. The amplitude equation (Proposition {\rm Re }\, f{propelas}), which determines the trace of the leading order term of the approximate solution, is a solvability condition for the Fourier transform with respect to $\theta$ of these corrector equations \eqref{correcteurv2}. In Chapter {\rm Re }\, f{chapter4} we solve the transformed equations \eqref{correcteurv2} for each $k\neq 0$ (here $k$ is the Fourier transform variable dual to $\theta$) and discover that the corrector is $O(1/k^2)$ near $k=0$. Roughly, these two factors of $1/k$ reflect the two integrations in $\theta$, each on the unbounded domain $\mathbb{R}$, that are needed to construct a corrector in this second order problem; on the $\theta$ side, each integration in $\theta$ introduces growth with respect to $\theta$ in the corrector. This illustrates the difficulty of constructing correctors in pulse problems.
Our solution of the transformed equations \eqref{correcteurv2} behaves too badly at $k=0$ to be inverse transformed, so we now regard it as a preliminary corrector. Using an idea we learned from \cite{AR}, we modify this object by multiplying it by a low frequency cutoff, $\chi(k/{\varepsilon}^b)$, where $\chi(s)$ is a smooth cutoff supported away from $0$ and equal to one on $|s|\geq 1$, and $b>0$ is a constant to be chosen. This modification introduces new errors of course, but we show in the error analysis of Chapter {\rm Re }\, f{chapter5} that the new errors are offset by the presence of the factor ${\varepsilon}^3$ on the corrector, provided $b$ is chosen correctly.
It is natural to wonder if one could construct correctors with better decay properties in $\theta$ if one started with boundary data given by a function $G(t,x_1,\theta)$ with stronger than $H^s$ type decay in $\theta$. It turns out that even
if one assumes $G(t,x_1,\theta)$ decays \emph{exponentially} as $|\theta|\to \infty$, the leading term of the approximate solution
generally exhibits no better than $H^s$ type decay in $\theta$; see Remark {\rm Re }\, f{baz}. This loss of $\theta$-decay from data to solution is \emph{}{}{}{typical of evanescent pulses}. It is a linear phenomenon and occurs even in problems where the uniform Lopatinskii condition is satisfied \cite{Willig}. Evanescent pulses are generally not even $L^1$ in $\theta$, so unlike evanescent wavetrains they have no well-defined mean.
With the exact and approximate solutions in hand we are ready in Chapter {\rm Re }\, f{chapter5} for the error analysis. As in Chapter {\rm Re }\, f{chapter3} the estimates, which involve some norms that cannot be localized in time, must be done on the full half-space $\Omega$. In particular, we need extensions of the approximate solutions, which at this point are only defined on a short time interval, to the full half space.
This extension process has to be done carefully and, in fact, it turns out to require most of theory of Chapter {\rm Re }\, f{chapter3}. Parallel to the trio of modified singular systems \eqref{c1}-\eqref{c3} that were estimated in the study of the exact solution in Chapter {\rm Re }\, f{chapter3}, we define a trio of approximate solution systems \eqref{p1}-\eqref{p3}
whose solutions provide the needed extensions; a causality argument (Remark {\rm Re }\, f{k2y}) shows that the solutions agree with the original approximate solutions for small times.
The exact solution trio \eqref{c1}-\eqref{c3} had solutions $(v^{\varepsilon},u^{\varepsilon})$ on $\Omega$, while the approximate solution trio \eqref{p1}-\eqref{p3} has solutions $(v_a^{\varepsilon},u^{\varepsilon}_a)$. Naturally, then, we consider the trio of error equations \eqref{p4}-\eqref{p6} satisfied by the differences $(w^{\varepsilon},z^{\varepsilon}):=(v^{\varepsilon}-v^{\varepsilon}_a,u^{\varepsilon}-u_a^{\varepsilon})$. As in the earlier cases, this trio must be estimated simultaneously in order to take advantage of the relation $w^{\varepsilon}={\nabla}bla_{\varepsilon} z^{\varepsilon}$ that holds for short times. These arguments are summarized in the introduction to Chapter {\rm Re }\, f{chapter5}. Ultimately, we are able to derive a rate of convergence to zero for the $E_{m,\gamma}$ norm \eqref{c00} of the difference between the exact and approximate solutions; this result is stated in Theorem {\rm Re }\, f{approxthm} and Corollary {\rm Re }\, f{corapprox}.\footnote{Several papers, for example \cite{JMR, CGW1, CW1,Hernandez}, in which singular systems were used to rigorously justify approximate solutions for quasilinear problems, used the method of simultaneous Picard iteration, which estimates the difference between exact solution iterates and approximate solution iterates. That method was not an option here, since we were unable to prove the uniform existence of exact solutions by any kind of iteration scheme. A variant of the method in section {\rm Re }\, f{end}, based on direct estimation of the difference between exact and approximate solutions, can be used to avoid the use of simultaneous Picard iteration in the earlier works.}
In Chapter {\rm Re }\, f{chapter6} we explain how the main theorems, Theorems {\rm Re }\, f{uniformexistence} and {\rm Re }\, f{approxthm}, extend to general isotropic hyperelastic materials
governed by an analytic stored energy function, and also how those theorems extend readily to the wavetrain case. In section {\rm Re }\, f{higherD} we discuss the only obstruction that remains to extending these theorems to dimensions $d\geq 3$. We show that in $d\geq 3$ the linearized problem has characteristics of variable multiplicity that fail to be algebraically regular in the sense of \cite{MZ}, and which are at the same time glancing. Thus, the problem falls outside the scope of existing Kreiss symmetrizer technology.
We have avoided any energy dissipation argument in the construction of exact solutions,
for this gives us hope to extend our work to free boundary problems for first order hyperbolic problems
that also give rise to surface waves. Such situations arise indeed in the modeling of liquid-vapor phase
transitions or in magnetohydrodynamics, see \cite{Benzoni1998,AliHunter}. We believe that the error
analysis in Chapter {\rm Re }\, f{chapter5} is flexible enough so that, provided one has an exact solution
in such a problem,
the
construction of a corrector and the estimates of the error terms involved in Chapter {\rm Re }\, f{chapter5}
could be adapted in order to yield a full justification of the high frequency asymptotics.
Chapters {\rm Re }\, f{chapter2}, {\rm Re }\, f{chapter3}, {\rm Re }\, f{chapter4} and {\rm Re }\, f{chapter5} are meant to be as independent as possible. Chapter {\rm Re }\, f{chapter5} is really the only chapter that depends substantially on others, those being mainly Chapters {\rm Re }\, f{chapter3} and {\rm Re }\, f{chapter4}.
We have included detailed and mostly non-technical introductions to these chapters in an effort to make the reader familiar with the main ideas before having to plunge into the estimates. We hope the reader will forgive a certain amount of repetition arising from this.
\paragraph{Notation}
Throughout this work, we let ${\mathcal M}_{n,N}({\mathbb K})$ denote the set of $n \times N$ matrices
with entries in ${\mathbb K} = {\mathbb R} \text{ or }{\mathbb C}$, and we use the notation ${\mathcal M}_N({\mathbb K})$
when $n=N$. The trace of a matrix $M \in {\mathcal M}_N({\mathbb K})$ is denoted $\text{\rm tr } M$.
The transpose of a matrix (or vector) $M$ is denoted $M^T$. We let $I$ denote the identity matrix, without
mentioning the dimension. The norm of a (column) vector $X \in {\mathbb C}^N$ is $|X| := (X^* \, X)^{1/2}$, where
the row vector $X^*$ denotes the conjugate transpose of $X$. If $X,Y$ are two vectors in ${\mathbb C}^N$, we let
$X \cdot Y$ denote the (bilinear) quantity $\sum_j X_j \, Y_j$, which coincides with the usual scalar product
in ${\mathbb R}^N$ when $X$ and $Y$ are real. We often use Einstein's summation convention in order to make
some expressions easier to read.
The Fourier transform of a function $f$ from ${\mathbb R}$ to ${\mathbb C}$ is defined as
$$
\forall \, k \in {\mathbb R} \, ,\quad \widehat{f}(k) := \int_{\mathbb R} {\rm e}^{-i\, k \, \theta} \, f(\theta) \, {\rm d}\theta \, .
$$
The Hilbert transform ${\mathcal H}f$ of $f$ is then defined by
$$
\widehat{{\mathcal H}f}(k) := -i \, \text{\rm sgn } \! \! (k) \, \widehat{f}(k) \, ,
$$
where $\text{\rm sgn}$ denotes the sign function.
The letter $C$ always denotes a positive constant that may vary from line to line or within the same line.
Dependence of the constant $C$ on various parameters is made precise throughout the text. The sign
$\lesssim$ means $\le$ up to a multiplicative constant.
To avoid having expressions like $v^{{\varepsilon},s}_{1,T}$ or $v^{{\varepsilon},s}_{T}$ appear repeatedly in Chapter {\rm Re }\, f{chapter3} and later, we shall often suppress the ${\varepsilon}$ and $T$ indices and write simply $v^s_1$ or $v^s$ instead. Here $v^{\varepsilon}_1$ is the first component of $v^{\varepsilon}=(v^{\varepsilon}_1,v^{\varepsilon}_2)$. The $s$ and $T$ on $v^{{\varepsilon},s}_{1,T}$ indicate that we are taking a Seeley extension (Proposition {\rm Re }\, f{c0e}) to all time of $v^{\varepsilon}_1|_{t<T}$. Almost every function that occurs in the study of singular systems has ${\varepsilon}$ dependence, so suppressing ${\varepsilon}$ should cause no trouble. The presence of the superscript $s$ should \emph{always} be taken to imply the presence of a suppressed subscript $T$.
The positive number $T$ will always be small, and sometimes will lie in a range that depends on ${\varepsilon}$, $0<T\leq T_{\varepsilon}$.
The coefficients in the linearized systems we study on the whole half-space, for example those appearing in \eqref{c1}-\eqref{c3}, will usually be functions of terms like $v^s$. Later reminders of this will be provided. Further notation is made precise in the body of
the text.
\chapter{Derivation of the weakly nonlinear amplitude equation}
\langle bel{chapter2}
\emph{\quad} In this Chapter, we revisit the weakly nonlinear asymptotic analysis for second order hyperbolic initial
boundary value problems that come from a variational principle. Our main goal is to derive an amplitude
equation that governs the evolution of small amplitude high frequency solutions in the case where the
so-called WKB ansatz incorporates all possible slow and fast variables. As expected from pioneering
works devoted to nonlinear elasticity, the amplitude equation we derive takes the form of a scalar
integro-differential equation taking place on the boundary of the space domain. This equation displays
two main features: propagation at an appropriate {\it group velocity} according to the slow spatial variables
along the boundary of the space domain, and nonlinearity due to the presence of a {\it bilinear Fourier
multiplier} that governs the evolution with respect to the fast variable. We unify previous works arising
either from the `applied' or more `theoretical' literature and clarify which solvability condition is actually
needed in order to derive our main amplitude equation.
Taking slow variables into account is crucial in the upcoming Chapters {\rm Re }\, f{chapter3}, {\rm Re }\, f{chapter4}, and {\rm Re }\, f{chapter5}
for providing a functional framework in which we are able to construct and analyze {\it exact} pulse
solutions. In the present Chapter, the analysis is mostly {\it formal} and we aim at identifying the leading
order term in the presumably valid asymptotic expansion of exact solutions. Showing that exact solutions
are indeed well approximated by this leading order term is the purpose of Chapter {\rm Re }\, f{chapter5}.
Though our main concern in this work is the system of nonlinear elasticity, we aim at keeping the discussion
in a rather general framework whenever possible, which might accelerate the adaptation of the present work
to related problems with surface waves arising for instance in the modeling of liquid vapor phase transitions,
magnetohydrodynamics and/or liquid crystals, see, e.g., \cite{Benzoni1998,AliHunter,Saxton,AustriaHunter}
and further references therein.
\section{The variational setting: assumptions}\langle bel{var}
\emph{\quad} In this Section, we revisit the analysis of \cite{BCproc} and derive the amplitude equation that governs
the evolution of weakly nonlinear surface (or Rayleigh) waves. Keeping the discussion in a rather general
framework, we start from a Lagrangian of the form:
\begin{equation*}
{\mathcal L} [{\bf u}]:= \int_0^T \! \! \! \int_{\Omega} \left( \dfrac{1}{2} \, |{\bf u}_t|^2 -W({\nabla}bla {\bf u}) \right) \,
{\rm d}x \, {\rm d}t \, .
\end{equation*}
Here the space domain $\Omega \subset {\mathbb R}^d$ is a half-space, ${\bf u} \in {\mathbb R}^N$ is the (possibly vector-valued)
unknown function, ${\bf u}_t$ denotes its time derivative and ${\nabla}bla {\bf u}$ denotes its spatial Jacobian matrix
(which we sometimes call its gradient). The function $W$ plays the role of a `stored elastic energy', and the total
amount of energy over the space domain $\Omega$ is then denoted
\begin{equation*}
{\mathbb E}n [{\bf u}] := \int_{\Omega} W({\nabla}bla {\bf u}) \, {\rm d}x \, .
\end{equation*}
Opposite to the
case considered in \cite{BCproc} we assume here for simplicity that $W$ only depends on the unknown ${\bf u}$
through its (spatial) gradient. Hence Assumptions (H1) and (H2) in \cite{BCproc} are trivially satisfied. In the case
of hyperelastic materials, which is our main concern here, there holds $N=d$, with either $d=2$ or $d=3$.
In the following calculations, Greek letters $\alpha, \beta,\gamma$ usually correspond to indices for the
coordinates of the vector ${\bf u}$ and therefore run through the set $\{ 1,\dots,N \}$. Roman letters $j,\ell,m$
refer to the coordinates of the space variable $x$ and therefore run through $\{ 1,\dots,d \}$. For instance, the
$(\alpha,j)$ coordinate of the matrix ${\nabla}bla {\bf u}$ is denoted $u_{\alpha,j}$, where the subscript `$,j$' is a
short cut for denoting partial differentiation with respect to $x_j$. We thus consider $W$ as a function from
$\cM_{N,d} ({\mathbb R})$ into ${\mathbb R}$ and tacitly assume that it is as smooth as we want (at least $\cC^3$ as far as the
derivation of the amplitude equation is concerned). We use from now on Einstein's summation convention
over repeated indices, unless otherwise stated.
Let us write the space domain as $\Omega = \{ x \cdot {\boldsymbol \nu} >0 \}$, with ${\boldsymbol \nu}$ the normal vector to $\partial
\Omega$ pointing inwards. We are then interested in {\it critical points} of the above Lagrangian ${\mathcal L}$. These
correspond to functions ${\bf u}$ that satisfy the interior equations:
\begin{equation}
\langle bel{eqint}
\forall \, \alpha=1,\dots,N \, ,\quad
\partial_t^2 u_\alpha -\left( \dfrac{\partial W}{\partial u_{\alpha,j}} ({\nabla}bla {\bf u}) \right)_{,j} =0 \, ,
\end{equation}
with boundary conditions:
\begin{equation}
\langle bel{cl}
\forall \, \alpha=1,\dots,N \, ,\quad
\nu_j \, \dfrac{\partial W}{\partial u_{\alpha,j}} ({\nabla}bla {\bf u}) {\mathbb B}ig|_{\partial \Omega} =0 \, ,
\end{equation}
In what follows, we assume that all constant states $\underline{\bf u} \in {\mathbb R}^N$ are critical points of ${\mathcal L}$,
independently of the choice of the space domain $\Omega$. We also normalize the stored energy so that it
vanishes at the origin. Equivalently, we make the following assumption:
\begin{itemize}
\item[(H1)] $\qquad W(0)=0$ and $\dfrac{\partial W}{\partial u_{\alpha,j}}(0) =0$ for all $\alpha,j$.
\end{itemize}
Assumption (H1) holds when the stored energy $W$ depends quadratically on ${\nabla}bla u$, and in that case the
equations \eqref{eqint}-\eqref{cl} are linear with respect to ${\bf u}$. Here we shall be interested in solutions ${\bf u}$
that are small perturbations of a constant state, say $0$ (up to translating in ${\bf u}$). In the context of nonlinear
elasticity, ${\bf u}(t,x)$ refers to the displacement with respect to an `equilibrium configuration'. The deformation
of the equilibrium configuration is given by the mapping $(x \mapsto x +{\bf u}(t,x))$.
For future use, we introduce the coefficients involved in the Taylor expansion of $W$ up to the third order
at the origin:
\begin{equation}
\langle bel{defcoeff}
c_{\alpha j \beta \ell} := \dfrac{\partial^2 W}{\partial u_{\alpha,j} \, \partial u_{\beta,\ell}}(0) \, ,\quad
d_{\alpha j \beta \ell \gamma m} :=
\dfrac{\partial^3 W}{\partial u_{\alpha,j} \, \partial u_{\beta,\ell} \, \partial u_{\gamma,m}}(0) \, .
\end{equation}
The linearization of \eqref{eqint}-\eqref{cl} at the constant solution ${\bf u} \equiv 0$ reads
\begin{equation}
\langle bel{eqlin}
\begin{cases}
\partial_t^2 v_\alpha -c_{\alpha j \beta \ell} \, v_{\beta,j\ell} =0 \, ,&
\forall \, \alpha=1,\dots,N \, ,\quad x \in \Omega \, ,\\
\nu_j \, c_{\alpha j \beta \ell} \, v_{\beta,\ell} \big|_{\partial \Omega} =0 \, ,& \forall \, \alpha=1,\dots,N \, ,
\end{cases}
\end{equation}
and the determination of formal and/or rigorous high frequency weakly nonlinear solutions to
\eqref{eqint}-\eqref{cl} heavily depends on the stability properties of \eqref{eqlin}. In what follows, we shall
assume that the linearized problem \eqref{eqlin} admits a one-dimensional space of `surface waves' which,
in the context of linearized elasticity, correspond to Rayleigh waves. Let us be a little bit more specific. We
first assume that the linearization of the stored energy $W$ at $0$ is strictly rank one convex (a strong form
of the Legendre-Hadamard condition), that is:
\begin{itemize}
\item[(H2)] There exists a constant $c>0$ such that for all ${\boldsymbol \xi} \in {\mathbb R}^d$ and all $v \in {\mathbb R}^N$, there holds
$$
c_{\alpha j \beta \ell} \, v_\alpha \, \xi_j \, v_\beta \, \xi_\ell \ge c \, |v|^2 \, |{\boldsymbol \xi}|^2 \, .
$$
\end{itemize}
Assumption (H2) ensures that the Cauchy problem
\begin{equation*}
\partial_t^2 v_\alpha -c_{\alpha j \beta \ell} \, v_{\beta,j\ell} =0 \, ,\quad \alpha=1,\dots,N \, ,\quad x \in {\mathbb R}^d \, ,
\end{equation*}
is well-posed in the homogeneous Sobolev space $\dot{H}^1({\mathbb R}^d;{\mathbb R}^N)$, see \cite{SerreJFA}. Hence \eqref{eqlin}
is a linear hyperbolic boundary value problem which should be supplemented with some initial data for $v$, which
we do not write for the moment. The analysis of \eqref{eqlin} follows the general theory of \cite{K,sakamoto} and
relies on the so-called normal mode analysis. We therefore look for solutions to \eqref{eqlin} of the form
\begin{equation*}
{\bf v}(t,x) = {\rm e}^{i \, (\tau-i\, \gamma) \, t+i \, {\boldsymbol \eta} \cdot x} \, V({\boldsymbol \nu} \cdot x) \, ,
\end{equation*}
with $\gamma>0$, ${\boldsymbol \eta}$ in the cotangent space to $\partial \Omega$ (which reduces here to assuming that
${\boldsymbol \eta}$ is orthogonal to ${\boldsymbol \nu}$ because $\Omega$ is a half-space) and a profile $V$ vanishing at $+\infty$.
Plugging the previous ansatz for ${\bf v}(t,x)$ in \eqref{eqlin}, we are led to determining the set of functions $V$,
from ${\mathbb R}^+$ into ${\mathbb C}^N$, that vanish at $+\infty$ and satisfy the second-order differential problem:
\begin{equation}
\langle bel{normalmode'}
\begin{cases}
(\tau-i\, \gamma)^2 \, V_\alpha +c_{\alpha j \beta \ell} \, (i\, \eta_j +\nu_j \, \partial_z) \,
(i\, \eta_\ell +\nu_\ell \, \partial_z) \, V_\beta=0 \, ,& \forall \, \alpha =1,\dots,N \, ,\quad z>0 \, ,\\
\nu_j \, c_{\alpha j \beta \ell} \, (i\, \eta_\ell +\nu_\ell \, \partial_z) \, V_\beta \big|_{z=0} =0 \, ,&
\forall \, \alpha =1,\dots,N \, .
\end{cases}
\end{equation}
It is convenient to rewrite \eqref{normalmode'} in a more compact form, and we therefore introduce the following
$N \times N$ matrices for all vector ${\boldsymbol \xi} \in {\mathbb R}^d$:
\begin{equation}
\langle bel{defASigma}
\Sigma ({\boldsymbol \xi}) := {\mathbb B}ig( c_{\alpha j \beta \ell} \, \xi_j \, \xi_\ell {\mathbb B}ig)_{\alpha,\beta=1,\dots,N} \, ,\quad
A_j({\boldsymbol \xi}) := {\mathbb B}ig( c_{\alpha j \beta \ell} \, \xi_\ell {\mathbb B}ig)_{\alpha,\beta=1,\dots,N} \, ,\quad \forall \, j=1,\dots,d \, .
\end{equation}
With the previous notation, \eqref{normalmode'} reads
\begin{equation}
\langle bel{normalmode}
\begin{cases}
\big( (\tau-i\, \gamma)^2 \, I_N -\Sigma({\boldsymbol \eta}) \big)\, V +i\, \big( \nu_j \, A_j({\boldsymbol \eta}) +\nu_j \, A_j({\boldsymbol \eta})^T \big) \,
\partial_z \, V +\Sigma({\boldsymbol \nu}) \, \partial_{zz}^2 V =0 & z>0 \, ,\\
i\, \nu_j \, A_j({\boldsymbol \eta}) \, V +\Sigma({\boldsymbol \nu}) \, \partial_z \, V \big|_{z=0} =0 \, ,&
\end{cases}
\end{equation}
where we have used the Schwarz relation which translates into symmetry properties for the coefficients
$c_{\alpha j \beta \ell}$ (namely, $c_{\alpha j \beta \ell}=c_{\beta \ell \alpha j}$). The strict rank one convexity
assumption (H2) shows that $\Sigma({\boldsymbol \nu})$ is symmetric positive definite, and therefore the second-order
boundary value problem \eqref{normalmode} can be equivalently recast as a first-order augmented system
\begin{equation}
\langle bel{hamiltonien}
\begin{cases}
\partial_z \begin{pmatrix}
V \\ W \end{pmatrix} = {\bf J} \, H(\tau-i\, \gamma,{\boldsymbol \eta}) \, \begin{pmatrix}
V \\ W \end{pmatrix} \, ,& z>0 \, ,\\
W(0) =0 \, ,&
\end{cases}
\end{equation}
where, using standard block matrix notation, we have set
\begin{equation}
\langle bel{defJH}
{\bf J} :=\begin{pmatrix}
0 & I_N \\
-I_N & 0 \end{pmatrix} \, ,\quad
H(\zeta,{\boldsymbol \eta}) := \begin{pmatrix}
K_0(\zeta,{\boldsymbol \eta}) & i\, \nu_j \, A_j({\boldsymbol \eta})^T \, \Sigma({\boldsymbol \nu})^{-1} \\
-i\, \Sigma({\boldsymbol \nu})^{-1} \, \nu_j \, A_j({\boldsymbol \eta}) & \Sigma({\boldsymbol \nu})^{-1} \end{pmatrix} \, ,
\end{equation}
and the upper-left block $K_0$ of $H$ in \eqref{defJH} is defined by:
\begin{equation}
\langle bel{defK0}
K_0(\zeta,{\boldsymbol \eta}) :=\zeta^2 \, I_N -\Sigma({\boldsymbol \eta}) +\big( \nu_j \, A_j({\boldsymbol \eta})^T \big) \,
\Sigma({\boldsymbol \nu})^{-1} \, \big( \nu_\ell \, A_\ell({\boldsymbol \eta}) \big) \, .
\end{equation}
The first-order `Hamiltonian' formulation \eqref{hamiltonien} was already introduced in \cite{SerreJFA} and it
was used in \cite{BC} in order to show structural properties for the amplitude equation which we shall derive
here in a slightly more general context. The second-order formulation \eqref{normalmode} was adopted in
\cite{BCproc} and we mainly follow this approach here. However, the equivalent formulation \eqref{hamiltonien}
will be used at some point, which is why we recall it here with notation similar to those in \cite{BC}.
The aim of the normal mode analysis is to determine the frequency pairs $(\tau-i\, \gamma,{\boldsymbol \eta})$ with
$\gamma \ge 0$ for which \eqref{normalmode} admits a nontrivial solution. The subtlety lies in the definition
of which solutions are admissible when $\gamma$ equals zero. However, the strict rank one convexity
assumption (H2) above yields the classical result that the matrix ${\bf J} \, H(\tau-i\, \gamma,{\boldsymbol \eta})$ is
{\it hyperbolic} (in the sense of dynamical systems, meaning that it has no purely imaginary eigenvalue)
as long as $\gamma \neq 0$ or\footnote{Here $\langle mbda_{\rm min} (S)$ denotes the smallest eigenvalue
of a real symmetric matrix $S$.}
\begin{equation*}
\gamma=0 \quad \text{\rm and} \quad \tau^2 <\min_{\omega \in {\mathbb R}} \langle mbda_{\rm min} \, \big( \Sigma
({\boldsymbol \eta} +\omega \, {\boldsymbol \nu}) \big) \, .
\end{equation*}
The latter case corresponds to `elliptic frequencies' $(\tau,{\boldsymbol \eta})$ of the cotangent bundle of ${\mathbb R}_t \times
\partial \Omega$. Observe that by the strict rank one convexity assumption (H2), this set contains a cone
of the form $\{ |\tau| < c \, |{\boldsymbol \eta}| \}$ for some $c>0$. For such elliptic frequencies, $H(\tau,{\boldsymbol \eta})$ in
\eqref{defJH} is Hermitian, and eigenvalues of ${\bf J} \, H(\tau,{\boldsymbol \eta})$ come in pairs
$(-\omega,\overline{\omega})$ because the adjoint matrix $({\bf J} \, H(\tau,{\boldsymbol \eta}))^*$ is conjugated to
$-{\bf J} \, H(\tau,{\boldsymbol \eta})$. Here, and from now on, we adopt the convention Re $\omega>0$. The stable
subspace ${\mathbb E}^s(\tau-i\, \gamma,{\boldsymbol \eta})$ of ${\bf J} \, H(\tau-i\, \gamma,{\boldsymbol \eta})$ is therefore well-defined,
has dimension $N$ and depends analytically on $(\tau-i\, \gamma,{\boldsymbol \eta})$ on the connected set that is the
union of $\{ \gamma>0\}$ and of the set of elliptic frequencies. Furthermore, it is known from the general
theory in \cite{K,sakamoto,Met} that, at least in the case where the hyperbolic operator in \eqref{eqlin} has
constant multiplicity, the stable subspace ${\mathbb E}^s$ admits a continuous extension up to $\gamma=0$ for all
$(\tau,{\boldsymbol \eta}) \neq (0,0)$. In what follows, ${\mathbb E}^s(\tau,{\boldsymbol \eta})$ denotes this continuous extension for all
$(\tau,{\boldsymbol \eta})$, which coincides with the `true' stable subspace of ${\bf J} \, H(\tau,{\boldsymbol \eta})$ for elliptic
frequencies\footnote{The analysis in this Chapter is only concerned with elliptic frequencies, which is
the reason why we do not emphasize too much the assumptions ensuring that the continuous extension
of ${\mathbb E}^s$ up to $\gamma=0$ is well-defined. Such assumptions will be enforced in Chapter {\rm Re }\, f{chapter3}
where continuous extension of ${\mathbb E}^s$ will play a major role.}.
Well-posedness of the linear initial boundary value problem \eqref{eqlin}, with prescribed initial data $v|_{t=0}$,
was investigated thoroughly by Serre in \cite{SerreJFA} where he obtained the striking result that strong
well-posedness holds {\it if and only if} the energy
\begin{equation*}
{\mathbb E}n_2 [{\bf u}]:= \int_{\Omega} W_2({\nabla}bla {\bf u}) \, {\rm d}x \, ,\quad
W_2(F) := c_{\alpha j \beta \ell} \, F_{\alpha,j} \, F_{\beta,\ell} \, ,\quad \forall \, F \in {\mathcal M}_{N,d}({\mathbb R}) \, ,
\end{equation*}
is coercive (and thereby strictly convex) over $\dot{H}^1(\Omega)$. The subscript 2 in ${\mathbb E}n_2,W_2$ refers to
the Taylor expansion at order $2$ of the original stored energy $W$ at the origin. When ${\mathbb E}n_2$ is coercive,
the analysis in \cite{SerreJFA} also shows that there exist {\it surface wave} solutions to \eqref{eqlin}, which
we shall assume here to be of {\it finite energy}. More precisely, in agreement with the results in \cite{SerreJFA},
we make the following assumption:
\begin{itemize}
\item[(H3)] For all ${\boldsymbol \eta} \neq 0$ in the cotangent space to $\partial \Omega$, there exists a real
$\tau_{\rm r}({\boldsymbol \eta})$ satisfying
$$
0 < \tau_{\rm r}({\boldsymbol \eta})
< \left( \min_{\omega \in {\mathbb R}} \langle mbda_{\rm min} \, \big( \Sigma ({\boldsymbol \eta} +\omega \, {\boldsymbol \nu}) \big) \right)^{1/2} \, ,
$$
such that for all $(\gamma,\tau,{\boldsymbol \eta}) \neq (0,0,0)$, there holds:
$$
\left\{ \begin{pmatrix}
V \\ W \end{pmatrix} \in {\mathbb E}^s(\tau -i\, \gamma,{\boldsymbol \eta}) \, | \, W=0 \right\} \neq \{ 0 \}
$$
if and only if $\gamma=0$ and $\tau =\pm \tau_{\rm r}({\boldsymbol \eta})$. Furthermore, for all $\underline{{\boldsymbol \eta}} \neq 0$, the
matrix ${\bf J} \, H(\tau,{\boldsymbol \eta})$ is geometrically regular near $(\pm \tau_{\rm r}(\underline{{\boldsymbol \eta}}),\underline{{\boldsymbol \eta}})$,
meaning that it admits a basis of eigenvectors $({\bf R}_1,\dots,{\bf R}_{2\, N})(\tau,{\boldsymbol \eta})$ that depends analytically
on $(\tau,{\boldsymbol \eta})$ near $(\pm \tau_{\rm r}(\underline{{\boldsymbol \eta}}),\underline{{\boldsymbol \eta}})$ with the convention that ${\bf R}_1,
\dots,{\bf R}_N$ span the stable subspace ${\mathbb E}^s$, and the $N \times N$ determinant
$$
\partialelta (\tau,{\boldsymbol \eta}) := \det \big( S_1(\tau,{\boldsymbol \eta}) \cdots S_N(\tau,{\boldsymbol \eta}) \big) \, ,\quad
{\bf R}_\alpha :=\begin{pmatrix}
R_\alpha \\ S_\alpha \end{pmatrix} \, ,
$$
has a simple root (with respect to $\tau$) at $(\pm \tau_{\rm r}(\underline{{\boldsymbol \eta}}),\underline{{\boldsymbol \eta}})$.
\end{itemize}
Let us recall that due to the fact that the so-called Lopatinskii determinant $\partialelta$ has a simple root at
$(\tau_{\rm r}(\underline{{\boldsymbol \eta}}),\underline{{\boldsymbol \eta}})$, the subspace
$$
\left\{ \begin{pmatrix}
V \\ W \end{pmatrix} \in {\mathbb E}^s(\tau_{\rm r}(\underline{{\boldsymbol \eta}}),\underline{{\boldsymbol \eta}}) \, | \, W=0 \right\}
$$
is one-dimensional. (Observe also that ${\mathbb E}^s(\tau,{\boldsymbol \eta})$ does not depend on the sign of $\tau$, which is why
we restrict here to positive values of $\tau$.) Consequently, for all tangential wave vector ${\boldsymbol \eta} \neq 0$, there
exists a one-dimensional family of surface waves
$$
{\rm e}^{\pm i \, \tau_{\rm r}({\boldsymbol \eta}) \, t+i \, {\boldsymbol \eta} \cdot x} \, V({\boldsymbol \nu} \cdot x) \, ,
$$
solution to \eqref{eqlin} with $V(+\infty) =0$ (the decay is actually exponential).
By rescaling, it is readily observed from \eqref{normalmode} that for all $k>0$, $\tau_{\rm r}(k\, \underline{{\boldsymbol \eta}})
=k\, \tau_{\rm r}(\underline{{\boldsymbol \eta}})$ and if
$$
{\rm e}^{\pm i \, \tau_{\rm r}({\boldsymbol \eta}) \, t+i \, {\boldsymbol \eta} \cdot x} \, V({\boldsymbol \nu} \cdot x) \, ,
$$
is a (nontrivial) surface wave solution to \eqref{eqlin}, then
$$
{\rm e}^{\pm i \, k\, \tau_{\rm r}({\boldsymbol \eta}) \, t+i \, k \, {\boldsymbol \eta} \cdot x} \, V(k\, {\boldsymbol \nu} \cdot x) \, ,
$$
is a (nontrivial) surface wave solution to \eqref{eqlin} for the rescaled frequencies $k\, (\tau_{\rm r}({\boldsymbol \eta}),{\boldsymbol \eta})$.
We have now made all our assumptions on the linearized problem \eqref{eqlin} and turn to the derivation of
the amplitude equation governing weakly nonlinear asymptotic solutions to the nonlinear problem \eqref{eqint},
\eqref{cl}.
\section{Weakly nonlinear asymptotics}\langle bel{wna}
\emph{\quad} In this Section, we show formally that high frequency weakly nonlinear solutions to the nonlinear equations
\eqref{eqint}, \eqref{cl} are governed by a nonlocal Burgers type equation that is similar to the ones derived
in \cite{BC} or \cite{BCproc} for second-order equations, or in \cite{H,Mar} for first-order hyperbolic systems. In
the case of elastodynamics, the derivation of such amplitude equations dates back at least to \cite{Lardner1} for
two-dimensional elasticity. Part of the analysis in \cite{Lardner1} was later simplified in \cite{ParkerTalbot,Parker}
though those last two references did not include any dependence of the weakly nonlinear solution on what we
call below `slow spatial variables'. Those slow variables were considered in \cite{Lardner1}, leading to rather
complicated calculations and a somehow mysterious relation (Equation (A17) in \cite{Lardner1}) stating that
weakly nonlinear Rayleigh waves actually propagate at the Rayleigh speed with respect to the slow tangential
variables along the boundary of the elastic material. The analogous relation in the case of anisotropic materials
(Equation (51) in \cite{Lardner2}) is explained in a more convincing way, though it seems to rely heavily on the
fact that one considers two-dimensional materials. Independently of one's ability to verify the accuracy of the
expressions given in \cite{Lardner1,Lardner2}, there is a puzzling fact arising, which is that the propagation
of the weakly nonlinear Rayleigh wave in the slow tangential space variables along the boundary seems to
be governed by the Rayleigh speed, which rather arises as a {\it phase velocity}, while one might reasonably
expect to see a {\it group velocity} arise. As we show below, the conclusions in \cite{Lardner1,Lardner2} are
indeed correct but they are linked to the fact that the boundary has only one spatial dimension so the dispersion
relation between $\tau_{\rm r}$ and ${\boldsymbol \eta}$ becomes linear. We also simplify below some of the calculations
in \cite{Lardner1,Lardner2} by showing that the substitution of normal derivatives in term of tangential ones
made in those two references is actually unnecessary, since one can directly derive the amplitude equation with
slow spatial derivatives taking place only along the boundary of $\Omega$. This `cancellation' property,
which seems to have been unnoticed in all above mentioned references, is our main improvement with
respect to \cite{BCproc} where only fast spatial variables were taken into account.
From now on, we fix a nonzero wave vector ${\boldsymbol \eta}$ in the cotangent space to $\partial \Omega$, and we fix,
as in \cite{Lardner1}, the frequency $\tau :=-\tau_{\rm r}({\boldsymbol \eta})>0$ (the case $\tau =\tau_{\rm r}({\boldsymbol \eta})$ is
entirely similar). We also fix a nonzero (exponentially decaying at $+\infty$) profile $V$ such that
$$
{\rm e}^{i \, \tau \, t+i \, {\boldsymbol \eta} \cdot x} \, V({\boldsymbol \nu} \cdot x) \, ,
$$
is a (surface wave) solution to \eqref{eqlin}.
We look for asymptotic solutions ${\bf u}^{\varepsilon}$ to the nonlinear equations \eqref{eqint} satisfying an
inhomogeneous version of the boundary conditions \eqref{cl}, namely
\begin{equation}
\langle bel{clinhom}
\forall \, \alpha=1,\dots,N \, ,\quad
\nu_j \, \dfrac{\partial W}{\partial u_{\alpha,j}} ({\nabla}bla {\bf u}^{\varepsilon}) {\mathbb B}ig|_{x \in \partial \Omega}
={\varepsilon}^2 \, G \left( t,x,\dfrac{\tau \, t +{\boldsymbol \eta} \cdot x}{{\varepsilon}} \right) \, ,
\end{equation}
where $G$ is defined on $(-\infty,T] \times \partial \Omega \times {\mathbb R}$ for some given time $T>0$, and decays
at infinity with respect to its last argument, which we denote $\theta$ from now on. We do not give a precise
meaning to the `decay at infinity' for $G$, but we tacitly assume at least that $G$ is $L^2$ with respect to
$\theta$ so that we can apply the Fourier transform\footnote{Unlike several previous works on pulse-like
solutions, we shall not consider here any polynomial decay with respect to the fast variable $\theta$, though
such technical considerations can be skipped in the framework of this Chapter and postponed to Chapter
{\rm Re }\, f{chapter3}.}. We expect the exact solution ${\bf u}^{\varepsilon}$ to \eqref{eqint}, \eqref{clinhom} to have an
asymptotic expansion of the form
\begin{equation}
\langle bel{wkbu}
{\bf u}^{\varepsilon} \sim {\varepsilon}^2 \, {\bf u}^{(1)} \left( t,x,\dfrac{\tau \, t +{\boldsymbol \eta} \cdot x}{{\varepsilon}},\dfrac{{\boldsymbol \nu} \cdot x}{{\varepsilon}} \right)
+{\varepsilon}^3 \, {\bf u}^{(2)} \left( t,x,\dfrac{\tau \, t +{\boldsymbol \eta} \cdot x}{{\varepsilon}},\dfrac{{\boldsymbol \nu} \cdot x}{{\varepsilon}} \right) +\cdots \, ,
\end{equation}
and we wish to determine the `amplitude equation' governing the evolution of the leading profile ${\bf u}^{(1)}$.
For convenience, we have considered here the regime of weakly nonlinear high frequency waves, which means
that the `slow' variables are $(t,x)$ and the `fast' variables are $(\theta,z) :=((\tau \, t +{\boldsymbol \eta} \cdot x)/{\varepsilon},{\boldsymbol \nu}
\cdot x/{\varepsilon})$, while in \cite{Lardner1,Lardner2,H,BC,BCproc} the authors considered the regime of weakly
nonlinear modulation on large times where the `slow' variables are $({\varepsilon} \, t,{\varepsilon} \, x)$ and the analogue of
the `fast' variables are $(\tau \, t +{\boldsymbol \eta} \cdot x,{\boldsymbol \nu} \cdot x)$. The link between the two regimes comes from
the scale invariance properties of \eqref{eqint}, \eqref{cl}. Namely, since the half-space $\Omega$ is invariant
by dilation, if ${\bf u}(t,x)$ is a solution to \eqref{eqint}, \eqref{cl}, then $\alpha \, {\bf u}(t/\alpha,x/\alpha)$ is also
a solution for any $\alpha>0$. This explains why the scaling in \eqref{wkbu} involves solutions of amplitude
$O({\varepsilon}^2)$ while the references \cite{Lardner1,Lardner2,H,BC,BCproc} consider solutions of amplitude
$O({\varepsilon})$. One advantage of our scaling is to deal with approximate and/or exact solutions to \eqref{eqint},
\eqref{clinhom} that are defined on a {\it fixed} time interval independent of the wavelength ${\varepsilon}$.
We follow the calculations in \cite{BCproc}, with the novelty here that the profiles ${\bf u}^{(1)},{\bf u}^{(2)}$ also
depend on the slow spatial variables $x$ while only a slow time variable was considered in \cite{BCproc}. Recalling
the definition \eqref{defASigma}, we introduce the so-called `fast-fast'/`fast-slow' operators in the interior and the
`fast'/`slow' operators on the boundary:
\begin{align*}
{\mathcal L}_{\rm ff} &:=\big( \tau^2 \, I -\Sigma({\boldsymbol \eta}) \big) \, \partial^2_{\theta \theta}
-\big( \nu_j \, A_j({\boldsymbol \eta}) +\nu_j \, A_j({\boldsymbol \eta})^T \big) \, \partial^2_{\theta z} -\Sigma({\boldsymbol \nu}) \, \partial^2_{zz} \, ,\\
{\mathcal L}_{\rm fs} &:= 2\, \tau \, \partial^2_{t \theta} -\big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, \partial^2_{j \theta}
-\big( A_j({\boldsymbol \nu}) +A_j({\boldsymbol \nu})^T \big) \, \partial^2_{j z} \, ,\\
\ell_{\rm f} &:= \nu_j \, A_j({\boldsymbol \eta}) \, \partial_\theta +\Sigma ({\boldsymbol \nu}) \, \partial_z \, ,\\
\ell_{\rm s} &:= A_j({\boldsymbol \nu})^T \, \partial_j \, ,
\end{align*}
as well as the `fast' quadratic operators:
\begin{align*}
{\mathcal Q}[{\bf u}]_\alpha &:= d_{\alpha j \beta \ell \gamma m} \, (\eta_j \, \partial_\theta +\nu_j \, \partial_z) \,
{\mathbb B}ig\{ \big( (\eta_\ell \, \partial_\theta +\nu_\ell \, \partial_z) \, u_\beta \big) \,
\big( (\eta_m \, \partial_\theta +\nu_m \, \partial_z) \, u_\gamma \big) {\mathbb B}ig\} \, ,\\
{\mathcal M}[{\bf u}]_\alpha &:= \nu_j \, d_{\alpha j \beta \ell \gamma m} \, {\mathbb B}ig\{
\big( (\eta_\ell \, \partial_\theta +\nu_\ell \, \partial_z) \, u_\beta \big) \,
\big( (\eta_m \, \partial_\theta +\nu_m \, \partial_z) \, u_\gamma \big) {\mathbb B}ig\} \, ,
\end{align*}
where $\alpha$ runs through $\{1,\dots,N\}$. The operator ${\mathcal Q}[\cdot ]$, resp. ${\mathcal M}[\cdot ]$, is
then defined as the vector valued operator in ${\mathbb R}^N$ whose $\alpha$-coordinate is ${\mathcal Q}[\cdot ]_\alpha$,
resp. ${\mathcal M}[\cdot ]_\alpha$.
Plugging the ansatz \eqref{wkbu} in \eqref{eqint}, \eqref{clinhom} and collecting the powers of ${\varepsilon}$, we are led to
constructing profiles ${\bf u}^{(1)},{\bf u}^{(2)}$ solutions to the equations
\begin{equation}
\langle bel{bkwordre1}
\begin{cases}
{\mathcal L}_{\rm ff} \, {\bf u}^{(1)} =0 \, ,& x \in \Omega \, ,\quad z>0 \, ,\\
\ell_{\rm f} \, {\bf u}^{(1)} {\mathbb B}ig|_{x \in \partial \Omega,z=0} =0 \, ,&
\end{cases}
\end{equation}
\begin{equation}
\langle bel{bkwordre2}
\begin{cases}
{\mathcal L}_{\rm ff} \, {\bf u}^{(2)} =-{\mathcal L}_{\rm fs} \, {\bf u}^{(1)} +\dfrac{1}{2} \, {\mathcal Q}[{\bf u}^{(1)}] \, ,&
x \in \Omega \, ,\quad z>0 \, ,\\
\ell_{\rm f} \, {\bf u}^{(2)} {\mathbb B}ig|_{x \in \partial \Omega,z=0} =G -{\mathbb B}ig( \ell_{\rm s} \, {\bf u}^{(1)}
+\dfrac{1}{2} \, {\mathcal M}[{\bf u}^{(1)}] {\mathbb B}ig) {\mathbb B}ig|_{x \in \partial \Omega,z=0} \, .&
\end{cases}
\end{equation}
One major difference here with respect to \cite{BCproc} is that the interior equations on ${\mathcal L}_{\rm ff}
\, {\bf u}^{(1,2)}$ hold for any $x \in \Omega$ and $z>0$ while the boundary conditions in
\eqref{bkwordre1}-\eqref{bkwordre2} correspond to a `double trace' $x \in \partial \Omega$ and $z=0$. However,
$x$ enters as a parameter in the interior equations since the fast-fast operator ${\mathcal L}_{\rm ff}$ only involves
differentiation with respect to $(\theta,z)$. We can therefore take the trace of the interior equation on $\partial
\Omega$. In particular, the leading profile ${\bf u}^{(1)}$ should satisfy
\begin{equation*}
\begin{cases}
{\mathcal L}_{\rm ff} \, {\mathbb B}ig( {\bf u}^{(1)} \big|_{x \in \partial \Omega} {\mathbb B}ig) =0 \, ,& z>0 \, ,\\
& \\
\ell_{\rm f} \, {\mathbb B}ig( {\bf u}^{(1)} \big|_{x \in \partial \Omega} {\mathbb B}ig) =0 \, ,& z=0 \, ,
\end{cases}
\end{equation*}
which is the fast problem considered in \cite{BCproc} (Equation (P1) there, though with slightly different notation
for the fast variables). The Fourier transform (with respect to the fast variable $\theta$) of the trace
${\bf u}^{(1)}|_{\partial \Omega}$ therefore satisfies
\begin{equation}
\langle bel{defw}
{\bf v}^{(1)} (t,x,k,z) := {\mathcal F}_\theta {\bf u}^{(1)}|_{x \in \partial \Omega} =\widehat{w}(t,x,k) \, \widehat{\bf r}(k,z)
\, ,\quad \widehat{\bf r}(k,z) := \begin{cases}
V(k\, z) \, ,& k>0 \, ,\\
\overline{V(-k\, z)} \, ,& k<0 \, .
\end{cases}
\end{equation}
We emphasize that, though we let $x$ denote one of the arguments of ${\bf v}^{(1)}$, it should be kept in mind
that $x$ here is restricted to the boundary of $\Omega$. The first and actually main task is to determine the
unknown scalar function $w$ (or equivalently its Fourier transform with respect to $\theta$), which will determine
the leading profile ${\bf u}^{(1)}$ at the boundary of $\Omega$. In Chapter {\rm Re }\, f{chapter4} we shall
go further in the construction of the asymptotic expansion of
${\bf u}^{\varepsilon}$.
Before going on, let us introduce two operators similar to the ones defined in \cite{BCproc}, and that arise after
performing the Fourier transform with respect to $\theta$ on ${\mathcal L}_{\rm ff}$ and $\ell_{\rm f}$:
\begin{align*}
{\bf L}^k &:= -(k\, \tau)^2 \, I +\Sigma(k\, {\boldsymbol \eta})
-i\, \big( \nu_j \, A_j(k\, {\boldsymbol \eta}) +\nu_j \, A_j(k\, {\boldsymbol \eta})^T \big) \, \partial_z -\Sigma({\boldsymbol \nu}) \, \partial^2_{zz} \, ,\\
{\bf C}^k &:= i\, \nu_j \, A_j(k\, {\boldsymbol \eta}) +\Sigma ({\boldsymbol \nu}) \, \partial_z \, ,
\end{align*}
The first corrector ${\bf u}^{(2)}$ to the leading profile ${\bf u}^{(1)}$ should satisfy the system \eqref{bkwordre2}.
In particular, the Fourier transform ${\bf v}^{(2)}$ with respect to $\theta$ of the trace ${\bf u}^{(2)}|_{x \in \partial
\Omega}$ should satisfy
\begin{equation}
\langle bel{correcteurv2}
\forall \, k \neq 0 \, ,\quad \begin{cases}
{\bf L}^k \, {\bf v}^{(2)} ={\mathbb F} (k,z) \, ,& z>0 \, ,\\
{\bf C}^k \, {\bf v}^{(2)} ={\mathbb G} (k) \, ,& z=0 \, ,
\end{cases}
\end{equation}
where the source terms ${\mathbb F},{\mathbb G}$ are defined by
\begin{equation}
\langle bel{defFG}
{\mathbb F} := {\mathcal F}_\theta {\mathbb B}ig( -{\mathcal L}_{\rm fs} \, {\bf u}^{(1)}
+\dfrac{1}{2} \, {\mathcal Q}[{\bf u}^{(1)}] {\mathbb B}ig) {\mathbb B}ig|_{x \in \partial \Omega} \, ,\quad
{\mathbb G} := \widehat{G} -{\mathcal F}_\theta \left( \ell_{\rm s} \, {\bf u}^{(1)}
+\dfrac{1}{2} \, {\mathcal M}[{\bf u}^{(1)}] \right) {\mathbb B}ig|_{x \in \partial \Omega,z=0} \, .
\end{equation}
The main problem at this stage is that both operators ${\mathcal L}_{\rm fs},\ell_{\rm s}$ involve a normal derivative
with respect to $\partial \Omega$, which led Lardner \cite{Lardner1,Lardner2} to perform substitutions of normal
derivatives in terms of tangential ones (enforcing a `compatibility' condition between the source terms ${\mathbb F},{\mathbb G}$
which precludes a secular growth phenomenon in $z$) and thereby deriving an amplitude equation for $w$
(called $\gamma$ in \cite{Lardner1,Lardner2}) along the boundary $\partial \Omega$. It turns out that these
manipulations in \cite{Lardner1,Lardner2} are {\it unnecessary} here. The only compatibility condition we need
between the source terms ${\mathbb F}$ and ${\mathbb G}$ aims at providing with the existence of a corrector ${\bf v}^{(2)}$
solution to \eqref{correcteurv2} that should be at least bounded in $z$ for all $k \neq 0$. Deriving accurate
bounds for such a corrector will be one of the main issues in the error analysis of Chapter {\rm Re }\, f{chapter4}.
Let us now recall the following duality relation which was exhibited in \cite{BCproc} (an analogous duality
relation was proved in \cite{BC} for the first order Hamiltonian formulation \eqref{hamiltonien}):
\begin{equation*}
\int_0^{+\infty} {\bf v} \cdot {\bf L}^k \, {\bf w} \, {\rm d}z - {\mathbb B}ig( {\bf v} \cdot {\bf C}^k \, {\bf w} {\mathbb B}ig) {\mathbb B}ig|_{z=0}
=\int_0^{+\infty} {\bf L}^{-k} \, {\bf v} \cdot {\bf w} \, {\rm d}z - {\mathbb B}ig( {\bf C}^{-k} \, {\bf v} \cdot {\bf w} {\mathbb B}ig) {\mathbb B}ig|_{z=0} \, ,
\end{equation*}
where we use the notation ${\bf v} \cdot {\bf v}'$ for the quantity $v_\alpha \, v'_\alpha$, and vectors have
indifferently real or complex coordinates. Actually, by translating in $z$, it is not hard to see that the same
duality relation holds on any interval $[z,+\infty)$, namely:
\begin{equation}
\langle bel{duality}
\int_z^{+\infty} {\bf v} \cdot {\bf L}^k \, {\bf w} \, {\rm d}Z - {\mathbb B}ig( {\bf v} \cdot {\bf C}^k \, {\bf w} {\mathbb B}ig) {\mathbb B}ig|_{Z=z}
=\int_z^{+\infty} {\bf L}^{-k} \, {\bf v} \cdot {\bf w} \, {\rm d}Z - {\mathbb B}ig( {\bf C}^{-k} \, {\bf v} \cdot {\bf w} {\mathbb B}ig) {\mathbb B}ig|_{Z=z}
\, .
\end{equation}
The surface wave profile $\widehat{\bf r}(k,\cdot)$ in \eqref{defw} satisfies
\begin{equation*}
\forall \, k \neq 0 \, ,\quad \begin{cases}
{\bf L}^k \, \widehat{\bf r}(k,\cdot) =0 \, ,& z>0 \, ,\\
{\bf C}^k \, \widehat{\bf r}(k,\cdot) =0 \, ,& z=0 \, ,
\end{cases}
\end{equation*}
and $\widehat{\bf r}(-k,z)=\overline{\widehat{\bf r}(k,z)}$. Applying the duality relation \eqref{duality} with $z=0$,
we find that for a `reasonable' solution ${\bf v}^{(2)}$ to the problem \eqref{correcteurv2} to exist, the source
terms ${\mathbb F},{\mathbb G}$ in \eqref{correcteurv2} should satisfy the Fredholm type condition
\begin{equation}
\langle bel{eqw1}
\forall \, k \neq 0 \, ,\quad
\int_0^{+\infty} \overline{\widehat{\bf r}(k,z)} \cdot {\mathbb F} (k,z) \, {\rm d}z -\overline{\widehat{\bf r}(k,0)} \cdot {\mathbb G} (k) =0\, .
\end{equation}
We do not make precise for the moment the meaning of `reasonable solution' for the corrector problem
\eqref{correcteurv2}. Let us merely say that for all fixed $k \neq 0$, the function ${\bf v}$ to which we apply
the duality relation \eqref{duality} is $\widehat{\bf r}(-k,z)$ and it thus has exponential decay in $z$ (with an
exponential decay rate that depends on $k$). For the integrals in \eqref{duality} to make sense, the corrector
${\bf v}^{(2)}$ should be for instance bounded in $z$ (with first two $z$-derivatives also bounded). The main
problem to avoid is to have a corrector ${\bf v}^{(2)}$ that displays exponential growth in $z$.
In what remains of this Section, we make Equation \eqref{eqw1} more explicit in terms of the scalar function $w$
that defines the trace of the leading profile ${\bf u}^{(1)}$ at the boundary $\partial \Omega$. Namely, our goal is
to prove the following result.
\begin{prop}
\langle bel{propamp}
Under Assumptions (H1), (H2), (H3), consider the source terms ${\mathbb F}$ and ${\mathbb G}$ defined by \eqref{defFG}, where
the profile ${\bf u}^{(1)}$ has the form \eqref{defw} on the boundary $\partial \Omega$, and satisfies
$$
\forall \, x \in \Omega \, ,\quad \begin{cases}
{\mathcal L}_{\rm ff} \, {\bf u}^{(1)} =0 \, ,\quad z>0 \, ,& \\
\lim_{z \to +\infty} {\bf u}^{(1)} (t,x,\theta,z) =0 \, .&
\end{cases}
$$
Then the compatibility condition \eqref{eqw1} is a closed evolution equation for the scalar function $w$. For
concreteness, let us assume furthermore that $\Omega$ is the half-space $\{ x_d >0\}$. Then \eqref{eqw1}
equivalently reads\footnote{Here the sign convention for $\tau$ plays a role. If we had chosen $\tau=\tau_{\rm r}$
rather than $\tau=-\tau_{\rm r}$, then the group velocity ${\nabla}bla_{{\boldsymbol \eta}} \tau_{\rm r}$ in \eqref{eqw} would have
been changed into its opposite.}
\begin{equation}
\langle bel{eqw}
\partial_t w +\sum_{j=1}^{d-1} \partial_{\eta_j} \tau_{\rm r} ({\boldsymbol \eta}) \, \partial_j w
+{\mathcal H} \, \big( {\mathcal B}(w,w) \big) =g \, ,
\end{equation}
where ${\mathcal H}$ denotes the Hilbert transform with respect to the variable $\theta$, ${\mathcal B}$
is the bilinear Fourier multiplier
\begin{equation}
\langle bel{defB}
\widehat{{\mathcal B}(w,w)} (k) :=-\dfrac{1}{4\, \pi \, c_0} \int_{\mathbb R} b(-k,k-\xi,\xi) \, \widehat{w}(k-\xi) \, \widehat{w}(\xi)
\, {\rm d}\xi \, ,
\end{equation}
where the constant $c_0$ is defined in Equation \eqref{defc0} below and the kernel $b$ is defined in \eqref{defnoyau}
(the slow variables $(t,x)$ enter as parameters in the definition of ${\mathcal B}$). Eventually the source term $g$ in
\eqref{eqw} depends linearly on $G$. Its expression is given in \eqref{defg}.
\end{prop}
\begin{proof}
The first main point is to clarify whether \eqref{eqw1} reads as a closed equation on the function $w$. This is
not obvious at first sight because both source terms ${\mathbb F}$ and ${\mathbb G}$ involve the trace of the normal derivative
of ${\bf u}^{(1)}$ at the boundary. It is only after taking the scalar product with $\overline{\widehat{\bf r}}$ in
\eqref{eqw1} that these normal derivatives will actually drop out. We thus first focus on the `linear' terms in
\eqref{eqw1}, meaning on all those terms in ${\mathbb F}$ and ${\mathbb G}$ where the leading profile ${\bf u}^{(1)}$ appears
linearly.
$\bullet$ \underline{The slow time derivative}. This term coincides with that obtained in \cite{BCproc}, and
we find that the only term on the left hand side of \eqref{eqw1} where a $t$-derivative of ${\bf u}^{(1)}$
appears is
\begin{align*}
\int_0^{+\infty} \overline{\widehat{\bf r}(k,z)} \cdot {\mathcal F}_\theta \left( -2\, \tau \, \partial^2_{t \theta} {\bf u}^{(1)}
\right) {\mathbb B}ig|_{x \in \partial \Omega} {\rm d}z &=-2\, i \, \tau \, \text{\rm sgn} (k) \int_0^{+\infty} |\widehat{\bf r}(1,z)|^2 \,
{\rm d}z \, \partial_t \widehat{w} \\
&=: i\, c_0 \, \text{\rm sgn} (k) \, \partial_t \widehat{w} \, ,
\end{align*}
with (recall that $\tau$ is nonzero due to Assumption (H3)):
\begin{equation}
\langle bel{defc0}
c_0 :=-2\, \tau \, \int_0^{+\infty} |\widehat{\bf r}(1,z)|^2 \neq 0 \, .
\end{equation}
$\bullet$ \underline{Slow spatial derivatives}. We now examine those terms in \eqref{eqw1} that involve a
partial derivative with respect to $x_j$. (Here $j$ is fixed and there is temporarily no summation over $j$
even though this index might be repeated.) One contribution comes from ${\mathbb F}$ and another contribution
comes from ${\mathbb G}$, giving the term
\begin{multline*}
\int_0^{+\infty} \overline{\widehat{\bf r}(k,z)} \cdot {\mathcal F}_\theta \left(
\big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, \partial^2_{j \theta} {\bf u}^{(1)}
+\big( A_j({\boldsymbol \nu}) +A_j({\boldsymbol \nu})^T \big) \, \partial^2_{j z} {\bf u}^{(1)} \right) {\mathbb B}ig|_{x \in \partial \Omega} {\rm d}z \\
+\overline{\widehat{\bf r}(k,0)} \cdot {\mathcal F}_\theta \left( A_j({\boldsymbol \nu})^T \, \partial_j \, {\bf u}^{(1)}
\right) {\mathbb B}ig|_{x \in \partial \Omega, z=0} \, .
\end{multline*}
Letting ${\bf v}_j$ denote the Fourier transform of the trace $\partial_j {\bf u}^{(1)} |_{\partial \Omega}$, the
latter term reduces to
\begin{multline}
\langle bel{eqw2}
\int_0^{+\infty} \overline{\widehat{\bf r}(k,z)} \cdot \left( i\, k \, \big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, {\bf v}_j
+\big( A_j({\boldsymbol \nu}) +A_j({\boldsymbol \nu})^T \big) \, \partial_z {\bf v}_j \right) \, {\rm d}z
+\overline{\widehat{\bf r}(k,0)} \cdot A_j({\boldsymbol \nu})^T \, {\bf v}_j |_{z=0} \\
=\int_0^{+\infty} \left( \overline{\widehat{\bf r}(k,z)} \cdot i\, k \, \big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, {\bf v}_j
+\overline{\widehat{\bf r}(k,z)} \cdot A_j({\boldsymbol \nu}) \, \partial_z {\bf v}_j -A_j({\boldsymbol \nu}) \, \partial_z \overline{\widehat{\bf r}(k,z)}
\cdot {\bf v}_j \right) \, {\rm d}z \, .
\end{multline}
The problem of course is that the $x_j$-derivative and the trace operator on $\partial \Omega$ do not
necessarily commute, so the latter term is not obviously computable in terms of the scalar function $w$.
More precisely, we can decompose the function $\partial_j {\bf u}^{(1)} |_{\partial \Omega}$ as
\begin{equation}
\langle bel{decompuj}
\partial_j {\bf u}^{(1)} |_{\partial \Omega} =\nu_j \, \partial_{\bf n} {\bf u}^{(1)} |_{\partial \Omega}
+\partial_{{\rm tan},j} {\bf u}^{(1)} |_{\partial \Omega} \, ,
\end{equation}
with $\partial_{\bf n} {\bf u}^{(1)}$ the normal derivative of ${\bf u}^{(1)}$ with respect to $\partial \Omega$ and
$\partial_{{\rm tan},j}$ a tangential vector field along $\partial \Omega$. It remains to verify that all the contributions
from the terms \eqref{eqw2} that involve the normal derivative of ${\bf u}^{(1)}$ at $\partial \Omega$ sum to zero.
Given any point $x \in \Omega$, the profile ${\bf u}^{(1)}$ should satisfy
$$
{\mathcal L}_{\rm ff} \, {\bf u}^{(1)} =0 \, ,\quad z>0 \, ,
$$
and ${\bf u}^{(1)}(z=+\infty) =0$. This means that for all $x \in \Omega$ and all $k \neq 0$, the Fourier transform
$\widehat{{\bf u}^{(1)}}(k,z)$ has exponential decay in $z$ and satisfies
$$
{\bf L}^k \, \widehat{{\bf u}^{(1)}}(k,z) =0 \, ,\quad z>0 \, .
$$
Taking the normal derivative on $\partial \Omega$, we find that the Fourier transform ${\bf v}_{\bf n}$ of
$\partial_{\bf n} {\bf u}^{(1)} |_{\partial \Omega}$ satisfies
$$
{\bf L}^k \, {\bf v}_{\bf n}(k,z) =0 \, ,\quad z>0 \, ,
$$
and ${\bf v}_{\bf n}(z=+\infty)=0$. Let us now apply the duality condition \eqref{duality} on an interval $[z,+\infty)$
with ${\bf v} :=\overline{\widehat{\bf r}(k,z)}$ and ${\bf w} :={\bf v}_{\bf n}(k,z)$. We get the relation
$$
\forall \, z >0 \, ,\quad \overline{\widehat{\bf r}(k,z)} \cdot {\bf C}^k \, {\bf v}_{\bf n}(k,z)
={\bf C}^{-k} \, \overline{\widehat{\bf r}(k,z)} \cdot {\bf v}_{\bf n}(k,z) \, ,
$$
that is
\begin{equation}
\langle bel{annulation}
\overline{\widehat{\bf r}(k,z)} \cdot {\mathbb B}ig( i\, k \, \big( \nu_\ell \, A_\ell({\boldsymbol \eta}) +\nu_\ell \, A_\ell({\boldsymbol \eta})^T \big)
+\Sigma({\boldsymbol \nu}) \, \partial_z \big) {\bf v}_{\bf n}
=\partial_z \overline{\widehat{\bf r}(k,z)} \cdot \Sigma ({\boldsymbol \nu}) \, {\bf v}_{\bf n} \, .
\end{equation}
We now use the decomposition \eqref{decompuj} in \eqref{eqw2}, and sum with respect to $j=1,\dots,d$ the
contributions involving the normal derivative ${\bf v}_{\bf n}$, which yields (here the summation convention
with respect to $j$ is used again):
\begin{multline}\langle bel{cancel}
\int_0^{+\infty} \left( \overline{\widehat{\bf r}(k,z)} \cdot i\, k \, \big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, \nu_j \, {\bf v}_{\bf n}
+\overline{\widehat{\bf r}(k,z)} \cdot \nu_j \, A_j({\boldsymbol \nu}) \, \partial_z {\bf v}_{\bf n}
-\nu _j \, A_j({\boldsymbol \nu}) \, \partial_z \overline{\widehat{\bf r}(k,z)} \cdot {\bf v}_{\bf n} \right) \, {\rm d}z \\
=\int_0^{+\infty} \left( \overline{\widehat{\bf r}(k,z)} \cdot i\, k \, \big( \nu_j \, A_j({\boldsymbol \eta}) +\nu_j \, A_j({\boldsymbol \eta})^T \big)
\, {\bf v}_{\bf n} +\overline{\widehat{\bf r}(k,z)} \cdot \Sigma({\boldsymbol \nu}) \, \partial_z {\bf v}_{\bf n}
-\Sigma ({\boldsymbol \nu}) \, \partial_z \overline{\widehat{\bf r}(k,z)} \cdot {\bf v}_{\bf n} \right) \, {\rm d}z =0 \, ,
\end{multline}
where we have used the relation (see \eqref{defASigma}):
$$
\forall \, {\boldsymbol \xi} \, ,\quad \xi_j \, A_j({\boldsymbol \xi}) =\Sigma({\boldsymbol \xi}) \, ,
$$
and the cancellation property \eqref{annulation}. In other words, we have proved that the sum with respect to
$j$ of the slow spatial derivative terms in \eqref{eqw1} reads
\begin{equation}
\langle bel{eqw3}
\partial_{{\rm tan},j} \widehat{w} \, \int_0^{+\infty}
\overline{\widehat{\bf r}(k,z)} \cdot \, i\, k \, \big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, \widehat{\bf r}(k,z)
+\overline{\widehat{\bf r}(k,z)} \cdot A_j({\boldsymbol \nu}) \, \partial_z \widehat{\bf r}(k,z)
-A_j({\boldsymbol \nu}) \, \partial_z \overline{\widehat{\bf r}(k,z)} \cdot \widehat{\bf r}(k,z) \, {\rm d}z \, .
\end{equation}
For concreteness, let us assume from now on that the half-space $\Omega$ is $\{ x_d>0\}$ so that for $j=d$,
one has $\partial_{{\rm tan},j} =0$, and for $j=1,\dots,d-1$, one has $\partial_{{\rm tan},j} =\partial_j$. (There
also holds that ${{\boldsymbol \nu}}$ is the last vector in the canonical basis of ${\mathbb R}^d$ though we do not simplify the
expressions that depend on ${\boldsymbol \nu}$ accordingly.) Then the slow spatial derivative terms arising in \eqref{eqw1}
read
$$
i\, \text{\rm sgn} (k) \, \sum_{j=1}^{d-1} c_j \, \partial_j \widehat{w} \, ,
$$
with
\begin{equation}
\langle bel{defcj}
c_j := \int_0^{+\infty} \overline{\widehat{\bf r}(1,z)} \cdot \, \big( A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T \big) \, \widehat{\bf r}(1,z)
\, {\rm d}z +2\, \text{\rm Im} \int_0^{+\infty} \overline{\widehat{\bf r}(1,z)} \cdot A_j({\boldsymbol \nu}) \,
\partial_z \widehat{\bf r}(1,z) \, {\rm d}z\, .
\end{equation}
$\bullet$ \underline{Quadratic terms}. All remaining terms in \eqref{eqw1} correspond to the contributions from the
quadratic operators ${\mathcal Q}$ in $\{ z>0 \}$ and ${\mathcal M}$ at $z=0$. These terms are exactly identical
to the ones considered in \cite{BCproc}, and we shall therefore not reproduce the (lengthy) calculations leading to
the final expression
$$
-\dfrac{1}{4\, \pi} \int_{\mathbb R} b(-k,k-\xi,\xi) \, \widehat{w}(t,x,k-\xi) \, \widehat{w}(t,x,\xi) \, {\rm d}\xi \, ,
$$
with
\begin{align}
b(\xi_1,\xi_2,\xi_3) :=& \int_0^{+\infty} d_{\alpha j \beta \ell \gamma m} \,
(\nu_j \, \rho_{\alpha,z} +i\, \xi_1 \, \eta_j \, \rho_\alpha) \, (\nu_\ell \, \rho_{\beta,z} +i\, \xi_2 \, \eta_\ell \, \rho_\beta) \,
(\nu_m \, \rho_{\gamma,z} +i\, \xi_3 \, \eta_m \, \rho_\gamma) \, {\rm d}z \, ,\langle bel{defnoyau}\\
&\rho_\alpha := \widehat{r}_\alpha (\xi_1,z) \, ,\quad \rho_\beta := \widehat{r}_\beta (\xi_2,z) \, ,\quad
\rho_\gamma := \widehat{r}_\gamma (\xi_3,z) \, ,\notag
\end{align}
and the subscript `$,z$' denotes differentiation with respect to $z$.
At this stage, we have found that the amplitude equation \eqref{eqw1} reads
\begin{equation*}
i\, c_0 \, \text{\rm sgn} (k) \, \partial_t \widehat{w} +i\, \text{\rm sgn} (k) \, \sum_{j=1}^{d-1} c_j \, \partial_j \widehat{w}
-\dfrac{1}{4\, \pi} \int_{\mathbb R} b(-k,k-\xi,\xi) \, \widehat{w}(t,x,k-\xi) \, \widehat{w}(t,x,\xi) \, {\rm d}\xi
=\overline{\widehat{\bf r}(k,0)} \cdot \widehat{G} (k)\, ,
\end{equation*}
where the constants $c_0,\dots,c_{d-1}$ are defined in \eqref{defc0}, \eqref{defcj}, and the kernel $b$ is given
in \eqref{defnoyau}. Multiplying by $-i \, \text{\rm sgn} (k) \, c_0^{-1}$, and taking the inverse Fourier transform
in $\theta$, we find that weakly nonlinear high frequency solutions to \eqref{eqint}, \eqref{clinhom} are governed,
at the leading order, by the amplitude equation
\begin{equation}
\langle bel{eqw5}
\partial_t w +\sum_{j=1}^{d-1} \dfrac{c_j}{c_0} \, \partial_j w +{\mathcal H} \, \big( {\mathcal B}(w,w) \big) =g \, ,
\end{equation}
where, as announced in the statement of Proposition {\rm Re }\, f{propamp}, ${\mathcal H}$ denotes the Hilbert transform
with respect to the fast variable $\theta$ (namely, $\widehat{{\mathcal H} \, f}(k) := -i \, \text{\rm sgn} (k) \, \widehat{f}
(k)$), and ${\mathcal B}$ is the bilinear Fourier multiplier defined in \eqref{defB}. The source term $g$ in \eqref{eqw5}
is obtained by setting
\begin{equation}
\langle bel{defg}
\widehat{g} (k):=-\dfrac{i\, \text{\rm sgn} (k)}{c_0} \, \overline{\widehat{\bf r}(k,0)} \cdot \widehat{G} (k)\, ,
\end{equation}
which defines a real valued function $g$ provided that $G$ is real valued, which was tacitly assumed in
\eqref{clinhom}.
$\bullet$ \underline{Simplifying the linear terms}. We shall have therefore proved Proposition {\rm Re }\, f{propamp}
provided that we get the relations
\begin{equation}
\langle bel{vitessegroupe}
\forall \, j=1,\dots,d-1 \, ,\quad \dfrac{c_j}{c_0} =\partial_{\eta_j} \tau_{\rm r} ({\boldsymbol \eta}) \, ,
\end{equation}
which express that the propagation of the amplitude $w$ along the boundary $\partial \Omega$ is governed by the
{\it group velocity} of the variety along which the Lopatinskii determinant $\partialelta$ vanishes. The proof of the relations
\eqref{vitessegroupe} follows some arguments that already appeared in \cite{BC} for the first-order Hamiltonian
formulation \eqref{hamiltonien}, and which we adapt\footnote{It is also the opportunity to correct some (harmless)
normalizing mistakes that appeared in \cite{BC}.} here to the second-order formulation \eqref{normalmode}.
From Assumption (H3), we know that the matrix ${\bf J} \, H(\tau,{\boldsymbol \eta})$ is diagonalizable and hyperbolic, the
diagonalization being locally analytic in the frequencies $(\tau,{\boldsymbol \eta})$. For clarity, we denote from now on the
reference frequency in \eqref{wkbu} $(\underline{\tau},\underline{{\boldsymbol \eta}})$ and keep the notation $(\tau,{\boldsymbol \eta})$
for real frequencies that are close to $(\underline{\tau},\underline{{\boldsymbol \eta}})$, though not necessarily linked by the
dispersion relation $\tau+\tau_{\rm r}({\boldsymbol \eta})=0$. We can choose a smooth basis $({\bf R}_1,\dots,{\bf R}_{2\, N})
(\tau,{\boldsymbol \eta})$ of ${\mathbb C}^{2\, N}$ such that
\begin{equation}
\langle bel{vecteurspropres}
\forall \, \alpha=1,\dots,N \, ,\quad \begin{cases}
{\bf J} \, H(\tau,{\boldsymbol \eta}) \, {\bf R}_\alpha (\tau,{\boldsymbol \eta}) =-\omega_\alpha (\tau,{\boldsymbol \eta})
\, {\bf R}_\alpha (\tau,{\boldsymbol \eta}) \, ,& \\
{\bf J} \, H(\tau,{\boldsymbol \eta}) \, {\bf R}_{N+\alpha} (\tau,{\boldsymbol \eta})
=\overline{\omega_\alpha (\tau,{\boldsymbol \eta})} \, {\bf R}_{N+\alpha} (\tau,{\boldsymbol \eta}) \, ,&
\end{cases}
\end{equation}
with the normalization convention
\begin{equation*}
\begin{pmatrix}
{\bf R}_1 & \cdots & {\bf R}_{2\, N} \end{pmatrix}^* \, {\bf J} \, \begin{pmatrix}
{\bf R}_1 & \cdots & {\bf R}_{2\, N} \end{pmatrix} =-{\bf J} \, .
\end{equation*}
Defining some vectors ${\bf L}_\alpha$ by
\begin{equation*}
\forall \, \alpha=1,\dots,2\, N \, ,\quad {\bf L}_\alpha (\tau,{\boldsymbol \eta}) :=\begin{cases}
{\bf R}_{N+\alpha} \, ,&\text{\rm if } \alpha \le N \, ,\\
-{\bf R}_{\alpha-N} \, ,&\text{\rm if } \alpha>N \, ,
\end{cases}
\end{equation*}
we get the orthogonality relations
\begin{equation}
\langle bel{orthogonalite}
\forall \, \alpha,\beta =1,\dots,2\, N \, ,\quad {\bf L}_\alpha^* \, {\bf J} \, {\bf R}_\beta =\delta_{\alpha \beta} \, ,
\end{equation}
which hold for all $(\tau,{\boldsymbol \eta})$ close to $(\underline{\tau},\underline{{\boldsymbol \eta}})$, though we shall only use
them at the reference frequency $(\underline{\tau},\underline{{\boldsymbol \eta}})$. The Lopatinskii determinant is then
defined\footnote{Observe that the value of $\partialelta$ depends on the basis of ${\mathbb E}^s$ with which it is defined
but the location of the roots and their multiplicity does not, which is why we can equivalently define $\partialelta$
with our basis $({\bf R}_\alpha)$.} as
$$
\partialelta (\tau,{\boldsymbol \eta}) := \det \big( S_1(\tau,{\boldsymbol \eta}) \cdots S_N(\tau,{\boldsymbol \eta}) \big) \, ,\quad
{\bf R}_\alpha :=\begin{pmatrix}
R_\alpha \\ S_\alpha \end{pmatrix} \, .
$$
In what follows, we decompose all vectors ${\bf R}_\alpha \in {\mathbb C}^{2\, N}$ as above, and not only those
corresponding to $\alpha=1,\dots,N$. Underlined quantities refer to evaluation at the frequency
$(\underline{\tau},\underline{{\boldsymbol \eta}})$. Since the vectors $\underline{S}_1,\dots,\underline{S}_N$ span
an $N-1$-dimensional subspace in ${\mathbb C}^N$, we can fix a nonzero vector $(\langle mbda_1,\dots,\langle mbda_N)$
in ${\mathbb C}^N$ such that (here we use the convention $\langle mbda_\gamma=0$ if $\gamma \ge N+1$):
\begin{equation}
\langle bel{lopfaible}
\langle mbda_\gamma \, \underline{S}_\gamma =0 \, .
\end{equation}
There is no loss of generality in assuming $\langle mbda_1 \neq 0$ and even in normalizing the $\langle mbda_\gamma$'s
by assuming $\langle mbda_1=1$, which means that $\underline{S}_2,\dots,\underline{S}_N$ are linearly
independent. In that case, the surface wave $V$ which appears in the definition \eqref{defw} of
$\widehat{{\bf r}}$ reads\footnote{Or at least we can fix it this way, since $V$ is defined up to a nonzero
multiplicative constant.}
\begin{equation}
\langle bel{decompV}
V(z) =\langle mbda_\gamma \, {\rm e}^{-\underline{\omega}_\gamma \, z} \, \underline{R}_\gamma \, .
\end{equation}
We now observe that the two linear forms
\begin{equation*}
X \in {\mathbb C}^N \longmapsto \det \big( X \, \, \underline{S}_2 \cdots \underline{S}_N \big) \, ,\quad
X \in {\mathbb C}^N \longmapsto (\langle mbda_\gamma \, \underline{R}_\gamma)^* \, X \, ,
\end{equation*}
are nonzero and vanish on the hyperplane spanned by $\underline{S}_1,\dots,\underline{S}_N$ (use
\eqref{lopfaible} and the orthogonality relations \eqref{orthogonalite} for the latter). Hence there exists
a nonzero constant $\kappa \in {\mathbb C}$ such that
\begin{equation*}
\forall \, X \in {\mathbb C}^N \, ,\quad \det \big( X \, \, \underline{S}_2 \cdots \underline{S}_N \big)
=\kappa \, (\langle mbda_\gamma \, \underline{R}_\gamma)^* \, X \, .
\end{equation*}
Let us assume for simplicity that the half-space $\Omega$ is $\{ x_d>0\}$, which means that the tangential
wave vectors ${\boldsymbol \eta}$ are parametrized by $\eta_1,\dots,\eta_{d-1}$. Then the partial derivative $\partial \partialelta$
of $\partialelta$ with respect to any of the variables $\tau,\eta_1,\dots,\eta_{d-1}$ is given by
$$
\partial \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =\det \big( \langle mbda_\alpha \,
\partial S_\alpha (\underline{\tau},\underline{{\boldsymbol \eta}}) \, \, \underline{S}_2 \cdots \underline{S}_N \big)
=\kappa \, \langle mbda_\alpha \, (\langle mbda_\gamma \, \underline{R}_\gamma)^* \,
\partial S_\alpha (\underline{\tau},\underline{{\boldsymbol \eta}}) \, .
$$
For each partial derivative $\partial$, with respect to either of the variables $\tau,\eta_1,\dots,\eta_{d-1}$,
we follow \cite{BC} and decompose
\begin{equation}
\langle bel{decompderivee}
\forall \, \alpha=1,\dots,N \, ,\quad
\partial {\bf R}_\alpha (\underline{\tau},\underline{{\boldsymbol \eta}}) =\mu_{\alpha \beta} \, \underline{{\bf R}}_\beta \, ,
\end{equation}
where summation with respect to $\beta$ includes all indices $\beta=1,\dots,2\, N$. We thus get the expression
$$
\partial \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}})
=\kappa \, \langle mbda_\alpha \, \mu_{\alpha \beta} \, (\langle mbda_\gamma \, \underline{R}_\gamma)^*
\, \underline{S}_\beta \, ,
$$
where summation runs over $\alpha,\gamma=1,\dots,N$ (recall $\langle mbda_\alpha=0$ for $\alpha \ge N+1$) and
$\beta=N+1,\dots,2\, N$ (because of the orthogonality relation $(\langle mbda_\gamma \, \underline{R}_\gamma)^* \,
\underline{S}_\beta=0$ for all $\beta=1,\dots,N$). We thus need to determine the coefficients $\mu_{\alpha \beta}$
in \eqref{decompderivee} for $\alpha \le N$ and $\beta \ge N+1$. These coefficients are obtained by differentiating
\eqref{vecteurspropres} and using the orthogonality relations \eqref{orthogonalite}. We get
$$
\forall \, \alpha =1,\dots,N \, ,\quad \forall \, \beta =N+1,\dots,2\, N \, ,\quad
\mu_{\alpha \beta} =\dfrac{\underline{{\bf L}}_\beta^* \, \partial H
(\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_{\beta-N}}}
=-\dfrac{\underline{{\bf R}}_{\beta-N}^* \, \partial H
(\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_{\beta-N}}} \, ,
$$
and therefore
$$
\partial \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =-\kappa \, \langle mbda_\alpha \,
\dfrac{\underline{{\bf R}}_\beta^* \, \partial H (\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}} \, (\langle mbda_\gamma \, \underline{R}_\gamma)^*
\, \underline{S}_{\beta+N} \, ,
$$
where summation now runs over $\alpha,\beta,\gamma=1,\dots,N$. It remains to observe that we have the relation
(use \eqref{lopfaible}):
$$
\forall \, \beta=1,\dots,N \, ,\quad
(\langle mbda_\gamma \, \underline{R}_\gamma)^* \, \underline{S}_{\beta+N}
= (\langle mbda_\gamma \, {\bf R}_\gamma)^* \, {\bf J} \, {\bf R}_{\beta+N}
= -(\langle mbda_\gamma \, {\bf L}_{\gamma+N})^* \, {\bf J} \, {\bf R}_{\beta+N} =-\overline{\langle mbda_\beta} \, ,
$$
and the partial derivative of the Lopatinskii determinant thus reduces to its final expression
\begin{equation}
\langle bel{deriveedelta}
\partial \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =\kappa \, \langle mbda_\alpha \, \overline{\langle mbda_\beta} \,
\dfrac{\underline{{\bf R}}_\beta^* \, \partial H (\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}} \, .
\end{equation}
We first use the relation \eqref{deriveedelta} for the $\tau$-partial derivative, and recall the expression
\eqref{defJH} of $H$:
\begin{equation*}
\partial_\tau \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =2\, \tau \, \kappa \, \langle mbda_\alpha \,
\overline{\langle mbda_\beta} \, \dfrac{\underline{R}_\beta^* \, \underline{R}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}}
= 2\, \tau \, \kappa \, \int_0^{+\infty} |\widehat{{\bf r}}(1,z)|^2 \, {\rm d}z
= -\kappa \, c_0 \, .
\end{equation*}
The computation of $\partial_{\eta_j} \partialelta$ is slightly more complicated but not too troublesome either.
We differentiate $H$ in \eqref{defJH} with respect to $\eta_j$ and use the definitions of $A_j,\Sigma$ in
\eqref{defASigma} to get
\begin{align*}
\underline{{\bf R}}_\beta^* \, \partial_{\eta_j} H (\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha
=&-\underline{R}_\beta^* \, (A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T) \, \underline{R}_\alpha \\
&+\underline{R}_\beta^* \, A_j({\boldsymbol \nu}) \, \Sigma({\boldsymbol \nu})^{-1} \, (\nu_\ell \, A_\ell({\boldsymbol \eta})) \, \underline{R}_\alpha
+\underline{R}_\beta^* \, (\nu_\ell \, A_\ell({\boldsymbol \eta})^T) \, \Sigma({\boldsymbol \nu})^{-1} \, A_j({\boldsymbol \nu})^T \, \underline{R}_\alpha \\
&+i\, \underline{R}_\beta^* \, A_j({\boldsymbol \nu}) \, \Sigma({\boldsymbol \nu})^{-1} \, \underline{S}_\alpha
-i\, \underline{S}_\beta^* \, \Sigma({\boldsymbol \nu})^{-1} \, A_j({\boldsymbol \nu})^T \, \underline{R}_\alpha \, .
\end{align*}
Recall that we are interested here in indices $\alpha,\beta$ between $1$ and $N$, so we can use the relations
(which express that ${\bf R}_\gamma$ is an eigenvector of ${\bf J}\, H$ for the eigenvalue $-\omega_\gamma$):
$$
\underline{S}_\alpha =(i\, \nu_\ell \, A_\ell({\boldsymbol \eta}) -\underline{\omega}_\alpha \, \Sigma({\boldsymbol \nu})) \,
\underline{R}_\alpha \, ,\quad
\underline{S}_\beta =(i\, \nu_\ell \, A_\ell({\boldsymbol \eta}) -\underline{\omega}_\beta \, \Sigma({\boldsymbol \nu})) \,
\underline{R}_\beta \, .
$$
This simplifies the above expression of $\underline{{\bf R}}_\beta^* \, \partial_{\eta_j} H
(\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha$ accordingly:
\begin{equation*}
\underline{{\bf R}}_\beta^* \, \partial_{\eta_j} H (\underline{\tau},\underline{{\boldsymbol \eta}}) \, \underline{{\bf R}}_\alpha
=-\underline{R}_\beta^* \, (A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T) \, \underline{R}_\alpha
-i\, \underline{\omega}_\alpha \, \underline{R}_\beta^* \, A_j({\boldsymbol \nu}) \, \underline{R}_\alpha
+i\, \overline{\underline{\omega}_\beta} \, \underline{R}_\beta^* \, A_j({\boldsymbol \nu})^T \, \underline{R}_\alpha \, .
\end{equation*}
Plugging this expression in \eqref{deriveedelta}, we obtain
\begin{align*}
\partial_{\eta_j} \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =& -\kappa \, \langle mbda_\alpha \, \overline{\langle mbda_\beta} \,
\dfrac{\underline{R}_\beta^* \, (A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T) \, \underline{R}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}} \\
&-i\, \kappa \, \langle mbda_\alpha \, \overline{\langle mbda_\beta} \, \underline{\omega}_\alpha \,
\dfrac{\underline{R}_\beta^* \, A_j({\boldsymbol \nu}) \, \underline{R}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}}
+i\, \kappa \, \langle mbda_\alpha \, \overline{\langle mbda_\beta} \, \overline{\underline{\omega}_\beta} \,
\dfrac{\underline{R}_\beta^* \, A_j({\boldsymbol \nu})^T \, \underline{R}_\alpha}
{\underline{\omega}_\alpha +\overline{\underline{\omega}_\beta}} \\
=&-\kappa \, \int_0^{+\infty} \widehat{{\bf r}}(1,z)^* \, (A_j({\boldsymbol \eta}) +A_j({\boldsymbol \eta})^T) \, \widehat{{\bf r}}(1,z) \, {\rm d}z
-2\, \kappa \, \text{\rm Im} \int_0^{+\infty} \widehat{{\bf r}}(1,z)^* \, A_j({\boldsymbol \nu}) \, \partial_z \widehat{{\bf r}}(1,z) \, {\rm d}z \, ,
\end{align*}
and we thus find the relation $\partial_{\eta_j} \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}}) =-\kappa \, c_j$.
In other words, we have found the relation
$$
\forall \, j=1,\dots,d-1 \, ,\quad \dfrac{c_j}{c_0} =
\dfrac{\partial_{\eta_j} \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}})}
{\partial_\tau \partialelta (\underline{\tau},\underline{{\boldsymbol \eta}})} \, ,
$$
and the claim of Proposition {\rm Re }\, f{propamp} follows from the factorization (which holds near the simple root
$(\underline{\tau},\underline{{\boldsymbol \eta}})$ of $\partialelta$, and here the sign convention for $\tau$ plays a role):
$$
\partialelta (\tau,{\boldsymbol \eta}) =\vartheta (\tau,{\boldsymbol \eta}) \, (\tau+\tau_{\rm r}({\boldsymbol \eta})) \, ,\quad
\vartheta (\underline{\tau},\underline{{\boldsymbol \eta}}) \neq 0 \, .
$$
We thus get $c_j/c_0 =\partial_{\eta_j} \tau_{\rm r}(\underline{{\boldsymbol \eta}})$, which yields the final form of the amplitude
equation \eqref{eqw}.
\end{proof}
Let us recall that our goal is to construct the asymptotic expansion \eqref{wkbu} of the solution ${\bf u}^{\varepsilon}$
to the high frequency problem \eqref{eqint}, \eqref{clinhom}. At this stage, we have shown that the leading order
amplitude ${\bf u}^{(1)}$ is given on the boundary $\partial \Omega$ by \eqref{defw} and that for a `reasonable'
corrector ${\bf u}^{(2)}$ in \eqref{wkbu} to exist, the amplitude function $w$ should solve the equation
\eqref{eqw}. In order to proceed and construct the first term in the asymptotic expansion \eqref{wkbu}, the main
question is to investigate the well-posedness properties of the equation \eqref{eqw}. This will rely on earlier
results by Hunter and Benzoni-Gavage \cite{Hunter2006,B}. Once we have $w$, the leading order term
${\bf u}^{(1)}$ is defined on the boundary $\partial \Omega$ and, following \cite{Le,Ma,MetivierCL}, one
possibility for defining ${\bf u}^{(1)}$ in the whole space domain $\Omega$ is simply to let
$$
{\bf u}^{(1)}(t,x,\theta,z) := \chi (x \cdot {\boldsymbol \nu}) \, {\bf u}^{(1)} (t,y,\theta,z) \, ,
$$
where $y$ denotes the orthogonal projection of $x$ on $\partial \Omega$, and the function $\chi \in
{\mathcal C}^\infty({\mathbb R})$ has compact support and satisfies $\chi(0)=1$. There are two questions then: can
we construct indeed a corrector ${\bf u}^{(2)}$ that satisfies \eqref{bkwordre2} ? and, provided that the exact
solution ${\bf u}^{\varepsilon}$ to \eqref{eqint}, \eqref{clinhom} exists on a fixed time interval independent of ${\varepsilon}$,
does \eqref{wkbu} provide with an accurate description of ${\bf u}^{\varepsilon}$ on this time interval ? These
questions are answered positively in Chapters {\rm Re }\, f{chapter3}, {\rm Re }\, f{chapter4}, and {\rm Re }\, f{chapter5}. But before entering
analytical issues, we are going to focus on how the previous analysis applies to the case of elastodynamics.
\section{Isotropic elastodynamics}\langle bel{isoe}
\emph{\quad} In this section we explain how the previous analysis applies to the system of elastodynamics for hyperelastic
materials. The results generalize those of \cite{Lardner1} to any space dimension, and put the leading order
amplitude equation in the form \eqref{eqw} which is more convenient in view of applying the well-posedness
results of \cite{Hunter2006,B}. We refer to \cite{ciarlet} for an introduction to elasticity and the physical
background. More specifically, we consider an elastic material in the reference domain $\Omega = \{ x \cdot
{\boldsymbol \nu} >0 \} \subset {\mathbb R}^d$. The deformation gradient is ${\nabla}bla \varphi =I+{\nabla}bla {\bf u}$, where ${\bf u}(t,x) \in
{\mathbb R}^d$ represents, at a given time $t$, the displacement of the material at a point $x \in \Omega$. The space
dimension $d$ equals either $2$ or $3$. Due to frame indifference, the elastic energy $W$ is a function of the
so-called Cauchy-Green strain tensor $C :={\nabla}bla \varphi^T \, {\nabla}bla \varphi \in {\mathcal M}_d({\mathbb R})$. We may
equivalently rewrite $W$ in terms of the Green - Saint Venant strain tensor $E :=(C-I)/2$. In terms of the
displacement gradient, this gives
$$
\forall \, \alpha,j =1,\dots,d \, ,\quad
E_{\alpha,j} = \dfrac{1}{2} \, (u_{\alpha,j} +u_{j,\alpha} +u_{\alpha,\ell} \, u_{j,\ell}) \, .
$$
When the material is {\it isotropic}, the energy $W$ only depends on the principal invariants of the Cauchy-Green
strain tensor $C$. Assuming that the energy $W$ is a smooth function of $E$, its Taylor expansion at $E=0$
then reads (see \cite[Chapter 1.4]{ciarlet}):
\begin{align}\langle bel{generalenergy}
W(E) =\dfrac{\langle mbda}{2} \, ({\rm tr } \, E)^2 +\mu \, {\rm tr } \, (E^2)
+\alpha_1 \, ({\rm tr } \, E)^3 +\alpha_2 \, ({\rm tr } \, E) \, {\rm tr } \, (E^2) +\alpha_3 \, {\rm tr } \, (E^3)
+O(\| E \|^4) \, .
\end{align}
In the above formula, $\langle mbda$ and $\mu$ stand for the so-called Lam\'e coefficients of the material,
and $\alpha_1,\alpha_2,\alpha_3$ are constants. For future use, we introduce the Frobenius norm of a
matrix $M \in {\mathcal M}_d({\mathbb R})$:
$$
\| M \|^2 := \sum_{\alpha,j=1}^d M_{\alpha j}^2 ={\rm tr } \, (M \, M^T) \, .
$$
Using the expression of $E$ in terms of the displacement gradient ${\nabla}bla {\bf u}$, we can rewrite the
above Taylor expansion of $W$ as
\begin{align}
W({\nabla}bla {\bf u}) =& \, \dfrac{\langle mbda}{2} \, ({\rm tr } \, {\nabla}bla {\bf u})^2
+\dfrac{\mu}{4} \, \left\| {\nabla}bla {\bf u} +{\nabla}bla {\bf u}^T \right\|^2 \notag \\
&+\beta_1 \, ({\rm tr } \, {\nabla}bla {\bf u})^3
+\beta_2 \, ({\rm tr } \, {\nabla}bla {\bf u}) \, \left\| {\nabla}bla {\bf u} +{\nabla}bla {\bf u}^T \right\|^2
+\beta_3 \, {\rm tr } \, ({\nabla}bla {\bf u} \, {\nabla}bla {\bf u}^T \, {\nabla}bla {\bf u})
+\beta_4 \, {\rm tr } \, ({\nabla}bla {\bf u}^3 ) +O(\| {\nabla}bla {\bf u} \|^4) \, .\langle bel{taylorW}
\end{align}
For the sake of completeness, the coefficients $\beta_1,\dots,\beta_4$ in \eqref{taylorW} are given by:
\begin{equation*}
\beta_1 := \alpha_1 \, ,\quad \beta_2 := \dfrac{\langle mbda}{2} +\dfrac{\alpha_2}{4} \, ,\quad
\beta_3 := \mu +\dfrac{3\, \alpha_3}{4} \, ,\quad \beta_4 := \dfrac{\alpha_3}{4} \, .
\end{equation*}
Their precise expression is not relevant for our purpose though. Let us note however that, even in the
simplest case of the so-called Saint Venant - Kirchhoff materials, for which one has
\begin{align}\langle bel{SVK}
\forall \, E \, ,\quad W(E) =\dfrac{\langle mbda}{2} \, ({\rm tr } \, E)^2 +\mu \, {\rm tr } \, (E^2) \, ,
\end{align}
the coefficients $\beta_1,\dots,\beta_4$ in the third order terms of the Taylor expansion of $W({\nabla}bla {\bf u})$
are not all zero. As a matter of fact, there is not much simplification in the final expression of the amplitude
equation if one assumes that one/some of the coefficients $\beta_1,\dots,\beta_4$ are zero, so we keep
them all in what follows\footnote{The only real simplification occurs when $\beta_2,\beta_3,\beta_4$ are all
zero, but this is incompatible with the definition of these coefficients since $\mu$ is nonzero and therefore
$\beta_3,\beta_4$ cannot vanish simultaneously.}.
From now on, we consider the previous derivation of the amplitude equation \eqref{eqw} when $N=d \ge 2$
and the energy $W$ satisfies \eqref{taylorW} near ${\nabla}bla {\bf u} =0$. Let us first take a look at assumptions
(H1), (H2), (H3). Assumption (H1) is trivially fulfilled since the Taylor expansion of $W$ starts at the second
order. Furthermore, we compute\footnote{Some of the expressions below already appeared in \cite{BC} so
we do not reproduce all the computations but rather give the expressions that are useful for our purpose.}:
$$
c_{\alpha j \beta \ell} =\langle mbda \, \delta_{\alpha j} \, \delta_{\beta \ell}
+\mu \, (\delta_{\alpha \beta} \, \delta_{j \ell} +\delta_{\alpha \ell} \, \delta_{\alpha j}) \, ,
$$
with $\delta$ the Kronecker symbol ($\delta_{n_1 n_2}=1$ if $n_1=n_2$, zero otherwise). The above
expression of the coefficients $c_{\alpha j \beta \ell}$ gives:
$$
\forall \, {\boldsymbol \xi},v \in {\mathbb R}^d \, ,\quad c_{\alpha j \beta \ell} \, v_\alpha \, \xi_j \, v_\beta \, \xi_\ell
=\mu \, |{\boldsymbol \xi}|^2 \, |v|^2 +(\langle mbda +\mu) \, (v \cdot {\boldsymbol \xi})^2 \, .
$$
The fulfillment of Assumption (H2) is then equivalent to the well-known inequalities
$$
\mu>0 \quad \text{ and } \quad \langle mbda+2\, \mu>0 \, .
$$
One can then define the velocity of `shear' and `pressure' waves by
$$
c_S := \sqrt{\mu} \, ,\quad c_P := \sqrt{\langle mbda +2\, \mu} \, .
$$
As has now long been known, see e.g. \cite{Lardner1,SerreJFA}, the fulfillment of Assumption (H3) is equivalent
to the additional requirement $c_P>c_S$, or equivalently $\langle mbda+\mu>0$. In that case, one may uniquely
define a velocity $c_R \in (0,c_S)$ by solving the polynomial equation
\begin{equation}
\langle bel{defcR}
\left( \dfrac{c_R^2}{2\, c_S^2} -1 \right)^4 =\left( 1-\dfrac{c_R^2}{c_S^2} \right) \, \left( 1-\dfrac{c_R^2}{c_P^2}
\right) \, .
\end{equation}
Assumption (H3) is then satisfied with $\tau_{\rm r}({\boldsymbol \eta}) :=c_R \, |{\boldsymbol \eta}|$ for all tangential wave vector
${\boldsymbol \eta}$ (which obviously satisfies the homogeneity property $\tau_{\rm r}(k\, {\boldsymbol \eta})=k\, \tau_{\rm r}({\boldsymbol \eta})$
for any $k>0$). Given a nonzero tangential wave vector ${\boldsymbol \eta}$, the one-dimensional family of surface waves
solution to \eqref{eqlin} is spanned by
\begin{equation}
\langle bel{Rayleighwaves}
{\rm e}^{\pm i \, \tau_{\rm r}({\boldsymbol \eta}) \, t+i \, {\boldsymbol \eta} \cdot x} \, V({\boldsymbol \nu} \cdot x) \, ,\quad
V(z) :={\rm e}^{-\omega_1 \, z} \, \big( -2\, i \, \omega_1 \, \omega_2 \, {\boldsymbol \eta}+2\, \omega_2 \, |{\boldsymbol \eta}|^2 \, {\boldsymbol \nu} \big)
+{\rm e}^{-\omega_2 \, z} \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, \big( i \, {\boldsymbol \eta} -\omega_2 \, {\boldsymbol \nu} \big) \, .
\end{equation}
The eigenmodes $\omega_{1,2}$ in \eqref{Rayleighwaves} are defined by:
\begin{align}\langle bel{emodes}
\omega_1 := |{\boldsymbol \eta}| \, \sqrt{1-\dfrac{c_R^2}{c_S^2}} \, ,\quad
\omega_2 := |{\boldsymbol \eta}| \, \sqrt{1-\dfrac{c_R^2}{c_P^2}} \, ,
\end{align}
and thus satisfy
\begin{equation}
\langle bel{dispersionRayleigh}
0<\omega_1<\omega_2 \, ,\quad 4\, |{\boldsymbol \eta}|^2 \, \omega_1 \, \omega_2 =\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big)^2 \, .
\end{equation}
The latter equality is an equivalent form of \eqref{defcR}.
Our goal now is to identify the amplitude equation \eqref{eqw} with the function $V$ given by \eqref{Rayleighwaves}
(and the corresponding $\widehat{r}(k,z)$ defined by \eqref{defw}). We first compute the group velocity
$$
\partial_{\eta_j} \tau_{\rm r}({\boldsymbol \eta}) =c_R \, \dfrac{\eta_j}{|{\boldsymbol \eta}|} \, .
$$
Specifying from now on to the case $\Omega = \{ x_d>0 \}$, that is ${\boldsymbol \nu} =(0,\dots,0,1)^T$, the amplitude
equation \eqref{eqw} reads:
$$
\partial_t w +c_R \, \dfrac{{\boldsymbol \eta}}{|{\boldsymbol \eta}|} \cdot {\nabla}bla_x w +{\mathcal H} \, \big( {\mathcal B}(w,w) \big) =g \, ,
$$
with ${\mathcal B}$ given by \eqref{defB}. Let us recall that the tangential wave vector ${\boldsymbol \eta}$ has the form
$(\eta_1,\dots,\eta_{d-1},0)^T$, and the amplitude $w$, which is defined on the boundary $\partial \Omega$
depends on $(t,x_1,\dots,x_{d-1},\theta)$. Let us examine more closely at the expression of the bilinear
Fourier multiplier ${\mathcal B}$.
$\bullet$ \underline{The constant $c_0$.} It is defined by \eqref{defc0}. With the definition \eqref{Rayleighwaves}
of $V$, we compute\footnote{Recall that we consider here the case $\tau=-\tau_{\rm r}({\boldsymbol \eta})=-c_R \, |{\boldsymbol \eta}|$.}
\begin{align}
c_0 &=-2\, \tau \, \int_0^{+\infty} |V(z)|^2 \, {\rm d}z \notag \\
&= -2\, \tau \, \left\{ \dfrac{4\, \omega_2^2 \, |{\boldsymbol \eta}|^2 \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big)}{2\, \omega_1}
+\dfrac{\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big)^2 \, \big( \omega_2^2 +|{\boldsymbol \eta}|^2 \big)}{2\, \omega_2}
-4\, \omega_2 \, |{\boldsymbol \eta}|^2 \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \right\} \notag \\
&=-4\, \tau \, |{\boldsymbol \eta}|^2 \, (\omega_2 -\omega_1) \left\{ |{\boldsymbol \eta}|^2 \, \left( \dfrac{\omega_2}{\omega_1} -1 \right)
+2\, \omega_1 \, \omega_2 \right\} \, ,\langle bel{expressionc0}
\end{align}
where we have used the dispersion relation \eqref{dispersionRayleigh} to obtain the final expression of $c_0$.
The expression \eqref{expressionc0} gives the constant in the definition \eqref{defB} of ${\mathcal B}$. It
remains to clarify the expression of the kernel $b$, which is given by \eqref{defnoyau}. The first task is to
compute the coefficients $d_{\alpha j \beta \ell \gamma m}$.
$\bullet$ \underline{The coefficients $d_{\alpha j \beta \ell \gamma m}$.} They are obtained as the third order
derivatives of $W$ at $0$, see \eqref{defcoeff}. Using the Taylor expansion \eqref{taylorW} of $W$, we get
the decomposition
$$
d_{\alpha j \beta \ell \gamma m} =6\, \beta_1 \, d_{\alpha j \beta \ell \gamma m}^1
+2\, \beta_2 \, d_{\alpha j \beta \ell \gamma m}^2 +\beta_3 \, d_{\alpha j \beta \ell \gamma m}^3
+3\, \beta_4 \, d_{\alpha j \beta \ell \gamma m}^4 \, ,
$$
with
\begin{align}
d_{\alpha j \beta \ell \gamma m}^1 &= \delta_{\alpha j} \, \delta_{\beta \ell} \, \delta_{\gamma m}
\, ,\langle bel{noyau1} \\
d_{\alpha j \beta \ell \gamma m}^2 &= \delta_{\alpha j} \, \delta_{\beta \gamma} \, \delta_{\ell m}
+\delta_{\beta \ell} \, \delta_{\alpha \gamma} \, \delta_{j m}
+\delta_{\gamma m} \, \delta_{\alpha \beta} \, \delta_{j \ell} \, ,\langle bel{noyau2} \\
d_{\alpha j \beta \ell \gamma m}^3 &=
\delta_{\alpha \beta} \, (\delta_{\gamma j} \, \delta_{\ell m} +\delta_{\gamma \ell} \, \delta_{j m})
+\delta_{\alpha \gamma} \, (\delta_{\beta j} \, \delta_{\ell m} +\delta_{\beta m} \, \delta_{j \ell})
+\delta_{\beta \gamma} \, (\delta_{\alpha \ell} \, \delta_{j m} +\delta_{\alpha m} \, \delta_{j \ell}) \, ,\langle bel{noyau3} \\
d_{\alpha j \beta \ell \gamma m}^4 &= \delta_{\alpha \ell} \, \delta_{\beta m} \, \delta_{\gamma j}
+\delta_{\alpha m} \, \delta_{\beta j} \, \delta_{\gamma \ell} \, .\langle bel{noyau4}
\end{align}
Each of these four coefficients is invariant through the permutations $(\alpha,j) \leftrightarrow (\beta,\ell)$,
$(\alpha,j) \leftrightarrow (\gamma,m)$ and $(\beta,\ell) \leftrightarrow (\gamma,m)$, and therefore through
any permutation of $(\alpha,j)$, $(\beta,\ell)$, $(\gamma,m)$. Associated with the above decomposition
for $d_{\alpha j \beta \ell \gamma m}$, we have a decomposition of the kernel $b(\xi_1,\xi_2,\xi_3)$ in
\eqref{defnoyau}. We thus find that in the case of isotropic elasticity, the kernel $b(\xi_1,\xi_2,\xi_3)$ is
a linear combination with real coefficients of the `elementary' kernels
\begin{equation}
\langle bel{defnoyauk}
b^k(\xi_1,\xi_2,\xi_3) := \int_0^{+\infty} d_{\alpha j \beta \ell \gamma m}^k \,
(\nu_j \, \rho_{\alpha,z} +i\, \xi_1 \, \eta_j \, \rho_\alpha) \,
(\nu_\ell \, \rho_{\beta,z} +i\, \xi_2 \, \eta_\ell \, \rho_\beta) \,
(\nu_m \, \rho_{\gamma,z} +i\, \xi_3 \, \eta_m \, \rho_\gamma) \, {\rm d}z \, ,
\end{equation}
where $k$ runs through $\{ 1,\dots,4\}$ and the coefficients $d_{\alpha j \beta \ell \gamma m}^k$ are given
in \eqref{noyau1}, \eqref{noyau2}, \eqref{noyau3}, \eqref{noyau4}. Let us recall eventually that the functions
$\rho_\alpha,\rho_\beta,\rho_\gamma$ are given in \eqref{defnoyau}.
In the remaining of this Section, we are going to make the kernels $b^1,\dots,b^4$ explicit, meaning that
we are going to express them as linear combinations of explicitly computable kernels. This will give the final
expression of $b$ as a linear combination of explicit kernels. In order to avoid lengthy and somehow useless
computations, we shall not try to make the coefficients in the linear combinations explicit. The interested reader
will achieve this by using the expressions below and expanding all trilinear expressions of $\exp(-\omega_{1,2}
\, |\xi| \, z)$ below explicitly.
From the definition \eqref{defnoyauk}, it appears that we need to compute the quantity
$$
\nu_j \, \dfrac{\partial}{\partial z} \big( \widehat{r}_\alpha (\xi,z) \big) +i\, \xi \, \eta_j \, \widehat{r}_\alpha (\xi,z) \, ,
$$
for all possible values of $\alpha,j$ and $\xi$. Using \eqref{Rayleighwaves}, we get (keeping the subscript
`$,z$' for denoting partial differentiation with respect to $z$):
\begin{equation*}
\nu_j \, \widehat{r}_{\alpha,z} (\xi,z) +i\, \xi \, \eta_j \, \widehat{r}_\alpha (\xi,z)
=\begin{cases}
|\xi| \, \eta_\alpha \, \eta_j \, {\mathbb T}_{11} (\xi,z) \, ,& \alpha,j \le d-1 \, ,\\
i\, \xi \, \eta_j \, \omega_2 \, {\mathbb T}_{d1} (\xi,z) \, ,& j \le d-1 \, ,\alpha=d \, ,\\
i\, \xi \, \eta_\alpha \, \omega_2 \, {\mathbb T}_{1d} (\xi,z) \, ,& \alpha \le d-1 \, ,j=d \, ,\\
-|\xi| \, \omega_2 \, {\mathbb T}_{dd} (\xi,z) \, ,& \alpha=j=d \, .
\end{cases}
\end{equation*}
with
\begin{equation}
\langle bel{rhoalphaj}
\begin{cases}
{\mathbb T}_{11} (\xi,z) := 2\, \omega_1 \, \omega_2 \, {\rm e}^{-\omega_1 \, |\xi| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} \, ,& \\
{\mathbb T}_{d1} (\xi,z) := 2\, |{\boldsymbol \eta}|^2 \, {\rm e}^{-\omega_1 \, |\xi| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} \, ,& \\
{\mathbb T}_{1d} (\xi,z) := 2\, \omega_1^2 \, {\rm e}^{-\omega_1 \, |\xi| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} \, ,& \\
{\mathbb T}_{dd} (\xi,z) := 2\, \omega_1 \, |{\boldsymbol \eta}|^2 \, {\rm e}^{-\omega_1 \, |\xi| \, z}
-\omega_2 \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} \, .&
\end{cases}
\end{equation}
In particular, there holds
\begin{equation}
\langle bel{rhojj}
\nu_j \, \widehat{r}_{j,z} (\xi,z) +i\, \xi \, \eta_j \, \widehat{r}_j (\xi,z) =|\xi| \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big)
\, \big( \omega_2^2 -|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} = -|\xi| \,
\dfrac{c_R^2}{c_P^2} \, |{\boldsymbol \eta}|^2 \, \big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi| \, z} \, .
\end{equation}
$\bullet$ \underline{The kernel $b^1$.} This is by far the simplest case. We start from the definition
\eqref{defnoyauk} and use the expression \eqref{noyau1} for the coefficients $d_{\alpha j \beta \ell \gamma m}^1$.
We get
$$
b^1(\xi_1,\xi_2,\xi_3) = \int_0^{+\infty} (\nu_j \, \rho_{j,z} +i\, \xi_1 \, \eta_j \, \rho_j) \,
(\nu_\ell \, \rho_{\ell,z} +i\, \xi_2 \, \eta_\ell \, \rho_\ell) \,
(\nu_m \, \rho_{m,z} +i\, \xi_3 \, \eta_m \, \rho_m) \, {\rm d}z \, ,
$$
where we warn the reader that, in the first term in the integral, $\rho_j$ is a short notation for $\widehat{r}_j(\xi_1,z)$,
in the second term, $\rho_\ell$ is a short notation for $\widehat{r}_\ell(\xi_2,z)$ and so on. We now use \eqref{rhojj},
and get (here the sum of products decouples as the product of sums over $j$, $\ell$ and $m$):
$$
b^1(\xi_1,\xi_2,\xi_3) =\star \, \int_0^{+\infty} |\xi_1| \, |\xi_2| \, |\xi_3| \,
\exp(-\omega_2 \, (|\xi_1|+|\xi_2|+|\xi_3|) \, z) \, {\rm d}z \, ,
$$
where, here and from now on, $\star$ denotes any {\it real} constant that depends only on ${\boldsymbol \eta}$ and that can
be explicitly computed from \eqref{rhoalphaj} or \eqref{rhojj} (though we shall not keep track of any such constant).
We thus obtain the expression:
\begin{equation}
\langle bel{expressionb1}
b^1(\xi_1,\xi_2,\xi_3) =\star \, \dfrac{|\xi_1| \, |\xi_2| \, |\xi_3|}{|\xi_1|+|\xi_2|+|\xi_3|} \, ,
\end{equation}
which corresponds to the simplified kernel introduced in \cite{HamiltonIlinskyZabolotskaya} and that also arises
in incompressible magnetohydrodynamics \cite{AliHunter}. As shown in \cite{AliHunterParker}, this kernel
corresponds in the physical space to the operator (here we forget about harmless multiplicative real constants):
$$
\dfrac{1}{2} \, \partial_{\theta \theta} {\mathbb B}ig( {\mathcal H} \big( ({\mathcal H} \, w)^2 \big) {\mathbb B}ig) +({\mathcal H} \, w)
\, \partial_{\theta \theta} \, w \, .
$$
$\bullet$ \underline{The kernel $b^2$.} We now use the expression \eqref{noyau2} and derive
\begin{align*}
b^2(\xi_1,\xi_2,\xi_3) =& \, \int_0^{+\infty} (\nu_j \, \rho_{j,z} +i\, \xi_1 \, \eta_j \, \rho_j) \,
(\nu_\ell \, \rho_{\beta,z} +i\, \xi_2 \, \eta_\ell \, \rho_\beta) \,
(\nu_\ell \, \rho_{\beta,z} +i\, \xi_3 \, \eta_\ell \, \rho_\beta) \, {\rm d}z \\
& +\int_0^{+\infty} (\nu_j \, \rho_{\alpha,z} +i\, \xi_1 \, \eta_j \, \rho_\alpha) \,
(\nu_\ell \, \rho_{\ell,z} +i\, \xi_2 \, \eta_\ell \, \rho_\ell) \,
(\nu_j \, \rho_{\alpha,z} +i\, \xi_3 \, \eta_j \, \rho_\alpha) \, {\rm d}z \\
& +\int_0^{+\infty} (\nu_j \, \rho_{\alpha,z} +i\, \xi_1 \, \eta_j \, \rho_\alpha) \,
(\nu_j \, \rho_{\alpha,z} +i\, \xi_2 \, \eta_j \, \rho_\alpha) \,
(\nu_m \, \rho_{m,z} +i\, \xi_3 \, \eta_m \, \rho_m) \, {\rm d}z \, .
\end{align*}
Changing indices in the last two integrals, we have written $b^2(\xi_1,\xi_2,\xi_3)$ under the form\footnote{Here,
and here only, $\varepsilon$ denotes the signature of a permutation.}
\begin{equation}
\langle bel{decompb2}
b^2(\xi_1,\xi_2,\xi_3) ={\mathbb B}^2(\xi_1,\xi_2,\xi_3) +{\mathbb B}^2(\xi_2,\xi_3,\xi_1)
+{\mathbb B}^2(\xi_3,\xi_1,\xi_2)
=\sum_{\underset{\varepsilon({\sigma})=1}{\sigma \in \mathfrak{S}_3 \, ,}}
{\mathbb B}^2 \big( \xi_{\sigma(1)},\xi_{\sigma(2)},\xi_{\sigma(3)} \big) \, ,
\end{equation}
where ${\mathbb B}^2$ is symmetric with respect to its last two arguments (that is,
${\mathbb B}^2(\xi_1,\xi_3,\xi_2) ={\mathbb B}^2(\xi_1,\xi_2,\xi_3)$ for all $\xi_1,\xi_2,\xi_3$), and the
formula \eqref{decompb2} shows that one `symmetrizes' the function ${\mathbb B}^2$ over the alternating
group $\mathfrak{A}_3 \subset \mathfrak{S}_3$. Using \eqref{rhoalphaj} and \eqref{rhojj}, we have
\begin{align*}
{\mathbb B}^2(\xi_1,\xi_2,\xi_3) =& \star \, |\xi_1| \, \int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
(\nu_\ell \, \rho_{\beta,z} +i\, \xi_2 \, \eta_\ell \, \rho_\beta) \,
(\nu_\ell \, \rho_{\beta,z} +i\, \xi_3 \, \eta_\ell \, \rho_\beta) \, {\rm d}z \\
=& \star \, |\xi_1| \, |\xi_2| \, |\xi_3| \, \int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
{\mathbb T}_{11}(\xi_2,z) \, {\mathbb T}_{11}(\xi_3,z) \, {\rm d}z \\
&+\star \, |\xi_1| \, \xi_2 \, \xi_3 \, \int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
{\mathbb T}_{d1}(\xi_2,z) \, {\mathbb T}_{d1}(\xi_3,z) \, {\rm d}z \\
&+\star \, |\xi_1| \, \xi_2 \, \xi_3 \, \int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
{\mathbb T}_{1d}(\xi_2,z) \, {\mathbb T}_{1d}(\xi_3,z) \, {\rm d}z \\
&+\star \, |\xi_1| \, |\xi_2| \, |\xi_3| \, \int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
{\mathbb T}_{dd}(\xi_2,z) \, {\mathbb T}_{dd}(\xi_3,z) \, {\rm d}z \, .
\end{align*}
Let us examine the first integral in the latter sum of four. We need to compute
$$
\int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \,
{\mathbb B}ig( 2\, \omega_1 \, \omega_2 \, {\rm e}^{-\omega_1 \, |\xi_2| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi_2| \, z} {\mathbb B}ig) \,
{\mathbb B}ig( 2\, \omega_1 \, \omega_2 \, {\rm e}^{-\omega_1 \, |\xi_3| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi_3| \, z} {\mathbb B}ig) \, {\rm d}z \, ,
$$
then multiply by $|\xi_1| \, |\xi_2| \, |\xi_3|$ and eventually `symmetrize' with respect to $\xi_1,\xi_2,\xi_3$
by summing as in \eqref{decompb2}, and this will give part of the kernel $b^2$. As reported in
\cite{H,Hunter2006}, this first integral in ${\mathbb B}^2$ gives rise in the expression of $b^2$ to a linear
combination with real coefficients of the following three kernels:
\begin{align}
& \dfrac{|\xi_1| \, |\xi_2| \, |\xi_3|}{|\xi_1|+|\xi_2|+|\xi_3|} \, ,\langle bel{noyauH1} \\
& |\xi_1| \, |\xi_2| \, |\xi_3| \, \left\{ \dfrac{1}{r \, |\xi_1|+|\xi_2|+|\xi_3|} +\dfrac{1}{|\xi_1|+r\, |\xi_2|+|\xi_3|}
+\dfrac{1}{|\xi_1|+|\xi_2|+r \, |\xi_3|} \right\} \, ,\langle bel{noyauH2} \\
& |\xi_1| \, |\xi_2| \, |\xi_3| \, \left\{ \dfrac{1}{|\xi_1|+r\, |\xi_2|+r \, |\xi_3|} +\dfrac{1}{r\, |\xi_1|+|\xi_2|+r\, |\xi_3|}
+\dfrac{1}{r\, |\xi_1|+r \, |\xi_2|+|\xi_3|} \right\} \, ,\langle bel{noyauH3}
\end{align}
where $r :=\omega_1 /\omega_2 \in (0,1)$ can be computed from the Lam\'e coefficients of the material.
The kernel in \eqref{noyauH1} already appears in \eqref{expressionb1}, so the $b^1$ part of the overall
kernel $b$ contributes to a term that is similar to one of the many in the decomposition of $b^2$.
Using the expression of ${\mathbb T}_{dd}$ in \eqref{rhoalphaj}, one can check that the last integral in
the above decomposition of ${\mathbb B}^2$ also gives rise, after multiplication by $|\xi_1| \, |\xi_2| \, |\xi_3|$
and symmetrization with respect to $\xi_1,\xi_2,\xi_3$, to a linear combination with real coefficients of the
three kernels in \eqref{noyauH1}, \eqref{noyauH2}, \eqref{noyauH3}.
We now examine the second and third integrals in the decomposition of ${\mathbb B}^2$, which involve
the expressions ${\mathbb T}_{d1}$, ${\mathbb T}_{1d}$ in \eqref{rhoalphaj}, and that are multiplied by
$|\xi_1| \, \xi_2 \, \xi_3$ (no absolute value for $\xi_2 \, \xi_3$ here !). For instance, we compute the integral
$$
\int_0^{+\infty} {\rm e}^{-\omega_2 \, |\xi_1| \, z} \, {\mathbb B}ig( 2\, |{\boldsymbol \eta}|^2 \, {\rm e}^{-\omega_1 \, |\xi_2| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi_2| \, z} {\mathbb B}ig) \, {\mathbb B}ig(
2\, |{\boldsymbol \eta}|^2 \, {\rm e}^{-\omega_1 \, |\xi_3| \, z}
-\big( \omega_1^2 +|{\boldsymbol \eta}|^2 \big) \, {\rm e}^{-\omega_2 \, |\xi_3| \, z} {\mathbb B}ig) \, {\rm d}z \, ,
$$
multiply by $|\xi_1| \, \xi_2 \, \xi_3$ and symmetrize with respect to $\xi_1,\xi_2,\xi_3$ as in \eqref{decompb2}.
This computation gives rise in the expression of $b^2$ to a linear combination with real coefficients of the
following kernels:
\begin{align}
& \dfrac{1}{|\xi_1|+|\xi_2|+|\xi_3|} \, {\mathbb B}ig\{ |\xi_1| \, \xi_2 \, \xi_3 +\xi_1 \, |\xi_2| \, \xi_3 +\xi_1 \, \xi_2 \, |\xi_3|
{\mathbb B}ig\} \, ,\langle bel{noyauH4} \\
& \dfrac{|\xi_1| \, \xi_2 \, \xi_3}{|\xi_1|+r\, |\xi_2|+r \, |\xi_3|} +\dfrac{\xi_1 \, |\xi_2| \, \xi_3}{r\, |\xi_1|+|\xi_2|+r\, |\xi_3|}
+\dfrac{\xi_1 \, \xi_2 \, |\xi_3|}{r\, |\xi_1|+r \, |\xi_2|+|\xi_3|} \, ,\langle bel{noyauH5} \\
& |\xi_1| \, \xi_2 \, \xi_3 \, \left\{ \dfrac{1}{|\xi_1|+r\, |\xi_2|+|\xi_3|} +\dfrac{1}{|\xi_1|+|\xi_2|+r\, |\xi_3|} \right\} \notag \\
&+\xi_1 \, |\xi_2| \, \xi_3 \, \left\{ \dfrac{1}{r\, |\xi_1|+|\xi_2|+|\xi_3|} +\dfrac{1}{|\xi_1|+|\xi_2|+r\, |\xi_3|}
\right\} \langle bel{noyauH6} \\
&+\xi_1 \, \xi_2 \, |\xi_3| \, \left\{ \dfrac{1}{r\, |\xi_1|+|\xi_2|+|\xi_3|} +\dfrac{1}{|\xi_1|+r\, |\xi_2|+|\xi_3|}
\right\} \, .\notag
\end{align}
The reader can check that, in the decomposition of ${\mathbb B}^2$, the integral that involves
${\mathbb T}_{1d}$ also gives rise, after multiplication by $|\xi_1| \, \xi_2 \, \xi_3$ and symmetrization,
to a linear combination with real coefficients of the three kernels in \eqref{noyauH4}, \eqref{noyauH5},
\eqref{noyauH6}.
At this stage, we have proved that, with the expression \eqref{noyau2} for $d^2_{\alpha j \beta \ell \gamma m}$,
the corresponding kernel $b^2$ in \eqref{defnoyauk} is a linear combination with real coefficients of the six
kernels given in \eqref{noyauH1}, \eqref{noyauH2}, \eqref{noyauH3}, \eqref{noyauH4}, \eqref{noyauH5} and
\eqref{noyauH6}.
$\bullet$ \underline{The kernels $b^3$ and $b^4$.} Using the expression of $d^3_{\alpha j \beta \ell \gamma m}$
in \eqref{noyau3}, we can decompose the kernel $b^3$ under the form
\begin{align*}
b^3(\xi_1,\xi_2,\xi_3)
&= {\mathbb B}^3(\xi_1,\xi_2,\xi_3) +{\mathbb B}^3(\xi_2,\xi_1,\xi_3) +{\mathbb B}^3(\xi_1,\xi_3,\xi_2) +{\mathbb B}^3(\xi_3,\xi_1,\xi_2)
+{\mathbb B}^3(\xi_2,\xi_3,\xi_1) +{\mathbb B}^3(\xi_3,\xi_2,\xi_1) \\
&= \sum_{\sigma \in \mathfrak{S}_3} {\mathbb B}^3 \big( \xi_{\sigma(1)},\xi_{\sigma(2)},\xi_{\sigma(3)} \big) \, ,
\end{align*}
where the function ${\mathbb B}^3$ is defined by
\begin{equation*}
{\mathbb B}^3(\xi_1,\xi_2,\xi_3) := \int_0^{+\infty} (\nu_j \, \rho_{\alpha,z} +i\, \xi_1 \, \eta_j \, \rho_\alpha) \,
(\nu_\ell \, \rho_{\alpha,z} +i\, \xi_2 \, \eta_\ell \, \rho_\alpha) \,
(\nu_\ell \, \rho_{j,z} +i\, \xi_3 \, \eta_\ell \, \rho_j) \, {\rm d}z \, ,
\end{equation*}
and is symmetrized over the symmetric group $\mathfrak{S}_3$ to obtain the kernel $b^3$. Considering
all possible indices $\alpha,j,\ell$, we end up with the following decomposition of ${\mathbb B}^3$:
\begin{align*}
{\mathbb B}^3(\xi_1,\xi_2,\xi_3) =& \star \, |\xi_1| \, |\xi_2| \, |\xi_3| \, \int_0^{+\infty} {\mathbb T}_{11}(\xi_1,z) \,
{\mathbb T}_{11}(\xi_2,z) \, {\mathbb T}_{11}(\xi_3,z) \, {\rm d}z \\
& +\star \, |\xi_1| \, |\xi_2| \, |\xi_3| \, \int_0^{+\infty} {\mathbb T}_{dd}(\xi_1,z) \,
{\mathbb T}_{dd}(\xi_2,z) \, {\mathbb T}_{dd}(\xi_3,z) \, {\rm d}z \\
& +\star \, \xi_1 \, \xi_2 \, |\xi_3| \, \int_0^{+\infty} {\mathbb T}_{d1}(\xi_1,z) \,
{\mathbb T}_{d1}(\xi_2,z) \, {\mathbb T}_{11}(\xi_3,z) \, {\rm d}z \\
& +\star \, \xi_1 \, \xi_2 \, |\xi_3| \, \int_0^{+\infty} {\mathbb T}_{1d}(\xi_1,z) \,
{\mathbb T}_{1d}(\xi_2,z) \, {\mathbb T}_{dd}(\xi_3,z) \, {\rm d}z \\
& +\star \, \xi_1 \, |\xi_2| \, \xi_3 \, \int_0^{+\infty} {\mathbb T}_{1d}(\xi_1,z) \,
{\mathbb T}_{11}(\xi_2,z) \, {\mathbb T}_{d1}(\xi_3,z) \, {\rm d}z \\
& +\star \, \xi_1 \, |\xi_2| \, \xi_3 \, \int_0^{+\infty} {\mathbb T}_{d1}(\xi_1,z) \,
{\mathbb T}_{dd}(\xi_2,z) \, {\mathbb T}_{1d}(\xi_3,z) \, {\rm d}z \\
& +\star \, |\xi_1| \, \xi_2 \, \xi_3 \, \int_0^{+\infty} {\mathbb T}_{11}(\xi_1,z) \,
{\mathbb T}_{1d}(\xi_2,z) \, {\mathbb T}_{1d}(\xi_3,z) \, {\rm d}z \\
& +\star \, |\xi_1| \, \xi_2 \, \xi_3 \, \int_0^{+\infty} {\mathbb T}_{dd}(\xi_1,z) \,
{\mathbb T}_{d1}(\xi_2,z) \, {\mathbb T}_{d1}(\xi_3,z) \, {\rm d}z \, .
\end{align*}
Computing the first two integrals in the latter decomposition of ${\mathbb B}^3$, we get linear combinations
with real coefficients of the kernels in \eqref{noyauH1}, \eqref{noyauH2}, \eqref{noyauH3} (the first two lines
are already symmetric with respect to $\xi_1,\xi_2,\xi_3$ so symmetrizing over $\mathfrak{S}_3$ only yields
the harmless multiplicative factor $6$).
We now compute the third and fourth integrals in the decomposition of ${\mathbb B}^3$, which are entirely
similar. Expanding the products in the integrals, then multiplying by $\xi_1 \, \xi_2 \, |\xi_3|$ and symmetrizing
over $\mathfrak{S}_3$ yields\footnote{As a matter of fact, symmetrizing over $\mathfrak{A}_3$ is sufficient
here because the third and fourth lines are already symmetric with respect to $\xi_1,\xi_2$.} a linear combination
of the kernels \eqref{noyauH4}, \eqref{noyauH5}, \eqref{noyauH6}, and the (hopefully last!) two kernels:
\begin{align}
& \dfrac{|\xi_1| \, \xi_2 \, \xi_3}{r\, |\xi_1|+|\xi_2|+|\xi_3|} +\dfrac{\xi_1 \, |\xi_2| \, \xi_3}{|\xi_1|+r\, |\xi_2|+|\xi_3|}
+\dfrac{\xi_1 \, \xi_2 \, |\xi_3|}{|\xi_1|+|\xi_2|+r\, |\xi_3|} \, ,\langle bel{noyauH7} \\
& |\xi_1| \, \xi_2 \, \xi_3 \,
\left\{ \dfrac{1}{r\, |\xi_1|+r\, |\xi_2|+|\xi_3|} +\dfrac{1}{r\, |\xi_1|+|\xi_2|+r\, |\xi_3|} \right\} \notag \\
&+\xi_1 \, |\xi_2| \, \xi_3 \, \left\{ \dfrac{1}{r\, |\xi_1|+r\, |\xi_2|+|\xi_3|} +\dfrac{1}{|\xi_1|+r\, |\xi_2|+r\, |\xi_3|}
\right\} \langle bel{noyauH8} \\
&+\xi_1 \, \xi_2 \, |\xi_3| \, \left\{ \dfrac{1}{r\, |\xi_1|+|\xi_2|+r\, |\xi_3|} +\dfrac{1}{|\xi_1|+r\, |\xi_2|+r\, |\xi_3|}
\right\} \, .\notag
\end{align}
The reader can check that the remaining four integrals in the decomposition of ${\mathbb B}^3$ yield,
after multiplication by either $\xi_1 \, |\xi_2| \, \xi_3$ or $|\xi_1| \, \xi_2 \, \xi_3$ and symmetrization
over $\mathfrak{S}_3$, a linear combination of the eight kernels in \eqref{noyauH1}, \eqref{noyauH2},
\eqref{noyauH3}, \eqref{noyauH4}, \eqref{noyauH5}, \eqref{noyauH6}, \eqref{noyauH7} and \eqref{noyauH8}.
We leave it as an exercise to the interested reader to verify that with the definition \eqref{noyau4},
the corresponding kernel $b^4$ can be also written as a linear combination with real coefficients
of the eight kernels given in \eqref{noyauH1}, \eqref{noyauH2}, \eqref{noyauH3}, \eqref{noyauH4},
\eqref{noyauH5}, \eqref{noyauH6}, \eqref{noyauH7} and \eqref{noyauH8}.
$\bullet$ \underline{Final simplifications.} In \cite{H,Hunter2006}, it is reported that only the kernels
in \eqref{noyauH1}, \eqref{noyauH2}, \eqref{noyauH3} are necessary for computing the leading order
amplitude equation. This is not necessarily in contradiction with the above computations since, according
to \eqref{eqw}, what we really need is the value of $b$ over the triplets $(\xi_1,\xi_2,\xi_3)$ verifying
$\xi_1+\xi_2+\xi_3=0$. Namely, we now introduce the symmetric kernel
$$
\Lambda (k,k') :=b \big( -(k+k'),k,k' \big) \, ,
$$
which allows us to rewrite ${\mathcal B}$ as
\begin{equation*}
\widehat{{\mathcal B}(w,w)} (k) :=-\dfrac{1}{4\, \pi \, c_0} \int_{\mathbb R} \Lambda (k-\xi,\xi) \, \widehat{w}(k-\xi) \,
\widehat{w}(\xi) \, {\rm d}\xi \, .
\end{equation*}
It remains to examine whether for the kernels we have found in \eqref{noyauH1}, \eqref{noyauH2},
\eqref{noyauH3}, \eqref{noyauH4}, \eqref{noyauH5}, \eqref{noyauH6}, \eqref{noyauH7} and \eqref{noyauH8},
the corresponding functions $\Lambda$ are linearly independent.
Let us start with the easiest case of the kernel in \eqref{noyauH4}. Let us first observe that for all $(k,k')$, we
have the relation
$$
|k+k'| \, \big( |k\, k'|+k\, k' \big) =(k+k') \, \big( |k|\, k' +k\, |k'| \big) \, ,
$$
and therefore
$$
\dfrac{|\xi_1| \, \xi_2 \, \xi_3 +\xi_1 \, |\xi_2| \, \xi_3 +\xi_1 \, \xi_2 \, |\xi_3|}{|\xi_1|+|\xi_2|+|\xi_3|} \,
{\mathbb B}ig|_{(\xi_1,\xi_2,\xi_3)=(-(k+k'),k,k')} =-\dfrac{|\xi_1| \, |\xi_2| \, |\xi_3|}{|\xi_1|+|\xi_2|+|\xi_3|}
{\mathbb B}ig|_{(\xi_1,\xi_2,\xi_3)=(-(k+k'),k,k')} \, ,
$$
which means that the kernel in \eqref{noyauH4} is useless in our list for decomposing the bilinear operator
\eqref{defB}. In the same spirit, if we let $b_r$, resp. $\widetilde{b}_r$, denote the kernel in \eqref{noyauH3},
resp. \eqref{noyauH5}, we then compute
$$
(b_r +\widetilde{b}_r)(-(k+k'),k,k') =\dfrac{4}{1+r} \, \dfrac{|\xi_1| \, |\xi_2| \, |\xi_3|}{|\xi_1|+|\xi_2|+|\xi_3|}
{\mathbb B}ig|_{(\xi_1,\xi_2,\xi_3)=(-(k+k'),k,k')} \, ,
$$
which means that the kernel in \eqref{noyauH5} is also useless. We now leave it as exercise to the
reader to verify that the kernels in \eqref{noyauH6}, \eqref{noyauH7}, \eqref{noyauH8}, when evaluated at
$(-k-k',k,k')$, can be written as linear combinations with real coefficients of the expressions in \eqref{noyauH1},
\eqref{noyauH2}, \eqref{noyauH3}.
We summarize our findings in the following Proposition. Let us emphasize that our result is independent
of the space dimension $d \ge 2$.
\begin{prop}
\langle bel{propelas}
Consider a hyperelastic isotropic material with Lam\'e coefficients satisfying $\mu>0$, $\langle mbda +\mu>0$.
Then weakly nonlinear Rayleigh waves traveling along the boundary of the half-space $\Omega := \{ x \cdot
{\boldsymbol \nu} >0 \}$ are governed on the boundary of $\Omega$ by the amplitude equation
\begin{align}\langle bel{ampeqn}
\partial_t w +c_R \, \dfrac{{\boldsymbol \eta}}{|{\boldsymbol \eta}|} \cdot {\nabla}bla_x w
+{\mathcal H} \, \big( {\mathcal B}(w,w) \big) =g \, ,
\end{align}
where the bilinear Fourier multiplier ${\mathcal B}$ in the variable $\theta$ has the expression
$$
\forall \, k \in {\mathbb R} \, ,\quad
\widehat{{\mathcal B}(w,w)} (k) =\int_{\mathbb R} \Lambda(k-k',k') \, \widehat{w}(k-k') \, \widehat{w}(k') \, {\rm d}k' \, ,
$$
and the kernel $\Lambda$ is a linear combination with real coefficients of the expressions in \eqref{noyauH1},
\eqref{noyauH2}, \eqref{noyauH3} evaluated at $(-k-k',k,k')$.
\end{prop}
\section{Well-posedness of the amplitude equation}
In this Section, we show a well-posedness result for the amplitude equation \eqref{eqw}, with the bilinear
operator ${\mathcal B}$ defined by \eqref{defB}. In the absence of slow spatial variables $y \in {\mathbb R}^{d-1}$,
well-posedness results for \eqref{eqw} have been obtained in \cite{Hunter2006} when the variable $\theta$
lies in the torus ${\mathbb R}/(2\, \pi \, {\mathbb Z})$ and in \cite{Secchi} when the variable $\theta$ lies in ${\mathbb R}$. Namely, in
both \cite{Hunter2006} and \cite{Secchi}, well-posedness is proved for an equation of the form
$$
\partial_ t w +\partial_\theta \, {\mathcal A}(w,w) =0 \, ,
$$
that is obtained from \eqref{eqw} by applying a half-derivative in $\theta$ (no slow spatial variable
$y \in {\mathbb R}^{d-1}$ is considered here). Applying such a half-derivative implies that the bilinear operator
${\mathcal A}$ is defined by means of a kernel $a$ that is homogeneous degree $1/2$ while the kernel
$b$ in \eqref{defB} is homogeneous degree $2$, see \cite{BC}. Let us also recall that for piecewise
smooth kernels homogeneous degree $0$, well-posedness has been proved in \cite{B} under a suitable
stability condition exhibited in \cite{H}. Our goal here is in some sense to encompass the results from these
previous works by considering the original form \eqref{eqw} of the amplitude equation and by including the
slow spatial variables $y \in {\mathbb R}^{d-1}$.
\begin{prop}
\langle bel{propwellposed}
Let ${\bf v} \in {\mathbb R}^{d-1}$ be a fixed velocity vector, and assume that the kernel $b$ in \eqref{defB} is
symmetric with respect to its arguments and that there exists a constant $C>0$ such that
\begin{equation}
\langle bel{hypotheseb}
\forall \, (\xi_1,\xi_2,\xi_3) \in {\mathbb R}^3 \, ,\quad |b(\xi_1,\xi_2,\xi_3)| \le C \, |\xi_1|^{1/2} \, |\xi_2|^{1/2} \,
|\xi_3|^{1/2} \, \min \big( |\xi_1|^{1/2},|\xi_2|^{1/2},|\xi_3|^{1/2} \big) \, .
\end{equation}
Then there exists an integer $\overline{m}$ that only depends on the space dimension $d$ such that for all
$m \in {\mathbb N}$ with $m \ge \overline{m}$ and for all $R>0$, there exists a time $T=T(m,R)$ such that if the initial
condition $u_0 \in H^m({\mathbb R}^{d-1}_y \times {\mathbb R}_\theta ;{\mathbb R})$ satisfies $\| u_0 \|_{H^m} \le R$, then there exists a
unique $u \in {\mathcal C}([0,T];H^m({\mathbb R}^{d-1}_y \times {\mathbb R}_\theta;{\mathbb R}))$ solution to the Cauchy problem
\begin{equation}
\langle bel{cauchyamplitude}
\partial_t u +\sum_{j=1}^{d-1} v_j \, \partial_j u +{\mathcal H} \, \big( {\mathcal B}(u,u) \big) =0 \, ,\quad
u|_{t=0}=u_0 \, .
\end{equation}
\end{prop}
When the initial condition for \eqref{eqw} vanishes but the source term $g$ in \eqref{eqw} is nonzero, one
solves \eqref{eqw} by using the Duhamel formula. We omit the details here and focus on the solvability of
the pure Cauchy problem for nonzero initial data and zero forcing term. In isotropic elastodynamics, the
kernel $b$ whose expression is given in Proposition {\rm Re }\, f{propelas} satisfies the bound \eqref{hypotheseb},
as shown in \cite{Hunter2006}.
\begin{proof}
Following the previous works \cite{Hunter2006,B,Secchi}, we only show here an a priori estimate for
the solutions to the Cauchy problem. By standard arguments, see for instance \cite{TaylorIII}, a priori
estimates can be turned into a well-posedness result as stated in Proposition {\rm Re }\, f{propwellposed} by
using convenient Fourier truncation approximations (recall here that the underlying space domain is
${\mathbb R}^{d-1}_y \times {\mathbb R}_\theta$ so Fourier analysis is readily available).
We therefore consider a solution $u \in {\mathcal C}([0,T];H^m({\mathbb R}^{d-1}_y \times {\mathbb R}_\theta;{\mathbb R}))$ to the
Cauchy problem \eqref{cauchyamplitude} and try to derive an estimate for the evolution of the $H^m$
norm of $u$. We shall make as if $u$ were sufficiently smooth for all manipulations below to be rigorous.
As a matter of fact, using the Fourier expression of the $H^m$ norm, we even only deal with the $L^2$
norm of the functions $u$, $\partial_\theta^m u$, $\partial_y^\alpha u$ with $|\alpha|=m$ ($\alpha \in
{\mathbb N}^{d-1}$). All other partial derivatives of $u$ can be dealt with by interpolating between such `extreme'
cases. Once again, we refer to \cite{TaylorIII} for all details on such arguments. Let us first prove the
following bounds on the operator ${\mathcal B}$.
\begin{lem}
\langle bel{lembornesB}
Under the assumptions of Proposition {\rm Re }\, f{propwellposed}, the bilinear operator ${\mathcal B}$ is symmetric.
It satisfies the Leibniz rule
$$
\partial_\theta {\mathcal B}(u,v) ={\mathcal B}(\partial_\theta u,v) +{\mathcal B}(u,\partial_\theta v) \, ,
$$
and more generally the Leibniz rule at any order of differentiation in $\theta$, as well as the
bounds\footnote{The bounds obviously extend by continuity to functions in appropriate Sobolev spaces
and are not restricted to functions in the Schwartz class.}
\begin{align*}
\forall \, u,v,w \in {\mathcal S}({\mathbb R}^{d-1} \times {\mathbb R};{\mathbb R}) \, ,\quad
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(v,w) \big) \, {\rm d}y \, {\rm d}\theta \right|
&\le C \, \| u \|_{L^2} \, \| v \|_{H^1} \, \| w \|_{H^{m_0}} \, ,\\
\forall \, u,v \in {\mathcal S}({\mathbb R}^{d-1} \times {\mathbb R};{\mathbb R}) \, ,\quad
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(u,v) \big)\, {\rm d}y \, {\rm d}\theta \right|
&\le C \, \| v \|_{H^{m_0+1}} \, \| u \|_{L^2}^2 \, ,
\end{align*}
for a suitable constant $C$ and any integer $m_0$ satisfying $m_0 >(d-1)/2 +2$. (The Sobolev norms refer
to the space domain ${\mathbb R}^{d-1}_y \times {\mathbb R}_\theta$.)
\end{lem}
\begin{proof}
The fact that ${\mathcal B}$ is symmetric comes from the symmetry of the kernel $b$ with respect to its
three arguments. We now consider three functions $u,v,w$ in the Schwartz space ${\mathcal S}({\mathbb R}^{d-1}_y
\times {\mathbb R}_\theta;{\mathbb R})$. Below the `hat' notation stands for the partial Fourier transform with respect to
$\theta \in {\mathbb R}$. Applying Plancherel's Theorem, we get\footnote{In the computations below, we do not use
the fact that $u,v,w$ are real valued.}
\begin{equation*}
\int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(v,w) \big) \, {\rm d}y \, {\rm d}\theta
=i \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} \overline{\widehat{u}(y,k)} \, \widehat{v}(y,k-k') \, \widehat{w}(y,k') \,
\text{\rm sgn} (-k) \, b(-k,k-k',k') \, {\rm d}y \, {\rm d}k \, {\rm d}k' \, .
\end{equation*}
We use the bound on $b$ together with the inequality
$$
|k|^{1/2} \le C \, {\mathbb B}ig( |k-k'|^{1/2} +|k'|^{1/2} {\mathbb B}ig) \, ,
$$
and obtain
\begin{multline*}
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(v,w) \big) \, {\rm d}y \, {\rm d}\theta \right|
\le C \, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} |\widehat{u}(y,k)| \, |\widehat{v}(y,k-k')| \, |\widehat{w}(y,k')| \, \times \\
\qquad \qquad \qquad {\mathbb B}ig(
|k-k'| \, |k'|^{1/2} +|k-k'|^{1/2} \, |k'| {\mathbb B}ig) \, \min (|k|^{1/2},|k-k'|^{1/2},|k'|^{1/2}) \, {\rm d}y \, {\rm d}k \, {\rm d}k' \\
\le C \, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} |\widehat{u}(y,k)| \, |k-k'| \, |\widehat{v}(y,k-k')| \, |k'| \, |\widehat{w}(y,k')|
\, {\rm d}y \, {\rm d}k \, {\rm d}k' \, .
\end{multline*}
We then apply the classical Young inequality $L^2 * L^1 \rightarrow L^2$, use Plancherel Theorem again and get
\begin{align*}
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(v,w) \big) \, {\rm d}y \, {\rm d}\theta \right|
&\le C \, \int_{{\mathbb R}^{d-1}} \| \widehat{u}(y,\cdot) \|_{L^2} \, \| k\, \widehat{v}(y,\cdot) \|_{L^2} \,
\| k\, \widehat{w}(y,\cdot) \|_{L^1} \, {\rm d}y \\
&\le C \, \int_{{\mathbb R}^{d-1}} \| u(y,\cdot) \|_{L^2} \, \| \partial_\theta v(y,\cdot) \|_{L^2} \,
\| (1+k^2)^{1/2} \, k\, \widehat{w}(y,\cdot) \|_{L^2} \, {\rm d}y \\
&\le C \, \| u \|_{L^2} \, \| \partial_\theta v \|_{L^2} \, \sup_{y \in {\mathbb R}^{d-1}} \| w(y,\cdot) \|_{H^2({\mathbb R})} \, .
\end{align*}
We then apply the Sobolev imbedding Theorem and obtain the first estimate of Lemma {\rm Re }\, f{lembornesB}.
Let us now turn to the case $u=v$ where we expect some cancelation arising from the skew-symmetric
operator ${\mathcal H}$. We compute
\begin{multline*}
\int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(u,v) \big) \, {\rm d}y \, {\rm d}\theta
=i\, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} \overline{\widehat{u}(y,k)} \, \widehat{u}(y,k-k') \, \widehat{v}(y,k') \,
\text{\rm sgn} (-k) \, b(-k,k-k',k') \, {\rm d}y \, {\rm d}k \, {\rm d}k' \\
=i\, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} \widehat{u}(y,-k) \, \widehat{u}(y,k-k') \, \widehat{v}(y,k') \,
\text{\rm sgn} (-k) \, b(-k,k-k',k') \, {\rm d}y \, {\rm d}k \, {\rm d}k' \\
=\dfrac{i}{2}\, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} \widehat{u}(y,-k) \, \widehat{u}(y,k-k') \, \widehat{v}(y,k') \,
{\mathbb B}ig( \text{\rm sgn} (-k) +\text{\rm sgn} (k-k') {\mathbb B}ig) \, b(-k,k-k',k') \, {\rm d}y \, {\rm d}k \, {\rm d}k' \, ,
\end{multline*}
where we have now used the fact that $u$ is real valued, and the symmetry of $b$. Let us observe
that if $-k$ and $k-k'$ have opposite signs, then the quantity $\text{\rm sgn} (-k) +\text{\rm sgn} (k-k')$
vanishes. If $-k$ and $k-k'$ have the same sign, then the sum of signs is either $2$ or $-2$, and there
holds
$$
|k| \le |k'| \, ,\quad \text{\rm and } \quad |k-k'| \le |k'| \, .
$$
This yields
\begin{multline*}
{\mathbb B}ig| \big( \text{\rm sgn} (-k) +\text{\rm sgn} (k-k') \big) \, b(-k,k-k',k') {\mathbb B}ig| \\
\le C \, {\mathbb B}ig| \text{\rm sgn} (-k) +\text{\rm sgn} (k-k') {\mathbb B}ig| \, |k|^{1/2} \, |k-k'|^{1/2} \, |k'|^{1/2} \,
\min (|k|^{1/2},|k-k'|^{1/2},|k'|^{1/2}) \le C \, |k'|^2 \, .
\end{multline*}
Using similar inequalities as above (convolution, Cauchy-Schwarz), we can then derive the bound
\begin{align*}
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, \big( {\mathcal B}(u,v) \big) \, {\rm d}y \, {\rm d}\theta \right|
&\le C \, \int_{{\mathbb R}^{d-1} \times {\mathbb R} \times {\mathbb R}} | \widehat{u}(y,-k)| \, |\widehat{u}(y,k-k')| \, |k'|^2 \, |\widehat{v}(y,k')|
\, {\rm d}y \, {\rm d}k \, {\rm d}k' \\
&\le C \, \int_{{\mathbb R}^{d-1}} \| \widehat{u}(y,\cdot) \|_{L^2} \, \| \widehat{u}(y,\cdot) \|_{L^2} \,
\| k^2\, \widehat{v}(y,\cdot) \|_{L^1} \, {\rm d}y \\
&\le C \, \| u \|_{L^2}^2 \, \sup_{y \in {\mathbb R}^{d-1}} \| v(y,\cdot) \|_{H^3({\mathbb R})} \, .
\end{align*}
Applying Sobolev imbedding Theorem completes the proof of Lemma {\rm Re }\, f{lembornesB}.
\end{proof}
Let us consider an integer $m \ge m_0+1$ with $m_0$ as in Lemma {\rm Re }\, f{lembornesB}. We consider a
sufficiently smooth solution $u$ to \eqref{cauchyamplitude} and compute (the transport terms with
respect to the variables $y$ are harmless here):
$$
\dfrac{{\rm d}}{{\rm d}t} \| u(t) \|_{L^2}^2 =-2 \, \int_{{\mathbb R}^{d-1} \times {\mathbb R}} u \, {\mathcal H} \, {\mathcal B}(u,u)
\, {\rm d}y \, {\rm d}\theta \, .
$$
Applying the first bound in Lemma {\rm Re }\, f{lembornesB}, we get
\begin{equation}
\langle bel{apriori1}
\left| \dfrac{{\rm d}}{{\rm d}t} \| u(t) \|_{L^2}^2 \right| \le C \, \| u(t) \|_{L^2} \, \| u(t) \|_{H^1} \, \| u(t) \|_{H^{m_0}}
\le C \, \| u(t) \|_{H^m}^3 \, .
\end{equation}
Let us now differentiate $m$ times \eqref{cauchyamplitude} with respect to $\theta$, and get
$$
\partial_t \partial_\theta^m u +\sum_{j=1}^{d-1} v_j \, \partial_j \partial_\theta^m u +2\, {\mathcal H} \, \big(
{\mathcal B}(\partial_\theta^m u,u) \big) =-\sum_{m'=1}^{m-1} \binom {m} {m'} \,
{\mathcal H} \, \big( {\mathcal B}(\partial_\theta^{m'} u,\partial_\theta^{m-m'} u) \big) \, .
$$
Taking the $L^2$ scalar product with $\partial_\theta^m u$, we get
\begin{align*}
\dfrac{{\rm d}}{{\rm d}t} \| \partial_\theta^m u(t) \|_{L^2}^2 =&-4 \, \int_{{\mathbb R}^{d-1} \times {\mathbb R}} \partial_\theta^m u \,
{\mathcal H} \, \big( {\mathcal B}(\partial_\theta^m u,u) \big) \, {\rm d}y \, {\rm d}\theta \\
&-2 \, \sum_{m'=1}^{m-1} \binom {m} {m'} \, \int_{{\mathbb R}^{d-1} \times {\mathbb R}} \partial_\theta^m u \,
{\mathcal H} \, \big( {\mathcal B}(\partial_\theta^{m'} u,\partial_\theta^{m-m'} u) \big) \, {\rm d}y \, {\rm d}\theta \, .
\end{align*}
For the first integral, we apply the second estimate of Lemma {\rm Re }\, f{lembornesB} and get
$$
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} \partial_\theta^m u \, {\mathcal H} \, \big( {\mathcal B}(\partial_\theta^m u,u)
\big) \, {\rm d}y \, {\rm d}\theta \right| \le C \, \| u(t) \|_{H^{m_0+1}} \, \| u(t) \|_{H^m}^2
\le C \, \| u(t) \|_{H^m}^3 \, .
$$
For the remaining terms, we apply the first estimate of Lemma {\rm Re }\, f{lembornesB}. Assuming without loss of
generality $m-1 \ge m' \ge m-m' \ge 1$, we get
\begin{align*}
\left| \int_{{\mathbb R}^{d-1} \times {\mathbb R}} \partial_\theta^m u \, {\mathcal H} \, \big( {\mathcal B}(\partial_\theta^{m'} u,
\partial_\theta^{m-m'} u) \big) \, {\rm d}y \, {\rm d}\theta \right| &\le C \, \| u(t) \|_{H^m} \, \| u(t) \|_{H^{m'+1}} \,
\| u(t) \|_{H^{m-m'+m_0}} \\
&\le C \, \| u(t) \|_{H^m}^2 \, \| u(t) \|_{H^{m-m'+m_0}} \, .
\end{align*}
We now choose $m \ge \overline{m}$ with $\overline{m} := 2\, m_0$, $m_0$ as in Lemma {\rm Re }\, f{lembornesB}.
Then $m-m' \le m/2$ and $m_0 \le m/2$ so that collecting all previous inequalities we get
\begin{equation}
\langle bel{apriori2}
\left| \dfrac{{\rm d}}{{\rm d}t} \| \partial_\theta^m u(t) \|_{L^2}^2 \right| \le C \, \| u(t) \|_{H^m}^3 \, .
\end{equation}
The $y$-partial derivatives of $u$ are estimated in an entirely similar way, using the Leibniz rule (with respect
to $y$) for the bilinear term ${\mathcal B}(u,u)$. We omit the details, which are entirely similar to what has been
done above for the $\theta$ $m$-th derivative. Collecting \eqref{apriori1}, \eqref{apriori2} and the analogous
estimates for the $y$-derivatives and all cross/intermediate $\theta,y$ derivatives, we end up with
$$
\left| \dfrac{{\rm d}}{{\rm d}t} \| u(t) \|_{H^m}^2 \right| \le C \, \| u(t) \|_{H^m}^3 \, ,
$$
as long as $m \ge \overline{m} =2\, m_0$, and $u$ is a sufficiently smooth solution to \eqref{cauchyamplitude}.
The latter differential inequality provides with a control of $ \| u(t) \|_{H^m}$ in terms of $ \| u_0 \|_{H^m}$
on a time interval that depends on $m$ (because the above constant $C$ does depend on $m$). Applying
standard regularization procedures as in \cite{TaylorIII}, we can prove well-posedness for
\eqref{cauchyamplitude} in the Sobolev space $H^m$, $m \ge \overline{m}$, as stated in Proposition
{\rm Re }\, f{propwellposed}.
\end{proof}
\chapter{Existence of exact solutions}
\langle bel{chapter3}
\section{Introduction}\langle bel{intro}
\emph{\quad}Our main focus in the remaining chapters is to provide rigorous answers to the basic questions of geometric optics for surface \emph{pulses} in isotropic hyperelastic materials. We will describe how the results extend to wavetrains in section {\rm Re }\, f{wavetrains}. Although many pieces of the argument work just as well in higher dimensions, there is one piece that requires us to assume $d=2$ in our main results, Theorems {\rm Re }\, f{uniformexistence} and {\rm Re }\, f{approxthm}.
At several points our estimates rely on the use of Kreiss symmetrizers and, as we explain in section {\rm Re }\, f{higherD}, there is a serious difficulty with the construction of Kreiss symmetrizers for linearized elasticity in $d\geq 3$.
We let the unknown $\phi=(\phi_1,\phi_2)(t,x)$ represent the deformation of an isotropic, hyperelastic, Saint Venant-Kirchhoff (SVK) material in the reference configuration $\omega=\{x=(x_1,x_2):x_2>0\}$, which is subjected to a surface force $g=g(t,x)$. Here $\phi(t,\cdot):\omega\to \mathbb{R}^2$ and $g(t,\cdot):\partial\omega\to \mathbb{R}^2$.
The equations are a second-order, nonlinear $2\times 2$ system\footnote{The interior equation is quasilinear, while the boundary equation is fully nonlinear.}
\begin{align}\langle bel{a0}
\begin{split}
&\partial_t^2\phi -\mathrm{Div}({\nabla}bla\phi\;\sigma({\nabla}bla\phi))=0\text{ in }x_2>0\\
& {\nabla}bla\phi\;\sigma({\nabla}bla\phi)n=g\text{ on }x_2=0,\; \;\\
&\phi(t,x)=x \text{ and }g=0 \text{ in }t\leq 0,
\end{split}
\end{align}
where $n=(-1,0)$ is the outer unit normal to the boundary of $\omega$, ${\nabla}bla\phi=(\partial_{x_j}\phi_i)_{i,j=1,2}$
is the spatial gradient matrix, $\sigma$ is the stress $\sigma({\nabla}bla\phi)=\langle mbda \mathrm{tr} E\cdot I+2\mu E$ with Lam\'e constants $\langle mbda$ and $\mu$ strictly positive, and $E$ is the strain $E({\nabla}bla\phi)=\frac{1}{2}({}^t{\nabla}bla\phi\cdot{\nabla}bla \phi-I)$. Here ${}^t{\nabla}bla\phi$ denotes the transpose of ${\nabla}bla\phi$, $\mathrm{tr} E$ is the trace of the matrix $E$, and
\begin{align}
\mathrm{Div}M=(\sum^2_{j=1}\partial_jm_{i,j})_{i=1,2}\text{ for a matrix } M=(m_{i,j})_{i,j=1,2}.
\end{align}
We recall that the stored energy function $W(E)$ for an SVK material \eqref{SVK} is the leading part, quadratic in $E$, of the general isotropic hyperelastic energy given by \eqref{generalenergy}.\footnote{When rewritten in terms of ${\nabla}bla\phi$ or ${\nabla}bla U={\nabla}bla \phi -I$, the stored energy function is fourth order in those arguments.} In section {\rm Re }\, f{generalisotropic} we explain how our results extend to this more general case. It will simplify the exposition to work initially with the SVK problem, which contains all the main difficulties.
The system \eqref{a0} has the form
\begin{align}\langle bel{a1}
\begin{split}
&\partial_t^2\phi +\sum_{|\alpha|=2} \mathcal{A}_\alpha({\nabla}bla\phi)\partial_x^\alpha\phi=0\text{ in }x_2>0\\
&B({\nabla}bla\phi)=g\text{ or, equivalently, }\partial _{x_2}\phi=\mathcal{H}(\partial_{x_1}\phi,g)\text{ on }x_2=0\\
&\phi(t,x)=x \text{ and }g=0 \text{ in }t\leq 0,
\end{split}
\end{align}
where the matrices $\mathcal{A}_\alpha$ are symmetric, and the real functions $\mathcal{A}_\alpha(\cdot)$, $B(\cdot)$, and $\mathcal{H}(\cdot)$ are $C^\infty$ (in fact, analytic) in their arguments. The second form of the boundary condition is obtained by an application of the implicit function theorem to the equation $B({\nabla}bla \phi)=g$, and is valid for $|{\nabla}bla\phi-I_{2\times 2}|_{L^\infty}$ small. Set $x'=(t,x_1)$.
Defining the displacement $U(t,x)=\phi(t,x)-x$, we rewrite \eqref{a1} as
\begin{align}\langle bel{a2}
\begin{split}
&\partial_t^2 U+\sum_{|\alpha|=2} A_\alpha({\nabla}bla U)\partial_x^\alpha U=0\text{ in }x_2>0\\
&h({\nabla}bla U)=g\text{ or }\partial _{x_2} U=H(\partial_{x_1} U,g)\text{ on }x_2=0\\
&U(t,x)=0 \text{ and }g(t,x_1)=0 \text{ in }t\leq 0,
\end{split}
\end{align}
where the functions $A_\alpha$, $h$, and $H$ are related to $\mathcal{A}_\alpha$, $B$, and $\mathcal{H}$ in the obvious way. Here $H$ is defined near $(0,0)$ and satisfies
\begin{align}\langle bel{a2a}
H(0,0)=0.
\end{align}
We take $g(t,x_1)$ to be an $H^s$ pulse with the weakly nonlinear scaling:
\begin{align}\langle bel{a3}
g=g^{\varepsilon}(t,x_1)={\varepsilon}^2 G\left(t,x_1,\frac{\beta\cdot (t,x_1)}{{\varepsilon}}\right),
\end{align}
where $G(x',\theta)\in H^s(\mathbb{R}^2\times \mathbb{R})$ for some large enough $s$ to be specified, and $\beta\in \mathbb{R}^2\setminus 0$ is a frequency in the elliptic region of (the linearization at $0$ of) \eqref{a2}, chosen so that the uniform Lopatinskii condition fails at $\beta$.\footnote{See definition {\rm Re }\, f{lopa}. We may take $\beta=(\pm\tau_r(\mathbf{\eta}),\mathbf{\eta})$, for $\tau_r(\mathbf{\eta})$ as in (H3) of chapter {\rm Re }\, f{chapter2} and $\bf{\eta}=1$.}
We will refer to $\beta$ as a \emph{Rayleigh frequency}. We expect the response $U^{\varepsilon}(t,x)$ to be a Rayleigh wave, or rather, a Rayleigh \emph{pulse}, propagating in the boundary.
The main step in constructing the leading term of a geometric optics approximation to the pulse $U^{\varepsilon}$ was given in Proposition {\rm Re }\, f{propwellposed} of Chapter {\rm Re }\, f{chapter2}. In chapter {\rm Re }\, f{chapter4} we complete the construction of the leading term and first corrector, thereby producing an approximate solution to the SVK system. The basic questions arise of
whether an exact pulse solution exists on a fixed time interval independent of ${\varepsilon}$, and, if so, whether the approximate solution is ``close" in some sense to the exact solution on such a time interval. These questions are answered in Theorems {\rm Re }\, f{uniformexistence} and {\rm Re }\, f{approxthm}.
In this chapter we resolve the first question in the affirmative and prove Theorem {\rm Re }\, f{uniformexistence}. In chapter {\rm Re }\, f{chapter5} we complete the proof of Theorem {\rm Re }\, f{approxthm}, which shows that the approximate solution is close in a precise sense to the exact solution for small ${\varepsilon}ilon$.\footnote{We note that for a fixed ${\varepsilon}$ the existence of $U^{\varepsilon}$ on a time interval $(-\infty,T_{\varepsilon}]$ that may depend on ${\varepsilon}$ follows from the main result of \cite{S-T}.}
We look for $U^{\varepsilon}(t,x)$ in the form
\begin{align}\langle bel{aa4}
U^{\varepsilon}(t,x)=u^{\varepsilon}(t,x,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}},
\end{align}
where $u^{\varepsilon}(t,x,\theta)$ satisfies the \emph{singular system} obtained from \eqref{a2} by plugging in the ansatz \eqref{aa4}:
\begin{align}\langle bel{a5}
\begin{split}
&\partial_{t,{\varepsilon}}^2 u^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha({\nabla}bla_{\varepsilon} u^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha u^{\varepsilon}=0\text{ in }\{(t,x,\theta):x_2>0\}\\
&\partial _{x_2} u^{\varepsilon}=H(\partial_{x_1,{\varepsilon}} u^{\varepsilon},{\varepsilon}^2 G(x',\theta))\text{ on }x_2=0\\
&u^{\varepsilon}(t,x)=0 \text{ and }G(x',\theta)=0 \text{ in }t\leq 0.
\end{split}
\end{align}
With $\beta=(\beta_0,\beta_1)$ and $\alpha=(\alpha_1,\alpha_2)$ the notation in \eqref{a5} is:
\begin{align}\langle bel{a6}
\begin{split}
&\partial_{t,{\varepsilon}}=\partial_t+\frac{\beta_0}{{\varepsilon}}\partial_\theta,\;\;\partial_{x_1,{\varepsilon}}=\partial_{x_1}+\frac{\beta_1}{{\varepsilon}}\partial_\theta\\
&{\nabla}bla_{\varepsilon}=(\partial_{x_1,{\varepsilon}},\partial_{x_2}),\;\;\partial_{x,{\varepsilon}}^\alpha=\partial_{x_1,{\varepsilon}}^{\alpha_1}\partial_{x_2}^{\alpha_2}.
\end{split}
\end{align}
The main effort of this paper is devoted to proving existence of a solution to the singular system \eqref{a5} on a fixed time interval independent of ${\varepsilon}$. We refer to the system as ``singular" not only because of the factors of $\frac{1}{{\varepsilon}}$ that appear, but also because $\partial_{x'}$ and $\partial_{\theta}$ derivatives occur in the linear combinations \eqref{a6}.
To study such systems we require pseudodifferential operators that are singular in the same sense; a calculus of such operators is described in the appendix.
Using an idea employed by \cite{S-T} in the nonsingular case, we set $v^{\varepsilon}=(v^{\varepsilon}_1,v^{\varepsilon}_2)$, formally make the substitutions
\begin{align}
v_1^{\varepsilon}=\partial_{x_1,{\varepsilon}}u^{\varepsilon}\text{ and }v^{\varepsilon}_2=\partial_{x_2}u^{\varepsilon},
\end{align}
in the system obtained by differentiating the interior and boundary equations of \eqref{a5} with respect to $\partial_{x_1,{\varepsilon}}$:
\begin{align}\langle bel{a7}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_1^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha(v^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha v_1^{\varepsilon}=-\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_1,{\varepsilon}}(A_\alpha(v^{\varepsilon}))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v_1^{\varepsilon}-\partial_{x_1,{\varepsilon}}(A_{(0,2)}(v^{\varepsilon}))\partial_{x_2}v_2^{\varepsilon}\\
&\partial _{x_2} v^{\varepsilon}_1-d_{v_1}H(v^{\varepsilon}_1,h(v^{\varepsilon}))\partial_{x_1,{\varepsilon}}v^{\varepsilon}_1=d_gH(v^{\varepsilon}_1,{\varepsilon}^2G)\partial_{x_1,{\varepsilon}}({\varepsilon}^2G)\text{ on }x_2=0.
\end{split}
\end{align}
Similarly, differentiating the interior equation of \eqref{a5} with respect to $x_2$ and making the same substitutions we obtain
\begin{align}\langle bel{a8}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_2^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha(v^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha v_2^{\varepsilon}=-\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_2}(A_\alpha(v^{\varepsilon}))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v^{\varepsilon}_1-\partial_{x_2}(A_{(0,2)}(v^{\varepsilon}))\partial_{x_2}v^{\varepsilon}_2\\
&v^{\varepsilon}_2=H(v^{\varepsilon}_1,{\varepsilon}^2 G)\text{ on }x_2=0.
\end{split}
\end{align}
Equations \eqref{a7} and \eqref{a8} are a coupled nonlinear system for the unknown $v^{\varepsilon}=(v^{\varepsilon}_1,v^{\varepsilon}_2)$. Once $v^{\varepsilon}$ is determined, one can solve the following linear system for $u^{\varepsilon}$. By Remark {\rm Re }\, f{a10} this system is equivalent to \eqref{a5}:
\begin{align}\langle bel{a9}
\begin{split}
&\partial_{t,{\varepsilon}}^2 u^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha(v^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha u^{\varepsilon}=0\\\
&\partial _{x_2} u^{\varepsilon}-d_{v_1}H(v^{\varepsilon}_1,h(v^{\varepsilon}))\partial_{x_1,{\varepsilon}}u^{\varepsilon}=H(v^{\varepsilon}_1,{\varepsilon}^2 G(x',\theta))-d_{v_1}H(v^{\varepsilon}_1,{\varepsilon}^2 G)v^{\varepsilon}_1\text{ on }x_2=0.
\end{split}
\end{align}
Recall that $u^{\varepsilon}$, $v^{\varepsilon}$ and $G$ all vanish in $t<0$.
\begin{rem}\langle bel{a10}
We show in section {\rm Re }\, f{local} that for each fixed ${\varepsilon}\in (0,1]$, the problems \eqref{a7}, \eqref{a8}, and \eqref{a9} have solutions on a time interval $(-\infty,T_{\varepsilon}]$ that may depend on ${\varepsilon}$.
Moreover, as a consequence of Proposition {\rm Re }\, f{localex}, we have $v^{\varepsilon}_1=\partial_{x_1,{\varepsilon}}u^{\varepsilon}$ and $v^{\varepsilon}_2=\partial_{x_2}u^{\varepsilon}$ on $(-\infty,T_{\varepsilon}]$. The work in sections {\rm Re }\, f{main} to {\rm Re }\, f{mainestimate} will show that $T_{\varepsilon}$ can be chosen independent of ${\varepsilon}$.
\end{rem}
\subsection{Assumptions}\langle bel{assumptions}
\emph{\quad} We make only one assumption, namely, that we are considering the equations of nonlinear elasticity with ``traction" boundary conditions for a Saint Venant-Kirchhoff material on a 2D half-space. More precisely, we assume:
\begin{itemize}
\item[(A1)] We study the equations \eqref{a2} with Lam\'e constants satisfying $\mu>0$, $\langle mbda+\mu>0$. The interior and boundary operators in these equations are the same as the operators appearing in \eqref{eqint}, \eqref{cl}, where $N=d=2$ and $W({\nabla}bla u)$ is given by \eqref{taylorW} for the Saint Venant-Kirchhoff stored energy $W(E)$ as in \eqref{SVK}.
\end{itemize}
In section \eqref{generalisotropic} we will replace assumption (A1) by the more general assumption:
\begin{itemize}
\item[(A1g)] We study the equations \eqref{a2} with Lam\'e constants satisfying $\mu>0$, $\langle mbda+\mu>0$. The interior and boundary operators in these equations are the operators appearing in \eqref{eqint}, \eqref{cl}, where $N=d=2$ and $W({\nabla}bla u)$ is given by \eqref{taylorW} for $W(E)$ as in \eqref{generalenergy}, the general isotropic hyperelastic stored energy. Moreover, we assume that $W(E)$ is an analytic function.\footnote{We consider only displacements with ${\nabla}bla u$ small, so it is enough to assume that $W(E)$ is analytic near $E=0$.}
\end{itemize}
\begin{rem}
The discussion in section {\rm Re }\, f{isoe} shows that under either of these assumptions, the hypotheses (H1), (H2), (H3) of chapter {\rm Re }\, f{chapter2} are satisfied.
\end{rem}
We recall from \eqref{a2} that the system satisfied by the displacement $U(t,x)=\phi(t,x)-x$ is:
\begin{align}\langle bel{d1}
\begin{split}
&\partial_t^2 U+\sum_{|\alpha|=2} A_\alpha({\nabla}bla U)\partial_x^\alpha U=0\text{ in }x_2>0\\
&\partial _{x_2} U=H(\partial_{x_1} U,g)\text{ on }x_2=0\\
&U(t,x)=0 \text{ and }g(t,x_1)=0 \text{ in }t\leq 0,
\end{split}
\end{align}
The ``background state" is ${\nabla}bla U=0$ and corresponds to the identity deformation $\phi(t,x)=x$. Let us write the arguments of $H$ as $(v_1,g)$.
The linearized system at the background state is
$(P^0,B^0)$, where
\begin{align}\langle bel{d2}
\begin{split}
&P^0=\partial_t^2+\sum_{|\alpha|=2}A_\alpha(0)\partial_x^\alpha\\
&B^0=\partial_{x_2}-d_{v_1}H(0,0)\partial_{x_1}.
\end{split}
\end{align}
The perturbed linearized system is, with $h(v)$ as in \eqref{a2},
\begin{align}\langle bel{d3}
\begin{split}
&P^v u=\partial_t^2 u+\sum_{|\alpha|=2}A_\alpha(v)\partial_x^\alpha u=f\\
&B^v u =\partial_{x_2}u-d_{v_1}H(v_1,h(v))\partial_{x_1}u = g.
\end{split}
\end{align}
\begin{rem}\langle bel{d4}
For $(v,G)$ near $(0,0)$ we have $h(v_1,v_2)=G\Leftrightarrow v_2=H(v_1,G)$. The linearized boundary operator at $v$ can be written
\begin{align}
h_{v_1}(v)\partial_{x_1}+h_{v_2}(v)\partial_{x_2}
\end{align}
By the chain rule applied to $h(v_1,H(v_1,G))=G$ we have
\begin{align}
(h_{v_2}(v))^{-1}h_{v_1}(v)=-H_{v_1}(v_1,h(v)),
\end{align}
which explains the form of $B^\mu$.
\end{rem}
\textbf{Lopatinskii determinant.}\footnote{The remainder of this section discusses some consequences of Assumption (A1), and is somewhat more technical. It may be read after the statement of the main results, section {\rm Re }\, f{mainr}, and after the survey of the proofs, section {\rm Re }\, f{survey}.}\;Below we let $(\sigma,\xi_1,\xi_2)$ denote real variables dual to $(t,x_1,x_2)$, set $\tau=\sigma-i\gamma$ for $\gamma\geq 0$, and
let $\Lambda_\cD$ denote the (nonsingular) operator defined by the Fourier multiplier $\Lambda(\sigma,\xi_1,\gamma):=(\sigma^2+\xi_1^2+\gamma^2)^{1/2}$.
In order to define the Lopatinskii determinant
we set $U=(e^{\gamma t}\Lambda_{\cD}e^{-\gamma t}u ,D_{x_2}u)=(\Lambda_{\cD,\gamma} u,D_{x_2} u)$, $U^\gamma=e^{-\gamma t}U$, and rewrite \eqref{d3} as a $4\times 4$ first-order system:
\begin{align}\langle bel{d4a}
D_{x_2}U^\gamma-\cA(v,\cD)U^\gamma=\tilde{f}^\gamma,\;\;\cB(v,\cD)U^\gamma=\tilde{g}^\gamma,
\end{align}
where, with the matrices $A_\alpha$ evaluated at $v$, we have
\begin{align}\langle bel{d4b}
\begin{split}
& \tilde{f}:=\begin{pmatrix}0\\-A_{(0,2)}^{-1}f\end{pmatrix},\;\;\tilde{g}=-ig\\
&\cA(v,\cD)=-\begin{pmatrix}0&\Lambda_{\cD}I_2\\\left(A_{(0,2)}^{-1}(D_{t}-i\gamma)^2+A_{(0,2)}^{-1}A_{(2,0)}D_{x_1}^2\right)\Lambda^{-1}_{\cD}&A_{(0,2)}^{-1}A_{(1,1)}D_{x_1}\end{pmatrix}\\
&\cB(v,\cD)=\begin{pmatrix}-d_{v_1}H(v_1,h(v))D_{x_1}\Lambda^{-1}_{\cD}&I_2\end{pmatrix}.
\end{split}
\end{align}
Denoting the matrix symbols of $\cA(v,\cD)$ and $\cB(v,\cD)$ by $\cA(v,\sigma,\xi_1,\gamma)$, $\cB(v,\sigma,\xi_1,\gamma)$ and setting $z=(\sigma,\xi_1,\gamma)$, when $\gamma>0$ we let $E^+(v,z)$ denote the direct sum of the generalized eigenspaces of $\cA(v,z)$ associated to the eigenvalues with positive imaginary part. This two-dimensional space has a continuous extension $E^+(\sigma,\xi_1,0)$ to points $(\sigma,\xi_1,0)\neq 0$.
\begin{defn}\langle bel{lopa}
Let $\mathcal{X}:=\{z=(\sigma,\xi_1,\gamma)\neq 0:\gamma\geq 0\}$.
1) The operators $(\cA(v,\cD),\cB(v,\cD))$ (or $(P^v,B^v)$) satisfy the Lopatinskii condition at $(v,z)$ provided
\begin{align}
\cB(v,z):E^+(v,z)\to \mathbb{C}^2
\end{align}
is an isomorphism.
2) One can locally choose a basis $\{r_1(v,z), r_2(v,z)\}$ for $E^+(v,z)$ that is $C^\infty$ in $\gamma>0$ and which extends continuously to $\gamma=0$.
We define the $(2\times 2)$ Lopatinskii determinant
\begin{align}\langle bel{lop}
D(v,z)=\det \left(\cB(v,z)r_1(v,z),\cB(v,z)r_2(v,z)\right):=\det b^+(v,z).
\end{align}
Clearly, $D(v,z)\neq 0$ if and only if the Lopatinskii condition holds at $(v,z)$.
3) Let $\partialelta(v,z,\xi_2)$ be the determinant of the principal symbol of $D_{x_2}-\cA(v,\cD)$. The \emph{elliptic region} at $v$ is the the set of $(\sigma,\xi_1,0)\in \mathcal{X}$ such that $\partialelta(v,\sigma,\xi_1,0,\xi_2)$ has no real roots in $\xi_2$.
\end{defn}
Here we list the properties of $(P^v,B^v)$ that are important for proving energy estimates and existence theorems.
(P1) The operator $P^0$ is strictly hyperbolic.
(P2) The boundary $x_2=0$ is noncharacteristic for $P^0$.
(P3) (LU) The Lopatinskii determinant $D(0,\sigma,\xi_1,\gamma)$ for $(P^0,B^0)$ is nonvanishing at all points $(\sigma,\xi_1,\gamma)$ with $\gamma>0$ and at all points $(\sigma,\xi_1,0)\in\mathcal{X}$ that lie outside the elliptic region at $v=0$.
(P4) (LC) If $\underline {z}=(\underline \sigma,\underline \xi_1,0)\neq 0$ is such that $D(0,\underline z)=0$, then $\partial_\sigma D(0,\underline z)\neq 0$.
(P5) (R$_v$) Suppose $D(0,\underline{z})=0$. There exists a $\mu_0>0$ such that if $|v|<\mu_0$ the following holds: there exist $2\times 2$ matrices $k_i(v,z)$, $i=1,2$, defined in a conic neighborhood ${\mathbb G}amma$ of $\underline{z}$, which are $C^\infty$, homogeneous of degree zero in $z$, and elliptic, such that for $z\in{\mathbb G}amma$
\begin{align}\langle bel{d4c}
b^+(v,z)=k_1(v,z)\begin{pmatrix}q(v,z)\Lambda^{-1}(z)&0\\0&1\end{pmatrix}k_2(v,z), \text{ where }q(v,\sigma,\xi_1,\gamma)=\sigma-i\gamma -\nu(v,\xi_1).
\end{align}
Here, $\nu(v,\xi_1)$ is a \emph{real}-valued $C^\infty$ function homogeneous of degree one in $\xi_1$; we note that $\underline{\xi}_1\neq 0$.
The above five properties, which are verified in [S-T], (pp. 268-270 and p. 283), are sufficient for proving the a priori estimates of section {\rm Re }\, f{main}.
These properties allow us to cover the half sphere $\Sigma=\{z=(\sigma,\xi_1,\gamma)\in \mathcal{X}: |z|^2=1\}$ by a finite number of open sets $\cO_j$, $j\in J_a\cup J_c$ such that for each $j\in J_a$, $D(0,z)$ is bounded away from $0$ in $\cO_j$, and for each $j\in J_c$, $D(0,z)=0$ at some point of $\cO_j$. Moreover, we can choose the $\cO_j$ such that if $\mu_1>0$ is small enough and $|v|\leq \mu_1$, we have:
(i) $D(v,z)$ is bounded away from zero for $z\in\cO_j$, $j\in J_a$, while for $j\in J_c$, we have $D(0,\underline{z})=0$ for some $\underline{z}\in\cO_j$ and $b^+(v,z)$ satisfies \eqref{d4c} for $z\in\cO_j$.
(ii) We can write $J_a=J_h\cup J_e$, where $J_h$ denotes the set of indices such that $\partialelta(0,z,\xi_2)$ has at least one real root $\xi_2$ for some $z\in\cO_j$, while $\partialelta(v,z,\xi_2)$ has no real roots for $j\in J_e$, $z\in\cO_j$.
(iii) For $j\in J_c$, $z\in\cO_j$ the symbol $\partialelta(v,z,\xi_2)$ has no real roots.
\begin{rem}\langle bel{diag}
1. Let $\cA(v,z)$ be the symbol of the operator $\cA(v,\cD)$ in \eqref{d4b}. The operator $P^0$ is strictly hyperbolic; consequently, for $|v|$ small the system \eqref{d4a} can be conjugated microlocally to the form called ``block structure" in the sense of Kreiss-Majda.\footnote{When $d\geq 3$, the operator $P^0$ fails to be strictly hyperbolic, and for $v$ small $P^v$ exhibits characteristics of variable multiplicity. As shown in \cite{MZ}, smooth Kreiss symmetrizers can sometimes be constructed in such situations. Unfortunately, Proposition {\rm Re }\, f{u2} shows that the results of \cite{MZ} do not apply to $(P^v,B^v)$ when $d\geq 3$.}
In particular, for $|v|$ small there exists for each $\underline{z}\in \mathcal{X}$ a $C^\infty$ invertible matrix $S(v,z)$, homogeneous of degree zero and defined in a conic neighborhood of $\underline{z}$, such that
\begin{align}\langle bel{conj}
S^{-1}\mathcal{A}S=\mathrm{diag}(a_1,\dots,a_k,a^+,a^-),
\end{align}
where the blocks $a^\pm$ satisfy $\pm\mathrm{Im}\;a^\pm>0$, and the blocks $a_j$ have Jordan form at $(0,\underline{z})$ with the single real eigenvalue $\langle mbda_j$. We refer to Chapter 7 of \cite{CP} for the full definition.
2. If $\underline{z}$ lies in the elliptic region at $v=0$, then for $|v|$ small and $z$ in a conic neighborhood of $\underline{z}$ the $4\times 4$ matrix $S(v,z)=[S^+ \;S^-]$ is chosen so that the 2 columns of $S^+(v,z)$ give a basis of $E^+(v,z)$. We define $b^\pm(v,z)=\cB(v,z)S^\pm(v,z)$; thus, in this case the vectors $r_1,r_2$ in \eqref{lop}satisfy $[r_1,r_2]=S^+$.
\end{rem}
We let $\{\phi_j(\sigma,\xi_1,\gamma), j\in J_e\cup J_h\cup J_c\}$ be a partition of unity subordinate to the covering $\{\cO_j\}$, consisting of symbols homogeneous of degree zero. The neighborhoods $\cO_j$ may be (and are) chosen so that each is a neighborhood in which some matrix $S(v,z)$ as above defined. In particular, we have
\begin{align}\langle bel{conj2}
S^{-1}\mathcal{A}S=\mathrm{diag}(a^+,a^-) \text{ for }|v|<\mu_1, z\in\cO_j, \text{ when }j\in J_e\cup J_c.
\end{align}
For each $S$ we will also denote by $S$ an elliptic extension of $S$ to all of $\mathcal{X}$.
The following additional properties, which are verified in [S-T], p. 286, are used in proving the existence theorems:\footnote{The labels F.G. (``formule de Green") and A.A. (``auto-adjoint"), like LU (``Lopatinskii uniform") and LC (``controle de Lopatinskii") above, are borrowed from \cite{S-T} and used here to facilitate comparison with that paper.}\\
(P6) (F.G.) : For $|v|$ small enough, $(P^v,B^v)$ satisfies the following Green's formula: there exist differential operators $\tilde B$, $A$, $\tilde A$ such that for all $u$, $v$ in $C^\infty_0(\overline{\mathbb{R}^3_+})$ (we drop the superscript $v$ here):
\begin{align}\langle bel{d5}
(Pu,w)_{L^2(t,x)}=(u,P^*w)_{L^2(t,x)}+\langle ngle Bu(0),\tilde A w(0)\rangle ngle_{L^2(t,x_1)}+\langle ngle Au(0),\tilde B w(0)\rangle ngle_{L^(t,x_1)}.
\end{align}
The operator $\tilde B$ has order $1$, while $A$ and $\tilde A$ have order $0$.
(P7) (A.A.) The problem $(P^v,B^v)$ satisfies (F.G.) and the operators $P^*$ and $\tilde B$ have the same principal parts as $P^v$ and $B^v$, respectively.
\begin{rem}\langle bel{d6}
1) An essential consequence of (P5) is that the Lopatinski determinant, when computed at states $v\neq 0$ but near $0$, can only vanish when $\gamma=0$. In other words for small enough perturbations $v$ the Lopatinskii determinant of the perturbed operators $(P^v, B^v)$, like that of $(P^0,B^0)$, vanishes only at points where $\gamma=0$.
In a weakly stable problem, where one has $D(0,\underline{z})=0$ for some $\underline{z}$ with $\gamma=0$, even if one knows that $D(0,z)$ can only vanish when $\gamma=0$, for the nonlinear theory one still has to rule out the possibility that $D(v,z)=0$ for some $z$ with $\gamma>0$ (an exponentially growing mode) for states $v$ near $0$. That is accomplished by verifying (P5) in this case.
Another consequence explained below is that $(R_v)$ allows us to avoid the need for a sharp Garding inequality in the singular calculus.\footnote{This is fortunate, since we have no such inequality.}
2) Caution: Glancing points as well as points in the hyperbolic and mixed hyperbolic-elliptic regions lie in the support of $\phi_j$ for some $j\in J_h$.
\end{rem}
The paper \cite{S-T} verified properties (P1)-(P7) for the SVK system with $\langle mbda>0$, $\mu>0$; in fact that verification works just as well when $\mu>0$, $\langle mbda+\mu>0$.
In section {\rm Re }\, f{generalisotropic} we explain why (P1)-(P7) also hold under the more general assumption (A1g).
\subsection{Main results}\langle bel{mainr}
\emph{\quad} Our first main result asserts the existence of exact solutions to the coupled singular problems \eqref{a7}-\eqref{a9} on a fixed time interval independent of ${\varepsilon}$. An immediate consequence is the existence of exact solutions to the original (nonsingular) problem \eqref{a2} on such a time interval.
\begin{theo}\langle bel{uniformexistence}
(a) Assume (A1) and let $\beta$ as in \eqref{a3} be a Rayleigh frequency. Suppose $m>3d+4+\frac{d+1}{2}$ and
consider the coupled singular problems \eqref{a7}, \eqref{a8}, \eqref{a9}, where $G(x',\theta)\in H^{m+3}(b\Omega)$ and vanishes in $t<0$.\footnote{The choice $m>3d+4+\frac{d+1}{2}$ is determined by the requirements of the singular calculus, Appendix {\rm Re }\, f{calculus}.} There exist positive constants ${\varepsilon}_0$ and $T_1$ and unique $C^2$ solutions $v^{\varepsilon}(t,x,\theta)=(v^{\varepsilon}_1,v^{\varepsilon}_2)$ and $u^{\varepsilon}(t,x,\theta)$ to the coupled problems on the time interval $[0,T_1]$ for ${\varepsilon}\in (0,{\varepsilon}_0]$; the constant $T_1$ is independent of ${\varepsilon}\in (0,{\varepsilon}_0]$. Moreover, we have $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_1}$.
(b) Consequently, for ${\varepsilon}\in (0,{\varepsilon}_0]$ the function $u^{\varepsilon}$ gives the unique $C^2$ solution to the system \eqref{a5} on the time interval $[-\infty,T_1]$, and
\begin{align}\langle bel{aaa4}
U^{\varepsilon}(t,x)=u^{\varepsilon}(t,x,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}}
\end{align}
gives the unique $C^2$ \emph{Rayleigh pulse} solution to the Saint Venant-Kirchhoff system \eqref{a2} with boundary data
\begin{align}\langle bel{aa3}
g=g^{\varepsilon}(t,x_1)={\varepsilon}^2 G\left(t,x_1,\frac{\beta\cdot (t,x_1)}{{\varepsilon}}\right).
\end{align}
(c) The $E_{m,T_1}(v^{\varepsilon})$ norm is uniformly bounded for ${\varepsilon}\in (0,{\varepsilon}_0]$; this norm is defined in \eqref{c00}.
\end{theo}
The main steps in the proof of Theorem {\rm Re }\, f{uniformexistence} are Propositions {\rm Re }\, f{mainprop} and {\rm Re }\, f{c5} (which give a priori estimates uniform with respect to ${\varepsilon}$) for the singular systems \eqref{a7}-\eqref{a9}, and Propositions {\rm Re }\, f{localex} and {\rm Re }\, f{continuation}, which are local existence and continuation theorems for a fixed ${\varepsilon}$.
The main technical tools used in the proof are the calculus of singular pseudodifferential operators for pulses constructed in \cite{CGW} and summarized in Appendix {\rm Re }\, f{calculus}, and the new estimates of singular norms of nonlinear functions (including microlocal ``Rauch-lemma"-type estimates for singular norms) given in section {\rm Re }\, f{nonlinear}. Section {\rm Re }\, f{commutator} gives several commutator estimates that extend the results of \cite{CGW}.
Our second main result gives a precise sense in which the approximate solution constructed in Chapter {\rm Re }\, f{chapter4} is close to the exact solution of Theorem {\rm Re }\, f{uniformexistence}, together with a rate of convergence. The proof of Theorem {\rm Re }\, f{approxthm} depends on the results of chapters {\rm Re }\, f{chapter3} and {\rm Re }\, f{chapter4} and is completed in chapter {\rm Re }\, f{chapter5}.
\begin{theo}\langle bel{approxthm}
We make the same assumptions as in Theorem {\rm Re }\, f{uniformexistence}, let $\delta>0$, and let $u^{\varepsilon}$, $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ be as in Theorem {\rm Re }\, f{uniformexistence}.
Let
$$
u^{\varepsilon}_a(t,x,\theta)={\varepsilon}^2u^{\varepsilon}_\sigma(t,x,\theta)+{\varepsilon}^3 u^{\varepsilon}_\tau(t,x,\theta)
$$
be the approximate solution to the singular system \eqref{a5} given in \eqref{p0} and let $v^{\varepsilon}_a={\nabla}bla_{\varepsilon} u^{\varepsilon}_a$. There exist positive constants ${\varepsilon}_1\leq {\varepsilon}_0$ and $T_2\leq T_1$ such that for ${\varepsilon}\in (0,{\varepsilon}_1]$ we have on $\Omega_{T_2}$,
\begin{align}\langle bel{az1}
E_{m,T_2}(v^{\varepsilon}-v^{\varepsilon}_a)\leq C_\delta {\varepsilon}^{\frac{1}{4}-\delta}.
\end{align}
\end{theo}
The norm $E_{m,T_2}$ \eqref{c00} is a sum of 18 terms.
As a corollary of Theorem {\rm Re }\, f{approxthm} and the estimates of $u^{\varepsilon}_\tau$ given in section {\rm Re }\, f{bblock} we obtain, for example, the following result for the exact solution $U^{\varepsilon}(t,x)$ to the original problem \eqref{a2}.
\begin{cor}\langle bel{corapprox}
Let $\delta>0$ and set
$$
U^{\varepsilon}(t,x)=u^{\varepsilon}(t,x,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}} \text{ and }U^{\varepsilon}_\sigma(t,x)={\varepsilon}^2u^{\varepsilon}_\sigma(t,x,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}} .
$$
There exist positive constants ${\varepsilon}_1\leq {\varepsilon}_0$ and $T_2\leq T_1$ such that for ${\varepsilon}\in (0,{\varepsilon}_1]$,
we have on $[-\infty,T_2]\times \mathbb R^2_+$:
\begin{align}
\begin{split}
& (a) \frac{1}{{\varepsilon}}|{\nabla}bla_{t,x}(U^{\varepsilon}(t,x)-{\varepsilon}^2U^{\varepsilon}_\sigma(t,x))|_{L^\infty}\leq C_\delta {\varepsilon}^{\frac{1}{4}-\delta} \text{ and }\\
& (b) |\partial^\alpha_{t,x}(U^{\varepsilon}(t,x)-{\varepsilon}^2U^{\varepsilon}_\sigma(t,x))|_{L^\infty}\leq C_\delta {\varepsilon}^{\frac{1}{4}-\delta}, \text{ where }\alpha=(\alpha_t,\alpha_{x_1},\alpha_{x_2})\text{ with } |\alpha|=2, \alpha_t\leq 1.
\end{split}
\end{align}
\end{cor}
\begin{proof}[Proof of Corollary]
The second estimate follows from \eqref{az1} and the presence of the terms $\left|\begin{pmatrix}\Lambda v_j\\partial_{x_2} v_j\end{pmatrix}\right|^{(2)}_{\infty,m,T_2}$, $j=1,2$ in the definition of
the $E_{m,T_2}$ norm \eqref{c0a}. Similarly, the first estimate follows from the presence of the term
$\left|\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{\infty,m,T_2}$ in the definition of the $E_{m,T_2}$ norm, together with Remark {\rm Re }\, f{extracontrol}.
\end{proof}
\subsection{Survey of the proofs. } \langle bel{survey} \emph{\quad} In the setting of the original nonsingular problem \eqref{a2} with non-oscillating data, Sable-Tougeron \cite{S-T} has shown that if sufficiently precise estimates can be obtained for the linearized problems corresponding to \eqref{a7}-\eqref{a9}, then the nonlinear problems can be solved by a standard fixed point iteration applied to (the nonsingular analogues of) the coupled problems \eqref{a7}-\eqref{a9}, even though the linearized estimates for the $v_1$ problem \eqref{a7} exhibit a loss of one derivative on the boundary.\footnote{With some abuse we shall refer to estimates for the linearized problems as ``the linearized estimates".} Roughly, the necessary precision is gained by taking advantage of the fact that the uniform Lopatinskii condition fails for \eqref{a7} in the \emph{elliptic} region to derive estimates that minimize the loss for pieces of the solution microlocalized to that region, and which exhibit no loss for pieces microlocalized away from the bad set. At the same time this argument takes advantage of two gains that derive from considering coupled differentiated problems like \eqref{a7}-\eqref{a9}: the fact that the coefficients now depend on the unknown $v$ itself rather that the gradient of the unknown as in \eqref{a5}, and the fact that the problem for $v_2$ \eqref{a8} is a Dirichlet problem, so the associated linearized problem satisfies the uniform Lopatinskii condition and solutions exhibit no loss of derivatives. In our study of the singular problems in this chapter we shall make use of all these ideas. Naturally, new difficulties arise due to the singular nature of the problems, and our purpose here is to summarize what is new in our approach to dealing with these.
The first step, carried out in section {\rm Re }\, f{main}, is to obtain the basic $L^2$ estimates that are satisfied by solutions of the linearized singular systems \eqref{b1}-\eqref{b2}.
This is done in Proposition {\rm Re }\, f{bb1}, whose proof is a fairly straightforward adaptation to the singular setting of an analogous result of \cite{S-T}.
To see the loss on the boundary in the estimates \eqref{bb2} for the Neumann-type problem \eqref{b1} relative to the estimates \eqref{bb3} for the Dirichlet problem \eqref{b2}, one must recall that $\cG_1$ is the forcing term in a first-order boundary condition, while $\cG_2$ is the forcing term in a zero-order boundary condition. Thus, there is a loss of one \emph{singular} derivative $(\Lambda_D)$ on the boundary in the estimate for $v_1$ relative to the corresponding estimate for $v_2$.\footnote{The singular operator $\Lambda_D$ and the singular norms that appear in the estimates of this chapter are defined in part (g) of Notations {\rm Re }\, f{spaces}.}
The next step is to obtain the ``slightly higher derivative" estimates of Proposition {\rm Re }\, f{basicest}.
A serious obstacle arises in the proof of \eqref{b4} and \eqref{b7}, where one seeks to control the $L^\infty(x_2,L^2(t,x_1,\theta))$ norm uniformly with respect to ${\varepsilon}$. Such control is essential for later getting $L^\infty(t,x,\theta)$ bounds. In a nonsingular problem one would use the equation and the fact that the boundary is noncharacteristic to control the $L^2$ norm of one normal derivative $\partial_{x_2}v$ using prior $L^2$ control of first-order tangential derivatives $\partial_{(t,x_1,\theta)}v$. In a singular problem the factors of $\frac{1}{{\varepsilon}}$ in the singular tangential derivatives $\partial_{t,{\varepsilon}}$ and $\partial_{x_1,{\varepsilon}}$ wreck this argument. The resolution is to use a singular operator in the extended calculus of section
{\rm Re }\, f{extended} to partition the solution into one piece with essential support in
the region $|\sigma,\xi_1,\gamma|\geq\delta |k/{\varepsilon}|$\footnote{Here $k$ is the Fourier variable dual to $\theta$ and $\delta>0$ is sufficiently small.} where the $\frac{1}{{\varepsilon}}$ factors are harmless and the above argument works, and another piece concentrated in the complementary region where $(\sigma,\xi_1)+\frac{\beta k}{{\varepsilon}}$ is nearly parallel to $\beta$. For the second piece one can use that fact that $\beta$ lies in the elliptic region to block-diagonalize the singular system and control the $L^\infty(x_2,L^2(t,x_1,\theta))$ norm by an energy estimate. This kind of difficulty always arises in singular boundary problems (see \cite{W, CGW2}, but this is the first time we have had to deal with it in a quasilinear weakly stable problem.
Another difficulty peculiar to singular problems occurs in the proof of the estimate \eqref{b3}, which one might hope to prove simply by commuting the singular operator $\Lambda^{\frac{1}{2}}_D$ through the linearized $v_1$ problem \eqref{b2} and applying the estimate \eqref{bb2}. It turns out that such an argument can be used only on certain microlocalized pieces of the solution, while other pieces require a separate new energy estimate. The two cases are distinguished by the size of the commutator errors that arise; those in the first case are controllable uniformly with respect to ${\varepsilon}$ (and they appear as the final term on the right in \eqref{b3}), while those in the second case are not. This analysis required a further development of the singular calculus of \cite{CGW} that is given in section {\rm Re }\, f{commutator}. The points discussed in this and the above paragraph are treated mainly in steps \textbf{2} and \textbf{4} of the proof of Proposition {\rm Re }\, f{basicest}. The estimates of Proposition {\rm Re }\, f{basicest} provide the foundation for Chapters 3 and 5.
These estimates introduce a technical problem into subsequent arguments that was already encountered by \cite{S-T} in the nonsingular setting. The presence of the microlocalized norms, that is the terms involving the singular operators $\phi_{j,D}$, forces us always to apply the linearized estimates to problems posed on the \emph{full} half-space $\Omega$, because there is no way to time-localize these norms. However, the other norms that appear in the linearized estimates, even the singular norms involving fractional powers of $\Lambda_D$, can be time-localized by interpolation, and because of that we are ultimately able to use the global estimates on $\Omega$ to prove a short-time nonlinear existence theorem.
In the proof of short-time existence for the nonlinear, nonsingular problem \eqref{a2} with non-oscillating data, \cite{S-T} used a clever iteration scheme to solve (the nonsingular analogues of) the coupled problems \eqref{a7}-\eqref{a9}. Calling the iterates $v^k=(v^k_1,v^k_2)$ and $u^k$, Sabl\'e-Tougeron found that the most difficult step was to get uniform (with respect to $k$) bounds for the $v^k$; for this she used her precise estimates and the other ``gains" described in the first paragraph of this section.\footnote{Here we refer to bounds in a Sobolev norm stronger than the $L^\infty$ norm.} It was possible to control the $v^k$ on a fixed time interval by working just with the two equations \eqref{a7}, \eqref{a8} and ignoring for the moment \eqref{a9}.
Once the $v^k$ were uniformly controlled, uniform bounds for the $u^k$ were readily obtained, and the existence of limits $v$ and $u$ satisfying $v={\nabla}bla u$ followed by relatively straightforward arguments.
We tried at first to prove the uniform (with respect to ${\varepsilon}$) existence theorem for the nonlinear singular systems, Theorem {\rm Re }\, f{uniformexistence}, by applying the iteration scheme of \cite{S-T} to the coupled problems \eqref{a7}-\eqref{a9}, and attempting to estimate the $v^k$, which now depend on ${\varepsilon}$, uniformly with respect to \emph{both} $k$ and ${\varepsilon}$, on a time interval independent of ${\varepsilon}$. The iteration scheme is the one given in \eqref{k0a}-\eqref{k0c}; the problem for $v^{k+1}$ is linear so one can apply the estimates of Proposition {\rm Re }\, f{basicest} to estimate $v^{k+1}$ provided one has the control of $v^k$ specified by \eqref{b3z}, where $v^k$ plays the role of $w$ in \eqref{b3z}. This approach failed. In particular, we
we were not able to bound the norm $|v^k/{\varepsilon}|_{\infty,T}$ of the $k$th iterate uniformly with respect to ${\varepsilon}$ and $k$ up to a fixed time $T$ independent of ${\varepsilon}$. If one knew $v^k={\nabla}bla_{\varepsilon} u^k$ (or could control the difference well enough), then this approach could have been made to work, but we had to abandon this idea. It seems that the relation
$v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$, which allows one to use the $u^{\varepsilon}$ equation \eqref{a9} to control $|v^{\varepsilon}/{\varepsilon}|_{\infty,T}$, holds only in the limit $k\to\infty$.
In order to take advantage of the relation $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ we use instead a continuous induction argument. This argument, given in the proof of Proposition {\rm Re }\, f{mainprop}, has three parts. First, one needs a \emph{short-time existence theorem} for ${\varepsilon}$ fixed for the nonlinear problems \eqref{a7}-\eqref{a9} on a domain $\Omega_{T_{\varepsilon}}$ that depends on ${\varepsilon}$; this is given by Proposition {\rm Re }\, f{localex}. This result also establishes the relation $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_{\varepsilon}}$.\footnote{The notation $\Omega_T$ is defined in Notations {\rm Re }\, f{spaces}(a).} Second, one needs a \emph{continuation theorem} for ${\varepsilon}$ fixed, Proposition {\rm Re }\, f{continuation}, which states that if one has a solution to \eqref{a7}-\eqref{a9} on some domain $\Omega_{T_{1,{\varepsilon}}}$ that is sufficiently small in an appropriate norm on that domain, then that solution can be extended with similar bounds to a later time $T_{2,{\varepsilon}}>T_{1,{\varepsilon}}$. The appropriate norm turns out to be the singular energy norm $E_{m,T_{1,{\varepsilon}}}(v)$ \eqref{c00} for $m$ sufficiently large. Finally, the argument requires an \emph{a priori estimate} for the $E_{m,\gamma}$ norm of solutions to the modified systems \eqref{c1}-\eqref{c3} on $\Omega$. This estimate is given in Proposition {\rm Re }\, f{c5}; the constants in the estimate depend only on $M_0$ as in \eqref{c0h}, a number that is independent of the parameters ${\varepsilon}$ and $T$ on which \eqref{c1}-\eqref{c3} depend.
A detailed explanation of how this estimate can be localized in time and combined with the other two results to obtain existence for \eqref{a7}-\eqref{a9} on a domain $\Omega_T$ with $T>0$ independent of ${\varepsilon}$ is given in section {\rm Re }\, f{strategy}.\footnote{We switched to a ``continuous induction" argument after studying the thesis of \cite{Pa}, where such a strategy is used on a quite different problem.}
To understand the form of the modified systems \eqref{c1}-\eqref{c3}, recall that although we are not using an iteration scheme, we still need to work with systems to which we can apply the linearized estimates. Observe that with one important exception (the right side of the boundary condition in \eqref{c2}; see Remark {\rm Re }\, f{c4a}), the coefficients in the modified systems all depend on $v^s$ rather than the unknown $v^{\varepsilon}$. Here $v^s$ is our notation with two parameters suppressed for $v^{{\varepsilon},s}_T$, a carefully chosen Seeley extension (Proposition {\rm Re }\, f{c0e}) to $t>T$ of $v^{\varepsilon}|_{\Omega_T}$, where $v^{\varepsilon}$ is the solution on $\Omega_{T_{\varepsilon}}$ to the nonlinear problems \eqref{a7}-\eqref{a8} provided by the local existence result, Proposition {\rm Re }\, f{localex}, and where $0<T<T_{\varepsilon}$. A consequence of the linear existence theorem, Theorem {\rm Re }\, f{exist}, is that the linearized singular problems exhibit \emph{causality}; roughly, solutions in $t<T$ are unaffected by changing the forcing terms and coefficients in $t>T$. Thus, for any choice of $T\in (0,T_{\varepsilon})$, we know that the solution $v^{\varepsilon}$ to the modified problems \eqref{c1}-\eqref{c2} on the full half space $\Omega$ agrees with the already given $v^{s,{\varepsilon}}_T=v^{\varepsilon}$ on $\Omega_T$. When we take into account the relations between the global norm $E_{m,\gamma}(v^{\varepsilon})$ and the time-localized norm $E_{m,T}(v^{\varepsilon})$ given by Proposition {\rm Re }\, f{c0e}, we are then able to estimate solutions of the nonlinear problems \eqref{a7}-\eqref{a8} on $\Omega_T$ by estimating solutions of the linear problems \eqref{c1}-\eqref{c2} on the full space $\Omega$.
The most difficult of the three steps is the proof of the a priori estimate. This is carried out in section {\rm Re }\, f{mainestimate}, which is the heart of the rigorous analysis.
To take advantage of the relation $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_{\varepsilon}}$, we must estimate solutions to the three systems \eqref{c1}-\eqref{c3} ``simultaneously" in the following sense. When we apply the linearized estimates of section {\rm Re }\, f{b1a} to the problem \eqref{c1} or \eqref{c2}, we find that terms arise that can only be controlled by estimating solutions to \eqref{c3}. Similarly, estimates of solutions to \eqref{c3} lead to terms that can only be controlled by estimating solutions to \eqref{c1} and \eqref{c2}. An example of this occurs in the first interior commutator estimate \eqref{e24az}, where the terms involving factors of $\frac{1}{{\varepsilon}}$ on the right can be controlled only by estimating solutions of \eqref{c3} and using the relation
$v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$. In a sense we have to estimate more than three systems simultaneously, since we estimate not only \eqref{c1}-\eqref{c3}, but also $\frac{1}{\sqrt{{\varepsilon}}}\eqref{c1}$, $\sqrt{{\varepsilon}}\eqref{c2}$, $\frac{1}{{\varepsilon}}\eqref{c3}$, and so on; the complete list is given in the outline of section {\rm Re }\, f{outline}. The norm $E_{m,\gamma}(v)$ that appears in the a priori estimate of Proposition {\rm Re }\, f{c5} must incorporate all the time-localizable norms that arise in estimating all these systems; thus, it is not surprising that it contains a rather large number of terms. In fact each term $E_{m,\gamma}(v_j)$ \eqref{c0a} in the definition of $E_{m,\gamma}$ is a sum of 18 terms. It is perhaps surprising that any \emph{finite} number of terms works!
We note that the interior and boundary commutator estimates of section {\rm Re }\, f{mainestimate} can be (and are) applied without change in the error analysis of section {\rm Re }\, f{chapter5}. However, the boundary and interior forcing estimates of chapter {\rm Re }\, f{chapter5} are substantially different.
Although we have used the singular calculus of \cite{CGW} in several earlier papers, for example \cite{CGW2,CW1,CW2}, this is the first time we have had to estimate singular norms
$\langle ngle \Lambda^r f(u)\rangle ngle_{m,\gamma}$ (defined below in Notations {\rm Re }\, f{spaces}(g)) of nonlinear functions of $u$. In fact, to take advantage of the extra microlocal precision in the linearized estimates of Proposition {\rm Re }\, f{basicest}, we also need to show that in some cases even \emph{microlocal} regularity of $u(t,x,\theta)$ in singular norms is preserved under nonlinear functions. The microlocal results are a singular analogue of the classical Rauch's lemma \cite{R}. Both types of results are new for singular norms and are proved in section {\rm Re }\, f{singular}, for example in Propositions {\rm Re }\, f{f3} and {\rm Re }\, f{f5}. We use these tools throughout chapters {\rm Re }\, f{chapter3} and {\rm Re }\, f{chapter5}. In estimates of $\langle ngle \Lambda^r f(u)\rangle ngle_{m,\gamma}$ where $r>0$, we have to assume that $f$ is real-analytic; we do not know if this can be weakened to, say, $C^\infty$. This restriction still allows us to handle the SVK system, and is the reason for the condition of analyticity of $W(E)$ in Assumption (A1g). Both tame and (simpler) non-tame versions of these estimates are given. The tame estimates are used mainly in the proof of the continuation result, Proposition {\rm Re }\, f{continuation}.
We conclude this survey with a few remarks on the local existence and continuation results for fixed ${\varepsilon}$ of section {\rm Re }\, f{local}. In this section we take advantage of the fact that when ${\varepsilon}$ is fixed, one \emph{can} use the equation to control normal derivatives $\partial_{x_2}v$ in terms of singular tangential derivatives $\partial_{t,{\varepsilon}}v$, $\partial_{x_1,{\varepsilon}}v$. However, even when ${\varepsilon}$ is fixed the nonlinear systems being studied are still singular, because $\partial_{x'}$ and $\partial_\theta$ derivatives occur in the linear combination $\partial_{x'}+\beta\frac{\partial_\theta}{{\varepsilon}}$. So the singular calculus is still needed to prove the linearized estimates on which the argument is based. Given those estimates and the results of section {\rm Re }\, f{singular}, the proof of the local existence theorem is a matter of adapting the argument of \cite{S-T} to singular problems. For example, the iteration scheme \eqref{k0a}-\eqref{k0c} is the singular analogue of the one used in \cite{S-T}. This argument is based on the non-tame linearized estimates \eqref{k1}-\eqref{k2}. The proof of the continuation result is based on a tame version of the linearized estimates given in Proposition {\rm Re }\, f{tame2}. We first prove existence of a low regularity continuation essentially by repeating the local existence proof; we then use the tame linearized estimates to show that this continuation has the desired higher regularity.
\section{The basic estimates for the linearized singular systems}\langle bel{main}
\emph{\quad} With apologies to the reader for the following long list, we begin by gathering in one place most of the notation for spaces and norms that is needed below.
\begin{nota}\langle bel{spaces}
We take $m$ to be a nonnegative integer, let $(\xi',k)=(\sigma,\xi_1,k)$ be the Fourier transform variables dual to $(x',\theta)=(t,x_1,\theta)$,
and set $\langle ngle \xi',k\rangle ngle=(|\xi'|^2+k^2+1)^{1/2}$, $\langle ngle \xi',k,\gamma\rangle ngle=(|\xi'|^2+k^2+\gamma^2)^{1/2}$.
(a)\;Let $\Omega:=\{(t,x_1,x_2,\theta)\in\mathbb{R}^{4}:x_2>0\}$, and for $T>0$ set $\Omega_T:=\Omega\cap\{-\infty<t<T\}$,
$b\Omega:=\{(t,x_1,\theta)\in\mathbb{R}^3$, and $b\Omega_T:=b\Omega\cap \{-\infty<t<T\}$.
(b)\;Let $H^m\equiv H^m(b\Omega)$ be the standard Sobolev space with norm $\langle ngle V(x',\theta)\rangle ngle_m=|\langle ngle \xi',k\rangle ngle^m \hat V(\xi',k)|_{L^2(\xi',k)}$.
For $\gamma\geq 1$ and $V\in H^m$ we define $|V|_{H^m_\gamma}=|\langle ngle \xi',k,\gamma\rangle ngle^m \hat V(\xi',k)|_{L^2(\xi',k)}$. For $V\in e^{\gamma t} \, H^m$ we set $\langle ngle V\rangle ngle_{m,\gamma} := |e^{-\gamma t} \, V |_{H^m_\gamma}$. We sometimes write $V^\gamma=e^{-\gamma t}V$.
(c)\;$L^2H^m\equiv L^2(\overline{\mathbb{R}}_+,H^m(b\Omega))$ with norm $|U(t,x,\theta)|_{L^2H^m} \equiv
|U|_{0,m}$ given by
\begin{equation*}
|U|_{0,m}^2=\int^\infty_0\langle ngle U(x',x_2,\theta)\rangle ngle_m^2 dx_2.
\end{equation*}
Similarly, for $U\in L^2e^{\gamma t}H^m:=L^2(\overline{\mathbb{R}}_+,e^{\gamma t}H^m(b\Omega))$ we set
\begin{align}
|U|_{0,m,\gamma}^2=\int^\infty_0\langle ngle U(x',x_2,\theta)\rangle ngle_{m,\gamma}^2 dx_2.
\end{align}
(d)\;$CH^m\equiv C(\overline{\mathbb{R}}_+,H^m(b\Omega))$ denotes the space of continuous bounded
functions of $x_2$ with values in $H^m(b\Omega)$, with norm $|U(t,x,\theta)|_{CH^m} =|U|_{\infty,m} :=
\sup_{x_2\geq 0} |U(.,x_2,.)|_{H^m(b\Omega)}$
The corresponding
norm on $CH^m\equiv C(\overline{\mathbb{R}}_+,e^{\gamma t}H^m(b\Omega))$is denoted $|V|_{\infty,m,\gamma}$.
(e) For a nonnegative integer $M$ and define $C^{0,M} :=C(\overline{\mathbb{R}}_+,C^{M}(b\Omega))$ as the space of
continuous bounded functions of $x_2$ with values in $C^{M}(b\Omega)$, with norm $|U(x',x_2,\theta)|_{C^{0,M}}
:= |U|_{L^\infty W^{M,\infty}}$. Here $L^\infty W^{M,\infty}$ denotes the space $L^\infty(\overline{\mathbb R}_+;
W^{M,\infty}(b\Omega))$.
(f)The corresponding spaces on $\Omega_T$ are denoted $L^2H^m_T$, $L^2e^{\gamma t}H^m_{T}$, $CH^m_T$,
$Ce^{\gamma t}H^m_{T}$ and $C^{0,M_0}_T$ with norms $|U|_{0,m,T}$, $|U|_{0,m,\gamma,T}$, $|U|_{\infty,m,T}$,
$|U|_{\infty,m,\gamma,T}$, and $|U|_{C^{0,M_0}_T}$ respectively. These norms have clear meanings when $m$ is a nonnegative integer; otherwise they are defined as usual by interpolation. On $b\Omega_T$ we use the spaces
$H^m_T$ and $e^{\gamma t}H^m_{T}$ with norms $\langle ngle U\rangle ngle_{m,T}$ and $\langle ngle U\rangle ngle_{m,\gamma,T}$.
(g) For $r\in\overline{\mathbb{R}}_+$ let $\Lambda^r_D$ be the singular operator associated to the singular symbol $\Lambda^r(\xi'+\frac{\beta k}{{\varepsilon}},\gamma):=(|\xi'+\frac{\beta k}{{\varepsilon}}|^2+\gamma^2)^{r/2}$ (see \eqref{singularpseudop}). With $\xi'=(\sigma,\xi_1)$ for $V=V(x',\theta)$ we define (with slight abuse of the notation $\langle ngle\cdot\rangle ngle_{m,\gamma}$ defined in (b)\footnote{Observe that for $\langle ngle \Lambda^r V \rangle ngle_{m,\gamma}$ as defined in \eqref{b0} and $\langle ngle w\rangle ngle_{m,\gamma}$ as in (b), we have
\begin{align}\notag
\langle ngle\Lambda^r V\rangle ngle_{m,\gamma}=\langle ngle w\rangle ngle_{m,\gamma}, \text{ where }w=e^{\gamma t}\Lambda^r_D(e^{-\gamma t}V).
\end{align}
}) the singular norms
\begin{align}\langle bel{b0}
\begin{split}
&\langle ngle\Lambda^r V\rangle ngle_{m,\gamma}:=|\Lambda^r_D (e^{-\gamma t}V)|_{H^m_\gamma}=\left|(|\xi'+\frac{\beta k}{{\varepsilon}}|^2+\gamma^2)^{r/2}\langle ngle \xi',k,\gamma\rangle ngle^m\hat V(\sigma-i\gamma,\xi_1,k)\right|_{L^2(\xi',k)}\\
&\langle ngle\Lambda^r_{1} V\rangle ngle_m:=\left|(|\xi'+\frac{\beta k}{{\varepsilon}}|^2+1)^{r/2}\langle ngle\xi',k\rangle ngle^m\hat V(\xi',k)\right|_{L^2(\xi',k)}.
\end{split}
\end{align}
(h)When $r$ is a nonnegative integer the norms defined in (g) can be localized to $b\Omega_T$ by replacing $\Lambda_D V$ by $((\partial_{x'}+\beta\frac{\partial_\theta}{{\varepsilon}})V,\gamma V)$. Denote the localized norms by $\langle ngle\Lambda^r V\rangle ngle_{m,\gamma,T}$ and $\langle ngle\Lambda^r_{1} V\rangle ngle_{m,T}$.
For non-integer $r>0$, the localized norms are defined by interpolation from the integral case.\footnote{A concise reference for the method of complex interpolation used here is \cite{T}, Chapter 4.}
(i)Having defined these singular norms of functions on $b\Omega$ and $b\Omega_T$, we define the corresponding norms for functions $U(x',x_2,\theta)$ on $\Omega$ and
$\Omega_T$ by obvious analogy with (c) and (d). We denote these norms $|\Lambda^r U|_{0,m,\gamma}$, $|\Lambda^r U|_{\infty,m,\gamma}$, $|\Lambda_1^r U|_{0,m}$, $|\Lambda^r_1 U|_{\infty,m}$; the corresponding time-localized norms are $|\Lambda^r U|_{0,m,\gamma,T}$, $|\Lambda^r U|_{\infty,m,\gamma,T}$, $|\Lambda_1^r U|_{0,m,T}$, $|\Lambda^r_1 U|_{\infty,m,T}$.
(j) If $\phi_D$ is a singular pseudodifferential operator associated to the singular symbol $\phi({\varepsilon} W(x',\theta), \xi'+\frac{\beta k}{{\varepsilon}},\gamma)$, we set\begin{align}\langle bel{b0a}
\begin{split}
&\langle ngle\phi V\rangle ngle_{m,\gamma}:=|\phi_D (e^{-\gamma t}V)|_{H^m_\gamma}.
\end{split}
\end{align}
We do not attempt to define a time-localized version of \eqref{b0a}.
(k)We will sometimes (for example, in convolutions) write $X:=\xi'+\beta\frac{k}{{\varepsilon}}$, $Y=\eta'+\beta\frac{l}{{\varepsilon}}$, where $(\eta',l)$ also denotes variables dual to $(x',\theta)$.
(l) All constants appearing in the estimates below are independent of ${\varepsilon}$, $\gamma$, and $T$ unless such
dependence is explicitly noted.
(m) We use $\partial$ to denote a tangential derivative with respect to one of the variables $(x',\theta)$.
(n) We set $D_{x_2,{\varepsilon}}=\frac{1}{i}\partial_{x_2}$, $D_{t,{\varepsilon}}=\frac{1}{i}\partial_{t,{\varepsilon}}$, $D_{x_1,{\varepsilon}}=\frac{1}{i}\partial_{x_1,{\varepsilon}}$, $D_{x',{\varepsilon}}=(D_{t,{\varepsilon}},D_{x_1,{\varepsilon}})$.
(o) Set $\Lambda^r_{D,\gamma}=e^{\gamma t}\Lambda^r_D e^{-\gamma t}$.
\end{nota}
\subsection{Statement of the estimates. }\langle bel{b1a}
\emph{\quad} Here we give the main estimates for the ``linearized versions" of \eqref{a7} and \eqref{a8}, namely,
\begin{align}\langle bel{b1}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_1^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha(w^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha v_1^{\varepsilon}=\mathcal{F}_1\text{ on }\Omega\\
&\partial _{x_2} v^{\varepsilon}_1-d_{v_1}H(w^{\varepsilon}_1,h(w^{\varepsilon}))\partial_{x_1,{\varepsilon}}v^{\varepsilon}_1=\mathcal{G}_1\text{ on }x_2=0
\end{split}
\end{align}
and
\begin{align}\langle bel{b2}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_2^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha(w^{\varepsilon})\partial_{x,{\varepsilon}}^\alpha v_2^{\varepsilon}=\mathcal{F}_2\text{ on }\Omega\\
&v^{\varepsilon}_2=\mathcal{G}_2\text{ on }x_2=0.
\end{split}
\end{align}
Here the functions $v^{\varepsilon}$, $w^{\varepsilon}$, $\mathcal{F}_i$, $\mathcal{G}_i$ all vanish in $t<0$.
The operators $\{\phi_{j,D}, j\in J_e\cup J_h\cup J_c\}$ that appear below form a (singular) pseudodifferential partition of the identity operator; they have symbols $\phi_j(X,\gamma)$, where the $\phi_j(\xi',\gamma)$ define the partition of unity chosen in section {\rm Re }\, f{assumptions}.
\begin{prop}\langle bel{basicest}
Suppose $n\geq 3d+4$, $s_0>\frac{d+1}{2}+2$, and let $\delta$ and $K$ be positive constants.\footnote{See, for example, Propositions {\rm Re }\, f{prop20} and {\rm Re }\, f{commutator4} to understand the restrictions on $n$ and $s_0$.} Assume that for ${\varepsilon}\in (0,1]$ the function $w^{\varepsilon}$ appearing in the coefficients of \eqref{b1}, \eqref{b2} satisfies
\begin{align}\langle bel{b3z}
\left|\frac{w^{\varepsilon}}{{\varepsilon}}\right|_{C^{0,n}}+\left|\frac{w^{\varepsilon}}{{\varepsilon}}\right|_{CH^{s_0}}+|\partial_{x_2}w^{\varepsilon}|_{L^\infty(\Omega)}<K , \; |w^{\varepsilon}|_{L^\infty(\Omega)}<\delta.
\end{align}
The following a priori estimates are valid provided $\delta$ is small enough; the constants $\gamma_0$ and $C$ that appear depend on $K$.
The first estimates are for the weakly stable Neumann-type problem \eqref{b1} satisfied by $v_1$: for $\gamma\geq \gamma_0$ we have
\begin{align}\langle bel{b3}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_1\\partial_{x_2}\Lambda^{\frac{1}{2}}v_1\end{pmatrix}\right|^2_{0,0,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{0,\gamma}+\sum_{J_h\cup J_e }\left < \phi_j \begin{pmatrix}\Lambda^{\frac{3}{2}} v_1\\partial_{x_2}\Lambda^{\frac{1}{2}} v_1\end{pmatrix} \right>^2_{0,\gamma}\leq \\
&\quad C \gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|^2_{0,0,\gamma}+\left< \Lambda \mathcal{G}_1\right>^2_{0,\gamma}\right)+ C\gamma^{-1}\left|\begin{pmatrix}\Lambda v_1/\sqrt{{\varepsilon}}\\partial_{x_2} v_1/\sqrt{{\varepsilon}}\end{pmatrix}\right|^2_{0,0,\gamma},
\end{split}
\end{align}
\begin{align}\langle bel{b4}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right|^2_{\infty,0,\gamma}\leq
C \gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|^2_{0,0,\gamma}+|\mathcal{F}_1|_{0,1,\gamma}^2+\left<\Lambda\mathcal{G}_1\right >^2_{0,\gamma}+\left < \Lambda^{\frac{1}{2}}\mathcal{G}_1\right>^2_{1,\gamma}\right)+\\
&\qquad \qquad \qquad \qquad C \gamma^{-1} \left|\begin{pmatrix}\Lambda v_1/\sqrt{{\varepsilon}}\\partial_{x_2} v_1/\sqrt{{\varepsilon}}\end{pmatrix}\right|^2_{0,0,\gamma}.
\end{split}
\end{align}
\begin{align}\langle bel{b5}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right|^2_{0,1,\gamma}
+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_1\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_1\end{pmatrix} \right>^2_{1,\gamma}+ \sum_{J_h \cup J_e}\left < \phi_j \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{1,\gamma}\leq\\
&\qquad\qquad\qquad C\gamma^{-1}\left(|\mathcal{F}_1|_{0,1,\gamma}^2+\left < \Lambda^{\frac{1}{2}}\mathcal{G}_1\right>^2_{1,\gamma}\right).
\end{split}
\end{align}
The next estimates are for the uniformly stable Dirichlet problem \eqref{b2} satisfied by $v_2$: for $\gamma\geq \gamma_0$
\begin{align}\langle bel{b6}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_2\\partial_{x_2}\Lambda^{\frac{1}{2}}v_2\end{pmatrix}\right|^2_{0,0,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix} \right>^2_{0,\gamma}
\leq\\
& C \left(\gamma^{-1}|\Lambda^{\frac{1}{2}}\mathcal{F}_2|^2_{0,0,\gamma}+\gamma\left< \Lambda \mathcal{G}_2\right>^2_{0,\gamma}+\sum_{J_h}\left<\phi_j\Lambda^{\frac{3}{2}}\mathcal{G}_2\right>^2_{0,\gamma}\right)+C\gamma^{-1}\left|\begin{pmatrix}\Lambda v_2/\sqrt{{\varepsilon}}\\partial_{x_2} v_2/\sqrt{{\varepsilon}}\end{pmatrix}\right|^2_{0,0,\gamma}.
\end{split}
\end{align}
\begin{align}\langle bel{b7}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix}\right|^2_{\infty,0,\gamma}\leq
C \left(\gamma^{-1}|\mathcal{F}_2|_{0,1,\gamma}^2+\gamma\left< \Lambda \mathcal{G}_2\right>^2_{0,\gamma}+\gamma\left< \Lambda^{\frac{1}{2}}\mathcal{G}_2\right>^2_{1,\gamma}+\sum_{J_h}\left<\phi_j\Lambda\mathcal{G}_2\right>^2_{1,\gamma}\right).
\end{split}
\end{align}
\begin{align}\langle bel{b8}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix}\right|^2_{0,1,\gamma}+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_2\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_2\end{pmatrix} \right>^2_{1,\gamma}
\leq
C \left(\gamma^{-1}|\mathcal{F}_2|_{0,1,\gamma}^2+\gamma\left< \Lambda^{\frac{1}{2}}\mathcal{G}_2\right>^2_{1,\gamma}+\sum_{J_h}\left<\phi_j\Lambda\mathcal{G}_2\right>^2_{1,\gamma}\right).
\end{split}
\end{align}
\end{prop}
\begin{rem}\langle bel{b9}
The terms on the right in \eqref{b3} and \eqref{b4} involving division by $\sqrt{{\varepsilon}}$ are needed to control the commutators with $\Lambda^{1/2}_D$ that appear when that operator is applied to the equation written as a first-order system.
\end{rem}
\subsection{Proofs of the estimates.}
\emph{\quad}Estimates \eqref{b3}, \eqref{b5} and \eqref{b6}, \eqref{b8} are proved by analyzing the singular systems \eqref{b1}, \eqref{b2} using the singular pulse calculus of \cite{CGW}. A first step is to rewrite the second-order differential systems as first order singular pseudo-differential systems to which we can readily apply singular Kreiss symmetrizers.
Estimates \eqref{b4} and \eqref{b7} require, in addition, a diagonalization argument using a cutoff $\chi$ \eqref{n31} in the extended calculus similar to that given in \cite{W}, but modified in the case of \eqref{b4} to take into account the weakly stable nature of the problem \eqref{b1}. The Dirichlet problem \eqref{b2} is uniformly stable, that is, the Lopatinskii determinant $D(0,z)$ (defined with $\cB_2=\begin{pmatrix}I_2&0\end{pmatrix}$ now) is nonvanishing for all $z\in\mathcal X$, so the set $J_c$ is empty in this case.
\emph{\quad} To prove the estimates for $v_1$ we set $U=(e^{\gamma t}\Lambda_{D}e^{-\gamma t} v_1,D_{x_2}v_1)=(\Lambda_{D,\gamma} v_1,D_{x_2} v_1)$, $U^\gamma=e^{-\gamma t}U$, and rewrite \eqref{b1} as a $4\times 4$ singular first-order system:
\begin{align}\langle bel{b10a}
D_{x_2}U^\gamma-\cA(w,D_{x',{\varepsilon}})U^\gamma=\tilde{\cF}_1^\gamma,\;\;\cB_1(w,D_{x',{\varepsilon}})U^\gamma=\tilde{\cG}_1^\gamma,
\end{align}
where, with the matrices $A_\alpha$ evaluated at $w$, we have
\begin{align}\langle bel{b10b}
\begin{split}
& \tilde{\cF}_1:=\begin{pmatrix}0\\-A_{(0,2)}^{-1}\cF_1\end{pmatrix},\;\;\tilde{\cG}_1=-i\cG_1\\
&\cA(w,D_{x',{\varepsilon}}):=-\begin{pmatrix}0&\Lambda_{D}I_2\\\left(A_{(0,2)}^{-1}(D_{t,{\varepsilon}}-i\gamma)^2+A_{(0,2)}^{-1}A_{(2,0)}D_{x_1,{\varepsilon}}^2\right)\Lambda^{-1}_{D}&A_{(0,2)}^{-1}A_{(1,1)}D_{x_1,{\varepsilon}}\end{pmatrix}\\
&\cB_1(w,D_{x',{\varepsilon}}):=\begin{pmatrix}-d_{v_1}H(w_1,h(w))D_{x_1,{\varepsilon}}\Lambda^{-1}_{D}&I_2\end{pmatrix}.
\end{split}
\end{align}
This is the singular analogue of \eqref{d4b}.
Writing the system \eqref{b2} as a first order system for $U=(e^{\gamma t}\Lambda_{D}e^{-\gamma t} v_2,D_{x_2}v_2)$, we obtain
\begin{align}\langle bel{b10d}
D_{x_2}U^\gamma-\cA(w,D_{x',{\varepsilon}})U^\gamma=\tilde{\cF}_2^\gamma,\;\;\cB_2U^\gamma=\tilde{\cG}_2^\gamma,
\end{align}
where $\tilde{\cF}_2$ is defined like $\tilde{\cF}_1$, but now
\begin{align}
\cB_2=\begin{pmatrix}I_2&0\end{pmatrix}\text{ and }\tilde{\cG}_2=\Lambda_{D,\gamma}\cG_2.
\end{align}
The following proposition gives the $L^2$ estimates for the singular systems \eqref{b10a}, \eqref{b10d}.
\begin{prop}\langle bel{bb1}
With notation and assumptions as in Proposition {\rm Re }\, f{basicest}, we have the following a priori estimates for the linearized singular Neumann and Dirichlet systems, \eqref{b10a} and \eqref{b10d}.
For $\gamma\geq \gamma_0$,
\begin{align}\langle bel{bb2}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right|^2_{0,0,\gamma}
+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_1\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_1\end{pmatrix} \right>^2_{0,\gamma}+ \sum_{J_h \cup J_e}\left < \phi_j \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{0,\gamma}\leq\\
&\qquad\qquad\qquad C\gamma^{-1}\left(|\mathcal{F}_1|_{0,0,\gamma}^2+\left < \Lambda^{\frac{1}{2}}\mathcal{G}_1\right>^2_{0,\gamma}\right)
\end{split}
\end{align}
and
\begin{align}\langle bel{bb3}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix}\right|^2_{0,0,\gamma}+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_2\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_2\end{pmatrix} \right>^2_{0,\gamma}
\leq
C \left(\gamma^{-1}|\mathcal{F}_2|_{0,0,\gamma}^2+\gamma\left< \Lambda^{\frac{1}{2}}\mathcal{G}_2\right>^2_{0,\gamma}+\sum_{J_h}\left<\phi_j\Lambda\mathcal{G}_2\right>^2_{0,\gamma}\right).
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }The following estimate holds for $\phi_{j,D}(e^{-\gamma t}U)$ with $j\in J_h\cup J_e$.
\begin{align}\langle bel{b10}
\begin{split}
&\gamma |\phi_j U|_{0,0,\gamma}^2+\langle ngle\phi_jU\rangle ngle^2_{0,\gamma}\leq \\
&\qquad C\left(\gamma^{-1}|\phi_j\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\phi_j\mathcal{G}_1\rangle ngle_{0,\gamma}^2+\gamma^{-1}|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-1}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
This Kreiss-type estimate is the singular analogue of estimate (1.6) in \cite{S-T}, p. 268, and it is proved by essentially the same argument; the only difference is that
all symbols, including the symbol of the Kreiss symmetrizer, are now quantized in the singular calculus by the procedure described in section {\rm Re }\, f{sect8}. For example, if we let $r(v,\xi',\gamma)$ denote the symbol of the (nonsingular) Kreiss symmetrizer used in the proof of (1.6) in \cite{S-T}, the corresponding singular operator that must be used in the proof of \eqref{b10} is the operator associated to $r(w^{\varepsilon}(x,\theta),X,\gamma)$.\footnote{Full details of how the singular calculus is implemented to prove estimates like \eqref{b10} are given in section 5 of \cite{W}.}
In view of the assumption \eqref{b3z} the function $w^{\varepsilon}(x,\theta)$ can play the role of ${\varepsilon} V(x,\theta)$ in the integral \eqref{singularpseudop}. The last two terms on the right in \eqref{b10} arise from commutators by application of Proposition {\rm Re }\, f{commutator3}.
\textbf{2. }
The following Kreiss-type estimate is a variant of \eqref{b10} that holds for $\phi_{j,D}U^\gamma$ with $j\in J_e$.
\begin{align}\langle bel{b11}
\begin{split}
&\gamma |\phi_j U|_{0,0,\gamma}^2+\gamma\langle ngle\phi_j\Lambda^{-\frac{1}{2}}U\rangle ngle^2_{0,\gamma}\leq \\
&\qquad C\left(\gamma^{-1}|\phi_j\mathcal{F}_1|_{0,0,\gamma}^2+\gamma\langle ngle\phi_j\Lambda^{-\frac{1}{2}}\mathcal{G}_1\rangle ngle_{0,\gamma}^2+\gamma^{-1}|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-1}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
This is the analogue of estimate (1.7) in \cite{S-T}, p. 268, and is again proved by using singular operators in the argument given there.
\textbf{3. } For $\phi_{j,D}U^\gamma$ with $j\in J_c$ (the ``bad" set where LU fails) we have
\begin{align}\langle bel{b12}
\begin{split}
&\gamma\left(|\phi_j U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-\frac{1}{2}}\phi_jU\rangle ngle^2_{0,\gamma}\right)\leq \\
&\qquad C\gamma^{-1}\left(|\phi_j\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda^{\frac{1}{2}}\phi_j\mathcal{G}_1\rangle ngle_{0,\gamma}^2+|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-\frac{1}{2}}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
This is the analogue of estimate (1.10) in \cite{S-T}, p. 272, and, except for an important simplification that we now discuss, is again proved by implementing the argument given there with singular operators.
Consider a zero order symbol $b^+(v,z)$ as in Remark {\rm Re }\, f{diag}, defined for $|v|$ small and $z\in\cO_j$, $j\in J_c$. Let $\tilde{b}^+(v,z)$ be an extension to $\mathcal{X}$ defined by taking extensions $\tilde{k}_1$, $\tilde{k}_2$, and $\tilde{q}$ of the factors in \eqref{d4c}, where the $\tilde{k}_i$ are elliptic of order zero and we still have $\tilde q$ of order one satisfying
\begin{align}\langle bel{b12z}
\Im \tilde q(v,z)=-\gamma.
\end{align}
Dropping tildes on extensions and letting $q_D$ denote the singular operator with symbol $q(w^{\varepsilon},X,\gamma)$, we see that the estimate
\begin{align}\langle bel{b12y}
{\mathbb R}e (iq_D\Lambda^{-1}_D v,v)\geq C\gamma |\Lambda^{-\frac{1}{2}}_D v|^2_{L^2}
\end{align}
follows immediately from \eqref{b12z}; there is no need for a singular sharp Garding inequality to replace the use of the standard sharp Garding inequality in \cite{S-T}.
With \eqref{b12y} one obtains the key estimate,
\begin{align}\langle bel{key}
|\Lambda^{\frac{1}{2}} b^+_D w|_{L^2}\geq C\gamma |\Lambda^{-\frac{1}{2}}_Dw|_{L^2},
\end{align}
which is analogous to (1.8) of \cite{S-T}, by arguing as on p. 270 of \cite{S-T}.
\textbf{4. }For later use we observe that since $q_D\Lambda^{-1}_D=(q\Lambda^{-1})_D$, we can apply \eqref{b12z} and Proposition {\rm Re }\, f{prop20}(b) to obtain
\begin{align}\langle bel{b12yy}
{\mathbb R}e (iq_D\Lambda^{-1}_D v,\Lambda_Dv)\geq C\gamma |v|^2_{L^2}.
\end{align}
A minor variation of the argument that gave \eqref{key} now yields\footnote{We are not able to prove \eqref{key2} simply by applying \eqref{key} with $w$ replace by $\Lambda^{\frac{1}{2}}_Dw$.}
\begin{align}\langle bel{key2}
|\Lambda_D b^+_D w|_{L^2}\geq C\gamma |w|_{L^2}.
\end{align}
\textbf{5. }Adding the estimates \eqref{b10}-\eqref{b12} and absorbing errors on the left yields for large enough $\gamma$:
\begin{align}\langle bel{b15}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right|^2_{0,0,\gamma}
+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_1\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_1\end{pmatrix} \right>^2_{0,\gamma}+ \sum_{J_h \cup J_e}\left < \phi_j \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{0,\gamma}\leq\\
&\qquad\qquad\qquad C\gamma^{-1}\left(|\mathcal{F}_1|_{0,0,\gamma}^2+\left < \Lambda^{\frac{1}{2}}\mathcal{G}_1\right>^2_{0,\gamma}\right).
\end{split}
\end{align}
\textbf{6. }To prove the estimates for $v_2$ we use the first-order system \eqref{b10d}.
For the Dirichlet problem \eqref{b2} the bad set $J_c$ is empty, so we can apply the estimates \eqref{b10}, \eqref{b11} with $\tilde{\cF}_2$, $\tilde{\cG}_2$ in place of $\cF_1$, $\cG_1$. Adding these estimates and absorbing terms yields
\begin{align}\langle bel{b16}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix}\right|^2_{0,0,\gamma}+\gamma \left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_2\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_2\end{pmatrix} \right>^2_{0,\gamma}
\leq
C \left(\gamma^{-1}|\mathcal{F}_2|_{0,0,\gamma}^2+\gamma\left< \Lambda^{\frac{1}{2}}\mathcal{G}_2\right>^2_{0,\gamma}+\sum_{J_h}\left<\phi_j\Lambda\mathcal{G}_2\right>^2_{0,\gamma}\right).
\end{split}
\end{align}
\end{proof}
Next we turn to the proof of Proposition {\rm Re }\, f{basicest}. The $L^2$ estimate of Proposition \eqref{bb1} for $v_1$ can easily be applied (as in step \textbf{1} of the proof below) to yield the estimates \eqref{b3} and \eqref{b5}. But these estimates by themselves provide no way to estimate $\left|\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right|_{\infty,0,\gamma}$; the presence of the factors of $1/{\varepsilon}$ in the components of $\cA(w,D_{x',{\varepsilon}})$ prevent us from employing the usual argument that uses the equation to control that norm by controlling $|D_{x_2}U|_{0,0,\gamma}$. Instead, we must adapt to this weakly stable problem the type of argument that was used in \cite{W} to control
$|U|_{\infty,0,\gamma}$ in the uniformly stable case. That is done for $v_1$ in steps \textbf{2} and \textbf{4} of the following proof.
\begin{proof}[Proof of Proposition {\rm Re }\, f{basicest}]
\textbf{1. Proof of \eqref{b5}. }The estimate \eqref{b5}) follows from \eqref{b15} applied to the problem satisfied by $\partial_{(t,x_1,\theta)} U$. However, we are \emph{not} able to prove \eqref{b3} simply by applying \eqref{b15} to the problem satisfied by $\cU:=\Lambda^{1/2}_{D,\gamma}U$. The norm of the commutator $\langle ngle\Lambda^{\frac{1}{2}}_D[\cB_1,\Lambda^{\frac{1}{2}}_D]U^\gamma\rangle ngle_{L^2}$, introduces error terms that are unacceptably large.\footnote{These error terms are exhibited in Proposition {\rm Re }\, f{commutator4}(b).} We get around this in the next step by microlocalizing the estimates.
\textbf{2. Proof of \eqref{b3}. } The next two estimates are proved by applying \eqref{b10} and \eqref{b11} to the problem satisfied by $\cU$; the less singular boundary norms in these microlocal estimates, as compared with \eqref{b15}, give rise to acceptable commutator errors. For $j\in J_h\cup J_e$ we have
\begin{align}\langle bel{b10aa}
\begin{split}
&\gamma |\Lambda^{\frac{1}{2}}\phi_j U|_{0,0,\gamma}^2+\langle ngle\Lambda^{\frac{1}{2}}\phi_jU\rangle ngle^2_{0,\gamma}\lesssim \\
&\qquad \gamma^{-1}|\Lambda^{\frac{1}{2}}\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda^{\frac{1}{2}}\mathcal{G}_1\rangle ngle_{0,\gamma}^2+\gamma^{-1}|\Lambda^{\frac{1}{2}}U|_{0,0,\gamma}^2+\gamma^{-1}|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2+\gamma^{-1}\langle ngle U(0)\rangle ngle^2_{0,\gamma},
\end{split}
\end{align}
while for $j\in J_e$ we obtain
\begin{align}\langle bel{b11aa}
\begin{split}
&\gamma |\Lambda^{\frac{1}{2}}\phi_j U|_{0,0,\gamma}^2+\gamma\langle ngle\phi_j U\rangle ngle^2_{0,\gamma}\lesssim \\
&\qquad \gamma^{-1}|\Lambda^{\frac{1}{2}}\mathcal{F}_1|_{0,0,\gamma}^2+\gamma\langle ngle \mathcal{G}_1\rangle ngle_{0,\gamma}^2+\gamma^{-1}|\Lambda^{\frac{1}{2}}U|_{0,0,\gamma}^2+\gamma^{-1}|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2+\gamma^{-1}\langle ngle U(0)\rangle ngle^2_{0,\gamma}.
\end{split}
\end{align}
The terms $|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2$ arise from Proposition {\rm Re }\, f{commutator4}(a) applied to the interior commutator, while the terms $\gamma^{-1}\langle ngle U(0)\rangle ngle^2_{0,\gamma}$ come from Proposition {\rm Re }\, f{commutator4}(d) applied to the boundary commutator.
Next we show that for $j\in J_c$
\begin{align}\langle bel{b12m}
\begin{split}
&\gamma\left(|\Lambda^{\frac{1}{2}}\phi_j U|_{0,0,\gamma}^2+\langle ngle \phi_jU(x_2)\rangle ngle^2_{0,\gamma}\right)\lesssim\\
&\gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda \mathcal{G}_1\rangle ngle_{0,\gamma}^2+|\Lambda^{\frac{1}{2}}U|_{0,0,\gamma}^2+|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2+\langle ngle U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
In this case \eqref{conj} becomes $S^{-1}\cA S=\mathrm{diag}(a^+,a^-)$ in $\cO_j$. Letting $a^\pm$ denote $C^\infty$, homogeneous of degree one extensions to $\mathcal{X}$ satisfying
$\pm\mathrm{Im} \;a^\pm>0$, we consider the problems
\begin{align}\langle bel{b12n}
\begin{split}
&D_{x_2}V_--a^-_DV_-=F_-\\
&D_{x_2}V_+-a^+_DV_+=F_+,\;\; b^+_DV_+(0)=G_+.
\end{split}
\end{align}
Writing
\begin{align}\langle bel{b12o}
\begin{split}
&\mathrm{Im} \int^\infty_{x_2} \langle ngle (D_{x_2}-a^-_D)V_-,\Lambda^2_D V_-\rangle ngle dx_2= \mathrm{Im}\int^\infty_{x_2}\langle ngle F_-,\Lambda_D^2 V_-\rangle ngle dx_2\\
&\mathrm{Im} \int^{x_2}_{0} \langle ngle (D_{x_2}-a^+_D)V_+, V_+\rangle ngle dx_2=\mathrm{Im} \int^{x_2}_{0}\langle ngle F_+, V_+\rangle ngle dx_2,
\end{split}
\end{align}
integrating terms involving $D_{x_2}$ by parts in $x_2$, and applying the singular Garding's inequality of Theorem {\rm Re }\, f{thm11} to terms involving $a^\pm_D$, we obtain
\begin{align}\langle bel{b16z}
\begin{split}
&\langle ngle \Lambda_D V_-(x_2)\rangle ngle_{L^2(x',\theta)}+|\Lambda^{\frac{3}{2}}_DV_-|_{L^2(x,\theta)}\lesssim |\Lambda^{\frac{1}{2}}F_-|_{L^2(x,\theta)}+|V_-/\sqrt{{\varepsilon}}|_{L^2(x,\theta)}\\
&\sqrt{\gamma} \langle ngle V_+(x_2)\rangle ngle_{L^2(x',\theta)}+\sqrt{\gamma}|\Lambda^{\frac{1}{2}}_DV_+|_{L^2(x,\theta)}\lesssim \sqrt{\gamma} \langle ngle V_+(0)\rangle ngle_{L^2(x',\theta)}+|F_+|_{L^2(x,\theta)}\\
\end{split}
\end{align}
Adding the estimates \eqref{b16z} and using \eqref{key2}, we find
\begin{align}\langle bel{b16p}
\begin{split}
&\sqrt{\gamma} \left(|\Lambda^{\frac{1}{2}}_DV|_{L^2}+\langle ngle V(x_2)\rangle ngle_{L^2}\right)\lesssim \sqrt{\gamma} \langle ngle V_+(0)\rangle ngle_{L^2}+\frac{1}{\sqrt{\gamma}}\left(|\Lambda^{\frac{1}{2}}_DF|_{L^2}+|V_-/\sqrt{{\varepsilon}}|_{L^2}\right)\\
&\lesssim \frac{1}{\sqrt{\gamma}}\left(\langle ngle \Lambda_Db^+_DV_+(0)\rangle ngle_{L^2}+|\Lambda^{\frac{1}{2}}_DF|_{L^2}+|V_-/\sqrt{{\varepsilon}}|_{L^2}\right)\lesssim\\\
&\frac{1}{\sqrt{\gamma}}\left(\langle ngle \Lambda_Db_DV(0)\rangle ngle_{L^2}+\langle ngle \Lambda_Db^-_DV_-(0)\rangle ngle_{L^2}+|\Lambda^{\frac{1}{2}}_DF|_{L^2}+|V_-/\sqrt{{\varepsilon}}|_{L^2}\right)\lesssim\\
&\frac{1}{\sqrt{\gamma}}\left(\langle ngle \Lambda_Db_DV(0)\rangle ngle_{L^2}+|\Lambda^{\frac{1}{2}}_DF|_{L^2}+|V_-/\sqrt{{\varepsilon}}|_{L^2}\right)\lesssim\\
\end{split}
\end{align}
By applying \eqref{b16p} to $V=(V_+,V_-):=S^{-1}_D\phi_{j,D}U^\gamma$ and using the commutator estimates of section 7, in particular Proposition {\rm Re }\, f{commutator5} and its corollary, we obtain \eqref{b12m}.
Combining the estimates \eqref{b10aa}, \eqref{b11aa}, and \eqref{b12m} yields \eqref{b3}.
\textbf{3. } For later use we observe that a similar argument yields for $j\in J_c$ the following extension of \eqref{b12}:
\begin{align}\langle bel{b16q}
\begin{split}
&\gamma\left(|\phi_j U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-\frac{1}{2}}\phi_jU(x_2)\rangle ngle^2_{0,\gamma}\right)\leq \\
&\qquad C\gamma^{-1}\left(|\phi_j\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda^{\frac{1}{2}}\phi_j\mathcal{G}_1\rangle ngle_{0,\gamma}^2+|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-\frac{1}{2}}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
For this one takes $L^2$ pairings with $\Lambda_D V_-$ and $\Lambda^{-1}_DV_+$ in \eqref{b12o}, instead of with $\Lambda^2_D V_-$ and $V_+$.
Similarly, one derives the following extension of \eqref{b11} for $j\in J_e$:
\begin{align}\langle bel{b16s}
\begin{split}
&\gamma |\phi_j U|_{0,0,\gamma}^2+\gamma\langle ngle\phi_j\Lambda^{-\frac{1}{2}}U(x_2)\rangle ngle^2_{0,\gamma}\leq \\
&\qquad C\left(\gamma^{-1}|\phi_j\mathcal{F}_1|_{0,0,\gamma}^2+\gamma\langle ngle\phi_j\Lambda^{-\frac{1}{2}}\mathcal{G}_1\rangle ngle_{0,\gamma}^2+\gamma^{-1}|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-1}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
\textbf{4. Proof of \eqref{b4}. }For $j\in J_c$ the estimate \eqref{b12m} implies
\begin{align}\langle bel{b12mm}
\begin{split}
&\gamma |\phi_jU|_{\infty,0,\gamma}^2\lesssim
\gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda \mathcal{G}_1\rangle ngle_{0,\gamma}^2+|\Lambda^{\frac{1}{2}}U|_{0,0,\gamma}^2+|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2+\langle ngle U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
Let $\chi_D$ denote a cutoff in the extended calculus, with the support property \eqref{n31}, chosen so that $\chi=\sum_{j\in J_c}\phi_j\chi$. The estimate \eqref{b12mm} yields
\begin{align}\langle bel{b16r}
\begin{split}
&\gamma |\chi U|_{\infty,0,\gamma}^2\lesssim \gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|_{0,0,\gamma}^2+\langle ngle\Lambda \mathcal{G}_1\rangle ngle_{0,\gamma}^2+|\Lambda^{\frac{1}{2}}U|_{0,0,\gamma}^2+|U/\sqrt{{\varepsilon}}|_{0,0,\gamma}^2+\langle ngle U(0)\rangle ngle^2_{0,\gamma}\right).
\end{split}
\end{align}
On the other hand from the support property of $\chi$ and the equation we obtain
\begin{align}\langle bel{b17}
|(1-\chi)U|_{\infty,0,\gamma}\leq |D_{x_2}(1-\chi)U|_{0,0,\gamma}+|U|_{0,0,\gamma}\leq C(|U|_{0,1,\gamma}+|\mathcal{F}_1|_{0,0,\gamma}).
\end{align}
The estimate \eqref{b4} for $v_1$ is then a consequence of \eqref{b16r}, \eqref{b3}, and \eqref{b5}.
\textbf{5. Proof of \eqref{b6} and \eqref{b8}. } The proof of \eqref{b8} is like that of \eqref{b5} but simpler, since boundary commutators are zero.
The estimate \eqref{b6} is proved by applying \eqref{bb3} to the problem satisfied by $\Lambda^{\frac{1}{2}}_DU^\gamma$, where $U^\gamma$ is defined using $v_2$ now.
\textbf{6. Proof of \eqref{b7}. }The argument is similar to step \textbf{2}, but the set $J_c$ is empty now. We show first that for $j\in J_e$ we have
\begin{align} \langle bel{b17z}
|\phi_j U|^2_{\infty,0,\gamma}
\leq C\left(\gamma^{-1}|\mathcal{F}_2|_{0,0,\gamma}^2+\langle ngle\Lambda\mathcal{G}_2\rangle ngle_{0,\gamma}^2+\gamma^{-1}|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-1}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{align}
In place of \eqref{b12n}, \eqref{b12o} we have
\begin{align}
\begin{split}
&\mathrm{Im} \int^\infty_{x_2} \langle ngle (D_{x_2}-a^-_D)V_-, V_-\rangle ngle dx_2=\mathrm{Im} \int^\infty_{x_2}\langle ngle F_-, V_-\rangle ngle dx_2\\
&\mathrm{Im} \int^{x_2}_{0} \langle ngle (D_{x_2}-a^+_D)V_+, V_+\rangle ngle dx_2=\mathrm{Im} \int^{x_2}_{0}\langle ngle F_+,V_+\rangle ngle dx_2
\end{split}
\end{align}
and
\begin{align}
\begin{split}
&\sqrt{\gamma}\langle ngle V_-(x_2)\rangle ngle_{L^2(x',\theta)}+\gamma|V_-|_{L^2(x,\theta)}\lesssim |F_-|_{L^2(x,\theta)}\\
&\sqrt{\gamma} \langle ngle V_+(x_2)\rangle ngle_{L^2(x',\theta)}+\gamma|V_+|_{L^2(x,\theta)}\lesssim \sqrt{\gamma}\langle ngle V_+(0)\rangle ngle_{L^2(x',\theta)}+|F_+|_{L^2(x,\theta)},
\end{split}
\end{align}
which implies
\begin{align}\langle bel{b12p}
\sqrt{\gamma}\langle ngle V(x_2)\rangle ngle_{L^2}+\gamma |V|_{L^2}\lesssim |F|_{L^2}+\sqrt{\gamma}\langle ngle b_DV(0)\rangle ngle_{L^2}.
\end{align}
Here in place of \eqref{key} we have used
\begin{align}
\langle ngle b^+_Dw\rangle ngle_{L^2}\geq C \langle ngle w\rangle ngle_{L^2}.
\end{align}
Applying \eqref{b12p} to the problem satisfied by $V=S^{-1}_D\phi_{j,D}U^\gamma$, we obtain \eqref{b17z}.
Let $\chi$ denote a cutoff in the extended calculus with the support property \eqref{n31}. Since $\chi$ can be chosen now so that $\chi=\sum_{j\in J_e}\phi_j\chi$, we have
\begin{align}
|\chi U|^2_{\infty,0,\gamma}
\leq C\left(\gamma^{-1}|\mathcal{F}_2|_{0,0,\gamma}^2+\langle ngle\Lambda\mathcal{G}_2\rangle ngle_{0,\gamma}^2+\gamma^{-1}|U|_{0,0,\gamma}^2+\langle ngle\Lambda^{-1}U(0)\rangle ngle^2_{0,\gamma}\right).
\end{align}
Together with \eqref{b17} (for $U$ corresponding to $v_2$ now) and \eqref{b8} this yields \eqref{b7}.
\end{proof}
\section{Uniform time of existence for the nonlinear singular systems}\langle bel{uniform}
\emph{\quad} In this section we describe our strategy for proving the existence of solutions to the coupled, nonlinear, singular systems \eqref{a7}-\eqref{a9} on a fixed time interval independent of small ${\varepsilon}>0$.
First, we define a ``singular energy norm" $E_{m,\gamma}(v)$, a time-localized variant of which, namely $E_{m,T}(v)$, will be shown to be uniformly bounded on a time interval independent of ${\varepsilon}\in (0,1]$.
\subsection{Singular energy norms}\langle bel{singular}
\emph{\quad} Let $m$ be a nonnegative integer.
Suppressing the epsilon superscript on $v$ and the subscript $D$ on the singular operator $\Lambda_D$, we
define for $v=(v_1,v_2)$ the singular energy norm =
\begin{align}\langle bel{c00}
E_{m,\gamma}(v):=E_{m,\gamma}(v_1)+E_{m,\gamma}(v_2),
\end{align}
where $E_{m,\gamma}(v_j)=$
\begin{align}\langle bel{c0a}
\begin{split}
&\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_j\\partial_{x_2}\Lambda^{\frac{1}{2}}v_j\end{pmatrix}\right|^{(2)}_{0,m,\gamma}+ \left|\begin{pmatrix}\Lambda v_j\\partial_{x_2} v_j\end{pmatrix}\right|^{(2)}_{\infty,m,\gamma}+\left|\begin{pmatrix}\Lambda v_j\\partial_{x_2} v_j\end{pmatrix}\right|^{(2)}_{0,m+1,\gamma}
+\left < \begin{pmatrix}\Lambda^{\frac{1}{2}} v_j\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_j\end{pmatrix} \right>_{m+1,\gamma}\\
&+\left|\begin{pmatrix}\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v_j\\\sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}D_{x_2} v_j\end{pmatrix}\right|^{(2)}_{0,m+1,\gamma}
+\left < \begin{pmatrix}\sqrt{{\varepsilon}}\Lambda v_j\\\sqrt{{\varepsilon}}D_{x_2} v_j\end{pmatrix} \right>_{m+1,\gamma}+\left|\begin{pmatrix}\Lambda v_j/\sqrt{{\varepsilon}}\\partial_{x_2} v_j/\sqrt{{\varepsilon}}\end{pmatrix}\right|^{(2)}_{0,m,\gamma}+\left \langle ngle \begin{pmatrix}\Lambda^{\frac{1}{2}} v_j/\sqrt{{\varepsilon}}\\partial_{x_2}\Lambda^{-\frac{1}{2}} v_j/\sqrt{{\varepsilon}}\end{pmatrix} \right\rangle ngle_{m,\gamma} \\
&+\left|\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{\infty,m,\gamma}+\left|\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{0,m+1,\gamma}+\left|\frac{\Lambda^{\frac{1}{2}}{\nabla}bla_{\varepsilon} u}{{\sqrt{{\varepsilon}}}}\right|_{0,m+1,\gamma}+\left\langle ngle\frac {{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}+\left|\frac{\Lambda^{\frac{1}{2}}{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{0,m,\gamma}.
\end{split}
\end{align}
Here the superscripts $(2)$ on some terms have the following meanings:\footnote{In each case the second term on the right is obtained from the first term by trading ${\varepsilon} D_{x_2}$ for a tangential derivative; this applies as well to the other three terms of \eqref{c0a} with $(2)$ as a superscript.}
\begin{align}\langle bel{c0bb}
\begin{split}
&\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_j\\partial_{x_2}\Lambda^{\frac{1}{2}}v_j\end{pmatrix}\right|^{(2)}_{0,m,\gamma}=\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_j\\partial_{x_2}\Lambda^{\frac{1}{2}}v_j\end{pmatrix}\right|_{0,m,\gamma}+\left|\begin{pmatrix}{\varepsilon} D_{x_2}\Lambda^{\frac{3}{2}}v_j\\{\varepsilon}\Lambda^{\frac{1}{2}}D_{x_2}^2v_j\end{pmatrix}\right|_{0,m-1,\gamma},\\
&\left|\begin{pmatrix}\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v_j\\\sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}D_{x_2} v_j\end{pmatrix}\right|^{(2)}_{0,m+1,\gamma}=\left|\begin{pmatrix}\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v_j\\\sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}D_{x_2} v_j\end{pmatrix}\right|_{0,m+1,\gamma}+\left|\begin{pmatrix}{\varepsilon}^{\frac{3}{2}}D_{x_2}\Lambda^{\frac{3}{2}} v_j\\{\varepsilon}^{\frac{3}{2}}\Lambda^{\frac{1}{2}}D^2_{x_2} v_j\end{pmatrix}\right|_{0,m,\gamma}.
\end{split}
\end{align}
\begin{rem}
1. It would be more proper to denote the norms in \eqref{c00} and \eqref{c0a} by $E_{m,\gamma}(v,{\nabla}bla_{\varepsilon} u)$ and $E_{m,\gamma}(v_j,{\nabla}bla_{\varepsilon} u)$, so there is an abuse of notation here that we have introduced in order to lighten many of the expressions that occur throughout the paper. The fact that ${\nabla}bla_{\varepsilon} u$ rather than $v$ appears in the third line of \eqref{c0a} is related to the fact that we will later have $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ \emph{only} on $\Omega_{T_{\varepsilon}}$.
2. Note that all interior norms in the first two lines of the definition of $E_{m,\gamma}(v)$ have ``total weight" $m+3$ if one assigns a weight of $1$ to ${\varepsilon} D_{x_2}$, $\infty$, $\Lambda^{\frac{1}{2}}$, $\frac{1}{\sqrt{{\varepsilon}}}$, and to each tangential derivative $\partial$, and assigns a weight of $2$ to $D_{x_2}$. All boundary norms in those lines have weight $m+2$.\footnote{In this weighting, ${\varepsilon} D_{x_2}$ must always be viewed like a tangential derivative of weight $1$, not as the ``composite" of ${\varepsilon}$ (weight $-2$) and $D_{x_2}$ (weight $2$)}
The same applies to the third line if ${\nabla}bla_{\varepsilon} u$ is treated like $v$.
3. The second terms on the right in \eqref{c0bb} arise in the interior commutator estimates. See, for example, part (b) of Proposition {\rm Re }\, f{e26}.
4. For each $j$, $E_{m,\gamma}(v_j)$ is a sum of $18$ terms. This is the smallest number of terms for which we have been able to obtain an estimate that ``closes" like the one in Proposition {\rm Re }\, f{c5}.
\end{rem}
We also define singular energy norms localized in time for functions defined on $\Omega_T$.
Set
\begin{align}\langle bel{c0c}
E_{m,T}(v)=E_{m,T}(v_1)+E_{m,T}(v_2),
\end{align}
where the norms
$E_{m,T}(v_i)$ are defined by the right side of \eqref{c0a}, except that norms $|\Lambda_1^r v_i|_{0,m,T}$, $|\Lambda^r_1v_i|_{\infty,m,T}$, etc., are now used in place of $|\Lambda^r v_i|_{0,m,\gamma}$, $|\Lambda^rv_i|_{\infty,m,\gamma}$, etc..\footnote{Recall parts (g)-(j) of Notations {\rm Re }\, f{spaces}.}Similarly, norms $E_{m}(v_i)$ are defined using norms $|\Lambda_1^r v_i|_{0,m}$, $|\Lambda^r_1v_i|_{\infty,m}$, etc..
Finally, we define in the obvious way $E_{m,\gamma,T}(v)$, the time-localized version of $E_{m,\gamma}(v)$, using $|\Lambda^r v_i|_{0,m,\gamma,T}$, $|\Lambda^rv_i|_{\infty,m,\gamma,T}$, etc..
\begin{prop}\langle bel{c0e}
a) Let $T>0$. For $v$ and $u$ defined on $\Omega$ we have
\begin{align}\langle bel{c0ee}
E_{m,\gamma}(v)\geq E_{m,\gamma,T}(v)\geq C e^{-\gamma T} E_{m,T}(v).
\end{align}
b) For $v$ and $u$ initially defined in $\Omega_T$ with $v=0$ in $t<0$, one can choose a Seeley extension $v^s_T$ to $\Omega$ \cite{CP} such that
\begin{align}\langle bel{c0g}
\begin{split}
&E_{m,\gamma}(v^s_T)\leq CE_{m,\gamma,T}(v^s_T)\leq C E_{m,T}(v^s_T),\\
&E_m(v^s_T)\leq C E_{m,T}(v^s_T),\text{ and }\\
&\text{the $t$-support of $v^s$ is contained in $[0,\frac{3}{2}T]$}.
\end{split}
\end{align}
Here all constants $C$ are independent of $\gamma$, ${\varepsilon}$, and $T$, and our Seeley extensions $v^s_T$ have $t$-support in a fixed compact subset of $[0,2)$ when $0<T\leq 1$.
(c) The exact analogues of (a) and (b) hold for the norms $|w|^*_{m,\gamma}$, $|w|^*_{m,\gamma,T}$, $|w|^*_{m,T}$, and $|w|^*_{m}$ defined in Definition {\rm Re }\, f{defnN}.
\end{prop}
\begin{proof}
\textbf{1. }
For $T>0$ the Seeley extension used here is defined first on sufficiently smooth functions $v(t,x,\theta)$, defined in $t\leq T$ and supported in $t\geq 0$,
by
\begin{align}\langle bel{c0gg}
v^s(t,x,\theta)=\begin{cases}v(t,x,\theta),\; t\leq T\\ \sum^M_{k=1}\langle mbda_k \;v(T+2^k(T-t),x,\theta),\;t>T\end{cases},
\end{align}
where $M$ is sufficiently large (depending on $m$) and the $\langle mbda_k$ are chosen to satisfy $\sum_{k=1}^M \langle mbda_k 2^{kj}=(-1)^j$ for $j=0,\dots,M-1$.
Observe that all terms in the sum have $t$-support in $[0,\frac{3}{2}T]$. The definition is extended by continuity to more general functions.
\textbf{2. } Inequalities corresponding to \eqref{c0ee} and \eqref{c0g} are proved for the individual norms appearing in the definition of $E_{m,\gamma}(v)$.
Since we can replace $\Lambda_D w$ by
$((\partial_{x'}+\beta\frac{\partial_\theta}{{\varepsilon}})w,\gamma w)$, such inequalities are obvious for the norms that involve only integral powers of $\Lambda_D$. The inequalities in the non-integral cases then follow by interpolation.
\end{proof}
\subsection{Strategy}\langle bel{strategy}
\emph{\quad} By Proposition {\rm Re }\, f{localex} for each fixed ${\varepsilon}\in (0,1]$ we have classical solutions $v^{\varepsilon}$, $u^{\varepsilon}$ to the quasilinear singular systems \eqref{a7}, \eqref{a8}, and \eqref{a9} on $\Omega_{T_{\varepsilon}}$ for some $T_{\varepsilon}>0$. Moreover, we have $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_{\varepsilon}}$. Below we often suppress the superscript ${\varepsilon}$ on $u$ and $v$.
Let $T_0<1$ and $M_0$ be positive constants that will be chosen later to be sufficiently small (independently of ${\varepsilon}$, $\gamma$, and $T$). For a fixed $m> 3d+4+\frac{d+1}{2}$ define for each ${\varepsilon}\in (0,1]$
\begin{align}\langle bel{c0h}
T^*_{\varepsilon}=\sup \{T\in(0,\min(T_0,\frac{T_{\varepsilon}}{2}]: E_{m,T}(v^{\varepsilon}) \leq M_0\},
\end{align}
a number that might converge to $0$ as ${\varepsilon}\to 0$, as far as we know now.
We will use the a priori estimate of Proposition {\rm Re }\, f{c5} and a continuation argument to prove the following proposition.
\begin{prop}\langle bel{mainprop}
We make the same assumptions as in Theorem {\rm Re }\, f{uniformexistence}, and let $v^{\varepsilon}$, $u^{\varepsilon}$ on $\Omega_{T_{\varepsilon}}$ satisfying $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ be solutions to the quasilinear singular systems \eqref{a7}, \eqref{a8}, and \eqref{a9} provided by Proposition {\rm Re }\, f{localex}. Define $T^*_{\varepsilon}$ as in \eqref{c0h}.
There exist ${\varepsilon}_0>0$ and $T_1$ independent of ${\varepsilon}\in (0,{\varepsilon}_0]$ such that
\begin{align}\langle bel{c0i}
T^*_{\varepsilon}\geq T_1>0\text{ for all }{\varepsilon}\in (0,{\varepsilon}_0].
\end{align}
\end{prop}
For any $T$ satisfying $0<T\leq T^*_{\varepsilon}$ we let $v^s_T$ denote a Seeley extension of $v|_{\Omega_T}$ to $\Omega$ chosen as in Proposition {\rm Re }\, f{c0e}. We suppress the $T$-dependence of $v^s_T$ below and write $v^s$ for $v^s_T$.
Consider now the following three \emph{linear} systems for the respective unknowns $v_1$, $v_2$, and $u$.
We choose $\chi_0(t)\geq 0$ to be a $C^\infty$ function that is equal to 1 on a neighborhood of $[0,1]$ and supported in $(-1,2)$.
\begin{align}\langle bel{c1}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_1+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha v_1=-\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_1,{\varepsilon}}(A_\alpha(v^s))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v_1^s-\partial_{x_1,{\varepsilon}}(A_{(0,2)}(v^s))\partial_{x_2}v_2^s\right]\\
&\partial _{x_2} v_1-d_{v_1}H(v^s_1,h(v^s))\partial_{x_1,{\varepsilon}}v_1=d_gH(v^s_1,{\varepsilon}^2G)\partial_{x_1,{\varepsilon}}({\varepsilon}^2G)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{c2}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_2+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha v_2=-\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_2}(A_\alpha(v^s))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v^s_1-\partial_{x_2}(A_{(0,2)}(v^s))\partial_{x_2}v^s_2\right]\\
&v_2=\chi_0(t)H(v_1,{\varepsilon}^2 G)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{c3}
\begin{split}
&\partial_{t,{\varepsilon}}^2 u+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha u=0\\\
&\partial _{x_2} u-d_{v_1}H(v^s_1,h(v^s))\partial_{x_1,{\varepsilon}}u=\left[H(v^s_1,{\varepsilon}^2 G(x',\theta))-d_{v_1}H(v^s_1,{\varepsilon}^2 G)v^s_1\right]\text{ on }x_2=0.
\end{split}
\end{align}
\begin{rem}\langle bel{c4}
1. The above three systems are solved on the full domain $\Omega$. Each system depends on the parameters ${\varepsilon}$ and $T$.
2. The definition of $v^s=v^s_T$ and causality (see Remark {\rm Re }\, f{k2y}) imply that the solutions $v_1$, $v_2$, and $u$ of \eqref{c1}, \eqref{c2}, and \eqref{c3} are equal to the corresponding solutions of the systems \eqref{a7}, \eqref{a8}, and \eqref{a9} on $\Omega_T$. The right sides of these equations depend on $v^s_T$, so the solutions
$v$ and $u$ change as $T$ changes, but we suppress this $T$-dependence in the notation.
3. We recall that the $t-$support of $v^s$ is contained in $[0,\frac{3}{2}T]$. Since we always have $0\leq 2T\leq T_{\varepsilon}$ and $v={\nabla}bla_{\varepsilon} u$ on $\Omega_{T_{\varepsilon}}$, it follows that $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\mathrm{supp }\;v^s$ and $v^s={\nabla}bla_{\varepsilon} u^s$.
\end{rem}
We will estimate solutions to \eqref{c1} and \eqref{c2} by applying the estimates of section {\rm Re }\, f{main}, where $\mathcal{F}_1$ and $\mathcal{G}_1$ are given by the right sides of the interior and boundary equations of \eqref{c1}, and where $\mathcal{F}_2$ and $\mathcal{G}_2$ are given by the right sides of \eqref{c2}. Note that the factor
$\partial_{x_1,{\varepsilon}}({\varepsilon}^2G)$ occurs in $\mathcal{G}_1$, and that the singular derivative here ``uses up" one of the two factors of ${\varepsilon}$ on $G$. The remaining factor is used up by $\Lambda_D$ in a term like $\langle ngle \Lambda \mathcal{G}_1\rangle ngle^2_{0,\gamma}$, which occurs on the right side of \eqref{b3}. On the other hand there is still a factor of ${\varepsilon}^{1/2}$ ``to spare" in a term like $\langle ngle \phi_j\Lambda^{\frac{3}{2}}\mathcal{G}_2\rangle ngle^2_{0,\gamma}$ which occurs on the right side of \eqref{b6}.
\begin{rem}\langle bel{c4a}
1. Observe that $v_1$, not $v^s_1$, occurs on the right in the boundary condition for \eqref{c2}; the reason is the following. When we apply the estimate \eqref{b6} to estimate
(say) $|\Lambda^{\frac{3}{2}}v_2 |_{0,m,\gamma}$, we need to bound $\langle ngle \phi_j\Lambda^{\frac{3}{2}}\mathcal{G}_2\rangle ngle_{m,\gamma}$, for $j\in J_h$. Since $\mathcal{G}_2$ depends on $v_1$, Proposition {\rm Re }\, f{f7} implies that to estimate $\langle ngle \phi_j\Lambda^{\frac{3}{2}}\mathcal{G}_2\rangle ngle_{m,\gamma}$ we need control of $\langle ngle\phi_j\Lambda^{\frac{3}{2}}v_1\rangle ngle_{m,\gamma}$, $j\in J_h$. That control comes from the estimate \eqref{b3}. If $v^s_1$ appeared as an argument of $\mathcal{G}_2$ in the boundary condition for \eqref{c2}, we would have to estimate $\langle ngle\phi_j\Lambda^{\frac{3}{2}}v^s_1\rangle ngle_{m,\gamma}$ instead, but we know of no way to do this. The estimate \eqref{b3} does not apply to $v^s_1$, since $v^s_1$ is not a solution of \eqref{c1} on $\Omega$ (just on $\Omega_T$). In addition, we cannot use a result like Proposition {\rm Re }\, f{c0e} to deduce control of
$\langle ngle\phi_j\Lambda^{\frac{3}{2}}v^s_1\rangle ngle_{m,\gamma}$ from control of $\langle ngle\phi_j\Lambda^{\frac{3}{2}}v_1\rangle ngle_{m,\gamma}$, since we know of no such result that
applies to norms involving $\phi_{j,D}$; we have no way to localize such norms in time.
On the other hand, since $v=v^s=v^s_T$ on $\Omega_T$, we have $E_{m,T}(v)=E_{m,T}(v^s)$, so we can use Proposition {\rm Re }\, f{c0e} to deduce control of $E_{m,\gamma}(v^s)$ from control of $E_{m,\gamma}(v)$. There is no version of Proposition {\rm Re }\, f{c0e} for norms involving $\phi_{j,D}$, and that is one reason we do not include such norms in the definition of $E_{m,\gamma}(v)$.
2. The estimate $E_{m,T}(v^{\varepsilon})\leq M_0$, valid for $0<T\leq T^*_{\varepsilon}$, and the fact that the coefficients of our systems are functions of $v^s=v^s_T$, imply that the constants appearing in the estimates will be uniform with respect to ${\varepsilon}$.
\end{rem}
We suppose that $G\in H^{m+3}(b\Omega)$ and for $T_0>0$ define the norm
\begin{align}\langle bel{c5a}
N_{m,T_0}({\varepsilon}^2 G):= \langle ngle \Lambda^2_1 {\varepsilon}^2 G\rangle ngle_{m,T_0}+\langle ngle\Lambda_1{\varepsilon} G\rangle ngle_{m,T_0}+\langle ngle \Lambda_1^{\frac{3}{2}}{\varepsilon}^{\frac{3}{2}}G\rangle ngle_{m+1,T_0}.
\end{align}
We choose a Seeley extension of $G|_{\Omega_{T_0}}$ (also denoted $G$) to $H^{m+3}(b\Omega)$, compactly supported in $t$, such that the obvious analogue of \eqref{c0g} is satisfied.
Observe that
\begin{align}\langle bel{MG}
N_{m,T_0}({\varepsilon}^2 G)\lesssim\langle ngle G\rangle ngle_{m+3,T_0}:=M_G\text{ for }{\varepsilon}\in (0,1].
\end{align}
\begin{rem}\langle bel{c5az}
We will eventually need to take $M_G$ small. This can be arranged for a given choice of $G$ by taking $T_0$ small, since $G$ vanishes in $t<0$. Alternatively, for a given $T_0$, one can adjust the choice of $G$ to make $M_G$ small. In this paper we use the first option. We can and do always suppose that $M_G\leq 1$.
\end{rem}
The main step in showing Proposition {\rm Re }\, f{mainprop} is the following a priori estimate, whose proof is concluded at the end of section {\rm Re }\, f{mainestimate}.
\begin{prop}\langle bel{c5}
Let $m> 3d+4+\frac{d+1}{2}$ and $G\in H^{m+3}(b\Omega)$. There exist positive constants $M_0$ as in \eqref{c0h} and $M_G$ as in \eqref{MG} such that the following is true.
There exist positive constants ${\varepsilon}_0$, $\gamma_0$, and there exist increasing functions $Q_i:\mathbb{R}_+\to\mathbb{R}_+$, $i=1,2$, with
$Q_i(z)\geq z$ such that for ${\varepsilon}\in (0,{\varepsilon}_0]$ and each $T$ with $0<T\leq T^*_{\varepsilon}$, the solution to \eqref{c1}-\eqref{c3} satisfies
\begin{align}\langle bel{c6}
E_{m,\gamma}(v)\leq \gamma^{-1}E_{m,\gamma}(v)Q_1(E_{m,T}(v^s))+(\gamma^{-\frac{1}{2}}+\sqrt{{\varepsilon}})Q_2(E_{m,T}(v^s))\text{ for }\gamma\geq \gamma_0.
\end{align}
\end{prop}
Assuming this proposition, we now prove Proposition {\rm Re }\, f{mainprop}.
\begin{proof}[Proof of Proposition {\rm Re }\, f{mainprop}]
Since $E_{m,T}(v^s)\leq M_0$ we can choose $\gamma_0$ so that
\begin{align}
\gamma^{-1}E_{m,\gamma}(v)Q_1(E_{m,T}(v^s))\leq E_{m,\gamma}(v)/2\text{ for }\gamma\geq \gamma_0.
\end{align}
Thus, for some $C>0$
\begin{align}\langle bel{c7}
Ce^{-\gamma T}E_{m,T}(v)\leq E_{m,\gamma}(v)\leq 2(\gamma^{-\frac{1}{2}}+\sqrt{{\varepsilon}})Q_2(E_{m,T}(v)) \text{ for }\gamma\geq \gamma_0,\;{\varepsilon}\in (0,{\varepsilon}_0].
\end{align}
For any $T\in (0,T^*_{\varepsilon}]$ such that $\frac{1}{T}\geq \gamma_0$, we can take $\gamma=1/T$ in \eqref{c7} to obtain
\begin{align}\langle bel{c8}
E_{m,T}(v)\leq 2\frac{e}{C}(\sqrt{T}+\sqrt{{\varepsilon}})Q_2(M_0)\text{ for }{\varepsilon}\in (0,{\varepsilon}_0].
\end{align}
Clearly, for $N\in\mathbb N$ one can choose ${\varepsilon}_0$ and $T_1=T_1(M_0,M_G)$ independent of ${\varepsilon}$ such that the right side of \eqref{c8} is $<M_0/N$. However, to finish we need to know that $T_1\leq T^*_{\varepsilon}$ for all ${\varepsilon}\in (0,{\varepsilon}_0]$. If not, there is some ${\varepsilon}\in (0,{\varepsilon}_0]$ such that $T_1>T^*_{\varepsilon}$. But then $E_{m,T^{*,-}_{\varepsilon}}(v^{\varepsilon}) <M _0/N$, where $T^{*,-}_{\varepsilon}<T^*_{\varepsilon}$ is as close as we like to $T^*_{\varepsilon}$. For $N$ large enough we can then use our local continuation result for fixed ${\varepsilon}$, Proposition {\rm Re }\, f{continuation}, to continue $v^{\varepsilon}$ to a time $T_c>T^*_{\varepsilon}$ such that $E_{m,T_c}(v^{\varepsilon}) < M_0$, a contradiction.\footnote{The proof of Prop. {\rm Re }\, f{continuation} shows that the size of $N$ depends on the norm of the Seeley extension operator used there.}
\end{proof}
Clearly, we need to study how the singular norms appearing on the right in \eqref{b3}-\eqref{b8} act on nonlinear functions of $v$, $v_2$, and ${\varepsilon}^2G$. We also need to show that microlocal regularity of functions like $\phi_{j,D}\Lambda_Dv_1^\gamma$ is preserved under nonlinear functions. This study is carried out in section {\rm Re }\, f{nonlinear}, and Proposition {\rm Re }\, f{c5} is proved in section {\rm Re }\, f{mainestimate}.
\section{Singular norms of nonlinear functions}\langle bel{nonlinear}
\emph{\quad} Here we take $x'=(x_0,x'')=(t,x'')\in\mathbb{R}^d$, $\theta\in \mathbb{R}$ and dual variables $\xi\in\mathbb{R}^d$, $k\in\mathbb{R}$, and (in this section) we drop the prime on $x$. We also fix $\beta\in\mathbb{R}^d\setminus 0$, set $X=\xi+\beta\frac{k}{{\varepsilon}}$, and write
\begin{align}\langle bel{f00}
\partial_{\varepsilon}:=\partial_x+\beta\frac{\partial_\theta}{{\varepsilon}},\;\partial_{{\varepsilon},\gamma}:=(\partial_t+\gamma,\partial_{x''})+\beta\frac{\partial_\theta}{{\varepsilon}}.
\end{align}
Observe that
\begin{align}\langle bel{f00a}
\partial_{{\varepsilon},\gamma}(e^{-\gamma t}u)=e^{-\gamma t}\partial_{\varepsilon} u.
\end{align}
We let $\Lambda$ denote the singular symbol $\Lambda=\sqrt{|X|^2+\gamma^2}$ and $\Lambda_1=\sqrt{|X|^2+1}$.
Sometimes, for example in convolutions, we will use the variables $y\in\mathbb{R}^d$, $\omega\in \mathbb{R}$, dual variables $\eta\in\mathbb{R}^d$, $l\in\mathbb{R}$, and set $Y=\eta+\beta\frac{l}{{\varepsilon}}$.
In this section we consider functions $u=u(x,\theta)$. Unless otherwise noted, all constants below are independent of $\gamma\geq 1$ and ${\varepsilon}\in (0,1]$. In every estimate we assume that the functions in question are such that the norms appearing on the right are finite.
The following proposition is used repeatedly in this section.
\begin{prop}[\cite{RR} , Lemma 1.2.2] \langle bel{f0a}
Let ${\varepsilon}\in (0,1]$ and take $\gamma\geq 1$.
Suppose $G_{{\varepsilon},\gamma}:\mathbb{R}^{d+1}\times \mathbb{R}^{d+1}\to\mathbb{C}$ is a locally integrable measurable function that can be decomposed into a finite sum (suppress ${\varepsilon}$,$\gamma$)
\begin{align}
G(\xi,k,\eta,l)=\sum_{j=1}^K G_j(\xi,k,\eta,l)
\end{align}
such that for each $j$ we have either
\begin{align}
\sup_{\xi,k}\int |G_j(\xi,k,\eta,l)|^2 d\eta dl< C\text{ or }\sup_{\eta,l}\int |G_j(\xi,k,\eta,l)|^2 d\xi dk< C,
\end{align}
where $C$ is independent of $({\varepsilon},\gamma)$. Then
\begin{align}
(f,g)\to \int G(\xi,k,\eta,l) f(\xi-\eta,k-l)g(\eta,l)d\eta dl
\end{align}
defines a continuous bilinear map of $L^2\times L^2\to L^2$, and
\begin{align}
|\int G(\xi,k,\eta,l) f(\xi-\eta,k-l)g(\eta,l)d\eta dl|_{L^2}\leq C |f|_{L^2} |g|_{L^2},
\end{align}
with $C$ independent of $({\varepsilon},\gamma)$.
\end{prop}
\subsection{Non-tame estimates}\langle bel{nontame}
We begin with the non-tame estimates that are used in the Proof of Proposition {\rm Re }\, f{c5}.
\begin{prop}\langle bel{f0}
Let $m>(d+1)/2$, $r\geq 0$, and set $\langle ngle u\rangle ngle_m:=|u|_{H^m}$. Assume the functions $u$ and $v$ are real-valued. Then
a) $\langle ngle u v\rangle ngle_m\leq C_1\langle ngle u\rangle ngle_m \langle ngle v\rangle ngle_m.$
b) $\langle ngle \Lambda^{r}_{1}(uv)\rangle ngle_m\leq C_2[\langle ngle \Lambda^{r}_{1}u\rangle ngle_m\langle ngle v\rangle ngle_m+\langle ngle u\rangle ngle_m\langle ngle \Lambda^{r}_{1}v\rangle ngle_m]$.
c) Let $C_3=\max (C_1,C_2)$. For $n\in\mathbb{N}$, $\langle ngle\Lambda^{r}_{1}(u^n)\rangle ngle_m\leq n\langle ngle\Lambda^{r}_{1} u\rangle ngle_m \cdot (C_3\langle ngle u \rangle ngle_m)^{n-1}$.
d) Let $f(u)$ be a real-valued, (real-)analytic function satisfying $f(0)=0$ with radius of convergence $R$ at $u=0$. If $C_3\langle ngle u\rangle ngle_m <R$, then
\begin{align}
\langle ngle \Lambda^{r}_{1} f(u)\rangle ngle_m\leq \langle ngle \Lambda^{r}_{1}u\rangle ngle_m \cdot h(C_3\langle ngle u \rangle ngle_m),
\end{align}
where $h$ is an analytic function with radius of convergence $R$ at $0$ with nonnegative coefficients.
\end{prop}
\begin{proof}
\textbf{(a,b)} Part (a) follows directly from Proposition {\rm Re }\, f{f0a} by taking
\begin{align}
G(\xi,k,\eta,l)=\frac{\langle ngle \xi,k\rangle ngle^m}{\langle ngle \xi-\eta,k-l\rangle ngle^m\langle ngle \eta,l\rangle ngle^m}
\end{align}
and consider the cases $|\eta,l|\leq \frac{1}{2}|\xi,k|$ and $|\eta,l|> \frac{1}{2}|\xi,k|$.
To prove (b), use Proposition {\rm Re }\, f{f0a} in a similar way, together with the inequality
\begin{align}
|X,1|^{r}\leq C (|X-Y,1|^{r}+ |Y,1|^{r}).
\end{align}
\textbf{(c,d)} Part (c) is a direct consequence of part (b), and (d) is proved by writing $f(z)=\sum^\infty_{n=1}a_nz^n$, applying (c), and taking $h(z)=\sum^\infty_{n=1}n|a_n|z^{n-1}$.
\end{proof}
\begin{prop}\langle bel{f1}
Let $m> (d+1)/2$, $r\geq0$. Assume that the functions $u$, $v$ are real-valued. Then
a) $|uv|_{H^m_\gamma}\leq D_1 |u|_{H^m_\gamma}|v|_{H^m}$.
b) $|\Lambda^{r}_D(uv)|_{H^m_\gamma}\leq D_2\left[| \Lambda^{r}_D u|_{H^m_\gamma}|v|_{H^m}+|u|_{H^m_\gamma}|\Lambda^{r}_{1,D}v|_{H^m}\right]$.
c) Let $D_3=\max (C_1,D_2)$. Let $f(u)$ be a real-valued, (real-)analytic function satisfying $f(0)=0$ with radius of convergence $R$ at $u=0$. If $D_3\langle ngle u\rangle ngle_m <R$, then
\begin{align}
|\Lambda^{r}_{D} f(u)|_{H^m_\gamma}\leq |\Lambda^{r}_{D}u|_{H^m_\gamma}h_1(D_3\langle ngle u\rangle ngle_m)+|u|_{H^m_\gamma} \langle ngle\Lambda^r_{1} u\rangle ngle_m h_2(D_3\langle ngle u \rangle ngle_m),
\end{align}
where the $h_i$ are analytic functions with radius of convergence $R$ at $0$ with nonnegative coefficients.
\end{prop}
\begin{proof}
\textbf{(a,b)} Part (a) is proved just like Proposition {\rm Re }\, f{f0} (a). To prove (b) use Proposition {\rm Re }\, f{f0a} with the inequality
\begin{align}
|X,\gamma|^{r}\leq C (|X-Y,\gamma|^{r}+ |Y,1|^{r}),
\end{align}
and proceed as in the proof of Proposition {\rm Re }\, f{f0}, part (b).
\textbf{(c)} Using part (b) we obtain
\begin{align}
\begin{split}
&|\Lambda^{r}_{D} f(u)|_{H^m_\gamma}=|\Lambda^{r}_{D} \left(f'(0)u+g(u)u^2\right)|_{H^m_\gamma}\leq \\
&\qquad\quad C |\Lambda^{r}_{D} u|_{H^m_\gamma}+D_2\left[|\Lambda^{r}_{D} u|_{H^m_\gamma}\langle ngle ug(u)\rangle ngle_m)+|u|_{H^m_\gamma}|\Lambda_{1,D}(ug(u))|_{H^m}\right].
\end{split}
\end{align}
To finish, apply Proposition {\rm Re }\, f{f0}(a),(d) to the $D_2$ term.\footnote{The proof shows that we could replace $D_3$ by $C_1$ in the $h_1$ term.}
\end{proof}
Recalling that $\langle ngle \Lambda^{r}u\rangle ngle_{m,\gamma}:=|\Lambda^{r}_D(e^{-\gamma t}u)|_{H^m_\gamma}$ and $\langle ngle \Lambda^{r}_1u\rangle ngle_{m}:=|\Lambda^{r}_{1,D}u|_{H^m}$ (as in Notations {\rm Re }\, f{spaces}), we have the following immediate corollary of Proposition {\rm Re }\, f{f1}.
\begin{cor}\langle bel{f2}
Let $m>(d+1)/2$, $r\geq 0$. Then
a) $\langle ngle \Lambda^{r}(uv)\rangle ngle_{m,\gamma}\leq C_2\left[ \langle ngle\Lambda^{r} u\rangle ngle_{m,\gamma }\langle ngle v\rangle ngle_m+\langle ngle u\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1v\rangle ngle_{m}\right]$.
(b) For $f$, $h_1$, $h_2$ as in Proposition {\rm Re }\, f{f1}:
\begin{align}
\langle ngle\Lambda^{r} f(u)\rangle ngle_{m,\gamma}\leq \langle ngle\Lambda^{r}u\rangle ngle_{m,\gamma}h_1(D_3\langle ngle u\rangle ngle_m)+\langle ngle u\rangle ngle_{m,\gamma} \langle ngle\Lambda^r_{1} u\rangle ngle_m h_2(D_3\langle ngle u \rangle ngle_m).
\end{align}
\end{cor}
It is now straightforward to generalize these results to real-valued, real-analytic functions $f(u)=f(u_1,\dots,u_K)=u_1g(u)$ of $K$ real variables converging in a polydisk $\mathbb{D}=\{u:|u_i|<r_i\}$ for some $r_i>0$.
\begin{prop}\langle bel{f3}
Let $f(u)=f(u_1,\cdots,u_K)$ be as just described. Let $C_3$, $D_3$ be as in Propositions {\rm Re }\, f{f0}, {\rm Re }\, f{f1} respectively.
Then
\begin{align}
\begin{split}
&(a) \langle ngle\Lambda^r_{1}f(u_1,\dots,u_K)\rangle ngle_m\leq \sum^K_{j=1}\langle ngle\Lambda^r_{1}u_j\rangle ngle_m h_j(\langle ngle C_3 u_1\rangle ngle_m,\dots,\langle ngle C_3 u_K\rangle ngle_m),\\
&(b) |\Lambda^{r}_{D} f(u)|_{H^m_\gamma}\leq |\Lambda^{r}_{D}u_1|_{H^m_\gamma}k(\langle ngle D_3u_1\rangle ngle_m,\dots,\langle ngle D_3u_K\rangle ngle_m)+|u_1|_{H^m_\gamma} \sum^K_{j=1}\langle ngle\Lambda^r_{1}u_j\rangle ngle_m k_j(\langle ngle D_3 u_1\rangle ngle_m,\dots,\langle ngle D_3 u_K\rangle ngle_m),\\
&(c) \langle ngle\Lambda^{r} f(u)\rangle ngle_{m,\gamma}\leq \langle ngle\Lambda^{r}u_1\rangle ngle_{m,\gamma}k(\langle ngle D_3u_1\rangle ngle_m,\dots,\langle ngle D_3u_K\rangle ngle_m)+\langle ngle u_1\rangle ngle_{m,\gamma} \sum^K_{j=1}\langle ngle\Lambda^r_{1}u_j\rangle ngle_m k_j(\langle ngle D_3 u_1\rangle ngle_m,\dots,\langle ngle D_3 u_K\rangle ngle_m),
\end{split}
\end{align}
where the functions $h_j$, $k$, $k_j$ are real-analytic functions of $u$ with nonnegative coefficients converging in the polydisk $\mathbb{D}$.
\end{prop}
Next we give microlocal estimates reminiscent of ``Rauch's Lemma" \cite{R}, except that they involve singular norms and microlocalization by singular pseudodifferential operators associated to symbols $\phi=\phi(X,\gamma)$. In these estimates we continue to assume that the functions in question are such that the norms appearing on the right are finite.
We also make the following assumption on the $\mathbb{R}$-valued functions $u$, $v$, or $u_j$ that appear.
\begin{assumption}\langle bel{f4c}
Let $u(x,\theta)$ be $\mathbb{R}$-valued. Suppose $r\geq 1$, $m>(d+1)/2$ and let $(\xi_0,\gamma_0)\in\mathbb{R}^{d+1}\setminus 0$. We suppose that there is a conic neighborhood ${\mathbb G}amma\subset \mathbb{R}^{d+1}\setminus 0$ of $(\xi_0,\gamma_0)$ such that if $\psi(X,\gamma)$ is any singular symbol of order $0$ (Definition {\rm Re }\, f{def4}) with $\mathrm{supp}\; \psi\subset\subset {\mathbb G}amma$, we have
\begin{align}\langle bel{f4a}
\psi_D\Lambda^r_D e^{-\gamma t}u\in H^m\text{ and }\Lambda_D^{r-\frac{1}{2}}e^{-\gamma t}u\in H^m.
\end{align}
\end{assumption}
\begin{nota}\langle bel{f4b}
In the microlocal estimates below we use operators defined by singular symbols $\psi(X,\gamma)$ and $\phi(X,\gamma)$ of order $0$ with the following properties. For ${\mathbb G}amma$ as in Assumption {\rm Re }\, f{f4c} there is a conic neighborhood ${\mathbb G}amma_1\subset\subset {\mathbb G}amma$ such that:
a) $\mathrm{supp}\;\psi\subset\subset {\mathbb G}amma$ and $\psi=1$ on ${\mathbb G}amma_1\cap \{|X,\gamma|\geq 1\}$.
b) $\mathrm{supp}\;\phi\subset\subset {\mathbb G}amma_1$.
\end{nota}
\begin{prop}\langle bel{f5}
Let $g(u)$ be a real-valued, real-analytic function with radius of convergence $R$ and set $f(u)=ug(u)$. Under Assumption {\rm Re }\, f{f4c} on $u$ and with $\phi$, $\psi$ as in Notations {\rm Re }\, f{f4b}, we have $\phi_D\Lambda^r_D e^{-\gamma t}f(u)\in H^m$ and there exists $C_1>0$ such that (suppress subscripts $D$ on $\phi$, $\psi$, $\Lambda$, $\Lambda_1$):
\begin{align}\langle bel{f6b}
\langle ngle\phi\Lambda^r f(u)\rangle ngle_{m,\gamma} \leq A+B,\;\text{ where }
\end{align}
\begin{align}\langle bel{f6c}
\begin{split}
&A= \langle ngle\Lambda^{r-\frac{1}{2}}u\rangle ngle_{m,\gamma}\langle ngle \Lambda^{r-\frac{1}{2}}_{1}u\rangle ngle_m k_1\\
&B=\langle ngle\psi\Lambda^r u\rangle ngle_{m,\gamma}k_2+\langle ngle\psi\Lambda u\rangle ngle_{m,\gamma}\langle ngle \Lambda^{r-1}_{1}u\rangle ngle_m k_3,
\end{split}
\end{align}
where the $k_i$ are analytic functions with power series having radius of convergence $R$ at $0$ and nonnegative coefficients, and evaluated at $(C_1\langle ngle u\rangle ngle_m)$.
\end{prop}
\begin{proof}
\textbf{1. } Using \eqref{f00a} we have $\langle ngle\phi\Lambda^r (ug(u))\rangle ngle_{m,\gamma}=$
\begin{align}
|\phi\Lambda^{r-1}\Lambda(e^{-\gamma t} ug(u))|_{H^m_\gamma}\sim |\phi\Lambda^{r-1}\partial_{{\varepsilon},\gamma}(e^{-\gamma t} ug(u))|_{H^m_\gamma}= |\phi\Lambda^{r-1}(e^{-\gamma t}\partial_{{\varepsilon}} (ug(u)))|_{H^m_\gamma},
\end{align}
which is $\leq A_1+A_2$, where
\begin{align}
A_1=|\phi\Lambda^{r-1}(e^{-\gamma t}\partial_{\varepsilon} u\; g(u))|_{H^m_\gamma},\;\;\;A_2=|\phi\Lambda^{r-1} (ug'(u)e^{-\gamma t}\partial_{\varepsilon} u)|_{H^m_\gamma}.
\end{align}
\textbf{2. }Write $e^{-\gamma t}\partial_{\varepsilon} u=\psi_D (e^{-\gamma t}\partial_{\varepsilon} u)+(1-\psi_D)(e^{-\gamma t}\partial_{\varepsilon} u):=w_1+w_2$. Then $A_1\leq A_{1,1}+A_{1,2}$, where
\begin{align}
\begin{split}
& A_{1,1}=|\phi\Lambda^{r-1}(w_1\; g(u))|_{H^m_\gamma}\leq |\Lambda^{r-1}(w_1\; g(u))|_{H^m_\gamma}\leq \langle ngle\psi\Lambda^r u\rangle ngle_{m,\gamma}k_1+\langle ngle\psi\Lambda u\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r-1}_1 u\rangle ngle_m k_2
\end{split}
\end{align}
by Proposition {\rm Re }\, f{f1}, and $A_{1,2}=|\phi\Lambda^{r-1}(w_2\; g(u))|_{H^m_\gamma}\leq$\footnote{We use $\phi(1-\psi)=0$ in \eqref{faa2}.}
\begin{align}\langle bel{faa2}
|\phi\Lambda^{r-1}(w_2\; g(0))|_{H^m_\gamma}+|\phi\Lambda^{r-1}(w_2\; uh(u))|_{H^m_\gamma}=|\phi\Lambda^{r-1}(w_2\; uh(u))|_{H^m_\gamma}.
\end{align}
We claim
\begin{align}\langle bel{fa2}
|\phi\Lambda^{r-1}(w_2\; uh(u))|_{H^m_\gamma}\leq C \langle ngle\Lambda_1^{r-\frac{1}{2}}(uh(u))\rangle ngle_m \langle ngle\Lambda^{r-\frac{1}{2}}u\rangle ngle_{m,\gamma}
\end{align}
To see this we apply Proposition {\rm Re }\, f{f0a} with $G_{{\varepsilon},\gamma}(\xi,k,\eta,l)=$
\begin{align}
\frac{\phi(X,\gamma)(1-\psi(Y,\gamma))\langle ngle X,\gamma\rangle ngle^{r-1}\langle ngle\xi,k,\gamma\rangle ngle^m}{\langle ngle X-Y,1\rangle ngle^{r-\frac{1}{2}}\langle ngle \xi-\eta,k-l,1\rangle ngle^m\langle ngle\eta,l,\gamma\rangle ngle^m|Y,\gamma|^{r-\frac{3}{2}}},
\end{align}
noting that $r-\frac{3}{2}\geq -\frac{1}{2}$ and that on the support of $G_{{\varepsilon},\gamma}$ we have
\begin{align}
|(X,\gamma)-(Y,\gamma)|=|(X-Y,0)|\geq C|X,\gamma|\text{ and } |(X-Y,0)|\geq C|Y,\gamma|.
\end{align}
By Proposition {\rm Re }\, f{f0}(d) the right side of \eqref{fa2} is bounded by $\langle ngle\Lambda^{r-\frac{1}{2}}u\rangle ngle_{m,\gamma}\langle ngle \Lambda^{r-\frac{1}{2}}_{1}u\rangle ngle_m k_3$.
\textbf{3. }The estimate of $A_2$ clearly yields terms of the same form.
\end{proof}
Easy modifications of the previous proof yield the following analogue of Proposition {\rm Re }\, f{f5} for $f(u_1,\dots,u_K)$.
\begin{prop}\langle bel{f7}
Let $g(u)=g(u_1,\dots,u_K)$ be a real-valued, real-analytic function of $K$ real variables that converges in a polydisk $\mathbb{D}=\{u:|u_i|<R_i\}$ for some $R_i>0$.
Let $r\geq1$ and suppose $m>(d+1)/2$. Set $f(u)=u_1g(u)$. Under Assumption {\rm Re }\, f{f4c} on the $u_j$ and with $\phi$, $\psi$ as in Notations {\rm Re }\, f{f4b}, we have $\phi_D\Lambda^r_D e^{-\gamma t}f(u)\in H^m$ and there exists $C>0$ such that (suppress subscripts $D$ on $\phi$, $\psi$, $\Lambda_1$, and $\Lambda$)
\begin{align}\langle bel{f9}
&\langle ngle\phi\Lambda^r f(u)\rangle ngle_{m,\gamma}\leq A+B, \text{ where }
\end{align}
\begin{align}
\begin{split}
&A=\langle ngle\Lambda^{r-\frac{1}{2}} u_1\rangle ngle_{m,\gamma}\left(\sum^K_{j=1}\langle ngle\Lambda^{r-\frac{1}{2}}_1 u_j\rangle ngle_m h_j\right)\\
&B=\sum^K_{i=1}\langle ngle\psi\Lambda^r u_i\rangle ngle_{m,\gamma}\;k_i+\sum^K_{i=1}\sum^K_{j=1}\langle ngle\psi\Lambda u_i\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r-1}_1u_j\rangle ngle_mk_{i,j}.
\end{split}
\end{align}
Here the functions $h_j$, $k_i$, and $k_{i,j}$ are real-analytic functions of $u$ with nonnegative coefficients, converging in the polydisk $\mathbb{D}$, and evaluated at $(\langle ngle Cu_1\rangle ngle_m,\dots,\langle ngle Cu_K\rangle ngle_m)$.
\end{prop}
\begin{rem}\langle bel{f10}
1.) We shall apply Proposition {\rm Re }\, f{f7} with $r=3/2$ or $r=1$.
2.) To estimate $\langle ngle \phi \Lambda^r f(u)\rangle ngle_{m,\gamma}$ for any real-analytic function $f(u)$ of $u=(u_1,\dots,u_K)$ such that $f(0)=0$, we write
\begin{align}
f(u)=\sum^K_{i=1} u_i g_i(u)
\end{align}
and apply Proposition {\rm Re }\, f{f7} to the individual terms in this sum, letting the various $u_i$ consecutively play the role of $u_1$.
3) Observe that the real-analyticity of the function $f(u)$ in Proposition {\rm Re }\, f{f7} is needed only for the parts of the proof that use Proposition {\rm Re }\, f{f3}. If one knew that Proposition {\rm Re }\, f{f3} held for (say) $f \in C^\infty$, then Proposition {\rm Re }\, f{f7} would be deducible by the same ``chain rule" proof for such $f$.
4) When $r\geq 1$, one can prove Proposition {\rm Re }\, f{f3} for $f\in C^\infty$ inductively by writing $|\Lambda^r_De^{-\gamma t}f(u)|
_{H^m_\gamma} \sim |\Lambda^{r-1} (\partial_{{\varepsilon},\gamma}e^{-\gamma t}f(u))|_{H^m_\gamma}$. However, we need Proposition {\rm Re }\, f{f3} for $r=1/2$ as well as $r=1$ and $3/2$ in the applications here. See, for example, the estimate \eqref{b3}.
\footnote{We have been unable to adapt either Gagliardo-Nirenberg type arguments or arguments involving (double) dyadic decompositions to estimate singular norms of nonlinear $C^\infty$ functions of $u$. The various estimates of this section work well, though, for singular norms of analytic functions of $u$.}
\end{rem}
In estimates of commutators we need the following refinement of Proposition {\rm Re }\, f{f1}(b).
\begin{prop}\langle bel{f12}
Let $r\geq 0$, and suppose $0\leq \sigma\leq \min(s,t)$, where $s+t-\sigma> \frac{d+1}{2}$. Then
\begin{align}
\begin{split}
(a)\;|\Lambda^r_D(uv)|_{H^\sigma_\gamma}\lesssim |\Lambda^r_D u|_{H^s_\gamma}|v|_{H^t}+|u|_{H^s_\gamma}|\Lambda^r_{1,D} v|_{H^t}\\
(b)\;\langle ngle\Lambda^r(uv)\rangle ngle_{\sigma,\gamma}\lesssim \langle ngle\Lambda^r u\rangle ngle_{s,\gamma}\langle ngle v\rangle ngle_t+\langle ngle u\rangle ngle_{s,\gamma}\langle ngle\Lambda^r_{1} v\rangle ngle_t.
\end{split}
\end{align}
\end{prop}
\begin{proof}
Using $|X,\gamma|^r\lesssim |X-Y,\gamma|^r+|Y,1|^r$, we have $|\Lambda^r_D(uv)|_{H^\sigma_\gamma}=$
\begin{align}
\begin{split}
&\left||X,\gamma|^r\langle ngle \xi,k,\gamma\rangle ngle^\sigma (\hat u * \hat v)\right|_{L^2(\xi,k)}\lesssim \\
&\left||X-Y,\gamma|^r\langle ngle \xi,k,\gamma\rangle ngle^\sigma (|\hat u| * |\hat v|)\right|_{L^2(\xi,k)}+\left||Y,1|^r\langle ngle \xi,k,\gamma\rangle ngle^\sigma (|\hat u| * |\hat v|)\right|_{L^2(\xi,k)} :=A+B.
\end{split}
\end{align}
We estimate $A$ and $B$ by applying Proposition {\rm Re }\, f{f0a} to $G(\xi,k,\eta,l)=$
\begin{align}
\frac{|X-Y,\gamma|^r\langle ngle\xi,k,\gamma\rangle ngle^\sigma}{|X-Y,\gamma|^r\langle ngle\xi-\eta,k-l,\gamma\rangle ngle^s\langle ngle\eta,l\rangle ngle^t}\text{ and }\frac{|Y,1|^r\langle ngle\xi,k,\gamma\rangle ngle^\sigma}{\langle ngle\xi-\eta,k-l,\gamma\rangle ngle^s|Y,1|^r\langle ngle\eta,l\rangle ngle^t}
\end{align}
respectively.
\end{proof}
In section {\rm Re }\, f{local} we work with the following norms on the half space that involve higher normal derivatives.
\begin{defn}\langle bel{normal}
Let $\mathbb{R}^{d+1}_+=\{x=(x_0,x_1,\dots,x_d):x_d\geq 0\}$ For functions $u(x,\theta)$ with $(x,\theta)\in \mathbb{R}^{d+1}\times \mathbb{R}$, $r\geq 0$, $m\in \mathbb{N}$, we define
\begin{align}
\begin{split}
&|\Lambda^r u|_{m,\gamma}:=\sum_{j=0}^m|\Lambda^r\partial^j_{x_d}u|_{0,m-j,\gamma},\\
&|\Lambda^r_1 u|_{m}:=\sum_{j=0}^m|\Lambda^r_1\partial^j_{x_d}u|_{0,m-j}.
\end{split}
\end{align}
Time localized versions of these norms, $|\Lambda^r u|_{m,\gamma,T}$, $|\Lambda^r_1 u|_{m,T},$ are defined in the usual way.
\end{defn}
We will need analogues of some of the above results for these norms.
\begin{prop}\langle bel{normalest}
Let $m>(d+2)/2$, $r\geq 0$. Then
a) $| \Lambda^{r}(uv)|_{m,\gamma}\lesssim |\Lambda^{r} u|_{m,\gamma }| v|_m+| u|_{m,\gamma}|\Lambda^{r}_1v|_{m}$.
(b) For $f$, $h_1$, $h_2$ as in Proposition {\rm Re }\, f{f1} and real-valued $u(x,\theta)$:
\begin{align}
|\Lambda^{r} f(u)|_{m,\gamma}\leq |\Lambda^{r}u|_{m,\gamma}h_1(C| u|_m)+|u|_{m,\gamma} |\Lambda^r_{1} u|_m h_2(C| u |_m).
\end{align}
(c) Similarly, the analogue of Proposition {\rm Re }\, f{f3} holds for $|\Lambda^rf(u_1,\dots,u_k)|_{m,\gamma}.$
\end{prop}
After taking extensions of $u$ and $v$ into $x_d<0$, one can use the Fourier transform in all variables to prove this Proposition just as before.
\subsection{Tame estimates} \langle bel{tames}
\emph{\quad}Next we provide the tame estimates that we need for the proof of the continuation argument of section {\rm Re }\, f{local}. In these estimates a lower Sobolev-type norm takes the role sometimes played by an $L^\infty$ or Lipschitz norm in tame estimates. We prove the first four results below after the statement of Proposition {\rm Re }\, f{j4}.
\begin{prop}\langle bel{j1}
Let $m_0>(d+1)/2$, $m\geq 0$, $r\geq 0$, and set $\langle ngle u\rangle ngle_m:=|u|_{H^m}$. Assume the functions $u$ and $v$ are real-valued. Then
a) $\langle ngle u v\rangle ngle_m\leq C_1(\langle ngle u\rangle ngle_m \langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_{m_0} \langle ngle v\rangle ngle_m).$
b) $\langle ngle \Lambda^{r}_{1}(uv)\rangle ngle_m\leq C_2[\langle ngle \Lambda^{r}_{1}u\rangle ngle_m\langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_m\langle ngle \Lambda^{r}_{1}v\rangle ngle_{m_0}+\langle ngle \Lambda^{r}_{1}u\rangle ngle_{m_0}\langle ngle v\rangle ngle_m+\langle ngle u\rangle ngle_{m_0}\langle ngle \Lambda^{r}_{1}v\rangle ngle_m]$.
c) For $n\in\mathbb{N}$, $\langle ngle\Lambda^{r}_{1}(u^n)\rangle ngle_m\leq [\langle ngle\Lambda^{r}_{1} u\rangle ngle_m \cdot n(C_3\langle ngle u \rangle ngle_{m_0})^{n-1}+\langle ngle\Lambda^{r}_{1} u\rangle ngle_{m_0}\langle ngle u\rangle ngle_m \cdot n(n-1)(C_3\langle ngle u \rangle ngle_{m_0})^{n-2}]$.
d) Let $f(u)$ be a real-valued, (real-)analytic function satisfying $f(0)=0$ with radius of convergence $R$ at $u=0$. If $C_3\langle ngle u\rangle ngle_m <R$, then
\begin{align}
\langle ngle \Lambda^{r}_{1} f(u)\rangle ngle_m\leq [\langle ngle \Lambda^{r}_{1}u\rangle ngle_m +\langle ngle u\rangle ngle_m\langle ngle \Lambda^{r}_{1}u\rangle ngle_{m_0}]\cdot h(C_3\langle ngle u \rangle ngle_{m_0}),
\end{align}
where $h$ is an analytic function with nonnegative coefficients and radius of convergence $R$ at $0$ .
\end{prop}
\begin{prop}\langle bel{j2}
Let $m_0>(d+1)/2$, $m\geq 0$, $r\geq0$. Assume that the functions $u$, $v$ are real-valued. Then
a) $|uv|_{H^m_\gamma}\leq C [|u|_{H^m_\gamma}\langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_{m_0}|v|_{H^m_\gamma}]$.
b) $|\Lambda^{r}_{D}(uv)|_{H^m_\gamma}\leq C[| \Lambda^{r}_{D}u|_{H^m_\gamma}\langle ngle v\rangle ngle_{m_0}+| u|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}v\rangle ngle_{m_0}+|\Lambda^{r}_{D}v|_{H^m_\gamma}\langle ngle u\rangle ngle_{m_0}+|v|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}u\rangle ngle_{m_0}]$.
c) Let $f(u)$ be a real-valued, (real-)analytic function satisfying $f(0)=0$ with radius of convergence $R$ at $u=0$. If $C\langle ngle u\rangle ngle_m <R$, then
\begin{align}
|\Lambda^{r}_{D} f(u)|_{H^m_\gamma}\leq [|\Lambda^{r}_{D}u|_{H^m_\gamma} +|u|_{H^m_\gamma}\langle ngle\Lambda^{r}_{1}u\rangle ngle_{m_0}]\cdot h(C\langle ngle u \rangle ngle_{m_0}),
\end{align}
where $h$ is an analytic function with nonnegative coefficients and radius of convergence $R$ at $0$.
\end{prop}
\begin{cor}\langle bel{j3}
Let $m_0>(d+1)/2$, $m\geq 0$, $r\geq 0$. Then
a) $\langle ngle \Lambda^{r}(uv)\rangle ngle_{m,\gamma}\leq C\left[ \langle ngle\Lambda^{r} u\rangle ngle_{m,\gamma }\langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1v\rangle ngle_{m_0}+\langle ngle\Lambda^{r} v\rangle ngle_{m,\gamma }\langle ngle u\rangle ngle_{m_0}+\langle ngle v\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1u\rangle ngle_{m_0}\right]$.
(b) For $f$ and $h$ as in Proposition {\rm Re }\, f{j2}:
\begin{align}
\langle ngle\Lambda^{r} f(u)\rangle ngle_{m,\gamma}\leq \left [\langle ngle\Lambda^{r}u\rangle ngle_{m,\gamma}+\langle ngle u\rangle ngle_{m,\gamma} \langle ngle\Lambda^r_{1} u\rangle ngle_{m_0}\right] h(C\langle ngle u \rangle ngle_{m_0}).
\end{align}
\end{cor}
\begin{prop}\langle bel{j4}
Let $m_0>(d+1)/2$, $m\geq 0$, $r\geq 0$.
Suppose $f(u)=f(u_1,\cdots,u_K)$ is a real-valued, real-analytic function of $K$ real variables converging in a polydisk $\mathbb{D}=\{u:|u_i|<r_i\}$ for some $r_i>0$ and such that $f(0)=0$.
Then
\begin{align}
\begin{split}
&(a) \langle ngle\Lambda^r_{1}f(u_1,\dots,u_K)\rangle ngle_m\leq \sum^K_{j=1}\langle ngle\Lambda^r_{1}u_j\rangle ngle_m h_j+\sum_{j,k=1}^K\langle ngle u_j\rangle ngle_m\langle ngle\Lambda^r_1u_k\rangle ngle_{m_0}h_{j,k},\\
&(b) |\Lambda^r_D f(u_1,\dots,u_K)|_{H^m_\gamma}\leq \sum^K_{j=1}|\Lambda^r_D u_j|_{H^m_\gamma} h_j+\sum_{j,k=1}^K | u_j|_{H^m_\gamma}\langle ngle\Lambda^r_1u_k\rangle ngle_{m_0}h_{j,k},\\
&(c) \langle ngle\Lambda^r f(u_1,\dots,u_K)\rangle ngle_{m,\gamma}\leq \sum^K_{j=1}\langle ngle\Lambda^r u_j\rangle ngle_{m,\gamma} h_j+\sum_{j,k=1}^K\langle ngle u_j\rangle ngle_{m,\gamma}\langle ngle\Lambda^r_1u_k\rangle ngle_{m_0}h_{j,k},\\
\end{split}
\end{align}
where the functions $h_j$, $h_{j,k}$ are real-analytic functions of $u$ with nonnegative coefficients converging in the polydisk $\mathbb{D}$,
and evaluated at $(\langle ngle C u_1\rangle ngle_{m_0},\dots,\langle ngle Cu_K\rangle ngle_{m_0})$ for some $C>0$. They may change from line to line.
\end{prop}
\begin{proof}
\textbf{1. }The previous four results all follow from Proposition {\rm Re }\, f{j2}(b). Proposition {\rm Re }\, f{j2}(a) follows by taking $r=0$, while Proposition {\rm Re }\, f{j1} (b) follows by taking $\gamma=1$; moreover, Proposition {\rm Re }\, f{j1}(b) implies Proposition {\rm Re }\, f{j1}(a). Corollary {\rm Re }\, f{j3})(a) follows from Proposition {\rm Re }\, f{j2}(b) because of our freedom to switch $e^{-\gamma t}$ from $u$ to $v$. The remaining parts of the above propositions, which concern $f(u)$, follow by repeated application of the parts already proved by using analyticity of $f$.
\textbf{2. Proof of Proposition {\rm Re }\, f{j2}(b). }We have
\begin{align}
|\Lambda^r(uv)|_{H^m_\gamma}\lesssim\sum^2_{i=1}\left ||X,\gamma|^r\langle ngle\xi,k,\gamma\rangle ngle^m\chi_i\;|\hat{u}(\xi-\eta,k-l)|*|\hat{v}(\eta,l)|\right|_{L^2(\xi,k)}=A_1+A_2,
\end{align}
where $\chi_1(\xi,k,\eta,l)$ (resp.$\chi_2$) is the characteristic function of $\{|\eta,l|\leq \frac{1}{2}|\xi,k|\}$ (resp. $\{|\eta,l|\geq \frac{1}{2}|\xi,k|\}$).
To see that $A_1\lesssim | \Lambda^{r}_{D}u|_{H^m_\gamma}\langle ngle v\rangle ngle_{m_0}+| u|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}v\rangle ngle_{m_0}$, we write $|X,\gamma|^r\lesssim |X-Y,\gamma|^r+|Y,1|^r$ and apply Proposition {\rm Re }\, f{f0a} with $G(\xi,k,\eta,l)$ first equal to
\begin{align}
\frac{|X-Y,\gamma|^r\langle ngle\xi,k,\gamma\rangle ngle^m\chi_1}{|X-Y,\gamma|^r\langle ngle\xi-\eta,k-l,\gamma\rangle ngle^m\langle ngle\eta,l\rangle ngle^{m_0}},\text{ and then }\frac{|Y,1|^r\langle ngle\xi,k,\gamma\rangle ngle^m\chi_1}{|Y,1|^r\langle ngle\xi-\eta,k-l,\gamma\rangle ngle^m\langle ngle\eta,l\rangle ngle^{m_0}}.
\end{align}
The proof is completed by showing similarly that $A_2\lesssim |\Lambda^{r}_{D}v|_{H^m_\gamma}\langle ngle u\rangle ngle_{m_0}+|v|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}u\rangle ngle_{m_0}$.
\end{proof}
The next estimates will be used in section {\rm Re }\, f{local} to get tame estimates of commutators for ${\varepsilon}$ fixed.
\begin{prop}\langle bel{j7}
Suppose $m_0>\frac{d+1}{2}$, $0\leq m\leq 2m_0$, and $m_1+m_2\leq m$ with $m_i\geq 0$ for $i=1,2$. Let $\partial^{k}$ denote any operator of the form
$\partial_{x',\theta}^\alpha$ with $|\alpha|=k$. Then for $r\geq 0$
\begin{align}\langle bel{j8}
\begin{split}
&(a) |\Lambda^r (\partial^{m_1}u\cdot \partial^{m_2}v)|_{L^2(x',\theta)}\lesssim | \Lambda^{r}_{}u|_{H^m_\gamma}\langle ngle v\rangle ngle_{m_0}+| u|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}v\rangle ngle_{m_0}+|\Lambda^{r}_{}v|_{H^m_\gamma}\langle ngle u\rangle ngle_{m_0}+|v|_{H^m_\gamma}\langle ngle \Lambda^{r}_{1}u\rangle ngle_{m_0}\\
&(b)\langle ngle\Lambda^r (\partial^{m_1}u\cdot \partial^{m_2}v)\rangle ngle_{0,\gamma}\lesssim \langle ngle\Lambda^{r} u\rangle ngle_{m,\gamma }\langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1v\rangle ngle_{m_0}+\langle ngle\Lambda^{r} v\rangle ngle_{m,\gamma }\langle ngle u\rangle ngle_{m_0}+\langle ngle v\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1u\rangle ngle_{m_0}\\
&(c) \langle ngle\Lambda^r (\partial^{m_1}u\cdot \partial^{m_2}v)\rangle ngle_{0,\gamma}\lesssim \langle ngle\Lambda^{r} u\rangle ngle_{m,\gamma }\langle ngle v\rangle ngle_{m_0}+\langle ngle u\rangle ngle_{m,\gamma}\langle ngle\Lambda^{r}_1v\rangle ngle_{m_0}+\langle ngle\Lambda^{r}_1 v\rangle ngle_{m}\langle ngle u\rangle ngle_{m_0,\gamma}+\langle ngle v\rangle ngle_{m}\langle ngle\Lambda^{r}u\rangle ngle_{m_0,\gamma}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
At least one of $m_1$, $m_2$ is $\leq m_0$. If $m_2\leq m_0$, estimate the left side of \eqref{j8}(a) by applying Proposition {\rm Re }\, f{f12}(a) with $\sigma =0$, $s=m-m_1$, $t=m_0-m_2$. If $m_1\leq m_0$, take $s=m_0-m_1$, $t=m-m_2$. Estimate (b) is proved by the same argument, writing
\begin{align}
e^{-\gamma t}(\partial^{m_1}u\cdot \partial^{m_2}v)=(e^{-\gamma t}\partial^{m_1}u)\cdot \partial^{m_2}v
\end{align}
in the case $m_2\leq m_0$, and switching the exponential to the other factor in the case $m_1\leq m_0$. The proof of (c) is similar.
\end{proof}
\begin{rem}\langle bel{j9}
1) We also need the exact analogues of Corollary {\rm Re }\, f{j3} and Propositions {\rm Re }\, f{j2}, {\rm Re }\, f{j4}, where the tangential norms are replaced by the norms involving powers of $\partial_{x_2}$ given in Definition {\rm Re }\, f{normal}. The statements and proofs are essentially identical, except that now we must have $m_0>\frac{d+2}{2}$.
2) There is some redundancy in the results of sections {\rm Re }\, f{nontame} and {\rm Re }\, f{tames}. One might think that the nontame estimates simply follow from corresponding tame ones by taking $m=m_0$, but that is not true in all cases. For example, Proposition {\rm Re }\, f{j1}(b) implies Proposition {\rm Re }\, f{f0}(b), but Proposition {\rm Re }\, f{j2}(b) does not imply Proposition {\rm Re }\, f{f1}(b). In section {\rm Re }\, f{mainestimate} we use only the simpler nontame estimates, while in section {\rm Re }\, f{local} we use both tame and nontame estimates.
\end{rem}
\section{Uniform higher derivative estimates and proof of Theorem {\rm Re }\, f{uniformexistence}}\langle bel{mainestimate}
\emph{\quad} In this section we prove Proposition {\rm Re }\, f{c5}.
First note that for ${\varepsilon}\in (0,{\varepsilon}_0]$ and $T\in(0,T^*_{\varepsilon}]$, $v^s=v^s_T$ satisfies
\begin{align}\langle bel{e1}
|v^s|_{\infty,m}\leq C{\varepsilon} M_0,
\end{align}
where $C$ depends on the Seeley extension but is independent of ${\varepsilon}$ and $T$. Thus
if ${\varepsilon}_0$ is small enough, the operators on the left in \eqref{c1}-\eqref{c3} will be admissible, that is, they will satisfy the properties listed in section {\rm Re }\, f{assumptions}.
Moreover, the bound $\eqref{e1}$ allows us to apply the singular calculus to \eqref{c1}-\eqref{c3} to obtain estimates involving constants that are uniform with respect to ${\varepsilon}$.
\begin{rem}\langle bel{e1z}
\emph{Throughout this section we make essential use of the fact that $v={\nabla}bla_{\varepsilon} u$ on the support of $v^s$ and that $v^s={\nabla}bla_{\varepsilon} u^s=({\nabla}bla_{\varepsilon} u)^s$}. This allows us to replace ${\nabla}bla_{\varepsilon} u$ by $v$, or $v$ by ${\nabla}bla_{\varepsilon} u$, in products where a factor of $v^s$ is present. This is important because, for example, we are able to control the norm
$|{\nabla}bla_{\varepsilon} u/{\varepsilon}|_{\infty,m,\gamma}$ but not the norm $|v/{\varepsilon}|_{\infty,m,\gamma}$. We are able to control the first norm by estimating solutions of $\frac{1}{{\varepsilon}}$\eqref{c3}, but we are not able to control the second norm by estimating solutions of $\frac{1}{{\varepsilon}}$\eqref{c1}, since that would require us to commute a negative power of $\Lambda_D$ through the equation, and the singular calculus gives inadequate uniform control of such commutators.
\end{rem}
\subsection{Outline of estimates}\langle bel{outline}
\emph{\quad}
When $j=1$\footnote{This brief outline is filled out in the remainder of this section.} the terms in the first line of \eqref{c0a} will be controlled by applying the estimates \eqref{b3}-\eqref{b5} to the problem \eqref{c1} (or rather, to the problem
$\partial_{x',\theta}^\alpha$\eqref{c1}, $|\alpha|\leq m$). The first two terms in the second line are controlled by applying estimate \eqref{b3} to the problem ``$\sqrt{{\varepsilon}}$\eqref{c1}."
When $j=2$ the terms in the first line of \eqref{c0a} and the first two terms in the second line will be controlled by similarly applying the estimates \eqref{b6}-\eqref{b8} to the problem \eqref{c2}.
The last two terms in the second line of \eqref{c0a} when $j=1$ (resp. j=2) are estimated by applying \eqref{b15} (resp. \eqref{b16}) to the problem $\frac{1}{\sqrt{{\varepsilon}}}$\eqref{c1} (resp. $\frac{1}{\sqrt{{\varepsilon}}}$\eqref{c2}).
Note that $u$ satisfies a Neumann-type problem of the same kind as \eqref{c1} for $v_1$. Thus we will control $|{\nabla}bla_{\varepsilon} u/{\varepsilon}|_{\infty,m,\gamma}$ and $|{\nabla}bla_{\varepsilon} u/{\varepsilon}|_{0,m+1,{\varepsilon}}$ by applying the estimates \eqref{b4} and \eqref{b5}, with $(\Lambda_{D,\gamma} v_1,D_{x_2} v_1)$ replaced by $(\Lambda_{D,\gamma} u,D_{x_2}u)$, to the problem $\frac{1}{{\varepsilon}}$\eqref{c3}
This will handle the first and second terms in the third line of \eqref{c0a}.
The term $\left|\Lambda^{\frac{1}{2}}{\nabla}bla_{\varepsilon} u/{\sqrt{{\varepsilon}}}\right|_{0,m+1,\gamma}$ is controlled by applying the estimate \eqref{b3} with $(\Lambda_{D,\gamma} v_1,D_{x_2} v_1)$ replaced by $(\Lambda_{D,\gamma} u,D_{x_2}u)$ to the problem $\frac{1}{\sqrt{{\varepsilon}}}$\eqref{c3}. These estimates
use the control of the first two terms in the second line of \eqref{c0a}. The estimate of $\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}$ is a byproduct of this estimate.
The term $|\Lambda^{\frac{1}{2}} {\nabla}bla_{\varepsilon} u/{\varepsilon}|_{0,m,\gamma}$ is controlled by applying \eqref{b3} to $\frac{1}{{\varepsilon}}\eqref{c3}$.
\begin{rem}\langle bel{extracontrol}
When the estimates \eqref{b3}-\eqref{b5} are applied to control singular norms of ${\nabla}bla_{\varepsilon} u$, note that these estimates give the same control of
$\partial_{t,{\varepsilon}}u$ that they do of $\partial_{x_1,{\varepsilon}}u$. Thus, the arguments below that give control of $E_{m,\gamma}(v)$ also give the same control of the norm that is defined by replacing every occurrence of ${\nabla}bla_{\varepsilon} u$ in the third line of \eqref{c0a} by $(\partial_{t,{\varepsilon}}u,{\nabla}bla_{\varepsilon} u)$.
\end{rem}
\subsection{Preliminaries}
\emph{} \;\; We denote the interior and boundary forcing terms in the $v_1$-problem \eqref{c1} by $\cF_1$ and $\cG_1$ respectively, in the $v_2$-problem \eqref{c2} by $\cF_2$ and $\cG_2$, and in the $u$-problem \eqref{c3} by $0$ and $\cG$.
Letting $\partial=(\partial_{x'},\partial_\theta)=(\partial_t,\partial_{x_1},\partial_\theta)$ and $\partial_\gamma=(\partial_t+\gamma,\partial_{x_1},\partial_\theta)$,
we have, for example, $|\Lambda^r v|_{0,m,\gamma}:=$
\begin{align}
|\Lambda^r_D (e^{-\gamma t}v)|_{H^m_\gamma}\sim \sum_{|\alpha|\leq m}|\partial^\alpha_\gamma\Lambda^r_D(e^{-\gamma t}v)|_{L^2(x_2,L^2(x',\theta))}=\sum_{|\alpha|\leq m}|\Lambda^r_D(e^{-\gamma t}\partial^\alpha v)|_{L^2(x_2,L^2(x',\theta))},
\end{align}
so we will prove Proposition {\rm Re }\, f{c5} by applying the estimates \eqref{b3}-\eqref{b5}, \eqref{b6}-\eqref{b8}, and \eqref{b15}-\eqref{b16} to the problems satisfied by $\partial^\alpha v$ or $\partial^\alpha u$, where $|\alpha|\leq m$ or $|\alpha|\leq m+1$, in some cases multiplied by a power of ${\varepsilon}$.
The functions $\cF_1$ and $\cF_2$ are sums of terms of the form
\begin{align}\langle bel{e2}
a(v^s)\partial_{x_{i,{\varepsilon}}}v^s_k\partial_{x_{j,{\varepsilon}}}v^s_l,
\end{align}
where $a=d_vA_\alpha$ is an analytic function of its arguments, $i,j,k,l$ take values in $\{1,2\}$, and we set $\partial_{x_{2,{\varepsilon}}}:=\partial_{x_2}$. In the estimates below it will be convenient to suppress
some subscripts and write terms like \eqref{e2} as
\begin{align}\langle bel{e3}
a(v^s)d_{\varepsilon} v^sd_{\varepsilon} v^s
\end{align}
The term $\cG_1$ has the form
\begin{align}\langle bel{e4}
d_gH(v^s_1,g_{\varepsilon})\partial_{x_{1,{\varepsilon}}}g_{\varepsilon},\text{ where }g_{\varepsilon}:={\varepsilon}^2G,
\end{align}
and, with $b(v^s_1,g_{\varepsilon})$ denoting an analytic function, we write this as\footnote{As explained in Remark {\rm Re }\, f{c4a}, it is important that $v_1$ and not $v^s_1$ occurs in $\cG_2$. In boundary terms $d_{\varepsilon}$ represents only $\partial_{x_{1,{\varepsilon}}}$, never $\partial_{x_2}$ of course.}
\begin{align}\langle bel{e5}
\cG_1= b(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon};\text{ also, }\cG_2= \chi_0(t)H(v_1,g_{\varepsilon}).
\end{align}
Since $H(0,0)=0$ we compute that
\begin{align}\langle bel{e6}
\cG=H(v^s_1,g_{\varepsilon})-d_{v_1}H(v^s_1,g_{\varepsilon})v^s_1=d_gH(0,0)g_{\varepsilon}+b(v^s_1,g_{\varepsilon})\left((v^s_1,g_{\varepsilon}),(v^s_1,g_{\varepsilon})\right),
\end{align}
and we write
\begin{align}\langle bel{e9}
\cG=C g_{\varepsilon}+ b(v^s_1,g_{\varepsilon})(v^s_1,g_{\varepsilon})^2.
\end{align}
\begin{rem}\langle bel{e9a}
In the estimates of interior and boundary forcing as well as the commutator estimates of this section, which depend only on the results of section {\rm Re }\, f{nontame} and do not make use of the basic energy estimates of section {\rm Re }\, f{b1a}, we take $m > \frac{d+1}{2}$, or sometimes slightly larger.
It is only in Propositions {\rm Re }\, f{g1} and {\rm Re }\, f{h1} that we require $m>3d+4+\frac{d+1}{2}$.
\end{rem}
\subsection{Estimates of interior forcing}\langle bel{if}
In this section we estimate the interior forcing terms needed to control the terms in \eqref{c0a}.
\begin{lem}\langle bel{e10}
Assume $m>\frac{d+1}{2}$ ($d=2$ now). We have
\begin{align}
|\Lambda^{\frac{1}{2}}(d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m,\gamma}\lesssim |\Lambda^{\frac{3}{2}}v^s|_{0,m,\gamma}|\Lambda_1 v^s|_{\infty,m}+|\Lambda v^s|_{\infty,m,\gamma}|\Lambda^{\frac{3}{2}}_1v^s|_{0,m}
\end{align}
in the case where both $d_{\varepsilon}=\partial_{x_1,{\varepsilon}}$; otherwise, $|\Lambda^{\frac{1}{2}}\partial_{x_2}v^s|_{0,m,\gamma}$, $|\Lambda_1^{\frac{1}{2}}\partial_{x_2}v^s|_{0,m}$, etc. appear in the obvious places on the right.
\end{lem}
\begin{proof}
Consider the case where both $d_{\varepsilon}=\partial_{x_1,{\varepsilon}}$.
Treating $x_2$ as a parameter and applying Corollary {\rm Re }\, f{f2} ``in the $(t,x_1,\theta)$ variables" we have
\begin{align}
\langle ngle\Lambda^{\frac{1}{2}}(d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m,\gamma}\lesssim \langle ngle\Lambda^{\frac{3}{2}}v^s\rangle ngle_{m,\gamma}\langle ngle\Lambda_1 v^s\rangle ngle_{m}+\langle ngle\Lambda v^s\rangle ngle_{m,\gamma}\langle ngle\Lambda^{\frac{3}{2}}_1v^s\rangle ngle_{m},
\end{align}
and the lemma follows immediately.
\end{proof}
\begin{nota}\langle bel{e10a}
1) In the statements and proofs below we will (with slight abuse) denote analytic functions with nonnegative coefficients like $k(\langle ngle D_3u_1\rangle ngle_m,\dots \langle ngle D_3 u_K\rangle ngle_m)$, where $u=(u_1,\dots,u_K)$,
simply by $h(\langle ngle u\rangle ngle_m)$, and the function $h$ may change from term to term.
In addition we will write sums like
\begin{align}
\langle ngle u_1\rangle ngle_{m,\gamma} \sum^K_{j=1}\langle ngle\Lambda^r_{1}u_j\rangle ngle_m k_j(\langle ngle D_3 u_1\rangle ngle_m,\dots,\langle ngle D_3 u_K\rangle ngle_m)
\end{align}
as in Proposition {\rm Re }\, f{f3} simply as $\langle ngle u\rangle ngle_{m,\gamma}\langle ngle\Lambda^r_1 u\rangle ngle h(\langle ngle u\rangle ngle_m)$.
2) The symbol $Q$ or $Q_j$, $j\in\mathbb{N}$, will always denote an increasing, continuous function from $\mathbb{R}_+$ to itself such that $Q(z)\geq z$. The symbol $Q^o$ or $Q_j^o$ will denote a function of this type such that $Q^o(0)=0$. The meaning of $Q$ or $Q^o$ may change from term to term.
\end{nota}
\begin{lem}\langle bel{e11}
Let $r>0$ and $m>\frac{d+1}{2}$.
\begin{align}
\langle ngle \Lambda^r (a(v^s)w)\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda^r w\rangle ngle_{m,\gamma}h(\langle ngle v^s\rangle ngle_m)+\langle ngle w\rangle ngle_{m,\gamma}\langle ngle \Lambda^r_1 v^s\rangle ngle_m h(\langle ngle v^s\rangle ngle_m).
\end{align}
\end{lem}
\begin{proof}
Write $a(v^s)=a(0)+b(v^s)v^s$, note that by Corollary {\rm Re }\, f{f2}
\begin{align}
\langle ngle\Lambda^r(b(v^s)v^s w)\rangle ngle_{m,\gamma}\lesssim \langle ngle\Lambda^r w\rangle ngle_{m,\gamma}\langle ngle b(v^s)v^s\rangle ngle_m+ \langle ngle w\rangle ngle_{m,\gamma} \langle ngle\Lambda^r_1(b(v^s)v^s)\rangle ngle_m,
\end{align}
and finish by applying Proposition {\rm Re }\, f{f3}(a).
\end{proof}
The next proposition gives control of $|\Lambda^{\frac{1}{2}}\cF_j|_{0,m,\gamma}$, $j=1,2$.
\begin{prop}\langle bel{e12}
Let $m>\frac{d+1}{2}$. For small enough $M_0>0$ as in \eqref{c0h}
we have
\begin{align}\langle bel{ee12}
|\Lambda^{\frac{1}{2}}(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m,\gamma}\lesssim E_{m,\gamma}(v^s)Q^o(E_{m}(v^s))\lesssim Q^o(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
Treating $x_2$ as a parameter, we have by Lemma {\rm Re }\, f{e11}
\begin{align}
\langle ngle\Lambda^{\frac{1}{2}}(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda^{\frac{1}{2}} (d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m,\gamma}h(\langle ngle v^s\rangle ngle_m)+\langle ngle d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m,\gamma}\langle ngle \Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_m h(\langle ngle v^s\rangle ngle_m)
\end{align}
Since $|d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m,\gamma}\lesssim |\Lambda v^s|_{0,m,\gamma}|\Lambda_1 v^s|_{\infty,m}$ and for $M_0>0$ small enough (depending on the polydisc of convergence of $h$),
\begin{align}\langle bel{e12b}
\sup_{x_2\geq 0}h(\langle ngle v^s\rangle ngle_m)\leq h(E_m(v^s))\leq h(CE_{m,T}(v^s)), \text{ where }E_m(v^s)\leq CE_{m,T}(v^s) \;\;(\text{recall } \eqref{c0g}),
\end{align}
the lemma now follows from Lemma {\rm Re }\, f{e10}. The presence of the factors involving $\Lambda_1$ allows us to use $Q^o$ instead of $Q$ in \eqref{ee12}. For the last inequality in \eqref{ee12} we use \eqref{c0g}.
\end{proof}
We need the following variant of Lemma {\rm Re }\, f{e11}.
\begin{lem}\langle bel{e12a}
Let $r\geq 0$ and $m>\frac{d+1}{2}$.
\begin{align}\langle bel{e13}
\begin{split}
&\langle ngle\Lambda^r (a(v^s)w)\rangle ngle_{m+1,\gamma}\lesssim \langle ngle \Lambda^r w\rangle ngle_{m+1,\gamma}h(\langle ngle v^s\rangle ngle_m)+\langle ngle w\rangle ngle_{m+1,\gamma}\langle ngle \Lambda^r_1v^s\rangle ngle_m h(\langle ngle v^s\rangle ngle_m)+\\
& \qquad \langle ngle \Lambda^r w\rangle ngle_{m,\gamma}h(\langle ngle v^s\rangle ngle_{m+1})+\langle ngle w\rangle ngle_{m,\gamma}\langle ngle \Lambda^r_1v^s\rangle ngle_{m+1} h(\langle ngle v^s\rangle ngle_{m+1}).
\end{split}
\end{align}
\end{lem}
\begin{proof}
Write $a(v^s)=a(0)+b(v^s)v^s$ and apply Proposition {\rm Re }\, f{j7}(c) with $``u"=w$, $``m"=m+1$, $``m_0"=m$, observing for example that
\begin{align}
\langle ngle \Lambda^r_1 (v^sb(v^s))\rangle ngle_m\lesssim \langle ngle \Lambda^r_1v^s\rangle ngle_m h(\langle ngle v^s\rangle ngle_m).
\end{align}
\end{proof}
The next proposition gives control of $|\cF_j|_{0,m+1,\gamma}$, $j=1,2$.
\begin{prop}\langle bel{e14}
Let $m>\frac{d+1}{2}$. For small enough $M_0>0$ as in \eqref{c0h}
we have
\begin{align}
|(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m+1,\gamma}\lesssim E_{m,\gamma}(v^s)Q^o(E_{m}(v^s))\lesssim Q^o(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
Treating $x_2$ as a parameter, we have by Lemma {\rm Re }\, f{e12a} (with $r=0$)
\begin{align}\langle bel{e14a}
\langle ngle a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m+1,\gamma}\lesssim \langle ngle d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m+1,\gamma}h(\langle ngle v^s\rangle ngle_m)+ \langle ngle d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m,\gamma}h(\langle ngle v^s\rangle ngle_{m+1}).
\end{align}
A standard Moser estimate gives
\begin{align}\langle bel{e14aa}
\langle ngle d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m+1,\gamma}\lesssim \langle ngle d_{\varepsilon} v^s\rangle ngle_{m+1,\gamma} |d_{\varepsilon} v^s|_{L^\infty}\lesssim \langle ngle \Lambda v^s\rangle ngle_{m+1,\gamma}|\Lambda_1 v^s|_{\infty,m}.
\end{align}
In addition,
\begin{align}
|v^s|_{\infty,m+1}\leq \left|\begin{pmatrix}\Lambda_1 v^s\\partial_{x_2} v^s\end{pmatrix}\right|_{0,m+1}\text{ implies }\;\; \mathrm{sup}_{x_2\geq 0} h(\langle ngle v^s\rangle ngle_{m+1})\leq h(E_m(v^s)),
\end{align}
so we can finish by taking the $L^2$ norm of \eqref{e14a} in $x_2$.
\end{proof}
To control the terms in the second line of \eqref{c0a} we must estimate $\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}\cF_j|_{0,m+1,\gamma}$, $j=1,2$.
\begin{lem}\langle bel{e14b}
Assume $m>\frac{d+1}{2}+1$. Then $\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}(d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m+1,\gamma}\lesssim$
\begin{align}
\left(\sqrt{{\varepsilon}}|\Lambda^{\frac{3}{2}}v^s|)_{0,m+1,\gamma}+|\Lambda v^s|_{0,m+1,\gamma}\right)|\Lambda_1v^s|_{\infty,m}+\sqrt{{\varepsilon}}|\Lambda v^s|_{\infty,m,\gamma}|\Lambda_1^{\frac{3}{2}}v^s|_{0,m}\lesssim E_{m,\gamma}(v_s)E_m(v_s).
\end{align}
\end{lem}
\begin{proof}
Treating $x_2$ as a parameter, we estimate a sum of terms of the form
\begin{align}
\sqrt{{\varepsilon}}\langle ngle \Lambda^{\frac{1}{2}}((\partial^{m_1}d_{\varepsilon} v^s)\cdot (\partial^{m_2}d_{\varepsilon} v^s))\rangle ngle_{0,0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$. In the case where one of $m_1$ or $m_2$ is $m+1$, say $m_1=m+1$, we apply Proposition {\rm Re }\, f{f12}(b) with $``u"=\partial^{m+1}d_{\varepsilon} v^s$ and $``v"=d_{\varepsilon} v^s$ to obtain
\begin{align}
A\lesssim \sqrt{{\varepsilon}}\langle ngle \Lambda^{\frac{3}{2}}v^s\rangle ngle_{m+1,\gamma}\langle ngle\Lambda_1 v^s\rangle ngle_m+\langle ngle \Lambda v^s\rangle ngle_{m+1,\gamma}\langle ngle\Lambda_1 v^s\rangle ngle_m.
\end{align}
On the last term we have used $\sqrt{{\varepsilon}}\langle ngle\Lambda^\frac{3}{2}_1v^s\rangle ngle_{m-\frac{1}{2}}\lesssim \langle ngle \Lambda_1 v^s\rangle ngle_m$.
In the other case where both $m_1$ and $m_2$ are $\leq m$ we apply Proposition {\rm Re }\, f{f12} with $s=m-m_1$ and $t=m-m_2$ to obtain
\begin{align}
A\lesssim \sqrt{{\varepsilon}}\langle ngle\Lambda^{\frac{3}{2}}v^s\rangle ngle_{m,\gamma}\langle ngle\Lambda_1v^s\rangle ngle_{m}+\sqrt{{\varepsilon}}\langle ngle\Lambda v^s\rangle ngle_{m,\gamma}\langle ngle\Lambda_1^{\frac{3}{2}}v^s\rangle ngle_{m}.
\end{align}
To finish we take the $L^2(x_d)$ norm in these inequalities.
\end{proof}
\begin{prop}[$\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}\cF_j|_{0,m+1,\gamma}$]\langle bel{e14c}
Assume $m>\frac{d+1}{2}+1$. For small enough $M_0$ we have .
\begin{align}
\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)|_{0,m+1,\gamma}\lesssim E_{m,\gamma}(v^s)Q^o(E_m(v^s))\lesssim Q^o(E_{m,T}(v^s))
\end{align}
\end{prop}
\begin{proof}
Treating $x_2$ as a parameter, we apply Lemma {\rm Re }\, f{e11} to get $\sqrt{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}}(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{0,m+1,\gamma}\lesssim$
\begin{align}
\sqrt{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}}(d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m+1,\gamma} h(\langle ngle v^s\rangle ngle_{m+1})+\sqrt{{\varepsilon}}\langle ngle d_{\varepsilon} v^s d_{\varepsilon} v^s\rangle ngle_{m+1,\gamma}\langle ngle\Lambda^{\frac{1}{2}}_1v^s\rangle ngle_{m+1} h(\langle ngle v^s\rangle ngle_{m+1}).
\end{align}
We finish by taking the $L^2(x_d)$ norm and by applying Lemma {\rm Re }\, f{e14b} and \eqref{e14aa}.\footnote{Control of $\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}v^s|_{\infty,m+1}$ comes from the first term in the second lines of \eqref{c0a}.}
\end{proof}
To control the term
\begin{align}\langle bel{e14dd}
\left|\begin{pmatrix}\Lambda v_j/\sqrt{{\varepsilon}}\\partial_{x_2} v_j/\sqrt{{\varepsilon}}\end{pmatrix}\right|^{(2)}_{0,m,\gamma}
\end{align}
we must estimate $\frac{1}{\sqrt{{\varepsilon}}}|\cF_j|_{0,m,\gamma}$.
\begin{prop}\langle bel{e14d}
Let $m>\frac{d+1}{2}$. For small enough $M_0>0$
we have
\begin{align}\langle bel{e14e}
\frac{1}{\sqrt{{\varepsilon}}}|a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s|_{0,m,\gamma}\lesssim Q^o(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
Treating $x_2$ as a parameter, we have by Corollary {\rm Re }\, f{f2}
\begin{align}
\frac{1}{\sqrt{{\varepsilon}}}\langle ngle(a(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m,\gamma}\lesssim \frac{1}{\sqrt{{\varepsilon}}}\langle ngle (d_{\varepsilon} v^s d_{\varepsilon} v^s)\rangle ngle_{m,\gamma}h(\langle ngle v^s\rangle ngle_m).
\end{align}
Since $|d_{\varepsilon} v^s d_{\varepsilon} v^s|_{0,m,\gamma}\lesssim |\Lambda v^s|_{0,m,\gamma}|\Lambda_1 v^s|_{\infty,m}$, the result follows.
\end{proof}
\subsection{Boundary forcing I}\langle bel{bf}
\emph{\quad\quad}Here we estimate the forcing terms given by $\cG_1$ and $\cG$ for the problems \eqref{c1} and \eqref{c3} respectively.
The next two Propositions, which are needed to estimate the terms in the first line of \eqref{c0a} when $j=1$, give control of $\langle ngle \Lambda \cG_1\rangle ngle_{m,\gamma}$ and $\langle ngle \Lambda^{\frac{1}{2}} \cG_1\rangle ngle_{m+1,\gamma}$; recall $\cG_1=b(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon} $.
\begin{prop}\langle bel{e15}
Let $m>\frac{d+1}{2}$. For small enough constants ${\varepsilon}_1>0$ and $M_0>0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$\footnote{Here $M_G$ is as in \eqref{MG}.}
\begin{align}
\langle ngle\Lambda\cG_1\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma} h(\langle ngle v^s_1\rangle ngle_m)+(\langle ngle \Lambda_1 v^s_1\rangle ngle_m+{\varepsilon}) h(\langle ngle v^s_1\rangle ngle_m)\lesssim (M_G+{\varepsilon})Q(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
Write $b(v^s_1,g_{\varepsilon})=b(0,0)+c(v^s_1,g_{\varepsilon})$ where $c(0,0)=0$. By Corollary {\rm Re }\, f{f2}
\begin{align}
\begin{split}
&\langle ngle\Lambda (c(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon})\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma} \langle ngle c(v^s_1,g_{\varepsilon})\rangle ngle_m+\langle ngle d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma}\langle ngle \Lambda_1 c(v^s_1,g_{\varepsilon})\rangle ngle_m\leq \\
&\qquad M_G h(\langle ngle v^s_1\rangle ngle_m)+{\varepsilon}(\langle ngle \Lambda_1 v^s_1\rangle ngle_m+{\varepsilon}) h(\langle ngle v^s_1\rangle ngle_m).
\end{split}
\end{align}
Here we have used Proposition {\rm Re }\, f{f3}(a) and the fact that, since $g_{\varepsilon}={\varepsilon}^2 G$ with $G\in H^{m+3}(b\Omega)$, we have for ${\varepsilon}_1>0$ small enough (depending on the polydisk of convergence of $c$)
\begin{align}
\langle ngle c(v^s_1,g_{\varepsilon})\rangle ngle_m\leq h(\langle ngle v^s_1\rangle ngle_m) \text{ and }\langle ngle d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma}\leq {\varepsilon} M_G\text{ for }{\varepsilon}\in (0,{\varepsilon}_1].
\end{align}
\end{proof}
\begin{prop}\langle bel{e16}
Let $m>\frac{d+1}{2}$. For small enough ${\varepsilon}_1>0$ and $M_0>0$ as in \eqref{c0h}
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}
\langle ngle\Lambda^{\frac{1}{2}}\cG_1\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}} h(\langle ngle v^s_1\rangle ngle_{m+1})+{\varepsilon} (\langle ngle \Lambda^{\frac{1}{2}}_1 v^s_1\rangle ngle_{m+1}+{\varepsilon}^{\frac{3}{2}}) h(\langle ngle v^s_1\rangle ngle_{m+1})\lesssim \sqrt{{\varepsilon}}Q(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
By Corollary {\rm Re }\, f{f2} we have $\langle ngle\Lambda^{\frac{1}{2}} (b(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon})\rangle ngle_{m+1,\gamma}\lesssim$
\begin{align}
\begin{split}
&\langle ngle \Lambda^{\frac{1}{2}} d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m+1,\gamma} h(\langle ngle v^s_1,g_{\varepsilon}\rangle ngle_{m+1})+\langle ngle d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m+1,\gamma}\langle ngle \Lambda^{\frac{1}{2}}_1 (v^s_1,g_{\varepsilon})\rangle ngle_{m+1} h(\langle ngle v^s_1,g_{\varepsilon}\rangle ngle_{m+1})\leq \sqrt{{\varepsilon}}Q(E_m(v^s)).
\end{split}
\end{align}
\end{proof}
To control the terms in the second line of \eqref{c0a} when $j=1$ we must estimate $\sqrt{{\varepsilon}}\langle ngle\Lambda\cG_1\rangle ngle_{m+1,\gamma}$.
\begin{prop}\langle bel{e22a}
Let $m>\frac{d+1}{2}$. For small enough ${\varepsilon}_1>0$ and $M_0>0$ as in \eqref{c0h}
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}
\sqrt{{\varepsilon}}\langle ngle\Lambda\cG_1\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}} h(\langle ngle v^s_1\rangle ngle_{m+1})+{\varepsilon}^{\frac{3}{2}} (\langle ngle \Lambda_1 v^s_1\rangle ngle_{m+1}+{\varepsilon}) h(\langle ngle v^s_1\rangle ngle_{m+1})\lesssim \sqrt{{\varepsilon}}Q(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
By Corollary {\rm Re }\, f{f2} we have $\sqrt{{\varepsilon}}\langle ngle\Lambda (b(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon})\rangle ngle_{m+1,\gamma}\lesssim$
\begin{align}
\begin{split}
&\sqrt{{\varepsilon}}\langle ngle \Lambda d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m+1,\gamma} h(\langle ngle v^s_1,g_{\varepsilon}\rangle ngle_{m+1})+\sqrt{{\varepsilon}}\langle ngle d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m+1,\gamma}\langle ngle \Lambda_1 (v^s_1,g_{\varepsilon})\rangle ngle_{m+1} h(\langle ngle v^s_1,g_{\varepsilon}\rangle ngle_{m+1}).
\end{split}
\end{align}
\end{proof}
To control the terms \eqref{e14dd} with $j=1$ we must estimate $\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}\cG_1\rangle ngle_{m,\gamma}$.
\begin{prop}\langle bel{e22f}
Let $m>\frac{d+1}{2}$. For small enough constants ${\varepsilon}_1>0$ and $M_0>0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}
\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}\cG_1\rangle ngle_{m,\gamma}\lesssim (M_G+\sqrt{{\varepsilon}})Q(E_{m,T}(v^s)).
\end{align}
\end{prop}
\begin{proof}
Parallel to Proposition {\rm Re }\, f{e15} we have
\begin{align}
\begin{split}
&\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}} (c(v^s_1,g_{\varepsilon})d_{\varepsilon} g_{\varepsilon})\rangle ngle_{m,\gamma}\lesssim \frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}} d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma} \langle ngle c(v^s_1,g_{\varepsilon})\rangle ngle_m+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle d_{\varepsilon} g_{\varepsilon}\rangle ngle_{m,\gamma}\langle ngle \Lambda^{\frac{1}{2}}_1 c(v^s_1,g_{\varepsilon})\rangle ngle_m\leq \\
&\qquad M_G h(\langle ngle v^s_1\rangle ngle_m)+\sqrt{{\varepsilon}}(\langle ngle \Lambda^{\frac{1}{2}}_1 v^s_1\rangle ngle_m+{\varepsilon}^{\frac{3}{2}}) h(\langle ngle v^s_1\rangle ngle_m).
\end{split}
\end{align}
\end{proof}
To control the terms
\begin{align}\langle bel{e26bb}
\left|\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{\infty,m,\gamma}, \;\left|\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{0,m+1,\gamma},\;\left|\frac{\Lambda^{\frac{1}{2}}{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right|_{0,m,\gamma}, \left|\frac{\Lambda^{\frac{1}{2}}{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right|_{0,m+1,\gamma},\text{ and }\left\langle ngle\frac {{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma},
\end{align}
in the third line of \eqref{c0a}, we use that fact that $v={\nabla}bla_{\varepsilon} u$ on the support of $v^s$ and apply the estimates \eqref{b3}-\eqref{b5} and \eqref{b15} to the problems $\frac{1}{{\varepsilon}}\eqref{c3}$ and $\frac{1}{\sqrt{{\varepsilon}}}\eqref{c3}$. For this we must estimate the forcing terms
\begin{align}\langle bel{e22bb}
\frac{1}{{\varepsilon}}\langle ngle\Lambda \cG\rangle ngle_{m,\gamma},\; \frac{1}{{\varepsilon}}\langle ngle \Lambda^{\frac{1}{2}}\cG\rangle ngle_{m+1,\gamma}, \text{ and }\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda \cG\rangle ngle_{m+1,\gamma}, \end{align}
where $\cG=Cg_{\varepsilon}+b(v^s_1,g_{\varepsilon})(v^s_1,g_{\varepsilon})^2$ in the notation of \eqref{e9}.
\begin{prop}\langle bel{e22cc}
Let $m>\frac{d+1}{2}$. For small enough ${\varepsilon}_1>0$ and $M_0>0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}
\begin{split}
&\;\frac{1}{{\varepsilon}}\langle ngle\Lambda \cG\rangle ngle_{m,\gamma}\lesssim M_G+\left(E_m^2(v^s)+{\varepsilon}^2 M_G^2\right) h(\langle ngle v_s\rangle ngle_m)\lesssim M_GQ(E_{m,T}(v^s))+Q^o(E_{m,T}(v^s))\\
\end{split}
\end{align}
\end{prop}
\begin{proof}
Clearly, $\frac{1}{{\varepsilon}}\langle ngle\Lambda g_{\varepsilon}\rangle ngle_{m,\gamma}\lesssim M_G$. Writing $v^s$ in place of $v^s_1$ and $b(v^s,g_{\varepsilon})=b(0,0)+c(v^s,g_{\varepsilon})$ with $c(0,0)=0$, we have by Corollary {\rm Re }\, f{f2} that $\frac{1}{{\varepsilon}}\langle ngle\Lambda [c(v^s,g_{\varepsilon})(v^s,g_{\varepsilon})^2]\rangle ngle_{m,\gamma}\lesssim $
\begin{align}\langle bel{e22d}
\begin{split}
&\qquad \frac{1}{{\varepsilon}}\langle ngle\Lambda(v^s,g_{\varepsilon})^2\rangle ngle_{m,\gamma}\langle ngle v^s,g_{\varepsilon}\rangle ngle_m h(\langle ngle v^s\rangle ngle_m)+\frac{1}{{\varepsilon}}\langle ngle(v^s,g_{\varepsilon})^2\rangle ngle_{m,\gamma}\langle ngle\Lambda_1(v^s,g_{\varepsilon})\rangle ngle_m h(\langle ngle v^s\rangle ngle_m):=A+B.
\end{split}
\end{align}
The estimate
\begin{align}\langle bel{e22dd}
\frac{1}{{\varepsilon}}\langle ngle\Lambda (v^s v ^s)\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda v^s\rangle ngle_{m,\gamma}\langle ngle v^s/{\varepsilon}\rangle ngle_m +\langle ngle v^s/{\varepsilon}\rangle ngle_{m,\gamma}\langle ngle\Lambda_1 v^s\rangle ngle_m\lesssim E^2_m(v^s)
\end{align}
and other similar ones yields,
\begin{align}
\frac{1}{{\varepsilon}}\langle ngle\Lambda(v^s,g_{\varepsilon})^2\rangle ngle_{m,\gamma}\lesssim E^2_m(v^s)+{\varepsilon}^2 M_G^2,
\end{align}
so $A\lesssim \left(E_m^2(v^s)+{\varepsilon}^2 M_G^2\right) h(\langle ngle v_s\rangle ngle_m).$ The term $B$ is treated similarly.
\end{proof}
\begin{prop}\langle bel{e22e}
Let $m>\frac{d+1}{2}$. For small enough ${\varepsilon}_1>0$ and $M_0>0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}
\begin{split}
(a)\;\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} \cG\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}}M_G+\left(E_m^2(v^s)+{\varepsilon}^{\frac{5}{2}} M_G^2\right) h(\langle ngle v_s\rangle ngle_{m+1})\lesssim Q^o(E_{m,T}(v^s))+\sqrt{{\varepsilon}}Q(E_{m,T}(v^s))\\
(b)\;\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda \cG\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}}M_G+\left(E_m^2(v^s)+{\varepsilon}^{\frac{5}{2}} M_G^2\right) h(\langle ngle v_s\rangle ngle_{m+1})\lesssim Q^o(E_{m,T}(v^s))+\sqrt{{\varepsilon}}Q(E_{m,T}(v^s)).
\end{split}
\end{align}
\end{prop}
\begin{proof}
(a) Clearly, $\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} g_{\varepsilon}\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}} M_G$.
By Proposition {\rm Re }\, f{j7}(c) we have in place of \eqref{e22dd}
\begin{align}
\begin{split}
&\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} (v^s v ^s)\rangle ngle_{m+1,\gamma}\lesssim \langle ngle \Lambda^{\frac{1}{2}} v^s\rangle ngle_{m+1,\gamma}\langle ngle v^s/{\varepsilon}\rangle ngle_m +\langle ngle v^s/\sqrt{{\varepsilon}}\rangle ngle_{m+1,\gamma}\langle ngle\Lambda^{\frac{1}{2}}_1 v^s/\sqrt{{\varepsilon}}\rangle ngle_m+\\
&\qquad \qquad\langle ngle \Lambda^{\frac{1}{2}} v^s/\sqrt{{\varepsilon}}\rangle ngle_{m,\gamma}\langle ngle v^s/\sqrt{{\varepsilon}}\rangle ngle_{m+1} +\langle ngle v^s/{\varepsilon}\rangle ngle_{m,\gamma}\langle ngle\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1}\lesssim E^2_m(v^s).
\end{split}
\end{align}
The rest of the proof is essentially the same as that of Proposition {\rm Re }\, f{e22cc}. The proof of (b) is identical, except that we use control of $\sqrt{{\varepsilon}}\langle ngle \Lambda v^s\rangle ngle_{m+1,\gamma}$ here.
\end{proof}
\subsection{Interior commutators}\langle bel{ic}
\emph{\quad}
To estimate terms in the first line of \eqref{c0a} we must estimate norms involving interior commutators of the form
\begin{align}\langle bel{e23}
|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^k]v|_{0,0,\gamma}\text{ and }|[a(v^s)d_{\varepsilon}^2,\partial^{k+1}]v|_{0,0,\gamma},
\end{align}
where $k\leq m$, $a=A_\alpha$, and $\partial^k$ denotes $(\partial_{x'},\partial_\theta)^\alpha$ for some multi-index such that $|\alpha|=k$. We will use $\partial$ to denote not only the tangential derivatives like $\partial_{t}$, $\partial_{x_1}$, and $\partial_{\theta}$ but also (with some abuse) ${\varepsilon} \partial_{x_1}+\beta_1\partial_\theta={\varepsilon} \partial_{x_1,{\varepsilon}}$.
\begin{prop}\langle bel{e24}
Suppose $m>\frac{d+1}{2}+1$.
(a) When $d_{\varepsilon}^2=\partial_{x_1,{\varepsilon}}^2$ or $\partial_{x_2}\partial_{x_1,{\varepsilon}}$ we have for $0\leq k\leq m$
\begin{align}\langle bel{e24az}
|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^k]v|_{0,0,\gamma}\lesssim |\Lambda^{\frac{3}{2}} v|_{0,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m})+|\Lambda v|_{\infty,m,\gamma}\left|\frac{\Lambda^{\frac{1}{2}}_1v^s}{{\varepsilon}}\right|_{0,m}h(|v^s|_{\infty,m}).
\end{align}
(b) When $d_{\varepsilon}^2=\partial_{x_2}^2$, we have the same estimate with $|\Lambda^{\frac{3}{2}} v|_{0,m,\gamma}$ and $|\Lambda v|_{\infty,m,\gamma}$ replaced respectively by
\begin{align}
|\Lambda^{\frac{1}{2}}{\varepsilon} \partial_{x_2}^2 v|_{0,m-1,\gamma} \text{ and } |{\varepsilon}\partial_{x_2}^2 v|_{\infty,m-1,\gamma}.
\end{align}
\end{prop}
\begin{proof}
(a) Writing $a(v^s)=a(0)+b(v^s)v^s$ and taking $k=m$ (the``main" case), we must estimate a sum of terms of the form
\begin{align} \langle bel{e24a}
|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 v)]|_{0,0,\gamma}=\frac{1}{{\varepsilon}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}\partial v)]|_{0,0,\gamma},
\end{align}
where $m_1+m_2=m$, $m_1\geq 1$. Treating $x_2$ as a parameter we apply Proposition {\rm Re }\, f{f12} ``in the $(x',\theta)$ variables" with $\sigma=0$, $s=m-m_2-1$ and $t=m-m_1$. Finish by taking the $L^2(x_2)$ norm of the inequality thus obtained. The proof of (b) is essentially the same.\footnote{The formulas in (a) and (b) agree if one regards ${\varepsilon}\partial_{x_2}$ (resp. $\partial_{x_2}$) as ``equivalent" to a tangential derivative $\partial$ (resp. $\Lambda_D$).}
\end{proof}
\begin{rem}
Observe that the right side of \eqref{e24az} is $\lesssim E_{m,\gamma}(v)Q(E_{m,T}(v^s))$. As we show below and in the next section, the same statement applies to \emph{all} interior and boundary commutators.
\end{rem}
\begin{prop}\langle bel{e25}
Suppose $m>\frac{d+1}{2}+1$.
(a) When $d_{\varepsilon}^2=\partial_{x_1,{\varepsilon}}^2$ or $\partial_{x_2}\partial_{x_1,{\varepsilon}}$ we have for $0\leq k\leq m+1$
\begin{align}
|[a(v^s)d_{\varepsilon}^2,\partial^k]v|_{0,0,\gamma}\lesssim |\Lambda v|_{0,m+1,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m})+|\Lambda v|_{\infty,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{0,m+1}h(|v^s|_{\infty,m+1}).
\end{align}
(b) When $d_{\varepsilon}^2=\partial_{x_2}^2$, we have the same estimate with $|\Lambda v|_{0,m+1,\gamma}$ and $|\Lambda v|_{\infty,m,\gamma}$ replaced respectively by
\begin{align}
|{\varepsilon} \partial_{x_2}^2 v|_{0,m,\gamma} \text{ and } |{\varepsilon}\partial_{x_2}^2 v|_{\infty,m-1,\gamma}.
\end{align}
\end{prop}
\begin{proof}
(a) Taking $k=m+1$, we must estimate a sum of terms of the form
\begin{align} \langle bel{e25a}
|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 v)]|_{0,0,\gamma}=\frac{1}{{\varepsilon}}|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}\partial v)]|_{0,0,\gamma}:= A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we obtain easily
\begin{align}\langle bel{e25bz}
A\lesssim |v^sb(v^s)/{\varepsilon}|_{0,m+1}|e^{-\gamma t}d_{\varepsilon}\partial v|_{L^\infty}\lesssim |\Lambda v|_{\infty,m,\gamma}|v^s/{\varepsilon}|_{0,m+1}h(|v^s|_{\infty,m+1}).
\end{align}
In the remaining case, we have $m_1\geq 1$, $m_2\geq 1$. Treating $x_2$ as a parameter, we estimate $A$ by applying Proposition {\rm Re }\, f{f12} with $r=0$, $\sigma=0$, $s=(m+1)-(m_2+1)$ and $t=m-m_1$, and taking the $L^2(x_2)$ norm of the inequality thus obtained. Again, the proof of (b) is the similar.
\end{proof}
The next estimate is needed to control terms in the second line of \eqref{c0a}.
\begin{prop}\langle bel{e26}
Suppose $m>\frac{d+1}{2}+2$.
(a) When $d_{\varepsilon}^2=\partial_{x_1,{\varepsilon}}^2$ or $\partial_{x_2}\partial_{x_1,{\varepsilon}}$ we have for $0\leq k\leq m+1$
\begin{align}
\begin{split}
&\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^k]v|_{0,0,\gamma}\lesssim \left(\left|\frac{\Lambda_1^{\frac{1}{2}}v^s}{\sqrt{{\varepsilon}}}\right|_{0,m+1}+\left|\frac{v^s}{{\varepsilon}}\right|_{0,m+1}\right)|\Lambda v|_{\infty,m,\gamma}h(|v^s|_{\infty,m+1})+\\
&\qquad \left(|\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v|_{0,m+1,\gamma}\;+|\Lambda v|_{0,m+1,\gamma}\right)\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m}).
\end{split}
\end{align}
(b) When $d_{\varepsilon}^2=\partial_{x_2}^2$, we have the same estimate with $|\Lambda v|_{\infty,m,\gamma}$, $|\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v|_{0,m+1,\gamma}$, and $|\Lambda v|_{0,m+1,\gamma}$ replaced by
\begin{align}
|{\varepsilon}\partial_{x_2}^2 v|_{\infty,m-1,\gamma},\; |{\varepsilon}^{\frac{3}{2}}\Lambda^{\frac{1}{2}}\partial_{x_2}^2 v|_{0,m,\gamma}, \text{ and }|{\varepsilon} \partial_{x_2}^2v|_{0,m,\gamma}.
\end{align}
respectively.
\end{prop}
\begin{proof}
Parallel to \eqref{e25a} we must estimate a sum of terms of the form
\begin{align} \langle bel{e26a}
\sqrt{{\varepsilon}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 v)]|_{0,0,\gamma}=\frac{1}{\sqrt{{\varepsilon}}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}\partial v)]|_{0,0,\gamma}:= A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we apply Proposition {\rm Re }\, f{f12}(b) (in the $(x',\theta)$ variables) with $m-\frac{3}{2}$ playing the role of $``s"$ and $``u"=d_{\varepsilon}\partial v$ to obtain
\begin{align}\langle bel{e26b}
A\lesssim \left(\left|\frac{\Lambda_1^{\frac{1}{2}}v^s}{\sqrt{{\varepsilon}}}\right|_{0,m+1}+\left|\frac{v^s}{{\varepsilon}}\right|_{0,m+1}\right)|\Lambda v|_{\infty,m,\gamma}h(|v^s|_{\infty,m+1}).
\end{align}
Here we have used $m-\frac{3}{2}>\frac{d+1}{2}$ to estimate
\begin{align}
|\sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}d_{\varepsilon}\partial v|_{\infty,m-\frac{3}{2},\gamma}\leq |\Lambda v|_{\infty,m,\gamma}.
\end{align}
In the case $m_1=m$ we similarly apply Proposition {\rm Re }\, f{f12}(b) with $``s"=m-\frac{5}{2}$ and $``t"=1$ to obtain \eqref{e26b} again.
In the case $m_1\leq m-1$ we apply Proposition {\rm Re }\, f{f12} with $\sigma=0$, $t=m-1-m_1$ and $s=(m+1)-(m_2+1)$ to obtain
\begin{align}
A\lesssim \left(|\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v|_{0,m+1,\gamma}\;\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m-1}+|\Lambda v|_{0,m+1,\gamma}\;\left|\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}\right|_{\infty,m-1}\right)h(|v^s|_{\infty,m}).
\end{align}
To finish we observe
$|\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}|_{\infty,m-1}\lesssim |\frac{v^s}{{\varepsilon}}|_{\infty,m}$.
\end{proof}
To control the terms \eqref{e26bb} we must estimate the interior commutators
\begin{align}
\frac{1}{{\varepsilon}}|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^m]u|_{0,0,\gamma}, \frac{1}{{\varepsilon}}|[a(v^s)d_{\varepsilon}^2,\partial^{m+1}]u|_{0,0,\gamma}, \text{ and }\;\frac{1}{{\varepsilon}^{1/2}}|\Lambda^{1/2}[a(v^s)d_{\varepsilon}^2,\partial^{m+1}]u|_{0,0,\gamma}.
\end{align}
In these estimates we sometimes write $v$ in place of $d_{\varepsilon} u$ when a factor of $v^s$ is present.
\begin{prop}\langle bel{e26c}
Suppose $m>\frac{d+1}{2}$. For $0\leq k\leq m$ we have
\begin{align}
\begin{split}
&\;\frac{1}{{\varepsilon}}|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^k]u|_{0,0,\gamma}\lesssim |\Lambda^{\frac{3}{2}} v|_{0,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m})+|\Lambda v|_{\infty,m,\gamma}\left|\frac{\Lambda^{\frac{1}{2}}_1v^s}{{\varepsilon}}\right|_{0,m}h(|v^s|_{\infty,m})\\
\end{split}
\end{align}
\end{prop}
\begin{proof}
Parallel to \eqref{e24a}, to prove (a) we must estimate a sum of terms of the form
\begin{align}
\frac{1}{{\varepsilon}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 u)]|_{0,0,\gamma}=\frac{1}{{\varepsilon}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]|_{0,0,\gamma},
\end{align}
where $m=m_1+m_2$, $m_1\geq 1$. To finish apply Proposition {\rm Re }\, f{f12} with $s=m-m_2$, $t=m-m_1$.
\end{proof}
\begin{prop}\langle bel{e26d}
Suppose $m>\frac{d+1}{2}$. For $0\leq k\leq m+1$ we have
\begin{align}
\frac{1}{{\varepsilon}}|[a(v^s)d_{\varepsilon}^2,\partial^k]u|_{0,0,\gamma}\lesssim |\Lambda v|_{0,m+1,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m})+|\Lambda v|_{\infty,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{0,m+1}h(|v^s|_{\infty,m+1}).
\end{align}
\end{prop}
\begin{proof}
(a) Parallel to \eqref{e25a} we must estimate a sum of terms of the form
\begin{align}
\frac{1}{{\varepsilon}}|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 u)]|_{0,0,\gamma}=\frac{1}{{\varepsilon}}|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]|_{0,0,\gamma}:= A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we obtain
\begin{align}\langle bel{e25b}
A\lesssim |v^sb(v^s)/{\varepsilon}|_{0,m+1}|e^{-\gamma t}d_{\varepsilon} v|_{L^\infty}\lesssim |\Lambda v|_{\infty,m,\gamma}|v^s/{\varepsilon}|_{0,m+1}h(|v^s|_{\infty,m+1}).
\end{align}
In the remaining case, we have $m_1\geq 1$, $m_2\geq 1$, and we estimate $A$ by applying Proposition {\rm Re }\, f{f12} with $r=0$, $\sigma=0$, $s=m+1-m_2$ and $t=m-m_1$, and taking the $L^2(x_2)$ norm of the inequality thus obtained.
\end{proof}
\begin{prop}\langle bel{e26h}
Suppose $m>\frac{d+1}{2}+1$.
For $0\leq k\leq m+1$ we have
\begin{align}
\begin{split}
&\frac{1}{\sqrt{{\varepsilon}}}|\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon}^2,\partial^k]u|_{0,0,\gamma}\lesssim \left(\left|\frac{\Lambda_1^{\frac{1}{2}}v^s}{\sqrt{{\varepsilon}}}\right|_{0,m+1}+\left|\frac{v^s}{{\varepsilon}}\right|_{0,m+1}\right)|\Lambda v|_{\infty,m,\gamma}h(|v^s|_{\infty,m+1})+\\
&\qquad \qquad \qquad |\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v|_{0,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m}).
\end{split}
\end{align}
\end{prop}
\begin{proof}
Parallel to \eqref{e26a} we must estimate a sum of terms of the form
\begin{align} \langle bel{e26i}
\frac{1}{\sqrt{{\varepsilon}}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 u)]|_{0,0,\gamma}=\frac{1}{\sqrt{{\varepsilon}}}|\Lambda^{\frac{1}{2}}[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]|_{0,0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$ and $m_1\geq 1$.
We can now repeat the proof of Proposition {\rm Re }\, f{e26} in the case $m_1=m+1$ with $``u"=d_{\varepsilon} v$ and using $``s"=m-\frac{1}{2}$ in place of $m-\frac{3}{2}$. In the case $m_1\leq m$ we apply Proposition {\rm Re }\, f{f12} with $t=m-m_1$ and $s=m-m_2$ to obtain
\begin{align}
A\lesssim \left(|\sqrt{{\varepsilon}}\Lambda^{\frac{3}{2}} v|_{0,m,\gamma} \left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}+|\Lambda v|_{\infty,m,\gamma}\left|\frac{\Lambda_1^{\frac{1}{2}}v^s}{\sqrt{{\varepsilon}}}\right|_{0,m}\right)h(|v^s|_{\infty,m}).
\end{align}
\end{proof}
To control the terms \eqref{e14dd} we must estimate the commutator $\frac{1}{\sqrt{{\varepsilon}}}|[a(v^s)d_{\varepsilon}^2,\partial ^k]v|_{0,0,\gamma}$, $k\leq m$.
\begin{prop}\langle bel{e26e}
Suppose $m>\frac{d+1}{2}+1$.
(a) When $d_{\varepsilon}^2=\partial_{x_1,{\varepsilon}}^2$ or $\partial_{x_2}\partial_{x_1,{\varepsilon}}$ we have for $0\leq k\leq m$
\begin{align}
\frac{1}{\sqrt{{\varepsilon}}}|[a(v^s)d_{\varepsilon}^2,\partial^k]v|_{0,0,\gamma}\lesssim \left|\frac{\Lambda v}{\sqrt{{\varepsilon}}}\right|_{0,m,\gamma}\left|\frac{v^s}{{\varepsilon}}\right|_{\infty,m}h(|v^s|_{\infty,m}).
\end{align}
(b) When $d_{\varepsilon}^2=\partial_{x_2}^2$, we have the same estimate with $\left|\frac{\Lambda v}{\sqrt{{\varepsilon}}}\right|_{0,m,\gamma}$ replaced by $\left|\frac{{\varepsilon}\partial^2_{x_2} v}{\sqrt{{\varepsilon}}}\right|_{0,m-1,\gamma}$.
\end{prop}
\begin{proof}
(a) Taking $k=m$, we must estimate a sum of terms of the form
\begin{align} \langle bel{e26f}
\frac{1}{\sqrt{{\varepsilon}}}|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}^2 v)]|_{0,0,\gamma}=\frac{1}{{\varepsilon}^{3/2}}|[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon}\partial v)]|_{0,0,\gamma}:= A,
\end{align}
where $m_1+m_2=m$, $m_1\geq 1$. Treating $x_2$ as a parameter, we estimate $A$ by applying Proposition {\rm Re }\, f{f12} with $r=0$, $\sigma=0$, $s=m-(m_2+1)$ and $t=m-m_1$, and taking the $L^2(x_2)$ norm of the inequality thus obtained. Again, the proof of (b) is similar.
\end{proof}
\subsection{Boundary commutators}\langle bel{bc}
The boundary commutators vanish for the Dirichlet problem \eqref{c2}, but not for the Neumann-type problem \eqref{c1}. To estimate terms in the first lines of \eqref{c0a}, we must estimate norms involving boundary commutators of the form
\begin{align}\langle bel{e27}
\langle ngle\Lambda [a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}\text{ and }\langle ngle \Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon},\partial^{k+1}]v\rangle ngle_{0,\gamma},
\end{align}
where $k\leq m$, $a(v^s)=d_{v_1}H(v^s_1,h(v^s))$, and $v=v_1$.
\begin{prop}\langle bel{e28}
Suppose $m>\frac{d+1}{2}+1$. We have for $0\leq k\leq m$
\begin{align}
\langle ngle\Lambda[a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}\lesssim \langle ngle\Lambda v\rangle ngle_{m,\gamma}\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m})+\left\langle ngle \frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right \rangle ngle_{m,\gamma}\langle ngle\Lambda_1v^s \rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m}).
\end{align}
\end{prop}
\begin{proof}
Writing $a(v^s)=a(0)+b(v^s)v^s$ and taking $k=m$, we must estimate a sum of terms of the form
\begin{align} \langle bel{e28a}
\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]\rangle ngle_{0,\gamma}=\frac{1}{{\varepsilon}}\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}\partial v)]\rangle ngle_{0,\gamma},
\end{align}
where $m_1+m_2=m$, $m_1\geq 1$. To finish apply Proposition {\rm Re }\, f{f12} with $r=1$, $\sigma=0$, $s=m-(m_2+1)$, and $t=m-m_1$.
\end{proof}
\begin{prop}\langle bel{e29}
Suppose $m>\frac{d+1}{2}+1$. We have for $0\leq k\leq m+1$
\begin{align}
\begin{split}
&\langle ngle\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}\lesssim \left(\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1} +\left\langle ngle\frac{\Lambda^{\frac{1}{2}}v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1})+\\
&\qquad \left(\langle ngle\Lambda^{\frac{1}{2}} v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m}).
\end{split}
\end{align}
\end{prop}
\begin{proof}
Taking $k=m+1$, we must estimate a sum of terms of the form
\begin{align} \langle bel{e29a}
\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]\rangle ngle_{0,\gamma}=\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}\partial v)]\rangle ngle_{0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we apply Proposition {\rm Re }\, f{f12} with $t=0$, $s=m-1$ to obtain
\begin{align}\langle bel{e30}
A\lesssim \left(\left\langle ngle\frac{\Lambda^{\frac{1}{2}}v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1}).
\end{align}
In the case $m_1\leq m$ we apply Proposition {\rm Re }\, f{f12} with $r=\frac{1}{2}$, $\sigma=0$, $s=m+1-(m_2+1)$, and $t=m-m_1$ to obtain
\begin{align}
A\lesssim \left(\langle ngle\Lambda^{\frac{1}{2}} v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m}).
\end{align}
\end{proof}
The next estimate is needed to control terms in the second line of \eqref{c0a}.
\begin{prop}\langle bel{e31}
Suppose $m>\frac{d+1}{2}+1$. We have for $0\leq k\leq m+1$
\begin{align}
\begin{split}
&\sqrt{{\varepsilon}}\langle ngle\Lambda[a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}\lesssim \left(\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle \sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1} +\langle ngle\Lambda v\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1})+\\
&\qquad \left(\langle ngle \sqrt{{\varepsilon}}\Lambda v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\langle ngle\Lambda_1 v^s\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m})
\end{split}
\end{align}
\end{prop}
\begin{proof}
Taking $k=m+1$, we must estimate a sum of terms of the form
\begin{align} \langle bel{e31a}
\sqrt{{\varepsilon}}\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]\rangle ngle_{0,\gamma}=\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}\partial v)]\rangle ngle_{0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we apply Proposition {\rm Re }\, f{f12} with $t=0$, $s=m-1$ to obtain
\begin{align}
A\lesssim \left(\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle \sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1} +\langle ngle\Lambda v\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1}).
\end{align}
In the case $m_1\leq m$ we apply Proposition {\rm Re }\, f{f12} with $s=m+1-(m_2+1)$, and $t=m-m_1$ to obtain
\begin{align}
A\lesssim \left(\langle ngle \sqrt{{\varepsilon}}\Lambda v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\langle ngle\Lambda_1 v^s\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m}).
\end{align}
\end{proof}
To control the terms \eqref{e26bb} we must estimate for $u$ in \eqref{c3} the boundary commutators
\begin{align}
\frac{1}{{\varepsilon}}\langle ngle\Lambda [a(v^s)d_{\varepsilon},\partial^m] u\rangle ngle_{0,\gamma},\;\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} [a(v^s)d_{\varepsilon},\partial^{m+1}] u\rangle ngle_{0,\gamma},\text{ and }\; \frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda [a(v^s)d_{\varepsilon},\partial^{m+1}] u\rangle ngle_{0,\gamma}.
\end{align}
\begin{prop}\langle bel{e32}
Suppose $m>\frac{d+1}{2}$. We have for $0\leq k\leq m$
\begin{align}
\begin{split}
&\;\frac{1}{{\varepsilon}}\langle ngle\Lambda[a(v^s)d_{\varepsilon},\partial^k]u\rangle ngle_{0,\gamma}\lesssim \langle ngle\Lambda v\rangle ngle_{m,\gamma}\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m})+\left\langle ngle \frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right \rangle ngle_{m,\gamma}\langle ngle\Lambda_1v^s \rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m})\\
\end{split}
\end{align}
\end{prop}
\begin{proof}
Parallel to \eqref{e28a}, to prove (a) we must estimate a sum of terms of the form
\begin{align}
\frac{1}{{\varepsilon}}\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} u)]\rangle ngle_{0,\gamma}=\frac{1}{{\varepsilon}}\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}v)]\rangle ngle_{0,\gamma},
\end{align}
where $m_1+m_2=m$, $m_1\geq 1$. To finish apply Proposition {\rm Re }\, f{f12} with $r=1$, $\sigma=0$, $s=m-m_2$, and $t=m-m_1$.
\end{proof}
\begin{prop}\langle bel{e33}
Suppose $m>\frac{d+1}{2}$. We have for $0\leq k\leq m+1$
\begin{align}
\begin{split}
&\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon},\partial^k]u\rangle ngle_{0,\gamma}\lesssim \left(\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1} +\left\langle ngle\frac{\Lambda^{\frac{1}{2}}v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1})+\\
&\qquad \left(\langle ngle\Lambda^{\frac{1}{2}} v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m}).
\end{split}
\end{align}
\end{prop}
\begin{proof}
Parallel to \eqref{e29a} we must estimate a sum of terms of the form
\begin{align}
\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} u)]\rangle ngle_{0,\gamma}=\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2} v)]\rangle ngle_{0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. In the case $m_1=m+1$ we apply Proposition {\rm Re }\, f{f12} to obtain
\begin{align}
A\lesssim \left(\left\langle ngle\frac{\Lambda^{\frac{1}{2}}v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1}).
\end{align}
In the case $m_1\leq m$ we apply Proposition {\rm Re }\, f{f12} with $r=\frac{1}{2}$, $\sigma=0$, $s=m+1-m_2$, and $t=m-m_1$ to obtain
\begin{align}
A\lesssim \left(\langle ngle\Lambda^{\frac{1}{2}} v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{\Lambda^{\frac{1}{2}}_1v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m}).
\end{align}
\end{proof}
\begin{prop}\langle bel{e36}
Suppose $m>\frac{d+1}{2}$. We have for $0\leq k\leq m+1$
\begin{align}
\begin{split}
&\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda[a(v^s)d_{\varepsilon},\partial^k]u\rangle ngle_{0,\gamma}\lesssim \left(\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right\rangle ngle_{m,\gamma}\;
\langle ngle \sqrt{{\varepsilon}}\Lambda^{\frac{1}{2}}_1 v^s\rangle ngle_{m+1} +\langle ngle\Lambda v\rangle ngle_{m,\gamma}\;\left\langle ngle\frac{v^s}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1}\right) h(\langle ngle v^s\rangle ngle_{m+1})+\\
&\qquad \left(\langle ngle \sqrt{{\varepsilon}}\Lambda v\rangle ngle_{m+1,\gamma}\;\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_m
+\left\langle ngle\frac{{\nabla}bla_{\varepsilon} u}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m+1,\gamma}\;\langle ngle\Lambda_1 v^s\rangle ngle_{m}\right) h(\langle ngle v^s\rangle ngle_{m})
\end{split}
\end{align}
\end{prop}
\begin{proof}
Taking $k=m+1$, we must estimate a sum of terms of the form
\begin{align}
\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} u)]\rangle ngle_{0,\gamma}=\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda[(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2} v)]\rangle ngle_{0,\gamma}:=A,
\end{align}
where $m_1+m_2=m+1$, $m_1\geq 1$. Comparison with \eqref{e31a} shows that we can repeat the proof of Proposition {\rm Re }\, f{e31}, now applying Proposition {\rm Re }\, f{f12} with $t=0$, $s=m$ in the case $m_1=m+1$, and with $t=m-m_1$, $s=m+1-m_2$ in the case $m_1\leq m$.
\end{proof}
To control the terms \eqref{e14dd} with $j=1$ we must estimate $\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}$, $k\leq m$ and $v=v_1$.
\begin{prop}\langle bel{e34}
Suppose $m>\frac{d+1}{2}+1$. We have for $0\leq k\leq m$ and $v=v_1$
\begin{align}
\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}[a(v^s)d_{\varepsilon},\partial^k]v\rangle ngle_{0,\gamma}\lesssim \left\langle ngle\frac{\Lambda^{\frac{1}{2}} v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m,\gamma}\left\langle ngle\frac{v^s}{{\varepsilon}}\right\rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m})+\left\langle ngle \frac{{\nabla}bla_{\varepsilon} u}{{\varepsilon}}\right \rangle ngle_{m,\gamma}\left\langle ngle\frac{\Lambda^{\frac{1}{2}}_1 v}{\sqrt{{\varepsilon}}}\right\rangle ngle_{m}h(\langle ngle v^s\rangle ngle_{m}).
\end{align}
\end{prop}
\begin{proof}
Parallel to \eqref{e28a} we must estimate a sum of terms of the form
\begin{align} \langle bel{e35}
\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}d_{\varepsilon} v)]\rangle ngle_{0,\gamma}=\frac{1}{{\varepsilon}}\langle ngle\Lambda^{\frac{1}{2}} [(\partial^{m_1}(b(v^s)v^s))\cdot (\partial^{m_2}\partial v)]\rangle ngle_{0,\gamma},
\end{align}
where $m_1+m_2=m$, $m_1\geq 1$. To finish apply Proposition {\rm Re }\, f{f12} with $r=\frac{1}{2}$, $\sigma=0$, $s=m-(m_2+1)$, and $t=m-m_1$.
\end{proof}
\subsection{Boundary forcing II}\langle bel{bf2}
\emph{\quad}Here we estimate the forcing given by $\cG_2$ in the problem \eqref{c2}. For this we require
\begin{prop}\langle bel{g1}
Suppose $m>3d+4+\frac{d+1}{2}$. There exist positive constants ${\varepsilon}_0$, $\gamma_0$
such that for $j\in J_h\cup J_e$, ${\varepsilon}\in (0,{\varepsilon}_0]$, and each $T$ with $0<T\leq T^*_{\varepsilon}$,
\begin{align}\langle bel{g2}
\begin{split}
&E_{m,\gamma}(v_1)+\langle ngle\phi_j\Lambda^{\frac{3}{2}}v_1\rangle ngle_{m,\gamma}+\langle ngle\phi_j\Lambda v_1\rangle ngle_{m+1,\gamma}+\sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}}v_1\rangle ngle_{m+1,\gamma}+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda v_1\rangle ngle_{m,\gamma}\lesssim\\
&\frac{1}{\sqrt{\gamma}}\{Q_1(E_{m,T}(v^s))\cdot E_{m,\gamma}(v_1)+Q^o_2(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q_3(E_{m,T}(v^s))\}\text{ for }\gamma\geq \gamma_0.
\end{split}
\end{align}
\end{prop}
\begin{proof}
For all terms in the definition of $E_{m,\gamma}(v_1)$,
domination by the right side of \eqref{g2} follows directly from the linearized estimates of section {\rm Re }\, f{b1a} and the propositions of sections {\rm Re }\, f{if}, {\rm Re }\, f{bf}, {\rm Re }\, f{ic}, and {\rm Re }\, f{bc}.
The ``right side" of every commutator proposition needed to estimate these terms is dominated by $Q_1(E_{m,T}(v^s))\cdot E_{m,\gamma}(v_1)$,
and the right side of every forcing proposition is dominated by $Q^o_2(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q_3(E_{m,T}(v^s))$. The estimates of section {\rm Re }\, f{b1a} show that $1/\sqrt{\gamma}$ is the correct factor in \eqref{g2}.
Consider for example the first term in the first line of \eqref{c0a} when $j=1$. By \eqref{b3} we have
\begin{align}\langle bel{g2a}
\left|\begin{pmatrix}\Lambda^{\frac{3}{2}} v_1\\partial_{x_2}\Lambda^{\frac{1}{2}} v_1\end{pmatrix}\right|_{0,m,\gamma}\lesssim \gamma^{-1}\left(|\Lambda^{\frac{1}{2}}\cF_1|_{0,m,\gamma}+\langle ngle\Lambda \cG_1\rangle ngle_{m,\gamma}+|\Lambda^{\frac{1}{2}}\cF_{1,com}|_{0,m,\gamma}+\langle ngle\Lambda \cG_{1,com}\rangle ngle_{m,\gamma}+\cE_{m,\gamma},\right),
\end{align}
where $\cF_{1,com}$ and $\cG_{1,com}$ denote the interior and boundary commutators estimated in Propositions {\rm Re }\, f{e24} and {\rm Re }\, f{e28}, and $\cE_{m,\gamma}$ denotes the error term
\begin{align}
\cE_{m,\gamma}:=\left|\begin{pmatrix}\Lambda v_1/\sqrt{{\varepsilon}}\\partial_{x_2} v_1/\sqrt{{\varepsilon}}\end{pmatrix}\right|_{0,m,\gamma}.
\end{align}
These propositions, along with the interior and boundary forcing estimates in Propositions {\rm Re }\, f{e12} and {\rm Re }\, f{e15}, show that the right side of \eqref{g2a} is dominated by the right side of \eqref{g2}. By the estimate \eqref{b3}, a term involving $\phi_j$ like $\langle ngle\phi_j\Lambda^{\frac{3}{2}}v_1\rangle ngle_{m,\gamma}$ is controlled as soon as the left side of \eqref{g2a} is. However, for this $\phi_j$ term we require the factor $\frac{1}{\sqrt{\gamma}}$ in place of $\gamma^{-1}$ on the right side of \eqref{g2a}.
\end{proof}
The next Corollary is immediate.
\begin{cor}\langle bel{g3}
Let $m>3d+4+\frac{d+1}{2}$.
There exist positive constants ${\varepsilon}_1={\varepsilon}_1(M_0)$, $\gamma_m=\gamma_m(M_0)$
such that for ${\varepsilon}\in (0,{\varepsilon}_1]$ and each $T$ with $0<T\leq T^*_{\varepsilon}$,
\begin{align}\langle bel{g4}
E_{m,\gamma}(v_1)\leq \frac{1}{\sqrt{\gamma}}[Q_2^o(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q_3(E_{m,T}(v^s))]\text{ for }\gamma\geq \gamma_m.
\end{align}
The function $\gamma(M_0)$ increases with $M_0$.
\end{cor}
\begin{rem}\langle bel{e17}
1) When we apply Proposition {\rm Re }\, f{f7} to estimate $\langle ngle\phi_j\Lambda^{\frac{3}{2}}\mathcal{G}_2\rangle ngle_{m,\gamma}$ for $j\in J_h$, we must
control norms like $\langle ngle \Lambda_1 (\chi^*(t)v_1)\rangle ngle_m$ (also $\langle ngle\chi^* v_1\rangle ngle_m$), where $v_1$ and not $v_1^s$ occurs, and $\mathrm{supp}\;\chi^*\subset (-1,2)$. Using Proposition {\rm Re }\, f{c0e} we have
\begin{align}
E_m(\chi^*(t)v_1)\lesssim E_{m,T=2}(v_1)\lesssim e^{\gamma_m(M_0)2}E_{m,\gamma_m(M_0)}(v_1)
\end{align}
Applying Corollary {\rm Re }\, f{g3} with $\gamma=\gamma_m(M_0)$ we obtain
\begin{align}\langle bel{e17a}
\begin{split}
&E_m(\chi^*(t)v_1)\lesssim e^{\gamma_m(M_0)2}[Q_2^o(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q_3(E_m(v^s))]\lesssim \\
&\qquad\qquad e^{\gamma_m(M_0)2}[Q_2^o(M_0)+(M_G+\sqrt{{\varepsilon}})Q_3(M_0)].
\end{split}
\end{align}
2) The right side of \eqref{e17a} is small when $M_G$, $M_0$, and ${\varepsilon}_1$ are small. The constant $M_G$ is small when $T_0$ as in \eqref{c0h} is small, since $G(x',\theta)=0$ in $t<0$.
\end{rem}
To control terms in the first line of \eqref{c0a} when $j=2$, we must estimate $\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}$ and $\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m+1,\gamma}$; recall $\cG_2=\chi_0(t)H(v_1,g_{\varepsilon})$.
\begin{prop}\langle bel{e18}
Let $m>\frac{d+1}{2}$. For small enough positive constants ${\varepsilon}_1$, $M_G$ \eqref{MG}, and $M_0$ \eqref{c0h}
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}\langle bel{e19}
\begin{split}
&(a)\;\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}\lesssim (\langle ngle \Lambda v_1\rangle ngle_{m,\gamma}+{\varepsilon})[Q_1(E_m(v^s))+Q_2(M_G)]\\
&(b)\;\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m+1,\gamma}\lesssim (\langle ngle \Lambda^{\frac{1}{2}} v_1\rangle ngle_{m+1,\gamma}+{\varepsilon}^{\frac{3}{2}})[Q_1(E_m(v^s))+Q_2(M_G)].
\end{split}
\end{align}
\end{prop}
\begin{proof}
Choose $\chi^*(t)$ supported in $(-1,2)$ such that $\chi_0\chi^*=\chi_0$ and write $\chi^*(t)v_1=v_1^*$.
By Proposition {\rm Re }\, f{f3} we have
\begin{align}\langle bel{e20}
\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}\leq \langle ngle \Lambda (v_1^*,g_{\varepsilon})\rangle ngle_{m,\gamma} h(\langle ngle v^*_1,g_{\varepsilon}\rangle ngle_m)+\langle ngle v^*_1,g_{\varepsilon}\rangle ngle_{m,\gamma}\langle ngle\Lambda_1(v^*_1,g_{\varepsilon})\rangle ngle_mh(\langle ngle v^*_1,g_{\varepsilon}\rangle ngle_m).
\end{align}
Then \eqref{e19}(a) follows since
\begin{align}
\langle ngle \Lambda (v_1^*,g_{\varepsilon})\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda v_1\rangle ngle_{m,\gamma}+{\varepsilon} M_G,
\end{align}
and the factors involving $\langle ngle\cdot\rangle ngle_m$ norms on the right in \eqref{e20} are controlled using \eqref{e17a}. In particular, ${\varepsilon}_1$, $M_0$, and $M_G$ must be taken small enough so that $\langle ngle v^*_1,g_{\varepsilon}\rangle ngle_m$ lies in the region of convergence of $h$. The commutator $\langle ngle [\Lambda,\chi^*]v_1\rangle ngle_{m,\gamma}$ is estimated using Proposition {\rm Re }\, f{commutator1}.
The inequality \eqref{e19}(b) is proved similarly.
\end{proof}
\begin{prop}\langle bel{e21}
Let $m>\frac{d+1}{2}$ and let $\phi_j$ and $\psi_j$, $j\in J_h$, be singular symbols related as in Notation {\rm Re }\, f{f4b}. For small enough positive constants ${\varepsilon}_1$, $M_G$, and $M_0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}\langle bel{e22}
\begin{split}
&(a)\;\langle ngle\phi_j\Lambda^{\frac{3}{2}} \cG_2\rangle ngle_{m,\gamma}\lesssim \left(\langle ngle \psi_j\Lambda^{\frac{3}{2}} v_1\rangle ngle_{m,\gamma}+\langle ngle \Lambda v_1\rangle ngle_{m,\gamma}+\sqrt{{\varepsilon}}\right)[Q_1(E_m(v^s))+Q_2(M_G)]\\
&(b)\;\langle ngle\phi_j\Lambda\cG_2\rangle ngle_{m+1,\gamma}\lesssim \left(\langle ngle \psi_j\Lambda v_1\rangle ngle_{m+1,\gamma}+\langle ngle \Lambda^{\frac{1}{2}} v_1\rangle ngle_{m+1,\gamma}+{\varepsilon}\right)[Q_1(E_m(v^s))+Q_2(M_G)]\\
\end{split}
\end{align}
\end{prop}
\begin{proof}
The argument is parallel to the proof of Proposition {\rm Re }\, f{e18}, except that Proposition {\rm Re }\, f{f7} is used in place of Proposition {\rm Re }\, f{f3}. Commutators like $\langle ngle[\psi_j\Lambda^{\frac{3}{2}},\chi^*(t)]v_1\rangle ngle_{m,\gamma}$ are estimated using Proposition {\rm Re }\, f{commutator1}.
\end{proof}
To control the terms in the second line of \eqref{c0a} when $j=2$, we must estimate
\begin{align}
\sqrt{{\varepsilon}}\langle ngle\Lambda\cG_2\rangle ngle_{m+1,\gamma},\; \sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}}\cG_2\rangle ngle_{m+1,\gamma},\; \frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m,\gamma}, \text{ and }\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda\cG_2\rangle ngle_{m,\gamma}, \text{ where }j\in J_h.
\end{align}
The proofs of the next two propositions are essentially repetitions of the proofs of Propositions {\rm Re }\, f{e18} and {\rm Re }\, f{e21}.
\begin{prop}\langle bel{e22b}
Let $m>\frac{d+1}{2}$. For small enough positive constants ${\varepsilon}_1$, $M_G$, and $M_0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}\langle bel{e22c}
\begin{split}
&(a)\;\sqrt{{\varepsilon}}\langle ngle\Lambda\cG_2\rangle ngle_{m+1,\gamma}\lesssim (\sqrt{{\varepsilon}}\langle ngle \Lambda v_1\rangle ngle_{m+1,\gamma}+{\varepsilon}^{\frac{3}{2}})[Q_1(E_m(v^s))+Q_2(M_G)]\\
&(b)\;\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m,\gamma}\lesssim (\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}} v_1\rangle ngle_{m,\gamma}+{\varepsilon})[Q_1(E_m(v^s))+Q_2(M_G)].
\end{split}
\end{align}
\end{prop}
\begin{prop}\langle bel{e23d}
Let $m>\frac{d+1}{2}$ and let $\phi_j$ and $\psi_j$, $j\in J_h$, be singular symbols related as in Notation {\rm Re }\, f{f4b}. For small enough positive constants ${\varepsilon}_1$, $M_G$, and $M_0$
we have for ${\varepsilon}\in (0,{\varepsilon}_1]$
\begin{align}\langle bel{e23e}
\begin{split}
&(a)\;\sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}} \cG_2\rangle ngle_{m+1,\gamma}\lesssim \left(\sqrt{{\varepsilon}}\langle ngle \psi_j\Lambda^{\frac{3}{2}} v_1\rangle ngle_{m+1,\gamma}+\sqrt{{\varepsilon}}\langle ngle \Lambda v_1\rangle ngle_{m+1,\gamma}+{\varepsilon}\right)[Q_1(E_m(v^s))+Q_2(M_G)]\\
&(b)\;\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda\cG_2\rangle ngle_{m,\gamma}\lesssim \left(\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \psi_j\Lambda v_1\rangle ngle_{m,\gamma}+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}} v_1\rangle ngle_{m,\gamma}+\sqrt{{\varepsilon}}\right)[Q_1(E_m(v^s))+Q_2(M_G)]\\
\end{split}
\end{align}
\end{prop}
The next proposition is the last step in the proof of Proposition {\rm Re }\, f{c5}.
\begin{prop}\langle bel{h1}
Suppose $m>3d+4+\frac{d+1}{2}$. There exist positive constants ${\varepsilon}_0$, $\gamma_0$
such that for ${\varepsilon}\in (0,{\varepsilon}_0]$ and each $T$ with $0<T\leq T^*_{\varepsilon}$,
\begin{align}\langle bel{h2}
\begin{split}
&E_{m,\gamma}(v_2)\lesssim \gamma^{-1}E_{m,\gamma}(v)Q(E_{m,T}(v^s))+(\gamma^{-\frac{1}{2}}+\sqrt{{\varepsilon}})Q(E_{m,T}(v^s))\text{ for }\gamma\geq \gamma_0.
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }For all terms in the definition of $E_{m,\gamma}(v_2)$,
domination by the right side of \eqref{h2} follows directly from the linearized estimates of section {\rm Re }\, f{b1a} and the propositions of sections {\rm Re }\, f{if}, {\rm Re }\, f{bf}, {\rm Re }\, f{ic}, and {\rm Re }\, f{bf2}.
The ``right side" of every interior or boundary commutator proposition needed to estimate these terms is dominated by $Q(E_{m,T}(v^s))\cdot E_{m,\gamma}(v)$,\footnote{Although the boundary commutators for the $v_2$ problem \eqref{c2} are zero, the boundary commutators for \eqref{c3} are not; such commutators are needed to control terms in the third row of \eqref{c0a} when $j=1$ or $j=2$.}
the right side of every interior forcing proposition is dominated by $ Q^o(E_{m,T}(v^s))$, and the boundary forcing terms involving $\cG$ \eqref{e22bb} are dominated by
\begin{align}
Q^o(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q(E_{m,T}(v^s))\leq Q(E_{m,T}(v^s))\text{ since }M_G\leq 1.
\end{align}
The boundary forcing terms involving $\cG_2$ have been estimated in this section. Those that do not involve $\phi_{j,D}$, $j\in J_h$ are dominated by
\begin{align}
(E_{m,\gamma}(v_1)+\sqrt{{\varepsilon}})[Q(E_{m,T}(v^s))+Q(M_G)]\leq (E_{m,\gamma}(v_1)+\sqrt{{\varepsilon}})Q(E_{m,T}(v^s)).
\end{align}
Using Proposition {\rm Re }\, f{g1}, we see that the boundary forcing terms involving $\phi_{j,D}$ on the left in Propositions {\rm Re }\, f{e21} and {\rm Re }\, f{e23d} are dominated by
\begin{align}
\begin{split}
&\frac{1}{\sqrt{\gamma}}\{Q_1(E_{m,T}(v^s))\cdot E_{m,\gamma}(v_1)+Q^o_2(E_{m,T}(v^s))+(M_G+\sqrt{{\varepsilon}})Q_3(E_{m,T}(v^s))\}[Q(E_{m,T}(v^s))+Q(M_G)]\leq\\
&\qquad \qquad \frac{1}{\sqrt{\gamma}}[E_{m,\gamma}(v_1)Q(E_{m,T}(v^s))+Q(E_{m,T}(v^s))].
\end{split}
\end{align}
The coefficients on the right in \eqref{h2} are explained below.
\textbf{2. }Consider for example the second term in the first line of \eqref{c0a} when $j=2$. By \eqref{b7} we have
\begin{align}\langle bel{h3}
\begin{split}
&\left|\begin{pmatrix}\Lambda v_2\\partial_{x_2}v_2\end{pmatrix}\right|_{\infty,m,\gamma}\lesssim \gamma^{-1}|\cF_2|_{0,m+1,\gamma}+\gamma^{-1}|\cF_{2,com}|_{0,m+1,\gamma}+\\
&\qquad \qquad (\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}+\langle ngle\Lambda^{\frac{1}{2}} \cG_{2}\rangle ngle_{m+1,\gamma})+\gamma^{-\frac{1}{2}}\sum_{j\in J_h}\langle ngle\phi_j\Lambda \cG_2\rangle ngle_{m+1,\gamma},
\end{split}
\end{align}
The discussion in step \textbf{1} shows that the right side of \eqref{h3} is dominated by
\begin{align}\langle bel{h4}
\begin{split}
&\gamma^{-1}Q(E_{m,T}(v^s))+\gamma^{-1}E_{m,\gamma}(v)Q(E_{m,T}(v^s))+(E_{m,\gamma}(v_1)+\sqrt{{\varepsilon}})Q(E_{m,T}(v^s))+\\
&\qquad\qquad \gamma^{-1}[E_{m,\gamma}(v_1)Q(E_{m,T}(v^s))+Q(E_{m,T}(v^s))].
\end{split}
\end{align}
By Corollary {\rm Re }\, f{g3} $E_{m,\gamma}(v_1)\leq \gamma^{-\frac{1}{2}}Q(E_{m,T}(v^s))$, so we see that the right side of \eqref{h4} is dominated by the right side of \eqref{h2}.
\end{proof}
\begin{proof}[Proof of Proposition {\rm Re }\, f{c5}]
The estimate of Proposition {\rm Re }\, f{c5} follows immediately from Corollary {\rm Re }\, f{g3} and Proposition {\rm Re }\, f{h1}.
\end{proof}
\section{Local existence and continuation for the singular problems with ${\varepsilon}$ fixed.}\langle bel{local}
\emph{\quad}For fixed ${\varepsilon}\in (0,1]$ we prove here existence and continuation theorems for solutions $v^{\varepsilon}_1$, $v^{\varepsilon}_2$, $u^{\varepsilon}$ to the triple of coupled systems \eqref{a7}, \eqref{a8}, \eqref{a9}.
We also establish the relation $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_{\varepsilon}}$. These results are used in the proof of Proposition {\rm Re }\, f{mainprop}, which establishes a uniform time of existence with respect to ${\varepsilon}$ for solutions of the singular nonlinear problems.
Recall that we describe these systems as singular both because of the factors of $\frac{1}{{\varepsilon}}$ that appear and because of the fact that $\partial_{x'}$ and $\partial_\theta$ derivatives occur in the combination $\partial_{x'}+\beta\frac{\partial_\theta}{{\varepsilon}}$. Even when we fix ${\varepsilon}$, the second singular feature is still present. Thus, we still need to use the singular calculus to prove estimates.
\begin{defn}
For $m\geq 0$ and $T\in \mathbb{R}$ set $\cN^m(\Omega_T)=\{w:|w|^*_{m,T}<\infty\}$, where
\begin{align}\langle bel{defnN}
|w|^*_{m,T}=\left|\begin{pmatrix}\Lambda_1^{\frac{3}{2}}w\\partial_{x_2}\Lambda_1^{\frac{1}{2}}w\end{pmatrix}\right|_{m,T}+\left < \begin{pmatrix}\Lambda_1 w\\partial_{x_2} w\end{pmatrix} \right>_{m,T}.
\end{align}
The norms $|w|^*_{m,\gamma}$, $|w|^*_{m,\gamma,T}$, and $|w|^*_m$ are defined by substituting $(m,\gamma)$, etc. for $(m,T)$ in \eqref{defnN}; recall Definition {\rm Re }\, f{normal}.
\end{defn}
\begin{rem}\langle bel{N}
1. Let $m>\frac{d+1}{2}+1$. Observe that if $(v^{\varepsilon},u^{\varepsilon})$ is a solution of \eqref{a7}, \eqref{a8}, \eqref{a9} such that $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_T$ and $E_{m,T}(v^{\varepsilon})<\infty$, then $v^{\varepsilon}\in\cN^{m+1}(\Omega_T)$ and we have
\begin{align}
|v^{\varepsilon}|^*_{m+1,T}\leq C_{\varepsilon} E_{m,T}(v^{\varepsilon}).
\end{align}
Indeed, the control of tangential derivatives is immediate, and since the boundary is noncharacteristic, the equation can be used in the standard way for ${\varepsilon}$ fixed to control $\partial_{x_2}$ derivatives. (Note: We need estimates as in section {\rm Re }\, f{nonlinear} involving higher $\partial_{x_2}$ derivatives for this.)
On the other hand for any $(v^{\varepsilon},u^{\varepsilon})\in\cN^{m+1}(\Omega_T)$ with $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ we have
\begin{align}
E_{m,T}(v^{\varepsilon})\leq C_{\varepsilon} |v^{\varepsilon}|^*_{m+1,T}.
\end{align}
2. Let $\delta>0$ and ${\varepsilon}>0$ be fixed. Suppose $(v^{\varepsilon},u^{\varepsilon})\in\cN^{m+1}(\Omega_T)$ for some $T>0$ with ${\nabla}bla_{\varepsilon} u^{\varepsilon}=v^{\varepsilon}$ and vanishes in $t<0$. For sufficiently small $T_{\varepsilon}$ with $0<T_{\varepsilon}< T$ we have
$E_{m,T_{\varepsilon}}(v)<\delta$. This is easily seen by first approximating $v$ in the $\cN^{m+1}(\Omega_T)$ norm by a $C^\infty$ function with compact support in $\overline{\Omega_T}$.
\end{rem}
\subsection{Local existence.}
\begin{prop}[Local existence for fixed ${\varepsilon}$.]\langle bel{localex}
Let $m>3d+4+\frac{d+1}{2}$ and
consider the problems \eqref{a7}, \eqref{a8}, \eqref{a9} for ${\varepsilon}$ fixed, where $G(x',\theta)\in H^{m+3}(b\Omega)$ (see section 4) and vanishes in $t<0$. There exist $T>0$ and unique solutions $v=(v_1,v_2)\in\cN^{m+1}(\Omega_T)$ to \eqref{a7}, \eqref{a8} and $u\in\cN^{m+1}(\Omega_T)$ to \eqref{a9} vanishing in $t<0$. Moreover, we have $v={\nabla}bla_{\varepsilon} u$ on $\Omega_T$.
\end{prop}
\begin{proof}
\textbf{1. A priori estimates and existence for the linearized problems. }Consider the linearized problems for ${\varepsilon}$ fixed
\begin{align}\langle bel{k-1}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_1+\sum_{|\alpha|=2} A_\alpha(w)\partial_{x,{\varepsilon}}^\alpha v_1=\mathcal{F}_1\text{ on }\Omega\\
&\partial _{x_2} v_1-d_{v_1}H(w_1,h(w))\partial_{x_1,{\varepsilon}}v_1=\mathcal{G}_1\text{ on }x_2=0
\end{split}
\end{align}
and
\begin{align}\langle bel{k0}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_2+\sum_{|\alpha|=2} A_\alpha(w)\partial_{x,{\varepsilon}}^\alpha v_2=\mathcal{F}_2\text{ on }\Omega\\
&v_2=\mathcal{G}_2\text{ on }x_2=0.
\end{split}
\end{align}
Here the functions $v_i$, $w$, $\mathcal{F}_i$, $\mathcal{G}_i$ all vanish in $t<0$.
Let $k\in\{0,1,\dots,m+1\}$. For the problem {\rm Re }\, f{k-1} we have: for $|w|^*_{m+1}$ sufficiently small, there exist constants $\gamma_{m+1}(k)$, $C=C_{m+1}(k)$ (depending on $|w|^*_{m+1}$, $k$, and ${\varepsilon}$) such that for $\gamma\geq \gamma_{m+1}(k)$
\begin{align}\langle bel{k1}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_1\\partial_{x_2}\Lambda^{\frac{1}{2}}v_1\end{pmatrix}\right|^2_{k,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{k,\gamma}+\sum_{J_h\cup J_e }\left < \phi_j \begin{pmatrix}\Lambda^{\frac{3}{2}} v_1\\partial_{x_2}\Lambda^{\frac{1}{2}} v_1\end{pmatrix} \right>^2_{k,\gamma}\leq \\
&\qquad\quad \frac{1}{\gamma}C\left(|\Lambda^{\frac{1}{2}}\mathcal{F}_1|^2_{k,\gamma}+\langle ngle\Lambda \mathcal{G}_1\rangle ngle^2_{k,\gamma}\right).
\end{split}
\end{align}
For the problem {\rm Re }\, f{k0} we have: for $|w|^*_{m+1}$ sufficiently small, there exist constants $\gamma_{m+1}(k)$, $C=C_{m+1}(k)$ (depending on $|w|^*_{m+1}$, $k$, and ${\varepsilon}$) such that for $\gamma\geq \gamma_{m+1}(k)$
\begin{align}\langle bel{k2}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_2\\partial_{x_2}\Lambda^{\frac{1}{2}}v_2\end{pmatrix}\right|^2_{k,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix} \right>^2_{k,\gamma}\leq \\
&\qquad\quad C\left(\frac{1}{\gamma}|\Lambda^{\frac{1}{2}}\mathcal{F}_2|^2_{k,\gamma}+ \gamma\langle ngle\Lambda \mathcal{G}_2\rangle ngle^2_{k,\gamma}+\sum_{j\in J_h}\langle ngle\phi_j\Lambda^{\frac{3}{2}} \mathcal{G}_2\rangle ngle^2_{k,\gamma}\right).
\end{split}
\end{align}
The estimates \eqref{k1} and \eqref{k2} are derived under the assumption of having sufficiently regular $v_i$. When $k=0$ they follow directly from \eqref{b3} and \eqref{b6}. Estimates of $|\cdot|_{0,k,\gamma}$ and $\langle ngle\cdot\rangle ngle_{k,\gamma}$ norms follow by applying the $k=0$ estimate to the tangentially differentiated problems $\partial^k\eqref{k-1}$, $\partial^k\eqref{k0}$. As in section {\rm Re }\, f{mainestimate} the singular norm estimates of section {\rm Re }\, f{nontame} are needed here; in particular, Proposition {\rm Re }\, f{f12}(b) is again used to estimate commutators.\footnote{As in section {\rm Re }\, f{mainestimate}, interior norms are estimated by first doing a tangential estimate in $(x',\theta)$ for fixed $x_2$, and then taking an $L^2(x_2)$ norm.} Estimates of $\left|\begin{pmatrix}D_{x_2}^j\left(\Lambda^{\frac{3}{2}}v_i\right)\\partial_{x_2}^j\left(D_{x_2}\Lambda^{\frac{1}{2}}v_i\right)\end{pmatrix}\right|_{0,k-j,\gamma}$ for $j>0$ are proved as usual by induction on $j$ using the equation and the noncharacteristic boundary assumption. Proposition {\rm Re }\, f{normalest} is needed here.
We have the following existence theorem for the singular linearized problems with ${\varepsilon}$ fixed:
\begin{theo}\langle bel{exist}
Consider the singular linear problems \eqref{k-1}, \eqref{k0} for ${\varepsilon}>0$ fixed and assume $d=2$.
Let $m>3d+4+\frac{d+1}{2}$ and suppose $|w|^*_{m+1}<\infty$. Suppose the data $\cF_i$, $\cG_i$, $i=1,2$ vanish in $t<0$ and are such that for $k\in\{0,1,\dots,m+1\}$ the right sides of \eqref{k1}, \eqref{k2} are finite.
For $|w|^*_{m+1}$ sufficiently small, there exist constants $\gamma_{m+1}(k)$, $C=C_{m+1}(k)$ (depending on $|w|^*_{m+1}$, $k$, and ${\varepsilon}$) such that for $\gamma\geq \gamma_{m+1}(k)$
the problems \eqref{k-1}, \eqref{k0} admit unique solutions $v_i$, $i=1,2$ vanishing in $t<0$ satisfying the estimates \eqref{k1}, \eqref{k2}.
\end{theo}
The theorem follows from the a priori estimates by a classical duality argument using the properties (P6) and (P7) described in section {\rm Re }\, f{assumptions}. We refer to \cite{S-T}, p. 279 or to \cite{CP}, Chapter 7 for this kind of argument.
\begin{rem}\langle bel{k2y}
An important consequence of Theorem {\rm Re }\, f{exist} that we use in the proof of Proposition {\rm Re }\, f{c5} and later in the error analysis
is that the linearized singular problems exhibit \emph{causality} \cite{CP}; solutions in $t<T$ are unaffected by changing the forcing terms and coefficients in $t>T$.
\end{rem}
\textbf{2. Iteration schemes. }For $T_0>0$ to be chosen, as before we let $\chi_0(t)\geq 0$ be a $C^\infty$ function that is equal to 1 on a neighborhood of $[0,1]$ and compactly supported in $(-1,2)$. We use iteration schemes similar to those in \cite{S-T}, initialized by $v^0=0$. In $t<0$ we have $G=0$ and $v_k=0$, $u_k=0$ for all $k$.
\begin{align}\langle bel{k0a}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_1^{k+1}+\sum_{|\alpha|=2} A_\alpha(v^k)\partial_{x,{\varepsilon}}^\alpha v_1^{k+1}=-\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_1,{\varepsilon}}(A_\alpha(v^k))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v_1^k-\partial_{x_1,{\varepsilon}}(A_{(0,2)}(v^k))\partial_{x_2}v_2^k\right]\\
&\partial _{x_2} v_1^{k+1}-d_{v_1}H(v^k_1,h(v^k))\partial_{x_1,{\varepsilon}}v_1^{k+1}=d_gH(v^k_1,{\varepsilon}^2G)\partial_{x_1,{\varepsilon}}({\varepsilon}^2G)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{k0b}
\begin{split}
&\partial_{t,{\varepsilon}}^2 v_2^{k+1}+\sum_{|\alpha|=2} A_\alpha(v^k)\partial_{x,{\varepsilon}}^\alpha v_2^{k+1}=-\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_2}(A_\alpha(v^k))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v^k_1-\partial_{x_2}(A_{(0,2)}(v^k))\partial_{x_2}v^k_2\right]\\
&v_2^{k+1}=\chi_0(t)H(v_1^{k+1},{\varepsilon}^2 G)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{k0c}
\begin{split}
&\partial_{t,{\varepsilon}}^2 u^{k+1}+\sum_{|\alpha|=2} A_\alpha(v^k)\partial_{x,{\varepsilon}}^\alpha u^{k+1}=0\\\
&\partial _{x_2} u^{k+1}-d_{v_1}H(v^k_1,h(v^k))\partial_{x_1,{\varepsilon}}u^{k+1}=\left[H(v^k_1,{\varepsilon}^2 G(x',\theta))-d_{v_1}H(v^k_1,{\varepsilon}^2 G)v^k_1\right]\text{ on }x_2=0.
\end{split}
\end{align}
This is actually a family of problems parametrized by $T$, where $0<T\leq T_0$. Each problem is solved on $\Omega$.
The function $v^k$ appearing in the coefficients of \eqref{k0a}-\eqref{k0c} is really $v^k_T$ (with the $T$ suppressed), where $v^k_T$ is a Seeley extension of $v_k|_{\Omega_T}$ to $\Omega$ chosen as in Proposition {\rm Re }\, f{c0e}. Similarly, $G$ here denotes a Seeley extension of $G|_{b\Omega_{T_0}}$ to $H^{m+3}(b\Omega)$. We can suppose $T_0$ is small enough so that $\langle ngle G\rangle ngle_{m+3,T_0}\leq 1$.
The function $v^{k+1}_1$ appearing in the boundary condition of \eqref{k0b} is taken to be the \emph{same} as that in \eqref{k0a}, not a Seeley extension of $v^{k+1}_1|_{\Omega_T}$, for the reasons explained in Remark {\rm Re }\, f{c4a}.
\textbf{3. High norm boundedness. }We make the following induction assumption:
$\bullet$ there exists $T>0$ independent of $k$ such that for $j=0,\dots,k$ we have
$|v_j|^*_{m+1,T}<\delta$
for $\delta$ to be chosen.\footnote{This choice is made precise later in this step; for now we claim that the estimates in this step are valid for $\delta$ and $T_0$ small enough.}
Using the estimates \eqref{k1}, \eqref{k2} together with Proposition {\rm Re }\, f{c0e} (which relates $|\cdot|^*_{m,\gamma}$ and $|\cdot|^*_{m,T}$ norms) and the estimates of nonlinear functions in section {\rm Re }\, f{nontame}, we can complete the induction step by an argument modeled on \cite{S-T}, pages 286-289.\footnote{We mainly use our Corollary {\rm Re }\, f{f2}, the singular Rauch-type lemma Proposition {\rm Re }\, f{f7}, and Proposition {\rm Re }\, f{normalest}. The use of Proposition {\rm Re }\, f{f7} permits a simplification of the argument in \cite{S-T}; for example, dyadic decompositions are not needed here. }
We denote the interior and boundary forcing terms in \eqref{k0a}, \eqref{k0b} by $\cF_1$, $\cF_2$ and $\cG_1$, $\cG_2$, respectively. By
Proposition {\rm Re }\, f{normalest} and Corollary {\rm Re }\, f{f2} we have
\begin{align}\langle bel{ka0}
|\Lambda^{\frac{1}{2}}\cF_i|_{m+1,\gamma}\lesssim C(\delta)\text{ and }\langle ngle\Lambda \cG_1\rangle ngle_{m+1,\gamma}\leq C(\delta,\langle ngle\Lambda^2_1G\rangle ngle_{m+1,T_0}).
\end{align}
Thus, estimate \eqref{k1} applied to the problem \eqref{k0a}, yields for $\gamma\geq \gamma_{m+1}(m+1):={\gamma}^*_{m+1}$ and $j\in J_h\cup J_e$:
\begin{align}\langle bel{ka1}
\gamma |v^{k+1}_1|^*_{m+1,\gamma}+\langle ngle\phi_j\Lambda^{\frac{3}{2}}v^{k+1}_1\rangle ngle_{m+1,\gamma}\leq K_1(\delta, \langle ngle\Lambda^2_1G\rangle ngle_{m+1,T_0}), \text{ where }\langle ngle\Lambda^2_1G\rangle ngle_{m+1,T_0}\lesssim \langle ngle G\rangle ngle_{m+3,T_0}.
\end{align}
The estimates giving \eqref{ka0} show that $K_1$ can be taken to be a continuous function such that $K_1(0,0)=0.$
To estimate $v^{k+1}_2$ with \eqref{k2} we write
\begin{align}\langle bel{ka2}
\chi_0(t)H(v_1^{k+1},{\varepsilon}^2 G)=h(\chi^*v^{k+1}_1, {\varepsilon}^2\chi^* G)(\chi^*v^{k+1}_1, {\varepsilon}^2\chi^* G),
\end{align}
where $\chi^*(t)$ is a smooth cutoff, compactly supported in $(-1,2)$, and equal to one on the support of $\chi_0$. Using \eqref{ka1} and
\begin{align}
\langle ngle\Lambda_1v^{k+1}_1\rangle ngle_{m+1,T=2} \lesssim e^{\gamma^*_{m+1}2}\langle ngle \Lambda v^{k+1}_1\rangle ngle_{m+1,\gamma^*_{m+1}},
\end{align}
we obtain
\begin{align}\langle bel{kaa2}
\langle ngle\Lambda_1(\chi^* v^{k+1}_1)\rangle ngle_{m+1}\lesssim e^{\gamma^*_{m+1}2}K_1\text{ and } \langle ngle \Lambda(\chi^* v^{k+1}_1)\rangle ngle_{m+1,\gamma}\lesssim \frac{1}{\gamma}K_1.
\end{align}
Applying Corollary {\rm Re }\, f{f2} to \eqref{ka2}, this gives
\begin{align}\langle bel{ka3}
\langle ngle\Lambda(\chi_0 H(v^{k+1}_1,{\varepsilon}^2 G))\rangle ngle_{m+1,\gamma}\lesssim C(K_1,\gamma^*_{m+1})(\gamma^{-1}+\langle ngle\Lambda G\rangle ngle_{m+1,T_0}).
\end{align}
Similarly, applying Proposition {\rm Re }\, f{f7} and using the simplified notation explained in Notation {\rm Re }\, f{e10a}, we obtain with $z=(\chi^*v^{k+1}_1, {\varepsilon}^2\chi^* G)$:
\begin{align}\langle bel{ka4}
\begin{split}
&\langle ngle\phi_j\Lambda^{\frac{3}{2}}(\chi_0 H(v^{k+1}_1,{\varepsilon}^2 G))\rangle ngle_{m+1,\gamma}\lesssim \\
&\quad\left(\langle ngle\Lambda z\rangle ngle_{m+1,\gamma}\langle ngle\Lambda_1 z\rangle ngle_{m+1}h(\langle ngle z\rangle ngle_{m+1})+ \langle ngle\psi\Lambda z\rangle ngle_{m+1,\gamma}\langle ngle\Lambda^{\frac{1}{2}}_1z\rangle ngle_{m+1}h(\langle ngle z\rangle ngle_{m+1})\right)+\langle ngle\psi\Lambda^{\frac{3}{2}}z\rangle ngle_{m+1,\gamma}h(\langle ngle z\rangle ngle_{m+1})\lesssim\\
&\quad\quad (\gamma^{-1}+\langle ngle\Lambda G\rangle ngle_{m+1,T_0}\rangle ngle)C(K_1,\gamma^*_{m+1})+ \left(\gamma^{-1/2}K_1+\langle ngle\Lambda^{\frac{3}{2}}G\rangle ngle_{m+1,T_0}\right).
\end{split}
\end{align}
In view of \eqref{ka0}, \eqref{ka3}, and \eqref{ka4} the estimate \eqref{k2} applied to the problem \eqref{k0b} gives
\begin{align}\langle bel{ka5}
|v^{k+1}_2|^*_{m+1,\gamma}\leq \gamma^{-1}K_2+K_3\langle ngle\Lambda G\rangle ngle_{m+1,T_0}+\gamma^{-\frac{1}{2}}K_4.
\end{align}
\textbf{Choice of $\delta$ and $T_0$.} We first fix $\delta$ sufficiently small so that the estimates \eqref{k1}, \eqref{k2} apply to the problems \eqref{k0a}, \eqref{k0b} when
$|v^k|^*_{m+1,T}<\delta$. We also need $\delta$ and $T_0$ small enough so that $|v^k|_{m+1}$, $\langle ngle \chi^* v^{k+1}_1,\chi^*{\varepsilon}^2G\rangle ngle_{m+1}$ and $\langle ngle v^k_1, {\varepsilon}^2 G\rangle ngle_{m+1}$ lie in the domain of convergence of the analytic functions (like $h$ in \eqref{ka4}, for example) that appear on the right in the estimates of section {\rm Re }\, f{nonlinear} that we use. To see that $\langle ngle\chi^* v^{k+1}_1\rangle ngle_{m+1}$ can be made small by taking $\delta$ and $T_0$ small, one uses the estimate \eqref{kaa2} and the fact that $K_1(0,0)=0$. A further possible reduction of $T_0$ occurs at the end of this step.
Taking $\gamma=\frac{1}{T}$ in \eqref{ka5} and using Proposition {\rm Re }\, f{c0e}, we find
\begin{align}\langle bel{k6a}
|v^{k+1}_2|^*_{m+1,T}\lesssim TK_2+K_3\langle ngle\Lambda G\rangle ngle_{m+1,T_0}+ T^{\frac{1}{2}}K_4.
\end{align}
After reducing $T_0$ if necessary, the right side of \eqref{k6a} will be $<\delta$ for small enough $T>0$.
Uniform boundedness of the iterates $|u^k|^*_{m+1,T}$ with respect to $k$ now follows by applying the estimate \eqref{k1} to the problem \eqref{k0c}.
\textbf{4. Low norm contraction. } Having the uniform boundedness of the iterates $v^k$ and $u^k$ in the norm $|\cdot|^*_{m+1,T}$ allows us to repeat the
low norm contraction argument of \cite{S-T}, p. 290, to show that the sequence $v^k$ is Cauchy in the $\left|\begin{matrix}\Lambda v^k\\partial_{x_2}v^k\end{matrix}\right|_{0,T}$ norm for a possibly smaller $T>0$. The same applies to the $u^k$. This argument uses the estimates
of Proposition {\rm Re }\, f{bb1} and, of course, some of the nonlinear estimates of section {\rm Re }\, f{nonlinear}.
A standard interpolation argument implies $v^k\to v$ and $u^k\to u$ in $\cN^m(\Omega_T)$, where $v$ and $u$ both lie in
$\cN^{m+1}(\Omega_T)$ and satisfy \eqref{a7}, \eqref{a8}, \eqref{a9} on $\Omega_T$.
\textbf{5. Show $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_T$. }By a computation similar to the one on p. 291 of \cite{S-T}, we see that the differences $v_1-\partial_{x_1,{\varepsilon}}u$ and $v_2-\partial_{x_2}u$ satisfy a coupled system of singular, linear equations on $\Omega_T$ with boundary and interior forcing terms that are identically zero. One can therefore apply the estimates of Proposition {\rm Re }\, f{bb1}
to see that these differences are both zero on $\Omega_T$.
\end{proof}
\subsection{Continuation. }
We will prove the continuation theorem using estimates of the following form that are tame with respect to the norm $|\cdot|^*_{k,\gamma}$.
\begin{prop}\langle bel{tame2}
Let $m_0>3d+4+\frac{d+1}{2}$ and suppose $k\in\{0,1,\dots,m_0+\left[\frac{m_0+1}{2}\right]\}$, where $[r]$ denotes the largest integer $\leq r$.
Suppose that the function $w$ occurring in the coefficients of the problem {\rm Re }\, f{k-1} has compact $t-$support contained in $[0,2)$ and let $\chi^*(t)$ be a $C^\infty$ function equal to one on $\mathrm{supp}\; w$ with support in $(-1,2)$.
For the problem {\rm Re }\, f{k-1} we have: there exist $\gamma_0>0$ and an increasing function $K:\mathbb{R}^+\to\mathbb{R}^+$ such that for $\gamma\geq \gamma_0$,
\begin{align}\langle bel{k3}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_1\\partial_{x_2}\Lambda^{\frac{1}{2}}v_1\end{pmatrix}\right|^2_{k,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix} \right>^2_{k,\gamma}+\sum_{J_h\cup J_e }\left < \phi_j \begin{pmatrix}\Lambda^{\frac{3}{2}} v_1\\partial_{x_2}\Lambda^{\frac{1}{2}} v_1\end{pmatrix} \right>^2_{k,\gamma}\leq \\
&\quad \frac{1}{\gamma}K^2(|w|^*_{m_0})\left(\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_1\\partial_{x_2}\Lambda^{\frac{1}{2}}v_1\end{pmatrix}\right|^2_{k,\gamma}+(|\chi^* v_1|^*_{m_0})^2\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}w\\partial_{x_2}\Lambda^{\frac{1}{2}}w\end{pmatrix}\right|^2_{k,\gamma}+|\Lambda^{\frac{1}{2}}\mathcal{F}_1|^2_{k,\gamma}\right)+\\
&\quad \quad \frac{1}{\gamma}K^2(|w|^*_{m_0})\left(\left\langle ngle\begin{pmatrix}\Lambda v_1\\partial_{x_2} v_1\end{pmatrix}\right\rangle ngle^2_{k,\gamma}+(|\chi^* v_1|^*_{m_0})^2\left\langle ngle\begin{pmatrix}\Lambda w\\partial_{x_2}w\end{pmatrix}\right\rangle ngle^2_{k,\gamma}+\langle ngle\Lambda \mathcal{G}_1\rangle ngle^2_{k,\gamma}\right).
\end{split}
\end{align}
For the problem {\rm Re }\, f{k0} we have: there exist $\gamma_0>0$ and an increasing function $K:\mathbb{R}^+\to\mathbb{R}^+$ such that for $\gamma\geq \gamma_0$,
\begin{align}\langle bel{k4}
\begin{split}
&\gamma \left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_2\\partial_{x_2}\Lambda^{\frac{1}{2}}v_2\end{pmatrix}\right|^2_{k,\gamma}+\gamma\left < \begin{pmatrix}\Lambda v_2\\partial_{x_2} v_2\end{pmatrix} \right>^2_{k,\gamma}\leq \\
&\quad \frac{1}{\gamma}K^2(|w|^*_{m_0})\left(\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}v_2\\partial_{x_2}\Lambda^{\frac{1}{2}}v_2\end{pmatrix}\right|^2_{k,\gamma}+(|\chi^* v_2|^*_{m_0})^2\left|\begin{pmatrix}\Lambda^{\frac{3}{2}}w\\partial_{x_2}\Lambda^{\frac{1}{2}}w\end{pmatrix}\right|^2_{k,\gamma}+|\Lambda^{\frac{1}{2}}\mathcal{F}_2|^2_{k,\gamma}\right)+\\
&\quad \quad K^2(|w|^*_{m_0})\left(\gamma\langle ngle\Lambda \mathcal{G}_2\rangle ngle^2_{k,\gamma}+\sum_{j\in J_h}\langle ngle\phi_j\Lambda^{\frac{3}{2}} \mathcal{G}_2\rangle ngle^2_{k,\gamma}\right).
\end{split}
\end{align}
In these estimates $|w|_{L^\infty(\Omega)}$ must be small enough, $\gamma_0$ depends on
\begin{align}\langle bel{k4z}
\left|\frac{w}{{\varepsilon}}\right|_{C^{0,n}}+\left|\frac{w}{{\varepsilon}}\right|_{CH^{s_0}}+|\partial_{x_2}w|_{L^\infty(\Omega)},\text{ where }n\geq 3d+4, \;s_0>\frac{d+1}{2}+2,
\end{align}
and $K(|w|^*_{m_0})$ depends also on ${\varepsilon}$ and $k$.
\end{prop}
\begin{proof}
\textbf{1. }One first proves the same estimates, but where the cutoffs $\chi^*$ are absent in the factors $|\chi^*v_j|^*_{m_0}$.
\textbf{2. }The proof is similar to that of estimates {\rm Re }\, f{k1} and {\rm Re }\, f{k2}. Again, the starting point is the $k=0$ estimate
given by \eqref{b3} and \eqref{b6}. Estimates of $|\cdot|_{0,k,\gamma}$ and $\langle ngle\cdot\rangle ngle_{k,\gamma}$ norms follow by applying the $k=0$ estimate to the tangentially differentiated problems $\partial^k\eqref{k-1}$, $\partial^k\eqref{k0}$. Estimates of commutators are now done using the tame estimate
of Proposition {\rm Re }\, f{j7}(b).
\textbf{3. }As before, estimates of $\left|\begin{pmatrix}D_{x_2}^j\left(\Lambda^{\frac{3}{2}}v_i\right)\\partial_{x_2}^j\left(D_{x_2}\Lambda^{\frac{1}{2}}v_i\right)\end{pmatrix}\right|_{0,k-j,\gamma}$ for $j>0$ are proved by induction on $j$ using the equation and the noncharacteristic boundary assumption.
We set $U=\begin{pmatrix}e^{\gamma t}\Lambda_De^{-\gamma t} v_i\\partial_{x_2}v_i\end{pmatrix}$ and estimate
$|\Lambda^{\frac{1}{2}}D_{x_2}^jU|_{0,k-j,\gamma}$ using the first order singular equation \eqref{b10d}. For $j$ such that $j\leq k$, $1\leq j\leq \left[\frac{m_0+1}{2}\right]$ (range I), we
apply the tame estimate of Proposition {\rm Re }\, f{j7}(b), while for $j$ such that $j\leq k$, $\left[\frac{m_0+1}{2}\right]\leq j\leq m_0+\left[\frac{m_0+1}{2}\right]$ (range II), we apply Proposition {\rm Re }\, f{f12}(b).
For example, in the estimate of $|\Lambda^{\frac{1}{2}}D_{x_2}^jU|_{0,k-j,\gamma}$ we must estimate terms of the form
\begin{align}
|\Lambda^{\frac{1}{2}}\left(a(w)d_{\varepsilon} D_{x_2}^{j-1}U\right)|_{0,k-j,\gamma}.
\end{align}
Since ${\varepsilon}$ is fixed, we treat $d_{\varepsilon}$ like a nonsingular tangential derivative. Thus, we are led to consider a sum of terms of form
\begin{align}\langle bel{k4y}
|\Lambda^{\frac{1}{2}}\left((\partial^{k_1}z)(\partial^{k_2} D_{x_2}^{j-1}U)\right)|_{0,0,\gamma},
\end{align}
where, with $a(w)=a(0)+b(w)w$,
\begin{align}
z:=b(w)w,\;k_1+k_2=k-(j-1), k_2\geq 1.
\end{align}
We first do a tangential estimate for $x_2$ fixed. For $j$ in range I we apply Proposition {\rm Re }\, f{j7}(b) with ``$m"=k-(j-1)$ and ``$m_0"=m_0-(j-1)$; the factors involving $z$ are estimated using Remark {\rm Re }\, f{j9}. For $j$ in range II we apply Proposition {\rm Re }\, f{f12}(b) with $``u"=D_{x_2}^{j-1}U$, $``v"=z$, $``s"=k-k_2-(j-1)$, $``t"=m_0-k_1$; factors involving $z$ are estimated using Proposition {\rm Re }\, f{normalest}. In both cases the remaining $L^2(x_2)$ norm
is easily controlled by arguments like those in section {\rm Re }\, f{mainestimate}
\textbf{4. }The cutoff functions $\chi^*$ appearing in the estimates \eqref{k3}, \eqref{k4}
can be inserted because the factors $|v_j|^*_{m_0}$ only arise from terms like \eqref{k4y} in which ``free" factors of $\partial^lw$ appear for some $l$. The support assumption on $w$ thus permits the insertion of $\chi^*$.
\end{proof}
\begin{prop}[Continuation of solutions for fixed ${\varepsilon}$] \langle bel{continuation}
(a) Let $m>3d+4+\frac{d+1}{2}$. For ${\varepsilon}$ fixed, suppose we have a solution $(v^{\varepsilon},u^{\varepsilon})\in \cN^{m+1}(\Omega_{T_1})$ of \eqref{a7}, \eqref{a8}, \eqref{a9} with $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ for some $0<T_1<T_0$, where $v^{\varepsilon}=0$ in $t<0$ and is such that $E_{m,T_1}(v^{\varepsilon})\leq \frac{M_0}{N}$
for $M_0$ as in \eqref{c0h}. For $N$ large enough there exists $T_2>T_1$ and an extension of $(v^{\varepsilon},u^{\varepsilon})$ to a solution on $\Omega_{T_2}$ with $v^{\varepsilon}\in \cN^{m+1}(\Omega_{T_2})$ and $v^{\varepsilon}={\nabla}bla_{\varepsilon} u^{\varepsilon}$ on $\Omega_{T_2}$, where $T_2$ depends on ${\varepsilon}$ and $M_0$. \footnote{The size of $N$ depends on the norm of the Seeley extension operator used in step \textbf{2} of the proof.}
(b) There exists $T_3$ with $T_1<T_3\leq T_2$ such that $E_{m,T_3}(v^{\varepsilon})<M_0$.
\end{prop}
\begin{proof}
As usual, we often suppress superscripts ${\varepsilon}$ below. We note that part (b) follows easily from part (a) by taking $T_3$ close enough to $T_1$.
\textbf{1. Translation.} By downward translation in time we reduce to the case where $T_1=0$ and where the given solution $v=(v_1,v_2)\in \cN^{m+1}(\Omega_{0})$ vanishes in $t<-T_1$ and satisfies $E_{m,0}(v)\leq \frac{M_0}{N}$.
\textbf{2. Iteration scheme.} Setting $g={\varepsilon}^2 G$ (which now vanishes in $t<-T_1$), let us write the translates of \eqref{k0a}, \eqref{k0b}, and \eqref{k0c} as
\begin{align}\langle bel{k5}
\begin{split}
&\cL(v^k)v^{k+1}_1=\cF_1(v^k,{\nabla}bla_{\varepsilon} v^k)\text{ on }\Omega\\
&\cB_1(v^k)v^{k+1}_1=\cG_1(v^k_1,g)\text{ on }x_2=0,\\
&v^{k_1+1}=v_1\text{ in }t<0.
\end{split}
\end{align}
\begin{align}\langle bel{k6}
\begin{split}
&\cL(v^k)v^{k+1}_2=\cF_2(v^k,{\nabla}bla_{\varepsilon} v^k)\text{ on }\Omega\\
&v^{k+1}_2=\cG_2(v^{k+1}_1,g)\text{ on }x_2=0,\\
&v^{k+1}_2=v_2\text{ in }t<0.
\end{split}
\end{align}
\begin{align}\langle bel{k6aa}
\begin{split}
&\cL(v^k)u^{k+1}=0\text{ on }\Omega\\
&\cB_1(v^k)u^{k+1}=\cG(v^k_1,g)\text{ on }x_2=0,\\
&u^{k+1}=u\text{ in }t<0.
\end{split}
\end{align}
We focus now on the continuation of $v$. The treatment of $u$ is similar.
Let $(v^e,u^e)\in \cN^{m+1}(\Omega)$ be an extension of $(v,u)|_{\Omega_0}$ to $\Omega$ supported in $[-T_1,2-T_1)$ and such that $v^e={\nabla}bla_{\varepsilon} u^e$ and $E_{m}(v^e)<\frac{M_0}{2}$. If we initialize the schemes \eqref{k5}, \eqref{k6} with $v^0=v^e$, the iterates may be written
$v^k=v^e+z^k$, where the $z^k$ satisfy
\begin{align}\langle bel{k7}
\begin{split}
&\cL(v^e+z^k)z^{k+1}_1=\cF_1(v^e+z^k,{\nabla}bla_{\varepsilon} (v^e+z^k))-\cL(v^e+z^k)v^e_1:={\mathbb F}_1.\\
&\cB_1(v^e+z^k)z^{k+1}_1=\cG_1(v^e_1+z^k_1,g)-\cB_1(v^e+z^k)v^e_1:={\mathbb G}_1\text{ on }x_2=0,\\
&z^{k+1}_1=0\text{ in }t<0.
\end{split}
\end{align}
\begin{align}\langle bel{k8}
\begin{split}
&\cL(v^e+z^k)z^{k+1}_2=\cF_2(v^e+z^k,{\nabla}bla_{\varepsilon} (v^e+z^k))-\cL(v^e+z^k)v^e_2:={\mathbb F}_2\\
&z^{k+1}_2=\cG_2(v^e_1+z^{k+1}_1,g)-v^e_2:={\mathbb G}_2\text{ on }x_2=0\\
&z^{k+1}_2=0\text{ in }t<0.
\end{split}
\end{align}
\textbf{3. Lower regularity continuation.} In the forcing terms on the right sides of \eqref{k7} and \eqref{k8} derivatives of $v^e$ occur, so we first obtain a lower regularity continuation.
Just as in the proof of Proposition {\rm Re }\, f{localex}, we introduce Seeley extensions $z^k_T$ with support in $[0,2-T_1)$ and regard \eqref{k7}, \eqref{k8} as a family of problems on $\Omega$ parametrized by $T$, where now $0<T\leq T_0-T_1$; thus, every $z^k$ (as opposed to $z^{k+1}$) inside a coefficient or forcing term is really $z^k_T$.
We make the induction assumption:
$\bullet$ there exists $T>0$ independent of $k$ such that for $j=0,\dots,k$ we have (dropping the subscript on $z^j$)
$|z^j|^*_{m,T}<\delta$ and thus $|z^j_T|^*_{m}<C_1\delta$ for a small enough $\delta$.\footnote{Here, as in the proof of Proposition {\rm Re }\, f{localex}, we require $\delta$ and $M_0$ to be small enough so that the estimates \eqref{k1}, \eqref{k2} apply.}
From this assumption and the regularity of $v^e$ we obtain (arguing as in the earlier proof)
\begin{align}
|\Lambda^{\frac{1}{2}}{\mathbb F}_i|_{m,\gamma}<\infty \text{ and }\langle ngle\Lambda{\mathbb G}_1\rangle ngle_{m,\gamma}<\infty;
\end{align}
moreover, ${\mathbb F}_i$ and ${\mathbb G}_i$ vanish in $t<0$.
Applying the argument used to prove Proposition {\rm Re }\, f{localex} to the scheme \eqref{k7}, \eqref{k8}, we obtain a solution $v=v^e+z \in \cN^{m}(\Omega_{T_2'})$ to \eqref{a7}, \eqref{a8} for some $T_2'>0$.
Henceforth, let $v^e$, $z^j$, $z^j_T$, and $z$ denote the upward translations by $T_1$ of the similarly denoted functions we have just defined.
We now have a continuation $v:=v^e+z\in \cN^{m}(\Omega_{T_2})$, where $T_2=T_1+T_2'>T_1$ and $v=0$ in $t<0$, and
the functions $v^j=v^e+z^j$ satisfy
\begin{align}\langle bel{k8a}
\begin{split}
&\cL(v^k)v^{k+1}_1=\cF_1(v^k,{\nabla}bla_{\varepsilon} v^k)\text{ on }\Omega_{}\\
&\cB_1(v^k)v^{k+1}_1=\cG_1(v^k_1,g)\text{ on }x_2=0,\\
&v^{k_1+1}=v_1\text{ in }t<T_1.
\end{split}
\end{align}
\begin{align}\langle bel{k8b}
\begin{split}
&\cL(v^k)v^{k+1}_2=\cF_2(v^k,{\nabla}bla_{\varepsilon} v^k)\text{ on }\Omega_{}\\
&v^{k+1}_2=\cG_2(v^{k+1}_1,g)\text{ on }x_2=0,\\
&v^{k+1}_2=v_2\text{ in }t<T_1,
\end{split}
\end{align}
where as before every $v^k$ (as opposed to $v^{k+1}$) appearing inside a coefficient or forcing term is really $v^k_T$. Recall that $v_1$ and $v_2$ vanish in $t<0$.
In the rest of the continuation argument the norm $|\cdot|^*_{m,T_2}$ will have the role usually played by a Lipschitz-type norm in such arguments.
\begin{rem}\langle bel{k9}
1) From our construction there exists a constant $R>0$ such that $v^k_{T_2}=v^e+z^k_{T_2}$ has support in a fixed compact subset of $[0,2)$ and satisfies
\begin{align}\langle bel{k10}
|v^k_{T_2}|^*_{m,T_2}\leq R \text{ and }|v^k_{T_2}|^*_{m}\leq C R\text{ for all }k.
\end{align}
2) Since $E_m(v^e)\leq \frac{M_0}{2}$,
and $|z^k_{T_2}|^*_{m}< C\delta$ for all $k$, we have in particular that
\begin{align}\langle bel{k11}
|v^k_{T_2}|_{\infty,m}\leq \frac{M_0}{2}+C\delta\text{ for all }k,
\end{align}
a fact that will be used in the next step.
3) Let $\chi^*(t)$ as usual denote a cutoff support in $(-1,2)$ and equal to one on the compact subset of $[0,2)$ in which the $v^k_{T_2}$ have support, and let $v^{k+1}_1$ be the solution of \eqref{k8a} (not a Seeley extension). By the argument that gave \eqref{kaa2}, but applied in the construction of $z^{k+1}_1$ satisfying \eqref{k7}, we obtain that there exists a constant $\cK$ independent of $k$ such that
\begin{align}\langle bel{k11a}
|\chi^* v^{k+1}_1|^*_m\leq \cK.
\end{align}
\end{rem}
\textbf{4. Higher regularity.}
To show $v\in\cN^{m+1}(\Omega_{T_2})$, it is enough to show that the iterates $v^j$ in \eqref{k8a}, \eqref{k8b} are uniformly bounded in $\cN^{m+1}(\Omega_{T_2})$, since we already have the iterates converging to $v$ in a lower norm. We make the following induction assumption:
$\bullet$ There exist positive constants $P>0$ and $\gamma_c$ independent of $k$ such that for $j=1,\dots, k$ we have
\begin{align}
|v^k|^*_{m+1,\gamma}\leq P\text{ for }\gamma\geq \gamma_c.
\end{align}
Once this is shown to hold for all $k$, it then follows from Proposition {\rm Re }\, f{c0e} that
\begin{align}
|v^k|^*_{m+1,T_2}\leq Ce^{\gamma_cT_2}|v^k|^*_{m+1,\gamma_c}\leq Ce^{\gamma_cT_2}P\text{ for all }k.
\end{align}
Part (b) of the Proposition follows as in part (2) of Remark {\rm Re }\, f{N}.
\textbf{5. Induction step.} Let $a^{k+1}_{j,\gamma}=|v^{k+1}_j|^*_{m+1,\gamma}$ for $j=1,2$. We will complete the induction step by applying the estimates \eqref{k3}, \eqref{k4} with $``k"=m+1$ and $``m_0"=m$, together with the estimates of sections {\rm Re }\, f{nontame} and {\rm Re }\, f{tames}, to the problems \eqref{k8a}, \eqref{k8b}.
In these estimates $K$, $K_j$ denote nonnegative continuous functions of one or more arguments which increase as any one argument increases. Also, $v^k=v^k_{T_2}$, but $v^{k+1}$ is not a Seeley extension.
We first apply Corollary {\rm Re }\, f{j3} (with Remark {\rm Re }\, f{j9}) to estimate $\cF_i$ in \eqref{k5}, \eqref{k6}, obtaining
\begin{align}\langle bel{k11aa}
|\Lambda^{\frac{1}{2}}\cF_i(v^k,{\nabla}bla_{\varepsilon} v^k)|_{m+1,\gamma}\leq K_1(R)P \text{ for }j=1,2.
\end{align}
Here and below we use \eqref{k11} to bound the analytic functions like $h(\langle ngle v^k, {\varepsilon}^2 G \rangle ngle_m$ that appear by constants $C$; we suppose that $M_0$, $\delta$, and $M_G$ have been chosen small enough so that the arguments of these functions lie in their domains of convergence. Since,
\begin{align}
\langle ngle \Lambda (v^k_1,{\varepsilon}^2 G)\rangle ngle_{m+1,\gamma}\lesssim P+M_G,
\end{align}
Corollary {\rm Re }\, f{j3} yields
\begin{align}
\langle ngle\Lambda\cG_1(v^k_1,{\varepsilon}^2 G)\rangle ngle_{m+1,\gamma}\leq C_1(P+PR+M_G).
\end{align}
Applying the estimate {\rm Re }\, f{k3} to \eqref{k8a} and using \eqref{k11a}, we obtain for $j\in J_h$ and $\gamma\geq \gamma_0$
\begin{align}\langle bel{k12}
\begin{split}
&a^{k+1}_{1,\gamma}+\gamma^{-\frac{1}{2}}\langle ngle \phi_j\Lambda^{\frac{3}{2}}v^{k+1}_1\rangle ngle_{m+1,\gamma}\leq \frac{1}{\gamma}K(R)\left(a^{k+1}_{1,\gamma}+\cK P+K_1(R)P\right)+\\
&\qquad \frac{1}{\gamma}K(R)\left(a^{k+1}_{1,\gamma}+\cK P+C_1(P+PR+M_G)\right)\leq \frac{1}{\gamma}(2K(R)a^{k+1}_{1,\gamma}+K_2(R)P+K_3(R)M_G).\\
\end{split}
\end{align}
Thus,
\begin{align}\langle bel{k12a}
\begin{split}
&a^{k+1}_{1,\gamma}+\gamma^{-\frac{1}{2}}\langle ngle \phi_j\Lambda^{\frac{3}{2}}v^{k+1}_1\rangle ngle_{m+1,\gamma}\leq \gamma^{-1}K_4(R)(P+M_G):=\gamma^{-1}K_5(R,P,M_G)\\&\qquad \qquad \text{ for }\gamma\geq \max {(4K(R),\gamma_0)}:=\gamma_a, \text{ where }K_5(R,0,0)=0.
\end{split}
\end{align}
This implies
\begin{align}\langle bel{k13}
|\Lambda(\chi^* v^{k+1}_1)|_{m+1,\gamma}\lesssim \frac{1}{\gamma}K_5(R,P,M_G)\text{ for }\gamma \geq \gamma_a,
\end{align}
and since $\mathrm{supp}\;\chi^*\subset (-1,2)$, this gives
\begin{align}\langle bel{k13b}
|\Lambda_1(\chi^* v^{k+1}_1)|_{m+1}\lesssim e^{\gamma_a 2}K_5(R,P,M_G):=K_6.
\end{align}
Application of the (nontame) estimate of Proposition {\rm Re }\, f{f3}(c) yields with $z=(\chi^*v^{k+1}_1, {\varepsilon}^2\chi^* G)$:
\begin{align}\langle bel{k13a}
\begin{split}
&\langle ngle\Lambda\cG_2\rangle ngle_{m+1,\gamma}=\langle ngle\Lambda(\chi_0 H(v^{k+1}_1,{\varepsilon}^2 G))\rangle ngle_{m+1,\gamma}\lesssim \left(\langle ngle\Lambda z\rangle ngle_{m+1,\gamma}+\langle ngle z\rangle ngle_{m+1,\gamma}\langle ngle\Lambda_1 z\rangle ngle_{m+1}\right)\;h(\langle ngle z\rangle ngle_{m+1})\lesssim \\
&\qquad \left(\frac{K_5}{\gamma}+ M_G\right)+\left(\frac{K_5}{\gamma}+M_G\right)(K_6+ M_G)\lesssim \frac{K_7}{\gamma}+M_G.
\end{split}
\end{align}
In the above estimate we have used \eqref{k13b}, noting that since $K_5(R,0,0)=0$ we will have $\langle ngle z\rangle ngle_{m+1}$ in the domain of convergence of $h$ provided $P$ and $M_G$ are small enough.
Similarly, applying Proposition {\rm Re }\, f{f7} and using \eqref{k12a} to estimate $\langle ngle \psi\Lambda^{\frac{3}{2}}v^{k+1}_1\rangle ngle_{m+1,\gamma}$, we obtain
\begin{align}\langle bel{k14}
\begin{split}
&\langle ngle\phi_j\Lambda^{\frac{3}{2}}(\chi_0 H(v^{k+1}_1,{\varepsilon}^2 G))\rangle ngle_{m+1,\gamma}\lesssim \\
&\quad \langle ngle\Lambda z\rangle ngle_{m+1,\gamma}\langle ngle\Lambda_1 z\rangle ngle_{m+1}h(\langle ngle z\rangle ngle_{m+1})+ \langle ngle\psi\Lambda z\rangle ngle_{m+1,\gamma}\langle ngle\Lambda^{\frac{1}{2}}_1z\rangle ngle_{m+1}h(\langle ngle z\rangle ngle_{m+1})+\langle ngle\psi\Lambda^{\frac{3}{2}}z\rangle ngle_{m+1,\gamma}h(\langle ngle z\rangle ngle_{m+1})\lesssim\\
&\qquad \left(\frac{K_5}{\gamma}+ M_G\right)(K_6+ M_G)+ \left(\frac{K_5}{\gamma}+ M_G\right)(K_6+\ M_G)+ \left(\frac{K_5}{\gamma^{\frac{1}{2}}}+ M_G\right)\lesssim \frac{K_8}{\gamma^{\frac{1}{2}}}+1.
\end{split}
\end{align}
With \eqref{k11a}, \eqref{k13a} and \eqref{k14} we can now apply the estimate \eqref{k4} to \eqref{k8b} to obtain
\begin{align}
\begin{split}
&a^{k+1}_{2,\gamma}\lesssim \frac{1}{\gamma}K(R)\left(a^{k+1}_{2,\gamma}+\cK P+K_1(R)P\right)+\\
&\qquad K(R)\left[\left(\frac{K_7}{\gamma}+M_G\right)+\frac{1}{\gamma^{\frac{1}{2}}}\left(\frac{K_8}{\gamma^{\frac{1}{2}}}+1\right)\right].
\end{split}
\end{align}
Thus, for large enough $\gamma$
\begin{align}
a^{k+1}_{2,\gamma}\lesssim \frac{1}{\gamma}K_9P+K_{10}M_G+\frac{1}{\gamma^\frac{1}{2}}K_{11}<\frac{P}{2},
\end{align}
if $\gamma$ is large enough and $M_G$ small enough.
Since \eqref{k12a} implies
$a^{k+1}_{1,\gamma}\leq \frac{P}{2}$ for $\gamma$ large enough, so this completes the induction step.
\end{proof}
\chapter{Approximate solutions}
\langle bel{chapter4}
\emph{\quad} This chapter is devoted to the construction of an approximate solution (leading term and corrector) to the coupled, singular nonlinear problems \eqref{a7}-\eqref{a9} on a short time interval. We will use the result of Proposition {\rm Re }\, f{propwellposed}, but otherwise this chapter can be read independently of chapter {\rm Re }\, f{chapter2} and all but the first few pages of chapter {\rm Re }\, f{chapter3}.
We introduce some notations here that differ from those used in chapter {\rm Re }\, f{chapter2}, but which we have found more suitable for estimating the nonlinear interactions that appear in the first corrector.
=
\section{Introduction}
We seek initially an approximate solution to the original nonlinear, nonsingular problem \eqref{a2} of the form
\begin{align}
U^{\varepsilon}_a(t,x)=u_a(t,x,\theta,z)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}, z=\frac{x_2}{{\varepsilon}}} \text{ on }\Omega_{T_1}
\end{align}
for some $T_1>0$, where\footnote{In chapter {\rm Re }\, f{chapter2} $u_\sigma$ and $u_\tau$ were denoted $u^{(1)}$ and $u^{(2)}$ respectively.}
\begin{align}\langle bel{o2a}
\begin{split}
&u_a(t,x,\theta,z)={\varepsilon}^2u_\sigma(t,x,\theta,z)+{\varepsilon}^3 u_\tau(t,x,\theta,z) \text{ with }\\
&u_\sigma(t,x,\theta,z)=\sum_{j=1}^4\sigma_j(t,x,\theta+\omega_j z) r_j\text{ and }u_\tau(t,x,\theta,z)= \sum_{j=1}^4 \left(\chi_{\varepsilon}(D_\theta)\tau_j(t,x,\theta,z)\right) r_j.
\end{split}
\end{align}
Here the $\sigma_j$ and $\tau_j$ are scalar profiles constructed in section {\rm Re }\, f{moreonc}, and the $\omega_j$, $r_j$ are characteristic roots and vectors satisfying
\begin{align}
\det L(\beta,\omega_j)=0 \text{ and } L(\beta,\omega_j)r_j=0,
\end{align}
where $L(\xi',\xi_2)$ is the $2\times 2$ matrix symbol defined below in \eqref{oo1}. The operator $\chi_{\varepsilon}(D_\theta)$ is a low-frequency cutoff operator that is discussed below. The functions
\begin{align}
u^{\varepsilon}_a(t,x,\theta):=u_a(t,x,\theta,\frac{x_2}{{\varepsilon}})\text{ and }v^{\varepsilon}_a={\nabla}bla_{\varepsilon} u^{\varepsilon}_a
\end{align}
will then turn out to be approximate solutions of the singular problems \eqref{a7}-\eqref{a9} on the same time interval.
When $U^{\varepsilon}_a(t,x)$ is plugged into the system \eqref{a2}, we obtain
\begin{align}\langle bel{o2}
\begin{split}
&\partial_{t}^2 U_a^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha({\nabla}bla U^{\varepsilon}_a)\partial_{x}^\alpha U_a^{\varepsilon}=F_a(t,x,\theta,z)_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}, z=\frac{x_2}{{\varepsilon}}}\text{ on }\Omega_{T_1}\\
&h({\nabla}bla U^{\varepsilon}_a)=g_a(t,x_1,\theta)|_{\theta=\frac{\beta\cdot (t,x_1)}{{\varepsilon}}}\text{ on }x_2=0,
\end{split}
\end{align}
for interior and boundary profiles $F_a$, $g_a$ described below. All functions here are zero in $t<0$.
The coefficients $A_\alpha(v)$ are polynomials in $v$ of order two, so we write
\begin{align}\langle bel{Asub}
A_\alpha(v)=A_\alpha(0)+L_\alpha(v)+Q_\alpha(v),
\end{align}
a sum of constant, linear, and quadratic parts. Let us set
\begin{align}\langle bel{o2aa}
D_{\varepsilon}=(\partial_{x_1}+\frac{\beta_1}{{\varepsilon}}\partial_\theta, \partial_{x_2}+\frac{1}{{\varepsilon}}\partial_z)
\end{align}
and define\footnote{Here, for example, we write $\partial_{x_1\theta}$ for $\partial_{x_1}\partial_\theta$.}
\begin{align}
\partial_{ss}^\alpha=\partial_x^\alpha,\;\partial^\alpha_{fs}=\begin{cases}2\beta_1\partial_{x_1\theta}, \;\alpha=(2,0)\\\beta_1\partial_{x_2\theta}+\partial_{x_1z},\;\alpha=(1,1)\\2\partial_{x_2z},\;\alpha=(0,2)\end{cases},\;\partial^\alpha_{ff}=\begin{cases}\beta^2_1\partial_{\theta\theta}, \;\alpha=(2,0)\\\beta_1\partial_{\theta z},\;\alpha=(1,1)\\\partial_{zz},\;\alpha=(0,2)\end{cases}.
\end{align}
Next set
\begin{align}\langle bel{o2b}
\begin{split}
&L_{ss}=\partial^2_t+\sum_{|\alpha|=2}A_\alpha(0)\partial^\alpha_{ss},\;\; L_{fs}=2\beta_0\partial_{t\theta}+\sum_{|\alpha|=2}A_\alpha(0)\partial^\alpha_{fs},\;\;L_{ff}=\beta^2_0\partial_{\theta\theta}+\sum_{|\alpha|=2}A_\alpha(0)\partial^\alpha_{ff}\\
&N(u_\sigma)=\sum_{|\alpha|=2}L_\alpha(\beta_1\partial_\theta u_\sigma,\partial_z u_\sigma)\partial^\alpha_{ff}u_\sigma.
\end{split}
\end{align}
The function $h(v)$ in the boundary condition of \eqref{o2} is a cubic polynomial in $v$ satisfying $h(0)=0$, so we write
\begin{align}\langle bel{hofv}
h(v)=\ell (v)+q(v)+c(v),
\end{align}
a sum of linear, quadratic, and cubic parts, and define
\begin{align}
\ell_s(u_\sigma)=\ell(\partial_{x_1}u_\sigma,\partial_{x_2}u_\sigma),\;\ell_f(u_\tau)=\ell(\beta_1\partial_\theta u_\tau,\partial_z u_\tau),\;n(u_\sigma)=q(\beta_1\partial_\theta u_\sigma,\partial_z u_\sigma).
\end{align}
One attempts to construct $u_\sigma$ and $u_\tau$ so that the terms in $F_a$ of orders $O({\varepsilon}^0)$ and $O({\varepsilon})$ vanish, and so that the terms in $g_a-g$, where $g={\varepsilon}^2 G$, of orders $O({\varepsilon})$ and $O({\varepsilon}^2)$ vanish. This leads directly to interior and boundary equations that one \emph{would like} $u_\sigma$ and $u_\tau$ to satisfy:
\begin{align}\langle bel{o3}
\begin{split}
&(a)\;L_{ff}u_\sigma=0\\
&(b)\;L_{ff}u_\tau+L_{fs}u_\sigma+N(u_\sigma)=0 \text{ in }x_2\geq 0, z\geq 0
\end{split}
\end{align}
and
\begin{align}\langle bel{o4}
\begin{split}
&(a)\;\ell_f(u_\sigma)=0\\
&(b)\;\ell_f u_\tau+\ell_su_\sigma+n(u_\sigma)=G(t,x_1,\theta)\text{ on }x_2=0,z=0.
\end{split}
\end{align}
We now introduce a new unknown $\cU_\tau(t,x,\theta,z)$ that will be related to $u_\tau$ by $u_\tau=\chi_{\varepsilon}(D_\theta)\cU_\tau$.
The amplitude equation of Proposition {\rm Re }\, f{propelas} is a solvability condition for the following problem (which is the same as \eqref{bkwordre2}):
\begin{align}\langle bel{o5}
\begin{split}
&(a)\;L_{ff}\cU_\tau=-(L_{fs}u_\sigma+N(u_\sigma))\text{ on }x_2=0,z\geq 0\\
&(b)\;\ell_f \cU_\tau+\ell_su_\sigma+n(u_\sigma)=G(t,x_1,\theta)\text{ on }x_2=0,z=0.
\end{split}
\end{align}
\emph{More precisely,} it is a solvability condition for the Fourier transform of this problem with respect to $\theta$. The traces $\hat\sigma_j(t,x_1,0,k)$ turn out to be constant multiples of $\hat w(t,x_1,k)$, where $w(t,x_1,\theta)$ is the solution of the amplitude equation {\rm Re }\, f{ampeqn}.
For each $k$ one is able to construct $\hat \cU_\tau(t,x,k,z)$ so that the transform of \eqref{o5}(b) holds and the transform of
\eqref{o5}(a) holds in $\{x_2\geq 0, z\geq 0\}$.
A difficulty is that $\hat \cU_\tau(t,x,k,z)$ is so singular at $k=0$ that one cannot take the inverse transform to define $\cU_\tau$; in fact, \eqref{o23a} shows that
$\hat \cU_\tau(t,x,k,z)=O(\frac{1}{k^2})$. The division by $k^2$ in \eqref{o23a} reflects the two integrations needed in this second order problem. Since $\cU_\tau(t,x,\theta,z)$ is not well-defined, we can not actually solve \eqref{o5}. Moreover, $\hat \cU_\tau(t,x,k,z)$ is too large to be of any direct use in the error analysis.
This is the reason for $\chi_{\varepsilon}(D_\theta)$ in the definition of $u_\tau$ \eqref{o2a}. For some $b>0$ to be chosen later and a $C^\infty$ cutoff $\chi(s)$ vanishing near $s=0$ with $\chi=1$ on $|s|\geq 1$, we define
\begin{align}
\hat u_\tau(t,x,k,z)=\chi\left(\frac{k}{{\varepsilon}^b}\right)\hat \cU_\tau(t,x,k,z):=\chi_{\varepsilon}(k)\hat \cU_\tau(t,x,k,z),
\end{align}
and observe that $u_\tau$, which is well-defined, satisfies
\begin{align}\langle bel{o6}
\begin{split}
&(a)\;L_{ff}u_\tau+\chi_{\varepsilon}(D_\theta)(L_{fs}u_\sigma+N(u_\sigma))=0\text{ on }x_2\geq 0,z\geq 0\\
&(b)\;\ell_f u_\tau+\chi_{\varepsilon}(D_\theta)(\ell_su_\sigma+n(u_\sigma))=\chi_{\varepsilon}(D_\theta)G(t,x_1,\theta)\text{ on }x_2=0,z=0.
\end{split}
\end{align}
The use of $\chi_{\varepsilon}(D_\theta)$ introduces new errors of course, but we show in chapter {\rm Re }\, f{chapter5} that if the exponent $\emph{}{}{}{b}$ is chosen correctly, the errors will converge to zero at a computable rate as ${\varepsilon}\to0$, \emph{}{}{}{because of the presence of the factor of ${\varepsilon}^3$ on $u_\tau$.}
This kind of \emph{}{}{}{low-frequency cutoff} was first used in the rigorous study of pulses propagating in the interior in work of Alterman-Rauch \cite{AR}.
Using \eqref{o3}(a), \eqref{o4}(a), and \eqref{o6}, we can now write the interior error profile in \eqref{o2} as
\begin{align}\langle bel{fa}
\begin{split}
&F_a(t,x,\theta,z):=L_{ss}u_a+{\varepsilon}^2 L_{fs}u_\tau+\sum_{|\alpha|=2}Q_\alpha(D_{\varepsilon} u_a)D_{\varepsilon}^\alpha u_a+\left(\sum_{|\alpha|=2}L_\alpha(D_{\varepsilon} u_a)D^\alpha_{\varepsilon} u_a-{\varepsilon} N(u_\sigma)\right)+\\
&\qquad {\varepsilon}(1-\chi_{\varepsilon}(D_\theta))(L_{fs}u_\sigma+N(u_\sigma)).
\end{split}
\end{align}
Similarly, we can write the boundary profile in \eqref{o2} as $g_a=g+h_a$, where the boundary error profile is
\begin{align}\langle bel{ba}
\begin{split}
&h_a(t,x_1,\theta):={\varepsilon}^3 \ell_s(u_\tau)+c(D_{\varepsilon} u_a)+(q(D_{\varepsilon} u_a)-{\varepsilon}^2n(u_\sigma))+\\
&\qquad {\varepsilon}^2\left[(\chi_{\varepsilon}(D_\theta)-1)G+(1-\chi_{\varepsilon}(D_\theta))(\ell_s(u_\sigma)+n(u_\sigma))\right]\text{ at } x_2=0,z=0.
\end{split}
\end{align}
\begin{rem}\langle bel{baz}
1. The terms in the first line of \eqref{fa} would have been the only terms to appear if there were no need to introduce $\chi_{\varepsilon}(D_\theta)$.
These terms are all (formally) $O({\varepsilon}^2)$ or smaller.
2. The second line represents the ``low frequency cutoff error" incurred by introducing $\chi_{\varepsilon}(D_\theta)$.
3. Remarks analogous to (1) and (2) apply to the first and second lines of \eqref{ba}.
4. The approximate solution exhibits \emph{}{}{}{amplification}, as often happens in weakly stable problems, in the following sense. The approximate solution is $O({\varepsilon}^2)$, and its gradient is of size $O({\varepsilon})$, given boundary data of size $O({\varepsilon}^2)$. If the uniform Lopatinskii condition were satisfied, we would expect the solution to be of size $O({\varepsilon}^3)$ and its gradient of size $O({\varepsilon}^2)$ in this second-order problem.
5. Even if one assumes $G(t,x_1,\theta)$ decays \emph{}{}{}{exponentially} as $|\theta|\to \infty$, the profiles $\sigma_j(t,x,\theta)$ defining $u_\sigma$
generally exhibit \emph{}{}{}{no better} than $H^s(t,x,\theta)$ ``decay" in $\theta$. This reflects that fact that $\hat\sigma_j(t,x,k)$ may be \emph{}{}{}{discontinuous} at $k=0$. This loss of $\theta$-decay from data to solution is \emph{}{}{}{typical of evanescent pulses}, and occurs even in problems where the uniform Lopatinskii condition is satisfied [Willig].
\end{rem}
\section{Construction of the leading term and corrector}\langle bel{moreonc}
\textbf{1. Some notation. }
As in Lardner \cite{Lardner1}, the operator $L_{ss}$ can be written
\begin{align}\langle bel{form}
L_{ss}=\partial_{tt}-\begin{pmatrix}r&0\\0&1\end{pmatrix}\partial_{x_1x_1}-\begin{pmatrix}0&r-1\\r-1&0\end{pmatrix}\partial_{x_1x_2}-\begin{pmatrix}1&0\\0&r\end{pmatrix}\partial_{x_2x_2},
\end{align}
where $r>1$ is the ratio of the squares of pressure $c_d$ and shear $c_s$ velocities.\footnote{We have $c_s^2=\mu$ and $c_d^2=(\langle mbda+2\mu)$, where $\langle mbda$, $\mu$ are the Lam\'e constants. The form \eqref{form} is obtained by taking units of time so that $c_s=1$. Observe $r=c_d^2/c_s^2>1$ since $\langle mbda+\mu>0$.}
The boundary frequency $\beta$ has the form $(-c,1)$ for a $c$ to be chosen, so the operators $L_{fs}$ and $L_{ff}$ are
\begin{align}
\begin{split}
&L_{fs}=-2c\partial_{t\theta}-\begin{pmatrix}2r&0\\0&2\end{pmatrix}\partial_{x_1\theta}-\begin{pmatrix}0&r-1\\r-1&0\end{pmatrix}(\partial_{x_1z}+\partial_{x_2\theta})-\begin{pmatrix}2&0\\0&2r\end{pmatrix}\partial_{x_2z},\\
&L_{ff}=\begin{pmatrix}c^2-r&0\\0&c^2-1\end{pmatrix}\partial_{\theta\theta}-\begin{pmatrix}0&r-1\\r-1&0\end{pmatrix}\partial_{\theta z}-\begin{pmatrix}1&0\\0&r\end{pmatrix}\partial_{zz},
\end{split}
\end{align}
and
\begin{align}
\begin{split}
&\ell_s=\begin{pmatrix}0&1\\r-2&0\end{pmatrix}\partial_{x_1}+\begin{pmatrix}1&0\\0&r\end{pmatrix}\partial_{x_2}\\
&\ell_f=\begin{pmatrix}0&1\\r-2&0\end{pmatrix}\partial_{\theta}+\begin{pmatrix}1&0\\0&r\end{pmatrix}\partial_{z}.
\end{split}
\end{align}
With $\xi'=(\sigma,\xi_1)=(\xi_0,\xi_1)$ the symbol of $L_{ss}$ is
\begin{align}\langle bel{oo1}
\begin{split}
&L(\xi',\xi_2)=\begin{pmatrix}\sigma^2-(r-1)\xi^2_1-|\xi|^2 &-(r-1)\xi_1\xi_2\\-(r-1)\xi_1\xi_2&\sigma^2-(r-1)\xi_2^2-|\xi|^2\end{pmatrix}
\end{split}
\end{align}
with characteristic roots $\omega_j$ and vectors $r_j$ satisfying
\begin{align}\langle bel{oo2}
\det L(\beta,\omega_j)=0\text{ and }L(\beta,\omega_j)r_j=0, j=1,\dots 4.
\end{align}
The numbers $\omega_1$ and $\omega_2$ are pure imaginary with positive imaginary part. We have $\omega_3=\overline{\omega_1}$, $\omega_4=\overline{\omega_2}$
and
\begin{align}\langle bel{ray1}
\omega_1^2=c^2-1, \; \omega_2^2= \frac{c^2}{r}-1,
\end{align}
the numbers on the right in \eqref{ray1} being negative since $\beta$ lies in the elliptic region.
If we define $q=q(c)>0$ by
\begin{align}\langle bel{ray2}
q^2=-\omega_1\omega_2,
\end{align}
then the condition for $\beta=(-c,1)$ to be a Rayleigh frequency is that\footnote{This is the condition for the $2\times 2$ matrix $\cB_{Lop}$ defined \eqref{o10a} to have vanishing determinant. Equation {\rm Re }\, f{ray} is equivalent to equation \eqref{defcR}. The $\omega_j$ appearing here are obtained from the $\omega_j$ in \eqref{emodes} by multiplying the latter by $i$.}
\begin{align}\langle bel{ray}
(2-c^2)^2=4q^2(c), \text{ or equivalently } 2-c^2=2q.
\end{align}
For the existence of $\beta=(-c,1)$ in the elliptic region satisfying \eqref{ray} we refer, for example, to \cite{T2}. We take
\begin{align}\langle bel{oo3}
r_1=\begin{pmatrix}-\omega_1\\1\end{pmatrix},\;r_2=\begin{pmatrix}1\\ \omega_2\end{pmatrix},\;r_3=\overline{r}_1, r_4=\overline{r}_2.
\end{align}
\textbf{2. First-order system for the $\sigma_j$. }
Consider a problem of the form
\begin{align}\langle bel{o6a}
L_{ff}u=f(t,x,\theta,z)\text{ in }\{x_2\geq 0, z\geq 0\},\;\;\; \ell_f u= g(t,x_1,\theta) \text{ on }x_2=z=0.
\end{align}
Taking the Fourier transform with respect to $\theta$ and setting $\hat U=\begin{pmatrix}\hat u\\ \partial_z\hat u\end{pmatrix}$, we obtain the $4 \times 4$ first order system
\begin{align}\langle bel{o6b}
\partial_z \hat U-G(\beta,k)\hat U=\hat \cF, \; C(\beta,k)\hat U=\hat g,
\end{align}
where $G(\beta,k)=\begin{pmatrix}0&1\\partial&B\end{pmatrix}$ with
\begin{align}\langle bel{o6c}
D=k^2\begin{pmatrix}r-c^2&0\\0&\frac{1-c^2}{r}\end{pmatrix},\; B=ik\begin{pmatrix}0&1-r\\\frac{1}{r}-1&0\end{pmatrix}\text{ and }\cF=\begin{pmatrix}0\\-\begin{pmatrix}1&0\\0&r\end{pmatrix}^{-1} f\end{pmatrix}
\end{align}
and
\begin{align}\langle bel{o6d}
C(\beta,k)=\begin{pmatrix}ik\begin{pmatrix}0&1\\r-2&0\end{pmatrix}&\begin{pmatrix}1&0\\0&r\end{pmatrix}\end{pmatrix}.
\end{align}
The matrix $G(\beta,k)$ has eigenvalues $ik\omega_j$, $j=1,\dots,4$
corresponding respectively to the right eigenvectors
\begin{align}
R_1=\begin{pmatrix}-\omega_1\\1\\-ik\omega_1^2\\ik\omega_1\end{pmatrix}, R_2=\begin{pmatrix}1\\\omega_2\\ik\omega_2\\ik\omega_2^2\end{pmatrix}, R_3=\begin{pmatrix}\omega_1\\1\\-ik\omega_1^2\\-ik\omega_1\end{pmatrix}, R_4=\begin{pmatrix}1\\-\omega_2\\-ik\omega_2\\ik\omega_2^2\end{pmatrix}.
\end{align}
Using the $R_k$ to diagonalize $G(\beta,k)$ we see that solutions of $\partial_z \hat U-G(\beta,k)\hat U=0$ that decay as $z\to +\infty$ must have the form
\begin{align}\langle bel{o10}
\hat U_j(t,x,k,z)=e^{ik\omega_j z}\hat \sigma_j(t,x,k)R_j,
\end{align}
where the $\hat \sigma_j$ are scalar functions to be determined such that
\begin{align}
k\mathrm{Im}\; \omega_j\geq 0 \text{ on supp }\hat \sigma_j.
\end{align}
This explains the form of $u_\sigma$ in \eqref{o2a}, where $r_j$ is the vector given by the first two components of $R_j$.
A short calculation using the relations \eqref{ray1}-\eqref{ray} shows that
\begin{align}\langle bel{o10a}
[C(\beta,k)R_1\;\; C(\beta,k)R_2]=ik\begin{pmatrix}2-c^2&2\omega_2\\2\omega_1&c^2-2\end{pmatrix}:=ik\cB_{Lop},
\end{align}
and that
\begin{align}\langle bel{o11}
\ker \cB_{Lop}=\mathrm{span}\;\begin{pmatrix}\omega_2\\-q\end{pmatrix},\;\;\mathrm{coker}\; \cB_{Lop}=\mathrm{span}\; \begin{pmatrix} q&\omega_2\end{pmatrix}.
\end{align}
In order for (the Fourier transform in $\theta$ of ) \eqref{o4}(a) to hold we must have, when $k>0$,
\begin{align}
C(\beta,k)(\hat U_1+\hat U_2)=0\text{ on }x_2=0, z=0 \text{ for }\hat U_i \text{ as in }\eqref{o10},
\end{align}
and thus in view of \eqref{o11}
\begin{align}\langle bel{o12}
\begin{pmatrix}\hat\sigma_1(t,x_1,0,k)\\\hat\sigma_2 (t,x_1,0,k)\end{pmatrix}=\alpha(t,x_1,k)\begin{pmatrix}\omega_2\\-q\end{pmatrix},
\end{align}
for some scalar amplitude $\alpha$. We take $\alpha$ to be $-2i \hat w(t,x_1,k)$, where $w$ is the function constructed using Proposition {\rm Re }\, f{propwellposed} to satisfy the solvability condition \eqref{eqw}.\footnote{The factor of $-2i$ could be replaced by one if we multiplied $r_1$ and $r_2$ in \eqref{oo3} by $-2i$.} With the traces of the $\hat\sigma_j$ thereby determined, we extend the $\hat\sigma_j$ into $x_2\geq 0$ by setting
\begin{align}\langle bel{o19a}
\hat\sigma_j(t,x_1,x_2,k):=\psi(x_2)\hat\sigma_j(t,x_1,0,k),
\end{align}
where $\psi\in C^\infty$ is compactly supported and equal to $1$ near $x_2=0$.
\textbf{3. First-order system for the $\tau_j$. }
To proceed further we must now consider the inhomogenous problem \eqref{o5} for $\cU_\tau$. We look for the solution as a sum of a homogeneous solution $\cU^h_\tau$ and a ``particular" solution $\cU^p_\tau$ as
\begin{align}\langle bel{o12a}
\begin{split}
&\cU_\tau=\cU_\tau^h+\cU^p_\tau, \text{ where }\\
&\cU_\tau=\sum^4_{j=1}\tau_j(t,x,\theta,z) r_j,\;\;\cU^h_\tau=\sum_{j=1}^4\tau_j^h(t,x,\theta,z)r_j,\;\;\cU^p_\tau=\sum_{j=1}^4\tau_j^p(t,x,\theta,z)r_j.
\end{split}
\end{align}
The Fourier transform of \eqref{o5} can be written as the first order system \eqref{o6b}-\eqref{o6d}, where now
\begin{align}\langle bel{o13}
\begin{split}
&(a)\;f=f_\sigma:=-(L_{fs}u_\sigma+N(u_\sigma))\\
&(b)\;g=-[\ell_su_\sigma+n(u_\sigma)]+G(t,x_1,\theta)\text{ on }x_2=0,z=0.
\end{split}
\end{align}
The interior forcing term $\hat{\cF}$ can be written
\begin{align}
\hat{\cF}=\sum^4_{j=1}\hat F_jR_j, \text{ with }\hat F_j=L_j\hat\cF,
\end{align}
where the $L_j$ are left eigenvectors of $G(\beta,k)$ associated to $\omega_j$ chosen so that $L_mR_n=\delta_{mn}$. The $L_j$ are given by
\begin{align}\langle bel{o14}
\begin{split}
&L_1(k)=(-ik(r-c^2),-ik\omega_1,\omega_1,-r)/(-2i\omega_1c^2k),\,\,L_2(k)=(ik\omega_2r, ik(c^2-1),1,r\omega_2)/2i\omega_2c^2k,\\
&L_3(k)=\overline{L}_1(-k), \;L_4(k)= \overline{L}_2(-k).
\end{split}
\end{align}
The decoupled interior system for the $\hat \tau^p_j$ is then
\begin{align}\langle bel{o15}
(\partial_z-ik\omega_j)\hat \tau^p_j=\hat F_j, \;j=1,\dots,4.
\end{align}
From \eqref{o14} and the form of $\cF$ in \eqref{o6c}, we see that
\begin{align}\langle bel{o15a}
\begin{split}
&\hat F_j=\ell_j(k) \hat f_\sigma \text{ for }f_\sigma\text{ as in \eqref{o13} and row vectors }\ell_j(k) \text{ given by: }\\
&\ell_1(k)= (\omega_1, -1)/(-2i\omega_1 c^2 k),\; \ell_2(k)=(1,\omega_2)/(2i\omega_2 c^2 k),\; \ell_3(k)=\overline{\ell}_1(-k),\; \ell_4(k)=\overline{\ell}_2(-k).
\end{split}
\end{align}
\textbf{4. Formulas for the $\hat\tau_j$. } Decaying solutions of \eqref{o15} are given by
\begin{align}\langle bel{o19aa}
\hat\tau^p_j(t,x,k,z)=\begin{cases}\int^z_0 e^{ik\omega_j(z-s)}\hat F_j(t,x,k,s)ds,\; k>0\\ \int^z_{+\infty}e^{ik\omega_j(z-s)}\hat F_j(t,x,k,s)ds,\; k<0\end{cases}\text{ for }j=1,2,
\end{align}
with the same formulas for $j=3,4$, except that $k>0$ (resp. $k<0$) is now associated with $\int_{+\infty}^z$ (resp. $\int^z_0$). For the homogeneous parts we have
\begin{align}
\hat\tau^h_j(t,x,k,z)= e^{ik\omega_jz}\hat \tau_j^*(t,x,k), \text{ where }\mathrm{supp}\;\hat\tau^*_j\subset\begin{cases} \{k\geq 0\},\;j=1,2\\ \{k\leq 0\},\;j=3,4\end{cases}
\end{align}
for functions $\hat\tau^*_j$ to be determined.
To complete the determination of the $\tau_j$ we now examine the Fourier transform in $\theta$ of the boundary equation in \eqref{o6a}.
As in \eqref{o6b} this can be written
\begin{align}\langle bel{o20}
C(\beta,k)\hat U_\tau=\hat g, \text{ where }g=-[\ell_su_\sigma+n(u_\sigma)]+G(t,x_1,\theta)\text{ and }\hat U_\tau:=\begin{pmatrix}\hat\cU_\tau\\\partial_z\hat\cU_\tau\end{pmatrix}
\end{align}
on $x_2=0$, $z=0$. We have
\begin{align}
\hat U_\tau=\hat U^h_\tau+\hat U^p_\tau, \text{ where }\hat U^h_\tau=\sum^4_{j=1}\hat\tau^*_j R_j,\;\;\hat U^p_\tau=\sum^4_{j=1}\hat \tau^p_j R_j\text{ on }x_2=0, z=0,
\end{align}
so \eqref{o20} becomes for $k>0$
\begin{align}\langle bel{o21}
C(\beta,k)\hat U^h_\tau=ik\cB_{Lop}\begin{pmatrix}\hat\tau^*_1\\ \hat\tau^*_2\end{pmatrix}=\hat g- C(\beta,k)\hat U^p_\tau\text{ on }x_2=0,z=0.
\end{align}
The matrix $\cB_{Lop}$ is singular, but the function $\hat w(t,x_1,k)$, which determines the traces of the $\hat\sigma_j$ on $x_2=0$ \eqref{o12}, was chosen precisely so that the right side of
\eqref{o21} lies in the range of $\cB_{Lop}$. Thus, we can solve for the $\hat \tau^*_j(t,x_1,0,k)$. We extend the $\hat\tau^*_j$ to $x_2\geq 0$ by setting
\begin{align}\langle bel{o21a}
\hat\tau^*_j(t,x_1,x_2,k)=\psi(x_2)\hat\tau^*_j(t,x_1,0,k)
\end{align}
for $\psi$ as in \eqref{o19a}.
This completes the construction of the $\hat\sigma_j$, $\hat\tau_j$. It remains to examine the regularity of these functions.
\textbf{5. Regularity of the $\hat\sigma_j$. }
We introduce the notation
\begin{align}
\hat H^s(t,x_1,k)=\{\hat u(t,x_1,k), \text{ where }u\in H^s(t,x_1,\theta)\},\;|\hat u|_{\hat H^s(t,x_1,k)}:=|\langle ngle k\rangle ngle^s\hat u(t,x_1,k)|_{L^2(k,H^s(t,x_1))}.
\end{align}
The space $\hat H^s(t,x,k)$ is defined similarly.
It follows from Proposition {\rm Re }\, f{propwellposed} that $\hat w(t,x_1,k)\in \hat H^s$ when $G\in H^{s}(t,x_1,\theta)$, provided $s>d+3$.
Thus, we have from \eqref{o12} and \eqref{o19a},
\begin{prop}
Assume $s>d+3$ and $G(t,x_1,\theta)\in H^{s}(t,x_1,\theta)$. Then
\begin{align}
\hat\sigma_j(t,x_1,0,k)\in \hat H^s(t,x_1,k)\text{ and }\hat\sigma_j(t,x_1,x_2,k)\in \hat H^s(t,x,k).
\end{align}
\end{prop}
\textbf{6. Regularity of the $\hat \tau_j$. }
\begin{prop}\langle bel{o23}
Assume $s>d+3$ and $G(t,x_1,\theta)\in H^{s}(t,x_1,\theta)$. We have
\begin{align}\langle bel{o23a}
\hat\tau_j(t,x,k,z)=\frac{1}{k^2}T_j(t,x,k,z)
\end{align}
where
\begin{align}
T_j(t,x,k,z)\in C_c(x_2;L^\infty(z,\hat H^{s-2}(t,x_1,k)))\cap C(x_2,z;\hat H^{s-2}(t,x_1,k)).
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }We take $j=1$ and using \eqref{o19aa} consider first
\begin{align}\langle bel{o23b}
\hat\tau^p_1(t,x,k,z)=\int^z_{+\infty}e^{ik\omega_1(z-s)}\hat F_1(t,x,k,s)ds,\; k<0
\end{align}
for $\hat F_1$ as in \eqref{o15a}. From \eqref{o15a} we see that $\hat F_1$ is a linear combination of terms of the form
\begin{align}\langle bel{o24}
\begin{split}
&(a)\frac{1}{k}\widehat{\partial\partial_\theta \sigma_j}, \\
&(b)\frac{1}{k}\widehat{\partial_\theta\sigma_m\partial_{\theta\theta}\sigma_n},
\end{split}
\end{align}
where $\sigma_j=\sigma_j(t,x,\theta+\omega_jz)$.
Let us consider first the case (b) when $m\neq n$. We then obtain for the corresponding term in $\hat\tau^p_1$ a scalar multiple of:
\begin{align}\langle bel{o24a}
\begin{split}
&\hat\tau^p_{mn}(t,x,k,z):= \frac{1}{k^2}T^p_{mn}(t,x,k,z)\text{ where }\\
&T^p_{mn}(t,x,k,z)=ke^{ik\omega_1z}\int_{+\infty}^z\int e^{-ik\omega_1s+i(k-k')\omega_ms+ik'\omega_n s}\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')dk'ds.\\
\end{split}
\end{align}
For a given $(x_2,z)$ the norm $|T^p_{mn}(t,x,k,z)|_{\hat H^{s-2}(t,x_1,k)}$ can be estimated by considering just the $L^2(k,H^{s-2}(t,x_1))$ and $L^2(t,x_1,\hat H^{s-2}(k))$ norms. Since $\mathrm{Im}\;(k-k')\omega_m\geq 0$ (resp. $\mathrm{Im}\;k'\omega_n\geq 0$) on $\mathrm{supp}\;\hat\sigma_m$ (resp. $\hat\sigma_n$), we obtain for the $H^{s-2}(t,x_1)$ norm:
\begin{align}\langle bel{o24aa}
\begin{split}
&|T^p_{mn}(t,x,k,z)|_{H^{s-2}(t,x_1)}\leq |k|\int^{+\infty}_z\int e^{ik\omega_1(z-s)}|\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')|_{H^{s-2}(t,x_1)}dk'ds=\\
&\qquad\qquad \frac{|k|}{ik\omega_1}\int |\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')|_{H^{s-2}(t,x_1)}dk',
\end{split}
\end{align}
which yields
\begin{align}\langle bel{o25}
|T^p_{mn}(t,x,k,z)|_{L^2(k,H^{s-2}(t,x_1))}\lesssim |\hat\sigma_m|_{\hat H^{s}(t,x_1,k)}|\hat\sigma_n|_{\hat H^{s}(t,x_1,k)}.
\end{align}
Here we have used a Moser estimate in the $(t,x_1)$ variables, and observed that, for example,
\begin{align}\langle bel{o25aa}
\begin{split}
&\left||\widehat{\partial_{\theta\theta}\sigma_n}(t,x_1,x_2,k)|_{L^\infty(t,x_1)}\right|_{L^1(k)}\lesssim \left| |\langle ngle k\rangle ngle^{\frac{1}{2}+\delta}\widehat{\partial_{\theta\theta}\sigma_n}(t,x_1,x_2,k)|_{H^{\frac{d}{2}+\delta}(t,x_1)}\right|_{L^2(k)}\lesssim\\
&\qquad\qquad |\hat\sigma_n|_{\hat H^{\frac{d}{2}+2+\frac{1}{2}+2\delta}(t,x_1,k)}.
\end{split}
\end{align}
The $L^2$ norm of the convolution in $k'$ can then be estimated by Young's inequality. The $L^2(t,x_1,\hat H^{s-2}(k))$ norm is estimated similarly, using
\begin{align}
\langle ngle k\rangle ngle^{s-2}\lesssim \langle ngle k-k'\rangle ngle^{s-2}+\langle ngle k'\rangle ngle^{s-2}
\end{align}
in place of the Moser estimate.
Continuity of $T^p_{mn}$ in $x_2$ is evident from the special $x_2$ dependence of the $\hat\sigma_j$ \eqref{o19a}, while continuity in $z$ follows from the dominated convergence theorem.
\textbf{2. }In the case $k>0$ we define $T^p_{mn}$ with $\int^z_{+\infty}...ds$ replaced by $\int^z_0...ds$, and find that $|T^p_{mn}(t,x,k,z)|_{H^{s-2}(t,x_1)}$ is again dominated by the right side of \eqref{o24aa}. The rest of the estimate goes as before. The estimates of the contributions to $\hat\tau^p_1$ in the case (b) when $m=n$ or case (a) are similar (or easier).
We observe that in the case \eqref{o24}(b) when $m=n=1$, the integral \eqref{o23b} is zero since $\hat\sigma_1(t,x,k)$ is supported in $k\geq 0$, so one just needs to estimate the
$\int^z_0\dots ds$ integral that defines $\hat\tau^p_1(t,x,k,z)$ when $k>0$. In this case the $ds$ integral produces a factor of $z$ (``secular growth").
Using the fact that
\begin{align}\langle bel{o25a}
\sup_{z\geq 0} ze^{-|k\omega_j|z}\thickapprox \frac{1}{|k|},
\end{align}
we see that this factor of $z$ has the same effect (introducing an extra factor of $1/|k|$) as the $ds$ integral in \eqref{o24aa}. Hence one obtains the same estimate for the contribution of \eqref{o24}(b) to $\hat\tau^p_1$ when $m=n=1$ as when $m\neq n$ or when $m=n\neq 1$.
\textbf{3. }From \eqref{o21} and the expression for $g$ \eqref{o20}, we see that the regularity of $\hat\tau^*_j(t,x_1,0,k)$, $j=1,2$ for $k>0$ is determined by the regularity of
\begin{align}
\frac{\widehat{\ell_s u_\sigma}}{k},\;\frac{\widehat{n(u_\sigma)}}{k},\; \frac{\widehat G}{k},\; \text{ and } \widehat U^p_\tau \text{ on }x_2=0,z=0.
\end{align}
The first three terms have the form $\frac{1}{k}h(t,x_1,k)$, where $h$ lies respectively in $\hat H^{s-1}$, $\hat H^{s-1}$, and $H^{s+1}$, while \eqref{o23a} for $\hat\tau^p_j$ implies
$k^2\hat U^p_\tau\in\hat H^{s-2}$ at $x_2=0,z=0$. With \eqref{o21a} we see that
\begin{align}
\hat\tau^h_j(t,x,k,z)=\hat\tau^*_j(t,x,k)e^{ik\omega_j z}
\end{align}
also satisfies \eqref{o23a}.
\end{proof}
\chapter{Error Analysis and proof of Theorem {\rm Re }\, f{approxthm}}\langle bel{chapter5}
\emph{\quad}In this chapter we show that the approximate solution is close in a precise sense to the exact solution constructed in Theorem {\rm Re }\, f{uniformexistence}.
We will use the notation and estimates of chapter {\rm Re }\, f{chapter3}, especially sections {\rm Re }\, f{b1a}, {\rm Re }\, f{uniform}, {\rm Re }\, f{nonlinear}, and {\rm Re }\, f{mainestimate}.
\section{Introduction}
The error analysis is performed on functions of $(t,x,\theta)$. We write
\begin{align}\langle bel{p0}
\begin{split}
&u^{\varepsilon}_a(t,x,\theta):={\varepsilon}^2 u^{\varepsilon}_\sigma(t,x,\theta)+{\varepsilon}^3u^{\varepsilon}_\tau(t,x,\theta),\text{ where }\\
&u^{\varepsilon}_\sigma(t,x,\theta):=u_\sigma(t,x,\theta,z)|_{z=\frac{x_2}{{\varepsilon}}}=\sum_{j=1}^4\left(\sigma_j(t,x,\theta+\omega_j z)|_{ z=\frac{x_2}{{\varepsilon}}}\right) r_j\text{ and }\\
&u^{\varepsilon}_\tau(t,x,\theta):=u_\tau(t,x,\theta,z)|_{z=\frac{x_2}{{\varepsilon}}} = \sum_{j=1}^4 \left(\chi_{\varepsilon}(D_\theta)\tau_j(t,x,\theta,z)|_{ z=\frac{x_2}{{\varepsilon}}}\right) r_j.
\end{split}
\end{align}
When $u_a^{\varepsilon}$ is plugged into the system \eqref{a5} we obtain
\begin{align}\langle bel{p0a}
\begin{split}
&\partial_{t,{\varepsilon}}^2 u_a^{\varepsilon}+\sum_{|\alpha|=2} A_\alpha({\nabla}bla_{\varepsilon} u^{\varepsilon}_a)\partial_{x,{\varepsilon}}^\alpha u_a^{\varepsilon}=F_a^{\varepsilon}:=F_a(t,x,\theta,\frac{x_2}{{\varepsilon}})\text{ on }\Omega_{T_1}\\
&h({\nabla}bla_{\varepsilon} u^{\varepsilon}_a)=g_a\text{ or }\partial _{x_2} u_a^{\varepsilon}=H(\partial_{x_1,{\varepsilon}} u_a^{\varepsilon}, g_a)\text{ on }x_2=0\\
\end{split}
\end{align}
for some $T_1>0$ and $F_a$, $g_a$ as in \eqref{fa}, \eqref{ba}. Here $u^{\varepsilon}_a$ is zero in $t<0$.
\begin{rem}\langle bel{real}
The functions $u^{\varepsilon}_\sigma$ and $u^{\varepsilon}_\tau$ must be constructed to be real-valued. The analysis of Chapter {\rm Re }\, f{chapter2} shows that the amplitude $w(t,x_1,\theta)$ is real-valued. The fact that the $\omega_j$ and $r_j$ occur in complex conjugate pairs permits one to construct $u^{\varepsilon}_\sigma$ as a real-valued function. Since the function $\chi(s)$ giving the low frequency cutoff in \eqref{p0} can be chosen to be an even function, one can similarly construct $u^{\varepsilon}_\tau$ to be real-valued for the same reason. This remark is used in solving the systems \eqref{p1}-\eqref{p3}.
\end{rem}
\subsection{Extension of approximate solutions to the whole space}
\emph{\quad} Recall that the estimates of $(v^{\varepsilon},{\nabla}bla u^{\varepsilon})$ leading to the proof of Theorem {\rm Re }\, f{uniformexistence} had to be performed initially on the whole half-space $\Omega$. We estimated functions $(v^{\varepsilon},{\nabla}bla u^{\varepsilon})$ on $\Omega$ that were solutions to the modified, singular \emph{linear} systems \eqref{c1}-\eqref{c3}, whose restrictions to $\Omega_T$ for $0<T<T_{\varepsilon}$ coincided (by causality, Remark \eqref{k2y}) with solutions provided by Theorem {\rm Re }\, f{localex} to the nonlinear singular problems \eqref{a7}-\eqref{a9}.
Similarly, the estimates in the error analysis must be done on the whole space. However, the approximate solutions $u^{\varepsilon}_a$ and $v^{\varepsilon}_a:={\nabla}bla_{\varepsilon} u^{\varepsilon}_a$ constructed in Chapter {\rm Re }\, f{chapter4} are defined just on $(-\infty, T_1]$ for some $ T_1>0$. Thus, we must extend these functions to the whole half-space in order to compare them to the functions $(v^{\varepsilon}, u^{\varepsilon})$. The form of the modified systems \eqref{c1}-\eqref{c3} suggests that we will need two kinds of extensions. First, we will need Seeley extensions $v^s_a$ to $\Omega$ of $v^{\varepsilon}_a|_{\Omega_T}$ for $0<T\leq T_1$, that we can hope to prove are close in the $E_{m,T}$ norm to the Seeley extensions $v^s$ appearing in the coefficients of the systems \eqref{c1}-\eqref{c3}. These Seeley extensions are good approximate solutions only for $t\leq T$; for later times they are useless as approximate solutions to the modified systems \eqref{c1}-\eqref{c3}. Thus, we also need extensions to $\Omega$ of $v^{\varepsilon}_a|_{\Omega_T}$ $u^{\varepsilon}_a|_{\Omega_T}$, that we can hope to prove are close in the $E_{m,\gamma}$ norm to the solutions on $\Omega$ of the modified systems \eqref{c1}-\eqref{c3}. These extensions will be constructed as solutions on $\Omega$ of the systems \eqref{p1}-\eqref{p3} below, which should be viewed as approximations to the systems \eqref{c1}-\eqref{c3}. We now explain how to obtain these extensions in more detail.
The first step is to extend the error terms. With $g(t,x_1,\theta)={\varepsilon}^2G$ as before,
we have
\begin{align}\langle bel{p0b}
\begin{split}
&F_a(t,x,\theta)=\cF(u_\sigma(t,x,\theta,z),u_\tau(t,x,\theta,z)|_{z=\frac{x_2}{{\varepsilon}}}\\
&g_a(t,x_1,\theta)=g(t,x_1,\theta)+h_a(t,x_1,\theta), \text{ where }h_a(t,x_1,\theta)=\cH(u_\sigma(t,x_1,0,\theta,0), u_\tau(t,x_1,0,\theta,0)),
\end{split}
\end{align}
where $\cF$ and $\cH$ are functions that may be read off from the formulas \eqref{fa}, \eqref{ba}.\footnote{The arguments of $\cF$ and $\cG$ should also involve derivatives of $(u_\sigma,u_\tau)$, but we have suppressed these in the notation.}
The functions $u_\sigma(t,x,\theta,z)$, $u_\tau(t,x,\theta,z)$, and $u_a(t,x,\theta,z)$ are built out of the component functions $\sigma_j(t,x,\theta)$, $j=1,\dots,4$.
For any $T$ satisfying $0<T\leq T_1$ we let $\sigma^s_{j,T}$ denote a Seeley extension of $\sigma_j|_{\Omega_T}$ to $\Omega$ defined as in \eqref{c0gg}, and we denote by
$u_\sigma^s$, the function of $(t,x,\theta,z)$ built out of the extended $\sigma_j$. This is the same as the Seeley extension to $\Omega$ of $u_\sigma|_{\Omega_T}$. Next we define
$u_\tau^s(t,x,\theta,z)$ to be the Seeley extension of $u_\tau|_{\Omega_T}$. This is \emph{not} the same as the extension obtained by replacing $\sigma_j$ by $\sigma_{j,T}^s$ in the definition of $u_\tau$.\footnote{The Seeley extension of a product is not the same as the product of the Seeley extensions.} We set
\begin{align}
u^s_a={\varepsilon}^2u^s_\sigma+{\varepsilon}^3u^s_\tau \text{ and }v_a^s(t,x,\theta,z)=D_{\varepsilon} u^s_a(t,x,\theta,z)
\end{align}
for $D_{\varepsilon}$ as in \eqref{o2aa} and observe that
\begin{align}
{\nabla}bla_{\varepsilon} u_a^{s,{\varepsilon}} (t,x,\theta)=v^{s,{\varepsilon}}_a(t,x,\theta).
\end{align}
Recalling that $g$ has already been extended, we now define (using superscript $e$ for ``extension")
\begin{align}\langle bel{p0bb}
\begin{split}
&F^e_a(t,x,\theta)=\cF(u^s_\sigma(t,x,\theta,z),u^s_\tau(t,x,\theta,z)|_{z=\frac{x_2}{{\varepsilon}}}\\
&g^e_a(t,x_1,\theta)=g(t,x_1,\theta)+h^e_a(t,x_1,\theta), \text{ where }h^e_a(t,x_1,\theta)=\cH(u^s_\sigma(t,x_1,0,\theta,0), u^s_\tau(t,x_1,0,\theta,0)).
\end{split}
\end{align}
We need additional extensions of $v_a^{\varepsilon}|_{\Omega_T}$ and $u^{\varepsilon}_a|_{\Omega_T}$ to $\Omega$ that are obtained by solving on $\Omega$ the following three {linear} systems for the respective unknowns $v^{\varepsilon}_{1a}$, $v^{\varepsilon}_{2a}$, and $u^{\varepsilon}_a$. In these systems all functions have arguments $(t,x,\theta)$ and we suppress superscripts ${\varepsilon}ilon$; as usual, subscripts $T$ are suppressed on Seeley extensions.
\begin{align}\langle bel{p1}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 v_{1a}+\sum_{|\alpha|=2} A_\alpha(v^s_a)\partial_{x,{\varepsilon}}^\alpha v_{1a}=\\
&\qquad -\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_1,{\varepsilon}}(A_\alpha(v^s_a))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v_{1a}^s-\partial_{x_1,{\varepsilon}}(A_{(0,2)}(v_a^s))\partial_{x_2}v_{2a}^s\right]+\partial_{x_1,{\varepsilon}}F^e_a,\\
&(b)\partial _{x_2} v_{1a}-d_{v_1}H(v^s_{1a},h(v^s_a))\partial_{x_1,{\varepsilon}}v_{1a}=d_gH(v^s_{1a},g^e_a)\partial_{x_1,{\varepsilon}}(g^e_a)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{p2}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 v_{2a}+\sum_{|\alpha|=2} A_\alpha(v^s_a)\partial_{x,{\varepsilon}}^\alpha v_{2a}=\\
&\qquad -\left[\sum_{|\alpha|=2,\alpha_1\geq 1}\partial_{x_2}(A_\alpha(v^s_a))\partial_{x_1,{\varepsilon}}^{\alpha_1-1}\partial_{x_2}^{\alpha_2}v^s_{1a}-\partial_{x_2}(A_{(0,2)}(v^s_a))\partial_{x_2}v^s_{2a}\right]+\partial_{x_2}F^e_a,\\
&(b)v_{2a}=\chi_0(t)H(v_{1a},g^e_a)\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{p3}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 u_a+\sum_{|\alpha|=2} A_\alpha(v^s_a)\partial_{x,{\varepsilon}}^\alpha u_a=F_a^e\\\
&(b)\partial _{x_2} u_a-d_{v_1}H(v^s_{1a},h(v^s_a))\partial_{x_1,{\varepsilon}}u_a=H(v^s_{1a},g_a^e)-d_{v_1}H(v^s_{1a},g^e_a)v^s_{1a}\text{ on }x_2=0.
\end{split}
\end{align}
In \eqref{p2} $\chi_0(t)\geq 0$ is the \emph{same} $C^\infty$ function (that is equal to 1 on a neighborhood of $[0,1]$ and supported in $(-1,2)$) as in \eqref{c2}.
The systems \eqref{p1}-\eqref{p3} for $(v_a,u_a)$ should be compared to the systems \eqref{c1}-\eqref{c3} for $(v,u)$. In view of the smallness of $F_a^e$ and $g^e_a-g$ and the expected, but still unproved, smallness of $v^s-v^s_a$, we can hope to show that $(v_a,u_a)$ is close to $(v,u)$ in the $E_{m,\gamma}$ norm on $\Omega$. We carry out this strategy in the remainder of this chapter by studying the error equations computed below.
\begin{rem}\langle bel{p3a}
1. The above three systems are solved on the full domain $\Omega$. Each system depends on the parameters ${\varepsilon}$ and $T$, which are usually suppressed in our notation.
2. Since the underlying linearized problem corresponding to both \eqref{p1} and \eqref{p3} is just weakly stable, it is important that the functions $v^s_a$ appearing in the coefficients here be real-valued. That is so as a consequence of remark {\rm Re }\, f{real}.
3. The definitions of $v^s_a=v^s_{a,T}$, $F^e_a$, $g^e_a$ and causality (see remark \eqref{k2y}) imply that the solutions $v_{1a}$, $v_{2a}$, and $u_a$ of \eqref{p1}, \eqref{p2}, and \eqref{p3} are equal to the already constructed, similarly denoted functions on $\Omega_T$. The various Seeley extensions depend on $T$, so the solutions
$v_a$ and $u_a$ change as $T$ changes.
\end{rem}
\subsection{Error equations}
\emph{\quad}By subtracting the equations \eqref{p1}, \eqref{p2}, \eqref{p3} from the equations \eqref{c1}, \eqref{c2}, \eqref{c3} we obtain the following equations on $\Omega$ for the error functions
\begin{align}\langle bel{p3b}
w=(w_1,w_2)=v-v_a \text{ and } z=u-u_a.
\end{align}
With slight abuse let us write the terms in brackets on the right sides of \eqref{p1}(a) and \eqref{p2}(a) respectively as $b_j(v^s_a)d_{\varepsilon} v^s_ad_{\varepsilon} v^s_a$, $j=1,2$ and similarly write the corresponding terms in \eqref{c1} and \eqref{c2} as $b_j(v^s)d_{\varepsilon} v^sd_{\varepsilon} v^s$, $j=1,2$.
\begin{align}\langle bel{p4}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 w_{1}+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha w_{1}=\\
&-\sum_{|\alpha|=2}(A_\alpha(v^s)-A_\alpha(v^s_a))\partial_{x,{\varepsilon}}^\alpha v_{1a}+b_1(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s -b_1(v^s_a)d_{\varepsilon} v^s_a d_{\varepsilon} v^s_a -\partial_{x_1,{\varepsilon}}F^e_a,\\
&(b)\partial _{x_2} w_{1}-d_{v_1}H(v^s_{1},h(v^s))\partial_{x_1,{\varepsilon}}w_{1}=-[d_{v_1}H(v^s_1,h(v^s))-d_{v_1}H(v^s_{1a},h(v^s_a))]\partial_{x_1,{\varepsilon}}v_{1a}+\\
& \qquad \qquad\qquad d_gH(v^s_{1},g)\partial_{x_1,{\varepsilon}}g - d_gH(v^s_{1a},g^e_a)\partial_{x_1,{\varepsilon}} g^e_a \text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{p5}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 w_{2}+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha w_{2}=\\
&-\sum_{|\alpha|=2}(A_\alpha(v^s)-A_\alpha(v^s_a))\partial_{x,{\varepsilon}}^\alpha v_{2a}+b_2(v^s)d_{\varepsilon} v^s d_{\varepsilon} v^s -b_2(v^s_a)d_{\varepsilon} v^s_a d_{\varepsilon} v^s_a -\partial_{x_2}F^e_a,\\
&(b)w_2=\chi_0(t)[H(v_{1},g) - H(v_{1a},g^e_a)]\text{ on }x_2=0.
\end{split}
\end{align}
\begin{align}\langle bel{p6}
\begin{split}
&(a)\partial_{t,{\varepsilon}}^2 z+\sum_{|\alpha|=2} A_\alpha(v^s)\partial_{x,{\varepsilon}}^\alpha z=
-\sum_{|\alpha|=2}(A_\alpha(v^s)-A_\alpha(v^s_a))\partial_{x,{\varepsilon}}^\alpha u_{a}-F^e_a,\\
&(b)\partial _{x_2} z-d_{v_1}H(v^s_{1},h(v^s))\partial_{x_1,{\varepsilon}}z=H(v^s_{1},g)-d_{v_1}H(v^s_{1},g)v^s_{1}\\
&-[H(v^s_{1a},g_a^e)-d_{v_1}H(v^s_{1a},g^e_a)v^s_{1a}]+[d_{v_1}H(v^s_1,h(v^s))-d_{v_1}H(v^s_{1a},h(v^s_a))]v_{1a}\text{ on }x_2=0.
\end{split}
\end{align}
\begin{rem}
1. We note again that these equations depend on ${\varepsilon}$ and $T$ as parameters, and so the same is true of the solutions. Just like the trios \eqref{c1}-\eqref{c3} and \eqref{p1}-\eqref{p3}, the equations \eqref{p4}-\eqref{p6} must be estimated simultaneously.
2. Smallness of $F^e_a$ and $h^e_a$ in appropriate norms on $\Omega$ will follow, provided one has smallness of $F_a$ and $h_a$ in the corresponding time-localized norms on $\Omega_T$, from continuity properties of Seeley extensions, and the rules of section {\rm Re }\, f{nonlinear} for computing norms of nonlinear functions of $u$ in terms of norms of $u$.
3. The smallness of $F^e_a$ and $h^e_a$ will imply the smallness of $(w,{\nabla}bla_{\varepsilon} z)$ on $\Omega_{T_2}$ for $T_2$ small enough, but independent of ${\varepsilon}$.
\end{rem}
The error analysis will be accomplished by estimating the error systems \eqref{p4}-\eqref{p6} on $\Omega$ in the $E_{m,\gamma}$ norm.
Since the right sides of these systems include terms that contain factors given by derivatives of $(v_a,u_a)$, we shall also need to estimate the approximate solution systems \eqref{p1}-\eqref{p3} in the $E_{m,\gamma}$ norm.
The next two sections are devoted to estimating the functions $F^e_a$ and $h^e_a=g^e_a-g$ that determine the size of the forcing in the approximate solution and error systems. Each of these functions is constructed from the building blocks $u^s_\sigma$ and $u^s_\tau$, so the first step is to estimate these components in a variety of singular norms; the estimates of $F^e_a$ and $h^e_a$ then follow by applying the results of section {\rm Re }\, f{nonlinear}.
Having estimates of $F^e_a$ and $h^e_a$, we estimate $(v_a,u_a)$ in section {\rm Re }\, f{estapprox}, and finally estimate the error $(w,z)$ in section {\rm Re }\, f{end}. The estimates of section {\rm Re }\, f{end} have much in common with the estimates of section {\rm Re }\, f{mainestimate}, the main difference being that new kinds of forcing terms such as, for example,
$$
-\sum_{|\alpha|=2}(A_\alpha(v^s)-A_\alpha(v^s_a))\partial_{x,{\varepsilon}}^\alpha v_{1a} \text{ in }\eqref{p4}
$$
are encountered.
\section{Building block estimates}\langle bel{bblock}
\emph{\quad} In these estimates the functions being estimated are evaluated at $(t,x_1,x_2,\theta,\frac{x_2}{{\varepsilon}})$ \emph{after} the indicated derivatives are taken.
Recall that
\begin{align}
u_\sigma(t,x,\theta,z)=\sum_{j=1}^4\sigma_j(t,x,\theta+\omega_j z) r_j\text{ and }u_\tau(t,x,\theta,z)= \sum_{j=1}^4 \left(\chi_{\varepsilon}(D_\theta)\tau_j(t,x,\theta,z)\right) r_j.
\end{align}
The functions $u_\sigma$ (resp., $u_\tau$) are built out of the constituent functions $\sigma_j(t,x,\theta)$, $j=1,...,4$ (resp., the $\sigma_j$ and $G(t,x_1,\theta)$).
The notation $C(m+r)$ indicates a constant that depends on norms
$$
\sup_{x_2\geq 0}|\sigma_j(t,x_1,x_2,\theta)|_{H^{m+r}(t,x_1,\theta)}\text{ or }|G(t,x_1,\theta)|_{H^{m+r}(t,x_1,\theta)}
$$
of the constituent functions.\footnote{The index $m+r$ can often be reduced in the case of $G$ norms, but we wish to lighten the notation by not indicating this.} In view of the special way $x_2$ dependence enters into the definitions of $u_\sigma$ and $u_\tau$ (recall \eqref{o19a}, the estimates take exactly the same form if the $\langle ngle \cdot\rangle ngle_{m,\gamma}$ norms on the left are replaced by
$|\cdot |_{0,m,\gamma}$ or $|\cdot|_{\infty,m,\gamma}$ norms.
The constant $p>0$ is the one that appears in the low-frequency cutoff $\chi_{\varepsilon}(k)=\chi(\frac{k}{p})|_{p={\varepsilon}^b}$, for $b>0$ to be chosen. We recall that $\chi\geq 0$ and
$\mathrm{supp}\;\chi(k/p)\subset \{k:|k|\geq p\}$.
We will sometimes use $\partial_f$ to represent $\partial_\theta$ or $\partial_z$ and $\partial_s$ to represent $\partial_t$, $\partial_{x_1}$, or $\partial_{x_2}$.
\begin{rem}\langle bel{qq1}
In the estimates of this section we write $u_\sigma$, $u_\tau$ to indicate the Seeley extensions $u^s_{\sigma,T}$, $u^s_{\tau,T}$. Recall that the Seeley extension of $u_{\tau,T}$ is not the same as the extension of $u_{\tau,T}$ obtained by replacing the constituent functions $\sigma_{j,T}$ by their Seeley extensions. Nevertheless, we need to estimate $u^s_{\tau,T}$ in terms of the $\sigma^s_{j,T}$. This is justified by noting that, for example,
\begin{align}
\langle ngle\Lambda^{\frac{1}{2}}\partial_f u^s_{\tau,T}\rangle ngle_{m,\gamma}\leq C \langle ngle\Lambda^{\frac{1}{2}}\partial_f u^s_{\tau,T}\rangle ngle_{m,T,\gamma}
\end{align}
and the time-localized norm on the right can be realized as an \emph{infimum} over norms of extensions of $u_{\tau,T}$ to the full-half space.
\end{rem}
We begin with estimates of $u_\sigma$.
\begin{prop}\langle bel{qq0}
Let $m\geq 0$.
\begin{align}
\begin{split}
&\langle ngle\Lambda^{\frac{k}{2}}\partial_{f } u_\sigma\rangle ngle_{m,\gamma}\lesssim \frac{C(m+1+\frac{k}{2})}{{\varepsilon}^{k/2}},\; k=0,1,2,3,4\\
&\langle ngle\Lambda^{\frac{k}{2}}\partial_{ff} u_\sigma\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2+\frac{k}{2})}{{\varepsilon}^{k/2}},\;k=0,1,2,3\\
&\langle ngle\Lambda^{\frac{k}{2}}\partial_{s } u_\sigma\rangle ngle_{m,\gamma}\lesssim \frac{C(m+1+\frac{k}{2})}{{\varepsilon}^{k/2}},\; k=0,1,2,3,4.
\end{split}
\end{align}
\end{prop}
\begin{proof}
Since $|X,\gamma |\lesssim |\xi',\gamma |+|\frac{k}{{\varepsilon}}|$, we can for $r\geq 0$ estimate norms of $\Lambda_D^r w$ by estimating pieces corresponding to
$ |\xi',\gamma |^r$ and $|\frac{k}{{\varepsilon}}|^r$. Let us write the operators associated to these pieces as $\Lambda_{x',D}^r$ and $\Lambda_{\theta,D}^r$ respectively.
As in the proof of Proposition {\rm Re }\, f{o23} we will use
\begin{align}\langle bel{qq2}
|u(t,x_1,k)|_{\hat H^m(t,x_1,k)}\lesssim |u(t,x_1,k)|_{L^2(k, H^m(t,x_1))}+|u(t,x_1,k)|_{L^2(t,x_1,\hat H^m(k))}.
\end{align}
Given the form \eqref{p0} of $u_\sigma$ and recalling that
\begin{align}\langle bel{qqq2}
\mathrm{supp}\;\hat\sigma_j(t,x,k)\subset\{k: k\mathrm{Im}\;\omega_j\geq 0\},
\end{align}
the estimates are immediate.
\end{proof}
The next proposition gives estimates of $\partial_f u_\tau$.
\begin{prop}Let $m>\frac{d+1}{2}$.
\begin{align}
\begin{split}
&\langle ngle\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim C(m+2)/p\\
&\langle ngle\Lambda^{\frac{1}{2}}\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2)}{{\varepsilon}^{1/2}p^{1/2}}+\frac{C(m+2+\frac{1}{2})}{p}\\
&\langle ngle\Lambda\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2)}{{\varepsilon}}+\frac{C(m+3)}{p}\\
&\langle ngle\Lambda^{\frac{3}{2}}\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2+\frac{1}{2})}{{\varepsilon}^{3/2}}+\frac{C(m+3+\frac{1}{2})}{p}\\
&\langle ngle\Lambda^{2}\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^2}+\frac{C(m+4)}{p}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }We give the proof of the second estimate, which is typical of the rest.
Consider, for example, $\langle ngle\Lambda^{1/2}\partial_\theta u_\tau\rangle ngle_{m,\gamma}$ and in particular $\langle ngle\Lambda^{1/2}\partial_\theta \;\chi_{\varepsilon}(D_\theta)\tau^p_1\rangle ngle_{m,\gamma}$.
We estimate first the part of $\tau^p_1$ considered in \eqref{o24a}, where $k<0$ and $m\neq n$:
Recall
\begin{align}\langle bel{qq4}
\begin{split}
&\hat\tau^p_{mn}(t,x,k,z)=e^{ik\omega_1z}\int_{+\infty}^z\int e^{-ik\omega_1s+i(k-k')\omega_ms+ik'\omega_n s}\frac{1}{k}\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')dk'ds:=\\
&\qquad \int^z_{+\infty}e^{ik\omega_1(z-s)}\hat F_{mn}(t,x,k,s)ds.
\end{split}
\end{align}
Here we will suppress some expressions involving $\gamma$ such as factors of $e^{-\gamma t}$.
Using \eqref{qq2} and \eqref{qqq2} we estimate $\langle ngle\Lambda^{1/2}_\theta\partial_\theta \;\chi_{\varepsilon}(D_\theta) \tau^p_{mn}\rangle ngle_{m,\gamma}$ as follows:
\begin{align}\langle bel{qq3}
\begin{split}
&\left|\frac{|k|^{3/2}}{{\varepsilon}^{1/2}}\;\chi(k/p)\int^{z}_{+\infty}\int e^{ik\omega_1(z-s)}| \frac{1}{k}\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')|_{H^{m}(t,x_1)}dk'ds\right|_{L^2(k)}\lesssim\\
&\left|\frac{1}{{\varepsilon}^{1/2}p^{1/2}}\;\int |\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')|_{H^{m}(t,x_1)}dk'\right|_{L^2(k)}\lesssim \frac{C(m+2)}{{\varepsilon}^{1/2}p^{1/2}}.
\end{split}
\end{align}
Here we have taken account of the factor of $1/k$ introduced by the $ds$ integral, and have
estimated the convolution in $k$ using the observation \eqref{o25aa} and Young's inequality.
When $\Lambda^{1/2}_{\theta,D}$ is replaced by $\Lambda^{1/2}_{x',D}$ a parallel argument shows that the right side of \eqref{qq3} is replaced by $\frac{C(m+2+\frac{1}{2})}{p}$.
The $L^2(t,x_1,\hat H^m(k))$ norm is estimated similarly after writing $\langle ngle k\rangle ngle^{1/2}\lesssim \langle ngle k-k'\rangle ngle^{1/2}+\langle ngle k'\rangle ngle^{1/2}$.
When $k>0$ the integral $\int^z_{+\infty}\dots ds$ in \eqref{qq3} is replaced by $\int^z_0\dots ds$, but the resulting estimate is the same. The cases $m=n$ also yield the same estimate\footnote{One actually gets a better estimate in this case, since $\partial_\theta\sigma_m\partial_{\theta\theta}\sigma_m=\frac{1}{2}\partial_\theta(\partial_\theta\sigma_m)^2$.}; this is true even in the case $m=n=1$ leading to secular growth for the reason given in step 2 of the proof of Proposition {\rm Re }\, f{o23}.
As in step 3 of the proof of Proposition {\rm Re }\, f{o23}, we find that $\tau^h_1$ satisfies the same estimate.
\textbf{2. }To estimate $\langle ngle\Lambda^{1/2}\partial_z\; \chi_{\varepsilon}(D_\theta)\tau^p_1\rangle ngle_{m,\gamma}$ we first use \eqref{qq4} to write
\begin{align}\langle bel{qq5}
\partial_z\hat\tau^p_{mn}(t,x,k,z)=\hat F_{mn}(t,x,k,z)+\int^z_{+\infty}ik\omega_1 e^{ik\omega_1(z-s)}\hat F_{mn}(t,x,k,z)ds.
\end{align}
The contribution of the second term can be estimated exactly as the corresponding term in step 1. Moreover, since the $ds$ integral introduces a factor of $1/k$ it is clear that the contribution of the first term satisfies the same estimate. The remaining details are straightforward.
\end{proof}
Next we consider estimates of $\partial_{ff}u_\tau$.
\begin{prop}\langle bel{qq6} Let $m>\frac{d+1}{2}$.
\begin{align}
\begin{split}
&\langle ngle\Lambda^{k/2}\partial_{\theta\theta} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2+\frac{k}{2})}{{\varepsilon}^{k/2}}, k=0,1,\dots,3\\
&\langle ngle\partial_{zz} u_\tau\rangle ngle_{m,\gamma}\lesssim C(m+3)/p\\
&\langle ngle\Lambda^{\frac{1}{2}}\partial_{zz} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^{1/2}p^{1/2}}+\frac{C(m+3+\frac{1}{2})}{p}\\
&\langle ngle\Lambda^{\frac{3}{2}}\partial_{zz} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3+\frac{1}{2})}{{\varepsilon}^{3/2}}+\frac{C(m+4+\frac{1}{2})}{p}\\
&\langle ngle\partial_{\theta z}u_\tau\rangle ngle_{m,\gamma}\lesssim C(m+2)\\
&\langle ngle\Lambda^{\frac{1}{2}}\partial_{\theta z} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2+\frac{1}{2})}{{\varepsilon}^{1/2}}\\
&\langle ngle\Lambda^{\frac{3}{2}}\partial_{\theta z} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3+\frac{1}{2})}{{\varepsilon}^{3/2}}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
The $\partial_{\theta\theta}$ estimates are obvious. For the estimates involving $\partial_{zz}\hat\tau^p_{mn}$ for $\hat\tau^p_{mn}$ as in \eqref{qq4}, we use \eqref{qq5} to obtain
\begin{align}
\begin{split}
&\partial_{zz}\hat\tau^p_{mn}=\int [i(k-k')\omega_m+ik'\omega_n ]e^{i(k-k')\omega_mz+ik'\omega_n z}\frac{1}{k}\widehat{\partial_\theta\sigma_m}(t,x,k-k')\widehat{\partial_{\theta\theta}\sigma_n}(t,x,k')dk+\\
&ik\omega_1\hat F_{mn}(t,x,k,z)+\int^z_{+\infty}(ik\omega_1)^2e^{ik\omega_1(z-s)}\hat F_{mn}(t,x,k,s)ds=A+B+C.
\end{split}
\end{align}
The term $C$ is estimated exactly like $\partial_{\theta\theta}\hat\tau^p_{mn}$, and $B$ satisfies the same estimates (since the $ds$ integral in $C$ introduces a factor of $1/k$). One shows by arguments already used in this section that in each $\partial_{zz}$ estimate the contribution of $A$ is dominated by the terms that appear on the right in Proposition {\rm Re }\, f{qq6}. The $\partial_{\theta z}$ estimates are similar but simpler.
\end{proof}
Finally, we have estimates of $\partial_s u_\tau$.
\begin{prop}Let $m>\frac{d+1}{2}$.
\begin{align}\langle bel{q6}
\begin{split}
&\langle ngle\partial_s u_\tau\rangle ngle_{m,\gamma}\lesssim C(m+3)/p^2\\
&\langle ngle\Lambda^{\frac{1}{2}}\partial_s u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^{1/2}p^{3/2}}+\frac{C(m+3+\frac{1}{2})}{p^2}\\
&\langle ngle\Lambda\partial_{s} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon} p}+\frac{C(m+4)}{p^2}\\
&\langle ngle\Lambda^{\frac{3}{2}}\partial_s u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^{3/2}p^{1/2}}+\frac{C(m+4+\frac{1}{2})}{p^2}\\
&\langle ngle\Lambda^2\partial_{s} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^2}+\frac{C(m+5)}{p^2}
\end{split}
\end{align}
\end{prop}
\begin{proof}
We omit the details of the proof, since it uses only the arguments given earlier in this section.
\end{proof}
\begin{rem}\langle bel{qq7}
Observe that $\partial_{sf}$ (resp. $\partial_{ss}$) estimates of $u_\sigma$ or $u_\tau$ can be obtained immediately from $\partial_f$ (resp. $\partial_s$) estimates by increasing the argument of $C(\cdot)$ by $1$ for each $C(\cdot)$ that appears. For example,
\begin{align}
\langle ngle\Lambda^{\frac{1}{2}}\partial_f u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+2)}{{\varepsilon}^{1/2}p^{1/2}}+\frac{C(m+2+\frac{1}{2})}{p}\text{ implies }\langle ngle\Lambda^{\frac{1}{2}}\partial_{sf} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^{1/2}p^{1/2}}+\frac{C(m+3+\frac{1}{2})}{p}.
\end{align}
Similarly, estimates of $u_\sigma$ or $u_\tau$ (without the $\partial_s$) can be obtained by decreasing the argument of $C(\cdot)$ in $\partial_s$ estimates.
\end{rem}
\section{Forcing estimates}\langle bel{forcing}
\emph{\qquad}\textbf{Interior forcing. }Here we give estimates of the terms appearing in $F_a^e(t,x,\theta)$ \eqref{p0bb}. The estimates starting in Proposition {\rm Re }\, f{r3} are based on the assumption that $1\geq p\geq {\varepsilon}^{\frac{1}{2}-\delta}$ for some $\delta>0$. In that case the building block estimates show that any given norm of $u_\sigma$ is always bigger than (i.e., satisfies an estimate with a bigger right hand side) than the same norm of ${\varepsilon} u_\tau$.\footnote{When constructing profiles it is not uncommon, but more importantly not circular, to assume at one stage that a profile has a given form, and then later to construct a profile with that form.} For example,
\begin{align}
\begin{split}
&\langle ngle\Lambda^{3/2}\partial_{zz}u_\sigma\rangle ngle_{m,\gamma}\lesssim \frac{1}{{\varepsilon}^{3/2}},\text{ while }\langle ngle\Lambda^{3/2}{\varepsilon}\partial_{zz}u_\tau\rangle ngle_{m,\gamma}\lesssim {\varepsilon}\left(\frac{1}{{\varepsilon}^{3/2}}+\frac{1}{p}\right)\\
&\langle ngle \partial_s u_\sigma\rangle ngle_{m,\gamma}\lesssim 1, \text{ while }\langle ngle\partial_s \;{\varepsilon} u_\tau\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}}{p^2}.
\end{split}
\end{align}
This observation allows us to greatly reduce the number of estimates that have to be done in estimating $F^e_a$ or $g^e_a$.
We have the following estimates for the ``low frequency cutoff" error in \eqref{fa}.
\begin{prop}\langle bel{r1}
Let $m>\frac{d+1}{2}$.
\begin{align}\langle bel{rr0}
\begin{split}
&(a)\;|\frac{1}{{\varepsilon}}\Lambda^{1/2}[{\varepsilon}(1-\chi_{\varepsilon}(D_\theta))(L_{fs}u_\sigma+N(u_\sigma))]|_{0,m,\gamma}\leq C(m+1)\frac{p^3}{{\varepsilon}^{1/2}}+ C(m+2+\frac{1}{2})p^{1/2}\\
&(b)\;|\Lambda^{3/2}[{\varepsilon}(1-\chi_{\varepsilon}(D_\theta))(L_{fs}u_\sigma+N(u_\sigma))]|_{0,m,\gamma}\leq C(m+2)\frac{p^2}{{\varepsilon}^{1/2}}+ C(m+3+\frac{1}{2}){\varepsilon}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. Preliminaries. }Since $|X,\gamma |\lesssim |\xi',\gamma |+|\frac{k}{{\varepsilon}}|$, we can for $r\geq 0$ estimate norms of $\Lambda_D^r w$ by estimating pieces corresponding to
$ |\xi',\gamma |^r$ and $|\frac{k}{{\varepsilon}}|^r$, thereby taking advantage of the helpful factor $|k|^r$ which is $\lesssim p^r$ on $\mathrm{supp }(1-\chi(k/p))\subset [-p,p]$.
We will again use \eqref{qq2} and we
recall from \eqref{o19a} that
\begin{align}\langle bel{neglect}
\hat\sigma_j(t,x_1,x_2,k)=\psi(x_2)\hat\sigma_j(t,x_1,0,k), \text{ where }\psi\in C^\infty_c \text{ and }\psi=1 \text{ near }x_2=0.
\end{align}
Below it will be convenient to suppress some arguments $(t,x_1,x_2)$ as well as some expressions involving $\gamma$ such as factors of $e^{-\gamma t}$.
\textbf{2. Part (a).} To estimate the $L^2(x_2,L^2(k, H^m(t,x_1)))$ norm of the term involving $N(u_\sigma)$ we consider first
\begin{align}\langle bel{rr2}
\left|\frac{\sqrt{|k|}}{\sqrt{{\varepsilon}}}(1-\chi(k/p))\int |\widehat{\partial_\theta\sigma}_m(k-k')|_{H^m(t,x_1)}e^{i(k-k')\omega_m\frac{x_2}{{\varepsilon}}}|\widehat{\partial_{\theta\theta}\sigma}_n(k')|_{H^m(t,x_1)}e^{ik'\omega_n\frac{x_2}{{\varepsilon}}}dk'\right|_{L^2(k,x_2)}:= A,
\end{align}
and recall that the support properties of the $\hat\sigma_j$ imply both $i(k-k')\omega_m\frac{x_2}{{\varepsilon}}\leq 0$ and $ik'\omega_n\frac{x_2}{{\varepsilon}}\leq 0$ on the support of the integrand in \eqref{rr2}. We
break up the $k'$ integral into pieces on $|k'|\leq p$, $|k'|\geq p$ that we call $A_1$, $A_2$ respectively. To estimate $A_1$ we ignore the exponentials (which are $\leq 1$),
write
\begin{align}
|\widehat{\partial_{\theta\theta}\sigma_n}(k')|=|k'^2\hat\sigma_n(k')|\leq p^2|\hat\sigma_n(k')|,
\end{align}
and use Cauchy-Schwarz to estimate the $dk'$ integral to obtain
\begin{align}
A_1\lesssim \frac{\sqrt{p}}{\sqrt{{\varepsilon}}}\cdot p^2\cdot C(m+1)\cdot \sqrt{p}.
\end{align}
Here the last $\sqrt{p}$ is $|1|_{L^2(\{|k|\leq p\})}$.
To estimate $A_2$ we ignore the first exponential and integrate the (square of the) second one to obtain
\begin{align}
|e^{ik'\omega_n\frac{x_2}{{\varepsilon}}}|_{L^2(x_2)}\lesssim \frac{\sqrt{{\varepsilon}}}{\sqrt{p}},
\end{align}
after using \eqref{neglect} to neglect the $x_2$ dependence in the $\hat\sigma_j$. Thus we find
\begin{align}
A_2\lesssim \frac{\sqrt{p}}{\sqrt{{\varepsilon}}} \cdot \frac{\sqrt{{\varepsilon}}}{\sqrt{p}} \cdot C(m+2)\cdot \sqrt{p},
\end{align}
where the final $\sqrt{p}$ arises as before.
For the analogue of $A$ obtained when $\Lambda^{1/2}_{\theta,D}$ is replaced by $\Lambda^{1/2}_{x',D}$, we clearly obtain $$A\leq C(m+2+\frac{1}{2})\sqrt{p},$$ so this completes the $L^2(x_2,L^2(k, H^m(t,x_1)))$ estimate in part (a) for the term that involves $N(u_\sigma)$. Writing
\begin{align}
\langle ngle k\rangle ngle^m\lesssim \langle ngle k-k'\rangle ngle^m+\langle ngle k'\rangle ngle^m,
\end{align}
we obtain the same estimate for the $L^2(x_2,L^2(t,x_1,\hat H^m(k)))$ norm of the term involving $N(u_\sigma)$ in part (a) by very similar arguments.
\textbf{3. } To estimate the $L^2(x_2,L^2(k, H^m(t,x_1)))$ norm of the term involving $L_{fs}u_\sigma$ in \eqref{rr0}(a), instead of \eqref{rr2} we first consider
\begin{align}\langle bel{rr3}
\left|\frac{\sqrt{|k|}}{\sqrt{{\varepsilon}}}(1-\chi(k/p))k|\widehat{\partial_{s}\sigma}_m(k)|_{H^m(t,x_1)}e^{ik\omega_m\frac{x_2}{{\varepsilon}}}\right|_{L^2(k,x_2)}:= A,
\end{align}
whence
\begin{align}
A\lesssim C(m+1)p.
\end{align}
Here again we have used \eqref{neglect} and computed $|e^{ik\omega_m\frac{x_2}{{\varepsilon}}}|_{L^2(x_d)}\lesssim \frac{\sqrt{{\varepsilon}}}{\sqrt{|k|}}$.
For the analogue of $A$ obtained when $\Lambda^{1/2}_{\theta,D}$ is replaced by $\Lambda^{1/2}_{x',D}$, we have $$A\leq C(m+1+\frac{1}{2})p,$$ so this completes the $L^2(x_2,L^2(k, H^m(t,x_1)))$ estimate in part (a) for the term that involves $L_{fs}u_\sigma$.
Again, we obtain the same estimate for the $L^2(x_2,L^2(t,x_1,\hat H^m(k)))$ norm.
\textbf{4. Part (b)}. This estimate is simpler than the one in part (a), since one can take advantage of the helpful factor $|k|^{3/2}$ in $\frac{|k|^{3/2}}{{\varepsilon}^{3/2}}$; in particular, there is no need to break up the $dk'$ integral as in step 2 or compute $L^2(x_d)$ norms of exponentials. Thus, we omit the details.
\end{proof}
We now omit some of the constants $C(m+r)$. Eventually we will take $r$ to be the largest such index appearing in the building block estimates.
\begin{prop}\langle bel{r3}
Let $m>\frac{d+1}{2}$. For $k=1,3$ we have
\begin{align}
\begin{split}
&(a)\;\langle ngle \Lambda^{k/2} L_{ss} u_a\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{2-\frac{k}{2}}\\
&(b)\;\langle ngle \Lambda^{k/2}({\varepsilon}^2 L_{fs}u_\tau)\rangle ngle_{m,\gamma} \lesssim {\varepsilon}^{2-\frac{k}{2}}\\
&(c)\;\langle ngle \Lambda^{k/2} [\sum_{|\alpha|=2}Q_\alpha(D_{\varepsilon} u_a)D_{\varepsilon}^\alpha u_a ]\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{2-\frac{k}{2}}\\
&(d)\;\langle ngle\Lambda^{k/2}[\sum_{|\alpha|=2}L_\alpha(D_{\varepsilon} u_a)D_{\varepsilon}^\alpha u_a -{\varepsilon} N(u_\sigma)]\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{2-\frac{k}{2}}
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. Preliminaries. }Recall
\begin{align}
\begin{split}
&u_a={\varepsilon}^2 u_\sigma+{\varepsilon}^3 u_\tau\\
&D_{\varepsilon}=(\partial_{x_1}+\frac{\beta_1}{{\varepsilon}}\partial_\theta,\partial_{x_2}+\frac{1}{{\varepsilon}}\partial_z).
\end{split}
\end{align}
With some abuse we will write $D_{\varepsilon}=\partial_s+\frac{\partial_f}{{\varepsilon}}$ and will, for example, write products of derivatives of possibly different components of $u_\sigma$ as
``$\partial_f u_\sigma \partial_f u_\sigma$".
We will use the observation explained at the beginning of this section that $u_\sigma$ is always bigger than ${\varepsilon} u_{\tau}$ in any given building block norm as long as
$0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$. Below we take $k=3$ to illustrate the proofs.
\textbf{2. (a). } The biggest contribution is by Proposition {\rm Re }\, f{qq0} and Remark {\rm Re }\, f{qq7}:
\begin{align}
{\varepsilon}^2\langle ngle\Lambda^{3/2}\partial_{ss}u_\sigma\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^2\cdot \frac{1}{{\varepsilon}^{3/2}}={\varepsilon}^{1/2}.
\end{align}
\textbf{3. (c). } For a given $\alpha$ the largest contribution comes from products ${\varepsilon}^6 \left(\frac{\partial_f u_\sigma}{{\varepsilon}} \frac{\partial_f u_\sigma}{{\varepsilon}} \frac{\partial_{ff} u_\sigma}{{\varepsilon}^2} \right)$. Using Corollary {\rm Re }\, f{f2}(a) and Proposition {\rm Re }\, f{qq0} we obtain
\begin{align}
{\varepsilon}^2\langle ngle\Lambda^{3/2}(\partial_f u_\sigma\partial_f u_\sigma \partial_{ff}u_\sigma)\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^2 \cdot \frac{1}{{\varepsilon}^{3/2}}={\varepsilon}^{1/2}.
\end{align}
\textbf{4. (d). }After ${\varepsilon} N(u_\sigma)$ is subtracted away and after examining the size of terms involving products $\partial_fu_\tau\partial_{ff}u_\tau$, one sees that the worst terms for a given $\alpha$ are ${\varepsilon}^4\left(\partial_s u_\sigma \frac{\partial_{ff}u_\sigma}{{\varepsilon}^2}\right)$ and ${\varepsilon}^4\left(\frac{\partial_f u_\sigma}{{\varepsilon}} \frac{\partial_{sf}u_\sigma}{{\varepsilon}}\right)$. For these we obtain the desired estimate as above.
\end{proof}
The next corollary is an immediate consequence of the previous two Propositions.
\begin{cor}\langle bel{r3a}
Let $m>\frac{d+1}{2}$.
\begin{align}
\begin{split}
&(a)\;\Lambda^{1/2}\partial_{x_1,{\varepsilon}}F_a^e|_{0,m,\gamma}\lesssim \sqrt{{\varepsilon}}+ (\frac{p^2}{\sqrt{{\varepsilon}}}+{\varepsilon})+\sqrt{{\varepsilon}}\lesssim \sqrt{{\varepsilon}}+ \frac{p^2}{\sqrt{{\varepsilon}}}\\
&(b)\;\frac{1}{{\varepsilon}}|\Lambda^{1/2}F^e_a|_{0,m,\gamma}\leq \frac{p^3}{{\varepsilon}^{1/2}}+p^{1/2}+\sqrt{{\varepsilon}}.
\end{split}
\end{align}
\end{cor}
\begin{prop}\langle bel{r3b}
There exists a constant $T_2>0$ such that $v^s_a={\nabla}bla_{\varepsilon} u^s_a$ satisfies
\begin{align}
E_{m,T}(v_a^s)\lesssim 1
\end{align}
uniformly for ${\varepsilon}\in (0,1]$ and $0\leq T\leq T_2$. \footnote{Recall that $v^s_a$ depends on $T$ as a parameter.}
\end{prop}
\begin{proof}
To prove this one must show that the quantities
$|\Lambda^{3/2}v^s_a|_{0,m,\gamma}$, $|\Lambda v^s_a|_{\infty,m,\gamma}$, $|\frac{{\nabla}bla_{\varepsilon} u^s_a}{{\varepsilon}}|_{\infty,m,\gamma}$, etc., appearing in the definition of $E_{m,T}(v_a^s)$, are all $\lesssim 1$.
Consider for example $|\Lambda^{3/2} v^s_a|_{0,m,\gamma}$. The worst terms are ${\varepsilon}^2 |\Lambda^{3/2}_\theta D_{\varepsilon} u_\sigma|_{0,m,\gamma}$, for which we perform the $L^2(x_2,k,H^m(t,x_1))$ estimate as follows. Writing $D_{\varepsilon}=\partial_s+\frac{\partial_f}{{\varepsilon}}$ we obtain for the worst ($\partial_f/{\varepsilon}$) part:
\begin{align}
\begin{split}
&{\varepsilon}\left|\frac{|k|^{3/2}}{{\varepsilon}^{3/2}}k\psi(x_2)|\hat\sigma_m(t,x_1,0,k)|_{H^m(t,x_1)}e^{ik\omega_m\frac{x_2}{{\varepsilon}}}\right|_{L^2(x_2,k)}\lesssim \\
&\qquad\qquad \left|k^2 |\hat\sigma_m(t,x_1,0,k)\right|_{H^m(t,x_1)}|_{L^2(k)}\lesssim C(m+2),
\end{split}
\end{align}
since $|e^{ik\omega_m\frac{x_2}{{\varepsilon}}}|_{L^2(x_2)}\lesssim \frac{{\varepsilon}^{1/2}}{|k|^{1/2}}$.
The remaining estimates are similar or simpler.
\end{proof}
\textbf{Boundary forcing estimates. }
Here we carry out the estimation of $g_a^e$ \eqref{p0bb}. The proof of the next proposition is similar to that of propositions {\rm Re }\, f{r1} and {\rm Re }\, f{r3} but simpler.
\begin{prop}\langle bel{r4}
Let $m>\frac{d+1}{2}$. For $k=0,\dots,4$ we have
\begin{align}
\begin{split}
&\langle ngle \Lambda^{k/2}{\varepsilon}^2 G\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{2-\frac{k}{2}}\\
&\langle ngle \Lambda^{k/2} {\varepsilon}^3 \ell_s(u_\tau)\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}^{3-\frac{k}{2}}}{p^{2-\frac{k}{2}}}\\
&\langle ngle \Lambda^{k/2}c(D_{\varepsilon} u_a)\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{3-\frac{k}{2}}\\
&\langle ngle\Lambda^{k/2}[q(D_{\varepsilon} u_a)-{\varepsilon}^2 n(u_\sigma)]\rangle ngle_{m,\gamma}\lesssim {\varepsilon}^{3-\frac{k}{2}}\\
&\langle ngle\Lambda^{k/2}[{\varepsilon}^2(\chi_{\varepsilon}(D_\theta)-1)G]\rangle ngle_{m,\gamma}\leq p^{k/2}{\varepsilon}^{2-\frac{k}{2}}C(m)+{\varepsilon}^2C(m+\frac{k}{2})\\
&\langle ngle\Lambda^{k/2}[{\varepsilon}^2(1-\chi_{\varepsilon}(D_\theta))(\ell_s(u_\sigma)+n(u_\sigma))]\rangle ngle_{m,\gamma}\leq p^{k/2}{\varepsilon}^{2-\frac{k}{2}}C(m+1)+{\varepsilon}^2C(m+1+\frac{k}{2}).
\end{split}
\end{align}
\end{prop}
This yields the immediate corollary:
\begin{cor}\langle bel{r4a}
Let $m>\frac{d+1}{2}$.
\begin{align}\langle bel{s9}
\begin{split}
&(a)\;\langle ngle g-g^e_a\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}^3}{p^2}+{\varepsilon}^3+{\varepsilon}^2\lesssim \frac{{\varepsilon}^3}{p^2}+{\varepsilon}^3+{\varepsilon}^2\lesssim {\varepsilon}^2\\
&(b)\;\langle ngle \Lambda^{1/2}(g-g^e_a)\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}^{5/2}}{p^{3/2}}+{\varepsilon}^{5/2}+p^{1/2}{\varepsilon}^{3/2}+{\varepsilon}^2 \lesssim \frac{{\varepsilon}^{5/2}}{p^{3/2}}+p^{1/2}{\varepsilon}^{3/2}\\
&(c)\;\langle ngle \Lambda(g-g^e_a)\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}^2}{p}+{\varepsilon}^2+p{\varepsilon} \lesssim p{\varepsilon} \\
&(d)\;\langle ngle \Lambda^{3/2}(g-g^e_a)\rangle ngle_{m,\gamma}\lesssim \frac{{\varepsilon}^{3/2}}{p^{1/2}}+{\varepsilon}^{3/2}+p^{3/2}{\varepsilon}^{1/2}+{\varepsilon}^2 \lesssim\frac{{\varepsilon}^{3/2}}{p^{1/2}}+p^{3/2}{\varepsilon}^{1/2}\\
&(e)\;\langle ngle \Lambda^2(g-g^e_a)\rangle ngle_{m,\gamma}\lesssim {\varepsilon}+p^2+{\varepsilon}^2 \lesssim p^2.
\end{split}
\end{align}
\end{cor}
\section{Estimates of the extended approximate solution}\langle bel{estapprox}
\emph{\quad}In order to estimate solutions of the error equations \eqref{p4}-\eqref{p6} we first need estimates uniform in ${\varepsilon}$ for solutions of \eqref{p1}-\eqref{p3} in the $E_{m,\gamma}(v_a)$ norm. Except for the boundary term in \eqref{p2} the coefficients of \eqref{p1}-\eqref{p3} depend on $v_a^s$ and the forcing is given by $F^e_a$ and $g^e_a$.
The next proposition and its proof are very similar to Proposition {\rm Re }\, f{c5}. Given the forcing estimates of section {\rm Re }\, f{forcing}, the proof is somewhat simpler, as we explain below.
\begin{prop}\langle bel{s1}
Let $m> 3d+4+\frac{d+1}{2}$ and $G\in H^{m+3}(b\Omega)$. There exist positive constants $M_G$ as in \eqref{MG} and $T_2$ such that the following is true.
There exist positive constants ${\varepsilon}_0$, $\gamma_0$, and there exist increasing functions $Q_i:\mathbb{R}_+\to\mathbb{R}_+$, $i=1,2$, with
$Q_i(z)\geq z$ such that for ${\varepsilon}\in (0,{\varepsilon}_0]$ and each $T$ with $0<T\leq T_2$, the solution to \eqref{p1}-\eqref{p3} satisfies
\begin{align}\langle bel{s2}
E_{m,\gamma}(v_a)\leq \gamma^{-1}E_{m,\gamma}(v_a)Q_1(E_{m,T}(v^s_a))+(\gamma^{-\frac{1}{2}}+\sqrt{{\varepsilon}})Q_2(E_{m,T}(v^s_a))\text{ for }\gamma\geq \gamma_0.
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }Contrary to the situation in Proposition {\rm Re }\, f{c5}, we already know (Prop. {\rm Re }\, f{r3b}) that there exist positive constants $T_2$, $M_0$ such that
\begin{align}\langle bel{s3}
E_{m,T}(v_a^s)\leq M_0 \text{ for } T\leq T_2,\; {\varepsilon}\in (0,1].
\end{align}
Moreover, since the constants $C(m+r)$ appearing in the estimates of section {\rm Re }\, f{bblock} can be made as small as desired by taking $T_2$ small, the same applies to $M_0$.
\textbf{2. } The two terms on the right side of \eqref{s2} correspond to the terms on the right in \eqref{c6}; the first comes from commutators, and the second from forcing.
The commutator analysis here is identical to that given for Prop. {\rm Re }\, f{c5}, since it depends only on the control of the coefficients given by \eqref{s3}.
Moreover, the forcing estimates of section {\rm Re }\, f{forcing} show that
\begin{align}
F_a^e=o(1) \text{ and }g^e_a= {\varepsilon}^2G+o(1) \text{ as } {\varepsilon} \to 0
\end{align}
in every one of the norms that is used in the forcing estimate.
Thus, the second term of \eqref{s2} has the same form as the second term of \eqref{c6}.
\end{proof}
\begin{cor}\langle bel{s4a}
Under the assumptions of Prop. {\rm Re }\, f{s1} we have, after enlarging $\gamma_0$ if necessary,
\begin{align}\langle bel{s4}
E_{m,\gamma}(v_a)\leq (\gamma^{-\frac{1}{2}}+\sqrt{{\varepsilon}})Q_2(E_{m,T}(v^s_a))\text{ for }\gamma\geq \gamma_0.
\end{align}
\end{cor}
\section{Endgame}\langle bel{end}
\emph{\quad} The next Proposition is the analogue of Proposition {\rm Re }\, f{g1} for the trio of error equations \eqref{p4}-\eqref{p6}. The main difference in the proof is that there are now new kinds of forcing terms to deal with.
Henceforth we will denote functions $Q_i(z)$ like those in Proposition {\rm Re }\, f{s1} simply by $Q(z)$, and we will allow $Q$ to change from term to term, or even from factor to factor within a given term. In the next proposition $T_1$ and $T_2$ are the positive constants appearing in Theorem {\rm Re }\, f{uniformexistence} and Proposition {\rm Re }\, f{s1} respectively.
\begin{prop}\langle bel{s5}
Suppose $m>3d+4+\frac{d+1}{2}$, $G\in H^{m+3}(b\Omega)$, and $0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$ for some $\delta>0$. There exist positive constants $T_3\leq\min\{T_1,T_2\}$, ${\varepsilon}_0$, $\gamma_0$
such that for $j\in J_h\cup J_e$, ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, and $T\leq T_3$ the solution to the error system \eqref{p4}-\eqref{p6} satisfies
\begin{align}\langle bel{s6}
\begin{split}
&E_{m,\gamma}(w_1)+\langle ngle\phi_j\Lambda^{\frac{3}{2}}w_1\rangle ngle_{m,\gamma}+\langle ngle\phi_j\Lambda w_1\rangle ngle_{m+1,\gamma}+\sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}}w_1\rangle ngle_{m+1,\gamma}+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda w_1\rangle ngle_{m,\gamma}\lesssim\\
&\frac{1}{\sqrt{\gamma}}\{Q_1(E_{m,T}(v^s))\cdot E_{m,\gamma}(w_1)+E_{m,T}(w^s)E_{m+1,\gamma}(v_a)Q_2(E_{m,T}(v^s_a,v^s))+\\
&\qquad E_{m,T}(w^s)Q_3(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
To prove \eqref{s6} we use \eqref{b3}-\eqref{b5} to estimate the solutions of \eqref{p4} and \eqref{p6}. The arguments have much in common with those of section {\rm Re }\, f{mainestimate}, so here we will emphasize what is different.
\textbf{1. }The first term on the right in \eqref{s6} arises by exactly the same commutator analysis as given in section {\rm Re }\, f{mainestimate}. See Proposition {\rm Re }\, f{g1}.
Let us denote the right sides of the interior equations \eqref{p4}(a)-\eqref{p6}(a) by $\cF_1$, $\cF_2$, $\cF$ respectively, and the right sides of the corresponding boundary equations
by $\cG_1$, $\cG_2$, $\cG$ respectively. These terms differ from the forcing terms estimated in section 5, and the error analysis depends on their careful estimation.
In particular, to estimate the terms which define $E_{m,\gamma}(w_1)$ in the first, second, and third lines of \eqref{c0a} respectively, one must respectively estimate
\begin{align}\langle bel{s6a}
\begin{split}
&(a)\;|\Lambda^{1/2}\cF_1|_{0,m,\gamma}, |\cF_1|_{0,m+1,\gamma}, \langle ngle\Lambda\cG_1\rangle ngle_{m,\gamma}, \langle ngle\Lambda^{1/2}\cG_1\rangle ngle_{m+1,\gamma}\\
&(b)\;\sqrt{{\varepsilon}}|\Lambda^{1/2}\cF_1|_{0,m+1,\gamma}, \frac{1}{\sqrt{{\varepsilon}}}|\cF_1|_{0,m,\gamma}, \sqrt{{\varepsilon}}\langle ngle\Lambda\cG_1\rangle ngle_{m+1,\gamma}, \frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{1/2}\cG_1\rangle ngle_{m,\gamma}\\
&(c)\;\frac{1}{{\varepsilon}}|\Lambda^{1/2}\cF|_{0,m,\gamma}, \frac{1}{{\varepsilon}}|\cF|_{0,m+1,\gamma}, \langle ngle\Lambda\cG\rangle ngle_{m,\gamma}, \frac{1}{{\varepsilon}} \langle ngle\Lambda^{1/2}\cG\rangle ngle_{m+1,\gamma}.
\end{split}
\end{align}
Below we will focus on estimating the terms in \eqref{s6a}(a) and (c). We will omit the (similar) details for (b).
\textbf{2. Interior forcing in \eqref{p4}. }We show that for $|\alpha|=2$
\begin{align}\langle bel{s7}
\begin{split}
&(a) \;|\Lambda^{1/2}[(A_\alpha(v^s)-A_\alpha(v^s_a))\partial^\alpha_{x,{\varepsilon}} v_{1a}]|_{0,m,\gamma}\leq E_{m,T}(w^s)E_{m+1,\gamma}(v_{1a})Q(E_{m,T}(v^s_a,v^s))\\
&(b)\;|[(A_\alpha(v^s)-A_\alpha(v^s_a))\partial^\alpha_{x,{\varepsilon}} v_{1a}]|_{0,m+1,\gamma}\leq E_{m,T}(w^s)E_{m+1,\gamma}(v_{1a})Q(E_{m,T}(v^s_a,v^s)).
\end{split}
\end{align}
We write $A_\alpha(v^s)-A_\alpha(v^s_a))=f(v^s_a,v^s)w^s$, and reduce to considering the case $\alpha=(2,0)$ by using the equation and the noncharacteristic boundary assumption to treat $\alpha=(1,1)$ or $(0,2)$ . Applying Corollary {\rm Re }\, f{f2}(a) with $x_2$ fixed, we have
\begin{align}
\begin{split}
&\langle ngle \Lambda^{1/2}[f(v^s_a,v^s)w^s\partial^\alpha_{x,{\varepsilon}}v_{1a}]\rangle ngle_{m,\gamma}\leq\\
&\quad \langle ngle\Lambda^{1/2}{\varepsilon}\partial^\alpha_{x,{\varepsilon}}v_{1a}\rangle ngle_{m,\gamma}\langle ngle f(v^s_a,v^s)\frac{w^s}{{\varepsilon}}\rangle ngle_m+\langle ngle{\varepsilon}\partial^\alpha_{x,{\varepsilon}}v_{1a}\rangle ngle_{m,\gamma}\langle ngle\Lambda^{1/2}_1[f(v^s_a,v^s)\frac{w^s}{{\varepsilon}}]\rangle ngle_m\leq\\
&\qquad \langle ngle\Lambda^{3/2}v_{1a}\rangle ngle_{m+1,\gamma}h(\langle ngle v^s_a,v^s\rangle ngle_m)\langle ngle\frac{w^s}{{\varepsilon}}\rangle ngle_m +\langle ngle\Lambda v_{1a}\rangle ngle_{m+1,\gamma}\langle ngle\Lambda^{1/2}_1(v^s_a,v^s)\rangle ngle_m h(\langle ngle v^s_a,v^s\rangle ngle_m)\langle ngle\frac{w^s}{{\varepsilon}}\rangle ngle_m+\\
&\qquad\qquad\langle ngle\Lambda v_{1a}\rangle ngle_{m+1,\gamma} h(\langle ngle v^s_a,v^s\rangle ngle_m) \langle ngle \Lambda^{1/2} \frac{w^s}{{\varepsilon}}\rangle ngle_m = A+B+C.
\end{split}
\end{align}
Next take the $L^2(x_d)$ norm of both sides. In the final estimate the factors $|\Lambda^{3/2}v_{1a}|_{0,m+1,\gamma}$, $|\Lambda v_{1a}|_{0,m+1,\gamma}$, and $| \Lambda^{1/2} \frac{w^s}{{\varepsilon}}|_{0,m}$ should respectively appear in each of the three terms on the right; the remaining factors in those terms all involve $L^\infty(x_d)$ norms. This gives \eqref{s7}(a).
After again writing $w^s\partial^\alpha_{x,{\varepsilon}}v_{1a}=\frac{w^s}{{\varepsilon}} {\varepsilon}\partial^\alpha_{x,{\varepsilon}}v_{1a}$, one can prove \eqref{s7}(b) by an argument similar to the proof of Proposition {\rm Re }\, f{e14}.
\textbf{3. }An argument similar to that in step 2 but more straightforward yields
\begin{align}
\begin{split}
&|\Lambda^{1/2}[b(v^s)d_{\varepsilon} v^sd_{\varepsilon} v^s-b(v^s_a)d_{\varepsilon} v^s_a d_{\varepsilon} v^s_a]|_{0,m,\gamma}\leq E_{m,T}(w^s)Q(E_{m,T}(v^s_a,v^s))\\
&|b(v^s)d_{\varepsilon} v^sd_{\varepsilon} v^s-b(v^s_a)d_{\varepsilon} v^s_a d_{\varepsilon} v^s_a]|_{0,m+1,\gamma}\leq E_{m,T}(w^s)Q(E_{m,T}(v^s_a,v^s))\\
\end{split}
\end{align}
\textbf{4. }Corollary {\rm Re }\, f{r3a}(a) gives
\begin{align}
|\Lambda^{1/2}\partial_{x_1,{\varepsilon}}F_a^e|_{0,m,\gamma}\lesssim \sqrt{{\varepsilon}}+ (\frac{p^2}{\sqrt{{\varepsilon}}}+{\varepsilon})+\sqrt{{\varepsilon}}\lesssim \sqrt{{\varepsilon}}+ \frac{p^2}{\sqrt{{\varepsilon}}}.
\end{align}
The same estimate clearly holds for $|\partial_{x_1,{\varepsilon}}F_a^e|_{0,m+1,\gamma}$.
\textbf{5. Boundary forcing in \eqref{p4}. }
An argument like that in step 2 yields
\begin{align}\langle bel{ss8}
\begin{split}
&\langle ngle\Lambda[d_{v_1}H(v^s_1,h(v^s))-d_{v_1}H(v^s_{1a},h(v^s_a))]\partial_{x_1,{\varepsilon}}v_{1a}]\rangle ngle_{m,\gamma}\leq E_{m,T}(w^s)E_{m+1,\gamma}(v_{1a})Q_2(E_{m,T}(v^s_a,v^s))\\
&\langle ngle\Lambda^{1/2}[d_{v_1}H(v^s_1,h(v^s))-d_{v_1}H(v^s_{1a},h(v^s_a))]\partial_{x_1,{\varepsilon}}v_{1a}]\rangle ngle_{m+1,\gamma}\leq E_{m,T}(w^s)E_{m+1,\gamma}(v_{1a})Q_2(E_{m,T}(v^s_a,v^s)).
\end{split}
\end{align}
We claim
\begin{align}\langle bel{s8}
\langle ngle\Lambda [d_gH(v^s_{1},g)\partial_{x_1,{\varepsilon}}g - d_gH(v^s_{1a},g^e_a)\partial_{x_1,{\varepsilon}} g^e_a]\rangle ngle_{m,\gamma}\lesssim E_{m,T}(w^s)Q(E_{m,T}(v^s_1,v^s_{1a}))+{\varepsilon}+p^2,
\end{align}
and the same estimate holds for $\langle ngle\Lambda^{1/2} [d_gH(v^s_{1},g)\partial_{x_1,{\varepsilon}}g - d_gH(v^s_{1a},g^e_a)\partial_{x_1,{\varepsilon}} g^e_a]\rangle ngle_{m+1,\gamma}$.
Writing $d_gH(v^s_{1},g)\partial_{x_1,{\varepsilon}}g - d_gH(v^s_{1a},g^e_a)\partial_{x_1,{\varepsilon}} g^e_a$ in the obvious way as a sum of three differences, we see that Corollary {\rm Re }\, f{r4a}(a),(c),(e) and Corollary {\rm Re }\, f{f2}(a) imply \eqref{s8}.
\textbf{6. Interior forcing in {\rm Re }\, f{p6}. }The above estimates show that the terms arising from the application of \eqref{b3}-\eqref{b5} to the system \eqref{p4} are dominated by the right side of
\eqref{s6}, so we now turn to the $z$-system, \eqref{p6}, which is needed to estimate terms like $|{\nabla}bla_{\varepsilon} z/{\varepsilon}|_{\infty,m,\gamma}$.\footnote{We will use the fact that a remark analogous to
Remark {\rm Re }\, f{e1z} applies to $w$ and ${\nabla}bla_{\varepsilon} z$. }
Corollary {\rm Re }\, f{r3a}(b) gives
\begin{align}
\frac{1}{{\varepsilon}}|\Lambda^{1/2}F^e_a|_{0,m,\gamma}\leq \frac{p^3}{{\varepsilon}^{1/2}}+p^{1/2}+\sqrt{{\varepsilon}}.
\end{align}
Next we show that for $|\alpha|=2$:
\begin{align}\langle bel{s10}
\begin{split}
&(a)\;\frac{1}{{\varepsilon}}|\Lambda^{1/2}[(A_\alpha(v^s)-A_\alpha(v^s_a))\partial^\alpha_{x,{\varepsilon}} u_{a}]|_{0,m,\gamma}\leq E_{m,T}(w^s)E_{m,\gamma}(v_{a})Q(E_{m,T}(v^s_a,v^s))\\
&(b)\;\frac{1}{{\varepsilon}}|[(A_\alpha(v^s)-A_\alpha(v^s_a))\partial^\alpha_{x,{\varepsilon}} u_{a}]|_{0,m+1,\gamma}\leq E_{m,T}(w^s)E_{m,\gamma}(v_{a})Q(E_{m,T}(v^s_a,v^s))\\
\end{split}
\end{align}
For fixed $x_2$ we estimate for $|\alpha|=1$
\begin{align}
\begin{split}
&\frac{1}{{\varepsilon}}\langle ngle \Lambda^{1/2}[f(v^s_a,v^s)w^s\partial^\alpha_{x,{\varepsilon}}v_{a}]\rangle ngle_{m,\gamma}\leq\\
&\quad \langle ngle\Lambda^{1/2}\partial^\alpha_{x,{\varepsilon}}v_{a}\rangle ngle_{m,\gamma}\langle ngle f(v^s_a,v^s)\frac{w^s}{{\varepsilon}}\rangle ngle_m+\langle ngle \partial^\alpha_{x,{\varepsilon}}v_{a}\rangle ngle_{m,\gamma}\langle ngle\Lambda^{1/2}_1[f(v^s_a,v^s)\frac{w^s}{{\varepsilon}}]\rangle ngle_m,
\end{split}
\end{align}
so applying rules of section {\rm Re }\, f{nonlinear} and taking the $L^2(x_2)$ norm of both sides gives \eqref{s10}(a). The proof of \eqref{s10}(b) is parallel to that of \eqref{s7}(b).
\textbf{7. Boundary forcing in {\rm Re }\, f{p6}. }As in \eqref{e6}, \eqref{e9} let us write
\begin{align}
H(v^s_1,g)-d_{v_1}H(v^s_1,g)v^s_1=Cg+b(v^s_1,g)(v^s_1,g)^2:=F(v^s_1,g).
\end{align}
We express the boundary forcing term in \eqref{p6} as
\begin{align}
\begin{split}
&\cG=[F(v^s_1,g)-F(v^s_{1a},g^e_a)]+[d_{v_1}H(v^s_1,h(v^s))-d_{v_1}H(v^s_{1a},h(v^s_a))]v_{1a}=A+B\\
\end{split}
\end{align}
By arguments similar to step 2 we obtain
\begin{align}
\frac{1}{{\varepsilon}}\langle ngle\Lambda B\rangle ngle_{m,\gamma}\leq E_{m,\gamma}(v_{1a})E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a)).
\end{align}
Writing $A$ as a sum of two differences, using the special form of $F$, and using \eqref{s9}(a),(c), we obtain
\begin{align}
\begin{split}
&\frac{1}{{\varepsilon}}\langle ngle\Lambda A\rangle ngle_{m,\gamma}\lesssim E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+(\frac{{\varepsilon}^2}{p^2}+{\varepsilon}^2+{\varepsilon})+(\frac{{\varepsilon}}{p}+{\varepsilon}+p) \lesssim \\
&\qquad\qquad E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+\frac{{\varepsilon}}{p}+p.
\end{split}
\end{align}
This gives
\begin{align}\notag
\frac{1}{{\varepsilon}}\langle ngle \Lambda(A+B)\rangle ngle_{m,\gamma}\lesssim E_{m,\gamma}(v_{1a})E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+\frac{{\varepsilon}}{p}+p
\end{align}
and similarly
\begin{align}\notag
\frac{1}{{\varepsilon}}\langle ngle \Lambda^{1/2}(A+B)\rangle ngle_{m+1,\gamma}\lesssim E_{m,\gamma}(v_{1a})E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+E_{m,T}(w^s) Q(E_{m,T}(v^s,v^s_a))+\frac{{\varepsilon}}{p}+p.
\end{align}
\textbf{8. }The same interior and boundary forcing estimates are satisfied by the terms in \eqref{s6a}(b).
Combining these results we obtain \eqref{s6}.
\end{proof}
By enlarging $\gamma_0$ if necessary and using Corollary {\rm Re }\, f{s4a}, we obtain
\begin{cor}\langle bel{s11}
Suppose $m>3d+4+\frac{d+1}{2}$ and $0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$ for some $\delta>0$. There exist positive constants $T_3$, ${\varepsilon}_0$, and $\gamma_0$
such that for $j\in J_h\cup J_e$, ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, and $T\leq T_3$:
\begin{align}\langle bel{s12}
\begin{split}
&E_{m,\gamma}(w_1)+\langle ngle\phi_j\Lambda^{\frac{3}{2}}w_1\rangle ngle_{m,\gamma}+\langle ngle\phi_j\Lambda w_1\rangle ngle_{m+1,\gamma}+\sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}}w_1\rangle ngle_{m+1,\gamma}+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda w_1\rangle ngle_{m,\gamma}\lesssim\\
&\frac{1}{\sqrt{\gamma}}\{E_{m,T}(w^s)Q(E_{m+1,T}(v^s_{a}))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\}.
\end{split}
\end{align}
\end{cor}
To complete the error analysis we will apply \eqref{b6}-\eqref{b8} to the Dirichlet problem \eqref{p5} in order to estimate $E_{m,\gamma}(w_2)$.
This requires us to estimate the terms in \eqref{s6a}(a),(b) where $\cF_1$, $\cG_1$ are replaced by $\cF_2$, $\cG_2$. In addition we must estimate norms of $\cG_2$ involving the singular pseudodifferential operators $\phi_{jD}$, $\psi_{jD}$ for $j\in J_h$ (recall Prop. {\rm Re }\, f{e21}, for example). The estimates of $\cF_2$ are identical to those already given of $\cF_1$. In the next two Propositions we give the required estimates of $\cG_2$. These estimates are quite similar to those of Propositions {\rm Re }\, f{e18}, {\rm Re }\, f{e21}, {\rm Re }\, f{e22b}, and {\rm Re }\, f{e23d} of section {\rm Re }\, f{mainestimate}, and are proved in the same way.
\begin{prop}\langle bel{s13}
Suppose $m>3d+4+\frac{d+1}{2}$ and $0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$ for some $\delta>0$. Let $\phi_j$ and $\psi_j$, $j\in J_h$ be singular symbols related as in Notation {\rm Re }\, f{f4b}. There exist positive constants ${\varepsilon}_0$, $\gamma_0$, $T_3$ such that
for ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, $T\leq T_3$ we have
\begin{align}\langle bel{s14}
\begin{split}
&(a)\;\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}\lesssim \langle ngle \Lambda w_1\rangle ngle_{m,\gamma}Q(E_{m,T}(v^s,v^s_a))+{\varepsilon}\\
&(b)\;\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m+1,\gamma}\lesssim \langle ngle \Lambda^{1/2} w_1\rangle ngle_{m+1,\gamma}Q(E_{m,T}(v^s,v^s_a))+{\varepsilon}\\
&(c)\;\langle ngle\phi_j\Lambda^{\frac{3}{2}} \cG_2\rangle ngle_{m,\gamma}\lesssim \left(\langle ngle \psi_j\Lambda^{\frac{3}{2}} w_1\rangle ngle_{m,\gamma}+\langle ngle \Lambda w_1\rangle ngle_{m,\gamma}\right)Q(E_{m,T}(v^s,v^s_a))+\sqrt{{\varepsilon}}\\
&(d)\;\langle ngle\phi_j\Lambda\cG_2\rangle ngle_{m+1,\gamma}\lesssim \left(\langle ngle \psi_j\Lambda w_1\rangle ngle_{m+1,\gamma}+\langle ngle \Lambda^{\frac{1}{2}} w_1\rangle ngle_{m+1,\gamma}\right)Q(E_{m,T}(v^s,v^s_a))+{\varepsilon}.
\end{split}
\end{align}
\end{prop}
\begin{prop}\langle bel{s17}
With notation as in the previous proposition we have for ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, $T\leq T_3$:
\begin{align}\langle bel{s18}
\begin{split}
&(a)\;\sqrt{{\varepsilon}}\langle ngle\Lambda\cG_2\rangle ngle_{m+1,\gamma}\lesssim \sqrt{{\varepsilon}}\langle ngle \Lambda w_1\rangle ngle_{m+1,\gamma} Q(E_{m,T}(v^s,v^s_a))+{\varepsilon}^{3/2}\\
&(b)\;\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\Lambda^{\frac{1}{2}}\cG_2\rangle ngle_{m,\gamma}\lesssim \frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}} w_1\rangle ngle_{m,\gamma}Q(E_{m,T}(v^s,v^s_a))+\sqrt{{\varepsilon}}\\
&(c)\;\sqrt{{\varepsilon}}\langle ngle\phi_j\Lambda^{\frac{3}{2}} \cG_2\rangle ngle_{m+1,\gamma}\lesssim \left(\sqrt{{\varepsilon}}\langle ngle \psi_j\Lambda^{\frac{3}{2}} w_1\rangle ngle_{m+1,\gamma}+\sqrt{{\varepsilon}}\langle ngle \Lambda w_1\rangle ngle_{m+1,\gamma}\right)Q(E_{m,T}(v^s,v^s_a))+{\varepsilon}\\
&(d)\;\frac{1}{\sqrt{{\varepsilon}}}\langle ngle\phi_j\Lambda\cG_2\rangle ngle_{m,\gamma}\lesssim \left(\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \psi_j\Lambda w_1\rangle ngle_{m,\gamma}+\frac{1}{\sqrt{{\varepsilon}}}\langle ngle \Lambda^{\frac{1}{2}} w_1\rangle ngle_{m,\gamma}\right)Q(E_{m,T}(v^s, v^s_a))+\sqrt{{\varepsilon}}.
\end{split}
\end{align}
\end{prop}
The following Proposition is the analogue for the error system of Proposition {\rm Re }\, f{h1}.
\begin{prop}\langle bel{endgame}
Suppose $m>3d+4+\frac{d+1}{2}$, $G\in H^{m+3}(b\Omega)$, and $0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$ for some $\delta>0$. There exist positive constants ${\varepsilon}_0$, $\gamma_0$, $T_3$ such that
for ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, and $T\leq T_3$ the solution to the error system \eqref{p4}-\eqref{p6} satisfies
\begin{align}\langle bel{s20}
\begin{split}
&E_{m,\gamma}(w_2)\lesssim \frac{1}{\gamma}E_{m,\gamma}(w_2)Q(E_{m,T}(v^s))+\\
&\qquad\qquad \frac{1}{\sqrt{\gamma}}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\right]+\sqrt{{\varepsilon}}.
\end{split}
\end{align}
\end{prop}
\begin{proof}
Given the earlier results of this section, the proof is quite similar to that of Proposition {\rm Re }\, f{h1}. Parallel to step 2 of the earlier proof, consider for example the term $\left|\begin{pmatrix}\Lambda w_2\\partial_{x_2}w_2\end{pmatrix}\right|_{\infty,m,\gamma}$. By \eqref{b7} applied to \eqref{p5} we have
\begin{align}\langle bel{s21}
\begin{split}
&\left|\begin{pmatrix}\Lambda w_2\\partial_{x_2}w_2\end{pmatrix}\right|_{\infty,m,\gamma}\lesssim \gamma^{-1}|\cF_{2,com}|_{0,m+1,\gamma}+ \gamma^{-1}|\cF_2|_{0,m+1,\gamma}+\\
&\qquad \qquad (\langle ngle\Lambda \cG_2\rangle ngle_{m,\gamma}+\langle ngle\Lambda^{\frac{1}{2}} \cG_{2}\rangle ngle_{m+1,\gamma})+\gamma^{-\frac{1}{2}}\sum_{j\in J_h}\langle ngle\phi_j\Lambda \cG_2\rangle ngle_{m+1,\gamma}:=A+B+C+D,
\end{split}
\end{align}
where $\cF_{2,com}$ is the interior commutator. By the interior commutator arguments of section {\rm Re }\, f{mainestimate} we have
\begin{align}
A\lesssim \frac{1}{\gamma}E_{m,\gamma}(w_2)Q(E_{m,T}(v^s)).
\end{align}
Corollary {\rm Re }\, f{s4a} and the estimates in steps 2, 3, and 4 of the proof of Proposition {\rm Re }\, f{s5} apply to $\cF_2$ to show
\begin{align}
B\lesssim \frac{1}{\gamma}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+\sqrt{p}+\frac{p^2}{\sqrt{{\varepsilon}}}\right].
\end{align}
From Proposition {\rm Re }\, f{s13}(a),(b) we have
\begin{align}
C\lesssim E_{m,\gamma}(w_1)Q(E_{m,T}(v^s_a,v^s))+\sqrt{{\varepsilon}},
\end{align}
while Proposition {\rm Re }\, f{s13}(d) gives
\begin{align}
D\lesssim \frac{1}{\sqrt{\gamma}}\left[ \sum_{j\in J_h} \left(\langle ngle \psi_j\Lambda w_1\rangle ngle_{m+1,\gamma}+\langle ngle \Lambda^{\frac{1}{2}} w_1\rangle ngle_{m+1,\gamma}\right)Q(E_{m,T}(v^s,v^s_a))+\sqrt{{\varepsilon}}\right].
\end{align}
Next apply Corollary {\rm Re }\, f{s11} to obtain
\begin{align}
C+D\lesssim \frac{1}{\sqrt{\gamma}}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\right]+\sqrt{{\varepsilon}}.
\end{align}
Thus, we see that $A+B+C+D$ is dominated by the right side of \eqref{s20}. The other terms making up $E_{m,\gamma}(w_2)$ are estimated in the same way using the earlier results of this section.
\end{proof}
Combining Corollary {\rm Re }\, f{s11} with Proposition {\rm Re }\, f{endgame} we immediately obtain the following corollary by enlarging $\gamma_0$ if necessary:
\begin{cor}\langle bel{s22}
Suppose $m>3d+4+\frac{d+1}{2}$ and $0<{\varepsilon}^{\frac{1}{2}-\delta}\leq p\leq 1$ for some $\delta>0$. There exist positive constants ${\varepsilon}_0$, $\gamma_0$, $T_3$ such that
for ${\varepsilon}\in (0,{\varepsilon}_0]$, $\gamma\geq \gamma_0$, and $T\leq T_3$ we have
\begin{align}\langle bel{s23}
\begin{split}
&E_{m,\gamma}(w)\lesssim \frac{1}{\sqrt{\gamma}}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\right]+\sqrt{{\varepsilon}}.
\end{split}
\end{align}
\end{cor}
\begin{cor}\langle bel{s24a}
Suppose $m>3d+4+\frac{d+1}{2}$ and $\delta>0$. There exist positive constants $T_4\leq T_3$, ${\varepsilon}_0$, and $C_\delta$ such that
for ${\varepsilon}\in (0,{\varepsilon}_0]$, $T\leq T_4$ the solution to the error system \eqref{p4}-\eqref{p6} satisfies
\begin{align}\langle bel{s24}
\begin{split}
&E_{m,T}(w)\leq C_\delta {\varepsilon}^{\frac{1}{4}-\delta}.
\end{split}
\end{align}
\end{cor}
\begin{proof}
Since $E_{m,\gamma}(w)\gtrsim e^{-\gamma T}E_{m,T}(w)$, we obtain from \eqref{s23}
\begin{align}\langle bel{s25}
\begin{split}
E_{m,T}(w)\lesssim e^{\gamma T} \left(\frac{1}{\sqrt{\gamma}}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\right]+\sqrt{{\varepsilon}}\right)
\end{split}
\end{align}
for $\gamma\geq \gamma_0$ as in Corollary {\rm Re }\, f{s22}. Thus, for $T\leq 1/\gamma_0$ we have
\begin{align}\langle bel{s26}
\begin{split}
E_{m,T}(w)\lesssim \sqrt{T}\left[E_{m,T}(w^s)Q(E_{m+1,T}(v^s_a))Q(E_{m,T}(v^s_a,v^s))+(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})\right]+\sqrt{{\varepsilon}}
\end{split}
\end{align}
Since $w=w^s_T$ on $\Omega_T$ for $T\leq T_3$, by
choosing $0<T_4\leq 1/\gamma_0$ small enough and setting $p={\varepsilon}^{\frac{1}{2}-2\delta}$, we deduce \eqref{s24} for $T\leq T_4$ from \eqref{s26}.
\end{proof}
\chapter{Some extensions}\langle bel{chapter6}
\section{Extension to general isotropic hyperelastic materials.}\langle bel{generalisotropic}
\emph{\quad} We now describe the minor changes in the analysis that are needed when assumption (A1) in chapter {\rm Re }\, f{chapter3} is replaced by assumption (A1g), that is, when we pass from a Saint Venant-Kirchhoff system to general isotropic hyperelastic materials. Theorems {\rm Re }\, f{uniformexistence} and {\rm Re }\, f{approxthm} both remain true as stated under assumption (A1g).
The amplitude equation in Chapter {\rm Re }\, f{chapter2} was already treated in this more general context; in particular, Proposition {\rm Re }\, f{propwellposed} applies to these more general materials.
The construction and estimation of the approximate solution in chapter {\rm Re }\, f{chapter4} goes through almost without change. In the definition of the coefficients $A_\alpha$ as in \eqref{Asub},
\begin{align}
A_\alpha(v)=A_\alpha(0)+L_\alpha(v)+Q_\alpha(v),
\end{align}
only the term $Q_\alpha(v)$ changes; it must now be defined as the sum of all terms in the expansion of $A_\alpha(v)$ that are quadratic or of higher order. Similarly, in the definition of the boundary function $h(v)$ as in \eqref{hofv},
\begin{align}
h(v)=\ell(v)+q(v)+c(v),
\end{align}
only the term $c(v)$ changes; it must now be defined as the sum of all terms in the expansion of $h(v)$ that are cubic or of higher order. We observe that these changes have no effect on the interior and boundary profile equations \eqref{o3} and \eqref{o4}. Thus, the construction of the $\sigma_j$ and $\tau_j$ and the building block estimates are unaffected.
However, these changes do affect the expressions for the interior and boundary error profiles
$F_a(t,x,\theta,z)$ \eqref{fa} and $h_a(t,x_1,\theta)$ \eqref{ba}, which respectively contain the terms
$$
\sum_{|\alpha|=2}Q_\alpha(D_{\varepsilon} u_a)D_{\varepsilon}^\alpha u_a \text{ and } c(D_{\varepsilon} u_a).
$$
Under assumption (A1) these terms were estimated in Proposition {\rm Re }\, f{r3}(c) and in the third estimate of Proposition {\rm Re }\, f{r4}. The estimates remain true as stated under assumption (A1g), but a small change is needed in the proofs. For example, in the proof of Proposition {\rm Re }\, f{r3}(c), instead of using Corollary {\rm Re }\, f{f2}(a) for products, one should use Proposition {\rm Re }\, f{f3}(c) for analytic functions $f(u)$.\footnote{Here we use the analyticity assumption on $W(E)$ in (A1g).} The cubic and higher order terms in $Q_\alpha(v)$ are readily seen to make contributions to the estimate that are negligible compared to the quadratic terms.
It remains to check that properties (P1)-(P7) of section {\rm Re }\, f{assumptions} continue to hold under the assumption (A1g). This is clear for (P1)-(P4), since the problem $(P^0,B^0)$ is unchanged.\footnote{Properties (P1)-(P7) were verified for the SVK system in \cite{S-T} under the assumption $\mu>0$, $\langle mbda>0$. The discussion in chapter
{\rm Re }\, f{chapter2} and an inspection of \cite{S-T} shows that her arguments apply just as well under the assumption $\mu>0$, $\langle mbda+\mu>0$.} The verification of (P5), given in \cite{S-T}, p. 283 for the SVK system, is based on showing that the Lopatinskii determinant $\det b^+(v,z)$ as in \eqref{lop} is real for $\gamma=0$. The argument is based on symmetry properties of the coefficient matrices $A_\alpha(v)$ defining $P^v$ \eqref{d3}, which arise as consequences of the fact that these matrices are derived from a stored energy function $W({\nabla}bla u)$. For example, the argument uses the fact that the $(\alpha,\beta)$ component of $A_{(1,1)}(v)$ is given by
\begin{align}
c_{\alpha1 \beta 2}(v)+c_{\alpha 2 \beta 1}(v), \text{ where } c_{\alpha j \beta \ell}(v)=\frac{\partial^2 W(v)}{\partial u_{\alpha,j}\partial u_{\beta,\ell}},
\end{align}
and thus we have the matrix equalities
\begin{align}
(c_{\alpha 2 \beta 1}(v))=(c_{\beta 1 \alpha 2}(v))=(c_{\alpha 1 \beta 2}(v))^t.
\end{align}
This property is clearly unaffected by the passage from (A1) to (A1g). Similarly, the verification of (P6) and (P7) on p. 286 of \cite{S-T} for the SVK system continues to hold under assumption (A1g) because the matrices $A_\alpha(v)$ are constructed from derivatives of $W(v)$.
\section{Extension to wavetrains. }\langle bel{wavetrains}
\emph{\quad} Next we summarize how the method used in this paper to treat surface pulses extends to treat surface wavetrains.
We explain in Remark {\rm Re }\, f{optimal} that another method is available for wavetrains that should yield more detailed qualitative information about the solutions.
We restrict attention to the 2D case. The boundary data \eqref{a3} is now given by ${\varepsilon}^2 G(t,x_1,\theta)$, where $\theta\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$.
Writing $G(t,x_1,\theta)=\sum_{n\in\mathbb{Z}} G^n(t,x_1)e^{in\theta}$, we assume $G^0=0$.
We work with exactly the same Sobolev spaces and singular operators as before, except now $\Omega=\{(t,x_1,x_2,\theta)\in\mathbb{R}^3\times \mathbb{T}:x_2>0\}$,
$k\in\mathbb{Z}$, where $k$ is the variable dual to $\theta$, and integrals $\int \dots dk$ are replaced by sums over $k\in\mathbb{Z}$. A version of the singular calculus for wavetrains parallel to the calculus for pulses used here was developed in \cite{CGW}; in fact the wavetrain calculus is ``better behaved", since $\theta$ now lies in a compact set.
Except for the changes just mentioned, the results of chapter {\rm Re }\, f{chapter3} on the existence of exact solutions hold with identical statements and proofs. The main changes in the treatment of wavetrains occur in the construction of approximate solutions. Unlike a pulse a wavetrain has a well-defined mean, and in the construction of approximate solutions one has to separate out the initial boundary value problems satisfied by the means of profiles. As in \eqref{o2a} we look for the profile of the approximate solution in the form $u_a={\varepsilon}^2 u_\sigma+{\varepsilon}^3 u_\tau$, where now
\begin{align}
\begin{split}
&u_\sigma(t,x,\theta,z)=\sum_{n\in\mathbb{Z}}u_\sigma^n(t,x,z)e^{in\theta}\\
&u_\sigma^0(t,x,z)=\underline{u}^0_\sigma(t,x)+u^{0,*}_\sigma(t,x,z) \text{ with }\lim_{z\to\infty}u^{0,*}_\sigma =0,
\end{split}
\end{align}
and a similar expansion holds for $u_\tau$. We define $\underline{u}^0_\sigma(t,x)$ to be the \emph{mean} of $u_\sigma$.
We seek $u_\sigma$ and $u_\tau$ as solutions of the interior and boundary profile equations \eqref{o3}, \eqref{o4} of the form
\begin{align}
u_\sigma(t,x,\theta,z)=\sum^4_{j=1}\sigma_j(t,x,\theta+\omega_jz)r_j \text{ and }u_\tau(t,x,\theta,z)=\sum^4_{j=1}\tau_j(t,x,\theta,z)r_j,
\end{align}
where the low frequency cutoff $\chi_{\varepsilon}(D_\theta)$ acting on the $\tau_j$ in \eqref{o2a} is now absent.
The determination of $u_\sigma-u_\sigma^0$ and of $u_\tau-u_\tau^0$ follows the procedure used in chapter {\rm Re }\, f{chapter4} to determine the $\hat\sigma_j(t,x,k)$ and the $\hat\tau_j(t,x,k,z)$ for $k\neq 0$. In particular, the traces $\hat\sigma_j(t,x_1,0,k)$, $k\neq 0$ are constant multiples of $\hat w(t,x_1,k)$, where $\hat w(t,x_1,k)$ is the solution of the amplitude equation \eqref{propelas}.
The building block estimates for $u_\sigma-u_\sigma^0$ and $u_\tau-u_\tau^0$ are readily seen to be the same as the estimates for $u_\sigma$ and $u_\tau$ given in section {\rm Re }\, f{bblock}, except for the change that every occurrence of $p$ is replaced by the number $1$. For example, parallel to the third estimate in Proposition {\rm Re }\, f{qq6} we now have instead
\begin{align}\langle bel{t1}
\langle ngle\Lambda^{\frac{1}{2}}\partial_{zz} (u_\tau-u_\tau^0)\rangle ngle_{m,\gamma}\lesssim \frac{C(m+3)}{{\varepsilon}^{1/2}}+C(m+3+\frac{1}{2}).
\end{align}
Additional work beyond what is given in chapter {\rm Re }\, f{chapter4} is needed to determine $u_\sigma^0$ and $u_\tau^0$, and this work has essentially been carried out in Chapter 2 of the thesis of Marcou \cite{Mar}. In that chapter Marcou considers a simplified version of the elasticity equations in which a number of the nonlinear terms, including quadratic terms, have been thrown away. Nevertheless, her method does apply directly to the full equations and shows that $u_\sigma^0=0$ and that $u_\tau^{0,*}$ is given by a simple integral formula.\footnote{This formula is provided in equations (2.6.5), (2.6.6) of \cite{Mar}, where $k=3$ and $H^0_{k-1}$, $K^0_{k-1}$ should be replaced by the analogous (quadratic) terms which appear in the full elasticity equations.} Since we are only concerned with solving the profile equations \eqref{o3}, \eqref{o4}, we have no need to construct the mean $\underline u^0_\tau(t,x)$; we can set it equal to $0$. To complete the building block estimates one needs to combine estimates like \eqref{t1} with corresponding estimates of $u^0_\tau=u_\tau^{0,*}$. Such estimates follow readily from the integral formulas given in \cite{Mar}. One finds in each case that the estimate already obtained (or a better one) continues to hold for $u^0_\tau$. Thus, for example, \eqref{t1} continues to hold when $u_\tau-u^0_\tau$ is replaced by $u_\tau$.
The forcing estimates of section {\rm Re }\, f{forcing} are adapted to the wavetrain case by using the building block estimates, modified in the way we have just described, as before, and observing that terms involving positive powers of $p$ on the right in the estimates of section {\rm Re }\, f{forcing} always arise from forcing terms in which the low frequency cutoff $\chi_{\varepsilon}(D_\theta)$ appears. Such $p$-terms are therefore absent now. Thus, for example, in the estimates of Corollary {\rm Re }\, f{r3a}, only the term $\sqrt{{\varepsilon}}$ should appear on the right. In estimates (c) and (d) of Corollary {\rm Re }\, f{r4a} the right sides are now ${\varepsilon}^2$ and ${\varepsilon}^{3/2}$, respectively. Similarly, the last term on the right in the estimate of Proposition {\rm Re }\, f{s5} is now ${\varepsilon}$, and on the right side of Proposition
{\rm Re }\, f{endgame},
\begin{align}
\frac{1}{\sqrt{\gamma}}(\frac{p^2}{\sqrt{{\varepsilon}}}+\sqrt{p}+\frac{{\varepsilon}}{p})+\sqrt{{\varepsilon}}
\end{align}
should be replaced by $\sqrt{{\varepsilon}}$. The same substitution should be made in the estimate of Corollary {\rm Re }\, f{s22}, and this yields the rate of convergence
\begin{align}
\begin{split}
&E_{m,T}(w)\lesssim \sqrt{{\varepsilon}}
\end{split}
\end{align}
in Corollary {\rm Re }\, f{s24a}.
\begin{rem}\langle bel{optimal}
We believe it should be possible to construct arbitrarily high order approximate solutions to the elasticity equations in the wavetrain case, and then to show that these approximate solutions are close to exact solutions on an ${\varepsilon}$-independent time interval by some variant of the method introduced by \cite{Gues}. With high order approximate solutions it should not be necessary to consider singular problems; one attempts to solve directly the error equation satisfied by the difference between exact and approximate solutions.
Such a result would yield more precise information than the result for wavetrains described above. For example, the second chapter of \cite{Mar} indicates that one should expect ``internal rectification" to occur in the leading corrector; in other words one expects $u_\tau$ to have in general a nonzero mean $\underline u_\tau^0(t,x)$. That property cannot be detected by our result for wavetrains, but could be detectable using this alternative method. As we have already mentioned, there is no hope of constructing high order approximate solutions in the case of pulses.
\end{rem}
\section{The case of dimensions $d\geq 3$. }\langle bel{higherD}
\emph{\quad}In this section we restrict attention to the Saint Venant-Kirchoff model \eqref{a0}. For $d\geq 3$ the solution of the amplitude equation and the construction of the approximate solution goes through as in $d=2$. However, there is a serious difficulty in $d\geq 3$ in constructing Kreiss symmetrizers for the linearized problem.
As we explain below, the linearized problem has characteristics of variable multiplicity; moreover, these include characteristics which fail to be \emph{algebraically regular} in the sense of \cite{MZ}, and which are at the same time \emph{glancing}.
We now let $\Omega=\{(t,x):x=(x_1,\dots,x_d), x_d>0\}$ and denote dual variables by $(\sigma,\xi)=(\sigma,\xi_1,\dots,\xi_d)$. We regard $\xi$ as a column vector, so $\xi\xi^t$ is the $d\times d$ matrix $(a_{ij}) = (\xi_i\xi_j)$. Writing $\phi(t,x)=x+U(t,x)$ as in \eqref{a2} and setting $\theta(t,x)={\nabla}bla U$, one obtains for the $d\times d$ principal matrix symbol of the interior equation in \eqref{a0} linearized at ${\nabla}bla\phi=I+\theta$:\footnote{Recall that $E(I+\theta)=\frac{1}{2}(\theta^t\theta+\theta+\theta^t)$.}
\begin{align}\langle bel{u0}
\begin{split}
&P(\theta,\sigma,\xi)=-\sigma^2I +(\langle mbda+\mu)(I+\theta)\xi\xi^t (I+\theta)^t+\mu|\xi|^2(I+\theta)(I+\theta^t)+\\
&\qquad \qquad \left[(\langle mbda \mathrm{tr}\;E(I+\theta)-\mu)|\xi|^2+\mu\xi^t(I+\theta)^t(I+\theta)\xi\right]I.
\end{split}
\end{align}
When $\theta=0$, we have $E(I)=0$ and this reduces to $-\sigma^2I+(\langle mbda+\mu)\xi\xi^t+\mu |\xi|^2I$, a system for which the characteristic equation is
\begin{align}\langle bel{u1}
(\sigma^2-\mu |\xi|^2)^{d-1}\;(\sigma^2-(\langle mbda+2\mu)|\xi|^2)=0.
\end{align}
The roots $\sigma=\pm\sqrt{\mu}|\xi|$, $\sigma=\pm\sqrt{\langle mbda+2\mu}\;|\xi|$ have multiplicities $d-1$ and $1$ respectively. Here the factors $\sigma-c|\xi|$ occurring in
\eqref{u1} have double roots in $\xi_d$ when $\sigma=c|(\xi_1,\dots,\xi_{d-1},0)|$. We then refer to $(\theta=0,\sigma,\xi_1,\dots,\xi_{d-1},0)$ as a \emph{glancing mode}.
\begin{prop}\langle bel{u2}
Assume $d=3$. Let $\underline{\xi}=(1,0,0)$ and
consider the glancing mode $(\theta=0,\sqrt{\mu},\underline{\xi})$, where $\sigma=\sqrt{\mu}$ is a root of multiplicity two of $P(0,\sigma,\underline{\xi})=0$.
For $\theta$ near $0$ the double root splits into roots $\sigma_j(\theta)$ satisfying $P(\theta,\sigma_j(\theta),\underline{\xi})=0$, where
\begin{align}
\sigma_j(\theta)=\sqrt{\mu}+n_j(\theta), \;j=1,2
\end{align}
and the $n_j(\theta)$ are distinct, continuous functions of $\theta$ which fail to be $C^1$ at $\theta=0$.
\end{prop}
\begin{proof}
The only things to check are that the $n_j$ are distinct and fail to be $C^1$ at $\theta=0$. We choose $\theta$ so that $\theta_{ij}=0$, except for the
$\theta_{23}$ and $\theta_{33}$ components which vary near $0$. Write
\begin{align}
P(\theta,\sigma,\xi)+\sigma^2I=M(\theta,\xi)+S(\theta,\xi)I,
\end{align}
where $S(\theta,\xi)I$ is given by the second line of \eqref{u0}. For this choice of $\theta$, the matrix $M(\theta,\underline{\xi})+S(\theta,\underline{\xi})I$ has the form
\begin{align}
\begin{pmatrix}\langle mbda+2\mu+S&0&0\\0&\mu+S+\mu\theta_{23}^2&\mu\theta_{23}\\0&\mu\theta_{23}&\mu+S+\mu(\theta_{33}^2+\theta_{33})\end{pmatrix}.
\end{align}
The eigenvalues of the lower $2\times 2$ block are computed to be
\begin{align}
\beta_{\pm}=(\mu+S(\theta,\underline{\xi}))+\frac{\mu}{2}(\theta_{23}^2+\theta_{33}^2+\theta_{33})\mp\frac{\mu}{2}\sqrt{[\theta_{23}^2-(\theta_{33}^2+\theta_{33})]^2+4\theta_{23}^2}.
\end{align}
The argument of the square root is $\theta_{33}^2+4\theta_{23}^2+O(|\theta|^3)$, so the square root term fails to be $C^1$ in $\theta$ near $\theta=0$.
\end{proof}
\begin{rem}
1. We refer to \cite{MZ} for the precise definition of algebraically regular multiple characteristics. Roughly, a characteristic mode of multiplicity $m\geq 2$ is algebraically regular when it splits smoothly with respect to small changes of parameters. Thus, the glancing mode in Proposition {\rm Re }\, f{u2} fails to be algebraically regular.
2. The state of the art in Kreiss symmetrizers is represented by the papers \cite{Met} and \cite{MZ}. The first paper constructs smooth Kreiss symmetrizers for symmetric hyperbolic systems, including high order systems, but only for systems with characteristics of constant multiplicity. Since we are dealing here with characteristics of variable multiplicity, the first paper does not apply.
The second paper treats only first order hyperbolic systems, and constructs smooth Kreiss symmetrizers in certain situations where characteristics of variable multiplicity are present.
The extension of the results of \cite{MZ} to higher order systems appears to be nontrivial, and has not yet been done as far as we know.
Moreover, the results of \cite{MZ} do not appear to cover first order problems with characteristic modes of multiplicity $m\geq 2$ that fail to be algebraically regular and are at the same time glancing.
Thus, Proposition {\rm Re }\, f{u2} indicates that even if we had an extension of \cite{MZ} to higher order systems, it might not apply to the kinds of variable multiplicity characteristics that we encounter in this problem. If Theorems {\rm Re }\, f{uniformexistence} and {\rm Re }\, f{approxthm} do actually extend to $d\geq 3$, it appears that further development of the theory of smooth Kreiss symmetrizers will be necessary to prove such an extension by our methods.
\end{rem}
\appendix
\chapter{Singular pseudodifferential calculus for pulses}
\langle bel{calculus}
In this Appendix, we summarize the parts of the singular pulse calculus constructed in \cite{CGW} that
are needed in Chapter {\rm Re }\, f{chapter3}. The calculus of \cite{CGW} was constructed with applications to
first-order hyperbolic systems in mind. The systems of elasticity equations considered in this work are
second-order, so some extensions of the calculus are needed; these are given in section {\rm Re }\, f{commutator}
below.
First we define the singular Sobolev spaces used to describe mapping properties. The variable in ${\mathbb R}^{d+1}$
is denoted $(x,\theta)$, $x \in {\mathbb R}^d$, $\theta \in {\mathbb R}$, and the associated frequency is denoted $(\xi,k)$. We
consider a fixed vector $\beta \in {\mathbb R}^d \setminus \{ 0\}$. Then for $s \in {\mathbb R}$ and ${\varepsilon} \in \, (0,1]$, the
anisotropic Sobolev space $H^{s,{\varepsilon}} ({\mathbb R}^{d+1})$ is defined by
\begin{equation*}
H^{s,{\varepsilon}}({\mathbb R}^{d+1}) := {\mathbb B}ig\{ u \in {\mathcal S}'({\mathbb R}^{d+1}) \, / \, \widehat{u} \in L^2_{\rm loc}({\mathbb R}^{d+1}) \,
\text{\rm and} \, \int_{{\mathbb R}^{d+1}} \left( 1+\left| \xi+\dfrac{k \, \beta}{{\varepsilon}} \right|^2 \right)^s
\, \big| \widehat{u}(\xi,k) \big|^2 \, {\rm d}\xi \, {\rm d}k <+\infty {\mathbb B}ig\} \, .
\end{equation*}
Here $\widehat{u}$ denotes the Fourier transform of $u$ on ${\mathbb R}^{d+1}$. The space $H^{s,{\varepsilon}}({\mathbb R}^{d+1})$ is
equipped with the family of norms
\begin{equation*}
\forall \, \gamma \ge 1 \, ,\quad \forall \, u \in H^{s,{\varepsilon}}({\mathbb R}^{d+1}) \, ,\quad
\| u \|_{H^{s,{\varepsilon}},\gamma}^2 := \dfrac{1}{(2\, \pi)^{d+1}} \, \int_{{\mathbb R}^{d+1}}
\left( \gamma^2 +\left| \xi+\dfrac{k \, \beta}{{\varepsilon}} \right|^2 \right)^s
\, \big| \widehat{u}(\xi,k) \big|^2 \, {\rm d}\xi \, {\rm d}k \, .
\end{equation*}
When $m$ is an integer, the space $H^{m,{\varepsilon}} ({\mathbb R}^{d+1})$ coincides with the space of functions $u \in L^2
({\mathbb R}^{d+1})$ such that the derivatives, in the sense of distributions,
\begin{equation*}
\left( \partial_{x_1} +\dfrac{\beta_1}{{\varepsilon}} \, \partial_\theta \right)^{\alpha_1} \dots
\left( \partial_{x_d} +\dfrac{\beta_d}{{\varepsilon}} \, \partial_\theta \right)^{\alpha_d} \, u \, ,\quad
\alpha_1+\dots+\alpha_d \le m \, ,
\end{equation*}
belong to $L^2 ({\mathbb R}^{d+1})$. In the definition of the norm $\| \cdot \|_{H^{m,{\varepsilon}},\gamma}$, one power of
$\gamma$ counts as much as one derivative.
\section{Symbols}
Our singular symbols are built from the following sets of classical symbols.
\begin{defn}\langle bel{n1}
Let $\cO\subset {\mathbb R}^N$ be an open subset that contains the origin. For $m\in{\mathbb R}$ we let ${\bf S}^m(\cO)$ denote
the class of all functions $\sigma:\cO\times {\mathbb R}^d\times [1,\infty)\to {\mathbb C}^{N \times N}$, $N \ge 1$, such that
$\sigma$ is $\cC^\infty$ on $\cO \times {\mathbb R}^d$ and for all compact sets $K\subset \cO$:
\begin{equation*}
\sup_{v\in K} \, \sup_{\xi'\in{\mathbb R}^d} \, \sup_{\gamma\geq 1} \, (\gamma^2+|\xi|^2)^{-(m-|\nu|)/2} \,
|\partial^\alpha_v\partial_{\xi'}^\nu \sigma(v,\xi,\gamma)| \leq C_{\alpha,\nu,K}.
\end{equation*}
\end{defn}
Let ${\mathcal C}^k_b({\mathbb R}^{d+1})$, $k \in {\mathbb N}$, denote the space of continuous and bounded functions
on ${\mathbb R}^{d+1}$, whose derivatives up to order $k$ are continuous and bounded. Let us first define the
singular symbols.
\begin{defn}[Singular symbols]
\langle bel{def4}
Fix $\beta\in{\mathbb R}^d\setminus 0$, let $m \in {\mathbb R}$, and let $n \in {\mathbb N}$. Then we let $S^m_n$ denote the set of
families of functions $(a^{{\varepsilon},\gamma})_{{\varepsilon} \in (0,1],\gamma \ge 1}$ that are constructed as follows:
\begin{equation}
\langle bel{singularsymbolp}
\forall \, (x,\theta,\xi,k) \in {\mathbb R}^{d+1} \times {\mathbb R}^{d+1} \, ,\quad a^{{\varepsilon},\gamma} (x,\theta,\xi,k) =
\sigma \left( {\varepsilon} \, V(x,\theta),\xi+\dfrac{k \, \beta}{{\varepsilon}},\gamma \right) \, ,
\end{equation}
where $\sigma \in {\bf S}^m({\mathcal O})$, $ V$ belongs to the space ${\mathcal C}^n_b ({\mathbb R}^{d+1})$ and
where furthermore $V$ takes its values in a convex compact subset $K$ of ${\mathcal O}$ that contains
the origin (for instance $K$ can be a closed ball centered round the origin).
\end{defn}
All results below extend to the case where in place of a function $V$ that is independent of ${\varepsilon}$, the
representation \eqref{singularsymbolp} is considered with a function $V_{\varepsilon}$ that is indexed by ${\varepsilon}$,
provided that we assume that all functions ${\varepsilon} \, V_{\varepsilon}$ take values in a {\it fixed} convex compact
subset $K$ of ${\mathcal O}$ that contains the origin, and $(V_{\varepsilon})_{{\varepsilon} \in (0,1]}$ is a bounded family
of ${\mathcal C}^n_b ({\mathbb R}^{d+1})$.
\section{Definition of operators and action on Sobolev spaces}
\langle bel{sect8}
To each symbol $a = (a^{{\varepsilon},\gamma})_{{\varepsilon} \in (0,1],\gamma \ge 1} \in S^m_n$ given by the formula
\eqref{singularsymbolp}, we associate a singular pseudodifferential operator $a^{{\varepsilon},\gamma}_D$, with
${\varepsilon} \in (0,1]$ and $\gamma \ge 1$, whose action on a function $u \in {\mathcal S} ({\mathbb R}^{d+1};{\mathbb C}^N)$ is
defined by
\begin{equation}
\langle bel{singularpseudop}
a^{{\varepsilon},\gamma}_D \, u \, (x,\theta) := \dfrac{1}{(2\, \pi)^{d+1}} \, \int_{{\mathbb R}^{d+1}} {\rm e}^{i\, (\xi \cdot x +k \, \theta)}
\, \sigma \left( {\varepsilon} \, V(x,\theta),\xi+\dfrac{k \, \beta}{{\varepsilon}},\gamma \right) \, \widehat{u} (\xi,k)
\, {\rm d}\xi \, {\rm d}k \, .
\end{equation}
Let us briefly note that for the Fourier multiplier $\sigma (v,\xi,\gamma) =i\, \xi_1$, the corresponding
singular operator is $\partial_{x_1} +(\beta_1/{\varepsilon}) \, \partial_\theta$. We now describe the action of
singular pseudodifferential operators on Sobolev spaces. Detailed proofs of all results stated below
can be found in \cite{CGW}.
\begin{rem}
\langle bel{8a}
\textup{We will usually write $a_D \, u$ instead of $a^{{\varepsilon},\gamma}_D \, u$ for the function defined in
\eqref{singularpseudop}. Also we often write $X$ for $\xi+k \, \beta/{\varepsilon}$.}
\end{rem}
\begin{prop}
\langle bel{prop13}
Let $n \ge d+1$, and let $a \in S^m_n$ with $m \le 0$. Then $a_D$ in \eqref{singularpseudop} defines
a bounded operator on $L^2 ({\mathbb R}^{d+1})$: there exists a constant $C>0$, that only depends on $\sigma$
and $V$ in the representation \eqref{singularsymbolp}, such that for all ${\varepsilon} \in (0,1]$ and for all
$\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad \left\| a_D \, u \right\|_0 \le \dfrac{C}{\gamma^{|m|}} \, \| u \|_0 \, .
\end{equation*}
\end{prop}
\noindent The constant $C$ in Proposition {\rm Re }\, f{prop13} depends uniformly on the compact set in which $V$
takes its values and on the norm of $V$ in ${\mathcal C}^{d+1}_b$. For operators defined by symbols of order
$m>0$, we have:
\begin{prop}
\langle bel{prop14}
Let $n \ge d+1$, and let $a \in S^m_n$ with $m>0$. Then $a_D$ in \eqref{singularpseudop} defines
a bounded operator from $H^{m,{\varepsilon}}({\mathbb R}^{d+1})$ to $L^2 ({\mathbb R}^{d+1})$: there exists a constant $C>0$, that
only depends on $\sigma$ and $V$ in the representation \eqref{singularsymbolp}, such that for all ${\varepsilon} \in
(0,1]$ and for all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad \left\| a_D \, u \right\|_0 \le C \, \| u \|_{H^{m,{\varepsilon}},\gamma} \, .
\end{equation*}
\end{prop}
\noindent The next proposition describes the smoothing effect of operators of order $-1$.
\begin{prop}
\langle bel{prop15}
Let $n \ge d+2$, and let $a \in S^{-1}_n$. Then $a_D$ in \eqref{singularpseudop} defines a bounded
operator from $L^2 ({\mathbb R}^{d+1})$ to $H^{1,{\varepsilon}}({\mathbb R}^{d+1})$: there exists a constant $C>0$, that only depends
on $\sigma$ and $V$ in the representation \eqref{singularsymbolp}, such that for all ${\varepsilon} \in (0,1]$ and for
all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad \left\| a_D \, u \right\|_{H^{1,{\varepsilon}},\gamma} \le C \, \| u \|_0 \, .
\end{equation*}
\end{prop}
\begin{rem}
\langle bel{a4}
\textup{In applications of the pulse calculus, we verify the hypothesis that for $V$ as in \eqref{singularsymbolp},
$V \in \mathcal{C}^n_b({\mathbb R}^{d+1})$, by showing $V\in H^s({\mathbb R}^{d+1})$ for some $s>\frac{d+1}{2}+n$.}
\end{rem}
\section{Adjoints and products}
\langle bel{sect9}
For proofs of the following results we refer again to \cite{CGW}. The two first results deal with adjoints of singular
pseudodifferential operators while the last two deal with products.
\begin{prop}
\langle bel{prop18}
Let $a=\sigma({\varepsilon} V,X,\gamma) \in S_n^0$, $n \ge 2\, (d+1)$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+1$, and let $a^*$ denote the conjugate transpose of the symbol $a$. Then $a_D$
and $(a^*)_D$ act boundedly on $L^2$ and there exists a constant $C \ge 0$ such that for all ${\varepsilon} \in
(0,1]$ and for all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
\left\| (a_D)^* \, u-(a^*)_D \, u \right\|_0 \le \dfrac{C}{\gamma} \, \| u \|_0 \, .
\end{equation*}
If $n \ge 3\, d +3$, then for another constant $C$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
\left\| (a_D)^* \, u-(a^*)_D \, u \right\|_{H^{1,{\varepsilon}},\gamma} \le C \, \| u \|_0 \, ,
\end{equation*}
uniformly in ${\varepsilon}$ and $\gamma$.
\end{prop}
\begin{prop}
\langle bel{prop19}
Let $a=\sigma({\varepsilon} V,X,\gamma) \in S_n^1$, $n \ge 3\, d +4$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+1$, and let $a^*$ denote the conjugate transpose of the symbol $a$. Then $a_D$
and $(a^*)_D$ map $H^{1,{\varepsilon}}$ into $L^2$ and there exists a family of operators $R^{{\varepsilon},\gamma}$
that satisfies
\begin{itemize}
\item there exists a constant $C \ge 0$ such that for all ${\varepsilon} \in (0,1]$ and for all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad \left\| R^{{\varepsilon},\gamma} \, u \right\|_0 \le C \, \| u \|_0 \, ,
\end{equation*}
\item the following duality property holds
\begin{equation*}
\forall \, u,v \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
\langle ngle a_D \, u,v \rangle ngle_{L^2} -\langle ngle u, (a^*)_D \, v \rangle ngle_{L^2}
=\langle ngle R^{{\varepsilon},\gamma} \, u,v \rangle ngle_{L^2} \, .
\end{equation*}
In particular, the adjoint $(a_D)^*$ for the $L^2$ scalar product maps $H^{1,{\varepsilon}}$ into $L^2$.
\end{itemize}
\end{prop}
\begin{prop}
\langle bel{prop20}
(a)\; Let $a,b \in S_n^0$, $n \ge 2\, (d+1)$, and suppose $b=\sigma({\varepsilon} V,X,\gamma)$ where $V \in
H^{s_0}({\mathbb R}^{d+1})$ for some $s_0>\frac{d+1}{2}+1$. Then there exists a constant $C \ge 0$ such that
for all ${\varepsilon} \in (0,1]$ and for all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
\left\| a_D \, b_D \, u -(a \, b)_D \, u \right\|_0 \le \dfrac{C}{\gamma} \, \| u \|_0 \, .
\end{equation*}
If $n \ge 3\, d +3$, then for another constant $C$, there holds
\begin{equation}\langle bel{prop20z}
\begin{split}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
& \left\| a_D \, b_D \, u -(a \, b)_D \, u \right\|_{H^{1,{\varepsilon}},\gamma} \le C \, \| u \|_0 \, ,\\
& \left\| a_D \, b_D \, u -(a \, b)_D \, u \right\|_0 \le C \, \| u \|_{H^{-1,{\varepsilon}},\gamma} \, ,
\end{split}
\end{equation}
uniformly in ${\varepsilon}$ and $\gamma$.
(b)\; Let $a \in S_n^1,b \in S_n^0$ or $a \in S_n^0,b \in S_n^1$, $n \ge 3\, d +4$, and in each case suppose
$b=\sigma({\varepsilon} V,X,\gamma)$ where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some $s_0>\frac{d+1}{2}+1$. Then there
exists a constant $C \ge 0$ such that for all ${\varepsilon} \in (0,1]$ and for all $\gamma \ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad \left\| a_D \, b_D \, u -(a\, b)_D \, u \right\|_0 \le C \, \| u \|_0 \, .
\end{equation*}
\end{prop}
\noindent The proof of the first estimate of \eqref{prop20z} in \cite{CGW} gives an explicit amplitude for
the remainder $a_D \, b_D \, u -(a\, b)_D \, u$, and from this it is clear that the adjoint of $a_D \, b_D \, u
-(a\, b)_D \, u$ has the same mapping property. Duality therefore implies the second estimate in
\eqref{prop20z}.
\begin{prop}
\langle bel{prop21}
Let $a \in S_n^{-1},b \in S_n^1$, $n \ge 3\, d +4$, and suppose $b=\sigma({\varepsilon} V,X,\gamma)$ where $V
\in H^{s_0}({\mathbb R}^{d+1})$ for some $s_0>\frac{d+1}{2}+1$. Then $a_D \, b_D$ defines a bounded operator
on $H^{1,{\varepsilon}}$ and there exists a constant $C \ge 0$ such that for all ${\varepsilon} \in (0,1]$ and for all $\gamma
\ge 1$, there holds
\begin{equation*}
\forall \, u \in {\mathcal S} ({\mathbb R}^{d+1}) \, ,\quad
\left\| a_D \, b_D \, u -(a\, b)_D \, u \right\|_{H^{1,{\varepsilon}},\gamma} \le C \, \| u \|_0 \, .
\end{equation*}
\end{prop}
\noindent Our next result is G{\aa}rding's inequality, either for symbols of degree $0$ or $1$.
\begin{theo}
\langle bel{thm11}
(a) Let $\sigma \in {\bf S}^0$ satisfy $\text{\rm Re} \, \sigma (v,\xi,\gamma) \ge C_K>0$ for all $v$ in a compact
subset $K$ of ${\mathcal O}$. Let now $a \in S^0_n$, $n \ge 2\, d+2$ be given by $a=\sigma({\varepsilon} V,X,\gamma)$,
where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some $s_0>\frac{d+1}{2}+1$ and is valued in a convex compact subset
$K$. Then for all $\delta >0$, there exists $\gamma_0$ which depends uniformly on $V$, the constant $C_K$
and $\delta$, such that for all $\gamma \ge \gamma_0$ and all $u \in {\mathcal S}({\mathbb R}^{d+1})$, there holds
\begin{equation*}
\text{\rm Re } \langle ngle a_D \, u ;u \rangle ngle_{L^2} \ge (C_K-\delta) \, \| u \|_0^2 \, .
\end{equation*}
(b) Let $\sigma \in {\bf S}^1$ satisfy $\text{\rm Re} \, \sigma (v,\xi,\gamma) \ge C_K \, \langle ngle\xi,\gamma\rangle ngle$
for all $v$ in a compact subset $K$ of ${\mathcal O}$. Let now $a \in S^1_n$, $n \ge 3d+4$ be given by
$a=\sigma({\varepsilon} V,X,\gamma)$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some $s_0>\frac{d+1}{2}+1$ and is valued
in a convex compact subset $K$. Then for all $\delta >0$, there exists $\gamma_0$ which depends uniformly
on $V$, the constant $C_K$ and $\delta$, such that for all $\gamma \ge \gamma_0$ and all $u \in {\mathcal S}
({\mathbb R}^{d+1})$, there holds
\begin{equation*}
\text{\rm Re } \langle ngle a_D \, u ;u \rangle ngle_{L^2} \ge (C_K-\delta) \, \| u \|_{H^{1/2,{\varepsilon}},\gamma}^2 \, .
\end{equation*}
\end{theo}
\section{Extended calculus}
\langle bel{extended}
At times we use a slight extension of the singular calculus. For given parameters $0<\delta_1<\delta_2<1$, we
choose a cutoff $\chi^e (\xi,k\, \beta/{\varepsilon},\gamma)$ such that
\begin{align}\langle bel{n31}
\begin{split}
&0\leq \chi^e \leq 1\, ,\\
&\chi^e \left( \xi,\dfrac{k\, \beta}{{\varepsilon}},\gamma \right) =1 \text{ on } \left\{
(\gamma^2 +|\xi|^2)^{1/2} \leq \delta_1 \, \left| \dfrac{k\, \beta}{{\varepsilon}} \right| \right\} \, ,\\
&\mathrm{supp }\chi^e \subset \left\{ (\gamma^2 +|\xi|^2)^{1/2} \leq \delta_2 \, \left| \dfrac{k\, \beta}{{\varepsilon}} \right|
\right\} \, ,
\end{split}
\end{align}
and define a corresponding Fourier multiplier $\chi_D$ in the extended calculus by the formula
\eqref{singularpseudop} with $\chi^e (\xi,k\, \beta/{\varepsilon},\gamma)$ in place of $\sigma({\varepsilon} V,X,\gamma)$.
Composition laws involving such operators are proved in \cite{CGW}, but here we need only the fact that
part {\it (a)} of Proposition {\rm Re }\, f{prop20} (composition of two zero order singular operators) holds when either
$a$ or $b$ is replaced by an extended cutoff $\chi^e$.
\section{Commutator estimates}\langle bel{commutator}
\emph{\quad} In the proofs of this section we ignore some constant factors such as powers of $2\pi$.
For a given amplitude $c(x,\theta,y,\omega,X,\gamma)$, $\widetilde{Op}(c)$ denotes the operator defined by
\begin{align}
\widetilde{Op}(c) u (x,\theta)=\int e^{i(x-y)\xi+i(\theta-\omega)k} c(x,\theta,y,\omega,X,\gamma)u(y,\omega)dyd\omega d\xi dk.
\end{align}
\begin{lem}\langle bel{n40}
Let $r\geq 0$ and let $M_{e^{iy\eta+i\omega l}}$ denote the operator that multiplies $u(y,\omega)$ by $e^{iy\eta+i\omega l}$. Then
\begin{align}
\|\Lambda_D^rM_{e^{iy\eta+i\omega l}}u\|_0\lesssim \|\Lambda^r_Du\|_0+ \langle ngle\eta\rangle ngle^r\|u\|_0+\left\langle ngle\frac{l}{{\varepsilon}}\right\rangle ngle^r\|u\|_0.
\end{align}
\end{lem}
\begin{proof}
This follows directly from the definition of the norm on the left and
\begin{align}
\left\langle ngle \xi+\eta+\frac{(k+l)\beta}{{\varepsilon}},\gamma \right\rangle ngle^r\lesssim \langle ngle X,\gamma\rangle ngle^r+\langle ngle\eta\rangle ngle^r+\left\langle ngle \frac{l}{{\varepsilon}}\right\rangle ngle^r.
\end{align}
\end{proof}
\begin{prop}\langle bel{commutator1}
i.) Set $x=(x_0,x'')=(t,x'')\in {\mathbb R}^{d+1}$ and $y=(y_0,y'')\in{\mathbb R}^{d+1}$, let $a(X,\gamma)$ be a singular symbol of order $r\geq 1$, and let $\chi(x_0)\in H^\infty({\mathbb R})$. Then for $m\in \{0,1,2,\dots\}$ we have
\begin{align}
\begin{split}
&(a)\; |[a_D,\chi] u|_{H^m_\gamma}\lesssim |\Lambda^{r-1}_D u|_{H^m_\gamma}\\
&(b)\;|[a_D,\chi] u|_{m,\gamma}\lesssim |\Lambda^{r-1}_D u|_{m,\gamma}.
\end{split}
\end{align}
ii.) The same estimates hold when $\chi(x)\in H^\infty({\mathbb R}^{d+1})$.
iii.) When $\chi(x,\theta)\in H^\infty({\mathbb R}^{d+1}\times{\mathbb R})$, we have\footnote{It is easy to give a version of this proposition for $\chi\in H^s$ for $s$ large enough.}
\begin{align}
|[a_D,\chi] u|_{m,\gamma}\lesssim |\Lambda^{r-1}_D u|_{m,\gamma}+\frac{|u|_{m,\gamma}}{{\varepsilon}^{r-1}}.
\end{align}
\end{prop}
\begin{proof}
\textbf{1. }It suffices to prove (a) for $m=0$, since derivatives with respect to $(x'',\theta)$ commute with the commutator and $\partial^k_t ([a_D,\chi]u)$ is a linear combination of terms of the form
\begin{align}
[a_D,\chi^{(k_1)}]\partial_t^{k_2}u, \text{ where }k_1+k_2=k.
\end{align}
Replacing $u$ by $e^{-\gamma t}u$ then yields (b).
\textbf{2. }The function $[a_D,\chi]u$ may be written as (some constant factors are ignored here and below)
\begin{align}\langle bel{n32}
\begin{split}
&\int e^{i(x-y)\xi+i(\theta-\omega)k}a(X,\gamma)[\chi(x_0)-\chi(y_0)]u(y,\omega)dyd\omega d\xi dk=\\
&\qquad i\int e^{i(x-y)\xi+i(\theta-\omega)k}\partial_{\xi_0}a(X,\gamma)\left(\int^1_0\chi'(y_0+s(x_0-y_0))ds\right)u(y,\omega)dyd\omega d\xi dk.
\end{split}
\end{align}
Write $\chi'(y_0+s(x_0-y_0))=\int\widehat{\chi'}(\eta_0)e^{i\eta_0(y_0+s(x_0-y_0))}d\eta_0$, let $\widetilde{Op}(A_{s,\eta_0})$ be the operator associated
to the amplitude
of order $r-1$ given by
\begin{align}
e^{i\eta_0s x_0}\cdot \partial_{\xi_0}a(X,\gamma)\cdot e^{i\eta_0(1-s)y_0},
\end{align}
and observe that
\begin{align}\langle bel{n33}
[a_D,\chi]=\int^1_0\int \widetilde{Op}(A_{s,\eta_0})\;\widehat{\chi'}(\eta_0)d\eta_0 ds.
\end{align}
We have
\begin{align}
\widetilde{Op}(A_{s,\eta_0})=M_{e^{i\eta_0s x_0}} \circ (\partial_{\xi_0}a)_D \circ M_{e^{i\eta_0(1-s)y_0}},
\end{align}
where $M$ denotes multiplication. Thus, by Lemma {\rm Re }\, f{n40}
\begin{align}\langle bel{n34}
|\widetilde{Op}(A_{s,\eta_0}) u|_{L^2}\lesssim |\Lambda^{r-1}_D( e^{i\eta_0(1-s)y_0}u)|_{L^2}\lesssim |\Lambda_D^{r-1}u|_{L^2}+|\eta_0|^{(r-1)}|u|_{L^2}.
\end{align}
Together with \eqref{n33} this gives the estimate (a) when $m=0$.
\textbf{3. }The proofs of (ii) and (iii) involve only minor changes.
\end{proof}
\begin{prop}\langle bel{commutator3}
For $j=1,0$ let $a^j=\sigma^j({\varepsilon} V,X,\gamma) \in S_n^j$, $n \ge 3\, d +4$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+1$, and let $\phi(X,\gamma)$ be a singular symbol of order $0$. We have
\begin{align}
\begin{split}
&(a)\;\| [a^1_D,\phi_D] u \|_0 \lesssim \| u\|_0\\
&(b)\; \|[a^0_D,\phi_D] u \|_0 \lesssim \| \Lambda^{-1}_D u\|_0.
\end{split}
\end{align}
\end{prop}
\begin{proof}
Part (a) follows directly from Proposition {\rm Re }\, f{prop20}(b) and part (b) follows from Proposition {\rm Re }\, f{prop20}(a).
\end{proof}
\begin{prop}\langle bel{commutator4}
Let the singular symbols $a^j$ have the product form $a^j=\sigma({\varepsilon} V)b^j(X,\gamma) \in S^j_n$, $n \ge 3\, d +4$, $j=0,1$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+2$.
\begin{align}
\begin{split}
&(a)\;\| [a^1_D,\Lambda_D^{\frac{1}{2}}] u \|_0 \lesssim \| \Lambda^{\frac{1}{2}} u\|_0+\|u/\sqrt{{\varepsilon}}\|_0\\
&(b)\|\Lambda_D^{\frac{1}{2}} [a^1_D,\Lambda_D^{\frac{1}{2}}] u \|_0 \lesssim \| \Lambda_Du\|_0+\|\Lambda^{\frac{1}{2}}_Du/\sqrt{{\varepsilon}}\|_0+\|u/{\varepsilon}\|_0\\
&(c)\;\| [a^1_D,\Lambda_D^{-\frac{1}{2}}] u \|_0 \lesssim \gamma^{-\frac{1}{2}}\| u\|_0\\
&(d)\; \|[a^0_D,\Lambda^{\frac{1}{2}}] u \|_0 \lesssim \gamma^{-\frac{1}{2}}\| u\|_0.
\end{split}
\end{align}
\end{prop}
\begin{proof}
\textbf{1. Proof of (a). }We write\footnote{Here $(a^*)_D^*$ denotes the adjoint of $(a^*)_D$.}
\begin{align}
\Lambda^{\frac{1}{2}}_Da_D=\Lambda^{\frac{1}{2}}_D[(a^*)_D^*+R],
\end{align}
where the operator $R$ is given by the amplitude
\begin{align}
\begin{split}
&(\sigma({\varepsilon} V(x,\theta)-\sigma({\varepsilon} V(y,\omega)))b(X,\gamma)=r_1+r_2,\text{ with }\\
&\quad r_1=\left(\int^1_0 d_v\sigma({\varepsilon} V(y+s(x-y),\theta)))\; \;{\varepsilon}\partial_yV(y+s(x-y),\theta)ds\right) (x-y)b(X,\gamma)\\
&\quad r_2=\left(\int^1_0d_v\sigma({\varepsilon} V(y, \omega+s(\theta-\omega)))\;\;{\varepsilon} \partial_\omega V(y,\omega+s(\theta-\omega))ds\right) (\theta-\omega)b(X,\gamma).
\end{split}
\end{align}
The operator $a_D\Lambda^{\frac{1}{2}}_D-\Lambda^{\frac{1}{2}}_D(a^*)_D^*$ is given by the amplitude
\begin{align}
[(\sigma({\varepsilon} V(x,\theta))-\sigma({\varepsilon} V(y,\omega)))b(X,\gamma)]\Lambda^{\frac{1}{2}}=(r_1+r_2)\Lambda^{\frac{1}{2}}.
\end{align}
Thus, we have
\begin{align}\langle bel{n35}
[a_D,\Lambda^{\frac{1}{2}}_D]= \widetilde{Op}\left((r_1+r_2)\Lambda^{\frac{1}{2}}\right)-\Lambda^{\frac{1}{2}}_D \circ \widetilde{Op}(r_1+r_2).
\end{align}
\textbf{2. }We focus on the ``worst" terms in \eqref{n35}, those involving $r_2$. Integration by parts with respect to $k$ using
$\partial_k(e^{i(x-y)\xi+i(\theta-\omega)k})=i(\theta-\omega)e^{i(x-y)\xi+i(\theta-\omega)k}$ shows that
\begin{align}\langle bel{n36}
\widetilde{Op}\left(r_2\Lambda^{\frac{1}{2}}\right)=\widetilde{Op}(f),\text{ where }f(x,\theta,y,\omega,X,\gamma)=\int^1_0 F(y,\omega+s(\theta-\omega))ds\cdot c(X,\gamma),
\end{align}
with $c$ of order $\frac{1}{2}$ and $F(y,\omega)$ satisfying
\begin{align}\langle bel{n37}
\int \langle ngle\eta,l\rangle ngle |\hat{F}(\eta,l)|d\eta dl\lesssim |F(y,\omega)|_{H^{s_0-1}(y,\omega)}.
\end{align}
We have
\begin{align}
F(y,\omega+s(\theta-\omega))=\int e^{ils\theta}\hat{F}(\eta,l) e^{i\eta y+il\omega(1-s)}d\eta dl.
\end{align}
Letting $\widetilde{Op}(A_{s,\eta,l})$ be the operator associated to the amplitude $e^{ils\theta}\hat{F}(\eta,l) c(X,\gamma) e^{i\eta y+il\omega(1-s)}$, we have with obvious notation
\begin{align}\langle bel{n38}
\widetilde{Op}(A_{s,\eta,l})=M_{e^{ils\theta}}\hat{F}(\eta,l)c_D M_{ e^{i\eta y+il\omega(1-s)}}.
\end{align}
We obtain
\begin{align}\langle bel{n38a}
\begin{split}
&\|\widetilde{Op}(A_{s,\eta,l}) u\|_{0}=\|\hat{F}(\eta,l)c_D\Lambda^{-\frac{1}{2}}_D\Lambda^{\frac{1}{2}}_DM_{ e^{i\eta y+il\omega(1-s)}}u\|_0\lesssim\\
& \qquad |\hat{F}(\eta,l)|\|\Lambda^{\frac{1}{2}}_DM_{ e^{i\eta y+il\omega(1-s)}}u\|_0\lesssim |\hat{F}(\eta,l)|\left(\|\Lambda^{\frac{1}{2}}_D u\|_0+\langle ngle\eta\rangle ngle^{\frac{1}{2}}\|u\|_0+\langle ngle l\rangle ngle^{\frac{1}{2}}\frac{\|u\|_0}{\sqrt{{\varepsilon}}}\right),
\end{split}
\end{align}
where the last inequality follows from Lemma {\rm Re }\, f{n40}.
With \eqref{n36} and \eqref{n37} this implies
\begin{align}\langle bel{n38b}
\|\widetilde{Op}\left(r_2\Lambda^{\frac{1}{2}}\right)u\|_0\lesssim \left(\|\Lambda^{\frac{1}{2}}_D u\|_0+\frac{\|u\|_0}{\sqrt{{\varepsilon}}}\right).
\end{align}
To see that the operator $\Lambda^{\frac{1}{2}}_D\widetilde{Op}(r_2)$ in \eqref{n35} satisfies
\begin{align}\langle bel{n39}
\|\Lambda^{\frac{1}{2}}_D\widetilde{Op}(r_2) u\|_0\lesssim \left(\|\Lambda^{\frac{1}{2}}_D u\|_0+\frac{\|u\|_0}{\sqrt{{\varepsilon}}}\right),
\end{align}
we first decompose $\widetilde{Op}(r_2)$ into operators like those in \eqref{n38}, where now $c_D$ in \eqref{n38} has order zero. The estimate \eqref{n39} then follows readily.
The terms involving $r_1$ in \eqref{n35} are treated similarly, but satisfy slightly better estimates since, for example, one integrates by parts with respect to $\xi$ instead of $k$ to define the analogue of $f$ in \eqref{n36}.
\textbf{3. Proof of (b). }With $\widetilde{Op}(A_{s,\eta,l})$ as in \eqref{n38}, two applications of Lemma {\rm Re }\, f{n40} yield
\begin{align}\langle bel{n41}
\begin{split}
&\|\Lambda^{\frac{1}{2}}\widetilde{Op}(A_{s,\eta,l}) u\|_{0}\lesssim\\
&\quad |\hat{F}(\eta,l)|\left(\|\Lambda_D u\|_0+\langle ngle l\rangle ngle^{\frac{1}{2}}\frac{\|\Lambda_D^{\frac{1}{2}}u\|_0}{\sqrt{{\varepsilon}}}+\langle ngle\eta\rangle ngle\|u\|_0+\langle ngle\eta\rangle ngle^{\frac{1}{2}}\langle ngle l\rangle ngle^{\frac{1}{2}}\frac{\|u\|_0}{\sqrt{{\varepsilon}}}+\left\langle ngle\frac{l}{{\varepsilon}}\right\rangle ngle\|u\|_0\right),
\end{split}
\end{align}
so \textbf{(b)} follows from \eqref{n37}.
\textbf{4. } The proof of (c) is similar to the proof of (b) but simpler. For example, the operators $c_D$ that appear as in \eqref{n38} but now in the decomposition of
$\widetilde{Op}\left(r_2\Lambda^{-\frac{1}{2}}\right)$ have order $-\frac{1}{2}$, so the right side of \eqref{n38b} is replaced by $\gamma^{-\frac{1}{2}}\|u\|_0$. The proof of (d) is almost the same as that of (c).
\end{proof}
\begin{prop}\langle bel{commutator5}
Let the singular symbols $a^j$ have the product form $a^j=\sigma^j({\varepsilon} V)b^j(X,\gamma) \in S^j_n$, $n \ge 3\, d +4$, $j=0,1$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+2$.
\begin{align}\langle bel{comm5}
\begin{split}
&\|\Lambda_D^{\frac{1}{2}} [a^1_D,a^0_D] u \|_0 \lesssim \| \Lambda^{\frac{1}{2}} u\|_0+\|u/\sqrt{{\varepsilon}}\|_0.
\end{split}
\end{align}
\end{prop}
\begin{proof}
Using a classical ``**-argument" to analyze compositions\footnote{See the proof of Proposition 11 of \cite{CGW}, for example.}, we obtain
\begin{align}
\begin{split}
&[a^1_D,a^0_D] =\widetilde{Op} \;[a^1({\varepsilon} V(x,\theta),X,\gamma)(a^0({\varepsilon} V(x,\theta),X,\gamma)-a^0({\varepsilon} V(y,\omega),X,\gamma))]-\\
&\quad \qquad \widetilde{Op} \;[a^0({\varepsilon} V(x,\theta),X,\gamma)(a^1({\varepsilon} V(x,\theta),X,\gamma)-a^1({\varepsilon} V(y,\omega),X,\gamma))]+\\
&a^1_D\widetilde{Op} \;[a^0({\varepsilon} V(x,\theta),X,\gamma)-a^0({\varepsilon} V(y,\omega),X,\gamma)]-a^0_D\widetilde{Op} \;[a^1({\varepsilon} V(x,\theta),X,\gamma)-a^1({\varepsilon} V(y,\omega),X,\gamma)].
\end{split}
\end{align}
The product form of the symbols allows us to analyze these operators using Fourier decompositions as in step \textbf{2} of the proof of Proposition {\rm Re }\, f{commutator4}. The result then follows by application of Lemma {\rm Re }\, f{n40}.
\end{proof}
We also need an estimate like \eqref{comm5} for symbols that are \emph{not} of product form.
\begin{cor}
Let the singular symbols $a^j$ have the form $a^j=\sigma^j({\varepsilon} V,X,\gamma) \in S^j_n$, $n \ge 3\, d +4$, $j=0,1$, where $V\in H^{s_0}({\mathbb R}^{d+1})$ for some
$s_0>\frac{d+1}{2}+2$, and where $\sigma^j(w,\xi,\gamma)$ is \emph{homogeneous} of degree $j$ in $(\xi,\gamma)$. Then
\begin{align}
\begin{split}
&\|\Lambda_D^{\frac{1}{2}} [a^1_D,a^0_D] u \|_0 \lesssim \| \Lambda^{\frac{1}{2}} u\|_0+\|u/\sqrt{{\varepsilon}}\|_0.
\end{split}
\end{align}
\end{cor}
\begin{proof}
One can reduce to the product case considered above by expanding the symbols $\sigma^j(w,\xi,\gamma)$ in terms of spherical harmonics.
\end{proof}
\end{document} |
\begin{document}
\title{Detecting genuine multipartite continuous-variable entanglement}
\author{Peter van Loock$^1$ and Akira Furusawa$^2$}
\affiliation{$^1$ Quantum Information Theory Group,
Zentrum f\"{u}r Moderne Optik,
Universit\"{a}t Erlangen-N\"{u}rnberg,
91058 Erlangen, Germany\\
$^2$ Department of Applied Physics,
University of Tokyo, Tokyo 113-8656, Japan}
\begin{abstract}
We derive necessary conditions in terms of the variances of
position and momentum linear combinations
for all kinds of separability of a
multi-party multi-mode continuous-variable state.
Their violations can be sufficient for genuine multipartite
entanglement, provided the combinations contain both conjugate
variables of all modes.
Hence a complete state determination,
for example by detecting the entire correlation matrix of a Gaussian
state, is not needed.
\end{abstract}
\maketitle
\section{Introduction}
Quantum entanglement shared by two parties enhances their capability
to communicate. In principle, it allows them to convey quantum information
reliably through a classical channel (quantum teleportation \cite{Benn93}),
to double the amount of classical information transmittable through a classical channel
(quantum dense coding \cite{Wiesner92}), or to prepare at a distance states from nonorthogonal
bases for secure communication (quantum key distribution
\cite{BB84,Ekert,BennMermin}).
These entanglement-assisted communication schemes are extendible to an arbitrary
number of parties sharing multipartite entanglement.
For instance, a sender may transfer quantum information through classical channels
to several receivers as reliably as allowed by optimal cloning
(telecloning \cite{Murao99}), or the parties may share quantum (or classical) information
retrievable only when all parties cooperate
(quantum secret sharing \cite{Hillqsecret99}).
A more recent proposal that exploits the multi-party quantum correlations of
multipartite entangled states is the so-called Byzantine agreement protocol
\cite{Fitzi01}.
In general, the presence of entanglement is verified
through the success of a quantum protocol that would fail otherwise
(e.g., quantum teleportation).
Such an operational criterion is only sufficient for
entanglement and failure does not necessarily rule out its presence.
In an experimental realization, however, before running through an entire
entanglement-based protocol, it is desirable first to confirm that the generation
of sophisticated multi-party entangled states has succeeded.
The aim of this paper is to provide a simple but unambiguous experimental test
to check for a particular kind of genuinely multipartite entangled states, namely
those described by continuous variables (cv) and produced with squeezed light and
linear optics.
Work in the direction of generating
tripartite cv entanglement has been carried out
already by sending one half of a two-mode two-party entangled state
through an extra beam splitter with a coherent state or a vacuum
state at its second input port \cite{Furusawa98,Jing02}.
The resulting three-mode state was a side product of the
Bell measurement for the teleportation of coherent states
using a preshared symmetric two-mode squeezed state
\cite{Furusawa98}.
Its tripartite entanglement was not further investigated in that
experiment.
In another experiment, reported recently \cite{Jing02},
the two-mode state was asymmetric, corresponding to
two independently squeezed states combined at an asymmetric $1:2$
beam splitter. The output three-mode state after an additional
symmetric beam splitter was then similar to the states proposed
in Ref.~\cite{PvLPRL00}.
Quantum communication, or more general quantum information with cv
has attracted a lot of interest due to the relative simplicity and high efficiency
in the generation, manipulation, and detection of optical cv
states. Although recent results suggest that these assets of Gaussian cv
operations (phase shifting, beam splitting, homodyne detections, phase-space
displacements, squeezing) are not extendible to more advanced quantum protocols
such as entanglement distillation \cite{Eisertdist,Fiurasekdist,Giedkedist},
the simple and efficient cv
approach still seems promising for many tasks and might be suitable for others too
when combined with discrete-variable (dv) strategies. On the other hand,
potential linear-optics implementations of quantum protocols solely based on dv
utilizing single photons are restricted by No-Go results such as the
impossibility of a complete distinction between the four
Bell states \cite{Luetkenhaus99}.
In order to perform such a Bell measurement
near perfectly with linear optics,
one has to employ complicated entangled states of many auxiliary
photons \cite{Knill01}.
In contrast, a Bell and also a GHZ state analyzer can be easily
constructed in the
cv setting using only beam splitters and homodyne detectors
\cite{SamKimble98,vanloockFdP02,vanloockcvbook02}.
How may one now verify experimentally the presence of entanglement without
implementing a full quantum protocol?
We are here particularly concerned about the experimental verification
of genuinely multipartite entangled states where none of the parties
is separable from the rest (in terms of the separability properties of the
total density matrix).
In general, theoretical tests might be as well applicable to the experimental
verification. For instance, the violation of inequalities imposed by local realism
confirms the presence of entanglement.
Proving genuine multipartite entanglement,
however, requires stronger violations \cite{Seevinck}
than those determined by the
commonly used Mermin-Klyshko $N$-party inequalities
\cite{Mermin90,KlyshkoGisin}.
Moreover, in any case, violations
of Bell-type inequalities using Gaussian cv entangled states
with always positive Wigner functions must rely on observables other than the
quadratures (i.e., position and momentum). Photon number parity may serve
as an appropriate dichotomic variable to reveal the nonlocality of the
cv entangled states \cite{Bana98}. This applies to
the two-party two-mode EPR-like \cite{Bana98} and to the $N$-party $N$-mode
GHZ-like cv states \cite{PvLnonlocal01}. Such an approach, however,
is not very feasible due to its need for detectors resolving large photon number.
The negative partial transpose (npt) criterion \cite{Peres96} is
sufficient and necessary for the bipartite inseparability of
$2\times 2$-dimensional, $2\times 3$-dimensional \cite{Horodecki96},
and $1\times N$ mode Gaussian states \cite{Simon00,WernerWolf01}.
A complete experimental determination of the state in question
would also enable an npt check.
In general, any theoretical test is applicable when the experimentalist has
full information about the quantum state after measurements on an ensemble
of identically prepared states (e.g. by quantum tomography
\cite{Leonhardtbook97}).
Such a {\it direct} verification of entanglement via a complete state
measurement is in general very demanding to the experimentalist,
in particular when the state to be determined is a potentially multi-party
entangled multi-mode state.
\section{Gaussian states}
The multi-party entanglement criteria
that we will derive here do not rely on the assumption
of Gaussian states. However, the states commonly produced
in the laboratory are indeed Gaussian and the theoretical
classification of different types of multipartite entanglement
becomes simpler for Gaussian states \cite{Giedke01}.
Since the entanglement properties of a multi-mode multi-party
state are invariant under local phase-space displacements,
the multi-mode states may have zero mean and their Wigner
function is of the form
\begin{eqnarray}\label{Gausswigndef}
W(\xi)=
\frac{1}{(2\pi)^N\sqrt{\det V^{(N)}}}\,\exp\left\{-\frac{1}{2}\,
\xi\left[V^{(N)}\right]^{-1}
\xi^{T}\right\}\;,\nonumber\\
\end{eqnarray}
with the $2N$-dimensional vector $\xi$ having
the quadrature pairs of all $N$ modes as its components,
\begin{eqnarray}
\xi&=&(x_1,p_1,x_2,p_2,...,x_N,p_N)\;,\nonumber\\
\hat\xi&=&(\hat x_1,\hat p_1,\hat x_2,\hat p_2,...,
\hat x_N,\hat p_N)\;,
\end{eqnarray}
and with the $2N\times 2N$ correlation matrix $V^{(N)}$ having
as its elements the second moments symmetrized according to
the Weyl correspondence \cite{Weyl50},
\begin{eqnarray}\label{corrdef}
{\rm Tr}[\hat\rho\,(\Delta\hat\xi_i\Delta\hat\xi_j+
\Delta\hat\xi_j\Delta\hat\xi_i)/2]
&=&\langle(\hat\xi_i\hat\xi_j+\hat\xi_j\hat\xi_i)/2\rangle\nonumber\\
&=&\int\,W(\xi)\,\xi_i \xi_j\, d^{2N}\xi\nonumber\\
&=&V^{(N)}_{ij}\;,
\end{eqnarray}
where $\Delta\hat\xi_i=\hat\xi_i-\langle\hat\xi_i\rangle=\hat\xi_i$ for
zero mean values.
Note that the correlation matrix of any physical state
must be real, symmetric, positive, and must obey
the commutation relation \cite{Simon00,WernerWolf01},
\begin{eqnarray}
[\hat\xi_k,\hat\xi_l]=\frac{i}{2}\,\Lambda_{kl}\;,
\quad\quad k,l=1,2,3,...,2N\;,
\end{eqnarray}
with the $2N\times 2N$ matrix $\Lambda$ having
the $2\times 2$ matrix $J$ as diagonal entry for each
quadrature pair, for example for $N=2$,
\begin{eqnarray}
\Lambda=
\left( \begin{array}{cc} J & 0 \\
0 & J
\end{array} \right)\;,\quad\quad
J=
\left( \begin{array}{cc} 0 & 1 \\
-1 & 0
\end{array} \right)\;.
\end{eqnarray}
A direct consequence of this commutation relation and the non-negativity
of the density operator $\hat\rho$ is the $N$-mode
uncertainty relation \cite{Simon00,WernerWolf01},
\begin{eqnarray}\label{Nmodeuncert}
V^{(N)}-\frac{i}{4}\,\Lambda\geq 0\;.
\end{eqnarray}
Note that this condition is equivalent to
$V^{(N)}+i\Lambda/4\geq 0$ by complex conjugation.
As for the direct verification of entanglement via a complete state
measurement, for Gaussian cv states,
the complete measurement of an $N$-party $N$-mode
quantum state is accomplished by determining the $2N\times 2N$
second-moment correlation matrix.
This corresponds to $N(1+2N)$ independent entries taking into
account the symmetry of the correlation matrix.
Kim et al. \cite{Kim02} recently demonstrated how to determine all these
entries in the two-party two-mode case using beam splitters and homodyne detectors.
Joint homodyne detections of the two modes yield the intermode correlations
such as $\langle\hat x_1\hat x_2\rangle - \langle\hat x_1\rangle\langle\hat x_2\rangle$,
$\langle\hat x_1\hat p_2\rangle - \langle\hat x_1\rangle\langle\hat p_2\rangle$, etc.
Determining the local intramode correlations such as
$\langle\hat x_1\hat p_1+\hat p_1\hat x_1\rangle/2
- \langle\hat x_1\rangle\langle\hat p_1\rangle$
is more subtle and requires
additional beam splitters and homodyne detections
(or, alternatively, heterodyne detections).
Once the $4\times 4$ two-mode correlation matrix is known, the npt criterion
can be applied as a sufficient and necessary condition for bipartite Gaussian
two-mode inseparability (where npt corresponds to a sign change
of all the momentum variables with positions unchanged \cite{Simon00}).
In fact, the entanglement can also be quantified for a given correlation matrix
\cite{Kim02,VidalWerner02}.
For three-party three-mode Gaussian states, one may pursue a similar strategy.
After measuring the 21 independent entries of the correlation matrix
(for example, by extending Kim et al.'s scheme \cite{Kim02} to the three-mode case),
the sufficient and necessary criteria by Giedke et al. \cite{Giedke01} can be
applied. Let us examine the separability properties of
(in particular, three-party three-mode) Gaussian states in more detail.
\section{Separability properties of Gaussian states}
The criteria by Giedke et al. \cite{Giedke01}
determine to which of five possible
classes of fully and partially separable, and fully inseparable states
a three-party three-mode Gaussian state belongs.
Hence genuine tripartite entanglement if present
can be unambiguously identified.
The classification is mainly based on the
npt criterion for cv states. Transposition
is a positive map that corresponds in phase space
to a sign change of all momentum
variables, $\xi^{T}\rightarrow \Gamma\xi^{T}=
(x_1,-p_1,x_2,-p_2,...,x_N,-p_N)^{T}$ \cite{Simon00}.
In terms of the correlation matrix, we have then
$V^{(N)}\rightarrow\Gamma V^{(N)}\Gamma$.
Since transposition is not a completely positive map,
its partial application to a subsystem only
may yield an unphysical state when the subsystem
was entangled to other subsystems.
Expressing partial transposition of a bipartite
Gaussian system by $\Gamma_a\equiv\Gamma\oplus
\mbox{1$\!\!${\large 1}}$ (where $A\oplus B$
means the block-diagonal matrix with the matrices
$A$ and $B$ as diagonal `entries', and $A$ and $B$
are respectively $2N\times 2N$ and $2M\times 2M$ square
matrices applicable to $N$ modes at $a$'s side
and $M$ modes at $b$'s side),
the condition that the partially transposed
Gaussian state described by $\Gamma_a V^{(N+M)}\Gamma_a$
is unphysical [see Eq.~(\ref{Nmodeuncert})],
$\Gamma_a V^{(N+M)}\Gamma_a\ngeq \frac{i}{4}\,\Lambda$,
is sufficient for the inseparability
between $a$ and $b$ \cite{Simon00,WernerWolf01}.
For Gaussian states with $N=1$ and arbitrary $M$,
this condition is sufficient and necessary \cite{WernerWolf01}.
The simplest example where the condition is no longer
necessary for inseparability involves two modes at each
side, $N=M=2$. In that case, states with positive partial
transpose (bound entangled Gaussian states) exist
\cite{WernerWolf01}.
For the general bipartite $N\times M$ case,
there is also a sufficient and necessary condition:
the correlation matrix $V^{(N+M)}$ corresponds to
a separable state iff a pair of correlation matrices
$V^{(N)}_a$ and $V^{(M)}_b$ exists such that
$V^{(N+M)}\geq V^{(N)}_a\oplus V^{(M)}_b$.
Since it is in general hard to find such a pair
of correlation matrices $V^{(N)}_a$ and $V^{(M)}_b$
for a separable state or to prove the non-existence
of such a pair for an inseparable state, this criterion
in not very practical.
A more practical solution was provided in
Ref.~\cite{Giedke01PRL}. The operational
criteria there, computable and testable via
a finite number of iterations,
are entirely independent of the npt criterion.
They rely on a nonlinear map between the correlation matrices
rather than a linear one such as the partial transposition,
and in contrast to the npt criterion,
they witness also the inseparability of bound entangled
states.
Thus, the separability
problem for bipartite Gaussian states with arbitrarily many
modes at each side is completely solved.
For three-party three-mode Gaussian states,
the only partially separable forms are those
with a bipartite splitting of $1 \times 2$ modes.
Hence already the npt criterion is sufficient and necessary.
The classification of tripartite three-mode Gaussian states
\cite{Giedke01},
\begin{eqnarray}
&&\quad{\rm class}\;1\,:\quad\quad
\bar V^{(3)}_1\ngeq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_2\ngeq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_3\ngeq \frac{i}{4}\,\Lambda\,,
\nonumber\\
&&\quad{\rm class}\;2\,:\quad\quad
\bar V^{(3)}_k\geq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_m\ngeq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_n\ngeq \frac{i}{4}\,\Lambda\,,
\nonumber\\
&&\quad{\rm class}\;3\,:\quad\quad
\bar V^{(3)}_k\geq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_m\geq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_n\ngeq \frac{i}{4}\,\Lambda\,,
\nonumber\\
&&{\rm class}\;4\;{\rm or}\;5\,:\quad
\bar V^{(3)}_1\geq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_2\geq \frac{i}{4}\,\Lambda\,,
\bar V^{(3)}_3\geq \frac{i}{4}\,\Lambda\,,
\nonumber\\
\end{eqnarray}
is solely based on the npt criterion, where
$\bar V^{(3)}_j\equiv\Gamma_j V^{(3)}\Gamma_j$ denotes
the partial transposition with respect to one mode $j$.
In classes 2 and 3, any permutation of modes ($k,m,n$)
must be considered.
Class 1 corresponds to the fully inseparable states.
Class 5 shall contain the fully separable states.
For the fully separable Gaussian states if described
by $V^{(3)}$, one-mode correlation matrices
$V^{(1)}_1$, $V^{(1)}_2$, and $V^{(1)}_3$ exist such that
$V^{(3)}\geq V^{(1)}_1\oplus V^{(1)}_2\oplus V^{(1)}_3$.
In general, fully separable quantum states can be written as
a mixture of tripartite product states,
$\sum_i \eta_i\, \hat\rho_{i,1}\otimes\hat\rho_{i,2}
\otimes\hat\rho_{i,3}$.
In class 2, we have the one-mode biseparable states,
where only one particular mode is separable from
the remaining pair of modes.
This means in the Gaussian case that
only for one particular mode $k$,
$V^{(3)}\geq V^{(1)}_k\oplus V^{(2)}_{mn}$
with some two-mode correlation matrix $V^{(2)}_{mn}$
and one-mode correlation matrix $V^{(1)}_k$.
In general, such a state can be written as
$\sum_i \eta_i\, \hat\rho_{i,k}\otimes\hat\rho_{i,mn}$
for one mode $k$.
Class 3 contains those states where two but not three
bipartite splittings are possible, i.e.,
two different modes $k$ and $m$
are separable from the remaining
pair of modes (two-mode biseparable states).
The states of class 4 (three-mode biseparable states)
can be written as a mixture of products
between any mode 1, 2, or 3
and the remaining pair of modes,
but not as a mixture of three-mode product states.
Obviously, classes 4 and 5 are not distinguishable
via the npt criterion.
An additional criterion for this distinction
of class 4 and 5 Gaussian states
is given in Ref.~\cite{Giedke01},
deciding whether one-mode correlation matrices
$V^{(1)}_1$, $V^{(1)}_2$, and $V^{(1)}_3$ exist such that
$V^{(3)}\geq V^{(1)}_1\oplus V^{(1)}_2\oplus V^{(1)}_3$.
For the identification of genuinely tripartite entangled
Gaussian states, only class 1 has to be distinguished
from the rest. Hence the npt criterion alone suffices.
What about more than three parties and modes?
Even for only four parties and modes, the separability
issue becomes more subtle.
The one-mode bipartite splittings,
$\sum_i \eta_i\, \hat\rho_{i,klm}\otimes\hat\rho_{i,n}$,
can be tested and possibly ruled out via the npt criterion
with respect to any mode $n$.
In the Gaussian language,
if $\bar V^{(4)}_n\ngeq\frac{i}{4}\,\Lambda$
for any $n$, the state cannot be written in the above form.
Since we consider here the bipartite splitting of
$1\times 3$ modes, the npt condition is sufficient and necessary
for Gaussian states.
However, also a state of the form
$\sum_i \eta_i\, \hat\rho_{i,kl}\otimes\hat\rho_{i,mn}$
leads to negative partial transpose with respect to any of the
four modes when the two pairs ($k,l$) and ($m,n$) are each
entangled. Thus, npt with respect to any individual mode
is necessary but not sufficient for genuine four-party entanglement.
One has to consider also the partial transposition
with respect to any pair of modes.
For this $2\times 2$ mode case, however, we know that
entangled Gaussian states with positive partial transpose exist
\cite{WernerWolf01}. But the npt criterion is still sufficient
for the inseparability between any two pairs.
As for a sufficient and necessary condition, one can use
those from Ref.~\cite{Giedke01PRL}.
In any case, in order to confirm genuine four-party
or even $N$-party entanglement, one has to rule out
any possible partially separable form.
In principle, this can be done by considering all possible
bipartite splittings (or groupings)
and applying either the npt criterion
or the stronger operational criteria from Ref.~\cite{Giedke01PRL}.
Although a full theoretical characterization including criteria for
entanglement classification has not been considered yet for more than
three parties and modes,
the presence of genuine multipartite entanglement can be confirmed,
once the complete $2N\times 2N$ correlation matrix is given.
Rather than detecting all the entries of the correlation matrix
we are aiming here at a simple check
based on only a few measurements, preferably efficient homodyne detections.
Even for larger numbers of parties, this check should remain simple.
Though it may not yield full information (e.g., the complete correlation
matrix) about the quantum state of interest, it should still unambiguously
verify the presence of genuine multipartite entanglement.
This check may prove the presence of entanglement {\it indirectly}
through measurements after transforming the relevant state first into
an appropriate form via linear optics.
\section{Detecting entanglement: bipartite case}
In the two-party two-mode case, the necessary separability
condition for any cv state \cite{Duan00}
\begin{equation}\label{2partycrit}
\langle[\Delta(\hat{x}_1-\hat{x}_2)]^2\rangle +
\langle[\Delta(\hat{p}_1+\hat{p}_2)]^2\rangle \geq 2\,|\langle[\hat x,\hat p]
\rangle | \;,
\end{equation}
can be tested, for example, with a single beam splitter.
The position and momentum variables $\hat x_l$ and $\hat p_l$
(units-free with $\hbar=\frac{1}{2}$,
$[\hat{x}_l,\hat{p}_k]=i\delta_{lk}/2$)
correspond to the quadratures of two electromagnetic
modes, i.e., the real and imaginary parts of the
annihilation operators of the two modes:
$\hat{a}_l=\hat{x}_l+i\hat{p}_l$.
The beam splitter provides the suitable
quadrature combinations for the positions and momenta simultaneously
detectable at the two output ports. Without beam splitter,
just by measuring first both positions and subtracting them electronically,
and in a second step detecting both momenta and combining these
electronically \cite{akiracvbook02}, a more direct test of the two-party
condition is possible. However, instead of a simultaneous detection
of the relevant combinations, it requires switching the two local oscillator
phases from position to momentum measurements.
For an ensemble of identically prepared states, this sequence of detections
would still enable the application of the two-party condition.
Note that the violation of Eq.~(\ref{2partycrit}) is only sufficient for
inseparability, i.e., there are (even Gaussian) cv entangled states
that satisfy Eq.~(\ref{2partycrit}). Any Gaussian cv state, however,
can be transformed via local operations into a standard form and the
presence of entanglement would then always yield a violation \cite{Duan00}
(alternatively, one may modify the inequality and leave the Gaussian state
unchanged to obtain a sufficient and necessary condition \cite{Giovannetti02}).
The point is that the entanglement of states already
in this standard form (such as two-mode squeezed states)
can, in principle, always (for any nonzero squeezing) be verified
experimentally by checking Eq.~(\ref{2partycrit}).
A full determination of the correlation matrix,
including elements such as
$\langle\hat x_1\hat p_2\rangle - \langle\hat x_1\rangle\langle\hat p_2\rangle$
which do not appear in the expressions of Eq.~(\ref{2partycrit}),
is not required.
Measuring also these elements may confirm that the
state is in standard form (when they are zero) and hence
render the condition Eq.~(\ref{2partycrit}) sufficient and necessary
for separability. In any case, it would also enable quantification
of the entanglement \cite{Kim02,VidalWerner02}.
The combinations in condition Eq.~(\ref{2partycrit}) are exactly
those detected in a cv Bell measurement of modes 1 and 2 \cite{SamKimble98}.
Thus, the verification of non-maximum two-mode cv entanglement
may rely on measurements of observables that are detected for
the projection onto the maximally entangled cv basis of two modes.
Now we investigate the $N$-party $N$-mode case
in that respect.
\section{The cv GHZ basis}
\label{cvGHZ}
Let us introduce the maximally entangled states
\begin{eqnarray}
|\Psi(v,u_1,u_2,...,u_{N-1})\rangle =
\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\, dx\, e^{2ivx}
\nonumber\\
\label{PVLmaxentGHZbasis}
\times |x\rangle\otimes|x-u_1\rangle
\otimes\,|x-u_1-u_2\rangle \quad\;\;\;\;\,
\nonumber\\
\otimes\!\!\cdots\!\!\otimes\,
|x-u_1-u_2-\cdots-u_{N-1}
\rangle\;.
\end{eqnarray}
Since $\int_{-\infty}^{\infty}\,|x\rangle\langle x|
=\mbox{1$\!\!${\large 1}}$ and
$\langle x|x'\rangle=\delta(x-x')$,
they form a complete,
\begin{eqnarray}
&&\int_{-\infty}^{\infty}\,dv\,du_1\,du_2\cdots du_{N-1}
\\
&&\times|\Psi(v,u_1,u_2,...,u_{N-1})\rangle\langle
\Psi(v,u_1,u_2,...,u_{N-1})|
=\mbox{1$\!\!${\large 1}}^{\otimes N},\nonumber
\end{eqnarray}
and orthogonal,
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!
\langle\Psi(v,u_1,u_2,...,u_{N-1})|
\Psi(v',u_1',u_2',...,u_{N-1}')\rangle
\\
&&=\delta(v-v')\delta(u_1-u_1')\delta(u_2-u_2')
\cdots\delta(u_{N-1}-u_{N-1}'),\nonumber
\end{eqnarray}
set of basis states for $N$ modes.
In a ``cv GHZ state analyzer'',
determining the quantities $v\equiv p_1+p_2+\cdots+p_N$,
$u_1\equiv x_1-x_2$, $u_2\equiv x_2-x_3$,...,
and $u_{N-1}\equiv x_{N-1}-x_N$ means
projecting onto the
basis $\{|\Psi(v,u_1,u_2,...,u_{N-1})\rangle\}$.
This can be accomplish with a sequence of beam splitters
and homodyne detections \cite{vanloockFdP02,vanloockcvbook02}.
Inferring from the two-party case, we may conjecture that
the $N$ quadrature combinations given by
$v,u_1,u_2,...,u_{N-1}$ provide a sufficient set of observables
for the verification of (possibly genuine) $N$-party entanglement.
Just as for two parties, the variances of these quantities could
then also be determined by combining the results of
direct $x$ and $p$ measurements electronically.
It was shown in Ref.~\cite{vanloockFdP02,vanloockcvbook02} that
conditions for genuine multipartite entanglement can be derived
based on the above $N$ combinations and additional assumptions
such as the purity and the total symmetry of the state in question.
Later we derive a set of $N-1$ conditions for those $N$ combinations
sufficient for the presence of genuine multipartite
entanglement. This set is well suited for the experimental
confirmation of the genuine multi-party entanglement
of cv GHZ-type states. No extra assumptions
about the state are needed in order to close the loophole of
partial separability.
First, we discuss now what the structure of
simple experimental criteria
for multipartite cv entanglement might be.
\section{Detecting entanglement: tripartite case}
Let us consider three parties and modes.
The goal is to extend the simple two-party two-mode entanglement check
to a simple test for genuine three-party three-mode entanglement.
The criteria are to be expressed in terms of the variances
of quadrature linear combinations for the modes involved.
Defining
\begin{eqnarray}\label{threemodecombin}
\hat u\equiv h_1\hat x_1 + h_2\hat x_2 + h_3\hat x_3 \;,
\hat v\equiv g_1\hat p_1 + g_2\hat p_2 + g_3\hat p_3 \;,
\end{eqnarray}
a fairly general ansatz is
\begin{eqnarray}\label{3partycritgenansatz}
\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}\geq
f(h_1,h_2,h_3,g_1,g_2,g_3)\;,
\end{eqnarray}
as a potential necessary condition for an at least partially
separable state.
The position and momentum variables $\hat x_l$ and $\hat p_l$
are the quadratures of the three electromagnetic modes.
The $h_l$ and $g_l$ are arbitrary real parameters.
We will prove the following statements
for (at least partially) separable states,
\begin{eqnarray}\label{3partystatements}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}
\nonumber\\
\label{statem1}
&& \rightarrow \;
f(h_l,g_l)=(|h_3 g_3| + |h_1 g_1 + h_2 g_2|)/2 \,,
\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,2}
\nonumber\\
\label{statem2}
&& \rightarrow \;
f(h_l,g_l)=(|h_2 g_2| + |h_1 g_1 + h_3 g_3|)/2 \,,
\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,23}\otimes\hat\rho_{i,1}
\nonumber\\
\label{statem3}
&& \rightarrow \;
f(h_l,g_l)=(|h_1 g_1| + |h_2 g_2 + h_3 g_3|)/2 \,.
\end{eqnarray}
Here, for instance, $\hat\rho_{i,12}\otimes\hat\rho_{i,3}$ indicates
that the three-party density operator is a mixture of states $i$
where parties (modes) 1 and 2 may be entangled or not, but party 3
is not entangled with the rest. Hence also the fully separable state
is included in the above statements.
In fact, for the fully separable state, we have
\begin{eqnarray}\label{3partyfullysep}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,1}\otimes\hat\rho_{i,2}
\otimes\hat\rho_{i,3}
\nonumber\\
&& \rightarrow \;
f(h_l,g_l)=(|h_1 g_1| + |h_2 g_2| + |h_3 g_3|)/2,
\end{eqnarray}
which is always greater or equal than any of the
boundaries in Eq.~(\ref{statem1}),
Eq.~(\ref{statem2}), or Eq.~(\ref{statem3}).
For the proof, let us assume that the relevant state
can be written as
\begin{eqnarray}\label{3partyassumption}
\hat\rho=\sum_i \eta_i\, \hat\rho_{i,km}\otimes\hat\rho_{i,n}\;.
\end{eqnarray}
For the combinations in Eq.~(\ref{threemodecombin}), we find
\begin{eqnarray}\label{derivation1}
&&\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}
\nonumber\\
&=&\sum_i \eta_i\; \left(\langle\hat{u}^2\rangle_i+
\langle\hat{v}^2\rangle_i\right)-
\langle\hat{u}\rangle_{\rho}^2-\langle\hat{v}\rangle_{\rho}^2
\nonumber\\
&=&\sum_i \eta_i\;
\Big[h_k^2\langle\hat{x}_k^2\rangle_i+
h_m^2\langle\hat{x}_m^2\rangle_i+
h_n^2\langle\hat{x}_n^2\rangle_i
\nonumber\\
&&\quad\quad\quad+g_k^2\langle\hat{p}_k^2\rangle_i+
g_m^2\langle\hat{p}_m^2\rangle_i+
g_n^2\langle\hat{p}_n^2\rangle_i
\nonumber\\
&&+2\Big(
h_k h_m\langle\hat{x}_k\hat{x}_m\rangle_i+
h_k h_n\langle\hat{x}_k\hat{x}_n\rangle_i+
h_m h_n\langle\hat{x}_m\hat{x}_n\rangle_i\Big)
\nonumber\\
&&+2\Big(
g_k g_m\langle\hat{p}_k\hat{p}_m\rangle_i+
g_k g_n\langle\hat{p}_k\hat{p}_n\rangle_i+
g_m g_n\langle\hat{p}_m\hat{p}_n\rangle_i\Big)\Big]
\nonumber\\
&&-\langle\hat{u}\rangle_{\rho}^2-\langle\hat{v}\rangle_{\rho}^2
\nonumber\\
&=&\sum_i \eta_i\;
\Big[
h_k^2\langle(\Delta\hat{x}_k)^2\rangle_i+
h_m^2\langle(\Delta\hat{x}_m)^2\rangle_i+
h_n^2\langle(\Delta\hat{x}_n)^2\rangle_i
\nonumber\\
&&\quad\quad\quad+
g_k^2\langle(\Delta\hat{p}_k)^2\rangle_i+
g_m^2\langle(\Delta\hat{p}_m)^2\rangle_i+
g_n^2\langle(\Delta\hat{p}_n)^2\rangle_i
\nonumber\\
&&\quad\quad\quad+2h_k h_m\Big(
\langle\hat{x}_k\hat{x}_m\rangle_i-
\langle\hat{x}_k\rangle_i\langle\hat{x}_m\rangle_i\Big)
\nonumber\\
&&\quad\quad\quad+2h_k h_n\Big(
\langle\hat{x}_k\hat{x}_n\rangle_i-
\langle\hat{x}_k\rangle_i\langle\hat{x}_n\rangle_i\Big)
\nonumber\\
&&\quad\quad\quad+2h_m h_n\Big(
\langle\hat{x}_m\hat{x}_n\rangle_i-
\langle\hat{x}_m\rangle_i\langle\hat{x}_n\rangle_i\Big)
\nonumber\\
&&\quad\quad\quad+2g_k g_m\Big(
\langle\hat{p}_k\hat{p}_m\rangle_i-
\langle\hat{p}_k\rangle_i\langle\hat{p}_m\rangle_i\Big)
\nonumber\\
&&\quad\quad\quad+2g_k g_n\Big(
\langle\hat{p}_k\hat{p}_m\rangle_i-
\langle\hat{p}_k\rangle_i\langle\hat{p}_m\rangle_i\Big)
\nonumber\\
&&\quad\quad\quad+2g_m g_n\Big(
\langle\hat{p}_m\hat{p}_n\rangle_i-
\langle\hat{p}_m\rangle_i\langle\hat{p}_n\rangle_i\Big)
\Big]
\nonumber\\
&&+\sum_i \eta_i\; \langle\hat{u}\rangle_i^2-\left(\sum_i \eta_i\;
\langle\hat{u}\rangle_i\right)^2
\nonumber\\
&&+\sum_i \eta_i\; \langle\hat{v}
\rangle_i^2-\left(\sum_i \eta_i\;
\langle\hat{v}\rangle_i\right)^2\,,
\end{eqnarray}
where $\langle\cdots\rangle_i$ means the average in the state
$\hat\rho_{i,km}\otimes\hat\rho_{i,n}$.
Note that in the derivation so far we have not used
the particular form in Eq.~(\ref{3partyassumption}) yet.
Exploiting this form of the state, we obtain
$\langle\hat{x}_k\hat{x}_n\rangle_i=
\langle\hat{x}_k\rangle_i\langle\hat{x}_n\rangle_i$,
$\langle\hat{x}_m\hat{x}_n\rangle_i=
\langle\hat{x}_m\rangle_i\langle\hat{x}_n\rangle_i$,
and similarly for the terms involving $p$.
Because modes $k$ and $m$ may be entangled
in the states $i$, we cannot replace
$\langle\hat{x}_k\hat{x}_m\rangle_i$ by
$\langle\hat{x}_k\rangle_i\langle\hat{x}_m\rangle_i$, etc.
By applying the Cauchy-Schwarz inequality as in the
two-party derivation of Ref.~\cite{Duan00},
$\sum_i P_i \langle\hat{u}\rangle_i^2 \geq \left(\sum_i P_i
|\langle\hat{u}\rangle_i|\right)^2$, we see that the last two lines
in Eq.~(\ref{derivation1}) are bounded below by zero.
Hence in order to prove
$\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}\geq
(|h_n g_n| + |h_k g_k + h_m g_m|)/2$,
it remains to be shown that for any $i$
[recall that the mixture in Eq.~(\ref{3partyassumption})
is a convex sum with $\sum_i\eta_i=1$],
\begin{eqnarray}\label{derivation2}
&&h_k^2\langle(\Delta\hat{x}_k)^2\rangle_i+
h_m^2\langle(\Delta\hat{x}_m)^2\rangle_i+
h_n^2\langle(\Delta\hat{x}_n)^2\rangle_i
\nonumber\\
&&+
g_k^2\langle(\Delta\hat{p}_k)^2\rangle_i+
g_m^2\langle(\Delta\hat{p}_m)^2\rangle_i+
g_n^2\langle(\Delta\hat{p}_n)^2\rangle_i
\nonumber\\
&&+2h_k h_m\Big(
\langle\hat{x}_k\hat{x}_m\rangle_i-
\langle\hat{x}_k\rangle_i\langle\hat{x}_m\rangle_i\Big)
\nonumber\\
&&+2g_k g_m\Big(
\langle\hat{p}_k\hat{p}_m\rangle_i-
\langle\hat{p}_k\rangle_i\langle\hat{p}_m\rangle_i\Big)
\nonumber\\
&&\geq (|h_n g_n| + |h_k g_k + h_m g_m|)/2\;.
\end{eqnarray}
By rewriting the left-hand-side of Eq.~(\ref{derivation2})
in terms of variances only, indeed we find
\begin{eqnarray}\label{derivation3}
&&h_n^2\langle(\Delta\hat{x}_n)^2\rangle_i+
g_n^2\langle(\Delta\hat{p}_n)^2\rangle_i
\nonumber\\
&&+\langle[\Delta(h_k\hat{x}_k + h_m\hat x_m)]^2\rangle_i
+\langle[\Delta(g_k\hat{p}_k + g_m \hat p_m)]^2\rangle_i
\nonumber\\
&&\geq
|\langle [h_n\hat x_n,g_n \hat p_n]\rangle |+
|\langle [h_k\hat x_k + h_m\hat x_m,g_k\hat{p}_k
+ g_m\hat p_m]\rangle |
\nonumber\\
&&=(|h_n g_n| + |h_k g_k + h_m g_m|)/2\;,
\end{eqnarray}
using the sum uncertainty relation
$\langle(\Delta\hat{A})^2\rangle+\langle(\Delta\hat{B})^2\rangle
\geq |\langle [\hat{A},\hat{B}]\rangle |$
and $[\hat{x}_l,\hat{p}_j]=i\delta_{lj}/2$.
Hence the statements in Eq.~(\ref{3partystatements})
are proven when we consider the corresponding
permutations of $(k,m,n)=(1,2,3)$.
The inequalities Eq.~(\ref{3partycritgenansatz}) with
Eq.~(\ref{statem1}), Eq.~(\ref{statem2}), and
Eq.~(\ref{statem3})
represent necessary conditions for all kinds
of (partial) separability in a tripartite three-mode
state. One may then prove the presence of genuine
tripartite entanglement through violations of these
inequalities, thus ruling out any (partially) separable
form. Whether there are really three different conditions
required for the verification depends on the choice
of the coefficients $h_l$ and $g_l$ in the linear
combinations. For a particular choice, some of the
conditions may coincide.
For example, consider
$h_1=g_1=1$ and $g_2=g_3=-h_2=-h_3=1/\sqrt{2}$
in Eq.~(\ref{threemodecombin}).
In this case, the boundaries in
Eq.~(\ref{statem1}) and Eq.~(\ref{statem2}) become
identical, $f(h_l,g_l)=1/2$.
The boundary of Eq.~(\ref{statem3}) is even larger,
$f(h_l,g_l)=1$, equivalent to that for a fully
separable state in Eq.~(\ref{3partyfullysep}).
Hence the violation of a {\it single} condition,
\begin{eqnarray}\label{threepartyexample}
&&\langle\{\Delta[\hat x_1-(\hat x_2+\hat x_3)/\sqrt{2}]\}^2
\rangle_{\rho}\nonumber\\
&&\;+\langle\{\Delta[\hat p_1+(\hat p_2+\hat p_3)/\sqrt{2}]\}^2
\rangle_{\rho}\geq 1/2\;,
\end{eqnarray}
is already sufficient for genuine tripartite entanglement.
These particular combinations are not only significant
for the reason that they yield nonzero boundaries
for all kinds of separable states.
Moreover, their commutator vanishes,
\begin{eqnarray}
[\hat x_1-(\hat x_2+\hat x_3)/\sqrt{2},
\hat p_1+(\hat p_2+\hat p_3)/\sqrt{2}]=0\;,
\end{eqnarray}
allowing for arbitrarily good violations of
Eq.~(\ref{threepartyexample})
and, in principle, the existence of a simultaneous eigenstate
of these two combinations.
Such a state corresponds to the three-mode state obtainable
by splitting one half of an infinitely squeezed two-mode squeezed
(EPR) state at a 50:50 beam splitter.
The EPR correlations, $\hat x_1 - \hat x_2\rightarrow 0$ and
$\hat p_1 + \hat p_2\rightarrow 0$, are then transformed into
the three-mode correlations
$\hat x_1 - (\hat x_2' + \hat x_3')/\sqrt{2}\rightarrow 0$ and
$\hat p_1 + (\hat p_2' + \hat p_3')/\sqrt{2}\rightarrow 0$.
Let us turn to an arbitrary number of parties (modes)
now.
\section{Detecting entanglement: multipartite case}
Inferring from the discussion of the previous section,
the recipe for verifying the genuine multipartite
entanglement between arbitrarily
many parties and modes is the following.
First, measure both quadratures $x$ and $p$ of all modes involved
and combine them in an appropriate linear combination.
The variances of these combinations may then yield violations
of conditions necessary for partial separability.
Appropriate combinations
are those where the total variances for all partially separable
states have nonzero lower bounds
and where the commutators of the combinations vanish.
As for the derivation of the corresponding entanglement
criteria, we employ the following steps.
1. Select a distinct pair of modes $(m,n)$.
2. Choose appropriate linear combinations of the quadratures
in order to rule out all possible separable splittings between this pair
of modes in the convex sum of the total density operator.
3. Consider different pairs $(m,n)$ to negate all partial separabilities;
if necessary add further conditions involving other linear combinations.
Below it will become clear that step 2 can be performed simply by using the
appropriate bipartite combinations, $\hat x_m - \hat x_n$ and
$\hat p_m + \hat p_n$, i.e., by taking all $h_l=g_l=0$ except $h_m=g_m=1$
and $h_n=-g_n=-1$ in the general combinations
\begin{eqnarray}\label{multimodecombin}
\hat u&\equiv& h_1\hat x_1 + h_2\hat x_2 +\cdots +
h_N\hat x_N \;,\nonumber\\
\hat v&\equiv& g_1\hat p_1 + g_2\hat p_2 + \cdots +
g_N\hat p_N \;.
\end{eqnarray}
The boundaries of the total variance conditions are then identical
for any pair $(m,n)$ separable in the convex sum,
namely $f(h_l,g_l)\equiv 1$ in
\begin{eqnarray}\label{multipartycritgenansatz}
\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}\geq
f(h_1,h_2,...,h_N,g_1,g_2,...,g_N)\;.\nonumber\\
\end{eqnarray}
However, in general, one obtains better multi-party conditions
when linear combinations for the quadratures of more than only
two modes are used. Through such multi-mode combinations the potential
multi-mode correlations are taken into account.
Before giving an example, let us first derive the general $N$-party
bounds in the condition Eq.~(\ref{multipartycritgenansatz}).
For any partially separable form, the total density operator
can be written as
\begin{eqnarray}\label{Npartyassumption}
\hat\rho=\sum_i \eta_i\, \hat\rho_{i,k_r \cdots m}
\otimes\hat\rho_{i,k_s\cdots n}\;,
\end{eqnarray}
with a distinct pair of ``separable modes'' $(m,n)$ and
the other modes $k_r\neq k_s$.
For the combinations in Eq.~(\ref{multimodecombin}), we find now
\begin{eqnarray}\label{multiderivation1}
&&\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}
\nonumber\\
&=&\sum_i \eta_i\; \left(\langle\hat{u}^2\rangle_i+
\langle\hat{v}^2\rangle_i\right)-
\langle\hat{u}\rangle_{\rho}^2-\langle\hat{v}\rangle_{\rho}^2
\nonumber\\
&=&\sum_i \eta_i\;\Big[
h_m^2\langle\hat{x}_m^2\rangle_i+
h_n^2\langle\hat{x}_n^2\rangle_i+
\sum_{j=1}^{N-2}h_{k_j} ^2
\langle\hat{x}_{k_j}^2\rangle_i
\nonumber\\
&&\quad\quad\quad+
g_m^2\langle\hat{p}_m^2\rangle_i+
g_n^2\langle\hat{p}_n^2\rangle_i+
\sum_{j=1}^{N-2}g_{k_j} ^2
\langle\hat{p}_{k_j}^2\rangle_i
\nonumber\\
&&+\sum_{j\neq j'=1}^{N-2}\Big(
h_{k_j} h_{k_{j'}}\langle\hat{x}_{k_j}\hat{x}_{k_{j'}}\rangle_i+
g_{k_j} g_{k_{j'}}\langle\hat{p}_{k_j}\hat{p}_{k_{j'}}\rangle_i
\Big)
\nonumber\\
&&+2\sum_{j=1}^{N-2}\Big(
h_{k_j} h_m\langle\hat{x}_{k_j}\hat{x}_m\rangle_i+
h_{k_j} h_n\langle\hat{x}_{k_j}\hat{x}_n\rangle_i
\nonumber\\
&&\quad\quad\quad\quad+
g_{k_j} g_m\langle\hat{p}_{k_j}\hat{p}_m\rangle_i+
g_{k_j} g_n\langle\hat{p}_{k_j}\hat{p}_n\rangle_i
\Big)
\nonumber\\
&&+2\Big(
h_m h_n\langle\hat{x}_m\hat{x}_n\rangle_i+
g_m g_n\langle\hat{p}_m\hat{p}_n\rangle_i\Big)\Big]
\nonumber\\
&&-\langle\hat{u}\rangle_{\rho}^2-\langle\hat{v}\rangle_{\rho}^2
\nonumber\\
&=&\sum_i \eta_i\;
\Big\{
h_m^2\langle(\Delta\hat{x}_m)^2\rangle_i+
h_n^2\langle(\Delta\hat{x}_n)^2\rangle_i
\nonumber\\
&&\quad\quad\quad+
g_m^2\langle(\Delta\hat{p}_m)^2\rangle_i+
g_n^2\langle(\Delta\hat{p}_n)^2\rangle_i
\nonumber\\
&&\quad\quad\quad+
\sum_{j=1}^{N-2}\Big(
h_{k_j}^2\langle(\Delta\hat{x}_{k_j})^2\rangle_i+
g_{k_j}^2\langle(\Delta\hat{p}_{k_j})^2\rangle_i
\Big)
\nonumber\\
&&\quad\quad\quad+
\sum_{r\neq r'}\Big[
h_{k_r} h_{k_{r'}}\Big(
\langle\hat{x}_{k_r}\hat{x}_{k_{r'}}\rangle_i-
\langle\hat{x}_{k_r}\rangle_i\langle\hat{x}_{k_{r'}}\rangle_i
\Big)
\nonumber\\
&&\quad\quad\quad\quad\quad\quad+
g_{k_r} g_{k_{r'}}\Big(
\langle\hat{p}_{k_r}\hat{p}_{k_{r'}}\rangle_i-
\langle\hat{p}_{k_r}\rangle_i\langle\hat{p}_{k_{r'}}\rangle_i
\Big)\Big]
\nonumber\\
&&\quad\quad\quad+
\sum_{s\neq s'}\Big[
h_{k_s} h_{k_{s'}}\Big(
\langle\hat{x}_{k_s}\hat{x}_{k_{s'}}\rangle_i-
\langle\hat{x}_{k_s}\rangle_i\langle\hat{x}_{k_{s'}}\rangle_i
\Big)
\nonumber\\
&&\quad\quad\quad\quad\quad\quad+
g_{k_s} g_{k_{s'}}\Big(
\langle\hat{p}_{k_s}\hat{p}_{k_{s'}}\rangle_i-
\langle\hat{p}_{k_s}\rangle_i\langle\hat{p}_{k_{s'}}\rangle_i
\Big)\Big]
\nonumber\\
&&\quad\quad\quad+
2\sum_{r}\Big[
h_{k_r} h_m\Big(
\langle\hat{x}_{k_r}\hat{x}_m\rangle_i-
\langle\hat{x}_{k_r}\rangle_i\langle\hat{x}_m\rangle_i
\Big)
\nonumber\\
&&\quad\quad\quad\quad\quad\quad+
g_{k_r} g_m\Big(
\langle\hat{p}_{k_r}\hat{p}_m\rangle_i-
\langle\hat{p}_{k_r}\rangle_i\langle\hat{p}_m\rangle_i
\Big)\Big]
\nonumber\\
&&\quad\quad\quad+
2\sum_{s}\Big[
h_{k_s} h_n\Big(
\langle\hat{x}_{k_s}\hat{x}_n\rangle_i-
\langle\hat{x}_{k_s}\rangle_i\langle\hat{x}_n\rangle_i
\Big)
\nonumber\\
&&\quad\quad\quad\quad\quad\quad+
g_{k_s} g_n\Big(
\langle\hat{p}_{k_s}\hat{p}_n\rangle_i-
\langle\hat{p}_{k_s}\rangle_i\langle\hat{p}_n\rangle_i
\Big)\Big]\Big\}
\nonumber\\
&&+\sum_i \eta_i\; \langle\hat{u}\rangle_i^2-\left(\sum_i \eta_i\;
\langle\hat{u}\rangle_i\right)^2
\nonumber\\
&&+\sum_i \eta_i\; \langle\hat{v}
\rangle_i^2-\left(\sum_i \eta_i\;
\langle\hat{v}\rangle_i\right)^2\,.
\end{eqnarray}
For the last equality, we exploited Eq.~(\ref{Npartyassumption}),
namely that modes $k_r$ through $m$ are separable from
modes $k_s$ through $n$ in the convex sum of the total density
operator.
Similar to the three-party case,
we can now apply the Cauchy-Schwarz inequality
to the last two lines of Eq.~(\ref{multiderivation1})
and express the remaining terms by variances only.
This leads for any $i$ to
\begin{eqnarray}\label{multiderivation2}
&&\Big\langle\Big[\Delta\Big(h_m\hat{x}_m +
\sum_r h_{k_r}\hat x_{k_r}\Big)\Big]^2\Big\rangle_i
\nonumber\\
&&+\Big\langle\Big[\Delta\Big(g_m\hat{p}_m +
\sum_r g_{k_r}\hat p_{k_r}\Big)\Big]^2\Big\rangle_i
\nonumber\\
&&+\Big\langle\Big[\Delta\Big(h_n\hat{x}_n +
\sum_s h_{k_s}\hat x_{k_s}\Big)\Big]^2\Big\rangle_i
\nonumber\\
&&+\Big\langle\Big[\Delta\Big(g_n\hat{p}_n +
\sum_s g_{k_s}\hat p_{k_s}\Big)\Big]^2\Big\rangle_i
\nonumber\\
&&\geq
\Big|\Big\langle \Big[h_m\hat x_m +
\sum_r h_{k_r}\hat x_{k_r},g_m \hat p_m +
\sum_r g_{k_r}\hat p_{k_r}\Big]\Big\rangle \Big|
\nonumber\\
&&\quad\,+\Big|\Big\langle \Big[h_n\hat x_n +
\sum_s h_{k_s}\hat x_{k_s},g_n \hat p_n +
\sum_s g_{k_s}\hat p_{k_s}\Big]\Big\rangle \Big|\;,
\nonumber\\
\end{eqnarray}
using again the sum uncertainty relation
$\langle(\Delta\hat{A})^2\rangle+\langle(\Delta\hat{B})^2\rangle
\geq |\langle [\hat{A},\hat{B}]\rangle |$.
Thus, by evaluating the commutators with
$[\hat{x}_l,\hat{p}_j]=i\delta_{lj}/2$,
we obtain for the total variance
\begin{eqnarray}\label{finalmultiparty}
&&\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}
\nonumber\\
&&\quad\geq\frac{1}{2}\,
\Big(\Big| h_m g_m +
\sum_r h_{k_r} g_{k_r} \Big| +
\Big| h_n g_n +
\sum_s h_{k_s} g_{k_s} \Big| \Big)\;.\nonumber\\
\end{eqnarray}
Any additional splitting of the parties
in the states $i$,
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,k_r \cdots m}
\otimes \cdots \otimes\hat\rho_{i,k_{r'}}
\otimes\hat\rho_{i,k_s\cdots n}
\otimes \cdots \otimes\hat\rho_{i,k_{s'}}$,
would in general make the bound larger,
eventually yielding the bound for the
fully separable state,
$\sum_j|h_j g_j|/2$ ($j=1...N$).
As mentioned previously, the well-known
bipartite combinations applied to modes $(m,n)$,
$\hat x_m - \hat x_n$ and $\hat p_m + \hat p_n$, mean
all $h_l=g_l=0$ except $h_m=g_m=1$
and $h_n=-g_n=-1$ in Eq.~(\ref{finalmultiparty})
and hence $\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}\geq 1$.
As for a simple example, we may extend that
from the previous section to $N$ modes and set
$h_1=g_1=1$ and $g_2=g_3=\cdots =g_N=
-h_2=-h_3=\cdots =-h_N=1/\sqrt{N-1}$.
Without loss of generality, we choose $m=1$
and obtain for a state of the form
Eq.~(\ref{Npartyassumption}),
\begin{eqnarray}\label{Npartyexampleonecond}
&&\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}
\nonumber\\
&&\quad\geq\frac{1}{2}\,
\Big(\Big| 1 - \frac{M_r}{N-1} \Big| +
\Big| \frac{1 + M_s}{N-1}\Big| \Big)\;,\nonumber\\
\end{eqnarray}
where $M_r$ is the number of modes potentially
entangled with mode $m=1$ in the convex sum
and $M_s$ is the number of
modes potentially entangled with mode $n$
in the convex sum.
Apart from the fully inseparable case $M_r=N-1$,
the boundary in Eq.~(\ref{Npartyexampleonecond})
is always greater than zero allowing for an
ultimate nonzero bound for all kinds of partial
separability. Since $[\hat u, \hat v]=0$,
genuine $N$-party entanglement can be verified
when $\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}$ is
sufficiently close to zero.
The ultimate (smallest) bound is given by the
state with the maximum number of modes $M_r$
inseparable from mode $m=1$ in the convex sum,
$M_r=N-2$, and hence $M_s=0$.
This bound is then $1/(N-1)$.
If none of the modes is inseparable from mode $m=1$,
$M_r=0$ and $M_s=N-2$, the boundary becomes simply that
of a fully separable state, namely one.
Thus, again the violation of a {\it single} condition,
\begin{eqnarray}\label{Npartyexample}
&&\langle\{\Delta[\hat x_1 - (\hat x_2 + \hat x_3 +\cdots
+\hat x_N)/\sqrt{N-1}]\}^2
\rangle_{\rho}\nonumber\\
&&\;+\langle\{\Delta[\hat p_1 + (\hat p_2 + \hat p_3 +\cdots
+\hat p_N)/\sqrt{N-1}]\}^2
\rangle_{\rho}\nonumber\\
&&\;\quad\quad\quad\quad\quad\quad\geq 1/(N-1)\;,
\end{eqnarray}
is sufficient for genuine $N$-partite entanglement.
As an example for the violation of the ultimate
bound for genuine $N$-party entanglement, consider
the $N$-mode state that emerges
after symmetrically splitting one half
of an infinitely squeezed two-mode squeezed state
by $N-2$ beam splitters. The output state is a simultaneous
eigenstate of $\hat u$ and $\hat v$. In this case,
the EPR correlations, $\hat x_1 - \hat x_2\rightarrow 0$ and
$\hat p_1 + \hat p_2\rightarrow 0$, are transformed into
the $N$-mode correlations
$\hat x_1 - (\hat x_2' + \hat x_3' +\cdots
+\hat x_N')/\sqrt{N-1}\rightarrow 0$ and
$\hat p_1 + (\hat p_2' + \hat p_3' +\cdots
+\hat p_N')/\sqrt{N-1}\rightarrow 0$.
A more symmetric example is where both halves
of an EPR state are symmetrically split at beam splitters.
The appropriate combinations are then for instance for four modes,
$\hat x_1' + \hat x_2' - \hat x_3' - \hat x_4'$ and
$\hat p_1' + \hat p_2' + \hat p_3' + \hat p_4'$, also having zero
commutator $[\hat u, \hat v]=0$.
As a further example, we will now discuss
the cv GHZ-type states with quadrature correlations
analogous to those of dv GHZ states.
\section{Example: cv GHZ-type states}
We consider a family of genuinely $N$-party entangled states.
The members of this family are those states that emerge from
a particular sequence of $N-1$
phase-free beam splitters (``$N$-splitter'')
with $N$ squeezed state inputs \cite{PvLPRL00}.
By choosing the squeezing direction of one distinct input mode
orthogonal to that of the remaining input modes
(mode 1 squeezed in $p$ and the other modes squeezed in $x$,
as shown in Figs.~\ref{fig1} and \ref{fig2} for $N=3$)
and the degree of squeezing $r_1$ of mode 1
potentially different from that of the other modes
(which are equally squeezed by $r_2$) \cite{Bowen01},
the output states have the following properties
\cite{vanloockFdP02,vanloockcvbook02}.
They are pure $N$-mode states,
totally symmetric under interchange of modes,
and retain the Gaussian character of the input states.
Hence they are entirely described by their second-moment
correlation matrix,
\begin{eqnarray}\label{GHZcorr}
V^{(N)}=\frac{1}{4}
\left( \begin{array}{ccccccc}
a & 0 & c & 0 & c & 0
& \cdots \\
0 & b & 0 & d & 0 & d
& \cdots \\
c & 0 & a & 0 & c & 0
& \cdots \\
0 & d & 0 & b & 0 & d
& \cdots \\
c & 0 & c & 0 & a & 0
& \cdots \\
0 & d & 0 & d & 0 & b
& \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\end{array} \right)\;,
\end{eqnarray}
where
\begin{eqnarray}\label{GHZcorrelements}
a&=&\frac{1}{N}e^{+2r_1}+\frac{N-1}{N}e^{-2r_2}\;,\nonumber\\
b&=&\frac{1}{N}e^{-2r_1}+\frac{N-1}{N}e^{+2r_2}\;,\nonumber\\
c&=&\frac{1}{N} (e^{+2r_1} - e^{-2r_2})\;,\nonumber\\
d&=&\frac{1}{N} (e^{-2r_1} - e^{+2r_2})\;.
\end{eqnarray}
For squeezed vacuum inputs, the multi-mode output states
have zero mean and their Wigner function is of the form
Eq.~(\ref{Gausswigndef}). The particularly simple form
of the correlation matrix in
Eq.~(\ref{GHZcorr}) is, in addition to the
general correlation matrix properties,
symmetric with respect to all modes and contains no
intermode or intramode $x$-$p$ correlations
(hence only four parameters $a$, $b$, $c$, and $d$ are
needed to determine the matrix). However,
the states of this form are in general biased with respect
to $x$ and $p$ ($a\neq b$).
Only for a particular relation between the squeezing
values $(r_1,r_2)$ \cite{vanloockFdP02,vanloockcvbook02},
\begin{eqnarray}\label{Bowenrelation}
e^{\pm 2 r_1}&=&(N-1)\\
&&\times \sinh 2r_2 \,\left[
\sqrt{1+\frac{1}{(N-1)^2\sinh^2 2r_2}} \pm 1 \right],
\nonumber
\end{eqnarray}
the states are unbiased
(all diagonal entries of the correlation matrix equal),
thus having minimum energy at a given degree of entanglement
or, in other words, maximum entanglement for a given mean photon number
\cite{Bowen01}. The other $N$-mode states
of the family can be converted into the minimum-energy state via
local squeezing operations \cite{Bowen01,vanloockFdP02,vanloockcvbook02}.
Only for $N=2$, we obtain $r=r_1=r_2$. In that case,
the matrix $V^{(N)}$ reduces to that of a two-mode squeezed state
which is the maximally entangled state of two modes at
a given mean energy with the correlation matrix entries
$a=b=\cosh 2r$ and $c=\sinh 2r=-d$.
For general $N$, the first squeezer with $r_1$ and the $N-1$
remaining squeezers with $r_2$ have different
squeezing. In the limit of large squeezing
($\sinh 2r_2\approx e^{+2r_2}/2$), we obtain approximately
\cite{vanloockFdP02,vanloockcvbook02}
\begin{eqnarray}\label{Bowenrelationlargesq}
e^{+ 2 r_1}\approx (N-1) e^{+ 2 r_2} \;.
\end{eqnarray}
For the whole family of $N$-party $N$-mode states with the
correlation matrix in Eq.~(\ref{GHZcorr}),
the quadrature combinations relevant
for detecting genuine multi-party entanglement
are
\cite{PvLPRL00,vanloockFdP02,vanloockcvbook02}
\begin{eqnarray}\label{corr3}
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad\;
\langle[\Delta(\hat{x}_m-\hat{x}_n)]^2\rangle=e^{-2r_2}/2\;,
\nonumber\\
&&\left\langle\left[\Delta
\left(\hat{p}_m+\hat{p}_n+g^{(N)}\sum^N_{j\neq m,n}
\hat{p}_j\right)\right]^2\right\rangle=\nonumber\\
&&\frac{[2+(N-2)g^{(N)}]^2}{4N}e^{-2r_1}
+\frac{(g^{(N)}-1)^2(N-2)}{2N}e^{+2r_2}\;.\nonumber\\
\end{eqnarray}
The total variances are then optimized (minimized) for
\begin{eqnarray}\label{optgain}
g_{\rm opt}^{(N)}=
\frac{e^{+2r_2}-e^{-2r_1}}{e^{+2r_2}+\frac{N-2}{2}\,e^{-2r_1}}
\;.
\end{eqnarray}
In the limit of infinite squeezing, $r_1,r_2\to\infty$, the
above correlations correspond to a simultaneous
eigenstate of the relative positions and the total momentum
such as the cv GHZ states in Eq.~(\ref{PVLmaxentGHZbasis}).
\begin{figure}
\caption{\label{fig1}
\label{fig1}
\end{figure}
\begin{figure}
\caption{\label{fig2}
\label{fig2}
\end{figure}
Let us now examine how to experimentally verify the genuine multipartite
entanglement of the cv GHZ-type states
(in any case, it may be verified in an operational way by doing quantum
teleportation between every pair of parties with the help of the remaining
party \cite{PvLPRL00}). Due to experimental imperfections,
we may assume that the entanglement of slightly degraded approximate versions
of the states generated according to a scheme as in Figs.~\ref{fig1}
and \ref{fig2} is to be verified.
We start again with only three parties and modes.
For a simple check, look at the following set of
inequalities,
\begin{eqnarray}\label{3partycritGHZ}
{\rm I.}\;\;\quad
\langle[\Delta(\hat{x}_1-\hat{x}_2)]^2\rangle +
\langle[\Delta(\hat{p}_1+\hat{p}_2+g_3\hat{p}_3)]^2\rangle\geq 1\;,
\nonumber\\
{\rm II.}\;\quad
\langle[\Delta(\hat{x}_2-\hat{x}_3)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+\hat{p}_2+\hat{p}_3)]^2\rangle\geq 1\;,
\nonumber\\
{\rm III.}\quad
\langle[\Delta(\hat{x}_1-\hat{x}_3)]^2\rangle +
\langle[\Delta(\hat{p}_1+g_2\hat{p}_2+\hat{p}_3)]^2\rangle\geq 1\;.
\nonumber\\
\end{eqnarray}
On the l.h.s. of condition I.,
we have $h_1=-h_2=g_1=g_2=1$ and $h_3=0$,
and hence the boundary for the total variance
in Eq.~(\ref{3partycritgenansatz})
becomes one with Eq.~(\ref{statem2}) and Eq.~(\ref{statem3}),
but zero with Eq.~(\ref{statem1}).
Similarly, using the l.h.s. of condition II., where
$h_2=-h_3=g_2=g_3=1$ and $h_1=0$,
the boundary is one for Eq.~(\ref{statem1}) and
Eq.~(\ref{statem2}), but zero for Eq.~(\ref{statem3}).
Finally, the l.h.s of condition III.
with $h_1=-h_3=g_1=g_3=1$ and $h_2=0$
corresponds to a boundary of one
in Eq.~(\ref{statem1}) and Eq.~(\ref{statem3}),
and a boundary of zero in Eq.~(\ref{statem2}).
Thus, the following statements
for (at least partially) separable states hold,
\begin{eqnarray}\label{3partystatementsGHZ}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}
\quad \rightarrow {\rm II.}\; {\rm and}\; {\rm III.}\;,
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,2}
\quad \rightarrow {\rm I.}\; {\rm and}\; {\rm II.}\;,
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,23}\otimes\hat\rho_{i,1}
\quad \rightarrow {\rm I.}\; {\rm and}\; {\rm III.}
\end{eqnarray}
The conditions in Eq.~(\ref{3partycritGHZ}) are necessary for
different kinds of partial separability. As a result,
the violation of {\it any pair} of inequalities in
Eq.~(\ref{3partycritGHZ}) is sufficient for genuine three-party
three-mode entanglement.
Violating only one condition in Eq.~(\ref{3partycritGHZ})
(for example, condition I.) means that the total density
operator cannot be written in two of the three forms
in Eq.~(\ref{3partystatementsGHZ})
(for example, neither in the form
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,2}$
nor in the form
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,23}\otimes\hat\rho_{i,1}$).
Using the classification of Ref.~\cite{Giedke01},
the classes 3 [two-mode biseparable states expressible
in two of the three forms in Eq.~(\ref{3partystatementsGHZ})],
4 [three-mode biseparable states expressible
in all of the three forms in Eq.~(\ref{3partystatementsGHZ})],
and 5 [fully separable states describable by
Eq.~(\ref{3partyfullysep})] are then ruled out.
The forms of the classes 1 (fully inseparable states) and 2
[one-mode biseparable states expressible
in one of the three forms in Eq.~(\ref{3partystatementsGHZ})]
remain both possible. In our example with the violation of I.,
the state might be genuinely tripartite entangled or
of the partially separable form
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}$.
Eventually, the violation of a second inequality
in Eq.~(\ref{3partycritGHZ}) (for instance, condition II.)
negates also the only remaining partially separable form (e.g.,
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}$),
thus proving the full inseparability of the state
\cite{noteonchineseexp}.
Note that even though pure and totally
symmetric multi-party entangled states are always genuinely multipartite
entangled \cite{vanloockFdP02,vanloockcvbook02}, asymmetric pure
or mixed entangled three-mode states
(e.g., from class 2 in Ref.~\cite{Giedke01}, the product state of a
bipartite entangled two-mode squeezed state and a vacuum state) and
symmetric mixed entangled three-mode states
(like the example for the three-mode
biseparable class, class 4, given in Ref.~\cite{Giedke01})
do not automatically exhibit genuine tripartite entanglement.
Due to the violation of {\it two} conditions in
Eq.~(\ref{3partycritGHZ}),
the two loopholes of partial separability, mixedness
and/or asymmetry, are ruled out.
The criteria here are only sufficient for full
inseparability and hence genuinely tripartite entangled states
may also satisfy all the conditions in Eq.~(\ref{3partycritGHZ})
(an example will be mentioned later). On the other hand,
note that we did not use the assumption of Gaussian states.
The derivation of the conditions relies only on the Cauchy-Schwarz
inequality and Heisenberg's (sum) uncertainty relation.
Alternatively, one could simply check the known
bipartite separability conditions \cite{Duan00}
for pairs of modes, i.e., $g_1=g_2=g_3=0$
in Eq.~(\ref{3partycritGHZ})
(or using products of variances \cite{Tan99} instead of sums).
Again, the statements in Eq.~(\ref{3partystatementsGHZ}) hold.
Hence two violations again verify genuine tripartite entanglement.
However, the significance of the more general conditions
in Eq.~(\ref{3partycritGHZ}) compared to those with
$g_1=g_2=g_3=0$ is that for the cv GHZ-type states,
as discussed later,
the former can be {\it always} violated
for any degree of multi-party entanglement
and the violations can steadily grow from
small towards ``perfect'' (that is all variances
of the combinations zero) as the three-mode entanglement
increases. In contrast, the bipartite conditions
with $g_1=g_2=g_3=0$ may be violated
for bad three-mode entanglement (small squeezing)
and satisfied for larger squeezing, thus not always
verifying genuine tripartite entanglement,
and in particular never verifying
good genuine tripartite entanglement.
Moreover, they might be always violated, but the violations
do not attain a significant amount
(e.g., three-mode states made from one squeezed state
\cite{vanloockFdP02,vanloockcvbook02}).
Similarly, using products of variances \cite{Tan99} instead of
sums in Eq.~(\ref{3partycritGHZ})
with $g_1=g_2=g_3=0$, violations may always
occur, but also only to a certain extent
\cite{vanloockFdP02,vanloockcvbook02}.
In Figs.~\ref{fig1} and \ref{fig2}, it is shown how to apply
the tripartite entanglement criteria experimentally
using homodyne detectors.
Let us also discuss the conditions for the $N=4$ case
in more detail.
We consider a set of six inequalities,
\begin{eqnarray}\label{4partycrit}
{\rm I.}\;\;\,
\langle[\Delta(\hat{x}_1-\hat{x}_2)]^2\rangle +
\langle[\Delta(\hat{p}_1+\hat{p}_2+g_3\hat{p}_3
+g_4\hat{p}_4)]^2\rangle\geq 1,
\nonumber\\
{\rm II.}\;\,
\langle[\Delta(\hat{x}_2-\hat{x}_3)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+\hat{p}_2+\hat{p}_3
+g_4\hat{p}_4)]^2\rangle\geq 1,
\nonumber\\
{\rm III.}\,
\langle[\Delta(\hat{x}_1-\hat{x}_3)]^2\rangle +
\langle[\Delta(\hat{p}_1+g_2\hat{p}_2+\hat{p}_3
+g_4\hat{p}_4)]^2\rangle\geq 1,
\nonumber\\
{\rm IV.}\,
\langle[\Delta(\hat{x}_3-\hat{x}_4)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+g_2\hat{p}_2+\hat{p}_3
+\hat{p}_4)]^2\rangle\geq 1,
\nonumber\\
{\rm V.}\;
\langle[\Delta(\hat{x}_2-\hat{x}_4)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+\hat{p}_2+g_3\hat{p}_3
+\hat{p}_4)]^2\rangle\geq 1,
\nonumber\\
{\rm VI.}\;
\langle[\Delta(\hat{x}_1-\hat{x}_4)]^2\rangle +
\langle[\Delta(\hat{p}_1+g_2\hat{p}_2+g_3\hat{p}_3
+\hat{p}_4)]^2\rangle\geq 1.
\nonumber\\
\end{eqnarray}
The position and momentum variables $\hat x_l$ and $\hat p_l$
are the quadratures of four electromagnetic modes this time.
The $g_l$ are again arbitrary real parameters.
Now the following statements
for (at least partially) separable states hold,
\begin{eqnarray}\label{4partystatements1}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,123}\otimes\hat\rho_{i,4}
\;\rightarrow {\rm IV.,}{\rm V.,}\;
{\rm and}\; {\rm VI.,}\;\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,124}\otimes\hat\rho_{i,3}
\;\rightarrow {\rm II.,}{\rm III.,}\;
{\rm and}\; {\rm IV.,}\;\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,134}\otimes\hat\rho_{i,2}
\quad\,\rightarrow {\rm I.,}{\rm II.,}\;
{\rm and}\; {\rm V.,}\;\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,234}\otimes\hat\rho_{i,1}
\;\,\rightarrow {\rm I.,}{\rm III.,}\;
{\rm and}\; {\rm VI.,}\;\nonumber\\
\end{eqnarray}
and,
\begin{eqnarray}\label{4partystatements2}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,34}
\;\rightarrow {\rm II.,}{\rm III.,}{\rm V.,}\;
{\rm and}\; {\rm VI.,}\;\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,24}
\quad\rightarrow {\rm I.,}{\rm II.,}{\rm IV.,}\;
{\rm and}\; {\rm VI.,}\;\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,14}\otimes\hat\rho_{i,23}
\quad\,\rightarrow {\rm I.,}{\rm III.,}{\rm IV.,}\;
{\rm and}\; {\rm V.,}\;\nonumber\\
\end{eqnarray}
and finally,
\begin{eqnarray}\label{4partystatements3}
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}
\otimes\hat\rho_{i,4}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm I.,}
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,2}
\otimes\hat\rho_{i,4}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm III.,}
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,14}\otimes\hat\rho_{i,2}
\otimes\hat\rho_{i,3}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm VI.,}
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,23}\otimes\hat\rho_{i,1}
\otimes\hat\rho_{i,4}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm II.,}
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,24}\otimes\hat\rho_{i,1}
\otimes\hat\rho_{i,3}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm V.,}
\nonumber\\
\hat\rho&=&\sum_i \eta_i\, \hat\rho_{i,34}\otimes\hat\rho_{i,1}
\otimes\hat\rho_{i,2}
\;\rightarrow {\rm all}\;{\rm except}\;{\rm IV.}
\nonumber\\
\end{eqnarray}
Note that again the fully separable state,
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,1}\otimes\hat\rho_{i,2}
\otimes\hat\rho_{i,3}\otimes\hat\rho_{i,4}$,
is included.
The above statements can be easily confirmed using
Eq.~(\ref{finalmultiparty}) for states of the general form
Eq.~(\ref{Npartyassumption}).
The different forms here are
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,klm}\otimes\hat\rho_{i,n}$,
including
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,kl}\otimes\hat\rho_{i,m}
\otimes\hat\rho_{i,n}$, and
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,km}\otimes\hat\rho_{i,ln}$,
including
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,km}\otimes\hat\rho_{i,l}
\otimes\hat\rho_{i,n}$,
with the two modes $m$ and $n$ always being separable.
For the combinations
$\hat u=\hat x_m - \hat x_n$ and
$\hat v= g_k\hat p_k + g_l\hat p_l + \hat p_m
+ \hat p_n$, the boundary of the total variance is
always one. The statements
Eq.~(\ref{4partystatements1}), Eq.~(\ref{4partystatements2}),
and Eq.~(\ref{4partystatements3}) become obvious then
by considering all possible pairs of modes $(m,n)$ of the
four modes $(k,l,m,n)$. Note that always when the two
modes $(m,n)$ are potentially entangled, the boundary for
the total variance drops to zero.
What kind of violations of the six inequalities
in Eq.~(\ref{4partycrit}) are now sufficient
to verify the full inseparability of a
four-mode four-party state?
The violations must rule out any of the
partially separable forms in
Eq.~(\ref{4partystatements1}),
Eq.~(\ref{4partystatements2}), and
Eq.~(\ref{4partystatements3}).
Let us, for example, consider violations of
the inequalities IV. and V.
These violations mean that all partially separable
forms in
Eq.~(\ref{4partystatements1}),
Eq.~(\ref{4partystatements2}), and
Eq.~(\ref{4partystatements3}) are excluded except
for the form
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,234}
\otimes\hat\rho_{i,1}$ in Eq.~(\ref{4partystatements1}).
In order to negate this form as well
a further violation is needed.
According to Eq.~(\ref{4partystatements1}),
one of the inequalities I., III., or VI.
should be violated in addition.
Here it is important to realize that
the conditions IV. and V. do not involve the $x$ quadrature
of mode 1, but that of all the other modes.
The additional test via any one of the conditions
I., III., or VI., of which all contain
both quadratures of mode 1, eventually
provides the missing information about mode 1. Hence
we learn that three conditions are sufficient here
to verify the full inseparability of a
four-mode four-party state.
We may choose
\begin{eqnarray}\label{4partycritfinal}
\langle[\Delta(\hat{x}_1-\hat{x}_2)]^2\rangle +
\langle[\Delta(\hat{p}_1+\hat{p}_2+g_3\hat{p}_3
+g_4\hat{p}_4)]^2\rangle < 1,
\nonumber\\
\langle[\Delta(\hat{x}_2-\hat{x}_3)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+\hat{p}_2+\hat{p}_3
+g_4\hat{p}_4)]^2\rangle < 1,
\nonumber\\
\langle[\Delta(\hat{x}_3-\hat{x}_4)]^2\rangle +
\langle[\Delta(g_1\hat{p}_1+g_2\hat{p}_2+\hat{p}_3
+\hat{p}_4)]^2\rangle < 1,
\nonumber\\
\end{eqnarray}
which involve both quadratures $x$ and $p$ of all four modes.
Note that apart from the coefficients $g_l$,
these four combinations correspond to those observables
measured in a four-party cv GHZ state analyzer.
Correspondingly, for $N$ parties and modes,
we may choose the following $N-1$ conditions
in terms of effectively $N$ combinations
(those of an $N$-party $N$-mode cv GHZ state analyzer),
\begin{eqnarray}\label{Npartycritfinal}
&&\langle[\Delta(\hat{x}_1-\hat{x}_2)]^2\rangle
\nonumber\\
&&+
\langle[\Delta(\hat{p}_1+\hat{p}_2+g_3\hat{p}_3
+\cdots +g_N\hat{p}_N)]^2\rangle < 1,
\nonumber\\
&&\langle[\Delta(\hat{x}_2-\hat{x}_3)]^2\rangle
\nonumber\\
&&+
\langle[\Delta(g_1\hat{p}_1+\hat{p}_2+\hat{p}_3
+g_4\hat{p}_4+\cdots +g_N\hat{p}_N)]^2\rangle < 1,
\nonumber\\
&&\quad\vdots\quad\quad\quad\vdots\quad\quad\quad
\vdots\quad\quad\quad
\vdots\quad\quad\quad\vdots\quad\quad\quad
\vdots\quad\quad\quad\vdots
\nonumber\\
&&\langle[\Delta(\hat{x}_{N-1}-\hat{x}_N)]^2\rangle
\nonumber\\
&&+
\langle[\Delta(g_1\hat{p}_1+g_2\hat{p}_2+\cdots +
g_{N-2}\hat{p}_{N-2}+\hat{p}_{N-1}+\hat p_N)]^2\rangle
\nonumber\\
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad
\quad\quad\quad\quad
\quad\quad\quad\quad< 1.
\end{eqnarray}
These conditions are sufficient to verify the
full inseparability (genuine $N$-party entanglement)
of an $N$-party $N$-mode state.
For arbitrary $N$, the proof relies on the fact that
in any partially separable form we may always select
a distinct pair of modes $(m,n)$ which are
separable in the states $i$ of the convex sum
of the density operator.
Only exploiting that modes $m$
and $n$ are separable, the combinations
\begin{eqnarray}\label{Npartycombin}
\hat u=\hat x_m - \hat x_n \;,\quad
\hat v=\sum_{j=1}^{N-2}
g_{k_j}\hat p_{k_j} + \hat p_m + \hat p_n\;,
\end{eqnarray}
always yield a boundary of one for the total variance
using Eq.~(\ref{finalmultiparty}) for states of the general
form Eq.~(\ref{Npartyassumption}).
By taking the pairs of modes $(1,2)$, $(2,3)$, ...,
$(N-1,N)$ for $(m,n)$, all partially separable forms
of the total density operator are covered
(as demonstrated explicitly for $N=4$) and hence
the $N-1$ conditions in Eq.~(\ref{Npartycritfinal}) are
sufficient for genuine $N$-party $N$-mode inseparability.
\begin{figure}
\caption{\label{fig3}
\label{fig3}
\end{figure}
The left-hand-side of the
inequalities in Eq.~(\ref{Npartycritfinal})
is shown in Fig.\ref{fig3}
for various cv GHZ-type $N$-mode states differing in the
relation between the squeezing $r_1$ and $r_2$
[Eq.~(\ref{GHZcorr}), Eq.~(\ref{GHZcorrelements}), and
Eq.~(\ref{corr3})].
Due to the total symmetry of all these states,
the left-hand-side of the conditions in Eq.~(\ref{Npartycritfinal})
becomes equal for all conditions (assuming $g_j\equiv g^{(N)}$).
Hence values below the boundary 1 here mean all inequalities
in Eq.~(\ref{Npartycritfinal}) are satisfied,
thus indicating genuine $N$-party entanglement.
In all cases in Fig.~\ref{fig3}, the optimal coefficients $g_j\equiv
g_{\rm opt}^{(N)}$ from Eq.~(\ref{optgain})
are used to minimize the total variances of Eq.~(\ref{corr3}).
If $N=30$, only for the unbiased states,
the conditions are always met (for any nonzero squeezing $r>0$) {\it and}
the total variances tend to zero for large squeezing.
Moreover, for the same squeezing $r$, the unbiased states with $N=30$
drop below the boundary 1 to a greater extent
than their unbiased tripartite counterparts.
In contrast, for the biased states (those with only
one squeezer, $r_2=0$ and $r=r_1$, and those with $N$ equally
squeezed states, $r=r_1=r_2$), the total variances approach
or even exceed the boundary 1 as the number of parties grows.
The example of the states with $N$ equal squeezers also demonstrates
that there are Gaussian states which are indeed genuinely
$N$-party entangled, but do not satisfy any of the
conditions in Eq.~(\ref{Npartycritfinal}).
It can be shown, however, taking into account the symmetry and purity
of the whole family of $N$-mode states (including those with $N$ equal
squeezers) that all these states are genuinely multi-party entangled
for any nonzero squeezing \cite{vanloockFdP02,vanloockcvbook02}.
Finally, we emphasize that one may use other conditions too
for verifying the genuine multipartite entanglement
of the cv GHZ-type states.
Even a single condition might be again sufficient.
For example, consider the combinations
$\hat u=2\hat x_1 -(\hat x_2 + \hat x_3)$ and
$\hat v=\hat p_1 + \hat p_2 + \hat p_3$ for three modes.
We have $[\hat u,\hat v]=0$, and indeed the GHZ-type three-mode
state becomes a simultaneous eigenstate of $\hat u$ and $\hat v$
in the limit of infinite squeezing, $r_1,r_2\to\infty$.
The boundaries of the total variance for these
combinations take on the value one when
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,12}\otimes\hat\rho_{i,3}$
or
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,13}\otimes\hat\rho_{i,2}$,
and the value two (corresponding to the fully separable state)
when
$\hat\rho=\sum_i \eta_i\, \hat\rho_{i,23}\otimes\hat\rho_{i,1}$.
Hence $\langle(\Delta\hat{u})^2\rangle_{\rho}+
\langle(\Delta\hat{v})^2\rangle_{\rho}<1$ is sufficient
for genuine tripartite entanglement.
The number of measurements required, however, remain the
same as for the criteria above expressed by $N-1$ conditions.
In any case, both quadratures of all modes must be detected
and combined in an appropriate way.
\section{Conclusions}
In summary, we proposed experimental criteria to detect
genuine multipartite continuous-variable entanglement.
These are expressed in terms of the variances of
particular combinations of all the quadratures involved.
The combinations are measurable with only a few simple
homodyne detections. For Gaussian states, it is then not necessary
to determine the entire correlation matrix in order to confirm the
genuine multipartite entanglement.
Furthermore, the conditions here do not rely on the
assumption of Gaussian states. An experimental confirmation
of the Gaussian character of the state in question is therefore
not needed either. Finally, we examined the applicability
of the conditions to a particular GHZ-type class of
genuinely multi-party entangled states.
These states are of Gaussian form, they are totally symmetric under
exchange of modes, and they have zero cross correlations between
the $x$ and the $p$ quadratures.
If they are in addition unbiased between the $x$ and the $p$
quadratures, they always (for any nonzero entanglement)
satisfy the conditions in terms of appropriately chosen linear
combinations. In the limit of perfect entanglement,
the variances of the combinations tend to zero
for the unbiased states and the conditions are perfectly met.
In an experiment, one normally has approximate a priori
knowledge about the state to be analyzed.
According to this a priori knowledge, one can then choose
appropriate linear combinations to be measured.
It would be desirable to know whether there is always,
for any given multi-party multi-mode state,
a single optimal condition to verify its genuine multipartite
entanglement and how to constructively derive this condition.
Inferring from the results here, such a condition may always
exist and the corresponding linear combinations must contain
both quadratures of all modes with optimized coefficients
$h_l$ and $g_l$.
A possible approach to this question is in terms
of so-called entanglement witnesses
\cite{Horodecki96,Terhal00}. One may then
interpret the inequalities for the total variances
as quantum expectation values of Hermitian operators
which take on negative values when they witness some kind
of partial inseparability.
{\it Acknowledgements:}
PvL is grateful to Masahide Sasaki, Masahiro Takeoka,
Marcos Curty, Norbert L\"{u}tkenhaus, and Samuel Braunstein for
useful discussions.
AF acknowledges the financial support of
MPHPT and MEXT of Japan. PvL thanks the Communications Research
Laboratory Tokyo for funding a research visit.
He also acknowledges the financial support of
the DFG under the Emmy-Noether programme.
\end{document} |
\begin{document}
\title{A general $q$-expansion formula based on matrix inversions and its applications}
\dedicatory{ \textsc{Jin Wang~\dag}\\[1mm]
Department of Mathematics, Soochow University\\
Suzhou 215006, P.~R.~China\\
Email:~\emph{[email protected]}
\thanks{\dag~Corresponding author. This work was supported by NSFC (Grant~No. 11471237)}}
\subjclass[2010]{Primary 33D15 ; Secondary 05A30.}
\keywords{$q$-Series; Expansion formula; Coefficient; Transformation; Summation; Matrix inversion; Lagrange--B\"{u}rmann inversion; Formal power series.}
\begin{abstract}
In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base
$$
\left\{z^n\frac{\poq{az}{n}}{\poq{bz}{n}}\bigg|n=0,1,2\cdots\right\}.$$
Some concrete expansion formulas and their applications to $q$-series identities are presented, including Carlitz's $q$-expansion formula, a new partial theta function identity and a coefficient identity of Ramanujan's ${}_1\psi_1$ summation formula as special cases.
\end{abstract}
\maketitle\thispagestyle{empty}
\markboth{ J. Wang}{A general $q$-expansion formula based on matrix inversions and its applications}
\section{Introduction}
Throughout the present paper, we adopt the standard notation and
terminology for $q$-series from the book \cite{10}. As customary, the $q$-shifted factorials of complex variable $z$
with the base $q: |q|<1$ are given by
\begin{eqnarray}
(z;q)_\infty
:=\prod_{n=0}^{\infty}(1-zq^n)\qquad\mbox{and}\quad (z;q)_n:=\frac{(z;q)_\infty}{(zq^n;q)_\infty}\label{guiding}
\end{eqnarray}
for all integers $n$. For integer $m\geq 1$, we use
the multi-parameter compact notation
\[(a_1,a_2,\ldots,a_m;q)_n:=(a_1;q)_\infty (a_2;q)_n\ldots (a_m;q)_n.\]
Also, the ${}_{r+1}\phi_r$ series with the
base $q$ and the argument $z$ is defined to be
\begin{align*}
{}_{r+1}\phi_r\bigg[\genfrac{}{}{0pt}{}{a_1,a_2,\cdots,a_{r+1}}{b_1,b_2,\cdots,b_{r}};q,z\bigg]&:=\sum _{n=0} ^{\infty}
\frac{\poq{a_1,a_2,\ldots,a_{r+1}}{n}}{\poq{q,b_1,b_2,\ldots,b_r}{n}}z^{n}.
\end{align*}
For any $f(z)=\sum_{n\geq 0}a_nz^n\in \mathbb{C}[[z]]$, $\mathbb{C}[[z]]$ denotes the ring of formal power series in variable $z$, we shall employ the coefficient functional
$$\boldsymbol\lbrack z^n\boldsymbol\rbrack \{f(z)\}:=a_n\,\,\mbox{and}\,\, a_0=f(0).$$
We also follow the summation convention that for any integers $m$ and $n$, $$\sum_{k=m}^{n}a_k=-\sum_{k=n+1}^{m-1}a_k.$$
In their paper \cite{ono}, G.H. Coogan and K. Ono presented the following identity which leads to the generating functions for values of
certain expressions of Hurwitz zeta functions at non-positive integers.
\begin{yl}[cf.\mbox{\cite[Proposition 1.1]{ono}}]\label{ma-id11} For $|z|<1$, it holds
\begin{align}
\sum_{n=0}^\infty\,z^n\frac{\poq{z}{n}}{\poq{-z}{n+1}}=
\sum_{n=0}^{\infty}(-1)^nz^{2n}q^{n^2}.\label{id11}
\end{align}
\end{yl}
The appearance of \eqref{id11} reminds us of the famous Rogers--Fine identity \cite[Eq. (17.6.12)]{dlmf}.
\begin{yl}For $|z|<1$, it holds
\begin{align}
(1-z)\sum_{n=0}^\infty\,z^n\frac{\poq{aq}{n}}{\poq{bq}{n}}=
\sum_{n=0}^{\infty}(1-azq^{2n+1})(bz)^nq^{n^2}\frac{\poq{aq,azq/b}{n}}{\poq{bq,zq}{n}}.\label{id117}
\end{align}
\end{yl}
In fact, Identity \eqref{id11} can be easily deduced from
\eqref{id117} via setting $aq=z=-b$.
Moreover, by setting $a=z=-b$ in \eqref{id117}, we obtain another similar identity.
\begin{yl}\label{ma-id1177}For $|z|<1$, it holds
\begin{align}
\sum_{n=0}^\infty\,z^n\frac{\poq{z}{n+1}}{\poq{-zq}{n}}=1+
2\sum_{n=1}^{\infty}(-1)^nz^{2n}q^{n^2}.\label{id1177}
\end{align}
\end{yl}
It is these identities, once treated as formal power series in $z$, that make us be aware of investigating in a full generality the problem of representations of formal power series in terms of the sequences
$$
\left\{z^n\frac{\poq{az}{n}}{\poq{bz}{n}}\bigg|n=0,1,2\cdots\right\},$$ which is just a base of the ring $\mathbb{C}[[z]]$. This fact asserts that for any $F(z)\in \mathbb{C}[[z]]$, there exists the series expansion
\begin{align}
F(z)=\sum_{n=0}^\infty\,c_nz^n\frac{\poq{az}{n}}{\poq{bz}{n}},\label{import1-0}
\end{align}
where the coefficients $c_n$ must be independent of $z$ but may depend on the parameters $a$ and $b$. In this respect, particularly
noteworthy is that in \cite{xrma}, X.R. Ma established a (formal) generalized Lagrange--B\"{u}rmann inversion formula.
We record it for direct reference.
\begin{yl}[cf.\mbox{\cite[Theorem 2.1]{xrma}}]\label{analogue}
Let $\{\phi_n(z)\}_{n=0}^{\infty}$ be arbitrary sequence of formal power series with $\phi_n(0)=1$. Then for any $F(z)\in \mathbb{C}[[z]]$, we have
\begin{subequations}\label{expan-one}
\begin{equation}
F(z)=\sum_{n=0}^{\infty}\frac{c_nz^{n}}{\prod_{i=0}^{n}\phi_i(z)},\label{expan15a}
\end{equation}
where the coefficients
\begin{eqnarray}
c_n=\sum_{k=0}^n\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\{F(z)\}\sum_{\stackrel{i\leq j_i}{k=j_{0}\leq j_1\leq j_2\leq\cdots\leq j_{n}
\leq j_{n+1}=n}}\prod_{i=0}^n\boldsymbol\lbrack z^{j_{i+1}-j_{i}}\boldsymbol\rbrack\{\phi_i(z)\}.\label{expan-one-ii}
\end{eqnarray}
\end{subequations}
\end{yl}
For further information on Lemma \ref{analogue}, we refer the reader to \cite{xrma}. As for the classical Lagrange--B\"{u}rmann inversion formula the reader might consult the book \cite[p. 629]{andrews4} by G.E. Andrews, R. Askey, and R. Roy. For its various $q$-analogues, we refer the reader to the paper \cite{andrews} by G.E. Andrews, \cite{carliz} by L. Carlitz, \cite{gessel-stan} by I. Gessel and D. Stanton, and \cite{kratt} by Ch. Krattenthaler, especially to the good survey \cite{11} of D. Stanton for a more comprehensive information.
A simple expression of the coefficients $c_n$ seems unlikely under the case \eqref{expan15a}. Without doubt, such an explicit formula is the key step to successful use of this expansion formula.
But in contrast, as far as \eqref{import1-0} is concerned, we are able to establish the following explicit expression of $c_n$ via the use of matrix inversions (see Definition 2.1 below). It is just the theme of the present paper.
\begin{dl}\label{analogue-two}
For $F(z)\in \mathbb{C}[[z]]$, there exists the series expansion
\begin{subequations}\label{expan-two}
\begin{equation}
F(z)=\sum_{n=0}^{\infty}c_nz^{n}\frac{\poq{az}{n}}{\poq{bz}{n}}\label{expan-two-1}
\end{equation}
with the coefficients
\begin{align}
c_n=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ F(z)\frac{\poq{bz}{n-1}}{\poq{az}{n}}\bigg\}-a\sum_{k=0}^{n-1}B_{n-k,1}(a,b)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{F(z)\frac{\poq{bz}{k}}{\poq{az}{k+1}}\bigg\},\label{coeformula}
\end{align}
where $B_{n,1}(a,b)$ are given by
\begin{align}
z=\sum_{n=1}^\infty B_{n,1}(a,b)z^n\frac{\poq{az}{n}}{\poq{bz}{n}}.\label{180}
\end{align}
\end{subequations}
\end{dl}
As a direct application of this expansion formula, we further set up a general transformation concerning the Rogers--Fine identity \eqref{id117}.
\begin{dl}\label{tlidentity} For $G(z)=\sum_{n=0}^{\infty}t_nz^n\in \mathbb{C}[[z]]$, it always holds
\begin{subequations}
\begin{align}
\frac{\poq{az}{\infty}}{\poq{bz}{\infty}}G(z)&=
\sum_{n=0}^{\infty}\frac{\poq{aq/b,az}{n}}{\poq{q,bz}{n}}(bz)^nq^{n(n-1)}\nonumber\\
&\qquad\times\big(\widetilde{G}(zq^n;a,b)-
azq^{2n}\widetilde{G}(zq^{n+1};a/q,b/q)\big),
\label{expan-two-lzlzlz}
\end{align}
where
\begin{align}
\widetilde{G}(z;a,b)&:=\sum_{n=0}^\infty t_nz^n\frac{\poq{az}{n}}{\poq{bz}{n}}.\label{dy}
\end{align}
\end{subequations}
\end{dl}
The rest of this paper is organized as follows. In Section 2, we shall prove Theorem \ref{analogue-two}. For this purpose, a series of preliminary results will be established. Section 3 is devoted to the proof of Theorem\ref{tlidentity}. Some applications of these two theorems to $q$-series are further discussed. Among these applications, there is a new partial theta function identity and a coefficient identity of Ramanujan's ${}_1\psi_1$ summation formula.
\section{The proof of Theorem \ref{analogue-two}}
In this section, we proceed to show Theorem \ref{analogue-two} which amounts to finding the coefficients $c_n$. For this purpose, we need the concept of matrix inversions and a series of preliminary lemmas.
\begin{dy}(cf.\cite[Chapters 2 and 3]{riodan} or
\cite[Definition 3.1.1]{egobook})\label{ddd000} A pair of infinite lower-triangular matrices $A=(A_{n,k})$ and $B=(B_{n,k})$ is said to be inverses to each other if and only if for any integers $n, k\geq 0,$
\begin{align}
\sum_{i=k}^n\,A_{n,i}B_{i,k}=\sum_{i=k}^n\,B_{n,i}A_{i,k}
=\left\{
\begin{array}{ll}
0, & n\neq k; \\
1, & n=k,
\end{array}
\right.
\end{align}
where $A_{n,k}=B_{n,k}=0$ if $n<k.$
As usual, we also say that $A$ and $B$ are invertible and write $A^{-1}$ for $B$.
\end{dy}
Consider now a particular matrix $A=(A_{n,k})$ with the entries $A_{n,k}$ given by
\begin{align}
z^k\frac{\poq{az}{k}}{\poq{bz}{k}}=\sum_{n=k}^\infty\,A_{n,k}z^n.\label{eq1-13}
\end{align}
It is easy to check that $A=(A_{n,k})$ is invertible. In what follows, let us assume its inverse
$$A^{-1}=(B_{n,k}(a,b)).$$ As such, we see that \eqref{eq1-13}
is equivalent to
\begin{align}
z^k=\sum_{n=k}^\infty B_{n,k}(a,b)z^n\frac{\poq{az}{n}}{\poq{bz}{n}}.\label{18}
\end{align}
Next, we shall focus on two kinds of generating functions of the entries $B_{n,k}(a,b)$ of the matrix $A^{-1}$.
\begin{yl}\label{ddd} Let $B_{n,k}(a,b)$ be the same as above.
Then we have
\begin{align}B_{n,k+1}(a,b)+(b-a)\sum_{i=k+2}^{n}b^{i-k-2} B_{n,i}(a,b)=q^{n-k-1}B_{n-1,k}(a,b).\label{impor-1-added}
\end{align}
\end{yl}
\noindent {\it Proof.}
At first, by replacing $z$ by $zq$ in \eqref{18}, we have
\begin{align*}
(zq)^k=\sum_{n=k}^\infty B_{n,k}(a,b)(zq)^n\frac{\poq{azq}{n}}{\poq{bzq}{n}}.
\end{align*}
Multiplying both sides with $z(1-az)/(1-bz)$ and shifting $n$ to $n-1$, we obtain
\begin{align}
q^k z^{k+1}\bigg(1+(b-a)z\sum_{i= 0}^{\infty}b^{i}z^{i}\bigg)=\sum_{n=k+1}^\infty B_{n-1,k}(a,b)q^{n-1}z^n\frac{\poq{az}{n}}{\poq{bz}{n}}.\label{1888}
\end{align}
Viewing \eqref{1888} from the definition of \eqref{18}, we find that \eqref{1888} can be recast as
\begin{align}
&q^k\sum_{n=k+1}^\infty B_{n,k+1}(a,b)z^n\frac{\poq{az}{n}}{\poq{bz}{n}}\nonumber\\
&+(b-a)q^k\sum_{i= 0}^{\infty}b^{i}\sum_{n=k+i+2}^\infty B_{n,k+i+2}(a,b)z^n\frac{\poq{az}{n}}{\poq{bz}{n}}
\nonumber\\
&=\sum_{n=k+1}^\infty B_{n-1,k}(a,b)q^{n-1}z^n\frac{\poq{az}{n}}{\poq{bz}{n}}.\label{18888}
\end{align}
Thus, by the uniqueness of the coefficients under the base $\{z^n\poq{az}{n}/\poq{bz}{n}\}_{n=0}^\infty$, it holds
\begin{align*}
q^k B_{n,k+1}(a,b)+(b-a)q^k\sum_{i=0}^{n-k-2}b^{i} B_{n,k+i+2}(a,b)=q^{n-1}B_{n-1,k}(a,b).
\end{align*}
After slight simplification, we obtain
\begin{align*}
B_{n,k+1}(a,b)+(b-a)\sum_{i=k+2}^{n}b^{i-k-2} B_{n,i}(a,b)=q^{n-k-1} B_{n-1,k}(a,b).
\end{align*}
Hence \eqref{impor-1-added} follows.
\rule{4pt}{7pt}
By use of Lemma \ref{ddd}, it is easy to set up a bivariate generating function of $\{B_{n,k}(a,b)\}_{n\geq k\geq 0}$.
\begin{yl} \label{bgf} Let $B_{n,k}(a,b)$ be defined by \eqref{18}. Then we have
\begin{align}
G(y,z)=\sum_{n=0}^\infty\frac{\poq{b/y}{n}}{\poq{a/y}{n}}(yz)^n+\sum_{n=0}^\infty (bzq^n-aG_1(zq^n))\frac{\poq{b/y}{n}}{\poq{a/y}{n+1}}(yz)^n,\label{impor-1}
\end{align}
where
\begin{align}
G(y,z)&:=\sum_{k=0}^\infty G_k(z)y^k,\label{impor-2}\\
G_k(z)&:=\sum_{n=k}^\infty B_{n,k}(a,b)z^n.\label{impor-3}
\end{align}
\end{yl}
\noindent {\it Proof.} It suffices to multiply both sides of \eqref{impor-1-added} with $b$. Then
we get
\begin{align}bB_{n,k+1}(a,b)+(b-a)\sum_{i=k+2}^{n}b^{i-k-1} B_{n,i}(a,b)= bq^{n-k-1}B_{n-1,k}(a,b).\label{sansan-5}
\end{align}
Shifting $k$ to $k-1$ in \eqref{impor-1-added} gives rise to
\begin{align}
B_{n,k}(a,b)+(b-a)\sum_{i=k+1}^{n}b^{i-k-1} B_{n,i}(a,b)= q^{n-k}B_{n-1,k-1}(a,b).\label{sansan-6}
\end{align}
By abstracting \eqref{sansan-5} from \eqref{sansan-6}, we come up with
\begin{align}
B_{n,k}(a,b)-aB_{n,k+1}(a,b)=q^{n-k}B_{n-1,k-1}(a,b)-bq^{n-k-1}B_{n-1,k}(a,b).\label{sansan-007}
\end{align}
After multiplying \eqref{sansan-007} by $z^{n}$ and summing over $n$ for $n\geq k$, then we have
\begin{align*}
\sum_{n=k}^\infty B_{n,k}(a,b)z^{n}&-a\sum_{n=k}^\infty B_{n,k+1}(a,b)z^{n}\\
&=zq^{1-k}\sum_{n=k}^\infty B_{n-1,k-1}(a,b)(qz)^{n-1}-bzq^{-k}\sum_{n=k}^\infty B_{n-1,k}(a,b)(qz)^{n-1}.
\end{align*}
In terms of $G_k(z)$ defined by \eqref{impor-3}, this relation can be expressed as
\begin{align}
G_k(z)-aG_{k+1}(z)=zq^{1-k}G_{k-1}(qz)-bzq^{-k}G_k(qz).\label{sansan-77777}
\end{align}
Then, by multiplying $y^{k+1}$ and then summing over $k$ for $k\geq 1$ on both sides of \eqref{sansan-77777}, we further obtain
\begin{align*}
(y-a)G(y,z)+(a-y)G_0(z)+ayG_1(z)=yz(y-b)G(y/q,qz)+byzG_0(qz),
\end{align*}
where $G(y,z)$ is given by \eqref{impor-2}. Observe that $G_0(z)=1$.
Then
\begin{align*}
(y-a)G(y,z)-yz(y-b)G(y/q,qz)=y-a+byz-ayG_1(z),
\end{align*}
namely,
\begin{align}
G(y,z)-yz\frac{1-b/y}{1-a/y}G(y/q,zq)=d(y,z),\label{123}
\end{align}
where, for clarity, we define
\[d(y,z):=1+\frac{y}{y-a}(bz-aG_1(z)).\]
Setting $(y,z)\to(y/q^n,zq^n)$ in \eqref{123}, we obtain its equivalent version as below
\begin{align}
X(n)-yz\frac{1-bq^n/y}{1-aq^n/y}X(n+1)=d(y/q^n,zq^n),\label{123-123}
\end{align}
where
\[X(n):=G(y/q^n,zq^n).\]
Iterating \eqref{123-123} $m$ times, we find
\begin{align*}
X(n)&=d(y/q^n,zq^n)+yz\frac{1-bq^n/y}{1-aq^n/y}X(n+1)\\
&=d(y/q^n,zq^n)+yz\frac{1-bq^n/y}{1-aq^n/y}d(y/q^{n+1},zq^{n+1})\\
&\qquad+(yz)^2\frac{(1-bq^n/y)(1-bq^{n+1}/y)}{(1-aq^n/y)(1-aq^{n+1}/y)}X(n+2)\\
&=\cdots\\
&=\sum_{k=0}^{m-1} d(y/q^{n+k},zq^{n+k})\frac{\poq{bq^n/y}{k}}{\poq{aq^n/y}{k}}(yz)^k+\frac{\poq{bq^n/y}{m}}{\poq{aq^n/y}{m}}(yz)^mX(n+m).
\end{align*}
Regarding the solution of this recurrence relation, we may guess and then show by induction on $m$ (set $n=0$) that
\begin{align*}
G(y,z)&=\sum_{k=0}^\infty d(y/q^k,zq^k)\frac{\poq{b/y}{k}}{\poq{a/y}{k}}(yz)^k\\
&=\sum_{k=0}^\infty\frac{\poq{b/y}{k}}{\poq{a/y}{k}}(yz)^k+\sum_{k=0}^\infty \frac{bzq^k-aG_1(zq^k)}{1-aq^k/y}\frac{\poq{b/y}{k}}{\poq{a/y}{k}}(yz)^k\\
&=\sum_{k=0}^\infty\frac{\poq{b/y}{k}}{\poq{a/y}{k}}(yz)^k+\sum_{k=0}^\infty (bzq^k-aG_1(zq^k))\frac{\poq{b/y}{k}}{\poq{a/y}{k+1}}(yz)^k.
\end{align*}
So the lemma is proved.
\rule{4pt}{7pt}
There also exists a finite univariate generating function of $\{B_{n,k}(a,b)\}_{k=0}^n$.
\begin{tl}\label{eee} Let $B_{n,k}(a,b)$ be defined by \eqref{18}. Then for integer $n\geq 1$, we have
\begin{align}\sum_{k=0}^nB_{n,k}(a,b)y^k=
\frac{\poq{b/y}{n-1}}{\poq{a/y}{n}}y^{n}-a\sum_{k=0}^{n-1} B_{n-k,1}(a,b)q^{(n-k)k} \frac{\poq{b/y}{k}}{\poq{a/y}{k+1}}y^k.\label{fff}
\end{align}
\end{tl}
\noindent {\it Proof.} It is an immediate consequence of Lemma \ref{bgf}.
To be precise, by Lemma \ref{bgf}, we see
\begin{align*}
G(y,z)&=\sum_{n\geq k\geq 0}B_{n,k}(a,b)y^kz^n\\
&=\sum_{n=0}^\infty\frac{\poq{b/y}{n}}{\poq{a/y}{n}}(yz)^n+\sum_{n=0}^\infty (bzq^n-aG_1(zq^n))\frac{\poq{b/y}{n}}{\poq{a/y}{n+1}}(yz)^n.
\end{align*}
By equating the coefficients of $z^n$ on both sides, it is easy to calculate that for $n\geq 1$,
\begin{align*}
\sum_{k=0}^nB_{n,k}(a,b)y^k&=\frac{\poq{b/y}{n}}{\poq{a/y}{n}}y^n+b\frac{\poq{b/y}{n-1}}{\poq{a/y}{n}}(qy)^{n-1}\\
&-a\sum_{k=0}^n B_{n-k,1}(a,b)q^{(n-k)k}\frac{\poq{b/y}{k}}{\poq{a/y}{k+1}}y^k\\
&=\frac{\poq{b/y}{n-1}}{\poq{a/y}{n}}y^{n}-a\sum_{k=0}^{n-1}B_{n-k,1}(a,b)q^{(n-k)k} \frac{\poq{b/y}{k}}{\poq{a/y}{k+1}}y^k.
\end{align*} The corollary is thus proved.
\rule{4pt}{7pt}
\begin{remark}
Evidently, the left-hand side of \eqref{fff} is a polynomial in $y$ while the right-hand side doesn't seem that case. In fact, the coefficients $B_{n-k,1}(a,b)$ given by \eqref{180}, i.e.,
$$
z=\sum_{n=1}^\infty B_{n,1}(a,b) z^n\frac{\poq{az}{n}}{\poq{bz}{n}},
$$
just satisfy
$$S_n(aq^k)=0\,\,\,(0\leq k\leq n-1),$$
where $S_n(y)$ is given by
\begin{align}
\frac{S_n(y)}{\prod_{k=0}^{n-1}(y-aq^k)}:=\frac{\poq{b/y}{n-1}}{\poq{a/y}{n}}y^{n}-a\sum_{k=0}^{n-1} B_{n-k,1}(a,b)q^{(n-k)k}\frac{\poq{b/y}{k}}{\poq{a/y}{k+1}}y^k.
\end{align}
This fact guarantees that the right-hand side of \eqref{fff} is really a polynomial in $y$.
\end{remark}
Corollary \ref{eee} leads us to a general matrix inversion, which will play a very crucial role in our main result, i.e., Theorem \ref{analogue-two}.
\begin{dl}[Matrix inversion]\label{matrixinversion}
Let $A=(A_{n,k})$ be the infinitely lower-triangular matrix with the entries
\begin{align}
A_{n,k}=\boldsymbol\lbrack z^{n-k}\boldsymbol\rbrack\bigg\{\frac{\poq{az}{k}}{\poq{bz}{k}}\bigg\}\label{eq1-13-inverse}
\end{align}
and assume $A^{-1}=(B_{n,k}(a,b))$. Then
\begin{align}B_{n,k}(a,b)&=\boldsymbol\lbrack z^{n-k}\boldsymbol\rbrack\bigg\{
\frac{\poq{bz}{n-1}}{\poq{az}{n}}\bigg\}-a\sum_{i=k}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i} \boldsymbol\lbrack z^{i-k}\boldsymbol\rbrack\bigg\{\frac{\poq{bz}{i}}{\poq{az}{i+1}}\bigg\}.\label{ssss}
\end{align}
\end{dl}
\noindent {\it Proof.} It is clear that \eqref{ssss} is valid for $n=k=0$ or $k=0$. Thus we only need to show \eqref{ssss} for $n\geq 1$. To that end, we first set $y\to 1/t$ in \eqref{fff} and then multiply both sides with $t^n$. All that we obtained is
\begin{align}\sum_{k=0}^nB_{n,k}(a,b)t^{n-k}=
\frac{\poq{bt}{n-1}}{\poq{at}{n}}-a\sum_{k=0}^{n-1}B_{n-k,1}(a,b) \frac{\poq{bt}{k}}{\poq{at}{k+1}}q^{(n-k)k}t^{n-k}.\label{hhh}
\end{align}
A comparison of the coefficients of $t^{n-k}$ yields
\begin{align*}B_{n,k}(a,b)&=\boldsymbol\lbrack t^{n-k}\boldsymbol\rbrack\bigg\{
\frac{\poq{bt}{n-1}}{\poq{at}{n}}\bigg\}-a\sum_{i=0}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i} \boldsymbol\lbrack t^{n-k}\boldsymbol\rbrack\bigg\{\frac{\poq{bt}{i}}{\poq{at}{i+1}}t^{n-i}\bigg\}\\
&=\boldsymbol\lbrack t^{n-k}\boldsymbol\rbrack\bigg\{
\frac{\poq{bt}{n-1}}{\poq{at}{n}}\bigg\}-a\sum_{i=k}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i} \boldsymbol\lbrack t^{i-k}\boldsymbol\rbrack\bigg\{\frac{\poq{bt}{i}}{\poq{at}{i+1}}\bigg\}.
\end{align*}
Thus \eqref{ssss} is confirmed.
\rule{4pt}{7pt}
As byproducts of our analysis, we find two interesting properties for $\{B_{n,k}(a,b)\}_{n\geq k\geq 0}$ as follows.
\begin{xinzhi} Let $B_{n,k}(a,b)$ be given by \eqref{18}.
Then for integer $n\geq 1$ and
$t\in \mathbb{C}$, we have
\begin{align}
B_{n,k}(at,bt)&=B_{n,k}(a,b)t^{n-k},\label{maxinrong}
\\
\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{
\frac{\poq{bz}{n-1}}{\poq{az}{n}}\bigg\}&=a\sum_{i=0}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i} \boldsymbol\lbrack z^{i}\boldsymbol\rbrack\bigg\{\frac{\poq{bz}{i}}{\poq{az}{i+1}}\bigg\}.\label{maxinrong-1}\end{align}
\end{xinzhi}
\noindent {\it Proof.} To establish \eqref{maxinrong}, it only needs to take $(a,b)\to (at,bt)$ in \eqref{18}. Then it follows
\begin{align*}
z^k=\sum_{n=k}^\infty\,B_{n,k}(at,bt)z^n\frac{\poq{atz}{n}}{\poq{btz}{n}}.
\end{align*}
On the other hand, on setting $z\to t\,z$ in \eqref{18}, we have
\begin{align*}
z^k=\sum_{n=k}^\infty\,B_{n,k}(a,b)t^{n-k}z^n\frac{\poq{atz}{n}}{\poq{btz}{n}}.
\end{align*}
By the uniqueness of series expansion, we obtain \eqref{maxinrong}.
Identity \eqref{maxinrong-1} is a special case $k=0$ of \eqref{ssss}, noting that for $n\geq 1,B_{n,0}=0$.
\rule{4pt}{7pt}
After these preliminaries we are prepared to show
Theorem \ref{analogue-two}.
\noindent {\it Proof.} The existence of \eqref{expan-two-1} is evident, because
$$
\left\{z^n\frac{\poq{az}{n}}{\poq{bz}{n}}\bigg|n=0,1,2\cdots\right\}$$
is the base of $\mathbb{C}[[z]]$.
Thus it only needs to evaluate the coefficients $c_n$ in \eqref{expan-two-1}. To do this, by Theorem \ref{matrixinversion}, we have
\begin{align*}
c_n&=\sum_{k=0}^n \boldsymbol\lbrack z^{k}\boldsymbol\rbrack\left\{F(z)\right\} B_{n,k}(a,b)\\
&=\sum_{k=0}^n\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\left\{F(z)\right\}\boldsymbol\lbrack z^{n-k}\boldsymbol\rbrack\bigg\{
\frac{\poq{bz}{n-1}}{\poq{az}{n}}\bigg\}\\
&-a\sum_{i=0}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i} \sum_{k=0}^i\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\{F(z)\}[z^{i-k}]\bigg\{\frac{\poq{bz}{i}}{\poq{az}{i+1}}\bigg\}\\
&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ F(z)\frac{\poq{bz}{n-1}}{\poq{az}{n}}\bigg\}-a\sum_{i=0}^{n-1}B_{n-i,1}(a,b)q^{(n-i)i}\boldsymbol\lbrack z^{i}\boldsymbol\rbrack\bigg\{ F(z)\frac{\poq{bz}{i}}{\poq{az}{i+1}}\bigg\}.
\end{align*}
The conclusion is proved.
\rule{4pt}{7pt}
\begin{remark}
It is worth mentioning that in \cite{garsia-1} A.M. Garsia and J. Remmel set up a $q$-Lagrange inversion formula, which asserts that for any formal power series $F(z)=\sum_{n=1}^{\infty}F_nz^n$ and $f(z)=\sum_{n=1}^{\infty}f_nz^n$ with
$F_1f_1\neq 0$,
\begin{eqnarray}
\sum_{n=1}^\infty f_nF(z)F(zq)\cdots F(zq^{n-1})=z\label{1}
\end{eqnarray}
holds if and only if
\begin{eqnarray}
\sum_{n=1}^\infty F_nf(z)f(z/q)\cdots f(z/q^{n-1})=z.\label{2}
\end{eqnarray}
However, to the author's knowledge, it is still hard to find out explicit expressions of $f_n:=B_{n,1}(a,b)$ from \eqref{1} even if $F(z)=z(1-az)/(1-bz)$.
\end{remark}
In the following, we shall examine a few specific formal expansion formulas covered by Theorem \ref{analogue-two}. As a first consequence, when $a=0$ we recover Carlitz's $q$-expansion formula {\cite[p. 206, Eq. (1.11)]{carliz}}.
\begin{tl}For any $F(z)\in \mathbb{C}[[z]]$, we have
\begin{subequations}\label{expan-three}
\begin{equation}
F(z)=\sum_{n=0}^{\infty}\frac{c_nz^{n}}{\poq{bz}{n}},\label{expan-three-1}
\end{equation}
where
\begin{equation}
c_n=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\big\{F(z)\poq{bz}{n-1}\big\}.\label{expan-three-2}
\end{equation}
\end{subequations}
\end{tl}
We remark that Carlitz's $q$-expansion formula is a useful $q$-analogue of the Lagrange--B\"{u}rmann inversion formula. The reader may consult the survey
\cite{11} of D. Stanton concerning this topic.
A second interesting consequence occurs when $b=0$.
\begin{tl} Let $B_{n,1}(a,0)$ be given by \eqref{180}. Then
\begin{subequations}\label{expan-two-11}
\begin{equation}
F(z)=\sum_{n=0}^{\infty}c_nz^{n}\poq{az}{n},\label{expan-two-22}
\end{equation}
where the coefficients
\begin{eqnarray}
c_n=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\frac{F(z)}{\poq{az}{n}}\bigg\}
-a\sum_{k=0}^{n-1}B_{n-k,1}(a,0)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{\frac{F(z)}{\poq{az}{k+1}}\bigg\}.
\end{eqnarray}
\end{subequations}
\end{tl}
As a third consequence, the special case $b=aq$ leads us to
\begin{tl}\label{coro310} Let $F(z)=\sum_{n=0}^{\infty}a_nz^n$ and $F_k(z)=\sum_{i=0}^ka_iz^i$, being the $k$-truncated series of $F(z)$. Suppose that
\begin{subequations}\label{expan-four}
\begin{equation}
\frac{F(z)}{1-az}=\sum_{n=0}^{\infty}\frac{c_nz^{n}}{1-azq^n}.\label{expan-four-1-0}
\end{equation}
Then $c_0=a_0$ and $n\geq1,$
\begin{equation}
c_n=\sum_{k=0}^{n-1}g_{n-k}(q)q^{(n-k)k}\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ \frac{F(z)-F_k(z)}{1-az}\bigg\},\label{expan-four-2-0}
\end{equation}
\end{subequations}
where $g_n(q)$ are polynomials in $q$ given recursively by
\begin{align*}g_n(q)=1
-\sum_{i=1}^{n-1}g_{n-i}(q)q^{(n-i)i}.
\end{align*}
\end{tl}
\noindent {\it Proof.} In such case, we first solve the recurrence relation \eqref{ssss} with $k=1$ for $B_{n,1}(a,aq)$, viz.
\begin{align*}B_{n,1}(a,aq)=a^{n-1}
-\sum_{i=1}^{n-1}B_{n-i,1}(a,aq)q^{(n-i)i} a^{i}.
\end{align*}
The solution is recursively given by
\begin{align}
\left\{
\begin{array}{ll}
&B_{n,1}(a,aq)=g_n(q)a^{n-1},\\
&\\
&g_n(q)=\displaystyle1-\sum_{i=1}^{n-1}g_{n-i}(q)q^{(n-i)i}.\label{crucial}
\end{array}
\right.
\end{align}
By virtue of \eqref{crucial}, we are now able to calculate $c_n$. To do this, by Theorem \ref{analogue-two} we have
\begin{align*}
c_n&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ \frac{F(z)}{1-az}\bigg\}-\sum_{k=0}^{n-1}g_{n-k}(q)q^{(n-k)k}a^{n-k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{\frac{F(z)}{1-az}\bigg\}\\
&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ \frac{F(z)}{1-az}\bigg\}-\sum_{k=0}^{n-1}g_{n-k}(q)q^{(n-k)k}\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\frac{F_k(z)}{1-az}\bigg\}\\
&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\sum_{k=0}^{n-1}g_{n-k}(q)q^{(n-k)k}\frac{F(z)-F_k(z)}{1-az}\bigg\}.
\end{align*}
In the penultimate equality we have used the fact that
\begin{align*}
a^{n-k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{\frac{F(z)}{1-az}\bigg\}=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\frac{F_k(z)}{1-az}\bigg\}
\end{align*}
and in the last equality, we have invoked \eqref{crucial} again.
The conclusion is proved.
\rule{4pt}{7pt}
It is also of interest to note that if $F(z)$ is a polynomial of degree $m+1$, say
$$
F(z)=(1-az)\prod_{n=1}^{m}(1-t_nz),
$$
and $F_k(z)=F(z)$ for $k\geq m+1$, then Corollary \ref{coro310} reduces to
\begin{tl} With the same notation as Corollary \ref{coro310}. Then we have
\begin{subequations}
\begin{equation}
\prod_{n=1}^{m}(1-t_nz)=\sum_{n=0}^\infty \frac{c_nz^{n}}{1-azq^n},\label{exexpan-two-1-0}
\end{equation}
where $c_0=1$ and $n\geq 1$,
\begin{equation}
c_n=\sum_{k=0}^{\min\{m,n-1\}} g_{n-k}(q)q^{(n-k)k}\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\prod_{i=1}^{m}(1-t_iz)-\frac{F_k(z)}{1-az}\bigg\}.\label{mistake-wangjin}
\end{equation}
\end{subequations}
\end{tl}
\section{Applications to $q$-series theory}
Unlike the preceding section, we now focus our attention on applications of Theorem \ref{analogue-two} to the $q$-series theory. In this sense, we assume that all results are subject to appropriate convergent conditions of rigorous analytic theory, unless otherwise stated.
Let us begin with the proof of Theorem \ref{tlidentity}.
\noindent {\it Proof.} We only need to make use of Theorem \ref{analogue-two} as well as the $q$-binomial theorem \cite[(II.3)]{10} to get
\begin{align*}
\frac{\poq{az}{\infty}}{\poq{bz}{\infty}}G(z)=\sum_{n=0}^{\infty}c_nz^{n}\frac{\poq{az}{n}}{\poq{bz}{n}}
=S_1-aS_2,
\end{align*}
where
\begin{align*}
S_1&:=\sum_{n=0}^{\infty}\sum_{i=0}^{n}\frac{\poq{aq/b}{i}}{\poq{q}{i}}b^iq^{(n-1)i}t_{n-i}z^{n}\frac{\poq{az}{n}}{\poq{bz}{n}},\\
S_2&:=\sum_{n=0}^{\infty}\sum_{k=0}^{n-1}B_{n-k,1}(a,b)q^{(n-k)k}\sum_{i=0}^k
\frac{\poq{aq/b}{i}}{\poq{q}{i}}(bq^k)^it_{k-i}z^{n}\frac{\poq{az}{n}}{\poq{bz}{n}}.
\end{align*}
After a mere series rearrangement, we get
\begin{align*}
S_1&=\sum_{i=0}^{\infty}\frac{\poq{aq/b}{i}}{\poq{q}{i}}b^iq^{i^2-i}z^{i}\frac{\poq{az}{i}}{\poq{bz}{i}}\sum_{n= 0}^{\infty}q^{ni}t_{n}z^{n}\frac{\poq{azq^i}{n}}{\poq{bzq^i}{n}}\\
&=\sum_{i= 0}^{\infty}\frac{\poq{aq/b,az}{i}}{\poq{q,bz}{i}}b^iq^{i^2-i}z^{i}\widetilde{G}(zq^i;a,b).
\end{align*}
Hereafter, as given by \eqref{dy},
\[\widetilde{G}(z;a,b)=\sum_{n=0}^{\infty}t_{n}z^{n}\frac{\poq{az}{n}}{\poq{bz}{n}}.\]
In a similar way, it is easily found that
\begin{align*}
S_2&=\sum_{i=0}^{\infty}\frac{\poq{aq/b}{i}}{\poq{q}{i}}b^i
\sum_{k=i}^{\infty}t_{k-i}z^k\frac{\poq{az}{k}}{\poq{bz}{k}}q^{ki}\Delta_{k},
\end{align*}
where
\[
\Delta_{k}:=\sum_{n= k+1}^{\infty}B_{n-k,1}(a,b)(zq^{k})^{n-k}\frac{\poq{azq^k}{n-k}}{\poq{bzq^k}{n-k}}=\sum_{n= 1}^{\infty}B_{n,1}(a,b)(zq^{k})^{n}\frac{\poq{azq^k}{n}}{\poq{bzq^k}{n}}=
zq^k.
\]
The last equality is based on \eqref{180}.
Therefore,
\begin{align*}
S_2&=z\sum_{i=0}^{\infty}\frac{\poq{aq/b,az}{i}}{\poq{q,bz}{i}}(bzq^{i+1})^i
\sum_{k=0}^{\infty}t_{k}\frac{\poq{azq^i}{k}}{\poq{bzq^i}{k}}(zq^{i+1})^{k}\\
&=z\sum_{i=0}^{\infty}\frac{\poq{aq/b,az}{i}}{\poq{q,bz}{i}}(bzq^{i+1})^i\widetilde{G}(zq^{i+1};a/q,b/q).
\end{align*}
Finally, we achieve
\begin{align*}
\frac{\poq{az}{\infty}}{\poq{bz}{\infty}}G(z)&=
\sum_{n=0}^{\infty}\frac{\poq{aq/b,az}{n}}{\poq{q,bz}{n}}(bz)^nq^{n(n-1)}\big(\widetilde{G}(zq^n;a,b)-
azq^{2n}\widetilde{G}(zq^{n+1};a/q,b/q)\big).
\end{align*}
This gives the complete proof of the theorem.
\rule{4pt}{7pt}
With regard to applications of Theorem \ref{tlidentity} to $q$-series, it is necessary to set up
\begin{tl}\label{tlnew}For integer $r\geq 0$ and $|cz|<1$, it holds
\begin{align}
&\frac{\poq{azq}{\infty}}{\poq{bz}{\infty}}{_{r+1}\phi_r}\bigg[\genfrac{}{}{0pt}{}{A_1,A_2,\ldots,A_{r+1}}{B_1,B_2,\ldots,B_{r}};q,cz\bigg]\nonumber\\
&=\lim_{x,y\to\infty}\sum_{n=0}^{\infty}\frac{\poq{az,(az)^{1/2}q,-(az)^{1/2}q,aq/b,x,y}{n}}{\poq{q,(az)^{1/2},-(az)^{1/2},bz,azq/x,azq/y}{n}}\bigg(\frac{bz}{xy}\bigg)^n\nonumber\\
&\qquad\quad\times{_{r+3}\phi_{r+2}}\bigg[\genfrac{}{}{0pt}{}{azq^n,azq^{2n+1},A_1,A_2,\ldots,A_{r+1}}{bzq^n,azq^{2n},B_1,B_2,\ldots,B_{r}};q,czq^n\bigg].\label{expan-two-5lz}
\end{align}
\end{tl}
\noindent {\it Proof.} It suffices to set in Theorem \ref{tlidentity}
$$G(z)={_{r+1}\phi_r}\bigg[\genfrac{}{}{0pt}{}{A_1,A_2,\ldots,A_{r+1}}{B_1,B_2,\ldots,B_{r}};q,cz\bigg],$$ which means
$$t_k=\frac{\poq{A_1,A_2,\ldots,A_{r+1}}{k}}{\poq{q,B_1,B_2,\ldots,B_{r}}{k}}c^k.$$
In the sequel, it is routine to compute
\begin{align*}
H_n(z;a,b)&:=\frac{\widetilde{G}(zq^n;a,b)-
azq^{2n}\widetilde{G}(zq^{n+1};a/q,b/q)}{1-azq^{2n}}\\
&=\sum_{k=0}^\infty\frac{\poq{azq^n,azq^{2n+1}}{k}}{\poq{bzq^n,azq^{2n}}{k}}(zq^n)^k t_k\\
&={_{r+3}\phi_{r+2}}\bigg[\genfrac{}{}{0pt}{}{azq^n,azq^{2n+1},A_1,A_2,\ldots,A_{r+1}}{bzq^n,azq^{2n},B_1,B_2,\ldots,B_{r}};q,czq^n\bigg].
\end{align*}
This reduces \eqref{expan-two-lzlzlz} of Theorem \ref{tlidentity} to
\begin{align}
&\frac{\poq{azq}{\infty}}{\poq{bz}{\infty}}{_{r+1}\phi_r}\bigg[\genfrac{}{}{0pt}{}{A_1,A_2,\ldots,A_{r+1}}{B_1,B_2,\ldots,B_{r}};q,cz\bigg]\nonumber\\
&\qquad=
\sum_{n=0}^{\infty}\frac{\poq{aq/b,az}{n}}{\poq{q,bz}{n}}(bz)^nq^{n(n-1)}\frac{1-azq^{2n}}{1-az}H_{n}(z;a,b).\label{xuyao}
\end{align}
Finally, using the basic relations
\begin{align*}
\frac{1-azq^{2n}}{1-az}
=\frac{\poq{(az)^{1/2}q,-(az)^{1/2}q}{n}}{\poq{(az)^{1/2},-(az)^{1/2}}{n}}
\end{align*}
and
\begin{align*}
\lim_{x,y\to\infty}
\frac{\poq{x,y}{n}}{\poq{azq/x,azq/y}{n}}\bigg(\frac{1}{xy}\bigg)^n
=q^{n(n-1)},
\end{align*}
we derive \eqref{expan-two-5lz} from \eqref{xuyao} directly.
\rule{4pt}{7pt}
The following are two special instances of Theorem \ref{tlidentity}.
\begin{lz}The following transformation formulas are valid.
\begin{align}
\frac{\poq{az}{\infty}}{\poq{bz}{\infty}}&=
\sum_{n=0}^{\infty}\frac{\poq{aq/b,az}{n}}{\poq{q,bz}{n}}(bz)^nq^{n(n-1)}(1-azq^{2n}),\label{expan-two-lzlz}\\
\frac{\poq{azq,ABz}{\infty}}{\poq{bz,Bz}{\infty}}&=
\sum_{n=0}^{\infty}\frac{\poq{(az)^{1/2}q,-(az)^{1/2}q,aq/b,az}{n}}{\poq{q,(az)^{1/2},-(az)^{1/2},bz}{n}}
(bz)^nq^{n(n-1)}\nonumber\\
&\qquad\times{_3\phi_2}\bigg[\genfrac{}{}{0pt}{}{azq^n,azq^{2n+1},A}{bzq^n,azq^{2n}};q,Bzq^n\bigg].\label{expan-two-4lz}
\end{align}
\end{lz}
\noindent {\it Proof.} Identity \eqref{expan-two-lzlz} comes from $G(z)=1$ in Theorem \ref{tlidentity} and \eqref{expan-two-4lz} does by taking $G(z)=\poq{ABz}{\infty}/\poq{Bz}{\infty}$, i.e., $t_k=\poq{A}{k}/\poq{q}{k}B^k$
in Theorem \ref{tlidentity} or $r=0$ in Corollary \ref{tlnew}.
\rule{4pt}{7pt}
The next conclusion shows how Theorem \ref{tlidentity} can be applied to known transformation formulas for finding new results.
\begin{tl} For $|z|<1$, we have
\begin{align} {}_{2}\phi _{1}\left[\begin{matrix}A,B\\ C\end{matrix}
; q, z\right]=&\sum_{n=0}^{\infty}
\frac{\poq{ABq/C,ABz/C}{n}}
{\poq{q,z}{n}} z^n q^{n(n-1)}\left(1-\frac{ABzq^{2n}}{C}\right)\nonumber\\
&\times{}_{4}\phi _{3}\left[\begin{matrix}ABzq^n/C,ABzq^{2n+1}/C,C/A,C/B\\ C,zq^n,ABzq^{2n}/C\end{matrix}
; q, \frac{ABzq^n}{C}\right].
\end{align}
\end{tl}
\noindent {\it Proof.} Performing as above, we choose in Theorem \ref{tlidentity}
\begin{align*}
G(z)={}_{2}\phi _{1}\left[\begin{matrix}C/A,C/B\\ C\end{matrix}
; q, \frac{ABz}{C}\right],\end{align*}
which corresponds to
\begin{align*}
t_k=\frac{\poq{C/A,C/B}{k}}{\poq{q,C}{k}}\left(\frac{AB}{C}\right)^k.
\end{align*}In this case, it is clear that
\begin{align*}
H_n(z;a,b):&=\frac{\widetilde{G}(zq^n;a,b)-
azq^{2n}\widetilde{G}(zq^{n+1};a/q,b/q)}{1-azq^{2n}}\\
&={}_{4}\phi _{3}\left[\begin{matrix}azq^n,azq^{2n+1},C/A,C/B\\ C,bzq^n,azq^{2n}\end{matrix}
; q, \frac{ABzq^n}{C}\right].
\end{align*}
As a result, from Theorem \ref{tlidentity} it follows
\begin{align*}
\frac{\poq{az}{\infty}}{\poq{bz}{\infty}}{}_{2}\phi _{1}\left[\begin{matrix}C/A,C/B\\ C\end{matrix}
; q, \frac{ABz}{C}\right]&=
\sum_{n=0}^{\infty}\frac{\poq{aq/b,az}{n}}{\poq{q,bz}{n}}(bz)^nq^{n(n-1)}(1-azq^{2n})H_n(z;a,b).
\end{align*}
In this form, taking $a=AB/C$ and $b=1$, we obtain
\begin{align}
&\frac{\poq{ABz/C}{\infty}}{\poq{z}{\infty}}{}_{2}\phi _{1}\left[\begin{matrix}C/A,C/B\\ C\end{matrix}
; q, \frac{ABz}{C}\right]\nonumber\\
&=\sum_{n=0}^{\infty}\frac{\poq{ABq/C,ABz/C}{n}}{\poq{q,z}{n}}z^nq^{n(n-1)}(1-ABzq^{2n}/C)H_n(z;AB/C,1).\label{need}
\end{align}
By combining \eqref{need} with Heine's third transformation \cite[(III.3)]{10}
\begin{align*}
{}_{2}\phi _{1}\left[\begin{matrix}A,B\\ C\end{matrix}
; q, z\right]=&\frac{\poq{ABz/C}{\infty}}{\poq{z}{\infty}}{}_{2}\phi _{1}\left[\begin{matrix}C/A,C/B\\ C\end{matrix}
; q, \frac{ABz}{C}\right],
\end{align*}
then we reformulate \eqref{need} in standard notation of $q$-series as
\begin{align*} {}_{2}\phi _{1}\left[\begin{matrix}A,B\\ C\end{matrix}
; q, z\right]=&\sum_{n=0}^{\infty}
\frac{\poq{ABq/C,ABz/C}{n}}
{\poq{q,z}{n}} z^n q^{n(n-1)}\left(1-\frac{ABzq^{2n}}{C}\right)\nonumber\\
&\times{}_{4}\phi _{3}\left[\begin{matrix}ABzq^n/C,ABzq^{2n+1}/C,C/A,C/B\\ C,zq^n,ABzq^{2n}/C\end{matrix}
; q, \frac{ABzq^n}{C}\right].
\end{align*}
The conclusion is proved.
\rule{4pt}{7pt}
Perhaps, the most interesting case is the following partial theta function identity. It can be derived from
Theorem \ref{tlidentity} with the help of two Coogan-Ono type identities \eqref{id11} and \eqref{id1177}.
\begin{tl}[Partial theta function identity]Let $\theta(z;q)$ be the partial theta function given by
\[\sum_{n=0}^\infty(-1)^nq^{n(n-1)/2}z^{n}.\] Then
\begin{align}
&\frac{\poq{zq}{\infty}}{\poq{-zq}{\infty}}+\sum_{n=0}^{\infty}\frac{\poq{-1,z}{n}}{\poq{q,-zq}{n}}(-z)^nq^{n^2+n}
\label{expan-two-lzlzlz-new}
\\
&\qquad=\sum_{n=0}^{\infty}\frac{\poq{-1,z}{n}}{\poq{q,-zq}{n}}\big(1+q^n+zq^{n}-zq^{2n}\big)(-z)^nq^{n^2}\theta(z^2q^{2n+1};q^2).\nonumber
\end{align}
\end{tl}
\noindent {\it Proof.} Recall that Lemma \ref{ma-id11} gives
\begin{align}
\sum_{k=0}^\infty z^{k}\frac{\poq{z}{k}}{\poq{-zq}{k}}=(1+z)\sum_{k=0}^\infty(-1)^kz^{2k}q^{k^2}.\label{idnewadded-1}
\end{align}
Lemma \ref{ma-id1177} can be restated as
\begin{align}
\sum_{k=0}^\infty\,z^k\frac{\poq{z}{k}}{\poq{-zq}{k}}(1-zq^k)=1+
2\sum_{k=1}^{\infty}(-1)^kz^{2k}q^{k^2}.\label{idnewadded-0}
\end{align}
Subtracting \eqref{idnewadded-0} from \eqref{idnewadded-1}, we obtain
\begin{align}
z\sum_{k=0}^\infty\,(qz)^k\frac{\poq{z}{k}}{\poq{-zq}{k}}&=z+(z-1)\sum_{k=1}^{\infty}(-1)^kz^{2k}q^{k^2},\nonumber\\
\mbox{i.e.,}\,\,\,\sum_{k=0}^\infty\,(qz)^k\frac{\poq{z}{k}}{\poq{-zq}{k}}&=\sum_{k=0}^{\infty}(-1)^kz^{2k}q^{k^2}-\sum_{k=1}^{\infty}(-1)^kz^{2k-1}q^{k^2}.
\label{idnewadded-2}
\end{align}
Using \eqref{idnewadded-1} and \eqref{idnewadded-2}, as well as referring to \eqref{dy} with $t_k=1$, we thus obtain
\begin{align}
\widetilde{G}(z;1,-q)&=(1+z)\sum_{k=0}^\infty(-1)^kz^{2k}q^{k^2},\\
\widetilde{G}(zq;1/q,-1)&=\sum_{k=0}^{\infty}(-1)^kz^{2k}q^{k^2}-\sum_{k=1}^{\infty}(-1)^kz^{2k-1}q^{k^2}.
\end{align}
Thus it is easy to check that
\begin{align*}\displaystyle
&\widetilde{G}(zq^n;1,-q)-
zq^{2n}\widetilde{G}(zq^{n+1};1/q,-1)\\
&=(1+zq^n)\sum_{k=0}^\infty(-1)^kz^{2k}q^{k^2+2kn}-zq^{2n}\sum_{k=0}^{\infty}(-1)^kz^{2k}q^{k^2+2kn}+q^n\sum_{k=1}^{\infty}(-1)^kz^{2k}q^{k^2+2kn}\\
&=-q^n+(1+q^n+zq^n-zq^{2n}
)\sum_{k=0}^{\infty}(-1)^kz^{2k}q^{k^2+2kn}.
\end{align*}
Note that the summation on the right-hand side can be recast in terms of $\theta(z;q)$. We thus obtain
\begin{align*}\displaystyle
&\widetilde{G}(zq^n;1,-q)-
zq^{2n}\widetilde{G}(zq^{n+1};1/q,-1)
=-q^n+(1+q^n+zq^n-zq^{2n}
)\theta(z^2q^{2n+1};q^2).
\end{align*}
This reduces the whole equation \eqref{expan-two-lzlzlz} to
\begin{align*}
\frac{\poq{zq}{\infty}}{\poq{-zq}{\infty}}&+\sum_{n=0}^{\infty}\frac{\poq{-1,z}{n}}{\poq{q,-zq}{n}}(-z)^nq^{n^2+n}
\\
&=\sum_{n=0}^{\infty}\frac{\poq{-1,z}{n}}{\poq{q,-zq}{n}}(-z)^nq^{n^2}\big(1+q^n+zq^{n}-zq^{2n}\big)\theta(z^2q^{2n+1};q^2).
\end{align*}
Thus Identity \eqref{expan-two-lzlzlz-new} is proved.
\rule{4pt}{7pt}
In the case that $z=q^{-m},m\geq 1$, \eqref{expan-two-lzlzlz-new} reduces to a finite summation of $\theta(z;q).$
\begin{lz}For $m\geq 1$, we have
\begin{align}
&\sum_{n=0}^{m}\frac{\poq{-1}{n}}{\poq{-q^{1-m}}{n}}
\bigg[\genfrac{}{}{0pt}{}{m}{n}\bigg]_qq^{3n^2/2+n/2-2nm}\label{qqq}\\
&=\sum_{n=0}^{m}\frac{\poq{-1}{n}}{\poq{-q^{1-m}}{n}}
\bigg[\genfrac{}{}{0pt}{}{m}{n}\bigg]_qq^{3n^2/2-n/2-2nm}
\big(1+q^n+q^{n-m}-q^{2n-m}\big)\theta(q^{2n-2m+1};q^2),\nonumber
\end{align}
where $\bigg[\genfrac{}{}{0pt}{}{m}{n}\bigg]_q$ is the usual $q$-binomial coefficient.
\end{lz}
\noindent {\it Proof.} It suffices to take $z=q^{-m}$ in \eqref{expan-two-lzlzlz-new} and simplify the obtained by using the facts that for integer $m\geq 1$, $\poq{q^{1-m}}{\infty}=0$ and
\begin{align*}
\frac{\poq{q^{-m}}{n}}{\poq{q}{n}}=
\bigg[\genfrac{}{}{0pt}{}{m}{n}\bigg]_q(-1)^nq^{n(n-1)/2-mn}.
\end{align*}
\rule{4pt}{7pt}
It would be natural to expect that Theorem \ref{analogue-two} can be applied to bilateral $q$-series. The reader is referred to \cite[Eq. (5.1.2)]{10} or \eqref{guiding} for the definition of bilateral $q$-series. As an interesting example, we now set up a coefficient identity of the famous Ramanujan ${}_1\psi_1$ summation formula \cite[(II.29)]{10}.
\begin{tl}Let $B_{n,1}(a,b)$ be given by \eqref{180}. For $|b/a|<|z|<1,$ and integer $n\geq 0$, it holds
\begin{align}
&\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\frac{(aqz^2,1/az^2;q)_{\infty}}
{(z,b/az,bq^nz,1/az;q)_{\infty}\poq{aqz}{n}}\bigg\}\label{313}=\frac{1}{(q,b/a;q)_{\infty}}\\
&\quad+aq\sum_{k=0}^{n-1}B_{n-k,1}(aq,bq)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{\frac{(aqz^2,1/az^2;q)_{\infty}}
{(z,b/az,bq^{k+1}z,1/az;q)_{\infty}\poq{aqz}{k+1}}\bigg\}.\nonumber
\end{align}
\end{tl}
\noindent {\it Proof.}
Observe that Ramanujan's $\,_1\psi_1$ sum states that for $|b/a|<|z|<1,$
\begin{eqnarray}\sum_{k=-\infty}^{\infty}\frac{(a;q)_k}{(b;q)_k}z^k=
\frac{(az,q/az,q,b/a;q)_{\infty}}
{(z,b/az,b,q/a;q)_{\infty}}.
\end{eqnarray}
Set $(a,b)\to (aqz,bqz)$. Then we arrive at
\begin{eqnarray}\sum_{k=-\infty}^{\infty}\frac{(aqz;q)_k}{(bqz;q)_k}z^k=
\frac{(aqz^2,1/az^2,q,b/a;q)_{\infty}}
{(z,b/az,bqz,1/az;q)_{\infty}},
\end{eqnarray}
which can be reformulated in the form
\begin{align}
f(z)+g(1/z)=F(z), \label{bilateral}
\end{align}
where we define
\begin{align*}
f(z)&:=\sum_{k=0}^{\infty}\frac{(aqz;q)_k}{(bqz;q)_k}z^k,\,\,g(z):=
\sum_{k=1}^{\infty}\frac{(z/b;q)_k}{(z/a;q)_k}\bigg(\frac{bz}{a}\bigg)^k,\\
F(z)&:=\frac{(aqz^2,1/az^2,q,b/a;q)_{\infty}}
{(z,b/az,bqz,1/az;q)_{\infty}}.
\end{align*}
Now we apply the expansion formula in Theorem \ref{analogue-two} to $f(z)$. It follows from \eqref{coeformula} that for $n\geq 0$,
\begin{align}
1=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ f(z)\frac{\poq{bqz}{n-1}}{\poq{aqz}{n}}\bigg\}-aq\sum_{k=0}^{n-1}B_{n-k,1}(aq,bq)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{f(z)\frac{\poq{bqz}{k}}{\poq{aqz}{k+1}}\bigg\}.\label{coeformula1890}
\end{align}
Next, observe that
\begin{align*}
\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{f(z)\frac{\poq{bqz}{n-1}}{\poq{aqz}{n}}\bigg\}=\sum_{k=0}^n\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\{f(z)\}\times\boldsymbol\lbrack z^{n-k}\boldsymbol\rbrack\bigg\{ \frac{\poq{bqz}{n-1}}{\poq{aqz}{n}}\bigg\},
\end{align*}
while for $k\geq 0$, due to \eqref{bilateral}, it holds
$$\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\{f(z)\}=\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\{F(z)\}.$$
We immediately obtain
\begin{align*}
\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{ f(z)\frac{\poq{bqz}{k-1}}{\poq{aqz}{k}}\bigg\}=\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{F(z)\frac{\poq{bqz}{k-1}}{\poq{aqz}{k}}\bigg\}.
\end{align*}
This simplifies \eqref{coeformula1890} to
\begin{align*}
1&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{ F(z)\frac{\poq{bqz}{n-1}}{\poq{aqz}{n}}\bigg\}-aq\sum_{k=0}^{n-1}B_{n-k,1}(aq,bq)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{F(z)\frac{\poq{bqz}{k}}{\poq{aqz}{k+1}}\bigg\}\\
&=\boldsymbol\lbrack z^{n}\boldsymbol\rbrack\bigg\{\frac{(aqz^2,1/az^2,q,b/a;q)_{\infty}}
{(z,b/az,bq^nz,1/az;q)_{\infty}\poq{aqz}{n}}\bigg\}\\
&-aq\sum_{k=0}^{n-1}B_{n-k,1}(aq,bq)q^{(n-k)k}\boldsymbol\lbrack z^{k}\boldsymbol\rbrack\bigg\{\frac{(aqz^2,1/az^2,q,b/a;q)_{\infty}}
{(z,b/az,bq^{k+1}z,1/az;q)_{\infty}\poq{aqz}{k+1}}\bigg\}.
\end{align*}
It turns out to be \eqref{313}.
\rule{4pt}{7pt}
We conclude our paper with a coefficient identity of the Coogan-Ono identity \eqref{id11} which can be easily derived by using \eqref{18}.
\begin{tl}
Let $B_{n,k}(a,b)$ be given by \eqref{18}. Then for any integer $n\geq 0$, we have
\begin{align}
\sum_{k=0}^{\lfloor n/2\rfloor}B_{n,2k}(1,-q)(-1)^kq^{k^2}+\sum_{k=0}^{\lfloor (n-1)/2\rfloor}B_{n,2k+1}(1,-q)(-1)^kq^{k^2}=1,
\end{align}
where $\lfloor x\rfloor$ denotes the usual floor function.
\end{tl}
\noindent {\it Proof.} It suffices to take $a=1,b=-q$ in Theorem \ref{analogue-two} and
$$F(z)=(1+z)\sum_{n=0}^{\infty}(-1)^nz^{2n}q^{n^2}:=\sum_{n=0}^{\infty}a_nz^n.$$
So we are back with the series expansion
\begin{align*}
F(z)=\sum_{n=0}^\infty\,z^n\frac{\poq{z}{n}}{\poq{-zq}{n}}.
\end{align*}
Thus, by \eqref{18} instead of \eqref{coeformula}, we obtain
\begin{align*}
1=\sum_{k=0}^nB_{n,k}(1,-q)a_{k}&=\sum_{2k=0}^nB_{n,2k}(1,-q)(-1)^kq^{k^2}\\
&+\sum_{2k+1=1}^nB_{n,2k+1}(1,-q)(-1)^kq^{k^2}.
\end{align*}
\rule{4pt}{7pt}
\end{document} |
\begin{document}
\title{Robust Singular Smoothers For Tracking Using Low-Fidelity Data}
\author{\authorblockN{Jonathan Jonker\authorrefmark{1},
Aleksandr Aravkin\authorrefmark{2},
James V. Burke\authorrefmark{1},
Gianluigi Pillonetto\authorrefmark{3} and
Sarah Webster\authorrefmark{4}}
\authorblockA{\authorrefmark{1}Department of Mathematics, University of Washington, Seattle, WA USA, \it{[email protected], [email protected]}}
\authorblockA{\authorrefmark{2}Department of Applied Mathematics, University of Washington, Seattle, WA USA,
\it{[email protected]} }
\authorblockA{\authorrefmark{3}Department of Information Engineering, University of Padova, Italy,
\it{[email protected]} }
\authorblockA{\authorrefmark{4}Applied Physics Laboratory, University of Washington, Seattle, WA USA,
{\it [email protected]} } }
\maketitle
\begin{abstract}
Tracking underwater autonomous platforms is often difficult because of noisy, biased, and discretized input data. Classic filters and smoothers based on standard assumptions of Gaussian white noise break down when presented with any of these challenges.
Robust models (such as the Huber loss) and constraints (e.g. maximum velocity) are used to attenuate these issues.
Here, we consider robust smoothing with singular covariance, which covers bias and correlated noise, as well as many specific model types, such as those used in navigation.
In particular, we show how to combine singular covariance models with robust losses and state-space constraints in a unified framework that can handle very low-fidelity data. A noisy, biased, and discretized navigation dataset from a submerged, low-cost inertial measurement unit (IMU) package, with ultra short baseline (USBL) data for ground truth, provides an opportunity to stress-test the proposed framework with promising results. We show how robust modeling elements improve our ability to analyze the data, and present batch processing results
for 10 minutes of data with three different frequencies of available USBL position fixes
(gaps of 30 seconds, 1 minute, and 2 minutes).
The results suggest that the framework can be extended to real-time tracking
using robust windowed estimation.
\end{abstract}
\IEEEpeerreviewmaketitle
\section{Introduction}
State-space models are ubiquitous in signal processing, and allow integration of disparate measurements to inform estimation, decisions, and control. Classic filtering~\cite{kalman1960new,KalBuc} and smoothing~\cite{rauch1965maximum,fraser1969optimum} are core tools used to estimate these models. Their dependence on high-fidelity data, driven by Gaussian assumptions on errors and innovations,
has demanded unequivocal attention from researchers and practitioners, and inspired robust dynamic inference methods.
While early robust approaches \citep{Kassam1985,Schick1994} sought to modify iterations of the Kalman filter (KF) and Rauch-Tung-Striebel (RTS) smoother, over the last 25 years researchers have used {\it robust formulations} to weave
assumptions on errors and innovations directly into the estimation problems themselves~\cite{Fahr1991,Bell1994,kim2009ell_1,aravkin2011ell,Farahmand2011,aravkin2013sparse,aravkin2014robust}. Constraints, when available, are also readily incorporated into the problem formulation~\cite{bell2009inequality}.
Specifying the formulation leaves one free to choose from a range of optimization algorithms; the survey~\cite{aravkin2017generalized} describes a general class of models as well as first- and second-order methods to solve them.
Our focus is on models with singular variances for process and measurement residuals. These models are excluded by the assumptions of generalized smoothing~\cite{aravkin2017generalized} and all of the various special cases cited in the survey.
In this paper, we build on the recently proposed framework of~\cite{jonker2019fast} for singular models,
and systematically develop complementary modeling elements: {\bf robust penalties, informative constraints, and singular models}.
The resulting approach exploits the structure of singular covariances head on rather than using workarounds such as pseudo-inverses or variance boosting that either
do not work in the general setting or introduce unnecessary changes to the fundamental model (see discussion in~\cite{jonker2019fast}).
\begin{figure}
\caption{\label{fig:intro}
\label{fig:intro}
\end{figure}
A simple synthetic tracking example with a singular process model shows how the common tack of replacing the inverse by the pseudo-inverse fails dramatically in the presence of outliers (Figure~\ref{fig:intro}). Any robust approach requires control of the null spaces associated to process and observations. We develop a direct practical formulation and method, and test it on a batch smoothing analysis of real field data, with a view towards real-time implementation in future work.
\noindent
{\bf Background.}
Our main goal is to infer an unobserved state sequence $x_1, \dots, x_N$ from
noisy observations $y_1, \dots, y_N$ using the model:
\begin{equation} \label{eq:statespace}
\begin{aligned}
x_1 & = x_0 + w_1
\\
x_k & = G_kx_{k-1} + w_k, \quad k = 2, \dots, N
\\
y_k & = H_kx_k + v_k, \quad k = 1, \dots, N,\\
x_k &\in X_k, \quad \mbox{each $X_k$ polyhedral,}
\end{aligned}
\end{equation}
where $x_0$ is a given initial state estimate, $G_k$ and $H_k$ are linear process and measurement models,
$y_1, \dots, y_N$ are observations, and $X_k$ specify additional information through constraints.
The framework of~\cite{aravkin2017generalized}
assumes that $w_k$ and $v_k$ are mutually independent random variables with known {\it nonsingular} covariances $Q_k$ and $R_k$, and that they follow from log-concave distributions; in particular they may be non-Gaussian.
Synthesizing all of this information gives the problem
\begin{equation}
\label{eq:original}
\begin{aligned}
\min_{x_1\in X_1, \dots, x_n \in X_N} &\sum_k\rho_p(Q_k^{-1/2}(G_kx_k-x_{k-1})) \\
&+ \rho_m(R_k^{-1/2}(H_kx_k- y_k)),
\end{aligned}
\end{equation}
with $\rho_p$ and $\rho_m$ convex penalties. Using~\eqref{eq:original} provides estimates that
are robust to outliers and can follow sudden changes in the state. Most of the inference-
or optimization-based work in the convex dynamic setting is a special case of~\eqref{eq:original}.
Many examples, including robust penalties and constraints, are collected in~\cite{aravkin2017generalized}.
We now extend to {\it singular covariances} $R_k$ (for errors $v_k$) and $Q_k$
(for innovations $w_k$). These models specify key use cases, particularly for innovations (process) modeling,
briefly summaried below (see~\cite{jonker2019fast} for a more detailed discussion).
\noindent
{\bf Deterministic integrals.} Most models in robotics, particularly in navigation, use integration to model process relationships between state variables (e.g. when position, velocity, and acceleration are part of the state). Any deterministic integral yields a singular process model. The simplest example
(with position a direct integral of stochastic velocity) is used to create Figure~\ref{fig:intro}.
\noindent
{\bf Nuisance parameters.} Unknown constants that need calibration (such as fixed
instrument biases) require special modeling in the nonsingular
paradigm~\eqref{eq:original}. With singular models, we can augment the state in order to infer these parameters.
\noindent
{\bf Auto-regressive models and correlated errors.}
State-space models are broadly used in auto-regressive, moving average,
and time series models~\cite{hyndman2008forecasting}. These elements
also appear in general smoothing models,
particularly to deal with correlated measurement errors~\cite{chui2017kalman}.
All three examples are accessible in the classic linear Gaussian setting. The KF need not invert $Q$ or $R$,
and provides the minimum variance estimate for both singular and nonsingular models~\cite{ansley1982geometrical}.
Some algorithms rely on precise knowledge of the error structure or explicit equality constraints~\cite{koopman1997exact,ko2007state,ait2011fixed}.
Correlated errors are dealt with by augmenting the state
and using a singular model~\cite{chui2017kalman}.
{\bf None of these techniques generalize to singular models in the setting of~\eqref{eq:original}}, and some naive generalizations fail dramatically (Figure~\ref{fig:intro}).
This paper builds on the reformulation of~\cite{jonker2019fast} for singular models.
We develop a systematic approach and test it using a navigation model with a real-world mooring dataset. We show how robust statistics, singular models, and constraints can be systematically used to overcome a range of challenges simultaneously present in the dataset: (1) outliers, (2) deterministic relationships between states, (3) measurement biases, and (4) coarsely discretized observations.
The paper proceeds as follows. Section~\ref{sec:summary} summarizes
the singular formulation of~\cite{jonker2019fast} and relevant optimization algorithms. Section~\ref{sec:modeling} develops the key modeling elements to address common data challenges. Section~\ref{sec:navmodels} presents the navigation models.
Section~\ref{sec:analysis} shows how the model elements come together to analyze the target mooring dataset, and obtain a high
fidelity track from low-fidelity observations.
Section~\ref{sec:conclusion} concludes with discussion and future work.
\section{Robust Singular Formulation and Algorithm}
\label{sec:summary}
We reformulate the robust smoothing problem to
seamlessly allow both singular and nonsingular covarinace models for errors and innovations.
The resulting problem can be solved with any primal-dual algorithm.
We show how to apply the classic Douglas-Rachford splitting (DRS) algorithm
(see e.g.~\cite{eckstein1992douglas,davis2016convergence}) to the reformulated problem.
Problem~\eqref{eq:original} can be reformulated by introducing auxiliary variables $u_k$
and $t_k$ to represent pre-whitened innovations and measurement residuals:
\begin{equation}
\label{eq:reform}
\begin{aligned}
\min_{x, u, t} & \quad \sum_k\rho_p(u_k) + \rho_m(t_k) + \rho_s(x_k) \\
\text{s.t.} & \quad Q_k^{1/2}u_k = G_kx_k - x_{k-1}\\
& \quad R_k^{1/2}t_k = H_kx_k-y_k
\end{aligned}
\end{equation}
where $\rho_s(x_k)$ may be taken as the convex indicator function to recover the constraints
in~\eqref{eq:original}:
\[
\rho_s(x_k) =
\begin{cases}
0 \quad x_k \in X_k\\
\infty \quad x_k \not \in X_k.
\end{cases}
\]
When $Q_k$ and $R_k$ are invertible, we can solve for $u_k,t_k$ and recover~(\ref{eq:original}).
Otherwise, problem~(\ref{eq:reform}) is well-posed while~(\ref{eq:original}) is not.
We can write~\eqref{eq:reform} in compact form
\begin{equation}
\label{eq:full}
\begin{aligned}
\min_{z} &\quad \rho(z) \quad \text{s.t. } Az = \hat{w}, \\
\rho(z) &= \sum_{k=1}^N \rho_p(u_k) + \rho_m(t_k) + \rho_s(x_k).
\end{aligned}
\end{equation}
where
\begin{equation}
\label{eq:zw}
\begin{aligned}
z^T &= \begin{pmatrix} u_1^T & t_1^T & x_1^T & \dots u_N^T & t_N^T & x_N^T\end{pmatrix} \\
\hat{w}^T &= \begin{pmatrix} x_0^T & y_1^T & 0 & y_2^T & \dots & 0 & y_N^T\end{pmatrix},
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:A}
A =
\begin{pmatrix} D_1 & 0 & \dots & 0 \\
B_1 & D_2 & 0 & \vdots \\
0& \ddots & \ddots & 0\\
0 & 0 & B_{N-1} & D_N\end{pmatrix},
\end{equation}
and
\begin{equation*}
\label{eq:DjBj}
\begin{aligned}
D_i = \begin{pmatrix}Q_i^{1/2} & 0 & I\\ 0 & R_i^{1/2} & H_i\end{pmatrix},
B_j = \begin{pmatrix}0 & \qquad 0 & -G_{j+1}\\ 0 & \qquad 0 & 0\end{pmatrix}.
\end{aligned}
\end{equation*}
The variables are ordered in such a way that $A$ is block bi-diagonal.
If all observations $z_i$ lie in the range of $H_i$, the constraint $Az = \hat{w}$ will
be feasible~\cite{jonker2019fast}.
The problem~\eqref{eq:full} is a convex optimization problem and can be solved using a
variety of techniques. We show that the DRS algorithm is straightforward to implement,
and preserves the computational complexity of the classic KF/RTS algorithms
because of the way $A$ is structured in~\eqref{eq:A}.
Given a convex function $f$,
its convex conjugate $f^*$ is given by
\[
f^*(y) = \sup_x \langle x, y \rangle - f(x),
\]
and its proximal operator with step $\alpha$,
denoted by $\prox_{\alpha f}$ (see e.g. \cite{combettes2011proximal}) is given by:
\begin{equation}
\label{eq:prox}
\prox_{\alpha f}(\zeta) = \arg\min_x \frac{1}{2\alpha} \|\zeta - x\|^2 + f(x).
\end{equation}
For a long list of objectives, prox operators
are available in closed form or are efficiently computable.
In particular this is the case when $\rho_p, \rho_m, \rho_s$ form any subset of the numerous elements briefly surveyed in Section~\ref{sec:modeling}.
It is actually the prox of $\rho^*$ that appears in the DRS iteration (Algorithm~\ref{alg:DRS}) rather than the prox of $\rho$, but these are linked by the simple formula
\[
\prox_\rho(z) + \prox_{\rho^*}(z)= z.
\]
To specify the algorithm, we let $g(z)$ be the indicator of the affine feasible region $Az = \hat w$:
\[
g(z) =
\begin{cases}
0 \quad Az=\hat{w}\\
\infty \quad Az \neq \hat{w}
\end{cases}
\]
Problem~\eqref{eq:full} can now be written simply as
\[
\min_{z} \rho(z) + g(z)
\]
which is a natural template for DRS, detailed in Algorithm~\ref{alg:DRS}.
\begin{algorithm}[H]
\caption[Caption]{Douglas-Rachford Splitting (DRS)
\label{alg:DRS}}
\begin{algorithmic}[1]
\Require{Initialize at any $z^0$, $\zeta^0$.}
\Loop
\State {$z^k = \prox_{\tau g}(z^{k-1}-\tau \zeta^{k-1})$}
\State {$\zeta^k = \prox_{\sigma \rho^*}(\zeta^{k-1}+\sigma(2z^{k} - z^{k-1}))$}
\EndLoop
\Return{$z^k$}
\end{algorithmic}
\end{algorithm}
To implement Algorithm~\ref{alg:DRS} we need proximal operators of $\rho_p, \rho_m$, and $\rho_s$. Eight common piecewise linear-quadratic (PLQ) penalties are shown in Figure~\ref{fig:PLQ}, and their proximal operators
are summarized in Table~\ref{table:PLQ}.
The proximal operator for $g$ is given by
\[
\prox_g(\eta) = \argmin_{Az=\hat{w}} \frac{1}{2}||\eta - z||^2
\]
which is a least squares problem with affine constraints.
Solving it efficiently leverages the structure of~\eqref{eq:A}. In particular we need to solve
a single structured linear system
\begin{equation}
\label{eq:linsystem}
\begin{bmatrix}I & A^T\\ 0 & AA^T\end{bmatrix}\begin{bmatrix}z\\ \nu \end{bmatrix} = \begin{bmatrix}\eta \\ A\eta - \hat{w}\end{bmatrix}
\end{equation}
where $AA^T$ is block tridiagonal and does not change between iterations. In our implementation, we need only compute a single block bidiagonal factorization once, which can then be used to solve~(\ref{eq:linsystem}) in $O(n^2N)$ operations
in each iteration, no more expensive than a single matrix-vector multiply.
For piecewise-linear quadratic $\rho$~\cite{rockafellar2009variational,aravkin2017generalized},
DRS converges to an optimal solution at a local linear rate~\cite{jonker2019fast}, which does not depend on the condition number of $A$.
A good initialization makes DRS competitive with the fastest
available solvers, even second order methods with quadratic local rates~\cite{aravkin2017generalized}.
\section{Modeling Elements}
\label{sec:modeling}
\begin{figure}
\caption{\label{fig:quadratic}
\label{fig:quadratic}
\caption{\label{fig:1norm}
\label{fig:1norm}
\caption{\label{fig:quantile}
\label{fig:quantile}
\caption{\label{fig:huber}
\label{fig:huber}
\caption{\label{fig:huberQ}
\label{fig:huberQ}
\caption{\label{fig:vapnik}
\label{fig:vapnik}
\caption{\label{fig:sel}
\label{fig:sel}
\caption{\label{fig:enet}
\label{fig:enet}
\caption{\label{fig:PLQ}
\label{fig:PLQ}
\end{figure}
The proposed framework has three complementary modeling elements:
singular covariance matrices $Q$ and $R$; process/measurement penalties $\rho_p, \rho_m$; and constraints
$\rho_s$ on the state.
In this section, we show a range of choices for each element,
and compute the operators required for Algorithm~\ref{alg:DRS}.
\noindent
{\bf Singular covariances}
can be used to capture affine constraints, auto-regressive structure, integrated errors, and bias.
\begin{itemize}
\item \emph{Affine constraints using singular $R$.}
the $i$th element of the state
at time $k$ is known exactly, add row
\[
\begin{bmatrix} 0 & \dots &0 & \underbrace{1}_{i} & 0 &\dots &0\end{bmatrix}
\]
to the measurement model $H_k$, a row and column of zeros to $R_k$, and the known value as the last element of $z_k$.
\item \emph{Bias with singular $Q$}. A common model for bias is to include it as a non-varying component
across the state:
\[
\widetilde x_k = \begin{bmatrix} x_k \\ b \end{bmatrix}, \quad \widetilde Q_k = \begin{bmatrix} Q & 0\\
0 & 0 \end{bmatrix}.
\]
\item \emph{Correlated noise using singular $Q$.}
Correlated noise $w_k$ is typically modeled by~\cite{Chui2009}
\[
w_k = M w_{k-1} + \beta_k, \quad \beta_k \sim N(0, Q).
\]
Here too we can augment the state and use a singular process variance:
\[
\widetilde x_k = \begin{bmatrix} x_k \\ w_k \end{bmatrix},
\widetilde G_k = \begin{bmatrix} G_k & I \\ 0 & M \end{bmatrix},
\quad \widetilde Q_k = \begin{bmatrix} 0 & 0 \\
0 & Q_k \end{bmatrix}.
\]
\end{itemize}
\noindent
{\bf Piecewise linear-quadratic (PLQ) Penalties.}
The proposed framework allows process innovations, measurement
residuals, and state regularization to come from any convex prox-friendly
penalty. To keep the exposition simple, we collect eight commonly used
convex piecewise linear-quadratic penalties in Figure~\ref{fig:PLQ},
and compute their prox operators in Table~\ref{table:PLQ}.
The penalties can be thought of in terms of three features:
\begin{itemize}
\item Behavior at origin: nonsmooth features encourage
exact fitting of the quantity being measured, while deadzones
are appropriate for discretized observations.
\item Tail growth: asymptotically linear penalties are more tolerant of large inputs. Applied to measurements, this gives robustness to outliers; applied to innovations, it gives an ability to quickly track evolving trends.
\item Asymmetry: allows handling of special cases where under-estimating is qualitatively different from over-estimating.
\end{itemize}
\noindent
{\bf Constraints.} It is very convenient to enforce simple constraints
on the state estimates $x_k$. If we take
\(
\rho_s(x) = \delta_X(x)
\)
then the prox operator $\prox_{\rho_s}$ is simply the projection onto the
set $X$. Box constraints are a very common type of constraints
that enforce known bounds on the state, and have a trivial projection. The proposed framework allows us to use any convex region that has a computationally efficient projection.
\begin{table}
\textcolor{black}{
\caption{\label{table:PLQ} Prox operators of common PLQ penalties.}
\begin{tabular}{|c|c|c|}\hline
{\bf Penalty $f$} & {\bf $\prox_{\alpha f}(z)$} & {\bf Ref.}\\ \hline
$\frac{1}{2}\|x\|^2$, Fig.~\ref{fig:quadratic}
& $ \frac{1}{1+\alpha}z$
& \cite{freedman2009statistical,SeberWild2003}\\
\hline
$\|x\|_1$, Fig.~\ref{fig:1norm}
&$ \mbox{sign}(z)\odot(|z|-\alpha)_+$
& \cite{Hastie90,LARS2004}\\
\hline
$q_\tau$, Fig.~\ref{fig:quantile}
&$\begin{cases} z_i-\alpha(1-\tau) & \quad z_i > \alpha(1-\tau)\\
z_i + \alpha\tau & \quad z_i < -\alpha\tau \\
0 & \quad \mbox{else}
\end{cases}$
& \cite{KB78,KG01}\\
\hline
$h_\kappa$, Fig.~\ref{fig:huber}
&$\frac{\alpha}{\alpha + \kappa}z + \frac{\kappa}{\alpha + \kappa} \prox_{(\alpha + \kappa)\|\cdot\|_1}(z)$
& \cite{Mar}\\
\hline
$q_{\tau,\kappa}$, Fig.~\ref{fig:huberQ}
&$\frac{\alpha}{\alpha + \kappa}z + \frac{\kappa}{\alpha + \kappa} \prox_{(\alpha + \kappa)q_\tau}(z)$
& \cite{aravkin2014qh}\\
\hline
$v_\epsilon$, Fig.~\ref{fig:vapnik}
&$\begin{cases} z_i-\alpha & \quad z_i > \epsilon + \alpha\\
\epsilon & \quad \epsilon < z_i \leq \alpha + \epsilon \\
z_i & \quad -\epsilon \leq z_i \leq \epsilon \\
-\epsilon & \quad -\epsilon-\alpha < z_i \leq - \epsilon \\
z_i + \alpha\tau & \quad z_i < -\alpha -\epsilon.
\end{cases}$
& \cite{Vapnik98}\\
\hline
hubnik-$\kappa$, Fig.~\ref{fig:sel}
&$\frac{\alpha}{\alpha + \kappa}z + \frac{\kappa}{\alpha + \kappa} \prox_{(\alpha + \kappa)v_\epsilon}(z)$
& \cite{chu2001unified,lee2005epsi}\\
\hline
e-net, Fig.~\ref{fig:enet}
&$\prox_{\frac{\alpha}{1 + 2\alpha} \|\cdot\|_1}\left(\frac{1}{1 + 2\alpha}z\right)$
& \cite{EN_2005}\\
\hline
\end{tabular}
}
\end{table}
\section{Navigation Model}
\label{sec:navmodels}
We use a
constant-velocity kinematic model that is
appropriate for many underwater vehicle applications, where
accelerations are heavily damped and trajectories are often long straight lines (e.g. for transit or survey work). When the attitude is known or changing slowly, the model can be linearized effectively.
For a vehicle that is well-instrumented in attitude, the uncertainty in position
(and the x-y states in particular) is typically orders of magnitude larger than
the uncertainty in attitude. Thus, in practice, we often simplify the full
nonlinear vehicle process model to track only position states
$( x, y, z)$, while assuming that
the attitude states $(r, p,h)$ are directly available from the most recent sensor
measurements. To make the model linear, the position and its derivatives are
referenced to the local-level frame.
An effective model must counteract biases, outliers, and data discretization in the IMU data. We develop this model using the elements of the proposed framework.
\noindent
\textbf{Process model.} To incorporate linear acceleration measurements from
an IMU, we track linear velocities
and linear acceleration in the state vector:
\begin{equation}
x_s = [x,y,z,\dot{x},\dot{y},\dot{z},\ddot{x},\ddot{y},\ddot{z}]^\top.
\end{equation} \label{lin_pm}
The linear kinematic process model is given by
\begin{align}
\dot{x}_s &=
\underbrace{\left[\begin{array}{ccc}
0 & I & 0 \\
0 & 0 & I \\
0 & 0 & 0 \end{array}\right]}_{\mbox{$F_s$}} x_s +
\underbrace{\left[\begin{array}{c}
0 \\
I \\
0 \end{array}\right]}_{\mbox{$G_s$}} w_s \label{x_s-dot},
\end{align}
where $w_s \sim \mathcal{N}(0,Q_s)$ is zero-mean
Gaussian noise.
The linear process model \eqref{x_s-dot} is
discretized using a Taylor series:
\begin{align}
x_{s_{k+1}} &= F_{s_k} x_{s_k} + w_{s_k} \label{disc_s}\\
F_{s_k} &= e^{F_s T} \approx
\approx \left[\begin{array}{ccc}
I & IT & \frac{1}{2}IT^2\\
0 & I & IT\\
0 & 0 & I \end{array} \right], \nonumber
\end{align}
where $I$ in~(\ref{disc_s}) denotes the $3\times 3$ identity matrix,
and the higher order terms are identically zero because of the
structure of $F_s$.
We model the process covariance as if the error were the next term in the Taylor series approximation,
a technique suggested by~\cite{YAA}. More precisely we set covariance to be the outer product, $\Gamma^T \Gamma$ where
\[
\Gamma = \begin{bmatrix}\frac{1}{3!}IT^3 & \frac{1}{2!}IT^2 & IT\end{bmatrix}
\]
This leads to a rank 3 covariance for a $9\times 9$ matrix for a model that comprises
position, velocity, and acceleration.
Given this covariance structure, the process model will penalize changes in acceleration. As the vehicle travels in a relatively straight line with small corrections, we expect to see acceleration mostly constant with a few small jumps. We use the $\ell_1$ norm for innovations, as it encourages exact fits while simultaneously allowing occasional sudden changes.
\noindent
\textbf{Measurement model.} The inertial measurement unit (IMU) measures linear and angular accelerations
relative to the physical frame of the
vehicle on which it is mounted, while
the state tracks linear acceleration relative to the navigation frame. We obtain the coordinate transformation between these frames using heading, pitch, and roll of the vehicle:
\begin{equation}
R(\varphi) = R^\top_h R^\top_p R^\top_r,
\end{equation}
where $R_h$, $R_p$, and $R_r$ are given by
\begin{equation}
\left[\begin{array}{ccc}
c{h} & s{h} & 0 \\
-s{h} & c{h} & 0 \\
0 & 0 & 1 \end{array}\right], \quad
\left[\begin{array}{ccc}
c{p} & 0 & -s{p} \\
0 & 1 & 0 \\
sp & 0 & cp \end{array}\right], \quad
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 &c{r} & s{r} \\
0 & -s{r} & c{r} \end{array}\right]
\end{equation}
with $c\cdot$ and $s\cdot$ shorthand for $\cos(\cdot)$
and $\sin(\cdot)$.
Position data from the USBL is sampled at a lower update rate than the IMU. For any $s_k$ where such position data is available, we have the measurement model
\[
H_{s_k} = \begin{bmatrix} I_{3 \times 3} & 0_{3 \times 6}\\ 0_{3 \times 6} & R(\varphi)\end{bmatrix}, \quad
z_{s_k} = \begin{bmatrix} \mathrm{usbl}^\top & \ddot{x}_{\textrm{meas}} & \ddot{y}_{\textrm{meas}} & \ddot{z}_{\textrm{meas}}\end{bmatrix}^\top
\]
If there is no position data measured at time $s$ then we use the model
\[
H_{s_k} = \begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 6}\\ 0_{3 \times 6} & R(\varphi)\end{bmatrix}, \quad
z_{s_k} = \begin{bmatrix} 0& \ddot{x}_{\textrm{meas}} & \ddot{y}_{\textrm{meas}} & \ddot{z}_{\textrm{meas}}\end{bmatrix}^\top.
\]
The covariance used for measurement data similarly depends on whether there is position data available:
\[
R_{s_k} = \begin{bmatrix} U_s & 0_{3 \times 3}\\ 0_{3 \times 3} & r_s I_{3 \times 3}\end{bmatrix}, \quad R_{s_k} = \begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 3}\\ 0_{3 \times 3} & r_s I_{3 \times 3}\end{bmatrix}
\]
where $U$ is a diagonal matrix reflecting position uncertainty, while $r_s$ captures uncertainty in IMU measurements.
\noindent
\textbf{Bias model.}
To compensate for the bias in acceleration data we augment the state vector to include bias variables:
\[
\bar{x}_s = [x_s^T, b_1, b_2, b_3]^\top
\]
where $b_1,b_2,b_3$ are bias terms for acceleration in the $x,y,z$ directions
in the local frame.
To pass the bias estimates forward in time the process matrix is augmented with an identity block.
\begin{align}
\bar{x}_{s_{k+1}} &= \bar{F}_{s_k}x_{s_k} + \bar{w}_{s_k}\\
\bar{F}_{s_k} &= \begin{bmatrix}F_{s_k} & 0\\ 0 & I\end{bmatrix}
\end{align}
At the first time point ($s_k = 1$) we augment the covariance matrix with an identity block,
and at all other time points we augment with a zero block. This adds equality constraints for the bias terms over all time points.
The approach generalized easily to model piecewise-constant biases over longer periods.
\[
\bar{Q}_{1} = \begin{bmatrix} \Gamma^T\Gamma & 0\\ 0 & I\end{bmatrix},
\quad \bar{Q}_{s_k} = \begin{bmatrix} \Gamma^T\Gamma & 0\\ 0 & 0\end{bmatrix} \quad s_k > 1.
\]
The measurement matrices are augmented with an identity block that shifts the acceleration measurements using the bias.
\noindent
\textbf{Discretization model.}
The measurement loss function is chosen to account for the level of discretization present in the data. The Vapnik loss (Figure~\ref{fig:PLQ}f) has a `deadzone' around the origin where small discrepancies are not penalized. This region is set using the level of discretization in the data, in this case $0.05$. The sharp corners of the Vapnik loss encourage errors to lie exactly in them, an unnecessary artifact. Thus we use the `Huberized' version dubbed `hubnik' (Figure~\ref{fig:PLQ} g).
The prox operators for both losses are computed in Table~\ref{table:PLQ}.
\section{Analysis of Mooring Data}
\label{sec:analysis}
We are interested in the ability to maintain an accurate position
estimate on-board an autonomous underwater vehicle in real-time using acceleration
measurements from a low-cost IMU, given periodic
position fixes.
To test this, we use the singular robust Kalman framework to analyze data collected from a surface mooring
equipped with an IMU on the subsea node.
The mooring node, which is drifting with the current, is used as a proxy for a slowly
moving underwater vehicle subject to unknown disturbances.
We look at the position uncertainty and error accrued over time between the
periodic, world-referenced position fixes provided by the USBL system.
\noindent
\textbf{Data description.}
Position fixes are available from a ship-based Sonardyne Ranger 2 USBL every 2 seconds, which we subsample to varying degrees for the analysis. Linear acceleration data from an LSM303D 3-axis accelerometer was collected at $\sim0.075$ m/s$^2$ precision using a Raspberry Pi Zero.
\begin{figure}
\caption{A snippet of the depth acceleration data, rotated into the world frame,
shows the discretization and bias of the acceleration data.}
\label{fig:acceldata}
\end{figure}
\begin{figure}
\caption{Comparison of classic Kalman smoothing applied to depth data with and without USBL position fixes.}
\label{fig:l2_depth}
\end{figure}
\begin{figure}
\caption{Acceleration estimates of classic Kalman smoother without USBL data (top) and with USBL data (bottom).}
\label{fig:l2_accel_depth}
\end{figure}
A small snippet of vertical acceleration data, Figure~\ref{fig:acceldata}, shows the relatively coarse discretization of the measurements along with a mean that is shifted away from zero indicating bias. Outliers also appear likely. Because the IMU was generally upright with small perturbations in pitch and roll, the discretizations in the z-axis of the instrument frame are still clearly visible when the data are rotated into the vertical world frame. The small perturbations in pitch and roll cause the slight variations visible within each discretization level.
To illustrate our model's usefulness on a data set such as in Figure~\ref{fig:acceldata}, we isolate a 25-second window of the data and add modeling elements one at a time, noting the improvements they provide. We finish by running the full model with varying amounts of infrequent position data on a 10-minute section of data as a more practical experiment.
\noindent
\textbf{Model elements applied to data.}
We consider 25 seconds of IMU data and apply a classic Kalman filter (using least squares measurement and process loss) with the singular navigation model detailed in section~\ref{sec:navmodels}. In practice, one would expect an underwater vehicle to be well instrumented in depth, but for illustration purposes we focus on depth and the vertical acceleration measurements. These measurements are the most biased and therefore improvements made by adding modeling elements are most clearly shown.
The 25 seconds of IMU data starts at a USBL measurement. Position is initialized to this starting USBL measurement, but for ease of comparison that position is treated as (0,0,0) and all subsequent USBL measurements are treated as relative offsets. DRS works better when its reasonably initialized. The initial state vector over the smoothing window is populated by setting acceleration to $0$, and then propagating forward the most recent position fix with a significantly damped measurement of the most recent velocity to prevent divergence of the initial vector.
We start by adding bias estimation (for the acceleration measurements). This requires a second USBL position fix, 19 seconds into the 25-second data series. Figure~\ref{fig:l2_depth} shows position and velocity estimates from a classic Kalman smoother applied to depth acceleration data with and without additional USBL position data. As expected due to the quality of the data we see poor performance in both cases although the small amount of additional position data offers slight improvement.
Figure~\ref{fig:l2_accel_depth} shows acceleration estimates of the classic Kalman smoother with and without additional USBL data. The two USBL measurements affect
acceleration estimates locally but are generally overpowered by the vastly more
complete set of (biased) acceleration data.
\begin{figure}
\caption{Comparison (position and velocity estimates) of debiasing least squares Kalman smoother vs. a robust debiasing smoother equipped with the hubnik loss (Figure~\ref{fig:PLQ}
\label{fig:l2_vs_robust}
\end{figure}
\begin{figure}
\caption{Acceleration estimates for least squares debiasing Kalman smoother (top) vs. robust debiasing smoother equipped with the smoothed Vapnik loss (Figure~\ref{fig:PLQ}
\label{fig:l2_vs_robust_accel}
\end{figure}
The picture improves significantly when we add in bias estimation (via singular process models), using the same two observations (compare Figures~\ref{fig:l2_depth} and ~\ref{fig:l2_vs_robust}). With the bias removed, we can now focus removing
the noticeable outliers form the acceleration data. For this purpose, we
use the robust hubnik (Figure~\ref{fig:PLQ}g) as our measurement loss. The `deadzone' is designed to work with the coarse discretization of the acceleration data.
Figure~\ref{fig:l2_vs_robust} compares the results of (debiased) fitting between the least squares
Kalman smoother and the robust version using the hubnik loss.
Figure~\ref{fig:l2_vs_robust_accel} shows the acceleration estimates with bias removed in both the classic and robust setting.
There are noticeable differences in the acceleration estimates at around 11 and 22 seconds.
When using the robust smoother,
the effect of the outliers on the model's estimate is greatly reduced (see
the velocity estimates at 11 and 22 seconds).
In the context of real-time tracking or forecasting, these sudden jumps will yield inaccurate predictions.
\noindent
\textbf{10 minutes of data.}
We apply the robust smoother with with bias, outliers, and discretization modeling elements
to 10 minutes of IMU data. USBL data is available approximately every $2$ seconds, but we test performance of the smoothing algorithm at larger gaps,
with USBL data supplied at $30, 60,$ and $120$ seconds.
\begin{figure}
\caption{Position estimates obtained with robust debiasing smoother for three frequencies of USBL fixes. Robust
smoothing allows reasonable tracking from infrequent USBL observations.}
\label{fig:posshort}
\end{figure}
\begin{figure}
\caption{Velocity estimates obtained with robust debiasing smoother for different frequencies of position data. Errors from acceleration measurements build up without USBL fixes, but infrequent USBL measurements still allow velocity estimation.}
\label{fig:velshort}
\end{figure}
Figure~\ref{fig:posshort} has the fitted position plots for all three frequencies.
We see that without any USBL fix data, the estimate suffers; but even
infrequent fixes give significant improvements when using the full capability of the smoother.
In fact there are diminishing returns in increasing the USBL frequency; this is a promising
result towards the future goal of a practical online implementation, particularly
in settings where high-quality USBL observations are unavailable (e.g. during dives).
Figure~\ref{fig:velshort} shows the fitted
velocity. Here the effect of additional USBL data are more apparent, as velocity is completely
inferred from position and acceleration. However, the smoothing estimates of velocity
at infrequent USBL fixes are still very good compared to those informed by frequent USBL fixes.
\section{Discussion and Future Work}
\label{sec:conclusion}
We have proposed a singular Kalman smoothing framework that can use
singular covariance models for process and measurements, convex robust
losses, and state-space constraints. The modeler can use any convex
loss that has an implementable prox, a class that includes the most common choices
used for inference in tracking and navigation.
The framework offers a wide range of flexibility that can be used to either counteract undesirable characteristics present in data or be used to increase model performance based off of relevant field knowledge. Future work will consider real-time implementation, as well as extension to nonlinear models.
Numerical experiments show that these modeling elements yield significant improvement on a noisy, challenging dataset. We also see that having
a robust model makes the smoother less reliant on frequent high-quality position updates,
which is a very promising development for underwater navigation.
This paper develops several tools required to move to robust singular
tracking in real-time. A promising aspect of singular noise models
is that they make it possible to do simple robust windowed smoothing,
where estimates are constrained between windows as the tracking proceeds.
Constraints on the state may play a bigger role in real-time estimation, since they can help
detect outliers faster. Finally, robust penalties that provide better
estimates may further improve performance of the DRS algorithm,
by providing an effective initialization
for each new window. We will focus on these developments in future work.
\end{document} |
\begin{document}
\title{Reply to Comment on `Spin Decoherence in Superconducting Atom Chips'}
\author{Bo-Sture K. Skagerstam}\email{[email protected]}
\affiliation{Complex Systems and Soft Materials Research Group, Department of Physics,
The Norwegian University of Science and Technology, N-7491 Trondheim, Norway}
\author{Ulrich Hohenester}
\affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria}
\author{Asier Eiguren}
\affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria}
\author{Per Kristian Rekdal}
\affiliation{Institut f\"ur Physik, Karl-Franzens-Universit\"at Graz, Universit\"atsplatz 5, A-8010 Graz, Austria}
\pacs{03.65.Yz, 03.75.Be, 34.50.Dy, 42.50.Ct}
\maketitle
In a recent paper \cite{skagerstam_06} we investigate spin decoherence in superconducting atom chips,
and predict a lifetime enhancement by more than five orders of magnitude in comparison to
normal-conducting atom chips. Scheel, Hinds, and Knight (SHK) \cite{scheel_06} cast doubt on these
results as they are seemingly an artifact of the two-fluid model used for the description of the superconductor,
and estimate a lifetime enhancement by a factor of ten instead. In this reply we show that this criticism
is unwarranted since neither our central result relies on the two-fluid model, nor the predictions of our
model strongly disagree with experimental data.
In Ref.~\cite{skagerstam_06} we employ a dielectric description of the superconductor based on a parameterization
of the complex optical conductivity $\sigma(T) \equiv \sigma_1(T) + i \sigma_2(T)$, viz.
\begin{equation} \label{sigma_eq}
\sigma(T) = 2/\omega\mu_0\delta^2(T)+i/\omega\mu_0\lambda_L^2(T) \, ,
\end{equation}
\noi
with $\delta(T)$ and $\lambda_L(T)$ the temperature dependent skin and London penetration depth, respectively.
The spin lifetime for $\lambda_L(T) \ll \delta(T)$ is then obtained by matching the electromagnetic fields at
the vacuum-superconductor interface, to arrive at our central result for the spin lifetime
$\tau \propto \sigma^{3/2}_2(T)/\sigma_1(T) \propto \delta^2(T)/\lambda_L^3(T)$. As our analysis is only based
on Maxwell's theory with appropriate boundary conditions, it is valid for $\delta(T)$ and $\lambda_L(T)$
values obtained from either a microscopic model description or from experimental data on $\sigma(T)$.
The specific choice of parameterization in \eq{sigma_eq} is motivated by the
two-fluid model, even though this model is not needed to justify it.
In order to obtain an estimate of the lifetime for niobium, we make use of the experimental value $\sigma_1(T_c) = \sigma_n$
\cite{casalbuoni_06} and consider for $T \leq T_c$ the Gorter-Casimir temperature dependence
$\sigma_1(T) = (T/T_c)^4 \sigma_n$ and $\sigma_2(T) = ( 1 - (T/T_c)^4 ) \, \sigma_2(0)$,
where $\sigma_2(0) = 1/\omega \mu_0 \lambda_L^2(0)$.
A decrease in temperature from $T_c$ to $T_c/2$ as considered in
Ref.~\cite{scheel_06}, then results in a reduction of $\sigma_1(T_c/2)$ by approximately one order of magnitude.
SHK correctly note that the modification of the quasi-particle dispersion in the superconducting state might
give rise to a coherence Hebel-Schlichter peak of $\sigma_1(T)$ below $T_c$ \cite{lifetime}.
To estimate the importance of this peak, we have computed $\sigma_1(T)$ using the Mattis-Bardeen formula.
At the atomic transition frequency of 560 kHz we obtain a peak height of approximately $5 \, \sigma_n$,
not hundred $\sigma_n$ \cite{scheel_05} as used by SHK.
From the literature it is well-known that this coherence peak becomes substantially reduced if lifetime effects
of the quasi-particles are considered \cite{lifetime,lifetime_2} and even disappears in the clean superconductor limit \cite{marsiglio:91}.
As a fair and conservative estimate we correspondingly assign an uncertainty of one order of magnitude to our spin lifetimes.
On the other hand, the major contribution of the $\tau$ enhancement in the superconducting state is due to the additional
$\lambda_L^3(T)$ contribution accounting for the efficient magnetic field screening in superconductors.
This factor is not considered by SHK and appears to be the main reason for the discrepancy between our results and those
of Ref.~\cite{scheel_06}.
The London length $\lambda_L(0) = 35$ nm as used in
\cite{skagerstam_06} corresponds to $\omega \sigma_2(T_c/2) \approx 6.1 \times 10^{20} \, ( \Omega \, $m$ \, $s$ )^{-1}$
and is in agreement with the experimental data of Ref.~\cite{casalbuoni_06}.
In conclusion, modifications of $\sigma(T)$ introduced by the details of the quasi-particle dispersion in the
superconducting state (BCS or Eliashberg theory) are expected to modify the estimated lifetime values by at
most one order of magnitude, but will by no means change the essentials of our findings, which only rely on
generic superconductor properties. Hence, our prediction for a lifetime enhancement by more than five orders
of magnitude prevails. Whether such high lifetimes can be obtained in superconducting atom chips
will have to be determined experimentally.
\end{document} |
\begin{document}
\begin{abstract}
Kechris and Martin showed that the Wadge rank of the $\omega$-th level of the decreasing difference hierarchy of coanalytic sets is $\omega_2$ under the axiom of determinacy.
In this article, we give an alternative proof of the Kechris-Martin theorem, by understanding the $\omega$-th level of the decreasing difference hierarchy of coanalytic sets as the (relative) hyperarithmetical processes with finite mind-changes.
Based on this viewpiont, we also examine the gap between the increasing and decreasing difference hierarchies of coanalytic sets by relating them to the $\Pi^1_1$- and $\Sigma^1_1$-least number principles, respectively.
We also analyze Weihrauch degrees of related principles.
\end{abstract}
\title{On some topics around the Wadge rank $\omega_2$}
\section{Introduction}
\subsection{Summary}
In this article, we investigate topological complexity of sets in the difference hierarchy of coanalytic sets.
For a finite sequence $(A_m)_{m<n}$ of sets, its difference $\diff_{m\leq n}A_m$ is defined as follows:
\[\diff_{m\leq n}A_m=A_n\setminus(A_{n-1}\setminus(\dots\setminus (A_1\setminus A_0))).\]
One important aspect of the finite difference operator is that one can use this to represent exactly all finite Boolean combinations, and another is that it has a natural algorithmic interpretation, as we will see later.
There are two ways of extending the difference operator to certain transfinite sequences of sets.
The first operator $\diff$ is applicable to increasing sequences of sets, and the second one $\diffd$ is applicable to decreasing sequences; see Section \ref{preliminaries}.
For a class $\Gamma$ of sets, let us define $D_\xi(\Gamma)$ as the collection of all sets of the form $\diff_{\eta<\xi}A_\eta$ for some increasing sequence $(A_\eta)_{\eta<\xi}$ of $\Gamma$-sets, and $D^\ast_\xi(\Gamma)$ as the the collection of all sets of the form $\diff_{\eta<\xi}^\ast B_\eta$ for some decreasing sequence $(B_\eta)_{\eta<\xi}$ of $\Gamma$-sets.
Then, $(D_\xi(\Gamma))_{\xi<\omega_1}$ is called the {\em increasing difference hierarchy} of $\Gamma$ sets, and $(D^\ast_\xi(\Gamma))_{\xi<\omega_1}$ is called the {\em decreasing difference hierarchy} of $\Gamma$ sets.
In this article, we give a detailed analysis of these hierarchies for the case where $\Gamma$ is the collection of all coanalytic sets, i.e., $\Gamma=\tpbf{\Pi}^1_1$.
The formal definition (see Section \ref{preliminaries}) of the transfinite levels of the difference hierarchy is rather non-intuitive.
In order to make the meaning of the definition clearer, we describe a computational interpretation of the difference hierarchy, which is much easier to understand.
It is well-known that $\Pi^1_1$ is a higher analog of computable enumerability (based on a certain kind of ordinal-step computability; see e.g.~\cite{HinmanBook,SacksBook}).
As $\Delta^1_1$ is also known as hyperarithmetic, let us call a higher analog of computability by {\em hyp-computability} (so, one may refer to $\Delta^1_1$ as hyp-finite and $\Pi^1_1$ as hyp-c.e.)
Then, roughly speaking:
\begin{itemize}
\item[(1)]
The $\eta$-th level $D_\eta(\Pi^1_1)$ of the increasing difference hierarchy can be viewed as {\em hyp-computability with finite mind-changes along a countdown starting from $\eta$}.
\end{itemize}
More precisely, $A\in D_\eta(\Pi^1_1)$ if and only if there exists a hyp-computable learner guessing whether $n\in A$ or not through the following trial-and-error process: At first the ordinal $\eta$ is displayed in the countdown indicator, and the learner guesses $n\not\in A$, but during the process, the learner can change her mind and make another guess.
Each time the learner changes her mind, the learner has to choose some smaller ordinal than the current value displayed in the countdown indicator.
This newly chosen ordinal will be the next value displayed in the indicator.
As there is no infinite decreasing sequence of ordinals, this guarantees that the learner changes her mind at most finitely often.
This is a higher analog of ``computability with finite mind-changes along an ordinal countdown,'' which has been studied in various contexts, such as computational learning theory, see e.g.~\cite{FS93,AJS99}.
This notion must not be confused with {\em hyp-computability with ordinal mind-changes}, which corresponds to the decreasing difference hierarchy.
Indeed:
\begin{itemize}
\item[(2)]
The $\eta$-th level $D_\eta^\ast(\Pi^1_1)$ of the decreasing difference hierarchy can be thought of as {\em hyp-computability with at most $\eta$ mind-changes}.
\end{itemize}
To be more precise, as before, the hyp-computable learner guesses $n\not\in A$ at first, but during the process, the learner can change her mind and make another guess.
However, since this is an ordinal step computation, the learner has the opportunity to change her mind ordinal many times.
At a limit step, the learner may have changed her mind unboundedly, in which case her guess is reset to state ``$n\not\in A$'' (as in infinite time Turing computation \cite{Carl19}).
During the computation, the number of mind-changes must be kept below $\eta$.
However, if it reaches $\eta$, the learner has to terminate the process with the final guess ``$n\not\in A$''.
In particular, the ambiguous class $\Delta(D_\omegaega^\ast(\Pi^1_1))$ of the $\omega$-th level of the decreasing difference hierarchy corresponds to {\em hyp-computability with finite mind-changes}, where $\Delta(\Gamma)=\Gamma\cap\neg\Gamma$.
Hence, ${\sf Diff}(\tpbf{\Pi}^1_1)\subseteq \Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$, where ${\sf Diff}(\Gamma)$ stands for the whole increasing difference hierarchy $\bigcup_{\xi<\omega_1}D_\xi(\Gamma)$.
Similarly, the whole decreasing difference hierarchy ${\sf Diff}^\ast(\Pi^1_1)=\bigcup_{\xi<\omega_1}D_\xi^\ast(\Gamma)$ can be interpreted as {\em hyp-computability with a fixed countable ordinal mind-changes}.
A higher analog of the limit lemma (due to Monin; see \cite[Proposition 6.1]{BGM17}) also shows that hyp-computability with ordinal mind-changes corresponds to the sets which are $\Delta^0_1$ relative to sets in $\Pi^1_1\cup\Sigma^1_1$.
In summary, we get the following inclusions:
\[D_n(\tpbf{\Pi}^1_1)=D^\ast_n(\tpbf{\Pi}^1_1)\subsetneq \dots\subsetneq{\sf Diff}(\tpbf{\Pi}^1_1)\subseteq\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))\subsetneq\dots\subsetneq{\sf Diff}^\ast(\tpbf{\Pi}^1_1)\subseteq\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1),\]
where $\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$ is the pointclass consisting of all sets which are $\tpbf{\Delta}^0_1$ relative to sets in $\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1$; see Section \ref{sec:beyond-decreasing}.
So far, we have introduced two hierarchies of length $\omega_1$; however, a question arises here:
Is it really the case that a hyp-computable procedure with finite mind-changes is always along some {\em countable} ordinal (i.e., some ordinal below $\omega_1$) countdown?
Surprisingly, the answer is no.
On the one hand, Fournier \cite{FoPhD} showed that the Wadge rank of $D_{1+\alpha}(\tpbf{\Pi}^1_1)$ is $\phi_{\omega_1}(\alpha)$ for $\alpha<\omega_1$, where $\phi_{\omega_1}$ is the $\omega_1$-st Veblen function of base $\omega_1$.
Hence, the Wadge rank of ${\sf Diff}(\tpbf{\Pi}^1_1)$ is $\phi_{\omega_1}(\omega_1)$.
On the other hand, according to Steel \cite{St81}, Kechris and Martin showed that the Wadge rank of $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is $\omega_2$ under the axiom of determinacy.
\begin{theorem}[Kechris-Martin (unpublished); see Steel \cite{St81}]\label{thm:main-theorem}
Under the axiom of determinacy {\sf AD}, the order type of the Wadge degrees of $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ sets is $\omega_2$.
\end{theorem}
This reveals the huge gap between ${\sf Diff}(\tpbf{\Pi}^1_1)$ and $D_\omega^\ast(\tpbf{\Pi}^1_1)$.
In other words, a hyp-computable procedure with finite mind-changes is not necessarily along a countable ordinal countdown.
\begin{fact}[see also Fournier \cite{Fo16}]
Under {\sf AD}, the increasing difference hierarchy of coanalytic sets is strictly included in the $\omega$th level of the decreasing difference hierarchy of coanalytic sets, i.e., ${\sf Diff}(\tpbf{\Pi}^1_1)\subsetneq\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$.
\end{fact}
The proof for the lower bound $\omega_2\leq{\rm otype}_W(\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1)))$ in Kechris-Martin's theorem has been written down in Steel \cite[Theorem 1.2]{St81} and Fournier \cite[Proposition 5.10]{FoPhD}.
For the upper bound, only a very rough idea, no more than two lines long, is commented on by Steel \cite{St81}.
According to Steel \cite{St81}, Martin's proof of the inequality ${\rm otype}_W(\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1)))\leq\omega_2$ is based on the analysis of the ordinal games associated to Wadge games involving sets in $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$.
In this article, we give a somewhat constructive alternative proof of Martin's upper bound which does not use any such techniques.
As a by-product of our constructive ideas, we can give a very clear solution to Fournier's problem,
which asks if the gap between the classes ${\sf Diff}(\tpbf{\Pi}^1_1)$ and $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ still exists even if we weaken the determinacy hypotheses (and may assume the axiom of choice).
\begin{question}[Fournier {\cite[Question 4.6]{Fo16}}]\label{four-main-question}
Is the equality between ${\sf Diff}(\tpbf{\Pi}^1_1)$ and $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ consistent under weaker determinacy hypothesis?
\end{question}
To solve Question \ref{four-main-question}, we give a natural set which belongs to the $\omega$-th level of the decreasing difference hierarchy, but not to the increasing difference hierarchy (see Section \ref{sec:Fournier-problem}).
\begin{theorem}\label{thm:solution-to-Fournier}
Without any extra set-theoretic hypothesis, the increasing difference hierarchy of coanalytic sets is strictly included in the $\omega$-th level of the decreasing difference hierarchy of coanalytic sets, i.e., ${\sf Diff}(\tpbf{\Pi}^1_1)\subsetneq\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ holds, constructively.
\end{theorem}
Beyond the decreasing difference hierarchy, we also turn our attention to the inclusion ${\sf Diff}^\ast(\tpbf{\Pi}^1_1)\subseteq\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$.
As mentioned above, the former corresponds to hyp-computability with fixed countable mind-changes, and the latter corresponds to hyp-computability with ordinal mind-changes by the relative higher limit lemma.
Then, it is natural to ask the following:
\begin{question}\label{question:beyond-difference}
Does the equality between ${\sf Diff}^\ast(\tpbf{\Pi}^1_1)$ and $\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$ hold?
\end{question}
Our answer to Question \ref{question:beyond-difference} is that there is a huge gap between ${\sf Diff}^\ast(\tpbf{\Pi}^1_1)$ and $\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$ (see Section \ref{sec:beyond-decreasing}), without assuming any extra set-theoretic hypothesis.
\begin{theorem}\label{thm:beyond-difference}
${\sf Diff}^\ast(\tpbf{\Pi}^1_1)\subsetneq\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$.
\end{theorem}
\subsection{Preliminaries}\label{preliminaries}
For the basics of (effective) descriptive set theory, we refer the reader to Moschovakis \cite{mos07}.
For background and basic facts about Wadge degrees, see \cite{AnLo12}.
For higher computability, see e.g.~\cite{HinmanBook,SacksBook,BGM17}.
We use $\varphi_e^x$ to denote the $e$th partial computable function relative to an oracle $x$.
The least non-computable ordinal is denoted by $\omega_1^{\rm ck}$.
Let ${\sf WO}\subseteq 2^{\omega\times\omega}\simeq 2^\omega$ be the set of well-orders on $\mathbb{N}$.
For each $y\in{\sf WO}$, we also write $(\mathbb{N},\leq_y)$ for the corresponding well-ordered set.
We use $|y|$ to denote the order type of $y$, and for each $a\in\mathbb{N}$, define $|a|_y$ as the order type of $\{b\in\mathbb{N}:b<_ya\}$.
It is known that ${\sf WO}$ is a $\tpbf{\Pi}^1_1$-complete set.
Indeed, if $P\subseteq\omega^\omega$ is $\Pi^1_1$, then there exists a computable function ${\bf o}_P$ such that, for any $x\in\omega^\omega$, $x\in P$ if and only if ${\bf o}_P(x)\in{\sf WO}$.
We often use this reduction to approximate a $\Pi^1_1$ set and a $\Pi^1_1$ function.
For instance, if $\psi\colon\!\!\!\subseteq\omega\to\omega$ is a partial $\Pi^1_1$ function (i.e., the graph of $\psi$ is $\Pi^1_1$), then for any $s<\omega_1^{\rm ck}$ the stage $s$ approximation of $\psi[s]$ can be defined as follows:
$\psi(n)[s]\downarrow=m$ if and only if the order type of ${\bf o}_P(n,m)$ is less than $s$, where $P$ is the graph of $\psi$.
For sets $A,B\subseteq\omega^\omega$, we say that {\em $A$ is Wadge reducible to $B$} ({\em written $A\leq_{\sf W}B$}) if there exists a continuous function $\theta\colon\omega^\omega\to\omega^\omega$, we have $A=\theta^{-1}[B]$.
A set $A$ is {\em selfdual} if $\neg A\leq_{\sf W}A$, where $\neg A$ is the complement of $A$.
For a pointclass $\Gamma$, we use $\neg\Gamma$ to denote its dual pointclass, that is, $\neg\Gamma=\{\neg A:A\in\Gamma\}$.
By Wadge's lemma \cite{Wadge83,AnLo12}, the Wadge degrees are semi-well-ordered under ${\sf AD}$, where ${\sf AD}$ stands for the axiom of determinacy.
Then, to each set $A\subseteq\omega^\omega$ one can assign the order type $|A|_{\sf W}$ of the collection of all nonselfdual sets $B\leq_{\sf W}A$, which is called the Wadge rank of $A$.
\section{Difference hierarchy}
\subsection{Difference of functions}
In this article, we deal with two {\em difference operators} $\diff$ and $\diff^\ast$.
However, the original definition of the increasing and decreasing difference operators is asymmetrical and rather hard to understand.
For the sake of clarity, we consider the difference operators for functions instead of sets, which yield a symmetric definition of the hierarchies.
Let $X$ and $Y$ be Polish spaces.
A sequence $(f_\xi)_{\xi<\eta}$ of partial functions $f_\xi\colon\!\!\!\subseteq X\to Y$ is {\em dom-increasing} if $({\rm dom}(f_\xi))_{\xi<\eta}$ is increasing; and {\em dom-decreasing} if $({\rm dom}(f_\xi))_{\xi<\eta}$ is decreasing.
Fix $c\in Y\cup\{{\uparrow}\}$, where the symbol ${\uparrow}$ stands for ``undefined''.
\begin{definition}\label{def:diff-for-functions}
For a dom-increasing sequence $(f_\xi)_{\xi<\eta}$ of partial functions, we define $c\diff_{\xi<\eta}f_\xi\colon\!\!\!\subseteq X\to Y$ as follows:
\[c\diff_{\xi<\eta}f_\xi(x)=
\begin{cases}
f_\gamma(x),&\mbox{ if $\gamma=\min\{\xi<\eta:x\in{\rm dom}(f_\xi)\}$},\\
c,&\mbox{ if no such $\gamma$ exists.}
\end{cases}
\]
For a dom-decreasing sequence $(f_\xi)_{\xi<\eta}$ of partial functions, we define $c\diff_{\xi<\eta}^\ast f_\xi\colon\!\!\!\subseteq X\to Y$ as follows:
\[c\diffd_{\xi<\eta}f_\xi(x)=
\begin{cases}
f_\gamma(x),&\mbox{ if $\gamma=\max\{\xi<\eta:x\in{\rm dom}(f_\xi)\}$},\\
c,&\mbox{ if no such $\gamma$ exists.}
\end{cases}
\]
\end{definition}
Note that if $c\in Y$ then the resulting function is always total.
The usual increasing and difference hierarchies of $\tpbf{\Pi}^1_1$ sets are obtained by putting $c=0$ and considering constant functions $f_\eta\colon x\mapsto i$ with $\tpbf{\Pi}^1_1$ domains where $i\in\{0,1\}$; see Section \ref{sec:diff-for-sets}.
Hereafter, to simplify our argument, we assume $Y\subseteq\omega$.
Let $cD_\eta(\tpbf{\Pi}^1_1)$ be the class of all functions of the form $c\diff_{\xi<\eta}f_\xi$ for a dom-increasing sequence $(f_\xi)_{\xi<\eta}$ of partial $\tpbf{\Pi^1_1}$ functions.
We also define $cD^\ast_\eta(\tpbf{\Pi}^1_1)$ in a similar manner.
To give a computability-theoretic interpretation of Definition \ref{def:diff-for-functions}, we also consider the lightface version of these classes.
For $\eta<\omega_1^{\rm ck}$, let $cD_\eta({\Pi}^1_1)$ be the class of all functions of the form $c\diff_{\xi<\eta}f_\xi$ for a uniform $\Pi^1_1$ dom-increasing sequence $(f_\xi)_{\xi<\eta}$ of partial $\Pi^1_1$ functions.
We also define $cD^\ast_\eta({\Pi}^1_1)$ in a similar manner.
Here, a sequence $(f_\xi)_{\xi<\eta}$ is uniformly $\Pi^1_1$ if $\{(\xi,n,m):f_\xi(n)\downarrow=m\}$ is $\Pi^1_1$, where a computable ordinal $\xi$ is always identified with its notation; see also \ref{approx-mind-changes}.
\subsection{Approximation with mind-changes}\label{approx-mind-changes}
To explain the intuitive meaning of two difference hierarchies, we first introduce the notion of finite-change approximations.
For a detailed study of approximations with mind-changes in the context of higher computability theory, we refer the reader to Bienvenu-Greenberg-Monin \cite{BGM17}.
The results in Sections \ref{approx-mind-changes} and \ref{sec:diff-for-sets} are only used for us to get an intuition about two difference hierarchies, and will not be used in later sections.
For this reason, readers without prior knowledge of higher computability may skip Sections \ref{approx-mind-changes} and \ref{sec:diff-for-sets}.
Fix a $\Pi^1_1$ path $O_1$ through Kleene's $\mathcal{O}$ whose order type is $\om_1^{\rm ck}$, and hereafter we identify $O_1$ with $\om_1^{\rm ck}$.
For a function $\varphi\colon\omega\times\omega_1^{\rm ck}\to\omega$, consider the set ${\sf mc}_\varphi(n)$ of all stages at which the value of $\varphi$ changes:
\[{\sf mc}_\varphi(n)=\{s<\om_1^{\rm ck}:\varphi(n,s)\not=\varphi(n,s+1)\}\]
We say that $\varphi$ is a {\em finite-change function} if ${\sf mc}_\varphi(n)$ is a finite set for any $n\in\omega$.
A function $\psi\colon\omega\times\om_1^{\rm ck}\to\eta$ is {\em antitone} if $s\leq t$ implies $\psi(n,s)\geq\psi(n,t)$ for any $n\in\omega$.
An antitone function is a {\em countdown for }$\varphi\colon\omega\times\omega_1^{\rm ck}\to\omega$ if for any $n\in\omega$ and $s\in\om_1^{\rm ck}$,
\[\varphi(n,s)\not=\varphi(n,s+1)\implies\psi(n,s)>\psi(n,s+1).\]
Observe that if $\varphi$ has a countdown, then $\varphi$ is a finite-change function.
If $\varphi$ changes at most finitely often, the limit value, $\lim_{s<t}\varphi(n,s)$, always exists, where
\[\lim_{s<t}\varphi(n,s)=m\iff \varphi(n,s)=m\mbox{ eventually holds for $s<t$}.\]
Here, we say that $A(s)$ eventually holds for $s<t$ if there exists $u<t$ such that $[u,t)\subseteq A$ holds, that is, for any $v$, $u\leq v<t$ implies $A(v)$.
We say that $\varphi$ is {\em continuous} if $\varphi(n,t)=\lim_{s<t}\varphi(n,s)$ for any limit ordinal $t<\om_1^{\rm ck}$.
Let $\eta$ be a computable ordinal.
A function $\varphi\colon\omega\times\om_1^{\rm ck}\to\eta$ is {\em hyp-computable} if its graph is $\Pi^1_1$, where recall that $\om_1^{\rm ck}$ is identified with the $\Pi^1_1$ set $O_1\subseteq\omega$, and note that $\eta=\{s<\om_1^{\rm ck}:s<\eta\}\subseteq O_1$.
Given $c\in\omega\cup\{{\uparrow}\}$, we now show that $cD_\eta(\Pi^1_1)$ is equivalent to hyp-computability with finite mind-changes along $(\eta+1)$-countdown with the initial value $c$.
\begin{prop}\label{prop:diff-characterization1}
A function $f\colon\!\!\!\subseteq\omega\to\omega$ belongs to $cD_\eta(\Pi^1_1)$ if and only if there exists a hyp-computable continuous function $\varphi\colon\omega\times\om_1^{\rm ck}\to\omega$ such that for any $n\in\omega$,
\begin{itemize}
\item $\varphi$ has an $(\eta+1)$-valued hyp-computable countdown,
\item $\varphi(n,0)=c$, and $f(n)=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{prop}
\begin{proof}
($\Rightarrow$)
Assume that $f=\diff_{\xi<\eta}f_\xi$ for a uniform sequence $(f_\xi)_{\xi<\eta}$ of partial $\Pi^1_1$ functions.
Fix a $\Delta^1_1$ approximation $(f_\xi[s])_{s<\om_1^{\rm ck}}$ of $f_\xi$, so that $f_\xi[s]$ is a $\Delta^1_1$ function uniformly in $s<\om_1^{\rm ck}$.
Then, define $\psi(n,s)=\min\{\xi<\eta:n\in{\rm dom}(f_\xi[s])\}$ if it exists; otherwise put $\psi(n,s)=\eta$.
It is clear that $\psi$ is a hyp-computable function, since given input $(n,s)$ we only need to simulate at most $\eta<\om_1^{\rm ck}$ many hyp-algorithms for $\Delta^1_1$ functions $(f_\xi[s])_{\xi<\eta}$.
Then we define $\varphi(n,s)=f_{\psi(n,s)}(n)$ if $\psi(n,s)<\eta$; otherwise put $\varphi(n,s)=c$.
The function $\varphi$ is also hyp-computable.
Clearly, $\psi$ is an $(\eta+1)$-valued antitone function, which is a countdown for $\varphi$.
Let $\gamma<\eta$ be the least ordinal such that $n\in{\rm dom}(f_\gamma)$ if it exists.
Then $f(n)=f_\gamma(n)$ by the definition of the difference operator $\mathbb{D}$.
For such a $\gamma$, there exists $s_0<\om_1^{\rm ck}$ such that $n\in{\rm dom}(f_\gamma[s_0])$, and for such an $s_0$, we have $\psi(n,s)=\gamma$ for any $s\geq s_0$ by minimality of $\gamma$.
Hence, $\varphi(n,s)=f_\gamma(n)=f(n)$ for any $s\geq s_0$.
This means that $\lim_{s<\om_1^{\rm ck}}\varphi(n,s)=f(n)$.
If there is no such a $\gamma$, we have $\psi(n,s)=\eta$ by the definition of $\psi$, and therefore $\varphi(n,s)=c$ for any $s<\om_1^{\rm ck}$.
Hence, $\lim_{s<\om_1^{\rm ck}}\varphi(n,s)=c=f(n)$.
($\Leftarrow$)
Let $\varphi$ be a function in the assumption, and $\psi$ be a countdown for $\varphi$.
Given $\xi<\eta$ and $n\in\omega$, if we see $\psi(n,s)\leq\xi$ for some $s<\om_1^{\rm ck}$, then for the least such an $s$, define $f_\xi(n)=\varphi(n,s)$.
If there is no such an $s$, then $f_\xi(n)$ remains undefined.
Clearly, $(f_\xi)_{\xi<\eta}$ is dom-increasing.
Note that $(f_\xi)_{\xi<\eta}$ is a $\Pi^1_1$ sequence since $\varphi$ and $\psi$ are both hyp-computable.
We claim that $\diff_{\xi<\eta}f_\xi(n)=f(n)$, where $f(n)=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$ by our assumption.
Let $\gamma<\eta$ be the least ordinal such that $n\in{\rm dom}(f_\gamma)$ if it exists.
Then $\diff_{\xi<\eta}f_\xi(n)=f_\gamma(n)$ by the definition of $\diff$.
By the definition of $f_\gamma$, the condition $n\in{\rm dom}(f_\gamma)$ implies that $\psi(n,s)\leq\gamma$ for some $s<\om_1^{\rm ck}$, and by minimality of $\gamma$, there is no $s<\om_1^{\rm ck}$ such that $\psi(n,s)<\gamma$.
Let $s_0<\om_1^{\rm ck}$ be the least ordinal such that $\psi(n,s_0)=\gamma$.
Then we have $f_\gamma(n)=\varphi(n,s_0)$ by our definition of $f_\gamma$.
Since there is no $t>s_0$ such that $\psi(n,t)<\psi(n,s_0)=\gamma$, by the countdown condition, we have $\varphi(n,t)=\varphi(n,s_0)$ for any $t>s_0$.
This means that $\diff_{\xi<\eta}f_\xi(n)=f_\gamma(n)=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
If there is no such a $\gamma$, $f_\xi(n)$ is undefined for all $\xi<\eta$, and thus, $\psi(n,s)=\eta$ for any $s<\om_1^{\rm ck}$.
Since $\psi$ is a countdown for $\varphi$, we have $\varphi(n,s)=\varphi(n,0)=c$ for any $s<\om_1^{\rm ck}$.
Therefore, $\diff_{\xi<\eta}f_\xi(n)=c=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{proof}
Next, let us move on to a function which may change infinitely often.
For such a function $\varphi$, in general, $\lim_{s<t}\varphi(n,s)$ does not necessarily exist.
Instead, for any constant $c\in\omega$ and ordinal $\beta<\omega_1^{\rm ck}$, we define
\[
c\lim_{s<t}\varphi(n,s)=
\begin{cases}
m,&\mbox{ if }\varphi(n,s)=m\mbox{ eventually holds for $s<t$},\\
c,&\mbox{ if there exists no such $m$}.
\end{cases}
\]
We say that $\varphi$ is {\em $c$-semicontinuous} if $\varphi(n,t)=c\lim_{s<t}\varphi(n,s)$ for any limit ordinal $t<\om_1^{\rm ck}$.
Note that any function $\varphi$ yields a semicontinuous function $\varphi^c$ by defining $\varphi(n,0)=c$; $\varphi^c(n,t+1)=\varphi(n,t)$ for any $t<\om_1^{\rm ck}$; and $\varphi^c(n,t)=c\lim_{s<t}\varphi(n,s)$ for any limit ordinal $t<\om_1^{\rm ck}$.
This is, for example, exactly the same as the behavior of infinite time Turing machines at limit steps.
Fix $c\in\omega\cup\{{\uparrow}\}$, and let $\eta$ be a computable ordinal.
We characterize $cD_\eta^\ast(\Pi^1_1)$ as hyp-computability with at most $\eta$ mind-changes with the initial and reset value $c$.
\begin{prop}\label{prop:diff-characterization2}
A function $f\colon\omega\to\omega$ belongs to $cD_\eta^\ast(\Pi^1_1)$ if and only if there exists a hyp-computable $c$-semicontinuous function $\varphi\colon\omega\times\om_1^{\rm ck}\to\omega$ such that for any $n\in\omega$,
\begin{itemize}
\item ${\rm otype}({\tt mc}_\varphi(n))\leq\eta$,
\item $\varphi(n,0)=c$, and $f(n)=c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{prop}
\begin{proof}
($\Leftarrow$)
Let $\varphi$ be a function in the assumption, and for each $n\in\omega$, let $(s^n_\xi)_{\xi<\lambda(n)}$ be the increasing enumeration of the set ${\tt mc}_\varphi(n)$ of all mind-change stages.
Since there is an order embedding of ${\tt mc}_\varphi(n)$ into $\eta$ by our assumption, we have $\lambda(n)\leq\eta$.
For any $n\in\omega$ and $\xi<\lambda(n)$, define $f_\xi(n)=\varphi(n,s^n_\xi+1)$.
If $\xi\geq\lambda(n)$, $f_\xi(n)$ is undefined.
Clearly $(f_\xi)_{\xi<\eta}$ is dom-decreasing since we have ${\rm dom}(f_\xi)=\{n\in\omega:\xi<\lambda(n)\}$.
Note also that $(f_\xi)_{\xi<\eta}$ is a $\Pi^1_1$ sequence since $\varphi$ is hyp-computable, and ${\tt mc}_\varphi$ has a hyp-computable increasing enumeration.
We claim that $c\diff^\ast_{\xi<\eta}f_\xi(n)=f(n)$, where $f(n)=c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$ by our assumption.
If $\lambda(n)$ is a successor ordinal, then $\gamma:=\lambda(n)-1$ is the greatest ordinal such that $n\in{\rm dom}(f_\gamma)$.
Then $c\diff^\ast_{\xi<\eta}f_\xi(n)=f_\gamma(n)$ by definition.
Then, $s_\gamma^n$ exists, and by maximality, there is no $t>s^n_\gamma$ such that $\varphi(n,t)\not=\varphi(n,t+1)$.
Hence, we have
\[f(n)=c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)=\varphi(n,s^n_\gamma+1)=f_\gamma(n)=c\diffd_{\xi<\eta}f_\xi(n).\]
If $\lambda(n)$ is a limit ordinal, there is no greatest ordinal $\gamma$ such that $n\in{\rm dom}(f_\gamma)$, so $c\diff^\ast_{\xi<\eta}f_\xi(n)=c$.
Moreover, for $s^n=\sup_{\xi<\lambda(n)}s^n_\xi$, we have $\varphi(n,s^n)=c\lim_{s<s^n}\varphi(n,s)=c$ since $\varphi(n,s^n_\xi)\not=\varphi(n,s^n_{\xi}+1)$ for any $\xi<\lambda(n)$.
Therefore, $c\diff^\ast_{\xi<\eta}f_\xi(n)=c=c\lim_{s<s^n}\varphi(n,s)=f(n)$.
($\Rightarrow$)
Assume that $f=c\diff^\ast_{\xi<\eta}f_\xi$ for a dom-decreasing sequence $(f_\xi)_{\xi<\eta}$ of partial $\Pi^1_1$ functions.
Then, we have a hyp-approximation $(f_\xi[s])_{s<\om_1^{\rm ck}}$ for $f$ for each $\xi<\eta$.
Fix $n\in\omega$.
Let $s_\xi$ be the least stage $s$ such that $f_\xi(n)[s]$ is defined if such an $s$ exists.
Clearly, we may assume that $\zeta\leq\xi$ implies $s_\zeta\leq s_\xi$ since $(f_\xi)_{\xi<\eta}$ is dom-decreasing.
Moreover, we claim that if we choose a hyp-approximation for $f_\xi$ appropriately, we may assume that $s_\xi$ is successor for each $\xi<\eta$, and $(s_\xi)_{\xi<\eta}$ is strictly increasing.
To see this, put $s(\xi,t)=(\eta+1)\cdot t+\xi+1$.
Then, $s\colon\eta\times\om_1^{\rm ck}\to\om_1^{\rm ck}$ is injective.
Fix $\xi<\eta$, and first declare that $f'_\xi(n)[0]$ is undefined.
If $s=s(\xi,t)$ for some ordinal $t<\om_1^{\rm ck}$, then put $f'_\xi(n)[s]=f_\xi(n)[t]$.
Assume that $s$ is not of the form $s(\xi,t)$.
If $s$ is successor, say $s=s'+1$, then put $f'_\xi(n)[s]=f'_\xi(n)[s']$.
If $s$ is limit, then put $f'_\xi(n)[s]=\lim_{t<s}f'_\xi(n)[t]$.
It is easy to see that $(f'_\xi[s])_{s<\om_1^{\rm ck}}$ is a hyp-approximation for $f_\xi$ for each $\xi<\eta$.
Moreover, since $s(\xi,t)$ is successor, and $s$ is injective, one can see that this approximation has the desired property.
Then, replace $(f_\xi[s])_{s<\om_1^{\rm ck}}$ with $(f'_\xi[s])_{s<\om_1^{\rm ck}}$.
For a successor ordinal $s<\om_1^{\rm ck}$, let $\psi(n,s)$ be the least ordinal $\xi<\eta$ such that $n\not\in{\rm dom}(f_\xi[s])$.
If there is no such $\xi$, put $\psi(n,s)=\eta$.
Note that $\psi(n,s)=\min(\{\xi<\eta:s<s_\xi\}\cup\{\eta\})$, so $\xi<\psi(n,s)$ if and only if $s_\xi\leq s$.
If $\psi(n,s)$ is successor, say $\psi(n,s)=\gamma+1$, then define $\varphi(n,s)=f_\gamma(n)$.
If $\psi(n,s)$ is limit, then define $\varphi(n,s)=c$.
For a limit ordinal $s<\om_1^{\rm ck}$, define $\varphi(n,s)=c\lim_{t<s}\varphi(n,s)$.
Obviously, $\varphi$ is $c$-semicontinuous.
One can also check that $\varphi$ is hyp-computable.
We inductively define an order embedding $h\colon{\tt mc}_\varphi(n)\to\eta$ which, given $t\in{\tt mc}_\varphi(n)$, returns an ordinal less than $\psi(n,t+1)$.
Put $s=t+1$.
If $\psi(n,s)$ is successor, define $h(t)=\psi(n,s)-1$.
If $\psi(n,s)$ is limit, note that $s^\ast:=\sup\{s_\xi:\xi<\psi(n,s)\}<s$ since $(s_\xi)_{\xi<\eta}$ is strictly increasing and $s$ is successor.
Note that if $u$ is a successor ordinal with $s^\ast<u\leq s$ then $\psi(n,u)=\psi(n,s)$ by the definitions of $s^\ast$ and $\psi$.
Moreover, $t\in{\tt mc}_\varphi(n)$ implies that $\varphi(n,t)\not=\varphi(n,s)$, so we must have $s^\ast=t$.
First suppose that, for any $\xi<\psi(n,s)$ there exists $\zeta$ such that $\xi<\zeta<\psi(n,s)$ and $f_\xi(n)\not=f_\zeta(n)$.
In this case, as $(s_\xi)_{\xi<\eta}$ is a strictly increasing sequence of successor ordinals, we have $\varphi(n,s_\xi-1)=f_\xi(n)\not=f_\zeta(n)=\varphi(n,s_\zeta-1)$.
This implies that $\varphi(n,t)=\varphi(n,s^\ast)=c\lim_{t<s^\ast}\varphi(n,t)=c$.
Moreover, $\varphi(n,s)=c$ since $\psi(n,s)$ is limit by our assumption.
This contradicts $t\in{\tt mc}_\varphi(n)$.
Thus, there exists $\xi<\psi(n,s)$ such that $\xi\leq\zeta<\psi(n,s)$ implies $f_\xi(n)=f_\zeta(n)$, for any $\zeta$.
Then, one might think that we can just define $h(t)$ as $\xi+1$; however recall that if $\psi(n,u)$ is a limit ordinal, then the value of $\varphi(n,u)$ is reset to $c$.
Thus, the value of $\varphi(n,u)$ may change even if $(f_\zeta(n))_{\xi\leq\zeta<\psi(n,s)}$ is constant.
Of course, if the value of $f_\xi(n)$ is $c$, there is no problem.
If $f_\xi(n)=c$, for any $u$ with $s_\xi\leq u\leq s$, we have $\varphi(n,u)=\varphi(n,s_\xi)=f_\xi(n)$.
In this case, we put $h(t)=\xi+1$, which implies $h(t)<\psi(n,s)$.
Note that $u<s$ and $u\in{\tt mc}_\varphi(n)$ implies $u<s_\xi$, so $u+1<s_{\xi+1}$ as $(s_\xi)_{\xi<\eta}$ is strictly increasing.
This implies $\psi(n,u+1)\leq\xi+1$ by the definition of $\psi$.
By the induction hypothesis, $h(u)<\psi(n,u+1)\leq\xi+1$.
Hence, $u<t$ implies $h(u)<h(t)$.
If $f_\xi(n)\not=c$, then there are two cases:
If $\psi(n,s)$ is a limit of limit ordinals, say $\psi(n,s)=\sup_{k\in\omega}\lambda_k$ where each $\lambda_k$ is limit, then we have $\varphi(n,s_{\lambda_k}-1)=c$ since $\psi(n,s_{\lambda_k}-1)=\lambda_k$, which is limit.
Then $s^\ast=\sup\{s_{\lambda_k}:k\in\omega\}$, and $t=s^\ast$ as seen before, so we have $\varphi(n,t)=c\lim_{u<s^\ast}\varphi(n,u)=c$.
However, $\varphi(n,u)=c$ as $\psi(n,s)$ is limit by our assumption.
Again, $t\in{\tt mc}_\varphi(n)$ implies that $\varphi(n,t)\not=\varphi(n,s)$, which is impossible.
Next, if $\psi(n,s)$ is not a limit of limit ordinals (while $\psi(n,s)$ is limit by our assumption), then $\psi(n,s)$ is of the form $\lambda+\omega$.
Then choose $k\in\omega$ such that $\xi\leq\lambda+k$, and define $h(t)=\lambda+k+1$, which implies $h(t)<\psi(n,s)$.
Note that for any successor ordinal $u$ with $s_{\lambda+k}\leq u<s^\ast=\sup\{s_{\lambda+\ell}:\ell\in\omega\}$ we have $\psi(n,u)=\lambda+\ell$ for some $k<\ell<\omega$.
In particular, $\psi(n,u)$ is successor, so $\varphi(n,u)=f_{\lambda+\ell}(n)=f_\xi(n)$.
Hence, for any $u$ with $s_{\lambda+k}\leq u\leq s$, we have $\varphi(n,u)=f_\xi(n)$.
Therefore, by the same argument as in the case $f_\xi(n)=c$, one can see that $u<t$ implies $h(u)<h(t)$.
Hence, $h$ is an order embedding.
We claim that $f(n)=c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
Let us consider $\gamma=\max\{\xi<\eta:n\in{\rm dom}(f_\xi)\}$ if it exists.
Then, $f(n)=f_\gamma(n)$ since $f=\diff^\ast_{\xi<\eta}f_\xi$.
Let $s$ be the least stage such that $n\in{\rm dom}(f_\xi[s])$.
By maximality of $\gamma$, for any successor ordinal $t\geq s$ we have $\psi(n,t)=\gamma+1$, and thus $\varphi(n,t)=f_\gamma(n)$ by definition.
Therefore, we have $c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)=f_\gamma(n)=f(n)$.
If there is no such a $\gamma$, then $f(n)=c$.
Put $\lambda=\min\{\xi<\eta:n\not\in{\rm dom}(f_\xi)\}$
Then, $\lambda$ must be a limit ordinal as $\gamma$ is undefined.
Let us consider $(s_\xi)_{\xi<\lambda}$.
Note that $\xi\mapsto s_\xi\colon\lambda\to\om_1^{\rm ck}$ is a total $\Pi^1_1$ function, and thus $\Delta^1_1$ since the domain is a computable ordinal.
Hence, by Spector's boundedness theorem (see e.g.~\cite[Corollary I.5.6]{SacksBook}), we have $s^\ast:=\sup\{s_\xi:\xi<\lambda\}<\om_1^{\rm ck}$.
For any successor ordinal $s\geq s^\ast$, we have $\psi(n,s)=\lambda$, and thus $\varphi(n,s)=c$ since $\lambda$ is limit.
Hence, we have $c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)=c=f(n)$.
\end{proof}
As a corollary, one can see that for any $n\in\omega$, and $\eta<\om_1^{\rm ck}$, we have
\[cD_n(\Pi^1_1)=cD_n^\ast(\Pi^1_1)\subseteq\dots\subseteq cD_\omega(\Pi^1_1)\subseteq\dots\subseteq cD_\eta(\Pi^1_1)\subseteq\dots\subseteq cD^\ast_\omega(\Pi^1_1).\]
By relativizing Propositions \ref{prop:diff-characterization1} and \ref{prop:diff-characterization2}, one can show the similar results for Baire space $\omega^\omega$.
\subsection{Difference hierarchy for sets}\label{sec:diff-for-sets}
Now let us return to the original unintuitive definition of difference operators for sets.
For a countable ordinal $\xi$, if $(A_\eta)_{\eta<\xi}$ is an increasing sequence of sets, then its difference $\diff_{\eta<\xi}A_\eta$ is defined as follows:
\[
\diff_{\eta<\xi}A_\eta=
\bigcup_{\substack{\eta<\xi\\{\sf par}(\eta)\not={\sf par}(\xi)}}\left(A_\eta\setminus\bigcup_{\gamma<\eta}A_\gamma\right),
\]
where ${\sf par}(\xi)=1$ if $\xi$ is odd; otherwise, ${\sf par}(\xi)=0$.
If $n$ is a natural number, one can see that $\diff_{m\leq n}A_m=A_n\setminus(A_{n-1}\setminus(\dots\setminus (A_1\setminus A_0)))$.
If $(B_\eta)_{\eta<\xi}$ is a decreasing sequence of sets, then its difference $\diff^\ast_{\eta<\xi}B_\eta$ is defined as follows:
\[
\diffd_{\eta<\xi}B_\eta=\bigcup_{\substack{\eta<\xi\\ \eta\text{ even}}}\left(B_\eta\setminus B_{\eta+1}\right),
\]
where if $\xi$ is odd, put $B_\xi=\emptyset$.
If $n$ is a natural number, one can see that $\diff^\ast_{m\leq n}B_m=B_0\setminus(B_1\setminus(\dots\setminus (B_{n-1}\setminus B_n)))$.
Let $D_\eta(\tpbf{\Pi}^1_1)$ be the class of all sets of the form $\diff_{\xi<\eta}A_\xi$ for an increasing sequence $(A_\xi)_{\xi<\eta}$ of $\tpbf{\Pi^1_1}$ sets.
Similarly, let $D_\eta({\Pi}^1_1)$ be the class of all sets of the form $\diff_{\xi<\eta}f_\xi$ for a uniform $\Pi^1_1$ increasing sequence $(A_\xi)_{\xi<\eta}$ of $\Pi^1_1$ sets.
We also define the classes $D^\ast_\eta(\tpbf{\Pi}^1_1)$ and $D^\ast_\eta({\Pi}^1_1)$ in a similar manner.
To understand the relationship between the difference operators for sets and function, it is useful to introduce the following hybrid version of difference operators.
Let $X$ and $Y$ be Polish spaces, and fix $c\in Y\cup\{{\uparrow}\}$.
\begin{definition}\label{def:diff-for-functions}
For an increasing sequence $(A_\xi)_{\xi<\eta}$ of sets and a sequence $(f_\xi)_{\xi<\eta}$ of partial functions, we define $c\diff_{\xi<\eta}[f_\xi/A_\xi]\colon\!\!\!\subseteq X\to Y$ as follows:
\[c\diff_{\xi<\eta}[f_\xi/A_\xi](x)=
\begin{cases}
f_\gamma(x),&\mbox{ if $\gamma=\min\{\xi<\eta:x\in A_\xi\}$},\\
c,&\mbox{ if no such $\gamma$ exists.}
\end{cases}
\]
For a decreasing sequence $(B_\xi)_{\xi<\eta}$ of sets and sequence $(f_\xi)_{\xi<\eta}$ of partial functions, we define $c\diff_{\xi<\eta}^\ast[f_\xi/B_\xi]\colon\!\!\!\subseteq X\to Y$ as follows:
\[c\diffd_{\xi<\eta}[f_\xi/B_\xi](x)=
\begin{cases}
f_\gamma(x),&\mbox{ if $\gamma=\max\{\xi<\eta:x\in B_\xi\}$},\\
c,&\mbox{ if no such $\gamma$ exists.}
\end{cases}
\]
\end{definition}
Let $cD_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$ be the class of all sets of the form $c\diff_{\xi<\eta}[f_\xi/A_\xi]$ for an increasing sequence $(A_\xi)_{\xi<\eta}$ of $\tpbf{\Pi^1_1}$ sets, and a sequence $(f_\xi)_{\xi<\eta}$ of continuous functions.
Similarly, let $cD_\eta(\Sigma^0_1/\Pi^1_1)$ be the class of all sets of the form $c\diff_{\xi<\eta}[f_\xi/A_\xi]$ for a uniform $\Pi^1_1$ increasing sequence $(A_\xi)_{\xi<\eta}$ of $\Pi^1_1$ sets, and a computable sequence $(f_\xi)_{\xi<\eta}$ of computable functions.
We also define the classes $cD^\ast_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$ and $cD^\ast_\eta(\Sigma^0_1/{\Pi}^1_1)$ in a similar manner.
Obviously, $cD_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)\subseteq cD_\eta(\tpbf{\Pi}^1_1)$ and $cD^\ast_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)\subseteq cD^\ast_\eta(\tpbf{\Pi}^1_1)$.
The lightface versions also hold.
These hybrid difference operators seem relevant for studying $\sigma$-continuous functions ($\omega$-decomposable functions; see e.g.~\cite{GKN21}).
As in Propositions \ref{prop:diff-characterization1} and \ref{prop:diff-characterization2}, the classes $cD_\eta(\Sigma^0_1/\Pi^1_1)$ and $cD^\ast_\eta(\Sigma^0_1/{\Pi}^1_1)$ are characterized as hyp-computability of an index $\gamma$ with mind-changes.
Such an index-guessing has been extensively studied in the theory of inductive inference (identification in the limit; see \cite{JOSW}).
Observe that the characteristic function of a set in $D_\eta(\tpbf{\Pi}^1_1)$ belongs to $0D_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$:
Given an increasing sequence $(A_\xi)_{\xi<\eta}$ of sets, define $f_\xi\colon A_\xi\to 2$ by $f_\xi(x)=1$ if ${\sf par}(\xi)\not={\sf par}(\eta)$; otherwise $f_\xi(x)=0$.
Similarly, the characteristic function of a set in $D^\ast_\eta(\tpbf{\Pi}^1_1)$ belongs to $0D_\eta^\ast(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$:
Given a decreasing sequence $(A_\xi)_{\xi<\eta}$ of sets, define $f_\xi\colon A_\xi\to 2$ by $f_\xi(x)=1$ if ${\sf par}(\xi)=0$; otherwise $f_\xi(x)=0$.
As a corollary of Proposition \ref{prop:diff-characterization1}, the class $D_\eta(\Pi^1_1)$ is characterized as hyp-computability with finite mind-changes along $(\eta+1)$-countdown with the initial value $0$.
\begin{cor}
A set $A\subseteq\omega$ belongs to $D_\eta(\Pi^1_1)$ if and only if there exists a hyp-computable continuous finite-change function $\varphi\colon\omega\times\om_1^{\rm ck}\to 2$ such that for any $n\in\omega$,
\begin{itemize}
\item $\varphi$ has an $(\eta+1)$-valued hyp-computable countdown,
\item $\varphi(n,0)=0$, and $A(n)=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{cor}
Similarly, as a corollary of Proposition \ref{prop:diff-characterization2}, the class $D^\ast_\eta(\Pi^1_1)$ is characterized as hyp-computability with at most $\eta$ mind-changes with the initial value $0$.
\begin{cor}
A set $A\subseteq\omega$ belongs to $D^\ast_\eta(\Pi^1_1)$ if and only if there exists a hyp-computable $c$-semicontinuous function $\varphi\colon\omega\times\om_1^{\rm ck}\to\omega$ such that for any $n\in\omega$,
\begin{itemize}
\item ${\rm otype}({\tt mc}_\varphi(n))\leq\eta$,
\item $\varphi(n,0)=0$, and $A(n)=0\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{cor}
It is easy to show the following analogues of Post's theorem.
\begin{prop}
A set $A\subseteq\omega$ belongs to $\Delta(D_\eta(\Pi^1_1))$ if and only if there exists a hyp-computable continuous finite-change function $\varphi\colon\omega\times\om_1^{\rm ck}\to 2$ such that for any $n\in\omega$,
\begin{itemize}
\item $\varphi$ has an $\eta$-valued hyp-computable countdown,
\item and $A(n)=\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{prop}
\begin{prop}
A set $A\subseteq\omega$ belongs to $\Delta(D^\ast_\eta(\Pi^1_1))$ if and only if there exists a hyp-computable $c$-semicontinuous function $\varphi\colon\omega\times\om_1^{\rm ck}\to\omega$ such that for any $n\in\omega$,
\begin{itemize}
\item ${\rm otype}({\tt mc}_\varphi(n))<\eta$,
\item and $A(n)=c\lim_{s<\om_1^{\rm ck}}\varphi(n,s)$.
\end{itemize}
\end{prop}
In particular, $\Delta(D^\ast_\omega(\Pi^1_1))$ corresponds to hyp-computability with finite mind-changes.
\section{Solution to Fournier's problem}\label{sec:3}
\subsection{Weihrauch lattice}\label{sec:Weihrauch}
Let us explain that the class $cD_\eta(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$ is to some extent a natural one in terms of the Weihrauch lattice.
This perspective will also be used to solve Fournier's Question \ref{four-main-question}.
The study of the Weihrauch lattice aims to measure the computability theoretic difficulty of finding a choice function witnessing the truth of a given $\forall\exists$-theorem (cf.\ \cite{pauly-handbook}) as an analogue of reverse mathematics \cite{SOSOA:Simpson}.
The notion of Weihrauch degree is used as a tool to classify certain $\forall\exists$-statements by identifying each $\forall\exists$-statement with a partial multivalued function.
Informally speaking, a (possibly false) statement $S\equiv\forall x\in X\;[Q(x)\rightarrow\exists yP(x,y)]$ is transformed into a partial multivalued function $f\colon\!\!\!\subseteq X\rightrightarrows Y$ such that ${\rm dom}(f)=\{x:Q(x)\}$ and $f(x)=\{y:P(x,y)\}$.
Then, measuring the degree of difficulty of witnessing the truth of $S$ is identified with that of finding a choice function for $f$.
Here, we consider choice problems for partial multivalued functions rather than relations in order to distinguish the hardest instance $f(x)=\emptyset$ and the easiest instance $x\in X\setminus{\rm dom}(f)$.
If one only considers subspaces of $\mathbb{N}^\mathbb{N}$, one can use the following version of Weihrauch reducibility:
For partial multivalued functions $f$ and $g$, we say that $f$ is {\em Weihrauch reducible to $g$} (written $f\leq_{\sf W}g$) if there are partial computable functions $h$ and $k$ such that the following holds:
Given an instance $x$ of $f$-problem (i.e., $x\in{\rm dom}(f)$), if we know a solution $y$ to the instance $h(x)$ of $g$-problem (i.e., $y\in g(h(x))$), then the algorithm $k$ tells us that $k(x,y)$ is a solution to the instance $x$ of $f$-problem (i.e., $k(x,y)\in f(x)$).
In other words,
\[(\forall x\in{\rm dom}(f))(\forall y)\;[y\in g(h(x))\implies k(x,y)\in f(x)].\]
The functions $h$ and $k$ are often called an {\em inner reduction} and an {\em outer reduction}, respectively.
To discuss Weihrauch reducibility in other spaces, we introduce some auxiliary concepts.
A {\em representation} of a set $X$ is a partial surjection $\delta_X\colon\!\!\!\subseteq\omega^\omega\to X$.
If $\delta_X(p)=x$, then $p$ is called a {\em $\delta_X$-name of $x$}\index{name} (or simply, a {\em name of $x$} if $\delta_X$ is clear from the context).
A pair of a set and its representation is called a {\em represented space}.
\begin{example}
Perhaps, one of the best known examples of represented spaces in descriptive set theory is the space ${\sf Bor}$ of Borel sets in a Polish space, where consider the representation $\delta_{\sf Bor}\colon\!\!\!\subseteq\omega^\omega\to{\sf Bor}$ defined by $\delta_{\sf Bor}(p)=A$ if and only if $p$ is a Borel code of $A$.
In other words, a $\delta_{\sf Bor}$-name of $A$ is exactly a Borel code of $A$.
\end{example}
\begin{definition}[see also \cite{pauly-handbook}]
Let $X$, $Y$, $Z$ and $W$ be represented spaces with representations $\delta_X$, $\delta_Y$, $\delta_Z$ and $\delta_W$, respectively.
For partial multivalued functions $f\colon\!\!\!\subseteq X\rightrightarrows Y$ and $g\colon\!\!\!\subseteq Z\rightrightarrows Y$, we say that $f$ is {\em Weihrauch reducible to $g$} (written $f\leq_{\sf W}g$) if there are partial computable functions $h$ and $k$ such that the following holds:
Given a $\delta_X$-name ${\tt x}$ of an instance $x$ of $f$-problem, the algorithm $h$ tells us a $\delta_Z$-name $h({\tt x})$ of an instance $x^\ast$ of $g$-problem, and if we know a $\delta_W$-name ${\tt y}$ of a solution $y$ to the instance $x^\ast$ of $g$-problem, then the algorithm $k$ tells us that $k({\tt x},{\tt y})$ is a $\delta_Y$-name of a solution to the instance $x$ of $f$-problem.
In other words,
\[(\forall {\tt x}\in{\rm dom}(f\circ\delta_X))(\forall {\tt y})\;[\delta_W({\tt y})\in g\circ\delta_Z(h({\tt x}))\implies \delta_Y\circ k({\tt x},{\tt y})\in f\circ\delta_X({\tt x})].\]
\end{definition}
We now consider the following $\forall\exists$-principles related to the difference hierarchy:
\begin{itemize}
\item {\em $\Gamma$-least number principle}: For any nonempty $\tpbf{\Gamma}$ set $A\subseteq\omega$, there exists the least element of $A$.
\item {\em $\Gamma$-counting}: For any finite $\tpbf{\Gamma}$ set $A\subseteq\omega$, the value $\# A$ exists.
\end{itemize}
We consider the case where $\Gamma$ is either ${\Pi}^1_1$ or ${\Sigma}^1_1$.
For such a $\Gamma$, note that if $X$ is a Polish space then the collection $\tpbf{\Gamma}(X)$ of all $\tpbf{\Gamma}$ subsets of $X$ has a total representation $\delta_\Gamma\colon\omega^\omega\to\tpbf{\Gamma}(X)$.
For instance, if $X=\mathbb{N}$ and $\Gamma={\Pi}^1_1$ then, for any $e\in\omega$ and $p\in\omega^\omega$, the concatenation $e{}^{\frown} p$ is a $\delta_{\Pi^1_1}$-name of $A\subseteq\omega$ if and only if $A$ is the $e$-th $\Pi^1_1(p)$ set.
Hereafter, we also use $P_x$ to denote $\delta_{\Pi^1_1}(x)$; that is, $P_x$ is the $\tpbf{\Pi}^1_1$ set coded by $x$.
\begin{definition}
We define the $\Gamma$-least number principle $\Gamma\lnp\colon\!\!\!\subseteq\tpbf{\Gamma}(\omega)\to\omega$ as follows:
\[
\Gamma\lnp(A)=
\begin{cases}
\min A,&\mbox{ if }A\not=\emptyset,\\
\mbox{undefined},&\mbox{ if }A=\emptyset.
\end{cases}
\]
We define the $\Gamma$-counting principle $\#\Gamma\colon\!\!\!\subseteq\tpbf{\Gamma}(\omega)\to\omega$ as follows:
\[
\#\Gamma(A)=
\begin{cases}
\# A,&\mbox{ if }\#A\mbox{ is finite},\\
\mbox{undefined},&\mbox{ otherwise}.
\end{cases}
\]
\end{definition}
Let $(X,\delta_X)$ be a represented space.
We say that a partial function $f\colon\!\!\!\subseteq X\to\omega$ is $cD_\omega(\Gamma)$-complete if $f\circ\delta_X\colon\!\!\!\subseteq \omega^\omega\to\omega$ belongs to $cD_\omega(\Gamma)$, and any $cD_\omega(\Gamma)$-function $g\colon\!\!\!\subseteq\omega^\omega\to\omega$ is Weihrauch reducible to $f$.
We define $cD^\ast_\omega(\Gamma)$-completeness in a similar manner.
We now consider the case $c={}\uparrow$ (indicating ``undefined'').
\begin{prop}
$\Pi^1_1\lnp$ is ${\uparrow} D_\omega(\Sigma^0_1/\Pi^1_1)$-complete.
\end{prop}
\begin{proof}
To see that $\Pi^1_1\lnp$ is in ${\uparrow} D_\omega(\Sigma^0_1/\Pi^1_1)$, define $A_n=\{x:(\exists k\leq n)\;k\in P_x\}$ where $P_x$ is the $x$th $\tpbf{\Pi}^1_1$ set, and consider the constant function $c_n\colon x\mapsto n$.
Then, $(A_n)_{n<\omega}$ is an increasing sequence of $\tpbf{\Pi}^1_1$ sets.
One can easily see that $\min P_x=\min\{n\in\omega:x\in A_n\}$ whenever $P_x$ is empty.
Recall from Definition \ref{def:diff-for-functions} that ${\uparrow}\diff_{n<\omega}[c_n/A_n](x)=\min\{n\in\omega:x\in A_n\}$ if it exists.
Therefore, ${\uparrow}\diff_{n<\omega}[c_n/A_n]$ is a realizer for $\Pi^1_1\lnp$.
To show the completeness, assume that a sequence $(A_n,f_n)_{n<\omega}$ of pairs of $\tpbf{\Pi}^1_1$ sets and continuous functions is given.
To see that ${\uparrow}\diff_{n<\omega}[f_n/A_n]$ is Weihrauch reducible to $\Pi^1_1\lnp$, let us consider the inner reduction $h$ which maps $x$ to a $\delta_{\Pi^1_1}$-name of $Q_x=\{n\in\omega:x\in A_n\}$, and the outer reduction $k$ which maps $(x,n)$ to $f_n(x)$.
If $m=\min Q_x$ exists, then ${\uparrow}\diff_{n<\omega}[f_n/A_n](x)=f_m(x)=k(x,\min Q_x)$.
If no such $m$ exists, ${\uparrow}\diff_{n<\omega}[f_n/A_n](x)$ is undefined.
This verifies the assertion.
\end{proof}
\begin{prop}
$\Sigma^1_1\lnp$ is ${\uparrow} D^\ast_\omega(\Sigma^0_1/\Pi^1_1)$-complete.
\end{prop}
\begin{proof}
To see that $\Sigma^1_1\lnp$ is in ${\uparrow} D^\ast_\omega(\Sigma^0_1/\Pi^1_1)$, define $B_n=\{x:(\forall k<n)\;k\in P_x\}$, and consider $c_n\colon x\mapsto n$.
Then, $(B_n)_{n<\omega}$ is an decreasing sequence of $\tpbf{\Pi}^1_1$ sets.
Put $S_x=\omega\setminus P_x$, and then one can see that $\min S_x=\max\{n<\omega:x\in B_n\}$ whenever $S_x\not=\emptyset$.
Therefore, ${\uparrow}\diff^\ast_{n<\omega}[c_n/B_n]$ is a realizer for $\Sigma^1_1\lnp$.
To show the completeness, assume that a sequence $(B_n,f_n)$ of pairs of $\tpbf{\Pi}^1_1$ sets and continuous functions is given.
Let us consider the inner reduction $h$ which maps $x$ to a $\delta_{\Sigma^1_1}$-name of $U_x=\{n\in\omega:x\not \in B_n\}$, and the outer reduction $k$ which maps $(x,n+1)$ to $f_n(x)$, where $k(x,0)\uparrow$.
Note that $\min U_x=m+1$ if and only if $\{n<\omega:x\in B_n\}=m$ as $(B_n)_{n<\omega}$ is decreasing.
If $\min U_x>0$, say $\min U_x=m+1$, then ${\uparrow}\diff_{n<\omega}^\ast[f_n/B_n](x)=f_m(x)=k(x,\min U_x)$.
If no such $m$ exists, ${\uparrow}\diff_{n<\omega}^\ast[f_n/B_n](x)$ is undefined.
This verifies the assertion.
\end{proof}
\begin{prop}
$\Pi^1_1\lnp\equiv_{\sf W}\#\Sigma^1_1$ and $\Sigma^1_1\lnp\equiv_{\sf W}\#\Pi^1_1$.
\end{prop}
\begin{proof}
Given $A\subseteq\omega$, define $A^\ast=\{n\in\omega:(\forall m<n)\;m\not\in A\}$.
Clearly, $\min A=\#A^\ast$.
If $A$ is $\tpbf{\Pi}^1_1$ then $A^\ast$ is $\tpbf{\Sigma}^1_1$, and moreover $A\mapsto A^\ast\colon\tpbf{\Pi}^1_1(\omega)\to\tpbf{\Sigma}^1_1(\omega)$ is computable, that is, given a $\tpbf{\Pi}^1_1$-code of $A$, one can effectively find a $\tpbf{\Sigma}^1_1$-code of $A^\ast$.
Similarly, if $A$ is $\Sigma^1_1$ then $A^\ast$ is $\Pi^1_1$, and moreover $A\mapsto A^\ast\colon\tpbf{\Sigma}^1_1(\omega)\to\tpbf{\Pi}^1_1(\omega)$ is computable.
Thus, the inner reduction $A\mapsto A^\ast$ witnesses that $\Pi^1_1\lnp\leq_{\sf W}\#\Sigma^1_1$ and $\Sigma^1_1\lnp\leq_{\sf W}\#\Pi^1_1$.
For the converse direction, assume that a ${\Sigma}^1_1$ set $A\subseteq\omega$ is given.
If $A$ is finite, then this fact is witnessed at some stage $<\om_1^{\rm ck}$ since $(\exists n)(\forall m>n)\;m\not\in A[s]$ is a $\Delta^1_1$ property, where $A[s]$ is the stage $s$ hyp-approximation of $A$.
Here, recall that $\Sigma^1_1$ is a higher analogue of ``co-c.e.,'' so $(A[s])_{s\in\omega}$ is a co-enumeration of $A$, that is, $s<t$ implies $A[s]\supseteq A[t]$.
At each stage $s$, check if $A[s]$ is finite.
If so, enumerate $\#A[s]$ into $B$.
Then, one can easily see $\min B=\#A$.
Moreover, given a $\Sigma^1_1$-code of $A$, one can easily find a $\Pi^1_1$-code of $B$.
This argument can be uniformly relativizable.
Thus, the inner reduction $A\mapsto B$ witnesses that $\#\Sigma^1_1\leq_{\sf W}\Pi^1_1\lnp$.
Assume that a $\Pi^1_1$ set $B\subseteq\omega$ is given.
If we see that the $n$th element is enumerated into $B$, i.e., $\#B[s]\geq n$, then co-enumerate $[0,n)$ from $A$.
Then, $\min A=\#B$.
Given a $\Pi^1_1$-code of $B$, one can easily find a $\Sigma^1_1$ code of $A$.
This argument can be uniformly relativizable.
The inner reduction $B\mapsto A$ witnesses that $\#\Pi^1_1\leq_{\sf W}\Sigma^1_1\lnp$.
\end{proof}
One can also consider the least number principle on a well-ordered set.
For a countable ordinal $\alpha$, let $\leq_\alpha$ be a well-order on $\mathbb{N}$ whose order type is $\alpha$.
Then, we use $\Gamma$-${\sf LNP}_\alpha$ to denote the least number principle with respect to $\leq_\alpha$; that is, $\Gamma$-${\sf LNP}_\alpha(A)$ is defined as the $\leq_\alpha$-smallest element of $A$ if it exists.
As in the above argument, one can observe that $\Pi^1_1$-${\sf LNP}_{\alpha}$ and $\Sigma^1_1$-${\sf LNP}_{\alpha}$ correspond to ${\uparrow} D_\alpha(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$ and ${\uparrow} D^\ast_\alpha(\tpbf{\Sigma}^0_1/\tpbf{\Pi}^1_1)$, respectively.
This idea leads to our solution to Question \ref{four-main-question}.
\subsection{Fournier's problem}\label{sec:Fournier-problem}
The increasing difference hierarchy can be defined by the combination of the parity function and the least number principle on countable well-orders.
Recall that the parity function ${\sf par}\colon{\sf Ord}\to 2$ returns $1$ if a given input is odd; otherwise, returns $0$.
For a countable ordinal $\eta$, let $(A_\xi)_{\xi<\eta}$ be an increasing sequence of subsets of $\omega^\omega$, and put $A_\eta=\omega^\omega$.
Then, it is not hard to check the following:
\[\diff_{\xi<\eta}A_\xi=\Big\{x\in\omega^\omega:{\sf par}\big(\min\{\alpha\leq\eta:x\in A_\alpha\}\big)\not={\sf par}(\eta)\Big\}.\]
Similarly, if $(B_\xi)_{\xi<\eta}$ is an decreasing sequence of subsets of $\omega^\omega$, then
\[
\left(\diffd_{\xi<\eta}B_\xi\right)(x)=
\begin{cases}
1&\mbox{ if }{\sf par}(\max\{\xi<\eta:x\in B_\xi\})=0,\\
0&\mbox{ if $\max\{\xi<\eta:x\in B_\xi\}$ does not exist}.
\end{cases}
\]
The {\em $\Pi^1_1$-least number principle} on a well-ordered set $(\omega,\preceq)$ states that any nonempty $\Pi^1_1$ set $P\subseteq\omega$ has the $\preceq$-smallest element.
We represent the $\Pi^1_1$-least number principle as a function as in Section \ref{sec:Weihrauch}.
Here, recall that we have a total representation $\delta_{\Pi^1_1}$ of $\tpbf{\Pi}^1_1(\mathbb{N})$.
A $\delta_{\Pi^1_1}$-name is often called a {\em $\tpbf{\Pi}^1_1$-code}.
Let us use $P_x$ to denote the subset of $\omega$ whose $\tpbf{\Pi}^1_1$-code is $x$, i.e., $P_x=\delta_{\Pi^1_1}(x)$.
For $y\in{\sf WO}$ and $A\subseteq\mathbb{N}$, we define $\min_yA$ as the $\leq_y$-least element of $A$, i.e., $a=\min_yA$ if and only if $a\in A$ and $b\not\in A$ for any $b<_ya$.
To be more precise, we define $\tpbf{\Pi}^1_1\lnpwo$ as the partial function which, given a $\tpbf{\Pi}^1_1$-code $x$ of $P\subseteq \mathbb{N}$ and a well-order $y=(\mathbb{N},\leq_y)$, returns the $\leq_y$-smallest element of $P$ whenever $P$ is nonempty, that is,
\[{\rm dom}(\tpbf{\Pi}^1_1\lnpwo)=\{(x,y):P_x\not=\emptyset\mbox{ and }y\in{\sf WO}\},\]
\[\tpbf{\Pi}^1_1\lnpwo(x,y)={\rm min}_yP_x.\]
We consider totalizations of $\tpbf{\Pi}^1_1\lnpwo$.
For each $c\in\omega$, define $c\ast\tpbf{\Pi}^1_1\lnpwo$ as follows:
\[
(c\ast\tpbf{\Pi}^1_1\lnpwo)(x,y)=
\begin{cases}
{\rm min}_{y} P_x &\mbox{ if }P_x\not=\emptyset\mbox{ and }y\in{\sf WO},\\
c&\mbox{ otherwise.}
\end{cases}
\]
Note that, contrary to Section \ref{sec:Weihrauch}, we deal with a realizer (i.e., a function on codes) rather than a function between represented spaces.
This ensures that $c\ast\tpbf{\Pi}^1_1\lnpwo$ is a total $\mathbb{N}$-valued function on $\omega^\omega$.
However, to discuss the Wadge degree, it must be restricted to a two-valued function.
To simplify our argument, we assume that $c=0$.
Then define $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$ as follows:
\[
(\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2})(x,y)=
\begin{cases}
{\sf par}_y({\rm min}_{y} P_x) &\mbox{ if }P_x\not=\emptyset\mbox{ and }y\in{\sf WO},\\
0&\mbox{ otherwise.}
\end{cases}
\]
Here, ${\sf par}_y(n)$ is the parity of the $\leq_y$-rank of $n$.
Then, $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$ is a two-valued function on $\omega^\omega$.
\begin{prop}\label{prop:lnpwo-non-approximable}
For any countable ordinal $\eta$, every $D_\eta(\tpbf{\Pi}^1_1)$ set is Wadge reducible to $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$.
\end{prop}
\begin{proof}
Assume that $\eta$ is even.
Fix a well-order $\leq_\eta$ on $\omega$ whose order type is $\eta$, and put $\bar{\eta}=(\omega,\leq_\eta)$.
Let $A=\diff_{\xi<\eta}A_\xi$ be a $D_\eta(\tpbf{\Pi}^1_1)$ set.
Then, the set $Q=\{(x,n):x\in A_{|n|_\eta}\}$ is $\tpbf{\Pi}^1_1$, where recall that $|n|_\eta$ is the $\leq_\eta$-rank of $n\in\mathbb{N}$ (see Section \ref{preliminaries}).
Thus, one can find a continuous function $\theta$ which, given $x$, returns a $\tpbf{\Pi}^1_1$-code of $Q_x=\{n\in\mathbb{N}:x\in A_{|n|_\eta}\}$.
We claim that $x\mapsto(\theta(x),\bar{\eta})$ is a Wadge reduction witnessing $A\leq_{\sf W}\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$.
Since $\eta$ is even, $x\in A$ if and only if $\min\{\xi\leq\eta:x\in A_\xi\}$ is odd if and only if $\min_\eta(\{n\in\mathbb{N}:x\in A_{|n|_\eta}\})$ is odd if and only if $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}(\theta(x),\bar{\eta})=1$.
This verify the claim.
The case where $\eta$ is odd can be proved in almost the same way.
\end{proof}
As a consequence, $c\ast\tpbf{\Pi}^1_1\lnpwo$ is not hyp-computable with finite mind-changes along any countable ordinal (since the hierarchy $(D_\eta(\tpbf{\Pi}^1_1))_{\eta<\omega_1}$ does not collapse).
On the other hand, it is intuitively clear that $c\ast\tpbf{\Pi}^1_1\lnpwo$ is hyp-computable with finite mind-changes.
Indeed, it is hyp-computable with finite mind-changes along the uncountable ordinal $\omega_1+1$.
To see this, let $(x,y)$ be an input.
Begin with the guess $c$ and ordinal counter $\omega_1<\omega_1+1$.
If $y$ is found to be ${\sf WO}$, then change the ordinal counter to the order type $|y|$ of $y$, which is smaller than $\omega_1$.
When something is first enumerated into $P_x$, we guess the current $\leq_y$-least element $n\in P_x$ as a correct answer, and change the ordinal counter to $|n|_y<|y|$.
If some number which is $\leq_y$-smaller than the previous guess is enumerated into $P_x$, then change the guess as above.
Continue this procedure.
This algorithm eventually guesses the correct output of $c\ast\tpbf{\Pi}^1_1\lnpwo(x,y)$.
Clearly, this procedure is hyp-computable with finite mind-changes along $\omega_1+1$.
Thus, we only need to formalize this argument as a $D_\omega^\ast(\tpbf{\Pi}^1_1)$ set.
\begin{prop}\label{prop:lnpwo-approximable}
$\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}\in D_\omega^\ast(\tpbf{\Pi}^1_1)$.
\end{prop}
\begin{proof}
Define $B_n$ as the set of all $(x,y)$ such that the parity (w.r.t.~the $\leq_y$-rank) of the $\leq_y$-least element of $P_x$ changes at least $n$ times under the cannonical hyp-computable guessing process.
In other words, $B_n$ is the set of all $(x,y)$ satisfying the following conditions:
\begin{align*}
y\in{\sf WO}\;\land\;(\exists s_1<\dots<s_n)\;&\big[{\sf par}_y({\rm min}_y P_x[s_1])=1\\
&\land\;(\forall i<n)\;{\sf par}_y({\rm min}_y P_x[s_i])\not={\sf par}_y({\rm min}_y P_x[s_{i+1}])\big].
\end{align*}
The standard hyperarithmetical quantification argument shows that $B_n$ is $\Pi^1_1$ since we only need to search for $x$-computable ordinals $s_i$.
To be more precise, first recall that the condition $n\in P_x$ is equivalent to ${\bf o}_P(n,x)\in{\sf WO}$.
In this case, ${\bf o}_P(n,x)$ is an $x$-computable well-order since ${\bf o}_P$ is computable.
Similarly, the condition $a=\min_y P_x[s]$ is equivalent to that $|{\bf o}_P(a,x)|<s$ and $|{\bf o}_P(b,x)|\geq s$ for any $b<_ya$.
This is a $\Delta^1_1$ condition on the $\Pi^1_1$ assumption that $s$ is an ordinal.
Putting it all together, the condition $(x,y)\in B_n$ can be written as follows:
\begin{align*}
y\in{\sf WO}\;\land\;(&\exists e_1,\dots,e_n)\;\big[(\forall i\leq n)\;\varphi_{e_i}^x\in{\sf WO}\;\land\;|\varphi_{e_1}^x|<\dots<|\varphi_{e_n}^x|\\
&\;\land\;(\exists a_1,\dots,a_n)[{\sf par}_y(a_1)=1\;\land\;(\forall i<n)\;{\sf par}_y(a_i)\not={\sf par}_y(a_{i+1})]\\
&\;\land \;(\forall i\leq n)\;[|{\bf o}_P(a_i,x)|<|\varphi_{e_i}^x|
\land\;(\forall b)\;(b<_ya_i\;\to\;|{\bf o}_P(b,x)|\geq|\varphi_{e_i}^x|)
]
\big]
\end{align*}
This only involves number quantification (with some $\Pi^1_1$ sets), so this property is $\Pi^1_1$.
It is clear that $(B_n)_{n\in\mathbb{N}}$ is decreasing.
Given $(x,y)$, let $n$ be the largest number such that $(x,y)\in B_n$ with witnesses $s_1<\dots<s_n$.
Then, we have ${\sf par}_y(\min_y P_x)={\sf par}_y(\min_y P_x[s_n])$; otherwise, we must find $s_{n+1}>s_n$ such that ${\sf par}_y(\min_y P_x[s_{n+1}])\not={\sf par}_y(\min_y P_x[s_n])$, which is impossible by the maximality of $n$.
Put $p_i={\sf par}_y(\min_y P_x[s_i])$.
Then, since $p_1=1$ and $p_i\not=p_{i+1}$, we have $p_i={\sf par}(i)$, and therefore, ${\sf par}_y(\min_y P_x)=p_n={\sf par}(n)$.
Consequently, if $n$ is the largest number such that $(x,y)\in B_n$ then ${\sf par}_y(\min_y P_x)={\sf par}(n)$.
Moreover, if there is no such an $n$ then $y\not\in{\sf WO}$ or $P_x$ is empty.
This shows that
\[
\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}(x,y)=\left(\diffd_{n<\omega}B_n\right)(x,y)=
\begin{cases}
{\sf par}(n)&\mbox{ if }n=\max\{n:(x,y)\in B_n\},\\
0&\mbox{ if there is no such an $n$}.
\end{cases}
\]
Hence, $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}\in D_\omega^\ast(\tpbf{\Pi}^1_1)$.
\end{proof}
Consequently, $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$ is contained in the $\omega$-th level of the decreasing difference hierarchy, but not in the increasing difference hierarchy.
This solves Fournier's question:
\begin{proof}[Proof of Theorem \ref{thm:solution-to-Fournier}]
By Proposition \ref{prop:lnpwo-approximable}, $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$ belongs to the $\omega$-th level of the decreasing difference hierarchy.
If $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}\in {\sf Diff}(\tpbf{\Pi}^1_1)$ would hold, then $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}\in D_\eta(\tpbf{\Pi}^1_1)$ for some $\eta<\omega_1$.
However, by Proposition \ref{prop:lnpwo-non-approximable}, every $D_{\eta+1}(\tpbf{\Pi}^1_1)$ set is Wadge reducible to $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}$.
A simple diagonalization argument shows the existence of a $D_{\eta+1}(\tpbf{\Pi}^1_1)$ set which is not Wadge reducible to a $D_\eta(\tpbf{\Pi}^1_1)$ set.
This implies a contradiction; hence, $\tpbf{\Pi}^1_1\lnpwo^{\upharpoonright 2}\not\in {\sf Diff}(\tpbf{\Pi}^1_1)$.
\end{proof}
\subsection{Beyond the decreasing difference hierarchy}\label{sec:beyond-decreasing}
The decreasing difference hierarchy over $\tpbf{\Pi}^1_1$ sets occupies a very small part of the smallest $\sigma$-algebra including all $\tpbf{\Pi}^1_1$ sets.
Let us turn our attention to the first level of the $\sigma$-algebra.
\begin{definition}[see e.g.~Becker {\cite[Page 719]{Becker88}}]
For a pointclass $\Gamma$, let $\tpbf{\Sigma}^0_1(\Gamma)$ be the smallest family including all $\Gamma$ sets and closed under countable union, finite intersection, and continuous preimage.
A set $A$ is in $\tpbf{\Delta}^0_1(\Gamma)$ if both $A$ and its complement is contained in $\tpbf{\Sigma}^0_1(\Gamma)$.
\end{definition}
Higher limit lemma \cite[Proposition 6.1]{BGM17} states that ${\Delta}^0_1(\Sigma^1_1\cup\Pi^1_1)$ is equivalent to hyp-computability with ordinal mind-changes.
Note that this result does not imply that ${\Delta}^0_1(\Sigma^1_1\cup\Pi^1_1)$ is equivalent to ${\rm Diff}^\ast(\tpbf{\Pi}^1_1)$.
This is because ${\rm Diff}^\ast({\Pi}^1_1)$ corresponds to hyp-computability with ordinal mind-changes involving some countable ordinal which bounds the number of mind-changes for all inputs, while in the case of ${\Delta}^0_1(\Sigma^1_1\cup\Pi^1_1)$, the number of mind-changes can be different for each input, and it is not always possible to give their upper bound by a single countable ordinal.
Indeed, using a similar argument as above, we show that ${\rm Diff}^\ast(\tpbf{\Pi}^1_1)$ is a proper subclass of $\tpbf{\Delta}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
The {\em $\Sigma^1_1$-least number principle} on a well-ordered set $(\omega,\preceq)$ states that any nonempty $\Sigma^1_1$ set $S\subseteq\omega$ has the $\preceq$-smallest element.
Let us use $S_x$ to denote the subset of $\omega$ whose $\tpbf{\Sigma}^1_1$-code is $x$, i.e., $S_x=\omega\setminus P_x$.
One can define the totalization $c\ast\tpbf{\Sigma}^1_1\lnpwo$ of the partial $\omega$-valued function $\tpbf{\Sigma}^1_1\lnpwo$ as above.
Then define its two-valued restriction $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$ as follows:
\[
(\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2})(x,y)=
\begin{cases}
{\sf par}_y({\rm min}_{y} S_x) &\mbox{ if }S_x\not=\emptyset\mbox{ and }y\in{\sf WO},\\
0&\mbox{ otherwise.}
\end{cases}
\]
Then, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$ is a two-valued total function on $\omega^\omega$.
\begin{prop}\label{prop:lnpwo-non-approximable2}
For any countable ordinal $\eta$, every $D_\eta^\ast(\tpbf{\Pi}^1_1)$ set is Wadge reducible to $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$.
\end{prop}
\begin{proof}
Fix a well-order $\leq_\eta$ on $\omega$ whose order type is $\eta$, and put $\bar{\eta}=(\omega,\leq_\eta)$.
Let $B=\diff_{\xi<\eta}B_\xi$ be a $D^\ast_\eta(\tpbf{\Pi}^1_1)$ set.
Then, the set $U=\{(x,n):x\not\in B_{|n|_\eta}\}$ is $\tpbf{\Sigma}^1_1$.
Thus, one can find a continuous function $\theta$ which, given $x$, returns a $\tpbf{\Sigma}^1_1$-code of $U_x=\{n\in\mathbb{N}:x\not\in B_{|n|_\eta}\}$.
We claim that $x\mapsto(\theta(x),\bar{\eta})$ is a Wadge reduction witnessing $B\leq_{\sf W}\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$.
If $\gamma=\max\{\xi\leq\eta:x\in B_\xi\}$ exists, $x\in B$ if and only if $\gamma$ is even.
In this case, the $\leq_\eta$-rank of $\min_\eta U_x$ is $\gamma+1$, which is odd, and therefore, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}(\theta(x),\bar{\eta})=1$.
If no such a $\gamma$ exists, then $x\not\in B$, and the $\leq_\eta$-rank of $\min_\eta U_x$ is a limit ordinal, which is even, and therefore, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}(\theta(x),\bar{\eta})=0$.
In either case, we have $B(x)=\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}(\theta(x),\bar{\eta})$.
\end{proof}
As a consequence, $c\ast\tpbf{\Sigma}^1_1\lnpwo$ is not hyp-computable with fixed countable ordinal mind-changes (since the hierarchy $(D^\ast_\eta(\tpbf{\Pi}^1_1))_{\eta<\omega_1}$ does not collapse).
On the other hand, it is intuitively clear that $c\ast\tpbf{\Sigma}^1_1\lnpwo$ is hyp-computable with ordinal mind-changes.
To see this, let $(x,y)$ be an input, and begin with the guess $c$.
If $y$ is found to be ${\sf WO}$, we guess the current $\leq_y$-least element $n\in S_x$ as a correct answer.
If all numbers which are $\leq_y$-smaller than or equal to the previous guess is removed from $S_x$, then change the guess as above.
Continue this procedure.
This algorithm eventually guesses the correct output of $c\ast\tpbf{\Sigma}^1_1\lnpwo(x,y)$.
Clearly, this procedure is hyp-computable with ordinal mind-changes.
Thus, we only need to formalize this argument as a $\tpbf{\Delta}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$ set.
\begin{prop}\label{prop:lnpwo-approximable2}
$\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}\in\tpbf{\Delta}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
\end{prop}
\begin{proof}
We first consider the set of stages at which the least value of $S_x$ changes later.
In other words, define $W$ as follows:
\begin{align*}
(x,y,s)\in W\iff &y,s\in{\sf WO}\;\land\exists t\leq_Tx\oplus y\;(t\in{\sf WO}\\
&\land\;|t|>|s|\;\land\;({\rm min}_yS_x[s]<{\rm min}_yS_x[t]\;\lor\;S_x[t]=\emptyset)).
\end{align*}
It is easy to see that $W$ is $\Pi^1_1$.
We claim that if $S_x\not=\emptyset$ and $y\in{\sf WO}$ then there exists an ordinal $s\leq_Tx\oplus y$ such that ${\rm min}_yS_x[s]={\rm min}_yS_x$.
To see this, assume that $a={\rm min}_yS_x$.
Then, for any $b<_ya$ there exists an $x$-computable ordinal $s_b\in{\sf WO}$ such that $b\in P_x[s_b]$.
Note that $A=\{b\in\omega:b<_ya\}$ is a $\Delta^1_1(y)$ set.
Then consider the map $b\mapsto e_b$, where $e_b$ is an $x$-computable index of such $s_b$, which is a $\Delta^1_1(x\oplus y)$ function.
The usual $\Sigma^1_1$-bounding argument (i.e., the relativized Spector boundedness theorem) ensures that $s=\sup\{s_b:b<_ya\}$ is an $(x\oplus y)$-computable ordinal.
This verifies the claim.
This claim shows that, for any $y\in{\sf WO}$ and $i<2$, the statement $S_x\not=\emptyset$ and ${\sf par}_y(\min_yS_x)=i$ holds if and only if there exists an ordinal $s\leq_Tx\oplus y$ such that $(x,y,s)\not\in W$ and ${\sf par}_y(\min_yS_x[s])=i$.
Therefore,
\begin{align*}
\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}(x,y)=1
\iff
&y\in{\sf WO}\;\land\;\exists s\leq_Tx\oplus y\\
&(s\in{\sf WO}\;\land\;(x,y,s)\not\in W\;\land\;{\sf par}_y({\rm min}_yS_x[s])=1),
\end{align*}
and similarly
\begin{align*}
\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}(x,y)=0
\iff
y&\not\in{\sf WO}\ \lor\ \forall s\leq_Tx\oplus y\;(x,y,s)\in W\\
\lor\ &[y\in{\sf WO}\;\land\;\exists s\leq_Tx\oplus y\\
&(s\in{\sf WO}\;\land\;(x,y,s)\not\in W\;\land\;{\sf par}_y({\rm min}_yS_x[s])=0)].
\end{align*}
The former formula is clearly $\tpbf{\Sigma}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
The latter formula contains a universal quantification, but the first line and the second line are separated, and the subformula ``$\forall s\leq_Tx\oplus y\;(x,y,s)\in W$'' is $\Pi^1_1$.
Hence, the latter formula is also $\tpbf{\Sigma}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
Note also that the first line in the latter formula is equivalent to the statement that either $y\not\in{\sf WO}$ or $S_x=\emptyset$ holds.
Consequently, we get $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}\in\tpbf{\Delta}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
\end{proof}
Consequently, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$ is contained in the first $\tpbf{\Delta}$-level of the $\sigma$-algebra containing $\tpbf{\Pi}^1_1$ sets, but not in the decreasing difference hierarchy.
That is, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$ witnesses the properness of the inclusion ${\sf Diff}^\ast(\tpbf{\Pi}^1_1)\subsetneq\tpbf{\Delta}^0_1(\tpbf{\Pi}^1_1\cup\tpbf{\Sigma}^1_1)$.
This solves Question \ref{question:beyond-difference}:
\begin{proof}[Proof of Theorem \ref{thm:beyond-difference}]
By Proposition \ref{prop:lnpwo-approximable2}, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$ belongs to $\tpbf{\Delta}^0_1(\tpbf{\Sigma}^1_1\cup\tpbf{\Pi}^1_1)$.
If $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}\in {\sf Diff}^\ast(\tpbf{\Pi}^1_1)$ would hold, then $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}\in D_\eta^\ast(\tpbf{\Pi}^1_1)$ for some $\eta<\omega_1$.
However, by Proposition \ref{prop:lnpwo-non-approximable2}, every $D_{\eta+1}^\ast(\tpbf{\Pi}^1_1)$ set is Wadge reducible to $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}$.
A simple diagonalization argument shows the existence of a $D_{\eta+1}^\ast(\tpbf{\Pi}^1_1)$ set which is not Wadge reducible to a $D_\eta^\ast(\tpbf{\Pi}^1_1)$ set.
This implies a contradiction; hence, $\tpbf{\Sigma}^1_1\lnpwo^{\upharpoonright 2}\not\in {\sf Diff}^\ast(\tpbf{\Pi}^1_1)$.
\end{proof}
\section{The $\omega$-th level of the decreasing difference hierarchy}\label{sec:4}
\subsection{$\omega_1$-prewellordered coproduct}
Next, we analyze the structure of $\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$ sets.
We first show the following useful characterization of $\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$ sets.
\begin{prop}
A set $P\subseteq\omega^\omega$ belongs to $\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$ if and only if there exists an infinite decreasing sequence $(P_n)_{n\in\omega}$ of $\tpbf{\Pi}^1_1$ sets such that $\bigcap_{n<\omega}P_n=\emptyset$ and $P=\diff^\ast_{n<\omega}P_n$.
\end{prop}
\begin{proof}
If $P\in\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$, then there exist infinite decreasing sequences $(A_n)_{n\in\omega}$ and $(B_n)_{n\in\omega}$ of $\tpbf{\Pi}^1_1$ sets such that $P=\diff^\ast_nA_n$ and $\neg P=\diff^\ast_nB_n$.
Note that $x\in\bigcap_n A_n$ implies $x\not\in\diff^\ast_nA_n$, so $x\in P$, and similarly, $x\in\bigcap_nB_n$ implies $x\not\in P$.
Hence, $\bigcap_nA_n\cap\bigcap_nB_n=\emptyset$.
Then, define $P_n=A_n\cap B_{n+1}$.
Then, $\bigcap_nP_n\subseteq\bigcap_n A_n\cap\bigcap_nB_n=\emptyset$.
Moreover, it is not hard to check that $P=\diff^\ast_nP_n$.
For the converse direction, let $(P_n)$ be such that $\bigcap_nP_n=\emptyset$ and $P=\diff^\ast_nP_n$.
Then, define $A_n=P_n$, $B_0=\omega^\omega$, and $B_n=P_{n+1}$.
It is easy to check that $P=\diff^\ast_nA_n$ and $\neg P=\diff^\ast_nB_n$.
\end{proof}
As we have already mentioned, the class $\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$ corresponds to hyp-computability with finite mind-changes.
As usual, the process of mind-changes can be represented by a well-founded tree.
We describe the details below.
Under {\sf AD}, recall that every nonselfdual subset of $\omega^\omega$ is Wadge equivalent to a subset of $2^\omega$; see e.g.~\cite[Lemma 1.5]{Kih19}, and any selfdual set is Wadge equivalent to the join of countably many nonselfdual set; see e.g.~\cite{AnLo12}.
Therefore, one may assume that everything is a subset of the $\sigma$-compact space $\mathcal{C}=\omega\times 2^\omega$.
Recall that ${\sf WO}\subseteq 2^{\omega\times\omega}\simeq 2^\omega$ is the set of all well-orders on $\omega$.
Let $(P_n)_{n\in\omega}$ be an infinite decreasing sequence of $\tpbf{\Pi}^1_1$ sets in $\mathcal{C}$ such that $\bigcap_{n\in\omega}P_n=\emptyset$.
Since ${\sf WO}$ is $\tpbf{\Pi}^1_1$-complete, there exists a continuous function $\theta_n$ witnessing $P_n\leq_{\sf W}{\sf WO}$.
Then, define $P_n[c,\alpha]=\theta_n^{-1}\{\alpha\}\cap(\{c\}\times 2^\omega)$ for any $c\in\omega$ and $\alpha\in{\sf WO}$.
Clearly, $P_n[c,\alpha]$ is compact, and we have $P_n=\bigcup_{c,\alpha}P_n[c,\alpha]$.
Hereafter we omit $c$ to simplify the notation.
\begin{definition}\label{def:tree-system}
Given such a sequence $P=(P_n)_{n\in\omega}$, one can define a system on a labeled ${\sf WO}$-branching well-founded tree $T_P\subseteq{\sf WO}^{<\omega}$ as follows:
To each node $\sigma$ of $T_P$ of length $n$, assign the sequence $(P_n[\alpha])_{\alpha\in{\sf WO}}$.
If the length $n$ is even, then the node is labeled by $0$; otherwise, it is labeled by $1$.
The domain on $\sigma$ is defined as $Q_\sigma:=\bigcap_{m<n}P_m[\sigma(m)]$.
We add the $\alpha$-th immediate successor of $\sigma$ whenever $Q_\sigma\cap P_n[\alpha]$ is nonempty.
In other words, define $T_P=\{\sigma\in{\sf WO}^{<\omega}:Q_\sigma\not=\emptyset\}$.
\end{definition}
Note that if $x\in Q_\sigma$ then $\sigma=\langle\theta_0(x),\theta_1(x),\dots,\theta_{|\sigma|-1}(x)\rangle$ since $x\in P_m[\sigma(m)]$ if and only if $\theta_m(x)=\sigma(m)$.
\begin{obs}
For $P=(P_n)_{n<\omega}$, if $\bigcap_{n\in\omega}P_n=\emptyset$
then $T_P$ is well-founded.
\end{obs}
\begin{proof}
If $T_P$ has an infinite path $p\in{\sf WO}^\omega$, then for any $n$, the compact set $\bigcap_{m\leq n}P_m[p(m)]$ is nonempty.
Therefore, by compactness, the whole intersection $\bigcap_{n<\omega}P_n[p(n)]\subseteq\bigcap_nP_n$ is also nonempty, which contradicts our assumption on $(P_n)_{n\in\omega}$.
\end{proof}
Note also that $T_P$ is Borel on ${\sf WO}^{<\omega}$.
One can recover the information on $\diff^\ast_nP_n$ in the following manner.
\begin{obs}\label{obs:tree-mind-change-x}
Let $P$ and $(\theta_n)$ be as above.
For $x\in\omega^\omega$, define $\sigma_x$ as the maximal initial segment of $(\theta_\ell(x))_{\ell<\omega}$ which is contained in $T_P$.
Then, $x\in\diff^\ast_{n<\omega}P_n$ if and only if $\sigma_x$ is labeled by $1$.
\end{obs}
\begin{proof}
Assume that $\sigma_x=(\theta_{\ell}(x))_{\ell<k}$.
Then, $x\in Q_{\sigma_x}\subseteq\bigcap_{m<k}P_m$.
Since this is maximal, $\sigma_x':=\sigma_x{}^{\frown}\theta_k(x)$ is not contained in $T_P$.
If $\theta_k(x)\in{\sf WO}$ then we have $x\in Q_{\sigma_x'}$, so $\sigma_x'$ must be contained in $T_P$.
Hence, $\theta_k(x)\not\in{\sf WO}$, so $x\not\in P_k$.
Therefore, $\max\{\ell:x\in P_\ell\}=k-1$.
Thus, $x\in\diff^\ast_nP_n$ if and only if ${\sf par}(k-1)=0$, so ${\sf par}(k)=1$.
This means that the length of $\sigma_x$ is odd.
In this case, $\sigma_x$ is labeled by $1$.
\end{proof}
In a more inductive manner, one can recover the information of $\diff^\ast_nP_n$.
For $\sigma\in T_P$, inductively define $f_\sigma\colon\omega^\omega\to\{0,1\}$ as follows:
If a leaf $\rho$ is labeled by $0$, define $f_\rho\colon x\mapsto 0$.
If a leaf $\rho$ is labeled by $1$, define $f_\rho\colon x\mapsto 1$.
If a node $\sigma$ is not a leaf, and is labeled by $i$, define
\[
f_\sigma(\alpha,x):=i\ast \bigsqcup_{\alpha\in{\sf WO}}f_{\sigma\alpha}(x):=
\begin{cases}
\ f_{\sigma\alpha}(x) &\mbox{ if }\alpha\in{\sf WO},\\
\ i &\mbox{ if }\alpha\not\in{\sf WO}.
\end{cases}
\]
\begin{lemma}\label{lem:from-tree-to-diff}
$x\in\diff^\ast_nP_n\iff f_{\langle\rangle}((\theta_\ell(x))_{\ell<\omega})=1$, where $\langle\rangle$ is the empty string.
\end{lemma}
\begin{proof}
Let $\sigma_x=(\theta_m(x))_{m<n}$ is a string as in Observation \ref{obs:tree-mind-change-x}.
Then $n$ be the least number such that $\theta_n(x)\not\in{\sf WO}$.
For $\sigma=\sigma_x$, by the definition of $f_\sigma$, note that
\[f_{\langle\rangle}(\theta_0(x),\theta_1(x),\theta_2(x),\dots)=f_\sigma(\theta_n(x),\theta_{n+1}(x),\dots).\]
If $\sigma$ is labeled by $i$ then $f_\sigma(\theta_n(x),y)=i$ for any $y$ since $\theta_n(x)\not\in{\sf WO}$.
Hence, $\sigma$ is labeled by $i$ if and only if $f_{\langle\rangle}((\theta_\ell(x))_{\ell<\omega})=i$.
By Observation \ref{obs:tree-mind-change-x}, $\sigma=\sigma_x$ is labeled by $1$ if and only if $x\in\diff^\ast_nP_n$.
This verifies the claim.
\end{proof}
Thus, $\diff^\ast_nP_n$ is constructed from constant functions and the ${\sf WO}$-indexed coproduct.
To formalize this idea, given a pointclass $\Gamma$, define $\Delta=\Gamma\cap\neg\Gamma$ as usual.
\begin{definition}\label{def:uniform-delta}
We say that $(A_\alpha)_{\alpha\in I}$ is a {\em uniform $\Delta$ collection} if
there are $B,C\in\Gamma$ such that for any $\alpha$ and $z$,
\[\alpha\in I\implies [z\in A_\alpha\iff (\alpha,z)\in B\iff (\alpha,z)\not\in C].\]
We say that a pointclass $\Gamma$ is {\em strictly closed under $\omega_1$-prewellordered ($\omega_1$-pwo) coproduct} if,
for any uniform $\Delta$ collection $(A_\alpha)_{\alpha\in{\sf WO}}$, we have
\[
\bigsqcup_{\alpha\in{\sf WO}}A_\alpha:=\{(\alpha,x):\alpha\in{\sf WO}\land x\in A_\alpha\}\in\Delta.
\]
\end{definition}
If we identify a set $A\subseteq\omega^\omega$ with its characteristic function $\chi_A\colon\omega^\omega\to 2$, then $\bigsqcup_\alpha A_\alpha$ and $0\ast\bigsqcup_\alpha A_\alpha$ are the same.
One can also see that if $\Gamma$ is strictly closed under $\omega_1$-pwo coproduct, then we have $1\ast\bigsqcup_{\alpha\in{\sf WO}}A_\alpha\in\Delta$.
To see this, first note that $A\in\Delta$ implies $\neg A\in\Delta$.
Similarly, if $(A_\alpha)_{\alpha\in{\sf WO}}$ is uniformly $\Delta$, so is $(\neg A_\alpha)_{\alpha\in{\sf WO}}$.
Thus,
\[1\ast\bigsqcup_{\alpha\in{\sf WO}}A_\alpha:=\{(\alpha,x):\alpha\not\in{\sf WO}\lor x\in A_\alpha\}=\neg\bigsqcup_{\alpha\in{\sf WO}}(\neg A_\alpha)\in\Delta.\]
\begin{obs}\label{obs:closed-pwo-coprod}
$D_\omega^\ast(\tpbf{\Pi}^1_1)$ is strictly closed under $\omega_1$-pwo coproduct.
\end{obs}
\begin{proof}
The algorithmic reason for this can be explained as follows:
Given an input $(\alpha,x)$, we have $\alpha\not\in{\sf WO}$ at the first stage, so the learner guesses that $(\alpha,x)\in\bigsqcup_{\alpha\in{\sf WO}}A_\alpha$ is false.
If the learner sees $\alpha\in{\sf WO}$ at some stage, change her mind, and then since $A_\alpha\in\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$, the learner only needs to simulate a guessing process to answer whether $x\in A_\alpha$ or not with finite mind-changes.
The set-theoretic reason for this is as follows:
Let a pair $(B,C)$ be a $\Delta$-definition of $(A_\alpha)_{\alpha\in{\sf WO}}$ as in Definition \ref{def:uniform-delta}.
It is easy to see that $\bigsqcup_{\alpha\in{\sf WO}}A_\alpha$ and its complement can be written as $\pi_0^{-1}[{\sf WO}]\cap B$ and $\pi_0^{-1}[\neg{\sf WO}]\cup C$, respectively.
Since $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is closed under finite union with $\tpbf{\Sigma}^1_1$ sets and finite intersection with $\tpbf{\Pi}^1_1$ sets, both sets belong to $D_\omega^\ast(\tpbf{\Pi}^1_1)$.
\end{proof}
A key basic fact on the closure property for $\Delta$ under ${\sf AD}$ is that, if $\Delta$ is closed under something, then it is closed {\em uniformly}, as shown by Becker \cite{Bec84}.
As a special case, we have the following:
\begin{fact}[Becker \cite{Bec84}, {\sf AD}]\label{lem:Becker-uniform}
If $\Gamma$ is strictly closed under $\omega_1$-pwo coproduct, then there exists a continuous function which, given a uniform $\Delta$-code of $(A_\alpha)_{\alpha\in{\sf WO}}$, returns a $\Delta$-code of $\bigsqcup_{\alpha\in{\sf WO}}A$.
\end{fact}
The Wedge reducibillity is too fine-grained to handle this level of pointclasses, and for this reason we first deal with a coarser reducibility.
For $A,B\subseteq\omega^\omega$, we say that $A$ is {\em Borel-Wadge reducible to $B$} ({\em written $A\leq_{\sf BW}B$}) if there exists a Borel function $\theta\colon\omega^\omega\to\omega^\omega$ such that, for any $x\in\omega^\omega$, $x\in A$ if and only if $\theta(x)\in B$.
The Borel-Wadge degrees are semi-well-ordered, and therefore, one can assign a Borel-Wadge rank $|A|_{\sf BW}$ to each set $A\subseteq\omega^\omega$.
A {\em Borel-Wadge pointclass} is a class of subsets of $\omega^\omega$ downward closed under Borel-Wadge reducibility, i.e., $A\in\Gamma$ and $B\leq_{\sf BW}A$ implies $B\in\Gamma$.
For basic information on Borel-Wadge reducibility, see Andretta-Martin \cite{AnMa03}.
Now we give a key result connecting the class $D_\omega^\ast(\tpbf{\Pi}^1_1)$ and the $\omega_1$-pwo coproduct.
\begin{prop}[{\sf AD}]\label{prop:omf-closed-coproduct}
$D_\omega^\ast(\tpbf{\Pi}^1_1)$ is a minimal nonselfdual Borel-Wadge pointclass which is strictly closed under $\omega_1$-pwo coproduct.
\end{prop}
\begin{proof}
By Observation \ref{obs:closed-pwo-coprod}, $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is strictly closed under $\omega_1$-pwo coproduct.
Thus, we only need to show the minimality.
Assume that $\Gamma$ is strictly closed under $\omega_1$-pwo coproduct.
It suffices to show that $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))\subseteq\Delta$.
As in Definition \ref{def:tree-system}, any $\diff^\ast_{n<\omega}P_n\in D_\omega^\ast(\tpbf{\Pi}^1_1)$ can be represented as a system on a labeled ${\sf WO}$-branching tree $T_P$, where $P=(P_n)_{n<\omega}$.
Then, assign a function $f_\sigma\colon\omega^\omega\to 2$ to each node $\sigma\in T_P$ as above, and define $Z_\sigma=f_\sigma^{-1}\{1\}$.
To be precise, if $\rho$ is a leaf then $Z_\rho$ is either $\emptyset$ or $\omega^\omega$ depending on the label of $\rho$, and if $\sigma\in T_P$ is not a leaf then $Z_\sigma=i\ast\bigsqcup_{\alpha\in{\sf WO}}Z_{\sigma\alpha}$, where $i$ is the label of $\sigma$.
\begin{claim}
$Z_{\sigma}\in\Delta$ for any $\sigma\in T_P$.
\end{claim}
\begin{proof}
By Fact \ref{lem:Becker-uniform}, there exists a continuous function which, given a uniform $\Delta$-code of $(A_\alpha)_{\alpha\in{\sf WO}}$, returns a $\Delta$-code of $i\ast\bigsqcup_{\alpha\in{\sf WO}}A_\alpha$.
We define a partial function $h\colon{\sf WO}^{<\omega}\to\omega^\omega$ such that $h(\sigma)$ is a $\Delta$-code of $Z_\sigma$.
The recursion theorem allows us to use a self-referential definition such as ``let $h(\sigma)$ be a $\Delta$-code of the $\omega_1$-pwo coproduct of the $\Delta$-sets $(Z_{\sigma\alpha})_{\alpha\in{\sf WO}}$ coded by $(h(\sigma\alpha))_{\alpha\in{\sf WO}}$.''
To discuss the complexity of $h$, we give the details of the above argument:
Given $\sigma\in{\sf WO}^{<\omega}$, first check whether $\sigma$ extends a leaf of $T_P$ or not.
This is a Borel property, so it is doable by a $\tpbf{\Pi}^1_1$-measurable way, and the recursion theorem holds for $\tpbf{\Pi}^1_1$, cf.~Moschovakis \cite[Theorem 7A.2]{mos07}.
If $\sigma$ extends a leaf $\rho$, then $h(\sigma)$ is a $\Delta$-code of either $\emptyset$ or $\omega^\omega$, depending on the length of the leaf $\rho$.
If $\sigma$ does not extend a leaf, calculate a $\tpbf{\Pi}^1_1$-code of $\alpha\mapsto h(\sigma\alpha)$.
Then, by applying Lemma \ref{lem:Becker-uniform} to this code, we hope to obtain the $\Delta$-code $c$ of of the $\omega_1$-pwo coproduct of the $\Delta$ sets coded by $(h(\sigma\alpha))_{\alpha\in{\sf WO}}$, and define $h(\sigma)=c$.
However, the problem is that since $h$ is $\tpbf{\Pi}^1_1$-measurable, it is not immediately guaranteed that $(Z_{\sigma\alpha})_{\alpha\in{\sf WO}}$ is a uniform $\Delta$ collection.
In order to overcome this difficulty, let us notice that $Q_\sigma:=\bigcap_{n<|\sigma|}P_n[\sigma(n)]$ is compact uniformly in $\sigma\in{\sf WO}^{<\omega}$ (even in $\sigma\in(\omega^\omega)^{<\omega}$).
In other words, we have a continuous function which, given $\sigma$, returns a $\tpbf{\Pi}^0_1$-code of $Q_\sigma$.
Hence, one can decide whether $\sigma$ extends a leaf by a partial stable Baire-one function $\psi$, where a function $f$ is stable Baire-one if there exists a partial continuous function $\tilde{f}$ such that for any $x\in{\rm dom}(f)$ we have $f(x)=\tilde{f}(n,x)$ for all but finitely many $n$.
In particular, such an $f$ is Baire-one, and therefore, the domain of $f$ can be extended to a Borel set.
The recursion theorem for partial stable Baire-one functions follows from the classical recursion theorem applied to the partial continuous function $\tilde{f}$.
Now, the definition of $h$ is given as follows:
If $\psi(\sigma)=1$ (i.e., $\sigma$ extends a leaf $\rho$), then $h(\sigma)$ is a code of $Z_\sigma$, which is either $\emptyset$ or $\omega^\omega$, depending on the length of the leaf $\rho$.
Otherwise, if $\sigma\alpha\in{\rm dom}(h)$ and $h(\sigma\alpha)$ is a $\Delta$-code of $Z_{\sigma\alpha}$, for a fixed $\Gamma$-universal set $G$, we have
\[x\in Z_{\sigma\alpha}\iff(\pi_0h(\sigma\alpha),x)\not\in G\iff(\pi_1h(\sigma\alpha),x)\in G.\]
Since $\Gamma$ is a Borel-Wadge pointclass, we have
\[G^i_\sigma:=\{(\alpha,x):\sigma\alpha\in{\rm dom}(h)\; \&\;(\pi_ih(\sigma\alpha),x)\in G\}\in\Gamma.\]
Moreover, a $\Gamma$-code $c^i_\sigma$ of $G^i_\sigma$ can be uniformly obtained from $\sigma$ and a code of $h$.
This ensures that, whenever $h(\sigma\alpha)$ is defined for all $\alpha\in{\sf WO}$, the collection $(Z_{\sigma\alpha})_{\alpha\in{\sf WO}}$ is uniformly $\Delta$, whose code is given by $(c^0_\sigma,c^1_\sigma)$.
Then, let $\tau_0$ be a partial continuous function obtained by Fact \ref{lem:Becker-uniform} and $\tau_1$ be its dual.
In particular, $\tau_i(c^0_\sigma,c^1_\sigma)$ is a code of $i\ast\bigsqcup_{\alpha\in{\sf WO}}Z_{\sigma\alpha}$.
Then, $h(\sigma)$ is defined as $\tau_i(c^0_\sigma,c^1_\sigma)$, where $i$ is the label of $\sigma$.
If $h(\sigma\alpha)$ is defined as a $\Delta$-code for all $\alpha\in{\sf WO}$, then $h(\sigma)$ is also defined, and gives a code of $Z_\sigma=i\ast\bigsqcup_{\alpha\in{\sf WO}}Z_{\sigma\alpha}$.
The recursion theorem ensures that $h$ is well-defined, and by transfinite recursion, we conclude that $h(\sigma)$ is a $\Delta$-code of $Z_{\sigma}$ for any $\sigma\in T_P$.
\end{proof}
It remains to show that $\diff^\ast_nP_n\in\Delta$.
By Lemma \ref{lem:from-tree-to-diff}, given $x$, we have
\[x\in \diffd_{n<\omega}P_n\iff (\theta_0(x),\theta_1(x),\theta_2(x),\dots)\in Z_{\langle\rangle}.\]
Consequently, $\diff^\ast_nP_n\leq_{\sf W}Z_{\langle\rangle}$ via $(\theta_0,\theta_1,\theta_2,\dots)$, and thus $\diff^\ast_nP_n\in\Delta$ by the above claim.
\end{proof}
\subsection{Lower bound}
A lower bound of the Wadge rank of $\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$ can be given by an argument explained in Steel \cite[Theorem 1.2]{St81}; see also Fournier \cite[Proposition 5.10]{FoPhD}.
\begin{lemma}[{\sf AD}]\label{lem:steel-fournier-game}
Assume that $\Gamma$ is strictly closed under $\omega_1$-pwo coproduct.
Then, the cofinality of the Wadge rank of $\Gamma$ is at least $\omega_2$.
\end{lemma}
\begin{proof}
Let $\psi\colon\omega_1\to\Delta$ be any function.
Consider the following Solovay game:
Player I chooses a large countable ordinal $\alpha$ and Player II chooses a $\Delta$ set whose Wadge rank is greater than $\psi(\alpha)$.
More precisely, Player I chooses $\alpha\in\omega^\omega$ and then Player II chooses $\Gamma$-codes of sets $D$ and $E$.
Player II wins if, whenever $\alpha\not\in{\sf WO}$, $D=\neg E$ and the Wadge rank of $D$ is greater than or equal to $\psi(|\alpha|)$.
Player I does not have a winning strategy $\tau$.
Otherwise, by $\Sigma^1_1$-bounding, there is an upper bound $\xi$ of ordinals in the image of $\tau$.
Then, $(\psi(\alpha))_{\alpha<\xi}$ gives countably many $\Delta$ sets, and by the closure property of $\Delta$, one can easily obtain a $\Delta$ set whose Wadge rank is greater than or equal to $\sup_{\alpha<\xi}\psi(\alpha)$.
Hence, Player II wins.
By the axiom of determinacy ${\sf AD}$, Player II has a winning strategy $\tau$.
Let $G$ be a universal $\Gamma$ set.
Then, define
\[A=\{(\alpha,x)\in\omega^\omega:x\in G_{\pi_0\tau(\alpha)}\}.\]
In other words, $A_\alpha=G_{\pi_0\tau(\alpha)}=\neg G_{\pi_1\tau(\alpha)}$.
Hence, $(A_\alpha)_{\alpha\in{\sf WO}}$ is uniformly $\Delta$.
By the closure property, $\bigsqcup_{\alpha\in{\sf WO}}A_\alpha\in\Delta$, whose Wadge rank is greater than or equal to $(\psi(\alpha))_{\alpha<\omega_1}$.
Hence, $\psi$ cannot be a cofinal sequence.
\end{proof}
Under ${\sf AD}$, it is known that ${\sf cf}(\omega_n)=\omega_2$ whenever $2\leq n<\omega$; see \cite[Corollary 28.8]{KanBook}.
\subsection{Upper bound}
By Proposition \ref{prop:omf-closed-coproduct}, $D^\ast_\omega(\tpbf{\Pi}^1_1)$ is the minimal Wadge pointclass which is strictly closed under $\omega_1$-pwo coproduct.
Therefore, for any $A\in\Delta(D^\ast_\omega(\tpbf{\Pi}^1_1))$, the pointclass $\Gamma_A=\{B\subseteq\omega^\omega:B\leq_{\sf W}A\}$ is not strictly closed under $\omega_1$-pwo coproduct.
In this section, we analyze the Wadge rank of such a pointclass.
Let $(A_\alpha)_{\alpha\in{\sf WO}}$ be a uniform $\Delta$ collection.
Then, for any $\xi<\omega_1$, put $A_{<\xi}:=\bigsqcup_{|\alpha|<\xi}A_\alpha$, where $|\alpha|$ is the order type of $\alpha$ if $\alpha$ is well-ordered.
We say that $\Gamma$ is {\em strictly closed under $(<\omega_1)$-coproduct} if, for any uniformly $\Delta$ collection $(A_\alpha)_{\alpha\in{\sf WO}}$ and any $\xi<\omega_1$, we have $A_{<\xi}\in\Delta$.
\begin{lemma}\label{lem:cofinal-omega-one}
Assume that $\Gamma$ is strictly closed under $(<\omega_1)$-coproduct, but not strictly closed under $\omega_1$-pwo coproduct, witnessed by $(A_\alpha)_{\alpha\in{\sf WO}}$.
Then, $(A_{<\xi})_{\xi<\omega_1}$ is a cofinal sequence in the Borel-Wadge degrees of $\Delta$ sets.
\end{lemma}
\begin{proof}
Put $A=\bigsqcup_{\alpha\in{\sf WO}}A_\alpha$.
Then, $A\not\in\Delta$ by our assumption.
If $B\in\Delta$, by Wadge's lemma, we have $B\leq_{\sf W}A$ via some $\theta$ and $B\leq_{\sf W}\neg A$ via some $\eta$.
Let $|x|_\theta$ be the rank of the $1$st corrdinate of $\theta(x)$.
In other words, $|x|_\theta=\alpha$ if and only if $\theta(x)\in A_\alpha$.
Define $|x|_\eta$ in the similar manner.
Then, since $(\neg{\sf WO})\times\omega^\omega\subseteq\neg A$ and $A\subseteq {\sf WO}\times\omega^\omega$, we have
\[|x|_\eta=\infty\implies x\in B\implies|x|_\theta<\infty.\]
Thus, there exists no $x$ such that both ``$|x|_\theta=\infty$'' and ``$|x|_\eta=\infty$'' hold.
Moreover, these properties are $\tpbf{\Sigma}^1_1$.
Hence, the properties ``$|x|_\theta=\infty$'' and ``$|x|_\eta=\infty$'' determine a disjoint pair of $\tpbf{\Sigma}^1_1$ sets.
Therefore, by Lusin's separation theorem, there exists a Borel set $C$ such that
\[|x|_\eta=\infty\implies x\in C\implies |x|_\theta<\infty.\]
In particular, $x\in C$ implies $|x|_\theta<\infty$ and $x\not\in C$ implies $|x|_\eta<\infty$.
Since $C$ is Borel, and $\theta$ and $\eta$ are continuous, by $\tpbf{\Sigma}^1_1$-boundedness, there exists $\xi<\omega_1$ such that, for any $x\in\omega^\omega$, $x\in C$ implies $|x|_\theta<\xi$ (i.e., $\theta(x)\in A_{<\xi}$), and $x\not\in C$ implies $|x|_\eta<\xi$ (i.e., $\eta(x)\in A_{<\xi}$).
Now, we define a Borel reduction $\gamma$ as follows:
\[
\gamma(x)=
\begin{cases}
(0,\theta(x))&\mbox{ if }x\in C,\\
(1,\eta(x))&\mbox{ if }x\not\in C.
\end{cases}
\]
Then, we claim that $B$ is Borel-Wadge reducible to $A_{<\xi}\sqcup \neg A_{<\xi}$ via $\gamma$, where $A_{<\xi}\sqcup \neg A_{<\xi}=(\{0\}\times A_{<\xi})\cup(\{1\}\times \neg A_{<\xi})$.
Since $\theta$ witnesses $B\leq_{\sf W}A$, $x\in B$ if and only if $\theta(x)\in A$.
Hence, if $x\in C$ then $x\in B$ if and only if $\gamma(x)=(0,\theta(x))\in\{0\}\times A$, and the latter is equivalent to $\gamma(x)\in\{0\}\times A_{<\xi}$ as we must have $|x|_\theta<\xi$.
Similarly, since $\eta$ witnesses $B\leq_{\sf W}\neg A$, $x\in B$ if and only if $\eta(x)\not\in A$.
Hence, if $x\not\in C$ then $x\in B$ if and only if $\gamma(x)=(1,\eta(x))\in\{1\}\times \neg A$, and the latter is equivalent to $\gamma(x)\in\{1\}\times \neg A_{<\xi}$ as we must have $|x|_\eta<\xi$.
This verifies the claim.
\end{proof}
By combining Lemma \ref{lem:cofinal-omega-one} and Proposition \ref{prop:omf-closed-coproduct}, the desired upper bound can be almost obtained: The Borel-Wadge rank of $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is at most $\omega_2$.
\subsection{Inside Borel-Wadge degrees}
Unfortunately, Lemma \ref{lem:cofinal-omega-one} only gives a result on Borel-Wadge degrees.
To prove Theorem \ref{thm:main-theorem}, this result has to be transformed into a result for Wadge degrees.
\begin{prop}[{\sf AD}]\label{lem:inside-Borel-Wadge-degrees}
The Wadge rank of $A$ is $\omega_2$ if and only if its Borel-Wadge rank is $\omega_2$.
\end{prop}
\begin{proof}
Clearly, the Wadge rank of $A$ is greater than or equal to its Borel-Wadge rank.
For the other direction, we claim that if the Wadge rank of $A$ has the cofinality at least $\omega_2$, so is its Borel-Wadge rank.
This claim implies that if the Wadge rank of $A$ is $\omega_2$ then its Borel-Wadge rank has to be at least $\omega_2$, so it concludes the proof.
Assume that the cofinality of the Borel-Wadge rank of $A$ is at most $\omega_1$.
Then, there exists a sequence $(A_\xi)_{\xi<\omega_1}$ such that $A_\xi<_{\sf BW}A$ for any $\xi<\omega_1$, and for any $B<_{\sf BW}A$ we have $B\leq_{\sf BW}A_\xi$ for some $\xi<\omega_1$.
Now, fix a total $\tpbf{\Sigma}^0_{\alpha+1}$-measurable function $\lambda_\alpha\colon\omega^\omega\to\omega^\omega$ such that for any $\tpbf{\Sigma}^0_\alpha$-measurable function $\theta\colon\omega^\omega\to\omega^\omega$ we have $\theta=\lambda_\alpha\circ\eta$ for some continuous function $\eta\colon\omega^\omega\to\omega^\omega$.
One can easily construct such a $\lambda_\alpha$; for instance, if $G\subseteq\omega^\omega\times\omega^2$ is a universal $\tpbf{\Sigma}^0_\alpha$ set, then define $\lambda_\alpha(z)(n)=m$ if $m$ is the least number such that $(z,n,m)\in G$; if such an $m$ does not exist, put $\lambda_\alpha(z)(n)=0$.
Note that $B\leq_{\sf BW}C$ if and only if there exists $\alpha<\omega_1$ such that $B=\theta^{-1}[C]$ for some $\tpbf{\Sigma}^0_\alpha$-measurable function $\theta$.
The last condition is equivalent to that $B=\eta^{-1}[\lambda_\alpha^{-1}[C]]$ for some continuous function $\eta$.
This means that $B\leq_{\sf W}\lambda_\alpha^{-1}[C]$.
Hence, $B\leq_{\sf BW}C$ if and only if $B\leq_{\sf W}\lambda_\alpha^{-1}[C]$ for some $\alpha<\omega_1$.
Put $A^\alpha_\xi=\lambda_\alpha^{-1}[A_\xi]$, and consider the sequence $(A^\alpha_\xi)_{\xi,\alpha<\omega_1}$.
Note that we have $A^\alpha_\xi\leq_{\sf W}A$; otherwise, $\neg A\leq_{\sf W}A^\alpha_\xi$ since $\leq_{\sf W}$ is semi-well-ordered under {\sf AD}, and this implies $\neg A\leq_{\sf BW}A_\xi$ by the above characterization of Borel-Wadge reducibility.
Then, however, we have $\neg A\leq_{\sf BW}A_\xi<_{\sf BW}A$, which is impossible (as $\neg A\leq_{\sf BW}A$ implies $\neg A\equiv_{\sf BW}A$).
Hence, $A_\xi^\alpha\leq_{\sf W}A$ for any $\xi,\alpha<\omega_1$.
Indeed, $A_\xi^\alpha<_{\sf W}A$ since $A\not\leq_{\sf BW}A_\xi$.
As $(A_\xi)_{\xi<\omega_1}$ is cofinal below the Borel-Wadge degree of $A$, for any $B<_{\sf W}A$ there is $\xi<\omega_1$ such that $B\leq_{\sf BW}A_\xi$, which means that $B\leq_{\sf W}A_\xi^\alpha$ for some $\alpha<\omega_1$.
Hence, $(A^\alpha_\xi)_{\xi,\alpha<\omega_1}$ is cofinal below the Wadge degree of $A$.
Consequently, the cofinality of the Wadge rank of $A$ is at most $\omega_1$.
\end{proof}
For a set $A\subseteq\omega^\omega$, recall that the pointclass $\Gamma_A$ is defined as $\{B\subseteq\omega^\omega:B\leq_{\sf W}A\}$.
\begin{lemma}[{\sf AD}]\label{lem:closed-under-Borel-coproduct}
If the Wadge rank of $A$ is $\omega_2$, then $\Gamma_A$ is strictly closed under $(<\omega_1)$-coproduct.
\end{lemma}
\begin{proof}
By Proposition \ref{lem:inside-Borel-Wadge-degrees}, if $|A|_{\sf W}=\omega_2$ then $|A|_{\sf BW}=\omega_2$.
In particular, $|A|_{\sf BW}$ has an uncountable cofinality.
Therefore, by Andretta-Martin \cite[Corollary 17 (a)]{AnMa03}, $A$ is Borel non-self-dual, i.e., $[A]_{\sf BW}\not=[\neg A]_{\sf BW}$.
Then, by \cite[Proposition 20]{AnMa03}, we have $[A]_{\sf W}=[A]_{\sf BW}$.
Let $(A_\alpha)_{\alpha\in{\sf WO}}$ be a uniformly $\Delta_A$ collection, where $\Delta_A=\Gamma_A\cap\neg\Gamma_A$.
Then, there exist $B,C\in\Gamma_A$ such that, whenever $\alpha\in{\sf WO}$, $x\in A_\alpha$ iff $(\alpha,x)\in B$ iff $(\alpha,x)\not\in C$.
We claim that, for any $\xi<\omega_1$, $A_{<\xi}$ is Borel-Wadge reducible to $B$ and $\neg C$.
To see this, first note that ${\sf WO}_{<\xi}=\{\alpha\in{\sf WO}:|\alpha|<\xi\}$ is Borel for any $\xi<\omega_1$.
Then, consider the reduction $\theta_B$ defined by $\theta_B(\alpha,x)=(\alpha,x)$ if $x\in{\sf WO}_{<\xi}$, and $\theta_B(\alpha,x)=z$ if $x\not\in{\sf WO}_{<\xi}$, where $z$ is an arbitrary element of $\omega^\omega$ which is not contained in $B$.
Then, $\theta_B$ witnesses that $A_{<\xi}\leq_{\sf BW}B$.
Similarly, one can construct a reduction $\theta_C$ witnessing $A_{<\xi}\leq_{\sf BW}\neg C$.
Since $B,C\leq_{\sf W}A$, we have $A_{<\xi}\leq_{\sf BW}A$ and $A_{<\xi}\leq_{\sf BW}\neg A$.
As discussed above, we have $[A]_{\sf W}=[A]_{\sf BW}\not=[\neg A]_{\sf BW}=[\neg A]_{\sf W}$.
Combining all of these, we obtain that $A_{<\xi}\leq_{\sf W}A$ and $A_{<\xi}\leq_{\sf W}\neg A$.
Therefore, $A_{<\xi}\in\Delta_A$.
This means that $\Gamma_A$ is strictly closed under $(<\omega_1)$-coproduct.
\end{proof}
Indeed, the above proof shows that if the Borel Wadge rank of $A$ has an uncountable cofinality, then $\Gamma_A$ is strictly closed under $(<\omega_1)$-coproduct.
Now, we give an alternative proof of the Kechris-Martin theorem saying that the Wadge rank of $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is $\omega_2$.
\begin{proof}[Proof of Theorem \ref{thm:main-theorem}]
By Proposition \ref{prop:omf-closed-coproduct}, $D_\omega^\ast(\tpbf{\Pi}^1_1)$ is strictly closed under $\omega_1$-pwo coproduct.
Then, by Lemma \ref{lem:steel-fournier-game}, the order type of the Wadge degrees of $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ sets is at least $\omega_2$.
If it is greater than $\omega_2$, then there exists a $\Delta(D_\omega^\ast(\tpbf{\Pi}^1_1))$ set $A\subseteq\omega^\omega$ whose Wadge rank is exactly $\omega_2$.
By Proposition \ref{lem:inside-Borel-Wadge-degrees}, the Borel Wadge rank of $A$ is also $\omega_2$.
The minimality of $D_\omega^\ast(\tpbf{\Pi}^1_1)$ ensured by Proposition \ref{prop:omf-closed-coproduct} implies that $\Gamma_A$ is not strictly closed under $\omega_1$-pwo coproduct.
Moreover, by Lemma \ref{lem:closed-under-Borel-coproduct}, $\Gamma_A$ is strictly closed under $(<\omega_1)$-coproduct.
Therefore, by Lemma \ref{lem:cofinal-omega-one}, there exists a cofinal sequence $(A_{<\xi})_{\xi<\omega_1}$ of length at most $\omega_1$ in the Borel-Wadge degrees of $\Delta_A$ sets.
This implies that the cofinality of $|A|_{\sf BW}$ is at most $\omega_1$.
However, since $|A|_{\sf BW}=\omega_2$, it contradicts the fact that ${\sf cf}(\omega_2)=\omega_2$.
\end{proof}
\section{Beyond $\omega_2$}
\subsection{$\Pi^1_1$-process with infinite mind-changes}
The relationships among key pointclasses mentioned in Sections \ref{sec:3} and \ref{sec:4} are summarized as in Figure \ref{fig:key-principles}.
\begin{figure}
\caption{Key principles}
\label{fig:key-principles}
\end{figure}
We now move to the $(\omega+1)$-st level, $(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$, of the decreasing difference hierarchy.
That is, we consider the following $\omega+1$ sequence $(P_\alpha)_{\alpha<\omega+1}$ of $\tpbf{\Pi}^1_1$ sets:
\[P_0\supseteq P_1\supseteq P_2\supseteq\dots\supseteq\bigcap_{n<\omega}P_n\supseteq P_\omega.\]
In this section, we deal with the following question:
\begin{question}
Calculate the Wadge rank of $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$.
\end{question}
To tackle this problem, we first show, perhaps somewhat surprisingly, that any infinite level of the decreasing difference hierarchy is strictly closed under $\omega_1$-pwo coproduct even if it is a successor level.
\begin{prop}\label{prop:inf-dd-pwo}
For any infinite ordinal $\eta\geq\omega$, $D^\ast_\eta(\tpbf{\Pi}^1_1)$ is strictly closed under $\omega_1$-pwo coproduct.
\end{prop}
\begin{proof}
Abbreviate $\Delta(D^\ast_\eta(\tpbf{\Pi}^1_1))$ as $\Delta$.
Let $(A_\alpha)_{\alpha\in{\sf WO}}$ be a uniform $\Delta$ collection.
Then, there exists a sequence $(P_\xi,\check{P}_\xi)_{\xi<\eta}$ of $\tpbf{\Pi}^1_1$ sets such that $A_\alpha=\diff^\ast_{\xi<\eta} P_\xi^{[\alpha]}=\neg\diff^\ast_{\xi_\eta}\check{P}_\xi^{[\alpha]}$ for any $\alpha\in{\sf WO}$, where $S^{[\alpha]}$ is the $\alpha$th section of $S$.
Then put $Q_\xi=\pi_0^{-1}[{\sf WO}]\cap P_\xi$.
Moreover, put $\check{Q}_0=\omega^\omega$, $\check{Q}_1=\pi_0^{-1}[{\sf WO}]$, and $\check{Q}_{2+\xi}=\pi_0^{-1}[{\sf WO}]\cap \check{P}_\xi$.
Note that $\xi<\eta$ implies $2+\xi<\eta$ since $\eta$ is infinite.
Moreover, $Q_\xi$ and $\check{Q}_\xi$ are $\tpbf{\Pi}^1_1$.
We claim that $\bigsqcup_{\alpha\in{\sf WO}}A_\alpha=\diff^\ast_{\xi<\eta}Q_\xi=\neg\diff^\ast_{\xi<\eta}\check{Q}_\xi$.
Given $(\alpha,x)$, if $\alpha\in{\sf WO}$ then one can easily see that $\max\{\xi:(\alpha,x)\in Q_\xi\}=\max\{\xi:x\in P^{[\alpha]}_\xi\}$ if it exists.
Therefore, $(\alpha,x)\in\diff^\ast_{\xi<\eta}Q_\xi$ if and only if $x\in \diff^\ast_{\xi<\eta}P^{[\alpha]}_\xi=A_\alpha$.
Moreover, if $\alpha\not\in{\sf WO}$ then $(\alpha,x)\not\in Q_0$, and therefore $(\alpha,x)\not\in\diff^\ast_{\xi<\eta}Q_\xi$.
Hence, $\bigsqcup_{\alpha\in{\sf WO}}A_\alpha=\diff^\ast_{\xi<\eta}Q_\xi$.
Again, given $(\alpha,x)$, if $\alpha\in{\sf WO}$ and $\{\xi:x\in \check{P}^{[\alpha]}_\xi\}=\emptyset$, then $x\not\in\diff^\ast_{\xi<\eta}\check{P}^{[\alpha]}_\xi=\neg A_\alpha$, and moreover $(\alpha,x)\in \check{Q}_1\setminus\check{Q}_2$; hence $(\alpha,x)\not\in\diff^\ast_{\xi<\eta}\check{Q}_\xi$.
If $\alpha\in{\sf WO}$ and $\{\xi:x\in \check{P}^{[\alpha]}_\xi\}\not=\emptyset$ then one can easily see that $\max\{\xi:(\alpha,x)\in \check{Q}_\xi\}=2+\max\{\xi:x\in \check{P}^{[\alpha]}_\xi\}$ if it exists.
In particular, both values have the same parity, and therefore, $(\alpha,x)\in\diff^\ast_{\xi<\eta}\check{Q}_\xi$ if and only if $x\in \diff^\ast_{\xi<\eta}\check{P}^{[\alpha]}_\xi=\neg A_\alpha$.
If $\alpha\not\in{\sf WO}$ then $(\alpha,x)\in\check{Q}_0\setminus\check{Q}_1$, and therefore $(\alpha,x)\in\diff^\ast_{\xi<\eta}\check{Q}_\xi$.
Hence, $\bigsqcup_{\alpha\in{\sf WO}}A_\alpha=\neg\diff^\ast_{\xi<\eta}\check{Q}_\xi$.
\end{proof}
As a consequence of Proposition \ref{prop:inf-dd-pwo}, combined with Lemma \ref{lem:steel-fournier-game}, one can see that the Wadge rank of $\Delta(D^\ast_{\omega+\eta}(\tpbf{\Pi}^1_1))$ is at least $\omega_2\cdot(1+\eta)$ for each $\eta<\omega_1$.
As a special case, we conclude that the Wadge rank of $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$ is at least $\omega_2\cdot 2$.
In fact, however, one can observe that the Wadge rank of $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$ is not such a small value.
For instance, one can obtain the following lower bound:
\begin{theorem}\label{thm:Wadge-rank-om1}
The Wadge rank of $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$ is greater than $\omega_2\cdot\omega_1$.
\end{theorem}
We will now prepare a proof of this theorem.
Let $(P_\alpha)_{\alpha<\omega+1}$ be a decreasing $\omega+1$ sequence of $\tpbf{\Pi}^1_1$ sets.
If moreover we have a $\tpbf{\Pi}^1_1$ set $\check{P}_\omega$ such that $\bigcap_{n<\omega}P_n=P_\omega\cup\check{P}_\omega$ and $P_\omega\cap\check{P}_\omega=\emptyset$, we call the sequence $(P_n,\check{P}_\omega)_{n\leq\omega}$ {\em type $\Delta(\omega+1)$.}
A decreasing $\omega+1$ sequence $(P_\alpha)_{\alpha<\omega+1}$ defines a set $P$ as in the usual difference hierarchy; that is, at the first $\omega$ levels, a hyp-computable learner proceeds as follows:
\[0\to 1\to 0\to \dots\]
If the guess changes infinitely many often, then the guess becomes $0$.
After that, we will be able to change the guess to $1$:
\[0\to 1\to 0\to \cdots(\omega\mbox{ changes})\cdots 0\to 1\]
A type $\Delta(\omega+1)$ sequence $(P_\alpha,\check{P}_\omega)_{\alpha<\omega+1}$ defines a set $P$ in a similar manner, where if the guess changes infinitely many often (which means $x\in\bigcap_{n<\omega}P_n$) then we soon decide the final value is $0$ or $1$ (which corresponds to either $x\in\check{P}_\omega$ or $x\in{P}_\omega$):
\[0\to 1\to 0\to \cdots(\omega\mbox{ changes})\cdots i\]
\begin{lemma}\label{lemma:om-plus-one}
A set $A$ is defined by a type $\Delta(\omega+1)$ sequence if and only if $A\in\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$.
\end{lemma}
\begin{proof}
The forward direction is trivial.
For the backward direction, we have two sequences $P,Q$ of type $\omega+1$ guessing $A$.
Given $x$, the first guessing process $P$ returns $0$ when the guess changes infinitely often but does not declare the $\omega$th mind-change, i.e., $x\in\bigcap_n P_n$ but $x\not\in P_\omega$.
Another guessing process $Q$ returns $1$ when $x\in\bigcap_n Q_n$ but $x\not\in Q_\omega$.
We construct a guessing sequence $D$ of type $\Delta(\omega+1)$.
At stage $\alpha$, if the number of changes of $P$ is smaller than $Q$, then the process $D$ emulates $P$; otherwise, $D$ emulates $Q$ whenever at least one of the numbers is finite.
In other words, compare $n_P=\sup\{n<\omega:x\in P_n[\alpha]\}$ and $n_Q=\sup\{n<\omega:x\in Q_n[\alpha]\}$.
If $t=\min\{n_P,n_Q\}$ is finite and even, returns $1$.
If $t=\min\{n_P,n_Q\}$ is finite and odd, returns $0$.
If $t$ is infinite, we have $x\in\bigcap_nP_n[\alpha]$ and $x\in\bigcap_nQ_n[\alpha]$.
In this case, either $P$ or $Q$ declare the $\omega$th mind-change; otherwise, $P$'s final guess is ``$A(x)=0$'' but $Q$'s final guess is ``$A(x)=1$'', which is impossible.
Thus, wait for seeing stage $\beta\geq\alpha$ such that either $P$ or $Q$ declare the $\omega$th mind-change, i.e., $x\in P_\omega[\beta]$ or $x\in Q_\omega[\beta]$.
In the former case, $D$'s final guess is ``$A(x)=1$'', i.e., $x\in D_\omega$.
In the latter case, $D$'s final guess is ``$A(x)=0$'', i.e., $x\in \check{D}_\omega$.
It is not hard to check that $D$ gives a process of type $\Delta(\omega+1)$ guessing $A$.
\end{proof}
\subsection{$\omega$-change matrix}
In order to prove Theorem \ref{thm:Wadge-rank-om1}, it suffices to show that there are at least $\omega_1$ many classes between $D^\ast_{\omega}(\tpbf{\Pi}^1_1)$ and $D^\ast_{\omega+1}(\tpbf{\Pi}^1_1)$.
First, we observe that there are at least $\omega$ many such classes.
A key observation is that, as we have seen above, $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$ corresponds to hyp-computability with at most $\omega$ mind-changes.
What we will show is that there is a finer hierarchy within hyp-computability with at most $\omega$ mind-changes.
The following definition is hard to understand, so we give an intuitive explanation after the definition.
\begin{definition}\label{def:om-change}
A double sequence $A=(A^j_n)_{(j,n)\in \ell\times\omega}$ of $\tpbf{\Pi}^1_1$ sets is called an {\em $\omega$-change $\ell\times\omega$ matrix} if the following holds (w.r.t.~some approximation of $(A^j_n)_{(j,n)\in \ell\times\omega}$):
\begin{enumerate}
\item For any $j<\ell$, $(A^j_n)_{n\in\omega}$ is a decreasing sequence.
\item For any $j<k<\ell$ and $\alpha\in{\sf WO}$, we have $A^k_0[\alpha]\cap A^j_n\subseteq A^j_n[\alpha]$ for any $n\in\omega$.
\end{enumerate}
Given $c$, we define a new difference operator $c\diff'_{\ell\times\omega}$, which takes an $\omega$-change $\ell\times\omega$ matrix $A$ and an $\ell\times\omega$ matrix $a=(a^j_n)$ as input.
To define this operator, we first introduce auxiliary parameters $v_k(x)$ for each $k\leq\ell$.
Then, we first put $v_0(x)=c$.
For each $k<\ell$, define $v_{k+1}$ as follows:
\[
v_{k+1}(x)=
\begin{cases}
a_m^k&\mbox{ if }m=\max\{n<\omega:x\in A_n^k\},\\
v_k(x)&\mbox{ if no such $m$ exists.}
\end{cases}
\]
Then we define $c\diff'_{\ell\times\omega}[a/A]$ as follows:
\[
c\diff_{\ell\times\omega}{}\!\!^\prime[a/A](x)=
\begin{cases}
v_k(x)&\mbox{ if }k=\min\left\{j<\ell:x\in\bigcap_{n<\omega}A^j_n\right\},\\
v_\ell(x)&\mbox{ if no such $k$ exists.}
\end{cases}
\]
Let $cD'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ be the class of all sets of the form $c\diff'_{\ell\times\omega}[a/A]$ for some $\omega$-change $\ell\times\omega$ matrix $A=(A^j_n)$ of $\tpbf{\Pi}^1_1$ sets and $\ell\times\omega$ matrix $a=(a^j_n)$ with $a_n^j\in\{0,1\}$.
If $c=0$ and $a_n^j={\sf par}(n)$ we simply write $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
\end{definition}
Let us explain an intuitive meaning of this definition.
Each row of an $\ell\times\omega$ matrix acts in the same way as the class $vD^\ast_\omega(\tpbf{\Pi}^1_1)$ for some $v$.
In other words, for each $k<\ell$, a hyp-computable process $\Psi_k$ is assigned to the $k$-th row, which may change the guess at most $\omega$ times, and when the $\omega$-th change occurs, the final guess is set to $v_k$.
The first guess is also set to $v_k$.
\begin{align*}
A^k_0\supseteq A^k_1\supseteq A^k_2\supseteq\cdots(\omega\mbox{ changes})&\cdots\\
v_k\to a^k_0\to a^k_1\to a^k_2\to \cdots(\omega\mbox{ changes})&\cdots v_k
\end{align*}
However, this value $v_k$ can vary if $k>0$.
A candidate $a^{k-1}_s$ for the value $v_k$ is determined by a guessing process $\Psi_{k-1}$ assigned to the row just one above it.
However, $\Psi_{k-1}$ may also change the guess $\omega$ times, so the final value $v_{k-1}$ depends on a guessing process $\Psi_{k-2}$ assigned to the $(k-1)$-th row if $k-1>0$.
Continue this argument, and if this process arrives the $0$-th row, and if the $\omega$-th change of $\Psi_0$ occurs, then the final guess is set to $c$.
Note, however, that although this explanation seems to proceed in order from the bottom row, the condition (2) in Definition \ref{def:om-change} guarantees that the process starts from the top row; that is, if we start the guessing process in some row, then the guessing processes in the rows above it has already terminated.
This assumption ensures that the guesses in each line can be integrated into a single hyp-computable process with at most $\omega$ mind-changes:
\begin{lemma}\label{lem:om-change-1}
For any $\ell<\omega$, $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)\subseteq \Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$.
\end{lemma}
\begin{proof}
Let $P\in D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ be given.
Then, $P$ is of the form $c\diff'_{\ell\times\omega}[a/A]$ for some $\omega$-change $\ell\times\omega$ matrix $A=(A^j_n)$ and $\ell\times\omega$ matrix $a=(a^j_n)$, where $c=0$ and $a^j_n={\sf par}(n)$.
To simplify our argument, one can assume that for any $x$ and $\alpha$ there are at most one $(j,n)$ such that $x$ is enumerated into $A^j_n$ at stage $\alpha$; that is, $x\in A^j_n[\alpha]$ but $x\not\in A^j_n[\beta]$.
For instance, one can assume that we only enumerate something into $A^j_n$ at stage $\omega^2\cdot\alpha+\omega\cdot j+n$ for some $\alpha$.
We construct a guessing sequence $D$ of type $\Delta(\omega+1)$.
Our guessing algorithm to compute $P(x)$ can be described as follows:
\begin{itemize}
\item At each stage $\alpha$, starting from the top row, one can calculate the current value $v_j(x)[\alpha]$ of $v_j$ for each $j\leq\ell$.
Indeed, it is easy to check that $(\alpha,k,x)\mapsto v_k(x)[\alpha]$ is Borel.
\item As the first case, if mind-changes occur infinitely often at some row $k$, then the guess is set to $v_k(x)[\alpha]$, and the computation terminates.
The condition (2) in Definition \ref{def:om-change} guarantees that at most one row is active at any stage $\alpha$, and thus, there is at most one row $j$ at which mind-changes occur infinitely often at $\alpha$.
Moreover, the condition (2) inductively ensures that the value of $v_j(x)$ will not change after stage $\alpha$ for any $j\leq k$, so the guess $v_k(x)[\alpha]$ matches the output value $P(x)$.
\item As the second case, if the mind-changes has not yet occurred infinitely many times at any row, then the algorithm currently guesses that the output value of $P(x)$ is $v_{\ell}(x)[\alpha]$.
Since $v_\ell(x)$ only changes when mind-changes occur at some row, and there are only a finite number of rows, the number of times of mind-changes has is kept finite in this case.
\end{itemize}
To be more precise, first check if there exists $k$ such that $x\in\bigcap_{n<\omega} A^k_m[\alpha]$.
If true, this is the first case.
If $\alpha$ is the least such stage, and $k$ is the least such row, then our algorithm returns $v_k(x)$.
By the condition (2) in Definition \ref{def:om-change}, since $x\in A^k_0[\alpha_0]$ we have $A^j_n[\alpha]=A^j_n$ for any $j<n$.
This means that $v_k(x)[\alpha]=v_k(x)$, and $k$ is the least row such that $x\in\bigcap_{n<\omega} A^k_m$.
Hence $c\diff'_{k<\omega}[a/A](x)=v_k(x)$ by definition.
If there exists no $k$ such that $x\in\bigcap_{n<\omega} A^k_m[\alpha]$, then this is the second case.
If this is true for any stage $\alpha$, then $x\not\in\bigcap_{n<\omega} A^k_m$ for any $k<\ell$, so the output of our guessing algorithm converges to $v_\ell(x)$.
In this case, by the definition of $\diff'$, we also have $c\diff'_{k<\omega}[a/A](x)=v_\ell(x)$.
Hence, our algorithm correctly guess the value of $P(x)=c\diff'_{k<\omega}[a/A](x)$.
As mentioned above, the mind-changes in the guess of our algorithm due to the second case occur only a finite number of times, and once the first case is reached, the guess never changes.
Also, in the first case, the guess is determined immediately.
Hence, this is a $\Delta(\omega+1)$ guessing process.
This completes the proof.
\end{proof}
To ensure that it is a reasonable pointclass, it should be closed under continuous substitution.
\begin{lemma}\label{lem:om-change-2}
$D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ is closed under continuous substitution, that is, $B\leq_{\sf W}A\in D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ implies $B\in D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
\end{lemma}
\begin{proof}
More generally, let $f\in cD'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ be given, and assume that $g=f\circ\theta$ for some continuous function $\theta$.
It suffices to show that $g\in cD'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
Then, $f$ is of the form $c\diff'_{\ell\times\omega}[a/A]$ for some $\omega$-change $\ell\times\omega$ matrix $A=(A^j_n)$ and $\ell\times\omega$ matrix $a=(a^j_n)$.
Then, put $B^j_n=\theta^{-1}[A^j_n]$, and then $B^j_n[\alpha]=\theta^{-1}[A^j_n[\alpha]]$ yields an approximation of $B^j_n$ for any $j<\ell$ and $n\in\omega$.
The property that $A$ is an $\omega$-change matrix is inherited by $B=(B^j_n)$.
Moreover, one can see that $g=c\diff'_{\ell\times\omega}[a/B]$ since $\theta^{-1}[\bigcap_{n<\omega}A^k_n]=\bigcap_{n<\omega}\theta^{-1}[A^k_n]$, and $\theta^{-1}[A^k_n\setminus A^k_{n+1}]=\theta^{-1}[A^k_n]\setminus\theta^{-1}[A^k_{n+1}]$.
Therefore, $g\in cD'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
\end{proof}
\begin{lemma}\label{lem:om-change-3}
For any $\ell<\omega$, $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ is strictly closed under $\omega_1$-pwo coproduct.
\end{lemma}
\begin{proof}
Put $\Delta=\Delta(D'_{\ell\times\omega}(\tpbf{\Pi}^1_1))$.
Let a uniform $\Delta$-collection $(P_\alpha)_{\alpha\in{\sf WO}}$ be given.
Then it is obtained by a collection of $\omega$-change matrices $A_\alpha=(A^{\alpha,j}_n)$.
Put $B_{n}^{j}=\bigsqcup_{\alpha\in{\sf WO}}A^{\alpha,j}_n$ for each $n<\omega$.
Then, $B_n^j$ is $\tpbf{\Pi}^1_1$ for each $n<\omega$ since the $\omega_1$-pwo-coproduct of a uniform collection of $\tpbf{\Pi}^1_1$ sets is $\tpbf{\Pi}^1_1$.
Moreover, $B=(B^{j}_n)_{j,n}$ is an $\omega$-change matrix:
For the condition (2), if $j<k$ and $(\alpha,x)\in B_0^k[\beta]\cap B^j_n$ then $x\in A_0^{\alpha,k}[\beta]\cap A^{\alpha,j}_n\subseteq A^{\alpha,j}_n[\beta]$, so $(x,\alpha)\in B^j_n[\beta]$.
To see the equality $\diff'_{\ell\times\omega}B=\bigsqcup_{\alpha\in{\sf WO}}P_\alpha$, let $(\alpha,x)$ be given.
Clearly, if $\alpha\not\in{\sf WO}$ then $(\alpha,x)\not\in B_0^j$, and therefore, $x\not\in\diff'_{\ell\times\omega}B$.
If $\alpha\in{\sf WO}$, then $(\alpha,x)\in B_n^j$ if and only if $x\in A_n^{\alpha,j}$.
Therefore, $(\alpha,x)\in\diff'_{\ell\times\omega}B$ if and only if $x\in\diff'_{\ell\times\omega}A_\alpha=P_\alpha$.
This completes the proof.
\end{proof}
\begin{lemma}\label{lem:om-change-4}
The hierarchy $(D'_{\ell\times\omega}(\tpbf{\Pi}^1_1))_{\ell<\omega}$ does not collapse; that is, for any $k<\ell<\omega$, $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)\setminus D'_{k\times\omega}(\tpbf{\Pi}^1_1)$ is nonempty.
\end{lemma}
\begin{proof}
We first construct a universal $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ set $G$.
The existence of a universal $\tpbf{\Pi}^1_1$ set clearly yields a total representation of all $\ell\times\omega$ matrices of $\tpbf{\Pi}^1_1$ sets which are not necessarily $\omega$-change matrices.
Given $\varepsilon\in\omega^\omega$, let $(A^j_n)$ be the $\ell\times\omega$ matrix coded by $\varepsilon$.
Then, define an $\ell\times\omega$ matrix $(B^j_n)$ as follows:
Given $\alpha<\omega_1$, if $x\not\in B_0^k[\alpha]$ for any $k>j$ then we declare that $x\in B_n^j[\alpha]$ if and only if $x\in A_n^j[\alpha]$.
If $x\in B_0^k[\alpha]$ for some $k>j$, declare that $x\in B_n^j[\alpha]$ if and only if $x\in B_n^j[\beta]$ for some $\beta<\alpha$.
That is, once a mind-change occurs in a lower row $k>j$, no more changes in the row $j$ will occur.
Then, put $B_n^j=\bigcup_{\alpha<\omega_1}B_n^j[\alpha]$, and then it is easy to see that $B_\varepsilon=(B_n^j)_{(n,j)\in\ell\times\omega}$ is an $\omega$-change matrix of $\tpbf{\Pi}^1_1$ sets.
Clearly, for any $\omega$-change $\ell\times\omega$ matrix $A$ there exists $\varepsilon$ such that $B_\varepsilon=A$.
We define $G$ as the set of all $(\varepsilon,x)$ such that $x\in\diff'_{\ell\times\omega} B_\varepsilon$.
Note that $G\in D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ since $G$ is of the form $\diff'_{\ell\times\omega} C$ for the $\omega$-change matrix $C=(C^j_n)$ defined by $C^j_n=\{(\varepsilon,x):x\in B^{\varepsilon,j}_n\}$, where $B_\varepsilon=(B^{\varepsilon,j}_n)$.
Next, it is easy to see that the dual class of $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$, i.e., $\neg D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$, is also included in $\Delta(D'_{(\ell+1)\times\omega}(\tpbf{\Pi}^1_1))$, by shifting the components of each row by one, and by adding the topmost row which always guesses $1$.
Hence, it remains to show that $\neg D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$ is not included in $D'_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
The rest of the proof is an easy diagonalization argument.
Let us consider $Q=\{x:(x,x)\not\in G\}$.
We claim that $Q$ does not belong to $D_{\ell\times\omega}(\tpbf{\Pi}^1_1)$.
Otherwise, there exists $x$ such that $Q=\diff'_{\ell\times\omega}B_x$.
However, $x\in Q$ if and only if $(x,x)\not\in G$ if and only if $x\not\in \diff'_{\ell\times\omega}B_x$, a contradiction.
This concludes the proof.
\end{proof}
Lemmas \ref{lem:om-change-1}, \ref{lem:om-change-2}, \ref{lem:om-change-3}, and \ref{lem:om-change-4}, combined with Lemma \ref{lem:steel-fournier-game}, imply that the Wadge rank of $\Delta(D^\ast_{\omega+1}(\tpbf{\Pi}^1_1))$ is at least $\omega_2\cdot \omega$.
It is straightforward to consider the transfinite version of this argument.
That is, for a limit ordinal $\lambda$, one may define $D'_{\lambda\times\omega}(\tpbf{\Pi}^1_1)$ as the class of all sets which can be written as countable disjoint unions of sets from $D'_{\lambda[n]\times\omega}(\tpbf{\Pi}^1_1)$, $n<\omega$, where $(\lambda[n])_{n<\omega}$ a fundamental sequence for $\lambda$.
For a successor ordinal $\xi=\zeta+1$, in order to define $D'_{\xi\times\omega}(\tpbf{\Pi}^1_1)$, one can simply add one more row to $D'_{\zeta\times\omega}(\tpbf{\Pi}^1_1)$.
As a consequence, inside $\Delta(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$, there are at least $\omega_1$ many classes strictly closed under $\omega_1$-pwo coproduct.
Hence, by Lemma \ref{lem:steel-fournier-game}, we conclude that the Wadge rank of $\Delta(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$ is at least $\omega_2\cdot\omega_1$.
However, by using $\omega_1$-pwo coproduct to combine these $\omega_1$ many classes, we can create a new class inside $\Delta(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$.
This concludes the proof of Theorem \ref{thm:Wadge-rank-om1}.
By repeating this process, it seems possible to construct $\omega_1+\omega_1$ many, $\omega_1^2$ many, or $\omega_2$ many different classes strictly closed under $\omega_1$-pwo coproduct.
If this is the case, by Lemma \ref{lem:steel-fournier-game}, one can show that the Wadge rank of $\Delta(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$ is at least $\omega_2^2$.
\begin{question}
Under ${\sf AD}$, is the Wadge rank of $\Delta(D_{\omega+1}^\ast(\tpbf{\Pi}^1_1))$ at least $\omega_2^2$?
\end{question}
One may also ask a similar question:
\begin{question}
Under ${\sf AD}$, is the Wadge rank of $\Delta(D_{\omega+n}^\ast(\tpbf{\Pi}^1_1))$ at least $\omega_2^{n+1}$?
\end{question}
We now move to the next level of $(D_{\omega+n}^\ast(\tpbf{\Pi}^1_1))_{n<\omega}$.
It is reasonable to ask the following question.
\begin{question}\label{que:rank-forward-backward}
Under {\sf AD}, calculate the Wadge rank of $\Delta(D^\ast_{\omega+\omega}(\tpbf{\Pi}^1_1))$.
\end{question}
However, we have the impression that answering this question is incredibly difficult.
This is because we feel that there is also a tremendously vast hierarchy between $(D_{\omega+n}^\ast(\tpbf{\Pi}^1_1))_{n<\omega}$ and $\Delta(D^\ast_{\omega+\omega}(\tpbf{\Pi}^1_1))$.
The first step is given by ``$\Pi^1_1$-processes with [forward $\omega$]$+$[backward $\omega$] mind-changes''.
More precisely, we consider the following $\omega+\omega$ sequence $(P_\alpha)_{\alpha<\omega+\omega}$ of $\tpbf{\Pi}^1_1$ sets:
\[P_0\supseteq P_1\supseteq P_2\supseteq\dots\supseteq\bigcap_{n<\omega}P_n\supseteq\bigcup_{n<\omega}P_{\omega+n}\supseteq\dots\supseteq P_{\omega+2}\supseteq P_{\omega+1}\supseteq P_\omega.\]
We call such a sequence {\em type $\omegaf+\omegab$}.
If moreover we have $\bigcap_{n<\omega}P_n=\bigcup_{n<\omega}P_{\omega+n}$, we call it {\em type $\Delta(\omegaf+\omegab)$.}
A type $\omegaf+\omegab$ sequence $(P_\alpha)_{\alpha<\omega+\omega}$ defines a set $P$ as in the usual difference hierarchy; that is, at the first $\omega$ levels, a $\Pi^1_1$ guess proceeds as follows:
\[0\to 1\to 0\to \cdots(\omega\mbox{ changes})\cdots 0\]
If the guess changes infinitely many often, then the guess becomes $0$.
After that, we will have a fresh mind-change counter $\omega$ controlling our next finite mind-changes.
\begin{question}
Under {\sf AD}, calculate the Wadge rank of $\Delta(\omegaf+\omegab)$.
\end{question}
In general, one can consider ``$\Pi^1_1$-processes with [forward $\omega$]$+$[backward $\alpha$] mind-changes'' for any $\alpha<\omega_1$.
Then we get the corresponding pointclass $\Delta(\omegaf+\alpha^{\leftarrow})$, and we still have $\Delta(\omegaf+\alpha^{\leftarrow})\subseteq\Delta(D^\ast_{\omega+\omega}(\tpbf{\Pi}^1_1))$ for any $\alpha<\omega_1$.
Based on these observations, we conjecture that the answer to Question \ref{que:rank-forward-backward} is at least $\omega_2^{\omega_2}$, but we do not have a method to calculate this at this time.
\end{ack}
\end{document} |
\begin{equation}gin{document}
\tilde{t}le{$\Omega$-theorem for short trigonometric sum}
\author{Jan Moser}
\address{Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA}
\email{[email protected]}
\keywords{Riemann zeta-function}
\begin{equation}gin{abstract}
We obtain in this paper new application of the classical E.C. Titchmarsh' discrete method (1934) in the theory of the Riemann
$\zf$ - function. Namely, we shall prove the first localized $\Omega$-theorem for short trigonometric sum. This paper is the English version
of the work of reference \cite{4}.
\end{abstract}
\title{$\Omega$-theorem for short trigonometric sum}
\section{Result}
\subsection{}
In this paper we shall study the following short trigonometric sum
\begin{equation}
S(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\cos(t\ln n),\
P_0=\sqrt{\frac{T}{2\pi}},\ t\in [T,T+U],
\end{equation}
where
\begin{equation}
U=T^{1/2}\psi\ln T,\ \psi\leq K\leq T^{1/6}\ln^2T,\ \psi <\ln T,
\end{equation}
and $\psi=\psi(T)$ stands for arbitrary slowly increasing function unbounded (from above). For example
\begin{equation}gin{displaymath}
\psi(T)=\ln\ln T,\ \ln\ln\ln T,\ \dots
\end{displaymath}
Let
\begin{equation}gin{displaymath}
\{ t_\nu\}
\end{displaymath}
be the Gram-Titchmarsh sequence defined by the formula
\begin{equation}gin{displaymath}
\vartheta(t_\nu)=\pi\nu,\quad \nu=1,2,\dots
\end{displaymath}
(see \cite{6}, pp. 221, 329) where
\begin{equation}gin{displaymath}
\vartheta(t)=-\frac t2\ln\pi+\mbox{Im}\left\{\ln\Gamma\left(\frac 14+i\frac t2\right)\right\}.
\end{displaymath}
Next, we denote by
\begin{equation}gin{displaymath}
G(T,K,\psi)
\end{displaymath}
the number of such $t_\nu$ that (see (1.1)) obey the following
\begin{equation}
t_\nu\in [T,T+U] \ \wedge \ |S(t_\nu,T,K)|>\frac 12 \sqrt{\frac{P_0}{K}}=AT^{1/4}K^{-1/2}.
\end{equation}
The following theorem holds true.
\begin{equation}gin{mydef1}
There are
\begin{equation}gin{displaymath}
T_0(K,\psi)>0, \ A>0
\end{displaymath}
such that
\begin{equation}
G(T,K,\psi)>AT^{1/6}K^{-1}\psi \ln^2T,\ T\geq T_0(K,\psi).
\end{equation}
\end{mydef1}
\subsection{}
Let us remind the following estimate by Karatsuba (see \cite{1}, p. 89)
\begin{equation}gin{displaymath}
\overset{*}{S}(x)=\sum_{1\leq n\leq x}n^{it}=\mathcal{O}(\sqrt{x}t^{\epsilon}),\ 0<x<t
\end{displaymath}
holds true on the Lindel\" of hypothesis ($0<\epsilon$ is an arbitrary small number in this). Of course,
\begin{equation}
\begin{equation}gin{split}
& \overset{*}{S}(t,T,K)=\sum_{e^{-1/K}P_0\leq n\leq P_0}n^{it}=\mathcal{O}(\sqrt{P_0}T^\epsilon)=\mathcal{O}(T^{1/4+\epsilon}), \\
& t\in [T,T+U],
\end{split}
\end{equation}
and
\begin{equation}gin{displaymath}
|\overset{*}{S}(t,T,K)|\geq |S(t,T,K)|.
\end{displaymath}
\begin{equation}gin{remark}
Since (see (1.1), (1.3), (1.4)) the inequality
\begin{equation}gin{displaymath}
|S(t,T,\ln\ln T)|> A\frac{T^{1/4}}{\sqrt{\ln\ln T}},\ T\to\infty
\end{displaymath}
is fulfilled for arbitrary big $t$, then the Karatsuba's estimate (1.5) is an almost exact estimate.
\end{remark}
\subsection{}
With regard to connection between short trigonometric sum and the theory of the Riemann zeta function see our paper \cite{3}.
\section{Main lemmas and proof of Theorem}
Let
\begin{equation}
w(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\frac{1}{\sqrt{n}}\cos(t\ln n),
\end{equation}
\begin{equation}
w_1(t,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{P_0}}\right)\cos(t\ln n).
\end{equation}
The following lemmas hold true.
\begin{equation}gin{mydef5A}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w^2(t_\nu,T,K)=\frac{1}{4\pi}UK^{-1}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation}
\end{mydef5A}
\begin{equation}gin{mydef5B}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_1^2(t_\nu,T,K)=\frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-3}\sqrt{T}\ln^2T).
\end{equation}
\end{mydef5B}
\begin{equation}gin{remark}
Of course, the formulae (2.3), (2.4) are asymptotic ones (see (1.2)).
\end{remark}
Now we use the main lemmas A and B for completion of the Theorem. Since (see \cite{2}, (23))
\begin{equation}
Q_0=\sum_{T\leq t_\nu\leq T+U}1=\frac{1}{2\pi}U\ln\frac{T}{2\pi}+\mathcal{O}\frac{U^2}{T},
\end{equation}
then we obtain (see (2.3), (2.4)) that
\begin{equation}
\frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w^2(t_\nu,T,K)\sim \frac{1}{2K},\ T\to\infty,
\end{equation}
\begin{equation}
\frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w_1^2(t_\nu,T,K)\sim \frac{1}{24K^3},
\end{equation}
\begin{equation}
\frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}w\cdot w_1=\mathcal{O}\left(\frac{1}{K^2}\right),
\end{equation}
(we used the Schwarz inequality in (2.8)). Next, we have (see (1.1), (2.1), (2.2)) that
\begin{equation}
w(t_\nu,T,K)=\frac{1}{\sqrt{P_0}}S(t_\nu,T,K)+w_1(t_\nu,T,K).
\end{equation}
Consequently we obtain (see (2.6) -- (2.9)) the following
\begin{equation}gin{mydef6}
\begin{equation}
\frac{1}{Q_0}\sum_{T\leq t_\nu\leq T+U}S^2\sim \frac{P_0}{2K},\ T\to\infty.
\end{equation}
\end{mydef6}
Next, we denote by $Q_1$ the number of such values
\begin{equation}gin{displaymath}
t_\nu\in [T,T+U],
\end{displaymath}
that fulfill the inequality (see (1.3))
\begin{equation}
|S|>\frac{1}{2}\sqrt{\frac{P_0}{K}};\quad Q_1=G(T,K,\psi),
\end{equation}
and
\begin{equation}gin{displaymath}
Q_0-Q_1=Q_2
\end{displaymath}
(see (2.5)). Since (see (1.1) and \cite{6}, p. 92)
\begin{equation}gin{displaymath}
|S(t,T,K)|<A \sqrt{P_0}T^{1/6},\ t\in [T,T+U],
\end{displaymath}
then we have (see (2.10), (2.11)) that
\begin{equation}gin{displaymath}
\frac{1}{3K}< AT^{1/3}\frac{Q_1}{Q_0}+\frac{1}{4K}\frac{Q_2}{Q_0}<A T^{1/3}\frac{Q_1}{Q_0}+\frac{1}{4K},
\end{displaymath}
i. e.
\begin{equation}
AQ_1>\frac{1}{12}Q_0T^{-1/3}K^{-1}.
\end{equation}
Consequently, we obtain (see (1.2), (2.5), (2.11), (2.12)) the following estimate
\begin{equation}gin{displaymath}
Q_1=G>A T^{1/6}K^{-1}\psi \ln^2T
\end{displaymath}
that is required result (1.4).
\section{Lemma 1}
Let
\begin{equation}
w_2=\ssum_{e^{-1/K}P_0<n<m<P_0}\frac{1}{\sqrt{nm}}\cos\left( t_\nu\ln\frac nm\right).
\end{equation}
The following lemma holds true.
\begin{equation}gin{mydef51}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_2=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation}
\end{mydef51}
\begin{equation}gin{proof}
The following inner sum (comp. \cite{5}, p. 102; $t_{\nu+1}\longrightarrow t_\nu$)
\begin{equation}
w_{21}=\sum_{T\leq T_\nu\leq T+U}\cos\{ 2\pi\psi_1(\nu)\},
\end{equation}
where
\begin{equation}gin{displaymath}
\psi_1(\nu)=\frac{1}{2\pi}t_\nu\ln\frac nm
\end{displaymath}
applies to our sum (3.2).
Now we obtain by method \cite{5}, pp. 102-103 the following estimate
\begin{equation}
w_{21}=\mathcal{O}\left(\frac{\ln T}{\ln\frac nm}\right).
\end{equation}
Since (see (1.2))
\begin{equation}gin{displaymath}
e^{-1/K}>1-\frac 1K\geq 1-\frac{1}{\psi}>\frac 12,
\end{displaymath}
then
\begin{equation}gin{displaymath}
2m>2e^{-1/K}P_0>P_0,\quad m\in (e^{-1/K}P_0,P_0),
\end{displaymath}
i. e. in our case (see (3.1)) we have that
\begin{equation}gin{displaymath}
2m>n.
\end{displaymath}
Consequently, the method \cite{6}, p. 116, $\sigma=\frac 12,\ m=n-r$ gives the estimate
\begin{equation}
\begin{equation}gin{split}
& \ssum_{e^{-1/K}P_0<n<m<P_0}\frac{1}{\sqrt{mn}\ln\frac nm}< \\
& <
A\sum_{e^{-1/K}P_0<n<P_0}\sum_{r\leq n/2}\frac 1r< AK^{-1}P_0\ln P_0<AK^{-1}\sqrt{T}\ln T,
\end{split}
\end{equation}
where
\begin{equation}
\sum_{e^{-1/K}P_0<n<P_0} 1\sim \frac{P_0}{K}.
\end{equation}
Now, required result (3.2) follows from (3.1), (3.3) -- (3.5).
\end{proof}
\section{Lemma 2}
Let
\begin{equation}
w_3=\ssum_{e^{-1/K}P_0<m<n<P_0}\frac{1}{\sqrt{mn}}\cos\{ t_\nu\ln(mn)\}.
\end{equation}
The following lemma holds true.
\begin{equation}gin{mydef52}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_3=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation}
\end{mydef52}
\begin{equation}gin{proof}
The following inner sum (comp. \cite{5}, p. 103; $t_{\nu+1}\longrightarrow t_\nu$)
\begin{equation}gin{displaymath}
w_{31}=\sum_{T\leq t_\nu\leq T+U}\cos\{ 2\pi \chi(\nu)\},
\end{displaymath}
where
\begin{equation}gin{displaymath}
\chi(\nu)=\frac{1}{2\pi}t_\nu\ln(nm)
\end{displaymath}
applies to our sum (4.2). Next, the method \cite{5}, pp. 103-104 gives us that
\begin{equation} \begin{equation}gin{split}
& w_{31}=\int_{\chi'(x)<1/2}\cos\{ 2\pi\chi(x)\}{\rm d}x+ \\
& + \int_{\chi'(x)>1/2}\cos[2\pi\{ \chi(x)-x\}]{\rm d}x+\mathcal{O}(1)=J_1+J_2+\mathcal{O}(1),
\end{split} \end{equation}
where
\begin{equation}gin{displaymath}
J_1=\mathcal{O}\left(\frac{\ln T}{\ln n}\right)=\mathcal{O}(1),\ n\in (e^{-1/K}P_0,P_0),
\end{displaymath}
and ($m<n<2m,\ n=m+r$)
\begin{equation}gin{displaymath}
J_2=\mathcal{O}\left( \frac{m\ln(m+1)}{r}\right).
\end{displaymath}
Now, the term $J_1$ contributes to the sum (4.2), (comp. (3.6)) as
\begin{equation}
\begin{equation}gin{split}
& \mathcal{O}\left(\ssum_{e^{-1/K}P_0<m<n<P_0}\frac{1}{\sqrt{mn}}\right) = \\
& = \mathcal{O}\left( \frac{1}{P_0}\ssum_{e^{-1/K}P_0<m<n<P_0} 1\right)= \\
& = \mathcal{O}\left(\frac{1}{P_0}\frac{P_0^2}{K^2}\right)=\mathcal{O}(K^{-2}\sqrt{T}),
\end{split}
\end{equation}
and the same contribution corresponds to the term $\mathcal{O}(1)$ in (4.3), while the contribution of the term $J_2$ is
\begin{equation}
\begin{equation}gin{split}
& \mathcal{O}\left(\sum_{e^{-1/K}P_0<m<P_0}\frac{1}{\sqrt{m}}\sum_{r=1}^m\frac{1}{\sqrt{m}}\frac{m\ln(m+1)}{r}\right)= \\
& = \mathcal{O}\left(\frac{P_0}{K}\ln^2P_0\right)=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{split}
\end{equation}
Now, the required result (4.2) follows from (4.1), (4.4), (4.5).
\end{proof}
\section{Lemma 3}
Let
\begin{equation}
w_4=\sum_{e^{-1/K}P_0<n<P_0}\frac 1n.
\end{equation}
The following lemma holds true.
\begin{equation}gin{mydef53}
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}w_4=\mathcal{O}(K^{-1}\sqrt{T}\ln^2T).
\end{equation}
\end{mydef53}
\begin{equation}gin{proof}
The following inner sum
\begin{equation}gin{displaymath}
w_{41}=\sum_{T\leq t_\nu\leq T+U}\cos\{ 2\pi\chi_1(\nu)\},
\end{displaymath}
where
\begin{equation}gin{displaymath}
\chi_1(\nu)=\frac{1}{\pi}t_\nu\ln n.
\end{displaymath}
applies to our sum (5.2). Since (comp. \cite{5}, p. 103)
\begin{equation}gin{displaymath}
\chi_1'(\nu)=\frac{\ln n}{\vartheta'(t_\nu)},
\end{displaymath}
next, (comp. \cite{5}, p. 100)
\begin{equation}gin{displaymath}
\begin{equation}gin{split}
& \vartheta'(t_\nu)=\frac 12\ln\frac{t_\nu}{2\pi}+\mathcal{O}\left(\frac{1}{t_\nu}\right)=
\frac 12\ln\frac{T}{2\pi}+\mathcal{O}\left(\frac UT\right)+\mathcal{O}\left(\frac 1T\right)\sim \\
& \sim \ln P_0,
\end{split}
\end{displaymath}
and
\begin{equation}gin{displaymath}
\ln P_0-\frac 1K<\ln n<\ln P_0,\ n\in (e^{-1/K}P_0,P_0),
\end{displaymath}
then
\begin{equation}gin{displaymath}
\chi_1'(\nu)\sim 1.
\end{displaymath}
Since, for example,
\begin{equation}gin{displaymath}
\frac 12<\chi_1'(\nu)<\frac 32,
\end{displaymath}
then we have (comp. \cite{5}, p. 104) that
\begin{equation}gin{displaymath}
w_{41}=\int\cos[2\pi\{\chi_1(x)-x\}]{\rm d}x+\mathcal{O}(1)=J_3+\mathcal{O}(1).
\end{displaymath}
Now, (comp. \cite{5}, p. 104)
\begin{equation}gin{displaymath}
\chi_1''(\nu)<-A\frac{\ln n}{T\ln^3T}<-\frac{B}{T\ln^2T},\ n\in (e^{-1/K}P_0,P_0),
\end{displaymath}
and
\begin{equation}gin{displaymath}
J_3=\mathcal{O}(\sqrt{T}\ln T),\quad w_{41}=\mathcal{O}(\sqrt{T}\ln T).
\end{displaymath}
Consequently, we get the required result (5.2)
\begin{equation}gin{displaymath}
\sum_{T\leq t_\nu\leq T+U}w_4=\mathcal{O}\left(\sqrt{T}\ln T\sum_{e^{-1/K}P_0<n<P_0}\frac 1n\right)=
\mathcal{O}(K^{-1}\sqrt{T}\ln T),
\end{displaymath}
where
\begin{equation}gin{displaymath}
\sum_{e^{-1/K}P_0<n<P_0}\frac 1n\sim \frac 1K
\end{displaymath}
by the well-known Euler's formula
\begin{equation}gin{displaymath}
\sum_{1\leq n<x}\frac 1n=\ln x+c+\mathcal{O}\left(\frac 1x\right),
\end{displaymath}
where $c$ is the Euler's constant.
\end{proof}
\section{Lemmas A and B}
\subsection{Proof of Lemma A}
First of all, we have (see (2.1)) that
\begin{equation}
\begin{equation}gin{split}
& w^2(t_\nu,T,K)=\\
& = \ssum_{e^{-1/K}P_0<m,n<P_0}\frac{1}{\sqrt{mn}}\cos( t_\nu\ln m)\cos( t_\nu\ln n)= \\
& = \frac 12 \sum_n \frac 1n+\ssum_{m<n}\frac{1}{\sqrt{mn}}\cos\left( t_\nu\ln\frac nm\right)+ \\
& + \ssum_{m<n}\frac{1}{\sqrt{mn}}\cos\{ t_\nu\ln (mn)\}+\frac 12\sum_{n}\frac 1n\cos( 2t_\nu\ln n)= \\
& = \frac{1}{2K}+\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)+w_2+w_3+w_4,
\end{split}
\end{equation}
(see (5.3), (3.1), (4.1), (5.1)). Consequently, we obtain the required result (2.3) from (6.1) by (2.5), (3.2), (4.2), (5.2).
\subsection{Proof of Lemma B} First of all we have (see (2.2)) that
\begin{equation}gin{displaymath}
w_1(t_\nu,T,K)=\sum_{e^{-1/K}P_0<n<P_0}\frac{\alpha(n)}{\sqrt{n}}\cos(t_\nu\ln n),
\end{displaymath}
where
\begin{equation}gin{displaymath}
\alpha(n)=1-\sqrt{\frac{n}{P_0}}.
\end{displaymath}
Of course, $\alpha(n)$ is decreasing and
\begin{equation}
0<\alpha(n)<\frac 1K,\quad n\in (e^{-1/K}P_0,P_0).
\end{equation}
Next, (comp. (6.1))
\begin{equation}
\begin{equation}gin{split}
& w_1^2(t_\nu,T,K)= \\
& = \frac 12\sum_n\frac{\alpha^2(n)}{n}+\ssum_{m<n}\frac{\alpha(m)\alpha(n)}{\sqrt{mn}}\cos\left( t_\nu\ln\frac nm\right)+ \\
& + \ssum_{m<n}\frac{\alpha(m)\alpha(n)}{\sqrt{mn}}\cos\left( t_\nu\ln(mn)\right)+\frac 12\sum_n
\frac{\alpha^2(n)}{n}\cos( 2t_\nu\ln n)= \\
& = \frac 12\bar{w}_1+\bar{w}_2+\bar{w}_3+\frac 12\bar{w}_4.
\end{split}
\end{equation}
Since (see (6.2))
\begin{equation}gin{displaymath}
\alpha(m)\alpha(n)<K^{-2}
\end{displaymath}
then we obtain by a similar way as in the case of the estimates (3.2), (4.2) and (5.2) that
\begin{equation}
\sum_{T\leq t_\nu\leq T+U}\left\{ \bar{w}_2+\bar{w}_3+\frac 12\bar{w}_4\right\}=
\mathcal{O}(K^{-3}\sqrt{T}\ln^2T).
\end{equation}
In the case of the sum
\begin{equation}
\frac 12\sum_{T\leq t_\nu\leq T+U}\bar{w}_1
\end{equation}
we use the following summation formula (see \cite{6}, p. 13)
\begin{equation}gin{displaymath}
\begin{equation}gin{split}
& \sum_{a\leq n<b}\varphi(n)=\int_a^b \varphi(x){\rm d}x+\int_a^b
\left( x-[x]-\frac 12\right)\varphi'(x){\rm d}x+ \\
& + \left( a-[a]-\frac 12\right)\varphi(a)-\left( b-[b]-\frac 12\right)\varphi(b)
\end{split}
\end{displaymath}
in the case
\begin{equation}gin{displaymath}
a=e^{-1/K}P_0, b=P_0, \varphi(x)=\frac{\alpha^2(x)}{x}=\frac 1x-\frac{2}{\sqrt{P_0x}}+\frac{1}{P_0}.
\end{displaymath}
Hence
\begin{equation}gin{displaymath}
\begin{equation}gin{split}
& \int_{e^{-1/K}P_0}^{P_0}\frac{\alpha^2(x)}{x}{\rm d}x=\frac{1}{K}-4\left( 1-e^{-1/(2K)}\right)+1-e^{-1/K}= \\
& = \frac{1}{12K^3}+\mathcal{O}(K^{-4}),
\end{split}
\end{displaymath}
and
\begin{equation}gin{displaymath}
\varphi'(x)=\mathcal{O}(P_0^{-2}),\
\varphi(e^{-1/K}P_0)=\mathcal{O}\left(\frac{1}{x^2P_0}\right),\ \varphi(P_0)=0.
\end{displaymath}
Consequently, we have (see (6.5))
\begin{equation}gin{displaymath}
\frac 12\bar{w}_1=\frac{1}{24K^3}+\mathcal{O}\left(\frac{1}{K^4}\right),
\end{displaymath}
and (see (1.2), (2.5))
\begin{equation}
\begin{equation}gin{split}
& \frac 12\sum_{T\leq t_\nu\leq T+U}\bar{w}_1=\frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+
\mathcal{O}\left(\frac{U\ln T}{K^4}\right)+\mathcal{O}\left(\frac{U^2}{K^3T}\right)= \\
& = \frac{1}{48\pi}UK^{-3}\ln\frac{T}{2\pi}+\mathcal{O}(K^{-3}\sqrt{T}\ln T).
\end{split}
\end{equation}
Finally, we obtain the required result (2.4) from (6.3) by (6.4) -- (6.6).
\begin{equation}gin{thebibliography}{29}
\bibitem{1}
A. A. Karatsuba, `\emph{Basic analytic number theory}`, Moscow, (1975), (in Russian).
\bibitem{2}
J. Moser, `On one theorem of Hardy-Littlewood in the theory of the Riemann zeta-function`, Acta Arith. 31,
(1976), 45-51; 40 (1981), 97-107, (in Russian).
\bibitem{3}
J. Moser, `A new estimate of short trigonometric sum`,
Acta Arith., 40 (1980), 357-367, (in Russian).
\bibitem{4}
J. Moser, `$\Omega$-theorem for short trigonometric sum`, Acta Arith. 42 (1983), 153-161, (in Russian).
\bibitem{5}
E. C. Titchmarsh, `On van der Corput's method and the zeta-function of Riemann, (IV)`,
Quart. J. Math. 5, (1934), 98-105.
\bibitem{6}
E. C. Titchmarsh, `\emph{The theory of the Riemann zeta-function}`, Clarendon Press, Oxford, 1951.
\end{thebibliography}
\end{document} |
\begin{document}
\title{The universal cover of a monomial triangular algebra without
multiple arrows }
\author{Patrick Le Meur
\footnote{\textit{e-mail:} [email protected]}
\footnote{\textit{adress:}
CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue du President Wilson, F-94230 Cachan}
}
\date{\today}
\maketitle
\abstract{
Let $A$ be a basic connected finite dimensional algebra over an
algebraically closed field $k$. Assuming that $A$ is monomial and that
the ordinary quiver $Q$ of $A$ has no oriented cycle and no multiple
arrows, we prove that $A$ admits a universal cover with group the
fundamental group of the underlying space of $Q$.
}
\section*{Introduction}
Let $A$ be a finite dimensional $k$-algebra where $k$ is an
algebraically closed field. In order to study the category $mod(A)$ of (left)
$A$-modules, one may assume that $A$ is basic and connected.
In \cite{riedtmann} (see also \cite{bongartz_gabriel}), C.~Riedtmann has introduced the covering
techniques which reduce the study of part of $mod(A)$ to
the easier one of $mod(\c C)$, where $\c C\to A$ is a Galois covering
and $\c C$ is locally bounded. These techniques
are based on the coverings of
translation quivers and their fundamental group, and therefore,
have been particularly efficient for representation-finite and
standard algebras $A$: In this case, P.~Gabriel (\cite{gabriel}) has constructed a
universal Galois covering of $A$, whose properties have led to a precise
description of the standard form of a representation-finite algebra
(\cite{bretscher_gabriel}). Unfortunately, the above construction of
\cite{gabriel} cannot be proceeded in the representation-infinite case precisely
because the Auslander-Reiten quiver is no longer connected.
In \cite{martinezvilla_delapena},
R.~Martinez-Villa and J.~A.~de~la~Pe\~na have constructed a Galois
covering $k\widetilde{Q}/\widetilde{I}\to A$ associated with each
presentation $kQ/I\simeq A$ (by quiver and admissible relations), for
any algebra $A$. This
Galois covering is induced by the universal cover
$(\widetilde{Q},\widetilde{I})\to (Q,I)$ with fundamental group
$\pi_1(Q,I)$ of the bound quiver $(Q,I)$, as a
generalisation of the universal cover of a translation quiver defined
in \cite{bongartz_gabriel} and \cite{riedtmann}. Like in topology, the group $\pi_1(Q,I)$ is defined
by means of a homotopy relation $\sim_I$ on the set of unoriented
paths of $Q$. When $A$ is
representation-finite and standard, this Galois covering coincides
with the one constructed by P.~Gabriel. Therefore, it is a natural
candidate for a universal cover of $A$ in the general
case. Alas, different presentations
may have non-isomorphic fundamental groups. So there may exist many
candidates for a universal cover. As an example, let
$A=kQ/<da>$, where
$Q$ is the quiver:
\begin{equation}
\xymatrix@1@=10pt{&\ar@{->}[rd]^c &&\\
\ar@{->}[rr]_a \ar@{->}[ru]^b &&\ar@{->}[r]_d&}\notag
\varepsilonnd{equation}
Then, $\pi_1(Q,<da>)\simeq\mathbb{Z}$. On the other hand,
$A\simeq kQ/<da-dcb>$, and $\pi_1(Q,<da-dcb>)=1$. Notice that $A$ is tilted of euclidean
an type and therefore belongs to a quite well-understood class of
algebras. This illustrates the fact that except for
representation-finite algebras there are quite a few classes of algebras
for which the existence of a universal cover is known.
In this text, we prove the existence of a universal cover for
certain monomial algebras, that is, quotients of paths algebras of
quivers by a monomial ideal (\textit{i.e.} generated by a set of
paths). More precisely, we prove the following main result.
\begin{thm}
\label{Thm1}
Let $A=kQ/I_0$, where $Q$ is a quiver without otiented cycle and without multiple
arrows, and $I_0$ is a monomial admissible ideal of $kQ$. Let $\widehat{\c C}\to
kQ/I_0$ be the Galois covering with group $\pi_1(Q)$ defined by the
presentation $kQ/I_0\simeq A$ (see
\cite{martinezvilla_delapena}). Then $\widehat{\c C}\to A$ is a
universal cover of $A$ in the following sense. For any Galois covering $\c C\to
A$ with group $G$ and with $\c C$ connected and locally bounded,
there exists a
commutative "factorisation diagram'':
\begin{equation}
\xymatrix@=8pt{
\widehat{\c C} \ar@{->}[rrd] \ar@{->}[dd] && \\
&&\c C \ar@{->}[d]\\
A \ar@{->}[rr]^{\sim} &&A
}\notag
\varepsilonnd{equation}
where the bottom arrow is an isomorphism of
$k$-algebras, extending the identity map on the set $Q_0$ of
vertices, and where $\widehat{\c C}\to \c C$ is a Galois covering with
group $N\vartriangleleft\pi_1(Q)$ such that there exists an
exact sequence of groups: $1\to N\to \pi_1(Q)\to G\to 1$.
\varepsilonnd{thm}
The author gratefully acknowledges an anonymous referee for pointing out
the following example from \cite[3.2]{geiss_delapena}. It shows
that the assumption on multiple arrows cannot be removed: Let
$A=k\left(\xymatrix@1{\cdot\ar@{->}@<2pt>[r]^x\ar@{->}@<-2pt>[r]_{x'} &
\cdot\ar@{->}@<2pt>[r]^y\ar@{->}@<-2pt>[r]_{y'} & \cdot}\right)/I_0$, where
$I_0=<yx',y'x>$. Then, $\pi_1(Q,I_0)=\pi_1(Q)\simeq
\mathbb{Z}\star\mathbb{Z}$, and if $char(k)\neq 2$, then $A\simeq kQ/I$
where $I=<yx-y'x',yx'-y'x>$, and $\pi_1(Q,I)\simeq
\mathbb{Z}/2\mathbb{Z}$. Now, let $\c C\to A$ be the Galois covering
with group $\pi_1(Q,I)$ defined by the presentation $A\simeq kQ/I$. Then, with the notations
of Theorem~\ref{Thm1}, it is easy to show that there is no $k$-linear
functor $\widehat{\c C}\to \c C$. Thus, in this example, $A$ admits no
universal cover in the sense of Theorem~\ref{Thm1}. From the very
definition of monomial algebras, one would expect that they all have a
universal cover. The above counter-example shows that this is not
always the case. It is all the more surprising as the involved algebra
is gentle, so its representation theory its fairly well-known.\\
We now explain the strategy of the proof of Theorem~\ref{Thm1}.
Our main tool is
the quiver $\Gamma$ of the homotopy
relations $\sim_I$ of the presentations $kQ/I\simeq A$. It was introduced in
\cite{lemeur2} to prove the existence of a universal cover for
algebras over a zero characteristic field, and whose ordinary quiver
have no double bypass. In general, if $kQ/I\simeq A$ and $kQ/J\simeq
A$, there is no simple relation between $\pi_1(Q,I)$ and
$\pi_1(Q,J)$ (and therefore between the associated Galois coverings of
$A$) unless $A$ is representation-finite (in which case $A$ is
schurian, so that $\pi_1(Q,I)=\pi_1(Q,J)$). This is the main difficulty in proving the existence
of a universal cover. Hopefully, such a relation exists when $I$ and $J$
are related by a transvection $\varphi_{\alpha,u,\tau}$, that is,
$J=\varphi_{\alpha,u,\tau}(I)$, where $(\alpha,u)$ is a bypass
(meaning that $\alpha$ is an arrow and $u$ is a path parallel to and different from
$\alpha$), $\tau\in k$ and $\varphi_{\alpha,u,\tau}\in Aut(kQ)$ is the
automorphism which maps $\alpha$ to $\alpha+\tau u$ and which fixes
any other arrow. In such a case, there is a natural quotient relation
between $\pi_1(Q,I)$ and $\pi_1(Q,J)$.
Besides, the Galois coverings of
$A$ with groups $\pi_1(Q,J)$ and $\pi_1(Q,I)$ are the vertical arrows
of a factorisation diagram like in Theorem~\ref{Thm1}, and the
associated exact sequence of groups is given by the above quotient relation. The quiver
$\Gamma$ is then defined as follows. Its vertices are the homotopy
relations $\sim_I$ of all the presentations $kQ/I\simeq A$, and there
is an arrow $\sim\to \sim'$ if there exist presentations $kQ/I\simeq
A$ and $kQ/J\simeq A$, and a transvection $\varphi$ such that:
$\sim=\sim_I$, $\sim'=\sim_J$, $J=\varphi(I)$, and $\pi_1(Q,J)$ is a
strict quotient of $\pi_1(Q,I)$. The quiver $\Gamma$ is then
finite, connected, and it has no oriented cycle. Notice that
$\Gamma$ is reduced to a point (with no arrow) when $A$ is schurian
(and in particular when $A$ is representation-finite). But, usually,
$\Gamma$ has many vertices and many arrows. We refer the reader
to Section~$1$ for more
details.
Roughly speaking, the existence of a universal cover is related to the
existence of a unique source in $\Gamma$. More precisely, assume
that there exists a presentation $kQ/I_0\simeq A$ (which in our case
will be the monomial presentation) such that for any other
presentation $kQ/I\simeq A$, there exists a sequence of ideals
$I_0,I_1=\varphi_1(I_0),\ldots,I_n=\varphi_n(I_{n-1})=I$, where
$\varphi_1,\ldots,\varphi_n$ are transvections defining
a path $\sim_{I_0}\to \sim_{I_1}\to\ldots\to \sim_{I_n}=\sim_I$ in $\Gamma$. Then,
the Galois covering of $A$ with group $\pi_1(Q,I_0)$
associated to the presentation $kQ/I_0\simeq A$ is a universal cover
of $A$.
As an example, assume that $A=kQ/I_0$, where $Q$ is the quiver
\begin{equation}
\xymatrix@=8pt{
&&3\ar@{->}[rd]^f&&&\\
&2\ar@{->}[ru]^e \ar@{->}[rr]^d&&4\ar@{->}[rd]^g&&\\
1\ar@{->}[ru]^c \ar@{->}[rrru]^b \ar@{->}[rrrr]_a&&&&5\ar@{->}[r]_h&6
}\notag
\varepsilonnd{equation}
and $I_0=<ha,gb,dc>$. Then $\Gamma$ has the following
shape:
\begin{equation}
\xymatrix@=10pt{
& \sim_{I_0} \ar@{->}[d] \ar@{->}[ld] \ar@{->}[rd]&\\
\bullet\ar@{->}[d] \ar@{->}[rd]&
\bullet\ar@{->}[ld] \ar@{->}[rd]&
\bullet\ar@{->}[d] \ar@{->}[ld]\\
\bullet\ar@{->}[rd]
&\bullet\ar@{->}[d]&\bullet\ar@{->}[ld]\\
&\bullet&}\notag
\varepsilonnd{equation}
We do not specify all the vertices, but one can check that for every
$\sim_I\in\Gamma$, the group $\pi_1(Q,I)$ is free over $3-l$
generators, where $l$ is the length of any path from $\sim_{I_0}$ to
$\sim_I$ (so two presentations may have distinct homotopy relations
yet isomorphic fundamental groups). For the
needs of the proof, we construct a specific total
order on the set of bypasses of $Q$. In our example, this order is:
$(d,fe)<(b,fec)<(b,dc)<(a,gfec)<(a,gdc)<(a,gb)$. Now, let
$I=<ha+hgfec,gb+gfec,dc>$, then:
\begin{enumerate}
\item $I=\varphi_{a,gfec,1}\varphi_{b,fec,1}(I_0)$. Moreover,
$\varphi_{a,gfec,1}\varphi_{b,fec,1}$ is the unique automorphism of
$kQ$ which transforms $I_0$ into $I$, and which maps every arrow
$\alpha$ to the sum of $\alpha$ and a linear combination of paths of
length greater than $1$, none of which lying in $I_0$ (indeed:
$gfec,fec\not\in I_0$). For that reason, we set
$\psi_I:=\varphi_{a,gfec,1}\varphi_{b,fec,1}$.
\item The equality $\psi_I=\varphi_{a,gfec,1}\varphi_{b,fec,1}$
expresses $\psi_I$ as a product of transvections
$\varphi_{a,gfec,1}, \varphi_{b,fec,1}$. The associated
sequence of bypasses is decreasing ($(a,gfec)>(b,fec)$). Actually,
the sequence $\varphi_{a,gfec,1}, \varphi_{b,fec,1}$ is unique for
this property.
\item We have a path $I_0\to \varphi_{b,fec,1}(I_0)\to
\varphi_{a,gfec,1}=\varphi_{b,fec,1}(I_0)=I$ in $\Gamma$.
\varepsilonnd{enumerate}
In this example, it is easy to check that for any other
presentation $A\simeq kQ/J$, the ideal $J$ defines a unique
automorphism $\psi_J$ (as in 1.) which decomposes uniquely (as in 2.),
giving rise to a path from $\sim_{I_0}$ to $\sim_J$ (as in 3.).
The proof of Theorem~\ref{Thm1} mimickes the three steps we proceeded
in that example. Indeed, we prove the three following
technical points:
\begin{enumerate}
\item If $kQ/I\simeq kQ/I_0$, then there exists a unique product
$\psi_I$ of
transvections
such that $\psi_I(I_0)=I$, and such that $\psi_I$ maps every arrow
$\alpha$ to the sum of $\alpha$ and a linear combination of paths
of length greater than $1$,
none of which lying in $I_0$.
\item There exists a suitable ordering on the set of bypasses such
that if $\psi\in Aut(kQ)$ is a
product of transvections, then $\psi$ can be written uniquely as
$\psi=\varphi_{\alpha_n,u_n\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$
with
$\tau_1,\ldots,\tau_n\in k^*$ and $(\alpha_n,u_n)>\ldots>(\alpha_1,u_1)$.
\item If $kQ/I\simeq kQ/I_0$, the unique ordered sequence of
transvections given by $1$. and $2$. yield a path in $\Gamma$
starting at $\sim_{I_0}$ and ending at $\sim_I$. Also, this sequence
gives rise to the factorisation diagram of Theorem~\ref{Thm1}.
\varepsilonnd{enumerate}
The text is therefore organised as follows. In Section~$1$ we recall all the
notions that we need to prove Theorem~\ref{Thm1}. In Section~$2$,
we prove some combinatorial facts on the paths in a quiver. These
lead to the order and to the decomposition of the second point
above. In Section~$3$ we prove the first point above. Finally, in
Section~$4$ we prove the last point and Theorem~\ref{Thm1}.
\section{Basic definitions}
A \textbf{$k$-category} is a category $\c C$
whose objects class $\c
C_0$ is a set, whose space of morphisms from $x$ to $y$ (denoted by $_y\c C_x$)
is a $k$-vector space for any $x,y\in\c C_0$ and whose
composition of morphisms is $k$-bilinear. All functors between
$k$-categories will be assumed to be $k$-linear
functors. In particular, $Aut(\c C)$ will denote the group of
$k$-linear automorphism of $\c C$, and $Aut_0(\c
C)$ will denote by for the subgroup $\{\psi\in Aut(\c C)\ |\ \psi(x)=x\ \text{for
any}\ x\in \c C_0\}$ of $Aut(\c C)$. The $k$-category $\c
C$ is called \textbf{connected} if it cannot be written as the
disjoint union of two full subcategories. An ideal $I$ of $\c C$ is the
data of subspaces $_yI_x\subseteq\ _y\c C_x$ (for any $x,y\in\c C_0$)
such that $fgh\in I$ whenever $f,g,h$ are composable morphisms
in $\c C$ such that $g\in I$.
The $k$-category $\c C$ is called \textbf{locally bounded} provided that:
$1$) for any $x\in\c C_0$, the vector spaces
$\bigoplus\limits_{y\in\c C_0}\ _y\c C_x$ and
$\bigoplus\limits_{y\in\c C_0}\ _x\c C_y$ are finite dimensional,
$2$) $_x\c C_x$ is a local algebra for any $x\in\c C_0$,
$3$) distinct objects are not isomorphic.
Let $A$ be a finite dimensional $k$-algebra and
let $\{e_1,\ldots,e_n\}$ be a complete set of
primitive orthogonal idempotents. Then $A$ is also a
$k$-category: $A_0:=\{e_1,\ldots,e_n\}$, $_{e_i}A_{e_i}:=e_jAe_i$ and the
product of $A$ induces the composition of morphisms.
Notice that different choices for the idempotents $e_1,\ldots,e_n$ give rise to
isomorphic $k$-categories. Also, $A$ is connected (resp. basic) as a $k$-algebra
if and only if it is connected (resp. locally bounded) as a
$k$-category. In the sequel we shall make no distinction between a
finite dimensional $k$-algebra and its associated $k$-category.
If $\c C$ is a locally bounded $k$-category, the radical of $\c C$ is
the ideal $\c R\c C$ of $\c C$ such that: $_y\c R\c C_x$ is
the space of non-isomorphisms $x\to y$ in $\c C$, for any $x,y\in\c C_0$.
The ideal of $\c C$ generated by compositions $gf$ where $f$ and $g$
lie in $\c R\c C$ will be denoted by $\c R^2\c C$.\\
A \textbf{Galois covering
with group $G$} of $\c C$ (by $\c C'$) is a functor $F\colon\c C'\to\c C$
endowed with a group morphism $G\to Aut(\c C')$ and such that: $1$) the
induced action of $G$ on $\c C'_0$ is free, $2$) $F\circ g=F$ for any $g\in
G$, $3$) for any $k$-linear functor $F'\colon \c C'\to \c C''$ such that
$F'\circ g=F'$ for any $g\in G$, there exists a unique
$\overline{F'}\colon \c C\to \c C''$ such that $\overline{F'}\circ
F=F'$ (in other words, $F$ is a quotient of $\c C'$ by $G$ in the
category of $k$-categories). For short, the Galois covering $F$ is
called connected if $\c C'$ is
connected and locally bounded (this implies that $\c C$ is
connected and locally bounded).
For more details on Galois coverings (in particular for the
connections with representations theory), we refer the reader to
\cite{bongartz_gabriel}.\\
\textbf{Quivers, paths, bypasses}. A quiver is a $4$-tuple
$Q=(Q_1,Q_0,s,t)$ where $Q_1$ and $Q_0$ are sets
and $s,t\colon Q_1\to Q_0$ are maps. The elements of $Q_1$ (resp. of
$Q_0$) are called the arrows (resp. the vertices) of $Q$. If
$\alpha\in Q_1$, the vertex $s(\alpha)$ (resp. $t(\alpha)$) is called
the source (resp. the target) of $\alpha$. The quiver
$Q$ is called \textbf{locally finite} if and only if any vertex is the
source (resp. the target) of finitely many arrows.
For example, if $\c C$ is a locally bounded $k$-category, the
\textbf{ordinary quiver of $\c C$} is the locally finite quiver
$Q$ such that: $Q_0:=\c C_0$ and for
any $x,y\in \c C_0$, the number of arrows starting at $x$ and
arriving at $y$ is equal to $dim_k\ _y\c R\c C_x/\,_y\c R^2\c C_y$.
A path in $Q$ of length $n$
($n\Gammaeqslant 0$)
with source $x\in Q_0$ (or starting at $x$) and target $y\in Q_0$ (or
arriving at $y$) is a sequence
of arrows $\alpha_1,\ldots,\alpha_n$ such that: $x=y$ if $n=0$,
$s(\alpha_1)=x$, $s(\alpha_{i+1})=t(\alpha_i)$ for any
$i\in\{1,\ldots,n-1\}$ and $t(\alpha_n)=y$. If $n\Gammaeqslant 1$ this
path will be written $\alpha_n\ldots\alpha_1$ and called non
trivial. If $n=0$ this path will be written $e_x$ and called
stationary at $x$. The length of this path is $|u|:=n$. The mappings $s,t$ are naturally extended to paths
in $Q$.
If $u$ and $v$ are paths, the concatenation $vu$ is defined if
and only if $t(u)=s(v)$ by the following rule:
$1$) $vu=v$ is $u$ is stationary,
$2$) $vu=u$ is $v$ is stationary,
$3$) $vu=\beta_m\ldots\beta_1\alpha_n\ldots\alpha_1$ if
$v=\beta_m\ldots\beta_1$ and $u=\alpha_n\ldots\alpha_1$ (with
$\alpha_i,\beta_j\in Q_1$).
Two paths in $Q$ are called \textbf{parallel} whenever they have the
same source and the same target.
An \textbf{oriented cycle} in $Q$ is a non trivial path whose source and target are
equal. The quiver $Q$ is said to have \textbf{multiple arrows} if and only if there
exist in $Q$ distinct parallel arrows. If $Q$ has no oriented cycle
and if $(\alpha,u,\beta,v)$ is a double bypass (see the introduction)
there exists two unique paths $u_1,u_2$ such that $u=u_2\beta
u_1$. In such a situation, the path $u_2vu_1$ will be called obtained from
$u=u_2\beta u_1$ after replacing $\beta$ by $v$.
\\
\textbf{Admissible presentations (see \cite[2.1]{bongartz_gabriel})}.
A quiver $Q$ defines the \textbf{path category} $kQ$ such that
$(kQ)_0=Q_0$, such that $_ykQ_x$ is the
$k$-vector space with basis the family of paths starting at $x$
and arriving at $y$, and the composition in $kQ$ is induced by the
concatenation of paths.
For short, \textbf{a normal form for $r\in\ _ykQ_x$} is an equality
$r=\sum\limits_{i=1}^n t_i u_i$ where $t_1,\ldots,t_n\in k^*$ and
$u_1,\ldots u_n$ are pairwise distinct paths in
$Q$. With this notation, \textbf{the support} of $r$
is the set $supp(r):=\{u_1,\ldots,u_n\}$ ($supp(0)=\varepsilonmptyset$). A
\textbf{subexpression of $r$} is a linear combination $\sum\limits_{i\in
E}t_i u_i$ with
$E\subseteq\{1,\ldots,n\}$.
Later, we will need the following fact: if $r=r_1+\ldots+r_n\in\ _ykQ_x$ is such that
$supp(r_1),\ldots,supp(r_n)$ are pairwise disjoint, then
$r_{i_1}+\ldots+r_{i_t}$ is a subsexpression of $r$, for any indices
$1\leqslant i_1<\ldots<i_t\leqslant n$.
An
ideal $I$ of $kQ$ is called \textbf{admissible} provided that:
$1$) any morphism in $I$ is a linear combination of paths of length at
least $2$,
$2$) the factor category $kQ/I$ is locally bounded.
A morphism in $I$ is called a \textbf{relation} (of $I$). In particular,
\textbf{a minimal relation of $I$} (see \cite{martinezvilla_delapena})
is a non zero relation $r$ of $I$ such that $0$ and $r$ are the only
subexpressions of $r$ which are relations. With this definition, any
relation of $I$ is the sum of minimal relations with pairwise disjoint
supports. A \textbf{monomial relation} is a path $u$ lying in $I$ and $I$ is called
monomial if it is generated by a set of monomial relations.
A pair $(Q,I)$ where $Q$ is a locally finite and $I$ is an admissible
ideal of $kQ$ is called a \textbf{bound quiver}. In such a case,
$kQ/I$ is locally bounded and it is connected if and only
if $Q$ is connected (i.e. the underlying graph of $Q$ is
connected). Conversely, if $\c C$ is a locally bounded $k$-category,
then there exists an isomorphism $kQ/I\xrightarrow{\sim}\c C$ where
$(Q,I)$ is a bound quiver such that $Q$ is the ordinary quiver of $\c
C$. Such an isomorphism is called \textbf{admissible presentation of
$\c C$}. If the ideal $I$ is monomial, the admissible presentation and
$\c C$ are called monomial. Notice that $\c C$ may have different admissible
presentations.\\
\textbf{Fundamental group of a presentation (see
\cite{martinezvilla_delapena})}. Let $(Q,I)$ be a bound quiver and let
$x_0\in Q_0$.
For every arrow $x\xrightarrow{a}y\in Q_1$
we define its formal inverse $a^{-1}$ with source
$s(a^{-1})=y$ and target $t(a^{-1})=x$. This defines a new quiver
$\overline{Q}=(Q_0,Q_1\cup\{a^{-1}\ |\ a\in Q_1\},s,t)$. With these notations, \textbf{a
walk in $Q$} is a path in $\overline{Q}$. The concatenation of walks
in $Q$ is by definition the concatenation of paths in $\overline{Q}$. The
\textbf{homotopy relation} of $(Q,I)$ is the equivalence relation on
the set of walks in $Q$, denoted by $\sim_I$ and generated by the
following properties:
\begin{enumerate}
\item $\alpha\alpha^{-1}\sim_I e_y$ and $\alpha^{-1}\alpha\sim_I e_x$
for any arrow $x\xrightarrow{\alpha}y$ in $Q$,
\item $u\sim_I v$ for any $u,v\in supp(r)$ where $r$ is a minimal
relation of $I$,
\item $wvu\sim_I wv'u$ for any walks $w,v,v',u$ such that $v\sim_I v'$
and such that the concatenations $wvu$ and $wv'u$ are well-defined
(i.e. $\sim_I$ is compatible with the concatenation).
\varepsilonnd{enumerate}
The $\sim_I$-equivalence class of a walk $\Gammaamma$ will be denoted by $[\Gammaamma]_I$.
Let $\pi_1(Q,I,x_0)$ be the set of equivalence classes of walks in $Q$
with source and target equal to $x_0$. The concatenation of walks
endows this set with a group structure (with unit $e_{x_0}$) and this group is called the
\textbf{fundamental group of $(Q,I)$ at $x_0$}. If $Q$ is connected, the
isomorphism class of this group does not depend on $x_0\in
Q_0$ and $\pi_1(Q,I,x_0)$ is denoted by $\pi_1(Q,I)$. If
$\c C$ is a connected locally bounded $k$-category and if $kQ/I\simeq
\c C$ is an admissible presentation, the fundamental group
$\pi_1(Q,I)$ is called the fundamental group of this presentation.\\
\textbf{Dilatations, transvections (see \cite{lemeur2})}. Let $Q$ be a
quiver. A \textbf{dilatation} of $kQ$ is an automorphism $D\in Aut_0(kQ)$ such that
$D(\alpha)\in k^*\alpha$ for any $\alpha\in Q_1$. The dilatations of $kQ$ form a
subgroup $\c D$ of $Aut_0(kQ)$. Let $(\alpha,u)$ be a
bypass in $Q$ and let $\tau\in k$. This defines
$\varphi_{\alpha,u,\tau}\in Aut_0(kQ)$ as follows:
$\varphi_{\alpha,u,\tau}(\alpha)=\alpha+\tau u$ and
$\varphi_{\alpha,u,\tau}(\beta)=\beta$ for any arrow $\beta\neq
\alpha$. The automorphism $\varphi_{\alpha,u,\tau}$ is called a
\textbf{transvection}. The composition of transvections is ruled as follows.
Let $\varphi_{\alpha,u,\tau}$ and $\varphi_{\alpha,u,\tau'}$, then
$\varphi_{\alpha,u,\tau}\varphi_{\alpha,u,\tau'}=\varphi_{\alpha,u,\tau+\tau'}\
\ \text{and}\ \ \varphi_{\alpha,u,\tau}^{-1}=\varphi_{\alpha,u,-\tau}$.
If $(\alpha,u,\beta,v)$ and $(\beta,v,\alpha,u)$ are not a double bypasses, then
$\varphi_{\alpha,u,\tau}\varphi_{\beta,v,\nu} =
\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau},$. If $(\alpha,u,\beta,v)$ is a double bypass and if $Q$ has no oriented cycle, then
$\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau} =
\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}
$, where $w$ is the path obtained from $u$ after replacing $\beta$ by
$v$.
The subgroup of $Aut_0(kQ)$ generated by all the
transvections is denoted by $\c T$. The dilatations and the transvections are useful to
compare admissible presentations of an algebra because of the following proposition:
\begin{prop}
\label{prop1.1}
(see \cite[Prop. 2.1, Prop. 2.2]{lemeur2})
Let $kQ/I\simeq A$ and $kQ/J\simeq A$ be admissible presentations of
the basic finite dimensional algebra $A$. If $Q$ has no oriented
cycle, then there exists $\psi\in Aut_0(kQ)$ such that $\psi(I)=J$.
Moreover,
$\c T$ is a normal subgroup of $Aut_0(kQ)$ and $Aut_0(kQ)=\c T\c D=\c
D\c T$.
\varepsilonnd{prop}
The dilatations and the transvections were introduced because they
allow comparisons between the fundamental groups of presentations of
the same locally bounded $k$-category. Notice that if $I,J$ are
admissible ideals of $kQ$ such that $\Gammaamma\sim_I\Gammaamma'\Rightarrow
\Gammaamma\sim_J\Gammaamma'$ for any walks $\Gammaamma,\Gammaamma'$, then the identity
map on the set of walks induces a surjective group morphism
$\pi_1(Q,I)\twoheadrightarrow\pi_1(Q,J)$. In particular, if $\sim_I$
and $\sim_J$ coincide, then $\pi_1(Q,I)=\pi_1(Q,J)$.
\begin{prop}
\label{prop1.2}
(see \cite[Prop. 2.5]{lemeur2})
Let $I$ be an
admissible ideal of $kQ$ (with $Q$ without oriented
cycle), let $\varphi\in
Aut_0(kQ)$ and set $J=\varphi(I)$. If $\varphi$ is a dilatation,
then $\sim_I$ and $\sim_J$ coincide. If
$\varphi=\varphi_{\alpha,u,\tau}$ is a transvection, then:
\begin{enumerate}
\item if $\alpha\sim_Iu$ and $\alpha\sim_Ju$ then $\sim_I$ and
$\sim_J$ coincide.
\item if $\alpha\not\sim_Iu$ and $\alpha\sim_J u$ then $\sim_J$
is generated by $\sim_I$ and $\alpha\sim_Ju$.
\item if $\alpha\not\sim_Iu$ and $\alpha\not\sim_Ju$ then $I=J$
and $\sim_I$ and $\sim_J$ coincide.
\varepsilonnd{enumerate}
If there exists a transvection $\varphi$ such that $\varphi(I)=J$ and
such the second point above occurs, then
\textbf{$\sim_J$ is called a direct successor of $\sim_I$}.
\varepsilonnd{prop}
Here the expression ``$\sim_J$ is generated by $\sim_I$ and
$\alpha\sim_J u$'' means that $\sim_J$ is the equivalence relation
on the set of walks in $Q$, compatible with the concatenation and
generated by the two following properties:
$1$) $\Gammaamma\sim_I\Gammaamma'\Rightarrow \Gammaamma\sim_J\Gammaamma'$,
$2$) $\alpha\sim_J u$. Following \cite[Def. 2.7]{lemeur2}, if $A$ is a
basic connected finite dimensional algebra with ordinary quiver $Q$
without oriented cycle, we define
\textbf{the quiver $\Gamma$ of the homotopy relations of $A$} to be
the quiver such that $\Gamma_0=\{\sim_I\ |\ kQ/I\simeq A\}$ and such
that there exists arrow $\sim_I\to \sim_J$ if and only if $\sim_J$ is
a direct successor of $\sim_I$. Recall (\cite[Rem. 5,
Prop. 2.8]{lemeur2}) that $\Gamma$ is finite, connected, without
oriented cycle and such that for any (oriented) path with source
$\sim_I$ and target $\sim_J$, the identity map on the set of walks in $Q$
induces a surjective group morphism $\pi_1(Q,I)\twoheadrightarrow \pi_1(Q,J)$.\\
\textbf{Gröbner bases} Let $E$ be a $k$-vector space with an ordered
basis $(e_1,\ldots,e_n)$, let $(e_1^*,\ldots,e_n^*)$ be the associated
dual basis of $E^*$, and let $F$ be a subspace of $E$. A Gröbner basis (see
\cite{adams_loustaunau} for the ususal definition) of $F$ is a basis
$(r_1,\ldots,r_d)$ such that:
\begin{enumerate}
\item $r_j\in e_{i_j}+Span(e_l\ ;\ l<i_j)$ for some $i_j$, for any
$j\in\{1,\ldots,r\}$,
\item $i_1<i_2<\ldots<i_r$,
\item $e_{i_j}^*(r_{j'})=0$ for any $j\neq j'$.
\varepsilonnd{enumerate}
It is well known that $F$ admits a unique Gröbner basis. Also, $r\in
F$ if and only if:
$r=\sum\limits_{j=1}^de_{i_j}^*(r)r_j$.
In the sequel, we will use this notion in the following setting: $E$
is the vector space with basis (for some order to be defined) the
family of non trivial paths in a
finite quiver $Q$ without oriented cycles and $F$ is the underlying
subspace of $E$ associated to an admissible ideal $I$ of $kQ$. Notice
that in this setting, the Gröbner basis of $F$ is made of minimal
relations of $I$. Also, if $r\in E$ and if $u$ is a non trivial path,
then: $u\in supp(r)\Leftrightarrow u^*(r)\neq 0$.\\
Until the end of the text, $Q$ will denote a finite quiver without
oriented cycle and without multiple arrows.
\section{Combinatorics on the paths in a quiver}
Recall from the previous section that if $(\alpha,u,\beta,v)$ is a
double bypass and if $\tau,\nu$ are scalars, then
$\psi:=\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}$ is equal to
$\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}$
where $w$ is the path obtained from $u$ by replacing $\beta$ by
$v$. Remark that
$\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}(\alpha)=\alpha+\tau
u+\tau\nu w$. Hence, the paths ($u$ and $w$) appearing in
$\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}(\alpha)-\alpha$ are axactely
those paths $\theta$ such that $(\alpha,\theta)$ is a bypass appearing
in one of the transvections of the product
$\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}$.
Moreover, the scalars ($\tau$ and $\tau\nu$) appearing with these
paths are exactely the scalars of the corresponding transvections in
this product. So, the computation
of $\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}(\alpha)$ can be done
just by looking for the occurences of $\alpha$ in the product
$\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}$.
From this point of view, the decomposition
$\psi=\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}$
is more useful than the decomposition $\psi=\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}$.
The aim of this section is to show that this phenomenon is a general one. In
this purpose the useful notion of derivation of a path and a total
order on the set of bypasses will be introduced.
\subsection{Derivation of paths}
\begin{definition}
\label{def2.1}
Let $u=\alpha_n\ldots\alpha_1$ and $v$ be paths in $Q$. Then $v$ is
called derived of $u$ (of order $t$) if there exist indices
$1\leqslant i_1<\ldots<i_t\leqslant n$ and bypasses
$(\alpha_{i_1},v_1),\ldots,(\alpha_{i_t},v_t)$ such that $v$ is
obtained from $u$ by replacing $\alpha_{i_l}$ by $v_l$, for each $l$:
\begin{equation}
v=\alpha_n\ldots\alpha_{i_t+1}v_t\alpha_{i_t-1}\ldots\alpha_{i_l+1}v_l\alpha_{i_l-1}\ldots
\alpha_{i_1+1}v_1\alpha_{i_1-1}\ldots\alpha_1\notag
\varepsilonnd{equation}
\varepsilonnd{definition}
\begin{rem}
\label{rem2.2}
If $\alpha\in Q_1$, then $u$ is derived of $\alpha$ if and only if
$(\alpha,u)$ is a bypass.
\varepsilonnd{rem}
With the above definition, the following lemma is easily
verified using the fact that $Q$ has no
multiple arrows and no oriented cycle.
\begin{lem}
\label{lem2.3}
\begin{enumerate}
\item If $v$ is derived of $u$ with both orders $t$ and $t'$, then
$t$=$t'$.
\item If $v$ is derived of $u$ of order $t$ then there exists a
sequence of paths $u_0=u,u_1,\ldots,u_t=v$ such that $u_i$ is derived of
$u_{i-1}$ of order $1$ for any $i$.
\item If $v$ is derived of $u$ of order $t$, then $|v|\Gammaeqslant
|u|+t$.
\item If $v$ is derived of $u$ of order $t$ and if $w$ is derived of
$v$ of order $t'$, then $w$ is derived of $u$ of order at least $t$.
\item Let $u,v,w$ be paths verifying:
\begin{itemize}
\item $v$ is derived of $u$,
\item $w$ is derived of $v$,
\item $w$ is derived of $u$ of order $1$,
\varepsilonnd{itemize}
then we have:
\begin{equation}
u=u_2\alpha u_1,\ \ v=u_2\theta u_1,\ \ w=u_2\theta' u_1\notag
\varepsilonnd{equation}
where $u_1,u_2$ are paths, $(\alpha,\theta)$ is a bypass and
$\theta'$ is derived of $\theta$.
\item If $v$ (resp. $v'$) is derived of $u$ (resp. of $u'$) of order
$t$ (resp. $t'$), then $v'v$ is derived of $u'u$ of order $t'+t$,
whenever these compositions of paths are well defined.
\varepsilonnd{enumerate}
\varepsilonnd{lem}
The following example shows that, in the preceding lemma, the inequality in the $4$-th point may be an equality.
\begin{ex}
\label{ex2.4}
Let $(\alpha,u,\beta,v)$ be a double bypass. Let $u_1,u_2$ be the paths
such that $u=u_2\beta u_1$. Then $u$ is derived of $\alpha$ of order
$1$, $w:=u_2v u_1$ is derived of $u$ of order $1$ and $w$ is
derived of $u$ of order $1$.
\varepsilonnd{ex}
\subsection{Order between paths, order between bypasses}
Now, we construct a total order on the set of non
trivial paths in $Q$. This construction is a particular case of the
one introduced in \cite{farkas_feustel_green}. Also it depends on an
arbitrary order $\vartriangleleft$ on $Q_1$. We assume that this order
$\vartriangleleft$ is fixed for this subsection. We shall write
$\vartriangleleft$ for the lexicographical order induced by
$\vartriangleleft$ on the set of nontrivial paths in $Q$. For details
on the correctness of the following definition we refer the reader to
\cite{farkas_feustel_green}.
\begin{definition}
\label{def2.5}
For $\alpha\in Q_1$, set:
\begin{equation}
W(\alpha)=Card(B(\alpha))\ \ \text{where}\ \ B(\alpha)=\{(\alpha,u)\
|\ (\alpha,u)\ \text{is a bypass in $Q$}\}\notag
\varepsilonnd{equation}
For $u=\alpha_n\ldots\alpha_1$ a path in $Q$ (with $\alpha_i\in Q_1$),
let us set:
\begin{equation}
W(u)=W(\alpha_n)+\ldots+W(\alpha_1)\notag
\varepsilonnd{equation}
These data define a total order $<$ on the set of non trivial paths in
$Q$ as follows:
\begin{equation}
u<v\Leftrightarrow \left\{
\begin{array}{rl}
&W(u) <W(v)\\
or&\\
&W(u)=W(v)\ and\ u\vartriangleleft v
\varepsilonnd{array}\right.\notag
\varepsilonnd{equation}
We shall write $<$ for the lexicographical order induced by $<$
on the set of couples of paths.
\varepsilonnd{definition}
\begin{rem}
\label{rem2.6}
If
$u$ and $v$ are (non trivial) paths such that $vu$ is well defined,
then $W(vu)=W(u)+W(v)$.
\varepsilonnd{rem}
\begin{ex}
Let $Q$ be the following quiver without oriented cycle and without
multiple arrows:
\begin{equation}
\xymatrix{
&&3\ar@{->}[rd]^f&&&\\
&2\ar@{->}[ru]^e \ar@{->}[rr]^d&&4\ar@{->}[rd]^g&&\\
1\ar@{->}[ru]^c \ar@{->}[rrru]^b \ar@{->}[rrrr]_a&&&&5\ar@{->}[r]_h&6
}\notag
\varepsilonnd{equation}
and let $\vartriangleleft$ be any total order on $Q_1$. Then, $B(a)=\{(a,bg),\,(a,gdc),\,(a,gfe)\}$,
$B(b)=\{(b,cd),\,(b,fec)\}$, $B(d)=\{(d,fe)\}$ and $B(x)=\varepsilonmptyset$
for $x\in Q_1\backslash\{a,b,d\}$. In particular, the paths with
source $1$ and target $5$ are ordered as follows:
\begin{equation}
gfec<gdc<gb<a\notag
\varepsilonnd{equation}
\varepsilonnd{ex}
\begin{lem}
\label{lem2.7}
\begin{enumerate}
\item If $u,v,u',v'$ are paths such that $v<u$
and $v'<u'$ then $v'v<u'u$ whenever these compositions are well
defined.
\item If $(\alpha,u)$ is a bypass, then $W(u)<W(\alpha)$. So
$u<\alpha$.
\item If $v$ is derived of $u$, then $v<u$.
\item If $(\alpha,u,\beta,v)$ is a double bypass and if $w$ is the
path obtained from $u$ after replacing $\beta$ by $v$, then:
\begin{equation}
(\beta,v)<(\alpha,w)<(\alpha,u)\notag
\varepsilonnd{equation}
\varepsilonnd{enumerate}
\varepsilonnd{lem}
\noindent{\textbf{Proof:}} $1)$ is a direct consequence of
Definition~\ref{def2.5} and Remark~\ref{rem2.6}.
$2)$ Let us write $u=a_n\ldots a_1$ with $a_i\in Q_1$ for each
$i$ (hence $a_i\neq a_j$ if $i\neq j$ because $Q$ has no oriented
cycle). Therefore:
\begin{enumerate}
\item[.] $B(a_1),\ldots,B(a_n)$ are pairwise disjoint,
\item[.] $W(u)=W(a_1)+\ldots+W(a_n)$
\varepsilonnd{enumerate}
Notice that if $(a_i,v)\in B(a_i)$, then
$(\alpha,a_n\ldots a_{i+1}v a_{i-1}\ldots
a_1)\in B(\alpha)$. Thus, we have a well defined mapping:
\begin{equation}
\begin{array}{crcl}
\theta\colon & B(a_1)\sqcup\ldots\sqcup B(a_n) & \longrightarrow &
B(\alpha)\\
& (a_i,v) & \longmapsto &(\alpha,a_n\ldots a_{i+1}v a_{i-1}\ldots a_1)
\varepsilonnd{array}\notag
\varepsilonnd{equation}
This mapping is one-to-one, indeed:
\begin{enumerate}
\item[.] if $\theta(a_i,v)=\theta(a_i,v')$ with $(a_i,v),(a_i,v')\in
B(a_i)$ then:
\begin{equation}
a_n\ldots a_{i+1}va_{i-1}\ldots a_1=a_n\ldots
a_{i+1}v'a_{i-1}\ldots a_1\notag
\varepsilonnd{equation}
and therefore $(a_i,v)=(a_i,v')$,
\item[.] if $\theta(a_i,v)=\theta(a_j,v')$ with $(a_i,v)\in B(a_i)$,
$(a_j,v')\in B(a_j)$ and $j<i$, then:
\begin{equation}
a_n\ldots a_{i+1}va_{i-1}\ldots a_1=a_n\ldots a_{j+1}v'a_{j-1}\ldots
a_1\notag
\varepsilonnd{equation}
So:
\begin{equation}
va_{i-1}\ldots a_1=a_i\ldots a_{j+1}v'a_{j-1}\ldots a_1\notag
\varepsilonnd{equation}
Since $v$ and $a_i$ are parallel and since $Q$ has no
oriented cycle, we infer that $v=a_i$ which is impossible because
$(a_i,v)\in B(a_i)$.
\varepsilonnd{enumerate}
On the other hand, $\theta$ is not onto. Indeed, if there exists
$(a_i,v)\in B(a_i)$ verifying $\theta(a_i,v)=(\alpha,u)$, then:
\begin{equation}
a_n\ldots a_1=u=a_n\ldots a_{i+1}v a_{i-1}\ldots a_1\notag
\varepsilonnd{equation}
which implies $a_i=v$, a contradiction. Since $\theta$ is one-to-one
and not onto, we deduce that:
\begin{equation}
W(\alpha)=Card(B(\alpha))>Card(B(a_1)\sqcup \ldots\sqcup B(a_n))=W(u)\notag
\varepsilonnd{equation}
This proves that $W(u)<W(\alpha)$ and that $u<\alpha$.
$3)$ is a direct consequence of $1)$ and of $2)$.
$4)$ Let us write $u=u_2\beta u_1$ (with $u_1,u_2$ paths) so that
$w=u_2vu_1$. From $2)$, we have:
\begin{equation}
W(\alpha)>W(u)=W(u_1)+W(\beta)+W(u_2)\Gammaeqslant W(\beta)\notag
\varepsilonnd{equation}
So $\beta<\alpha$ and therefore
$(\beta,v)<(\alpha,w)$.
Using $2)$ again, we also have:
\begin{equation}
W(w)=W(u_2)+W(v)+W(u_1)<W(u_2)+W(\beta)+W(u_1)=W(u)\notag
\varepsilonnd{equation}
So $w<u$ and therefore
$(\alpha,w)<(\alpha,u)$
$\blacksquare$\\
Unless otherwise specified, $<$ will always denote an order on the set
of paths as in Definition~\ref{def2.5}.
\subsection{Image of a path by a product of transvections}
In this paragraph, we apply the previous constructions to find an easy
way to compute $\psi(u)$ when $\psi\in\c T$ and $u$ is a path in
$Q$. We begin with the following lemma on the description of
$\psi(\alpha)$ when $\psi\in\c T$ and $\alpha\in Q_1$. Recall that $Q$ has no
multiple arrows and no oriented cycle.
\begin{lem}
\label{lem2.8}
Let $\psi\in\c T$ and let $\alpha\in Q_1$. Then $\psi(\alpha)-\alpha$
is a linear combination of paths parallel to $\alpha$ and of length
greater than or equal to $2$. In particular, $\alpha\in
supp(\psi(\alpha))$ and $\alpha^*(\psi(\alpha))=1$.
\varepsilonnd{lem}
\noindent{\textbf{Proof:}}
The conclusion is immediate if $\psi$ is a transvection because $Q$
has no multiple arrows. The conclusion in the general case is obtained
using an easy induction on the number of transvections whose product
equal $\psi$.
$\blacksquare$\\
The preceding lemma gives the following description of $\psi(u)$ when
$\psi\in\c T$ and $u$ is a path. We omit the proof which is immediate
thanks to Lemma~\ref{lem2.8} and to point $6)$ of Lemma~\ref{lem2.3}.
\begin{prop}
\label{prop2.9}
Let $\psi\in\c T$ and let $u=\alpha_n\ldots\alpha_1$ be a path in $Q$
(with $a_i\in Q_1$ for any $i$). For each $i$, let:
\begin{equation}
\psi(\alpha_i)=\alpha_i+\sum\limits_{j=1}^{m_i}\lambda_{i,j}u_{i_j}\notag
\varepsilonnd{equation}
be a normal form for $\psi(\alpha_i)$. Then $supp(\psi(u))$ is the set
of the paths in $Q$ described as follows. Let $r\in \{0,\ldots,n\}$.
Let $1\leqslant i_1<\ldots<i_r\leqslant n$ be indices. For each
$l\in\{1,\ldots,r\}$, let $j_l\in\{1,\ldots, m_{i_l}\}$. Then the
following path obtained from $u$ after replacing $\alpha_{i_l}$ by
$u_{j_l}$ for each $l$ belongs to $supp(\psi(u))$:
\begin{equation}
\alpha_n\ldots\alpha_{i_r+1}u_{j_r}\alpha_{i_r-1}
\ldots\alpha_{i_l+1}u_{j_l}\alpha_{i_l-1}\ldots
\alpha_{i_1+1}u_{j_1}\alpha_{i_1-1}\ldots\alpha_1\notag
\varepsilonnd{equation}
Moreover, this path appears in $\psi(u)$ with coefficient:
\begin{equation}
\lambda_{i_1,j_1}\ldots \lambda_{i_r,j_r}\notag
\varepsilonnd{equation}
As a consequence, $\psi(u)-u$ is a linear combination of paths derived
of $u$.
\varepsilonnd{prop}
\begin{ex}
The previous proposition does not hold if $Q$ has multiple arrows. For
example, if $Q$ is the Kronecker quiver
$\xymatrix{1\ar@/^/@{->}[r]^a\ar@/_/@{->}[r]_b&2}$ and if
$\psi=\varphi_{a,b,1}\varphi_{b,a,-1}\varphi_{a,b,1}$, then
$\psi(a)=b$ and $\psi(b)=-a$.
\varepsilonnd{ex}
\begin{rem}
\label{rem2.10}
If $(\alpha,u)$ is a bypass and if $v\in supp(\psi(u)-u)$, then
$(\alpha,v)$ is also a bypass and $(\alpha,v)<(\alpha,u)$.
\varepsilonnd{rem}
Now we are able to state the main result of this paragraph. It describes
$\psi(\alpha)$ $(\alpha\in Q_1)$ using a particular writing of $\psi$
as a product of transvections. Notice that the following proposition
formalises the phenomenon observed at the begining of the section.
\begin{prop}
\label{prop2.11}
Let $(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)$ be an increasing sequence of
bypasses, let
$\tau_1,\ldots,\tau_n\in k^*$ and set
$\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$.
For any $\alpha\in Q_1$, there is a normal form for $\psi(\alpha)$:
\begin{equation}
\psi(\alpha)=\alpha+\sum_{i\ such\ that\ \alpha=\alpha_i}\tau_iu_i\notag
\varepsilonnd{equation}
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} Let us prove that the conclusion of the
proposition is true using an induction on $n\Gammaeqslant 1$. By
definition of a transvection, the proposition holds of $n=1$. Assume
that $n\Gammaeqslant 2$ and that the conclusion of the proposition holds
if we replace
$\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$
by
$\varphi_{\alpha_{n-1},u_{n-1},\tau_{n-1}}\ldots\varphi_{\alpha_1,u_1,\tau_1}$.
Therefore, for $\alpha\in Q_1$, we have a normal form:
\begin{equation}
\varphi_{\alpha_{n-1},u_{n-1},\tau_{n-1}}\ldots\varphi_{\alpha_1,u_1,\tau_1}(\alpha)=
\alpha +\sum\limits_{i\leqslant n-1,\ \alpha=\alpha_i}\tau_iu_i\notag
\varepsilonnd{equation}
So:
\begin{equation}
\psi(\alpha) = \varphi_{\alpha_n,u_n,\tau_n}(\alpha)+
\sum\limits_{i\leqslant n-1,\
\alpha=\alpha_i}\tau_i\varphi_{\alpha_n,u_n,\tau_n}(u_i)\tag{$i$}
\varepsilonnd{equation}
Let $i\in\{1,\ldots,n-1\}$. Thanks to Lemma~\ref{lem2.7}, the
inequality $(\alpha_i,u_i)<(\alpha_n,u_n)$ implies that
$(\alpha_i,u_i,\alpha_n,u_n)$ is not a double bypass. Thus,
$\alpha_n$ does not appear in the path $u_i$. This proves that:
\begin{equation}
(\forall i\in\{1,\ldots,n-1\})\ \
\varphi_{\alpha_n,u_n,\tau_n}(u_i)=u_i\tag{$ii$}
\varepsilonnd{equation}
The definition of $\varphi_{\alpha_n,u_n,\tau_n}$, together with $(i)$
and $(ii)$, imply the equality:
\begin{equation}
\psi(\alpha) = \alpha+
\sum\limits_{
\alpha=\alpha_i}\tau_iu_i\tag{$iii$}
\varepsilonnd{equation}
It only remains to prove that the equality $(iii)$ is a normal
form. Remark that all the scalars which appear in the right-hand side
of $(iii)$ are non zero. Moreover, if $i\in\{1,\ldots,n\}$ verifies
$\alpha=\alpha_i$, then $\alpha\neq u_i$, because $(\alpha,u_i)$ is a
bypass. Finally, if $1\leqslant i<j\leqslant n$ verify
$\alpha=\alpha_i=\alpha_j$, then $(\alpha,u_i)=(\alpha_i,u_i)<(\alpha_j,u_j)=(\alpha,u_j)$ so
$u_i\neq u_j$. Therefore, $(iii)$ is a normal form for
$\psi(\alpha)$.
$\blacksquare$\\
When $\psi\in \c T$ is like in Proposition~\ref{prop2.11}, we
shall say that $\psi$ is written as a decreasing product of
transvections. Later we will prove that any $\psi\in\c T$ can be
written uniquely as a decreasing product of transvections.
The description in Proposition~\ref{prop2.11} will be particularly
useful in the sequel.
We end this paragraph with two propositions concerning the description
of $\psi(r)$ when $\psi\in\c T$ and $r$ is a linear combination of
paths. The following proposition gives conditions for $\psi^{-1}(r')$
to be a subexpression of $r$ when $r'$ is a subexpression of
$\psi(r)$.
\begin{prop}
\label{prop2.12}
Let $\psi\in \c T$, let $r\in\ _ykQ_x$ and let $r'$ be a subexpression
of $\psi(r)$. Let $\simeq$ be the equivalence relation on the set of
paths in $Q$ generated by:
\begin{equation}
v\in supp(\psi(u))\Rightarrow u\simeq v\notag
\varepsilonnd{equation}
Assume that for any $u,v\in supp(\psi(r))$ verifying $u\simeq v$ we
have:
\begin{equation}
u\in supp(r')\Leftrightarrow v\in supp(r')\notag
\varepsilonnd{equation}
Then $\psi^{-1}(r')$ is a subexpression of $r$.
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} Let $\simeq'$ be the trace of $\simeq$ on
$supp(r)$ and let us write $supp(r)=c_1\sqcup \ldots \sqcup c_n$ as a disjoint union of its
$\simeq'$-classes. This partition of $supp(r)$ defines a decomposition of $r=r_1+\ldots+r_n$
where $r_i$ is the subexpression of $r$ verifying $supp(r_i)=c_i$. For
each $i$, let us fix a normal form:
\begin{equation}
r_i=\sum\limits_{j=1}^{n_i}t_{i,j}u_{i,j}\notag
\varepsilonnd{equation}
so that we have the following normal form for $r$:
\begin{equation}
r=\sum\limits_{i=1}^n\sum\limits_{j=1}^{n_i}t_{i,j}u_{i,j}\notag
\varepsilonnd{equation}
Let us set $r_i':=\psi(r_i)$. In order to prove that $\psi^{-1}(r')$
is a subexpression of $r$, we will prove that there exist indices
$1\leqslant i_1<\ldots<i_t\leqslant n$ verifying
$r'=r_{i_1}'+\ldots+r_{i_t}'$ (so that
$\psi^{-1}(r')=r_{i_1}'+\ldots+r_{i_t}'$). In this purpose, we will
successively prove the following
facts:
\begin{enumerate}
\item[$1)$] $u,v\in supp(r_i')\Rightarrow u\simeq v$, for any $i$,
\item[$2)$] $supp(r_1'),\ldots,supp(r_n')$ are pairwise disjoint,
\item[$3)$] for each $i$, $r_i'$ is a subexpression of $\psi(r)$,
\item[$4)$] if $i\in\{1,\ldots,n\}$ verifies $supp(r')\cap
supp(r_i')\neq\varepsilonmptyset$, then $supp(r_i')\subseteq supp(r')$,
\varepsilonnd{enumerate}
$1)$ Let $i\in\{1,\ldots,n\}$ and let $u,v\in supp(r_i')$. So there
exist $u',v'\in supp(r_i)$ such that
$u\in supp(\psi(u'))$ and $v\in supp(\psi(v'))$.
By definition of $\simeq$ and of $r_i$, we deduce that:
\begin{equation}
u,v\in supp(r_i')\Rightarrow u\simeq v\tag{$i$}
\varepsilonnd{equation}
$2)$ Let $i,j\in \{1,\ldots,n\}$ be such that there exists $v\in
supp(r_i')\cap supp(r_j')$. So there exist $u\in supp(r_i)$ and $u'\in
supp(r_j)$ such that
$v\in supp(\psi(u))$ and $v\in supp(\psi(u'))$.
This implies that
$u\simeq v\simeq u'$.
Since $u\in c_i=supp(r_i)$ and $u'\in c_j=supp(r_j)$, we deduce that
$c_i=c_j$ and therefore $i=j$. So:
\begin{equation}
i\neq j\Rightarrow supp(r_i')\cap supp(r_j')=\varepsilonmptyset\tag{$ii$}
\varepsilonnd{equation}
$3)$ We have $\psi(r)=r_1'+\ldots +r_n'$ so $(ii)$ implies that:
\begin{equation}
\text{$r_i'$ is a subexpression of $\psi(r)$ for any $i$}\tag{$iii$}
\varepsilonnd{equation}
$4)$ Let $i\in\{1,\ldots,n\}$ and assume that there exists $u\in
supp(r_i')\cap supp(r')$. If $v\in supp(r_i')$ then $u\simeq v$ thanks
to $(i)$. So, by assumption on $r'$, we have $v\in supp(r')$. This
proves that:
\begin{equation}
supp(r_i')\cap supp(r')\neq \varepsilonmptyset\Rightarrow supp(r_i')\subseteq
supp(r')\tag{$iv$}
\varepsilonnd{equation}
Now, we can prove that $\psi^{-1}(r')$ is a subexpression of
$r$. Thanks to $(iii)$, the elements $r',r_1',\ldots,r_n'$ are
subexpressions of $\psi(r)$. So $(iv)$ and the equality
$\psi(r)=r_1'+\ldots +r_n'$ imply that there exist indices $1\leqslant
i_1<\ldots < i_t\leqslant n$ such that
$r'=r_{i_1}'+\ldots+r_{i_t}'$.
So
$\psi^{-1}(r')=r_{i_1}+\ldots+r_{i_n}$.
This proves that $\psi^{-1}(r')$ is a subexpression of
$r$.
$\blacksquare$\\
The last proposition of this subsection gives a sufficient condition
on $u\in supp(r)$ to verify $u\in supp(\psi(r))$.
\begin{prop}
\label{prop2.13}
Let $\psi\in\c T$, let $r\in\ _ykQ_x$ and let $u\in supp(r)$. Then, at
least one of the two following facts is verified:
\begin{enumerate}
\item $u\in supp(\psi(r))$,
\item there exists $v\in supp(r)$ such that $u\neq v$ and such that
$u\in supp(\psi(v))$.
\varepsilonnd{enumerate}
As a consequence, if $u$ is not derived of $v$ for any $v\in supp(r)$,
then:
\begin{equation}
u\in supp(\psi(r))\ \ \text{and}\ \ u^*(\psi(r))=u^*(r)\notag
\varepsilonnd{equation}
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} Let us fix a normal form
$r=\sum\limits_{i=1}^nt_iu_i$
where we may assume that $u=u_1$. Let us assume that $u\not\in
supp(\psi(r))$, i.e. $u^*(\psi(r))=0$. Recall from
Proposition~\ref{prop2.9} that $u^*(\psi(u))=1$, so:
\begin{equation}
0=u^*(\psi(r))=t_1+\sum\limits_{i=2}^nt_iu^*(\psi(u_i))\tag{$i$}
\varepsilonnd{equation}
Therefore, there exists $i_0\in\{2,\ldots,n\}$
such that $u^*(\psi(u_{i_0}))\neq 0$. So:
\begin{equation}
u_{i_0}\in supp(r),\ \ u_{i_0}\neq u_1=u\ \ \text{and}\ \
u_1^*(\psi(u_{i_0}))\neq 0\notag
\varepsilonnd{equation}
This proves the first assertion of the proposition.
Now let us assume that $u$ is not derived of $v$ for any $v\in
supp(r)$. Let $i\in\{2,\ldots,n\}$. Since $u=u_1\neq u_i$,
Proposition~\ref{prop2.9} gives the following implications:
\begin{equation}
u\in supp(\psi(u_i))\Rightarrow u\in supp(\psi(u_i)-u_i)\Rightarrow
\text{$u$ is derived of $u_i$}\notag
\varepsilonnd{equation}
By assumption on $u$, this implies that $u^*(\psi(u_i))=0$ for any
$i\Gammaeqslant 2$. Using $(i)$, we deduce the announced conclusion:
$u^*(\psi(r))=t_1=u^*(r)\neq 0$
$\blacksquare$\\
\subsection{Ordering products of transvections}
In Proposition~\ref{prop2.11} we have seen that $\psi(\alpha)$ may be
computed easily when $\psi\in \c T$ and $\alpha\in Q_1$ provided that
$\psi$ is written as a decreasing product of transvections. The main
result of this
subsection proves that any $\psi\in\c T$ can be uniquely written
that way. Recall that $<$ is an order on the set of non trivial
paths in $Q$ defined in Definition~\ref{def2.5}. The following
notations will be useful.
\begin{definition}
\label{def2.14}
Let $(\alpha,u)$ be a bypass. We set $\c T_{<(\alpha,u)}$ and $\c
T_{\leqslant (\alpha,u)}$ to be the subgroups of $\c T$ generated by
the following sets of transvections:
\begin{equation}
\begin{array}{l}
\{\varphi_{\beta,v,\tau}\ |\ (\beta,v)<(\alpha,u)\ \text{and}\ \tau\in
k\}\ \ for\ \c T_{<(\alpha,u)}\\
\{\varphi_{\beta,v,\tau}\ |\ (\beta,v)\leqslant (\alpha,u)\ \text{and}\ \tau\in
k\}\ \ for\ \c T_{\leqslant(\alpha,u)}
\varepsilonnd{array}\notag
\varepsilonnd{equation}
Also, we define $\c T_{(\alpha,u)}$ to be the following subgroup of
$\c T$:
\begin{equation}
\c T_{(\alpha,u)}=\{\varphi_{\alpha,u,\tau}\ |\ \tau\in k\}\notag
\varepsilonnd{equation}
\varepsilonnd{definition}
\begin{rem}
\label{rem2.15}
\begin{enumerate}
\item[.] $\c T_{(\alpha,u)}$ is indeed a subgroup of $\c T$ because
$\varphi_{\alpha,u,\tau}\varphi_{\alpha,u,\tau'}=\varphi_{\alpha,u,\tau+\tau'}$
for any $\tau,\tau'\in k$. Actually, the following mapping is an
isomorphism of abelian groups:
\begin{equation}
\begin{array}{rcl}
k&\longrightarrow & \c T_{(\alpha,u)}\\
\tau&\longmapsto & \varphi_{\alpha,u,\tau}
\varepsilonnd{array}\notag
\varepsilonnd{equation}
\item[.] $\c T_{\leqslant(\alpha,u)}$ is generated by $\c
T_{<(\alpha,u)}\cup \c T_{(\alpha,u)}$.
\item[.] If $(\alpha,u)<(\beta,v)$, then $\c
T_{\leqslant(\alpha,u)}\subseteq \c T_{\leqslant(\beta,v)}$ and
$T_{<(\alpha,u)}\subseteq T_{<(\beta,v)}$.
\item[.] $\c T=\bigcup\limits_{(\alpha,u)}\c T_{\leqslant (\alpha,u)}$
and if $(\alpha_m,u_m)$ is the greatest bypass in $Q$, then $\c T=\c
T_{\leqslant {(\alpha_m,u_m)}}$ (recall that $Q$ has finitely many
bypasses because it has no oriented cycle).
\varepsilonnd{enumerate}
\varepsilonnd{rem}
The following lemma proves that any $\psi\in \c T$ is a decreasing
product of transvections.
\begin{lem}
\label{lem2.16}
\begin{enumerate}
\item[.] $\c T_{<(\alpha,u)}$ is a normal subgroup of $\c T_{\leqslant
(\alpha,u)}$, for any bypass $(\alpha,u)$.
\item[.] Let $(a_1,v_1)<\ldots<(a_N,v_N)$ be the (finite) increasing
sequence of all the bypasses in $Q$. Then:
\begin{enumerate}
\item[-] $\c T_{ <(a_i,v_i)}=\c T_{\leqslant(a_{i-1},v_{i-1})}$ if
$i\Gammaeqslant 1$,
\item[-] $\c T_{<(a_1,v_1)}=1$,
\item[-] $\c T_{\leqslant (a_i,v_i)}=\c T_{(a_i,v_i)}\c
T_{(a_{i-1},v_{i-1})}\ldots\c T_{(a_1,v_1)}$.
\varepsilonnd{enumerate}
\varepsilonnd{enumerate}
\varepsilonnd{lem}
\noindent{\textbf{Proof:}} Thanks to Remark~\ref{rem2.15}, it is
sufficient to prove that if $\tau,\nu\in k$ and if $(\beta,v),(\alpha,u)$
are bypasses such that $(\beta,v)<(\alpha,u)$, then:
\begin{equation}
\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}\in
\varphi_{\alpha,u,\tau}\c T_{<(\alpha,u)}\tag{$\star$}
\varepsilonnd{equation}
There are two situations wether $(\alpha,u,\beta,v)$ is a double
bypass or not.
If $(\alpha,u,\beta,v)$ is a double bypass, then
Section~$1$ gives:
\begin{equation}
\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}=
\varphi_{\alpha,u,\tau}\varphi_{\alpha,w,\tau\nu}\varphi_{\beta,v,\nu}\notag
\varepsilonnd{equation}
where $w$ is the path obtained from $u$ after replacing $\beta$ by
$v$. Moreover, Lemma~\ref{lem2.7} implies that
$(\beta,v)<(\alpha,w)<(\alpha,u)$. Therefore, $(\star)$ is satisfied when
$(\alpha,u,\beta,v)$ is a double bypass.
If $(\alpha,u,\beta,v)$ is not a double bypass, then
Section~$1$ gives (notice that thanks to Lemma~\ref{lem2.7} and to
the inequality $(\beta,v)<(\alpha,u)$ we know that $(\beta,v,\alpha,u)$ is not a
double bypass):
\begin{equation}
\varphi_{\alpha,u,\tau}\varphi_{\beta,v,\nu}=\varphi_{\beta,v,\nu}\varphi_{\alpha,u,\tau}\notag
\varepsilonnd{equation}
So $(\star)$ is also satisfied when $(\alpha,u,\beta,v)$ is not a double
bypass.
$\blacksquare$\\
Using the preceding lemma and Proposition~\ref{prop2.11}, it is now
possible to prove that any $\psi\in \c T$ is
uniquely a decreasing product of transvections.
\begin{prop}
\label{prop2.17}
Let $(\alpha,u)$ be a bypass and let $\psi\in\c
T_{\leqslant(\alpha,u)}$. Then, there exist a non negative integer $n$,
a sequence of bypasses $(\alpha_1,u_1),\ldots,(\alpha_n,u_n)$ and non
zero scalars $\tau_1,\ldots,\tau_n\in k^*$ verifying:
\begin{enumerate}
\item[(i)]
$\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$,
\item[(ii)] $(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)\leqslant (\alpha,u)$.
\varepsilonnd{enumerate}
Moreover, the integer $n$ and the sequence
$(\alpha_1,u_1,\tau_1),\ldots,(\alpha_n,u_n,\tau_n)$ are unique for
these properties.
\varepsilonnd{prop}
\noindent{\textbf{Proof:}}
The existence is given by Lemma~\ref{lem2.16}. So it suffices to
characterise the triples $(\alpha_i,u_i,\tau_i)$ using $\psi$ only. Let
$A,B$ and $T$ be the following sets:
\begin{equation}
\begin{array}{l}
A:=\{\alpha\in Q_1\ |\ \psi(\alpha)\neq \alpha\}\\
B:=\{(\alpha,u)\ |\ \text{$(\alpha,u)$ is a bypass, $\alpha\in A$ and
$u\in supp(\psi(\alpha))$}\}\\
T:=\{(\alpha,u,\tau)\ |\ (\alpha,u)\in B\ \text{and}\
\tau=u^*(\psi(\alpha))\}
\varepsilonnd{array}\notag
\varepsilonnd{equation}
Notice that the definition of $A,B,T$ depend on $\psi$ only (and not
on the triples $(\alpha_i,u_i,\tau_i)$). Let $\beta\in Q_1$. Then
Proposition~\ref{prop2.11} gives a normal form:
\begin{equation}
\psi(\beta)=\beta+\sum\limits_{i\ \text{such that}\
\beta=\alpha_i}\tau_iu_i\notag
\varepsilonnd{equation}
By definition of a normal form and because of ($i$) and ($ii$), the
following equalities hold:
\begin{equation}
\begin{array}{l}
A=\{\alpha_1,\ldots,\alpha_n\}\\
B=\{(\alpha_1,u_1),\ldots,(\alpha_n,u_n)\}\\
T=\{(\alpha_1,u_1,\tau_1),\ldots,(\alpha_n,u_n,\tau_1)\}
\varepsilonnd{array}\notag
\varepsilonnd{equation}
This proves that $n$ and
$(\alpha_1,u_1,\tau_1),\ldots,(\alpha_n,u_n,\tau_n)$ are uniquely
determined by the sets $A,B,T$ (which depend on $\psi$ only) and by
the total order $<$.
$\blacksquare$\\
\section{Comparison of the presentations of a monomial algebra}
Let $A=kQ/I_0$ with $I_0$ a monomial admissible ideal of $kQ$ and let
$kQ/I\simeq A$ be an admissible presentation of $A$. Thanks to
Proposition~\ref{prop1.1}, there exists $\psi$ a
product of transvections and of a dilatation such that
$\psi(I_0)=I$. The aim of this section is to exhibit $\psi_I$ the
``simplest'' possible among all the $\psi$'s verifying
$\psi(I_0)=I$. It will appear that $\psi_I$ verifies a property which
makes it unique. The construction of $\psi_I$ will use specific
properties of the Gröbner basis of $I$, due to the fact that $I_0$ is
monomial. So, throughout the section, $<$ will denote a total order
on the set of non trivial paths in $Q$, as in
Definition~\ref{def2.5}. Before studying the Gröbner basis of $I$, it
is useful to give some properties on
the automorphisms $\psi\in AUt_0(kQ)$ verifying $\psi(I_0)=I_0$.
\begin{lem}
\label{lem3.1}
Let $D\in \c D$ be a dilatation. Then $D(I_0)=I_0$. As a consequence,
if $kQ/I\simeq A$ is an admissible presentation, then there exists
$\psi\in\c T$ such that $\psi(I_0)=I$.
\varepsilonnd{lem}
\noindent{\textbf{Proof:}} The first assertion is due to the fact that
$D(u)\in k^*u$ for any path $u$ and to the fact that $I_0$ is
monomial. The second one is a consequence of the first one and of
Proposition~\ref{prop1.1}.
$\blacksquare$\\
\begin{lem}
\label{lem3.2}
Let $(\alpha,u)$ be a bypass in $Q$. Then exactly one of the two
following assertions is satisfied:
\begin{enumerate}
\item[.] $\varphi_{\alpha,u,\tau}(I_0)=I_0$ for any $\tau\in k$.
\item[.] $\varphi_{\alpha,u,\tau}(I_0)\neq I_0$ for any $\tau\in
k^*$.
\varepsilonnd{enumerate}
\varepsilonnd{lem}
\noindent{\textbf{Proof:}} Assume that $\tau\in k^*$ verifies
$\varphi_{\alpha,u,\tau}(I_0)=I_0$ and let $\mu\in k$. Let $v\in I_0$
be a path. If $\alpha$ does not appear in $v$, then
$\varphi_{\alpha,u,\nu}(v)=v\in I_0$. Assume that $\alpha$ appears
in $v$, i.e. $v=v_2\alpha v_1$ with $v_1,v_2$ paths in which $\alpha$
does not appear (because $Q$ has no oriented cycle). Therefore,
$\varphi_{\alpha,u,\tau}(v)=v+\tau v_2uv_1\in I_0$.
Thus, $v_2uv_1\in
I_0$. This implies that $\varphi_{\alpha,u,\nu}(v)=v+\nu v_2uv_1\in
I_0$. Since $I_0$ is monomial,
$\varphi_{\alpha,u,\nu}(I_0)=I_0$.
$\blacksquare$\\
\begin{lem}
\label{lem3.3}
Let $(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)$ be an increasing sequence
of bypasses, let $\tau_1,\ldots,\tau_n\in k^*$ and set
$\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$.
Then:
\begin{equation}
\psi(I_0)=I_0\ \ \Leftrightarrow\ \ \varphi_{\alpha_i,u_i,\tau_i}(I_0)=I_0\
\text{for any $i$}\notag
\varepsilonnd{equation}
\varepsilonnd{lem}
\noindent{\textbf{Proof:}} Assume that $\psi(I_0)=I_0$. Let
$i\in\{1,\ldots,n\}$, let $u=a_r\ldots a_1\in I_0$ be a path (with
$a_i\in Q_1$) and fix $i\in \{1,\ldots,n\}$. If $a_j\neq \alpha_i$ for any
$j\in\{1,\ldots,r\}$ then
$\varphi_{\alpha_i,u_i,\tau_i}(u)=u\in I_0$. Now assume that there
exists $j\in\{1,\ldots,r\}$ such that $a_j=\alpha_i$ ($j$ is
necessarily unique because $Q$ has no oriented cycle). Therefore:
\begin{equation}
\varphi_{\alpha_i,u_i,\tau_i}(u)=u+\tau_ia_r\ldots
a_{j+1}u_ia_{j-1}\ldots a_1\tag{$i$}
\varepsilonnd{equation}
On the other hand, Proposition~\ref{prop2.9} and
Proposition~\ref{prop2.11} imply that $a_r\ldots
a_{j+1}u_ia_{j-1}\ldots a_1\in supp(\psi(u))$. Thus (recall
that $\psi(u)\in I_0$ and that $I_0$ is monomial):
\begin{equation}
a_r\ldots a_{j+1}u_ia_{j-1}\ldots a_1\in I_0\tag{$ii$}
\varepsilonnd{equation}
From $(i)$ and $(ii)$ we deduce that
$\varphi_{\alpha_i,u_i,\tau_i}(u)\in I_0$ for any path $u\in I_0$. So
$\varphi_{\alpha_i,u_i,\tau_i}(I_0)=I_0$ for any $i$. The remaining
implication is immediate.
$\blacksquare$\\
\begin{rem}
\label{rem3.4}
The three preceding lemmas imply that the group $Aut_0(kQ,I_0)$
defined as follows:
\begin{equation}
Aut_0(kQ,I_0):=\{\psi\in Aut(kQ)\ |\ \psi(x)=x\ for\ any\ x\in
Q_0,\ and\ \psi(I_0)=I_0\}\notag
\varepsilonnd{equation}
is generated by the dilatations and by all the transvections preserving
$I_0$:
\begin{equation}
Aut_0(kQ,I_0)=<\c D\cup \{\varphi\ |\ \varphi\ is\ a\ transvection\
such\ that\ \varphi(I_0)=I_0\}>\notag
\varepsilonnd{equation}
\varepsilonnd{rem}
\begin{ex}
The preceding remark does not hold for any ideal $I$, even if $kQ/I$
is monomial. For example, let $Q$ be the quiver:
\begin{equation}
\xymatrix{
&2\ar@{->}[rd]^c &&\\
1\ar@{->}[ru]^b \ar@{->}[rr]_a & & 3\ar@{->}[r]_d &4
}\notag
\varepsilonnd{equation}
and let $I=<da-dcb>$. Notice that $kQ/I\simeq kQ/I_0$ where
$I_0=\varphi_{a,cb,1}(I)=<da>$. On the other hand:
\begin{enumerate}
\item $Id=\varphi_{a,cb,0}$ is the only transvection lying in
$Aut_0(kQ,I)$,
\item for $t\in k\backslash\{0,1\}$, the dilatation $D_t$ such that $D_t(a)=ta$ and $D_t(x)=x$ for any
other arrow $x$ does not belong to $Aut_0(kQ,I)$,
\item $D_t\varphi_{a,cb,t}\in Aut_0(kQ,I)$ for any $t\in k^*$.
\varepsilonnd{enumerate}
So $Aut_0(kQ,I)$ is not generated by $\c D$ and by the transvections
it contains.
\varepsilonnd{ex}
The following proposition gives the announced properties on the
Gröbner bases of the admissible
ideals $I$ of $kQ$ such that $kQ/I\simeq A$. Recall that for such an
$I$, there exists $\psi\in \c T$ such that $\psi(I_0)=I$ (see
Lemma~\ref{lem3.1}).
\begin{prop}
\label{prop3.5}
Let $\psi\in\c T$ and let let $I=\psi(I_0)$. Let $B_0$ (resp. $B$) be
the Groebner basis of $I_0$ (resp. of $I$). Then $B_0$ is made of all
the paths in $Q$ which belong to $I_0$. Moreover, the mapping:
\begin{equation}
\begin{array}{rcl}
B&\longrightarrow & B_0\\
r&\longmapsto & max(supp(r))
\varepsilonnd{array}\tag{$\star$}
\varepsilonnd{equation}
is well defined and bijective. For $u\in B_0$, let $r_u\in B$
be the inverse image of $u$ under $(\star)$. Then $supp(r_u-u)$ is a
set of paths derived of $u$.
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} Let $u_1<\ldots<u_n$ be the increasing
sequence of all the non trivial paths in $Q$. Let $(r_1,\ldots,r_d)$
be the Gröbner basis of $I$ and for each $j\in\{1,\ldots,d\}$, let
$i_j\in\{1,\ldots,n\}$ be such that:
\begin{equation}
r_j\in u_{i_j}+Span(u_l\ ;\ l<i_j)\notag
\varepsilonnd{equation}
Since $I_0$ is monomial, $B_0$ is made of all the paths in $Q$
belonging to $I_0$.
Let $j\in \{1,\ldots,d\}$. Since $u_{i_j}=max(supp(r_j))$, the path
$u_{i_j}$ is not derived of $u$ for any $u\in supp(r_j)$ (thanks to
Lemma~\ref{lem2.7}). So Proposition~\ref{prop2.13} implies that
$u_{i_j}\in supp(\psi^{-1}(r_j))\in I_0$. Because $I_0$ is monomial,
this proves that $u_{i_j}\in I_0$. Therefore, the mapping $(\star)$ is
well defined. It is also one-to-one because of the definition of the
Groebner basis of $I$.
Let $u\in
B_0$. Proposition~\ref{prop2.9} implies that
$u=max(supp(\psi(u))$. Since $\psi(u)\in I$, there
exists $j\in\{1,\ldots,d\}$ such that $u=u_{i_j}=max(supp(r_j))$. This
proves that $(\star)$ is onto and therefore bijective.
It remains to prove the last assertion of the proposition. This will
be done by proving by induction on $j\in\{1,\ldots,d\}$ that the following
assertion is true:
\begin{equation}
H_j:"\text{$supp(r_j-u_{i_j})$ is a set of paths derived of
$u_{i_j}$}"\notag
\varepsilonnd{equation}
Remark that Proposition~\ref{prop2.9} implies that for any $j$:
\begin{equation}
u_{i_j}=max(supp(\psi(u_{i_j})))\ \ \text{and}\
u_{i_j}^*(\psi(u_{i_j}))=1\tag{$i$}
\varepsilonnd{equation}
Moreover, $\psi(u_{i_j})\in I$ because $(\star)$ is well defined and
because $\psi(I_0)=I$.
Now begins the induction. Both $r_1$ and $\psi(u_{i_1})$ lie in
$I$. Moreover, $u_{i_1}=max(supp(r_1))$ by definition of $u_{i_1}$
and $u_{i_1}=max(supp(\psi(u_{i_1})))$ because of
Proposition~\ref{prop2.9}. So $H_1$ is true.
Assume that $j\Gammaeqslant 2$ and that $H_1,\ldots,H_{j-1}$ are
true. Since $\psi(u_{i_j})\in I$ and because of $(i)$, the following holds:
\begin{equation}
\psi(u_{i_j})=r_j+\sum\limits_{
\begin{array}{c}
j'<j,\\
u_{i_{j'}}\in supp(\psi(u_{i_j}))
\varepsilonnd{array}
}
u_{i_{j'}}^*(\psi(u_{i_j}))r_{j'}\notag
\varepsilonnd{equation}
So:
\begin{equation}
r_j-u_{i_j}=\psi(u_{i_j})-u_{i_j}-\sum\limits_{
\begin{array}{c}
j'<j,\\
u_{i_{j'}}\in supp(\psi(u_{i_j}))
\varepsilonnd{array}
}
u_{i_{j'}}^*(\psi(u_{i_j}))\left[(r_{j'}-u_{i_{j'}})+u_{i_j'}\right]
\tag{$ii$}
\varepsilonnd{equation}
Notice that in the above equality:
\begin{enumerate}
\item[$(iii)$] $supp(\psi(u_{i_j})-u_{i_j})$ is a set of paths derived of
$u_{i_j}$ (thanks to Proposition~\ref{prop2.9}),
\item[$(iv)$] if $j'<j$ verifies $u_{i_{j'}}\in supp(\psi(u_{i_j}))$, then:
\begin{enumerate}
\item[$(v)$] $u_{i_{j'}}$ is derived of $u_{i_j}$ (see $(iii)$ above),
\item[$(vi)$] $supp(r_{j'}-u_{i_{j'}})$ is a set of paths derived of
$u_{i_{j'}}$ (because $H_{j'}$ is true) and therefore derived of
$u_{i_j}$ (thanks to $(v)$ and to Lemma~\ref{lem2.3}).
\varepsilonnd{enumerate}
\varepsilonnd{enumerate}
The points $(ii)-(vi)$ prove that $H_j$ is true. Hence,
$H_j$ is true for any $j\in\{1,\ldots,d\}$. This finishes the proof of
the proposition.
$\blacksquare$\\
Now it is possible to define precisely the automorphism $\psi_I$ mentionned at the
beginning of the section.
\begin{prop}
\label{prop3.6}
Let $kQ/I\simeq A$ be an admissible presentation. Then there exists a
unique $\psi_I\in\c T$ verifying the following conditions:
\begin{enumerate}
\item[$1)$] $\psi_I(I_0)=I$,
\item[$2)$] if $(\alpha,u)$ is a bypass such that $u\in
supp(\psi_I(\alpha))$ then $\varphi_{\alpha,u,\tau}(I_0)\neq I_0$
for any $\tau\in k^*$ (see Lemma~\ref{lem3.2}).
\varepsilonnd{enumerate}
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} $\bullet$ First, the existence of
$\psi_I$. Thanks to Lemma~\ref{lem3.1}, there exists
$\psi\in\c T$ verifying $1)$. Set:
\begin{equation}
\c A := \{\psi\in\c T\ |\ \psi(I_0)=I\}\notag
\varepsilonnd{equation}
and assume that for any $\psi\in\c A$, the condition $2)$ is not
verified. So, for any $\psi\in\c A$, there is a finite (recall that $Q$
has no oriented cycle) and non empty
set of bypasses (see Lemma~\ref{lem3.2}):
\begin{equation}
B_{\psi} = \left\{
(\alpha,u)\ \left|\
\begin{array}{l}
\text{$(\alpha,u)$ is a bypass}\\
u\in supp(\psi(\alpha))\\
\varphi_{\alpha,u,\tau}(I_0)=I_0\ \text{for any $\tau\in k$}
\varepsilonnd{array}\right\}\right.\notag
\varepsilonnd{equation}
For each $\psi\in\c A$, let $(\alpha_{\psi},u_{\psi})=max\ B_{\psi}$ and
let $\psi\in A$ be such that:
\begin{equation}
(\alpha_{\psi},u_{\psi})=min\ \{(\alpha_{\psi'},u_{\psi'})\ |\ \psi'\in\c
A\}\notag
\varepsilonnd{equation}
For simplicity, set $(\alpha,u):=(\alpha_{\psi},u_{\psi})$, $\tau:=u^*(\psi(\alpha))$
$\psi':=\psi\varphi_{\alpha,u,-\tau}$.
Notice that $\psi'\in\c A$ because $(\alpha,u)\in B_{\psi}$. In order to
get a contradiction, let us prove that
$(\alpha_{\psi'},u_{\psi'})<(\alpha,u)$. To do this, let us
prove first that $(\alpha,u)\not\in B_{\psi'}$. Thanks to
Proposition~\ref{prop2.17}, the following equality holds:
\begin{equation}
\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}\notag
\varepsilonnd{equation}
where $(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)$ and where
$\tau_1,\ldots,\tau_n\in k^*$. On the other hand, since
$u^*(\psi(\alpha))=\tau\neq 0$,
Proposition~\ref{prop2.11} gives:
\begin{equation}
(\varepsilonxists ! i\in\{1,\ldots,n\})\ \
(\alpha_i,u_i,\tau_i)=(\alpha,u,\tau)\notag
\varepsilonnd{equation}
Let us set:
\begin{equation}
\psi_1:=\varphi_{\alpha_{i-1},u_{i-1},\tau_{i-1}}\ldots\varphi_{\alpha_1,u_1,\tau_1}
\in \c T_{<(\alpha,u)}\notag
\varepsilonnd{equation}
Hence, the following equality holds:
\begin{equation}
\psi'=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_{i+1},u_{i+1},\tau_{i+1}}
\varphi_{\alpha,u,\tau}\psi_1\varphi_{\alpha,u,\tau}^{-1} \notag
\varepsilonnd{equation}
Since $\psi_1\in\c T_{<(\alpha,u)}$, Lemma~\ref{lem2.16} implies that
$\varphi_{\alpha,u,\tau}\psi_1\varphi_{\alpha,u,\tau}^{-1}\in \c
T_{<(\alpha,u)}$. Therefore, Proposition~\ref{prop2.17} gives the
equality:
\begin{equation}
\varphi_{\alpha,u,\tau}\psi_1\varphi_{\alpha,u,\tau}^{-1} =
\varphi_{\beta_m,v_m,\nu_m}\ldots\varphi_{\beta_1,v_1,\nu_1}\notag
\varepsilonnd{equation}
where $(\beta_1,v_1)<\ldots<(\beta_m,v_m)<(\alpha,u)$ and
$\nu_1,\ldots,\nu_m\in k^*$. As a consequence:
\begin{equation}
\psi'=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_{i+1},u_{i+1},\tau_{i+1}}
\varphi_{\beta_m,v_m,\nu_m}\ldots\varphi_{\beta_1,v_1,\nu_1}\notag
\varepsilonnd{equation}
where
$(\beta_1,v_1)<\ldots<(\beta_m,v_m)<(\alpha,u)<(\alpha_{i+1},u_{i+1})<\ldots<(\alpha_n,u_n)$
and where $\tau_{i+1},\ldots,\tau_n,\nu_1,\ldots,\nu_m\in
k^*$. In particular, Proposition~\ref{prop2.11} implies that $u\not\in
supp(\psi'(\alpha))$. Therefore,
$(\alpha,u)\not\in B_{\psi'}$ and in particular,
$(\alpha,u)\neq (\alpha_{\psi'},u_{\psi'})=max\ B_{\psi'}$.
Thus, in order to prove that $(\alpha_{\psi'},u_{\psi'})<
(\alpha,u)$, it suffices to pove that the following
implication holds for any bypass $(\beta,v)$:
\begin{equation}
v\in supp(\psi'(\beta))\ \text{and}\ (\alpha,u)<(\beta,v)\ \Rightarrow
\varphi_{\beta,v,t}(I_0)\neq I_0\ \text{for any $\tau\in k^*$}\tag{$i$}
\varepsilonnd{equation}
Let $(\beta,v)$ be a bypass such that $v\in supp(\psi'(\beta))$ and
such that $(\alpha,u)<(\beta,v)$. Since
$\psi'=\psi\varphi_{\alpha,u,-\tau}$, the following holds:
\begin{equation}
\psi'(\beta)=\left\{
\begin{array}{ll}
\psi(\beta)& \text{if $\beta\neq \alpha$}\\
\psi(\beta)-\tau\psi(u)&\text{if $\beta=\alpha$}
\varepsilonnd{array}\right.
\notag
\varepsilonnd{equation}
Therefore, $v\in supp(\psi'(\beta))\subseteq supp(\psi(\beta))\cup
supp(\psi(u))$. Remark that if $v\in supp(\psi(u))\backslash supp(\psi(\beta))$, then
$\alpha=\beta$ and Proposition~\ref{prop2.9} implies that $v$ is derived of $u$ (we have
$u\neq v$ because $\beta=\alpha$ and $(\alpha,u)<(\beta,v)$) and
therefore $(\alpha,u)>(\alpha,v)=(\beta,v)$ whereas we assumed that
$(\alpha,u)<(\beta,v)$. This proves that $v\in
supp(\psi(\beta))$. Since
$(\beta,v)>(\alpha,u)=(\alpha_{\psi},u_{\psi})= max\ B_{\psi}$ we deduce that
$\varphi_{\beta,v,\tau}(I_0)=I_0$ for any $\tau\in k$. This proves
that the implication $(i)$ is satisfied. Thus:
\begin{equation}
(\alpha_{\psi'},u_{\psi'})< (\alpha,u)=(\alpha_{\psi},u_{\psi})\notag
\varepsilonnd{equation}
This contradicts the minimality of $(\alpha_{\psi},u_{\psi})$ and
proves the existence of $\psi$.
$\bullet$ It remains to prove the uniqueness of $\psi_I$. Assume
that $\psi,\psi'\in\c T$ verify the conditions $1)$ and $2)$. In order
to prove that $\psi=\psi'$, it is sufficient to prove that
$\theta^*(\psi(\alpha))=\theta^*(\psi'(\alpha))$ for any bypass
$(\alpha,\theta)$. Let $\alpha\in Q_1$ and assume that there exists a
minimal path $\theta$ such that $(\alpha,\theta)$ is bypass and such that
$\theta^*(\psi(\alpha))\neq\theta^*(\psi'(\alpha))$. We may assume
that $\theta^*(\psi(\alpha))\neq 0$, i.e. $\theta\in
supp(\psi(\alpha))$. Since $\psi$ verifies $2)$, we deduce that there
exist paths $u$ and $v$ such that:
\begin{equation}
u\in I_0,\ v\not\in I_0\ \text{and}\
\varphi_{\alpha,\theta,1}(u)=u+v\not\in I_0\notag
\varepsilonnd{equation}
Notice that Proposition~\ref{prop2.9} gives:
\begin{equation}
\left\{
\begin{array}{l}
v^*(\psi(u))=\theta^*(\psi(\alpha))\ \text{and}\ u^*(\psi(u))=1\\
v^*(\psi'(u))=\theta^*(\psi'(\alpha))\ \text{and}\ u^*(\psi'(u))=1
\varepsilonnd{array}\right.\tag{$ii$}
\varepsilonnd{equation}
Moreover, $\psi(u),\psi'(u)\in I_0$ because $u\in I$. Therefore,
Proposition~\ref{prop3.5} gives, the same notations concerning the
Groebner bases, we have:
\begin{equation}
\left\{
\begin{array}{l}
\psi(u)=r_u+\sum\limits_{w\in \c A_{\psi}}w^*(\psi(u))r_w\\
\psi'(u)=r_u+\sum\limits_{w\in \c A_{\psi'}}w^*(\psi'(u))r_w
\varepsilonnd{array}
\right.\notag
\varepsilonnd{equation}
where $\c A_{\psi}$ is equal to:
\begin{equation}
\c A_{\psi}:=\{w\in supp(\psi(u))\ |\ w\neq u\ \text{and}\ w\in
I_0\}\notag
\varepsilonnd{equation}
So:
\begin{equation}
\left\{
\begin{array}{l}
v^*(\psi(u))=v^*(r_u)+\sum\limits_{w\in \c A_{\psi}}w^*(\psi(u))v^*(r_w)\\
v^*(\psi'(u))=v^*(r_u)+\sum\limits_{w\in \c A_{\psi'}}w^*(\psi'(u))v^*(r_w)
\varepsilonnd{array}\right.\tag{$iii$}
\varepsilonnd{equation}
Let $w\in \c A_{\psi}$ be such that
$v^*(r_w)\neq 0$, i.e. $v\in supp(r_w)$. Remark
that $v\in supp(r_w-w)$ because $v\not\in I_0$ and
$w\in I_0$. So:
\begin{enumerate}
\item[.] $v$ is derived of $w$ (thanks to Proposition~\ref{prop3.5}
and because $v\in supp(r_w-w)$).
\item[.] $v$ is derived of $u$ of order $1$ (because
$\varphi_{\alpha,\theta,1}(u)=u+v$).
\item[.] $w$ is derived of $u$ (because $w\in\c A_{\psi}$ and thanks
to Proposition~\ref{prop2.9}).
\varepsilonnd{enumerate}
Using Lemma~\ref{lem2.3}, these three facts imply that:
\begin{equation}
u=u_2\alpha u_1,\ v=u_2\theta u_1\ \text{and}\ w=u_2\theta'u_1\tag{$iv$}
\varepsilonnd{equation}
where $u_1,u_2$ are paths and where $\theta'$ is a path derived of
$\theta$. In particular, $(\alpha,\theta')$ is a bypass such that
$\theta'<\theta$ (see Lemma~\ref{lem2.7}). Therefore, the minimality
of $\theta$ forces $\theta'^*(\psi(\alpha))=\theta'^*(\psi'(\alpha)))$. Moreover,
$(iv)$ and Proposition~\ref{prop2.9} imply that
\begin{equation}
w^*(\psi(u))=\theta'^*(\psi(\alpha))=\theta'^*(\psi'(\alpha))=w^*(\psi'(u))\notag
\varepsilonnd{equation}
Therefore we have proved the following implication:
\begin{equation}
w\in\c A_{\psi}\ \text{and}\ v^*(r_w)\neq 0\ \Rightarrow\ w^*(\psi(u))v^*(r_w)=w^*(\psi'(u))v^*(r_w)\tag{$v$}
\varepsilonnd{equation}
After exchangeing the roles of $\psi$ and $\psi'$, the arguments used
to prove $(v)$ also give the following implication:
\begin{equation}
w\in\c A_{\psi'}\ \text{and}\ v^*(r_w)\neq 0\ \Rightarrow\ w^*(\psi(u))v^*(r_w)=w^*(\psi'(u))v^*(r_w)\tag{$vi$}
\varepsilonnd{equation}
Then, $(iii)$, $(v)$ and $(vi)$ give
$v^*(\psi(u))=v^*(\psi'(u))$. This and $(ii)$ imply that
$\theta^*(\psi(\alpha))=\theta^*(\psi'(\alpha))$, a
contradiction. This proves that $\psi=\psi'$.
$\blacksquare$\\
\section{Proof of the main theorem}
Let $A=kQ/I_0$ where $I_0$ is a monomial admissible ideal of $kQ$. The
aim of this section is to prove that the quiver $\Gamma$ of the
homotopy relations of the admissible presentations of $A$ admits
$\sim_{I_0}$ as unique source. This fact will be used in order to
the existence of the universal cover of $A$. Notice that $\sim_{I_0}$ is
a source of $\Gamma$. Indeed, all minimal relations in $I_0$ are monomial
relations so, for any $\sim_I \in \Gamma_0$ we have
$\Gammaamma\sim_{I_0}\Gammaamma'\Rightarrow \Gammaamma\sim_I\Gammaamma'$. In order to
prove that $\sim_{I_0}$ is the unique source in $\Gamma$ it will be proved
that for any admissible presentation $kQ/I\simeq A$, the decomposition
of $\psi_I$ (given by Proposition~\ref{prop3.6}) into a decreasing
product of transvections (see Proposition~\ref{prop2.17}) defines a
path in $\Gamma$ starting at $\sim_{I_0}$ and ending at $\sim_I$. In this
purpose, the following proposition will be useful.
\begin{prop}
\label{prop4.1}
Let $kQ/I\simeq A$ be an admissible presentation. Then, for any bypass
$(\alpha,u)$:
\begin{equation}
u\in supp(\psi_I(\alpha))\Rightarrow u\sim_I\alpha\notag
\varepsilonnd{equation}
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} For simplicity, set $\psi:=\psi_I$. Thanks
to Proposition~\ref{prop2.17} there is an equality:
\begin{equation}
\psi=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}\notag
\varepsilonnd{equation}
with $(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)$ and
$\tau_1,\ldots,\tau_n\in k^*$. Thanks Proposition~\ref{prop2.11}
it suffices to prove that $\alpha_i\sim_I u_i$ for any
$i$. This will be done using a decreasing induction on
$m\in\{1,\ldots,n\}$. Let $H_m$ be the assertion:
\begin{equation}
H_m:"\text{$\alpha_i\sim_Iu_i$ for any $i\in\{m,m+1,\ldots,n\}$}"\notag
\varepsilonnd{equation}
$H_{n+1}$ is true because $\{i\ |\ n+1\leqslant i\leqslant n\}$ is
empty. So assume that $H_{m+1}$ is true ($m\in\{1,\ldots,n\}$). In
order to prove that $H_m$ is true, it thus suffices to prove that
$\alpha_m\sim_I u_m$. From
Proposition~\ref{prop2.11}, the path $u_m$ lies in $supp(\psi(\alpha_m))$. Hence,
Proposition~\ref{prop3.6} provides a path $u\in I_0$ such that
$\varphi_{\alpha_m,u_m,1}(u)\not\in
I_0$. Therefore, there exist paths $v_1,v_2$ such that:
\begin{equation}
u=v_2\alpha_mv_1,\ \ v:=v_2u_mv_1\not\in I_0\ \text{and}\ \varphi_{\alpha_m,u_m,1}(u)=u+v\tag{$i$}
\varepsilonnd{equation}
Since $\psi(u)\in I$, there exists a decomposition:
\begin{equation}
\psi(u)=r_1+\ldots+r_N\notag
\varepsilonnd{equation}
where $r_1,\ldots,r_N$ are minimal relations in $I$ with pairwise
disjoint supports. Remark that $u,v\in supp(\psi(u))$ thanks to
Proposition~\ref{prop2.9} and to Proposition~\ref{prop2.11}. Without
loss of generality, it may be
assumed that $v\in supp(r_1)$. Let $i\in\{1,\ldots,N\}$ be such that
$u\in supp(r_i)$. If $i=1$ then $u\sim_I v$ and $(i)$ gives
$\alpha_m\sim_I u_m$. So assume that $i\neq 1$.
Remark that $\psi^{-1}(r_1)\in I_0$ because $r_1\in I$. Since $I_0$ is
monomial, this also implies
that $v\not\in supp(\psi^{-1}(r_1))$. And thanks to
Proposition~\ref{prop2.13}, this proves that:
\begin{equation}
\text{there exists $w\in supp(r_1)$ such that $v$ is derived from
$w$}\tag{$ii$}
\varepsilonnd{equation}
Therefore:
\begin{enumerate}
\item[.] $w$ is derived of $u$ since $w\in supp(r_1)\subseteq
supp(\psi(u))$ (see Proposition~\ref{prop2.9}, notice that $u\neq w$ because
$u\not\in supp(r_1)$),
\item[.] $v$ is derived of $w$ (see $(ii)$),
\item[.] $v$ is derived of $u$ of order $1$ (because of $(i)$).
\varepsilonnd{enumerate}
Thanks to Lemma~\ref{lem2.3}, these three points imply that:
\begin{equation}
w=v_2\theta v_1\ \text{and $u_m$ is derived of $\theta$}\tag{$iii$}
\varepsilonnd{equation}
Since $w\in supp(\psi(u))$, the equalities $w=v_2\theta v_1$,
$u=v_2\alpha_m v_1$ and Proposition~\ref{prop2.9} imply that
$\theta\in supp(\psi(\alpha_m))$. Hence, there exists
$j\in\{1,\ldots,n\}$ such that:
\begin{equation}
(\alpha_m,\theta)=(\alpha_j,u_j)\notag
\varepsilonnd{equation}
Since $u_m$ is derived of $\theta$ (see $(iii)$), this last equality gives
$u_j=\theta>u_m$ (see Lemma~\ref{lem2.7}) and therefore $j>m$. On the
other hand,
$H_{m+1}$ is true, so:
\begin{equation}
\alpha_m=\alpha_j\sim_I u_j=\theta\tag{$iv$}
\varepsilonnd{equation}
Finally, $v\sim_I w$, because $r_1$ is a minimal relation in $I$ such
that $v,w\in supp(r_1)$. This together with $(i)$, $(iii)$ and $(iv)$
imply that
$\alpha_m\sim_Iu_m$.
So $H_m$ is true and the induction is finished.
$\blacksquare$\\
\begin{rem}
\label{rem4.2}
The preceding proposition proves that $\alpha\sim_I u$ for
any $u\in supp(\psi(\alpha))$. On the other hand, $\sim_{I_0}$ is
weaker than $\sim_I$ (i.e. $\Gammaamma\sim_{I_0}\Gammaamma'\Rightarrow
\Gammaamma\sim_I\Gammaamma'$). These two properties are linked in
general. Indeed, in \cite[Prop. 4.2.35, Prop. 42.36]{lemeur_thesis}
the author has proved that if $I$ is an admissible ideal (non
necessarily monomial) of $kQ$ and if $\psi\in\c T$ is such that
$\alpha\sim_{\psi(I)}u$ for any bypass $(\alpha,u)$ such that $u\in
supp(\psi(\alpha))$, then $\sim_I$ is weaker than $\sim_{\psi(I)}$.
\varepsilonnd{rem}
Now it is possible to provethe existence of a path in $\Gamma$ starting at $\sim_{I_0}$ and ending
at $\sim_I$, whenever $kQ/I\simeq A$.
\begin{prop}
\label{prop4.3}
Let $kQ/I\simeq A$ be an admissible presentation. Let
$(\alpha_1,u_1)<\ldots<(\alpha_n,u_n)$ be the bypasses and
$\tau_1,\ldots,\tau_n\in k^*$ the scalars such that
$\psi_I=\varphi_{\alpha_n,u_n,\tau_n}\ldots\varphi_{\alpha_1,u_1,\tau_1}$
(see
Proposition~\ref{prop2.17}).
For each $i\in\{1,\ldots,n\}$, set:
\begin{equation}
I_i:=\varphi_{\alpha_i,u_i,\tau_i}\ldots\varphi_{\alpha_1,u_1,\tau_1}(I_0)\notag
\varepsilonnd{equation}
then, for each $i$, exactly one of the two following situations occurs:
\begin{enumerate}
\item[.] $\sim_{I_{i-1}}$ and $\sim_{I_i}$ coincide,
\item[.] $\varphi_{\alpha_i,u_i,\tau_i}$ induces an arrow
$\sim_{I_{i-1}}\to \sim_{I_i}$ in $\Gamma$.
\varepsilonnd{enumerate}
In particular, there exists a path in $\Gamma$ starting at $\sim_{I_0}$
and ending at $\sim_{I_n}=\sim_I$.
\varepsilonnd{prop}
\noindent{\textbf{Proof:}} Let $i\in\{1,\ldots,n\}$ and set
$\psi_i:=\varphi_{\alpha_i,u_i,\tau_i}\ldots\varphi_{\alpha_1,u_1,\tau_1}$.
Thus $I_i=\psi_i(I_0)$. Using Proposition~\ref{prop2.11} and
Proposition~\ref{prop3.6} it is easily verified that
$\psi_i=\psi_{I_i}$.
Therefore, Proposition~\ref{prop4.1} applied to $I_i$ gives
$\alpha_i\sim_{I_i}u_i$.
Since $I_i=\varphi_{\alpha_i,u_i,\tau_i}(I_{i-1})$, this proves that
(see Proposition~\ref{prop1.2}) either $\sim_{I_{i-1}}$ and $\sim_{I_i}$ coincide or
$\varphi_{\alpha_i,u_i,\tau_i}$ induces an arrow $\sim_{I_{i-1}}\to
\sim_{I_i}$ in $\Gamma$. Thus, the vertices
$\sim_{I_0},\sim_{I_1},\ldots,\sim_{I_n}=\sim_I$ of $\Gamma$
are the vertices of a path in $\Gamma$ (maybe with repetitions) starting
at $\sim_{I_0}$ and ending at $\sim_I$.
$\blacksquare$\\
The preceding proposition and the fact that $\Gamma$ has no oriented
cycle gives immediately the following corollary
which was proved by the author in \cite{lemeur2} in the case of
algebras without double bypass over an algebraically
closed field of characteristic zero.
\begin{cor}
\label{cor4.4}
Let $Q$ be a quiver without oriented cycle and without multiple
arrows. Let $I_0$ be an admissible and monomial ideal of $kQ$ and let
$A=kQ/I_0$. Then the quiver $\Gamma$ of the homotopy relations of the
admissible presentations of $A$ admits $\sim_{I_0}$ as unique source.
\varepsilonnd{cor}
The following example shows that the preceding corollary does not hold
if $Q$ has multiple arrows.
\begin{ex}
Let $A=kQ/I_0$ where $Q$ is the quiver
$\xymatrix{1\ar@/^/@{->}[r]^a\ar@/_/@{->}[r]_b&2\ar@{->}[r]^c&3}$
and $I_0=<ca>$. Then $\Gamma$ is equal to:
\begin{equation}
\xymatrix{
\sim_{I_0} \ar@{->}[rd] &&\sim_{I_1}\ar@{->}[ld]\\
&\sim_{I_2}&
}\notag
\varepsilonnd{equation}
where $I_1=<cb>$ and $I_2=<ca-cb>$. In particular, $\Gamma$ has two
distinct sources. Notice however, that the mapping $a\mapsto b,\
b\mapsto a,\ c\mapsto c$ defines a group isomorphism
$\pi_1(Q,I_0)\simeq \pi_1(Q,I_1)$. One has $\pi_1(Q,I_0)\simeq
\pi_1(Q,I_1)\simeq\mathbb{Z}$ and $\pi_1(Q,I_2)=1$.
\varepsilonnd{ex}
Proposition~\ref{prop4.3} also allows one to prove
Theorem~\ref{Thm1}. It extends \cite[Thm. 2]{lemeur2} to monomial
triangular algebras
without multiple arrows. Notice that Theorem~\ref{Thm1} makes
no assumption on the characteristic of $k$. Also recall that
$\pi_1(Q,I_0)=\pi_1(Q)$.
\noindent{\textbf{Proof of Theorem~\ref{Thm1}:}} The proof is
identical to the proof of \cite[Thm. 2]{lemeur2} except that one uses
Proposition~\ref{prop4.3} instead of
\cite[Lem. 4.3]{lemeur2}.
$\blacksquare$\\
\varepsilonnd{document} |
\begin{document}
\title{Catalan Numbers and Jacobi Polynomials}
\begin{abstract}
We prove that the inverse of the Hankel matrix of
the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.
\end{abstract}
\section{Introduction}
We prove the that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We prove the same result for an infinite family of generalized Catalan numbers. Based on the analogy of these matrices with the Hilbert matrix, we call the Hankel matrix of reciprocals of the Catalan numbers the {\it Catbert matrix}, a portmanteau of Catalan and Hilbert matrix.
All matrices in this paper are indexed starting with 0, and are infinite unless otherwise stated. For an infinite matrix $M$, denote the $n\times n$ upper left submatrix by $M(n)$.
The original inspiration for this work is the the Hilbert matrix $H$, defined by $H=[ h_{i,j} ]$ with $h_{i,j}=\frac{1}{i+j+1}$, and the interesting fact that the inverse of the $n \times n$ Hilbert matrix has integer entries. Choi asked what sort of coincidence it is if the inverse of a matrix of reciprocals of integers has integer entries \cite{TRICKS}. The most general collection of sequences for which we prove that the corresponding Hankel matrices have integer inverses are sequences with generating functions of the form $g(x) = q(1-p^2x)^{q/p},$ where $p$ is an integer with $p\ge 2$, and $q$ is an integer relatively prime to $p$. The proof extends to their subsequences $(g_a, g_{a+1}, g_{a+2}, \ldots).$
Other matrices of reciprocals of integers that have been shown to have inverses with integer entries include the Filbert matrix and generalizations, based on reciprocals of Fibonacci numbers and generalized Fibonacci numbers \cite{FILBERT,KILICPRODINGER}; matrices based on reciprocals of sequences of binomial coefficients $b_n=\frac{1}{\alpha\binom{n+\alpha}{\alpha}}$ for a natural number $\alpha$ \cite{BERG}; and the matrices of reciprocals of the super Catalan and super Patalan numbers \cite{SUPERPAT_PUBLISHED}.
Approaches to proving that the inverse matrices of reciprocals of integers have integer entries include orthogonal polynomials \cite{BERG}, and Wilf-Zeilberger (WZ) theory \cite{AeqB,FILBERT}. This paper will use both of these approaches in its proofs.
\section{Generalized Catbert Matrices}
First we define the sequences that we will show give rise to Hankel matrices of reciprocals of integers, whose inverses have integer entries.
\begin{definition}
\label{GENPATSEQ}
Let $p$ be an integer with $p\ge2$, and let $q$ be an integer that is relatively prime to $p$. Define the sequence of generalized Catalan numbers $g$ by
\(g_n = p^{2n}\binom{n+q/p}{n}=(-p^2)^{n}\binom{-1-q/p}{n}\).
We use $g^{(q/p)}=g$ when we want to be explicit about the values of the parameters $p$ and $q$.
\end{definition}
The sequences $g$ can be found as the non-zero entries of columns in a recursive matrix of super Catalan or super Patalan numbers \cite{LMMS, SUPERPAT_PUBLISHED}. The sequence of central binomial coefficients is $g^{(-1/2)}$ in this notation. The sequence of Catalan numbers is related to the sequence $g^{(-3/2)}$ as follows; if $g=g^{(-3/2)}$ and $C$ is the sequence of Catalan numbers, then $C_n = -\frac{1}{2}g_{n+1}.$
Theorems 9-11 of \cite{SUPERPAT_PUBLISHED} prove that the elements of $g$ are integers, with the qualification that the notation here is different, and the cited proofs only explicitly apply to a subset of the sequences that we consider here. The approach extends easily to the current context.
\begin{theorem}
The elements of the sequences $g^{(q/p)}$ defined in Definition \ref{GENPATSEQ} are integers.
\end{theorem}
\begin{proof}
Let $p$ be as in definition \ref{GENPATSEQ}, let $q=-(p+1)$, and let $g=g^{(q/p)}$. Now define the sequence $a$ for $n\ge 0$ by $a_n=-\frac{1}{p}g_{n+1}$. Then by Theorem 9 of \cite{SUPERPAT_PUBLISHED}, the sequence $a$ satisfies the recurrence relation
\[ a_n = \sum_{k=2}^p(-p)^{k-2}\binom{p}{k} \prod_{i_1+\ldots+i_k=n-k+1} a(i_j).
\]
Since $a_0=1$, this implies that $a_n$ is an integer, and therefore the elements of $g$ are integers, as $g_0=1$ and $g_n=-pa_{n-1}$.
Similarly, Theorem 10 of \cite{SUPERPAT_PUBLISHED} implies that the elements of $g^{(q/p)}$ are integers for $q=-(p+r)$ and $1<r<p$, as they are convolutional powers of $g^{(-(p+1)/p)}$.
Finally, the well known identity
\[\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1} \]
implies the equations
\begin{align}
\label{GINT1}
g^{(q/p)}_n &= g^{((q-p)/p)}_n +p^2g^{(q/p)}_{n-1} \\
\label{GINT2}
g^{(q/p)}_n &= g^{((q+p)/p)}_n-p^2g^{((q+p)/p)}_{n-1}.
\end{align}
Equations \eqref{GINT1} and \eqref{GINT2} allow us to extend the proof that the elements of $g^{(q/p)}$ are integers to all $p,q$ satisfying the conditions in Definition \ref{GENPATSEQ}.
\end{proof}
Now we define the Hankel matrices based on generalized Catalan numbers. We define these Hankel matrices based on the subsequences of such a sequence $g$, starting with the term $g_a$ where $a\ge 0$.
\begin{definition}
\label{GENPATHANKEL}
Let $g$ be a sequence of generalized Catalan numbers, and let $a$ be a non-negative integer. Let the matrix $G$ be the Hankel matrix of reciprocals of $g$, offset by $a$, so that $G_{i,j} = 1/g_{i+j+a}$, for $0\le i, j$. We use $G(n)$ to denote the $n \times n$ upper left submatrix of $G$. We also use $G^{(a,q/p)}$ and $G^{(a,q/p)}(n)$ for $G$ and $G(n)$, respectively, when we want to make the parameters $a$, $p$, and $q$ explicit.
\end{definition}
The sequences $g^{(q/p)}$ are based on sequences of binomial coefficients $\binom{n+q/p}{n}$. Berg described shifted Jacobi polynomials that are orthogonal polynomials for the Hankel matrix with entries $ \displaystyle{ \frac{1}{\alpha\binom{i+j+\alpha}{\alpha}}}$, where $\alpha$ is a natural number \cite[Theorem 4.2]{BERG}. He used the relationship to express the inverse of the Hankel matrix as a product involving the coefficient matrix of the orthogonal polynomials. We consider a more general situation than Berg. The sequences of generalized Catalan numbers that we consider use the rational parameter $q/p$ instead of a natural number parameter $\alpha$, and also use an additional scaling factor $p^{2n}$. We also consider subsequences starting at an offset from the beginning of the sequence. We still use his general approach, based on the coefficient matrix of orthogonal polynomials. Our proofs will use computer algebra to show that the polynomials for the sequence $g^{(q/p)}$ are orthogonal.
\section{Orthogonal polynomials for G}
We define a lower triangular matrix $L$, whose rows are the coefficients of orthogonal polynomials for the bilinear form defined by $G$. The coefficients are based on the coefficients of shifted Jacobi polynomials as defined by Hetyei \cite[equation (4)]{HETYEI}, and are then scaled by powers of $p^2$.
\begin{definition}
\label{CATBERTOPLOWERTRI}
Let $a$, $p$, and $q$ be as in definition \ref{GENPATHANKEL}. Define $L=L^{(a,q/p)}$ to be the lower triangular matrix with
\begin{equation}
L_{n,k} = (-1)^{n+k}p^{2k}\binom{n+k+a+q/p-1}{k}\binom{n+a}{k+a}.
\end{equation}
\end{definition}
To show that $L$ is a coefficient matrix of orthogonal polynomials for the bilinear form defined by $G$, we show that the product $LGL^T$ is a diagonal matrix.
We show this in two steps. First, we show that the rows of $L$ satisfy a three term recurrence. Second, we prove that the rows are orthogonal with respect to the bilinear form defined by $G$.
\section{Three Term Recurrence and Orthogonality}
We state and prove a three term recurrence relation for the rows of L.
\begin{lemma}
\label{CATBERTTHREETERM}
Let
\begin{equation}
\alpha_n=-\frac{(n+a+1)(q+np)(q+2np++ap+3p)}{(n+2)(q+np+ap+p)(q+2np+ap+p)},
\end{equation}
let
\begin{equation}
\beta_n=-\frac{(q+2np+ap+2p)(2nq+aq+3q+2n^2p+2anp+4np+(a+1)^2p)}{(n+2)(q+np+ap+p)(q+2np+ap+p)},
\end{equation}
and let
\begin{equation}
\gamma_n=\frac{p(q+2np+ap+2p)(q+2np+ap+3p)}{(n+2)(q+np+ap+p)}.
\end{equation}
Then
\begin{equation}
\label{THREETERMEQN}
\alpha_nL(n,k)+\beta_nL(n+1,k)+\gamma_nL(n+1,k-1)=L(n+2,k).
\end{equation}
\end{lemma}
The proof of Lemma \ref{CATBERTTHREETERM} follows from calculations performed with the wxMaxima computer algebra system \cite{WXMAX,MAX}. The wxMaxima script to perform the calculations is in the ancillary file `CatalanJacobiTHREETERM.wxm' included with this article.
Next we show that the rows of $L$ are orthogonal with respect to the bilinear form defined by $G$, by showing that $LGL^T$ is a diagonal matrix. Specifically, we show that $LG$ is upper triangular, and that implies that $LGL^T$ is diagonal.
\begin{lemma}
\label{CATBERTORTHOGLEM}
Let $n$ be an integer with $n \ge 2$. The matrices $L$ and $G$ satisfy
\[\sum_{k=0}^n L_{n,k}G_{k,n-1}=0\] and
\[\sum_{k=0}^n L_{n,k}G_{k,n-2}=0\].
\end{lemma}
\begin{proof}
The wxMaxima script to verify these equations is in the ancillary file `CatalanJacobiORTHOGONAL.wxm' included with this article.
\end{proof}
\begin{theorem}
\label{CATBERTOPTHEOREM}
The product $LGL^T$ is a diagonal matrix.
\end{theorem}
\begin{proof}
The matrices $L$ and $G$ satisfy
$\sum_{k=0}^n L_{n,k}G_{k,m}=0$ for all $m$ with $0\le m < n$,
by Lemma \ref{CATBERTORTHOGLEM}, Lemma \ref{CATBERTTHREETERM}, and mathematical induction. This says that $LG$ is an upper triangular matrix. Since $G$ is symmetric, this implies that $GL^T$ is lower triangular. As $L$ is lower triangular, this implies that $LGL^T$ is both upper triangular and lower triangular, and thus $LGL^T$ is diagonal.
\end{proof}
\section{The Diagonal of the Product}
For the forthcoming proofs, we will need to know the value of the diagonal entries of $LGL^T$.
\begin{definition}
\label{NORMDEFIN}
Let $a$, $p$, and $q$ be as in definition \ref{GENPATHANKEL}.
Define $N$ to be the diagonal matrix with \[N^{(a,q/p)}_{n,n} = \frac{p^{2a}(2np+ap+q)\binom{n+a+q/p-1}{a}}{q\binom{n+a}{a}}.\]
\end{definition}
\begin{theorem}
\label{CATBERTNORM}
The matrices $L$, $G$, and $N$ satisfy
\begin{equation}
\label{CATBERTNORMEQN}
LGL^T=N^{-1}.
\end{equation}
\end{theorem}
We prove this theorem by a calculation with the Zeilberger algorithm as implemented in wxMaxima. By Lemma \ref{CATBERTORTHOGLEM}, the entries of the diagonal of $LGL^T$ are given by $L_{n,n}\sum_{k=0}^n L_{n,k}G_{k,n}$, so the theorem follows from the summation identity
\begin{equation}
\sum_{k=0}^n L_{n,k}G_{k,n}L_{n,n} = N^{-1}_{n,n}.
\end{equation}
For an explanation of how the Zeilberger algorithm proves such a summation identity, see Section 2.3 of \cite{AeqB}, particularly the itemized list before Example 2.3.1. For the computer calculation, we divide the summand by the conjectured sum value $N^{-1}_{n,n}$, as in step 2 of the referenced list. In our case, the output from the function `Zeilberger' contains a first order recurrence with non-constant terms. As the coefficients of the recurrence are negatives of each other, it immediately shows that the sum over $k$ is constant as a function of $n$. The wxMaxima script to perform the calculations is in the ancillary file `CatalanJacobiDIAGONAL.wxm' included with this article.
\section{Lucas's Theorem and Variations}
For the next proofs, we need to use Lucas's theorem on binomial coefficients modulo a prime number, and some variations.
\begin{lemma}
\label{LUCASLEM}
Let $p$ be a prime number, and let $n$ and $k$ satisfy $n \equiv 0 \pmod{p}$, and $k \not\equiv 0 \pmod{p}$. Then $\binom{n}{k} \equiv 0 \pmod{p}$.
\end{lemma}
Lemma \ref{LUCASLEM} is a simple consequence of Lucas's theorem \cite[Theorem 1]{BAILEY}.
We next consider three variations of Lucas's theorem that involve binomial coefficients with non-integer parameters.
\begin{lemma}
\label{LUCASVAR1}
Let $q$ be a prime number, let $p$ be an integer with $\gcd(q,p)=1$, let $n$ be an integer such that
$p^2n+pq \equiv 0 \pmod{q}$,
and let $k$ be a positive integer such that $k \not\equiv 0 \pmod{q}$.
Then $p^{2k}\binom{n+q/p}{k} \equiv 0 \pmod{q}$.
\end{lemma}
\begin{proof}
We may express
\begin{equation}
\label{LUCASVAR1EQN}
p^{2k}\binom{n+q/p}{k}=
\frac{\prod_{i=0}^{k-1}\bigl(p^2(n-i)+pq\bigr)}{\prod_{i=0}^{k-1}(i+1)}.
\end{equation}
By induction on $k$, it follows that the numerator of equation (\ref{LUCASVAR1})
is always divisible by more powers of $q$ than the denominator, except when
$k \equiv 0 \pmod{q}.$.
\end{proof}
\begin{lemma}
\label{LUCASVAR2}
Let $q=-3$, let $p=2$,
let $n$ satisfy $n \equiv 2 \pmod{3}$,
and let $k$ satisfy
$k \equiv 2 \pmod{3}$.
Then $4^k\binom{n+k-3/2}{k} \equiv 0 \pmod{3}$.
\end{lemma}
\begin{proof}
We may express
\begin{equation}
\label{LUCASVAR2EQN}
4^k\binom{n+k-3/2}{k} = 4^k\binom{n+k-1-3/2}{k}+4^k\binom{n+k-1-3/2}{k-1}.\end{equation}
Now by Lemma \ref{LUCASVAR1},
$4^k\binom{n+k-1-3/2}{k}$ and $4^k\binom{n+k-1-3/2}{k-1}$
are both divisible by $3$, therefore
$4^k\binom{n+k-3/2}{k}$ is also divisible by $3$.
\end{proof}
\begin{lemma}
\label{LUCASVAR3}
Let $p$ be a prime number, let $q$ be an integer with $\gcd(q,p)=1$, let $n$ be an integer, and let $k$ be a positive integer.
Then $p^{2k}\binom{n+q/p}{k} \equiv 0 \pmod{p}$.
\end{lemma}
\begin{proof}
We express $p^{2k}\binom{n+q/p}{k}$ as
\[\prod_{i=0}^k \frac{p^2n+pq}{i+1}. \]
In this expression, the numerator is divisible by $p^k$, while the largest power of $p$ that divides the denominator is at most $p^{(k-1)/(p-1)}$ by Legendre's formula \cite[Equation (1.2)]{STRAUB}.
\end{proof}
\section{Main Theorems}
\begin{theorem}
\label{MAINTHEOREM}
Let $a$, $p$, $q$, and $G^{(a,q/p)}(n)$ be as in Definitions \ref{GENPATSEQ} and \ref{GENPATHANKEL}. Then
\noindent
(1) if $q=\pm 1$, the entries of the inverse of the matrix $G^{(a,q/p)}(n)$ are integers.
\noindent
(2) if $q=\pm 2$, the entries of the inverse of the matrix $G^{(a,q/p)}(n)$ are integers.
\end{theorem}
From equation \eqref{CATBERTNORMEQN}, it follows that
the $n\times n$ upper left submatrix $G(n)$ satisfies
\begin{equation}
\label{GEQN}
G(n) = L(n)^{-1}N(n)^{-1}(L(n)^T)^{-1}.
\end{equation}
Thus for the $n\times n$ upper left submatrix $G(n)$ we have
\begin{equation}
\label{GINVEQUATION}
G(n)^{-1}=L(n)^{T}N(n)L(n).
\end{equation}
We now introduce an alternate expression for the product $N(n)L(n)$ that will allow us to remove the binomial coefficient term in the denominators of the entries of $N(n)$.
\begin{definition}
\label{MKDEFIN}
Define the diagonal matrix $M$ by
\[M^{(a,q/p)}_{n,n} = \frac{(2np+ap+q)}{q}.\]
Define the lower triangular matrix $K$ by
\[
K^{(a,q/p)}_{n,k} = (-1)^{n+k}p^{2(k+a)}\binom{n+k+a+q/p-1}{k+a}\binom{n}{k}.
\]
\end{definition}
\begin{lemma}
\label{MKLEMMA}
The matrices $K$, $L$, $M$, and $N$ satisfy
\begin{equation}
\label{NLMKEQN}
NL = MK.
\end{equation}
\end{lemma}
\begin{proof}
Since $N$ and $M$ are diagonal, it is sufficient to prove that
\begin{equation}
\label{NLMKEQN2}
N_{n,n}L_{n,k} = M_{n,n}K_{n,k}.
\end{equation}
The $(n,k)$ entry of $NL$ is given by
\begin{align}
N_{n,n}&L_{n,k}
\\ = &
\frac{p^{2a}(2np+ap+q)\binom{n+a+q/p-1}{a}}{q\binom{n+a}{a}}
(-1)^{n+k}p^{2k}\tbinom{n+k+a+q/p-1}{k}\tbinom{n+a}{k+a}
\\ = &
(-1)^{n+k}p^{2(k+a)}(2np+ap+q)\frac{\binom{n+a+q/p-1}{a}\binom{n+k+a+q/p-1}{k}\binom{n+a}{k+a}}{q\binom{n+a}{a}}
\\ = &
(-1)^{n+k}p^{2(k+a)}(2np+ap+q)\frac{\binom{n+k+a+q/p-1}{k+a}\binom{n}{k}}{q}
\\ = &
\frac{2np+ap+q}{q}(-1)^{n+k}p^{2(k+a)}\tbinom{n+k+a+q/p-1}{k+a}\tbinom{n}{k}
\\ = &
M_{n,n}K_{n,k},
\end{align}
and the last expression is the $(n,k)$ entry of $MK$.
\end{proof}
From Lemma \ref{MKLEMMA} we can rewrite the factorization of $G(m)^{-1}_{n,k}$ in equation (\ref{GINVEQUATION}) as
\begin{equation}
\label{LMKEQN}
G(m)^{-1}=L(m)^TM(m)K(m).
\end{equation}
We can express the $(n,k)$ entry of $G(m)^{-1}$ as
\begin{equation}
\label{LMKEQN3}
G(m)^{-1}_{n,k}=\sum_{i=n}^{m-1} L(m)_{i,n}M(m)_{i,i}K(m)_{i,k}.
\end{equation}
By symmetry and equation \eqref{NLMKEQN2}, we have
\begin{equation}
\label{LMKEQN4}
L(m)_{i,n}M(m)_{i,i}K(m)_{i,k}=K(m)_{i,n}M(m)_{i,i}L(m)_{i,k}.
\end{equation}
Writing out the full expression of the products in equation \eqref{LMKEQN4}, the left side expands as
\begin{equation}
\label{LMKPRODEQN}
\begin{split}
L_{i,n}M_{i,i}K_{i,k} &\\
= &(-1)^{n+k}(p^2)^{n+k+a}
{\textstyle \frac{(2ip+ap+q)}{q}
\binom{i+n+q/p-1}{n}\binom{i+a}{n+a}
\binom{i+k+q/p-1}{k+a}\binom{i}{k}}, \\
\intertext{and the right side expands as}
K_{i,n}M_{i,i}L_{i,k} &\\
= &(-1)^{n+k}(p^2)^{n+k+a}
{\textstyle \frac{(2ip+ap+q)}{q}
\binom{i+n+q/p-1}{n+a}\binom{i}{n}
\binom{i+k+q/p-1}{k}\binom{i+a}{k+a}}.
\end{split}
\end{equation}
\begin{proof} (of Theorem \ref{MAINTHEOREM}.)
Using equations \eqref{LMKEQN}-\eqref{LMKPRODEQN} to prove the theorem we have to show that $q$ divides one of the terms of the numerator of a right-hand side of equation \eqref{LMKPRODEQN}. Case (1) is obvious. For case (2), we show that $2$ divides the numerator of one side of equation \eqref{LMKEQN4}, in particular, that $2$ divides either the numerator of $M_{i,i}$, $K_{i,k}$, or $L_{i,k}.$
If $a$ is even, then $2$ divides the numerator of $M(m)_{i,i}$, so we may assume that $a$ is odd. Now if $i$ and $k$ have different parity, then one of $\binom{i}{k}$ or $\binom{i+a}{k+a}$ is divisible by $2$ by Lemma \ref{LUCASLEM}. Hence $2$ divides either $K_{i,k}$ or $L_{i,k},$ respectively.
Now we may assume that $i$ and $k$ have the same parity, still assuming that $a$ is odd. Now either $k$ or $k+a$ is odd. By Lemma \ref{LUCASVAR1} this implies that $2$ divides either
$p^{2k}\binom{i+k+a+q/p-1}{k}$ or $p^{2(k+a)}\binom{i+k+a+q/p-1}{k+a}$, hence $2$ divides either $L_{i,k}$ or $K_{i,k},$ respectively.
Thus in every case, we have shown that $2$ divides the summand in equation \eqref{LMKEQN3}, so $2$ divides $G^{-1}_{n,k}$, and hence $G(m)^{-1}$ is an integer matrix when $q=\pm 2$.
\end{proof}
\begin{theorem}
\label{MAINTHEOREM2}
Let $a$, $p$, $q$, and $G^{(a,q/p)}(n)$ be as in Definitions \ref{GENPATSEQ} and \ref{GENPATHANKEL}. Then the entries of the inverse of the matrix $\frac{1}{q}G^{(a,q/p)}(n)$ are integers.
\end{theorem}
\begin{proof}
By Lemma \ref{MKLEMMA}, it follows that
\[ \bigl( \frac{1}{q}G(m)\bigr)^{-1}
= qG(m)^{-1} = L(m)^T\bigl(qM(n)\bigr)K(m).\]
Since $qM$, $L$, and $K$ are integer matrices, it follows that
\[\bigl( \frac{1}{q}G(m)\bigr)^{-1}\] is an integer matrix.
\end{proof}
We now use equation \eqref{LMKEQN} to express the determinant of
$\bigl(G(n)^{(a,q/p)}\bigr)^{-1}$ as a product of the diagonal elements of $M{(a,q/p)}$, $L{(a,q/p)}$, and $K{(a,q/p)}$, omitting the factors that are identically equal to $1$.
\begin{theorem}
\label{DETTHM}
The determinant of the matrix $\bigl(G(n)^{(a,q/p)}\bigr)^{-1}$ is given by
\begin{multline}
\label{DETEQN1}
\det \Bigl(\bigl(G(n)^{(a,q/p)}\bigr)^{-1}\Bigr)
\\ =
\prod_{k=0}^{n-1}
(p^2)^{2k+a} \frac{2kp+ap+q}{q}\binom{2k+a+q/p-1}{k}\binom{2k+a+q/p-1}{k+a},
\end{multline}
and the determinant of the matrix $\bigl(\frac{1}{q}G(n)^{(a,q/p)}\bigr)^{-1}$ is given by
\begin{multline}
\label{DETEQN2}
\det \Bigl(\bigl(\frac{1}{q}G(n)^{(a,q/p)}\bigr)^{-1}\Bigr)
\\ =
\prod_{k=0}^{n-1}
(p^2)^{2k+a} (2kp+ap+q)\binom{2k+a+q/p-1}{k}\binom{2k+a+q/p-1}{k+a}.
\end{multline}
\end{theorem}
\begin{proof}
Equation \eqref{LMKEQN} implies that the determinant of \(\bigl(G(n)^{(a,q/p)}\bigr)^{-1}\)
is the product of the diagonals of $L(n)^{(a,q/p)}$, $M(n)^{(a,q/p)}$, and $K(n)^{(a,q/p)}$. Similarly, the determinant of \(\bigl(\frac{1}{q}G(n)^{(a,q/p)}\bigr)^{-1}\)
is the product of the diagonals of $L(n)^{(a,q/p)}$, $qM(n)^{(a,q/p)}$, and $K(n)^{(a,q/p)}$.
\end{proof}
\section{The Catbert Matrix}
The Catbert matrix is the Hankel matrix of reciprocals of the Catalan numbers. The sequence \(g^{(-3/2)} = 1,-2,-2,-4,-10,\ldots\) is related to the Catalan numbers; \(g^{(-3/2)}\) is the sequence of Catalan numbers, multiplied by $-2$, and prepended by \(1\). We express the Catbert matrix in terms of a matrix $G^{(a,q/p)}$ by using the offset $1$ and multiplying by $-2$.
\begin{definition}
\label{CATBERTDEFIN}
The {\em Catbert matrix} is the matrix \( C \) given by
\( C=-2G^{(1,-3/2)}=-2G \).
\end{definition}
It follows that
\( L^{(1,-3/2)} \)
is a coefficient matrix of orthogonal polynomials for the Catbert matrix \( C \).
\begin{theorem}
\label{CATBERTTHM}
The inverse of the Catbert matrix has integer entries.
\end{theorem}
\begin{proof}
Let $L = L^{(1,-3/2)}$, $K = K^{(1,-3/2)}$, and $M = M^{(1,-3/2)}$. By definitions \ref{CATBERTOPLOWERTRI}, \ref{NORMDEFIN}, and \ref{MKDEFIN}, we have
\begin{equation}
L_{i,k} = (-1)^{i+k}4^{k}\binom{i+k-3/2}{k}\binom{i+1}{k+1},
\end{equation}
\begin{equation}
K_{i,k} = (-1)^{i+k}4^{k+1}\binom{i+k-3/2}{k+1}\binom{i}{k},
\end{equation}
and
\begin{equation}
M_{i,i} = \frac{4i-1}{3}.
\end{equation}
Now \( C(m)^{-1} = \frac{1}{2}\bigl(G^{(1,-3/2)}\bigr)^{-1} = \frac{1}{2}L(m)^TM(m)K(m) \).
We need to show that both $2$ and $3$ divde some factor of the numerator of
\[G(m)^{-1}=L(m)^TM(m)K(m)=K(m)^TM(m)L(m).\] Lemma \ref{LUCASVAR3} implies that $2$ divides $K_{i,k}.$
To prove divisibility by $3$, we use equations \eqref{LMKEQN3} and \eqref{LMKEQN4} as in the proof of Theorem \ref{MAINTHEOREM}. If $i\equiv 1 \pmod{3}$ then $3$ divides $4i-1$, and so $3$ divides the numerator of $M_{i,i}$.
If $i\equiv 0 \pmod{3}$ and $k \not\equiv 0 \pmod{3}$, then $3$ divides $\binom{i}{k}$ by Lemma \ref{LUCASLEM}, and thus $3$ divides $K_{i,k}$.
If $i\equiv 2 \pmod{3}$ and $k \not\equiv 2 \pmod{3}$, then $3$ divides $\binom{i+1}{k+1}$ by Lemma \ref{LUCASLEM}, and thus $3$ divides $L_{i,k}$.
If $i\equiv 0 \pmod{3}$ and $k \equiv 0 \pmod{3}$, then $3$ divides $4^{k}\binom{i+k-3/2}{k+1}$ by Lemma \ref{LUCASVAR1}, and thus $3$ divides $K_{i,k}$.
If $i\equiv 2 \pmod{3}$ and $k \equiv 2 \pmod{3}$, then $3$ divides $4^{k}\binom{i+k-3/2}{k}$ by Lemma \ref{LUCASVAR2}, and thus $3$ divides $L_{i,k}$.
So for all possible values of $i\pmod{3}$ and $k\pmod{3}$, we have shown that $3$ divides the numerator of \(L(m)^TM(m)K(m)\). Hence \(C^{-1}\), the inverse of the Catbert matrix, has integer entries.
\end{proof}
We can now express the determinant of the inverse Catbert matrix $C(n)^{-1}$ as a product of the diagonal elements of $M$, $L$, and $K$, omitting the factors that are identically equal to $1$.
\begin{corollary}
The determinant of the inverse Catbert matrix $C(n)^{-1}$ is given by
\begin{equation}
\label{CATBERTDET}
det\bigl(C(n)^{-1}\bigr)=\prod_{k=0}^{n-1}
4^{2k+1}\frac{4k-1}{6}\binom{2k-3/2}{k}\binom{2k-3/2}{k+1}.
\end{equation}
\end{corollary}
The sequence of determinants of the inverse Catbert matrix is sequence \seqnum{A296056}
in the {\em On-Line Encyclopedia of Integer Sequences} \cite{OEIS}.
\end{document} |
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